Gauge Structure of the Standard Model (2024)

11.1 Introduction

In this chapter we make an overview of the gauge structure of the standard model (SM), with the purpose of establishing notation and writing down the Feynman rules for all vertices and propagators in a general ’t Hooft gauge. The part of the SM Lagrangian involving the fermions will be studied in the next chapter.

This chapter does not intend to be either pedagogical or self-contained. For a thorough derivation of the SM Lagrangian, the reader is advised to consult one of the texts existent on the subject—for instance Abers and Lee (1973); Fritzsch and Minkowski (1981); Leader and Predazzi (1982); Cheng and Li (1994). Some readers may prefer to skip this chapter altogether.

11.2 SU(2)

The Glashow–Weinberg–Salam (Glashow 1961; Weinberg 1967; Salam 1968) model for the electroweak interactions, termed the standard model, has gauge group $SU (2) \otimes U (1)$ ⁠. The generators of SU(2) are denoted $T_{1}, T_{2}$ ⁠, and $T_{3}$ ⁠. They are Hermitian and obey the commutation relations

$[T_{k}, T_{l}] = i ϵ_{klm} T_{m} .$

(11.1)

One defines

$T_{\pm} \equiv \frac{T_{1} \pm {iT}_{2}}{\sqrt{2}} .$

(11.2)

Then, $T_{-} = T_{+}^{†}$ and the commutation relations are

$\begin{array}{l} [T_{+}, T_{-}] & = T_{3}, \\ [T_{3}, T_{\pm}] & = \pm T_{\pm} . \end{array}$

(11.3)

In the doublet representation, the $T_{k}$ are represented by the Pauli matrices $τ_{k}$ divided by 2, i.e.,

$T_{1}^{(2)} = \frac{1}{2} (\begin{array}{l} 0 & 1 \\ 1 & 0 \end{array}), T_{2}^{(2)} = \frac{1}{2} (\begin{array}{c} 0 & - i \\ i & 0 \end{array}), T_{3}^{(2)} = \frac{1}{2} (\begin{array}{c} 1 & 0 \\ 0 & - 1 \end{array}) .$

(11.4)

Then,

$T_{+}^{(2)} = \frac{1}{\sqrt{2}} (\begin{array}{l} 0 & 1 \\ 0 & 0 \end{array}), T_{-}^{(2)} = \frac{1}{\sqrt{2}} (\begin{array}{l} 0 & 0 \\ 1 & 0 \end{array}) .$

(11.5)

In the triplet representation,

$T_{1}^{(3)} = \frac{1}{\sqrt{2}} (\begin{array}{l} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}), T_{2}^{(3)} = \frac{1}{\sqrt{2}} (\begin{array}{c} 0 & - i & 0 \\ i & 0 & - i \\ 0 & i & 0 \end{array}), T_{3}^{(3)} = (\begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{array}) .$

(11.6)

Then,

$T_{+}^{(3)} = (\begin{array}{l} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}), T_{-}^{(3)} = (\begin{array}{l} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}) .$

(11.7)

11.3 Covariant derivative

The covariant derivative is

$\partial^{μ} - ig (W_{1}^{μ} T_{1} + W_{2}^{μ} T_{2} + W_{3}^{μ} T_{3}) - i g' B^{μ} Y .$

(11.8)

Here, g denotes the SU(2) coupling constant and $g'$ is the U(1) coupling constant. The U(1) charge Y is termed (weak) hypercharge; it is, for each irreducible representation of $SU (2) \otimes U (1)$ ⁠, a real multiple of the unit matrix. In our normalization, the electric charge Q is given by $Q = T_{3} + Y$ ⁠. The three $W_{k}^{μ}$ are the gauge fields of SU(2), while $B^{μ}$ is the U(1) gauge field. We shall often omit the Lorentz indices on the gauge fields and on the derivatives, in order not to overload the notation.

