]>PLB32577S03702693(17)30064310.1016/j.physletb.2017.01.046The AuthorsPhenomenologyFig. 1x−Q2 kinematical plane of bins for SoLID data with HERMES [28,29] and COMPASS [30–32] data sets.Fig. 1Fig. 2Upper panels: uquark and dquark transversity distributions at Q2=2.4 (GeV2) as a function of x with existing errors from KPSY15 (light shade area) and the estimated errors after the SoLID data (both statistical and systematical errors are included in quadrature) are taken into account. The acceptance region in x of the SoLID experiment is indicated by the green horizontal line. Left plot: only the proton target data are taken into account, central plot: only the neutron target data are taken into account, right plot: combination of proton and neutron targets data are taken into account. Bottom panels: The ratio of the estimated errors and the current errors of transversity, δh1SoLID/δh1KPSY15, for u (solid line) and d (dashed line) quarks. Left plot: the proton target, central plot: the neutron target, right plot: combination of proton and neutron targets. The “bumps” around x≃0.2 of the dquark ratio plots are artifacts of usage of Soffer positivity bound [56] when parameterizing transversity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 2Fig. 3The isovector nucleon tensor charge gT after the pseudodata of SoLID is taken into account is compared with result of Kang et al. 2015 [23] at Q2=10 GeV2, result from Ref. [42] (Radici et al. 2015) at 68% C.L. and Q2=4 GeV2, and result from Ref. [40] at 95% C.L. (Anselmino et al. 2013) at Q2=0.8 GeV2, and Ref. [57] (Gamberg, Goldstein 2001) at Q2=1 GeV2. Lattice computation are at Q2=4 GeV2 of Bali et al. Ref. [15], Gupta et al. Ref. [16], Green et al. Ref. [11], Aoki et al. Ref. [18], Bhattacharya et al. Ref. [12,13], Gockeler et al. Ref. [19]. Pitschmann et al. is DSE calculation Ref. [21] at Q2=4 GeV2. Model calculations include QCD sum rule estimate by He, Ji Ref. [58], Chiral Quark Soliton Model by Schweitzer et al. [59], Light Cone Wave Functions by Pasquini et al. [60], and bag models and CQSM results by Wakamatsu from Ref. [61]. Two SoLID points are the truncated and full tensor charges from Eq. (21).Fig. 3Table 1Fitted parameters of the transversity distributions for u and dquark, and Collins fragmentation functions. The table is from Ref. [23].Table 1Nuh=0.85±0.09au=0.69±0.04bu=0.05±0.04
Ndh=−1.0±0.13ad=1.79±0.32bd=7.00±2.65
Nuc=−0.262±0.025αu=1.69±0.01βu=0.00±0.54
Ndc=0.195±0.007αd=0.32±0.04βd=0.00±0.79
gc=0.0236±0.0007(GeV2)
Table 2Kinematic limits of SoLID. The binsize for PT is doubled when number of total events <5×106, and the bin size in x varies to keep number of events in one bin ∼106. The actual bin size of the last bin with the center at x=0.6 will extend up to x∼0.7.Table 2VariableMinMaxBin SizeBins
Q21.0 GeV28.0 GeV2∼1.0 GeV26 bins
z0.30.70.058 bins
PT0.0 GeV1.6 GeV0.2 GeV≤8 bins
x0.050.6NA≤8 bins
Table 3The statistical and systematic uncertainties of the simulated data. The values are averaged over all π+ bins. Similar results hold for π− bins. The typical maximal asymmetry size is of order ≲10% for proton and neutron targets, thus maximal systematical uncertainty is of order ≲0.006.Table 3Statistical (abs.)Systematic (abs.)Systematic (rel.)
