In the high-energy limit of QCD, scattering off nucleons and nuclei can be described in terms of Wilson-line correlators whose energy dependence is perturbative. The energy dependence of the two-point correlator, called the dipole amplitude, is governed by the Balitsky-Kovchegov (BK) equation. The initial condition for the BK equation can be fitted to the experimental data, which requires evolving the dipole amplitude for a large set of different parameter values. In this work, we train a transformer model to learn the energy dependence of the dipole amplitude, thereby replacing repeated time-consuming numerical BK evolutions during the fit by fast emulator predictions. The transformer predicts the learned dipole amplitude and the leading order inclusive deep inelastic scattering cross section very accurately, allowing for efficient fitting of the initial condition to the experimental data. Using this setup, we fit the initial condition of the BK equation to the inclusive deep inelastic scattering data from HERA and consider two different starting points $\textit{x}$ for the evolution. We find better agreement with the experimental data for a smaller $\textit{x}$ . This work paves the way for future studies involving global fits of the dipole amplitude at leading order and beyond.

China

Meisen Gao

United States

Zhong-Bo Kang

Jani Penttala

Ding Shao

Determination of the initial condition for the Balitsky-Kovchegov equation with transformers

We determine the initial condition of the Balitsky-Kovchegov (BK) evolution equation at next-to-leading order (NLO) accuracy using HERA deep inelastic scattering data. We use Bayesian inference to compare model values, derived from NLO impact factors and the BK evolution equation with kinematical constraint resumming most important higher order corrections, to precise data and to extract posterior distributions characterizing the initial condition. The total cross section and charm quark production data from HERA are found to provide stringent constraints on the posterior. These distributions quantify the uncertainty in the initial condition and serve as necessary input for propagating uncertainties to all NLO calculations within the color glass condensate framework.

Finland

Carlisle Casuga

Henri Hänninen

Heikki Mäntysaari

Initial condition for the Balitsky-Kovchegov equation at next-to-leading order

We perform analysis of the small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> nonlinear evolution equation formulated in momentum space supplemented by higher order terms. The equation is defined in the wide range of transverse momentum and longitudinal momentum fractions extending previous studies performed in Kutak and Kwiecinski [Screening effects in the ultrahigh-energy neutrino interactions, <a href="http://dx.doi.org/10.1140/epjc/s2003-01236-y">Eur. Phys. J. C 29, 521 (2003)</a>.] and in Kutak and Stasto [Unintegrated gluon distribution from modified BK equation, <a href="http://dx.doi.org/10.1140/epjc/s2005-02223-0">Eur. Phys. J. C 41, 343 (2005)</a>.]. The linear part of the equation is motivated by the renormalization group improved small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> approach, which accounts for resummation of higher orders, and includes collinear splitting functions and kinematical constraints. The solution to the equation is then used to perform the fit to deep inelastic scattering reduced cross section data.

Poland

Krzysztof Kutak

Wanchen Li

Anna Stasto

Robert Straka

Revisiting the gluon density from the Balitsky-Kovchegov equation with kinematical constraints and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> terms

A stable numerical solution of the impact-parameter-dependent next-to-leading order Balitsky-Kovchegov equation is presented for the first time. The rapidity evolution of the dipole amplitude is discussed in detail. Dipole amplitude properties, such as the evolution speed or anomalous dimension behavior, are studied as a function of the impact parameter and the dipole size and compared to solutions of the impact-parameter-dependent leading-order Balitsky-Kovchegov equation with the collinearly improved kernel. The next-to-leading evolution presented here also strongly suppresses the Coulomb tails compared to the collinearly improved and leading order solutions. The origin of this behavior is explored and potential avenues to improve the next-to-leading order kernel are pointed out.

Czechia

J. Cepila

J.G. Contreras

M. Matas

M. Vaculciak

Impact-parameter-dependent solutions to the Balitsky-Kovchegov equation at next-to-leading order

The ongoing Run 3 at the Large Hadron Collider (LHC) is substantially increasing the luminosity delivered to the experiments during Run 1 and Run 2. The advent of the high-luminosity upgrade of the LHC (Run 4 to 6), as well as the improvements to all detectors, will allow for the collection of an unprecedented amount of data in the next decade. This opens the possibility of performing measurements which have been limited by the smallness of the available data samples. This is the case of multidifferential studies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">J</mi><mo>/</mo><mi>ψ</mi></math>, as well as of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi></math> excited states, in exclusive diffractive photon-induced interactions. Here, we present predictions for the cross sections of these processes utilizing the dipole amplitude from the Balitsky-Kovchegov (BK) equation solved in the target rapidity and including the full impact-parameter dependence. Cross sections are computed as a function of the photon-proton center-of-mass energy as well as a function of Mandelstam-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>. Ratios of cross sections for different states and for the same state at different Mandelstam-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math> values are also presented. The contribution to these observables of the nonlinear terms in the BK equation is discussed.

