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. Additionally, we determine the large-N c small-x asymptotics of the OAM distributions to be L q +¯ q ( x, Q 2 ) ∼ L G ( x, Q 2 ) ∼ ∆Σ( x, Q 2 ) ∼ ∆ G ( x, Q 2 ) ∼ (cid:18) 1 x (cid:19) α h , with the intercept α h the same as obtained in the small-x helicity evolution [3], which can be approximated as α h ≈ 3 . 66074 (cid:113) α s N c 2 π . This result is in complete agreement with [4]. Additionally, we calculate the ratio of the quark and gluon OAM distributions to the flavor-singlet quark and gluon helicity parton distribution functions respectively in the small-x region.


I. Introduction
II. Evolution equations for the moment amplitudes in the large-N c limit III.Solution of the moment amplitude evolution equations IV.Summary of our solution and the small-x asymptotics A. Summary of our solution B. Small-x asymptotics of the OAM distributions V. OAM to helicity PDF ratios VI. Conclusions and outlook

I. INTRODUCTION
One of the most important open questions in hadronic physics is the proton spin puzzle [5][6][7][8][9][10][11][12][13].The proton spin puzzle is best described by spin sum rules, due to Ji [14] and Jaffe and Manohar [15].The latter reads where S q+q (S G ) is the quark (gluon) helicity contribution to the proton spin, and L q+q (L G ) is the quark (gluon) contribution to the proton spin from the orbital angular momentum (OAM).The reader may find reviews of the proton spin puzzle in [6-9, 11, 12].The helicity contributions can be expressed as integrals over the longitudinal momentum fraction x S q+q (Q 2 ) = 1 2 where Q 2 is the renormalization scale and the flavor-singlet quark helicity parton distribution function (hPDF) is ∆Σ(x, Q 2 ) = f =u,d,s,...
∆q f (x, Q 2 ) and ∆q f (x, Q 2 ) are respectively the quark and anti-quark hPDFs of a given flavor f .∆G(x, Q 2 ) is the gluon hPDF.The current values of the spin carried by the quarks and gluons as extracted from experimental data are S q+q (Q 2 = 10 GeV 2 ) = 0.15 ÷ 0.20 for x ∈ [0.001, 1] and S G (Q 2 = 10 GeV 2 ) = 0.13 ÷ 0.26 for x ∈ [0.05, 1] [ 6-9, 11, 12].The fact that these two numbers sum to less than 1/2 is the proton spin puzzle.The remaining spin could reside at lower values of x and/or the OAM contributions.
The OAM contributions can also be written as integrals over the longitudinal momentum fraction x [16-20]1 , Both the helicity PDFs and OAM distributions receive contributions from the small x region.This region is important to study because of the limited amount of data as well as the finite acceptance in x in any given experiment.Since x scales with the inverse of the center-of-mass energy squared, any experiment, even future experiments at the Electron-Ion Collider [6,10,11,13], can only probe down to some x min > 0. Therefore, theoretical input is needed to constrain the net amount of spin for values of Bjorken x < x min .
The helicity PDFs, ∆Σ(x, Q 2 ) and ∆G(x, Q 2 ), have been well studied in recent years; they have been experimentally extracted in the accessible region of x and Q 2 [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].The OAM contributions have been studied much less.Even though their evolution in Q 2 is known [17,37], and there have been some proposals on how to access these distributions experimentally [38,39], there is currently no experimental extraction of L q+q (x, Q 2 ) and L G (x, Q 2 ).In order to fully understand the proton spin puzzle, more work is needed to understand the OAM distributions.As mentioned above, the small x region is particularly important for study.
Using the LCOT formalism, small x evolution equations were derived for the polarized dipole amplitudes, which are related to the hPDFs, in [48,50,53,54] (KPS).The KPS equations were derived in the DLA.Recently, these equations received important corrections.It was discovered that additional sub-eikonal operators mix with the ones studied in the original KPS papers under evolution, and thus resulted in revised evolution equations [60] (KPS-CTT) (see [49] also).These revised evolution equations for the polarized dipole amplitudes were solved numerically in the large-N c [60], and large-N c &N f [74] limits (where N f is the number of quark flavors) as well as analytically in the large-N c limit [3].Additionally, the large-N c &N f solution has been used for a successful analysis of the polarized deep inelastic scattering (DIS) and semi-inclusive DIS (SIDIS) small-x data [75].
