# Dark Matter in the Private Higgs Model

C. B. Jackson<EMAIL_ADDRESS>Physics Department, Brookhaven National
Laboratory, Upton, NY 11973-5000, USA

###### Abstract

The extremely large hierarchy observed in the fermion mass spectrum remains as
one of the most puzzling and unresolved issues in particle physics. In a
recent proposal, however, it was demonstrated that by introducing one Higgs
doublet (or Private Higgs) per fermion this hierarchy could be made natural by
making the Yukawa couplings between each fermion and its respective Higgs
boson of order unity. Among the interesting predictions of the Private Higgs
scenario is a variety of scalars which could be probed at future collider
experiments and a possible dark matter candidate. In this paper, we study in
some detail the dark matter sector of the Private Higgs model. We first
calculate the annihilation cross sections of dark matter in this model and
find that one can easily account for the observed density of dark matter in
the Universe with relatively natural values of the model’s parameters.
Finally, we investigate the possibility of detecting Private Higgs dark matter
indirectly via the observation of anomalous gamma rays originating from the
galactic halo. We show that a substantial flux of photons can be produced from
the annihilation of Private Higgs dark matter such that, if there is
considerable clumping of dark matter in the galactic halo, the flux of these
gamma rays could be observed by ground-based telescope arrays such as VERITAS
and HESS.

††preprint: BNL-HET-08/10

##  I Introduction

One of the most puzzling issues in the Standard Model is the large hierarchy
observed in the masses of fermions. For example, in the quark sector alone,
the masses of the heaviest (top) and lightest (up) quarks are separated by
nearly five orders of magnitude. Conversely, if one assumes that all fermions
receive their mass via interactions with the same Higgs doublet (as in the
Standard Model (SM)), the large hierarchy of masses observed in the fermion
sector translates into a large hierarchy in the Yukawa couplings of the
fermions.

Recently, it has been proposed that the hierarchy of fermion masses can be
made natural by extending the scalar sector of the SM to include one Higgs
doublet (or Private Higgs (PH)) per fermion Porto and Zee (2007). In this
scenario, all of the Yukawa couplings can be made of ${\cal{O}}(1)$ by tuning
parameters of the model. In other words, the vacuum expectation values (vev’s)
of each respective PH field can be made to satisfy $v_{f}\sim m_{f}$ such that
the hierarchy in the fermion mass spectrum becomes natural.

The approach to electroweak symmetry breaking (EWSB) in the PH model is quite
different than those of other multi-Higgs models. First, one introduces one
gauge singlet scalar $S_{q}$ per quark flavor $q$ and uses the vev’s of these
fields along with certain interactions between these fields and the various PH
fields to induce “negative-mass-squared” instabilities. By using different
terms in the Lagrangian for the top PH and non-top PH fields, one can easily
explain the hierarchy in vev’s by tuning certain parameters of the model. As a
consequence of this approach, the lighter the fermion is, the heavier its
associated PH particle must be in order to explain the smallness of the
respective vev. In particular, the mass of the PH particle associated with the
up quark can be shown to lie in the $10^{2}-10^{3}$ TeV range which is
definitely beyond the reach of the Large Hadron Collider (LHC). However, there
is interesting phenomenology originating from the sector of the top and bottom
PH fields along with the singlet scalars. In this work, we study a scenario
where the physical spectrum of this sector contains a light SM-like Higgs
boson, a heavy scalar Higgs boson, a pair of charged Higgs bosons and a
pseudoscalar Higgs boson. The last three of these arise mainly from the bottom
PH field and all have masses in the $\sim$ TeV range. In addition to these,
there are also two light scalars which are admixtures of the singlet states
associated with the top and bottom quarks ($S_{t}$ and $S_{b}$). By
construction, $S_{t}$ and $S_{b}$ are dark to interactions with SM gauge
fields and fermions. While we will focus mainly on the light scalars in this
work, the heavier Higgs bosons could be probed at the LHC via production with
bottom quarks (since the Yukawa coupling between the bottom quarks and the
bottom PH field is of order unity)111Presumably, the PH partner of the $\tau$
could also provide interesting phenomenology; however, we will focus on the
quark sector here..

In order to avoid cross-talk between different quarks, the PH model contains a
set of six discrete symmetries (one for each quark flavor). Under these
symmetries, the right-handed quarks, their respective PH fields and the gauge
singlet scalars $S_{q}$ are all odd, while all other SM fields are even. The
existence of these discrete symmetries provides one of the most interesting
features of the PH scenario which is the possibility of a dark matter (DM)
candidate. Scalar DM was originally proposed over twenty years ago in Ref.
Silveira and Zee (1985) and has been studied more recently in several
different scenarios including singlet scalar DM Holz and Zee (2001); McDonald
(1994); Patt and Wilczek (2006); Bertolami and Rosenfeld (2007); He et al.
(2007); Davoudiasl et al. (2005) and in the so-called Inert Doublet Model Ma
(2006); Barbieri et al. (2006); Cirelli et al. (2006); Deshpande and Ma
(1978); Majumdar and Ghosal (2006); Casas et al. (2007); Calmet and Oliver
(2007). However, as we will demonstrate, the features of PHDM can be quite
different from previously studied scenarios.

The remainder of the paper is structured as follows. First, in Section II, we
review the structure of the PH model and demonstrate how EWSB is realized in
this model. In section Section III, utilizing the observations from WMAP
Spergel et al. (2003), we show that the PH model is able to account for all of
the observed dark matter in the Universe for relatively natural values of the
model’s parameters. In addition, in Section IV, we consider the possibility of
detecting PHDM via its annihilation into anomalous gamma rays in the galactic
halo. We show that, with a favorable distribution of DM in the halo, PHDM
could be detected by ground-based telescopes, but is probably beyond the reach
of the space-based GLAST telescope Morselli et al. (2002). Finally, in Section
V, we conclude.

##  II The Model

The main goal of the Private Higgs model is to account for the extremely large
hierarchy observed in the fermion mass spectrum Porto and Zee (2007). For
purposes of this paper, we will focus on the quark sector. In contrast to the
SM, where one introduces a single scalar doublet which couples to all quarks,
the PH scenario democratically introduces one Higgs doublet $\phi_{q}$
$(q=u,d,s,c,t,b)$ per quark. All of the PH fields are assumed to have
identical $SU(2)\times U(1)$ quantum numbers as the SM Higgs. In addition to
the PH fields, the scalar sector of the PH model also contains a set of gauge
singlet scalars $S_{q}$. In order to avoid cross talk between quarks of
different flavors, a set of six discrete symmetries $K_{q}$ is imposed on the
model. Under the $K_{q}$ symmetries, the right-handed quarks ($U_{q},D_{q}$)
along with the PH fields and $S_{q}$ are all odd, i.e.:

$U_{q}\to-U_{q}\,(D_{q}\to-
D_{q})\,\,\,,\,\,\,\phi_{q}\to-\phi_{q}\,\,\,,\,\,\,S_{q}\to-S_{q}\,,$ (1)

while all other fields are considered even. The Lagrangian which is symmetric
under the $K_{q}$ symmetries is then given by:

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{SM-H}-\sum_{q}(Y_{D}^{PH}\,\overline{Q}_{L}\phi_{D}D_{q}+Y_{U}^{PH}\,\overline{Q}_{L}\tilde{\phi}_{U}U_{q})$
(2)
$\displaystyle+\sum_{q}\biggl{[}\partial_{\mu}S_{q}\partial^{\mu}S_{q}+(D_{\mu}\phi_{q})^{\dagger}D^{\mu}\phi_{q}\biggr{]}-V(S_{q},\phi_{q})\,,$

where $\tilde{\phi}_{U}=i\sigma_{2}\phi_{U}$, ${\cal{L}}_{SM-H}$ is the SM
Lagrangian without the Higgs terms and $Y_{D}^{PH},Y_{U}^{PH}$ are Yukawa
matrices. The scalar potential $V(S_{q},\phi_{q})$ takes the form:

$\displaystyle V(S_{q},\phi_{q})$ $\displaystyle=$
$\displaystyle\sum_{q}\biggl{\\{}\frac{1}{2}M_{S_{q}}^{2}S_{q}^{2}+\frac{\lambda_{S}^{q}}{4}S_{q}^{4}+\frac{1}{2}M_{\phi_{q}}^{2}\phi_{q}^{\dagger}\phi_{q}+\lambda_{q}(\phi_{q}^{\dagger}\phi_{q})^{2}-g_{sq}S_{q}^{2}\phi_{q}^{\dagger}\phi_{q}\biggr{\\}}$
$\displaystyle+$ $\displaystyle\sum_{q\neq
q^{\prime}}\biggl{\\{}a_{qq^{\prime}}^{S}S_{q}^{2}S_{q^{\prime}}^{2}+\gamma_{qq^{\prime}}S_{q}S_{q^{\prime}}\phi_{q}^{\dagger}\phi_{q^{\prime}}+\chi_{qq^{\prime}}S_{q^{\prime}}^{2}\phi_{q}^{\dagger}\phi_{q}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,+\,a_{qq^{\prime}}\phi_{q}^{\dagger}\phi_{q^{\prime}}\phi_{q}^{\dagger}\phi_{q^{\prime}}+b_{qq^{\prime}}\phi_{q}^{\dagger}\phi_{q}\phi_{q^{\prime}}^{\dagger}\phi_{q^{\prime}}+c_{qq^{\prime}}\phi_{q}^{\dagger}\phi_{q^{\prime}}\phi_{q^{\prime}}^{\dagger}\phi_{q}\biggr{\\}}+h.c.\,,$

where, for stability of the potential,
$a_{qq^{\prime}},b_{qq^{\prime}},c_{qq^{\prime}}<0$. In our analysis, we will
assume these terms are small and neglect them in the following.

In the PH model, instead of inducing EWSB through the usual “negative-mass-
squared” approach where $M_{\phi_{q}}^{2}<0$, one utilizes the vev’s of the
singlet fields $S_{q}$ and the interactions between the $S_{q}$’s and the PH
fields. In particular, for the top PH one assumes $M_{\phi_{t}}^{2}>0$ and
induces EWSB through the $g_{st}$ and $\chi_{qt}$ couplings as well as the
vev’s of the $S_{q}$ fields. Thus, taking $g_{st},\chi_{qt}>0$ and
$\frac{1}{2}M_{\phi_{t}}^{2}-g_{st}\langle S_{q}\rangle^{2}-\sum_{q\neq
t}\chi_{qt}\langle S_{q}\rangle^{2}\equiv\mu_{t}^{2}<0$, the top PH is forced
to develop a negative-mass-squared instability which, in turn, spontaneously
breaks the $SU(2)_{L}\times U(1)_{Y}$ gauge symmetry. Therefore, in a sense,
the top PH plays the role of the SM Higgs.

