Computing Nucleon Electric Dipole Moment from lattice QCD
CComputing Nucleon Electric Dipole Moment fromlattice QCD
Taku Izubuchi a , b , Hiroshi Ohki ∗ a , c , Sergey Syritsyn a , d , a RIKEN/BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA b Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA c Department of Physics, Nara Women’s University, Nara 630-8506, Japan d Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USAE-mail: [email protected]
Electric dipole moments (EDMs) of nucleons and nuclei are actively considered as direct evidenceof the CP violation. Calculations of nucleon EDMs on lattice are required to connect the quark-and hadron- level effective CP violating interactions within QCD or other CP violating sources innew physics beyond the standard model. Among them, the theta-induced nucleon EDM, that isthe only such renormalizable interaction, has widely been investigated on a lattice. In the report,we review recent developments of the lattice calculations of nucleon EDM induced QCD thetaterm. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] A p r omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki
1. Introduction
Observation of a non-zero nucleon electric dipole moment (nEDM) would be direct evidencefor violation of CP(T)-symmetry. The Standard Model (SM) has a CP-violating phase in the CKMmatrix. However, there is at least one substantial motivation to search for CP violation beyondthe SM: the magnitude of the CP violation effect in CKM is insufficient to explain the observedbaryon asymmetry of the universe as required by one of the famous Sakharov’s conditions forthe universe’s matter origin. Currently the most sensitive probes for the CP-violating phenomenaare EDM searches in hadronic, atomic, and molecular systems. The best limits on EDMs comefrom experiments on neutrons (ILL) [1] and
Hg [2] which constrain the nucleon EDM (nEDM) | d n | ≤ . × − [ e cm]. This bound is ∼ larger than the prediction ∼ − [ e cm] from theCP phase of the CKM matrix in the SM. Observation of non-zero nEDMs would be a discovery offundamental importance, and however, even a null result would make serious impact on cosmologyand high-energy theory. Violations of CP symmetry at the quark level are represented by a numberof effective quark and gluon operators. Among them, the only such renormalizable interaction isthe QCD " θ -term", iS θ = i θ Q = i θ ∑ x q ( x ) , q ( x ) = π Tr (cid:2) G µν ˜ G µν ( x ) (cid:3) (1.1)where Q is the topological charge and q ( x ) is the topological charge density operator. Assumingthat the θ -term is the only source of the CP violation, we have a strong constraint on the parameter | θ | (cid:46) − . The problem of why θ is so small is known as the strong CP problem . The precisionof EDM measurements of nucleons and nuclei will increase in the future experiments using neutronsources, which plan to improve neutron EDMs bounds by 1-2 orders of magnitude. However,quantitative connection between magnitudes of EDMs and such CP violation operators at the quark-gluon level is very limited and model-dependent (see Refs. [4], [5] for a recent review of theEDM phenomenology). Therefore connecting the quark- and hadron-level effective interactionsthat include CP violating sources is an urgent task for lattice QCD.In this proceedings, we review recent progress on the lattice calculations of the nEDM inducedby the θ term. As for other CP-violating matrix elements that arise from the operators beyond theSM, these are discussed in the plenary lecture [6] .
2. CP odd Form factor and parity mixing
In this section, we briefly review the methods for the lattice computations of the nEDM. The θ -induced nEDMs have been calculated on a lattice from the CP-odd electric dipole form factor(EDFF) F ( Q ) with the Q → (cid:104) p (cid:48) , σ (cid:48) | J µ | p , σ (cid:105) (cid:26)(cid:26) CP = ¯ u p (cid:48) , σ (cid:48) (cid:2) F ( Q ) γ µ + (cid:0) F ( Q ) + iF ( Q ) (cid:1) i σ µν q ν M N (cid:3) u p , σ , (2.1) A dynamical solution to the strong CP problem is proposed in a parallel talk by G. Schierholz [3]. See also [7] for a recent review. omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki where Q = − q and q = p (cid:48) − p , and F and F are the Dirac and Pauli form factors. The