Local non-Abelian symmetries in Lagrangian quantum field theory

13.3.1 Local SU(2) and SU(3) Lagrangians

We consider here only the particular examples relevant to the strong and elec-troweak interactions of quarks: namely, a (weak) SU(2) doublet of fermions in-teracting with SU(2) gauge fields W _i μ, and a (strong) SU(3) triplet of fermions interacting with the gauge fields A^μ _a . We follow the same steps as in the U(1) case of chapter 7, noting again that for quantum fields the sign of the expo-nents in (13.2) and (13.52) is reversed, by convention; thus (12.89) is replaced

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 53 in which our local SU(2) Lagrangian is not suitable (yet) for describing weak interactions. First, weak interactions violate parity, in fact ‘maximally’, by which is meant that only the ‘left-handed’ part ˆψL of the fermion field enters the interactions with the W^μ fields, where ˆψL ≡ (1−γ5

2

)ψ; for this reasonˆ the weak isospin group is called SU(2)L. Secondly, the physical W^± are of course not massless, and therefore cannot be described by propagators of the form (13.69). And thirdly, the fermion mass term violates the ‘left-handed’

SU(2) gauge symmetry, as the discussion in section 12.3.2 shows. In this case, however, the chiral symmetry which is broken by fermion masses in the Lagrangian is a local, or gauge, symmetry (in section 12.3.2 the chiral flavour symmetry was a global symmetry). If we want to preserve the chiral gauge symmetry SU(2)L – and it is necessary for renormalizability – then we shall have to replace the simple fermion mass term in (13.66) by something else, as will be explained in chapter 22.

The locally SU(3)c-invariant Lagrangian for one quark triplet (cf (12.137))

ˆ

where ‘f’ stands for ‘flavour’, and ‘r, b, and g’ for ‘red, blue, and green’, is

¯ˆ

q_f(i ˆD/ −mf)ˆqf−1

4FˆaμνFˆ_a^μν− 1

2ξ(∂μAˆ^μ_a)(∂νAˆ^ν_a) (13.71) where ˆD^μ is given by (13.53) with A^μ replaced by ˆA^μ, and the footnote before equation (13.68) also applies here. This leads to the interaction term (cf (13.59))

−gs¯ˆq_fγ^μλ/2ˆqf·Aˆμ (13.72) and the Feynman rule (13.60) for figure 13.2. Once again, the gluon quanta must be massless, and their propagator is the same as (13.69), with δij → δab (a, b = 1,2, . . .8). The different quark flavours are included by simply repeating the first term of (13.71) for all flavours:

E

f

¯ˆ

q_f(i ˆD/ −mf)ˆqf, (13.73) which incorporates the hypothesis that the SU(3)_c-gauge interaction is ‘flavour-blind’, i.e. exactly the same for each flavour. Note that although the flavour masses are different, the masses of different ‘coloured’ quarks of the same flavour are the same (mu/=md, mu,r=mu,b=mu,g).

The Lagrangians (13.66)–(13.68), and (13.71), though easily written down after all this preparation, are unfortunately not adequate for anything but tree graphs. We shall indicate why this is so in section 13.3.3. Before that, we want to discuss in more detail the nature of the gauge-field self-interactions contained in the Yang-Mills pieces.

' '

52 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.3 Correspondingly, the E in (13.23) and theη’s in (13.56) become field operators, with a reversal of sign.

The globally invariant Lagrangian (12.87) becomes locally SU(2)-invariant if we replaced ∂^μ by D^μ of (13.10), with Wˆ^μ now a quantum field:

ˆ = ¯ˆ Dˆ − m)ˆq

= ¯qˆ(i /∂ − m)ˆq − g ¯qγ^μτ /2ˆq · Wˆ _μ LD,local SU(2) q(i /

ˆ (13.66)

with an interaction of the form ‘symmetry current (12.109) dotted into the gauge field’. To this we must add the SU(2) Yang-Mills term invariant under the gauge transformations (13.26) or (13.34) of the W-fields.

