MSUHEP-22-033 Gluon Parton Distribution of the Nucleon from 211-Flavor Lattice QCD in the Physical-Continuum Limit

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MSUHEP-22-033

Gluon Parton Distribution of the Nucleon from 2+1+1-Flavor Lattice QCD in the

Physical-Continuum Limit

Zhouyou Fan,1William Good,1, 2 and Huey-Wen Lin1, 2

1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824

2Department of Computational Mathematics, Science and Engineering,

Michigan State University, East Lansing, MI 48824

We present the ﬁrst physical-continuum limit x-dependent nucleon gluon distribution from lattice

QCD using the pseudo-PDF approach, on lattice ensembles with 2+ 1 + 1 ﬂavors of highly improved

staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence

action on three lattice spacings a≈0.9, 0.12 and 0.15 fm and three pion masses Mπ≈220, 310

and 690 MeV, with nucleon two-point measurements numbering up to O(106) and nucleon boost

momenta up to 3 GeV. We study the lattice-spacing and pion-mass dependence of the reduced

pseudo-ITD matrix elements obtained from the lattice calculation, then extrapolate them to the

continuum-physical limit before extracting xg(x)/hxig. We use the gluon momentum fraction hxig

calculated from the same ensembles to determine the nucleon gluon unpolarized PDF xg(x) for the

ﬁrst time entirely through lattice-QCD simulation. We compare our results with previous single-

ensemble lattice calculations, as well as selected global ﬁts.

I. INTRODUCTION

Many precision phenomenology and theoretical pre-

dictions for hadron colliders rely on accurate estimates

of the uncertainty in Standard-Model (SM) predictions.

Among these predictions, the parton distribution func-

tions (PDFs), the nonperturbative functions quantifying

probabilities for ﬁnding quarks and gluons in hadrons

with particular momentum fraction, are particularly im-

portant inputs in high-energy scattering [1–11]. The

gluon PDF g(x) needs to be known precisely to calculate

the cross section for these processes in pp collisions, such

as the cross section for Higgs-boson production and jet

production at the Large Hadron Collider (LHC) [12,13],

and direct J/ψ photoproduction at Jeﬀerson Lab [14].

The future U.S.-based Electron-Ion Collider (EIC) [15],

planned to be built at Brookhaven National Lab, will fur-

ther our knowledge of the gluon PDF [16–18]. In Asia,

the Electron-Ion Collider in China (EicC) [19] is also

planned to impact the gluon and sea-quark distributions.

Although signiﬁcant eﬀorts to extract the gluon distribu-

tion g(x) have been made in the last decade, there are

still problems in obtaining a precise g(x) in the large-x.

Lattice quantum chromodynamics (QCD) is a non-

perturbative theoretical method for calculating QCD

quantities that has full systematic control. Calculations

of x-dependent hadron structure in lattice QCD have

multiplied since the proposal of Large-Momentum Ef-

fective Theory (LaMET) [20–22]. Many lattice works

have been done on nucleon and meson PDFs, and

generalized parton distributions (GPDs) based on the

quasi-PDF approach [23–55]. Alternative approaches

to lightcone PDFs in lattice QCD are the Compton-

amplitude approach (or “OPE without OPE”) [56–68],

the “hadronic-tensor approach” [69–74], the “current-

current correlator” [48,63,75–80] and the pseudo-PDF

approach [78,81–97]. A few works have started to include

lattice-QCD systematics, such as ﬁnite-volume eﬀects, in

their calculations [39,80]. However, most these calcula-

tions are still, at the current stage, done with a single

lattice spacing. Most lattice calculations of PDFs use

next-to-leading-order (NLO) matching or, equivalently,

NLO Wilson coeﬃcients [22,98–100], and some lattice

calculations of the valence pion PDF [101] have incorpo-

rated NNLO matching [46,102]. More work is needed

to reduce high-twist systematics and improve the lattice

determination of small-xand antiquark PDFs with very

large boost momenta.

Recently, progress has been made in the most-

calculated isovector quark distribution of nucleon by

MSULat [49], ETMC [51] and HadStruc Collabora-

tions [103], who studied lattice-spacing dependence.

MSULat studied three lattice spacings (0.09, 0.12 and

0.15 fm) and pion masses (135, 220, 310 MeV) and per-

formed a simultaneous continuum-physical extrapolation

using a third-order z-expansion on renormalized LaMET

matrix elements [49] with nucleon boost momenta around

2.2 and 2.6 GeV. ETMC also uses three lattice spac-

ings, 0.06, 0.08, and 0.09 fm, but with heavier pion mass

(370 MeV) and investigated the continuum extrapola-

tion of the data on renormalized LaMET matrix ele-

ments with boost momentum around 1.8 GeV [51]. Had-

Struc Collaboration studied three lattice spacings, 0.048,

0.065, and 0.075 fm with two-ﬂavor 440-MeV lattice en-

sembles using the continuum pseudo–Ioﬀe-time distribu-

tion (ITD) [103]. Most of the works above found mild

nonzero dependence on lattice spacing (varying with the

Wilson-link displacement) in the nucleon case for LaMET

or pseudo-ITD matrix elements.

