エネルギー運動量テンソルを用いたキンクの力学的構造の量子論的解析

阪大理

伊藤広晃
京大基研

北沢正清

エネルギー運動量テンソル (EMT)⚫ 𝑇𝜇𝜈 ：並進対称性に付随したネーターカレント
エネルギー密度𝑇𝜇𝜈 =

𝑇00
𝑇01
𝑇02
𝑇03

𝑇01
𝑇11
𝑇21
𝑇23

運動量密度

𝑇02
𝑇12
𝑇22
𝑇32

𝑇03
𝑇13
𝑇23
𝑇33

EMTの良いところ ゲージ不変・保存量
 物理的意味が明瞭
 近接作用なので直観的応力⚫ ハドロン内部のEMT分布の実験的測定
Pressure : proton
Burkert, et al 2018

EMT distribution : pion
Kumano, et al 20182/19

Quantum fluctuation of EMT
ഥ⚫ Distribution of EMT in QQ
ഥ on Lattice QCD
Stress distribution around QQ

Pulling force
Pushing force(parallel to field) (perpendicular to field)

Describing stressYanagihara, et al. 2019ഥ⚫ Quantum effect in QQ𝑤𝑟ሷሷ
Luscher,Munster,and Weisz 1981

Viewing fluxtube
Theoretical model
Fattening due tostring vibration
2𝑤 ~ log𝑟

Motivate theoretical analysis of quantum
ഥ systemcorrection to EMT distribution in 𝐐𝐐

3/19

キンクの量子振動
目的
1+1次元ソリトン近傍のEMT分布の量子補正を解析的に計算する。
1+1次元𝝓𝟒 模型のキンク
𝜙kink𝑥 場の方程式の安定な古典解
 局所的で非自明な構造

Classical EMT: Trivial
Quantum EMT: Nontrivial, First trial
4/19

今日話すこと

1 Introduction
2 𝝓𝟒 模型のEMT
3 Sine-Gordon模型のEMT
5/19

𝝓𝟒 Model in 1+1 d21𝜆𝑚
𝑆 = න 𝑑𝑥 2 𝜕𝜇 𝜙𝜕𝜇 𝜙 −𝜙2 −24𝜆𝑚22𝑈= 𝜙 −𝜆2vacuum − 𝑚+√𝜆Classical ◼ Kink solution of EOM
𝜙𝑐𝑙 = +𝑚𝑚 𝑥−𝑋tanh√𝜆21-loop 𝐸𝐿𝑂 ~𝑂(𝝀𝟎 )√𝜆𝐿𝑂𝑇00~𝑂(𝝀𝟎 )+𝑚√𝜆𝑿𝑥~𝑚−1𝑿𝑥−
X: location of kink√𝜆◼ 𝑇00 under kink solution 𝑇00

𝑚4𝑚 𝑥−𝑋−4𝑇00 =cosh𝝀2

𝜙𝑚𝜙𝑐𝑙𝑚2Dashen, et al 1974
Goldhaber, et al. 2003
𝑳𝑶𝟎We calculate 𝑻𝑳𝑶
𝟎𝟎 (𝒙) and 𝑻𝟏𝟏 (𝒙) in 𝑂(𝝀 )!

(Comparison with previous studies ➡ last slide)

6/19

Eigenmode of quantum fluctuation
Dashen, Hasslacher and Neveu 1974

Expanding 𝜙 4 action S 𝜙 around a kink
Substituting 𝜙(𝑥) = 𝜙𝑐𝑙 + 𝜂(𝒙)𝜂:quantum fluctuation
2211𝜕3𝜙𝜆𝑐𝑙
𝑆 𝜂 = 𝑆𝑐𝑙 + ∫ 𝑑𝑥 2 [ 𝜕0 𝜂 2 − 𝜂 − 2 − 𝑚2 +
𝜂 − 𝜆𝜙𝑐𝑙 𝜂3 − 𝜂4 ]22𝜕𝑥
𝜆42𝜕23𝜙𝑐𝑙− 2 − 𝑚2 +𝜂𝑛 = 𝜔𝑛2 𝜂𝑛𝜕𝑥
𝜆𝜔

