Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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10.10 The LHC and the Search for the Higgs Boson 309
reason and since the Higgs mass is unknown, different search strategies are needed
to allow an efficient Higgs selection and identification in a broad mass interval. Note
that the funny shape of the BR to Z
0
bosons for 130GeV <M
H
<2M
Z
is due
to a threshold effect where the decay into two on-mass shell W bosons becomes
possible.
As the Higgs couples to all fermions proportionally to their mass, the decay into
top quarks is dominant. If M
H
<2M
top
' 350GeV, the dominant fermionic decay
will be to bottom quarks. The decays into
C
, the heaviest charged leptons, is an
order of magnitude below that of b
b. Above the WW and ZZ kinematic thresholds
(200 GeV), the Higgs boson should mainly decay into a pair of massive gauge
bosons. For large enough M
H
, when the phase space factors can be ignored, the
branching ratio into WW bosons is two times larger than the decay branching ratio
into ZZ bosons.
Since the photon is massless, there is no coupling between the Higgs boson and
the photon. However, the Higgs decay into is possible through a virtual fermion
or boson loop. As can be seen in Fig. 10.28,theH ! branching ratio is at
most 0.3% of the dominant H ! b
b channel. However, this is an important decay
channel studied at the LHC because the two photon final state has a clean signature,
especially in the low Higgs mass region.
Finally, it should be noted that the total width
H
is strongly dependent on M
H
:
it is
H
D 3 10
3
, 0.1 and 200 GeV at M
H
D120, 160 and 400 GeV, respectively.
This fact has important consequences on the search strategies, as discussed in
Sect. 10.10.3.
10.10.3 Search Strategies at LHC
To estimate the number of expected signal Higgs events, the total cross-sections
showninFig.10.27 must be multiplied by the BRs of Fig. 10.28.Todiscover
the Higgs boson at LHC, many different signatures are searched for, but lepton
and photon final states are favored with respect to hadronic final states because
of the large QCD background at hadron colliders. In addition, the lepton and
photon energy is better measured than the energy of hadronic jets, and thus clearer
signatures are expected.
Low Higgs mass (M
H
< 130 GeV/c
2
). Precision electroweak measurements at
LEP and Fermilab seems to favor Higgs masses below 155 GeV/c
2
[10A09]. The
mass range M
H
< 130 GeV/c
2
is particularly interesting, but it is the most difficult
to explore. The dominant H ! b
b channel is overwhelmed by the background due
to the huge QCD double-jet rate (more than six orders of magnitude higher).
The most promising channel is the decay H ! . The signal in this channel
is represented by a resonant peak in the invariant mass distribution. A very
good photon resolution is required to discriminate the signal peak from the large
continuum from direct photon production and fake photons from hadron jets.
310 10 High Energy Interactions and the Dynamic Quark Model
Since the Higgs boson width is small in this mass range (a few MeV for Higgs
masses between 110 and 140 GeV), the measured mass peak is entirely dominated
by the experimental resolution.
Intermediate mass (130 GeV/c
2
< M
H
< 2M
Z
). In this mass region, the branch-
ing ratio for the Higgs decay into a vector boson pair becomes important (see
Fig. 10.28). The fully leptonic decay H ! ZZ
! `
C
`
`
0C
`
0
has the cleanest
experimental signature below M
H
' 150 GeV/c
2
, particularly in the four muon
channel. Only one vector boson is on-mass shell. The signal selection is based on
the reconstruction of the Z
0
mass from one of the lepton pairs. The main irreducible
background arises from the ZZ continuum.
For 150 < M
H
< 180 GeV/c
2
, the most promising detection mode is represented
by the leptonic decays of both W bosons: H ! WW
! `
C
`
`
0
`
0
.The
BR(W ! `) is about 32%, where ` D e; ; (68% into hadrons). The simulta-
neous leptonic decay of both W occurs in about 10% of the cases. Experimentally,
a good detection of isolated high transverse momentum muons and electrons and
an accurate calorimeter with hermetic coverage to measure the transverse energy of
the missing neutrinos is needed. Since the Higgs mass cannot be reconstructed as a
narrow mass peak (the energy resolution is poor in this channel and
H
O(1 GeV)
in this M
H
range), the signal should be observed as an excess of events above
backgrounds which therefore must be known as accurately as possible.
High Higgs mass (M
H
> 2M
Z
). If M
H
>2M
Z
,theH ! ZZ
! 4` provides a
very clear signature, as a good mass peak can be reconstructed with the four leptons.
