Uniform Transcendentality Research Articles

Two-loop five-point two-mass planar integrals and double Lagrangian insertions in a Wilson loop

We consider the complete set of planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the calculation of the second-order QCD corrections to the production of two heavy vector bosons in association with a jet or a photon at a hadron collider. We construct pure bases for these integrals and reconstruct their analytic differential equations in canonical form through numerical sampling over finite fields. The newly identified symbol alphabet, one of the most complex to date, provides valuable data for bootstrap methods. We then apply our results to initiate the study of double Lagrangian insertions in a four-cusp Wilson loop in planar maximally supersymmetric Yang-Mills theory, computing it through two loops. We observe that it is finite, conformally invariant in four dimensions, and of uniform transcendentality. Furthermore, we provide numerical evidence for its positivity within the amplituhedron region through two loops.

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter ϵ. The results are expressed as one-fold integrals over classical polylogarithms. We provide a set of supplementary files containing our results in machine-readable form, including a proof-of-concept implementation for numerical evaluations of the one-fold integrals valid within a subset of the Euclidean region.

The two-loop eight-point amplitude in ABJM theory

In this paper, we present the two-loop correction to scattering amplitudes in three-dimensional mathcal{N} = 6 Chern-Simons matter theory. We use the eight-point case as our main example, but the method generalizes to all multiplicities. The integrand is completely fixed by dual conformal symmetry, maximal cuts, constraints from soft-collinear behavior, and from the vanishing of odd-multiplicity amplitudes. After performing integrations with Higgs regularizations, the integrated results demonstrate that the infrared divergence is again identical to that of mathcal{N} = 4 super Yang-Mills. After subtracting divergences, the finite part is dual conformal invariant, and respects various symmetries; it has uniform transcendentality weight two and exhibits a nice analytic structure.

Integrating three-loop modular graph functions and transcendentality of string amplitudes

Modular graph functions (MGFs) are SL(2, ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2, ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.

Generating series of all modular graph forms from iterated Eisenstein integrals

We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes. By analysing the differential equation of the generating series we construct a solution for their low-energy expansion to all orders in the inverse string tension α′. Our solution is expressed through initial data involving multiple zeta values and certain real-analytic functions of the modular parameter of the torus. These functions are built from real and imaginary parts of holomorphic iterated Eisenstein integrals and should be closely related to Brown’s recent construction of real-analytic modular forms. We study the properties of our real-analytic objects in detail and give explicit examples to a fixed order in the α′-expansion. In particular, our solution allows for a counting of linearly independent modular graph forms at a given weight, confirming previous partial results and giving predictions for higher, hitherto unexplored weights. It also sheds new light on the topic of uniform transcendentality of the α′-expansion.

Deriving canonical differential equations for Feynman integrals from a single uniform weight integral

Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to Höschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-planar two-loop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper.

Exploring transcendentality in superstring amplitudes

It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of uniform transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region mathrm{mathcal{M}} R surrounding the cusp and its complement mathrm{mathcal{M}} L, and an exact expression is obtained for the contribution to the amplitude from mathrm{mathcal{M}} R. The low-energy expansion of the mathrm{mathcal{M}} R contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from mathrm{mathcal{M}} L is computed in terms of modular graph functions up to order D12 mathrm{mathcal{R}} 4 in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the mathrm{mathcal{M}} L contributions. Uniform transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sum functions, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.

Perturbative four-point functions in planar mathcal{N}=4 SYM From hexagonalization

We use hexagonalization to compute four-point correlation functions of long BPS operators with special R-charge polarizations. We perform the computation at weak coupling and show that at any loop order our correlators can be expressed in terms of single value polylogarithms with uniform and maximal transcendentality. As a check of our computation we extract nine-loop OPE data and compare it against sum rules of (squared) structures constants of non-protected exchanged operators described by hundreds of Bethe solutions.

