Table Of ContentLattice QCD Methods for Hadronic Polarizabilities
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J B. C.Tiburzi †
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0 DepartmentofPhysics,TheCityCollegeofNewYork,NewYork,NY,USA
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GraduateSchoolandUniversityCenter,TheCityUniversityofNewYork,NewYork,NY,USA
] RIKENBNLResearchCenter,BrookhavenNationalLaboratory,Upton,NY,USA
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a E-mail:[email protected]
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e Chiraldynamicsmakesdefinitivepredictionsfortheelectromagneticpolarizabilitiesofhadrons
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near the chirallimit; but, agreementwith experimentis tenuousin some cases. We providean
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overviewoflatticeQCDmethodstocomputetheelectricandmagneticpolarizabilitiesofhadrons.
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v Centraltothesemethodsisthelatticesimulationofquarksinuniform,classicalelectromagnetic
2 fields.Along-termgoalisthedeterminationofpolarizabilitiesdirectlyfromlatticecomputations,
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6 however,inthenearterm,onemayneedtorelyonpartiallyquenchedchiralperturbationtheory.
4 Nonethelessthesamestrikingpredictionsforthepionmassdependenceofelectricandmagnetic
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1 polarizabilitiescanbemadefromchiraldynamics,andtestedwithlatticeQCD.Aparticularfocus
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3 isanovelnewmethodtohandlechargedparticlecorrelationfunctionsinmagneticfields.
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The7thInternationalWorkshoponChiralDynamics,
August6-10,2012
JeffersonLab,NewportNews,Virginia,USA
Speaker.
∗
†Iamgratefultomyinsightfulcollaboratorsforfruitfulcollaborationonthevariousprojectsthataresummarized
inthistalk.WorksupportedinpartbyajointCCNY–RBRCfellowship,anawardfromtheProfessionalStaffCongress
oftheCUNY,andbytheU.S.NationalScienceFoundation,underGrantNo.PHY-1205778.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
LatticeQCDMethodsforHadronicPolarizabilities B.C.Tiburzi
1. Electromagnetic Polarizabilities
The electric and magnetic polarizabilities provide an opportunity for a stringent test of the
chiraldynamicsinsidehadrons. Essentialtothestoryofchiraldynamicsisthespontaneous break-
downofchiralsymmetry. Forthecaseoftwomassless quark flavors, theQCDfunctional integral
possesses an SU(2) SU(2) symmetry, which is broken to the vector subgroup by the vacuum
L R
⊗
expectation value of the chiral condensate, < y y >=0. In this picture, the pions emerge as
L R 6
Goldstonemodes,andhavemassesthatvanishintheabsenceofexplicitchiralsymmetrybreaking,
thatism2p =0+mq <yy > F2,wheretheexplicitchiralsymmetrybreakingisparameterizedby
| |
m ,thequarkmass. Thispictureisaneffectivedescriptionoflow-energyQCDprovidedthequark
q
masses are small compared to the scale of strong interactions. In terms of hadronic parameters
that are free from QCD renormalization scale and scheme dependence, this condition translates
into mp /(4p F) 1; and, when met, allows for the quark mass dependence of low-energy QCD
≪
observables todeterminedsystematically inanexpansion aboutthechirallimit, m =0.
q
In chiral perturbation theory, the interactions of hadrons with pions are constrained by the
form ofspontaneous andexplicit chiral symmetrybreaking. Asaresult, thebarehadron fieldsare
dressed withpionsinaFockstateexpansion, schematically oftheform
P 0 = c p 0 +c p 0p +p +..., N =c n +c pp +..., (1.1)
0 1 − 0 1 −
| i | i | i | i | i | i
where the bare fields are denoted with lower-case lettering, c ’s are related to the wave-function
0
renormalization, and c ’saredetermined bytheinteractions ofthetheory. Depending onkinemat-
1
ics, the higher Fockcomponents mayonly bevirtual states. Abovewehave chosen states that are
electrically neutraltoemphasizethatdynamicalfluctuations produce Fockcomponents containing
charged hadrons. The electromagnetic interaction thus serves as a probe sensitive to the higher
Fockcomponentsofthebarehadronfields. Withinchiralperturbationtheory,theelectricandmag-
neticpolarizabilities(a andb ,respectively)aredeterminedtoleadingorderentirelyfromhigher
E M
Fock components. For the poins [1], and nucleons [2], there is a characteristic singularity in the
chirallimitofthesequantities,
1 1 1 1
a p ,b p , a N,b N . (1.2)
E M ∼ mp ∼ √mq E M ∼ mp ∼ √mq
Experimental determination ofpolarizabilities canbeachieved through theanalysis oflow-energy
Compton scattering data. Inprinciple, this iseasiest forthe proton, however, the low-energy limit
is dominated by thetotal charge interaction, theThomson cross section; and, while increasing the
energy leads toincreased sensitivity to polarizabilities, it also introduces higher-order response of
the nucleon. For the most recent comprehensive analysis of proton Compton scattering, see [3].
