| Summary: | I discuss how the axial current coupling constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>A</mi></msub></semantics></math></inline-formula> renormalized in scale symmetric chiral EFT defined at a chiral matching scale impacts on the axial current matrix elements on beta decays in nuclei with and without neutrinos. The “quenched” <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>A</mi></msub></semantics></math></inline-formula> observed in nuclear superallowed Gamow–Teller transitions, a long-standing puzzle in nuclear physics, is shown to encode the emergence of chiral-scale symmetry hidden in QCD in the vacuum. This enables one to explore how trace-anomaly-induced scale symmetry breaking enters in the renormalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>A</mi></msub></semantics></math></inline-formula> in nuclei applicable to certain non-unique forbidden processes involved in neutrinoless double beta decays. A parallel is made between the roles of chiral-scale symmetry in quenching <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>A</mi></msub></semantics></math></inline-formula> in highly dense medium and in hadron–quark continuity in the EoS of dense matter in massive compact stars. A systematic chiral-scale EFT, presently lacking in nuclear theory and potentially crucial for the future progress, is suggested as a challenge in the field.
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