On the fourdimensional formulation of dimensionally regulated amplitudes
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Abstract
Elaborating on the fourdimensional helicity scheme, we propose a pure fourdimensional formulation (FDF) of the \(d\)dimensional regularization of oneloop scattering amplitudes. In our formulation particles propagating inside the loop are represented by massive internal states regulating the divergences. The latter obey Feynman rules containing multiplicative selection rules which automatically account for the effects of the extradimensional regulating terms of the amplitude. We present explicit representations of the polarization and helicity states of the fourdimensional particles propagating in the loop. They allow for a complete, fourdimensional, unitaritybased construction of \(d\)dimensional amplitudes. Generalized unitarity within the FDF does not require any higherdimensional extension of the Clifford and the spinor algebra. Finally we show how the FDF allows for the recursive construction of \(d\)dimensional oneloop integrands, generalizing the fourdimensional openloop approach.
Keywords
Clifford Algebra Dimensional Regularization Feynman Rule Helicity Amplitude Loop Momentum1 Introduction
The recent development of novel methods for computing oneloop scattering amplitudes has been highly stimulated by a deeper understanding of their multichannel factorization properties in special kinematic conditions enforced by onshellness [1, 2, 3] and generalized unitarity [4, 5], strengthened by the complementary classification of the mathematical structures present in the residues at the singular points [6, 7, 8, 9, 10, 11].
The unitaritybased methods, reviewed in [12, 13, 14, 15, 16, 17, 18, 19], use two general properties of scattering amplitudes such as analyticity and unitarity. The former grants that the amplitudes can be reconstructed from their singularitystructure while the latter grants that the residues at the singular points factorize into products of simpler amplitudes.
Integrandreduction methods [6, 20], instead, allow one to decompose the integrands of scattering amplitudes are into multiparticle poles, and the multiparticle residues are expressed in terms of irreducible scalar products formed by the loop momenta and either external momenta or polarization vectors constructed out of them. The polynomial structure of the multiparticle residues is a qualitative information that turns into a quantitative algorithm for decomposing arbitrary amplitudes in terms of master integrals (MIs) by polynomial fitting at the integrand level. In this context the onshell conditions have been used as a computational tool reducing the complexity of the algorithm. A more intimate connection among the idea of reduction under the integral sign and analyticity and unitarity has been pointed out recently. Using basic principles of algebraic geometry [7, 8, 21, 22, 23], have shown that the structure of the multiparticle poles is determined by the zeros of the denominators involved in the corresponding multiple cut. This new approach to integrand reduction methods allows for their systematization and for their allloop extension.
Moreover, the proper understanding of the integrands of the amplitudes paved the way to the recent proposal of a fourdimensional renormalization scheme, which allows one the recognize and subtract UVdivergent contributions already at the integrand level [24, 25, 26].
Dimensionally regulated amplitudes are constituted by terms containing (poly)logarithms, also called cutconstructible terms, and rational terms. The former may be obtained by the discontinuity structure of integrals over the fourdimensional loop momentum. The latter ones, instead, escape any fourdimensional detectability and require one to cope with integrations including also the \((d4)\) components of the loop momentum.
Within generalizedunitarity methods both terms can in principle be obtained by performing \(d\)dimensional generalized cuts [27, 28, 29, 30, 31, 32]. In this context, the issue of addressing factorization in conjunction with regularization clearly emerges, since \(d\)dimensional unitarity requires to work with treelevel amplitudes involving external particles in arbitrary, noninteger dimensions. Polarization states, dimensionality of the onshell momenta, and the completeness relations for the particles wavefunctions have to be consistently handled since the number of spin eigenstates depends on the spacetime dimension. Therefore, in many cases generalized unitarity in arbitrary noninteger dimensions is avoided and cutconstructible and rational terms are obtained in separate steps. The former are computed by performing fourdimensional generalized cuts in the unregularized amplitudes. If possible the rational terms are obtained by using special properties of the amplitude under consideration, like the supersymmetric decomposition [33, 34].
Within integrandreduction methods, different approaches are available, according to the strategies adopted for the determination of cutconstructible and rational terms.
In some algorithms, the computation of the two ingredients proceeds in two steps [35]: the cutconstructible piece and part of the rational contributions, the socalled \(R_1\) term, are obtained by reducing fourdimensional part of the integrand. The remaining contribution to the rational part, \(R_2\), is instead computed by introducing new countertermlike diagrams which depend on the model under consideration [35, 36]. Alternatively, the term \(R_2\) can be be obtained by using fourdimensional Feynman rules, as described in [37].
