Non-Abelian global symmetries in Lagrangian quantum field theory

12.4.1 SU(2)_f and SU(3)_f

As may already have begun to be apparent in chapter 7, Lagrangian quantum field theory is a formalism which is especially well adapted for the description of symmetries. Without going into any elaborate general theory, we shall now give a few examples showing how global flavour symmetry is very easily built into a Lagrangian, generalizing in a simple way the global U(1) symmetries considered in section 7.1 and section 7.2. This will also prepare the way for the (local) gauge case, to be considered in the following chapter.

Consider, for example, the Lagrangian

Lˆ= ¯ˆu(i∂−m)ˆu+¯ˆd(i ∂−m) ˆd (12.85) describing two free fermions ‘u’ and ‘d’ of equal mass m, with the overbar now meaning the Dirac conjugate for the four-component spinor fields. Note carefully that we are suppressing the space-time arguments of the quantum fields ˆu(x),d(x). As in (12.50), we are using the convenient shorthand ˆˆ ψu= ˆu and ˆψd= ˆd. Let us introduce

ˆ q= uˆ

dˆ (12.86)

so that ˆL can be compactly written as

Lˆ= ¯ˆq(i ∂−m)ˆq. (12.87) In this form it is obvious that ˆL– and hence the associated Hamiltonian ˆH– is invariant under the global U(1) transformation

ˆ

q = e^iαqˆ (12.88)

'

'

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 25 (cf (12.1)) which is associated with baryon number conservation. It is also invariant under global SU(2)f transformations acting in the flavour u-d space (cf (12.32)):

qˆ = exp(−iα · τ /2)ˆq (12.89) (for the change in sign with respect to (12.51), compare section 7.1 and section 7.2 in the U(1) case). In (12.89), the three parameters α are independent of x. What are the conserved quantities associated with the invariance of Lˆ under (12.89) ? Let us recall the discussion of the simpler U(1) cases studied in sections 7.1 and 7.2. Considering the complex scalar field of section 7.1, the

−iα ˆ

analogue of (12.89) was just φˆ→ φˆ^' = e φ, and the conserved quantity was the Hermitian operator Nˆφ which appeared in the exponent of the unitary operator Uˆ that effected the transformation φˆ→ φˆ^' via

φˆ^' = UˆφˆUˆ^† , (12.90) with

Uˆ = exp(iαNˆ_φ). (12.91)

For an infinitesimal α, we have

φ, Nφ,

φˆ^' ≈ (1 − iE) ˆ Uˆ ≈ 1 + iE ˆ (12.92)

so that (12.90) becomes

(1 − iE)φˆ = (1 + iENˆ_φ)φˆ(1 − iENˆ_φ)≈ φˆ + iE[Nˆ_φ, φˆ]; (12.93) hence we require

[Nˆ_φ, φˆ] = −φˆ (12.94)

for consistency. Insofar as Nˆ_φ determines the form of an infinitesimal version of the unitary transformation operator Uˆ , it seems reasonable to call it the generator of these global U(1) transformations (compare the discussion after (12.27) and (12.35), but note that here Nˆ_φ is a quantum field operator, not a matrix).

Consider now the SU(2)f transformation (12.89), in the infinitesimal case:

qˆ = (1 − iE · τ /2)ˆq. (12.95) Since the single U(1) parameter E is now replaced by the three parameters E = (E1, E2, E3), we shall need three analogues of Nˆ_φ, which we call

1

2 1

2 1 2

1

ˆ 2

T ^{( )} = (Tˆ₁ ⁾, T ˆ₂ ^{( )},T ˆ₃ ^{( )}), (12.96) corresponding to the three independent infinitesimal SU(2) transformations.

The generalizations of (12.90) and (12.91) are then

1

2 1

' = Uˆ^{( )}qˆUˆ^{( )}2 ^† (12.97) qˆ

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 27 operators certainly ‘do the job’ expected of field theoretic isospin operators, in this isospin-1/2 case.

