An Introduction to the Standard Model

Transcription

An Introduction to the Standard Model
Gianluigi Fogli
Dipartimento di Fisica & INFN - Bari
Gianluigi Fogli
An Introduction to the Standard Model, Canfranc, July 2013
Outline
1.  A critical approach to the Standard Model
2.  The Electromagnetic Interaction
3.  Weak Interactions
4.  Electroweak Interactions
5.  Gauge Symmetry
6.  The Standard Model
7.  Renormalization and Running Coupling Constants
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Reference textbooks
Main text of reference for these lectures:
Francis Halzen and Alan D. Martin,
“Quarks and Leptons: An Introductory Course in Modern Particle Physics”
A more theoretical approach:
Giovanni Costa and Gianluigi Fogli,
“Symmetries and Group Theory in Particle Physics”
A more experimental approach:
Alessandro Bettini,
“Introduction to Elementary Particle Physics”
Texts on Quantum Field Theory:
James D. Bjorken and Sidney Drell,
“Relativistic Quantum Fields”
Michael E. Peskin and Daniel V. Schroeder,
“An Introduction to Quantum Field Theory”
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1. A critical approach to the Standard Model
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1. About Fundamental Interactions
•  Scope of Fundamental Physics: description all natural phenomena in terms of
basic theoretical laws, able to reproduce and
predict experimental observations.
•  At the microscopic level all natural phenomena understood in terms of three
basic fundamental interactions
Electromagnetic
Weak
Strong
interactions
•  All these three interactions studied within the same framework characterized
by
Quantum Mechanics
Special Relativity
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Local Relativistic Quantum Field Theory
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2. The Gauge Invariance Principle
Each particle is considered point-like and is described by a field with specific
transformation properties under the Lorentz Group.
The properties of the three interactions are understood within a common,
general principle, the
Gauge Invariance Principle
In other words, they are asked to satisfy a “gauge” symmetry invariance, which
means that they are supposed to be invariant under phase transformations that
rotate the basic internal degrees of freedom (internal quantum numbers), with
rotation angles dependent on the specific space-time coordinates (Local Gauge
Invariance).
Field Theories with gauge symmetry in a 4-dimensional space-time are
“renormalizable”, which means that they are completely determined in terms of
the gauge group of symmetry and the representations of the interacting fields.
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3. “The Standard Model”
Indeed, a complete description of the three interactions is possibile in terms of a
well-defined gauge theory, characterized by
12 gauged non-commuting charges
known as “The Standard Model” of the fundamental interactions.
It is remarkable that at present only a subgroup of the Standard Model
symmetry is reflected by the spectrum of the physical states.
The part of the electroweak symmetry related to the Higgs mechanism and to
the spontaneous symmetry breaking of the gauge symmetry is still hidden to us.
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4. Where is Gravity ?
For all material bodies on the Earth and in all astrophysical and cosmological
phenomena a fourth interaction,
the gravitational interaction
has an important role, however negligible in atomic and nuclear physics.
The theory of General Relativity is the classical (non quantistic) description of
gravity, going beyond the statical approximation described by the Newton Law
and including dynamical phenomena (for example, gravitational waves).
As one can easily understand, the problem of formulating a quantum theory of
gravitational interactions is one of the central problems of the contemporary
theoretical physics.
On the other hand, quantum effects in gravity are expected to become important
only for energy concentrations in space-time that in practice are not accessible
to the experimentation in laboratories.
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5. The Planck Mass
So, gravity has no measurable effects on a subatomic scale and no manifestations
that can guide us to a quantum field theory of it.
It is possible however to estimate the distance r at which the gravitational force
between two particles become significant.
Let us remind how we obtain a dimensionless measure of the strenght of
electromagnetic interactions: we compare the electrostatic energy of repulsion
between two electrons at a distance equal to the natural unit of lenght (the
Compton wavelenght) with the rest mass energy of the electron:
1
e2
α = 4π
ħ/mc
mc2
1 e2
=
4π
ħc
1
~
~ 137
4π adopting the rationalized
Heaviside-Lorentz system:
it reduces Maxwell eqs. to
their simplest form and is
usual in particle physics
which represents the so-called fine structure constant.
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In a similar way, we can estimate that gravitational effects become of order 1
(the interaction becomes a strong interaction) when the masses of the two
particles that interact gravitationally (m1 = m2 = M) are such that the
gravitational potential is comparable to the rest mass energies of the particles,
i.e. when
GM2
r
Mc2
~ 1
for masses separated by a natural unit of lenght
r
=
ħ
Mc
This gives
Mc2 =
ħc5
G
1/2
= 1.22 x 1019 GeV
Planck Mass
The Planck Mass is the only dimensional quantity appearing in gravity. It indicates
the mass scale at which gravitational effects become significant.
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It is interesting to compare the Planck mass to the mass scale at present under
observation in the LHC experiments at CERN: 7-14 TeV (later we will compare
also with the mass scale coming from GUT’s, the Grand Unified Theories).
Instead of considering the mass scale, we can look to the distance at which the
interaction takes place, making use of the Heisemberg indetermination principle,
p Δr ≥ ħ.
In the case of the LHC, we are testing the interaction at distances of the order
rLHC ≥ 10-18 cm
On the basis of the experiments performed until now, we can say that down to
distances of this order of magnitude the subatomic particles do not show an
appreciable internal structure and behave as elementary and point-like.
Using the Planck Mass to estimate the distances at which the quantum effects
due to gravity become significant, we find
rPM ≥ 10-33 cm
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6. String theories
At these distances, r ~ 10-33 cm, the particles so far appeared as point-like could
well reveal a structure, like strings, and could require a completely different
theoretical framework.
A theoretical framework of which the local quantum field theory description of
the Standard Model would be just a “low energy”/”large distance” limit.
At present the most complete and plausible description of quantum gravity is a
theory formulated in terms of non-point-like objects, called “strings”, extended
over distances much shorter than those experimentally accessible, objects that
live in a space-time with 10 or 11 dimensions.
The additional dimensions beyond the usual 4 are, typically, compactified. In
principle, string theories constitute an all-comprehensive framework that
suggests a unified description of all interactions. However, at present they
represent only a purely speculative framework, not accessible to experiment.
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7. More on implications of a description in terms of relativistic
local fields
The description in terms of relativistic local fields has important implications
that we attempt to review here.
Since the photon emerges in a natural way from the quantization of the Maxwell
field Aµ(x), it is reasonable to ask whether also the other particles observed in
nature, primarily the electron, are also related to force fields by the same
quantization procedure.
When this point of view is assumed, it becomes natural to associate with each
kind of observed particles a field φ(x) which satisfies an assumed wave equation.
The particle interpretation of the field φ(x) is obtained when we apply the
canonical quantization program.
An implication of such a program is that we are led to a theory with differential
wave propagation. The fields φ(x) are continuous functions of continuous x and t,
and the values of φ(x) at x are determined by properties of the fields
infinitesimally close to x.
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For most wave fields (for example, sound waves or the vibrations of strings and
membranes) such a description is an idealization which is valid for distances
larger than the characteristic lenght which measures the granularity of the
medium (the air for sound waves). For smaller distances these theories are then
modified in a profound way.
Apparently, the electromagnetic field is a notable exception. Indeed, until the
special theory of relativity obviated the necessity of a mechanical interpretation,
physicists made great efforts to discover evidence for such a mechanical
description of the radiation field.
After the requirement of an “ether” propagating light waves had been abandoned,
it has been considerably less difficult to accept the same idea when the observed
wave properties of the electron suggested the introduction of a new field ψ(x).
From that moment, the present description of the subatomic world in terms of
relativistic local fields follows.
However, it is a gross and profound extrapolation of our present experimental
knowledge to assume that a wave description successful at “large” distances (as
we have seen, 10-18 cm at LHC) may be extended to distances an indefinite
number of order of magnitude smaller.
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In the relativistic theory the assumption that the field description is correct in
arbitrarily small space-time intervals leads – in perturbation theory – to
divergent expressions for the electron self-energy and the “bare charge”.
These divergence difficulties have been sidestepped with the renormalization
theory. However, it is widely felt that the divergences, that enter through the
estimate of the quantum effects of the theory, are symptomatic of a chronic
desease in the small-distance behaviour of the theory.
At this point it is legitime to ask why local field theories, that is theories of
fields which can be described by differential laws of wave propagation, have been
so extensively used and accepted.
There are several reasons of that.
An important reason is that with their aid we have found a significant agreement
with observations.
But the foremost reason is brutally simple. There exists no convincing form of a
theory which avoids differential field equations.
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The Standard Model is indeed a theory based on a relativistic local field
description. Because of the existence of creation and annihilation processes, it is
at once a theory of the many-body problem.
The prescription of a quantization strongly involves the existence of a
Hamiltonian H. However, since it generates infinitesimal time displacements, we
are led to a description with differential development in time. Lorentz invariance
then requires a differential development in space as well.
The notion of a Lorentz invariant microscopic description in terms of continuous
coordinates x and t implies that the influence of the interaction should not
propagate through space-time with velocity faster than c. This notion of
“microscopic causality” strongly forces us again into the field concepts.
Of course, a Hamiltonian may well not exist for a non-local “granular” theory: if it
does not, the link connecting us with the quantization method of non-relativistic
theories is fatally broken.
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There is no concrete experimental evidence of a “granularity” at small distances.
There is likewise nothing but positive evidence that special relativity is correct in
the high-energy domain, and, furthermore, there is positive evidence that the
notion of microscopic causality is a correct hypothesis.
Since there exists no alternative theory which is any more convincing, we are
forced to restrict ourselves to the formalism of relativistic local causal fields. It
is undoubtedly true that a modified theory must have local field theory as an
appropriate large-distance approximation or correspondence.
However, we again emphasize that the formalism of the Standard Model may well
describe only the large-distance limit (at present, distances ≥ 10-18 cm) of a
physical world of considerably different submicroscopic properties.
But let us now abandon these speculations and go to the concrete formulation of
the Standard Model.
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2. The electromagnetic interaction
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1. A spinless electron in an electromagnetic field
A free “spinless” electron satisfies the Klein-Gordon equation
(∂µ∂µ + m2 ) φ(x) = 0
which is nothing else that the relativistic energy-momentum relation
E2 + →
p2 = m2
with the introduction of the differential quantum operators pµ → iħ
∂ .
∂xµ
In classical electrodynamics the motion of a particle of charge –e in an
electromagnetic potential Aµ is obtained by the substitution
pµ → pµ + eAµ
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quantum-mechanically
corresponding to
i∂µ → i∂µ + eAµ
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so that the Klein-Gordon equation becomes
(∂µ∂µ + m2 ) φ(x) = -Vφ(x)
sign chosen according to the relative
sign of kinetic energy and potential
in the Schrodinger equation
with
V = -ie (∂µAµ + Aµ∂µ) – e2A2
e related to the coupling: in natural units
e2
α =
4π
In the usual non-relativistic perturbation theory the transition amplitude for the
scattering of a “spinless” electron from a state φi to the state φf off an
electromagnetic potential Aµ is given by
Tfi = -i ∫ φf*(x) V(x) φi(x) d4x = i
∫ φf*(x) ie (∂µAµ + Aµ∂ ) φi(x) d4x
µ
Integrating by parts
∫ φf* ∂µ(Aµ φi) d4x = - ∫ ∂ (φf*) Aµ φi d4x
µ
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we obtain
φi
Tfi = -i ∫ jµfi(x) Aµ(x) d4x
jµfi
φf
with
jµfi = - ie [φf*∂µ(φi) - ∂µ(φf*) φi]
electromagnetic
current
Aµ
If the ingoing and outgoing electrons have momenta pi and pf, respectively, we can
write, with Ni and Nf normalization constants,
φi = Ni e-ip x
i
φf = Nf
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e-ipf x
jµfi = - e NiNf (pi + pf)µ ei(pf-pi) x
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1. “Spinless” electron-muon scattering
We are now able to calculate the scattering of the electron from another particle,
for example a muon. The graph is simple: pA and pC are the initial and final momenta
of the electron, pB and pD of the muon.
jµ(1)
epA
epC
The calculation is similar to the previous one: we have
to identify Aµ with its source, the charged “spinless”
muon of the lower vertex.
