Hidenori Fukaya (Osaka U.)
for JLQCD Collaboration
JLQCD collaboration, arXiv:1111.0417
S. Aoki (Tsukuba) & HF, PRD 84, 014501 (2011) [arXiv:1105.1606][P.H. Damgaard (NBI,NBIA) & HF, JHEP0901:052, 2009]1

1. Introduction
Q.
What is the “epsilon regime” ?
A.
Quantum Chromo Dynamics (QCD)in (vicinity of) thelimit
in a finite volume.
2

1. Introduction [Nambu, 1961 ] (2008 Nobel Prize)
Chiral symmetry is important near
• Chiral symmetry breaking and constituent mass
→ “effective” quark actionacquires constituent mass
→ hadron masses are ~ O(1) GeV.

• Pion effective theory（pseudo) Nambu-Goldstone boson = pion
described by Chiral perturbation theory （ChPT)[Weinberg 1979]3

1. Introduction	
Origin of mass = chiral symmetry breaking.
 Hadrons consist of quarks.

uud

 But

proton mass >> quark mass ×3(1GeV)(3-6MeV) Chiral symmetry breaking generate~90% of mass. 	
4

1. Introduction	JLQCD (+ TWQCD) collaboration
is simulating lattice QCD
with exact chiral symmetry
using overlap fermion action.

51. Introduction	JLQCD (+ TWQCD) collaboration
is simulating lattice QCD
with exact chiral symmetry
using overlap fermion action.

6

1. Introduction	
Chiral symmetry is expensive.
differenct action = different errors and cost.
Fermion action	

ChiralDiscretizatsymmetry	 ion error	

Numerical cost	Overlap

Exact	O(a2)	

Very expensive	Domain-wall	

Weaklybroken	O(a2)	

Expensive	Wilson 	Broken	O(a)	

Marginal	

Staggered	Broken

O(a2)	

Cheap(U(1) remains)	

7

1. Introduction	
Chiral symmetry is expensive.
differenct action = different errors and cost.
Fermion action	

ChiralDiscretizat
symmetry	 ion error	

Numerical cost	

Lattice size	Overlap

Exact	

O(a2)	

Very expensive	< 2.4 fm	Domain-wall	

Weaklybroken	O(a2)	

Expensive	~4fm	Wilson 	Broken	O(a)	

Marginal	~5fm	

Staggered	Broken

O(a2)	

Cheap~6fm	

O(a2)	

Cheap

(U(1) remains)	

New-type(min. doubling)	

Exact	JLQCD = Small volume QCD…	

8

1. Introduction
Finite volume = Pion physics.
Correlation length (1/M) of QCD particles
Pions(~140MeV) ~ 1.4fm
Kaons(~500MeV)~ 0.4fm
Rho (~800MeV)~0.26fm
Proton (~1GeV) ~0.2fm
AtQCD = pion (+ kaon) theory.
Finite volume correction in QCD =chiral perturbation theory (ChPT)
weakly coupled = analytically calculable.

91. Introduction
Our strategy
Numerical calculation

Analytic calculation

Small lattice QCD
with exact chiral symmetry	

Finite volumecorrection by pion
theory	Chiral symmetry isimportant but expensive.

QCD at V=∞	10Contents✓ 1. Introduction

2. Analytic calculation of finite V effects(on the pions)
3. Numerical lattice QCD results
4. Summary11

2. Analytic calculation of finite V
effects
Pion correlator at V=(in Euclidean space-time,)�d x�P (x)P (0)� = Aδ
3abab

�

dp4 eip4 t
2π p24 + Mπ2∝ exp(−Mπ t)
12

2. Analytic calculation of finite V
effects
Pion correlator at finite V ( in the p-expansion )(periodic boundary for t-direction)

13

2. Analytic calculation of finite V
effects
BUT…

In the limit,

Infra-red divergence due to finite V ???
despite we have IR cut-off 1/V1/4?

