Irreducible Multiplets of Three-Quark Operators on the Lattice: Controlling Mixing under Renormalization

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by  Thomas Kaltenbrunner

EPJmanuscriptNo.
(willbeinsertedbytheeditor)
Irreducible Multiplets of Three-Quark Operators on the Lattice
Controlling Mixing under Renormalization⋆
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ThomasKaltenbrunner,MeinulfGo¨ckeler,andAndreasScha¨fer
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0 Institutfu¨rTheoretischePhysik,Universita¨tRegensburg,93040Regensburg,Germany
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n Received:date/Revisedversion:date
a
J
Abstract. High luminosity accelerators have greatly increased the interest in semi-exclusive and exclusive reac-
5 tions involving nucleons. The relevant theoretical information is contained in the nucleon wavefunction and can be
2
parametrizedbymomentsofthenucleondistributionamplitudes,whichinturnarelinkedtomatrixelementsofthree-

quarkoperators.ThesecanbecalculatedfromfirstprinciplesinlatticeQCD.However,onthelatticetheproblemsof
]
t operatormixingunderrenormalizationareratherinvolved.Inasystematicapproachweinvestigatethisissueindepth.
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Usingthespinorialsymmetrygroupofthehypercubiclatticewederiveirreduciblytransformingthree-quarkoperators,
l
- whichallowustocontrolthemixingpattern.
p
e
Keywords. irreducible–three-quarkoperator–renormalization–mixing–spinorial–hypercubic–lattice
h
[

1 1 Introduction malizationisavitalingredientforanylatticecalculation.This
v
paperwill focusona detailedanalysisofthe operatormixing
2
In the investigation of the internal nuclear structure, distribu- underrenormalization,whichisgenerallyconstrainedbysym-
3
tion amplitudes play an essential role. Generally, in calcula- metries.
9
tions dealing with exclusive high-energy processes, one can In the Euclideancontinuumtheory,mixingbetween oper-
3
. factorize the associated diagrams into hard and soft subpro- atorsis restrictedby their transformationpropertiesunderthe
1
cesses.Whileahardsubprocesscanbeevaluatedperturbatively symmetrygroupO .Operatorsthattransformaccordingtoin-
0 4
andischaracteristicforthereaction,thedistributionamplitudes equivalent irreducible representations of the symmetry group
8
describingthenonperturbativesoftsubprocessareuniversal[1, cannotmixunderrenormalization.However,onthelatticethis
0
: 2]. Thus, all these computations need the distribution ampli- symmetrygroupisreducedtoitsdiscretizedcounterpartH(4),
v tudes as input to produce quantitative results. Presently there whichmeansthatingeneralmoreoperatorswillparticipatein
i
X existonlymodeldependentcalculationsandtheQCDsumrule themixingprocess.In[8]agenericstudyforquark-antiquark
approach[3,4,5],whilethemethodofchoiceforthedetermi- operators was performed along these lines. We will modify
r
a nationofdistributionamplitudesfromfirstprinciplesislattice thisapproachtodealwiththehalf-integerspinassignedtoour
QCD[6,7]. three-quarkoperators.Inordertogaincontroloftherenormal-
Afterperforminganexpansionnearthelightcone,moments ization properties, we will construct irreducibly transforming
ofthesenucleondistributionamplitudesareexpressedinterms multiplets of three-quark operators with respect to the spino-
ofmatrixelementsoflocalthree-quarkoperatorsthatareeval- rialhypercubicgroupH(4).
uatedbetweenabaryonstateandthevacuumandcanbecom-
puted on the lattice. Apart from isospin symmetrization and
colorantisymmetrization,thesethree-quarkoperatorstypically
2 The Symmetry of the Hypercubic Lattice
looklike
Asthepositionsofthecovariantderivativeswithinanoperator
D ...D f (x) D ...D g (x) D ...D h (x),
µ1 µm α · ν1 νn β · λ1 λl γ donotaffectanyof the followingarguments,we will assume
(1.1)
foreaseofnotationthatallofthemactonthelastquarkfield.
wheref,g andhdenotethethreequarkfieldswithspinorin-
Given that we will work with linear combinations of the ele-
dicesα,βandγ,respectively,locatedatsomespace-timepoint
mentary three-quarkoperatorsin eq. (1.1), let us furthermore
x.
introduce tensors T(i) that represent their coefficients. Sup-
In orderto enable fully quantitativepredictionsfor exclu-
pressingthecolorindicesandomittingthecommonspace-time
sivebaryonicprocessesadetailedunderstandingofthesethree-
coordinate a local three-quark operator then generally looks
quarkoperatorsis essential. As they pick up radiativecorrec-
like:
tionsandaresubjecttomixingwithotheroperators,theirrenor-
⋆ worksupportedbyBMBF O(i) =Tα(iβ)γµ1...µnfαgβDµ1...Dµnhγ. (2.1)
2 ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice
Theregularizedbareoperatorisrelatedtoitsrenormalizedcoun- Table2.1.Symmetriesofthehypercubiclattice.
terpart (i),ren byarenormalizationmatrixZ,
O e1 e1
−
e e
(i),ren =Z (j),bare, (2.2) 2 − 2
O ijO e3 e3
−
e e
and mixingunderrenormalizationshowsup in non-vanishing 4 − 4
off-diagonalelementsofZ.Giventhelargenumberofdegrees
of freedom (note that j = 1,...,43+n), controlling mixing
is obviously a complex problem. However, by appropriately setofgeneratingrelations[10]:
choosingthe coefficientsT(i), it can be achievedthat for any
giveni the numberof bareoperatorscontributingwith a non- Ii2 =1, IiIj =IjIi tI1 =I1t,
vanishingcoefficientZ isrestrictedtoaminimum.Asmen- tI =I t, tI =I t, tI =I t,
ij 2 4 3 2 4 3
tionedintheintroduction,thisisdonebystudyingthetransfor-
γI =I , γI =I γ, γI =I γ,
1 3 2 2 4 4
mationpropertiesofthegivenoperatorsunderrotationsandre-
flectionsinspace-time.Three-quarkoperatorsthatdonottrans- γ2 =1, t3 =1, (tγ)4 =1. (2.5)
formidenticallytooneanotherdonotmix.Asanytwoopera-
The generatorsI represent inversionswhereas t and γ stand
tors (i)and (j)thatbelongtoinequivalentirreduciblerepre- j
O O for a combined reflection and interchange of the axes. Each
sentationsofH(4)fulfillthisrequirement,theirrelatedrenor-
elementG H(4)canbeexpressedasaproductofthegener-
malization matrix elements Z and Z vanish. Once all ir- ∈
ij ji ators.
reduciblytransformingmultipletsofthree-quarkoperatorsare
Forthis groupthereareall in alltwentyinequivalentirre-
known,theidenticallytransformingoperatorscanbereadoff. ducible representations. They are labeled by τn with the su-
ThenZ decomposesintoablockdiagonalformwithoneblock k
perscriptngivingthedimensionofthisrepresentationandthe
assignedtoeachsetofidenticallytransformingoperators.This
optional subscript k counting inequivalent representations of
greatlyfacilitateskeepingtrackofthemixing.
thesamedimension,ifexistent[9].Everyirreduciblerepresen-
tation is uniquelyidentified by the traces of its representation
matricesτn(G)forthegroupelements,calledcharactersχof
k
2.1 TheHypercubicGroup therepresentation:
Inthissectionweintroducethesymmetrygroupofthehyper- χn(G)= τn(G) . (2.6)
k k ii
cubic lattice (see, e.g., [9]). This so-called hypercubic group Xi
H(4)determines,howobjectswith integerspinbehaveunder
transformationsofthediscretizedspace-time. Forus,τ14isofparticularinterestbecauseits4×4-matricesde-
In terms of group theory, the hypercubic lattice can be scribe,howLorentzvectorssuchascovariantderivativestrans-
thoughtofasasetofsymmetrytransformationsofitsaxes.Let formunderthegroupaction.
e , j = 1,...,4 denote unit vectors pointing in the direction
j
of the four canonical axes and let us arrange them as shown
2.2 TheSpinorialHypercubicGroup
in Table 2.1. The symmetry group of a lattice consists of all
transformationsthatleavethesymmetryitself untouched,i.e.,
Asalreadymentionedintheintroduction,weareinterestedin
the lattice looksthesame beforeand afterthetransformation.
thetransformationpropertiesofobjectswithhalf-integerspin.
