4-quark operator contributions to neutron electric dipole moment

1. 4-quark operator contributions to neutron electric dipole moment Haipeng An, University of Maryland; PHENO 2009 In collaboration with Xiangdong Ji, Fanrong Xu

2. Intrinsic EDM interacting with the electric field A system in an external electric field the potential energy generated by the EDM is Therefore, for elementary particles T-transformation: P-transformation: So if a particle has an intrinsic EDM, its interaction with the photon is odd under parity and time reversal (CP) transformations. The effective vertex of the electric dipole moment for spin-1/2 particle

3. Experimental upper bound • The current upper bound is |den| < 2.9 x 10-26e cm (Institut Laue Langevin) C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006) • Current experiment at the Oak Ridge National Lab will give two orders of magnitude improvement. Takeyasu M. Ito, J.Phys.Conf.Ser, 69:012037, (2007)

4. Motivations • Neutron EDM is flavor conserving; • It is difficult for electroweak theory to generate flavor-conserving CP-violations; • New sources of the CP violations are needed for the sake of baryogenesis; • The new sources of CP violations may generate larger flavor-conserving CP-violation sources. Quark EDM from the electroweak sector is about 10-34 e cm. E.P. Shabalin, Sov. J. Nucl. Phys. 28, 75 (1978) A. Czarnecki, B. Krause, Phys. Rev. Lett. 78, 4339(1997) Two-body interaction contribution den≈10-32-10-31 e cm Nanopoulos et al., Phys. Lett. B87, 53 (1979)

5. P-odd, CP-odd, Flavor neutral operators Dim-3 and 4, related by U(1)A transformation Dim-5 QEDM and QCDM Dim-6 three-gluon operator

6. Method • Matching the CP-odd operators to the hadronic operators in the chiral perturbation theory 1. Decompose 4-quark operators into irreducible representations of SU(3)L x SU(3)R chiral symmetry; 2. Find all the leading order corresponding hadronic operators. 3. Use hadronic models to calculate the Wilson coefficients of hadronic operators by calculating some simple matrix elements. • Calculate the chiral loop diagrams to get the Neutron EDM 1. Get the CP-odd nucleon-pion couplings and CP-odd neutron mass from the hadronic operators; 2. Calculate the chiral loop diagrams to get the Neutron EDM; 3. CP-odd neutron mass transforms the neutron magnetic moment to neutron EDM. • Meson condensate effects

7. Meson-condensate contribution • Flavor neutral, P-odd, CP-odd operators have the same quantum numbers as the neutral meson fields π0, η, η’. For example, we can use factorization method B0=1.3 GeV C4 is the wilson coefficient of the 4-quark operator The meson condensates are proportional inversely to the quark masses.

8. Meson-condensate contribution • In the chiral perturbation theory, the chiral Lagrangian is constructed by • Mesons condensate • Redefine meson and baryon fields

9. Meson condensate contribution • Baryon fields in the chiral perturbation theory are collected as • It transforms nonlinearly under the chiral transformation; • is introduced to make it transforms linearly • Baryon fields also need to be redefined in the presence of the meson condensates. • The redefinition is equivalent to a chiral transformation, so it only bothers the terms explicitly breaking chiral symmetry.

10. Meson-condensate contribution Corrections of Baryon masses due to the nonzero light quark masses (σ-term)

11. CP odd mass of neutron Transform the magnetic dipole to electric dipole Meson-condensate contribution Corrections of Goldberger-Treiman relation

12. Direct contribution • Decompose the 4-quark operators into irreducible representations of the SU(3)LxSU(3)R chiral group, • Collect all the leading order hadronic operators in the same representations, take the case as an example

13. P-odd, CP-odd hadronic operators

14. Direct contribution • Match the quark operators to the hadronic operators and get the wilson coefficients • Leading terms

15. Direct contribution • Calculate the simplest matrix elements and determine the wilson coefficients

16. Direct contribution • We used nonrelativistic quark model and the MIT bag model to do the matching; • The weakness of using quark models is that it is difficult to calculate the scale dependence, so they can only be used as an order estimate; • The operators without tilde can generate CP-odd pion-nucleon vertices, • The operators with a tilde can give neutron CP-odd mass,

17. Axion field Induced Strong CP • The θ-term in QCD violates CP and contributes to neutron EDM, • Peccei-Quinn symmetry was invented to cure this problem;

18. Photopion-production counter term of NEDM Contributions to neutron EDM Direct matching

19. Numerical upper bound for Wilson coefficients of four-quark operators

20. Conclusion • All the leading order P-odd, CP-odd, flavor-conserving chiral operators are collected; • Meson condensate contributions and direct matching contributions are estimated; • The contribution from the induced theta-term can be added in the result if the Peccei-Quinn symmetry is used to cure the strong CP-problem.