The Contribution of Novel CP Violating Operators to the nEDM using Lattice QCD
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The Contribution of Novel CP Violating Operators to the nEDMusing Lattice QCD
Rajan Gupta , a , Tanmoy Bhattacharya , Vicenzo Cirigliano , and Boram Yoon Los Alamos National Laboratory, Theoretical Division, T-2, Los Alamos, NM 87545
Abstract.
In this talk, we motivate the calculation of the matrix elements of novel CPviolating operators, the quark EDM and the quark chromo EDM operators, within thenucleon state using lattice QCD. These matrix elements, combined with the bound on theneutron EDM, would provide stringent constraints on beyond the standard model physics,especially as the next generation of neutron EDM experiments reduce the current bound.We then present our lattice strategy for the calculation of these matrix elements, in par-ticular we describe the use of the Schrodinger source method to reduce the calculation ofthe 4-point to 3-point functions needed to evaluate the quark chromo EDM contribution.We end with a status report on the quality of the signal obtained in the lattice calculationsof the connected contributions to the quark chromo EDM operator and the pseudoscalaroperator it mixes with under renormalization.
The observed universe has 6 . + . − . × − baryons for every black body photon [1], whereas in abaryon symmetric universe, we expect no more that about 10 − baryons for every photon [2]. It isdi ffi cult to include such a large excess of baryons as an initial condition in an inflationary cosmologicalscenario [3]. The way out of the impasse lies in generating the baryon excess dynamically during theevolution of the universe.In the early history of the universe, if the matter-antimatter symmetry was broken post inflationand reheating, then one is faced with Sakharov’s three necessary conditions [4]: the process has toviolate baryon number, evolution has to occur out of equilibrium, and charge-conjugation and T (orequivalently CP if CPT remains a good symmetry) invariance has to be violated.CP violation ( (cid:8)(cid:8) CP) exists in the standard model (SM) of particle interactions due to a phase in theCabibo-Kobayashi-Maskawa (CKM) quark mixing matrix [5], and possibly by a similar phase in theleptonic sector, given that the neutrinos are not massless [6]. (cid:8)(cid:8)
CP arising from the CKM matrix in theSM contributes at O (10 − ) e-cm, much smaller than the current experimental bound d n < . × − e-cm. Thus the nEDM puts no constrain on the SM and the strength of the (cid:8)(cid:8) CP in the CKM matrix ismuch too small to explain Baryogenesis.In principle, the SM has an additional source of (cid:8)(cid:8)
CP arising from the e ff ect of QCD instantons. Thepresence of these finite action non-perturbative configurations in a non-Abelian theory often leads toinequivalent quantum theories defined over various ‘ Θ ’-vacua [7]. Because of asymptotic freedom, a Speaker. e-mail: [email protected]. Los Alamos Report LA-UR-16-29579 a r X i v : . [ h e p - l a t ] J a n PJ Web of Conferences all non-perturbative configurations including instantons are strongly suppressed at high temperatureswhere baryon number violating processes occur. Because of this, (cid:8)(cid:8)
CP due to such vacuum e ff ects donot lead to appreciable baryon number production. In short, additional much larger (cid:8)(cid:8) CP is needed fromphysics beyond the SM (BSM).To determine whether such additional (cid:8)(cid:8)
CP exists, a very promising approach is to measure thestatic electric dipole moments of elementary particles, atoms and molecules, which are necessarilyproportional to their spin. Since under time-reversal the direction of spin reverses but the electricdipole moment does not, a non-zero measurement confirms T violation or equivalently (cid:8)(cid:8)
CP. Of theelementary particles, atoms and nuclei that are being investigated, the electric dipole moment of theneutron (nEDM) is the laboratory where lattice QCD can provide the theoretical part of the calculationneeded to “connect” the experimental bound (value) on the nEDM to the strength of (cid:8)(cid:8)
CP in a givenBSM theory.Most extensions of the SM have new sources of (cid:8)(cid:8)