Eigenvalues

Eigenfunctions

𝜔𝑞2 = 𝑞 2 + 2𝑚2

𝜂𝑞 𝑥

3 2𝑚2

𝜂1 (𝑥)

𝜔12 =𝝎𝟐𝟎 = 𝟎𝑥→±∞𝑥

1exp[𝑖 𝑞𝑥 ± 𝜹 𝒒 ]2Phase shift
𝜙𝑐𝑙𝑿 𝑥
𝜼𝟎 𝒙 = 𝝏𝒙 𝝓𝒄𝒍Translational mode➡IR divergence

7/19

集団座標法
ラグランジアンの書き換え
𝑿 → 𝑿(𝒕)𝐿 𝜋, 𝜙 → 𝐿[𝜋,෤ 𝜂,෤ 𝑋, 𝑃]=𝜋,෤ 𝜂:without෤translational mode
場の並進モードの自由度 →ソリトンの重心𝑋の運動の自由度⚫ 並進対称性⚫ ローレンツ対称性

Gervais, Jevicki, Sakita 1975
Goldstone and Jackiw 1975
Tomboulis 1975

計算する物理量𝟏𝝁෥ 𝒆−𝒊∫ 𝒅𝒕 𝑳[෥𝝅,෥𝜼]
෥, 𝜼𝑻 𝝂 ≡ ∫ 𝓓෥𝝅𝓓෥
𝜼𝑻 𝝂 𝝅𝒁𝝁𝐢𝐧 𝑶(𝝀𝟎 )8/19Expectation value of EMT distribution
Diagram211𝑚
2⟨𝑇00 ⟩ = 𝑇00 𝜙𝑐𝑙 + ⟨ 𝜕0 𝜂 2 ⟩ + ⟨ 𝜕1 𝜂 2 ⟩ + 𝜕1 𝜙𝑐𝑙 𝜕1 𝜂 + 𝜆𝜙𝑐𝑙 𝜙𝑐𝑙−22𝜆12−1𝜆𝑚𝑂(𝜆 2 )−2221𝑂(𝜆 )+ 3𝜙𝑐𝑙 −𝜂 + 𝑂(𝜆 )2𝜆𝜂𝑂(𝜆0 )

𝜂2 = 𝜂 𝑥 𝜂 𝑥= 𝐺 𝑥, 𝑥=𝑥~𝑂(𝜆0 )UV divergent!

𝜂 =𝑥𝜆𝜙𝑐𝑙 (𝑦)= ∫ 𝑑𝑦 𝜆𝜙𝑐𝑙 𝑦 𝐺 𝑥, 𝑦 𝐺(𝑦, 𝑦)
1~𝑂(𝜆2 )UV divergent!
9/19

EMTの正則化
全エネルギーの正則化 Dashen, et al 1974
𝑬𝐩𝐡𝐲𝐬 = 𝐸soliton − 𝐸vac + (𝛿𝐸soliton − 𝛿𝐸vac )⚫ Vacuum subtraction⚫ Mass renormalizationCounter terms

EMTの正則化⟨𝑻𝝁𝝂 ⟩𝐩𝐡𝐲𝐬 = ⟨𝑇𝜇𝜈 ⟩soliton −⟨𝑇𝜇𝜈 ⟩vac +(⟨𝛿𝑇𝜇𝜈 ⟩soliton −⟨𝛿𝑇𝜇𝜈 ⟩vac )10/19

Vacuum subtraction in 𝝓𝟒 model
Rebhan and Nieuwenhuizen 1997

⟨𝑇𝜇𝜈 ⟩soliton −⟨𝑇𝜇𝜈 ⟩vac +(⟨𝛿𝑇𝜇𝜈 ⟩soliton −⟨𝛿𝑇𝜇𝜈 ⟩vac ) = finite

Mode number cutoff (MNC)✓ 有限サイズ系、離散運動量のもとで計算✓ カットオフ数を有限に取って(有限) − (有限)に-ソリトンと真空の総モード数が同じになるようにカットオフ数を決める
𝐸𝐸Soliton continuous mode𝑵−𝟏

