Table Of ContentCalculation of Nucleon Electric Dipole Moments
Induced by Quark Chromo-Electric Dipole Moments
7
1
0 Taku Izubuchi a, Michael Abramczykb, Tom Bluma,b, Hiroshi Ohkia,
2 ∗
Sergey Syritsyna,c,d
n
a aRIKEN/BNLResearchCenter,BrookhavenNationalLaboratory,Upton,NY11973,USA
J bPhysicsDepartment,UniversityofConnecticut,Storrs,CT06269,USA
1 cDepartmentofPhysicsandAstronomy,StonyBrookUniversity,StonyBrook,NY11794,USA
3
d ThomasJeffersonNationalAcceleratorFacility,12000JeffersonAvenue,NewportNews,VA
] 23606,USAE-mail: [email protected]
t
a
l
-
p Wepresentinitialresultsofcomputingnucleonelectricdipolemomentinducedbyquarkchromo-
e
EDM, CP-violating quark-gluon coupling. Using chirally-symmetric domain wall and Möbius
h
[ fermions with pion mass m =172MeV, we calculate the connected part of the electric dipole
π
1 formfactorF3(Q2).Inaddition,weperformanexploratorystudyofthemethodtocalculateEDM
v usinguniformbackgroundelectricfieldonalatticeintroducedwithoutbreaking theperiodicity
2
5 inthetimedirection.
0
0
0
.
2
0
7
1
:
v
i
X
r
a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
NeutronEDMinducedbychromo-EDM TakuIzubuchi
1. Introduction
Theoriginofnuclearmatter,ortheexcessofnucleonsoverantinucleonsintheearlyUniverse,
isoneofthegreatestpuzzlesinphysicsknownasthebaryonicasymmetryoftheUniverse(BAU).
OneoftherequiredconditionsfortheBAUisviolationoftheCPsymmetry. StandardModel(SM)
(cid:8)C(cid:8)P from CKM matrix phases is not sufficient to explain BAU, and signs of additional (cid:8)C(cid:8)P are
actively sought in experiments. The most promising ways to look for(cid:8)C(cid:8)P are measurements of
electric dipole moments (EDM) of atoms, nucleons and nuclei. Planned EDM experiments will
improve their bounds by 2 orders of magnitude. Knowledge of nucleon structure and interactions
isrequiredtointerprettheseexperimentsintermsofquarkandgluonfieldsandputconstraintson
manypotentialextensionsofparticlephysicsbeyondtheStandardModel(BSM)suchasSUSYand
GUTassourcesofadditional(cid:8)C(cid:8)P. Connectingthequark-andhadron-leveleffective(cid:8)C(cid:8)Pinteractions
isataskforlatticeQCD(anextensivereviewofEDMphenomenologycanbefoundinRef.[]).
Protonandneutronelectricdipolemomentsmaybeinducedby(cid:8)C(cid:8)Poperators. Theonlydim-
4 operator is the QCD θ¯-angle. The θ¯-induced nucleon EDMs have been calculated on a lattice
previously from energy shifts in background electric field [] or extracting(cid:8)C(cid:8)P electric dipole form
factor(EDFF)F (Q2)[]. NucleonEDMs(nEDMs)inducedbydim-6operatorshavebeenstudied
3
usingQCDsumrulesandchiralperturbationtheory[]. Anotherimportantdim-6operatorisquark
chromo-electricdipolemoment(cEDM)
˜
δ
L =i ∑ ψψ¯(TaGq )σµνγ ψ, (1.1)
cEDM 2 µν 5
ψ=u,d
andcalculationsofcEDM-inducednEDMshavestartedusingWilsonfermions[].
In this work, we present initial calculations of nEDM induced by quark chromo-EDM using
chirally-symmetric Domain wall fermions. We compute connected parts of the EDFF F from 4-
3
pointcorrelationfunctionsofnucleonwithquarkcurrentandcEDMquark-bilinearoperator(1.1).
In addition, we test the method to compute EDM from energy shift in background electric field,
which is introduced without violating boundary conditions following Ref. [], on a small 163 32
×
latticeandfindballparkagreementbetweenthetwomethods1.
