Macroscopic smoothing scale and functional form for the Kronecker-delta term governing CDW–homogeneous interpolation

Ascertain the macroscopic length scale associated with chiral density wave domain sizes and construct a smooth functional replacement for the Kronecker-delta term in the effective potential ΔU(φ,q) that interpolates between the homogeneous (q=0) and CDW (q≠0) states, enabling a physically consistent modeling of the nucleation barrier and bubble dynamics.

Background

In the nucleon–meson model used to study the CDW phase, the effective potential includes a Kronecker-delta term that distinguishes the homogeneous and inhomogeneous configurations. For nucleation calculations, a discontinuous step is unphysical; it should be replaced by a smooth interpolation governed by a macroscopic length scale (e.g., CDW domain size).

The authors emphasize that this macroscopic scale is unknown, preventing a reliable treatment of the nucleation barrier and bubble growth dynamics in the CDW–hadronic phase transition.

References

Moreover, there is a subtlety in treating the Kronecker delta term; this should actually be a smooth function connecting from zero to one with a macroscopic length scale, such as the size of the CDW domain. This scale is completely unknown to us.

— Revisiting QCD-induced little inflation with chiral density wave state and its implications on pulsar timing array gravitational-wave signals  (2603.29772 - Jung et al., 31 Mar 2026) in Subsection "Spinodal line of the CDW phase" (Section 4.2)