¨ 2$%t Óé ½hep4.nucl.phys.titech.ac.jp/workshop/jps2011a/16psb4... · 2011. 10. 5. ·...

シンポジウム：核子構造の３次元的な理解に向けて2011年9月16日、日本物理学会（弘前大学）

格子QCDによるアプローチ佐々木　勝一（東京大学）

このトークでは触れません

• フレーバー１重項に関連する物理量• 核内クォーク軌道角運動量の寄与• 一般化されたパートン分布関数• 横方向運動量依存のパートン分布

このトークで触れる物理量

• 一般化されたパートン分布関数の前方極限 (t→0かつξ→0)での振る舞い

• 非偏極、縦偏極パートン分布関数の低次モーメント：

平均クォーク運動量分布、平均クォークヘリシティ分布

• 核子の形状因子の零運動量移行近傍の振る舞い：

軸性電荷、異常磁気能率、荷電半径

アイソスピン対称な計算(mu=md)では、これらのアイソベクター成分はいわゆるベンチマーク計算に相当する。

• 格子QCDのイロハ

• 格子QCDシミュレーションの現状

• 格子QCDによる核子構造研究

• 理研-BNL-コロンビア(RBC)の結果

• 問題点

• 現在進行中の計算

格子QCDのイロハ

格子上の場の理論４次元ユークリッド空間上の格子点上で経路積分を用いて場の理論を定義

• 格子間隔 a :紫外発散の有限化

• 格子点(site)は４組の整数

• 微分は差分に置き換え

� � � �� ✻

❄

L

✲✛a

✲site

✲link

n = (n1, n2, n3, n4)

(−π/a < kµ ≤ π/a)

フェルミオンの運動項の格子化

格子化

Scon =�

d4xΨ(x) (γµ∂µ + m) Ψ(x)

Ψ(x)→ Ψ(n)

∂µΨ(x)→ 12

�∆µΨ(n) + ∆�

µΨ(n)�

=12a

(Ψ(n + µ)−Ψ(n− µ))

格子上の自由なフェルミオン作用

差分

中央差分⇒エルミート性の保持

前進差分後進差分

Slat = a4�

n

��

µ

Ψ(n)γµΨ(n + µ)−Ψ(n− µ)

2a+ mΨ(n)Ψ(n)

�

ゲージ場の格子化　-リンク変数の導入-Ψ(n)∂µΨ(n) の差分化に伴う非局所項はΨ(n)Ψ(n± µ)

ゲージ変換に対して不変でない。Ψ(n)→ V (n)Ψ(n)

格子理論の運動項においても同様なゲージ不変な取り扱いが考えられる。

Ψ(n)Uµ(n)Ψ(n + µ)! ! ! !! ! ! !

!

Uµ(n)

n ! µ

リンク変数はこのように方向を持った量として定義されているので、その逆向きの変数をU †

µ(n) = U!1µ (n) = U!µ(n + µ) (31)

と定義する。（図も参照）

! ! ! !! ! ! !

"

U †µ(n)

n ! µ

従って、フェルミオンとの相互作用は、ウィルソンフェルミオンの場合

SF = !a4 1

2

!

n

!

µ

!(n)"(r ! !µ) Uµ(n)!(n + ") + (r + !µ) U †

µ(n ! µ)!(n ! µ)#

+(m + 4r)a4!

n

!(n)!(n) (32)

で与えられる。ここで # = 12(m+4r) という量を定義して、クォーク場を!" =

$a3

2!!と再定義すると、格子作用は

SF = !#!

n

!

µ

!(n)"(r ! !µ) Uµ(n)!(n + ") + (r + !µ) U †

µ(n ! µ)!(n ! µ)#

+!

n

!(n)!(n) (33)

と書き直せる。#はホッピングパラメタと呼ばれ、定義から分かるようにクォークの質量が大きいとき、#は小さくなる。r = 1のとき自由なフェルミオン系ではクォークの質量がゼロは # = 1/8に相当するが、相互作用を受けると繰り込みを受けるので# = 1/8からずれる。ただし、格子QCDにおいては漸近的自由性より連続極限で裸の結合定数はゼロになるから、$ " #で # " 1/8 になることが予想される。また、逆の極限、強結合極限$ = 0では、パイオンの質量がゼロになる点として # = 1/4が知られている。従って、パイオンの質量がゼロになる #が $の全ての領域で存在し、そのような臨界 #c($)が $の関数として、強結合極限と連続極限の間 1/4 $ #c($) $ 1/8を単調に結んでいると信じられている。

1.5.2 プラケット変数の導入リンク変数によるゲージ不変量は任意の閉じたループ (C)に沿ってリンク変数の積を

とって、そのトレースをとったもの,　

!C

!

"

"

11

リンク変数 Uµ(n) ≡ eiagAµ(n)

連続理論においても非局所的な　　　　に対してゲージ不変性を保つために、ゲージ変換に対して、と変換する関数をのように挟んでゲージ不変な非局所演算子を作る。このとき　　　　はU(1)理論においてはなじみがあり、

Ψ(x)Ψ(y)U(x, y)→ V (x)U(x, y)V †(y)

Ψ(x)U(x, y)Ψ(y)U(x, y)

U(x, y) = eigR x

y Aµ(z)dzµ → PeigR x

y Aaµ(z)T adzµ

格子上でゲージ場の運動項はどのようになるべきか？

作用決定の指導原理として

• ゲージ不変

• a → ０で連続理論に帰着

• ゲージ対称性以外の連続理論の持つべき対称性を最大限共有

ゲージ場の運動項 -プラケット変数の導入-

※格子上の定義には任意性がある

リンク変数によるゲージ不変量は任意のループCに沿ってリンク変数の積をとって、そのトレースをとればよい

ゲージ場の運動項 -プラケット変数の導入-

リンク変数によるゲージ不変量は任意のループCに沿ってリンク変数の積をとって、そのトレースをとればよい。

! ! ! !! ! ! !

!

Uµ(n)

n ! µ

リンク変数はこのように方向を持った量として定義されているので、その逆向きの変数をU †

µ(n) = U!1µ (n) = U!µ(n + µ) (31)

と定義する。（図も参照）

! ! ! !! ! ! !

"

U †µ(n)

n ! µ

従って、フェルミオンとの相互作用は、ウィルソンフェルミオンの場合

SF = !a4 1

2

!

n

!

µ

!(n)"(r ! !µ) Uµ(n)!(n + ") + (r + !µ) U †

µ(n ! µ)!(n ! µ)#

+(m + 4r)a4!

n

!(n)!(n) (32)

で与えられる。ここで # = 12(m+4r) という量を定義して、クォーク場を!" =

$a3

2!!と再定義すると、格子作用は

SF = !#!

n

!

µ

!(n)"(r ! !µ) Uµ(n)!(n + ") + (r + !µ) U †

µ(n ! µ)!(n ! µ)#

+!

n

!(n)!(n) (33)

と書き直せる。#はホッピングパラメタと呼ばれ、定義から分かるようにクォークの質量が大きいとき、#は小さくなる。r = 1のとき自由なフェルミオン系ではクォークの質量がゼロは # = 1/8に相当するが、相互作用を受けると繰り込みを受けるので# = 1/8からずれる。ただし、格子QCDにおいては漸近的自由性より連続極限で裸の結合定数はゼロになるから、$ " #で # " 1/8 になることが予想される。また、逆の極限、強結合極限$ = 0では、パイオンの質量がゼロになる点として # = 1/4が知られている。従って、パイオンの質量がゼロになる #が $の全ての領域で存在し、そのような臨界 #c($)が $の関数として、強結合極限と連続極限の間 1/4 $ #c($) $ 1/8を単調に結んでいると信じられている。

1.5.2 プラケット変数の導入リンク変数によるゲージ不変量は任意の閉じたループ (C)に沿ってリンク変数の積を

とって、そのトレースをとったもの,　

!C

!

"

"

11

Uµ(n)→ V (n)Uµ(n)V †(n + µ)

リンク変数のゲージ変換

閉じたループとして一番簡単なものは、最小の正方形で

つまりTr (P!CU) (34)

これが、ゲージ不変なことはリンク変数のゲージ変換

Uµ(n) ! V (n)Uµ(n)V †(n + µ) (35)

により明らか。閉じたループとして一番簡単なものは、最小の正方形で!

"# $n

!!!! = Uµ(n)U!(n + µ)U †µ(n + !)U †(n) (36)

これをベースに作用を構成するゲージ不変な要素としてプラケット変数

Pµ!(n) =1

NcTr

! !"# $

!!!! "(37)

を定義する。このゲージ不変なプラケットで格子上のゲージ作用を記述すると

S = "C

g2

#

n

#

µ !=!

Pµ!(n) (38)

と与えられる。a ! 0としたときに、この格子上のゲージ不変な作用が連続理論のYang-Mills作用に一致すること、またそのためには C = Ncであることを次に示す。まず、計算を見やすくするためにBµ # gaAµとおく。プラケットは定義から

!"# $n

!!!! = eiBµ(n)eiB!(n+µ)e"iBµ(n+!)e"iB!(n) (39)

と書ける。ここでB!(n + µ)やBµ(n + !)が格子間隔 aが小さいとして

B!(n + µ) = B!(n) + a"µB!(n) + O(a2)

Bµ(n + !) = Bµ(n) + a"!Bµ(n) + O(a2)

となるとこと、さらにBaker-Campbell-Husdor"(ベーカー・キャンベル・ハウスドルフ）の公式8を用いて、

!"# $

!!!! = exp{iBµ + iB!$ %& 'O(a)

+ ia"µB! "1

2[Bµ, B! ]

$ %& 'O(a2)

" 1

2[Bµ, a"µB! ]

$ %& 'O(a3)

+ · ··}

$ exp{"iBµ " iB! " ia"!Bµ " 1

2[Bµ, B! ] "

1

2[B! , a"!Bµ] + · · ·}

= exp{ia"µB! " ia"!Bµ " [Bµ, B! ]$ %& 'O(a2)

+a3X3 + a4X4 + · · ·}

= exp{iga2Fµ! + a3X3 + a4X4 + O(a5)} (40)

ここで a2の項はFµ! = "µA! " "!Aµ + ig[Aµ, A! ] (41)

8exp{X} exp{Y } = exp{X + Y + 12 [X,Y ] + 1

12 ([X, [X,Y ]] + [Y, [Y,X]]) + · · ·}

12



Uµ(n) ! V (n)Uµ(n)V †(n + µ) (35)


"# $n

!!!! = Uµ(n)U!(n + µ)U †µ(n + !)U †(n) (36)


Pµ!(n) =1

NcTr

! !"# $

!!!! "(37)


S = "C

g2

#

n

#

µ !=!

Pµ!(n) (38)


!"# $n



B!(n + µ) = B!(n) + a"µB!(n) + O(a2)

Bµ(n + !) = Bµ(n) + a"!Bµ(n) + O(a2)


!"# $

!!!! = exp{iBµ + iB!$ %& 'O(a)

+ ia"µB! "1

2[Bµ, B! ]

$ %& 'O(a2)

" 1

2[Bµ, a"µB! ]

$ %& 'O(a3)

+ · ··}


2[Bµ, B! ] "

1

2[B! , a"!Bµ] + · · ·}


+a3X3 + a4X4 + · · ·}

= exp{iga2Fµ! + a3X3 + a4X4 + O(a5)} (40)



12 ([X, [X,Y ]] + [Y, [Y,X]]) + · · ·}

12

作用を構成するゲージ不変な要素としてプラケット変数を定義する

ゲージ場の運動項 -連続理論との整合性-

ゲージ不変なプラケット変数で格子上のゲージ作用を記述する

Slat = −2Nc

g2

�

n

�

µ>ν

Pµν(n)

一見すると連続理論のゲージ作用と似て非なるものの様であるが、a→０の連続極限を考えると

と表されることが分かる。a3や a4の項の係数X3,X4の具体的な計算は省くが、X3、X4

のいづれも SU(3)の Rie環の元であるので、トレースをとると Tr!a = 0で Tr{X3} =Tr{X4} = 0 となることを次で使う。

Tr{!

