γ 5 in the four-dimensional helicity scheme

We investigate the regularization-scheme dependent treatment of γ 5 in the framework of dimensional regularization, mainly focusing on the four-dimensional helicity scheme ( FDH ). Evaluating distinctive examples, we find that for one-loop calculations, the recently proposed four-dimensional formulation ( FDF ) of the FDH scheme constitutes a viable and efficient alternative compared to more traditional approaches. In addition, we extend the considerations to the two-loop level and compute the pseudoscalar form factors of quarks and gluons in FDH . We provide the necessary operator renormalization and discuss at a practical level how the complexity of intermediate calculational steps can be reduced in an efficient way.


I. INTRODUCTION
The success of quantum-field theoretical predictions over the past decades was enabled, among other things, by the applicability of dimensional regularization as the method provides a mathematically consistent tool to handle ultraviolet (UV) and infrared (IR) divergences in the multi-loop regime. From the very moment of the introduction of dimensional regularization in Ref.
[1], however, special attention had to be paid to the treatment of γ 5 since the object is closely related to concepts that are only valid in integer dimensions. In a series of publications [2-15] that cover a time span of more than 40 years, different approaches have been developed in order to find consistent rules for the treatment of γ 5 in the dimensional framework. Irrespective of this effort, in the overwhelming majority of computations that have been performed so far, the original γ 5 definition of Ref.
[1] has been used, giving expression to the fact that even today no efficient alternatives are available that are well suited for all kinds of calculations.
Parallel to the development of γ 5 schemes, the search for new efficient calculational methods has focused on finding regularization prescriptions that reduce the technical complexity at the practical level. Recently, the current status of the most prominent schemes has been summarized in Ref. [16]. Among the considered dimensional schemes are the 't Hooft-Veltman scheme (HV) [1], conventional dimensional regularization (CDR) [17], dimensional reduction (DRED) [18], the four-dimensional helicity scheme (FDH) [19,20], and its recently proposed four-dimensional formulation (FDF) [21] at one loop.
In this article, we investigate the treatment of γ 5 in the aforementioned dimensional schemes, mainly concentrating on the FDH scheme. As prescriptions for γ 5 we consider the original one of 't Hooft/Veltman and an anticommuting γ 5 . Having the practitioner in mind, we perform distinctive one-and two-loop calculations and show which of the γ 5 schemes is the more efficient alternative for the respective process under consideration. In order to enable a step-by-step comparison between the different γ 5 schemes and the different dimensional schemes, the outline of the letter is the following: In Sec. II A, we provide the definitions of γ 5 in CDR/HV and extend them to FDH/DRED in Sec. II B. To illustrate practical consequences of these definitions, we evaluate characteristic one-loop examples in Secs. II C and II D, putting emphasis on differences and similarities of the various approaches. The extension of these considerations to the two-loop level is discussed in Sec. III by computing the pseudo-scalar form factors of quarks and gluons in massless QCD. The necessary operator renormalization as well as the UV-renormalized results are provided in Sec. IV.

A. CDR and HV
One main reason for the recurrent appearance of seeming inconsistencies related to γ 5 is the fact that for a consistent formulation of d-dimensional integration, the fourdimensional Minkowski space S ½4 has to be embedded into an infinite-dimensional space QS ½d 1 [17], S ½4 ⊂ QS ½d : ð2:1Þ Although QS ½d and the related quantities formally have finite-dimensional properties, common concepts of S ½4 like index counting are no longer applicable. Regarding γ 5 , this interplay between finite-and infinite-dimensional aspects has caused quite a lot of confusion in the past and led to the introduction of different γ 5 schemes (GS). Depending on which GS is chosen, special attention has to be paid to the evaluation of the Lorentz algebra, to the breaking of symmetries, to the treatment of anomalies, and to the UV renormalization at higher perturbative orders. According to the different characteristics regarding these points, it is useful to distinguish the following two classes of GS: (i) The first class contains schemes where γ 5  (ii) The second class contains schemes where γ 5 is defined algebraically, for example as anticommuting (AC) with (quasi) d-dimensional γ matrices [4,5], AC∶ fγ AC 5 ; γ μ ½d g ≡ 0: ð2:3Þ In Eq. (2.2), γ BM 5 is defined via the totally antisymmetric Levi-Civita pseudotensor ε μνρσ which is closely related to the concept of index counting in strictly four dimensions. While the dimensionality of the γ matrices is treated differently in various dimensional schemes, it is mandatory to consider ε μνρσ as a strictly four-dimensional object. Only in this way it is possible to avoid ambiguous results and mathematical inconsistencies found before e. g. in Ref. [22]. Usually, the mismatch between the dimensionality of ε μνρσ and other algebraic objects is circumvented by workarounds whose ranges of validity are often not obvious, at least not at first sight. More details regarding this issue will be given in Sec. III B.
A direct consequence of Eq. (2.2) is that all (anti) commutation relations of γ BM 5 are implicitly part of the definition and therefore fixed, e.g.

B. FDH and DRED
So far, the algebraic behavior of γ 5 has been considered in the quasi d-dimensional space QS ½d which is the natural domain of CDR and of d-dimensional integration. In Ref. [23], it is shown that in order to consistently formulate FDH and DRED, this space has to be enlarged to QS ½d s via a direct (orthogonal) sum with the so-called "evanescent" space QS ½n ϵ , Although d s is usually taken to be 4 in FDH and DRED, it is clear that QS ½d s is an infinite-dimensional space with finitedimensional algebraic properties. 3 According to the structure of the vector spaces in Eq. (2.5), quasi d s -dimensional metric tensors and γ matrices can be split as g μν ½d s ¼ g μν ½d þ g μν ½n ϵ and γ μ ½d s ¼ γ μ ½d þ γ μ ½n ϵ , resulting in with dim ∈ f4; d; d s ; n ϵ g.
Due to the even number of γ matrices in Eq. (2.2), γ BM 5 commutes with the evanescent degrees of freedom in FDH and DRED. Moreover, from Eq. (2.7a) it directly follows that the structure of the (anti)commutation relation in d and d s dimensions is the same, 1 Following Ref. [16], we denote the (quasi)dimensionality dim of a quantity by a subscript [dim]. Throughout this article, the modified space-time dimension is always defined as d ≡ 4 − 2ϵ.
2 In order to distinguish this prescription from other aspects of the original HV scheme, we solely use the abbreviation BM to denote a scheme for the treatment of γ 5 .

