Sivers Effect in Dihadron Electroproduction

The Sivers effect in polarized SIDIS can be measured in two hadron production as sine modulations involving the azimuthal angles $\varphi_T$ and $\varphi_R$ of both the total and the relative transverse momenta of the hadron pair, complementary to the conventional single hadron studies. In this talk we briefly present the results obtained in our recent work [1] and [2]. We also present the leading order parton model expression for the two hadron SIDIS cross section for different choices of the relative transverse momentum that dismiss the seeming contradiction of our results with previous work. Finally, we show the numerical predictions for the corresponding single spin asymmetries in the kinematics of COMPASS experiment obtained using the modified version of the $\tt{LEPTO}$ Monte Carlo event generator that includes the Sivers effect.


Introduction
One of the most interesting phenomena related to internal 3D structure of polarized nucleons is the Sivers effect [3]. The corresponding SSA in one hadron SIDIS have been measured in HERMES [4], COMPASS [5], and JLab [6] experiments and allow to extract the Sivers PDF, e.g. [7].
Recently, the two hadron production in polarized SIDIS (2h SIDIS) and in the jets from e + e − annihilation attracted a lot of interest both in experimental [8][9][10] and theoretical [11][12][13] studies. The main accent has been on studies of transversity PDFs and so called interference fragmentation functions.
Here we present the phenomenology of Sivers effect in 2h SIDIS. We also discuss the different choices for the relative transverse momentum of the hadrons pair and their implications on the observable Sivers asymmetry.

Cross section expression
The process of two hadron electro-production in the current fragmentation region of SIDIS is schematically depicted in Fig. 1. In this work we adopt the γ * − N center of mass frame, where the z axis is along the direction of the virtual photon momentum q. The x-z plane is the lepton scattering plane. In this frame, we will define the transverse components of the momenta with respect to the z axis with subscript T and the transverse momenta with respect to the fragmenting quark's direction with subscript ⊥ , as demonstrated in Fig. 2. We use that standard SIDIS kinematical variables as described in Ref. [7]. It is easy to see that in this system k T = k T and (similar to single hadron SIDIS [7]) in the DIS limit for small transverse momenta of the produced hadrons We assume that the cross section of the process of interest factorizes into convolutions of transverse polarization dependent TMD PDFs, f q ↑ (x, k T ), unpolarized dihadron fragmentation functions (DiFFs), D h 1 h 2 q (z 1 , z 2 , P 1⊥ , P 2⊥ ) and a hard lepton-quark scattering cross section. The polarized PDF f q ↑ is composed of contributions from the unpolarized f q 1 (x, k T ) and the Sivers term describing the correlation between the active quark's transverse momentum, k T and the transverse polarization of the nucleon, S T , where f ⊥q 1T (x, k T ) denotes the Sivers PDF, M is the nucleon mass and the subscript 3 denotes the z component of the vector. Then, using Eq. (1), the 2h SIDIS cross section can be separated into the usual unpolarized part, σ U , and a spin-dependent part induced by the Sivers effect, σ S : where C(x, Q 2 ) = α 2 Q 4 (1+(1−y) 2 ) and α is the fine-structure coupling constant.
It is easy to see using rotational and parity invariance, that the most general dependence of σ S on the azimuthal angles ϕ 1,2 (of the transverse momenta P 1,2T ) and ϕ S is given by two "Sivers-like" terms where σ S , σ 1 and σ 2 depend on x, Q 2 , z 1 , z 2 , P 1T , P 2T and P 1T · P 2T (or cos(ϕ 1 − ϕ 2 )). The explicit expressions for structure functions σ U , σ 1 and σ 2 are presented in Ref. [2] using Gaussian parameterizations for PDFs, similar to Ref. [7], and the following form for DiFFs: where the term cP 1⊥ · P 2⊥ takes into account the experimentally established transverse momentum correlations (see, e.g. Ref. [14], where a −2 coefficient is used instead of c) in unpolarized 2h SIDIS.

