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Single-Logarithmic Corrections to Small-$x$ Helicity Evolution
The small- x x quark helicity evolution equations at double-logarithmic order, with the kernel \sim\alpha_s\ln^2(1/x) ∼ α s ln 2 ( 1 / x ) , have been derived previously. In this work, we derive the single-logarithmic corrections to the equations, to order \alpha_s\ln(1/x) α s ln ( 1 / x ) of the evolution kernel. The new equations include the effects of the running coupling and the unpolarized small- x x evolution, both of which are parametrically significant at single-logarithmic order. The large- N_c N c and large- N_c\ N_f N c N f approximations to the equation are computed. (Here, N_c N c and N_f N f are the numbers of quark colors and flavors, respectively.) Their solutions will provide more precise estimates of the quark helicity distribution at small x x , contributing to the resolution of the proton spin puzzle.
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 [1, 2], we are interested in revisiting the sub-eikonal correction considered by us earlier in [3]. 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 :$$ {\displaystyle \begin{array}{l}{f}_{1T}^{\perp NS}\left(x\ll 1,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{3.4\sqrt{\frac{\alpha_s{N}_c}{4\pi }}}\\ {}{h}_1^{\perp \textrm{NS}}\left(x\ll 1,{k}_T^2\right)=C\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{-1}.\end{array}} $$ 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 [3] due to the new contributions identified recently in [4].
Small $x$ physics beyond eikonal approximation: an effective Hamiltonian approach
Understanding the spin structure of hadrons in the small x regime is an important direction to unravel the spin puzzle in hadronic physics. To include spin degrees of freedom in the small x regime requires going beyond the usual eikonal approximation in high energy QCD. We developed an effective Hamiltonian approach to study spin related observables in the small x regime using the shockwave formalism. The small-x effective Hamiltonian incorporates both quark and gluon propagators in the background fields and the background field induced interaction vertices up to next-to-eikonal order. A novel feature of sub-eikonal interactions is the background gluon field induced gluon radiation inside the shockwave. Its relation to chromo-electrically polarized Wilson line correlator is established both in small x helicity evolution and in longitudinal double-spin asymmetry for gluon production.
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 first 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 $\mathscr{O}$, whose evolution is also known in the literature. We then show that the effect of non-zero skewness ξ ≠ 0 is to modify the value of the evolution parameter (rapidity) in the arguments for the dipole amplitudes N and $\mathscr{O}$ from Y = ln(1/x) to Y = ln min{1/|x|, 1/|ξ|}.
Proceedings of RIKEN BNL Research Center Workshop: Small-x Physics in the EIC Era [Slides]
Understanding the high energy limit of hadronic and nuclear collisions is at the forefront of nuclear and particle physics. When boosted to ultrahigh energies, all hadrons and nuclei eventually transform into a universal form of matter called the Color Glass Condensate (CGC). The CGC is characterized by the high density (saturation) of small-x gluons which leads to distinct experimental signatures. Tantalizing hints of the CGC have been observed at HERA, RHIC and the LHC, but the prospects for the discovery of the CGC are more promising at future experiments such as the Electron-Ion Collider (EIC) at BNL Indeed, according to the National Science Academy report published in 2019, one of the three major goals of the EIC is to address the nature of the gluon saturation. (The other two are the mass and spin structure of the nucleons.) With this in mind, the small-x community is gearing up to meet the challenges of the EIC era. Over the past several years, there has been impressive progress in the next-to-leading order (NLO) calculations in the CGC framework of various observables such as single hadron production in proton-nucleus collisions, inclusive and exclusive dijet and trijet production in Deep Inelastic Scattering (DIS), jet-plus-photon production in DIS, etc. We expect that NLO calculations will be the standard tool to confront future experimental data at the EIC. Another emerging trend of the community is the interplay between small-x physics and spin physics. The RHIC result for the gluon helicity ΔG has underscored the necessity to understand the longitudinal spin structure of the proton at small-x. There have been theoretical indications that a significant fraction of spin and orbital angular momentum is stored in the small-x region. As for the transversely polarized proton, a surprising new connection between the gluon Sivers function at small-x and the QCD Odderon has been pointed out and its implications at the EIC has been discussed. In view of these developments, we think it is timely to organize a dedicated workshop on small-x physics to summarize the present status of the field and to discuss future directions. A major focus of this workshop will be to identify outstanding problems that could significantly benefit from collaborative efforts amongst scientists working on formal, phenomenological, and computational aspects of small-x physics.
