# A vector leptoquark interpretation of the muon $g-2$ and $B$ anomalies

Mingxuan Du  Department of Physics, Nanjing University, Nanjing 210093, China
Jinhan Liang  Department of Physics, Nanjing University, Nanjing 210093, China
Zuowei Liu  Department of Physics, Nanjing University, Nanjing 210093, China
CAS Center for Excellence in Particle Physics, Beijing 100049, China  Van Que
Tran  Department of Physics, Nanjing University, Nanjing 210093, China

###### Abstract

We show that a single vector leptoquark can explain both the muon $g-2$
anomaly recently measured by the Muon g-2 experiment at Fermilab, and the
various B decay anomalies, including the $R_{D^{(*)}}$ and $R_{K^{(*)}}$
anomalies which have been recently reported by the LHCb experiment. In order
to provide sizeable positive new physics contributions to the muon $g-2$, we
assume that the vector leptoquark particle couples to both left-handed and
right-handed fermions with equal strength. Our model is found to satisfy the
experimental constraints from the large hadron collider.

Introduction:–Recently, the Muon g-2 experiment at Fermilab, E989, has
announced its new measurement on the anomalous magnetic moment of muon

$a_{\mu}={g_{\mu}-2\over 2}.$ (1)

The result of the previous muon g-2 experiment at BNL, E821, is $3.7\sigma$
from the standard model (SM) expectation. The new E989 Run 1 data give a
smaller $a_{\mu}$ with better accuracy which is $3.4\sigma$ away from the SM
value. The combined results of the Fermilab E989 Run 1 data and the BNL E821
data give rise to a slightly smaller $a_{\mu}$ with a better precision,
yielding a $4.24\sigma$ deviation from the SM value fermilab-muon-g2 ;
Aoyama:2020ynm

$\Delta a_{\mu}=a_{\mu}^{\rm Exp}-a_{\mu}^{\rm SM}=(251\pm 59)\times
10^{-11}.$ (2)

Recently, various $B$ meson decays have shown significant deviations from the
SM predictions, most of which are related to muon final states. The lepton
flavour universality in $B$ meson decays can be tested by measuring the ratios
of the $b\to s\ell\ell$ transitions

$\displaystyle R_{K}=\frac{{\rm BR}(B^{+}\to K^{+}\mu^{+}\mu^{-})}{{\rm
BR}(B^{+}\to K^{+}e^{+}e^{-})},$ (3) $\displaystyle R_{K^{*}}=\frac{{\rm
BR}(B^{0}\to K^{*0}\mu^{+}\mu^{-})}{{\rm BR}(B^{0}\to K^{*0}e^{+}e^{-})},$ (4)

and the ratios of the $b\to c\ell\bar{\nu_{\ell}}$ decays

$R_{D^{(*)}}=\frac{{\rm BR}(B\to D^{(*)}\tau\bar{\nu})}{{\rm BR}(B\to
D^{(*)}\ell\bar{\nu})},$ (5)

where $\ell=e,\mu$. Recently, LHCb has updated the measurement on $R_{K}$
Aaij:2021vac

$R_{K}^{\rm{Exp}}=0.846^{+0.042+0.013}_{-0.039-0.012},$ (6)

in the region of $q^{2}=[1.1,6]$ GeV2, which is 3.1$\sigma$ away from the SM
prediction Bobeth:2007dw ; Bordone:2016gaq  $R_{K}^{\rm SM}=1.0003\pm 0.0001$.
The LHCb has also reported the results of $R_{K^{*}}$ Aaij:2017vbb ,

