We compute the three-loop banana integral with four unequal masses in dimensional regularisation. This integral is associated to a family of K3 surfaces, thus representing an example for Feynman integrals with geometries beyond elliptic curves. We evaluate the integral by deriving an $\textit{ε}$-factorised differential equation, for which we rely on the algorithm presented in a recent publication [1]. Equipping the space of differential forms in Baikov representation by a set of filtrations inspired by Hodge theory, we first obtain a differential equation with entries as Laurent polynomials in $\textit{ε}$. Via a sequence of basis rotations we then remove any non-$\textit{ε}$-factorising terms. This procedure is algorithmic and at no point relies on prior knowledge of the underlying geometry.

Switzerland

Sebastian Pögel

Germany

Toni Teschke

China

Xing Wang

Stefan Weinzierl

The unequal-mass three-loop banana integral

Feynman integrals are very often computed from their differential equations. It is not uncommon that the $\textit{ε}$-factorised differential equation contains only dlog-forms with algebraic arguments, where the algebraic part is given by (multiple) square roots. It is well-known that if all square roots are simultaneously rationalisable, the Feynman integrals can be expressed in terms of multiple polylogarithms. This is a sufficient, but not a necessary criterium. In this paper we investigate weaker requirements. We discuss under which conditions we may use different rationalisations in different parts of the calculation. In particular we show that we may use different rationalisations if they correspond to different parameterisations of the same integration path. We present a non-trivial example — the one-loop pentagon function with three adjacent massive external legs involving seven square roots — where this technique can be used to express the result in terms of multiple polylogarithms.

United Kingdom

Georgios Papathanasiou

Konglong Wu

Yang Zhang

Rationalisation of multiple square roots in Feynman integrals

We study a recently identified four-loop Feynman integral that contains a three-dimensional Calabi-Yau geometry and contributes to the scattering of black holes in classical gravity at fifth post-Minkowskian and second self-force order (5PM 2SF) in the conservative sector. In contrast to previously studied Calabi-Yau Feynman integrals, the higher-order differential equation that this integral satisfies in dimensional regularization exhibits $\textit{ε}$-dependent apparent singularities. We introduce an appropriate ansatz which allows us to bring such cases into an $\textit{ε}$-factorized form. As a proof of principle, we apply it to the integral at hand.

Denmark

Hjalte Frellesvig

Roger Morales

Matthias Wilhelm

Calabi-Yau Feynman integrals in gravity: $\textit{ε}$-factorized form for apparent singularities

It has long been known that the maximal cut of the equal-mass four-loop banana integral is a period of a family of Calabi-Yau threefolds that depends on the kinematic variable $\textit{z}$ = $\textit{m}$ /$\textit{p}$ . We show that it can also be interpreted as a period of a family of genus-two curves. We do this by introducing a general Calabi-Yau-to-curve correspondence, which in this case locally relates the original period of the family of Calabi-Yau threefolds to a period of a family of genus-two curves that varies holomorphically with the kinematic variable $\textit{z}$. In addition to working out the concrete details of this correspondence for the equal-mass four-loop banana integral, we outline when we expect a correspondence of this type to hold.

Hans Jockers

Sören Kotlewski

Pyry Kuusela

Andrew McLeod

Maik Sarve

A Calabi-Yau-to-curve correspondence for Feynman integrals

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an $\textit{ϵ}$-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by-product, we also obtain $\textit{ϵ}$-resummed results for the self-energy in the on-shell limit $\textit{p}$ = $\textit{m}$ , which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Claude Duhr

Federico Gasparotto

Christoph Nega

Lorenzo Tancredi

On the electron self-energy to three loops in QED

We compute the family of Feynman integrals related to the H graph with unequal masses in relativistic quantum field theory. We present an <math><mi>ϵ</mi></math>-factorized differential equation for the 40 master integrals. The alphabet consists of 29 dlog forms with algebraic arguments, involving six square roots. As these square roots are not simultaneously rationalizable, we express the master integrals in terms of iterated integrals. In addition, we express the H graph with unit powers of the propagators up to weight 4 in terms of multiple polylogarithms in compact form.

Philipp Alexander Kreer

H graph with unequal masses in quantum field theory

It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper we show that such sectors may have additional symmetries. First of all, self-duality, which was first observed in Feynman integrals related to Calabi-Yau geometries, often carries over to non-Calabi-Yau Feynman integrals. Secondly, we show that in addition there can exist Galois symmetries relating integrals. In the simplest case of two master integrals within a sector, whose definition involves a square root r, we may choose a basis (I 1, I 2) such that I 2 is obtained from I 1 by the substitution r → −r. This pattern also persists in sectors, which a priori are not related to any square root with dependence on the kinematic variables. We show in several examples that in such cases a suitable redefinition of the integrals introduces constant square roots like   <math> <msqrt> <mn>3</mn> </msqrt> </math>  $$ \sqrt{3} $$ . The new master integrals are then again related by a Galois symmetry, for example the substitution   <math> <msqrt> <mn>3</mn> </msqrt> </math>  $$ \sqrt{3} $$  →   <math> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math>  $$ -\sqrt{3} $$ . To handle the case where the argument of a square root would be a perfect square we introduce a limit Galois symmetry. Both self-duality and Galois symmetries constrain the differential equation.

Xiaofeng Xu

Self-dualities and Galois symmetries in Feynman integrals

The Moller experiment and the P2 experiment aim at measuring the weak mixing angle at low scales. The Moller experiment uses  $$e^- e^- \rightarrow e^- e^-$$  <math> <mrow> <msup> <mi>e</mi> <mo>-</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> <mo>→</mo> <msup> <mi>e</mi> <mo>-</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> </mrow> </math>  -scattering, the P2 experiment uses  $$e^- N \rightarrow e^- N$$  <math> <mrow> <msup> <mi>e</mi> <mo>-</mo> </msup> <mi>N</mi> <mo>→</mo> <msup> <mi>e</mi> <mo>-</mo> </msup> <mi>N</mi> </mrow> </math>  -scattering. In both cases, two-loop electroweak corrections have to be taken into account, and here in particular diagrams which give rise to large logarithms. In this paper we compute a set of two-loop electroweak Feynman integrals for point-like particles, which are obtained from a box integral by the insertion of a light fermion loop. By rationalising all occurring square roots we show that these Feynman integrals can be expressed in terms of multiple polylogarithms. We present the results in a form, which makes the large logarithms manifest. We provide highly efficient numerical evaluation routines for these integrals.

Nico Böttcher

Niklas Schwanemann

Box integrals with fermion bubbles for low-energy measurements of the weak mixing angle

The maximal cut of the nonplanar crossed box diagram with all massive internal propagators was long ago shown to encode a hyperelliptic curve of genus 3 in momentum space. Surprisingly, in Baikov representation, the maximal cut of this diagram only gives rise to a hyperelliptic curve of genus 2. To show that these two representations are in agreement, we identify a hidden involution symmetry that is satisfied by the genus 3 curve, which allows it to be algebraically mapped to the curve of genus 2. We then argue that this is just the first example of a general mechanism by means of which hyperelliptic curves in Feynman integrals can drop from genus <math><mi>g</mi></math> to <math><mo>⌈</mo><mi>g</mi><mo>/</mo><mn>2</mn><mo>⌉</mo></math> or <math><mo>⌊</mo><mi>g</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math>. We find an algorithm to test for the presence of genus drop, and highlight further instances of this mechanism in Feynman integrals.

Robin Marzucca

CERN

United States

Andrew J. McLeod

Ben Page

Genus drop in hyperelliptic Feynman integrals

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.