PHYS 5326 – Lecture #15, 16, 17.5
Monday, Apr. 16, 2007
Dr. Jae Yu
1.
2.
3.
4.
5.
6.
7.
8.
Spontaneous Symmetry Breaking
Higgs Mechanism
Introducing SUSY
Super Symmetry Breaking
MSSM Higgs and Their Masses
Higgs Mass Theoretical Upper Bounds
SM Higgs Production Processes in Hadron Colliders
Winter 07 Experimental Results
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
Jae Yu
1
Announcements
• This is the last make up class
• Venkat’s thesis analysis lecture on Wednesday
– Time: 1 – 2:30pm
– Location: CPB126
• Given the shortage of time, we will not have 2nd term
exam
• Evaluation
–
–
–
–
–
Term exam: 20%
Homework: 20%
Project report: 25%
Presentation: 15%
Total 80%
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Spontaneous Symmetry Breaking
While the collection of ground states does preserve the
symmetry in L, the Feynman formalism allows to work
with only one of the ground states.  Causes the
symmetry to break.
This is called “spontaneous” symmetry breaking,
because symmetry breaking is not externally caused.
The true symmetry of the system is hidden by an
arbitrary choice of a particular ground state. This is
the case of discrete symmetry w/ 2 ground states.
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Spontaneous Breaking of a Continuous Symmetry
A lagrangian, L, for two fields, f1 and f2 can be written
1
1


L    f1    f1     f2    f2 
2
2
1 2 2
1 2 4
2
4
  f1  f2  -  f1  f2 
2
4
is even and is invariant under f1, f2  -f1, -f2 .
The potential energy term becomes
1 2 2
1 2 4
2
U  -  f1  f2    f1  f24 
2
4
w/ the minima on the circle:
Monday, Apr. 16, 2007
f
2
PHYS 5326, Spring1,min
2007
Jae Yu
f
2
2,min
  /
2
2
4
Spontaneous Breaking of a Continuous Symmetry
To apply Feynman calculus,
we need to expand about a
particular ground state (the
“vacuum”). Picking
f1,min   /  and f2,min  0
And introduce two new fields, h and x, which are the
fluctuations about the vacuum:
h  f1 -  / 
Monday, Apr. 16, 2007
and
x  f2
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Spontaneous Breaking of a Continuous Symmetry
The new L becomes
1
1


2 2

L     h    h  -  h      x    x  
2
 2

2
2


1 2
3
2
4
4
2 2 
   h  hx  - h  x  2h x     /  
4

 4
3point
vtx
LKG of the field h w/
the mass mh  2 / c
Lfree of the
massless field x
4point
vtx
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
Jae Yu
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Spontaneous Breaking of Continuous
Global Symmetry
One of the fields is automatically massless.
Goldstone’s theorem says that breaking of continuous
global symmetry is always accompanied by one or
more massless scalar (spin=0) bosons, called the
Goldstone Bosons.
This again poses a problem because the effort to
introduce mass to weak gauge fields introduces a
massless scalar boson which has not been observed.
This problem can be addressed if spontaneous SB is
applied to the case of local gauge invariance.
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Higgs Mechanism
L in page 3 can be simplified by combining two real
fields into one complex field f  f1  if2 ; f *f  f12  f22
Using this new form of the field, the L looks exactly
like that of a single scalar field
*
1
1 2 *
1 2 * 2

L     f    f    f f -  f f 
2
2
4
Now the rotational symmetry becomes invariance
i
f

e
f.
under U(1) gauge transformation,
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Higgs Mechanism
L can be made invariant under local gauge
transformation by introducing a vector field, A, and
replacing the partial derivatives with covariant ones.
The new L then becomes
1 
iq  *    iq  
L      f    
f 
2 
c   
c 
1 2 *
1 2 * 2
1

  f f -  f f  F F
2
4
16
We then can break the symmetry as we did before by
introducing new fields h  f1 -  /  and x  f2 .
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Higgs Mechanism
The new L is written in terms of h and x as
1
1


2 2

L     h    h  -  h      x    x  
2
 2

Lfree of the massless
Goldstone boson x
Bad term!!
2
 1
1
iq


  
 q 


 F F  
A
A
2
i

x
A



  
2 c  
 c
 16

2

1 2
q 
 q  



  h   x  - x   h   A     h  x  h 2    A A 
2
 c  


 c
2
2

1 2
3
2
4
4
2 2 
- h  hx h  x  2h x    / 
4
 4

LKG of the scalar field h
w/ the mass mh  2 / c
Monday, Apr. 16, 2007





Lproca of the free gauge field A
w/ the mass mA  2  q / c
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Higgs Mechanism
Issues with the new L are the unwanted Goldstone
boson x and the term
x
 q 

