The Contribution of Novel CP Violating Operators to the nEDM using Lattice QCD

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by  Rajan Gupta

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0 The Contribution of Novel CP Violating Operators to the nEDM
2
using Lattice QCD

n
a
J
RajanGupta1,a,TanmoyBhattacharya1,VicenzoCirigliano1,andBoramYoon1
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1 1LosAlamosNationalLaboratory,TheoreticalDivision,T-2,LosAlamos,NM87545

]
t Abstract. In this talk, we motivate the calculation of the matrix elements of novel CP
a
l violating operators, the quark EDM and the quark chromo EDM operators, within the
-
p nucleonstateusinglatticeQCD.Thesematrixelements,combinedwiththeboundonthe
e neutronEDM,wouldprovidestringentconstraintsonbeyondthestandardmodelphysics,
h especiallyasthenextgenerationofneutronEDMexperimentsreducethecurrentbound.
[ Wethenpresentourlatticestrategyforthecalculationofthesematrixelements,inpar-

ticularwedescribetheuseoftheSchrodingersourcemethodtoreducethecalculationof
1
the4-pointto3-pointfunctionsneededtoevaluatethequarkchromoEDMcontribution.
v
2 Weendwithastatusreportonthequalityofthesignalobtainedinthelatticecalculations
3 oftheconnectedcontributionstothequarkchromoEDMoperatorandthepseudoscalar
1 operatoritmixeswithunderrenormalization.
4
0
.
1
1 Introduction: Baryogenesis and the need for novel CP violation
0
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1 The observed universe has 6.1+−00..32 × 10−10 baryons for every black body photon [1], whereas in a
: baryon symmetric universe, we expect no more that about 10−20 baryons for every photon [2]. It is
v
difficulttoincludesuchalargeexcessofbaryonsasaninitialconditioninaninflationarycosmological
i
X scenario[3]. Thewayoutoftheimpasseliesingeneratingthebaryonexcessdynamicallyduringthe
r evolutionoftheuniverse.
a
In the early history of the universe, if the matter-antimatter symmetry was broken post inflation
and reheating, then one is faced with Sakharov’s three necessary conditions [4]: the process has to
violatebaryonnumber, evolutionhastooccuroutofequilibrium, andcharge-conjugationandT(or
equivalentlyCPifCPTremainsagoodsymmetry)invariancehastobeviolated.
CPviolation((cid:8)C(cid:8)P)existsinthestandardmodel(SM)ofparticleinteractionsduetoaphaseinthe
Cabibo-Kobayashi-Maskawa(CKM)quarkmixingmatrix[5],andpossiblybyasimilarphaseinthe
leptonicsector,giventhattheneutrinosarenotmassless[6].(cid:8)C(cid:8)ParisingfromtheCKMmatrixinthe
SMcontributesatO(10−32)e-cm,muchsmallerthanthecurrentexperimentalboundd <2.9×10−26
n
e-cm. ThusthenEDMputsnoconstrainontheSMandthestrengthofthe(cid:8)C(cid:8)PintheCKMmatrixis
muchtoosmalltoexplainBaryogenesis.
Inprinciple,theSMhasanadditionalsourceof(cid:8)C(cid:8)ParisingfromtheeffectofQCDinstantons.The
presenceofthesefiniteactionnon-perturbativeconfigurationsinanon-Abeliantheoryoftenleadsto
inequivalent quantum theories defined over various ‘Θ’-vacua [7]. Because of asymptotic freedom,
aSpeaker.e-mail:[email protected]
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allnon-perturbativeconfigurationsincludinginstantonsarestronglysuppressedathightemperatures
wherebaryonnumberviolatingprocessesoccur. Becauseofthis,(cid:8)C(cid:8)Pduetosuchvacuumeffectsdo
notleadtoappreciablebaryonnumberproduction.Inshort,additionalmuchlarger(cid:8)C(cid:8)Pisneededfrom
physicsbeyondtheSM(BSM).