QCD θ term is introduced either as a modification of the lattice CP-even QCD action with imaginary θ term [10, 11] or as a Taylor expansion of nucleon correlation functions, S QCD → S QCD + iS θ . In thelatter case the nucleon-current correlation functions in (cid:8)(cid:8) CP QCD vacuum are modified as (cid:104) N [ ¯ q γ µ q ] ¯ N (cid:105) (cid:26)(cid:26) CP = Z (cid:90) D U D ¯ ψ D ψ N [ ¯ q γ µ q ] ¯ Ne − S − iS θ = C NJ ¯ N − i θ C QNJ ¯ N + O ( θ ) , (2.2)where C NJ ¯ N = (cid:104) N [ ¯ q γ µ q ] ¯ N (cid:105) and C QNJ ¯ N = (cid:104) N [ ¯ q γ µ q ] ¯ N ∑ x [ q ( x )] (cid:105) are the nucleon-current correlationfunctions evaluated in the CP -even QCD vacuum. To obtain the EDFF F ( Q ) in Eq. (2.1) we haveto calculate the nucleon two and three point functions C Q ( (cid:126) p , t ) = ∑ (cid:126) y e − i (cid:126) p · (cid:126) y (cid:104) N ( (cid:126) y , t ) ¯ N ( (cid:126) , ) ∑ x [ q ( x )] (cid:105) , (2.3) C Q ( t ,(cid:126) p ; t op ,(cid:126) q ) = ∑ (cid:126) y ,(cid:126) z e − i (cid:126) p · (cid:126) y + (cid:126) q · (cid:126) z (cid:104) N ( (cid:126) y , t ) J µ ( (cid:126) z , t op ) ¯ N ( (cid:126) , ) ∑ x [ q ( x )] (cid:105) . (2.4)In general the nucleon ground states as well as their overlaps with the positive-parity nucleonground state are modified in (cid:8)(cid:8) CP QCD vacuum as (cid:104) | N | p , σ (cid:105) (cid:26)(cid:26) CP = Z N e i αγ u p , σ = Z N ˜ u p , σ , (2.5)where ˜ u p , σ is a spinor wave function for the nucleon state | p , σ (cid:105) (cid:26)(cid:26) CP and Z N is a normalizationconstant. The spinor ˜ u p , σ satisfies the following free Dirac equation with CP-violating γ mass ( / p − m N e − i αγ ) ˜ u p , σ = ( / p − m N e − i αγ ) e i αγ u p , σ = , (2.6)where u p , σ is a wave function spinor in CP-even QCD vacuum. Thus when the CP violating effectexists in the QCD vacuum, one should use the modified nucleon spinor ˜ u p , σ in lattice calculationswhich affects the kinematic coefficients. For example, ignoring excite states, the nucleon two-pointcorrelation function with CP violating operator can be represented as C Q ( (cid:126) p , t ) = | Z N | e − E p t E p ∑ σ ˜ u p , σ ¯˜ u p , σ = | Z N | e − E p t E p (cid:0) m N e i αγ − i / p (cid:1) , (2.7)where we use the completeness condition for the free Dirac spinor, ∑ σ ˜ u p , σ ¯˜ u p , σ = e i αγ (cid:32) ∑ σ u p , σ ¯ u p , σ (cid:33) = m N e i αγ − i / p . (2.8)The modification also affects the nucleon matrix elements (cid:104) p (cid:48) , σ (cid:48) | J µ | p , σ (cid:105) (cid:26)(cid:26) CP as¯˜ u p (cid:48) , σ (cid:48) (cid:20) ˜ F γ µ + ( ˜ F + i ˜ F γ ) i σ µν q ν m N (cid:21) ˜ u p , σ = ¯ u p (cid:48) , σ (cid:48) (cid:20) ˜ F γ µ + e i αγ ( ˜ F + i ˜ F γ ) i σ µν q ν m N (cid:21) u p , σ = ¯ u p (cid:48) , σ (cid:48) (cid:20) F γ µ + ( F + iF γ ) i σ µν q ν m N (cid:21) u p , σ . omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki
In the original lattice calculation [8], while the CP violation effect on the kinematic coefficient inEq. (2.8) have been correctly taken into account, the inadequate definition of these form factors of˜ F , , has been used prior to [14]. As a result, all previous lattice results has a spurious contributionsto the EDFF computed with ˜ F from the Pauli form factor F as˜ F = F − α F . (2.9)Thus if the "Old" formula ( ˜ F ) is used for extracting the nEDM ˜ d n = ˜ F ( ) / ( m N ) , there is acorrection from the spurious mixing, which becomes significant when α becomes large.The inconsistency between "Old" and "New" formula can be directly and numerically con-firmed by comparing with computing nEDM from the energy shift method. The uniform electricfield that preserves translation invariance and the (anti-) periodic boundary conditions on a latticewas first introduced in [20] to study the nEDM, and also applied to CP-even quantities such aselectric polarizability and magnetic moments of the nucleon [21, 22]. The uniform backgroundelectric field is analytically continued to an imaginary value, so that the nucleon energy shift dueto nEDM becomes imaginary. Expanding the two-point function up to the first order in θ we candirectly extract the nEDM contribution that is linear in t . For simplicity we only consider the neu-tral particles, since the correlation