-2i 
   x  A
 c
A
which can be interpreted as one point vertex interaction
between scalar field x and vector field A.
This kind of terms indicate that the fundamental particles
in the theory are identified incorrectly. Both problems
can be resolved exploiting gauge invariance of L.
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Higgs Mechanism
Since the complex field in L is f  f1  if2 ,
The U(1) gauge transformed field is written
f  f '  eif   cos  i sin  f1  if2 
 f1 cos - f2 sin    i f2 cos  f1 sin 
The transformed field, f’, becomes real by picking 
that makes the complex term 0.
f2 cos  f1 sin   0    - tan f2 / f1 
-1
'
'
'
f

f

i
f
Under this condition, since
1
2 , f’2 is 0, making
the transformed x become 0, eliminating it from L.
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Higgs Mechanism
In this gauge, the new L becomes
2

1 
1  iq    
1

2 2
L     h    h  -  h    F F  
 A A 
2 c  
2
  16

2
 q 2  

1 2 

3
4
    h  h   A A  - h - h 
2 
4 
 c   
Lproca of the free gauge field A
1 2 2
  / 
w/ the mass mA  2  q / c
4
LKG of the scalar field h
w/ the mass mh  2 / c
Monday, Apr. 16, 2007
This L has all the necessary properties
1.
2.
3.
No massless Goldstone boson (x) & no bad single point
interaction term
Massive gauge field, A
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A single massive
scalar
boson,
h,
(the
Higgs
boson)
Jae Yu
Higgs Mechanism
• The new L describes the same physical system
– A particular gauge (  - tan f2 / f1  )has been chosen
– Keep the Feynman calculus still valid
-1
• Symmetry has been spontaneously broken, giving masses to
gauge field and producing a massive scalar boson
– The massive gauge vector boson picks up another degree of freedom,
the longitudinal polarization, compared to two transverse ones
originally.
– The longitudinal polarization came from Goldstone boson
– The original ghost Goldstone boson has been eaten by the gauge
boson, giving mass to the gauge vector boson and the longitudinal
polarization.
• In SM, the Higgs boson is responsible for mass of weak
vector bosons, leptons and quarks
Monday, Apr. 16, 2007
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The Higgs Particle
To understand the Higgs mechanism,
imagine that a room full of physicists
chattering quietly is like space filled
with the Higgs field.
... a well-known scientist walks in,
creating a disturbance as he moves
across the room and attracting a
cluster of admirers with each step ...
... if a rumor crosses the room, ...
... this increases his resistance
to movement, in other words,
he acquires mass, just like a particle
moving through the Higgs field...
... it creates the same kind of clustering, but this
time among the scientists themselves. In this
analogy, these clusters are the Higgs particles.
Courtesy of CERN. The concept was inspired by Prof. David J. Miller of University College London.
Monday, Apr. 16, 2007
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Issues with Higgs
• Haven’t been observed at any laboratories yet.
• The Higgs potential is completely unknown.
• Does not explain why fermion masses are what
they are.
• Minimal SM loop corrections give quadratic
divergences to the mass of Higgs.
• Other Symmetry breaking models…
• How many Higgs?
• What are the couplings (Yukawa?)? How strong
are they?
Monday, Apr. 16, 2007
PHYS 5326, Spring 2007
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Introduction to Super Symmetry
• An alternate solution to resolve mass hierarchy issue caused by the
quadratic divergences.
• A symmetry that relates particles of differing spins
• Particles are combined in superfields which contain the fields differing
by spin ½.
• Scalars and fermions in superfields have the same coupling to gauge
bosons and cause the quadratic divergence to cancel
• The LSUSY contains scalar and fermion pairs of equal mass
– SUSY connects particles of different spins but all other
