<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi></mml:math> boson interpolating between a generalized dark photon or dark <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Z</mml:mi></mml:math>, an axial boson, and an axionlike particle
P. Fayet
Abstract
A light boson $U$ from an extra $U(1)$ interpolates between a generalized dark photon coupled to $Q$, $B$, and ${L}_{i}$ (or $B\ensuremath{-}L$), plus possibly dark matter, a dark $Z$ coupled to the $Z$ current, and one axially coupled to quarks and leptons. We identify the corresponding $U(1{)}_{F}$ symmetries, with $F={\ensuremath{\gamma}}_{Y}Y+{\ensuremath{\gamma}}_{B}B+{\ensuremath{\gamma}}_{{L}_{i}}{L}_{i}+{\ensuremath{\gamma}}_{A}{F}_{A}+{\ensuremath{\gamma}}_{{F}^{\ensuremath{'}}}{F}^{\ensuremath{'}}+{\ensuremath{\gamma}}_{d}{F}_{d}$, ${F}_{d}$ acting in a dark sector and ${F}^{\ensuremath{'}}$ on possible semi-inert Brout-Englert-Higgs (BEH) doublets uncoupled to quarks and leptons. The $U$ current is obtained from the $U(1{)}_{F}$ and $Z$ currents, with a mixing determined by the spin-0 BEH fields. The charge ${Q}_{U}$ of chiral quarks and leptons is a combination of $Q$, $B$, ${L}_{i}$, and ${T}_{3L}$ with the axial ${F}_{A}$. It involves in general isovector and isoscalar axial terms, in the presence of two BEH doublets. A longitudinal $U$ with axial couplings has enhanced interactions and behaves much as an axionlike particle. Its axial couplings ${g}_{A}$, usually restricted to $\ensuremath{\lesssim}\text{ }2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\text{ }{m}_{U}(\mathrm{MeV})$, lead to effective pseudoscalar ones ${g}_{P}={g}_{A}\ifmmode\times\else\texttimes\fi{}2{m}_{q,l}/{m}_{U}={2}^{1/4}{G}_{F}^{1/2}{m}_{q,l}\text{ }{A}_{\ifmmode\pm\else\textpm\fi{}}$. ${A}_{\ifmmode\pm\else\textpm\fi{}}$ is proportional to an invisibility parameter $r=\mathrm{cos}{\ensuremath{\theta}}_{A}$ induced by a singlet v.e.v., possibly large and allowing the $U$ to be very weakly interacting. This allows for a very small gauge coupling, expressed with two doublets and a singlet as ${g}^{\ensuremath{'}\ensuremath{'}}/4\ensuremath{\simeq}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }{m}_{U}(\mathrm{MeV})\text{ }r/\mathrm{sin}2\ensuremath{\beta}$. We discuss phenomenological implications for meson decays, neutrino interactions, atomic-physics parity violation, naturally suppressed ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}U$ decays, etc. The $U$ boson fits within the grand unification framework, in symbiosis with a $SU(4{)}_{\mathrm{es}}$ electrostrong symmetry broken at the grand unification scale, with ${Q}_{U}$ depending on $Q$, $B\ensuremath{-}L$, ${F}_{A}$, and ${T}_{3A}$ through three parameters ${\ensuremath{\gamma}}_{Y}$, ${\ensuremath{\gamma}}_{A}$, and $\ensuremath{\eta}$.