Lattice Parton Collaboration ($\rm{\bf LPC}$)

Introduction: Parton distribution functions (PDFs) characterize the internal structure of hadrons in terms of the number densities of their quark and gluon constituents. They are crucial inputs for interpreting the experimental data collected at high-energy colliders such as the LHC. At leading-twist accuracy, there exist three quark PDFs: the unpolarized, the helicity and the transversity PDF Ralston and Soper (1979). Among them, the transversity PDF describes the correlation between the transverse polarizations of the nucleon and its quark constituents, and thus play an important role in describing the transverse spin structure of the nucleon Jaffe and Ji (1991). In contrast to the unpolarized and helicity PDFs, the transversity PDF is much less constrained from experiments. This is because it is a chiral-odd quantity, and has to couple to another chiral-odd quantity, such as the chiral-odd fragmentation or distribution function, in order to be measurable experimentally Jaffe and Ji (1992); Cortes et al. (1992); Jaffe and Ji (1993). Currently, our knowledge of the transversity PDF mainly comes from measuring certain spin asymmetries in semi-inclusive deep-inelastic scattering or electron-positron annihilation to a hadron pair and Drell-Yan processes Gamberg et al. (2022); Constantinou et al. (2021). Based on these data, various global analyses of the transversity PDF have been performed Anselmino et al. (2007, 2009, 2013); Kang et al. (2015, 2016); Bacchetta et al. (2011, 2013); Radici et al. (2015); Lin et al. (2018a); Radici and Bacchetta (2018); Cammarota et al. (2020); Gamberg et al. (2022). Such analyses are expected to be greatly improved when more accurate data are accumulated at ongoing and future experiments at, e.g., the JLab 12 GeV upgrade and the Electron-Ion Collider (EIC).

On the other hand, recent theoretical developments Braun and Müller (2008); Ji (2013, 2014); Ma and Qiu (2018); Lin et al. (2018b); Radyushkin (2017) have allowed us to calculate/fit the Bjorken $x$-dependence of PDFs from first-principle lattice QCD. Such calculations are particularly important for quantities like the nucleon transversity PDF which are hard to extract from experiments. In the past few years, several lattice calculations Alexandrou et al. (2018); Liu et al. (2018); Egerer et al. (2022) of the nucleon transversity PDF have been carried out using either the large-momentum effective theory (LaMET) Ji (2013, 2014); Ji et al. (2021a) or the short-distance expansion (pseudo-PDF) Radyushkin (2017). However, they were all done at a single lattice spacing, while a reliable extrapolation to the continuum is required to make a comparison with experimental measurements. Moreover, the non-perturbative renormalization used in these calculations has been shown to suffer from undesired infrared (IR) effects arising from the improper renormalization at long distances Ji et al. (2021b). Therefore, it is highly desirable to obtain a reliable prediction that uses a proper renormalization and is valid in the continuum and physical mass limit. This is the purpose of the present work.

In this work, we present a state-of-the-art lattice calculation of the isovector quark transversity PDF $\delta u(x)-\delta d(x)$ of the proton, using the LaMET approach. The lattice matrix elements of the transversity quasi-PDF are calculated at four lattice spacings $a=\{0.098,0.085,0.064,0.049\}$ fm and various pion masses ranging between $220$ and $350$ MeV, with proton momenta up to $2.8$ GeV. The result is non-perturbatively renormalized in the hybrid scheme Ji et al. (2021b) with self renormalization Huo et al. (2021) proposed recently, which is a viable IR-safe renormalization approach, and extrapolated to the continuum, physical mass and infinite momentum. We also make a comparison between our results and recent global analyses on the isovector quark transversity PDF of the proton Cammarota et al. (2020); Gamberg et al. (2022).

Theoretical Framework: The leading-twist quark transversity PDF $\delta{q}(x)$ of the proton is defined as Jaffe and Ji (1991),

where $|PS_{\perp}\rangle$ denotes a transversely polarized proton with momentum $P$ along the $z$-direction and polarization $S_{\perp}$ along the transverse direction, $x$ is the momentum fraction carried by the quark, $\mu$ is the renormalization scale in the $\overline{\rm MS}$ scheme, $\xi^{\pm}=(\xi^{t}\pm\xi^{z})/\sqrt{2}$ are light-cone coordinates and $\mathcal{W}[0,\xi^{-}]=\mathcal{P}\,{\rm exp}\big{[}ig\int_{\xi_{-}}^{0}du~{}{n}\cdot A(un)\big{]}$ is the gauge link along the light-cone direction ensuring gauge invariance of the non-local quark bilinear correlator.

