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Abstract 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 3 $$ \sqrt{3} $$ . The new master integrals are then again related by a Galois symmetry, for example the substitution 3 $$ \sqrt{3} $$ → − 3 $$ -\sqrt{3} $$ . 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.

Abstract The Moller experiment and the P2 experiment aim at measuring the weak mixing angle at low scales. The Moller experiment uses $$e^- e^- \rightarrow e^- e^-$$ e - e - → e - e - -scattering, the P2 experiment uses $$e^- N \rightarrow e^- N$$ e - N → e - N -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.

This paper proposes a linear algebra-based supervised machine learning algorithm for the symbolic representation of arbitrarily non-linear and recursive systems. It introduces multiple extensions to the algorithmic class of “Forward Orthogonal Least Squares Regressions” (FOrLSR), which performs dictionary-based sparse symbolic regressions. The regression, being only provided with the system’s input and output, performs variable combinations and non-linear transformations from a given dictionary of analytic expressions and selects the optimal ones to represent the unknown system. This yields a “symbolic” system representation, having the minimum number of terms to enforce sparsity ( $L_{0}$ -norm), while keeping the highest possible precision. The first proposed algorithm (rFOrLSR) restructures the FOrLSR to be in matrix form (for large scale GPU and BLAS-like optimizations), recursive (to reduce the computational complexity from quadratic in model length to linear) and allows regressors to be imposed (to include user expertise and perform tree-searches). Furthermore, the dictionary search is restructured into a breadth-first arborescence traversal kept sparse by five proposed theorems, four corollaries and one pruning mechanism, while adding a validation procedure for the final model selection. The proposed arborescence (AOrLSR) scans large search-space segments, significantly increasing the probability of finding an optimal system representation, while only computing a marginal fraction of the search-space. The regression and arborescence are solvers for arbitrarily-determined linear equation systems which maximize sparsity in the solution vectors.

Solar neutrinos have played a central role in the discovery of the neutrino oscillation mechanism. They still are proving to be a unique tool to help investigate the fusion reactions that power stars and further probe basic neutrino properties. The Borexino neutrino observatory has been operationally acquiring data at Laboratori Nazionali del Gran Sasso in Italy since 2007. Its main goal is the real-time study of low energy neutrinos (solar or originated elsewhere, such as geo-neutrinos). The latest analysis of experimental data, taken during the so-called Borexino Phase-II (2011-present), will be showcased in this talk—yielding new high-precision, simultaneous wide band flux measurements of the four main solar neutrino components belonging to the “pp„ fusion chain (pp, pep, 7 Be, 8 B), as well as upper limits on the remaining two solar neutrino fluxes (CNO and hep).

Abstract We report on an exact calculation of lattice correlation functions on a finite four-dimensional lattice with either Euclidean or Minkowskian signature. The lattice correlation functions are calculated by the method of differential equations. This method can be used for Euclidean and Minkowskian signature alike. The lattice correlation functions have a power series expansion in 1/ λ $$ \sqrt{\lambda } $$ , where λ is the coupling. We show that this series is convergent for all non-zero values of λ. At small coupling we quantify the accuracy of perturbative approximations. At the technical level we systematically investigate the interplay between twisted cohomology and the symmetries of the twist function.

Abstract We describe a systematic approach to cast the differential equation for the l-loop equal mass banana integral into an ε-factorised form. With the known boundary value at a specific point we obtain systematically the term of order j in the expansion in the dimensional regularisation parameter ε for any loop l. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

Abstract We consider the full set of master integrals with internal top-and W-propagators contributing to the three-loop Higgs self-energy diagrams of order O $$ \mathcal{O} $$ (α 2 α s ). We split the master integrals into a system relevant to the Feynman diagrams proportional to the product of Yukawa couplings y b y t and the complement. For both systems we define master integrals of uniform weight, such that the associated differential equation is in ε-factorised form. The occurring square roots are rationalised and all master integrals are expressible in multiple polylogarithms.

Abstract We show that the differential equation for the three-loop equal-mass banana integral can be cast into an ε-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The ε-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter ε. The alphabet of the iterated integrals contains six letters.

Abstract We study a two-loop four-point function with one internal mass. This Feynman integral is one of the simplest Feynman integrals depending on two elliptic curves. We transform the associated differential equation into an ε-form. We study the entries of the differential equation, and in particular the entries which depend on both elliptic curves.

Abstract Measurements of the directivity of acoustic sound sources must be interpolated in almost all cases, either for spatial upsampling to higher resolution representations of the data, for spatial resampling to another sampling grid, or for use in simulations of sound propagation. The performance of different interpolation techniques applied to sparsely sampled directivity measurements depends on the sampling grid used but also on the radiation pattern of the sources themselves. Therefore, we evaluated three established approaches for interpolation from a low-resolution sampling grid using high-resolution measurements of a representative sample of musical instruments as a reference. The smallest global error on average occurs for thin plate pseudo-spline interpolation. For interpolation based on spherical harmonics (SH) decomposition, the SH order and the spatial sampling scheme applied have a strong and difficult to predict influence on the quality of the interpolation. The piece-wise linear, spherical triangular interpolation provides almost as good results as the first-order spline approach, albeit with on average 20 times higher computational effort. Therefore, for spatial interpolation of sparsely sampled directivity measurements of musical instruments, the thin plate pseudo-spline method applied to absolute-valued data is recommended and, if necessary, a subsequent modeling of the phase.