In general, the PH scenario can contain many new free parameters in addition
to those of the SM. In order to simplify our analysis in the following
sections, we will make a succession of approximations. Thus, our results will
not probe the full parameter space of the PH scenario, but should be viewed as
a first step in this direction. To begin, we follow Ref. Porto and Zee (2007)
and assume that $M_{\phi_{t}}^{2}\ll g_{st}v_{s}^{2}$ which is in accordance
with the symmetry breaking pattern discussed above. We also consider the case
where $g_{st}\sim\chi_{qt}$ and $a_{qq^{\prime}}^{S}\ll 1$. To give the
$S_{q}$ fields a vev, one introduces an instability $M_{S_{q}}^{2}<0$ such
that, under our assumptions, the potential in the $S_{q}-\phi_{t}$ sector
reduces to:

$V(S_{q},\phi_{t})=\frac{\lambda_{S}^{q}}{4}\biggl{(}S_{q}^{2}-\frac{(v_{s}^{q})^{2}}{2}\biggr{)}^{2}+\lambda_{t}(\phi_{t}^{\dagger}\phi_{t})^{2}-g_{st}S_{q}^{2}\phi_{t}^{\dagger}\phi_{t}\,,$
(3)

where summation over quark flavor $q$ is implicit and, in principle, the
quantity $v_{s}^{q}$ is a bare parameter. Minimizing this potential, we find
the conditions:

$\frac{\partial V(S_{q},\phi_{t})}{\partial S_{q}}\biggl{|}_{\langle
S_{q}\rangle,\langle\phi_{t}\rangle}=\lambda_{S}^{q}\biggl{(}\langle
S_{q}\rangle^{2}-\frac{(v_{s}^{q})^{2}}{2}\biggr{)}-2g_{st}\langle\phi_{t}\rangle^{2}=0\,,$
(4)

and:

$\frac{\partial V(S_{q},\phi_{t})}{\partial\phi_{t}}\biggl{|}_{\langle
S_{q}\rangle,\langle\phi_{t}\rangle}=2\lambda_{t}\langle\phi_{t}\rangle^{2}-g_{st}\langle
S_{q}\rangle^{2}=0\,.$ (5)

Solving these equations for the individual vev’s we find:

$\langle
S_{q}\rangle^{2}=\frac{(v_{s}^{q})^{2}}{2}\biggl{(}\frac{\lambda_{S}^{q}\lambda_{t}}{\lambda_{S}^{q}\lambda_{t}-g_{st}^{2}}\biggr{)}\equiv\frac{v_{s}^{2}}{2}\,,$
(6)

for the vev of $S_{q}$ and for the top PH vev:

$\langle\phi_{t}\rangle^{2}=\frac{(v_{s}^{q})^{2}}{4}\biggl{(}\frac{g_{st}\lambda_{S}^{q}}{\lambda_{S}^{q}\lambda_{t}-g_{st}^{2}}\biggr{)}\equiv\frac{v_{h}^{2}}{2}\,.$
(7)

Note that, for simplicity, we have identified the individual vev’s $v_{s}^{q}$
with one common parameter $v_{s}$. Finally, we also note the relationship
between $v_{s}$ and $v_{h}$:

$v_{h}^{2}=\frac{g_{st}}{2\lambda_{t}}v_{s}^{2}\,.$ (8)

Next, we consider the non-top PH fields which acquire their vev’s in a
slightly different manner. First, as in the case of the top PH, the mass
parameter $M_{\phi_{q}}^{2}$ is assumed to be positive. However, for the
$\phi_{q}$ fields (where $q\neq t$), one imposes the condition
$M_{\phi_{q}}^{2}>g_{sq}v_{s}^{2}$ in contrast to the case of the top PH.
Then, vev’s for the non-top PH fields are induced through the cubic term
$\gamma_{qq^{\prime}}$ and the vev’s $v_{s}$ and $v_{h}$. Again, to simplify
our analysis, we will make some assumptions. Specifically, we will assume
that:

$M_{\phi_{q}}^{2}\gg g_{sq}v_{s}^{2}\,\,,\,\,\lambda_{q}\,$ (9)

which is consistent with the symmetry breaking pattern discussed above. Then,
after $S_{q}$ and $\phi_{t}$ pick up vev’s, the relevant part of the
$\phi_{q}$ potential is:

$\frac{1}{2}M_{\phi_{q}}^{2}\phi_{q}^{\dagger}\phi_{q}-\frac{\gamma_{qt}}{\sqrt{2}}\frac{v_{h}v_{s}^{2}}{2}\phi_{q}\,.$
(10)

Minimizing this potential, the vev’s for the non-top Higgs fields are found to
be:

$\langle\phi_{q}\rangle=\frac{\gamma_{qt}}{\sqrt{2}}\frac{v_{h}v_{s}^{2}}{M_{\phi_{q}}^{2}}\equiv
v_{q}\,.$ (11)

Eq. (11) summarizes the main result of the PH scenario. By having the
parameter $\gamma_{qt}$ to be small while keeping $M_{\phi_{q}}^{2}$ large,
one is able to make all Yukawa couplings (which are given by $m_{q}/v_{q}$) of
${\cal{O}}(1)$ without fine-tuning. As a consequence of this relation, one can
show from Eq. (11) that the lighter quarks have associated PH particles in the
$10^{2}-10^{3}$ TeV range which are definitely beyond the reach of current or
future experiments Porto and Zee (2007). However, the masses from the
$\phi_{t}-S_{q}$ sector can be naturally light (100’s GeV), while the bottom
PH particle can have masses in the TeV range.

Finally, inserting Eqs. (7) and (11) into the Lagrangian of Eq. (2), it is
easy to show that the $W^{\pm}$ mass is given in the PH model by:

$m_{W}^{2}=\frac{1}{2}gv_{h}^{2}\biggl{[}1+\sum_{q\neq
t}\biggl{(}\frac{v_{q}^{2}}{v_{h}^{2}}\biggr{)}\biggr{]}\,.$ (12)

Obviously, the leading term in the sum comes from the bottom PH; however, even
in this case, the contribution is of order $m_{b}^{2}/m_{t}^{2}\sim 0.001$.
Thus, the contributions to EWSB from quarks lighter than the top are
negligible and our statement above that the role of the SM Higgs boson is
being played by the top PH is verified.

###  II.1 Mass Eigenstates and Their Interactions

In this section, we study the top-bottom sector of the PH model in some
detail. In particular, we will consider the case where
$\lambda_{S}^{t}=\lambda_{S}^{b}\equiv\lambda_{S}$ and
$\lambda_{S}\ll\lambda_{S}^{q}$ for $q\neq t,b$. Thus, the gauge singlet
scalars associated with the lighter quarks become heavy and effectively
decouple from our analysis. Then, under our assumptions, the scalar potential
in the top-bottom sector reduces to:

$\displaystyle V(S_{q},\phi_{t},\phi_{b})$ $\displaystyle=$
$\displaystyle\frac{\lambda_{S}}{4}\biggl{[}\biggl{(}S_{t}^{2}-\frac{v_{s}^{2}}{2}\biggr{)}^{2}+\biggl{(}S_{b}^{2}-\frac{v_{s}^{2}}{2}\biggr{)}^{2}\biggr{]}+\lambda_{t}(\phi_{t}^{\dagger}\phi_{t})^{2}+\frac{1}{2}M_{\phi_{b}}^{2}\phi_{b}^{\dagger}\phi_{b}$
$\displaystyle-$ $\displaystyle
a_{tb}^{S}(S_{t}^{2}S_{b}^{2})-\gamma_{tb}S_{t}S_{b}(\phi_{b}^{\dagger}\phi_{t}+\phi_{t}^{\dagger}\phi_{b})-g_{st}\biggl{[}S_{t}^{2}\phi_{t}^{\dagger}\phi_{t}+S_{t}^{2}\phi_{b}^{\dagger}\phi_{b}+S_{b}^{2}\phi_{t}^{\dagger}\phi_{t}\biggr{]}\,.$

To begin, we expand the PH Higgs fields in the usual way:

$\displaystyle\phi_{t}~{}=~{}\left(\begin{array}[]{c}\omega^{+}\\\
\frac{1}{\sqrt{2}}(v_{h}+h_{t}+i\chi^{0})\end{array}\right)\,,$ (16)
$\displaystyle\phi_{b}~{}=~{}\left(\begin{array}[]{c}H^{+}\\\
v_{b}+H_{b}+iA_{b}\end{array}\right)\,,$ (19)

while the singlet fields are expanded as:

$S_{t,b}=\frac{1}{\sqrt{2}}(v_{s}+\sigma_{t,b})\,.$ (20)

In the above expansions, $\omega^{\pm}$ and $\chi^{0}$ are assumed to play the
roles of the usual Goldstone bosons which are eaten by the $W^{\pm}$ and $Z$,
while $H^{\pm}$ and $A_{b}$ are charged and pseudoscalar Higgs bosons,
respectively. Both the $H^{\pm}$ and $A_{b}$ will have masses on the order of
$M_{\phi_{b}}\sim$ TeV and could provide interesting phenomenology at the LHC.
Note that we are neglecting mixing between the “pure Goldstones”
($\omega^{\pm},\chi^{0}$) and the “physical Higgs bosons” ($H^{\pm},A_{b}$).
These mixings are typically of order
$\frac{\gamma_{tb}v_{b}^{2}}{M_{\phi_{b}}^{2}}$ and, thus, are extremely
small.

Inserting the expansions of the Higgs fields from Eqs. (16) - (20) into
(II.1), we first extract the mass terms of the Goldstone bosons which we
require to vanish:

$m_{\omega^{\pm}}^{2}=m_{\chi^{0}}^{2}=\lambda_{t}v_{h}^{2}-\frac{1}{2}g_{st}v_{s}^{2}=0\,.$
(21)

Note that this equation is in agreement with Eq. (8).

Next, we could attempt to diagonalize the full $4\times 4$ mass matrix in the
$(h_{t},H_{b},\sigma_{t},\sigma_{b})$ basis. However, as shown in Ref. Porto
and Zee (2007), for values of the model parameters that we consider in our
analysis most of the mixings between the various scalars are negligible. In
particular, the mixing between $h_{t}$ and $\sigma_{q}$ is negligible
provided:

$8g_{st}^{3}\ll(2g_{st}-\lambda_{S})^{2}\lambda_{t}\,\,\,\,\mbox{(for
$q=t$)}\,,$ (22)

and

$\gamma_{bt}\biggl{(}\frac{m_{b}}{m_{t}}\biggr{)}\ll g_{st}\,\,\,\,\mbox{(for
$q=b$)}\,.$ (23)

Similarly, the mixing between $\sigma_{q}$ and $H_{b}$ can be neglected
provided:

$v_{h}\biggl{(}\frac{m_{b}}{m_{t}}\biggr{)}\ll v_{s}\,.$ (24)

All of these conditions are satisfied for the parameter choices in our
analysis, hence we choose to neglect the above mixings. However, it should be
noted that the mixing between $h_{t}$ and $H_{b}$ serves to reproduce the
small SM coupling between bottom quarks and the SM-like Higgs boson. In the
following, we identify $h_{t}$ and $H_{b}$ with approximate mass eigenstates
$h^{0}$ and $H^{0}$ respectively and assume the coupling between $h^{0}$ and a
pair of bottom quarks takes its SM value.

