Two-Loop QCD Corrections to the Heavy Quark Form Factors: Axial Vector Contributions

We consider the Z Q Qbar vertex to second order in the QCD coupling for an on-shell massive quark-antiquark pair and for arbitrary momentum transfer of the Z boson. We present closed analytic expressions for the two parity-violating form factors of that vertex at the two-loop level in QCD, excluding the contributions from triangle diagrams. These form factors are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.


Introduction
This paper is the second of a series that is devoted to the computation of the electromagnetic and neutral current form factors of heavy quarks Q at the two-loop level in QCD [1]. These form factors are relevant for a number of applications. For instance, they are part of the order α 2 s QCD corrections to the differential electroproduction cross sections of heavy quarks e + e − → γ * , Z * → QQX and, in particular, part of the second order QCD corrections to the forward-backward asymmetry A Q f b . As far as b quarks are concerned, an order α 2 s calculation with a non-zero b quark mass is of interest in view of the discrepancy between experimental measurement and theoretical expectations -see [2] and the references given in [1] concerning the state of the theoretical predictions. A review of the present status of electroweak corrections to the forward-backward asymmetry is given in [3]. At a future linear collider, forwardbackward asymmetries will play a prominent role in very precise measurements of the neutral current couplings of bottom and of top quarks [4]. Clearly, predictions will be required taking the mass of the heavy quark fully into account.
In [1] we presented closed analytic expressions to order α 2 s of the heavy-quark electromagnetic vertex form factors for arbitrary momentum transfer. Up to an overall coupling factor these are identical to the corresponding vector, i.e., parityconserving form factors that appear in the amplitude of the decay of a virtual Z boson into a heavy quark-antiquark pair. In this paper, we compute the axial vector form factors G 1 and G 2 , excluding the anomalous triangle graph contributions, Fig. 1 (a) and (b), which contribute only through the mass splitting of a quark isospin doublet in the triangle loop. These terms deserve a separate discussion and will be given elsewhere [5].
The paper is organized as follows. In Section 2 we fix our conventions and describe how the form factors can be obtained from the vertex amplitude by appropriate projections. In Section 3 the renormalization constants in the scheme that we use -which is the same as the one chosen in [1] -are collected for the convenience of the reader. Sections 4 and 5 contain, for spacelike momentum transfer, the unsubtracted and renormalized axial vector form factors at one-loop and two-loop order, respectively, for the cases of the renormalization scale being both equal and different from the heavy quark mass. In Section 6 the form factors are analytically continued above theQQ threshold, and we give there also their threshold and asymptotic expansions. We conclude in Section 7.

