Nucleon generalized form factors from lattice QCD with nearly physical quark masses

0.37 MB·

by  Gunnar Bali

Nucleon generalized form factors from lattice QCD
with nearly physical quark masses
6
1
0
2

n
a Gunnar Bali,a Sara Collins,a Meinulf Göckeler,a Rudolf Rödl∗,a Andreas Schäfer,a
J
Andre Sternbeckb
9
aInstitutfürTheoretischePhysik,UniversitätRegensburg,93040Regensburg,Germany
1
bTheoretisch-PhysikalischesInstitut,Friedrich-Schiller-UniversitätJena,07743Jena,Germany

] E-mail: [email protected]
t
a
l
-
p WedeterminegeneralizedformfactorsofthenucleonfromlatticesimulationswithN =2mass-
f
e
h degeneratenon-perturbativelyimprovedWilson-Sheikholeslami-Wohlertfermionsdowntoapion
[
mass of 150 MeV. We also present the resulting isovector quark angular momentum. Possible

1 excited-statecontaminationsareinvestigatedwithcorrelatedsimultaneousfits.
v
8
1
8
4
0
.
1
0
6
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/
Nucleongeneralizedformfactors RudolfRödl
1. Introduction
GeneralizedPartonDistributions(GPDs)wereintroducedinthe’90s[1,2]. Theydescribethe
innerstructureofthenucleon. Thestartingpointistheparametrizationoftheoff-diagonalmatrix
elementsofthelightconeoperator[3,4,5,6]
1 (cid:20) iσµνn ∆ (cid:21)
(cid:104)N(P(cid:48))|O (x)|N(P)(cid:105)tw=ist2 u(P(cid:48)) H (x,ξ,t)/n+E (x,ξ,t) µ ν u(P), (1.1)
q q q
2 2m
N
 
1 (cid:90)+∞dλ (cid:18) λ (cid:19) (cid:90)+λ2 (cid:18) λ (cid:19)
O (x)= eiλxψ − n /nPexp−ig dα Aβ(αn)n ψ + n . (1.2)
q 2 2π q 2  β q 2
−∞ −λ
2
The path ordering operator P in eq. (1.2) ensures gauge invariance. Further, it depends on the
momentumfractionxandthequarkflavorq. Thespinorsψ(·)andψ(·)areparametrizedalongthe
lightcone(n2=0)bytheparameterλ.
Key ingredients of the parameterization of the matrix element in eq. (1.1) are the constraint
givenbytheLorentzstructureandthedefinitionsofH (·)andE (·),whichdependonthemomen-
q q
tumfractionxandthekinematicvariablesξ =−nµ∆ /2andt =∆2 with∆µ =P(cid:48)µ−Pµ.
µ
The directcalculation ofGPDs by virtueof latticeQCD is notpossible sinceO (x)is alight
q
coneoperator. However,onecanexpandeq.(1.2)intowersoflocaltwisttwooperators
(cid:104)N(P(cid:48))|ψ γ{µ i↔Dµ1 ···i↔Dµn}ψ |N(P)(cid:105)=
q q
(cid:110) n
=u(P(cid:48)) ∑δ γ{µ∆µ1···∆µiPµi+1···Pµn} Aq (t)
i,even n+1,i
i=0
−i n
+ ∑δ ∆ σα{µ∆µ1···∆µiPµi+1···Pµn} Bq (t)
2m i,even α n+1,i
i=0
1 (cid:111)
+ δ ∆µ···∆µnCq (t) u(P). (1.3)
m n,odd n+1
q q q
The coefficient functions A (t), B (t) andC (t) are so called (vector) Generalized Form
n+1,i n+1,i n+1
Factors(GFFs)whicharerelatedto(vector)GPDsby
(cid:90) +1 n
dxxnHq(x,ξ,t)= ∑δ (−2ξ)iAq (t)+δ (−2ξ)n+1Cq (t), (1.4)
i,even n+1,i n,odd n+1
−1 i=0
(cid:90) +1 n
dxxnEq(x,ξ,t)= ∑δ (−2ξ)iBq (t)−δ (−2ξ)n+1Cq (t). (1.5)
i,even n+1,i n,odd n+1
−1 i=0
The knowledge of Au−d(t), Bu−d(t) is of particular interest since they encode the total angular
2,0 2,0
momentum
1(cid:16) (cid:17)
Ju−d = Au−d(t =0)+Bu−d(t =0) . (1.6)
2 2,0 2,0
Moreover,wedetermineA˜u−d(t)andB˜u−d(t)whicharetheGFFscorrespondingtotheaxialGPDs.
