Neutron Electric Dipole Moments with Clover Fermions
Boram Yoon, Tanmoy Bhattacharya, Vincenzo Cirigliano, Rajan Gupta
NNeutron Electric Dipole Moments with CloverFermions
Boram Yoon ∗ Los Alamos National Laboratory, Computer, Computational, and Statistical Sciences CCS-7,Los Alamos, NM 87545E-mail: [email protected]
Tanmoy Bhattacharya
Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM 87545E-mail: [email protected]
Vincenzo Cirigliano
Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM 87545E-mail: [email protected]
Rajan Gupta
Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM 87545E-mail: [email protected]
We present preliminary results for the contributions to the neutron EDM arising from the QCD θ -term, the Weinberg three-gluon and the quark chromo-EDM operators from our ongoing latticecalculations using clover valence quarks on the MILC HISQ lattices. We use the gradient-flowtechnique to smooth the lattices and renormalize the gluonic operators, and use the Schwingersource method to incorporate the quark chromo-EDM interactions in the quark propagator. Forthe QCD θ -term and the Weinberg three-gluon operator, we report results in the gradient-flowscheme from 8 ensembles at four lattice spacings and three pion masses, including 2 physicalpion mass ensembles described in Table 1. For the quark chromo-EDM, unrenormalized resultsare presented at two lattice spacings, a = .
12 and 0 .
09 fm, and two pion masses, M π =
310 MeVand 220 MeV.
The 37th International Symposium on Lattice Field Theory - LATTICE201916-22 June 2019Wuhan, China. ∗ 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 ] M a r eutron EDM with Clover Fermions Boram Yoon
1. Introduction
Neutrons can have nonvanishing electric dipole moment (EDM) if the theory has broken P andT symmetries, or CP violation (CPV). Since CPV in the standard model (SM) is small or stronglysuppressed at high temperature, new CPV from beyond the SM (BSM) is needed to explain matter-antimatter asymmetry via baryogenesis, and EDMs of elementary particles are good probes of it.The CPV interactions of interest in the low-energy effective Lagrangian are of dimension 4–6: L d = , , = − g π ¯ θ G ˜ G − i ∑ q = u , d , s d q ¯ q ( σ µν F µν ) γ q − i ∑ q = u , d , s ˜ d q ¯ q ( σ µν G µν ) γ q + d w g6 f abc G a µν ˜ G νρ , b G µ , c ρ + ∑ i C ( q ) i O ( q ) i , (1.1)where ˜ G µν , b = ε µναβ G b αβ /
2. Here, the terms on the r.h.s are the QCD θ -term (d=4); quark EDM(qEDM) and quark chromo-EDM (CEDM) (d=5), and the Weinberg’s three-gluon operator ( W ggg )and various four-quark operators (d=6). In this paper, we will discuss the calculation of the neutronEDM induced by the QCD θ , W ggg , and the CEDM terms.
2. Neutron EDM from QCD θ -term and Weinberg’s three-gluon operator Expectation values of observables in the presence of the θ -term can be calculated using stan-dard lattices generated without the θ -term in the action by exploiting the small- θ expansion [1] (cid:104) O ( x ) (cid:105) ¯ θ = Z ¯ θ (cid:90) d [ U , q , q ] O ( x ) e − S QCD − i θ Q = (cid:104) O ( x ) (cid:105) ¯ θ = − i ¯ θ (cid:104) O ( x ) Q (cid:105) ¯ θ = + O ( ¯ θ ) , (2.1)where Q is the topological charge Q = (cid:82) d x G ˜ G π . Since the phenomenological estimate, θ (cid:46) O ( − ) [2], is tiny, the leading order term in θ suffices.We calculate the topological charge using the O ( a ) -improved field-strength tensor [3] withgradient flow [4] on the MILC HISQ lattices [5]. After analyzing 10 different ensembles with a = . − .