$\displaystyle R_{K^{*}}^{\rm{Exp}}=0.660^{+0.110}_{-0.070}\pm{0.03},$ (7)
$\displaystyle R_{K^{*}}^{\rm{Exp}}=0.685^{+0.113}_{-0.069}\pm{0.05},$ (8)

in the regions of $q^{2}=[0.045,1.1]$ GeV2 and $q^{2}=[1.1,6]$ GeV2
respectively, indicating $2.2\sigma$ and $2.4\sigma$ deviations from the SM
predictions, which are $R_{K^{*}}^{\rm SM}=0.92\pm 0.02$ and $R_{K^{*}}^{\rm
SM}=1.00\pm 0.01$ in these two regions. Together with the recent results from
Belle Abdesselam:2019dgh , the world averages for $R_{D^{(*)}}$ measurements
are Amhis:2019ckw  $R_{D}^{\rm{Exp}}=0.340\pm 0.030$ and
$R_{D^{*}}^{\rm{Exp}}={0.295}\pm 0.014$, whereas the SM expectations are
Amhis:2019ckw  $R_{D}^{\rm SM}={0.299\pm 0.003}$ and $R_{D^{*}}^{\rm
SM}={0.258\pm 0.005}$, yielding a $\sim 3\sigma$ deviation when these two are
combined.

Recently, LHCb has also released its measurement on $B_{s}\to\mu^{+}\mu^{-}$
with the full run 2 data LHCb:Bsmumu

$\rm{BR}(B_{s}\to\mu^{+}\mu^{-})=(3.09^{+0.46+0.15}_{-0.43-0.11})\times
10^{-9}.$ (9)

By updating the world average of $B_{s}\to\mu^{+}\mu^{-}$ branching ratio and
the correlated $B^{0}\to\mu^{+}\mu^{-}$, Ref. Altmannshofer:2021qrr  found a
$2.3\sigma$ deviation from the SM prediction.

In this paper, we use a single vector leptoquark (LQ) to explain both the muon
$g-2$ anomaly and the various B decay anomalies, in particular the
$R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies, which have been measured more
precisely recently and shown significant deviations from the SM. LQs are new
physics particles that couple simultaneously to a lepton and a quark; see e.g.
Dorsner:2016wpm  for a recent review. LQ models have been proposed to explain
various $B$ anomalies. A single scalar LQ has been proposed to explain both
the muon $g-2$ and the $B$ anomalies Bauer:2015knc  111Models with multiple
scalar leptoquarks have also been explored Chen:2017hir ; Nomura:2021oeu ..
However, recently, Ref. Angelescu:2021lln  shows that it is difficult for a
single scalar LQ to explain both $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies
because the measured $R_{K^{(*)}}$ values are smaller than the SM expectations
whereas $R_{D^{(*)}}$ are larger. It is also found that a vector LQ that
transforms as $({\bf 3},{\bf 1},2/3)$ under the standard model gauge group
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ can explain both $R_{K^{(*)}}$ and
$R_{D^{(*)}}$ anomalies Angelescu:2021lln . In this study, we further show
that such a vector LQ model can also explain the muon g-2 anomaly, in addition
to the B meson decay anomalies which satisfying various LHC constraints. Our
model is different from that considered in Ref. Angelescu:2021lln  because we
consider the couplings to both left-handed and right-handed fermions, whereas
only left-handed couplings are assumed in Ref. Angelescu:2021lln . Our
analysis shows that both left-handed and right-handed couplings between the
$U_{1}$ boson and the SM fermions are essential for a sizeable new physics
contribution to explain the muon g-2 anomaly.

Leptoquark model:–We consider the vector LQ $U_{1}=({\bf 3},{\bf 1},2/3)$ that
couples to both left-handed and right-handed standard model fermions. The
interaction Lagrangian between the vector LQ $U_{1}^{\mu}$ and SM fermions in
the weak basis is given by

${\cal
L}=x_{ij}^{L}\bar{Q}_{L}^{i}\gamma_{\mu}L_{L}^{j}U_{1}^{\mu}+x^{R}_{ij}\bar{d}_{R}^{i}\gamma_{\mu}\ell_{R}^{j}U_{1}^{\mu}+h.c.,$
(10)

where ${Q}_{L}^{\rm T}=(u_{L},d_{L})$ is the left-handed quark doublet,
${L}_{L}^{\rm T}=(\nu_{L},\ell_{L})$ is the left-handed lepton doublet,
${d}_{R}$ ($\ell_{R}$) is the right-handed quark (lepton), $x^{L}_{ij}$ and
$x^{R}_{ij}$ are couplings with $i$ and $j$ as the generation index of the SM
fermions. Here we assume that the down quarks and the charged leptons are
diagonal. Rotating the up quark fields to the mass basis, the interaction
Lagrangian becomes

