Quantum Field Theory Equations

Classical Field Theory

Action Functional

S[\phi] = \int d^4x \, \mathcal{L}(\phi,\, \partial_\mu\phi)

The action is the spacetime integral of the Lagrangian density ℒ. The principle of least action (δS = 0) yields the field equations of motion.

Euler-Lagrange Equation for Fields

\partial_\mu \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)} - \frac{\partial\mathcal{L}}{\partial\phi} = 0

Field equation of motion obtained by extremizing the action. Generalizes the particle E-L equation to continuous fields.

Klein-Gordon Lagrangian

\mathcal{L} = \tfrac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - \tfrac{1}{2}m^2\phi^2

Lagrangian density for a real scalar field φ of mass m. The first term is kinetic; the second is the mass term.

Klein-Gordon Equation

(\Box + m^2)\phi = 0, \qquad \Box \equiv \partial_\mu\partial^\mu

Relativistic wave equation for a massive scalar field, derived from the KG Lagrangian via the Euler-Lagrange equation.

Dirac Lagrangian

\mathcal{L} = \bar{\psi}\bigl(i\gamma^\mu\partial_\mu - m\bigr)\psi

Lagrangian for a spin-½ Dirac spinor field ψ. The Dirac adjoint is ψ̄ = ψ†γ⁰.

Noether Current

j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\delta\phi, \qquad \partial_\mu j^\mu = 0

To every continuous symmetry δφ of the action there corresponds a conserved current j^μ (Noether's theorem).

Conserved Noether Charge

Q = \int d^3x \, j^0(\mathbf{x},t)

Spatial integral of the time component of the Noether current. dQ/dt = 0 follows from current conservation.

Canonical Energy-Momentum Tensor

T^{\mu\nu} = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\partial^\nu\phi - g^{\mu\nu}\mathcal{L}

Conserved tensor (∂_μ T^{μν} = 0) arising from invariance under spacetime translations. T^{00} is the Hamiltonian density.

Canonical Quantization — Scalar Field

Canonical Momentum Density

\pi(x) = \frac{\partial\mathcal{L}}{\partial\dot{\phi}} = \dot{\phi}(x)

Canonical momentum conjugate to the scalar field φ. For the KG Lagrangian this equals the time derivative of the field.

Equal-Time Commutation Relations

[\phi(\mathbf{x},t),\, \pi(\mathbf{y},t)] = i\,\delta^{(3)}(\mathbf{x}-\mathbf{y})

The fundamental quantum condition promoting φ and π to operators. The other equal-time commutators vanish.

Mode Expansion (Scalar Field)

\phi(x) = \int \frac{d^3p}{(2\pi)^3}\,\frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \Bigl(a_{\mathbf{p}}\,e^{-ip\cdot x} + a_{\mathbf{p}}^\dagger e^{ip\cdot x}\Bigr)

Expansion of the scalar field in plane-wave modes. p·x = ω_p t − p·x in the exponent. a and a† are annihilation and creation operators.

On-Shell Dispersion Relation

\omega_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2 + m^2}

Energy of a mode with momentum p for a field of mass m. Sets p² = m² (on-shell condition).

Ladder Operator Commutator

[a_{\mathbf{p}},\, a_{\mathbf{q}}^\dagger] = (2\pi)^3\,\delta^{(3)}(\mathbf{p}-\mathbf{q})

Creation and annihilation operators satisfy canonical commutation relations. All other combinations commute.

Normal-Ordered Hamiltonian

H = \int \frac{d^3p}{(2\pi)^3}\,\omega_{\mathbf{p}}\,a_{\mathbf{p}}^\dagger a_{\mathbf{p}}

After normal ordering (:…:) to discard the infinite zero-point energy. Each a†a counts the occupation of mode p with energy ω_p.

Vacuum State

a_{\mathbf{p}}|0\rangle = 0 \quad \forall\,\mathbf{p}, \qquad \langle 0|0\rangle = 1

The Fock vacuum is annihilated by all lowering operators. Single-particle states are |p⟩ = a†_p|0⟩.

