arXiv subjects

Yuri V. Kovchegov

Publications and source records attributed to Yuri V. Kovchegov.

At least 19 recordsLinked to original sources

arXiv2026

Orbital angular momentum at small $x$ in the large $N_c\&N_f$ limit

We extend the small-$x$ analysis of the quark and gluon orbital angular momentum (OAM) distributions in the proton from the large-$N_c$ limit considered in our earlier works to the large-$N_c\&N_f$ limit, in which the numbers of quark colors $N_c$ and flavors $N_f$ are large with their ratio held fixed. Working in the double-logarithmic approximation (DLA), summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling and $x$ the proton's momentum fraction carried by a parton, we correct the small-$x$ operator expression for the quark OAM distribution suggested earlier in \cite{Kovchegov:2024wjs} and relate both the quark and gluon OAM distributions to the impact-parameter moments of the polarized dipole amplitudes. We derive the large-$N_c\&N_f$ evolution equations for these moment amplitudes in the DLA; these include a new equation for the moment amplitude governing the quark OAM. We then solve these evolution equations numerically together with the helicity evolution for $N_f = 2,3,4,5,6$ and $N_c =3$. We find that, similar to the large-$N_c$ case, the OAM distributions share a common small-$x$ intercept with the helicity distributions, $L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim ΔΣ(x,Q^2) \sim ΔG(x,Q^2) \sim \left(\frac{1}{x} \right)^{α_h}$ with the intercept $α_h \approx {3.48 \sqrt{α_s N_c/2π}}$ for $N_f = N_c = 3$ (cf.~\cite{Borden:2025ehe}): this result, along with $α_h$ for other values of $N_f \neq 0$ that we studied, is smaller than the intercept of $3.66\sqrt{α_s N_c/2π}$ found in the large-$N_c$ limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as $x\to 0$, obtaining $L_{q+\bar{q}}(x,Q^2)/ΔΣ(x,Q^2) \approx -1.01$ and $L_G(x,Q^2)/ΔG(x,Q^2) \approx -1.94$ at $Q^2=10\, \mathrm{GeV}^2$, with both ratios being nearly independent of $N_f$.

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arXiv2026

Unpolarized GPDs at small $x$ and non-zero skewness

We study the small-$x$ asymptotics of unpolarized generalized parton distributions (GPDs) and generalized transverse momentum distributions (GTMDs). Unlike the previous works in the literature, we consider the case of non-zero (but small) skewness while allowing for non-linear contributions to the evolution equations. We show that unpolarized GPDs and GTMDs at small $x$ are related to the eikonal dipole amplitude $N$, whose small-$x$ evolution is given by the BK/JIMWLK evolution equations, and to the odderon amplitude $\cal O$, whose evolution is also known in the literature. We show that the effect of non-zero skewness $ξ\neq 0$ is to modify the value of the evolution parameter (rapidity) in the arguments for the dipole amplitudes $N$ and $\cal O$ from $Y = \ln (1/x)$ to $Y = \ln \min \left\{ 1/|x| , 1/|ξ| \right\}$.

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arXiv2026

On the Two $R$-Factors in the Small-$x$ Shockwave Formalism

There are two $R$-factors frequently used in the phenomenology of exclusive processes at small values of the Bjorken $x$ variable. One $R$-factor takes into account the effects of non-zero longitudinal momentum transfer, which is assumed to be zero in the dipole scattering amplitude. Another $R$-factor accounts for the real part of the elastic scattering amplitude which is often neglected, with the standard dipole scattering amplitude giving only the imaginary part of the elastic amplitude. In this work we present two new theoretical developments aimed at eliminating the need for the two $R$-factors. We argue that the $R$-factors can be replaced by (i) modifying the argument of the dipole scattering amplitude and by (ii) augmenting the initial conditions for its non-linear small-$x$ evolution. Specifically, we show that to account for the effects of non-zero skewness $ξ$, one has to replace the rapidity argument $Y = \ln (1/x)$ of the eikonal dipole amplitude $N$ and the odderon dipole amplitude $\cal O$ by $Y = \ln \min \left\{ 1/|x|, 1/|ξ|\right\}$. The prescription applies to the elastic scattering cross sections, as well as for calculations of the Generalized Parton Distributions and Generalized Transverse Momentum Dependent parton distributions at small $x$ and at small but non-zero skewness $ξ$. We also show that the real part of the scattering amplitude, proportional to Im~$N$, which is intimately connected to the signature factor of the amplitude, can be accounted for by a more careful evaluation of the initial condition for the evolution and by writing the non-linear evolution equation in an integral form. One can similarly construct Im~$\cal O$ for the odd-signature odderon amplitude. We hope that future implementation of our prescriptions presented here will eliminate the need for both phenomenological $R$-factors.