Exclusive quarkonium photoproduction: Predictions with the Balitsky-Kovchegov equation including the full impact-parameter dependence

We apply Bayesian inference to determine the posterior likelihood distribution for the parameters describing the initial condition of the small-<math><mi>x</mi></math> Balitsky-Kovchegov evolution equation at leading logarithmic accuracy. The HERA structure function data is found to constrain most of the model parameters well. In particular, we find that the HERA data prefers an anomalous dimension <math><mi>γ</mi><mo>≈</mo><mn>1</mn></math> unlike in previous fits where <math><mi>γ</mi><mo>&gt;</mo><mn>1</mn></math> which led to, e.g., the unintegrated gluon distribution and the quark-target cross sections not being positive definite. The determined posterior distribution can be used to propagate the uncertainties in the nonperturbative initial condition when calculating any other observable in the color glass condensate framework. We demonstrate this explicitly for the inclusive quark production cross section in proton-proton collisions and by calculating predictions for the nuclear modification factor for the <math><msub><mi>F</mi><mn>2</mn></msub></math> structure function in the EIC and LHeC/FCC-he kinematics.

Mikko Karhunen

Inferring the initial condition for the Balitsky-Kovchegov equation

In this study, we employ the homogeneous balance method to obtain an analytical solution to the Balitsky-Kovchegov equation with running coupling constant. We utilize two distinct prescriptions of the running coupling scale, namely the saturation scale dependent running coupling and the dipole momentum dependent running coupling. By fitting the proton structure function experimental data, we determine the free parameters in the analytical solution. The resulting <math><msup><mi>χ</mi><mn>2</mn></msup><mo>/</mo><mrow><mi>d</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>f</mi><mo>.</mo></mrow></math> values are determined to be 1.07 and 1.43, respectively. With these definitive solutions, we are able to predict exclusive <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production and demonstrate that analytical solutions with running coupling are in excellent agreement with <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> differential and total cross section. Furthermore, our numerical results indicate that the analytical solution of the BK equation with running coupling constant can provide a reliable description for both the proton structure function and exclusive vector meson production.

Yanbing Cai

Xiaopeng Wang

Xurong Chen

Analytic solution of the Balitsky-Kovchegov equation with a running coupling constant using the homogeneous balance method

Solutions of the target-rapidity Balitsky-Kovchegov (BK) equation are studied considering, for the first time, the complete impact-parameter dependence, including the orientation of the dipole with respect to the impact-parameter vector. In our previous work [1] it has been demonstrated that the spurious Coulomb tails could be tamed using the collinearly-improved kernel and an appropriate initial condition in the projectile-rapidity BK equation. Introducing a different interpretation of the evolution variable, the target-rapidity formulation of the BK equation brings non-locality in rapidity and a kernel modification, removing the term that previously helped to suppress the Coulomb tails. To address this newly emerged non-locality, three different prescriptions are explored here to take into account the rapidities preceding the initial condition. Two of these approaches induce mild Coulomb tails, while the other is free from this effect within the studied rapidity range. The range is chosen to correspond to that of interest for existing and future experiments. To demonstrate that this set up can be used for phenomenological studies, the obtained solutions are used to compute the <math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> structure function of the proton and the diffractive photo- and electro-production of J/ψ off protons. The predictions agree well with HERA data, confirming that the target-rapidity Balitsky-Kovchegov equation with the full impact-parameter dependence is a viable tool to study the small Bjorken-x limit of perturbative QCD at current facilities like RHIC and LHC as well as in future colliders like the EIC.

Solutions to the Balitsky-Kovchegov equation including the dipole orientation

Effects of non-linear small-x evolution of the gluon distribution given by the Balitsky–Kovchegov equation are analyzed within the collinear approximation framework. We perform a twist decomposition of the proton structure functions $$F_2$$ <math> <msub> <mi>F</mi> <mn>2</mn> </msub> </math> and $$F_{\textrm{L}}$$ <math> <msub> <mi>F</mi> <mtext>L</mtext> </msub> </math> obtained from the Balitsky–Kovchegov equation using the Mellin representation of the scattering cross-sections at high energies. In both the structure functions we find strong corrections coming from the non-linear effects in the gluon evolution at twist 2, and strongly suppressed higher twist effects. This implies that unitarization effects of high energy scattering amplitudes are mostly the leading twist effect. Furthermore we consider the double logarithmic limit of the Balitsky–Kovchegov equation for the collinear gluon distribution, and compare the result to the Gribov–Levin–Ryskin equation. We find that these two equations differ by two powers of the hard scale logarithm for the large scales.

Leszek Motyka

Mariusz Sadzikowski

Twist decomposition of non-linear effects in Balitsky–Kovchegov evolution of proton structure functions

Exclusive vector meson production is an excellent probe for describing the structure of protons. In this study, based on the dipole model, the differential cross sections, total cross sections, and ratios of the longitudinal to transverse cross section of the   $ J/\psi $   and   $ J/\psi $   productions are calculated with the analytical solution of the Balitsky-Kovchegov (BK) equation. In addition, we consider the influences of two meson wave function models on the results. Our predictions, which are slightly sensitive to meson wave functions, agree with the experimental data. The analytical solution of the BK equation is reliable for description of exclusive vector meson productions in a certain range of   $ J/\psi $  .

Xiao-Peng Wang

Wei Kou

Gang Xie

Ya-Ping Xie

Xu-Rong Chen

Exclusive vector meson production with the analytical solution of Balitsky-Kovchegov equation  