Helicity Evolution at Small x: Summary of Recent Developments

We construct small-$x$ evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \&\, N_f$ limits. We construct a numerical solution of the helicity evolution equations in the large-$N_c$ limit. Employing the extracted intercept, we are able to predict directly from theory the behavior of the quark helicity PDFs at small $x$, which should have important phenomenological consequences. We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called"spin crisis."Based on JHEP 1601 (2016) 072 (arXiv:1511.06737), arXiv:1610.06197 and arXiv:1610.06188.


I. INTRODUCTION
These proceedings are based on the work done in [1][2][3]. Our aim is to derive perturbative QCD prediction for the asymptotic small Bjorken x behavior of the quark and gluon helicity distribution functions and for related observables. In these proceedings we will concentrate on the flavor-singlet quark helicity distribution ∆q(x, Q 2 ). We will derive helicity evolution equations resumming powers of α s ln 2 (1/x) with α s the strong coupling constant: this resummation is referred to as the double-logarithmic approximation (DLA). These evolution equations allow one to determine the leading perturbative behavior of the small-x asymptotics of ∆q(x, Q 2 ) (see [2,3]). Such theoretical input is necessary to assist the efforts to determine the small-x part of the quark contribution to proton spin ∆Σ(x, Q 2 ) = ∆u + ∆ū + ∆d + ∆d + . . . (x, Q 2 ), (1) where the helicity parton distribution functions (hPDFs) are The ultimate goal of determining the proton spin carried by quarks [and the spin carried by the gluons, S G (Q 2 )] is to resolve the proton spin crisis. * Email: kovchegov.1@osu.edu † Email: dap67@psu.edu ‡ Email: sievertmd@lanl.gov

II. THE OBSERVABLES
The small-x helicity observables can be obtained by studying the cross section for semi-inclusive deep inelastic scattering (SIDIS) on a longitudinally polarized target, γ * + p → q + X. The contributions are shown diagrammatically in Fig. 1 (see [4] for a derivation).
The corresponding flavor-singlet quark helicity transverse momentum-dependent parton distribution function (TMD) is [1] The notation is explained in Fig. 1. Here k = (k x , k y ) denotes transverse vectors, with k ⊥ = |k|. Variable z denotes the fraction of the virtual photon's longitudinal momentum carried by the anti-quark with z i = Λ 2 /s, where Λ is the infra-red (IR) cutoff, and s is the SIDIS center-of-mass energy squared. The object G is the polarized dipole amplitude, which is defined by [1] where b = (1/2)(x 1 + x 0 ). The propagator of an eikonal quark with polarization σ in the background quark or gluon field of the target proton is written as Diagrams contributing to the small-x SIDIS process on a longitudinally polarized target, and to quark helicity TMD g q 1L (x, kT ). where is the light-cone Wilson line, and V pol is the helicitydependent sub-eikonal correction. The double angle brackets indicate averaging in the target wave function, with an inverse factor of center-of-mass energy squared scaled out: The polarized dipole amplitude can be used to obtain other helicity observables. The flavor-singlet quark helicity PDF, at small-x is equal to The g 1 structure function is where ψ T and ψ L are the well-known light cone wave functions for the γ * → qq splitting (see e.g. [1]).
Our aim is to find the small-x evolution equations for the polarized dipole amplitude G 10 (z). Once G 10 (z) is found, we can use Eqs. (3), (9) and (10) to construct the flavor-singlet quark helicity TMD, PDF and the g 1 structure function.