Raw asymmetry0.0014Target polarization3%
Detector resolution<0.0001Nuclear effect4∼5%
Random coincidence0.2%
Radiative correction2∼3%
Diffractive meson3%
0.0067Total0.0014Total6∼7%
Table 4Table of tensor charges computed using Eq. (11). Tensor charges are calculated at Q2=2.4 GeV2 and Q2=10 GeV2 and in four regions of x corresponding to the acceptance of SoLID, 0.05<x<0.6; the full region, 0<x<1; and the regions outside of acceptance, 0<x<0.05, 0.6<x<1. The errors are computed at 90% C.L. The isovector nucleon tensor charge gT is calculated using the full region 0<x<1 and a truncated region 0.05<x<0.6, see Eq. (21).Table 4ObservableQ2 (GeV2)KPSY15δKPSY15δSoLIDδSoLID/δKPSY15(%)
δu[0,0.05]2.40.0460.0100.00549
δu[0.05,0.6]2.40.3490.1220.01512
δu[0.6,1]2.40.0180.0070.00114
δu[0,1]2.40.4130.1330.01814

δu[0,0.05]100.0510.0110.00546
δu[0.05,0.6]100.3320.1170.01412
δu[0.6,1]100.01260.00480.000714
δu[0,1]100.3950.1280.01814

δd[0,0.05]2.4−0.0290.0280.00310
δd[0.05,0.6]2.4−0.2000.0730.0069
δd[0.6,1]2.4−0.000040.000090.0000113
δd[0,1]2.4−0.2290.0940.0089

δd[0,0.05]10−0.0350.0300.00310
δd[0.05,0.6]10−0.1840.0670.0069
δd[0.6,1]10−0.000020.000060.0000114
δd[0,1]10−0.2190.0900.0089

gT(truncated)2.40.550.140.01813
gT(full)2.40.640.150.02114

gT(truncated)100.510.130.01713
gT(full)100.610.140.02014
Unveiling the nucleon tensor charge at Jefferson Lab: A study of the SoLID caseZhihongYeabyez@anl.govNobuoSatocnsato@jlab.orgKalyanAlladadkalyan@jlab.orgTianboLiubliutb@jlab.orgJianPingChencjpchen@jlab.orgHaiyanGaobgao@phy.duke.eduZhongBoKangefzkang@physics.ucla.eduAlexeiProkudingc⁎prokudin@jlab.orgPengSunhpsun@msu.eduFengYuanifyuan@lbl.govaMedium Energy Group, Physics Division, Argonne National Lab, Lemont, IL 60439, USAMedium Energy GroupPhysics DivisionArgonne National LabLemontIL60439USAbDepartment of Physics, Duke University, Durham, NC 27708, USADepartment of PhysicsDuke UniversityDurhamNC27708USAcJefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USAJefferson Lab12000 Jefferson AvenueNewport NewsVA23606USAdLaboratory of Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USALaboratory of Nuclear ScienceMassachusetts Institute of TechnologyCambridgeMA02139USAeDepartment of Physics and Astronomy, University of California, Los Angeles, CA 90095, USADepartment of Physics and AstronomyUniversity of CaliforniaLos AngelesCA90095USAfTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USATheoretical DivisionLos Alamos National LaboratoryLos AlamosNM87545USAgDivision of Science, Penn State Berks, Reading, PA 19610, USADivision of SciencePenn State BerksReadingPA19610USAhDepartment of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USADepartment of Physics and AstronomyMichigan State UniversityEast LansingMI48824USAiNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USANuclear Science DivisionLawrence Berkeley National LaboratoryBerkeleyCA94720USA⁎Corresponding author.Editor: J.P. BlaizotAbstractFuture experiments at the Jefferson Lab 12 GeV upgrade, in particular, the Solenoidal Large Intensity Device (SoLID), aim at a very precise data set in the region where the partonic structure of the nucleon is dominated by the valence quarks. One of the main goals is to constrain the quark transversity distributions. We apply recent theoretical advances of the global QCD extraction of the transversity distributions to study the impact of future experimental data from the SoLID experiments. Especially, we develop a simple strategy based on the Hessian matrix analysis that allows one to estimate the uncertainties of the transversity quark distributions and their tensor charges extracted from SoLID data simulation. We find that the SoLID measurements with the proton and the effective neutron targets can improve the precision of the u and dquark transversity distributions up to one order of magnitude in the range 0.05<x<0.6.KeywordsSemiinclusive deep inelastic scatteringTensor chargeTransversityJefferson Lab 12 GeV upgradeSoLIDJLABTHY1623281IntroductionThe nucleon tensor charge is a fundamental property of the nucleon and its determination is among the main goals of existing and future experimental facilities [1–7]. It also plays an important role in constraining new physics beyond the standard model [8–10] and has been an active subject of lattice QCD [9,11–19] and Dyson–Schwinger Equation (DSE) [20,21] calculations. In terms of the partonic structure of the nucleon, the tensor charge, δq for a particular quark type q, is constructed from the quark transversity distribution, h1(x,Q2), which is one of the three leadingtwist quark distributions that describe completely spin1/2 nucleon [1–5]:(1)δq(Q2)≡∫01dx(h1q(x,Q2)−h1q¯(x,Q2)).It is extremely