FDR, an easier way to NNLO calculations: a two-loop case study

In this paper we illustrate the simplifications produced by FDR in NNLO computations. We show with an explicit example that - due to its four-dimensionality - FDR does not require an order-by-order renormalization and that, unlike the one-loop case, FDR and dimensional regularization (DR) generate intermediate two-loop results which are no longer linked by a simple subtraction of the ultraviolet (UV) poles in epsilon. Our case study is the two-loop amplitude for H ->gamma gamma, mediated by an infinitely heavy top loop, in the presence of gluonic corrections. We use this to elucidate how gauge invariance is preserved with no need of introducing counterterms in the Lagrangian. In addition, we discuss a possible four-dimensional approach to the infrared (IR) problem compatible with the FDR treatment of the UV infinities.


Introduction
Computing radiative corrections has become of uppermost importance in particle phenomenology [1]. The present lack of unexpected signals at the LHC pulls the effects of New Physics in a domain where small discrepancies have to be searched via detailed comparisons between experimental results and precise calculations of the Standard Model background. Due to the large QCD coupling constant, precise predictions at the LHC often require NNLO accuracy. On the other hand, two-(or more)-loop calculations in the complete Electroweak (EW) model will be mandatory at the future International Linear Collider to meet the experimental accuracy foreseen, for example, in Higgs Physics [2].
In this paper we investigate the possibility of further simplifying NNLO computations by abandoning dimensional regularization [29]. Despite its known virtues, DR requires a heavy analytic work aimed at subtracting powers of 1/ǫ of UV or IR origin even before attacking the calculation of the finite physical part. For instance, DR forces an orderby-order iterative renormalization, which is especially cumbersome when computing loop corrections in the EW model or in SUSY: the full set of one-loop counterterms has to be determined and added in a two-loop computation, and so on. Furthermore, loop functions used at a given perturbative level must be further expanded in ǫ -when appearing at higher orders -to include terms generating O(ǫ 0 ) contributions when multiplied by the new poles. Such complications arise in DR because constants needed to preserve the symmetries of the Lagrangian are often produced by ǫ/ǫ terms, which are kept under control by the iterative renormalization.
This has driven us to study the performances of FDR [30] as a simpler four-dimensional approach beyond one loop 1 . The key point of FDR is that the use of counterterms is avoided by defining a four-dimensional and UV-free loop integration in a way compatible with shift and gauge invariance. Having done this, the correct results automatically emerge once the theory is fixed in terms of physical observables by means of a finite renormalization relating the parameters of the Lagrangian to measured quantities. In addition, IR infinities can be naturally accommodated within the same four-dimensional framework used to cope with the UV divergences. This is why we envisage in FDR a great potential to reduce the complexity of the NNLO calculations, especially when used together with numerical techniques.
In this paper we present, as the first example of a two-loop FDR calculation, the QCD corrections to the top-loop-mediated Higgs decay into two photons, in the limit m top → ∞. This computation gives the opportunity to fully appreciate the simplifications due to the four-dimensionality of the approach in a realistic two-loop case study 2 . In the next section we review the general FDR idea with special emphasis on the two-loop case. We discuss, in particular, the shift and gauge invariance properties of the FDR integration, the main differences with DR, and the IR problem. The two-loop FDR calculation of H → γγ is presented in section 3 and the technical details are collected in the final appendices.