characteristics are the same
• SUSY is a broken symmetry because there is no partner of particles
with the same mass but different spin non-zero mass splitting
between partners is an indication of broken symmetry
Monday, Apr. 16, 2007
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Introduction to Supersymmetry
•
•
•
•
•
Motivation for Supersymmetry (SUSY):
– Hierarchy Problem: SUSY stabilizes Higgs mass against loop correction.
– Grand Unification: SUSY modifies running of SM gauge couplings “just enough” to
give
Grand Unification at single scale. (SUSY is needed in String theories).
– Dark Matter: R-parity conservation causes the Lightest SUSY Particle (LSP) to be
stable
SUSY particles (gluinos and squarks) are expected to be produced in the proton-proton
collision.
Long decay chains and large mass differences between SUSY states; many high PT
objects are observed (lepton, jets, b-jets).
If R-Parity is conserved, cascade decays to stable undetected LSP; large ETmiss
signatures.
A typical decay chain of SUSY particles:
p
p
0
~
c
0
~
q
~
~
1
~
c
g
l
q
q 2
l
l
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SUSY Particle Spectrum
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•
•
•
•
Some SUSY Models
MSSM: Minimal SUSY extension of the SM (MSSM) brings 105 additional free parameters
into the theory, thus making a systematic study of the full parameter space difficult. Since
SUSY is a broken symmetry one can reduce the number of parameters by constraining the
SUSY breaking. The origin of the SUSY breaking and its mediation to the MSSM sector are
key features of SUSY models.
mSUGRA: In the minimal Supergravity (mSUGRA) model which is used as a standard
benchmark model, the SUSY breaking is transmitted from the hidden sector to the observable
sector by gravity. The gauginos, scalars masses and the trilinear couplings are assumed to
be unified at the Grand Unification (GUT) scale leading to only five free parameters: m0, m1/2,
A0, tan(β), sgn(µ).
GMSB: In the Gauge Mediated SUSY breaking models the breaking is mediated by gauge
interactions. LSP are taken to be nearly massless gravitinos. The NLSP can be either neutral
or charged. In the case it is neutral, it is the lightest combination of Higgsinos and gauginos,
and behaves in the same way as the neutralino in the mSUGRA model, except for its decay.
If the lifetime is short the events will contain two hard photons and missing energy, which
provide a distinguished signature against any other SUSY model.
AMSB: Another possibility is that the hidden sector of the theory does not have the right
structure to provide masses through either the mSUGRA or GMSB mechanisms. Instead, the
leading contributions come from a combination of gravity and anomalies. This model is known
as Anomaly Mediated SUSY Breaking (AMSB) and predicts a different pattern of
signatures and masses.
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Minimal Supersymmetric Model (MSSM)
Uses the same SU(3)xSUL(2)xUY(1) gauge symmetry
as the Standard Model and yields the following list of
particles
Chiral Superfield
Vector Superfield
Monday, Apr. 16, 2007
Majorana fermion partners
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Higgs Sector in MSSM
In SM L for EW interactions, fermion masses are
generated by the Yukawa terms in L
L  -d QLFd R - u QLF uR  h.c.
c
Higgs coupling to d quark
Higgs coupling to u quark
In MSSM, the term proportional to Fc =-it2F* is not
allowed, requiring an introduction of another scalar
doublet to give t3=1 for SU(2)L fermion doublet mass.
Thus, MSSM has two higgs doublets, F1 and F2.
Monday, Apr. 16, 2007
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Supersymmetric Scalar Potential
Through the requirement of supersymmetric gauge
invariance and demand for perturbative algebra to
be valid, the scalar potential in MSSM is
V
2
F
1
2
 F2
2



