Table Of ContentSome continuum physics results from the lattice
3 V-A correlator
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2
n
a
J
1 P.A. Boyle,a,L. DelDebbio,a N.Garron,b, R.J.Hudspith,a,E.Kerranec, K.Maltman∗
2
,d,e andJ.M.Zanottie
] aPhysicsandAstronomy,UniversityofEdinburgh,EdinburghEH93JZ,UK
t
a bSchoolofMathematics,TrinityCollege,Dublin2,Ireland
l
- cInstitutodeFìsicaTèoricaUAM/CSIC,UniversidadAutònomadeMadrid,Cantoblanco
p
e E-28049Madrid,Spain
h dMathematicsandStatistics,YorkUniversity,TorontoM3J1P3Canada
[
eCSSM,UniversityofAdelaide,Adelaide5005Australia
2 E-mail:[email protected],[email protected],
v
[email protected],[email protected],
5
6 [email protected],[email protected],
5 [email protected]
2
.
1
0 We presentpreliminaryresultsonextractionsofthechiral LECsL10 andC87 andconstraintson
3 the excitedpseudoscalarstate p (1300)and p (1800)decayconstantsobtainedfrom an analysis
1
: of lattice data for the flavor ud light quark V-A correlator. A comparisonof the results for the
v
i correlatorto the correspondingmildly-model-dependentcontinuum results (based primarily on
X
experimentalhadronict decaydata)isalsogiven.
r
a
The30thInternationalSymposiumonLatticeFieldTheory
June24-29,2012
Cairns,Australia
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
1. The V-A correlator
Wefocusonthedifferenceofflavorud vector(V)andaxialvector(A)current-current 2-point
functions, P mn ,andtheirJ=0,1scalarcomponents, P (J) ,definedinMinkowskispaceby
V/A V/A
P mn (q2) ≡i d4xeiq·xh0|T Jm (x)J†n (0) |0i
V/A Z (cid:16) V/A V/A (cid:17)
= qm qn −q2gmn P (1) (Q2)+qm qn P (0) (Q2) (1.1)
V/A V/A
(cid:0) (cid:1)
where, as usual, Q2 = −q2. In what follows, we denote DP (J) ≡P (J)−P (J). TheP (J) (Q2), for
V A V/A
Q2>0,alsodeterminethecorresponding Euclidean2-pointfunctions
P mn (Q2) = Q2d mn −Qm Qn P (1) (Q2)−Qm Qn P (0) (Q2), (1.2)
V/A V/A V/A
h iEucl
(cid:0) (cid:1)
makingtheP (J) (Q2)accessiblefromlatticesimulations. P (0) andP (1) bothhavekinematicpoles
V/A A A
at Q2 = 0 while the J = 0+1 sum does not. Since, beyond NLO in the chiral expansion, the
pole residues involve at-present-unknown NNLO LECs, we focus on DP (Q2) ≡ P (0+1)(Q2)−
V
P (0+1)(Q2), which satisfies an unsubtracted dispersion relation with only physical singularities.
A
Thecorresponding spectral function, D r (s),consists ofad -function ats=m2p andcontinuum be-
ginning at 4m2p . The “continuum part”, D P (Q2), of DP (Q2) results from removing the p pole:
DP (Q2) = D P (Q2)−2fp2/(Q2+m2p ). Fors<m2t ,D r (s)canbedeterminedexperimentally from
hadronic t decay data [1]. Public versions are available from both ALEPH [2] and OPAL [3],
with ayet-to-be-corrected problem affecting the covariance matrix oftheformer. Beyond s=mt2,
an alternate representation results from fitting a physically motivated model for duality violations
(DVs)[5]tointegratedversionsofthedata[4]. TheOPALdataandfittedDVmodelprovideadis-
persivedetermination ofDP (Q2)atspacelikeQ2>0,whereitcanalsobemeasuredonthelattice.
