Neutron Electric Dipole Moment from Beyond the Standard Model on the Lattice

Introduction Quark EDM Dirac Equation Operator Mixing Lattice Results Conclusions

Neutron Electric Dipole Moment from Beyond the Standard Model on the lattice

Tanmoy Bhattacharya

Los Alamos National Laboratory

2019 Lattice Workshop for US-Japan Intensity Frontier Incubation Mar 27, 2019 Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Eﬀective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction Standard Model CP Violation

Two sources of CP violation in the Standard Model. Complex phase in CKM quark mixing matrix. Too small to explain baryon asymmetry Gives a tiny (∼ 10−32 e-cm) contribution to nEDM CP-violating mass term and eﬀective ΘGG˜ interaction related to QCD instantons Eﬀects suppressed at high energies −10 nEDM limits constrain Θ . 10 Contributions from beyond the standard model Needed to explain baryogenesis May have large contribution to EDM

Experimental landscape ExperimentalExperimental landscape landscape nEDM upper limits (90% C.L.) 4 nEDMnEDM upper upper limits limits (90% (90% C.L.) C.L.) 10−19 19 19 Beam System Current SM 1010− −20 −20−20 BeamBraggBeam scattering SystemSystem Current Current29 SM SM38 10 cm) 10−21 e BraggUCNBragg scattering (Sussex-RAL-ILL) scattering e 10 10 cm) cm) −21−2221 2929 3838 10 ( 10− e e 10n UCNUCN (Sussex-RAL-ILL) (Sussex-RAL-ILL) ee 1010 1010 22 22 UCN (PNPI) 19 35 ( ( − − d −23

10n 10n µ 101919 103535 UCNUCN (PNPI) (PNPI) Experimentald d −23−2423 landscape µµ 101016 101034 1010 −24−24 ⌧ 101616 103434 nEDM upper10 limits10−25 (90% C.L.) −25−25 ⌧⌧ 101026 101031 1010−26 BSM 19 LANL and European n 10 10 − −26−2726 physics 2626 3131 10 1010 Current limit LANLLANL and and European European BSMBSM nn 101023 101031 −20 −27−2827 Beam physicsphysics System Current SM p 10 10 10 1010 CurrentCurrent limit limit SNS nEDM 2323 3131 −28−2928 BraggStandard scattering model 199pp 101029 101033 cm) −21 1010 SNSSNS nEDM nEDM 29 38 10 −29−3029 StandardStandard model model Hg 10 10 e 1010 UCN (Sussex-RAL-ILL) e 10 10 199199 2929 3333 22 10