The identification is performed through the Maxwell
equations
q
Aµ = jµ(2)
pB
µ-
pD
jµ(2)
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with
µ-
From
jµ(2) = - e NBND (pD + pB)µ ei(pD-pB) x
eiqx = -q2eiqx
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it follows
Aµ = - 12 jµ(2)
q
with
q = pD - pB
In conclusion
Tfi = -i ∫ jµ
(1) (x)
- 12
q
jµ(2) (x) d4x
Using the currents introduced before we can write
Tfi = -i NANBNCND (2π)4 δ(4)(pD+pc-pA-pB)M
with
M = ie (pA+pC) µ
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-i
gµν
q2
ie (pB+pD) ν
invariant
amplitude
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The invariant amplitude is a relativistic quantity and represents the Feynman
diagram of the process. It contains the dynamics of the interaction. In M we
distinguish
-i
e-
-i
jµ(1)
ie (pA+pC) µ
gµν
q
q2
µ-
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e-
ie (pB+pD) ν
jν(2)
µ-
gµν
q2
is the propagator of the photon (a
spin
1
particle)
exchanged
between electron and muon.
The photon is “virtual”, or off-mass-shell: the
changed particle carries the quantum number
of the photon, but not the mass.
Each
vertex
factor
contains
the
electromagnetic coupling e and the fourvector index of a current.
In conclusion, the graph describes the
exchange of a virtual photon between the two
currents associated to the two particles,
electron and muon.
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1. Electrodynamics of spin- 21 particles
1
It is relatively simple to extend the previous approach to spin 2 particles. A free
electron of four-momentum pµ is described by a four-dimensional Dirac spinor
ψ (x) = u(p) e-ipx
which satisfies the Dirac equation
(γµpµ ‒ m) ψ = 0
The substitution
pµ → pµ + eAµ
describes the electron in an electromagnetic field Aµ: we write the equation in the
form
(γµpµ ‒ m) ψ = γ0V ψ
with
γ0V = - e γµAµ
where γ0 is introduced to make the
equation relativistically invariant.
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We can repeat the previous procedure and calculate the scattering amplitude:
Tfi = -i ∫ ψf†(x) V(x) ψi(x) d4x = ie ∫ ψf†(x) γµAµ ψi(x) d4x = -i
with
jµfi = -e ψf γµ ψi = - e uf γµ ui ei(p -p )x
f
∫ jµfiAµ d4x
ψ = ψ†γ0
where
i
Jµfi
The vertex factor is now a
4x4 matrix in spin space
1
Jµfi
1
ui
uf
ie(pf+pi)µ
ieγµ
The Gordon decomposition of the current shows that the spin
interacts via both its charge and its magnetic moment:
uf γ µ ui =
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1
uf (pf+pi)µ + iσµν(pf-pi)ν ui 2m
with
1
2
electron
σµν = i (γµγν – γνγµ)
2
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We can now estimate the scattering amplitude of the process e-µ- → e-µ- Tfi = -i ∫ jµ(1) (x)
- 12
q
jµ(2) (x) d4x
which implies
Tfi = -i (- e uC γµ uA)
- 12
q
(- e uD γµ uB) (2π)4 δ(4) (pA+pB-pC-pD)
so that the invariant amplitude is
-iM = (ie uC γµ uA)
-i
gµν
(ie uD γν uB)
2
q
e-
gµν
-i 2
q
µ-
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jµ(1)
e-
ieγµ
ieγν
jν(2)
µ-
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Now we have to estimate |M|2 for all possible spin configurations, which means
average over the spins of the incoming particles and sum over the spins of the
particles in the final state:
|M|2 → |M|2 =
1
(2sA+1)(2sB+1)
∑
all spin
states
|M|2
Introducing explicitly the particle momenta
M = - e2 u(k’) γµ u(k) 12 u(p’) γµ u(p)
q
e-
e-
k
we can write
|M|2 =
e4
q
µνL muon
L
µν
4 e
where
Leµν
=
1
∑
[
u(k’) γµ u(k)][u(k’) γν u(k)]*
2 e spins
gµν
-i 2
q
p
µ-
k’
ieγµ
q
ieγν
P’
µ-
with a similar expression for Lmuon
µν .
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The spin summation requires some care, but can be done with some trace
techniques.
First of all, let us write (using γν γ0 = γ0γν) [u(k’) γν u(k)] * =
[u(k’) γ0 γν u(k)] = [u(k) γν γ0 u(k’)] = [u(k)γνu(k’)]
The sum in Leµν can now be done. Writing explicitly the indeces and using the
completeness relations of the Dirac spinors (m being the electron mass) Leµν
=
(s’)
(s)
(s)
(s’)
1
∑
uα (k’) γµαβ ∑ uβ (k) uγ (k) γνγδ uδ (k’)
2
s
s’
(k’ + m)δα
i.e.
Leµν
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=
(k + m)βγ
1
Tr [(k’ + m) γµ (k + m) γν ]
2
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Traces are calculated making use of the commutation algebra of the γ matrices γµγν + γνγµ = 2gµν
Accordingly
Leµν =
1 Tr
2
(k’ γµ k γν ) +
1 m2 Tr
2
(γµ γν ) =
2 [k’µkν + k’νkµ – (k’ k – m2) gµν]
•
In similar way
muon
Lµν
=
1 Tr
2
(p’γµ p γν ) +
1
2
M2 Tr (γµ γν ) =
2 [p’µpν + p’νpµ – (p’ p – M2) gµν]
•
So that, in conclusion
|M|2
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=
e4
8 4 [(k’•p’)(k•p) + (k’•p)(k•p’) – m2(p’•p) – M2(k’•k) + 2m2M2]
q
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3. Weak Interactions
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1. The V-A structure of weak interactions
Typical weak processes
β decay
n → p + e- + νe
mean life 820 s
µ- decay
µ- → e- + νe + νµ
τ = 2.2 x 10-6 s
π- decay
π- → µ- + νµ
τ = 2.6x 10-8 s
Fermi explanation of β-decay (1932) inspired by the structure of the
electromagnetic interactions: the β-decay in its “crossed” form
p + e- → n + νe
no propagator:
point-like int.
described by the invariant amplitude
M = G ( un γµ up) ( uν γµ ue)
Fermi constant
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charged weak currents
p
jµ(1)
e-
n
νe
jµ(2)
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2. Parity violation
In 1956 Lee and Yang made a critical survey of all the weak interaction data and
argued persuasively that parity was not conserved in weak interactions. The Wu
experiment in the same year studied β-transition of polarized cobalt nuclei
60Co
→ 60Ni* + e- + νe
z
-νR
and observed that the electron is emitted
preferentially in the direction opposite to that of
the spin of the 60Co nucleus.
The observation is consistent with the
explanation that the required Jz = 1 is formed by
a right-handed antineutrino νR and a left-handed
electron eL.
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+
eLJz = 5
Jz = 4
Jz = 1
60Co
60Ni*
- R
+ (e-)L+ (ν)
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The cumulative evidence of many experiments led to the conclusion that only
νL
and
-νR
are involved in weak interactions. The absence of νR and νL is a clear indication of
parity violation:
Γ(π+ → µ+νL) ≠ Γ(π+ → µ+νR) = 0
P violation
Not only parity is maximally violated in weak interaction, but also charge
conjugation, i. e. the interchange particle-antiparticle. Indeed, C transforms a νL
into a νL so that
Γ(π+ → µ+νL) ≠ Γ(π- → µ-νL) = 0
C violation
However, the combination of the two symmetry operations is not violated, at least in
principle, in weak interactions
Γ(π+ → µ+νL) = Γ(π- → µ-νR)
CP invariance
We will discuss later the problem of CP violation in weak interactions.
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At the end, we arrive to the conclusion that weak interactions phenomena are
described by a V-A current-current interaction with a universal coupling G.
Accordingly for β- and µ-decay
M( p → n e+νe) = G
√2
[ un γµ (1-γ5) up][uν γµ(1-γ5)ue]
M(µ- → e- νeνµ) =
[ uνµ
γµ (1-γ5) uµ][ue γµ (1-γ5) uν e]
G
√2
e
Weak interactions are then of the general form
M = 4G Jµ Jµ
√2
Jµ = uν γµ 1 (1-γ5)ue
with
2
Jµ = ue γµ 1 (1-γ5)uν
2
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charge-raising
current
charge-lowering
current
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3. Interpretation of the coupling G: the intermediate vector
boson theory
The comparison between electromagnetic and weak amplitudes shows that G
essentially replaces e2/q2. Thus G has dimensions GeV-2. It is tempting to extend the
analogy by assuming that weak interactions are characterized by the exchange of
charged vector bosons, W±, with an amplitude of the form (for the µ-decay)
-eνµ) =
M(µ- → e- ν
g
g
1
uνµ
γµ (1-γ5) uµ
ue γµ (1-γ5) uνe
2
2
MW – q √2
√2
νµ
where g/√2 is a dimensionless variable
and q the momentum carried by the
vector boson W±.
g
W±
√2
g
µ-
√2
e-
-e
ν
At the present level the introduction of W± simply leads to a reinterpretation of the
Fermi constant G. This is the so-called “intermediate vector boson hypothesis”.
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In situations in which q2 « M2W (e.g. β-decay and µ-decay) we have
G
√2
=
g2
8M2W
and the propagator disappears: the interaction is point-like. From the previous
equation we are induced to suspect that weak interactions are weak since M2W is
large, while it is reasonable to expect g ≈ e.
With the previous structure of the invariant amplitude, a large number of processes
can be explicitly calculated and compared with the experimental measurements.
Let us only mention an important experimental results, concerning the constant G as
measured in µ-decay and in β-decay. One obtains
Gµ = (1.16632 ± 0.00002) × 10-5 GeV-2
Gβ = (1.136 ± 0.003) × 10-5 GeV-2
The reason of this difference is important and will be discussed later.
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3. Weak neutral currents
Detection in 1973 of neutrino events of the type
-
νµ e - → -
νµ e - clean events, but small cross-section
νµ N → νµ X -
νµ N
deep-inelastic scattering (DIS)
→ -
νµ X !
neutral
currents
A quantitative comparison with the strenght of neutral currents (NC) to charged
currents (CC) weak processes was performed. For example, for DIS processes the
following experimental values were obtained
Rν ≡
R-ν ≡
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σNC(ν)
σCC(ν)
σNC(ν)
-
σCC(ν)
≡
≡
σ (νµ N → νµ X) σ (νµ N →
µ- X)
-µ N → -νµ X) σ (ν
-µ N → µ+ X) σ (ν
= 0.31 ± 0.01
= 0.38 ± 0.02
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Data can be understood in terms of neutral current-current interactions of
amplitude (at the quark level)
M=
GN
√2
[ uν γµ (1- γ5) uν] [uq γµ (cVq - cAq γ5)uq]
GN cV cA
new parameters
The conventional normalization of the weak neutral current is then of the type
NC µ = 4GN 2 JNC JNC µ
M = 4G 2ρ JNC
J
µ
µ
√2
√2
Jµ
NC (ν) = 1
2
with
Jµ
NC (q) =
-uν γµ
1 (1- γ ) u 5 ν
2
q
q
u-q γµ 1 (cV - cAγ5) uq
2
Neutral currents, unlike charged currents, are not pure V-A (cV ≠ cA): they have
right-handed components. However the neutrino is left-handed with cV=cA=1/2.
