# On the Standard Model of Particle Physics

### Prelude

Since our discussion of the Standard Model will center around its Lagrangian, it seems prudent to remember some of the reasons we are so interested in a Lagrangian formulation in the first place. From a Lagrangian function
L(φi(x), ∂μ φi(x))
and the associated action
S = ∫ L d4x
we can obtain (see, i.e., Itzykson and Zuber, Quantum Field Theory):

• the Euler-Lagrange equations of motion, by requiring the action to remain stationary under variations in the fields
δ S / δ φi(x) = 0 →
∂L / ∂φi(x) - ∂μ (∂L / ∂(∂μ φi(x))) = 0;
• the canonical energy-momentum tensor, from requiring the action to be stationary under xμ → xμ + εμ:
Tμν = ∂L / ∂(∂μ φi(x)) (∂ν φi(x)) - L
which is conserved (∂μ Tμν = 0);
• conserved currents from Noether's Theorem, which requires the action to be stationary under field variations with respect to an internal symmetry group (defined by generators Ta):
δS (φi(x) → εa(x) Ta φi(x)) = 0
→ jμa(x) = ∂L(φ + δφ) / ∂(∂μ εa(x));
• and the accompanying conserved charges:
Qa = ∫ d3x j0a(x);
• and perhaps most importantly (at least for perturbation theory), the Feynman rules for the primitive interaction vertices: each term in the Lagrangian specifies either propagators (kinetic terms) or possible fundamental interactions between the various field components.
It is for this last reason that we will be particularly interested in examining the types of terms included in:

### The Standard Model Lagrangian

The fields in the Standard Model transform under the gauge group SU(3) ⊗ SU(2)left ⊗ U(1):

• 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.
The particle content is:

• 3 families of left-handed SU(2) lepton doublets: (e-, ν), (μ-, νμ) and (τ-, ντ)
• 6 right-handed SU(2) singlets: e-, ν, μ-, νμ, τ- and ντ (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.
Our version of the Lagrangian assumes Dirac spinors and minimal Higgs couplings, with neutrino masses:
L = F1μ ν F1μ ν + Tr F2μ ν F2μ ν + Tr F3μ ν F3μ ν +
(((-iμ - g1 Aμ) δab - g2 W μ ab) φ* b) (((iμ - g1 Aμ) δac - g2 Wμ ac) φ c) - μ2 φ* a φ a - λ (φ* a φ a)2 +

i Le* u a α γμαβ ((∂μ + i g1 Aμ) Le u a β + i Le* u a α γμαβ ( i g2 Wμ ab) Le u b β -

MLea uv (Le* u a α φ*a Le v 1 α + Le* u 1 α φT a Le v a α) -
MLeb uv (Le* u a α φa Le v 2 α + Le* u 2 α φ* T a Le v a α) +
i Qu* u r a α γμαβ ((∂μ + i g1 Aμ δrs) + i g3 Gμ rs) Qu u s a β + i Qu* u r a α γμαβ ( i g2 Wμ ab δrs) Qu u s b β -
MQua uv (Qu* u r a α φ*a Qu v r 1 α + Qu* u r 1 α φT a Qu v r a α) -
MQub uv (Qu* u r a α φa 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:

• Leu a α → ((1 - i ε1) δba - i ε2ab) Leu b α
• Quu r a α → ((1 - i ε1) δba δsr - i ε2ab δsr - i ε3rs δba) Quu s b α
• φa → ((1 - i ε1) δba - i ε2ab) φb
• Aμ → Aμ + ∂μ ε1 / g1
• Wμ ab → Wμ ab + ∂μ ε2 ab / g2 - i2, Wμ]ab
• Gμ rs → Gμ rs + ∂μ ε3 rs / g3 - i3, Gμ]rs
If one expands this Lagrangian in every index, one obtains 12,793 terms (detailed in the
appendix). Of these,

• 18 are photon kinetic terms;
• 72 are W kinetic terms;
• 270 are 3- and 4-W terms (which are isospin neutral);
• 246 are gluon kinetic terms;
• 5886 are 3- and 4-gluon terms (which are color neutral);
• 8 are Higgs kinetic terms;
• 384 are fermion kinetic terms;
• 5 are Higgs potential terms;
• 144 are Higgs/gauge terms (64 of which change isospin);
• 1440 are fermion mass terms;
• 384 are fermion/photon vertex terms, preserving isospin and color;
• 2304 are fermion/W terms (2/3 of which change isospin); and
• 1632 are quark/gluon terms (1152 of which change color).
It is probably not possible to overestimate the importance of the covariant derivative and how gauge invariance limits the types of terms which are possible. Just as in General Relativity the covariant derivative allows us to define a consistent idea of the tangent space, in the standard model it allows us to consistently define isospin and color space, and hypercharge. And it is responsible for:

• the existence of bosonic derivative couplings (labeled "momentum factor" in the table below), and the lack of such couplings for fermions; and
• the fact that all 2W, 3W and 4W vertices are isospin neutral, and that all 3- and 4-gluon vertices are color neutral (note that no 2-gluon vertices occur, because the Higgs are SU(3) scalars).

