Two-loop off-shell QCD amplitudes in FDR

We link the FDR treatment of ultraviolet (UV) divergences to dimensional regularization up to two loops in QCD. This allows us to derive the one-loop and two-loop coupling constant and quark mass shifts necessary to translate infrared finite quantities computed in FDR to the MSbar renormalization scheme. As a by-product of our analysis, we solve a problem analogous to the breakdown of unitarity in the Four Dimensional Helicity (FDH) method beyond one loop. A fix to FDH is then presented that preserves the renormalizability properties of QCD without introducing evanescent quantities.


Introduction
When using customary approaches [1][2][3][4][5] to handle the ultraviolet (UV) problem in quantum field theory (QFT), intermediate steps are necessary to extract physical answers from loop calculations. In particular, the Lagrangian L of the theory is modified by adding UV counterterms (CTs). They absorb the divergences generated by the high-frequency part of the loop integrals and as one moves up the orders in the loop expansion one must include lower loop CT calculations in order to be consistent. When carrying out this renormalization program UV infinities are usually regulated via dimensional regularization [4] (DReg) and renormalized quantities are defined by specifying what is subtracted from the bare ones. A particularly convenient subtraction scheme is MS, in which only UV poles and universal constants are dropped.
The FDR 1 approach of [6] deals with UV infinities in a different way. A new kind of loop integration 2 is introduced that coincides with Riemann integration in UV finite cases, but produces a finite and regulator free answer also when acting on divergent integrands. 3 In this way no CTs need to be incorporated into L: they are traded for a change in the definition of the loop integration. Moreover, FDR directly generates renormalized amplitudes since it is independent of any UV cutoff.
The main aim of this paper is to construct the one-and two-loop transition rules between FDR and MS in the framework of QCD. As a perturbative treatment of a renormalizable QFT is unique up to a renormalization scheme dependence, we have achieved this task by studying off-shell 4 ℓ-loop QCD correlation functions G ℓ and by searching for a DReg renormalization scheme which reproduces the FDR answer when ℓ = 1 and ℓ = 2, i.e.
We dub such a scheme D Reg FDR and its renormalization constants Z ℓ,FDR . These can then be used to extract the coupling constant and quark mass shifts that relate FDR to MS. We emphasize that in a typical FDR calculation there are no Z ℓ,FDR , since one directly computes eq. (1.1b). We introduce them in the context of this paper only because we want to work out the correspondence of FDR with a canonical renormalization approach based on counterterms.
The second insight of our work concerns the nature of self contractions of Lorentz indices and γ-matrices explicitly appearing in G ℓ bare , that we denote by n s . When studying eq. (1.1) one needs to reconcile the value n s = n dictated by DReg in (1.1a) with n s = 4 used in (1.1b). This has to be done without spoiling the renormalizability of QCD. Following FDR as a guidance, we have been able to establish the correct n s → 4 limit of DReg, compatible with a local subtraction parametrized in terms of the Z ℓ,FDR . A mismatch between the structure of G ℓ bare and the CTs has been recently recognized [7][8][9] to cause the failure of the renormalization program beyond one-loop in the original formulation of the Four Dimensional Helicity scheme (FDH) [10,11]. Since we observe that terms restoring the right cancellations are produced by FDR, we argue that the FDR approach provides a natural solution to this problem. In contrast with methods that require the introduction of ǫ-scalars [12], in our case no new fields nor evanescent couplings need to be added to the QCD Lagrangian. Moreover, a shift linking the MS quark mass to a fixed version of FDH can be worked out. So far we have tested this approach for off-shell correlation functions. What happens in the presence of IR divergences needs to be further investigated.
The outline of the paper is as follows. In the next section we review FDR. Sections 3 and 4 describe our computational strategy. Our results are collected in section 5 and appendix A. Furthermore, in section 6 we discuss our fix to FDH. Finally, appendices B-D contain explicit examples of our algorithm.