As we will see, in the Standard Model all the couplings cV, cA are all given in
terms of only one parameter, sin2θW.
The parameter ρ “measures” the relative strenght of neutral to charged
current processes. In the minimal version of the Standard Model (MSM) ρ = 1.
This value is confirmed by the experiment within the experimental errors.
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4. The Cabibbo angle
We have seen that charged currents are constructed considering transition
between states coupled in the following pairs of (left-handed) fermionic doublets:
νe
e-
u
d
νµ
µ-
They are coupled with the same universal coupling G. However, there are also
processes that implies transitions u → s. For example the decay
u
K+ → µ+ νµ
µ+
K+
s
similarly
to
u
µ+
d
νµ
π+
νµ
In 1963 Cabibbo suggests to accomodate observation as K+ decay maintaining
universality but modifying the quark doublets: charged currents couple “rotated”
quark states as
u
u
d’ = d cosθc + s sinθc
with
d'
s'
s’ = -d sinθ + s cosθ
c
Gianluigi Fogli
c
40
The quarks d and s are mixed in terms of the arbitrary parameter θc, the Cabibbo
angle, to be measured experimentally. This gives rise to a comparison of ΔS = 0 and
ΔS = 1 processes, with a suppression ~ sin2θc of the decay of strange particles
(being θc ≈ 13°).
The introduction of Cabibbo angle leads to a new form of charged currents, always
in the form
M = 4G Jµ Jµ
√2
CC
but with now
- ) 1 γµ (1- γ5) U d
Jµ = ( u c
s
2
being
U =
cosθc
sinθc
- sinθc
cosθc
It is now clear the reason of the small difference between Gµ and Gβ: Gβ includes
the cosine of the Cabibbo angle. The relation is
Gβ = Gµ cosθc
Gianluigi Fogli
41
4. The GIM Mechanism
The original motivation of the proposal of the GIM mechanism has been to
understand why there are no transitions s ⇔ d, which change flavor but not
charge.
Indeed, the experimental evidence for the absence of strangeness-changing
neutral currents is compelling. Absent or strongly suppressed are decays of strange
particles as
K0 → µ+µ-
K+ → π+e+e-
K+ → π+νν
This leads to the conclusions that direct transitions
s ⇔ d
d
µ+
s
µ-
K0
are forbidden.
When the GIM mechanism has been proposed (1970), only the three lightest quarks
(u, d, s) were supposed to exist.
Gianluigi Fogli
42
However, even excluding direct neutral current transitions s ⇔ d, the K0 → µ+µdecay would occur at a rate far in excess of what is observed
Γ(K0 → µ+µ-)
Γ(K0 →
all)
= (9.1 ± 1.9) × 10-9
because of charged current u ⇔ d’ transition
due to the box diagram in which the
transition is mediated by a quark u.
d
u
s
µ-
νµ
W
µ+
M ~ cosθcsinθc
The GIM proposal was to introduce a fourth
quark c, of charge 2/3, such that a second
box diagram occurs, which would cancel
exactly the first if it were not for the mass
difference between c and u.
d
The existence of the quark c was predicted
together with an estimate of its mass, in good
agreement with the next estimates from
experiment.
s
Gianluigi Fogli
W
W
c
µ-
νµ
W
µ+
M ~ -sinθccosθc
43
It is important to understand how the GIM mechanism works, as an effect of the
unitarity of the matrix U which describes the quark rotation in the charged
current structure
d’
s’
=
cosθc
-sinθc
sinθc
cosθc
d
s
that we rewrite
synthetically
d’i = ∑j Uijdj
-
Starting from neutral currents of the form d’id’i and summing up we have
-
-
-
∑i d’id’i = ∑ijk djU jiUikdk = ∑j djdj which shows that only diagonal transitions are allowed.
Q:
Why the mixing is taken in the down-type quarks?
A:
The reason is historical. The mixing could equally be taken in the up-type
quarks.
Q:
Why no Cabibbo mixing in the leptonic sector?
A:
The reason is the mass degeneracy of neutrinos, that are massless in the
SM. Indeed, as well known, there is a mixing also in the leptonic sector.
Gianluigi Fogli
44
4. The Kobayashi Maskawa matrix
As well-known, there exists a third generation of leptons and quarks, so that we
have to consider three quark doublets
c
t
u
d
b
s
with their mixing.
In the case of two doublets the mixing is described in terms on only one parameter,
the Cabibbo mixing angle θc.
How many parameters we have in the general case of N doublets?
The number of parameter (angles and phases) of a N×N unitary matrix U is N2. We
can subtract (2N-1) phases since we can absorb them in the 2N quark fields, apart
from 1 overall phase. We can also subtract the number of parameters (angles) that
characterize an orthogonal N × N matrix. In conclusion we have
N2 – (2N - 1) -
Gianluigi Fogli
1
2
N (N – 1) =
1
(N
2
– 1)(N – 2)
residual phases
45
There are no phases for N=2, but there is a phase if N=3. Then, for three
generations we have 3 angles and 1 phase factor, eiδ.
We rewrite charged currents in terms of all the three quark doublets
- --
Jµ = ( u c t )
1 γµ (1- γ )
5
2
U
d
s
b
The unitary matrix U describes the charged currents transitions according to
Uud
Uus
Uub
Ucd
Ucs
Ucb
Utd
Uts
Utb
c1
=
-s1c3
-s1s3
s1c2
c1c2c3-s2s3eiδ
c1c2s3-s2c3eiδ
s1s2
c1s2c3+c2s3eiδ
c1s2s3+c2c3eiδ
where one of the most usual parametrizations is also indicated (with ci, si cosine and
sine of the angles θ1, θ2, θ3).
Gianluigi Fogli
46
5. CP violation
An analysis of CP invariance requires the comparison of the amplitudes of a generic
process and of the process conjugate under CP.
Let us consider the charged current process ab → cd.
The invariant amplitude is given by
a
M ∼ Jca Jµ bd ∼ uc γµ (1- γ5) Uca ua u-b γµ (1- γ5) Ubd ud =
c
W
µ
= Uca Udb* uc γµ (1- γ5) ua ud γµ (1- γ5) ub
Uca
b
Udb
*
d
On the basis of the correspondence between particle solutions going backward in
time and antiparticle solutions going forward in time, the previous amplitude
describes both the two processes
ab → cd
Gianluigi Fogli
--
--
cd → ab
47
Let us now consider the process
-ab- → -cdIts amplitude is obtained taking the conjugate
currents, so that it corresponds to
M ∼
µ
Jca
µ
Jµ bd ∼
U * U ca
db ua γ (1- γ5) uc ub γµ (1- γ5) ud
-a
Uca*
-c
W
-
b
-
Udb
d
This process is described by the same Hamiltonian H: indeed H is hermitian, so it
contains M + M .
In order to understand if the process ab → cd is invariant under CP, we have to
calculate the process conjugate under CP and compare its invariant amplitude,
MCP, with M .
If
MCP = M
occurs, then the process ab → cd is CP invariant. Otherwise CP is violated.
Gianluigi Fogli
48
We omit the explicit calculations, essentially based on the transformation
properties under C and P of Dirac spinors and gamma matrices.
We find
MCP ∼ Uca Udb* ua γµ (1- γ5) uc ub γµ (1- γ5) ud
The comparison shows that, provided that the elements of the mixing matrix U are
real, we find
MCP = M
This is the case in which U is a 2×2 matrix and only 4 quarks (u, d, c, s) exist.
If however we have three generations of quarks, the mixing matrix becomes the
3×3 matrix of Kobayashi-Maskawa. It now contains a complex phase eiδ. Then in
general it is
MCP ≠ M
and CP invariance is violated.
Gianluigi Fogli
49
4. Electroweak Interactions
Gianluigi Fogli
50
1. Weak Isospin and Hypercharge
The Standard Model leads to a description of fundamental interactions in terms of
a gauge invariant renormalizable theory.
A first step in this direction is to find within the weak interaction phenomenology
an underlying symmetry group.
Let us start from the charged currents written in terms of the left-handed fields
νe
Jµ = Jµ(+) = uν γµ 1 (1-γ5) ue = ν γµ 1 (1-γ5) e = νL γµ eL W+
2
2
e-
Jµ = Jµ(-) = u-e γµ
1
2
- γµ
(1-γ5) uν = e
1
2
(1-γ5) ν = eL γµ νL e-
W-
νe
By introducing
the doublet
Gianluigi Fogli
χL =
νe
e-
L
and the usual “step-up”
and “step-down” operators
τ± =
1
2
(τ1 ± iτ2)
51
the two currents can be rewritten in a two-dimensional form
-Lγµτ+χL
Jµ(+) (x) = χ
Jµ(-) (x) = χLγµτ-χL
If now we add a third current in the form of a neutral current
Jµ(3) (x) = -χLγµ
1τ χ
2 3 L
=
1
2
-νLγµνL - 1 -eLγµeL
2
e- (νe)
W0
e- (νe)
we have thus constructed an “isospin” triplet of weak currents
Jµ(i) (x) = χLγµ
1τ χ
2 i L
(i = 1, 2, 3)
whose corresponding “charges”
T i = ∫ J0(i)(x) d3x
satisfy an SU(2)L algebra
[Ti,Tj] = ieijkTk
and can be assumed as generators of a new quantum number, the “Weak Isospin”.
Gianluigi Fogli
52
Question: can Jµ(3) (x) be identified with the weak neutral current introduced in the
study of the Deep Inelastic Scattering νµ N → νµ X ? The answer is negative, since Jµ(3) (x) is a pure left-handed current, whereas the
observed phenomenological current contains a (small) right-handed component.
However, we know a current which contains a right-handed component, the vectorial
electromagnetic current, which contains a right- as well a left-handed component.
Taking apart, for simplicity, the multiplicative factor e (the electric charge), we can
write the current
- µe = - -eLγµeL - -eRγµeR
jµem(x) = - eγ
so that the current appearing in the electromagnetic interaction cam be written as
- µQψ
jµ = e jµem = e ψγ
where Q is the electric charge generator, with eigenvalue Q = -1 for the electron.
In other words, Q is the generator of the U(1)em symmetry group of the
electromagnetic interactions.
Gianluigi Fogli
53
At this point, we have two symmetry groups
SU(2)L
U(1)em
with generators T i and currents Jµ(i)
with generator Q and current
jµem
But we have also a phenomenological neutral currents JµNC which needs to be
interpreted in terms of them.
The two currents JµNC and jµem do not “respect” the SU(2)L symmetry, but we can
expect that this is the case for a suitable linear combination of them. This
combination is a neutral current that can be identified with the member Jµ(3) of the
isospin triplet.
The corresponding orthogonal combination is independent of SU(2)L, i.e. it must be
an isospin singlet under SU(2)L and can be connected to the generator of a new U(1)
group, whose generator is a linear combination of T(3) and Q.
The explicit form of this linear combination of T(3) and Q is quite arbitrary. We can
adopt the same relation which characterizes the third component of the isotopic
spin and the electric charge in the definition of the hypercharge through the Gell
Mann-Nishijima scheme (i.e. the scheme adopted for the arrangement of the
strange particles in the SU(2)I symmetry of the hadronic isospin multiplets).
Gianluigi Fogli
54
So, we adopt the well-known Gell Mann-Nishijima relation
Q = T (3) +
Y
2
to connect the third component of the “weak isospin” to a new generator, the “weak
hypercharge” Y. Going to the currents associated to these generators, we can write
jµem = Jµ(3) + 1 jµY
2
The hypercharge generator Y can be taken as the generator of a new abelian group
U(1)Y, so that the complete symmetry group is now enlarged to
SU(2)L ⊗ U(1)Y
U(1)em will appear as a subgroup of it, to be properly identified as the subgroup
whose generator is the electric charge Q.