"# $!!!! } = Tr

!1 + iga2Fµ! + a3X3 + a4X4 +

g2a4

2Fµ!Fµ! + O(a5)

"

= NC +g2a4

2Tr{Fµ!Fµ!} + O(a6) (42)

右辺一行目の第２項から４項までは、それぞれ Rie環の元のトレースレスの性質から落ちる。第５項のように、二つ以上のRie環の元の積はRie環の元ではないのでトレースをとっても残る。物理に関係ない定数項を無視すれば a ! 0で

S ! "a4#

n

1

2

#

µ !=!

Tr{Fµ!Fµ!} (43)

望ましい連続理論の作用、QCDの作用に帰着することが見て取れた。厳密には SU(2)では Tr{

!"# $

!!!! }は実であるが、NC # 3では必ずしも実とならないのでエルミート共役（反対向きのループに対応するプラケット）との平均、つまりプラケットの実部によって作用を定義する。

1 " 1

2NCTr{

!"# $

!!!! +"!$ #

!!!! } = 1 " 1

NCReTr{

!"# $

!!!! }$ %& '

=Pµ!

= "a4 g2

2NCTr{Fµ!Fµ!} + O(a6) (44)

上記を用いて、格子上のゲージ理論は最終的にSlat(U) = "

#

n

#

µ>!

[1 " Pµ!(U)] (45)

のように与えられる。ここで "は" = 2Nc/g

2 (46)

で定義される9。

1.6 コンパクトな変数とエリツァーの定理経路積分は作用を構成する基本的要素、リンク変数について行なえばよいので

Z =

(!ldUl exp{"Slat(U)} (47)

を得る。連続理論の場合との大きな違いは、積分する自由度がリンク変数（群の元、U $SU(Nc)）であるために、積分が

)!ldUl = 1のように１に規格化されている点である。こ

のことによって、格子ゲージ理論では連続理論のときのようにゲージ固定する必要性がない10。そして、リンク変数に対する経路積分の測度は、直接ゲージ群の多様体の上の測度（不変ハール測度）

d(V U) = d(UV ) = dU for "V $ SU(Nc) (48)9! の定義に含まれる因子 2は*

µ!=! = 2*

µ>! による。10あるいは連続理論との連続性を考えて、ゲージ固定の必要がない理由は、ゲージ場Aµが、連続理論において"% < Aµ < %に対して、格子理論では""/(ag) < Aµ < "/(ag)であるため、“ゲージ体積”が有限なためとも言い換えることができる。

13

Slat = Scon + a2S2 + a4S4 + · · ·

+a6

12Tr{Fµν(D2

µ + D2ν)Fµν}+O(a8)

�+O(g4a6)



Uµ(n) ! V (n)Uµ(n)V †(n + µ) (35)


"# $n

!!!! = Uµ(n)U!(n + µ)U †µ(n + !)U †(n) (36)


Pµ!(n) =1

NcTr

! !"# $

!!!! "(37)


S = "C

g2

#

n

#

µ !=!

Pµ!(n) (38)


!"# $n



B!(n + µ) = B!(n) + a"µB!(n) + O(a2)

Bµ(n + !) = Bµ(n) + a"!Bµ(n) + O(a2)


!"# $

!!!! = exp{iBµ + iB!$ %& 'O(a)

+ ia"µB! "1

2[Bµ, B! ]

$ %& 'O(a2)

" 1

2[Bµ, a"µB! ]

$ %& 'O(a3)

+ · ··}


2[Bµ, B! ] "

1

2[B! , a"!Bµ] + · · ·}


+a3X3 + a4X4 + · · ·}

= exp{iga2Fµ! + a3X3 + a4X4 + O(a5)} (40)



12 ([X, [X,Y ]] + [Y, [Y,X]]) + · · ·}

12

with

}lattice artifact

= NC −g2

2�a4Tr{FµνFµν}

格子QCDにおけるパラメータ

• クォーク質量(mu=md, ms)と格子間隔 a(g)INPUT: π, K中間子の質量とΩバリオンの質量

CP-PACS collaboration

u,d,s (2+1 flavor)u,d (2 flavor)quench approx.

格子QCDにおける系統誤差４つの起源

• クォーク真空偏極：動的クォークの数

• 有限格子間隔 a :紫外発散の有限化

• 有限体積効果 L :格子点は有限

• クォーク質量の物理点へのカイラル外挿

動的クォークの数

!O(U, !)" =1Z

!D!D!DU O(U, !)e!SG(U)!!M(U)!

=1Z

!DU O(U, M!1(U))(det{M(U)})Nf e!SG(U)

=1Z

!DU O(U, M!1(U))e!SG(U)+Nf TrLnM(U)

det{M(U)} = 1 !" Nf = 0

シミュレーションの現状

現実的な格子QCDシミュレーションの確立

ストレンジネスを含む動的クォーク効果を完全に含んだシミュレーション（２＋１フレーバー）

厳密なカイラル対称性を持つ（JLQCD, RBC)

physical pointでの計算（PACS-CS）

数値計算の精密化

物理量によっては実験値との誤差が数%以下

格子QCDシミュレーションの現状

• Wilson-type fermion (Wilson, O(a)-improved Wilson)

✓ No chiral symmetry

• Staggered-type fermion (Staggered, Asqtad etc)

✓ No flavor symmetry

• Chiral (Ginsparg-Wilson) fermion (Overlap, DWF)

✓ Exact chiral symmetry and complete flavor symmetry

Collaboration Nf a (fm) La(fm) mπ (MeV) Fermion action

MILC 2+1 0.18 2.9 > 260 O(a2)-improved Staggered



MILC 2+1 0.09 2.4-5.8 > 164 O(a2)-improved Staggered

MILC 2+1 0.06 2.9-3.8 > 310 O(a2)-improved Staggered


RBC-UKQCD 2+1 0.114 2.7 > 330 Domain wall fermion

2+1 0.084 3.1 > 300 Domain wall fermion

PACS-CS 2+1 0.091 3.0 > 156 O(a)-improved Wilson

JLQCD 2+1 0.11 1.8 > 300 Overlap fermion

ETMC 2+1+1 0.078, 0.086 2.8 > 280 Twisted mass Wilson

動的クォークの数２＋１フレーバー（mu=md≠ms, mc,b,t=∞）

格子間隔（カットオフ）1.7 GeV - 2.2 GeV

有限体積（空間のサイズ）2.0 fm - 3.0 fm

クォークの質量（π中間子の質量)最も軽いπ中間子が350 MeV 以下

格子QCDシミュレーションの現状

現実的シミュレーションの取り組み

日本における主な拠点

筑波大学（PACS-CS collaboration)‣ Wilson形式による物理的クォーク質量近傍でのシミュレーション

KEK（JLQCD collaboration)‣ Overlap形式による厳密なカイラル対称性を持つシミュレーション

理研BNL（RBC collaboration）‣ DWF形式による現実的な対称性を持つシミュレーション

0 500 1000

m! [MeV]

500

1000

mK

[M

eV]

CP-PACS/JLQCD

PACS-CSPhysical Point

simulation points

unitary points with mπ∼<400MeV

7

PACS-CS collaboration


• Nf=2+1 simulations

• O(a)-improved Wilson quark

• 323 x 64 lattice

• a = 0.09 fm (a-1 = 2.2 GeV)

• V=(2.9 fm)3

CP-PACSからのアップグレード

Physical pointに向けて

Phys. Rev. D79 (2009) 034503Phys. Rev. D81 (2010) 074503



• Nf=2+1 simulations

• O(a)-improved Wilson quark

• 323 x 64 lattice

• a = 0.09 fm (a-1 = 2.2 GeV)

• V=(2.9 fm)3

27!ud = 0.137785,!s = 0.13660

0

0.5

1

1.5

2

m[G

eV]

ρK*

φ NΛ

ΣΞ

ΔΣ∗Ξ∗Ω

meson octet baryon decuplet baryon

π

K chpt fse ( mπ,mK,m

Ω-input )

κ ud=0.137785

ChPT experiment !ud = 0.137785

mMSud [MeV] 2.53(5) ! 3.5(3)

mMSs [MeV] 72.7(8) ! 73.4(2)

f" [MeV] 134.0(4.2) 130.7 ± 0.1 ± 0.36 129.0(5.4)

fK [MeV] 159.4(3.1) 159.8 ± 1.4 ± 0.44 160.6(1.4)

Phys. Rev. D79 (2009) 034503Phys. Rev. D81 (2010) 074503

質量の起源　　　

　　　⇒　カイラル対称性の自発的破れ　(Nambu, 1961)

✓ 厳密なカイラル対称性を持つシミュレーションで検証

‣ Dirac演算子の固有値分布

‣ Banks-Casher関係式

‣ ε-regimeのパイオン相関関数

‣ トポロジカル感受率

‣ GMOR関係式

�0|qq|0� �= 0

!"" #$%&'

!!""! !!

!"#$%&&!"'($

!!"## #$%

!!"&

'!()*+,-*

M2π/m ∝ �qq�

χt/m ∝ �qq�

limm→0

πρ(λ, m) ∝ �qq�

５つの異なる方法による　　の測定�qq� �0|qq|0�13µ=2 GeV [MeV]

JLQCD collaborationPhys. Rev. Lett. 104 (2010) 122002

RBC collaboration

K meson physics Nucleon structure

QCD thermodynamics

Precise fK / fπStrange quark massε (indirect CP violation)ε’(direct CP violation)ΔI=1/2 ruleCKM matrix |Vus|η’ mass problem

Charges gA, gT

Nucleon form factorsNucleon structure functionHyperon beta decayNeutron EDMProton decay

Phase transition temperature Tc

Order of phase transitionEquation of state for QGPFate of J/ψ (heavy quarkonium)Transport coefficients of QGP

Nucleon structure from 2+1f DWF QCD

Y. Aoki, T. Blum, H.-W. Lin, M.-F. Lin, S. Ohta, S. Sasaki, T. YamazakiR.J. Tweedie, J.M. Zanotti (RBC+UKQCD collaboration)

• Nucleon axial charge (gA)

- Phys. Rev. Lett. 100 (08) 171602

• Isovector nucleon form factors (mean-squared radius)- Phys. Rev. D79 (09) 114505

• Isovector nucleon structure function ( , )- Phys. Rev. D82 (09) 014501

�x�u−d �x�∆u−∆d

β=2.13 (a-1=1.7 GeV), V=243 x 64 x 16

格子QCDのこれまでの成果は？Hadron Spectrum

(measured at or extrapolated to the physical point)

• Durr et al. (BMW) Science 322 (2008) 1224

2+1 impr. Wilson fermions,

mPS ≥ 190 MeV

• Aoki et al. (PACS-CS) Phys. Rev. D79(2009) 034503

2+1 impr. Wilson fermions,

mPS ≥ 160 MeV no systematic error incl.

• Lin et al. (HSC) Phys. Rev. D79 (2009) 034502

2+1 anistropic Clover fermions,


• Alexandrou et al. (ETMC) Phys. Rev. D78 (2008)

014509

2 twisted mass fermions,

mPS ≥ 300 MeV

• MILC prelim. arXiv:0903.3598[hep-lat],. . .

