Moments of nucleon isovector structure functions in $2+1+1$-flavor QCD

We present results on the isovector momentum fraction, $\langle x \rangle_{u-d}$, helicity moment, $\langle x \rangle_{\Delta u-\Delta d}$, and the transversity moment, $\langle x \rangle_{\delta u-\delta d}$, of the nucleon obtained using nine ensembles of gauge configurations generated by the MILC collaboration using $2+1+1$-flavors of dynamical highly improved staggered quarks (HISQ). The correlation functions are calculated using the Wilson-Clover action and the renormalization of the three operators is carried out nonperturbatively on the lattice in the RI${}^\prime$-MOM scheme. The data have been collected at lattice spacings $a \approx 0.15,\ 0.12,\ 0.09,$ and 0.06 fm and $M_\pi \approx 310,\ 220$ and 135 MeV, which are used to obtain the physical values using a simultaneous chiral-continuum-finite-volume fit. The final results, in the $\overline{MS}$ scheme at 2 GeV, are $\langle x \rangle_{u-d} = 0.173(14)(07)$, $\langle x \rangle_{\Delta u-\Delta d} = 0.213(15)(22)$ and $\langle x \rangle_{\delta u-\delta d} = 0.208(19)(24)$, where the first error is the overall analysis uncertainty and the second is an additional systematic uncertainty due to possible residual excited-state contributions. These are consistent with other recent lattice calculations and phenomenological global fit values.


I. INTRODUCTION
The elucidation of the hadron structure in terms of quarks and gluons is evolving from determining the charges and form factors of nucleons to including more complex quantities such as parton distribution functions (PDFs) [1], transverse momentum dependent PDFs (TMDs) [2], and generalized parton distributions (GPDs) [3] as experiments become more precise [4,5]. These distributions are not measured directly in experiments, and phenomenological analyses including different theoretical inputs are needed to extract them from experimental data. Input from lattice QCD is beginning to play an increasingly larger role in such analyses [6]. In cases where both lattice results and phenomenological analyses of experimental data (global fits) exist, one can compare them to validate the control over systematics in the lattice calculations, and on the other hand provide a check on the phenomenological process used to extract these observables from experimental data. In other cases, lattice results are predictions. The list of * santanu@lanl.gov † rajan@lanl.gov ‡ sungwoo@lanl.gov § boram@lanl.gov ¶ tanmoy@lanl.gov * * hwlin@pa.msu.edu quantities for which good agreement between lattice calculations and experimental results, and their precision, has grown very significantly as discussed in the recent Flavor Averaging Group (FLAG) 2019 report [7]. While steady progress has been made in developing the framework for calculating distribution functions using lattice QCD [8,9], even calculations of their moments have had large statistical and/or systematic uncertainties prior to 2018. This was the case even for the best studied quantity, the isovector momentum fraction x u−d [6]. In this work, we show that the lattice data for the momentum fraction, helicity and transversity moments are now of quality comparable to that for nucleon charges (zeroth moments). Together with much more precise data from the planned electron-ion collider [4] and the Large Hadron Collider, which will significantly improve the phenomenological global fits, we anticipate steady progress towards a detailed description of the hadron structure.
In this paper we present results on the three moments from high statistics calculations done on nine ensembles generated using 2+1+1-flavors of Highly Improved Staggered Quarks (HISQ) [10] by the MILC collaboration [11]. The data at four values of lattice spacings a, three values of the pion mass, M π , including two ensembles at the physical pion mass, and on a range of large physical volumes, characterized by M π L, allow us to carry  Table I. Lattice parameters, nucleon mass MN , number of configurations analyzed, and the total number of high precision (HP) and low precision (LP) measurements made. For the a06m310W ensemble, HP data were not collected, however, we note that the bias correction factor on all other eight ensemble was negligible. out a simultaneous fit in these three variables to address the associated systematics uncertainties. We also investigate the dependence of the results on the spectra of possible excited states included in the fits to remove excited-state contamination (ESC), and assign a second error to account for the associated systematic uncertainty. Our final results are x u−d = 0.173 (14)(07), x ∆u−∆d = 0.213 (15) (22) and x δu−δd = 0.208 (19) (24) in the MS scheme at 2 GeV. On comparing these with other lattice and phenomenological global fit results in Sec. VI, we find a consistent picture emerging.
