Electroweak $\eta_{\rm w}$ meson

We argue that the Standard Model is accompanied by a new pseudo-scalar degree of freedom, $\eta_{\rm w}$-meson, which cancels the topological susceptibility of the electroweak vacuum and gets its mass from this effect. The prediction is based on the analyticity properties of the Chern-Simons correlator combined with the basic features of gravity. Depending on the quality level of the $U(1)_{B+L}$-symmetry, $\eta_{\rm w}$ emerges as a $B+L$ pseudo-Goldstone boson or as a St\"uckelberg $2$-form of the electroweak gauge redundancy. An intriguing scenario of the first category is the emergence of $\eta_{\rm w}$ in the form of the phase of a $U(1)_{B+L}$-violating fermion condensate triggered by the instantons, somewhat similar to $\eta'$-meson in QCD. Regardless of its origin, the presence of $\eta_{\rm w}$-meson in the theory appears to be a matter of consistency.