Transverse momentum distributions of quarks from the lattice using extended gauge links

TL;DR

This work demonstrates a lattice QCD approach to access intrinsic transverse momentum distributions of quarks in the nucleon by employing non-local quark bilinears linked by straight Wilson lines. By computing ratios of nucleon three-point to two-point functions for isovector operators, the authors extract matrix elements related to TMDPDFs and observe a Gaussian-like dependence on quark separation, enabling a first estimate of the first x-moment f1_n1(k_T). The preliminary, unrenormalized results yield root-mean-square transverse momenta of roughly 0.56–0.70 GeV, compatible with phenomenological values within uncertainties and highlighting a viable path toward lattice-based TMDPDFs. The study also outlines future work to incorporate SIDIS-like gauge links to infinity and back and to address operator renormalization and systematic effects.

Abstract

We present preliminary numerical studies in Lattice QCD related to the intrinsic transverse momentum distribution of partons in the nucleon. We employ non-local operators, consisting of spatially separated quark creation and annihilation operators connected by a straight Wilson line. A clear signal is already obtained from a small number of configurations at a pion mass of about 600 MeV. As an example, we demonstrate that we can obtain the first x-moment of the transverse momentum dependent parton distribution function f_1^{n=1}(k_T) from our data. Our results, which are not renormalized, show a Gaussian-like distribution. The root mean squared transverse momentum is about 560 MeV for a Gaussian fit, close to phenomenological values.

Figures (6)

• Figure 1: a) Factorized tree level diagram of semi-inclusive deep inelastic scattering (SIDIS), b) Gauge link to infinity and back as in SIDIS, c) Straight gauge link html:<A name="ref-fig-softblobs">html:</A> LAB: fig-softblobs
• Figure 2: a) Evaluation of the three-point function in the numerator of eq. (\ref{['eq-ratiodef']}) on the lattice (schematic), here for an operator $\mathcal{O}^{\Gamma}_\text{lat}$ with d-quarks. Only one of the two possible connected contractions of quark fields is shown. All-to-all propagators are avoided by combining three of the quark propagators into a sequential propagator (dark area). b) Overview of quark separations $\vec{\ell}$ in the x-y-plane used in this investigation. We have calculated all quark separations which lie on the x- or y-axis, up to a length of 20 lattice units. In the first quadrant, we have included all quark separations up to a length of 8 lattice units (inner grey circle) and a selection of longer ones. html:<A name="ref-fig-method">html:</A> LAB: fig-method
• Figure 3: Sample plateau plots: $R_{\Gamma} (\tau; \vec{P},\vec{\ell})$ is plotted versus $\tau$. The horizontal line and the error band indicate the plateau value and its error, extracted from the points (marked red), at $\tau = 4,5$ and $6$. a) Real part of $R_\Gamma (\tau; \vec{P},\vec{\ell})$ for $\Gamma = \gamma_4$, nucleon momentum $\vec{P}=(0,0,0)$, and a link path five units long in $x$ direction, i.e. $|\vec{\ell}|=5$. b) Imaginary part of $R_{\Gamma} (\tau, \vec{P},\vec{\ell})$ for $\Gamma = \gamma_3 \gamma_5$, nucleon momentum $\vec{P}=(0,0,0)$, and the link path shown in fig. \ref{['fig-steplike']}, i.e. $|\vec{\ell}|=6.7$. html:<A name="ref-fig-plateau">html:</A> LAB: fig-plateau
• Figure 4: a) Results for $\Gamma=\gamma_4$, $\vec{P}=(0,0,0)$. We plot $\mathrm{Re}\, R_{\gamma_4}(\vec{P},\vec{\ell})$ for all link paths versus the separation $|\vec{\ell}|$ of quark creation and annihilation operator. b) $\mathrm{Re}\, R_{\gamma_4}(\vec{P} = 0,\vec{\ell})$ versus $|\vec{\ell}|$ for link paths in the x-y-plane. Results for link paths which transform into one another under rotation or reflection have been averaged. Dashed turquoise curve: fit to the data with a single Gaussian function $H_1(|\vec{\ell}|)$, see eq. ( \ref{['eq-gauss']}). Solid red curve: fit with the superposition of two Gaussian functions $H_2(|\vec{\ell}|)$. The parameters determined from the fits are listed in tables \ref{['tab-fitresultssingle']} and \ref{['tab-fitresultsdouble']}. html:<A name="ref-fig-ratios">html:</A> LAB: fig-ratios
• Figure 5: a) $\mathrm{Im}\, R_{\Gamma}$ versus $|\vec{\ell}|$ for $\Gamma=\gamma_3 \gamma_5$ (axial vector), momentum $\vec{P}=(0,0,0)$ and link paths in the x-y-plane. Results for link paths which transform into one another under rotation or reflection have been averaged. b) $\frac{1}{2} \mathrm{Re}\, \{R_\Gamma(\vec{\ell}) + R_\Gamma(-\vec{\ell}) \}$ for $\Gamma=\gamma_4$, $\vec{\ell}$ on the positive $x$-axis, and non-zero nucleon momentum $\vec{P}=(-1,0,0)$. In both plots, the data have been fitted with a single Gaussian function $H_1(|\vec{\ell}|)$, see eq. ( \ref{['eq-gauss']}). Parameters determined from the fits are listed in table \ref{['tab-fitresultssingle']}. html:<A name="ref-fig-moreratios">html:</A> LAB: fig-moreratios