2+1 impr. staggered fermions,


• LHPC Phys.Rev. D79 (2009) 054502

MA: 2+1 stagg. sea/DWF valence,

mPS ≥ 300 MeV

0

200

400

600

800

1000

1200

1400

1600

1800

! K " # K$ % a0 a1 b1 N & ' ( ) '$($ *

mH [MeV]

BMWPACS-CS

HSCETMCMILCLHPC

E. E. Scholz — Light Hadron Masses and Decay Constants 1

2+1 flavor staggered QCD (HPQCD/MILC/UKQCD/Fermilab) 2+1 flavor light hadron spectroscopy

From Scholz@lattice2010

現実的に近いシミュレーションによる計算結果の精密化

ただし、精密計算は主に、、、

クォーク質量やカイラル凝縮

結合定数αs

ハドロンの質量（基底状態）

π、K中間子などのweak matrix element

核子を含むバリオンの物理ではoctet, decuplet　バリオンの質量のみ

格子QCDによる核子構造理解の現状

Nucleon structure from 2+1f DWF QCD

Y. Aoki, T. Blum, H.-W. Lin, M.-F. Lin, S. Ohta, S. Sasaki, T. YamazakiR.J. Tweedie, J.M. Zanotti (RBC+UKQCD collaboration)

• Nucleon axial charge (gA)

- Phys. Rev. Lett. 100 (08) 171602

• Isovector nucleon form factors (mean-squared radius)- Phys. Rev. D79 (09) 114505

• Isovector nucleon structure function ( , )- Phys. Rev. D82 (09) 014501

�x�u−d �x�∆u−∆d

β=2.13 (a-1=1.7 GeV), V=243 x 64 x 16

possesses two types of the Wick contraction

t t’’

t’

t’

t t’’

connected contribution disconnected contribution

!!N (t)O(t!)!N

(t!!)"

Difficulty for flavor singlet quantity

(required for flavor singlet quantity)Isovector (u-d)

expens

ive ca

lculati

on

Calculation of Matrix Elements (1)

no dependence of t

t

t’’t’

�ψN (t�)O(t)ψN

(t��)� =�

n,m

e−En(t�−t)�ψN |n��n|O|m��m|ψ

N�e−Em(t−t��)

→ �ψN |N��N |O|N��N |ψN�e−EN (t�−t��)

t� � tand

t� t��

• Matrix element can be extracted from the following ratio�ψN (t�)O(t)ψ

N(t��)�

�ψN (t�)ψN

(t��)�→ �N |O|N�

核子の構造に関しては

2+1 flavor DWF 実験値核子軸性電荷 gA 1.19(6) 1.2695(29)核子異常磁気能率 2.75(28) 3.70589

核子平均２乗半径(Dirac) 0.584(23) fm 0.797(4) fm核子平均２乗半径(Pauli) 0.636(57) fm 0.879(18) fm平均クォーク運動量分布 0.218(19) 0.154(3)

平均クォークヘリシティ分布 0.256(23) 0.196(4)

実験値を再現できているとは言えない

大きな有限体積効果の問題

核子軸性電荷 gA

0 0.1 0.2 0.3 0.4 0.5m!

2[GeV2]

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Nf=2+1(2.7fm)Nf=2+1(1.8fm)L"#2.7fm1.8fm

gA DWF results

T. Yamazaki et al., Phys. Rev. Lett. 100 (2008) 171602.

mπ=330MeVで有限体積効果を１％以下に押さえるためには3.5～4.1 fmの空間サイズが必要

ただし、他の物理量ではそれほど問題なさそう。

大きな有限体積効果の問題

0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.15

0.2

0.25

0.3

0.35

0.4

2.7 fm1.8 fm〈x〉

Δu-Δdbare

0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.1

0.15

0.2

0.25

0.3

0.35

2.7 fm1.8 fm〈x〉u-d

bare

●　2+1 flavor DWF (2.7 fm)■　2+1 flavor DWF (1.8 fm)

Not yet renormalized Not yet renormalized平均クォーク運動量分布平均クォークヘリシティ分布

Y. Aoki et al., Phys. Rev. D82 (2010) 014501.

gA = �1�∆u−∆d注：

核子の大きさの問題

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

FV

(q2)/

FV

(0)

m!=0.33[GeV] (N

f=2+1 DWF)

experiment

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0.5

1

1.5

2

2.5

3

3.5

4

F2re

n(q

2)

m!=0.33[GeV] (N

f=2+1 DWF)

experiment

10

dipole form:

F1(q2) =F1(0)

(1 + q2/M21 )2

�(r1)2�12 =

�12M2

1

平均自乗半径

F and ρ are related through the three-dimensional Fourier integral

Form factor F (q2) Probability density ρ(r)1 δ(r)

(1 + q2a2)−1 ρ0r exp(−r/a)

(1 + q2a2)−2 ρ0 exp(−r/a)exp(−q2b2/4) ρ0 exp[−(r/b)2]

3[sin(|q|R)−|q|R cos(|q|R)](|q|R)3 ρ0θ(R− r)

1

(Breit frame)F (q2) =!

d3r!(r) exp(iq · r)

= 1 ! 16q2"r2# + · · ·

: mean-square radius

dipole form

!r2"

Form factor and Probability density

dipole form:

平均自乗半径0 0.1 0.2 0.3 0.4 0.5

mπ

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

experimentNf=2+1 DWF (2.7fm)(〈r1

2〉)1/2[fm]

�(r1)2�12 =

�12M2

1

�(rv1)2� = �(rv

E)2� − 32

F v2 (0)M2

N


Dirac rms

T. Yamazaki et al., Phys. Rev. D79 (2009) 114505.

0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

experimentNf=2+1 DWF (2.7fm)LOHBChPT

(〈r12〉)1/2[fm]

Dirac rms

�r21�(mπ,lat) = �r2

1�exp −1 + 5g2

A,exp

(4πFπ,exp)2ln

�m2

π,lat

µ2

�Baryon ChPT(LO)

Infrared divergence



0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)HBChPT (LO)

(〈r12〉)1/2[fm]

experiment

Dirac rms

�r21�(mπ,lat) = �r2

1�exp −1 + 5g2

A,exp

(4πFπ,exp)2ln

�m2

π,lat

µ2

�Baryon ChPT(LO)

Infrared divergence


Pion cloud effect?


π中間子の大きさにも同様の問題が

to its large uncertainty. From these observations, we onlyuse results for hr2iS in the following analysis, and leave aprecise determination of cS for future studies.

V. CHIRAL EXTRAPOLATION

A. Fit based on one-loop ChPT

Since the form factors FV;S!q2" are independent of q2 atLO in ChPT, the chiral expansion of the radii hr2iV;S startsfrom the one-loop order of ChPT. We first compare ourlattice results with the one-loop expressions [14]

hr2iV # $ 1

NF2 !1% 6Nlr6" $1

NF2 ln!M2

!

"2

"; (39)

hr2iS #1

NF2

#$ 13

2% 6Nlr4

$$ 6

NF2 ln!M2

!

"2

"; (40)

where N # !4!"2, and F is the decay constant in the chirallimit. We adopt the normalization of the decay constantF! # 92 MeV at the physical quark mass. The renormal-

ization scale " is set to 4!F in our analysis. At this orderof the chiral expansion, F is the only LEC appearing in theM! dependent terms. We fix this important parameter toF # 79:0!%5:0

$2:6" MeV, which has been determined from ourdetailed analysis of the pion mass and decay constant [5].Each one of Eqs. (39) and (40), therefore, has a single fitparameter, namely, LECs lr6 or l

r4 in their constant term.

We find that the NLO fits of lattice data are not quitesuccessful as seen in Fig. 15 and Table XII. While the dataof hr2iV can be fitted with reasonable #2=d:o:f:& 0:14, thevalue extrapolated to the physical quark mass hr2iV #0:3637!43" fm2 is significantly smaller than the experi-mental value 0:437!16" fm2 based on Nf # 2 ChPT [16]and 0:452!11" fm2 quoted by PDG [49]. As for the scalarradius, the one-loop formula fails to reproduce our data ofhr2iS as indicated by the quite large value of #2=d:o:f:& 9:the data have a mild quark mass dependence in contrast tothe 6 times enhanced chiral logarithm compared to hr2iV .This failure of the NLO fits is not due to our choice of F.

If F is treated as a free parameter, the fit to hr2iS results inan unacceptably large value F ’ 200 MeV to achieve rea-

TABLE XI. Parametrization Eq. (38) for scalar form factor FS!q2". Results for the vectorradius hr2iS and curvature cS in lattice units are also listed.

m #2=d:o:f: aS;1 aS;2 aS;3 aS;4 hr2iS cS

0.050 1.3(1.0) 3.04(31) 6.6(1.3) 5.1(1.4) ' ' ' 18.2(1.9) 6.6(1.3)0.050 1.3(1.1) 3.51(49) 10.5(3.5) 1.6(9.2) 9.6(7.7) 21.1(3.0) 10.5(3.5)0.035 1.8(1.0) 2.79(31) 5.6(1.4) 4.3(1.6) ' ' ' 16.8(1.9) 5.6(1.4)0.035 1.9(1.2) 2.60(51) 3.5(4.8) $3!15" $6!13" 15.6(3.1) 3.5(4.8)0.025 1.9(1.1) 3.37(32) 6.1(1.6) 3.6(1.8) ' ' ' 20.2(1.9) 6.1(1.6)0.025 1.9(1.3) 2.97(53) 1.8(4.6) $9!13" $12!11" 17.8(3.2) 1.8(4.6)0.015 1.5(1.0) 3.51(51) 6.6(2.7) 3.6(3.3) ' ' ' 21.0(3.1) 6.6(2.7)0.015 1.7(1.1) 3.0(1.0) 1.0(9.1) $14!26" $16!22" 18.1(6.0) 1.0(9.1)

0.0 0.1 0.2 0.3

M!2 [GeV

2]

0.2

0.3

0.4

0.5

<r2 > V

[fm

2 ]

expr’t (Nf=2 ChPT)

expr’t (PDG)

0.0 0.1 0.2 0.3

M!2 [GeV

2]

0.0

0.2

0.4

0.6

0.8

<r2 > S [

fm2 ]

expr’t

FIG. 15. Chiral fit of hr2iV (left panel) and hr2iS (right panel) using one-loop ChPT formulas. Filled squares are the lattice data andthe value extrapolated to the physical point. In the left panel, we also plot the experimental value hr2iV # 0:437!16" fm2 from ananalysis based on Nf # 2 ChPT [16] (open circle) and 0:452!11" fm2 quoted by Particle Data Group [49] (star). The star symbol in theright panel represents hr2iS # 0:61!4" fm2 obtained from an indirect determination through !! scattering [51].

S. AOKI et al. PHYSICAL REVIEW D 80, 034508 (2009)

034508-14

�r2π�(mπ,lat) =

1(4πFπ,exp)2

�mπ,lat

µ2

�ChPT(LO) Infrared divergence

Expt. JLQCD collaboration, Phys. Rev. 80 (2009) 034508

2-flavor dynamical overlap simulations→　exact chiral symmetryVolume L~1.9 fmmπ ＞290 MeV

0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)HBChPT (LO)

(〈r22〉)1/2[fm]

experiment

Baryon ChPT(LO)

Infrared divergence

もっと強い赤外発散の例：パウリ半径

Pion cloud effect?