important to extend the experimental study of the quark transversity distribution to both large and small Bjorken x to constrain the total tensor charge contributions. The Jefferson Lab 12 GeV program [6] is going to explore the region of relatively largex dominated by valence quarks while the planned Electron Ion Collider [5,7,22] is going to extend the range to unexplored lower values of x, providing a possibility to study the antiquark transversity distributions.In this paper we analyze the impact of future proposed SoLID experiment at Jefferson Lab 12 GeV on the determination of tensor charge and transversity distributions for u and dquarks. Our studies are based on the QCD global fit of the available SemiInclusive Deep Inelastic Scattering (SIDIS) data and e+e− annihilation into hadron pairs performed in Ref. [23] which we will refer as KPSY15. The current available experimental data suggests that antiquark transversities are very small compared to u and dquark transversities. In this study we assumed that antiquark transversities are negligible. Using the best fit of transversity distributions of Ref. [23] we simulated pseudodata for SoLID experiment and estimate the improvement of u and dquark transversity distributions with respect to our present knowledge. In order to perform a reliable estimate of improvement we develop a simple method based on Hessian error analysis described in Section 4.This study also provides information on contribution of tensor charge from kinematical region of Jefferson Lab 12 GeV and will serve as a guide in planning future experiments.2Present status of extraction of transversity from experimental dataTransversity is a chiral odd quantity and thus in order to be measured in a physics process it should couple to another chiral odd distribution. There are several ways of accessing transversity. It can be studied in SIDIS process where it couples, for instance, to the Collins TMD fragmentation functions [24], and produces the socalled Collins asymmetries. Transversity can also couple to the dihadron interference fragmentation functions in SIDIS [25] and thus collinear transversity can be studied directly. Transversity can be studied in the Drell–Yan process in polarized hadron–hadron scattering [26,27] where it couples either to antiquark transversity or to the socalled the Boer–Mulders functions.SIDIS experimental measurements have been made at HERMES [28,29], COMPASS [30–32], and JLab HALL A [33] experiments. The BELLE, BABAR and the BESIII collaborations have studied the asymmetries in e+e− annihilation into hadron pairs at the center of mass energy around s≃10.6 GeV [34–36], and s≃3.6 GeV [37], respectively.The effort to extract transversity distributions and Collins fragmentation functions has been carried out extensively in the last few years [38–41,23]. QCD analysis of the data where transversity couples to the socalled dihadron interference fragmentation functions was performed in Ref. [42]. These results have demonstrated the powerful capability of the asymmetry measurements in constraining quark transversity distributions and hence the nucleon tensor charge in high energy scattering experiments. The first extraction of the transversity distributions and Collins fragmentation functions with TMD evolution was performed in Refs. [43,23].Collins asymmetries in SIDIS are generated by the convolution of the transversity function h1 and Collins function H1⊥. The relevant contributions to the SIDIS crosssections are(2)d6σdxBdydzdψd2PT=σ0[FUU+sin(ϕh+ϕs)2(1−y)1+(1−y)2FUTsin(ϕh+ϕs)+...], where σ0=2παem2Q21+(1−y)2y, and ϕs and ϕh are the azimuthal angles for the nucleon spin and the transverse momentum of the outgoing hadron with respect to the lepton plane, respectively, dψ≃dϕs. FUU and FUTsin(ϕh+ϕs) are the unpolarized and transverse spindependent polarized structure functions respectively, and the ellipsis represents other polarized structure functions not relevant for this analysis. The polarized structure function FUTsin(ϕh+ϕs) contains the convolution of transversity distributions with the Collins fragmentation functions, h1⊗H1⊥, and unpolarized structure function FUU is the convolution of the unpolarized TMD distributions and the unpolarized fragmentation functions, f1⊗D1. The Collins asymmetry is defined as(3)AUTsin(ϕh+ϕs)(x,y,z,PT)=2(1−y)1+(1−y)2FUTsin(ϕh+ϕs)FUU.Neglecting sea quark contributions, the structure function FUTsin(ϕh+ϕs) for the proton (P) and the neutron (N) targets can be written as:(4)FUTsin(ϕh+ϕs)(P,π+)=eu2h1u⊗H1⊥,fav+ed2h1d⊗H1⊥,unf,(5)FUTsin(ϕh+ϕs)(P,π−)=eu2h1u⊗H1⊥,unf+ed2h1d⊗H1⊥,fav,(6)FUTsin(ϕh+ϕs)(N,π+)=eu2h1d⊗H1⊥,fav+ed2h1u⊗H1⊥,unf,(7)FUTsin(ϕh+ϕs)(N,π−)=eu2h1d⊗H1⊥,unf+ed2h1u⊗H1⊥,fav. Here