The three-loop singlet contribution to the massless axial-vector quark form factor
ZZU-TH 09/21
The three-loop singlet contributionto the massless axial-vector quark form factor
T. Gehrmann and
A. Primo
Physik-Institut, Universit¨at Z¨urich, Winterthurerstrasse 190,CH-8057 Z¨urich, Switzerland
Abstract
We compute the three-loop corrections to the quark axial vector form factor in masslessQCD, focusing on the pure-singlet contributions where the axial vector current couples to aclosed quark loop. Employing the Larin prescription for γ , we discuss the UV renormaliza-tion of the form factor. The infrared singularity structure of the resulting singlet axial-vectorform factor is explained from infrared factorization, defining a finite remainder function. a r X i v : . [ h e p - ph ] F e b he quark form factors describe the coupling of a quark-antiquark pair to an external cur-rent, which can be a vector, scalar, axial-vector or pseudo-scalar. Higher order perturbativecorrections to these form factors provide important universal information [1–3] on anomalousdimensions, and they constitute the purely virtual corrections to important collider processessuch as gauge boson production, Higgs boson decay or deeply inelastic scattering. After renor-malization of ultraviolet (UV) divergences, the form factors remain divergent due to infrared(IR) poles.For massless quarks, the three-loop QCD corrections to vector [4–6], scalar [7] and pseudo-scalar [8] form factors were derived in the literature, and important progress has been maderecently towards the four-loop corrections to the vector form factor [9–11]. All massive formfactors are known to two-loop order in QCD [12–15], supplemented by partial three-loop re-sults [16].Owing to chirality conservation, the massless vector and axial-vector form factors can differfrom each other only through contributions where the external current couples to a closedquark loop, which is then connected to the external quark-antiquark pair through virtual gluonexchanges. These so-called pure-singlet contributions (PS) occur for the first time at two loops,where they vanish for the vector form factor while yielding an IR-finite contribution for theaxial-vector form factor [14]. The three-loop pure-singlet contributions to the vector form factorwere computed earlier [5, 6] and found to be IR-finite. All contributions where the axial vectorcurrent insertion couples to the external quarks are denoted as non-singlet (NS), and the sum ofnon-singlet and pure singlet contributions yields the singlet (S) form factor. In the present letter,we derive the three-loop pure singlet contributions to the axial-vector form factor in masslessQCD, thereby completing the full set of massless three-loop quark form factors.Using dimensional regularization to handle both infrared and ultraviolet divergences, onemust extend the four-dimensional chirality projection operator γ to symbolic d = 4 − (cid:15) space-time dimensions. For this purpose, we follow the prescription introduced by Larin [17], whichreplaces the symmetrized axial vector vertex factor as follows: γ µ γ →
12 ( γ µ γ − γ γ µ ) → i (cid:15) µν ν ν γ ν γ ν γ ν . (1)It is based on the original γ formulation of t’Hooft and Veltman [18], which was further re-fined by Breitenlohner and Maison [19], but has the further advantage that the Lorentz indexspace does not need to be split into 4-dimensional and ( d − d dimensions throughout. In the Larin scheme,a finite renormalization of the axial vector current is required (besides the conventional UV-renormalization) to restore chirality conservation of massless quarks and to ensure the validityof the chiral anomaly. The axial renormalization constants in the Larin scheme were known tothree-loop order for the non-singlet contributions for a long time [17, 20], while the finite contri-bution to the renormalization of the singlet current has been derived only most recently [21].The axial vector vertex function is obtained by the insertion of an off-shell axial-vectorcurrent with virtuality q = ( p + p ) between a quark-antiquark pair with on-shell momenta p and p , yielding the Born-level expression¯ u ( p )Γ µA, u ( p ) = ¯ u ( p ) γ µ γ u ( p ) . (2)1he axial vector form factor is then obtained by applying a projection operator on the all-ordervertex function Γ µA : F A = − (cid:15) )(1 − (cid:15) )(1 − (cid:15) )(3 − (cid:15) ) q T r ( p / Γ µA p / γ µ γ ) , (3)where the Larin prescription (1) is to be applied throughout. With the above normalization, theBorn-level axial vector form factor is equal to unity: F (0) A = 1. The amplitude-level calculationsclosely follow those of the three-loop vector form factor [6].The MS renormalization of the axial vector vertex function Γ µA involves [17, 20, 21] the renor-malization of the coupling constant Z g and of the axial vector current insertion Z ms Z f , where Z ms and Z f denote the divergent and finite parts of the axial vector renormalization constant.They depend on the prescription used for γ in dimensional regularization. Different axial vectorrenormalization constants are required for the singlet and non-singlet axial vector form factors.They have been