Lattice QCD calculations of nucleon transverse momentum-dependent parton distributions using clover and domain wall fermions

1.5 MB·

by  B. Yoon

Lattice QCD calculations of nucleon transverse
momentum-dependent parton distributions using
clover and domain wall fermions
6
1
0 B. Yoon∗a,†, T. Bhattacharyaa, M. Engelhardtb, J. Greenc, R. Guptaa, P. Häglerd,
2
B. Muschd, J. Negelee, A. Pochinskye, S. Syritsynf
n
a aLosAlamosNationalLaboratory,MSB283,P.O.Box1663,LosAlamos,NM87545,USA
J bDepartmentofPhysics,NewMexicoStateUniversity,LasCruces,NM88003-8001,USA

1 cInstitutfürKernphysik,JohannesGutenberg-UniversitätMainz,55099Mainz,Germany
2
dInstitutfürTheoretischePhysik,UniversitätRegensburg,93040Regensburg,Germany

] eCenterforTheoreticalPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,
t
a USA
l
- fTheoryCenter,ThomasJeffersonNationalAcceleratorFacility,NewportNews,VA23606,USA
p
†E-mail: [email protected]
e
h
[
WepresentalatticeQCDcalculationoftransversemomentumdependentpartondistributionfunc-

1
tions(TMDs)ofprotonsusingstaple-shapedWilsonlines. Fortime-reversaloddobservables,we
v
7 calculatethegeneralizedSiversandBoer-MulderstransversemomentumshiftsinSIDISandDY
1
cases, and for T-even observables we calculate the transversity related to the tensor charge and
7
5 thegeneralizedworm-gearshift. Thecalculationisdoneontwodifferentn =2+1ensembles:
f
0
domain-wallfermion(DWF)withlatticespacing0.084fmandpionmassof297MeV,andclover
.
1
fermion with lattice spacing 0.114fm and pion mass of 317MeV. The results from those two
0
6 differentdiscretizationsareconsistentwitheachother.
1
:
v
i
X
r
a
The33rdInternationalSymposiumonLatticeFieldTheory,
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
1. IntroductionandMethodology
Intrinsicmotionofpartonsinsidenucleonsgivesanimportantpictureofthenucleonstructure.
The intrinsic motion of partons in the transverse plane in momentum space is described by the
transverse momentum-dependent parton distributions (TMDs). TMDs have been studied actively
both theoretically, i.e., by using the QCD factorization [1, 2, 3] , and experimentally, i.e., COM-
PASS, HERMES and JLab experiments [4, 5, 6]. Here we report a lattice QCD calculation of the
TMDs.
ThemethodologyweuseforthecalculationoftheTMDsiselaboratelydescribedinthepre-
viouspapers[7,8]. Itcanbesummarizedasfollows. Consideringoff-light-conekinematicsfrom
the beginning, TMD correlators are parametrized in terms of the Lorentz-invariant amplitudes, so
that they can be calculated on the lattice. Only the ratios of isovector (u−d) TMD observables,
thatcancelsoftfactors,multiplicativerenormalizationfactorsanddisconnectedquarkloopcontri-
butions,arecalculated. TheSemi-InclusiveDeep-InelasticScattering(SIDIS)andDrell-Yan(DY)
scatteringprocessesaredescribedbythestaple-shapegaugelinksconnectingtwoseparatedquarks
intheTMDcorrelators,whosedirectionandextentaredenotedbyvandη,respectively. Herethe
stapledirectionvistakentobespace-likeinordertoavoidtherapiditydivergences[9];light-cone
ˆ
can be approached in the limit of the Collins-Soper parameter ζ →∞, guided by a perturbative
evolution[10,11].
Inthisstudy,weanalyzetwoensembles;wecallthemtheCloverandDWFensembles. Lattice
parametersaregiveninTable1. UnlikethepreviousstudyRef.[8],weuseunitarycombinationof
seaandvalencequarks,i.e.,weusethesamefermiondiscretizationsfortheseaandvalencequarks.