Definition of the FDR loop integral
FDR subtracts UV divergences at the integrand level. This is obtained in two steps. Firstly, the propagators of the particles flowing in the loops are given a common additional term −µ 2 , formally identified with the +i0 propagator prescription. For example, vector-boson and fermion propagators with momentum (q +p i ) and mass M i read 3 , in the unitary gauge, respectively, withD Secondly, UV infinities are isolated by a repeated use of the identity In fact -being d i is at most linear in q -the second term in the r.h.s. of Eq. (2.3) is less UV divergent than the original denominator, so that UV divergences can be systematically moved to terms such as 1/q 2 , which depend only on µ, and directly subtracted from the integrand. Schematically, dubbing J the original integrand of an ℓ-loop function, one has J(q 1 , . . . , q ℓ ) = J INF (q 1 , . . . , q ℓ ) + J F,ℓ (q 1 , . . . , q ℓ ) , (2.4) where J INF collects the UV divergent integrands. Then, the FDR integral over J is defined as 4 where, due to the limit µ → 0, only a logarithmic dependence on µ remains, which can be traded for a dependence on the renormalization scale 5 . Thus, FDR and normal integration coincide in a convergent integral, since no divergent part J INF can be extracted from its integrand. Furthermore, the space-time is kept strictly four-dimensional also in divergent integrals -with g αβ = diag(1, −1, −1, −1) -because µ 2 is nothing but the infinitesimal deformation needed to define the loop integrals 6 and it is not generated by higher-dimensional components of the integration momenta. This allows one to perform, in particular, the Dirac gamma algebra in four dimensions, with extra rules needed to keep gauge invariance, as explained in subsection 2.4.
An explicit example of integrand FDR expansion 7 at one loop is given by where p 0 = 0 and the terms in square brackets are proportional to UV divergent integrands. A two-loop example with . (2.9) Note that identities such as are needed to extract the sub-divergences. Then, the one-and two-loop FDR integrals over the integrands in Eqs. (2.6) and (2.8) read (2.11) 7 We denote the expansion of an integrand J needed to bring it in the form of Eq. (2.4) as its FDR defining expansion.
It is important to realize that divergent tensor structures are fully subtracted from the original integrand, as in Eq. (2.6) 8 . Owing to the Lorentz invariance and fourdimensionality of this definition, irreducible tensors can be decomposed in terms of scalars. For example 9 can be rewritten as Finally, polynomials in the integration variables represent a limiting case of Eq. (2.4), in which J F,ℓ (q 1 , . . . , q ℓ ) = 0 .

Shift invariance and uniqueness
FDR integrals are invariant under the shift of any integration variable. This can be easily proven by using the fact that they can be thought as finite differences of shift-invariant dimensionally-regulated 10 divergent integrals (see Eq. (2.4)) (2.17) 8 It can be shown that FDR tensors are equivalent to DR tensors at one loop, but differences start at two loops and beyond [39]. 9 The FDR defining expansion of is given in appendix C. 10 Here and in the following n = 4 + ǫ and µR is the renormalization scale.
The explicit demonstration is given in appendix A. A corollary to this theorem is the uniqueness of the definition in Eq. (2.5). In fact, the subtracted integrands in J INF (q 1 , . . . , q ℓ ) are unambiguously determined by the UV content of the original integrand, the only possible ambiguity being shifts of the loop momenta in J(q 1 , . . . , q ℓ ), which, however, produce the same FDR integral.
Eq. (2.17) also demonstrates that whenever DR loop integrals are known, their FDR counterparts can also be computed.

Independence of the cutoff
As a result of the subtraction of the divergent integrands, non integrable powers of 1/q 2 are developed in J F,ℓ (q 1 , . . . , q ℓ ). Such IR poles get regulated by the µ 2 propagator prescription, which gives a meaning to the the r.h.s. of Eq. (2.5). Thus, the original UV cutoff is traded for an IR one: µ. Here we show that FDR integrals depend at most logarithmically on µ. Furthermore, µ can be traded for the renormalization scale µ R , rendering the definition of the FDR integration independent of any cutoff.
We start from Eq. (2.17). Since the first term in its r.h.s. is the original DR regulated integral it does not depend on µ, in the limit µ → 0 11 . On the other hand, polynomially divergent integrands in J INF cannot contribute either, because they generate polynomials in µ. Therefore, the µ dependence in the l.h.s. is entirely due to powers of ln(µ/µ R ) created by the subtraction of the logarithmically divergent integrals. If one redefines FDR integrals without subtracting such logarithms, no dependence on µ is produced. This is equivalent to the operation of adding back all ln(µ/µ R )s to the l.h.s. of Eq. (2.17). Then, the limit µ → 0 can be taken, µ becomes µ R and no cutoff is left. The identification µ = µ R after lim µ→0 is understood in all FDR integrals appearing in this paper.

