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Standard Model III: Higgs and QCD
Rogério Rosenfeld
Instituto de Física Teórica
UNESP
Physics Beyond SM – 06/12/2006
UFRJ
• Standard model (SU(3)cxSU(2)LxU(1)Y ) passed
all experimental tests!
It is based on three main principles:
1. Quantum field theory (renormalizability)
2. Gauge symmetry (fundamental interactions)
3. Spontaneous symmetry breaking (mass generation)
Precision measurements at the 0.1% level allowed to test
the model at the quantum level (radiative corrections).
Top quark mass was predicted before its actual detection!
• However, symmetry breaking mechanism has
not yet been directly tested.
M H  114.4 GeV
LEPII 
direct searches
 
e e  ZH
M H  199 GeV @ 95% CL - indirect searches
[Robert Clare 2003]
[LEPEWWG Summer 2006]
2006-07-24: Summer 2006 =======================
Contact: [email protected] Summer 2006:
Changes in experimental inputs w.r.t. winter 2006
* New Tevatron Mtop
* New LEP-2 MW and GW combination (ADLO final, but combination preliminary)
Hence new world averages for MW and GW
Blue-band studies: ==================
For ZFITTER 6.41 and later (currently 6.42), and flag AMT4=6 fixed,
two flags govern the theory uncertainties in the complete two-loop calculations
of MW (DMWW=+-1: +-4 MeV) and fermionic two-loop calculations of
sin2teff (DSWW=+-1: +-4.9D-5). The theory uncertainty for the Higgs-mass
prediction is dominated by DSWW.
The blue band will be the area enclosed by the two
ZFITTER DSWW=+-1 \Delta\chi^2 curves. The one-sided 95%CL (90% two-sided)
upper limit on MH is given by ZFITTER's DSWW=-1 curve:
MH <= 166 GeV (one-sided 95%CL incl. TU)
(increasing to 199 GeV when including the LEP-2 direct search limit).
Direct searches for the Higgs
SM Higgs branching ratios
Direct searches for the Higgs
LEPII
SM Higgs Tevatron production cross section
M. Spira
hep-ph/9810289
SM Higgs LHC production cross section
M. Spira
hep-ph/9810289
Luminosity required to find the Higgs
Carena & Haber
hep-ph/0208209
Significance of the Higgs signal at LHC
Gianotti & Mangano
hep-ph/0504221
There could be surprises...
• Is this the end of particle theory?
Program: find the Higgs, study its properties
(mass, couplings, widths) and go home??
• NO!
Standard Model is incomplete:
• fermion masses (Yukawas, see-saw)
• fermion mixings (CKM and the like)
• dark matter (new physics)
• dark energy (new physics)
• grand unification (SUSY-GUT?)
• gravity!
• ...
Furthermore, the SM has conceptual
problems related to the scalar sector:
• Triviality
• Stability
• Hierarchy and Naturalness
• Unitarity
Conceptual problems of the SM
I. Triviality
Running of :

yt
yt
yt
yt


d  
3 2
 2
2
d ln 
8
2
for large :
Running of :
   
 v 2 
2
  
3 v
2
2
1
ln  / v
2
8
2

Landau pole: the coupling constant diverges at an
energy scale  where:
  
3 v
2
2
ln  / v  1
2
8
2

The only way to have a theory defined at all energy
scales without divergences is to have zero coupling:
theory is trivial!
Lesson to be learned:
Higgs sector is an effective theory, valid only up to a
certain energy scale .
Given a cut-off scale  there is an upper bound on
the Higgs mass:
2
  
3 v
2
2
ln  / v  1 
2
8
2 2
16 v
2
2
2
M H  2 v v 
2
2
3 ln  / v
 



II. Stability
Higgs boson can’t be too light (small ):
Running of :


yt
yt
yt
yt

[small ]
4
d  2 
3 4
3 yt
2
2
2
2


y











v

ln

/
v
2
2 t
2
d ln 
8
8
Vacuum stability (0)
implies a lower bound:
4
t
2

3y 2
M 
v ln 2 / v 2
4
2
H

Triviality and stability bounds on the Higgs mass
Riesselmann, hep-ph/9711456
LEPII limit
Triviality and stability bounds on the Higgs mass
Kolda&Murayama,
hep-ph/0003170
III. Hierarchy and naturalness
Higgs boson mass (Higgs two-point function) receives
quantum corrections:
yt
yt
g

