Four-dimensional Helicity Scheme Research Articles

One-loop hexagon integral to higher orders in the dimensional regulator

The state-of-the-art in current two-loop QCD amplitude calculations is at five-particle scattering. Computing two-loop six-particle processes requires knowledge of the corresponding one-loop amplitudes to higher orders in the dimensional regulator. In this paper we compute analytically the one-loop hexagon integral via differential equations. In particular we identify its function alphabet for general D-dimensional external states. We also provide integral representations for all one-loop integrals up to weight four. With this, the one-loop integral basis is ready for two-loop amplitude applications. We also study in detail the difference between the conventional dimensional regularization and the four-dimensional helicity scheme at the level of the master integrals and their function space.

γ5 in the four-dimensional helicity scheme

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

Regularization-scheme dependence of QCD amplitudes in the massive case

We investigate QCD amplitudes with massive quarks computed in the four-dimensional helicity scheme (FDH) and dimensional reduction at NNLO and describe how they are related to the corresponding amplitudes computed in conventional dimensional regularization. To this end, the scheme dependence of the heavy quark and the velocity-dependent cusp anomalous dimensions is determined using soft-collinear effective theory. The results are checked against explicit computations of massive form factors in FDH at NNLO. Our results complete the description of the scheme dependence of QCD amplitudes at NNLO.

SCET approach to regularization-scheme dependence of QCD amplitudes

We investigate the regularization-scheme dependence of scattering amplitudes in massless QCD and find that the four-dimensional helicity scheme (FDH) and dimensional reduction (DRED) are consistent at least up to NNLO in the perturbative expansion if renormalization is done appropriately. Scheme dependence is shown to be deeply linked to the structure of UV and IR singularities. We use jet and soft functions defined in soft-collinear effective theory (SCET) to efficiently extract the relevant anomalous dimensions in the different schemes. This result allows us to construct transition rules for scattering amplitudes between different schemes (CDR, HV, FDH, DRED) up to NNLO in massless QCD. We also show by explicit calculation that the hard, soft and jet functions in SCET are regularization-scheme independent.

Computation of $$H\rightarrow gg$$ H → g g in fdh and dred: renormalization, operator mixing, and explicit two-loop results

The $H\to gg$ amplitude relevant for Higgs production via gluon fusion is computed in the four-dimensional helicity scheme (FDH) and in dimensional reduction (DRED) at the two-loop level. The required renormalization is developed and described in detail, including the treatment of evanescent $\epsilon$-scalar contributions. In FDH and DRED there are additional dimension-5 operators generating the $H g g$ vertices, where $g$ can either be a gluon or an $\epsilon$-scalar. An appropriate operator basis is given and the operator mixing through renormalization is described. The results of the present paper provide building blocks for further computations, and they allow to complete the study of the infrared divergence structure of two-loop amplitudes in FDH and DRED.

On the four-dimensional formulation of dimensionally regulated amplitudes

Elaborating on the four-dimensional helicity scheme, we propose a pure four-dimensional formulation (FDF) of the \(d\)-dimensional regularization of one-loop 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 extra-dimensional regulating terms of the amplitude. We present explicit representations of the polarization and helicity states of the four-dimensional particles propagating in the loop. They allow for a complete, four-dimensional, unitarity-based construction of \(d\)-dimensional amplitudes. Generalized unitarity within the FDF does not require any higher-dimensional extension of the Clifford and the spinor algebra. Finally we show how the FDF allows for the recursive construction of \(d\)-dimensional one-loop integrands, generalizing the four-dimensional open-loop approach.

The infrared structure of QCD amplitudes and H → gg in FDH and DRED

We consider variants of dimensional regularization, including the four-dimensional helicity scheme (fdh) and dimensional reduction (dred), and present the gluon and quark form factors in the fdh scheme at next-to-next-to-leading order. We also discuss the generalization of the infrared factorization formula to fdh and dred. This allows us to extract the cusp anomalous dimension as well as the quark and gluon anomalous dimensions at next-to-next-to-leading order in the fdh and dred scheme, using MS‾ and DR‾ renormalization. To obtain these results we also present the renormalization procedure in these schemes.

Regularization schemes and higher order corrections

I apply commonly used regularization schemes to a multiloop calculation to examine the properties of the schemes at higher orders. I find complete consistency between the conventional dimensional regularization scheme and dimensional reduction, but I find that the four-dimensional helicity scheme produces incorrect results at next-to-next-to-leading order and singular results at next-to-next-to-next-to-leading order. It is not, therefore, a unitary regularization scheme.

