Gauge Theories

In Chapter 3 we considered continuousglobal symmetries, parametrized by real constants βⁱ. In localor gaugesymmetries¹ theβⁱ are promoted to arbitrary differentiable functions βⁱ(x) of space and time. Gauge invariance is sometimes motivated on esthetic grounds. For example, why should the phase of the electron field on the Earth be correlated with its phase on Mars? This is not entirely compelling because the standard model does involve global symmetries (though they may derive from gauge symmetries in an underlying theory). In any case, we will take the pragmatic view that gauge invariance is a powerful tool for constructing well-behaved field theories and are the unique renormalizable field theories for spin-1 particles. In particular, each generator of a gauge invariant theory must correspond to an (apparently) massless spin-1gauge boson, which mediates an (apparently) long-range force, and the diagonal generators of an unbroken gauge theory correspond to conserved charges (Weyl, 1929). The gauge interactions are uniquely prescribed once one specifies the gauge group, the representations of the matter fields, and a gauge coupling constant g for each group factor. This approach is opposite to historical development: Maxwell’s equations of classical electrodynamics were first derived from observation and consistency, and then it was noticed that they were invariant under gauge transformations, i.e., that the vector and scalar potential involved redundant degrees of freedom. In this chapter, we outline the construction of gauge invariant Lagrangian densities. More detailed treatments include (Abers and Lee, 1973; Weinberg, 1973c,d, 1995; Peskin and Schroeder, 1995).

Let Φa, a= 1· · ·n, representnspin-0 or ¹₂ fields that transform as Φa(x)→

e^iβ(x)·L

abΦb(x)≡

Uβ(x)

abΦb(x) (4.1)

under a gauge transformation (generalizing (3.30)).Lⁱ, i= 1·· ·N, aren×nrepresentation matrices of the Lie algebra of the gauge groupG. Equation (4.1) is equivalent to

Φ(x)→Φ(x)≡e^iβ(x)·LΦ(x) =Uβ(x)

Φ(x), (4.2)

where Φ(x) is anncomponent column vector with components Φa(x). If a theory involves nψ fermion fieldsψa in a column vectorψ andnφreal or complex scalarsφc in a vectorφ, then

ψ(x)→e^iβ(x)·Lψψ(x)≡Uψβ(x) ψ(x) φ(x)→e^iβ(x)·Lφφ(x)≡Uφβ(x)

φ(x),

(4.3)

1Gauge transformations are often referred to as redundancies rather than symmetries because they refer to unobservable degrees of freedom rather than relating different systems.

135

where L_ψ and L_φ are respectively the fermion and scalar representation matrices. In the case of a chiral symmetry,

Lⁱ_ψ=Lⁱ_LP_L+Lⁱ_RP_R withLⁱ_L6=Lⁱ_R, (4.4) so that

ψ_L(x)→e^i~β(x)·~LLψ_L(x)≡U_L β(x)~ ψ_L(x) ψR(x)→e^i~β(x)·~LRψR(x)≡UR β(x)~

ψR(x).

(4.5)

The transformations of the necessary gauge fields will be detailed below.

4.1 THE ABELIAN CASE

As a first example, we take G= U(1). We already considered the electromagnetic inter-actions of charged pions and electrons in Sections 2.6and 2.8, but repeat the key results.

Under a (non-chiral) gauge transformation

ψ→e^−iβ(x)ψ, φ_π^±→e^±iβ(x)φ_π^±, Aµ→Aµ−1

e∂µβ, (4.6) whereψ is the electron field,φ_π+ andφ_π⁻=φ^†_π+ are respectively theπ⁺ andπ⁻ fields, A is the photon (gauge) fieldγ, ande >0. The gauge covariant derivatives transform as

D^µψ≡(∂^µ−ieA^µ)ψ→e^−iβ(x)D^µψ

D^µφ_π± ≡(∂^µ±ieA^µ)φ_π±→e^±iβ(x)D^µφ_π±. (4.7) Thus,

L= ¯ψ(iD6 −m)ψ+ (Dµφ_π+)^†D^µφ_π+−1

4F_µνF^µν−µ²φ^†_π+φ_π+−λ φ^†_π+φ_π+

2

, (4.8) where F_µν = ∂_µA_ν −∂_νA_µ is gauge invariant. However, an explicit photon mass term

M_A²

2 A_µA^µisnotgauge invariant, so theγmust be massless. The Feynman rules for the gauge vertices are given in Figures 2.10and 2.13. They are unique except for e and the charge assignments. However, the mass parameters are arbitrary. Only one pion self-interaction is consistent withU(1), but the coefficientλis arbitrary.

