On the Standard Model of Particle Physics

The Standard Model Lagrangian

• This U(1) is hypercharge; there is also a global U(1) symmetry associated with lepton number conservation.
• The spontaneously broken SU(2) ⊗ U(1) results in a Higgs doublet, three massive flavor-changing SU(2) gauge bosons and the massless photon, associated with electromagnetic U(1). These gauge bosons are linear combinations of the 4-vector isospin-changing W field described below.
• The eight gluons (G, described below), are massless; SU(3) is an unbroken symmetry.

(See Huang, Quarks, Leptons and Gauge Fields, for more detail.)

The gauge groups are topologically equivalent to S¹, S³ and an S³ bundle over S⁵.

• 3 families of left-handed SU(2) lepton doublets: (e^-, ν), (μ^-, ν_μ) and (τ^-, ν_τ)
• 6 right-handed SU(2) singlets: e^-, ν, μ^-, ν_μ, τ^- and ν_τ (because neutrino mixing has been observed, we assume massive neutrinos; therefore right-handed neutrinos are included here; however, they have never been observed)
• 3 families of left-handed quarks (u, d), (c, s) and (b, t), which transform under the full gauge group
• 6 right-handed SU(2) singlets which transform under SU(3) ⊗ U(1): u, d, c, s, b, t
• one scalar Higgs SU(2) doublet: (φ₊, φ₀)
• vector gauge bosons A (U(1)), W (SU(2)) and G (SU(3))

In all cases, antiparticles have identical mass, but opposite electric charge and helicity.

L = F₁^{μ ν} F_{1μ ν} + Tr F₂^{μ ν} F_{2μ ν} + Tr F₃^{μ ν} F_{3μ ν} +

(((-i ∂^μ - g₁ A^μ) δ^a_b - g₂ W ^{μ a}_b) φ^{* b}) (((i ∂_μ - g₁ A_μ) δ_a^c - g₂ W_{μ a}^c) φ _c) - μ² φ^{* a} φ _a - λ (φ^{* a} φ _a)² +

i Le^{* u a α} γ^μ_α^β ((∂_μ + i g₁ A_μ) Le _{u a β} + i Le^{* u a α} γ^μ_α^β ( i g₂ W_{μ a}^b) Le _{u b β} -

M_{Lea u}^v (Le^{* u a α} φ^*_a Le _{v 1 α} + Le^{* u 1 α} φ^{T a} Le _{v a α}) -
M_{Leb u}^v (Le^{* u a α} φ_a Le _{v 2 α} + Le^{* u 2 α} φ^{* T} _a Le _{v a α}) +

i Qu^{* u r a α} γ^μ_α^β ((∂_μ + i g₁ A_μ δ_r^s) + i g₃ G_{μ r}^s) Qu _{u s a β} + i Qu^{* u r a α} γ^μ_α^β ( i g₂ W_{μ a}^b δ_r^s) Qu _{u s b β} -

M_{Qua u}^v (Qu^{* u r a α} φ^*_a Qu _{v r 1 α} + Qu^{* u r 1 α} φ^{T a} Qu _{v r a α}) -
M_{Qub u}^v (Qu^{* u r a α} φ_a Qu _{v r 2 α} + Qu^{* u r 2 α} φ^{* T} _a Qu _{v r a α})

Our notation is defined as follows:

μ and ν are spacetime indices	"Tr" denotes the trace over generator indices
α and β are spinor indices	"*" denotes the complex conjugate
a, b and c are SU(2) isospin indices	a superscript "T" denotes the transpose
r and s are SU(3) color indices	blue denotes the left-hand projection of a fermion field
u and v are family indices (assuming 3 families each of leptons and quarks)	red denotes the right-hand projection of a fermion field
Le represents the lepton isospin doublet	λ (> 0) and μ (imaginary) parameterize the Higgs potential
Qu represents the (colored) quark isospin doublet	MLea, MLeb, MQua and MQub are each 3 x 3 mass/isospin mixing matrices
φ represents the Higgs isospin doublet (φ+, φ0)	g1, g2 and g3 are the coupling constants for U(1), SU(2) and SU(3), respectively
φ* represents the conjugate Higgs isospin doublet (φ0, - φ*+)
A is the U(1) hypercharge gauge boson	F1μ &nu = ∂μ Aν - ∂ν Aμ
W is the SU(2) isospin gauge boson	F2μ &nu = ∂μ Wν - ∂ν Wμ - g2 [Wμ, Wν]
G is the SU(3) color gauge boson	F3μ &nu = ∂μ Gν - ∂ν Gμ - g3 [Gμ, Gν]

