Large Momentum Effective Theory Research Articles

Transverse-momentum-dependent wave functions of the pion from lattice QCD

We present a lattice QCD calculation of the transverse-momentum-dependent wave function (TMDWF) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks with lattice spacing a=0.121 fm from the MILC collaboration, and one with 2+1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS collaboration with a=0.098 fm. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWF simulated on the lattice, we extract the TMDWF. The results for both lattice ensembles are compatible and a comparison with a phenomenological parametrization is made. Our studies provide a first attempt of calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization. Published by the American Physical Society 2024

Nucleon helicity parton distribution function in the continuum limit with self-renormalization

We present the first lattice calculation of the nucleon isovector helicity parton distribution function (PDF) in the framework of large-momentum effective theory (LaMET) that uses the hybrid scheme with self-renormalization. We use ensembles generated by the MILC collaboration at lattice spacings a={0.1207,0.0888,0.0582} fm, with Nf=2+1+1 flavors of highly improved staggered quarks at sea pion mass of Mπ≈315 MeV. We use clover-improved action for our valence quarks with nucleon boost momentum Pz≈1.75 GeV and high-statistics measurements for the LaMET matrix elements. We perform an extrapolation to the continuum limit and improve the handling of systematic errors using renormalization-group resummation (RGR) and leading-renormalon resummation (LRR). Our final nucleon helicity PDF is renormalized in the MS‾ scheme at energy scale μ=2.0 GeV. We compare our results with and without the two systematic improvements of RGR and LRR at each lattice spacing as well as the continuum limit, and we see that the application of RGR and LRR greatly reduces the systematic errors across the whole x range. Our continuum results with both RGR and LRR show a small positive antiquark region for the nucleon helicity PDF as well as a change of as much as a factor of two in the central values compared to results with neither RGR or LRR. By contrast, the application of RGR and LRR only changes the central values by about 5% in the quark region. We compare our lattice results with the global fits by the JAM, NNPDF and DSSV collaborations, and we observe some tension between our results.

Logarithmic moments of B-meson quasidistribution amplitude

It was demonstrated that the lattice simulation of B-meson light-cone distribution amplitude (LCDA) is feasible via the quasi-distribution amplitude (quasi-DA) in large momentum effective theory (LaMET). The structures of logarithmic moments (LMs) of B-meson quasi-DA are explored in this work. The one-loop results indicate mixing in the matching: the n-th LM would be not only factorized into the n-th LM of LCDA, but also other moments with different power, accompanied by short distance coefficients. These results supply the understanding of the matching in LaMET and may provide guidance to the lattice study of LMs or other parameters of B-meson LCDA.

Pion valence quark distribution at physical pion mass of N f = 2 + 1 + 1 lattice QCD

We present a state-of-the-art calculation of the unpolarized pion valence-quark distribution in the framework of large-momentum effective theory (LaMET) with improved handling of systematic errors as well as two-loop perturbative matching. We use lattice ensembles generated by the MILC collaboration at lattice spacing a ≈ 0.09 fm, lattice volume 643 × 96, N f = 2 + 1 + 1 flavors of highly-improved staggered quarks and a physical pion mass. The LaMET matrix elements are calculated with pions boosted to momentum P z ≈ 1.72 GeV with high-statistics of O(106) measurements. We study the pion PDF in both hybrid-ratio and hybrid-regularization-independent momentum subtraction (hybrid-RI/MOM) schemes and also compare the systematic errors with and without the addition of leading-renormalon resummation (LRR) and renormalization-group resummation (RGR) in both the renormalization and lightcone matching. The final lightcone PDF results are presented in the modified minimal-subtraction scheme at renormalization scale μ = 2.0 GeV. We show that the x-dependent PDFs are compatible between the hybrid-ratio and hybrid-RI/MOM renormalization with the same improvements. We also show that systematics are greatly reduced by the simultaneous inclusion of RGR and LRR and that these methods are necessary if improved precision is to be reached with higher-order terms in renormalization and matching.

Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD.