The Axial Vector Form Factors
We consider the amplitude V µ c 1 c 2 (p 1 , p 2 ) for the decay of a virtual Z boson of fourmomentum q = p 1 + p 2 into a massive quark-antiquark pair of momenta p 1 , p 2 and colors c 1 , c 2 . The quarks Q,Q are on their mass shell, p 2 1 = p 2 2 = m 2 , where m denotes the pole mass of Q. The squared center-of-mass energy is S = (p 1 + p 2 ) 2 .
A general decomposition of the vertex function V µ (p 1 , p 2 ) involves 6 form factors, two of which odd under a CP transformation. As we consider here, besides QCD interactions, Standard Model (SM) neutral current interactions to lowest order, CP invariance holds. This implies that V µ depends only on 4 form factors, and we use the decomposition where s = S/m 2 , σ µν = i 2 [γ µ , γ ν ],ū c 1 (p 1 ), v c 2 (p 2 ) are Dirac spinors and v Q , a Q are the SM vector and axial vector couplings of the massive quark Q to the Z boson: where s w (c w ) is the sine (cosine) of the weak mixing angle, T Q 3 the third component of the weak isospin and Q Q is the charge of the heavy quark in units of the positron charge e > 0.
The dimensionless form factors F i , G i can be obtained from V µ c 1 c 2 (p 1 , p 2 ) by appropriate projections. We consider the spinor traces where t µ = p µ 2 −p µ 1 . Since we are working in D = 4−2ǫ dimensions we calculate these traces in D dimensions as well. Inserting Eq. (2) into Eqs. (4)(5)(6)(7) and performing the traces one obtains: The trace of the unit matrix is kept equal to four also in D dimensions. In calculating the diagrams considered in this paper we use an anticommuting γ 5 in D dimensions.
(This prescription is also used in the derivation of the formulae (10), (11).) This prescription is appropriate as the diagrams below correspond to the order α 2 S "nonsinglet" contributions to the matrix element of the axial vector current, and it is well known that for these contributions a canonical, i.e., non-anomalous Ward identity must hold to this order. Within dimensional regularisation this is most conveniently implemented with an anticommuting γ 5 . In a subsequent paper [5], the contributions involving closed triangle loops, Fig. 1, will be computed for a mass-split doublet of fermions in the triangle loop. In that context, the naive anticommuting γ 5 used here is problematic in D = 4. However, it will be shown there that using a different γ 5 prescription [6,7] does not affect the non-anomalous contributions presented here.
The formulae Eqs. (8)(9)(10)(11) show that with the above projections the computation of the vector form factors F 1,2 and the axial vector (i.e., parity-violating) form factors G 1,2 decouple from each other. The vector form factors in Eqs. (8,9) were computed in [1]. Here we determine the form factors in Eqs. (10,11) to the second order in the strong coupling constant α S , excluding the contributions from the triangle diagrams shown in Fig. 1. Expanding the renormalized form factors to the second order in α S , we have: where the superscripts (1l) and (2l) denote the one-and two-loop contributions. The subscript "R" labels the renormalization scheme specified in the next Section. After having performed the renormalization, the form factors still depend on the parameter ǫ, which regularizes the remaining infrared divergencies. We keep α S dimensionless also in D = 4 dimensions.
The form factors are represented as series in ǫ and expressed in terms of 1dimensional harmonic polylogarithms H( a; x) up to weight 4 [8,9], which are functions of the dimensionless variable x defined by We give our results firstly in the kinematical region in which s is spacelike (0 ≤ x ≤ 1), where the form factors are real. In Section 6 we shall perform the analytical continuation to the physical region above threshold, s > 4, −1 < x ≤ 0, and explicitly decompose the form factors into real and imaginary parts.
In what follows N f denotes the number of light quarks (which we take to be massless) running in the loops Fig. 4 (g), and where N c is the number of colors.

Renormalization Scheme
As in our previous paper [1], we use renormalized perturbation theory with α S = g 2 s /(4 π) being defined as the standard MS coupling in QCD with N f massless and one massive quark, while we define the mass m and the wave-function of the heavy quark Q in the on-shell (OS) scheme.
For the renormalization procedure we need the coupling renormalization and the gluon wave function to one-loop: where and, in the Feynman gauge, The renormalization constants concerning the heavy quark are defined in the on-shell scheme. Here we need to one-loop order. Here and in the following µ denotes the mass scale of dimensional regularization/renormalization. The wave function Z OS 2 is needed to two-loop order. The latter was computed in [10,11]. Using the result of [11] and expressing it in terms of the renormalized MS coupling α S we have where Z (2) 2 is: Further we need the renormalization constant Z 1F for the QQ gluon vertex to one loop. Using a Slavnov-Taylor identitiy and Eqs. (15,17,19) we get: For the counterterm contributions to the renormalized form factors it is convenient to define (here our notation differs slightly from that in [1]) where δ can be read off from Eq. (19). In this renormalization scheme the renormalized vertex function Γ µ to order α 2 S is given by the contributions from the 1-particle irreducible diagrams, Figs. 2 and 4, which we call the unsubtracted contributions, and the counterterms given below.

One-loop Unsubtracted and UV-Renormalized Form Factors
In this Section we give the results for G contribution from the diagram Fig. 2 including the terms of order ǫ. They are needed for computing the order α 2 S counterterms shown in Figs. 5 (c), (d) and (f) of Section 5.2 below. We obtain: The coefficients a i and b i (i = 1 . . . 3) are: respectively, Figure 3: Subtraction term for the one-loop renormalization.
The counterterm of Fig. 3 contributes to G 1 :   Performing the γ algebra we obtain, as explained in [12,13,14], the form factors in Eqs. (10,11) expressed in terms of hundreds of different scalar integrals. These integrals are reduced to a small set of master integrals by means of the socalled Laporta algorithm [15] with the help of integration-by-parts identities [16] and Lorentz-invariance identities [17]. Symmetry relations which one can establish between different integrals are also used during the reduction. The master integrals themselves were evaluated with the method of differential equations [17,18,19,20] in [12,14]. The master integrals, and thus the form factors are represented as series in the regularization parameter ǫ and expressed in terms of 1-dimensional harmonic polylogarithms up to weight 4 [8,9], which are functions of the dimensionless variable x defined in Eq. (14). As in the one-loop case we take s to be space-like.
The two-loop unsubtracted form factors are found to be: where C(ǫ) is defined in Eq. (16) and N f , C F , C A , and T R are given at the end of Section 2. One finds for the c i (i = 1 . . . 12): The d i (i = 1 . . . 12) are:  (2) x 5 + 320 with ζ(n) being Riemann's ζ-function.