CP. Each of these contributes to the nEDM andfor some models it can be as large as 10 − e-cm. Planned experiments are aiming to reduce the boundon d n < . × − e-cm to d n (cid:46) × − e-cm. The strategy for finding out which class of BSMtheories are viable candidates is as follows: As the bound on the nEDM is lowered in current andplanned experiments, BSM theories with (cid:8)(cid:8) CP giving rise to a nEDM larger than this bound get ruledout provided the “connection” between the bound and the couplings is known su ffi ciently accurately.This “connection” is the matrix elements of the novel CP violating interactions within the neutronstate that simulations of lattice QCD are gearing up to provide. Many candidate BSM theories have been proposed by theorists. While the true BSM theory is notknown, one can write down all possible (cid:8)(cid:8)
CP interactions at the energy scale of hadronic matter ( ∼ few GeV) in terms of quark and gluon fields based on symmetry and organized by their dimension;operators with higher dimension being, in general, suppressed. At the lowest dimension five, thereare two leading operators called the quark EDM (qEDM) and the quark chromo EDM (CEDM). TheQCD Lagrangian in the presence of the Θ -term, the qEDM and the CEDM operators is L QCD −→ L (cid:26)(cid:26)
CPQCD = L QCD + i Θ G µν ˜ G µν + i (cid:88) q d γ q q Σ µν ˜ F µν q + i (cid:88) q d Gq q Σ µν ˜ G µν q (1)where the sum is over all the quark flavors, Σ µν = [ γ µ , γ ν ] /
2, and F µν and G µν are the electromagneticand chromo field strength tensors. The couplings d γ q are the qEDMs and the d Gq are the CEDMs.They parameterize new (cid:8)(cid:8) CP in BSM theories. The goal is to constrain BSM models by boundingthese couplings using the experimental bound on the neutron EDM and lattice QCD calculationsof the matrix elements of the corresponding operators. In other words, the matrix elements of theelectromagnetic current J EM µ between neutron states in the presence of CP violation provides the“connection” between the BSM couplings and the nEDM.In the presence of (cid:8)(cid:8) CP, the electromagnetic current, defined as δ L /δ A µ , gets an additional term: e (cid:88) q q γ µ q −→ e (cid:88) q q γ µ q + (cid:15) ρσνµ p ν (cid:88) q d γ q q Σ µν q . (2)The matrix elements of this leading qEDM (second) term are the flavor diagonal tensor charges g qT : (cid:104) N | J EM µ | N (cid:105) (cid:12)(cid:12)(cid:12) qEDM , (cid:54) CP = (cid:15) µνκλ q ν (cid:28) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) d γ u ¯ u Σ κλ u + d γ d ¯ d Σ κλ d + d γ s ¯ s Σ κλ s + . . . (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:29) , lim q → d γ u g uT + d γ d g dT + d γ s g sT + . . . . (3) ONF12
QEDM operator Θ -term σ µν γ σ µν γ γ µ Q γ µ Q Figure 1.
Calculating nEDM requires evaluating the interaction of the em current with the neutron. The contri-bution of connected and disconnected diagrams from the (left) qEDM and (right) Θ -term. The connected and disconnected (with u , d , s and c quark loops) Feynman diagrams for the qEDMcontribution are shown in Fig. 1 (left). The first lattice results for the qEDM were given in [8] andphenomenological consequences for a particular BSM theory, Split SUSY, were presented in Ref. [9].The key computational challenge to evaluating Eq. (3) is the calculation of the disconnected contri-butions to g qT . These were shown to be small and noisy, and decrease with the quark mass [8]. Theerrors in the strange disconnected contribution were small enough to yield the continuum limit esti-mate g sT = . (cid:8)(cid:8) CP interactions are Yukawa like,i.e., the couplings are proportional to the quark mass. Consequently, the impact of the small value of g sT with O (1) error, gets magnified by the quark mass ratio 2 m s / ( m u + m d ) =
27. Thus, it is importantto improve the estimate of g sT and, for the same reason, calculate g cT .Contributions from the CEDM and the next order in quark EDM arise due to the change inthe action, L QCD −→ L (cid:26)(cid:26)
CPQCD . In an ideal world, the best way to calculate these e ff ects would beto simulate the lattice theory using a discretized version of L (cid:26)(cid:26) CPQCD and compute the matrix element (cid:104) N | J EM µ | N (cid:105) (cid:12)(cid:12)(cid:12) (cid:26)(cid:26) CP . This ideal approach is not practical because (cid:8)(cid:8) CP interactions are complex and latticesimulations of theories with a complex action are not yet realistic.The method of choice is to treat these (cid:8)(cid:8)
