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QCD, Background Field Method, and Anomalies

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QCD, Background Field Method, and Anomalies 10.12.2020 QCD, background field method, and anomalies (Lecture notes - a.a. 2020/21) Fiorenzo Bastianelli 1 Feynman rules for perturbative QCD Having obtained the gauge-fixed action for the non-abelian gauge theories, one can write down the QCD gauge-fixed lagrangian by adding the fermions (the quark fields). Using an arbitrary Rξ, gauge one obtains the following gauge-fixed lagrangian 6 1 ξ X L = − F a F aµν − Ba@µAa + (Ba)2 − @µc¯aD ca − (γµD + m ) : (1) tot 4 µν µ 2 µ f µ f f f=1 Eliminating the auxiliary field Ba one finds 6 1 1 X L = − F a F aµν − (@µAa )2 − @µc¯aD ca − (γµD + m ) : (2) tot 4 µν 2ξ µ µ f µ f f f=1 This lagrangian can be split as Ltot = L2 + Lint, with L2 the quadratic part used to find the propagators, and Lint the interacting part used to get the vertices. Recalling that a a a abc b c Fµν = @µAν − @νAµ + gf AµAν (3) one finds 6 1 1 1 X L = − (@ Aa)2 + ( − )(@µAa )2 − @µc¯a@ ca − (γµ@ + m ) (4) 2 2 µ ν 2 2ξ µ µ f µ f f f=1 and g2 L = − gf abc@µAνaAb Ac − f abcf adeAb Ac AµdAνe int µ ν 4 µ ν 6 (5) abc µ a b c a X µ a − gf @ c¯ Aµc + igAµ f γ T f : f=1 Propagators It is immediate to extract the propagators from (4). We recall that the perturbative propagators are the inverse of the kinetic terms: in an hypercondensed notation (for real fields φi and i complex fields ; i) we have the formulae 1 i j i j ij jk k S[φ] = − φ Kijφ ! hφ φ i = −iG (KijG = δ ) 2 i (6) ¯ ¯ i j i ¯ i i j i S[ ; ] = − iK j ! h ji = −iG j (K jG k = δk) valid for either commuting or anticommuting fields. 1 Thus, from L2 we get Z d4p δab p p hAa (x)Ab (y)i = −i eip(x−y) η − (1 − ξ) µ ν µ ν (2π)4 p2 − i µν p2 Z d4p δab hca(x)¯cb(y)i = −i eip(x−y) (7) (2π)4 p2 − i Z d4p −ip= + m h i − ip·(x−y) f f (x) f (y) = i 4 e 2 2 (2π) p + mf − i with no sum over f (the quark flavours), from which one extracts the propagators in momentum space δab p p − i (η − (1 − ξ) µ ν p2 − i µν p2 δab − i (8) p2 − i −ip= + m − f i 2 2 p + mf − i Vertices As for the vertices, one can obtain the elementary Feynman rules by Fourier transforming the various monomials contained in Lint. Let us review our conventions by presenting the definition of the Fourier transform of the effective action for a field ' (the quantum action in Srednicki), and the proper vertices it gives rise to (the amputated 1PI Feynman diagrams). The effective action Γ['] is expanded in a Taylor series as 1 X 1 Z Γ['] = dDx : : : dDx Γ(x ; :::; x )'(x ):::'(x ) n! 1 n 1 n 1 n n=0 1 Z D D X 1 d p1 d pn = ··· Γ(~ p ; :::; pn)' ~(p ):::'~(pn) (9) n! (2π)D (2π)D 1 1 n=0 1 Z D D X 1 d p1 d pn = ··· (2π)Dδ(p + :: + p )Γ(p ; ::; p )' ~(p ):::'~(p ) n! (2π)D (2π)D 1 n 1 n 1 n n=0 where we have defined the Fourier transform of the fields with outgoing momenta as Z Z dDp '~(p) = dDx e−ipx'(x) ;'(x) = eipx'~(p) (10) (2π)D and the Fourier transform of the proper vertices with ingoing momenta as Z D D ip1x1+::+ipnxn Γ(~ p1; ::; pn) = d x1::d xn e Γ(x1; ::; xn) (11) D = (2π) δ(p1 + :: + pn)Γ(p1; ::; pn) the second line following from momentum conservation (translational invariance of Γ(x1; ::; xn)). Graphically, we denote these correlation functions by x1 x4 p1 p4 Γ(x1; ::; x4) = ; Γ(p1; ::; p4) = (12) p2 p3 x2 x3 2 1 1 where in the second graph conservation of momentum is understood, and the external lines indicate only the flow of momentum and do not include external propagators (the 1PI graphs are amputated). n n 0 1 Denoting Γ = Γ (p1; ::; pn), one finds that Γ is the zero point function, Γ (p) is the one- point function, often set to vanish by adjusting the vev of the quantum field '(p), Γ2(−p; p) is the effective kinetic term appearing in 1 Z dDp dDp Γ2(') = − 1 2 (2π)Dδ(p + p )Γ2(p ; p )' ~(p )' ~(p ) 2 (2π)D (2π)D 1 2 1 2 1 2 1 Z dDp = − '~(−p)Γ2(−p; p)' ~(p) (13) 2 (2π)D 1 Z dDp = − '~(−p) p2 + m2 − Π(p2) '~(p) 2 (2π)D with Π(p2) the self-energy contribution: the remaining part in Γ2 is the leading kinetic term, −i whose inverse (times −i) gives the perturbative propagator in momentum space p2+m2 . The