Pauli rms

�r22�(mπ,lat) = C0 +

g2A,exp

8πF 2π,expκV,exp

MN,exp

mπ,exp

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7mπ

2[GeV2]

0.15

0.2

0.25

0.3

0.35

0.4

Nf=2+1 DWF (2.7 fm)Nf=0 DWF (2.4fm)LO HBChPT

experiment

〈x〉Δu-Δd

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7mπ

2[GeV2]

0.1

0.15

0.2

0.25

0.3

0.35

Nf=2+1 DWF (2.7 fm)Nf=0 DWF (2.4 fm)LO HBChPT

experiment

〈x〉u-dMS at µ = 2.0 GeV MS at µ = 2.0 GeV

Chen & Ji, PLB523, 171 (2001)Detmold-Melnitchouk-Thomas, PRD66, 054501 (2002)

�x�u−d = C0

�1 − 1 + 3g2

A

(4πFπ)2M2

π lnM2

π

µ2

��x�∆u−∆d = C0

�1 − 1 + 2g2

A

(4πFπ)2M2

π lnM2

π

µ2

�

µ = MN = 940MeVgA = 1.26, Fπ = 92.8MeV

注：青点線はフィットではありません

他の物理量もchiral logが必要？


平均クォーク運動量分布平均クォークヘリシティ分布

.(Un-)Renormalization

• normalizing by �x�refu−d at mrefπ = 500 MeV eliminates Z factors

0 0.1 0.2 0.3 0.4 0.5

m!

2 [GeV

2]

0.2

0.4

0.6

0.8

1

1.2<

x>

u-d

/ <

x>

u-d

ref

RBC NF=2+1 DWF

RBC NF=2 DWF

LHPC NF=2+1 DWF/MILC

ETMC NF=2 TMF

QCDSF NF=2 IWF

CTEQ6.6C and QCDSF

• artificial but suggests the groups might agree on the shape (up to norm.)

• renormalization, among other systematics, must be checked

7/22

Figure from D. Renner (Lat09)

From Pleiter@Lattice2010 (arXiv:1101.2326)

.(Un-)Renormalization

• normalizing by �x�refu−d at mrefπ = 500 MeV eliminates Z factors

0 0.1 0.2 0.3 0.4 0.5

m!

2 [GeV

2]

0.2

0.4

0.6

0.8

1

1.2

<x

>u

-d /

<x

>u

-d

ref

RBC NF=2+1 DWF

RBC NF=2 DWF

LHPC NF=2+1 DWF/MILC

ETMC NF=2 TMF

QCDSF NF=2 IWF

CTEQ6.6C and QCDSF

• artificial but suggests the groups might agree on the shape (up to norm.)

• renormalization, among other systematics, must be checked

7/22

QCDSF results (updated)

Nucleon form factors and structure functions D. Pleiter

4. n= 2moments of PDFs

The lowest moment of the unpolarized PDF !x"q = v2 corresponds to the momentum fractioncarried by the quarks in the nucleon. Lattice results from different collaborations tend to be signifi-cantly larger than the phenomenological value. Fig. 4a and 4b show our most recent results for theiso-vector and iso-scalar channel. In the latter case disconnected contributions have been ignored.

Also shown are the results from a fit to results utilizing methods of covariant Baryon Chi-ral Perturbation Theory (BChPT) [7]. Fits have been performed with most parameters fixed tophenomenologically known values. The iso-vector (iso-scalar) channel data is fitted with only 2free parameters: v2 in the chiral limit and the coupling c8 (c9). Near the physical light quarkmasses, BChPT predicts v2 to become larger when the quark mass becomes heavier. In our data formPS ! 250MeV we do not see any indication for a bending down when approaching the physicalpion mass. It thus does not seem that a lack of results at sufficiently small quark masses couldexplain the large discrepancy between the phenomenological value and the lattice results. Thereare some indications that part of the discrepancy can be explained by excited state contamination[8].

0 0.1 0.2 0.3 0.4

m2PS [GeV2]

0.1

0.15

0.2

0.25

0.3

v 2MS (2

GeV

)

!=5.20!=5.25!=5.29!=5.40

<x>u-dMRST06

(a)

0 0.1 0.2 0.3 0.4

m2PS [GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

v 2MS (2

GeV

)

!=5.20!=5.25!=5.29!=5.40

<x>u+dMRST06

(b)Figure 4: The left and right panel show results for the second moment of the iso-vector and iso-scalar unpolarized PDFs, respectively, as a function of m2

PS. The solid lines show the fits to anexpression from ChEFT.

In Fig. 5a the results for the second moment of the polarized PDF !x""q = a1 is shown. Dis-cretization effects again seem to be absent in data. From a comparison of the results for differentvolumes it seems that also finite size effects are small. Results from Heavy Baryon Chiral Pertur-bation Theory (HBChPT) [9] lead to the following expression:

a(u#d)1 (mPS) =C!

1#4g2

A +12(4# fPS)2m

2PS ln

"

m2PSµ2

#$

+ · · · (4.1)

In Fig. 5a we plot this expression using µ =mN andC chosen such that it matches the phenomeno-logical value. The bending down which we observe in our data for mPS ! 0.5MeV is much lessthan one would expect from this HBChPT result.

5

‣ Nf=2 NP O(a) improved Wilson fermions

‣ mπ > 180 MeV

‣ Max volume: 2.9 fm (lightest 3pts.)

‣ Four lattice spacings (1/a=2.4-3.3 GeV)

‣ non-relativistic nucleon operator

異常磁気能率

0 0.1 0.2 0.3 0.4 0.5

m!

2[GeV2]

0

1

2

3

4

5

6

Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)

F2(0)

experiment

‣ mπ依存性が弱く、重いmπの計算では実験値をよく再現している- SU(6)クォーク模型における核子異常磁気能率の説明の成功と関係？

‣ mπが軽くなると実験値の下方へシフトー軸性電荷と似ている- 有限体積効果を示唆か？


パウリ形状因子 γµF1(q2) + iσµνqν

2MNF2(q2)

find the lightest point to be slightly smaller than the resultsat the other quark masses, albeit with a large error. Thus,we consider this pion mass dependence is due to statistics,not a finite-volume effect as in the axial charge, and this isconfirmed by our results from the smaller volume simula-tions in Fig. 20. The results are reasonably fitted by a linearfunction of the pion mass squared, and we obtain hr22i1=2 !0:64"6# fm at the physical pion mass. This result again is27% smaller than the experimental value, 0.88(2) fm.

Here again the quantity is expected to diverge as 1=!!!!!!!m!

pin the chiral limit in HBChPT [64–66], however our resultsdo not indicate such divergence. In contrast to the Diracradius case, perhaps because of the larger statistical errors,HBChPT can simultaneously fit the experiment and ourdata. The fit inspired by a prediction [64],

A!!!!!!!m!

p"1$ Bm! $ Cm! log

"m!

"

##; (28)

where A, B, and C are free parameters (A !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g2AMN=8!F

2!F2"0#

qin HBChPT), " is the scale and fixed

to 1 GeV for simplicity, gives a larger#2=d:o:f:"degrees of freedom# ! 3:4. We cannot obtain areasonable #2=d:o:f: without fixing the coefficient A using

the experimental values. We need further light quark masscalculation with better statistics to test the prediction in thelattice QCD calculation.

C. Form factors of the axial-vector current

In this subsection we show the form factors obtainedfrom the axial-vector currents, FA"q2# and FP"q2#. Theyare extracted from the ratios of three- and two-point func-tions defined in Eqs. (14) and (15). Figure 21 shows that thetypical plateaus of the ratios with the A3 component of thecurrent at mf ! 0:01 are reasonably flat in the middle timeregion between the source and sink operators. We plot theratios !A

L"q3 ! 0; t# and !AL"q3 ! 0; t# separately, since

!AL"q3 ! 0; t# contains both form factors, while !A

L"q3 !0; t# contains only FA"q2#. It is worth noting that there is no!A

L"q3 ! 0; t# in the case of ~q / "1; 1; 1#. !AT"q; t# has a

slope in the range t ! 1–8 with large statistical errors asshown in the bottom panel of Fig. 21. We consider theslope to be caused by poor statistics in the data. The valuesof the matrix elements for all the ratios are determined byconstant fits with the range of t ! 4–8.Using the relations Eqs. (18) and (19), the two form

factors are determined through the following equationswhich depend on the spatial momentum transfer in thethree-point function:

FA"q2# !(!A

L"q3 ! 0# for n ! 0; 1; 2; 4

!AL"q3 ! 0# $ q23

MN"MN$E"q##!AT"q# for n ! 3

; (29)

FP"q2# !(!A

T"q#=MN for n ! 2; 3MN$E"q#

q23"!A

L"q3 ! 0# %!AL"q3 ! 0## for n ! 1; 4 ; (30)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

q2[GeV

2]

0

0.5

1

1.5

2

2.5

3

3.5

4mf=0.01(2.7fm)mf=0.02(2.7fm)mf=0.03(2.7fm)mf=0.01(1.8fm)mf=0.02(1.8fm)mf=0.03(1.8fm)

F2(q2)

FIG. 20 (color online). Comparison of F2 with larger andsmaller volumes denoted by closed and open symbols, respec-tively, at each quark mass.

0 0.1 0.2 0.3 0.4

m!2[GeV

2]

0.4

0.6

0.8

1

1.2Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)HBChPTexperiment

"r22#1/2

[fm]

FIG. 19 (color online). Same as Fig. 15 except Pauli rms radiushr22i1=2 determined from the dipole fit. Prediction of HBChPTwith the experimental result [12,23], and fit result with HBChPTprediction (dash-dot line) are also plotted. The striped symbol isshifted to the minus direction in the x-axis.

NUCLEON FORM FACTORS WITH 2$ 1 FLAVOR . . . PHYSICAL REVIEW D 79, 114505 (2009)

114505-13

rithmic effects are to be seen in lattice results of the Diracradius.

2. Pauli form factor F2!q2"Figure 16 shows the momentum-transfer dependence of

our results for the Pauli form factor at each quark mass.These values are tabulated in Table IV. The form factor isrenormalized by F1!0".

This form factor can also be described by the conven-tional dipole form,

F2!q2" #F2!0"

!1$ q2=M22"2

; (26)

with M2 # 0:78!2" GeV and F2!0" # 3:705 89 extractedfrom fits to experimental data. In contrast to the Dirac formfactor, there are two parameters, the overall strength F2!0"and the dipole massM2: the former gives the isovector partof the anomalous magnetic moment,!p %!n % 1, and thelatter the Pauli mean-squared radius, hr22i # 12=M2

2, as inthe Dirac case. We fit the form factor with these twoparameters.

To check reliability of the dipole fit, we measure theratio of the Sachs electric and magnetic form factors,Eqs. (2) and (3),

GM!q2"GE!q2"

# !VT !q"

!V4 !q"

; (27)

which exhibits a mild q2 dependence [1,49]. At zero mo-mentum transfer, we obtain 1$ F2!0" from the ratio.Figure 17 shows that the result for GE!q2"=GM!q2" % 1at q2 # 0, obtained via a linear fit in q2, is consistent withthe determination from a dipole fit of F2!q2".

In Fig. 18 we present the anomalous magnetic momentof the nucleon, determined by the dipole fit presented in

Table V, together with some other lattice QCD calculationsand the experimental value. Our present results slightlydecrease with the pion mass, in agreement with previouslattice calculations [1,23]. They extrapolate well linearly inthe pion mass squared and result in a value 26% smallerthan the experiment. This result at the physical pion mass isconsistent with those of previous calculations [1,19] usinga linear fit.We present in Fig. 19 the result of the Pauli rms radius.

These results are obtained from a dipole fit and summa-rized in Table V. Some other lattice QCD calculations[1,23] are also plotted in the figure for comparison. We

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0

1

2

3

4

5 F2(q2)

GM(q2)/GE(q

2)-1

experiment

FIG. 17 (color online). Dipole fit with F2!q2" and linear fitwith the ratio of electric and magnetic form factorsGM!q2"=GE!q2" % 1 at mf # 0:01. The result of the ratio atq2 # 0 is shifted to the minus direction in the x-axis.

