Neutron EDM with external electric field

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

2. Introduction • parameter BothCKM 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: Neutron EDM with external electric field

3. Neutron electric dipole moment (NEDM) • 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. • Recent experimental upper bound Harris, et al. (1999) Neutron EDM with external electric field

4. Strong CP problem • In order to obtain , we calculate the first term of expansion of EDM up to : • From some model estimations • From the definition of • Unnatural for two theoretical parameters. There may be some physical meaning Need fine tune as order of 0.0000001% !! “Strong CP problem” Neutron EDM with external electric field

5. Some model estimations • 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. Neutron EDM with external electric field

6. Lattice works • Aoki, Gocksch, PRL63 (1989) 1125 • E. Shintani, S. Aoki, N. Ishizuka, K. Kanaya, Y. Kikukawa, Y. Kuramashi, M. Okawa, Y. Taniguchi, A. Ukawa and T. Yoshie, PRD72, 014504 (2005) • Berruto, Blum, Orginos, Soni, hep-lat/0512004 Neutron EDM with external electric field

7. Contents • Introduction • NEDM from form factors • NEDM with external electric field • Method • Numerical results • Mass dependence of EDM in quenched approximation • Summary Neutron EDM with external electric field

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

9. EDM form factor in 16^3x32 lattice with DW • possibility of calculation of EDM form factor from our formulation • Fitting results Neutron EDM with external electric field

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

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

12. 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 t=1 t=0=NT Periodicity is broken Source point Neutron EDM with external electric field

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

14. Numerical results • Lattice parameters • quenched approximation, #configuration = 1000 • Lattice size : , RG Iwasaki : • Domain-wall quark, • Nucleon mass : corresponding to Neutron EDM with external electric field

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

16. E=0.004 • Results of E=0.004 E=-0.004 • There is no signal in case • We can observe expected E oddness: Neutron EDM with external electric field

17. Results of effective mass of R • Reduction of theta=0 and E^2 contribution • Effective mass of R fitting results in Neutron EDM with external electric field

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

19. 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 Neutron EDM with external electric field

20. 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. Neutron EDM with external electric field

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

22. Lattice size , clover fermion • Effective mass plot for nucleon • Smearing source, and point sink. source point: t=1 • #configs. =2000 • K=0.1320 • Nucleon mass ~1.8 GeV Neutron EDM with external electric field

23. 24^3x32 lattice, (0,0,E) only • Fitting values are not so different. Size effect not so large. Neutron EDM with external electric field

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

25. 24^3x32 lattice, source point t=8 • (0,0,E) only, K=0.1320 plot of In left figure (source point t=8) the signal seems to be different. effect is large. source point t=8 source point t=1 Neutron EDM with external electric field

26. 24^3x32 lattice, source point t=8 • (0,0,E) only, K=0.1320 plot of After reduction of contributions, it is clear to see the EDM signal in the different sign of E in both cases. source point t=8 source point t=1 Neutron EDM with external electric field

27. 24^3x32 lattice, source point t=8 • (0,0,E) only, K=0.1320 effective mass plot of Fitting results source point t=8 source point t=1 Combined fitting results • In source point t=8, the EDM signal begins from 8+6. • The plateau in [5,7] may be the enhancement • from boundary. Neutron EDM with external electric field

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

29. The average over E • Spin component in electric field • In E=(E,0,0) (0,E,0) Neutron EDM with external electric field

30. Effective mass plot for nucleon • Lattiece size : , clover fermion • Source point t=1 • #configs. K=0.1320: 2000 K=0.1330: 1800 K=0.1340: 2000 • Nucleon mass K=0.1320: ~1.8 GeV K=0.1330: ~1.6 GeV K=0.1340: ~1.4 GeV Neutron EDM with external electric field

31. K=0.1320 , #configs.=2000 Fitting results • A little reduction of statistical error correlation between each direction of E Neutron EDM with external electric field

32. K=0.1330 , #configs.=1800 Results of EDM: Neutron EDM with external electric field

33. K=0.1340 , #configs.=2000 Results of EDM: Neutron EDM with external electric field

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

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

36. Summary • We try to extract EDM from energy difference in the constant electric field. • 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. • We check the mass dependence of CP-odd objects from theta vacuum. • We observe non-vanishing mass dependence of EDM in quenched approximation. • This behavior is also observed in CP-odd phase factor. Neutron EDM with external electric field

37. Future works • Application to 2, 2+1 flavor configs. generated by CP-PACS collaboration • 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