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V. N. Velizhanin

Publications and source records attributed to V. N. Velizhanin.

At least 19 recordsLinked to original sources
arXiv2026

Four-loop beta function in N=2 supersymmetric Yang-Mills theory, dimensional reduction, and treatment of $γ_5$

Adopting the dimensional-reduction scheme, we diagrammatically compute, at four loops, the beta function of N=2 supersymmetric Yang-Mills theory, which is well known to vanish beyond one loop. This provides a formidable testbed for both the treatment of the problematic $γ_5$ Dirac matrix in D dimensions and the implementation of the dimensional-reduction scheme. In the use of the reading-point method of $γ_5$, being preferable for convenience, specific details of implementation start to matter at four loops. We thus rigorously validate these implementation issues, which have previously been advocated for the calculation of the gauge-coupling beta functions at four loops in the Standard Model.

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arXiv2026

Three-loop renormalization of the N=1, N=2, N=4 supersymmetric Yang-Mills theories

We calculate the renormalization constants of the N=1, N=2, N=4 supersymmetric Yang-Mills theories in an arbitrary covariant gauge in the dimensional reduction scheme up to three loops. We have found, that the beta-functions for N=1 and N=4 SYM theories are the same from the different triple vertices. This means that the dimensional reduction scheme works correctly in these models up to third order of perturbative theory.

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arXiv2026

Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

We consider the anomalous dimension $γ_{gg}^{(3)}(N)$ of the twist-two gluon operator of arbitrary Lorentz spin $N$ in the quark flavor singlet sector of a general gauge theory at four loops and construct its contribution proportional to $ζ(3)$ in analytic form by applying the Lenstra-Lenstra-Lovász algorithm to the available low-$N$ moments. We exploit generalized Gribov-Liptov reciprocity, establish new self-tuning relations for the anomalous dimension matrix of the singlet sector, and inject information from $\mathcal{N}=1,4$ supersymmetric Yang-Mills theories. We also present the contribution to the rational part of $γ_{gg}^{(3)}(N)$ with color factor $C_F^2n_f^2$. Exact contributions to the four-loop splitting function $P_{gg}^{(3)}(x)$ hence resulting via inverse Mellin transformation help us to reduce theoretical uncertainties in scaling violations of parton distribution functions in QCD.

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arXiv2026

Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part

Both for quark flavor asymmetry and valence, we consider the anomalous dimension of the non-singlet twist-two quark operator of arbitrary Lorentz spin N at four loops in SU(nc) color gauge theory and present its term proportional to zeta(3) in closed form. These results have been extracted from published Mellin moments, for N=1,...,16 and N=3,...,15, respectively, by analytic reconstruction using advanced methods of number theory. Via Mellin transformation, we obtain the exact functional forms in x of the respective pieces of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi splitting functions. This allows us to reduce the theoretical uncertainties in the approximations of these splitting functions otherwise amenable from the first few low-N values.

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arXiv2025

Flavour Non-Singlet Splitting Functions at Four Loops in QCD -- The Fermionic Contributions

We have determined the fourth-order $n_f$ contributions to the two splitting functions governing the evolution of all flavor differences of quark distributions of hadrons in perturbative quantum chromodynamics with $n_f$ light flavors. The analytic forms of these functions are presented in both Mellin $N$-space and momentum-fraction $x$-space for a general gauge group. In the small-$x$ limit double logarithms occur, but the small-$x$ rise of both splitting functions is confined to extremely small $x$-values, $x \lesssim 10^{-6}$. The large-$x$ limit includes the $n_f$-part of the four-loop quark virtual anomalous dimension. Using this result we obtain also the $n_f$ contributions to the corresponding gluonic quantity and the complete threshold-enhanced logarithms from soft-gluon emission for a large class of inclusive observables, including Higgs boson production in gluon-gluon fusion.

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arXiv2025

Four-loop anomalous dimension of flavor non-singlet twist-two operator of general Lorentz spin in QCD: zeta(3) term

We consider the anomalous dimension of the flavor non-singlet twist-two quark operator of arbitrary Lorentz spin N at four loops in QCD and construct its contribution proportional to zeta(3) in analytic form by applying advanced methods of number theory on the available knowledge of low-N moments. In conjunction with similar results on the zeta(5) and zeta(4) 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-N form of the rational part. Via Mellin transformation, we thus obtain the exact functional form in x of the respective piece of the non-singlet 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-N moments.