Note that we do not have a simple group, but the product of two groups, which
means that we need to introduce, in addition to the electric charge, another
coupling constant. Or, which is equivalent, we have to introduce two couplings, one
for SU(2)L and one for U(1)Y, by identifying the electric charge as a proper
combination of them.
Gianluigi Fogli
55
The SU(2)L ⊗ U(1)Y proposal was first made by Glashow in 1961, long before the
discovery of the weak neutral currents, and, as we will see later, was extended to
accomodate massive vector bosons (W±, Z0) by Weinberg in 1967 and Salam in 1968.
This is the Standard Model of the electroweak interactions.
Since we have to do with the product of two symmetry groups, the generator Y
must commute with the generators T(i). Accordingly, all the members of an isospin
multiplet must have the same value of the hypercharge. For example, for the
electron doublet (ν e-) it is
jµY = 2jµem + 2Jµ(3) = -2 (e-RγµeR + e-LγµeL) – (ν-LγµνL – e-LγµeL) = -2 e-RγµeR – 1 χ-LγµχL
so that the left-handed doublet (ν e-)L has hypercharge -1 and the isospin singlet
eR has hypercharge -2.
We are now able to provide the assignment of weak isospin and hypercharge to
leptons and quark of the first generation.
Gianluigi Fogli
56
Weak isospin and hypercharge assignment to leptons and quarks of the first
generation.
Particle
T
T(3)
Q
Y
νe
1
2
1
2
0
-1
e-L
1
2
1
2
-1
-1
e-R
0
0
-1
-2
uL
1
2
1
2
2
3
1
3
dL
1
2
1
3
uR
0
0
2
3
1
3
4
3
dR
0
0
1
3
- 23
Gianluigi Fogli
-
-
1
2
-
57
2. The basic electroweak interaction
We can now modify the current-current form of the weak interactions assuming
that it corresponds to an effective interaction resulting from the exchange of
massive vector bosons (in the limit of small momentum transfer).
Exactly in the same way used to develop QED from the basic interaction
- i e jµem Aµ
coupling
current
vector boson
We can introduce an isotriplet of vector bosons Wµ(i) and a single vector boson Bµ,
with coupling g and g’, respectively, to describe the SU(2)L ⊗ U(1)Y interaction, which
takes the form
- i g Jµ(i) Wµ(i) - - i g’
Wµ(i)
Gianluigi Fogli
g
Jµ(i)
1
2
j µY B µ
Bµ
1
2
g’
j µY
58
The electromagnetic interaction, described in terms of the photon Aµ, and the
phenomenological neutral current interaction, described in terms of the vector
boson Zµ, are embedded in the previous form of the electroweak interaction.
We have only to extract them explicitly, taking into account that they must appear
as two orthogonal combinations of the neutral vector bosons associated to Jµ(3) and
jµY. Accordingly, we can introduce the physical states (the mass eigenstates) in the
form
Aµ = Bµ cosθw + Wµ(3) sinθw
(massless)
Zµ = - Bµ sinθw + Wµ(3) cosθw
(massive)
The parameter θw, the Weinberg angle, “measures” the mixing. It is a
phenomenological parameter, to be determined experimentally. Introducing
explicitly Aµ and Zµ in the structure of the electroweak interaction we obtain
- i g Jµ(3) Wµ(3) - i g’
1 Y µ
j B
2 µ
-i
Gianluigi Fogli
1
2
( g cosθw Jµ(3) - 21
= - i ( g sinθw Jµ(3) +
g’ cosθw jµY ) Aµ +
g’ sinθw jµY ) Zµ
59
The first term is to be identified with the electromagnetic interaction: using
e jµem = e (Jµ(3) +
1 Y
j )
2 µ
it follows the important relation between couplings and mixing angle
g sinθw = g’ cosθw = e
tg θw =
g’
g
The second term contains the neutral current JµNC. It can be rewritten
1
2
- i ( g cosθw Jµ(3) -
g’ sinθw jµY ) Zµ = - i
= -i
g
cosθw
( Jµ(3) -
1
2
g’
g
sinθw cosθw jµY ) Zµ =
g
( Jµ(3) – sin2θw jµem )
cosθw
So, one identifies
Jµ
Gianluigi Fogli
NC
= Jµ
(3)
–
sin2θ
w jµ
em
coupled to Zµ
with coupling
Zµ = - i
g
JµNC
cosθw
Zµ
g
cosθw
60
3. The effective current-current interactions
Let us recall the charged current invariant amplitude
Mcc = 4G Jµ Jµ
√2
CC
with, in the isospin notation,
-Lγµτ+χL
Jµ = Jµ(+) (x) = χ
On the other hand, introducing the charged vector bosons, we can rewrite the basic
interaction in the form
-i
g
√2
which leads to
Mcc =
Jµ
Jµ Wµ(+) + Jµ Wµ(-)
g
√2
Jµ
1
M2w
g
√2
Jµ
CC
where 1 2 is the approximation of the W propagator at low q2.
Mw
Gianluigi Fogli
W(±)
Jµ
61
Comparing the two expressions of Mcc we obtain
G
√2
=
g2
8M2w
In an analogous way for the neutral current interaction in terms of Z0 exchange
MNC
=
g
cosθw
Jµ
NC
1
M2Z
g
cosθw
JµNC
comparing with the corresponding current-current form
4GN
MNC = 4G 2ρ Jµ
NC JNC µ =
2 Jµ
NC NC µ
√2
√2
we can identify
g2
G
ρ
=
2 2
8M z cos θw
√2
JµNC
Z0
JµNC
By comparing the expressions for charged and neutral currents, we find for the
parameter ρ, which measures the relative strenght of the two weak interactions
ρ
=
M2w
M2z cos2θw
We shall see that the minimal version of the Standard Model (MSM) predicts ρ = 1.
This value is confirmed experimentally within small errors.
Gianluigi Fogli
62
4. Feynman rules for electroweak interactions
For the electromagnetic interaction
- µQψ ) Aµ
-i e jµem Aµ = -i e ( ψγ
γ
vertex factor
f
-i e Qf γµ
-
f
For the charged current interaction
-i
( χ-Lγµτ+χL ) Wµ(+) = -i √2 ( -νLγµeL ) Wµ(+)
g
g
√2
W(+)
e+
νe
-i
g
√2
( χ-Lγµτ-χL ) Wµ(-) = -i
(
e-LγµνL ) Wµ(-)
√2
g
W(-)
-i
g
( Jµ(3) – sin2θw jµem ) Zµ =
cosθw
Z0
g ψfγµ 1 (1-γ5)T(3) – sin2θw Q ψf Zµ
2
cosθw
Gianluigi Fogli
e-
g
√2
γµ
1
(1-γ5)
2
-νe
For the neutral current interaction
-i
-i
f
f
-i
g
cosθw
γµ
1 (c f-γ cf )
2 V 5 A
63
It follows that all the couplings cV and cA are determined in the Standard Model in
terms of only one parameter, sin2θw, being for each fermion f
f
f
cA = T(3)f
cV = T(3)f – 2 sin2θw Qf
-
The explicit values of the different Z → f f vertex factors are reported in the
Table (sin2θw = 0.234)
fermion
Qf
cAf
cVf
νe , νµ , …
0
1
2
1
2
e- , µ- , …
-1
Gianluigi Fogli
-
1
3
1
2
1
2
2
3
u,c,…
d,s,…
-
-
1
2
-
1
+ 2 sin2θw
2
≈ - 0.03
1
- 4 sin2θw ≈ + 0.19
2
3
- 21
+ 2 sin2θw ≈ - 0.34
3
64
A large number of experiments have been performed in the ‘80 to test the
Standard Model, more specifically to measure sin2θw in different processes. Let us
mention:
Deep inelastic scattering
Neutrino-electron scattering
One-pion production from neutrino scattering
Electroweak interference in e+e- annihilation
Parity violating effects in atomic transitions
Parity violating asymmetry in the inelastic scattering of longitudinally
polarized electrons or muons
In all these processes, a perfect agreement with the Standard Model has been
found.
Gianluigi Fogli
65
5. Gauge symmetries
Gianluigi Fogli
66
1. Lagrangians and single particle wave equation
The fundamental belief is that all particle interactions may be dictated by the socalled “local gauge symmetries”. We shall see that this is intimately connected with
the idea that conserved physical quantities (such as electric charge, color, …) are
conserved in local regions of space and not just globally.
The framework in which these principles are discussed is the Lagrangian Field
Theory. It is well-known that in classical mechanics the particle equations of motion
can be obtained from the Lagrange equations
d
dt
∂L
.
∂ qi
-
∂L = 0
∂ qi
with
where
L=T-V
with
qi generalized coordinates
.
qi = dqi/dt
T kinetic energy
V potential energy
It is possible to extend the formalism to a continuous system, that is a system φ
with continuously varying coordinates: φ( x , t).
Gianluigi Fogli
67
The Lagrangian becomes a Lagrangian density
.
L ( qi, qi, t) → L ( φ, ∂φ
, xµ)
∂xµ
with
L = ∫ L d 3x
which satisfies the equation of motion
∂
∂xµ
∂L
∂φ
∂ µ
∂x
- ∂∂Lφ
=0
What is the relation between the Lagrangian approach and the perturbative method
based on Feynman rules? To each Lagrangian there corresponds a set of Feynman
rules, so that, once we identify these rules, the connection is established. The
identification proceeds as follows:
We associate to the various terms in the Lagrangian a set of propagators and
vertex factors.
The propagators are determined by the terms quadratic in the fields, i.e. the
- ψ, etc.
terms in the Lagrangian containing for example φ2, ψ
The other terms in L are associated to interaction vertices.
Gianluigi Fogli
68
Let us start from the Lagrangian
- γµ∂µ ψ ‒ m ψ- ψ
L = iψ
The Euler-Lagrange equation is easily derived to be the Dirac equation of a free
particle.
The Lagrangian is clearly invariant under the phase transformation
ψ(x) → eiα ψ(x)
with α real constant
The family of phase transformations U(α) = eiα forms a unitary Abelian group
U(1)
This invariance, through the Noether theorem, implies the existence of a conserved
current and, at the integral level, of a conserved “generalized charge”, which acts
as the generator of the group U(1).
Gianluigi Fogli
69
By estimating δL and imposing the invariance in the form δL = 0, one easily obtain
∂ µj µ = 0
jµ(x) = ie
with
1
2
∂L
- γµ ψ
- ∂L
ψ–ψ
=
e
ψ
∂(∂µψ)
∂(∂µψ)
where the proportionality factor has been chosen in such a way to match up the
electromagnetic current of an electron of charge –e.
From the conservation of the current one easily derives that
Q=
∫ j0 (x) d3x
is a conserved quantity, the electromagnetic charge.
In conclusion, the existence of an invariance under this kind of transformation,
which is referred to as a “global phase transformation”, leads to a conserved
current and a conserved generalized charge.
Gianluigi Fogli
70
2. U(1) local gauge invariance and QED
A more subtle, and more general, invariance can be invoked by requiring invariance
even when α differs from space-time point to point, i.e. α = α(x), under the
transformation
ψ(x) → eiα(x) ψ(x)
This is a “local gauge invariance” requirement. However, it is easy to verify that
the Lagrangian is not invariant, because of the derivative term ∂µ ψ.
In order to insist in the requirement of invariance, we have to modify the
derivative term, going from the ordinary derivative to the so-called
∂µ → Dµ = ∂µ - ieAµ
covariant derivative
defined in terms of an arbitrary vector field Aµ(x), by requiring a well-defined
transformation property for Aµ(x)
Aµ → Aµ + 1e ∂µα(x)
Gianluigi Fogli
71
It is easily checked that
Dµψ → eiα(x) Dµψ
so that the Lagrangian is invariant if expressed in terms of the covariant
derivative.