function of charged particles is more complicated. To introducethe background electric field we consider the following Euclidean U ( ) vector potential A x , j = n i j Φ i j x i , A x , i | x i = L i − = − n i j Φ i j L i x j , (2.10)where Φ µν = π L µ L ν is the quantum field flux on a plaquette ( µν ) and n µν is the correspondingnumber of quanta. The (anti-) periodicity on a lattice in both space and time requires the Diracquantization conditions Q q Φ µν L µ L ν = π n µν , (cid:18) Q u = , Q d = − (cid:19) (2.11)The electric field vector is then given as (cid:126) E = ( n Φ , n Φ , n Φ ) . The corresponding effectiveDirac equation for nucleon field ˜ N is given as (cid:18) / p − ( ˜ κ + i ˜ ζ γ ) F µν σ µν m N − m N e − i αγ (cid:19) ˜ N = , (2.12)where ˜ κ = ˜ F ( ) and ˜ ζ = ˜ F ( ) are the effective anomalous magnetic and electric moments in thebasis of ˜ N with γ mass. It is obvious that the (cid:8)(cid:8) CP phase e i αγ can be completely rotated away bya field redefinition N = e i αγ ˜ N , where the two couplings also transform e i α ( ˜ κ + i ˜ ζ ) = ( κ + i ζ ) .Considering an electric field in z -direction (cid:126) E = ( , , E z ) in the rest frame p µ = ( (cid:126) , E s ) , we obtainan on-shell solution spinor u E z , s for the spin polarized along z -direction with s = ± , which hasa spin-dependent energy eigenvalue E s = m N − ζ m N ( sE z ) + O ( E z ) . From the result we see thatthe resulting energy shift is consistent with the EDFF obtained in the basis of N without γ mass,and κ = F ( ) and ζ = F ( ) are the nucleon magnetic and electric dipole moment coefficients.By taking the analytic continuation of the electric field E z to the imaginary value we obtain theenergy shift of a nucleon on lattice as ˜ E ± = m N ± δ E , with δ E = − ζ / ( m N ) iE z . To extract the3 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki nucleon energy shift we calculate the following nucleon two point functions in the presence of thebackground electric fields C Q ,(cid:126) E ( (cid:126) p , t ) = ∑ (cid:126) y e − i (cid:126) p · (cid:126) y (cid:104) N ( (cid:126) y , t ) | ¯ N ( (cid:126) , ) ∑ x [ q ( x )] (cid:105) (cid:126) E . (2.13)Using the solution spinor u E z , s and ignoring excited states we can expand C Q ,(cid:126) E ( (cid:126) p , t ) as C Q ,(cid:126) E ( (cid:126) , t ) = | Z N | ∑ s = ± ˜ u E z , s ¯˜ u E z , s e − ˜ E s t E s = | Z N | (cid:20) + γ ( − Σ z δ Et ) + i α γ + Σ z κ E z m γ (cid:21) e − m N t + O ( δ E , E z ) , (2.14)where Σ z = − i γ x γ y . Using the standard ratio method we define an “effective” energy shift ζ eff n ( t ) = m N d n = − m N E z ( R ( t + ) − R ( t )) , and R ( t ) is defined as a ratio of two-point functions with twodifferent spin projections R ( t ) = Tr [( T S z + − T S z − ) C (cid:26)(cid:26) CP2pt ,(cid:126) E ( (cid:126) , t )] [ T p C (cid:26)(cid:26) CP2pt ,(cid:126) E ( (cid:126) , t )] , (2.15)where we use spin polarization projection operators T S z ± = + γ ( ± Σ z ) , and T p = ( + γ ) . Thus thenEDM in the energy shift method is independent from the parity mixing ambiguity, from which wenumerically verify the consistency with the “New” formula ( F ). For more detail on the analyses,see Ref. [14].We show some nEDM results in comparison between the form factors and the energy shiftmethods. Fig. 1 shows how the spurious mixing affects the result for the θ -EDM. As shown in thefigure, the magnitudes of F ( Q ) with "New" are smaller than the "Old" values. We also show theresult for nEDM ζ eff n computed from the effective energy shift in Fig. 2, where we computed withtwo values of flux quanta n = n = ± ±
2. A plateau for the effective energy ζ eff n at t = ∼ | n | = | d θ n | (cid:46) . e m N at m π =
330 MeV, we estimate an extrapolated value to thephysical point based on a naive scaling of the ChPT expectation d n ∼ m q ∼ m π , which suggeststhat the signal becomes weaker as the quark mass is approaching to the physical point and wewould obtain a smaller value of | d θ n ( m phys π ) | (cid:46) . e m N . From our findings in order to promote adirect calculation of nEDM at the physical point, various noise reduction techniques that work inparticular for gluon operator are required in addition to a significant increase in statistics.
