Theory of Elementary Particles

Objectives

The main theme of the course is non-Abelian gauge theory in 3+1 dimensions.
Yang-Mills action, BRST quantization, renormalization, renormalization group (beta-function),
anomalies, theta angle, instantons, solitons, will be discussed.

Special Announcement:
There will be an additional lecture on July 29.

Hours and Rooms: Mondays 13:00-14:45 at Room 285 of Rigakubu-1-Go-Kan

Instructor:

Kentaro Hori
Kavli IPMU, Kashiwa Campus
kentaro.hori _at_ ipmu.jp

Course Plan:

Non-Abelian gauge theory: the classical theory

Quantization

Renormalization and Beta function

Anomalies

Theta Vacua

Yang-Mills instantons

Solitons

References:

Homepage of QFTII 2023-2024
Peskin and Schroeder, An Introduction to Quantum Field Theory
Weinberg, The Quantum Theory of Fields, I & II
Coleman, Quantum Field Theory: Lectures of Sidney Coleman (book), Notes from Sidney Coleman's Physics 253a (arXiv)
Kugo, Geijiba no Ryoushiron, I & II (J = in Japanese)

Grading Scheme: Reports.
The report problem will be announced soon. The deadline for submission will be August 2, 23:59.

Progress:

Lecture 1 (April 8): Conventions; Yang-Mills theory, Coupling to matter field: scalars, fermions.
Lecture 2 (April 15): Symmetry and Ward identity; Quantization of gauge theories.
Lecture 3 (April 22): Introduction to perturabation theory: Feynman diagrams, connected vs disconnected.
Lecture 4 (May 7): More on diagrams, 1PI effectiove action, Ward identity for 1PI effective action, Ward identities in gauge theories.
Lecture 5 (May 20): Regularization and renormalization.
Lecture 6 (revised) (May 27): Renormalizability of gauge theories.
Lecture 7 (revised) (June 3): Renormalization group; Renormalization group flow of 4d non-Abelian gauge theories (preparation).
Lecture 8 (June 10): Renormalization group flow of 4d non-Abelian gauge theories: computation of the one-loop diagrams.
Lecture 9 (June 17): Renormalization group flow of 4d non-Abelian gauge theories: the one-loop beta function of the gauge coupling constant.
Lecture 10 (June 24): Anomaly: statement and computation. (A correction to the lecture is included.)

Additional notes:

For Lecture 1: Some math exercises
For Lecture 2: Path integrals ((a) examples, (b) symmetry-twisted partition functions, (c) Ward identities in a general dimensions)
                        Fermionic path-integrals ((a) amplitudes and partition functions, (b) multiple pairs of fermions, (c) ghost systems)
                        Right and left actions; Hermiticity of gauge fixed system
                        Quantization of gauge theories in operator formalism (symplectic view)
For Lecture 3: (a) Free theories, (b) Perturbative expansion is an asymptotic expansion, (c) Decomposition to connected parts.
For Lecture 4: (a) Proof of the properties of the 1PI effective action.
For Lecture 5: (a) Computation of one loop integrals.
For Lecture 6: (a) Possible form of divergent variation (some math), (b) New counter terms at the N-th loop.
For Lecture 7: (a) Constraints on the coefficients of 1PI effective action.
For Lecture 8: (a) Evaluation of the one-loop diagrams.
For Lecture 10: (a) Axial anomaly as a chiral anomaly.

Some useful formulae.
Notes on Lie groups and Lie algebras: The standard normalization of Yang-Mills action is explained inside.

Plan:

April 8: Conventions; Yang-Mills theory, Coupling to matter field: scalars, fermions; Quantization of gauge theories.
April 15: Symmetry and Ward identity; Quantization of gauge theories.
April 22: Introduction to perturabation theory.
May 7 (Tuesday!): 1PI effectiove action, Zinn-Justin equation.
May 13: No lecture due to an event of the School of Science.
May 20: Regularization and renormalization.
May 27: Renormalizability of gauge theories.
May 31 (Friday): No lecture.
June 3: Renormalization group.
June 10: Renormalization group of 4d non-Abelian gauge theories: computation of the one-loop diagrams.
June 17: Renormalization group of 4d non-Abelian gauge theories: the one-loop beta function of the gauge coupling constant.
June 24: Anomaly: statement and computation.
July 1: Anomaly: computation and general structure ....
July 29: An additional lecture.