- January 23 (Wednesday):
- Syllabus of the spring semester.

Loop diagrams: amputating the external leg bubbles.

Calculating a one-loop diagram: Feynman trick for denominators; Wick rotation to the Euclidean momentum space; cutting off the UV divergence; explaining the cutoff;~~physical vs. bare couplings~~. - January 28 (Monday):
- Bare and physical couplings:
The bare coupling and the cutoff; physical coupling in terms of a physical amplitude;
resumming the perturbation theory in terms of physical coupling; cutoff independence.

The ultraviolet regulators: Wilson's hard-edge cutoff; Pauli–Villars; higher-derivative regulator; covariant higher derivative for gauge theories; mentioned the dimensional regularization and the lattice. - January 30 (Wednesday):
- Dimensional regularization:
*D*<4 as a UV regulator; Gaussian and non-Gaussian integrals in non-integer dimensions; taking the*D*→4 limit.

The optical theorem: origin in the unitary**S**matrix; one-loop example. - February 4 (Monday):
- Correlation functions of quantum fields:
correlation functions; relation to the free fields and evolution operators; Feynman rules;
connected correlation functions.

The two-point correlation function: Källén–Lehmann spectral representation; features of the spectral density function; analytic two-point function F_{2}(*p*^{2}): poles and particles; pole mass = physical particle mass; branch cuts and the multi-particle continuum; physical and un-physical sheets of the Riemann surface; resonances. Begin perturbation theory for the F_{2}: re-summing the 1PI bubbles; Σ(*p*^{2}) and renormalization of the mass and of the field strength; mass renormalization in the φ^{4}theory; fine tuning problem. - February 6 (Wednesday):
- Quadratic UV divergences:
regulator dependence; dimensional regularization of quadratic divergences.

Field strength renormalization in the Yukawa theory: calculating the one-loop diagram: the trace, the denominator and the numerator, Wick rotation; the UV divergence structure: Σ(*p*^{2})=(div.constant)+(div.constant)×*p*^{2}+finite_*f*(*p*^{2}); calculating the Σ(*p*^{2}); dΣ/d*p*^{2}and the scalar field strength renormalization.

Counterterms perturbation theory: ℒ_{bare}=ℒ_{physical}+counterterms; Feynman rules for the counterterms; adjusting δ^{Z}, δ^{m}, and δ^{λ}order by order in λ; one-loop examples. - Extra lecture on February 11 (Monday):
- Lehmann–Symanzik–Zimmermann reduction formula:
relating the correlation functions
F
_{n}(*p*) to the scattering amplitudes: the amputated core and the external leg bubbles; the poles for the on-shell_{1},…p_{n}*p*_{i}^{0}→±*E*(**p**_{i}) and their relations to the asymptotic*x*→±∞ limits; the asymptotic |in⟩ and ⟨out| states and the LSZ (Lehmann–Symanzik–Zimmermann) reduction formula; scattering amplitudes and the amputated diagrams.^{0}_{i} - Regular lecture on February 11 (Monday):
- Counting the divergences:
superficial degree of divergence; graphs and subgraphs; classifying the divergences;
counterterms and canceling the divergences.

Subgraph divergences: in situ cancelation of subgraph divergences by counterterms; nested divergences; overlapping divergences; BPHZ theorem.

Divergences and renormalizability: supeficial degree of divergence in the φ^{k}theories; super-renormalizable, renormalizable, and non-renormalizable theories. - February 13 (Wednesday):
- Troubles with non-renormalizable theories.

Dimensional analysis and renormalizability: canonical dimensions of fields and couplings; power-counting renormalizability; renormalizable theories in 4D; other dimensions.

QED perturbation theory: the counterterms and the Feynman rules; the divergent amplitudes and their momentum dependence; missing counterterms. - February 18 (Monday):
- QED perturbation theory:
missing counterterms and the Ward–Takahashi identities.

Σ^{μν}(*k*) at one loop order: calculation, checking the WT identity, the divergence and the δ_{3}counterterm.