3. Noise reduction technique for θ -EDM As explained in the previous section, θ -induced nEDM would be extremely challenging inthe physical point, because the topological charge fluctuation dominates the large statistical noisegrowing with lattice volume V as δ Q = (cid:104) Q (cid:105) ∝ V , while the signal becomes small. Since the4 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki op -0.4-0.200.20.40.6 F ( Q ) New |Q |=0.22 [GeV ]New |Q |=0.44 [GeV ]New |Q |=0.66 [GeV ]OldOldOld Figure 1:
Plateaus for the nucleon EDFF F ( Q ) from QCD θ term for t op =
8. The results with "Old"include spurious mixing with the F . Results areshown for a lattice ensemble of domain wall fermionconfigurations of 24 ×
64 for m π =
330 MeV. ζ e ff ( t ) |n|=1|n|=2 Figure 2:
Effective nEDM ζ eff n from QCD θ term.Results are shown for a lattice ensemble of domainwall fermion configurations of 24 ×
64 for m π = global topological charge is zero on average because our QCD action is CP-even, F ( Q ) is thesignal of the correlation between the gluon operator and the fermionic (nucleon) functions as shownin Eq. (2.4). Thus it has been originally suggested that truncation of the topological charge sum at alarge distance from the nucleon position can reduce fluctuations of the nEDM [13] (also known asa cluster decomposition of disconnected diagrams [23]), since contributions of Q at large distancemay be neglected in computing the nEDM, while its correlation has a large noise which is notsuppressed with space-time distance due to the global nature of the topological charge. To extendthis method, in Ref. [24] we consider a generalized reduced topological charge density whichseparately restrict time and space to a cylindrical volume V Q ,˜ Q ( ∆ t Q , r Q ) = π ∑ x ∈ V Q Tr (cid:2) ˆ G µν ˜ˆ G µν (cid:3) x , ( (cid:126) x , t ) ∈ V Q : (cid:40) | (cid:126) x − (cid:126) x | ≤ r Q , t − ∆ t Q < t < t + t sep + ∆ t Q , (3.1)where t and t + t sep are the positions of the nucleon source and sink. This setup is illustratedin Fig. 3, where the three-point functions are inside entirely the region V Q and the truncation in t -direction for ˜ Q is symmetric with respect to the nucleon sources and sinks for both two- andthree-point functions. We also set (cid:126) x to the nucleon source position to further reduce the noise atlarge distances. We should note that a spatial restriction may introduce another bias for nEDM. Inorder to avoid such ambiguity, the convergence with r Q and ∆ t Q in Eq. (3.1) must be verified ateach nucleon momenta, especially in computing the Q dependence of the EDFFs. Figure 3:
Truncated sampling of the topologicalcharge density (3.1) for reducing the noise in the CP-odd nucleon correlation functions (2.3) and (2.4). Thecorrelation with the points outside V Q is expected tobe suppressed but gives a large noise. omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki
We use the lattice QCD ensemble of dynamical domain wall fermion with heavy pion mass m π =
330 MeV. We calculate 64 low-precision and 1 high-precision samples using the AMA sam-pling method [25]. The topological charge density is calculated using the “5-loop-improved” fieldstrength ˆ G µν [26] with the gradient flow ( t g f = a ). The r Q and ∆ t Q dependence of the mixingangle α and the EDFF for the neutron (connected diagram only) are shown in Figs. 4 and 5, wherewe observe error reduction for smaller values of r Q and ∆ Q and convergence for r Q ≥ ∆ t Q ≥ α and F ( Q ) . We have also performed a calculation using ensembles at the physical point on48 ×
96 lattice with 33,000 statistics. Unfortunately we have found no signal for neutron EDFFs(See Fig. 6), and the results are consistent with zero with the statistical uncertainty. Our result ofsignal-to-noise ratio ∼ . m π =
330 MeV indicates that the expected signal-to-noise ratio at thephysical point has to be improved by a factor of ∼
10 which requires × O ( ) more statistics toconfirm the existence of the strong CP problem. t − . − . . . . . . . . α r Q = 8 ∆ t Q = 2∆ t Q = 4∆ t Q = 8 ∆ t Q = 12∆ t Q = 32 t α r Q = 12 0 2 4 6 8 10 12 14 t α r Q = 16 0 2 4 6 8 10 12 14 t α r Q = ∞ Figure 4:
Dependence of the spatial and temporal cuts ( r Q , ∆ t Q ) in the reduced topological charge (3.1) onthe nucleon parity mixing angle α . . . . . . . F p , ✓ | t | = , | r | F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] . . . . . . . . . . F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . 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F p , ✓ | t | = , | r | T =7 aT =8 aT =9 a F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] . . . . . . . . . . F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . F p , ✓ | t | = , | r | T =7 aT =8 aT =9 a F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] . . . . . . . . . . F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . F p , ✓ | t | = , | r | T =7 aT =8 aT =9 a F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | Q [GeV ] F p , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] . . . . . . . . . . F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | . . . . . . . . . Q [GeV ] F n , ✓ | t | = , | r | Figure 5:
Neutron EDFF induced by θ -term and its dependence of the spatial and temporal cuts ( r Q , ∆ t Q )at m π =