We report a state-of-the-art lattice QCD calculation of the isovector quark transversity distribution of the proton in the continuum and physical mass limit using large-momentum effective theory. The calculation is done at four lattice spacings a={0.098,0.085,0.064,0.049} fm and various pion masses ranging between 220 and 350MeV, with proton momenta up to 2.8GeV. The result is nonperturbatively renormalized in the hybrid scheme with self-renormalization, which treats the infrared physics at large correlation distance properly, and extrapolated to the continuum, physical mass, and infinite momentum limit. We also compare with recent global analyses for the nucleon isovector quark transversity distribution.

Towards double parton distributions from first principles using Large Momentum Effective Theory

In double parton scattering (DPS), two partonic collisions take place between one pair of colliding hadrons. The effect of DPS can be significant for precision measurements due to the additional radiation from secondary partonic collisions, and especially for specific processes such as same-sign WW production. Its effect is usually included through Monte Carlo parton showers. In a factorization approach to DPS, the initial state is described by double parton distributions (DPDs). These are currently poorly constrained by experiment, but provide a view on interesting correlations between partons in the hadron. Here we show that the Large Momentum Effective Theory approach can be applied to DPDs. Specifically, we present a general matching relation between DPDs and lattice-calculable quasi-DPDs for general flavor, spin and color structures. We furthermore calculate the one-loop matching coefficients for the quark-quark DPDs, verifying that the infrared logarithms and divergences cancel in the matching. While we restrict to the flavor-non-singlet case, we do take color and spin correlations into account. Interestingly, quasi-DPDs combines nontrivial features from both the collinear and transverse momentum dependent quasi-parton distribution functions. This represents a first step in extending the quasi-PDF approach to DPDs, opening up a new way to constrain these distributions using lattice QCD.

Pion valence-quark generalized parton distribution at physical pion mass

We present the first lattice-QCD x-dependent pion valence-quark generalized parton distribution (GPD) calculated directly at physical pion mass using the Large-Momentum Effective Theory (LaMET) with next-to-next-to-leading order perturbative matching correction. We use clover fermions for the valence action on 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, with lattice spacing a≈0.09 fm and box size L≈5.5 fm; the pion two-point measurements number up to O(106) with boost momentum 1.73 GeV. The pion valence distribution is renormalized in hybrid scheme with Wilson-line mass subtraction at large distances in coordinate space, followed by a procedure to match it to the MS‾ scheme. We focus on the zero-skewness limit, where the GPD has a probability-density interpretation in the longitudinal Bjorken x and the transverse impact-parameter distributions. We take the integral of our GPD functions to generate leading moment so that we can make comparisons with past lattice-QCD and experimental determinations of the pion form factors and found consistent agreement among them. We predict the higher GPD moments and reveal x-dependent tomography of the pion for the first time using lattice QCD.

Selected lattice QCD results on hadron structure

We present recent results on hadron structure using lattice QCD simulations with physical values of the light, strange and charm quark masses. We overview progress on quantities that probe the three-dimensional structure of the nucleon focusing on recent results on Mellin moments and the direct computation of the x-dependence of parton distribution functions employing mainly the quasi-parton distribution approach within the large momentum effective theory.

Lattice calculation of the intrinsic soft function and the Collins-Soper kernel

We calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudoscalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton.

Threshold resummation for computing large-x parton distribution through large-momentum effective theory

Parton distribution functions (PDFs) at large x are poorly constrained by high-energy experimental data, but extremely important for probing physics beyond standard model at colliders. We study the calculation of PDFs at large-x through large-momentum Pz expansion of the lattice quasi PDFs. Similar to deep-inelastic scattering, there are two distinct perturbative scales in the threshold limit where the matching coefficient can be factorized into a space-like jet function at scale Pz|1 − y| and a pair of heavy-light Sudakov form factors at scale Pz. The matching formula allows us to derive a full renormalization group resummation of large threshold logarithms, and the result is consistent with the known calculation to the next-to-next to leading order (NNLO). This paves the way for direct large-x PDFs calculations in lattice QCD. As by-products, we find that the space-like jet function is related to a time-like version calculated previously through analytic continuation, and the heavy-light Sudakov form factor, calculated here to NNLO, is a universal object appearing as well in the large momentum expansion of quasi transverse-momentum-dependent PDFs and quasi wave-function amplitudes.