The dispersive result is nominally quite precise, but has some (mild) model-dependence from the
useoftheDVmodel. Italsoinvolvest →4pn t contributions intheV channelwhoseuncertainties
may have been underestimated, given that the corresponding 4p branching fractions differ from
expectations basedonCVCandmeasurede+e−→4p cross-sections bymuchmorethanistypical
for isospin-breaking corrections [6, 7]. The mildness of the model-dependence follows from the
fact that, intherange Q2<(500 MeV)2 expected tobeofrelevance tothedetermination ofchiral
LECs,DVcontributions toD r (s)accountforafewtoseveral%ofD P (Q2),theprecisevaluesde-
pending onthepoint inthespectrum atwhichoneswitchesfromdatatothefittedDVmodel. The
cross-check on the lattice results for D P provided by the dispersive representation is also useful
in light of the freedom to vary m on the lattice, which, in principle, provides access to NNLO
u,d,s
chiralLECscurrently unknownand/ordifficulttoextractreliably withcontinuum methods.
2. Lattice data forDP (Q2)
DP (Q2)hasbeen determined forthe fine1/a=2.28 GeV,mp =289, 345 and394 MeV,and
coarse1/a=1.37GeV,mp =171and248RBC/UKQCDDWFensemblesdetailedinRefs.[8,9].
Thelatterprovideanincreasednumberoflow-Q2points,improvingthedeterminationofthechiral
2
SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
LECs. The values of fp and mp needed to convert DP (Q2) to D P (Q2) are given in Refs. [8, 9].
Thesimulationm valuesare,inallcases,close,butnotexactlyequaltothephysicalvalue. Asthe
s
SU(2) LECscorrespond to physical m , we analyze the data in the chiral SU(3) framework. The
s
ensemble m values required forthispurpose arealsogiveninRefs.[8,9]. Thespectral function,
K
D r ,ofD P ,andhence alsoD P itself, areO(m0)inthechiralexpansion. Wethusexpectthelattice
ℓ
D P (Q2) to approach the physical results for sufficiently light m . We find that, within errors, for
ℓ
the low-Q2 region ofinterest to ushere, the lattice D P (Q2)agree wellwith one another, and with
thecontinuum OPAL+DVmodelresults,forallbutthemp =394MeVcase,asshowninFig.1.
1/a=1.37 GeV, m =171 MeV
0.1 p
1/a=1.37 GeV, m =248 MeV
p
1/a=2.28 GeV, m =289 MeV
p
0.08
1/a=2.28 GeV, m =345 MeV
p
1/a=2.28 GeV, m =394 MeV
) p
2Q0.06 OPAL+DV model, central
_( OPAL+DV model, 1s errors
DP 0.04
0.02
0
0 0.2 0.4 0.6 0.8
2 2
Q [GeV ]
Figure1: LatticeandOPAL+DVmodelresultsforD P (Q2)
3. The chiral LECs
At NLO in the chiral expansion, DP is controlled by the single NLO LEC Lr (m ). Two
10
previous lattice studies determined Lr (m ) by analyzing DP (1) [10, 11] at NLO. With the lattice
10
spacingsavailable,thesecond-smallestnon-zeroQ2werefoundtobetoolarge(∼(650MeV)2and
∼(460MeV)2 forRefs. [10]and [11]respectively) toallow forasuccessful NLOfit. FinalNLO
analysisresultswerethusbasedonthesinglelowestQ2values,(320MeV)2and∼(230MeV)2,re-
spectively. Thecurrentanalysisimprovesonthepreviousonesinanumberofways. First,thenew
coarser lattices allows access to an increased number of low-Q2 points. Second, the statistics for
the mp =289 MeV ensemble considered previously [11] have now been doubled. Third, because
theresidueoftheQ2=0kinematicpoleinDP (1)involvesanunknownNNLOcontribution, whose
contribution, relative tothat ofthe term involving the NLO constant Lr , gets enhanced when one
10
3
SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
goestothelowQ2desirableforextractingLECs,weswitchtoanalyzing D P ratherthanDP (1),the
p pole contribution which must be subtracted to obtain the former having both an exactly known
residue and being farther from the region of the lattice data than is the kinematic pole in DP (1).
The NLO results for Lr (m ), for m =m =0.77 GeV, obtained for each Q2 <0.25 GeV2, and all
10 0
buttheheaviestmp =394MeV ensemble,areshowninFig.2. Alsoshownforcomparisonarethe
results ofanNLOanalysis oftheOPAL+DVmodeloverasimilar rangeofQ2.
