Symmetries Ii

The remainder of this book is devoted to the study of dynamical symmetries and gauge field theories. We use the term "dynamical symmetry" to refer to any symmetry which is so restrictive that it completely determines the structure of the Lagrangian. The discovery that such symmetries exist, and that they actually seem to correctly describe the physical world, is one of the most remarkable, exciting, and successful of the recent developments in modern physics.

The most striking example of such symmetries are the local, non-Abelian gauge symmetries. Quantum Chromodynamics, or QCD, is a theory based on the SU (3) gauge group. This is the modern theory of the strong interactions, which has the remarkable property that the coupling constant approaches zero as the momentum flowing through the interaction vertex approaches infinity, a property r; ferred to as asymptotic freedom (which will be discussed in Chapter 17). Largely because of this, QCD has led to a plethora of successful predictions at high energies, which give us great faith in the theory even though predictions at low energies are hard to make. Another local gauge symmetry, based on the product of two groups, SU(2) x {/(1), leads to the unification of the electromagnetic and weak interactions into a single electroweak theory, referred to as the Standard Model. Here a new feature is added; the gauge symmetry is "hidden" because the ground state of the theory, the vacuum, spontaneously breaks the symmetry. The breaking is said to be "spontaneous" because the mathematical form of the Lagrangian forces the vacuum to break the symmetry; no additional assumption is needed. In spite of its modest name, the Standard Model has also been a major success, but because there are variations of this model, based on larger gauge groups with more parameters, which are also consistent with the data, it is less clear that the Standard Model will survive into the next century without changes.

Both QCD and the Standard Model are formulated directly in terms of the elementary constituents of nature: the quarks and leptons. This makes it hard to extract predictions for the observed hadrons, which are complex composites of quarks. In this respect a third symmetry, chiral symmetry, which is only an approximate symmetry, has been very successful because it gives effective Lagrangians which can be expressed directly in terms of the observed hadrons.

The most successful version of chiral symmetry is also spontaneously broken.

In this chapter we will discuss gauge theories, chiral symmetry, and spontaneous symmetry breaking. The quantization of gauge fields will be the topic of Chapters 14 and 15. Discussion of the standard electroweak theory is also postponed until Chapter 15. Finally, we return to a more detailed discussion of renormalization in Chapter 16 and a derivation and discussion of asymptotic freedom in Chapter 17.

We begin this chapter by showing how the familiar gauge invariance of QED can be understood to be a consequence of a local L/(l) gauge symmetry. The U(\) symmetry group is Abelian, and as a result QED has a rather simple structure, and the incredible power of the idea of local gauge invariance is not clearly illustrated by QED alone. Only when we consider non-Abelian gauge groups does the rich structure of such theories become apparent, and we discuss these theories in the middle sections of this chapter. The chapter concludes with a study of chiral symmetry and how this symmetry can be spontaneously broken.


We introduced gauge transformations very briefly in Eq. (8.10). There are two types of Abelian gauge transformations depending on whether or not 9 is a function of x:

9 = constant (global gauge transformation)

Consider global gauge transformations first.

If the Lagrangian is invariant under the gauge transformation, Noether's theorem tells us that there is a conserved quantity associated with this invariance. Gauge transformations always leave the space-time coordinates unchanged, and hence the A^ of Eq. (8.20) is always zero, and the conserved quantity associated with the gauge transformation, always referred to as a current, has the form

Note that, by convention, the sign of the current is opposite to the sign of öß in Eq. (8.20).

Now, a global Abelian gauge transformation is defined by

complex fields real fields,


where q can be a different number for each complex field (later, q will be associated with the charge). Note that there is only one infinitesimal parameter and that the group consists of multiplying complex fields by a complex number with unit modulus, which is the unitary group of one dimension, U( 1). The infinitesimal form of the transformations are i/>'a{x) = {l-iq0)1>a(x)

and hence f2Q = —iqi>a(x), Q.a = iqipa(x), and the conserved quantity associated with this [/(l) symmetry is

Ipa-Tpa dC

In order for £ to be invariant under the global gauge transformation, it is sufficient that it be bilinear in ijj and tj}, or for scalar fields, $ and 0. For a free spinor theory, where

1> , the conserved quantity is

which we recognize as the EM current, Eq. (10.2), provided we identify q = e . (13.7)

Hence we see that conservation of charge can be "understood" as a consequence of a global gauge symmetry of the theory.

Local Gauge Invariance

We now generalize the gauge transformation, permitting the phase 6 to depend on the local space-time point, i.e., 9 — 8(x). This means that a gauge transformation can be carried out in one region of space-time without "knowing" what is taking place elsewhere. If it were the case that the value of the gauge phase angle had any physical significance, it would be essential that the gauge transformation be local, in order to allow time for information about any changes in the phase angle to propagate from one locality to another. However, the phase angle probably contains no information, in which case the requirement of local gauge invariance is merely the (very powerful) requirement that this phase angle can be completely arbitrary from point to point, except for the requirement that it be a smooth, differentiable function.

