Download A T S O U

A THEORETICAL STUDY ON
ULTRA HIGH ENERGY COSMIC RAY
INTERACTIONS USING MONTE CARLO
SIMULATION TECHNIQUE
A THESIS SUBMITTED TO
THE UNIVERSITY OF GAUHATI AS THE FULFILMENT
FOR THE DEGREE OF DOCTOR OF
PHILOSOPHY IN PHYSICS
YEAR : 2006
BY
UMANANDA DEV GOSWAMI
DEPARTMENT OF PHYSICS
GAUHATI UNIVERSITY
ASSAM, INDIA
Dedicated to my father
LATE KUSHA DEV GOSWAMI
and my mother
SRIMATI MUKHYADA GOSWAMI
DECLARATION
A THEORETICAL STUDY ON ULTRA HIGH ENERGY
COSMIC RAY INTERACTIONS USING MONTE CARLO SIMULATION TECHNIQUE
I am submitting this thesis :
to the Gauhati University for the Degree of Doctor of Philosophy in Physics. I hereby declare that,
I have fulfilled the requirements for the degree and the whole work incorporated in this thesis is
the genuine research work done by me. This thesis work or any part of it has not been submitted
by me or anybody else for any other degree to the Gauhati University or any other institution of
the world.
(Umananda Dev Goswami)
Dr. (Mrs) Kalyanee Boruah
Ph: +91-361-2570531 ( O )
Professor
+91-361-2674537 ( R)
Department of Physics
GAUHATI UNIVERSITY
Guwahati – 781 014
Assam, INDIA
Mobile: 9435543920
E-mail : [email protected]
[email protected]
Residence : Qr. No. 45, GU Campus
Ref:
Date: 09.08.2006..........
TO WHOM IT MAY CONCERN
This is to certify that Mr. Umananda Dev Goswami is a research student under
my guidance from March, 2001 to August, 2006, and he has
been continuously doing
theoretical work in this Department. He also visited the Institute of Kernphysik,
Forchungszentrum, Karlsruhe, Germany, during November to December, 2002 and carried
theoretical simulation work on Charmed particle, under the supervision of Dr. Ralf Engel.
The whole research work incorporated in his thesis is original work done by him
and no part is submitted to any University for any other degree.
( KALYANEE BORUAH )
Professor & Research Guide
ACKNOWLEDGEMENT
First of all I would like to express my sincere gratitude to my esteemed guide, Professor Kalyanee Boruah of the Department of Physics, Gauhati University for her indefatigable and indelible
efforts in carrying forward my research work up to this level. I will not hesitate to confess that,
Professor Kalyanee Boruah is only person who kept open the door for me to enter into the world
of physics research, when all other doors were virtually closed to me in Gauhati University. In this
reference, I should next remember with a deep sense of gratefulness, Dr. Pratyush Purkayasthya,
the research scholar of the same department for his selfless endeavour to introduce me first with
Professor Kalyanee Boruah to initiate my research work. I am also indebted to Professor Pradip
Kumar Boruah, Department of Instrumentation and USIC, Gauhati University, for his help in his
various capacities and encouragement during the whole period of this work.
It is my pleasure to acknowledge my indebtedness to my respected teachers Professor Hiralal
Doura, former Vice-Chancellor of Gauhati University, Professor S. A. S. Ahmed, Head of the Department of Physics, Professor Pranayee Datta, Head of the Department of Electronic Science,
Professor Hiralal Das, Professor Barindra Kumar Sarma, Professor Minakhi Devi, Dr. N. Nimai
Singh, Dr. Kalpana Doura and Dr. Madhujhya Prasad Bora of the Department of Physics, Gauhati
University for their valuable comments and inspiring attitude towards my work. I want to render
my thanks to Dr. Tulshi Bezborah of Department of Electronic Science, Gauhati University, Mr.
Bhim Prasad Sarmah of Department of Physics, Tezpur University, Dr. Budhadev Bhattacharjee,
Mr. Basanta Ranjan Borauh, and Mrs. Banti Tiru of Department of Physics, Gauhati University for
the same reason. My thanks also goes to all the staff members of the Department of Physics and
the Computer Centre, Gauhati University for their helps in various occasions. I offer my special
thanks to Anil, Bhupen and Nayan for their service in providing me foods during my working period in the department. I should also express my thankfulness to the watchmen of the department,
Hareswar, Pravat, Gajen and Narayan for keeping open the departmental door for my convenience
at any time.
I am obliged to the research scholars of Gauhati University Dr. Rajanish Saikia, Mr. Debajyoti Boruah, Mr. Jyoti Prasad Phukan, Dr. Biman Jyoti Medhi, Dr. Saradee Boruah, Dr. Ranjan
Sarmah, Mr. Nayanmoni Saikia, Mr. Mrinal Kumar Das, Mrs. Kalpana Roy Sinha, Dr. Julee
Saikia, Mrs. Gitanjali Devi, Mrs. Ranjita Sarmah, Mrs. Namita Bordoloi, Mrs. Dipshikha Kalita,
Mr Subash Rajbongshi, Mr. Ranjit Choudhary, and all the members (including cooks and watchmen) of the V. V. Rao Research Scholars’ Hostel, Guahati University, where I was staying for the
period of 1999 - 2003, for their happy association with me during the period of my work. Moreover, Mr. Debajyoti Boruah, Dr. Saradee Boruah, Mrs. Kalpana Roy Sinha and Dr. Julee Saikia
deserve special thanks for their moral assistance to me during my tough time and Mrs. Gitanjali
Devi also deserves the same for providing me shelter in Guwahati for a long period.
I take opportunity to convey my gratefulness to Professor S. K. Datta and Dr. S. Mukherjee of
Nuclear Science Center, New Delhi, for giving the opportunity to learn and introducing me with
the programming techniques in that institution.
Sometimes it is hard to express the gratefulness with words because of boundless help and
kindness of some persons. Under one of such situation at this moment I may say that, I am immensely thankful to Dr. Dieter Heck of Institut für Kernphysik, Forschungzentrum Karlsruhe,
Germany, Dr. Varsha R. Chitnis, Professor P. N. Bhat, Professor B. S. Acharya, and Mr. B. B. Sing
of Tata Institute of Fundamental Research, Mumbai for their painstaking help in various aspect of
using CORSIKA code and analyzing data with the same. In this regard the guidance of Dr. Dieter
Heck and Dr. Varsha R. Chitnis was significant. Professor B. S. Acharya is always very much kind
to me. Because of this I could visit and work with the group members of High Energy Gamma
Ray Observatory, Pachmarhi, Madhya Pradesh, twice for long periods. Here I should remember
with thanks for the helpful and nice company of Mr. Prasanna N. Purohit, Mr. Sanjay Sarmah, Mr.
Sandeep Kumar Duhan, Mr. Manoj Kumer Mishra, Mr. Mahesh Pose and all other staff members
of the group. Furthermore these persons along with Professor B. S. Acharya and Dr. Varsha R.
Chitnis inspired me to write this thesis.
Another such situation where the effect of words becomes less to express the gratitude is related with my works in the Institut für Kernphysik, Forschungszentrum Karlsruhe, Germany. I
gratefully want to mention that, I started my work on charmed particles in this institution under
the initiating guidance of Dr. Ralph Engel with the assistance of Dr. Sergey Ostapchenko, Dr. T.
Thouw, and Dr. Dieter Heck during my fourty days visit to that institute in the year 2002. So I
am deeply indebted to these persons particularly to Dr. Ralph Engel for the contribution to my
research life. I would like to thank Professor Johannes Blümer, Head of the Institut für Kernphysik, Dr. Andreas Haungs, the leader of the KASCADE group, and the higher authority of the
Forschungszentrum Karlsruhe, Germany for making my visit possible and for kind hospitality during my visit to that institute. Specially I will never forget Dr. Andreas Haungs for his brother like
care to me, who was looking after my visit. It is also a sweet memory for me the friendly companies of Dr. Till Bergmen, Dr. Taunguy Pierog, Dr. Toma Gabriel, Dr. Heinrich Rebel, and all
other group members of that institute. Although, I am a very small person in the field of research,
the great scientists like Dr. Dieter Heck, Dr. T. Thouw, Dr. Ralph Engel, Dr. Sergey Ostapchenko,
and Dr. Andreas Haungs along with Dr. Till Bergmen and Dr. Taunguy Pierog took me always
together with them for the lunch during my stay there. This is really a unforgettable memory for
me. Besides all, I gratefully remember Dr. Sailendra Sing Chouhan, Indian Post Doctoral Fellow
of Forschungszentrum Karlsruhe for his lively company in Karlsruhe.
I am very happy to express my gratitude to my friends Dr. Monojit Chakraborty, Scientist,
Centre of Plasma Physics, Guwahati, and Miss Kabita Patowari, Research Scholar, Department
of Chemistry, Gauhati University, and honourable persons Prof. Durlov Dev Goswami, Head, Department of Physics, North Lakhimpur College, Lakhimpur, Mr. Maloy Kumar Dutta, Sub-Teacher
of English, Udalguri Higher Secondary School, Udalguri, and Dr. Naresh Sarmah, Director of
Swadeshi Academy of Guwahati for their help and encouragement in different respects, especially
for their financial help to visit to Germany, without which it would have been impossible for me
to visit that country. In the same sense I intend to convey thanks to my other friends Mr. Umesh
Bora, JTO, Guwahati, Mr. Nayan Jyoti Sarmah, Lecturer of Physics, Swadeshi Academy, Guwahati, Mr. Rajib Mahanta, Department of Physics, Dibrugarh University, Dibrugarh, Mr. Aditya
Dahal, Lecturer, Department of Physics, D. H. S. K. College, Dibrugarh and Miss Nabalata Bora
of Guwahati for their cordial support during my work.
For a brief period within 2003 to 2004, covering this work period, I was serving as a Lecturer
in Physics in the Central Institute of Himalayan Culture Studies, Dahung, West Kameng District,
Arunachal Pradesh. It is my moral duty to express my gratitude to the O.S.D Mr. Ranjit Kumar
Bhattacharjee, Principal Mr. Nawang Tashi Bapu, Lecturers Mr. Jitendra Kumar Tiwari, Miss
Yasmine Hazratji Kharshiing, Mr. Mohan Mishra, Mr. Pema Tsering, Mr. Tashi Tsering, Mr. Brajbhusan Ojha, the Office assistant Mr. Golap Gogoi, and all the students of the institute for their
affections to me and the solidarity with my research activity during that period. They born with all
inconveniences arising out of my frequent absence in that institution within that period as I had to
came out frequently for my research and other domestic purposes.
I am greatly indebted to the principal Dr. Tapan Chandra Bhuyan, Head of the Department of
Physics Mr. Pradip Kumar Saikia, the faculty members of the same department Dr. Udayan Mcfarlane and Mr. Dinesh Gogoi of Debraj Roy College, Golaghat for their helping attitude to continue
my research work and to write this thesis, where I am presently serving as a Lecturer in Physics.
I should gratefully remember the faculty members of the same college Dr. Padmeswar Gogoi, Dr.
Anil Kumur Boruah, Dr. Ananda Bharali, Dr. Yamini Baishya, Dr. Bidyadhar Borthakur, Mr.
Jyotishprakash Datta, Mr. Bedanta Bora, and Mr. Bhaskar Jyoti Boruah for their moral support to
my work here. Also I am grateful to Mr. Dipak Hiloidari and Mrs. Rupa Hiloidari of Golaghat for
their support and help in needy times in Golaghat.
Before finishing, I want to state with a deep honour from my heart that, I am fortunate to
express my nonreturnable indebtedness to my Father Late Kusha Dev Goswami, Mother Srimati
Mukhyada Goswami whose nursing with an unlimited care grown up me to this position. But
I feel inconsolable sorrow that my father could not see this work of his child. My elder brothers Mr. Joykrishna Dev Goswami, and Mr. Ramdev Goswami, elder sisters Srimati Nilaprava
Goswami Mahanta, Srimati Dharmaprava Goswami Bharali, and Srimati Nirada Goswami, Sisterin-law Mrs. Mousumi Goswami, Brother-in-laws Mr. Khagen Mahanta, and Mr. Khagen Sarmah
Bharali are very much kind to me and so they take all pains, provide me with their moral help and
co-operations to continue my research work. So I have no language to convey my gratefulness
to them. At this moment I also remember my younger brother Late Sadananda Dev Goswami
with tearful eyes, who left us in the year 1997. He was always with me to promote my academic
and all other interests. All of my nephews and nieces Sudipta, Jyotsna, Sumsumi, Kanchanmoni,
Binod, Tolokon, Utpal, Munukon and Topokon are the sources of inspiration for me. I am happy
to convey my affections to them here. My maternal uncles Late Nabin Chandra Dev Goswami and
Mr. Durlove Dev Goswami were always concerned with my work and also encouraging me to go
ahead. Untimely demise of uncle Late Nabin Chandra Dev Goswami was another big blow to me
during this period. I offer my hurtful gratitude to them.
Finally, I desire to convey my cordial thanks to those persons who were helping me in different
respects and moments in different places relating to my research work, whose names I forgot at
this moment and also not possible to mention considering the space. I beg apology to them for my
inability.
ABSTRACT
Energetic particles (mostly charged) radiations with energy ranging from ∼ 106 eV to ≥ 1020
eV impinging continuously in the earth atmosphere from outer space are known as the Cosmic
Rays (CR). The study of CR, especially in Ultra High Energy (UHE) range (≥ 1017 eV) opens
some important fields of research in astronomy and particles physics. Interactions of Ultra High
Energy Cosmic Rays (UHECR) in the atmosphere continuously develop a number of possible processes, of which some of them may lead to the production of different short-lived heavy particles.
These might include some discovered and undiscovered epoch changing particles which have a
particular significance in the world of physics. In resemblance of interactions, the detection of CR
in this range is also a subject of great challenge and worth noting. It is entirely based on the indirect
method of detection by measuring different measurable parameters of EAS, produced by primary
particles in the atmosphere, with a very large ground based detectors array. Both, the study of interaction processes for the production of heavy short-lived particles and the detection of UHECR
are relied heavily on the detailed Monte Carlo (MC) simulation of the processes responsible for
concerned particles production and development of EAS respectively, with the help of a reliable
physical model. Thus MC simulation is a powerful and essential numerical method for the study
of CR in general by virtually producing complete course of development of a CR event in the same
manner as the real event.
In this work, the first part incorporates the study of the production of Standard Model (SM)
Higgs boson in UHECR interactions with air nuclei. Higgs boson is the most significant, but undiscovered component of SM. Here we developed and discussed a new approach for the observation
of production of SM Higgs boson in UHECR interactions, based on the idea that, during UHECR
interactions with air nuclei, this scalar particle is produced through vacuum excitation due to a
fraction of interaction energy transfer to vacuum. Models are developed for hadronic interaction
based on the GENCL code of the UA5 experiment of CERN and for the process of Higgs boson production due to vacuum excitation. We consider the high energy (HE) muons or prompt
muons multiplicities as probes for the production of Higgs boson. The prompt muons multiplicities distributions with and without this effect for different fractions of energy transfer to vacuum
are compared to check the evidence of production of Higgs boson during UHECR interactions. It
is found that the Higgs boson production mechanism is significant starting from E0 ∼ 1018 eV. The
study reveals that, the average prompt muon numbers become independent of threshold energies at
higher values of primary energy E0 (≥ 1018 eV) of the projectile particle, since these numbers for
all threshed energies approaches to same value at higher primary energy E0 . This nature of prompt
muons is an indication of Higgs boson production in UHECR interactions according to our model.
The second part of this work involves in the investigation of the production of charmed hadrons
−
(D+ , D− , D0 , D̄0 , Λ+
c , Λ̄c ) in pp collisions as a function of
√
s, xF , p2⊥ , and p⊥ in the framework
of the QGSJET model. The study of charmed hadrons production characteristics in pp collision is
particularly important for CR physics in the context of atmospheric prompt leptons fluxes. Here
our aim is to check the reliability of QGSJET model to be used to study the charmed hadrons
production in CR hadronic interactions with air nuclei. It is found that, the charmed hadrons production cross sections or the average multiplicities in pp collisions are relatively very small. The
maximum production of all charmed hadrons take place with low values of xF , p2⊥ , and p⊥ within a
√
small range for all values of s under study. Charmed hadroproduction cross sections as a function
of xF and p2⊥ are compared with the LEBC-EHS and LEBC-MPS experiments’ data for D-mesons
production. The agreement is quite satisfactory for smaller values of p2⊥ (≤ 2 GeV2 c−2 ). The
behaviour of charmed hadrons production as a function of p⊥ is not consistent with expectation.
It is also seen that there is an asymmetry in charmed hadroproduction in pp collisions. For all
√
xF , asymmetry is prominent in the low value of s. There is strong preference for producing Λ+
c
√
−
rather than Λ̄c baryons, while that for producing D̄ rather than D mesons for this range of s.
√
For low s range, asymmetry increases from zero to one at xF ≤ 0.3, whereas for higher value of
√
s, the asymmetry becomes one at xF ≥ 0.3 . The patterns of asymmetric production of different
√
charmed hadrons with xF are approximately same as that with s. we compare our calculations
−
with data from Fermilab experiment E781 (SELEX) for Λ+
c and Λ̄c production. Agreement is
quite good. The asymmetry of charmed hadrons production with p⊥ does not follows any definite
pattern. Thus the QGSJET model is not well versed to explain the charmed hadrons production
characteristics as a function of p⊥ .
The final phase of this study is the reanalysis of GU Miniarray data using the standard MC
code CORSIKA. The GU Miniarray is a UHECR detector consisting of eight plastic scintillators
of carpet area 2 m2 , each viewed by fast PMTs. It is used to detect giant EAS by the method
of time spread measurement of secondary particles produced in the atmosphere. Because of its
unconventional features, its old data are subjected to reanalysis with CORSIKA to show its feasibility and to gain greater acceptability. In this reanalysis the energies of the air showers have
been re-estimated by the simulation with CORSIKA. In the miniarray method, the CR energy is
determined via its relation to the ground level parameter Ns , the shower size. This relation was
obtained previously through a best fitted relation in agreement with QGS model and Yakutsk data.
In this work we use CORSIKA code with QGSJET model of HE hadronic interactions to simulate
miniarray data leading to a modified relation between primary energy and shower size. We get a
new energy spectrum in this reanalysis for 1017 eV to 1019 eV primary energy.
Contents
List of Figures
iii
List of Tables
vi
1 INTRODUCTION
1.1 The Discovery and General Nature of Cosmic Rays . . . . . . . .
1.1.1 Definition of terms . . . . . . . . . . . . . . . . . . . . .
1.1.2 Detection methods at different energies . . . . . . . . . .
1.1.3 Observed primary energy spectrum . . . . . . . . . . . .
1.1.4 Primary composition . . . . . . . . . . . . . . . . . . . .
1.1.5 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.6 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 High Energy Interactions . . . . . . . . . . . . . . . . . . . . . .
1.2.1 General characteristics of high energy interactions . . . .
1.2.2 Production of short-lived heavy particles in CR interaction
1.3 Scientific Motivation . . . . . . . . . . . . . . . . . . . . . . . .
1.4 This Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
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2 MONTE CARLO SIMULATION TECHNIQUE
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.2 Basic Concept of Monte Carlo Simulation Technique .
2.3 Random Variables . . . . . . . . . . . . . . . . . . . .
2.3.1 Discrete random variables . . . . . . . . . . .
2.3.2 Continuous random variables . . . . . . . . .
2.4 Generators of Random Variables . . . . . . . . . . . .
2.4.1 Pseudo-random numbers . . . . . . . . . . . .
2.5 Transformation of Random Variables . . . . . . . . . .
2.6 Monte Carlo Simulation of EAS . . . . . . . . . . . .
2.7 CORSIKA : A Standard EAS Monte Carlo Simulation
Package . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Interaction models in CORSIKA . . . . . . . .
2.7.2 Particles in CORSIKA . . . . . . . . . . . . .
2.7.3 Coordinate system in CORSIKA . . . . . . . .
2.7.4 Model of atmosphere in CORSIKA . . . . . .
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40
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3 PRODUCTION OF HIGGS BOSON THROUGH VACUUM EXCITATION IN UHECR
HADRONIC INTERACTION
48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 The Physics of Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Basic concept of symmetry principle . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Standard model and electroweak theory . . . . . . . . . . . . . . . . . . . 53
3.2.3 Spontaneous symmetry breaking : The Higgs mechanism . . . . . . . . . . 54
3.2.4 Present experimental status of Higgs boson search . . . . . . . . . . . . . 58
3.3 Higgs boson Production Through Vacuum Excitation . . . . . . . . . . . . . . . . 58
3.3.1 Temperature dependence of vacuum . . . . . . . . . . . . . . . . . . . . . 58
3.3.2 Vacuum excitation and bubble formation . . . . . . . . . . . . . . . . . . 60
3.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 Higgs boson production model . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2 Interaction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Historical Events in Support of Higgs bosons Production . . . . . . . . . . . . . . 71
3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 CHARMED HADRON PRODUCTION IN pp INTERACTION
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 A Short History of Charmed Particles : The November Revolution
4.3 A Description on QGSJET Model . . . . . . . . . . . . . . . . .
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 General features of charmed hadron production . . . . . .
4.4.2 Asymmetry in charmed hadron production . . . . . . . .
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 USE OF CORSIKA TO REANALYSE GU MINIARRAY DATA
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Miniarray Experiment . . . . . . . . . . . . . . . . . . .
5.3 Simulation and Data Analysis . . . . . . . . . . . . . . . . .
5.3.1 Numerical equations . . . . . . . . . . . . . . . . . .
5.3.2 Detector response . . . . . . . . . . . . . . . . . . . .
5.3.3 Energy calibration . . . . . . . . . . . . . . . . . . .
5.4 Data Selection Criteria . . . . . . . . . . . . . . . . . . . . .
5.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . .
5.5.1 Shower rate and shower size spectrum . . . . . . . . .
5.5.2 Energy spectrum . . . . . . . . . . . . . . . . . . . .
5.6 Conclusion and Remarks . . . . . . . . . . . . . . . . . . . .
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6 SUMMARY AND FUTURE OUTLOOK
114
6.1 Higgs boson Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Charmed Hadron production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 GU Miniarray Data Analysis Using CORSIKA . . . . . . . . . . . . . . . . . . . 117
LIST OF PUBLICATIONS
120
List of Figures
1.1
1.2
1.3
1.4
1.5
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
The Cosmic Rays all particle energy spectrum indicating approximate integral
fluxes at different energy range [2]. . . . . . . . . . . . . . . . . . . . . . . . . .
All particle energy spectra of Cosmic Rays flux. To enhance the features visible in
the spectrum, the flux is multiplied by E02.5 [44]. . . . . . . . . . . . . . . . . . .
HiRes data versus AGASA data. The fit to the HiRes is done by a two component
model taking into account the GZK effect [37]. . . . . . . . . . . . . . . . . . .
The nucleon interaction length (dashed line) and attenuation length (solid line) for
photon-pion production and the proton attenuation length for pair production (thin
solid line) in the combined CMBR [2]. . . . . . . . . . . . . . . . . . . . . . . .
The Hillas diagram showing size and magnetic field strength of possible sites of
particle acceleration. Objects below the corresponding diagonal lines cannot accelerate protons (iron nuclei) to 1020 eV. βc is the characteristics velocity of the
magnetic scattering centers [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Physical meaning of a probability density distribution function p(x). . . . . . . .
Uniformly distributed random variable in (0, 1). . . . . . . . . . . . . . . . . .
Method of transformation of discrete random variables. . . . . . . . . . . . . . .
Inverse transformation method. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Neumann’s method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monte Carlo Simulation of Gaussian Distribution. . . . . . . . . . . . . . . . .
The hybrid calculation (solid line) and the full Monte Carlo simulation (dotted line)
of the muon number distribution at sea level for vertical, proton induced showers
with E0 = 1015 eV [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inelastic proton-air cross sections for models (excluding NEXus) used in CORSIKA for high energy hadronic interactions as a function of projectile momentum.
The shaded band represents the results of a fit to the data with p < 105 GeV/c [6].
Coordinate system in CORSIKA. . . . . . . . . . . . . . . . . . . . . . . . . . .
Possible vacuum states of the field φ. The case for m < 0 indicates the SSB condition of the field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated temperature dependent Higgs boson mass as a function of central temperature of the bubble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Volume of the excited region of vacuum or bubble varies with bubble energy. The
filled circles denote simulated data and the dashed line indicates the best fit. . . .
Variation of central temperature of the excited region of vacuum or bubble with
bubble energy. The half-filled circles denote simulated data and the dashed line
indicates the best fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
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7
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8
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9
. 13
. 14
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31
32
34
35
37
38
. 41
. 43
. 45
. 55
. 62
. 63
. 64
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Average Higgs boson number with bubble energy. The filled squares denote simulated data and the dashed line indicates the best fit obtained by using equation
(3.44). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Muon multiplicity distributions at first interaction level for 5000 proton induced
showers for different fractions of energy transfer. The data is taken for muon
threshold energies (Eµthr ) of 0.01 TeV, 0.1 TeV, and 1 TeV. The panel of the figures
on the left hand side is for primary energy (E0 ) 1019 eV and on the right hand side
is for 1020 eV respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Average muon number at first interaction level versus bubble energy for different
muon energy thresholds. Data is taken from 5000 proton induced showers. . . .
Variation of average number of muons with primary energy of Cosmic Rays particle at first interaction level for different muon threshold energies and for fraction
of energy transfer fe = 0.3. The average number is calculated for 5000 proton
induced showers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dispersion of muon multiplicity distribution for different fe %= 0 from fe = 0. . .
Few basic processes of charmed production in pQCD. (a), (b), (c) are leading order
processes and (d) is an important NLO process. . . . . . . . . . . . . . . . . . .
cc̄
√ production cross sections data with theoretical predictions at different values of
s [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A general multi-Pomeron contribution to hadron-hadron scattering amplitude. Elementary scattering processes (vertically thick lines) are described as Pomeron
exchanges [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A general Pomeron (L.H.S.) consists of the soft and semihard Pomerons, represented by the 1st and the 2nd contributions on the R.H.S. [28]. . . . . . . . . . .
Inelastic and production cross sections√of different charmed hadrons in pp collisions with the centre of mass energy ( s). The cross sections are calculated with
the QGSJET are compared with the experimental data [34] for D/D̄ production in
the low energy range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Average multiplicities of different charmed hadrons produced in pp collisions with
√
s. The average multiplicities are calculated with the QGSJET for 106 numbers
of events. The lines in the figure are least square fits for different hadrons obtained
with the equation (4.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusive cross sections for the production
of charmed hadrons in pp collisions at
√
different centre of mass energy ( s) as a function of Feynman xF . The results
from QGSJET calculations at 27.4 GeV and 38.8 GeV centre of mass energies are
compared √
with the experimental data [35] for D/D̄ production. The line in the
figure for s = 100000 GeV is the best fitted result with the simulated data that is
obtained by using equation (4.9) is shown as an example. . . . . . . . . . . . . .
Inclusive cross sections for the production
of charmed hadrons in pp collisions at
√
different centre of mass energy ( s) as√a function of p2⊥ . The
√ cross sections are
calculated with the QGSJET model at s = 27.4 GeV and s = 38.8 GeV are
compared √
with the experimental data [35] for D/D̄ production. The line in the
figure for s = 100000 GeV is the best fitted result with the simulated data that is
obtained by using equation (4.10) is shown as an example. . . . . . . . . . . . .
. 65
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. 80
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. 84
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. 88
4.9
Inclusive cross sections for the production
of charmed hadrons in pp collisions at
√
different
centre of mass energy ( s) as a function of p⊥ . The line in the figure for
√
s = 100000 GeV is the best fitted result with the simulated data that is obtained
by using equation (4.11) is shown as an example. . . . . . . . . √. . . . . . . . .
4.10 Asymmetry in charmed hadrons production in pp collisions with s as calculated
with QGSJET model. . . . . . . . . . . . . . . . . . . . . . . . . . .√. . . . . .
4.11 Asymmetry in charmed hadrons production in pp collisions at different s as function of Feynman
√ xF . Results for Λc particles are compared with the experimental
data [37] at s = 31.9 GeV. . . . . . . . . . . . . . . . . . . . . . . .√. . . . . .
4.12 Asymmetry in charmed hadrons production in pp collisions at different s as function of p⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Block diagram of the GU Miniarray data acquisition system. . . . . . . . . . . .
Shower disk thickness versus distance from the shower axis. Linsley’s relation is
shown in comparison with CORSIKA simulation. . . . . . . . . . . . . . . . .
Lateral particle density distribution obtained from the simulation with CORSIKA
and from the old relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effective area of GU Miniarray versus primary energy for proton and iron showers
simulation using CORSIKA compared with the results from old relation. . . . .
Shower size versus primary energy for proton and iron shower simulation using
CORSIKA. Solid line represents energy calibration curve for proton and the dotted
line for iron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event number distribution as a function of shower disk thickness (σ) as recorded
by the GU Miniarray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integral shower rate spectrum with ρ1 = 1.5m−2 and σ1 = 100 ns. Data for proton and iron are obtained from reanalysis of experimental data using CORSIKA
assuming proton and iron as primary CR particles. . . . . . . . . . . . . . . . .
Integral shower size spectrum with ρ1 = 1.5 m−2 . Data for proton and iron are
obtained from reanalysis of experimental data using CORSIKA assuming proton
and iron as primary CR particles. . . . . . . . . . . . . . . . . . . . . . . . . .
Miniarray differential energy spectrum. The differential flux is multiplied by E02.5 .
Data for proton and iron are obtained from reanalysis of experimental data using CORSIKA assuming proton and iron as primary CR particles. Region between solid lines give the flux as compiled from AGASA and Haverah Park data
by Nagano and Watson (2000). . . . . . . . . . . . . . . . . . . . . . . . . . .
