The Collins asymmetry in electroproduction of Kaon at the electron ion colliders within TMD factorization

We apply the transverse momentum dependent factorization formalism to investigate the transverse single spin dependent Collins asymmetry with a $\sin(\phi_h+\phi_s)$ modulation in the semi-inclusive production of Kaon meson in deep inelastic scattering process. The asymmetry is contributed by the convolutions of the transversity distribution function $h_1(x)$ of the target proton and the Collins function of the Kaon in the final state. We adopt the available parametrization of $h_1(x)$ as well as the recent extracted result for the Kaon Collins function. To perform the transverse momentum dependent evolution, the parametrization of the nonperturbative Sudakov form factor of the proton and final state Kaon are utilized. We numerically predict the Collins asymmetry for charged Kaon production at the electron ion colliders within the accuracy of next-to-leading-logarithmic order. It is found that the asymmetry is sizable and could be measured. We emphasize the importance of planned electron ion colliders in the aspect of constraining sea quark distribution functions as well as accessing the information of the nucleon spin structure and the hadronization mechanism.

fragmentation functionH(z) may be used as the future probe [18] to access the valence quark transversity through the sin φ S asymmetry in single transversely polarized SIDIS process, in which the transverse momentum of the final state hadron is integrated out. Although much progress has been achieved, the information for the sea quark transversity distribution function is almost unknown due to the lack of the experimental data. The Kaon production SIDIS process may be an ideal analyzer to study the sea quark distribution function due to the strange constitute of the Kaon meson. Thereby, through the sin(φ h + φ S ) Collins asymmetry in the Kaon meson production, there might be an opportunity to obtain the information of the sea quark transversity.
The purpose of this work is to evaluate the Collins asymmetry in Kaon production SIDIS process at the kinematics region of the planned electron ion colliders (EICs), such as the proposed EIC in US and the EIC in China (EicC), in which the transversely polarized proton target will be available. Since it is supposed that high luminosity can be realized at the EICs, the sea quark content of the nucleon may be explored with unprecedent accuracy. The theoretical tool adopted in this study is the TMD factorization formalism [19][20][21][22], which has been widely applied to various high energy processes, such as SIDIS [19,21,[23][24][25][26], e + e − annihilation [21,27,28], Drell-Yan [21,29], and W/Z production in hadron collision [20,21,30]. In this framework, the differential cross section can be written as the convolution of the well-defined TMD PDFs and/or FFs. The energy dependence of the TMD PDFs and FFs is encoded in the TMD evolution functions, the solution of which is usually given in b space, which is conjugate to the transverse momentum space [20,21] through Fourier transformation. After solving the TMD evolution equation, the scale dependence of the TMDs may be included in the exponential form of the so-called Sudakov-like form factor [20,21,24,31]. The Sudakov form factor can be further separated into perturbatively calculable part and the nonperturbative part, the latter one can not be calculated through perturbative theory and may be obtained by parameterizing experimental data. We will investigate the effect of nonperturbative Sudakov form factor on the corresponding TMD PDFs and FFs when performing the TMD evolution to obtain the numerical results for the Collins asymmetry in Kaon production in SIDIS process.
The rest of the paper is organized as follows. In Sec. II, we provide a detailed review on the Collins asymmetry in SIDIS process within the TMD factorization formalism. Particularly, we present the procedure for the TMD evolution of the unpolarized and polarized TMDs involved in our calculations. In Sec. III, we perform the numerical estimate of the Collins asymmetry at the kinematics of two planned EICs. We summarize the paper and discuss the results in Sec. IV.