But with the introduction of the covariant derivative the Lagrangian becomes
- ψ = -ψ (i γµ∂µ ‒ m ) ψ + e -ψγµψ Aµ
L = iψ γµDµ ψ ‒ m ψ
which is the Lagrangian of an electron in the presence of the electromagnetic field
Aµ: in other words, the Lagrangian of QED.
The result is to some extent rather surprising: the requirement of “local gauge
invariance” applied to the Lagrangian of a free particle leads to a Lagrangian which
describes the interaction of the particle with a vector field (gauge field) that we
can interpret as the electromagnetic field Aµ.
In other words, we can derive QED from the requirement of local gauge invariance.
Gianluigi Fogli
72
Regarding the new field Aµ as the physical photon field, we add to the Lagrangian a
term corresponding to the kinetic energy of the photon field. It is usually
expressed in terms of the gauge invariant field strenght tensor
Fµν = ∂µ Aν - ∂ν Aµ
Accordingly, the complete QED Lagrangian becomes
L= ψ (i γµ∂µ ‒ m ) ψ + e ψγµψ Aµ -
1
4
FµνFµν
It is important to observe that the addition of a mass term for Aµ
1
2
M2 AµAµ
is prohibited by gauge invariance because of the assumed transformation
properties of the field Aµ. The gauge vector boson must be massless.
Gianluigi Fogli
73
3. Non-Abelian gauge invariance and QCD
We attempt now to extend the requirement of local gauge invariance to non-Abelian
gauge groups. For example, we can take the SU(3) group of phase transformations
on the quark color fields. The free Lagrangian is (for a single quark q)
L0 = qi (i γµ∂µ ‒ m ) qi
(i = 1, 2, 3)
with q1, q2 , q3 the three color fields of a quark of given flavor.
The phase transformations are of the form
q(x) → U q(x) = eiα a (x)T a q(x)
(a = 1, …, 8)
with U arbitrary 3×3 unitary matrix. Ta are the eight generators of SU(3) in the
three-dimensional representation (the Gell Mann matrices λa/2). They satisfy the
commutation relations
[Ta, Tb] = i fabc Tc
Gianluigi Fogli
fabc structure functions of SU(3)
74
We can repeat the step of the Abelian case: by using infinitesimal phase
transformations we have
q(x) → [1+i αa(x)Ta] q(x)
∂µ q → (1+i αaTa) ∂µ q + i Ta q ∂µ αa
We introduce the (eight) gauge fields Gµa transforming according to
1
Gµa → Gµa - g ∂µ αa
(where g is the coupling)
and form the covariant derivative
Dµ = ∂µ + i g TaGµa
Then we make the replacement ∂µ → Dµ in the Lagrangian and obtain
- γµTa q) Gµa L0 → L = q (i γµ∂µ ‒ m ) q – g ( q
Gianluigi Fogli
75
But for non-Abelian gauge transformations this does not appear an invariant
Lagrangian, since
- γµ (TaTb – TbTa) q → ( -q γµTa q) + fabcαb ( -q γµTc q) ( q- γµTa q) → ( q γµTa q) + iαb q
as an effect of the commutator of the generators Ta.
In order to obtain gauge invariance, we have to re-write the transformation
properties of the gauge fields in the form
1
Gµa → Gµa - g ∂µ αa ‒ fabc ab Gµc
Adding to L the gauge invariant kinetic energy term of the gauge fields we have for
the QCD gauge invariant Lagrangian the final form
a
L= q (i γµ∂µ ‒ m ) q ‒ g ( q γµTa q) Gµa ‒ Gµν
Gµν
a with
a
a
a
b
c
Gµν = ∂µGν - ∂νGµ ‒ g fabcGµ Gν in order to satisfy the requirement of gauge invariance of the kinetic energy of the
gauge fields.
Gianluigi Fogli
76
The QCD Lagrangian describes the interaction between colored quarks q and vector
gluons Gµ with coupling g. Gauge invariance requires the eight gluons to be massless.
a
Note that the particular expression of the terms Gµν, due to the non-Abelian
character of the gauge group, introduces, together with the kinetic energy terms,
also a self-interaction effect between the gauge fields.
In a symbolic form we can distinguish
L = “qq” + “G2” + g ”qqG” + g ”G3” + g2 ”G4”
qa
qa
qa
gab
ab
gab
gab gab
qb
qa gc a
gc a
gc a
gacgc a
gac
gab
gbc
gbagab
gba
gab
qb g
bc
The first three terms have their analogue in QED. They describe the free
propagators of quarks and gluons and the quark-gluon interaction. The remaining
two terms correspond to three and four gluon vertices and reflect the fact that
gluons themselves carry color.
Gianluigi Fogli
77
4. Massive gauge bosons ?
The gauge bosons are required to be massless in gauge invariant theories. How can
we justify the existence of massive vector bosons, as they appear in weak and
electroweak interactions?
Of course, a possibility is to forget about gauge invariance and include the vector
boson mass terms as well.
However, the point is that the introduction or not of mass terms is not merely an
aesthetic problem. The introduction of mass terms not only spoils gauge invariance,
but also introduces unrenormalizable divergences that make the theory
meaningless.
This is essentially due to the more divergent behaviour of the propagators of
massive vector bosons. They are of the form
i
Gianluigi Fogli
- gµν +
Q2
-
qµ qν
M2
M2
qµ qν
~
q2 →
∞
q2M2
78
This behaviour for q2 → ∞ no longer prevents the loop integrals
∫ d4q (propagator) ….
from diverging for large loop momenta.
Even the introduction of a cut-off does not work, since the inspection of the
diagrams containing more loops shows that new, even more severe divergences
appear in each order, and ultimately an infinite number of unknown parameters has
to be introduced.
The point is: it is possible to introduce masses without breaking gauge invariance?
The answer is yes, it is possible through an intriguing procedure known as
Higgs mechanism
Gianluigi Fogli
79
5. Spontaneous Symmetry Breaking
Let us consider the following simple Lagrangian describing the self interaction of a
scalar particle associated to the field φ(x)
L=T–V=
1
2
∂µφ∂µφ ‒ ( 21 µ2φ2 + 41 λφ4)
with λ > 0
The symmetry here is a discrete reflection symmetry:
φ
-φ
We have two possible forms of the potential depending on the parameter µ2.
The case µ2 > 0 is rather familiar. It describes a scalar
field with mass µ in a confining potential.
V(q)
The ground state (“vacuum” in the quantum language)
corresponds to φ = 0.
The solution satisfies the reflection symmetry of the
Lagrangian.
Gianluigi Fogli
q
80
The case µ2 < 0 is more interesting. The Lagrangian has a mass term with a wrong
sign, moreover the potential has two minima. From
V(q)
∂V
= φ (µ2 + λφ4) = 0
∂φ
we see that they correspond to
φ=±v
with
v=
-
√
µ2
-v
v
q
λ
The extremum φ = 0 is not a minimum.
Perturbative calculations should involve expansion from a minimum of the potential,
either φ = v or φ = - v. We have to choose one of them. Note that this does not
implies a loss of generality, since the other minimum can be always reached through
a symmetry operation: the symmetry is not lost.
Let us choose φ0 = v and expand φ(x) around the minimum by writing
φ(x) = v + η(x)
Gianluigi Fogli
81
We perform the substitution of φ(x) in the Lagrangian (shift of the field) and
discuss the new Lagrangian in terms of the new field η(x), whose minimum
corresponds to η0 = 0. After a few calculations
L’=
1 ∂ η ∂ µη
2 µ
‒ λ v 2 η2 ‒ λ v η3 -
1
4
λ η4 + const
We see that the new field η(x) has a mass term of the right sign, with
mη = √ 2λv2 = √-2µ2
whereas the higher order terms in η(x) represent the interaction of the field η(x)
with itself.
What about L and L ’ ? The two Lagrangians describe the same physics. If we were
able to solve the two Lagrangians exactly, we would find the same results.
Gianluigi Fogli
82
But we are forced to use a perturbative approach and calculate the fluctuations
around the vacuum, the point where we have the minimum energy. Under this
profile, L ’ seems the right choice: we expand around a minimum and we can derive
the spectrum of states.
In particular the scalar particles under study when described by the Lagrangian L ’
have a well defined mass!
What about the reflection symmetry? Since the two Lagrangians are equivalent,
the symmetry is not lost. Whereas in L the symmetry is manifest, in L ’ it is hidden,
but still present. In particular we find exactly the same results independently of
the specific choice of the minimum: φ = v or φ = - v.
We say that a “spontaneous symmetry breaking” has occurred, which has
“generated” the mass of the particles.
Gianluigi Fogli
83
5. Spontaneous Breaking of a global gauge symmetry
We can adopt the previous approach in the case of a complex scalar field described
by the Lagrangian
L = (∂µφ)*(∂µφ) – µ2φ*φ – λ(φ*φ)2
with
φ=
1
√2
(φ1 ± i φ2)
invariant under the global symmetry
φ → eiα φ
Considering the case λ > 0 and µ2 < 0, we can re-write L in terms of φ1 and φ2
L=
1
2
(∂µφ1)2 +
1 ( φ )2
µ 2
2
∂
–
1 2 2
µ (φ1
2
+ φ22 ) – 41 λ(φ21 + φ22 )2
with the potential which is now a function of the two fields φ1 and φ2:
V = V(φ1, φ2)
Gianluigi Fogli
84
For µ2 < 0 there is now a circle of minima of V(φ) in the plane (φ1,φ2) of radius v, such
that
V(q)
2
µ
2 =2
2
2
v
φ 1 + φ2 = v
with
λ
q2
Without loss of generality we choose the point
φ1 = v
q1
φ2 = 0
j
d
as the minimum of V(φ) and expand L around the vacuum in terms of the fields η(x)
and ξ(x) through the substitution
φ(x) =
so obtaining
L’=
Gianluigi Fogli
1
2
∂ µξ
2
+
1
2
1
√2
[v + η(x) + i ξ(x)]
2
∂µη + µ 2η2 + const + cubic and quartic terms in η and ξ
85
The third term in L ‘ has the form of a mass term leading to
1
2
mη2η2 for the field η(x),
mη = √- 2µ2
The first term in L ‘ is the kinetic term of ξ(x), but there is no a corresponding mass
term for ξ(x). This is the result of the
Goldstone theorem
which states that massless scalar particles occur whenever a continuous symmetry
of a physical system is “spontaneously broken”.
In conclusion, the attempt of generating a massive gauge boson through the
spontaneous symmetry breaking of a global continuous symmetry leads to a theory
“plagued” by the presence of massless scalar particles to worry about.
Neverthless, let us proceed from a global to a local gauge theory. A miracle is about
to happen.
Gianluigi Fogli
86
5. The Higgs mechanism
We can now consider the spontaneous symmetry breaking of a “local gauge
symmetry”. The simplest case is the U(1) gauge symmetry
φ(x) → eiα(x) φ(x)
with
φ=
1
√2
(φ1 ± i φ2)
We introduce in the lagrangian
L = (∂µφ)*(∂µφ) – µ2φ*φ – λ(φ*φ)2
the covariant derivative
∂µ → Dµ = ∂µ – i e Aµ
with the gauge field Aµ transforming according to
Aµ → Aµ + 1e ∂µ α(x)
Gianluigi Fogli
87
The Lagrangian takes then the form
L = (∂µ+ i e Aµ) φ* (∂µ - i e Aµ) φ – µ2φ*φ – λ(φ*φ)2 -
1
4
FµνFµν
If µ2 > 0, then this is just the QED Lagrangian for a charged scalar particle of mass
m, with the addition of a φ4 self-interaction term.