The decay into electrons and muons is considered to be a “gold-plated” channel.
Both lepton pairs will have a on-mass shell Z which makes it possible to reduce
many types of backgrounds. The limitations for possibly detecting the Higgs boson
in this decay channel are due to the reduced production rate (the BR(H ! ZZ !
4`/ 0:1%for` D e; ) and to the large width of the Higgs boson in this mass
range. For example, in 1 year at high luminosity (10
34
cm
2
s
1
), the production
rate for a Higgs boson with M
H
D 700 GeV/c
2
and decaying into 4` is of the order
of 100 events. The large width
H
of heavy Higgs (
H
D 200 GeV for M
H
D
400 GeV) makes the observation of a mass peak very difficult.
For large M
H
, the decays to vector bosons is dominant and the main detection
channel is the H ! WW ! `jj where j denotes a jet originated from a quark.
This WW decay has a 30% branching ratio, yielding a rate 50 times higher than
the four lepton channel from H ! ZZ decays.
Discovery expectations at the LHC. From March of 2010 to March 2011, the
LHC delivered an integrated luminosity of 50 pb
1
at
p
s D 7 TeV. This
luminosity is smaller by a factor of 20 with respect to that planned in the first
year of data taking. The detection mostly relies on the gg ! H production
mechanism with the subsequent decays H ! ;WW
and ZZ
.Thevalueof
the cross-section (Fig. 10.27) times the BR (Fig. 10.28) ranges from a few fb (for
H ! ZZ ! 4`)to1 pb (for H ! WW ! `
C
`
for a high Higgs mass).
Including the detection efficiencies, and with a luminosity of a few fb
1
, only the
10.10 The LHC and the Search for the Higgs Boson 311
gg ! H ! WW ! `
C
`
channel is accessible. While for other mass regions,
some hints are possible. No signal evidence has been reported [10C11].
The SM Higgs boson is expected to be found at the LHC, provided that an
integrated luminosity L 30 fb
1
is collected and that the detectors perform as
expected. For higher luminosities, more channels can be used, thus strengthening
the signal and providing definite answers to the question if indeed a scalar Higgs
boson exists.
Chapter 11
The Standard Model of the Microcosm
11.1 Introduction
The Standard Model (SM) of the microcosm describes the fundamental constituents
of matter and the interactions amongst them. The Standard Model includes 12
fermionic fundamental constituents (see Tables 1.1 and 1.2):
6 Leptons with Spin D
1
2
6 Quarks with Spin D
1
2
9
=
;
and their corresponding antiparticles: (11.1)
The fundamental fermions are classified into three families; each family is made of
a quark doublet, a charged lepton and its associated neutrino plus their antiparticle
counterparts. The six quarks each appear in three different colors. The fermions
and the corresponding antifermions have quantum numbers of opposite sign. The
total number of fundamental fermions is therefore equal to 24 fermions plus 24
antifermions.
An important problem in physics is to understand the ultimate constituents of
matter and the fundamental interactions that occur amongst them; it is of equal
importance to establish whether these interactions are, in reality, different manifes-
tations of a single underlying (or unified) interaction which has yet to be discovered.
The unification question was already addressed by Einstein at the beginning of the
last century, though without reaching a conclusion. More recently, progress has been
made. Four fundamental interactions have been identified at the common energy
scale of our laboratories: weak, electromagnetic, strong and gravitational. The
weak and electromagnetic interactions are unified in the electroweak theory of the
Standard Model at an energy scale already attained at accelerators. The electroweak
Standard Model and the Quantum ChromoDynamics (QCD), describing the strong
interaction, together form the Standard Model of the electroweak and strong
interactions (SM). The Standard Model of the microcosm refers to this definition.
S. Braibant et al., Particles and Fundamental Interactions: An Introduction to Particle
Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-94-007-2464-8
11,
© Springer Science+Business Media B.V. 2012
313
314 11 The Standard Model of the Microcosm
The force fields are quantized and the force carriers correspond to 12 fundamental
vector bosons: the photon , the intermediate gauge bosons W
C
, W
, Z
0
,and8
gluons.
Besides the ultimate fermionic constituents and the bosons that mediate the
fundamental interactions, the electroweak SM predicts the existence of the scalar
Higgs boson, with spin D 0, necessary for the spontaneous symmetry breaking
mechanism through which fermions and W
C
, W
, Z
0
bosons acquire mass. The
symmetry is spontaneously broken through a scalar field. In the SM, the self-
interacting Higgs field is chosen as a complex doublet. Massive particles acquire
their mass through interaction with the Higgs field. Three of the four degrees of
freedom connected to the complex doublet are “absorbed,” giving mass to the
W
C
, W
and Z
0
bosons, while the last degree of freedom gives origin to a new
boson, the Higgs boson. The Higgs boson interacts with a fermion-antifermion pair,
giving them masses. In total, there are 61 fundamental particles: 24 fermions C24
antifermions C12 vector bosons C1 Higgs boson.