Thus, just as in the U(1) (electromagnetic) case, the W-quanta of this theory are massless. We presumably also need a gauge-fixing term for the gauge

The Feynman rule for the fermion-W vertex is then the same as already given in (13.41), while the W-propagator is (figure 13.3)

i[

−g^μν + (1 − ξ)k^μk^ν /k²]

δ^ij . (13.69)

k² + iE

Before proceeding to the SU(3) case, we must now emphasize three respects

1We shall see in section 13.5.3 that in the non-Abelian case this gauge-fixing term does not completely solve the problem of quantizing such gauge fields; however, it is adequate for tree graphs.

52 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.3

SU(2) gauge-boson propagator.

by its local version

ˆ

q^'= exp(−igα(x)ˆ ·τ/2)ˆq (13.64) and (12.132) by

ˆ

q^'= exp(−igsα(x)ˆ ·λ/2)ˆq. (13.65) Correspondingly, theEin (13.23) and theη’s in (13.56) become field operators, with a reversal of sign.

The globally invariant Lagrangian (12.87) becomes locally SU(2)-invariant if we replaced∂^μ byD^μ of (13.10), with ˆW^μ now a quantum field:

LˆD,local SU(2) = ¯ˆq(iD/ˆ −m)ˆq

= ¯ˆq(i/∂−m)ˆq−g¯ˆqγ^μτ/2ˆq·Wˆ μ (13.66) with an interaction of the form ‘symmetry current (12.109) dotted into the gauge field’. To this we must add the SU(2) Yang-Mills term

L^Y−M,SU(2)=−1

4Fˆμν·Fˆ^μν (13.67) to get the local SU(2) analogue ofL^QED. It isnotpossible to add a mass term for the gauge fields of the form ¹₂Wˆ ^μ·Wˆ μ, since such a term would not be invariant under the gauge transformations (13.26) or (13.34) of the W-fields.

Thus, just as in the U(1) (electromagnetic) case, the W-quanta of this theory are massless. We presumably also need a gauge-fixing term for the gauge fields, as in section 7.3.2, which we can take to be¹

L^gf=−1 2ξ

(∂μWˆ ^μ·∂νWˆ ^ν)

. (13.68)

The Feynman rule for the fermion-W vertex is then the same as already given in (13.41), while the W-propagator is (figure 13.3)

i[

−g^μν+ (1−ξ)k^μk^ν/k²]

k²+ iE δ^ij. (13.69)

Before proceeding to the SU(3) case, we must now emphasize three respects

1We shall see in section 13.5.3 that in the non-Abelian case this gauge-fixing term does notcompletely solve the problem of quantizing such gauge fields; however, it is adequate for tree graphs.

E

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 53 in which our local SU(2) Lagrangian is not suitable (yet) for describing weak interactions. First, weak interactions violate parity, in fact ‘maximally’, by which is meant that only the ‘left-handed’ part ψˆ_L of the fermion field enters the interactions with the W ^μ fields, where ψˆ_L ≡ (1−γ₅ )

ψˆ; for this reason the weak isospin group is called SU(2)L. Secondly, the physical W2 ^± are of course not massless, and therefore cannot be described by propagators of the form (13.69). And thirdly, the fermion mass term violates the ‘left-handed’

SU(2) gauge symmetry, as the discussion in section 12.3.2 shows. In this case, however, the chiral symmetry which is broken by fermion masses in the Lagrangian is a local, or gauge, symmetry (in section 12.3.2 the chiral flavour symmetry was a global symmetry). If we want to preserve the chiral gauge symmetry SU(2)L – and it is necessary for renormalizability – then we shall have to replace the simple fermion mass term in (13.66) by something else, as will be explained in chapter 22.

