arXiv subjects

Alexander Prygarin

Publications and source records attributed to Alexander Prygarin.

4 recordsLinked to original sources

arXiv2026

The next-to-next-to-leading order BFKL eigenvalue at odd conformal spin in planar N=4 super Yang-Mills

We give the next-to-next-to-leading order color-singlet BFKL eigenvalue of planar N=4 super Yang-Mills at odd conformal spin n in closed form. At the two lower orders the known expressions contain nested harmonic sums only at the two conjugate points z=(|n|-1)/2+i nu and zbar. At three loops the sums are evaluated on a ladder of integer-shifted arguments from the reflected point -zbar up to z, with one point past it, the coefficient at a rung fixed by its two distances to the ends. Seven families are the exception, their coefficients written through ladder sums whose summand shifts with the summation index. For one of the seven the coefficient lies outside the fixed-coefficient algebra of nested harmonic sums of the two distances, an arithmetic obstruction in the denominators. That exclusion reaches the eigenvalue coefficient only through an agreement of rule and coefficient verified and not proved. Rules uniform in the conformal spin produce all forty-seven families at every odd n>=3; no per-spin coefficient is tabulated in the closed form, and the n=1 boundary block is supplied separately. Those rules are a reconstruction from the computed spins, exact at every one of them, verified over the range n<=99 and at the holdout spin n=101, and not proved at arbitrary odd n. Each atom-table coefficient is a rational combination of 1, pi^2 and zeta_3, coefficient and atom together carrying transcendental weight five. Along nu=0 the intercepts match the Quantum Spectral Curve values at each computed odd spin through n=91, the extent of those values. The comparison tests the integrand and the reduction together at one point of each spin, for most of them for the first time. Away from that line the closed form gives the collinear behavior of the block uniformly in the spin, which the intercepts alone do not fix.

hep-th↗

arXiv2026

The next-to-next-to-leading order BFKL eigenvalue at odd conformal spin in planar N=4 super Yang-Mills: a reconstructed closed form, coefficient structure, and arithmetic

We give the colour-singlet eigenvalue of the three-loop BFKL kernel of planar N=4 super Yang-Mills in closed form at odd conformal spin n. It splits into a holomorphic block and its conjugate, chi_2(n,nu)=(1/4)[F_n(z)+F_n(zbar)] with z=(|n|-1)/2+i nu, and the block is a finite combination of nested harmonic sums of one variable, rational functions and transcendental constants, of uniform transcendental weight five. At the two lower orders the sums are evaluated at the two conjugate points only. At three loops they are evaluated at every point of the integer ladder running from the reflected endpoint -zbar up to z, and one step beyond it, and for all but seven of the slots the coefficient at each rung is fixed by the two distances to the ends. Summing over the one-variable functions before summing along the ladder replaces the rules of those slots by one function of one variable under each ladder weight. The remaining seven are carried by a sum along the ladder whose summand moves with the summation index. All forty-seven coefficient slots follow from rules uniform in the conformal spin at every odd n>=3, with no per-spin coefficient tabulated in the definition; the n=1 boundary block is supplied separately. Those rules are a reconstruction from the computed spins, exact at every one of them, verified over the range n<=99 and at the holdout spin n=101, and not proved at arbitrary odd n. Along nu=0 the intercepts agree with the Quantum Spectral Curve values at every computed odd spin through n=91. That comparison tests the three-loop integrand and the reduction performed here together, at one point of each spin, and for most of those spins it is the first comparison of its kind.

hep-th↗

arXiv2019

The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five

We analyze the structure of the eigenvalue of the color-singlet Balitsky-Fadin-Kuraev-Lipatov~(BFKL) equation in N=4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N=4 SYM , i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how non-linear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

hep-th↗

arXiv2012

The Multi-Regge limit of NMHV Amplitudes in N=4 SYM Theory

We consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2->4 and 3->3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2->4 and 3->3 NMHV amplitudes, as well as a particular sub-set of two loop 2 ->2 +n N^kMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.

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