QCD Downunder 2017@Cairns, QLD, Australia 13/7/2017

Electromagnetic pion form factor at physical point from Nf = 2+1 QCD

K.-I. Ishikawa, N. Ishizuka, Y. Kuramashi,Y. Nakamura, Y. Namekawa, Y. Taniguchi, N. Ukita,T. Yamazaki, and T. Yoshie

Junpei Kakazu University of Tsukuba

(PACS Collaboration)

1

Motivation• Electromagnetic form factor =deviation from charged point particle 　→Hadron’s structure (cf. charge radius)

previous results of radii in lattice calculation at physical point and experimental value

• chiral extrapolation → 　　　　　　　 　　　error increases

• small momentum transfer and suppressing finite size effect → need large box size

calculation near physical point on the large volume +extrapolation to physical point

Purpose

However, errors of charge radii in lattice calculation are larger than experimental one

2

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

<r2 >(

fm2 ) experimental value

’16 ETMC Nf=2+1+1 TM’16 JLQCD Nf=2+1 Overlap’15 HPQCD Nf=2+1+1 HISQ’13 Mainz Nf=2 Clover’11 PACS-CS Nf=2+1 Clover’09 ETMC Nf=2 TM’08 RBC/UKQCD Nf=2+1 DWF

Electromagnetic form factor⌦⇡+(pf )

��Vµ

��⇡+(pi)↵= (pf + pi)µf⇡⇡(q

2)

electromagnetic current Vµ =X

f

Qf ̄f�µ f :charge of flavor

(normalization condition relating to pion’s charge)q2 = �(pf � pi)

2 � 0 (space-like momentum transfer →calculating directly )

Definition

calculating 3-pt. function and 2-pt. function by Lattice QCD

C⇡V ⇡ = ZV

D0���O⇡(tf ,

�!p f =�!0 )V4(t,

�!p = �!p f ��!p i)O†⇡(0,

�!p i)���0E

=ZV Z⇡(

�!0 )

2m⇡Z⇡(�!p )

⌦⇡(�!0 )

��V4(�!p )

��⇡(�!p )↵e�m⇡(tf�t)

=ZV Z⇡(

�!0 )

2m⇡Z⇡(�!p )

(E⇡(p) +m⇡)f⇡⇡(q2)e�m⇡(tf�t)

R(t, p) =2m⇡Z⇡(0)

(E⇡(p) +m⇡)Z⇡(p)

C⇡V ⇡(t, tf ; p)

C⇡V ⇡(t, tf , 0)e(E⇡(p)�m⇡)t

in we can extract form factor from R0 ⌧ t ⌧ tf

= ZVZ⇡(0)Z⇡(p)

4E⇡(p)m⇡h⇡(0) |V4(0, p)|⇡(p)i e�E⇡(p)te�m⇡(tf�t)

= ZVZ⇡(0)Z⇡(p)

4E⇡(p)m⇡(E⇡(p) +m⇡)f⇡⇡(q

2)e�E⇡(p)te�m⇡(tf�t)

|Z⇡(p)|2(e�E⇡(p)t + e�E⇡(p)(T�t))|Z⇡(p)|2(e�E⇡(p)t + e�E⇡(p)(T�t))

Z⇡(p) = h0 |O⇡(0, 0)|⇡(p)i

3

⌦⇡+(pf )

��Vµ

��⇡+(pi)↵= (pf + pi)µf⇡⇡(q

2)

⌦⇡+(pf )

��Vµ

��⇡+(pi)↵= (pf + pi)µf⇡⇡(q

2)

• connected 3-point function is consist of ３quark propagators

• １ random wall source

• ２ sequential source

• ３ random wall source

Calculation of 3-point function

random wall source(A,B:color&spinor index )

calculation cost is reduced by random wall source RBC&UKQCD:JHEP( 0807 (2008)112)

disconnected term is vanished by charge symmetry

�!p f =�!0

�!p i =�!0

�!p i 6=�!0

4

light pion mass＋ periodic boundary condition in temporal direction →current time dependence appears in R