H1⊥,fav and H1⊥,unf, are the favored and the unfavored Collins fragmentation functions, respectively. In this context, favored refers to fragmentation of struck quarks of the same type as the constituent valence quarks of the produced pion while the unfavored being the opposite case. Previous global analysis [23,40] have found that both the favored and unfavored Collins functions have approximately similar magnitude (with opposite signs). Therefore, since eu2=4ed2, the uquark transversity is more constrained in the proton sample than dquark transversity and the situation is reversed in the neutron case. One expects from these considerations that only the neutron target can help to reach the same relative impact on determination of dquark transversity compared to improvement of uquark transversity from the proton target data.In the KPSY15 analysis the transversity distributions was parametrized as at the input scale Q0=2.4 GeV as(8)h1q(x,Q0)=Nqhxaq(1−x)bq(aq+bq)aq+bqaqaqbqbq⋅12(f1q(x,Q0)+g1q(x,Q0)), where f1q and g1q are the collinear unpolarized [44] and polarized [45] quark distributions for q=u and dquark, respectively.On the other hand, the twist3 Collins fragmentation functions were parametrized in terms of the unpolarized fragmentation functions,(9)Hˆfav(3)(z,Q0)=Nuczαu(1−z)βuDπ+/u(z,Q0),(10)Hˆunf(3)(z,Q0)=Ndczαd(1−z)βdDπ+/d(z,Q0), which correspond to the favored and unfavored Collins fragmentation functions, respectively. For Dπ+/q we use the recent extraction from Ref. [46].In summary, the analysis of KPSY15 used a total of 13 parameters in their global fit: Nuh, Ndh, au, ad, bu, bd, Nuc, Ndc, αu, αd, βd, βu, gc (GeV2), where gc is a parameter to model the width of the Collins fragmentation function. The parameters are shown in Table 1.Since the existing experimental data have only probed the limited region 0.0065<x<0.35, the following partial contribution to the tensor charge, neglecting antiquark contributions, was defined [23](11)δq[xmin,xmax](Q2)≡∫xminxmaxdxh1q(x,Q2).At this point let us discuss the systematical uncertainties in the determination of the tensor charge coming from the theory side. Ref. [23] used the best uptodate knowledge of the TMD evolution in order to take into account scale dependence of TMDs. However, the TMD evolution formalism at lowQ2 region is currently not sufficiently under control [47], for instance the socalled nonperturbative part of TMD evolution kernel has to be extracted from experimental data and presently no satisfactory extraction exists. Possible ways of overcoming these issues were discussed in Ref. [48]. Progress in this area is expected in future. We use the current extraction of Ref. [23] as a proxy to carry out the SoLID analysis and the goal of the current work is to estimate the ratio of the improvement and this estimate is largely model independent.3Simulated data for SoLIDSeveral SIDIS experiments have been approved at Jefferson Lab 12 GeV to measure the asymmetries from proton and neutron targets with polarization in both the transverse and longitudinal directions. Among those, three Hall A experiments, E1210006 [49] (90 days), E1211007 [50] (35 days), and E1211108 [51] (120 days), plan to take data using the proposed high intensity and large acceptance device named SoLID [52,53], and measure both the singlespin asymmetries (SSA) and doublespin asymmetries (DSA) on polarized NH3 (proton) and 3He (effective neutron) targets. These experiments can produce an extensive set of SIDIS data with very high accuracy and thus provide unique opportunity to study TMD structure functions in the valence quark region.In these experiments, the electron beam energy will be set at two different values, 8.8 GeV and 11 GeV. The momentum of the detected electrons and hadrons can range from 1 GeV/c up to their maximum values. The SoLID configuration dedicated to the SIDIS measurements provides a full 2π coverage in azimuthal angle and a coverage of the polar angle from 8∘ up to 24∘. The polarized luminosities of the proton target and the 3He target are 1035 cm−2s−1 and 1036 cm−2s−1, respectively. The polarization and dilution factor of the proton (3He) target are 70% (60%) and 0.13 (0.3), respectively.For the purpose of the present analysis, we simulate the Collins asymmetries using the KPSY15 parametrization at the kinematic settings presented in the proposals of these experiments [49–51]. The high luminosity allows us to bin the data in four dimensions, e.g. x, z, Q2, and PT. The acceptance of the proposed SoLID measurements are summarized in Table 2. There are in total 1014 bins for H3e(e,e′)π+, 879 bins for H3e(e,e′)π−, 612 bins for p(e,e′)π+, and 488 