. 89
. 91
. 92
. 93
. 100
. 102
. 103
. 104
. 105
. 106
. 107
. 108
. 109
List of Tables
4.1
4.2
A comparison of inclusive production cross sections of different
√ D-mesons as obtained
from
the
QGSJET
calculations
with
experiments
[35]
at
s = 27.4 GeV and
√
s = 38.8 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Values of parameters n0 , n1 and α of equation (4.7) for different charmed hadrons. 85
5.1
5.2
A comparison of slope before the dip of the differential energy spectrum. . . . . . 110
A comparison of overall slope of the differential energy spectrum. . . . . . . . . . 110
vi
Chapter 1
INTRODUCTION
The physics of Cosmic Rays (CR), especially in the Ultra High Energy (UHE) range plays
a significant role in the present day scientific world. It’s study over more than ninety four years
gradually opens up wide to a wider and new to a newer horizon of scientific research, not only in the
field of particle physics, astronomy (γ-ray astronomy, x-ray astronomy, radio-astronomy, neutrinoastronomy etc.) and astrophysics, but also it opens gradually a avenue to the physics beyond the
standard model. Moreover, we may say that CR related phenomena play an important part in our
everyday public life, as a significant fraction of the natural radioactivity on earth is caused by the
fragments of CR induced Extensive Air Showers (EAS). As claimed, some scientists even found
evidence for a direct link between the CR flux reaching the earth and the global climate [1].
The study of the nature of CR above 1015 eV is entirely based on the indirect methods of
measurements (ref section 1.1.2) of different characteristics of EAS produced by primary particles
in the earth’s atmosphere with the help of ground based large array of detectors, as the flux of CR
in this region of energies is very low. The low flux of CR also demands detectors of considerably
large exposure area. This factor is very much prominent in the case of Ultra High Energy Cosmic
Rays (UHECR) (≥ 1017 eV) because of their extreme low flux. For example around 1019 eV the
flux of UHECR is estimated to one event per square kilometer per year per steradian and above 1020
eV it is 0.5 to 1 event per square kilometer per century per steradian [2, 3]. So to assist these types
of indirect experiments to arrive at a conclusion on the nature of CR primary particles, there must
be some particle interaction models based on extrapolation of accelerator data at available energies
to CR energies and those are governed by Monte Carlo (MC) simulation technique. Precisely, the
MC simulation of EAS is very much important and powerful numerical tool to study the nature of
CR primaries above 1015 eV.
In the UHECR interactions with air nuclei there is a glorious possibility of production of some
short-lived heavy particles such as, Higgs bosons and charmed hadrons (ref section 1.2.2). These
particles introduce some special signatures in the EAS besides conventional ones. As in the case for
detection, to study the phenomena of production of these types of short-lived heavy particles in the
1
1.1 Discovery and General Nature of Cosmic Rays
2
UHECR interactions, the MC simulation of EAS based on some specialised hadronic interaction
models (i.e. the models which include the production features of heavy short-lived particles) is
the most necessary first step. The study of the possible production of short-lived heavy particles
such as Higgs boson in UHECR interactions with air nuclei is also important in the context of both
CR and the standard model of particle physics. In this thesis work, we developed and studied the
necessary theoretical foundations for the phenomenon of the production of Higgs boson and the
charmed hadrons in UHECR interactions in the atmosphere by the MC simulation technique and
also to use this technique to reanalyse the giant EAS data taken by GU Miniarray [4] which was
operating in the Department of Physics, Gauhati University to study UHECR.
In the following section for the completeness we shall discuss briefly the salient features of
the CR as a whole with a special emphasis on UHECR along with the related phenomena and
observing techniques. In the section 1.2 we shall also discuss briefly about the high energy (HE)
CR interactions, the scientific motivations of this work and the plan of this thesis.
1.1 The Discovery and General Nature of Cosmic Rays
In the history of physics, the day of August 7, 1912 is one of a luminous days, because
that day itself illuminates a new horizon of physics research with the discovery by Viktor Hess [5]
that some energetic particles radiations are continuously impinging the earth. He discovered using
an electrometer during balloon flight that, the rate at which a static charge on the electrometer
diminished grows as a function of increasing height and inferred that these ionising radiations are
coming from space. This discovery is the result of searches beginning with a mystery surrounding the continuous and uncontrollable leakage of electrical charge from a well insulated gold leaf
electroscope. The mystery remained unexplained almost from the time of Henry Coulomb who
had noticed this phenomenon in 1785. Further experiments by a variety of workers all over the
world established that these radiations are not only extra-terrestrial, but also extra-solar. The name
Cosmic Radiations was given to these penetrating radiations that are coming from depths of space
by Millikan in 1925. Viktor Hess was awarded the Nobel Prize in 1936 for this noble discovery.
CR mainly consists of protons and ionised nuclei and constitute important astrophysical and
astronomical windows. These particles which are coming from outside the solar system are called
Primary CR. When a primary CR particle enters the earth’s atmosphere, it interacts with an air nucleus and produces a large number of pions. The charged pions as they travel down either interact
with other nuclei and produce more secondaries or decay into muons and neutrinos, which constitute the penetrating charged and neutral components, called hard component of secondary CR. The
neutral pions decay into γ-rays and these in turn initiate large electromagnetic cascades and give
rise to the soft component of secondary CR observed in the lower atmosphere. A fraction of the
soft component also comes from the decay of low energy muons [6]. The number of particles pro-
1.1 Discovery and General Nature of Cosmic Rays
3
duced in the first few collisions as well as the number of generations that contribute to the nuclear
cascade increase with the energy of the primary particle. As the longitudinal development of the
combined nuclear and electromagnetic cascades progresses downwards, due to the multiple scattering with air nuclei of gradually increasing density towards earth surface, the secondaries spread
out laterally. At very high energies (> 1014 eV) the cascade development results in the simultaneous incidence of very large number of particles, viz. electrons, positron, photons, π-mesons,
nucleons, antinucleons, µ-mesons, neutrinos, K-mesons, Hyperons etc, the number amounting to
several thousand particles spread over hundreds of square meters at the observational level. At primary energies of ≥ 1019 eV, the number could be a billion particles over several square kilometers.
This spectacular showers of secondary particles have been given the name Extensive Air Showers
(EAS). It was Bruno Rossi [7] who first discovered air showers as early as in 1934 when he had
noticed coincidences between several counters placed in a horizontal plane, far in excess of chance
coincidence. The most systematic investigations on EAS were undertaken by Pierre Auger and his
collaborators [8]. So Auger and his collaborators have been given the credit for discovering EAS.
In the context of interaction energy range, air shower studies assumes increasing prominence from
mid 40’s when it became clear that, the primaries responsible for some EAS development were
much higher energetic than what could be produced at man made accelerators [6]. Interestingly,
although CR research has been going on for last more than 94 years, many fundamental questions
such as that of the nature of the source, composition etc. still remain unanswered.
To have a general overview of CR physics and to highlight its importance, we give a brief
introduction in the following sections.
1.1.1 Definition of terms
The CR have a wide range of energy spectrum, from 106 eV to more than 1020 eV [2, 3, 9].
The part of the spectrum with energy ≥ 1012 eV is only important in connection with air shower
phenomena [6]. Considering the different terminologies used by different authors for different part
of CR spectrum, we define CR in between 1012 eV to 1014 eV as High Energy (HE). Above 1014
eV to ≤ 1017 eV can be termed as Very High Energy (VHE). But it has worth to mention that
about the definition of the term of Ultra High Energy (UHE), there are different views. In the
reference [2] it defines as the energy range ≥ 1018 eV, in [10] it is considered as the energy range
> 1019 eV and in [3] as the energy range > 1020 eV. On the other hand in [2] the CR above 1020
eV is termed as Extremely High Energy (EHE). Clearly UHE includes EHE, but not vice versa.
However, here we define UHE as the energy range of CR particles ≥ 1017 eV.
1.1 Discovery and General Nature of Cosmic Rays
4
1.1.2 Detection methods at different energies
The methods of detection of CR depend on the energy of the primary particles. The direct observation of CR primaries is only possible from space by flying detectors with balloon or spacecraft
at energies below 1014 eV [42], because of very limited size of such detectors and steeply falling
differential CR spectrum [2]. As the flux of the γ-rays (neutral component of CR) at a given energy is lower than the charged CR flux by several orders of magnitude, this energy threshold for
the statistical limit occurs at even lower energies for γ-rays. As an example this threshold is only
100 GeV for the instruments on board of the Compton Gamma Ray observatory (CGRO) [12].
However very recently RUNJOB experiment [13] and the experiment in [14] have claimed that
they measured directly the CR primary particles up to energy 1015 eV with improved techniques.
The space based detectors of charged CR traditionally use nuclear emulsion stakes such as
in the JACEE experiment [15]. Spectrometer techniques are also used which are advantageous
for measuring the chemical composition. In the Energetic Gamma Ray Experiment Telescope
(EGRET) on board for γ-rays, the CGRO uses spark chambers combined with a NaI calorimeter
[2]. Direct measurement of the primary particles allows detailed analysis, e.g., of the individual
particles’ energies, their chemical composition, and other properties.
Beyond the energy threshold as mentioned above, where the direct detection is not possible
for very low statistics, indirect detection techniques are used taking the advantage of EAS produced by the primary CR particles at these energies (> 1014 eV) on the earth surface. In these
techniques, charged hadronic particles, as well as electrons and muons in the EAS are recorded
on the ground based detectors arrays such as water cherenkov detectors used in the old Volcano
Ranch [16] and Haverah Park [17] and scintillation detectors used in the GU Miniarray experiment
[4] (ref chapter 5). A multidetector experiment, whose one detectors layer is scintillation counters
is the KASCADE experiment at the Forschungszentrum Karlsruhe in Germany, which has now
been incorporated into KASCADE-Grande [18] for the energy range of 5×1014 − 2×1017 eV.
Currently operating large ground arrays for UHECR EAS ≥ 1020 eV are the Yakutsk experiment
in Russia [19] and the Akeno Giant Air Shower Arrays (AGASA) [20] near Tokyo, Japan. AGASA
is at present the largest fully operational EAS detectors array covering an area of roughly 100 km2
with about 100 detectors each separated by about 1 km. The Sydney University Giant Air Shower
Recorder (SUGAR) [21] operated until 1979 was the largest array in the southern hemisphere.
Although these ground based arrays have the advantage of huge effective collecting areas and
thus able to measure even very low CR fluxes, the information gathered on the primary particles
is of only very indirect nature. For instance, the energy of the shower initiating primary particle is
estimated or reconstructed by appropriately parametrizing it in terms of some measurable parameters such as shower size at the observational level, density at a particular observation point with
distance from the shower core at which it is quite insensitive to the primary composition and the
interaction model used to simulate air showers[2]. It is, however, a quite difficult process because
1.1 Discovery and General Nature of Cosmic Rays
5
it require the computer simulation of EAS development with an appropriate particle interaction
model at energies far beyond the regime that can be experimentally tested with the particle accelerator experiment.
At UHE range (≥ 1018 eV), another technique for the observation of CR is presently used,
the measurement of fluorescence light in optical to ultraviolet, emitted by atmospheric nitrogen
molecules that are excited during passage of the air shower. This fluorescence technique has the
advantages of being used equally well for both charge and neutral primaries and of yielding very
direct information about the deposition of energy in the atmosphere. Thus the energy of the primary particles can be estimated from this energy deposition information and then composition of
the primary particles can be extracted easily from the measured depth of shower maximum [2].
The main drawback of this technique, however, is that of overall low duty cycle, around 10%.
Additionally, this technique requires elaborate modelling of the atmospheric conditions. The most
prominent example of this technique is the High Resolution Fly’s Eye (HiRes) experiment [22] in
Utah, USA.
The very much complementary results around the GZK cut-off (ref section 1.1.6) yield by the
experiments based on these two techniques, i.e. by the HiRes and the AGASA [23] provides a
ground base for setting up a hybrid experiment known as Pierre Auger Observatory [24] which incorporates both techniques. Presently the southern part of the observatory being setup in Malarüe,
located at an elevation of 1400 m in the province of Mendoza, Argentina will combine a huge number of surface particle detectors (1600 covering an area of 3000 km2 ) with 24 optical fluorescence
detectors in stereoscopic configuration. After completion the Pierre Auger Observatory will be the
largest experimental setup ever built by man on the earth and will provide the best statistics for
UHECR ≥ 1019 eV. This observatory will therefore provide the necessary information to address
some of the important questions regarding the nature and origin of UHECR.
Cherenkov radiation from the charged particles that travel faster than the phase velocity of
light is also explored as another method for detection of HE primary particles. However the output
in Cherenkov light is much larger for γ-ray primary than for charged CR primaries. Because of
this nature, the Cherenkov technique is considered as one of the best tools available to discriminate γ-rays from point sources against the strong background of charged CR and to pinpoint the
source and its location responsible for its emission. Consequently this technique leads to develop
a promising branch of astronomy known as γ-ray astronomy to address the astrophysical problems
[25]. The High Energy Gamma Ray Astronomy (HEGRA) experiment [26], High Energy Gamma
Ray Observatory (HEGRO) [27], 10 meter Wipple Telescope [28], Gamma Ray observatory in the
Outback (CANGROO) [2] are some major experimental setup in this field.
As muons of a few hundred GeV and above penetrate into the depth of the order of a kilometer
even in rock, so they can be detected underground. For example the Monopole Astrophysics and
Cosmic Ray Observatory (MACRO) experiment [29], located in the deep underground Gran Sasso
1.1 Discovery and General Nature of Cosmic Rays
6
Laboratory near Rome, Italy, having a rock overburden of about 1.5 km, consists of ( 600 tons
of liquid scintillator and acts as a giant time of flight counter. Operated in coincidence with the
Cherenkov telescope array EAS-TOP located above it, it is used to study the primary CR composition around the knee (ref section 1.1.3) region [30]. A similar combination is represented by the
Antarctic Muon And Neutrino Detector Array (AMANDA) and the South Pole Air Shower Array
(SPASE) [31] of scintillation detector. AMANDA consists of strings of photomultiplier tubes of
a few hundred meters in earth deployed in the antarctic ice at depth of up to 2 kilometer to look
for electromagnetic showers and Cherenkov light from the leptons produced by neutrino-induced
charged current reactions.
The emission of pulsed radio signal by CR air showers provides another opportunity to use
this phenomenon to study the nature of CR primaries. The radio technique bears many of the
advantages of the fluorescence technique in addition with 100% duty circle. LOPES experiment
based on LOFAR [32], and the GMRT [33] experiment are currently engaged in this field.
1.1.3 Observed primary energy spectrum
We have already mentioned in the section 1.1.1 that the CR primaries have a wide range
of energy spectra, span many orders of magnitude. Figure 1.1 & 1.2 show observed differential
spectrum of the primary CR fluxes from energies of the order of 109 eV up to ≥ 1020 eV as
contributed by direct as well as indirect experimental strategies as discussed in the section 1.1.2.
To enhance the features in the spectrum, the flux in the Figure 1.2 is multiplied by E02.5 . The most
intriguing feature is that, the spectrum exhibits power law behaviour over a wide range of energies
with the index of the order of − 2.75 on average, moreover it also shows some more interesting
prominent features as the so called knee at ( 4×1015 eV and the ankle at ( 5×1018 eV [2].
The knee feature was first pointed out by Kulikov and Kristiansen in 1959 [34]. The Fly’s Eye
experiment first discussed the ankle features in detail [35]. It was reported that the slope between
the knee and up to ( 4×1017 eV is very close to 3.0, then it seems to steepen to about 3.2 up to the
dip at 3×1018 eV, after which it flattens to about 2.7 above the dip. The origin of these particular
features is still not quite clear and hence is the subject of ongoing scientific discussion.
The situation at the UHE end of the CR spectrum is more puzzling specially because of the
CR absolute flux is rather unclear to date at this end and so, much attention is currently focused on
the UHECR with increasing activities of theoretical as well as experimental point of view [2, 36].
Theoretically established GZK cutoff (ref section 1.1.6) predicts a diminishing of the CR flux at
energies above ∼ 5×1019 eV . While the HiRes data indeed seem to indicate a flux depression
in this energy region, the AGASA data show a continuation of the spectrum up to energies of >
1020 eV, as shown in the Figure 1.3 [36]. A more recent report from HiRes [39] based on fitting
to the UHECR spectrum measurements with broken power laws indicates the previously observed
feature of ankle at 1018.5 eV and the evidence for a suppression at higher energies above 1019.8 eV.
1.1 Discovery and General Nature of Cosmic Rays
7
Figure 1.1: The Cosmic Rays all particle energy spectrum indicating approximate integral fluxes
at different energy range [2].
The energy for this high energy suppression agrees with the expected GZK cut-off. However, the
statistics are very low at these energies to really decide whether there is a flux depression or not.
It is expected that with its huge effective collecting area, the Pierre Auger Observatory will finally
provide the necessary statistics to address this issue very shortly [24, 36].
Events above 1020 eV
Volcano Ranch experiment [16] first reported the detection of the CR event above 1020 eV.
After that the Haverah Park experiment reports of detecting eight events around 1020 eV [17] and
the Yakutsk array saw one event above this energy [19]. The SUGAR array in Australia reported
eight events above 1020 eV [21], the highest one at 2×1020 eV. The predecessor of HiRes, the Fly’s
1.1 Discovery and General Nature of Cosmic Rays
8
Figure 1.2: All particle energy spectra of Cosmic Rays flux. To enhance the features visible in the
spectrum, the flux is multiplied by E02.5 [44].
Eye experiment [35] observed only one event on 15 October 1991, which is still the world record
holder highest energy event at 3.2×1020 eV. Probably the second highest event at ( 2.1×1020 eV
in the data set was seen by the AGASA experiment [20] which meanwhile detected a total of eleven
events above 1020 eV [3]. HiRes sees only two such events with exposure that is estimated twice
the AGASA [3, 38]. The theoretical and astrophysical implications of these events are particularly
significant.
1.1.4 Primary composition
Primary CR composition studies provide vital information that is related with the questions of
their origin and mechanism of acceleration and propagation (ref section 1.1.6). The composition
of CR up to the energy ∼ 1014 eV, those are accessible for direct measurements with detectors on
satellites and balloons (ref section 1.1.2) is known relatively well. The composition in this region
of energies is very similar to the composition of elements in solar system (i. e. Galactic origin)
with some deviations [40].
For CR above ∼ 1014 eV, the traditional methods for obtaining the information about the
chemical composition are the study of correlation between the electron and muon densities, in-
terpretation of size and lateral distribution patterns of electrons and muons in EAS at the detector
level in case of ground arrays and the depth of shower maximum in case of optical observation of
1.1 Discovery and General Nature of Cosmic Rays
9
Figure 1.3: HiRes data versus AGASA data. The fit to the HiRes is done by a two component
model taking into account the GZK effect [37].
EAS [2, 41]. For a given primary, a heavier nucleus produces EAS with a higher muon content
and the shower maximum higher up in the atmosphere as compared to that produces by a proton
primary on average. The higher muon content in the shower of heavy primary is due to the fact
that, as the shower of this primary develops relatively higher up in the atmosphere where the atmosphere is less dense, it is comparatively easier for the charged pions of this shower to decay to
muons before interacting with air nuclei. Another reason of higher content of muon is that when
the primary particle is a heavy nucleus, the first generation mesons are approximately 1/A less
energetic than in proton shower, and have higher decay probability [2, 3]. There is another well
established technique for measurement of CR composition, that is the analysis of the slope of the
Cherenkov light from the EAS [41].
There is an overall indication that, the chemical composition becomes heavier with increasing energies below the knee [42, 43, 44]. Around the knee at ( 4×1015 eV, where compositional
behaviour may play crucial role in attempt to understand the origin and nature of CR in this energy range, the situation becomes less clear and most of the experimental results, such as from the
SOUDAN-2 [45], HEGRA [46], KASCADE [47, 44], EAS-TOP [43, 44], and other experiments
[44] seems to indicate a substantial proton component and no significant increase in primary mass,
whereas the Tibet-Hybrid AS [48] strongly suggested that the knee region is dominated by heavy
1.1 Discovery and General Nature of Cosmic Rays
10
components. Similarly GRAPES-3 [49] also reports the increase of mean mass number gradually
through the knee region. The results from the Dual Imaging Cherenkov Experiment (DICE) [50]
indicates a lighter composition above knee, but evidence for an increasingly heavy composition
above the knee has been reported by the KASCADE [51] and by the HEGRA [52].
Based on the energy dependence of the depth of shower maximum, the predecessor of HiRes,
the Fly’s Eye [35] collaboration reported a composition change from a heavy component below
the ankle to a light component above. This indicates the flux of heavy galactic CR are decreasing
and the new CR component that is responsible for the change of the spectral index at the ankle
consists of proton and possibly He-nuclei. This observation is also reported by the new results of
HiRes working together with CASA/MIA [53] air shower array. However, this is not confirmed by
the AGASA experiment [20, 10]. AGASA collaboration still claims a gradual decrease of the Fe
fraction between energies of 1017.5 eV and 1019 eV. Alternative methods for estimation of the CR
composition, applied by other experimental groups also give a relatively large fraction of heavy
nuclei around 1018 eV [3]. Interestingly there have been suggestion that the observed energy dependence of the depth of shower maximum could be caused by air shower physics rather than an
actual composition change [54].
Observed EAS at the highest energies favours for nucleon primaries, but due to too poor statistics and large fluctuations from shower to shower, the issue is not concluded yet. Consequently
different scenarios have been developed to explain composition of CR in this energy range. According to the top-down scenario (ref section1.1.6) of EHECR (> 1020 eV) origin, the EHECR
primaries to be dominated by photons and neutrinos rather than nucleons. In this regard two special
studies have been performed by the AGASA and the Haverah Park groups using different approach.
AGASA [55] observes the particle density at 100 meters from the shower core that is expected to
be dominated by muons. This density should be much lower in γ-initiated showers. With a new
analysis the Haverah park [56] studied highly inclined air showers expecting fast declining γ-flux
with zenith angle, as the absorption of γ-shower is stronger than nuclei. Both experiments limit the
fraction of γ-rays above 1019 eV at 30% of all CR, which is at higher energies (3 − 4×1019 eV)
is less strict (i.e. 67% − 55%) because of the declining statistics. On the other hand, the highest
energy Fly’s Eye event is claimed to be inconsistent with γ-ray primary [57]. It should be noted,
however, that at least for electromagnetic showers, EAS simulation at EHE is complicated by the
Landou-Pomeranchuk Migdal (LPM) effect and by the influence of geomagnetic field [2]. Furthermore, in the simulations, EAS development depends to some extent on the hadronic interaction
models [2]. To draw a definite conclusion on the composition of the EHECR, we have to wait for
sufficient statistics from the next generation experiments such as Pierre Auger Observatory.
1.1 Discovery and General Nature of Cosmic Rays
11
1.1.5 Anisotropy
The location or concentration of the arrival direction distributions of CR in a particular region
of celestial sphere is specified by the term anisotropy. It is basically connected with UHECR arrival
direction distributions. Because of energy limit and possible sources of UHECR it was assumed
that they must arrive from regions of outer space where there are higher concentration of matter
in the form of astrophysical compact objects. This presumed anisotropy of the arrival direction of
UHECR is another subject of much debate in connection with their origin because of contradictory
experimental findings. AGASA [58] reported on the basis of UHECR data that the distribution is
isotropic on large scale and nonisotropic on small (one degree) scale. There is no preference of
higher event rate coming from the galactic plane or any other known astrophysical concentration
of matter, although an association with the supergalactic plane was reported [59] on the basis of
earlier smaller event sample. However, there are five doublets and a triplet of events arriving at
less than 2.5o from each other. The angular resolution of the detector is below 2o and the statistical
significance of this clustering is ≈ 3σ. Interestingly the clusters do not point any known luminous
astrophysical objects.
The result of AGASA about the clustering of events is not confirm by HiRes [53] and by
the preliminary data of Auger [36]. The much more accurate stereoscopic event reconstruction of
HiRes does not find small scale anisotropy [3]. But according to recent report, the HiRes stereo
event with slightly lower than 4×1019 eV arrives from the direction overlapping on the AGASA’s
triplet region [36] and the arrival directions of these events with energy > 1019 eV are correlated
with BL Lacs [36]. However no conclusion can be drawn from these low statistics’ results, so the
improvement of the statistics and the energy regulation on the relevant data from HiRes and Auger
will lead to solve the controversy over anisotropy of UHECR in near future.
1.1.6 Origin
The origin and the associated acceleration and propagation mechanism of CR ≥ 1014 eV is
still an unsolved problem as a whole and the seriousness of the problem increases as a function of
energy of CR, particularly in EHE range it is a mystery. The issue of the origin of CR between ∼
109 eV to at least near to the knee region is almost settled, as the various studies suggested that
their bulk is confined to the galaxy and is probably produced in Supernova Remnants (SNRs) [2].
The intensities of CR at energies below ∼ 109 eV are temporally correlated with the solar activity
which is a direct evidence for their origin at the sun [2]. The picture dims around the knee and
the ankle. As this region of CR spectrum is very much important regarding the possible sources
of CR which in turn is most significant in astrophysical aspects, many theoretical works have been
concentrating on it and have been developing numbers of theories or models in this connection.
Currently, the most favoured theory of CR acceleration is the so-called Diffusion Shock Ac-
1.1 Discovery and General Nature of Cosmic Rays
12
celeration Mechanism (DSAM) [60], which is based on classical theory of Fermi mechanism of
CR acceleration [61]. According to DSAM, acceleration of CR charged particles occurs through
diffusion shock in astronomical sources. An important feature of DSAM is that particles emerge
out of the acceleration site with a characteristic power law spectrum with an index that depends
only on the shock compression ratio, and not on the shock velocity. One of the most attractive
scenarios in this connection is the acceleration in shock fronts of SNRs [62]. Recently it has received strong support from the direct observation of HE photons originating from SNRs shells by
the HESS γ-ray telescope [63]. A general class of models proposes the knee features as the super
position of components differently accelerated in the source (e.g. [62]), whereas other models explain the spectral changes as a consequence of diffusion effects during the CR propagation in the
Galaxy (e.g. [64]). Erlykin and Wolfendale [65] proposes a single nearby SNR as the origin of
the knee feature. Most recently they reported [66] a number of possibilities to explain the origin
of CR above the knee, such as variety of supernova and hypernovae, pulsars, a giant Galactic halo
and an extragalactic origin by way of shock in the galactic halo. Similarly a number of models
propose pulsars as a source of CR acceleration (e.g. [67]), and some of them proposing a strong
electromagnetic fields for direct particle acceleration (e.g. [68]). Furthermore, there is a class of
exotic models postulating new particles or charged interaction properties explaining the knee, e.g.
by changes in the development of the EAS rather than the CR flux itself [86]. The class of models predicting a rigidity (momentum per unit charge) dependent scaling of the knee position, i.e.,
models based on acceleration effects at the source or diffusion/drift effects during propagation in
the Galaxy getting support from the recent results of the KASCADE experiment [70].
The real mystery is looming with the source of EHECR although there is a general impression
that they have an extragalactic origin. There are lots of setbacks to meet with to arrive at a final
conclusion in this part of CR energies with this idea of their source, some of which are as mentioned below.
Shortly after the discovery of Cosmic Microwave Background Radiation (CMBR), Greisen
in USA [71] and Zatsepin & Kuzmin [72] in the erstwhile USSR published that the CR spectrum
should have an end around energy of 5×1019 eV, because during their propagation in extragalactic
space nucleons lose energy drastically due to photo production of pions when they collide with
photons of CMBR. The mean free path for this collision is ∼ few Mpc [73] as shown in the Figure
1.4. This process effectively limits the path lengths or the possible distances of the sources of
UHECR (> 5×1019 eV) to ≤ 100 Mpc from the earth [2]. This limit is now known as GreisenZatsepin-Kuzmin (GZK) cut-off. That is why, the apparent flux of CR with energy beyond the
GZK cut-off observed from the AGASA (ref section 1.1.3) is a mystery.
The conventional scenario (i.e. within the framework of standard model) of theories of origin
of UHECR is the so-called bottom-up acceleration scenario [2], where it is argued that charged
particles are accelerated from lower energies to the requisite high energies in certain special astro-
1.1 Discovery and General Nature of Cosmic Rays
13
Figure 1.4: The nucleon interaction length (dashed line) and attenuation length (solid line) for
photon-pion production and the proton attenuation length for pair production (thin solid line) in
the combined CMBR [2].
physical environments as mentioned above. Besides the different problems encountered in trying
to explain UHECR acceleration such as coming out of a particle with an energy of the order of
1020 eV from the dense regions in and/or around the source without losing much energy, the main
difficulty of this bottom-up scenario is that, the distance of the most favourable sources of CR at
≥ 1020 eV, the powerful radio galaxies, are located at large cosmological distance >> 100 Mpc
from earth. This is the distance beyond the GZK cut-off distance [2]. Thus there are no obvious
candidate sources at this energy in the distance range permitted by the GZK cut-off [74].
In an attempt to overcome these difficulties in explaining the presence of UHECR at energies
beyond the GZK cut-off, a different scenario known as top-down decay scenario, in contrast to
bttom-up has been postulated and is in developing stage. According to this picture, the UHECR
particles are the decay products of some supermassive X particles of mass >> 1020 eV, which
themselves are either emitted by topological defects such as magnetic monopoles or cosmic strings
that could be produced in the early Universe during symmetry breaking phase transition envisaged
in Grand Unified Theories (GUT) or may have been created directly in the early Universe and
survive until today [2]. The possible decay products of X-particles are photons and neutrinos. So
1.2 High Energy Interactions
14
Figure 1.5: The Hillas diagram showing size and magnetic field strength of possible sites of particle acceleration. Objects below the corresponding diagonal lines cannot accelerate protons (iron
nuclei) to 1020 eV. βc is the characteristics velocity of the magnetic scattering centers [2].
an important signature of these processes would be a specific ratio of CR to neutrinos as well as
TeV γ-rays. The main problem of this scenario in general is its highly model dependent and involvement with untested physics beyond the standard model [2]. Conversely its relation with latter
field brings the ideas of new physics as well as ideas of early Universe cosmology into the realm
of UHECR, which is most exiting and welcome.
