The 1984 Moody-Wilczek analysis established that a light pseudoscalar such as the axion mediates a distinct parity- and time-reversal-violating macroscopic force between an unpolarized mass and a polarized spin (the monopole-dipole interaction), estimated its strength in terms of an effective CP-violating angle and the pion-nucleon sigma term, and proposed resonant mechanical experiments with small high-Q oscillators aimed at a Brownian-noise-limited measurement sensitivity sufficient to test the entire suggested range of the CP-violating angle θ\theta for the axion monopole-dipole force (accelerations of order 10−2010^{-20} cm/s2^2, approximate). Two linked issues from that work remain open. Experimentally, torsion pendula, comagnetometers, levitated force sensors, and proposed atom-interferometric and resonant-NMR experiments have improved bounds on the scalar-pseudoscalar coupling product gsgpg_sg_p by many orders of magnitude, but none has detected the force or attained the sensitivity projected in the source. Theoretically, the mapping from the underlying CP violation (the effective θˉ\bar\theta parameter and the P- and T-violating four-quark operators induced by the weak interactions) to the effective scalar axion-nucleon coupling is still not reliably calculable: the pion-nucleon sigma term remains in tension between lattice QCD (about 42 MeV) and phenomenology (about 60 MeV), and the first direct lattice determination of the theta-induced CP-violating pion-nucleon coupling appeared only in 2026 with a combined uncertainty of roughly 40%. The open problem is to complete the program: attain the projected monopole-dipole sensitivity over laboratory force ranges, and determine the CP-violating axion-nucleon coupling from first principles, so that a detection or bound translates into constraints on fundamental parameters.

Background

Quantum chromodynamics (QCD) admits a physical, CP-violating interaction term proportional to an angle θˉ\bar\theta; the observed smallness of nuclear electric dipole moments (EDMs) bounds ∣θˉ∣|\bar\theta| below about 10−1010^{-10}, which is the strong CP problem. The Peccei-Quinn mechanism makes θ\theta dynamical: a spontaneously broken global U(1) symmetry produces a light pseudoscalar, the axion, whose vacuum value cancels θˉ\bar\theta. The axion is the pseudo-Nambu-Goldstone boson of that symmetry (a light weakly coupled particle emerging from spontaneous symmetry breaking), with mass mA2=(mπ2fπ2/F2) mumd/(mu+md)2m_A^2=(m_\pi^2f_\pi^2/F^2)\,m_um_d/(m_u+m_d)^2 set by the pion mass mπm_\pi, the pion decay constant fπf_\pi, the light-quark masses, and the Peccei-Quinn scale FF: numerically mA≈10−5m_A\approx10^{-5} eV and a Compton wavelength λA≈2\lambda_A\approx2 cm for F=1012F=10^{12} GeV. Moody and Wilczek (1984) derived the three nonrelativistic potentials from single exchange of a light spin-0 boson with scalar coupling gSg_S and pseudoscalar coupling gPg_P: a monopole (scalar-scalar) Yukawa potential, a dipole (pseudoscalar-pseudoscalar) spin-spin potential, and the mixed monopole-dipole potential

VSP(r)=gS(1)gP(2)8πM2 (σ^2⋅r^)(mφr+1r2)e−mφr,V_{SP}(r)=\frac{g_S^{(1)}g_P^{(2)}}{8\pi M_2}\,(\hat\sigma_2\cdot\hat r)\left(\frac{m_\varphi}{r}+\frac{1}{r^2}\right)e^{-m_\varphi r},