According to LaMET, we can extract the transversity PDF from the following transversity quasi-PDF on the lattice

where $\tilde{h}(z,P_{z},1/a)$ is the equal-time or quasi-light-front (quasi-LF) correlation that can be calculated directly on the lattice, $a$ denotes the lattice spacing, and $N=P_{z}/P_{t}$ is a normalization factor. Throughout the paper, we take the flavor combination $\delta\tilde{u}(x)-\delta\tilde{d}(x)$ to eliminate disconnected contributions.

The bare quasi-LF correlation above contains both linear and logarithmic ultraviolet (UV) divergences, which need to be removed by a proper non-perturbative renormalization. Various approaches have been suggested and implemented in the literature Chen et al. (2017); Izubuchi et al. (2018); Alexandrou et al. (2017); Radyushkin (2018); Braun et al. (2019); Li et al. (2018). However, they all suffer from the problem that the renormalization factor introduces undesired non-perturbative contributions distorting the IR behavior of the original quasi-LF correlation. This is avoided in the so-called hybrid scheme Ji et al. (2021b), where one separates the quasi-LF correlations at short and long distances and renormalize them separately. At short distances, the renormalization is done by dividing by the same hadron matrix element in the rest frame, as is done in the ratio scheme Radyushkin (2018); while at long distances one removes the UV divergences of the quasi-LF correlation only, which are determined by the so-called self renormalization Huo et al. (2021) through fitting the bare matrix elements at multiple lattice spacings to a physics-dictated functional form. The renormalized quasi-LF correlation then takes the following form

where $z_{s}$ is introduced to separate the short and long distances, and has to lie in the perturbative region. $Z_{R}(z,1/a)$ denotes the renormalization factor extracted from the self renormaliztaion procedure using the same transversity three-point correlator in the rest frame. Details are given in the Supplemental Material sup (2022). We have included a factor $\eta_{s}$ which is similar to a scheme conversion factor and determined by requiring continuity of the renormalized quasi-LF correlation at $z=z_{s}$. Of course, one needs to check the stability of the final result with respect to the choice of $z_{s}$. We vary $z_{s}$ in a certain perturbative range and include the difference in the systematic uncertainties. Note that all singular dependence on $a$ has been canceled in the above equation so that $\tilde{h}_{R}$ is independent of $a$ (up to some finite discretization effects).

After performing a Fourier transform to momentum space, we can match the transversity quasi-PDF to the transversity PDF through the following perturbative matching

where $C$ is the perturbative matching kernel whose explicit expression to $O(\alpha_{s})$ is given in the Supplemental Material sup (2022). The transversity at negative $y$ can be interpreted as the antiquark transversity via the relation $\delta\bar{q}(y,\mu)=-\delta q(-y,\mu)$. $\mathcal{O}\Big{(}{\Lambda^{2}_{\rm{QCD}}\over(xP_{z})^{2}},{\Lambda^{2}_{\rm{QCD}}\over((1-x)P_{z})^{2}}\Big{)}$ denotes higher-twist contributions suppressed by the nucleon momentum $P_{z}$.

Lattice calculation: To improve the signal-to-noise ratio of calculations with high-momentum nucleon states, we employ the momentum smearing source technique Bali et al. (2016). Besides, we apply two steps APE smearing to further improve the signal. We also use the sequential source method with fixed sink to calculate the quark three-point correlator, as illustrated in Fig. 1. In order to perform the continuum and chiral extrapolation, we use six different lattice ensembles generated by the CLS collaboration Bruno et al. (2015) with lattice spacing $a=\{0.098,0.085,0.064,0.049\}~{}\mathrm{fm}$ and pion masses ranging between $354$ and $222$ MeV. For each ensemble, we calculate several source-sink separations with hundreds to thousands of measurements among 500 gauge configurations (for X650 it is 1500 because the autocorrelation time is larger for this rather coarse lattice). Details of the lattice setup and parameters are collected in Table 1.

