# Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$
Mass

Marcel Algueró<EMAIL_ADDRESS>Grup de Fisica Teòrica (Departament de
Fisica), Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona)
Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science
and Technology, Campus UAB, E-08193 Bellaterra (Barcelona)  Andreas Crivellin
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  Claudio Andrea Manzari
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  Joaquim Matias<EMAIL_ADDRESS>Grup de
Fisica Teòrica (Departament de Fisica), Universitat Autònoma de Barcelona,
E-08193 Bellaterra (Barcelona)  Institut de Fisica d’Altes Energies (IFAE),
The Barcelona Institute of Science and Technology, Campus UAB, E-08193
Bellaterra (Barcelona)

###### Abstract

The discrepancies between the measurements of rare (semi-)leptonic $B$ decays
and the corresponding Standard Model predictions point convincingly towards
the existence of new physics for which a heavy neutral gauge boson
($Z^{\prime}$) is a prime candidate. However, the effect of the mixing of the
$Z^{\prime}$ with the SM $Z$, even though it cannot be avoided by any
symmetry, is usually assumed to be small and thus neglected in
phenomenological analyses. In this letter we point out that a mixing of the
naturally expected size leads to lepton flavour universal contributions,
providing a very good fit to $B$ data. Furthermore, the global electroweak fit
is affected by $Z-Z^{\prime}$ mixing where the tension in the $W$ mass,
recently confirmed and strengthened by the CDF measurement, prefers a non-zero
value of it. We find that a $Z^{\prime}$ boson with a mass between $\approx
1-5\rm{TeV}$ can provide a unified explanations of the $B$ anomalies and the
$W$ mass. This strongly suggests that the breaking of the new gauge symmetry
giving raise to the $Z^{\prime}$ boson is linked to electroweak symmetry
breaking with intriguing consequences for model building.

††preprint: PSI-PR-22-02, ZU-TH 02/22

##  I Introduction

Even though the LHC has not discovered any particles beyond the ones of the
Standard Model (SM) yet, in the last years intriguing hints for the violation
of lepton flavour universality (LFU) have been accumulated (see e.g. Refs.
Crivellin and Hoferichter (2021); Fischer  _et al._ (2021); Crivellin and
Matias (2022) for recent reviews). Among them, the updated measurement of the
ratios of semi-leptonic rare $B$ meson decay $R_{K^{+}}={\cal B}(B^{+}\to
K^{+}\mu^{+}\mu^{-})/{\cal B}(B^{+}\to K^{+}e^{+}e^{-})$ Hiller and Kruger
(2004) by LHCb Aaij  _et al._ (2021a) is particularly prominent since it
provides first evidence for LFU violation (LFUV) in a single observable.
Furthermore, when combining all tests of LFUV (like $R_{K^{+}}$) Aaij  _et
al._ (2017); Capdevila  _et al._ (2016); Wehle  _et al._ (2017); Aaij  _et
al._ (2021b) with $B$ decays involving muon pairs (most prominently
$P_{5}^{\prime}$ Descotes-Genon  _et al._ (2013); Aaij  _et al._ (2020) and
$B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij  _et al._ (2021c, d)), one finds a
preference for new physics (NP) hypotheses of more than $7\,\sigma$ Algueró
_et al._ (2021) compared to the SM 111Very close results and pulls were found
in Ref. Hurth  _et al._ (2022) using also a complete set of observables but a
different treatment of hadronic uncertainties. See also Ref. Altmannshofer and
Stangl (2021) for an analysis using a smaller subset of the available data as
well as Refs. Kowalska  _et al._ (2019); Blake  _et al._ (2020); Geng  _et
al._ (2021); Ciuchini  _et al._ (2019) and Ref. London and Matias (2021) for a
detailed comparison.. Note that such a high significance is only possible
since all measurements are compatible with each other, i.e. they form a
coherent picture.

Simple patterns where NP couples solely to muons can in fact explain the
discrepancies between the SM and experiment in rare semi-leptonic $B$ decays
very well. However, it turns out that structures with additional LFU
contributions can describe data even better Algueró  _et al._ (2019a, 2022).
This means that allowing simultaneously for presence of LFUV and LFU NP
effects, one can further improve the goodness of the global fits. Indeed, some
of these hypotheses exhibit the highest significance among all studied
scenarios Capdevila  _et al._ (2018); Algueró  _et al._ (2019b, 2021);
Altmannshofer and Stangl (2021).222In fact, several models giving raise to
combined LFU and LFUV contributions, including 2HDMs Crivellin  _et al._
(2019a), leptoquarks Crivellin  _et al._ (2019b, 2020), $SU(2)_{L}$ triplets
vector bosons Capdevila  _et al._ (2021) and models with vector-like quarks
Bobeth  _et al._ (2017); Crivellin  _et al._ (2021), have been proposed in the
literature.

