Pion form factors
in the ε regime
Hidenori Fukaya (Osaka Univ.)
for JLQCD collaboration :S. Aoki, HF, S. Hashimoto, T. Kaneko,
H. Matsufuru, J. Noaki, T. Onogi, N. Yamada
arXiv:1211.0743

1. Introduction
0.5

Experiment value
This work (Nf = 2+1)!"#$%#$&'()**+
RBC/UKQCD (Nf = 2+1)
ETMC (Nf = 2)

 r2 (fm2)

0.4

JLQCD/TWQCD (Nf = 2)�∂FV (q 2 ) ���r�V = 6∂q 2 �q2 =00.35

0.3

0.25

0.2
00.1

0.2

0.3

M2/ (GeV2)

0.4

For calculation of pion form factors,
we need
1. Very small pion mass
2. Good control of chiral symmetry-> large chiral logarithm∼ ln m2π

We simulate (exactly chiral symmetric)
overlap quarks with mud 〜 3 MeV.

1. Introduction

Eletctro-magnetic form factor of pion
0.45

1. Introduction

0.5,-./01&2034&!"#$%#$&'()**+

JLQCD (& TWQCD) project [2006-2012]= QCD with overlap quarks (exact chiral sym.).
1/a ~ 1.8 GeV, L ~ 1.8 fm
p regime lattices : mπ = 290-780 MeVε regime lattice : mπ ~ 100 MeV
mπ L ∼ 0.9 7&890.1&:;-<1&=&1>1?<@New project with Hitachi SR16K
5&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&6
& IBM BG/Q started.

1. Introduction

1. Introduction

Our strategy

Good chirality = high numerical costquark action

ChiralDiscretizat
symmetry ion error

Numerical calculation

Numerical costVolume sizeOverlap

Exact

O(a2)

Very expensive

~2fm

Domain-wall

WeaklybrokenO(a2)

Expensive~4fm

Wilson Broken

O(a)

Marginal~5fmBrokenO(a2)

Cheap~6fm

Staggered(U(1) remains)

Analytic calculation$49EE&E9]?1&B#C&&[-<I&1^9?<&?I-09E&@W441<0W,-;-<1&F3E/41&?3001?H3;&&ZW&L-3;&<I130W

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A8B#C&D&$49EE&F3E/41&B#CG
5

1. Introduction

1. Introduction
Whatʼs new in this talk ?
Automatic cancellation of ﬁnite V eﬀects

Finite volume = Pion physics.
#3001E9H3;&E1;.<I&5*JK6&32&B#C&L90H?E1@&&!-3;@5~*M)K1=6&~&*NM24&&O93;@5~P))K1=6~)NM24&&QI3&5~R))K1=6~)N(S24&&!03<3;&5~*T1=6&~)N(24&7

1. non-zero momenta
2. take an appropriate ratio-> We can eliminate LO ﬁnite V eﬀects(= we can forget about Bessel functions)even in the ε regime.

(for 2fm < L < 4fm)"<& & &&&&B#C&D&L-3;&5U&V93;6&<I130WN&X941EWY&:;-<1&F3E/41&?3001?H3;&-;&B#C&D&&& &&&&&&&&&&&&&&&&&&&&&&&&&L-3;&<I130W&5?I-09E&L10</0Z9H3;&<I130W6&&[19VEW&?3/LE1\&D&9;9EWH?9EEW&?9E?/E9ZE1N&6=> Pion form factors

8

2. Pion eﬀective theory

CONTENTS

Pion theory at ﬁnite V䘟 1. Introduction`103%43\1&D&$a5X6&530&a5X66&& & & &49<0-^&43\1E

2. Pion eﬀective theory
3. 3pt functions at ﬁnite V
4. Preliminary lattice results
5. Summary

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2. Pion eﬀective theory

2. Pion eﬀective theory

What is the ε regime ?

