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 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. A system in an external electric field the potential energy generated by the EDM is Therefore, for elementary particles The effective vertex of the electric dipole moment for spin-1/2 particle

3. Experimental upper bound The current upper bound is |d e n | < 2.9 x 10 -26 e 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 d e n ≈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 π 0, η, η’. For example, we can use factorization method C 4 is the wilson coefficient of the 4- quark operator The meson condensates are proportional inversely to the quark masses. B 0 =1.3 GeV

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

12. Direct contribution Decompose the 4-quark operators into irreducible representations of the SU(3) L xSU(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. Induced Strong CP The θ-term in QCD violates CP and contributes to neutron EDM, Peccei-Quinn symmetry was invented to cure this problem; Axion field

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

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.