In this letter, we point out that, extending the SM by a new heavy neutral
gauge boson ($Z^{\prime}$), one has, in addition to the usually considered
direct LFUV effect in $b\to s\ell^{+}\ell^{-}$ Buras and Girrbach (2013);
Gauld  _et al._ (2014a, b); Altmannshofer  _et al._ (2014a); Crivellin  _et
al._ (2015a, b); Niehoff  _et al._ (2015); Aristizabal Sierra  _et al._
(2015); Carmona and Goertz (2016); Falkowski  _et al._ (2015); Celis  _et al._
(2016, 2015); Crivellin  _et al._ (2015c); Boucenna  _et al._ (2016a);
Altmannshofer  _et al._ (2016); Boucenna  _et al._ (2016b); Crivellin  _et
al._ (2017); Garcia Garcia (2017); Faisel and Tandean (2018); King (2017);
Chiang  _et al._ (2017); Di Chiara  _et al._ (2017); Ko  _et al._ (2017);
Sannino  _et al._ (2018); Carmona and Goertz (2018); Raby and Trautner (2018);
Falkowski  _et al._ (2018); Benavides  _et al._ (2020); Maji  _et al._ (2019);
Singirala  _et al._ (2019); Guadagnoli  _et al._ (2018); Allanach and Davighi
(2018); Kohda  _et al._ (2018); King (2018); Duan  _et al._ (2019); Rocha-
Moran and Vicente (2019); Dwivedi  _et al._ (2020); Foldenauer (2019); Ko  _et
al._ (2019); Allanach and Davighi (2019); Kawamura  _et al._ (2019);
Altmannshofer  _et al._ (2020); Calibbi  _et al._ (2020); Aebischer  _et al._
(2020); Kawamura  _et al._ (2020); Crivellin  _et al._ (2021); Allanach
(2021); Greljo  _et al._ (2021); Davighi (2021); Bause  _et al._ (2022);
Allanach  _et al._ (2021a); Navarro and King (2021); Ko  _et al._ (2021);
Allanach  _et al._ (2021b), also a LFU effect, which is generated via
$Z-Z^{\prime}$ mixing. In fact, because both bosons have the same quantum
numbers, this mixing cannot be avoided by any symmetry. Furthermore, in the
case that electroweak (EW) symmetry breaking and the breaking of the symmetry
giving rise to the $Z^{\prime}$ mass are connected, one even expects a mixing
of the order of $m_{Z}^{2}/m_{Z^{\prime}}^{2}$. Importantly, $Z-Z^{\prime}$
mixing has also an impact on the global EW fit, in particular on
$Z\ell^{+}\ell^{-}$ and $Z\nu\nu$ couplings and if the $Z^{\prime}$ is an
$SU(2)_{L}$ singlet (i.e. not the neutral component of an $SU(2)_{L}$
multiplet), in addition the prediction of the $W$ mass is altered compared to
the SM. The latter is very important since the global EW fit displayed a
tension of $1.8\,\sigma$ de Blas  _et al._ (2021) in this observable. This
discrepancy was recently confirmed and strengthened by the CDF measurement
Aaltonen  _et al._ (2022) whose central value is $7\,\sigma$ above the SM
prediction Zyla  _et al._ (2020a). Combining this new measurement with the
existing ones from the LHC Aaboud  _et al._ (2018a); Chatrchyan  _et al._
(2011); Aaij  _et al._ (2015, 2022), one finds $m_{W}=(80.4133\pm 0.0080)$GeV
and $m_{W}=(80.413\pm 0.015)$GeV, where in the second formula the error has
been inflated to reflect the tensions between the different measurements. The
SM prediction is given by $m_{W}^{\rm SM}=(80.3499\pm 0.0056)$GeV, and
$m_{W}^{\rm SM}=(80.3505\pm 0.0077)$GeV for a conservative error estimate de
Blas  _et al._ (2022). This corresponds to a $6.5\,\sigma$ and $3.7\,\sigma$
tension for the standard and the conservative scenario, respectively.333Note
that this article was submitted before the $W$ mass measurement of CDF was
released such that this result can be considered as a confirmation of the
prediction of the original version of the manuscript.

Therefore, in $Z^{\prime}$ models an interesting interplay between $b\to
s\ell^{+}\ell^{-}$ processes and the global EW fit arises if the
$Z-Z^{\prime}$ mixing angle is non-zero Allanach  _et al._ (2021a). While this
mixing has usually been assumed to be negligibly small 444Note that the effect
of $Z-Z^{\prime}$ mixing in the $W$ mass in the context of $b\to
s\ell^{+}\ell^{-}$ was already pointed out in Ref. Allanach  _et al._ (2021a).
However, the impact on, and the correlations with, the $b\to
s\ell^{+}\ell^{-}$ fit were not shown. , the goal of this letter is to assess
the size and impact of $Z-Z^{\prime}$ mixing via a combined analysis of
flavour and EW data, providing a unified explanation of both anomalies.

##  II Setup

We extend the SM by adding a heavy neutral $SU(2)_{L}$ singlet gauge boson.
Following the notation of Ref. del Aguila  _et al._ (2010); de Blas  _et al._
(2013) the kinetic term and the mass term of this new boson, before EW
symmetry breaking, are

$\displaystyle\begin{aligned}
\mathcal{L}_{Z^{\prime}_{0}}=&-\frac{1}{4}Z^{\prime}_{0,\mu\nu}Z_{0}^{\prime\mu\nu}+\frac{\mu_{Z}^{\prime
2}}{2}Z^{\prime}_{0\mu}Z^{\prime\mu}_{0}\,\\\
&+g_{Z^{\prime}}Z^{\prime}_{0\mu}Z^{\prime\mu}_{0}\phi^{\dagger}\phi-
ig_{Z^{\prime}}^{\phi}Z^{\prime\mu}_{0}\phi^{\dagger}\overleftrightarrow{D}_{\mu}\phi\,,\end{aligned}$
(1)

where
$Z^{\prime}_{0,\mu\nu}\equiv\partial_{\mu}Z^{\prime}_{0\nu}-\partial_{\nu}Z^{\prime}_{0\mu}$
is the field strength tensor,
$\overset{\leftrightarrow}{D}_{\mu}\;=\overset{\rightarrow}{D}_{\mu}-(\overset{\leftarrow}{D}_{\mu})^{\dagger}$,
$\phi$ is the SM Higgs $SU(2)_{L}$ doublet and we use
$D_{\mu}=\partial_{\mu}+ig_{2}W^{a}_{\mu}T^{a}+ig_{1}YB_{\mu}$ as the
definition of the SM part of the covariant derivative and
$g_{Z^{\prime}}^{\phi}$ is real by hermicity. The physical $Z$ and
$Z^{\prime}$ masses are obtained from diagonalizing the mass matrix

