Light meson spectrum with 2 dynamical N f overlap fermions · Light meson spectrum with 2 dynamical overlap fermions Jun Noaki (IPNS, KEK) for JLQCD Collaboration: S. Aoki, H. Fukaya,

Light meson spectrum with dynamical Light meson spectrum with dynamical overlap fermionsoverlap fermions

Jun Noaki (IPNS, KEK)

for JLQCD Collaboration:

S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko,

H. Matsufuru, T. Onogi, E. Shintani, N. Yamada

N f=2

LATTICE 2007, RegensburgJ. Noaki, 7/30/2007

Approach to the low energy QCD with exact chiral symmetry

● Theoretically clean● Any #flavor is possible

JLQCD's project Matsufuru's review

●

● 6 sea quark masses,● >10,000 trajs for each

This talk: “Check of consistency with ChPT?”

● light meson observables:

Dynamical overlap fermions

Dov =1R

[ 15 H W ]

Neuberger, 1998

m & f

2/16

N f= 2

ms /6 ≤ mq ≤ ms


At a glance

Profile

163×32 1.93×3.8fm4

SovS IwasakiSex-Wilson N f = 2

a−1≃1.6722GeV r0=0.49fm

290 MeV ≤ mPS ≤ 750 MeV

#config ≥ 500

⟨P t ptsP 0 pts ⟩ , ⟨P t ptsP 0 smr ⟩ m , f (ren)

P-P correlators with smeared/local sources

Qtop = 0

(130 MeV normalization)

3/16


Plan

New techniques

● analysis with low eigenmodes

Finite size effects

● ordinary FSE correction● fixed topology correction

Non-perturbative renormalization

ChPT tests

● chiral extrapolation at NNLO● low energy constants

Summary

4/16


Low mode preconditioning

With lowlying eigenmodes

Dovui=iui

S q x , y =∑i=1

50

ui x ui y

imq

S qHigh

x , y

C t = C HH t C LH t

C HL t C LL

t

Lowest 50 eigenpairs stored on disk

Meson correlator

5/16

Giusti et al., 2003




Dovui=iui

S q x , y =∑i=1

50

ui x ui y

imq

S qHigh

x , y

C t = C HH t C LH t

C HL t C LL

t


Meson correlator

5/16

Giusti et al., 2003

CG Is x8 faster




Dovui=iui

S q x , y =∑i=1

50

ui x ui y

imq

S qHigh

x , y

C t = C HH t C LH t

C HL t C LL

t


Meson correlator

Average over source locations (low-mode averating)DeGrand & Schaefer, 2004; Giusti et al., 2004

5/16

Giusti et al., 2003

CG is x8 faster




Dovui=iui

S q x , y =∑i=1

50

ui x ui y

imq

S qHigh

x , y

C t = C HH t C LH t

C HL t C LL

t


Meson correlator

Average over source locations (low-mode averating)DeGrand & Schaefer, 2004; Giusti et al., 2004

5/16

Giusti et al., 2003

CG is x8 faster


Finite size effects

is not large enough

● for the lightest mass:

Correction with ChPT exact chiral symm.

● Standard FSE

● Fixed topology

6/16

L≃ 1.9 fm

m L≃ 2.8


Standard FSE correction

Resummed Luscher's formulae + NNLO ChPT

mLm∞

= 1−∑k=1

∞ nk

2k[2 I m2k , 4 2 I m

4 k , ,1,2,3,4 O 3 ]

f Lf ∞

= 1∑k=1

∞ nk

k[2 I f2k , 42 I f

4 k , ,1,2,3,4 O 3]

= m

4 f 2

, = m L

Colangelo, Durr and Haefeli, 2005

m

2 f

NLO −4.46% 6.31%−2.1% 4.4%

NNLO

for the lightest mass

errors from phenomenological LECs

applicable only to full theory (not PQ)

..

..

