arXiv subjects

Alex Prygarin

Publications and source records attributed to Alex Prygarin.

At least 19 recordsLinked to original sources

arXiv2026

The signature-changing soft-gluon term as a possible source of non-planar $ζ_2$ terms at small $x$ in the non-singlet quark-plus-antiquark splitting function

A relation underlying resummations at small momentum fraction $x$ predicts the highest logarithms of quark-plus-antiquark flavor-difference splitting functions from lower orders. Complete three- and four-loop results violate it at third-highest logarithms by pi-squared terms suppressed for many colors, of unexplained origin and color structure. At four loops Moch and Vogt suggested a squared imaginary phase. We retain the signature-changing soft-gluon term also in positive signature in Kirschner and Lipatov's double-logarithmic equations for forward quark-antiquark scattering, without derivation, with their printed negative-signature sign and adjoint Born term. We show that the proposed equations reproduce both violations in full color for quark-plus-antiquark flavor differences at zero power of $x$, fixed coupling and first order in pi squared, for defining-representation quarks of special unitary groups with at least three colors. The proposed equations at third-highest logarithms use one wave per exchanged color and signature, the retained term at lowest-order strength as the positive-signature color-neutral channel's only new source of these violations, no such terms from gluons across the softest exchanged quark-antiquark pair at comparable or larger transverse momentum, and unchanged quark-pair normalization in the splitting function. This agreement gives three independent tests of one system with no continuous coefficient adjusted to the violations, although corresponding non-quartic quark-minus-antiquark terms are missed. In the proposed equations, the retained term weighted by this phase is a possible origin of both violations. The equations predict these full-color five-loop quark-plus-antiquark terms if the relation otherwise holds and no new source first acts at five loops. Confirmation would support the retained term as the origin of the violations through five loops.

hep-ph↗

arXiv2026

Information geometry of DGLAP evolution

Quark and gluon splittings shift proton momentum toward smaller fractions with increasing resolution. Momentum conservation makes the summed momentum-weighted quark and gluon densities, when nonnegative, a probability distribution. We apply information geometry, the geometry of probability distributions, to this probabilistic form of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution. We evaluate, to our knowledge for the first time, the speed at which the shape of the proton's momentum distribution changes, measured by how well distributions at neighboring resolutions can be told apart statistically. We show that this speed has a closed-form lower bound at leading order, with one-loop splitting functions and running coupling, five fixed quark flavors, nonnegative quark and gluon densities and sufficiently regular inputs. Three numbers of the input (the distribution at the starting resolution) fix the bound at every scale without computing the evolved distribution, once the coupling is specified. The squared bound is at least 94.8 percent of the squared speed for three of the five model inputs at four scales from 2 GeV to 10 TeV. The bound is reached when the distribution's relative rate of change is linear in the logarithm of the inverse momentum fraction. The relative rate of change is close to this linear form in mean square weighted by momentum for these three model inputs at all four scales, and closer still to linear in the square root of the logarithm at the three higher scales. These descriptions are compatible because the logarithm and its square root are strongly correlated when weighted by these momentum distributions.

hep-ph↗

arXiv2026

An information bound for multiplicities in rapidity windows of the dipole cascade behind the entanglement entropy picture

In Mueller's dipole cascade without transverse dimensions, with one counted particle for each dipole produced in a window, the multiplicities in three consecutive rapidity windows are Poisson counts of one Gamma-distributed source. We show that at fixed middle multiplicity the outer two share less information than the Gaussian value $-\tfrac12\ln(1-ρ^2)$ of their partial correlation, a value that second moments alone determine. For windows of one common width at a constant splitting rate this value never exceeds $\tfrac12\ln(4/3)$. The Gaussian value is not a bound on mutual information in general, and for a source of the same mean and variance with another law the information can exceed it. The law of the counts keeps its form under migration of particles across window edges and detection losses when both act on each particle independently of the others and of the source, and its factorial cumulants of second and third order test necessary conditions for it. In Monte Carlo simulations of the cascade with transverse dimensions at leading logarithmic accuracy and fixed coupling, the counts are not Poisson counts of one source, so the bound is a result of the model without them. In the model without transverse dimensions the law also fixes the normalized second factorial cumulant of one window at $1/k$, with $k$ the number of initial dipoles. The values of this cumulant formed from the mean and variance that H1 publishes for deep inelastic scattering lie far below the value one of a cascade from a single initial dipole with one counted particle for each dipole produced in a window. The quantity bounded is a classical conditional mutual information between counted multiplicities.