Therearetwoclassesofoperationsfulfillingthisrequest.The
The appropriate symmetry group was studied in [10] and is
firstoneistheinterchangeoftwoaxes:
calledspinorialhypercubicgroupH(4).Incontrasttoitsnon-
spinorialcounterpart,thisgrouphastocontainfurtherfeatures
(e , e ) (e , e ). (2.3)
i − i ↔ j − j likethephase-factorsforthelatticeanalogueofafullrotation.
Thiscorrespondstoexchangingtworowsinthetable.Asthere ThereforethedefiningrelationsofH(4)mustbemodifiedby
is a total of four rows, it is readily seen that these operations twistingthemwithasetofZ factors:
2
representthepermutationgroupwithfourelements,S .Invert-
inganaxisistheothersymmetryoperationonecanth4inkof: Ii2 =−1, IiIj =−IjIi tI1 =I1t,
tI =I t, tI =I t, tI =I t,
2 4 3 2 4 3
(e , e ) ( e ,e ). (2.4)
i i i i γI = I , γI = I γ, γI = I γ,
− 7→ − 1 3 2 2 4 4
− − −
Inthediagramthismeanstoflipthetwoentrieswithinonerow. γ2 = 1, t3 = 1, (tγ)4 = 1. (2.7)
− − −
ThecorrespondingsymmetryisZ .Ifoneagaintakesintoac-
2 Beyondtheirreduciblerepresentationsdirectlyinheritedfrom
countallfourrows,onearrivesatZ 4.Workingoutthecom-
2 H(4), five more irreducible representationsare found. Hence
mutationrelationsbetweentheexchangeandreflectionofaxes,
theseare“purelyspinorial”andmarkedwithanunderscorebe-
anisomorphismbetweenthe symmetrygroupofthehypercu-
biclatticeandasemidirectproductZ24⋊S4 (wreathproduct neaththeirdimension:τ14,τ24,τ8,τ112 andτ212.Herethefour-
ofZ andS )isfound.Thegrouporderis4! 24 =384. dimensionalrepresentation τ4 is importantas it describes the
2 4 1
Letusnowturntoamoreabstractapproa·ch.Thehypercu- transformationoffour-spinorsunderthegroupaction.Bycon-
bicgroupcanbedefinedbysixgeneratorst,γ,I ,...,I anda structionthegrouporderofH(4)istwicethatofH(4).
1 4
ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice 3
3 Construction of Irreducible Three-Quark Table3.1.Relationofquarkfieldswithdottedandundottedindicesto
Operators theWeylrepresentation
In this section we will explain, how irreducibly transforming Weylrepresentation ψ1 ψ2 ψ3 ψ4
three-quarkoperatorscanbeconstructed.Underthegroupac- (un)dottedindices Φ0 Φ1 Σ Σ
0˙ 1˙
tionofH(4)anoperatorisconvertedintoalinearcombination
ofotheroperators.Amajorsteponthewaytowardsirreducible chirality + +
− −
multipletsis to determine,which operatorsmay be presentin
theselinearcombinations.
Knowing the representation matrices for spinors and Lo- Thuswe can deduceirreduciblytransformingthree-quarkop-
rentzvectorsoneisinprincipleabletodeducethetransforma- erators for the latter group by constructing irreducible repre-
tionofanythree-quarkoperator(2.1)underanygroupelement sentations in SU(2) SU(2), which is accomplishedby ap-
×
G H(4).TothisendeachspinorandLorentzindexistrans- propriately symmetrizing the SU(2) indices according to the
for∈med separately with the representation matrices of τ4 and correspondingstandard Young tableaux [12]. In leading-twist
τ4,respectively,resultingintheG-transformedthree-qua1rkop- this enforces independent total symmetrization of the dotted
1 andundottedindices.
erator
(j),G transformed =G (i). (3.1) Tobespecific,letusassumethataparticularcombination
− ij
O O ofquarkchiralitiesisgiven,i.e.,thespinorindicesarechosen
However,giventhelargeamountofindependentoperators (i),
tobeeitherdottedorundotted.ThenanSO irreduciblytrans-
O 4
thiswouldyieldtransformationmatricesGjiofratherunhandy formingmultipletisconstructedasfollows:
dimension.
metArysgtrhoeusppoinfotrhiealEhuycpliedrecaunbiccognrtoinuupuimseOmb,eidrrdeedducinibtlhyetrsaynms-- fa˙gbDµ1...Dµnhc → fa˙gb(Dσ)d1e˙1...(Dσ)dne˙nhc
4
formingoperatormultipletsofthelatteroneformaclosedset → f{a˙g{b(Dσ)d1e˙1...(Dσ)dne˙n}hc},
with respect to the groupaction of H(4). In other words: the (3.3)
H(4) representationmatricesG are block-diagonalwith re-
ij
spect to multiplets of three-quark operators transforming ir- with ... denotingthe symmetrizationof the indices on the
{ }
reducibly under the continuum group.When choosing appro- samelevel.Lookingatthedottedandundottedindices,which
priatelinearcombinationswithinthesemultipletstheirblocks each can take the values zero and one, we immediately read
may however decompose into even smaller blocks under the offthatthismultipletconsistsof(n+3) (n+2)three-quark
·
operators. Operators with other chirality combinations of the
spinorialhypercubicgroup,resultinginthedesiredH(4)irre-
quarkfieldsaretreatedinthesamemanner,sothatthespaceof
duciblerepresentations.UsingthesymmetrygroupoftheEu-
elementary three-quark operators (1.1) decomposes into sub-
clideancontinuumthussubdividestheproblemofsearchingfor
spacesofSO irreduciblemultiplets.
H(4) irreducible three-quark operators in the whole operator 4
Sofaronlyfour-dimensionalrotationsweretakenintoac-
space into the task of decomposingO irreduciblemultiplets.
4 count. The link to the full symmetry group of the Euclidean
Thisreducesthedimensionoftheproblemconsiderably.
continuumO isgivenbyreflectionoperations:letrrepresent
Wewillthereforefirstderivemultipletsofirreduciblytrans- 4
some reflection in four dimensions, then O = SO rSO
formingthree-quarkoperatorsinO4.Inasecondstepaprojec- [13], which also holds for the covering gro4ups O 4an∪d SO4.
4 4
torforthedecompositionintoH(4)irreducibleoperatormul-
ThereforeanO irreduciblemultipletofthree-quarkoperators
4
tiplets is constructed.Thatfixes our choicefor the coefficient
canbeconstructedbycombininganyofthejustdeducedSO
tensorsT(i)ineq.(2.1). 4
irreduciblemultipletswithitsparitypartnertoalargerone.
In the next subsection we will exploit the fact that in the
basisoftheseO irreduciblemultipletsallrepresentationma-
3.1 IrreducibilityinSO andO 4
4 4
tricesG inequation(3.1)aresimultaneouslyblock-diagonal.
ij
Unless stated otherwise, we will focus on the leading-twist ThisfacilitatesthefurtherdecompositionintothedesiredH(4)
case from now on (i.e., twist 3). For ease of notation let us irreduciblerepresentations.
write all quark fields with dotted and undotted indices in the
chiralWeyl representation(cf.,e.g., [11]). Then a four-spinor
naturallydecomposesintotwoWeyl-spinorsofdefinitechiral-
3.2 IrreducibilityinH(4)
ity (Table 3.1) whose transformationpropertiesare character-
izedbyanSU(2)representation.Analogously,weconvertthe
Beforeweexplainhowtheactualdecompositionworks,letus
covariantderivativesto an SU(2) SU(2) representationby
contracting them with the Pauli m×atrices σ . Then the whole findoutwhichH(4)irreduciblerepresentationsmayshowup.