Keeping gauge invariance
Now we discuss how gauge invariance is preserved in FDR. Our starting point is the existence of graphical proofs of the Ward-Slavnov-Taylor identities [40], in which the correct relations among Green's functions are demonstrated -at any loop order -directly at the level of Feynman diagrams. Such proofs are valid under two circumstances: • divergent loop integrals should be defined in a way that shifting the integration momenta is possible as if they were convergent ones [41]; • cancellations between numerators and denominators should be preserved 12 .
Since the first property has been already proven, we concentrate here on the second requirement, which we study by means of a two-loop example.
Consider the scalar integral (2.18) To define it in FDR, it is necessary to make explicit the µ 2 dependence in its denominators 13 , which amounts to the replacement However, this change should be performed without altering the cancellations which ensure that the same result is obtained both by simplifying the reducible numerators before computing the integrals and by working out the integrals without simplifying the numerators. That happens only if 1. any q 2 i generated by Feynman rules in the numerator of a diagram 14 is also changed as in Eq.
2. simplifications at the integrand level are possible, such as . (2.20) Either way, integrals with µ in the numerator appear -which we dub extra integrals -that need to be properly defined. For instance, since an explicit computation gives (2.23) 13 See Eq. (2.7). 14 Such q 2 terms are created, for instance, when (q + pi) α (q + pi) β /M 2 i and / q + / p1 + Mi in Eq. (2.1) are multiplied by g αβ and / q respectively, before tensor reduction. 15 The r.h.s. of Eq. (2.22) vanishes because FDR integrals are at most logarithmic in µ. 16 It is interesting to study how a finite contribution is generated by the definition in Eq. (2.23). In JD(q1, q2) produces a finite constant when µ → 0. The value of this integral is given in subsection 3.1.
By using this definition, Eq. (2.20) directly follows from the FDR defining expansion of its two sides. Note that the index 1 in µ 2 | 1 only denotes the expansion to be used: although only one kind of µ 2 exists are in general different, because they are defined by expanding respectively. The described procedure is completely general: the extra integrals are defined by the FDR expansion of the integrals obtained by replacing µ| i → q i . As a consequence, the µ| i in the numerator are sensitive to changes of variables. For example, if q 1 → q 1 − q 2 and (2.26) Extra integrals can be computed either directly, by considering the finite part J D (q 1 , q 2 ) of the relevant denominator expansion -as done in Eq. (2.23) -or indirectly, by rewriting J D (q 1 , q 2 ) as a difference between the original integrand and its subtracted pieces. This second way is usually more convenient, because the original integral does not contribute in the limit µ → 0. For example, Eq. (C.1) gives which coincides with the result in footnote 16. Finally, extra integrals give the possibility to rewrite tensors in terms of scalars plus constants. For instance, Eq. (2.13) produces (2.28) Decompositions like this will be extensively used in the calculation presented in section 3.
Having studied the general mechanism of the gauge cancellations in FDR, we further elucidate it by means of the process investigated in this paper, namely H → γγ mediated by a fermion with mass m. In this case the proof of gauge invariance relies on the graphical equivalence depicted in Fig. 1, which, in turn, is realized by the Feynman identity Consider now the generic ℓ-loop amplitude in Fig. 2. Its integrand reads where the sum is over all contributing Feynman diagrams and Γ i o (Γ i e ) is proportional to a product of an odd (even) number of gamma matrices. Gauge invariance requires that whereJ α is the integrand in Eq. (2.31) regulatedà la FDR by replacing q 2 i → q 2 i −µ 2 in both numerators and denominators. Eq. (2.32) can be directly proven at the integrand level.
With this aim, we first concentrate on the replacements responsible for the conservation of the specific current in Fig. 2: where the loop denominators in Γ i o,e are also barred. In the previous equation has the effect of changing q 2 toq 2 in the first trace. Thus, when contracting with p, it is possible to reconstruct and cancel denominators in agreement with the Feynman identity in Eq. (2.29). After that directly follows from the shift invariance properties of the loop integrals, as in DR. We explicitly tested Eq. (2.36) up to two loops in H → γγ.
With more photons, replacements as in Eq. (2.34) have to be performed for all integration momenta appearing in the trace 17 . The one-loop prescription is that defined in [38]: given a fermionic string, one chooses arbitrarily the sign of µ within the first / q; the sign of the subsequent one is opposite, if an even number of γ-matrices occur between the two / qs, and it is the same otherwise 18 . This rule is sufficient in the presence of one fermion line only, as in the calculation at hand. With two or more lines, and no summation indices among them, each fermion string can be separately treated as described. If sums occur, after applying the above algorithm, extra µ 2 terms need to be extracted according to the following procedure where Γ (k) represents a string of k gamma matrices. Eq. (2.37) is proven by noting that n (m) anticommutation are needed to bring / q near to γ α (γ α ) and can be easily checked by taking the traces and substituting q 2 → q 2 − µ 2 . As an example of such rules, the integrand of the one-loop H → γ(p 1 )γ(−p 2 ) amplitude is proportional to and its FDR regulated version reads which satisfies the Ward identities We emphasize that there is nothing mysterious in Eq. (2.39): the same result is obtained by computing the trace in Eq. (2.38) and replacing q 2 →q 2 . The advantage of Eq. (2.39) is that it permits a trivial proof of the Ward identities at the integrand level.
The corresponding procedure at two loops is better explained with an example. Consider the trace which contributes to the second diagram of Fig. 5. Its FDR counterpart reads Eq. (2.42) is obtained from Eq. (2.41) by using -one after the other -the one-loop replacements / q 1 → / q 1 and / q 2 → / q 2 , which generate the terms proportional to µ 2 | 1 and µ 2 | 2 . Thẽ µ 2 12 contribution originates, instead, from the substitution 44) and is obtained by simultaneously barring / q 1 and / q 2 in Eq. (2.41) (with the same rule used at one loop to determine the sign of µ| i inside each / q i , without distinguishing between / q 1 and / q 2 ) and subtracting the µ 2 | i terms already calculated. What is left is, by construction, proportional to powers of µ| 1 µ| 2 ≡μ 2 12 and gives the last term in Eq. (2.42). Once again, T αβ is equivalent to the replacements in the original trace T αβ . Thus, the µ 2 | i ensure the right cancellations leading to the fulfillment of the Ward identities. We explicitly checked that the two-loop H → γ(p 1 )γ(−p 2 ) integrandJ αβ (q 1 , q 2 ), constructed as described, satisfies In practical cases it is often convenient to simplify reducible numerators before computing the loop integrals. In that way, only irreducible tensors appear and extra integrals are just produced by tensor decomposition, as in Eq. (2.28). This is the strategy of the calculation presented in section 3.