Dressed propagators in QED: the electron's F_{2}(*p̸*); the photon's F_{2}^{μν}(*k*); equations for the finite parts of the δ^{2}, δ^{m}, and δ^{3}counterterms. - February 20 (Wednesday):
- Effective coupling at high energies:
Loop corrections to Coulomb scattering; effective QED coupling α
_{eff}(*E*) and it's running with log(energy); running of other coupling types.

Ward–Takahashi identities: the identities; Lemma 1 (the tree-level two-electron amplitudes); Lemma 2 (the one-loop no-electron amplitudes); identities for the (bare) higher-loop amplitudes; Z_{1}=Z_{2}; generalizing to multiple charged fields. - Extra lecture on February 25 (Monday):
- Spontaneous Symmetry breaking:
scalar poantial, 〈Φ〉≠0, and spontaneous symmetry breaking;
complex field example; linear sigma model; Wigner and Goldstone–Nambu modes of symmetries;
Goldstone theorem; broken-symmetry currents in the linear sigma model;
~~pion scattering~~; spontaneous symmetry breaking in d≠4 dimensions. - Regular lecture on February 25 (Monday):
- Ward–Takahashi identities:
δ
_{1}=δ_{2}; in the counterterm perturbation theory; general Ward–Takahashi identities and their relation to the current conservation.

Form factors: probing nuclear and nucleon structure with electrons; the form factors; the on-shell form-factors*F*_{1}(*q*^{2}) and*F*_{2}(*q*^{2}); the gyromagnetic ratio.

Began the dressed QED vertex at one loop: the diagram and the denominator. - February 25 (Wednesday):
- Dressed QED vertex at one loop:
numerator algebra;
calculating the
*F*_{2}form factor and the anomalous magnetic moment; the experimental and the theoretical electron's and muon's magnetic moments at high precision.

The electric from factor*F*_{1}(*q*^{2}) at one loop: calculating the integrals; the infrared divergence and its regulation; momentm dependence of the IR divergence; the δ_{1}counterterm.

Began IR finiteness of the*measureable*quantities: similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung. - March 4 (Monday):
- IR finiteness of the measureable quantities:
similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung;
detectable vs. undetectable photons and the observed cross-sections;
IR finiteness of the observed cross-sections.

Briefly: higher lops and/or more soft photons; optical theorem for the finite observed cross-sections; no Fock space for QED and other gauge theories; jets in QCD.

Gauge dependence in QED: δ_{1}(ξ) and δ_{2}(ξ).

Began symmetries and counterterms. - March 6 (Wednesday):
- Finish symmetries and counterterms.

Intro to renormalization group: large log problem for*E*≫*m*; adjusting the counterterms to avoid large logs; off-shell renormalization schemes; running counterterms and running couplings.

Renormalization group basics: the anomalous dimension of a quantum field; the β function and the running coupling. - Extra lecture on March 11 (Monday):
- Higgs mechanism:
semiclassical abelian example; propagator of the massive photon; Higgs mechanism for general (abelian) SSB;
~~non-abelian examples; general gauge theory~~. - Regular lecture on March 11 (Monday):
- Renormalization group equations:
RGE and its solution for the λφ
^{4}theory; renormalization of QED: anomalous dimensions, β_{e}to one-loop order, solving the RGE for QED; β–functions for general couplings; coupled RGE foe the Yukawa theory (briefly).

Begin types of RG flows: β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom; IR price of asymptotic freedom in QCD; confinement;~~chiral symmetry breaking~~. - March 13 (Wednesday):
- Finish types of RG flows — fixed points:
β(g*)=0; UV stability vs. IR stability; scale invariance and conformal symmetry;
RG flows for multiple couplings; flow to UV versus flow to IR.

