Pentagon Integrals Research Articles

Multiple Mellin-Barnes integrals with straight contours

We show how the conic hull method, recently developed for the analytic and non-iterative evaluation of multifold Mellin-Barnes (MB) integrals, can be extended to the case where these integrals have straight contours of integration parallel to the imaginary axes in the complex planes of the integration variables. MB integrals of this class appear, for instance, when one computes the $\epsilon$-expansion of dimensionally regularized Feynman integrals, as a result of the application of one of the two main strategies (called A and B in the literature) used to resolve the singularities in $\epsilon$ of MB representations. We upgrade the Mathematica package MBConicHulls.wl which can now be used to obtain multivariable series representations of multifold MB integrals with arbitrary straight contours, providing an efficient tool for the automatic computation of such integrals. This new feature of the package is presented, along with an example of application by calculating the $\epsilon$-expansion of the dimensionally regularized massless one-loop pentagon integral in general kinematics and $D=4-2\epsilon$.

Proving the dimension-shift conjecture

We prove the conjecture made by Bern, Dixon, Dunbar, and Kosower that describes a simple dimension shifting relationship between the one-loop structure of \mathcal{N}=4𝒩=4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. The proof captures all orders in dimensional regularisation using unitarity cuts, by combining massive spinor-helicity with Coulomb-branch supersymmetry. The form of these amplitudes can be given in terms of pentagon and box integrals using a generalised DD-dimensional unitarity technique which captures the full amplitude to all multiplicities.

Feynman Integrals and Scattering Amplitudes from Wilson Loops.

We study Feynman integrals and scattering amplitudes in N=4 super-Yang-Mills theory by exploiting the duality with null polygonal Wilson loops. As the main application, we compute for the first time the symbols of the general double pentagon integrals, which give the finite part of two-loop maximally helicity violating (MHV) amplitudes and finite components of next-to-MHV (NMHV) amplitudes to all multiplicities. The rational parts of the symbol consist of 164 letters, while the algebraic part contains 96 algebraic letters and cancel in MHV amplitudes and NMHV components which are free of square roots.

Pentagon integrals to arbitrary order in the dimensional regulator

We analytically calculate one-loop five-point Master Integrals, pentagon integrals, with up to one off-shell leg to arbitrary order in the dimensional regulator in d = 4−2\U0001d716 space-time dimensions. A pure basis of Master Integrals is constructed for the pentagon family with one off-shell leg, satisfying a single-variable canonical differential equation in the Simplified Differential Equations approach. The relevant boundary terms are given in closed form, including a hypergeometric function which can be expanded to arbitrary order in the dimensional regulator using the Mathematica package HypExp. Thus one can obtain solutions of the canonical differential equation in terms of Goncharov Polylogartihms of arbitrary transcendental weight. As a special limit of the one-mass pentagon family, we obtain a fully analytic result for the massless pentagon family in terms of pure and universally transcendental functions. For both families we provide explicit solutions in terms of Goncharov Polylogartihms up to weight four.

Multi-Regge limit of the two-loop five-point amplitudes in mathcal{N} = 4 super Yang-Mills and mathcal{N} = 8 supergravity

In previous work, the two-loop five-point amplitudes in mathcal{N} = 4 super Yang-Mills theory and mathcal{N} = 8 supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the mathcal{N} = 4 super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.

Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory

We apply the Landau equations, whose solutions parameterize the locus of possible branch points, to the one- and two-loop Feynman integrals relevant to MHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory. We then identify which of the Landau singularities appear in the symbols of the amplitudes, and which do not. We observe that all of the symbol entries in the two-loop MHV amplitudes are already present as Landau singularities of one-loop pentagon integrals.

One-loop pentagon integral in d dimensions from differential equations in ϵ-form

We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to ϵ-form, we manage to obtain a simple one-fold integral representation exact in space-time dimensionality. The expansion of the obtained result in ϵ and the analytical continuation to physical regions are discussed.

