In the minimal left-right symmetric model (MLRSM) with parity as the left-right symmetry, the smallness of the QCD vacuum angle combination θˉ\bar{\theta} ceases to be a mere fine-tuning: it constrains the right-handed-neutrino spectrum, lepton-flavour violation, leptonic electric dipole moments (EDMs), and neutrinoless double beta decay. This problem asks for the completion of that program: (1) a complete map of the viable parity-MLRSM parameter space consistent with the strong-CP bound at tree level and under radiative corrections, together with all correlated EDM, flavour, neutrinoless-double-beta-decay, collider and cosmological constraints; and (2) a resolution of the cosmological domain-wall problem of spontaneously broken parity — proof or exclusion of parity non-restoration at high temperature in this model, or a quantified explicit parity-breaking alternative. The literature finds both cores open: global analyses since 2021 tighten but do not complete the map, and the non-restoration question raised in the 1990s remains unanswered for this model.

Background

The Standard Model of particle physics (SM) contains an accidental CP-violating parameter of the strong interaction: the QCD vacuum angle θ\theta combines with the phases of the quark mass matrices into the single physical quantity θˉ=θ+arg⁡det⁡(MuMd)\bar{\theta}=\theta+\arg\det(M_uM_d). A nonzero θˉ\bar{\theta} induces electric dipole moments (EDMs — permanent charge separations of a particle that violate parity and time-reversal symmetry) of the neutron and of diamagnetic atoms such as 199^{199}Hg. Experimental limits, e.g. dn≲10−26 e⋅cmd_n\lesssim 10^{-26}\,e\cdot\mathrm{cm}, imply the astonishing bound ∣θˉ∣≲10−10|\bar{\theta}|\lesssim 10^{-10}, with no SM symmetry requiring such smallness: this is the strong CP problem. The canonical solution, the Peccei–Quinn (PQ) mechanism, promotes θˉ\bar{\theta} to a dynamical axion field relaxed to zero; it removes rather than predicts the small value, postulates a global symmetry expected to be violated by quantum gravity, and predicts an ultralight axion still being sought.

The minimal left-right symmetric model (MLRSM) extends the SM gauge group to SU(2)L×SU(2)R×U(1)B−LSU(2)_L\times SU(2)_R\times U(1)_{B-L}, with a discrete symmetry — parity P (or alternatively charge conjugation C) — interchanging the two weak groups and exchanging left- and right-handed fermions. When the left-right symmetry is parity, hermiticity of the quark Yukawa couplings requires θ\theta to vanish separately in the P-symmetric limit; however, since the Higgs bi-doublet vacuum expectation value is in general complex, the quark mass matrices are Hermitian only when the induced spontaneous CP violation vanishes, and a nonzero residual CP-odd parameter sαt2βs_\alpha t_{2\beta} — with sα=sin⁡αs_\alpha=\sin\alpha and t2β=tan⁡2βt_{2\beta}=\tan 2\beta, where α\alpha and β\beta are the Higgs bi-doublet vacuum-expectation-value angles — generates at tree level θˉ≃sαt2β mt/mb\bar{\theta}\simeq s_\alpha t_{2\beta}\,m_t/m_b (here mbm_b and mtm_t are the bottom- and top-quark masses). Keeping θˉ\bar{\theta} within its experimental bound therefore requires sαt2βs_\alpha t_{2\beta} to be practically vanishing, parametrically sαt2β≲10−10 mb/mts_\alpha t_{2\beta}\lesssim 10^{-10}\,m_b/m_t (assuming no fine-tuning against θ\theta); a separate, weaker bound sαt2β≲2mb/mts_\alpha t_{2\beta}\lesssim 2m_b/m_t is also quoted in the literature. The theory predicts heavy right-handed gauge bosons WRW_R, ZRZ_R, heavy Majorana right-handed neutrinos NN whose seesaw relation Mν=−MDTMN−1MDM_\nu=-M_D^TM_N^{-1}M_D explains light-neutrino masses, and relations between left- and right-handed mixing matrices that allow the seesaw to be disentangled. Because parity is a discrete symmetry broken spontaneously, the early universe would form domain walls (sheet-like topological defects separating degenerate vacua), which would come to dominate the energy density; they must be removed by small explicit parity breaking or by parity failing to be restored in thermal equilibrium above the symmetry-breaking scale (symmetry non-restoration, first discussed by Mohapatra, Senjanović and Weinberg in the 1970s). Whether non-restoration actually occurs is clouded by higher-order effects in the high-temperature expansion, as shown by Bimonte and Lozano.