SincetheQ2-dependence ofP (Q2)isknowntobepoorlyreproduced bytheNLOrepresen-
V
tation[12],onemightbesurprisedbytherelativestabilityoftheresultsfordifferent Q2. However,
themissingintermediater contributionbelievedresponsiblefortheNLOP slopeproblem[12]is
V
encodedintheNNLOLECCr [13],andthecontribution toP (Q2)proportional toCr isexactly
93 V 93
cancelledbythatproportionaltoCr intheNNLOexpressionforP (0+1)(Q2). Nonetheless,thecen-
93 A
tral value forthe average slope of D P (Q2)with respect to Q2 is significantly larger than expected
from the NLO expression, albeit at only the ∼2s level. The structure of the full NNLO result,
knownfromtheresultsofRef.[13],isverylinearinQ2 fortheQ2consideredhere,soasignificant
portion oftheNNLOcontribution iseasily removedbyfitting theresultsofFig.2toalinearform
and using this to extrapolate to Q2 =0. The only NNLO contributions remaining to be removed
are then those entering D P (0). These involve two NNLO LEC combinations, Cr −Cr −Cr ,
61 12 80
whichisnotlarge-N suppressed, andCr −Cr −Cr ,whichis[13,14]. Thefirstcombinationhas
c 62 13 81
beenestimatedinRef.[14]usingtheresultsofpreviouscontinuumworks. Thesecondiscurrently
unknown, and has only been loosely bounded, using rough large-N -suppression arguments [14].
c
The resulting NNLOLECcombination assessments wereused in Ref. [14]to obtain acontinuum
extraction of Lr andCr (the NNLO LEC expected to dominate the slope of D P ). The analysis
10 87
employed the ALEPH data and was based on two additional, not explicitly tested, assumptions,
namely(i)thattheNNLOformwillsuccessfully represent D P (Q2)and(ii)thattheVandAchan-
nel DV spectral contributions which, being governed by the resonance structure in the channel in
question, are expected to be channel-dependent, can be assumed to be approximately the same in
formandhencecombinedintoasingleDVansatzfortheV-Adifference. Whilethelatterassump-
tion isnot borne out by the combined Vand Achannel fitsofRefs. [4], the contribution to D P of
theDVpartof D r issmallinthelow-Q2 region. Itisthusofinterest tocomparetheresults ofour
fittothoseofthismildlymodel-dependent continuum analysis, whichare[14]
Lr (m ) = −0.0041(4) ; Cr (m ) = 0.0049(2) GeV−2, (3.1)
10 0 NNLO 87 0 NNLO
withtheerrordominatedentirelybytheuncertaintyintheestimateforunknownlarge-N -suppressed
c
NNLOLECcombination. ResultsobtainedusinginsteadtheOPALdata,andincorporating there-
sultsoftheDVmodelfitsofRefs.[4]inordertoremovethesecond ofthetwoassumptions noted
above,areinextremelycloseagreement. Anadditional unknownsystematic error,associated with
the two additional assumptions noted above, of course also exists forthe results of Eq. (3.1). Our
finalgoalistoperformaNNLOanalysisofthelatticedata,includingarangeofm sufficienttoput
q
constraintsonthecurrentlyunknownNNLOLECs(somethinglikelytobefeasiblegiventhediffer-
enceoftheresultsforD P formp =394MeV fromthoseforthesmallermp seeninFig.1),butthis
analysis has not yet been completed. The results of following the continuum estimates/arguments
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SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
ofRef.[14]fortheunknownNNLOLECcombinations entering D P (0)are
Lr (m ) = −0.0038(4) (4) ; Cr (m ) = 0.0040(21) (2) GeV−2, (3.2)
10 0 latt NNLO 87 0 latt NNLO
ingoodagreementwith,thoughlessprecisethantheNNLO-LEC-inducedpartoftheerrorobtained
explicitly inthecontinuum analysis ofRef.[14].
0.002
1/a=1.37 GeV, m =171 MeV
p
1/a=1.37 GeV, m =248 MeV
p
0 1/a=2.28 GeV, m =289 MeV
p
) 1/a=2.28 GeV, m =345 MeV
V p
e -0.002 OPAL data+DV model
G
7
7 -0.004
0.