3
For more comments on the definition and the structure of the vector spaces in Eq. (2.5) we refer to [23][24][25] and references therein. Here it should only be mentioned that setting d s ¼ 4 results in n ϵ ¼ 2ϵ.
As a consequence, in practical calculations it is possible to either use a quasi d s -dimensional Lorentz algebra or to explicitly perform the split of Eq. (2.5).
In contrast, the (anti)commutation relations of γ AC 5 are not fixed a priori but have to be part of the definition. We therefore define  [23].
To illustrate the implications of the different schemes for γ 5 , we consider the following simple one-loop examples in the FDH scheme: the correlator γ μ γ 5 → e þ e − and the (anomalous) correlator of an axial-vector current and two vector currents (AVV correlator). Each of the examples is evaluated by using γ BM 5 and γ AC 5 as defined in Eqs. (2.2) and (2.3), respectively. In addition we apply FDF, a recently proposed genuine four-dimensional formulation of the FDH algebra at the one-loop level. In the analytical results, the fermion mass is denoted by m and p 1 , p 2 are the (outgoing) momenta of the external fermions/gauge fields. For simplicity we consider QED and set e ¼ 1 for the gauge coupling.

5
The application of γ BM 5 in a d-dimensional framework with d ≠ 4 results in different algebraic properties compared to the unregularized theory which can be easily seen from Eq. (2.4). The (d-dimensional) axial-vector operator is therefore usually symmetrized "by hand" and written as [6] Using this relation together with Eqs. (2.2) and (2.6), and multiplying with q μ ≡ ðp 1 þ p 2 Þ μ then yields for the left diagram in Fig. 1 The (on-shell) renormalization of the external fermion fields as well as the prediction for the structure of the IR divergences in the FDH scheme are given in Ref. [26], Subtracting the IR divergence, it follows that field renormalization is not sufficient to obtain the correct result since the (scheme-dependent) UV divergence does not cancel. The general reason is that symmetries of the unregularized theory like chiral and Lorentz invariance FIG. 1. One-loop contributions to the correlator γ μ γ 5 → e þ e − . The diagrams contain a gauge field (left) and an associated FDFscalar (right). The latter diagram is only present in FDF. 4 In this example, Lorentz indices related to vector fields are treated in d s ¼ d þ n ϵ dimensions, see also Eq. (2.5). The case d s ¼ 4 (and therefore n ϵ ¼ 2ϵ) then corresponds to FDH and DRED, whereas results in CDR and HV are obtained for n ϵ ¼ 0.
Here and in the following, the irrelevant dimension of the external momenta is set to d and terms of Oðϵ 0 n ϵ Þ are omitted since they vanish after setting n ϵ ¼ 2ϵ and taking the subsequent limit ϵ → 0. The regularization scale is fixed via μ 0 ≡ m. Note further, that the ε pseudotensor is considered outside dimensional regularization and treated in strictly four dimensions. is used in a d-dimensional framework. 5 As a consequence, initial symmetries have to be restored by means of additional counterterms. In Sec. IVA, it will be shown that for the one-loop example at hand, this renormalization reads δZ BM;ð1Þ ðn ϵ Þ ¼ δZ It is given by a pure MS pole term δZ BM MS which is finite after setting n ϵ ¼ 2ϵ and by a regularization-scheme independent constant δZ 5 . In CDR (n ϵ ¼ 0), the latter is usually determined through relations that are valid in strictly fourdimensional schemes like the Pauli-Villars setup, see e.g. Ref. [6]. In Sec. IV we present an alternative approach that is based on a comparison between results obtained with γ BM 5 and γ AC 5 . Combining Eqs. (2.10)-(2.12) and taking the subsequent limit d → 4, we obtain for the UV-renormalized and IRsubtracted correlator Since all evanescent terms ∼n ϵ drop out through UV renormalization, this final result does not depend on the applied dimensional scheme.

5
For the case of an anticommuting γ AC 5 we write the FDH one-loop amplitude as ð2:14Þ The result has been obtained by (anti)commuting γ AC 5 to the right and evaluating the remaining algebra by means of Eq. (2.6). Due to the absence of an explicit symmetrization and the reduced number of γ matrices in the numerator, the evaluation of the algebra is much simpler compared to Eq. (2.10). Moreover, the consequent use of an anticommutator in Eq. (2.8) leads to a sign change of the n ϵ term. Applying the field renormalization of Eq. (2.11a) and subtracting the IR divergence we then directly recover the result in Eq. (2.13). In contrast to γ BM 5 therefore no symmetry-restoring counterterms are needed to get the correct result.

