arXiv subjects

Stefan Weinzierl

Publications and source records attributed to Stefan Weinzierl.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the electron self-energy to three loops in QED

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 $ε$-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 $ε$-resummed results for the self-energy in the on-shell limit $p^2 = m^2$, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

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arXiv2024

The electron self-energy in QED at two loops revisited

We reconsider the two-loop electron self-energy in quantum electrodynamics. We present a modern calculation, where all relevant two-loop integrals are expressed in terms of iterated integrals of modular forms. As boundary points of the iterated integrals we consider the four cases $p^2=0$, $p^2=m^2$, $p^2=9m^2$ and $p^2=\infty$. The iterated integrals have $q$-expansions, which can be used for the numerical evaluation. We show that a truncation of the $q$-series to order ${\mathcal O}(q^{30})$ gives numerically for the finite part of the self-energy a relative precision better than $10^{-20}$ for all real values $p^2/m^2$.

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arXiv2024

The H-graph with unequal masses in quantum field theory

We compute the family of Feynman integrals related to the H-graph with unequal masses in relativistic quantum field theory. We present an $\varepsilon$-factorised 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 four in terms of multiple polylogarithms in compact form.

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arXiv2024

The H-graph with equal masses in terms of multiple polylogarithms

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.

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arXiv2024

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

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

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arXiv2024

On $\varepsilon$-factorised bases and pure Feynman integrals

We investigate $\varepsilon$-factorised differential equations, uniform transcendental weight and purity for Feynman integrals. We are in particular interested in Feynman integrals beyond the ones which evaluate to multiple polylogarithms. We show that a $\varepsilon$-factorised differential equation does not necessarily lead to Feynman integrals of uniform transcendental weight. We also point out that a proposed definition of purity works locally, but not globally.

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arXiv2023

Recent developments from Feynman integrals

This talk reviews recent developments in the field of analytical Feynman integral calculations. The central theme is the geometry associated to a given Feynman integral. In the simplest case this is a complex curve of genus zero (aka the Riemann sphere). In this talk we discuss Feynman integrals related to more complicated geometries like curves of higher genus or manifolds of higher dimensions. In the latter case we encounter Calabi-Yau manifolds. We also discuss how to compute these Feynman integrals.

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arXiv2023

Lattice Correlation Functions from Differential Equations

We discuss how methods developed in the context of perturbation theory can be applied to the computation of lattice correlation functions, in particular in the non perturbative regime. The techniques we consider are integration-by-parts identities (supplemented with symmetry relations) and the method of differential equations, cast in the framework of twisted co-homology. We report on calculations of correlation functions for a scalar $λϕ^4$ theory and lattices of small size, both in Euclidean and Minkowskian signature.

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arXiv2023

Feynman integrals, geometries and differential equations

In this talk we discuss the construction of a basis of master integrals for the family of the $l$-loop equal-mass banana integrals, such that the differential equation is in an $\varepsilon$-factorised form. As the $l$-loop banana integral is related to a Calabi-Yau $(l-1)$-fold, this extends the examples where an $\varepsilon$-factorised form has been found from Feynman integrals related to curves (of genus zero and one) to Feynman integrals related to higher-dimensional varieties.

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arXiv2023

Genus Drop in Hyperelliptic Feynman Integrals

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 $g$ to $\lceil g/2 \rceil$ or $\lfloor g/2 \rfloor$, which can be checked for algorithmically. We use this algorithm to find further instances of genus drop in Feynman integrals.

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arXiv2023

A Database with Directivities of Musical Instruments

We present a database of recordings and radiation patterns of individual notes for 41 modern and historical musical instruments, measured with a 32-channel spherical microphone array in anechoic conditions. In addition, directivities averaged in one-third octave bands have been calculated for each instrument, which are suitable for use in acoustic simulation and auralisation. The data are provided in SOFA format. Spatial upsampling of the directivities was performed based on spherical spline interpolation and converted to OpenDAFF and GLL format for use in room acoustic and electro-acoustic simulation software. For this purpose, a method is presented how these directivities can be referenced to a specific microphone position in order to achieve a physically correct auralisation without colouration. The data is available under the CC BY-SA 4.0 licence.