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arXiv2025

Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations, where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-$N_c\&N_f$ limit and are double-logarithmic, resumming powers of $α_s\ln^2(1/x)$. Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, while $α_s$ is the strong coupling constant and $x$ is the Bjorken-$x$ variable. Using our solution, we obtain analytic expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function as double-inverse Laplace transforms. We also extract analytic expressions for the four DGLAP polarized anomalous dimensions $Δγ_{qq}, Δγ_{qG}, Δγ_{Gq}$, and $Δγ_{GG}$: these expressions resum powers of $α_s/ω^2$ to all orders at large-$N_c\&N_f$ (with $ω$ the Mellin moment variable). We extract the leading small-$x$ growth of the helicity distributions, \begin{align} ΔΣ(x,Q^2) \sim ΔG(x,Q^2)\sim g_1(x,Q^2) \sim \left(\frac{1}{x}\right)^{α_h}, \end{align} where the intercept $α_h$ satisfies an algebraic equation. We determine $α_h$ numerically for various values of $N_c$ and $N_f$. We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-$x$ asymptotic limit (for different $N_f/N_c$). We find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-$N_c$ case, our intercept $α_h$ exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework.

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arXiv2025

Weizsäcker-Williams Gluon Helicity Distribution and Inclusive Dijet Production in Longitudinally Polarized Electron-Proton Collisions

It is well-known that the back-to-back (correlation) limit of inclusive quark-antiquark dijet production in unpolarized high energy electron-proton collisions can probe the Weizsäcker-Williams (WW) gluon transverse momentum-dependent distribution (TMD) at small $x$ \cite{Dominguez:2010xd, Dominguez:2011wm}. In this paper, we consider a helicity-dependent version of the same process: we study the double-spin asymmetry for inclusive quark-antiquark dijet production in longitudinally polarized electron-proton scattering at high energies. We show that in the back-to-back limit this process probes the WW gluon helicity TMD. Furthermore, we derive the small-$x$ evolution equation for the operator related to the WW gluon helicity distribution. We find that in the double-logarithmic approximation and in the large-$N_c$ limit, the small-$x$ asymptotics of the WW gluon helicity distribution is governed by exactly the same evolution equation as that for the dipole gluon helicity distribution. The longitudinal double-spin asymmetry for inclusive dijet production in the longitudinally polarized electron-proton collisions can thus test the small-$x$ helicity evolution equations and facilitate constraining the initial conditions for phenomenology based on these equations.

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arXiv2025

First study of polarized proton-proton scattering with small-$x$ helicity evolution

We perform a phenomenological study of helicity-dependent parton distribution functions (PDFs) using small-$x$ helicity evolution equations, incorporating for the first time single-inclusive jet production data in polarized proton-proton ($pp$) scattering at parton momentum fractions $x < 0.1$. We also simultaneously include double-longitudinal spin asymmetries in inclusive and semi-inclusive deep-inelastic scattering probing $x < 0.1$. Employing the polarized small-$x$ pure-glue calculation of $pp\to gX$ for the jet production cross section, we modify the large-$N_c\&N_f$ KPS-CTT evolution equations by setting $N_f = 0$ to replicate the large-$N_c$ (pure-glue) limit, while retaining external quark flavors for the spinor field operators. We find that the $pp$ data have a considerable impact on the helicity PDFs at small $x$, reducing their uncertainties and leading to a total quark and gluon helicity in the proton for $x < 0.1$ of $-0.04 \pm 0.23$. Combining our analysis with the a recent JAM helicity PDF analysis of the world polarized data, which includes $x > 0.1$, we find a total quark and gluon helicity contribution for $x > 10^{-7}$ of between 0.02 and 0.51.