Algebra in genuine four dimensions-FDF
FDF is a novel regularization approach that was introduced to reproduce FDH results at the one-loop level [21].
Starting from unregularized analytical expressions, loop momenta in FDF are shifted as = k ½4 → = k ½d ≡ = k ½4 þ iμγ 5 before any other algebraic manipulation is performed. The scale μ corresponds to the (d − 4)-dimensional components of the loop momentum and serves as a regulator for the in general divergent quasi d-dimensional loop integrals. By definition, odd powers of μ are set to zero, resulting in the useful relation One main advantage of the FDF approach is that the Lorentz algebra is realized in strictly four dimensions; Eqs. (2.2) and (2.3) are therefore equivalent, i.e. γ BM 5 ¼ γ AC 5 ≡ γ 5 . Applying this setup, the analytical expression for the left diagram in Fig. 1 ð2:16Þ For the evaluation of the algebra we used Eq. (2.15) to cancel against the denominator, resulting in the μ 2 -dependent "extra integral" [16] Although only strictly four-dimensional quantities and an anticommuting γ 5 have been used to obtain the result in Eq. (2.16), the γ BM 5 result in Eq. (2.10) for n ϵ ¼ 2ϵ is recovered. The conceptual reason is that within FDF, similar relations as in Eq. (2.4b) hold, e.g. 7 FDF∶ fγ 5 ; = k ½d g ¼ 2iμ: ð2:18Þ To obtain a physical result that is compatible with the symmetries of the underlying theory we therefore have to add the same counterterms as for the case of γ BM 5 . Compared to Eq. (2.10), however, the evaluation of the analytical expressions is significantly simplified. D. One-loop example 2: AVV triangle As a second example we consider the AVV triangles in Fig. 2 for the case of massless fermions. In the present case of an NLO fermion loop, the only difference between the dimensional schemes CDR, HV, FDH, and DRED is the dimensionality of the external gauge-field momenta. Since the final result of the amplitude is finite, as will be shown below, the limit d → 4 can be taken without any UV renormalization. After having taken the physical limit, the virtual one-loop amplitudes are therefore the same in all these dimensional schemes.

5
Applying the same setup as in the previous example we obtain in CDR where, as before, the ε pseudotensor is considered outside dimensional regularization throughout the calculation. Taking the limit ϵ → 0, the result in Eq. (2.19) coincides with the well-known (anomalous) axial Ward identity (AWI) given e.g. in Refs. [27][28][29].

5
One important characteristic related to the treatment of γ AC 5 in dimensional schemes is that traces including odd numbers of γ AC 5 either vanish or are not cyclic anymore. Demanding, for example, cyclicity of traces including γ AC 5 leads to relations like [9] ðd − 4ÞTr½γ μ γ ν γ ρ γ σ γ AC 5 ½d ¼ 0: ð2:20Þ For d ≠ 4, this equation can only be fulfilled for a vanishing trace. Since similar relations hold for other numbers of γ matrices in the trace we get and gauge invariance is broken explicitly. Different solutions have been proposed e.g. in Refs. [4,5] and [12,13,15] by modifying the trace operation in such a way that the result in Eq. (2.19) is recovered. These modified traces, however, lead to significant complications in practical calculations, in particular at higher perturbative orders. In this paper we therefore refrain from the explicit evaluation of γ AC 5 -odd traces. Instead, in Sec. IV B we show how this can be avoided at the practical level.

FDF
Finally we evaluate the triangle diagrams by utilizing the FDF approach. Using the same four-dimensional Feynman rules as in Sec. II C, the analytical expression reads FIG. 2. One-loop contributions to the (anomalous) AVV correlator T μαβ AVV including one axial-vector and two vector vertices. 7 This relation follows from It is important to notice that in practical computations, relations like in Eq. (2.18) are not used explicitly since quasi d-dimensional quantities are in FDF split into a strictly four-dimensional and a μ-dependent part. The γ 5 matrix is therefore effectively an anticommuting one.
A crucial difference compared to other dimensional schemes is the appearance of rank two tensor integrals with strictly four-dimensional loop momenta in the numerator, Using Eq. (2.15) and neglecting odd powers of μ, the relevant coefficient is given by The first integrand is given by d-dimensional quantities only and the integral can be evaluated without any complication. In contrast, the second integral contains strictly fourdimensional components of the loop momentum. Using Eq. (2.15) to cancel against the denominator gives rise to the integral in Eq. (2.17) for m ¼ 0. It turns out that this integral is the only one that contributes to the AVV correlator in the FDF approach. In other words, the anomaly is entirely given by a μ 2 integral that stems from the evaluation of the tensor integrals, In this way, the result in Eq. (2.19) is recovered, including higher terms in the ϵ expansion. Again, the computational effort is significantly reduced compared to the case of γ BM 5 .

Comment on Bose symmetry
Recently it has been shown [30,31] that special care has to be taken when using an anticommuting γ AC 5 since gauge invariance and Bose symmetry may not be maintained simultaneously, even if the dimension of the underlying space-time remains unchanged during the regularization process. At the root of this symmetry breaking are γ AC 5 -odd traces which yield different contributions compared to the case of γ BM 5 . In Ref. [31], the interplay between gauge invariance and Bose symmetry is investigated in the framework of implicit regularization (IREG). Using γ BM 5 as defined in Eq. (2.2) together with the right-and left-handed chiral operators V μ R ≡ 1 2 γ μ ðIþγ BM 5 Þ and V μ L ≡ 1 2 γ μ ðI−γ BM 5 Þ at the vertices, the following results for the different correlators are provided 8 In contrast, the same correlators read for the case of an anticommuting γ AC ð2:26aÞ The crucial difference between these two results is that only Eq. (2.25) are likewise compatible with gauge invariance and Bose symmetry since in this case Bose symmetry does not impose any additional restrictions on the distribution of the anomaly on the pseudo-scalar and the vector current [31]. It is therefore possible to entirely shift the anomaly away from the vector current in order to preserve gauge invariance.
Using the FDF approach, we computed the aforementioned chiral correlators and find agreement with Eq. (2.25), i.e.
In FDF, the results are entirely generated by extraintegrals like in Eq. (2.17). Although using a strictly four-dimensional algebra in combination with an anticommuting γ 5 , FDF is therefore compatible with Bose symmetry and gauge invariance at the same time. This finding is confirmed by the validity of the vector Ward identities for which we find in FDF It should be mentioned explicitly that these findings are a result of the algebraic rules within FDF. If we were to evaluate the algebra in the unregularized theory and apply the rules of FDF only afterwards, we would obtain vanishing results for the "mixed" correlators RRL, RLR, RLL like in Eq. (2.26) [although Eqs. (2.24b) and (2.28) would still hold]. Since the analytical expressions are in general divergent, however, it is clear that the application of a proper regularization has to be the initial step that is necessary to avoid ambiguous results.