In the U(1) case, considering now the fermionic example of section 7.2 for variety, we could go further and associate the conserved operator ˆNψ with a conserved current Nˆ_ψ^μ:

are of course functions of the space-time co-ordinate x, via the (suppressed) dependence of the ˆq-fields onx. Indeed one can verify from the equations of motion that

∂μTˆ⁽

is a conserved isospin current operatorappropriate to theT =¹₂ (u, d) system; it transforms as a 4-vector under Lorentz transformations, and as aT = 1 triplet under SU(2)f transformations.

Clearly there should be some general formalism for dealing with all this more efficiently, and it is provided by a generalization of the steps followed, in the U(1) case, in equations (7.6)–(7.8). Suppose the Lagrangian involves a set of fields ˆψr (they could be bosons or fermions) and suppose that it is invariant under the infinitesimal transformation

δψˆr=−iETrsψˆs (12.111) for some set of numerical coefficientsTrs. Equation (12.111) generalizes (7.5).

Then since ˆLis invariant under this change, 0 =δLˆ= ∂Lˆ

26 12. Global Non-Abelian Symmetries

and ₁

theoretic representation of the generators of SU(2)f , an interpretation we shall shortly confirm. In the infinitesimal case, (12.97) and (12.98) become

1 which is the analogue of (12.94). Equation (12.100) expresses a very specific

1 sug-gested, a field theoretic representation of the generators of SU(2), appropriate to the case T = 1₂ , follows from the fact that they obey the SU(2) algebra

26 12. Global Non-Abelian Symmetries seem reasonable in this case too to regard the ˆT⁽

1 2)

’s as providing a field theoretic representationof the generators of SU(2)f, an interpretation we shall shortly confirm. In the infinitesimal case, (12.97) and (12.98) become

(1−iE·τ/2)ˆq= (1 + iE·Tˆ⁽

using the Hermiticity of the ˆT⁽

1 2)

’s. Expanding the right-hand side of (12.99) to first order inE, and equating coefficients ofEon both sides, (12.99) reduces to (problem 12.9)

[ ˆT⁽

1 2)

,q] =ˆ −(τ/2)ˆq, (12.100) which is the analogue of (12.94). Equation (12.100) expresses a very specific commutation property of the operators ˆT⁽

1 as can be checked (problem 12.10) from the anticommutation relations of the fermionic fields in ˆq. We shall derive (12.101) from Noether’s theorem (Noether 1918) in a little while. Note that if ‘τ/2’ is replaced by 1, (12.101) reduces to the sum of the u and d number operators, as required for the one-parameter U(1) case. The ‘ˆq^†τq’ combination is precisely the field-theoreticˆ version of theq^†τqcoupling we discussed in section 12.1.3. It means that the three operators ˆT⁽

1 2)

themselves belong to aT = 1 triplet of SU(2)f. It is possible to verify that these ˆT⁽

1

so that their eigenvalues are conserved. That the ˆT⁽

1 2)

are, as already sug-gested, a field theoretic representation of the generators of SU(2), appropriate to the case T = ¹₂, follows from the fact that they obey the SU(2) algebra (problem 12.11):

[ ˆT_i⁽¹²⁾,Tˆ_j⁽¹²⁾] = iEijkTˆ_k⁽¹²⁾. (12.103) For many purposes it is more useful to consider the raising and lowering operators

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 27 which destroys a d quark and creates a u, or destroys a ¯u and creates a d, in ¯ operators certainly ‘do the job’ expected of field theoretic isospin operators, in this isospin-1/2 case.

The obvious generalization appropriate to (12.101) is

1 in the U(1) case, in equations (7.6)–(7.8). Suppose the Lagrangian involves a set of fields ψˆ_r (they could be bosons or fermions) and suppose that it is invariant under the infinitesimal transformation

δψˆ_r = −iETrsψˆ_s (12.111)

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 29 the ˆφ1−φˆ2system of section 7.1. An infinitesimal such rotation is (cf (12.64), and noting the sign change in the field theory case)

