Axionlike and Pseudoscalar Portals

An axionlike particle (ALP) is a pseudoscalar whose leading interactions are organized by an approximate shift symmetry. Derivative fermion currents and topological gauge couplings are related by integration by parts, anomalous Ward identities, and chiral field redefinitions, so Wilson coefficients are basis dependent while on-shell amplitudes are not. A QCD axion is the special case whose QCD potential dynamically relaxes the strong-CP angle; a generic ALP need not obey its mass–coupling relation.

Required background. Consistency Checklist for Standard Model Extensions supplies the anomaly, stability, and validity gates. Strong CP and the Axion Interface supplies the QCD $\theta$ convention and the QCD-axion mechanism.

Helpful background. Local Field Redefinitions and the Equivalence Theorem explains why complete on-shell amplitudes survive the basis changes below.

Shift-symmetric effective interactions

Let $a$ have periodicity $a\simeq a+2\pi f_a$ when it is a compact field, and define

$$\widetilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}, \qquad \epsilon^{0123}=+1.$$

A low-energy interaction basis is

$$\begin{aligned} \mathcal L_a={}& \frac12(\partial a)^2-\frac12m_a^2a^2 +\sum_\psi\frac{c_\psi}{2f_a}(\partial_\mu a) \bar\psi\gamma^\mu\gamma_5\psi\\ &+\frac{a}{f_a}\left[ c_G\frac{\alpha_s}{8\pi}G^A_{\mu\nu}\widetilde G^{A\mu\nu} +c_W\frac{\alpha_2}{8\pi}W^I_{\mu\nu}\widetilde W^{I\mu\nu} +c_B\frac{\alpha_1}{8\pi}B_{\mu\nu}\widetilde B^{\mu\nu} \right]. \end{aligned}$$

The derivative and topological terms respect a continuous shift up to total derivatives in perturbation theory. The mass and any nonderivative potential break it. For a compact field, periodicity and the global gauge group constrain anomaly coefficients; arbitrary real coefficients may be adequate for a local noncompact EFT but do not by themselves define a consistent compact ultraviolet axion.

Above electroweak breaking the $B$ and $W$ coefficients are the gauge-invariant data. In the displayed normalization the photon coefficient after breaking is

$$c_{\gamma\gamma}=c_W+c_B,$$

because $\alpha_2s_W^2=\alpha_1c_W^2=\alpha$. The correlated $Z\gamma$, $ZZ$, and $WW$ couplings must be retained; assigning only an $aF\widetilde F$ vertex at high energy generally violates electroweak gauge completion.

Basis translation by anomalous Ward identities

Define each renormalized axial-current anomaly in the same scheme by

$$\partial_\mu J_{5,\psi}^\mu =2im_\psi\bar\psi\gamma_5\psi +\sum_i\frac{g_i^2}{16\pi^2} \mathcal A_i(\psi)F^i_{\mu\nu}\widetilde F_i^{\mu\nu}.$$

Then, up to a total derivative,

$$\frac{c_\psi}{2f_a}(\partial_\mu a)J_{5,\psi}^\mu =-\frac{c_\psi a}{2f_a} \left[ 2im_\psi\bar\psi\gamma_5\psi +\sum_i\frac{g_i^2}{16\pi^2} \mathcal A_i(\psi)F^i\widetilde F_i \right].$$

This identity translates a derivative coupling into a pseudoscalar Yukawa interaction plus shifts of the topological coefficients. The same translation follows from a local chiral rotation of $\psi$ only when the Jacobian/anomaly term and the phase of every mass or Yukawa coupling are included. Dropping the anomaly shift changes $a\to\gamma\gamma$ or $a\to gg$ amplitudes and is not a harmless basis choice. The low-energy axion construction and its anomalous field redefinitions are developed in Georgi, Kaplan, and Randall 1986, pp. 73–78.

An amplitude provides the independent check. Compute a process such as $a\to\gamma\gamma$ once with the derivative fermion coupling and its triangle graph, and once after the field redefinition with the shifted local coefficient and pseudoscalar coupling. The sum agrees when masses, anomaly normalization, and renormalization scheme are translated together.

