Search

The Balitsky-Kovchegov (BK) evolution equation is an equation derived from perturbative Quantum Chromodynamics that allows one to calculate the scattering amplitude of a pair of quark and antiquark off a hadron target, called the dipole amplitude, as a function of the collision energy. The initial condition, being a non-perturbative object, usually has to be modeled separately. Typically, the model contains several tunable parameters that are determined by fitting to experimental data. In this contribution, we propose an implementation of the BK solver using differentiable programming. Automatic differentiation offers the possibility that the first and second derivatives of the amplitude with respect to the initial condition parameters are automatically calculated at all stages of the simulation. This fact should considerably facilitate and speed up the fitting step. Moreover, in the context of Transverse Momentum Dis- tributions (TMD), we demonstrate that automatic differentiation can be used to obtain the first and second derivatives of the amplitude with respect to the quark-antiquark separation. These derivatives can be used to relate various TMD functions to the dipole amplitude. Our C++ code for the solver, which is available in a public repository [1], includes the Balitsky one-loop running coupling prescription and the kinematic constraint. This version of the BK equation is widely used in the small-x evolution framework., Comment: 17 pages, 6 figures, source code is published in the repository

In the high-energy limit of DIS experiments the effective degrees of freedom of QCD are Wilson line operators. Their evolution in the rapidity variable is predicted by the set of Balitsky-JIMWLK evolution equations. We analyze a new class of two-point correlation functions of Wilson line operators where the Wilson lines are taken at different values of the rapidity variable. Such correlation functions can appear in the discussion of the kinematical constraint. We find that in the Langevin formulation of the JIMWLK equation, such correlation functions are affected by an infrared divergence. We discuss a possible regularization of this divergence and its consequences for the implementation of the kinematical constraint for the JIMWLK equation.

We propose a scalable framework for deciding, proving, and explaining (in)equivalence of context-free grammars. We present an implementation of the framework and evaluate it on large data sets collected within educational support systems. Even though the equivalence problem for context-free languages is undecidable in general, the framework is able to handle a large portion of these datasets. It introduces and combines techniques from several areas, such as an abstract grammar transformation language to identify equivalent grammars as well as sufficiently similar inequivalent grammars, theory-based comparison algorithms for a large class of context-free languages, and a graph-theory-inspired grammar canonization that allows to efficiently identify isomorphic grammars.

Exclusive production of a $\chi_{c1}$ axial vector quarkonia in photon-proton scattering at high energies requires a $C$-odd $t$-channel exchange. In the limit of vanishing momentum transfer this occurs either via the exchange of a photon, the Primakoff process, where the spin of the proton does not change. For axial-vector meson production, as a consequence of the Landau-Yang theorem, the Primakoff cross section is finite as $t \to 0$. Alternatively, a $C$-odd spin dependent Odderon can be exchanged, which involves a spin flip of the proton. The resulting cross section is related to the square of the gluon Sivers function or its $\boldsymbol{k}_\perp^2$-moment. Using two models for the gluon Sivers function from the literature we compute the ratio of Sivers to Primakoff cross sections and the angular coefficient $\lambda_\theta$ governing the angular distribution of the $\chi_{c1} \to J/\psi + \gamma$ decay as functions of $x$. We point out that these observables constrain the magnitude of the gluon Sivers function at small $x$ which could be accessed in electron-proton scattering and ultraperipheral proton-proton and nucleus-proton collisions., Comment: 6 pages (with references), 3 figures

We describe a method to estimate R\'enyi entanglement entropy of a spin system, which is based on the replica trick and generative neural networks with explicit probability estimation. It can be extended to any spin system or lattice field theory. We demonstrate our method on a one-dimensional quantum Ising spin chain. As the generative model, we use a hierarchy of autoregressive networks, allowing us to simulate up to 32 spins. We calculate the second R\'enyi entropy and its derivative and cross-check our results with the numerical evaluation of entropy and results available in the literature., Comment: 10 pages, 7 figures