III. PSEUDOSCALAR FORM FACTORS IN FDH
In the following, we extend the previous findings to the two-loop level by computing the pseudoscalar form factors of quarks and gluons in the FDH scheme. The results of the form factors that are currently available have been obtained by using CDR and γ BM 5 as defined in Eq. (2.2), see e.g. [32] and references therein. In the following we consider the form factors up to two loops for (i) different dimensional schemes, i.e. CDR/HV and FDH, and (ii) different γ 5 schemes, i.e. γ BM 5 and γ AC 5 . In principle, also the FDF scheme is a viable candidate for treating γ 5 in the framework of dimensional regularization. However, since it is (currently) unclear how this approach can be consistently formulated beyond the one-loop level, we do not consider FDF here.

A. Effective Lagrangian
The coupling strength of a pseudo-scalar Higgs boson A to quarks is directly proportional to the respective quark mass. Denoting the pseudoscalar current by j 5;k ≡ iψ k γ 5 ψ k , the corresponding Lagrangian can be written as where v and y i denote the Higgs vacuum expectation value and dimensionless Yukawa couplings which depend on the underlying theory, respectively, the sum runs over all light quark flavors q ∈ fd; u; s; c; bg, and t corresponds to the top quark. One way to obtain an effective Lagrangian corresponding to Eq. (3.1) is to consider the (all-order) anomalous relation [29] between the pseudoscalar current j 5;k and the axial-vector current j μ 5;k ≡ψ k γ μ γ 5 ψ k in the full theory, where G a μν is the gluonic field strength tensor and α s ¼ g 2 s =ð4πÞ denotes the strong coupling. In the limit of a large top mass, m 2 t ≫ p 2 , the derivative ∂ μ j μ 5;t and the masses of the light quarks can be neglected. The (unregularized) effective Lagrangian can then be written as [33] where the ψ q are now quark fields in the effective theory. One important feature of the effective Lagrangian is that it does not carry any mass dependence anymore. Although the interaction between a (pseudoscalar) Higgs and quarks vanishes in the full theory if the quark masses are set to zero, in the effective theory we consider the case of N F massless quarks which are described by the field ψ. The implications of this choice will be discussed below.
In a next step, we study the effective Lagrangian (3.3) in the framework of the aforementioned dimensional schemes. For this, it is useful to envision some universal characteristics of dimensionally regularized quantities. In any dimensional scheme, derivatives and loop momenta are treated as (quasi) d-dimensional objects. In contrast, for the dimensionality of metric tensors, γ matrices, and vector fields there is some freedom which is fixed by the choice of a specific regularization scheme. In CDR, for example, all Lorentz indices (except for the ones of the ε pseudotensor) are treated in d dimensions. The CDR-regularized version of the first curly bracket in Eq. (3.3) therefore reads O G;CDR ≡ fε μνρσ g ½4 fG a μν G a ρσ g ½d : ð3:4aÞ The corresponding Feynman rules are given in Appendix A 1.
One key feature of the Feynman rules stemming from operator (3.4a) is that all of them contain (quasi) ddimensional momenta with uncontracted Lorentz indices. Due to permutations in μ, ν, ρ, σ, the metric tensors in Eqs. (A1c) and (A1d) also have to be considered in d dimensions. The dimensionality of the indices in Eq. (3.4a) is therefore valid in all realizations of dimensional regularization, i.e. This in particular means that in FDH and DRED no evanescent operators related to ϵ-scalar-Higgs interactions arise at the tree level. The regularization of the second curly bracket in Eq. (3.3) is more involved due to the treatment of γ 5 . According to the discussion in Sec. II we obtain the regularized operators 9 BM∶ O BM J;CDR ¼ i 3! fε μνρσ g ½4 f∂ μ ðψγ ν γ ρ γ σ ψÞg ½d and ð3:5aÞ In analogy to the discussion of operator O G it follows that Eq. (3.5) are valid in all implementations of dimensional regularization, As for operator O G , the corresponding Feynman rules are given in Appendix A 1.