The Lagrangian will no longer be invariant under the local gauge transformation unless it has a particular form. Consider the free fermion part first. We have

where 3^6 = dö{x)/dxß. To eliminate the extra term and make £ invariant, we need a vector field Aß which interacts with the current and transforms in a special way. To find this special transformation law for the vector field, add an interaction of the form J^A^:

£' = C'p - -J"A' =CF + Jßdße - -J*A' q * q *

From now on we will take q/e = 1, so that the gauge transformation of the vector field is precisely what we wrote down for electromagietism in Sec. 2.2 [Eq. (2.15) with 9 = -Ac].

Next, consider the Lagrangian for the fields AA general form for the Lagrangian is

¿field = AiF^F"" + A2 G^G"" + ro* , (13.12)

where my is a possible mass term for the gauge fields and G^ is a possible symmetric combination of fields and derivatives,

which, together with the antisymmetric combination FM„, insures that the hypothetical Lagrangian (13.12) contains any combination of the independent terms dtlAvd'iAv and d^A^A^. Now the gauge transformation leaves invariant, but


Hence neither of these terms is gauge invariant, and £geld = £fieid requires that both m\ = 0 and À2 = 0. The vector field must be massless and have a free Lagrangian of the familiar form where Ai = corresponds to the conventional normalization used for the EM field. We conclude that the requirement of local gauge invariance dictates the form of QED.


Next, we discuss how the concept of gauge invariance can be extended to non-Abelian groups. Assume the "charged" fields have several components, such as isospin or color, describing some internal degree of freedom, and consider a unitary transformation which transforms them into each other. This gauge transformation can be written rl>'(x) = IJip(x) , (13.16)

where, for n degrees of freedom, U is an n x n matrix which is unitary. The group of such matrices can always be written as a product of the f/(l) group (which in n dimensions is the product of a complex number with unit modulus and the n x n unit matrix) and the SU (n) group of unitary matrices with unit determinant (a condition which fixes the phase). In this section we will limit discussion to the SU(n) group, so that we may assume det U = 1, and when a detailed example is needed, we will use the familiar 5/7(2) group. For SU(2) the U matrix is jj _

where are the familiar Pauli matrices (xthe generators of SU{2), g is the coupling constant, and ej(x) are three independent rotation "angles." To simplify the formulas, we will use the notation e(x) = inei(x)

If the Lagrangian is independent of the internal degree of freedom (which, for example, could be isospin or color), the free Lagrangian will be a sum over the Lagrangians for each component of the internal degree of freedom (denoted by the subscript £), and


d dx"

where, in the second expression, a unit matrix in n-dimensional space is implied. If the gauge transformation is global (et = const), the free Fermi Lagrangian is

Fig. 13.1 Three of the Feynman rules for QCD. These, and the others, are summarized in Appendix B.

• There is only one coupling constant, which enters into the ffg coupling, the 3g coupling, and the 4g coupling. These theories will not be gauge invariant if these three couplings take on arbitrary values.

Quantum Chromodynamics

As previously noted, QCD can be defined as the Yang-Mills theory with SU(3) local gauge invariance. The fermions are called quarks, and each flavor or type of quark has three internal degrees of freedom called color. There are n2 - 1 = 8 vector gauge fields called gluons.

The interactions between the quarks and gluons in QCD are a special example of those already given in Eqs. (13.55) and (13.57) and suggest the additions to Feynman Rule 1 shown in Fig. 13.1. While these rules are some of the interactions which arise in QCD, it turns out that a complete discussion of the quantization of QCD will lead to several additional Feynman rules, which we are not equipped to introduce now. We will return to this topic in Chapter 15, where a systematic discussion and derivation of the Feynman rules for QCD will be given.


The last of the continuous symmetries which will be discussed in this chapter is chiral symmetry. Unlike the gauge symmetries discussed so far, chiral symmetry is believed to be only an approximate symmetry of the strong interactions (but a very good approximate symmetry). A simple model which displays this symmetry, .he sigma model, will be discussed in the next section. Using this model, we will also discuss spontaneous symmetry breaking, a mechanism of great importance in physics. It turns out that a combination of gauge symmetry with spontaneous symmetry breaking is the basis for our present understanding of the electroweak forces.

We begin by considering the following transformations (referred to as chiral transformations) of a fermion field with two internal degrees of freedom (different from color):

This looks like an SU{2) gauge transformation but is quite different because of the presence of the 75. The 75 operates on the Dirac components of ip, while the Ti operates on the additional two-dimensional space corresponding to two internal degrees of freedom of the fermion, which could be isospin but more generally is referred to as a flavor space. In applications, the flavor space is usually isospin, which can be

for nucléons or for nucléons or for quarks .

for quarks .