In contrast with the quark PDFs, the gluon PDFs

calculations are less calculated, due to their notoriously

noisier matrix elements on the lattice. To date, there

have only been a few exploratory gluon-PDF calcula-

tions for unpolarized nucleon [36,92,95], pion [96] and

kaon [104], and polarized nucleon [97] using the pseudo-

PDF [105] and quasi-PDF [38,106] methods. Most of

these calculations, like many exploratory lattice calcula-

arXiv:2210.09985v1 [hep-lat] 18 Oct 2022

tions, are done only using one lattice spacing at heavy

pion mass.

In this work, we report the ﬁrst continuum-limit un-

polarized gluon PDF of nucleon study using three lat-

tice spacings: 0.09, 0.12 and 0.15 fm with pion mass

ranging from 220 to 700 MeV using the pseudo-PDF

method. The remainder of this paper is organized as

follows: In Sec. II, we present the procedure for how

the lattice correlators are calculated and analyzed to ex-

tract the ground-state matrix elements for the pseudo-

PDF method. We then study the lattice-spacing de-

pendence of matrix elements at 310 and 700-MeV pion

mass in Sec. III, checking both O(a) and O(a2) forms

using multiple continuum-extrapolation strategies. We

perform a physical-continuum extrapolation to obtain

continuum reduced pseudo-ITDs (RpITDs) matrix ele-

ment, before ﬁnal determination of the nucleon unpolar-

ized gluon PDF xg(x) is obtained from the xg(x)/hxig

and hxigresults. Using the gluon momentum fraction

calculated on the same ensemble, we obtain the gluon

PDF and compare with the phenomenological global-ﬁt

PDF results. We consider the quark-mixing systematics,

but they are found to be small. The ﬁnal conclusion and

future outlook can be found in Sec. IV.

II. LATTICE SETUP, CORRELATORS AND

MATRIX ELEMENTS

This calculation is carried out using four ensem-

bles with Nf= 2 + 1 + 1 highly improved staggered

quarks (HISQ) [107], generated by the MILC Collabora-

tion [108], with three diﬀerent lattice spacings (a≈0.9,

0.12 and 0.15 fm) and three pion masses (220, 310,

690 MeV); see Table Ifor more details. We apply

ﬁve steps of hypercubic (HYP) smearing [109] to the

gauge links to reduce short-distance noise. Wilson-clover

fermions are used in the valence sector, and the valence-

quark masses are tuned to reproduce the lightest light

and strange sea pseudoscalar meson masses (which cor-

respond to pion masses 310 and 690 MeV, respectively).

A similar setup is used by PNDME collaboration [110–

121] with local operators, such as isovector and ﬂavor-

diagonal charges, form factors and moments; the results

from this mixed-action setup are consistent with the same

physical quantities calculated using diﬀerent fermion ac-

tions [55,122–126].

On each lattice conﬁguration, we calculate the nucleon

two-point correlators using multiple sources:

C2pt

N(Pz;t) = h0|ΓZd3y e−iyPzχ(~y, t)χ(~

0,0)|0i,(1)

with the nucleon interpolation operator χas

lmn[u(y)lTiγ4γ2γ5dm(y)]un(y) (where {l, m, n}are

color indices, u(y) and d(y) are quark ﬁelds), the

projection operator Γ = 1

2(1 + γ4), tis lattice Euclidean

time, and Pzis the nucleon boost momentum along

the spatial z-direction. We use Gaussian momentum

Ensemble a09m310 a12m220 a12m310 a15m310

a(fm) 0.0888(8) 0.1184(10) 0.1207(11) 0.1510(20)

L3×T323×96 323×64 243×64 163×48

Mval

π(GeV) 0.313(1) 0.2266(3) 0.309(1) 0.319(3)

Mval

ηs(GeV) 0.698(7) N/A 0.6841(6) 0.687(1)

Pz(GeV) [0,3.05] [0,2.29] [0,2.14] [0,2.56]

Ncfg 1009 957 1013 900

N2pt

meas 387,456 1,466,944 324,160 259,200

tsep [6,10] [6,10] [5,9] [4,8]

TABLE I. Lattice spacing a, valence pion mass (Mval

π) and

ηsmass (Mval

ηs), lattice size (L3×T), number of conﬁgura-

tions (Ncfg), number of total two-point correlator measure-

ments (N2pt

meas), and source-sink separation times tsep used in

the three-point correlator ﬁts of Nf= 2 + 1 + 1 clover va-

lence fermions on HISQ ensembles generated by the MILC

Collaboration and analyzed in this study.

smearing [127] on the quark ﬁeld to improve the signal

for nucleon boost momenta up to 3.0 GeV. Hundreds

of thousands of measurements are made, varying for

diﬀerent ensembles. Compared to our previous nucleon

gluon PDF calculation on one a12m310 ensemble with

105measurements [92], this study uses more measure-

ments and varies the lattice spacing. We then calculate

the three-point gluon correlator by combining the gluon

loop with nucleon two-point correlators,

C3pt

N(z, Pz;tsep, t) =

h0|ΓZd3y e−iyPzχ(~y, tsep)Og(z, t)χ(~

0,0)|0i,(2)

where tis the gluon-operator insertion time, tsep is the

source-sink time separation. Og(z, t) is the gluon opera-

tor introduced in Ref. [105]:

O(z)≡X

i6=z,t

O(Fti, F ti;z)−1

i,j6=z,t

O(Fij , F ij ;z),(3)

where the operator O(Fµν , F αβ ;z) =

Fµ

ν(z)U(z, 0)Fα

β(0), and zis the Wilson link length.