෍−(𝑵−𝟏)

𝑵

𝑞𝑛2 + 2𝑚2 −෍𝑛=−𝑵vacuum２N+1

𝑘𝑛2 + 2𝑚2 ２N-1
離散モード✓ 引き算の後、カットオフ数を無限に

𝐸=0soliton vacuum
反周期境界でのモードの様子
11/19

Mass renormalization
EMT regularization

Counter terms

Dashen, Hasslacher and Neveu 1974

⟨𝑇𝜇𝜈 ⟩soliton −⟨𝑇𝜇𝜈 ⟩vac +(⟨𝛿𝑇𝜇𝜈 ⟩soliton −⟨𝛿𝑇𝜇𝜈 ⟩vac ) = finite

Mass renormalization✓ Only 1-loop mass renormalization✓ Mass renormalization in vacuum sector➡Counter terms appear in soliton sector
Vacuum sector 𝑚2 → 𝑚2 + 𝛿𝑚2
𝛿𝑚2+Soliton sector

𝜆𝜙kink=0𝜆

𝛿𝑚2 𝜙kink++2
𝛿𝑚2 𝜙kink= finite= finite

12/19

Result in 𝝓𝟒 ModelIn finite system whoselength is R
3√2𝑚 1𝑅∞
𝑇00 𝑥 = −+ 𝑇00(𝑥)2𝜋 𝑅
𝑅𝑇11=∞𝑇00𝑚2=1𝜆3√2𝑚 12𝜋 𝑅◼ 𝑇00 distribution

𝑅/2𝑅lim න 𝑑𝑥 𝑇00=𝐸𝑅→∞ −𝑅/2∞𝑅න 𝑑𝑥 lim 𝑇00−∞

𝑅→∞

reproduce the 𝐸𝑀𝑁𝐶 calculated by Dashen et al.

◼ 𝑇11 distribution

𝑅𝑇11is constant➡ 𝜕0 𝑇 01 + 𝜕1 𝑇 11 = 0

EMT conservation law is satisfied

13/19

Comparison with Goldhaber et al.
Different regularization

Goldhaber, et al. 2003

Our study: finite system

Previous study: infinite system

𝟑√𝟐𝒎 𝟏𝑅∞
𝑇00 𝑥 = −+ 𝑇00(𝑥)𝟐𝝅 𝑹𝑃𝑟𝑒𝑣𝑇00(𝑥) = −𝟑√𝟐𝒎 𝟏𝑅𝑇11 = −𝟐𝝅 𝑹𝑃𝑟𝑒𝑣
𝑇11𝑥𝟑𝒎𝟐𝟏∞+𝑇(𝑥)00
𝟒𝝅 𝐜𝐨𝐬𝐡𝟐 (𝒎𝒙/√𝟐)𝟑𝒎𝟐
𝟏=−𝟒𝝅 𝐜𝐨𝐬𝐡𝟐 (𝒎𝒙/√𝟐)◆ Different result for “finite” and “infinite” regularizations∞∞𝑃𝑟𝑒𝑣𝑅◆ ∫−∞ 𝑑𝑥 𝑇00 = ∫−∞ 𝑑𝑥 𝑇00= 𝐸Dashen+◆ 𝝏𝒙 𝑻𝑷𝒓𝒆𝒗≠𝟎𝟏𝟏𝑻𝑷𝒓𝒆𝒗
is inconsistent with EMT conservation law.
𝟏𝟏

Our result is more reliable.