Quantum corrections to Higgs boson mass depend
quadratically on a cut-off energy scale :
=10 TeV as an example
Fine tuning of the bare Higgs mass is required to keep
the Higgs boson light with respect to :
M. Schmaltz
hep-ph/0210415
=10 TeV as an example
Hierarchy problem: in the SM there is no symmetry
that protects the Higgs boson to pick up mass of the
order of the cut-off!
If we want the SM to be valid up to Planck scale, how
can one generate the hierarchy MH << MPl??
Roughly we have:
2  phys
2
MH
Example:
2  bare 


MH
2  bare 
2

   MH
  1 
2
 

 phys
MH
 100 GeV;   10 GeV 
15
Large amount of
 M H2bare  
 26
1 


10
! fine tuning. It is not
2

 

NATURAL.
IV. Unitarity
The Higgs boson has another important role in the SM:
it makes the scattering of gauge bosons to have a good
high energy behaviour.
Scattering matrix
S in  out

Conservation of probability: S matrix is unitary S S  1
S-matrix can be written in terms of scattering amplitudes

S  1  2     p f   pi  pi  p f 
i
 f

4
4  

The 22 scattering amplitude can be expanded in terms
of Legendre polynomials of the scattering angle.
This is called the partial wave expansion:
2  2  16  2l  1 al s  Pl cos  
Partial waves:
al s 
l
s: center-of-mass energy2
Unitarity of S-matrix implies:
Im al s   al s   al s   e
2
i l  s 
sin  l s 
Hence, there is a unitary limit for the lth partial wave:
al s   1
Re al s   1 / 2
The 22 WW scattering amplitude is given by:
For example, the WWZZ l=0 partial wave is:

s
2
al 0 s  

 O MW / s
2
2
2
32 v 32 v s  M H
s
s  M W2
M 2H

2
32 v
s

Low energy theorem
Unitarity of l=0 partial wave for coupled channel
VVVV scattering implies:
M H  780 GeV
Higgs summary
• Higgs potential is responsible for electroweak
symmetry breaking
• Higgs couplings and vacuum expectation value
generates masses for fermions and EW gauge bosons
• Higgs restores partial wave unitarity in WW scattering
• SM is an effective theory: either the SM Higgs boson
or new physics will be found at the LHC
Quantum Chromo Dynamics
• QCD is an unbroken gauge theory based on
the SU(3)c gauge group.
• Quarks come in 3 colors and transform as the
fundamental representation of SU(3)c .
• There are 8 gluons that transform as the
adjoint representation of SU(3)c .
QCD Lagrangian
are the 8 Gell-Mann matrices
In principle, neglecting light quark masses, QCD
has only one free parameter:
Given as one should be able to compute
everything in QCD (like hadron spectrum and
form factors)!
However, at low energies the coupling is large
and perturbative methods can’t be used...
Lattice QCD
Hadron spectra from lattice QCD
CP-PACS Collaboration
hep-lat/0206009
quenched fermions!
1st evidence for gluons
1979
Gluons have self-interactions!
QCD has the property of asymptotic freedom:
its coupling becomes weak at large energies.
We define an effective energy dependent
coupling constant as (q) :
virtual corrections
The lowest order result for the running of the
QCD effective coupling is:
a s q  
as
 0a s
1
log q / q0 
2
2
 0  11  n f  0
3
asymptotic freedom a s q   0
q  q0
Today the so-called QCD beta function is known up to 4 loops!
Running of the QCD coupling constant
Bethke
hep-ex/0606035
QCD summary
• QCD at high energies can be treated perturbatively;
many calculations (NLO,NNLO,...) have been done.
• Low energy QCD is much harder: recent progress with
dynamical fermions. Hadron spectra (pentaquarks??).
• QCD is essential for the calculation of high energy
cross sections: Particle Distribution Functions (PDF).
• QCD at finite temperature and chemical potential is
being actively studied: new states of matter (QGP,CGC)
Careful with pentaquarks!
THANKS