2. Latticedetails
In our preliminary calculation, we employ two lattice ensembles with N =2+1 dynamical
f
domainwallfermionslistedinTab.1. ThefirstisaDSDRensemble[]withlightpionmassm
π
≈
172MeV,onwhichwecalculatetheelectricdipoleformfactor(EDFF)F (Q2)fromvectorcurrent
3
matrixelementsofthenucleonin(cid:8)C(cid:8)Pvacuum. TheforwardlimitoftheEDFFgivestheEDM,d =
N
F (0)/(2m ). ThesecondisaIwasaki+DWFensemblewithheavierpionmassm 420MeV[],
3 N π
≈
on which we calculate the nucleon EDFF from vector current matrix elements and the nucleon
EDM from nucleon energy shifts in background electric field. In large-statistics calculations on
the second ensemble, we aim to compare the EDFF and EDM computed with the two different
methods.
1Preliminary results presented in the conference poster did not show the form factor dependence on momentum
transferandenergyshiftmethod.Weomitgradientflowstudyfromthismanuscriptbecauseofspaceconsiderations.
1
NeutronEDMinducedbychromo-EDM TakuIzubuchi
Table1: Latticeensemblesonwhichthesimulationswereperformed. Thestatisticsareshownfor“sloppy”
(low-precision)samples.
L L L a[fm] am am m [MeV] conf stat N N
x t 5 l s π ev CG
× ×
DSDR 323 64 32 0.1432(7) 0.001 0.045 172.3(9) 28 896 1400 125
× ×
Iwasaki 163 32 16 0.114(2) 0.01 0.032 422(7) 500 16500 150 100
× ×
We use all-mode-averaging [] framework to optimize sampling, in which we approximate
quark propagators with truncated-CG solutions to a Möbius operator []. We use the Möbius op-
erator with short 5th dimension L and complex s-dependent coefficients b +c = ω 1 (later
5s s s s−
referredtoas“zMobius”)thatapproximatesthesame4deffectiveoperatorastheShamiroperator
withthefullL =32(DSDR)orL =16(Iwasaki). Theapproximationisbasedonthedomain
5f 5f
wall-overlapequivalence
1+m 1 m D/
[D/DWF] = q − qγ ε (H ), H =γ W , (2.1)
4d 2 − 2 5 L5 T T 52+D/
W
∏L5s(1+ω 1x) ∏L5s(1 ω 1x)
εMöbius(x)= s s− − s − s− εShamir(x). (2.2)
L5s ∏Ls5s(1+ωs−1x)+∏sL5s(1 ωs−1x) ≈ L5f
−
wherethecoefficientsω arechosensothatthefunctionεMöbius(x)istheminmaxapproximationto
s L5s
theεShamir(x). WefindthatL =10isenoughforefficient4doperatorapproximation. Shortened
L5f 5s
5th dimension reduces the CPU and memory requirements, which is especially significant for the
lighterpionmassensembleL =32isreducedtoL =10saving70%ofthecost. Wedeflatethe
5f 5s
low-lyingeigenmodesofthetheinternaleven-oddpreconditionedoperator,tomaketruncated-CG
AMA efficient. The numbers of deflation eigenvectors N and truncated CG iterations N are
ev CG
giveninTab.1. Figure1(left)showsthedeviationoftruncatedsolutionstothezMöbiusoperator
from the exact Shamir solution, demonstrating the effect of deflating the eigenmodes between the
lightandthestrangequarkmasses(Fig.1,right). Wecompute32sloppysamplesperconfiguration.
To correct any potential bias due to the approximate D/ operator and truncated CG, we compute
oneexactsamplewiththeShamiroperatorperconfiguration. Thelatteriscomputediterativelyby
refiningthesolutionofthe“zMobius”toapproachthesolutionoftheShamiroperator,againtaking
advantageoftheshortL anddeflation.
