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Lecture 15 – Next steps
●
The Higgs boson
●
Review of the Standard Model

Problems of the Standard Model

Proposed Solutions
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The Standard Model
Goal: a theory which describes all of the fundamental constituents of nature and their
interactions with the minimum of assumptions and free parameters. Ultimately describe
all interactions over small distance scales and cosmological observations.
The Standard Model is our best attempt at this - assess how successfult in this lecture.
6 quarks, 6 leptons, 3 exchange bosons
+ antiparticles.
Two independent forces (electroweak and QCD).
19 free parameters: particle masses, mixing angles,
CP-violating term, couplings....
Consistent method of introducing interactions via
so-called gauge invariance and Feynam diagram
formalism (next lecture course).
The Standard Model assumes massless neutrinos
but this is easily fixed.
Barring neutrino oscillations, the Standard Model has never failed a single experimental test.
There is still one test left to pass - finding the Higgs boson.
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The Higgs boson
The missing particle in the Standard Model. Explains mass generation of
the fundamental particles.
The Higgs mechanism is a way of explaining why, in an apparently unified
electroweak theory, the W  and Z 0 are heavy and the  is massless.
Some consequences:
A spin-0 massive boson, the Higgs particle H 0 , is required.
A Higgs field pervades space: fermions interacting with the field acquire mass.
A fermion with mass m f can also couple to the Higgs boson with strength g Hff .
 mf 
g Hff  2 gW 
 (15.01)
 mW 
Couplings to other particles, with strength proportional to particle mass.
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How do we look for the Higgs ?
How is it produced and how does it decay ?
At LEP: e   e   H 0  Z 0
208 GeV centre-of-mass energy
Sensitive to Higgs masses up to
120 GeV.
Production mechanism
b
H bb
0
bb
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Have we already found it ?
Lots of excitement around 2000/2001 as LEP reached the
end of its life.
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Observation of a Higgs ?
An excess of events was seen at mass 115 GeV but reanalysis of
data and rigorous statistical calculation of significance means it is
impossible (and stupid) to conclude a Higgs was seen.
Lower mass limit MH > 113.5 GeV (15.02)
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Been here before - top quark nondiscovery…
●
1984 CERN
●
UA1 experiment
●
pp (630 GeV cm energy)
●
Something they would
rather forget
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Race for the Higgs
The Tevatron (pp at 2 TeV centre-of-mass energy) is now hunting the Higgs.
The LHC (pp at 14 TeV centre-of-mass energy) will take up the chase in 2009.
Different production mechanisms compared with LEP and different decays sought.
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Where is the Higgs ?
Excluded by direct search.
Most likely Higgs mass value from fits to
measured electroweak quantities in the
Standard Model.
The Higgs is either just around the corner or
nature is more complicated than we suppose.
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How good is the Standard Model ?
Criteria
Predictivity and testability
U,G or VG
VG – the only ’failure’ is neutrino
masses and we can patch that up by
adding extra parameters.
Higgs yet to be found.
The SM can be killed but is still v.
much alive!
Completeness*
U – no quantum theory of gravity ?
Dark matter ? ….
Compactness
G - Based on 19 free parameters –
not bad for describing EM,weak and
strong forces below  1TeV.
* The focus of the rest of this lecture
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Speculation strategy
We have few answers but that doesn't mean we can't ask sensible questions.
(1) At which energies can we expect that the Standard model will not
describe subatomic particle interactions ?
(2) In which areas is the Standard Model incomplete and which
theories have been proposed address these problems ?
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How well can we localise a particle ?
To what precision can we know the position of
a particle, eg electron ?
In quantum mechanics the position can be known
to infinite accuracy if we accept we have no knowledge
of its momentum.
Eg from basic quantum mechanics: Heisenberg's microscope.
1
Resolution in position x   
(2.36) ;   probing photon wavelength.
p
p =photon momentum
 xpx
px maximum change in momentum in x-direction of particle.
1 (2.12)
Above picture assumes reaction:   e     e 
Quantum field theory changes this picture. If p  2me (me =electron particle)
 kinematically feasible reaction:   e     e   e   e 
Two identical particles in final state. No longer possible to say anything about electron
position for p  2me .
 Fundamental limitation on knowledge of position: x 
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(15.03)
2m
12
Compton Wavelength
More formally (and don't worry about factors of  )
2
Compton wavelength of a particle: c 
(15.04)
m
Introduced in lecture 12 as the distance below which the electromagnetic
coupling constant starts to change i.e. the distance at which
quantum field theory below important in describing particle behaviour.
2
Electron: c 
 2.43 1010 m. (12.03)
me
Different ways to think about this number but the point is that c that a
quantum description of matter says that we can localise a particle of
2
mass m to a region of size: c 
.
m
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Gravity
From general relativity: any object of mass m contained within its Scharzschild radius leads to a
gravitational singularity (black hole): Scharzschild radius : rs  2Gm.
G  Gravitational constant.
Quantum description of nature implies that
a particle position be known to accuracy: C 
2
.
m
However, for C  rc the particle is contained within
such a small size that a gravitational singularity
occurs.
The quantum prediction of a particle localised to
a certain distance must be invalid if that localisation
is taking place inside a black hole :).
 (naively) quantum gravity becomes important at: rc  C  2Gm 
2