III. LARGE-Nc LIMIT
Similar to the case of JIMWLK evolution and Balitsky hierarchy, the general evolution equation for G 10 (z) does not close: on its right-hand side it contains operator expectation values other than G 10 (z). The operators on the right-hand side contain higher number of Wilson lines than G 10 (z). This leads to the helicity evolution analogue of the Balitsky hierarchy.
However, also similar to the unpolarized (BK) case, the evolution equations become closed equations involving G 10 (z) in the large-N c limit. In addition, specific to the helicity evolution case at hand, evolution equations also close in the large-N c & N f limit. Below we will first discuss the large-N c case.
Here we simply quote the results, referring the reader to the derivation details in [1,3]. Similar to [5], our evolution equations also resum the leading-logarithmic (LLA) powers of α s ln(1/x) by including the BK/JIMWLK evolved unpolarized dipole S-matrix where we assume that which is true at LLA. Note that LLA terms of the pure helicity evolution are not systematically included in this approach: hence we do not have a complete LLA calculation, and our results are strictly correct only in the DLA limit where S 01 (z) = 1. The evolution equation for G 10 (z) is illustrated in the top line of Fig. 2. Note that the large-N c limit is gluondominated: hence the dipole 10 is made out of quark and anti-quark lines of the large-N c gluon. The equation where Γ 02, 21 (z ) is the new object (as compared to the unpolarized evolution), characteristic of helicity evolution. Γ 02, 21 (z ) is the "neighbor dipole" amplitude. Its evolution is described in the bottom line of Fig. 2. As shown in the figure, the "neighbor" dipole evolution continues in dipole 02, but the information about the dipole 21 comes in through the transverse size integration limit.
(This is in contrast to unpolarized evolution, where the evolution in, say, dipole 02 does not depend on the size of the dipole 21 or on anything else outside the dipole 02.) The evolution for the neighbor dipole amplitude reads (ρ 2 = 1/(z s)) Eqs. (13) and (14), when augmented by the BK evolution for S, present a closed system of equations. The initial conditions G (0) and Γ (0) are given by the Born-level interactions, enhanced by multiple rescatterings which bring in saturation effects.
In the strict DLA limit we can simplify Eqs. (13) and (14) by putting S = 1 and assuming that G 21 = G 12 . We Helicity evolution equations also close in the large-N c & N f limit. To write down these new evolution equations we need to define a couple of new objects. In addition to G 10 (z) defined in Eq. (4a) above, which is made out of quark and anti-quark lines of gluons (with x 1 line polarized), let us define with x 1 being a true quark or anti-quark polarized line and x 0 being the (anti-)quark line of the gluon, and with both x 0 and x 1 being true quark and anti-quark lines and x 1 polarized. Equation (18) is illustrated diagrammatically in the first line of Fig. 3. The equation for G is now dz z Note a new object,Γ 02, 21 , which is the neighbor dipole amplitude with line 2 being an actual polarized quark (or anti-quark), and, unlike in Γ 02, 21 , not a quark (or anti-quark) line of a large-N c gluon. Equation (19) is illustrated diagrammatically in the second line of Fig. 3.

Finally, the evolution for A 01 (z) reads
A 10 (z) = A It is depicted in the last line of Fig. 3.
Note that Eq. (14) for the neighbor dipole amplitude also has to be modified yielding  In the pure DLA limit we linearize all these equations by putting S = 1 in them (we again assume that G 01 = G 10 , which is true for a large, longitudinally polarized target): dz z The linearized equations for Γ andΓ in the large- as well as rescaling all η's and s ij 's, Using these variables, we write the large-N c helicity evolution equations (15) Since the equations at hand are linear, and we are mainly interested in the highenergy intercept, we can scale out α 2 s π C F /N c . In order to solve Eqs. (27), we first write down a discretized version of them where G ij ≡ G(s i , η j ), Γ ijk ≡ Γ(s i , s k , η j ), and with η max the maximum η value and N η the number of grid steps in the η direction, and likewise for s max , N s . The discretized equations (29) are exact in the limit ∆η , ∆s → 0 and η max , s max → ∞. To optimize the numerics, we set η max = s max . With the discretized evolution equations (29) in hand (along with the initial conditions (28) suitably discretized), we first choose values for η max = s max and ∆η = ∆s. We then systematically go through the η-s grid in such a way that each G ij (and Γ ijk ) only depends on G, Γ values that have already been calculated. Thus, we can determine G ij for each i, j. Our numerical solution (for η max = 40, ∆η = 0.05) is plotted in Fig. 5. We find where we have reinstated the factor α s N c /2π originally scaled out by Eq. (26). We note that the value in Eq. (32) is in disagreement with the "pure glue" intercept of 3.66 α s N c /2π obtained by BER [6] by about 35%. In [3] we identify DLA diagram contributions not included by the authors of [6] in their treatment of the problem. We believe that omitting those diagrams limited the resummation of [6] to the leading-twist contribution only. In comparison, our result (32) resums all twists at small x.
Interestingly, the leading twist approximation to α P −1 in BFKL evolution is larger than the exact all-twist intercept by about 30% [7]; it is possible that something similar is occurring for helicity evolution. In Ref. [3], we have explored this possibility, performed various analytical cross-checks of our helicity evolution equations, and compared to BER where possible; we have not found any inconsistencies in our result.