Extraction of form factor

5

pion mass is 0.145 GeVtf � ti = 36

we should consider wrapping around effect

0 10 20 30 40t

0.92

0.94

0.96

0.98

1R(t,p)

q2=0.019GeV2

q2=0.033GeVq2=0.046GeV2

form factor＋wrapping around effect

light pion mass＋ periodic boundary condition in temporal direction →current time dependence appears in R

f0 + f1 ⇥e�E2⇡⇥te�E⇡f

⇥(T�tf ) � e�E2⇡⇥(tf�t)e�E⇡i⇥(T�tf )

e�E⇡i⇥te�E⇡f⇥(tf�t)

,f0 + f1 ⇥e�E2⇡⇥te�E⇡f

⇥(T�tf ) � e�E2⇡⇥(tf�t)e�E⇡i⇥(T�tf )

e�E⇡i⇥te�E⇡f⇥(tf�t)

,＋

Extraction of form factor

tfti

f0 + f1 ⇥e�E2⇡⇥te�E⇡f

⇥(T�tf ) � e�E2⇡⇥(tf�t)e�E⇡i⇥(T�tf )

e�E⇡i⇥te�E⇡f⇥(tf�t)

,

E2⇡ = Ei + Ef :sum of single pion energy

we should consider wrapping around effect

6

h0|Vµ|⇡⇡if1 is related to

tfti tfti

fits including wrapping around effect work well in our data

red : fit including wrapping around effect blue : form factor

7

R(t, p) =2m⇡Z⇡(0)

(E⇡(p) +m⇡)Z⇡(p)

C⇡V ⇡(t, tf ; p)

C⇡V ⇡(t, tf , 0)e(E⇡(p)�m⇡)twe fit

fit form

the extraction form factor of the smallest and second smallest momentum transfer

10 15 20 25t

0.93

0.94

0.95

0.96

0.97

0.98

0.99

R(t,p)

Simulation details

m⇡L ⇡ 6 (ud,s) = (0.126117, 0.124790)

gauge configuration (HPCI Strategic Program Field 5)

Iwasaki gauge+stout smeared link Wilson clover action N f = 2 +1

� = 1.82, nstout

= 6, ⇢ = 0.1, csw

= 1.11

p1 (1, 0, 0) (0, 1, 0) (0, 0, 1)p2 (1, 1, 0) (1, 0, 1) (1, 1, 0)p3 (1, 1,�1) (1,�1, 1) (1, 1,�1)p4 (2, 0, 0) (0, 2, 0) (0, 0, 2)p5 (2, 1, 0) (0, 2, 1) (1, 0, 2)p6 (1, 1, 2) (1, 2, 1) (1, 1, 2)

4 sources × 4 directions (x, y, z, t) ×2 random sources = 32 meas. per config.

measurement parameter

tf � ti = 36

All the results are preliminary

resources PRIMRGY cx400 (tatara), RIIT, Kyusyu University and HAPACS, CCS, University of Tsukuba

a�1 = 2.333(GeV) ! a = 0.084(fm)L3 ⇥ T = 963 ⇥ 96 mK ⇡ 0.525GeV

directions of source momenta �!n

�!p =2⇡

L�!n

100 configurations (every 20 traj.)

periodic boundary condition for all directions

m⇡ = 0.145(GeV), L = 8.1(fm)

q2 = 2m⇡(E⇡ �m⇡)

bin size:100 traj.sink momentum : 0

strange mass reweighting factors of , s = 0.124768s = 0.124812

8

NLO ChPT fit works well

preliminary result: NLO SU(2) ChPT fit of generation point f⇡⇡(q2) = 1 +

1

f2

h2lr6q

2+ 4

˜H(m2⇡, q

2, µ2)

i

⌦r2↵=

�12l6f2

� 1

8⇡2f2

✓log

✓m2

⇡

µ2

◆+ 1

◆

µ = m⇢ = 0.77(GeV), f = 0.12925(GeV)

µ = m⇢ = 0.77(GeV), f = 0.12925(GeV) (mu,md ! 0)

there is one fit parameter l6: Low Energy Constant

In FLAG’s line (mu,md ! 0)f = 0.122553(GeV)