But we take µ2 < 0, since we want to generate mass terms through the spontaneous
symmetry breaking mechanism. In this case we have to translate the field φ(x) to
the ground state. With the same substitution as before
φ(x) =
the Lagrangian becomes
L’=
1
2
∂ µξ
Gianluigi Fogli
2
+
1
2
1
√2
[v + η(x) + i ξ(x)]
mass term
2
∂ µ η - v 2 λ η2 +
1
2
mass term
e2v2AµAµ - e vAµ∂µξ -
strange off-diagonal term
1
F Fµν
4 µν
+ interaction terms
88
The particle spectrum in L ‘ contains
a massless Goldstone boson ξ(x)
mξ = 0
a massive scalar field η(x)
mη = √- 2µ2 =
a massive vector field Aµ(x)
mA = ev
√2λv2
We have obtained a massive vector field, but we have still the occurrence of a
massless Goldstone boson.
However, the occurrence of a strange term off-diagonal in the vector field induces
to be careful in the interpretation of L ’.
Indeed, giving mass to Aµ we have raised the number of degrees of freedom of the
system: the polarization degrees of freedom go from 2 to 3 (addition of the
longitudinal polarization of the massive vector boson) whereas the number of
degrees of freedom of the scalar fields seems to be the same.
Since we have one more degree of freedom in L ’, this means that the fields in L ’ not
all correspond to distinct physical particles.
Gianluigi Fogli
89
Which field in L ‘ is unphysical ? Can we make use of gauge invariance to find a
particular gauge transformation that eliminates one degree of freedom from the
Lagrangian ?
Let us note that at the lowest order in ξ(x)/v we can write
φ(x) =
1
√2
[v + η(x) + i ξ(x)] ≅
1
√2
[v + η(x)] e i ξ(x)/v
This suggests the use of a different specific set of fields in the original Lagrangian:
h(x) , θ(x) , Aµ(x)
assuming for φ(x) and Aµ(x) the following transformation properties
φ(x) →
1
√2
[v + h(x) + i θ(x)] ≅ √21 [v + h(x)] e i θ(x)/v
Aµ → Aµ + 1e ∂µ θ(x)
Gianluigi Fogli
90
This corresponds to a specific choice of the gauge, chosen in such a way to make the
field h(x) real. Gauge invariance, i.e. the arbitrariness in the choice of α(x), is lost
since we have identified the, in principle arbitrary, α(x) with the specific field θ (x)
which makes h (x) real.
The substitution in the original Lagrangian L leads to
L =
1
2
∂ µh
2
- λv 2h2 +
1 2 2
e v A µA µ
2
- λv h3 -
1
4
λh4 +
1
2
e2v2AµAµ h2 + e2vAµAµ h -
1
F Fµν
4 µν
The Goldstone boson has disappeared. The extra degree of freedom is indeed
spurious, it corresponds to the freedom of making a gauge transformation.
L describes two interacting massive particles, a massive scalar field h(x) and a
massive gauge vector boson Aµ(x).
The field h(x) is called “Higgs particle”. The unwanted massless Goldstone boson has
been turned into the longitudinal polarization of the massive vector boson. It is a
“would be” Goldstone boson and it has been “eated” by the vector boson.
Gianluigi Fogli
91
6. Spontaneous Symmetry Breaking of a local SU(2) gauge symmetry
We start from the Lagrangian
L = (∂µφ) (∂µφ) – µ2φ φ – λ(φ φ)2
where φ(x) is an SU(2) doublet of complex fields
φ(x) =
φα
φβ
=
1
√2
φ1 + i φ2
φ3 + i φ4
L is invariant under the global SU(2) phase transformations
φ → φ = ei α a τa/2 φ
But if we require invariance under a local phase transformation, then we have to
introduce the covariant derivative
τa a
∂µ → Dµ = ∂µ + i g Wµ
2
a
τa
in terms of a triplet of vector fields Wµ and the coupling constant g, the
being
2
the SU(2) generators in the doublet representation.
Gianluigi Fogli
92
Under infinitesimal local gauge transformations
φ(x) → φ’(x) = 1 + i αa(x)
and
τa
2
φ(x)
→
→
→ →
Wµ → Wµ – g1 ∂µ→
α – α × Wµ
last term due to the nonAbelian character of SU(2)
The Lagrangian then becomes
L=
∂ µφ + i g
→
1→
τ • Wµφ
2
∂ µφ + i g
→
1→
τ • Wµ φ
2
– V(φ) -
1 → →µν
Wµν•W
4
with the usual potential term
V(φ) = µ2φ φ – λ(φ φ)2
and the kinetic energy of the vector fields given by
→
→
→
→ →
Wµν = ∂µWν - ∂νWµ ‒ g Wµ × Wν
Gianluigi Fogli
last term due to the nonAbelian character of SU(2)
93
We are interested to the spontaneous symmetry breaking of L. Accordingly we
assume λ > 0 and µ2 < 0. The minimum of V(φ) is assumed in all the points that satisfy
φ φ=
1
2
2
2
2
2
( φ1 + φ2 + φ3 + φ4 ) = -
µ2
λ
We choose a specific minimum
2
φ3
φ1 = φ2 = φ4 = 0
µ2
== v2
λ
The choice introduces the spontaneous breaking of the SU(2) symmetry. We expand
around the specific vacuum
φ0 =
1
√2
0
v
Gauge invariance, with the specific choice of the gauge, allows to simply substitute
in L the expansion
φ(x) =
1
√2
0
v + h(x)
with only the Higgs field surviving.
Gianluigi Fogli
94
It may appear rather surprising, but, according with the approach used before in
the U(1) case, we can imagine to parametrize the fluctuations around φ0 in terms of
four real fields θ1(x), θ2(x), θ3(x) and h(x) using the expression
φ(x) =
1
√2
e
0
→→
i τ•θ(x)/v
v + h(x)
Let us expand the expression in terms of small perturbations: we obtain
φ(x) =
1
√2
1 + iθ3/v
i(θ1 - i θ2)/v
0
i(θ1 + i θ2)/v
1 - iθ3/v
v + h(x)
=
1
√2
θ2 + i θ1
v + h – iθ3
so fully parametrizing the deviation from the vacuum. On the other hand, the
Lagrangian is locally invariant: we can then “gauge” the three would-be Goldstone
bosons θ1(x), θ2(x), θ3(x), with the appropriate choice of the gauge. We arrive to the
previous form of the “shift”, which is so justified
φ(x) =
1
√2
0
v + h(x)
At this point, in L ‘ there remains only the Higgs field without any trace of the gauge
fields. They have been “gauged away” and disappear from the theory.
Gianluigi Fogli
95
The three gauge vector bosons are now massive. We can isolate the mass terms
ig
1
2
→→
τ•Wµφο
2
=
g2
8
(3)
Wµ
(1)
Wµ
(1)
(2)
(2)
Wµ – i Wµ
-
+ i Wµ
(3)
Wµ
0
v
2
=
g2v2
8
∑ (Wµ(i) ) 2
i
and find their common mass to be
MW =
1
2
gv
In conclusion, we have a theory characterized by one massive scalar field h(x) and
three massive gauge fields. The Goldstone bosons have been “eated” by the gauge
fields when they become massive.
The choice of the scalar field representation is crucial. If φ(x) instead than a
complex doublet is chosen to be a SU(2) triplet of real scalar fields, then, always
for λ > 0 and µ2 < 0, we find that only two gauge bosons acquire a mass, whereas the
third remain massless.
Gianluigi Fogli
96
6. The Weinberg-Salam Standard Model
Gianluigi Fogli
97
1. Revisiting electroweak interactions
Electromagnetic amplitudes have been described as due to the interaction
- µQ ψ) Aµ
- i e jµ Aµ = - i e (ψγ
em
Q generator of U(1)em
We have seen that the interaction derives from demanding gauge invariance of the
Lagrangian of a free fermion
- (iγ ∂µ ‒ m) ψ
L=ψ
µ
under U(1)em local gauge transformations
ψ → ψ’= eia(x)Q ψ
We then obtain the Lagrangian
- (iγ ∂µ ‒ m) ψ ‒ e (ψ- γ Q ψ) Aµ L=ψ
µ
µ
kinetic energy and mass of ψ
Gianluigi Fogli
1
4
interaction
FµνFµν
kinetic energy of Aµ
98
In order to include electroweak interactions, we have to introduce the interactions
due to SU(2)L and U(1)Y. The first through the coupling of the isotriplet of lefthanded weak currents Jµ(i) with a triplet of vector bosons Wµ(i)
- i g Jµ•Wµ = - i g χLγµT•Wµ χL
→
→ →
→
The second interaction with the hypercharge current coupled to a vector boson Bµ
- i g’
1 Y µ
j B =
2 µ
- γ 1 Y ψ Bµ
-i g’ ψ
µ
2
The transformation properties of left-handed and right-handed fields are then
→
→
χL → χL’ = eiα(x)•T + iβ(x)Y χL
ψR → ψR’ = eiβ(x)Y ψL
For example, in the case of the (νe e-) lepton pair we have
νe
eψR = eR
χL =
Gianluigi Fogli
L
isospin doublet with T = 1 , Y = - 1 2
isospin singlet with T = 0 , Y = - 2 An Introduction to the Standard Model, Canfranc, July 2013
99
The electromagnetic interaction is embedded in SU(2)L ⊗ U(1)Y being
Q = T(3) +
1
Y
2
jµem = Jµ(3) +
and then
1 Y
j
2 µ
Accordingly, the neutral current of the SU(2)L ⊗ U(1)Y interaction can be rewritten
by introducing Wµ(3) and Bµ in terms of Aµ and Zµ through the mixing angle θw
- i g Jµ(3) Wµ(3) - i g’
1 Y µ
j B
2 µ
= - i ( g sinθw Jµ(3) +
- i ( g cosθw Jµ(3) = - i e jµem Aµ - i
1
g’ cosθw jµY ) Aµ +
2
1
g’ sinθw jµY ) Zµ =
2
g
JµNC
cosθw
Zµ
so obtaining
JµNC = Jµ(3) – sin2θw jµem
Gianluigi Fogli
and
100
From the requirement of gauge invariance under SU(2)L ⊗ U(1)Y it is possible to
derive the electroweak gauge invariant lagrangian. Always in the case of the (νe e-)
lepton pair, we have
L1 = χLγµ i ∂µ - g 21 →
τ•Wµ - g (→
1
)
2
Bµ χL + eRγµ i ∂µ - g (- 1) Bµ eR -
→ →
1 W Wµν
µν
4
-
1
B Bµν
4 µν
where the values of the hypercharge have been explicitly inserted. The last two
terms correspond to kinetic energy and self-coupling of the Wµ(i) fields and to the
kinetic energy of the Bµ field, respectively.
L neutral. Gauge invariance requires the Lagrangian to be neutral, i.e. it must
transform as a singlet under U(1)em in order to imply charge conservation.
Fermion masses. L1 describes massless gauge bosons. Mass terms for the gauge
bosons are not gauge invariant and cannot be added. The same is true also for
the fermions fields: a mass term, for example for the electron, is given by
- e = - m e- me e
e
1
2
(1 – γ5) +
1
(1
2
- e + -e e )
+ γ5) e = - me (e
R L
L R
But this term breaks gauge invariance since eL is the member of an isospin
doublet and eR is an isospin singlet. The fermion masses will be introduced by
the spontaneous symmetry breaking of the theory.
Gianluigi Fogli
101
2. The Higgs field
We want to introduce the Higgs mechanism in such a way to have W± and Z0 massive,
whereas the photon Aµ remains massless.
This is obtained adding to L1 a second SU(2)L ⊗ U(1)Y invariant Lagrangian containing
the scalar fields necessary to induce the Higgs mechanism itself:
L2 =
→→
i ∂µ ‒ g T•Wµ - g
1
2
YBµ φ
2
- V(φ)
Minimal Standard Model
(Weinberg 1967)
where
φ(x) =
φ+
φ0
=
1
√2
φ1 + i φ2
φ3 + i φ4
is an isospin doublet with weak hypercharge Y = 1.