One defines

$W^{\pm} \equiv \frac{W_{1} \mp {iW}_{2}}{\sqrt{2}} .$

(11.9)

Instead of g and $g'$ it is useful to introduce e (the electric-charge unit) and the angle $θ_{w}$ defined through

$\begin{array}{l} g & = \frac{e}{s_{w}}, \\ g' & = - \frac{e}{c_{w}}, \end{array}$

11.4 Self-interactions of the gauge bosons

Interactions among the gauge bosons are typical of a non-abelian gauge theory. In the SM, they arise because of the presence of the non-abelian SU(2) in the gauge group. If we denote

$\begin{array}{l} F_{1}^{μ ν} \equiv \partial^{μ} W_{1}^{ν} - \partial^{ν} W_{1}^{μ} + g (W_{2}^{μ} W_{3}^{ν} - W_{3}^{μ} W_{2}^{ν}), \\ F_{2}^{μ ν} \equiv \partial^{μ} W_{2}^{ν} - \partial^{ν} W_{2}^{μ} + g (W_{3}^{μ} W_{1}^{ν} - W_{1}^{μ} W_{3}^{ν}), \\ F_{3}^{μ ν} \equiv \partial^{μ} W_{3}^{ν} - \partial^{ν} W_{3}^{μ} + g (W_{1}^{μ} W_{2}^{ν} - W_{2}^{μ} W_{1}^{ν}), \\ F_{Y}^{μ ν} \equiv \partial^{μ} B^{ν} - \partial^{ν} B^{μ}, \end{array}$

(11.13)

then, the gauge-kinetic Lagrangian can be written

$\begin{array}{l} - \frac{1}{4} (F_{1}^{μ ν} F_{1 μ ν} + F_{2}^{μ ν} F_{2 μ ν} + F_{3}^{μ ν} F_{3 μ ν} + F_{Y}^{μ ν} F_{Y μ ν}) \\ = & - (\partial_{μ} W_{ν}^{+}) (\partial^{μ} W^{ν -}) + (\partial_{μ} W_{ν}^{+}) (\partial^{ν} W^{μ -}) \\ - \frac{1}{2} (\partial_{μ} A_{ν}) (\partial^{μ} A^{ν}) + \frac{1}{2} (\partial_{μ} A_{ν}) (\partial^{ν} A^{μ}) \\ - \frac{1}{2} (\partial_{μ} Z_{ν}) (\partial^{μ} Z^{ν}) + \frac{1}{2} (\partial_{μ} Z_{ν}) (\partial^{ν} Z^{μ}) \\ + non-quadratic terms . \end{array}$

(11.14)

The non-quadratic terms in the last line of eqn (11.14) yield the vertices in Fig. 11.1.

fig. 11.1.

Open in new tabDownload slide

Self-interactions of the gauge bosons.

11.5 Gauge interactions of the scalars

The scalar sector of the SM consists of only one doublet, $ϕ$ ⁠, which has $Y = 1 / 2$ ⁠. If one chooses a negative sign for the coefficient $μ$ in the quadratic term ${μϕ}^{†} ϕ$ of the Higgs potential—see eqn (11.30)—then the $SU (2) \otimes U (1)$ gauge symmetry is broken into U(1), which is identified with the gauge group of the electromagnetic interaction. Without loss of generality, one can make an SU(2) rotation so that it is the lower component of $ϕ$ which acquires a vacuum expectation value (VEV) v, which is a c-number, constant over the whole of Minkowski space. We write

$ϕ = (\begin{array}{c} φ^{+} \\ φ^{0} \end{array}) = (\begin{array}{c} φ^{+} \\ v + (H + iχ) / \sqrt{2} \end{array}) .$

(11.15)

Here, H and $χ$ are Hermitian Klein–Gordon fields, $φ^{\pm}$ are the Goldstone bosons to be absorbed in the longitudinal components of $W^{\pm}, χ$ is the Goldstone boson to be absorbed in the longitudinal component of Z, and the physical Higgs particle is H. One introduces the conjugate SU(2) doublet

$\tilde{ϕ} \equiv i τ_{2} ϕ^{†^{T}} = (\begin{array}{c} φ^{0^{†}} \\ - φ^{-} \end{array}) = (\begin{array}{c} v + (H - iχ) / \sqrt{2} \\ - φ^{-} \end{array}),$

(11.16)

which has $Y = - 1 / 2$ ⁠.