To determine whether such additional(cid:8)C(cid:8)P exists, a very promising approach is to measure the
static electric dipole moments of elementary particles, atoms and molecules, which are necessarily
proportional to their spin. Since under time-reversal the direction of spin reverses but the electric
dipole moment does not, a non-zero measurement confirms T violation or equivalently(cid:8)C(cid:8)P. Of the
elementaryparticles,atomsandnucleithatarebeinginvestigated,theelectricdipolemomentofthe
neutron(nEDM)isthelaboratorywherelatticeQCDcanprovidethetheoreticalpartofthecalculation
needed to “connect” the experimental bound (value) on the nEDM to the strength of(cid:8)C(cid:8)P in a given
BSMtheory.
MostextensionsoftheSMhavenewsourcesof(cid:8)C(cid:8)P. EachofthesecontributestothenEDMand
forsomemodelsitcanbeaslargeas10−26e-cm.Plannedexperimentsareaimingtoreducethebound
ond < 2.9×10−26 e-cmtod (cid:46) 3×10−28 e-cm. ThestrategyforfindingoutwhichclassofBSM
n n
theories are viable candidates is as follows: As the bound on the nEDM is lowered in current and
plannedexperiments,BSMtheorieswith(cid:8)C(cid:8)PgivingrisetoanEDMlargerthanthisboundgetruled
outprovidedthe“connection”betweentheboundandthecouplingsisknownsufficientlyaccurately.
This “connection” is the matrix elements of the novel CP violating interactions within the neutron
statethatsimulationsoflatticeQCDaregearinguptoprovide.
2 QCD Lagrangian and the EM current in the presence of CP Violation
Many candidate BSM theories have been proposed by theorists. While the true BSM theory is not
known, one can write down all possible(cid:8)C(cid:8)P interactions at the energy scale of hadronic matter (∼
fewGeV)intermsofquarkandgluonfieldsbasedonsymmetryandorganizedbytheirdimension;
operators with higher dimension being, in general, suppressed. At the lowest dimension five, there
aretwoleadingoperatorscalledthequarkEDM(qEDM)andthequarkchromoEDM(CEDM).The
QCDLagrangianinthepresenceoftheΘ-term,theqEDMandtheCEDMoperatorsis
L −→L(cid:26)(cid:26)CP =L +iΘG G˜ + i(cid:88)dγqΣµνF˜ q + i(cid:88)dGqΣµνG˜ q (1)
QCD QCD QCD µν µν q µν q µν
q q
wherethesumisoverallthequarkflavors,Σµν =[γµ,γν]/2,andF andG aretheelectromagnetic
µν µν
and chromo field strength tensors. The couplings dγ are the qEDMs and the dG are the CEDMs.
q q
They parameterize new(cid:8)C(cid:8)P in BSM theories. The goal is to constrain BSM models by bounding
these couplings using the experimental bound on the neutron EDM and lattice QCD calculations
of the matrix elements of the corresponding operators. In other words, the matrix elements of the
electromagnetic current JEM between neutron states in the presence of CP violation provides the
µ
“connection”betweentheBSMcouplingsandthenEDM.
Inthepresenceof(cid:8)C(cid:8)P,theelectromagneticcurrent,definedasδL/δAµ,getsanadditionalterm:
(cid:88) (cid:88) (cid:88)
e qγµq−→e qγµq+(cid:15)ρσνµpν dγqΣµνq. (2)
q
q q q
ThematrixelementsofthisleadingqEDM(second)termaretheflavordiagonaltensorchargesgq:
T
(cid:104)N | JµEM | N(cid:105)(cid:12)(cid:12)(cid:12)q(cid:54)CEPDM,1 = (cid:15)µνκλqν(cid:28)N(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)duγu¯Σκλu+ddγd¯Σκλd+dγss¯Σκλs+...(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)N(cid:29) ,
lim dγgu +dγgd +dγgs +... . (3)
q→0 u T d T s T
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QEDMoperator Θ-term
σµνγ5 γµ
Q Q
σµνγ5 γµ
Figure1. CalculatingnEDMrequiresevaluatingtheinteractionoftheemcurrentwiththeneutron. Thecontri-
butionofconnectedanddisconnecteddiagramsfromthe(left)qEDMand(right)Θ-term.