hep-ph↗

arXiv2026

Lévy structure of the forward fixed-coupling BFKL kernel and a fixed-order obstruction to positivity in the symmetric scheme

We establish a Levy interpretation of normalized forward BFKL small-$x$ evolution at leading order and fixed coupling, and exclude a probability law at fixed-order next-to-leading accuracy in the symmetric scheme. After the Marchesini-Onofri conjugation and growth subtraction, one positive step measure in closed form covers all conformal spins, on the cylinder of logarithmic transverse momentum and azimuth. The process is pure jump, and the Pomeron intercept is the relaxation rate of its first azimuthal harmonic. With $s_0=q_1q_2$ and the coupling at that argument, the negative cubic collinear pole of the truncated eigenvalue excludes a probability law at every positive coupling and rapidity. Its leading coefficient is independent of $N_c$ and $n_f$. The pole dominates the leading-order simple pole within $\sqrt{\barα/2}$ of the edge in the leading collinear approximation. Translation-preserving multiplicative conjugations cannot restore positivity. No non-negative step measure generates that truncated evolution, while a positive radial completion at zero conformal spin matches the computed order, so the obstruction is a property of the fixed-order truncation. The tested symmetric-scheme resummations also fail positivity, as shown in closed form for the pure and matched all-poles forms, except for a degenerate zero process, and by computation at the examined couplings for the full prescription. The improved finite-rapidity Green function fails at the displayed parameters under the contour assumption. A resummed kernel in another rapidity scheme has a non-negative step measure at tested couplings. An unweighted transverse walk describes the leading-order evolution with no approximation, but a probabilistic resummation of the symmetric next-to-leading kernel must establish positivity beyond perturbative matching, and whether one exists is left open.

hep-ph↗

arXiv2026

Higher-order local constraints from reciprocal symmetry and entanglement entropy of charged-particle multiplicity distributions in $pp$ collisions

The KNO-violating term $f_s$ of the charged-particle multiplicity distribution in $pp$ collisions measures the relative deviation of $\langle n\rangle P_n$ from $e^{-z}$, with $z=n/\langle n\rangle$, and is reported to obey the reciprocal symmetry $f_s(z)=f_s(1/z)$, taken here as input. Being an evenness condition in $\ln z$, it generates a tower of local constraints on the derivatives of $P_n$ at the mean. The lowest member holds for the ATLAS data at $7$, $8$ and $13$~TeV at the few-per-cent level, while the third-derivative residual testing the next member is not determined with a controlled uncertainty by the present binning and stays inconclusive. The global test is consistent at $7$ and $8$~TeV, while at $13$~TeV a residual deviation remains, which a closure test shows is not a binning artefact. Multiplicative noise in the Mueller colour-dipole cascade, the negative binomial and the dipole cascades with recombination each give an $f_s$ that is not invariant under $z\to1/z$, so none of them carries the symmetry. We further obtain a dynamics-independent entropy in the KNO continuum, $S=\ln\langle n\rangle+I_0-\tfrac12\int e^{-z}f_s^2\,dz$, with $I_0$ a support factor tending to unity in the continuum limit. The linear term cancels by normalisation and unit mean, independently of the symmetry, and the remaining correction is quadratic and negative. It reproduces the Shannon entropy of the ATLAS data at the $10^{-3}$ level.

hep-ph↗

arXiv2026

Reciprocal symmetry and KNO scaling violation in proton-proton collisions

We analyze the charged particle multiplicity distributions in $p-p$ collisions and discuss the violation of the Koba--Nielsen--Olesen (KNO) scaling. We extract the deviations from the leading exponential behavior of the KNO scaled probability and identify a reciprocal symmetry $z\leftrightarrow 1/z$ in the KNO violating corrections observed in the ATLAS and CMS data at $\sqrt{s}=7,\,8,\,13$~TeV. The symmetry imposes a local constraint on the multiplicity distribution at $n=\langle n\rangle$, namely $P'(\langle n\rangle)=-P(\langle n\rangle)/\langle n\rangle$, which we verify directly in the data. We use this constraint to extract the entanglement entropy from the well-measured region $n\simeq\langle n\rangle$, avoiding the large uncertainties associated with the distribution tail.