0 0.1 0.2 0.3 0.4

m!2[GeV

2]

0

1

2

3

4

5

6


F2(0)

experiment

FIG. 18 (color online). Same as Fig. 15 except anomalousmagnetic moment, F2!0" # !p %!n % 1, determined fromthe dipole fit. The experimental result [12] is also shown.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0

0.5

1

1.5

2

2.5

3

3.5

4mf=0.005mf=0.01mf=0.02mf=0.03experiment

F2(q2)

FIG. 16 (color online). The Pauli form factor, F2!q2", renor-malized by ZV # 1=F1!0". The dashed curve is a fit to experi-mental data.

TAKESHI YAMAZAKI et al. PHYSICAL REVIEW D 79, 114505 (2009)

114505-12

‣ F2(0)は直接計算できないため(運動学的な制限) q2の外挿を行う

‣ 有限体積でアクセスできるq2は制限されている(q=2π/L*n)

‣ F2(0) の有限体積効果の有無を判断するのは難しい

パウリ形状因子

find the lightest point to be slightly smaller than the resultsat the other quark masses, albeit with a large error. Thus,we consider this pion mass dependence is due to statistics,not a finite-volume effect as in the axial charge, and this isconfirmed by our results from the smaller volume simula-tions in Fig. 20. The results are reasonably fitted by a linearfunction of the pion mass squared, and we obtain hr22i1=2 !0:64"6# fm at the physical pion mass. This result again is27% smaller than the experimental value, 0.88(2) fm.

Here again the quantity is expected to diverge as 1=!!!!!!!m!

pin the chiral limit in HBChPT [64–66], however our resultsdo not indicate such divergence. In contrast to the Diracradius case, perhaps because of the larger statistical errors,HBChPT can simultaneously fit the experiment and ourdata. The fit inspired by a prediction [64],

A!!!!!!!m!

p"1$ Bm! $ Cm! log

"m!

"

##; (28)

where A, B, and C are free parameters (A !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g2AMN=8!F

2!F2"0#

qin HBChPT), " is the scale and fixed

to 1 GeV for simplicity, gives a larger#2=d:o:f:"degrees of freedom# ! 3:4. We cannot obtain areasonable #2=d:o:f: without fixing the coefficient A using

the experimental values. We need further light quark masscalculation with better statistics to test the prediction in thelattice QCD calculation.

C. Form factors of the axial-vector current

In this subsection we show the form factors obtainedfrom the axial-vector currents, FA"q2# and FP"q2#. Theyare extracted from the ratios of three- and two-point func-tions defined in Eqs. (14) and (15). Figure 21 shows that thetypical plateaus of the ratios with the A3 component of thecurrent at mf ! 0:01 are reasonably flat in the middle timeregion between the source and sink operators. We plot theratios !A

L"q3 ! 0; t# and !AL"q3 ! 0; t# separately, since

!AL"q3 ! 0; t# contains both form factors, while !A

L"q3 !0; t# contains only FA"q2#. It is worth noting that there is no!A

L"q3 ! 0; t# in the case of ~q / "1; 1; 1#. !AT"q; t# has a

slope in the range t ! 1–8 with large statistical errors asshown in the bottom panel of Fig. 21. We consider theslope to be caused by poor statistics in the data. The valuesof the matrix elements for all the ratios are determined byconstant fits with the range of t ! 4–8.Using the relations Eqs. (18) and (19), the two form

factors are determined through the following equationswhich depend on the spatial momentum transfer in thethree-point function:

FA"q2# !(!A

L"q3 ! 0# for n ! 0; 1; 2; 4

!AL"q3 ! 0# $ q23

MN"MN$E"q##!AT"q# for n ! 3

; (29)

FP"q2# !(!A

T"q#=MN for n ! 2; 3MN$E"q#

q23"!A

L"q3 ! 0# %!AL"q3 ! 0## for n ! 1; 4 ; (30)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

q2[GeV

2]

0

0.5

1

1.5

2

2.5

3

3.5

4mf=0.01(2.7fm)mf=0.02(2.7fm)mf=0.03(2.7fm)mf=0.01(1.8fm)mf=0.02(1.8fm)mf=0.03(1.8fm)

F2(q2)

FIG. 20 (color online). Comparison of F2 with larger andsmaller volumes denoted by closed and open symbols, respec-tively, at each quark mass.

0 0.1 0.2 0.3 0.4

m!2[GeV

2]

0.4

0.6

0.8

1

1.2Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)HBChPTexperiment

"r22#1/2

[fm]

FIG. 19 (color online). Same as Fig. 15 except Pauli rms radiushr22i1=2 determined from the dipole fit. Prediction of HBChPTwith the experimental result [12,23], and fit result with HBChPTprediction (dash-dot line) are also plotted. The striped symbol isshifted to the minus direction in the x-axis.

NUCLEON FORM FACTORS WITH 2$ 1 FLAVOR . . . PHYSICAL REVIEW D 79, 114505 (2009)

114505-13

rithmic effects are to be seen in lattice results of the Diracradius.

2. Pauli form factor F2!q2"Figure 16 shows the momentum-transfer dependence of

our results for the Pauli form factor at each quark mass.These values are tabulated in Table IV. The form factor isrenormalized by F1!0".

This form factor can also be described by the conven-tional dipole form,

F2!q2" #F2!0"

!1$ q2=M22"2

; (26)

with M2 # 0:78!2" GeV and F2!0" # 3:705 89 extractedfrom fits to experimental data. In contrast to the Dirac formfactor, there are two parameters, the overall strength F2!0"and the dipole massM2: the former gives the isovector partof the anomalous magnetic moment,!p %!n % 1, and thelatter the Pauli mean-squared radius, hr22i # 12=M2

2, as inthe Dirac case. We fit the form factor with these twoparameters.

To check reliability of the dipole fit, we measure theratio of the Sachs electric and magnetic form factors,Eqs. (2) and (3),

GM!q2"GE!q2"

# !VT !q"

!V4 !q"

; (27)

which exhibits a mild q2 dependence [1,49]. At zero mo-mentum transfer, we obtain 1$ F2!0" from the ratio.Figure 17 shows that the result for GE!q2"=GM!q2" % 1at q2 # 0, obtained via a linear fit in q2, is consistent withthe determination from a dipole fit of F2!q2".

In Fig. 18 we present the anomalous magnetic momentof the nucleon, determined by the dipole fit presented in

Table V, together with some other lattice QCD calculationsand the experimental value. Our present results slightlydecrease with the pion mass, in agreement with previouslattice calculations [1,23]. They extrapolate well linearly inthe pion mass squared and result in a value 26% smallerthan the experiment. This result at the physical pion mass isconsistent with those of previous calculations [1,19] usinga linear fit.We present in Fig. 19 the result of the Pauli rms radius.

These results are obtained from a dipole fit and summa-rized in Table V. Some other lattice QCD calculations[1,23] are also plotted in the figure for comparison. We

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0

1

2

3

4

5 F2(q2)

GM(q2)/GE(q

2)-1

experiment

FIG. 17 (color online). Dipole fit with F2!q2" and linear fitwith the ratio of electric and magnetic form factorsGM!q2"=GE!q2" % 1 at mf # 0:01. The result of the ratio atq2 # 0 is shifted to the minus direction in the x-axis.

0 0.1 0.2 0.3 0.4

m!2[GeV

2]

0

1

2

3

4

5

6


F2(0)

experiment

FIG. 18 (color online). Same as Fig. 15 except anomalousmagnetic moment, F2!0" # !p %!n % 1, determined fromthe dipole fit. The experimental result [12] is also shown.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q2[GeV

2]

0

0.5

1

1.5

2

2.5

3

3.5

4mf=0.005mf=0.01mf=0.02mf=0.03experiment

F2(q2)

FIG. 16 (color online). The Pauli form factor, F2!q2", renor-malized by ZV # 1=F1!0". The dashed curve is a fit to experi-mental data.

TAKESHI YAMAZAKI et al. PHYSICAL REVIEW D 79, 114505 (2009)

114505-12

๏ 異常磁気能率 κ=F2(0)を計算するためのトリック

✓ ツイストした周期境界条件を使ってq2~0のデータにアクセスする

✓ 外磁場を掛けて(Background method)、核子の質量変化を測る

γµF1(q2) + iσµνqν

2MNF2(q2)

異常磁気能率（続き）

0 0.1 0.2 0.3 0.4 0.5

m!

2[GeV2]

1

2

3

4

5


"vexperiment

0 0.1 0.2 0.3 0.4

m!

2[GeV2]

0

1

2

3

4

5

6


F2(0)

experiment

κv =(mN )phys

mN× F2(0)

RBC plot QCDSF plot

こちらの量でChPT-typeフィットを行っている

カイラル領域でのπ中間子のループ効果によるエンハンスメントを期待？

One-loopのカイラル摂動論は不充分

0 0.2 0.4 0.6 0.8m![GeV]

0.5

1

1.5

2Nf=2+1 DWF (2.7fm)Nf=2+1 Asqtad+DWF (2.5fm)Nf=2+1 Asqtad+DWF (3.5fm)

gA

experiment

HBChPT (One-loop)

HBChPT (Two-loop)

HBChPT formula

Two-loop ChPT for gA(ver.1.0)

Shoichi Sasaki

Mar. 14, 2008

gA = g0

�

1 +

�α2

(4πF )2ln

Mπ

λ+ β2

�

M2π + α3M

3π

+

�α4

(4πF )4ln2 Mπ

λ+

γ4

(4πF )2ln

Mπ

λ+ β4

�

M4π + α5M

5π +O(M6

π)

�

(1)

where

α2 = −2− 4g20 (2)

α3 =1

(4πF )2

�2 + 2g2

0

m0−

8

3(c3 − 2c4)

�

(3)

α4 = −16

3−

11

3g20 + 16g4

0 (4)

α5 =1

(4πF )2

�c3

m0− 2α3l4

�(5)

β2 =4

g0

�d16 − 2g0d28

�−

1 + 2g20

4(4πF )2ln

Mπ

λ−

g20

(4πF )2(6)

β4 =4c4

m0

1

(4πF )2+

2g20

(4πF )4

�l4 + 2 ln

Mπ

λ

�(7)

γ4 =4(c4 − c3)

m0−

2α2

(4πF )2

�l4 + 2 ln

Mπ

λ

�

−1

(4πF )2

(5 + 3g20)(4− g2

0)

2ln

Mπ

λ−

20 + 12g20

g0d16 (8)

Fπ = F

�

1 +M2

π

(4πF )2l4 +O(M4

π)

�

(9)

mN = m0 − 4c1M2π +O(M3

π) (10)

1

One loop: Kambor-Mojzis JHEP 9904, 031 (99)

Two loop: Bernard-Meissner PLB639, 278 (06)

格子QCDによる核子構造の最新計算

QCDSF Collaboration, arXiv:1106.3580

• Nf=2 NP O(a) improved Wilson fermions

• mπ > 180 MeV

• Max volume: 2.9 fm

• Two lattice spacings: (a=0.06, 0.072 fm)

ETM Collaboration, Phys. Rev. D83 (2011) 045010

• Nf=2 twisted mass fermions

• mπ > 260 MeV

• Max volume: 2.8 fm

• Three lattice spacings: (a=0.056, 0.070, 0.089 fm)

• Continuum limit

0 0.1 0.2 0.3 0.4

m!

