We present a method for high-precision numerical evaluations of Lauricella functions whose indices are linearly dependent on some parameter <italic>&#949;</italic> in terms of their Laurent series expansions at zero. This method is based on finding analytic continuations of these functions in terms of Frobenius generalized power series. Being one-dimensional, these series are much more suited for high-precision numerical evaluations than multi-dimensional sums arising in approaches to analytic continuations based on re-expansions of hypergeometric series or Mellin&#8211;Barnes integral representations. To accelerate the calculation procedure further, the <italic>&#949;</italic> dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of <italic>&#949;</italic>, which, in addition, allows for efficient parallel implementation. The method has been implemented in the <monospace>PrecisionLauricella</monospace> package, written in Wolfram Mathematica language.

Germany

M.A. Bezuglov

B.A. Kniehl

Russia

A.I. Onishchenko

O.L. Veretin

High-precision numerical evaluation of Lauricella functions

We present updated results for the wave function normalization constants and the first moments of the light-cone distribution amplitudes for the lowest-lying baryon octet. The analysis is carried out on a large number of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>n</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> lattice gauge ensembles, including ensembles at physical pion (and kaon) masses. These are spread across five different lattice spacings, enabling a controlled continuum limit. The main differences with respect to our earlier work [G. S. Bali et al. (RQCD Collaboration), Light-cone distribution amplitudes of octet baryons from lattice QCD, <a href="http://dx.doi.org/10.1140/epja/i2019-12803-6">Eur. Phys. J. A 55, 116 (2019)</a>.] are the use of two-loop conversion factors to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>MS</mi><mo stretchy="true">¯</mo></mover></math>-like scheme and of an updated set of low-energy constants in the parametrization of the quark mass dependence. As a byproduct, for the first time, the anomalous dimensions for local leading-twist three-quark operators with one derivative are obtained.

G.S. Bali

V.M. Braun

S. Bürger

M. Göckeler

M. Gruber

F. Kaiser

P. Wein

Updated determination of light-cone distribution amplitudes of octet baryons in lattice QCD

We consider the anomalous dimension of the flavor nonsinglet twist-two quark operator of arbitrary Lorentz spin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> at four loops in QCD and construct its contribution proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> in analytic form by applying advanced methods of number theory on the available knowledge of low-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> moments. In conjunction with similar results on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></math> contributions, this completes our knowledge of the transcendental part of the considered anomalous dimension. This also provides important constraints on the as-yet elusive all-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> form of the rational part. Via Mellin transformation, we thus obtain the exact functional form in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> of the respective piece of the nonsinglet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi splitting function at four loops. This allows us to appreciably reduce the theoretical uncertainty in the approximation of that splitting function otherwise amenable from the first few low-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> moments.

United States

V.N. Velizhanin

Four-Loop Anomalous Dimension of Flavor Nonsinglet Twist-Two Operator of General Lorentz Spin in QCD: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math> Term

We consider the nonplanar universal anomalous dimension of twist-two operators at four loops in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></math> supersymmetric Yang-Mills theory and push its direct diagrammatic calculation through Lorentz spin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi><mo>=</mo><mn>20</mn></math>, one unit beyond the state of the art, so as to confirm the correctness of the general, all-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math> result conjectured previously by us [Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> SYM theory, <a href="http://dx.doi.org/10.1016/j.nuclphysb.2021.115429">Nucl. Phys. B968, 115429 (2021)</a>.] imposing certain constraints on its analytic form. Thanks to our new result, such constraints can be eliminated altogether. By the same token, this allows us to rederive, in a completely independent way, the nonplanar four-loop cusp anomalous dimension by taking the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math> limit of the general result.

Nonplanar four-loop anomalous dimensions of twist-two operators in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> supersymmetric Yang-Mills theory: Higher moment, general result, and cusp anomalous dimension

We perform direct diagrammatic calculations of the anomalous dimensions of twist-two operators in extended <math><mi>N</mi><mo>=</mo><mn>2</mn></math> and <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theories (SYM). In the case of <math><mi>N</mi><mo>=</mo><mn>4</mn></math> SYM, we compute the four-loop anomalous dimension of the twist-two operator for several fixed values of Lorentz spin. This is the first direct diagrammatic calculation of this kind, and we confirm results previously obtained by means of integrability. For <math><mi>N</mi><mo>=</mo><mn>2</mn></math> SYM, we obtain the general result for the anomalous dimension at third order of perturbation theory and find the three-loop Cusp anomalous dimension.