FDR
FDR integration was first introduced in [6] and several examples of one-and two-loop computations can be found in [13][14][15][16][17]. Here we briefly review its definition, which we need to make contact with our calculation.
Consider an ℓ-loop integrand J(q 1 , . . . , q ℓ ) depending on ℓ loop momenta q 1 , . . . , q ℓ . The multi-loop FDR integration over J is defined as 5 where J F (q 1 , . . . , q ℓ , µ 2 ) is the UV finite part of J(q 1 , . . . , q ℓ , µ 2 ), specified below, and µ a vanishing mass required to extract J F from J. J(q 1 , . . . , q ℓ , µ 2 ) and J F (q 1 , . . . , q ℓ , µ 2 ) are read off from J(q 1 , . . . , q ℓ ) according to the following rules: i) Squares of integration momenta appearing both in the denominators of J(q 1 , . . . , q ℓ ) and in contractions generated by Feynman rules are shifted by µ 2 This replacement is called global prescription; ii) A splitting is performed in such a way that UV divergences are entirely parametrized in terms of divergent integrands 6 contained in [J INF ], that solely depend on µ 2 ; iii) The global prescription of eq. (2.2) should be made compatible with a key property of multi-loop calculus: 7 in an ℓ-loop diagram, one should be able to calculate a sub-diagram, insert the integrated form into the full diagram and get the same answer. (2.4) We dub this sub-integration consistency.
Finally, after lim µ→0 is taken, ln µ → ln µ R is understood in the r.h.s. of eq. (2.1), where µ R is an arbitrary renormalization scale. This definition preserves shift invariance (2.5) and the possibility of canceling numerators and denominators , (2.6) which are properties needed to retain the symmetries of L. From eqs. (2.5) and (2.6) it follows that algebraic manipulations in FDR integrands are allowed as if they where convergent ones. This authorizes one to reduce complicated multi-loop integrals to a limited set of Master Integrals (MI) by using four-dimensional tensor decomposition [14] or integrationby-parts identities [17]. In other words, the definition in eq. (2.1) can be applied just at the end of the calculation, when the actual value of the MIs is needed. An important subtlety implied by eq. (2.6) is that the needed cancellation works only if integrands involving explicit powers of µ 2 in the numerator are also split via eq. (2.3) as if µ 2 = q 2 i , where q 2 i is the momentum squared which generates µ 2 . As a consequence, although only one kind of µ 2 exists, one has to keep track of its origin when it appears in the numerator of J(q 1 , . . . , q ℓ , µ 2 ). For this we use the notation µ 2 | i , which understands the same splitting required for q 2 i . FDR integrals with powers of µ 2 | i in the numerator are called "extra integrals". Their computation is elementary. One-and two-loop examples can be found in [6,14] and appendix D.

FDR and DReg on the same footing
Eq. (1.1) defines the D Reg FDR scheme we are looking for, i.e. the DReg scheme reproducing the FDR correlation functions. As the r.h.s. of (1.1a) is computed in DReg while the r.h.s. of (1.1b) in FDR, we need a common framework which accommodates both approaches. Here we illustrate this framework at two loops, but the same considerations apply at any loop order.
Our starting point is eq. (2.3) when J(q 1 , q 2 ) is the sum of all integrands contributing to G 2 bare . If J(q 1 , q 2 ) is free of IR infinities we can rewrite the DReg integration over it as 8 where the absence of IR divergences authorizes us to extract lim µ→0 out of the integral. The integral over J F (q 1 , q 2 , µ 2 ) is G 2 FDR and through eq. (3.1) we have isolated it within 8 n is the number of dimensions defined as n = 4 − 2ǫ.
G 2 bare , so that the D Reg FDR scheme can be determined by solely looking at the pieces which differ between eq. (1.1a) and (1.1b), namely J INF (q 1 , q 2 , µ 2 ) , (1-loop-CTs) and (2-loop-CTs). (3.2) This defines our strategy. For instance the (1-loop-CTs) are known at two loops, so that D Reg FDR is defined by choosing which sets the finite part of G 2 bare to the FDR value. An important condition should be fulfilled by eq. (3.3). It should be local and proportional to the the Born correlation function G 0 : which is the correct form to be re-absorbed into Z 2,FDR . We dub eq. (3.4) our renormalizability condition. To understand its consequences it is convenient to split [J INF ] as follows: is a factorizable contribution in which only one sub-integration is UV divergent. It is called the sub-vacuum and depends on the kinematic scales entering the UV finite sub-integration. The second piece is independent of kinematics and it is called the global vacuum. 9 Eqs. INF ] should cancel one-loop counterterms. In section 4 we will describe the subtleties of this cancellation.