µ
three-quarkoperator transformsas a direct productof SU(2) AsstatedinSections2.1and2.2,thecovariantderivativesand
representations.Now,thereexistsa homomorphismthatlinks quark fields transform according to τ14 and τ14, respectively.
theirreduciblerepresentationsofSU(2) SU(2)to thoseof Thereforethethree-quarkoperatorin(1.1)transformsasadi-
× rectproductoftheserepresentations:
SO :
4
SU(2) SU(2) SO4. (3.2) τ4 τ4 τ4 τ4 τ4. (3.4)
× ≃ 1 ⊗ 1 ⊗ 1 ⊗···⊗ 1 ⊗ 1
4 ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice
This product is reducible. Knowing the characters χα for a Here Gα denotesthe lk elementof the representationmatrix
lk
givenirreduciblerepresentationτα,itcanbedecomposedwith τα(G). Acting with P˜α on the set in question results in m
11
thehelpoftheidentity independentthree-quark operators, where m is the multiplic-
ity of the representation τα. If we now apply the projectors
τα τβ = cγτγ, (3.5) P˜α, j = 1,...,d to these m operators separately we
⊗ Xγ g{en1ejratemirreducibleαm}ultipletsofthree-quarkoperators.That
results in the requested separation of the m equivalent irre-
where
duciblemultiplets.
c = 1 χγ(G) χα(G) χβ(G) (3.6) After performing these steps all irreducibly transforming
γ ∗
H(4) · · three-quarkoperatorsofthespinorialhypercubicgroupareknown.
X
| |G H(4)
∈
and H(4) denotesthegrouporder.Applyingthisformulait-
| | 4 Three-Quark Operators and
eratively,we derivethe followingcontentofH(4)irreducible
Renormalization
multiplets for three-quark operators with zero to two deriva-
tives(includinghighertwist):
In the previous section we have explained how multiplets of
zeroderivatives: τ14⊗τ14⊗τ14 = H(4) irreducibly transforming three-quark operators of lead-
5τ4 τ8 3τ12, ing twist can be constructed. Starting from different Young
1 ⊕ ⊕ 1 tableaux in the SO case the very same concept applies to
4
onederivative: τ4 τ4 τ4 τ4 = highertwist.TheresultsaresummarizedinAppendixA.There
1 ⊗ 1 ⊗ 1 ⊗ 1
8τ4 4τ8 12τ12 4τ12, we give a full list of all leading-twist irreducible three-quark
1 ⊕ ⊕ 1 ⊕ 2 operatorswithuptotwoderivativesandallhigher-twistopera-
twoderivatives: τ4 τ4 τ4 τ4 τ4 = torswithoutderivatives.
1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1
20τ4 3τ4 18τ8 41τ12 23τ12. In the following we want to discuss the consequencesfor
1 ⊕ 2 ⊕ ⊕ 1 ⊕ 2 the mixing propertiesof the lattice operatorsunderrenormal-
(3.7)
ization. It was already stated in Section 2 that mixing is pro-
As expected, only spinorial representations show up after the hibited between three-quark operators belonging to inequiva-
reductionprocess. lentirreduciblerepresentations.InTable4.1wesortourresults
We nowproceedwith the decompositionof the O multi- according to their representation and mass dimension. Here
4
plets from the previoussection. Therefore,their 768 transfor- (i) denotesthei-thoperatorwithinthej-thH(4)irreducible
mation matrices Gij in (3.1) must be known explicitly.To be mOujltiplet. Then the above statement means that renormaliza-
more precise: the diagonal blocks of these matrices are suffi- tiononlymixesoperatorswithinthesamerow.Moreprecisely:
cient. we sort the operatorswithin any multiplet in such a way that
Foreverygroupelementthismatrixblockcanbeconstruc- their transformationmatricesundergroupaction are identical
tedbytransformingeveryquarkfieldandderivativeofanop- forequivalentrepresentations.Thus,thei-thoperatorsinmul-
erator separately as explained above and writing the result in tiplets of equivalent representationstransform identically and
terms of the original operators. The coefficients involved are thereforemixonlywitheachother.
identified with the representation matrix elements Gij. Then, Whenworkingwithdimensionalregularizationinthecon-
with the knowledgeof the charactersofthe irreduciblerepre- tinuumtheory,mixingisalsoforbiddenforoperatorswithdif-
sentations, a projector is constructed. When applied to an O4 ferentmassdimensions,i.e.,differentcolumnscannotmix.On
multipletit projectsouta usuallysmaller multipletthattrans- the lattice, however, this last statement is not valid anymore.
formsaccordingtothedesiredirreducibleH(4)representation Due to the existence of a dimensionful quantity, namely the
(see,e.g.,[14]): lattice spacing a, lower-dimensional operators may mix with
higher-dimensionaloneswithcoefficientsproportionaltopow-
d
Pα = α χα(G)∗ G. (3.8) ersof1/a,e.g.:
H(4) ·
X
| |G H(4) 1
∈ (i),ren =Z (j),bare+Z bare,lowerdim. (4.1)
ij ′
Here d denotes the dimension of the irreducible representa- O O · a ·O
α
tiontobeprojectedout.
Inpracticeitprovesdifficulttoproperlyextracttherenormal-
Some O irreduciblemultipletscontainseveralequivalent
4 izationcoefficientsZ ofmixinglower-dimensionaloperators.
H(4) irreducible representations τα. Then, the action of Pα ′
Thereforethis situation should be avoided whereverpossible.
yieldsasetofthree-quarkoperatorsthatactuallycontains smal-
To thisend onecan try to restrictoneself to those representa-
ler multiplets, each closed under the group action on its own
tionsthatdonotpossesslower-dimensionalcounterpartssuch
andirreducible.Toseperatethesemultipletsasecondprojector asτ12(τ4)forthree-quarkoperatorswithone(two)derivatives.
P˜α isintroduced(see,e.g.,[15]): 2 2
lk We can summarize these statements: the i-th operator of
a multiplet may mix with any i-th operator of the same or
d
P˜lαk = α (Gαlk)∗G. (3.9) lower dimension from the same row in Table 4.1. All opera-
H(4)
X torswithinonemultipletsharethesamerenormalizationcoef-
| |G H(4)
∈
ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice 5
Table4.1. Irreducibly transforming multipletsof three-quark opera- derivatives,asmentionedabove:
torssortedbytheirmass-dimension.
(i) (i) (i)
, , ,
Off... Ofg... Ofh...
dimension9/2 dimension11/2 dimension13/2
(i) , (i) , (i) . (4.3)
Ogg... Ogh... Ohh...
(0derivatives) (1derivative) (2derivatives)
Ofcoursetherearethreeclassesofoperatorswithonederiva-
(i), tive:
O1 (i) ,
τ4 (i), (i), ODD1 (i) , (i), (i) . (4.4)
1 O2 O3 (i) , (i) Of... Og... Oh...
(i), (i) ODD2 ODD3 Referringto the restrictionsof mixingunder renormaliza-
O4 O5
tion,alloperatorsstillobeythepatterndisplayedinTable4.1.
(i) ,
τ4 ODD4 One only has to keep in mind that a multiplet-index D may
2
(i) , (i) actuallyrepresentan f,g orh as explainedabove.Thiscom-
ODD5 ODD6 pletes our study of the symmetry properties for leading-twist
(i) , three-quark operators with up to two derivatives. In the next
τ8 (i) (i) ODD7
O6 OD1 (i) , (i) sectionwegivefurtheridentitiesforthespecialcaseofisospin
ODD8 ODD9 1/2symmetrizedoperators.