For the neutron, one must use Compton scattering off deuterium or other light nuclei to extract
polarizabilities, which introduces additional theoretical uncertainty. For pions, experiments have
resorted to photo-pion production off the nucleon [4], and new results are anticipated from recent
COMPASSmeasurements usingpionscattering offPrimakoffphotons [5].
2. Methods forParticles inElectric Fields
Theelectromagnetic polarizabilities provide an opportunity forlattice QCDcomputations, as
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LatticeQCDMethodsforHadronicPolarizabilities B.C.Tiburzi
theyarequantitiesthathavebeensubjecttodebate. Alatticedeterminationofthepolarizabilitiesof
thedeuteron, forexample,wouldbeamajorcontribution, as onewouldbeabletostudyfew-body
dynamics in addition to the chiral behavior. Needless to say, lattice QCD is not yet at the point
of such studies, however, there has been progress in treating strong interactions in the presence
of uniform, classical electromagnetic fields, see [6] for overviews. One might think the natural
starting pointforlatticecomputations ofpolarizabilities wouldbetheComptonscattering tensor,
Tmn (k′,k)= d4xd4ye−ik′m xm +ikm ym H T Jm (x)Jn (y) H , (2.1)
Z h | { }| i
however, thispresents twomajorcomplications. Thelow-energy limitinEq.2.1isconstrained by
spatial momentum quantization conditions resulting from periodic boundary conditions imposed
onthequarkfields. Progresshasbeenmadeonthisfront. Thedesired termsatsecondorderinthe
photonfrequencycanbeisolatedbystudyingzeromomentumderivativesofthequarkpropagators,
and these can be computed approximately by using partially twisted boundary conditions, and
takingthelimitofvanishingtwistangle[7]. Thisprocedureleadsonetothecomputationofcertain
hadronic four-point functions for which nearby intermediate states between the current insertions
presentanessentialcomplication. Whiletherehasbeenprogressincomputingfour-pointfunctions
inthecaseofthemesons,namelytheK –K massdifference[8],itislikelythatsuchmethodswill
L S
notbepracticable fornucleons.
The external field approach provides an alternate method to access electromagnetic polariz-
abilities. One adds a classical electromagnetic field to QCD computations, and studies the sub-
sequent external field dependence of hadronic correlation functions. Such dependence gives one
accesstohadroniccouplingstotheexternalfield. Toincludeanexternalelectromagnetic field,one
appendsU(1)-valued linkstotheSU(3)colorgaugelinks
Um (x) Ume.m.(x)Um (x). (2.2)
−→
This must be done for valence quark propagators, and in the quark determinant used to generate
gauge ensembles. The latter encompasses contributions due to the electric charges of sea quarks,
and has only been achieved on lattices studying thermodynamics with staggered quarks. In the
near term, calculations in weak external fields will exclude contributions from the sea quarks in
gauge field generation. Gauge ensemble re-weighting techniques are a promising way to include
sea quark charges [9]. Thelight quark massregime isproblematic forthe quenching ofsea quark
chargesaltogether,duetotheso-calledexceptionalityofgaugeconfigurations. Exceptionalconfig-
urations create anessential roadblock forpost-multiplying electromagnetic links toexisting gluon
gauge configurations. Inasmuch as such configurations are not encountered, one can address the
quenching ofseaquark electric charges using chiral perturbation theory, foradiscussion see[10],
and predictions exist for pion and nucleon polarizabilities as a function of the sea quark electric
charges[11]. Asourmethodsareaddressedwithexploratory latticestudies,wecoupletheexternal
fieldstovalencequarksonly.
Afinalwordontheinclusionofuniform,externalelectromagneticfieldsonalattice. Theperi-
odicityoflatticequarkfieldsleadstoaquantizationconditiononthestrengthofexternalfields[12],
because the hyper-torus forms a closed surface that does not leak any flux. Forelectric and mag-
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LatticeQCDMethodsforHadronicPolarizabilities B.C.Tiburzi
neticfields,thequantization conditions areoftheform1
qE =2p n/Lb , qB=2p n/L2, (2.3)
where L is the length of a spatial direction, with all three spatial directions assumed to be of the
same length, and b is the length of the temporal direction of the lattice. We write E for the
electric field tomake clear that weare in Euclidean space, where the action density would appear
as 1Fmn Fmn = 1 ~B2+E~2 . AnalyticcontinuationisnecessaryforresultsinMinkowskispace;but,
4 2
(cid:16) (cid:17)
as weare interested in quantities that are perturbative in the strength of the field, the continuation
istrivial. TheSchwingermechanism [15],whichisanon-perturbative phenomenon, isfortunately
absentinEuclideanspace. Inthissection, weconcern ourselveswithexternalelectricfields.