330 MeV.
A systematic analysis of the truncation method based on spectrum decomposition is proposedin Refs. [15, 16]. This study uses a truncation of the topological charge density operator summedover spatial directions ¯ Q defined as¯ Q ( t Q ) = (cid:90) d xq ( x , t Q ) , q ( x , t Q ) = π G ˜ G ( x , t Q ) (3.2)where the topological charge density q ( x , t Q ) is again calculated using the gradient flow method.To calculate the nucleon mixing angle, the following modified two-point function ∆ C ( t , t Q ) = (cid:104) T { N ( T ) ¯ Q ( t Q ) ¯ N ( ) }(cid:105) , (3.3)is considered. The spectrum decomposition for ∆ C ( t , t Q ) reveals its t Q dependence and provide asystematic way to estimate the truncation error. For example, in the case of 0 < t Q < t , its spectral6 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki − . − . − . . . F p , θ | ∆ t | = , | r | ≤ T = 8 aT = 9 aT = 10 a F p , θ | ∆ t | = , | r | ≤ .
00 0 .
05 0 .
10 0 .
15 0 . Q [GeV ] − . − . . . . . F n , θ | ∆ t | = , | r | ≤ .
00 0 .
05 0 .
10 0 .
15 0 . Q [GeV ] F n , θ | ∆ t | = , | r | ≤ Figure 6:
Preliminary results for the proton and neutron EDFF induced by θ -term and its dependence ofthe spatial and temporal cuts ( r Q , ∆ t Q ) at m π =
139 MeV [24]. decomposition has the form ∆ C ( t , t Q ) = (cid:104) N ( t ) ¯ Q ( t Q ) ¯ N ( ) (cid:105) ∼ ∑ n , m e − E n ( t − t Q ) − E m t Q (cid:104) | N | n (cid:105)(cid:104) n | ¯ Q | m (cid:105)(cid:104) m | ¯ N | (cid:105)∼ ∑ m (cid:54) = n cosh ( ∆ m mn ( t Q − t / )) , (3.4)where ∆ mn = E m − E n . Note that two states of n and m should have different intrinsic parities,otherwise the contribution vanishes since the matrix element should be zero, e.g., (cid:104) n | ¯ Q | n (cid:105) = n has even or odd parity. On the other hand, in the case of t < t Q , it has ∆ C ( t , t Q ) = (cid:104) ¯ Q ( t Q ) N ( t ) ¯ N ( ) (cid:105) ∼ ∑ n , m e − E n t Q − E m t (cid:104) | ¯ Q | n (cid:105)(cid:104) n | N | m (cid:105)(cid:104) m | ¯ N | (cid:105)∼ ∑ n (cid:48) e − E n (cid:48) t Q , (3.5)where the state n (cid:48) should be a P-odd state that couples to a nucleon state with a non-zero value ofthe matrix element (cid:104) | ¯ Q | n (cid:48) (cid:105) (cid:54) =
0. From the spectrum decomposition we see that the nucleon mixingangle α is a mixing parameter between the ground state nucleon and CP-odd excited states. Tosee the truncation artefacts, the authors of Refs. [15, 16] consider the partial summed two-pointcorrelation function C ¯ Q ( t s ) = t s ∑ t Q = − t s ∆ C ( t , t Q ) . From Eqs. (3.4) and (3.5) its asymptotic formbehaves like C ¯ Q ( t s ) = A + Be − Et s for t s (cid:38) t , where the contribution for large t s is expected to beexponentially suppressed. This is numerically checked as shown in the left panel of Fig. 7, wherea plateau is obtained for t s ∼ t and the contributions from t s > t seem to be below the statisticalfluctuation. Neglecting unnecessary noise from t s (cid:38) t , an improvement of α up to a factor 2 isobtained. The same spectrum decomposition can be applied to the modified nucleon three pointfunction ∆ C ¯ Q ( t , t Q , t op ) = (cid:104) T { N ( T ) J µ ( t op ) ¯ Q ( t Q ) ¯ N ( ) }(cid:105) , where a fit analysis using its asymptoticform with the α -improvement yields a factor of 2 ∼ F ( Q ) . The right panel of Fig. 7 shows a chiral (and continuum) extrapolation for d θ n using7 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki six ensembles with m π >
410 MeV for several lattice spacings. A ChPT fit ansatz d θ n ∼ m q ∼ m π is used to further constrain the nEDM towards the chiral limit, which yields d θ n = − . ( ) θ e fm at the physical point with ∼ σ deviation from zero. This result is consistent with theaforementioned naively scaled value | d θ n ( m phys π ) | (cid:46) . e m N [24]. Their fitted data, however, are inheavy pion mass region and do not clearly show the chiral behavior d θ n ∝ m π , so that the fit resultsseem to be less convincing. To avoid model dependence, more accurate results near the physicalpoint are needed. t s [ fm ] α N t s = tt s = t t s = t Nucleon Mixing Angle α N over t s m π = 410 MeV t = 0 . fmm π = 570 MeV t = 0 . fmm π = 700 MeV t = 0 . fm m π [ MeV ] d n [ e f m ] Continuum
