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

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) Brock et al. (1995), transverse momentum dependent PDFs (TMDs) Yoon et al. (2017a), and generalized parton distributions (GPDs) Diehl (2003) as experiments become more precise Accardi et al. (2016a); Boer et al. (2011). 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 Lin et al. (2018). 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 quantities for which good agreement between lattice calculations and experimental results, and their precision, has grown very significantly as discussed in the recent Flavor Lattice Averaging Group (FLAG) 2019 report Aoki et al. (2020). While steady progress has been made in developing the framework for calculating distribution functions using lattice QCD Cichy and Constantinou (2019); Karthik (2019), 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 $\langle x\rangle_{u-d}$ Lin et al. (2018). 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 Accardi et al. (2016a) and the Large Hadron Collider, which will significantly improve the phenomenological global fits, we anticipate steady progress toward 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) Follana et al. (2007) by the MILC Collaboration Bazavov et al. (2013). The data at four values of lattice spacings $a$, three values of the pion mass, $M_{\pi}$, including two ensembles at the physical pion mass, and on a range of large physical volumes, characterized by $M_{\pi}L$, allow us to carry 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 $\langle x\rangle_{u-d}=0.173(14)(07)$, $\langle x\rangle_{\Delta u-\Delta d}=0.213(15)(22)$ and $\langle x\rangle_{\delta u-\delta d}=0.208(19)(24)$ in the $\overline{\rm MS}$ 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 two- and 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 chiral-continuum-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 excited-state contamination are shown in Appendix A and the results for renormalization factors, $Z_{VD,AD,TD}$, for the three operators in Appendix B.

The parameters of the nine HISQ ensembles are summarized in Table 1. They cover a range of lattice spacings ($0.057\leq a\leq 0.15$ fm), pion masses ($135\leq M_{\pi}\leq 310$) MeV and lattice sizes ($3.7\leq M_{\pi}L\leq 5.5$). Most of the details of the lattice methodology, the strategies for the calculations and the analyses are already given in Refs. Bhattacharya et al. (2015, 2016); Gupta et al. (2018). We construct the correlation functions needed to calculate the matrix elements using Wilson-clover fermions on these HISQ ensembles. This mixed-action, clover-on-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. Gupta et al. (2018). The Sheikholeslami-Wohlert coefficient Sheikholeslami and Wohlert (1985) 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 Hasenfratz and Knechtli (2001) HISQ lattices.

The masses of light clover quarks were tuned so that the clover-on-HISQ pion masses, $M_{\pi}^{\rm val}$, match the HISQ-on-HISQ Goldstone ones, $M_{\pi}^{\rm sea}$. $M_{\pi}^{\rm val}$ values are given in Table 1. $M_{\pi}^{\rm sea}$ values are available in Ref. Gupta et al. (2018). All fits in $M_{\pi}^{2}$ to study the chiral behavior are made using the clover-on-HISQ $M_{\pi}^{\rm val}$ 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 $M_{\pi}$. The number of high precision (HP) and low precision (LP) measurements made on each configuration in the truncated solver bias corrected method Bali et al. (2010); Blum et al. (2013) for cost-effective increase in statistics are specified in Table 1.

In this work, we calculate the first moments of spin independent (or unpolarized), $q=q_{\uparrow}+q_{\downarrow}$, helicity (or polarized), $\Delta q=q_{\uparrow}-q_{\downarrow}$, and transversity, $\delta q=q_{\top}+q_{\perp}$ distributions, defined as

	$\displaystyle\langle x\rangle_{q}$	$\displaystyle=$	$\displaystyle\int_{0}^{1}\leavevmode\nobreak\ x\leavevmode\nobreak\ [q(x)+\overline{q}(x)]\leavevmode\nobreak\ dx\,,$		(1)
	$\displaystyle\langle x\rangle_{\Delta q}$	$\displaystyle=$	$\displaystyle\int_{0}^{1}\leavevmode\nobreak\ x\leavevmode\nobreak\ [\Delta q(x)+\Delta\overline{q}(x)]\leavevmode\nobreak\ dx\,,$		(2)
	$\displaystyle\langle x\rangle_{\delta q}$	$\displaystyle=$	$\displaystyle\int_{0}^{1}\leavevmode\nobreak\ x\leavevmode\nobreak\ [\delta q(x)+\delta\overline{q}(x)]\leavevmode\nobreak\ dx\,,$		(3)

where $q_{\uparrow(\downarrow)}$ corresponds to quarks with helicity aligned (antialigned) with that of a longitudinally polarized target, and $q_{\top(\perp)}$ corresponds to quarks with spin aligned (antialigned) 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 $\langle x\rangle_{q}$ and $\langle x\rangle_{\Delta q}$ of the nucleon are also obtained from phenomenological global fits while a computation of the nucleon transversity $\langle x\rangle_{\delta q}$ using lattice QCD is still a prediction due to the lack of sufficient experimental data Lin et al. (2018).