9

l6 �

2/d.o.f

n=1 -0.01236(51)n=2 -0.01240(49) 0.08n=3 -0.01238(53) 0.17n=6 -0.01216(57) 0.55

FLAG -0.01233(127)

almost physical pion mass small enough

n : the number of q used in the fit in ascending order from smallest q

arXiv:1607.00299 [hep-lat]

H(m2, q2, µ2) =

m2

32⇡2

0

@�4

3

+

5

18

x� x� 4

6

rx� 4

xlog

0

@

qx�4x

+ 1

qx�4x

� 1

1

A

1

A� q2

192⇡2log

m2

µ2(x = � q2

m2).

q2

fSU(2)⇡⇡ (q2) = 1 +

1

f2

⇥2l6(µ)q

2 + 4H(m2⇡, q

2, µ2)⇤

0 0.02 0.04 0.06 0.08

q2[GeV2]

0.85

0.9

0.95

1

f ππ(q

2 )

su(2) chpt n=2su(2) chpt n=6FLAG

preliminary result: NLO SU(2) ChPT fit with reweighted

(mu,md ! 0)f = 0.12956(GeV)

(mu,md ! 0)f = 0.12875(GeV)

s = 0.124812

s = 0.124768

l6 �

2/d.o.f

n=1 -0.01227(72)n=2 -0.01199(64) 0.7n=3 -0.01229(55) 1.4n=6 -0.01197(59) 0.95

FLAG -0.01233(127)

l6 �

2/d.o.f

n=1 -0.01158(62)n=2 -0.01173(63) 0.35n=3 -0.01106(65) 3.0n=6 -0.01101(73) 1.8

FLAG -0.01233(127)Including third smallest momentum transfer, of these fits increase�

2/d.o.f = 0.7

we decide the result of n=2 as central value in our analysis of strange mass extrapolation.

⌦⇡+(pf )

��Vµ

��⇡+(pi)↵= (pf + pi)µf⇡⇡(q

2)

mK = 0.526(GeV)

mK = 0.523(GeV)

ms

10

0 0.02 0.04 0.06 0.08

q2[GeV2]

0.85

0.9

0.95

1

f ππ(q

2 )

n=6 κs=0.124768n=6 κs=0.124812n=2 κs=0.124768n=2 κs=0.124812

preliminary result: physical strange mass extrapolation

l6(�1s ) = c1�1

s + c0

constant fit(blue)

linear fit(green)

s = 0.124902

In all s = 0.124902

11

1st error : statistical 2nd : combined error by choice 2pt fit range & choice of q fit range of NLO SU(2) ChPT 3rd : model dependance of strange mass extrapolation

our result with systematic error l6 = �0.01211(48)(+830 )(0�68)

l6 (const.) l6 (linear)

n=2 -0.01211(48) -0.01279(206)

n=6 -0.01182(53) -0.01390(189)

FLAG -0.01233(127)

8.006 8.008 8.01 8.012 8.014 8.016κs

-1

-0.016

-0.014

-0.012

-0.01

l 6

n=2n=6n=2 constant fit n=6 constant fitn=2 linear fitn=6 linear fitphysical κsour resut(stat. error)our resut (total error)

preliminary result: NLO SU(3) ChPT fit

there is one fit parameter L9: Low Energy Constant

previous SU(3) ChPT studies

1.078 ff0

1.229

12

NLO SU(3) ChPT fit works in our data our result is consistent with other calculations.

extrapolation with NLO SU(3) ChPT formula to the physical π, K mass

fSU(3)⇡⇡

(q2) = 1 +

1

f20

⇥�4L9(µ)q

2+ 4H(m2

⇡

, q2, µ2) + 2H(m2

K

, q2, µ2)

⇤,

H(m2, q2, µ2) =

m2

32⇡2

0

@�4

3

+

5

18

x� x� 4

6

rx� 4

xlog

0

@

qx�4x

+ 1

qx�4x

� 1

1

A

1

A� q2

192⇡2log

m2

µ2(x = � q2

m2).

