Schedule for: 24w5189 - Beyond Elliptic Polylogarithms

In this talk, the Brown-Levin construction of elliptic polylogarithms is generalized to Riemann surfaces of arbitrary genus. Homotopy-invariant iterated integrals on a higher-genus surface are generated from a flat connection with simple poles in two marked points. The integration kernels of the flat connection consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials. The closure of our higher-genus polylogarithms under taking primitives in marked points necessitates bilinear identities among the integration kernels that generalize the Fay identities among Kronecker-Eisenstein kernels at genus one. I will illustrate the derivation and use of higher-genus Fay identities, present all-weight formulae and comment on their echos on the meromorphic integration kernels of Enriquez.

I will discuss the recent progress on the computation of two-loop master integrals for leading-color ttH scattering amplitudes. In the first part, I will report on the recent calculation of master integrals involving closed light fermions loops. I will elaborate on the techniques used to obtain an epsilon factorized differential equation and its analytic form. We find novel analytic structures at the two-loop level that have not been encountered so far. In the second part of the talk, I will give an outlook with emphasis on the expected complexity for the remaining integral families with special emphasis on the elliptic sectors.

Feynman integrals for collider physics are known to contain intricate geometries and to evaluate to complicated transcendental numbers and functions. In this talk, I will first introduce Feynman integrals contributing to the emission of gravitational waves through the post-Minkowskian expansion. Then, using differential equations and the Baikov representation we identify and classify the geometries that appear order-by-order in the expansion, revealing a novel Calabi-Yau three-fold at four loops.

In this talk I present the linear interactions between the two metrics of bigravity derived for a generalized Kerr-Schild ansatz in the context of the classical double copy. By contracting the equations of motion using Killing vectors, we arrive to the single and zeroth copy equations corresponding to this family of solutions. The expressions for the stationary and time-dependent solutions are obtained, and by decoupling the time-dependent solutions, we recover a massless and a massive field whose mass is proportional to the Fierz-Pauli mass and depends on the coefficients of the interaction potential between the metrics. This result, which was previously documented in literature, is reinterpreted in the classical double copy framework, and some examples are presented.

One of the advantages of string perturbation theory over point-particle quantum field theory is that, for topological reasons, the proliferation of Feynman diagrams at high loop orders is much reduced. In principle, this property is inherited by the worldline formalism as the infinite string-tension limit of string theory. It allows one to derive integral representations summing up large numbers of Feynman diagrams. However, evaluating such integrals analytically leads to a non-standard integration problem for which existing integration tables, algorithms or algebraic manipulation programs are of litle help. I summarize the state-of-the-art of a long-term effort to develop methods specifically for this type of integrals, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.

I will present a new method for working out the implications of the hierarchical principle, which places strong restrictions on the analytic structure of Feynman integrals. This method does not require working out any algebraic blowups, and can be applied to integrals involving any configuration of massive or massless particles. The constraints that it allows us to derive hold to all orders in dimensional regularization.

The coaction for multiple polylogarithms played an important role in a better understanding of this class of functions, which ultimately was used in the computation of observables in different areas of theoretical physics. However, it is now well-known that at higher orders in perturbative quantum field theory special functions defined on geometries of higher dimension or higher genus show up. Most prominently elliptic multiple polylogarithms appear. Specific values of these functions are closely connected to iterated Eisenstein integrals. In this talk, I will discuss work in progress on a candidate-coaction for iterated Eisenstein integrals that acts on the generating series of these special functions and comment on the interplay of the candidate-coaction with modular transformations, derivatives and evaluations.

One of the most crucial ingredients in quantum field theory is the S matrix encompassing all the scattering information that can occur in the universe with each matrix element denoting a scattering process. Using perturbative methods, it is possible to derive Feynman diagrams from matrix elements. Over the years computing amplitudes of interactions by means of Feynman rules seemed irreplaceable but this point of view may change due to on-shell methods which includes helicity spinor formalism to describe massless particles, spin-spinors formalism to describe massive particles and BCFW recursion relations, just to name a few. In this talk we review the traditional method of calculating amplitudes using Feynman rules and compare it with those obtained through on-shell methods of some processes.

We consider the 5-mass kite family of self-energy Feynman integrals and present a systematic approach for constructing an eps-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the locations of relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be extracted from integrals over maximal cuts.

I will discuss a few examples of maximally cut integrals of elliptic and Calabi-Yau type. For such geometries we always have one distinguished holomorphic differential form and one distinguished integration cycle. I will describe the explicit geometric construction and the topology of this cycle in a few examples. Finally, I will discuss the construction of the remaining cycles in the middle homology, by using Picard-Lefschetz theory. Integration along these other cycles yield further "cuts" of the "maximal cut".

In this talk I will describe an algorithm to determine the minimal order differential equations associated to a given Feynman integral in dimensional or analytic regularisation. The algorithm is an extension of the Griffiths-Dwork pole reduction. After a brief review of the Griffiths-Dwork method I will present the features of its extension and give some examples of its use.

ε-factorised differential equations are needed to solve Feynman integrals order by order in ε. In this talk, I will first introduce geometries beyond multiple polylogarithms, and then discuss how to derive an ε-factorized differential equation for integrals with an underlying Calabi-Yau geometry. I will show some examples of where our method applies and I will finally conclude with some outlooks.

Recent developments in the study of quantum field theory amplitudes and gravitational wave physics have established the need for a generalization of elliptic multiple polylogarithms (eMPLs) to more complicated geometries. In this talk I will introduce a basis of hyperelliptic multiple polylogarithms (heMPLs), which allow to compute multiple integrals over rational functions on hyperelliptic curves, higher genus generalizations of elliptic curves. I will then motivate a geometric description using the Jacobian variety and Schottky parametrization and explain how the previously constructed basis can be translated into the geometric language. Finally I will give an outlook on ongoing work, applying the previous formalism to the maximal cut of the non-planar crossed box.