ZZ + jet production via gluon fusion at the LHC

Pair production of Z bosons in association with a hard jet is an important background for Higgs particle or new physics searches at the LHC. The loop-induced gluon-fusion process gg > ZZg contributes formally only at the next-to-next-to-leading order. Nevertheless, it can get enhanced by the large gluon flux at the LHC, and thus should be taken into account in relevant experimental searches. We provide the details and results of this calculation, which involves the manipulation of rank-5 pentagon integrals. Our results show that the gluon-fusion process can contribute more than 10% to the next-to-leading order QCD result and increases the overall scale uncertainty. Moreover, interference effects between Higgs and non-Higgs contributions can become large in phase-space regions where the Higgs is far off-shell.

Star integrals, convolutions and simplices

Abstract We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop n-gon integrals in n dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli’s formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the 6d hexagon and 8d octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.

Photon-pair jet production via gluon fusion at the LHC

Photon-pair direct or jet-associated production is important for relevant standard model measurement, Higgs and new physics searches at the Large Hadron Collider (LHC). The loop-induced gluon-fusion process gg → γγg, which formally contributes only at the next-to-next-to-leading order to γγj productions, may get enhanced by the large gluon flux at the LHC. We have checked and confirmed previous results on gg → γγ (Dicus and Willenbrock 1988 Phys. Rev. D 37 1801), γγg (de Florian and Kunszt 1999 Phys. Lett. B 460 184) at one loop, using the traditional Feynman diagram based approach, taking into account the quark mass effects, and updated them further for the 7 and 14 TeV LHC with new inputs and settings. We provide the details and results of the calculations, which involve manipulation of rank-5 pentagon integrals. Our results show that the gluon-fusion process can contribute about 10% of the Born result, especially at small Mγγ and PTγγ, and increase further the overall scale uncertainty. Top quark loop effects are examined in detail, which become important near or above the threshold Mγγ ≳ 2mt.

The one-loop pentagon to higher orders in ϵ

We compute the one-loop scalar massless pentagon integral I_5^{6-2 eps} in D=6-2\eps dimensions in the limit of multi-Regge kinematics. This integral first contributes to the parity-odd part of the one-loop N=4 five-point MHV amplitude m_5^{(1)} at O(eps). In the high energy limit defined, the pentagon integral reduces to double sums or equivalently two-fold Mellin-Barnes integrals. By determining the O(eps) contribution to I_5^{6-2 eps}, one therefore gains knowledge of m_5^{(1)} through to O(eps^2) which is necessary for studies of the iterative structure of N=4 SYM amplitudes beyond one-loop. One immediate application is the extraction of the one-loop gluon-production vertex through to O(eps^2) and the iterative construction of the two-loop gluon-production vertex through to finite terms which is described in a companion paper. The analytic methods we have used for evaluating the pentagon integral in the high energy limit may also be applied to the hexagon integral and may ultimately give information on the form of the remainder function.

Unitarity cuts with massive propagators and algebraic expressions for coefficients

In the first part of this paper, we extend the $d$-dimensional unitarity cut method of Anastasiou et al. to cases with massive propagators. We present formulas for integral reduction with which one can obtain coefficients of all pentagon, box, triangle and massive bubble integrals. In the second part of this paper, we present a detailed study of the phase space integration for unitarity cuts. We carry out spinor integration in generality and give algebraic expressions for coefficients, intended for automated evaluation.

Dimensionally-regulated pentagon integrals

We present methods for evaluating the Feynman parameter integrals associated with the pentagon diagram in 4-2∈ dimensions, along with explicit results for the integrals with all masses vanishing or with one non-vanishing external mass. The scalar pentagon integral can be expressed as a linear combination of box integrals, up to O(∈) corrections, a result which is the dimensionally-regulated version of a D = 4 result of Melrose, and of van Neerven and Vermaseren. We obtain and solve differential equations for various dimensionally-regulated box integrals with massless internal lines, which appear in one-loop n-point calculations in QCD. We give a procedure for constructing the tensor pentagon integrals needed in gauge theory, again through O(∈ 0).