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

experimentNf=2+1 DWF (RBC-UKQCD)Nf=2 Clover (QCDSF)Nf=2 Clover (QCDSF)Nf=2 tmWilson (ETMC)

"r12#1/2[fm]

Volume L~2.9 fmmπ = 180 MeV

核子の大きさの問題（未だ解決せず）

0 0.1 0.2 0.3 0.4 0.5

m!

2[GeV2]

1

2

3

4

5

Nf=2+1 DWF (RBC-UKQCD)Nf=2 Clover (QCDSF)Nf=2 Clover (QCDSF)Nf=2 tmWilson (ETMC)

"vexperiment

異常磁気能率も小さい

QCDSF plot

FIG. 11 (color online). Continuum results for the isovectorDirac and Pauli form factors F1 and F2 at Q2 ! 357 GeV2.The dashed line is a fit to a constant, whereas the dotted line is afit to a line.

FIG. 12 (color online). Continuum results for the isovector

Dirac and Pauli mean squared radii r2p"n1 and r2p"n

2 . Thenotation is the same as in Fig. 11.

NUCLEON ELECTROMAGNETIC FORM FACTORS IN . . . PHYSICAL REVIEW D 83, 094502 (2011)

094502-11

ETMC collaboration, Phys. Rev. D83 (2011) 094502

the constant fit. Therefore, we conclude that finite a effectsare negligible and for the intermediate pion masses weobtain the values in the continuum by fitting our data at! ! 3:9 and ! ! 4:05 to a constant.

In Figs. 12 and 13 we show the continuum extrapolationof the r.m.s Dirac and Pauli radii and the anomalousmagnetic moment, respectively. The corresponding valuesof "p"n at the six reference pion masses used in the figures

are given in Table IVand those of the Dirac and Pauli meansquare radii in Table V.Having results in the continuum limit we can now

perform the chiral fits described in the previous section.We show these chiral fits to the continuum results for theanomalous magnetic moment and Dirac and Pauli meansquare radii in Figs. 14–16.The behavior observed is similar to that obtained when

using the raw lattice data. Namely, chiral fits to the Diracand Pauli form factors F1 and F2 bring agreement withexperiment at low Q2-values, and therefore the values for"p"n and r2p"n

1 derived using the parameters of the chiralfit to F1 and F2 agree with the experimental values. Thedescription of r2p"n

2 is also reasonable bringing latticeresults close to the value obtained at the physical point,although not fully reproducing the experimental value. Inthe figures we also include the curves obtained by fittingseparately the anomalous magnetic moment and radii. Forthe former the mean value obtained at the physical point islower as compared to the value obtained from fitting F1

andF2, with, however, almost overlapping errors. For thesefits we used the physical value of gA. As discussed in theprevious section in connection with Fig. 10, had we used

FIG. 13 (color online). Continuum results for the nucleonanomalous magnetic moment "p"n. The notation is the sameas in Fig. 11.

TABLE IV. In the second, third and fourth column we give theinterpolated values of "p"n at the value of m#r0 given in the firstcolumn. We used r0=a ! 5:22#2$, 6.61(3) and 8.31(5) for ! !3:9, 4.05 and 4.2, respectively. In the fifth column we give thevalue of "p"n after extrapolating to a ! 0 using a constant fit. Inthe parenthesis we give the corresponding values when using alinear fit.

r0m# "p"n

(! ! 3:9) (! ! 4:05) (! ! 4:2) (a ! 0)

1.1019 2.12(13) 2.13(15) 1.84(17) 2.05(8) [1.79(26)]1.0 2.08(18) 2.36(23) 2.18(14)0.95 2.09(21) 2.42(22) 2.25(15)0.85 2.27(27) 2.53(26) 2.41(19)0.686 2.34(36) 2.54(51) 2.40(29)0.615 2.49(23) 2.57(53) 2.62(24) 2.55(16) [2.71(41)]

TABLE V. In the second, third and fourth raws we give the interpolated values of r2p"n1 and r2p"n

2 in fm2 at the value of m#r0 givenin the first column. We used r0=a ! 5:22#2$, 6.61(3) and 8.31(5) for ! ! 3:9, 4.05 and 4.2, respectively. In the fifth raws we give the

value of r2p"n1 and r2p"n

2 after extrapolating to a ! 0 using a constant fit. In the parenthesis we give the corresponding values whenusing a linear fit.

r0m# r2p"n1 r2p"n

2 r2p"n1 r2p"n

2 r2p"n1 r2p"n

2 r2p"n1 r2p"n

2

(! ! 3:9) (! ! 4:05) (! ! 4:2) (a ! 0)

1.1019 0.236(17) 0.300(39) 0.229(17) 0.319(45) 0.183(16) 0.238(43) 0.214(9)[0.160(39)] 0.285(24) [0.226(70)]1.0 0.226(36) 0.379(79) 0.258(33) 0.326(62) 0.243(24) 0.347(49)0.95 0.234(41) 0.382(89) 0.259(32) 0.345(59) 0.249(25) 0.356(49)0.85 0.286(30) 0.282(81) 0.261(35) 0.382(68) 0.276(23) 0.340(52)0.686 0.378(41) 0.409(99) 0.221(50) 0.494(126) 0.442(78) 0.442(78)0.615 0.307(27) 0.446(61) 0.210(53) 0.534(132) 0.266(32) 0.441(50) 0.280(17) [0.220(54)] 0.450(37) [0.445(91)]

C. ALEXANDROU et al. PHYSICAL REVIEW D 83, 094502 (2011)

094502-12

荷電半径異常磁気能率

連続極限（a→0）連続極限（a→0）

そもそもの疑問

✓ 典型的な空間サイズ　L ～ 2 - 3 fm

✓ 最小の有限な運動量：

- |p| ~ 2π/ L = 0.6 , 0.4 GeV

cf: Λχ～４πFπ～１ GeV

‣ 低エネルギー有効理論が成り立つ条件があいまい

有限体積中でカイラル摂動論はどれだけ正しいか？

有効理論・カイラル摂動論のアプローチ

• 低エネルギー(長波長)では複合粒子であるハドロンも場の理論(点粒子)で記述できる

• カイラル対称性の自発的な破れに伴う擬南部・ゴールドストン粒子(パイ中間子など)は長波長極限で最も意味のある自由度

• ただし、擬南部・ゴールドストン粒子の相互作用は対称性によって強い制約がかかる

0 0.1 0.2 0.3 0.4 0.5mπ

2[GeV2]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Nf=2+1 DWF (2.7fm)Nf=2 DWF (1.9fm)Nf=0 DWF (3.6fm)Nf=2 Wilson (1.9fm)Nf=0 Wilson (3.0fm)HBChPT (LO)

(〈r12〉)1/2[fm]

experiment

その疑問に答えるには少なくともmπが250 MeV以下で空間サイズが4-5 fm程度(|pmin|~mπ)の規模で、且つカイラル対称性を保持した数値計算が必要。

先の疑問に答えるには

核子の大きさ例）RBC-UKQCDの取り組み2+1 f DWF simulationsβ=1.75 (a-1≈1.4 GeV),323x64x32 latticeLa ≈ 4.6 fmmπ＝250 and 170 MeV ターゲット領域

RBC-UKQCDの最近の取り組み

空間サイズ：　4.6 fm

π中間子の質量：　170, 250 MeV

核子質量 M. Lin, Y. Aoki, T. Blum, C. Dawson, T. Izubuchi, C. Jung, S. Ohta, S. Sasaki, T. Yamazaki

0 0.1 0.2m!

2[GeV2]

0.8

0.9

1

1.1

1.2

1.3

mN

[GeV

]

Nf=2+1 DWF (L=2.8fm, 1/a=1.7GeV)Nf=2+1 DWF (L=2.8fm, 1/a=2.3GeV)Nf=2+1 DWF (L=4.6fm, 1/a=1.4GeV)experiment

mπ = 250 MeV

mπ = 170 MeV

RICC@理研和光RIKEN Integrated Cluster of Clusters

preliminary

荷電半径、異常磁気能率など

只今、計測中もうしばらくお待ち下さい

軸性電荷：gA/gV

0 0.1 0.2 0.3 0.4 0.5m!

2[GeV2]

0.7

0.8

0.9

1

1.1

1.2

1.3

Nf=2+1 DWF (L=2.7fm, 1/a=1.7GeV)Nf=2+1 DWF (L=1.8fm, 1/a=1.7GeV)Nf=2+1 DWF (L=4.6fm, 1/a=1.4GeV)

gAexperiment

preliminary

Summary from lattice 2011Comparison

III-8

from H. Wittig’s talk

Expt.

}2 flavor2+1 flavor

軸性電荷 gA

Collaboration Nf a (fm) La(fm) mπ (MeV) Fermion actionRBC-UKQCD 2+1 0.14 4.6 > 170 Domain wall fermionCLS/Mainz 2 0.05, 0.07, 0.08 3.3 > 290 O(a)-improved Wilson

ETMC 2 0.056, 0.070, 0.089 2.8 > 260 Twisted mass Wilson

Collaboration Nf a (fm) La(fm) mπ (MeV) Fermion actionRBC-UKQCD 2+1 0.14 4.6 > 170 Domain wall fermion

QCDSF 2 0.060, 0.072 2.9 > 180 O(a)-improved WilsonCLS/Mainz 2 0.05, 0.07, 0.08 3.3 > 290 O(a)-improved Wilson

ETMC 2 0.056, 0.070, 0.089 2.8 > 260 Twisted mass Wilson

2

CLS/Mainz (Brandt et al.), arXiv:1105.1554ETMC (Alexandrou et al.), arXiv: 1012.0857

まだ体積が足りないのか？

gA vs m2π gA vs mπL

2 3 4 5 6 7 8 9 10 11 12m!L

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Nf=2+1 (L=2.7fm, 1/a=1.7GeV)Nf=2+1 (L=1.8fm, 1/a=1.7GeV)Nf=2+1 (L=4.6fm, 1/a=1.4GeV)

experiment

gA(DWF)

0 0.1 0.2 0.3 0.4 0.5m!

2[GeV2]

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Nf=2+1 DWF (L=2.7fm, 1/a=1.7GeV)Nf=2+1 DWF (L=1.8fm, 1/a=1.7GeV)Nf=2+1 DWF (L=4.6fm, 1/a=1.4GeV)

gAexperiment

preliminary preliminary

0.70.80.9

11.11.21.3

Nf=2 Wilson β=5.50Nf=2 Wilson β=5.60

0.70.80.9

11.11.21.3

Nf=2+1(2.7fm)Nf=2+1(1.8fm)Nf=2(1.9fm)Nf=0(2.4fm)Nf=2+1 Mix(2.5fm)

Nf=2 Imp. Wilson β=5.20Nf=2 Imp. Wilson β=5.25Nf=2 Imp. Wilson β=5.29Nf=2 Imp. Wilson β=5.40

gA (Wilson)

gA (DWF)

Scaling in mπL T. Yamazaki et al, (RBC+UKQCD) PRL100, 171602 (08)

Wilson: D. Dolgov et al., (LHPC-SESAM) PRD66, 034506 (2002)Wilson: C. Alexandrou et al., (MIT-Cyprus) PRD76, 094511 (2007)Clover: A. Khan et al., (QCDSF) PRD74, 094508 (2006)

クォーク運動量分布とヘリシティ分布の比

0 0.1 0.2 0.3 0.4 0.5m!