The paper is organized as follows: In Sec. II, we briefly summarize the lattice parameters and methodology. The definitions of moments and operators investigated are given in Sec. III. The twoand three-point functions calculated, and their connection to the moments, are specified in Sec. IV, and the analysis of excited state contributions to extract ground state matrix elements is presented in Sec. V. Results for the moments after the chiralcontinuum-finite-volume (CCFV) extrapolation are given in Sec. VI, and compared with other lattice calculations and global fits. We end with conclusions in Sec. VII. The data and fits used to remove excitedstate contamination are shown in Appendix A and the results for renormalization factors, Z V D,AD,T D , for the three operators in Appendix B.

II. LATTICE METHODOLOGY
The parameters of the nine HISQ ensembles are summarized in Table I. They cover a range of lattice spacings (0.057 ≤ a ≤ 0.15 fm), pion masses (135 ≤ M π ≤ 310) MeV and lattice sizes (3.7 ≤ M π L ≤ 5.5). Most of the details of the lattice methodol-ogy, the strategies for the calculations and the analyses are already given in Refs. [12][13][14]. We construct the correlation functions needed to calculate the matrix elements using Wilson-clover fermions on these HISQ ensembles. This mixed-action, cloveron-HISQ, formulation is nonunitary and can suffer from the problem of exceptional configurations at small, but a priori unknown, quark masses. We have not found evidence for such exceptional configurations on any of the nine ensembles analyzed in this work.
For the parameters used in the construction of the 2-and 3-point functions with Wilson-clover fermion see Table II of Ref. [14]. The Sheikholeslami-Wohlert coefficient [15] used in the clover action is fixed to its tree-level value with tadpole improvement, c sw = 1/u 0 , where u 0 is the fourth root of the plaquette expectation value calculated on the hypercubic (HYP) smeared [16] HISQ lattices.
The masses of light clover quarks were tuned so that the clover-on-HISQ pion masses, M val π , match the HISQ-on-HISQ Goldstone ones, M sea π . M val π values are given in Table I. M sea π values are available in Ref. [14]. All fits in M 2 π to study the chiral behavior are made using the clover-on-HISQ M val π since the correlation functions, and thus the chiral behavior of the moments, have a greater sensitivity to it. Henceforth, for brevity, we drop the superscript and denote the clover-on-HISQ pion mass as M π . The number of high precision (HP) and low precision (LP) measurements made on each configuration in the truncated solver bias corrected method [17,18] for cost-effective increase in statistics are specified in Table I.

III. MOMENTS AND MATRIX ELEMENTS
In this work, we calculate the first moments of spin independent (or unpolarized), q = q ↑ + q ↓ , helicity (or polarized), ∆q = q ↑ − q ↓ , transversity, δq = q + q ⊥ distributions, defined as where q ↑(↓) corresponds to quarks with helicity aligned (anti-aligned) with that of a longitudinally polarized target, and q (⊥) corresponds to quarks with spin aligned (anti-aligned) with that of a transversely polarized target. These moments, at leading twist, can be extracted from the hadron matrix elements of one-derivative vector, axial-vector and tensor operators at zero momentum transfer. The unpolarized and polarized moments x q and x ∆q of the nucleon are also obtained from phenomenological global fits while a computation of the nucleon transversity x δq using Lattice QCD is still a prediction due to lack of sufficient experimental data [6].
We are interested in extracting the forward nucleon matrix elements N (p)|O|N (p) , with the nucleon initial and final momenta, p, taken to be zero in this work. The complete set of one-derivative vector, axial-vector, and tensor operators are: where q = {u, d} is the isodoublet of light quarks and σ µν = (γ µ γ ν − γ ν γ µ )/2. The derivative consists of four terms: It is also implicit that, where relevant, the traceless part of the above operators is taken. Their renormalization is carried out nonperturbatively in the regularization independent RI -MOM scheme as discussed in Appendix B. A more detailed discussion of these twist-2 operators and their renormalization can be found in Refs. [19] and [20].
In this work, we consider only isovector quantities. These are obtained from Eq. (4) by choosing τ a = τ 3 for the Pauli matrix. The decomposition of the matrix elements of these operators in terms of the generalized form factors at zero momentum transfer is: The relation between the momentum fraction, helicity moment, and the transversity moment, and the generalized form factors is x q = A 20 (0), x ∆q = A 20 (0) and x δq = A T 20 (0) respectively. We end this discussion by mentioning that other approaches have been proposed to calculate the moments of PDFs from Lattice QCD in recent years [21][22][23].