Light-cone distribution amplitudes of a light baryon in large-momentum effective theory

Momentum distributions of quarks/gluons inside a light baryon in a hard exclusive process are encoded in the light-cone distribution amplitudes (LCDAs). In this work, we point out that the leading twist LCDAs of a light baryon can be obtained through a simulation of a quasi-distribution amplitude calculable on lattice QCD within the framework of the large-momentum effective theory. We calculate the one-loop perturbative contributions to LCDA and quasi-distribution amplitudes and explicitly demonstrate the factorization of quasi-distribution amplitudes at the one-loop level. Based on the perturbative results, we derive the matching kernel in the overline{textrm{MS}} scheme and regularization-invariant momentum-subtraction scheme. Our result provides a first step to obtaining the LCDA from first principle lattice QCD calculations in the future.

Leading power accuracy in lattice calculations of parton distributions

In lattice-QCD calculations of parton distribution functions (PDFs) via large-momentum effective theory, the leading power (twist-three) correction appears as O(ΛQCD/Pz) due to the linear-divergent self-energy of Wilson line in quasi-PDF operators. For lattice data with hadron momentum Pz of a few GeV, this correction is dominant in matching, as large as 30% or more. We show how to eliminate this uncertainty through choosing the mass renormalization parameter consistently with the resummation scheme of the infrared-renormalon series in perturbative matching coefficients. An example on the lattice pion PDF data at Pz=1.9 GeV shows an improvement of matching accuracy by a factor of more than 3∼5 in the expansion region x=0.2∼0.5.

Hadronic structure on the light front. VII. Pions and kaons and their partonic distributions

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front, motivated by the QCD vacuum structure and lattice results. The spontaneous breaking of chiral symmetry on the light front, is shown to parallel that in the rest frame, where the non-local instanton induced $^\prime$t Hooft interaction plays a central role. By rewriting this interaction solely in terms of the good component of the fermionic field, a scalar chiral condensate emerges in the mean-field approximation, which is identical to the one obtained in the rest frame. The pions and kaons emerge as deeply bound Goldstone modes in the chiral limit, with the scalar-isoscalar sigma meson mode as a threshold state with zero binding. We explicitly derive the light front distribution amplitudes (DAs) and partonic functions (PDFs) for these mesons. The DAs and PDFs are in good agreement with those extracted from the QCD instanton vacuum in the rest frame, using the large momentum effective theory (LaMET). The QCD evolved DAs and PDFs compare well with available measurements, as well as recent lattice results.

Resumming quark's longitudinal momentum logarithms in LaMET expansion of lattice PDFs

In the large-momentum expansion for parton distribution functions (PDFs), the natural physics scale is the longitudinal momentum (pz) of the quarks (or gluons) in a large-momentum hadron. We show how to expose this scale dependence through resumming logarithms of the type lnn⁡pz/μ in the matching coefficient, where μ is a fixed renormalization scale. The result enhances the accuracy of the expansion at moderate pz>1 GeV, and at the same time, clearly shows that the partons cannot be approximated from quarks with pz∼ΛQCD which are not predominantly collinear with the parent hadron momentum, consistent with power counting of the large-momentum effective theory. The same physics mechanism constrains the coordinate space expansion at large distances z, the conjugate of pz, as illustrated in the example of fitting the moments of the PDFs.

Gluon transverse-momentum-dependent distributions from large-momentum effective theory

We demonstrate that gluon transverse-momentum-dependent parton distribution functions (TMDPDFs) can be extracted from lattice calculations of appropriate Euclidean correlations in large-momentum effective theory (LaMET). Based on perturbative calculations of gluon unpolarized and helicity TMDPDFs, we present a matching formula connecting them and their LaMET counterparts, where the latter are renormalized in a scheme facilitating lattice calculations and converted to the overline{textrm{MS}} scheme. The hard matching kernel is given up to one-loop level. We also show that the perturbative result is independent of the prescription used for the pinch-pole singularity in the relevant correlations. Our results offer a guidance for the extraction of gluon TMDPDFs from lattice simulations, and have the potential to greatly facilitate perturbative calculations of the hard matching kernel.