Choice of the relative transverse momentum
It is often convenient to use linear combinations of P 1T and P 2T as independent transverse momentum variables. One vector is chosen as the transverse component of the pair's total momentum, For the second vector, the following choices have been considered 1. In Refs. [1,2,9] -the transverse component of the relative momentum 2. In Refs. [10,11] -the weighted difference where 3. In Ref. [8,9,12] -the transverse to γ * -direction component of the transverse to pair total momentum, P, component of R: where andû denotes the unit vector in the direction of vector u.
The explicit calculations in the leading order approximation where we neglect mass terms and for small transverse momenta compared to large Q then and It is useful to define the total and relative transverse momenta also in terms of transverse momenta acquired in fragmentation: The remarkable feature of the R T,A is that at the leading order approximations it is independent on the quark intrinsic transverse momentum, k T , due to relations Eq. (1): whereas depend on k T .

Cross section in terms of relative and total hadron momenta
Let us first rewrite Eq. (3) in terms of P T and R T,A where we defined Note that, the product of δ-functions in Eq.(20) can be written as δ 2 (P T − P ⊥ − zk T ) δ 2 R T,A − R ⊥,A . Then, after integrating over R ⊥,A , we can write Eq.(20) as The P T -integrated cross section becomes and does not depend on target polarization. This is what we have expected for the second choice of relative transverse momentum, since for SIDIS it is note sensitive to quark intrinsic transverse momentum. This feature of Sivers effect in terms of the third choice was well known [12] and can be easily understood by considering Eq. (16). In our studies [1,2] of the Sivers effect in 2h SIDIS we have chosen the first definition of the relative transverse momentum. As for the previous choice, we can rewrite the Eq. (3) in terms of P T and R T : where For the P T -integrated cross section now we have We can see that with this choice of the relative transverse momentum the quark intrinsic k T appears also in the argument of DiFF and. Thus, in general, the Sivers effect in R T -dependent cross section will be nonzero for asymmetric (z 1 z 2 ) pair production. The model independent form of the unintegrated 2h SIDIS cross section that includes the Sivers effect in terms of structure functions is The explicit expressions for σ T and σ R with adopted parametrization of PDFs and DiFFs are given in [2], where it is also shown that in general σ R 0 for asymmetric pair production.
Having in mind the future experimental extractions of the Sivers asymmetries, it is useful to calculate the cross section in Eq. (28) after integrating over the azimuthal angle of the relative or total transverse momentum, ϕ R or ϕ T respectively:  where σ U,i , σ T,i and σ R,i are the zeroth (i = 0) and the first (i = 1) harmonics of the cos(n(ϕ T − ϕ R )) Fourier expansions of the corresponding structure functions. We see that both the sin(ϕ T − ϕ S ) and sin(ϕ R − ϕ S ) modulations have contributions from both the σ T and σ R unintegrated cross section terms.