Dirac Field & Spinors

Dirac Equation

(i\gamma^\mu\partial_\mu - m)\psi = 0

First-order, relativistic wave equation for a spin-½ particle. Factorizes the Klein-Gordon equation.

Clifford Algebra

\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbf{1}

The defining relation of the gamma matrices in metric signature (+−−−). The γ^μ are 4×4 matrices.

Dirac Adjoint

\bar{\psi} = \psi^\dagger\gamma^0

Lorentz-covariant adjoint. Products of the form ψ̄ψ, ψ̄γ^μψ are Lorentz scalars and four-vectors respectively.

Feynman Slash Notation

\not{a} \equiv \gamma^\mu a_\mu, \qquad \not{p}^2 = p^2

Contraction of a four-vector a_μ with the gamma matrices. The Dirac equation becomes (i∂̸ − m)ψ = 0.

Equal-Time Anticommutation Relations

\{\psi_\alpha(\mathbf{x},t),\,\psi_\beta^\dagger(\mathbf{y},t)\} = \delta_{\alpha\beta}\,\delta^{(3)}(\mathbf{x}-\mathbf{y})

Fermi-Dirac statistics require anticommutators. The spin-statistics theorem guarantees this for half-integer spin fields.

Dirac Field Mode Expansion

\psi(x) = \int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\sum_s\!\Bigl(a_{\mathbf{p}}^s\,u^s(p)\,e^{-ip\cdot x} + b_{\mathbf{p}}^{s\dagger}v^s(p)\,e^{ip\cdot x}\Bigr)

Expansion in u-spinors (particles, operator a) and v-spinors (antiparticles, operator b†). Sum over spins s = ±½.

Spin-Sum Completeness Relations

\sum_s u^s(p)\bar{u}^s(p) = \not{p}+m, \qquad \sum_s v^s(p)\bar{v}^s(p) = \not{p}-m

Used to average or sum over unobserved spin states when computing |M|². Essential for squaring amplitudes.

Gordon Identity

\bar{u}(p^\prime)\gamma^\mu u(p) = \bar{u}(p^\prime)\!\left[\frac{(p+p^\prime)^\mu}{2m} + \frac{i\sigma^{\mu\nu}q_\nu}{2m}\right]\!u(p)

Decomposes the vector current into a charge-current term and a magnetic-moment term. q = p′ − p is the momentum transfer.

Gauge Fields & QED

Electromagnetic Field Tensor

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu

Antisymmetric tensor built from the four-potential A^μ. Encodes E and B: F^{0i} = −E^i, F^{ij} = −ε^{ijk}B_k.

Maxwell Lagrangian

\mathcal{L}_{\text{Maxwell}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}

Free electromagnetic field Lagrangian; invariant under U(1) gauge transformations A_μ → A_μ − ∂_μα.

U(1) Gauge Transformation

\psi \to e^{i\alpha(x)}\psi, \qquad A_\mu \to A_\mu - \frac{1}{e}\partial_\mu\alpha(x)

Local gauge symmetry of QED. Invariance under this transformation forces the introduction of the photon field.

QED Covariant Derivative

D_\mu = \partial_\mu + ieA_\mu

Replaces ∂_μ in the Dirac Lagrangian to enforce local gauge invariance. The coupling constant e is the electric charge.

QED Lagrangian

\mathcal{L}_{\text{QED}} = \bar{\psi}(i\not{D} - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}

Full QED Lagrangian. Expanding D̸ = ∂̸ + ieÃ generates the three-point interaction vertex −eψ̄γ^μ A_μ ψ.

Lorenz Gauge Condition

\partial_\mu A^\mu = 0

Covariant gauge-fixing condition. Does not uniquely fix the gauge but simplifies the equations of motion to □A^μ = ej^μ.

Covariant Maxwell Equations

\partial_\nu F^{\mu\nu} = e\,j^\mu, \qquad \partial_{[\lambda}F_{\mu\nu]} = 0

Inhomogeneous equation (sources ej^μ) and homogeneous Bianchi identity (automatically satisfied by F = dA).

Feynman Propagators

Scalar Propagator (position space)

D_F(x-y) = \langle 0|\,\mathcal{T}\{\phi(x)\phi(y)\}\,|0\rangle

Time-ordered two-point function of the scalar field. The fundamental building block of perturbation theory.