2,0 2,0
2
Nucleongeneralizedformfactors RudolfRödl
2. Latticeset-up
WeuseN =2mass-degeneratenon-perturbativelyimprovedWilson-Sheikholeslami-Wohlert
f
fermionswithWilsonglue. ThenecessarygaugeconfigurationswereproducedbytheQCDSFcol-
laboration and RQCD (Regensburg QCD). The lattice spacing was set as described in [7]. To en-
sureground-statedominanceweuseoptimizedWuppertalsmearingasdescribedin[8]. Allresults
arecomputedatarenormalizationscale µ =2GeVusingnon-perturbativelyimprovedconversion
factors. Intable1wegiveanensembleoverview.
Ensemble β a[fm] κ V m [GeV] Lm N t /a
π π conf f
I 5.20 0.081 0.13596 323×64 0.2795(18) 3.69 1986(4) 13
II 5.29 0.071 0.13620 243×48 0.4264(20) 3.71 1999(2) 15
III 0.13620 323×64 0.4222(13) 4.90 1998(2) 15,17
IV 0.13632 323×64 0.2946(14) 3.42 2023(2) 7(1),9(1),11(1)
13,15,17
V 403×64 0.2888(11) 4.19 2025(2) 15
VI 643×64 0.2895(07) 6.71 1232(2) 15
VII 0.13640 483×64 0.1597(15) 2.78 3442(2) 15
VIII 643×64 0.1497(13) 3.47 1593(3) 9(1),12(2),15
IX 5.40 0.060 0.13640 323×64 0.4897(17) 4.81 1123(2) 17
X 0.13647 323×64 0.4262(20) 4.18 1999(2) 17
XI 0.13660 483×64 0.2595(09) 3.82 2177(2) 17
Table1: N =2latticeset-up. Numberofsourcesperconfigurationinbrackets.
f
3. ExtractingGeneralizedFormFactors
GFFsareobtainedbysolvingan(ingeneral)overdeterminedsystemofequations. Inthecase
ofthevectorGFFstheequationsystemreads
   T    
A (t) A (t)
2,0 2,0
ε(t,τ )=MB (t)−(cid:126)c(t,τ ) cov−1((cid:126)c(t,τ ))MB (t)−(cid:126)c(t,τ ). (3.1)
sink   2,0  sink  sink   2,0  sink 
C (t) C (t)
2 2
We extract the GFFs of interest by minimizing eq. (3.1). The coefficient matrix M in eq. (3.1) is
fullydeterminedbyeq.(1.3). Thematrixelements(cid:126)c(t,τ )areextractedfromlatticethree-point
sink
functions. The dependency on the source-sink separation τ 1 is examined in the next section,
sink
whereinthelimitτ →∞thegroundstateGFFsareobtained.
sink
Agoodsignalisaprerequisiteforexcited-statefitssincewehavetofixmanyfitparameters. It
turnsoutthatexcited-statefitsarenotpossiblefort<0ifwenaivelyimplementeq.(3.1). However,
wecandramaticallyimprovethesignalifweaveragethree-pointfunctionswhichleadtothesame
rowsinM.
1Wesetτsource=0.