We are interested in extracting the forward nucleon matrix elements $\langle N(p)|{\cal O}|N(p)\rangle$, with the nucleon initial and final 3-momenta, $\vec{p}$, taken to be zero in this work. The complete set of one-derivative vector, axial-vector, and tensor operators is the following:

	$\displaystyle{\cal O}^{\mu\nu}_{V^{a}}$	$\displaystyle=$	$\displaystyle\overline{q}\gamma^{\{\mu}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{\nu\}}\tau^{a}q\,,$		(12)
	$\displaystyle{\cal O}^{\mu\nu}_{A^{a}}$	$\displaystyle=$	$\displaystyle\overline{q}\gamma^{\{\mu}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{\nu\}}\gamma^{5}\tau^{a}q\,,$		(21)
	$\displaystyle{\cal O}^{\mu\nu\rho}_{T^{a}}$	$\displaystyle=$	$\displaystyle\overline{q}\sigma^{[\mu\{\nu]}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{\rho\}}\tau^{a}q\,,$		(30)

where $q=\{u,d\}$ is the isodoublet of light quarks and $\sigma^{\mu\nu}=(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})/2$. The derivative $\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}_{\nu}\equiv\frac{1}{2}(\overrightarrow{D}_{\nu}-\overleftarrow{D}_{\nu})$ consists of four terms:

	$\displaystyle\overline{\psi}(\Gamma\overrightarrow{D}_{\nu}-\Gamma\overleftarrow{D}_{\nu})\psi(x)$	$\displaystyle\equiv$	$\displaystyle\frac{1}{2}\big{[}\overline{\psi}(x)\Gamma U_{\nu}(x)\psi(x+\nu)$		(31)
		$\displaystyle-$	$\displaystyle\overline{\psi}(x)\Gamma U^{\dagger}_{\nu}(x-\nu)\psi(x-\nu)$
		$\displaystyle+$	$\displaystyle\overline{\psi}(x-\nu)\Gamma U_{\nu}(x-\nu)\psi(x)$
		$\displaystyle-$	$\displaystyle\overline{\psi}(x+\nu)\Gamma U^{\dagger}_{\nu}(x)\psi(x)\big{]}\,.$

Lorentz indices within $\{\leavevmode\nobreak\ \}$ in Eq. (30) are symmetrized and within $[\,]$ are antisymmetrized. 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. Gockeler et al. (1996); Harris et al. (2019).

In this work, we consider only isovector quantities. These are obtained from Eq. (30) by choosing $\tau^{a}=\tau^{3}$ 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 as follows:

	$\displaystyle\langle N(p,s^{\prime})|{\cal O}^{\mu\nu}_{V^{a}}$	$\displaystyle|N(p,s)\rangle=$
		$\displaystyle{}\overline{u}_{N}^{p}(s^{\prime})A_{20}(0)\gamma^{\{\mu}p^{\nu\}}u_{N}^{p}(s)$		(32)
	$\displaystyle\langle N(p,s^{\prime})|{\cal O}^{\mu\nu}_{A^{a}}$	$\displaystyle|N(p,s)\rangle=$
		$\displaystyle{}i\overline{u}_{N}^{p}(s^{\prime})\tilde{A}_{20}(0)\gamma^{\{\mu}p^{\nu\}}\gamma^{5}u_{N}^{p}(s)$		(33)
	$\displaystyle\langle N(p,s^{\prime})|{\cal O}^{\mu\nu\rho}_{T^{a}}$	$\displaystyle|N(p,s)\rangle=$
		$\displaystyle{}i\overline{u}_{N}^{p}(s^{\prime})A_{T20}(0)\sigma^{[\mu\{\nu]}p^{\rho\}}u_{N}^{p}(s)$		(34)

The relation between the momentum fraction, the helicity moment, and the transversity moment, and the generalized form factors is $\langle x\rangle_{q}=A_{20}(0)$, $\langle x\rangle_{\Delta q}=\tilde{A}_{20}(0)$ and $\langle x\rangle_{\delta q}=A_{T20}(0)$ 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 Davoudi and Savage (2012); Detmold and Lin (2006); Detmold et al. (2018).