B. Common definition of the form factors
The regularized operators in Eqs. (3.4) and (3.5) give rise to different pseudoscalar form factors of quarks and gluons. So far, in the literature these quantities have been considered in the framework of CDR, using γ BM 5 as defined in Eq. (2.2). The quark form factor related to contributions from operator (3.5), for example, is usually defined via squares of the absolute value of the corresponding matrix elements, where n denotes the loop order in the perturbative expansion. By definition, each term in the sum contains products of ε pseudotensors. Although the ε μνρσ are strictly fourdimensional objects, in the literature their products are usually treated in d dimensions [35,36], 10 fE μ 1 μ 2 μ 3 μ 4 E ν 1 ν 2 ν 3 ν 4 g ½d ≡ f−g μ 1 ν 1 g μ 2 ν 2 g μ 3 ν 3 g μ 4 ν 4 AE perm:g ½d ; ð3:8Þ where "perm." denotes terms originating from further permutations in the Lorentz indices. Even though the application of Eq. (3.8) in general leads to ambiguous results [22], we consider the implications of this choice by using it to evaluate the numerators in Eq. (3.7). If p 1 , p 2 denote the momenta of the external quarks with q ≡ p 1 þ p 2 and p 2 1 ¼ p 2 2 ≡ p 2 , we obtain for the first numerator in the perturbative expansion hM BM;ð0Þ q;J jM BM;ð0Þ q;J i ¼ i 3! 2 q μ 1 q ν 1 fE μ 1 μ 2 μ 3 μ 4 E ν 1 ν 2 ν 3 ν 4 g ½d fTr½γ μ 4 γ μ 3 γ μ 2 = p 1 γ ν 2 γ ν 3 γ ν 4 = p 2 g ½d . This regularization has to be introduced since the r.h.s. of Eq. (3.9) vanishes for d ¼ 4.
Since the regulator drops out in the definition of the form factors, however, the effects of vanishing quark masses are eliminated. In this way, the Lorentz structure related to the pseudoscalar vertex is effectively disentangled from the kinematics of the process and only the (anti)commutation property of the ε pseudotensor is kept. In contrast, a separation between the Lorentz structure and the mass dependence of the effective Lagrangian is not possible when using an anticommuting γ AC 5 . In this case, the square of the absolute value vanishes in the massless onshell case, even for arbitrary d, An on-shell definition of the form factor for the case of massless quarks similar to the one in Eq. (3.7) is therefore not possible for γ AC 5 . The reason is that using Eq. (2.3) as the defining property of γ AC 5 , the ε pseudotensor is implicitly treated in strictly four dimensions. Like in Eq. (3.9) for d ¼ 4, the square of the tree-level amplitudes then vanishes for p 2 ¼ 0. In the next section we provide alternative definitions of the pseudoscalar form factors, 9 Eq. (3.5) is obtained by starting from the unregularized Lagrangian (3.3) and applying the shift of Eq. (2.9) together with Def. (2.2). The structure of operator O BM J;CDR has first been discussed in Ref. [34]. 10 In order to distinguish this d-dimensional treatment of the ε pseudotensor from a strictly four-dimensional one we use the symbol E.
avoiding the use of Eq. (3.8) and including the case of an anticommuting γ AC 5 .
C. Alternative definition and bare results for γ BM 5 Like in Secs. II C and II D, in the following we consider the ε pseudotensor outside dimensional regularization and treat it in strictly four dimensions. In this way it is only the remainder that is dimensionally regularized. Following Ref. [33], we write, for example, the (all-order) contribution of operator O BM J to the quark form factor as 11 M BM q;J ¼ fε μνρσ g ½4ū ðp 1 ÞfðR BM q;J Þ μνρσ g ½d vðp 2 Þ; ð3:11Þ where u and v denote spinors of the external quarks. By construction, the remainder in the second curly bracket is totally antisymmetric in μ, ν, ρ, σ. Regarding its Lorentz decomposition there is only one structure that is linear in the external momentum q. Making the (anti)symmetrization explicit, we write ðR BM q;J Þ μνρσ ¼ ðq μ γ ν γ ρ γ σ − q ν γ μ γ ρ γ σ AE perm:ÞR BM q;J ≡ ðP BM q Þ μνρσ R BM q;J : ð3:12Þ For the extraction of the remainder without indices we define the normalization factor The coefficient of the remainder is then obtained by In practical calculations we directly implement the quantity ðR BM q;J Þ μνρσ . In other words, we use the Feynman rule in Eq. (A1a) and suppress the ε pseudotensor. Using the projection in Eq. (3.14), this modified Feynman rule is then used to compute one-and two-loop contributions of operator O BM J to the form factor. In general, these results are UV and IR divergent. After UV renormalization and IR subtraction, however, the limit d → 4 can be taken. A contraction with the four-dimensional indices of ε μνρσ is then possible.
Using this approach and γ BM 5 as given in Eq. (2.2), we define the (regularization-scheme dependent) pseudoscalar form factors of quarks The notationR ðnÞ a;A for the remainders is chosen such that the index n denotes the loop order in the perturbative expansion, a ∈ fq; gg indicates a contribution to the quark or the gluon form factor, and A ∈ fJ; Gg specifies whether the respective contribution originates from operator O J or O G . The explicit definition of the remainders is given in Appendix A 1. To distinguish the underlying regularization we use a bar for quantities in the FDH scheme and no bar for quantities in CDR/HV. Note that contributions related to operator O J depend on the applied γ 5 scheme which is indicated by the superscript BM.
The lowest-order contributions to the form factors are shown in Fig. 3. As discussed in Sec. III A, they do not depend on the applied version of dimensional regularization, i.e.R BM;ð0Þ q;G and similar for amplitudes with external gluons. At higher perturbative orders, however, FDH results differ from the ones in CDR/HV due to the different treatment of the Lorentz algebra. For the practical calculations in the FDH scheme we follow the guideline given in Sec. 4 of Ref. [26]. More precisely, at the one-loop level we perform the split of Eq. (2.5) and distinguish the evanescent coupling α e which is related to ϵ-scalar-fermion interactions from the gauge FIG. 3. Lowest-order contributions to the pseudoscalar form factorsF GS q;J ,F q;G ,F g;G , andF GS g;J (from left to right). Note, that the "mixed" amplitudesM q;G andM g;J are loop induced and are therefore at least of Oðα s Þ. 11 All other form factors involving ε pseudotensors are treated in the same way. The corresponding definitions are given in Appendix A 1. coupling α s . 12 The two-loop calculations are performed by using a (quasi) d s -dimensional Lorentz algebra as given in Eq. (2.6). Throughout the calculation, d s is identified with 4. The one-and two-loop results of the (bare) form factors in FDH are given in Appendix A 2. They have been obtained in the following way: The generation of the diagrams and analytical expressions has been done with the Mathematica package FEYNARTS [37]. In order to cope with the Lorentz structure in the FDH scheme we used a modified version of TRACER [38]. The subsequent integral reduction and evaluation has been done with an in-house-algorithm that is based on integration-by-parts identities and the Laporta algorithm [39].

5
As shown in Sec. II C, the evaluation of the Lorentz algebra using an anticommuting γ AC Suppressing the spinors and using ðγ AC 5 Þ 2 ¼ I, the remainder can be extracted viā Anticommuting γ AC 5 to the left, the evaluation of the algebra only yields integrals that are scaleless for p 2 1 ¼ p 2 2 ¼ 0. Like in Eq. (3.10), a separation between the Lorentz structure and the mass dependence of the effective Lagrangian is then not possible. An anticommuting γ AC 5 can therefore not be used to obtain the quark form factor related to Lagrangian (3.1) in a massless framework. 13 However, in Sec. IV B we consider Eq. (3.18) in the massless off-shell case to determine so far unknown UV renormalization constants. In this case, the amplitude has a nonvanishing value.