It is rather likely that at much higher energies the electroweak and strong
interactions unify in the framework of the Grand Unified Theories. At even higher
energies, the gravitation should also enter this unification scheme. An intermediate
energy scale corresponding to solutions beyond the SM, e.g., supersymmetry (see
Sect. 13.2), may also exist.
In this chapter, we shall recall some considerations leading to the electroweak
unification and the electroweak model will be described. A few concepts of QCD
will then be presented. Some fundamental notions will be repeated even if they were
already introduced in previous chapters.
11.2 Weak Interaction Divergences and Unitarity Problem
The theory of weak interactions described thus far works well at low energies and at
the first order. At higher orders, it presents divergences that cannot be canceled
without introducing an indefinitely large number of arbitrary constants; in this
way, any predictive capacity of the theory is essentially lost. The V A theory
is divergent. We recall that in the Fermi theory, the involved leptons are assumed to
have a point-like interaction whose strength depends on the coupling constant G
F
.
Consider, for example, the process
e
Ce
!
e
Ce
. The weak matrix element
of the point-like interaction can be written as
M
fi
D
G
F
p
2
Œ
.1
5
/
e
Œ
e
.1
5
/
: (11.2)
The cross-section of this elastic process is (for E
cm
m
e
)
.
e
e
!
e
e
/ '
G
2
F
q
2
max
D
2G
2
F
m
e
E
lab
D
G
2
F
s
D
4G
2
F
p
2
: (11.3)
11.2 Weak Interaction Divergences and Unitarity Problem 315
Fig. 11.1 Feynman diagrams
of the
e
e
!
e
e
process:
(a) for the contact Fermi
interaction and (b)forthe
charged current (CC) weak
interaction with a W boson
exchange
e
–
e
–
e
–
e
–
W
–
ν
e
ν
e
ν
e
ν
e
G
F
g
g
ab
E
lab
is the
e
energy in the laboratory system, q
2
max
D 2m
e
E
lab
, s D E
2
cm
and p
is the momentum of the
e
(orofthee
) in the center-of-mass system. The cross-
section depends on G
2
F
and on the phase space factor. It increases with p
squared
and exceeds the unitarity limit (this limit is determined, in analogy with optics, from
the condition that for any partial angular momentum wave function `, the intensity
of the scattered wave cannot be larger that of the incident wave). For spin s D 1=2
particles, the wave analysis for the cross-section leads to
`D0
D
¯
2
2
D
2p
2
: (11.4)
The cross-section (11.3) exceeds (11.4)for
4G
2
F
p
2
>
2p
2
,thatis,for
p
>
2
8G
2
F
1=4
D
p
8G
F
1=2
D
p
8 1:17 10
5
1=2
' 300 GeV=c:
Therefore, for p
> 300 GeV/c, the weak interaction Fermi cross-section (11.3)
exceeds the unitarity limit.
The problem is solved by the introduction of the vector bosons W
˙
. Massive
bosons lead to the presence of a propagator of the type
1
q
2
Cm
2
W
(see Fig.11.1b). The
matrix element becomes
M
fi
D
g
p
2
.1
5
/
2
e
1
q
2
C m
2
W
g
p
2
e
.1
5
/
2
:
(11.5)
The change G
F
!
G
F
m
2
W
q
2
Cm
2
W
modifies the cross-section (11.3)by
.
e
e
!
e
e
/ D
4G
2
F
m
4
W
p
2
.q
2
C m
2
W
/
2
: (11.6)
For q
2
p
2
m
2
W
, one has
.
e
e
!
e
e
/ '
4G
2
F
m
4
W
p
2
: (11.7)
316 11 The Standard Model of the Microcosm
Therefore, the cross-section for
e
e
!
e
e
increases with p
2
up to p
300 GeV/c, and then becomes almost constant and finally decreases with increasing
p
2
. The unitarity limit violation is consequently solved.