The bare matrix elements are calculated with the nucleon carrying different spatial momenta on each ensemble: $P_{z}=\{1.84,2.37,2.63\}$ GeV on X650, $P_{z}=\{1.82,2.27,2.73\}$ GeV on H102, $P_{z}=1.82$ GeV on H105 and C101, $P_{z}=\{1.62,2.02,2.43,2.83,3.24\}$ GeV on N203 and $P_{z}=\{2.09,2.62\}$ GeV on N302. We also calculate the zero-momentum bare matrix elements for each ensemble, which provide the renormalization factor at short distance and the inputs to extract the renormalization factor at long distance through self renormalization.

To extract the ground state matrix element, we decompose the two-point correlator $C^{2\text{pt}}(P_{z},t_{\text{sep}})$ and three-point correlator $C_{\Gamma}^{3\text{pt}}(P_{z},t,t_{\text{sep}})$ (with $\Gamma=\gamma^{t}\gamma^{\perp}\gamma_{5}$) as in Bhattacharya et al. (2014),

where $\left\langle 0\left|\mathcal{O}_{\Gamma}\right|0\right\rangle=\tilde{h}(z,P_{z},1/a)$ is the ground state matrix element, $t$ denotes the insertion time of $O_{\Gamma}$. The ellipses denote the contribution from higher excited states of the nucleon which decay faster than the ground state and first-excited state. We extract the ground state matrix element by performing a two-state combined fit with $C^{2\text{pt}}(P_{z},t_{\text{sep}})$ and the ratio $R_{\Gamma}(z,P_{z},t_{\text{sep}},t)={C_{\Gamma}^{3\text{pt}}(z,P_{z},t_{\text{sep}},t)}/{C^{2\text{pt}}(P_{z},t_{\text{sep}})}$. Details of the fit can be found in the Supplemental Material sup (2022).

In Fig. 2, we plot the renormalized quasi-LF correlation in the hybrid scheme, as a function of the quasi-LF distance $\lambda=zP_{z}$. The data points are obtained on ensembles with nearly the same pion mass. As can be seen from the figure, the results show a good convergence as $P_{z}$ increases for all ensembles. However, as we increase the nucleon boost momentum, the excited-state contamination worsens so that uncertainties increase. The $P_{z}=2.63$ GeV data on X650 and $P_{z}=3.23$ GeV data on N203 have much larger uncertainties compared to other data sets due to the relatively larger momentum on these ensembles. We exclude them in our analysis below. In Fig. 3, we show the pion mass dependence of the renormalized quasi-LF correlations on ensembles with the same lattice spacing $a=0.085$ fm, where the pion masses are $m_{\pi}=\{354,281,222\}$ MeV, respectively. As shown in the figure, the results only exhibit a very mild dependence on the pion mass.

From the figures above, we can see that the uncertainty of the renormalized quasi-LF correlation grows at large distance, while a Fourier transform to momentum space requires the quasi-LF correlation at all distances. If we do a brute-force truncation and Fourier transform, unphysical oscillations will appear in the momentum space distribution. To resolve this issue, we adopt a physics-based extrapolation form Ji et al. (2021b) at large quasi-LF distance

where the algebraic terms in the square bracket account for a power law behavior of the transversity PDFs in the endpoint region, and the exponential term comes from the expectation that at finite momentum the correlation function has a finite correlation length (denoted as $\lambda_{0}$) Ji et al. (2021b), which becomes infinite when the momentum goes to infinity. The detail of the extrapolation is expected to affect the final results in the small and large $x$ region where LaMET expansion breaks down Ji et al. (2021a). An example of the extrapolation is given in the Supplemental Material sup (2022).