The connected and disconnected (with u, d, s and c quark loops) Feynman diagrams for the qEDM
contribution are shown in Fig. 1 (left). The first lattice results for the qEDM were given in [8] and
phenomenologicalconsequencesforaparticularBSMtheory,SplitSUSY,werepresentedinRef.[9].
The key computational challenge to evaluating Eq. (3) is the calculation of the disconnected contri-
butionstogq. Thesewereshowntobesmallandnoisy, anddecreasewiththequarkmass[8]. The
T
errorsinthestrangedisconnectedcontributionweresmallenoughtoyieldthecontinuumlimitesti-
mategs = 0.008(9)[8]. InmostpromisingBSMtheories,thenew(cid:8)C(cid:8)PinteractionsareYukawalike,
T
i.e.,thecouplingsareproportionaltothequarkmass. Consequently,theimpactofthesmallvalueof
gs withO(1)error,getsmagnifiedbythequarkmassratio2m /(m +m )=27. Thus,itisimportant
T s u d
toimprovetheestimateofgs and,forthesamereason,calculategc.
T T
Contributions from the CEDM and the next order in quark EDM arise due to the change in
the action, L −→ L(cid:26)(cid:26)CP . In an ideal world, the best way to calculate these effects would be
QCD QCD
to simulate the lattice theory using a discretized version of L(cid:26)(cid:26)CP and compute the matrix element
(cid:104)N | JµEM | N(cid:105)(cid:12)(cid:12)(cid:12)(cid:26)(cid:26)CP. Thisidealapproachisnotpracticalbecause(cid:8)C(cid:8)QPCDinteractionsarecomplexandlattice
simulationsoftheorieswithacomplexactionarenotyetrealistic.
Themethodofchoiceistotreatthese(cid:8)C(cid:8)Pinteractionsasperturbationsinthesmallcouplingsdγ
q
anddG,andexpandthetheoryaboutL (seeSec.3). Then,theleadingtermsthatcontributetothe
q QCD
nEDM are the matrix elements of the product of the electromagnetic current JEM and the 4-volume
µ
integraloftheoperators. FortheCEDMoperator,thematrixelementsthathavetobecalculatedare
(cid:42) (cid:12) (cid:90) (cid:12) (cid:43)
(cid:12) (cid:12) (cid:16) (cid:17) (cid:12)
(cid:104)N | JµEM | N(cid:105)(cid:12)(cid:12)(cid:26)(cid:26)CCEPDM = N(cid:12)(cid:12)(cid:12)JµEM d4x[ duGu¯Σνκu+ddGd¯Σνκd+dGs s¯Σνκs+... G˜νκ](cid:12)(cid:12)(cid:12)N , (4)
asdiscussedinSec.3. SimilarlyforthenextorderqEDM
(cid:42) (cid:12) (cid:90) (cid:12) (cid:43)
(cid:104)N | JµEM | N(cid:105)(cid:12)(cid:12)(cid:12)(cid:26)(cid:26)qCEPDM,2 = N(cid:12)(cid:12)(cid:12)(cid:12)JµEM d4x[(cid:16)duFu¯Σνκu+ddFd¯Σνκd+dsFs¯Σνκs+...(cid:17)F˜νκ](cid:12)(cid:12)(cid:12)(cid:12)N .(5)
ThissecondqEDMtermhastwoelectromagneticinteractionsandisthereforeexpectedtobesmaller
thantheleadingtermgiveninEq.(3). Forthisreasonweneglectitscontribution. Finally,thecontri-
butionoftheΘ-termrequirescalculatingthematrixelement:
(cid:42)(cid:12) (cid:90) (cid:12) (cid:43)
(cid:104)N | JEM | N(cid:105)(cid:12)(cid:12)(cid:12)Θ = (cid:12)(cid:12)(cid:12)JEM d4xΘGµνG˜µν(cid:12)(cid:12)(cid:12)n . (6)
µ (cid:54)CP (cid:12) µ (cid:12)
ThiscorrelationofJ withtopologicalchargeQgivesrisetothe2diagramsshowninFig.1(right).
