arXiv subjects

Vladimir A. Smirnov

Publications and source records attributed to Vladimir A. Smirnov.

At least 19 recordsLinked to original sources

arXiv2026

Five legs @ three loops: slightly off-shell dual conformal integrals

We calculate the three-loop master integrals contributing to the three-loop five-point amplitude on the special Coulomb branch of $\mathcal{N}=4$ SYM theory. For the genuine pentagon integrals, we follow the approach of Ref. [JHEP 12 (2025) 107], which includes a regularization preserving dual conformal invariance (DCI). As a new ingredient, we introduce a simple method, allowing to factor out the dependence on the DCI cross ratios from the contribution of each region. The remaining integrals are then essentially simplified by taking successive limits of vanishing external invariants. For 3 out of 82 regions contributing to the most complicated integral $\mathcal{I}_5^{(3)}$ we were not able to perform the integration even after these simplifications. For these three regions, we perform the integration-by-parts (IBP) reduction in parametric representation and evaluate the resulting locally finite integrals using HyperInt.

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arXiv2026

Five legs @ three loops: N=4 sYM amplitude near mass-shell

We present a three-loop analysis of the scattering amplitude of five nearly massless W-bosons in planar maximally supersymmetric Yang-Mills theory. The basis of the master integrals is established, making use of the unitarity-cut sewing technique in six-dimensional N=(1,1) super-Yang-Mills theory. Its dimensional reduction down to four allows us to generate masses for internal and external states. We descend on the special Coulomb branch of maximally supersymmetric Yang-Mills theory by setting all propagator masses to zero. Employing explicit expressions for all integrals that we calculated in a companion paper, we find a concise representation for this infrared-sensitive observable. We confirm its exponentiation, both for infrared and finite terms. The infrared double logarithm manifests the anticipated universality through the octagon anomalous dimension as its governing coefficient. Unlike our previous two-loop result, this consideration reveals that each of the three independent kinematic structures furnishing the amplitude possesses its own function of 't Hooft coupling.

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arXiv2026

Second-order self-force potential-region binary dynamics at $O(G^5)$ in supergravity

We compute the potential-graviton contributions to the conservative scattering angle of two non-spinning bodies in maximal supergravity at fifth order in Newton's constant, including second-order self-force effects. Our goal is to tackle the challenging integrals arising at this order in Einstein gravity, but within the technically simpler framework of supergravity. The calculation employs the scattering-amplitude framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. The final result is expressed as a series expansion around the static limit, thereby avoiding the explicit evaluation of intricate special functions. This series solution for the master integrals applies, as well, to the corresponding computation in general relativity. Remarkably, we observe nontrivial cancellations among contributions associated with Calabi-Yau integrals, alongside a distinct contribution governed by a Heun differential equation.

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arXiv2025

Amplitudes, Supersymmetric Black Hole Scattering at $\mathcal{O}(G^5)$, and Loop Integration

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in N = 8 supergravity at the fifth post-Minkowskian order to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present all master integrals and solve their associated differential equations necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in N = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in N = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

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arXiv2025

On multi-propagator angular integrals

We study multi-propagator angular integrals, a class of phase-space integrals relevant to processes with multiple observed final states and a test-bed for transferring loop-integral technology to phase space integrals without reversed unitarity. We present an Euler integral representation similar to Lee-Pomeransky representation and explicitly describe a recursive IBP reduction and dimensional shift relations for the general case of $n$ denominators. On the level of master integrals, applying a differential equation approach, we explicitly calculate the previously unknown angular integrals with four denominators for any number of masses to finite order in $\varepsilon$. Extending the idea of dimensional recurrence, we explore the decomposition of angular integrals into branch integrals reducing the number of scales in the master integrals from $(n+1)n/2$ to $n+1$. To showcase the potential of this method, we calculate the massless three denominator integral to establish all-order results in $\varepsilon$ including a resummation of soft logarithms.

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arXiv2024

Expansion by regions: an overview

A short review of expansion by regions is presented. It is a well-known strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in powers of a given small parameter. Prescriptions of this strategy are formulated in a simple geometrical language and are illustrated through simple examples.

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arXiv2024

Expansion by regions meets angular integrals

We study the small-mass asymptotic behavior of so-called angular integrals, appearing in phase-space calculations in perturbative quantum field theory. For this purpose we utilize the strategy of expansion by regions, which is a universal method both for multiloop Feynman integrals and various parametric integrals. To apply the technique to angular integrals, we convert them into suitable parametric integral representations, which are accessible to existing automation tools. We use the the code \texttt{asy.m} to reveal regions contributing to the asymptotic expansion of angular integrals. To evaluate the contributions of these regions in an epsilon expansion we apply the method of Mellin-Barnes representation. Our approach is checked against existing results on angular integrals revealing a connection between contributing regions and angular integrals constructed from an algebraic decomposition. We explicitly calculate the previously unknown asymptotics for angular integrals with three and four denominators and formulate a conjecture for the leading asymptotics and the pole part for a general number of denominators and masses.

