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

Germany

B.A. Kniehl

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.

Russia

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

We present a simple representation for analytically continued nested harmonic sums for the arbitrary complex arguments. This representation can be obtained for a wide range of nested harmonic sums from a precomputed database for the pole expressions of these sums near negative integers. We describe the procedure for the precise numerical evaluations of the corresponding results from the dispersion representation.

Analytic continuation of harmonic sums: Dispersion representation

We compute the contribution to the anomalous dimension of the twist-2 operators in <math><mi>N</mi><mo>=</mo><mn>4</mn></math> SYM theory, which is proportional to the number of fermion loops inside Feynman diagrams or, formally, to the number of fermions. The result was obtained by the method based on the calculation of critical indices at the critical point by analogy with previous similar computations in scalar theories and in QCD. The obtained result is much simpler with compare to analogous results in QCD and almost satisfies the maximal transcendentality principle. A possible relation between the obtained result and integrability is discussed.

Fermionic contribution to the anomalous dimension of twist-2 operators in SYM theory, critical indices and integrability

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 present the extension of our previous results for three-loop Yukawa coupling beta-functions to the case of complex Yukawa matrices describing the flavour structure of the SM. The calculation is carried out in the context of unbroken phase of the SM with the help of the MINCER program in a general linear gauge and cross-checked by means of MATAD / BAMBA codes. In addition, ambiguities in Yukawa matrix beta-functions are studied.

JINR

A.V. Bednyakov

A.F. Pikelner

Three-loop SM beta-functions for matrix Yukawa couplings

We compute the non-planar contribution to the anomalous dimension of the eighth moment of the twist-2 operators in &lt;math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"&gt;&lt;mi mathvariant="script"&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt; supersymmetric Yang–Mills theory at four loops. This result was obtained from the calculations of some elements of the anomalous dimension matrix for twist-2 operators with the help of the relations between its elements and eigenvalues. The properties of the obtained result and possible relations with a planar results are discussed.

Non-planar anomalous dimension of twist-2 operators: Higher moments at four loops

Using known all-loop results from the BFKL and generalized doublelogarithmic equations and large spin limit we have recomputed the five-loop planar anomalous dimension of twist-2 operators without consideration of any wrapping effects. One part of the anomalous dimension was calculated in an usual way with the help of Asymptotic Bethe Ansatz. The rest part, related with the wrapping effects, was reconstructed from a known constraints with the help of methods from the numbers theory.

Twist-2 at five loops: wrapping corrections without wrapping computations

Three-loop renormalization group equations for the Higgs self-coupling and Higgs mass parameter are recalculated in the case of complex Yukawa matrices which encompass the general flavor structure of the Standard Model. In addition, the anomalous dimensions for both the quantum Higgs field and its vacuum expectation value are presented in the MS¯ -scheme. A numerical study of the latter quantities is carried out for a certain set of initial parameters.