Sea quark Sivers distribution

We compute sea quark Sivers distribution within color glass condensate(CGC) framework. It has been found that up to the leading logarithm accuracy, the collinear twist-3 approach and the CGC calculation yield the same result for sea quark Sivers distribution in the dilute limit. We further verify that transverse momentum dependent factorization is consistent with CGC treatment at small $x$ for the case of transverse spin asymmetry in open charm quark production in semi-inclusive deeply inelastic scattering process in an overlap kinematical region.


II. SEA QUARK SIVERS DISTRIBUTION IN CGC
Since the number density of gluons rises very rapidly towards small x region, sea quark production is dominated by gluon splitting channel. The spin independent sea quark distribution has been computed in the dipole model [47] and in the CGC formalism [48]. The calculation of sea quark Sivers function can be formulated in a similar manner. The target polarization dependence enters the formula through the spin dependent odderon. We start our calculation with introducing the operator definition for the odderon. Within the CGC formalism, one can identify the following operator as the dipole odderon operator [30], whereD with the Wilson line in the fundamental representation being given by, Here, the plus and minus light cone components are defined in a common way. Obviously, odderon operator changes sign when exchanging the transverse positions of two Wilson lines. As a consequence odderon contribution is the k ⊥ -odd part of the dipole amplitude in momentum space. Motivated by this observation, one can alternatively identify odderon contribution by parameterizing dipole amplitude in the following way, where the first term in the second line is the spin independent gluon distribution. The second and the third terms are spin independent odderon [16] and spin dependent odderon [19] respectively. F E (x g , k 2 ⊥ ) is the so-called gluon elliptical distribution [22]. It is common practice to compute various gluon distributions using the MV model in which valence quark inside nucleon/nuclei is treated as color source. It has been found that odderon contribution vanishes if valence quark/color source is uniformly distributed in the transverse plane of nucleon/nucleci. For the spin independent case, the odderon contribution arises from the transverse gradient of color source distribution [16], while an axial asymmetrical valence quark distribution in the transverse plane of a transversely polarized nucleon gives rise to a spin dependent odderon [19].
In the following we compute small x quark Sivers function in terms of the two point function and relate it to spin dependent odderon. As well known, the Sivers function is process dependent. We thus specify the gauge link in the matrix element definition of sea quark Sivers function to be the future pointing one which is built up through final state interaction in SIDIS process. There are three diagrams contributing to the quark production amplitude in CGC formalism. The calculation is rather straightforward. Here we only would like to mention one interesting feature that four point function shows up in the intermediate step of the calculation. However, after some algebraic manipulation introduced in Ref. [61], four point function collapses into the two point one. The final expression can be cast into the following form, where ξ = x/x g with x g standing for the momentum fraction of the incoming gluon. The coefficient A is given by, One can easily read off the Sivers function from the Eq. 4, or alternatively, which will be used as the initial condition when evolving quark Sivers function to higher scale with the Collins-Soper evolution equation.
We notice that sea quark Sivers function has been computed in the collinear twist-3 approach [45,46] as well. It would be interesting to check whether the CGC calculation and the collinear twist-3 approach produce the same result in the dilute limit(l 2 ⊥ ≫ Q 2 s ) where they both apply. To this end, one can extrapolate the CGC result to the dilute limit by Taylor expanding A(l ⊥ , k ⊥ ) in terms of the power of |k ⊥ |/|l ⊥ |.
where the first two terms vanish due to the Ward identity, and the third term only contributes to the spin independent distribution. The leading power contribution to the sea quark Sivers distribution comes from the fourth term. Inserting this expansion into Eq. 4, one obtains, The next step is to invoke the relation between the k ⊥ moment of the spin dependent odderon and the collinear C-odd tri-gluon correlation function first derived in [19], where the C-odd tri-gluon correlation function is defined as, with x g being the total momentum transfer carried by gluons, which can be conveniently chosen to be x g ≡ Max{x 1 , x 2 }. Note that our convention for the tri-gluon correlation function differs from that defined in Ref. [45,46] by a factor −8πM . If the symmetric tensor d abc is replaced with the antisymmetric one f abc in the above matrix element, one can correspondingly define the N type C-even tri-gluon correlation function.
