3. Chiral Perturbation Theory

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3. Chiral Perturbation Theory

• 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).
• ???????? ?????? (???).
• 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 ???????????????
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3.1. QCD???????????? ???????
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? QCD???????
???? ??????
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? ?????????
????????
??????????
(vector charge)
(axial-vector charge)
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? ?????????????
?????????
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? ????
?????????? mu, md 5 - 10 MeV ?
???????? explicit ?????? ???(constituent)??????
??(uud), ???(udd)??? 1 GeV ?
?????????Mu, Md 300 MeV Mu, Md
QCD???????????????????
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? ??-???????????????????
NG boson ?????????????????? ??????????????????????
????
? Chiral Perturbation Theory
?????????????? ????????????????
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3.2. Basic Concept of the ChPT
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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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3.3. Derivative Expansion
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? Derivative expansion
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3.4. Order Counting
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? M matrix element with Ne external p lines
(Ni internal p 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 scale E a common
energy scale
? Chiral order
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? Examples of chiral order
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3.5. Lagrangian (leading order)
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? Building blocks
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p kinetic term
p mass term
p Interaction terms
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3.6. Particle Asignment and Masses at Leading
Order
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? pseudoscalar mesons
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? masses at leading order (Nf 3 ???)
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3.7. pp???????????
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? ?????????
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pp????
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3.8. Lagrangian (next order)
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0
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3.9. Background Field Method in pure Yang-Mills
Theory
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? SU(N)?????????YM Lagrangian
???

? background field ? quantum field ???
background field
quantum field
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???
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? Gauge fixing term (Feynman gauge)

? Fadeev-Popov ghost term

GF FP terms ? SU(N)???? keep ? ?????????GF
FP ??????????
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? Lagrangian
tree contribution
quantum correction at one loop
equations of motion for background fields
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?????
tree contribution
equations of motion for background fields
quantum correction at one loop
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? Feynman rules
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2???????????log ??????
? ????
? ????????
asymptotic free
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3.10. Background Field Method
ChPT ?? loop ????????
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? Background fields ??? (1)
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? Background fields ??? (2)
quantum field
background fields
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? Background fields including external gauge
fields
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? ???????
?1???2???2?????????? ? ??????????F?Fp???
?
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generator Ta ??????????????????
?
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? Feynman Rules
propagator
vertices ??
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3.11. Pion Decay Constant ????
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? tree contribution
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? 1-loop correction
?????? dimensional regularization
log??
2???
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? ????
? Pion decay constant
chiral ??(Mp0)?? decay constant
chiral???????
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3.12. Pion Decay Constant ????
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(gmn part ??)
? tree contribution
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? 1-loop correction
? ????
? Decay constants

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? QCD??????????????????????
QCD(?????????)???? ?????????????
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3.13. Renormalizat ion of Low Energy Constants
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? Renormalization in the dimensional
regularization
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3.14. Vector Form Factors and L9
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? Vector form factors
? Charge radii from ChPT
independent of L9
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? predictions experiments
PDG (2006) 0.4520.011 0.3140.035 -
0.0770.010
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3.15. p? e ?? and L10
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? p ? e ??
?
? Axial-vector form factor
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3.16. Values of low energy constants
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3.17. Chiral Anomaly and Wess-Zumino-Witten Term
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? Chiral anomaly
SU(3)L SU(3)R infinitesimal transformation
Lagrangian is invariant (classical level).
? Axial part, aR -aL ß, 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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? p0 ? ??
photon field
charge matrix
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3.18. U(1)A Anomaly and ?
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? U(1)A Anomaly
change of effective action
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? Effective Action with ?

? Anomaly
? ? ???