µ
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 
eiε  ududdd

  PP
 ε
  PPPP
 ε
 ε

 PP
 ε
  ududdd

  ududdd

  PPε

  PPPPεε

 PP
 ε
  ududdd

  + ududdd       PP ε  PPPP ε ε   PPss ‐eeεqq   ududdd       ududdd       PP ε  PPPPεε     Ps‐eεq   ududdd       
Ps eq 
Figure2.CEDMcontributionisthereweightingfactortimesthesumofconnectedanddisconnecteddiagrams.
(cid:8)
3 Simulating QCD Including Complex(cid:8)CP CEDM and γ operators
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TheQCDLagrangianbecomescomplexwiththeadditionofanyCPviolatinginteraction. Arobust
cost-effective method for simulating theories with complex actions does not exist. To calculate the
contribution of the(cid:8)C(cid:8)P interactions to the nEDM, we work within the framework of the Schwinger
sourcemethod. WestartbynotingthatthefermionpartoftheLagrangian,L(cid:26)(cid:26)CP inEq.(1),involves
QCD
onlyquarkbilinearoperatorsinpresenceoftheqEDMandCEDMinteractions. Thefermionfieldsin
thepathintegralcan, therefore, stillbeintegratedout. Thereare, however, twomodificationstothe
standard calculations of 3-point functions due to the presence of(cid:8)C(cid:8)P interactions treated as sources.
First,theDiracoperator,inthepresenceofsaytheCEDMterm,becomes
r r
D/ +m− D2+c ΣµνG −→ D/ +m− D2+Σµν(c G +i(cid:15)G˜ ) . (7)
sw µν sw µν µν
2 2
Quark propagators calculated with this modified Dirac operator include the insertion of the CEDM
operatoratallspace-timepoints.Second,theBoltzmannweightofallgaugeconfigurationsgenerated
without the CEDM term in the action need to be modified by reweighting them by the ratio of the
determinantsoftheDiracoperatorsforthetwotheories–withandwithout(cid:8)C(cid:8)Pterms:
det(D/ +m− rD2+Σµν(c G +i(cid:15)G˜ )
2 sw µν µν
det(D/ +m− rD2+c ΣµνG )
2 sw µν
(cid:20) r (cid:21)
= expTrln 1+i(cid:15)ΣµνG˜ (D/ +m− D2+c ΣµνG )−1 (8)
µν sw µν
2
(cid:20) r (cid:21)
≈ exp i(cid:15)TrΣµνG˜ (D/ +m− D2+c ΣµνG )−1 . (9)
µν sw µν
2
Since the coupling (cid:15) for the CEDM term for all quark flavors is small, we expect the leading order
reweightingfactortobeagoodapproximation.This‘reweightingfactor’foreachgaugeconfiguration
is,toleadingorder,theintegraloverallspacetimepointsofthevalueofaclosedquarkloopwiththe
insertionoftheCEDMoperator.
