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Pion Correlators in the ε- regime. Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai). 0. Contents. Introduction Lattice Simulations Results ( quenched ) Conclusion. 1. Introduction. 1-1. Our Goals Lattice QCD - PowerPoint PPT Presentation
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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).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
quenched ChPT (in small V). Full QCD simulation⇒ those of ChPT (in small V).In particular, dependence on topological charge Qand X ≡ mΣV is important .
J.Gasser,H.Leutwyler(‘87),F.C.Hansen(‘90),
H.Leutwyler,A.Smilga(92)…
S.R.Sharpe(‘01)P.H.Damgaard et al.(‘02)…
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)
P.H.Damgaard et al. (02)
1-3. Pion correlators in the ε-regime
Quenched ChPT in small V
Pion correlators are not exponential but
ChPT in small V (Nf=2)
where
and
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
We calculate lowest 100 eigenvalues and eigen functions so that we deform D as
⇒ costs for at m=0 ~ costs for at m=100MeV !
L.Giusti et al.(03)
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
for large |x-y|.The difference < 0.5% for 3 t 7 .≦ ≦
2-2. Low-mode contribution in pion correlators
Pion source averaging over space-timeNow 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 (quenched).
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.
|Q| 0 1 2 3
# of conf.
50 76 57 19
3. Results (quenched QCD)
3-1. Pion correlators m = 5 MeV
Q =1
Q =2
Q =3
m = 8 MeV
Q =1
Q =2
Q =3
m = 12.8 MeV
Our data show remarkable Q and m dependences.
preliminarypreliminary
Using
we simultaneously fit all of our data
(15 correlators ) with the function;
← Ogawa’s talkP.H.Damgaard (02)
3-2. Low energy parameters
m=2.6MeVm=5 MeVm=10.2MeV
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.
preliminarypreliminary
4. ConclusionIn quenched QCD in the ε-regime, using Overlap operator ‘exact’ ⇒ chiral symmetry, 2.6 MeV m ≦ ≦ 12.8 MeV , lowest 100 eigenmodes (dominance~99.5%), Pion source averaging over space-time,
( equivalent to 100 times statistics )
we compare the pion correlators with ChPT .
⇒ The correlators show remarkable Q and
m dependences.
⇒ Σ=(307±23 MeV)3, Fπ=111.1±5.2 MeV,
α=0.07±0.65, m0=958±44 MeV.
まとめ
(実質)100倍の統計をためると
できなかったことができた。
4. ConclusionAs 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.