Neutron, d n ( a, m π ) Fit
Neutron d n = − . × − ¯ θ e fm Figure 7:
Results presented in Ref. [16]. (Left) The improved nucleon mixing angle α plotted against thesum parameter t s . (Right) A chiral and continuum extraplation of nEDM d θ n plotted as a function of m π .The bands are the fit results. New results near the physical point ( m π = O ( a ) -improved field strength[27] with the gradient flow on MILC HISQ ensembles with a = . − .
15 fm. This study alsouses the truncation method in t -direction. While the convergence properties in partial sum can beseen in both two- and three- point functions, due to the slow convergence at the physical point nosignificant improvement is observed. It is, however, remarkable that the number of measurementsis O (100k), which gives a statistically significant signal for F ( Q ) with non-zero Q near thephysical point (see the left panel of Fig. 8). It is also reported that a variance reduction techniqueintroduced in [28] has about 25% error reduction. In this calculation the excited state contaminationis removed by using the two-state fits with multiple source-sink separations for each momentum Q . The results for F ( Q ) at heavier m π with non-zero Q are consistent with the previous latticeresults [24, 16]. The result from the chiral (interpolation) and continuum extrapolation fit for d θ n are also presented in the right panel of Fig. 8, which yields a non-zero signal of | d θ n | = . ( ) θ e fm at the physical point. While this result is also consistent with estimates from ChPT analysis andthe QCD sum rules [4], it is not sufficient to constrain the θ parameter. We also notice that eventhough the values of d θ n at finite a are all positive except for the data at m π =
135 MeV, these valuesin the continuum limit become negative, which may indicate a sizable discretization effect on d θ n .In addition, there is an increasing tendency of F ( Q ) towards Q , m π → d θ n in the simulation mass region, and understandingthe Q dependence of F ( Q ) would be important to precisely determine d θ n at the physical point.8 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki
We have shown in this section that there are several ongoing studies of the θ -induced nEDM.Even having O (100k) statistics with employing noise reduction techniques at the physical point,it still is not sufficient to constrain the θ parameter due to the large fluctuation of the topologicalcharge density, which should become even worse when approaching to the continuum and largervolume limit. There also are several systematic uncertainties due to finite lattice spacing and Q → F ( Q ) , which need to be further explored. Thus θ -induced EDM at thephysical point will be extremely challenging and will require more special techniques that workwell in particular for gluon operators. In the next section, we would like to propose a new approachusing a matrix element of the nucleon with background electric field. Since this approach is basedon the energy shift method, we can directly obtain d θ n without Q extrapolation, which may bepotentially advantageous over the form factor method. -0.020-0.0100.0000.0100.0200.0300.0400.050 0 0.1 0.2 0.3 0.4 0.5 F , n θ ( Q ) / M N (f m ) Q (GeV ) a06m135a09m310a09m220 a09m130a12m310a12m220 a12m220La15m310 -0.04-0.020.000.020.040.06 0 0.03 0.06 0.09 0.12 χ /dof=0.57 F , n θ ( Q = ) / M N (f m ) M π [GeV ] a06m135a09m310a09m220 a09m130a12m310a12m220 a15m310Extrap Figure 8:
Preliminary results presented in Ref. [17]. (Left) The Q dependence of F ( Q ) / ( m N ) . (Right)A chiral and continuum extraplation of nEDM d θ n plotted as a function of m π .