Anomalous dimensions of twist-two operators in extended And super Yang-Mills theories

We compute the non-planar contribution to the universal anomalous dimension of twist-two operators in <math><mi>N</mi><mo>=</mo><mn>4</mn></math> supersymmetric Yang-Mills theory at four loops through Lorentz spin eighteen. Exploiting the results of this and our previous calculations along with recent analytic results for the cusp anomalous dimension and some expected analytic properties, we reconstruct a general expression valid for arbitrary Lorentz spin. We study various properties of this general result, such as its large-spin limit, its small-x limit, and others. In particular, we present a prediction for the non-planar contribution to the anomalous dimension of the single-magnon operator in the β-deformed version of the theory.

Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in SYM theory

We consider two-loop two-, three-, and four-point diagrams with elliptic subgraphs involving two different masses, m and M. Such diagrams generally arise in matching procedures within nonrelativistic QCD and QED and are relevant, e.g., for top-quark pair production at threshold and parapositronium decay. We present the obtained results in several different representations: series solution with binomial coefficients, integral representation, and representation in terms of generalized hypergeometric functions. The results are valid up to terms of <math><mi>O</mi><mo>(</mo><mi>ε</mi><mo>)</mo></math> in <math><mi>d</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>2</mn><mi>ε</mi></math> space-time dimensions. In the limit of equal masses, <math><mi>m</mi><mo>=</mo><mi>M</mi></math>, the obtained results are written in terms of elliptic constants with explicit series representations.

JINR

A.V. Kotikov

Two-loop diagrams in nonrelativistic QCD with elliptics

We consider electroweak corrections to the relation between the running MS‾ mass mb of the b quark in the five-flavor QCD×QED effective theory and its counterpart in the Standard Model (SM). As a bridge between the two parameters, we use the pole mass Mb of the b quark, which can be calculated in both models. The running mass is not a fundamental parameter of the SM Lagrangian, but the product of the running Yukawa coupling yb and the Higgs vacuum expectation value. Since there exist different prescriptions to define the latter, the relations considered in the paper involve a certain amount of freedom. All the definitions can be related to each other in perturbation theory. Nevertheless, we argue in favour of a certain gauge-independent prescription and provide a relation which can be directly used to deduce the value of the Yukawa coupling of the b quark at the electroweak scale from its effective QCD running mass. This approach allows one to resum large logarithms ln⁡(mb/Mt) systematically. Numerical analysis shows that, indeed, the corrections to the proposed relation are much smaller than those between yb and Mb .

A.V. Bednyakov

A.F. Pikelner

On the b -quark running mass in QCD and the SM

We calculate the full one-loop electroweak radiative corrections, of O(α2αs) , to the cross section of single Z -boson inclusive hadroproduction at finite transverse momentum ( pT ). This includes the O(α) corrections to Z+j production, the O(αs) corrections to Z+γ production, and certain QCD-electroweak interference contributions involving a single quark trace. We recover the QCD and purely weak corrections and study the QED corrections and the QCD-electroweak interference contributions for the first time. We also consider direct and resolved photoproduction in elastic and inelastic scattering. We present pT and rapidity distributions for the experimental conditions at the Fermilab Tevatron and the CERN LHC and assess the significance of the various contributions.

W. Hollik

E.S. Scherbakova

Electroweak corrections to Z -boson hadroproduction at finite transverse momentum

We calculate the cross section for the inclusive production of B mesons in pp and pp¯ collisions at next-to-leading order in the general-mass variable-flavor-number scheme and show that a suitable choice of factorization scales leads to a smooth transition to the fixed-flavor-number scheme. Our numerical results are in good agreement with data from the Tevatron and LHC experiments at small and at large transverse momenta.