1.2 High Energy Interactions
The physics of CR EAS and mechanisms of interactions between different kind of particles
are intimately and inherently related. This relation certainly demands a special attention in the
1.2 High Energy Interactions
15
regions of VH and UH energies, where the direct detection of CR primaries is an impossible task
(ref section 1.1.2). However, for proper modeling of interaction mechanisms at CR energies, lower
energy region is also significant for extrapolation purpose. In the process of EAS development due
to interaction of CR primaries with air particles, a series of secondary interactions take place again
in the atmosphere which include hadron-hadron, hadron-nucleus and nucleus-nucleus interactions,
that lead to the generation of thousands to billions of particles on the observation level, depending
upon the energy of primary particles. As the observable parameters of these EAS generated by CR
in the earth’s atmosphere are the only way to study the nature of CR primaries above ∼ 1014 eV (ref
section 1.1.2), proper study of particle interactions at different CR energies is very much important
to guide the shower prediction strategies. Moreover, with the advent of increasingly sophisticated
air shower experiment with increasing capacity towards UHE region, the need for more detailed
treatment of particles interactions (basically hadronic) becomes urgent [41]. For this purpose, to
fulfil the demand, different particle interaction models or event generators have been developed
over years by different groups, those are based on the extrapolation of the data obtained from the
fixed target and accelerator experiments at highest available energies to CR energies. Such widely
used models of particle interaction in the field of CR are, e.g., QGSJET [75], SIBYLL [76], and
DPMJET [77] (ref section 2.7.1). Although these models account all possible information from
fixed target and collider experiments for their development in association with their basic theory
(pQCD) to be used for EAS study at CR energies, there is however an important difference, that
is, for studying the CR cascade, the main interest is in the forward fragmentation region of particle
collision, on the other hand at accelerator the best study is made in the central region. So far
it is found that QGSJET model is extremely successful model in explaining the HE interactions,
although not fully consistent with the description of data [41]. In the following we shall discuss
some general features of CR interaction at high energies.
1.2.1 General characteristics of high energy interactions
When an astroparticle enters the earth’s atmosphere, it interacts strongly with air particles
at random and the position of interaction can be decided on a statistical basis by its inelastic cross
section on the target particle [6]. In HE collision a part of the primary particle energy is lost in
the production of secondary particles. The secondary particles are predominantly pions, kaons
and nucleons-antinucleons pairs and in a small measure other exotic particles. The fraction of the
energy lost by the primary particle is called the inelasticity. These secondaries themselves being
hadrons, collide further with air nuclei and give rise to succeeding generations of nuclear interacting particles. The number of secondaries called the total multiplicity, slowly increases with the
energy of the interaction. Since the density of the atmosphere is low at high altitudes where high
energy interaction takes place, the particles traverse a considerable distance before encountering
with other air nuclei, some of them like charged pions and kaons decay and give rise to µ-mesons,
1.2 High Energy Interactions
16
electrons and neutrinos. The neutral pions decay into γ-rays which initiate electromagnetic cascade alone. The large distances between the successive interactions also results the lateral spread
of the particles to tens of meters to hundreds of meters [6]. This lateral spread of EAS particles
can be measured with the transverse momenta, p⊥ as they acquire during their development. The
direction of primary particle can be determined from another parameter, known as the longitudinal
momenta pL of the EAS particles. If θ is the angle by which a secondary particle is deflected
with respect to primary particle direction, then this is related to momenta by tanθ = p⊥ /pL . The
√
distribution of pL is expressed in terms of the Feynman variable xF = 2pc.m
L / s, where s is the
square of the centre of mass (c.m) energy of the interacting particles. This variable also gives
a relation between the laboratory energy Elab of the secondary particles with primary energy E0
of the projectile particle by Elab ( E0 xF . There is another variable, known as the rapidity, fre-
quently used to describe the behaviour of particles in inclusively measured reactions, defined by
y =
1
0 +pL
log E
,
2
E0 −pL
which corresponds to tanh(y) = pL /E0 . The handy variable to approximate
the rapidity is the pseudorapidity, if the mass and momentum of a particle are not known. It is
an angular variable defined by η = − log tan(θ/2). There is spread in the arrival times of the
particles at the observational level, because of different velocities of particles of different mass and
the momentum distribution. The photons which traverse with the velocity of light will arrive first,
then the electrons, then the muons and then the nucleons and anti-nucleons and the other heavy
particles. The spread in the arrival time distribution is also controlled by the extent of scattering
process (i.e. on primary particle energy and density of the atmosphere) and the distance from the
core at which the times are measured. Furthermore, the details of the lateral distribution, the time
structure, the number and energy spectra of the different components naturally depend on the collision characteristics at different energies for different types of primaries, as well as on the decay
characteristics of the unstable particles [6].
There are many simple features of hadron-hadron interactions at fixed target energies, some of
them are [77] :
(i) The average charged multiplicity increases logarithmically with energy.
(ii) The multiplicity distribution follows Koba-Nielsen-Olesen (KNO) scaling, which implies
that the shape of multiplicity distribution should asymptotically become energy independent [89].
(iii) Approximately constant height of rapidity plateau.
(iv) Feynman scaling of the longitudinal momentum.
(v) Average p⊥ of secondary particles is approximately independent of energy.
On the other hand the data from the collider experiments show a more complicated picture of
hadron-hadron interactions characteristics, which are very much important in CR studies as stated
above, such as [77] :
(i) The charged multiplicity increases faster than in proportion with energy.
(ii) The height of the rapidity distribution in the central region increases approximately loga-
1.2 High Energy Interactions
17
rithmically with energy.
(iii) KNO scaling of the multiplicity distribution is violated. The distribution becomes wider
with increasing importance of the tail of high multiplicity events.
(iv) Average p⊥ increases with energy. There is a positive correlation between p⊥ of charged
particles and the multiplicity of the events.
(v) There are high p⊥ jets. The production of small (several GeV) jets, called minijet, becomes
important as the interaction energy increases. These jets contribute to the rise of average p⊥ , to rise
of the central rapidity density, to total multiplicity, and to observed correlation between < p⊥ >
and multiplicity with energy.
All these characteristics of interactions have been confirmed also by CR experiments, with an
additional property that, very high energy collision is not totally inelastic and a leading particle
that carried away considerable amount of the colliding particle energy persists in the core of the
air showers [6]. Interaction characteristics of CR beyond 1015 eV could not be supported by collider experiments, because this range is above their highest energy end. From the CR experiments
it was observed that the interaction characteristics changes drastically beyond 1015 eV, e.g. very
steep spectra of gamma and electron at energy > 1016 eV compared to the spectra at the lower
energies [6]. This type of behaviour of interaction, obviously demands the development of special models of the particles interaction based on the theory of modern HE particle physics and the
available collider data to explain the phenomena of VHECR to UHECR physics.
1.2.2 Production of short-lived heavy particles in CR interaction
With the fluxes of CR primaries whose kinetic energy goes beyond 1020 eV (although the
flux with energy > 1019 eV are extremely small), the interactions of these celestial particles in
the earth’s atmosphere at various energies provide an idea to consider the earth itself as a supergiant universal accelerator which is able to produced numerous physical processes, most of which
are beyond human capabilities to produced. Notwithstanding, thorough and careful studies of the
signatures of theses physical processes may lead to solve some mysterious problems of existing
physics and physics beyond that. Obviously for such purpose, we must have to develop more efficient detection techniques to keep track of the signatures of various physical processes that take
place in the atmosphere which may appear most significant for us.
One of such processes, we claim, that may be taking place in the earth’s atmosphere due to interactions of UHECR with air nuclei, is the production of Higgs bosons through vacuum excitation
(ref chapter 3). The fundamental idea of this mechanism is that during UHECR interaction with
air nuclei, a fraction of interaction energy is transferred to a microscopic volume of neighbouring
vacuum. This process excites that part of vacuum leading to formation of a bubble of thermal energy. This bubble immediately decays to Higgs bosons, due to Salam-Weinberg phase transition,
the signature of which is the production of excessive prompt muon multiplicity in UHECR interac-
1.3 Scientific Motivation
18
tions [78, 79]. There are some examples of unusual historical events in support of this mechanism
(ref section 3.6).
The basic theoretical formalism of Higgs boson production through vacuum excitation was
done by Mishra et al. [80] based on the nonperturbative mechanism of thermofield theory [81].
Kalaynee Boruah [79] first used this idea to study the production of this scalar particle in UHECR
interactions with air nuclei as a consequence of a part of energy transfer to nearby vacuum, by
using the method of MC simulation (ref chapter 2). The details of this process is discussed in the
chapter 3.
Another process of our interest is the production of charmed hadrons in CR interactions with
air particles. The process of the production of charmed hadrons at CR energies is already established by many experiments (e.g. [84, 85]) and by many theoretical works (e.g. [82, 83]) as
a source of prompt muons and neutrinos. In chapter 4 we have discussed in detail the charmed
hadroproduction in pp collisions in the framework of QGSJET model [75].
1.3 Scientific Motivation
It has already become vivid that, the UHECR and their interactions in the atmosphere is a
subject of high potential, and perhaps this subject has lots of hidden capacity to deal with the most
of the existing unsolved problems of physics. To explore these possibilities with gradually improving experimental techniques, the MC simulation is only most powerful, trusted and recognised but
challenging numerical method that make a bridge between pure theory and experiment to show
the real path to be followed by the experiment in this energy range of CR spectrum in particular
(ref chapter 2). One of the most challenging unsolved problem of particle physics till today is the
discovery of Standard Model (SM) Higgs boson. This particle is responsible for the spontaneous
symmetry breaking of electroweak interaction (ref chapter 3). So far efforts of various missions
to discover this scalar became fruitless and some mission in this regards are still continuing (ref
chapter 3).
As already mentioned in the section 1.2.2 that there are very high chances of production of
this scalar particle in the earth atmosphere due to various processes of interactions of UHECR with
air particles. The muons of very high energy can be considered as a signatures of the production
of this illusive particle as it decays finally into leptons of very high energy. Thus another aspect of
search of Higgs boson in UHECR interactions is that, it may appear as one of the potential sources
of very HE leptons or prompt leptons (muons and neutrinos), which is also not solved yet completely (ref chapter 3). So in connection with the Higgs boson production in UHECR interactions
a complete theoretical works with MC simulation is most necessary.
Production of charmed particles in CR interactions is another important context as they are
recognised as notable candidates or sources for prompt muons and neutrinos [82] in the atmo-
1.4 This Work Plan
19
sphere. However a considerable attention is not focused on this important part of interaction in
view of most rare process, and as a consequence this process has not been even included in the
most popular CR MC simulation code, e.g. in CORSIKA [87]. But this process is being included
in the various HE interaction models such as QGSJET [75] and DPMJET [77]. It should be noted
that, although the charmed particle production (as well as Higgs boson) in atmospheric CR interactions is a rare process, it may change the traditional shower behaviour completely whenever it
occurs. So the process of charmed particle production should be included in the CR simulation
codes available at present. In this consideration, the study of the reliability of the models predictions as compared to available data is also very important.
In the data of UHECR flux there is a wide range of uncertainty, particularly it becomes prominent in case of small detector setup, because of extreme low flux of CR in this range. That is why
it is wisely advisable and utmost necessary to analyse these types of data using a standard MC
simulation code like CORSIKA [87]. GU Miniarray [4] was a small detectors setup employed to
detect UHECR from 1996 to 1998 and was reported of detection of CR EAS of energy up to ∼
1019 eV by using a old method of analysis (ref chapter 5). In spite of having a very small setup, its
claim of detection of CR at that UHE is quite interesting and so its data should better be reanalysed
using the standard simulation code CORSIKA [87]. This process of analysis will provide also an
opportunity to use CORSIKA for the data analysis of such small detectors setup, which is actually
meant for and used so far for huge detectors setups (ref chapter 2).
Above three are our points of motivation of the present thesis work.
1.4 This Work Plan
The present thesis is the conglomeration of our work on UHECR and the physical processes
associated with their interactions in the atmosphere leading to the production of Higgs bosons and
charmed hadrons that been have done using MC simulation method. Here we have developed and
discussed the related theoretical framework for the Higgs boson and charmed hadron production,
and for the reanalysis of GU miniarray data using CORSIKA [87]. We organise our thesis work as
follows :
In chapter 2 we have discussed the basic idea of MC method and its effectiveness for analysis
EAS development. Also we have discussed here about the CORSIKA [87] simulation code with
associated HE interaction models.
In chapter 3, as published in [78], we have included basic ideas of Higgs boson theory, thermofield theory and process of vacuum excitation that leads to the production of Higgs bosons in
UHECR according to our picture. For MC simulation of this Higgs boson production signature,
a Higgs boson production model is also introduced along with an interaction model. The results
of the Higgs boson model and the simulation code for the signature of this particle production are
1.4 This Work Plan
20
discussed in the relevant part of this chapter.
The charmed hadrons production in pp collision in the framework of QGSJET model [75]
is incorporated in the chapter 4, a part of which is already published in [88]. Here a brief introduction is given about the formulation of QGSJET model. The general features of the charmed
hadrons that are produced in pp collision using QGSJET model are discussed in detail long with
their asymmetric production behaviour.
The chapter 5, as published in [4], describes the detail strategies used in the reanalysis of GU
Miniarray data along with a brief idea of the experiment. At the end we show our reanalysis results
in comparison with the results of other CR research groups in the world.
The summary of the whole work is given in the chapter 6.
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Chapter 2
MONTE CARLO SIMULATION
TECHNIQUE
2.1 Introduction
The Monte Carlo (MC) simulation technique is an extremely versatile universal numerical
method of solving various problems of science and engineering by means of random sampling of
variables distributed uniformly between 0 and 1. This technique is formally assumed to born in
1949 after the publication of the paper The Monte Carlo method by N. Metropolis and S. Ulam
[1] in the journal of American Statistical Association. However the theoretical foundation of it has
been known from long time. American mathematician John Von Neumann and Stanislav Ulam are
considered as pioneers to have developed this method [2]. The term Monte Carlo was introduced
by Von Neumann and Ulam during second world war as a code word for the secret work at Los
Alamos, because this method was then applied to problems related to the atom bomb, like random
neutron diffusion in a fissionable materials. The Monte Carlo term is due to the name of the city
of Monte Carlo in Monaco, which was famous for its casinos and so the term was suggested by the
gambling casinos of the city. However until the advent of electronic computer this technique could
not be used on any significant scale as of today because, manual simulation of random variables is
a very time and labour consuming procedure.
The most important aspect of the physics research is the agreement between the experimental
findings and the theoretical predictions. Many times experimental physicists find difficult to do this
job because of the very complex nature of a theory in the process of its implementation, although
it may look simple from the formal point of view, especially in the case of current HE physics
panorama. Moreover, for the comparison of an experimental result with a theory it is essential to
consider the experimental situation including response of detector(s) used in the experiment and
the nature of the material(s) in the detector(s). Under this situation, the MC simulation gives a way
to overcome the compelling hurdle by allowing us to reproduce similar effects from the theory that
26
2.2 Basic Concept of Monte Carlo Simulation Technique
27
a real experiment could produce. Thus the MC simulation consists of two processes of simulations,
the simulation of the physical process (theory) and the detection process (detector) making use of
known properties of matter [3].
There are two distinctive features of the MC technique. First, its simple structure of the
computation algorithm. As a rule, a program is prepared to perform only one random trial, that is
repeated N times, each trial being independent of all others and the results of all trials are averaged.
That is why, the MC method is sometimes called as method of statistical trials. The second feature
of this technique is that, the error of calculation is proportional to
!
k/N , where k is a constant and
N is the number of trials [2]. Since the number of trials can not be increased after certain limit,
because of amount of work factor connected with it, thus the accuracy of this technique can be
considerably improved for many problems, only by a proper choice of the computation technique
with an appreciably smaller values of k [2].
The MC method has a wide range of applicability covering almost all branches of science and
technology. According to the general impression, this technique allows to simulate any process,
whose development is affected by random factors, but the fact is that it can be applied to many
mathematical problems not affected by any random influences, as they can be connected with an
artificially constructed probabilistic model (even more than one) making possible the solution of
these problems [2].
2.2 Basic Concept of Monte Carlo Simulation Technique
Simulation is a process of generating artificial (or pseudo or fictitious or virtual) events or phenomena in resemblance of real physical one(s) on the basis of the model(s) of a complex system
that is (are) governed by a sequence of choices selected with random variables (ref section 2.3).
The concept of simulation is based on the fact that many processes, real physical or imaginary
ones, can be idealizes as a sequence of choices or distributions. If it is possible to approximate
each choice or distribution by some probability function, then one can formulate a model which
can generate (i.e. simulate) the process. Many real processes of nature can be approximated in
this way. For specification, we consider the case of HE physics, where relativistic and quantum
effects become essential and the last fact introduces a basic uncertainty in the observation. In other
words, for a given initial state, a process in HE physics has no deterministic solution for the final
state and has to be answered in terms of probabilities. This implies the stochastic nature of real
physical processes in which the studies are made based on probabilities [4]. As an example, if we
make an experiment searching for final state of the process e+ e− → µ+ µ− and we consider the
angular distribution of the µ− in the centre of mass frame of the colliding e+ e− , it comes out an
A + Bcos2 θ distribution. This is one of the typical random distributions that we can find in HE
physics. It states what is the probability of finding a µ− produced in a given angular range. The
2.3 Random Variables
28
repetition of the experiment will end up with similar distributions. Similarly, some other simple
examples of natural processes whose final states are based on probabilistic distribution are : the
decay of an unstable particle in a particular time, the emission of some particles from a bombarded
target, the multiple scattering of a fast moving charged particle in a dense medium etc. If in the
process of simulation with such model(s), the selection of the particular sequences of choices or
distributions is done by random variables distributed uniformly between 0 and 1, the simulation
technique is known as MC simulation technique.
Thus in MC simulation one generates many virtual or artificial events according to the simulating model. Each event is the results of a particular sequence of choices. These events are then
counted, classified and distributed according to criteria as if they were real events, then one can
make comparisons with existing real data, or predictions of what should be the outcome of a given
real experiments, provided that the established model simulates nature sufficiently well. For comparison with any real data, the statistical error in both the real events as well as pseudo MC events
must be always considered. In this respect, the more number of pseudo events are required than
the number of available real events, as the error in MC events is calculated by
!
k/N , where k is a
constant, depends upon the type of model and N is number of events (ref section 2.1). In general
one has to make various approximation in the model for the sake of simplicity and for the limited
knowledge of a natural process. The deviation from similarity to nature is thus compensated by a
great number of generated events and smaller statistical error.
Since the type of choice of a variable in MC simulation is a random choice in accordance with
the chosen probability function for that variable, in the following sections we shall discuss about
random variables, the related probability functions and generators of random variables.
2.3 Random Variables
In the common sense, as the name implies, random variables are those, which assume values
that can not be specified or predicted previously. On the other hand, in mathematics this term
is used in a broad sense. In this sense, although it is not possible to know what values will this
variables take in a given case, it is possible to know what values it can assume and what are the
respective probabilities. Thus, to present a random variable we must indicate what values it can
assume and what are the probabilities of these values. However, although the probability function
may be modified or parametrized by previous choices, the values of the variables must bot be
predictable from the results of the previous choices. To satisfy this characteristics, the probability
interval is compared with a uniform random number distribution,
P(ν)dν = kdx,
(2.1)
i.e. the probability that the variable has a value between ν and ν + dν is proportional to the
probability that the random number RN has a value between x and x + dx. Equation (2.1) is the
2.3 Random Variables
29
fundamental formula of the MC simulation technique, where k is simply a constant of normalisation found by integrating equation (2.1) over the allowed range of ν and x, since we choose a
uniform random number distribution. It should be noted that, a particular set of values of a random
variable is called random numbers denoted by RN .
It is convenient to consider the following two types of random variables with associated probability distributions as determined from their relation with the random number distribution.
2.3.1 Discrete random variables
Discrete random variable ν takes values from a discrete set of numbers and it is specified by
the table [2],

ν=
x1
x2
. . . .
xn
p(x1 ) p(x2 ) . . . . p(xn )

,
(2.2)
where x1 , x2 , ...., xn are the possible values of the variable ν and p(x1 ), p(x2 ), ...., p(xn ) are the
corresponding probabilities, i.e. the probability that the random variable ν will be equal to xi is
p(xi ). In other words p(xi ) also gives the frequency density of ν at each point xi . The table 2.2
is called the distribution table of the random variable. In general, the values x1 , x2 , ...., xn can be
arbitrary. But the probabilities p(x1 ), p(x2 ), ...., p(xn ) must satisfy two conditions :
(i) all p(xi ) are positive, i.e. p(xi ) > 0, and
(ii) the sum of all p(xi ) is equal to unity, known as normalization condition,
n
&
p(xi ) = 1.
i=1
The second condition means that in each trial ν necessarily takes one of the values x1 , x2 , ....,
xn . We can define the cumulative distribution function of discrete random variable ν as,
P{xl ≤ ν ≤ xm } =
m
&
p(xi ).
(2.3)
i=l
This equation gives the probability of finding the discrete random variable ν in between xl and
xm . The mathematical expectation, or the expected values of this variable is expressed as,
+ν, =
n
&
xi p(xi ).
(2.4)
i=1
Its physical meaning is that the values xi associated with greater probabilities are included into
the sum with larger weights. If c is an arbitrary non-random number, the expected value follows
that,
+ν + c, = +ν, + c,
(2.5)
+cν, = c +ν, .
(2.6)
and,
2.3 Random Variables
30
Similarly if ν and η are two arbitrary random variables, then,
+ν + η, = +ν, + +η, .
(2.7)
The expected value of the square of the deviation of the random variable ν from its mean value
+ν, is called the variance of the random variable ν and is expressed as,
'
(
'
(
∆ν = (ν − +ν,)2 ≈ ν 2 − +ν,2 > 0.
(2.8)
In case of many observations of the variables ν, the arithmetic mean of the set of values ν1 , ν2 ,
....., νN (each of these will be equal to one of the numbers x1 , x2 , ...., xn ) will be close to +ν,, i.e.,
1
(ν1 + ν2 + ...... + νN ) ≈ +ν, .
N
(2.9)
The variance ∆ν characterizes the spreading of individual values ν around the mean value +ν,.
For any nonarbitrary number variance obeys the properties,
∆ (ν + c) = ∆ν,
∆ (cν) = c2 ∆ν.
(2.10)
The concept of independence of random variables plays an important role in the probability
theory. The simple form of this concept can be expressed for two independent random variables ν
and η as,
+νη, = +ν, +η, ,
∆ (ν + η) = ∆ν + ∆η.
(2.11)
The most common probability distribution of discrete random variable is the Binomial Distribution (BD), in which the outcome of each independent trial is dichotomous, i.e. success or
failure, e.g. yes or no, head or tail, hit or miss etc. The probability of r successes in N trials
regardless of the order in which they occur, is given by the BD function,
p(r) =
N!
ρr (1 − ρ)N −r ,
r!(N − r)!
(2.12)
where ρ is the probability of success in a single trial. The mean or expected value and the variance
of this distribution are respectively,
µ=<ν>=
&
rp(r) = N ρ.
(2.13)
r
σ 2 = ∆ν =
&
r
(r − µ)2 p(r) = N ρ(1 − ρ).
(2.14)
In the limiting case, when ρ → 0 and N → ∞, the BD is referred as Poisson Distribution
(PD) and the distribution function is,
p(r) =
µr e−µ
.
r!
The PD is one of the most frequently encountered distribution in HE physics.
(2.15)
2.3 Random Variables
31
2.3.2 Continuous random variables
A random variable ν is said to be continuous if it takes any value out of a certain interval, say
(a, b). A continuous random variable ν is defined by specifying the interval (a, b) of its variation,
and the function p(x), called the probability density of the random variable ν or the distribution
density of ν [2].
The physical meaning of p(x) is that, if (a# , b# ) be an arbitrary interval contained within (a, b)
(i.e. a# ≤ a, b# ≤ b), the probability that ν falls inside (a# , b# ) is given by the integral,
#
#
P{a ≤ ν ≤ b } =
) b!
a!
p(x)dx.
(2.16)
The integral (2.16) is equal to the shaded area as shown in the Figure 2.1. It is the cumulative
distribution function of the continuous random variable ν. The set of values of ν can be any
interval, from a = −∞ to b = ∞, but the density p(x) must satisfy two conditions, viz.,
y
y = p(x)
o
a
/
/
a b
b
x
Figure 2.1: Physical meaning of a probability density distribution function p(x).
(i) the density p(x) is positive, i.e.
p (x) > 0,
(2.17)
and
(ii) the integral of the density p(x) over the whole interval (a, b) is equal to 1 :
) b
a
p(x)dx = 1.
(2.18)
2.3 Random Variables
32
This is the normalisation condition of the continuous random variable ν. The expected value
of the continuous random variable is given by,
*b
xp(x)dx
+ν, = *a b
=
a p(x)dx
) b
a
xp(x)dx.
(2.19)
This equation implies that any value of x from the interval (a, b), entering the integral with
the weight p(x) can be the value of ν. All the results of section 2.3.1 starting with equation (2.5)
to equation (2.11) are valid for continuous random variables also. A random variable, defined in
the interval (0, 1) with the density p(x) = 1 is said to be uniformly distributed in (0, 1) as shown
in Figure 2.2. The expected value of uniformly distributed continuous random variables is
the variance is
1
.
12
1
2
and
The uniformly distributed random variables have particular interest in MC
simulation.
y
y = p(x)
1
o
1
x
Figure 2.2: Uniformly distributed random variable in (0, 1).
A continuous random variable ν, defined on the entire axis (−∞, ∞) and characterized by the
density,
(x−µ)2
1
e− 2σ2 ,
(2.20)
2πσ
where µ and σ > 0 are numerical parameters, is said to be Normal (Gaussian) random variable.
p (x) = √
The distribution given by above equation is known as Gusssian (Normal) Distribution (GD). It
is actually the approximation of the BD given by equation (2.12) for large number of trail N and
moderate value of ρ. The parameter µ does not affect the shape of the p(x) curve, its variation
only shifts the curve as a whole along x-axis. On the contrary, variation of σ changes the shape
of the curve, as it is easily seen that a decrease of σ is accompanied by an increase of maximum
value of p(x). But according to the condition (2.18) the entire area below the p(x) curve is equal
2.4 Generators of Random Variables
33
to unity. Therefore the curve will stretch upward in the vicinity of x = µ, but will decrease at all
sufficiently large values of x. It can be proved that, +ν, = µ, and ∆ν = σ 2 .
Normal random variables are frequently encountered in investigation of most diverse prob-
lems as these variables have tendency that are correlated with natural phenomena. For example,
the experimental error (δ) is usually a normal random variable. If there is no systematic error (bias)
√
then µ = +δ, = 0. The value σ = ∆δ, called the standard deviation, characterizes the error of the
method of measurement. For this reason the normal random variables are used in most of the MC
models.
There are two types of distribution functions of continuous random variables, one is conveniently analytically integrable and other is not conveniently analytically integrable. Among second
type relation we find many of the continuous variables in a physical process viz., cross-sections,
angular distributions, centre of mass spectra for production of, or decay into three or more bodies
etc.
2.4 Generators of Random Variables
The random variables described above represent an ideal mathematical concept. They will
have physical significance only, when they can be used to describe satisfactorily some natural
phenomenon found out experimentally. However such a description is always approximate. Moreover a random variable which describes quite satisfactorily a certain physical quantity in one type
of phenomena may prove quite an unsatisfactory when some other phenomena are being investigated. Thus the generation of proper random variables for a particular phenomenon is an important
task of MC simulation [2].
The method of generating random variables are usually divided into three types, viz., tables of
random number, random number generators and the method of pseudo-random numbers. First two
methods are manual and initially these methods were used to generate truly random numbers, e.g.,
coin flopping, roulette wheel, noise signals in electric tubes etc. These methods were too slow for
general use (second one is however more efficient than first one) and could not be reproduced [2].
So we shall not discuss these two methods here and we will concentrate only on pseudo-random
numbers, which is nowadays used entirely in simulation works all over the world because of its
advantage of quick reproduction and very easy arithmetic operation in computers.
2.4.1 Pseudo-random numbers
The numbers calculated by means of some specific formula and simulating the values of the
random variables RN are called pseudo-random numbers. They are called pseudo because they are
not truly random as they are generated by a completely systematic arithmetic process. The series
of numbers obtained in this way are uniquely determined by the starting value [2].
2.5 Transformation of Random Variables
34
The most widespread algorithm for generation of Pseudo-random numbers was suggested by
D. Lehmer, known as the Congruential Method or the Method of Residues. The most frequently
used functional form of this algorithm (some times known as mixed congruential method) is,
Rk+1 = (mRk + µ) (modN ) ,
(2.21)
where Rk is the k-th random number, m is a multiplier, N is a positive integer, µ is another positive
integer. The (k +1)th element is obtained as a remainder when (mRk + µ) is divided by N . For the
given values of m, N and the first member of Rk , a sequence of numbers, fairly evenly distributed
over the range (0, N ) is obtained. A proper choice of the constants can give random sequence with
period as large as desirable. Such modular multiplier methods are most commonly used as random
number generators at present.
2.5 Transformation of Random Variables
Transformation of random variables is the most essential part of MC simulation as already
mentioned that the solution of various problems requires simulation of various random variables.
This can be achieved by transforming or mapping (sometimes also called dragging) a selected (i.e.
standard) random variable on to the parameter space. Usually this role is played by the random
variable RN , uniformly distributed in (0, 1) (ref section 2.3) [2].
To draw the value of the discrete random variable ν with the distribution given by the table
2
1
0
p{x1}
n
p{x1}+p{x2}
1 − p{xn}
1
Figure 2.3: Method of transformation of discrete random variables.