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arXiv2025

Nonplanar Four-Loop Anomalous Dimensions of Twist-Two Operators in N=4 Super Yang-Mills Theory: Higher Moment, General Result, and Cusp Anomalous Dimension

We consider the nonplanar universal anomalous dimension of twist-two operators at four loops in N=4 supersymmetric Yang-Mills theory and push its direct diagrammatic calculation through Lorentz spin j=20, one unit beyond the state of the art, so as to confirm the correctness of the general, all-j result conjectured previously by us [1] 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 re-derive, in a completely independent way, the nonplanar four-loop cusp anomalous dimension by taking the large-j limit of the general result.

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arXiv2024

Anomalous dimensions of twist-two operators in extended N=2 and N=4 super Yang-Mills theories

We perform direct diagrammatic calculations of the anomalous dimensions of twist-two operators in extended N=2 and N=4 super Yang-Mills theories (SYM). In the case of N=4 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 N=2 SYM, we obtain the general result for the anomalous dimension at third order of perturbation theory and find the three-loop Cusp anomalous dimension.

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arXiv2022

Analytic continuation of harmonic sums: dispersion representation

We present a simple representation for analytically continued nested harmonic sums for the arbitrary complex argument. 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.

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arXiv2022

Fermionic contribution to the anomalous dimension of twist-2 operators in N=4 SYM theory, critical indices and integrability

We compute the contribution to the anomalous dimension of the twist-2 operators in N=4 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 transcedentality principle. A possible relation between the obtained result and integrability is discussed.

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arXiv2022

Analytic continuation of harmonic sums with purely imaginary indices near the integer values

We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in ABJM model, which contains the nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in N=4 SYM and QCD, however, we did not find a general expression for the ``BFKL Pomeron'' eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of the nested harmonic sums.

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arXiv2022

Exact result in N=4 SYM theory: Generalised double-logarithmic equation

We present the new results for the generalised double-logarithmic equation, obtained from the analytical continuation of the seven-loop anomalous dimension of twist-2 operators in the planar N=4 SYM theory. The double-logarithmic equation is related to the special asymptotic of the scattering amplitudes, when the large logarithms of the energy of scattering particles are appeared and should be summed in all order of perturbative theory. These large logarithms correspond to the poles of the analytically continued anomalous dimension. The generalised double-logarithmic equation includes the subleading logarithms. We have found, that the expansion of the generalised double-logarithmic equation can be ressumed in the form of rational functions with simple denominator. The solution of the generalised double-logarithmic equation provides a lot of information about the poles of the analytically continued anomalous dimension in all orders of perturbative theory. We have found also the generalised double-logarithmic equation for the analytically continued anomalous dimension near the value, which is related with BFKL-equation.

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arXiv2021

NNNLLA BFKL pomeron eigenvalue in the planar N=4 SYM theory

We find the eigenvalue of the kernel of BFKL equation in the next-to-next-to-next-to-leading logarithm approximation (NNNLLA) in the planar N=4 SYM theory by using the quantum spectral curve to compute values at fixed spin, and reconstructing the general result using the LLL-algorithm, which also was used for the reconstruction of the general result for the intercept function for arbitrary conformal spin at fourth order, computed earlier in paper 1802.06908 up to the rational part. We have found, that a new type of harmonic sums enter into results. Those sums have the argument multiplied by factor two with compare to usual harmonic sums and contain the last imagine index, what reflect their relations with the multiple polylogarithms, generated by ``fourth root of unity''. The final result contains Catalan number, which did not appear early in the calculations for the anomalous dimension of twist-2 operators.

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arXiv2021

Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in N=4 SYM theory

We compute the non-planar contribution to the universal anomalous dimension of twist-two operators in N=4 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 beta-deformed version of the theory.

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arXiv2021

Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory

We compute the nonplanar contribution to the universal anomalous dimension of the SU(4)-singlet twist-two operators in N=4 supersymmetric Yang-Mills theory at four loops through Lorentz spin 18. From this, we numerically evaluate the nonplanar contribution to the four-loop lightlike cusp anomalous dimension and derive the transcendental $ζ_3$ and $ζ_5$ parts of the universal anomalous dimension for arbitrary Lorentz spin in analytic form. As for the lightlike cusp anomalous dimension and the $ζ_5$ part of the universal anomalous dimension, we confirm previous results.

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arXiv2017

Generalised double-logarithmic equation in QCD

We present a generalisation of the double-logarithmic equation for the anomalous dimension of the non-singlet unpolarized twist-2 operators in QCD. Using the known three-loop result, this generalisation allows to predict a small x expansion of the four-loop non-singlet splitting functions in QCD for all powers of logarithms up to the single-logarithm term.

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