Abstract We analyse the dependence of the peak position of the thrust distribution on the cutoff value in the Nagy–Soper dipole shower. We compare the outcome of the parton shower simulations to a relation of the dependence from an analytic computation, derived within soft-collinear effective theory. We show that the result of the parton shower simulations and the analytic computation are in good agreement.

The initial phase of the inspiral process of a binary system producing gravitational waves can be described by perturbation theory. At the third post-Minkowskian order a two-loop double box graph, known as H-graph contributes. We consider the case where the two objects making up the binary system have equal masses. We express all master integrals related to the equal-mass H-graph up to weight four in terms of multiple polylogarithms. We provide a numerical program which evaluates all master integrals up to weight four in the physical regions with arbitrary precision.

Abstract We present the two-loop master integrals relevant to the O $$ \mathcal{O} $$ (αα s )-corrections to the decay H → b b ¯ $$ b\overline{b} $$ through a H t t ¯ $$ Ht\overline{t} $$ -coupling. We keep the full dependence on the heavy particle masses, but neglect the b-quark mass. The occurring square roots can be rationalised and the result is expressed in terms of multiple polylogarithms.

Sound and music are well-studied aspects of the quality of experience in restaurants; the role of the room acoustical conditions, their influence on the visitors’ soundscape evaluation and their impact on the overall customer satisfaction in restaurants, however, has received less scientific attention. The present field study therefore investigated whether sound pressure level, reverberation time, and soundscape pleasantness can predict factors associated with overall restaurant quality. In total, 142 persons visiting 12 restaurants in Berlin rated relevant acoustical and non-acoustical factors associated with restaurant quality. Simultaneously, the A-weighted sound pressure level (LA,eq,15) was measured, and the reverberation time in the occupied state (T20,occ) was obtained by measurements performed in the unoccupied room and a subsequent calculation of the occupied condition according to DIN 18041. Results from linear mixed-effects models revealed that both the LA,eq,15 and T20,occ had a significant influence on soundscape pleasantness and eventfulness, whereby the effect of T20,occ was meditated by the LA,eq,15. Also, the LA,eq,15 as well as soundscape pleasantness were significant predictors of overall restaurant quality. A comprehensive structural equation model including both acoustical and non-acoustical factors, however, indicates that the effect of soundscape pleasantness on overall restaurant quality is mediated by the restaurant’s atmosphere. Our results support and extend previous findings which suggest that the acoustical design of restaurants involves a trade-off between comfort and liveliness, depending on the desired character of the place.

We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transformation in the fibre. Only in the combination of the two transformations we stay within the same class of functions.

We study linear relations among correlation functions on a lattice obtained from integration-by-parts identities. We use the framework of twisted cocycles and determine for a scalar theory a basis of correlation functions, in which all other correlation functions may be expressed.

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter ε. In order to do so, we transform the system of differential equations for the master integrals to an ε-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M1,3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M‾1,3. On the hypersurface τ=const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

Abstract In this article we give the details on the analytic calculation of the master integrals for the planar double box integral relevant to top-pair production with a closed top loop. We show that these integrals can be computed systematically to all order in the dimensional regularisation parameter ε. This is done by transforming the system of differential equations into a form linear in ε, where the ε 0-part is a strictly lower triangular matrix. Explicit results in terms of iterated integrals are presented for the terms relevant to NNLO calculations.

Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.

Abstract We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms — the scattering forms — on the moduli space of a Riemann sphere with n marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor ℳ ¯ 0 , n \ ℳ 0 , n $$ {\overline{\mathrm{\mathcal{M}}}}_{0,n}\backslash {\mathrm{\mathcal{M}}}_{0,n} $$ . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.

We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of t∈R. Furthermore, the nome q of the elliptic curve satisfies over the complete range in t the inequality |q|≤1, where |q|=1 is attained only at the singular points t∈{m2,9m2,∞}. This ensures the convergence of the q-series expansion of the ELi-functions and provides a fast and efficient evaluation of these Feynman integrals.

We show that a recently discovered relation, which expresses tree-level single trace Einstein–Yang–Mills amplitudes with one graviton and (n−1) gauge bosons as a linear combination of pure Yang–Mills tree amplitudes with n gauge bosons, can be derived from the CHY representation. In addition we show that there is a generalisation, which expresses tree-level single trace Einstein–Yang–Mills amplitudes with r gravitons and (n−r) gauge bosons as a linear combination of pure Yang–Mills tree amplitudes with n gauge bosons. We present a general formula for this case.