( − 129HgHg 101027 101033 −30−−3031 10n 1010 UCN (PNPI) 19 35 Xe 10 10 −31−31 129129 2727 3333 d 10−23 1010−32 µ 10 10 225XeXe 101023 101033 −32−32 1960 1970 1980 1990 2000 2010 2020 2030 Ra 10 10 −24 1010 16 34 225225 2323 3333 10 1960196019701970198019801990199020002000Publication20102010202020202030 2030year RaRa 1010 1010 −25 ⌧ 10 10 10 3 PublicationPublication year year −26 3 3 26 31 TABLE I. Approximate limits on EDM’s of10 LANL and European BSM n 10 10 TABLETABLE I. I. Approximate Approximate limits limits on on EDM’s EDM’s of of −27 FIG. 2. Progress has slowedphysics down in nEDM various systems and their predictions from10 Current limit 23 31 −28 FIG.FIG. 2. 2. Progress Progress has has slowed slowed down down in in nEDM nEDM p 10 10 variousvarious systems systems and and their their predictions predictions from from10 measurements,SNS but nEDM future experiments are the standard model CKM CP-violation. Standard model 199 29 33 thethe standard standard model model CKM CKM CP-violation. CP-violation. 10−29 measurements,plannedmeasurements, that will but but increase future future experiments theexperiments precision are are by Hg 10 10 10−30 plannedplanned that that will will increase increase the the precision precision by by 129 27 33 −31 up to two orders of magnitude Xe 10 10 10 upup to to two two orders orders of of magnitude magnitude 10−32 225Ra 10 23 10 33 1960 1970 1980 1990 2000 2010 2020 2030 and matched from nuclear level to quark/gluon e↵ective CP-violatingPublication operators, year as discussed in andand matched matched from from nuclear nuclear level level to to quark/gluon quark/gluon e↵ e↵ectiveective CP-violating CP-violating operators, operators, as as discussed discussed in in the introduction. Since the structure and interactions of the nucleon 3 are described by QCD in the thethe introduction. introduction. Since Since the the structure structure and and interactions interactions of of the the nucleon nucleon are are described described by by QCD QCD in in the the TABLE I. Approximate limits on EDM’snonperturbative of regime, LQCD calculations are essential to carry out this matching in a model- nonperturbativenonperturbative regime, regime, LQCD LQCD calculations calculationsFIG. 2. are are Progress essential essential has to to carry slowedcarry out out down this this matching matching in nEDM in in a a model- model- various systems and their predictionsindependent from way. At the quark-gluon level, there are several e↵ective operators that may be orga- independentindependent way. way. At At the the quark-gluon quark-gluonmeasurements, level, level, there there are are several several but efuture↵ e↵ectiveectiveexperiments operators operators that that mayare may be be orga- orga- the standard model CKM CP-violation.nized by their dimension. From the lowest-dimension dimension-4 QCD ✓ -term,1 to dimension- nizednized by by their their dimension. dimension. From From the theTanmoy lowest-dimension lowest-dimension Bhattacharya dimension-4 dimension-4nEDM from QCD QCD BSM✓ on✓QCD the-term,-term, lattice1 1toto dimension- dimension- 5(6) quark-EDM and quark-gluon chromo-EDMplanned that (cEDM), will2 increaseto the dimension-6 theQCDQCD precision CP-violating by 4-quark 5(6)5(6) quark-EDM quark-EDM and and quark-gluon quark-gluon chromo-EDM chromo-EDM (cEDM), (cEDM),2 2toto the the dimension-6 dimension-6 CP-violating CP-violating 4-quark 4-quark and the 3-gluon interactions (Weinbergup to operator, two orders or gluon of cEDM),magnitude these e↵ective operators rep- andand the the 3-gluon 3-gluon interactions interactions (Weinberg (Weinberg operator, operator, or or gluon gluon cEDM), cEDM), these these e↵ e↵ectiveective operators operators rep- rep- resent BSM CP-violating interactions that are increasingly suppressed by the energy scale of the resentresent BSM BSM CP-violating CP-violating interactions interactions that that are are increasingly increasingly suppressed suppressed by by the the energy energy scale scale of of the the underlying new physics. Quark/gluon CP-violating interaction can manifest itself at the nuclear underlyingunderlying new new physics. physics. Quark/gluon Quark/gluon CP-violating CP-violating interaction interaction can can manifest manifest itself itself at at the the nuclear nuclear and matched from nuclear leveland to atomic quark/gluon level in two e ways.↵ective First, CP-violating they induce intrinsic operators, EDMs in as the discussed proton and in the neutron. andand atomic atomic level level in in two two ways. ways. First, First, they they induce induce intrinsic intrinsic EDMs EDMs in in the the proton proton and and the the neutron. neutron. the introduction. Since the structureSecond, quarkand interactions and gluon CP-violating of the nucleoninteractions are induce described CP-violating by QCDnucleon in and the nucleon-pion Second,Second, quark quark and and gluon gluon CP-violating CP-violating interactions interactions induce induce CP-violating CP-violating nucleon nucleon and and nucleon-pion nucleon-pion nonperturbative regime, LQCDcouplings calculations that also are contribute essential to nuclear to carry EDMs. out Quantification this matching of the in nuclear a model- EDMs, even for couplingslightcouplings nuclei, that that will also also require contribute contribute low-energy to to nuclear nuclear nuclear EDMs. EDMs. e↵ective Quantification Quantification theories and ab-initioof of the the nuclear nuclear nuclear EDMs, EDMs, many-body even even for cal- for independent way. At the quark-gluonlightculationslight nuclei, nuclei, level, that will will are require require there based low-energy low-energy are on nuclear several nuclear nuclear EFTs. e↵ective e↵ e↵ective Forective heavy operators theories theories nuclei, and and suchthat ab-initio ab-initio as may199 nuclear nuclearHg be and orga- many-body many-body255Ra, nuclear cal- cal- culationsculations that that are are based based on on nuclear nuclear EFTs. EFTs. For For heavy heavy nuclei, nuclei, such such1 as as199199HgHg and and255255Ra,Ra, nuclear nuclear nized by their dimension. Frommodification the lowest-dimension to EDM are expected dimension-4 to be especially QCD large.✓QCD-term, to dimension- modificationmodification to to EDM EDM are are expected expected to to be be especially especially large. large. 