In leptophilic scenarios, dark matter interactions with nuclei, relevant for direct detection experiments and for the capture by celestial objects, could only occur via loop-induced processes. If the mediator is a scalar or pseudo-scalar particle, which only couples to leptons, the dominant contribution to dark matter-nucleus scattering would take place via two-photon exchange with a lepton triangle loop. The corresponding diagrams have been estimated in the literature under different approximations. Here, we present new analytical calculations for one-body two-loop and two-body one-loop interactions. The two-loop form factors are presented in closed analytical form in terms of generalized polylogarithms up to weight four. In both cases, we consider the exact dependence on all the involved scales, and study the dependence on the momentum transfer. We show that some previous approximations fail to correctly predict the scattering cross section by several orders of magnitude. Moreover, we quantitatively show that form factors in the range of momentum transfer relevant for local galactic dark matter, can be significantly smaller than their value at zero momentum transfer, which is the approach usually considered.

Italy

Raghuveer Garani

Federico Gasparotto

Pierpaolo Mastrolia

Henrik Munch

Spain

Sergio Palomares-Ruiz

Switzerland

Amedeo Primo

Two-photon exchange in leptophilic dark matter scenarios

We complete the analytic evaluation of the master integrals for the two-loop non-planar box diagrams contributing to the top-pair production in the quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals are determined from their differential equations, which are cast into a canonical form using the Magnus exponential. The analytic expressions of the Laurent series coefficients of the integrals are expressed as combinations of generalized polylogarithms, which we validate with several numerical checks. We discuss the analytic continuation of the planar and the non-planar master integrals, which contribute to q q → tt in QCD, as well as to the companion QED scattering processes ee → μμ and eμ → eμ.

Stefano Vita

Thomas Gehrmann

Stefano Laporta

United States

Ulrich Schubert

Master integrals for the NNLO virtual corrections to   <math> <mi>q</mi> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mo>→</mo> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ q\overline{q}\to t\overline{t} $$  scattering in QCD: the non-planar graphs

In this paper we present the results of the exact computation of contributions to the Higgs boson decay into bottom quarks that are proportional to the top Yukawa coupling. Our computation demonstrates that approximate results already available in the literature turn out to be particularly accurate for the three physical mass values of the Higgs boson, the bottom and top quarks. Furthermore, contrary to expectations, the impact of these corrections on differential distributions relevant for the searches of the Higgs boson decaying into bottom quarks at the Large Hadron Collider is rather small.

Gianmarco Sasso

Hungary

Gábor Somogyi

Francesco Tramontano

Exact top Yukawa corrections to Higgs boson decay into bottom quarks

We evaluate the master integrals for the two-loop non-planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to-leading order in QED. We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e + e − colliders, as well as for the QCD corrections to top-pair production at hadron colliders. In particular our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to μe → μe and e + e − → μ + μ −.

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs

We evaluate the master integrals for the two-loop, planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to leading-order in QED. We adopt the method of differential equations and the Magnus exponential series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e + e −-colliders, as well as for the QCD corrections to top-pair production at hadron colliders.

Massimo Passera

Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs

We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a 3×3 matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants.

Germany

Lorenzo Tancredi

Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

We compute the two-loop master integrals required for the leading QCD corrections to the interaction vertex of a massive neutral boson X 0 , e.g. H, Z or γ ∗ , with a pair of W bosons, mediated by a SU(2) L quark doublet composed of one massive and one massless flavor. All the external legs are allowed to have arbitrary invariant masses. The Magnus exponential is employed to identify a set of master integrals that, around d = 4 space-time dimensions, obey a canonical system of differential equations. The canonical master integrals are given as a Taylor series in ϵ = (4 − d ) / 2, up to order four, with coefficients written as combination of Goncharov polylogarithms, respectively up to weight four. In the context of the Standard Model, our results are relevant for the mixed EW-QCD corrections to the Higgs decay to a W pair, as well as to the production channels obtained by crossing, and to the triple gauge boson vertices ZWW and γ ∗ WW .

Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZW W and γ ∗ W W

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ϵ=(4−d)/2 , where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.

On the maximal cut of Feynman integrals and the solution of their differential equations

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, d = d ∥ + d ⊥ , being d ∥ the dimension of the parallel space spanned by the legs of the diagrams. When the number n of external legs is n ≤ 4,thecorrespondingrepresentationofmultiloopintegralsexposesasubsetofintegration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external ones. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to n = 8 legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.

United Kingdom

Tiziano Peraro

Adaptive integrand decomposition in parallel and orthogonal space

We present a set of relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. Within dimensional regularization, the combined use of color-kinematics duality and integrand reduction yields the existence of relations between the integrand residues of partial amplitudes with different orderings of the external particles. These relations can be established for the cut-constructible contributions as well for the ones responsible for rational terms. Starting from the general parametrization of one-loop residues and applying Laurent expansion in order to extract the coefficients of the amplitude decomposition in terms of master integrals, we show that the full set of relations can be obtained by considering the BCJ identities between d -dimensional tree-level amplitudes. We provide explicit examples for multi-gluon scattering amplitudes at one loop.