The most complete high-energy evolution of Wilson line operators is described by the set of equations called Balitsky-JIMWLK evolution equations. It is known from the studies of the linear - the BFKL - evolution equation that the leading corrections come from the kinematically enhanced double collinear logarithms. A method for resumming such logarithmic corrections to all orders for the Balitsky-Kovchegov equation is known under the name of kinematical constraint. In this work, we discuss the progress in implementing these corrections into the Langevin formulation of the JIMWLK equation. In particular, we introduce a set of correlation functions which are nonlocal in the rapidity variable. They appear in the construction of the kinematical constraint, however, their behavior with rapidity has not been investigated numerically so far. We derive their large-$N$ evolution equations, solve them numerically, and comment on their implications for the implementation of the full kinematical constraint., Comment: Proceedings of the XXX Cracow Epiphany Conference on Precision Physics at the High Energy Colliders, Jan 8 - 12, 2024

Exclusive $C=+1$ scalar, axial-vector, and tensor quarkonium production in high-energy electron-proton scattering requires a $C$-odd $t$-channel exchange of a photon or a three gluon ladder. We derive the expressions for the corresponding amplitudes. The relative phase of the photon vs. three gluon exchange amplitudes is determined by the sign of the light-front matrix element of the eikonal color current operator $d^{abc}J^{+a}J^{+b}J^{+c}$ at moderate $x$, and is not affected by small-$x$ QCD evolution. Model calculations predict constructive interference, which is particularly strong for momentum transfer $|t|\sim 1$~GeV$^2$ where the cross section for $\chi_{cJ}$ production exceeds that for pure photon exchange by up to a factor of 4. We find that exclusive $\chi_{cJ}$ electroproduction at the Electron-Ion Collider should occur with well measurable rates and measurements of these processes should allow to find an evidence of the perturbative Odderon exchange. We also compute the total electroproduction cross section as a function of energy and provide first estimates of the number of $\chi_{cJ}$ events per month at the Electron-Ion Collider design luminosity., Comment: minor updates, matches the published version

The paper presents a numerical method for simulating flow and mechanics in fractured rock. The governing equations that couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the contact conditions prevent fractures from self-penetration. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of dualization and an optimal quadratic programming algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for tunnel excavation with hundreds of fractures in 3D. The paper's novelty consists in a combination of three crucial ingredients: (i) application of discrete fracture-matrix approach to poroelasticity, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems, and (iii) efficient solution of its mechanical quadratic programming part with a large number of fractures in mutual contact by means of own solvers implemented into an in-house software library.

Machine learning techniques, in particular the so-called normalizing flows, are becoming increasingly popular in the context of Monte Carlo simulations as they can effectively approximate target probability distributions. In the case of lattice field theories (LFT) the target distribution is given by the exponential of the action. The common loss function's gradient estimator based on the "reparametrization trick" requires the calculation of the derivative of the action with respect to the fields. This can present a significant computational cost for complicated, non-local actions like e.g. fermionic action in QCD. In this contribution, we propose an estimator for normalizing flows based on the REINFORCE algorithm that avoids this issue. We apply it to two dimensional Schwinger model with Wilson fermions at criticality and show that it is up to ten times faster in terms of the wall-clock time as well as requiring up to $30\%$ less memory than the reparameterization trick estimator. It is also more numerically stable allowing for single precision calculations and the use of half-float tensor cores. We present an in-depth analysis of the origins of those improvements. We believe that these benefits will appear also outside the realm of the LFT, in each case where the target probability distribution is computationally intensive., Comment: 16 pages, 5 figures, 6 tables, 3 listings. Revised version as published in CPC. Added results for a other values of hoping parameter

We compute the back-to-back dijet cross-section in deep inelastic scattering (DIS) at small $x$ to next-to-leading order (NLO) in the Color Glass Condensate effective field theory. Our result can be factorized into a convolution of the Weizs\"{a}cker-Williams gluon transverse momentum dependent distribution function (WW gluon TMD) with a universal soft factor and an NLO coefficient function. The soft factor includes both double and single logarithms in the ratio of the relative transverse momentum $P_\perp$ of the dijet pair to the dijet momentum imbalance $q_\perp$; its renormalization group (RG) evolution is resummed into the Sudakov factor. Likewise, the WW TMD obeys a nonlinear RG equation in $x$ that is kinematically constrained to satisfy both lifetime and rapidity ordering of the projectile. Exact analytical expressions are obtained for the NLO coefficient function of transversely and longitudinally polarized photons. Our results allow for the first quantitative separation of the dynamics of Sudakov suppression from that of gluon saturation. They can be extended to other final states and provide a framework for precision tests of novel QCD many-body dynamics at the Electron-Ion Collider., Comment: 41 pages, 9 figures, 5 supplemental materials, to be published in Physical Review Letters

