Parity violating semi-inclusive deeply inelastic scattering at the Electron-Ion Collider

We present a systematic calculation of the current jet production semi-inclusive deeply inelastic scattering process at the Electron-Ion Collider energy. Contributions from weak interactions are considered which give rise to parity violating effects. We consider the general form of the polarized electron beam scattering off the polarized target which has spin 1. The calculations are carried out up to twist-3 level in the quantum chromodynamics parton model by applying the collinear expansion where multiple gluon scattering is taken into account and gauge links are obtained automatically. We present complete results for structure functions and spin/azimuthal asymmetries in terms of the gauge invariant transverse momentum dependent parton distribution functions. Both the unpolarized and the polarized electron beam cases correspond to 24 azimuthal asymmetries, in which 6 of them appear at the leading twist, while 18 of them contribute at twist-3 level. In addition we also calculate the parity-violating asymmetries which arise from the interference of the electromagnetic and weak interactions.


I. INTRODUCTION
Thanks to the asymptotic freedom of the strong interaction, many high energy reactions can be studied within the formalism of quantum chromodynamics (QCD) factorization theorems [1], which separate the calculable hard parts from the non-perturbative soft parts in the cross sections. These soft parts often involve parton distribution functions (PDFs) and fragmentation functions (FFs). Both of them are important quantities in describing high energy reactions. When three dimensional, i.e., the transverse momentum dependent (TMD) PDFs and FFs are considered, the sensitive quantities studied in experiments are often different azimuthal asymmetries. These asymmetries are measurable quantities which can be used to extract TMD PDFs and FFs which give information about the nucleon structure and the hadronization mechanism. If only TMD PDFs are taken into account, one of the best reactions to study them is the semi-inclusive deeply inelastic scattering (SIDIS) with current jet production process. Usually one photon approximation is used to calculate the (SI)DIS processes. When weak interaction is taken into account, considering the neutral current interactions, the intermediate propagator can either be virtual photon (γ * ) or Z 0 boson. Since weak interaction does not respect parity conservation, we can study the asymmetries induced by the parity-violating effects through weak interaction. This process is called as parity-violating deeply inelastic scattering (PVDIS). Parity-violating asymmetries [2,3], arising from the interference of electromagnetic (EM) and weak interactions, were first observed in DIS experiments carried out at SLAC [4,5] and have been studied widely. Recently, measurements have been carried out [6][7][8][9][10][11][12][13][14][15][16][17][18]. Proposals for precise measurements in the future are available [19,20]. We further extend the consideration into the SIDIS at the Electron-Ion Collider (EIC) [21] energies in this paper.
The EIC is a high-energy, high-luminosity collider with the capability to accelerate polarized electron and nucleon/ions. * Corresponding author The high energy and luminosity combined with polarized beams will provide a wealth of data in an area never explored before. Therefore, it offers many opportunities to study spin effects and different azimuthal asymmetries. Though, the EIC is proposed mainly for the study of strong interactions, it has the ability to measure parity violating quantities when weak interaction is taken into account. Electro-weak inclusive and semi-inclusive DIS processes have been studied extensively [3,[22][23][24][25][26][27]. However, systematic researches about structure functions and spin/azimuthal asymmetries are still lacking. This includes a full kinematic analysis for the cross section, QCD parton model calculations beyond the leading power accuracy and the study of hadron polarization effects, etc.
Higher twist effects are often significant for semi-inclusive reaction processes and TMD observables. Especially for the case of twist-3 (sub-leading power) corrections, they often lead to azimuthal asymmetries which are different from the leading twist ones [28][29][30]. Thus, the studies of higher twist effects will give complementary or even direct access to the nucleon structure or hadronization mechanism. It has been shown that the collinear expansion is a powerful tool to calculate higher twist effects systematically by taking into account multiple gluon exchange contributions. By using collinear expansion, on the one hand gauge links will be generated automatically which make the calculation explicitly gauge invariant. On the other hand the formalism takes a very simple factorization form which consists of calculable hard parts and TMD PDFs/FFs. This will greatly simplify the systematic calculation of higher twist contributions. Based on the collinear expansion formalism, higher twist contributions to DIS and electron positron annihilation processes have been studied extensively [31][32][33][34][35][36][37][38][39][40][41].
The rest of this paper is organized as follows. In Sec. II, we make kinematic analysis for e − N → e − q(jet)X process and present the differential cross section in terms of structure functions. In Sec. III and IV, we present detailed calculations of the hadronic tensor and the cross section respectively up to twist-3 level in terms of the gauge invariant TMD PDFs in the QCD parton model. The results including structure functions and spin/azimuthal asymmetries are given in Sec. V. Finally, a summary is given in Sec. VI.