2.2, let us now consider the interval 0 < y < 1 and divide it into n intervals with lengths equal
to p(x1 ), p(x2 ), ......, p(xn ). The coordinates of division points apparently will be y1 = p(x1 ),
y2 = p(x1 ) + p(x2 ), y3 = p(x1 ) + p(x2 ) + p(x3 ), ........, yn = p(x1 ) + p(x2 ) + ......+ p(xn−1 ). Now
enumerating these intervals by the numbers 1, 2,........, n as shown in the Figure 2.3 for drawing
a value of ν, we shall have to select a value of RN and fix the point with y = RN . If this point
falls into the ith interval we shall assume that ν = xi , since the random variable RN is distributed
uniformly within (0, 1), the probability of RN lying within one of the interval is equal to the length
of this interval. Hence,
P{0 ≤ RN ≤ p(x1 )} = p(x1 ),
P{p(x1 ) ≤ RN ≤ p(x1 ) + p(x2 )} = p(x2 ),
2.5 Transformation of Random Variables
35
............................................
P{p(x1 ) + p(x2 ) + ........ + p(xn−1 ) ≤ RN ≤ 1} = p(xn ).
According to the procedure ν = xi if,
p(x1 ) + p(x2 ) + ........ + p(xi−1 ) ≤ RN ≤ p(x1 ) + p(x2 ) + ........ + p(xi ),
and the probability of this event is equal to p(xi ). It should be noted that the order of enumerating
the numbers x1 , x2 , x3 , ......., xn in the distribution of ν can be arbitrary, but it must be fixed before
the drawing.
For example, the discrete random variables are chosen for MC simulation of the number of
particles produced in a collision, or the decay modes when more than one is possible. The random
number used here is still continuously distributed, so that the n discrete choices, each with relative
probability p(xi ) (i = 1, 2, ...n) are to be compared with the partial integrals on the RN distribution.
Since the random number must fall in between 0 and 1, so we will get,
p(x1 ) + p(x2 ) + ........ + p(xn ) =
) p(x1 )
0
dx +
) p(x1 )+p(x2 )
p(x1 )
dx + ........ +
) 1
1−p(xn )
dx = 1
= {p(x1 ) − 0} + {p(x1 ) + p(x2 ) − p(x1 )} + .....
In other words if a RN is found between 0 and p(x1 ), the variable has its first value, between
p(x1 ) and p(x1 ) + p(x2 ), the second value, and so on. However, the list of values may depend upon
previous choices or calculations, such as the multiplicity of pion production as a function of center
of mass energy.
y
1
R
o a
b
Figure 2.4: Inverse transformation method.
x
2.5 Transformation of Random Variables
36
For generating values of a continuous random variable with an analytically integrable probability density distribution function, let us assume a continuous random variable ν distributed in the
interval (a, b) with the density p(x). The values of ν can be obtained from the equation,
F (ν) = Nr
) ν
a
p (x) dx = RN ,
(2.22)
where Nr is the normalisation constant, can be evaluated from the equation,
Nr
) b
a
p (x) dx = 1.
(2.23)
It follows from the general properties of density, expressed by equations (2.17) and (2.18) that
F (a) = 0, F (b) = 1 and F# (x) = p(x) > 0. From equations (2.22) and (2.23) it is clear that all
values of ν can be obtained by the inverse transformation of the function F (ν), i.e.,
ν = F −1 (R) .
(2.24)
So this method is also sometimes known as inverse transformation method. This method can
be applied for generation of continuous random variable ν for any interval in between (a, b).
If the density function is not integrable analytically Neumann’s method is used. To explain
this method let us assume that the random variable ν is specified on a finite interval (a, b) and its
density is limited by,
p (x) ≤ ym .
The variable ν may be drawn by first selecting two values R1 and R2 of the random variable
RN and generate a random point T(k1 , k11 ) with coordinates,
k1 = a + R1 (b − a), and k11 = R2 ym .
Then if the point T lies below the curve y = p(x), assume ν = k1 , if the point T is above
the curve y = p(x), reject the pair (R1 , R2 ) and repeat the procedure again. The efficiency of this
method depends on the ratio of the area under the curve to unity. If the curve is too narrow and
steep, the method is very inefficient and takes too many trials [5, 2].
If the result of integration is transcendental equation, iterative procedures have to be used.
For example, in case of the transverse momentum distribution, the solution has to be obtained by
iterative procedure. The transverse momentum distribution is given by,
p (x) dx = xe−x dx.
(2.25)
Here x = p⊥ /p0 . The equation in this case is,
) x
0
xe−x dx = R,
(2.26)
2.5 Transformation of Random Variables
37
y
y
m
T
k11
o
a
k1
b
x
Figure 2.5: The Neumann’s method.
which gives, x = ln
method.
+
x+1
R
,
. This is solved by iterative procedure, e.g., using Newton-Raphson
Another method that is generally used is to create a look-up table of values of the random
variables in the required range such that a cell picked with a random numbers gives the values of
the random variable. This method necessarily yield discrete values and the steps have to be taken
carefully so that the tail of the function is also sampled adequately [5, 2].
In case of normal random variable, the values of variable can be generated by the equation (for
µ = 0, σ = 1),
1 ) ν − x2
√
e 2 dx = R.
2π −∞
(2.27)
However, this equation can not be solved explicitly because of the complex form of the Gaussian density function and for the infinite interval of possible values of ν. But this function can be
generated by choosing two random numbers R1 and R2 from which two independent values of the
parameter can be obtain as [5],
x1 =
x2 =
√
√
−2lnR1 sin(2πR2 ),
−2lnR1 cos(2πR2 ).
(2.28)
Both values x1 and x2 are normally distributed. This method is known as Box-Muller method.
Figure 2.6 shows the simulated result of the Gaussian distribution using Box-Muller method.
In the following section we shall discuss the general picture of the MC simulation of EAS
2.6 Monte Carlo Simulation of EAS
38
60
Frequency
50
40
30
20
10
0
-4
-3
-2
-1
0
1
2
3
4
Arbitrary variable
Figure 2.6: Monte Carlo Simulation of Gaussian Distribution.
without going into details of processes for the same. We refer to the chapter 3 for detailed MC
simulation of the process of our interest.
2.6 Monte Carlo Simulation of EAS
It is already mentioned in the chapter 1 that, the only way to study CR of energy above 1015 eV
is the EAS generated by them in the earth atmosphere. In EAS experiments the measurable quantities such as, secondary particles numbers at detector level, their lateral and energy distributions,
and arrival times, and Cherenkov light production (usually shower maximum) etc. are used for
mapping of primary particle proprieties, e.g. energy, particle type and mass number, and the angle
of incidence. The analysis and interpretation of such experimental EAS data requires the detailed
MC simulation of EAS, to see the detailed course of EAS production processes with respect to
primary energy, particle type and the direction of incidence. The detailed MC simulation of EAS
development in the atmosphere is a very complicated process because it requires the consideration of numerous factors affecting the air showers of CR, so needs more time effective effort. In
general, such a simulation includes the simulation of high and low energy hadronic multiparticle
production, electromagnetic processes in air such as bramsstrahlung and pair production and also
the calculation of the corresponding detector response.
The backbone of the MC simulation of EAS is the hadronic interaction models, because these
2.6 Monte Carlo Simulation of EAS
39
models only generate necessary events required for multiparticle production in the atmosphere as
produced by the CR primaries. Moreover, the complete calculation of air showers involves the
description of hadronic multiparticle production at energies extending from the particle production threshold to the energy of the incident primary particle. Similarly the interpretation of the
observed inclusive flux of different particle types in the atmosphere is impossible without detailed
knowledge of hadronic multiparticle production. The hadronic interaction models are based on the
different phenomenological theories of hadronic interactions at different energies. The HE interaction part of these models rely on the extrapolation of available experimental data of the present
accelerator experiments to a wide range to cover the highest recorded energies of CR events.
Currently most extensively used standard MC EAS simulation packages are CORSIKA [6]
(ref section 2.7.1) and AIRES [7], and for simulating detector response is GEANT [8]. Moreover,
there is an older package known as MOCCA [9], on whose predictions, many of the available
primary CR composition and all particle flux measurements are still based [10].
The AIRES (Air-shower Extended Simulation) system originally designed on the basis of the
MOCCA code [9], where the hadronic collisions are processed by means of three models, depending on the energy of the projectile. For collisions with energy less than ≈ 100 GeV, an extension of
the Hillas Splitting Algorithm (EHAS) [19] is used, while for higher energies it offers for possible
selection between QGSJET [11] and SIBYLL [12]. More recently AIRES [7] is used to estimate
the influence of diffractive processes in the final shower observations [20] and to study in detail
the characteristics of shower initiated by photons in connection with CR composition analysis at
the highest energies [21].
The MC simulation of EAS faces some drawbacks also. For example, in the collision experiments, whose data are used for extrapolating to the higher energies in the hadronic interaction
models, the particles produced in the forward region of the interaction are not registered, while
they are responsible for most of the shower characteristics. Similarly the atmospheric target are
light nuclei which have not been studied in collider experiments. Moreover, air shower simulation
becomes a very difficult technical problem at the end of the CR spectrum, i.e. at energies above
1019 eV, because of the number of charged particles that have to be followed in the MC scheme
is proportional to the shower energy. For example in this energy range CR shower can have more
than 1011 charged particles at the shower maximum [22]. That is why, when a large number of
showers have to be simulated, it becomes practically impossible to follow each individual particle in the process of direct simulation. Traditionally, the widely used technique as a solution of
this problem is the thinning technique [9], where a weight is assigned to shower particle below a
certain energy threshold, so that the average number of particles at the detector level is correctly
reproduced. This technique introduces artificial fluctuation to shower parameters even when small
energy thresholds are used. However, various methods have been proposed recently to reduce these
artificial fluctuations (e.g. [23]).
2.7 CORSIKA
40
In keeping view of the drawbacks mentioned above of MC simulation of EAS at highest energy
end, a new approach of simulation has been proposed, usually known as Hybrid Simulation [22].
It is a fast, one dimensional calculation which provides production for longitudinal shower profiles
(e.g. total number of charged particles and muons along the shower axis). It can also be used to
calculate lateral distributions and arrival times of the shower. This method allows the collection of
sufficiently high statistics without losing information about shower fluctuations. The hybrid simulations are based on the key idea to follow the development of air showers in detail above a certain
energy threshold and to replace subthreshold particles by a simplified and efficient approximation
of the subshowers initiated by them. Typically in this method, the first few interactions of CR in
the atmosphere are initiated by MC method to generate high energy secondary particles as initial
distribution and then the particle densities observed at detector level are calculated by solving the
corresponding transport equations [22]. Figure 2.7 shows a comparative longitudinal shower development profiles obtained by CORSIKA MC and hybrid simulation procedures.
Alternative methods of simulation of EAS are the analytical and parametrization procedures.
These procedures have been getting a good response from different CR researchers [10] because
of advantage of making the physics of various process transparent in analytical methods in due
course of solving cascade equations and giving guidelines for finding efficient parametrization
for shower variables. One of the extensively used parametrization function for the lateral shower
distribution is the Nishimura-Kamata-Greisen (NKG) function [10]. The more recent advance
formula for the longitudinal shower profile is the Greisen-Il’ina-Linsley (GIL) formula [24]. The
GIL formula is based on a synthesis of the Greisen parametrization [25] for electromagnetic showers and Il’ina’s formula for primary nuclei [26]. Incorporating explicitly the role of the hadronic
shower component by implementing superposition model predictions, this formula gives a simple
and transparent parametrization of the shower evolution.
For a better introduction of the overall physical situation and organisational structure, in the
following section, we shall discuss about CORSIKA simulation package only, on which we have
a particular interest (ref chapter 5).
2.7 CORSIKA : A Standard EAS Monte Carlo Simulation
Package
CORSIKA [6] is a detailed four dimensional MC simulation package to study the evolution
and proprieties of EAS initiated by CR primaries in the atmosphere. It’s name CORSIKA is the
acronym for COsmic Ray SImulation for KASCADE, as it was developed originally to perform
simulations for the KASCADE (KArlsruhe Shower Core and Array DEtector) experiment [27]
at Karlsruhe in Germany. The CORSIKA program is capable to simulate EAS in the atmosphere
up to energies of some 1020 eV, the highest possible CR primary energies, by simulating the in-
2.7 CORSIKA
41
Figure 2.7: The hybrid calculation (solid line) and the full Monte Carlo simulation (dotted line) of
the muon number distribution at sea level for vertical, proton induced showers with E0 = 1015 eV
[22].
teractions and decays of nuclei, hadrons, muons, electrons, and photons at different stages of their
development in the atmosphere. It provides all necessary information related with important measurable parameters in CR EAS experiments such as type, energy, location, direction and arrival
times of all secondary particles that are created in air shower and arrived at a selected observation
level.
Basically the CORSIKA program consists of four parts. The first part is a general program
frame handling the in- and output, performing decay of unstable particles, and tracking of the particles taking into account ionisation energy loss and deflection by multiple scattering and the earth
magnetic field. The second part treats the hadronic interactions of nuclei and hadrons with air nuclei at energies above 80 GeV. The third part simulates the hadronic interactions at energies below
80 GeV and the fourth part describes the interactions of electrons, positrons, and photons. For the
2.7 CORSIKA
42
hadronic interactions parts it contains several models (ref section 2.7.1) corresponding to energy
range of interactions, that may be activated optionally with varying precision of the simulation and
consumption of computer time. CORSIKA is a complete set of standard Fortran routines. It uses
no additional program libraries for the simulation of air showers. Therefore, it runs on (almost)
every computer where Fortran is available. But the CORSIKA package is under a continuous developing process with modification of limitations and incorporation of new ideas by the CORSIKA
development group.
2.7.1 Interaction models in CORSIKA
The CORSIKA MC simulation package incorporates a variety of interaction models in accordance with the range of interaction energies and type of projectiles. For the treatment of HE
hadronic interactions following models are used.
QGSJET : The Quark Gluon String Model with JETs, known as QGSJET model [11] is being
developed in the framework of Gribov’s reggeon approach [28] and is used for HE hadronic interactions using the quasi-eikonal Pomeron parametrization for the elastic hadron-nucleon scattering
amplitude. The hadronization process is treated in the quark gluon string model. This model is also
used for simulation of ultra relativistic heavy ion collisions with detailed simulation of creation,
interaction and fragmentation of color strings (ref section 4.3).
SIBYLL : SIBYLL [12] is a QCD based minijet model used to simulate hadronic interactions
at extreme high energies. Like QGSJET, this model is also used for simulation of ultra relativistic
heavy ion collision.
VENUS : The Very Energetic NUclear Scattering, popularly known as VENUS [16] is a model
program used in CORSIKA specially for simulation of Ultra relativistic heavy ion collision. It handles nucleus-nucleus collisions with an up to-date theoretical approach.
DPMJET : Dual Parton Model with JETs, termed as DPMJET [13] is a HE hadronic interaction model of hadron-hadron, hadron-nucleus and nucleus-nucleus collisions developed using
the two component Dual Parton Model with soft chains and multiple minijets at each elementary
interactions.
HDPM : The simple MC generator HDPM [14] is developed based on Dual parton model and
is used to simulate fairly HE hadronic interactions up to experimentally available energy range of
interaction. Proton-proton interaction simulated with HDPM and other models agree fairly well
with each other, but for HE nucleon-nucleus collisions this model predictions start to disagree with
other models due to the simple modeling in HDPM. HDPM routines are default in CORSIKA activation process.
NEXus : NEXus [15], the acronym for NEXt generation of unified scattering approach, is
a completely new model program combining features of the former VENUS and QGSJET with
extensions enabling a save extrapolation up to higher energies, using the universality hypothesis
2.7 CORSIKA
43
to treat the HE interactions. It handles nucleus-nucleus collisions with an up to-date theoretical
approach.
In CORSIKA the hadronic interaction cross-sections at HE (above 80 GeV) may be adopted
according to the available model, independently of the selected interaction model. Among the
above mentioned HE interactions models, it is found that, QGSJET model prediction is quite satisfactory in connection with experimental observations [29]. Figure 2.8 shows the inelastic protonair cross sections for models (excluding NEXus) used in CORSIKA for HE hadronic interactions
as a function of projectile momentum.
Figure 2.8: Inelastic proton-air cross sections for models (excluding NEXus) used in CORSIKA
for high energy hadronic interactions as a function of projectile momentum. The shaded band
represents the results of a fit to the data with p < 105 GeV/c [6].
In CORSIKA the simulation of low energy hadronic interactions are done with the GHEISHA
code [17]. GHEISHA (Gamma Hadron Electron Interaction code) is an interactive package widely
used in the detector MC program GEANT [8], and is used in describing hadronic collisions up to
some 100 GeV in many experiments. But in CORSIKA it is used to calculate the elastic and inelastic cross-sections of hadrons below 80 GeV in air and their interactions and particle production.
Recently as an alternative the UrQMD (Ultra relativistic Quantum Molecular Dynamics) [30] has
been coupled to treat low energy hadron-nucleus and especially nucleus-nucleus interactions.
Electrons and photons interactions can be treated either with the EGS4 code [18] follow-
2.7 CORSIKA
44
ing each particle and its reactions explicitly, or using the analytic NKG formulae [31] to obtain
electron densities at selected locations and the total number of electrons at up to two observation
levels. However, it should be mentioned that, at the highest electron and γ-energies above 1017
eV the NKG option does not contain the Landau-Pomeranchuk-Migdal effect (EGS4 contains it),
which may alter the shower development by the decrease of the pair formation and bremsstrahlung
cross-section with increasing energy. Therefore NKG treatment deviate more and more with energy from the results gained with the EGS4 option.
Furthermore, in CORSIKA, it is optionally possible to explicitly generate Cherenkov light
in the atmosphere, to handle electronic and muonic neutrinos and anti-neutrinos, and to simulate
showers with flat incidence. The routines treating the Cherenkov radiation have been supplied by
the HEGRA collaboration [32] and considerable modification for improvement has been made by
K. Bernlöhr [33]. Thin sampling option [9] to reduce the computation times, for UHE showers
above 1016 eV also exists. Thin option shorten the computing time by allowing only a fraction of
the secondary particles to be followed having energy above certain threshold in shower development, as we have discussed in the section 2.6.
With all these models and associated options (also there are numbers of options in CORSIKA which are not mentioned here as they are out of our scope), the CORSIKA becomes an
extremely versed MC simulation code in present times to analyze the data of different types of
EAS experiments and hence a fertile ground to test the interaction models credibility to predict
EAS characteristics at different energies along the line of the experimental evidence.
2.7.2 Particles in CORSIKA
CORSIKA treats almost all particles found in CR interactions except charmed particles. These
together consists of 50 elementary particles with the γ-photon; leptons e± , µ± ; the mesons π 0 ,
0
π ± , K ± , KS/L
, η; the baryons p, n, Λ, Σ± , Σ0 , Ξ0 , Ξ− , Ω− ; the corresponding anti-baryons; the
resonance states ρ± , ρ0 K ∗± , K ∗0 , K¯∗0 , ∆++ , ∆+ , ∆0 ∆− and the corresponding anti-baryonic
resonances. Optionally the neutrinos νe and νµ and their anti-particles resulting from π, K, and
µ decay may be generated explicitly. Furthermore, the nuclei up to A = 56 can be treated by
identifying their numbers of protons and neutrons. All these particles are tracked through the
atmosphere under various processes encountered in.
2.7.3 Coordinate system in CORSIKA
To locate the arrival direction and point of interaction of a primary CR particle and to study the
consequent shower development profile with respect to a detector setup at any particular position
on the earth surface, the definition of a coordinate system is very much important. Figure 2.9
shows the coordinate system used in CORSIKA.
2.7 CORSIKA
45
z−axis (upward)
particle momentum
y−axis (west)
x−axis (north)
o
Figure 2.9: Coordinate system in CORSIKA.
It is defined with respect to a Cartesian coordinate system with the positive x-axis pointing to
the magnetic north, the positive y-axis to the west, and the z-axis upwards. The origin is located at
sea level. The zenith angle θ of a particle trajectory is measured between the particle momentum
vector and the negative z-axis, and the azimuthal angle φ between the positive x-axis and x-y
component of the particle momentum vector proceeding counterclockwise as shown in the Figure
2.9.
2.7.4 Model of atmosphere in CORSIKA
CORSIKA adopted atmosphere consists of N2 , O2 , and Ar with volume fractions of 78.1%,
21.0%, and 0.9% respectively. The atmosphere is considered as flat for the incidence angle nearly
up to 75o . The density of variation of the atmosphere is modeled with 5 layers with first four layers
following an exponential dependence and the last layer following a linear dependence. The top of
the atmosphere is considered at 112.8 km from the ground.
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Chapter 3
PRODUCTION OF HIGGS BOSON
THROUGH VACUUM EXCITATION IN
UHECR HADRONIC INTERACTION
3.1 Introduction
Peter W. Higgs [1] in 1964 published a research paper in the Physical Review Letters, reporting about the development of a theory which provides an explanation for the factor responsible
for spontaneous breaking of the symmetry of electroweak interaction, due to which, the W and Z
bosons, the carriers of weak force are heavy particles, while the electromagnetic photon remains
massless. This theory, known as Higgs Theory establishes a profound influence on the HE particle physics. According to the theory, a field and its particle also known after the name of Peter
Higgs as the Higgs field and Higgs boson respectively, are responsible for breaking of the symmetry of the Salam-Weinberg interaction (i.e. electroweak interaction) spontaneously. Unfortunately
this Standard Model (SM) Higgs boson does not show its face so far and all direct experimental
searches for this scalar proved fruitless till today. So the discovery of this illusive particle is one of
the main mission of the present and future HE particle physics experiments. It is expected that, this
particle will be discovered at the Large Hadron Collider (LHC) of CERN [2] or at the Fermilab
Tevatron [3].
In this chapter, we will explore an idea of possible production of Higgs bosons in UHECR
interactions with air nuclei [4, 27]. In UHECR interactions, there is a possibility that due to a
fraction of interaction energy transfer to the nearby vacuum of microscopic volume, it becomes
locally hot and bubbles are formed by phase transition. These bubbles contain Higgs bosons, and
Higgs bosons decay very fast to heavy farmion pairs as bubbles cools. This effect is manifested in
a very rapid increase of the multiplicity of charged hadrons with energy. The possibility of vacuum excitation and bubble formation depends on the fraction of total energy of collision (centre
48
3.1 Introduction
49
of mass energy) that goes to local vacuum. However this fraction of energy is not known [5]. We
have conceived this notion from the theory of Mishra et al. [5], who have claimed based on the
thermofield theory of vacuum [6] that, the Higgs particles are produced through vacuum excitation
when enough energy is pumped into a microscopic volume of vacuum. In this connection, considering all the above mentioned ideas and theories, we have developed a model (ref section 3.4.1)
of the Higgs boson production in UHECR interactions in the earth atmosphere. This model is
incorporated with the conventional hadronic interaction model (ref section 3.4.2) being developed
based on GENCL code of UA5 experiment [7] of CERN to generate CR events of our interest. We
consider that, the very HE muons (prompt muons) produced at very first collision bear the signature of Higgs boson production. Following paragraph elaborates the reason for this consideration.
In recent times considerable experimental and theoretical interests have been focused on the
neutrino and muon fluxes from CR interactions with the earth’s atmosphere, since they reflect primary interactions at energies that far exceed the highest available accelerator energies. The range
of energies of the neutrinos and muons determine their source. At low energies, atmospheric muon
and neutrino fluxes are dominated by conventional sources, i.e., decays of relatively long-lived particles such as π and K mesons. With increasing energy, the probability increases that such particles
interact in the atmosphere before decaying. This implies that in these energies, the sources of HE
muons and neutrinos, called prompt muons and neutrinos are not the conventional ones, instead,
must be some short-lived heavy particles, produced in CR interactions. Thus in other words, these
prompt fluxes can be considered as probes for production of some new short-lived heavy particles in CR interactions [8]. More precisely, an increase in the multiplicity of muons (decays from
charged hadrons) with energy in CR interactions with air nuclei is an indication of the production
and decay of some new short-lived heavy particles in those interactions. Moreover, there is highest
probability that these short-lived heavy particles are produced at the first hadronic (hadron-hadron,
hadron-nucleus and nucleus-nucleus) interactions only, because in the subsequent secondary interactions energies would be insufficient for this type of particles production (particularly for the
case of Higgs boson). Thus in our present analysis, we are interested only with first hadronic collision and we consider an excessive HE muon multiplicity in comparison with the multiplicity from
conventional sources as the probe for Higgs bosons production in UHECR first hadronic collision.
We therefore incorporate this factor in the conventional hadronic cascade simulation program that
is developed as mentioned above [4]. The interaction and decay processes in the atmosphere are
simulated only for hadrons (π, K, and N) and muons above 0.01 TeV, 0.1 TeV and 1 TeV. For the
production and decay processes of the hadrons and Higgs boson, subroutines are developed and
coupled to the cascade program in order to generate muons, to which most of the hadrons and
Higgs particles finally decay.
Since the mechanism of Higgs boson production through vacuum excitation would be possible only at the very first UHECR interactions with air nuclei, so to observed this effect we have to
3.2 The Physics of Higgs boson
50
detect young showers at high altitude taking very high energetic muons as probes. These muons
will obviously follow high transverse momentum distributions. Similarly, another possibility to
observed the effect of Higgs bosons production is the detection of underground muons at very high
depth under the earth surface [4].
We organised this chapter as follows. The basic concept of the development of the physics of
Higgs is most important to understand the thermofield theory of Higgs boson production. So in
the section 3.2 we brief a discussion on the physics of Higgs under different headings. In the section 3.3 we describe the theoretical basis of Higgs boson production through vacuum excitation.
The section 3.4 is devoted to explain the structure of our model and MC algorithm to simulate
the hadronic cascade. The results of this work are discussed in section 3.5. Some historical experimental evidences in support of possibility of Higgs particle production in CR interactions are
mentioned in the section 3.6. The chapter is concluded with the section 3.7 by making some
relevant remarks.
3.2 The Physics of Higgs boson
The concept of physics of Higgs boson has a wide ranging consequence and significance
in particle physics, because it provides an explanation of the origin of mass or gives a theoretical
solution of mass problem within the periphery of SM. However, since the Higgs boson is still not
discovered experimentally, the mass problem persists as unsolved so far and it develops even the
question of the validity of the SM. Any way, the Higgs boson physics has conceptual relations
with the principle of symmetry to electroeweak theory. So to have an overall understanding and
development of the subject we consider the following headings.
3.2.1 Basic concept of symmetry principle
Symmetry means invariance of some physical system under some transformation operations. If a physical system undergoes certain transformations (e.g. translation, rotations etc.), and
if the transformed system looks identical (on the basis of physical observables) to the untransformed one, then those transformations are called symmetry transformation and the system is said
to posses those symmetries. Generally symmetries are of two types, viz., continuous and discrete.
Continuous symmetry transformations are labelled by parameters each of which can take any value
in a given range, e.g., translation of empty space is a continuous symmetry. On the other hand,
discrete symmetry transformation are labelled by a set of integers or discrete numbers. Rotation
of a square is a discrete symmetry as only n×90o , for all integer n, leave the space unchanged.
Similarly Parity, Charge conjugation and Time reversal are discrete symmetry operations.
Symmetry principles have very significant role in physics, because the invariance of symmetry of a physical system is always associated with a conservation law. There are lots of familiar
3.2 The Physics of Higgs boson
51
examples in classical physics [9], for instance, the translational invariance implies the momentum
conservation. In quantum mechanics, observables are associated with operators, so the conservation law in Heisenberg picture is equivalent to the statement that, the corresponding operator
commutes with the Hamiltonian.
Symmetries and conservation laws are related in a similar manner in case of field theory also.
Emi Noether in 1918 [10] made this connection more precise by her famous, the Noether Theorem. The simple form of the statement of this theorem is that, every symmetry of nature yields a
conservation law, conversely, every conservation law reveals an underlying symmetry [9].
When the symmetries of a field are characterized by the space-time independent parameters or
phase factors, then these types of symmetries are called global symmetries. Under global symmetry or global transformation, the complex field φ(x) are transformed in exactly the same way or by
the same amount for all space-time point x, and such transformation can be expressed as,
φ(x) −→ eiθ φ(x),
(3.1)
where θ is the space time independent parameter representing the transformation. The Dirac Lagrangian,
L = φ̄(x)(iγ µ ∂µ − m)φ(x)
(3.2)
is invariant under this transformation. On the other hand, when the symmetry transformations are
space time dependent, i.e., θ = θ(x), they are called local symmetries or gauge symmetries [11],
are transformed as,
φ(x) −→ eiθ(x) φ(x).
(3.3)
The Dirac Lagrangian (3.2) is not invariant under this transformation [9]. So to develop a
theory which will be invariant under a space-time dependent phase change, we need to form a
gauge covariant derivative Dµ , to replace ∂µ , which follows the transformation as,
Dµ φ(x) −→ [Dµ φ(x)]# = e−iθ(x) Dµ φ(x),
(3.4)
Dµ φ(x) = (∂µ + ieAµ )φ(x),
(3.5)
and is given by,
where Aµ (x) is the gauge field, which has the transformation property,
1
Aµ (x) −→ A#µ (x) = Aµ (x) + ∂µ θ(x).
e
(3.6)
The gauge invariant Lagrangian will now take the form,
L = φ̄(x)iγ µ (∂µ + ieAµ )φ(x) − mφ̄(x)φ(x).
(3.7)
The gauge symmetries may be used to generate dynamics, known as gauge interactions. The
prototype gauge theory is the quantum electrodynamics. In principle, all fundamental interactions
3.2 The Physics of Higgs boson
52
are described by some form of gauge theory [9].
