Vector Portals and Kinetic Mixing

Two Abelian gauge fields can communicate through a renormalizable kinetic-mixing operator. The physical couplings are obtained only after the positive kinetic form is canonicalized, the resulting mass matrix is diagonalized, and the same transformations are applied to all currents. Kinetic mixing does not excuse an anomalous gauged current, supply a missing mass-generation mechanism, or justify a zero-width resonance approximation.

Required background. Consistent New Matter and Gauge Sectors supplies charges, anomalies, and gauge-boson mass generation. Gauge-Boson Masses and Electroweak Mixing supplies the neutral electroweak mass-basis construction.

Helpful background. Vacuum Polarization, Running Charge, and Screening supplies the mixed two-point-function logic.

Canonical Abelian kinetic mixing

For two $U(1)$ gauge fields, begin with

$$\mathcal L= -\frac14F_{\mu\nu}F^{\mu\nu} -\frac14X_{\mu\nu}X^{\mu\nu} -\frac{\epsilon}{2}F_{\mu\nu}X^{\mu\nu} +eA_\mu J^\mu +\frac12m_X^2X_\mu X^\mu.$$

The kinetic matrix is

$$K=\begin{pmatrix}1&\epsilon\\\epsilon&1\end{pmatrix},$$

with eigenvalues $1\mp\epsilon$. Positivity therefore requires $|\epsilon|<1$. Define $r=\sqrt{1-\epsilon^2}$ and

$$A_\mu=A_{c\mu}-\frac{\epsilon}{r}X_{c\mu}, \qquad X_\mu=\frac{1}{r}X_{c\mu}.$$

The transformation satisfies $S^{\mathsf T}KS=I$ and gives

$$\mathcal L\supset eA_{c\mu}J^\mu -\frac{\epsilon e}{r}X_{c\mu}J^\mu +\frac12\frac{m_X^2}{r^2}X_{c\mu}X_c^\mu.$$

Thus the induced visible-current coupling is $-\epsilon e/r$ and the canonical hidden-vector mass parameter is $m_X/r$ in this convention. A different triangular or symmetric canonicalization redistributes couplings differently, but after masses and currents are transformed consistently every amplitude agrees. Kinetic mixing and the associated charge shifts were derived in Holdom 1986, pp. 196–198.

For the exact fixture $\epsilon=3/5$, $r=4/5$, the kinetic eigenvalues are $2/5$ and $8/5$, the induced coupling is $-3e/4$, and the mass is $5m_X/4$. Any calculation that obtains these couplings while retaining the unrescaled mass has transformed only part of the Lagrangian.

Mass mixing and current rotation

After canonical normalization, allow an independent symmetric mass matrix

$$\mathcal L_{\rm mass}=\frac12V_c^{\mathsf T}M_c^2V_c, \qquad V_c=(A_c,X_c)^{\mathsf T}.$$

An orthogonal matrix $O$ gives $O^{\mathsf T}M_c^2O=\operatorname{diag}(m_1^2,m_2^2)$. The current vector transforms as $j_m=O^{\mathsf T}j_c$ under the same convention. Diagonalizing $M^2$ in the original noncanonical basis is not an ordinary orthogonal eigenproblem and generally gives wrong residues and couplings.

For

$$M_c^2=\begin{pmatrix}4&1\\1&9\end{pmatrix},$$

the positive minors are $4$ and $35$, the eigenvalues are

$$m_{1,2}^2=\frac{13\mp\sqrt{29}}{2},$$

and $|\tan2\theta|=2/5$. Applying the same rotation to $j_c=(e,-3e/4)$ fixes both mass-eigenstate couplings. When the off-diagonal mass entry tends to zero, only the mass-mixing rotation disappears; the separately generated kinetic-mixing current remains. Exact matrix arithmetic reproduces the full transformation.

For hypercharge mixing, perform this construction before electroweak mass diagonalization. The resulting photon- and $Z$-current couplings depend on whether the hidden vector is massless, Higgsed, or Stückelberg massive. Statements about “millicharge” can be basis dependent when two vectors are massless; scattering amplitudes and the global charge lattice are not.

Origin and running of the mixing

The operator $F_Y^{\mu\nu}F^X_{\mu\nu}$ is gauge invariant for two Abelian factors. Matter charged under both produces a mixed vacuum polarization, schematically

$$\mu\frac{d\epsilon}{d\mu} \propto\frac{g_Yg_X}{(4\pi)^2} \sum_i d_iY_iX_i,$$

where the exact coefficient depends on Weyl/Dirac and kinetic-term conventions. Threshold matching is required when bi-charged matter is integrated out. A vanishing charge trace can remove one-loop running over a complete degenerate multiplet, but split thresholds can still generate a finite mixing.

If $X_\mu$ also gauges a direct current $J_X^\mu$, its charges must satisfy all local and global anomaly tests. A field redefinition can move kinetic mixing among currents; it cannot cancel the nonconservation of an anomalous gauge current. An anomalous low-energy description requires explicit compensating degrees of freedom or Wess–Zumino terms and a declared cutoff Preskill 1991, §§2–4.

Mass generation, longitudinal control, and widths

A massive vector must inherit a consistent longitudinal mode from a hidden Higgs or Abelian Stückelberg sector. Coupling a bare Proca field to a nonconserved current generally produces amplitudes that grow with energy and reveal the model’s cutoff. Verify a representative longitudinal partial wave and include the symmetry-breaking scalar when it is kinematically or unitarily relevant.

For a mass eigenstate $X$ with a pure vector coupling $g_fX_\mu\bar f\gamma^\mu f$, the tree-level fermionic partial width is

$$\Gamma(X\to f\bar f)= \frac{N_cg_f^2m_X}{12\pi} \left(1+\frac{2m_f^2}{m_X^2}\right) \sqrt{1-\frac{4m_f^2}{m_X^2}}.$$

Axial couplings, chiral masses, radiative corrections, and other open channels change this expression. Near the pole, the amplitude has the form

$$\mathcal A(s)=\mathcal A_{\rm SM}(s) +\frac{g_{\rm in}g_{\rm out}} {s-s_p}, \qquad s_p=M_p^2-iM_p\Gamma_p,$$

and the interference $2\operatorname{Re}(\mathcal A_{\rm SM}^*\mathcal A_X)$ is part of the observable. Production times branching ratio is justified only when the pole is narrow, nonresonant terms vary slowly, and the measurement definition does not spoil factorization. A fixed ad hoc width cannot repair a gauge-incomplete current.

Checks and failure modes

• Require $K>0$ before any field redefinition and positive residues afterward.
• Verify $S^{\mathsf T}KS=I$, then rotate kinetic terms, masses, gauge fixing, and all currents consistently.
• Check that $\epsilon\to0$ removes the induced current and that independent mass mixing vanishes with its own off-diagonal entry.
• Distinguish a radiatively induced current from a fundamental gauged current with its own anomaly conditions.
• Near a resolved pole, retain the width and Standard Model interference; far below it, match to a contact operator and check the power residual.

Non-Abelian hidden dynamics belongs to the gauge-dynamics and nonperturbative volumes. The mediator-to-EFT transition belongs to Effective, Simplified, and Mediator Descriptions, and any exclusion or coupling status belongs to Effective Field Theory and Tests of the Standard Model.

References

• Holdom, Bob. “Two U(1)’s and Epsilon Charge Shifts.” Physics Letters B 166 (1986): 196–198. DOI.
• Preskill, John. “Gauge Anomalies in an Effective Field Theory.” Annals of Physics 210 (1991): 323–379. DOI.