II. THE PROCESS AND GENERAL FORM OF THE CROSS SECTION
A. The semi-inclusive PVDIS process To be explicit, we consider the current jet production SIDIS process at EIC energies, where N can be a nucleon with spin-1/2 or an ion, e.g., a deuteron with spin-1. q denotes a quark which corresponds to a jet of hadrons observed in experiments. In this paper, we consider the case of the electron scattering off a spin-1 target. This gives us the opportunity to access also the tensor polarization effects. We consider the neutral current interaction at the tree level of electroweak theory, i.e., the exchange of a virtual photon γ * or a Z 0 boson with momentum q = l − l ′ between the electron and the target. The standard variables for SIDIS are The differential cross-section is given by The symbol r can be γγ, ZZ and γZ, for EM, weak and interference terms, respectively. A summation over r in Eq. (2.3) is understood, i.e., the total cross section is given by A r 's are defined as The leptonic tensors are given by where c e 1 = (c e V ) 2 + (c e A ) 2 and c e 3 = 2c e V c e A . c e V and c e A are defined in the weak interaction current J µ (x) =ψ(x)Γ µ ψ(x) with Γ µ = γ µ (c e V − c e A γ 5 ). Similar notations are also used for quarks where the superscript e is replaced by q. The hadronic tensors are given by It is convenient to consider the k ′ ⊥ -dependent cross section, i.e., where the k ′ z integrated TMD semi-inclusive hadronic tensor is given by (2.14) In terms of the variables in Eq. (2.2), we have where ψ is the azimuthal angle of l ′ around l, M is the target mass which will be often neglected at high energy limit. Therefore the cross section can be written as (2.16)

B. The general form of the cross section in terms of structure functions
In considering the polarized reactions, the general form of the hadronic tensor is divided into a symmetric and an antisymmetric part, W µν = W S µν + iW Aµν , where we have omitted the subscript r = γγ, ZZ, γZ for simplicity. Furthermore, we have where h µν σ j 's andh µν σ j 's represent the space reflection even and odd basic Lorentz tensors (BLTs), respectively. They are con-structed from available kinematical variables in the reaction process. The subscript σ specifies the polarizations.
It has been shown that a distinct feature for BLTs in semiinclusive reactions is that the polarization dependent BLTs can be taken as a product of the unpolarized BLTs and polarization dependent Lorentz scalar(s) or pseudo-scalar(s), see [33] for the detailed discussions about the description of polarizations for spin-1 hadron and the construction of BLTs. We repeat the results here for completeness. There are 9 unpolarized BLTs given by The subscript U denotes the unpolarized part, and p q ≡ p − q(p · q)/q 2 satisfies p q · q = 0. We have also used notations As mentioned above, the vector polarization dependent BLTs can be constructed from the unpolarized BLTs and be written as a unified form given by where ε k ′ S ⊥ = ε αβ ⊥ k ′ ⊥α S T β , ε αβ ⊥ = ε µναβn µ n ν ; λ h is the hadron helicity while S T is the transverse polarization component. There are 27 such vector polarized BLTs in total.
The tensor polarized part is composed of S LL -, S LT -and S T T -dependent parts. There are 9 S LL -dependent BLTs, they are given by For the S T T part, we have There are 81 such BLTs in total.
In expressing the cross section, we choose a coordinate system so that the momenta related to this SIDIS process take the following forms: And the transverse vector polarization is parameterized as For the tensor polarization dependent parameters, we parameterize and define them as in Ref. [42], i.e., After making Lorentz contractions with the leptonic tensor, we obtain the general form for the cross section. We give the general form of the cross section through weak interaction channel. The cross section is given by The total cross section including the electromagnetic and interference terms will formally take the same structure. We define functions of y which will be often used: In expressing the cross section, these functions are equivalent to variables K and ε used in Ref. [29].
The explicit results for the cross section in terms of structure functions for different polarization configurations are: For both unpolarized and polarized electron cases, there exist 81 structure functions respectively, which correspond to the number of independent BLTs. For unpolarized electron beam, there are 39 structure functions correspond to parity conserved terms and the other 42 are parity violated. While for polarized electron beam, it's just opposite, i.e., 42 structure functions are parity conserved and 39 are parity violated.