(i), (i), (i) , (i) ,
τ12 O7 OD2 ODD10 ODD11
1
O8(i), O9(i) OD(i)3, OD(i)4 OD(i)D12, OD(i)D13 5 Isospin Symmetrized Operators
(i) ,
(i), (i), ODD14 Ifoneisinterestedinnucleonphysics,onehastocareaboutthe
τ12 OD5 OD6 (i) , (i) , appropriateisospinsymmetrizationoftheoperatorsused.Due
2 (i), (i) ODD15 ODD16 tothepresenceoftwoequalquarkflavors,identitieswillthen
OD7 OD8 (i) , (i)
ODD17 ODD18 showupwhichreducethenumberofindependentthree-quark
operators.Wewanttobrieflydiscussthisinthefollowing.Our
resultsaresummarizedinAppendixB.
ficients. Hence it is sufficientto renormalizeoneoperatorper There are two possible symmetry classes for isospin 1/2:
multipletonly,e.g.,i=1. mixed antisymmetric, denoted by MA in the following, and
Recall, however, that without loss of generality we have mixedsymmetric(MS).Dealingwith threequarksofisospin
discussed three-quarkoperators with all derivativesacting on I = 1/2, m = 1/2, one can first couple two of them to
I
±
the last quark. That was possible, because the actual position eitherm =0orm =1.Fortheprotonthethirdquarkfieldis
I I
of the derivativehas no influenceonthe transformationprop- thenaddedinsuchawaythattheresultantthree-quarkoperator
erties and thus on the classification for renormalization.Mix- hasI =1/2,m =+1/2:
I
ingbetweenoperatorswithmerelyinterchangedpositionofthe
derivativesisnotprohibited.Hence,itisimportanttonotehow 2 1
MS = uud + (udu + duu ),
theadditionaloperatorsnotlistedexplicitlyinAppendixAare | i −r3| i r6 | i | i
generated.Foragivenoperatoronlythepositionofthederiva-
1
tivesischangedtoanyquarkfieldwithouttouchingthespinor MA = (udu duu ). (5.1)
andvectorindicesquoted.Thisyieldsthreetimesasmanypos- | i r2 | i−| i
siblymixingmultipletsinthecaseofoneandsixtimesasmany Theirreducibleoperatorsdiscussedintheprevioussectionand
inthecaseoftwoderivatives. listed in Appendix A are converted to isospin 1/2 operators
Letusintroduceanotationthatcharacterizestheoperators whenreplacingthe f,g andh quarkfieldsbythe appropriate
uniquely.WereplacetheDinthesubscriptofamultipletwith MS or MA linear combinationsof uud, udu and duu given
anf,ifthederivativeactsonthefirstquark,ag(h)ifitactson above. The spinor and vector indices as well as the positions
thesecond(third)quark.E.g.,theoperator (4) withderiva- ofthe covariantderivativesremainunchanged.Letus givean
tivesactingonthefirstandsecondquarkthOenDDlo1o7kslike: exampleforthecaseofanMAsymmetrization:
5i 3
Of(4g)17 = 45√i6(cid:18)35(Dσ){0{0˙f0(Dσ)00˙}g0h0} Of(4g)1,7MA = 8√3(cid:18)5(Dσ){0{0˙u0(Dσ)00˙}d0u0}
3
−(Dσ){1{0˙f1(Dσ)10˙}g1h0} − 5(Dσ){0{0˙d0(Dσ)00˙}u0u0}
−2·(Dσ){0{1˙f1(Dσ)01˙}g1h0}(cid:17). (4.2) −(Dσ){1{0˙u1(Dσ)10˙}d1u0}
+(Dσ) 1 d1(Dσ)1 u1u0
Duetothetotalsymmetrizationofthespinorindicesaninter- { {0˙ 0˙} }
changeofthetwoderivativeshasnoeffectontheoperator,i.e., −2·(Dσ){0 1˙u1(Dσ)01˙ d1u0}
(4) = (4) .Thuswecanalwaysordertheindicesf,gand { }
Ohgaflp17habeOticfagl1l7yleavingonlysixclassesofoperatorswithtwo +2·(Dσ){0{1˙d1(Dσ)01˙}u1u0}(cid:17). (5.2)
6 ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice
Table5.1. Irreducibly transforming multipletsof three-quark opera- identities among the MA operators. That leads to a minimal
torswithisospin1/2sortedbytheirmassdimension. setoflinearlyindependentthree-quarkoperatormultipletswith
isospin 1/2. These multiplets are summarized in Table 5.1,
dimension9/2 dimension11/2 dimension13/2 where, just as in Table 4.1, the allowed mixings under renor-
malizationcanbereadoff.
(0derivatives) (1derivative) (2derivatives)
Asalloperatorswithinonemultipletsharethesamerenor-
(i),MA, (i),MA, malizationcoefficients,onlyanorderoftendifferentrenormal-
τ4 O1 Off1 ization matricesof dimensioneight by eightor lower need to
1
(i),MA (i),MA, (i),MA be calculated. The nonperturbative evaluation of these coef-
O3 Off2 Off3
ficients for two flavors of clover fermions is in progress and
(i),MA, (i),MA,
Off4 Off5 willyieldafullsetofrenormalizationconstantsforisospin1/2
τ4 (i),MA, (i),MA, symmetrizedoperatorsofleadingtwistwithuptotwoderiva-
2 Off6 Ogh4 tives.Theseupcomingresultswillallowustorenormalizethe
(i),MA, (i),MA firstfewmomentsoftheprotondistributionamplitude.Tothis
Ogh5 Ogh6
end, one relates matrix elements of identically transforming
(i),MA, (i),MA,
Off7 Off8 three-quark operators, e.g., = (i),MA, = (i),MA
τ8 Of(i1),MA Of(if),9MA, Og(ih)7,MA, and O3 = Of(i7),MA with i fiOx1ed, toOmf5omentsOΦ21, Φ2Oafn6d Φ3
(i),MA, (i),MA oftheprotondistributionamplitude.Schematicallywecanex-
Ogh8 Ogh9 pressthe renormalizedmomentsin terms ofthe renormalized
(i),MA, (i),MA, operatorsby
(i),MA, Off10 Off11
Of2 (i),MA, (i),MA, Φren 0 ren P ,
τ12 (i),MA (i),MA, Off12 Off13 1 ∼h |O1 | i
1 O7 OOff(3i4),MA Og((ihi))1,,M0MAA,,Og((ihi))1,,M1MAA, ΦΦr3r2eenn ∼∼hh00||OO32rreenn||PPii,,
Ogh12 Ogh13
where P denotes a proton state of definite momentum (for
(i),MA, (i),MA, (i),MA, | i
Of5 Off14 Off15 detailssee [16]). With the renormalizationand mixingcoeffi-
τ12 Of(i6),MA, Of(if),1M6A, Of(if)1,M7A, acrierinvtesoatftohuerftohlrleoew-qinugarrkeloaptieornatobrest,wOeeirennth=eZreinjoOrjm,awliezefidnaalnlyd
2
(i),MA, (i),MA, (i),MA, thebaremoments:
Of7 Ogh14 Ogh15
(i),MA (i),MA, (i),MA Φren =Z Φ +Z Φ +Z Φ ,
Of8 Ogh16 Ogh17 1 11 1 12 2 13 3
Φren =Z Φ +Z Φ +Z Φ ,
2 21 1 22 2 23 3
Φren =Z Φ +Z Φ +Z Φ .
The presence of two u quarks leads to operator identities 3 31 1 32 2 33 3
when use is made of the anticommutation relation for Grass- Thisemphasizesoncemoretheimportanceofthedetailedstudy
mann variables and the symmetry of the coefficient tensors ofoperatormixingunderrenormalizationonthelatticethatwe
T(i). Again we want to clarify the procedureby a simple ex- havepresentedhere.
ample.TheMAthree-quarkoperatorswithoutderivativeread
aftercolorantisymmetrization:
6 Summary and Outlook
1
(i),MA =T(i) (uadbuc daubuc)ǫ
O αβγ√2 α β γ − α β γ abc Three-quarkoperatorsplayanimportantroleinthelatticede-
1 terminationofnon-perturbativecontributionstohardexclusive
= √2(Tα(iβ)γ −Tβ(iα)γ)·uaαdbβucγ ·ǫabc. (5.3) processesinvolvingbaryons.Inordertogetcontinuumresults
fromcalculationsusingtheseoperators,theymustberenormal-
In AppendixA.1 we find thatfor the operators (i) and (i) izedanditiscrucialtostudythemixingbehavior.