bins for p(e,e′)π−, respectively. The number of events in each bin is calculated by integrating over the cross sections and acceptance of individual events in this bin, and then accounting for the detector efficiencies and the target related characteristics, such as the luminosity, target polarization, effective neutron polarization as well as the dilution factor. The average values of x, z, Q2, and PT are recorded in each bin together with the statistical uncertainty.We also estimate the overall systematic uncertainty, summarized in Table 3, to the experimental measurement, such as the raw asymmetry, target polarization, detector resolution, nuclear effects, random coincidence, and radiative corrections. The average statistical errors and the systematic errors are comparable in size in most bins. The final uncertainties of the simulated Collins asymmetries are given as statistical and systematic uncertainties added in quadrature.The distribution of bins in x−Q2 plane for SoLID and the comparison to HERMES [28,29] and COMPASS [30–32], bins are presented in Fig. 1. The SoLID experiment plans to extend mainly into the larger x region with Q2 coverage comparable with HERMES. A direct comparison of the statistical precision of SoLID and the existing data is not possible due to different binning criteria between experiments, but an estimate of the level of precision can be given. For example, the average statistical precision of each bin for SoLID is about 1% consisting of more than 600 bins for p(e,e′)π+ channel, compared to 37.1% (relative to the size of the asymmetry) for HERMES consisting of 7 bins in x shown in Fig. 1 for the same channel. Note that SoLID implements W2 cut at around 5.5 GeV2. We leave the feasibility of implementing target mass corrections and usage of low W region in the analysis of the experimental data for future developments of the theory and phenomenology.4Error estimation methodology from simulated dataIn this section we describe the new method to estimate the impact of the future SoLID data to the transversity distribution of u and dquarks. Our method follows Bayesian statistics where the new information is added sequentially on top of the prior knowledge without requiring a combined analysis of the old data and the new data. We provide a simple strategy to quantify the impact of new measurements on the transversity distribution using the Hessian approach.In general the information of the best fit parameters and their uncertainties is encoded in the likelihood function(12)L(Dα)∼exp(−12χ2(a,D)) where a represents a vector of the model parameters and D denotes collectively the experimental data points and their uncertainties. χ2 is the standard Chisquared function defined as(13)χ2=∑i(Di−Ti(a)δDi)2, where Ti(a) is the theoretical calculation for experimental measurement of Di and δDi is the experimental error of the measurement. The probability density of the parameters can be constructed from the likelihood function using the Bayes' theorem:(14)P(aD)∼L(Da) π(a), where π(a) is the prior distribution. Typically the latter is set to be normalized theta functions to remove unphysical regions in the parameter space. The expectation value and variance for an observable O (i.e. h1u,d,δu,δd) can be estimated as(15)E[O]=∫dnaP(aD)O(a),V[O]=∫dnaP(aD)[O(a)−E[O]]2. In most of the situations the evaluation of the above integrals are not practical due to the large number of parameters needed in the model as well as numerical cost in evaluating P(aD) or equivalently the χ2 function. A traditional method to estimate Eq. (15) is the maximum likelihood (ML). First the parameters a0 that maximizes the likelihood (or minimized the χ2 function) is determined so that one can write(16)E[O]≈O(a0).A very simple method to estimate the variance is the Hessian approach [54,55]. The idea is to compute the covariance matrix of the parameters using the Hessian of the χ2 function:(17)Ci,j−1≈Hi,j=12∂χ2(a,D)∂ai∂aja0. From the eigen values λk and their corresponding normalized eigen vectors vk of the covariance matrix one can estimate the variance on O as(18)V[O]=Δχ24∑k(O(a0+λkvk)−O(a0−λkvk))2. The factor of Δχ2 (commonly known as the tolerance factor) is introduced in order to accommodate possible tensions among the data sets. In the ideal Gaussian statistics, 68% CL corresponds to Δχ2=1. In the present analysis we use the value of Δχ2=29.7 quoted in the KPSY15 analysis. We stress however that our analysis focuses on the relative improvement after inclusion of the future SoLID data for which the tolerance factor drops out.A simple Bayesian strategy to estimate the impact of the future measurements on the existing uncertainties is to