2.1 NeutralParticles
Forneutral hadrons, the method is simple to explain. One includes an external electric field,
and measures two-point correlation functions of hadrons. In the long Euclidean time limit, these
correlation functions shouldfollowanexponential falloff,
GE(t )=ZE e−E(E)t , (2.4)
where E(E) is the energy of the hadron in the external electric field. For a neutron in an external
electric field,theenergyisgivenby[16]
E(E)=M + a m 2/4M2 E2/2+..., (2.5)
n E n n
−
(cid:0) (cid:1)
where termsoforder E4 have been dropped. Thecontribution involving thesquare of theneutron
magneticmomentarisesfromtreatingtheneutronspinrelativistically. IntermsofneutronCompton
scattering,theanalogouscontributionappearsasaBornterm,namelythesecond-ordertermarising
from two interactions of leading-order. The exponential falloff of neutron correlators leads to an
extraction ofthequantity inparentheses. Ifoneisinterested inisolating theelectric polarizability,
one requires a method to determine the neutron magnetic moment, and this can be achieved by
looking at the amplitudes inoff-diagonal spin components ofthe correlator [16]. In that study, an
ensemble of anisotropic clover fermions [17]was successfully used todemonstrate the technique.
TheanalogueoftheBorntermwasshowntoaffectextractionoftheelectricpolarizability by50%.
2.2 ChargedParticles
The study of charged particles in external electric fields is obviously complicated, however,
the same philosophy can be applied. One determines hadronic correlation functions for various
fieldstrengths,andthenmatchesontothebehaviorexpected fromasingle-particle effectiveaction.
Thisbehavior isnotasimpleexponential falloffinEuclidean time. Forexample, thecharged pion
propagator shouldbehaveas[18]
¥ ds
GE(t )=ZE Z0 √sinhQEse−21QEt2cothQEs−21Ep2(E)s, (2.6)
1Strictlyspeakingsuchquantizationconditionsdonotleadtojustuniformelectromagneticfields. Therearealso
gauge-invariant,finitevolumeartifactsinvolvingthenon-trivialholonomyoftheexternalfield[13]. Whilesuchcontri-
butionsshouldbeexponentiallysuppressed, e mpL,latticeresultssuggestthattheeffectmightbenon-negligibleeven
−
∼
atlargepionmasses[14].Thatstudyemploysmethodsdifferentthanthoseoutlinedhere.
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LatticeQCDMethodsforHadronicPolarizabilities B.C.Tiburzi
with Ep (E)=mp + 12a EE2+···. While fits to the non-standard t -behavior can be challenging,
the technique has been successfully demonstrated in an exploratory lattice calculation of charged
pionandkaoncorrelation functions [19]. Similarsuccess wasalsoachieved inthestudy ofproton
correlators usingageneralization ofthemethodtospin-half particles[16].
3. A Method forChargedParticles inMagneticFields
The quantization condition for uniform magnetic fields has so far proven restrictive for the
studyofperturbatively smalleffectsforhadrons. Withincreasedlatticevolumes,smallermagnetic
field strengths can be accessed. The energy eigenstates of charged particles in external magnetic
fields are described by Landau levels. For the charged pion, the long-time limit of the standard
latticecorrelation function shouldproducethestandard exponential behavior
G (t )=(cid:229) p (~x,t )p †(0,0) =Z e E0(B)t + , (3.1)
B B −
h i ···
~x
where E0 is the pion energy in the lowest Landau level, E0(B)=mp + |2QmBp|+ 21b MB2+···. The
omitted terms in the long-time limit of the correlation function include excited hadronic states, as
wellasthehigherLandaulevels. Forlargevaluesoftheexternalmagneticfield,theLandaulevels
will be widely separated in energy, and only the lowest Landau level will survive the long-time
limit. As we are interested in perturbatively small magnetic fields, however, the narrow Landau
level spacing, D E/M = QB /M2, will lead to a pileup. For smaller values of the magnetic field,
| |
onewillrequirelongertimestoseparate outthecontribution fromthelowestLandaulevel.