Conformal anomaly of super Wilson loop

Classically supersymmetric Wilson loop on a null polygonal contour possesses all symmetries required to match it onto non-MHV amplitudes in maximally supersymmetric Yang–Mills theory. However, to define it quantum mechanically, one is forced to regularize it since perturbative loop diagrams are not well defined due to presence of ultraviolet divergences stemming from integration in the vicinity of the cusps. A regularization that is adopted by practitioners by allowing one to use spinor helicity formalism, on the one hand, and systematically go to higher orders of perturbation theory is based on a version of dimensional regularization, known as Four-Dimensional Helicity scheme. Recently it was demonstrated that its use for the super Wilson loop at one loop breaks both conformal symmetry and Poincaré supersymmetry. Presently, we exhibit the origin for these effects and demonstrate how one can undo this breaking. The phenomenon is alike the one emerging in renormalization group mixing of conformal operators in conformal theories when one uses dimensional regularization. The rotation matrix to the diagonal basis is found by means of computing the anomaly in the Ward identity for the conformal boost. Presently, we apply this ideology to the super Wilson loop. We compute the one-loop conformal anomaly for the super Wilson loop and find that the anomaly depends on its Grassmann coordinates. By subtracting this anomalous contribution from the super Wilson loop we restore its interpretation as a dual description for reduced non-MHV amplitudes which are expressed in terms of superconformal invariants.

Four-dimensional helicity scheme and dimensional reconstruction

The four-dimensional helicity regularization scheme is often used in one-loop QCD computations. It was recently argued in Ref. [1] that this scheme is inconsistent beyond the one-loop order in perturbation theory. In this paper, we clarify the reason for this inconsistency by studying the perturbative expansion of the vector current correlator in one-flavor QED through three-loop order. We develop a simple, practical way to fix the four-dimensional helicity scheme using the idea of dimensional reconstruction, and demonstrate its application in several illustrative examples.

Regularization schemes and higher order corrections

I apply commonly used regularization schemes to a multiloop calculation to examine the properties of the schemes at higher orders. I find complete consistency between the conventional dimensional regularization scheme and dimensional reduction, but I find that the four-dimensional helicity scheme produces incorrect results at next-to-next-to-leading order and singular results at next-to-next-to-next-to-leading order. It is not, therefore, a unitary regularization scheme.

Two-Loop Helicity Amplitudes for Quark-Quark Scattering in QCD and Gluino-Gluino Scattering in Supersymmetric Yang-Mills Theory

We present the two-loop QCD helicity amplitudes for quark-quark and quark-antiquark scattering. These amplitudes are relevant for next-to-next-to-leading order corrections to (polarized) jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity (FDH) variants of dimensional regularization and present the scheme dependence of the results. We verify that the finite remainder, after subtracting the divergences using Catani's formula, are in agreement with previous results. We also provide the amplitudes for gluino-gluino scattering in pure N=1 supersymmetric Yang-Mills theory. We describe ambiguities in continuing the Dirac algebra to D dimensions, including ones which violate fermion helicity conservation. The finite remainders after subtracting the divergences using Catani's formula, which enter into physical quantities, are free of these ambiguities. We show that in the FDH scheme, for gluino-gluino scattering, the finite remainders satisfy the expected supersymmetry Ward identities.

Two-loop helicity amplitudes for quark-gluon scattering in QCD and gluino-gluon scattering in supersymmetric Yang-Mills theory

We present the two-loop QCD helicity amplitudes for quark-gluon scattering, and for quark-antiquark annihilation into two gluons. These amplitudes are relevant for next-to-next-to-leading order corrections to (polarized) jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity (FDH) variants of dimensional regularization. The transition rules for converting the amplitudes between the different variants are much more intricate than for the previously discussed case of gluon-gluon scattering. Summing our two-loop expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Anastasiou, Glover, Oleari and Tejeda-Yeomans. We describe the amplitudes for 2→2 scattering in pure N = 1 supersymmetric Yang-Mills theory, obtained from the QCD amplitudes by modifying the color representation and multiplicities, and verify supersymmetry Ward identities in the FDH scheme.

Supersymmetric regularization, two-loop QCD amplitudes, and coupling shifts

We present a definition of the four-dimensional helicity (FDH) regularization scheme valid for two or more loops. This scheme was previously defined and utilized at one loop. It amounts to a variation on the standard 't Hooft-Veltman scheme and is designed to be compatible with the use of helicity states for "observed" particles. It is similar to dimensional reduction in that it maintains an equal number of bosonic and fermionic states, as required for preserving supersymmetry. Supersymmetry Ward identities relate different helicity amplitudes in supersymmetric theories. As a check that the FDH scheme preserves supersymmetry, at least through two loops, we explicitly verify a number of these identities for gluon-gluon scattering (gg to gg) in supersymmetric QCD. These results also cross-check recent non-trivial two-loop calculations in ordinary QCD. Finally, we compute the two-loop shift between the FDH coupling and the standard MS-bar coupling, alpha_s. The FDH shift is identical to the one for dimensional reduction. The two-loop coupling shifts are then used to obtain the three-loop QCD beta function in the FDH and dimensional reduction schemes.