Thereisanadditionalchallenge: TheCEDMoperatorhasanUVdivergentmixingwiththe“γ -
5
operator,qγ q/a2 [10]. Thus,onehasto(i)calculatethematrixelementofthe“γ -operator”,which
5 5
we do in the same way as for the CEDM operator, (ii) calculate a similar reweighting factor, and
(iii) handle the 1/a2 divergent mixing. Since the reweighting is an overall phase, and configuration
to configuration fluctuations can be large, it is therefore important to demonstrate the quality of the
signalinF (0)afterreweightingforboththeCEDMandγ terms. InSec.5,wedescribetheprogress
3 5
CONF12
wehavemadetowardsaddressingthesechallenges.ThereisalsoamixingoftheCEDMoperatorwith
theGµνG˜µνoperator. AddressingthismixingispartofthecalculationoftheΘ−termperEq.(6)[10].
The Schwinger source method has allowed us to recast the challenging calculation of 4-point
functionsgiveninEq.(4)tothedifficultcalculationof3-pointfunctions[11].TheFeynmandiagrams
needed to calculate (cid:104)N|JEM(q)|N(cid:105) in the presence of CEDM and γ interactions are illustrated in
µ 5
Fig.2. Startingwithanensembleofgaugeconfigurationsgeneratedwithastandardlatticeaction,for
exampletheWilson-cloveraction,thestepsinthecalculationare:
• Calculate propagators, labeled P in Fig. 2, using the standard Wilson-clover Dirac operator. We
assumeisospinsymmetrysothepropagatorsforuanddquarksarenumericallythesame.
• CalculateasecondsetofpropagatorswiththeCEDMtermincludedwithcoefficient(cid:15)intheWilson-
clovermatrix:
(cid:18) r (cid:19)−1 (cid:18) r (cid:19)−1
D/ +m− D2+c ΣµνG −→ D/ +m− D2+Σµν(c G +i(cid:15)G˜ ) . (10)
sw µν sw µν µν
2 2
Thesepropagators,labeledP ,includethefulleffectofinsertingtheCEDMoperatoratallpossible
(cid:15)
pointsalongthem. Thecostofinversionincreasesbyabout7%withrespectto P,however,using
PasthestartingguessintheinversionforP reducesthenumberofiterationsrequiredby20–40%
(cid:15)
dependingonthequarkmass. TheoverallaveragecostofP isfoundtobeabout80%ofP.
(cid:15)
• Using Pand P ,construct4kindsofsequentialsources,labeled Pseq, Pseq, Pseq ,and Pseq ,atthe
(cid:15) u d −(cid:15),u −(cid:15),d
sinktime-slicecorrespondingtotheinsertionofaneutronatzeromomentum.Thesubscriptsuord
inPseqandPseq,andsimilarlyinPseq andPseq ,denotetheflavoroftheuncontractedspinorinthe
u d −(cid:15),u −(cid:15),d
neutronsource. Theminussigninthecoupling,−(cid:15),accountsforthebackwardmovingpropagator.
• Ifusingthecoherentsequentialsourcemethod,thenNsourcesatdifferenttime-slicesfortheNdif-
ferentcalculationsbeingdonesimultaneouslyonthatconfiguration,areaddedtogether. Sequential
propagatorswiththese(coherent)sequentialsourcesarecalculatedbyinvertingtheDiracoperator
withandwithouttheCEDMtermasappropriate. The4typesof(coherent)sequentialpropagators
areshownbytheblockof4correlationfunctionsintherighthalfofFig.2.
• Usingthetwooriginalandthefoursequentialpropagators,calculatetheconnected3-pointfunction.
Thesecontractionsgivethefour3-pointfunctionsshownontherightinFig.2. Notethataseparate
insertionofJEMisdoneontheuanddquarklines.
µ
• Calculatethedisconnectedquarkloopwiththeinsertionofelectromagneticcurrentatzeromomen-
tumforeachofthequarkflavors,u, d, s, c(andbifneeded)usingtheDiracpropagatorwithand
withouttheCEDMterm.
• Calculatethecorrelationofthesequarkloopswiththeappropriatenucleon2-pointcorrelationfunc-
tions. The4typesofdisconnectedFeynmandiagramsrequiredareillustratedintheleftofFig.2.