2
g 2  g '2
g2 *
2
2 2

F1 - F 2

F1 F 2
8
2
This potential has its minimum at <F10>=< F20 >=0,
giving <V>=0, resulting in no EW symmetry breaking.
It is difficult to break supersymmetry but we do know
it must be broken.
Monday, Apr. 16, 2007
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Soft Supersymmetry Breaking
The simplest way to break SUSY is to add all possible
soft (scale ~ MW) supersymmetry breaking masses for
each doublet, along with arbitrary mixing terms, keeping
quadratic divergences under control.
The scalar potential involving Higgs becomes

VH    m
2
2
1
g 2  g '2

8



g
F - F   2
F1    m
1
2
2
2
2 2
2
2
2
F2 -  B ij  F1i  F2j  h.c 
2
2
F  F2
*
1
2
The quartic terms are fixed in terms of gauge couplings
therefore are not free parameters.
Monday, Apr. 16, 2007
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Higgs Potential of the SUSY
The Higgs potential in SUSY can be interpreted as to
be dependent on three independent combinations of
parameters
 m ;
2
2
1
 m ;
2
2
2
B
Where B is a new mass parameter.
If B is 0, all terms in the potential are positive, making
the minimum, <V>=0, back to <F10>=< F20 >=0.
Thus, all three parameters above should not be zero
to break EW symmetry.
Monday, Apr. 16, 2007
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SUSY Breaking
Symmetry is broken when the neutral components of
the Higgs doublets get vacuum expectation values:
F1  v1;
F2  v2
The values of v1 and v2 can be made positive, by
redefining Higgs fields.
When the EW symmetry is broken, the W gauge boson
gets a mass which is fixed by v1 and v2.
M
Monday, Apr. 16, 2007
2
W
g 2
2
  v1  v2 
2
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SUSY Higgs Mechanism
Before the EW symmetry was broken, the two complex
SU(2)L Higgs doublets had 8 DoF of which three have
been observed to give masses to W and Z gauge
bosons, leaving five physical DoF.
These remaining DoF are two charged Higgs bosons
(H+/-), a CP-odd neutral Higgs boson, A0, and 2 CPeven neutral higgs bosons, h0 and H0.
It is a general prediction of supersymmetric models to
expand physical Higgs sectors.
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SUSY Higgs Mechanism
After fixing v12+ v22 such that W boson gets its correct
mass, the Higgs sector is then described by two
additional parameters. The usual choice is
v2
tan  
v1
And MA, the mass of the pseudoscalar Higgs boson.
Once these two parameters are given, the masses of
remaining Higgs bosons can be calculated in terms of
MA and tan.
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The  Parameter
The  parameters in MSSM is a concern, because this
cannot be set 0 since there won’t be EWSB. The mass
of Z boson can be written in terms of the radiatively
corrected neutral Higgs boson masses and ;
 M - M tan  
2
M  2
-2 

2
tan  - 1


2
Z
2
h
2
H
2
This requires a sophisticated cancellation between Higgs
masses and . This cancellation is unattractive for SUSY
because this kind of cancellation is exactly what SUSY
theories want to avoid.
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The Higgs Masses
The neutral Higgs masses are found by diagonalizing
the 2x2 Higgs mass matrix. By convention, h is taken
to be the lighter of the neutral Higgs.
At the tree level the neutral Higgs particle masses are:
M
2
h,H
1 2
  M A  M Z2
2
M  M
2
A

2 2
Z

- 4 M Z2 M A2 cos 2 2  

The pseudoscalar Higgs
particle mass is:
2 B
M 
sin 2
Charged scalar Higgs
particle masses are:
M H2   MW2  M A2
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2
A
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Relative Size of SUSY Higgs Masses
The most important predictions from the masses given
in the previous page is the relative magnitude of Higgs
masses
M H   MW
M H0  M Z
M h0  M A
M h0  M Z cos 2
However, the loop corrections to these relationship are large.
For instance, Mh receives corrections from t-quark and tsquarks, getting the correction of size ~ GFMt4
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Loop Corrections to Higgs Masses
The neutral Higgs boson masses become
M
2
h ,H
1
 M A2  M Z2
2

2




2
2
2
2
2
h
   M A  M Z  cos 2 
   M A  M Z  sin 2 
2
sin  



2
~ 2 

3GF
m 
Where h is the one4

h 
M t log 1  4
2


loop correction
M
2
t 



Mh has upper
limit for tan>1.
Monday, Apr. 16, 2007
M h2  M Z2 cos 2 2    h
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Mass of CP-even
0
h
vs MA and tan
Mh plateaus with
MA>300GeV/c2
For given values of
tan and the squark
masses, there is an
upper bound on the
lightest higgs mass
at around 110GeV for
a small mixing and
130 GeV for large
mixing.
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Suggested Reading
•S. Dawson, “Introduction to Electroweak Symmetry Breaking,”
hep-ph/9901280: sections 10 & 11.
•Higgs search at TeVatron:
http://fnth37.fnal.gov/higgs/higgs.html
•J. Gunion, H. Haber, G. Kane, S. Dawson, “The Higgs
Hunter’s Guide,” Perseus Publishing: Ch. 4.
•G. Kane “The Supersymmetry Soft-Breaking Lagrangian –
Where Experiment and String Theory Meet”
•M. Spira and P. Zerwas, “Electroweak Symmetry Breaking and
Higgs Physics,” hep-ph/9803257
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Higgs Particle Searches
• What are the Higgs particles we are looking for?
– Standard Model Higgs: Single neutral scalar
– MSSM Higgs: Five scalar and pseudoscalar particles
• h0, H0, H+/- and A0
– Higgs in Other Models
• What are the most distinct characteristics of Higgs
particles?
– In both SM and MSSM, the Higgs particles interact
with fermions through Yukawa coupling whose
strength mostly is set by the fermion masses.
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Theoretical MH Upper Bound in SM
From the SU2xU1 L