FDR versus DR
The proof that DR preserves gauge invariance and unitarity relies on the possibility of introducing order-by-order local counterterms in the Lagrangian L. On the contrary, FDR makes no reference to L. In this subsection we use the simple two-loop QED example of [43] to comment on the conceptual differences between the two approaches.
Consider a DR calculation of the one-loop photon self-energy (2.47) Then, at two loops and up to terms O(ǫ), Simply removing the poles from the last expression gives Π 2 0 + 2Π −1 Π 1 , which is not the right result because it violates unitarity. As is well known, the correct procedure to undertake in DR is to renormalize order by order, i.e. to add one-loop counterterms in L such that In FDR, the divergences are subtracted at the level of the definition of the loop integration, so that the product of two one-loop diagrams is simply the product of the two finite parts, with no need of introducing extra interactions in L. Thus, one directly obtains with Π FDR (p 2 ) = Π 0 . This difference can be also understood from the DR ↔ FDR naive correspondence which gives lim ǫ→0 ǫ/ǫ = 1, while lim µ→0 µ ln µ = 0. From all of that it is manifest that spurious ǫ/ǫ terms such as Π −1 Π 1 -which need to be kept under control in DR by the order-by-order renormalization -never appear in FDR. The result of an FDR calculation typically depends on the parameters contained in L, and a (finite) global renormalization is needed only to link them to experimental measurements at the desired perturbative accuracy. In particular -and in contrast with DR -no renormalization is necessary when no parameter appears in the final result. This is the case of the calculation presented in section 3.