Senjanović and Tello (2020, published 2023) argued that whereas in the SM the smallness of θˉ\bar{\theta} constrains nothing, in the parity-MLRSM it acts as a 'blessing': keeping θˉ≲10−10\bar{\theta}\lesssim 10^{-10} forces either nearly real neutrino Dirac Yukawa couplings — predicting small leptonic EDMs, as first noticed by Kuchimanchi — or relatively light right-handed neutrinos; their estimate MN≲5×10−6/5(MWR/GeV)4/5M_N\lesssim 5\times 10^{-6/5}(M_{W_R}/\mathrm{GeV})^{4/5} GeV gives MN≲500M_N\lesssim 500 GeV for MWR≃10M_{W_R}\simeq 10 TeV, with suppressed lepton-flavour violation (LFV) and enhanced neutrinoless double beta decay (0νββ0\nu\beta\beta) prospects. Crucially, in their account no model — MLRSM with P or C, PQ, or any other — predicts the numerical value of θˉ\bar{\theta}; it is an input to be separated experimentally from weak-interaction CP violation by measuring several EDMs. What their analysis leaves open is whether this strong-CP-guided, predictive version of the theory is actually viable and complete, and whether spontaneous parity breaking is cosmologically consistent.

Problem Statement

For the minimal left-right symmetric model with generalized parity PP as the left-right symmetry (minimal scalar sector; self-contained fermion-mass sector including neutrinos), determine whether the strong-CP-guided theory is viable, predictive and cosmologically consistent, by completing two linked deliverables:

(1) Construct the map of the viable parameter space — the flavour structure of the quark and lepton Yukawa sectors, the right-handed-neutrino mass spectrum including the light-sterile (warm-dark-matter) regime, the scale MWRM_{W_R} and the heavy-scalar masses — consistent with the strong-CP bound at tree level (θˉ≃sαt2β mt/mb≲10−10\bar{\theta}\simeq s_\alpha t_{2\beta}\,m_t/m_b\lesssim 10^{-10}, i.e. a practically vanishing sαt2βs_\alpha t_{2\beta}) and under radiative corrections, and with the correlated set of constraints and predictions: EDMs of the neutron, diamagnetic atoms and charged leptons; charged-lepton flavour violation; K- and B-meson CP violation; beta decays; neutrinoless double beta decay; and collider searches for WRW_R, ZRZ_R, heavy scalars and sterile neutrinos. Decide in particular whether the two branches emphasized in the source — relatively light MNM_N with suppressed lepton-flavour violation and enhanced 0νββ0\nu\beta\beta, and nearly real neutrino Dirac Yukawa couplings with suppressed leptonic EDMs — remain viable.

(2) Settle the cosmological consistency of spontaneously broken parity: prove or exclude parity non-restoration at high temperature within the MLRSM, including the higher-order thermal effects that make existing arguments inconclusive, or else determine the magnitude and operator origin of the explicit parity breaking required for the domain-wall network to disappear consistently with cosmology.

The common objective is the verdict on the source paper's thesis: that the smallness of strong CP violation narrows the MLRSM parameter space and correlates its observables rather than constituting a fine-tuning problem. The numerical value of θˉ\bar{\theta} itself is an unpredicted input; separating its strong and weak contributions through correlated EDM measurements is part of the experimental route, not something the theory is asked to predict.

References

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