IV. CORRELATION FUNCTIONS AND MOMENTS
We use the following interpolating operator N to create/annihilate the nucleon state where {a, b, c} are color indices, q 1 , q 2 ∈ {u, d} and C = γ 0 γ 2 is the charge conjugation matrix. The nonrelativistic projection (1 ± γ 4 )/2 is inserted to improve the signal, with the plus and minus signs applied to the forward and backward propagation in Euclidean time, respectively [19]. At zero momentum, this operator couples only to the spin 1 2 state. The zero momentum 2-point and 3-point nucleon correlation functions are defined as where α, β are spin indices. The source is placed at time slice 0, the sink is at τ and the one-derivative operators, defined in Sec. III, are inserted at time slice t. Data have been accumulated for the values of τ specified in Table I, and in each case for all intermediate times 0 ≤ t ≤ τ . To isolate the various operators, projected 2-and 3-point functions are constructed as The projector P 2pt = 1 2 (1 + γ 4 ) in the nucleon correlator gives the positive parity contribution for the nucleon propagating in the forward direction. For the connected 3-point contributions P 3pt = 1 2 (1 + γ 4 )(1 + iγ 5 γ 3 ) is used. With these spin projections, the explicit operators used to calculate the forward matrix elements are: Our goal is to obtain the matrix elements (M E), of these operators within the ground state of the nucleon. These M E are related to the moments as: where M N is the nucleon mass. The three moments are dimensionless, and their extraction on a given ensemble does not require knowing the value of the lattice scale a. It enters only when performing the chiral-continuum extrapolation to the physical point as discussed in Sec. VI.

V. CONTROLLING EXCITED STATE CONTAMINATION
To calculate the matrix elements of the operators defined in Sec. III between ground-state nucleons, contributions of all possible excited states need to be removed. The lattice nucleon interpolating operator N given in Eq. (9), however, couples to the nucleon, all its excitations and multiparticle states with the same quantum numbers. Previous lattice calculations have shown that the ESC can be large [24][25][26]. In our earlier works [12][13][14]27], we have shown that this can be controlled to within a few percent. We use the same strategy here. In particular, we use HYP smearing of the gauge links before calculating Wilson-clover quark propagators with optimized Gaussian smeared sources using the multigrid algorithm [28,29]. Correlation functions constructed from these smeared source propagators have smaller excited state contamination [27]. To extract the ground state matrix elements from these, we fit the three-point data at several τ -values (listed in Table I) simultaneously using the spectral decomposition given in Eq. (21).
Fits to the zero-momentum two-point functions, C 2pt , were carried out keeping up to four states in the spectral decomposition: Fits are made over a range {τ min − τ max } to extract M i and A i , the masses and the amplitudes for the creation/annihilation of these states by the interpolating operator N . In fits with more than two states, estimates of the amplitudes A i and masses M i for i ≥ 2 were sensitive to the choice of the starting time slice τ min . We used the largest time interval allowed by statistics, i.e., by the stability of the covariance matrix. We perform two types of 4-state fits. In the fit denoted {4}, we use the empirical Bayesian technique described in the Ref. [30] to stabilize the three excited-state parameters. In the second fit, denoted {4 N π }, we use as prior for M 1 either the non-interacting energy of N (−1)π(1) or the N (0)π(0)π(0) state, which are both lower than the M 1 obtained from the {4} fit, and roughly equal for the six ensembles. The lower energy N (−1)π(1) state has been shown to contribute in the axial channel [31], whereas for the vector channel the N (0)π(0)π(0) state is expected to be the relevant one. We find that these two fits to the two-point function cannot be distinguished on the basis of the χ 2 /DOF, in fact the full range of M 1 between the two estimates from {4} and {4 N π } are viable firstexcited-state masses on the basis of χ 2 /DOF alone. The same is true of the values for M 2 . We therefore, investigate the dependence of the results for moments on the exited-state spectra by doing the full analysis with multiple strategies as discussed below. The ground-state nucleon mass obtained from the various fits is denoted by the common symbol M N ≡ M 0 and the mass gaps by The analysis of the zero-momentum three-point functions, C 3pt O , is performed retaining upto three states |i in the spectral decomposition: The operators, O, are defined in Eqs. (14), (15) and (16). By fixing the momentum at the sink to zero and inserting the operator at zero momentum   (8) (21)   transfer we get the forward matrix element. The practical challenge discussed above is determining the relevant M 1 and M 2 to use, and failing that, to investigate the sensitivity of 0|O|0 to possible values of M 1 and M 2 and including that variation as a systematic uncertainty. For a given strategy for determining M 1 and M 2 , we extract the desired ground state matrix element 0|O|0 by fitting the three-point correlators C 3pt O (t; τ ) for a subset of values of t and τ simultaneously. This subset is chosen to reduce ESC-we select the largest values of τ and discard t skip number of points next to the source and sink for each τ . These values of τ and of t skip are given in Table II.