The calculation
As outlined in the previous section the core of our strategy is calculating FDR vacua in DReg instead of throwing them away. As a first step we write down 10 the twelve integrands J G ℓ i (i = 1, . . . , 6 and ℓ = 1, 2) corresponding to the one-particle irreducible one-and twoloop QCD correlation functions drawn in figure 1, excluding CT diagrams. Then, we split them as in eq. (2.3) and compute the integrals over their vacuum parts, in the symmetric off-shell kinematic point With this choice all integrals are made free of IR divergences, so eq. (3.1) applies. Furthermore, the complexity of the calculation is reduced to a one-scale problem. An important As I 1 i and I 2 i are regulated in DReg, n s = n should be used in eq. (4.1). The knowledge of the I ℓ i allows us to parametrize the FDR subtraction in terms of renormalization constants Z ℓ,FDR i . However, not all of them are independent, as they are related by QCD Slavnov-Taylor (S-T) identities. 11 In the following we consider, in turn, the ℓ = 1 and ℓ = 2 contributions to the Z ℓ,FDR i .

The one-loop case
At one loop the calculation of I 1 i is simple. Schematically: • The global prescription of eq. (2.2) ensures that reducibleq 2 s in the numerator simplify with denominators; • The integrands are then split as in eq. (2.3) by using the propagator identity which allows one to express I 1 i in terms of tensors of the kind 11 See section 5.
• Finally, by virtue of lim µ→0 , polynomially divergent integrals vanish, so any tensor structure is reduced to a fundamental scalar and where g α 1 ···α 2r is completely symmetric and only made of products of metric tensors.
Eventually, the logarithm in eq. (4.7) has to be combined with the one-loop analogue of the J F term in eq. (3.1) to compensate the µ dependence of the finite part. 12 As a result, the UV divergent part of Now we study Fin(I 1 i ), namely the UV convergent contribution to I 1 i . It determines the finite part of the one-loop subtraction term in eq. (1.1a) and appears due to the explicit presence of n s in J we see that when λǫ multiplies UV single poles, constants are generated that are fully subtracted in D Reg FDR but contribute to the finite part of G 1 i in MS. Thus, the requirement of eq. (1.1b) causes a deviation from the minimality of the Z 1,FDR i proportional to (1 − λ), such that λ = 0 (1) in D Reg FDR (MS). As one finds that both Inf(I 1 i ) and Fin(I 1 i ) factorize the Born, one computes In summary, at one loop it is possible to perform a calculation in DReg and consistently renormalize the result to reproduce the FDR answer. In other words, FDR can be interpreted as a particular renormalization scheme of DReg, i.e. the D Reg FDR scheme we are looking for.