II. THE COLLINS ASYMMETRY IN SIDIS PROCESS WITHIN TMD FACTORIZATION
The process under study is the SIDIS process with Kaon production using an unpolarized electron beam scattering off a transversely polarized proton target where ℓ and ℓ ′ stand for the four-momenta of the incoming and outgoing electrons, respectively, whereas P and P h denote the four-momenta of the proton target and the final-state hadron (which is Kaon in this work), respectively. The reference frame of the studied SIDIS process is shown in Fig. 1, in which the momentum direction of virtual photon is defined as z−axis according to the Trento conventions [32]. P hT and S T are the transverse component of P h and the spin vector S, respectively. φ h denotes the the azimuthal angle of the final hadron around the virtual photon, and φ S stands for the angle between the lepton scattering plane and the direction of the transverse spin of the nucleon target. We define the following invariants to express the differential cross section as well as the physical observables: where s is the total center of mass energy, q = ℓ − ℓ ′ denotes the momentum of the virtual photon with invariant mass Q 2 . The 5-fold differential cross section (x B , y, z h , P hT ) with a transversely polarized target has the following general form at twist-2 level [3,33,34], where σ 0 =  the transverse spin-dependent structure function, which contributes by the coupling of the transversity distribution function of the proton target and the Collins function of the fragmenting of a transversely polarized quark to the final-state Kaon. The Collins asymmetry with sin (φ h + φ s ) modulation can be written in terms of F UU and F Here, f 1 (x B , p T ) and h 1 (x B , p T ) are the unpolarized TMD distribution function and the transversity distribution function of the proton target, respectively. They depend on the Bjorken variable x B and the transverse momentum p T of the quark inside the proton. On the other hand, D 1 (z h , k T ) and H ⊥ 1 (z h , k T ) are respectively the unpolarized fragmentation function and the Collins function, which depend on the longitudinal momentum fraction z h and the transverse momentum k T of the final-state quark. The notation C represents the convolution: Substituting Eq. (6) into Eq. (4), one can obtain the unpolarized structure function as Here, K T = −z h k T is the perpendicular momentum of the hadron h with respect to the quark momentum, and we have applied the Fourier transformation to give the unpolarized TMD distributions and fragmentation functions in b space (denoted by tildes): We should point out that the energy dependence of the distribution functions and fragmentation functions has been neglected in Eq. (7) and will be discussed in detail in the following. Similarly, the spin-dependent structure function F sin(φ h +φs) UT in Eq. (5) can be rewritten as Here, the transversity distribution function and Collins function in b space are defined as A. TMD evolution formalism The evolution of the TMDs is usually performed in b space (which is conjugated to the transverse momentum space [20,21] through Fourier Transformation), since the cross section in b space can be expressed as the simple production instead of the complicate convolution of functions in transverse momentum space. After performing a reverse Fourier transformation from the b space back to the transverse momentum space, we can obtain the TMD physical observables that can be measured experimentally. Thus it is important to understand the b-behavior of the TMD functions.
Specifically, in TMD factorization the distributionsF (x B , b; µ, ζ F ) and fragmentation functionsD(z h , b; µ, ζ D ) in b space have two kinds of energy dependence. The first one is µ that is the renormalization scale related to the corresponding collinear PDFs/FFs, and the other one is ζ F (ζ D ) which is the energy scale serving as a cutoff to regularize the light-cone singularity in the operator definition of the TMDs. The µ and ζ F (ζ D ) dependences are encoded in different TMD evolution equations. The energy evolution for the ζ F (ζ D ) dependence of the TMD distributions (fragmentation functions) is given by the Collins-Soper (CS) [20,21,35] equation while the µ dependence is given by the renormalization group equation with α s the strong coupling at the energy scale µ,K the CS evolution kernel, and γ K , γ F and γ D the anomalous dimensions. Hereafter, we will set µ = Solving these TMD evolution equations, one can obtain the solution of the energy dependence for TMDs, which has the general form asF where F is the factor related to the hard scattering, S is the Sudakov-like form factor. Eq. (17) shows that the energy evolution of TMD distributions from an initial energy µ to another energy Q is encoded in the Sudakov-like form factor S by the exponential form exp(−S). Similarly, for fragmentation functions, they have the same solution structure asD where D is the factor related to the hard scattering. The coefficients F and D depend on the factorization schemes, which have been studied in details in Ref [36].