The two ensembles have different types of fermion discretizations: one is the RBC/UKQCD lat-
ticeswithdomain-wallfermionsata=0.084fm,andtheotheristheJLab/William&Marylattices
with clover fermions at a = 0.114fm. Both are at approximately the same pion mass, close to
300MeV. By comparing results on those two ensembles, we can see the dependence of the TMD
observables on discretization effects, including chiral symmetry breaking, and scaling violations.
When we define the TMD ratio observables, we assume that soft factors are purely multiplica-
tive, such that they cancel in ratios. We furthermore assume that the ratios also cancel the quark
fieldrenormalizationfactors,sothattheyarescaleinvariant. Bycomparingthetwocalculationsat
differentscales,wecanseetowhatextentthisassumptionholds.
ID FermionType Geometry a(fm) m (MeV) #confs. #meas.
π
Clover Clover 323×96 0.114 317 967 23208
DWF Domain-wall 323×64 0.084 297 533 4264
Table1:LatticeparametersoftheRBC/UKQCDdomain-wallensembleandtheJLab/William&Maryclover
ensembleusedinthiswork. n =2+1forbothensembles.
f
2. Results
Inthisstudy,wecalculatefourTMDobservables: (1)generalizedSiversshift,(2)generalized
Boer-Muldersshift,(3)transversityh and(4)generalizedworm-gearshiftfromg .
1 1T
2
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
0.6 0.6
Sivers(cid:45)Shift,u(cid:45)d(cid:45)quarks Sivers(cid:45)Shift,u(cid:45)d(cid:45)quarks
(cid:76)V 0.4 (cid:76)V 0.4
e e
G G
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 00..02 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 00..02
(cid:144) (cid:96) (cid:144) (cid:96)
(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)111T (cid:45)0.2 Ζ(cid:200)b(cid:61)T(cid:200)0(cid:61).302.,34fm, (cid:64)(cid:68)(cid:72)(cid:76)(cid:166)111T (cid:45)0.2 Ζ(cid:200)b(cid:61)T(cid:200)0(cid:61).401.,25fm,
(cid:142)fN (cid:45)0.4 Clover (cid:142)fN (cid:45)0.4 DWF
m m
(cid:152)DY SIDIS(cid:153) (cid:152)DY SIDIS(cid:153)
(cid:45)0.6 (cid:45)0.6
(cid:45)(cid:165) (cid:45)10 (cid:45)5 0 5 10 (cid:165) (cid:45)(cid:165) (cid:45)10 (cid:45)5 0 5 10 (cid:165)
Η(cid:200)v(cid:200) (cid:72)latticeunits(cid:76) Η(cid:200)v(cid:200) (cid:72)latticeunits(cid:76)
0.6 0.6
Sivers(cid:45)Shift,u(cid:45)d(cid:45)quarks Sivers(cid:45)Shift,u(cid:45)d(cid:45)quarks
(cid:76)V 0.4 (cid:76)V 0.4
e e
G G
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 00..02 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 00..02
(cid:144) (cid:96) (cid:144) (cid:96)
(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)111T (cid:45)0.2 Ζ(cid:200)b(cid:61)T(cid:200)0(cid:61).302.,46fm, (cid:64)(cid:68)(cid:72)(cid:76)(cid:166)111T (cid:45)0.2 Ζ(cid:200)b(cid:61)T(cid:200)0(cid:61).401.,34fm,
(cid:142)fN (cid:45)0.4 Clover (cid:142)fN (cid:45)0.4 DWF
m m
(cid:152)DY SIDIS(cid:153) (cid:152)DY SIDIS(cid:153)
(cid:45)0.6 (cid:45)0.6
(cid:45)(cid:165) (cid:45)10 (cid:45)5 0 5 10 (cid:165) (cid:45)(cid:165) (cid:45)10 (cid:45)5 0 5 10 (cid:165)
Η(cid:200)v(cid:200) (cid:72)latticeunits(cid:76) Η(cid:200)v(cid:200) (cid:72)latticeunits(cid:76)
Figure 1: Dependence of the generalized Sivers shift on the staple extent η|v| for Clover (left) and DWF
(right)ensemblesat|bbb |=3a(top),and4a(bottom). Collins-Soperparameterisfixedatthehighestvalue
T
ζˆ =0.41and0.32forDWFandCloverensembles,respectively.