The small-x asymptotics of the hPDFs have been calculated by BER [41], and by solving the KPS-CTT equations [3,60,74].It was shown that there is less than a 1% difference between the results for the intercept (power of x) in the large-N c limit and less than a 3% difference between the results in the large-N c &N f limit.The origin of the discrepancy is speculated on in the Appendix of [3] (see also [50]).(Additionally, see [76][77][78] for a discrepancy due to scheme dependence between the IREE and the small-x limit of the exact 3-loop calculations of spin-dependent DGLAP anomalous dimensions.) The OAM distributions have also been studied in the LCOT framework.An analysis based on the KPS equations was done in [2], but did not include the corrections found in [60].In [1], these corrections were accounted for and the OAM distributions were expressed in terms of the polarized dipole amplitudes and their first impact-parameter moments.These impact-parameter moments were dubbed the "moment amplitudes" in [1].Novel evolution equations for the moment amplitudes based on the KPS-CTT equations were constructed and solved numerically in [1].In the previous study based on the KPS equations, the small-x asymptotics of the OAM distributions were significantly different from each other.In the large-N c limit, more than a 50% difference was found between the intercepts for the quark and gluon OAM distributions [2].This result was consistent with the discrepancies between the helicity small x asymptotics obtained via the KPS equations and those obtained by BER.After including the corrections which resulted in the KPS-CTT evolution, it was found that the small x asymptotics of the OAM distributions agree with the result from [4], at least within the precision of the numerical solution in [1].Furthermore, the ratios of the OAM distributions to the hPDFs were studied in [1] and compared to the results in [4].The ratios in the quark and gluon ratios were found to be in good numerical agreement, with a small discrepancy reported in the gluon sector.
In this paper, we seek to elucidate the numerical findings of [1] by solving the large-N c evolution equations for the moment amplitudes, Eq. ( 8) below, analytically.In Section II, we recall these evolution equations and the relations between the OAM distributions and the polarized dipole amplitudes and their first impact-parameter moments.We present the solution of the moment amplitude evolution equations in Section III, which is based on the double Laplace transform method of [3].A summary of our solution and the resulting small-x asymptotics of the quark and gluon OAM distributons is presented in Section IV.Importantly, we find the OAM distributions have the same asymptotics as the hPDFs and the g 1 structure function, given in Eq. ( 40) as well as in the Abstract above.We trace the origin of this homogeneity to the mixing of the polarized dipole amplitudes with the moment amplitudes in the moment evolution equations.
In Section V, we use the solution of the moment amplitudes and the KPS-CTT equations to calculate the ratios of the OAM distributions to the hPDFs in the quark and gluon sectors analytically.We confirm the numerical results of [1] and speculate on the small discrepancy with the results in [4].We conclude in Section VI.

