All orders structure and efficient computation of linearly reducible elliptic Feynman integrals

We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts identities. We then elaborate on the differential equations method. Specifically, we show that the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in $\epsilon$-form. In the examples we consider the canonical differential equations can be directly solved in terms of eMPLs up to arbitrary order of the dimensional regulator. The remaining 1-dimensional integral may be performed to express such integrals completely in terms of eMPLs. We apply these methods to solve two- and three-points integrals in terms of eMPLs. We analytically continue these integrals to the physical region by using their 1-dimensional integral representation.Comment: The differential equations method is applied to linearly reducible elliptic Feynman integrals, the solutions are in terms of elliptic polylogarithms, JHEP version, 50 page

Constructing polylogarithms on higher-genus Riemann surfaces

An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini.Comment: 54 pages, 2 figures; v2: references added, expanded the discussion of modular properties in sections 3 and

The complete set of two-loop master integrals for Higgs + jet production in QCD

In this paper we complete the computation of the two-loop master integrals relevant for Higgs plus one jet production initiated in arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234. We compute the integrals by defining differential equations along contours in the kinematic space, and by solving them in terms of one-dimensional generalized power series. This method allows for the efficient evaluation of the integrals in all kinematic regions, with high numerical precision. We show the generality of our approach by considering both the top- and the bottom-quark contributions. This work along with arXiv:1609.06685, arXiv:1907.13156, arXiv:1907.13234 provides the full set of master integrals relevant for the NLO corrections to Higgs plus one jet production, and for the real-virtual contributions to the NNLO corrections to inclusive Higgs production in QCD in the full theory.Comment: 32 pages, references added, minor revisio

Modular graph forms from equivariant iterated Eisenstein integrals

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown’s alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown’s conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown’s construction fully explicit to all orders

Computational and mathematical aspects of Feynman integrals

This thesis covers a number of different research projects which are all connected to the central topic of computing Feynman integrals efficiently through analytic methods. Improvements in our ability to evaluate Feynman integrals allow us to increase the order in perturbation theory at which we are able to produce theoretical predictions for various processes in the Standard Model, which can be tested at the Large Hadron Collider. In the first part of this thesis, we cover novel research on the analytic computation of elliptic Feynman integrals. We will show how certain elliptic Feynman integrals can be written as one-fold integrals over polylogarithmic Feynman integrals, which can be solved from systems of differential equations in a canonical dlog-form, or by using the method of direct integration. Thereafter, we discuss a method for computing Feynman integrals from their differential equations in terms of one-dimensional series expansions along contours in phase-space. By connecting series expansions along multiple line segments, the method allows us to obtain high precision numerical results for various Feynman integrals at arbitrary points in phase-space. We will also present a novel Mathematica package called DiffExp, which provides a general implementation of these series expansion methods. As an illustrative example, we apply the package to obtain high precision results for the unequal-mass banana graph family in the physical region. Next, we present the computation of the complete set of non-planar master integrals relevant for Higgs plus jet production at next-to-leading order with full heavy quark mass dependence. The non-planar integrals fit into two integral families. We provide a choice of basis that puts many of the sectors in a canonical dlog-form, and we show that high precision numerical results can be obtained for all integrals using series expansion methods. Lastly, we discuss work on the diagrammatic coaction of the equal-mass elliptic sunrise family

Feynman parameter integration through differential equations

We present a new method for numerically computing generic multi-loop Feynman integrals. The method relies on an iterative application of Feynman's trick for combining two propagators. Each application of Feynman's trick introduces a simplified Feynman integral topology which depends on a Feynman parameter that should be integrated over. For each integral family, we set up a system of differential equations which we solve in terms of a piecewise collection of generalized series expansions in the Feynman parameter. These generalized series expansions can be efficiently integrated term by term, and segment by segment. This approach leads to a fully algorithmic method for computing Feynman integrals from differential equations, which does not require the manual determination of boundary conditions. Furthermore, the most complicated topology that appears in the method often has less master integrals than the original one. We illustrate the strength of our method with a five-point two-loop integral family

Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation

Abstract The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers

Evaluation of multiloop multiscale Feynman integrals for precision physics

Modern particle physics is increasingly becoming a precision science that relies on advanced theoretical predictions for the analysis and interpretation of experimental results. The planned physics program at the LHC and future colliders will require three-loop electroweak and mixed electroweak-QCD corrections to single-particle production and decay processes and two-loop electroweak corrections to pair-production processes. This article presents a new seminumerical approach to multiloop multiscale Feynman integrals calculations which will be able to fill the gap between rigid experimental demands and theory. The approach is based on differential equations with boundary terms specified at Euclidean kinematic points. These Euclidean boundary terms can be computed numerically with high accuracy using sector decomposition or other numerical methods. They are then mapped to the physical kinematic configuration by repeatedly solving the differential equation system in terms of series solutions. An automatic and general method is proposed for constructing a basis of master integrals such that the differential equations are finite. The approach also provides a prescription for the analytic continuation across physical thresholds. Our implementation is able to deliver 8 or more digits of precision, and has a built-in mechanism for checking the accuracy of the obtained results. Its efficacy is illustrated with state-of-the-art examples for three-loop self-energy and vertex integrals and two-loop box integrals