06 fm and M π = −
130 MeV, we find that (i) Q converges to a stable distributionafter the gradient flow time τ F ≈ .
34 fm; however, (ii) it requires much longer τ F for Q to convergeto an integer, and this τ F depends on a and M π ; coarser a or smaller M π lattices need longer τ F ;and (iii) very long autocorrelations length longer than 30 configurations are found in the a m a m
220 ensembles, so we do not include those two in our analysis. As an example, we showvarious aspects of the topological charge measured on the a m
130 ensemble in Fig. 1.After correlating the topological charge with the neutron 2- and 3-point functions as perEq. (2.1), the CPV phase α arising in the neutron state is obtained by solvingIm C P ( t ) Re C ( t ) ≡ Im Tr (cid:2) γ ( + γ ) (cid:104) N ( t ) N ( ) (cid:105) (cid:3) Re Tr (cid:2) ( + γ ) (cid:104) N ( t ) N ( ) (cid:105) (cid:3) = M N sin (cid:0) α ( t ) (cid:1) E N + M N cos (cid:0) α ( t ) (cid:1) , (2.2) Throughout this paper, we will use the notation aABmXY Z to denote an ensemble, where AB represents the ap-proximated lattice spacing in units of 0 .
01 fm, and
XY Z represents the pion mass in units of MeV. eutron EDM with Clover Fermions Boram Yoon
40 20 0 20 40Q0.000.010.020.030.04 P r o b a b ili t y a09m130 F =0.00 fm8 F =0.17 fm8 F =0.34 fm8 F =0.51 fm8 F =0.76 fm P r o b a b ili t y a09m130 F =0.00 fm8 F =0.17 fm8 F =0.34 fm8 F =0.51 fm8 F =0.76 fm
40 20 0 20 40NearestInteger(Q)0.000.010.020.030.04 P r o b a b ili t y a09m130 F =0.76 fm t acf A u t o c o rr e l a t i o n a09m130 8 F =0.00 fm8 F =0.17 fm8 F =0.34 fm8 F =0.51 fm8 F =0.76 fm Q a09m1308 F =0.76 fm Figure 1: Various aspects of the topological charge calculated on the a m
130 HISQ ensemble forfive different gradient flow times τ F : (left top) distribution of Q , (center top) distribution of thenon-integer part of Q , (right top) distribution of the integer part of Q , (left bottom) autocorrelationfunction, and (right bottom) Q as a function of the lattice trajectory time. Ensemble a (fm) M π (MeV) L × T M π L N conf N meas a15m310 0.1510(20) 320(5) 16 ×
48 3.9 1920 123ka12m310 0.1207(11) 305.3(4) 24 ×
64 4.54 1013 65ka12m220 0.1184(10) 216.9(2) 32 ×
64 4.29 1156 74ka12m220L 0.1189(09) 217.0(2) 40 ×
64 5.36 1000 128ka09m310 0.0888(08) 312.7(6) 32 ×
96 4.50 2196 140ka09m220 0.0872(07) 220.3(2) 48 ×
96 4.71 961 123ka09m130 0.0871(06) 128.2(1) 64 ×
96 3.66 1289 165ka06m135 0.0570(01) 135.5(2) 96 ×
192 3.70 453 29k
Table 1: List of MILC HISQ ensembles analyzed for Q and W ggg . and the electric dipole form-factor F is extracted from (cid:104) N | V µ ( q ) | N (cid:105) CPV = u N ( p (cid:48) ) (cid:20) F ( q ) γ µ + i F ( q ) M N σ µν q ν − F ( q ) M N σ µν q ν γ (cid:21) u N ( p ) , (2.3)where V µ is the electromagnetic current, M N is the neutron mass, u N ( p ) is free neutron spinor, q = p − p (cid:48) , and F and F are the Dirac and Pauli form-factors. The anapole form factor F A isirrelevant, assuming PT -conservation. Extracting the form factors from the real and imaginaryparts of the lattice three-point functions at multiple combinations of the momentum transfer for thesame q is an over-constrained problem. We solve the equations simultaneously, weighting each byits statistical variance. Details of the extraction of F are given in Refs. [6]. The lattice ensemblesanalyzed and the number of configurations/measurements are listed in Table 1.To understand errors, we investigated correlations in the spatial and temporal directions be-2 eutron EDM with Clover Fermions Boram Yoon [8t WF ] = 0.34 fm α θ R T a09m130a09m220a09m310 = 0.18 GeV [8t WF ] = 0.34 fm τ =10, t=5 F , n θ R T a09m130 a09m220 a09m310 Figure 2: CPV phase α (left) and electric dipole form-factor F (right) calculated using the topo-logical charges calculated from the timeslices near the neutron source | t Q − t src | ≤ R T (for α ) orthose from the timeslices near the current insertion | t Q − t ins | ≤ R T (for F ).tween Q and 2- and 3-point functions. Fig. 2 shows the phase α θ and F θ versus R T , the numberof timeslices over which Q ( t ) is summed about the neutron source or the current insertion time, | t Q − t src, ins | ≤ R T . For the physical pion mass, where the need for reducing error is the largest, con-vergence requires Q ( x , t ) to be summed over almost all { x , t } . Therefore, we do find any significantadvantage to using Q ( x , t ) summed over a limited volume to reduce errors.The size of CPV observables α and F depend on the parameter ¯ θ used in Eq. (2.1). We findthat the dependence is linear within errors for the values of θ used. We, therefore, report resultsdivided by θ , with θ = .
2. We also use the variance reduction technique (VRT) introduced in[6] by calculating (cid:104) O V R
CPV (cid:105) θ = (cid:104) O CPV | θ − c · O CPV | θ = (cid:105) . Since (cid:104) O CPV (cid:105) θ = =
0, adding this does notchange the result, but, the error is reduced due to the large correlations. Here c is the coefficientdetermined following [6], and it turns out to be c ≈
1. This VRT is not useful when θ (cid:38)
1, butbecomes crucial when θ (cid:28)
1; for θ = .
2, we find about 25% reduction in the error of F .After calculating F for multiple source-sink separations for each Q , we remove the excitedstate contamination using the two-state fit ansatz [7] and extrapolate to Q → Q ansatz to obtain d n = | e | F ( Q = ) / M N on each ensemble. We repeated the same procedure for W ggg . The chiral-continuum extrapolation is done using the leading chiral term [8] and linear in a : d θ N = c θ M π + c θ M π log ( M π / M N , phys ) + c θ a , d WN = c W + c W M π + c W M π log ( M π / M N , phys ) + c W a . Preliminary results, presented in Fig. 3 in the gradient-flow scheme, are consistent with zero within2 σ : d θ N = − . ( ) ¯ θ | e | fm and d WN = − . ( ) d w | e | fm. The extrapolation results fluctuate aroundzero for different fit ansatze. The results on M π ≈
310 MeV ensembles are similar to those fromrecent lattice calculations [9, 10], however, our data show significant discretization corrections.
3. Neutron EDM from CEDM term
Since the CEDM operator is a quark bilinear, we use the Schwinger source method (SSM) toinclude the CEDM interactions by changing the Dirac operator: D clov → D clov + ( i / ) εσ µν γ G µν .This is implemented by shifting the Sheikholeslami-Wohlert coefficient, c sw → c sw + i εγ [11, 12]. F is then extracted from (cid:104) N | V µ ( q ) | N (cid:105) CPV calculated with the modified valence quark propagators.In this study, we ignored the contributions from the disconnected diagrams and the reweightingfactor due to the change in the fermion determinant det (cid:2) D clov + ( i / ) εσ µν γ G µν (cid:3) / det [ D clov ] [11].3 eutron EDM with Clover Fermions Boram Yoon -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 -1.0-0.50.00.51.01.52.02.53.0 0 0.1 0.2 0.3 0.4 0.5 F , n W ( 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 0.15 χ /dof=0.57 F , n θ ( Q = ) / M N (f m ) a [fm] a06m135a09m310a09m220 a09m130a12m310a12m220 a15m310Extrap -3-2-1012345 0 0.03 0.06 0.09 0.12 0.15 χ /dof=0.94 F , n W ( Q = ) / M N (f m ) a [fm] a06m135a09m310a09m220 a09m130a12m310a12m220 a15m310Extrap -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 -3-2-1012345 0 0.03 0.06 0.09 0.12 χ /dof=0.94 F , n W ( Q = ) / M N (f m ) M π [GeV ] a06m135a09m310a09m220 a09m130a12m310a12m220 a15m310Extrap Figure 3: Preliminary results on the neutron F / M N from the QCD θ -term (left) and the W ggg (right). Top row shows Q -dependence, and bottom two rows show the continuum (middle) andchiral (bottom) extrapolations. Gray data points in the bottom two rows show the Q → F data for Q < .