FDR versus FDH
In this subsection we discuss the differences between FDR and the Four Dimensional Helicity scheme (FDH) of [44], which is equivalent to Dimensional Reduction [45] at one-loop. FDH is a variant of DR, in which gauge cancellations are kept by integrating all momentum integrals over n-component momenta and considering any g αβ resulting from the integration as n-dimensional. Observed external states are treated in four dimensions (preserving supersymmetry) and unobserved internal ones are defined in such a way that the contraction q α q β g αβ = q 2 gives rise to an n-dimensional object when q is an integration momentum. If q 2 is split into four-dimensional (q 2 4 ) and ǫ-dimensional (q 2 ) components andq 2 is identified with −µ 2 , there is a formal equivalence -at the integrand level -between the procedures used by FDR and FDH to determine the µ 2 pieces [46,47]. However differences start when integrating. FDR integration is defined in a way that non-local sub-divergences are subtracted right away, as in the second line of Eq. (2.8), while in FDH sub-divergences are compensated by counterterms added at a previous renormalization stage, as in any conventional subtraction scheme. It is exactly this peculiarity that makes possible to avoid an order-by-order renormalization in FDR.
As a consequence of this dissimilarity, integrals containing µ 2 give different results, at two loops and beyond, when computed in FDR and FDH. For example Only at one loop, because no sub-divergences are present, FDR and FDH coincide, as observed in [37]. For instance, . (2.52)

Infrared divergences
Although the process H → γγ is free of IR infinities, we devote this subsection to illustrate how soft and collinear singularities can be treated compatibly with FDR. We first discuss divergences in the virtual contribution, and then show how they are matched by a particular treatment of the real radiation. As for the loop integration, the definition in Eq. (2.5) can be maintained also in the presence of IR singularities. For instance, the FDR versions of the massless one-and two-loop scalar integrals in Fig. 3 read , Note that the on-shell conditions p 2 1 = p 2 2 = 0 are used at the integrand level. Thus, infrared virtual divergences get regulated by the µ 2 -deformed propagators 20 , which generates powers of logarithms of µ 2 , upon integration. A particularly interesting situation is when the integral is also UV divergent. In this case it is easy to see that UV divergent scale-less ℓ-loop FDR integrals vanish, as in DR. In fact, the only allowed external variable is a momentum p such that p 2 = 0, whose fate is to appear in the numerator of J F,ℓ (q 1 , . . . , q ℓ ) in Eq. (2.4) to improve the UV convergence of the original integrand. Therefore, J F,ℓ (q 1 , . . . , q ℓ ) is entirely made of integrands proportional to positive powers of (q i · p), that vanish, by Lorentz invariance, after integration. The simplest case is the fully massless one-loop 2-point scalar function . (2.55) The FDR expansion of its integrand reads The same result is obtained by a direct computation

