Gauge-Boson Masses and Electroweak Mixing

The Higgs kinetic term turns one electroweak gauge-field combination into the charged $W^\pm$, another into the neutral $Z$, and leaves the orthogonal photon combination exactly massless. At tree level this gives

$$m_W=\frac{gv}{2},\qquad m_Z=\frac{v}{2}\sqrt{g^2+g'^2},\qquad A_\mu=s_WW_\mu^3+c_WB_\mu ,$$

with $s_W=g'/\sqrt{g^2+g'^2}$ and $c_W=g/\sqrt{g^2+g'^2}$. Beyond tree level, masses and the weak angle require a declared pole and input-scheme definition.

Required background. The Higgs doublet and electroweak symmetry breaking supplies the vacuum orbit and $H\sim(\mathbf1,\mathbf2)_{1/2}$. Normal forms, spectra, and projectors supplies the matrix diagonalization used for the neutral fields.

Helpful background. Forms, adjoints, and isometries clarifies why the neutral-field rotation must preserve canonical kinetic terms.

Mass terms from the Higgs covariant derivative

Use $Q=T_3+Y$, $T^a=\tau^a/2$, and

$$D_\mu H= \left(\partial_\mu-i g\,\frac{\tau^a}{2}W_\mu^a -i g'\frac12B_\mu\right)H, \qquad H_0=\frac{1}{\sqrt2}\begin{pmatrix}0\\v\end{pmatrix}.$$

The quadratic gauge-field terms in $(D_\mu H_0)^\dagger D^\mu H_0$ are

$$\mathcal L_{\rm mass} =\frac{g^2v^2}{8}\left[(W_\mu^1)^2+(W_\mu^2)^2\right] +\frac{v^2}{8}\left(gW_\mu^3-g'B_\mu\right)^2 .$$

Introducing

$$W_\mu^\pm=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt2}$$

puts the charged term in the canonical form $m_W^2W_\mu^+W^{-\mu}$, so $m_W^2=g^2v^2/4$. This normalization is independently fixed by the $SU(2)$ ladder generators in the charged current.

For $V_\mu=(W_\mu^3,B_\mu)^{\mathsf T}$, the neutral term is

$$\mathcal L_{\rm neutral\,mass} =\frac12V_\mu^{\mathsf T}M_0^2V^\mu, \qquad M_0^2=\frac{v^2}{4} \begin{pmatrix} g^2&-gg'\\ -gg'&g'^2 \end{pmatrix}.$$

The determinant vanishes and the trace is $v^2(g^2+g'^2)/4$. Thus the two eigenvalues are

$$m_A^2=0, \qquad m_Z^2=\frac{v^2}{4}(g^2+g'^2).$$

This derivation, including its normalization from the Higgs kinetic term, is given in Schwartz 2014, §29.1, pp. 584–588.

The photon and the weak mixing rotation

Define the orthogonal rotation

$$\begin{pmatrix}Z_\mu\\A_\mu\end{pmatrix} = \begin{pmatrix} c_W&-s_W\\ s_W&c_W \end{pmatrix} \begin{pmatrix}W_\mu^3\\B_\mu\end{pmatrix}, \qquad s_W=\frac{g'}{\sqrt{g^2+g'^2}}, \quad c_W=\frac{g}{\sqrt{g^2+g'^2}}.$$

The massive eigenvector is proportional to $(g,-g')$, while the photon eigenvector is proportional to $(g',g)$. The latter is the gauge field of the unbroken generator $Q=T_3+Y$: substituting the inverse rotation into

$$gT_3W_\mu^3+g'YB_\mu$$

gives

$$eQA_\mu+\frac{g}{c_W}\left(T_3-s_W^2Q\right)Z_\mu, \qquad e=gs_W=g'c_W .$$

Masslessness is therefore not an accidental cancellation. It follows because the Higgs vacuum is neutral under $Q$, so the covariant derivative cannot generate an $A_\mu A^\mu$ term.

The tree-level mass relation

$$\rho_{\rm tree}\equiv \frac{m_W^2}{m_Z^2c_W^2}=1$$

follows from a single scalar doublet with this hypercharge and a canonical kinetic term. It is not a representation-independent identity: other scalar multiplets or vacuum expectation values can change it.

Gauge fixing and physical meaning

Before gauge fixing, expanding $H$ also produces bilinear terms of the form $m_VV^\mu\partial_\mu\phi$, where $\phi$ is a would-be Goldstone field. An $R_\xi$ gauge uses

$$\mathcal F^\pm=\partial_\mu W^{\pm\mu}\mp i\xi_Wm_W\phi^\pm, \qquad \mathcal F^Z=\partial_\mu Z^\mu-\xi_Zm_Z\chi$$

to cancel those mixings. The Goldstone and ghost masses then depend on $\xi$, but physical pole positions and on-shell amplitudes do not. Unitary gauge removes the Goldstone fields from the displayed spectrum but does not provide a different symmetry-breaking mechanism.

At loop level, three distinctions matter:

• the complex pole of a propagator is not generally identical to a line-shape mass parameter;
• $s_W^2=1-m_W^2/m_Z^2$ defines an on-shell angle only after the mass convention is fixed;
• a short-distance weak angle and an on-shell weak angle differ by finite radiative corrections.

The tree formulas remain useful algebraic checks, not scheme-free definitions to be inserted into every higher-order calculation.

Checks and limiting cases

Zero-mode check. Multiplying $M_0^2$ by $(g',g)^{\mathsf T}$ must give zero. A nonzero answer usually signals the wrong sign in the neutral covariant derivative or a mixed hypercharge convention.

Canonical-rotation check. The $2\times2$ mixing matrix is orthogonal because the $W^3$ and $B$ kinetic terms are canonically normalized. Kinetic mixing or noncanonical normalization must be removed before the same rotation can be used.

Coupling limits. As $g'\to0$, $A_\mu\to B_\mu$ and the three $SU(2)$ vectors have the common mass $gv/2$. As $g\to0$, $A_\mu\to W_\mu^3$; the formulas remain algebraically continuous even though the identification of the interacting electromagnetic sector changes.

Degree-of-freedom check. Three massless gauge fields acquire one longitudinal polarization each by absorbing three scalar directions; one scalar radial mode remains. The total number of physical degrees of freedom is unchanged.

Common pitfalls

Reading the weak angle directly from a current fit. A fitted effective angle, an on-shell angle, and a short-distance running angle are distinct beyond tree level. State which one enters before comparing results.

Diagonalizing only the potential. Gauge-boson masses come from the Higgs kinetic term. The potential determines $v$ and the scalar mass but contains no gauge-field quadratic term.

Calling the photon massless because a determinant happened to vanish. The determinant is the matrix expression of the unbroken $Q$ generator. That group-theoretic reason is the check that survives a change of basis.

Handoff

The current sector receives the typed mass-basis data

$$\left\{W^\pm,\ Z,\ A;\ m_W,m_Z;\ s_W,c_W;\ A=s_WW^3+c_WB;\ e=gs_W=g'c_W\right\}.$$

These data feed the charged and neutral weak currents, the longitudinal-vector consistency analysis, and, after a renormalization prescription is chosen, electroweak input schemes.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.1, pp. 584–588. DOI.