Since the symmetry transformations which leave the physical system invariant form a group,
therefore the set of symmetry transformations are usually represented by a group [12]. So by the
group theoretical analysis, independent of any detailed dynamical consideration, the consequence
of a symmetry can be deduced. This is the great relevance of the group theory in physics. The
most familiar groups in particle physics are the unitary group and special unitary group. The
group of all unitary matrices of order n is known as unitary group and is denoted by U(n). It has
n2 real parameters (since a unitary matrix of order n has n2 independent elements), so it has also
n2 generators, those can be obtained from the expression of a unitary matrix of order n, viz.,


2
U = exp i
n
&
j=1
κ j Jj  ,
(3.8)
i.e. all elements of U(n) can be generated from the right side of equation (3.8) by giving all possible
values to the n2 real parameters κj . The n2 independent harmitian matrices Jj are thus generators
of U(n). Obviously, they are not unique, and any n2 independent linear combinations of these could
equally well be used as the generators of U(n) [12]. Thus the U(1) group has only one parameter
and one generator . That is why U(1) group is used to represent the electromagnetic interaction
symmetry in particle physics.
The special unitary group SU(n), the group of n×n unitary matrices with unit determinant are
most frequently encountered in particle physics theories. For example, SU(2) in isospin invariance,
SU(3) in the eight fold way etc. [9]. In SU(n) there are (n2 - 1) traceless hermitian n×n matrices,
thus an element of it can be written as in the case of U(n) as,

U = exp i
2 −1
n&
j=1

κ j Jj  .
(3.9)
Here the group generators Jj are represented by traceless hermitian matrices. It should be
noted that only (n - 1) of (n2 - 1) generators are diagonal. Thus in SU(2) group there are three
group parameters, it is a group of 2×2 unitary unimodular matrices, we can write from equation
(3.9) as,

U (κ1 , κ2 , κ3 ) = exp i
3
&
j=1

κ j σj  ,
(3.10)
where σj ’s are the 2×2 traceless hermitian matrices, usually we choose the basis to be the standard
Pauli matrices [12],






0 1 
0 −i 
1 0 
σ1 = 
, σ2 = 
, σ3 = 
.
1 0
i 0
0 −1
3.2 The Physics of Higgs boson
53
3.2.2 Standard model and electroweak theory
In particle physics, the combination of the gauge theories of weak and electromagnetic interactions, and quantum chromodynamics (QCD) is known as the Standard Model (SM). This model
demands the existence of 12 fermions, with 6 leptons and 6 quarks, and their antiparticles, which
can be categorized into 3 generations, and 6 gauge bosons, viz., γ, W± , Z0 , and 8 gluons. Despite
a number of unanswered problems and shortcomings, this model has earned impressive successes
in dealing with the particle physics phenomena [13]. It should be pointed out in our context that,
the nondiscovery of Higgs boson, which is closely connected with the question of origin of mass,
is the greatest challenge to this model reliability.
The most elegant and profound part of SM is the Glashow-Winberg-Salam Theory (also
referred as Winberg-Salam Theory or generally Electroweak Theory) of electroweak interaction. It is a unified standard theory of weak and electromagnetic interactions. The symmetry of
the electroweak interaction is represented by the group SU(2)L ×U(1)Y and the mediators of this
combined field are the massive W± , Z0 gauge bosons and massless γ photons [9, 14].
He was Schwinger [15] who first advanced the idea of weak and electromagnetic unification.
In 1961, Glashow [16] published first paper on unification of weak and electromagnetic interactions by proposing a model having SU(2)L ×U(1)Y gauge symmetry with intermediate vector
boson (IVB) masses were inserted by hand. This work also discussed subsequently by Salam and
Ward [17] and finally the complete theory in its present form with IBM masses generated by the
Higgs mechanism, responsible for the spontaneous breaking of SU(2)L ×U(1)Y symmetry (ref section 3.2.3), was proposed by Winberg [18] and Salam [19] independently. ’t Hooft’s [20] demonstration of the renormalizability (a prescription which allows to consistently isolate and remove
all infinities from the physically measurable quantities) of gauge theories and the experimental
supports [9, 14] have given a solid background to the Weinberg-Salam theory of electroweak interaction and that leads to bestowing the Nobel prize for physics to Glashow, Weinberg and Salam
in the year 1979.
The choice of the group SU(2)L ×U(1)Y for the symmetry of elecrtoweak interaction is due to
the fact that, the left handed leptons (as well as quarks) are doublet with three generations, such
as (νe , e− )L , (νµ , µ− )L , and (ντ , τ − )L (for the case of leptons). Thus the rotational symmetry of
this isospin doublet structure in weak isospin space is represented by the group SU(2)L with weak
isospin quantum number Iw = 12 . On the other hand right handed leptons are always singlet, so to
bring all right and left handed leptons in a single group with a symmetry property, weak hypercharge quantum number Y is introduced, such that left doublet must have same value of charge.
The weak hypercharge is defined by the Gell-Mann-Nishijima relation, Q = Iw3 + Y2 , and it gives
for left handed Y = − 1, and for right handed Y = − 2. This resulting weak hypercharge symmetry
is represented by U(1)Y group. Thus the underlying symmetry group of electroweak interaction is
called SU(2)L ×U(1)Y , SU(2)L refers to the weak isospin, with subscript L to indicate the involve-
3.2 The Physics of Higgs boson
54
ment of left handed states only and U(1)Y refers to weak hypercharge involving both chiralities
[14]. In the following we shall discuss the spontaneous breaking of SU(2)L ×U(1)Y symmetry of
electroweak interaction due to Higgs mechanism which provides masses to IVB.
3.2.3 Spontaneous symmetry breaking : The Higgs mechanism
In ferventive attempts to accommodate massive gauge field in the gauge theory to apply it to
the weak interaction, explore the impressive phenomena − the Spotaneous Symmetry Breaking
(SSB) and the Higgs Mechanism. Non invariance of a physical system under certain symmetry transformation is called the symmetry breaking under that transformation. When a symmetry
breaks down without influence of any external factors or agency, then the process of symmetry
breakdown is referred as spontaneous. In SSB, there is a hidden symmetry because the Lagrangian
is fully invariant under the symmetry transformations, but the dynamics are such that the vacuum
state (ground state) is not a singlet of the symmetry group. The choice of one from all possible degenerate vacuum states as the physical vacuum, breaks the symmetry and spoils the usual
symmetry consequence of energy-level degeneracies [9, 14]. Thus the non-invariance or the degeneracy of the vacuum state or equivalently nonvanishing vacuum expectation value of field operators
in certain symmetry transformation, is the condition of SSB. Therefore, in other words, SSB can
be defined as the non invariance of vacuum expectation value under symmetry transformation.
To illustrate the circumstance in which the symmetry breaking condition takes place, let us
consider the Lagrangian density,
1
1
λ
L = (∂µ φ)2 − m2 φ2 − φ4 ,
2
2
4
(3.11)
which has discrete symmetry,
φ −→ φ# = −φ.
From equation (3.11), the potential term can be identified as,
1
U (φ) = (∆φ)2 + V (φ),
2
(3.12)
with
1
λ
V (φ) = m2 φ2 + φ4 .
(3.13)
2
4
Since the term (∆φ)2 is non-negative, so the minimum of U(φ) will have the property ∆φ = 0
with the constant value of φ, given by minimizing V(φ) of equation (3.13) directly. As the coupling
constant λ is also positive (so that energy is bound from below), therefore for minimization of V(φ),
we get, two possible cases,
(i) for m2 > 0, φ = 0, and
(ii) m2 < 0, φ = 0, ±(−m2 /λ)1/2 as shown in the Figure 3.1. Thus for m2 < 0, the ground
3.2 The Physics of Higgs boson
55
state of the field is nonvanishing, with the vacuum expectation value (VEV) of the field operator
φ is
+0 | φ | 0, = ξ0 ,
(3.14)
with
+
ξ0 = ± −m2 /λ
v(
,1/2
.
(3.15)
)
v(
o
)
o
2
2
m >0
m<0
Figure 3.1: Possible vacuum states of the field φ. The case for m < 0 indicates the SSB condition
of the field.
The two possible values in (3.15) corresponds to the two possible vacuua. The choice of either
one (and only one) to build the theory, say ξ0 = + (−m2 /λ)1/2 , clearly breaks the original refection
symmetry φ −→ − φ of the theory. This is the condition of broken symmetry. Usually to build
the theory a small oscillations around the true vacuum is considered such that a new quantum field
variable with zero VEV is developed say,
φ # = φ − ξ0 .
(3.16)
In terms of this shifted field, the Lagrangian density lost its original symmetry, and we say
that, the symmetry is broken spontaneously [9, 14].
The spontaneous breaking of a continuous global symmetry is always accompanied by the
appearance of mass less scalar (spin-0) particles, known as Goldstone bosons and the statement is
called as the Goldstone Theorem [21]. On the other hand, when the symmetry of local gauge field
breaks down spontaneously, it generates masses to the IVB by disappearing Goldstone bosons and
acquiring mass by the gauge field itself (i.e. photon has become massive, or in other words gauge
field has eaten the Goldstone boson and becomes heavy). This is the famous Higgs Mechanism
[1]. That is, the local gauge symmetry breaks down spontaneously due to Higgs mechanism.
3.2 The Physics of Higgs boson
56
Accordingly, the Higgs mechanism is responsible for the masses of IVB of weak interaction. The
pattern of spontaneous breakdown of SU(2)L ×U(1)Y symmetry of electroweak interaction can be
represented by [9, 14],
SU (2)L × U (1)Y −→&φ'0 U (1)em ,
(3.17)
where +φ,0 is the symmetric vacuum expectation value of electromagnetic field represented by
U(1)em group. Three of the original four gauge bosons corresponding to the total four generators
of the SU(2)L ×U(1)Y group will become massive and one, corresponding to the photons, remains
massless. For a brief introduction of the process of Higgs mechanism, let us consider the weak
interaction Lagrangian of a charged scalar field φ [5, 9, 22],
1
L = (Dµ φ)† (Dµ φ) − V (φ) − Wµν Wµν ,
2
(3.18)
σ
Dµ φ = ∂µ + ig Wµ φ,
2
(3.19)
where,
1
2
+
,
+
,
V (φ) = −m2 φ† φ + λ φ† φ ,
(3.20)
Wµν = ∂µ Wν − ∂ν Wµ − gWµ × Wν .
(3.21)
and,
The Wµν is the field tensor for the weak gauge bosons Wµ , g is the coupling constant and σ =
(σ1 , σ2 , σ3 ) are the usual Pauli matrices. The charged and the neutral W bosons form a SU(2)
vector, reflecting the nonabelian (elements are not commutating) nature of this gauge group, leads
to gauge boson self-interaction term in equation (3.21). Accordingly the gauge transformation on
Wµ has an extra term and φ transforms as usual, such as,
1
φ −→ e−iα(x)σ φ, Wµ −→ Wµ − ∂µ α(x) − α(x) × Wµ .
2
(3.22)
It can be shown that, the last term in the Lagrangian (3.18) representing gauge boson kinetic
energy and self interaction, the middle term representing scalar mass and self interaction and the
first term representing scalar kinetic energy and gauge interaction are invariant under the simultaneous gauge transformation (3.22). On the other hand, the addition of a mass term −m2g Wµ Wµ ,
to provide the mass to gauge boson, would clearly break the gauge invariance of the Lagrangian, in
contrast with the scalar mass term, mφ† φ, which is gauge invariant. This phenomenon is exploited
to give mass to the gauge bosons through back door without breaking the gauge invariance of the
Lagrangian, which is the renowned Higgs mechanism of SSB [22]. SU(2) gauge theory with a
complex doublet of scalar field,


φ1 
φ=
,
φ2
(3.23)
3.2 The Physics of Higgs boson
57
where φ1 and φ2 are real fields, and with m2 > 0, the minimum of the scalar potential V(φ), moves
out from the origin to a finite value, with a minimum potential at,
'
(
ξ0 =
3
φ† φ
0
=
where
m2
λ
ξ02
,
2
41/2
.
This means that the field develops a non zero vacuum expectation value as,


0 
1
+φ,0 = √ 
.
2 ξ0
(3.24)
For the stable quantum perturbative expansion, to develop a valid perturbative field theory, the
field has to be translated only around a local minimum into the form,
φ# = φ − +φ,0 .
(3.25)
This gives the valid perturbative field theory in terms of the redefined field, representing the
physical Higgs boson. The three other components of the complex doublet field are absorbed
to give mass and hence to give longitudinal components to the gauge boson [9, 14, 22]. The
substitution of (3.25) in the Lagrangian (3.18) leads to mass term for the gauge bosons as,
1
mW = gξ0 ,
2
ξ0 (g 2 + g #2 )1/2
,
2
tanθw = g # /g, θw is the Weinberg angle. The middle term of the Lagrangian finally leads to a real
mZ 0 =
mass for the physical Higgs boson, after substituting (3.25) in it, i.e.,
√
√
mH = ξ0 2λ = mW (2 2λ/g).
Inserting mW = 80 GeV and g = 0.65 along with a perturbative limit on the scalar self coupling
λ ≤ 1, implies that the Higgs boson mass is bounded by mH < 1000 GeV [22]. But the results
of different searches for Higgs boson indicates that it should be bounded by mH ≤ 200 GeV (ref
section 3.2.4). On the other hand, the experimentally observed accurate values of masses of gauge
bosons are mW = 80.423 ± 0.039 GeV and mZ 0 = 91.1876 ± 0.0021 GeV [23].
In the next section we incorporate a very brief description on the status of Higgs boson search
at present.
3.3 Higgs boson Production Through Vacuum Excitation
58
3.2.4 Present experimental status of Higgs boson search
It is already pointed out that, inspite of numbers of intense experimental efforts to discover SM
Higgs boson, no breakthrough is seen so far. However, these experiments provide some indirect
evidences to arrive at some conclusions about the possible range of Higgs boson mass lying within.
Among the various missions in this regard, the Large Electron Positron Collider (LEP) of CERN
contributes significantly to the present status of SM Higgs boson mass range. In an earlier analysis
LEP place the mass of a SM like Higgs state, with a significant decay branching ratio into bottom
(b) quarks, above approximately 115 GeV. An alternative analysis based only on the assumption of
Higgs boson decay into hadronic jets, without b-tagging leads to a bound of about 113 GeV [25].
A most recent report combining the final results of the four LEP experiments : ALEPH, DELPHI,
L3, and OPAL, a lower bound of 114.4 GeV is set on the mass of the SM Higgs boson at the
95% confidence level. For Higgs boson masses which are relevant at LEP, the SM Higgs boson is
expected to decay mainly into bb̄ quark pairs (the branching ratio is 74% for a mass of 115 GeV)
and the rest of the decay width constitutes with the decays of τ τ̄ , W W ∗ (≈ 7% each), and cc̄ (≈
4%) [35].
A Linear Collider (LC) in USA, which has the capability of performing precision measure√
ments of both Higgs boson production and decay rates, in the energy range s ≈ (350 - 500)
GeV, provided that the Higgs boson mass lies in the range mH ≤ 200 GeV. The dominant pro-
duction mechanism for such a light Higgs boson is e+ e− → hZ, with the largest decay channel
being h → bb̄ or h → W W ∗ [24, 25]. Indirect evidence from precision measurements at e+ e−
collider suggests the existence of a light Higgs boson in the mass range of (95 - 235) GeV at 95%
confidence level with a statistical preference towards the lower end [26].
3.3 Higgs boson Production Through Vacuum Excitation
So far we have discussed the Higgs mechanism and related theories in a temperature independent situation. In this section we are going to introduce the temperature dependent theories for
Higgs boson production with the following sections.
3.3.1 Temperature dependence of vacuum
The theory of temperature dependence of vacuum is discussed in [5] in an elaborate manner
by using the methodology of thermofield theories [6]. Here we follow the same theory to develop
the required formulation for the temperature dependence of vacuum in out context. In general, in
the nonperturbative methodology of field theory, the coherent state of vacuum at zero temperature
is defined as [5],
| vac# > ≡ U | vac >,
(3.26)
3.3 Higgs boson Production Through Vacuum Excitation
59
where U is a unitary operator, can be expressed as,

U = exp ξ
) 3
λz
2
41/2
+
,

a (z)† − a (z) dz  .
(3.27)
Here a and a† are the annihilation and creation operators in the Hilbert space and satisfy the
quantum condition [a, a† ] = 1. For the complex field φ, VEV is given as,
1
< vac# | φ(z) | vac# > = √ ξ.
2
(3.28)
The expectation value of the Hamiltonian density is calculated as,
1
λ
10 = < vac# | τ 00 | vac# > = − m2 ξ 2 + ( )ξ 4 .
2
4
(3.29)
A minimization of energy density 10 with respect to ξ gives the result ξ = ξ0 = [m2 /λ]1/2 .
This concept can be generalized to finite temperature by using the methodology of thermofield
dynamics [5, 6]. According to thermofield dynamics the temperature dependent vacuum is given
by,
| vac# , β > = U (β) | vac# >,
(3.30)
with
+
,
1
, U (β) = exp B† − B ,
kT
where thermal modes are created with
β=
†
B =
)
θ(k, β)a(k)#† ǎ(−k)† dk.
(3.31)
(3.32)
In the above equation the operators ǎ and ǎ† are the annihilation and creation operators in
the extra Hilbert space and satisfy the same quantum algebra as above. For bosons, the function
θ(k, β) is given by,
sinh2 θ(k, β) =
1
.
exp [βω(k, β)] − 1
(3.33)
It should be noted that, | vac# > is the temperature dependent stable state of | vac > after
Salam-Weinberg phase transition at T = 0. On the other hand the state | vac# , β > is a thermal
00
vacuum state at temperature T %= 0. If τef
f is effective Hamiltonian density, then energy density at
temperature β becomes,
#
V (ξ, β) ≡ 1(β) = < vac , β |
5
00
τef
f
1
| vac , β > = (2π)−3
2
#
)
)
3λ
1
−3
+
(2π)
dk
4
ω(k, β){exp[βω(k, β)] − 1}
ω(k, β)2 + k2 + 3λξ 2 − m2
dk
ω(k, β){exp[βω(k, β)] − 1}
62
λ
m2 2
+ ξ4 −
ξ ,
4
2
(3.34)
3.3 Higgs boson Production Through Vacuum Excitation
60
where ω(k, β) = [k2 + mH (β)2 ]1/2 , with mH (β) being the Higgs boson mass at temperature 1/β.
For numerical evaluations it is useful to rewrite equation (3.34) in terms of the dimensionless
quantities with substitutions :
z=
ξ
mH (β)
k
, µ=
, y = βξ0 , x = ,
ξ0
ξ0
ξ0
where ξ0 is the value of ξmin = (m2 /λ)1/2 for zero temperature. Now the expression for effective
potential becomes,
V (z, y) =
ξ04
5
6
λ 4 λ 2 1
3λ
z − z + I1 (z, y) + [I2 (z, y)]2 ≡ ξ04 V1 (z, y),
4
2
2
4
where,
I1 (z, y) =
and
1
2π 2
) ∞ 2
x [ω(x)2 + x2 + λ(3z 2 − 1)]
ω(x){exp[yω(x)] − 1}
0
dx,
(3.35)
(3.36)
1 )∞
x2
I2 (z, y) = 2
dx,
2π 0 ω(x){exp[yω(x)] − 1}
(3.37)
with ω(x) = (x2 + µ2 )1/2 . The gap in energy density of the thermal vacuum with respect to the
vacuum at zero temperature is given by,
∆1(β) = V (ξmin , β) − V (ξmin , β = ∞) =
ξ04
5
6
λ
V1 (zmin , y) +
.
4
(3.38)
3.3.2 Vacuum excitation and bubble formation
With vacuum as the medium in which collision takes place, a local destabilization of it can
occur if enough energy is pumped into a microscopic volume. This destabilization thermalizes
locally that part of vacuum and forms bubble in it with a nonzero temperature and local thermal
equilibrium. The total energy of such a locally excited region or the bubble can be given as [5],
Eb =
)
∆1 [β (r)] dr,
(3.39)
where ∆1 [β (r)] is the gap in energy density of the thermal vacuum with respect to the vacuum at
zero temperature, given by,
∆1 [β (r)] =
with ξ0 = (m2 /λ)
1/2
ξ04
5
6
λ
V1 (zmin , y (r)) +
,
4
(3.40)
being the field expectation value corresponding to the minimum potential
V1 (zmin , y (r)) as given in equation (3.35). V1 and β = 1/kT are now spatially dependent.
Such a locally excited region of vacuum or bubble will contain Higgs particles due to SalamWeinberg phase transition of the vacuum. These particles are coupled to fermions and get converted to quark or lepton pairs as bubble cools. The Higgs particles in the bubble will primarily
3.4 Models
61
decay to heavy fermion-antifermion pairs [5]. The number of Higgs boson inside the bubble is
given by,
nH =
)
N [β (r)] dr,
(3.41)
where N (β) is the number density of the Higgs particle at temperature β and is given by,
N (β) = (2π)−3
with ω (k, β) = (k2 + m2H )
1/2
)
1
dk,
exp [βω (k, β)] − 1
(3.42)
, mH being the Higgs boson mass. The temperature distribution
inside the bubble is taken as,
β (r)−1 = T (r) = T0 exp(−ar2 ),
(3.43)
where T0 is the temperature at the centre of the bubble and the parameter a decides the region over
which the vacuum is excited, with bubble volume approximately a−3/2 .
3.4 Models
3.4.1 Higgs boson production model
Considering the aforesaid thermofield theory of vacuum, we developed a model of hadronic
interaction in the light of possibility of Higgs boson production through vacuum excitation due to
a fraction of centre of mass or interaction energy transfer to the vacuum during UHECR collision.
√
The fraction of centre of mass energy ( s) that goes to excited region of vacuum or bubble is un√
known [5]. For definiteness we consider the various fraction of s that goes to bubble formation
as fe = 0.0, 0.01, 0.02,. . . . , 0.5. The gap in energy density of thermal vacuum with respect to
the vacuum at zero temperature and the number density of Higgs boson at different temperature
T is calculated using equations (3.40) and (3.42) for the simulated mass of Higgs boson at the
corresponding temperatures. Figure 3.2 shows the variation of the mass of Higgs boson with the
central temperature T0 of the bubble. At temperature T = Tc ≈ 2.1ξ0 ( 525 GeV, called critical
temperature, where the shape of the potential changes to have zero VEV, Higgs boson mass goes
to zero as expected and it again rises for temperature T > Tc [5]. Accordingly the simulated Higgs
boson masses varies as a function of T0 as shown in the Figure 3.2.
Here we consider the interacting particles as the proton primary and air nucleus. Assuming
a given primary energy of proton (E0 ) from 1015 eV to 1020 eV and a given fraction of the interaction energy transfer for the vacuum excitation and bubble formation (fe ) from 0 to 0.5, first we
√
calculated bubble energy Eb = s.fe for each event. Then the volume of the bubble Vb = Eb /∆1
(ref equation 3.40 for ∆1) i.e. excited region for each event is calculated. Figure 3.3 shows the
variation of bubble volume or excited region of vacuum with bubble energy of the vacuum. The
3.4 Models
62
mH (GeV)
1000
100
10
100
1000
T0 (GeV)
10000
Figure 3.2: Simulated temperature dependent Higgs boson mass as a function of central temperature of the bubble.
central temperature of the bubble is plotted against the bubble energy of vacuum in the Figure 3.4.
Finally the mean number of Higgs boson is evaluated for the corresponding bubble volume. We
parametrized approximately the relation between average number of Higgs boson < nH > with
bubble energy Eb . We have used this equation in our simulation program to calculate average number of Higgs boson for a particular event with a particular primary energy of proton. The actual
number nH for a particular event and primary energy is chosen from a Poisson Distribution (PD)
with this mean. In the Figure 3.5, the simulated Higgs boson numbers are plotted for different
values of bubble energies. The Higgs boson number follows bubble energy as.
nH = n0 (Eb )γ ,
(3.44)
where n0 = 2.45 ± 0.04 and γ = 0.99 ± 0.004, and Eb is measured in TeV.
Considering the present experimental status of the Higgs boson mass (ref section 3.2.4), we
have used the following decay channels [28] of the Higgs boson for our model to develop subroutines which are incorporated with the main cascade program to observe the possible signature of
Higgs production through vacuum excitation in UHECR interactions :
(i) mH < mW , H → bb̄(≈ 90%), τ + τ − (≈ 10%),
(ii) mW < mH < mZ , H → bb̄(≈ 85%), τ + τ − (≈ 10%), cc̄(≈ 4%),
3.4 Models
63
Vb (GeV-3)
0.001
0.0001
1e-05
1
10
Eb (TeV)
100
Figure 3.3: Volume of the excited region of vacuum or bubble varies with bubble energy. The filled
circles denote simulated data and the dashed line indicates the best fit.
(iii) mZ < mH < 2mW ,
0% − 10%),
H → bb̄(≈ 80% − 1%),
W W ∗ (≈ 0.01% − 97%),
ZZ ∗ (≈
(iv) 2mW < mH < 2mZ , H → W W (≈ 94% − 100%),
(v) mH > 2mZ , H → W W (≈ 75%), ZZ(≈ 25%),
The WW and ZZ channels further decay to electrons, muons and neutrinos according to W
→ eν, µν or Z → e+ e− , µ+ µ− . The bb̄ channel either directly decays to muons or electrons
(10%) or indirectly via intermediate states τ + τ − , π + π − , 2π0 , etc. and cc̄ channel also either di-
rectly decays to muons or electrons (≈ 6%) or indirectly via various intermediate states [23]. The
τ + τ − decay modes are decided by individual channels :
τ ± → µ± ν ν̄(≈ 18%), e± ν ν̄(≈ 17%), h± ν ν̄(≈ 52%).
We have calculated the muon multiplicity by considering only the various decay channels leading to muons. In the hadron-air interaction model the particles are produced in clusters according
to the theoretical [29] and experimental [7] considerations. In our present simulation for UHECR
interactions, we consider only non-diffractive (ND) events (actually non-single diffractive), where
particles are produced in the central region, flat in rapidity, and in two fragmentation regions. Here
an event is built up of two leading and a varying number of central clusters. Each cluster is given
3.4 Models
64
350
T0 (GeV)
300
250
200
150
100
50
0
10
20
30 40 50
Eb (TeV)
60
70
Figure 3.4: Variation of central temperature of the excited region of vacuum or bubble with bubble
energy. The half-filled circles denote simulated data and the dashed line indicates the best fit.
a transverse momentum p⊥ and a rapidity y. After transforming the rapidities to conserve energy
and momentum, the clusters are made to decay isotropically.
3.4.2 Interaction model
We followed the MC algorithm of the GNCL code of the UA5 experiment of CERN [7] for
the hadron-air interaction model of non-single-diffractive (NSD) events [30] to prepare a simulation program for generating events to counting of HE muons from conventional sources due to
UHECR interactions with air nuclei. As the conventional standard hadronic interactions models
(e.g. DPMJET, QGSJET, SIBYLL etc.) does not include the effect of production of short-lived
heavy particles like Higgs bosons through vacuum excitation, it is not possible to use directly these
models to study the unconventional mechanism of Higgs production. Moreover, any model or code
to be used to study this mechanism should be modified in such a way that, a part of interaction energy must go to the vacuum for Higgs production and rest of it is used for conventional events
generation. Accordingly we used the algorithm of the GNCL code for the development of our
program as it is convenient in this connection in comparison with the standard interaction models
mentioned above. The algorithm of this model is as follows :
(i) The number of charged hadrons nch are chosen from a Negative Binomial (NB) distribution :
3.4 Models
65
nH
100
10
1
1
10
Eb (TeV)
100
Figure 3.5: Average Higgs boson number with bubble energy. The filled squares denote simulated
data and the dashed line indicates the best fit obtained by using equation (3.44).

5
nch + k − 1 
< nch > /k
P (nch ) = 
1+ < nch > /k
nch
6nch 5
1
1+ < nch > /k
6k
,
(3.45)
with the following parameters,
and
< nch > = −7.0 + 7.25s0.127 ,
(3.46)
√
k −1 = −0.104 + 0.058ln( s),
(3.47)
where s is in GeV2 .
(ii) Cluster formation and decay is the basic multiparticle production mechanism. Out of six
different clusters we consider only three, viz. pion, kaon and the leading cluster, excluding the less
frequent nucleon, hyperon and Ξ-pairs. The nature of the leading particles (proton or neutron) is
chosen considering the charge exchange probability as given in [7].
(iii) Number of kaons produced are grouped into pairs (cluster) of zero strangeness including
neutral kaon and kaon resonance pairs [7, 30]. All pairs have same production probability and each
kaon is a K∗ with 60% probability, according to CERN intersecting storage rings (ISR) measurements [31]. The actual number of kaon clusters is drawn from PD with a mean deduced from the
3.4 Models
66
K/π ratio,
RK = < K ± > / < π ± > = 0.024 + 0.0062ln(s).
(3.48)
The K∗ ’s decay into Kπ pairs as follows :
1
2
K ∗0 → K 0 π 0 ( ), K + π − ( ),
3
3
(3.49)
1
2
K ∗+ → K 0 π + ( ), K + π 0 ( ),
(3.50)
3
3
K 0 (and K̄ 0 ) are considered to be KS0 or KL0 with equal probability. All K 0 , K ± , and pions finally
decay into muons and electrons and their decay are governed by standard branching ratios.
(iv) The remaining charged particles are π + and π − , which are grouped into clusters including
π 0 . The algorithm is based on drawing the number of charged pions from a PD with an average
of 1.8, repeatedly until there are no charged particle left and then drawing π 0 from an independent
PD with the following parameter [7],
µπ0 = [0.5(2 + 1.03nch ) − 0.4µk ] /nc ,
(3.51)
where µk = [Rk /(1 + Rk )]nch! , nch! being the number of charged particles left to be simulated,
and nc is the number of pion clusters.