The locally SU(3)c-invariant Lagrangian for one quark triplet (cf (12.137))

qˆ_f =

⎛

⎝ fˆ_r fˆb

fˆ_g

⎞

⎠, (13.70)

where ‘f’ stands for ‘flavour’, and ‘r, b, and g’ for ‘red, blue, and green’, is

¯ ˆ Fˆ^μν ˆ ˆ

qˆ_f (i/ˆ 1

Faμν − 1

(∂μA^μ)(∂ν A^ν ) (13.71) D− mf )ˆqf −4 a 2ξ a a

where Dˆ^μ is given by (13.53) with A^μ replaced by Aˆ , and the footnote μ

before equation (13.68) also applies here. This leads to the interaction term (cf (13.59))

−gsq¯ˆ_f γ^μλ/2ˆqf · Aˆ_μ (13.72) and the Feynman rule (13.60) for figure 13.2. Once again, the gluon quanta must be massless, and their propagator is the same as (13.69), with δij → δab (a, b = 1,2, . . .8). The different quark flavours are included by simply repeating the first term of (13.71) for all flavours:

¯qˆ_f (iD/ˆ − mf )ˆqf , (13.73)

f

which incorporates the hypothesis that the SU(3) -gauge interaction is ‘flavour-_c blind’, i.e. exactly the same for each flavour. Note that although the flavour masses are different, the masses of different ‘coloured’ quarks of the same flavour are the same (m_u /= md, m_u,r = m_u,b = mu,g).

The Lagrangians (13.66)–(13.68), and (13.71), though easily written down after all this preparation, are unfortunately not adequate for anything but tree graphs. We shall indicate why this is so in section 13.3.3. Before that, we want to discuss in more detail the nature of the gauge-field self-interactions contained in the Yang-Mills pieces.

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 55 necessity for finding room in the scheme for the neutral weak boson Z⁰ as well. We shall see how this works in chapter 19; meanwhile we continue with this X−γmodel. We shall show that when the X−γ interaction contained in (13.79) is regarded as a 3−X vertex in a local SU(2) gauge theory, the value of δhas to equal 1; for this value the theory is renormalizable. In this interpretation, the X^μ wave function is identified with ‘^√1₂(X₁^μ+ iX₂^μ)’ and X¯^μ with ‘^√1₂(X₁^μ−iX₂^μ)’ in terms of components of the SU(2) triplet X_i^μ, whileA^μis identified with X₃^μ.

Consider then equation (13.79) written in the form²

0X^μ−∂^ν∂^μXν = ˆV X^μ (13.80) where

V Xˆ ^μ = −ie{[∂^ν(AνX^μ) +A^ν∂νX^μ]

−(1 +δ) [∂^ν(A^μXν) +A^ν∂^μXν]

+δ[∂^μ(A^νXν) +A^μ∂^νXν]}, (13.81) and we have dropped terms ofO(e²) which appear in the ‘D²’ term; we shall come back to them later. The terms inside the{ }brackets have been written in such a way that each [ ] bracket has the structure

∂(AX) +A(∂X) (13.82)

which will be convenient for the following evaluation.

The lowest-order (O(e)) perturbation theory amplitude for ‘X→X’ under the potential ˆV is then

−i /

X_μ^∗(f) ˆV X^μ(i)d⁴x. (13.83) Inserting (13.81) into (13.83) clearly gives something involving two ‘X ’-wave-functions and one ‘A’ one, i.e. a triple-X vertex (with A^μ ≡X₃^μ), shown in figure 13.4. To obtain the rule for this vertex from (13.83), consider the first [ ] bracket in (13.81). It contributes

−i(−ie) /

X_μ^∗(2){∂^ν(X3ν(3)X^μ(1)) +X₃^ν(3)∂νX^μ(1)}d⁴x (13.84) where the (1), (2), (3) refer to the momenta as shown in figure 13.4, and for reasons of symmetry are all taken to be ingoing; thus

X₃^μ(3) =E^μ₃exp(−ik3·x) (13.85)

2The sign chosen for ˆV here apparently differsfrom that in the KG case (3.101), but it does agree when allowance is made, in the amplitude (13.83), for the fact that the dot product of the polarization vectors is negative (cf (7.87)).