m
(15.05)
m
G
1
 1.2  1019 GeV (15.06) (drop the  )
G
The Standard Model must fail for masses and energies > Planck mass and a theory of quantum
gravity is needed.
Formally define the Planck mass 
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Question
Compare the values of the electromagnetic and gravitational attractive forces between
two stationary massive particles with charges  e and - e if the particles have (a) mass=1 GeV and
(b) Planck mass. The particles are separated by a macroscopic distance.
e2
Fem
4 0 r 2
e2
R


m 2G
Fgrav
4m 2 0G
r2
G  6.67 1011 m3kg-1s-2  0  8.85 10 12 Fm1 e  1.602 10 19 C
m  1 GeV  1.5  10 27 kg
1.602 10 
R
4  3.14  1.5  10   8.85 10
19 2
27 2
12
 6.67 10 11
 1036
m  m p  1019 GeV  1.5  10 8 kg
 R  102
 Gravity is extremely weak until we get to the Planck scale.
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Eg strong force becomes weak at short distances (<1fm)
 asymptotic freedom.
measurements
From lecture 12:
The coupling constants vary with momentum transfer
(or distance)
1/coupling
Other possible energy scales
E
Electromagnetic
Weak
GUT scale
Strong
Log(Momentum transfer, Q(GeV) )
  33  2 N f 
 Q
 s  Q    s  M Z  1 
 s  M Z  ln 
6