⌦r2↵= �6

df(q2)

dq2

����q

2=0

=

24L9(µ)

f20

� 1

8⇡2f20

✓log

✓m2

⇡

µ2

◆+ 1

◆� 1

16⇡2f20

✓log

✓m2

K

µ2

◆+ 1

◆.

fSU(3)⇡⇡

(q2) = 1 +

1

f20

⇥�4L9(µ)q

2+ 4H(m2

⇡

, q2, µ2) + 2H(m2

K

, q2, µ2)

⇤,

H(m2, q2, µ2) =

m2

32⇡2

0

@�4

3

+

5

18

x� x� 4

6

rx� 4

xlog

0

@

qx�4x

+ 1

qx�4x

� 1

1

A

1

A� q2

192⇡2log

m2

µ2(x = � q2

m2).

⌦r2↵= �6

df(q2)

dq2

����q

2=0

=

24L9(µ)

f20

� 1

8⇡2f20

✓log

✓m2

⇡

µ2

◆+ 1

◆� 1

16⇡2f20

✓log

✓m2

K

µ2

◆+ 1

◆.

�

2/d.o.f ⇡ 0.7

⌦r2↵= 0.410(71)(+118

0 )(fm)

but there is the large uncertainty of the decay constant

f0 = 0.10508(GeV) 1000L9

3ms(n=6) 3.603(0.656)(+1.4960 )

JLQCD(15A) 4.6(1.1)(+0.1�0.5)(0.4)

JLQCD(14) 2.4(0.8)(1.0)RBC/UKQCD(08A) 3.08(23)(51)

f0min

= 0.10508 f0 0.11990 = f0max

(mu

,md

,ms

! 0)

0.10508 f0 0.11990(GeV)

0 0.02 0.04 0.06 0.08

q2[GeV2]

0.85

0.9

0.95

1

f ππ(q

2 )

n=6 f0=0.11990(GeV)n=6 f0=0.10508(GeV)

mphysK = 0.493(GeV)

mphys⇡ = 0.140(GeV)

comparison with other calculation at physical point

consistent with previous results and consistent with experiment

f⇡⇡(q2) = 1 +

1

f2

h2lr6q

2+ 4

˜H(m2⇡, q

2, µ2)

i

⌦r2↵=

�12l6f2

� 1

8⇡2f2

✓log

✓m2

⇡

µ2

◆+ 1

◆

m⇡ = 0.13957(GeV)

+extrapolation to the physical strange mass

extrapolation with NLO SU(2) ChPT formula

13

our result with systematic error

⌦r2↵= 0.422(14)(0�24)(

+200 )(fm2)

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

<r2 >(

fm2 ) experimental value

’16 ETMC Nf=2+1+1 TM’16 JLQCD Nf=2+1 Overlap’15 HPQCD Nf=2+1+1 HISQ’13 Mainz Nf=2 Clover’11 PACS-CS Nf=2+1 Clover’09 ETMC Nf=2 TM’08 RBC/UKQCD Nf=2+1 DWFour result(stat. error)our result(total error)

summaryWe calculated pion electromagnetic form factor

in N=2+1 Lattice QCD• • fits including wrapping around effect to extract form factor • strange quark mass reweighting technique • NLO SU(2) ChPT

preliminary result:

consistent with experiment and consistent previous results

m⇡ = 0.145(GeV), L = 8.1(fm)

future works

(NLO SU(2) ChPT fit at physical pion mass and physical strange quark mass)

• curvature by NNLO SU(2) ChPT fit • excited state contribution • estimate error from finite lattice spacing • higher precision calculation (using finer and larger lattice, on the physical point quark masses, including QED )

14

• NLO SU(3) ChPT fit works in our data

⌦r2↵= 0.422(14)(0�24)(

+200 )(fm2)

• back up

15

Charge radiusIn non-relativistic limit , ff is regarded as 3D Fourier transformation of charge density

f

⇡⇡

(q2) =

Zd

3xe

i

�!q ·�!x

⇢

⇡

(�!x )

Assuming spherical symmetry of density andZ

d

3x⇢⇡(x) = 1

�!q ·�!x ⌧ 1Expand ff by

f⇡⇡(q2) =

Z 1

0r

2dr

Z 1

�1d(cos✓)

Z 2⇡

0d�(1� 1

2q

2 · r2cos2✓ + . . .)⇢⇡(r)

= 1� 1

6q2

Z 1

04⇡r2drr2⇢⇡(r) + . . .

f⇡⇡(q2) = 1 �

⌦r2⇡↵

6q2 + . . .