54 13. Local Non-Abelian (Gauge) Symmetries

13.3.2 Gauge field self-interactions

We start by pointing out an interesting ambiguity in the prescription for

‘covariantizing’ wave equations which we have followed, namely ‘replace ∂^μ by D^μ’. Suppose we wished to consider the electromagnetic interactions of charged massless spin-1 particles, call them X’s, carrying charge e. The stan-dard wave equation for such free massless vector particles would be the same as for A^μ, namely

oX^μ − ∂^μ∂^ν Xν = 0. (13.74) To ‘covariantize’ this (i.e. introduce the electromagnetic coupling) we would replace ∂^μ by D^μ = ∂^μ + ieA^μ so as to obtain

D²X^μ − D^μD^ν X_ν = 0. (13.75) But this procedure is not unique: if we had started from the perfectly equiv-alent wave equation

oX^μ − ∂^ν ∂^μXν = 0 (13.76) we would have arrived at

D²X^μ − D^ν D^μX_ν = 0 (13.77) which is not the same as (13.75), since (cf (13.45))

[D^μ, D^ν ] = ieF ^μν . (13.78) The simple prescription ∂^μ → D^μ has, in this case, failed to produce a unique wave equation. We can allow for this ambiguity by introducing an arbitrary parameter δ in the wave equation, which we write as

D²X^μ − D^ν D^μXν + ieδF ^μν Xν = 0. (13.79) The δ term in (13.79) contributes to the magnetic moment coupling of the X-particle to the electromagnetic field, and is called the ‘ambiguous magnetic moment’. Just such an ambiguity would seem to arise in the case of the charged weak interaction quanta W^± (their masses do not affect this argu-ment). For the photon itself, of course, e = 0 and there is no such ambiguity.

It is important to be clear that (13.79) is fully U(1) gauge-covariant, so that δ cannot be fixed by further appeal to the local U(1) symmetry. Moreover, it turns out that the theory for arbitrary δ is not renormalizable (though we shall not show this here): thus the quantum electrodynamics of charged massless vector bosons is in general non-renormalizable.

However, the theory is renormalizable if – to continue with the present terminology – the photon, the X-particle, and its antiparticle the X¯ are the members of an SU(2) gauge triplet (like the W’s), with gauge coupling con-stant e. This is, indeed, very much how the photon and the W^± are ‘unified’, but there is a complication (as always!) in that case, having to do with the

54 13. Local Non-Abelian (Gauge) Symmetries 13.3.2 Gauge field self-interactions

We start by pointing out an interesting ambiguity in the prescription for

‘covariantizing’ wave equations which we have followed, namely ‘replace ∂^μ by D^μ’. Suppose we wished to consider the electromagnetic interactions of charged massless spin-1 particles, call them X’s, carrying chargee. The stan-dard wave equation for such free massless vector particles would be the same as forA^μ, namely

oX^μ−∂^μ∂^νXν= 0. (13.74)

To ‘covariantize’ this (i.e. introduce the electromagnetic coupling) we would replace∂^μ byD^μ=∂^μ+ ieA^μ so as to obtain

D²X^μ−D^μD^νXν = 0. (13.75) But this procedure is not unique: if we had started from the perfectly equiv-alent wave equation

oX^μ−∂^ν∂^μXν= 0 (13.76)

we would have arrived at

D²X^μ−D^νD^μXν = 0 (13.77) which is not the same as (13.75), since (cf (13.45))

[D^μ, D^ν] = ieF^μν. (13.78) The simple prescription ∂^μ → D^μ has, in this case, failed to produce a unique wave equation. We can allow for this ambiguity by introducing an arbitrary parameterδin the wave equation, which we write as

D²X^μ−D^νD^μXν+ ieδF^μνXν = 0. (13.79) The δ term in (13.79) contributes to the magnetic moment coupling of the X-particle to the electromagnetic field, and is called the ‘ambiguous magnetic moment’. Just such an ambiguity would seem to arise in the case of the charged weak interaction quanta W^± (their masses do not affect this argu-ment). For the photon itself, of course,e= 0 and there is no such ambiguity.