5s
3. ElectricDipoleFormfactorF
3
The EDFF F is a parity-odd quantity induced by(cid:8)C(cid:8)P interactions. To compute the effect of
3
chromo-EDM(1.1),weTaylor-expandQCD+(cid:8)C(cid:8)PvacuumaveragesasS S+i∑ δ˜ OCP
→ x ψ ψ,x
(cid:104)N[q¯γµq]N¯(cid:105)(cid:26)C(cid:26)P=CNJN¯ −iδ˜ψδψCPCNJN¯ +O(δ˜ψ2), (3.1)
1
with CNJN¯ =(cid:104)N[q¯γµq]N¯(cid:105), δψCPCNJN¯ =(cid:104)N[q¯γµq]N¯ ·∑OψCP,x(cid:105), OψCP= 2ψ¯(TaGaµν)σµνγ5ψ,
x
where in the second line stand for vacuum averages computed with CP-even action S. In
(cid:104)·(cid:105)
this work, we calculate only fully connected contributions to the correlation functions C and
NJN¯
2
NeutronEDMinducedbychromo-EDM TakuIzubuchi
101 ����
ShamirL5=32
Mo¨biusL5=12
100 zMo¨biusL5=10
zMo¨biusL5=10withdeflation ����
10−1
x/x∗∗|||10−2 � ����
−
x|
10−3 ����
10−4 �����
�������
����
10−50 2000 4000 6000 iter8a0t0i0ons10000 12000 14000 16000 �� ���� ���� ���� ������� ����� ����� ����� �����
Figure 1: (Left) Comparison of 4d solution vectors (quark propagators) between approximate 5D Dirac
operators(L =10complex“zMobius”andL =12regularMobius)andtheexactShamirquarkpropagator
5 5
withL =32vstheiterationcount. (Right)Low-modeeigenvaluesofzMobiusL =10computedwithlight
5 5
andstrangequarkmasses.
u u d u u d u u d u u d u u d u u d u u d
u u d u u d u u d u u d u u d u u d u u d
Figure2: Quark-connectedcontractionsofnucleon,quarkcurrent,andcEDMoperators.
δCPC . To compute the latter, the quark-bilinear cEDM density (1.1) is inserted once in every
NJN¯
ψ-quark line ofC diagrams, generating the four-point functions shown in Fig.2. We evaluate
NJN¯
these 4-point contractions using sequential propagators. In addition to the usual one forward and
two backward (sink-sequential) propagators, we compute one cEDM-sequential and four doubly-
sequential ({cEDM, sink}-sequential) propagators. This is the minimum number inversions re-
quired to compute cEDM-induced nEDM for all combinations of two degenerate flavors u and d,
with one source-sink separation t and one sink momentum (cid:126)p. Compared to the method used
sep (cid:48)
in Ref. [], in effect we compute the first derivative in theCP-odd coupling ε at ε =0, obviating
any higher-order dependence on ε. We use (cid:126)p =0 and t =8a=1.14fm for the 323 64 and
(cid:48) sep
×
t = 8,10 a=0.91,1.15fmforthe163 32ensemble. TheCP-evenpartofthenucleon-current
sep
{ } ×
correlator(3.1)requiresonlycontractionsbutnoadditionalinversions. BothC andδCPC are
NJN¯ NJN¯
computedwiththepolarizationprojectorT = 1+γ4Σ = 1+γ4( iγ γ ). Thisprojectorissufficient
toextractformfactorsF andF from(cid:8)C(cid:8)Pponlucleo2nm3atrixe2lem−ent1s2ofthevectorcurrent2
1,2 3
N q¯γµq N =u¯ (cid:2)F (Q2)γµ+F (Q2)σµνqν +F (Q2)iγ5σµνqν(cid:3)u . (3.2)
p p p 1 2 3 p
(cid:104) (cid:48)| | (cid:105) (cid:48) 2m 2m
N N
WeemploytheusualC /C ratiostocanceloverlapsandexponentialfactorsandweextract
3pt 2pt
matrix elements of the ground state as the central plateau average (see Fig. 4, right). Excited
state contaminations are neglected in this preliminary analysis. The P-even form factors F are
1,2
2AllconventionsforformfactorsandmomentaareEuclidean
3
NeutronEDMinducedbychromo-EDM TakuIzubuchi
20 5
0 0
20
− 5
40 −
α5 −60 UD α5−10 UD
− 15
80 −
−
−100 −20
120 25
− 0 2 4 6 8 10 12 − 0 2 4 6 8 10 12 14
t t
Figure3: Theparity-mixingphaseoftheprotonpropagatorinducedbyu-andd-cEDM,forthem =172
π
(left)and422MeV(right). ThemixingisnoticeablylargerwhentheunpairedquarkisaffectedbycEDM.