5(6) quark-EDM and quark-gluonZD chromo-EDM comments: What (cEDM), is the state-of-the2 to the dimension-6 art? An exemplary CP-violating plot? Did we 4-quark reach the level of precision/advancementZDZD comments: comments: What What we is is promisedthe the state-of-the state-of-the in the 2013art? art? whitepaper? An An exemplary exemplary plot? plot? Did Did we we reach reach the the level level of of and the 3-gluon interactions (Weinbergprecision/advancementprecision/advancement operator, we we orpromised promised gluon in in cEDM), the the 2013 2013 whitepaper? whitepaper? these e↵ective operators rep- resent BSM CP-violating interactions5-year goals that and are plans increasingly: LQCD calculation suppressed of nucleon by EDMs the is energy similar to scale calculations of the of nucleon underlying new physics. Quark/gluon5-yearform5-year factors, goals goals CP-violating and and and plans plans its methodology:: LQCD LQCD interaction calculation calculation is straightforward of ofcan nucleon nucleon manifest. However, EDMs EDMs is itself is similarthe similar required at to to the calculations calculations statistical nuclear of precision of nucleon nucleon is formform factors, factors, and and its its methodology methodology is isstraightforwardstraightforward.. However, However, the the required required statistical statistical precision precision is is and atomic level in two ways.more First, di theycult to induce achieve because intrinsic of additional EDMs (CP-violating) in the proton interactions and the added neutron. as perturbation to moreCPmore-even di dicultcult QCD to toachieve interactions. achieve because because The of of di additional additionalculty may (CP-violating) (CP-violating) vary substantially interactions interactions depending added added on as as theperturbation perturbation CP-violating to to Second, quark and gluon CP-violatingCPoperatorCP-even-even QCD in QCD interactions question, interactions. interactions. and in induce The Thesome di di casesculty CP-violatingculty the may may calculation vary vary substantially substantially nucleon may become and depending dependingchallenging nucleon-pion on on the theand CP-violating CP-violating require new couplings that also contributeoperatormethodsoperator to nuclear in infor question, question, evaluating EDMs. and and LQCD in inQuantification some some correlators. cases cases the the calculation Thecalculation of status the may nuclear mayof LQCD become become EDMs,calculationschallengingchallenging even withandand for di require require↵erent new CP-new methodsviolatingmethods for foroperators evaluating evaluating is summarized LQCD LQCD correlators. correlators. below. The The status status of of LQCD LQCD calculations calculations with with di di↵↵erenterent CP- CP- light nuclei, will require low-energyviolatingviolating nuclear operators operators e↵ is isective summarized summarized theories below. below. and ab-initio nuclear many-body calculations that are based on nuclear– Isovector EFTs. quark For EDM-induced heavy nuclei, p,nEDM such calculations as 199Hg are straightforward and 255Ra, nuclearsince they are given ––IsovectorbyIsovector the tensor quark quark EDM-induced charge EDM-induced up to p,nEDM electromagnetic p,nEDM calculations calculations corrections. are arestraightforwardstraightforward Isoscalar quarksincesince they theyEDM-induced are are given given modification to EDM are expectedby top,nEDMby the be the tensor especially tensor calculations charge charge large. up are up to also to electromagnetic electromagneticstraightforward corrections., corrections. eventhough Isoscalar the Isoscalar disconnected quark quark EDM-induced EDM-induced contributions ZD comments: What is the state-of-thep,nEDM(thosep,nEDM arising calculations calculations art? from An the are are exemplaryinteractions also alsostraightforwardstraightforward of plot? sea quarks,, eventhough Did eventhough with we the reach the the CP-violating disconnected disconnected the level currents) contributions contributions of lead to (those(those arising arising from from the the interactions interactions of of sea sea quarks quarks with with the the CP-violating CP-violating currents) currents) lead lead to to 1 precision/advancement we promisedThe ✓QCD in theterm 2013is allowed whitepaper? in QCD as part of the SM, although its smallness is dicult to reconcile without 1 1 TheextendingThe✓QCD✓QCDterm SM.term is is allowed allowed in in QCD QCD as as part part of of the the SM, SM, although although its its smallness smallness is is di dicultcult to to reconcile reconcile without without 2extendingDimension-5extending SM. SM. operators arise from dimension-6 operators during electroweak symmetry breaking. In many explicit 5-year goals and plans: LQCD2 2Dimension-5 calculationBSMDimension-5 models, operators operators however, of nucleonarise arisethese from from operators dimension-6 dimension-6 EDMs are additionally operators is operators similar duringsuppressed during to electroweak electroweak calculations due to the symmetry symmetry chirality ofbreaking. violationbreaking. nucleon Inthat In many many accompanies explicit explicit BSMthemBSM models, models,and their however, however, contribution these these operators isoperators no larger are are than additionally additionally the dimension-6 suppressed suppressed operators. due due to to the the chirality chirality violation violation that that accompanies accompanies form factors, and its methodologythemthem is and andstraightforward their their contribution contribution is is no. no However, larger larger than than the the the dimension-6 dimension-6 required operators. operators. statistical precision is more dicult to achieve because of additional (CP-violating) interactions added as perturbation to CP-even QCD interactions. The diculty may vary substantially depending on the CP-violating operator in question, and in some cases the calculation may become challenging and require new methods for evaluating LQCD correlators. The status of LQCD calculations with di↵erent CP- violating operators is summarized below. – Isovector quark EDM-induced p,nEDM calculations are straightforward since they are given by the tensor charge up to electromagnetic corrections. Isoscalar quark EDM-induced p,nEDM calculations are also straightforward, eventhough the disconnected contributions (those arising from the interactions of sea quarks with the CP-violating currents) lead to