14/19

Sine-Gordon (SG) Model in 1+1 d𝑆 = න 𝑑𝑥 241𝑚𝜆𝜇𝜕 𝜙𝜕 𝜙 −(cos( 𝜙) − 1)2 𝜇𝜆𝑚𝑚4
𝜆𝑈=(cos( 𝜙) − 1)𝜆𝑚𝜙vacuumClassical ◼ Kink solution of EOM
𝜙kink = +4𝑚

arctan 𝑒2𝜋𝑚√𝜆𝑚(𝑥−𝑋)

𝜙kink√𝜆X: location of kink

◼ 𝑇00 under kink solution

𝐿𝑂𝑇00~𝑂(𝝀𝟎 )

𝑥~𝑚−1
𝑿

𝑥

𝑇00

2𝑚4
𝑇00 =
cosh−2 𝑚(𝑥 − 𝑋)
𝝀Weak interaction
Perturbation

1-loop 𝐸𝐿𝑂 ~𝑂(𝝀𝟎 )

𝑿

15/19

Eigenmode of quantum fluctuation
Dashen, Hasslacher and Neveu 1974

Expanding SG action S 𝜙 around a kink
Substituting 𝜙(𝑥) = 𝜙kink + 𝜂(𝒙)𝜂:quantum fluctuation
11𝜕2
𝜆22𝑆 𝜂 = 𝑆𝑐𝑙 + ∫ 𝑑𝑥 [ 𝜕0 𝜂 − 𝜂 − 2 + 𝑚 cos( 𝜙kink ) 𝜂 − ⋯ ]22𝜕𝑥𝑚2𝜔𝜕2𝜆− 2 − 𝑚2 cos( 𝜙kink ) 𝜂𝑛 = 𝜔𝑛2 𝜂𝑛𝜕𝑥𝑚Eigenvalues

𝜔𝑞2 = 𝑞 2 + 𝑚2
𝝎𝟐𝟎 = 𝟎𝑥

Eigenfunctions1
𝜂𝑞 𝑥
exp[𝑖 𝑞𝑥 ± 𝜹 𝒒 ]2Phase shift
𝜼𝟎 𝒙 = 𝝏𝒙 𝝓𝒄𝒍𝑥→±∞

離散モードが一つしか無い！！

𝜙𝑐𝑙𝑿𝑥
16/19

Vacuum subtraction in SG model
Rebhan and Nieuwenhuizen 1997
Dashen, et al 1974

⟨𝑇𝜇𝜈 ⟩soliton −⟨𝑇𝜇𝜈 ⟩vac +(⟨𝛿𝑇𝜇𝜈 ⟩soliton −⟨𝛿𝑇𝜇𝜈 ⟩vac ) = finite

Mode number cutoff (MNC)

𝝓𝟒 模型 𝐸

SG模型 𝐸𝐸

２N+1２N-1２N

無い

離散モード

𝐸=0soliton

𝐸

離散モード

２N+1𝐸=0soliton
vacuum反周期境界のモードの様子

vacuum離散モードが一つでもMNCは機能する
17/19

Result in SG modelIn finite system whose
length is R
𝑚1𝑅∞
𝑇00 𝑥 = −+ 𝑇00
(𝑥)
𝜋𝑅
𝑚1
𝑅
𝑇11 =𝜋𝑅◼ 𝑇00 distribution

∞
𝑇00

𝑅/2

𝑅lim න 𝑑𝑥 𝑇00=𝐸
𝑅→∞ −𝑅/2

𝑚2=1
𝜆∞𝑅න 𝑑𝑥 lim 𝑇00−∞

𝑅→∞

reproduce the 𝐸𝑀𝑁𝐶 calculated by Dashen et al.

◼ 𝑇11 distribution

𝑅𝑇11is constant➡ 𝜕0 𝑇 01 + 𝜕1 𝑇 11 = 0EMT conservation law is satisfied

18/19

Summary
1+1d 𝝓𝟒 模型・SG模型でEMT分布の1-loop計算
・集団座標法でIR発散を除去
・Vacuum subtraction(Mode number cutoff)と質量くりこみでUV発散を除去・得られた𝑇00 の空間積分は既知の全エネルギーを再現
・𝑇11 はEMT保存則と無矛盾
Future work:
2+1d 𝝓𝟒 model・有限温度・ブリーザー状態
19/19