CP interactions as perturbations in the small couplings d γ q and d Gq , and expand the theory about L QCD (see Sec. 3). Then, the leading terms that contribute to thenEDM are the matrix elements of the product of the electromagnetic current J EM µ and the 4-volumeintegral of the operators. For the CEDM operator, the matrix elements that have to be calculated are (cid:104) N | J EM µ | N (cid:105) (cid:12)(cid:12)(cid:12) CEDM (cid:26)(cid:26) CP = (cid:42) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J EM µ (cid:90) d x [ (cid:16) d Gu ¯ u Σ νκ u + d Gd ¯ d Σ νκ d + d Gs ¯ s Σ νκ s + . . . (cid:17) ˜ G νκ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:43) , (4)as discussed in Sec. 3. Similarly for the next order qEDM (cid:104) N | J EM µ | N (cid:105) (cid:12)(cid:12)(cid:12) qEDM , (cid:26)(cid:26) CP = (cid:42) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J EM µ (cid:90) d x [ (cid:16) d Fu ¯ u Σ νκ u + d Fd ¯ d Σ νκ d + d Fs ¯ s Σ νκ s + . . . (cid:17) ˜ F νκ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:43) . (5)This second qEDM term has two electromagnetic interactions and is therefore expected to be smallerthan the leading term given in Eq. (3). For this reason we neglect its contribution. Finally, the contri-bution of the Θ -term requires calculating the matrix element: (cid:104) N | J EM µ | N (cid:105) (cid:12)(cid:12)(cid:12) Θ (cid:54) CP = (cid:42)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J EM µ (cid:90) d x Θ G µν ˜ G µν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:43) . (6)This correlation of J µ with topological charge Q gives rise to the 2 diagrams shown in Fig. 1 (right). PJ Web of Conferences e i ε u d d u d d P ε P ε P u d d u d d P P P ε u d d u d d P P ε P ε u d d u d d P P P ε P P ε P ε P + u d d u d d P ε P ε P P seq u d d u d d P P P ‐ε seq P ε u d d u d d P P ε P ε P ‐ε seq u d d u d d P P P seq P ε Figure 2.
CEDM contribution is the reweighting factor times the sum of connected and disconnected diagrams. (cid:8)(cid:8)
CP CEDM and γ operators The QCD Lagrangian becomes complex with the addition of any CP violating interaction. A robustcost-e ff ective method for simulating theories with complex actions does not exist. To calculate thecontribution of the (cid:8)(cid:8) CP interactions to the nEDM, we work within the framework of the Schwingersource method. We start by noting that the fermion part of the Lagrangian, L (cid:26)(cid:26) CPQCD in Eq. (1), involvesonly quark bilinear operators in presence of the qEDM and CEDM interactions. The fermion fields inthe path integral can, therefore, still be integrated out. There are, however, two modifications to thestandard calculations of 3-point functions due to the presence of (cid:8)(cid:8)
CP interactions treated as sources.First, the Dirac operator, in the presence of say the CEDM term, becomes / D + m − r D + c s w Σ µν G µν −→ / D + m − r D + Σ µν ( c s w G µν + i (cid:15) ˜ G µν ) . (7)Quark propagators calculated with this modified Dirac operator include the insertion of the CEDMoperator at all space-time points. Second, the Boltzmann weight of all gauge configurations generatedwithout the CEDM term in the action need to be modified by reweighting them by the ratio of thedeterminants of the Dirac operators for the two theories–with and without (cid:8)(cid:8) CP terms:det( / D + m − r D + Σ µν ( c s w G µν + i (cid:15) ˜ G µν )det( / D + m − r D + c s w Σ µν G µν ) = exp Tr ln (cid:20) + i (cid:15) Σ µν ˜ G µν ( / D + m − r D + c s w Σ µν G µν ) − (cid:21) (8) ≈ exp (cid:20) i (cid:15) Tr Σ µν ˜ G µν ( / D + m − r D + c s w Σ µν G µν ) − (cid:21) . (9)Since the coupling (cid:15) for the CEDM term for all quark flavors is small, we expect the leading orderreweighting factor to be a good approximation. This ‘reweighting factor’ for each gauge configurationis, to leading order, the integral over all spacetime points of the value of a closed quark loop with theinsertion of the CEDM operator.There is an additional challenge: The CEDM operator has an UV divergent mixing with the “ γ -operator, q γ q / a [10]. Thus, one has to (i) calculate the matrix element of the “ γ -operator”, whichwe do in the same way as for the CEDM operator, (ii) calculate a similar reweighting factor, and(iii) handle the 1 / a divergent mixing. Since the reweighting is an overall phase, and configurationto configuration fluctuations can be large, it is