Interactions in Modern Particle Theory - Weak Interactions

188 Pages·2010·1.316 MB·English

by  H. Georgi

Howard Georgi
Weak Interactions
2010
Contents
1 — Classical Symmetries 2
1.1 Noether’s Theorem – Classical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1a — Quantum Field Theory 11
1a.1 Local Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1a.2 Composite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1b — Gauge Symmetries 21
1b.1 Noether’s Theorem – Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1b.2 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1b.3 Global Symmetries of Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 — Weinberg’s Model of Leptons 29
2.1 The electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 SU(2)×U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Renormalizability? An Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Spontaneous Symmetry Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 The Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 The σ-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.8 Neutral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9 e+e− → µ+µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 — Quarks and QCD 49
3.1 Color SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 A Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Quark Doublets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 GIM and Charm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 The Standard Six–Quark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 — SU(3) and Light Hadron Semileptonic Decays 63
4.1 Weak Decays of Light Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Isospin and the Determination of V . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ud
4.3 f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
π
4.4 Strangeness Changing Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 PT Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
i
ii
4.6 Second Class Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 The Goldberger-Treiman Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.8 SU(3)−D and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 — Chiral Lagrangians — Goldstone Bosons 76
5.1 SU(3)×SU(3) → SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Effective Low–Momentum Field Theories . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Symmetry breaking and light quark masses . . . . . . . . . . . . . . . . . . . . . . . 81
5.5 What Happened to the Axial U(1)? . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Light Quark Mass Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 The Chiral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8 Semileptonic K Decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.9 The Chiral Symmetry–Breaking Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.10 Important Loop Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.11 Nonleptonic K Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 — Chiral Lagrangians — Matter Fields 93
6.1 How Do the Baryons Transform? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 A More Elegant Transformation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Nonlinear Chiral Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Successes of the Nonrelativistic Quark Model . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Hyperon Nonleptonic Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6a - Anomalies 105
6a.1 Electromagnetic Interactions and π0 → 2γ . . . . . . . . . . . . . . . . . . . . . . . . 105
6a.2 The Steinberger Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6a.3 Spectators, gauge invariance and the anomaly . . . . . . . . . . . . . . . . . . . . . . 111
7 — The Parton Model 123
7.1 Mode Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Heavy Quark Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Deep Inelastic Lepton-Hadron Scattering. . . . . . . . . . . . . . . . . . . . . . . . . 127
7.4 Neutrino-Hadron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 Neutral Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.6 The SLAC Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8 — Standard Model Precision Tests 134
8.1 Choosing a Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Effective Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3 The Symmetries of Strong and Electroweak Interactions . . . . . . . . . . . . . . . . 140
8.4 The ρ Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.5 M and M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
W Z
8.6 Neutrino-hadron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.7 Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
1
9 — Nonleptonic Weak Interactions 150
9.1 Why We Can’t Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.2 The Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.3 Charm Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.4 Penguins and the ∆I = 1 Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2
10 — The Neutral K Mesons and CP Violation 161
10.1 K0−K0 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.2 The Box Diagram and the QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . 163
10.3 The Gilman-Wise ∆S = 2 Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4 CP Violation and the Parameter (cid:15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5 K → ππ and the Parameter (cid:15)(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
L
A Review of Dimensional Regularization 175
A.1 n Dimensional Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B Background Field Gauge 179
B.1 The β function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Chapter 1
— Classical Symmetries
The concept of symmetry will play a crucial role in nearly all aspects of our discussion of weak
interactions. At the level of the dynamics, the fundamental interactions (or at least that subset of
the fundamental interactions that we understand) are associated with “gauge symmetries”. But
more than that, the underlying mathematical language of relativistic quantum mechanics — quan-
tum field theory — is much easier to understand if you make use of all the symmetry information
that is available. In this course, we will make extensive use of symmetry as a mathematical tool to
help us understand the physics. In particular, we make use of the language of representations of
Lie algebras.
1.1 Noether’s Theorem – Classical
At the classical level, symmetries of an action which is an integral of a local Lagrangian density are
associated with conserved currents. Consider a set of fields, φ (x) where j = 1 to N, and an action
j
(cid:90)
S[φ] = L(φ(x),∂ φ(x))d4x (1.1.1)
µ
where L is the local Lagrangian density. The index, j, is what particle physicists call a “flavor”
index. Different values of j label different types, or “flavors”, of the field φ. Think of the field,
φ, without any explicit index, as a column vector in flavor space. Assume, for simplicity, that the
Lagrangian depends only on the fields, φ, and their first derivatives, ∂ φ. The equations of motion
µ
are
δL δL
∂ = . (1.1.2)
µ
δ(∂ φ) δφ
µ
Note that (1.1.2) is a vector equation in flavor space. Each side is a row vector, carrying the flavor
index, j.