From eqns (11.12), (11.4), and (11.5) it follows that the gauge-kinetic Lagrangian of $ϕ$ is

$\begin{array}{l} [\partial φ^{-} - i e A φ^{-} + i \frac{g}{\sqrt{2}} W^{-} φ^{0^{†}} + i \frac{g}{2 c_{w}} Z (c_{w}^{2} - s_{w}^{2}) φ^{-}] \\ \times & [\partial φ^{+} + i e A φ^{+} - i \frac{g}{\sqrt{2}} W^{+} φ^{0} - i \frac{g}{2 c_{w}} Z (c_{w}^{2} - s_{w}^{2}) φ^{+}] \\ + & (\partial φ^{0^{†}} + i \frac{g}{\sqrt{2}} W^{+} φ^{-} - i \frac{g}{2 c_{w}} Z φ^{0^{†}}) \\ \times & (\partial φ^{0} - i \frac{g}{\sqrt{2}} W^{-} φ^{+} + i \frac{g}{2 c_{w}} Z φ^{0}) . \end{array}$

(11.17)

We use the relationship between v and the masses of the W and Z,

$v = \frac{\sqrt{2}}{g} m_{W} = \frac{\sqrt{2} c_{w}}{g} m_{Z},$

(11.18)

and find that the expression in eqn (11.17) is

$\begin{array}{l} (\partial φ^{-}) (\partial φ^{+}) + \frac{1}{2} [{(\partial H)}^{2} + {(\partial χ)}^{2}] + m_{W}^{2} W^{-} W^{+} + \frac{1}{2} m_{Z}^{2} Z^{2} \\ + {im}_{W} (W^{-} \partial φ^{+} - W^{+} \partial φ^{-}) + m_{Z} Z \partial χ \\ + non ‐ quadratic terms . \end{array}$

11.6 Gauge-fixing Lagrangian

A gauge fixing is required in order to define the propagators of the gauge bosons and of the Goldstone bosons. Here we shall consider only ’t Hooft gauges. Let us first study the case of the gauge boson Z and the Goldstone boson $χ$ ⁠. The quadratic terms in the Lagrangian containing Z and $χ$ are—see eqns (11.19) and (11.14)—

$- \frac{1}{2} (\partial_{μ} Z_{ν}) (\partial^{μ} Z^{ν}) + \frac{1}{2} (\partial_{μ} Z_{ν}) (\partial^{ν} Z^{μ}) + \frac{1}{2} m_{Z}^{2} Z_{μ} Z^{μ} + \frac{1}{2} (\partial_{μ} χ) (\partial^{μ} χ) + m_{Z} Z_{μ} \partial^{μ} χ .$

(11.20)

One uses the gauge-fixing Lagrangian

$- \frac{1}{2 ξ_{Z}} {(\partial_{μ} Z^{μ} - ξ_{Z} m_{Z} χ)}^{2} = - \frac{1}{2 ξ_{Z}} (\partial_{μ} Z^{μ}) (\partial_{ν} Z^{ν}) + m_{Z} χ \partial_{μ} Z^{μ} - \frac{ξ_{Z}}{2} m_{Z}^{2} χ^{2},$

(11.21)

where $ξ_{Z}$ is an arbitrary real non-negative number, whose value is $ξ_{Z} = 0$ in the Landau gauge, $ξ_{Z} = 1$ in the Feynman gauge, and $ξ_{Z} = \infty$ in the unitary gauge. Physically meaningful quantities are independent of $ξ_{Z}$ ⁠. Adding eqns (11.20) and (11.21) we find that $χ$ has the usual propagator for a scalar boson, with squared mass $ξ_{Z} m_{Z}^{2}$ ⁠. Also, after an integration by parts, the $Z - χ$ mixing terms $m_{Z} Z_{μ} \partial^{μ} χ$ and $m_{Z} χ \partial_{μ} Z^{μ}$ cancel out. The remaining terms yield, after some integration by parts, the Z propagator. This propagator has two parts—see Fig. 11.3. The first part has its pole on the physical squared mass $m_{Z}^{2}$ ⁠. The second part has its pole on the unphysical squared mass $ξ_{Z} m_{Z}^{2}$ ⁠. The effects of this second part of the Z propagator must cancel out with the effects of the propagation of $χ$ —and of the Z ghost, as we shall see in the next section—which have the same fictitious squared mass $ξ_{Z} m_{Z}^{2}$ ⁠.

fig. 11.3.

Open in new tabDownload slide

Propagators of the gauge bosons.