Finally, there can be substantial mixing between the two singlet scalars
$\sigma_{t}$ and $\sigma_{b}$ via the $a_{tb}^{s}$ and $\gamma_{tb}$ terms.
Diagonalizing the $2\times 2$ mass matrix in this sector, we find two mass
eigenstates ($\Sigma_{1}$ and $\Sigma_{2}$) with mass eigenvalues:

$\displaystyle m_{\Sigma_{1}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\lambda_{S}^{2}-g_{st}v_{h}^{2}-a_{tb}^{s}v_{s}^{2}\biggr{)}-\biggl{(}a_{tb}^{s}v_{s}^{2}+\frac{\gamma_{tb}}{\sqrt{2}}v_{h}v_{b}\biggr{)}\sin
2\alpha\,,$ (25) $\displaystyle m_{\Sigma_{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}\lambda_{S}^{2}-g_{st}v_{h}^{2}-a_{tb}^{s}v_{s}^{2}\biggr{)}+\biggl{(}a_{tb}^{s}v_{s}^{2}+\frac{\gamma_{tb}}{\sqrt{2}}v_{h}v_{b}\biggr{)}\sin
2\alpha\,.$ (26)

Note that for $a_{tb}^{s},\gamma_{tb}>0$ and $0<\alpha<\pi/2$, $\Sigma_{1}$
plays the role of the lightest PH particle (LPHP) which is stable against
decay and, thus, provides a candidate for DM. Using Eqs. (25) and (26), we can
exchange two of the free parameters (e.g., $\lambda_{S}$ and $a_{tb}^{s}$) for
the masses of the two singlet scalars. This is the approach we will take.
Therefore, in the analysis to follow, we will take as our free parameters the
masses $m_{\Sigma_{1}}$, $m_{\Sigma_{2}}$ as well as the couplings $g_{st}$
and $\gamma_{tb}$ and the mixing angle $\alpha$. Note that the conditions for
small mixings between the $\sigma_{q}$’s and the PH fields forces $g_{st}$ to
take small values.

##  III Private Higgs Dark Matter

As mentioned earlier, one of the most interesting aspects of the PH scenario
is the prospect of a Weakly Interacting Massive Particle (WIMP) with masses in
the expected natural range for DM. In this context, the PH model is similar to
other scalar DM models such as the gauge singlet models of Refs. Silveira and
Zee (1985); Holz and Zee (2001); McDonald (1994); Patt and Wilczek (2006);
Bertolami and Rosenfeld (2007); He et al. (2007); Davoudiasl et al. (2005) and
the Inert Doublet Model (IDM) Ma (2006); Barbieri et al. (2006); Cirelli et
al. (2006); Deshpande and Ma (1978); Majumdar and Ghosal (2006); Casas et al.
(2007); Calmet and Oliver (2007). In this section, we calculate the
annihilation cross sections of PHDM into SM particles and show that, for
relatively natural values of the model parameters, one can account for all of
the observed dark matter in the Universe. In the next section, we investigate
the possibility of indirectly detecting PHDM via its annihilation into
anomalous gamma rays in the galactic halo.

First, let us consider the present relic abundance of PHDM in the Universe. In
the following, we will assume that the mass splitting
$m_{\Sigma_{2}}-m_{\Sigma_{1}}$ is large enough that coannihilation reactions
between $\Sigma_{1}$ and $\Sigma_{2}$ do not significantly affect the relic
abundance. These effects will be considered in future work. In the early
Universe, the singlet scalar $\Sigma_{1}$ would have been in equilibrium with
the rest of the cosmic fluid. This equilibrium is maintained via $\Sigma_{1}$
pair-annihilation and pair-creation reactions which proceed through the
$s$-channel exchange of the SM-like Higgs $h^{0}$. The leading $2\to 2$
$s$-channel reactions which contribute to these processes are shown in Fig. 1.

Figure 1: Leading $s$-channel processes which maintain the singlet scalar
$\Sigma_{1}$ in equilibrium with the rest of the cosmic fluid.

The present relic abundance of PHDM is determined by the pair-annihilation
rates in the non-relativistic limit. The rates for each allowed channel are
given in the non-relativistic limit as:

$a(X)\equiv\lim_{u\to 0}\sigma(\Sigma_{1}\Sigma_{1}\to X)\,u\,$ (27)

where $u$ is the relative velocity of the annihilating particles. The total
annihilation cross section is then given by summing over each of the allowed
channels. Computing the cross sections for the diagrams in Fig. 1, we find:

$\displaystyle a(W^{+}W^{-})$ $\displaystyle=$
$\displaystyle\frac{g_{\Sigma_{1}\Sigma_{1}h^{0}}^{2}}{2\pi
v_{h}^{2}}\frac{\sqrt{1-\mu_{w}}}{(4m_{\Sigma_{1}}^{2}-m_{h^{0}}^{2})^{2}+\Gamma_{h^{0}}^{2}m_{h^{0}}^{2}}\biggl{(}1-\mu_{w}+\frac{3}{4}\mu_{w}^{2}\biggr{)}\,,$
(28) $\displaystyle a(ZZ)$ $\displaystyle=$
$\displaystyle\frac{g_{\Sigma_{1}\Sigma_{1}h^{0}}^{2}}{4\pi
v_{h}^{2}}\frac{\sqrt{1-\mu_{z}}}{(4m_{\Sigma_{1}}^{2}-m_{h^{0}}^{2})^{2}+\Gamma_{h^{0}}^{2}m_{h^{0}}^{2}}\biggl{(}1-\mu_{z}+\frac{3}{4}\mu_{z}^{2}\biggr{)}\,,$
(29) $\displaystyle a(f\bar{f})$ $\displaystyle=$
$\displaystyle\frac{g_{\Sigma_{1}\Sigma_{1}h^{0}}^{2}}{4\pi
v_{h}^{2}}\frac{(1-\mu_{f})^{\frac{3}{2}}}{(4m_{\Sigma_{1}}^{2}-m_{h^{0}}^{2})^{2}+\Gamma_{h^{0}}^{2}m_{h^{0}}^{2}}\,,$
(30)

where $\mu_{i}=m_{i}^{2}/m_{\Sigma_{1}}^{2}$, $\Gamma_{h^{0}}$ is the width of
the $h^{0}$ for which we use SM values and the expression for the coupling
$g_{\Sigma_{1}\Sigma_{1}h^{0}}$ is given by:

$g_{\Sigma_{1}\Sigma_{1}h^{0}}=-g_{st}v_{h}+\frac{2\gamma_{tb}v_{b}}{\sqrt{2}}c_{\alpha}s_{\alpha}\,.$
(31)

The WMAP collaboration Spergel et al. (2003) provides a very precise
determination of the present DM abundance which, at the two-sigma level, is
given by:

$\Omega_{DM}h^{2}=0.111\pm 0.018\,.$ (32)

As shown in Ref. Birkedal et al. (2004), for a generic model of DM, the
present abundance of DM is mainly determined by $J_{0}$ (the angular momentum
of the dominant partial wave contributing to DM annihilation) and the
annihilation cross section. In contrast, the relic abundance depends only
weakly on the mass or spin of the DM particle. Thus, the very precise
constraints from WMAP on $\Omega_{DM}h^{2}$ translate into very precise
constraints on the quantity $a\equiv\sum_{X}a(X)$ depending on the value of
$J_{0}$. In particular, for an “$s$-wave annihilator” ($J_{0}=0$) such as the
case considered here, the WMAP measurement translates into the bounds:

$a=0.8\pm 0.1\,\,\mbox{pb}\,,$ (33)

nearly independent of the mass or spin of the DM particle (see Fig. 1 of Ref.
Birkedal et al. (2004)).

In Fig. 2, we plot the values of $a(X)$ for two different values of the
coupling $g_{st}$ and several different values of the mixing angle $\alpha$.
In these plots, we have set the bottom PH vev equal to the bottom quark mass
and $\gamma_{tb}=1$. The horizontal dashed lines indicate the limits on $a(X)$
from Eq. (33). Clearly, from Eqs. (28)- (31), we see that the annihilation
cross sections depend quadratically on $g_{st}$. This is evident in the plot
of Fig. 2 where we see a small shift in $g_{st}$ results in a large shift in
the ranges of $m_{\Sigma_{1}}$ allowed by the WMAP data. Finally, we note that
these plots are only meant to show that for relatively natural values of the
model parameters it is indeed possible to account for the observed density of
DM in the Universe. A full scan of the PH parameter space would probably find
other choices of parameters which could fulfill the constraints from Eq. (33).

Figure 2: The annihilation cross section as a function of the $\Sigma_{1}$
mass and the mixing parameter $\alpha$. The dashed horizontal lines indicate
the WMAP constraints on the annihilation cross sections given by Eq. 33.

##  IV Indirect Detection of PHDM

Next, we would like to investigate the possibility of detecting PHDM. We will
focus here on indirect detection and save an analysis of direct detection for
future work.

As we have seen, the annihilation rates for PHDM are approximately velocity-
independent in the non-relativistic regime. In general, this implies that DM
collected in galactic halos has a substantial probability to pair-annihilate
resulting in anomalous high-energy cosmic rays which can be distinguished from
astrophysical backgrounds. In particular, gamma rays from these annihilations
provide a chance to extract information about DM, since they can travel over
galactic scales without scattering.

Figure 3: Diagrams which dominate photon pair-production in the $\Sigma_{1}$
annihilation in the galactic halo.

The production of gamma rays from $\Sigma_{1}\Sigma_{1}$ annihilation can
originate from several different processes (including hadronization,
factorization and radiation from final-state particles). However, for
simplicity, we will assume the dominate source is from direct annihilation
into a two-body final state as shown in Fig. 3222Here, we concentrate on the
dominant $\gamma\gamma$ signal and save a discussion of the $Z\gamma$ and/or
$h^{0}\gamma$ channels for future work.. Note that, under our assumptions,
only SM particles circulate the loop. In the full parameter space of the PH
model, it would be possible to have charged Higgs circulating the loop.
However, their couplings to $h^{0}$ are always of order $v_{b}/v_{h}$ or
$v_{b}/v_{s}$ and, thus, can be safely ignored in comparison to the SM loops.
The cross section for photon pair-production in the PH scenario can be written
as:

$\sigma_{\gamma\gamma}u=\frac{2g_{\Sigma_{1}\Sigma_{1}h^{0}}^{2}}{(s-m_{h^{0}}^{2})^{2}+\Gamma_{h^{0}}m_{h^{0}}^{2}}\frac{\hat{\Gamma}(h^{0}\to\gamma\gamma)}{\sqrt{s}}\,,$
(34)

where the expression for $g_{\Sigma_{1}\Sigma_{1}h^{0}}$ is given above and,
in the non-relativistic regime, $s\simeq 4m_{\Sigma_{1}}^{2}$. The hat on
${\hat{\Gamma}}$ indicates that one should replace $m_{h^{0}}\to\sqrt{s}$ in
the standard expressions for on-shell Higgs decays. The expressions needed to
construct $\hat{\Gamma}(h^{0}\to\gamma\gamma)$ can be found in several reviews
(e.g., see Ref. Djouadi (2008)) and, hence, we will not repeat them here.