II. EVOLUTION EQUATIONS FOR THE MOMENT AMPLITUDES IN THE LARGE-Nc LIMIT
The large-N c DLA OAM evolution equations, as derived in [1], evolve the first impact-parameter moments of the polarized dipole amplitudes G 10 (zs) and G i 10 (zs).The operator definitions of the polarized dipole amplitudes, given in [60], are in terms of infinite light-cone Wilson lines and polarized light-cone Wilson lines.The latter are expressed as sub-eikonal operators inserted between semi-infinite light-cone Wilson lines.The large-N c DLA helicity evolution equations involve the impact-parameter integrated polarized dipole amplitudes, G(x2 10 , zs) and G 2 (x 2 10 , zs) defined by As we will see below, G 1 (x 2 10 , zs) does not contribute to the helicity PDFs or the evolution of G(x 2 10 , zs) and G 2 (x 2 10 , zs).We use x = (x 1 , x 2 ) to denote two-dimensional transverse vectors, and x ij = x i − x j for i, j = 0, 1, 2, . . .labeling the partons.The amplitudes above depend on x 2 10 = |x 10 | 2 , the transverse size squared of the dipole.Additionally, the amplitudes depend on the center of mass energy between the original projectile and the target, s, multiplied by the smallest momentum fraction of the two partons making up the dipole, z. 2 The OAM distributions depend not only on G(x 2 10 , zs) and G 2 (x 2 10 , zs), but also on the first impact-parameter moments of G 10 (zs) and G i 10 (zs) defined as where the ellipses denote additional tensor structures that do not contribute to the OAM distributions.The amplitudes I 3 , I 4 , I 5 and I 6 were dubbed the "moment" amplitudes in [1].The evolution of the moment amplitudes mixes them with G(x 2 10 , zs) and G 2 (x 2 10 , zs) and was derived in [1].In addition to the amplitudes in Eqs. ( 5) and ( 6), the evolution for the moment amplitudes (as well the helicity evolution) depends on the so-called neighbor dipole amplitudes, denoted by Γ 10,21 (zs) and Γ i 10,21 (zs), which are auxiliary amplitudes necessary to enforce lifetime ordering in the evolution [48].Their operator definitions are the same as those for G 10 (zs) and G i 10 (zs), except for a difference in the light-cone lifetime cutoff, which depends on the adjacent dipole size [54,55,60].One can write decompositions analogous to Eqs. ( 5) and ( 6) for the neighbor amplitudes, where again the ellipses denote additional tensor structures that do not contribute to the helicity PDFs, OAM distributions or the evolution of the dipole and moment amplitudes.Neither the helicity PDFs nor the OAM distributions depend on the neighbor amplitudes directly.The neighbor amplitudes only contribute to the evolution of the dipole and moment amplitudes.As we will see below, the helicity PDFs and OAM distributions depend only on the polarized dipole and moment amplitudes.
The DLA evolution equations for the moment amplitudes in the large-N c limit are [1]    where the inhomogeneous terms are the initial conditions of the evolution.The DLA evolution equations for the neighbor moment amplitudes are [1]    where Γ p (x 2 10 , x 2 21 , z ′ s) for p = 2, 3, 4, 5, 6 are only defined for x 10 ≥ x 21 , and 1/Λ is an infrared (IR) cutoff for all dipole sizes.
Finally, it is more useful to work in terms of the rescaled variables [51,56,60] Utilizing Eqs. ( 10)-( 13) in Eqs. ( 8) and ( 9), we may write the moment amplitude and neighbor evolution equations as where 0 ≤ s 10 ≤ s 21 ≤ η ′ , and we have changed the order of integration in the third lines of Eqs. ( 8) and ( 9).Additionally, we have defined The task is now to solve Eqs.(14).Furthermore, since the moment equations do not close on their own as mentioned above, we will need the solution for G, G 2 and Γ 2 from [3].This solution is expressed in double Laplace transforms, and reads (see Eqs. ( 53) and ( 54) of [3]) where the double Laplace images are Here, the initial conditions of the helicity evolution have also been expressed in terms of double Laplace transforms and, following [3], we have also defined As usual, the contours in Eqs. ( 16) and ( 18) run parallel to the imaginary axis to the right of all the singularities of the integrands.Note, for the contours in Eqs. ( 16) and ( 18), Re ω > Re γ [3].The solution of Eqs. ( 14), along with Eqs.(16), will give us both the canonical OAM distributions and helicity PDFs at small-x and large-N c , by employing the following relations derived in [1,2,60] where we have defined ᾱs = αsNc 2π .Here we have assumed all flavors contribute equally so that summing over flavors results in a factor of N f in Eqs.(20a) and (20c).This assumption will need to be revised for phenomenology, where G and I 3 would depend on flavor (see [75] for example).Note we have neglected the derivatives from Eq. ( 36) of [1], and Eq. ( 42) of [60].Such derivatives remove one logarithm of energy and are therefore outside of our DLA approximation.

