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

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

Alice Maria Donati and Roberto Pittau
Departamento de Física Teórica y del Cosmos and CAFPE, Campus Fuentenueva s. n., Universidad de Granada, E-18071 Granada, Spain
###### Abstract

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 . Our case study is the two-loop amplitude for , 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.

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

Alice Maria Donati and Roberto Pittau

Departamento de Física Teórica y del Cosmos and CAFPE, Campus Fuentenueva s. n., Universidad de Granada, E-18071 Granada, Spain

\abstract@cs

## 1 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].

While NLO techniques are very well established [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], work is ongoing to solve the NNLO problem in its full generality [16]. As for the virtual sector, progress has been recently achieved by extending generalized unitarity techniques at two-loops [17, 18, 19, 20, 21], while the antenna subtraction [22, 23] and sector decomposition [24, 25, 26, 27] methods look promising tools to deal with IR divergences beyond NLO [28].

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 of UV or IR origin even before attacking the calculation of the finite physical part. For instance, DR forces an order-by-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 – when appearing at higher orders – to include terms generating 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 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 111Other four-dimensional treatments are listed in [31, 32, 33, 34, 35, 36].. 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 . 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 222One-loop examples have been worked out in [37, 38].. 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 is presented in section 3 and the technical details are collected in the final appendices.

## 2 FDR and the importance of working in four dimensions

### 2.1 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 , formally identified with the propagator prescription. For example, vector-boson and fermion propagators with momentum and mass read 333 denotes a generic integration momentum and and external momentum., in the unitary gauge,

 gαβ−(q+pi)α(q+pi)β/M2i¯Dpi\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak and\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak to0.0pt/q+to0.0pt/pi+Mi¯Dpi, (2.0)

respectively, with

 ¯Dpi = (q+pi)2−M2i−μ2\leavevmode\nobreak =\leavevmode\nobreak ¯q2−di, ¯q2 ≡ \leavevmode\nobreak q2−μ2,\leavevmode\nobreak \leavevmode\nobreak di\leavevmode\nobreak ≡\leavevmode\nobreak M2i−p2i−2(q⋅pi). (2.0)

Secondly, UV infinities are isolated by a repeated use of the identity

 1¯Dpi=1¯q2+di¯q2¯Dpi. (2.0)

In fact – being is at most linear in  – the second term in the r.h.s. of Eq. (2.0) is less UV divergent than the original denominator, so that UV divergences can be systematically moved to terms such as , which depend only on , and directly subtracted from the integrand. Schematically, dubbing the original integrand of an -loop function, one has

 J(q1,…,qℓ)=JINF(q1,…,qℓ)+JF,ℓ(q1,…,qℓ), (2.0)

where collects the UV divergent integrands. Then, the FDR integral over is defined as 444Throughout the paper FDR integration is denoted by the symbol .

 ∫[d4q1]…[d4qℓ]J(q1,…,qℓ)≡limμ→0∫d4q1…d4qℓJF,ℓ(q1,…qℓ), (2.0)

where, due to the limit , only a logarithmic dependence on remains, which can be traded for a dependence on the renormalization scale 555See subsection 2.3.. Thus, FDR and normal integration coincide in a convergent integral, since no divergent part can be extracted from its integrand. Furthermore, the space-time is kept strictly four-dimensional also in divergent integrals – with  – because is nothing but the infinitesimal deformation needed to define the loop integrals 666Unlike in DR, the limit is taken outside integration (see Eq. (2.0)). 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 777We denote the expansion of an integrand needed to bring it in the form of Eq. (2.0) as its FDR defining expansion. at one loop is given by

 qαqβ¯Dp0¯Dp1 = [qαqβ¯q4]+(d0+M21−p21)[qαqβ¯q6]−2p1γ[qαqβqγ¯q6] +4p1γp1δ[qαqβqγqδ¯q8]+JαβF,1(q), JαβF,1(q) = qαqβ(4(q⋅p1)2d1¯q8¯Dp1+(M21−p21)d0+d1−2(q⋅p1)¯q6¯Dp1 (2.0) −2d0(q⋅p1)¯q6¯Dp1+d20¯q4¯Dp0¯Dp1),

where and the terms in square brackets are proportional to UV divergent integrands. A two-loop example with