In the small b region (b ≪ 1/Λ QCD ), the b dependence of TMDs is perturbative and can be calculated by QCD. However, the dependence in large b region turns to non-perturbative, since the operators are separated by a large distance. It is convenient to include a nonperturbative Sudakov-like form factor S NP to take into account the TMD evolution effect in the large b region, and the form factor which is usually given in a parametrization form and must be obtained by analyzing experimental data, given the present lack of non-perturbative calculations. To combine the perturbative information at small b with the nonperturbative part at large b, a matching procedure should be introduced with a parameter b max serving as the boundary between the two regions.
which was introduced in the original CSS prescription [20]. The prescription also allows for a smooth transition from perturbative to nonperturbative regions and avoids the Landau pole singularity in α s (µ b ). The typical value of b max is chosen around 1 GeV −1 to guarantee that b * is always in the perturbative region.
In the small b region, the TMDs can be expressed as the convolutions of the perturbatively calculable hard coefficients and the corresponding collinear counterparts at fixed energy µ, which could be the collinear PDFs/FFs or the multiparton correlation functions [19,37] where ⊗ stands for the convolution in the momentum fraction x, and F i/H (ξ, µ) is the corresponding collinear counterpart of flavor i in hadron H at the energy scale µ, which could be a dynamic scale related to b * by µ b = c 0 /b * , with c 0 = 2e −γE and the Euler constant γ E ≈ 0.577 [19]. In addition, the sum Σi runs over all parton flavors. Independent on the type of initial hadrons, the perturbative hard coefficients C q←i have been calculated for the parton-target case [24,38] as the series of (α s /π) and the results have been presented in Ref. [37] (see also Appendix A of Ref. [24]). Thus, the general expression of TMDs in b space in Eqs. (17), (18) can be rewritten asF The Sudakov-like form factor S can be separated into the perturbatively calculable part S pert (Q; b * ) and the nonperturbative part S NP (Q; b) According to the intensive study in Refs. [26,[39][40][41][42], the perturbative part of the Sudakov-like form factor S pert (Q; b * ) has the general form For different kinds of TMDs, S pert (Q; b * ) is universal and has the same result, namely, S pert (Q; b * ) is spin independent.
The coefficients A and B in Eq. (26) can be expanded as the series of α s /π: Here, we list A (n) to A (2) and B (n) to B (1) up to the accuracy of next-to-leading-logarithmic (NLL) order [20,24,26,40,43,44]: However, the nonperturbative Sudakov-like form factor S NP (Q; b) can not be obtained from perturbative calculation, and is usually parameterized from experimental data. The general form of S NP (Q; b) was suggested as [20] The functions g 1 (b) and g 2 (b) contain different information. g 2 (b) gives the evolution information of the CS kernel K in large b region, which does not depend on the particular process and has universal expression for all kinds of TMDs [21,24,33,39]. It has no dependence on momentum fractions and energy scale. Also, g 2 (b) shall follow the power behavior as b 2 at small-b region, according to the power counting analysis in Ref. [45], which can be an essential constraint for the parametrization of g 2 (b). A Gaussian form parametrization of g 2 (b) was given by the Brock-Landry-Nadolsky-Yuan (BLNY) fit [43] as g 2 (b) = g 2 b 2 with g 2 being a free parameter g 2 = 0.184, which is reliable to describe SIDIS and Drell-Yan data. While, the nonperturbative function g 1 (b) contains information on intrinsic nonperturbative transverse motion of bound partons, namely, it depends on the type of hadron and quark flavor as well as x B for TMD distributions. As for the TMD fragmentation functions, it may depend on z h , the type of the produced hadron, and quark flavor. In other words, g 1 (b) should depend on specific TMDs. Assuming the universal BLNY fitting form of g 2 (b) and a Gaussian form of g 1 (b), one can obtain the nonperturbative Sudakov form factor for the PDF and FF like Here, the factor 1/2 in front of g 2 comes from the fact that only one hadron is involved in the PDF or FF. The parameter g pdf/ff 1 depend on the type of TMDs, which can be regarded as the width of intrinsic transverse momentum for the relevant TMDs at the initial energy scale Q 0 [24,44,46] as where k 2 ⊥ Q0 and p 2 ⊥ Q0 represent the relevant averaged intrinsic transverse momenta squared for TMD distributions and TMD fragmentation functions at the initial scale Q 0 , respectively.