Matching the B -meson quasidistribution amplitude in the RI/MOM scheme

Within the framework of large-momentum effective theory, the light-cone distribution amplitude (LCDA) of the $B$ meson in heavy-quark effective theory can be extracted from lattice calculations of the quasidistribution amplitude through the hard-collinear factorization formula. This quasiquantity can be renormalized in a regularization-independent momentum subtraction (RI/MOM) scheme. In this work, we derive the matching coefficient which connects the renormalized quasidistribution amplitude in the RI/MOM scheme and the standard LCDA in the $\overline{\mathrm{MS}}$ scheme at one-loop accuracy. Our numerical analysis verifies the feasibility of the RI/MOM scheme for renormalizing the $B$-meson quasidistribution amplitude. These results will be crucial for exploring the partonic structure of heavy-quark hadrons.

Generalized parton distributions through universal moment parameterization: zero skewness case

We present a global analysis program for the generalized parton distributions (GPDs) based on conformal moment expansion. We apply the strategy of universal moment parameterization to fit both the collinear parton distribution functions (PDFs) from phenomenology and generalized form factors from lattice calculations, and show that the parameterization is flexible enough to accommodate these constraints. In addition, we can also fit direct lattice calculations of GPDs from large-momentum effective theory. In this work we focus on the analysis of t-dependent PDFs which correspond to GPDs in the ξ → 0 limit. The strategy also applies to the ξ ≠ 0 region with extra parameters, and therefore can be fitted to experimental observables in the future. With a demonstrative example of fitted GPDs, we exhibit the quark transverse angular momentum densities of the proton as well as the impact parameter space distributions of quarks in both unpolarized and transversely polarized protons.

Pion and Kaon Distribution Amplitudes from Lattice QCD.

We present a state-of-the-art lattice QCD calculation of the pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory. The calculation is done at three lattice spacings a≈{0.06,0.09,0.12} fm and physical pion and kaon masses, with the meson momenta P_{z}={1.29,1.72,2.15} GeV. The result is nonperturbatively renormalized in a recently proposed hybrid scheme with self-renormalization, and extrapolated reliably to the continuum as well as the infinite momentum limit. We find a significant deviation of the pion and kaon DAs from the asymptotic form, and a large SU(3) flavor breaking effect in the kaon DA.

Transverse-momentum-dependent wave functions and soft functions at one-loop in large momentum effective theory

In large-momentum effective theory (LaMET), the transverse-momentum-dependent (TMD) light-front wave functions and soft functions can be extracted from the simulation of a four-quark form factor and equal-time correlation functions. In this work, using expansion by regions we provide a one-loop proof of TMD factorization of the form factor. For the one-loop validation, we also present a detailed calculation of mathcal{O} (αs) perturbative corrections to these quantities, in which we adopt a modern technique for the calculation of TMD form factor based the integration by part and differential equation. The one-loop hard functions are then extracted. Using lattice data from Lattice Parton Collaboration on quasi-TMDWFs, we estimate the effects from the one-loop matching kernel and find that the perturbative corrections depend on the operator to define the form factor, but are less sensitive to the transverse separation. These results will be helpful to precisely extract the soft functions and TMD wave functions from the first-principle in future.

Nonperturbative determination of the Collins-Soper kernel from quasitransverse-momentum-dependent wave functions

In the framework of large-momentum effective theory at one-loop matching accuracy, we perform a lattice calculation of the Collins-Soper kernel, which governs the rapidity evolution of transverse-momentum-dependent (TMD) distributions. We first obtain the quasi-TMD wave functions at three different meson momenta, on a lattice with valence clover quarks on a dynamical highly improved staggered quark sea and lattice spacing $a=0.12\text{ }\text{ }\mathrm{fm}$ from the MILC Collaboration, and renormalize the pertinent linear divergences using Wilson loops. Through one-loop matching to the light-cone wave functions, we determine the Collins-Soper kernel with transverse separation up to 0.6 fm. We study the systematic uncertainties from operator mixing and scale dependence, as well as the impact from higher power corrections. Our results potentially allow for a determination of the soft function and other transverse-momentum-dependent quantities at one-loop accuracy.