7/16


Fixed topology correction

Topology fixing:

8/16

O topV −1 a4top

Susceptibility

⟨mq P0t ⋅mqP

00 ⟩Q=0

t=large=A⋅e−m ' t−

top

VO V−2

Chiu's talk

Onogi's talk

top = mq/N f

Brower et al, 2003;

saddle pt. expansion + U(1) axial WTI


Fixed topology correction

NLO ChPT Aoki et al, 2007

m ,Q=0

m =0 = 1−

116V top

[1 1 ln m /3 2 ]

f ,Q=0

f =0 = 1

14V top

1 ln m /4 2

Topology fixing:

Susceptibility

⟨mq P0t ⋅mqP

00 ⟩Q=0

t=large=A⋅e−m ' t−

top

VO V−2

Chiu's talk

Onogi's talk

8/16

O topV −1

top = mq/N f

Brower et al, 2003; a4top

saddle pt. expansion + U(1) axial WTI


Corrected data

- Two corrections cancel. - Fixed top. correction is negligible.

m

2/mq

(ren) f (ren)

- Suppression by small volume.

9/16


RI/MOM condition:

Quark mass renormalization

Z q−1Z p = 1

S p = A⋅ ∂∂mq

⟨ ⟩Z q ZmBS⋅mq2

⟨ ⟩ = ⟨1V∑i=1

50 2mq

mq2i

2 ⟩

Landau gauge

Mass dependence

WTI+OPE:

with real data:

Martinelli et al., 1995

10/16

P p = A⋅⟨ ⟩

mq

Z q ZmBP⋅mq2


RI/MOM condition:

P p = A⋅⟨ ⟩

mq

Z q ZmBP⋅mq2

S p = A⋅ ∂∂mq

⟨ ⟩Z q ZmBS⋅mq2

⟨ ⟩ = ⟨1V∑i=1

50 2mq

mq2i

2 ⟩

Z q=Z AWTI×A

Mass dependence

with real data:

Martinelli et al., 1995

quark field renorm.:

P

S

10/16


WTI+OPE:

simul. fit for same color

Z q−1Z p = 1

Landau gauge



Matching (3-loop level)

Z mMS2GeV=0.74212

Chiral limit

Z mMS2GeV = wMS

−1 2GeV⋅ZmRGI

Z mRGI = wRI/MOM p⋅Z m

RI/MOM p

msea = mval = 0 :

Z mRGI

Z mRI/MOM

11/16

Chetyrkin & Retey, 2000


Chiral extrapolation: NNLO analysis

NLO m2 /mq = 2B0 1 ln L3

f = f 1−2 ln L4independent fits

= m

4 f 2

Expansion in: composed by actual data

12/16



NLO m2/mq = 2B0 1 ln L3 f = f 1−2 ln L4

NNLOm2

mq

= 2B0 [1 ln72

ln 22 L4f −

43 L16 2 ln ]

L3 −92 lnK 1

2

f = f [1−2 ln5 ln 2−32 L532 2 ln ]

L4 −102 lnK 2

2

input: L= 7 ln 1

4 f 2

8 ln 2

4 f 2

independent fits

= m

4 f 2

simultaneous fit

Expansion in: composed by actual data

12/16

from phenomenology


In target range,


NNLO'm2

mq

= 2B0 1 ln72

ln 2L3K 1 '

2

f = f 1−2 ln5 ln 2 L4K 2 '

2

A⋅2 lnB⋅2≈ B' 2

independent fits, again

13/16

Instability of A and B

NNLO terms are delicate


Fit curvesm

2/mq

(ren) f (ren)

- Reasonably fit to the data.

- Vartical errors taken into account.

14/16


Low energy parameters=B0⋅f

2/2f =122.2MeV

15/16

Gasser & Leutwyler,1984


l 4phys

= ln 4

mphys

2

= ln4 f

mphys

2

L42 f

Low energy parameters

l 3phys

= ln 3

mphys

2

= ln4 f

mphys

2

−L32B0

=B0⋅f2/2f =122.2MeV

15/16

Gasser & Leutwyler,1984


Summary

dynamical overlap fermions

NNLO ChPT analysis of

●

●

● The scale uncertainty yet to be investigated for systematic error.

● NLO is not sufficient in our mass region.

Next:

● 2+1 flavor Hashimoto's poster

● larger volume check of the FSE correction

16/16

N f=2

m & f

mK& f K

f =111 4 2 , 1 / 3=242 6 6

l 3phys

=2.9 0.4 2.6 , l 4phys

=4.3 5 2 (stat.)(|NNLO'-NNLO|)

Documents

Light meson spectrum with 2 dynamical N f overlap fermions · Light meson spectrum with 2 dynamical overlap fermions Jun Noaki (IPNS, KEK) for JLQCD Collaboration: S. Aoki, H. Fukaya,