IV. UV RENORMALIZATION
To obtain UV-renormalized Green functions it is useful to distinguish two classes of contributions, (i) renormalization of the couplings, fields, and the gauge parameter, (ii) renormalization of the effective operators O G and O GS J . The renormalization of evanescent couplings in the FDH scheme is well known [40,41]. In any l-loop calculation, the coupling α e describing the interaction of ϵ-scalars and quarks has to be distinguished from the gauge coupling α s in (l − 1)-loop contributions [26], see also Fig. 4. The multiplicative coupling renormalization is given by where μ r and μ 0 denote the renormalization scale and the regularization scale, respectively. In the following we set μ r ≡ μ 0 and suppress the explicit scale dependence of the renormalized couplings; as renormalization prescription we use the MS scheme. The corresponding renormalization constants in FDH are given in Appendix A 3.

5
To describe the UV behavior of the operators O G and O GS J , multiplicative renormalization transformations similar to Eq. (4.1) are not sufficient since the operators mix under renormalization. As shown in Sec. III A, the operator basis remains unchanged when using the FDH scheme instead of CDR due to the absence of evanescent operators at the treelevel. The related renormalization constants, however, are different in both schemes. In analogy to the CDR result [6], we therefore write the operator mixing in FDH as 14

FIG. 4. One-loop diagrams contributing to the form factorF GS
q;J including a gluon (left) and an associated ϵ-scalar (right). The right diagram is proportional to the evanescent coupling α e and only contributes in the FDH scheme. 12 For the definition of α e we refer to Ref. [24]. The only oneloop diagram ∼α e that is relevant for the present computation is the right one in Fig. 4. 13 The fact that the amplitude vanishes for γ AC 5 is not a characteristic of the AC scheme itself but of the observable under consideration. Even using γ BM 5 , the square of the absolute values in Eq. (3.9) vanishes in the massless on-shell case if the ε pseudotensors are treated in strictly four dimensions. 14 Compared to the original reference we added the superscript GS indicating the dependence on the applied γ 5 scheme. The renormalization constants in CDR are defined in the same way without a bar. ð4:2Þ The "mixed" constantsZ GJ andZ GS JG are related to UV divergences of the second and the rightmost diagram in Fig. 3, respectively, and to perturbative corrections thereof. As shown in Ref. [42], the latter constant vanishes to all orders in perturbation theory, i. e.Z GS JG ¼ 0. The former, on the other hand, is at least of Oðα s Þ. Due to the absence of evanescent contributions to the second topology in Fig. 3, its one-loop coefficient is regularization-scheme independent, As discussed in Sec. II C, the use of γ BM 5 in a dimensional framework spoils properties of the axial-vector current and the Ward identities. In this case an additional finite renormalizationZ BM 5 has to be introduced to restore the initial properties [43]. We therefore definē The values ofZ GG ,Z GJ , andZ BM MS in the FDH scheme can be obtained by making use of the fact that they are the only so far unknown quantities entering the UV-renormalized and IR-subtracted form factors. Using Eq. (4.3) and the structure of the IR divergences given in Eq. (A9), the particular structure of the operator mixing allows one to determine the one-and two-loop renormalization coefficients in a unique way.
To illustrate the determination of the renormalization constants we consider the renormalized form factorF BM;ð1Þ q;J given by Eqs. (A6a) and (A8a) as an example. At the oneloop level, any UV renormalization constant has at most single ϵ poles in the framework of dimensional regularization. Depending on which specific scheme is used, the coefficients of these poles differ by terms ∼n ϵ , depending on the treatment of metric tensors and γ matrices. The scheme-dependent part of a one-loop renormalization constant is therefore finite for n ϵ ¼ 2ϵ, ðδZ ð1Þ − δZ ð1Þ Þ ¼ Oðn ϵ =ϵÞ ¼ Oðϵ=ϵÞ ¼ Oðϵ 0 Þ: ð4:6Þ In order to make the scheme-dependent terms explicit, however, we leave n ϵ as an arbitrary variable in the following results. Identifying the (renormalized) couplings, α e ¼ α s , a comparison of Eq. (A8a) with prediction (A9a) for the IR divergences then yieldsF  [6],Z BM MS receives a nonvanishing one-loop contribution in the FDH scheme which is finite for n ϵ ¼ 2ϵ, All other renormalization coefficients can be obtained in the same way. The explicit calculation yields þ Oðα 3 s Þ; ð4:9aÞ þ Oðα 3 s Þ: ð4:9cÞ For n ϵ ¼ 0, Eq. (4.9) agree with the well-known CDR results given e.g. in Ref. [6]. Like in CDR,Z GG coincides with the renormalization of the gauge coupling, see also Eq. (A7a). The results in Eq. (4.9) have been cross-checked with an explicit calculation of the form factors in the off-shell case, including a renormalization of the external parton fields and the gauge parameter. The CDR value of the finite renormalization constant in Eq. (4.5b) is known up to the two-loop level [6], ð4:10aÞ In general, UV renormalized and IR subtracted FDH results differ at most by terms of Oðϵ 0 n ϵ Þ from the corresponding quantities in CDR. Setting n ϵ ¼ 2ϵ and taking the subsequent limit ϵ → 0, these differences then vanish. The value of Z BM 5 is therefore a regularization-scheme independent quantity to all orders in perturbation theory, The regularization-scheme dependent renormalization of operator O BM J at the one-loop level has first been studied in Ref. [3]. 15 For the finite renormalization, the following results are provided, which are valid in CDR (left) and FDH (right). At first sight, there seems to be a contradiction to Eq. (4.10b). However, in Ref.
[3] n ϵ is identified with 2ϵ throughout the calculation. In this way, contributions fromZ BM MS and Z BM 5 are combined. The results in Eq. (4.11) are therefore in agreement with a combination of Eqs. (4.9a) and (4.10).