• TocorrectfortheomissionofthechromoEDMoperatorintheactionduringthegenerationofthe
gaugeconfigurations,calculatethevolumeintegralofthequarkloopwiththeCEDMinsertionfor
eachflavor. Multiplythesumoftheconnectedanddisconnectedcontributionsbythereweighting
factor–exponentialoftheestimateofthe“CEDMloop”fortheconfigurationwiththeappropriate
couplingfactori(cid:15). ThisisshownbytheoverallfactorinFig.2.
• Repeat the calculation for different values of (cid:15) that bracket the expected numerical values of the
couplingsdG forthevariousquarkflavors.
q
• Repeatthewholecalculationfortheγ -operatorinsteadoftheCEDMoperator.
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4 Matrix Elements and Form Factors from 3-point Functions
Thematrixelements,definedinEqs.(3),(4)and(6),areextractedfromthevacuumtovacuum3-point
correlation function: insertion of the electromagnetic current at times t between the neutron source
andsinkoperatorsattime0andT:
(cid:88)
(cid:104)Ω|N((cid:126)0,0)JµEM((cid:126)q,t)N†((cid:126)p,T)|Ω(cid:105)= uNe−MNt (cid:104)N|JµEM(q)|N(cid:48)(cid:105)e−EN(cid:48)(T−t)u(cid:48)N. (11)
N,N(cid:48)
whereintherighthandsidewehaveusedtheexpansioninacompletebasisofstates|N(cid:105),|N(cid:48)(cid:105),...that
coupletotheneutroninterpolatingoperatorN. WeuseN ≡(cid:15)abc[daTCγ (1+γ )ub]dcwhereC =γ γ
5 4 0 2
isthechargeconjugationmatrix,a, b,carethecolorindices,u, darethequarkflavors,u isthefree
N
neutronspinorandM istheneutronmass. Inthepresenceof(cid:8)C(cid:8)P,thenucleon2-pointfunctionis
N
(cid:104)Ω|N((cid:126)0,0)N†((cid:126)p,t)|Ω(cid:105)|limt→∞ = A2uNe−p4tuN = A2e−p4teiαNγ5(i/p+mN)eiαNγ5, (12)
wherethephaseangleα arisesduetothe(cid:8)C(cid:8)Pcoupling(cid:15)anddependsonit. Todeterminethedesired
N
regionoflinearityofαversus(cid:15) weshowtheimaginarypartoftheneutron2-pointfunctioninFig.3.
It demonstrates the quality of the signal for α for appropriately small values of (cid:15) on two different
ensembles. Fig.4showstheexpectedlinearbehaviorofαversus(cid:15) overarangeofsmall(cid:15).
0.07 -0.10
ε = 0.004 -0.11 ε = 0.004
0.06 γ5
-0.12
α 0.05 γ5 -0.13
α
-0.14
0.04
-0.15
0.03 -0.16
0 2 4 6 8 10 12 0 2 4 6 8 10 12
t t
0.07 -0.10
ε = 0.003 -0.11 ε = 0.003
0.06 -0.12 γ5
-0.13
α 0.05 γ5
α -0.14
0.04 -0.15
-0.16
0.03 -0.17
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16
t t
Figure 3. The phase α extracted from the plateau in the imaginary part of the neutron 2-point function with
(cid:15) = 0.004onthea12m310ensemble(top)and(cid:15) = 0.003onthea09m310ensemble(bottom). Theplotsonthe
leftarefortheCEDMoperatorandontherightfortheγ mixingoperator.
5
Oncethematrixelementoftheelectromagneticcurrent, JEM(q)definedinEq.2, withinthenu-
µ
cleon state is extracted using Eq. (11), it is parameterized in terms of four Lorentz covariant form
CONF12
Figure4.Linearbehaviorofthephaseαversus(cid:15)withtheinsertionoftheCEDMandtheγ operators.