The data for the ratio C 3pt O (τ ; t)/C 2pt (τ ) are shown in Figs. 5 and 6 in the Appendix A for all nine ensembles. The signal in the three-point correlators decreases somewhat from momentum fraction to helicity moment to transversity moment. Nevertheless, we are able to make 3 * state (3-state with 2|O|2 = 0) fits in all cases. The spectral decomposition predicts that the data for all three quantities is symmetric about t = τ /2, however, on some of the ensembles, and for some of the larger values of τ , the data show some asymmetry, which is indicative of the size of statistical fluctuations that are present.
The fits to C 2pt (τ ) and C 3pt O (τ ; t) are carried out within a single-elimination jackknife process, which is used to get both the central values and the errors.
We have investigated five fit types, {4, 2}, {4 N π , 2}, {4, 3 * }, {4 N π , 3 * } and {4, 2 free }, based on the spectral decomposition to understand and control ESC. The labels {m, n} denote an m-state fit to the two-point function and an n-state fit to the three-point function. In the 2 free -fit to the threepoint function, M 1 is left as a free parameter, while a 3 * -fit is a 3-state fit with 2|O|2 = 0. The results from the five strategies for the momentum fraction, x u−d , in Table III, for the helicity moment, x ∆u−∆d , in Table IV, and for the transversity moment, x δu−δd , in Table V illustrate the observed behavior for the a09m310 ensemble, which has the highest statistics, and the physical mass ensemble a06m135 at the smallest value of a.
For all three observables, the five results in Tables III, and IV, and V for the ground state matrix element, 0|O|0 , are consistent within 2σ on the a09m310 ensemble. On the a06m135 ensemble, the difference in ∆M 1 ≡ M 1 − M 0 between {4} and {4 N π } analyses becomes roughly a factor of two, and ∆M 1 from the {2 free } fit is larger than even the {4} value, i.e., the {2 free } fit does not prefer the small ∆M 1 given by {4 N π }. On the other hand, the ∆M 1 from a two-state fit is expected to be larger since it is an effective combination of the mass gaps of the full tower of excited states. Due to a small ∆M 1 , fits with the spectrum from {4 N π } fail on a06m135, whereas, on both ensembles, the {4, 3 * } and {4, 2 free } fits gives results consistent within 2σ. The estimates from these two fit types are given in Table II. To summarize, our overall strategy is to keep as many excited states as possible without overparameterization of the fits. We, therefore, choose, for the central values, the {4, 3 * } results, and to take into account the spread due to the fit type, we add a second, systematic, uncertainty to the final results in Table VII. This is taken to be the difference between the results obtained by doing the full analysis with the {4, 3 * } and {4, 2 free } strategies.
The renormalization of the matrix elements is carried out using estimates of Z V D , Z AD , and Z T D calculated on the lattice in the RI −MOM scheme and then converted to the MS scheme at 2 GeV as described in the Appendix B. The final values of Z V D , Z AD , and Z T D used in the analysis are given in Table IX.

VI. CHIRAL, CONTINUUM AND INFINITE VOLUME EXTRAPOLATION
To obtain the final, physical results at M π = 135 MeV, M π L → ∞ and a = 0, we make a simultaneous CCFV fit keeping only the leading correction term in each variable: Note that, since the operators are not O(a) improved and we used the Clover-on-HISQ formulation, we take the discretization errors to start with a term linear in a. The fits to the {4, 3 * } data from the nine ensembles are shown in Figs. 1, 2 and 3. The fit parameters are summarized in Table VI.
