Single-spin asymmetry in dihadron production in SIDIS off the longitudinally polarized nucleon target

We study the single longitudinal-spin asymmetry of dihadron production in semi-inclusive deep inelastic scattering process. We consider the collinear picture in which the transverse momentum of the final-state hadron pair is integrated out, such that the $\sin \phi_R$ azimuthal asymmetry arises from the coupling $h_L\, H_{1}^{\sphericalangle}$ as well as the coupling $g_1 \,\widetilde{G}^{\sphericalangle}$. We calculate the unknown twist-3 dihadron fragmentation function $\widetilde{G}^{\sphericalangle}$ using a spectator model which is successful in describing the dihadron production in the unpolarized process. Using the spectator model results for the quark distributions and dihadron fragmentation functions, we estimate the $\sin \phi_R$ asymmetry of dihadron production in SIDIS at the kinematics of COMPASS and compare it with the COMPASS preliminary data. In addition, the prediction on the $\sin \phi_R$ asymmetry at the typical kinematics of the future Electron Ion Collider is also presented. In order to test the reliability of the spectator model estimate, we compare the model result for the distribution $h_L$ with the Wandzura-Wilczek approximation for that distribution, and compare $H_{1}^{\sphericalangle}$ with the existing parametrization. Although the asymmetry is dominated by the $h_L H_{1}^{\sphericalangle}$ term, we find that the contribution from the $g_1\, \widetilde{G}^{\sphericalangle}$ term should also be taken into account in certain kinematical region.

We study the single longitudinal-spin asymmetry of dihadron production in semi-inclusive deep inelastic scattering process. We consider the collinear picture in which the transverse momentum of the final-state hadron pair is integrated out, such that the sin φR azimuthal asymmetry arises from the coupling hL H ∢ 1 as well as the coupling g1 G ∢ . We calculate the unknown twist-3 dihadron fragmentation function G ∢ using a spectator model which is successful in describing the dihadron production in the unpolarized process. Using the spectator model results for the quark distributions and dihadron fragmentation functions, we estimate the sin φR asymmetry of dihadron production in SIDIS at the kinematics of COMPASS and compare it with the COMPASS preliminary data. In addition, the prediction on the sin φR asymmetry at the typical kinematics of the future Electron Ion Collider is also presented. In order to test the reliability of the spectator model estimate, we compare the model result for the distribution hL with the Wandzura-Wilczek approximation for that distribution, and compare H ∢ 1 with the existing parametrization. Although the asymmetry is dominated by the hLH ∢ 1 term, we find that the contribution from the g1 G ∢ term should also be taken into account in certain kinematical region.

I. INTRODUCTION
Understanding the partonic structure of the nucleon and the fragmentation mechanism of hadrons are the main tasks in QCD and hadronic physics. The azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS) process have been recognized as useful tools for these quests. The full description of SIDIS includes a set of parton distribution functions (PDFs) and fragmentation functions (FFs). [1][2][3]. In recent years, the study of hadron pair production in SIDIS has received a lot of attention. The dihadron FF (DiFF), which describes the probability that a quark fragmented into two hadrons: q → H 1 H 2 X, appears in this process.
The leading-twist differential cross-section in polarized SIDIS, containing different azimuthal modulations involving dihadron fragmentation, was presented in Ref. [12]. The study was then extended to the subleading twist in Ref. [28], where the issue of gauge invariance of the DiFFs was also discussed. In this formalism, the structure functions are expressed as the convolution of the distribution function and the DiFF. Experimentally, hadron pair productions off both the unpolarized target and the transversely polarized target have been measured by the HERMES collaboration [29] and the COMPASS collaboration [30,31]. Very recently, preliminary results on the azimuthal spin asymmetries in hadron pair production off the longitudinally polarized proton target were also obtained by the COMPASS collaboration [32]. When the transverse momentum of the hadron pair is integrated out and the incident lepton beam is unpolarized, only one modulation -the sin φ R azimuthal angle dependence -remains. Here φ R is the angle between the lepton plan and the two-hadron plane. The preliminary COMPASS measurement showed a clearly positive sin φ R asymmetry within experimental precision. In the parton model, two sources [28] contribute to this asymmetry, one is the coupling of the twist-3 distribution h L and the twist-2 DiFF H ∢ 1 , the other is the twist-3 DiFFG ∢ combined with the helicity distribution g 1 .