 ¯D1=¯q21−m21,\leavevmode\nobreak \leavevmode\nobreak ¯D2=¯q22−m22,\leavevmode\nobreak \leavevmode\nobreak ¯D12=¯q212−m212,\leavevmode\nobreak \leavevmode\nobreak q12=q1+q2 (2.0)

 1¯D1¯D2¯D12 = [1¯q21¯q22¯q212]+m21[1¯q41¯q22¯q212]+m22[1¯q21¯q42¯q212]+m212[1¯q21¯q22¯q412] (2.0) + m41(¯D1¯q41)[1¯q42]+m42(¯D2¯q42)[1¯q41]+m412(¯D12¯q412)[1¯q41] + JF,2(q1,q2),

where

 JF,2(q1,q2) = −m41q21+2(q1⋅q2)(¯D1¯q41)¯q42¯q212−m42q22+2(q1⋅q2)¯q41(¯D2¯q42)¯q212−m412q212−2(q1⋅q12)¯q41¯q22(¯D12¯q412) (2.0) + m21m22(¯D1¯q21)(¯D2¯q22)¯q212+m21m212(¯D1¯q21)¯q22(¯D12¯q212)+m22m212¯q21(¯D2¯q22)(¯D12¯q212) + m21m22m212(¯D1¯q21)(¯D2¯q22)(¯D12¯q212).

Note that identities such as

 1¯q212=1¯q22−q21+2(q1⋅q2)¯q22¯q212,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak 1¯q22=1¯q21−q212−2(q1⋅q12)¯q21¯q22 (2.0)

are needed to extract the sub-divergences. Then, the one- and two-loop FDR integrals over the integrands in Eqs. (2.1) and (2.0) read

 ∫[d4q]qαqβ¯Dp0¯Dp1 = limμ→0∫d4qJαβF,1(q), ∫[d4q1][d4q2]1¯D1¯D2¯D12 = limμ→0∫d4q1d4q2JF,2(q1,q2). (2.0)

It is important to realize that divergent tensor structures are fully subtracted from the original integrand, as in Eq. (2.1) 888It can be shown that FDR tensors are equivalent to DR tensors at one loop, but differences start at two loops and beyond [39].. Owing to the Lorentz invariance and four-dimensionality of this definition, irreducible tensors can be decomposed in terms of scalars. For example 999The FDR defining expansion of is given in appendix C.

 ∫[d4q1][d4q2]qα1qβ1¯D31¯D2¯D12=limμ→0∫d4q1d4q2qα1qβ1JD(q1,q2),\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak where JD(q1,q2)={(1¯q61−1¯D31)q21+2(q1⋅q2)¯q42¯q212 +1¯D31¯q22¯D12(m22¯D2+m212¯q212)}, (2.0)

can be rewritten as

 ∫[d4q1][d4q2]qα1qβ1¯D31¯D2¯D12=gαβ4∫[d4q1][d4q2]q21¯D31¯D2¯D12, (2.0)

with

 ∫[d4q1][d4q2]q21¯D31¯D2¯D12=limμ→0∫d4q1d4q2q21JD(q1,q2). (2.0)

Finally, polynomials in the integration variables represent a limiting case of Eq. (2.0), in which

 JF,ℓ(q1,…,qℓ)=0. (2.0)

As a consequence

 ∫[d4q](¯q2)α=0, (2.0)

for any integer .

### 2.2 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 101010Here and in the following and is the renormalization scale. divergent integrals (see Eq. (2.0))

 ∫[d4q1]…[d4qℓ]J(q1,…,qℓ) = limμ→0μ−ℓϵR(∫dnq1…dnqℓJ(q1,…,qℓ) (2.0) −∫dnq1…dnqℓJINF(q1,…,qℓ)).