Something wrong ! Exp->Cosh is not enough !
14

2. Analytic calculation of finite V
effects
Many vacua contribute at finite V
This fake IR divergence is due to a fixed vacuum:

but at finite V, the vacuum is not uniquelydetermined: vacuum= moduli = dynamical variable

U0 should be non-perturbatively treated.
→ ε-expansion (is needed for)15

2. Analytic calculation of finite V
effects	
ε expansion of Chiral LagrangianZero-mode = SU(N) matrix model	
Non-zero-mode = massless bosons	(perturbative) interactions	= a hybrid system ofmatrix model and massless bosonic fields
16

2. Analytic calculation of finite Veffects
ε and p expansions are really useful ?
ε expansion

p expansion

Group	Nf	Action	

a(fm)	L	Mπ (MeV)	

ETMC

2	

Twisted mass	

0.05-0.100	

~3fm	

280	

MILC	

2+1	

Staggered	

0.045-0.12	

3~6 fm	250	

RBC/UKQCD	

2+1	

Domain wall	

0.085-0.11	

3~4fm	290	JLQCD	

2+1	Overlap	

0.11	

1.8fm	

310	PACS-CS	

2+1	

Wilson	0.09	~3fm	

140	

BMW	

2+1	

Wilson	0.065-0.125	

3~5fm	

190	

ALV	

2+1	

DW on MILC	

0.06-0.12	

3~4fm	

250	HPQCD	

2+1	

HISQ	

0.045-0.15	

3~4fm	360	But on the lattice,	

No requirement in original theory ( if):both expansions are bad. → Better way of expansion
17(covering both regimes) ?

2. Analytic calculation of finite Veffects
New expansion
p-expansion
ε-expansion

18

2. Analytic calculation of finite Veffects
New expansion
p-expansion
ε-expansion

New i (interpolating)- expansion

19[Damgaard & HF, 2008]	

2. Analytic calculation of finite Veffects
The pion (chiral) Lagrangian at finite V
Zero-mode = SU(N) matrix model	
Non-zero-mode = massive bosons	(perturbative) interactions	[Damgaard & HF, 2008]	

= a hybrid system ofmatrix model and massive bosonic fields

20

2. Analytic calculation of finite V
effects
Pion correlator at 1-loop[Aoki & HF, 2011]	

Because of the mixing of zero and non-zero modes,
the calculation is fairly tedious :21

2. Analytic calculation of finite V
effects
where[Aoki & HF, 2011]	

22

2. Analytic calculation of finite V
effects	
The zero-mode integrals“simplest” example (in 2+1 flavor theory)23

2. Analytic calculation of finite Veffects
Pion correlator at 1-loop[Aoki & HF, 2011]	

cancels IR divergence:
disappears in the p-regime:(functions of)	24

2. Analytic calculation of finite Veffects
Summary of Sec.2
1. Finite V effects = Pion physics(Other hadrons are heavy and insensitive to finite V. )2. Pion physics = Chiral perturbation theory (ChPT)

weakly coupled → analytically calculable.
3. ChPT in finite V = matrix model + bosonic fields

computations are tedious… but the result is beautiful:

Smooth interpolation to infinite V in a infra-red finite way. 25

3. Numerical lattice QCD results(preliminary)
QCD simulation with exact chiral symmetry[JLQCD & TWQCD collaborations, 2006-2011]	

2+1-flavor overlap Dirac fermions [Neuberger 98]
Iwasaki gauge action, β=2.3, 1/a ~ 1.759 GeV.
Lattice size : L=16 [1.8 fm], T=48.
Topology fixed: Q=0 (or 1)
Quark masses : ms=0.08, 0.100,
mud= 0.002, 0.015,0.025,0.035,0.050,0.080,0.100(~3 MeV)ε-regime,(30MeV <)p-regime	26

3. Numerical lattice QCD results(preliminary)Really inside the ε-regime?→ Scalar channel knows.

mud=0.002	

mud=0.015	

27

3. Numerical lattice QCD results(preliminary)Really inside the ε-regime? → Yes.
ε expansion analysis :mud=0.002	

[Damgaard et al, 2002]	

PP →SS →consistent with our previous value
from Dirac spectrum.

Next, let us extract

with the “i” expansion formula.
( Naïve GMOR relation suggests)	 28

3. Numerical lattice QCD results(preliminary)
Pion mass and decay constant in the ε regime
Fitting withafter 1-loop (of non-zeromodes) volumecorrection, we obtain29

3. Numerical lattice QCD results(preliminary)
Chiral “interpolation” forbigger than ourprevious analysis using
p-regime data only :Linear fit looks better
than NLO ChPT fit, though…
30

3. Numerical lattice QCD results(preliminary)
Chiral “interpolation” for
ChPT fit looks bad :deviation.
2-loop (1/V) effects fromnon-zero modes undercontrol ?

~7.5%	

31

3. Numerical lattice QCD results(preliminary)
Bad convergence of ChPT for
In the limit?

,Bad.
(big correction/LO)
Good.

32

4. Summary
Small lattice QCD with exact chiral symmetry+Finite V correction from pion effective theory(ChPT)

can extract physics at V=∞.
33