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

2. Neutron EDM with external electric field
Introduction
parameter
Both CKM 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:
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Neutron EDM with external electric field

3. Neutron EDM with external electric field
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)
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Neutron EDM with external electric field

4. Neutron EDM with external electric field
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
% !!
“Strong CP problem”
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Neutron EDM with external electric field

5. Neutron EDM with external electric field
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.
Model
EDM (e fm)
Ref
Current algebra
CHPT (tree)
(one loop)
QCD sum rule
Crewther, et al. (1979)
Aoki, Hatsuda (1992)
Cheng (1991)
Pospelov, Ritz (1999)
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Neutron EDM with external electric field

6. Neutron EDM with external electric field
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, (2005)
Berruto, Blum, Orginos, Soni, hep-lat/
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Neutron EDM with external electric field

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

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

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

12. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

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

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

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

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

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

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

19. on the same gauge configurations
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
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Neutron EDM with external electric field

20. Neutron EDM with external electric field
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.
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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
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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
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Neutron EDM with external electric field

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

24. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

25. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

26. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

27. Neutron EDM with external electric field
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.
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Neutron EDM with external electric field

28. Neutron EDM with external electric field
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.
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Neutron EDM with external electric field

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

30. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

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

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

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

34. Neutron EDM with external electric field
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
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Neutron EDM with external electric field

35. Neutron EDM with external electric field
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.
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Neutron EDM with external electric field

36. Neutron EDM with external electric field
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.
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Neutron EDM with external electric field

37. Neutron EDM with external electric field
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
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Neutron EDM with external electric field