[Paper Review] Absence of $CP$ violation in the strong interactions

Wen-Yuan Ai, Juan S. Cruz+2 more|arXiv (Cornell University)|Jan 20, 2020

Particle physics theoretical and experimental studies24 references22 citations

TL;DR

This paper resolves the strong CP problem by demonstrating that CP violation in QCD is absent when the infinite spacetime volume limit is taken before summing over topological sectors. Using one-instanton background calculations and the Atiyah-Singer index theorem, it shows that chiral phases from complex quark masses and vacuum angle θ align in the infinite-volume limit, canceling CP-violating effects. The key result is that no observable CP violation arises in strong interactions without extending the Standard Model, provided the correct order of limits is applied.

ABSTRACT

We derive correlation functions for massive fermions with a complex mass in the presence of a general vacuum angle. For this purpose, we first build the Green's functions in the one-instanton background and then sum over the configurations of background instantons. The quantization of topological sectors follows for saddle points of finite Euclidean action in an infinite spacetime volume and the fluctuations about these. For the resulting correlation functions, we therefore take the infinite-volume limit before summing over topological sectors. In contrast to the opposite order of limits, the chiral phases from the mass terms and from the instanton effects then are aligned so that, in absence of additional phases, these do not give rise to observables violating charge-parity symmetry. This result is confirmed when constraining the correlations at coincident points by using the index theorem instead of instanton calculus.

Motivation & Objective

• To resolve the strong CP problem by clarifying the role of the vacuum angle θ and quark mass phases in CP violation.
• To clarify the order of limits—volume infinity before topological sector interference—as crucial for determining whether CP violation appears in QCD.
• To demonstrate that CP-violating effects from complex quark masses and the θ-term cancel when the correct limit order is applied.
• To confirm the result using both instanton calculus and the Atiyah-Singer index theorem, ensuring it holds beyond perturbation theory.

Proposed method

• Constructs Green's functions in the one-instanton background to compute fermion correlation functions with complex quark masses and vacuum angle θ.
• Performs a summation over instanton configurations after taking the infinite spacetime volume limit, ensuring topological quantization via saddle points of finite Euclidean action.
• Applies the Atiyah-Singer index theorem to constrain the coincident-point limit of fermion correlations, confirming the absence of CP violation independently of instanton calculus.
• Uses a ϑ-adjoint inner product formalism to handle complex eigenvalues of the Dirac operator with complex mass terms, preserving unitarity and spectral properties.
• Demonstrates that field redefinitions cannot remove the phase θ + ∑αj, but this combination remains unobservable when the infinite-volume limit is taken first.
• Analyzes the Hamiltonian formulation in the wave functional approach, showing that the physical predictions are invariant under field redefinitions that shift θ, confirming the robustness of the result.

Experimental results

Research questions

• RQ1 Does the vacuum angle θ in QCD lead to observable CP violation when combined with complex quark masses?
• RQ2 What is the physical consequence of the order in which the infinite-volume limit and topological sector interference are performed?
• RQ3 Can the absence of CP violation in the strong interactions be rigorously shown beyond perturbation theory?
• RQ4 Is the CP-odd combination θ + ∑αj physically observable, or does it cancel due to the limit order?
• RQ5 Does the Atiyah-Singer index theorem confirm the absence of CP violation in the coincident-point limit of fermion correlations?

Key findings

• CP violation in the strong interactions is absent when the infinite spacetime volume limit is taken before summing over topological sectors.
• The chiral phases from complex quark masses and the vacuum angle θ align in the infinite-volume limit, preventing the formation of observable CP-violating phases.
• The result is confirmed independently using the Atiyah-Singer index theorem, showing that the coincident-point limit of fermion correlations does not exhibit CP violation.
• The combination θ + ∑αj, which is invariant under field redefinitions, remains unobservable due to the correct limit order, resolving the strong CP problem without new physics.
• The path integral formulation projects onto pure gauge configurations at infinity only when the infinite-volume limit is taken first, ensuring topological quantization and the absence of CP violation.
• The spectral decomposition of the fermionic propagator in Minkowski space with complex mass terms shows that the propagator remains CP-conserving when the correct limit order is applied.