${\cal
L}=(Vx^{L})_{ij}\bar{u}^{i}_{L}\gamma_{\mu}\nu_{L}^{j}U^{\mu}_{1}+x^{L}_{ij}\bar{d}_{L}^{i}\gamma_{\mu}\ell_{L}^{j}U_{1}^{\mu}+x^{R}_{ij}\bar{d}_{R}^{i}\gamma_{\mu}\ell_{R}^{j}U_{1}^{\mu}+h.c.{\color[rgb]{0,0,1}{,}}$
(11)

where V is the CKM matrix. We do not consider couplings with right-handed
neutrinos. We assume that the coupling between the vector LQ $U_{1}$ and the
SM gauge bosons (photon and gluon) are standard gauge couplings. The $U_{1}$
coupling to photon contributes to the muon $g-2$. The $U_{1}$ coupling to
gluons are important for the production cross section of $U_{1}$ at the LHC.

Figure 1: New contributions to muon $g-2$ from the vector LQ $U_{1}$.

Muon g-2 anomaly:– In our vector LQ model, the main new physics contributions
to the muon $g-2$ arise from the loop diagrams shown in Fig. 1. The vector LQ
contributions are given by Leveille:1977rc ; Jegerlehner:2009ry ;
Lindner:2016bgg ; Biggio:2016wyy

$\displaystyle\Delta
a_{\mu}={-}\frac{{N_{c}Q_{q}}}{8\pi^{2}}\int_{0}^{1}dx\frac{f^{2}_{v}F_{v}(x,\epsilon)+f^{2}_{a}F_{a}(x,\epsilon)}{(1-x)(\lambda^{-2}-x)+\epsilon^{2}x}$
$\displaystyle{+}\frac{{N_{c}Q_{U_{1}}}}{8\pi^{2}}\int_{0}^{1}dx\frac{f^{2}_{v}F_{v}^{\prime}(x,\epsilon)+f^{2}_{a}F_{a}^{\prime}(x,\epsilon)}{(1-x)(\epsilon^{2}-x)+\lambda^{-2}x},$
(12)

where $N_{c}=3$ is color number, $Q_{q}$ is the electric charge of the SM
quark, and $Q_{U_{1}}=2/3$ is electric charge of $U_{1}$,
$\epsilon=m_{q}/m_{\mu}$, $\lambda=m_{\mu}/m_{U_{1}}$,
$f_{v}=(x^{R}+x^{L})/2$, $f_{a}=(x^{R}-x^{L})/2$,
$F_{v}(x,\epsilon)=2x(1-x)\left(x-2(1-\epsilon)\right)+\lambda^{2}(1-\epsilon)^{2}x^{2}(1+\epsilon-x)$,
$F_{v}^{\prime}(x,\epsilon)=-2x^{2}(1+x-2\epsilon)+\lambda^{2}(1-\epsilon)^{2}x(1-x)(x+\epsilon)$,
$F_{a}(x,\epsilon)=F_{v}(x,-\epsilon)$, and
$F_{a}^{\prime}(x,\epsilon)=F_{v}^{\prime}(x,-\epsilon)$.

We consider four different types of couplings: $x^{L}=x^{R}$, $x^{L}=-x^{R}$,
$x^{L}=0$, and $x^{R}=0$; we find that only the $x^{L}=x^{R}$ case can give
rise to a sizeable positive new physics contribution to the muon $g-2$. Thus
we assume $x^{L}=x^{R}$ hereafter. We denote $x\equiv x^{L}=x^{R}$. We note
that the $U_{1}$ boson has pure vector couplings to fermions in the
$x^{L}=x^{R}$ case. To explain the muon $g-2$, $R_{K^{(*)}}$ and $R_{D^{(*)}}$
anomalies, we assume nonzero $x_{s\mu}$, $x_{b\mu}$ and $x_{b\tau}$ values.

Figure 2: New contributions to the $B^{+}\to K^{+}l^{+}l^{-}$ process due to
the vector LQ $U_{1}$.