The two-loop case
Here we illustrate the calculation for massless QCD. When dealing with a non vanishing quark mass m q the formulae complicate a bit, but the reasoning remains unchanged. For simplicity, we start from cases 14 in which we see that the sub-integration consistency (2.4) does not play any role, and postpone to section 4.3 the study of more complicated situations.
After scalarization by means of tensor reduction and integration-by-parts, one finds two types of contributions to I 2 i : • An integral over the global vacuum of the kind where F ij (n, n s ) are rational functions and q 12 = q 1 + q 2 ; • An integral over the factorizable sub-vacuum 15 of the form By judiciously using the identity can be split into a piece which develops a ln µ 2 upon integration and a term where µ can be set to zero at the integrand level With our special kinematics we find where A i (n, n s ), A ′ i (n, n s ) and B i (n, n s ) are rational functions. Notice that the UV finiteness of J i (q 1 , µ) ensures that the the pole parts of the integrals over J ai (q 1 , µ) and J bi (q 1 ) cancel each other. Furthermore, B i (n, n s ) = 0 in the case of two-point correlation functions. An explicit example of the procedure yielding eqs. (4.12)-(4.17) is reported in appendix B.
The additional complexity at two loops is that, unlike the integral 15 This sub-vacuum is generated by the application of the identities (with i = j) needed to disentangle the sub-divergences of I 2 i from the finite part.
depends on the kinematics. Thus, it generates logarithms of physical scales that cannot be absorbed into the Z 2,FDR i . Therefore eq. (3.3) tells us that, in order to establish the connection between FDR and the standard renormalization approach, we must demonstrate that such non-local terms are compensated by the sum of all diagrams containing insertions of one-loop counterterms. 16 This contribution is dubbed CT i in the following. To achieve this cancellation, we have to recast eq. (4.19) into a form suitable to be combined with the CT i . Observing that 17 leads us to consider instead of eqs. (4.18) and (4.19). Now SV ′ bi has the same structure of a counterterm diagram, as it factorizes V 0 . An explicit calculation of the CT i gives and we obtain, for correlation functions without external fermions, the following two-loop renormalization constants (4.23) in accordance with the renormalizability condition of eq. (3.4). As in eq. (4.9), the choice of the point µ = µ R has the effect of removing the dependence on µ from the last integral in eq. (3.1). As in the one-loop case, n s in eqs. (4.12) and (4.17) generates finite terms proportional to (1−λ) in the Z 2,FDR i . In addition, finite contributions are created by two-loop integrands that do not multiply powers of λǫ, e.g.
To denote their origin we multiply by a parameter δ any finite combination which is fully subtracted in D Reg FDR but contributes to the finite part of G 2 i in MS, so that δ = 1 (0) in D Reg FDR (MS). 16 For consistency with the form of the Z 1,FDR i the one-loop insertions have to be computed with λ = 0. 17 V0 is defined in eq. (4.9).