The gauge-fixing term for the $W^{\pm} ‐ φ^{\pm}$ sector reads

$\begin{array}{l} - \frac{1}{ξ_{W}} (\partial^{μ} W_{μ}^{+} - i ξ_{W} m_{W} φ^{+}) (\partial^{ν} W_{ν}^{-} + i ξ_{W} m_{W} φ^{-}) \\ = & - \frac{1}{ξ_{W}} (\partial^{μ} W_{μ}^{+}) (\partial^{ν} W_{ν}^{-}) + {im}_{W} (φ^{+} \partial^{μ} W_{μ}^{-} - φ^{-} \partial^{μ} W_{μ}^{+}) \\ - ξ_{W} m_{W}^{2} φ^{+} φ^{-} . \end{array}$

(11.22)

From eqns (11.19) and (11.22) we see that $φ^{\pm}$ has the usual propagator for a charged scalar, with unphysical squared mass $ξ_{W} m_{W}^{2}$ ⁠. The $W^{\pm} - φ^{\pm}$ mixing terms cancel out. The gauge boson $W^{\pm}$ has a propagator with two parts, one with pole on the physical squared mass $m_{W}^{2}$ ⁠, the other one with pole on the unphysical squared mass $ξ_{W} m_{W}^{2}$ ⁠. Any physical quantity must be independent of the real non-negative number $ξ_{W}$ ⁠. Note that $ξ_{W}$ does not need to be equal to $ξ_{Z}$ ⁠; we may choose different ’t Hooft gauges for the $Z - χ$ sector and for the $W^{\pm} - φ^{\pm}$ sector.

The gauge-fixing term for the photon is

$- \frac{1}{2 ξ_{A}} (\partial^{μ} A_{μ}) (\partial^{ν} A_{ν}) .$

(11.23)

Once again, the unphysical gauge parameter $ξ_{A}$ does not have to be equal to either $ξ_{W}$ or $ξ_{Z}$ ⁠. From eqns (11.14) and (11.23) it follows that the photon propagator is as given in Fig. 11.3. It has a $k_{μ} k_{ν}$ part with gauge-dependent coefficient $1 - ξ_{A}$ ⁠; that part of the propagator must give a vanishing contribution to any physical quantity.

The propagators of the scalars are collected in Fig. 11.4. The mass $m_{H}$ of H originates in the scalar potential, to be treated in § 11.8.

fig. 11.4.

Open in new tabDownload slide

Propagators of the scalars.

11.7 Ghosts

The ghost Lagrangian depends on the gauge-fixing conditions and on the gauge transformations. An infinitesimal gauge transformation of the scalar doublet reads

$\begin{array}{l} δϕ = i (g α_{k} T_{k}^{(2)} + g' \frac{β}{2}) ϕ, \\ δ \tilde{ϕ} = i (g α_{k} T_{k}^{(2)} - g' \frac{β}{2}) \tilde{ϕ}, \end{array}$

(11.24)

where the $α_{k}$ are the three infinitesimal parameters of the SU(2) gauge transformation, and $β$ is the infinitesimal parameter of the U(1) gauge transformation. Writing

$\begin{array}{c} α^{\pm} & = \frac{α_{1} \mp i α_{2}}{\sqrt{2}}, \\ (\begin{array}{c} β \\ α_{3} \end{array}) & = (\begin{array}{c} c_{w} & s_{w} \\ - s_{w} & c_{w} \end{array}) (\begin{array}{l} α_{A} \\ α_{Z} \end{array}) \end{array}$

(11.25)

—cf. eqns (11.9) and (11.11)—we find

$\begin{array}{l} δ φ^{+} & = i m_{W} α^{+} + i [(- e α_{A} + g \frac{c_{w}^{2} - s_{w}^{2}}{2 c_{w}} α_{Z}) φ^{+} + \frac{g}{2} (H + i χ) α^{+}], \\ δ φ^{-} & = - i m_{W} α^{-} - i [(- e α_{A} + g \frac{c_{w}^{2} - s_{w}^{2}}{2 c_{w}} α_{Z}) φ^{-} + \frac{g}{2} (H - i χ) α^{-}] \\ δ χ & = - m_{Z} α_{Z} - \frac{g}{2 c_{w}} H α_{Z} + \frac{g}{2} (φ^{-} α^{+} + φ^{+} α^{-}) . \end{array},$

(11.26)

The same infinitesimal transformation of the gauge fields reads

$\begin{array}{l} δ A_{μ} = \partial_{μ} α_{A} + ie (α^{-} W_{μ}^{+} - α^{+} W_{μ}^{-}), \\ δ Z_{μ} = \partial_{μ} α_{Z} - {igc}_{w} (α^{-} W_{μ}^{+} - α^{+} W_{μ}^{-}), \\ δ W_{μ}^{\pm} = \partial_{μ} α^{\pm} \pm {igW}_{μ}^{\pm} (- s_{w} α_{A} + c_{w} α_{Z}) \mp ig (- s_{w} A_{μ} + c_{w} Z_{μ}) α^{\pm} . \end{array}$

