Role of the equivalence principle in gauge and axial symmetries of Yukawa coupling, and the strong CP problem

Abstract: It is demonstrated the fundamental role of the equivalence principle in gravity for the Yukawa coupling between scalar and fermion fields. The Kibble-Zurek mechanism for formation of topological defects as vortexes and monopoles breaks down in system with a global gauge symmetry only. At the same time, the different vacuums can occur, which are separated be domain walls. The equivalence principle makes the strong violation of CP invariance impossible. Thus the axion hypothesis becomes redundant.

• The paper demonstrates that the equivalence principle enforces a unique vacuum phase in Yukawa interactions, thus eliminating strong CP violation.
• The analysis employs analogies with condensed matter systems to show how local gauge symmetry and phase dynamics govern mass generation.
• The work challenges conventional axion-based solutions by suppressing topological defects and enforcing CP symmetry through gravitational constraints.

Role of the Equivalence Principle in Gauge and Axial Symmetries of Yukawa Coupling, and the Strong CP Problem

Introduction

This paper offers a detailed theoretical investigation into the interrelation of the equivalence principle, gauge/axial symmetries, and the structure of Yukawa couplings, with particular emphasis on the strong CP problem. The analysis exposes a fundamental—rather than merely formal—role for the equivalence principle in enforcing CP symmetry within quantum field theory, thus challenging conventional motivations for axion scenarios. The discussion leverages analogies between condensed matter systems and particle physics to clarify how certain symmetry breakings and topological defect mechanisms are governed by these principles.

Analogy between Condensed Matter and Particle Physics

The parallel between the mechanisms of spontaneous symmetry breaking in condensed matter physics (CMP) and the Higgs mechanism in particle physics is explored in detail. The Ginzburg-Landau (GL) theory is recognized as an Abelian Higgs model analogue, where the complex order parameter maps to scalar fields, the superconducting phase $\theta$ is the would-be Goldstone boson, and magnetic vector potentials acquire mass analogously to the Higgs effect. The author extends this analogy to multi-band superconductors where Josephson couplings induce additional collective excitations, associated with Leggett modes—a connection previously utilized for modeling extensions of the electroweak sector.

A central distinction is highlighted: in particle physics, Yukawa couplings introduce chiral interactions between scalar and fermion fields, yielding Dirac masses and CP-violating structures absent from CMP. This structure forms the core of subsequent analysis.

Yukawa Coupling, Gauge Symmetry, and the Equivalence Principle

The paper rigorously examines the behavior of Yukawa interactions under global and local gauge transformations, first within a $U(1)$ context and then generalized to $SU(2) \times U(1)$. The principal observations are:

• Global Gauge Non-invariance: The Yukawa coupling is not invariant under global phase shifts of scalar fields unless the corresponding phases of the Dirac fields are synchronously rotated, a coordination enforced only dynamically in presence of the gauge fields.
• Role of Local Gauge Transformations: The local gauge symmetry enables the gauge field to absorb phase fluctuations (Higgs mechanism), transferring the scalar phase into the gauge sector and ensuring physical observables (e.g., Dirac masses) are determined by the modulus of the scalar field, up to a discrete choice of vacuum.
• Equivalence Principle as Physical Selector of Vacuum Phase: Critically, the author demonstrates that global phase misalignments of the Yukawa coupling result in physical consequences: they would yield fermion masses containing pseudoscalar contributions, breaking P and CP symmetries and, more fundamentally, generating parity-violating gravitational potentials (via the Einstein equations for the energy source).

This is shown to prevent simultaneous removal of Christoffel symbols in left- and right-handed sectors, violating the existence of local inertial frames and, therefore, the equivalence principle. To uphold the equivalence principle, the vacuum phase must be universally fixed (e.g., $\theta_0 = 0$), which dynamically restores P and CP invariance. The upshot is that the equivalence principle, not just gauge symmetry, is responsible for selecting CP-conserving vacua in the presence of Yukawa couplings.

Topological Defects and the Kibble-Zurek Mechanism

A key implication follows for cosmological topological defect formation. In systems with only global gauge symmetry, the Kibble-Zurek mechanism predicts the formation of domain structures (e.g., cosmic strings, domain walls) due to uncorrelated vacuum choices in causally disconnected regions. However, if the equivalence principle enforces a universal phase assignment, these mechanisms are suppressed; the vacuum phase is homogeneous everywhere, precluding the spontaneous formation of such defects.

Topological defect formation is thereby permitted only in the presence of local gauge symmetry, where nontrivial field configurations (supported by thermal fluctuations of magnetic flux) can become topologically stabilized during symmetry breaking. The legacy cosmological inflation hypothesis for eliminating relic defects is rendered unnecessary if this property is fundamental.

Multiple Vacua and Domain Walls in Combined Gauge Groups

For combined gauge groups such as $SU(2)\times U(1)$, the vacuum structure is richer: several discrete vacua exist, distinguished by their equilibrium phases, leading to the possibility of domain walls separating regions with different vacua. Here, the equivalence principle restricts all local vacua to select identical phases, but the presence of several physically equivalent vacua introduces the possibility of cosmologically extended domain walls under specific circumstances. The Yukawa/gauge dynamics in the electroweak theory are thus constrained not just by symmetry but by the geometric requirements of gravity.

Resolution of the Strong CP Problem

The strong CP problem in QCD—arising from the apparent noninvariance of the Lagrangian under CP due to $\theta$-vacuum terms and complex phases in the quark mass matrix—is critically reexamined. The conventional Peccei-Quinn solution, introducing axions to dynamically cancel the effective $\bar{\theta}$ parameter, is argued to be unnecessary. According to the paper:

• The equivalence principle enforces $\beta=0$ for the chiral mass phase and, via a permitted chiral rotation, $\theta_0=0$ for the $\theta$-term.
• This mechanism eliminates the parameter space yielding observable CP violation in strong interactions, and predicts that strong CP violation must be vanishingly small, in line with current experimental neutron EDM constraints.
• The absence of axion signals in contemporary searches is asserted to be a natural consequence; the axion field is rendered redundant without additional theoretical constructs.

The framework ensures that weak CP violation via the CKM matrix remains consistent: only strong eigenstates, corresponding to the diagonal mass basis, are relevant for gravitational interactions.

Theoretical and Practical Implications

The discussion elevates the status of the equivalence principle within quantum field theory, attributing it a dynamically constraining role not only for gravity but across the entire vacuum selection mechanism of the Standard Model and beyond. The elimination of axion requirements simplifies model building for baryogenesis, dark matter, and precision EDM calculations. The suppression of cosmological topological defects has direct consequences for early universe phase transitions and the absence of relic signals.

Future directions may include explicit computation of gravitational corrections to vacuum selection, generalized treatment in nonminimal scalar-tensor theories or in beyond-standard-model settings, and reanalysis of mechanisms for baryogenesis or leptogenesis that depend on nontrivial vacuum structures.

Conclusion

This work articulates a comprehensive theoretical argument that the equivalence principle mandates universal phase synchronization across scalar and fermion fields coupled via Yukawa interactions. This synchronization enforces CP-symmetric vacua, suppresses topological defect formation in purely global gauge systems, and precludes observable CP violation in strong interactions without recourse to additional axion fields. The equivalence principle hence emerges as a physically operative constraint not just in gravity, but in the selection and structure of particle physics vacua.