Electromagnetic Form Factors of Excited Nucleons via Parity-Expanded Variational Analysis

0.36 MB·

by  Finn M. Stokes

Electromagnetic Form Factors of Excited Nucleons
via Parity-Expanded Variational Analysis
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1 Finn M. Stokes , Waseem Kamleh, Derek B. Leinweber and Benjamin J. Owen
∗
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CentrefortheSubatomicStructureofMatter,
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DepartmentofPhysics,
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UniversityofAdelaide,SA5005
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J E-mail: [email protected]

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VariationalanalysistechniquesinlatticeQCDarepowerfultoolsthatgiveaccesstotheexcited

]
t statespectrumofQCD.Atzeromomentum,thesetechniquesarewellestablishedandcancleanly
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l isolateenergyeigenstatesofeitherpositiveornegativeparity.Inordertocomputetheformfactors
-
p of a single energy eigenstate, we must perform a variational analysis at non-zero momentum.
e
When we do this with baryons, we run into issues with parity mixing, as boosted baryons are
h
[ not eigenstates of parity. The parity-expanded variational analysis (PEVA) technique is a novel

1 methodforensuringthesuccessfulandconsistentisolationofboostedbaryoneigenstates. This
v
is achieved through a parity expansion of the operator basis used to construct the correlation
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7 matrix.World-firstcalculationsofexcitedstatenucleonformfactorsusingthisnewtechniqueare
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presented,showingtheimprovementoverconventionalmethods.
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34thannualInternationalSymposiumonLatticeFieldTheory
24–30July2016
UniversityofSouthampton,UK
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
1. Introduction
In order to evaluate the form factors and transition moments of baryon excitations in lattice
QCD,itisnecessarytoisolatethesestatesatfinitemomentum. Excitedbaryonshavebeenisolated
onthelatticethroughacombinationofparityprojectionandvariationalanalysistechniques[1,2,
3, 4, 5, 6, 7, 8, 9]. At zero momentum, these techniques are well established and can isolate the
states of interest. However, at non-zero momentum, these techniques are vulnerable to opposite
paritycontaminations.
To resolve this issue, we developed the Parity-Expanded Variational Analysis (PEVA) tech-
nique [10]. By introducing a novel Dirac projector and expanding the operator basis used to con-
struct the correlation matrix, we are able to cleanly isolate states of both parities at finite momen-
tum.
UtilisingthePEVAtechnique,weareabletopresentheretheworld’sfirstlatticeQCDcalcu-
lationsofnucleonexcitedstateformfactorsfreefromoppositeparitycontaminations. Specifically,
the Sachs electromagnetic form factors of a localised negative parity nucleon excitation are ex-
amined. Furthermore, we clearly demonstrate the efficacy of variational analysis techniques at
providing access to ground state form factors with extremely good control over excited state ef-
fects.
2. ConventionalVariationalAnalysis
Webeginbybrieflyhighlightingwhereoppositeparitycontaminationsenterintoconventional
variationalanalysistechniquesandmotivatethePEVAtechnique. Intheseproceedingsweusethe
PaulirepresentationforDiracmatrices. Inordertodiscussoppositeparitycontaminations,weneed
tobeabletocategorisestatesbytheirparity. However,eigenstatesofnon-zeromomentumarenot
eigenstatesofparity,sowemustcategoriseboostedstatesbytheirrest-frameparity.
Toperformaconventionalvariationalanalysisonspin-1/2baryons, wetakeabasisofncon-
ventionalbaryonoperatorsχi,whichcoupletostatesofbothparities,
(cid:114) (cid:114)
m m
(cid:104)Ω|χi|B+(cid:105)=λBi+ EBB++ uB+(p,s), (cid:104)Ω|χi|B−(cid:105)=λBi− EBB− γ5uB−(p,s),
−
andusethisbasistoformann nmatrixoftwo-pointcorrelationfunctions
×
Gij(p;t):=∑eipx Ω χi(x)χj(0) Ω .
·
(cid:104) | | (cid:105)
x
This correlation matrix contains states of both parities, so we introduce the ‘parity projector’
Γ =(γ I)/2,andtakethespinortrace,definingtheprojectedcorrelationmatrixGij(Γ ;p,t):=
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tr±(cid:0)Γ Gij±(p,t)(cid:1). By inserting a complete set of states between the two operators, and ±noting the
±
useofEuclideantime,wecanrewritethisprojectedcorrelationmatrixas
E (p) m E (p) m
Gij(Γ±;p,t)=∑B+e−EB+(p)tλBi+λBj+ B+2EB+±(p)B+ +∑B−e−EB−(p)tλBi−λBj− B−2EB−∓(p)B− ,
Atzeromomentum,E (0)=m andtheprojectedcorrelationmatriceswilleachcontainterms
B B
ofasingleparity. However,atnon-zeromomentumE (p)=m andtheprojectedcorrelationma-
B B
(cid:54)
trices contain O((E m)/2E) opposite parity contaminations. These opposite parity contamina-
−
tionswereinvestigatedinRef.[11].