These considerations are easily generalized to an arbitrary non-chiral U(1), with nψ

fermionsψ_aandn_φcomplex scalarsφ_b, with chargesq_aandq_b⁰ in units of the gauge coupling g. (The simple QED example hadq_e− =−1, q_π⁰₊= +1,andg=e.) The fields transform as

ψ_a→e^iqaβ(x)ψ_a, φ_b→e^iqb0β(x)φ_b, A_µ →A_µ−1

g∂_µβ(x), (4.9) whereAis the gauge boson. One can think of the charges as the elements of the (reducible) diagonal fermion and scalar representation matrices L_ψ = diag q₁q₂· · ·q_nψ

and L_φ = diag

q⁰₁q⁰₂· · ·q_n⁰

φ

. The gauge covariant derivatives are

D^µψ_a= (∂^µ+igq_aA^µ)ψ_a→e^iqaβ(x)D^µψ_a

D^µφb= (∂^µ+igq_b⁰A^µ)φb →e^iqb0β(x)D^µφb, (4.10)

or in column vector notation,

D^µψ= (∂^µI+igA^µL_ψ)ψ→e^iβ(x)LψD^µψ

D^µφ= (∂^µI+igA^µL_φ)φ→e^iβ(x)LφD^µφ, (4.11) whereI is then_ψ×n_ψ orn_φ×n_φ identity. The kinetic terms

L⁰= ¯ψiDψ6 + (Dµφ)^†

| {z }

φ^†←−

∂µI−igAµLφ

D^µφ−1

4F_µνF^µν (4.12)

are gauge invariant. These lead to vertices with the massless gauge boson similar to those in Figures 2.10 and 2.13, except e → gq_b⁰ for φb and −e → gqa for ψa. (There are no off-diagonal transitions such asψ1→ψ2.)

It is clear that onlygqaandgq_b⁰ are physical. One can always rescalegprovidedqa, q_b⁰ are rescaled accordingly. For example, this freedom is used in QED to set the electron charge to −1. In a pureU(1) theory, the relative values of the charges are arbitrary. However, if theU(1) is a subgroup of a simple group, then the relative charges are fixed by the higher symmetry.

It is straightforward to extend these considerations to a chiralU(1), i.e., in which the L and R charges q_aL,R of ψ_a are different (this is not the case for QED). Define L_ψ = L_LP_L+L_RP_R, where L_L,R are the diagonal charge matrices of ψ_L,R and P_L,R are the chiral projections in (2.196) on page 40. Then, the fermion term in (4.12) becomes

ψi¯ Dψ6 = ¯ψ(i6∂I−gAL6 _ψ)ψ= ¯ψ(i6∂I−gA[L6 LP_L+L_RP_R])ψ, (4.13) and theψa vertex inFigure 2.13is

−igγ_µ

q_aL+q_aR 2

−

q_aL−q_aR 2

γ⁵

. (4.14)

One can add mass and additional non-derivative interaction terms toL⁰ provided they are invariant under the globalU(1). For example,²

L=L⁰−ψ¯_am_abψ_b−φ^†_cµ²_cdφ_d +h

ψ¯_aΓ^c_abψ_bφ_c+ ¯ψ_aiΓˆ^c_abγ⁵ψ_bφ_c+h.c.i

+λ_abcdφ^†_aφ^†_bφ_cφ_d (4.15) would be invariant provided qa =qb for nonzeromab, q_c⁰ =q_d⁰ for nonzeroµ²_cd,qa=qb+q⁰_c for nonzero Γ^c_ab or ˆΓ^c_ab, andq⁰_a+q⁰_b=q_c⁰ +q_d⁰ for nonzeroλ_abcd.

4.2 NON-ABELIAN GAUGE THEORIES

Now consider a non-abelian gauge symmetry (Yang and Mills, 1954), in which the spin-0 and

1

2 fields transform according to (4.3), whereGis a simple group. Any mass terms, Yukawa interactions, or non-derivative scalar self-interactions that are invariant under the corre-sponding global symmetry are automatically gauge invariant as well. However, the fermion or scalar kinetic energy terms contain derivatives and are therefore not gauge invariant,

∂^µΦ→∂^µh

U ~β(x) Φi

=U ∂^µΦ + [∂^µU] Φ. (4.16)

2In (4.15) the matrixµ²must be Hermitian, and there are constraints on theλabcdfrom Hermiticity and Bose statistics.m_abis assumed Hermitian, but non-Hermitian fermion mass matrices can be introduced by rewriting in terms ofψL,R(or allowingγ⁵ terms). Other types of cubic or quartic scalar self-interactions could be allowed if they are globally invariant, such asφaφbφc+h.c.forq_a⁰ +q_b⁰+q_c⁰= 0.

