1. Neutron Electric Dipole Moment at Fixed Topology
Why fixed topology?
N-point functions for CP-even and CP-odd quantities
NEDM and Magnetic Moment
Zero modes and quenched divergence
Lattice07, Regensburg, Aug. 2, 2007

2. θVacuum and Topological Sectors
QCD has θvacuum
Cluster decomposition and locality require the form
Topological sectors
Q has integer eigenvalues (4 torus, P.B.C., Q from the overlap operator)
Physics is coded in fixed topological sectors with finite volume correction.

3. Why Study Fixed Topological Sector ?
The rate of tunneling between different topological sector is diminished as
the gauge link gets smoother, e.g. Iwasaki --> DBW2.
the quark mass approaches the physical u/d mass.
the lattice is approaching the continuum limit.
The pseudofermion force can diverge in HMC
for overlap fermions when changing topology.

4. Recent Approaches to Simulation of Overlap Fermion
JLQCD
Add an action (P. Vranas) to quench the small eigenmodes in the overlap kernel
Tunneling of global topology is suppressed
QCD (Horvath, hep-lat/ ; KFL, hep-lat/ )
Gauge action from the overlap operator
MC Simulation (talk by Thomas Streuer) on 84 lattice, ma=0.5,
a = – 0.16 fm
has not encountered change of topological sector.

5. QCD at Fixed Topology R. Brower, S. Chandrasekharan, J. Negele, and
U.-J. Wiese (hep-lat/ ) --- Hadron mass
S. Aoki, H. Fukaya, S. Hashimoto, and T. Onogi
(arXiv: ) --- topological susceptibility and
CP-odd quantities

6. Neutron Electric Dipole Moment
Basics of θ vacuum and topological sectors
Saddle point approximation (V large, fixed)

7. Green’s Function n-point Green’s function
Saddle point expansion ( fixed)

8. Mass
Mass and topological susceptibility can be obtained in fixed topological sector with several V.

9. NEDM
EM form factors
In QCD, θis small (< 10-9)

10. Question: Since , one might naively think that
will diverge?
Expanding at small
Therefore,

11. NEDM
Consider the 3-point function
Fixed topological sector

12. Remarks
Usually the zero mode contributions to chiral condensate and correlation functions are proportional to and are thus considered as finite volume artifacts. In the CP-odd case, they are part of the physical spectrum.
In the quenched approximation, the zero modes lead to 1/m divergence for NEDM (1/m2V2, 1/m3V3 are finite volume artifacts.)
Electric dipole and Pauli form factors:
Brodsky, Gardner, and Hwang (hep-ph/ )
Recall

13. Conclusion
CP-odd quantities, such as the neutron electric dipole moment and anapole moment in QCD, can be obtained in one or more topological sectors as are CP-even quantities, as long as the finite volume corrections are taken into account.