It is important to be clear that (13.79) is fully U(1) gauge-covariant, so that δcannot be fixed by further appeal to the local U(1) symmetry. Moreover, it turns out that the theory for arbitraryδisnot renormalizable(though we shall not show this here): thus the quantum electrodynamics of charged massless vector bosons is in general non-renormalizable.

However, the theory is renormalizable if – to continue with the present terminology – the photon, the X-particle, and its antiparticle the ¯X are the members of an SU(2) gauge triplet (like the W’s), with gauge coupling con-stante. This is, indeed, very much how the photon and the W^±are ‘unified’, but there is a complication (as always!) in that case, having to do with the

/

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 55 necessity for finding room in the scheme for the neutral weak boson Z⁰ as well. We shall see how this works in chapter 19; meanwhile we continue with this X −γ model. We shall show that when the X−γ interaction contained in (13.79) is regarded as a 3 −X vertex in a local SU(2) gauge theory, the value of δ has to equal 1; for this value the theory is renormalizable. In this

μ 1 1 μ

√

μ μ μ

interpretation, the X^μ wave function is identified with ‘ ₂(X + iX₂)’ and X¯^μ

while A^μ is identified with X

√ 1 −iX

with ‘ ₂(X_{1 2})’ in terms of components of the SU(2) triplet X_i ,

μ .

3

Consider then equation (13.79) written in the form²

0X^μ −∂^ν ∂^μX_ν = V Xˆ ^μ (13.80) where

V Xˆ ^μ = −ie{[∂^ν (Aν X^μ) + A^ν ∂ν X^μ]

−(1 + δ) [∂^ν (A^μXν ) + A^ν ∂^μXν ]

+ δ [∂^μ(A^ν X_ν ) + A^μ∂^ν X_ν ]}, (13.81) and we have dropped terms of O(e²) which appear in the ‘D²’ term; we shall come back to them later. The terms inside the { }brackets have been written in such a way that each [ ] bracket has the structure

∂(AX) + A(∂X) (13.82) which will be convenient for the following evaluation.

The lowest-order (O(e)) perturbation theory amplitude for ‘X →X’ under the potential Vˆ is then

/ X ^∗

−i _μ(f) ˆV X^μ(i)d⁴ x. (13.83) Inserting (13.81) into (13.83) clearly gives something involving two

‘X’-wave-3

3

≡X^μ

figure 13.4. To obtain the rule for this vertex from (13.83), consider the first [ ] bracket in (13.81). It contributes

μ(2){∂^ν(X_3ν (3)X^μ(1)) + X^ν

functions and one ‘A’ one, i.e. a triple-X vertex (with A^μ ), shown in

X ^∗ (3)∂_ν X^μ(1)}d⁴

−i(−ie) x (13.84)

where the (1), (2), (3) refer to the momenta as shown in figure 13.4, and for reasons of symmetry are all taken to be ingoing; thus

μ μ

exp(−ik3

X₃(3) = E₃ ·x) (13.85)

2The sign chosen for Vˆ here apparently differs from that in the KG case (3.101), but it does agree when allowance is made, in the amplitude (13.83), for the fact that the dot product of the polarization vectors is negative (cf (7.87)).

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 57

FIGURE 13.5

Tree graphs contributing to X + d→X + d.

the positively charged X, and τ₋ = (τ1−iτ2)/√

2 for the emission of the X. Then figure 13.5(a) is

(−ie)²ψ¯⁽¹²⁾(p2)τ₋

2 /e2 i /p1+/k1−m

τ+

2 /e1ψ⁽¹²⁾(p1) (13.91) where

ψ⁽¹²⁾= ( u

d )

, (13.92)

and we have chosen real polarization vectors. Using the explicit forms (12.25) for theτ-matrices, (13.91) becomes

