Neutron EDM with external electric field Eigo Shintani

Neutron EDM with external electric field Eigo Shintani (Univ. of Tsukuba) In collaboration with S. Aoki, Y. Kuramashi and CP-PACS collaboration 11/27/2020 Neutron EDM with external electric field 1

Introduction parameter Both CKM matrix phase and QCD vacuum effects contribute to CP violation (T and P violation) parameter. According to chiral rotation for fermion field, two terms correspond to one term. Definition: 11/27/2020 Neutron EDM with external electric field 2

n Neutron electric dipole moment (NEDM) Ø Directly measurement of CP violation Hamiltonian (spin in electromagnetic field ): electric dipole moment: , magnetic dipole moment: From measurement of the Larmor frequency with ultra-cold neutron, EDM can be measured. Ø Recent experimental upper bound Harris, et al. (1999) 11/27/2020 Neutron EDM with external electric field 3

n Strong CP problem Ø In order to obtain , we calculate the first term of expansion of EDM up to : Ø From some model estimations Ø From the definition of Need fine tune as order of 0. 0000001% !! Ø Unnatural for two theoretical parameters. There may be some physical meaning “Strong CP problem” 11/27/2020 Neutron EDM with external electric field 4

n Some model estimations Model EDM (e fm) Current algebra CHPT (tree) (one loop) QCD sum rule n Ref Crewther, et al. (1979) Aoki, Hatsuda (1992) Cheng (1991) Pospelov, Ritz (1999) Lattice calculation Reliable and accurate estimation from first principles of QCD precise determination of If more accurate experiment will be established, this value may be the most strict estimation. 11/27/2020 Neutron EDM with external electric field 5

n Lattice works q q q Aoki, Gocksch, PRL 63 (1989) 1125 E. Shintani, S. Aoki, N. Ishizuka, K. Kanaya, Y. Kikukawa, Y. Kuramashi, M. Okawa, Y. Taniguchi, A. Ukawa and T. Yoshie, PRD 72, 014504 (2005) Berruto, Blum, Orginos, Soni, hep-lat/0512004 11/27/2020 Neutron EDM with external electric field 6

n Contents Ø Introduction Ø NEDM from form factors Ø NEDM with external electric field Method p Numerical results p Ø Mass dependence of EDM in quenched approximation Ø Summary 11/27/2020 Neutron EDM with external electric field 7

NEDM from form factors n Nucleon electromagnetic form factors Ø Matrix element: with momentum transfer n Shintani et al. (2005) CP even CP odd Electric dipole moment To obtain EDM we have to carry out the momentum extrapolation. 11/27/2020 Neutron EDM with external electric field 8

n EDM form factor in 16^3 x 32 lattice with DW Ø possibility of calculation of EDM form factor from our formulation Ø Fitting results 11/27/2020 Neutron EDM with external electric field 9

New strategy for obtaining EDM extrapolations in EDM form factor definition Ø Zero momentum limit Ø Chiral limit Ø Continuum limit New strategy Ø Definition with external electric field p p p 11/27/2020 Aoki, Gocksch (1989) Not need momentum extrapolation Simple formulation Periodicity is broken, boundary effects Neutron EDM with external electric field 10

NEDM with electric field n Definition Spin dependent energy difference in static and uniform electric field , and CP-odd vacuum angle : : spin up or down nucleon energy on : spin direction vacuum Ø We need to calculate nucleon energy in electric field at zero momentum only. p p 11/27/2020 this method is simple and we can obtain EDM directly !! may be more advantageous than form factor case Neutron EDM with external electric field 11

n Method on the lattice Real electric field is included in link variables : quark charge We can choose arbitrary value for E but periodicity in time direction is broken. t=NT Periodicity is broken t=1 t=0=NT Source point 11/27/2020 Neutron EDM with external electric field 12

n The ratio of nucleon propagator with theta In to reduce E=0 contribution where Sampling of topological charge is important ! 11/27/2020 Neutron EDM with external electric field 13

Numerical results n Lattice parameters q q quenched approximation, #configuration = 1000 Lattice size : , RG Iwasaki : Domain-wall quark, Nucleon mass : corresponding to 11/27/2020 Neutron EDM with external electric field 14

n Topological charge Measurement of bosonic definition in cooling config. #cooling=20 Histogram with 1000 configs. Topological charge in each configurations 11/27/2020 Neutron EDM with external electric field 15

n Results of E=0. 004 E=-0. 004 Ø There is no signal in case Ø We can observe expected E oddness: 11/27/2020 Neutron EDM with external electric field 16

n Results of effective mass of R Ø Reduction of theta=0 and E^2 contribution Ø Effective mass of R fitting results in 11/27/2020 Neutron EDM with external electric field 17

n E and theta dependence E dependence Theta dependence Obtained signal has a good linear behavior of E and theta. expected NEDM signal in this method. 11/27/2020 Neutron EDM with external electric field 18

n Comparison of clover and DW In order to apply this calculation to dynamical configurations generated by CP-PACS, we try calculation with clover fermion in similar nucleon mass parameters on the same gauge configurations 11/27/2020 Neutron EDM with external electric field 19

n Comparison of clover and DW Clover fermion DW fermion Fitting results: This method is also successful in clover fermion. chirality of fermion is not so important for this method. 11/27/2020 Neutron EDM with external electric field 20