φˆ^'= ˆφ+E×φˆ (12.120) which implies

δφˆr=−iEaT_ars⁽¹⁾φˆs, (12.121) with

T_ars⁽¹⁾=−iEars (12.122) as in (12.48). There are of course three conserved ˆToperators again, and three Tˆ^μ’s, which we call ˆT⁽¹⁾and ˆT^(1)μrespectively, since we are now dealing with a T = 1 isospin case. The a= 1 component of the conserved current in this case is, from (12.116),

Tˆ₁^(1)μ= ˆφ2∂^μφˆ3−φˆ3∂^μφˆ2. (12.123) Cyclic permutations give us the other components which can be summarised as

Tˆ^(1)μ= i( ˆφ^(1)trT⁽¹⁾∂^μφˆ⁽¹⁾−(∂^μφˆ⁽¹⁾)^trT⁽¹⁾φˆ⁽¹⁾) (12.124) where we have written

φˆ⁽¹⁾=

⎛

⎝ φˆ1

φˆ2

φˆ3

⎞

⎠ (12.125)

and ^tr denotes transpose. Equation (12.124) has the form expected of a bosonic spin-0 current, but with the matrices T⁽¹⁾ appearing, appropriate to theT = 1 (triplet) representation of SU(2)f.

The general form of such SU(2) currents should now be clear. For an isospinT-multiplet of bosons we shall have the form

i( ˆφ^(T)†T^(T)∂^μφˆ^(T))−(∂^μφˆ^(T))^†T^(T)φˆ^(T)) (12.126) where we have put the†to allow for possibly complex fields; and for an isospin T-multiplet of fermions we shall have

ψ¯ˆ^(T)γ^μT^(T)ψˆ^(T) (12.127) where in each case the (2T + 1) components of ˆφ or ˆψ transforms as a T -multiplet under SU(2), i.e.

ψˆ^(T)'= exp(−iα·T^(T)) ˆψ^(T) (12.128) and similarly for ˆφ^(T), whereT^(T)are the 2T+1×2T+1 matrices representing the generators of SU(2)f in this representation. In all cases, the integral over

28 12. Global Non-Abelian Symmetries

But

∂Lˆ

∂^μ ∂Lˆ

= (12.113)

∂ψˆ_r ∂(∂^μψˆ_r) from the equations of motion. Hence

∂Lˆ

∂^μ δψˆ_r = 0 (12.114)

∂(∂^μψˆ_r)

which is precisely a current conservation law of the form

∂^μˆjμ = 0. (12.115)

Indeed, disregarding the irrelevant constant small parameter E, the conserved current is

∂Lˆ

ˆ ˆ

jμ = −i Trsψs. (12.116)

∂(∂^μψˆ_r) Let us try this out on (12.87) with

δqˆ = (−iE · τ /2)ˆq. (12.117) As we know already, there are now three E’s, and so three Trs’s, namely

1(τ1)rs, 1(τ2)rs,1 (τ3)rs. For each one we have a current, for example

2 2 2

ˆ^{( 1 )} ∂Lˆ τ₁ τ1

T_1μ ² = −i q = ˆˆ ¯qγ_μ qˆ (12.118)

∂(∂^μqˆ) 2 2

and similarly for the other τ’s, and so we recover (12.109). From the invari-ance of the Lagrangian under the transformation (12.117) there follows the conservation of an associated symmetry current. This is the quantum field theory version of Noether’s theorem.

This theorem is of fundamental significance as it tells us how to relate symmetries (under transformations of the general form (12.111)) to ‘current’

conservation laws (of the form (12.115), and it constructs the actual currents for us. In gauge theories, the dynamics is generated from a symmetry, in the sense that (as we have seen in the local U(1) of electromagnetism) the symmetry currents are the dynamical currents that drive the equations for the force field. Thus the symmetries of the Lagrangian are basic to gauge field theories.