The results of the CCFV fits show that the finite volume correction term, c 4 , is not constrained at all. We therefore, also present results from a CC fit with c 4 = 0 in Eq. (22). Results for c 1 from the two fit ansatz overlap and there is a small positive slope in both a and M 2 π for all three quantities. The data for both {4, 3 * } and {4, 2 free }, given in Table II, are very similar, but with a systematic shift of about 0.01-0.02 in all three cases. This difference arises because ∆M 1 for {4, 2 free } is larger (except in a09m310) and because the convergence with respect to τ is from above as shown in Figs. 5 and 6  the CCFV fits is undetermined. The CC results with the two strategies, {4, 3 * } and {4, 2 free }, are summarized in Table VII. For our best estimates, we take the {4, 3 * } results and add a second, systematic, error that is the difference between these two strategies and represents the uncertainty in controlling the excited-state contamination. A comparison of these results with other lattice QCD calculations on ensembles with dynamical fermions is presented in the top half of Table VIII. Our results agree with those from the Mainz group [20] that have also been obtained using data on a comparable number of ensembles, but all with M π > 200 MeV, to perform a chiral and continuum extrapolation. The one difference is the slope c 3 of the chiral correction. For our clover-on-HISQ formulation, we find a small positive value while the Mainz data show a small negative value [20]. Our results are also consistent within 1σ with the ETMC 20 [32] and ETMC 19 [33] values that are from a single physical mass ensemble. Results for momentum fraction and the helicity moment from RQCD 18 [34] are taken from their Set A with the difference between Set A and B values quoted as a second systematic uncertainty. The result for the transversity moment is from a single 150 MeV ensemble. These values are larger, especially for the helicity and transversity moment. Other earlier lattice results show a spread, however, in each of these calculations, the systematics listed in the last column of Table VIII have not been addressed or controlled and could, therefore, account for the differences.
Estimates from phenomenological global fits, most of which have also been reviewed in Ref. [6], are summarized in the bottom of Table VIII. We find that results for the momentum fraction from global fits are, in most cases, 1-2σ smaller and have much smaller errors. Results for the helicity moment are consistent and the size of the errors comparable. Lattice estimates of the transversity moment are a prediction.

VII. CONCLUSIONS
In this paper, we have presented results for the isovector quark momentum fraction, x MS u−d , helicity moment, x MS ∆u−∆d , and transversity moment, x MS δu−δd , from a high statistics lattice QCD calculation. Attention has been paid to the systematic uncertainty associated with excited-state contamination. We have carried out the full analysis with different estimates of the mass gaps of possible excited states, and use the difference in results between the two strategies that give stable fits on all ensembles as an additional systematic uncertainty to account for possible residual excited-state contamination.
The behavior versus M π , the lattice spacing a and finite volume parameter M π L have been investigated using a simultaneous fit that includes the leading correction in all three variables as given in Eq. (22). The nine data points cover the range 0.057 < a < 0.15 fm, 135 < M π < 320 MeV and 3.7 < M π L < 5.5. Over this range, all three moments, x MS u−d , x MS ∆u−∆d and x MS δu−δd , do not show a large dependence on a or M π or M π L. As shown

Collaboration
Ref.  [40] 0.156(7) JAM17 † [6,41] 0.241(26) NNPDF3.1 [42] 0.152(3) ABMP2016 [43] 0.167(4) CJ15 [44] 0.152(2) HERAPDF2.0 [45] 0.188(3) CT14 [46] 0.158(4) MMHT2014 [47] 0.151(4) NNPDFpol1.1 [48] 0.195(14) DSSV08 [49,50] 0.203 (9) Table VIII. Our Lattice QCD results are compared with other lattice calculations with N f flavors of dynamical fermions in rows 2-9, and with results from phenomenological global fits in the remainder of the table. In both cases, the results are arranged in reverse chronological order. All results are in the MS scheme at scale 2 GeV. For a discussion and comparison of lattice and global fit results up to 2017, see Ref. [6] and a more recent comparison in [40] for x u−d . The JAM17 † estimate for x ∆u−∆d is obtained from [6], where, as part of the review, an analysis was carried out using the data in [41]. The following abbreviations are used in the remarks column for various sources of systematic uncertainties in lattice calculations-DIS: Discretization effects, CE: Chiral extrapolation, FV: Finite volume effects, NR: Nonperturbative renormalization, ES: Excited state contaminations. A prefix "N-" means that the systematic uncertainty was not adequately controlled or not estimated.   