In this work, we study the sin φ R asymmetry by adopting the spectator model results for the distribution functions and fragmentation functions. We not only take into account the coupling h L H ∢ 1 , but also investigate the role of the T-odd DiFFG ∢ , which encodes the quark-gluon-quark correlation and has not been considered in previous studies. It was suggested in Ref. [33] that the fragmentation contribution in the twist-3 collinear framework may be also important for the SSA in pp collision. Later phenomenological analysis [34] showed that, besides the contribution of the twist-3 collinear distribution functions, twist-3 fragmentation functions are also necessary for describing the SSA data in both SIDIS and pp collision [35][36][37][38] in a consistent manner [39]. To calculateG ∢ , we apply a spectator model in Ref. [23] where the parameters of the model is tuned to the output of the PYTHIA event generator. Similar model has also been applied to calculate the single hadron fragmentation functions [40]. we adopt the approach in Refs. [41,42] in order to generate the gluon rescattering effect needed for nonzero T-odd fragmentation functions. Using the model results for the distributions and DiFFs, we estimate the sin φ R asymmetry at COMPASS kinematics and compare it with the COMPASS preliminary data.
This paper is organized in the following way. In Sec. II, we review the theoretical framework of the sin φ R azimuthal asymmetry of dihadron production in unpolarized lepton beam scattered off a longitudinally polarized proton target. In Sec. III, we use a spectator model to calculate the twist-3 dihadron fragmentation function G ∢ . In Sec. IV, We make the numerical estimate of the sin φ R azimuthal asymmetry at the kinematics of COMPASS as well as EIC. We summarize this work in Sec. V.

II. FORMALISM OF THE sin φR ASYMMETRY OF DIHADRON PRODUCTION IN SIDIS
As displayed in Fig.1, the process under study is the dihadron production in SIDIS off a longitudinally polarized proton target: where the four-momenta of the incoming and the outgoing leptons are denoted by ℓ and ℓ ′ , P is the momentum of the target with mass M . In this process, the active quark with momentum p is struck by the virtual photon with momentum q = ℓ − ℓ ′ . The final-state quark with momentum k = p + q then fragments into two final-state hadrons, h + and h − , plus unobserved state X. The momenta of the pair are denoted by P 1 , P 2 , respectively. In order to express the differential cross section as well as to calculate the DiFFs, we adopt the following kinematical variables Here, we have used the light-cone coordinates a µ = (a + , a − , a T ), where a ± = (a 0 ± a 3 )/ √ 2 and a T is the transverse component of the vector. Therefore, x represents the longitudinal momentum fraction of the initial quark, z i is the longitudinal momentum fraction of hadron h i found in the fragmented quark. Furthermore, M h , P h and R are the invariant mass, the total momentum and the relative momentum of the hadron pair, respectively.
The momenta P µ h , k µ and R µ thus can be written as [23] where θ is defined as the polar angle between the direction of P 1 in the center of mass frame of the hadron pair and the direction of P h in the lab frame [43], | R| = M 2 h /4 − m 2 π and φ R is the angle between the lepton plan and the two-hadron plane, as shown in Fig.1. There are several useful expression of the scalar products as follows In the following we will consider the condition the lepton beam is unpolarized and the nucleon target is longitudinally polarized. In this case, if the transverse momentum of the dihadron is integrated out, the differential cross section for an unpolarized (spin-averaged) target and a longitudinally polarized target can be cast to Here, the first subscript and the second subscript in σ XY denote the polarization states of the beam and the target, respectively. In Eq. (9), f a 1 (x) and D a 1 z, M 2 hh , cos θ are the unpolarized PDF and unpolarized DiFF for flavor a; while in Eq. (10), h a L (x) is the twist-3 distribution coupled with the chiral-odd DiFF H ∢,a 1 z, M 2 h , cos θ , g a 1 (x) is the helicity distribution coupled with the twist-3 T-odd DiFF G ∢,a z, M 2 h , cos θ . As shown in Eq. (10), both of these couplings contribute to the sin φ R azimuthal asymmetry in SIDIS.