After the renormalization and extrapolation, we can Fourier transform the quasi-LF correlation to momentum space and extract the transversity PDF by applying perturbative matching. The transversity PDF extracted this way still contains lattice artifacts which shall be removed by performing a continuum extrapolation. Also, our calculations are not done at infinite momentum and the physical point, so we have to extrapolate to infinite momentum and physical pion mass. To this end, we perform a simultaneous extrapolation using the following functional form including $a$, $P_{z}$, and $m_{\pi}$,

where $\delta q(x,P_{z},a,m_{\pi})$ on the l.h.s. denotes the transversity PDF results obtained in our calculation on different ensembles. The extrapolation of the pion mass dependence follows from the study in Ref. Chen and Ji (2001) (with $\mu_{0}=1$ GeV and $g^{\prime}=-(4g_{A}^{2}+1)/[2(4\pi f_{\pi})^{2}]$ and $g_{A}$ the axial charge of the nucleon) and for the CLS ensembles in Ref. Bali et al. (2019). The $a^{2}f(x)$ term denotes the leading discretization error which begins at $\mathcal{O}(a^{2})$ as the lattice action is already $\mathcal{O}(a)$ improved. The $a^{2}P_{z}^{2}$ term represents the momentum-dependent discretizaton error. The last term in the square bracket accounts for the leading higher-twist contribution, where in the numerator we also include a potential $a$-dependence. Choosing $g(x,a)$ as a generic function or parametrizing it with a quadratic form in $a$ yields similar results with slightly different errors. We refer the readers to the Supplemental Material sup (2022) for details of this extrapolation. Eventually the desired transversity PDF is given by

where $m_{\pi,\rm phys}=135$ MeV.

Numerical result: Our final result for the physical isovector quark transversity PDF $\delta u(x)-\delta d(x)$ (normalized by nucleon isovector tensor charge $g_{T}$) is shown in Fig. 4, together with the results of recent global analyses from the JAM collaboration (JAM20 Cammarota et al. (2020) and JAM22 Gamberg et al. (2022)). JAM20 is the global analysis that finds, for the first time, agreement between phenomenology and lattice calculation of all nucleon tensor charges $\delta u,\delta d$ and the isovector combination $g_{T}$. JAM22 provides an update to JAM20 by including certain new data sets and constraints from the Soffer bound and lattice $g_{T}$ result. Given the sensitivity of the results to the choice of data sets, we tend to view the difference between the two curves as an indication of systematic uncertainties from global fits. As can be seen from the figure, our result lies between the two global analyses and agrees with both within $1\sim 2\sigma$ error. Note that we have plotted two shaded bands at the endpoint regions to indicate that LaMET predictions are not reliable there (taken as $x\in[-0.1,0.1]$ and $x\in[0.9,1]$), which are estimated from the observation that the higher-twist terms in LaMET factorization Eq. (Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD) are of $\mathcal{O}(1)$, i.e., ${\Lambda_{\mathrm{QCD}}/(xP_{z})}\sim 1$ and ${\Lambda_{\mathrm{QCD}}/((1-x)P_{z})}\sim 1$ with the highest momentum in this work $P_{z}=2.83\mathrm{GeV}$. This estimation is consistent with the recent approach incorporating the renormalization group resummation (RGR) procedure Su et al. (2022), as can be seen from the Supplemental Material sup (2022). Our error band includes both statistical and systematic uncertainties, where the latter have four different sources. The first is the renormalization scale dependence, which is estimated by varying the scale to 3 GeV. The second error comes from the extrapolation to continuum, infinite momentum, and physical mass, which is relatively small. The third error is from the choice of $z_{s}$ in the hybrid renormalization scheme. We choose $z_{s}=0.3$ fm, vary it down to $z_{s}=0.18$ fm and take the difference as a systematic error. Lastly, the large $\lambda$ extrapolation also introduces some error that mainly affects the small-$x$ region $-0.2\lesssim x\lesssim 0.2$. We have chosen different regions to do the extrapolation to estimate this error. More details of the systematic uncertainties can be found in the Supplemental Material sup (2022).

In the negative $x$ region, our result is consistent with zero, which puts a strong constraint on the sea flavor asymmetry. JAM plans to update their global analysis by including spin asymmetry data from STAR Gamberg et al. (2022). It will be very interesting to see how their new analysis compares with our lattice result.

Summary: We present a state-of-the-art calculation of the isovector quark transversity PDF with LaMET. The calculation is done at various lattice spacings, pion masses and large nucleon momenta, and extrapolated to the continuum, physical mass and infinite momentum limit. With high statistics, we have performed multi-state analyses with multiple source-sink separations to remove the excited-state contamination, and applied state-of-the-art renormalization, matching and extrapolation. Our result provides the most reliable lattice prediction of the isovector quark transversity PDF in the proton so far, and will offer guidance to measurements at JLab and EIC.