hep-ph↗

arXiv2025

KNO scaling, memorylessness and maximal entanglement at universal fixed point

We analyze the experimental data of $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions by the ATLAS and confront it with the AGK model developed by two of the authors, the Kharzeev-Levin~(KL) model and the simple exponential behavior for the Koba, Nielsen and Olesen~(KNO) scaling function. We show that all three models virtually coincide with all available experimental curves crossed at value $z=n/\langle n \rangle$ of the KNO scaling parameter in the broad range of the center-of-mass energies. The theoretical results and the experimental data show that the KNO violating terms are highly suppressed in the vicinity of $z=2$. The special status of the universal scaling point $z=2$, where the KNO scaling is restored for $\mathtt{p}\mathtt{-}\mathtt{p}$ collisions, suggests that it originates from statistical saturation of the optical theorem. We claim that any unitary model that approximates memoryless distribution in the high energy limit must have the same KNO scaling function $e^{-z}$ in the vicinity of $z=2$ with KNO violating corrections of the order of one over average multiplicity squared. This reflects the maximal entanglement of the final states.

hep-ph↗

arXiv2025

Pomeron evolution, entanglement entropy and Abramovskii-Gribov-Kancheli cutting rules

We use Pomeron evolution equation in zero transverse dimension based on the Abramovskii-Gribov-Kancheli~(AGK) cutting rules to calculate the von Neumann entropy and $q$-moments that used as an experimental test for the Koba-Nielsen-Olesen~(KNO) scaling. In order to avoid the negative probabilities that emerge from the negative AGK weights in the Minkowski space, we reformulate the Pomeron evolution in the Euclidean space. The resulting positive definite probabilities for cut and uncut Pomerons are used in calculating the $q$-moments. The comparison to the experimental data shows that our AGK based model successfully describes $q$-moments dependence on the mean multiplicity without any adjustable parameter in the experimental data of the $\mathtt{p-p}$ collisions by $\mathtt{ALICE}$ Collaboration.

hep-ph↗

arXiv2025

Pomeron Evolution and Squeezed States in Quantum Optics

We apply the formalism of coherent states in quantum optics to pomeron evolution and show that evolving squeezed pomeron states are equivalent to pomeron fan diagrams at the leading order of perturbative expansion. Based on our results, we interpret the action of the displacement operator as pomeron propagation and the action of the squeeze operator as pomeron interaction.

hep-ph↗

arXiv2023

Pole decomposition of BFKL eigenvalue at zero conformal spin and the real part of digamma function

We consider the powers of leading order eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov~(BFKL) equation at zero conformal spin. Using reflection identities of harmonic sums we demonstrate how involved generalized polygamma functions are introduced by pole separation of a rather simple digamma function. This generates higher weight generalized polygamma functions at any given order of perturbative expansion. As a byproduct of our analysis we develop a general technique for calculating powers of the real part of digamma function in a pole separated form.

hep-th↗

arXiv2021

Universal transcendentality limit of BFKL eigenvalue

We consider a special limit of the BFKL eigenvalue at $ν\to 0$ and odd values of the conformal spin $n$. We show that in this limit the NLO BFKL eigenvalue can be expressed in terms of a limited set of transcendental constants with rational coefficients. We show that the leading transcendentality term in this limit is universal and does not depend on a specific value of $n$.

hep-th↗

arXiv2021

Real valued functions for BFKL eigenvalue

We consider known expressions for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in $N=4$ super Yang-Mills theory as a real valued function of two variables anomalous dimension and the conformal spin. We define new real valued functions of two complex conjugate variables that have a definite complexity analogous to the weight of the nested harmonic sums. We argue that those functions span a general space of functions for the BFKL eigenvalue at any order of the perturbation theory.