We will denote the chiral transformation matrix by

Us = e~i95755T,e' . Note that U5 is unitary, but because {7M,75} = 0,

[This can be proved by expanding U5 in a power series, and noting that (75)"7M = (~l)n7M (75)"' and resuming the power series.] Hence V and ?/> transform in the same way, l// =

so that under a global transformation, the kinetic energy term is invariant under a chiral transformation

but the mass term is not, mip'tp' — 771^1)51)5^ m%j)tp .

Therefore, in constructing a Lagrangian which is invariant under chiral transformations, one usually begins by assuming that the fermion masses are zero. The presence of a small fermion mass term provides a mechanism for breaking chiral symmetry.

In discussing chiral symmetry, it is customary to also include the usual (global) SU(2) gauge symmetry, which is also a symmetry once we have ip fields with two internal degrees of freedom (two flavors). Hence the overall symmetry is designated SU{2) x SU{2); one SU{2) with a 75 in the generator and the other without a 75. Alternatively, these combined transformations are equivalent to the transformations

where the operators V± in the exponents are projection operators,

and the infinitesimal parameters tL and tR are independent. This group is designated SU{2)l x SU(2)r, with one SU{2) group containing the "left-handed"

projection operator V_ and the other the "right-handed" operator V+ (recall the discussion of right- and left-handed spinors in Sec. 5.11).

An understanding of the physical meaning of chiral symmetry follows from the combined form (13.64). Chiral invariance says that an SU(2) gauge symmetry can be independently realized on the two spaces projected out by the V± operators; i.e., the gauge transformations on these two subspaces can have different parameters eR and eL as written in Eq. (13.64). For this to be true, the helicity of any particle must be conserved by all interactions, and sufficient conditions for this to be true are (1) the fermions are massless, so that their helicity cannot be changed by bringing them to rest and reversing their direction of motion, and (2) the interactions do not explicitly flip the spin. We see immediately that helicity conservation places strong restrictions on the interactions. However, in Sec. 13.6 we will show that interactions can be constructed in which the mass of the fermions is non-zero.

The conserved current associated with chiral symmetry is found from the infinitesimal transformations, for which

Hence the conserved current is

For massless fields the fermion Lagrangian is i 7 d ,

and the conserved current is an axial current:

The ordinary SU{2) gauge symmetry gave a conserved vector current [recall Eq. (13.21)]

nit so that together we have conserved vector and axial vector currents.


We saw in the last section that chiral symmetry seems to imply that the fermions of the theory must be massless. However, this restriction can be eliminated by the construction of a theory in which the fermion mass arises as part of the interaction. There are many ways to do this. One way is to construct a theory in which the fermions start off as massless and then to generate fermion mass through spontaneous symmetry breaking. This is the route we will follow in the next two sections. In this section we will construct the linear sigma model, which is a theory of massless "nucleons" interacting with mesons, and then, in the next section, discuss the mechanism of spontaneous symmetry breaking and show how this mechanism can generate nucleón mass. Another way to build a chirally invariant theory with massive fermions is to construct the interaction in such a way that mass is included from the start. An example of such a model is the non-linear sigma model, which will be discussed in Sec. 13.7. Both the linear and the non-linear sigma models are very useful in understanding the interactions of nucleons with light mesons and in understanding the pion, which plays a special role in the description of the strong interactions at low energies.

We therefore begin by considering the interactions of the Fermi fields with mesons, particularly the pion. To be slightly more general, assume the mesons are self-conjugate scalar or pseudoscalar mesons. The interaction is therefore of the form

where M is a superposition of scalar and pseudoscalar meson submatrices FGL 60]:

where ga and gn are real constants, and are 2 x 2 Hermitian matrix fields in the flavor space, and the i multiplying the term comes from the fact that hermiticity implies

so that

It is the fact that M depends linearly on the meson fields which is the origin of the term linear sigma model.

Next, we determine the transformation laws of the meson field matrix from the requirement that SU(2) x SU(2) be a good symmetry of the interaction. The transformations in the SU(2) gauge group will be denoted U and can be written [recall Eq. (13.17)]

U = e-'95T'£' = , where in this section we adopt a new convention £ = r¿e¿. Symmetry under the gauge group implies that

which gives

Hence <j)a and (j)^ can be expanded in terms of 1 and the SU(2) generators. It turns out to be sufficient to choose (j)^ to be pure isoscalar and (j)^ to be pure isovector, so that

[This is the reason for the names a and w. It is also possible to make the Dirac scalar interaction pure isovector and the Dirac pseudoscalar interaction pure isoscalar, but this will not be discussed here. ]

Now, examine the implications of chiral symmetry. Require

Hence, under the chiral transformation U5 we require

we can obtain a compact form for U5,

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