$\mathcal{M}^{2}=\begin{pmatrix}m_{Z_{0}}^{2}&-\frac{y}{c_{W}}\\\
-\frac{y}{c_{W}}&m_{Z^{\prime}_{0}}^{2}\end{pmatrix}\,,\qquad
y\equiv\frac{v^{2}}{2}\,g_{2}\,g_{Z^{\prime}}^{\phi}\,,$ (2)

in the $Z_{0},\,Z_{0}^{\prime}$ basis, where $Z_{0}$ coincides with the the SM
$Z$ for $g_{Z^{\prime}}^{\phi}=0$ with
$m_{Z_{0}}^{2}=\frac{v^{2}}{4}\left(g_{1}^{2}+g_{2}^{2}\right)$,
$\frac{v}{\sqrt{2}}\approx 174\,$GeV and $c_{W}$ is the cosine of the Weinberg
angle. At leading order in $v/m_{Z_{0}^{\prime}}$ we have

$\displaystyle m_{Z}^{2}\simeq$
$\displaystyle\;m_{Z_{0}}^{2}-\frac{y^{2}}{c_{W}^{2}m_{Z_{0}^{\prime}}^{2}}\equiv
m_{Z_{0}}^{2}\left(1+\delta m_{Z}^{2}\right)\,.$ (3)

Note that the corrections to the mass of the $Z$ with respect to the SM value
$m_{Z_{0}}$ can only be negative. The mass eigenstates $Z^{(\prime)}$ can then
be expressed as

$\left({\begin{array}[]{*{20}{c}}Z\\\
{Z^{\prime}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}{Z_{0}^{\prime}\sin\xi+Z_{0}\cos\xi}\\\
{Z_{0}^{\prime}\cos\xi\;-Z_{0}\sin\xi}\end{array}}\right)\,,$ (4)

where $\sin\xi\simeq\frac{y}{c_{W}m_{Z_{0}^{\prime}}^{2}}$ describes the
$Z-Z^{\prime}$ mixing.

The interactions with the SM fields are given by

$\displaystyle\begin{split}\mathcal{L}_{Z^{\prime}_{0}}^{\rm
fermions}=&{\bar{u}}_{j}{\gamma_{\mu}}(g_{ji}^{uL}{P_{L}}+g_{ji}^{uR}{P_{R}}){u_{i}}\,Z^{\prime\mu}_{0}\\\
&+{\bar{d}}_{j}{\gamma_{\mu}}(g_{ji}^{dL}{P_{L}}+g_{ji}^{dR}{P_{R}}){d_{i}}\,Z^{\prime\mu}_{0}\\\
&+g_{ji}^{\ell
L}({\bar{\nu}}_{j}{\gamma_{\mu}}{P_{L}}{\nu_{i}})\,Z^{\prime\mu}_{0}\\\
&+\bar{\ell}_{j}{\gamma_{\mu}}(g_{ji}^{\ell L}{P_{L}}+g_{ji}^{\ell
R}{P_{R}}){\ell_{i}}\,Z^{\prime\mu}_{0}\,,\end{split}$ (5)

where, in the down basis,
$g_{ji}^{uL}=V_{jk}g_{kk^{\prime}}^{dL}V_{ik^{\prime}}^{*}$. Note that the
couplings to left-handed charged leptons and neutrinos (up and down quarks)
are the same (up to a CKM rotation), due to $SU(2)_{L}$ invariance and that
only the relative phase between $\sin\xi$ and $g^{L,R}_{ij}$ is physical, such
that one can assume $\sin\xi$ to be positive without loss of generality. In
the following, we will assume flavour diagonal coupling to leptons and in the
quark sector disregard all couplings except left-handed $b-s$ couplings.

##  III Observables

###  III.1 $b\to s\ell^{+}\ell^{-}$

| Best-fit point |  $1\,\sigma$ CI |  $2\,\sigma$ CI
---|---|---|---
${\cal C}_{9\mu}^{\rm V}$ | $-0.96$ | $[-1.11,-0.80]$ | $[-1.25,-0.64]$
${\cal C}_{10}^{\rm U}=-k{\cal C}_{9}^{\rm U}$ | $+0.30$ | $[+0.15,+0.45]$ | $[+0.00,+0.61]$
Table 1:  $1\,\sigma$ and $2\,\sigma$ confidence intervals for the NP scenario
in Eq. (7) with a PullSM of $6.9\,\sigma$ and p-value=28.3%. Figure 1:
Preferred 1$\,\sigma$, 2$\,\sigma$ and 3$\,\sigma$ regions in the $({\cal
C}_{9\mu}^{\rm V},{\cal C}_{10}^{\rm U}=-k{\cal C}_{9}^{\rm U})$ plane for the
scenario discussed in the paper, including all available $b\to
s\ell^{+}\ell^{-}$ data and using the most updated version of ACDMN code
Algueró  _et al._ (2021). Note that the SM case corresponds to the (0,0)
point.

In $Z^{\prime}$ models without $Z-Z^{\prime}$ mixing, the simple one
dimensional scenario with the best fit to data is obtained from a left-handed
$b-s$ coupling and a vectorial muon coupling, i.e. the ${\cal C}_{9\mu}^{\rm
V}$ scenario Algueró  _et al._ (2021). Allowing in addition for $Z-Z^{\prime}$
mixing we have

$\displaystyle{\cal C}_{9\mu}^{\rm V}$
$\displaystyle=-\frac{\pi^{2}}{e^{2}}\frac{4\sqrt{2}g_{23}^{dL}g_{22}^{\ell
V}}{G_{F}m_{Z^{\prime}}^{2}V_{tb}V_{ts}^{*}}\,,$ (6) $\displaystyle{\cal
C}_{10}^{{\rm U}}$ $\displaystyle=-k{\cal C}_{9}^{\rm
U}=\dfrac{\sqrt{2}\pi^{2}}{e^{2}}\frac{g_{2}\,g_{23}^{dL}\sin\xi}{c_{W}G_{F}m_{Z}^{2}V_{tb}V_{ts}^{*}}\,,$

using the effective Hamiltonian of Ref. Grinstein  _et al._ (1988); Buchalla
_et al._ (1996) where $g_{22}^{\ell V}=({g_{22}^{\ell L}+g_{22}^{\ell
R}})/{2}$. This corresponds to the scenario

$\\{{\cal C}_{9\mu}^{\rm V}\,,\,{\cal C}_{10}^{\rm U}=-k{\cal C}_{9}^{\rm
U}\\}\,,$ (7)

with $k=1/(1-4s_{w}^{2})$ (see the appendix for the definitions of the
operators). The superscript ${\rm V}$ (${\rm U}$) in the Wilson coefficient
stands for a LFUV (LFU) contribution.