5
factorsF ,F ,F andF :
1 2 A 3
(cid:34)
F (q2)
(cid:104)N|JEM(q)|N(cid:105) = u γ F (q2) + iσµν q 2
µ N µ 1 ν 2M
N

+ (2iMNγ5qµ−γµγ5q2) FAM(q22) + σµν qνγ5 F23M(q2)uN, (13)
N N
F and F arethestandardDiracandPauliformfactors. Theanapoleformfactor F violatesparity
1 2 A
Pandtheelectricdipoleformfactor F violatesPandCP. F isextractedfromthedifferentmatrix
3 3
elementsbyusingdifferentcombinationsofthemomentumtransferq andspinprojections. Thezero
µ
momentum limit of these form factors gives the charges and dipole moments: the electric charge is
F (0)=1andtheanomalousmagneticdipolemomentisF (0)/2M . Thecontributionofthematrix
1 2 N
elementofeach(cid:8)C(cid:8)PinteractiondefinedinEqs.(3),(4), and(6)totheelectricdipolemomentofthe
neutronisgivenbyd =lim F (q2)/2M .
n q2→0 3 n
5 Status of Numerical Calculations
So far, we have performed numerical calculations on two 2+1+1-flavor HISQ ensembles generated
by the MILC collaboration [14]. The first, labeled a12m310, has a = 0.12 fm and M = 310 MeV
π
andthesecond,labeleda09m310,hasa=0.09fmand M =310MeV.Thecorrelationfunctionsare
π
constructedusingWilson-cloverfermions. Onthea12m310(a09m310)ensemblewehaveanalyzed
400 (270) configurations with 64 measurements on each configuration. The ensembles used and
our strategy for the calculation of 2- and 3-point functions with this clover-on-HISQ approach are
describedinRef.[12].
In Fig. 5, we show the data for F from the connected diagrams in the presence of the CEDM
3
term, and in Fig. 6, the data for F in the presence of the γ term. The data are presented for three
3 5
values of the source-sink separation t = 8,10 and 12. The excited-state contamination in the
sep
matrix elements, and thus in F , is removed by taking the t →∞ limit. These figures show that
3 sep
anacceptablesignal-to-noiseratiocanbeobtainedtoestimate F withO(1)errors,ourfirstgoalfor
3
theCEDMcalculations.
Therearetheoreticalreasonstoexpectthattheconnectedcontributionoftheγ operatoris,with
5
smallcorrections,proportionaltothatoftheCEDMoperator[13]. InFig.7,weshowtheratioofthe
twocontributionsandfindthatthisexpectationisactuallyrealized.
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1 1 1 1
0.5 0.5 0.5 0.5
U, d-uF3-0 .05 D, d-uF3-0 .05 U, d-uF3-0 .05 D, d-uF3-0 .05
-1-.15 ttssteesppe==p=11802 q2=1 -1-.15 ttssteesppe==p=11802 q2=1 -1-.15 tttssseeeppp===111024 q2=1 -1-.15 tttssseeeppp===111024 q2=1
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
1 1 1 1
0.5 0.5 0.5 0.5
U, d-uF3-0 .05 D, d-uF3-0 .05 U, d-uF3-0 .05 D, d-uF3-0 .05
-1-.15 ttssteesppe==p=11802 q2=4 -1-.15 ttssteesppe==p=11802 q2=4 -1-.15 tttssseeeppp===111024 q2=4 -1-.15 tttssseeeppp===111024 q2=4
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
Figure5. IllustrationofthesignalinF withtheinclusionoftheCEDMterm. Thedatainthetoprowarefor
3
(cid:126)p=(1,0,0)×2π/Lainthefollowingorder:insertiononuanddquarksforthea12m310ensemblefollowedby
insertiononuanddquarksforthea09m310ensemble.Theplotsinthesecondrowareinthesameorderexcept
with(cid:126)p=(2,0,0)×2π/La.