Other methods, instead, aim at the combined determination of the two ingredients by reducing the dimensionally regulated integrand. Therefore the numerator of the integrand has to be generated and manipulated in \(d\) dimensions and acquires a dependence on \((d4)\) and on the square of the \((d4)\)dimensional components of the loop momentum, \(\mu ^2\) [31, 32, 38]. The multiparticle residues are finally determined by performing generalized cuts by setting \(d\)dimensional massive particles on shell. This is equivalent to have onshell fourdimensional states whose squared mass is shifted by \(\mu ^2\).
If the integrand at a generic multiple cut is obtained as a product of treelevel amplitudes, the issues related to factorization in presence of dimensional regularization have to be addressed. An interesting approach [31] uses the linear dependence of the amplitude on the spacetime dimensionality to compute the \(d\)dimensional amplitude. In particular the latter is obtained by interpolating the values of the oneloop amplitude in correspondence to two different integer values of the spacetime. When fermions are involved, the spacetime dimensions have to admit an explicit representation of the Clifford algebra [32]. More recently, this idea has been combined with the sixdimensional helicity formalism [39] for the analytic reconstruction of oneloop scattering amplitudes in QCD via generalized unitarity.
In this article, we elaborate on the fourdimensional helicity (FDH) scheme [28, 40, 41] and we propose a fourdimensional formulation (FDF) of the \(d\)dimensional regularization scheme which allows for a purely fourdimensional regularization of the amplitudes. Within FDF, the states in the loop are described as fourdimensional massive particles. The fourdimensional degrees of freedom of the gauge bosons are carried by massive vector bosons of mass \(\mu \) and their \((d4)\)dimensional ones by real scalar particles obeying a simple set of fourdimensional Feynman rules. A \(d\)dimensional fermion of mass \(m\) is instead traded for a tardyonic Dirac field with mass \(m +i \mu \gamma ^5\). The \(d\)dimensional algebraic manipulations are replaced by fourdimensional ones complemented by a set of multiplicative selection rules. The latter are treated as an algebra describing internal symmetries.
Within integrandreduction methods, our regularization scheme allows for the simultaneous computation of both the cutconstructible and the rational terms by employing a purely fourdimensional formulation of the integrands. As a consequence, an explicit fourdimensional representation of generalized states propagating around the loop can be formulated. Therefore, a straightforward implementation of \(d\)dimensional generalized unitarity within exactly four spacetime dimensions can be realized, avoiding any higherdimensional extension of either the Dirac [31, 32] or the spinor algebra [42].
Another interesting consequence of our framework is the possibility to extend to \(d\) dimensions the recursive generation of the integrand from offshell currents and open loops, now limited to four dimensions [43, 44, 45].
The paper is organized as follows. Section 2 is devoted to the description of our regularization method, while Sect. 3 describes how generalized unitarity method can be applied in presence of a FDF of oneloop amplitudes. Sections 4, 5 and 6, collect the applications of generalizedunitarity methods within the FDF. Section 7 describes how the integrand of the FDF of oneloop amplitudes can be generated recursively within the openloop approach.
2 Fourdimensional Feynman rules

The loop momenta are considered to be \(d\)dimensional. All observed external states are considered as four dimensional. All unobserved internal states, i.e. virtual states in loops and intermediate states in trees, are treated as \(d_s\)dimensional.

Since \(d_s > d > 4\), the scalar product of any \(d\) or \(d_s\)dimensional vector with a fourdimensional vector is a fourdimensional scalar product. Moreover, any dot product between a \(d_s\)dimensional tensor and a \(d\)dimensional one is a \(d\)dimensional dot product.

The Lorentz and the Clifford algebra are performed in \(d_s\) dimensions, which has to be kept distinct from \(d\). The matrix \(\gamma ^5\) is treated using the ’t Hooft–Veltman prescription, i.e. \(\gamma ^5\) commutes with the Dirac matrices carrying \(2\epsilon \) indices.

After the \(\gamma \)matrix algebra has been performed, the limit \(d_s \rightarrow 4\) has to be performed, keeping \(d\) fixed. The limit \(d\rightarrow 4\) is taken at the very end.
The rules (12) constitute an abstract algebra which is similar to the algebras implementing internal symmetries. For instance, in a Feynman diagrammatic approach the (\(2\epsilon \))SRs can be handled as the color algebra and performed for each diagram once and for all. In each diagram, the indices of the (\(2\epsilon \))SRs are fully contracted and the outcome of their manipulation is either \(0\) or \(\pm 1\).