III. SOLUTION OF THE MOMENT AMPLITUDE EVOLUTION EQUATIONS
Inspired by the success of the double Laplace transform method used to solve the helicity evolution equations [3] (see also [52]), we also use double inverse Laplace representations here to solve Eqs.(14).To start, let us write the moment amplitudes, Ī3 (s 10 , η), ⃗ I(s 10 , η), and their inhomogeneous terms, Ī(0) 3 (s 10 , η), ⃗ I (0) (s 10 , η), as As mentioned above, we take the contours to the right of all the singularities of the integrands in Eqs.(21).Using Eqs. ( 21), we will solve Eqs.(14b) and (14d) first.To start, we can relate the different double Laplace images in Eqs. ( 21) by using them in Eq. (14b).Doing the s 21 and η ′ integrals, we get Treating η − s 10 and s 10 as separate variables, we can perform the forward Laplace transforms in Eq. ( 22) to see or, solving for ⃗ I ωγ , Next, we observe the following scaling relation from Eqs. (14b) and (14d), which, after using Eqs.(21), gives Eqs. (14b) and (14d) have several boundary conditions that need to be satisfied.We explicitly check that our solution satisfies these boundary conditions in Appendix A. Therefore, we conclude Eqs.(14b) and (14d) are completely solved.
Let us now solve Eqs.(14a) and (14c).One can show, after differentiating Eq. (14c), that Γ3 obeys the following second-order partial differential equation This differential equation has a homogeneous and a particular solution, which we label Γ(h) Γ3ωγ (s 10 ) yields the condition The two solutions to Eq. ( 29) are γ = δ + ω and γ = δ − ω as defined above in Eq. (19a).The complete homogeneous solution to Eq. ( 27) is then where Γ± 3ω (s 10 ) are arbitrary functions of s 10 that can be constrained by boundary conditions.We explicitly determine these functions in Appendix B; they are listed in Eq. (B6) and below in Eq. (33a).

IV. SUMMARY OF OUR SOLUTION AND THE SMALL-x ASYMPTOTICS A. Summary of our solution
In the previous Section, we constructed an exact solution to Eqs. (14).We reiterate the solution here for convenience for f ∈ {G, G 2 , Ī3 , ⃗ I}, and, as defined above, we have With the exact expressions for the polarized dipole and moment amplitudes in hand, we can now write down the helicity PDFs and OAM distributions.Plugging Eqs.(34) with Eq. ( 35) into Eqs.(20) gives where, as above in Eqs. ( 20), we have defined ᾱs ≡ αsNc 2π .In arriving at Eqs. (37a) and (37c), we have done the s 10 and η integrals in Eqs.(20a) and (20c).We emphasize here that, while Eqs.(37a) and (37b) were derived in [1], here we have explicitly found the double Laplace images.As mentioned above, Eqs.(37c) and (37d) were derived in [60] and the double Laplace images were found in [3].
All together, Eqs.(37) give the exact small-x, large-N c DLA expressions for the quark and gluon OAM distributions and the quark and gluon helicity PDFs.

B. Small-x asymptotics of the OAM distributions
From Eqs. (37a) and (37b), we can see that the small-x asymptotics of the OAM distributions are governed by the rightmost singularity in the ω-plane.One can show this singularity is given by setting the large square root in γ − ω to 0 [3], which gives The solution to Eq. ( 38) with the largest real part is [3] Via Eqs.(37a) and (37b), the small-x asymptotics of the quark and gluon OAM distributions are then Eq. ( 40), along with the solution of the small-x, large-N c evolution equations for the moment amplitudes, Eqs.(34), represent the main result of this work.Importantly, the OAM distributions have the same intercept α h ≡ ω b √ ᾱs as the helicity PDFs.In fact, one can explicitly remove (by hand) the mixing with G, G 2 , and Γ 2 and solve Eqs.(14).The resulting solution has small-x asymptotics that are also governed by the rightmost singularity in the ω-plane.In this case however, one finds the rightmost singularity to be ω ′ b = 2 < ω b .This is similar to the study of the OAM distributions based on the KPS equations in [2].There, the "moment amplitudes" were sub-leading at small-x and did not mix with G, G 2 , or Γ 2 under evolution.