Let us look at another example, this time involving spin-0 fields. Suppose we have three spin-0 fields all with the same mass, and take

1 1 1 1

ˆ ∂μφˆ₁∂^μφˆ₁ + ∂μφˆ₂∂^μφˆ₂ + ∂μφˆ₃∂^μφˆ₃ −

L = m ²(φˆ²₁ +φˆ²₂ +φˆ²

3). (12.119)

2 2 2 2

It is obvious that Lˆ is invariant under an arbitrary rotation of the three φˆ’s among themselves, generalizing the ‘rotation about the 3-axis’ considered for

28 12. Global Non-Abelian Symmetries But

∂Lˆ

∂ψˆr

=∂^μ ∂Lˆ

∂(∂^μψˆr) (12.113)

from the equations of motion. Hence

∂^μ ∂Lˆ

∂(∂^μψˆr)δψˆr = 0 (12.114) which is precisely a current conservation law of the form

∂^μˆjμ= 0. (12.115)

Indeed, disregarding the irrelevant constant small parameterE, the conserved current is

ˆjμ=−i ∂Lˆ

∂(∂^μψˆr)Trsψˆs. (12.116) Let us try this out on (12.87) with

δˆq= (−iE·τ/2)ˆq. (12.117) As we know already, there are now threeE’s, and so threeTrs’s, namely

1

2(τ1)rs,¹₂(τ2)rs,¹₂(τ3)rs. For each one we have a current, for example Tˆ_1μ⁽¹²⁾=−i ∂Lˆ

∂(∂^μq)ˆ τ1

2 qˆ= ¯ˆqγμ

τ1

2qˆ (12.118)

and similarly for the otherτ’s, and so we recover (12.109). From the invari-ance of the Lagrangian under the transformation (12.117) there follows the conservation of an associated symmetry current. This is the quantum field theory version of Noether’s theorem.

This theorem is of fundamental significance as it tells us how to relate symmetries (under transformations of the general form (12.111)) to ‘current’

conservation laws (of the form (12.115), and it constructs the actual currents for us. In gauge theories, the dynamics is generated from a symmetry, in the sense that (as we have seen in the local U(1) of electromagnetism) the symmetry currents are the dynamical currents that drive the equations for the force field. Thus the symmetries of the Lagrangian are basic to gauge field theories.

Let us look at another example, this time involving spin-0 fields. Suppose we have three spin-0 fields all with the same mass, and take

Lˆ=1

2∂μφˆ1∂^μφˆ1+1

2∂μφˆ2∂^μφˆ2+1

2∂μφˆ3∂^μφˆ3−1

2m²( ˆφ²₁+ ˆφ²₂+ ˆφ²₃). (12.119) It is obvious that ˆL is invariant under an arbitrary rotation of the three ˆφ’s among themselves, generalizing the ‘rotation about the 3-axis’ considered for

'

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 29 the φˆ₁ −φˆ₂ system of section 7.1. An infinitesimal such rotation is (cf (12.64), and noting the sign change in the field theory case)

φˆ = φˆ + E × φˆ (12.120)

which implies

T ⁽¹⁾ ˆ

δφˆ_r = −iEa _ars φs, (12.121) with

T _ars ⁽¹⁾ = −iE_ars (12.122) as in (12.48). There are of course three conserved Tˆ operators again, and three

μ (1) (1)μ

Tˆ ’s, which we call Tˆ and Tˆ respectively, since we are now dealing with a T = 1 isospin case. The a = 1 component of the conserved current in this case is, from (12.116),

Tˆ₁ ^(1)μ = φˆ₂∂^μφˆ₃ − φˆ₃∂^μφˆ₂. (12.123) Cyclic permutations give us the other components which can be summarised as

ˆ^(1)μ φ^(1)tr T⁽¹⁾∂^μφˆ⁽¹⁾ − (∂^μφˆ⁽¹⁾)^tr T⁽¹⁾φˆ⁽¹⁾)

T = i( ˆ (12.124)

where we have written

φˆ⁽¹⁾ =

⎛

⎝ φˆ₁ φˆ₂ φˆ3

⎞

⎠ (12.125)

and ^tr denotes transpose. Equation (12.124) has the form expected of a bosonic spin-0 current, but with the matrices T⁽¹⁾ appearing, appropriate to the T = 1 (triplet) representation of SU(2)_f .