functions Γn for n ≥ 3 give the proper vertices. From the above conventions it is clear that the propagator and the vertices are obtained by substituting the effective action Γ['] the classical action S['] suitably gauge fixed. a1a2a3 In particular, the trilinear gluon coupling in (5) give rises to the vertex iVµ1µ2µ3 (p1; p2; p3) a1a2a3 a1a2a3 iVµ1µ2µ3 (p1; p2; p3) = gf p1,µ2 ηµ1µ3 + permutations of external lines a1a2a3 = −gf (p1 − p2)µ3 ηµ1µ2 + (p2 − p3)µ1 ηµ2µ3 + (p3 − p1)µ2 ηµ3µ1 (14) = iStot which is extracted from iStot (as e enters the path integral) with Γ ! S in the above formulae (i.e. the vertex iV n corresponds to iΓn for a vertex with n fields when the classical action S substitutes the effective action Γ in the Fourier transform formulae given above: the classical action gives the tree-level term of the effective action, which in addition contains also ~-corrections (i.e. the loop correction inlcuding the counterterms)). 3 Similarly, we get the 4-gluon vertex a1a2a3a4 2 ba1a2 ba3a4 iVµ1µ2µ3µ4 (p1; p2; p3; p4) = −ig f f ηµ1µ3 ηµ2µ4 + permutations of lines 2 3 4 2 ba1a2 ba3a4 = −ig [f f (ηµ1µ3 ηµ2µ4 − ηµ1µ4 ηµ2µ3 ) ba1a3 ba4a2 +f f (ηµ1µ4 ηµ3µ2 − ηµ1µ2 ηµ3µ4 ) ba1a4 ba2a3 +f f (ηµ1µ2 ηµ4µ3 − ηµ1µ3 ηµ4µ2 )] (15) = the ghost-antighost-gluon vertex abc abc iVµ (k; p; q) = gf kµ (16) and the quark-quark-gluon vertex (for a fixed flavour) µa i 0 µ a i i(V ) j(q; p; q ) = −gγ (T ) j (17) These rules are somewhat more complex then those for QED, nevertheless now one can start computing perturbatively various scattering processes, renomalize the theory loopwise by introducing counterterms, and compute the beta function. 2 Perturbative calculation and the beta function To renomalize the theory one must introduce the counterterms that will cancel the infinities and match the renomalization conditions needed to define the theory in terms of a fixed number of observables. Naively, looking at (4) and (5) one would expect at least nine Z factors (here we consider 4 only one quark flavour) 1 a 2 µ a 2 1 µ a 2 µ a a µ L = − Z [(@ A ) − (@ A ) ] − Z (@ A ) − Z 0 @ c¯ @ c − Z γ @ − Z m 2 3 µ ν µ 2ξ ξ µ 2 µ 2 µ m g2 (18) − Z f abc@µAνaAb Ac − Z f abcf adeAb Ac AµdAνe 3g µ ν 4g 4 µ ν abc µ a b c a µ a − Z10 gf @ c¯ Aµc + iZ1gAµ γ T : however, gauge invariance (or more properly BRST invariance) can be used to derive Ward identities (that in this particular context are known as Slavnov-Taylor identites), which show that the Z factors are not all independent. In particular, one finds that Zξ = 1 as in QED, and 2 2 2 2 Z1 2 Z10 2 Z3g 2 Z4g 2 g0 = 2 g µ~ = 2 g µ~ = 3 g µ~ = 2 g µ~ (19) Z2 Z3 Z20 Z3 Z3 Z3 where d = 4 − . As in the classical lagrangian, this means that there is only one independent coupling constant g. Thus, to compute the β-function one could start calculating Z1;Z2 and Z3 at one-loop, and use the first relation in (19). From the correction to the quark propagator one obtains Z2 in the MS scheme g2 1 ! Z = 1 − C(R) + O(g4) : 2 8π2 The study of the quark-quark-gluon vertex gives g2 1 ! Z = 1 − [C(R) + T (A)] + O(g4) : 1 8π2 Finally, to renomalize the gluon wave function one must study the divergences in the gluon propagator and a laborious calculation produces 5 4 g2 1 Z = 1 + T (A) − n T (R) + O(g4) : 3 3 3 F 8π2 Then, one has all the elements to compute the one-loop beta function 11 4 g3 β(g) = − T (A) − n T (R) + O(g5) (20) 3 3 F 16π2 1 which for QCD (where T (A) = 3 and T (R) = 2 ) shows asymptotic freedom for a number of quark flavours nF ≤ 16. If interested, these one-loop calculations can be followed in detail in chapter 73 of Srednicki. 5 3 Ward identities Ward identites arising from the BRST symmetry can be used to deduce several properties of the theory. One particular application is to show that scattering amplitudes are transversal in the polarization of the gluons. This means that if in a given total amplitude M we extract the µ physical polarization µ(p) of one gluon, say by setting M = µ(p)M , then after substituting this physical polarization µ(p) with a longitudinal one, µ(p) ! pµ, one finds a vanishing result if all the other states are physical (i.e.
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