$R_{K^{(*)}}$ anomaly:–The semileptonic process $b\to s\ell^{+}\ell^{-}$ is
responsible for the $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decay. The new physics
contributions due to the new vector LQ $U_{1}$ are shown in Fig. 2. The
effective Hamiltonian describing the $b\to s\ell^{+}\ell^{-}$ process can be
parameterized as follows Grinstein:1987vj ; Buchalla:1995vs ; Buras:1998raa ;
Cornella:2019hct

${\cal H}=-\frac{G_{F}\alpha}{\sqrt{2}\pi}V_{tb}V_{ts}^{*}\sum_{i}{{\cal
C}_{i}{\cal O}_{i}+h.c.{\color[rgb]{0,0,1}{,}}}$ (13)

where ${\cal O}_{i}$ are the four-fermion operators, ${\cal C}_{i}$ are the
Wilson coefficients, $G_{F}$ is the Fermi constant, $\alpha\sim 1/137$,
$V_{tb}$ and $V_{ts}$ are the CKM matrix element. The operators that are of
importance to the $R_{K^{(*)}}$ analysis in our model include ${\cal
O}_{9,9^{\prime}}^{\ell\ell}=\bar{s}\gamma_{\mu}P_{L,R}b(\bar{\ell}\gamma^{\mu}\ell)$,
and ${\cal
O}_{10,10^{\prime}}^{\ell\ell}=\bar{s}\gamma_{\mu}P_{L,R}b(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)$,
where $\ell=e,\mu,\tau$ and $P_{R,L}=(1\pm\gamma_{5})/2$.222Operators such as
${\cal O}_{S,S^{\prime},P,P^{\prime}}$ also arise in the $x^{L}=x^{R}$ case.
These operators are neglected in our analysis, because they do not give rise
to significant contributions to $R_{K^{(*)}}$ as the ${\cal
O}_{9,9^{\prime},10,10^{\prime}}$ operators, as shown in Ref. Ghosh:2017ber .
The Wilson coefficients can be parameterized as ${\cal C}_{i}={\cal
C}_{i}^{\rm SM}+\Delta{\cal C}_{i}$ where ${\cal C}_{i}^{\rm SM}$ represents
the SM contribution and $\Delta{\cal C}_{i}$ denotes the new physics
contributions. For the vector LQ $U_{1}$ model where nonzero $x_{b\mu}$ and
$x_{s\mu}$ values are introduced, the new physics contributions to the Wilson
coefficients are given by

$\Delta{\cal C}\equiv\Delta{\cal C}_{9}^{\mu\mu}=\Delta{\cal
C}_{9^{\prime}}^{\mu\mu}=\Delta{\cal C}_{10^{\prime}}^{\mu\mu}=-\Delta{\cal
C}_{10}^{\mu\mu},$ (14)

where

$\Delta{\cal C}=-\frac{\pi
v^{2}}{V_{tb}V_{ts}^{*}\alpha}\frac{x_{s\mu}x_{b\mu}^{*}}{m^{2}_{U_{1}}}.$
(15)

where $v=246$ GeV.

For the $R_{K}$ calculations, operators ${\cal O}_{i}$ and ${\cal
O}_{i^{\prime}}$ yield nearly the same effects, for $i=9,10$. Because our
model predicts $\Delta{\cal C}_{9}^{\mu\mu}=\Delta{\cal
C}_{9^{\prime}}^{\mu\mu}$ and $\Delta{\cal C}_{10}^{\mu\mu}=-\Delta{\cal
C}_{10^{\prime}}^{\mu\mu}$ in the $x^{L}=x^{R}$ case, the effects on $R_{K}$
due to ${\cal O}_{10}$ cancel with ${\cal O}_{10^{\prime}}$, and we only need
to consider the effects due to the operator ${\cal O}_{9}$ with the
coefficient $2\Delta{\cal C}$. We determine the $\Delta{\cal C}$ value needed
to interpret the recent LHCb $R_{K}$ result Aaij:2021vac  using the
theoretical analysis on $R_{K}$ in Ref. Ghosh:2017ber  and find that