O7 O8 Here we have investigated the constraints imposed by
the role of the spinor indices on the first and second quark is
grouptheory.Thehypercubicgroupanditsspinorialanalogue
exchanged,i.e.,
were used to determine the behavior of three-quarkoperators
T(i) =T(i) . (5.4) under transformationsof the discretized space-time. In a first
7,αβγ 8,βαγ stepwesubstantiallyreducedthedimensionsofthe occurring
Whenusingthisrelationin(5.3)thefollowingidentitybetween representation matrices by studying the continuum behavior.
thetwoisospinsymmetrizedmultipletsisderived: Then we used a projectorto derivethe multiplets of H(4) ir-
reduciblytransformingthree-quarkoperators.Whengrouping
(i),MA = (i),MA. (5.5) themaccordingtotheirrepresentationandmassdimension,the
O7 −O8 possiblemixingpatternscanbereadoff.Thisprovidestheba-
A list of all identities inducedby the isospin symmetriza- sisfortherenormalizationofisospin-symmetrizedthree-quark
tion is givenin AppendixB. Therewe systematicallyexpress operatorsneededfortheevaluationofnucleondistributionam-
allMS operatorsintermsofMAoperatorsandthengiveall plitudes,whichwillbepublishedinaforthcomingpaper.
ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice 7
A Irreducibly Transforming Three-Quark combinationofthethreequarkfields, (i)and (i)comefrom
O4 O5
Operators + +and++ combinations,respectively(cf.(3.3)).
− −
Weproceedwiththeleading-twistoperators.Theoperators
IdnuctihbilsyAtrpapnesnfodrimxiwngemlisutlttihpeletesxpolficthitrefeo-rqmuaorfktohpeerHat(o4r)s.iTrrhee- aOc6(cio)rcdoinngtationtτh8r:eequarkfieldsofequalchiralityandtransform
operatorsareconstructedsuchthatundergroupactionofH(4)
anyi-thoperatorwithinonemultiplettransformsidenticallyto O6(1) =f0g0h0,
the i-th operatorof another multiplet belongingto an equiva- 1
(2) = (f0g0h1+f0g1h0+f1g0h0),
lentrepresentation. O6 √3
Ondemand,interestedgroupsmayalsoreceivethecoeffi-
1
cienttensorsinelectronicformtofacilitatetheimplementation (3) = (f0g1h1+f1g0h1+f1g1h0),
ofthecorrespondingoperators. O6 √3
(4) =f1g1h1,
O6
(5) =f g h ,
A.1 OperatorswithoutDerivatives O6 0˙ 0˙ 0˙
1
(6)
= (f g h +f g h +f g h ),
For three-quark operators without derivatives we present the O6 √3 0˙ 0˙ 1˙ 0˙ 1˙ 0˙ 1˙ 0˙ 0˙
fullsetofirreducibleoperators,includingthoseofnon-leading 1
(7)
twist(cid:16)O1(i),O2(i),O3(i),O4(i)and O5(i)(cid:17).Thespinorindicesare O6 = √3(f0˙g1˙h1˙ +f1˙g0˙h1˙ +f1˙g1˙h0˙),
giveninthechiralWeylrepresentation. (8) =f g h . (A.4)
Thefirstfivemultipletsbelongtotheirreduciblerepresen- O6 1˙ 1˙ 1˙
tationτ14: Finally,therearethreemoremultipletsthatbelongtoτ12:
1
1
(1) = (f1g0h0 2 f0g1h0+f0g0h1), (1) =f0g h ,
O1 √6 − · O7 0˙ 0˙
1
O1(2) = √16(2·f1g0h1−f0g1h1−f1g1h0), O7(2) = √2(f0g0˙h1˙ +f0g1˙h0˙),
(3) = 1 (f g h 2 f g h +f g h ), O7(3) =f0g1˙h1˙,
O1 √6 1˙ 0˙ 0˙ − · 0˙ 1˙ 0˙ 0˙ 0˙ 1˙ (4) =f1g h ,
1 O7 0˙ 0˙
O1(4) = √6(2·f1˙g0˙h1˙ −f0˙g1˙h1˙ −f1˙g1˙h0˙), (A.1) O7(5) = √12(f1g0˙h1˙ +f1g1˙h0˙),
(6) =f1g h ,
1 O7 1˙ 1˙
(1) = (f1g0h0+f0g1h0 2 f0g0h1), (7) =f g0h0,
O2 √6 − · O7 0˙
1
O2(2) = √16(2·f1g1h0−f0g1h1−f1g0h1), O7(8) = √2(f0˙g0h1+f0˙g1h0),
1 (9) =f g1h1,
(3) = (f g h +f g h 2 f g h ), O7 0˙
O2 √6 1˙ 0˙ 0˙ 0˙ 1˙ 0˙ − · 0˙ 0˙ 1˙ (10) =f g0h0,
1 O7 1˙
O2(4) = √6(2·f1˙g1˙h0˙ −f0˙g1˙h1˙ −f1˙g0˙h1˙), (A.2) O7(11) = √12(f1˙g0h1+f1˙g1h0),
(12) =f g1h1. (A.5)
1 O7 1˙
(1) = (f0g h f0g h ),
O3 √2 0˙ 1˙ − 1˙ 0˙ The operators (i) ( (i)) are chirality partners of (i).
(2) = 1 (f1g h f1g h ), TheyfollowwheneOxc8hanOgi9ngtheindexonquarksoneandOt7wo
O3 √2 0˙ 1˙ − 1˙ 0˙ (three).