We perform the most recent fit Algueró  _et al._ (2021) to the scenario in Eq.
(7), including 254 observables and the latest measurements by LHCb of LFUV
observables, namely, $R_{K^{0}_{S}}$ Aaij  _et al._ (2021b) and $R_{K^{*+}}$
Aaij  _et al._ (2021b) as well as the new branching ratio and angular
distribution of $B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij  _et al._ (2021c, d). We
obtain the best fit point and confidence level regions in Table 1. The results
of the global fit in our scenario are shown in Fig. 1.

###  III.2 $B_{s}-\bar{B}_{s}$ Mixing

The most important constraint on $Z^{\prime}-b-s$ couplings, i.e.
$g_{23}^{dL}$, comes from $B_{s}-\bar{B}_{s}$ mixing where the contribution to
the Hamiltonian ${\cal H}_{\rm eff}={\cal C}_{1}{\cal O}_{1}$, with

${\cal
O}_{1}=\big{(}\bar{b}\gamma^{\mu}P_{L}s\big{)}\times\big{(}\bar{b}\gamma_{\mu}P_{L}s\big{)},$

is given by

${{\cal
C}_{1}}=\frac{1}{2}\bigg{(}\frac{g_{23}^{dL}}{m_{Z^{\prime}}}\bigg{)}^{2}\left(1+\frac{\alpha_{s}}{4\pi}\frac{11}{3}\right)\,,$
(8)

including the NLO matching corrections of Ref. Buras and Girrbach (2012). Note
that the effect of the mixing induced $Z-b-s$ couplings can be neglected as it
corresponds to a dimension 8 contribution. Employing the 2-loop
renormalization group evolution Ciuchini  _et al._ (1998); Buras  _et al._
(2000), this leads to an effect, normalized to the SM one, of

$\left(\frac{g^{dL}_{23}}{0.52}\right)^{2}\left(\frac{10{\rm
TeV}}{m_{Z^{\prime}}}\right)^{2}=0.110\pm 0.090$

using the bag factor of Ref. Aoki  _et al._ (2020) and the global fit to NP in
$\Delta F=2$ observables of Ref. Bona  _et al._ (2008).

###  III.3 LFUV in tau decays

Assuming lepton flavour conservation, $Z^{\prime}-W$ boxes contribute to
$\tau\to\mu\,\nu_{\tau}\overline{\nu}_{\mu}$ as Altmannshofer  _et al._
(2014a):

$\displaystyle\begin{split}\frac{\mathcal{A}(\tau\to\mu\,\nu_{\tau}\overline{\nu}_{\mu})}{\mathcal{A}(\tau\to\mu\,\nu_{\tau}\overline{\nu}_{\mu})_{SM}}=&1-\frac{3}{8\pi^{2}}\,g^{\ell
L}_{22}\,g^{\ell
L}_{33}\frac{\ln\left(\frac{m_{W}^{2}}{m_{Z^{\prime}}^{2}}\right)}{1-\frac{m_{Z^{\prime}}^{2}}{m_{W}^{2}}}\,,\end{split}$
(9)

and analogously for $\tau\to e\,\nu_{\tau}\overline{\nu}_{e}$ and $\mu\to
e\,\nu_{\mu}\overline{\nu}_{e}$. Note that at vanishing momentum transfer the
$Z^{\prime}$ induced correction to the $W$-$\ell$-$\nu$ vertex vanishes as
$SU(2)_{L}$ gauge invariance is not broken. This we compared to the
experimental results Amhis  _et al._ (2021) (see Ref. Bryman  _et al._ (2021)
for an overview on LFUV):

$\displaystyle\begin{split}\left.\frac{\mathcal{A}\left[\tau\to\mu\nu\bar{\nu}\right]}{\mathcal{A}\left[\mu\to
e\nu\bar{\nu}\right]}\right|_{\rm EXP}&=1.0029\pm 0.0014\,,\\\
\left.\frac{\mathcal{A}\left[\tau\to\mu\nu\bar{\nu}\right]}{\mathcal{A}\left[\tau\to
e\nu\bar{\nu}\right]}\right|_{\rm EXP}&=1.0018\pm 0.0014\,,\\\
\left.\frac{\mathcal{A}\left[\tau\to
e\nu\bar{\nu}\right]}{\mathcal{A}\left[\mu\to
e\nu\bar{\nu}\right]}\right|_{\rm EXP}&=1.0010\pm 0.0014\,,\end{split}$ (10)

with the correlation matrix given in Ref. Amhis  _et al._ (2021).555Here we
neglected semi-leptonic tau decays as well as other probes of LFUV in the
charged current which are not affected in the absence of quark coupling (see
Ref. Bryman  _et al._ (2021) for a recent review).