1 The ✓QCD term is allowed in QCD as part of the SM, although its smallness is dicult to reconcile without extending SM. 2 Dimension-5 operators arise from dimension-6 operators during electroweak symmetry breaking. In many explicit BSM models, however, these operators are additionally suppressed due to the chirality violation that accompanies them and their contribution is no larger than the dimension-6 operators. Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Eﬀective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction Eﬀective Field Theory

Energy fundamental CP−odd phases TeV

QCD d ~ e θ ,d q , d q , w Cqe ,C qq

nuclear C S,P,T g π NN neutron EDM

EDMs of EDMs of atomic paramagnetic diamagnetic atoms (Tl) atoms (Hg)

Pospelov and Ritz, Ann. Phys. 318 (2005) 119.

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Eﬀective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction BSM Operators Standard model CP violation in the weak sector. Strong CP violation from dimension 3 and 4 operators anomalously small. Dimension 3 and 4: ¯ CP violating mass ψγ5ψ. µν Toplogical charge Gµν G˜ . 2 Suppressed by vEW/MBSM: ¯ µν Electric Dipole Moment ψΣµν F˜ ψ. ¯ µν Chromo Dipole Moment ψΣµν G˜ ψ. 2 Suppressed by 1/MBSM: Weinberg operator (Gluon chromo-electric moment): Gµν Gλν G˜µλ. Various four-fermi operators.