(v) All clusters made up of more than one particle are given some excitation energy in terms
of an additional mass. For the kaon clusters the excitation energy follows the distribution,
dn
−2E
∝ exp
,
2
dE
b
1
2
(3.52)
where b is a free parameter having the value 0.75 GeV for kaon clusters. The pion clusters are
given masses m from the following distribution :
dN
= 1.1 [1 + N0 (0, 0.2)] exp
dm
1
nπ − 1 ,
3
71 2
8
(3.53)
where nπ is the number of pions in the cluster and N0 (0,0.2) is a number drawn from a Gaussian
distribution with mean 0 and standard deviation 0.2 GeV.
(vi) Transverse momenta p⊥ and longitudinal momenta pL are given to the clusters in two
steps. The transverse momenta are randomized from either an exponential distribution :
dN
∝ exp (−bp⊥ ) ,
dp2⊥
(3.54)
or from an inverse power-law distribution,
dN
1
∝
,
2
dp⊥
(p⊥ + p0 )α
(3.55)
where b = 6 GeV/c, p0 = 3 GeV/c, α = 3 + 1/[0.01 + 0.011 ln (s)]. For the single pions (≈ 10%
of all clusters), p⊥ is always sampled from the exponential distribution. In other cases the relative
3.5 Result and Discussion
67
amount of the two distributions is made to depend on the multiplicity of the event. For proton-air
interactions, these distributions are multiplied by the parameter,
R (p⊥ ) = 0.0363p⊥ + 0.057,
(3.56)
for p⊥ ≤ 4.52 GeV/c.
The azimuthal angles of the leading nucleons and of the meson clusters are chosen randomly
between 0 and 2π. To conserve momenta in the xy-plane perpendicular to the beam axis (zdirection) we make two independent linear translations in the components of p⊥ ,
pnew
= pold
i
i −
+&
,
pold
/N,
i
i = x, y
(3.57)
where the summation is over the N clusters in the event. Here p⊥ is generated independent of
rapidity. Longitudinal momentum is given to a cluster by assigning to it a rapidity y,
pL = mT sinh (y) ,
where mT =
(3.58)
!
(m2 + p2⊥ ) is the transverse mass. Rapidity distribution has a central plateau and a
fall off at higher values of |y|, and can be described analytically by two Gaussian peaks [32], with
the following parameters [33],
s1 = 0.146 ln (E0 ) + 0.164,
(3.59)
σ1 = 0.120 ln (E0 ) + 0.255,
(3.60)
Box-Muller method is used to generate the rapidities and the two leading clusters are given the
highest and lowest rapidities. They are converted to longitudinal momenta pL and so adjusted as
to conserve total momentum and energy, after assigning a fraction < k > (inelasticity parameter
≈ 0.5) of available energy to leading nucleons.
(vii) Each cluster with given energy is made to decay via the available channels with a prob-
ability proportional to their respective branching ratios. The numbers of muons above threshold
energies of 0.01 TeV, 0.1 TeV, and 1 TeV are counted for each event.
3.5 Result and Discussion
The MC simulation program developed for the first interaction is run for the primary energies
E0 = 1015 eV to 1020 eV and for different fractions of energy transfer to bubble formation fe (0.0
-0.5). The resulting muon multiplicity distributions for 5000 showers for different E0 and fe are
compared with corresponding simulations with fe = 0. In order to derive signature of Higgs boson
production, muon multiplicity distributions for muon energy thresholds of 0.01 TeV, 0.1 TeV and
1 TeV are selected as the probes. As an example, the Figure 3.6 shows the comparison of the
3.5 Result and Discussion
68
600
600
500
dN/dNµEµ
thr
400
300
200
100
100
300
500
Nµ
Eµ
thr
700
850
100
700
1300 1900 2500 3100
Eµ
thr
thr
550
400
250
= 0.01TeV
fe = 0.0
fe = 0.3
fe = 0.5
700
700
550
400
250
100
300
500
Eµ
Nµ
thr
700
900
1100
600
1100 1600 2100 2600
Nµ
Eµ
= 1TeV
thr
dN/dNµEµ
2300
1200
thr
= 0.1TeV
fe = 0.0
fe = 0.3
fe = 0.5
1600
thr
3400
100
= 0.1TeV
fe = 0.0
fe = 0.3
fe = 0.5
4500
= 1TeV
200
Nµ
fe = 0.0
fe = 0.3
fe = 0.5
100
Eµ
300
100
100
dN/dNµ
400
1100
dN/dNµEµ
= 0.1TeV
thr
Eµ
900
fe = 0.0
fe = 0.3
fe = 0.5
500
= 0.01TeV
1000
dN/dNµ
= 0.01TeV
fe = 0.0
fe = 0.3
fe = 0.5
= 0.1TeV
dN/dNµ
Eµ
thr
= 0.01TeV
700
1300
1000
700
400
100
100
100
300
E
Nµ µ
thr
500
700
900
100
700
= 1TeV
1300
Nµ
thr
Eµ
1900
2500
= 1TeV
Figure 3.6: Muon multiplicity distributions at first interaction level for 5000 proton induced showers for different fractions of energy transfer. The data is taken for muon threshold energies (Eµthr )
of 0.01 TeV, 0.1 TeV, and 1 TeV. The panel of the figures on the left hand side is for primary energy
(E0 ) 1019 eV and on the right hand side is for 1020 eV respectively.
3.5 Result and Discussion
69
muon multiplicity distributions for primary energies of 1019 eV and 1020 eV, with fe = 0.0, 0.3
and 0.5 for different muon threshold energies. It is observed that, the separation between muon
multiplicity distributions increases, moving towards larger multiplicity side for the higher values
of primary energy and for the higher values of fraction of energy transfer with a particular value of
muon threshold energy. Similarly, for a particular value of the primary energy as the muon threshold energy increases, the separation between muon multiplicity distributions for different fractions
of energy transfer fe becomes more and more distinct, and the muon multiplicity distribution with
fe = 0 (i.e. without Higgs effect) moves towards lower multiplicity side gradually. This indicates
that, with the rise of primary energy E0 and the fraction of energy transfer fe , very HE muon or
prompt muon multiplicity increases considerably pointing to the possibility of Higgs signature.
<Nµ>
10000
Muon threshold energy (Eµthr)
0.01 TeV
0.10 TeV
1.00 TeV
1000
100
10
100
Eb (TeV)
1000
Figure 3.7: Average muon number at first interaction level versus bubble energy for different muon
energy thresholds. Data is taken from 5000 proton induced showers.
In the Figure 3.7 we have shown the variation of average muons number with the bubble energy for different threshold energies of muon. It is clear from this figure that the average muons
number increases remarkably with the increase of the value of the bubble energy for all values of
the muon threshold energies. On the other hand the difference in average muons number decreases
for different muon threshold energies as the bubble energy is increasing and at a considerable high
value of bubble energy this difference vanishes. Exactly same observation can be also made from
the Figure 3.8, which gives a plot between primary energy of the CR particle and the average muon
number that are produced during interactions with air nuclei for a fraction of energy transfer fe =
3.5 Result and Discussion
70
0.3 for different muon threshold energies. From these observations it can be inferred that, at higher
values of bubble energies or primary energies, the number of prompt muons increases considerably
and that is why, at these energies the average muons multiplicities are almost independent of muon
threshold energies. This shows that, at higher values of bubble energies or primary energies only
prompt or very HE muons dominate the total muon flux, indicating the possible signature of Higgs
boson production in those interactions.
10000
<Nµ>
thr
1000
Muon threshold energy (Eµ
0.01 TeV
0.10 TeV
1.00 TeV
)
100
0.1
1
10
E0 (EeV)
100
Figure 3.8: Variation of average number of muons with primary energy of Cosmic Rays particle at
first interaction level for different muon threshold energies and for fraction of energy transfer fe =
0.3. The average number is calculated for 5000 proton induced showers.
The relation between average muons number and primary energy can be expressed by a generalised equation as,
< Nµ >= N0 (E0 )β ,
(3.61)
where the parameters N0 and β are obtained by fitting the equation (3.61) with the data for E0 and
Nµ for different muon energy thresholds.
The significance of our mechanism is studied by calculating χ2 value,
χ2 =
n
&
{(Nµfe (=0 )i − (Nµfe =0 )i }2
i=1
(Nµfe =0 )i
(3.62)
where Nµfe (=0 is the number of muons for fe %= 0 and Nµfe =0 is the same for fe = 0. The result of
this study is shown in the Figure 3.9. It is found from this study that, the Higgs boson production
3.6 Historical Events
71
1e+06
18
E0 = 10 eV
E0 = 1019 eV
20
E0 = 10 eV
χ
2
100000
10000
1000
0
0.1
0.2
0.3
fe
0.4
0.5
0.6
Figure 3.9: Dispersion of muon multiplicity distribution for different fe %= 0 from fe = 0.
mechanism is significant from E0 ∼ 1018 eV for fractions of energy transfer fe ≥ 0.1. For further
higher energies this mechanism gives significant contributions starting from much lower values of
fe .
3.6 Historical Events in Support of Higgs bosons Production
There are some unusual historical events which could not be explained on the basis of
simulation of known physics. Now these events can be reexamined on the light of Higgs boson
production through the present mechanism. Those events are :
(i) Chiron events : In 1986, the Brasil-Japan Collaboration in Chacaltaya Emulsion Chamber
experiment had observed events [36] in chambers 19 and 21 with exceptional proprieties of having
total p⊥ ∼ 5 GeV and inside them there were miniclusters associated with as small a p⊥ as ∼ 10 −
20 MeV. These were referred to as Chiron events because of their unusual nature. The production
of miniclusters along with absence of π 0 could thus be interpreted as signature of Higgs particles
productions in these events due to the mechanism as stated above.
(ii) Halo events : Halo events [37] in CR were observed by Japan-Russian Collaboration,
where there were excessive multiplicities with uncountable particle number. As in our present
mechanism, it is clear from the Eb versus nH plot that, the Higgs particle multiplicity rises almost
linearly with the bubble energy, hence, for energies beyond the threshold for the present process,
3.7 Concluding Remarks
72
the multiplicity will increase linearly with energy which is much faster than what can be expected
from ordinary physics. Thus the halo events may indicate the start of new physics through an
unusual rise in the multiplicity resulting from excitation of vacuum. Further, the multiple cores in
halos just look like multiple bubble formation [5].
(ii) Cygnus X-3, Hercules X-1 signals : There were some signals with high muon content
from giant EAS originating from the directions of Cygnus X-3 and of Hercules X-1 [5]. These
indicate hadronic interactions responsible for high muon content, which was originally explained
by assuming some neutral stable hadronic particles, e.g., cygnets, quark nuggets, etc. that are
coming from Cygnus X-3 and Hercules X-1 are responsible for these signals. Instead, we may say
that, the Higgs boson production through vacuum excitation as stated above could give rise to the
excess muon signals through the preferential heavy flavour production [5].
3.7 Concluding Remarks
The mechanism of Higgs boson production through vacuum excitation consists of three parts:
(i) bubble formation, (ii) Higgs particle production by temperature dependent vacuum depending
on quantum mechanical behavior, and (iii) dissipation of the bubble through particle production via
conversion of Higgs boson to fermion pairs. The characteristic features of Higgs boson production
by the said mechanism would be (a) relatively high p⊥ for Higgs particle decay, (b) preferential
production of heavy flavors, and (c) an unusual rise in multiplicity with energy [5, 25]. In the
present analysis we have concentrated on the last part of the features mentioned above, by applying the parametrization to the phenomenological model of UHECR interactions in the atmosphere.
It should be pointed out that, other possible sources of HE excessive muon multiplicity are
the semi-leptonic decays of hadrons containing heavy quarks, most notably charm and bottom. It
is reported in [8] that charm contribution to the atmospheric muon flux becomes dominant over
the conventional contribution at energies of about 105 GeV. FREJUS Collaboration [34] has recently reported enhanced muon flux in TeV energy range, indicating that perhaps prompt muon
flux from charm decay is larger than expected in the standard atmospheric neutrino and muon
calculation. Workers of this group have been looking into prompt muons and neutrinos production in CR interactions with nuclei in the atmosphere which arise through semi-leptonic decays of
hadrons containing heavy quarks, mostly charm and bottom. Most recently the LEP working group
for Higgs boson searches [35] reported that the SM Higgs boson is expected to decay via bb̄ (main
channel, the branching ratio is 74% for a mass of 115 GeV) and cc̄ with 4% branching ratio besides
decays to τ + τ − , WW∗ and gg channels. So it can be expected that bottom and charmed mesons
from Higgs boson also contribute to the prompt muon and neutrino fluxes or to the excessive muon
and neutrino multiplicity with energy, besides from the conventional bottom and charmed mesons
produced in the CR interactions with air nuclei. However to disentangle Higgs contributions to
3.7 Concluding Remarks
73
prompt muon flux from the charmed or other heavy particles contributions, the study of transverse
momentum distribution spectra of muons would be helpful. The muons from Higgs bosons will
obviously follow high p⊥ distribution than from other sources because of the mechanism we reported here. A further study will shed more light over the subject of prompt flux in connection
with its source and nature.
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[21] J. Goldstone, Nuovo Cimento 19, 154 (1961); J. Goldstone, A. Salam, and S. Winberg, Phys.
Rev. 127, 965 (1962).
[22] D. P. Roy, Pramana 51(1 & 2), 7 (1998).
[23] K. Hagiwara, Phys. Rev. D 66, 010001 (2002).
[24] S. Dawson and S. Heinemeyer, Phys. Rev. D 66, 055002 ((2002);
[25] Edmond L. Berger et al., Phys. Rev. D 66, 095001 (2002).
[26] J. H.Field, Phys. Rev. D 61, 013010 (2000).
[27] Kalyanee Boruah, Phys. Rev. D 56, 7376 (1997).
[28] R. M. Barnett et al., Phys. Rev. D 54, 225 (1996).
[29] A. Giovannini and L. Van Hove, Acta Phys. Pol., B19, 495(1998).
[30] C. Forti et al., Phys. Rev. D 42, 3668 (1990).
[31] T. Akessen et al., Nucl. Phys. B203, 27 (1982).
[32] A. Klar and J. Huffer, Phys. Rev. D 31, 491 (1985).
[33] J. N. Capdevielle, Nucl. Phys. B (Pro. Supply) 39A, 154 (1995).
[34] http://physics.arizona.edu/∼ina/charm atm.html.
[35] Search for the Standard Model Higgs Boson at LEP, ALEPH, DELPHI, L3 and OPAL Collaborations, CERN-EP/2003-011, 13 March 2003.
[36] S. Hasegawa, Proc. Int. Symp. on Cosmic Ray Super high Energy Interactions, Beijing China
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Moscow, Russia (1989); E. H. Shubuya, Proc. 20th ICRC, Moscow, Russia (1987), 8, 258.
Chapter 4
CHARMED HADRON PRODUCTION IN
pp INTERACTION
4.1 Introduction
Hadrons those occupy the place in between light hadrons and heavy hadrons and whose all
or a part of quark constituents are charmed quarks or quark are termed as Charmed Hadrons.
Charmed hadron production in hadronic collision is a subject of growing interest for HE (prompt)
atmospheric muon and neutrino fluxes associated with the CR. An important contribution to the
fluxes of muons and neutrinos on the earth surface is coming from the decays of particles that
are produced through the interactions of CR in the atmosphere. These fluxes have a particular
importance as they reflect primary interactions at energies that far exceed the highest accelerator
energies available. At lower energies (up to GeV) the conventional sources of these fluxes are π
and K mesons (relatively long lived particles). In the HE range (up to multi-TeV), the most notable
sources of these fluxes are considered as charmed particles those are produced in the atmosphere
in CR interactions [1, 2]. However to draw a definite conclusion on the fluxes of prompt muons
and neutrinos from the available data obtained with surface and underground detectors, we need a
sound physical model which should be based on extrapolating of proper charmed production data
obtained at accelerator energies to the orders of magnitude higher energies for the relevant CR
collisions. So it is important to study the charmed production characteristics of a given physical
model to understand its reliability. Here is our motivation in this context.
Conventionally charmed quarks are considered to be produced in pQCD processes. To leading order (LO) and next-to-leading order (NLO) these are the gluon-gluon fusion process gg
→ cc̄ and the quark-antiquark annihilation process q q̄ → cc̄. Figure (4.1) shows few basic pro-
cesses of charmed production in pQCD. The charmed production cross section is calculated using
the usual convolution of parton densities in the colliding hadrons and the hard parton level cross
section from pQCD [1]. There was a wide variety of theoretical models which were more or less
76
4.1 Introduction
77
satisfactorily applied to the description of charmed particle hadroproduction. These models can
be divided into three main groups : the models based on pQCD and parton model [3], the models
based on Lund string model [4] and the models which incorporate Dual Topological Unitarization
Scheme [5]. On the other hand in the hadronisation models charmed particle production is strongly
¯
suppressed. The Lund model gives a production probability of different quark flavours as uū:dd:cc̄
≈ 1:1:0.3:10−11 , i.e. charm production in the hadronisation phase can be neglected, as reported in
[6].
The quark-gluon string (QGS) model of the supercritical Pomeron [7] is based on the 1/N ex-
Figure 4.1: Few basic processes of charmed production in pQCD. (a), (b), (c) are leading order
processes and (d) is an important NLO process.
pansions in QCD and allows to consider hadron interactions at large distances and small transverse
momenta. A close approach to the QGS model is a dual topological unitarization scheme developed in [8]. To solve the problem of proper accounting for semihard processes [7] in QGS model, a
most successful model in the field of HE hadronic interactions, called Quark Gluon String model
with JETs (QGSJET) [7] is developed. This model incorporates the charmed production events in
hadronic collisions. However, it is not being used in this respect in a noticeable degree in consideration of rare events of charmed hadroproduction. Notwithstanding, charmed particle production
in CR interactions is very much important in connection with prompt leptons as mentioned above.
So it is better to check the reliability of this model to be used to study the charmed hadrons production in CR hadronic interactions with air nuclei. QGSJET01 [9] is a MC code developed recently
4.2 The November Revolution
78
based on the QGSJET model. We have used this code to study the charmed hadroproduction in pp
collisions.
In this chapter, a short history of charmed particles is introduced in the section 4.2, in the
section 4.3 we incorporated a brief discussion about the QGSJET model, results and discussions
are included in the section 4.4 and the section 4.5 is devoted for the conclusion of this work.
4.2 A Short History of Charmed Particles : The November
Revolution
Till the mid seventies, there are only three established quark flavours : u, d, and s, against
four leptons : e, νe , µ, and νµ . This asymmetry between quarks and leptons leads to the prediction
of a new 2/3-charged heavy fourth quark flavour by Bjorken and Glashow in 1964 [11]. Latter on
this prediction was also consistent with the requirement to suppress strangeness changing neutral
current effect [10]. In 1974, two research groups, one led by C. C. Ting at Brookhaven and the
other led by Burton Richter at SLAC, published simultaneously [12, 13] the completely unexpected
discovery of something entirely different kind of heavy (∼ 3100 MeV) neutral particle. Ting
naming the particle as J and Richter calling it as ψ. This J/ψ(3100) particle has rather unusual
property of having a width (only about 70 KeV) much narrower than the widths of typical hadrons
(e.g. Γp ∼ 150 MeV, Γω ∼ 10 MeV) [14]. The interpretation was that J/ψ is a bound state of a
new heavy quark as predicted by Bjorken and Glashow, the charmed quark c and its antiparticle
c̄, i.e. J/ψ ∼ cc̄, which is now universally accepted. This discovery gives a solid background to
enhance the quark model and precipitated a series of events and discoveries [15], and this came to
be known as the November Revolution [16].
It should be noted that, the charmness of J/ψ particle is hidden as it is the bound state of c with
charmness + 1 and c̄ with charmness − 1. So for the confirmation of fourth quark existence or to
confirm the charmed hypothesis, all kinds of new baryons and mesons, carrying various amount of
naked charmed quark should be there [16]. The first evidence for charmed baryons (Λ+
c = udc and
¯
possibly Σ++ = uuc) appeared in 1975 [17], the first charmed mesons (D0 = cū and D+ = cd)
c
were found in 1976 [18] and the charmed strange meson (Ds = cs̄) in 1977 [19]. With these
discoveries the interpretation of J/ψ as cc̄ was established beyond reasonable doubt and the quark
model itself was put back on its feet.
Furthermore it should be mentioned that, well ahead of the discovery of J/ψ particle, there are
reports of possible charmed events in CR experiments. In 1971, K. Niu and his co-workers detected
first such events in nuclear emulsion chamber exposed at balloons [20]. Similarly in 1974, V. S.
Aseikin et al. of the Tien-Shan experiment published very first data on several events with slow
attenuated component, later known as Long Flying Component (LFC), detected in the ionization
calorimeter [21]. In 1975, E. L. Feinberg proposed that charmed particles could be responsible for
4.3 A Description on QGSJET Model
79
LFC production [22]. After the discovery of D-mesons this idea was abandoned for several years
because the life-time of mesons was wrongly estimated (18 times less). Latter in the year 1982, I.
M. Dremin and V. Yakovlev estimated the charmed production cross-section at 10 -20 TeV (Figure
4.2, 1.4 - 2.8 mb/nucl) [23]. PAMIR collaboration confirmed about the LFC in 1987 and RHIC
[24] confirmed 1982 Tien-Shan data in 2005 [25].
Figure 4.2: cc̄ production cross sections data with theoretical predictions at different values of
√
s
[25].
4.3 A Description on QGSJET Model
The demonstration of increasing influence of the semihard processes on HE hadronic interactions by collider data [26] forced to modify the existing hadronic interaction models for application
in the super high energy regime. As the QGS model was very successful to reproducing numerous
CR experimental data [7], so this model was considered as the base one for calculations at super
high energies including semihard process. Moreover, the QGS model offered a relatively easy approach to the simulation of CR interactions at super high energies and ensured a good agreement
to the accelerator at lower energies. The new model based on QGS model which includes the
treatment of the semihard interactions with the eikonal approach and not only includes the hard
interactions of partons but the preceding soft preevolution also known as QGSJET model [7].
4.3 A Description on QGSJET Model
80
Hadronic and nuclear collisions in QGSJET model are treated in the framework of Gribov’s
reggeon approach [27] as multiple scattering processes (Figure 4.3), where individual scattering contributions are described phenomenologically as Pomeron exchanges. The Pomeron corresponds to microscopic partons (quark and gluon) cascades, which mediate the interaction between
the projectile and the target hadrons, and consists of two parts : soft and semihard Pomerons (Figure 4.4). The soft Pomeron is described by the corresponding eikonal χs (s, b), represents a purely
soft cascade of partons, whereas the semihard Pomeron with the eikonal χh (s, b), corresponds to a
cascade, typically represented by a piece of QCD ladder sandwiched between two soft Pomerons
[28]. So in this approach appropriate cross sections for the inelastic interaction between hadrons
Figure 4.3: A general multi-Pomeron contribution to hadron-hadron scattering amplitude. Elementary scattering processes (vertically thick lines) are described as Pomeron exchanges [28].
soft Pomeron
QCD ladder
soft Pomeron
Figure 4.4: A general Pomeron (L.H.S.) consists of the soft and semihard Pomerons, represented
by the 1st and the 2nd contributions on the R.H.S. [28].
i and j may be calculated using the expression [7] :
χij (s, b) = χsij (s, b) + χhij (s, b),
(4.1)
4.3 A Description on QGSJET Model
81
where the soft part of the eikonal is given by,
χsij (s, b) =
γi γj
b2
exp(∆y
−
).
2
2
Rij
4Rij
(4.2)
Here s is the c.m. energy squared, b is the impact parameter, y = ln(s), ∆ = αp (0) − 1,
2
and Rij
= Ri2 + Rj2 + αp# (0)y. Parameters of the Pomeron trajectory (∆ and αp# (0)) and another
parameters (γ and R2 ) describing the Pomeron-hadron vertices are to be determined from the data
on the total cross section and on the slop of the diffraction cone [7].
The term χhij (s, b), the semihard part of the eikonal can be expressed as [7] :
χhij (s, b)
)
r2 )
=
dy1 dy2 χsij (s, b)(eyi +yj , b)
2
×σh (ey−y1 −y2 , Q0 ),
(4.3)
where y is the total rapidity interval, y1(2) are the rapidities of the Pomeron ends, σh (ey−y1 −y2 , Q0 )
is the hard parton interaction cross section, r2 is the adjustable parameter associated with parton
density [7].
The cross sections of various processes may be obtained as in the usual QGS model if one
takes the eikonal (4.1) into consideration. For instance, the total cross section is given by the
formula [7] :
σijt (s)
1
=
eij
)
d2 b[1 − exp{−eij (χsij (s, b) + χhij (s, b))}],
(4.4)
where eij is so-called the shower enhancement co-efficient, for pp interactions epp = 1.5. For the
cross section of n cut soft Pomerons and m semihard blocks one may get [7] :
σijn,m (s)
1
=
eij n!m!
)
d2 b(2eij χsij (s, b))n (2eij χhij (s, b))m
×exp{−2eij (χsij (s, b) + χhij (s, b))}.
(4.5)
The old set of soft Pomeron parameters [29] ∆ = 0.07, αp# (0) = 0.21 GeV−2 , R2pp = 3.56 GeV−2 ,
γp2 = 3.64 GeV−2 are consistent with collider data when semihard processes are taken into account
(Q0 = 2 GeV, r2 = 0.6 GeV−2 ). The QGSJET reproduces collider data quite satisfactory [7, 9].
The MC simulation for hadron cascade in the atmosphere is analogous to usual soft case [31]
and secondary particles arise from both usual soft strings and semihard jets hadronization [30].
Charmed particles production in QGSJET has been included via fragmentation of strings (LUNDtype algorithm). In this type of algorithm cc̄-pairs are created from vacuum, if one of them couples
¯ quark, it produces a D-meson, being coupled to a di-quark makes Λc [32]. Charmed
to u (ū, d, d)
hadron production in pp collision is expressed as a sum of the contributions from the valance
quark-antiquark or antiquark-diquark chains and 2(n − 1) sea quark-antiquark chains :
hd
hd
hd
hd
ϕhd
n (s, x) = a0 {Fqq (x+ , n)Fq (x− , n) + Fq (x+ , n)
4.4 Results and Discussion
82
hd
hd
hd
×Fqq
(x− , n) + 2(n − 1)Fsea
(x+ , n)Fsea
(x− , n),
(4.6)
!
where x± = 12 [ x2 + x2⊥ ± x] and ahd
0 is the density of charmed hadron formation in the centre of
the chain [33].
4.4 Results and Discussion
To study the production characteristics of charmed hadrons in pp collisions under the QGSJET
√
model we have generated 106 numbers of events with the centre of mass energy ( s) ranging from
10 GeV to 100000 GeV with the help of the MC code QGSJET01. In this study, we only consider
−
hadrons D+ , D− , D0 , D̄0 , Λ+
c , Λ̄c . We have discussed the production characteristics of these
hadrons under the following two headings.
4.4.1 General features of charmed hadron production
Figure 4.5 shows the inelastic and the production cross sections of different charmed hadrons
√
in pp collisions as a function of s. Calculated production cross sections for D/D̄ are compared
with the experimental data from [34]. No data is available with us for comparison with experiment
for Λc /Λ̄c charmed hadrons production in study. Although for D/D̄ also experimental data are
insufficient for comparison, it is observed that the calculated production cross sections are in close
agreement with experiments. Table 4.1 also shows the comparison of inclusive production cross
sections of different D-mesons with the experimental results from LEBC-EHS and LEBC-MPS
√
√
Collaboration [35] at s = 27.4 GeV and s = 38.8 GeV respectively. The agreement is very good
with the LEBC-MPS results. Average multiplicities for different charmed hadrons in study that are
√
produced for different values of s in pp collisions are as shown in the Figure 4.6. Experimental
data is not available to compare with these results. Although the average multiplicities of different
charmed hadrons are very small, they are not inconceivably inconsiderable, because regarding CR
showers in the atmosphere, the extremely rare cases of the charmed hadron production is also very
significant, as it can change the traditional shower behaviour completely [7]. This is due to the fact
that, as discussed earlier, the charmed hadrons are most notable sources of atmospheric prompt
leptons which are very important to understand the nature of CR as a whole [1, 2]. The average
multiplicities of different charmed hadrons under study are not much different and approaches to
√
approximately the same value as the values of s goes on increasing beyond 1000 GeV (Figure
4.6). The average multiplicities of different charmed hadrons under study can be expressed as a
√
function of s as,
< n >= n0 e−n1 /(
√ α
s)
,
(4.7)
where the parameters n0 , n1 and α have slightly different values for different charmed hadrons as
shown in the Table 4.2.
4.4 Results and Discussion
83
100000
σ(µb)
10000
1000
100
σ
10
inel.
–
D/D
σ –
Λc/Λc
σtotal
σ –
exp.(D/D)
σ
10
100
1000 10000 100000
√s (GeV)
Figure 4.5: Inelastic and production cross sections of different charmed hadrons in pp collisions
√
with the centre of mass energy ( s). The cross sections are calculated with the QGSJET are
compared with the experimental data [34] for D/D̄ production in the low energy range.
√
The Feynman x variable (xF = 2pc.m
|| / s) describes the relation of the laboratory energy Elab
of final particles with the projectile energy E0 by :
Elab
where m⊥ =
9
E0
4m2⊥
=
( x2F +
+ xF ) ( E 0 xF ,
2
s
(4.8)
!
m2 + p2⊥ , which is valid in the forward hemisphere (xF ≥ 0) for xF >> 4m2⊥ /s.
Particles in the backward hemisphere (xF ≤ 0) are very soft in the laboratory, and therefore of
less importance for the development of showers [36]. This variable also describes the mode of the
projectile energy distribution in final particles of shower. This variable has significant role in CR
physics to study the rank of particle production in shower development by the string fragmentation
process and hence to explain charge excess of final particles in the shower. In the context of charge
excess study, the high value of xF is of particular importance because these values contribute more
to the charge excess [36]. In view of this we have studied the inclusive cross sections for the pro√
duction of charmed hadrons in pp collisions at different centre of mass energy s as a function
of Feynman xF . Some of the results are shown in the Figure 4.7. It is observed that, the maxi√
mum charmed hadron production takes place with low values of xF (≤0.3) for all values s under
4.4 Results and Discussion
84
0.1
<n>
0.01
D+
DD0
–0
D
Λc
–
Λc
0.001
0.0001
10
100
1000 10000 100000
√s (GeV)
Figure 4.6: Average multiplicities of different charmed hadrons produced in pp collisions with
√
s.