With all the ingredients, the scale-dependent TMDs in b space as functions of x B /z h , b, and Q can be rewritten as Performing a Fourier transformation from the b space back to the k ⊥ can lead to the TMD distribution functions and fragmentation functions.

B. Unpolarized structure function
In this subsection, we will turn to the expression of the unpolarized structure function F UU in Eq. (4) in terms of the unpolarized distribution function f 1 and the unpolarized fragmentation function D 1 . Using the general representation of the distribution function and fragmentation function in Eqs. (35) and (36), the unpolarized distribution functioñ f 1 (x B , b, Q) and the unpolarized fragmentation functionD 1 (z h , b, Q) can be written as Substituting them into the unpolarized structure function in Eq. (7), one can have the following expression with the structure function in b-space as Here nonperturbative Sudakov form factor S SIDIS NP (Q; b) is the combination of the unpolarized distribution function part and the unpolarized fragmentation function part where g f1 1 and g D1 1 , given by Eq. (34), can be extracted from experimental data in SIDIS, Drell-Yan, and e + e − annihilation. And the structure function in the perturbative b-region can be written as We would like to point out that the coefficients and factors related to the perturbative hard scattering above depend on the TMD factorization scheme used to define the TMD operators. However, one can defined scheme-independent coefficients C (SIDIS) which absorb all the scheme-dependent hard factors and coefficients. The expressions of them can be found in Ref. [33]: with the splitting functions P q←q and P g←q have the general form where C F = 4/3, T R = 1/2, and the subscript symbol "+" denotes "+ function". Using the scheme-independent coefficients C (SIDIS) q←i andĈ (SIDIS) j←q , the F UU (b * ) can be written as Therefore, the final result of the unpolarized structure function in Eq. (4) can be obtained as C. Spin-dependent structure function Using the general representation of the distribution function in Eq. (35), the proton transversity distribution in b spaceh 1 (x B , b, Q) can be written as where H(α s (Q)) is the hard scattering factor, δC q←i is the coefficient related to the transversity distribution, and h i/p is the collinear transversity distribution of flavor i in the proton at the energy scale µ b . In the perpurbative b-region, the Collins function may be expressed as the convolution of the perturabtively calculable coefficients δĈ j←q and the corresponding twist-3 collinear correlation functionĤ (3) h/j : Furthermore,Ĥ K/j (z h , µ b ) is related to the first-p T momentum of the Collins function as [33] Therefore, utilizing the general representation of the fragmentation function in Eq. (36), the Collins function in b-space can be rewritten as with H Collins (α s (Q)) being the hard factor. Substituting the transversity distribution function and the Collins function in Eqs. (51) and (54) into the spindependent structure function in Eq. (10), one can obtain where F α UT (Q; b) is the spin-dependent structure function in b-space and has the evolved form The nonperturbative part of the Sudakov-like form factor receives contributions from the transversity distribution and the Collins function: and F UT (b * ) is the structure function in the perturbative b * region as Similar to the unpolarized case, all of the hard scattering factor and coefficients can be absorbed in a new C-coefficients to remove the scheme dependence, which leads to the new C-coefficients in the spin-dependent case as where the functionP c q←q (z) is given byP Thus, F UT (b * ) can be rewritten as with which the final result of the spin-dependent structure function is obtained as

III. NUMERICAL ESTIMATE
Based on the formalism set up above, we will present the numerical estimate for the Collins asymmetry in Kaon production SIDIS process at the kinematics range of EicC and EIC in this section.