3
Nucleongeneralizedformfactors RudolfRödl
4. ExtractionofmatrixelementsfromlatticeQCDandexcitedstates.
As mentioned in section 2 we use optimized Wuppertal smearing on our quark field interpo-
lators. Possibleremainingexcitedstatescontributionsaretreatedwithsimultaneouscombinedfits
totwo-andthree-pointfunctions. Wethereforeparametrizethetwoandthree-pointfunctions
(cid:113)
C ((cid:126)p,τ,τ )=c(τ ) z(cid:126)0 z(cid:126)pi e−mN(τsink−τ)e−E((cid:126)pi,mN)τ
3 i sink sink N N
+ x1 e−mN(τsink−τ)e−E(cid:48)((cid:126)pi)τ+ x2 e−m(cid:48)N(τsink−τ)e−E((cid:126)pi,mN)τ+ x3 e−m(cid:48)N(τsink−τ)e−E(cid:48)((cid:126)pi)τ,
E((cid:126)p, m )+m
C1e(p,τ)=z(cid:126)p N N e−E((cid:126)p,mN)τ,
2 N E((cid:126)p, m )
N
E((cid:126)p, m )+m E(cid:48)((cid:126)p)+m
C2e(p,τ)=z(cid:126)p N N e−E((cid:126)p,mN)τ+z(cid:126)p N(cid:48) e−E(cid:48)((cid:126)p)τ. (4.1)
2 N E((cid:126)p, m ) N(cid:48) E(cid:48)((cid:126)p)
N
Above we denote the nucleon mass as m and its energy as E((cid:126)p,m ). Further, we assume the
N N
continuum dispersion relation for the ground state E((cid:126)p,m )=m2 +(cid:126)p2 . All expressions corre-
N N
sponding to the first excited-state are indicated by a prime. The energy of the first excited-state
E(cid:48)((cid:126)p) is left as a fit parameter since this could be a mulit-hadronic state. Our kinematic set-up
is chosen such that the final momentum is always zero (cid:126)p =(cid:126)0. To accomplish the ground state
f
extraction(parameterc(τ ))wefixz(cid:126)0 andz(cid:126)pi byvirtueofthetwo-pointfunctions.
sink N N
5. Assessmentoftheexcited-statefit
WesimultaneouslyfitallparametersaccordingtothecoefficientmatrixM. Thisisnottrivial
buthastheadvantagethatrowswithapoorsignalarestabilizedbyothers. Thefitparameterx in
3
eq. (4.1) is only resolvable if multiple τ per ensemble are available. Therefore, we perform in
sink
general a 3-exponent fit where we set x =0. To check whether this is justified or not we utilize
3
ensembleVIIIwhichhasthreedifferentsourcesinkseparationsτ /a=9,12,15,seefig.1.
sink
0.030
Combined
0.028 Ratio Ratio Ratio 4 exp
3 exp 3 exp 3 exp
0.026
0.024
0.022
0.020
0.018
0.016
0.014
6 4 2 0 2 4 6 8 9 10 11 12 13 14 15 16
(τ τ /2)/a τ /a
− sink sink
Figure1: Extractionmethodanalysis: Datapointsintheleftpanelareratiosofthree-pointandtwo-point
functionsfordifferentsourcesinkseparationsofensembleVIIIforacertainrowr inthecoefficientmatrix
1
Matt=−0.211GeV2. Thecorrespondingsolidlinesareratioscreatedwiththefitfunctionsusingeq.(4.1)
andsettingx =0. Rightpanelshowsmatrixelementsc(τ )obtainedwithdifferentextractionmethods.
3 sink
4
Nucleongeneralizedformfactors RudolfRödl
In the right panel of fig. 1 we compare different values for the matrix element as a function
of τ and as a function of different extraction methods. More specifically, we compare the ratio
sink
method and the 3 exponent excited-state fit to the full excited-state fit with 4 exponents which is
shownasagrayband.
Weconcludefromtherightpanelthatatgivenstatisticsadistinctionbetweenthedifferentfit
methods is not possible if the source sink separation is τ /a=15 which is about 1fm. This is
sink
consistentwithwhatwehavefoundin[8]. Inthisworkweusethe3-exponentfit.