Therefore, one can trace the leading small-x, large-N c asymptotic behavior of the OAM distributions to the mixing of the moment amplitudes with the helicity amplitudes G, G 2 , and Γ 2 under evolution. 3This situation is similar to the polarized DGLAP evolution equations for the OAM distributions in the Wandzura-Wilczek approximation [17,37].There, the (twist-2 part of the) OAM distributions mix with the helicity PDFs.The leading asymptotic behavior of the (twist-2 part of the) OAM distributions is given by the anomalous dimensions of the helicity PDFs.As we do not employ the Wandzura-Wilczek approximation in this work, it appears the inclusion of genuine twist-3 terms, at least in the case of the small-x evolution presented here, does not affect this conclusion.
We should compare the result in Eq. ( 40) to the one obtained in [4].In [4], the BER IREE formalism was extended to the OAM distributions.The resulting OAM distributions were found to obey the same small-x asymptotics as the helicity PDFs.Namely, they found (see Eq. ( 7) of [4]) where α BER h is the intercept of both the quark and gluon helicity PDFs.The analytic expression for α BER h in the large-N c limit, given in [50] and for which the detailed solution is given in [3], is Comparing Eqs. ( 40) and ( 41), we see a slight difference between the intercepts of the OAM distributions found above in the LCOT formalism and found in [4] using the BER IREE formalism.This small discrepancy is the same as the one between the helicity PDF intercepts obtained in the LCOT formalism [3,60] and the intercepts obtained by BER [40,41].See the Appendix of [3] for an explanation of the potential origin of this discrepancy.

V. OAM TO HELICITY PDF RATIOS
Now that we have the analytic expressions for the quark and gluon OAM distributions at small-x and large-N c , we can explicitly calculate the ratio of the OAM distributions to the helicity PDFs.These ratios have previously been studied in [1,2,4,49,79,80], with a discrepancy in the quark ratio between [4] and the numerical solution of [1].We seek here to investigate this claim and reveal any insight that can be obtained from the exact solution of the moment amplitude evolution.
To start, we evaluate Eqs.(37) for specific initial conditions.Then we take the ratio of Eq. (37a) to Eq. (37c) and Eq.(37b) to Eq. (37d) to obtain the quark and gluon ratios respectively.In [1], it was found the asymptotic (x → 0) behavior of these ratios is independent of the initial conditions of both the helicity and moment evolution.Therefore, without loss of generality, we choose the following initial conditions Using Eqs.(43) in Eqs.(35) gives where we have defined Now, using Eqs.(44) in Eqs.(37) gives where we have expressed both the OAM distributions and hPDFs with the arguments y = √ ᾱs ln 1/x and t = √ ᾱs ln Q 2 /Λ 2 .We will use these arguments for the rest of this Section, since such variables appear more natural in our expressions as can be seen from Eqs. (37).Note from Eqs. (46c) and (46d), we have ∆Σ(y, t) = −(N f /4N c )∆G(y, t).