Renormalisation schemes: scheme dependence of the couplings and the β-functions; the*minimal subtraction*schemes MS and MS-bar; extracting β-functions from residues of the 1/ε poles. - March 18 and March 20:
- Spring break, no classes.
- March 25 (Monday):
- Non-abelian gauge symmetries: review of the abelian gauge theory; non-abelian covariant derivatives, matrix-valued connections, and gauge transforms; non-abelian vector fields and tension tensors; Yang–Mills theory; QCD; tree-level QCD Feynman rules.
- March 27 (Wednesday):
- Non-abelian gauge symmetries:
general gauge groups; matter fields in general multiplets;
products of gauge symmetries; the Standard Model.

Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral; derivation of the Hamiltonian form.

Gave out the midterm exam. - April 1 (Monday):
- Introduction to path integrals:
Lagrangian path integrals — derivation and normalization;
the partition function; harmonic oscillator example.

Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; free fields and propagators; perturbation theory and Feynman rules; sources and the generating functionals.

Convergence of path integrals and imaginary time; Euclidean 4D for QFT path integrals. - April 3 (Wednesday):
- Functional integrals in Euclidean spacetime:
QFT↔StatMech corresponcence; Euclidean lattice as a UV cutoff;
restoration of the SO(4)→Lorentz symmetry in the continuum limit;
coupling↔temperature analogy.

Begin fermionic functional integrals: Grassmann numbers; Berezin integrals over fermionic variables.

Collect midterm exams. - Extra lecture on April 8 (Monday):
- Glashow–Weinberg–Salam theory:
Higgsing the
SU(2) down to the_{W}×U(1)_{Y}U(1) ; the photon and the massive W_{EM}^{±}and Z^{0}vectors; the mixing angle; the weak currents; the effective Fermi theory; ρ=1 for the neutral current.

Began Fermions in the GWS theory: Quantum numbers of the quarks~~and the leptons~~; the Yukawa couplings giving rise to the masses. - Regular lecture on April 8 (Monday):
- Functional integrals for fermions:
fermionic gaussian integrals; free Dirac field with sources; Dirac propagator;
FI for fermions in EM background; Det(D̸+m) and the electron loops;
1/(D̸+m) and the tree diagrams.

Functional integral for the EM field: gauge transforms as redundancies; gauge-fixing conditions and the Faddeev–Popov determinants; Landau-gauge propagator from the functional integral; gauge-averaging and the Feynman gauge.

Briefly functuonal integrals at finite temperature: periodicity in Euclidean time, period=β; Det(−∂^{2}+m^{2}) over periodic fields; antiperiodic fermionic fields; Fadde'ev–Popov determinant and the number of photonic degrees of freedom. - April 10 (Wednesday):
- Quantizing the Yang--Mills theory:
fixing the non-abelian gauge symmetry;
Faddeev–Popov ghost fields; gauge-fixed YM Lagrangian.

QCD: Quantum Lagrangian and the Feynman rules; generalization to other gauge theories; weakened Ward identities in QCD: the qq̄→gg example; ghost pair production and unitarity.

Intro to BRST symmetry: BRST symmetry action on the fields; nilpotency of the BRST operator; BRTS invariance of the QCD Lagrangian. - Extra lecture on April 15 (Monday):
- Quarks and leptons of the Standard Model: Yukawa couplings and masses; the EM current and the weak currents (charged and neutral) in terms of the fermion fields; flavor mixing and the Cabibbo–Kobayashi–Maskawa matrix.
- Regular lecture on April 15 (Monday):
- BRTS symmetry:
the Fock space of QCD — the physical and the unphysical states;
BRST cohomology and getting rid of the unphysical states;
BRST symmetries of the amplitudes; cancellation of unphysical processes.

QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; allowed counterterms and renormalizability; Ward-like (but weaker) identities for the counterterms. - April 17 (Wednesday):
- Basic group theory: the Casimir and the Index.

QCD β–function at the one-loop order: the beta function and the counterterms; calculating the δ_{2}for quarks; calculating the δ_{1}for quarks — the QED-like loop and the non-abelian loop; calculating the δ_{3}— the quark loop, the gluon loop, the sideways loop, and the ghost loop;~~assembling the β–function~~. - April 22 (Monday):
- β–functions for QCD and other gauge theories:
assembling QCD beta function; other gauge theories; allowing for Majorana and Weyl fermions;
allowing for scalar fields; general formula.