Next, we would like to compute the flux of photons observed on Earth from
$\Sigma_{1}$ annihilation in the galactic halo. The monochromatic flux due to
the $\gamma\gamma$ final state, observed by a telescope with a line of sight
parameterized by $\Psi=(\theta,\phi)$ and a field of view $\Delta\Omega$ can
be written as Bergstrom et al. (1998):

$\Phi=(1.1\times
10^{-9}\,\mbox{s}^{-1}\mbox{cm}^{-2})\biggl{(}\frac{\sigma_{\gamma\gamma}u}{1\,\,\mbox{pb}}\biggr{)}\biggl{(}\frac{100\,\,\mbox{GeV}}{m_{\Sigma_{1}}}\biggr{)}\bar{J}(\Psi,\Delta\Omega)\Delta\Omega\,,$
(35)

where the dependence of the flux on the halo dark matter density distribution
is contained in $\bar{J}$. Many models predict a large spike in the DM density
in the neighborhood of the galactic center, making the line of sight towards
the center of the galaxy the preferred one. However, the features of the peak
are highly model-dependent resulting in values of $\bar{J}$ ranging from
$10^{3}$ to $10^{7}$ for $\Delta\Omega=10^{-3}$ sr (typical for ground-based
atmospheric Cerenkov telescopes) Navarro et al. (1997); Moore et al. (1998,
1999); Gnedin et al. (2004).

The monochromatic photon flux predicted from PHDM annihilation for the two
values of $g_{st}$ studied previously are shown in Fig. 4. For these plots, we
have assumed there is no substantial spiking in the galactic center (i.e.,
$\bar{J}\Delta\Omega=1$). In the energy range considered in these plots,
ground-based atmospheric Cherenkov telescopes (such as VERITAS Weekes et al.
(2002) and HESS Hinton (2004)) typically have a flux sensitivity down to the
$10^{-12}$ s-1 cm-2 level. On the other hand, the upcoming space-based
telescope GLAST Morselli et al. (2002) is limited by statistics to the
$10^{-10}$ s-1 cm-2 level over the energy range considered. From these plots,
it is clear that without a substantial spike in the galactic center, PHDM will
be difficult to observe in either ground- or space-based observatories.
However, if the halo does exhibit a substantial spike or strong clumping
(e.g., if $\bar{J}\geq 10^{5}$ at $\Delta\Omega\simeq 10^{-}3$), PHDM could be
observed at ground-based telescopes assuming small values of $g_{st}$ and
relatively light masses ($m_{\Sigma_{1}}\simeq 100-120$ GeV).

Figure 4: The flux of monochromatic photons from the reaction
$\Sigma_{1}\Sigma_{1}\to\gamma\gamma$ for $\bar{J}\Delta\Omega=1$ for two
different values of the coupling $g_{st}$.

##  V Conclusions

The Private Higgs model attempts to address the large hierarchy observed in
the fermion mass spectrum by introducing one Higgs doublet for each fermion.
EWSB is achieved not by the usual “negative-mass-squared” approach, but by
introducing a set of gauge singlet scalars and using the vev’s of these fields
and their interactions with the PH fields to induce instabilities. In order to
avoid cross-talk between quarks of different flavors, one also introduces a
set of discrete symmetries. This provides one of the most interesting features
of the Private Higgs model: a possible dark matter candidate.

In this paper, we have begun an investigation of the PH dark matter sector. We
found that for relatively natural values of the model’s parameters the PH
model provides a candidate which can account for the relic density of dark
matter observed in the present Universe. To show this, we calculated the
annihilation cross section for PHDM into SM particles and compared to limits
on the cross section which can be obtained from the WMAP observations.

Finally, we investigated the possibility of detecting PHDM via anomalous gamma
rays originating from the annihilation of PHDM in the galactic halo. While the
observation of these gamma rays may be difficult for the space-based GLAST
observatory, we showed that evidence of PHDM could be observed at ground-based
atmospheric Cerenkov telescopes such as VERITAS and HESS if there is
substantial clustering of dark matter in the galactic halo.

###### Acknowledgements.

The author is very grateful to Sally Dawson for a careful reading of this
manuscript and Rafael Porto for extremely useful discussions on the Private
Higgs model. This manuscript has been authored by employees of Brookhaven
Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S.
Department of Energy. The publisher by accepting the manuscript for
publication acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes.

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††institutetext:  1 Department of Physics and CTC, National Tsing Hua
University, Hsinchu 300, Taiwan
2 Division of Quantum Phases and Devices, School of Physics, Konkuk
University, Seoul 143-701, Republic of Korea
3 Department of Physics, University of Illinois at Chicago, Illinois 60607 USA

# Iso-doublet Vector Leptoquark solution to the Muon $g-2$, $R_{K,K^{*}}$,
$R_{D,D^{*}}$, and $W$-mass Anomalies

Kingman Cheung1,2, Wai-Yee Keung3, and Po-Yan Tseng1

###### Abstract

We investigate the iso-doublet vector leptoquark $V_{2}$ as a solution to the
$B$ anomalies $R_{K,K^{*}}$ and $R_{D,D^{*}}$, as well as explaining the muon
and electron anomalous magnetic moments, and the very recent $W$-mass anomaly.

##  1 Introduction

Recent trends in hunting for new physics beyond the standard model (SM) can
divided into (i) high energy frontier, (ii) precision frontier, and (iii)
cosmology frontier. While the high energy frontier has not been finding
anything new other than the discovery of the Higgs boson and the cosmology
frontier involves large uncertainties associated with observations, the
precision frontier, on the other hand, has shown some surprising results.
Namely, there are a number of anomalies in $B$ meson decays, the muon
anomalous moment, and the very recent $W$-boson mass measurement CDF:2022hxs .

After accumulating data for a number of years, there exist persistent
discrepancies between the SM predictions and the experimental results for the
flavor-changing neutral current rare decays of $B$ mesons in $b\to s\ell\ell$.
In particular, the lepton-flavor universality violation in $B\to K$ transition
observed by LHCb

$R_{K}=\frac{{\rm BR}(B\to K\mu^{+}\mu^{-})}{{\rm BR}(B\to
Ke^{+}e^{-})}\;,\,\;\;\;R_{K*}=\frac{{\rm BR}(B\to K^{*}\mu^{+}\mu^{-})}{{\rm
BR}(B\to K^{*}e^{+}e^{-})}\;,$ (1)

with the measurements LHCb:2021trn ; LHCb:2017avl

$R_{K}=0.846\,^{+0.042}_{-0.039}\,{}^{+0.013}_{-0.012}\;,\;\;\;\mbox{for 1.1
GeV${}^{2}<q^{2}<6$ GeV${}^{2}$}\;,$ (2)
$R_{K^{*}}=\left\\{\begin{array}[]{lr}0.66\,^{+0.11}_{-0.07}\,\pm
0.03&0.045\;{\rm GeV}^{2}<q^{2}<1.1\;{\rm GeV}^{2}\,,\\\
0.69\,^{+0.11}_{-0.07}\,\pm 0.05&1.1\;{\rm GeV}^{2}<q^{2}<6.0\;{\rm
GeV}^{2}\,,\end{array}\right.$ (3)

which deviate from the SM predictions by as large as $3\sigma$. The advantage
of using ratios is that the ratio can have a lot of hadronic uncertainties in
each branching ratio measurement to be eliminated. Precise measurements of
these ratios with significant deviations from the SM predictions can hint at
new physics. The same short distance process $b\to s\ell\ell$ is also
responsible for $B_{s}^{0}\to\ell^{+}\ell^{-}$.

Another set of observables is related to the short-distance process $b\to
c\ell\nu$ and the observables are HFLAV:2019otj

$\displaystyle R_{D}$ $\displaystyle=$ $\displaystyle\frac{{\rm BR}(B\to
D\tau\nu)}{{\rm BR}(B\to D\ell\nu)}=0.340\pm 0.027\pm 0.013\,,$ $\displaystyle
R_{D^{*}}$ $\displaystyle=$ $\displaystyle\frac{{\rm BR}(B\to
D^{*}\tau\nu)}{{\rm BR}(B\to D^{*}\ell\nu)}=0.295\pm 0.011\pm 0.008\;,$ (4)

and the combined discrepancy to SM prediction is at the $3.1\sigma$ level.

Another long-standing experimental anomaly is the muon anomalous moment (aka.
$g-2$). The most recent muon $g-2$ measurement was performed by the E989
experiment at Fermilab, which reported the new result Muong-2:2021ojo

$\Delta a_{\mu}=(25.1\pm 5.9)\times 10^{-10}\;,$ (5)

which deviates at the level of $4.2\sigma$ from the most recent SM prediction.
111 There was a recent lattice calculation of the eading hadronic contribution
to the muon magnetic moment Borsanyi:2020mff , which brought the SM prediction
within $1\sigma$ of the experimental result. Yet, one has to wait further for
the lattice community to settle on the calculation. We will focus on Eq. (17)
for the deviation of muon $g-2$.  On the other hand, the $g-2$ measurements
for electron also show discrepancy with the SM prediction. The electron $g-2$
was used to be the most precise determination of the fine-structure constant
$\alpha$. Nevertheless, there were two contradicting determinations of
$\alpha$ Aoyama:2012wj ; Aoyama:2019ryr  resulting in two theory predictions,
which deviate from the experimental measurement Hanneke:2008tm  in opposite
direction, given by

$\displaystyle\Delta a_{e}^{\rm LKB}$ $\displaystyle=a^{\rm exp}_{e}-a^{\rm
LKB}_{e}=(4.8\pm 3.0)\times 10^{-13}\;,$ $\displaystyle\Delta a_{e}^{\rm B}$
$\displaystyle=a^{\rm exp}_{e}-a^{\rm B}_{e}=(-8.8\pm 3.6)\times 10^{-13}\;.$
(6)

In the following analysis, we show the results for both cases of $\Delta
a_{e}$. Note that if we want to explain the $\Delta a_{\mu}$ with new physics,
it is very likely subject to constraints coming from lepton-flavor-violating
(LFV) decays, such as $\mu\to e\gamma$.

There has been a vast literature in explaining all or part of the anomalies.
Especially, the leptoquark (LQ) provides viable explanations for the
$R_{K,K^{*}}$ and/or $R_{D,D^{*}}$ anomalies. It was illustrated in Ref.
Angelescu:2021lln  that only the iso-singlet vector LQ $U_{1}$ with the SM
quantum numbers $({\bf 3},{\bf 1},2/3)$ can explain both $R_{K,K^{*}}$ and
$R_{D,D^{*}}$, while the scalar LQ’s $S_{1},S_{3},R_{2}$ and the iso-triplet
vector LQ $U_{3}\,({\bf 3},{\bf 1},2/3)$ can only explain one of the
anomalies. There has been very few studies on the isospin-doublet vector LQ,
denoted by $V_{2}\,({\bf 3},{\bf 2},5/6)$. In this work, we attempt to fill
the gap by showing that the iso-doublet vector LQ $V_{2}$ can satisfy the
anomalies $R_{K,K^{*}}$ and $R_{D,D^{*}}$, as well as it can satisfy the
$\Delta a_{\mu}$ and $\Delta a_{e}$ subject to LFV constraints.