ThegeneralizedSiversshiftaddressesthedistributionofunpolarizedquarksinatransversely
polarized proton, and it is defined by (cid:104)kkk (cid:105) (bbb2;...)≡m f˜⊥[1](1)(bbb2;...)/f˜[1](0)(bbb2;...), where
y TU T N 1T T 1 T
m is the nucleon mass, f⊥ is the Sivers function, and f is the unpolarized function. Here tilde
N 1T 1
representstheFouriertransformationofafunctioninthetransversemomentumkkk -spaceontothe
T
bbb -space, superscript [1] denotes the first Mellin moment in the longitudinal momentum fraction
T
(cid:16) (cid:17)n
of the quark, x, and superscript (n) are defined as f˜(n)(x,bbb2...) ≡ n! − 2 ∂ f˜(x,bbb2;...).
T m2N bbb2T T
DetaileddescriptionsandphysicalinterpretationsofthenotationsaregiveninRef.[8].
InordertoobtainthegeneralizedSiversshiftinSIDISorDYprocess,thestapleextent|η||v|
ofthegaugelinkconnectingthetwoseparatedquarksneedstobeextrapolatedtoinfinity. Theη|v|-
dependence of the generalized Sivers shift is shown in Fig. 1. In our setup, DY process is given
bythelimitη|v|→−∞,andSIDISbythelimitη|v|→∞. AspredictedintheQCDfactorization,
DY and SIDIS results show the opposite sign and have the same magnitude. We find that η|v|-
dependence forms a plateau when |η||v|≥7a, regardless of the lattice spacing. Hence we obtain
theresultsinDYandSIDISprocessesbytakingaverageover|η||v|=7a∼η , simultaneously
max
intheDYandSIDISlimit,imposinganti-symmetricconditioninη|v|.
Althoughtheyformaplateauastheylieonaconstantwithin1∼2σ deviation, wefindnon-
zero slope in some cases, which indicates that there could be remaining systematic error when
|η||v| is close to 7a. As one can see, in addition, the statistical uncertainty increases as the staple
extent increases. Hence, if we use weighted average to obtain the asymptotic value, it will be
governed by the results at small |η||v|, where the systematic error is relatively larger than other
3
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
(cid:45)0.10 (cid:45)0.15
SiversShift(cid:72)SIDIS(cid:76),
SiversShift(cid:72)SIDIS(cid:76),u(cid:45)d(cid:45)quarks
(cid:76)GeV (cid:45)0.15 u(cid:45)d(cid:45)quarks (cid:76)GeV (cid:45)0.20
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 (cid:45)(cid:45)00..2250 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 (cid:45)(cid:45)00..3205
(cid:144) (cid:144)
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)11mfN1T (cid:45)(cid:45)00..3350 Ζ(cid:96)(cid:187)0.3 DClWovFer (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)11mfN1T (cid:45)(cid:45)00..4305 (cid:200)bT(cid:200)(cid:61)0.34fm DClWovFer
(cid:45)0.40 (cid:45)0.45
0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5
(cid:96)
(cid:200)bT(cid:200) (cid:72)fm(cid:76) Ζ
Figure 2: Dependence of the generalized Sivers shift on |bbb | (left) and ζˆ (right) for the two different
T
ensembles.In|bbb |-dependenceplotζˆ isfixedaround0.35,andinζˆ-dependenceplot|bbb |isfixedat0.34fm.