(11.27)

The Fadeev–Popov ghost Lagrangian for a general ’t Hooft gauge reads, from eqns (11.21), (11.22), and (11.23),

$\begin{array}{l} L_{FP} & = - \sum_{i} [{\bar{c}}_{Z} \frac{δ (\partial_{μ} Z^{μ} - ξ_{Z} m_{Z} χ)}{δ α_{i}} c_{i} + {\bar{c}}_{A} \frac{δ (\partial_{μ} A^{μ})}{δ α_{i}} c_{i} \\ + {\bar{c}}^{+} \frac{δ (\partial^{μ} W_{μ}^{+} - i ξ_{W} m_{W} φ^{+})}{δ α_{i}} c_{i} + {\bar{c}}^{-} \frac{δ (\partial^{μ} W_{μ}^{-} + i ξ_{W} m_{W} φ^{-})}{δ α_{i}} c_{i}] . \end{array}$

(11.28)

The notation of the sum over i is the following: $c_{i}$ denotes $c_{Z}, c_{A}, c^{+}$ ⁠, and $c^{-}$ ⁠, and the corresponding $α_{i}$ are $α_{Z}, α_{A}, α^{+}$ ⁠, and $α^{-}$ ⁠, respectively. Notice that there are two distinct charged ghosts, $c^{+}$ and $c^{-}$ ⁠, together with their distinct antighost fields, ${\bar{c}}^{+}$ and ${\bar{c}}^{-}$ ⁠, respectively. We easily find

$\begin{array}{l} ℒ_{FP} & = - {\bar{c}}_{Z} (\partial_{μ} \partial^{μ} + ξ_{Z} m_{Z}^{2}) c_{Z} - {\bar{c}}_{A} \partial_{μ} \partial^{μ} c_{A} - {\bar{c}}^{+} (\partial_{μ} \partial^{μ} + ξ_{W} m_{W}^{2}) c^{+} \\ - {\bar{c}}^{-} (\partial_{μ} \partial^{μ} + ξ_{W} m_{W}^{2}) c^{-} + non ‐ quadratic terms . \end{array}$

(11.29)

From eqn (11.29) we find the ghost propagators in Fig. 11.5. The non-quadratic terms yield the vertices in Fig. 11.6. Note that some of those vertices are proportional to the gauge parameters $ξ_{W}$ and $ξ_{Z}$ ⁠.

fig. 11.5.

Open in new tabDownload slide

Propagators of the ghosts.

fig. 11.6.

Open in new tabDownload slide

Ghost vertices.

11.8 Self-interactions of the scalars

The self-interactions of the scalars originate in the scalar potential—which appears in the Lagrangian density $ℒ$ with an overall minus sign, $ℒ = \dots - V$ ⁠:

$V = {μϕ}^{†} ϕ + λ {(ϕ^{†} ϕ)}^{2},$

(11.30)

where

$ϕ^{†} ϕ = v^{2} + \sqrt{2} vH + \frac{H^{2} + χ^{2}}{2} + φ^{-} φ^{+} .$

(11.31)

The vacuum stability condition is equivalent to the condition that the terms linear in H vanish: $μ = - 2 λ v^{2}$ ⁠. We trade $λ$ by the Higgs mass $m_{H}$ through $4 λ v^{2} = m_{H}^{2}$ ⁠, and use the W mass $m_{W}$ instead of v by having recourse to eqn (11.18). We obtain

$\begin{array}{l} V & = - \frac{m_{W}^{2} m_{H}^{2}}{2 g^{2}} + \frac{m_{H}^{2}}{2} H^{2} + \frac{{gm}_{H}^{2}}{2 m_{W}} (\frac{H^{2} + χ^{2}}{2} + φ^{-} φ^{+}) H \\ + \frac{g^{2} m_{H}^{2}}{8 m_{W}^{2}} {(\frac{H^{2} + χ^{2}}{2} + φ^{-} φ^{+})}^{2} . \end{array}$

(11.32)

We find that, besides the mass term for H and the vacuum energy density $- m_{W}^{2} m_{H}^{2} / (2 g^{2})$ ⁠, the potential V contains cubic and quartic terms which yield the vertices in Fig. 11.7.

fig. 11.7.

Open in new tabDownload slide

Self-interactions of the scalars.

Download all slides