Gianluigi Fogli
102
V(φ) is the usual potential. We assume λ > 0 and µ2 < 0 choosing the vacuum
expectation value
φ0 =
1
√2
0
v
with
T=
1
2
, Y=1 , Q=0
which breaks T(3) and Y, but leaves Q unbroken:
Q φ0 = 0
φ0 → φ0 = eiα(x) Q φ0 = φ0
Since the vacuum φ0 is still left invariant by some subgroup of the gauge group, then
the gauge bosons associated with this subgroup will remain massless.
With the previous choice, this is indeed the case of U(1)em with the photon expected
massless after the occurrence of the spontaneous symmetry breaking.
The other three vector bosons associated to the remaining three generators of
SU (2)L ⊗ U(1)Y will become massive and will be identified with W± and Z0 .
Gianluigi Fogli
103
2. Masses of the gauge bosons
The gauge boson masses come from L2 when the usual shift is applied
φ(x) →
1
√2
0
= φ0 +
v + h(x)
1
√2
0
h(x)
From the terms proportional to φ0 we obtain
(3)
gWµ + g’ Bµ
→
2
1
ig 1 →
τ•Wµ - i g’ 1 Bµ φ0 =
2
2
8
=
=
Gianluigi Fogli
(2)
(Wµ(1)) 2 + (Wµ )2
1 v2g2
2
1
2
vg
(1)
g (Wµ +
2
(-)
(+)
Wµ W µ
+
(2)
i Wµ )
(1)
(2)
g (Wµ - i Wµ )
-
(3)
gWµ +
g’ Bµ
0
v
2
=
+ v2 (g’Bµ – gWµ(3) ) (g’Bµ – gW (3) µ) =
1 2
(3)
v
(
W
µ
2
Bµ )
g2 - gg’
- gg’
g’2
Bµ
W (3) µ
104
By comparing the first term with the expectation for the mass of a charged vector
boson M2 W (+)W µ(-) we find
W µ
MW = 1 v g
2
The second term can be rewritten
1
8
v2 g2 Wµ(3) Wµ(3) - 2 g g’ Wµ(3)Bµ + g’2 BµBµ =
1 2
v
8
(3)
gWµ - g’ Bµ
2
(3)
+ 0 g’ Wµ + g Bµ
2
which shows that the eigenvalue of the combination (g’ Wµ + g Bµ) is zero. The
orthogonal combination has eigenvalue different from zero. We identify the first
combination as the massive Zµ and the orthogonal one as the photon, with masses
1 M 2 Z 2 + 1 M 2 A 2
Z
µ
A µ
2
2
Normalizing the fields
Aµ =
Zµ =
Gianluigi Fogli
g’ Wµ(3) + g Bµ
√
g2
+
g’2
g Wµ(3) - g’ Bµ
√g2 + g’2
with
MA = 0
with
MZ =
1
v
2
√g2 + g’2
105
By taking into account that
i.e.
tg θw =
g’
g
we find
Aµ = Bµ cosθw + Wµ(3) sinθw
Zµ = - Bµ sinθw + Wµ(3) cosθw
and
MW = cosθw MZ
The two vector boson masses are different, and this appears an effect of the
mixing.
Note that the previous relation has been obtained in the Minimal Standard Model
(one Higgs doublet) and is specific of the Minimal Standard Model. Different
choices of the Higgs sector lead in general to different relations, even though the
existence of more than one Higgs doublet leads to the same numerical results.
Gianluigi Fogli
106
Making use of the relation which connects the vector boson masses to the low
energy Fermi coupling constant
G
√2
=
g2
8M2w
and comparing the the result obtained for MW, MW =
1
2v2
=
g2
8M2w
=
G
√2
1
2
v g, one derives
Making use of the experimental value of the Fermi coupling constant one obtains for
the vev v
v = 246 GeV
Similarly, making use of the value at low energy of θw one can predict the vector
boson masses
MW =
37.3
sinθw
GeV
MW =
74.6
sin2θw
GeV
These relations are well verified experimentally. All the phenomenology strongly
supports the Standard Model in its minimal version (even though more Higgs
doublets leads to the same relations).
Gianluigi Fogli
107
3. The parameter ρ
We have just established the relation between MZ and MW, that we can rewrite in
the form
MW
= cosθw
MZ
Let us remind that the parameter ρ, introduced to specify the relative strenght of
the neutral and charged current, is expressed in terms of the vector boson
masses. Accordingly, for the Minimal Standard Model
ρ
=
M2w
M2z cos2θw
=
1
whereas for a general Higgs contributions
ρ
Gianluigi Fogli
=
M2w
M2z cos2θw
=
∑ v i2
Ti(Ti+1) -
∑
1
2
1 Y2
2 i
v i2 Y i2
vi , Ti , Yi are vev, weak isospin
and weak hypercharge of the
generic Higgs representation
108
4. Masses for the fermions
A direct fermion mass term – m ψψ
cannot be assumed since it breaks gauge
invariance. But terms corresponding to fermion masses come from the piece of the
Lagrangian L3 which couples fermions to the Higgs sector.
Assuming the minimal version of the Standard Model, i.e. only one Higgs doublet, we
have for the fermion doublet (νe e-) the following gauge invariant couplings
L3 = - Ge
-e e- ) L
(ν
φ+
φ0
eR +
-R (φ - -φ0)
e
νe
e
L
The spontaneous symmetry breaking, introduced through the usual shift
φ(x) →
1
√2
0
v + h(x)
= φ0 +
1
√2
0
h(x)
transforms L3 into
L3 = Gianluigi Fogli
1
Gv
√2 e
- LeR + e-ReL) (e
-
1
G (e LeR +
√2 e
-eReL) h
109
i.e.
-e L3 = - me e
me e e h
v
Since v = 246 GeV, the coupling of the
electron to the Higgs particle is very
small.
with
me =
1
Gv
√2 e
eR (T= 0 , Y= -2)
eL (T=
he+e- vertex factor
-i
h0 (T=
1
, Y= 1)
2
Ge
√2
1
, Y= -1)
2
= -i
g
2
me
MW
Quark masses are generated in the same way. In order to give mass also to the
upper member of the doublet (the u quark) we have to introduce the Higgs doublet
conjugate to φ
- φ0
v + h(x)
1
φc = -i τ2 φ* =
√2
φ
0
after SSB
which transforms as φ under SU(2), but has opposite weak hypercharge.
Gianluigi Fogli
110
We can then construct the following Lagrangian
L4 =
- Gd ( u d) L
-
φ+
- - φ0
d - Gu ( u d)L
u + h.c. φ- R
φ0
R
which can be rewritten in the form:
-
L4 = - md d d – mu
-u u -
md mu ddh uuh
v
v
Of course, the formalism can be extended to the case of more generations of
quarks and leptons. In the case of quarks, because of the mixing, the Yukawa
couplings take a matrix form.
It is however interesting to observe that the Higgs coupling to the different
generations is always flavor conserving because of the GIM mechanism. In
conclusion, no flavor changing neutral currents (FCNC) appear in the theory.
Gianluigi Fogli
111
4. The Higgs mass
The Higgs mass cannot be predicted from the theory. Starting from the potential
and considering it as an effective potential
V(φ) = µ2φ φ – λ(φ φ)2 + ….
the use of the first two terms of it leads to
mh2 = 2v2λ
Since v is fixed, large values of mh corresponds to large value of λ. A meaningful
perturbative approach requires λ < 1 and then the Higgs mass cannot larger than a
few hundred GeV.
On the other hand, mh cannot be smaller than, say, ~10 GeV, otherwise radiative
corrections would wash out the minimum at v ≠ 0.
It is matter of facts that the Higgs mass has been calculated through its quantum
effects well before its recent measurement at LHC.
Gianluigi Fogli
112
7. Renormalization and running coupling constants
Gianluigi Fogli
113
1. A loop diagram and the photon propagator
Let us consider the Feynman graph for the Rutherford scattering of an electron
from a static charge Ze (a nucleus, typically).
The same procedure adopted in the study of the scattering
e-µ- → e-µ- leads to the invariant amplitude
Jµ
e-
ui
-iM = (ie uf γµ ui)
-i
gµν
→
(i
j
(
ν q ))
q2
-i
i.e.
e-
ieγµ
gµν
uf
q2
•
-i(Ze,0)
-i
(- i Ze)
-iM = (ie uf γ0 ui)
q2
being for a static charge
j0( x ) = ρ( x ) = Ze δ( x )
→
Gianluigi Fogli
→
→
and
→→
j( x ) = 0
114
Let us now introduce a loop along the photon propagator. We can apply the
Feynman rules and obtain
(-1)n for a diagram with n fermion loops
Jµ
e-
ui
e-
ieγµ
gµρ
-i 2
q
uf
q
-iM = (-1) (ie uf γµ ui)
•
∫ (2π)
•
-i
ieγρ
q-p
p
ieγσ
gσν
-i 2
q
•→
-i jν( q )
Gianluigi Fogli
d 4p
4
gµρ
q2
-i
(ieγρ)αβ
•
i(p + m)βλ
p2 –
m2
(ieγσ)λτ
i(q - p + m)τα
(q-p)2 – m2
•
gσν
→
(
i
j
(
q
))
ν
q2
where p is the four-momentum circulating around the
loop. Since p is not observable, we have to sum over all
possible values of p.
115
Comparing with the lowest order, the effect of the loop can be regarded as a
modification of the propagator: we can write
-i
gµν
q2
-i
gµν
q2
+
-i
gµρ
ρσ
gσν
I
-i
q2
q2
-i
gµν
q2
(-i)
(-i)
+ q2 Iµν
q2
where
Iµν
(q2)
=
(-1)1
d 4p
∫ (2π)
Tr (ieγρ)
4
i(p + m)
p2 – m2
(ieγσ)
i(q - p + m)
(q-p)2 – m2
Symbolically
-
Gianluigi Fogli
116
In the integral we have terms that diverge as p → ∞. It can be shown that Iµν can
be written (after a lenghty calculation)
Iµν = -i gµν q2 I(q2) + (terms ∼ qµqν)
these terms vanish when coupled
with external currents because
of charge conservation
with
I(q2)
= α
3π
∫
∞
m2
dp2
p2
-
2α
π
∫
0
1
q2z(1-z)
dz z (1-z) ln 1 m2
where m is the electron mass.
In I(q2):
The first term is divergent, but only logarithmically because of the
“conspiracy” of the divergent terms, with cancellation of the terms
quadratically divergent.
The second term is a finite part, properly parametrized.
Gianluigi Fogli
117
In the following we will be interested to the contribution of Iµν only for small
values and large values of (-q2). So, with M introduced as a cut-off to “cure” the
divergence, we have
for
(-q2)
for
(-q2)
small
I(q2)
2
α
q
M2
α
=
ln
+
15π
m2
m2
3π
large
I(q2)
-q2
α
M2
α
=
ln
ln
3π
m2
m2
3π
=
α
M2
ln
3π
-q2
Apparently, the divergence spoils of any meaning the result. But let us consider
the invariant amplitude including the loop contribution in the small (-q2) limit
-iM =
- γ0 u)
(ie u
(-i)
q2
2
α
1ln M2
3π
m 2
α
q
+ O (e4) (-iZe)
2
15π
m We can re-write it in the form
-iM =
Gianluigi Fogli
- γ0 u)
(ieR u
(-i)
q2
1-
eR2
60 π2
q2
(-iZeR)
m2
118
if we assume
eR = e 1 -
e2
12π2
ln
M2
1
2
m2
It is easy verified that the two expressions of the invariant amplitude are
equivalent at the O(e2).
We assume eR as the physically measured electric charge in any long range Coulomb
experiment as the Thomson scattering or the Rutherford scattering
γ
γ
e
e
Thomson scattering
So, interpreting
e
e
• Z√α
Rutherford scattering
eR2
α = 4π
we introduce a measurable parameter into the play, in terms of which the invariant
amplitude is finite.