T T
When|bbb |issmall,thelatticeresultsaresupposedtobecontaminatedbythelatticecutoffeffect. In|bbb |-
T T
dependenceplot,theshadedareaisthesmall|bbb |regionwhere|bbb |≤3a ≈0.25fm.
T T DWF
data points. In order to avoid this problem, we calculate the asymptotic value by using only the
meanvaluesofthedatapoints,andcalculatethestatisticalerrorbyusingtheJackknifemethod.
Here we use different η for different measurements. For Clover, η is 12 up to |bbb |=
max max T
0.23fm, 11 for |bbb | = 0.34fm, 10 for |bbb | = 0.45fm, and so on. For DWF, η is 12 up to
T T max
|bbb |=0.42fm, 10 for |bbb |=0.50fm, 9 for |bbb |=0.59fm, and so on. Therefore, the results at
T T T
large |bbb | region are obtained only from a few data points |η||v| close to 7a. Hence, they have
T
smaller statistical error than those at small |bbb | region, and they may have unresolved systematic
T
errorbecausetherearetoosmallnumberofdatapointstoconfirmifweareinanasymptoticregion
thatformsplateauornot.
Fig.2showsthedependenceofthegeneralizedSiversshifton|bbb |(left)andζˆ (right)forthe
T
twodifferentensembles. WhenwedefinethegeneralizedSiversshiftbytheratio,weassumethat
the two quarks in the non-local quark bilinear operators are separated far enough so that renor-
malization constant is independent of the gamma structure of the operator. This assumption is
correct only when |bbb | is large enough. Furthermore, when |bbb | is small, the lattice results are
T T
contaminated by the lattice cutoff effect. In |bbb |-dependence plot, the shaded area represents the
T
small |bbb | region where |bbb |≤3a ≈0.25fm. Here the minimum separation 3a is chosen
T T DWF DWF
so that Clover and DWF ensembles of different fermion discretizations and cutoff scales give the
consistentresultsforallthe|bbb |-dependencesoftheobservablesgiveninFigs.2,3,and4.
T
ˆ
Theright-handsideplotonFig.2showstheCollins-Soperevolutionparameterζ-dependence
ofthegeneralizedSiversshiftatfixed|bbb |=0.34fm. Theζˆ isascaleparameterintroducedtoavoid
T
the rapidity divergences by taking the Wilson line off the light-cone to the space-like region. The
ˆ ˆ
light-conelimitisζ →∞,andTMDswithfiniteζ canbeevolvedtothelight-coneperturbatively.
ˆ
In order to make the perturbative evolution reliable, however, ζ needs to be large enough so that
ˆ ˆ
ζ =2m ζ (cid:29)Λ . In this study, we calculate only up to ζ =0.41, which is too small to use
N QCD
ˆ
the perturbativative evolution. The major problem of the lattice QCD calculation at large ζ is the
ˆ
statisticaluncertainty;largerζ calculationrequireslargerprotonmomenta,whichintroduceslarge
statisticalerror[8].
Another T-odd TMD we calculate is the generalized Boer-Mulders shift, which is defined
4
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
0.1 0.00
Boer(cid:45)MuldersShift(cid:72)SIDIS(cid:76),
Boer(cid:45)MuldersShift(cid:72)SIDIS(cid:76),u(cid:45)d(cid:45)quarks
(cid:76)GeV 0.0 u(cid:45)d(cid:45)quarks (cid:76)GeV (cid:45)0.05
(cid:72) (cid:72) (cid:45)0.10
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10f1 (cid:45)0.1 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10f1 (cid:45)0.15
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)11(cid:144)mhN1 (cid:45)(cid:45)00..32 Ζ(cid:96)(cid:187)0.3 DClWovFer (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)(cid:166)11(cid:144)mhN1 (cid:45)(cid:45)00..2250 (cid:200)bT(cid:200)(cid:61)0.34fm DClWovFer
(cid:45)0.4 (cid:45)0.30
0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5
(cid:96)
(cid:200)bT(cid:200) (cid:72)fm(cid:76) Ζ
Figure3:DependenceofthegeneralizedBoer-Muldersshifton|bbb |(left)andζˆ (right)forthetwodifferent
T
ensembles. NotationsandotherparametersarethesameasthoseinFig.2.