Size effects and boundary effects In using clover fermion the computational cost is reasonable for the study of Ø reduction of finite size effects large lattice size Ø reduction of boundary effects shifted source point 11/27/2020 Neutron EDM with external electric field 21

Lattice size , clover fermion Effective mass plot for nucleon Ø Smearing source, and point sink. source point: t=1 Ø #configs. =2000 Ø K=0. 1320 Ø Nucleon mass ~1. 8 Ge. V 11/27/2020 Neutron EDM with external electric field 22

n 24^3 x 32 lattice, (0, 0, E) only Ø Fitting values are not so different. Size effect not so large. 11/27/2020 Neutron EDM with external electric field 23

n Shift source point Ø The nucleon state receives the boundary effects from the broken time periodicity of link variable by the electric field. Ø Separating a source point far away from boundary, the boundary effect can be reduced. t=NT Periodicity is broken t=1 t=0=NT Source point 11/27/2020 Neutron EDM with external electric field 24

n 24^3 x 32 lattice, source point t=8 Ø (0, 0, E) only, K=0. 1320 plot of source point t=8 source point t=1 In left figure (source point t=8) the signal seems to be different. effect is large. 11/27/2020 Neutron EDM with external electric field 25

n 24^3 x 32 lattice, source point t=8 Ø (0, 0, E) only, K=0. 1320 plot of source point t=8 source point t=1 After reduction of contributions, it is clear to see the EDM signal in the different sign of E in both cases. 11/27/2020 Neutron EDM with external electric field 26

n 24^3 x 32 lattice, source point t=8 Ø (0, 0, E) only, K=0. 1320 effective mass plot of source point t=8 source point t=1 Fitting Combined fittingpoint results Ø Inresults source t=8, the EDM signal begins from 8+6. Ø The plateau in [5, 7] may be the enhancement from boundary. 11/27/2020 Neutron EDM with external electric field 27

Mass dependence of EDM n Quenched approximation Ø Partition function with p term Full QCD In zero quark mass limit: CP-odd contribution is absent. p Quenched QCD because the second term dose not depend on quark mass. CP-odd contribution does not vanish in chiral limit. 11/27/2020 Neutron EDM with external electric field 28

The average over E Ø Spin component in electric field Ø In E=(E, 0, 0) (0, E, 0) 11/27/2020 Neutron EDM with external electric field 29

Effective mass plot for nucleon Ø Lattiece size : Ø Source point t=1 Ø #configs. K=0. 1320: 2000 K=0. 1330: 1800 K=0. 1340: 2000 Ø Nucleon mass K=0. 1320: ~1. 8 Ge. V K=0. 1330: ~1. 6 Ge. V K=0. 1340: ~1. 4 Ge. V 11/27/2020 , clover fermion Neutron EDM with external electric field 30

n K=0. 1320 , #configs. =2000 Fitting results Ø A little reduction of statistical error correlation between each direction of E 11/27/2020 Neutron EDM with external electric field 31

n K=0. 1330 , #configs. =1800 Results of EDM: 11/27/2020 Neutron EDM with external electric field 32

n K=0. 1340 , #configs. =2000 Results of EDM: 11/27/2020 Neutron EDM with external electric field 33

n Quark mass dependence for EDM Ø These results show that EDM does not vanish in chiral limit. Ø It is different with full QCD situation ( in ) Quenched effects 11/27/2020 Neutron EDM with external electric field 34

n Quark mass dependence for CP-odd phase Ø Next leading term of nucleon propagator Ø employ with clover fermion. Ø Another CP-odd objects on theta vacuum. Ø does not vanish in chiral limit in quenched app. 11/27/2020 Neutron EDM with external electric field 35

Summary Ø We try to extract EDM from energy difference in the constant electric field. p p We can observe NEDM signals in this method. This method works well in both domain-wall and clover The size effect is small. The boundary effect may be large so we have to take it account into the determination of fitting range. Ø We check the mass dependence of CP-odd objects from theta vacuum. p p We observe non-vanishing mass dependence of EDM in quenched approximation. This behavior is also observed in CP-odd phase factor. 11/27/2020 Neutron EDM with external electric field 36

Future works Ø Application to 2, 2+1 flavor configs. generated by CP-PACS collaboration p p 11/27/2020 Firstly we perform with Nf=2 flavor configuration of clover fermion. Several source points to accumulate more statistics. Check the valence or sea quark mass dependence Chiral limit and continuum limit Neutron EDM with external electric field 37