2[GeV2]

0.5

0.6

0.7

0.8

0.9

1

Nf=2+1 DWF (L=4.8 fm)Nf=2+1 DWFNf=2+1 Asqtad+DWF (LHPC)

"x#u-d"x#

$u-$d

experiment

‣ 個々のmπ依存性を議論できるほど統計が溜まっていない

‣ 非摂動論的手法による繰り込み定数の計算も別途必要

preliminary

簡単なまとめ

物理点に近づいたシミュレーションができるが、充分に大きい体積で計算できているか？：e.g. ４x(1/mπ)~5.8 fm

核子の大きさの問題：”π中間子の雲”の効果は見えるか？

他の物理量にもカイラルlogの効果が現れるか？

格子QCDアプローチによる核子構造の理解の現状を例えるなら

ご清聴ありがとうございました。

Backup slides

ディラック／パウリ平均自乗半径

paper. Recent reviews on the experimental situation can befound in Ref. [27]. The slopes of the form factors at q2 ! 0determine mean-squared radii, which can be related todipole masses as hr2i i ! 12=M2

i (i ! E or M) in the dipoleform Gi"q2# ! Gi"0#="1$ q2=M2

i #. The experimental val-ues of the electric root mean-squared (rms) radius for theproton and the magnetic rms radii of the proton and neu-tron are compiled in Table I. These rms radii are all equalwithin errors and are in agreement with the empiricaldipole parameter !. On the other hand, the slope of theneutron electric form factor Gn

E"q2# is determined withhigh precision from double-polarization measurements ofneutron knockout from a polarized 2H or 3He target, whileonly a small deviation from zero is observed for Gn

E"q2# atlow q2 [27]. Combined with all four of the electric chargeand magnetization radii of the proton and neutron, wefinally evaluate the rms radii for the weak vector form

factor and induced tensor form factor as!!!!!!!!!!!!!!h"rV#2i

p!

0:797"4# fm and!!!!!!!!!!!!!!h"rT#2i

p! 0:879"18# fm, which corre-

spond to the dipole masses, MV ! 0:857"8# GeV andMT ! 0:778"23# GeV. See Appendix A for details.

The axial-vector form factor at zero momentum transfer,namely, the axial-vector coupling gA ! FA"0#, is preciselydetermined by measurements of the beta asymmetry inneutron decay. The value of gA ! 1:2695"29# is quotedin the 2006 PDG [22]. Nevertheless, kinematics of neutronbeta decay are quite limited due to a very small massdifference of the proton and neutron. Other experimentalmethods are utilized for determination of the q2 depen-dence of FA"q2#. For this purpose, there are basically twotypes of experiments, namely, quasi-elastic neutrino scat-tering and charged pion electroproduction experiments.The former suffers from severe experimental uncertaintiesconcerning the incident neutrino flux and the backgroundsubtraction of elastic events, while model-dependentanalysis is somewhat inevitable for the latter [9]. Bothmethods reported that the dipole form FA"q2# !FA"0#="1$ q2=M2

A# is a good description for low andmoderate momentum transfer q2 < 1 GeV2. The resultingworld average of the dipole mass parameter MA is quotedasMA ! 1:026"21# GeV from neutrino scattering orMA !1:069"16# GeV from pion electroproduction in Ref. [9]. Asfor a small discrepancy between two averages, it has been

argued that within heavy-baryon chiral perturbation theorythe finite pion mass correction of%0:055 GeV to the lattervalue may resolve this discrepancy [9]. Therefore, one cantranslate the axial dipole mass into the axial rms radius of!!!!!!!!!!!!!!h"rA#2i

p! 0:67"1# fm, which is consistently obtained

from quasi-elastic neutrino scattering experiments andcharged pion electroproduction experiments [9].On the other hand, the induced pseudoscalar form factor

FP"q2# is less well known experimentally [10]. The mainsource of information on FP"q2# stems from OMC on theproton, !% $ p ! "! $ n. One measures the inducedpseudoscalar coupling gP ! m!FP"q20# at the specific mo-mentum transfer for the muon capture by the proton at restas q20 ! 0:88m2

!. The induced pseudoscalar coupling gP isalso measured in RMC, !% $ p ! #$ "! $ n. Before2006, the Saclay OMC experiment, which was the mostrecent OMC experiment at that time, reported"gOMC

P #Saclay;original ! 8:7& 1:9 [29]. Combining with theolder OMC experiments including bubble chamber mea-surements, the world average for OMC is obtained as"gOMC

P #oldAve ! 8:79& 1:92, which is given inRefs. [9,29]. Surprisingly, this value is close to the theo-retically predicted value by heavy-baryon chiral perturba-tion theory gChPTP ! 8:26& 0:16 [9]. However, the novelRMC experiment at TRIUMF [30,31] is puzzling: theirmeasured value of gRMC

P ! 12:4& 1:0 is quite higher thanthe theoretical value as is the OMC value as gRMC

P '1:4gOMC

P . This disagreement is reduced by reanalysiswith the updated!$ lifetime [10]. Then, the updated resultof the Saclay OMC experiment yields "gOMC

P #Saclay;updated !10:6& 2:7. Accordingly, the weighted world average forOMC, "gOMC

P #updatedAve: ! 10:5& 1:8 given in Ref. [10], isshifted away from the theoretical expected value, while theupdated average value is in agreement with the RMC resultwithin its error. Indeed, there is a caveat that the ortho-paratransition rate in !-molecular Hydrogen, to which eitherOMC and RMC results are very sensitive, is poorly knowndue to mutually inconsistent results among two experi-ments [29,32] and theory [33]. Comprehensive reviews ofa history of gP have been given in Refs. [9,10].Recently, a new OMC experiment has been done by the

MuCap Collaboration [11]. The MuCap result is nearlyindependent of !-molecular effects in contrast with theprevious OMC experiments and the RMC experiment.After the electroweak radiative corrections, which wereunderestimated in the old literature, are correctly takeninto account [34], the new precise OMC measurementyields

gMuCapP ! 7:3& 1:1: (15)

Including the new MuCap result and taking into accountthe electroweak radiative corrections, the new world aver-age of the OMC results becomes "gOMC

P #newAve ! 8:7& 1:0[34]. As for other q2 data of FP"q2#, only a few data points

TABLE I. Experimental values of magnetic moments, electriccharge, and magnetization radii of the proton and neutron.

Observable Experimental value Reference

!p $2:792 847 351"28# [22]!n %1:91 304 273"45# [22]h"rpE#2i1=2 0.8750(68) fm [22]h"rnE#2i %0:1161"22# fm2 [22]h"rpM#2i1=2 0.855(35) fm [27]h"rnM#2i1=2 0.873(11) fm [28]

SHOICHI SASAKI AND TAKESHI YAMAZAKI PHYSICAL REVIEW D 78, 014510 (2008)

014510-4

ACKNOWLEDGMENTS

It is a pleasure to acknowledge S. Choi for his privatecommunication providing actual values of the form factorFP!q2" in his experiment. We would like to thank ourcolleagues in the RBC collaboration and especiallyT. Blum for helpful suggestions and his careful readingof the manuscript, and H.-W. Lin and S. Ohta for fruitfuldiscussions. We also thank RIKEN, Brookhaven NationalLaboratory and the U.S. DOE for providing the facilitiesessential for the completion of this work. The results ofcalculations were performed by using QCDOC at RIKENBNL Research Center. S. S. is supported by the JSPS for aGrant-in-Aid for Scientific Research (C) 19540265). T. Y.is supported by U.S. DOE Grant No. DE-FG02-92ER40716 and the University of Connecticut.

Note added.—After the completion of this work, webecame aware of a paper [7] where the nucleon axial-vector form factor FA!q2" and induced pseudoscalar formfactor FP!q2" are calculated in quenched and unquenchedlattice QCD using Wilson fermions.

APPENDIX A: VARIOUS RMS RADII IN THEVECTOR CHANNEL

In Table I, the electric charge and magnetization radii forthe proton and neutron are summarized. Using these ex-perimental values, the isovector electric charge and iso-vector magnetization radii can be evaluated by thefollowing relations [4,8]:

h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)

where GvE!M"!q2" % Gp

E!M"!q2" $GnE!M"!q2" and !v %

!p $!n. Then, one obtains""""""""""""""h!rvE"2i

q% 0:939!5" fm and

"""""""""""""""h!rvM"2i

q% 0:862!14" fm. Similarly, the rms radii for the

isovector Dirac form factor Fv1 !q2" % Fp

1 !q2" $ Fn1 !q2"

and the isovector Pauli form factor Fv2 !q2" % Fp

2 !q2" $Fn2 !q2" can be given through the following relations [4,8]:

h!rv1 "2i % h!rvE"2i$3

2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)

which yield""""""""""""""h!rv1 "2i

q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.

APPENDIX B: GENERALIZED GOLDBERGER-TREIMAN RELATION AND PION-POLE

DOMINANCE

The generalized Goldberger-Treiman relation is derivedfrom the nucleon matrix elements of the currents on bothsides of the axial Ward-Takahashi identity [24];@"A

a"!x" % 2mPa!x" where the exact isospin symmetry is

considered as m % mu % md. The nucleon matrix elementof the divergence of the axial-vector current is representedin the following form:

hN!p0"j@"Aa"!0"jN!p"i

% !uN!p0"&i!p6 $ p6 0"FA!q2" $ q2FP!q2"'#5tauN!p"

% &2MNFA!q2" $ q2FP!q2"' !uN!p0"#5tauN!p": (B1)

Here, it is worth mentioning that we have used the Diracequation for the nucleon !uN!p"!ip6 (MN" % !ip6 (MN"uN!p" % 0 to get from the first line to the secondline. Then, one easily finds that the q2 dependences ofthree form factors are constrained by the following rela-tion:

2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)

which is a consequence of the axial Ward-Takahashi iden-tity. This expression may be referred to as the generalizedGoldberger-Treiman relation [24].Here, we discuss the case where the limits m ! 0 and

q2 ! 0 are taken on Eq. (B2). Of course, the left-hand side(l.h.s.) of Eq. (B2) yields a nonzero value in the doublelimit. First, we consider the case where the chiral limit isfirst taken before the limit of q2 ! 0.

limq2!0

! limm!0

2MNFA!q2"" % limq2!0

!q2 limm!0

FP!q2""; (B3)

which requires the massless pion pole in FP!q2" in thechiral limit [36] as limm!0FP!q2" / 1

q2for nonvanishing of

the l.h.s. of Eq. (B3). Secondly, the chiral limit is takenafter the limit of q2 ! 0:

limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)

which requires the 1=m singularity in GP!q2" at q2 % 0 aslimq2!0GP!q2" / 1

m ) 1m2

$for nonvanishing of the l.h.s. of

Eq. (B4). As a result, FP!q2" and GP!q2" must have thepion-pole structure, which should become dominant at lowq2 [36]. Therefore, one can deduce that FP!q2" and GP!q2"are described by the following forms, at least, in thevicinity of the pole position q2 % $m2

$ [36,53]:


014510-28

ACKNOWLEDGMENTS





h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)




q% 0:939!5" fm and

"""""""""""""""h!rvM"2i



1 !q2" $ Fn1 !q2"




2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)


q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.


DOMINANCE








2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)



limq2!0

! limm!0


!q2 limm!0

FP!q2""; (B3)




limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)


m ) 1m2



$ [36,53]:


014510-28

アイソベクトル要素

ACKNOWLEDGMENTS





h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)




q% 0:939!5" fm and

"""""""""""""""h!rvM"2i



1 !q2" $ Fn1 !q2"




2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)


q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.


DOMINANCE








2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)



limq2!0

! limm!0


!q2 limm!0

FP!q2""; (B3)




limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)


m ) 1m2



$ [36,53]:


014510-28

ACKNOWLEDGMENTS





h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)




q% 0:939!5" fm and

"""""""""""""""h!rvM"2i



1 !q2" $ Fn1 !q2"




2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)


q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.