Scalar Propagator (momentum space)

\widetilde{D}_F(p) = \frac{i}{p^2 - m^2 + i\epsilon}

Feynman propagator in momentum space. The iε prescription selects the Feynman contour and enforces causal boundary conditions.

Fermion Propagator (momentum space)

\widetilde{S}_F(p) = \frac{i(\not{p}+m)}{p^2 - m^2 + i\epsilon}

Propagator for an internal Dirac fermion line with four-momentum p. The numerator is the spin sum ∑_s u u̅.

Photon Propagator — Feynman Gauge

\widetilde{D}_F^{\mu\nu}(k) = \frac{-i\,g^{\mu\nu}}{k^2 + i\epsilon}

Massless boson propagator in Feynman (Lorenz) gauge ξ = 1. The most common gauge choice for QED calculations.

Photon Propagator — General Rξ Gauge

\widetilde{D}_F^{\mu\nu}(k) = \frac{-i}{k^2+i\epsilon}\!\left(g^{\mu\nu} - (1-\xi)\frac{k^\mu k^\nu}{k^2}\right)

General covariant gauge family. ξ = 1: Feynman gauge; ξ = 0: Landau gauge. Physical observables are ξ-independent.

Källén-Lehmann Spectral Representation

\langle 0|\mathcal{T}\{\phi\phi\}|0\rangle = \int_0^\infty d\mu^2\,\frac{\rho(\mu^2)}{p^2 - \mu^2 + i\epsilon}

Exact non-perturbative form of the full propagator. The spectral density ρ(μ²) ≥ 0; its single-particle pole gives the field-strength renormalization Z.

Perturbation Theory & S-Matrix

S-Matrix (Time-Ordered Exponential)

S = \mathcal{T}\exp\!\left(-i\int d^4x\,\mathcal{H}_I(x)\right)

Unitary operator encoding all scattering amplitudes. ℋ_I is the interaction Hamiltonian density in the interaction picture.

Dyson Series

S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int d^4x_1\cdots d^4x_n\,\mathcal{T}\{\mathcal{H}_I(x_1)\cdots\mathcal{H}_I(x_n)\}

Perturbative expansion of S in powers of the coupling constant. The n = 0 term is the identity (no scattering); n = 1 gives first-order processes.

Scattering Amplitude Definition

\langle f|S|i\rangle = \delta_{fi} + i(2\pi)^4\delta^{(4)}(p_f - p_i)\,\mathcal{M}_{fi}

The invariant amplitude M_{fi} encodes dynamics; the delta function enforces four-momentum conservation.

Wick's Theorem

\mathcal{T}\{\phi_1\cdots\phi_n\} = {:\!\phi_1\cdots\phi_n\!:} + \text{all contractions}

Reduces time-ordered products to normal-ordered terms plus Feynman propagators (contractions). Essential for extracting Feynman diagrams.

Field Contraction

\overline{\phi(x)\,\phi(y)} = D_F(x-y)

A Wick contraction of two scalar fields equals the Feynman propagator. Analogously for fermion and photon fields.

LSZ Reduction Formula (scalar)

\langle p_1\cdots|\cdots k_m\rangle = \prod_j\!\left[i\!\int d^4x_j\,e^{ik_j\cdot x_j}(\Box_{x_j}+m^2)\right]\cdots\langle\Omega|\mathcal{T}\{\phi(x_1)\cdots\}|\Omega\rangle

Relates S-matrix elements to residues of poles in the full n-point Green's function. Each external leg amputates the propagator and puts the particle on-shell.

Feynman Rules — QED

QED Vertex Factor

-ie\gamma^\mu

One factor per QED vertex connecting two fermion lines (momenta p, p′) and one photon line (index μ). Momentum is conserved at each vertex.

Internal Fermion Propagator

\frac{i(\not{p}+m)}{p^2-m^2+i\epsilon}

One factor per internal fermion line with four-momentum p. Arrow direction follows fermion-number flow.