We can consider mean square of charge radius as 1st differential coefficient of ff

(so we need large box size for small momentum)16

R in Dirichlet BC

-10 -5 0 5 10t-(tf-ti)/2

0.94

0.96

0.98

1

R(t,p)

periodicDirichlet

Dirichlet

11conf, q2 ⇡ 0.019(GeV2)

|tboundary

� t

i

| = |tbounary

� t

f

|= 35

periodic

19conf, q2 ⇡ 0.019(GeV2)

|tf � ti| = 36

there is no t dependence in Dirichlet BC

|tf � ti| = 25

17

4 ts < t < Tでの結果 (0.51GeV)

|tf − ti| = 20 乱数の数 8個 tf < t < T でフィットレンジは t = 31− 36 ti = 0 < t < tf でフィットレンジはt = 9− 12以下の様なRを構成する

R(t, q = −p) =2mπZπ(0)

(Eπ(p) +mπ)Zπ(p)

CπV π(t, tf ,−→p ,−→0 )

CπV π(t, tf ,−→0 ,

−→0 )

e(Eπ(−→p )−mπ)t(ti = 0 < t < tf )

R(t, q = −p) =2mπZπ(0)

(Eπ(p) +mπ)Zπ(p)

CπV π(t, tf ,−→p ,−→0 )

CπV π(t, tf ,−→0 ,

−→0 )

e(Eπ(−→p )−mπ)(T−t)(tf < t < T )

形状因子の抽出には以下の式を使用した

f = f0 + f1 ×e−E2π×te−Eπf×(T−tf ) − e−E2π×(tf−t)e−Eπi×(T−tf )

e−Eπi×te−Eπf×(tf−t)(ti = 0 < t < tf )

f = f0 + f1 ×e−E2π×(t−tf )e−Eπi×tf − e−E2π×(T−t)e−Eπf×tf

e−Eπi×(T−t)e−Eπf×(t−tf )(tf < t < T )

f0 f1 × 106

q21 0.8362(47) 1.350(172)q22 0.7603(122) 1.122(215)q23 0.6851(264) 1.0342(366)

Table 12: ２πのフィット(tf < t < T )

f0 f1q21 0.8383(22) 0.4714 (4104)q22 0.7479 (64) -0.7725(1.7470)q23 0.6679(147) 3.9180(3.6210)

Table 13: ２πのフィット(0 < t < ts)

f0q21 0.8388(20)q22 0.7467(43)q23 0.6744(91)

Table 14: 定数フィット (0 <t < ts)

6

18

heavy pion mass

ff is extracted by fitting plateau of R as constant

when

Extraction of form factor

lattice calculation of R

one pion from source the other pion from sink

(using periodic boundary condition in temporal direction)

Diagrammatically,

0 ⌧ t ⌧ tf

tf � ti = 20

preliminary result: pion mass dependance

f⇡⇡(q2) = 1 +

1

f2

h2lr6q

2+ 4

˜H(m2⇡, q

2, µ2)

i

⌦r2↵=

�12l6f2

� 1

8⇡2f2

✓log

✓m2

⇡

µ2

◆+ 1

◆

by NLO SU(2) ChPT formula

l6 = �0.01216(57)!

⌦r2↵= 0.407(16)fm2

20

the result of n=6 NLO SU(2) ChPT fit

NO physical strange mass extrapolation

0 0.1 0.2M2

π(GeV2)

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

<r2 >(

fm2 )

’16 ETMC Nf=2+1+1 TM’16 JLQCD Nf=2+1 Overlap’15 HPQCD Nf=2+1+1 HISQ’11 PACS-CS Nf=2+1 Clover’09 ETMC Nf=2 TM’08 RBC/UKQCD Nf=2+1 DWFthis work(preliminary)FLAGexperiment

disconnect diagram of 3-point function DRAPER and WOLOSHYN,Nucl. Phys., B318 (1989), p. 319-336

from charge conjugation

Cdisc⇡Jµ⇡(U) =

1

2Cdisc

⇡Jµ⇡(U) +1

2Cdisc

⇡Jµ⇡(U⇤)

disconnected term is vanished by charge symmetry

21