2
v 
L  Df -  f -  - g d d Lf d R - g u u Lf cu R  h.c.
2
2
2
2
2
Where, the scale v is the EWSB scale, v  1 2G  246GeV,
and the mass of the Higgs particle and fermions are
gfv
2
mf 
M H  v
2
F
Where  is the quartic coupling and gf is the Yukawa coupling
Monday, Apr. 16, 2007
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36
Theoretical MH Upper Bound in SM
While MH cannot be predicted in SM, internal consistency and
extrapolation to high energies can provide upper and lower
bounds.
Based on the general principle of t-E uncertainty, particles
become unphysical if their masses grow indefinitely. Therefore
MH must be bound to preserve the unitarity in the perturbative
regime.
From an asymptotic expansion of a WLWL S-wave scattering, an
upper limit on MH can be obtained:
M  2 2 GF   850GeV 
2
H
Monday, Apr. 16, 2007
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2
37
Theoretical MH Upper Bound in SM
The SM tells that there is no new physics between EWSB
scale (~1TeV) and the GUT scale (1019GeV). This can
provide a restrictive upper limit because the SM can extend
to a scale L before a new type of strong, short range
interaction can occur between fundamental particles.
From the variation of quartic Higgs coupling, , and the topHiggs Yukawa coupling, gt, with energy parameterized by
t=log(2/v2), and requiring L to be finite, one can obtain
the Higgs mass upper bound
M
Monday, Apr. 16, 2007
2
H
 8 v 3log  v / 
2 2
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2
2

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Theoretical MH Upper Bound in SM
LSM
For the central mt=175GeV/c2
Monday, Apr. 16, 2007
MH
1 TeV
55  M H  700 GeV c 2
1019GeV
130  M H  190 GeV c 2
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SM Higgs Properties
• Profiles of Higgs Particles determined by its mass
• The Yukawa coupling of Higgs to fermions set by the fermion
mass, mj, and to the electroweak gauge bosons by their
masses, MV.
1/ 2
g ffH   2GF  m f
1/ 2
2


gVVH  2  2GF  M V
Physical observables, the total decay width, lifetime and branching
ratio to specific final states are determined by these parameters.
Monday, Apr. 16, 2007
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40
Higgs Decay to Fermions
Higgs partial decay
width to fermions are

 H f f

GF
 Nc
g ffH m2f  M H2  M H
4 2
Number of color quantum
numbers, 1 or 3
For MH=100GeV,
mb(MH2)~3GeV,
mc(MH2)~0.6GeV
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Higgs Decay to Gauge Boson Pairs
  H  VV 
 V
GF
M H3 1 - 4 x  12 x 2  V
16 2
Where x  M V2 / M H2 and
 V  2 or 1 for W or Z
  H  VV * 
3GF2 M V4
'

M
R
(
x
)

H
V
16 3
Where W'  1 and
 Z'  7 /12 - 10sin 2 W / 9  40sin 4 W / 27
  H   

3G 
3 4
2
M
N
e
-7
H
c
t
3
3
128 2
2
F
2
2
Valid in the limit M H2  4 M W2 , 4 M t2
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42
Summary of
SM Higgs
Branching
Ratio
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43
Higgs Production Processes at
Hadron Colliders
Gluon fusion:
gg  H

-
WW, ZZ Fusion:
W W , ZZ  H
Higgs-strahlung
off W,Z:
qq  W , Z  W , Z  H
*
*
Higgs Bremsstrahlung qq, gg  tt  H
off top:
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Higgs Production X-Sec at 2TeV and 14TeV
e+ (+)
e- (-)
Experiment
e+
e-
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What do we know as of Winter 07?

M W ~ M t2  ln M H2
2
M H  76-33
24 GeV/c
LEP EWWG: http://www.cern.ch/LEPEWWG
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114.4<MH<182 GeV
46

Homework Reminder
•Derive the new L in page 6 in this lecture.
•Derive the new L for two fields in page 10 in this
lecture
•Show that one of the two scalar fields could be
massless when the choice of minima were made at
f1 

2
; f2  -

2
•Derive the new L in page 13 of this lecture
•Due Wednesday, Apr. 25
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Homework Assignment
• Study the summary SM Higgs branching ratio
plot in slide 43 and plan experimental strategies
to search for Higgs particles in the following two
scenarios
– MH=115GeV
– MH >150GeV
• Due: Monday, Apr. 30
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