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 in [1]. 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 ${L}_{q+\overline{q}}\left(x,{Q}^2\right)\sim {L}_G\left(x,{Q}^2\right)\sim \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {\left(\frac{1}{2}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2_π}}}$in agreement with [2] 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.
Proton Spin from Small-x With Constraints from the Valence Quark Model
We apply the valence quark model to constrain the non-perturbative initial condition for the small-x helicity evolution. The remaining free parameters are constrained by performing a global analysis to the available polarized small-x deep inelastic scattering data. A good description of the world data is obtained with only 8 free parameters. The model parameters are tightly constrained by the data, allowing us to predict the proton polarized structure-function to be negative at small x. Furthermore, we obtain the small-x quark and gluon spins, depending on the applied running coupling prescription.
Proton spin from small-x with constraints from the valence quark model
We apply the valence quark model [1] to constrain the non-perturbative initial condition for the small-x helicity evolution. The remaining free parameters are constrained by performing a global analysis akin to [2] to the available polarized small-x deep inelastic scattering data. A good description of the world data is obtained with only 8 free parameters. The model parameters are tightly constrained by the data, allowing us to predict the proton polarized structure function $g^p_1$ to be negative at small x. Furthermore, we obtain the small-x quark and gluon spins to give a contribution $\int^{0.1}_{10^{-5}} dx (\frac{1}{2}\Delta\Sigma + \Delta{G}) = 0.63 ± 0.10$ or $1.35 ± 0.16$ (or to the proton spin, depending on the applied running coupling prescription.
Back-to-back inclusive dijets in DIS at small x : Sudakov suppression and gluon saturation at NLO
Back-to-back dijet cross-sections in deeply inelastic scattering (DIS) at small xBj are suppressed by many-body multiple scattering and screening effects arising from gluon saturation at high parton densities. They are similarly sensitive in these kinematics to large Sudakov logarithms from soft gluon radiation. Uncovering novel physics in this DIS channel therefore requires understanding the interplay of the two phenomena. In this work, we compute the small $x_\text{Bj}$ inclusive dijet DIS cross-section in back-to-back kinematics at next-to-leading order (NLO) in the Color Glass Condensate effective field theory (CGC EFT). Our result includes, for the first time, all real and virtual NLO contributions to the impact factor. These include all Sudakov double and single logarithm contributions, as well as all other finite $\mathcal{O}(α_s)$ terms that contribute at this order. We demonstrate explicitly that resummations of small x and Sudakov logarithms can be performed simultaneously in the CGC EFT. This requires that the JIMWLK kernel for small x evolution of the Weizsäcker-Williams (WW) gluon distribution satisfies a kinematic constraint imposed by lifetime ordering of successive gluon emissions; the corresponding modifications to the kernel, corresponding to resummations of large double transverse logarithms, are precisely of the type required to stabilize JIMWLK evolution beyond leading logarithmicaccuracy. We compute the azimuthal harmonics of the NLO back-to-back distributions and show their sensitivity to both the unpolarized and linearly polarized WW gluon distributions. Finally, we discuss how TMD factorization is broken by an emergent saturation scale at small $x$.
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 re- cent 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. 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 ${f}_{1T}^{\perp q}\left(x,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left({k}_T^2\right){\left(\frac{1}{x}\right)}^0+\cdots$ with the function C O ( x, ${k}_T^2$) varying slowly with x and the ellipsis denoting the subasymptotic and sub-sub-eikonal (order- x ) corrections.
Single inclusive hadron production in DIS at small x: next to leading order corrections
We calculate the one-loop corrections to single inclusive hadron production in Deep Inelastic Scattering (DIS) at small x in the forward rapidity region using the Color Glass Condensate formalism. We show that the divergent parts of the next to leading order (NLO) corrections either cancel among each other or lead to x (rapidity) evolution of the leading order (LO) dipole cross section according to the JIMWLK evolution equation and DGLAP evolution of the parton-hadron fragmentation function. The remaining finite parts constitute the NLO (α s ) corrections to the LO single inclusive hadron production cross section in DIS at small x.
