FR02
Gauge fields and effective theories
A low-energy description can predict the influence of particles too heavy to produce. To use it well, first learn which changes merely relabel a state, then calculate what a heavy field leaves behind and how omitted terms affect a prediction.
Choose a starting point
The first chapter starts with complex arrows. Chapters 2–3 move into fields, covariant derivatives and the Higgs mechanism. Chapters 4–6 follow one mediator model through matching, inference and published benchmarks. The anomaly chapter is a separate branch with its own prerequisites.
Use the textbook plan and prerequisite route to find the background skills a chapter needs. You can start once its own prerequisites are met. Textbook plan · FR01
L1 · 1
Can changing local phase scales change a particle’s motion?
Derive link variables and Wilson loops on four sites, then distinguish a change of gauge from a change of magnetic flux.
L2 · 2
Why does an ordinary derivative miss the same physical state?
Derive the covariant derivative, then use charge conservation to connect gauge fixing with Ward identities.
L2 · 3
Why can the comparison rule interact with itself?
Derive Yang–Mills self-interactions from noncommuting matrices, then calculate vector masses in the Abelian Higgs model.
L3 · 4
How do you calculate the influence of a particle you cannot produce?
Match a heavy-scalar model at tree level, derive successive low-energy terms, and calculate their exact truncation errors.
L3 · 5
Why can a new operator basis appear to tighten a constraint?
Transform coefficients, predictions and covariance together, then diagnose false changes of physical uncertainty.
L4 · 6
Does more precise data make a low-energy theory more reliable?
Recalculate published truncation, operator-running and forward-positivity benchmarks to separate precision from theory accuracy.
L4 · 7
Does a level crossing zero prove a quantum anomaly?
Recalculate a one-dimensional Wilson-fermion spectrum and distinguish doubling, explicit chiral breaking, spectral flow and a continuum anomaly.
Four questions you can manipulate
- Change local phases and compare one closed loop
- Raise the energy until a low-energy series loses accuracy
- Shear coefficient space and follow a physical prediction
- Compare low-energy levels and the lattice regulator
Existing field-mode and partial-wave experiments provide optional context for modes and scattering. The derivations here include independent tables and examples. Field modes · Partial-wave scattering
Sources and reproduced scope
Derivations, tables and the plot are authored for this course. The reading locations below separate source results from our teaching extensions. Numerical examples are model references or labeled synthetic observations. No experimental dataset or source figure is copied.
- David Tong, Gauge Theory (2018 lecture notes)
Sections 2.1–2.1.3 (connections and Wilson loops), 2.7.3 (Higgs phase), 3.1.1–3.2.2 (two-dimensional anomaly and Ward identities), 4.2.1 (link endpoints) and 4.3.1–4.3.3 (lattice fermions). Printed pages 26–37, 105–108, 122–139, 207–210, 220–228.
- Aneesh V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863v1 (2018)
Sections 3.5 and 4 (integrating out and power counting), 5.5–5.7 (matching), 5.10.1, PDF equations (5.51)–(5.60) (QCD running and operator mixing), and 6.2–6.3 (field redefinitions). The scalar-channel examples are derived for this course. The known-log boundary expansion checks truncation of the published running factors; it does not calculate finite hard matching.
- R. J. Furnstahl, N. Klco, D. R. Phillips and S. Wesolowski, Quantifying truncation errors in effective field theory, arXiv:1506.01343v1; Physical Review C 92, 024005 (2015)
Equations (1)–(2) define the expansion and remainder. Section II and Table I specify coefficient priors; equation (29) retains several omitted terms. The present numerical benchmark checks these definitions with a geometric series, not the paper’s neutron–proton calculations.
- Kazuo Fujikawa, Characteristics of chiral anomaly in view of various applications, Physical Review D 97, 016018 (2018), DOI 10.1103/PhysRevD.97.016018
Section III A, equations (45)–(50), PDF page 10, defines the Wilson regulator and doubling spectrum; section III B discusses finite-spacing spectral flow. The tunable mass and finite antiperiodic ring are our stated extensions. The publisher licenses the paper under CC BY 4.0; no original figure is copied.
- A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, Analyticity and an IR Obstruction to UV Completion, arXiv:hep-th/0602178v2; JHEP 10 (2006) 014
Section 4, equations (19)–(23), printed pages 14–16: a tree-level linear-sigma-model amplitude and its forward poles. We reproduce equation (20) at zero momentum transfer and its small-energy coefficient. Figures 5–6 are reading references; no source artwork is copied.