Nontriviality of dynamical Chern-Simons gravity and the standard model

Given the growing interest in gravitational-wave and cosmological parity-violating effects in dynamical Chern-Simons (dCS) gravity, it is crucial to investigate whether the scalar-gravitational Pontryagin term in dCS gravity persists when formulated in the context of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mrow><a:mi mathvariant="normal">U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:mo stretchy="false">)</a:mo></a:mrow><a:mrow><a:mi mathvariant="normal">B</a:mi><a:mo>−</a:mo><a:mi mathvariant="normal">L</a:mi></a:mrow></a:msub></a:math> anomaly in the standard model (SM). In particular, it has been argued that dCS gravity can be reduced to Einstein gravity after “rotating away” the gravitational-Pontryagin coupling into the phase of the Weinberg operator—analogous to the rotation of the axion zero mode into the quark mass matrix. We find that dCS gravity is nontrivial if the scalar field <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>ϕ</h:mi></h:math> has significant spacetime dependence from dynamics. We provide a comprehensive consideration of the dCS classical and quantum symmetries relevant for embedding a dCS sector in the SM. We find that, because of the baryon/lepton number chiral gravitational anomaly, the scalar-Pontryagin term cannot be absorbed by a field redefinition. Assuming a minimal extension of the SM, we also find that a coupling of the dCS scalar with right-handed neutrinos induces both the scalar-Pontryagin coupling and an axionlike phase in the dimension-five Weinberg operator. We comment on the issue of gauging the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mrow><j:mi mathvariant="normal">U</j:mi><j:mo stretchy="false">(</j:mo><j:mn>1</j:mn><j:mo stretchy="false">)</j:mo></j:mrow><j:mrow><j:mi mathvariant="normal">B</j:mi><j:mo>−</j:mo><j:mi mathvariant="normal">L</j:mi></j:mrow></j:msub></j:math>, the observational effects with these two operators present for upcoming experiments, and the origin of dCS gravity in string theory.