update the covariance matrix. Since the only information provided is the projected statistical and systematic uncertainties, the expectation values (or equivalently a0) remain the same. To update the covariance matrix we note that the χ2 function is additive and one can write the new Hessian matrix as(19)Hi,jNew=12∂χ2(a,Dold)∂ai∂aja0+12∂χ2(a,Dnew)∂ai∂aja0, where Dold is the data set used in a previous analysis (i.e. KPSY15) and the Dnew is the simulated data set for the future experiment. In our analysis only the covariance matrix from the KPSY15 analysis was provided. The new covariance matrix with the projected SoLID measurements was calculated as(20)CNew−1=Hi,jNew=CKPSY15−1+12∂χ2(a,DSoLID)∂ai∂aja0. Using the new covariance matrix one can determine the impact of future data sets by estimating the uncertainties for the observables O, such as transversity or tensor charges, using Eq. (18).5Tensor charge and transversity from SoLIDOur results for u and dquark transversity distributions at Q2=2.4 GeV2 are presented in Fig. 2 along with results from KPSY15. The uncertainties of KPSY15 are given as light shaded bands, while the projected errors after the SoLID data are taken into account are shown as dark shaded bands. To quantify the improvement of adding the future SoLID data, we show in the bottom plots of Fig. 2 the ratio of the estimated errors relative to the current errors. The results are shown using only the proton target data (left panels), the neutron data (central panels), and combination of the proton and the neutron data sets (right panels). In KPSY15 the uncertainty bands for transversity was calculated using the envelop method with a tolerance of Δχ2=29.7 which differs somehow from our Hessian error analysis. We stress that while the absolute error bands can differ depending on the error analysis, the ratio of the errors is independent of the error analysis.One can see that, the proton target data improves uquark transversity uncertainty (as can be seen from the left plot of the bottom panel of Fig. 2) while dquark transversity improvement remains at a modest ∼60% level. The effective neutron target data as expected allows for a much better improvement of dquark transversity uncertainty (as can be seen from the central plot of the bottom panel of Fig. 2) and a relatively good improvement of uquark (up to 80% reduction of errors) as well. It happens because of a higher statistics on the effective neutron target in comparison to the proton target. The right plot of the bottom panel of Fig. 2 shows that in the kinematical region of SoLID, 0.05<x<0.6, the errors will be reduced by approximately 90%, i.e. one order of magnitude, for both u and dquark transversities if measurements are performed on both the proton and effective neutron targets.Notice that the maximal improvements are attained in region covered by the SoLID data 0.05<x<0.6 and the impact decreases outside of this region as expected. One may notice the “bump” around x≃0.2 of the dquark transversity in all three bottom plots. It appears to be an artifact of usage of Soffer positivity bound [56] in the parametrization of transversity for u and dquarks. Indeed, around x≃0.2 the error corridor saturates the bound and it shows up as a “bump” in the ratio plot.The tensor charges can be calculated using Eq. (11) if one neglects seaquark contributions. In Table 4 we present the estimated improvements for the truncated tensor charges at Q2=2.4 GeV2 and Q2=10 GeV2 separated into three kinematical regions of x: the region of SoLID acceptance (0.05<x<0.6) and the regions outside of SoLID coverage. For the region where SoLID has the maximum impact we find the improvement of about 90% (up to one order of magnitude) for both u and dquark tensor charges.Finally we present our estimates for the precision of extraction of isovector nucleon tensor charge gT=δu−δd, after the data of SoLID is taken into account:(21)gT(truncated)=+0.55−0.018+0.018,gT(full)=+0.64−0.021+0.021, at Q2=2.4 GeV2 where truncated means contribution from the region covered by the SoLID data 0.05<x<0.6, and full is the contribution from 0<x<1. See Table 4 for a detailed comparison. The precision of this result can be readily compared to precision of the lattice QCD calculations. As studied in Ref. [42], parameterizations of transversity that are substantially different in the region not covered by experimental data but similar in the region covered by the data lead to the growth of uncertainties of gT in the full kinematical region 0<x<1. While the uncertainty of the gT(full) could be underestimated, our estimate of the uncertainty of gT(truncated) is reliable. As one can see from Eq. (21) and Fig. 3 