2.0 0.35
h1(cid:72)SIDIS(cid:76), g1TShift,
1.8 u(cid:45)d(cid:45)quarks Clover (cid:76)V 0.30 u(cid:45)d(cid:45)quarks Clover
DWF e DWF
G
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:144)f1 11..46 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10(cid:72)f1 0.25
(cid:142)(cid:64)(cid:68)(cid:72)(cid:76)10h1 1.2 (cid:142)(cid:64)(cid:68)(cid:72)(cid:76)11(cid:144)g1T 0.20
1.0 (cid:96) mN 0.15 (cid:96)
Ζ(cid:187)0.3 Ζ(cid:187)0.3
0.8 0.10
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8
(cid:200)bT(cid:200) (cid:72)fm(cid:76) (cid:200)bT(cid:200) (cid:72)fm(cid:76)
Figure4:Dependenceofthetransversityratioh˜[1](0)/f˜[1](0)(left)andthegeneralizedg (right)on|bbb |for
1 1 1T T
thetwodifferentensembles. NotationsandotherparametersarethesameasthoseinFig.2.
by (cid:104)kkk (cid:105) (bbb2;...)≡m h˜⊥[1](1)(bbb2;...)/f˜[1](0)(bbb2;...), which describes the distribution of trans-
y UT T N 1 T 1 T
verselypolarizedquarksinanunpolarizedproton. Hereh⊥ istheBoer-Muldersfunction.
1
ThedependenceoftheBoer-Muldersshifton|bbb |andζˆ fortwodifferentensemblesisshown
T
in Fig. 3. We find that the results on the two different Clover and DWF ensembles are consistent
ˆ
within their statistical uncertainty. In the generalized Boer-Mulders shift case, ζ dependence has
ˆ
beenstudieduptoζ =2.03withpionsinRef.[12],exploitingthefactthatsignal-to-noiseratiois
muchlargerinpionsthaninprotons. Weexpectsimilarscalingbehaviorintheproton,butitisnot
ˆ
clearlyshownbythisstudybecausecalculationextendsonlyuptoζ =0.41.
Fortherestofthissection,webrieflydescribetheresultsoftheT-evenTMDs: thetransversity
h andthegeneralizedworm-gearshiftg . UnliketheT-oddTMDs,suchastheSiversandBoer-
1 1T
Muldersdistributions,T-evenTMDsarepredictedtobeprocessindependent,i.e.,sameinDYand
SIDIS processes. Hence they have been studied with lattice QCD by using the straight Wilson
line [13, 7], and it has been shown that the difference between straight gauge link and staple-
shaped gauge link is small for these T-even TMDs [8]. In this study, we reconfirm this argument
for different lattice discretizations and lighter pion masses. We found that most of the results
from the two lattice discretizations are consistent with each other, except in the small |bbb | region
T
of the generalized worm gear shift g . In Fig. 4, we show the |bbb | dependence of the two T-
1T T
even TMDs. As one can see, the Clover and DWF results of g shift are different when |bbb |≤
1T T
5
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
V) 0.1 Extraction from experiments, ζˆ≈ 0.77
e DWF-on-AsqTad; 0.12 fm, 518 MeV
G
Clover; 0.114 fm, 317 MeV
d; 0.0 DWF; 0.084 fm, 297 MeV
-
u
S, -0.1
DI
SI
hift ( -0.2
S
s -0.3
r
e
v
Si -0.4
en. |bT| ≈ 0.35 fm
G
-0.5
0 0.2 0.4 0.6 0.8
ζˆ
Figure5: ExperimentalestimateandlatticeQCDcalculationsofthegeneralizedSiversshiftinSIDISlimit
ˆ
for different Collins-Soper parameter values ζ with various lattice setups. The results of DWF-on-Asqtad
are obtained from the previous study of Ref. [8]. The extraction from the experiments has been done at
ζˆ ≈0.77.