In the small x limit, one has O(x 1 , in the leading logarithm approximation. We then proceed by expressing the Sivers distribution in terms of tri-gluon correlation function O(x g ), Within the leading logarithm approximation, O(x g ) can be viewed as the function that is independent of x g . 1 It is straightforward to carry out the integration over ξ. One ends up with, On the other hand, sea quark Sivers function computed in the collinear twist-3 approach reads [50,51], where the contribution from the C-even tri-gluon correlation function N (x g , x g ) and N (x g , 0) has been neglected. This is justified to do so because the O type function is enhanced by the power of 1 xg when taking into account small x evolution with respect to the N type function in the small x limit [52]. 0) and carrying out ξ integration, we find that the collinear twist-3 result is in complete agreement with that we derived in CGC formalism. However, it is worth to emphasize again that such equivalence only can be established up to the leading logarithm accuracy.
Let's close discussions on the analytical calculations with one final remark. Following the similar procedure, one also can evaluate small x anti-quark Sivers function, which turns out to be the same as the quark Sivers function in magnitude, but with a reversed sign. This is in sharp contrast to the spin independent case where small x quark and anti-quark distributions computed in CGC formalism are identical. The relative minus sign can be best understood by noticing the C-odd nature of the spin dependent odderon. It might be promising to determine sea quark Sivers function by fitting to the experimental data on the SSA in W ± production in pp collisions [54]. Theū andd Sivers function extracted from this observable [55] seems to be compatible with our result in terms of sign.
We are now ready to present some numerical results for sea quark Sivers distribution by taking into account TMD evolution effect. At tree level, both the unpolarized gluon distribution and the spin dependent odderon can be computed in the MV model. They are given by, which will be used as the initial conditions when implementing TMD evolution. In the above expression, κ u p and κ d p are the contributions from up and down quarks to the anomalous magnetic moment of proton, respectively. The color factor is given by To facilitate numerical estimation, we reexpress the transverse area of nucleon as [47], where G(x g , µ 0 ) is the standard gluon PDF in a nucleon, for which we will employ the MSTW 2008 LO PDF set. In the following numerical estimations, the initial scale µ 0 is chosen to be µ 0 = 0.55 GeV. In an earlier work [11], we determined the initial scale as the saturation scale. However, in the current case, only if one chooses a fixed initial scale which doesn't change with varying x g , the ratio between O ⊥ 1T (x g , k 2 ⊥ ) and F (x g , k 2 ⊥ ) roughly behaves as x 0.3 that is compatible with the predication from the BFKL and the BKP equations. The saturation scale is further fixed using the GBW model [ The Collins-Soper evolution that governs the energy dependence of parton TMDs has been well established in the moderate and large x region [2,3]. Recent progress [57][58][59][60][61] suggests that small x gluon TMDs satisfy the same Collins-Soper equation. One would expect that the same analysis applies to the small x quark TMD case. After solving the Collins-Soper equation for quark TMDs, all large logarithm terms are resummed into an exponential, known as the Sudaokv factor. The evolved quark TMDs take form [3,62], where µ b = 2e −γE /|b ⊥ |, the standard ζ parameter is chosen identical to the renormalization scale µ b and not shown here. The unpolarized quark TMDs and the derivative of quark Sivers TMD at the initial scale µ 0 in b ⊥ space are given by, The standard treatment for the non-perturbative part applies to the Sudakov factor S(µ 2 b , Q 2 ) which at one-loop order reads, where µ 2 b * is defined as The parametrization for the non-perturbative Sudakov factor S N P (b 2 ⊥ , Q 2 ) is taken as [63], with b max = 1.5 GeV −1 , g 1 = 0.201 GeV 2 , g 2 = 0.184 GeV 2 , g 3 = −0.129, x 0 = 0.009, Q 0 = 1.6 GeV. In our numerical estimation, we used the one-loop running coupling constant α s , with n f = 3 and Λ QCD = 216 MeV. With these ingredients, we are able to reproduce all numerical results for unpolarized small x quark TMDs presented in Ref. [64]. We further evolve sea quark Sivers function from the initial scale where the MV model result is used as the initial input up to scales Q 2 = 30 GeV 2 and Q 2 = 100 GeV 2 . As the spin asymmetry is determined by the ratio between unpolarized quark TMD and quark Sivers TMD, we plot the ratio as the function of l ⊥ at x = 10 −3 and x = 10 −4 in Fig.1 and Fig.2, respectively. As expected, the ratio is suppressed with increasing energy and decreasing x. One further observes that the ratios grows with increasing transverse momentum at low |l ⊥ |, until it reaches a maximal value around l ⊥ = 1.8GeV. The maximal value of the ratio is on the percentage level. It is worthy to mention that the absolute size of the Sivers function critically depends on the initial scale we used. However, the transverse momentum dependence and the energy dependence of the Sivers function we predicated is less model dependent.

