4-quark operator contributions to neutron electric dipole moment

Haipeng An,

University of Maryland;

PHENO 2009

In collaboration with Xiangdong Ji, Fanrong Xu

Intrinsic EDM interacting with the electric field

, E E s s

T-transformation:

P-transformation: , E E s s

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

Experimental upper bound

The current upper bound is

|den| < 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)

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)

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

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

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

C4 is the wilson coefficient of the 4-quark operator

The meson condensates are proportional inversely to the quark masses.

B0=1.3 GeV

Meson-condensate contribution

In the chiral perturbation theory, the chiral Lagrangian is constructed by

Mesons condensate

Redefine meson and baryon fields

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.

Meson-condensate contribution

Corrections of Baryon masses due to the nonzero light quark masses (σ-term)

Meson-condensate contribution

Corrections of Goldberger-Treiman relation

CP odd mass of neutron

Transform the magnetic dipole to electric dipole

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

P-odd, CP-odd hadronic operators

Direct contribution Match the quark operators to the hadronic operators and get the

wilson coefficients

Leading terms

Direct contribution

Calculate the simplest matrix elements and determine the wilson coefficients

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,

Induced Strong CP The θ-term in QCD violates CP and contributes to neutron EDM,

Peccei-Quinn symmetry was invented to cure this problem;

1010 Strong CP problem!!

Axion field

Contributions to neutron EDM

Direct matching

Photopion-production

counter term of NEDM

Numerical upper bound for Wilson coefficients of four-quark operators

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.