The average multiplicities are calculated with the QGSJET for 106 numbers of events. The lines
in the figure are least square fits for different hadrons obtained with the equation (4.7).
consideration. Although there is no considerable difference, the production cross sections for Dmesons are little higher than that for Λc -baryons for almost all values of xF . For higher values of
√
s, experimental data are not available to compare with the results of the QGSJET calculations.
√
√
However, the results with s = 27.4 GeV and s = 38.8 GeV are compared with the experimental
data from [35] for only D/D̄ production. The agreement is not quite good.
At this point our motivation is that, the study of charmed hadroproduction cross sections distributions with xF up to high value will help to cheek the reliability of the model prediction in
the field of prompt leptons data from the surface and underground detectors. Since the primary
energies of CR are estimated from the ground based observations guided by some physical model,
exactly in the same way it would also be possible to identify the parent charmed hadrons of atmospheric prompt leptons from the surface and underground observations if proper guidance is given
by the concerned hadronic interaction model.
4.4 Results and Discussion
85
Table 4.1: A comparison of inclusive production cross sections of different D-mesons as obtained
√
√
from the QGSJET calculations with experiments [35] at s = 27.4 GeV and s = 38.8 GeV.
√
s (GeV)
Particles
D+ /D−
27.4 (LEBC-EHS)
D0 /D̄0
D/D̄
D+ /D−
38.8 (LEBC-MPS)
0
D /D̄
0
D/D̄
QGSJET σ (µb) Experimental σ (µb) [35]
20.4 ± 2.0
11.9 ± 1.2
39.7 ± 3.9
30.2 ± 2.2
19.3 ± 1.9
18.3 ± 1.9
24.3 ± 2.4
26 ± 4
46.8 ± 4.7
48+10
−8
22+9
−7
22.6 ± 2.3
Table 4.2: Values of parameters n0 , n1 and α of equation (4.7) for different charmed hadrons.
n0
n1
α
n0
n1
α
D+
D−
D0
0.548 ± 0.040
0.998 ± 0.058
0.350 ± 0.023
11.693 ± 0.288
10.920 ± 0.180
12.267 ± 0.323
D̄0
Λc
Λ̄c
1.265 ± 0.072
0.656 ± 0.035
0.329 ± 0.023
0.130 ± 0.001
10.835 ± 0.164
0.100 ± 0.001
0.107 ± 0.002
10.928 ± 0.187
0.120 ± 0.001
0.140 ± 0.001
13.336 ± 0.378
0.16 ± 0.002
In keeping view of further use of the QGSJET model predictions in the simulation works for
atmospheric prompt leptons to compare with detector results at any energy (up to 100000 GeV) we
parametrized the cross sections of charmed hadrons production in pp collisions. The distributions
of the production cross sections with xF follows the inverse power law as,
dσ
= σx0 x−β
F ,
dxF
(4.9)
√
where σx0 and β are two constants which are different for different values of s and for different
√
charmed hadrons. The line in the Figure 4.7 for s = 100000 GeV is the best fitted result obtained
by using equation (4.9) with the simulated data and is shown as an example.
We have also studied the charmed hadroproduction as a function of p2⊥ and p⊥ as it is very
important for our purpose to have a good knowledge of distributions of these particles with respect
to these variables. Figure 4.8 shows the distributions of charmed hadrons production cross sections
4.4 Results and Discussion
86
1e+06
–
D/–D
Λc/Λc
100000
total
√s = 100000 GeV
10000
1000
100
0
100
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
10000
–
D/–D
Λc/Λc
√s = 1000 GeV
1000
100
0
–
√s = 100 GeV
100
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
D/–D
Λc/–Λc
total
–
exp(D/D)
√s = 38.8 GeV
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
–
D/–D
Λc/Λc
total
–
exp.(D/D)
1000
dσ/dxF(µb)
dσ/dxF(µb)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
10
10
100
0
D/D
Λc/–Λc
total
1000
total
dσ/dxF(µb)
dσ/dxF(µb)
10000
1000
√s = 10000 GeV
1000
10
100000
1
D/–D
Λc/Λc
total
10000
10
1
–
100000
dσ/dxF(µb)
dσ/dxF(µb)
1e+06
√s = 27.4 GeV
100
10
10
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
Figure 4.7: Inclusive cross sections for the production of charmed hadrons in pp collisions at
√
different centre of mass energy ( s) as a function of Feynman xF . The results from QGSJET
calculations at 27.4 GeV and 38.8 GeV centre of mass energies are compared with the experimental
√
data [35] for D/D̄ production. The line in the figure for s = 100000 GeV is the best fitted result
with the simulated data that is obtained by using equation (4.9) is shown as an example.
4.4 Results and Discussion
87
as function of p2⊥ for different values of
√
s in pp collisions. The production cross sections for D-
mesons are slightly higher than that for Λc -baryons within the observable range of p2⊥ (i. e. within
which production cross section follows some order). The maximum production of all charmed
√
√
hadrons take place within very low values of p2⊥ , for all values of s, but as values of s increases
the production of charmed hadrons with higher and higher values of p2⊥ also increases. We compare
√
our calculated results with the experimental data from [35] for D/D̄ production at s = 27.4GeV
√
and s = 38.8GeV. Agreement is good in the lower range of p2⊥ . The production cross sections as
√
function of p2⊥ obey the second order exponential distribution for all values of s, which can be
express as,
dσ
2
2
= σ0p2⊥ + a1 e−p⊥ /γ1 + a2 e−p⊥ /γ2 ,
2
dp⊥
(4.10)
√
where σ0p2⊥ , a1 , a2 , γ1 and γ2 are constants having different values for different values of s and
√
for different charmed hadrons. As an example, the line as shown in the Figure 4.8 for s = 100000
GeV is the best fitted result obtained by using equation (4.10) with the calculated data.
The distributions of charmed hadrons production cross sections as a function of p⊥ for differ√
ent values of s are shown in the Figure 4.9. It is seen that from very small value (∼ 0.05 GeV/c)
to about 0.5 GeV/c of p⊥ , the production cross sections increases gradually starting with higher
values and after that it starts to decrease following a definite pattern up to some higher value of
√
√
p⊥ depending upon the value of s. For s = 100000 GeV this value of p⊥ is approximately 6
√
GeV/c and for s = 27.4 GeV it is about 2.5 GeV/c. So the range of p⊥ for the production of
√
charmed hadrons with very high and relatively very low values of s is not very wide. This is an
inconsistent result with expectation and is a major shortcoming of the model. This might be due
to not inclusion of NLO diagrams in the model. On the other hand, the pattern of production of all
charmed hadrons with p⊥ are same as that with xF and p2⊥ . However, the difference of production
of D-mesons and Λc -baryons is considerable within a small range and for low values of p⊥ (≤ 1
GeV/c) only (where cross section is high for D-mesons) and for rest values of p⊥ the pattern is
√
same as that for xF and p2⊥ for all values of s. The production cross sections as a function of p⊥
√
is found to follow the Lorentzian distribution for all s, which can be expressed as,
d3 σ
2σ1p⊥
σ2p⊥
E 3 = σ0p⊥ +
2
dp
π 4(p⊥ − p0⊥ )2 + σ2p
⊥
where σ0p⊥ , σ1p⊥ , σ2p⊥ and p0⊥ are constants for the particular value of
(4.11)
√
s and for the particular
charmed Hadron. The fitting of this function (4.11) with our calculated data is shown in Figure 4.9
√
for s = 100000 GeV as a line for an example.
4.4.2 Asymmetry in charmed hadron production
The study of characteristics of charmed hadroproduction asymmetry is another important
aspect in connection with the charge excess of atmospheric prompt leptons. The charge excess
4.4 Results and Discussion
88
–
2
100
10
1
0
2
4
6
p⊥
2
8
1
2
4
6
2
8
10
14
–
D/–D
Λc/Λc
total
100
√s = 100 GeV
10
2
2
10
12
2
(GeV/c)
2
2
10
1000
√s = 1000 GeV
1
0
2
4
6
2
p⊥
1000
8
10
(GeV/c)
12
0
1
2
2
2
4
5
6
7
8
2
(GeV/c)
–
2
dσ/dp⊥ [µb/(GeV/c) ]
√s = 38.8 GeV
10
3
p⊥
–
100
D/–D
Λc/Λc
total
–
exp.(D/D)
100
10
√s = 27.4 GeV
1
2
1
0.1
0.01
1
0.1
14
D/–D
Λc/Λc
total
–
exp.(D/D)
2
2
√s = 10000 GeV
100
0
0.1
dσ/dp⊥ [µb/(GeV/c) ]
1000
p⊥
–
100
–
D/–D
Λc/Λc
total
2
D/–D
Λc/Λc
total
1000
10000
10 12 14 16
(GeV/c)
10000
dσ/dp⊥ [µb/(GeV/c) ]
2
√s = 100000 GeV
1000
dσ/dp⊥ [µb/(GeV/c) ]
2
10000
dσ/dp⊥ [µb/(GeV/c) ]
D/–D
Λc/Λc
total
2
dσ/dp⊥ [µb/(GeV/c) ]
100000
0
1
2
3
4
5
6
2
2
p⊥ (GeV/c)
7
8
0.1
0.01
0
1
2
p⊥
2
3
2
(GeV/c)
4
5
Figure 4.8: Inclusive cross sections for the production of charmed hadrons in pp collisions at
√
different centre of mass energy ( s) as a function of p2⊥ . The cross sections are calculated with
√
√
the QGSJET model at s = 27.4 GeV and s = 38.8 GeV are compared with the experimental
√
data [35] for D/D̄ production. The line in the figure for s = 100000 GeV is the best fitted result
with the simulated data that is obtained by using equation (4.10) is shown as an example.
4.4 Results and Discussion
89
–
3
100
Ed3σ/d3p (µb/GeV2)
√s = 100000 GeV
10000
1000
–
D/–D
Λc/Λc
total
100000
3
2
Ed σ/d p (µb/GeV )
1e+06
1
2
3
4
p⊥ (GeV/c)
5
1000
100
10
6
total
√s = 1000 GeV
1000
3
100
Ed3σ/d3p (µb/GeV2)
10000
0
1
2
3
p⊥ (GeV/c)
1000
2
3
4
p⊥ (GeV/c)
5
–
D/–D
Λc/Λc
total
√s = 100 GeV
1000
100
4
–
0
Ed3σ/d3p (µb/GeV2)
2
D/–D
Λc/Λc
total
3
100
√s = 38.8 GeV
3
1
10
10
Ed σ/d p (µb/GeV )
0
10000
–
D/–D
Λc/Λc
3
2
Ed σ/d p (µb/GeV )
100000
10
0.5
1
1.5
p⊥ (GeV/c)
2
0.5
1
1.5
2
p⊥ (GeV/c)
1000
2.5
–
D/–D
Λc/Λc
total
100
√s = 27.4 GeV
10
0
√s = 10000 GeV
10000
10
0
D/–D
Λc/Λc
total
100000
0
0.5
1
1.5
p⊥ (GeV/c)
2
Figure 4.9: Inclusive cross sections for the production of charmed hadrons in pp collisions at
√
√
different centre of mass energy ( s) as a function of p⊥ . The line in the figure for s = 100000
GeV is the best fitted result with the simulated data that is obtained by using equation (4.11) is
shown as an example.
4.4 Results and Discussion
90
of secondary particles are due to the flavour contents of the colliding hadrons, because based
on the flavour contents, the type (positive, negative, direct or anti) and the rank (first, second,
third etc.) of secondary particles are produced [36]. So in the QGSJET model the asymmetry in
charmed hadroproduction is incorporated via string fragmentation process (i.e. as a features of
the hadronization process) [33]. Actually asymmetry is a manifestation of an intrinsic charmed
content of the colliding hadrons [42]. The experiment E769 [38] observed a Λc asymmetry for a
250 GeV/c proton beam and in the same experiment D meson asymmetry for a 250 GeV/c pion
beam was also measured. The WA89 experiment [39] studied the charmed particles produced
−
by a Σ− 340 GeV/c beam. Considerable production asymmetry between D+ , D− and Λ+
c , Λ̄c
was observed in this experiment. The WA92 [39] and E791 [40] experiments show D meson
asymmetry in 350 GeV/c pion beam and a 500 GeV/c pion beam respectively. In our reference,
−
we have studied the hadroproduction asymmetry for charmed hadrons D+ , D− , D0 , D̄0 , Λ+
c , Λ̄c
√
in pp collisions as function of s, xF , and p⊥ using QGSJET model. We have calculated the
hadroproduction asymmetry As by [37],
As =
σc − σc̄
,
σc + σc̄
(4.12)
where σc (σc̄ ) is the production cross section for the charmed particle (anti-charmed particle) in
study. Asymmetry in charmed hadrons production in pp collisions as calculated with the QGSJET
√
model as a function of s is shown in the Figure 4.10. It is observed that, for all xF asymmetry
√
√
is prominent in the low value of s. This is obvious because as the value of s increases, the
difference in numbers of different charmed hadrons production decreases. It is also observed that
−
there is strong preference for producing Λ+
c baryon rather than Λ̄c , while that for producing D̄
√
√
mesons rather than D mesons for this range of s (≤ 200 GeV). Moreover, for whole range of s
under study this tendency follows by produced charmed hadrons with varying degrees. In the low
energy range, this tendency is much strong in case of Λc -baryons than D-mesons. As the value of
√
s increases beyond 200 GeV asymmetric tendency of production of charmed hadrons decreases
√
and at sufficiently high value of s (≥ 100000 GeV) this tendency almost vanishes.
Some results of the study of asymmetry in charmed hadrons production in pp collision at
√
different s as a function of Feynman xF is shown in Figure 4.11. It is clear from the figure
√
that for low s range (≤ 100 GeV) asymmetry increases from zero to one at xF ≤ 0.3, whereas
√
for higher value of s, it increases to the same value at xF ≥ 0.3. The patterns of production
of different charmed hadron groups as a function xF are that, more number of Λ+
c baryons are
produced than that of Λ̄−
c baryons, whereas the reverse is the case for D-mesons for all values of
√
√
s. This behaviour of production of charmed hadrons is similar to that with s for all values of
√
xF . We have compared the results of Λc particles at s = 31.9 GeV with the experimental data
from [37]. It is observed that agreement is quite satisfactory.
The asymmetric production of charmed hadrons as a function of p⊥ does not follow any well
√
defined pattern for all values of s as shown in the Figure 4.12. However, we can infer that for
4.5 Conclusion
91
0.8
D+ - D0 –0
D -D
–
Λc - Λc
Asymmetry
0.6
0.4
0.2
0
-0.2
-0.4
10
100
1000
10000
100000
√s (GeV)
Figure 4.10: Asymmetry in charmed hadrons production in pp collisions with
√
s as calculated
with QGSJET model.
high values of
√
s, asymmetry increases from zero at the value of p⊥ ≥ 3 GeV/c to one at p⊥ ∼
√
4.5 GeV/c. For low values of s this tendency is more even for very low value of p⊥ . At these
√
low values of s pattern of asymmetric production of charmed hadrons with p⊥ is approximately
√
same as that with s and xF . After all, the charmed hadroproduction asymmetry as a function of
p⊥ could not be explained well with the QGSJET model.
4.5 Conclusion
From the study of charmed hadroproduction in pp collisions with the QGSJET model [7]
it is found that the average multiplicities of charmed hadrons under study are comparatively very
low for the whole energy range of our interest and the difference in multiplicities of different
√
charmed hadrons is also very small and decreases considerably with increasing s. Notwithstanding these rare events are very much important in the context of CR physics as they are able to
change the traditional shower behaviour completely in the form of prompt leptons excessive multiplicity. Comparison of production cross sections for D/D̄ with the available experimental data
√
[35] at s < 40 GeV shows almost good agreement. From the study of distributions of charmed
hadrons production cross sections as functions of xF and p2⊥ it is seen that the maximum number of
√
charmed hadrons are produced with small values of xF and p2⊥ for all values of s. However with
4.5 Conclusion
92
1.5
1.5
1
+
D
0
-D
–0
D -D
–
Λc - Λc
0.5
0
√s = 100000 GeV
-0.5
+
Asymmetry
Asymmetry
1
-1
-1.5
-
D -D
0 –0
D -D
–
Λc - Λc
0.5
0
√s = 10000 GeV
-0.5
-1
0
-1.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
XF
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
XF
1.5
1.5
1
Asymmetry
+
-
D -D
0 –0
D -D
–
Λc - Λc
0.5
0
√s = 1000 GeV
-0.5
Asymmetry
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
XF
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
XF
1.5
1
D+0 - D
–0
D -D
Λc - –Λc
0.5
0
√s = 50 GeV
-0.5
-1
Asymmetry
1
Asymmetry
√s = 100 GeV
-0.5
-1.5
0
1.5
-1.5
0
-1
-1
-1.5
D+0 - D
–0
D -D
Λc - –Λc
0.5
0.5
0
-0.5
+
-
D0 - D
D - –D0
Λc -– –Λc
exp.(Λc-Λc)
√s = 31.9 GeV
-1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
XF
-1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
xF
Figure 4.11: Asymmetry in charmed hadrons production in pp collisions at different
√
s as function
√
of Feynman xF . Results for Λc particles are compared with the experimental data [37] at s =
31.9 GeV.
4.5 Conclusion
D+0 - D
–0
D -D
Λc - –Λc
0.5
D+0 - D
–0
D -D
Λc - –Λc
1
Asymmetry
1
Asymmetry
93
0
-0.5
0.5
0
-0.5
√s = 100000 GeV
-1
0
1
2
3
4
p⊥ (GeV/c)
5
6
D+ - D0
D0 - –D
Λc - –Λc
0.5
0
0
-0.5
0.5
2
3
p⊥ (GeV/c)
4
√s = 100 GeV
0
D+0 - D
–0
D -D
Λc - –Λc
1
1
0.5
0
-0.5
0
0.5
1
1.5
p⊥ (GeV/c)
-0.5
√s = 31.9 GeV
0
0.5
1
1.5
p⊥ (GeV/c)
Figure 4.12: Asymmetry in charmed hadrons production in pp collisions at different
of p⊥ .
2.5
0
-1.5
2
1
1.5
2
p⊥ (GeV/c)
0.5
-1
√s = 50 GeV
-1
0.5
D+ - D0
D0 - –D
Λc - –Λc
1.5
Asymmetry
Asymmetry
1
5
-0.5
-1
0
2
3
4
p⊥ (GeV/c)
0
√s = 1000 GeV
-1
1
D+ - D0
D0 - –D
Λc - –Λc
1
Asymmetry
1
Asymmetry
√s = 10000 GeV
-1
√
2
s as function
4.5 Conclusion
increasing
√
94
s the number of charmed hadrons with the increasing values of xF and p2⊥ are also
produced. The agreement of D/D̄ production cross sections as function of xF with experimental
√
data [35] at s < 40 GeV are not satisfactory, whereas the same as function of p2⊥ is almost good
√
in the lower range of p2⊥ . Asymmetric behaviour of charmed hadroproduction as function of s,
xF , and p⊥ are also studied. This study reveals that, the asymmetry of charmed hadrons produc√
tion decreases with increasing s and asymmetry increases from zero to one at xF ≤ 0.3 for low
√
√
s range (≤ 100 GeV) whereas for higher values s it increases to the same value at xF ≥ 0.3.
There is a strong preference for the production of charmed baryons over anti-baryons, while that
for charmed anti-mesons over mesons. The asymmetric behaviour of Λc particles production is
√
compared with experiment [37] at s = 31.9 GeV and gives a satisfactory agreement. Unfortunately the QGSJET model predictions of charmed hadrons production characteristics as a function
of p⊥ is not satisfactory and is a serious problem with it to be used it in the charmed field.
References
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[17] E. G. Cazzoli et al., Phys. Rev. Lett. 34, 1125 (1975).
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97
Chapter 5
USE OF CORSIKA TO REANALYSE GU
MINIARRAY DATA
5.1 Introduction
GU Miniarray [1] is an unconventional detector array consisting of eight plastic scintillation
counters covering a very small carpet area (ref section 5.2). This is used to detect giant EAS
initiated by UHECR (≥ 1017 eV) in the earth’s atmosphere based on the Linsley’s effect [2].
According to the Linsley’s effect, with the increasing distance from the shower core the arrival
time spread in a particle sample from a given shower increases. Thus the measured time spread of
particles striking on a localized detector system gives an estimate of the distance to the shower core.
The number of particles give the measure of the local particle density. From these measurements,
the shower size is estimated from the parametrized lateral density distribution function of shower
size and distance to shower core. Finally, the primary energy corresponding to an event with the
estimated shower size is derived using the best fitted relation between shower size and the primary
energy. So, the GU Miniarray detectors system is designed specially to measure both charged
particle density and arrival time at the detector level [1]. Previously the primary energy of the EAS
detected by GU Miniarray was derived using the best fitted relation in agreement with QGS model
and Yakutsk data [3].
It is well known that, the flux of UHECR is so low that, the direct detection of the primary
particles is almost impossible (ref chapter 1). That is why, in this region, indirect methods of
detection are employed by exploiting the phenomenon of EAS. Traditionally these methods use
ground based particle and optical detectors arrays spread over a wide area (sometimes several
km2 ) (ref section 1.1.2). On the other hand, in contrast to this traditional means of detection of
UHECR, the GU Miniarray exposes relatively a very small effective area (carpet area of 2 m2 ) for
giant EAS particles. Thus it is advisable to check the effectiveness of the GU Miniarray detector
to detect the giant EAS, by simulating and analysing its data with a standard MC code. This is
98
5.2 Miniarray Experiment
99
due to the fact that, to study the cascade development in the atmosphere, it is best to use MC
code to simulate the same taking into account all knowledge of HE hadronic and electromagnetic
interactions involved (ref section 2.6). In this connection we present in this chapter a reanalysis
of old data of GU Miniarray, taken during the period from October 1996 to April 1998, using the
widely used standard MC code CORSIKA [4] (ref section 2.7) with the HE hadronic interaction
model QGSJET [5] (ref section 4.3). Here we re-estimated the primary energy of giant EAS from
the parametrized relation between the shower size and primary energy obtained from the new
analysis [6].
For the completeness, in the next section we discuss briefly the experimental setup of the GU
Miniarray detector.
5.2 The Miniarray Experiment
The GU Miniarray consists of eight plastic scintillators of carpet area 2 m2 , each viewed by
fast PMTs [7]. It is located on the rooftop of the Physics building, Gauhati University (26o 10# N,
91o 45# E and altitude 51.8 m). Figure 5.1 shows the block diagram of the GU Miniarray data acquisition system. The signals from the eight detectors are amplified and then carried to the control
room via co-axial cables. In the control room all the eight signals are discriminated to provide
corresponding logic signals. The discriminated output signals are then individually shaped into
narrow pulses of 20 ns width and OR’ed together to give a serial pulse train. The serial pulse train
is branched into two individual channels, one fed to the Digital Storage Oscilloscope (DSO) and
the other to the trigger unit. The trigger unit senses the incoming pulse train and generates the
necessary trigger pulse. Once triggered, the oscilloscope trace is stopped and the recorded data is
transferred to the PC via GPIB interface. The microprocessor monitors the status of the detectors
by recording count rates at regular interval and also transfers count rate data to the PC via RS232C
serial interface for storage [1].
Each detector unit consists of a plastic scintillator block (size 50 × 50 × 5 cm3 ) having
polyvinyltoluene base (with resolution 20% and decay time 4 ns), a fast photomultiplier tube (EMI
9807B) and a preamplifier unit enclosed into a light tight enclosure. The PMT is a round faced
end window type with semitransparent bialkali photocathode having maximum sensitivity in the
blue region of the spectrum. It operates at an anode potential of 1800 V and has a gain of 7.1×106
with rise time 2 ns. The anode pulse is coupled to the preamplifier unit which is a double stage
differential amplifier with rise time 2 ns, gain 10 and overall band width 200 MHz [1]. All the eight
detectors are housed in a hut constructed on the roof top of the Physics building, Gauhati University. The detector pulses after preamplification, are transmitted to the control room via co-axial
cables (RG58U). A microcomputer based on 8086 microprocessor is used to monitor the operation of the detectors by recording count rate at predetermined intervals. Only genuine pulses are
5.2 Miniarray Experiment
100
SCINTILATION
DETECTORS
D1 D2
D3
D4
D5
8086
MICROCOMPUTER
RAM
8 CHANNEL
D6
D7 D8
COUNTER
MULTIPLEXER
I/O
CONTROL
8
PRE
AMP
RS232C
PULSE
WIDTH
TRIGGER
TRIG IN
OR
500 MHz
D. S. O.
TEKTRONIX
TDS 520A
Y IN
8 DISCRIMINATORS
GPIB
PC
COM PORT
AT
COMPUTER
Figure 5.1: Block diagram of the GU Miniarray data acquisition system.
considered for analysis and noise pulses are effectively rejected by triggering criteria and visual
inspection.
The integral CR flux of the secondary charged particles is 1.8×102 m−2 s−1 , and thus the number of charged particles crossing the scintillator block of area 0.25 m2 is 45 s−1 . Therefore, the
single particle rate for one channel of the detector array can be considered as 45 Hz. The calibration of the detectors for single particle pulse height is performed using a standard single channel
analyzer (ECIL, SC604B) and a counter. All the eight discriminator biases are adjusted at the individual single particle peak position. The single particle peak of a scintillation detector corresponds
to a mean count rate of about 50 s−1 . This corresponds to a flux of ∼ 200 particles/m2 /s [1]. The
minimum energy of the detected particle is about 100 MeV. For recording an event, a trigger pulse
is generated under some prerequisite criteria, viz.,
(i) A hardware trigger requiring particle in the range 2 or above within the time window.
(ii) The minimum time spread between the particles must be 100 ns.
Trigger rates were monitored daily over the length of the experiment and were found to be
stable. The data for the period of 3.633×107 s (≈ 421 days) include for each event, the charged
particle densities and arrival times of these particles with reference to the leading particle (first
arriving particle hitting the miniarray) [1].
5.3 Simulation and Data Analysis
101
In the following section the theoretical background of simulation and reanalysis of GU Miniarray detector data are presented.
5.3 Simulation and Data Analysis
For the purpose of reanalysis of old data of the GU Miniarray, we have generated EAS by
CORSIKA version 6019 [4] (ref section 2.7) with the QGSJET [5, 9] model (ref section 4.3) based
event generator code QGSJET01 [4, 5] for HE hadronic interaction. For our MC data library we
have simulated 500 vertical showers in the energy range from 1017 eV to 1020 eV with an energy
slope of − 2.65 taking proton and iron as primary particles. Statistical thinning [10, 4, 11] of
shower particles was applied at the level of 10−4 E0 with a maximum particle weight limit of 1030 .
The threshold energies are 0.1 GeV for hadrons, muons and electromagnetic component.
5.3.1 Numerical equations
The relation between shower disk thickness σ (ns) and the core distance r (m) has been
derived by J. N. Capdevielle et al. [12] from their simulation with CORSIKA for near vertical
shower, which is given by,
r
σ(r) = B(1 + )β ,
(5.1)
c
where B = 2.6, c = 25 and β = 1.4. Figure 5.2 shows the comparison of shower disk thickness at
different distances from shower axis as calculated from equation (5.1) with results from Linsley’s
original equation [1, 13] for the same. The figure shows a reasonable agreement between these two
equations with a slight discrepancy in the higher core distances. The particle density distribution
function for large shower and medium core distances (100 m < r < 1000 m) has been parametrized
by using our simulated miniarray data for proton and iron initiated showers separately which is as
follows,
ρ(Ns , r) = 1Ns r−n ,
(5.2)
where Ns is the size of the shower. For proton primary 1 = 0.00053, n = 1.5 and for iron primary
1 = 0.0086, n = 1.9. Figure 5.3 shows the lateral density distributions for proton and iron initiated
showers simulated for the miniarray using CORSIKA in comparison with earlier relation [1, 14]
that was used for miniarray data analysis previously. There is a noticeable disagreement between
relations obtained with CORSIKA and the earlier one. The integral and the differential shower
size spectra are respectively given by [3],
J(Ns ) = DNs−γ ,
(5.3)
j(Ns ) = − γDNs−γ−1 ,
(5.4)
where the constants have values D = 318 and γ = 1.65. With the help of the above relations
5.3 Simulation and Data Analysis
102
900
800
Linsley
Corsika
700
σ (ns)
600
500
400
300
200
100
200 400 600 800 1000 1200 1400
r (m)
Figure 5.2: Shower disk thickness versus distance from the shower axis. Linsley’s relation is
shown in comparison with CORSIKA simulation.
we derived the frequencies of Linsley’s events as a function of the minimum time spread (σ1 ), the
threshold density (ρ1 ) and the shower size (Ns ) as,
FL (> σ1 , ρ1 ) =
) ∞
Nmin
FL (> Ns > σ1 , ρ1 ) =
where,
A(Ns )j(Ns )dNs ,
) ∞
Ns
A(Ns )j(Ns )dNs ,
2
2
A(Ns ) = π(rmax
− rmin
).
(5.5)
(5.6)
(5.7)
Here, the effective area A(Ns ) is an annular ring with outer radius (rmax ) determined by the density
threshold ρ1 = 1.5 m−2 and its inner radius (rmin ) determined by the minimum time spread σ1 as
selected. From equations (5.1) and (5.2) we can show that,
:
;
rmin = c (σ1 /B)1/β − 1 ,
3
41/n
(5.8)
1Ns
rmax =
.