The general form of such SU(2) currents should now be clear. For an isospin T -multiplet of bosons we shall have the form

φ^{(T )†}T^{(T )}∂^μφˆ^{(T )})− (∂^μφˆ^{(T )})^†T^{(T )}φˆ^{(T )})

i( ˆ (12.126)

where we have put the † to allow for possibly complex fields; and for an isospin T -multiplet of fermions we shall have

ψ¯ˆ^{(T )}γ^μT^{(T )}ψˆ^{(T )} (12.127) where in each case the (2T + 1) components of φˆ or ψˆ transforms as a T -multiplet under SU(2), i.e.

ψˆ^{(T )'} = exp(−iα · T^{(T )})ψˆ^{(T )} (12.128) and similarly for φˆ^{(T )}, where T^{(T )} are the 2T +1×2T +1 matrices representing the generators of SU(2)_f in this representation. In all cases, the integral over

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 31 SU(3)-invariant interactions can also be formed. A particularly impor-tant one is the ‘SU(3) dot-product’ of two octets (the analogues of the SU(2) triplets), which arises in the quark-gluon vertex of QCD (see chapters 13 and 14): In (12.136), ˆqf stands for the SU(3)c colour triplet

ˆ current·gauge field’ characteristic of all gauge interactions.

12.4.2 Chiral symmetry

As our final example of a global non-Abelian symmetry, we shall introduce the idea of chiral symmetry, which is an exact symmetry for fermions in the limit in which their masses may be neglected. We have seen that the u and d quarks have indeed very small masses (≤ 5 MeV) on hadronic scales, and even the s quark mass (∼100 MeV) is relatively small. Thus we may certainly expect some physical signs of the symmetry associated with mu ≈md ≈ 0, and possibly also of the larger symmetry holding when mu≈md≈ms≈0.

As we shall see, however, this expectation leads to a puzzle, the resolution of which will have to be postponed until the concept of ‘spontaneous symmetry breaking’ has been developed in Part VII.

We begin with the simplest case of just one fermion. Since we are interested in the ‘small mass’ regime, it is sensible to use the representation (3.40) of the Dirac matrices, in which the momentum part of the Dirac Hamiltonian is

‘diagonal’ and the mass appears as an ‘off-diagonal’ coupling:

α=

30 12. Global Non-Abelian Symmetries

all space of the μ= 0 component of these currents results in a triplet of isospin operators obeying the SU(2) algebra (12.47), as in (12.103).

The cases considered so far have all been free field theories, but SU(2)-invariant interactions can be easily formed. For example, the interaction g1ψ¯ˆτ ψˆ· φˆ describes SU(2)-invariant interactions between a T = ¹ ₂ isospinor (spin- ¹ ₂ ) field ψˆ, and a T = 1 isotriplet (Lorentz scalar) φˆ . An effective

inter-¯ˆ ˆ ˆ

action between pions and nucleons could take the form gπψτ γ5ψ· φ, allowing for the pseudoscalar nature of the pions (we shall see in the following section that ψγ¯ˆ ˆ5ψ is a pseudoscalar, so the product is a true scalar as is required for a parity-conserving strong interaction). In these examples the ‘vector’ analogy for the T = 1 states allows us to see that the ‘dot product’ will be invariant.

as will be discussed in the following chapter. This is just the SU(2) dot product of the symmetry current (12.109) and the gauge field triplet, both of which are in the adjoint (T = 1) representation of SU(2). as well as the usual global U(1) transformation associated with quark number conservation. The associated Noether currents are (in somewhat informal notation)

30 12. Global Non-Abelian Symmetries all space of theμ= 0 component of these currents results in a triplet of isospin operators obeying the SU(2) algebra (12.47), as in (12.103).

The cases considered so far have all been free field theories, but SU(2)-invariant interactions can be easily formed. For example, the interaction g1ψτ¯ˆ ψˆ·φˆ describes SU(2)-invariant interactions between a T = ¹₂ isospinor (spin- ¹₂) field ˆψ, and aT = 1 isotriplet (Lorentz scalar) ˆφ. An effective inter-action between pions and nucleons could take the formgπψτ¯ˆ γ5ψˆ·φ, allowingˆ for the pseudoscalar nature of the pions (we shall see in the following section thatψγ¯ˆ 5ψˆis a pseudoscalar, so the product is a true scalar as is required for a parity-conserving strong interaction). In these examples the ‘vector’ analogy for theT = 1 states allows us to see that the ‘dot product’ will be invariant.