The DiFFs D a 1 and H ∢,a 1 are encoded in the integrated quark-quark correlator ∆(z, R) for fragmentation [28]: where U a [b,c] is the gauge-link running from b to c along a to ensure the gauge invariance of the operator. The twist-3 DiFF G ∢ arises from the multiparton correlation during the quark fragmentation, described by the quark-gluon-quark correlator [28,41]: Here, F −α ⊥ is the field strength tensor of the gluon. After integrating out k T , one obtains The DiFF G ∢ thus can be extracted from ∆ α A (z, k T , R) by the trace As shown in Ref. [23], The DiFFs D 1 and H ∢ 1 can be expanded in the relative partial waves of the hadron pair system: in which the expansion is truncated at the p-wave level. Similarly to H ∢ 1 , we can also expand the twist-3 DiFF G ∢ up to the p-wave level as Here, G ∢ ot originates from the interference of s and p waves, and G ∢ lt comes from the interference of two p waves with different polarization.
In this work, following the similar procedure in Ref. [13], we will not consider the cos θ-dependent terms in the expansion of DiFFs because of two reasons. Firstly, cos θ-dependent terms correspond to the higher order contribution in the partial wave expansion and can only be significant when the two hadrons produce via a spin-1 resonance. Secondly, when integrating out the angular θ in the interval [−π, π] which is our case, the cos θ-dependent terms should vanish. Therefore, we focus on the functions D a 1,oo , H ∢ 1,ot and G ∢ ot . In this scenario, the sin φ R asymmetry of dihadron production in the single longitudinally polarized SIDIS may be expressed as [32], Following the COMPASS convention, the depolarization factors are not included in the numerator and denominator.
Before actually calculating the unknown DiFF G ∢ ot in the spectator model, we briefly review the calculation of twist-2 DiFFs D 1,oo and H ∢ 1,ot in the same model in Ref. [23]. In that paper, the twist-2 DiFFs D 1 and H ∢ 1 were expanded in terms of the relative partial waves of the hadron pair system up to the p-wave level, similar to the expressions in Eqs. (15), (16). The DiFF D 1,oo receives the pure s-wave contribution (terms with vertex |F s | 2 ) and the pure p-wave contribution (terms with vertex |F p | 2 ), For H ∢ 1,ot the gauge link (gluon exchange) would not contribute and therefore one needs vertex factors (F s and F p ) that are complex. Thus the DiFF H ∢ 1,ot was calculated from the the interference of s and p waves which is proportional to Im(F s * F p ) or Im(F s F p * ) [23]. Here, the interference of F s and F p generates the necessary phase for the nonzero H ∢ 1,ot . Furthermore, in the model the background of s wave was assumed to be free of the resonances which means the vertex F s is real while the p-wave amplitude contains the contributions from the ρ and the ω mesons. By fitting the output of the PYTHIA Monte Carlo generator for the dihadron count in SIDIS, the parameters of the spectator model were fixed and the numerical results of the twist-2 DiFFs D 1,oo and H ∢ 1,ot were given. In the following, we present the calculation of the DiFF G ∢ ot in the same spectator model. The DiFF G ∢ ot comes from the quark-gluon-quark correlation at the twist-3 level. Here, different from the calculation of the twist-2 T-odd distribution H ∢ 1,ot in Ref. [23], the gluon degree of freedom shows up explicitly, as given in the operator definition (12). The corresponding diagram for the calculation in the spectator model is shown in Fig. 2. The left hand side of Fig. 2 corresponds to the quark-hadron vertex P h ; X|ψ(0)|0 , while the right hand side corresponds to the vertex containing gluon rescattering 0|igF −α ⊥ (η + )ψ(ξ + )|P h ; X . Here, we apply the Feynman gauge, in which the transverse gauge links U ξT and U 0T can be neglected [44,45]. Therefore, the s and p wave contributions to the quark-gluon-quark correlator for dihardon fragmentation in the spectator model can be written as where m and m s are the masses of the fragmenting quark and the spectator, and where the factor (l − g αµ T − l α T g −µ ) comes from the Feynman rule corresponding to the gluon field strength tensor, as denoted by the open circle in Fig. 2. F s , F p are the vertices refer to the s-wave contribution and p-wave contribution [23] and have the following forms: Here,λ(M 2 , and Θ denotes the unit step function. The first two terms of F p can be identified with the contributions of the ρ and the ω resonances decaying into two pions. The masses and widths of the two resonances are adopted from the PDG [46]: M ρ = 0.776GeV, Γ ρ = 0.150GeV, M ω = 0.783GeV, Γ ω = 0.008GeV. In Eq. (19), in principle one of the exponential from factors should depend on the loop momentum l. Here we follow the choice in Ref. [40] to replace (k − l) 2 in the form factor to k 2 . The reason for this choice is that the form factor is introduced to the purpose of cutting off the high-k T region. Maintain the form factor depending only on k 2 can pull the form factor out of the integral and simplify the calculation.