200 160
F3P,U-cedm(T=8) q=[0,0,1]
=1] 150 FFF333PNN,,,DUD---ccceeedddmmm(((TTT===888))) EDM112400 qqq===[[[010,,,110,,,112]]]
˜2BareFQd()()[3u,d 10550000 Fq()plateau,D-c3N104680000
− 20
100 0
− 0.00 0.05 0.10 0.15 0.20 0.25 0 2 4 6 8
Q2[GeV2] τ
Figure4: Formfactorscomputedonm =172ensemble: protonandneutronEDFFF (Q2)inducedbyu-
π 3
andd-cEDM(left)andformfactorplateausforneutronF inducedbyd-quarkcEDM(right). Source-sink
3
separationist =8a=1.15fm.
sep
extractedfromoverdeterminedfit(e.g.,Ref.[])ofC combiningallorientationsofthecurrent
NJµN¯
andmomenta. TheEDFFF isextractedfromthetimelike-componentofδCPC .(cid:8)C(cid:8)Pinteraction
3 NJ4N¯
leadstoparitymixingofthenucleonspinors
u˜p=eiα5γ5up, (cid:104)Nδ(t,(cid:126)p)N¯δ(cid:48)(0)(cid:105)t→=∞ [−i/p+m2NEe2iα5γ5]δδ(cid:48)e−ENt (3.3)
N
that complicates analysis of F , as δCPC receives contributions from F due to “polarization
3 NJ4N¯ 1,2
mixing” α γ ,T []. To subtract them, we determine the cEDM flavor-dependent mixing
5 5 pol
∼ { }
anglefromtheCP-oddtwo-pointfunctionδCPC as(seeFig.3)
NN¯
α5(t)= ReTr(cid:2)T+γ5δCPCNN¯(t)(cid:3)/ReTr(cid:2)T+CNN¯(t)(cid:3), (3.4)
−
where T+ = (1+γ )/2 is the positive-parity projector. Remarkably, connected cEDM leads to
4
parityrotationonlywhen(cid:8)C(cid:8)Pinteractionaffectsthe“spectator”(non-diquark)flavorinthenucleon.
Weusethevalueα (t )andτ-dependentplateauvaluesofF forthesubtraction. Thesubtracted
5 sep 1,2
formulaforF is[]
3
F3ψ = E2+mNm (cid:104)2EN(EqN+mN)Tr(cid:2)T+ΣzδψCPRNJ4N¯(cid:3)−α5GE(cid:105), (3.5)
N N 3
wherespinΣz= iγ1γ2andδCPRNJ4N¯ istheδCPC3pt/C2pt ratiocancelingnucleonfieldnormaliza-
−
tionandtimeexponentials. TheF resultsforthem =172MeVensembleareshowninFig.43
3 π
3Ononeoutof30analyzedconfigurations,wehaveencounteredanoutliersamplethathadvaluesforthe2-and
3-pointfunctionsstronglydeviatingfromtherestoftheensemble. Wehaveomittedtheentiregaugeconfiguraionas
wellasoneadjacenttoitfromthepreliminaryanalysis,andweareinvestigatingwhatcausedthisoutlier.
4
NeutronEDMinducedbychromo-EDM TakuIzubuchi
4. Energyshiftinbackgroundelectricfield
ThebackgroundfieldmethodprovidesopportunitytocomputeEDMsdirectly,whereasQ2
→
0 extrapolation of F (Q2) may lead to systematic uncertainties. We have done an exploratory
3
calculationofcEDM-inducednEDMonthe163 32m =422MeVensembleinuniformelectric
π
×
fieldtocomparethesetwomethods. WeusetheWick-rotatedelectricfield,which,ifquantizedas
Q E =n 2π , does not break (a)periodic boundary conditions []. The Wick-rotated electric field
q k kLkLt
cannot create pion pairs out of vacuum, but it can “accelerate” charged particles and complicate
calculationoftheirrestmass. Forthisreason,weapplythisanalysisonlytotheneutron.