Thiscomplication canbesidestepped altogether[20]. Inspecting Eq.3.1,weseethatthesum
overalllattice sites projects the correlator onto zero spatial momentum,~p=0. Inthe presence of
a magnetic field, it isimpossible forall components of three-momentum toremain good quantum
numbers. Consequently the correlator contains all Landau levels. A more judicious choice of
correlation function isgivenby
G (t )=(cid:229) y (x) p (~x,t )p †(0,0) , (3.2)
B 0∗
h i
~x
where y (x) is the coordinate wave-function of the lowest Landau level. The long-time limit of
0
thecorrelation function hasthesameexponential falloff, however, thefirstomittedtermsarefrom
higher lyinghadronic states justasintheabsence ofthemagnetic field. Thiscanbedemonstrated
with the Schwingerproper-time trick. Explicit projection ofthe lowest Landau level should bean
economic technique inperturbatively smallmagneticfields.
As the technique must be practicable on the lattice, we investigated the effects of discretiza-
tion on the lowest Landau level, and the effects of finite volume. For the former, we found that
discretization effectsonthewave-functionofthelowestLandaulevelrequiredinEq.3.2wereneg-
ligible. The discretization corrections to the energy of the lowest Landau level could have amore
substantial effectcompetitive withthemagneticpolarizability,
1
E02(B,a)=m2p +|QB|−(cid:18)8a2Q2+mp b M(cid:19)B2. (3.3)
Finitevolumecorrections whereshowntobeimportant;but, canbetreatedasanapplicationofthe
magneticperiodicity oftheaction[21].
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LatticeQCDMethodsforHadronicPolarizabilities B.C.Tiburzi
References
[1] B.R.Holstein,CommentsNucl.Part.Phys.A19,221(1990).
[2] V.Bernard,N.KaiserandU.G.Meissner,Phys.Rev.Lett.67,1515(1991);
V.Bernard,N.Kaiser,A.SchmidtandU.G.Meissner,Phys.Lett.B319,269(1993);
T.R.Hemmert,B.R.HolsteinandJ.Kambor,Phys.Rev.D55,5598(1997);
S.R.Beane,M.Malheiro,J.McGovern,D.PhillipsandU.vanKolck,Nucl.Phys.A747,311(2005).
[3] J.A.McGovern,D.R.PhillipsandH.W.Griesshammer,arXiv:1210.4104[nucl-th].
[4] J.Ahrens,etal.,Eur.Phys.J.A23,113(2005).
[5] P.Abbonetal.[COMPASSCollaboration],Nucl.Instrum.Meth.A577,455(2007).
[6] B.C.Tiburzi,PoSLATTICE2011,020(2011);
M.D’Elia,arXiv:1209.0374[hep-lat].
[7] G.M.deDivitiis,R.PetronzioandN.Tantalo,Phys.Lett.B718,589(2012).
[8] J.Yu,PoSLATTICE2012,129(2012).
[9] W.Freeman,A.Alexandru,F.LeeandM.Lujan,PoSLATTICE2012,015(2012).
[10] B.C.Tiburzi,Phys.Rev.D79,077501(2009).
[11] W.Detmold,B.C.TiburziandA.Walker-Loud,Phys.Rev.D73,114505(2006);
J.Hu,F.-J.JiangandB.C.Tiburzi,Phys.Rev.D77,014502(2008).
[12] G.’tHooft,Nucl.Phys.B153,141(1979);
J.SmitandJ.C.Vink,Nucl.Phys.B286,485(1987);
P.H.DamgaardandU.M.Heller,Nucl.Phys.B309,625(1988).
[13] B.C.Tiburzi,Phys.Lett.B674,336(2009);
W.Detmold,B.C.TiburziandA.Walker-Loud,arXiv:0908.3626[hep-lat].
[14] M.Engelhardt[LHPCCollaboration],Phys.Rev.D76,114502(2007).
[15] J.S.Schwinger,Phys.Rev.82,664(1951).
[16] W.Detmold,B.C.TiburziandA.Walker-Loud,Phys.Rev.D81,054502(2010).
[17] R.G.Edwards,B.JooandH.-W.Lin,Phys.Rev.D78,054501(2008);
H.-W.Linetal.[HadronSpectrumCollaboration],Phys.Rev.D79,034502(2009).
[18] B.C.Tiburzi,Nucl.Phys.A814,74(2008).
[19] W.Detmold,B.C.TiburziandA.Walker-Loud,Phys.Rev.D79,094505(2009).
[20] B.C.TiburziandS.O.Vayl,arXiv:1210.4464[hep-lat].
[21] M.H.Al-HashimiandU.-J.Wiese,AnnalsPhys.324,343(2009).
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