One must therefore introduce a gauge covariant derivative

∂^µ →D^µ ≡∂^µ+ig ~A^µ·L,~ (4.17) where Aⁱ_µ, i = 1· · ·N, are real vector gauge fields (one per generator), g is an arbitrary (real) gauge coupling, A~^µ·~L=A^iµLⁱ, and Lⁱ are the representation matrices for Φ, i.e., Lⁱ=Lⁱ_φ, Lⁱ_ψ,or Lⁱ_LP_L+Lⁱ_RP_R. Of course,∂^µ≡∂^µI, whereI is then×nidentity matrix.

In terms of components,

(D^µΦ)_a=h

∂^µδ_ab+ig ~A^µ·~L_abi

Φb. (4.18)

These gauge covariant derivatives specify the interactions of the spin-0 or spin-¹₂ fields with the gauge bosons in terms of a single gauge coupling g (or one per group factor for a non-simple group), once the group and representations are specified. The Feynman rules for the gauge interactions are shown inFigure 4.1. For example, the fermion kinetic energy term

L^KEψ = ¯ψiDψ6 (4.19)

implies that the amplitude forψ_b to absorb or emit gauge bosonAⁱ and turn intoψ_a is just

−igγ^µ

Lⁱ_Lab1−γ⁵

2 +Lⁱ_Rab1 +γ⁵ 2

−−−−−−−→

LL=LR=L −igγ^µLⁱ_ab. (4.20) This must be sandwiched between appropriate uor vspinors and contracted with a gauge polarization vector, as described in Chapter 2. Unlike the abelian case, the non-diagonal generators imply transitions between the different members of the fermion (or scalar) IRREP that are related by the symmetry.

Similarly, the kinetic energy term for a complex scalar representation becomes (D^µφ)^†D_µφ= (∂^µφ)^†∂_µφ−igAⁱ_µ

φ^†←→∂^µLⁱφ

+g²A^iµA^j_µφ^†LⁱL^jφ. (4.21) The second term on the right yields the three-point vertex inFigure 4.1since∂^µφ→ −ip¯^µφ (∂^µφ^†→+ip¯^µφ^†), where ¯pis the momentum entering (exiting) the vertex for φ(φ^†), i.e.,

¯

palways flows in the direction of the arrow. The four-point vertex fromiL^KEφ follows from the last term, taking into account that there are two ways to contract A^iµA^j_µ with the external gauge fields. The vertices for Hermitian scalars are considered inProblem 4.6.

We must still determine the transformation ofAⁱ_µ. The first requirement is

D^µΦ→U D^µΦ. (4.22)

This implies that the fermion and scalar kinetic energy terms L^KEψ,φ= ¯ψiD6 _ψψ+ D^µ_φφ†

D_φµφ

→ψiU¯ _ψ^†U_ψD6 _ψψ+ D^µ_φφ^†

| {z }

φ^†←−

∂^µI−ig ~A^µ·~Lφ

U_φ^†U_φD_φµφ (4.23)

are gauge invariant, sinceU^†U =I. The necessary transformation is given by A~_µ·L~ →A~_µ⁰ ·~L≡U ~A_µ·~L U⁻¹+ i

g(∂µU)U⁻¹, (4.24)

b a

i, µ

b, pb a, pa

i, µ

b, pb a, pa

j, ν i, µ

q p

r j, ν

i, µ

k, σ

j, ν i, µ

k, σ l, ρ

−igγ^µLⁱ_ψab=−igγ^µ

Lⁱ_LabPL+Lⁱ_RabPR

−ig(¯pa+ ¯pb)^µLⁱ_φab

ig²g^µν

Lⁱ_φL^j_φ+L^j_φLⁱ_φ

ab

+gcijk[g^µν(q−p)^σ+g^µσ(p−r)^ν

+g^νσ(r−q)^µ]≡gc_ijkC^µνσ(p, q, r)

−ig²cijmcklm(g^µσg^νρ−g^µρg^νσ)

−ig²c_ikmc_jlm(g^µνg^ρσ−g^µρg^νσ)

−ig²c_ilmc_jkm(g^µνg^ρσ−g^µσg^νρ)

Figure 4.1

Feynman rules for the interactions of gauge bosons with fermions, scalars, and gauge self-interactions. The scalar vertices apply to both complex or Hermitian scalars. ¯

p_a

and ¯

p_b

in the second vertex are the momenta flowing in the direction of the arrows, which in some cases are the negative of the physical momenta. For example, ¯

p_a

=

−p_a

if the

a

is twisted downward to represent an incident antiparticle or Hermitian scalar, while ¯

p_b

=

−p_b

for an outgoing antiparticle or Hermitian scalar, as in

Figure 2.10. Recall that Lⁱ_φab

=

−Lⁱ_φba

for a Hermitian scalar. In the triple gauge vertex the momenta

p, q,

and

r

all flow into the vertex and satisfy

p+q+r

= 0.

The function

C^µνσ

(p, q, r) is totally antisymmetric if the indices and corresponding

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