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ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
3. Parity-ExpandedVariationalAnalysis
To solve the problem of opposite parity contaminations at finite momentum, we developed
thePEVAtechnique[10]. Inthissection,wesummarisethePEVAtechnique,anddescribehowit
appliestoformfactorcalculations.
ThePEVAtechniqueworksbyexpandingtheoperatorbasisofthecorrelationmatrixtoisolate
energyeigenstatesofbothrest-frameparitiessimultaneouslywhilestillretainingasignatureofthis
parity. By considering the Dirac structure of the unprojected correlation matrix, we construct the
novel momentum-dependent projector Γ := 1(I+γ )(I iγ γ pˆ ). This allows us to construct
p 4(cid:110) 4 − 5 k k (cid:111)
a set of “parity-signature” projected operators χpi =Γpχi, χpi(cid:48) =Γpγ5χi , where the primed in-
dicesdenotetheinclusionofγ ,invertingthewaytheoperatorstransformunderparity. Unlikethe
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conventional opertors χi, the inclusion of Γp ensures that the operators χpi and χpi(cid:48) have definite
parityatzeromomentumwithoutrequiringprojectionbyΓ .
±
By performing a variational analysis with this expanded basis [10], we construct optimised
operatorsφα(x)thatcoupletoeachstateα. Wecanthenusetheseoperatorstocalculatethethree
p
pointcorrelationfunction
G+µ(p(cid:48),p;t2,t1;α):= ∑ e−ip(cid:48)·x2ei(p(cid:48)−p)·x1(cid:104)φpα(cid:48)(x2)|Jµ(x1)|φαp(0)(cid:105),
x2,x1
where Jµ is the O(a)-improved [12] conserved vector current used in [13], inserted with some
momentum transfer q= p p. We can take the spinor trace of this with some projector Γ to get
(cid:48)
theprojectedthreepointco−rrelationfunctionGµ(p,p;t ,t ;Γ;α):= tr(cid:0)ΓGµ(p,p;t ,t ;α)(cid:1).
+ (cid:48) 2 1 + (cid:48) 2 1
There is an arbitrary sign choice in the definition of Γ , so it is convenient to define Γ :=
p p(cid:48)
1(I+γ )(I+iγ γ pˆ )=Γ , which is equally valid. We can then use this alternate projector in
4 4 5 k k p
−
constructing an alternate sinkoperator φ α(x), while leaving the sourceoperator unchanged. This
p(cid:48)
gives us an alternate three point correlation function, Gµ(p,p;t ,t ;α), leading to an alternate
(cid:48) 2 1
projectedthreepointcorrelationfunction,Gµ(p,p;t ,t ;−Γ;α).
(cid:48) 2 1
−
Wecanthenconstructthereducedratio,
(cid:118)
R±(p(cid:48),p;α;r,s):=(cid:117)(cid:117)(cid:116)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)rµG±µ(p(cid:48),p;t2,t1G;(spν(cid:48)Γ;νt2;;αα))rGρ(Gpρ±;t(2p;,αp)(cid:48);t2,t1;sσΓσ;α)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:115) (cid:115)
sign(cid:0)r Gγ (p,p;t ,t ;s Γ ;α)(cid:1) 2Eα(p) 2Eα(p(cid:48)) ,
× γ ± (cid:48) 2 1 δ δ Eα(p)+mα Eα(p(cid:48))+mα
wherer ands arecoefficientsselectedtodeterminetheformfactors. Byinvestigatingther and
µ µ µ
s dependenceofR ,wefindthattheclearestsignalsaregivenby
µ
±
RT = 2 R (cid:0)p,p;α;(1,0),(1,0)(cid:1), and
± 1 pˆ pˆ(cid:48) ± (cid:48)
± ·
RS = 2 R (cid:0)p,p;α;(0,rˆ),(0,sˆ)(cid:1),
∓ 1 pˆ pˆ(cid:48) ∓ (cid:48)
± ·
wheresˆ ischosensuchthatp sˆ=0=p sˆ,rˆ isequaltoqˆ sˆ,andthesign ischosensuchthat
(cid:48)
· · × ±
1 pˆ pˆ ismaximised.