In the limit of small q
2
.q
2
! 0/, the interaction through the exchange of a W
boson gives the same result as the point-like Fermi interaction. For q
2
m
2
W
,the
cross-section (11.6) reduces to the Fermi cross-section (11.3). Comparing matrix
elements (11.2)and(11.5), one finds:
lim
q
2
!0
g
2
8
1
.q
2
C m
2
W
/
'
g
2
8m
2
W
D
G
F
p
2
: (11.8)
Even after introducing the W
˙
bosons, there are still divergences in the weak
interactions, e.g., the t-channel diagram for the
e
e
! W
C
W
process shown
in Fig. 8.19a has a quadratically divergent cross-section. The divergence is precisely
canceled by the s-channel diagram shown in Fig. 8.19b. The cancellation is due
to the presence of Z
0
exchange neutral currents and a relationship between the
Z
0
`` and the W
˙
`` coupling constants. Historically, this was one of the reasons
for introducing the Z
0
boson.
The lowest order diagram for the electromagnetic process e
C
e
! W
C
W
shown in Fig. 9.16b is also divergent; in this case, the divergence is eliminated
by adding the diagram shown in Fig. 9.16c where a Z
0
boson is exchanged. This
cancellation occurs only if the weak coupling constant g is approximately equal to
the electromagnetic constant, that is, if g ' e. This implies the unification of weak
and electromagnetic fields.
The latter consideration can be applied to the
e
e
!
e
e
process described
by the diagrams of Fig. 11.1. Placing g ' e in (11.8), one obtains m
W
' m
Z
0
'
100 GeV. This was the first estimate of the masses of the W
˙
;Z
0
bosons, that is,
only such high masses can insure the unification with the same coupling constant
and the reduction to the Fermi theory in the low energy limit.
11.3 Gauge Theories
The Standard Model includes the description of the electroweak and strong interac-
tions. The electroweak and QCD theories are both gauge theories (see below), each
with a symmetry group that characterizes the interaction [11A84].
Here, we show that in order to construct a gauge theory, it is necessary to:
• Choose the group describing the symmetry that characterizes the interactions in
question
• Require local gauge invariance for the group symmetry transformations
• Choose the Higgs field that introduces the spontaneous symmetry breaking.
This allows particles to acquire masses without explicitly breaking the gauge
invariance
11.3 Gauge Theories 317
• Renormalize the couplings and the masses of the theory so that they correspond
to the known experimental data. The study of renormalization leads to the
concept of “running,” that is, the dependence on the energy scale of the coupling
constants
11.3.1 Choice of the Symmetry Group
Some particles exhibit very similar properties; this suggests the existence of
symmetries. For example, quarks appear in three colors and the properties of the
weak interaction suggest the grouping of particles in doublets. These properties
naturally lead to adopting the group structures SU(2) and SU(3) respectively for
the weak and strong interactions. The electromagnetic interaction does not modify
the quantum numbers of interacting particles and therefore can be described by the
group U(1).
The standard model for electroweak and strong interactions is based on the
symmetry of unitary groups
1
:
SU.3/
C
˝ ŒSU.2/
L
˝ U.1/
Y
:
Historically, the necessity for the quarks to appear in three colors was introduced
in order to maintain the Pauli exclusion principle in the case of hadrons made of
three quarks (Sect. 7.8.1). Later, it became clear that the role of color was more
important than what was originally thought; it was indeed understood afterwards
that the “color charge” acts as the source of the strong interaction (the color field),
just as the electric charge is the source of the electric field.
The “charge” for the weak interaction is the third component of the weak
isospin,I
3
. The charged current weak interaction affects all left-handed particles,
i.e., particles with spin antiparallel to the momentum (negative elicity). A weak
isospin equal to ˙1=2 is therefore assigned to left-handed particles, while right-
handed particles are organized in isospin singlets (see Table 11.1). In the limit of
zero mass, the V A nature of the charged current weak interaction only involves
left-handed fermions. For massive particles, the interaction preferentially involves
left-handed particles. Assuming that neutrinos have zero mass,
2
the weak couplings
of right-handed neutrinos and left-handed antineutrinos would be equal to zero.
Therefore, for every generation, we have 15 matter fields: 2 left-handed leptons
and 1 right-handed, 2 3 left-handed quarks and 2 3 right-handed ones (the factor
3 takes color into account).
1
The unitary transformations rotate the vectors, but do not modify their length. The unitary
symmetry groups SU(N) are Special Unitary groups with a determinant equal to C1.
2
Recent experimental results on neutrino oscillations favor the hypothesis that neutrinos have a
very small nonzero mass(Sect. 12.6).