###  III.4 Electroweak fit

The EW sector of the SM has been tested with a very high precision at LEP
Schael  _et al._ (2013, 2006) but also at the Tevatron Aaltonen  _et al._
(2013) and the LHC Aaboud  _et al._ (2018a); Chatrchyan  _et al._ (2011); Aaij
_et al._ (2015). Since it can be parametrized by only three Lagrangian
parameters, we choose as usual the set with the smallest experimental error
consisting of the Fermi constant ($G_{F}=1.1663787(6)\times 10^{-5}\,{\rm
GeV}^{-2}$ Zyla  _et al._ (2020b)), the mass of the $Z$ boson
($m_{Z}=91.1875(21)$ GeV Schael  _et al._ (2006)) and the fine structure
constant $\alpha_{em}=7.2973525664(17)\times 10^{-3}$ Zyla  _et al._ (2020b);
Mohr  _et al._ (2016); Bouchendira  _et al._ (2011); Parker  _et al._ (2018).

In our model, the relation between the Lagrangian values and the measurements
of $G_{F}$ and $m_{Z}$ is shifted with respect to the SM. While the effect in
$\mu\to e\nu\bar{\nu}$ is analogous to the one in $\tau\to\mu\nu\bar{\nu}$
discussed above we have $\frac{m_{Z}^{2}}{m_{Z_{0}}^{2}}\approx
1-\sin\xi^{2}\frac{m_{Z^{\prime}_{0}}^{2}}{m_{Z_{0}}^{2}}$. However, since the
$Z$ mass is used as an input, this translates into a shift in the $W$ mass
prediction of approximately

$\displaystyle\frac{m_{W}^{2}}{m_{W_{0}}^{2}}\approx
1+\sin\xi^{2}\frac{m_{Z^{\prime}_{0}}^{2}}{m_{Z_{0}}^{2}}\,.$ (11)

Note that this shift is positive definite such that the corresponding tension
can be explained.

This modification of the $W$ mass as well as $Z\ell\ell$ and $Z\nu\nu$ Schael
_et al._ (2006) are implemented in HEPfit De Blas  _et al._ (2020) (including
the $Z^{\prime}$ vertex corrections Altmannshofer  _et al._ (2014a); Haisch
and Westhoff (2011)). In addition, the Higgs mass ($m_{H}=125.16\pm 0.13$ GeV
Aaboud  _et al._ (2018b); Sirunyan  _et al._ (2020)), the top mass
($m_{t}=172.80\pm 0.40$ GeV Tev (2016); Aaboud  _et al._ (2019); Sirunyan  _et
al._ (2019)), the strong coupling constant ($\alpha_{s}(m_{Z})=0.1181\pm
0.0011$ Zyla  _et al._ (2020b)) and the hadronic contribution to the running
of $\alpha_{\rm em}$ ($\Delta\alpha_{\rm had}=276.1(11)\times 10^{-4}$ Zyla
_et al._ (2020b)) have been used as input parameters, since they enter EW
observables indirectly via loop effects. The complete set of observables used
are listed in the appendix.

Figure 2: Global fit to EW data, neutrino trident production, LEP bounds on
4-lepton contact interactions and $\tau\to\mu\nu\nu$ data with vectorial
flavour diagonal couplings $g^{L}_{ii}=g^{R}_{ii}=g^{V}_{i}$. Here we
marginalized over the $Z-Z^{\prime}$ mixing angle $\xi$. The 68% and 95%
confidence level regions are shown for a $Z^{\prime}$ mass of $2\,$TeV. Note
that a preference for the $L_{\mu}-L_{\tau}$ scenario emerges.

###  III.5 Neutrino Trident Production

The production of a $\mu^{+}\mu^{-}$ pair from the scattering of a muon-
neutrino off the Coulomb field of a nucleus, known as neutrino trident
production, constitutes a sensitive probe of new neutral current interactions
in the lepton sector Altmannshofer  _et al._ (2014a, b). Generalizing the
formula of Ref. Altmannshofer  _et al._ (2014b) we find

$\displaystyle\begin{split}&\dfrac{{{\sigma_{\rm SM+NP}}}}{{{\sigma_{\rm
SM}}}}=1\\\ &+8\dfrac{g_{22}^{\ell
L}}{g_{2}^{2}}\dfrac{m_{W}^{2}}{m_{Z^{\prime}}^{2}}\dfrac{\left({1+4s_{W}^{2}}\right)\left({g_{22}^{\ell
L}+g_{22}^{\ell R}}\right)+\left({g_{22}^{\ell L}-g_{22}^{\ell
R}}\right)}{\left({1+4s_{W}^{2}}\right)^{2}+1}\,.\end{split}$ (12)

This ratio is bounded by the weighted average $\sigma_{\rm
exp}/\sigma_{\mathrm{SM}}=0.83\pm 0.18$ obtained from averaging the CHARM-II
Geiregat  _et al._ (1990), CCFR Mishra  _et al._ (1991) and NuTeV results
Adams  _et al._ (1998).

###  III.6 Direct searches

LEP-II sets stringent bounds on 4-lepton operators from
$e^{+}e^{-}\to\ell^{+}\ell^{-}$ (with $\ell=e,\mu,\tau$) Schael  _et al._
(2013) for specific chiralities. A general approach to derive the constraints
for any $Z^{\prime}$ model is discussed in Refs. Falkowski and Mimouni (2016);
Buras  _et al._ (2021) which provides the formula used in our analysis. In the
limit in which the only quark couplings of the $Z^{\prime}$ are to $b-s$, LHC
searches are not very constraining and assuming a lower limit of $2\,$TeV is
not in conflict with ATLAS and CMS searches.