A symmetry of the action is some infinitesimal change in the fields, δφ, such that
S[φ+δφ] = S[φ], (1.1.3)
or
L(φ+δφ,∂ φ+δ∂ φ) = L(φ,∂ φ)+∂ Vµ(φ,∂ φ,δφ), (1.1.4)
µ µ µ µ µ
where Vµ is some vector function of the order of the infinitesimal, δφ. We assume here that we can
throw away surface terms in the d4x integral so that the Vµ terms makes no contribution to the
action. But
δL δL
L(φ+δφ,∂ φ+δ∂ φ)−L(φ,∂ φ) = δφ+ ∂ δφ, (1.1.5)
µ µ µ µ
δφ δ(∂ φ)
µ
2
Weak Interactions — Howard Georgi — draft - March 25, 2010 — 3
because δ∂ φ = ∂ δφ. Note that (1.1.5) is a single equation with no j index. The terms on the
µ µ
right hand side involve a matrix multiplication in flavor space of a row vector on the left with a
column vector on the right. From (1.1.2), (1.1.4) and (1.1.5), we have
∂ Nµ = 0, (1.1.6)
µ
where
δL
Nµ = δφ−Vµ. (1.1.7)
δ(∂ φ)
µ
Often, we will be interested in symmetries that are symmetries of the Lagrangian, not just
the action, in which case Vµ = 0. In particular, our favorite symmetry will be linear unitary
transformations on the fields, for which
δφ = i(cid:15) Taφ, (1.1.8)
a
where the Ta for a = 1 to m are a set of N ×N hermitian matrices acting on the flavor space, and
the (cid:15) are a set of infinitesimal parameters. We can (and sometimes will) exponentiate (1.1.8) to
a
get a finite transformation:
φ → φ(cid:48) = ei(cid:15)aTaφ, (1.1.9)
which reduces to (1.1.8) for small (cid:15) . The Ta’s are then said to be the “generators” of the trans-
a
formations, (1.1.9)
We will be using the Ta’s so much that it is worth pausing to consider their properties system-
atically. The fundamental property of the generators is their commutation relation,
[Ta,Tb] = if Tc, (1.1.10)
abc
where f are the structure constants of the Lie algebra, defined in any nontrivial representation (a
abc
trivialrepresentationisoneinwhichTa = 0,forwhich(1.1.10)istriviallysatisfied). Thegenerators
can then be classified into sets, called simple subalgebras, that have nonzero commutators among
themselves, but that commute with everything else. For example, there may be three generators,
Ta for a = 1 to 3 with the commutation relations of SU(2),
[Ta,Tb] = i(cid:15) Tc, (1.1.11)
abc
and which commute with all the other generators. Then this is an SU(2) factor of the algebra. The
algebra can always be decomposed into factors like this, called “simple” subalgebras, and a set of
generators which commute with everything, called U(1)’s.
The normalization of the U(1) generators must be set by some arbitrary convention. However,
the normalization of the generators of each simple subgroup is related to the normalization of the
structure constants. It is important to normalize them so that in each simple subalgebra,
(cid:88)
f f = kδ . (1.1.12)
acd bcd ab
c,d
Then for every representation,
tr(TaTb) ∝ δ . (1.1.13)
ab
The mathematician’s convention is to grant a special status to the structure constants and choose
k = 1 in (1.1.12). However, we physicists are more flexible (or less systematic). For SU(n), for
example, we usually choose k so that
1
tr(TaTb) = δ . (1.1.14)
ab
2
Weak Interactions — Howard Georgi — draft - March 25, 2010 — 4
for the n dimensional representation. Then k = n.
A symmetry of the form of (1.1.8) or (1.1.9) which acts only on the flavor space and not on
the space-time dependence of the fields is called an “internal” symmetry. The familiar Poincare
symmetry of relativistic actions is not an internal symmetry.
In this book we will distinguish two kinds of internal symmetry. If the parameters, (cid:15) , are
a
independent of space and time, the symmetry is called a “global” symmetry. Global symmetries
involve the rotation of φ in flavor space in the same way at all points in space and at all times.
This is not a very physically appealing idea, because it is hard to imagine doing it, but we will
see that the concept of global symmetry is enormously useful in organizing our knowledge of field
theory and physics.
Later on, we will study what happens if the (cid:15) depend on x. Then the symmetry is a “local”
a
or “gauge” symmetry. As we will see, a local symmetry is not just an organizing principle, but is
intimately related to dynamics.
For now, consider (1.1.7) for a global internal symmetry of L. Because the symmetry is a
symmetry of L, the second term, Vµ is zero.