III. THE SSA IN OPEN CHARM PRODUCTION IN SIDIS: CGC V.S. TMD
One of observables that allows us to access small x charm quark Sivers function is the SSA for open charm in SIDIS process. The spin independent differential cross section for this process has been calculated in the dipole model [47] and in the CGC formalism [48]. It is straightforward to extend the calculation to the spin dependent case which reads [19], Here the common kinematical variables in SIDIS process are defined as Q 2 = −q · q, x B = Q 2 /2P · q, y = q · P/P e · P , l ⊥ = p h⊥ /z andξ = z h /z = l · P/P · q where l, P e , P , and q are momenta for produced charm quark, incoming lepton and proton, and virtual photon, respectively. The term proportional to the spin dependent odderon in the second line is responsible for the SSA for charm production. For simplicity, we only taken into account the transverse polarized virtual photon contribution to the differential cross section. The hard part H(ξ, k ⊥ , l ⊥ , Q 2 ) is given by, where ρ is defined as ρ =ξ(1 −ξ)Q 2 .
Extracting charm quark Sivers function in this process relies on the use of TMD factorization. To establish TMD factorization, there must exist an additional large scale that is much larger than parton transverse momentum. In the current case, TMD factorization can be applied in the kinematical region where Q 2 is much larger than l 2 ⊥ . To recover TMD factorization formula, one has to extrapolate the CGC result to this kinematical region by isolating the leading power contribution in terms of l 2 ⊥ /Q 2 . This analysis actually has been carried out for the unpolarized case in Ref. [64]. In the same manner, one can show that the CGC result reduces to that obtained in TMD factorization for the polarized case as well.
One first notices that the integral in Eq.24 is dominated by the end point contributionξ → 1. We thus should make power expansion around this end point. To this end, we insert a delta function dξδ(ξ − 1/(1 + Λ 2 /ρ)) into Eq.24 with [64]. One proceeds by expanding this delta function, where the second delta function gives rise to power suppressed contribution. We then can carry out the integration overξ using the first delta function, and obtain the spin dependent differential cross section, On the other hand, the SSA also can be formulated in the TMD factorization framework, To make contact with the results derived in the CGC formalism, we ignore p ⊥ in the above formula. This approximation is justified because p ⊥ is typically of order of Λ QCD , while k ⊥ is of order of Q s which is much larger than Λ QCD . Using the expression for the Sivers function given in Eq.6, the spin dependent differential cross section Eq.27 is reduced to the TMD formula Eq.28 as expected. In the kinematical region where Q 2 ∼ l 2 ⊥ ≫ k 2 ⊥ , the SSA for open charm production can be calculated in collinear twist approach [46]. One would expect that the equivalence between the collinear twist-3 approach and CGC framework in describing this observable only can be achieved in the leading logarithm approximation as discussed in the previous section.

IV. SUMMARY
To summarize, we compute sea quark Sivers function in terms of the C-odd part of the dipole amplitude which is identified as spin dependent odderon inside a transversely polarized target. Due to the C-odd nature of odderon, the computed quark Sivers function and anti-quark Sivers function are the same in size, but differ by a minus sign. As a consistency check, we verified that in the dilute limit, the CGC result for sea quark Sivers function is reduced to that obtained using collinear twist-3 approach in the leading logarithm approximation. We further show that sea quark Sivers function can be accessed through the SSA in SIDIS by justifying TMD factorization formula from a full CGC calculation. It is worth to mention again that spin dependent odderon is related to three T-odd gluon TMDs inside a transversely polarized nucleon and the C-odd tri-gluon correlation [19,20]. In view of these findings and the fact that the spin dependent odderon is the dynamical origin of sea quark Sivers function, one may conclude that spin dependent odderon plays a central role in describing single spin asymmetries phenomenology at small x.