which violates parity and time reversal (and hence CP): it couples the spin of one fermion (mass M2M_2, Pauli spin operator σ^2\hat\sigma_2, mediator mass mφm_\varphi) to the position of another, so it acts between an unpolarized macroscopic mass and a polarized spin. For the axion, the pseudoscalar nucleon coupling follows from chiral symmetry and is of order mN/Fm_N/F, but the scalar coupling requires CP violation: gaNN=(θ/F) [2mumd/(mu+md)2] ⟨N∣σ∣N⟩g_{aNN}=(\theta/F)\,[2m_um_d/(m_u+m_d)^2]\,\langle N|\sigma|N\rangle, where the pion-nucleon sigma term ⟨N∣σ∣N⟩\langle N|\sigma|N\rangle (the nucleon matrix element of the light-quark scalar density, taken from pion-nucleon scattering) was taken as 60 MeV. Moody and Wilczek proposed detecting this force with small mechanical oscillators of extreme intrinsic quality factor QQ (sapphire or niobium crystals at kHz frequencies, or Meissner-levitated superconducting masses, i.e., masses floating by magnetic-field expulsion from a superconductor), adapting gravitational-wave-detection techniques: millikelvin temperatures, SQUID (superconducting quantum interference device) or interferometric displacement readout, active antiseismic shielding, and velocity-feedback 'electronic refrigeration', projecting Brownian-noise-limited sensitivity down to accelerations of order 10−2010^{-20} cm/s2^2 (approximate) — the level needed to test the entire suggested range of θ\theta over the astrophysically allowed Peccei-Quinn window, from the stellar energy-loss lower bound to the cosmological overclosure upper bound (F≈109F\approx10^{9}–101210^{12} GeV). Dobrescu and Mocioiu (2006) later generalized the classification to sixteen spin- and velocity-dependent long-range potentials between fermions, of which the monopole-dipole term is the leading P,T-violating one. The modern experimental toolkit includes torsion balances (rotating pendula sensitive to anomalous torques on polarized electron spins), comagnetometers (co-located spin species whose precession-frequency ratios cancel ordinary magnetic fields and isolate anomalous spin couplings), levitated optomechanical force sensors, and atom interferometry; atomic and molecular EDMs constrain the same P,T-odd hadronic couplings indirectly. On the theory side, the CP-even axion-nucleon couplings are now computed to better than 10% from chiral perturbation theory (the low-energy effective field theory of QCD) combined with lattice QCD (discrete space-time simulation of the QCD path integral), but the CP-odd scalar coupling depends on nonperturbative matrix elements that remain hard to compute: the sigma term itself, the strange-quark scalar content, and the CP-odd pion-nucleon couplings induced by θˉ\bar\theta and by the dimension-six four-quark CP-violating operators generated by weak interactions.

Problem Statement

The problem left open by the source is the completion of the program it proposed: a laboratory test of the axion-mediated P- and T-violating monopole-dipole force at the sensitivity projected in the source, together with the hadronic theory needed to interpret it. It has two jointly necessary components that share one objective — establishing whether this new macroscopic force exists at the predicted strength and what it implies for CP violation.

(i) Experimental: a quantitative experimental design or feasibility analysis, not a performed measurement, of a laboratory measurement of the monopole-dipole interaction between an unpolarized source mass and a polarized spin — a detection at stated statistical significance, or an upper bound — that attains the Brownian-noise-limited measurement sensitivity projected in the source for the proposed high-Q mechanical experiments (monocrystal oscillators, hybrid low-frequency oscillators, Meissner-levitated masses), sufficient to test the entire suggested range of the CP-violating angle θ\theta for the axion monopole-dipole force over the force ranges set by the axion Compton wavelength for Peccei-Quinn scale FF in the astrophysically allowed window identified in the source, from the stellar energy-loss lower bound to the cosmological overclosure upper bound (the traditional axion window F≈109F\approx10^{9}–101210^{12} GeV targeted by modern dedicated experiments, with cosmologically preferred F≈1012F\approx10^{12} GeV). Achieving this requires overcoming the obstacles identified there: uncontrolled dissipation channels (surface, suspension, defect-motion, thermoelastic, and fluxoid losses, the last being dissipation from the motion of quantized magnetic-flux lines that penetrate superconducting films) that limit the intrinsic quality factors and relaxation times of macroscopic oscillators; seismic and tilt coupling, requiring active shielding; magnetic backgrounds from residual impurities in the nominally unpolarized source mass; and readout backaction.

(ii) Theoretical: a first-principles determination of the effective CP-violating scalar axion-nucleon coupling gsNg_s^N in terms of the underlying CP violation — the effective θˉ\bar\theta parameter and the coefficients (Wilson coefficients) of the P- and T-violating four-quark operators induced by the weak interactions — including the nonperturbative QCD matrix elements it depends on (the pion-nucleon sigma term with its strange-quark content, and the CP-odd pion-nucleon couplings), with quantified uncertainties.

The two components are jointly interpreted: the calculation fixes the QCD-axion target line and its uncertainty band, and the proposed experiment reaches the sensitivity at which it would detect the force or bound it at that level, with the result translated into constraints on fundamental parameters (θˉ\bar\theta, Wilson coefficients of CP-violating operators, FF, axion model class). The problem does not reduce to the CP-even axion-nucleon couplings (already computed to about ten percent) or to adjacent spin-spin or velocity-dependent potentials, which are separately constrained.

References

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