1
(3) = (f g0h1 f g1h0),
O3 √2 0˙ − 0˙
A.2 OperatorswithoneDerivative
1
(4) = (f g0h1 f g1h0). (A.3)
O3 √2 1˙ − 1˙ We list the operators with one and two derivatives using the
dottedand undottedindices forthe quarkfieldsas introduced
The operatorsO4(i) (O5(i)) are constructedfrom O3(i) by inter- in Sec. 3.1 and denote separate total symmetrization in the
changeoftheindicesonthequarksf andg (h).Theyarechi- (un)dottedindicesbycurlybrackets.Theproductofthecovari-
ralitypartners:whereas (i) originatesfroma ++chirality ant derivatives with the Pauli matrices reads in the Euclidean
O3 −
8 ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice
formulation The last four multiplets of operators transform according
to the irreducible representation τ12. The operators (i) are
(Dσ)00˙ =+D1−iD2, constructedfromquarkchiralities−2 ++: OD5
(Dσ)0 = D +iD ,
1˙ − 3 4 (1) = 1 f g{0(Dσ)0 h0} 3 f g{1(Dσ)1 h0},
(Dσ)10˙ =−D3−iD4, OD5 2√2 {0˙ 0˙} − 2√2 {1˙ 1˙}
(Dσ)11˙ =−D1−iD2. (A.6) OD(25) = 2√32f{0˙g{1(Dσ)00˙}h0}− 2√12f{1˙g{1(Dσ)11˙}h1},
3 1
The eight operators OD(i)1 belong to the irreducible repre- OD(35) = 2√2f{0˙g{1(Dσ)10˙}h0}− 2√2f{1˙g{0(Dσ)01˙}h0},
sentationτ8 andareconstructedfromthreequarkswithequal
1 3
chiralities: (4) = f g{1(Dσ)1 h1} f g{1(Dσ)0 h0},
OD5 2√2 {0˙ 0˙} − 2√2 {1˙ 1˙}
OD(11) =+12 ·`f{0˙g0˙(Dσ)10˙h0˙}+f{1˙g1˙(Dσ)11˙h1˙}´, OD(55) =f{1˙g{0(Dσ)00˙}h0},
OD(21) =−√3·f{1˙g1˙(Dσ)00˙h0˙}, OD(65) =f{1˙g{1(Dσ)10˙}h1},
1 3
OD(31) =+√3·f{1˙g1˙(Dσ)10˙h0˙}, OD(75) = 2√2f{0g{0˙(Dσ)0}0˙h0˙}− 2√2f{1g{1˙(Dσ)1}1˙h0˙},
1
OD(41) =−2 ·`f{0˙g0˙(Dσ)00˙h0˙}+f{1˙g1˙(Dσ)01˙h1˙}´, OD(85) = 2√32f{0g{1˙(Dσ)0}0˙h0˙}− 2√12f{1g{1˙(Dσ)1}1˙h1˙},
1
(5) = f{0g0(Dσ)0 h0}+f{1g1(Dσ)1 h1} , 3 1
OD1 −2 ·“ 1˙ 1˙ ” OD(95) = 2√2f{0g{1˙(Dσ)0}1˙h0˙}− 2√2f{1g{0˙(Dσ)1}0˙h0˙},
(6) =+√3 f{1g1(Dσ)0 h0},
OD1 · 0˙ (10) = 1 f{0g (Dσ)0} h 3 f{1g (Dσ)1} h ,
OD(71) =−√3·f{1g1(Dσ)01˙h0}, OD5 2√2 {1˙ 1˙ 1˙}− 2√2 {1˙ 0˙ 0˙}
OD(81) =+12 ·“f{0g0(Dσ)00˙h0}+f{1g1(Dσ)10˙h1}”. (A.7) OD(151) =f{1g{0˙(Dσ)0}0˙h0˙},
(12) =f{1g (Dσ)0} h . (A.9)
OD5 {1˙ 1˙ 1˙}
The twelve operators (i) generate a τ12 irreduciblerep-
resentationandarisefromOquDa2rkchiralities−1 ++: tOheD(i)fi6r(sOtaD(in)7d)sreecsounltdf(rtohmirdO)D(qiu)5abrky.iFnitnearlclhyawngeinhgavteh:e indices on
√3
OD(12) =−2√2 ·(f{0˙g{1(Dσ)00˙}h0}+f{1˙g{1(Dσ)11˙}h1}), OD(18) =+√2f{1˙g1˙(Dσ)11˙h0˙},
OD(22) =√3·f{1˙g{1(Dσ)00˙}h0}, OD(28) =−√2f{1˙g0˙(Dσ)00˙h0˙},
(3) = √3 (f g{1(Dσ)1 h1}+f g{1(Dσ)0 h0}), OD(38) =+√2f{1˙g0˙(Dσ)10˙h0˙},
OD2 −2√2 · {0˙ 0˙} {1˙ 1˙} (4) = √2f g (Dσ)0 h ,
OD8 − {1˙ 1˙ 1˙ 0˙}
OD(42) = 2√√32 ·(f{0˙g{1(Dσ)10˙}h0}+f{1˙g{0(Dσ)01˙}h0}), OD(58) =+21f{1˙g1˙(Dσ)11˙h1˙}− 21f{0˙g0˙(Dσ)10˙h0˙},
OD(52) =−√3f{1˙g{1(Dσ)10˙}h0}, OD(68) =+21f{0˙g0˙(Dσ)00˙h0˙}− 21f{1˙g1˙(Dσ)01˙h1˙},
OD(62) = 2√√32 ·(f{0˙g{0(Dσ)00˙}h0}+f{1˙g{1(Dσ)11˙}h0}), OD(78) =+√2f{1g1(Dσ)11˙h0},
(8) = √2f{1g0(Dσ)0 h0},
√3 OD8 − 0˙
OD(72) = 2√2 ·(f{0g{1˙(Dσ)0}0˙h0˙}+f{1g{1˙(Dσ)1}1˙h1˙}), OD(98) =+√2f{1g0(Dσ)01˙h0},
OD(82) =−√3·f{1g{1˙(Dσ)0}0˙h0˙}, OD(180) =−√2f{1g1(Dσ)10˙h0},
OD(92) = 2√√32 ·(f{0g{1˙(Dσ)0}1˙h1˙}+f{1g{1˙(Dσ)1}0˙h0˙}), OD(181) =+21f{1g1(Dσ)11˙h1}− 21f{0g0(Dσ)01˙h0},
1 1
(10) = √3 (f{0g (Dσ)0} h +f{1g (Dσ)1} h ), OD(182) =+2f{0g0(Dσ)00˙h0}− 2f{1g1(Dσ)10˙h1}. (A.10)
OD2 −2√2 · {1˙ 1˙ 0˙} {0˙ 0˙ 0˙}
(11) =√3 f{1g (Dσ)0} h ,
OD2 · {1˙ 1˙ 0˙} A.3 OperatorswithtwoDerivatives
√3
(12) = (f{0g (Dσ)0} h +f{1g (Dσ)1} h ).
OD2 −2√2 · {0˙ 0˙ 0˙} {1˙ 1˙ 0˙} Here we display irreducible multiplets of three-quark opera-
(A.8) tors with two covariant derivatives. As stated in the text, the
positionsofthederivativesdonotinfluencethetransformation
The chirality partners (i) ( (i)) are derived from (i) by properties.Henceonecanproducefurthermultipletsbyassign-
OD3 OD4 OD2
exchangingtheindicesassignedtothef quarkwiththoseofg ingthe derivativesto anyquarkfield one likes. Theycan also
(h). actontwodifferentquarks.
ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice 9
Thefirstthreemultipletstransformaccordingtoτ14: OD(7D)7=√43f{1g{0˙(Dσ)00˙(Dσ)0}0˙h0˙}+ √43f{1g{1˙(Dσ)01˙(Dσ)0}1˙h1˙}
OD(1D)1=+ 32f{1g{1˙(Dσ)01˙(Dσ)0}0˙h0˙}+ 14f{1g{0˙(Dσ)10˙(Dσ)1}0˙h0˙} + √23f{1g{1˙(Dσ)11˙(Dσ)1}0˙h0˙},
+ 14f{1g{1˙(Dσ)11˙(Dσ)1}1˙h1˙}, OD(8D)7=23f{1g{1˙(Dσ)01˙(Dσ)0}0˙h0˙}− 14f{1g{0˙(Dσ)10˙(Dσ)1}0˙h0˙}
OD(2D)1=− 14f{0g{0˙(Dσ)00˙(Dσ)0}0˙h0˙}− 14f{0g{1˙(Dσ)01˙(Dσ)0}1˙h1˙} − 14f{1g{1˙(Dσ)11˙(Dσ)1}1˙h1˙}. (A.13)
3
− 2f{1g{1˙(Dσ)11˙(Dσ)0}0˙h0˙}, Oncemore, (i) ( (i) )areobtainedafterexchangeofthe
ODD8 ODD9
OD(3D)1=+ 32f{1˙g{1(Dσ)10˙(Dσ)00˙}h0}+ 14f{1˙g{0(Dσ)01˙(Dσ)01˙}h0} indeTxhoenntehxetfiorpsetrqautoarrkmwulittihptlhetastboenlothnegsteocothnedi(rtrheidrudc)iqbulearrke.p-
+ 41f{1˙g{1(Dσ)11˙(Dσ)11˙}h1}, resentationτ112:
OD(4D)1=− 14f{0˙g{0(Dσ)00˙(Dσ)00˙}h0}− 41f{0˙g{1(Dσ)10˙(Dσ)10˙}h1} OD(1D)10=− 23f{1g{1˙(Dσ)00˙(Dσ)0}0˙h0˙}− 12f{1g{1˙(Dσ)11˙(Dσ)1}1˙h0˙},
3
− 2f{1˙g{1(Dσ)11˙(Dσ)00˙}h0}. (A.11) OD(2D)10=2√12“f{1g{0˙(Dσ)10˙(Dσ)1}0˙h0˙}−f{1g{1˙(Dσ)11˙(Dσ)1}1˙h1˙}”,
TheoperatorsOD(i)D2(OD(i)D3)aregeneratedbyexchangingthe OD(3D)10=32f{1g{1˙(Dσ)01˙(Dσ)0}1˙h0˙}+ 21f{1g{1˙(Dσ)10˙(Dσ)1}0˙h0˙},
indexonthefirstwiththatonthesecond(third)quarkfield.