(−ie)²d(p¯ 2) 1

√2 /e2 i /p1+/k1−m

√1

2 /e1d(p1). (13.93) We must now discuss how to implement gauge invariance. In the QED case of electron Compton scattering (section 8.6.2), the test of gauge invariance was that the amplitude should vanish if any photon polarization vectore^μ(k) was replaced byk^μ– see (8.165). This requirement was derived from the fact that a gauge transformation on the photonA^μtook the formA^μ→A^'μ=A^μ−∂^μχ, so that, consistently with the Lorentz condition, e^μ could be replaced by e^'μ=e^μ+βk^μ(cf 8.163) without changing the physics. But the SU(2) analogue of the U(1) gauge transformation is given by (13.26), for infinitesimale’s, and although there is indeed an analogous ‘−∂^μE’ part, there is also an additional part (with g → e in our case) expressing the fact that the X’s carry SU(2) charge. However this extra part does involve the couplinge. Hence, if we were to make the fullchange corresponding to (13.26) in a tree graph of ordere², the extra part would produce a term of ordere³. We shall take the view that gauge invariance should hold at each order of perturbation theory separately;

thus we shall demand that the tree graphs for X-d scattering, for example, should be invariant undere^μ→k^μ for anye.

The replacemente1→k1in (13.93) produces the result (problem 13.9) (−ie)²i

2d(p¯ 2)/e2d(p1) (13.94)

56 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.4 Triple-X vertex.

for example. The first term in (13.84) can be easily evaluated by a partial integration to turn the ∂^ν onto the X_μ^∗(2), while in the second term ∂_ν acts straightforwardly on X^μ(1). Omitting the usual (2π)⁴ δ⁴ energy-momentum conserving factor, we find (problem 13.8) that (13.84) leads to the amplitude ie 1 · 2 (k1 − k2)· ³. (13.86) In a similar way, the other terms in (13.83) give

−ieδ( 1 · ³ 2 · k2 − 2 · ³ 1 · k1) (13.87) and

+ie(1 + δ)( ₂ · ³ 1 · k2 − 1 · ³ 2 · k1). (13.88) Adding all the terms up and using the 4-momentum conservation condition

k1 + k2 + k3 = 0 (13.89) we obtain the vertex

+ie{ 1 · 2 (k₁ −k2)· 3 + ₂ · 3 (δk₂ − k3)· 1 + ₃ · 1 (k₃ −δk1)· ²}. (13.90) It is quite evident from (13.90) that the value δ = 1 has a privileged role, and we strongly suspect that this will be the value selected by the proposed SU(2) gauge symmetry of this model. We shall check this in two ways: in the first, we consider a ‘physical’ process involving the vertex (13.90), and show how requiring it to be SU(2)-gauge invariant fixes δ to be 1; in the second, we

‘unpack’ the relevant vertex from the compact Yang-Mills Lagrangian −¹ 4Xˆ_μν · ˆμν

X .

The process we shall choose is X + d → X + d where d is a fermion (which we call a quark) transforming as the T₃ = −¹ ₂ component of a doublet under the SU(2) gauge group, its T₃ = + partner being the u. There are two 1₂

contributing Feynman graphs, shown in figure 13.5(a) and (b). Consider first the amplitude for figure 13.5(a). We use the rule of figure 13.1, with the √ τ-matrix combination τ₊ = (τ₁ + iτ2)/ 2 corresponding to the absorption of

56 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.4 Triple-X vertex.

for example. The first term in (13.84) can be easily evaluated by a partial integration to turn the∂^ν onto theX_μ^∗(2), while in the second term ∂ν acts straightforwardly onX^μ(1). Omitting the usual (2π)⁴ δ⁴ energy-momentum conserving factor, we find (problem 13.8) that (13.84) leads to the amplitude ie 1· ² (k1−k2)· ³. (13.86) In a similar way, the other terms in (13.83) give