To summarize, the QCD \(d\)dimensional Feynman rules in the ’t Hooft–Feynman gauge, collected in [47], may have the following FDF:
Our prescriptions, Eq. (11), can be related to a fivedimensional theory characterized by \(g^{55}=1\), \(\ell ^5 = \mu \) and a \(4\times 4\) representation of the Clifford algebra, \(\{\gamma ^0, \ldots , \gamma ^3, i \gamma ^5 \}\). Regularization methods in five dimensions have been proposed as an alternative formulation of the Pauli–Villars regularization [53] or as regulators of massless pure Yang–Mills theories at one loop [54]. Our method distinguishes itself by the presence of the (\( 2\epsilon \))SRs, a crucial ingredient for the correct reconstruction of dimensionally regularized amplitudes.
It is worth to notice that the possibility to obtain the rational part of oneloop amplitudes by using fourdimensional Feynman rules has been already investigated in [37]. The method presented there computes the \(\mu ^2\)dependent part of the numerator only, thus its Feynman rules are different from the ones presented in Eq. (13l). In particular our method does not introduces any additional scalar particle for each fermion flavor since the replacement of \(\tilde{\gamma }^\alpha \) with \(\gamma ^5\) takes care of the \(d_s\)dimensional Clifford algebra automatically. Moreover, the presence of the (\( 2\epsilon \))SRs guarantee the proper reconstruction of the \(\mu ^2\)independent part of the numerator. Finally the propagators of the FDF, Eqs. (13a)–(13d), depend on \(\mu ^2\), thus all particles are massive. Therefore in the FDF formulation the \(d\)dimensional cuts needed by both integrand reduction and generalized unitarity become fourdimensional massive cuts. In particular, as we show momentarily, a treelevelbased construction of the integrand has to involve amplitudes built by using \(\mu \)dependent spinors and polarizations vectors, fulfilling massive completeness relations.
3 Generalized unitarity
Generalizedunitarity methods in \(d\) dimensions require an explicit representation of the polarization vectors and the spinors of \(d\)dimensional particles. The latter ones are essential ingredients for the construction of the treelevel amplitudes that are sewn along the generalized cuts. In this respect, the FDF scheme is suitable for a four dimensional realization of the \(d\)dimensional generalized unitarity. The main advantage of the FDF is that the fourdimensional expression of the propagators of the particles in the loop admits an explicit representation in terms of generalized spinors and polarization expressions, whose expression is collected below.
The FDF within generalized unitarity may be seen as a massive implementation of \(d\)dimensional regularization. However, the FDF is different from the most commonly used massive regularization prescriptions, i.e. the one introducing a massive scalar particle [59] and the sixdimensional helicity method [39]. Indeed the former relies on the supersymmetric decomposition of the amplitude in terms of cutconstructible supersymmetric amplitudes and an amplitude involving a scalar. The original amplitude is computed in two steps. The cutconstructible part is obtained by using fourdimensional unitarity while the rational one is computed by using the amplitude involving a \(d\)dimensional scalar, which is traded with a massive fourdimensional ones. The FDF does not rely on existence of the supersymmetric decomposition and computes the full amplitude without splitting it.
The sixdimensional helicity method casts \(d\)dimensional onshell momenta into a sixdimensional massless spinor and, on the cuts, uses sixdimensional helicity spinors to compute efficiently the relevant treelevel amplitudes. However, since dimensional regularization cannot be achieved in finite dimensions, the sixdimensional helicity method deliver a result that has to be corrected by hand with the help of topologies involving sixdimensional scalars along the lines of [31]. The FDF, instead, splits the \(d\)dimensional objects into their fourdimensional and \((d4)\)dimensional parts and finds a fourdimensional representation for both of them. Moreover, it introduces the \((2\epsilon )\)SRs to account for the orthogonality of the subspaces and for the effects of the \((d_s4) \rightarrow 0\) limit. No further corrections are needed since FDF properly takes care of the peculiar features of \(d\)dimensional regularization. Therefore, in the context of onshell and unitaritybased methods, they are a simple alternative to approaches introducing explicit higherdimensional extension of either the Dirac [31, 32] or the spinor [39, 42] algebra.
4 The \(\mathbf {gggg}\) amplitude
As a first example we consider the fourgluon colorordered helicity amplitude \(A_{4}\left( 1_{g}^{+},2_{g}^{+},3_{g}^{+},4_{g}^{+}\right) \). The latter vanishes at treelevel, while the oneloop contribution is finite, rational and can be obtained from the quadruple cut \(C_{1234}\) [28, 40, 59, 60, 61]. The relevant treelevel threepoint amplitudes are computed by using the colorordered Feynman rules collected in Appendix C and collected in Appendix E.
For clarity reasons, in this example we have computed the \((2\epsilon )\)SRs factors, \(\mathcal {T}_i\), explicitly. It is worth to notice that in practice the \((2\epsilon )\)SRs can easily be automated and can be performed cutbycut once and for all, even before the treelevel amplitudes are computed. Therefore the cut topologies which vanish because of the \((2\epsilon )\)SRs can be discarded at the beginning of the computation without affecting its complexity.