(
r 0
1-0.006
L
-0.008
-0.01
0 0.05 0.1 0.15 0.2 0.25
2 2
Q [GeV ]
Figure2: NLOresultsforLr (m )fromanalysesoflatticeandOPAL+DVmodelversionsofD P
10 0
4. Constraints onthe p ′ and p ′′ Decay Constants
Excited I =1 pseudoscalar mesons, P, couple with strengths 2f m2 to the divergence of the
P P
flavorud axialcurrent. Theirdecayconstants, f ,entertheconventional determination ofm +m
P u d
employingsumrulesforthetwo-pointfunctionofthisdivergence[16]andarecurrentlydetermined
aspartoftheanalysis. Asimilarinternaldeterminationisrequiredintheextractionofm +m from
u s
sumrulesforthetwo-pointfunctionofthedivergenceoftheflavorusaxialcurrent. Finally,flavor-
breaking sum rules used to determine |V | from hadronic t decay [17, 18], or a combination of
us
hadronic t decayandelectroproduction cross-section data[19],encounteraproblemwiththevery
bad convergence of the OPE representation of J =0 contributions, necessitating the subtraction
of the chirally suppressed, but not totally negligible, strange excited state scalar and pseudoscalar
contributions to the differential t decay spectrum. The strange pseudoscalar subtraction relies on
theexcitedK decayconstants obtained inthesumruleanalysis.
The lattice data allows us to test the reliability of the sum rule determination of such de-
cay constants, as follows. The quantity P(Q2) ≡Q2DP (0)(Q2), which, for m =m is equal to
u d
5
SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
−Q2P (0)(Q2),isfreeofkinematicsingularities andsatisfiesaonce-subtracted dispersion relation.
A
Since P (0)(Q2) and the quantities mp , fp which determine the pion pole contribution to the dis-
A
persive representation are all measurable on the lattice, the following rearranged version of this
relation provides constraints on the continuum contribution, and hence on the excited state decay
constants, fp ′ and fp ′′,foreachpairof Q2 andsubtraction point Q20:
P(Q2)−P(Q2)+ (Q2−Q20)2fp2m2p = −(Q2−Q2) ¥ ds sr A(0)(s) . (4.1)
0 (s+Q2)(s+Q20) 0 Z9mp2 (s+Q2)(s+Q20)
SpectralpositivityensuresthattheLHSprovidesanupperboundonthecontributionsfromanysub-
setofthefullsetofexcitedpseudoscalar states. Inthenarrowwidthapproximation, thisconstraint
represents astraight lineinthe f2-f2 plane foreach pair (Q2,Q2). Thefactthat theexcited state
p ′ p ′′ 0
decayconstantsscaleasm2 canbeusedtoscaleeachsuchboundfromthemassesusedinthesim-
p
ulation downtophysicalmp . Itturnsoutthat,atpresent, onlythehigh-statistics, 1/a=2.28GeV,
mp = 289 MeV ensemble provides data sufficiently accurate for this purpose. The envelope of
the resulting set of constraint lines, scaled down to physical mp , is shown in Fig. 3. Also shown,
for comparison, are the results obtained/used in Refs. [16]. These are obviously in good agree-
ment withthe lattice constraints, leaving room forsmall additional contributions to the dispersive
representation fromyethigherexcitedpseudoscalar resonances.
0.006
Lattice bound (central)
Lattice bound (data-1s )
0.005 Lattice bound (data-1s )
PS sum rule, 5-loop D=0
) 0.004
V
e
G
(0)0.003
0
8
1
(
fp 0.002
0.001
0
0 0.001 0.002 0.003 0.004 0.005
f (GeV)
p (1300)
Figure3: Latticeconstraintsonthep ′andp ′′decayconstants
5. Acknowledgements
Thecomputations weredoneusingtheSTFC’sDiRACfacilities atSwanseaandEdinburgh.
PAB,LDD,NGandRJHaresupported byanSTFCConsolidated Grant,andbytheEUunder
6
SomecontinuumphysicsresultsfromthelatticeV-Acorrelator K.Maltman
GrantAgreementPITN-GA-2009-238353 (ITNSTRONGnet). EKwassupported bythe
ComunidadAutònomadeMadridundertheprogram HEPHACOSS2009/ESP-1473andthe
European UnionunderGrantAgreementPITN-GA-2009-238353 (ITNSTRONGnet). KM
acknowledges thehospitality oftheCSSM,UniversityofAdelaide, andsupportofNSERC
(Canada). JMZissupported bytheAustralian ResearchCouncilgrantFT100100005.
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