therefore important to demonstrate the quality of thesignal in F (0) after reweighting for both the CEDM and γ terms. In Sec. 5, we describe the progress ONF12 we have made towards addressing these challenges. There is also a mixing of the CEDM operator withthe G µν ˜ G µν operator. Addressing this mixing is part of the calculation of the Θ − term per Eq. (6) [10].The Schwinger source method has allowed us to recast the challenging calculation of 4-pointfunctions given in Eq. (4) to the di ffi cult calculation of 3-point functions [11]. The Feynman diagramsneeded to calculate (cid:104) N | J EM µ ( q ) | N (cid:105) in the presence of CEDM and γ interactions are illustrated inFig. 2. Starting with an ensemble of gauge configurations generated with a standard lattice action, forexample the Wilson-clover action, the steps in the calculation are: • Calculate propagators, labeled P in Fig. 2, using the standard Wilson-clover Dirac operator. Weassume isospin symmetry so the propagators for u and d quarks are numerically the same. • Calculate a second set of propagators with the CEDM term included with coe ffi cient (cid:15) in the Wilson-clover matrix: (cid:18) / D + m − r D + c s w Σ µν G µν (cid:19) − −→ (cid:18) / D + m − r D + Σ µν ( c s w G µν + i (cid:15) ˜ G µν ) (cid:19) − . (10)These propagators, labeled P (cid:15) , include the full e ff ect of inserting the CEDM operator at all possiblepoints along them. The cost of inversion increases by about 7% with respect to P , however, using P as the starting guess in the inversion for P (cid:15) reduces the number of iterations required by 20–40%depending on the quark mass. The overall average cost of P (cid:15) is found to be about 80% of P . • Using P and P (cid:15) , construct 4 kinds of sequential sources, labeled P seq u , P seq d , P seq − (cid:15), u , and P seq − (cid:15), d , at thesink time-slice corresponding to the insertion of a neutron at zero momentum. The subscripts u or d in P seq u and P seq d , and similarly in P seq − (cid:15), u and P seq − (cid:15), d , denote the flavor of the uncontracted spinor in theneutron source. The minus sign in the coupling, − (cid:15) , accounts for the backward moving propagator. • If using the coherent sequential source method, then N sources at di ff erent time-slices for the N dif-ferent calculations being done simultaneously on that configuration, are added together. Sequentialpropagators with these (coherent) sequential sources are calculated by inverting the Dirac operatorwith and without the CEDM term as appropriate. The 4 types of (coherent) sequential propagatorsare shown by the block of 4 correlation functions in the right half of Fig. 2. • Using the two original and the four sequential propagators, calculate the connected 3-point function.These contractions give the four 3-point functions shown on the right in Fig. 2. Note that a separateinsertion of J EM µ is done on the u and d quark lines. • Calculate the disconnected quark loop with the insertion of electromagnetic current at zero momen-tum for each of the quark flavors, u , d , s , c (and b if needed) using the Dirac propagator with andwithout the CEDM term. • Calculate the correlation of these quark loops with the appropriate nucleon 2-point correlation func-tions. The 4 types of disconnected Feynman diagrams required are illustrated in the left of Fig. 2. • To correct for the omission of the chromo EDM operator in the action during the generation of thegauge configurations, calculate the volume integral of the quark loop with the CEDM insertion foreach flavor. Multiply the sum of the connected and disconnected contributions by the reweightingfactor–exponential of the estimate of the “CEDM loop” for the configuration with the appropriatecoupling factor i (cid:15) . This is shown by the overall factor in Fig. 2. • Repeat the calculation for di ff erent values of (cid:15) that bracket the expected numerical values of thecouplings d Gq for the various quark flavors. • Repeat the whole calculation for the γ -operator instead of the CEDM operator. PJ Web of Conferences