2pt functions in ε expansion of ChPTIn the p regime, the vacuum is ﬁxed:

Before d3x integral

but near the chiral limit at ﬁnite V, Mπ L < 1vacuum= zero-mode = dynamical variable

U0 should be non-perturbatively treated.
→ ε regime1110

�P (x)P (0)� = A + B

� 1 eip� xp� �=0V p�2+C� eip� xp� �=0p�4+ ···A, B, C · · · : matrix integrals
�(Bessel functions of mΣV(· · · ) : non-zero modeʼsp� �=0)

2. Pion eﬀective theory

3. 3pt functions at ﬁnite V

2pt functions in ε expansion of ChPT

2pt functions in ε expansion of ChPT

After d3x integral → power function of t.
1L3

�

1d x�P (x)P (0)� = A + B
V3� ��
�2
1T
1t−−+ C [· · · ] + · · ·2
224

(with periodic boundary conditions)cf. large V result:B � cosh(mπ (t − T /2))

3. 3pt functions at ﬁnite V

1L3�d3 xe−ipx �P (x)P (0)� =1L3

�

d3 xe−ipx A + B(w/ periodic b.c.)� 1 e−ip�0 t+ ···2V p�20 +p�
p0

where E(p) = |p| . the same form as the p-regime :�B � cosh( |p|2 + m2π (t − T /2))

B

1
4πF 2 L2∼

3pt functions in ε regime
�CP3pt
V0 P (∆t, ∆t ; pi , pf )= B 3pt (mΣV ) [E(pi ) + E(pf )] FV (q 2 )B=0+cosh(E(p)(t − T /2)) + · · · ,2E(p)L3 sinh(E(p)T /2)

But Bessel functions are still contained in

LO ﬁnite V eﬀect is eliminated !
NLO

3. 3pt functions at ﬁnite V

2pt functions in ε expansion of ChPT
After d3x integral with p �= 0Ratio of 2pt functions with p �= 0
� 3 −ipx1
�P (x)P (0)�cosh(E(p)(t − T /2))
L3 � d xe+ ··· ,=
�1� )(t − T /2))
3 xe−ip x �P (x)P (0)�cosh(E(pd3L.

× cosh(E(pi )(∆t − T /2)) cosh(E(pf )(∆t� − T /2)) + · · ·��CP3pt
1V P (t, t ; pi , pf )3pt
E(p)+E(pf )N|pii |,|pf | fixed|pi |,|pf |,q 2
RV (t, t� ; |pi |, |pf |, q 2 ) ≡ 
�
�
1
12pt
2pt

CP P (t, pi )  2pt
CP P (t� , pf )
2pt
N|pN|pf | fixed|p |i | fixed|p |if= B 3pt/2pt (mΣV )FV (q 2 ) + · · ·3. 3pt functions at ﬁnite V
Correlator with zero momentum

The constant term can be cancelled by3pt
3pt
��∆t� CP3pt
V P (t, t ; pi , 0) ≡ CP V P (t, t ; pi , 0) − CP V P (t, tref ; pi , 0)

3pt
3pt
�
�∆t ∆t� CP3pt
V P (t, t ; 0, 0) ≡ CP V P (t, t ; 0, 0) − CP V P (t, tref ; 0, 0)

3pt
�− CP3pt
V P (tref , t ; 0, 0) + CP V P (tref , tref ; 0, 0)

3. 3pt functions at ﬁnite V
Ratio of 3pt functions

�RV (t, t� ; |pi |, |pf |, q 2 )�RV1 (t, t� ; |pi |, 0, q 2 )FV (t, t , q ) ≡or
RV2 (t, t� ; 0, 0, 0)
RV2 (t, t� ; 0, 0, 0)
�2= FV (q ) + O2

�

14πF 2 L2�

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 &&&&&&230&?9;?1E-;.&<I1&@4190-;.&1>1?<Y&01;3049E-b9H3;&9;\&@3&3;N �2 FV (t, t� ; q 2 ) = RV (t,t ;|p�i |,|pf |,q )RV (t,t ;0,0,0)��= FV (q 2 ) + O e−mπ L'c9@I-43<3&1<&9EN&()))+

3. 3pt functions at ﬁnite VCorrelator with zero momentum
The both ratios
1RV1 (t, t� ; |pi |, 0, q 2 ) ≡RV2 (t, t� ; 0, 0, 0) ≡1

�3pt
N|pi | fixed|pi |,q 2�

�

�
3pt
�
CP3pt
V P (t, t ; pi , 0) − CP V P (t, tref ; pi , 0)CP2pt
P (t, pi )2pt
N|pi | fixed|pi |��,2pt ��−∆t� ∂CP2pt
P (t , 0) + E(pi )∆t� CP P (t , 0)