 MZ


 
1
(12.05)
Couplings appear to unify for Q  1016 GeV.
 Grand unified theories (GUTs) unify em, weak and strong forces
(to come).
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Speculation strategy
We have few answers but that doesn't mean we can't ask sensible questions.
(1) At which energies can we expect that the Standard model will not
describe subatomic particle interactions ?
Quantum gravity effects must play a role for masses and energies at and
above the Planck scale (
1019 GeV). The GUT scale ( 1016 GeV) looks
a promising energy for "new physics" to appear.
(2) In which areas is the Standard Model incomplete and which
theories have been proposed address these problems ?
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Problems of the Standard Model
A subjective selection of three open areas in particle physics about which the Standard Model has nothing to say.
(i) Cosmology: Dark matter.
22% of universe's energy budget in the form of "dark matter".
Current evidence suggests that WIMPs: electrically neutral and weakly interacting
massive particles with masses 1  10 TeV may be responsible ( LHC energies)
(ii) Forces: unification and gravity
Is there hope for a theory which unifies all of the fundamental forces or at least
the strong, em and weak forces ? Why is gravity weak until the Planck mass
(the hierarchy problem) ?
(iii) Properties of particles: electric charge quantisation
Why do we never observe particles with charge, eg, 1.5234e ?
If the ultimate aim is a theory of everything which predicts particles, forces and
cosmological measurements from a single principle/equations then solutions to
one of the above problems should address in some way the other problems.
*There's loads more, eg matter - antimatter asymmetry, the strong CP problem (why is there no observed
CP violation in the strong processes), neutrino masses, dark energy etc. but we'll take (i), (ii) and (iii) as
opportunities to show how a problem is defined and solutions proposed.
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Supersymmetry
Every Standard Model has a supersymmetry partner.
Symmetry between bosons and fermion
Quarks (fermions)  Squarks (bosons) ; W , Z ,  , g (bosons)  W , Z ,  , g (fermions)
Symmetry is broken otherwise SM and SUSY particles (sparticleS) would have the
same mass.
SM and SUSY particles have different R-parity. Conservation of R-parity stops SUSY
sparticles decaying to SM particles.
R=(-1)3 B  L  2 S  1 SM particles
(15.07)
= -1 SUSY partner particles.
B=baryon number, L=lepton number, S =Spin quantum number.
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Why look for SUSY ?
Many reasons for looking for SUSY, amongs them...
(1) It predicts a dark matter candidate: i.e. a WIMP with mass
TeV.
Neutralino:  0 a mixed state of SUSY partners of the Higgs, Z and  .
(2) Unification of the couplings is more exact if SUSY sparticles exist.
Can develop SUSY grand unified theories (GUTs) which unify the electromagnetic,
Standard Model
Electromagnetic
E
1/coupling
1/coupling
weak and strong forces.
Weak
Standard Model+SUSY
Electromagnetic
Weak
Strong
Strong
Log(Momentum transfer, Q(GeV) )
Log(Momentum transfer, Q(GeV) )
(3) Solves the hierarchy problem (beyond this course)
Lecture 17 - explore how to look for SUSY at a LHC experiment.
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Grand Unified Theories
Incorporate strong, electromagnetic and weak forces into a GUT.
Simplest model: SU(5) (Georgi-Glashow).
Introduce new heavy exchange bosons X and Y : mass 1016 GeV.
Prediction of proton decay.
Violation of lepton and baryon number.
Eg p   0  e 
Predictions for lifetime 
1030 years.
Current limits (SuperK- lecture)   1033 years.
Other GUTS predict   1033 years.
GUTs also predict heavy magnetic monopoles m  1016 GeV
and explain charge quantisation.
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Extra spatial dimensions
Original ideas on extra dimensions from T. Kaluza and O. Klein (1921).
Several different models incorporating extra dimensions on the market
today.
Large Extra Dimensions.
Hierarchy problem  gravity is weak since it
propagates in extra dimensions (bulk) and we see
a diluted form of it in our 3+1 dimension world (brane).
1
Gravitational potential V  r   n 1 (15.08) where r  R
r
n  number of extra dimensions.
R  distance scale for interactions at which the effects of
extra dimensions are observed. n  2  R  1 mm (15.09)
In general, many extra dimensions theories often predict "new" heavy
particles with masses
TeV and provide dark matter candidates.
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Micro Black Holes at the LHC
In general, when two particles pass
each other with enough energy, a micro
black hole can be formed.
For three spatial dimensions, gravity is
too weak. With extra dimensions gravity
becomes stronger, micro black holes
can be created.
"Normal" black hole: size km,
mass m sun , temperature
"Micro" blackhole: size
temperature
1016K, 
0.01K, 

10 18m, mass 1 TeV,
1027 s (evaporate through
Hawking radition.)
The world won't end when we turn on the LHC.
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Electric charge quantisation
Maybe its better not to be too ambitious and just focus on one specific problem.
Electric charge quantisation.
Why is electric charge always meaured in integer multiples of the elementary
charge e ?
Why are the electron and proton charges the same (barring a sign) ?
qelectron  q proton
The best limits state:
 1020 (15.10)
qelectron
Is there any way to accommodate electric charge quantisation within
quantum mechanics ?
For clarity - use practical units for following derivation.
Also, we'll derive from start to finish...
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Maxwell’s equations
Electric and magnetic fields from electric charges and currents  qe , e , je  and
magnetic charges and currents  qm ,  m , jm 
e
B
E 
(15.11) ;   B  0  m (15.12) ;   E  
-0 jm (15.13)
0
t
  B   0 0
E
 je (15.14)
t
1


Lorentz force law: F  qe E  v  B  qm  B  2 v  E  (15.15)
c




vB
E
qe
E
v
B
qm
v
No magnetic monopoles have ever been observed  qm   m  0, jm  0
E 
e
B
E
(15.11) ;   B  0 (15.16) ;  E  
(15.17) ;  B  0 0
 je (15.14)
0
t
t
Lorentz force law: F  qE  v  B (15.18)
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Monopoles and charge quantisation
Alternative version of Dirac's argument (1931)
Electric charge qe at origin monopoe of charge qm
z
r
a distance d away on the z - axis.
Electric and magnetic fields from qe and qm ,
qm
respectively, at point P :
qe r
0 qm r 
E
(15.19) ; B 
(15.20)
3
3
4 0 r
4 r 
1
2
r '  r  dzˆ ; r    r  d  2rd cos    B 
2
2
P

d
r
qe
0 qm
4
x
r  dzˆ
r
2
 d 2  2rd cos  
(15.21)
3
2
Momentum density in electromagnetic field :
 d  r  zˆ 
0 qe qm
p  0  E  B 
(15.22)
2
3
 4  r 3  r 2  d 2  2rd cos   2
Angular momentum density
=r  p=