Note that g₁, g₂ and g₃ are functions of the distance scale; at large scales, g₃ > g₁ > g₂; at small scales, g₁ grows while g₃ gets smaller. At very small scales, the three couplings seem to converge.

This Lagrangian is symmetric under the following local infinitesimal transformations:

• Le^{u a α} → ((1 - i ε₁) δ_b^a - i ε₂^a_b) Le^{u b α}
• Qu^{u r a α} → ((1 - i ε₁) δ_b^a δ_s^r - i ε₂^a_b δ_s^r - i ε₃^r_s δ_b^a) Qu^{u s b α}
• φ^a → ((1 - i ε₁) δ_b^a - i ε₂^a_b) φ^b
• A_μ → A_μ + ∂_μ ε₁ / g₁
• W_{μ a}^b → W_{μ a}^b + ∂_μ ε_{2 a}^b / g₂ - i [ε₂, W_μ]_a^b
• G_{μ r}^s → G_{μ r}^s + ∂_μ ε_{3 r}^s / g₃ - i [ε₃, G_μ]_r^s

Electroweak Symmetry Restoration

In that regime:

• all elementary fermions and gauge fields are massless, and therefore move with velocity c;

  (composite particles still can have dynamically-generated mass due to their mutual interactions)

• fermion isospin families become degenerate and therefore can be treated as identical;
• the remaining spacetime conserved quantities are 4-momentum and helicity;
• assuming that the electroweak scale is around 1 TeV, this corresponds cosmologically to z > 10¹⁶; or alternatively, during late stage stellar collapse when energy densities are in excess of neutron star densities, i.e., > ≈ 10¹⁹ J/m³;
• the only free constants are now the U(1), SU(2) and SU(3) couplings.

We would like to consider an "unbroken standard model" Lagrangian as an effective field theory obtained when the mean energy rises above μ⁴/λ (the local maximum at φ=0). Here is that Lagrangian:

L = F₁^{μ ν} F_{1μ ν} + Tr F₂^{μ ν} F_{2μ ν} + Tr F₃^{μ ν} F_{3μ ν} +

(-i ∂^μ φ^{* a})(i ∂_μ φ_a) +

i Le^{* a α} γ^μ_α^β (∂_μ + i g₁ A_μ) Le _{a β} + i Le^{* a α} γ^μ_α^β ( i g₂ W_{μ a}^b) Le _{b β} +

i Qu^{* r a α} γ^μ_α^β ((∂_μ + i g₁ A_μ δ_a^b δ_r^s) + i g₂ W_{μ a}^b δ_r^s + i g₃ G_{μ r}^s δ_a^b) Qu _{s b β}

All Higgs terms have gone to zero except the kinetic terms, and all fermion terms are reduced in number by 2/3 due to family degeneracy.

• ∂² φ_a = 0
• ((i ∂_μ - g₁ A_μ) δ_a^b - g₂ W_{μ c} (τ₂^c_a^b)) γ^μ_α^β Le _{b β} = 0
• (i ∂_μ - g₁ A_μ) δ_a^b γ^μ_α^β Le _{b β} = 0
• (((i ∂_μ - g₁ A_μ) δ_a^b - g₂ W_{μ c} (τ₂^c_a^b)) δ_r^s - g₃ G_{μ t} (τ₃^t_r^s) δ_a^b) γ^μ_α^β Qu _{s b β} = 0
• ((i ∂_μ - g₁ A_μ) δ_r^s - g₃ G_{μ t} (τ₃^t_r^s)) γ^μ_α^β Qu _{s a β} = 0
• ∂² A_μ - ∂_μ∂^ν A_ν + g₁ γ^μ_α^β (Le^{* a α} Le _{a β} + Qu^{* r a α} Qu _{r a β}) = 0
• ∂² W_{μ a} - ∂_μ∂^ν W_{ν a} - f_{2 a b c} (∂^ν (W_μ^b W_ν^c) + W_ν^c ∂^ν W_μ^b - W_ν^c ∂_μ W^{ν b}) -