Table VIII. The left panel compares results for the momentum fraction, the middle for the helicity moment, and the right for the transversity moment. The PNDME 20 result is also shown as the blue band to facilitate comparison. . Data and fits for a06m135 (top row), a06m310W (second row), a09m130 (third row) and a09m220 (last row). In each row, the three panels shows the ratio C 3pt O (τ ; t)/C 2pt (τ ) scaled according to Eq. (17)- (19) to give x u−d (left), x ∆u−∆d (middle), and x δu−δd (right). For each τ , the line in the same color as the data points is the result of the fit used (see Sec. V) to obtain the ground state matrix element. The ensemble ID, the final result x (also shown by the blue band and summarized in Table II), the values of τ , and χ 2 /DOF of the fit are also given in the legends. The interval of the y-axis is selected to be the same for all the panels to facilitate comparison.   in Table VI, possible dependence on the lattice size, characterized by M π L, is not resolved by the data, i.e., the coefficient c 4 is unconstrained. We, therefore, take for our final results those obtained from just the chiral-continuum fit. The small increase with a and M 2 π , evident in Figs 1-3, is well fit by the leading correction terms that are linear in these variables. Also, for all three observables, the chirally extrapolated value is consistent with the data from the two physical mass ensembles. In short, the observed small dependence in all three variables, and having two data points at M π ∼ 135 MeV to anchor the chiral fit, allows a controlled extrapolation to the physical point, M π = 135 MeV and a = 0.
Our final results, given in Table VII, are compared with other lattice calculations and phenomenological global fit estimates in Table VIII. Estimates of all three quantities are in good agreement with those from the Mainz collaboration [20], also obtained using a chiral-continuum extrapolation, and the ETMC collaboration [32,33] that are from a single physical mass ensemble. On the other hand, most global fit estimates for the momentum fraction are about 10% smaller and have much smaller errors, while those for the helicity moment are consistent within 1σ. Lattice estimates for the transversity moment are a prediction. The overall consistency of results suggests that lattice QCD calculations of these isovector moments are now mature and future calculations will steadily reduce the statistical and systematic uncertainties in them. In this appendix, we describe the calculation of the renormalization factors, Z V D,AD,T D , for the three one-derivative operators. These are determined nonperturbatively on the lattice in the RI −MOM scheme [52,53] as a function of the lattice scale p 2 = p µ p µ , and then converted to MS scheme using 3-loop perturbative factors calculated in the continuum in Ref. [54]. For data at each p, we perform horizontal matching by choosing the MS scale µ = |p|. These numbers are then run in the continuum MS scheme from scale µ to 2 GeV using three-loop anomalous dimensions [54]. This calculation of Z V D,AD,T D is done for one value of M π at each a. Based on our experience with local operators [13], where we found insignificant dependence of results on M π , we assume that these values, within errors, give the mass-independent renormalization factors at each a. Also, the decomposition of the three operators into irreducible representations given in Refs. [19,20], show that they can only mix with higher dimensional operators. In our CCFV fits, such O(a) effects would also be taken into account and removed by the continuum extrapolation.
In Fig. 7, we show the behavior of the renormalization factors Z V D,AD,T D in the MS scheme at µ = 2 GeV for the four ensembles as a function of |p|-the scale of the RI −MOM scheme on the lattice. In Fig. 8 we compare Z V D used to renormalize x u−d for the four ensembles, one at each lattice spacing.
For all three operators, the data do not show a window in |p| where the results are independent of |p|. Thus, for the final estimates we use the strategy labeled "Method B" in Ref. [13]. This corresponds to taking an average over the data points in an interval of 2 GeV 2 about p 2 = Λ/a, where the scale Λ = 3 GeV is chosen to be large enough to avoid nonperturbative effects and at which perturbation theory is expected to be reasonably well behaved. This choice satisfies both pa → 0 and Λ/p → 0 in the continuum limit as desired. The window over which the data are averaged and the error (half the height of the band) are shown by shaded bands in Figs. 7 and 8. To be conservative, and noting the variation with p 2 , we take twice the error (full height of the band) for the error estimate for all three Z s in the final analysis.
These final estimates of Z V D , Z AD and Z T D used to renormalize the momentum fraction, the helicity moment and the transversity moment, respectively, are given in Table IX.