Figure 3: Global fit EW precision observables, neutrino trident production, LEP bounds on 4-lepton contact interactions and $\tau\to\mu\nu\nu$ data (orange) and $b\to s\ell^{+}\ell^{-}$ data (blue) in the $g^{\prime}$ \- $\sin\xi$ plane for $m_{Z^{\prime}}=2\;\rm TeV$ and $m_{Z^{\prime}}=3\;\rm TeV$. One can see that both regions overlap nicely and that a non-zero value of the mixing angle is preferred. Observable | Scenario 1 | Experiment | Pull
---|---|---|---
$R_{K^{+}}^{[1.1,6]}$ | $+0.79\pm 0.01$ | $+0.85\pm 0.04$ | $-1.3$
$R_{K^{0}_{S}}^{[1.1,6]}$ | $+0.79\pm 0.01$ | $+0.66\pm 0.20$ | $+0.7$
$R_{K^{*}0}^{[1.1,6]}$ | $+0.87\pm 0.08$ | $+0.69\pm 0.12$ | $+1.3$
$R_{K^{*+}}^{[0.045,6]}$ | $+0.84\pm 0.04$ | $+0.70\pm 0.18$ | $+0.8$
$Q_{5}^{[1.1,6]}$ | $+0.28\pm 0.02$ | $+0.66\pm 0.50$ | $-0.8$
$\langle P_{5}^{\prime}\rangle^{[4,6]}$ | $-0.57\pm 0.11$ | $-0.44\pm 0.12$ | $-0.8$
$\langle P_{5}^{\prime}\rangle^{[6,8]}$ | $-0.79\pm 0.11$ | $-0.58\pm 0.09$ | $-1.4$
$10^{7}\times{\cal B}_{B_{s}\to\phi\mu^{+}\mu^{-}}^{[4,6]}$ | $+0.78\pm 0.15$ | $+0.62\pm 0.06$ | $+1.0$
$10^{9}\times{\cal B}_{B_{s}\to\mu^{+}\mu^{-}}$ | $+3.08\pm 0.14$ | $+2.85\pm 0.34$ | $+0.6$
Table 2: Predictions for some of the most relevant observables in the $b\to
s\ell^{+}\ell^{-}$ fit within the scenario of Eq. (7). The pulls are given in
units of standard deviations.

##  IV Phenomenology

Let us now study the combined phenomenological consequences of $Z-Z^{\prime}$
mixing in rare semi-leptonic $B$ decays and the global EW fit with the aim of
obtaining a combined explanation. For this purpose we will focus on an
illustrative simplified scenario with an $SU(2)_{L}$ singlet $Z^{\prime}$,
such that $Z-Z^{\prime}$ mixing can account for the discrepancy in the $W$
mass. Furthermore, $b\to s\ell^{+}\ell^{-}$ data motivates vectorial couplings
to leptons, i.e. $g_{ii}^{\ell L}=g_{ii}^{\ell R}=g_{ii}^{\ell V}$ which also
allow for simple configurations without gauge anomalies such as
$L_{\mu}-L_{\tau}$ Altmannshofer  _et al._ (2014a); Crivellin  _et al._
(2015c) or $B_{3}-L_{2}$ Allanach (2021). In addition $g_{11}^{\ell V}=0$ and
$g_{22}^{\ell V}=-g_{33}^{\ell V}=g^{\prime}$, i.e. a $L_{\mu}=-L_{\tau}$
symmetry He  _et al._ (1991a); Foot (1991); He  _et al._ (1991b), is motivated
by the EW fit since the effect of $Z-Z^{\prime}$ mixing in $Z\to\nu\nu$ will
cancel to leading order. Therefore, larger lepton couplings are possible (see
Fig. 2) and $\tau\to\mu\nu\nu$ receives the desired constructive contribution
via $W-Z^{\prime}$ box diagrams.666Note that our analysis would to a good
approximation also apply to other scenarios, such as $B_{3}-L_{2}$. In
addition to these couplings to leptons, we assume only the presence of left-
handed $Z^{\prime}-b-s$ couplings.777Note that such a scenario could be
generated in models with vector-like quarks Altmannshofer  _et al._ (2014a);
Bobeth  _et al._ (2017) where absence of $Z^{\prime}$ couplings to light
quarks avoids problems with direct LHC searches as well as larger effects in
the total $Z$ width from mixing.

Importantly, as discussed in the introduction, the current experimental
average for the mass of the $W$ boson, shows at least a $3.7~{}\sigma$
discrepancy with the value predicted from the EW fit within the SM de Blas
_et al._ (2022). This prediction is changed in our model according to Eq. (11)
such that one accounts for data with a non-zero mixing angle of
$|\sin\xi|\simeq 3.5\times 10^{-3}\times 1{\rm TeV}/m_{Z^{\prime}}$. Moving to
the complete EW fit (including also LFUV in tau decays, LEP bounds on 4-lepton
operators and neutrino trident production) we have
$m_{Z^{\prime}},\,g^{\prime}$ and $\sin\xi$ as free parameters. However, since
all expressions depend on $g^{\prime 2}/m_{Z^{\prime}}^{2}$ despite
logarithmic terms we set $m_{Z^{\prime}}=2\,$TeV. The resulting preferred
regions from the EW fit and LFUV in tau decays are shown in Fig. 3 for
$m_{Z^{\prime}}=2$ and $3$ TeV. Including $b\to s\ell^{+}\ell^{-}$ as well as
$B_{s}-\bar{B}_{s}$ mixing, in addition $g^{dL}_{23}$ enters as a free
parameter. Marginalizing over $g^{dL}_{23}$ we find the $1\,\sigma$ and
$2\,\sigma$ regions shown in blue in Fig. 3. Note that all $2\sigma$ regions
nicely overlap, showing that both the EW fit and $b\to s\ell^{+}\ell^{-}$ data
prefer a non-zero $Z-Z^{\prime}$ mixing angle such that the $W$ mass can be
explained.