We describe a new direct method to estimate bipartite mutual information of a classical spin system based on Monte Carlo sampling enhanced by autoregressive neural networks. It allows studying arbitrary geometries of subsystems and can be generalized to classical field theories. We demonstrate it on the Ising model for four partitionings, including a multiply-connected even-odd division. We show that the area law is satisfied for temperatures away from the critical temperature: the constant term is universal, whereas the proportionality coefficient is different for the even-odd partitioning., Comment: 12 pages, 8 figures

In JHEP 11 (2022) 169, we performed the first complete computation of the back-to-back inclusive di-jet cross-section in Deeply Inelastic Scattering (DIS) at small $x_{\rm Bj}$ to next-to-leading order (NLO) in the Color Glass Condensate effective field theory (CGC EFT). We demonstrate here that for di-jets with relative transverse momentum $P_\perp$ and transverse momentum imbalance $q_\perp$, to leading power in $q_\perp/P_\perp$, the cross-section for longitudinally polarized photons can be fully factorized into the product of a perturbative impact factor and the nonperturbative Weizs\"{a}cker-Williams (WW) transverse momentum dependent (TMD) gluon distribution to NLO accuracy. The impact factor can further be expressed as the product of a universal soft factor which resums Sudakov double and single logs in $P_\perp/q_\perp$ and a coefficient function given by a remarkably compact analytic expression. We show that in the CGC EFT the WW TMD satisfies a kinematically constrained JIMWLK renormalization group evolution in rapidity. This factorization formula is valid to all orders in $Q_s/q_\perp$ for $q_\perp, Q_s \ll P_\perp$, where $Q_s$ is the semi-hard saturation scale that grows with rapidity., Comment: 73 pages, 8 figures, 1 table, revised version with extended discussion of running coupling effect to CSS evolution at small x

We apply the Hierarchical Autoregressive Neural (HAN) network sampling algorithm to the two-dimensional $Q$-state Potts model and perform simulations around the phase transition at $Q=12$. We quantify the performance of the approach in the vicinity of the first-order phase transition and compare it with that of the Wolff cluster algorithm. We find a significant improvement as far as the statistical uncertainty is concerned at a similar numerical effort. In order to efficiently train large neural networks we introduce the technique of pre-training. It allows to train some neural networks using smaller system sizes and then employing them as starting configurations for larger system sizes. This is possible due to the recursive construction of our hierarchical approach. Our results serve as a demonstration of the performance of the hierarchical approach for systems exhibiting bimodal distributions. Additionally, we provide estimates of the free energy and entropy in the vicinity of the phase transition with statistical uncertainties of the order of $10^{-7}$ for the former and $10^{-3}$ for the latter based on a statistics of $10^6$ configurations., Comment: 14 pages, 12 figures, published version

It was recently proposed that neural networks could be used to approximate many-dimensional probability distributions that appear e.g. in lattice field theories or statistical mechanics. Subsequently they can be used as variational approximators to asses extensive properties of statistical systems, like free energy, and also as neural samplers used in Monte Carlo simulations. The practical application of this approach is unfortunately limited by its unfavorable scaling both of the numerical cost required for training, and the memory requirements with the system size. This is due to the fact that the original proposition involved a neural network of width which scaled with the total number of degrees of freedom, e.g. $L^2$ in case of a two dimensional $L\times L$ lattice. In this work we propose a hierarchical association of physical degrees of freedom, for instance spins, to neurons which replaces it with the scaling with the linear extent $L$ of the system. We demonstrate our approach on the two-dimensional Ising model by simulating lattices of various sizes up to $128 \times 128$ spins, with time benchmarks reaching lattices of size $512 \times 512$. We observe that our proposal improves the quality of neural network training, i.e. the approximated probability distribution is closer to the target that could be previously achieved. As a consequence, the variational free energy reaches a value closer to its theoretical expectation and, if applied in a Markov Chain Monte Carlo algorithm, the resulting autocorrelation time is smaller. Finally, the replacement of a single neural network by a hierarchy of smaller networks considerably reduces the memory requirements., Comment: 14 pages, 6 figures