Expanding Eq. (19), we arrive at where Λ s and Λ p are the z-dependent Λ-cutoffs having the form [23] Λ s,p = α s,p z βs,p (1 − z) γs,p , and 2/Λ 2 sp = 1/Λ 2 s + 1/Λ 2 p . The on-shell condition of the spectator gives the relation between k 2 and the transverse momentum k T : [22], The first and second lines of Eq. (22) provide the pure s-wave and p-wave contributions, respectively. Therefore, they will not contribute to the interference of s and p-waves functions G ∢ ot , only the third and fourth lines have contribution. At one loop level, in principle there are two sources for nonzero G ∢ ot . One is the imaginary part of the loop integral over l, combined with the real part of F s * F p . The other is the imaginary part of F s * F p , combined with the real part of the loop integral over l. For the imaginal part of the integral, we apply the Cutkosky cutting rules; while for the real part of the integral, we adopt the Feynman parametrization. Thus, the final result for G ∢ ot (z, M 2 h ) has the form Here, the coefficients A and B come from the decomposition of the integral [41,42], and have the expressions The functions I i appearing in the above equations are defined as [47] where . The coefficient C in Eq. (25) has the form which is proportional to the fragmenting quark mass m.

IV. NUMERICAL ESTIMATE
In order to obtain the numerical result for G ∢ ot (z, M 2 h ), we choose the values for the parameters m, m s , α s,p , β s,p and γ s,p from Ref. [23], where the model parameters were tuned to the output of PYTHIA event generator for dihadron production in SIDIS: Particularly, in Ref. [23] the quark mass m is fixed as 0 GeV. Therefore, in this model, only the Re(F s * F p ) term in Eq. (25) contributes to G ∢ ot numerically, since the Im(F s * F p ) term is proportional to the quark mass m. As for the strong coupling, we choose α s ≈ 0.3.
In the left panel of Fig. 3, we plot the DiFF G ∢ ot (normalized by D 1,oo ) as function of z with M h integrated over the region 0.3GeV < M h < 1.6GeV. In the right panel of Fig. 3, we plot G ∢ ot (normalized by D 1,oo ) as function of M h with z integrated over the region 0.2 < z < 0.9. We find that G ∢ ot is negative in the entire z region when M h is integrated out, while it is positive in the region M h < 0.8 GeV and negative in the region M h > 0.8 GeV. The size of G ∢ ot is less than one percent compared to the leading-twist DiFF D 1,oo . In the following, we numerically estimate the sin φ R azimuthal asymmetry in the dihadron production off a longitudinally polarized proton by considering both the h L H ∢,a 1,ot term and the g 1 G ∢ ot term. Using Eq. (18), we can obtain the expressions of the x-dependent, z-dependent and M h -dependent sin φ R asymmetry as follows For the other DiFFs H ∢,a 1,ot (z, M 2 h ) and D 1,oo (z, M 2 h ) needed in the calculation, we apply the same spectator model results from Ref. [23]. For the twist-3 distribution h L , we choose the result in Ref. [48], as for the twist-2 PDFs f 1 and g 1 , we adopt the results calculated from the same model [49] for consistency.
As there are several model inputs in the expression of the sin φ R asymmetry, it is necessary to check the reliability of the model calculations of the distributions and DiFFs. For the twist-3 distribution function h L (x), we compare the spectator model result in Ref. [48] with that from the Wandzura-Wilczek approximation [50] h L (x) ≈ 2x 1 x dy y 2 h 1 (y).