WecomputetheenergyshiftfromtheCP-oddcorrectionδCPCNN¯,Ez totheneutronpropagator
inpresenceoftheuniformelectricfieldE
z
U U eiQAµ, A (z,t)=zE (cid:0) z=0...(L 1)(cid:1), A (z,t)= tL E (cid:0) t =L 1). (4.1)
µ µ tˆ z z zˆ z t
→ ∀ − − ∀ −
δCPCNN¯,Ez iscomputedsimilarlytoFig.2butwithoutthequarkcurrent. CP-oddinteractionleads
tospin-dependentenergyshiftδE = d Sˆ iE(cid:126),fromwhichweextractd as
N N
− ·
2m deff(t)= 2mN(cid:2)R (t+1) R(t)(cid:3), R (t)= Tr(cid:2)T+ΣzδCPCNN¯,Ez(t)(cid:3). (4.2)
N N − Ez z − z Tr(cid:2)T+CNN¯,Ez(t)(cid:3)
We have computed deff(t) for two values of the electric field (1 and 2 E ), and averaged the
N × × min
resultoverE andS directions. Thestatisticsarethesameasintheformfactorcalculation.
z z
In Figure 5 we compare F (Q2) and 2m deff(t), where the EDM “units” are adjusted for a
3 N N
direct comparison. The EDM value from the energy shift appears to reach plateau for t =4...6.
However, the comparison EDFF values computed with t =8a and 10a shows that F requires
sep 3
detailed analysis of excited state contributions. For the d-cEDM, there appears to be 2... 2.5
× ×
differencebetweendeff andF (Q2 ). ConsideringthetrendofF (Q2)withQ2 0,theresultsof
N 3 min 3 →
formfactorandenergyshiftcalculationsappeartobecompatible. Wenotethattheminimalelectric
field is E = 0.097GeV2 = 490MeV/fm on the small 163 32 lattice. Such strong electric
min
×
field may distort the nucleon and introduce systematic shift to deff, complicating the comparison.
N
Nevertheless,wefindthedatasuggestencouragingagreementbetweenthetwomethods.
Q2[GeV2] Q2[GeV2]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
30 30
M) EDFFF3(Q2)(T=8) M) EDFFF3(Q2)(T=8)
ED 25 EDFFF3(Q2)(T=10) ED 25 EDFFF3(Q2)(T=10)
d-c 20 u-c 20
( (
2Q) 15 2Q) 15
F(3 10 F(3 10
vs 5 vs 5
effmdt()NN −50 ((SSSS))UUEE==±12 effmdt()NN −50 ((SSSS))DDEE==±12
2 10 ± 2 10 ±
− 0 2 4 6 8 10 12 − 0 2 4 6 8 10 12
t t
Figure5:ComparisonofneutronEDFFF (Q2)vs.Q2(topscale)effectiveEDM2m deff(t)(4.2)vs.t(bot-
3 N N
tom scale) computed from energy shift on the 422MeV ensemble, for d-cEDM (left) and u-cEDM(right).
5
NeutronEDMinducedbychromo-EDM TakuIzubuchi
5. Summary
We have obtained preliminary results for the nucleon electric dipole form factors induced by
light quark chromo-EDM interaction using chirally-symmetric domain wall fermions. Our initial
calculationdoesnottakeintoaccountrenormalizationandmixingofchromo-EDMoperatorswith
other(cid:8)C(cid:8)P operators, and neglects effects of excited states. In addition, we have performed an ex-
ploratory study of electric dipole moments using the background electric field method and have
comparedthemtotheformfactorresults. Whilethestatisticalandsystematicprecisionisinsuffi-
cienttodrawdefiniteconclusions,theseresultsappeartobecomparable. Usingmethodsdeveloped
inthiswork,thestudieswillbeextendedtothephysicalpointandwillincludeanalysisofexcited
states,aswellasrenormalizationandmixingof(cid:8)C(cid:8)Poperators.
ACKNOWLEDGEMENTS
We are grateful to the RBC/UKQCD collaboration for the gauge configurations and to the
RIKENACCCfacilityforthecomputingresourcesprovidedforthiswork. T.B.issupportedbyUS
DOEgrantDE-FG02-92ER40716. T.I.issupportedinpartbytheJapaneseMinistryofEducation
Grant-in-Aid, No. 26400261 The work of H.O. is supported by the RIKEN Special Postdoctoral
Researcher program. S.S. thanks RBRC for support under its joint tenure track fellowship with
StonyBrookUniversityS.S.alsothanksKITPUCSBforhospitalityatthe“Nuclear16”workshop
andsupportbytheNationalScienceFoundationunderGrantNo. NSFPHY11-25915.
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