Strong coupling study of Aoki phasein Staggered-Wilson fermion

T. Z. Nakano (YITP/Kyoto Univ.)
Collaborators:
T. Misumi (YITP), T. Kimura (Univ. of Tokyo/RIKEN),
A. Ohnishi (YITP), M. Creutz (BNL)
PoS (LAT2011), 108 (2011) [arXiv:1110.1231].
1/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Contents
 Introduction
Aoki phase in Wilson fermion

 Aoki phase in Wilson fermionStrong coupling study Aoki phase in Staggered-Wilson fermionStrong coupling study Summary & Future works

2/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Phase structure in Wilson Fermion
 Aoki Phase

Parity-flavor symmetry is spontaneously broken in some quarkmass region. Aoki (1984)

 Quark mass tuning  Chiral limit

3/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Analysis for Aoki Phase Strong coupling analysis, Gross-Neveu Model

indicate Gross-Neveu Model : Aoki (1984)
 Strong coupling (expansion) : Aoki : (1986); (1989)

 Numerical calculation
Vacuum : High T :Aoki (1987); (1989)

Aoki, Gocksch (1990),
Aoki, Ukawa, Umemura (1996), Aoki, Kaneda, Ukawa (1997), Aoki(1996)

 Chiral perturbation theory
4/35

Sharpe, Singleton (1998), Sharpe (2009)"NTFL workshop'' T. Z. Nakano

2012/02/10

Staggered-Wilson Fermion
 Flavored Mass : Generalized Wilson terms
Adams :
 Hoelbling :Adams (2010, 2011), Hoelbling (2011).
Creutz, Kimura, Misumi (2010)

 Aoki phase in Staggered-Wilson fermion ?

5/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Purpose Applicability of Staggered-Wilson (Overlap) fermion
Chiral limit

 Previous studyLattice Gross-Neveu model with flavored mass [Kimura’s talk]Creutz, Kimura, Misumi (2011)

6/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Purpose We study Aoki phase of Staggered-Wilson fermion instrong coupling lattice QCD.
 MethodHopping parameter expansion
 Effective potential analysis
 Staggered-Wilson Fermion

7/35

Adams type , Hoelbling type

"NTFL workshop'' T. Z. Nakano

2012/02/10

Contents
 Introduction
Aoki phase in Wilson fermion

 Aoki phase in Wilson fermion
Strong coupling study (Condensate, Pion mass) Hopping Parameter Expansion

 Effective Potential Analysis

 Aoki phase in Staggered-Wilson fermionStrong coupling study Summary & Future works

8/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Hopping Parameter Expansion (HPE) Strong coupling study


Aoki (1984)

Hopping Parameter Expansion (HPE)

Hopping Parameter
9/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Hopping Parameter Expansion (HPE) Feynman rule

10/35

"NTFL workshop'' T. Z. Nakano

2012/02/10

1 pt. Function
 Diagrams (1 pt. function) Expansion for hopping parameter (K) Link variables appear as a pair of Summation for K. (K2 , K4 ,…)

11/35

.

"NTFL workshop'' T. Z. Nakano

2012/02/10

1 pt. Function
 Self-Consistent Equation for

12/35"NTFL workshop'' T. Z. Nakano

2012/02/10

1 pt. Function
 Mean-field treatment


Scalar Pesudo-scalar Two solution

Parity conserved

Parity broken

For
13/35

, Parity-broken?
"NTFL workshop'' T. Z. Nakano

2012/02/10

2 pt. Function
 Diagrams (2 pt. function)  Self-consistent equation14/35"NTFL workshop'' T. Z. Nakano

2012/02/10

2 pt. Function
 Pion massF.T.
 Insert

For? ,?

Tachyon : instability of ground stateAnother lower energy state
15/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Effective Potential (Condensate) Effective Potential for Meson (large Nc)

We can identify the Parity-broken phase

For
16/35

,
"NTFL workshop'' T. Z. Nakano

2012/02/10

Effective Potential (Pion Mass) Critical Mass (Mc) corresponds to the chiral limit.