Contributions from the Counterterms
The terms in Fig. 5 with and The contributions from these two diagrams to the axial vector form factors are: • Figs. 5 (c) and 5 (d): with These diagrams yield • Fig. 5 (e): with This leads to the contributions with N 2 as defined in Eq.(66). They contribute • Fig. 5 (g) is defined as the two-loop renormalization constant δ (2l) 2 times the Born level amplitude and reads: Its contribution to G 1 is:

UV-Renormalized Two-Loop Form Factors
Adding the terms given in Sections 5.1 and 5.2 we obtain the UV-renormalized axial vector form factors to second order in QCD. They still contain terms proportional to ǫ −2 and ǫ −1 due to infrared and collinear singularities in the loops. In this Subsection we put the renormalization scale µ equal to the on-shell mass m (The logarithms that are present if µ = m are given in Section 5.4.). Then we get: where: −570

Form Factors for µ = m
In this Section we give the expressions for the renormalized axial vector two-loop form factors for the case of µ = m. At the one-loop level we do not have an explicit dependence on the logarithm of the ratio of the renormalization scale and the mass of the heavy quark, because an overall factor (µ 2 /m 2 ) ǫ can be taken out, see Eqs. (35,36). At the two-loop level, such a dependence results from the coupling constant renormalization, first appearing at this level. Factoring an overall (µ 2 /m 2 ) 2ǫ , we have: where the functions G i (ǫ, s) and P (2l) i (s) can be obtained either from the renormalization group [1] or by explicit expansion. We find:

Analytical Continuation above Threshold
The results for the renormalized form factors can be analytically continued into the timelike region s > 0 and in particular above the physical threshold s > 4 by using the substitution [1] x → −y + iǫ, with For s > 4 the form factors become complex due to absorptive parts. We write: These imaginary parts arise from the analytical continuation of the harmonic polylogarithms with rightmost index 0. In the following two Subsections we shall give the real and imaginary parts of G 1,2 at one and two loops, putting µ = m.

One-Loop Form Factors above Threshold
The The above expressions for the singular term and the term of order ǫ 0 of G 1 agree with those of [21].

Two-Loop Form Factors above Threshold
The real and imaginary parts of the coefficientsc i andd i in Eq. (78) are: (142)

Threshold Expansions
In this Section we expand the one and two loop form factors near threshold S ∼ 4m 2 , i.e. y → 1 in powers of The limit r → ∞ corresponds to the massless limit m → 0. Therefore the chiralityflipping form factors F 2 and G 2 must be of order 1/r, and those terms in the chiralityconserving form factor F 1 that survive the limit r → ∞ must be equal to the corresponding terms in G 1 . The asymptotic expansions given above and in [1] satisfy these constraints.
All the results of this Section can be obtained in an electronic form by downloading the source of this manuscript from http://www.arxiv.org.

Summary
In this paper we calculated the axial vector form factors G 1 and G 2 to second order in the QCD coupling, excluding the contributions from Fig. 1 (a) and (b). The results for the two form factors were obtained keeping the full dependence on the mass of the heavy quark, as well as on the arbitrary momentum transfer.
The renormalized form factors are expressed in terms of the MS coupling α s for (N f + 1) quark flavors and of the on-shell mass of the heavy quark. The expressions for the unsubtracted as well as the UV-renormalized form factors are given in closed analytic form as a Laurent expansion in ǫ = (4 − D)/2. The coefficients of this expansion have a suitable representation in terms of 1-dimensional harmonic polylogarithms. Poles in ǫ, which correspond to soft and collinear divergences, are still present in G 1,2,R , like in F 1,2,R . Depending on the observable considered, these divergences must cancel either among each other or against the divergences arising from the real radiation, which in this paper was not taken into account.
As already discussed in the introduction and in our preceeding paper [1] our results for the vector and axial vector form factors are part of the order α 2 S QCD corrections to the differential electroproduction cross sections of heavy quarks and, moreover, have a number of immediate applications. These include studies of the extrapolation of the continuum amplitudes to theQQ production threshold and studies concerning the generic singularity structure of QCD amplitudes with massive quarks. Of general interest is the computation of the NNLO QCD corrections to the forward-backward asymmetry of heavy quarks. We shall report on this in a future publication.