Thus we can write the conserved current as
δL
Nµ = i(cid:15) Taφ. (1.1.15)
a
δ(∂ φ)
µ
Because the infinitesimal parameters are arbitrary, (1.1.15) actually defines m conserved currents,
δL
Jµ = −i Taφ for a = 1 to m. (1.1.16)
a δ(∂ φ)
µ
I stress again that this discussion is all at the level of the classical action and Lagrangian. We
will discuss later what happens in quantum field theory.
1.2 Examples
Example 1
Let Φ , for j = 1 to N, be a set of real scalar boson fields. The most general possible real quadratic
j
term in the derivatives of Φ in the Lagrangian is
1
L (Φ) = ∂ ΦT S∂µΦ. (1.2.1)
KE µ
2
where S is a real symmetric matrix. Note the matrix notation, in which, the Φ are arranged
j
into an N-component column vector. In physical applications, we want S to be a strictly positive
matrix. Negative eigenvalues would give rise to a Hamiltonian that is not bounded below and zero
eigenvalues to scalars that do not propagate at all. If S is positive, then we can define a new set of
fields by a linear transformation
Lφ ≡ Φ, (1.2.2)
such that
LT SL = I. (1.2.3)
In terms of φ, the Lagrangian becomes
1
L (φ) = ∂ φT ∂µφ. (1.2.4)
KE µ
2
Weak Interactions — Howard Georgi — draft - March 25, 2010 — 5
This is the canonical form for the Lagrangian for a set of N massless free scalar fields. Under an
infinitesimal linear transformation,
δφ = Gφ, (1.2.5)
where G is an N ×N matrix, the change in L is
KE
1
δL (φ) = ∂ φT (G+GT)∂µφ. (1.2.6)
KE µ
2
If G is antisymmetric, the Lagrangian is unchanged! We must also choose G real to preserve the
reality of the fields. Thus the Lagrangian is invariant under a group of SO(N) transformations,
δφ = i(cid:15) Taφ, (1.2.7)
a
where the Ta, for a = 1 to N(N −1)/2 are the N(N −1)/2 independent, antisymmetric imaginary
matrices. These matrices area representation of theSO(N) algebra.1 When exponentiated, (1.2.7)
produces an orthogonal transformation, a representation of the group SO(N)
φ → φ(cid:48) = Oφ, (1.2.8)
where
OT = O−1. (1.2.9)
This is a rotation in a real N-dimensional space.
Notice that we have not had to do anything to impose this SO(N) symmetry ex-
cept to put the Lagrangian into canonical form. It was an automatic consequence of
the physical starting point, the existence of N free massless scalar fields. The canon-
ical form of the derivative term in the kinetic energy automatically has the SO(N)
symmetry.
The corresponding Noether currents are
Jµ = −i(∂µφT)Taφ. (1.2.10)
a
The symmetry, (1.2.8), is the largest internal rotation symmetry that a set of N real spinless
bosons can have, because the kinetic energy term, (1.2.4), must always be there. However, the
symmetry may be broken down to some subgroup of SO(N). This can happen trivially because of
the mass term if the scalars are not all degenerate. The mass term has the general form:
1
L (φ) = − φT M2φ, (1.2.11)
mass
2
where M2 is a real symmetric matrix called the mass matrix. Its eigenvalues (if all are positive)
are the squared masses of the scalar particles. The mass term is invariant under an orthogonal
transformation, O, if
OT M2O = M2 or equivalently [O,M2] = 0. (1.2.12)
If M2 is proportional to the identity matrix, then the entire SO(N) is unbroken. In general,
the transformations satisfying (1.2.12) form a representation of some subgroup of SO(N). The
subgroup is generated by the subset of the generators, Ta, which commute with M2.
1Usually, I will not be careful to distinguish an algebra or group from its representation, because it is almost
always the explicit representation that we care about.
Weak Interactions — Howard Georgi — draft - March 25, 2010 — 6
The mass matrix can be diagonalized by an orthogonal transformation that leaves the deriva-
tive term, (1.2.4), unchanged. Then the remaining symmetry is an SO((cid:96)) for each (cid:96) degenerate
eigenvalues. For example, if K of the fields have mass m and the other N −K have mass m , the
1 2
diagonal mass matrix can be taken to have to form
m2 
1
 ... 0 
 
 m2 
M2 =  1  . (1.2.13)
 m2 
 2 
 0 ... 
