3. Chiral Perturbation Theory
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Transcript of 3. Chiral Perturbation Theory
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•S.Weinberg, Physica 96A, 327 (1979).•J.Gasser and H.Leutwyler, Ann. of Phys. 158, 142 (1984).•J.Gasser and H.Leutwyler, Nucl. Phys. B 250, 465, (1985).•A.Pich, lecture note for Les Houches Summer School (hep-ph/9806303).•A.Manohar and H.Georgi, Nucl. Phys. B 234, 189 (1984).•“ ゲージ場の量子論” 九後汰一郎著 (培風館) .•C.Rosenzweig, J.Schechter and C.G.Trahern, Phys. Rev. D 21, (1980).•P.Di Vecchia and G.Veneziano, Nucl. Phys. B 171, 253 (1980).•E.Witten, Ann. of Phys. 128, 363 (1980).
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☆ QCD のパラメータの極限・対称性の情報
摂動的 QCD ( 漸近的自由性 )
(ミクロな領域 ; 高エネルギー領域)
ヘビークォーク対称性
カイラル対称性
ハドロンによる量子補正が小さくなる
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☆ QCDラグランジアン
カレント・クォーク質量
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☆ カレントと保存電荷
; ベクトル・カレント; 軸性ベクトル・カレント
(vector charge) (axial-vector charge)
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☆ カイラル対称性の自発的破れ・オーダー・パラメータ
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☆ 現実世界・カレント・クォーク質量 ; mu, md ・・・ 5 - 10 MeV ⇒ カイラル対称性は explicit に破れている・構成子 (constituent) クォーク質量 陽子 (uud), 中性子 (udd) の質量 ・・・ 1 GeV → ハドロンの中では、Mu, Md ・・・ 300 MeV Mu, Md ; QCD の強い相互作用により生成された有効質量
◎ QCDラグランジアンでの近似的カイラル対称性
◎・ π 中間子 ・・・ 近似的 南部 - ゴールドストーン粒子
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◎ 南部 - ゴールドストーン定理の低エネルギー定理
NG boson の低エネルギー極限における散乱振幅は対称性の要求から、力学系の詳細によらずに決定される。
◎ Chiral Perturbation Theory
低エネルギー定理の系統的記述高いエネルギー領域への組織的拡張
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☆ Generating Functional of QCD
・ current quark masses ・・・ VEV of S
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☆ Basic Concept of the ChPT
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☆ Derivative expansion
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☆ M ・・・ matrix element with Ne external π lines
(Ni internal π lines and NL loops)
・ general form of an interaction
・・・ dimension carried by the coupling constants
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☆ General expression of the matrix element
μ : a common renormalization scaleE : a common energy scale
☆ Chiral order
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☆ Examples of chiral order
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☆ Building blocks
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π kinetic term π mass term
π Interaction terms
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◎ pseudoscalar mesons
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☆ masses at leading order (Nf = 3 の場合 )
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☆ ラグランジアン ・・・ 低エネルギー極限では D = 2 のみ寄与
◎ ファインマンルール
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ππ 散乱振幅
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・ Eq. of motion
・ trivial relation = O(p4)
= 0
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=0
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・ Eq. of motion
・ trivial relation = O(p4)
= 0
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☆ SU(N) 局所対称性に基づく YM Lagrangian
background field
quantum field
◎ background field と quantum field に分離
変換性 : ;
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・変換性
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◎ Gauge fixing term (Feynman gauge)
◎ Fadeev-Popov ghost term
;
;
GF + FP terms は SU(N) 対称性を keep ⇔ 普通の量子化では、 GF + FP がゲージ対称性を破る
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tree contribution
quantum correction at one loop
◎ Lagrangian
equations of motion for background fields
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・ 具体的計算
tree contribution
quantum correction at one loop
equations of motion for background fields
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☆ Feynman rules
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・2次発散はキャンセルし、 log 発散のみ存在
◎ くりこみ
◎ くりこみ群方程式
asymptotic free
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ChPT での loop 計算のための準備
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☆ Background fields の導入 (1)
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☆ Background fields の導入 (2)quantum field
background fields
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☆ Background fields including external gauge fields
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☆ ラグランジアン
tree contribution
quantum correction at one loop
equations of motion for background fields
第1項と第2項への2次発散の補正が異なる ⇒ くりこみを行った後、 Fχ=Fπ とおく
⇒
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・ generator Ta は質量が対角化されるようにとっておく
例 :
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☆ Feynman Rules
・ propagator
・ vertices の例
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◎ tree contribution
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◎ 1-loop correction
・積分の正則化 ・・・ dimensional regularization
log 発散2次発散
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◎ くりこみ
◎ Pion decay constant
chiral 極限 (Mπ=0) での decay constant
chiral 極限からのズレ
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◎ tree contribution
(g part のみ )
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◎ 1-loop correction
◎ くりこみ
◎ Decay constants
・・・
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注: QCDのテストというよりは、カイラル対称性のテスト
・ QCD(カイラル対称性以外)のテスト・ もしくは、計算手法のテスト
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☆ Renormalization in the dimensional regularization
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3.14. Vector Form Factors
and L9
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◎ Charge radii from ChPT
☆ Vector form factors
・・・ independent of L9
π, K
π, K
π
W
K
e
ν
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◎ predictions & experiments
PDG (2006) 0.452±0.011 ; 0.314±0.035 ; - 0.077±0.010
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3.15. π→ e νγ and L10
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☆ π → e νγγ
W
π
e
ν ⇔
independent of μ
◎ Axial-vector form factor
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3.16. Values of low energy constants
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☆ Chiral anomaly
SU(3)L × SU(3)R infinitesimal transformation
Lagrangian is invariant (classical level).
◎ Axial part, αR = - αL = β, is broken at quantum level.
+
change of effective action
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☆ Wess-Zumino-Witten Lagrangian
◎ Wess-Zumino anomaly equation for effective Lagrangian of NG bosons
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☆ Solution for Wess-Zumino anomaly equation
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☆ π0 → γγ
;photon field charge matrix
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3.18. U(1)A Anomaly and η’
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+
☆ U(1)A Anomaly
change of effective action
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☆ Effective Action with η’
・・・
◎ Anomaly
◎ η’ の質量