5 The Open image in new window amplitude
The triple cuts are given by
The double cuts read as follows:
Rightturning amplitude The computation of the coefficients of \(A_{4}^{\text {R}} \) is similar to the one leading to the computation of the ones of \(A_{4}^{\text {L}}\). The explicit expression of the corresponding coefficients \(c^{\mathrm{\left[ R\right] }}_{i_1\ldots i_k; \, n }\) are shown in Appendix F.
6 The \(\mathbf {gggH}\) amplitude
The oneloop amplitude can be decomposed according to Eq. (D.13), in terms of three different ordering of the external particles, i.e. \(123H\), \(12H3\) and \(1H23\). In the case of the first ordering the coefficients \(c_{i_1\ldots i_k; \; n }\) and the corresponding cut \(C_{i_1\ldots i_k}\) read as follows:
The cut \(C_{123  H}\) does not give any contribution. The remaining coefficients are collected in Appendix G. The oneloop amplitude can be obtained by using the coefficients collected in Eqs. (43) and (G.25) and the decomposition (D.13). The result agrees with the literature [63].
7 Generalized open loop
The FDF of \(d\)dimensional oneloop amplitudes is compatible with methods generating recursively the integrands of oneloop amplitudes [64, 65] and leads to the complete reconstruction of the numerator of Feynman integrands as a polynomial in the loop variables, \(\ell ^\nu \) and \(\mu \). Our scheme allows for a generalization of the current implementations of these techniques [43, 44, 45]. Indeed, currently the latter can reconstruct only the fourdimensional part the numerator of the integrands, which is polynomial in \(\ell ^\nu \) only. In the following we focus on the generalization of the openloop technique [43] within the FDF scheme.
The recursive construction of the colorstripped treelevel diagrams, \(\mathcal{A} ^{(\mathrm{diag})}\), is not affected by the new Feynman particles and Feynman rules, which enter at looplevel only.
8 Conclusions
We introduced a fourdimensional formulation (FDF) of the \(d\)dimensional regularization of oneloop scattering amplitudes. Within our FDF, particles that propagate inside the loop are represented by massive particles regularizing the divergences. Their interactions are described by generalized fourdimensional Feynman rules. They include selection rules accounting for the regularization of the amplitudes. In particular, massless spin1 particles in \(d\)dimensions were represented in four dimensions by a combination of massive spinone particle and a scalar particle. Fermions in \(d\)dimensions were represented by fourdimensional fermions obeying the Dirac equation for tardyonic particles. The integrands of oneloop amplitudes in the FDF and in the FDH scheme differ by spurious terms which vanish upon integration over the loop momentum. Therefore the two schemes are equivalent.
In the FDF, the polarization and helicity states of the particles inside the loop admit an explicit fourdimensional representation, allowing for a complete, fourdimensional, unitaritybased construction of \(d\)dimensional amplitudes. The application of generalizedunitarity methods within the FDF has been described in detail by computing the NLO QCD corrections to helicity amplitudes of the processes \(gg \rightarrow q{\bar{q}}\) and \(gg \rightarrow gH\).
Mutual cancelations among the contributions of the longitudinal gluons and the ones of the scalar particles suggest a connection among them that deserves further investigations.
The FDF Feynman rules are compatible with methods generating recursively the integrands of oneloop amplitudes. In this context we have proposed a generalization to the openloop method, which allows for a complete reconstruction of the integrand, currently limited to four dimensions only.
The FDF approach is suitable for analytic as well as numerical implementation. Its main asset is the use of purely fourdimensional ingredients for the complete reconstruction of dimensionally regulated oneloop amplitudes. We plan to investigate its applicability beyond one loop. In particular we aim at using explicit fourdimensional representations to avoid the complications emerging from the formal manipulations of the \((d4)\)dimensional degrees of freedom.
Notes
Acknowledgments
We wish to thank Francesco Buciuni for crosschecking parts of the results. A.R.F. and W.J.T. thank the MaxPlanckInstitute for Physics in Munich for the kind hospitality at several stages of this project. For the same reasons, E.M. wishes to thank the Department of Mathematics and Physics of the University of Salento. A.R.F. is partially supported by the UNALDIB Grant No. 20629 of the “Convocatoria del programa nacional de proyectos para el fortalecimiento de la investigaciòn, la creaciòn y la innovaciòn en posgrados de la Universidad Nacional de Colombia 2013–2015”. The work of P.M. is supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovaleskaja Award Project “Advanced Mathematical Methods for Particle Physics”, endowed by the German Federal Ministry of Education and Research. W.J.T. is supported by Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO).
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