(cid:48)
± ·
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ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
WecanthenfindtheSachselectricandmagneticformfactors
GE(Q2)=(cid:2)Q2(cid:0)Eα(p(cid:48))+Eα(p)(cid:1)(cid:0)(Eα(p)+mα)(cid:0)Eα(p(cid:48))+mα(cid:1) (cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:1)RT
2(cid:12)(cid:12)q(cid:12)(cid:12)(cid:0)1 pˆ pˆ(cid:48)(cid:1)(cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:0)(Eα(p)+mα)(cid:0)Eα(p(cid:48))+∓mα(cid:1) (cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p±(cid:48)(cid:12)(cid:12)(cid:1)RS(cid:3)
/(cid:104)4±mα(cid:104)(cid:0)Eα∓(p)E·α(p(cid:48))+m2α∓(cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:1)(cid:12)(cid:12)q(cid:12)(cid:12)2+4(cid:12)(cid:12)p(cid:12)(cid:12)2(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)2(cid:0)1±∓pˆ·pˆ(cid:48)(cid:1)(cid:105)(cid:105),∓and
GM(Q2)=(cid:2) 2(cid:0)1 pˆ pˆ(cid:48)(cid:1)(cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:0)(Eα(p)+mα)(cid:0)Eα(p(cid:48))+mα(cid:1) (cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:1)RT
± (cid:12)(cid:12)q(cid:12)(cid:12)∓(cid:0)Eα·(p(cid:48))+Eα(p)(cid:1)(cid:0)(Eα(p)+mα)(cid:0)Eα(p(cid:48))+m±α(cid:1) (cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)±(cid:1)RS(cid:3)
/(cid:104)2−(cid:104)(cid:0)Eα(p)Eα(p(cid:48))+m2α∓(cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)(cid:1)(cid:12)(cid:12)q(cid:12)(cid:12)2+4(cid:12)(cid:12)p(cid:12)(cid:12)2(cid:12)(cid:12)p(cid:48)(cid:12)(cid:12)2(cid:0)1∓∓pˆ·pˆ(cid:48)(cid:1)(cid:105)(cid:105). ∓
ThedetailsofthisprocedurewillbepresentedinfullinRef.[14].
4. Results
Inthissection,wepresentworld-firstlatticeQCDcalculationsoftheSachselectricandmag-
neticformfactorsofthegroundstatenucleonandfirstnegative-parityexcitationwithgoodcontrol
over opposite parity contaminations. We compare the results obtained by the PEVA technique to
ananalysisusingconventionalparityprojection.
These results are calculated on the second heaviest PACS-CS (2+1)-flavour full-QCD en-
semble [15], made available through the ILDG [16]. This ensemble uses a 323 64 lattice, and
×
employs an Iwasaki gauge action with β = 1.90 and non-perturbatively O(a)-improved Wilson
quarks. We use the m =570MeV PACS-CS ensemble, and set the scale using the Sommer pa-
π
rameterwithr =0.49fm,givingalatticespacingofa=0.1009(23)fm. Withthisscale,ourpion
0
massis515(8)MeV. Weused343gaugefieldconfigurations,withasinglesourcelocationoneach
configuration. χ2/dofiscalcularedwiththefullcovariancematrix,andallfitshaveχ2/dof <1.2.
For the analyses in this section, we start with a basis of eight operators, by taking two con-
ventional spin-1/2 nucleon operators (χ1 =εabc[ua(cid:62)(Cγ5)db]uc, and χ2 =εabc[ua(cid:62)(C)db]γ5uc),
andapplying16,35,100,and200sweepsofgaugeinvariantGaussiansmearingwhencreatingthe
propagators[2]. Fortheconventionalvariationalanalysis,wetakethisbasisofeightoperatorsand
project with Γ , and for the PEVA analysis, we parity expand the basis to sixteen operators and
±
projectwithΓ andΓ .
p p(cid:48)
Toextracttheformfactors,wefixthesourceattimeslice16,andthecurrentinsertionattime
slice21. Wechoosetimeslice21byinspectingthetwopointcorrelationfunctionsassociatedwith
eachstateandobservingthatexcitedstatecontaminationsarestronglysuppressedbytimeslice21.
We then extract the form factors from the ratios given in Sec. 3 for every possible sink time and
lookforaplateauconsistentwithasingle-stateansatz.
Beginningwiththegroundstate,inFig.1weplotG (Q2)andinFig.2weplotG (Q2)with
E M
respect to sink time, at Q2 =0.144GeV2. There are only slight differences between the results
extractedbyPEVAandtheresultsextractedbyaconventionalvariationalanalysisinthiscase. We
believethisisbecausetheoppositeparitycontaminationsaresmall,andcomefromheavierstates
thataresuppressedbyEuclideantimeevolution.