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arXiv2023

Real time lattice correlation functions from differential equations

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 $1/\sqrtλ$, 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.

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arXiv2023

Bananas of equal mass: any loop, any order in the dimensional regularisation parameter

We describe a systematic approach to cast the differential equation for the $l$-loop equal mass banana integral into an $\varepsilon$-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 $\varepsilon$ for any loop $l$. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

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arXiv2023

Magnitude-Corrected and Time-Aligned Interpolation of Head-Related Transfer Functions

Head-related transfer functions (HRTFs) are essential for virtual acoustic realities, as they contain all cues for localizing sound sources in three-dimensional space. Acoustic measurements are one way to obtain high-quality HRTFs. To reduce measurement time, cost, and complexity of measurement systems, a promising approach is to capture only a few HRTFs on a sparse sampling grid and then upsample them to a dense HRTF set by interpolation. However, HRTF interpolation is challenging because small changes in source position can result in significant changes in the HRTF phase and magnitude response. Previous studies greatly improved the interpolation by time-aligning the HRTFs in preprocessing, but magnitude interpolation errors, especially in contralateral regions, remain a problem. Building upon the time-alignment approaches, we propose an additional post-interpolation magnitude correction derived from a frequency-smoothed HRTF representation. Employing all 96 individual simulated HRTF sets of the HUTUBS database, we show that the magnitude correction significantly reduces interpolation errors compared to state-of-the-art interpolation methods applying only time alignment. Our analysis shows that when upsampling very sparse HRTF sets, the subject-averaged magnitude error in the critical higher frequency range is up to 1.5 dB lower when averaged over all directions and even up to 4 dB lower in the contralateral region. As a result, the interaural level differences in the upsampled HRTFs are considerably improved. The proposed algorithm thus has the potential to further reduce the minimum number of HRTFs required for perceptually transparent interpolation.

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arXiv2023

Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral

Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these integrals can be computed with the method of differential equations. The four-loop equal-mass banana integral is the simplest Feynman integral whose geometry is a non-trivial Calabi-Yau manifold. We show that its differential equation can be cast into an $\varepsilon$-factorised form. This allows us to obtain the solution to any desired order in the dimensional regularisation parameter $\varepsilon$. The method generalises to other Calabi-Yau Feynman integrals. Our calculation also shows that the four-loop banana integral is only minimally more complicated than the corresponding Feynman integrals at two or three loops.

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arXiv2022

Theory Techniques for Precision Physics -- Snowmass 2021 TF06 Topical Group Report

The wealth of experimental data collected at laboratory experiments suggests that there is some scale separation between the Standard Model (SM) and phenomena beyond the SM (BSM). New phenomena can manifest itself as small corrections to SM predictions, or as signals in processes where the SM predictions vanish or are exceedingly small. This makes precise calculations of the SM expectations essential, in order to maximize the sensitivity of current and forthcoming experiments to BSM physics. This topical group report highlights some past and forthcoming theory developments critical for maximizing the sensitivity of the experimental program to understanding Nature at the shortest distances.

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arXiv2022

Non-perturbative computation of lattice correlation functions by differential equations

We show that methods developed in the context of perturbative calculations can be transferred to non-perturbative calculations. We demonstrate that correlation functions on the lattice can be computed with the method of differential equations, supplemented with techniques from twisted cohomology. We derive differential equations for the variation with the coupling or -- more generally -- with the parameters of the action. Already simple examples show that the differential equation with respect to the coupling has an essential singularity at zero coupling and a regular singularity at infinite coupling. The properties of the differential equation at zero coupling can be used to prove that the perturbative series is only an asymptotic series.

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arXiv2022

Three-loop master integrals for the Higgs boson self-energy with internal top-quarks and W-bosons

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 ${\mathcal O}(α^2 α_s)$. 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 $\varepsilon$-factorised form. The occurring square roots are rationalised and all master integrals are expressible in multiple polylogarithms.

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