(5.9)
ρ1
The minimum detectable shower size of the GU Miniarray detectors is given by the condition,
A(Nmin ) = 0,
which gives,
:
Nmin = (ρ1 /1) c{(σ1 /B)1/β − 1}
;n
.
(5.10)
5.3 Simulation and Data Analysis
-2
Density (m )
1000
103
Proton
Iron
Old Relation
100
10
1
100
1000
Distance (m)
Figure 5.3: Lateral particle density distribution obtained from the simulation with CORSIKA and
from the old relation.
5.3.2 Detector response
Simulation of detector response for all the charged particles in the vertical direction of GU
Miniarray detectors has been performed. Particles are collected for 500 simulated showers within
the annular area ranging from rmin to rmax and for primary energies from 1017 eV to 1020 eV
with minimum detectable energy of the particles as 100 MeV. From this, mean particle density at
different core distances have been calculated. For the minimum detectable shower size and the
threshold density, events are counted for all those simulated showers for different primary energy
bins. Figure 5.4 shows the variation of effective area of the GU Miniarray with primary energy.
The effective area increases with increasing energy. The simulated data have lower effective area
as compared to the old relation [1].
5.3.3 Energy calibration
The primary energy for each shower is reconstructed using the estimated value of Ns . The
calibration curves for proton and iron primaries are plotted in the Figure 5.5. This figure shows
the comparison of these curves with the best fitted relation with Yakutsk data [1, 3], that was used
104
10
8
10
7
10
6
10
5
Proton
Iron
Old Relation
2
Effective annular area (m )
5.4 Data Selection Criteria
17
10
10
Energy (eV)
18
Figure 5.4: Effective area of GU Miniarray versus primary energy for proton and iron showers
simulation using CORSIKA compared with the results from old relation.
previously for data analysis. From this new analysis the energy calibration can be parametrized as,
E0 (eV ) = aNsb ,
(5.11)
where a = 2.217×1011 , b = 0.798 with an approximate percentage error of 12% for proton and
a = 6.194×1011 , b = 0.898 with an approximate percentage error of 8% for iron. There is a
noticeable difference between the old and the new calibration, which leads to the reconstruction of
energies.
5.4 Data Selection Criteria
From numerical calculations it is found that for a given threshold density (ρ1 = 1.5 m−2 ),
the minimum detectable shower size increases and the shower rate decreases with increasing time
spread. The miniarray should be able to pick out at least few large air shower events from a swarm
of irrelevant events including the counter noises, the background soft radiations and small air showers. In order to eliminate the large number of small air showers a minimum time spread has to be
assigned. Present analysis shows that for the GU Miniarray of 2 m2 area, the minimum acceptable
shower sizes are 1.57×107 for proton and 9.67×106 for iron against previously calculated value
Shower size
5.5 Results and Discussion
10
10
105
8
Proton
Iron
Old Relation
7
10
17
18
10
Energy (eV)
Figure 5.5: Shower size versus primary energy for proton and iron shower simulation using CORSIKA. Solid line represents energy calibration curve for proton and the dotted line for iron.
of 7.5×106 with a minimum time spread σ = 100 ns. In view of the small particle density encountered, each scintillator is not expected to receive more than one particle at a time from a shower.
Figure 5.6 shows the event distribution as a function of shower disk thickness (σ) as recorded by
the GU Miniarray during the said period of operation.
Most of the data collected by the miniarray do not belong to true EAS events. The data were
reduced by the selection process and by visual inspection. True EAS events with a time spread of
shower front σ ≥ 100 ns and with the local particle densities ρ ≥ 1.5 m−2 were only collected and
analyzed. All these criteria are imposed in the simulation of the miniarray data with CORSIKA.
The results of the reanalysis of the miniarray data that were collected from October 1996 to April
1998 are discussed in the section below.
5.5 Results and Discussion
In this reanalysis of the GU Miniarray data as mentioned above, the following new results are
derived.
Number of events
5.5 Results and Discussion
106
100
10
100
1000
σ (ns)
Figure 5.6: Event number distribution as a function of shower disk thickness (σ) as recorded by
the GU Miniarray.
5.5.1 Shower rate and shower size spectrum
The integral shower rate spectrum of the selected events with ρ1 = 1.5 m−2 as function of time
spread is shown in the Figure 5.7. The errors are estimated by considering the ± 10 ns instrumental
error. Here unfolding of data is done by first parametrizing the integral flux as a function of time
spread (σ) from the simulation with the CORSIKA. Then the parametrized equation is used to
unfold the values of the integral flux for different experimental values of the σ. The corresponding
parametrized equation is,
FL = F0 exp(b/σ c ),
(5.12)
where F0 = 0.00015, b = 974.284, c = 0.91 with an approximate percentage error of 19% for
proton flux and F0 = 0.0029, b = 72866.7, c = 1.78 with an approximate percentage error of 16%
for iron flux. This figure also shows the comparison of the result of the old analysis with the present
reanalysis under pure proton and pure iron assumptions. A significant disagreement is observed for
higher values of time spreads. This indicates a more rapid decrease of shower rate with increasing
time spread than as observed in the earlier analysis [1]. Thus it shows a remarkable drawback
of earlier analysis which had counted more showers with higher time spread than the actual case.
There is also another possibility to encounter more shower number with time that is due to inherent
drawback of the experimental setup to measure the time spread of shower disk thickness. It should
be noted that, the error introduced in the data due to improper time measurement is independent
-2
-1
-1
FL(>σ1) (m sr day )
5.5 Results and Discussion
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
107
Proton
Iron
Old analysis
100 200 300 400 500 600 700
σ (ns)
Figure 5.7: Integral shower rate spectrum with ρ1 = 1.5m−2 and σ1 = 100 ns. Data for proton and
iron are obtained from reanalysis of experimental data using CORSIKA assuming proton and iron
as primary CR particles.
of methods of analysis. So it will also effect the new analysis and hence will effect the structure of
differential energy spectrum.
Integral shower size spectrum considering events with threshold density ρ1 = 1.5 m−2 is shown
in the Figure 5.8. Data are unfolded using the parametrized equation,
FL = F0 Nsb ,
(5.13)
where F0 = 2.557×108 , b = − 0.94 with an approximate percentage error of 22% for proton flux
and F0 = 3.773×107 , b = − 0.83 with an approximate percentage error of 24% for iron flux.
Reanalyzed data obtained using CORSIKA is compared with the data from earlier analysis. In
this case the disagreement with old method of analysis is less significant compared with shower
rate spectrum. However, there is a disagreement and it is observed that this is more for small
shower size. From the new analysis it is observed that shower flux decreases almost linearly with
increasing shower size for both proton and iron primaries.
10
2
10
1
10
0
108
Proton
Iron
Old analysis
-2
-1
-1
FL(>N) (m sr day )
5.5 Results and Discussion
10
7
10
Shower size
8
Figure 5.8: Integral shower size spectrum with ρ1 = 1.5 m−2 . Data for proton and iron are obtained
from reanalysis of experimental data using CORSIKA assuming proton and iron as primary CR
particles.
5.5.2 Energy spectrum
Figure 5.9 shows the differential energy spectrum derived from the reanalysis of the GU
Miniarray data assuming proton and iron as CR primary particles. We also compare here the new
differential energy spectrum with the old analysis and results of Yakutsk [15], AGASA [16] and
HiRes [17, 19]. A compilation of spectrum derived from AGASA and Haverah Park [18] data by
Nagano and Watson [20] is also shown in the Figure 5.9 for comparison. The following important
results have been observed from the present analysis,
(1) The primary energy estimated by the new analysis is significantly higher than the previous
analysis. Energy spectrum after reanalysis is found to span from 1017 eV to 1019 eV for proton
primary and from 1017 eV to 1019.4 eV for iron primary and further confirms the irregular behavior
of energy spectrum at UHE region with a prominent dip.
(2) The differential energy spectrum shows structure similar to that observed by other world
groups. Spectral breaks are found to occur at higher energies compared with old method of analysis. The spectrum becomes steeper around 1017.5 eV and 1017.7 eV and flattens around 1018.7 eV
and 1019.1 eV for proton and iron primaries respectively forming a dip. Earlier analysis showed a
dip around 1018.2 eV. A comparison of different features with other world data are shown in the
5.5 Results and Discussion
109
17
10
Proton
Iron
Old analysis
Yakutsk
AGASA
HiRes
16
10
j(E) E0
2.5
-2
-1
-1
1.5
(m sec sr eV )
1018
1015
1014
1017
1018
1019
E0 (eV)
Figure 5.9: Miniarray differential energy spectrum. The differential flux is multiplied by E02.5 . Data
for proton and iron are obtained from reanalysis of experimental data using CORSIKA assuming
proton and iron as primary CR particles. Region between solid lines give the flux as compiled from
AGASA and Haverah Park data by Nagano and Watson (2000).
5.5 Results and Discussion
110
Tables 5.1 and 5.2.
(3) There is a significant difference between spectra predicted by pure proton and pure iron
Table 5.1: A comparison of slope before the dip of the differential energy spectrum.
Experiment
Slope before the dip Energy range (eV)
Yakutsk
AGASA
HiRes-I
Miniarray(old analysis)
Miniarray(new analysis,p)
Miniarray(new analysis,Fe)
-3.195 ± 0.009
1017.5 - 1019.2
-3.185 ± 0.059
1017.2 - 1018.5
-2.733 ± 0.094
1017.5 - 1018.7
-2.981 ± 0.058
1017.6 - 1018.9
-3.468 ± 0.131
1017.4 - 1018.2
-2.538 ± 0.081
1017.7 - 1019.1
assumptions. However, proton assumption results agree more with Nagano and Watson compilation within spectral range from 1017.5 eV to 1018.5 eV. It needs to be mentioned that, beyond
1019 eV, results of different giant arrays are contradictory. AGASA provides strong evidence for
the existence of CR with energies beyond Greisen-Zatzepin-Kuzmin (GZK) cut-off (ref section
1.1.3). By contrast, data from the HiRes detector in US are compatible with the existence of the
GZK cut-off [21]. However the statistics are too low in this range of CR spectrum to arrive at
a definite conclusion. So more statistics is necessary to resolve this important feature related to
Astrophysics. This matter will definitely be taken up by the Pierre Auger Observatory [22] in a
very near future.
(4) The differential energy spectrum corresponding to best fit in the energy region from 1017.0
Table 5.2: A comparison of overall slope of the differential energy spectrum.
Experiment
Slope
Energy range (eV)
Yakutsk
-3.031 ± 0.047
1017.5 - 1019.8
-3.151 ± 0.036
1017.2 - 1019.2
-2.360 ± 0.075
1017.0 - 1019.0
-2.884 ± 0.059
1017.6 - 1019.2
-2.938 ± 0.108
1017.0 - 1018.8
Miniarray(new analysis,Fe) -2.207 ± 0.067
1017.0 - 1019.4
AGASA
HiRes-I
Miniarray(old analysis)
Miniarray(new analysis,p)
eV to 1019.0 eV is derived as,
j(E0 ) = j0 E0−n
(5.14)
5.6 Conclusion and Remarks
111
For proton primary j0 = 7.107×1012 , n = 2.360 ± 0.075 and for iron primary j0 = 2.772×1010 ,
n = 2.207 ± 0.067.
(5) A relatively higher flux is seen in the reanalysis of the data for both proton and iron pri-
maries assumption, beyond the dip, as compared with other world data. This overestimation of the
flux by the GU Miniarray indicates some inherent shortfall of the detectors array (ref section 5.6).
5.6 Conclusion and Remarks
The contribution of the GU Miniarray detectors to the UHECR world data set is not negligible
as the structure of its differential energy spectrum in this part of CR is comparable to other renown
groups of CR research of the world, those are engaged with giant detectors arrays for this purpose.
In this sense, the GU Miniarray is a unique type of detectors array. It has three main advantages
: low cost, small size and less time factor with low labour demands. So this method is really a
boon for scientists of under developed countries to engage with that part of CR research where
traditionally big funding was necessary for such research activities.
The structure of the differential energy spectrum of GU Miniarray detector is comparable with
the same of the other world groups. However, it does not agree well with the results of these
groups and it overestimates the event rate around and above ankle. This implies some short of
serious shortcoming in the miniarray detector setup. Since the effectiveness of GU Miniarray
detector is based on time spread measurement of shower front of particle sample of a given shower
(Linsley’s Effect), so this overestimation of shower rate at highest energy part is mainly due to the
error in the measurement of time by the miniarray. Other drawbacks are inclusion of some delayed
particles, small detector area, and associated triggering criterion. Thus to minimize these errors,
the size of the miniarray must be increased such that more particles can be detected per event and
unwanted events may be distinguished by careful investigation with improved triggering criterion.
References
[1] T. Bezboruah, K. Boruah and P. K. Boruah, Astropart. Phys. 11, 395 (1999).
[2] J. Linsley , G. Phys. G12, 51 (1986).
[3] A. M. Hillas, Phys. Report 20C, 79 (1975).
[4] D. Heck and J. Knapp, Report FZKA 6019 (1998), Forschungszentrum Karlsruhe.
[5] N. N. Kalmykov, S. S. Ostapchenko and A. I. Pavlov, Nucl. Phys. B (Proc. Suppl.) 52B, 17
(1997).
[6] U. D. Goswami, K. Boruah, P. K. Boruah, T. Bezboruah, Astropart. Phys. 22, 421 (2005); U.
D. Goswami, K. Boruah, and P. K. Boruah, Proc. 29th ICRC, Pune (2005), 7, 159.
[7] T. Bezboruah, K. Boruah and P. K. Boruah, Nucl. Inst. Meth. in Phys. Res. A410, 206 (1998).
[8] K.-H. Kampert et al., (KASCADE Collaboration), Proc. 26th ICRC, Salt Lake City, USA
(1999), 3, 159.
[9] V. N. Gribov, Sov. Phys. JETP 26, 414 (1968); V. N. Gribov, Sov. Phys. JETP 29, 483 (1969).
[10] W. R. Nelson, H. Hirayama and D. W. O. Rogers, Report SLAC 265 (1986), Stanford Linear
Accelerator Centre.
[11] M. Hillas, Nucl. Phys. B (Proc. Supply) 52B, 29 (1997).
[12] J. N. Capdevielle et al., Proc. 28th ICRC, Tsukuba, Japan (2003), 2, 217.
[13] J. Linsley and L. Scarsi, Phys. Rev. Lett. 9, 123 (1962).
[14] T. Hara et al. Proc. 25th ICRC, Durban, South Africa (1997), 6, 229.
[15] N. N. Efimov et al. (Yakutsk collaboration), Proc. of Int. Sym. on Astrophysical Aspect of
the Most Energetic Cosmic Rays, Singapure (1991), 20; B. N. Afnasiv (Yakutsk collaboration), Proc. of Int. Sym. on Extremely High Energy Cosmic Rays : Astrophysics and future
observations, Tokyo (1996), 32.
112
5 References
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[16] N. Hayashida et al. (AGASA collaboration), Phys. Rev. Lett. 73, 3491 (1994); S. Yoshida
et al. (AGASA collaboration), Astropart. Phys. 3, 105 (1995); M. Takeda et al. (AGASA
collaboration), Phys. Rev. Lett. 81, 1163 (1998).
[17] R. W. Spinger et al. (HiRes collaboration), Proc. 29th ICRC, Pune (2005), 7, 391.
[18] M. A. Lawrence, R. J. O. Reid, and A. A. Watson, J. Phys. G: Nucl. Part. Phys., 17, 733
(1991).
[19] A. V. Glushkov et al. Proc. 28th ICRC, Tsukuba, Japan (2003), 1, 389.
[20] M. Ave et al. Proc. 27th ICRC, Hamberg, Germany (2001), 1, 381.
[21] M. Nagano and A. A. Watson, Rev. Mod. Phys. 72, 863 (2000); M. Takeda et al. (AGASA
collaboration), Astropart. Phys. 19, 135 (2003); R. U. Abbasi et al. (HiRes collaboration),
Phys. Rev. Lett. 92, 151101 (2004); Astropat. Phys. 22, 139 (2004).
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Mantsch, (Pierre Auger collaboration), Proc. 29th ICRC, Pune (2005), 10, 115.
Chapter 6
SUMMARY AND FUTURE OUTLOOK
Cosmic Rays (CR) consisting mainly of charged particles coming from outer space are continuously striking the earth’s atmosphere from all directions with energies ranging from ∼ 106 eV to
≥ 1020 eV. They provide important astrophysical and astronomical windows to the scientific community of the world, as their study constitutes some important fields of research in both astronomy
and particle physics. Particularly, the study of Ultra High Energy Cosmic Rays (UHECR) (≥ 1017
eV) is getting a special focus from physicists because of various aspects of their interactions, nature and origin.
The UHECR interactions in the atmosphere unfold a large number of different possible phenomena. These phenomena are associated with the possible production of different particles which
obviously serve as signatures of the same. Different theoretical and experimental methodologies
are to be followed or to be developed to study the signatures of these phenomenon. One of the
most significant theoretical methodology is the Monte Carlo (MC) simulation based on the phenomenological models of the HE hadronic interactions. These models are to be founded on the
latest theoretical knowledge of HE physics and the data of accelerator experiments, which are to
be extrapolated to required HE on the basis of the theory. Also these models must include that
phenomenon (phenomena) of UHECR interactions on which particular interest is (are) focused.
On the other hand, the testing of these models predictions with the available experimental data of
the concerned phenomenon is very much urgent to rely on the models for further study.
6.1 Higgs boson Production
One of the most important possible phenomenon that we have studied in this work is the
production of standard model (SM) Higgs boson in UHECR interactions in the earth’s atmosphere.
The SM Higgs boson is the most important ingredient of this model which is not discovered so far,
however, expected to be discovered at the CERN Large Hadron Collider (LHC) or at the Fermilab
Trevarton. It’s discovery is crucial as the test of the SM mass problem is based on this illusive
114
Summary and Future Outlook
115
particle’s existence. According to SM, the Higgs boson is responsible for the spontaneous breaking
of symmetry of electroeweak interaction by providing masses to the intermediate vector bosons
(IVB) of weak interactions. This phenomenon is known as Higgs mechanism. Here we conjecture
on the basis of the thermofield theory that in the UHECR interactions with air nuclei a part of
the interaction energy is transferred to the neighbouring microscopic volume of vacuum, which
excites it locally and forming a bubble of vacuum with thermal energy. This bubble formation
is equivalent to the Salam-Weinberg phase transitions of gauge field and contains Higgs bosons.
These Higgs bosons decay primarily to charged hadrons very fast as the bubble cools, which in turn
get converted to high energy (prompt) heavy fermion pairs. Thus the signature of the Higgs boson
production in UHECR interactions is the very rapid increase of the multiplicity of heavy prompt
fermion pairs with energy. We consider in this work, the excessive prompt muons multiplicity
as the signature of the Higgs boson production in the said mechanism. It may be noted that,
the possibility of Higgs boson production through vacuum excitation depends on the fraction of
total energy of collision that goes to local vacuum, which is an unknown factor. A model is
being developed based on the above idea of the Higgs boson production which is incorporated
in the hadronic interaction model that is compiled from the algorithm of the GENCL code of
the UA5 experiment of CERN to generate a necessary event for Higgs production. We used this
interaction model because of its simplicity to incorporate Higgs production factor in it and for the
fact that the conventional hadronic interaction models (such as QGSJET, SIBYLL etc.) does not
incorporate the Higgs production phenomenon. 5000 events are generated with the primary energy
of proton from 1015 eV to 1020 eV using the simulation program based on these model. The Higgs
production model indicates that the central temperatures of the bubble or excited region of vacuum
follow power law with the centre of mass energy or the bubble energy (the part of energy that
goes to vacuum for bubble formation). Similarly the number of Higgs bosons produced in the
process also follow the same law with the bubble energy. To see the signature of Higgs boson
production in UHECR interactions, we consider the prompt muon multiplicities produced in the
first hadronic interaction (because at this interaction energy only the production Higgs is possible)
with the energy thresholds of 0.01 TeV, 0.1 TeV and 1 TeV as probes and found that the average
prompt muon number increases considerably for all threshold energies and the difference in the
numbers of prompt muons with different threshold energies decreases with the increasing primary
or bubble energies. Thus with increasing primary energies all muons observed from the UHECR
interactions belong to very high energy group, which is the indication of the of possible production
of Higgs boson as far as the model is concerned. The average number of prompt muons follow
a power law with primary energy of CR particles. The study of this mechanism reveals that,
the Higgs bosons production through vacuum excitation is only significant from E0 ∼ 1018 eV
with fractions of energy transfer to the vacuum fe ≥ 0.1. It needs to be mentioned that, already
established most prominent sources of prompt leptons are the charmed hadrons. So to differentiate
Summary and Future Outlook
116
prompt muons arising from Higgs bosons of our mechanism and from charmed hadrons, the study
of their transverse momentum distributions would be helpful. Obviously muons from Higgs bosons
would have high transverse momentum because of the nature of the mechanism we report here.
So to observed Higgs production effect we have to detect young showers at high altitude taking
prompt muons as probes, which will follow high transverse momentum distributions. Similarly
it is possible to observe the Higgs signature by the detection of underground muons at very high
depth under the earth surface. The observation of high transverse momentum in Chiron events and
excessive multiplicity in Halo events as well as hadronic signal associated with the direction of
Cygnus X-3 and Hercules X-1 may be associated with this mechanism.
6.2 Charmed Hadron production
Second important phenomenon in CR interactions which we have studied here is the production of charmed hadrons in hadronic collisions. Presently a considerable attention is focused on
the charmed particle production in hadronic collisions in view of prompt leptons fluxes associated
with the CR. The study of prompt lepton sources is very important to understand the physics of CR
as a whole. It is believed that, the most notable sources of prompt leptons are the charmed hadrons
that are produced in the atmosphere in CR interactions. However to draw a definite conclusion in
this connection from the available data obtained with surface and underground detectors, we need
a sound physical models that is based on the extrapolation of charmed production data of accelerator experiments to the CR energies. QGSJET is a most successful HE hadronic interaction model
whose predictions are quite satisfactory for experimental observations of CR. This model incorporates the charmed hadron production part in it. But so far, this important part is not being included
in the standard MC code for CR EAS such as in CORSIKA, considering it as a very rare process.
Although charmed hadrons production in CR interactions is a rare process, whenever it occurs, it
may change the traditional shower behaviour drastically. So it needs to be included for the study
of charmed interactions part in the standard MC code for CR EAS with proper charmed hadrons
production model. With this view we have studied the charmed hadroproduction characteristics of
QGSJET model in pp interactions. In this study we have considered production characteristics of
−
charmed hadrons D+ , D− , D0 , D̄0 , Λ+
c , and Λc with the centre of mass energy from 10 GeV to
105 GeV for 106 number of events. The important findings of our investigation are :
(1) The production cross sections or the average multiplicities of charmed hadrons in pp collisions are relatively very small. Comparison of production cross sections with the available experimental data at low energies for D-mesons shows a good agreement. Moreover the average
multiplicities of different charmed hadrons are not much different and approaches approximately
√
the same value as the s goes on increasing beyond 1000 GeV. The average multiplicities of
√
charmed hadrons follows an inverse power law with s.
Summary and Future Outlook
117
(2) The study of production cross sections as a function of Feynman variable xF shows that
the maximum charmed hadrons production takes place with low values of xF (≤ 0.3) for all values
√
s under consideration. The production cross section for D-mesons is slightly higher than that for
√
Λc -baryons. It is comparatively more distinct for low values of s. Available experimental data
√
at low values of s are compared with the model predictions. The agreement is not good. These
distributions also follows the inverse power law.
(3) The pattern of charmed hadron production as a function of p2⊥ is almost same as that with
xF . Here the comparison with the available experimental data at low energies shows a good agreement in the lower range of p2⊥ . The production cross sections as function of p2⊥ obeys the second
order exponential distribution.
(4) The study of charmed hadron production cross sections as a function of p⊥ shows that
√
the range of p⊥ for the production of charmed hadrons with very high and low values of s is
relatively small. This is an unexpected implication of the model. The production cross sections as
a function of p⊥ is found to follow the Lorentzian distribution.
(5) The study of the asymmetry in charmed hadrons production characteristics is an important
aspect for charge excess of atmospheric prompt leptons. This study reveals that for all values of
√
√
xF , asymmetry is most prominent in the low values of s and as values of s increases beyond
200 GeV this tendency of production of charmed hadrons decreases and vanishes at sufficiently
√
high values of s (≥ 100000 GeV). Moreover, there is strong preference for producing Λ+
c rather
than Λ−
c , while that for producing D̄ mesons rather than D mesons. This tendency is much strong
in case of Λc -baryons than D-mesons.
√
√
(6) The asymmetry of charmed hadrons production as a function of xF indicates that for low
s (≤ 100 GeV) asymmetry increases from zero to one at xF ≤ 0.3, whereas for higher value of
s, it increases to the same value at xF ≥ 0.3. The pattern of production of charmed hadrons in
√
this case is similar that with s. The agreement with the experimental data is quite satisfactory.
(7) The asymmetric production of charmed hadrons as a function of p⊥ does not follow any
pattern as a whole. Thus the QGSJET model is not well versed to explain the behaviour of charmed
hadroproduction as a function p⊥ and needs modification in this connection.
6.3 GU Miniarray Data Analysis Using CORSIKA
The detection of UHECR was a challenge because of its extremely low flux. This challenge
could be overcome by using ground based detector array covering a very large area to detect giant EAS produced by UHECR primaries in the earth atmosphere. These detector arrays involve
a considerable man power and cost. Hopefully, the detection of UHECR can be done by a very
low cost (with minimum man power) detector setup based on the Linsely effect. GU Miniarray is
such a detector with only 2 m2 carpet area consisting of eight plastic scintillators, each viewed by
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fast PMTs. It was fully operating in Gauhati University Physics Department from 1996 to 1998.
Previously the data of the miniarray was analysed using the best fitted relation in agreement with
QGS model and Yakutsk data. As it is an unconventional detector for detecting UHECR, so it is
most necessary to check its reliability in this respect by reanalysing its data using a standard MC
code. Furthermore, in the case of indirect method of detection of CR, the MC simulation of EAS is
very important to see the detail course of development of air shower for a particular energy, arrival
direction and mass number of the primary particle. Here we use CORSIKA with the QGSJET
model to reanalyse GU Miniarray data taken during the said period under this motivation and provides that,
(1) New analysis gives the estimation of primary energy significantly higher than the previous
analysis. Energy spectrum after reanalysis is found to span from 1017 eV to 1019 eV for proton
primary and from 1017 eV to 1019.4 eV for iron primary and further confirms the irregular behavior
of energy spectrum at UHE region with a prominent dip.
(2) The differential energy spectrum shows structure similar to that observed by other world
groups. Spectral breaks are found to occur at higher energies compared with old method of analysis. The spectrum becomes steeper around 1017.5 eV and 1017.7 eV and flattens around 1018.7 eV
and 1019.1 eV for proton and iron primaries respectively forming a dip. Earlier analysis showed a
dip around 1018.2 eV.
(3) There is a significant difference between spectra predicted by pure proton and pure iron assumptions. However, proton assumption results agree more with Nagano and Watson compilation
within spectral range from 1017.5 eV to 1018.5 eV. Beyond 1019 eV results of different giant arrays
are contradictory. A new parametrized equation for the differential energy spectrum corresponding
to best fit in the energy region 1017.0 eV − 1019.0 eV is derived from this reanalysis.
The future prospect of the work
(1) The present work of Higgs boson production through vacuum excitation in UHECR
hadronic interactions can be extended by incorporating this effect into standard QCD based HE
hadronic interaction models such as QGSJET, SIBYLL, DPMJET etc. for better acceptability of
this mechanism. Higgs boson production model can be improved by considering the findings of
LHC of CERN on Higgs boson mass after it is fully operational in 2007. The study of the Higgs
boson’s decay to heavy flavours such as charmed and bottom quarks and the transverse momentum
distribution of the final decay products of Higgs are other aspects of this mechanism.
(2) The charmed hadroproduction can be studied more broadly by using different standard HE
hadronic interactions models (incorporating charmed production factor) and for different colliding
particles. This would be helpful for finding the best model for the charmed part by comparing with
the available experimental data over a wide range (e.g. π − p data from Fermilab and the CERN
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SPS).
(3) More improve analysis for GU Miniarray may be made by studying the simulated arrival
time structure of the air shower over the miniarray detector using the CORSIKA. With improve
triggering criterion and increased size, the giant EAS detection efficiency of miniarray may be
adjusted with the detector arrays of other well known world group.
LIST OF PUBLICATIONS
(1) Simulation for signature of Higgs boson in UHE Cosmic ray interactions through vacuum excitation, published in Indian Journal of Physics, Vol. 78(11), pp 1253-1259, November 2004.
Authors : U. D. Goswami and K. Boruah.
(2) Reanalysis of GU Miniarray data using CORSIKA, published in Astroparticle Physics, Vol.
22/5-6, pp 421-429, January 2005.
Authors : U. D. Goswami, K. Boruah, P. K. Boruah and T. Bezboruah.
(3) Search for Higgs boson in UHE cosmic rays, published in Czechoslovak Journal of Physics,
Vol. 55, No. 6, pp 657 - 672, June 2005.
Authors : U. D. Goswami and K. Boruah.
(4) Asymmetry in charmed hadron production in pp collisions, published in Proceedings of 29th
International Cosmic Ray Conference, Pune, India (2005), Vol. 9, pp 21 - 23.
Authors : U. D. Goswami and K. Boruah.
(5) New results from Gauhati University miniarray detector, published in Proceeding of 29th
International Cosmic Ray Conference, Pune, India (2005), Vol. 7, pp 159 - 162.
Authors : U. D. Goswami, K. Boruah and P. K. Boruah.
(6) Higgs boson production in UHECR interactions with air nuclei, published in Proceeding of
29th International Cosmic Ray Conference, Pune, India (2005), Vol. 9, pp 25 - 28.
Authors : U. D. Goswami and K. Boruah.
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