Now we close the γ-contours in Eqs. ( 46) to the left.(Note that Re ω > Re γ along the integration contours.)The result is Due to the complex branch structure of the integrands in Eqs.(47), we approximate the ω-integrals.Since we would like to determine the OAM to hPDF ratios in the asymptotic limit, where there is little to no dependence on the initial conditions, it suffices to evaluate the integrals in Eqs.(47) to leading order in y only.The leading order behavior is given by the rightmost singularity in the ω-plane, as mentioned above.Some of the branch structure beyond this rightmost singularity is shown in Fig. 6 of [1].There is a branch cut along the real axis from ω b to some ω ′ b where ω ′ b < ω b .Therefore, to evaluate any one of the integrals in Eqs.(47), one can wrap the contour across this branch cut and approximate the integrals as the integrals of the discontinuity across the branch cut.If we label the integrands (including pre-factors) of Eqs.(47) via L q+q,ω , L G,ω , ∆Σ ω , and ∆G ω , we may write this approximation as ∆Σ(y, t) ≈ lim ∆G(y, t) ≈ lim where, due to the y-dependent exponentials in Eqs.(47), we have extended the ξ-integration to infinity, and discarded the rest of the contribution from closing the contour as subleading.One can also think of this as sending ω ′ b → −∞.Now it is a simple matter to plug Eqs.(47) into Eqs.(48) and compute the integrals.However, since we are only interested in the leading order (in y) result, one can expand the integrands (excluding the large exponentials e −ξy ) around ξ = 0, and keep only the leading terms.The resulting integral is then a power series in 1/y multiplied by the leading exponential.For example, this series is explicitly constructed for Eqs.(48c) and (48d) in the Appendix of [1].When one takes the ratio of Eq. (48a) to Eq. (48c) or Eq.(48b) to Eq. (48d), the result is again a series in 1/y.We can then write the ratios of Eq. (48a) to Eq. (48c) and Eq.(48b) to Eq. (48b) as The explicit expressions for the asymptotic ratios, R q (t) and R G (t), are given in Appendix C. In Fig. 1, we show R q (t), B q (t), R G (t), and B G (t) for the branch cut discontinuity approximation in Eqs. ( 48) as a function of t.In Fig. 1, we plot R q (t), B q (t), R G (t), B G (t) in panels (a), (b), (c), and (d) respectively.Additionally, we give the explicit FIG. 1: Plots of the coefficients R q (t), B q (t), R G (t), B G (t) from Eqs. ( 49) as a function of t resulting from the branch cut discontinuity approximation in Eqs.(48).Here N c = 3 and α s = 0.25.
values of R q (t), B q (t), R G (t), and B G (t) for some values of t in Table I.
We should compare the results in Fig. 1 and Table I to the predictions from [4].From Eqs. ( 6) and ( 7) in [4] (see also Eq. ( 24) there), after using the fact that the parameter α ∼ √ α s is perturbatively small, we see, in the DLA, [4] predicts Comparing these results to Fig. 1 and Table I, we see that we predict nearly the same value for the asymptotic quark and gluon ratios.Furthermore, in both the quark and gluon sectors, we note from Fig. 1, our results are dependent on t = √ ᾱs ln Q 2 /Λ 2 , and as t increases, the asymptotic ratios, R q (t) and R G (t), get further away from the results of [4] (this is true even for asymptotically large t, see Appendix C).Therefore, we conclude that both the quark and gluon ratios determined here disagree with the results of [4].This is contrast to [1], where the numerically small difference in the gluon sector could have been attributed to underestimation of the numerical error.
The ratios in Fig. 1 are consistent with the numerical findings of [1].Other than the difference between LCOT helicity evolution and BER IREE-based helicity evolution (see [3] for details), there are a few potential sources of discrepancy between the ratios of [4] and ours in Fig. 1 and Table I.One potential source of discrepancy is the large-N c limit we take here.For the quark ratio, our implementation of the large-N c limit is only approximately correct, since one needs to assume the existence of at least one external hard quark to calculate ∆Σ and L q+q .This hard quark line may result in soft quark emissions from evolution, which are omitted here (see the Conclusions section of [56] for more details) .One could hope that taking the large-N c &N f limit, as done in [60], might account for the discrepancy.However, a preliminary numerical solution of the large-N c &N f moment amplitude evolution equations for N f /N c → 0 has been constructed and has not been able to account for the discrepancy in the quark ratio.A full analysis of the large-N c &N f limit is left for future work.Additionally, the Wandzura-Wilczek approximation [81], which neglects the genuine twist-3 contributions to the OAM distributions, is employed in [4].Although the twist-3 contributions, particularly their evolution in Q 2 , have been studied in [82], a dedicated small-x study has not yet been performed.So far in this work, we have not identified the twist-3 contribution to the OAM distributions explicitly.To examine their effect on the OAM to hPDF ratios, let us attempt to isolate the twist-3 contributions explicitly.
We start with the following relations derived in [19], where H q(g) (x, Q 2 ), E q(g) (x, Q 2 ) are the standard unpolarized quark (gluon) twist-2 generalized parton distributions (GPDs) at zero skewness and in the limit of zero momentum transfer.L 3 q+q (x, Q 2 ) and L 3 G (x, Q 2 ) are the twist-3 contributions to the quark and gluon OAM distributions respectively.They can be expressed using twist-3 GPDs [19], and they are the neglected contributions in the Wandzura-Wilczek approximation.Since we are working in the DLA, we can neglect the unpolarized twist-2 GPDs, as their evolution is single-logarithmic.