Very recently, there was a new $W$-boson mass measurement by the CDF
Collaboration CDF:2022hxs :

$M_{W}=80.4335\pm 0.0094\;{\rm GeV}$

which is about $7\sigma$ above the SM prediction of $M_{W,SM}=80.361\pm 0.006$
GeV ParticleDataGroup:2016lqr . Additional corrections to the $W$-boson mass
can come from new physics, which can be conveniently parameterized in terms of
the Peskin-Takeuchi parameters $S,T,U$ Peskin:1991sw . The $W$-boson mass
correction is most sensitive to the $T$ parameter. It is well known that
electroweak doublet with a mass splitting between the upper and lower
components contributes positively to the $T$ parameter, thus gives a positive
contribution to the $W$ mass.

We highlight the capabilities of the LQ $V_{2}$ in solving the anomalies:

  1. 1.

The more popular explanation of $R_{K,K^{*}}$ anomaly is by decreasing the
$b\to s\mu^{+}\mu^{-}$ using the left-handed coupling to the muon, such that
the combination of the Wilson coefficients $C_{9}$ and $C_{10}$ is in the form
$C_{9}=-C_{10}$ in accordance with the best fit Altmannshofer:2021qrr . This
is the case of the iso-singlet vector LQ $U_{1}$, On the other hand, the iso-
doublet vector LQ $V_{2}$ couples to the right-handed lepton. such that it
contribute to $C_{9}$ and $C_{10}$ in the combination of $C_{9}=C_{10}$. We
therefore require $V_{2}$ to couple to electron and to increase $b\to
se^{+}e^{-}$.

  2. 2.

$V_{2}$ can enhance substantially the $\Delta a_{\mu}$ and change $\Delta
a_{e}$ in both directions using different combinations of couplings.
Nevertheless, in order to reach with $2\sigma$ of $\Delta a_{\mu}$ one of the
couplings has to be very large. This is due to the constraint from the $\mu\to
e\gamma$.

  3. 3.

$R_{D,D^{*}}$ can be easily satisfied.

  4. 4.

The iso-doublet LQ $V_{2}$ can couple to the weak gauge bosons and thus
modifies the $S,T,U$ parameters. It can explain the very recent $W$-mass
anomaly measured by CDF.

The organization of this work is as follows. In the next section, we introduce
the interactions of the vector LQ. In Sec. 3, we describe the effects of the
vector LQ on various observables that we used in this analysis. In Sec. 4, we
give the results and the valid parameter space. We conclude in Sec. 5.

##  2 Vector Leptoquark Interactions

The weak iso-doublet vector LQ is denoted by $V_{2}$ with the SM quantum
numbers $({\bf\bar{3}},\,{\bf 2},\,5/6)$. The $V_{2}$ is written as

$V_{2}=\left(\begin{array}[]{c}V^{+4/3}\\\ V^{+1/3}\end{array}\right)$

For simplicity we drop the subscript of $V_{2}$ from now on. The gauge
interactions of the $V_{2}$ is given by

$\displaystyle{\cal L}_{V_{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}V_{\mu\nu}^{\dagger}\,V^{\mu\nu}+M_{V}^{2}V^{\dagger}_{\mu}\,V^{\mu}$
(7) $\displaystyle+$ $\displaystyle
ig_{3}V^{\dagger}_{\mu}\frac{\lambda^{A}}{2}V_{\nu}\,G^{A,\mu\nu}+ig_{2}V^{\dagger}_{\mu}\frac{\tau^{k}}{2}V_{\nu}W^{k,\mu\nu}+ig_{1}V^{\dagger}_{\mu}YV_{\nu}B^{\mu\nu}$

where $V_{\mu\nu}=\sum_{i=1,2}D_{\mu}V^{i}_{\nu}-D_{\nu}V^{i}_{\mu}$ and

$D_{\mu}=\partial_{\mu}+ig_{1}YB_{\mu}+ig_{2}\frac{\tau^{k}}{2}W^{k}_{\mu}+ig_{3}\frac{\lambda^{A}}{2}G_{\mu}^{A}$
(8)

where the $SU(2)_{L}$ index $k=1,2,3$ and the color index $A=1,...,8$. 222
Here we have take all $\kappa=1$ and $\tilde{\kappa}=0$ as in the convention
of Ref. Altmannshofer:2020ywf  with the assumption $V_{2}$ coming from
spontaneous breakdown of a gauge symmetry. Note the major difference is that
$U_{1}$ discussed in Ref. Altmannshofer:2020ywf  always connects down-type
quarks and charged leptons of the same chiralities, while here the $V_{2}$
always connects them with opposite chiralities.  In order to extract the
electromagnetic interaction with $V_{2}$ we transform $B_{\mu}$ and
$W^{3}_{\mu}$ into $A_{\mu}$ and $Z_{\mu}$ by the usual transformation:

$B_{\mu}=c_{w}A_{\mu}-s_{w}Z_{\mu}\,,\qquad
W^{3}_{\mu}=s_{w}A_{\mu}+c_{w}Z_{\mu}\,,$

where $c_{w}=\cos\theta_{W},\,s_{w}=\sin\theta_{W}$ and $\theta_{W}$ is the
weak mixing angle. As long as the electromagnetic interaction is concerned, we
simplify the Lagrangian as

${\cal
L}_{V_{2},em}=-\frac{1}{2}V_{\mu\nu}^{\dagger}V^{\mu\nu}+ieQ_{V}V^{\dagger}_{\mu}V_{\nu}F^{\mu\nu}+...$
(9)

where $D_{\mu}=\partial_{\mu}+ieQ_{V}A_{\mu}+...$. Extracting the triple
vertex $V^{\dagger}VA$ we obtain

$\displaystyle{\cal L}_{V^{\dagger}VA}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[ieQ_{V}(\partial_{\mu}V_{\nu}^{\dagger}-\partial_{\nu}V^{\dagger}_{\mu})(A^{\mu}V^{\nu}-A^{\nu}V^{\mu})-ieQ_{V}(A_{\mu}V^{\dagger}_{\nu}-A_{\nu}V_{\mu}^{\dagger})(\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu})\right]$
$\displaystyle+ieQ_{V}V^{\dagger}_{\mu}V_{\nu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})$
$\displaystyle=$ $\displaystyle-
ieQ_{V}(\partial_{\mu}V_{\nu}^{\dagger}-\partial_{\nu}V_{\mu}^{\dagger})A^{\mu}V^{\nu}+ieQ_{V}A_{\mu}V_{\nu}^{\dagger}(\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu})+ieQ_{V}V^{\dagger}_{\mu}V_{\nu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})$

We assign the 4-momenta and polarization vectors for $V^{\dagger},\,V,A$ as

$V^{\dagger}:p^{\prime},\;\lambda^{\prime}\,;\qquad V:p,\;\lambda\,;\qquad
A:k,\;\epsilon\,.$

then we obtain the triple vertex as

${\cal L}_{\rm
V^{\dagger}VA}=eQ_{V}\left[g_{\alpha\beta}(p-p^{\prime})_{\gamma}+g_{\beta\gamma}(p^{\prime}-k)_{\alpha}+g_{\gamma\alpha}(k-p)_{\beta}\right]\,\lambda^{\alpha}\lambda^{{}^{\prime}\beta}\epsilon^{\gamma}$
(10)

Therefore, the interaction of $V$ with the photon is the same as the
conventional charged vector boson.

The interactions of $V_{2}$ with SM fermions are given by 333In principle,
there could be diquark couplings to $V_{2}$ but they would lead to dangerous
proton decay Greljo:2021npi . Therefore, we do not include them here.

$\displaystyle{\cal L}_{Vff}$ $\displaystyle=$ $\displaystyle
X^{RL}_{ij}\epsilon^{ab}\,\overline{d_{R}^{c,i}}\,\gamma^{\mu}V^{a}_{\mu}\,L^{j,b}_{L}+X^{LR}_{ij}\epsilon^{ab}\,\overline{Q_{L}^{c,i,a}}\,\gamma^{\mu}V^{b}_{\mu}\,e^{j}_{R}+{h.c.}$
(11) $\displaystyle=$ $\displaystyle
X^{RL}_{ij}\left[\overline{d_{R}^{c,i}}\,\gamma^{\mu}\ell_{L}^{j}\,V^{+4/3}_{\mu}-\overline{d_{R}^{c,i}}\,\gamma^{\mu}\nu^{j}_{L}V_{\mu}^{+1/3}\right]$
$\displaystyle+X^{LR}_{ij}\left[\overline{u_{L}^{c,i}}\,\gamma^{\mu}e_{R}^{j}\,V^{+1/3}_{\mu}-\overline{d_{L}^{c,i}}\,\gamma^{\mu}e_{R}^{j}\,V^{+4/3}_{\mu}\right]+h.c.\,,$

where we assume that the down-type quarks and charged leptons are in the mass
eigenstates while the CKM matrix is associated with the up-type quarks. To
convert the up-type quarks of the interaction basis in Eq.(11) into the mass
eigenstates, we need to replace $\overline{u^{c,i}_{L}}$ in Eq.(11) by
$V_{ij}\overline{u^{c,i}_{L}}$ via CKM matrix $V_{ij}$.

##  3 Various Observables

In this section, we summarize various observables that we are going to include
in this analysis, namely, the muon anomalous magnetic moment (aka $g-2$) and
lepton-flavor violating radiative decays, and $B$ anomalies $R_{K,K^{*}}$ and
$R_{D,D^{*}}$. The $W$-boson mass is accounted for the mass splitting the
isospin components of the $V_{2}$, which has negligible effects on the low-
energy observables.

###  3.1 $g-2$ and $\ell_{i}\to\ell_{j}\gamma$

The amplitudes for the lepton anomalous magnetic moment $\Delta a_{\ell}$ and
lepton-flavor violating radiative decays $\ell_{i}\to\ell_{j}\gamma$ are
related. We can write the transition amplitude for

$\ell_{i}(p)\to\ell_{j}(p-q)\;\gamma(q)$

where the 4-momentum $p$ is coming and $q$ is outgoing, as

$\displaystyle
T=e\epsilon^{\mu*}(q)m_{\ell_{i}}\,\bar{u}(p-q)i\sigma_{\mu\nu}q^{\nu}\,\left[A^{\ell_{i}\ell_{j}}_{L}P_{L}+A^{\ell_{i}\ell_{j}}_{R}P_{R}\right]\,u(p)\;.$
(12)

Then the partial width of $\Gamma(\ell_{i}\to\ell_{j}\gamma)$ can be expressed
as

$\Gamma(\ell_{i}\to\ell_{j}\gamma)=\frac{\alpha_{\rm
em}}{4}m_{\ell_{i}}^{5}\left(|A^{\ell_{i}\ell_{j}}_{L}|^{2}+|A^{\ell_{i}\ell_{j}}_{R}|^{2}\right)\;,$
(13)

where the mass of the daughter lepton $\ell_{j}$ is ignored.