To extract the ground-state matrix element, we use a

two-state ﬁt on the two-point correlators and a two-sim

ﬁt on the three-point correlators:

C2pt

N(Pz, t) =

|AN,0|2e−EN,0t+|AN,1|2e−EN,1t+. . . , (4)

C3pt

N(z, Pz, t, tsep) = (5)

|AN,0|2h0|Og|0ie−EN,0tsep

+|AN,0||AN,1|h0|Og|1ie−EN,1(tsep−t)e−EN,0t

+|AN,0||AN,1|h1|Og|0ie−EN,0(tsep−t)e−EN,1t

+|AN,1|2h1|Og|1ie−EN,1tsep

+. . . ,

a09m310

two-sim

tsep=10

tsep=8

tsep=9

tsep=6

tsep=7

-4-2 0 2 4

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

t-t

sep

RRatio(z2, Pz=2)

two-sim fit

two-sim

4567

sep

min

two-sim fit

two-sim

7 8 9 10 11

sep

max

a12m220

two-sim

tsep=10

tsep=8

tsep=9

tsep=6

tsep=7

-4-2 0 2 4

-0.2

0.0

0.2

0.4

0.6

t-t

sep

RRatio(z2, Pz=2)

two-sim fit

two-sim

4567

sep

min

two-sim fit

two-sim

7 8 9 10 11

sep

max

a12m310

two-sim

tsep=9

tsep=7

tsep=8

tsep=5

tsep=6

-4-2 0 2 4

-0.5

0.0

0.5

1.0

t-tsep/2

RRatio(z2, Pz=2)

two-sim fit

two-sim

3 4 5 6

tsep

min

two-sim fit

two-sim

6 7 8 9 10

tsep

max

a15m310

two-sim

tsep=8

tsep=6

tsep=7

tsep=4

tsep=5

-4-2 0 2 4

-0.4

-0.2

0.0

0.2

0.4

0.6

t-t

sep

RRatio(z2, Pz=2)

two-sim fit

two-sim

2345

sep

min

two-sim fit

two-sim

56789

sep

max

FIG. 1. Example ratio plots (left) and two-sim ﬁts (last 2 columns) of the light nucleon correlators at pion masses Mπ≈

{310,220,310,310}MeV from the a09m310, a12m220, a12m310 and a15m310 ensembles. The gray band shown on each plot is

the extracted ground-state matrix element from the two-sim ﬁt that we use as our best value. From left to right, the columns

are: the ratio of the three-point to two-point correlators with the reconstructed ﬁt bands from the two-sim ﬁt using the ﬁnal

tsep inputs, shown as functions of t−tsep/2, the one-state ﬁt results for the three-point correlators at diﬀerent tsep values, the

two-sim ﬁt results using tsep ∈[tmin

sep , tmax

sep ] varying tmin

sep and tmax

sep .

where the |AN,i|2and EN,i are the ground-state (i= 0)

and ﬁrst excited state (i= 1) amplitude and energy,

respectively.

To visualize our ﬁtted matrix-element extraction, we

compare to ratios of the three-point to the two-point cor-

relator

RN(z, Pz, tsep, t) = C3pt

N(z, Pz, t, tsep)

C2pt

N(Pz, t).(6)

The left-hand side of Fig. 1shows example ratios for the

gluon matrix elements from all four ensembles at pion

masses Mπ∈ {220,310}MeV at selected momenta Pz

and Wilson-line length z. The left column shows the ra-

tio plots with data points of Rfrom diﬀerent source-sink

separation, tsep, along with the reconstructed bands from

the ﬁt, showing how well the ﬁt describing the data in

Eq. 6; the ﬁnal ground-state matrix elements are shown

in grey bands. We observe that the ratios increase with

increasing source-sink separation tsep and continuously

to approach the ground-state matrix elements obtained

from the simultaneous two-state ﬁt to three-point cor-

relators with ﬁve inputs of tsep. The middle and right

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MSUHEP-22-033GluonPartonDistributionoftheNucleonfrom2+1+1-FlavorLatticeQCDinthePhysical-ContinuumLimitZhouyouFan,1WilliamGood,1,2andHuey-WenLin1,21DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI488242DepartmentofComputationalMathematics,ScienceandEngineering,MichiganStateUnive...

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MSUHEP-22-033 Gluon Parton Distribution of the Nucleon from 211-Flavor Lattice QCD in the Physical-Continuum Limit

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