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arXiv2025

Elastic Dijet Production in Electron Scattering on a Longitudinally Polarized Proton at Small $x$: A Portal to Orbital Angular Momentum Distributions

We calculate the elastic production of dijets from electron collisions with a longitudinally polarized proton target at small values of the Bjorken $x$ variable. Building on the pioneering proposals of \cite{Hatta:2016aoc,Bhattacharya:2022vvo, Bhattacharya:2023hbq, Bhattacharya:2024sck} for measuring the quark and gluon orbital angular momentum (OAM) distributions, our focus is on both the longitudinal double spin asymmetry (DSA) and longitudinal single spin asymmetry (SSA). We compute the numerators of these asymmetries in the small-$x$ formalism of the light-cone operator treatment. Utilizing the small-$x$ expressions for the OAM distributions derived in our earlier paper, we demonstrate that the DSA provides a robust probe for both the quark and gluon OAM distributions within the proton. In contrast, we find that while the SSA is also sensitive to the OAM distributions, extraction of the latter from the SSA would require new developments in small-$x$ theory and phenomenology, and is probably not feasible at this point in time. These findings highlight the potential of DSA measurements in elastic dijet production at the future Electron-Ion Collider to provide the first-ever direct access to the quark and gluon OAM distributions at small $x$, paving the way for new insights into the proton spin puzzle.

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arXiv2024

Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated

We revisit the problem of small Bjorken-$x$ evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave ($s$-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength $F^{12}$ and quark "axial current" ${\barψ}γ^+γ^5ψ$ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of $α_s\,\ln^2(1/x)$ with $α_s$ the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, $\overleftarrow{D}^i\,{D}^i$, also sandwiched between the light-cone Wilson lines (with the repeated index $i=1,2$ summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, $D^i-\overleftarrow{D}^i$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this operator into small-$x$ helicity evolution, and construct a novel evolution mixing all three operators ($D^i-\overleftarrow{D}^i$, $F^{12}$, and ${\barψ}γ^+γ^5ψ$), generalizing the previous results. We also construct closed DLA evolution equations in the large-$N_c$ and large-$N_c\& N_f$ limits, with $N_c$ and $N_f$ the numbers of quark colors and flavors, respectively. Solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics of the quark and gluon helicity distributions $ΔΣ$ and $ΔG$, along with the $g_1$ structure function, \[ΔΣ(x,Q^2)\simΔG(x,Q^2)\sim g_1(x,Q^2)\sim\left(\frac{1}{x}\right)^{3.66\,\sqrt{\frac{α_s\,N_c}{2π}}},\] in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.

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arXiv2024

Orbital Angular Momentum at Small $x$ Revisited

We revisit the problem of the small Bjorken-$x$ asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised small-$x$ helicity evolution derived recently. We relate the quark and gluon OAM distributions at small $x$ to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the OAM distributions, we derive novel small-$x$ evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling constant). We solve these evolution equations numerically and extract the leading large-$N_c$, small-$x$ asymptotics of the quark and gluon OAM distributions, which we determine to be \begin{align} L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim ΔΣ(x, Q^2) \sim ΔG(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{α_s N_c}{2π}}}, \notag \end{align} in agreement with the existing results in the literature within the precision of our numerical evaluation. (Here $N_c$ is the number of quark colors.) We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-$x$ region.

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arXiv2024

Helicity Evolution at Small $x$: Quark to Gluon and Gluon to Quark Transition Operators

We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken-$x$ evolution equations for helicity in the flavor-singlet channel derived earlier. While such transitions do not affect the large-$N_c$ version of the evolution equations for helicity, the large-$N_c \& N_f$ equations are affected. ($N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large-$N_c \& N_f$ equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of $α_s \, \ln^2 (1/x)$ with $α_s$ the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results. Similar to the large-$N_c$ helicity evolution in the shock-wave approach, our large-$N_c \& N_f$ small-$x$ splitting functions agree with those obtained in the infrared evolution equations framework up to three loops, but appear to slightly disagree at four loops.

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arXiv2024

Novel Cross Section Ratios as Possible Signals of Saturation in UPCs

We propose new cross section ratios in ultra-peripheral A$+$A and $p+$A collisions (UPCs) as a potential experimental signal of saturation physics. We consider the ratio $R_1$ of elastic vector meson photoproduction cross section to the inclusive hadron or jet photoproduction cross section. The ratio can be measured in the $γ+$A and $γ+ p$ collisions taking place in the UPCs. We label the ratios $R_1 ({\rm A})$ and $R_1 (p)$, respectively. Constructing the double ratio $R_{\rm{UPC}} ({\rm A}) = R_1 ({\rm A})/R_1 (p)$, and performing a small-$x$ calculation both in the quasi-classical approximation and by including small-$x$ evolution, we observe that $R_{\rm{UPC}} ({\rm A})$ exhibits a markedly different dependence on the nuclear atomic number inside and outside the saturation region. This result indicates that $R_{\rm{UPC}} ({\rm A})$ and $R_1 ({\rm A})$ measurements in UPCs may help in the experimental searches for the evidence of saturation physics.