On the other hand, the anomalous magnetic moment form factor of the lepton
$\ell$ is given by

$\displaystyle{\cal L}_{g-2}$ $\displaystyle=$ $\displaystyle\frac{e\,\Delta
a_{\ell}}{4m_{\ell}}\bar{\ell}\sigma_{\mu\nu}\ell\,F^{\mu\nu}$ (14)
$\displaystyle=$ $\displaystyle\frac{e\,\Delta
a_{\ell}}{2m_{\ell}}\bar{\ell}i\sigma_{\mu\nu}(q^{\nu})\,\ell\,A^{\mu}$

where $\Delta a_{\ell}$ is the anomalous magnetic moment of the lepton and
here $q$ is coming into the vertex. Comparing Eq. (12) and Eq. (14) we have

$\Delta a_{\ell}=-m_{\ell}^{2}(A^{\ell\ell}_{L}+A^{\ell\ell}_{R})\;.$ (15)

We adopted the formulas here from Ref. Hati:2019ufv  for
$\ell_{i}\to\ell_{j}\gamma$

$\displaystyle A^{\ell_{i}\ell_{j}}_{L}$ $\displaystyle=$
$\displaystyle-\frac{N_{C}}{16\pi^{2}M_{V}^{2}}\,\sum_{k}\biggr{[}-2X^{LR*}_{k\ell_{j}}X^{RL}_{k\ell_{i}}\frac{m_{k}}{m_{\ell_{i}}}\,(Q_{V}+Q_{b^{c}})$
(16)
$\displaystyle+\left(X^{LR*}_{k\ell_{j}}X^{LR}_{k\ell_{i}}+X^{RL*}_{k\ell_{j}}X^{RL}_{k\ell_{i}}\frac{m_{\ell_{j}}}{m_{\ell_{i}}}\right)\;\left(-\frac{5}{6}Q_{V}-\frac{2}{3}Q_{b^{c}}\right)\biggr{]}\,,$
$\displaystyle A^{\ell_{i}\ell_{j}}_{R}$ $\displaystyle=$ $\displaystyle
A^{\ell_{i}\ell_{j}}_{L}(L\leftrightarrow R)$

where $Q_{b^{c}}=-Q_{b}$ denotes the electric charge of the charge conjugate
of the $b$ quark, and $Q_{V}=+4/3$ refers to the upper component of $V_{2}$.
We have made use of the fact that all down-type quark masses are much smaller
than the LQ mass $m_{k}/M_{V}\ll 1$ such that the loop functions $f,g,,h,j$
approach constant values Hati:2019ufv .

We can express the contribution of the LQ to the lepton anomalous moment as

$\Delta a_{\ell}=-\frac{N_{C}}{16\pi^{2}}\biggr{[}4{\cal
R}e(X^{LR*}_{3\ell}X^{RL}_{3\ell})\frac{m_{b}m_{\ell}}{M_{V}^{2}}(Q_{V}+Q_{b^{c}})+2(|X^{LR}_{3\ell}|^{2}+|X^{RL}_{3\ell}|^{2})\frac{m_{\ell}^{2}}{M_{V}^{2}}\left(\frac{5}{6}Q_{V}+\frac{2}{3}Q_{b^{c}}\right)\biggr{]}$
(17)

where we assume that the contribution from $b$ quark dominates over the $s$
and $d$ quarks. The partial width for the radiative decay of, such as $\mu\to
e\gamma$, is given by

$\Gamma(\mu\to e\gamma)=\frac{\alpha_{\rm em}}{4}m_{\mu}^{5}\left(|A_{L}^{\mu
e}|^{2}+|A_{R}^{\mu e}|^{2}\right)$ (18)

###  3.2  $R_{K,K^{*}}$

The effective Lagrangian for a generic exclusive decay of $b\to
s\ell^{-}\ell^{+}$ , with $\ell=e,\,\mu,\,\tau$ can be written as

${\cal
L}_{bs\ell\ell}\supset\frac{4G_{F}}{\sqrt{2}}\,V_{tb}V^{*}_{ts}\,\frac{e^{2}}{16\pi^{2}}\,\sum_{i}\,\left[C_{i}{\cal
O}_{i}+C^{\prime}_{i}{\cal O}^{\prime}_{i}\right]+h.c.\,,$ (19)

where

$\displaystyle{\cal
O}_{9}=(\bar{s}\gamma_{\mu}P_{L}b)\,(\bar{\ell}\gamma^{\mu}\ell)\,,\qquad{\cal
O}^{\prime}_{9}=(\bar{s}\gamma_{\mu}P_{R}b)\,(\bar{\ell}\gamma^{\mu}\ell)\,,$
$\displaystyle{\cal
O}_{10}=(\bar{s}\gamma_{\mu}P_{L}b)\,(\bar{\ell}\gamma^{\mu}\gamma^{5}\ell)\,,\qquad{\cal
O}^{\prime}_{10}=(\bar{s}\gamma_{\mu}P_{R}b)\,(\bar{\ell}\gamma^{\mu}\gamma^{5}\ell)\,,$
$\displaystyle{\cal O}_{S}=(\bar{s}\ P_{R}b)\,(\bar{\ell}\ell)\,,\qquad{\cal
O}^{\prime}_{S}=(\bar{s}P_{L}b)\,(\bar{\ell}\ell)\,,$ $\displaystyle{\cal
O}_{P}=(\bar{s}P_{R}b)\,(\bar{\ell}\gamma_{5}\ell)\,,\qquad{\cal
O}^{\prime}_{P}=(\bar{s}P_{L}b)\,(\bar{\ell}\gamma_{5}\ell)\,,$

Since the term responsible for the left-handed down-type quark $V_{2}$
interaction is
$-X_{ij}^{LR}\,\overline{d_{L}^{c,i}}\,\gamma^{\mu}e_{R}^{j}\,V^{+4/3}_{\mu}$,
which involves only the right-handed charged lepton $e_{R}^{j}$, it will give
rise to the relation $C_{9}=C_{10}$ between $C_{9}$ and $C_{10}$. Therefore,
we focus the interactions with the electron instead of muon, as the best fit
results preferred $C_{9}=-C_{10}$ for muon but $C_{9}=+C_{10}$ for electron.
The contributions to the Wilson coefficients from $V_{2}$ focusing on the
interactions with electron are given by

$\displaystyle C^{bsee}_{9}=+C_{10}^{bsee}$ $\displaystyle=$
$\displaystyle-\frac{4\pi^{2}}{e^{2}}\,\frac{v^{2}}{M^{2}_{V_{2}}}\,\frac{X^{LR}_{31}X^{LR*}_{21}}{V_{ts}^{*}V_{tb}}\;,$
$\displaystyle C^{{}^{\prime}bsee}_{9}=-C_{10}^{{}^{\prime}bsee}$
$\displaystyle=$
$\displaystyle-\frac{4\pi^{2}}{e^{2}}\,\frac{v^{2}}{M^{2}_{V_{2}}}\,\frac{X^{RL}_{31}X^{RL*}_{21}}{V_{ts}^{*}V_{tb}}\;,$
$\displaystyle C^{bsee}_{S}=-C_{P}^{bsee}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{e^{2}}\,\frac{2v^{2}}{M^{2}_{V_{2}}}\,\frac{X^{RL}_{31}X^{LR*}_{21}}{V_{ts}^{*}V_{tb}}\;,$
$\displaystyle C^{{}^{\prime}bsee}_{S}=+C_{P}^{{}^{\prime}bsee}$
$\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{e^{2}}\,\frac{2v^{2}}{M^{2}_{V_{2}}}\,\frac{X^{LR}_{31}X^{RL*}_{21}}{V_{ts}^{*}V_{tb}}\;,$

Strictly speaking, the above Wilson coefficients are calculated at the
electroweak scale by integrating out the heavy degrees of freedom such as the
LQ. One has to evolve them down to the $m_{b}$ scale. However, giving that the
evolution effect is very small because of small mixings between the operators
and ${\cal
O}_{2}\equiv(\bar{s}\gamma_{\mu}(1-\gamma_{5})c)(\bar{c}\gamma^{\mu}(1-\gamma_{5})b)$,
444We specially thank Wolfgang Altmannshofer for explaining this point. we
directly employ the above expressions in our analysis, similarly for the
Wilson coefficient $C_{S_{R}}$.

###  3.3 $R_{D,D^{*}}$

The effective Lagrangian for a generic exclusive decay of $b\to
c\tau^{-}\bar{\nu}_{\tau}$ can be written as

$\displaystyle{\cal L}_{bc\ell\nu}$ $\displaystyle=$
$\displaystyle-2\sqrt{2}G_{F}V_{cb}\,\biggr{[}(1+C_{V_{L}})(\overline{c_{L}}\gamma^{\mu}b_{L})\,(\overline{\tau}\gamma^{\mu}\nu_{\tau_{L}})+C_{V_{R}}(\overline{c_{R}}\gamma^{\mu}b_{R})\,(\overline{\tau}\gamma^{\mu}\nu_{\tau_{L}})$
(20) $\displaystyle+$ $\displaystyle
C_{S_{R}}(\overline{c_{L}}b_{R})\,(\overline{\tau}\nu_{\tau_{L}})+C_{S_{L}}(\overline{c_{R}}b_{L})\,(\overline{\tau}\nu_{\tau_{L}})+C_{T}(\overline{c_{R}}\sigma^{\mu\nu}b_{L})\,(\overline{\tau}\sigma_{\mu\nu}\nu_{\tau_{L}})\;\biggr{]}$

The isospin $-1/2$ component of the $V_{2}$ contributes to the process $b\to
c\tau\nu_{\tau}$ via the operator
$(\overline{c_{L}}b_{R})\,(\overline{\tau}\nu_{\tau_{L}})$. and thus modifies
the coefficient $C_{S_{R}}$:

$C_{S_{R}}=-\frac{1}{2\sqrt{2}G_{F}}\,\frac{1}{V_{cb}}\,\sum_{k}\left(V_{k2}\,\frac{2}{M_{V}^{2}}\,X^{RL}_{33}X^{LR*}_{k3}\right)$
(21)

###  3.4 $B_{s}\to\ell^{+}\ell^{-}$

Including the new physics contributions, the general expression of the
$B_{s}\to\ell^{+}\ell^{-}$ in terms of Wilson coefficients is Kosnik:2012dj

$\displaystyle{\rm Br}(B_{s}\to\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle\tau_{B_{s}}f^{2}_{B_{s}}m^{3}_{B_{s}}\frac{G^{2}_{F}|V_{tb}V^{*}_{ts}|^{2}e^{4}}{(4\pi)^{5}}\sqrt{1-4m^{2}_{\ell}/m^{2}_{B_{s}}}$
$\displaystyle\times$
$\displaystyle\left[\frac{m^{2}_{B_{s}}}{m^{2}_{b}}\left|C_{S}-C^{\prime}_{S}\right|^{2}\left(1-\frac{4m^{2}_{\ell}}{m^{2}_{B_{s}}}\right)+\left|\frac{m_{B_{s}}}{m_{b}}\left(C_{P}-C^{\prime}_{P}\right)+\frac{2m_{\ell}}{m_{B_{s}}}\left(C^{\rm
SM}_{10}+C_{10}-C^{\prime}_{10}\right)\right|^{2}\right]\,,$

where $C^{\rm SM}_{10}=-4.1$ stems from the SM contribution,
$\tau_{B_{s}}=1.52\times 10^{-12}~{}{\rm s}$ and $f_{B_{s}}=228.4~{}{\rm MeV}$
are the lifetime and decay constant of the $B$-meson, respectively. The
current measurements are Altmannshofer:2021qrr

$\displaystyle{\rm Br}(B_{s}\to\mu^{+}\mu^{-})=(3.09^{+0.48}_{-0.44})\times
10^{-9}\,,$ (23) $\displaystyle{\rm Br}(B_{s}\to e^{+}e^{-})<9.4\times
10^{-9}~{}~{}\text{at $90\%$ C.L from PDG},$ (24)

and consistent with the SM.