OD(4D)10=12f{0g{1˙(Dσ)01˙(Dσ)0}1˙h0˙}+ 23f{1g{1˙(Dσ)10˙(Dσ)0}0˙h0˙},
OD(1D)4=√23f{0˙g{1(Dσ)10˙(Dσ)00˙}h0}− √43f{1˙g{0(Dσ)01˙(Dσ)00˙}h0} OD(5D)10=2√12“f{0g{1˙(Dσ)01˙(Dσ)0}1˙h1˙}−f{0g{0˙(Dσ)00˙(Dσ)0}0˙h0˙}”,
− √43f{1˙g{1(Dσ)11˙(Dσ)10˙}h1}, OD(6D)10=− 12f{0g{1˙(Dσ)00˙(Dσ)0}0˙h0˙}− 32f{1g{1˙(Dσ)11˙(Dσ)0}1˙h0˙},
OD(2D)4=√43f{1˙g{0(Dσ)00˙(Dσ)00˙}h0}+ √43f{1˙g{1(Dσ)10˙(Dσ)10˙}h1} OD(7D)10=− 23f{1˙g{1(Dσ)00˙(Dσ)00˙}h0}− 12f{1˙g{1(Dσ)11˙(Dσ)11˙}h0},
− √23f{1˙g{1(Dσ)11˙(Dσ)01˙}h0}, OD(8D)10=2√12“f{1˙g{0(Dσ)01˙(Dσ)01˙}h0}−f{1˙g{1(Dσ)11˙(Dσ)11˙}h1}”,
OD(3D)4=√23f{0g{1˙(Dσ)01˙(Dσ)0}0˙h0˙}− √43f{1g{0˙(Dσ)10˙(Dσ)0}0˙h0˙} OD(9D)10=32f{1˙g{1(Dσ)10˙(Dσ)10˙}h0}+ 21f{1˙g{1(Dσ)01˙(Dσ)01˙}h0},
− √43f{1g{1˙(Dσ)11˙(Dσ)0}1˙h1˙}, OD(1D0)10=12f{0˙g{1(Dσ)10˙(Dσ)10˙}h0}+ 23f{1˙g{1(Dσ)01˙(Dσ)00˙}h0},
OD(4D)4=√43f{1g{0˙(Dσ)00˙(Dσ)0}0˙h0˙}+ √43f{1g{1˙(Dσ)01˙(Dσ)0}1˙h1˙} OD(1D1)10=2√12“f{0˙g{1(Dσ)10˙(Dσ)10˙}h1}−f{0˙g{0(Dσ)00˙(Dσ)00˙}h0}”,
1 3
− √23f{1g{1˙(Dσ)11˙(Dσ)1}0˙h0˙}. (A.12) OD(1D2)10=− 2f{0˙g{1(Dσ)00˙(Dσ)00˙}h0}− 2f{1˙g{1(Dσ)11˙(Dσ)10˙}(Ah.01}4.)
Again, (i) ( (i) ) result from (i) by exchanging the Theoperators (i) ( (i) ) resultuponinterchangingthe
indexonOtDheDf5wOithDDth6atontheg(h)qOuDarDk4field.Theybelongto indicesonf anOdDgD(h11). ODD12
τ4,whereasthefollowingthreemultipletstransformaccording
to2τ8: OD(1D)13=4√√52“−2·f{1g1(Dσ)0{0˙(Dσ)00˙}h0}
OD(1D)7=41f{0˙g{0(Dσ)00˙(Dσ)00˙}h0}+ 41f{0˙g{1(Dσ)10˙(Dσ)10˙}h1} −f{0g0(Dσ)0{1˙(Dσ)01˙}h0}−f{1g1(Dσ)1{1˙(Dσ)11˙}h0}”,
− 23f{1˙g{1(Dσ)11˙(Dσ)00˙}h0}, OD(2D)13=+√√55f{1g1(Dσ)0{1˙(Dσ)00˙}h0},
OD(2D)7=√23f{0˙g{1(Dσ)10˙(Dσ)00˙}h0}+ √43f{1˙g{0(Dσ)01˙(Dσ)00˙}h0} OD(3D)13=4√2“−f{0g0(Dσ)0{0˙(Dσ)00˙}h0}
OD(3D)7=√++43√√f42{331˙ffg{{{11˙˙0gg({{D11((σDD)0σσ0˙))(11D11˙˙((σDD)0σσ0˙))}10h01˙˙0}}}hh+10}},,√43f{1˙g{1(Dσ)10˙(Dσ)10˙}h1} OODD((45DD))1133==√4−−√√5ff5f2{{{11“1ggg−101((2(DDD·σσfσ)){)1011{{g{01˙˙11˙((((DDDDσσσσ))))101001˙˙{0˙}}0}˙hh(hD000}}}σ−−,)02f0˙}{·h1fg0{1}1(gD1σ(D)1σ{)1˙0({D1˙σ(D)1σ1˙)}0h1˙1}}h”0},”,
OD(4D)7=32f{11˙g{1(Dσ)10˙(Dσ)00˙}h0}−41f{1˙g{0(Dσ)01˙(Dσ)01˙}h0} OD(6D)13=4√√52“−·f{1g0(Dσ)0{0˙(Dσ)00˙}h0}
OD(5D)7=41−f{340fg{{1˙0˙g({D1(σD)0σ0˙)(1D1˙(σD)0σ})0˙1h1˙0}˙}h+1},14f{0g{1˙(Dσ)01˙(Dσ)0}1˙h1˙} OD(7D)13=4−√√f52{1“g−1(2D·σf{)11˙g{01˙˙((DDσσ))1{00˙}0˙h(D1}σ−)0}20˙·hf0˙{}1g1(Dσ)1{1˙(Dσ)01˙}h0}”,
√−32f{1g{1˙(Dσ)11˙(Dσ)0}0˙h0˙}, √3 −f{0˙g0˙(Dσ){10˙(Dσ)1}0˙h0˙}−f{1˙g1˙(Dσ){11˙(Dσ)1}1˙h0˙}”,
OD(6D)7= 2 f{0g{1˙(Dσ)01˙(Dσ)0}0˙h0˙}+ 4 f{1g{0˙(Dσ)10˙(Dσ)0}0˙h0˙} OD(8D)13=√5f{1˙g1˙(Dσ){10˙(Dσ)0}0˙h0˙},
+ √43f{1g{1˙(Dσ)11˙(Dσ)0}1˙h1˙}, OD(9D)13=4√√52“−f{0˙g0˙(Dσ){00˙(Dσ)0}0˙h0˙}
10 ThomasKaltenbrunneretal.:IrreducibleMultipletsofThree-QuarkOperatorsontheLattice
i
−f{1˙g1˙(Dσ){01˙(Dσ)0}1˙h0˙}−2·f{1˙g1˙(Dσ){10˙(Dσ)1}0˙h0˙}”, OD(5D)17=2−√3“5·f{1g0(Dσ)0{1˙(Dσ)00˙}h0}
OD(1D0)13=4√√52“−2·f{1˙g1˙(Dσ){01˙(Dσ)0}0˙h0˙} +f{1g1(Dσ)1{1˙(Dσ)10˙}h1}”,
i
−f{1˙g0˙(Dσ){10˙(Dσ)1}0˙h0˙}−f{1˙g1˙(Dσ){11˙(Dσ)1}1˙h1˙}”, OD(6D)17=2√3“f{0g0(Dσ)0{1˙(Dσ)00˙}h0}
OD(1D1)13=+√5f{1˙g1˙(Dσ){11˙(Dσ)0}0˙h0˙}, +5·f{1g1(Dσ)1{1˙(Dσ)10˙}h0}”,
OD(1D2)13=4√√52“−f{1˙g0˙(Dσ){00˙(Dσ)0}0˙h0˙} OD(7D)17=45√i6“−2·f{1˙g1˙(Dσ){11˙(Dσ)1}0˙h0˙}
−f{1˙g1˙(Dσ){01˙(Dσ)0}1˙h1˙}−2·f{1˙g1˙(Dσ){11˙(Dσ)1}0˙h0˙}”. +35f{1˙g1˙(Dσ){01˙(Dσ)0}1˙h1˙}−f{1˙g0˙(Dσ){00˙(Dσ)0}0˙h0˙}«,