##  V Conclusions and Outlook

In this article we systematically studied the impact of $Z-Z^{\prime}$ mixing
on the global fit to $b\to s\ell^{+}\ell^{-}$ data and EW precision
observables. Concerning the former, we observe that a LFU effect is generated
while in the latter the mixing leads to modified $Z$ couplings and to an
enhancement in the predicted $W$ mass w.r.t. the SM, which accommodates the
new experimental average (including the recent one from CDF). Therefore, while
in previous analyses in the literature the effect of $Z-Z^{\prime}$ mixing was
usually assumed to be small and was therefore mostly neglected, we stress that
both $b\to s\ell^{+}\ell^{-}$ data and the EW fit even prefers a small but
non-zero value of the order of $10^{-3}$ for $m_{Z^{\prime}}\approx 1{\rm
TeV}-5{\rm TeV}$. Note that this is in agreement with the expectation
$\sin\xi\approx g_{2}g^{\prime}m_{Z}^{2}/m_{Z^{\prime}}^{2}$ for a TeV scale
$Z^{\prime}$ with order one couplings in case $U(1)^{\prime}$ and EW symmetry
breaking are related.

If $b\to s\ell^{+}\ell^{-}$ data is in fact explained by a $Z^{\prime}$ with
non-vanishing $Z-Z^{\prime}$ mixing, one predicts a pattern for the main
observables driving the anomaly as shown in Table 2. We observe that all
tensions with experiment reduce significantly below the 1.5$\,\sigma$ level in
the scenario analyzed. Because $b\to s\ell^{+}\ell^{-}$ ratios testing LFUV
depend naturally (and almost entirely) on ${\cal C}_{9\mu}^{\rm V}$ and thus
do not carry information on $\sin\xi$, angular observables are necessary for a
distinctive study of $Z^{\prime}$ models. It will therefore be important to
verify with more precise LHCb data together with future Belle II analysis if
this scenario gets reinforced.

Furthermore, forthcoming LHC measurements of the $W$ mass may reinforce the
current tension and any improvement in the global EW fit (e.g. in the top mass
or in $Z$ decays) would lead to a more precise $W$ mass predictions which
could be very precisely measured with future lepton colliders such as FCC-ee
Abada  _et al._ (2019), ILC Bae (2013), CEPC An  _et al._ (2019) or CLIC Aic
(2012). Importantly, if a non-zero $Z-Z^{\prime}$ mixing is established in the
future, like e.g. predicted in the model of Ref. Crivellin  _et al._ (2015b),
this would imply that $SU(2)_{L}$ and $U(1)^{\prime}$ are broken by a field
charged under both symmetries with important consequences for model building.

###### Acknowledgements.

We thank Joe Davighi for bringing a missing factor 1/2 in Eq. (6) to our
attention. The work of A.C. and C.A.M. is supported by a Professorship Grant
(PP00P2_176884) of the Swiss National Science Foundation. JM gratefully
acknowledges the financial support by ICREA under the ICREA Academia
programme. JM and MA received financial support from Spanish Ministry of
Science, Innovation and Universities (project PID2020-112965GB-I00/AEI/
10.13039/501100011033) and from the Research Grant Agency of the Government of
Catalonia (project SGR 1069).

## Appendix

Observable | Experimental value
---|---
$m_{W}\,[\text{GeV}]$ | $80.379(12)$
$\Gamma_{W}\,[\text{GeV}]$ | $2.085(42)$
${\cal B}(W\to\text{had})$ | $0.6741(27)$
${\cal B}(W\to\text{lep})$ | $0.1086(9)$
$\text{sin}^{2}\theta_{\rm eff,\,e}^{\rm CDF}$ | $0.23248(52)$
$\text{sin}^{2}\theta_{\rm eff,\,e}^{\rm D0}$ | $0.23146(47)$
$\text{sin}^{2}\theta_{\rm eff,\,\mu}^{\rm CDF}$ | $0.2315(10)$
$\text{sin}^{2}\theta_{\rm eff,\,\mu}^{\rm CMS}$ | $0.2287(32)$
$\text{sin}^{2}\theta_{\rm eff,\,\mu}^{\rm LHCb}$ | $0.2314(11)$
$P_{\tau}^{\rm pol}$ | $0.1465(33)$
$A_{e}$ | $0.1516(21)$
$A_{\mu}$ | $0.142(15)$
$A_{\tau}$ | $0.136(15)$
$\Gamma_{Z}\,[\text{GeV}]$ | $2.4952(23)$
$\sigma_{h}^{0}\,[\text{nb}]$ | $41.541(37)$
$R^{0}_{e}$ | $20.804(50)$
$R^{0}_{\mu}$ | $20.785(33)$
$R^{0}_{\tau}$ | $20.764(45)$
$A_{\rm FB}^{0,e}$ | $0.0145(25)$
$A_{\rm FB}^{0,\mu}$ | $0.0169(13)$
$A_{\rm FB}^{0,\tau}$ | $0.0188(17)$
$R_{b}^{0}$ | $0.21629(66)$
$R_{c}^{0}$ | $0.1721(30)$
$A_{\rm FB}^{0,b}$ | $0.0992(16)$
$A_{\rm FB}^{0,c}$ | $0.0707(35)$
$A_{b}$ | $0.923(20)$
$A_{c}$ | $0.670(27)$
Table 3:  Electroweak observables Schael  _et al._ (2006); Zyla  _et al._
(2020b) used in our fit performed using HEPfit De Blas  _et al._ (2020) with
$m_{Z_{0}}$, $\alpha$ and $G_{F}$ as input.