A similar dot product occurs in the interaction between the isospinor ˆψ⁽¹²⁾ and the weak SU(2) gauge field ˆWμ, which has the form

g¯ˆqγ^μτ

2qˆ·Wˆ μ (12.129)

as will be discussed in the following chapter. This is just the SU(2) dot product of the symmetry current (12.109) and the gauge field triplet, both of which are in the adjoint (T = 1) representation of SU(2).

All of the foregoing can be generalized straightforwardly to SU(3)f. For example, the Lagrangian

describes free u, d and s quarks of equal massm. ˆLis clearly invariant under global SU(3)f transformations

ˆ

q^' = exp(−iα·λ/2)ˆq, (12.132) as well as the usual global U(1) transformation associated with quark number conservation. The associated Noether currents are (in somewhat informal notation)

Gˆ^(q)μ_a = ¯ˆqγ^μλa

2 qˆ a= 1,2, . . .8 (12.133) (note that there are eight of them), and the associated conserved ‘charge operators’ are

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 31 SU(3)-invariant interactions can also be formed. A particularly impor-tant one is the ‘SU(3) dot-product’ of two octets (the analogues of the SU(2) triplets), which arises in the quark-gluon vertex of QCD (see chapters 13 and 14):

We begin with the simplest case of just one fermion. Since we are interested in the ‘small mass’ regime, it is sensible to use the representation (3.40) of the Dirac matrices, in which the momentum part of the Dirac Hamiltonian is

‘diagonal’ and the mass appears as an ‘off-diagonal’ coupling:

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 33 In the massless limit, the chirality of ˜φ and ˜χis a good quantum number (γ5 commuting with the energy operator), and we may say that ‘chirality is conserved’ in this massless limit. On the other hand, the massive spinorω is clearlynotan eigenstate of chirality:

Referring to (12.140) and (12.141), we may therefore regard the mass terms as ‘coupling the states of different chirality’.

‘R’, ‘L’ is used for thechiralityeigenvalue.

We now reformulate the above in field-theoretic terms. The Dirac La-grangian for a single massless fermion is

Lˆ⁰=ψi¯ˆ /∂ψ.ˆ (12.151) This is invariant not only under the now familiar global U(1) transformation ψˆ→ψˆ^'= e^−iαψ, but also under the ‘globalˆ chiralU(1)’ transformation

ψˆ→ψˆ^' = e^−iθγ5ψˆ (12.152) where θ is an arbitrary (x-independent) real parameter. The invariance is easily verified: using {γ⁰, γ5}= 0 we have

32 12. Global Non-Abelian Symmetries

We now recall the matrix γ5 introduced in section 4.2.1 φ, χ become helicity eigenstates (cf problem 9.4), having definite ‘handedness’.

As m →0 we have E → |p|, and (12.140) and (12.141) reduce to a massless particle. Also in this limit, the Dirac energy operator is

( σ ·p 0 )

32 12. Global Non-Abelian Symmetries We now recall the matrixγ5 introduced in section 4.2.1

γ5= iγ⁰γ¹γ²γ³, (12.142) in this representation. The matrixγ5plays a prominent role in chiral symme-try, as we shall see. Its defining property is that it anticommutes with theγ^μ matrices:

{γ5, γ^μ}= 0. (12.144)

‘Chirality’ means ‘handedness’, from the Greek word for hand, χEiρ. Its use here stems from the fact that, in the limitm→0 the 2-component spinors φ, χbecome helicity eigenstates (cf problem 9.4), having definite ‘handedness’.