 
m2
2
The symmetry is an SO(K)×SO(N −K) which rotates the degenerate subsets, of the form
(cid:18)O 0 (cid:19)
O = 1 , (1.2.14)
0 O
2
where O and O act on the fields with mass m and m respectively. The finite group elements,
1 2 1 2
(1.2.14), are generated by exponentiation of the K(K −1)/2+(N −K)(N −K −1)/2 generators
(cid:18)S 0(cid:19) (cid:18)0 0 (cid:19)
1 and , (1.2.15)
0 0 0 S
2
where S and S are antisymmetric imaginary matrices.
1 2
If the mass matrix is non-degenerate, that is with no pair of eigenvalues equal, the SO(N)
symmetry is completely broken and there is no continuous symmetry that remains, although the
Lagrangian is still invariant under the discrete symmetry under which any component of φ changes
sign.
ThesymmetrycanalsobebrokendownbyinteractiontermsintheLagrangian. Withcubicand
quartic terms in φ, the symmetry can be broken in more interesting ways. Consider, as a simple
example that will be of use later, a Lagrangian with N = 8 scalars, with K = 4 with mass m (the
1
φ for j = 1 to 4) and the rest with mass m (the φ for j = 5 to 8). The kinetic energy and mass
j 2 j
terms then have the symmetry, SO(4)×SO(4). Arrange the 8 real fields into two complex doublet
fields as follows: √ √
(cid:18)(φ +iφ )/ 2(cid:19) (cid:18)(φ +iφ )/ 2(cid:19)
ξ ≡ 1 2 √ , ξ ≡ 5 6 √ . (1.2.16)
1 (φ +iφ )/ 2 2 (φ +iφ )/ 2
3 4 7 8
Now consider an interaction term of the form
L = λξ†ξ ξ†ξ . (1.2.17)
int 1 2 2 1
This interaction term is not invariant under the SO(4) × SO(4), but it is invariant under an
SU(2)×U(1) symmetry under which
ξ → U ξ for j = 1 to 2, (1.2.18)
j j
where the 2×2 matrix, U, is unitary. The U can written as a phase (the U(1) part) times a special
unitary matrix V (detV = 1), so it is a representation of SU(2)×U(1). This is the subgroup of
SO(4)×SO(4) left invariant by the interaction term, (1.2.17).
The symmetry structure of a field theory is a descending hierarchy of symmetry,
from the kinetic energy term down through the interaction terms. The kinetic energy
term has the largest possible symmetry. This is broken down to some subgroup by the
mass term and the interaction terms. This way of looking at the symmetry structure is
particularly useful when some of the interactions terms are weak, so that their effects
can be treated in perturbation theory.
Weak Interactions — Howard Georgi — draft - March 25, 2010 — 7
Example 2
Letψ forj = 1toN befree,massless,spin-1,four-componentDiracfermionfieldswithLagrangian
j 2
L(ψ) = iψ∂/ψ. (1.2.19)
The Dirac fermion fields are complex, thus (1.2.19) has an obvious SU(N)×U(1) symmetry under
which, for infinitesimal changes,
δψ = i(cid:15) Taψ, (1.2.20)
a
where the Ta are the N2 hermitian N ×N matrices, which generate the defining (or N) represen-
tation of SU(N)×U(1). When exponentiated this gives
ψ → U ψ, (1.2.21)
where U is a unitary N ×N matrix
U = ei(cid:15)aTa. (1.2.22)
The SU(N) part of U is generated by the N2−1 traceless hermitian matrices, while the generator
of the U(1) part, which commutes with everything, is proportional to the identity. The U(1) is just
fermion number.
In (1.2.19),
δL
= iψγµ, (1.2.23)
δ(∂ ψ)
µ
hence the Noether current is
Jµ = ψγµTaψ. (1.2.24)
a
With Ta = 1, the conserved Noether current associated with the U(1) is just the fermion number
current.
The transformation, (1.2.21), is by no means the largest internal symmetry of (1.2.19). To
see the rest of the symmetry, and for many other reasons later on, we will make extensive use of
two-component, Weyl fermion fields, which are related to ordinary spin-1, four-component fermion
2
fields, ψ, by projection with the projection operators
P ≡ (1±γ )/2. (1.2.25)
± 5
Define the left-handed (L) and right-handed (R) fields as
ψ ≡ P ψ, ψ ≡ P ψ. (1.2.26)
L + R −
Then
ψ = ψP , ψ = ψP . (1.2.27)
L − R +
But then because
P γµ = γµP , (1.2.28)
+ −
we can write
L(ψ) = iψ∂/ψ = iψ ∂/ψ +iψ ∂/ψ . (1.2.29)
L L R R
Evidently, we have the freedom to make separate SU(N)×U(1) transformations on the L and R
fermions:
δψ = i(cid:15)LTaψ , δψ = i(cid:15)RTaψ , (1.2.30)
L a L R a R