###  3.5 $W$ boson mass

The $W$-boson mass due to new physics expressed in terms of the $\Delta
S,\Delta T,\Delta U$ is given by

$M_{W}^{2}=M^{2}_{W,SM}+\frac{\alpha
c_{W}^{2}M_{Z}^{2}}{c_{W}^{2}-s_{W}^{2}}\,\left[-\frac{\Delta
S}{2}+c_{W}^{2}\Delta T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}\Delta U\right]$
(25)

where $c_{W},s_{W}$ are, respectively, the cosine and the sine of the Weinberg
angle. $\alpha$ is the fine-structure constant, and $M_{Z}$ is the $Z$-boson
mass. In the SM, $M_{W,SM}=80.361\pm 0.006$ GeV ParticleDataGroup:2016lqr ,
while the most recent measurement by CDF is $M_{W}=80.4335\pm 0.0094$ GeV
CDF:2022hxs . Such a large deviation can be most easily accommodated by a
positive $\Delta T$. It is well know that an additional weak doublet with a
mass splitting gives a contribution to $\Delta T$ as 555This expression is
based on the formulas listed in the Appendix D of the Stefan Pokorski’s book
Pokorski:1987ed .

$\displaystyle\Delta T$ $\displaystyle=$ $\displaystyle\frac{C}{4\pi
M_{W}^{2}s_{W}^{2}}\,F(m_{1},m_{2})\;,$ (26)

where $C=1(3)$ for color singlet (triplet) and

$F(m_{1},m_{2})=m_{1}^{2}+m_{2}^{2}-\frac{2m_{1}^{2}m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\log\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\;.$

Note that $\Delta T$ is proportional to the mass splitting $\Delta
m=|m_{1}-m_{2}|$ between the upper and the lower component of the doublet, and
thus always positive. With this $\Delta T$ the change in $M_{W}^{2}$ is given
by

$\Delta M_{W}^{2}=\frac{\alpha c_{W}^{4}M_{Z}^{2}}{c_{W}^{2}-s_{W}^{2}}\Delta
T$ (27)

##  4 Parameter scans and Results

Table 1: Best fitted results of the relevant Wilson coefficients for $R_{K,K^{*}}$ Altmannshofer:2021qrr  and $R_{D,D^{*}}$ Bhol:2021iow , muon and electron anomalous magnetic moments $\Delta a_{\mu}$ Muong-2:2021ojo  and $\Delta a_{e}$ Aoyama:2012wj ; Aoyama:2019ryr ; Hanneke:2008tm . Note that there is a controversy for $\Delta a_{e}$ due to two different experimental measurements.  $C_{9}^{bsee}=C_{10}^{bsee}$ | $-1.28\,^{+0.24}_{-0.23}$
---|---
$C^{\ell}_{S_{R}}$ | $0.027\,^{+0.025}_{-0.026}$
$\Delta a_{\mu}$ | $(251\pm 59)\times 10^{-11}$
$\Delta a^{\rm LKB}_{e}$ | $(4.8\pm 3.0)\times 10^{-13}$
$\Delta a^{\rm B}_{e}$ | $(-8.8\pm 3.6)\times 10^{-13}$

The best fitted results on the Wilson coefficients were updated in early 2021
Altmannshofer:2021qrr . We list them in Table 1. The LQ couplings can involve
different leptonic flavors in the same Feynman diagram, and so it will also
give rise to the lepton-flavor-violating processes, notably, the radiative
leptonic decays $\ell_{i}\to\ell_{j}\gamma$. While these processes have not
been observed, the upper limits are quite restrictive:

$\displaystyle B(\tau\to\mu\gamma)<4.4\times 10^{-8}\,,$ $\displaystyle
90\%\;{\rm C.L.}$ $\displaystyle B(\tau\to e\gamma)<3.3\times 10^{-8}\,,$
$\displaystyle 90\%\;{\rm C.L.}$ $\displaystyle B(\mu\to e\gamma)<4.2\times
10^{-13}\,,$ $\displaystyle 90\%\;{\rm C.L.}$

Figure 1:  Scan: The chi-square distribution includes $R_{K,K^{*}}$,
$R_{D,D^{*}}$, $B_{s}\to\ell^{+}\ell^{-}$, and $\ell_{j}\to\ell_{i}\gamma$
data. The best-fit point, BP-1, gives $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm
LFV}}=22.48$ with pull 3.8$\sigma$ comparing to the SM,
$\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}}=36.92$.  Table 2:  Benchmark points
with $m_{V_{2}}=2~{}{\rm TeV}$. SM gives $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm
LFV}}=36.92$, $\chi^{2}|_{R_{K}+R_{D}+{\rm LFV}+a_{\mu}+a^{\rm
LKB}_{e}}=57.58$, and $\chi^{2}|_{R_{K}+R_{D}+{\rm LFV}+B_{s}+a_{\mu}+a^{\rm
B}_{e}}=61.00$.

BP-1 BP-2 BP-3 $X^{\rm LR}_{21}$ -0.445 -17.85 -0.0115 $X^{\rm LR}_{31}$
0.0271 $6.44\times 10^{-4}$ 0.927 $X^{\rm RL}_{31}$ $1.58\times 10^{-4}$
$-2.48\times 10^{-8}$ -0.0145 $X^{\rm LR}_{22}$ -0.0245 -$7.18\times 10^{-5}$
-0.0606 $X^{\rm LR}_{32}$ $-2.01\times 10^{-3}$ 6.820 $1.11\times 10^{-5}$
$X^{\rm RL}_{32}$ $-8.25\times 10^{-3}$ -0.183 $-1.19\times 10^{-5}$ $X^{\rm
LR}_{23}$ -0.367 $-2.35\times 10^{-3}$ 0.990 $X^{\rm RL}_{33}$ 0.217
$-1.87\times 10^{-4}$ -0.0252 $C^{bs\mu\mu}_{9}=C^{bs\mu\mu}_{10}$ $7.77\times
10^{-3}$ $-7.73\times 10^{-3}$ $-1.06\times 10^{-4}$
$C^{bsee}_{9}=C^{bsee}_{10}$ -1.90 -1.82 -1.68 $C^{\ell}_{S_{R}}$ 0.0276
$-1.55\times 10^{-4}$ 0.0141 $C^{bs\mu\mu}_{S}=-C^{bs\mu\mu}_{P}$ -0.0638
$-4.14\times 10^{-3}$ $-2.27\times 10^{-4}$ $C^{bsee}_{S}=-C^{bsee}_{P}$
0.0222 $-1.40\times 10^{-4}$ -0.053 $\Delta a^{V_{2}}_{\mu}$ $-2.11\times
10^{-13}$ $2.12\times 10^{-9}$ $-5.19\times 10^{-13}$ $\Delta a^{V_{2}}_{e}$
$-8.02\times 10^{-16}$ $-1.05\times 10^{-12}$ $4.52\times 10^{-13}$ ${\rm
Br}(\tau\to\mu\gamma)$ $3.36\times 10^{-11}$ $7.94\times 10^{-12}$ $2.04\times
10^{-9}$ ${\rm Br}(\tau\to e\gamma)$ $1.86\times 10^{-8}$ $1.01\times 10^{-9}$
$3.65\times 10^{-9}$ ${\rm Br}(\mu\to e\gamma)$ $1.27\times 10^{-13}$
$6.37\times 10^{-14}$ $8.68\times 10^{-14}$ $\chi^{2}|_{R_{K}}$ 21.34 22.15
21.63 $\chi^{2}|_{R_{D}}$ 0.001 1.09 0.245 $\chi^{2}|_{B_{s}\to\mu\mu}$ 0.002
0.567 0.0408 $\chi^{2}|_{B_{s}\to ee}$ 0.018 0.000 0.548 $\chi^{2}|_{\rm LFV}$
1.12 0.065 0.156 $\chi^{2}|_{a_{\mu}}$ 18.10 0.434 18.11 $\chi^{2}|_{a^{\rm
LKB}_{e}}$ 2.57 26.12 0.009 $\chi^{2}|_{a^{\rm B}_{e}}$ 5.96 0.232 13.69
$\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}}$ 22.48 23.88 22.62
$\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a_{\mu}+a^{\rm LKB}_{e}}$ 43.15 50.43
40.74 $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a_{\mu}+a^{\rm B}_{e}}$ 46.54
24.54 54.42 $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a^{\rm LKB}_{e}}$ 25.05
50.00 22.63 $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a^{\rm B}_{e}}$ 28.44
24.11 36.31

We perform the global chi-square fit to the observables, including
$R_{K,K^{*}}$, $R_{D,D^{*}}$, $B_{s}\to\ell^{+}\ell^{-}$, and
$\ell_{i}\to\ell_{j}\gamma$, by scanning the couplings of LQ $V_{2}$ with the
scanning ranges of various couplings:

$\displaystyle\text{\bf Scan}:~{}~{}-20$ $\displaystyle\leq$ $\displaystyle
X^{\rm LR}_{21}\leq 20\,,~{}~{}0\leq X^{\rm
LR}_{31}\leq\sqrt{4\pi}\,,~{}~{}-1\leq X^{\rm RL}_{31}\leq 1\,,$
$\displaystyle-\sqrt{4\pi}$ $\displaystyle\leq$ $\displaystyle X^{\rm
LR}_{22}\leq\sqrt{4\pi}\,,~{}~{}-2\sqrt{4\pi}\leq X^{\rm LR}_{32}\leq
2\sqrt{4\pi}\,,~{}~{}-1\leq X^{\rm RL}_{32}\leq 1\,,$ $\displaystyle-1$
$\displaystyle\leq$ $\displaystyle X^{\rm LR}_{23}\leq 1\,,~{}~{}-2\leq X^{\rm
RL}_{33}\leq 2\,,~{}~{}m_{V_{2}}=2~{}{\rm TeV}\,.$ (28)

