# Naturalising the Third Family Hypercharge Model for Neutral Current $B-$Anomalies

## TL;DR

This paper proposes a natural deformation of the Third Family Hypercharge Model with a family-dependent $U(1)_F$ gauge symmetry, allowing a $Z'$ boson to explain $B$-meson decay anomalies while satisfying experimental constraints.

## Contribution

It introduces a modified $U(1)_F$ charge assignment that enables a $Z'$ to address $B$-anomalies without large lepton mixing or violating constraints.

## Key findings

- The model fits $B$-decay anomalies with $Z'$ mass between 1.2 and 12.5 TeV.
- It avoids large lepton flavor violation by specific charge assignments.
- The $Z'$ mediates effective operators explaining the $B$-decay discrepancies.

## Abstract

We consider a deformation of the Third Family Hypercharge Model, which arguably makes the model more natural. Additional non-zero charges of the spontaneously broken, family-dependent $U(1)_F$ gauge symmetry are assigned to the second family leptons, and the third family leptons' charges are deformed away from their hypercharges in such a way that the $U(1)_F$ gauge symmetry remains anomaly-free. Second family $U(1)_F$ lepton charges allow a $Z^\prime$ coupling to muons without having to assume large charged lepton mixing, which risks violating tight lepton flavour violation bounds. In this deformed version, only the bottom and top Yukawa couplings are generated at the renormalisable level, whereas the tauon Yukawa coupling is absent. The $Z^\prime$ mediates a beyond the Standard Model contribution to an effective $(\bar b s) (\bar \mu \mu)$ vertex in the combination $C_9=-9C_{10}$ and is able to fit the apparent discrepancy between Standard Model predictions in flavour changing neutral-current $B-$meson decays and their measurements, whilst simultaneously avoiding current constraints from direct $Z^\prime$ searches and other measurements, when $1.2\ \text{TeV} < M_{Z^\prime} < 12.5\ \text{TeV}$.