The matrix elements, defined in Eqs. (3), (4) and (6), are extracted from the vacuum to vacuum 3-pointcorrelation function: insertion of the electromagnetic current at times t between the neutron sourceand sink operators at time 0 and T : (cid:104) Ω | N ( (cid:126) , J EM µ ( (cid:126) q , t ) N † ( (cid:126) p , T ) | Ω (cid:105) = (cid:88) N , N (cid:48) u N e − M N t (cid:104) N | J EM µ ( q ) | N (cid:48) (cid:105) e − E N (cid:48) ( T − t ) u (cid:48) N . (11)where in the right hand side we have used the expansion in a complete basis of states | N (cid:105) , | N (cid:48) (cid:105) , . . . thatcouple to the neutron interpolating operator N . We use N ≡ (cid:15) abc [ d a T C γ (1 + γ ) u b ] d c where C = γ γ is the charge conjugation matrix, a , b , c are the color indices, u , d are the quark flavors, u N is the freeneutron spinor and M N is the neutron mass. In the presence of (cid:8)(cid:8) CP, the nucleon 2-point function is (cid:104) Ω | N ( (cid:126) , N † ( (cid:126) p , t ) | Ω (cid:105)| lim t →∞ = A u N e − p t u N = A e − p t e i α N γ ( i / p + m N ) e i α N γ , (12)where the phase angle α N arises due to the (cid:8)(cid:8) CP coupling (cid:15) and depends on it. To determine the desiredregion of linearity of α versus (cid:15) we show the imaginary part of the neutron 2-point function in Fig. 3.It demonstrates the quality of the signal for α for appropriately small values of (cid:15) on two di ff erentensembles. Fig. 4 shows the expected linear behavior of α versus (cid:15) over a range of small (cid:15) . ε = 0.004 α t -0.16-0.15-0.14-0.13-0.12-0.11-0.10 0 2 4 6 8 10 12 ε γ = 0.004 α γ t ε = 0.003 α t -0.17-0.16-0.15-0.14-0.13-0.12-0.11-0.10 0 2 4 6 8 10 12 14 16 ε γ = 0.003 α γ t Figure 3.
The phase α extracted from the plateau in the imaginary part of the neutron 2-point function with (cid:15) = . on the a m ensemble (top) and (cid:15) = . on the a m ensemble (bottom). The plots on theleft are for the CEDM operator and on the right for the γ mixing operator. Once the matrix element of the electromagnetic current, J EM µ ( q ) defined in Eq. 2, within the nu-cleon state is extracted using Eq. (11), it is parameterized in terms of four Lorentz covariant form ONF12
Figure 4.
Linear behavior of the phase α versus (cid:15) with the insertion of the CEDM and the γ operators. factors F , F , F A and F : (cid:104) N | J EM µ ( q ) | N (cid:105) = u N (cid:34) γ µ F ( q ) + i σ µν q ν F ( q )2 M N + (2 i M N γ q µ − γ µ γ q ) F A ( q ) M N + σ µν q ν γ F ( q )2 M N u N , (13) F and F are the standard Dirac and Pauli form factors. The anapole form factor F A violates parityP and the electric dipole form factor F violates P and CP. F is extracted from the di ff erent matrixelements by using di ff erent combinations of the momentum transfer q µ and spin projections. The zeromomentum limit of these form factors gives the charges and dipole moments: the electric charge is F (0) = F (0) / M N . The contribution of the matrixelement of each (cid:8)(cid:8) CP interaction defined in Eqs. (3), (4), and (6) to the electric dipole moment of theneutron is given by d n = lim q → F ( q ) / M n . So far, we have performed numerical calculations on two 2 + + a m a = .
12 fm and M π =
310 MeVand the second, labeled a m a = .
09 fm and M π =
310 MeV. The correlation functions areconstructed using Wilson-clover fermions. On the a m
310 ( a m F from the connected diagrams in the presence of the CEDMterm, and in Fig. 6, the data for F in the presence of the γ term. The data are presented for threevalues of the source-sink separation t sep = ,
10 and 12. The excited-state contamination in thematrix elements, and thus in F , is removed by taking the t sep → ∞ limit. These figures show thatan acceptable signal-to-noise ratio can be obtained to estimate F with O (1) errors, our first goal forthe CEDM calculations.There are theoretical reasons to expect that the connected contribution of the γ operator is, withsmall corrections, proportional to that of the CEDM operator [13]. In Fig. 7, we show the ratio of thetwo contributions and find that this expectation is actually realized. PJ Web of Conferences -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =1 F U , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =1 F D , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =1 F U , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =1 F D , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =4 F U , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =4 F D , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =4 F U , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =4 F D , d - u τ - t sep /2 t sep =10t sep =12t sep =14 Figure 5.