58)
19 J (1) and J (2) are not UV subtracted since they produce UV convergent integrals. 20 A different µ 2 can be used to regulate UV divergences (µ 2 UV ) and IR ones (µ 2 IR ). However, a common µ 2 simplifies the calculation, as will be shown later. Since IR infinities are more easily understood in terms of µ 2 IR > 0, it is convenient to choose µ 2 UV = µ 2 IR = µ 2 > 0.
from which it is manifest that, in the limit p 2 → 0, a cancellation occurs between two logarithms of UV and IR origin, respectively 21 . In summary, IR divergent loop integrals are defined by taking the limit µ → 0 outside integration, after subtracting -when necessary -UV divergent integrands. In order to preserve the cancellation of the IR logarithms in physical quantities, this definition should be accompanied by a consistent treatment of the infinities appearing in the real emission, which we discuss in the following.
Consider how the divergent 1 → 2 splitting is regulated in the loop integrals. The situation is depicted in Fig. 4(a), where thick lines represent unobserved loop particles -whose propagator is made massive by the addition of µ 2 -and the cut line is an external observed massless particle. This is matched by the real radiation pattern of Fig. 4(b), where thick lines are unobserved external particles merging into an internal observed massless one. In both situations unobserved particles get a mass µ and unitarity relates the two cases as follows (2.61) Therefore, would-be-massless external particles with momenta p i should be given a mass µ. This is achieved by trading the original massless m-body phase space dΦ m for a massive one, denoted by dΦ m , in which In this way, singular configurations produce logarithms which cancel the IR dependence on µ 2 of the virtual contribution. However, this strategy should be carried out without breaking gauge invariance. To illustrate the way to proceed we consider m-jet production at NNLO in e + e − annihilation. The building blocks of the calculation depend on the set 21 It is instructive to study the same case in DR, where B DR (p 2 , 0, 0) = µ −ǫ R d n q 1 q 2 (q+p) 2 . Now B DR (0, 0, 0) vanishes because IR and UV poles in ǫ compensate. In fact, by introducing an arbitrary separation scale M , the two divergences can be disentangled (2.59) cancel each other. However, this argument has a potential problem, because it requires the cancellation of two analytic continuations, IUV and IIR, originally defined in domains that do not overlap [48] (ǫ < 0 and ǫ > 0): since no value of ǫ exists where they are defined simultaneously, it is not obvious whether their difference represents the original function B DR (0, 0, 0). A possible mathematically consistent solution can be formulated in terms of modified Gaussian integrals in the n-dimensional Euclidean space [48]. In contrast, the FDR derivation in Eq. (2.57) is straightforward.
(a) (b) Figure 4: Splitting regulated by massive (thick) unobserved particles. The one-particle cut in (a) contributes to the virtual part, the two-particle cut of (b) to the real radiation.
of invariants They are: • the virtual and real NLO corrections, • the one-loop corrections to the NLO real radiation, dσ V,1 NNLO {s i 1 ÷im }, • the double radiation dσ R NNLO {s i 1 ÷i m+1 }.
After α S renormalization, they give a m-jet cross section accurate up to NNLO The integrands behave as therefore the integrations over single-and double-unresolved massless phase-spaces ( dΦ m+1 and dΦ m+2 , respectively) generate logarithmic IR divergences which have to be regulated.
In DR, the last two lines of Eq. (2.65) are interpreted as a limit to ǫ → 0 of integrals computed in n = 4 + ǫ dimensions. We instead define a mapping from massless to massive invariants as follows and rewrite where µ is the same regulator used in the IR divergent loop integrals, and The proof that Eq. (2.68) converges to the right results is simple. First note that the mapping in Eq. Therefore, the only possible singular behavior is which, being the internal propagator massless, matches the virtual IR poles, as in Fig. 4  . The additional integrals, generated when each term in W NNLO does not meet its corresponding pole, vanish in the limit µ → 0. This last property follows from the fact that the integral is at most logarithmically divergent. The reason why a pole cannot be changed by hand only in the terms where it appears, is that the logarithmic behavior is reached only after gauge cancellations, which should not be altered. This can be easily understood with a toy model: . (2.73) The correct procedure gives To summarize, IR infinities can be safely treated in four dimensions. An explicit oneloop example, involving both IR and UV infinities, can be found in [37]. Furthermore, the outlined strategy opens the possibility of a numerical treatment of NNLO calculations similar to the phase-space slicing method at NLO [49]. The advantage is that all singularities are automatically expressed in terms of powers of a logarithmic regulator -ln µ -with no need of subtracting 1/ǫ poles. An investigation of the numerical performance of such a strategy is outside the scope of this work, although currently under study. We think that it is a promising one because, owing to the four-dimensionality of the calculation, we envisage that the bulk of the cancellations can be easily arranged to happen at the integrand level.