Small- x factorization from effective field theory
We derive a factorization theorem that allows for resummation of small-x logarithms by exploiting Glauber operators in the soft collinear effective field theory. Our analysis is carried out for the hadronic tensor W μν in deep inelastic scattering, and leads to the definition of a new gauge invariant soft function S μν that describes quark and gluon emission in the central region. This soft function provides a new framework for extending resummed calculations for coefficient functions to higher logarithmic orders. Our factorization also defines impact factors by universal collinear functions that are process independent, for instance being identical in small-x DIS and Drell-Yan.
Orbital angular momentum small- x evolution: exact results in the large- N c limit
We construct an exact solution to the revised small-x orbital angular momentum (OAM) evolution equations derived in [1], based on an earlier work [2]. These equations are derived in the double logarithmic approximation (summing powers of α s ln 2 (1/x) with α s the strong coupling constant and x the Bjorken x variable) and the large-N c limit, with N c the number of quark colors. From our solution, we extract the small-x, large-N c expressions of the quark and gluon OAM distributions.
Dihadron correlations in small- x DIS at NLO: transverse momentum dependent fragmentation
We compute the inclusive dihadron cross-section in Deep Inelastic Scattering at next-to-leading order (NLO) and small x in the Color Glass Condensate. We focus on the kinematic limit where the hadrons are produced at forward rapidities (in the direction of the virtual photon) and back-to-back in the transverse plane. Our calculation demonstrates that the coefficient of the Sudakov double logarithm for this process is –$\frac{αs}{2π}$ [C F + $\frac{Nc}{2}$] instead of –$\frac{αsNc}{4π}$ when back-to-back jets are measured in the final state. To preserve the universality of the Sudakov soft factor associated with the Weizsäcker-Williams transverse momentum dependent (TMD) gluon distribution, we promote the collinear fragmentation functions into TMD fragmentation functions. We then perform the resummation of the Sudakov logarithms through Collins-Soper-Sterman evolution of the TMD fragmentation functions and the Weizsäcker-Williams TMD gluon distribution. Finally, analytic expressions are obtained for the NLO coefficient functions in the $\overline{MS}$-scheme. These results pave the way towards numerically calculating dihadron correlations at small x at the future Electron-Ion Collider with full NLO accuracy.
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 [1–3] 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 from [1–3] 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.
Helicity at small x: oscillations generated by bringing back the quarks
We construct a numerical solution of the recently-derived large-N c &N f small-x helicity evolution equations with the aim to establish the small-x asymptotics of the quark helicity distribution beyond the large-N c limit explored previously in the same framework. (Here N c and N f are the numbers of quark colors and flavors.) While the large-N c helicity evolution involves gluons only, the large-N c &N f evolution includes contributions from quarks as well. We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of x. Our numerical results in the large-N c &N f limit lead to the x-dependence of the flavor-singlet quark helicity distribution which is well-approximated by $$ {\left.\Delta \Sigma \left(x,{Q}^2\right)\right|}_{\mathrm{large}\hbox{-} {N}_c\&{N}_f}\sim {\left(\frac{1}{x}\right)}^{\alpha_h^q}\cos \left[{\omega}_q\ln \left(\frac{1}{x}\right)+{\varphi}_q\right]. $$. The power $α^{q}_{h}$ exhibits a weak N f -dependence, and, for all Nf values considered, remains very close to $α^{q}_{h}$(N f =0)=$\left(4/\sqrt{3}\right)\sqrt{\alpha_s{N}_c/\left(2\pi \right)}$ obtained earlier in the large-N c limit. The novel oscillation frequency ω q and phase shift φ q depend more strongly on the number of flavors N f (with ω q = 0 in the pure-glue large-N c limit). The typical period of oscillations for ΔΣ is rather long, spanning many units of rapidity. We speculate whether the oscillations we find are related to the sign variation with x seen in the strange quark helicity distribution extracted from the data.
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” $\overline{\psi}\gamma$ + γ 5 ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (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, ${}_D{}^{\leftarrow i} {}_D{}^{i}$, also sandwiched between the light-cone Wilson lines (with the repeated transverse 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, ${}_D{}^{i} {}_D{}^{\leftarrow i}$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (${}_D{}^{i} {}_D{}^{\leftarrow i}$, F 12 , and $\overline{\psi}\gamma$ + γ 5 ψ), generalizing the results of . We also construct closed double-logarithmic 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, $$ \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {g}_1\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\pi }}} $$ in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.