Then, via Eqs.(51), we can write the DLA expressions for the twist-3 contributions to the quark and gluon OAM distributions as Similarly, we write the DLA twist-2, Wandzura-Wilczek (WW) contribution to the OAM distributions as [4,19] Then, using Eqs.( 52) and ( 53), we may write the asymptotic limit of Eqs.(49) as where we have again used the more convenient variables y = √ ᾱs ln(1/x), t = √ ᾱs ln(Q 2 /Λ 2 ) in the arguments of the OAM distributions and the hPDFs.
As first observed in [4], using the asymptotic form for helicity PDFs and the OAM distributions in Eqs.(53) gives where α h = ω b √ ᾱs for ω b given above in Eq. (39).To compute the twist-3 part of the ratios in Eqs. ( 54), we substitute the exact expressions for the OAM distributions and the helicity PDFs from Eqs. (37) into Eqs.(52).We then perform the procedure described above to extract the OAM to hPDF ratios separately for the twist-2 (WW) and twist-3 OAM contributions.The results for the asymptotic ratios for the quark and gluon sector are shown in Fig. 2. As expected, the twist-2 contributions are exactly the values given in Eqs.(55). 4 We see from Fig. 2 that the twist-3 contributions vary with t and are similar in magnitude to the twist-2 contributions.However, as done in [4], it may be the case that one needs to expand in α h to compute expressions in the formal DLA limit.In this case, the twist-3 contributions would be the difference between the ratios, R q and R G , here and those of [4], given in Eqs.(50).The resulting numerical values for the twist-3 contributions would then be roughly two orders of magnitude smaller than in Fig 2 .However, above we assume no such expansion is taken.Therefore, according to the indirect analysis performed here, it seems as though the twist-3 contributions may be the source of the discrepancy between the OAM to hPDF ratios computed here and those computed in [4].

VI. CONCLUSIONS AND OUTLOOK
Let us summarize what we have accomplished here.We have found an exact solution of the large-N c equations for the small-x moment amplitude evolution derived in [1], given in Eqs. ( 8) and ( 9).This solution is given in Eqs.(34) and (35).With our solution, we have written down expressions for the quark and gluon OAM distributions at large-N c and small-x, given in Eqs.(37a) and (37b) respectively.We have determined the small-x asymptotics of the quark and gluon OAM distributions in Eq. (40).Notably, we find the small-x behavior is largely driven by the mixing of the moment amplitudes I 3 , I 4 , I 5 , I 6 with the dipole amplitudes G, G 2 , and neighbor amplitude Γ 2 .Therefore, we find the discrepancy for the intercepts of the OAM distributions obtained here and the intercept obtained in [4] to be the same as the discrepancy between the helicity PDF intercepts obtained in the LCOT formalism [3] and the BER formalism [40,41].We have also studied the ratios of the quark and gluon OAM distributions to their helicity PDF counterparts in the small-x region.In [1], the quark and gluon ratios in the WW approximation from Eqs. (55) were approximated by R W W q ≈ −1 and R W W G ≈ −2, with the numerical results of [1] indicating an agreement of the net quark ratio R q = R W W q + R 3 q and R W W q as well as an agreement between the net ratio gluon ratio within the accuracy of the numerical approximation.We have found small, but non-zero disagreement between R G and R W W G as well as R q and R W W q .We have tentatively traced these discrepancies to the twist-3 contributions to the OAM distributions.However, a more complete study is warranted to verify this conclusion.
since we may we close either the ω-or the γ-contour to the right.Therefore, we see Eqs. (A1) are satisfied by use of Eq. ( 23) and Eqs.(A3).

ω.Ī
Now we close the contour to the right, picking up the pole at ω + 1 ω .We are left with (s 10 ) e δ + ω s10 .(B10)Employing Eq. (B6), and doing the forward Laplace transform over s 10 , we find