Internal Photon Propagator

\frac{-ig_{\mu\nu}}{k^2+i\epsilon}

One factor per internal photon line with four-momentum k (Feynman gauge). Indices μ, ν contract with adjacent vertex factors.

External Fermion Lines

u^s(p) \;(\text{in}), \quad \bar{u}^s(p) \;(\text{out})

Spinor factors for external (on-shell) fermion legs. Use v and v̄ for antifermions.

External Antifermion Lines

\bar{v}^s(p) \;(\text{in}), \quad v^s(p) \;(\text{out})

Spinor factors for external antifermion legs (e.g., positrons). Reversed convention from fermion lines.

External Photon Lines

\epsilon_\mu^{(\lambda)}(k) \;(\text{in}), \quad \epsilon_\mu^{*(\lambda)}(k) \;(\text{out})

Polarization vector for external on-shell photons with momentum k and polarization λ.

Loop Momentum Integration

\int\!\frac{d^4\ell}{(2\pi)^4}

Integrate over the unconstrained four-momentum ℓ for each independent loop. Loop integrals are generally divergent and require regularization.

Fermion Loop Sign

(-1)^{\text{\# fermion loops}}

Each closed fermion loop contributes an extra factor of −1 from the anticommuting nature of Grassmann (fermionic) fields.

Cross Sections & Decay Rates

General Decay Rate

d\Gamma = \frac{1}{2M}\,|\mathcal{M}|^2\,d\Phi_n

Differential decay rate for a particle of mass M decaying into n final-state particles. |M|² is the spin-summed, spin-averaged amplitude squared.

Lorentz-Invariant Phase Space (nLIPS)

d\Phi_n = \left(\prod_{i=1}^n \frac{d^3p_i}{(2\pi)^3\,2E_i}\right)(2\pi)^4\delta^{(4)}\!\left(P - \textstyle\sum_i p_i\right)

Lorentz-invariant measure over all final-state momenta subject to four-momentum conservation.

Two-Body Phase Space (CM frame)

\int d\Phi_2 = \frac{|\mathbf{p}_{\rm cm}|}{8\pi\sqrt{s}}

Phase space integrated over angles for 2-body final states in the CM frame with √s = total CM energy.

Differential Cross Section

d\sigma = \frac{1}{4E_1 E_2\,|v_1-v_2|}\,|\mathcal{M}|^2\,d\Phi_n

For 2 → n scattering; |v₁ − v₂| is the relative velocity of the beams. In the CM frame the flux factor = 2√s.

Mandelstam Variables

s = (p_1+p_2)^2,\quad t = (p_1-p_3)^2,\quad u = (p_1-p_4)^2

Lorentz-invariant combinations of momenta for 2 → 2 scattering (particles 1,2 in; 3,4 out).

Mandelstam Sum Rule

s + t + u = m_1^2 + m_2^2 + m_3^2 + m_4^2

Constraint relating the three Mandelstam variables. In the massless limit, s + t + u = 0.

Optical Theorem

2\,\mathrm{Im}\,\mathcal{M}(p \to p) = 2E_{\rm cm}\,|\mathbf{p}_{\rm cm}|\,\sigma_{\rm tot}

Relates the imaginary part of the forward elastic amplitude to the total cross section. Follows from unitarity of the S-matrix.

Renormalization

Superficial Degree of Divergence (QED)

D = 4 - E_\gamma - \tfrac{3}{2}E_\psi

Power counting in QED: D is the degree of UV divergence of a diagram with E_γ external photon lines and E_ψ external fermion lines.

Callan-Symanzik Equation

\left[\mu\frac{\partial}{\partial\mu} + \beta(g)\frac{\partial}{\partial g} - n\gamma\right]G^{(n)}(x_1,\ldots,x_n;\mu,g) = 0

Renormalization group equation: physical Green's functions are independent of the arbitrary scale μ. β and γ encode the running of coupling and fields.

QED Beta Function (one loop)

\beta(e) = \mu\frac{de}{d\mu} = \frac{e^3}{12\pi^2} > 0

QED is infrared-free: the coupling grows with energy scale. Leads to a Landau pole at extremely high energies.