Finite remainders of the Konishi at two loops in mathcal{N}=4 SYM

We present three point form factors (FF) in mathcal{N}=4 Super Yang Mills theory for both the half-BPS and the Konishi operators at two loop level in the ‘t Hooft coupling using Feynman diagrammatic approach. We have chosen on shell final states consisting of gϕϕ and ϕλλ, where ϕ, λ, g are scalar, Majorana fermion and gauge fields respectively. The computation is done both in the modified dimensional reduction as well as in the four dimensional helicity scheme. We have studied the universal structure of infrared (IR) singularities in these FFs using Catani’s IR subtraction operators. Exploiting the factorisation property of the IR singularities and following BDS like ansatz for the IR sensitive terms in FFs, we determine the finite remainders of them. We find that the finite remainders of FFs of the half-BPS for both the choices of final states give not only identical results but also contain terms of uniform transcendentality of weight two and four at one and two loop levels, respectively. In the case of the Konishi operator, the finite remainders depend on the external states and do not exhibit uniform transcendentality. However, surprisingly, the leading transcendental terms for gϕϕ agree with that of the half-BPS. We have demonstrated the role of on shell external states for the FFs in the context of maximum transcendentality principle.

Two-loop SL(2) form factors and maximal transcendentality

Form factors of composite operators in the SL(2) sector of N=4 SYM theory are studied up to two loops via the on-shell unitarity method. The non-compactness of this subsector implies the novel feature and technical challenge of an unlimited number of loop momenta in the integrand's numerator. At one loop, we derive the full minimal form factor to all orders in the dimensional regularisation parameter. At two loops, we construct the complete integrand for composite operators with an arbitrary number of covariant derivatives, and we obtain the remainder functions as well as the dilatation operator for composite operators with up to three covariant derivatives. The remainder functions reveal curious patterns suggesting a hidden maximal uniform transcendentality for the full form factor. Finally, we speculate about an extension of these patterns to QCD.

On-shell methods for the two-loop dilatation operator and finite remainders

We compute the two-loop minimal form factors of all operators in the SU(2) sector of planar N=4 SYM theory via on-shell unitarity methods. From the UV divergence of this result, we obtain the two-loop dilatation operator in this sector. Furthermore, we calculate the corresponding finite remainder functions. Since the operators break the supersymmetry, the remainder functions do not have the property of uniform transcendentality. However, the leading transcendentality part turns out to be universal and is identical to the corresponding BPS expressions. The remainder functions are shown to satisfy linear relations which can be explained by Ward identities of form factors following from R-symmetry.

Singularity structure of maximally supersymmetric scattering amplitudes.

We present evidence that loop amplitudes in maximally supersymmetric (N=4) Yang-Mills theory (SYM) beyond the planar limit share some of the remarkable structures of the planar theory. In particular, we show that through two loops, the four-particle amplitude in full N=4 SYM has only logarithmic singularities and is free of any poles at infinity--properties closely related to uniform transcendentality and the UV finiteness of the theory. We also briefly comment on implications for maximal (N=8) supergravity theory (SUGRA).

Nonglobal logarithms at three loops, four loops, five loops, and beyond

We calculate the coefficients of the leading nonglobal logarithms for the hemisphere mass distribution analytically at 3, 4, and 5 loops at large ${N}_{c}$. We confirm that the integrand derived with the strong-energy-ordering approximation and fixed-order iteration of the Banfi-Marchesini-Syme (BMS) equation agree. Our calculation exploits a hidden $\mathrm{PSL}(2,\mathbb{R})$ symmetry associated with the jet directions, apparent in the BMS equation after a stereographic projection to the Poincar\'e disk. The required integrals have an iterated form, leading to functions of uniform transcendentality. This allows us to extract the coefficients, and some functional dependence on the jet directions, by computing the symbols and coproducts of appropriate expressions involving classical and Goncharov polylogarithms. Convergence of the series to a numerical solution of the BMS equation is also discussed.

Single soft gluon emission at two loops

Abstract We study the single soft-gluon current at two loops with two energetic partons in massless perturbative QCD, which describes, for example, the soft limit of the two-loop amplitude for gg → Hg. The results are presented as Laurent expansions in ϵ in D = 4 − 2ϵ spacetime dimension. We calculate the expansion to order ϵ 2 analytically, which is a necessary ingredient for Higgs production at hadron colliders at next-to-next-to-next-to-leading order in the soft-virtual approximation. We also give two-loop results of the single soft-gluon current in $ \mathcal{N}=4 $ Super-Yang-Mills theory, and find that it has uniform transcendentality. By iteration relation of splitting amplitudes, our calculations can determine the three-loop single soft-gluon current to order ϵ 0 in $ \mathcal{N}=4 $ Super-Yang-Mills theory in the limit of large N c .