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arXiv2023

Astrocyte control bursting mode of spiking neuron network with memristor-implemented plasticity

A mathematical model of a spiking neuron network accompanied by astrocytes is considered. The network is composed of excitatory and inhibitory neurons with synaptic connections supplied by a memristor-based model of plasticity. Another mechanism for changing the synaptic connections involves astrocytic regulations using the concept of tripartite synapses. In the absence of memristor-based plasticity, the connections between these neurons drive the network dynamics into a burst mode, as observed in many experimental neurobiological studies when investigating living networks in neuronal cultures. The memristive plasticity implementing synaptic plasticity in inhibitory synapses results in a shift in network dynamics towards an asynchronous mode. Next,it is found that accounting for astrocytic regulation in glutamatergic excitatory synapses enable the restoration of 'normal' burst dynamics. The conditions and parameters of such astrocytic regulation's impact on burst dynamics established.

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arXiv2023

Total Born cross section of $e^+e^-$-pair production by an electron in the Coulomb field of a nucleus

We calculate the total Born cross section of the $e^+e^-$-pair production by an electron in the field of a nucleus (trident process) using the modern multiloop methods. For general energies we obtain the cross section in terms of converging power series. The threshold asymptotics and the high-energy asymptotics are obtained analytically. In particular, we obtain additional contribution to the Racah formula due to the identity of the final electrons. Besides, our result for the leading term of the high-energy asymptotics reveals a typo in an old Racah paper [Racah1937].

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arXiv2023

Master Integrals for Four-Loop Massless Form Factors

We present analytical results for all master integrals for massless three-point functions, with one off-shell leg, at four loops. Our solutions were obtained using differential equations and direct integration techniques. We review the methods and provide additional details.

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arXiv2023

Conservative binary dynamics at order $O(α^5)$ in electrodynamics

We compute the potential-photon contributions to the classical relativistic scattering angle of two charged non-spinning bodies in electrodynamics through fifth order in the coupling. We use the scattering amplitudes framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. At fifth order, the result is expressed in terms of cyclotomic polylogarithms. Our calculation demonstrates the feasibility of the corresponding calculations in general relativity, including the evaluation of the encountered four-loop integrals.

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arXiv2022

The $Hb\bar{b}$ vertex at four loops and hard matching coefficients in SCET for various currents

We compute the four-loop corrections to the Higgs-bottom vertex within massless QCD and present analytic results for all color structures. The infrared poles of the renormalized form factor agree with the predicted four-loop pattern. Furthermore, we use the results for the Higgs-bottom, photon-quark, and Higgs-gluon form factors to provide hard matching coefficients in soft-collinear effective theory up to four-loop accuracy.

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arXiv2022

Quark and gluon form factors in four-loop QCD

We compute the photon-quark and Higgs-gluon form factors to four-loop order within massless perturbative Quantum Chromodynamics. Our results constitute ready-to-use building blocks for N${}^4$LO cross sections for Drell-Yan processes and gluon-fusion Higgs boson production at the LHC. We present complete analytic expressions for both form factors and show several of the most complicated master integrals.

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arXiv2021

The Four-Loop $\mathcal{N}=4$ SYM Sudakov Form Factor

We present the Sudakov form factor in full color $\mathcal{N}=4$ supersymmetric Yang-Mills theory to four loop order and provide uniformly transcendental results for the relevant master integrals through to weight eight.

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arXiv2021

Analytic results for two-loop planar master integrals for Bhabha scattering

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equa- tions with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic mul- tiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

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arXiv2021

Fermionic corrections to quark and gluon form factors in four-loop QCD

We analytically compute all four-loop QCD corrections to the photon-quark and Higgs-gluon form factors involving a closed massless fermion loop. Our calculation of non-planar vertex integrals confirms a previous conjecture for the analytical form of the non-fermionic contributions to the collinear anomalous dimensions of quarks and gluons.

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arXiv2021

Glue-and-Cut at Five Loops

We compute epsilon-expansions around 4 dimensions of a complete set of master integrals for momentum space five-loop massless propagator integrals in dimensional regularization, up to and including the first order with contributions of transcendental weight nine. Our method is the glue-and-cut technique from Baikov and Chetyrkin, which proves extremely effective in that it determines all expansion coefficients to this order in terms of recursively one-loop integrals and only one further integral. We observe that our results are compatible with conjectures that predict $π$-dependent contributions.

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arXiv2020

Two-loop mixed QCD-EW corrections to $gg \to Hg$

We compute the two-loop mixed QCD-Electroweak (QCD-EW) corrections to the production of a Higgs boson and a gluon in gluon fusion. The relevant four-point functions with internal massive propagators are expressed as multiple polylogarithms with algebraic arguments. We perform the calculation by integration over Feynman parameters and, independently, by the method of differential equations. We compute the two independent helicity amplitudes for the process and we find that they are both finite. Moreover, we observe a weight drop when all gluons have the same helicity. We also provide a simplified expression for the all-plus helicity amplitude, which is optimised for fast and reliable numerical evaluation in the physical region.

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