Modified LEPTO (mLEPTO) results
In this section we present numerical results obtained using a modified version (mLEPTO) [15,16]   function of Ref. [7], that were fitted to the experimental data [4,5].
In mLEPTO simulations we generated 10 11 DIS events in the kinematical region of the the COMPASS experiment [5], where E µ = 160 GeV. The following kinematic cuts were applied: Q 2 > 1 GeV 2 , 0.1 < y < 0.9, 0.03 < x < 0.7, W > 5 GeV. The single hadron SSAs calculated in this simulations then can be directly compared with those measured in Ref. [5].
We present the results of our mLEPTO simulations with only a single sin(ϕ − ϕ S ) modulation kept (see Eqs. (29,30)), where ϕ = ϕ h for one hadron production, and ϕ = ϕ R or ϕ = ϕ T for two hadron production (see Eqs. (29,30)). Then the cross section can be written as where σ U andσ S are the corresponding unpolarized and Sivers terms, and the asymmetry is defined as In Fig. 3 we compare mLEPTO results for the SSAs in single hadron SIDIS production off transversely polarized proton target with the measurements by the COM-PASS collaboration [5], with the additional kinematic cuts P T > 0.1 GeV and z > 0.2 used in the experiment. This allows us to validate mLEPTO by demonstrating that it provides a good description of the data used in extracting the parametrizations of Sivers PDFs.
In mLEPTO we can determine the flavor of the struck quark in each SIDIS event. This proves to be a very useful information for understanding the underlying phenomenology of our results. In Fig. 4 the results for the relative rates for the struck quark flavor in events that produce a given type hadron satisfying the kinematic cuts imposed in our analysis are depicted. We see that the production of the positively charged hadrons, h + , is dominated by the u quarks, while in production of the negatively charged hadrons, h − , both u andū quark contribute almost equally,  Figure 5. mLEPTO predictions for the dependence of the Sivers asymmetry on: (a) x, and (b) the total energy fraction z, in oppositely charged hadron pair production off proton target for both ϕ R and ϕ T asymmetries integrated over P T and R, respectively. The bands labeled "Cut" are the results with the additional cut on the positively charged hadron's momentum.
along with a significant input by the d quark. We can easily interpret these by by examining the relative sizes of the quark PDFs and the fragmentation functions within the kinematical limits of the simulations. Our predictions for the dependence of the Sivers SSAs for a SIDIS production of a hadron pair with opposite charges h + h − are presented in Fig. 5, where in the pairs we choose h + as the first hadron and h − as the second hadron. We imposed the following kinematic cuts on the momenta of the hadrons P 1(2)T > 0.1 GeV, z 1(2) > 0.1. Here we extract the SSAs corresponding to both sin(ϕ T − ϕ S ) and sin(ϕ R − ϕ S ) modulations (labelled "T" and "R" on the plots). We note that the SSAs, particularly those for the "R" modulations are significantly smaller than those for the single hadron case. This might make them difficult to measure experimentally, and thus might be harder to access experimentally. Fortunately, the SSAs can be enhanced by imposing asymmetric cuts, z 1 > 0.3 and P T 1 > 0.3 GeV, (bands labeled "Cut" in Fig. 5) on the momenta of the hadrons in a pair, as expected from the results in the previous Section.
We can also generate non zero "R" type SSAs for same-charged pairs by ordering the hadrons in the pairs according to their energy fraction: the hadron h 1 would be the one with the larger value of z (i.e. z-order the hadrons in a pair  a different set of dihadron fragmentation functions than those used in SSAs of oppositely charged hadrons. These differences can be significant, analogous to the results of Ref. [18] for the NJL-jet calculations of dihadron fragmentation functions that depend on z h and M h . This expectation is confirmed by analyzing the relative rates of the different hadron pair production and their dependence on the struck quark's flavor from the plots in Fig. 6. Both for the pairs with no additional cuts depicted in Fig. 6(a) and for those with asymmetric cuts on the momenta of the hadrons in the pair in Fig. 6( Figure 7. mLEPTO predictions for the dependence of the Sivers asymmetry on: (a) x, and (b) the total energy fraction z, in positively charged hadron pair production off a proton target for both ϕ R and ϕ T asymmetries integrated over P T and R, respectively. The bands labeled "Cut" are the results with the additional cut on the first hadron's momentum, as described in the text.

Conclusions
In our recent work [1,2], we proposed a new approach for measuring the Sivers function in the 2h SIDIS process. We presented the results for the relevant cross section calculation that involve the Sivers PDF, with two non-vanishing SSAs with respect to sin(ϕ R − ϕ S ) and sin(ϕ T − ϕ S ), respectively. In contrast, the leading twist expression for the cross-section of the two-hadron SIDIS presented in Ref. [12] with different choice of the relative transverse momentum contains only a sin(ϕ T − ϕ S ) modulation term. The sin(φ R,P − ϕ S ) modulation term is absent there, as well as in the subleading twist expression of Ref. [13] when integrated over the total transverse momentum of the pair.
Here we have presented the detailed explanation of origin of the 'unexpected' nonzero sin(ϕ R −ϕ S ) modulation in 2h SIDIS. We showed that, using the leading order approximations the sin(ϕ R −ϕ S ) type modulations with the choice of the relative transverse momenta of Refs. [12,13] should indeed vanish, while they should be in general nonzero with our definition of ϕ R . The actual measurement of sine modulations in terms of above two different choices of the relative transverse momentum can serve as a good test of the applicability of factorization formalism and various leading order approximation in 2h SIDIS at the energy scale of that experiment.
We conclude that the magnitude of Sivers SSAs in the dihadron SIDIS process should be comparable to those for the single hadron SIDIS. Thus the experimental measurements of the Sivers SSAs for various hadron pairs using the data collected at COMPASS and the future SIDIS experiments, such as those planned at JLAB12 and EIC, are feasible and would contribute a large amount of new information for extracting the Sivers PDFs.