4. New approach based on the matrix element with the background electric field
The idea is to simply apply the truncation technique to the energy shift method in the presenceof the background electric field, in which we find that the energy shift δ E in Eq. (2.14) is givenas a nucleon matrix element of the topological charge density operator ¯ Q in Eq. (3.2). Throughoutthe section, the state’s momenta are set to zero and arguments of (cid:126) p are suppressed. Performing thespectrum decomposition of ∆ C ( t , t Q ) as in Eq. (3.4) but now in the presence of the backgroundelectric field, we obtain the following result for t > t Q as ∆ C ,(cid:126) E ( t , t Q ) = (cid:104) N ( t ) ¯ Q ( t Q ) ¯ N ( ) (cid:105) (cid:126) E = ∑ n , m e − E n ( t − t Q ) − E m t Q (cid:104) | N | n , (cid:126) E (cid:105)(cid:104) n , (cid:126) E | ¯ Q | m , (cid:126) E (cid:105)(cid:104) m , (cid:126) E | ¯ N | (cid:105)∼ | Z N + | e − m N + t (cid:104) N + , (cid:126) E | ¯ Q | N + , (cid:126) E (cid:105) , (4.1)where | N + , (cid:126) E (cid:105) is the ground state nucleon in the presence of the background electric field. Weshould note that in contrast to Eq. (3.4), there is a leading order contribution from the groundstate nucleon given as a matrix element (cid:104) N + , (cid:126) E | ¯ Q | N + , (cid:126) E (cid:105) . Again taking the partial summation of9 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki ∆ C ,(cid:126) E ( t , t Q ) over sink-source separation t , we obtain C ¯ Q ,(cid:126) E ( t ) = t ∑ t Q = ∆ C ,(cid:126) E ( t , t Q ) ∼ | Z N + | e − m N + t ( t (cid:104) N + , (cid:126) E | ¯ Q | N + , (cid:126) E (cid:105) ) , (4.2)where the matrix element is given the coefficient of linear in t . Comparing Eq. (2.14) with Eq. (4.2),we find that the matrix element should correspond to the energy shift (cid:104) N + , (cid:126) E | ¯ Q | N + , (cid:126) E (cid:105) = − ζ m N ¯ u (cid:104) (cid:126) Σ · (cid:126) E (cid:105) u + O ( (cid:126) E ) . (4.3)This formula is analogous to the leading order energy correction in the perturbation theory ofquantum mechanics, c.f., ∆ E n = (cid:104) n | ∆ ˆ H | n (cid:105) for Hamiltonian ˆ H = ˆ H + ∆ ˆ H . In this case we regardthe electric field as a perturbation in addition to the CP-odd Hamiltonian ( θ -term), and also takeinto account a leading order perturbation effect on the state | N + , (cid:126) E (cid:105) . Even without θ -term, theground state | N + , (cid:126) E (cid:105) could mix with CP-odd states [29] due to the background electric field as | N + , (cid:126) E (cid:105) = | N + (cid:105) + c (cid:126) E · (cid:126) D | N − (cid:105) + · · · , (4.4)where | N + (cid:105) is the parity-even ground state nucleon in (lattice) CP-even vacuum. The leading ordercorrection should come from a parity-odd nucleon | N − (cid:105) with an overlap coefficients of c and theelectric field (cid:126) E and an expectation value of the dipole operator (cid:126) D [29]. Substituting Eq. (4.4) intoEq. (4.3), we obtain (cid:104) N + , (cid:126) E | ¯ Q | N + , (cid:126) E (cid:105) = (cid:126) E · (cid:126) D (cid:104) N + | ¯ Q | N − (cid:105) + ( c . c . ) + · · · , (4.5)where we note (cid:104) N ± | ¯ Q | N ± (cid:105) = | N ± (cid:105) is defined in CP-even QCD vacuum. Thus the contribution of the matrixelement (cid:104) N + | ¯ Q | N − (cid:105) is exactly the same as the parity mixing effect that appears in calculation of α in Eq. (3.4). In the perturbation theory point of view, the EDM is an interplay of the electric fieldand the CP-odd operator both in the first order perturbation. Using the standard ratio method as inEq. (2.15) and the relation to the matrix element in Eq. (4.3), we obtain the modified energy shiftformula R S z ± ( t , t Q ) = Tr [ T S z ± ∆ C ,(cid:126) E ( t , t Q )] Tr [ T p C pt ,(cid:126) E ( t )] → ∓ ζ m N E z , ( t → ∞ ) , (4.6)From this formula, it is clear that we do not need to extend t Q outside the sink-source position,since there is no other term that is proportional to t in t Q > t . In fact the contributions from t Q > t should be excited state contaminations that should disappear in the limit t → ∞ . Thus, without Q extrapolation, the EDFF F ( ) can be directly extracted from the ratio by dividing by the electricfield as | F ( ) | = lim t → ∞ m N | R S z ± ( t , t Q ) || E z | .We show our preliminary results on the θ -induced nEDM from the matrix element approach.Since we compute the matrix element with the electric fields along z -direction in both positiveand negative, we have four results for each component of ± E z and projections T S z ± (see Fig. 9).Obviously these data are correlated with each other and differences between spin up (or positive10 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki E z ) and spin down (or negative E z ) reduce the error. Fig. 10 shows the gradient flow time t g f dependence of F ( ) for each sink-source time separation t = T . As expected, the signal becomesbetter as increasing the flow time, and the result for F ( ) becomes stable at t g f ≥ T ≥ (cid:126) E . Fig. 11 shows the comparison of two resultsfor F ( ) with different electric field strength with | n | = | n | = T ≥