1 1 1 1
U, , d-uγF53- 00 ..055 ttssteesppe==p=11802 D, , d-uγF53- 00 ..055 ttssteesppe==p=11802 U, , d-uγF53- 00 ..055 tttssseeeppp===111024 q2=1 D, , d-uγF53- 00 ..055 tttssseeeppp===111024 q2=1
-1 q2=1 -1 q2=1 -1 -1
-1.5 -1.5 -1.5 -1.5
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
1 1 1 1
U, , d-uγF53- 00 ..055 ttssteesppe==p=11802 D, , d-uγF53- 00 ..055 ttssteesppe==p=11802 U, , d-uγF53- 00 ..055 tttssseeeppp===111024 D, , d-uγF53- 00 ..055 tttssseeeppp===111024
-1 q2=4 -1 q2=4 -1 q2=4 -1 q2=4
-1.5 -1.5 -1.5 -1.5
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
Figure 6. Illustration of the signal in F with the inclusion of the γ term. The data in the top row are for
3 5
(cid:126)p=(1,0,0)×2π/Lainthefollowingorder:insertiononuanddquarksforthea12m310ensemblefollowedby
insertiononuanddquarksforthea09m310ensemble.Theplotsinthesecondrowareinthesameorderexcept
with(cid:126)p=(2,0,0).
Toconclude, thedata sofarshowthatan acceptablesignal-to-noiseratiocanbe obtainedinthe
connected contributions to F for both the CEDM and γ operator insertion. We also find that The
3 5
ratio of the connected contribution of the γ to CEDM term is a constant. This implies that the
5
mixingoftheγ termwiththeCEDMtermcanbecastaspartofthemultiplicativerenormalizationof
5
theCEDMoperator. ControllingitsO(1/a2)UVdivergentcoefficientisahurdleforlatticetheories
thatrespectchiralsymmetry(domainwalloroverlapfermions)andthosethatdon’tsuchasWilson-
clover fermions. Work to address the full mixing of the CEDM operator and the calculation of the
disconnecteddiagramsandthereweightingfactorsisunderprogress.
CONF12
6 6 6 6
U, , d-uU, d-uγF / F533-- 42024 q2=1 ttssteesppe==p=11802 D, , d-uD, d-uγF / F533-- 42024 q2=1 ttssteesppe==p=11802 U, , d-uU, d-uγF / F533-- 42024 tttssseeeppp===111024 q2=1 D, , d-uD, d-uγF / F533-- 42024 tttssseeeppp===111024 q2=1
-6 -6 -6 -6
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
6 6 6 6
U, d-uF3 24 q2=4 tstespe=p=180 D, d-uF3 24 q2=4 tstespe=p=180 U, d-uF3 24 q2=4 tttssseeeppp===111024 D, d-uF3 24 q2=4 tttssseeeppp===111024
U, , d-uγF / 53-- 420 D, , d-uγF / 53-- 420 U, , d-uγF / 53-- 420 D, , d-uγF / 53-- 420
-6 -6 -6 -6
-4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2
Figure7. Theratiooftheconnectedcontributionoftheγ termtotheCEDMtermintheF formfactor. The
5 3
restisthesameasinFig.5.
Acknowledgments
We thank the MILC Collaboration for providing the 2+1+1-flavor HISQ ensembles. Simulations
were carried out on computer facilities of (i) Institutional Computing at Los Alamos National Lab;
(ii)theUSQCDCollaboration,whicharefundedbytheOfficeofScienceoftheU.S.Departmentof
Energy; (iii) the National Energy Research Scientific Computing Center, a DOE Office of Science
User Facility supported by the Office of Science of the U.S. Department of Energy under Contract
No. DE-AC02-05CH11231. ThecalculationsusedtheChromasoftwaresuite[15]. Worksupported
bytheU.S.DepartmentofEnergyandtheLANLLDRDprogram.
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