Recently a machine learning approach to Monte-Carlo simulations called Neural Markov Chain Monte-Carlo (NMCMC) is gaining traction. In its most popular form it uses neural networks to construct normalizing flows which are then trained to approximate the desired target distribution. In this contribution we present new gradient estimator for Stochastic Gradient Descent algorithm (and the corresponding \texttt{PyTorch} implementation) and show that it leads to better training results for $\phi^4$ model. For this model our estimator achieves the same precision in approximately half of the time needed in standard approach and ultimately provides better estimates of the free energy. We attribute this effect to the lower variance of the new estimator. In contrary to the standard learning algorithm our approach does not require estimation of the action gradient with respect to the fields, thus has potential of further speeding up the training for models with more complicated actions., Comment: 20 pages, 5 figures, v2: improved algorithm for calculating gradients

We provide a deepened study of autocorrelations in Neural Markov Chain Monte Carlo (NMCMC) simulations, a version of the traditional Metropolis algorithm which employs neural networks to provide independent proposals. We illustrate our ideas using the two-dimensional Ising model. We discuss several estimates of autocorrelation times in the context of NMCMC, some inspired by analytical results derived for the Metropolized Independent Sampler (MIS). We check their reliability by estimating them on a small system where analytical results can also be obtained. Based on the analytical results for MIS we propose a new loss function and study its impact on the autocorelation times. Although, this function's performance is a bit inferior to the traditional Kullback-Leibler divergence, it offers two training algorithms which in some situations may be beneficial. By studying a small, $4 \times 4$, system we gain access to the dynamics of the training process which we visualize using several observables. Furthermore, we quantitatively investigate the impact of imposing global discrete symmetries of the system in the neural network training process on the autocorrelation times. Eventually, we propose a scheme which incorporates partial heat-bath updates which considerably improves the quality of the training. The impact of the above enhancements is discussed for a $16 \times 16$ spin system. The summary of our findings may serve as a guidance to the implementation of Neural Markov Chain Monte Carlo simulations for more complicated models., Comment: 20 pages, 11 figures

Quarkonium production mechanism in high multiplicity small collision systems has recently been pursued in the color-glass-condensate (CGC) effective theory combined with non-relativistic QCD (NRQCD) factorization, allowing to study initial state interactions. Quarkonium polarization, potentially measured in future experiments, would help elucidate the quarkonium production mechanism at high multiplicities. In this paper, we provide predictions on $J/\psi$ polarization parameters in high multiplicity proton-proton ($pp$) and proton-nucleus ($pA$) collisions within the CGC+NRQCD framework. Theoretical predictions are given for $J/\psi$ rapidity $2.5< y_{J/\psi}<4$, charged-particle multiplicity pseudorapidity $|\eta_{ch} | <1$ and energies $\sqrt{S}=13\mathrm{~TeV}$ for $pp$, $\sqrt{S}=8.16\mathrm{~TeV}$ for $pA$ collisions. Considering two leptonic frame choices (Collins - Soper and helicity) we find a weak polarization of $J/\psi$ that additionally decreases with growing event activities. No significant system size dependence between $pp$ and $pA$ collisions is obtained - this could be a new constraint to initial state interactions in small collision systems., Comment: 9 pages, 1 figure

We study the prompt photon hadroproduction at the LHC with the $k_T$-factorization approach and the $qg^* \to q\gamma$ and $g^*g^* \to q\bar q\gamma$ partonic channels, using three unintegrated gluon distributions which depend on gluon transverse momentum. They represent three different theoretical schemes which are usually considered in the $k_T$-factorization approach, known under the acronyms: KMR, CCFM and GBW gluon distributions. We find sensitivity of the calculated prompt photon transverse momentum distribution to the gluon transverse momentum distribution. The predictions obtained with the three approaches are compared to data, that allows to differentiate between them. We also discuss the significance of the two partonic channels, confronted with the expectations which are based on the applicability of the $k_T$-factorization scheme in the high energy approximation., Comment: 23 pages, 10 figures, version accepted for publication in PRD

The studies of the associated production processes of a top-quark pair with a colour-singlet boson, e.g. Higgs, W or Z, are among the highest priorities of the LHC programme. Correspondingly, improvements in precision of theoretical predictions for these processes are of central importance. In this talk, we review our latest results on resummation of soft gluon corrections. The resummation is carried out using the direct QCD Mellin space technique in three-particle invariant mass kinematics. We discuss the impact of the soft gluon corrections on predictions for total cross sections and differential distributions., Comment: 12 pages, 9 figures, proceedings of the XXVI Cracow Epiphany Conference on LHC Physics: Standard Model and Beyond, to be published in Acta Physica Polonica B