"Extra"-extra integrals and sub-prescription
In the case of QCD Green's functions with external fermions eq. (4.22) does not hold true 18 and leads to results incompatible with eq. (3.4). In this section, we show that the FDR subintegration consistency (2.4) requires the introduction in the finite part of the correlation functions of "extra"-extra integrals (EEIs) with the same structure of SV ′ i , namely EEI i (µ) = EEI ai (µ) + EEI bi . (4.26) They are just what is needed to restore the renormalizability condition, i.e. 19 and with EEI ai independent of physical scales. The FDR origin of the EEI i is the need of introducing a sub-prescription to cure the mismatch between the global prescription of eq. (2.2) and the consistency condition (2.4). In fact, although essential to preserve gauge invariance at two loops, the shift may clash with the analogous replacement required to ensure (2.4) at the level of divergent one-loop sub-diagrams. This is better explained with an example. Consider the two-loop diagram 20 of figure 2. Before applying eq. (4.29) its numerator reads 2 2q 2 12 (q 2 1 + k 2 1 + 2(q 1 · k 1 )) Figure 2: The two-loop diagram contributing the N f corrections to G 2 3 . N (q 1 , q 2 ) is given in the text dropping irrelevant constants. The replacement q 2 i →q 2 i is performed only in the denominators.
The sub-prescription is defined as the effect of eq. (4.29) on N (q 1 , q 2 ) from the point of view of the divergent sub-diagram pulled out from the rest of the diagram, as in figure 3. From the perspective of the q 2 sub-integration one has to distinguish internal and external parts. We denote the separation of the parts external to the sub-diagram by placing hats on the Lorentz indices. Algebraically the hats do not make any difference other than to denote the fact that they are of an origin which is external to the subdiagram, so all standard identities apply, for example γαγα = γ α γα = 4, γαq 2 α = / q 2 .
(4.31) 18 Interestingly, it applies when computing I 2 3 and I 2 6 with ns = 4, which is the value prescribed by FDH, as discusses in section 6. Nevertheless, we emphasize that we must set ns to n because our strategy is to reproduce the FDR result by regularizing FDR vacua in DReg, which dictates ns = n. 19 When calculating the vacuum EEIs contribute with a minus sign and should be computed with ns = n. 20 This is the only possible contribution to G 2 3 proportional to N f .
We then study how the part 21 of N (q 1 , q 2 ) divergent in q 2 N (2) (q 1 , q 2 ) = 8(/ q 1 + / k 1 )q 2 2 + 8/ q 2 (/ q 1 + / k 1 )/ q 2 , (4.32) transforms under eq. (4.29). From the perspective of the sub-prescription the second term is inert, since it has an origin external to the sub-diagram. Thus On the other hand, the two-loop global prescription dictates that also the second term must transform as 22 leading to no change in N (2) : . (4.36) Here, we mark theμ 2 | 2 with a hat because it is acting only on the q 2 sub-integral. One finds 24 where the last representation is suitable to be used to prove eq. (4.27). A further example of computation of an EEI via sub-prescription is given in appendix C. Finally, it is important to realize that β αα β  figure 1 we infer that one must find zero when summing up the EEIs from all contributing diagrams. Indeed, we have explicitly checked that this happens in the case of the N f corrections to the gluon and ghost propagators. 21 Here this is quadratic because less-than-quadratic terms do not generate extra integrals. 22 This is obtained by anticommuting until theq2 meet. 23 By doing so, we reintroduce the correct one-loop behaviour leading to (2.4). 24 See appendix D. 25 See section 5.

Results
The main result of our calculation is the list of QCD renormalization constants in appendix A. When setting (λ, δ) = (1, 0) the known MS formulae [20][21][22] are recovered. The choice (λ, δ) = (0, 1) corresponds to D Reg FDR , i.e. the renormalization scheme which reproduces the FDR QCD Green's functions up to two loops.
As required by the QCD S-T identity we have verified that the three ratios produce the same coupling constant renormalization, namely Z α S (λ, δ) in eq. (A.9). In the case of Z α S (0, 1) this provides a stringent test on the universality of FDR.
Using m q = 0 in G ℓ 3 gives the quark mass renormalization constant in eq. (A.10). Again the correct MS result [21] is reproduced with λ = 1 and δ = 0. As an extra check, we have verified that any m q dependence drops in Z ΨΨ , as should be.
Eq. (A.9) can be used to infer the coupling constant shift between FDR and MS In an analogous way eq. (A.10) produces the quark mass shift

Eqs. (5.3) and (5.4) provide the transition rules from IR finite QCD quantities computed in FDR and their analogue in MS.
Finally, it is well known that one can use the coupling constant shift between two schemes at a given order to relate the two beta functions at one order higher. In the following we sketch out the two-loop proof of this [11] and use it to derive the three-loop beta function in FDR. Let us consider two different renormalization schemes defined in terms of two coupling constants α A and α B related by a shift The two beta functions are linked by the chain rule Writing both sides in terms of the same α through eq. (5.5) we find the standard result of scheme independence up to two loops, i.e. b A 0 = b B 0 and b A 1 = b B 1 as well as an expression for the three-loop beta function in scheme B, using only two-loop information from scheme To calculate the three-loop beta function in FDR we use the three-loop MS beta function in [23] together with the values from from eq. (5.3). This gives