0.25fm. However,weexpectlargesystematicuncertaintiesregardingthelatticeregularizationand
renormalizationwhen|bbb |issmall, andtheresultsfromdifferentdiscretizationsmaydifferinthe
T
region.
3. ComparisonwithPhenomenologicalEstimateandConclusion
In this study, we compared the lattice QCD calculation from two different discretizations:
a=0.114fm Clover vs. a=0.084fm DWF at m ≈300MeV. We found that the results of the
π
CloverandDWFensemblesareconsistentwitheachotherintheregionwherelatticeartifactsare
expectedtobesmall.
InFig.5,wecomparethecurrentlatticeQCDcalculationsofthegeneralizedSiversshiftwith
the experimental estimate. The generalized Sivers shift is defined by the ratio of the Sivers func-
tionandtheunpolarizedfunction. Inordertocalculatetheexperimentalestimateofthegeneralized
Siversshift,weusetheSiversshiftresultsofHERMES,COMPASSandJeffersonLabexperiments
extracted in Ref. [14]. In the comparison plot, it is clearly shown that the three lattice ensembles
withdifferentpionmassanddifferentdiscretizationgiveconsistentresults. Onealsocanfindthat
thediscrepancybetweenthecentralvaluesofphenomenologicalvalueandlatticeresultsbecomes
ˆ ˆ
smaller as ζ increases. Here the ζ is a scale introduced to regulate the rapidity divergences, and
ˆ
thefactorizationworksonlywhenζ islargeenoughsothatitliesintheperturbativeregime. The
ˆ
extraction from the experiments has been done at ζ ≈0.77, which corresponds to ζ ≈1.5GeV,
ˆ ˆ
while precise lattice results are obtained at ζ ≤0.41, and the lattice data points at ζ >0.41 have
huge uncertainty. Going by the behavior we see in the pion study [12], we expect that extrapo-
ˆ
lating the lattice results to ζ →0.77 brings the lattice result close to the extracted value from the
experiments.
6
LatticeQCDcalculationofnucleonTMDsat300MeVpionmass B.Yoon
Acknowledgments
Computations were performed using resources provided by the U.S. DOE Office of Science
through the National Energy Research Scientific Computing Center (NERSC), a DOE Office of
Science User Facility, under Contract No. DE-AC02-05CH11231, as well as through facilities of
the USQCD Collaboration, employing the Chroma software suite [15]. The RBC/UKQCD col-
laboration is gratefully acknowledged for providing gauge configurations analyzed in this work,
as are K. Orginos (supported by DOE grant DE-FG02-04ER41302) and the Jefferson Lab lattice
group(supportedbyDOEgrantDE-AC05-06OR23177,underwhichJeffersonScienceAssociates,
LLC,operatesJeffersonLaboratory). SupportbytheHeisenberg-FellowshipprogramoftheDFG
(P.H.), the PRISMA Cluster of Excellence at the University of Mainz (J.G.), and the RIKEN For-
eignPostdoctoralResearcherProgram(BNL)aswellastheNathanIsgurFellowship(JLab)(S.S.)
is acknowledged. This work was furthermore supported by the U.S. DOE and the Office of Nu-
clearPhysicsthroughgrantsDE-FG02-96ER40965(M.E.), DE-SC0011090(J.N.)andDE-FC02-
06ER41444 (A.P.). R.G., T.B. and B.Y. are supported by DOE grant DE-KA-1401020 and the
LDRDprogramatLANL.
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