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA 30 July 2014 | UNCLASSIFIED | 17

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Standard Model CP Violation Quark EDM Experimental situation Dirac Equation Eﬀective Field Theory Operator Mixing BSM Operators Lattice Results Impact on BSM physics Conclusions Form Factors Introduction Form Factors Vector form-factors Dirac F1, Pauli F2, Electric dipole F3, and Anapole FA 2 2 Sachs electric GE ≡ F1 − (q /4M )F2 and magnetic GM ≡ F1 + F2 " 2 2 [γµ, γν] F2(q ) hN|Vµ(q)|Ni = uN γµ F1(q ) + i qν 2 2mN 2 2 FA(q ) + (2i mN γ5qµ − γµγ5q ) 2 mN 2 # [γµ, γν] F3(q ) + qνγ5 uN 2 2mN

The charge GE (0) = F1(0) = 0. GM (0)/2MN = F2(0)/2MN is the (anomalous) magnetic dipole moment. F3(0)/2mN is the electric dipole moment. FA and F3 violate P; F3 violates CP. Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Tensor Charge Dirac Equation FLAG: Isovector Tensor Charge Operator Mixing FLAG: Isoscalar Tensor Charge Lattice Results Impact on BSM Conclusions Quark EDM Tensor Charge

The only nEDM contribution known with certainty from the lattice is that due to the quark EDM. In this case, the ratio of nEDM to qEDM is just the tensor charge.

u d s gT = 0.784(28)(10) gT = −0.204(11)(10) gT = −0.027(16)

The isovector combinations are known better:

u−d gT = 0.989(32)(10)

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Tensor Charge Dirac Equation FLAG: Isovector Tensor Charge Operator Mixing FLAG: Isoscalar Tensor Charge Lattice Results Impact on BSM Conclusions Quark EDM FLAG: Isovector Tensor Charge

−

FLAG average for =++ PNDME 18 PNDME 16 =++

PNDME 15 PNDME 13

Mainz 18 JLQCD 18

=+ LHPC 12 RBC/UKQCD 10D

ETM 17 ETM 15D =

RQCD 14 RBC 08

Radici 15 Kang 15 Goldstein 14 Pheno. Pitschmann 14 0.4 0.6 0.8 1.0 1.2

FLAG average for =++ FLAG average for =++ FLAG average for =++ =++

PNDME 18B PNDME 18B

PNDME 18B =++ =++

PNDME 16 PNDME 16 PNDME 15 PNDME 15 PNDME 15

=+ JLQCD 18

JLQCD 18 JLQCD 18 =+ =+

= ETM 17 = ETM 17 = ETM 17 0.7 0.8 0.9 −0.25 −0.20 −0.15 −0.02−0.01 0.00 0.01 0.02

7000 dn/de = 4 5000 10 000 dn/de = 3 3000 7000 dn/de = 4 L 5000 2000 3.5

GeV e = H /d 1500 dn sind φdn/dn/de e= = 3 2 m .5 3000 e = 3 = 1 /d = 2 dn e 1000 L n/d 2000 d e = 2

GeV n/d H d 1500 1.7 dn/de = 2 m e = n/d 1000 d = 2 /de dn 500 1000 2000 5000 1 ¥104

M2 GeV

500 1000 2000 5000 1 ¥104 H L M2 GeV

H L Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Free neutrons Dirac Equation Phase conventions Operator Mixing State-dependent phases Lattice Results Electric Dipole Moment Conclusions Dirac Equation Free neutrons

1 By Lorentz invariance, an on-shell spin- 2 massive ﬁeld ψ obeys (eβγ5 p/ − eiαγ5 m)ψ = 0 . This ‘free’ equation has space-time discrete symmetries:

(β−iα)γ5 P : ψ(~x,t) → e γ0 ψ(−~x,t) βγ5 ∗ C : ψ(~x,t) → i e γ2 ψ (~x,t) −iαγ5 ∗ T : ψ(~x,t) → − e γ1γ3 ψ (−~x,t)

Even when theory does not have these symmetries, asymptotic states always do, but operators have extra γ5 phases. e(−β+iα)γ5/2 ψ has standard phases.

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Free neutrons Dirac Equation Phase conventions Operator Mixing State-dependent phases Lattice Results Electric Dipole Moment Conclusions Dirac Equation Phase conventions

c Consider N = (¯u γ5d)u with usual α = β = 0 phases for u and d. If theory has C, P, T , then N has the same phases. Otherwise, by Lorentz invariance, we still have

h 2 2 i−1 P = e2β(p )γ5 Π(p2)p/ − e2iα(p )γ5 Σ(p2)m ,

2 2 which means e(−β(p )+iα(p ))γ5 N has the standard propagator (and hence the standard asymptotic symmetry representation).