−ieδ( 1· ^{3 2}·k2− ²· ^{3 1}·k1) (13.87) and

+ie(1 +δ)(2· ^{3 1}·k2− ¹· ^{3 2}·k1). (13.88) Adding all the terms up and using the 4-momentum conservation condition

k1+k2+k3= 0 (13.89)

we obtain the vertex

+ie{ ¹· ²(k1−k2)· ³+ 2· ³(δk2−k3)· ¹+ 3· ¹(k3−δk1)· ²}. (13.90) It is quite evident from (13.90) that the value δ= 1 has a privileged role, and we strongly suspect that this will be the value selected by the proposed SU(2) gauge symmetry of this model. We shall check this in two ways: in the first, we consider a ‘physical’ process involving the vertex (13.90), and show how requiring it to be SU(2)-gauge invariant fixesδto be 1; in the second, we

‘unpack’ the relevant vertex from the compact Yang-Mills Lagrangian−¹4Xˆμν· Xˆ^μν.

The process we shall choose is X + d→X + d where d is a fermion (which we call a quark) transforming as theT3=−¹2 component of a doublet under the SU(2) gauge group, its T3 = +¹₂ partner being the u. There are two contributing Feynman graphs, shown in figure 13.5(a) and (b). Consider first the amplitude for figure 13.5(a). We use the rule of figure 13.1, with the τ-matrix combination τ+ = (τ1+ iτ2)/√

2 corresponding to the absorption of

/ /

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 57

FIGURE 13.5

Tree graphs contributing to X + d → X + d.

the positively charged X, and τ− = (τ1 − iτ2)/ √2 for the emission of the X.

Then figure 13.5(a) is

τ− i τ+

1 2

1

(−ie)²ψ¯^{( )}(p2) e2 e1ψ^{( )}2 (p1) (13.91) /p1+ /k1 − m

2 2

where ( u )

1

ψ^{( )} 2 = d , (13.92)

and we have chosen real polarization vectors. Using the explicit forms (12.25) for the τ-matrices, (13.91) becomes

1 i 1

(−ie)²d¯(p2)√ /e2 √ /e1d(p1). (13.93) 2 /p1+ /k1 − m 2

We must now discuss how to implement gauge invariance. In the QED case of electron Compton scattering (section 8.6.2), the test of gauge invariance was that the amplitude should vanish if any photon polarization vector e^μ(k) was replaced by k^μ – see (8.165). This requirement was derived from the fact that a gauge transformation on the photon A^μ took the form A^μ → A^'μ = A^μ −∂^μχ, so that, consistently with the Lorentz condition, e^μ could be replaced by e^'μ = e^μ+βk^μ (cf 8.163) without changing the physics. But the SU(2) analogue of the U(1) gauge transformation is given by (13.26), for infinitesimal e’s, and although there is indeed an analogous ‘−∂^μE’ part, there is also an additional part (with g → e in our case) expressing the fact that the X’s carry SU(2) charge. However this extra part does involve the coupling e. Hence, if we were to make the full change corresponding to (13.26) in a tree graph of order e , the extra part would produce a term of order e3. We shall take the view that gauge invariance should hold at each order of perturbation theory separately;

thus we shall demand that the tree graphs for X-d scattering, for example, should be invariant under e^μ → k^μ for any e.

The replacement e1 → k1 in (13.93) produces the result (problem 13.9) (−ie)² i

d¯(p2)/e2d(p1) (13.94) 2

2

13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 59

γ γ

X X

FIGURE 13.7 γ−γ−X−X vertex.

applied to figures 13.6 and 13.7, but it has to be admitted that this approach is becoming laborious. It is, of course, far more efficient to deduce the vertices from the compact Yang-Mills Lagrangian−¹4Xˆμν·Xˆ^μν, which we shall now

applied to figures 13.6 and 13.7, but it has to be admitted that this approach is becoming laborious. It is, of course, far more efficient to deduce the vertices from the compact Yang-Mills Lagrangian−¹4Xˆμν·Xˆ^μν, which we shall now

Im Dokument Non-Abelian Gauge Theories (Seite 66-85)