The explicit demonstration is given in appendix A. A corollary to this theorem is the uniqueness of the definition in Eq. (2.0). In fact, the subtracted integrands in are unambiguously determined by the UV content of the original integrand, the only possible ambiguity being shifts of the loop momenta in , which, however, produce the same FDR integral.

Eq. (2.0) also demonstrates that whenever DR loop integrals are known, their FDR counterparts can also be computed.

### 2.3 Independence of the cutoff

As a result of the subtraction of the divergent integrands, non integrable powers of are developed in . Such IR poles get regulated by the propagator prescription, which gives a meaning to the the r.h.s. of Eq. (2.0). Thus, the original UV cutoff is traded for an IR one: . Here we show that FDR integrals depend at most logarithmically on . Furthermore, can be traded for the renormalization scale , rendering the definition of the FDR integration independent of any cutoff.

We start from Eq. (2.0). Since the first term in its r.h.s. is the original DR regulated integral it does not depend on , in the limit 111111This is true in the absence of IR divergences. However, UV and IR infinities simultaneously occur only in scale-less integrals, which vanish in FDR (see subsection 2.7).. On the other hand, polynomially divergent integrands in cannot contribute either, because they generate polynomials in . Therefore, the dependence in the l.h.s. is entirely due to powers of created by the subtraction of the logarithmically divergent integrals. If one redefines FDR integrals without subtracting such logarithms, no dependence on is produced. This is equivalent to the operation of adding back all s to the l.h.s. of Eq. (2.0). Then, the limit can be taken, becomes and no cutoff is left. The identification after is understood in all FDR integrals appearing in this paper.

### 2.4 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 121212Quoting Martinus Veltman [42]: Gauge invariance implies a tight interplay between the numerator of an integrand and its denominator. Changing either of the two will generally destroy gage invariance .

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

 ∫[d4q1][d4q2]1¯D21¯D2¯D12. (2.0)

To define it in FDR, it is necessary to make explicit the dependence in its denominators 131313See Eq. (2.0)., which amounts to the replacement

 q2i→q2i−μ2. (2.0)

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 generated by Feynman rules in the numerator of a diagram 141414Such terms are created, for instance, when and in Eq. (2.1) are multiplied by and respectively, before tensor reduction. is also changed as in Eq. (2.0);

2. simplifications at the integrand level are possible, such as

 ∫[d4q1][d4q2]q21−m21−μ2¯D31¯D2¯D12 = ∫[d4q1][d4q2]1¯D21¯D2¯D12. (2.0)

Either way, integrals with in the numerator appear – which we dub extra integrals – that need to be properly defined. For instance, since an explicit computation gives

 ∫[d4q1][d4q2]q21−m21¯D31¯D2¯D12 ≠ ∫[d4q1][d4q2]1¯D21¯D2¯D12, (2.0)

one deduces that 151515The r.h.s. of Eq. (2.0) vanishes because FDR integrals are at most logarithmic in .

 ∫[d4q1][d4q2]μ2¯D31¯D2¯D12≠limμ→0μ2∫[d4q1][d4q2]1¯D31¯D2¯D12=0. (2.0)

In fact, a non-zero contribution must be added to the l.h.s. of Eq. (2.0) to produce the r.h.s. of Eq. (2.0). The right cancellation occurs if the denominators are expanded in front of as if it was a , namely as in Eq. (2.1) 161616It is interesting to study how a finite contribution is generated by the definition in Eq. (2.0). In thus produces a finite constant when . The value of this integral is given in subsection 3.1.:

 ∫[d4q1][d4q2]μ2|1¯D31¯D2¯D12=limμ→0∫d4q1d4q2μ2JD(q1,q2). (2.0)

By using this definition, Eq. (2.0) directly follows from the FDR defining expansion of its two sides. Note that the index in only denotes the expansion to be used: although only one kind of exists

 ∫[d4q1][d4q2]μ2|1¯D31¯D2¯D12,\leavevmode\nobreak ∫[d4q1][d4q2]μ2|2¯D31¯D2¯D12 \leavevmode\nobreak and\leavevmode\nobreak ∫[d4q1][d4q2]μ2|12¯D31¯D2¯D12 (2.0)

are in general different, because they are defined by expanding

 ∫[d4q1][d4q2]q21¯D31¯D2¯D12,\leavevmode\nobreak ∫[d4q1][d4q2]q22¯D31¯D2¯D12 \leavevmode\nobreak and\leavevmode\nobreak ∫[d4q1][d4q2]q212¯D31¯D2¯D12, (2.0)

respectively.