275 GeV . The F results from a12m220 and a12m220L ensemblesare averaged in the continuum/chiral extrapolation, because no volume dependence is observed.The α is calculated by solving Eq. (2.2) and F from Eq. (2.3). In addition to the CEDMoperator, we also calculate α and F for O γ ≡ − i ¯ q γ q , as it mixes with the CEDM operator underrenormalization. The ensemble, number of configurations and measurements analyzed are{(Ens, N conf , N meas )} = {( a m a m L , 475, 61K), ( a m ε , simply a parameter, is small so that we can ignore the contributions from O ( ε ) . This is checked by the linearity of the CPV observables in ε as shown in Ref. [12]. Fig. 4shows that F / ε is constant for ε (cid:46) . ε give better signal in F ,while for smaller ε the VRT described above improves the signal significantly. On a m F U , CEDM / ε ( τ = a , t = a , Q = . ) is 0.9(2.2) at ε = .
002 and0.84(55) at ε = . ε = .
002 and 0.867(82) at ε = . eutron EDM with Clover Fermions Boram Yoon τ =8a, t=4a, Q = 0.51 GeV F , n ε / ε ε CEDM@D
Figure 4: Linearity of CEDM F in ε . τ /2 F , n C E D M / ε Q (GeV ) U, τ =11 a U, τ =13 a U, τ =15 a D, τ =11 a D, τ =13 a D, τ =15 a Figure 5: CEDM F calculated on a m
310 atthree different source-sink separations. τ = 1.2 fm, t = 0.6 fm F , n C E D M / ε Q (GeV ) U, a12m310 U, a12m220LU, a09m310 D, a12m310 D, a12m220LD, a09m310 τ = 1.2 fm, t = 0.6 fm F , n γ / ε Q (GeV ) U, a12m310 U, a12m220LU, a09m310 D, a12m310 D, a12m220LD, a09m310
Figure 6: Neutron F / M N from CEDM as a function of Q . Note that these are unrenormalized.We find relatively small excited state contamination when the source and sink separation τ (cid:38) . F obtained at τ = . a -dependence between a m
310 and a m M π -dependence by comparing a m
310 and a m L results. Similar large M π -dependence is also observed in Refs. [9, 13]. Since we calculate only the connected diagrams, F induced by O γ is a lattice artifact that should disappear in exact chiral symmetry limit [14].
4. Conclusion
Preliminary results of the neutron EDM induced by the QCD θ -term, W ggg , and the CEDMinteractions, calculated at multiple values of pion masses and lattice spacings are presented. The a , M π and the Q dependencies of the neutron EDM from θ - and Weinberg-term need furtherinvestigation. For the renormalization of the CEDM operator, we are investigating the gradientflow scheme. 5 eutron EDM with Clover Fermions Boram Yoon
Acknowledgments
We thank MILC Collaboration for providing the 2+1+1-flavor HISQ lattices. Simulations werecarried out on computer facilities at (i) the National Energy Research Scientific Computing Centersupported by the U.S. Department of Energy (DOE) Office of Science (OS) under Contract No.DE-AC02-05CH11231; and, (ii) the Oak Ridge Leadership Computing Facility at the Oak RidgeNational Laboratory supported by the DOE OS under Contract No. DE-AC05-00OR22725; (iii)the USQCD Collaboration, which are funded by the DOE OS, (iv) Institutional Computing at LosAlamos National Laboratory. This work was supported by the DOE OS, Office of High EnergyPhysics under Contract No. 89233218CNA000001, and by the LANL LDRD program.
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