hep-th↗

arXiv2020

Hermitian Separability of BFKL eigenvalue in Bethe Salpeter approach

We consider the Bethe Salpeter approach to the BFKL evolution in order to naturally incorporate the property of the Hermitian Separability in the BFKL approach. We combine the resulting all order ansatz for the BFKL eigenvalue together with reflection identities for harmonic sums and derive the most complicated term of the next-to-next-to-leading order BFKL eigenvalue in SUSY N=4. We also suggest a numerical technique for reconstructing the unknown functions in our ansatz from the known results for specific values of confomal spin.

hep-th↗

arXiv2020

Reflection Identities of Harmonic Sums and pole decomposition of BFKL eigenvalue

We analyze known results of next-to-next-to-leading(NNLO) singlet BFKL eigenvalue in $N=4$ SYM written in terms of harmonic sums. The nested harmonic sums building known NNLO BFKL eigenvalue for specific values of the conformal spin have poles at negative integers. We sort the harmonic sums according to the complexity with respect to their weight and depth and use their pole decomposition in terms of the reflection identities to find the most complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the conformal spin. The obtained result is compatible with the Bethe-Salpeter approach to the BFKL evolution.

hep-ph↗

arXiv2019

BFKL Eigenvalue and Maximal Alternation of Harmonic Sums

We analyze the known results for the eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation in the perturbative regime using the analytic continuation of harmonic sums from even positive arguments to the complex plane. The resulting meromorphic functions have poles at negative integer values of the argument. The typical classification of harmonic sums is determined by two major parameters: $a)$ the \textit{weight} - a sum of inverse powers of the summation indices; $b)$ the \textit{depth} - a number of nested summations. We introduce the third parameter: the \textit{alternation} - a number of nested sign-alternating summations in a given harmonic sum. We claim that the maximal alternation of the nested summation in the functions building the BFKL eigenvalue is preserved from loop to loop in the perturbative expansion. The BFKL equation is formulated for arbitrary color configuration of the propagating states in the $t$-channel. Based on known results one can state that color adjoint BFKL eigenvalue be can written using only harmonic sums with positive indices, maximal alternation zero, and at most depth one, whereas the singlet BFKL eigenvalue is constructed of harmonic sums with maximal sign alternation being equal one. We also note that for maximal alternation being equal unity the harmonic sums can be expressed through alternation zero harmonic sums with half-shifted arguments.

hep-th↗

arXiv2018

Reflection identities of harmonic sums of weight four

We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive 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 bilinear set of reflection identities at weight four which present the main result of the paper. We also discuss how other trilinear and quartic reflection identities can be easily constructed from our result with the use of well known shuffle relations for harmonic sums.

math.NT↗

arXiv2018

Reflection identities of harmonic sums up to weight three

We discuss reflections identities of harmonic sums up to weight three. The need for this kind of identities emerges in analysis of the general structure of eigenvalue of the BFKL equation. The reflection identities decompose a product of two harmonic sums with pole singularities at real integer points into a linear combination of other functions with pole singularities at either negative integers or zero and positive integers. This provides a pole separation of expressions with a mixed pole structure.

hep-th↗

arXiv2016

On a residual freedom of the next-to-leading BFKL eigenvalue in color adjoint representation in planar $\mathcal{N} = 4$ SYM

We discuss a residual freedom of the next-to-leading BFKL eigenvalue that originates from ambiguity in redistributing the next-to-leading~(NLO) corrections between the adjoint BFKL eigenvalue and eigenfunctions in planar $\mathcal{N}=4$ super-Yang-Mills~(SYM) Theory. In terms of the remainder function of the Bern-Dixon-Smirnov~(BDS) amplitude this freedom is translated to reshuffling correction between the eigenvalue and the impact factors in the multi-Regge kinematics~(MRK) in the next-to-leading logarithm approximation~(NLA). We show that the modified NLO BFKL eigenvalue suggested by the authors can be introduced in the MRK expression for the remainder function by shifting the anomalous dimension in the impact factor in such a way that the two and three loop remainder function is left unchanged to the NLA accuracy.

hep-th↗