QCD axion versus a generic ALP

For a QCD axion, the combination

$$\left(\bar\theta+c_G\frac{a}{f_a}\right) \frac{\alpha_s}{8\pi}G\widetilde G$$

generates a nonperturbative potential whose minimum relaxes the physical strong-CP angle. The same QCD dynamics determines the leading axion mass in terms of $f_a$, quark masses, and low-energy QCD inputs. Those relations, including mixing with neutral mesons, are calculated in di Cortona et al. 2016, §§2–3.

A generic ALP can receive its dominant mass from another explicit breaking sector; then $m_a$ and $f_a$ are independent EFT parameters, and $c_G$ may vanish. Calling every pseudoscalar with $aF\widetilde F$ an “axion” incorrectly imports the strong-CP solution and QCD mass relation. Conversely, a QCD axion model must state the anomaly/domain-wall coefficient and ultraviolet charge assignment, not only its photon coupling.

Decays, CP, and the EFT domain

If the low-energy photon interaction is written

$$\mathcal L\supset-\frac14g_{a\gamma\gamma} aF_{\mu\nu}\widetilde F^{\mu\nu},$$

then

$$\Gamma(a\to\gamma\gamma)= \frac{|g_{a\gamma\gamma}|^2m_a^3}{64\pi}.$$

In the coefficient convention above, $|g_{a\gamma\gamma}|=\alpha|c_{\gamma\gamma}|/(2\pi f_a)$ before low-energy threshold and meson-mixing corrections. Fermionic decays depend on the translated pseudoscalar coupling and open only above threshold. Hadronic decays require low-energy QCD rather than a partonic formula near confinement. A systematic electroweak ALP EFT and its decay matching are given by Bauer, Neubert, and Thamm 2017, §§2–4.

With $a$ CP odd, $aF\widetilde F$ and $(\partial a)J_5$ can conserve CP, whereas $aFF$ or incompatible scalar couplings signal additional CP violation. This classification assumes the stated phases and transformation law; complex fermion mass matrices must be treated together with the anomalous rotations.

The scale $f_a$ is not automatically the EFT cutoff. A strongly coupled compact realization often suggests a scale of order $4\pi f_a$, while weakly coupled completions can introduce charged states at a lower scale. For each process require all invariant momentum transfers and $m_a$ to lie below the lightest omitted state, check coefficient-weighted partial waves, and retain any resolved ultraviolet mediator rather than extrapolating the dimension-five vertex through it.

Checks and failure modes

• Declare $\widetilde F$, $f_a$, periodicity, gauge-coupling factors, and anomaly normalization before comparing coefficients.
• Translate derivative, pseudoscalar, and topological terms as one basis change and verify one on-shell amplitude.
• Recover the exact shift symmetry as every explicit-breaking parameter tends to zero.
• Distinguish an electroweak-gauge-invariant coefficient set above $v$ from photon and $Z$ couplings below $v$.
• Do not infer a QCD-axion mass relation, cosmological abundance, or current exclusion for a generic ALP.

Strong-CP dynamics remains with Strong CP and the Axion Interface, nonperturbative QCD inputs with the QCD chapters, cosmology with the thermal/cosmology volume, and versioned searches with Effective Field Theory and Tests of the Standard Model.

References

• Bauer, Martin, Matthias Neubert, and Andrea Thamm. “Collider Probes of Axion-Like Particles.” Journal of High Energy Physics 2017, no. 12 (2017): 044. DOI.
• di Cortona, Giovanni Grilli, Edward Hardy, Javier Pardo Vega, and Giovanni Villadoro. “The QCD Axion, Precisely.” Journal of High Energy Physics 2016, no. 1 (2016): 034. DOI.
• Georgi, Howard, David B. Kaplan, and Lisa Randall. “Manifesting the Invisible Axion at Low Energies.” Physics Letters B 169 (1986): 73–78. DOI.