A. The collinear expansion
In the QCD parton model, we can calculate the hadronic tensor in terms of gauge-invariant TMD PDFs. At the tree level, we need to consider the contributions from the series of diagrams shown in Fig. 1, i.e., the multiple gluon scattering contributions.
After collinear expansion, the hadronic tensor is expressed in terms of the gauge-invariant quark-quark and quark-jgluon(s)-quark correlators and calculable hard parts [34,35,38], where j denotes the number of gluons exchanged and c denotes different cuts. After integration over k ′ z ,W ( j,c) r,µν 's are simplified as up to the relevant twist-3 level. The hard parts h r 's arê The gauge-invariant quark-quark and quark-gluon-quark correlators are defined aŝ where D ρ (y) = −i∂ ρ + gA ρ (y) is the covariant derivative. L(0, y) is the gauge link obtained from the collinear expansion procedure, which guarantees the gauge invariance of the correlators.

B. Decomposition of quark-quark and quark-gluon-quark correlators
The quark-quark and quark-gluon-quark correlators are 4 × 4 matrices in Dirac space which can be decomposed in terms of the Dirac Gamma-matrices {I, iγ 5 , γ α , γ α γ 5 , iσ αβ γ 5 }. In the SIDIS process e − N → e − qX, where the fragmentation is not considered, only the chiral even PDFs are involved. Thus we only need to consider the γ α -and the γ α γ 5 -terms in the decomposition of the correlators. We havê (3.10) The TMD PDFs are defined through the decomposition of the correlation functions. Following the convention in Ref. [39], we have For the quark-gluon-quark correlator, we have By inserting Eqs. (3.11)-(3.14) into Eqs. (3.15) and (3.16), we can get the relationships between the twist-3 TMD PDFs defined via the quark-quark correlator and those defined via the quark-gluon-quark correlator. They can be written in a unified form, i.e., where k =null, ⊥, S =null, L, T , LL, LT and T T whenever applicable.

IV. THE CROSS SECTION UP TO TWIST-3
Substituting the leading twist hadronic tensor in Eq. (3.30) and the leptonic tensor into Eq. (2.13) yields the leading twist cross section. Here, we also give the expressions explicitly for the weak interaction part where we have defined k ⊥M = | k ⊥ |/M, and to simplify the expressions. T q i (y)'s andT q i (y)'s are related to space reflection even and odd structure respectively in the cross section. For EM interaction, it requires c e/q 3 = 0 and c e/q 1 = 1. In this case, only T q 0 (y) and T q 1 (y) are left, and T q 0 (y) = A(y), T q 1 (y) = C(y). For the interference terms, we need to set c e/q 3 = c e/q A and c e/q 1 = c e/q V . The kinematic factors are also different. To make it transparent, we can get the EM and interference cross sections by replacing the parameters in the weak interaction cross section according to Tab. I Similarly, substituting the twist-3 hadronic tensor in Eq. (3.31) and the leptonic tensor into Eq. (2.13) yields the twist-3 cross section. It is given by It is also straightforward to obtain the interference and EM contributions by doing the corresponding replacements. To further unify the notations, we define T q i,r (y)'s andT  We see that only half of the terms will survive if only EM interaction is considered.

V. STRUCTURE FUNCTIONS AND AZIMUTHAL ASYMMETRIES RESULTS UP TO TWIST-3
In Sec. II, we have presented the general form of the cross section in terms of structure functions. In the previous section we have also presented the cross section in terms of the gauge invariant TMD PDFs. They match to each other. In this section, we present the structure functions and azimuthal asymmetries results in terms of the TMD PDFs.