2 2 But e(−β(p )+iα(p ))γ5 is not local!

By Källen-Lehman spectral representation, the propagator is

2 2 2β(µ )γ5 −2iα(µ )γ5 2 X 2 2 e p/ + e Zm(µ )µ ρ(µ )ZN (µ ) 2 2 2 . p − (Zm(µ )µ) When there is no overall symmetry operator, phases are state-dependent. (−β(m2 )+iα(m2 ))γ Nst ≡ e N N 5 N has standard transformations and equation of motion, but only for the neutron; the excited states have non-standard phases. Note that nontrivial α violates CP, whereas nontrivial β violates PT.

eσ · B is even under C, P and T , eσ · E is odd under P and T . ! σ · B iσ · E Σ · F ∝ , iσ · E σ · B

which is σ · B in the rest frame iﬀ p/ ± m = 0. In general, we need to use eiαγ5 Σ · F . So, important to use Nst instead of N in analyses. At the Green’s function level, this is

2 2 2 2 (−β(m )+iα(m ))γ5/2 (β(m )+iα(m ))γ5/2 hTNstON¯sti = e N N hTNON¯ie N N .

Abramczyk, Aoki, Blum, Izubichi, Ohki, and Syritsyn arXiv:1701.07792 [hep-lat]

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Divergent mixing Dirac Equation Nonperturbative renormalization condition Operator Mixing Gradient ﬂow Lattice Results Lattice implementation Conclusions Operator Mixing Divergent mixing

The quark EDM and the CPV mass-term have no lower dimensional operators they can mix with. Anomaly relates the Θ-term to the CPV mass-term. The chromoEDM operator has a divergent mixing with the CPV mass-term. The Weinberg operator has a divergent mixing with the Θ-term. We need to take a → 0 limit to control discretization errors n O(ΛQCDa) from the low scales inside the hadron.

2 The divergent mixing, e.g. OCEDM − (c/a )OCPV , means that in 2 matrix elements, large O(1/(ΛQCDa) ) cancel with the correct choice of c. Can use nonperturbative renormalization to remove divergence and relate to MS scheme. RI-sMOM mixes with gauge-variant and Equation-of-motion operators (about 60 for Weinberg). RI-sMOM with Background gauge-ﬁxing still needs the EOM operators. Position space renormalization needs multiloop calculations (four-loop for cEDM to O(αs)).

Regulate the theory before discretizing! √ Gradient ﬂow smears the point operators on a length scale 8tWF and commutes with the continuum limit along the line of constant physics.

−1 a→0 Zψ (a)CEDM(a, tWF ) → CEDM(tWF ) a→0 Weinberg(a, tWF ) → Weinberg(tWF )

Continuum perturbation links these to MS quantities. Power 2 subtractions are proportional to 1/8ΛQCDt. 2 −2 Need a window a 8t ΛQCD for being able to do this perturbatively.

Use Runge-Kutta integrator for gradient flow. Save intermediate configurations in memory for the propagator adjoint-flow. Use large step-size for much of the calculation. Use bias-correction with a few small step-size calculations.

Compare results at multiple tWF .

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Wilson flow on MILC lattices Dirac Equation Signal in α from Weinberg Operator Mixing Signal in F3 from Weinberg Lattice Results Signal in α from CEDM Conclusions Lattice Results Wilson flow on MILC lattices Large fluctuations from configuration to configuration.

30 20 10

ggg 0 O -10 -20 NHYP = 100 a12m310 -30 1000 2000 3000 4000 5000 Trajectory

Smearing reduces the ﬂuctuations; scale dependence remains.