H → γγ at two loops
The diagrams contributing to the QCD corrections of the top-loop-mediated Higgs decay into two photons are depicted in Fig. 5.
The amplitude reads where p 1 and −p 2 are the momenta of the outgoing photons. One has with v being the vev of the Higgs boson and M is well defined in the limit m → ∞ we are interested in. This means that, order by order, the form factor F(η) can be written as In this section, we re-derive 22 -within the FDR framework -the known result [50] which implies that the QCD corrections factorize the one-loop amplitude (3.9) 22 We use the Feynman rules of appendix B.

The building blocks
Since we are working in the large top mass limit, denominators can be expanded as follows where the on-shell condition p 2 j = 0 for the photons has been used. An expansion to the second order, as the one above, is sufficient to the level of accuracy we are interested in, i.e. O(1/η). All external momenta can then be neglected and the top mass is the only relevant scale. As a consequence, we only have to deal with vacuum integrals.
After cancelling between numerator and denominator theq 2 1 ,q 2 2 ,q 2 12 terms generated by the Feynman rules 23 , tensor integrals up to rank 4 contribute to the amplitude. Because there is no dependence on external momenta, odd rank integrals vanish and the tensor reduction gives: A αβρσ abrs + A αρβσ arbs + A ασβρ asbr at two loops , (3.11) where g αβρσ = g αβ g ρσ + g αρ g βσ + g ασ g βρ , and A αβρσ abrs = 5(q a · q b )(q r · q s ) − (q a · q r )(q b · q s ) − (q a · q s )(q b · q r ) g αβ g ρσ . (3.12) Denominators can then be reconstructed by rewriting During this tensor decomposition, the µ 2 | 1 , µ 2 | 2 , µ 2 | 12 terms are kept only when they generate a non-zero contribution. This means that they should be power-counted as the corresponding squared loop momenta, and contribute only if the integral is divergent. The final result can then be completely expressed in terms of scalar two-loop integrals, products of two one-loop integrals and extra integrals containing µ 2 | j (j = 1, 2, 12). For convenience, we introduce the notation (3.14) 16) 23 Remember the discussion at the beginning of subsection 2.4. and αm (µ 2 ) = [d 4 q] µ 2 (q 2 − m 2 ) α , (3.17) (3.18) The one-loop and factorizable integrals of Eqs. (3.14) and (3.15) can be computed as derivatives of the quadratically divergent one-loop tadpole [30] [ , while those in Eq. (3.16) are obtained by deriving with respect to the mass parameters the basic integral All extra integrals relevant for our calculation can be expressed in terms of three fundamental objects with f given in Eq. (D.2). We need derived in footnote 16, and The first of Eqs. (3.25) is computed indirectly from the FDR expansion

The result
By summing all Feynman diagrams and performing the tensor reduction we end up with The final result in Eq. (3.8) follows by inserting the expressions of the scalar and extra integrals computed in subsection 3.1.
A few remarks are in order. At two loops the one-to-one correspondence between DR and FDR is lost and it is no longer true that FDR integrals are obtained from DR ones after subtracting poles (and universal constants). For example, if we were to interpret the integrals appearing in Eq. (3.27) as dimensionally regulated ones, we would not get zero and a 1/ǫ pole would even remain! Differences already start at the level of the basic two-loop scalar integral. The DR counterpart of Eq. (D.1) reads [51] with a different coefficient in front of the ln 2 . This can be understood because two different mechanisms to preserve gauge invariance are used by DR and FDR, the latter avoiding an order-by-order renormalization. Another advantage of FDR is that the same formulae for the scalar one-loop functions can be used also when they combine to form a factorizable two-loop integral. Differently stated, Eq. (3.15) is simply the product of two integrals of the kind given in Eq. (3.14). This does not happen in DR, where terms of O(ǫ) must be added to the one-loop functions appearing in a two-loop calculation. Note also that there is no need of renormalizing F (2) 0 and F 1 . This directly follows from the discussion in subsection 2.5. FDR renormalization amounts to the mere operation of fixing results in terms of physical quantities, and since the top mass disappears due to the limit m top → ∞, no fixing is needed. This is not the case when using DR, where renormalization is required in order to compensate spurious ǫ/ǫ constants generated in the limit n → 4. The situation is analyzed in the next subsection.