6. ExtractionofAu−d,Bu−d andJu−d
2,0 2,0
WeutilizeBaryonChiralPerturbationTheorytoextractJu−d atphysicalpionmass. Ouraim
istoquoteanupperandlowerboundofAu−d,Bu−d andJu−d includingthestatisticalerrorandthe
2,0 2,0
systematical error from the fit range choice. To that end we perform global combined fits to our
latticeensembleswithfitfunctionsAu−d(t,m2,(cid:126)Θ )andBu−d(t,m2,(cid:126)Θ )knownfrom(BChPT)[9].
2,0 π A 2,0 π B
Morespecifically,
 (1+3g2)m2 log(m2π)
Au2−,0d(t,mπ) = 1− 1A6f2ππ2 µ2 L+m2πM2A +m3πM3A +t(T0A+m2πT1A) (6.1)
π
g2m2 log(m2π)  (1+2g2)m2log(m2π)
Bu2−,0d(t,mπ) = A 16π f2π2µ2 L+1− 1A6f2ππ2 µ2 LB +m2πM2B+t(T0B+m2πT1B).
π π
(6.2)
withfitparameters(cid:126)Θ =(L,MA,MA,TA,TA)and(cid:126)Θ =(L,LB,MB,TB,TB). ThefitparametersTA
A 2 3 0 1 B 2 0 1 1
andTB areintroducedbyhandbuttheynaturallyappearinnextorderofBChPT.Wedothissince
1
weconsiderrelativelylargevirtualitieswith|t |=0.6GeV(cid:29)m .
max π
In order to get an upper and lower bound for Au−d and Bu−d we randomly sample fit range
2,0 2,0
choices(toestimatethesystematicalerror). Foreachsamplewefitthebootstrapensemble(toesti-
matethestatisticalerror). Thenwegeneratethedistributionsofthefitparameters(cid:126)Θ and(cid:126)Θ which
A B
containthecombinederror. Inthelaststepwedrawfitparametersfromthedistributions(cid:126)Θ and(cid:126)Θ
A B
andevaluateeq.(6.1)andeq.(6.2)witht=0GeVandfixedpionmassm . Thisyieldsdistributions
π
forAu−d(t =0,m ),Bu−d(t =0,m )andJu−d(m )=(Au−d(t =0,m )+Bu−d(t =0,m ))/2.
2,0 π 2,0 π π 2,0 π 2,0 π
Wedefinetheloweranduppererroras0.16and0.84quantilesrespectively. Byrepeatingthis
process for many pion masses m we are able to draw error bands as shown in the left panel of
π
fig.2whiletherightpanelshowstheoutcomeform =mphy.
π π
7. LatticeresultsfortheAxial-GFFsA˜u−d andB˜u−d
2,0 2,0
We carefully study fit range dependencies (induced by the combined fit to two- and three-
pointfunctions). Theerrorbarsinfig.3showthe(statisticandsystematic)one-sigmaerror. Each
singleerrorbarisconstructedfromahistogramwithsample-sizes=(#offitrangecombinations)
· (#number of bootstrap ensembles). Here we have used s=75·300=22500. A more detailed
analysiswillbepublishedinthenearfuture.
5
Nucleongeneralizedformfactors RudolfRödl
0.30
=0) 0.25 Drop E: VII & VIII & IX Drop E: IX
2Q 0.20
(
udA−2,00.15
0.10
0.00 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20 0.22 0.24
mπ2 in [GeV]2 A2u,−0d(Q2 =0,mπphy)
0.35
0) Drop E: VII & VIII & IX Drop E: IX
=
2Q 0.30
(
udB−2,00.25
0.00 0.02 0.04 0.06 0.08 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31
mπ2 in [GeV]2 B2u,−0d(Q2 =0,mπphy)
0.30
0) Drop E: VII & VIII & IX Drop E: IX
=
2Q0.25
(
d−
u
J0.20
0.00 0.02 0.04 0.06 0.08 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27
mπ2 in [GeV]2 Ju−d(Q2 =0,mπphy)
Figure2: ResultsforAu−d andBu−d andJu−d. Thedashedblacklineindicatesthephysicalpionmass. We
2,0 2,0
alwaysexcludetheheaviestensembleIXduetotheapplicabilityofBChPT(blue). Toprobethestabilityof
ourAnsatz,wealsoshowresultswhereweadditionallydropthetwolightestensemblesVII&VIII(green).