we predict an order of magnitude improvement of the error. Future data from Electron Ion Collider will extend the region of the data and allow to explore lowx region.In Fig. 3 we compare our result with extraction of Radici et al. Ref. [42] at Q2=4 GeV2, Anselmino et al. Ref. [40] at Q2=0.8 GeV2; Gamberg, Goldstein 2001 Ref. [57] at Q2=1 GeV2. Our result is also compared to a series of lattice computations, at Q2=4 GeV2 of Bali et al. Ref. [15], Gupta et al. Ref. [16], Green et al. Ref. [11], Aoki et al. Ref. [18], Bhattacharya et al. Refs. [12,13], Gockeler et al. Ref. [19]. Pitschmann et al. [21] is a DSE calculation at Q2=4 GeV2. There are many model calculations of transversity and tensor charge, QCD sum rules were used to estimate tensor charge in Ref. [58], Chiral Quark Soliton Model (CQSM) calculations are presented in Ref. [59], Lightcone wave functions were used in Ref. [60], bag models and CQSM results were reported in Ref. [61].The value of gT extracted from the data may influence searches beyond the standard model [8–10].6Summary and conclusionsWe have studied impact of future SoLID data on both the proton and the effective neutron targets on extraction of transversity for u and dquarks and tensor charge of the nucleon. A new method based on Hessian error analysis was developed in order to estimate the impact of future new data sets on TMD distributions. Based on the global QCD analysis with TMD evolution of the current data of Ref. [23] we estimated that the combination of both the proton and the effective neutron targets is essential for the appropriate extraction of tensor charge. As one can clearly see in Fig. 2 we predict a balanced improvement in the precision of extraction for both u and dquarks up to one order of magnitude in the range 0.05<x<0.6 with such a combination of measurements.We would like to emphasize that it is also important to investigate other possible contributions to asymmetries that may influence extraction of the quark transversity distributions from the experimental data. One particular example is the highertwist contributions, which can be thoroughly studied when the future data are available from Jefferson Lab 12 GeV upgrade, including both spinaveraged and spindependent cross section measurements. In addition, with the wide kinematic coverage in Q2, the planed Electron Ion Collider will provide valuable information on higher twist contributions as well.Under assumptions of Ref. [23] we also predict an impressive improvement in the extraction of tensor charge as can be seen in Table 4 in the presence of SoLID measurements. It appears that the acceptance region of SoLID will reveal most of contribution from u and d quarks to the tensor charge of the nucleon. The contribution from the region of highx not covered by SoLID (x>0.6) appear to be small for both u and d quarks, see Table 4. The same seems to be true for the contribution from lowx region, (x<0.05). However, with different functional forms in this region, the uncertainties could be larger. Such behavior could be best studied with the future Electron Ion Collider, in addition to contributions to the tensor charge from antiquarks which was omitted in the present analysis. We leave this for a future study.The precision at which isovector tensor charge gT can be extracted from the SoLID data will be comparable to the precision of lattice QCD calculates, as can be seen from Fig. 3, and will provide a unique opportunity for searches beyond the standard model. Our results demonstrate the powerful capabilities of future measurements of SoLID apparatus at Jefferson Lab 12 GeV Upgrade.AcknowledgementsWe are grateful to John Arrington, Leonard Gamberg, Wally Melnitchouk, and Ernst Sichtermann for useful discussions. This work was partially supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the TMD Topical Collaboration (A.P., F.Y.), and under Contracts No. DEAC0506OR23177 (A.P., N.S., J.C.), DEAC0206CH11357 (Z.Y.), DEFG0294ER40818 (K.A.), No. DEAC0205CH11231 (F.Y.), No. DEAC5206NA25396 (Z.K.), DEFG0203ER41231 (H.G., Z.Y., T.L.), by the National Science Foundation under Contract No. PHY1623454 (A.P.), and by the National Natural Science Foundation of China under Contract No. 11120101004 (H.G., Z.Y., T.L.), and in part by Duke Kunshan University (H.G., T.L.).References[1]J.P.RalstonD.E.SoperProduction of dimuons from highenergy polarized proton–proton collisionsNucl. Phys. B1521979109[2]R.L.JaffeX.JiChiral odd parton distributions and polarized Drell–YanPhys. Rev. Lett.671991552555[3]J.L.CortesB.PireJ.P.RalstonMeasuring the transverse polarization of quarks in the protonZ. Phys. 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