5
In order to determine the so far unknown renormalization of operator O AC J , we consider contributions toM AC q;J up to the two-loop level in the off-shell case. Following Ref. [27], it is useful to distinguish two classes of contributions: (i) Type A: Contributions where the γ AC 5 vertex is attached to an external quark line, see the left diagram in Fig. 5. (ii) Type B: Contributions where the γ AC 5 vertex is attached to a quark loop, see the right diagram in Fig. 5.

Type A contributions
Type A contributions toM AC q;J can be evaluated in a particular simple way by applying the setup described in Sec. III D. Using ðγ AC 5 Þ 2 ¼ I, all traces can be reduced to expressions without any appearance of γ AC 5 . In this way, no difficulties related to the evaluation of the trace arise. In particular, Type A amplitudes do not contribute to the anomaly. In analogy to the case of γ BM 5 , we therefore write the renormalized operator as Type A∶ ðO J Þ ren ¼Z AC MS ðO AC J Þ bare : ð4:12Þ As before,Z AC MS contains pure poles in ϵ for arbitrary n ϵ . In contrast to Eq. (4.5b), however, we do not include a finite renormalization which is due to the fact that Type A amplitudes are not related to the anomalous contributions toM AC q;J . The fact that there is no need for the introduction of symmetry-restoring counterterms at the one-loop level when using γ AC 5 has first been discussed in Ref. [44]. Further evidence for the validity of Eq. (4.12) beyond the one-loop level will be given below.
The so far unknown renormalization constant can be obtained from an off-shell computation of the amplitudes M AC q;J . We performed the explicit calculation up to the twoloop level and obtain the simple result The renormalization of operator O AC J is therefore trivial, at least up to two loops. This result is closely related to the use of an anticommutator in the right definition of Eq. (2.8a). If we were to define ½γ AC 5 ; γ μ ½n ϵ ≡ 0 instead, δZ AC MS would have a nonvanishing value starting at one loop. In the same way it is the different treatment of strictly 4-and (d − 4)dimensional quantities in Eq. (2.4) that results in the nonvanishing constant δZ BM MS given in Eq. (4.9a).  15 In this reference, the underlying regularization is called "supersymmetric dimensional regularization" (SDR) which in our nomenclature corresponds to dimensional reduction (DRED). Since, however, contributions with external ϵ-scalars vanish for the pseudoscalar form factors, the results coincide with the ones in FDH. Fig. 6. These diagrams either yield direct contributions to the gluon form factor at the one-loop level or they contribute as subdiagrams at higher loop orders. Their one-loop result has been obtained in Sec. II D by using γ BM 5 and the FDF framework, respectively. Generalizing to the case of QCD, we write the corresponding amplitude as

Type B contributions
ABJ ðα s Þδ ab fϵ αβμν g ½4 fl 1 μ l 2 ν g ½d þ OðϵÞ; ð4:14bÞ where the l 1 , l 2 are line momenta attached to the loop. Since momenta do not contain evanescent degrees of freedom it follows that quark loops with external ϵ-scalars vanish. The fact that the result in Eq. (4.14) is regularization-scheme independent has first been found in Ref. [3].
To obtain a similar result with γ AC 5 it is in principle necessary to modify the trace operation. These redefinitions, however, are usually made in such a way that they reproduce Eq. (4.14). Instead of rederiving the already known result in a different framework we directly use it in practical computations. This is done by realizing that the Lorentz and the color structure in Eq. (4.14) are exactly the same as in the Feynman rule given in Eq. (A1c). Accordingly, for Type B contributions the renormalization of operator O J is closely related to the one of O G , see Fig. 7. Up to the two-loop level we therefore write ABJ ðα s Þ × ðO G Þ ren þ Oðα 3 s Þ: ð4:15Þ In this way, γ 5 is effectively removed from the computation. The necessary one-loop renormalization of operator O G does not depend on the treatment of γ 5 and is known from Sec. IVA. 16

Comparison of BM and AC
With the results of the previous sections it is possible to compare the UV-renormalized off-shell values ofF GS q;J obtained in BM and AC, The subscript "ren" indicates that a coupling, gauge parameter, and field (sub)renormalization is applied to the bare coefficients. Taking the limit ϵ → 0, we find that both results in Eq. (4.16) coincide, This provides further evidence for the fact that there is no need for the introduction of finite counterterms when using γ AC 5 . Compared to BM, therefore not only the evaluation of the algebra is much simpler but also the renormalization of operator O AC J . Extending these considerations to higher loop orders, it is possible to determine the so far unknown three-loop value of Z BM 5 from a genuine three-loop calculation. So far, the standard way to obtain Z BM 5 is to consider the (anomalous) relation between the axial-vector and the pseudo-scalar current in the effective theory for the case of N F massless quarks and to evaluate it between two gluon states (see e.g. Ref. [6]). Since the anomaly itself is of Oðα s Þ, however, the l-loop coefficient of Z BM 5 has to be obtained from an (l þ 1)-loop calculation. In contrast, using an extension of Eqs. (4.16) and (4.17) beyond the two-loop level allows one to determine the same coefficient from an l-loop calculation.

C. UV renormalized form factors
Using the results of the renormalization constants from the previous sections together with Eq. (A8), the UV renormalized form factors in the FDH scheme finally read FIG. 7. Equivalence between anomalous two-loop contributions toM AC q;J and δ ð1Þ ABJM ð1Þ q;G . 16 The approach of evaluating Type A contributions using γ AC 5 and Type B contributions using γ BM 5 has been discussed before in Ref. [45]. In this reference, however, the right diagram in Fig. 7 is evaluated as a whole by using projections that lead to similar expressions as in Eq. (3.8). Accordingly, the ε pseudotensor is treated in d ≠ 4 dimensions and additional finite counterterms have to be added to obtain the correct result. In Eq. (4.15), on the other hand, the known Oðα s Þ value of the anomaly is used to effectively reduce the evaluation of the two-loop diagram to a one-loop problem that does not depend on the specific treatment of γ 5 .
Compared to the CDR results which are given e.g. in Ref. [32], the one-loop coefficients differ by terms of Oðϵ 0 Þ, whereas at the two-loop level these differences are of Oðϵ −2 Þ. After subtracting the IR divergences and taking the physical limit ϵ → 0, however, we obtain the same (regularization-scheme independent) results.