In order to obtain the numerical result of the unpolarized structure function in Eq. (50), we need to utilize the collinear unpolarized distribution function f 1 (x B , µ b ) and the collinear unpolarized fragmentation function D 1 (z h , µ b ) as the inputs of the evolution, for which we resort to the existed parametrizations. For the collinear unpolarized distribution function f 1 (x B ) of the proton, we adopt the NLO set of the CT10 parametrization [47] (central PDF set), while for the fragmentation function D 1 (z h ) we apply the next-leading-order DSS parametrization [48]. Besides, for f 1 and D 1 , the free parameter g 1 in Eq. (34) is a very important ingredient in the evolution of the TMDs. Here, we adopt k 2 ⊥ = 0.57GeV 2 , p 2 ⊥ = 0.12GeV 2 given in Ref. [49]. For the universal parameter g 2 in the nonperturbative Sudakov form factor, it has been extracted in the BLNY parametrization [43] as g 2 = 0.184.
In the case of the spin-dependent structure function in Eq. (62), the collinear transversity function and the collinear correlation function of Collins function turn to be the inputs of the TMD evolution. For the collinear transversity, we adopt the standard parameterization at the initial scale Q 2 0 = 2.4 GeV 2 from Ref. [33], where g q 1 is the helicity distribution function, for which we adopt the DSSV parametrization from Ref. [50]. For the q = u and d quarks, the parameterized results of N h q , a q , and b q are taken from the Table I of Ref. [33]. However, there is no information of the sea quark transversity distribution function, the high precision quantitative measurement of which is an important goal of the planned electron ion collider. Thus, we assume the transversity of the sea quark at the initial energy scale has the form of where N s ≤ 1 is the factor taking into account the positivity bound. To perform the DGLAP evolution of the transversity distribution function from the initial scale Q 2 0 = 2.4GeV 2 to the scale of µ b = c 0 /b * , the evolution package HOPPET [51] is applied and modified to include the evolution kernel of the transversity. The value of the strong coupling α s are consistently obtained at 2-loop order as with fixed n f = 5 and Λ QCD = 0.225 GeV. We note that the running coupling in Eq. (65) satisfies α s (M 2 Z ) = 0.118 and the initial value α s (Q 0 ) = 0.327.
Although there is no direct parametrization for the collinear correlation functionĤ (3) h/q (z h ), we can obtain it from the parametrization of the Collins function for the Kaon meson. In Ref. [16], the Kaon Collins function is extracted from the semi-inclusive hadron pair production in e + e − annihilation, which turns out to be in good agreement with the measurements performed by the HERMES [7] and COMPASS Collaborations [52,53]. The Collins function was parameterized in Ref. [16] as Using the relation between H ⊥ 1,h/q (z h , p ⊥ ) and D h/q ↑ (z h , p ⊥ ): and Eq. (53), one can obtain the expression ofĤ (3) h/q (z h ) as followŝ For the kaon Collins function, N C q (z h ) is set as the favored and disfavored constant in the parametrization [16], For the g 1 parameter needed in performing the TMD evolution of the transversity distribution function in Eq. (34), we assume the same Gaussian width of the transverse momentum value as k 2 ⊥ = 0.57GeV 2 [15,16]. For the g 1 parameter of the Kaon Collins function in the non-perturbative Sudakov form factor, we adopt it from the parametrization in Ref. [15] g H ⊥ where M 2 C = 0.28 GeV 2 [15]. We shall note that the initial energy scale in the non-perturbative Sudakov form factor in Eq. (32) is chosen as Q 2 0 = 2.4GeV 2 . Also, the DGLAP evolution ofĤ (3) h/q (z h ) was assumed to be the same as transversity, which means the homogenous term of the evolution kernel is adopted.