Fixing the UV behaviour of FDH
Recently it has been observed that the original formulation of the FDH method breaks unitarity [7]. This was seen as the standard renormalization program fails to remove all UV poles at high enough perturbative orders. During our work, it proved of interest to try and investigate this problem from our renormalization scheme perspective. Using the technology developed in the previous sections we can easily perform an analysis of the validity of the interpretation of FDH as a perturbative description of QFT. We begin by setting up the problem as in eqs. (1.1). We first recall eq. (1.1a) for an arbitrarily renormalized, dimensionally regulated correlation function. Next, we equate it to correlation functions calculated in the FDH scheme up to two loops, i.e. 26 Here and in the following we explicitly write the values of ns to highlight if contributions are regulated in FDH (ns = 4) or DReg (ns = n). In a fully general FDH scheme ns is an arbitrary parameter, but we restrict to the ns = 4 case.
If FDH amounts to a different renormalization scheme to MS, then the ℓ-loop-CTs should satisfy the renormalizability condition (3.4), i.e. they should be proportional to the Born and local. We can proceed to calculate these at each order in a similar way to section 4, making use of information from the FDR vacuum part. At one loop, the calculation is simple and provides the exact same Z 1 as FDR, indicating that FDH exhibits the same coupling shift from MS. At two loops the computation is more complicated but in general we can write the following expression for what should be the two-loop counterterms: We now split SV and EEI as in eqs. (4.21) and (4.26). As n s = n in the second line, we can make use of eq. (4.27). A direct calculation of the first line gives instead a striking different result for n s = 4: the CT part cancels the b piece of the sub-vacuum, but not the EEI. In summary This is problematic for FDH as the EEI b term does not in general satisfy the renormalizability condition. That is, we see that for EEI b = 0, FDH cannot be interpreted as a perturbative description of QFT. Nevertheless, as we have seen previously, the EEI b contribution is zero for external gauge states and so the only expected problems come with external fermions, just as experienced in [7]. The form of equation (6.3) naturally suggests a new definition of FDH which satisfies the renormalizability condition. Let us consider changing the bare two-loop FDH correlation function in the following way 27 This now allows us to write down a working FDH analogue of eq. (4.28) Here we call this modified definition FDH ′ . Its renormalization constants Z ℓ,FDH ′ are given in appendix A and correspond to the case (λ, δ) = (0, 0). From these we are able to calculate the analogous version of eq. (5.1) in this scheme, indicating that the QCD S-T identity is respected, even with external fermions 28 . What's more, we verify that in all cases the coupling constant shift agrees with the literature value [11], i.e. FDH ′ is equivalent to FDH when the latter scheme makes sense and provides consistent predictions in all the other cases. Together these results suggest that, at least off-shell, this definition does not face the renormalization difficulties of the original FDH formulation. Furthermore, as we newly have control over the fermion sector, we are able to calculate a mass renormalization constant, and thereby a mass shift between MS and FDH ′ Our new definition offers a different perspective than the recently proposed approaches to the unitarity based difficulties of FDH. Roughly speaking, any solution requires a connection between internal and external states. In the approach of [12], evanescent operators are introduced, as in dimensional reduction, to make the external states behave like the internal ones, thereby introducing a series of evanescent couplings. Conversely, our approach based upon sub-integration consistency makes internal states behave like external ones, and so does not require the introduction of any new fields whilst maintaining n s = 4 spin degrees of freedom.

Summary and outlook
As the FDR UV subtraction is consistently encoded in the definition of a four-dimensional and finite loop integration, the FDR approach to QFT does not require the introduction of counterterms in the Lagrangian. In particular, an order-by-order renormalization is avoided: the ℓ th perturbative order is computed by only looking at ℓ-loop Feynman diagrams.
We have proven, up to two loops in QCD, that FDR is equivalent to a particular renormalization scheme of Dimensional Regularization, dubbed D Reg FDR , whose renormalization constants, extracted from two-and three-point vertices, obey the Slavnov-Taylor identities. D Reg FDR and MS are related by shifts in α s and m q that we have explicitly computed. The transition rules derived in this paper can be used to translate calculations of IR finite quantities from FDR to MS.
During our analysis, we have identified a bottom-up interpretation of the failure of the naive FDH scheme beyond one loop. FDR provides a natural fix to this: it automatically generates the finite pieces needed to restore the renormalizability of FDH. These extra terms are computed without introducing ǫ-scalars nor evanescent quantities in the Lagrangian and can be directly read off from two-loop diagrams. Including such FDR inspired terms in FDH defines a consistent renormalizable scheme that we have called FDH ′ .
In this paper we have concentrated our focus on IR finite quantities. The possibility of consistently using FDR to regulate final state soft/collinear divergences at one-loop has been proven in ref. [15]. The study of FDR in the presence of NNLO IR singularities is left for future investigations.