Here, h 1 (x) is the transversity distribution function and has been extracted [51] phenomenologically. It is known that for the twist-3 distribution g T (x) the Wandzura-Wilczek approximation works reasonably well [52]. In Fig. 4, we plot the numerical results of (h u L + h d L /4)/(f u 1 + f d 1 /4) which is the relevant one appearing in the expression for the sin φ R asymmetry. The solid line denotes the model result, while the dashed line corresponds to the result from the Wandzura-Wilczek approximation. The comparison shows that the former one qualitatively agrees with the latter one. To compare estimate the sin φ R asymmetry in SIDIS at COMPASS, we adopt the following kinematical cuts [32] 0.003 < x < 0.4, 0.1 < y < 0.9, 0.2 < z < 0.9, Here, W is invariant mass of the virtual photon-nucleon system and W 2 = (P + q) 2 ≈ 1−x x Q 2 . We note that we have not considered the evolution effect of the distribution functions and DiFFs, as the evolution of the twist-3 functions h L and G ∢ ot remain unknown. In Fig. 6, we plot the sin φ R asymmetry in dihadron production off the longitudinally polarized proton at the kinematics of COMPASS. The x-, z-and M h -dependent asymmetries are depicted in the left panel, central, and right panels of the figure, respectively. The dashed lines represent the contribution from the h L H ∢,a 1,ot term and the dotdashed lines depict the contribution from the g 1 G ∢ term. The solid lines display the sum of the two contributions. The full circles with error bars show the preliminary measurement by the COMPASS collaboration for comparison. We find that in the large x region and in the small M h region, the contribution from the h L H ∢,a 1,ot term dominates the asymmetry. The g 1 G ∢ becomes important in the small x region and large M h region. Combining the contributions from the two terms, our calculation agrees with the COMPASS preliminary data on the sin φ R asymmetry. Particularly, in the region M h > 1 GeV, the contribution from the h L H ∢,a 1,ot underestimates the asymmetry, whereas the inclusion of the g 1 G ∢ term can yield a better description of the COMPASS preliminary data.
In addition, we also make the prediction on the sin φ R asymmetry in the single-longitudinally polarized SIDIS at the future EIC. Such a facility would be ideal to study this observable. We adopt the following EIC kinematical cuts [53] √ s = 45GeV, 0.001 < x < 0.4, 0.01 < y < 0.95, 0.2 < z < 0.8, The x-, z-and M h -dependent asymmetries are plotted in the left, central, and right panels of Fig. 7. We find that the overall tendency of the asymmetry at the EIC is similar to that at COMPASS. Although the size of the asymmetry is smaller due to the higher-twist nature of the asymmetry, it is still measurable at the kinematics of EIC.

V. CONCLUSION
In this work, we have studied the single longitudinal-spin asymmetry with a sin φ R modulation of dihadron production in SIDIS. We not only considered the contributions from the coupling of the twist-3 distributions h L and the DiFF H ∢ 1,ot , but also took into account the coupling of the helicity distribution g 1 and the twist-3 DiFF G ∢ . Using the same spectator model which has been adopted to calculate DiFFs D 1,oo and H ∢ 1,ot , we calculated the twist-3 T-odd DiFF G ∢ ot by considering the gluon rescattering effect. We found that the contribution to G ∢ ot comes from the interference of the s and p waves. We also test the reliability of the spectator model by comparing the model results of the dihadron fragmentation function H ∢ 1 with the existed parametrization as well as the model results of the twist-3 distribution function h L with the Wandzura-Wilczek approximation. Using the numerical results of the DiFFs, we estimated the sin φ R asymmetry and compared it with the COMPASS measurement. Our calculation shows that the h L H ∢ 1,ot term dominates in the most of the kinematical region. However, the inclusion of the g 1 G ∢ ot contribution yields a better description of the COMPASS data, especially in the large M h region. In addition, we also made a prediction on the sin φ R asymmetry in SIDIS at the typical kinematics of a future EIC. Our study shows that the twist-3 DiFF G ∢ should be considered in phenomenological analysis in order to provide a better understanding of the sin φ R asymmetry in dihadron production in SIDIS.