�∆t ∆t� CP3pt
V P (t, t ; 0, 0),2pt
2pt �2pt �−∆t CP P (t, 0)∆t� ∂CP P (t , 0) − ∆t ∂CP2pt
P (t, 0)∆t� CP P (t , 0)

= B 3pt/2pt (mΣV )FV (q 2 ) + · · ·3. Preliminary lattice results
Summary of numerical simulations

Iwasaki gauge action with β=2.30
2+1 dynamical overlap quarks
1/a ~ 1.759 GeV, L=16348 (L~1.8 fm)
mud =0.002(~3 MeV), ms=0.08
Q=0 ﬁxed
Smearing is used for PS operators φs (r) = e−0.4r
All-to-all propagators with 120 exact low-modes andnoise method for the high modes.
Dispersion relation is used for E with mπ = 98(5)MeV
148 samples from 2500 trj.

3. Preliminary lattice results
RV (t, t� ; |pi |, |pf |, q 2 )
RV2 (t, t� ; 0, 0, 0)1
pi2=2, pf2=1, q2=1

0.4

0
1

01

t=5
t=6
t=70.6
0.4

0.4
0.2

0101

pi2=2, pf2=1, q2=3

t=5
t=6
t=70.4

0.8

0.2
010

pi2=2, pf2=0, q2=2

t=6
t=7
t=8

pi2=3, pf2=0, q2=3

0.4

05tv

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0.60.2
0

t=6
t=7
t=8

0.6

0.2

0.6

pi2=1, pf2=0, q2=1

0.8
FVv(t, tv,q2)

FV(t, tv,q2)

0.4
0.2

0.8

t=6
t=7
t=805tv

10

3. Preliminary lattice results
FV (q 2 )1V(q )

0.6

FV (q 2 ) =0.4

1+ a1 q 2 + a2 (q 2 )21 − q 2 /m2ρ

�r2 �V = 60.20�∂FV (q 2 ) ��∂q 2 �q2 =0= 0.53(4) fm2 .

-2-1.5-12-0.5
2q [GeV ]3. Preliminary lattice results
NOTES

mud=0.002
VMD + pol.
pol.
mud=0.015

0.8

X1[&\9<9&-;&f&01.-41

0.60.2pi2=1, pf2=1, q2=2

RV1 (t, t� ; |pi |, 0, q 2 )
RV2 (t, t� ; 0, 0, 0)

0.8

0.6

0.8

FV(t, tv,q2)

1

t=5
t=6
t=7

FVv(t, tv,q2)

FV(t, tv,q2)

0.8

FV (t, t� , q 2 ) ≡FVv(t, tv,q2)

FV (t, t� , q 2 ) ≡3. Preliminary lattice results0

1. We DONʼT need Bessel functions in the analysis.
2. The result still contain NLO
1∼∼ 7% ﬁnite V eﬀects.
4πF 2 V 1/2
To do list :1. Check the stability of q2 ﬁt.
2. Check the dispersion relation.
3. 1-loop ChPT corrections [HF & T. Suzuki, in progress]
4. Twisted boundary conditions.
5. Expand the volume in our new simulations.

4. Summary
Our strategy

Numerical calculation

Achknowledgements

Analytic calculation$49EE&E9]?1&B#C&&[-<I&1^9?<&?I-09E&@W441<0W,-;-<1&F3E/41&?3001?H3;&&ZW&L-3;&<I130W

This work is supported by
Grant-in-Aid for Scientiﬁc Research on
Innovative Areas “Research on theEmergence of Hierarchical Structureof Matter by Bridging Particle,
Nuclear and Astrophysics in
Computational Science”(No.2004, 23105710).

B#C&9<&=D_25

4. SummaryComarison with others

Whatʼs new in this talk :Automatic cancellation of ﬁnite V eﬀects

Physical point simulation by PACS-CS [2011]

1. non-zero momenta (or subtraction)
2. take an appropriate ratio-> We can eliminate LO ﬁnite V eﬀects(= we can forget about Bessel functions)even in the ε regime.
=> Pion form factors

26

Zero-mode subtracted correlators :FV (t, t� , q 2 ) ≡RV (t, t� ; |pi |, |pf |, q 2 )
RV2 (t, t� ; 0, 0, 0)

may be helpful ?