computed to three-loop order in the Larin scheme for the non-singlet [17, 20]and singlet [21] axial vector current.After renormalization, the massless non-singlet form factors for axial vector and vector agreewith each other in their finite parts that are obtained by subtracting their universal infrared polestructure [4, 22–24], as required by chirality conservation for massless fermions and as obtainedfor a naively anti-commuting γ . To extract the pure singlet axial vector form factor, one takesthe difference of the renormalized singlet and non-singlet axial vector form factors: F A, PS = F A, S − F A, NS . (4)By taking the difference of the singlet and non-singlet renormalization constants [17, 20, 21], wedefine pure-singlet renormalization constants which turn out to be useful in arranging [14] thedifferent contributions in terms of non-singlet and pure-singlet type: Z ms , PS = Z ms , S − Z ms , NS (5)= C F N F,J (cid:18) (cid:15) a + ( −
66 + 109 (cid:15) ) C A − (cid:15)C F + (12 + 2 (cid:15) ) N F (cid:15) a (cid:19) + O ( a ) ,Z f , PS = Z f , S − Z f , NS (6)= C F N F,J (cid:18) a + ( −
326 + 1404 ζ ) C A + (621 − ζ ) C F + 176 N F a (cid:19) + O ( a ) , where C A = N , C F = ( N − / (2 N ) are the QCD colour factors and a = α s ( µ ) / (4 π ) denotesthe renormalized QCD coupling constant. The overall power of N F,J can be identified withthe number of quark flavours that couple to the external axial vector current, while N F is thenumber of massless quark flavours. In the following, we take N F,J = 1 throughout.Expanding out the pure singlet axial vector form factor (4) in powers of the renormalizedstrong coupling constant, one finds that up to three loops it receives the following types ofcontributions: pure-singlet amplitudes at three loops, pure-singlet amplitudes at two loops withrenormalization of coupling constant and non-singlet current, non-singlet amplitudes at tree-leveland one-loop with pure-singlet renormalization, as well as the tree-level non-singlet amplitude2ith products of pure-singlet and non-singlet renormalization. All form factor amplitudes arecomputed using the Larin prescription throughout.The resulting renormalized pure-singlet axial vector form factor can be written as expansionin the renormalized coupling constant F A, PS = a F (2) A, PS + a F (3) A, PS + O ( a ) . (7)The two-loop contribution has been computed previously. We reproduce the result of [14] andprovide higher order terms in the (cid:15) expansion, as required for the study of the infrared singularitystructure at higher loop orders: F (2) A, PS = C F (cid:34) −
18 + 6 L µ + 2 π (cid:15) (cid:32) − L µ + 6 L µ + 4 ζ + 59 π (cid:33) + (cid:15) (cid:32) − L µ + 4 L µ + 1463 ζ + 1469 π
108 + 4 π (cid:33) + O ( (cid:15) ) (cid:35) , (8)where we have introduced L µ = log( − q /µ ). It should be noted that only the finite term inthe above expression is independent on the prescription used for γ , while all (cid:15) -type terms arespecific to the Larin scheme.The three-loop pure singlet axial vector form factor is our main new result. It reads F (3) A, PS = 1 (cid:15) C F (cid:32) − L µ − π (cid:33) + 1 (cid:15) C F (cid:32) − L µ − L µ − ζ − π (cid:33) + C F (cid:32) − L µ + π L µ − L µ − L µ − ζ − π − π (cid:33) + C F N F (cid:32) − L µ + 8 π L µ + 4 L µ − π (cid:33) + C F C A (cid:32) − L µ − π L µ − L µ + 62 ζ + 385 π − π (cid:33) . (9)The divergent contributions in the above expression are of infrared origin. They can be expressedby applying the one-loop infrared singularity operator [22]I (1) q ¯ q = − C F (cid:18) (cid:15) + 3 (cid:15) (cid:19) (cid:32) − (cid:15) π (cid:33) , (10)such that F (3 , finite) A, PS = F (3) A, PS − I (1) q ¯ q F (2) A, PS (11)= C F (cid:32) − L µ − π − π (cid:33) + C F N F (cid:32) − L µ + 8 π L µ + 4 L µ − π (cid:33) + C F C A (cid:32) − L µ − π L µ − L µ + 62 ζ + 385 π − π (cid:33) (12)3s finite and independent on the γ -scheme used, while (9) is valid only in the Larin scheme.The three-loop pure-singlet axial vector form factor contributes to coefficient functions forobservables in polarized hadron collisions and to the three-loop coefficient function for hadronic Z -boson production. In the latter, its contribution cancels for mass-degenerate quark isospindoublets in the loop, such that a sizable effect can be expected only from the third-generationquarks. For these, the massless form factor computed here provides a reliable description of thebottom quark contribution, while the top quark contribution remains to be derived.In this letter, we completed the calculation of the three-loop quark form factors in mass-less QCD by deriving the pure-singlet axial vector form factor at this order. Our computationemployed the Larin scheme for γ throughout in all amplitudes and projectors. The three-looppure-singlet axial vector form factor is infrared-divergent. Its singularity structure in accor-dance with the expectation from infrared factorization, such that a scheme-independent finiteremainder can be defined. Acknowledgements
This research was supported by the Swiss National Science Foundation (SNF) under contract200020-175595.
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