ForboththePEVAandconventionalvariationalanalysis,weseeveryclearandcleanplateaus
in the form factors, indicating very good control over excited state contaminations. This supports
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ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
1.0
0.8
0.6
0.4
E 0.2
G
0.0
0.2
− PEVA (up) Conv. (up)
0.4
PEVA (d ) Conv. (d )
− p p
16 18 20 22 24 26 28 30
t/a
Figure 1: G (Q2) for the ground state nucleon at Q2 =0.144GeV2. We plot the conventional analysis
E
withopenmarkersandthenewPEVAanalysiswithclosedmarkers. Ourfitstotheplateausareillustrated
by shaded bands, with the central value indicated by dashed lines for the conventional analysis, and solid
lines for the PEVA analysis. We plot the contributions from the singly represented quark sector with blue
triangles,anddoublyrepresentedquarksectorwithorangesquares,forsinglequarksofunitcharge.
1.5
1.0
0.5
M 0.0
G
0.5
−
PEVA (u ) Conv. (u )
1.0 p p
− PEVA (d ) Conv. (d )
p p
1.5
− 16 18 20 22 24 26 28 30
t/a
Figure2:G (Q2)forthegroundstatenucleonatQ2=0.144GeV2. Weplottheconventionalanalysiswith
M
openmarkersanddashedlinesandthePEVAanalysiswithclosedmarkersandsolidlines.Thecontributions
from the singly represented quark sector are plotted with red hexagons, and from the doubly represented
quarksectorwithgreenpentagons,forsinglequarksofunitcharge.
previous work demonstrating the utility of variational analysis in calculating baryon matrix ele-
ments[17,18]. ByusingsuchtechniquesweareabletocleanlyisolateprecisevaluesfortheSachs
electricandmagneticformfactorsofthegroundstatenucleon.
Movingontothefirstnegativeparityexcitedstate,inFig.3weplotG (Q2)atQ2=0.146GeV2
E
vs the sink time. We see that the PEVA analysis (closed points) allows us to fit a plateau (shaded
bands with solid lines) at a much earlier time, and with a significantly different value to the con-
ventional analysis (open points, dashed lines). This demonstrates the effectiveness of the PEVA
techniqueatremovingoppositeparitycontaminations. Thelocalisednatureofthisstateismanifest
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ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
1.0
0.8
0.6
0.4
E 0.2
G
0.0
0.2
− PEVA (up) Conv. (up)
0.4
PEVA (d ) Conv. (d )
− p p
16 18 20 22 24 26 28
t/a
Figure3: G (Q2)forthefirstnegativeparityexcitationofthenucleonatQ2=0.146GeV2. Weplotthethe
E
conventionalanalysiswithopenmarkersanddashedlines,andthenewPEVAanalysiswithclosedmarkers
andsolidlines. Weplotthecontributionsfromthesinglyrepresentedquarksectorwithbluetriangles,and
doublyrepresentedquarksectorwithorangesquares,forsinglequarksofunitcharge. ThePEVAanalysis
plateausatamuchearliertimewithasignificantlydifferentvalue,demonstratingitseffectiveness.
1.5
1.0
0.5
0.0
M
G
0.5
−
1.0
−
PEVA (u ) Conv. (u )
p p
1.5
− PEVA (dp) Conv. (dp)
2.0
− 16 18 20 22 24 26
t/a
Figure4:G (Q2)forthegroundstatenucleonatQ2=0.146GeV2. Weplottheconventionalanalysiswith
M
openmarkersanddashedlinesandthePEVAanalysiswithclosedmarkersandsolidlines.Thecontributions
from the singly represented quark sector are plotted with red hexagons, and from the doubly represented
quark sector with green pentagons, for single quarks of unit charge. The significantly different values of
G (Q2)forthesinglyrepresentedquarksectordemonstratetheimportanceofthePEVAtechnique.
M
inthelargevaluesforG (Q2),similartothatfortheproton. Wealsoseesimilarimprovementsin
E
G (Q2),aspresentedinFig.4.
M
5. Conclusion
We have demonstrated the effectiveness of the PEVA technique specifically and variational
analysis techniques in general at controlling excited state effects. For the ground state nucleon,
conventional variational analysis techniques are sufficient to provide clean plateaus that allow for
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ElectromagneticFormFactorsofExcitedNucleons FinnM.Stokes
theeffectiveextractionofformfactors. However,forexcitedstates,oppositeparitycontaminations
haveaclearandsignificanteffect. ThePEVAtechniqueallowsustoremovethesecontaminations
andcalculatetheformfactorsofexcitednucleonsforthefirsttime.
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