ABJM amplitudes and WL at finite N

We evaluate ABJM observables at two loops, for any value of the rank N of the gauge group. We compute the color subleading contributions to the four-point scattering amplitude in ABJM at two loops. Contrary to the four dimensional case, IR divergent N-subleading contributions are proportional to leading poles in the regularization parameter. We then exploit the non-planar calculation for the amplitude to derive an expression for the two-loop Sudakov form factor at any N. In the planar limit the result coincides with the one recently obtained in literature by using Feynman diagrams and unitarity. Finally, we analyze the subleading contributions to the light-like four-cusps Wilson loop and interpret the result in terms of the non-abelian exponentiation theorem. All these perturbative results satisfy the uniform transcendentality principle, hinting at its validity in ABJM beyond the planar limit.

On DIS Wilson coefficients in [formula omitted] super Yang–Mills theory

In this Letter we evaluate Wilson coefficients for “deep inelastic scattering” (DIS) in N=4 SYM theory at NLO in perturbation theory, using as a probe an R-symmetry conserved current. They exhibit uniform transcendentality and coincide with the piece of highest transcendentality in the corresponding QCD Wilson coefficients. We extract from the QCD result a NNLO prediction for the N=4 SYM Wilson coefficient, and comment on the features of its Regge limit asymptotics.

Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation

We apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. After a set of master integrals has been found using the integration-by-parts method, the crucial point of this strategy is to introduce a new basis where all master integrals are pure functions of uniform transcendentality. In this paper, we apply this method to all planar three-loop four-point massless on-shell master integrals. We explicitly find such a basis, and show that the differential equations are of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all orders in the dimensional regularization parameter epsilon, including all boundary constants, in a purely algebraic way. The solution is expressed in terms of harmonic polylogarithms. We explicitly write out the Laurent expansion in epsilon for all master integrals up to weight six.

The three-loop form factor in $ \mathcal{N} = {4} $ super Yang-Mills

In this paper we study the Sudakov form factor in N=4 super Yang-Mills theory to the three-loop order. The latter is expressed in terms of planar and non-planar loop integrals. We show that it is possible to choose a representation in which each loop integral has uniform transcendentality. We verify analytically the expected exponentiation of the infrared divergences with the correct values of the three-loop cusp and collinear anomalous dimensions in dimensional regularisation. We find that the form factor in N=4 super Yang-Mills can be related to the leading transcendentality part of the quark and gluon form factors in QCD. We also study the ultraviolet properties of the form factor in D>4 dimensions, and find unexpected cancellations, resulting in an improved ultraviolet behaviour.

Subleading-color contributions to gluon-gluon scattering in 𝒩 = 4 SYM theory and relations to 𝒩 = 8 supergravity

We study the subleading-color (nonplanar) contributions to the four-gluon scattering amplitudes in = 4 supersymmetric SU(N) Yang-Mills theory. Using the formalisms of Catani and of Sterman and Tejeda-Yeomans, we develop explicit expressions for the infrared-divergent contributions of all the subleading-color L-loop amplitudes up to three loops, and make some conjectures for the IR behavior for arbitrary L. We also derive several intriguing relations between the subleading-color one- and two-loop four-gluon amplitudes and the four-graviton amplitudes of = 8 supergravity. The exact one- and two-loop = 8 supergravity amplitudes can be expressed in terms of the one- and two-loop N-independent = 4 SYM amplitudes respectively, but the natural generalization to higher loops fails, despite having a simple interpretation in terms of the 't Hooft picture. We also find that, at least through two loops, the subleading-color amplitudes of = 4 SYM theory have uniform transcendentality (as do the leading-color amplitudes). Moreover, the = 4 SYM Catani operators, which express the IR-divergent contributions of loop amplitudes in terms of lower-loop amplitudes, are also shown to have uniform transcendentality, and to be the maximum transcendentality piece of the QCD Catani operators.