V. CONCLUSIONS
In this article we discussed the regularizationscheme dependent treatment of γ 5 within dimensional regularization. So far, CDR in combination with γ BM 5 as defined in Eq. (2.2) has been the most commonly used approach to perform perturbative computations in the dimensional framework. One main reason might be that the approach is based on an explicit construction prescription which enables the use of standard calculational techniques like cyclicity of the trace. At the practical level, however, the evaluation of the algebra is cumbersome due to the increased number of γ matrices and the ad hoc (anti) symmetrization of γ BM 5 operators. Moreover, since initial symmetries are broken explicitly there is an immanent need for the introduction of additional counterterms to obtain correct results. In comparison, the application of an anticommuting γ AC 5 simplifies the evaluation of the Lorentz algebra significantly which is due to the fact that algebraic properties remain unchanged compared to the unregularized theory. This, however, is not the case for γ AC 5 -odd traces. Since these either vanish or do not exhibit cyclicity, special attention has to be paid to the nonbreaking of gauge invariance and other symmetries of the underlying theory. At the one-loop level, the FDF approach avoids all these complications related to the treatment of γ 5 since, using a strictly four-dimensional algebra, the matrices γ BM 5 and γ AC 5 as well as their algebraic behavior are identical. In Secs. II C and II D, FDF has proven as an effective implementation of the Lorentz algebra that reduces the technical complexity significantly, even including contributions to the axial anomaly. At the same time the results are compatible with gauge invariance and Bose symmetry. In the examples considered, the FDF results are entirely given by so-called extra integrals which can be evaluated in a particular simple way. The question whether FDF can be extended beyond the one-loop level, such that it leads to a facilitation compared to more traditional schemes remains to be answered. At the two-loop level, we investigated the possibility of utilizing the benefits of different γ 5 schemes and computed the pseudo-scalar form factors of quarks and gluons in the FDH scheme. We have shown explicitly that evanescent Higgs-ϵ-scalar interactions are absent and determined the so far unknown UV renormalization of the corresponding operators. The results of the UVrenormalized form factors are compatible with the general prediction for IR divergences in FDH. As a general recommendation for the treatment of γ 5 we find that the use of γ BM 5 should be avoided whenever γ AC 5 leads to an obvious and immediate simplification. This clearly applies to Type A contributions to the pseudoscalar form factors where not only the evaluation of analytical expressions is simplified but also the operator renormalization when using γ AC 5 . It should be mentioned explicitly that these simplifications are not restricted to FDH but apply to all considered dimensional schemes. For the evaluation of (anomalous) γ 5 -odd expressions (like Type B contributions in Sec. IV B), however, it is not clear at all, if the use of γ AC 5 leads to a perceptible simplification due to the aforementioned complications. In this case, the use of γ BM 5 therefore still constitutes a viable alternative. Moreover, seizing a suggestion of Ref. [9], Type B contributions can be obtained by removing γ 5 in analytical expressions altogether. This can be done by using the well-known and scheme-independent results of the anomalies. In this way not only the evaluation of the amplitudes is significantly simplified but also the related operator renormalization.
Finally it should be mentioned that observables related to Lagrangian (3.1) are usually obtained in an effective theory for the case of massless quarks. One requirement for this option is that the Lorentz structure related to the γ 5 vertex can be effectively disentangled from the kinematics of the underlying process. As it turned out, for the schemes considered in this article this is only possible for γ BM 5 . For the choice of a particular γ 5 scheme one therefore has to compare the complexity of a calculation with massless quarks and the extended γ BM 5 algebra with the complexity of a calculation with massive quarks and a simplified γ AC 5 algebra. The decision which of these alternatives is the more efficient one remains to be made on an individual basis.

ACKNOWLEDGMENTS
It is a pleasure to thank Dominik Stôckinger and Michael Spira for fruitful discussions and helpful comments on the manuscript. In Eq. (A7b), the renormalized couplings are set equal, i.e. α e ¼ α s . For the calculations in the off-shell case, also a UV renormalization of the external quark and gluons fields and the gauge parameter is needed. The corresponding renormalization constants can be found in Refs. [26,46]. According to operator renormalization in BM, the first perturbative coefficients of the UV-renormalized form factors in the FDH scheme are given bȳ The subscript "ren" indicates that the coupling renormalization (4.1) is applied to the bare one-loop amplitudes. After UV renormalization, the evanescent coupling α e is identified with the gauge coupling, i.e. α e ¼ α s .

IR divergence structure
The IR divergence structure of one-and two-loop FDH amplitudes has been investigated in Ref. [24]. Specifying to the case of massless form factors with two external quarks and gluons, respectively, a Z factor subtracting all IR divergences is given by : The relation between the perturbative coefficients of ln Z and the UV-renormalized form factors is given by The Z factor is written in terms of the IR anomalous dimensionsΓ 0 mn ¼ −2γ cusp mn C q=g andΓ mn ¼ 2γ q=g mn with C q ¼ C F for the quark form factor and C g ¼ C A for the gluon form factor. In FDH, the values of the partonic IR anomalous dimensionsγ cusp mn ,γ q mn , andγ g mn are known up to the two-loop level [24]. Together with the known values of the one-loop β coefficients it is therefore possible to predict the entire IR divergence structure of the FDH form factors up to the two-loop level. Since Eq. (A9a) is written in terms of UV renormalized couplings, they can be set equal (α e ¼ α s ).