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Theory bridge

Coupling conventions and translations

Connect the dimensionless fermion couplings used in the SD5thF limits to effective energy scales, axion and ALP parameters, and conventions commonly used in neighboring fields.

Reference table

Scalar and pseudoscalar interactions

Here \(\Lambda_X\) is an effective interaction scale, \(f_a\) is the axion or ALP decay constant, and \(C_f\) is a dimensionless, model-dependent fermion coefficient.

InteractionLagrangian termCoupling relationDimensions
Scalar–photon \(\displaystyle \mathcal L\supset\frac{\phi}{4\Lambda_\gamma}F_{\mu\nu}F^{\mu\nu}\) \(\displaystyle g_s^\gamma\equiv\frac{1}{\Lambda_\gamma}\) \([g_s^\gamma]=E^{-1}\)
Scalar–electron \(\displaystyle \mathcal L\supset\frac{\phi}{\Lambda_e}m_e\bar e e\) \(\displaystyle g_s^e=\frac{m_e}{\Lambda_e}\) \([g_s^e]=1\)
Scalar–proton \(\displaystyle \mathcal L\supset\frac{\phi}{\Lambda_p}m_p\bar p p\) \(\displaystyle g_s^p=\frac{m_p}{\Lambda_p}\) \([g_s^p]=1\)
Scalar–neutron \(\displaystyle \mathcal L\supset\frac{\phi}{\Lambda_n}m_n\bar n n\) \(\displaystyle g_s^n=\frac{m_n}{\Lambda_n}\) \([g_s^n]=1\)
Pseudoscalar–fermion \(\displaystyle \mathcal L\supset i g_p^f a\,\bar\psi_f\gamma^5\psi_f\) \(\displaystyle |g_p^f|=\frac{|C_f|m_f}{f_a}\) \([g_p^f]=1\)
Derivative axion–fermion \(\displaystyle \mathcal L\supset\frac{C_f}{2f_a}(\partial_\mu a)\bar\psi_f\gamma^\mu\gamma^5\psi_f\) \(\displaystyle |g_{aff}|=\frac{|g_p^f|}{2m_f}=\frac{|C_f|}{2f_a}\) \([g_{aff}]=E^{-1}\)
Pseudoscalar–photon \(\displaystyle \mathcal L\supset\frac{g_{a\gamma\gamma}}{4}aF_{\mu\nu}\widetilde F^{\mu\nu}\) \(\displaystyle g_{a\gamma\gamma}\equiv\frac{1}{\Lambda_\gamma}\)
Ni, Balakin & Mei (2010) ↗
 \([g_{a\gamma\gamma}]=E^{-1}\)
01 / FIFTH FORCE → AXION → SPIN PRECESSION

From SD5thF couplings to axion conventions

The review and this database start from the dimensionless spin-0 fermion couplings

\[ \mathcal L_\phi=\phi\sum_\psi\bar\psi\left(g_s^\psi+i\gamma^5g_p^\psi\right)\psi . \]

For an axion or ALP with fermion coefficient \(C_f\) and decay constant \(f_a\), the pseudoscalar coupling is connected to the gradient coupling used in spin-precession searches through

SD5thF\(\displaystyle |g_p^f|\)
Axion / ALP\(\displaystyle \frac{|C_f|m_f}{f_a}\)
Spin precession\(\displaystyle |g_{aff}|=\frac{|g_p^f|}{2m_f}=\frac{|C_f|}{2f_a}\)

In the review’s gradient-Hamiltonian notation, \(|g_{aNN}|=|g_p^N|/(2m_N)\) and \(|g_{aee}|=|g_p^e|/(2m_e)\). This is the bridge between the coupling products plotted by SD5thF and the parameters commonly reported by axion spin-precession experiments.

02 / FORCE RANGE

Mass–range conversion

The mediator mass \(M\) and interaction range \(\lambda\) are related by its reduced Compton wavelength:

\[ \lambda=\frac{\hbar}{Mc}. \]

This conversion links the boson-mass axes used in particle and axion searches to the interaction-range axes used throughout the SD5thF explorer.

03 / QCD AXION

Model dependence

For a generic ALP, mass and couplings can be treated as independent parameters. For the canonical QCD axion, \(m_a\) and \(f_a\) are related approximately by

\[ m_a\simeq5.7\,\mu{\rm eV}\left(\frac{10^{12}\,{\rm GeV}}{f_a}\right). \]

The coefficients \(C_f\), and therefore the translation from \(f_a\) to \(g_p^f\), depend on the axion model and renormalization scale.

Important qualifications

Before translating a published bound

Selected literature

References behind the translations

Grouped by the physical connection they support. The definitions in the cited paper should always be checked before applying a numerical conversion.

Axions, ALPs, and fermion couplings

  1. Peccei & Quinn, “Constraints Imposed by CP Conservation in the Presence of Pseudoparticles,” Phys. Rev. D 16, 1791 (1977).
  2. Peccei & Quinn, “CP Conservation in the Presence of Pseudoparticles,” Phys. Rev. Lett. 38, 1440 (1977).
  3. Weinberg, “A New Light Boson?” Phys. Rev. Lett. 40, 223 (1978).
  4. Wilczek, “Problem of Strong P and T Invariance in the Presence of Instantons,” Phys. Rev. Lett. 40, 279 (1978).
  5. Moody & Wilczek, “New Macroscopic Forces?” Phys. Rev. D 30, 130 (1984).
  6. Marsh, “Axions and ALPs: A Very Short Introduction,” arXiv:1712.03018 (2017).
  7. Grilli di Cortona et al., “The QCD Axion, Precisely,” JHEP 01, 034 (2016), for the quoted QCD-axion mass–decay-constant relation.

Pseudoscalar–photon coupling

  1. Ni, Balakin & Mei, “Pseudoscalar-Photon Interactions, Axions, Non-Minimal Extensions, and Their Empirical Constraints from Observations,” in Proceedings of the Conference in Honour of Murray Gell-Mann’s 80th Birthday, 526–535 (2010).Directly relevant to the \(aF\widetilde F\) interaction, photon–axion conversion, birefringence, and cosmic polarization rotation; its normalization should be matched before numerical reuse. arXiv:1109.0581 · BibTeX key: ni_pseudoscalar-photon_2010
  2. Sikivie, “Experimental Tests of the Invisible Axion,” Phys. Rev. Lett. 51, 1415 (1983).
  3. Raffelt & Stodolsky, “Mixing of the Photon with Low-Mass Particles,” Phys. Rev. D 37, 1237 (1988).

Axion-gradient coupling and spin precession

  1. Graham & Rajendran, “New Observables for Direct Detection of Axion Dark Matter,” Phys. Rev. D 88, 035023 (2013).
  2. Stadnik & Flambaum, “Axion-Induced Effects in Atoms, Molecules, and Nuclei,” Phys. Rev. D 89, 043522 (2014).
  3. Budker et al., “Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr),” Phys. Rev. X 4, 021030 (2014).
  4. Abel et al., “Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields,” Phys. Rev. X 7, 041034 (2017).
Closely related work

From effective scales to laboratory searches

Leefer et al. connect scalar couplings and effective energy scales to atomic-spectroscopy and macroscopic fifth-force constraints. The RMP review supplies the spin-dependent interaction and axion-gradient conventions used by SD5thF.

Leefer et al., Phys. Rev. Lett. 117, 271601 (2016) ↗

Cong et al., Rev. Mod. Phys. 97, 025005 (2025) ↗

This page is a convention map, not a substitute for the definitions and assumptions in the cited primary analysis.