Gianluigi Fogli
119
2. The g – 2
Can we verify that this approach is meaningfull and leads to a physically observable
effect?
The answer is yes. There are several examples. One of the most significant concerns
the magnetic moment of the electron, the so-called g-2.
Coming back to the Gordon decomposition
uf γ µ ui =
1
uf (pf + pi)µ + iσµν(pf - pi)ν ui 2m
with
σµν = i (γµγν – γνγµ)
2
it is possible to verify that the σµνqν term represents the contribution of the
magnetic moment of the electron
→
→
→
→
e →
µ= - e σ
often written as
µ =-g
S
with S = 1 →
σ
2
2m
2m
g is called gyromagnetic ratio, it satisfy
g=2
Gianluigi Fogli
120
If we now consider the correction to the vertex related to the
loop diagram in the figure, it is possible to verify that the
diagram modifies the structure of the electromagnetic current
in the following form
- e u-fγµui
→
-f γµ
-eu
q2
m 3
α
1+
ln
mγ
8
3π
m2
-
α
1
i σµνqν
2π
2m
ui
This implies a contribution of order α to the gyromagnetic ratio coming from the
second term: i.e.
→
α
α
→
µ= - e 1+
σ
g=2+
π
2m
2π
We can now calculate higher order contributions to g – 2: we find
g–2
2
theor
g–2
2
exper
=
1
2
α
π
- 0.32848 α
π
2
+ (1.49 ± 0.2) α
π
3
+ … = (1159655.4 ± 3.3) × 10-9
= (1159657.7 ± 3.5) × 10-9
which proves relativistic quantum field theory at the level of quantum effects.
Gianluigi Fogli
121
3. Renormalization in QED
Renormalization is a rather complex procedure and cannot be developed here. The
various diagrams lead to the redefinition of charge, mass and wave function of the
electron, by re-absorbing the divergent parts of the diagrams.
The radiative corrections come from the finite terms and can be tested
experimentally.
In particular the loop diagrams in the photon propagator, known as
vacuum polarization
modify the charge of the electron. We can include the loops at higher order in the
form
e
e0
1 -
=
e
Gianluigi Fogli
2
+
…
e0
122
which corresponds to the expansion
e2 = e20
1 – I(q2 ) + O(e40 )
The geometric series can be summed up
e
e0
1
=
e
e0
1 +
and we can write
e2 = e20
Gianluigi Fogli
1
1 + I(q2 )
123
This shows that we can introduce a renormalized coupling constant α(Q2) = e2(Q2)
for different values of Q2 = - q2, where Q2 is the physical value of the momentum
transfer of the process we are considering. The coupling behaves as a
1
α(Q2) = α0
1+
running coupling constant
I(q2 )
In the large Q2 = - q2 limit we can use the expression obtained for I(q2) and write
α0
2
α(Q ) =
α0
Q2
1ln
3π
M2
There remains the dependence on the cut-off M2. But it can be eliminated by taking
a renormalization momentum µ as reference and subtracting α(µ2) from α(Q2):
α(Q2)
=
α(µ2)
α(µ2)
1ln
3π
Q2
µ2
The renormalization procedure is completed: only physical measurable quantities
appear.
Gianluigi Fogli
124
We interpret the result of a running coupling constant a(Q2) as the effect of the
distance at which we measure the charge of the electron.
The presence of the pairs e+e- in the perturbative expansion of the electron
propagator gives rise to a electromagnetic screening when we “measure” the electric
charge of the electron with a test charge.
-
+
+
R
-
- +
-
+
+
test charge
+
Therefore, the closer one approaches the
electron, the larger is the charge one
measures. One expects a behaviour of the
Coulomb charge as that shown in the figure.
electron
charge
high energy
probe
low energy
probe
α = Gianluigi Fogli
1
137
R
125
3. Running coupling constant in QCD
A similar approach can be applied in QCD, but with very different results. This
depends on the effects of the additional graphs that contain the self-interaction
effects of gluons.
The running coupling constant as(Q2) for QCD is characterized by a coefficient
αs(µ2)
4π
2
n –5
3 f
+ 16
to be compared with
the QED coefficient
α(µ2)
4π
- 43
The first term is essentially the same: in the QED case we have the photon which
- to be
can fluctuate in the pair e+e-, in QCD the gluon can fluctuate in the pair qq,
multiplied for the number of flavors. Moreover, there is a factor 2 in the definition
of the two coupling constants.
The factor – 5 comes from the fermion loops of transverse gluons. It can be shown
that these terms contribute always with a negative sign.
Finally, we have a factor + 16, related to the introduction of “ghost” particles that
cancel the unphysical polarizations but do not lead to the production of physical
particles, and then contribute with a positive term.
Gianluigi Fogli
126
In conclusion, the QCD “running coupling constant” is given by
αs(Q2)
=
αs(µ2)
αs(µ2)
1+
33 – 2nf ln
12π
Q2
µ2
The coefficient is positive. It needs to have more than 16 quark flavors to change
the sign and then the behaviour of as(Q2), which decreases with increasing Q2 and
becomes small for short-distance interactions. We refer to this property as
asymptotic freedom
Conversely, at sufficiently low Q2 the coupling becomes large, i.e. of order O(1). It is
usual to denote the value of Q2 at which this occurs as Λ2. It follows that
Λ2 = µ2 exp
Gianluigi Fogli
- 12π
(33 - 2nf) as(µ2)
127
It follows that αs(Q2) can be written in terms of Λ2 in the form
αs(Q2)
=
12π
33 – 2nf ln
Q2
Λ2
For Q2 much larger than Λ2, αs(Q2) is small and a perturbative description of quarks
and gluons is possible. This is the regime of perturbative QCD.
For Q2 of order Λ2, the perturbative approach does not make sense: quarks and
gluons arrange themselves into hadrons. This is the regime of confinement.
color
charge
high energy
probe
confinement
region
αs = 1
1 Fermi
R
Gianluigi Fogli
128
4. Grand Unification
Is it possible to go beyond the Standard Model searching for an unifying group
G ⊂ SU(3)c ⊗ SU(2)L ⊗ U(1)Y
??
The dependence on Q of the three couplings,
g , g’ , gs
(where we assume αs = g2s /4π) seems to agree with the possibility that for some
large-momentum (or short-distance) scale Q = MX the three couplings merge into a
single coupling gG, so that the group G describes a unified interactions with coupling
gG(Q)
at
Q = MX
Indeed, the two non-Abelian groups are asymptotically free, whereas the coupling of
the Abelian group increases with Q, so suggesting a convergence towards a common
value.
Gianluigi Fogli
129
Let us identify the couplings according to
running coupling constants
g1(Q) = C g’(Q)
α3
g2(Q) = g(Q)
and
g3(Q) = gs(Q)
gi2
αi =
4π
α2
α1
MX
Z mass
where C is a Clebsh-Gordan coefficient of
the group G.
102
Energy Scale (GeV)
1015
It is convenient to make use of the quantities 1/αi = 1/g2i since they depend linearly
on ln Q. For example, in the case of g3 = gs we obtain, by rewriting the relation
previously found for αs,
1
g32(µ)
Gianluigi Fogli
=
1
g32(Q)
+ 2b3 ln
Q
µ
with
b3 =
1
4π2
2
3
nf – 11
130
Similar expressions can be found for the other two couplings, so that, by identifying
Q with MX and gi(MX) = gG, we can write
1
gi2 (µ)
=
1
gG2
+ 2bi ln
MX
µ
with
b1 =
1
4π2
2n
3 f
b2 =
1
4π2
22
3
-
+ b1
b3 =
1
4π2
– 11
+ b1
A straightforward calculation allows to eliminate nf and gG from the three equations.
We have M
6π
ln X =
µ
11(1 + 3C2)
1
α
-
1 + C2
αs
with MX expressed in terms of α and αs estimated at the momentum of reference µ,
C being dependent on the unification group G.
Gianluigi Fogli
131
Assuming µ = 10 GeV, we can use the actual values α = 1/137 and αs = 0.1. Concerning
C, it can be taken C2 = 5/3, the value relative to the unification group SU(5). We
obtain
MX = 5 × 1014 GeV
with a weak dependence on the different parameters.
The Weinberg angle is determined in a grand unified theory, since it is given by
sin2 θw =
g12(Q)
g12(Q) + C2g22 (Q)
It follows sin2θw = 3/8 at Q = MX.
On the other hand, it is possible to derive, with a simple calculation, the value of
sin2θw at Q = 10 GeV, to be compared with its actual value. We have
sin2 θW =
1
1 + 3C2
1 + 2C2
α
αs
which gives sin2θw = 0.2, close to the experimental value.
Gianluigi Fogli
132
4. SU(5) and proton decay
The simplest grand unification group is SU(5), proposed by Georgi and Glashow in
1974. SU(5) can accomodate all the known fermions (leptons and quarks) in two distinct
irreducible representations of the group, according to -
(1 , 1) + (3 , 1) + (3 , 2)
-
5 = (1 , 2) + (3 , 1) = (νe , e-)L + dL
10 =
= e+L + uL + (u ,d)L
in terms of IR’s of
SU(3)c ⊗ SU(2)L
On the other hand, in SU(5) there are 24 vector bosons. We distinguish
-
24 = (8 + 1) + (1 , 3) + (1 , 1) + (3 , 2) + (3 , 2)
gluons
W±, Z0, γ
leptoquarks X, Y
The leptoquarks X and Y are two colored heavy gauge bosons, SU(2)L doublets. They
mediate interactions that turn quarks into leptons and viceversa, with violation of
leptonic and baryonic number. Gianluigi Fogli
133
In particular leptoquarks are expected to mediate proton decay. We can estimate
the decay rate, by comparing the process to the µ-decay: from
G
√2
=
g2
νµ
8M2w
it follows
Γ(µ- → e-νeνµ) = …. G2mµ5 = …. mµ5
4
MW
g
µ-
√2
W±
e-
g
ν
e
√2
In a similar way we expect
GG
√2
=
gG2
u
2
8MX
so that we can estimate
2
5
Γ(p → πe-) = …. GGmp = ….
Gianluigi Fogli
gG
X
p
mp5
MX4
gG
e+
-
u
d
d
d
π0
134
We can compare in the range of values of αs going from 0.1 to 0.2 the corresponding
values of MX, sin2θw, and τp. We find it difficult to reconcile the results: either
sin2θw is too small, or it is the proton which decays too fast.
as
MX (GeV)
sin2θw
τp (years)
0.1
5 × 1014
0.21
~ 1027
0.2
2 × 1016
0.19
~ 1034
However, an accurate prediction of the proton lifetime requires a more
sophisticated calculation. The measurement of the proton lifetime would be an important experimental result
in favor of GUT’s, since this process is allowed in any theory of grand unification. As said before, there are other, larger, groups that are even better candidates for
the unification. In particular, they have the possibility of including also right handed
neutrinos. But it is clear that we need more experimental information to say
something of more convincing on grand unification. Gianluigi Fogli
135
5. The quantization of the electric charge
This is one of the more interesting results related to GUT’s, in particular to SU(5). Indeed, within SU(5) the photon is one of the gauge bosons of the group and, as a
consequence, the electric charge Q is one of the generators. Since the group is a
simple group, the trace of each generator, and then also of the electric charge Q, is
zero for any representation of the group.
-
For the representation 5 this means that Tr Q = 3Q d- + Qν + Qe- = 0
Qd =
1
3
Q e-
an amazing results, since implies that charge is quantized.
A similar calculation for the representation 10 leads to
Qu = -2Qd
The combination of the two results solves one of the most intriguing mistery of
particle physics: why it is
Qp = - Q eGianluigi Fogli
136

An Introduction to the Standard Model

Transcription

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