### Electroweak Symmetry Restoration

Above the electroweak symmetry breaking scale, the Higgs decouples since <φ> = 0 is then a stable classical minimum of the potential (Kolb & Turner, The Early Universe, p. 198).

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 = F1μ ν F1μ ν + Tr F2μ ν F2μ ν + Tr F3μ ν F3μ ν +
(-iμ φ* a)(iμ φa) +

i Le* a α γμαβ (∂μ + i g1 Aμ) Le a β + i Le* a α γμαβ ( i g2 Wμ ab) Le b β +

i Qu* r a α γμαβ ((∂μ + i g1 Aμ δab δrs) + i g2 Wμ ab δrs + i g3 Gμ rs δab) 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.
Let us consider this effective theory as a purely classical field theory (albeit one with scalar, spinor and vector fields). The Euler-Lagrange equations of motion (obtained for the unbarred fields by varying with respect to the barred ones) are (before gauge fixing)

• 2 φa = 0
• ((iμ - g1 Aμ) δab - g2 Wμ c2cab)) γμαβ Le b β = 0
• (iμ - g1 Aμ) δab γμαβ Le b β = 0
• (((iμ - g1 Aμ) δab - g2 Wμ c2cab)) δrs - g3 Gμ t3trs) δab) γμαβ Qu s b β = 0
• ((iμ - g1 Aμ) δrs - g3 Gμ t3trs)) γμαβ Qu s a β = 0
• 2 Aμ - ∂μν Aν + g1 γμαβ (Le* a α Le a β + Qu* r a α Qu r a β) = 0
• 2 Wμ a - ∂μν Wν a - f2 a b c (∂ν (Wμb Wνc) + Wνcν Wμb - Wνcμ Wν b) -
f2 a b c f2cd e Wμd Wνb Wν e + 1/2 γμαβ2 a bc) (Le* b α Le c β + Qu* r b α Qu r c β)
• 2 Gμ r - ∂μν Gν r - f3 r s t (∂ν (Gμs Gνt) + Gνtν Gμs - Gνtμ Gν s) -
f3 r s t f3tu v Gμu Gνs Gν v + 1/2 γμαβ3 r st) Qu* s a α Qu t a β
Note that gauge generators (τi) have been inserted explicitly since most generators have been traced over. Symmetry group structure constants are designated fi a b c.

### Aspects of Quantum Field Theory

Several aspects of Quantum Field Theory are particularly relevant to the Standard Model:

• 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

Gauge group generators and Higgs terms have been marked by function (i.e., UbarD changes isospin from "down" to "up", and RbarG changes color from green to red), and the resulting Lagrangian terms are summarized below:
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).
 marker number of terms type of terms notes Yphot2 18 photon kinetic terms Yphot 384 fermion/photon vertices preserves isospin, color Higgs 1440 fermion mass terms Higgs2 2 Higgs potential terms (μ) Higgs2 8 Higgs kinetic terms Higgs2 Yphot 16 Higgs/photon vertices preserves isospin, φ momentum factor Higgs2 Yphot2 8 Higgs/photon vertices preserves isospin Higgs2 Yphot UbarU 4 Higgs/photon/W vertices isospin neutral Higgs2 Yphot UbarD 16 Higgs/photon/W vertices changes isospin Higgs2 Yphot DbarU 16 Higgs/photon/W vertices changes isospin Higgs2 Yphot DbarD 4 Higgs/photon/W vertices isospin neutral Higgs2 UbarU 8 Higgs/W vertices isospin neutral, φ momentum factor Higgs2 UbarD 16 Higgs/W vertices changes isospin, φ momentum factor Higgs2 DbarU 16 Higgs/W vertices changes isospin, φ momentum factor Higgs2 DbarD 8 Higgs/W vertices isospin neutral, φ momentum factor Higgs2 UbarU2 4 Higgs/2 W vertices isospin neutral Higgs2 UbarD2 12 Higgs/2 W vertices isospin neutral Higgs2 DbarU2 12 Higgs/2 W vertices isospin neutral Higgs2 DbarD2 4 Higgs/2 W vertices isospin neutral subtotal 144 terms Higgs4 3 Higgs potential terms (λ) DbarU UbarD UbarU2 36 4 W vertices isospin neutral DbarD DbarU UbarD UbarU 36 4 W vertices isospin neutral DbarU2 UbarD2 18 4 W vertices isospin neutral DbarD2 DbarU UbarD 36 4 W vertices isospin neutral subtotal 126 terms DbarU UbarD UbarU 72 3 W vertices isospin neutral, W momentum factor DbarD DbarU UbarD 72 3 W vertices isospin neutral, W momentum factor subtotal 144 terms UbarU2 18 W kinetic terms DbarU UbarD 36 W kinetic terms DbarD2 18 W kinetic terms subtotal 72 terms UbarU 384 fermion/W vertices isospin neutral, preserves color UbarD 768 fermion/W vertices changes isospin, preserves color DbarU 768 fermion/W vertices changes isospin, preserves color DbarD 384 fermion/W vertices isospin neutral, preserves color subtotal 2304 terms BbarG GbarB, BbarB2 36 4 gluon vertices color neutral BbarR RbarB, BbarB2 36 4 gluon vertices color neutral BbarG GbarB, GbarG2 120 4 gluon vertices color neutral GbarR RbarG, GbarG2 120 4 gluon vertices color neutral BbarR RbarB, RbarR2 120 4 gluon vertices color neutral GbarR RbarG, RbarR2 120 4 gluon vertices color neutral BbarG GbarB, BbarB GbarG 84 4 gluon vertices color neutral BbarR RbarB, BbarB RbarR 84 4 gluon vertices color neutral GbarR RbarG, GbarG RbarR 72 4 gluon vertices color neutral BbarG GbarR RbarB, BbarB 288 4 gluon vertices color neutral BbarR RbarG GbarB, BbarB 288 4 gluon vertices color neutral BbarG GbarR RbarB, GbarG 576 4 gluon vertices color neutral BbarR RbarG GbarB, GbarG 576 4 gluon vertices color neutral BbarG GbarR RbarB, RbarR 576 4 gluon vertices color neutral BbarR RbarG GbarB, RbarR 576 4 gluon vertices color neutral BbarG2 GbarB2 18 4 gluon vertices color neutral BbarR2 RbarB2 18 4 gluon vertices color neutral GbarR2 RbarG2 18 4 gluon vertices color neutral BbarG GbarB BbarR RbarB 96 4 gluon vertices color neutral BbarG GbarB GbarR RbarG 96 4 gluon vertices color neutral BbarR RbarB GbarR RbarG 96 4 gluon vertices color neutral subtotal 4014 terms BbarG GbarB, BbarB 72 3 gluon vertices color neutral, G momentum factor BbarR RbarB, BbarB 72 3 gluon vertices color neutral, G momentum factor BbarG GbarB, GbarG 144 3 gluon vertices color neutral, G momentum factor GbarR RbarG, GbarG 144 3 gluon vertices color neutral, G momentum factor BbarR RbarB, RbarR 144 3 gluon vertices color neutral, G momentum factor GbarR RbarG, RbarR 144 3 gluon vertices color neutral, G momentum factor BbarG GbarR RbarB 576 3 gluon vertices color neutral, G momentum factor BbarR GbarB RbarG 576 3 gluon vertices color neutral, G momentum factor subtotal 1872 terms BbarB2 18 gluon kinetic terms GbarG2 60 gluon kinetic terms RbarR2 60 gluon kinetic terms BbarG GbarB 36 gluon kinetic terms BbarR RbarB 36 gluon kinetic terms GbarR RbarG 36 gluon kinetic terms subtotal 270 terms BbarB 96 quark gluon vertices preserves isospin, color neutral GbarG 192 quark gluon vertices preserves isospin, color neutral RbarR 192 quark gluon vertices preserves isospin, color neutral BbarG 192 quark gluon vertices preserves isospin, changes color BbarR 192 quark gluon vertices preserves isospin, changes color GbarB 192 quark gluon vertices preserves isospin, changes color GbarR 192 quark gluon vertices preserves isospin, changes color RbarB 192 quark gluon vertices preserves isospin, changes color RbarG 192 quark gluon vertices preserves isospin, changes color subtotal 1632 terms (none) 384 fermion kinetic terms 12793 total number of terms
(the "momentum factor" comes from the derivative of one gauge boson or one Higgs)