Illustration of the signal in F with the inclusion of the CEDM term. The data in the top row are for (cid:126) p = (1 , , × π/ La in the following order: insertion on u and d quarks for the a m
310 ensemble followed byinsertion on u and d quarks for the a m
310 ensemble. The plots in the second row are in the same order exceptwith (cid:126) p = (2 , , × π/ La . -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =1 F U , γ , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =1 F D , γ , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =1 F U , γ , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =1 F D , γ , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =4 F U , γ , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -4 -2 0 2 4q =4 F D , γ , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =4 F U , γ , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -1.5-1-0.5 0 0.5 1 -6 -4 -2 0 2 4 6q =4 F D , γ , d - u τ - t sep /2 t sep =10t sep =12t sep =14 Figure 6.
Illustration of the signal in F with the inclusion of the γ term. The data in the top row are for (cid:126) p = (1 , , × π/ La in the following order: insertion on u and d quarks for the a m
310 ensemble followed byinsertion on u and d quarks for the a m
310 ensemble. The plots in the second row are in the same order exceptwith (cid:126) p = (2 , , To conclude, the data so far show that an acceptable signal-to-noise ratio can be obtained in theconnected contributions to F for both the CEDM and γ operator insertion. We also find that Theratio of the connected contribution of the γ to CEDM term is a constant. This implies that themixing of the γ term with the CEDM term can be cast as part of the multiplicative renormalization ofthe CEDM operator. Controlling its O (1 / a ) UV divergent coe ffi cient is a hurdle for lattice theoriesthat respect chiral symmetry (domain wall or overlap fermions) and those that don’t such as Wilson-clover fermions. Work to address the full mixing of the CEDM operator and the calculation of thedisconnected diagrams and the reweighting factors is under progress. ONF12 -6-4-2 0 2 4 6 -4 -2 0 2 4q =1 F U , γ , d - u / F U , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -6-4-2 0 2 4 6 -4 -2 0 2 4q =1 F D , γ , d - u / F D , d - u τ - t sep /2 t sep =8t sep =10t sep =12 -6-4-2 0 2 4 6 -6 -4 -2 0 2 4 6q =1 F U , γ , d - u / F U , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -6-4-2 0 2 4 6 -6 -4 -2 0 2 4 6q =1 F D , γ , d - u / F D , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -6-4-2 0 2 4 6 -4 -2 0 2 4q =4 F U , γ , d - u / F U , d - u τ - t sep /2 t sep =8t sep =10 -6-4-2 0 2 4 6 -4 -2 0 2 4q =4 F D , γ , d - u / F D , d - u τ - t sep /2 t sep =8t sep =10 -6-4-2 0 2 4 6 -6 -4 -2 0 2 4 6q =4 F U , γ , d - u / F U , d - u τ - t sep /2 t sep =10t sep =12t sep =14 -6-4-2 0 2 4 6 -6 -4 -2 0 2 4 6q =4 F D , γ , d - u / F D , d - u τ - t sep /2 t sep =10t sep =12t sep =14 Figure 7.
The ratio of the connected contribution of the γ term to the CEDM term in the F form factor. Therest is the same as in Fig. 5. Acknowledgments
We thank the MILC Collaboration for providing the 2 + + ffi ce of Science of the U.S. Department ofEnergy; (iii) the National Energy Research Scientific Computing Center, a DOE O ffi ce of ScienceUser Facility supported by the O ffi ce of Science of the U.S. Department of Energy under ContractNo. DE-AC02-05CH11231. The calculations used the Chroma software suite [15]. Work supportedby the U.S. Department of Energy and the LANL LDRD program. References [1] C. Bennett et al., Astrophys.J.Suppl. , 1-27 (2003).[2] E. W. Kolb and M. S. Turner, Front. Phys. , 1-547 (1990).[3] P. Coppi, eConf C040802 , L017 (2004).[4] A.D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. , 338-402 (2008).[7] A. D. Dolgov, Phys. Rept. , 309–386 (1992).[8] T. Bhattacharya, et al., Phys. Rev. D92 , 094511 (2015).[9] T. Bhattacharya, et al., Phys. Rev. Lett,
D92 , 114026 (2015).[11] T. Bhattacharya, et al., PoS LATTICE2015, 238 (2016).[12] T. Bhattacharya, et al., Phys. Rev.
D94 , 054508 (2016).[13] T. Bhattacharya, et al., PoS LATTICE2016, 001 (2016).[14] A. Bazavov, et al., Phys. Rev.
D92 , 114026 (2015).[15] R. G. Edwards and B. Joo, Nucl. Phys. Proc. Suppl.140