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arXiv2024

Gluon Double-Spin Asymmetry in the Longitudinally Polarized $p+p$ Collisions

We derive the first-ever small-$x$ expression for the inclusive gluon production cross section in the central rapidity region of the longitudinally polarized proton-proton collisions. The cross section depends on the polarizations of both protons, therefore comprising the numerator of the longitudinal double-spin asymmetry $A_{LL}$ for the produced gluons. The cross section is calculated in the shock wave formalism and is expressed in terms of the polarized dipole scattering amplitudes on the projectile and target protons. We show that the small-$x$ evolution corrections are included into our cross section expression if one evolves these polarized dipole amplitudes using the double-logarithmic helicity evolution derived in \cite{Kovchegov:2015pbl, Kovchegov:2016zex, Kovchegov:2018znm, Cougoulic:2022gbk}. Our calculation is performed for the gluon sector only, with the quark contribution left for future work. When that work is complete, the resulting formula will be applicable to longitudinally polarized proton-proton and proton-nucleus collisions, as well as to polarized semi-inclusive deep inelastic scattering (SIDIS) on a proton or a nucleus. Our results should allow one to extend the small-$x$ helicity phenomenology analysis of \cite{Adamiak:2023yhz} to the jet/hadron production data reported for the longitudinally polarized proton-proton collisions at RHIC and to polarized SIDIS measurements at central rapidities to be performed at the EIC.

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arXiv2023

Global analysis of polarized DIS & SIDIS data with improved small-$x$ helicity evolution

We analyze the world polarized deep-inelastic scattering (DIS) and semi-inclusive DIS (SIDIS) data at low values of $x < 0.1$, using small-$x$ evolution equations for the flavor singlet and nonsinglet helicity parton distribution functions (hPDFs). The hPDFs for quarks, antiquarks, and gluons are extracted and evolved to lower values of $x$ to make predictions for the future Electron-Ion Collider (EIC). We improve on our earlier work by employing the more realistic large-$N_c\, \& N_f$ limit of the revised small-$x$ helicity evolution, and incorporating running coupling corrections along with SIDIS data into the fit. We find an anti-correlation between the signs of the gluon and $C$-even quark hPDFs as well as the $g_1$ structure function. While the existing low-$x$ polarized DIS and SIDIS data are insufficient to constrain the initial conditions for the polarized dipole amplitudes in the helicity evolution equations, future EIC data will allow more precise predictions for hPDFs and the $g_1$ structure function for $x$ values beyond those probed at the EIC. Using the obtained hPDFs, we discuss the contributions to the proton spin from quark and gluon spins at small $x$.

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arXiv2023

Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\& N_f$

We present a numerical solution of the revised version of the small-$x$ helicity evolution equations at large $N_c$ and $N_f$. (Here $N_c$ and $N_f$ are the numbers of quark colors and flavors, respectively.) The evolution equations are double-logarithmic in the Bjorken $x$ variable, resumming powers of $α_s \, \ln^2 (1/x)$ with $α_s$ the strong coupling constant. The large-$N_c \& N_f$ evolution we consider includes contributions of small-$x$ quark emissions and is thus more realistic than the large-$N_c$ one, which only involves gluon emissons. The evolution equations are written for the so-called ``polarized dipole amplitudes", which are related to the helicity distribution functions and the $g_1$ structure function. Unlike the previously reported solution of the earlier version of helicity evolution equations at large $N_c \& N_f$, our solution does not exhibit periodic oscillations in $\ln (1/x)$ for $N_f < 2 N_c$, while only showing occasional sign reversals. For $N_f = 2 N_c$, we report oscillations with $\ln (1/x)$, similar to those found earlier. We determine the intercept of our evolution for $N_f < 2 N_c$ as well as the parameters of the oscillatory behavior for $N_f = 2 N_c$. We compare our results to the existing resummation and finite-order calculations for helicity-dependent quantities in the literature.