We use the following interpolating operator ${\mathcal{N}}$ to create/annihilate the nucleon state

$\displaystyle{\mathcal{N}}=\epsilon^{abc}\Big{[}q_{1}^{aT}(x)C\gamma^{5}\frac{(1\pm\gamma_{4})}{2}q_{2}^{b}(x)\Big{]}q^{c}_{1}(x)\,,$

(35)

where $\{a,b,c\}$ are color indices, $q_{1},q_{2}\in\{u,d\}$ and $C=\gamma_{0}\gamma_{2}$ is the charge conjugation matrix. The nonrelativistic projection $(1\pm\gamma_{4})/2$ is inserted to improve the signal, with the plus and minus signs applied to the forward and backward propagation in Euclidean time, respectively Gockeler et al. (1996). At zero momentum, this operator couples only to the spin $\frac{1}{2}$ state. The zero momentum 2-point and 3-point nucleon correlation functions are defined as

	$\displaystyle\bm{C}^{2pt}_{\alpha\beta}(\tau)$	$\displaystyle=\sum_{\bm{x}}\langle 0|{\mathcal{N}}_{\alpha}(\tau,{\bm{x}})\overline{{\mathcal{N}}}_{\beta}(0,{\bm{0}})|0\rangle$		(36)
	$\displaystyle\bm{C}^{3pt}_{\mathcal{O},\alpha\beta}(\tau,t)$	$\displaystyle=\sum_{{\bm{x}}^{\prime},{\bm{x}}}\langle 0|{\mathcal{N}}_{\alpha}(\tau,{\bm{x}}){\cal O}(t,{\bm{x}}^{\prime})\overline{{\mathcal{N}}}_{\beta}(0,{\bm{0}})|0\rangle$		(37)

where $\alpha$ and $\beta$ are spin indices. The source is placed at time slice 0, the sink is at $\tau$ and the one-derivative operators, defined in Sec. III, are inserted at time slice $t$. Data have been accumulated for the values of $\tau$ specified in Table 1, and in each case for all intermediate times $0\leq t\leq\tau$.

To isolate the various operators, projected $2$- and $3$-point functions are constructed as

	$\displaystyle C^{2pt}$	$\displaystyle=$	$\displaystyle{\rm Tr}\big{(}{\cal P}_{2pt}\bm{C}^{2pt}\big{)}$		(38)
	$\displaystyle C_{\mathcal{O}}^{3pt}$	$\displaystyle=$	$\displaystyle{\rm Tr}\big{(}{\cal P}_{3pt}\bm{C}^{3pt}_{\mathcal{O}}\big{)}\,.$		(39)

The projector ${\cal P}_{2pt}=\frac{1}{2}\,(1+\gamma_{4})$ in the nucleon correlator gives the positive parity contribution for the nucleon propagating in the forward direction. For the connected $3$-point contributions ${\cal P}_{3pt}=\frac{1}{2}(1+\gamma_{4})(1+i\gamma^{5}\gamma^{3})$ is used. With these spin projections, the explicit operators used to calculate the forward matrix elements are:

	$\displaystyle\langle x\rangle_{u-d}$	$\displaystyle:$	$\displaystyle{\cal O}^{44}_{V^{3}}=\overline{q}(\gamma^{4}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{4}-\frac{1}{3}{\bm{\gamma}}\cdot\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\bf D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\bf D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\bf D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\bf D\hfil$\crcr}}})\tau^{3}q$		(56)
	$\displaystyle\langle x\rangle_{\Delta u-\Delta d}$	$\displaystyle:$	$\displaystyle{\cal O}^{34}_{A^{3}}=\overline{q}\gamma^{\{3}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{4\}}\gamma^{5}\tau^{3}q$		(65)
	$\displaystyle\langle x\rangle_{\delta u-\delta d}$	$\displaystyle:$	$\displaystyle{\cal O}^{124}_{T^{3}}=\overline{q}\sigma^{[1\{2]}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle D\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle D\hfil$\crcr}}}^{4\}}\tau^{3}q\,.$		(74)

Our goal is to obtain the matrix elements, ($ME$), of these operators within the ground state of the nucleon. These $ME$ are related to the moments as follows:

	$\displaystyle\langle 0|{\cal O}^{44}_{V^{3}}|0\rangle$	$\displaystyle=$	$\displaystyle-M_{N}\,\langle x\rangle_{u-d}\,,$		(75)
	$\displaystyle\langle 0|{\cal O}^{34}_{A^{3}}|0\rangle$	$\displaystyle=$	$\displaystyle-\frac{iM_{N}}{2}\,\langle x\rangle_{\Delta u-\Delta d}\,,$		(76)
	$\displaystyle\langle 0|{\cal O}^{124}_{T^{3}}|0\rangle$	$\displaystyle=$	$\displaystyle-\frac{iM_{N}}{2}\,\langle x\rangle_{\delta u-\delta d}\,,$		(77)

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.