A. Structure functions results
We first present the structure functions in terms of gauge invariant PDFs. For the leading twist part, we have, 14) = c e 1 c q 1 k ⊥M g ⊥ 1LT , (5.28) For the twist-3 part, we have One can get the full structure functions results measured in experiments by summing the weak, EM and interference terms together. To this end, we would better redefine the structure functions to include also the kinematic factor A r 's. The results is simple to get, e.g., Azimuthal asymmetries from unpolarized electron beam In addition to structure functions, we also calculate the azimuthal asymmetries results. We consider both the unpolarized beam (λ e = 0) and the polarized beam (λ e = ±1) cases. They contribute to different azimuthal asymmetries results. We first consider the unpolarized case. The azimuthal asymmetry is defined as, e.g., for the unpolarized or longitudinally polarized target case, and for the transversely polarized target case. dσ is used to denote dσ dxdydψd 2 k ′ ⊥ , and dϕ S ≈ dψ which integration corresponds to take the average over the out going electron's azimuthal angle [29,43]. The subscripts such as (U, T ) denote the polarizations of the lepton beam and the target, respectively. At the leading twist, there are six polarization dependent azimuthal asymmetries which are given by (the sum over r = ZZ, γZ and γγ is implicit in the numerator and the denominator respectively) We find that there are three parity-violating azimuthal asymmetry modulations among the six in total. If only electromagnetic interactions are considered only three parity conserved modulations are left. At twist-3, we have 18 azimuthal asym-metries. They are given by  There are only ten azimuthal asymmetries left if the weak interactions are excluded.

C. Azimuthal asymmetries form polarized electron beam
For the case of the polarized electron beam, we obtain similar results as the unpolarized case. They have one-to-one correspondence. At the leading twist we have six kinds of asymmetries, We find that three parity-violating azimuthal asymmetry mod- There are only ten azimuthal asymmetries left if the weak interactions are excluded.

D. Parity-violating asymmetries
With the advent of highly-polarized electron beams, parity violation measurements have become a standard tool for probing a variety of phenomena, for example, the Standard Model, the role of strange quarks in the proton and the neutron distribution in nuclei. The parity violating asymmetry in DIS offers a unique window into the interesting physics. This asymmetry is sensitive to the hadronic structure of the nucleon and to the Standard Model couplings, e.g., c A , c 3 . To be explicit, it is convenient to consider the inclusive DIS. Integrating over − T q 1 (y) − λ e T q 1 (y) λ h g 1L (x) +|S T | sin ϕ S T q 2 (y) − λ eT q 2 (y) f T (x) − cos ϕ S T q 3 (y) − λ e T q 3 (y) g T (x) +|S LT | sin ϕ LT T q 3 (y) − λ e T q 3 (y) g LT (x) + cos ϕ LT T q 2 (y) − λ eT q 2 (y) f LT (x) . (5.160) The complete differential cross section is given by To calculate the parity-violating asymmetries, we assume that the lepton is longitudinal polarized. Here we introduce the definition of the parity-violating asymmetry, where the subscript σ denotes the target polarization, superscript PV denotes parity-violating. First of all, we take the target as unpolarized. According to the definition, we have .

(5.169)
Parity-violating asymmetries given in this part combine the electroweak and QCD theories. Measuring these asymmetries can be important ways to examine electroweak and QCD theories simultaneously.

VI. SUMMARY
In this paper, we present a complete and systematic calculation of the parity-violating current jet production SIDIS process at the EIC. We consider both the EM and weak interactions. We presented the general form of the differential cross section of this process in terms of structure functions by making full kinematical analysis. In QCD parton model the calculations are carried out by applying the collinear expansion where the multiple gluon scattering is taken into account and gauge links are obtained systematically and automatically.
We consider both unpolarized and polarized electron beams scattering off polarized spin-1 target. There are in total 36 structure functions contribute at twist-2 and 72 structure functions contribute at twist-3 for different polarization configurations. We also presented the azimuthal asymmetries results. For both unpolarized and polarized electron beams cases, there are 24 azimuthal asymmetries up to twist-3, in which 6 of them correspond to the leading twist TMD PDFs while the other 18 correspond to the twist-3 TMD PDFs. Among these structure functions and azimuthal asymmetries results, only one fourth of them will left if only electromagnetic interaction is take into account. The remaining others are all generated through weak interaction and its interference with EM interaction. We also calculate the parity-violating asymmetries for weak interaction is considered. Though, the EIC is being proposed mainly for the study of strong interactions, it has a unique ability to measure parity violating quantities.