A. The renormalization constants of the QCD vertices
In the following we list the renormalization constants of the QCD correlation functions of figure 1 up to two loops in the Feynman-'t Hooft gauge The three renormalization schemes we are interested in are parametrized by the values of λ and δ given in table 1. N c is the number of colours, N f the number of active fermions, 2Nc and α S = α S (λ, δ) is the QCD coupling constant of each scheme. The constant originates from the evaluation of the two-loop global vacuum in DReg, Z GΨΨ = Z 6 (λ, δ) The above results can be used to derive the renormalization constant of the QCD coupling through eqs. (5.2) Finally, computing G ℓ ΨΨ with m q = 0 gives the renormalization constant associated with the quark mass

B. A two-loop diagram
As an example of the algorithm we use to extract and compute the FDR vacuum we work out in detail one among the two-loop diagrams contributing to G 2 2 in figure 1, namely the ghost-loop correction depicted in figure 4.
The integrand of the corresponding am- plitude is proportional to with q 12 = q 1 + q 2 and D 1 = (q 1 + k 1 ) 2 . To read off J(q 1 , q 2 , µ 2 ) from J(q 1 , q 2 ) we apply the shift of eq. (2.2) in both the numerator and the denominator and simplify reducible numerators. This gives Notice that the integrand manipulations we have performed so far are allowed both in FDR and DReg. In the following we concentrate on the first tensor and explicitly derive the splitting The other terms in eq. (B.2) are easier and can be treated analogously.
• J 2 (q 1 , q 2 , µ 2 ) : The the odd-rank tensor in the r.h.s. of eq. (B.10) gives zero upon integration. On the other hand µ 2 → 0 is allowed in the q 1 integrand multiplying the rank-two tensor. Thus The first term in the curly bracket contributes to SV 2 (µ) with A ′ 2 = −A 2 = g αβ /4, while the second to GV 2 (µ) with F 22 = 0 and F 21 = g αβ /4.
The integrand depending on q 1 in eq. (B.18) behaves as 1/q 4 1 when q 1 → 0. To disentangle the term which develops a ln µ 2 we rewrite 29 q 1γ q 1δ (p 2 + 2(p · q 1 )) which results in Since the integral over q 1 is UV finite only contributes to SV 2 (µ). By using tensor reduction and integration-by-parts identities one finds A ′ 2 = − 1 24 g µν + n−4 24 p µ p ν and A 2 = 1 24 g µν for this term.

C. Sub-prescription example
Here we discuss a further example of the sub-prescription in order to aid the understanding of the reader for future FDR calculations. Consider the FDR integral corresponding to the contribution to G 2 3 in figure 5, where D i = q 2 i + k 2 1 + 2(q i · k 1 ). We wish to discuss how to extract the q 1 q 2 k 1 q 1 +k 1 q 12 +k 1 q 2 +k 1 Figure 5: Diagram contributing to G 2 3 . The fermion momenta follows the fermion line.
EEI resulting from the sub-prescription, so as in figure 2 we consider the un-promoted numerator in order to find the relevant terms. In this diagram we have two sub-divergences, one for fixed q 1 and another fixed q 2 . The terms from the sub-prescription of each can be extracted considering the sub-divergences independently. First we shall consider q 1 fixed. Let us disconnect the divergent subdiagram, as in figure 6.
30 See appendix D.