Asm→0 we haveE→ |p|, and (12.140) and (12.141) reduce to

(σ·p/|p|) ˜φ = φ˜ (12.145) (σ·p/|p|) ˜χ = −χ,˜ (12.146) so that the limiting spinor ˜φhas positive helicity, and ˜χ negative helicity (cf (3.68) and (3.69)). In thism→0 limit, the two helicity spinors aredecoupled, reflecting the fact that no Lorentz transformation can reverse the helicity of a massless particle. Also in this limit, the Dirac energy operator is

α·p=

( σ·p 0 0 −σ·p

)

(12.147) which is easily seen to commute withγ5. Thus the massless states may equiva-lently be classified by the eigenvalues ofγ5, which are clearly±1 sinceγ₅²=I.

Consider then a massless fermion with positive helicity. It is described by the ‘u’-spinor

( φ˜ 0

)

which is an eigenstate of γ5 with eigenvalue +1.

Similarly, a fermion with negative helicity is described by ( 0

˜ χ

)

which has γ5 =−1. Thus for these states chirality equals helicity. We have to be more careful for antifermions, however. A physical antifermion of energy E and momentumpis described by a ‘v’- spinor corresponding to−E and−p; but with m = 0 in (12.140) and (12.141) the equations forφ and χ remain the same for −E,−p as for E,p. Consider the spin, however. If the physical antiparticle has positive helicity, withp along thez-axis say, then sz= +¹₂. The correspondingv-spinor must then have sz =−¹2 (see section 3.4.3) and must therefore be of ˜χ type (12.146). So the v-spinor for this antifermion of positive helicity is γ5 eigenvalue is equal to the helicity, and for antifermions it is equal to minus the helicity. It is theγ5 eigenvalue that is called the ‘chirality’.

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 33 In the massless limit, the chirality of φ˜ and ˜χ is a good quantum number (γ5 commuting with the energy operator), and we may say that ‘chirality is conserved’ in this massless limit. On the other hand, the massive spinor ω is clearly not an eigenstate of chirality:

(

φ ) ( φ )

γ5ω = /= λ . (12.148)

−χ χ

Referring to (12.140) and (12.141), we may therefore regard the mass terms as ‘coupling the states of different chirality’.

as required. The corresponding Noether current is

ˆμ ¯ˆ ˆ

j₅ = ψγ^μγ5ψ, (12.155)

12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 35 Equation (12.158) then implies that ˆPQˆ5Pˆ⁻¹ =−Qˆ5, following the normal rule for operator transformations in quantum mechanics. Consider now the state ˆQ5|+>. We have

PˆQˆ5|+> = (

PˆQˆ5Pˆ⁻¹) Pˆ|+>

= −Qˆ5|+> (12.160)

showing that ˆQ5|+>is an eigenstate of ˆPwith the opposite eigenvalue, -1.

A very important physical consequence now follows from the fact that (in this simple m = 0 model) ˆQ5 is a symmetry operator commuting with the Hamiltonian ˆH. We have

HˆQˆ5|ψ>= ˆQ5Hˆ|ψ>=EQˆ5|ψ>. (12.161) Hence for every state|ψ>with energy eigenvalueE, there should exist a state Qˆ5|ψ> with the same eigenvalue E and the opposite parity: that is, chiral symmetry apparently implies the existence of ‘parity doublets’.

Of course, it may reasonably be objected that all of the above refers not only to the massless, but also the non-interactingcase. However, this is just where the analysis begins to get interesting. Suppose we allow the fermion field ˆψto interact with a U(1)-gauge field ˆA^μvia the standard electromagnetic coupling

Lˆ^int =qψγ¯ˆ ^μψˆAˆμ. (12.162) Remarkably enough, ˆL^int is also invariant under the chiral transformation (12.152), for the simple reason that the ‘Dirac’ structure of (12.162) is exactly the same as that of the free kinetic term ψ¯ˆ /∂ψ: the ‘covariant derivative’ˆ prescription∂^μ→D^μ=∂^μ+ iqAˆ^μautomatically means that any ‘Dirac’ (e.g.

γ5) symmetry of the kinetic part will be preserved when the gauge interaction is included. Thus chirality remains a ‘good symmetry’ in the presence of a

γ5) symmetry of the kinetic part will be preserved when the gauge interaction is included. Thus chirality remains a ‘good symmetry’ in the presence of a

Im Dokument Non-Abelian Gauge Theories (Seite 40-53)