(A.15)
Furthermoretherearethefiveτ12multiplets.Westartwiththe OD(8D)17=45√i6“−2·f{1˙g1˙(Dσ){00˙(Dσ)0}0˙h0˙}
2 3
multipletDD14: +5f{0˙g0˙(Dσ){10˙(Dσ)1}0˙h0˙}−f{1˙g1˙(Dσ){11˙(Dσ)1}1˙h0˙}«,
OD(1D)14=√23“f{1g{1˙(Dσ)11˙(Dσ)0}1˙h0˙}−f{0g{1˙(Dσ)00˙(Dσ)0}0˙h0˙}”, OD(9D)17=45√i6“2·f{1˙g1˙(Dσ){01˙(Dσ)0}0˙3h0˙}
OOOODDDD((((2345DDDD))))11114444====2√√√√222√33332““““ffff{{{111{ggg1{{{g111{˙˙˙(((0˙DDD(Dσσσσ)))010)0011˙˙˙(((0˙DDD(Dσσσσ)))000)}}}0}001˙˙˙0hhh˙h000˙˙˙}}}0˙}−−−−ffff{{{101{ggg1{{{g111{˙˙˙(((1˙DDD(Dσσσσ)))101)1110˙˙˙(((1˙DDD(Dσσσσ)))101)}}}0}110˙˙˙hhh1˙h000˙˙˙}}}1˙}””””,,,, OODD((11DD10))1177==42+++5√√ifffi63{{{111˙˙˙„“ggg5011−˙˙˙(((·DDD35ff{σσσ1{˙)))0g˙{{{g0˙1100(˙011˙˙˙D((((DDDDσσ)σσσ{))))1{10000}}˙}0(˙011˙˙˙D(hhhD010σ˙˙˙}}}σ)0)−+}0,}0˙250h˙·hf0˙f{}0˙{1}˙1g˙g1˙1(˙D(Dσσ){){111˙0(˙D(Dσσ)1)}1}1˙0h˙h1˙}0˙}«”,,
OD(6D)14=2√√32“f{1g{0˙(Dσ)00˙(Dσ)0}0˙h0˙}−f{1g{1˙(Dσ)01˙(Dσ)0}1˙h1˙}”, OD(1D2)17=2−√i3“f{0˙g0˙(Dσ){10˙(Dσ)0}0˙”h0˙}
OD(7D)14=√23“f{0˙g{1(Dσ)00˙(Dσ)00˙}h0}−f{1˙g{1(Dσ)11˙(Dσ)10˙}h0}”, +5·f{1˙g1˙(Dσ){11˙(Dσ)0}1˙h0˙}” (A.17)
OD(8D)14=√23“f{1˙g{1(Dσ)11˙(Dσ)11˙}h0}−f{1˙g{1(Dσ)00˙(Dσ)00˙}h0}”, and
OD(9D)14=√23“f{0˙g{1(Dσ)10˙(Dσ)10˙}h0}−f{1˙g{1(Dσ)01˙(Dσ)00˙}h0}”, OD(1D)18=r56“f{1g0(Dσ)0{0˙(Dσ)00˙}h0}−f{1g1(Dσ)1{1˙(Dσ)01˙}h0}”,
OD(1D0)14=√23“f{1˙g{1(Dσ)01˙(Dσ)01˙}h0}−f{1˙g{1(Dσ)10˙(Dσ)10˙}h0}”, OD(2D)18=r56“f{1g1(Dσ)1{1˙(Dσ)11˙}h0}−f{1g1(Dσ)0{0˙(Dσ)00˙}h0}”,
OD(1D1)14=2√√32“f{1˙g{1(Dσ)11˙(Dσ)10˙}h1}−f{1˙g{0(Dσ)01˙(Dσ)00˙}h0}”, OD(3D)18=r56“f{1g1(Dσ)1{0˙(Dσ)00˙}h0}−f{1g0(Dσ)0{1˙(Dσ)01˙}h0}”,
OD(1D2)14=2√√32“f{1˙g{1(Dσ)10˙(Dσ)10˙}h1}−f{1˙g{0(Dσ)00˙(Dσ)00˙}h0}”. OD(4D)18=r56“f{1g1(Dσ)0{1˙(Dσ)01˙}h0}−f{1g1(Dσ)1{0˙(Dσ)10˙}h0}”,
(A.16) OD(5D)18=2√√53“f{1g1(Dσ)1{1˙(Dσ)10˙}h1}−f{1g0(Dσ)0{1˙(Dσ)00˙}h0}”,
Aexgcahiann,gtheeocfhthirealiintydepxaortnnethrseOfirD(sit)Dw15ith(OthD(ai)Dto1n6)thaeresegceonnedra(ttehdirbdy) OD(6D)18=2√√53“f{1g1(Dσ)1{1˙(Dσ)10˙}h0}−f{0g0(Dσ)0{1˙(Dσ)00˙}h0}”,
quarFkinfiaellldy,.twoequivalentmultipletsexistthatoriginatefrom OD(7D)18=r56“f{1˙g1˙(Dσ){11˙(Dσ)1}0˙h0˙}−f{1˙g0˙(Dσ){00˙(Dσ)0}0˙h0˙}”,
pornoejeOct4orisrrPe˜dαucinibtrloedmucueltdipinleet.qT.(h3e.y9)w: ere separated using the OD(8D)18=r56“f{1˙g1˙(Dσ){00˙(Dσ)0}0˙h0˙}−f{1˙g1˙(Dσ){11˙(Dσ)1}1˙h0˙}”,
lk
OD(1D)17=45√i36“2·f{1g1(Dσ)1{1˙(Dσ)01˙}h0} OODD((91DD)0)1188==rr5656““ff{{11˙˙gg01˙˙((DDσσ)){{1001˙˙((DDσσ))10}}01˙˙hh00˙˙}}−−ff{{11˙˙gg11˙˙((DDσσ)){{0110˙˙((DDσσ))01}}00˙˙hh00˙˙}}””,,
OD(2D)17=4−−5√i5536ff“{{210gg·10f(({DD1σσg1))10(D{{01˙˙σ((DD)0σσ{0))˙10(D01˙˙}}σhh)100}}0˙}++hff0}{{11gg01((DDσσ))01{{01˙˙((DDσσ))0101˙˙}}hh00}}««,, OODD((11DD21))1188==22√√√√5533““ff{{10˙˙gg00˙˙((DDσσ)){{1100˙˙((DDσσ))00}}00˙˙hh00˙˙}}−−ff{{11˙˙gg11˙˙((DDσσ)){{1111˙˙((DDσσ))00}}11˙˙(Ahh.10˙˙1}}8””) ,.
OD(3D)17=45√i6“−2·f{1g1(Dσ)1{0˙(Dσ)00˙}h0}
B Isospin induced Identities
3
−f{1g0(Dσ)0{1˙(Dσ)01˙}h0}+ 5f{1g1(Dσ)1{1˙(Dσ)11˙}h1}«,
OD(4D)17=45√i6„35f{0g0(Dσ)0{0˙(Dσ)00˙}h0} IsnymthmisetArizpepdentdhirxeew-qeuasrukmompaerriazteoros.urExrepsluolittsinfgoridiesnotsiptiiens1b/e2-
−f{1g1(Dσ)1{0˙(Dσ)10˙}h0}−2·f{1g1(Dσ)0{1˙(Dσ)01˙}h0}”, tween them we arrive at a minimal independent set of mul-