VI. IMPACT ON THE PROTON SPIN
In order to determine the quark and gluon spin based on Eq. (1), one needs to extract the helicity PDFs. There are several groups who have performed such analyses, e.g., DSSV [8,9], JAM [10,11], LSS [12][13][14], NNPDF [15,16]. While the focus at small x has been on the behavior of ∆G(x, Q 2 ), there is actually quite a bit of uncertainty in the size of ∆Σ(x, Q 2 ) in that regime as well. Let us define the truncated integral One finds for DSSV14 [9] that the central value of the full integral ∆Σ [0] (10 GeV 2 ) is about 40% smaller than ∆Σ [0.001] (10 GeV 2 ). The NNPDF14 [16] helicity PDFs lead to a similar decrease, although, due to the nature of neural network fits, the uncertainty in this extrapolation is 100%. On the other hand, for JAM16 [11] helicity PDFs the decrease from the truncated to the full integral of ∆Σ(x, Q 2 ) seems to be at most a few percent. The origin of this uncertainty, and more generally the behavior of ∆Σ(x, Q 2 ) at small x, is mainly due to varying predictions for the size and shape of the sea helicity PDFs, in particular ∆s(x, Q 2 ) [8][9][10][11][15][16][17]. So far, the only constraint on ∆s(x, Q 2 ), and how it evolves at small x, comes from the weak neutron and hyperon decay constants. Therefore, there is a definite need for direct input from theory on the small-x intercept of ∆Σ(x, Q 2 ): this is what we have provided in this Letter. We now will attempt to quantify how the small-x behavior of ∆Σ(x, Q 2 ) derived here affects the integral in Eq. (1). We take a simple approach and leave a more rigorous phenomenological study for future work. First, we attach a curve ∆Σ(x, Q 2 ) = N x −α h (with α h given in (32)) to the DSSV14 result for ∆Σ(x, Q 2 ) at a particular small-x point x 0 . Next, we fix the normalization N by requiring ∆Σ(x 0 , Q 2 ) = ∆Σ(x 0 , Q 2 ). Finally, we calculate the truncated integral (33) of the modified quark helicity PDF for different x 0 values. The results are shown in Fig. 6 for Q 2 = 10 GeV 2 and α s ≈ 0.25, in which case α h ≈ 0.80. We see that the small-x evolution of ∆Σ(x, Q 2 ) could offer a moderate to significant enhancement to the quark spin, depending on where in x the effects set in and on the parameterization of the helicity PDFs at higher x.
Thus, it will be important to incorporate the results of this work, and more generally the small-x helicity evolution equations discussed here, into future extractions of helicity PDFs.

VII. SUMMARY
In [1,3] we have derived small-x evolution equations for the polarized dipole amplitude. The equations close in the large-N c and large-N c & N f limits. The large-N c equations are presented above. The solution of these equations provides theoretical input on the perturbative value of the small-x intercept for the quark helicity TMD and PDF, and for the g 1 structure function.
We have also numerically solved the small-x helicity evolution equations (15) of Ref. [1] in the large-N c limit. We found an intercept of α h = 2.31 α s N c /2π, which, from Eq. (31), is a direct input from theory on the behavior of ∆Σ(x, Q 2 ) at small x. Although a more rigorous phenomenological study is needed, we demonstrated in a simple approach that such an intercept could offer a moderate to significant enhancement of the quark contribution to the proton spin. Therefore, it appears imperative to include the effects of the small-x helicity evolution discussed here in future fits of helicity PDFs, especially those to be obtained at an Electron-Ion Collider.