We write the interactions of the SM $Z$ with fermions as:

$\displaystyle\begin{split}\mathcal{L}_{Zff}=&\;\overline{\ell}_{j}\gamma_{\mu}\left(\Delta_{ji}^{\ell
L}P_{L}+\Delta_{ji}^{\ell
R}P_{R}\right)\ell_{i}Z^{\mu}+\overline{\nu}_{j}\gamma_{\mu}\Delta_{ji}^{\nu
L}P_{L}\nu_{i}Z^{\mu}\\\
&+\overline{u}_{j}\,\gamma_{\mu}\left(\Delta_{ji}^{uL}P_{L}+\Delta_{ji}^{uR}P_{R}\right)u_{i}\,Z^{\mu}\\\
&+\overline{d}_{j}\,\gamma_{\mu}\left(\Delta_{ji}^{dL}P_{L}+\Delta_{ji}^{dR}P_{R}\right)d_{i}\,Z^{\mu}\,,\end{split}$
(13)

with $i,j=1,2,3$ and

$\displaystyle\begin{split}&\Delta_{ji}^{\ell L(R)}\simeq\sin\xi\,g_{ji}^{\ell
L(R)}+\,g_{\rm SM}^{\ell L(R)}\delta_{ji}\,,\\\ &\Delta_{ji}^{\nu
L}\simeq\sin\xi\,g_{ji}^{\ell L}+\,g_{\rm SM}^{\nu L}\delta_{ji}\,,\\\
&\Delta_{ji}^{uL}\simeq\sin\xi\,V_{jk}g_{kk^{\prime}}^{q}V_{ik^{\prime}}^{*}+\,g_{\rm
SM}^{uL}\delta_{ji}\,,\\\ &\Delta_{ji}^{uR}\simeq\sin\xi\,g_{ji}^{u}+\,g_{\rm
SM}^{uR}\delta_{ji}\,,\\\
&\Delta_{ji}^{dL(R)}\simeq\sin\xi\,g_{ji}^{q(d)}+\,g_{\rm
SM}^{dL(R)}\delta_{ji}\,,\\\ \end{split}$ (14)

where $g_{\rm SM}^{iL(R)}$ are the SM couplings given by

$\displaystyle g_{\rm SM}^{\nu L}$ $\displaystyle=$
$\displaystyle-\frac{e}{2s_{W}c_{W}}\,,$ $\displaystyle g_{\rm SM}^{\ell L}$
$\displaystyle=$
$\displaystyle\frac{e}{2s_{W}c_{W}}\left(1-2s_{W}^{2}\right)\,,\qquad g_{\rm
SM}^{\ell R}=-\frac{e\,s_{W}}{c_{W}}\,,$ $\displaystyle g_{\rm SM}^{uL}$
$\displaystyle=$
$\displaystyle-\frac{e}{s_{W}c_{W}}\left(\frac{1}{2}-\frac{2}{3}s_{W}^{2}\right)\,,\qquad
g_{\rm SM}^{uR}=\frac{2}{3}\frac{e\,s_{W}}{c_{W}}\,,\,\,\,\,\,\,\,$ (15)
$\displaystyle g_{\rm SM}^{dL}$ $\displaystyle=$
$\displaystyle\frac{e}{s_{W}c_{W}}\left(\frac{1}{2}-\frac{1}{3}s_{W}^{2}\right)\,,\qquad
g_{\rm SM}^{dR}=-\frac{1}{3}\frac{e\,s_{W}}{c_{W}}\,,$

with $e=g_{1}g_{2}/\sqrt{g_{1}^{2}+g_{2}^{2}}=g_{1}c_{W}=g_{2}s_{W}$ being the
electric charge. Moreover, taking into account the $Z-Z^{\prime}$ mixing in
Eq. (14) and the vertex corrections Altmannshofer  _et al._ (2014a); Haisch
and Westhoff (2011), we have the following modified $Z$ couplings to leptons

$\displaystyle\Delta_{ij}^{\ell L}$ $\displaystyle=g_{\rm SM}^{\ell
L}\left(\delta_{ij}+\sin\xi\frac{g_{ij}^{\ell L}}{g_{\rm SM}^{\ell
L}}+\sum\limits_{k}\frac{g_{ik}^{\ell L}g_{kj}^{\ell
L}}{(4\pi)^{2}}{K_{F}}\left(\frac{m_{Z}^{2}}{m_{Z^{\prime}}^{2}}\right)\right)\,,$
(16) $\displaystyle\Delta_{ij}^{\nu L}$ $\displaystyle=g_{\rm SM}^{\nu
L}\left({{\delta_{ij}}+\sin\xi\frac{{g_{ij}^{\ell L}}}{{g_{{\rm{SM}}}^{\nu
L}}}+\sum\limits_{k}{\frac{{g_{ik}^{\ell L}g_{kj}^{\ell
L}}}{{{{(4\pi)}^{2}}}}}{K_{F}}\left({\frac{{m_{Z}^{2}}}{{m_{Z^{\prime}}^{2}}}}\right)}\right)\,,$
$\displaystyle\Delta_{ij}^{\ell R}$ $\displaystyle=g_{\rm SM}^{\ell
R}\left({{\delta_{ij}}+\sin\xi\frac{{g_{ij}^{\ell R}}}{{g_{{\rm{SM}}}^{\ell
R}}}+\sum\limits_{k}{\frac{{g_{ik}^{\ell R}g_{kj}^{\ell
R}}}{{{{(4\pi)}^{2}}}}}{K_{F}}\left({\frac{{m_{Z}^{2}}}{{m_{Z^{\prime}}^{2}}}}\right)}\right)\,,$

at the $Z$ pole with

$\displaystyle\begin{split}{K_{F}}\left(x\right)=&-\frac{{2{{(x+1)}^{2}}({\rm{L}}{{\rm{i}}_{2}}(-x)+\ln(x)\ln(x+1))}}{{{x^{2}}}}\\\
&-\frac{{7x+4}}{{2x}}+\frac{{(3x+2)\ln(x)}}{x}.\end{split}$ (17)

The effective Hamiltonian Grinstein  _et al._ (1988); Buchalla  _et al._
(1996) in which heavy degrees of freedom have been integrated out is given by:

${\cal H}_{\rm eff}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\sum_{i}{\cal
C}_{i}{\cal O}_{i}$ (18)

The relevant operators for this paper are: where $P_{L,R}=(1\mp\gamma_{5})/2$.

The set of observables used in the EW fit are given in Table 3.

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