318 11 The Standard Model of the Microcosm
Table 11.1 Summary of fermion multiplets of the electroweak interaction. Left-handed
weak isospin doublets are shown in parentheses; the right-handed singlets are shown
separately. For left-handed quarks, the u;c;t quarks of the strong interaction and the
“rotated” d
0
;s
0
;b
0
are used. The rotation is done according to the CKM matrix, that is,
a generalization of the “Cabibbo rotation.” The electric charge Q of the two states of each
doublet differs by one unit; the difference Q I
3
D Y
W
=2 is the same within each doublet
(1=2 for left-handed leptons, C1=6 for quarks)
Fermion multiplets II
3
QY
W
Leptons
e
e
L
L
L
1=2
C1=2
1=2
0
1
1
1
e
R
R
R
00 1 2
Quarks
u
d
0
L
c
s
0
L
t
b
0
L
1=2
C1=2
1=2
C2=3
1=3
C1=3
C1=3
u
R
c
R
t
R
00 C2=3 C4=3
d
0
R
s
0
R
b
0
R
00 1=3 2=3
The electromagnetic interaction takes place through the exchange of both the
neutral gauge boson of the SU(2)
L
group and that of the U(1)
Y
group; thus, the
“charge” of the U(1)
Y
group cannot simply coincide with the electric charge;
it instead corresponds to the weak hypercharge Y
W
defined by the Gell–Mann–
Nishijima relation
Q D I
3
C
1
2
Y
W
: (11.9)
Y
W
is equal to B L for the left-handed doublets, and to 2Q for the right-handed
singlets (B is the baryonic number equal to that 1=3 for the quarks and to 0 for the
leptons; L is the leptonic number equal to 1 for the leptons and to 0 for the quarks).
Since I
3
and Q are both conserved, Y
W
is a conserved quantum number.
11.3.2 Gauge Invariance
11.3.2.1 Gauge Invariance in QED
A global gauge transformation is a phase transformation (i.e., a phase rotation) of
the material field ; it can be written as
!
0
D
e
i˛=„c
! .1 C i˛=„c/
(11.10a)
!
0
D
e
i˛=„c
! .1 i˛=„c/
; (11.10b)
where the relations on the right are valid for infinitesimal transformations and ˛
is a scalar that has the same value in each space-time point. The invariance for
this transformation implies that the phase of the wave function is arbitrary and not
11.3 Gauge Theories 319
observable. The derivative of the wave function transforms as the wave function
itself, @
0
=@x
D @
0
D e
i˛
@
.
The gauge transformation can be generalized considering ˛ as a function of
space-time, ˛ D ˛.x/. In this case, one has a local gauge transformation that can be
different from point to point. The electromagnetic theory is a local gauge invariant
theory. If the wave function and its derivative @ =@x
D @
transform in the
same way, the Lagrangian is invariant under gauge transformations since it includes
Œ
.x/@
.x/ terms. The gauge transformation can be written in the form
!
0
D e
i˛.x/
: (11.11)
The derivative of .x/ is then
@
0
.x/
@x
D @
0
.x/ D e
i˛.x/
Œ@
.x/ C i .x/@
˛.x/ (11.12)
¤ e
i˛.x/
@
.x/: (11.13)
The derivative does not transform as the wave function, and therefore the Lagrangian
is not invariant under gauge transformation. The interaction of fermions with the
electromagnetic field B
is introduced through the covariant derivative of the
electromagnetic field
3
@
! D
D @
C ieB
: (11.14)
Furthermore, to ensure the gauge invariance of the Lagrangian, the quadrivector
potential of the electromagnetic field B
must transform as
B
! B
0
.x/ D B
.x/
1
e
@
˛.x/: (11.15)
With the transformations defined in (11.11), (11.14)and(11.15), the covariant
derivative of the electromagnetic field transforms as
D
0
.x/ D e
i˛.x/
Œ@
.x/ C i .x/@
˛.x/ C ieB
.x/ .x/ i .x/@
˛.x/
D e
i˛.x/
D
.x/: (11.16)
Thus, the term Œ
.x/D
.x/ is invariant under local gauge transformations.
The fact that we use the covariant derivative is connected to the nature of the
interaction of an electric charge e with the field B
,i.e.,eB
. The gauge invariance
for the electromagnetic field leads to a conserved current (and therefore to the
conservation of the electric charge). The infinite set of phase transformations (11.11)
forms the abelian unitary group U(1). QED is invariant under gauge transformations
of this group.
3
The notation B
instead of A
is used to distinguish the electromagnetic field from the isovector
field A
introduced for the weak interaction in the next paragraph.