DOMINANCE








2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)



limq2!0

! limm!0


!q2 limm!0

FP!q2""; (B3)




limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)


m ) 1m2



$ [36,53]:


014510-28

ACKNOWLEDGMENTS





h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)




q% 0:939!5" fm and

"""""""""""""""h!rvM"2i



1 !q2" $ Fn1 !q2"




2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)


q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.


DOMINANCE








2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)



limq2!0

! limm!0


!q2 limm!0

FP!q2""; (B3)




limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)


m ) 1m2



$ [36,53]:


014510-28

ACKNOWLEDGMENTS





h!rvE"2i # $61

GvE!q2"

dGvE!q2"dq2

!!!!!!!!q2%0% h!rpE"2i$ h!rnE"2i;

(A1)

h!rvM"2i # $61

GvM!q2"

dGvM!q2"dq2

!!!!!!!!q2%0

% !p

!vh!rpM"2i$

!n

!vh!rnM"2i; (A2)




q% 0:939!5" fm and

"""""""""""""""h!rvM"2i



1 !q2" $ Fn1 !q2"




2

Fv2 !0"M2

N

; (A3)

h!rv2 "2i %1

!v $ 1!!vh!rvM"2i$ h!rv1 "2i"; (A4)


q% 0:797!4" fm and

""""""""""""""h!rv2 "2i

q%

0:879!18" fm.


DOMINANCE








2MNFA!q2" % q2FP!q2" ( 2mGP!q2"; (B2)



limq2!0

! limm!0


!q2 limm!0

FP!q2""; (B3)




limm!0

! limq2!0

2MNFA!q2"" % limm!0

!2m limq2!0

GP!q2""; (B4)


m ) 1m2



$ [36,53]:


014510-28

ディラック平均自乗半径

パウリ平均自乗半径

= 0.797(4) fm

= 0.879(18) fm

tors at low q2 are discussed with great interest. At the endof this section, we discuss the consequence of the axialWard-Takahashi identity among the axial-vector form fac-tor, the induced pseudoscalar form factor and the pseudo-scalar form factor. In Sec. V, we discuss the finite volumeeffects on the form factors using results from three differ-ent volumes. Meanwhile, we also check whether approxi-mated forms of q2 dependence of form factors, which areobserved at low q2, are still valid even in the relatively highq2 region, up to at least q2 ! 1:0 GeV2, apart from con-sideration of the finite volume effects. In Sec. VI, wecompare our results with previous works. Finally, inSec. VII, we summarize the present work and discussfuture directions.

II. WEAK NUCLEON FORM FACTORS ANDEXPERIMENTAL STATUS

In general, the nucleon matrix elements of the weakcurrent are given by a linear combination of the vectorand axial-vector matrix elements. Here, let us introduce thevector and axial-vector currents, which are expressed interms of the isospin doublet of quark fields " #u; d$T

Va!#x$ " ! #x$"!ta #x$; (2)

Aa!#x$ " ! #x$"!"5ta #x$; (3)

where ta are the SU#2$ flavor matrices normalized to obeyTr#tatb$ " #ab. Then, the nucleon matrix elements aregiven by

hN#P0$jJwk! #x$jN#P$i" hN#P0$jVa!#x$ % Aa!#x$jN#P$i; (4)

" !uN#P0$#OV!#q$ %OA

!#q$$tauN#P$eiq&x; (5)

where q ' P( P0 is the momentum transfer between theinitial (P) state and the final state (P0) and N represents thenucleon isospin doublet as N " #p; n$T . Four form factorsare needed to describe these matrix elements: the weakvector and induced tensor (weak magnetism) form factorsfor the vector current

O V!#q$ " "!FV#q2$ % $!%q%FT#q2$ (6)

and the weak axial-vector and induced pseudoscalar formfactors for the axial-vector current [25]

O A!#q$ " "!"5FA#q2$ % iq!"5FP#q2$; (7)

which are here given in the Euclidean metric convention[26]. Thus, q2 denoted in this paper, which stands forEuclidean four-momentum squared, corresponds to thetimelike momentum squared as q2M " (q2 < 0 inMinkowski space.

The weak matrix elements are related to the electromag-netic matrix elements if the strange contribution is ignored

under the exact isospin symmetry. A simple exercise inSU#2$ Lie algebra leads to the following relation betweenthe vector part of the weak matrix elements of neutron betadecay and the difference of proton and neutron electro-magnetic matrix elements [8,15]

hpj !u"!djni " hpj !u"!u( !d"!djpi" hpjjem! jpi( hnjjem! jni; (8)

where jem! " 23 !u"!u( 1

3!d"!d. This relation gives a con-

nection between the weak vector and induced tensor formfactors and the isovector part of electromagnetic nucleonform factors

Fv1 #q2$ " FV#q2$; (9)

Fv2 #q2$ " 2MNFT#q2$; (10)

where Fv1 (Fv2 ) denotes the isovector combination of theDirac (Pauli) form factors of the proton and neutron, whichare defined by

hN#P0$jjem! #x$jN#P$i " !uN#P0$!"!F

N1 #q2$

% $!%

q%2MN

FN2 #q2$"uN#P$; (11)

where MN denotes the nucleon mass, which is defined asthe average of neutron and proton masses, andN representsp (proton) or n (neutron). Experimental data from elasticelectron-nucleon scattering is usually presented in terms ofthe electric GE#q2$ and magnetic GM#q2$ Sachs formfactors, which are related to the Dirac and Pauli formfactors [8,27]

GNE #q2$ " FN1 #q2$ (

q2

4M2N

FN2 #q2$; (12)

GNM#q2$ " FN1 #q2$ % FN2 #q2$: (13)

Their normalization at q2 " 0 are given by the proton(neutron) charge and magnetic moment [22]:

Proton: GpE#0$" 1; Gp

M#0$"&p"%2:792847351#28$;Neutron: Gn

E#0$" 0; GnM#0$"&n"(1:91304273#45$:

(14)

Therefore, one finds FV#0$ " GpE#0$ (Gn

E#0$ " 1 and2MNFT#0$ " Gp

M#0$ (GnM#0$ ( 1 " 3:70 589. As for the

q2 dependence of the form factors, it is experimentallyknown that the standard dipole parametrization GD#q2$ ""2=#"2 % q2$ with " " 0:84 GeV (or "2 " 0:71 GeV2)describes well the magnetic form factors of both the protonand neutron and also the electric form factor of the proton,at least, in the low q2 region [27]. Here, the currentinteresting issues of the q2 dependence of the electromag-netic form factors at higher q2 are beyond the scope of this

NUCLEON FORM FACTORS FROM QUENCHED LATTICE . . . PHYSICAL REVIEW D 78, 014510 (2008)

014510-3

tors at low q2 are discussed with great interest. At the endof this section, we discuss the consequence of the axialWard-Takahashi identity among the axial-vector form fac-tor, the induced pseudoscalar form factor and the pseudo-scalar form factor. In Sec. V, we discuss the finite volumeeffects on the form factors using results from three differ-ent volumes. Meanwhile, we also check whether approxi-mated forms of q2 dependence of form factors, which areobserved at low q2, are still valid even in the relatively highq2 region, up to at least q2 ! 1:0 GeV2, apart from con-sideration of the finite volume effects. In Sec. VI, wecompare our results with previous works. Finally, inSec. VII, we summarize the present work and discussfuture directions.

II. WEAK NUCLEON FORM FACTORS ANDEXPERIMENTAL STATUS

In general, the nucleon matrix elements of the weakcurrent are given by a linear combination of the vectorand axial-vector matrix elements. Here, let us introduce thevector and axial-vector currents, which are expressed interms of the isospin doublet of quark fields " #u; d$T

Va!#x$ " ! #x$"!ta #x$; (2)

Aa!#x$ " ! #x$"!"5ta #x$; (3)

where ta are the SU#2$ flavor matrices normalized to obeyTr#tatb$ " #ab. Then, the nucleon matrix elements aregiven by

hN#P0$jJwk! #x$jN#P$i" hN#P0$jVa!#x$ % Aa!#x$jN#P$i; (4)

" !uN#P0$#OV!#q$ %OA

!#q$$tauN#P$eiq&x; (5)

where q ' P( P0 is the momentum transfer between theinitial (P) state and the final state (P0) and N represents thenucleon isospin doublet as N " #p; n$T . Four form factorsare needed to describe these matrix elements: the weakvector and induced tensor (weak magnetism) form factorsfor the vector current

O V!#q$ " "!FV#q2$ % $!%q%FT#q2$ (6)

and the weak axial-vector and induced pseudoscalar formfactors for the axial-vector current [25]

O A!#q$ " "!"5FA#q2$ % iq!"5FP#q2$; (7)

which are here given in the Euclidean metric convention[26]. Thus, q2 denoted in this paper, which stands forEuclidean four-momentum squared, corresponds to thetimelike momentum squared as q2M " (q2 < 0 inMinkowski space.

The weak matrix elements are related to the electromag-netic matrix elements if the strange contribution is ignored

under the exact isospin symmetry. A simple exercise inSU#2$ Lie algebra leads to the following relation betweenthe vector part of the weak matrix elements of neutron betadecay and the difference of proton and neutron electro-magnetic matrix elements [8,15]

hpj !u"!djni " hpj !u"!u( !d"!djpi" hpjjem! jpi( hnjjem! jni; (8)

where jem! " 23 !u"!u( 1

3!d"!d. This relation gives a con-

nection between the weak vector and induced tensor formfactors and the isovector part of electromagnetic nucleonform factors

Fv1 #q2$ " FV#q2$; (9)

Fv2 #q2$ " 2MNFT#q2$; (10)

where Fv1 (Fv2 ) denotes the isovector combination of theDirac (Pauli) form factors of the proton and neutron, whichare defined by

hN#P0$jjem! #x$jN#P$i " !uN#P0$!"!F

N1 #q2$

% $!%

q%2MN

FN2 #q2$"uN#P$; (11)

where MN denotes the nucleon mass, which is defined asthe average of neutron and proton masses, andN representsp (proton) or n (neutron). Experimental data from elasticelectron-nucleon scattering is usually presented in terms ofthe electric GE#q2$ and magnetic GM#q2$ Sachs formfactors, which are related to the Dirac and Pauli formfactors [8,27]

GNE #q2$ " FN1 #q2$ (

q2

4M2N

FN2 #q2$; (12)

GNM#q2$ " FN1 #q2$ % FN2 #q2$: (13)

Their normalization at q2 " 0 are given by the proton(neutron) charge and magnetic moment [22]:

Proton: GpE#0$" 1; Gp

M#0$"&p"%2:792847351#28$;Neutron: Gn

E#0$" 0; GnM#0$"&n"(1:91304273#45$:

(14)

Therefore, one finds FV#0$ " GpE#0$ (Gn

E#0$ " 1 and2MNFT#0$ " Gp

M#0$ (GnM#0$ ( 1 " 3:70 589. As for the

q2 dependence of the form factors, it is experimentallyknown that the standard dipole parametrization GD#q2$ ""2=#"2 % q2$ with " " 0:84 GeV (or "2 " 0:71 GeV2)describes well the magnetic form factors of both the protonand neutron and also the electric form factor of the proton,at least, in the low q2 region [27]. Here, the currentinteresting issues of the q2 dependence of the electromag-netic form factors at higher q2 are beyond the scope of this

NUCLEON FORM FACTORS FROM QUENCHED LATTICE . . . PHYSICAL REVIEW D 78, 014510 (2008)

014510-3

¨ 2$%t Óé ½hep4.nucl.phys.titech.ac.jp/workshop/jps2011a/16psb4... · 2011. 10. 5. ·...

Documents