Pion Correlators
in the ε- regime
Hidenori Fukaya (YITP)
collaboration with
S. Hashimoto (KEK)
and K.Ogawa (Sokendai)
0. Contents
1. Introduction
2. Lattice Simulations
3. Results （quenched）
4. Conclusion
1. Introduction
1-1. Our Goals

Lattice QCD
- 1st principle and non-perturbative calculation.

Chiral perturbation theory (ChPT)
- Low energy effective theory of QCD (pion theory).
- Free parameters Fπ and Σ.
It is important to determine Fπ and Σ from
1-st principle calculation but simulations at
m~0 (m<30MeV) and large V (V>2fm) are difficult...
⇒ Consider fm universe (ε-regime).
1. Introduction
1-1. Our Goals
In the ε- regime ( mπL < 1 , LΛQCD>>1), we have
ChPT with finite V correction.
 Quenched QCD simulation
⇒ low energy constants (Σ, Fπ, α…) of
et
quenched ChPT (in small V). S.R.Sharpe(‘01)P.H.Damgaard
al.(‘02)…
 Full QCD simulation
J.Gasser,H.Leutwyler(‘87),F.C.Hansen(‘90),
H.Leutwyler,A.Smilga(92)…
⇒ those of ChPT (in small V).
In particular, dependence on topological charge Q
and X ≡ mΣV is important .
1-2. Setup
 To simulate m～0 region, ’Exact’ chiral
symmetry is required.
⇒ Overlap operator (Chebychev
polynomial (of order～150 )) which
satisfies Ginsparg-Wilson relation.
 Fitting pion correlators in small V at different Q
and m with ChPT in the ε-regime ⇒ extract Σ,
Fπ, α, m0
P.H.Ginsparg,K.G.Wilson(‘82), H.Neuberger(‘98)
1-3. Pion correlators in the ε-regime
where
 Quenched ChPT in small V
P.H.Damgaard et al. (02)
Pion correlators are not exponential but
and
 ChPT in small V (Nf=2)
Fitting
the coefficient of H1(t) and H2(t) with
lattice data at various Q and m, we extract
Σ, Fπ, α, m0.
2. Lattice Simulations
2-1. Calculation of D -1
Overlap at m~0 ⇒ Large numerical costs !

Low mode preconditioning
L.Giusti et al.(03)
We calculate lowest 100 eigenvalues and eigen
functions so that we deform D as
⇒ costs for
at m=100MeV !
at m=0 ~ costs for
2-2. Low-mode contribution in pion
correlators

Is the low-mode contribution dominant ?
As m→0 ⇒ low-modes must be important.
We find the contribution from
is negligible
( ~ only 0.5 %.) for m<0.008 (12.8MeV)
and Q ≠ 0 at large t , so we can approximate
The difference < 0.5%
for large |x-y|. for 3 ≦ t ≦ 7 .
2-2. Low-mode contribution in pion
correlators

Pion source averaging over space-time
Now we know
at all x. ⇒ we know
at any x and y. Averaging over x0 and t0;
reduces the noise almost 10 times !
2-3. Numerical Simulations
Size :β=5.85, 1/a = 1.6GeV, V=104 (1.23fm)4
 Gauge fields: updated by plaquette action
|Q|
0
1
2
3
(quenched).

# of conf. 50

76
57
19
Fermion mass: m=0.016,0.032,0.048,0.064,0.008
( 2.6MeV ≦ m ≦12.8 MeV !!)



100 eigenmodes are calculated by ARPACK.
Q is evaluated from # of zero modes.
Source pion is averaged over x=odd sites for
Q ≠ 0.
preliminary
3. Results (quenched QCD)
3-1. Pion correlators
=
mm
m
= 12.8
=8
5 MeV
MeV
MeV
Q =3
Q =2
Q =3
Q =2
=1
Q =1
Our data show
remarkable Q and m
dependences.
preliminary
3-2. Low energy parameters
Using
m=5 MeV ← Ogawa’s talk
m=10.2MeV
m=2.6MeV
P.H.Damgaard (02)
we simultaneously fit all of our data
(15 correlators ) with the function;
We obtain
Σ = (307±23 MeV)3, Fπ= 111.1±5.2MeV,
α = 0.07±0.65, m0 = 958±44 MeV, χ2/dof=1.5.
4. Conclusion
In quenched QCD in the ε-regime, using
 Overlap operator ⇒ ‘exact’ chiral symmetry,
まとめ
 2.6 MeV ≦ m ≦ 12.8 MeV ,
（実質）１００倍の統計をためると
 lowest
100 eigenmodes (dominance~99.5%),
 Pion source averaging over space-time,
できなかったことができた。
( equivalent to 100 times statistics )
we compare the pion correlators with ChPT .
3, F remarkable
⇒ Σ=(307±23
The correlators
MeV)
show
MeV,
and
π=111.1±5.2Q
m dependences.
α=0.07±0.65,
m0=958±44 MeV.
4. Conclusion
As future works,
 a → 0 limit and renormalization,
 isosinglet meson correlators,
 full QCD ( → Ogawa’s talk),
 consistency check with p-regime results,
will be important.
A. Full QCD

Lowest 100 eigenvalues
A. Full QCD

Truncated determinant
The truncated determinant is equivalent to
adding a Pauli-Villars regulator as
where, for example,



γ→0 limit ⇒ usual Pauli-Villars (gauge inv,local).
Λ→0 limit ⇒ quench QCD (good overlap config. ?)
If Λa is fixed as a→0, unitarity is also restored.

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西村淳氏（総研大＆KEK）、青木一氏（佐賀大）との共同研究