Running Coupling (QED)

\alpha(\mu^2) = \frac{\alpha(\mu_0^2)}{1 - \dfrac{\alpha(\mu_0^2)}{3\pi}\ln\!\dfrac{\mu^2}{\mu_0^2}}

QED coupling constant at scale μ. α(m_e) ≈ 1/137; at the Z-pole α(m_Z) ≈ 1/128 — a measurable logarithmic running.

Ward-Takahashi Identity

q_\mu\,\Gamma^\mu(p,p+q) = S_F^{-1}(p+q) - S_F^{-1}(p)

QED Ward identity relating the vertex function Γ^μ to the fermion self-energy. Implies Z₁ = Z₂, protecting charge renormalization.

Dyson Resummation (Full Propagator)

\widetilde{D}(p^2) = \frac{i}{p^2 - m^2 - \Sigma(p^2) + i\epsilon}

Resums the geometric series of self-energy insertions Σ(p²). The pole of D̃ defines the physical mass and field-strength renormalization Z.

Anomalous Dimension

\gamma_\phi(\mu) = \frac{\mu}{2Z_\phi}\frac{dZ_\phi}{d\mu}

Measures the deviation of the field scaling dimension from its classical value. Non-zero γ leads to anomalous scaling in critical phenomena.

Path Integrals

Scalar Field Path Integral

Z = \int\!\mathcal{D}\phi\;\exp\!\left(i\int d^4x\,\mathcal{L}(\phi,\partial_\mu\phi)\right)

Sum over all field configurations weighted by e^{iS/ℏ}. Equals the vacuum persistence amplitude ⟨0|0⟩ in the absence of sources.

Generating Functional

Z[J] = \int\!\mathcal{D}\phi\;\exp\!\left(i\int d^4x\bigl[\mathcal{L} + J(x)\phi(x)\bigr]\right)

Source J(x) coupled linearly to φ generates all Green's functions by functional differentiation.

n-Point Green's Function

G^{(n)}(x_1,\ldots,x_n) = \frac{1}{Z[0]}\left.\frac{(-i)^n\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)}\right|_{J=0}

Full (disconnected) n-point correlator extracted by taking n functional derivatives of Z[J] and setting J = 0.

Free-Field Generating Functional

Z_0[J] = \exp\!\left(-\frac{1}{2}\int d^4x\,d^4y\;J(x)\,D_F(x-y)\,J(y)\right)

Closed-form result of the Gaussian path integral for free scalar theory. Encodes all free propagators.

Connected Green's Functions

W[J] = -i\ln Z[J], \qquad G_c^{(n)} = \frac{(-i)^{n-1}\delta^n W}{\delta J^n}\bigg|_{J=0}

W[J] generates only connected diagrams. The 2-point function from W gives the full connected propagator.

1PI Effective Action

\Gamma[\phi_c] = W[J] - \int d^4x\,J(x)\phi_c(x), \qquad \phi_c(x) = \frac{\delta W}{\delta J(x)}

Legendre transform of W[J]. Γ[φ_c] generates only one-particle-irreducible (1PI) diagrams. Its second functional derivative is the inverse full propagator.

Fermion Path Integral

Z[\eta,\bar{\eta}] = \int\!\mathcal{D}\bar{\psi}\,\mathcal{D}\psi\;\exp\!\left(i\int d^4x\bigl[\mathcal{L} + \bar{\eta}\psi + \bar{\psi}\eta\bigr]\right)

Path integral over Grassmann-valued spinor fields. η and η̄ are Grassmann-valued sources for ψ and ψ̄.

Non-Abelian Gauge Theories

Lie Algebra Structure Constants

[T^a, T^b] = if^{abc}T^c

The generators T^a of a compact Lie group obey this algebra. For SU(N): T^a = λ^a/2 (Gell-Mann matrices for N=3).

Non-Abelian Covariant Derivative

D_\mu = \partial_\mu - igT^a A_\mu^a

Minimal coupling for a matter field in representation T^a. The index a runs over the adjoint representation of the gauge group.

Non-Abelian Field Strength

F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A_\mu^b A_\nu^c

Unlike QED, F contains a nonlinear term from the gauge algebra — the origin of gluon self-interactions in QCD.