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arXiv2023

Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations derived recently. The equations we solve are obtained in the large-$N_c$ limit (with $N_c$ the number of quark colors) and are double-logarithmic (summing powers of $α_s \ln^2(1/x)$ with $α_s$ the strong coupling constant and $x$ the Bjorken $x$ variable). Our solution provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function, with their leading small-$x$ asymptotics given by \begin{align} ΔΣ(x, Q^2) \sim ΔG (x, Q^2) \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{α_h} , \notag \end{align} where the exact analytic expression we obtain for the intercept $α_h$ can be approximated by $α_h = 3.66074 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. Our solution also yields an all-order (in $α_s$) resummed small-$x$ anomalous dimension $Δγ_{GG} (ω)$ which agrees with all the existing fixed-order calculations (to three loops). Notably, our anomalous dimension is different from that obtained in the infrared evolution equation framework developed earlier by Bartels, Ermolaev, and Ryskin (BER), with the disagreement starting at four loops. Despite the previously reported agreement at two decimal points based on the numerical solution of the same equations, the intercept of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ given by a different analytic expression from ours with the numerical value of $α_h^{BER} = 3.66394 \, \sqrt{\frac{α_s \, N_c}{2 π}}$. We speculate on the origin of this disagreement.

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arXiv2022

Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution

We apply the formalism developed earlier for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken $x$ to construct the small-$x$ asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental "polarized Wilson line" operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small-$x$. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the "polarized Wilson line" and show that the dominant (eikonal) term which contributes to the quark Sivers function at small $x$ is the spin-dependent odderon, confirming the recent results of Dong, Zheng and Zhou. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou (see also the work by Szymanowski and Zhou). We also analyze the sub-eikonal corrections to the quark Sivers function using the constructed "polarized Wilson line" operator. We derive new small-$x$ evolution equations re-summing double-logarithmic powers of $α_s \, \ln^2 (1/x)$ with $α_s$ the strong coupling constant. We solve the corresponding novel evolution equations in the large-$N_c$ limit, obtaining a sub-eikonal correction to the spin-dependent odderon contribution. We conclude that the quark Sivers function at small $x$ receives contributions from two terms and is given by \begin{align} f_{1 \: T}^{\perp \: q} (x, k_T^2) = C_O (x, k_T^2) \, \frac{1}{x} + C_1 (k_T^2) \, \left( \frac{1}{x} \right)^0 + \ldots \end{align} with the function $C_O (x, k_T^2)$ varying slowly with $x$ and the ellipsis denoting the sub-asymptotic and sub-sub-eikonal (order-$x$) corrections.

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arXiv2022

T-odd Leading-Twist Quark TMDs at Small $x$

We study the small-$x$ asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-$x$ asymptotics of the quark Sivers function is given by the spin-dependent odderon, we are interested in revisiting the sub-eikonal correction considered by us earlier. We first simplify the expressions for both TMDs at small Bjorken $x$ and then construct small-$x$ evolution equations for the resulting operators in the large-$N_c$ limit, with $N_c$ the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter $α_s \, \ln^2 (1/x)$, where $α_s$ is the strong coupling constant, which is assumed to be small. Solving these evolution equations numerically (for the Sivers function) and analytically (for the Boer-Mulders function) we arrive at the following leading small-$x$ asymptotics of these TMDs at large $N_c$: \begin{align} f_{1 \: T}^{\perp \: NS} (x \ll 1 ,k_T^2) & = C_O (x, k_T^2) \, \frac{1}{x} + C_1 (x, k_T^2) \, \left( \frac{1}{x} \right)^{3.4 \, \sqrt{\frac{α_s \, N_c}{4 π}}} , \notag \\ h_1^{\perp \, \textrm{NS}} (x \ll 1, k_T^2) & = C (x, k_T^2) \left( \frac{1}{x} \right)^{-1}. \notag \end{align} The functions $C_O (x, k_T^2)$, $C_1 (x, k_T^2)$, and $C (x, k_T^2)$ can be readily obtained in our formalism: they are mildly $x$-dependent and do not strongly affect the power-of-$x$ asymptotics shown above. The function $C_O$, along with the $1/x$ factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with $C_1$), our result shown above supersedes the one obtained in our previous work due to the new contributions identified recently.

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arXiv2022

Helicity Evolution at Small $x$: the Single-Logarithmic Contribution

We calculate single-logarithmic corrections to the small-$x$ flavor-singlet helicity evolution equations derived recently in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of $α_s \ln(1/x)$, which are an important correction to the dominant powers of $α_s \ln^2(1/x)$ summed up by the double-logarithmic kernel at small values of Bjorken $x$ and with $α_s$ the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive employing the light-cone perturbation theory simultaneously include the small-$x$ evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.

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