The described procedure is completely general: the extra integrals are defined by the FDR expansion of the integrals obtained by replacing . As a consequence, the in the numerator are sensitive to changes of variables. For example, if and ,

 ∫[d4q1][d4q2]μ2|12¯D31¯D2¯D12\leavevmode\nobreak \leavevmode\nobreak →\leavevmode\nobreak \leavevmode\nobreak ∫[d4q1][d4q2]μ2|1¯D1¯D2¯D312. (2.0)

Extra integrals can be computed either directly, by considering the finite part of the relevant denominator expansion – as done in Eq. (2.0) – or indirectly, by rewriting 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 . For example, Eq. (C.0) gives

 (2.0)

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. (2.0) produces

 ∫[d4q1][d4q2]qα1qβ1¯D31¯D2¯D12 = gαβ4{∫[d4q1][d4q2]1¯D21¯D2¯D12 (2.0) +m21∫[d4q1][d4q2]1¯D31¯D2¯D12 +∫[d4q1][d4q2]μ2|1¯D31¯D2¯D12}.

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 mediated by a fermion with mass . 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

 to0.0pt/q+mDto0.0pt/pto0.0pt/q+to0.0pt/p+mDp=to0.0pt/q+mD−to0.0pt/q+to0.0pt/p+mDp, (2.0)

where

 D=q2−m2\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak and\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak Dp=(q+p)2−m2. (2.0)

Consider now the generic -loop amplitude in Fig. 2.

 ϵα(p)Jα(q,⋯,qℓ) = ϵα(p)∑i1DDp (2.0) ×Tr[(to0.0pt/q+m)γα(to0.0pt/q+to0.0pt/p+m)(Γio+Γie)],

where the sum is over all contributing Feynman diagrams and () is proportional to a product of an odd (even) number of gamma matrices. Gauge invariance requires that

 pα∫[d4q]⋯[d4qℓ]¯Jα(q,⋯,qℓ)=0, (2.0)

where is the integrand in Eq. (2.0) regulated à la FDR by replacing in both numerators and denominators. Eq. (2.0) 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:

 Jα→J′α = (2.0) +mTr[γα(to0.0pt/q+to0.0pt/p)Γie]+mTr[to0.0pt/qγαΓie]),

where the loop denominators in are also barred. In the previous equation

 to0.0pt/¯q=to0.0pt/q±μ (2.0)

has the effect of changing to in the first trace. Thus, when contracting with , it is possible to reconstruct and cancel denominators

 pαJ′α = ∑i1¯D(Tr[to0.0pt/qΓio]+mTr[Γie]) (2.0) −1¯Dp(Tr[(to0.0pt/q+to0.0pt/p)Γio]+mTr[Γie]),

in agreement with the Feynman identity in Eq. (2.0). After that

 pα∫[d4q]⋯[d4qℓ]J′α(q,⋯,qℓ)=0 (2.0)

directly follows from the shift invariance properties of the loop integrals, as in DR. We explicitly tested Eq. (2.0) up to two loops in .

With more photons, replacements as in Eq. (2.0) have to be performed for all integration momenta appearing in the trace 171717Sums over internal indices have to be previously worked out.. The one-loop prescription is that defined in [38]: given a fermionic string, one chooses arbitrarily the sign of within the first ; the sign of the subsequent one is opposite, if an even number of -matrices occur between the two s, and it is the same otherwise 181818If chirality matrices are involved, a gauge invariant treatment requires their anticommutation at the beginning (or the end) of open strings before replacing