17/35"NTFL workshop'' T. Z. Nakano

2012/02/10

2 Flavor Case Possibility (1)Parity symmetry breaking

 (2)Parity-flavor symmetry breaking

 Parity-flavor symmetry breaking
 Vafa-Witten’s theorem 

18/35: NG boson for spontaneous breaking of parity-flavor
symmetry"NTFL workshop'' T. Z. Nakano

2012/02/10

Aoki Phase in Wilson Fermion Hopping Parameter Expansion

 Effective Potential Analysis

19/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Contents
 Introduction
Aoki phase in Wilson fermion

 Aoki phase in Wilson fermion
Strong coupling study (Condensate, Pion mass)

 Aoki phase in Staggered-Wilson fermion
Strong coupling study
 Hoelbling type
 Adams type



Discussion

 Summary & Future works

20/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Staggered-Wilson Fermion
 Lattice action (Hoelbling type) C. Hoelbling (2011) Flavored mass : 2 hopping

21/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Hopping Parameter Expansion (HPE) Feynman rule

1 hopping

2 hopping

22/35"NTFL workshop'' T. Z. Nakano

2012/02/10

1 pt. Function
 Diagrams (1 pt. function) 

Expansion for K
 Link variables appear as a pair of Summation for K. (K2 , K4 ,…).

I discuss later
23/35"NTFL workshop'' T. Z. Nakano

2012/02/10

1 pt. Function
 Diagrams (1 pt. function) Expansion for K
 Link variables appear as a pair of Summation for K. (K2 , K4 ,…)24/35"NTFL workshop'' T. Z. Nakano.

2012/02/10

1 pt. Function
 Mean-field treatment


Scalar Pesudo-scalar Two solution

Parity conserved

Parity broken

For
25/35, Parity-broken?
"NTFL workshop'' T. Z. Nakano

2012/02/10

2 pt. Function
 Diagrams (2 pt. function)  Self-consistent equation

26/35"NTFL workshop'' T. Z. Nakano

2012/02/10

2 pt. Function
 Pion massF.T.
 Insert
 Another method
 Calculate

For, Insert? ,?

This result implies Aoki phase
27/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Effective Potential (Condensate) Effective Potential for Meson (large Nc)

We can identify the Parity-broken phase

For28/35,"NTFL workshop'' T. Z. Nakano

2012/02/10

Effective Potential (Pion Mass) Critical Mass (Mc) corresponds to the chiral limit.

29/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Aoki Phase in Staggered-Wilson Fermion
 Hoelbling type

 We can perform the latticesimulation with staggered-Wilson fermionsby tuning mass parameter.

30/35

?

"NTFL workshop'' T. Z. Nakano

2012/02/10

2 Flavor (Taste) Case Possibility (1)Parity symmetry breaking

 (2)Parity-flavor symmetry breaking

 Parity-flavor symmetry breaking
 Vafa-Witten’s theorem 

31/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Staggered-Wilson Fermion
 Lattice action (Adams type) D. H. Adams (2011) Flavored mass : 4 hopping

32/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Analysis in Adams type Similar way as Hoelbling type
e.g. HPE (1 pt. function)

33/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Discussion
 This HPE analysis is exact up to Other higher order terms in HPE

 Is Summation possible ?
34/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Summary & Future Works
 We have studied Aoki phase in Staggerd-Wilson fermion

in strong coupling limit.

 We can perform the lattice simulation with staggered-Wilson fermions by tuning mass parameter.
 Future Works
35/35

Evaluation of the summation for other higher diagrams"NTFL workshop'' T. Z. Nakano

2012/02/10

Back Up

36/35

"NTFL workshop'' T. Z. Nakano

2012/02/10

2 pt. Function
 Self-consistent equation for

37/35"NTFL workshop'' T. Z. Nakano

2012/02/10

Aoki Phase in Staggered-Wilson Fermion
 Adams type

38/35

"NTFL workshop'' T. Z. Nakano

2012/02/10