Yang-Mills Lagrangian

\mathcal{L}_{\rm YM} = -\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}

Kinetic Lagrangian for non-Abelian gauge bosons. Generates 3- and 4-point self-interaction vertices absent in QED.

QCD Lagrangian

\mathcal{L}_{\rm QCD} = -\frac{1}{4}G_{\mu\nu}^a G^{a\mu\nu} + \sum_f \bar{q}_f(i\not{D} - m_f)q_f

G^a_{μν} is the gluon field strength for SU(3)_c. Sum over quark flavors f; the covariant derivative couples quarks to gluons.

QCD Beta Function (one loop)

\beta(g_s) = -\frac{g_s^3}{16\pi^2}\!\left(\frac{11}{3}C_A - \frac{4}{3}T_F n_f\right)

β < 0 for SU(3) with n_f < 16.5 quark flavors, giving asymptotic freedom. C_A = 3, T_F = ½ for SU(3).

Running QCD Coupling (asymptotic freedom)

\alpha_s(\mu^2) = \frac{2\pi}{\beta_0\ln(\mu/\Lambda_{\rm QCD})}, \qquad \beta_0 = 11 - \frac{2n_f}{3}

α_s → 0 as μ → ∞ (asymptotic freedom). Λ_QCD ≈ 200 MeV is the QCD scale where the coupling becomes non-perturbative.

Faddeev-Popov Ghost Lagrangian

\mathcal{L}_{\rm ghost} = \bar{c}^a\bigl(-\partial^\mu D_\mu^{ab}\bigr)c^b

Anticommuting ghost fields c, c̄ must be introduced in covariant gauges to cancel unphysical gauge-boson polarizations in loop diagrams.

Spontaneous Symmetry Breaking & Higgs Mechanism

Higgs Potential (Mexican Hat)

V(\phi) = -\mu^2|\phi|^2 + \lambda|\phi|^4, \quad \mu^2 > 0,\; \lambda > 0

Double-well (or Mexican-hat for complex φ) potential. The minimum is a circle of radius v, not at the origin — the symmetry is spontaneously broken.

Vacuum Expectation Value

v = \langle|\phi|\rangle = \sqrt{\frac{\mu^2}{2\lambda}}

Non-zero VEV that breaks the U(1) symmetry. In the Standard Model, v ≈ 246 GeV sets the electroweak scale.

Goldstone's Theorem

\text{dim}\bigl(G/H\bigr) \;=\; n_{\rm Goldstone\; bosons}

For every spontaneously broken continuous symmetry generator there is a massless Nambu-Goldstone boson. G is the original symmetry group, H the residual (unbroken) group.

Higgs Boson Mass

m_H^2 = 2\mu^2 = 4\lambda v^2

Mass of the physical Higgs boson — the radial fluctuation around the vacuum. Measured at m_H ≈ 125 GeV.

W Boson Mass (Higgs Mechanism)

m_W = \frac{1}{2}gv

W± bosons acquire mass by "eating" the would-be Goldstone bosons. g is the SU(2) weak gauge coupling.

Z Boson Mass

m_Z = \frac{m_W}{\cos\theta_W} = \frac{v}{2}\sqrt{g^2 + g^{\prime 2}}

Z⁰ mass from mixing SU(2) and U(1)_Y gauge bosons. θ_W is the Weinberg mixing angle; m_Z ≈ 91 GeV.

Weinberg Angle

\sin^2\theta_W = 1 - \frac{m_W^2}{m_Z^2} \approx 0.231

Electroweak mixing angle relating gauge couplings: tan θ_W = g′/g. Its precise value is measured in electroweak precision tests.

Yukawa Coupling (Fermion Mass)

\mathcal{L}_{\rm Yukawa} = -y_f\,\bar{\psi}_L H\psi_R + \text{h.c.}, \qquad m_f = \frac{y_f v}{\sqrt{2}}

Fermion masses arise from Yukawa couplings to the Higgs doublet H after SSB. The top quark has y_t ≈ 1, while the electron has y_e ≈ 3 × 10⁻⁶.

Quantum Field Theory — Equation Sheet