Color charge correlations in the proton at moderately small $x\sim 0.1$ are extracted from its light-cone wave function. The charge fluctuations are far from Gaussian and they exhibit interesting dependence on impact parameter and on the relative transverse momentum (or distance) of the gluon probes. We provide initial conditions for small-$x$ Balitsky-Kovchegov evolution of the dipole scattering amplitude with impact parameter and $\hat r \cdot \hat b$ dependence, and with non-zero $C$-odd component due to three-gluon exchange. Lastly, we compute the (forward) Weizsaecker-Williams gluon distributions, including the distribution of linearly polarized gluons, up to fourth order in $A^+$. The correction due to the quartic correlator provides a transverse momentum scale, $q > 0.5$ GeV, for nearly maximal polarization., Comment: 19 pages, 10 figures; tables of dipole scattering amplitude via two- and three-gluon exchange are available as ancillary files

We present theoretical predictions for selected differential cross sections for the process $pp \to t \bar{t} B$ at the LHC, where $B$ can be a Higgs ($H$), a $Z$ or a $W$ boson. The predictions are calculated in the direct QCD framework up to the next-to-next-leading logarithmic (NNLL) accuracy and matched to the complete NLO results including QCD and electroweak effects. Additionally, results for the total cross sections are provided. The calculations deliver a significant improvement of the theoretical predictions, especially for the $t \bar{t} H$ and the $t \bar{t} Z$ production. In these cases, predictions for both the total and differential cross sections are remarkably stable with respect to the central scale choice and carry a substantially reduced scale uncertainty in comparison with the complete NLO predictions., Comment: 28 pages, 23 figures; Published version

We discuss the Drell-Yan dilepton production using the transverse momentum dependent parton distributions evolved with the Catani-Ciafaloni-Fiorani-Marchesini-Kwieci\'nski (CCFM-K) equations in the single loop approximation. Such equations are obtained assuming angular ordering of emitted partons (coherence) for $x\sim 1$ and transverse momentum ordering for $x \ll 1$. This evolution scheme also contains the Collins-Soper-Sterman (CSS) soft gluon resummation. We make a comparison with a broad class of data on transverse momentum spectra of low mass Drell-Yan dileptons., Comment: 29 pages, 9 figures

Production of a forward Drell-Yan lepton pair accompanied by a jet separated by a large rapidity interval is proposed to study the BFKL evolution at the LHC. Several observables to be measured are presented including the azimuthal angle dependence of the lepton pair which allows to determine Drell-Yan structure functions., Comment: in proceedings of XXVII International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2019), 8-12 Apr, Torino (Italy)

In this talk we present results of our recent calculations of cross sections and distributions for the associate production of top-antitop quark pairs with a heavy boson (Higgs, W, Z) at the LHC, obtained using threshold resummation in direct QCD, i.e. in the Mellin-space approach., Comment: 4 pages, 6 figures, contribution to the 2019 QCD session of the 54th Rencontres de Moriond

Exclusive production of a $\eta_c$ pseudo-scalar meson in $\gamma^{(*)} + p \to \eta_c + p$ scattering at high energies involves a ${\cal C}$-odd exchange in the $t$-channel. We formulate the description of this process within the high-energy framework of eikonal dipole scattering. We obtain expressions for the light-cone wave function of the $\eta_c$ required in this framework as well as for the ${\cal C}$-odd amplitudes due to exchange of a single photon, of a photon plus two gluons, and of three gluons. We relate these amplitudes to correlators of the $+$ component of the quark current in the light-cone wave function of the proton. For high transverse momenta these correlators correspond to Generalized Parton Distributions (GPDs) given by diagrams where all exchanged gauge bosons attach to a single quark in the proton. Diagrams involving multi-quark matrix elements potentially sensitive to correlations, screen infrared singularities. Moreover, they are numerically important for configurations where the exchanged bosons nearly share the total momentum transfer. Using two simple models for the three quark Fock state of the proton at $x\simeq0.1$, we find that single photon exchange dominates for $|t|<1.5$~GeV$^2$. Here, the quark GPD could be measured cleanly in $\gamma^{(*)} + p \to \eta_c + p$ via single photon exchange. For higher momentum transfer three gluon (`Odderon') exchange is dominant., Comment: 16 pages, 3 figures