 0 qe qm d
4 

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r   r  zˆ 
r 3  r 2  d 2  2rd cos  
3
2
(15.23)
26
r   r  zˆ   r  r  zˆ   r 2 zˆ  r 2 cos  rˆ  r 2 zˆ (15.24)
The xˆ and yˆ co-ordinates will integrate to zero. Use:  rˆ  z  cos 
 Angular momentum in the field: L 
 0 qe qm d
 4 
2
zˆ 
(15.25)
r 2  cos 2   1 r 2 sin  drd d
r 3  r 2  d 2  2rd cos  
3
2
(15.26)
1

r 1  u  dr
0 qe qm d
Set u  cos   L 
zˆ  2   du 
(15.27)
2
3
 4 
1
0
 r 2  d 2  2rdu  2
2



0
rdr
r
2
 d  2rdu 
2
3
2
 ru  d 

0

d 1  u

 r
2
 d  2rdu 
2
1
2
u
d

d 1  u 2  d 1  u 2  d
u 1
1

(15.28)
2
d 1  u  d 1  u 
1  u  du
qq d 1
 L  0 e m zˆ 
8
d 1 1  u 
1
2

2
1
0 qe qm
0 qe qm
u2

zˆ  1  u  du =
zˆ u 
8
8
2
1
1
1
0 qe qmQ
zˆ (15.29)
4
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Dirac’s quantisation condition
z
Angular momentum in the field: L=

0 qe qm
zˆ (15.29)
4
Obs! Independent of separation d !
Angular momentum is quantised: 
 qe 
n 4
nh

(15.31)
20 qm 0 qm
0 qe qm
 n (15.30)
4
2
r
P
qm

d
r
qe
x
 If there's one type of magnetic charge in the universe, anywhere in the universe, this "explains"
why electric charge is quantised ; its a consequence of angular momentum quantisation.
This is one reason why we look for them. In addition they also turn up just about everywhere else in
physics (except in experiments), eg GUTs (m  1015 GeV), quark confinement models..
Possible monopole charge: qe  elementary charge e ; n  1  qD 
h
= "Dirac monopole" charge. (15.32)
0 e
0 qD2
Coupling constant for Dirac monopoles:  m 
 34 (15.33)
4
 m  1  (1) field theory/Feynman diagram formalism impossible ;
(2) several thousand times greater ionisation energy loss than, eg, proton with same
momentum (lecture 16).
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Speculation strategy
We have few answers but that doesn't mean we can't ask sensible questions.


(1) At which energies can we expect that the Standard model will not
describe subatomic particle interactions ?
Quantum gravity effects must play a role for masses and energies at and
above the Planck scale ( 1019 GeV). The GUT scale ( 1016 GeV) looks
a promising energy for "new physics" to appear.
(2) In which areas is the Standard Model incomplete and which
theories have been proposed address these problems ?
Dark matter, hierarchy problem, force unification, charge quantisation
(to name but four)
SUSY, extra dimensions, magnetic monopoles are just some of the things
we've been speculating..But this is a game - we need data!
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So how close are we to a unified theory of all the forces ?
At present string theory offers the best hope. It is the most promising
candidate theory for quantum gravity.
However, its been the most promising theory for over 20 years now...
Lecture 9 - hadron masses can be calculated using a picture of hadrons
as excitations of string. This formed part of the early ideas which led to
string theory.
Point-like particles are tiny quantised one-dimensional strings.
Extra dimensions and supersymmetry accommodated within string theory.
Extremely challenging to come up with a quantitative prediction from string
theory which can be tested.
Time will tell.
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Summary
●
●
●
Higgs discovery would be confirmation of the
Standard Model
Standard Model is incomplete
A range of proposed solutions exist which
postulate the existence of ”new” particles which
could be ”around the corner” at LHC energies.
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