As for the EicC, we adopt the following kinematical cuts 0.01 < x < 0.5, 0.07 < y < 0.9, 0.2 < z < 0.7, 1 GeV 2 < Q 2 < 100 GeV 2 , W > 2 GeV, s = 280 GeV 2 , P hT < 0.5 GeV, where W 2 = (P + q) 2 ≈ 1−x x Q 2 is invariant mass of the virtual photon-nucleon system. Since TMD factorization is proved to be valid to describe the physical observables in the region P hT ≪ Q, P hT < 0.5 GeV is chosen to guarantee the applicability of TMD factorization. Using the above kinematical configurations and applying Eqs. (3), (50), and (62), we calculate the transverse single-spin dependent Collins asymmetry of Kaon production in SIDIS process at EicC and EIC. The corresponding numerical results are plotted in Fig. 2, in which the left, middle, and right panels show the Collins asymmetry as functions of x, z, and P hT , respectively. The upper six panels show the predictions on Collins asymmetry at EicC for K + production and K − production; while the lower panels give the results at EIC for K + production and K − production. In each figure, we plot the Collins asymmetry with the adjusting factor N s =0 (no sea quark contribution in the initial energy scale of the transversity parametrization), 0.5, and 1 (saturate positivity bound) in the parametrization of the collinear transversity function for the sea quark using solid, dashed, and dotted lines, respectively.
As shown in Fig. 2, the Collins asymmetry from our calculation is negative in all cases and sizable both at EicC and EIC. The magnitude of x, P hT dependent asymmetries can reach around 0.05. For z dependent asymmetry, the magnitude can reach 0.1 at EIC and 0.01 at EicC. Our estimate also shows that the magnitude of the asymmetry increases with increasing N s in Eq. (64) of the collinear sea quark transversity function and the magnitudes at N s = 1 are about twice as large as those at N s = 0.5, which indicates that the sea quark transversity distribution function plays an important role in the Collins asymmetry of Kaon meson production. The effect of the transversity of the sea quarks turns out to be smaller in the K + production process than that in the K − process. The reason may be that the valence quarks for K + is us, for which Collins function needs to be convoluted with the transversity of the proton. There is only one sea quark transversity distribution function hs /p 1 coupled with the favored Collins function in the case. While for the K − production process, the valence quarks of K − turn to beūs, the relevant transversity distribution coupled to the favored Collins function becomes hū /p 1 and h s/p 1 , which are both sea quark transversity. For the asymmetry as the function of x, there is a clear peak at x ≈ 0.05 at EicC when considering the non-zero sea quark transversity, while the peak vanishes with zero sea contribution of transversity. Although the peak turns to be vague at EIC, the tendency still remains. The z, P hT dependent asymmetries are rather small in the case of vanishing sea quark transversity. Thus, there is a great opportunity to access the sea quark transversity by utilizing the electron ion colliders to measure the Collins asymmetry of Kaon production in SIDIS process.

IV. CONCLUSION
In this work, we have applied the TMD factorization formalism to study the single transverse-spin Collins asymmetry sin (φ h + φ s ) modulation of Kaon production in SIDIS process at the kinematics configurations of EIC and EicC. The asymmetry arises from the coupling of the target proton transversity and the Collins fragmentation function of the Kaon meson. We have taken into account the TMD evolution of distributions and fragmentation functions by including the Sudakov form factors. For the nonperturbative Sudakov form factor associated with the TMDs, we have adopted the traditional Gaussian form. The hard coefficients associated with the corresponding collinear functions are kept in the next-to-leading-logarithmic order. For the transversity distribution of the proton needed, we have employed a recent parametrization for which the TMD evolution effect is considered. For the Collins function of the producing Kaon, we adopted the available parametrization, which can well describe the SIDIS data. Our results demonstrated that, the Collins asymmetry of Kaon production in SIDIS process is sizable at the kinematics configurations of both EicC and EIC.Thereby it could be measured by these experiments in the future. Furthermore, we have considered the contribution of sea quark transversity to the asymmetry. The numerical calculation showed that different choices of the sea quark transversity (N s = 0, 0.5, and 1) lead to different sizes of the asymmetry, particularly in the case of K − production. Therefore, the measurement on the Collins asymmetry of semi-inclusive Kaon production at future electron ion colliders can provide useful constraints on the sea quark transversity.