  f_{2 a b c} f₂^c_{d e} W_μ^d W_ν^b W^{ν e} + 1/2 γ^μ_α^β (τ_{2 a b}^c) (Le^{* b α} Le _{c β} + Qu^{* r b α} Qu _{r c β})

• ∂² G_{μ r} - ∂_μ∂^ν G_{ν r} - f_{3 r s t} (∂^ν (G_μ^s G_ν^t) + G_ν^t ∂^ν G_μ^s - G_ν^t ∂_μ G^{ν s}) -

  f_{3 r s t} f₃^t_{u v} G_μ^u G_ν^s G^{ν v} + 1/2 γ^μ_α^β (τ_{3 r s}^t) Qu^{* s a α} Qu _{t a β}

Note that gauge generators (τ_i) have been inserted explicitly since most generators have been traced over. Symmetry group structure constants are designated f_{i a b c}.

Aspects of Quantum Field Theory

• All interactions take place in a Lorentz-invariant spacetime; curved backgrounds can be accommodated if a complete set of mode expansions exist.
• Before and after an interaction, the particles are essentially isolated; this is required for the computation of scattering amplitudes, which require well-defined "in" and "out" states.
• Particles are quantum excitations of fields; each field (electron, muon, etc.) is defined on the entire spacetime, so in principle non-local interactions are possible.
• Particles can have both continuous and discrete quantum numbers:

  • Continuous quantum numbers include mass, momentum and angular momentum (by virtue of the Poincare group of which the particles are representations).

    As a scalar, mass is frame-independent; the momentum (including energy) and angular momentum are frame-dependent; an observer which is accelerated relative to a Lorentz vacuum will observe a thermal bath of particles.

  • Discrete quantum numbers include spin and helicity (again due to the Poincare group); helicity (handedness) can change via Lorentz transformations, but only for massive particles.
  • Discrete quantum numbers also include hypercharge, isospin and color, by virtue of the internal symmetry groups of which the particles are representations (U(1), SU(2)_left and SU(3), respectively).

• Particles can be thought of as either having a definite location or a definite momentum, but not both simultaneously; this arises from the representation of fields in terms of Fourier conjugate variables.

  Similarly, a gauge boson cannot have both a definite angular momentum and a definite polarization, because angular momentum implies a rotating polarization.

  Representation as Fourier series in a Lorentz-invariant background has another consequence. "C" symmetry interchanges positive and negative energy modes; "P" symmetry is spatial inversion, and "T" symmetry is time reversal. U(1) and SU(3) interactions conserve C, P and T separately, while SU(2) conserves C, but can violate P and T. CPT is always conserved.

• Particles are indistinguishable, and pairs can behave either symmetrically or antisymmetrically; particles with antisymmetric behavior are described by spinor fields, and particles that behave symmetrically are described by either scalar or vector fields.
• It is in principle possible to calculate the probability of a specific interaction taking place.

  • The probability amplitude for a given process is the sum of every topologically distinct elementary process which obeys the relevant conservation laws.
  • The probability is a function of the product of the fields describing the interacting particles.
  • Gauge degrees of freedom must be fixed before computing the probability.

Appendix

Note that all index summation was carried out explicitly before counting terms (the Pauli and Gell-Mann matrices were used as SU(2) and SU(3) generators, respectively, and the Dirac representation was used for the γ matrices).

©2018, Kenneth R. Koehler. All Rights Reserved.