Since the theoretical uncertainties are still ambiguous among $\Delta
a_{e,\mu}$, we have not included the observables in the global fit at this
stage, but rather treat them as posterior predictions. Due to the facts that
all the observables and Wilson coefficients are originated from products of
two couplings, such that only the relative sign between couplings would be
revealed from the chi-square fitting, we scanned the positive value of $X^{\rm
LR}_{31}$ and both signs for the other couplings. Note that the $X^{\rm
LR}_{21}$ and $X^{\rm LR,RL}_{31}$ ($X^{\rm LR}_{22}$ and $X^{\rm
LR,RL}_{32}$) are related to $C^{bsee}_{9,10}$ ($C^{bs\mu\mu}_{9,10}$). The
$X^{\rm LR}_{23}$ combining with $X^{\rm RL}_{3\ell}$ dominately contributes
to $C^{\ell}_{S_{R}}$, while other combinations are suppressed due to the off-
diagonal elements of CKM matrix. Since the flavors of neutrino are
indistinguishable in the process $b\to c\tau\nu_{\ell}$, we sum over the
neutrino flavors. The $\Delta a_{e}$ ($\Delta a_{\mu}$) are generated through
a pair of bottom-quark and $V_{2}$ running in the loop, since the heavy
$b$-quark mass enhances flipping the chiralities of external muons, thereby
they are strongly correlated to $X^{\rm LR,RL}_{31}$ ($X^{\rm LR,RL}_{32}$).
We adopted the two-dimensional chi-square statistics, i.e $\Delta\chi^{2}\leq$
2.30 (6.18) corresponds to 1-$\sigma$ (2-$\sigma$) regions. The chi-square
distributions are shown in Fig. 1, based on projections of two selected
parameters whereas marginalizing the others.

Among the scanning points using the data $R_{K,K^{*}}$, $R_{D,D^{*}}$,
$B_{s}\to\ell^{+}\ell^{-}$, and $\ell_{i}\to\ell_{j}\gamma$, we further select
three benchmark points, which yield minima of chi-square with respect to
various groups of observables, and more details are listed in Table 2:

  * •

BP-1: the best fit to $R_{K,K^{*}}$, $R_{D,D^{*}}$,
$B_{s}\to\ell^{+}\ell^{-}$, and LFV data and gives
$\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}}=22.48$ with a pull 3.8$\sigma$
comparing to the SM, $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}}=36.92$.

  * •

BP-2: gives $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a_{\mu}+a^{\rm
B}_{e}}=24.54$, and thus provides a simultaneous solution to $(g-2)_{\mu}$ and
$(g-2)^{\rm B}_{e}$.

  * •

BP-3: gives $\chi^{2}|_{R_{K}+R_{D}+B_{s}+{\rm LFV}+a^{\rm LKB}_{e}}=22.63$,
thus provides an explanation for $(g-2)^{\rm LKB}_{e}$.

Because the iso-doublet vector LQ $V_{2}$ couples to the right-handed lepton
and thus induces $C^{bsee}_{9}=C^{bsee}_{10}$, the
$(C^{bsee}_{9},C^{bsee}_{10})$ panel in Fig. 1. The BP-1 in Table 2 shows that
the $C^{bsee}_{9}=C^{bsee}_{10}=-1.90$ from $V_{2}$ provides the best solution
for $R_{K,K^{*}}$ anomaly, which increases $b\to se^{+}e^{-}$ to reduce the
values of $R_{K,K^{*}}$. It implies the correlation

$\displaystyle\frac{X^{\rm LR}_{31}(X^{\rm
LR}_{21})^{*}}{m^{2}_{V_{2}}}\simeq\frac{-0.0030}{(1~{}{\rm TeV})^{2}}$ (29)

and is shown in the $(-X^{\rm LR}_{21},X^{\rm LR}_{31})$ panel in Fig. 1.

The $(X^{\rm LR}_{23},X^{\rm RL}_{33})$ panel in Fig. 1 indicates mild
correlation between $(X^{\rm LR}_{23}$ and $X^{\rm RL}_{33})$, that came from
the $R_{D,D^{*}}$ observables and thus Wilson coefficient $C^{\ell}_{S_{R}}$.
According to the preferred value of $C^{\ell}_{S_{R}}$ from Table 1, Eq.(21)
gives

$\displaystyle\frac{X^{\rm RL}_{33}(X^{\rm
LR}_{23})^{*}}{m^{2}_{V_{2}}}\simeq\frac{-0.019}{(1~{}{\rm TeV})^{2}}\,,$ (30)

where the minus sign is originated from the $V_{cd}\simeq-0.041$. Since
$C^{\ell}_{S_{R}}=0$ is still compatible with observation within 1-$\sigma$
($\Delta\chi^{2}\leq 2.3$), the chi-square regions in $(X^{\rm LR}_{23},X^{\rm
RL}_{33})$ panel connected.

The $B_{s}\to\mu^{+}\mu^{-}$ observable dictates the chi-square regions in
$(C^{bs\mu\mu}_{S},C^{bs\mu\mu}_{P})$ panel, and there are two solutions,
$C^{bs\mu\mu}_{S}=-C^{bs\mu\mu}_{P}\simeq-0.064$ and $\simeq 0.00$ correspond
respectively to non-SM and SM solutions, which also exhibit in the $(X^{\rm
LR}_{22},X^{\rm RL}_{32})$ and $(\Delta a_{\mu},X^{\rm LR}_{22})$ panels. The
non-SM and SM solutions require

$\displaystyle\frac{X^{\rm RL}_{32}(X^{\rm
LR}_{22})^{*}}{m^{2}_{V_{2}}}\simeq\frac{-5.1\times 10^{-5}}{(1~{}{\rm
TeV})^{2}}\,,~{}\text{and}~{}~{}~{}~{}\frac{|X^{\rm RL}_{32}(X^{\rm
LR}_{22})^{*}|}{m^{2}_{V_{2}}}\lesssim\frac{10^{-5}}{(1~{}{\rm TeV})^{2}}\,,$
(31)

respectively. On the other hand, the $B_{s}\to e^{+}e^{-}$ observable, which
is consistent with SM, set limit on

$\displaystyle\frac{|X^{\rm RL}_{31}(X^{\rm
LR}_{21})^{*}|}{m^{2}_{V_{2}}}\lesssim\frac{10^{-4}}{(1~{}{\rm TeV})^{2}}\,,$
(32)

explaining the region in $(X^{\rm LR}_{21},X^{\rm RL}_{31})$ panel in Fig. 1.

BP-1 fits well to $R_{K,K^{*}}$, $R_{D,D^{*}}$, $B_{s}\to\ell^{+}\ell^{-}$,
and LFV observables, however does not induce sizable $\Delta a_{\mu}$ or
$\Delta a_{e}$. BP-2 intriguingly provides simultaneous solutions to $\Delta
a_{\mu}$ and $\Delta a^{\rm B}_{e}$, as well as a decent fit to B-physics
observables. The severe restriction from $\mu\to e\gamma$ is the main
difficulty to a generate sizable $\Delta a_{\mu}$. Specifically, the former is
related to the linear combinations of $X^{\rm LR}_{31}X^{\rm RL}_{32}$ and
$X^{\rm LR}_{32}X^{\rm RL}_{31}$, while the latter is related to the product
$X^{\rm LR}_{32}X^{\rm RL}_{32}$. Therefore, suppressing the $X^{\rm
LR,RL}_{31}$ to avoid the $\mu\to e\gamma$ constraint may help to unleash a
large enough $\Delta a_{\mu}$. According to Eq.(29) and (32), apparently, the
above scenario can be achieved by a large enough $|X^{\rm LR}_{21}|$, and this
is main result of BP-2. Unfortunately, the minimum requirement of $|X^{\rm
LR}_{21}|$ is significantly above the perturbative limit, $\sqrt{4\pi}$. For
example, it needs $X^{\rm LR}_{21}\lesssim-8$ from $(X^{\rm LR}_{21},\Delta
a_{\mu})$ panel in order to start overlapping with 2-$\sigma$ region of
$(g-2)_{\mu}$, and it is $X^{\rm LR}_{21}=-17.85$ for BP-2. Finally, BP-3
provides an alternative solution for $\Delta a^{\rm LKB}_{e}$ and B anomalies,
meanwhile is consistent with LFV and B-physics limits.

Figure 2:  The resulting $W$-boson mass due to the mass splitting between the
upper and lower isospin component of the vector LQ $V_{2}$ around 2 TeV. Note
that the lower band in green is the SM prediction while the upper band in the
latest CDF measurement.

The raising of the $W$-boson mass due to the mass splitting of the iso-doublet
$V_{2}$ is given in Eq. (27). The raise only depends on the absolute of the
mass difference and is shown in Fig. 2. It can be seen that a mass splitting
of $25\text{-}30$ GeV provides a viable solution to the $W$-boson anomaly.
Such a mass splitting corresponds to $1.25\%-1.5\%$ of a 2 TeV LQ.

##  5 Conclusions

Most works in literature on solving the $R_{K,K^{*}}$ anomaly rely on reducing
$b\to s\mu^{+}\mu^{-}$ with the LQ couplings to the left-hand muon.
Nevertheless, it remains an almost equally viable solution of increasing $b\to
se^{+}e^{-}$. Here in this work, we have attempted to use the iso-doublet
vector LQ $V_{2}$ that couples to the right-handed electron to increase $b\to
se^{+}e^{-}$ and found parameter space to explain the $R_{K,K^{*}}$.
Simultaneously, it can also explain the $R_{D,D^{*}}$ and consistent with the
$B_{s}$ decays.

We have also investigated the possibility of explaining the muon anomalous
magnetic moment. Such a possibility is severely constrained by the leptonic
radiative decay $\ell_{i}\to\ell_{j}\gamma$. We have successfully found some
parameter space points that can explain all $R_{K,K^{*}}$, $R_{D,D^{*}}$,
$B_{s}$ decays, LFV, and $\Delta a_{\mu}$ and $\Delta a^{\rm B}_{e}$ (see
BP-2), though one of the couplings is close to or larger than the perturbative
limit.

Furthermore, the iso-doublet vector leptoquark $V_{2}$ naturally explains the
$W$-boson anomaly with a mass splitting of order $25\text{-}30$ GeV between
the isospin components.

## Acknowledgment

We specially thank to Wolfgang Altmannshofer for useful discussion on
renormalization of Wilson coefficients, and to Chih-Ting Lu for discussion on
$W$-boson anomaly. K.C. also thanks Wai-Yee and Gum-see Keung for their great
hospitality. The research was supported in part by the Ministry of Sciences
and Technology with grant number MoST-110-2112-M-007-017-MY3 and
MoST-111-2112-M-007-012-MY3.

Note Added: We came across a number of works in attempt to explain the
$W$-boson anomaly Fan:2022dck ; Zhu:2022tpr ; deBlas:2022hdk ; Yang:2022gvz ;
Athron:2022qpo ; Strumia:2022qkt ; Arias-Aragon:2022ats ; Athron:2022isz ;
Heckman:2022the ; Sakurai:2022hwh ; Lu:2022bgw .

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