Renormalization
Here we demonstrate that if we insist with an order-by-order renormalization we obtain a vanishing contribution to F where Σ( / p) is the top self-energy depicted in Fig. 6 and This gives the one-loop counterterms and diagrams of Fig. 7, which generate a contribution to F  It is worth mentioning that in DR C 0,ct DR = 0 and C 1,ct DR = O(ǫ) , (3.34) so that C 1,ct DR contributes to the amplitude when multiplied against the 1/ǫ pole contained in δm DR (the DR counterpart of δm 25 ), and renormalization is necessary.

Conclusions
We have presented the first two-loop calculation ever performed in FDR. The O(α S ) corrections to the H → γγ amplitude -mediated by an infinitely heavy top loop -have been computed in a fully four-dimensional fashion. This example has allowed us to show that FDR is an approach to loop calculations in which • gauge invariance is preserved; • order-by-order renormalization is avoided; • a finite renormalization is only needed to fix the parameters of the theory in terms of experimental observables; • ℓ-loop integrals are directly re-usable in (ℓ+1)-loop calculations, with no need of further expanding in ǫ.
In addition, we have described how infrared divergences can be dealt with within the same four-dimensional framework used to cope with the ultraviolet infinities.
We have also demonstrated that DR and FDR are not related in a direct way -beyond one loop -since FDR integrals cannot be interpreted any longer as DR ones devoid of the pole part. Due to its four-dimensionality we envisage a great potential of FDR in further simplifying NNLO computations. More investigation is needed in this direction, that we plan to undertake in the near future. and where q 12 = q 1 + q 2 and p 12 = p 1 + p 2 . Since integrals of polynomials in the integration variables vanish, the divergent parts of any one-or two-loop FDR integral can be written -after expanding in the external momenta -in terms of the four cases α = 1 , α = 2 , β = γ = δ = 1 , and β = γ = 1 with δ > 1 .
In all the other cases Eqs. (A.1) and (A.2) coincide with finite integrals, for which shift invariance trivially holds. We start proving Eq. (A.1) with α = 1. By using the shorthand notation one writes the FDR expansions of the two sides of the equation as where the shift invariance of the dimensionally regulated integrals over 1/D and 1/q 2 has been used. By now expanding the last integrand one obtains Since the l.h.s. of Eq. (A.7) vanishes upon integration at any order in p, the same happens for the combination − p 2 + 2(q · p) q 4 + 4 (q · p) 2 q 6 . (A.8) The last integral in Eq. (A.6) can then be rewritten as   Finally, deriving with respect to m 2 12 demonstrates the last case. With more loops the proof follows the same reasoning: the mismatch between the FDR expansion of shifted and unshifted integrands is cured by vanishing integrals obtained by expanding the polynomially divergent integrals in J INF (q 1 , . . . , q ℓ ) at the relevant order in p, as in Eq. (A.7).

B. Feynman rules
For completeness we list, in Fig. 8, the Feynman rules used in the calculation. Q t , m 0 and v are the top quark charge, the top bare mass and the vacuum expectation value of the Higgs field, respectively.

C. A few FDR defining expansions
We collect here the two-loop FDR defining expansions used throughout the paper. Denominators are defined in Eq. (2.7) and divergent integrands are written in square brackets. By power counting -due to the presence of 1/q 4 i terms -a logarithmic dependence on µ is expected in I 1 (m 1 ), while µ can be immediately set to zero in I 2 (m 1 , m 2 ). A natural split is then obtained in FDR: I 2 (m 1 , m 2 ) only depends on The relevant DR integrals can be found in the appendix of [51].