Quark*confinement*in QCD: chromo-electric field lines and flux tubes; condensation of chromo-magnetic monopoles; flux tubes as hadronic strings; mesons and baryons; spinning hardonic string and the Regge trajectories. - April 24 (Wednesday):
- Chiral symmetry:
axial symmetry of a massless fermion; QED anomaly of the axial symmetry;
SU(N
_{f})_{L}×SU(N_{f})_{R}×U(1)_{V}exact symmetry of QCD with massless quarks; approximate SU(2)_{L}×SU(2)_{L}and SU(3)_{L}×SU(3)_{L}of real-life QCD.

Spontaneous breaking of chiral symmetry: overview of SSB and of the Goldstone symmetry; pions as pseudo–Goldstone bosons; generalize to 3 light flavors.

Linear and non-linear sigma models: linear sigma model for the SU(2)×SU(2)→SU(2) SSB; taking the λ→∞ limit to the non-linear sigma model; general NLΣM's; the SU(2)–valued non-linear field and its symmetries; vector and axial currents of the NLΣM; pions as Goldstone bosons of the NLΣM; generalize to 3 flavors. - Extra lecture on April 29 (Monday):
**CP**Violation: weak interactions break C and P but might preserve CP; neutral K mesons and their mixing; GIM mechanism; decays of K_{S}and K_{L}; imaginary part of ΔM and the CP violation; CP violation for the neutral D and B mesons.

CP violation in the Standard Model: CP action on W^{±}, on the charged currents, and on the CKM matrix; un-removable CKM phase for 3 families.- Regular lecture on April 29 (Monday):
- Non-linear sigma models and chiral symmetry breakdown:
NLΣMs in general; NLΣM for the SU(N)×SU(N)→SU(N) ΧSB;
symmetry currents of the NLΣM; quark-qntiquark condensate in QCD and the NLΣM for the condensate;
perturbation by the quark masses; U(3) NLΣM and η↔η′ mixing;
constituent quark mass and the non-relativistic quark model.

Started axial anomaly in QED: Formal Ward-like identities for the axial current; hidden momentum shift and the regularization trouble. - May 1 (Wednesday):
- Axial anomaly in QED:
limiting the anomaly to the one-loop triangle diagrams; regulation analysis for the Pauli–Villars cutoff;
⟨∂
_{μ}*J*^{μ5}_{reg}⟩= −2iM_{PV}⟨η̄γ^{5}η⟩; evaluating the triangle diagrams: relation to the PV regulator loops, numerator algebra, taking the integrals, summary; anomalies for QED with multiple fermions; decay of the neutral pion to 2 photons. - Extra lecture on May 6 (Monday):
- Neutrino masses: neutrino oscillations; Dirac and Majorana masses for the neutrinos; SM origin of neutrino masses; seesaw mechanism.
- Regular lecture on May 6 (Monday):
- Anomalous decay of the neutral pion to 2 photons.

Axial anomaly in QCD: 2-gluon and 3-gluon anomalies; anomaly cancelation for the SU(N_{f}) currents and the η/π problem.

Chiral gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; trace=0 conditions for anomaly cancelation; lepton and baryon anomalies in the Standard Model. - May 8 (Wednesday):
- Gauge anomalies: triangle anomaly in chiral QED; anomalous gauge variance of log(det(̸D));
anomaly in non-abelian chiral theories; Wess–Zumino consistency conditions;
anomaly coefficients A
^{abc}and traces over chiral fermions; rules for anomaly cancellation; checking anomaly cancellation for the Standard Model; relations between quark and lepton charges; anomaly cancelation in general chiral theories; massive fermions do not affect the anomalies; cubic Casimirs and cubic anomaly indices for simple gauge groups; applications to Grand Unification.

Gave out the final exam.

Last Modified: May 8, 2019. Vadim Kaplunovsky

vadim@physics.utexas.edu