8, This result indicates that O ( (cid:126) E ) corrections on F ( ) aresmall. From the plateau at T = t g f = F ( ) = . ( ) for the neutron at m π = Q = Q Sz+, E+Sz+, E-Sz-, E+Sz-, E-
T=8, |n|=2 Q ( T=8, |n|=2
Figure 9:
Preliminary results of F ( ) from the matrix element approach. Results are obtained on a 24 × m π =
330 MeV. 4 different components (Left) and their linear combinations for obtaining bettersignals (Right).
5. Summary
Lattice calculations of nEDM are important for interpreting CP-violation effects in EDM ex-periments and cosmological observations. A number of groups are putting in the effort requiredfor computing nEDM at the physical point using the form factor method. The nEDM induced by θ -term has a large statistical noise in its correlation to the topological charge density, which is notsuppressed at a large distance due to its global nature. To reduce the error several noise reductiontechniques using the truncation of (space and) time region of the topological charge density havebeen employed. It is found that while the truncation reduces the error by a factor of 2 at a heavierpion mass, due to its poor convergence no significant improvement is observed at the physical pionmass. There also are several systematic uncertainties in F ( Q ) . The uncertainties of the discretiza-tion effect, Q → F ( Q ) have not been wellunderstood, which seem to become significant near the physical point. To control the systematicerrors and to further improve the statistical signal, we have proposed a new approach using the ma-trix element with background electric fields. This method only requires a local topological charge11 omputing Nucleon Electric Dipole Moment from lattice QCD Hiroshi Ohki . . . . . . . t gf = 1 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 2 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 4 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 8 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 1 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 2 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 4 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 8 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 1 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 2 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 4 . a T=5T=6 T=7T=8 T=9T=10 . . . . . . . t gf = 8 . a T=5T=6 T=7T=8 T=9T=10
Figure 10:
Gradient flow time dependence of F ( ) for each sink-source time separation T . The horizontalaxis represents t Q − T /
2. Results are obtained on a 24 ×
64 lattice with m π =
330 MeV and | n | = -6 -4 -2 0 2 4 600.050.10.150.20.25 T=5T=6T=7T=8T=9T=10 |n|=1 -6 -4 -2 0 2 4 600.050.10.150.20.25
T=5T=6T=7T=8T=9T=10 |n|=2
Figure 11:
Electric field dependence of F ( ) for | n | = t Q − T /
2. Results are obtained on a 24 ×
64 lattice with m π =
430 MeV. density operator between sink and source positions, and thus can avoid the large topological noiseat a large distance. In addition, no Q extrapolation is required since the forward matrix elementis directly obtained from the energy shift. Our preliminary results have demonstrated that we canachieve statistically-significant signal at heavier pion masses that are consistent with the previousresults. This method can in principle be applied to any (cid:8)(cid:8) CP operators, in which the Weinberg’s three-gluon operator especially is beneficial for this method, since there is no additional computation costfor the gluonic operator. We need further investigations at the physical point.12 omputing Nucleon Electric Dipole Moment from lattice QCD
Hiroshi Ohki
Acknowledgments
The speaker would like to thank the organizers of Lattice2019 in Wuhan for the invitation.We would also like to thank Tanmoy Bhattacharya and Boram Yoon for sending us their mate-rials and the fruitful discussions. We are grateful for the gauge configurations provided by theRBC/UKQCD collaboration. This research used resources of the Argonne Leadership ComputingFacility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357, and Hokusai supercomputer of the RIKEN ACCC facility. H.O. is supported in partby JSPS KAKENHI Grant Numbers 17K14309 and 18H03710. S.S. is supported by the NationalScience Foundation under CAREER Award PHY-1847893.
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