0.23 2.4 a12m310 a12m310 0.22 2.2 0.21 2.0 Weinberg’s Three-gluon Op 0.20 QCD Θ-term 0.19 1.8 0.18 1.6 [GeV] [GeV] 0.17 1.4 1/4 gg

0.16 1/8 ggg χ 0.15 χ 1.2 0.14 1.0 0.13 0.8 0 1 2 3 4 5 0 5 10 15 20 t WF tWF

a ≈ 0.12fm; Mπ ≈ 310MeV; 128 measurements×1012 conﬁgurations

-0.20 -0.30 a12m310 -0.25 a12m310 Q2 = 0 -0.32 -0.30 -0.34 -0.35 top top

Q Q -0.36 -0.40 α α 1/2 -0.38 -0.45 [8tWF] = 0.34 fm [8t ]1/2 = 0.48 fm -0.50 WF 1/2 -0.40 [8tWF] = 0.68 fm 1/2 [8tWF] = 0.48 fm -0.55 -0.42 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 t/a Q2 (GeV2)

0.2 Qtop 0.0 Wggg -0.2 -0.4 CPV α -0.6 2 -0.8 Q = 0 a12m310 -1.0 0 0.2 0.4 0.6 0.8 1/2 [8tWF] (fm)

1.0 1.0 Q2 = 0.18 GeV2 Q2 = 0.18 GeV2 Q2 = 0.35 GeV2 Q2 = 0.35 GeV2 Q2 = 0.51 GeV2 Q2 = 0.51 GeV2 0.5 0.5 V V tsep = 8a tsep = 10a

/ g 0.0 / g 0.0 W W n, 3 n, 3

F -0.5 F -0.5 a12m310 a12m310 1/2 1/2 [8tWF] = 0.48 fm [8tWF] = 0.48 fm -1.0 -1.0 -4 -2 0 2 4 -4 -2 0 2 4 t - tsep / 2 t - tsep / 2

1.0 1.0 Q2 = 0.18 GeV2 Q2 = 0.18 GeV2 Q2 = 0.35 GeV2 Q2 = 0.35 GeV2 Q2 = 0.51 GeV2 Q2 = 0.51 GeV2 0.5 0.5 V tsep = 8a V tsep = 10a

/ g 0.0 / g 0.0 Q Q n, 3 n, 3 F -0.5 F -0.5 a12m310 a12m310 1/2 1/2 [8tWF] = 0.48 fm [8tWF] = 0.48 fm -1.0 -1.0 -4 -2 0 2 4 -4 -2 0 2 4 t - tsep / 2 t - tsep / 2

0.028 0.028 0.026 a12m310 0.026 a12m310 0.024 0.024 0.022 0.022 (d) 0.020 (d) 0.020 0.018 0.018

CEDM 0.016 CEDM 0.016

α 0.014 α 0.014 2 2 G5.5 2 2 G5.5 0.012 Q = 0.18 GeV G3.0 0.012 Q = 0.35 GeV G3.0 0.010 0.010 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t

0.004 0.004 a12m310 a12m310 Q2 = 0.18 GeV2 Q2 = 0.35 GeV2 0.003 0.003 (u) (u) 0.002 0.002 CEDM CEDM

α 0.001 α 0.001 G5.5 G5.5 G3.0 G3.0 0.000 0.000 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t

0.028 0.028 a12m310 a12m310 0.026 0.026 0.024 0.024 (d)

0.022 (d) 0.022 5 γ α

CEDM 0.020 0.020

α G5.5 G5.5 0.018 G3.0 0.018 G3.0 Point Point 0.016 0.016 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Q2 (GeV2) Q2 (GeV2)

0.006 0.005 G5.5 0.005 G3.0 0.004 Point 0.004

(u) 0.003

0.003 (u) 5 γ 0.002 α

CEDM 0.002 a12m310

α G5.5 0.001 0.001 G3.0 a12m310 Point 0.000 0.000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Q2 (GeV2) Q2 (GeV2)

Tanmoy Bhattacharya nEDM from BSM on the lattice Introduction Quark EDM Dirac Equation Exciting future Operator Mixing Lattice Results Conclusions Conclusions Exciting future

Gradient Flow seems to be a good next thing to try. Have to establish signal and window. Have to check excited states and extrapolation. Perturbative calculations in progress. Next one needs the four-fermion operators

Tanmoy Bhattacharya nEDM from BSM on the lattice