0.28
0.26
0.24
) 0.22
2
Q
0.20
(
0
2,0.18
˜A
0.16
0.14
0.12
1.4
1.2
1.0
)
2 0.8
Q
(
00.6
2,
˜B
0.4
0.2
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Q2 in [ GeV ]2
Figure 3: Lattice results for A˜u−d and B˜u−d. For both GFFs we do not see a strong volume or pion mass
2,0 2,0
dependence. MostnotablyarethesmallerrorsofensembleVIwitha643×64volumeandLm =6.71.
π
6
Nucleongeneralizedformfactors RudolfRödl
8. Conclusion
WedetermineAu−d,Bu−d andJu−d byvirtueoflatticeQCD.Further,utilizingBaryonChiral
2,0 2,0
PerturbationTheory,wecalculatetheobservablesatthephysicalpionmass,where,wefocusedon
theconstructionofreliableerrors. Thisisachievedbythehistogrammethodwhichpropagatesthe
staticalerrorandthesystematicerrorofthefitrangechoice. Thefinalvaluesareshownintable2.
Observable MS µ =2GeV
Au−d(Q2=0, mphy) 0.200(-7/+9)
2,0 π
Bu−d(Q2=0, mphy) 0.275(-11/+8)
2,0 π
Ju−d(mphy) 0.238(-8/+5)
π
Table 2: Table of results: The numbers correspond to the blue distributions shown in fig. 2. They have a
systematicerrorfromtheapplicationofBaryonChiralPerturbationTheorytoourlatticeensembles.
Further, we determine A˜u−d and B˜u−d which are the GFFs corresponding to the Axial-GPD,
2,0 2,0
however,moreworkhastobedoneespeciallytoextrapolateB˜u−d toQ2=0.
2,0
Acknowledgements
This work is funded by Deutsche Forschungsgemeinschaft (DFG) within the transregional col-
laborativeresearchcentre55(SFB-TRR55). WeacknowledgecomputertimeonSuperMUCatthe
LeibnizRechenzentruminGarching.
References
[1] D.Müller,D.Robaschik,B.Geyer,F.M.Dittes,andJ.Hoˇrejši,“Wavefunctions,evolutionequations
andevolutionkernelsfromlightrayoperatorsofQCD,”Fortsch.Phys.,vol.42,pp.101–141,1994.
[2] X.-D.Ji,“Gauge-InvariantDecompositionofNucleonSpin,”Phys.Rev.Lett.,vol.78,pp.610–613,
1997.
[3] X.-D.Ji,“Offforwardpartondistributions,”J.Phys.,vol.G24,pp.1181–1205,1998.
[4] X.-D.Ji,“DeeplyvirtualComptonscattering,”Phys.Rev.,vol.D55,pp.7114–7125,1997.
[5] LHPC,P.Hagler,J.W.Negele,D.B.Renner,W.Schroers,T.Lippert,andK.Schilling,“Transverse
structureofnucleonpartondistributionsfromlatticeQCD,”Phys.Rev.Lett.,vol.93,p.112001,2004.
[6] M.Diehl,“Generalizedpartondistributions,”Phys.Rept.,vol.388,pp.41–277,2003.
[7] G.S.Balietal.,“NucleonmassandsigmatermfromlatticeQCDwithtwolightfermionflavors,”
Nucl.Phys.,vol.B866,pp.1–25,2013.
[8] G.S.Bali,S.Collins,B.Glässle,M.Göckeler,J.Najjar,R.H.Rödl,A.Schäfer,R.W.Schiel,
W.Söldner,andA.Sternbeck,“NucleonisovectorcouplingsfromN =2latticeQCD,”Phys.Rev.,
f
vol.D91,no.5,p.054501,2015.
[9] P.Wein,P.C.Bruns,andA.Schäfer,“Firstmomentsofnucleongeneralizedpartondistributionsin
chiralperturbationtheoryatfullone-looporder,”Phys.Rev.,vol.D89,no.11,p.116002,2014.
7