${\Delta{\cal C}(R_{K})\simeq-0.35\pm 0.11}$ (16)

For the $R_{K^{*}}$ calculations, because operators ${\cal O}_{i}$ and ${\cal
O}_{i^{\prime}}$ give rise to nearly the same contributions for $i=9,10$ but
with opposite signs, the net effects in our model can be approximated by the
operator ${\cal O}_{10}$ with the coefficient -$2\Delta{\cal C}$. The required
$\Delta{\cal C}$ value for the $R_{K^{*}}$ Aaij:2017vbb  interpretation is
also obtained via the analysis in Ref. Ghosh:2017ber

${\Delta{\cal C}(R_{K^{*}})\simeq-0.63^{+0.28}_{-0.22}}$ (17)

Thus one can explain both $R_{K}$ and $R_{K^{*}}$ anomalies within the 1
$\sigma$ (2 $\sigma$) error corridor in the range $-0.46<\Delta{\cal C}<-0.35$
($-0.57<\Delta{\cal C}<-0.13$).333Our results are similar to the global fit
(for the $x^{R}=0$ case) $\Delta{\cal C}(R_{K})=-0.41\pm 0.09$ where both
$R_{K}$ and $B_{s}\to\mu^{+}\mu^{-}$ measurements are taken into account
Angelescu:2021lln .

$R_{D^{(*)}}$ anomaly:– Here we compute $R_{D^{(*)}}$ for the LQ model we
consider. The relevant low-energy EFT Lagrangian describing the $b\to
c\ell\nu$ is given by Shi:2019gxi ; Murgui:2019czp ; Alonso:2015sja

${\cal
L}=-2\sqrt{2}G_{F}V_{cb}\Big{[}\left(1+y^{\ell}_{V_{L}}\right)\mathcal{O}_{V_{L}}+y^{\ell}_{S_{R}}\mathcal{O}_{S_{R}}\Big{]}+h.c.{\color[rgb]{0,0,1}{,}}$
(18)

where $y^{\ell}_{V_{L}}$ and $y^{\ell}_{S_{R}}$ are the Wilson coefficients
and the four-fermion operators are given by

$\mathcal{O}_{V_{L}}=\left(\bar{c}_{L}\gamma^{\mu}b_{L}\right)\left(\bar{\ell}_{L}\gamma_{\mu}\nu_{L}\right),\quad\mathcal{O}_{S_{R}}=\left(\bar{c}_{L}b_{R}\right)\left(\bar{\ell}_{R}\nu_{L}\right).$
(19)

The new physics contributions to the Wilson coefficients at tree level from
the vector LQ model can be obtained as

$\displaystyle y^{\ell}_{V_{L}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}y^{\ell}_{S_{R}}=\frac{v^{2}}{2V_{cb}}\frac{(Vx)_{c\ell^{\prime}}(x_{b\ell})^{*}}{m^{2}_{U_{1}}}$
(20) $\displaystyle=$
$\displaystyle\frac{v^{2}}{2V_{cb}m_{U_{1}}^{2}}(V_{cb}x_{b\ell^{\prime}}+V_{cs}x_{s\ell^{\prime}})(x_{b\ell})^{*},$

where the sum over $\ell^{\prime}$ for generations of the neutrino is implicit
on the right hand side. We assume that the non-zero matrix elements in our
model are $x_{s\mu}$, $x_{b\mu}$ and $x_{b\tau}$ values. In our numerical
analysis, we set $x_{b\tau}=5x_{b\mu}$ so that the new physics contributions
to the $b\to c\mu\nu$ transitions are much smaller than the new physics
contributions to the $b\to c\tau\nu$ transitions. Thus We neglect the new
physics contributions to the $b\to c\mu\nu$ transitions and adopt the analysis
in Ref. Shi:2019gxi  to obtain

$y^{\tau}_{V_{L}}=0.08^{+0.03}_{-0.03},\quad
y^{\tau}_{S_{R}}=-0.05^{+0.09}_{-0.10},$ (21)

which further give rise to

$y^{\tau}_{V_{L}}=-\frac{1}{2}y^{\tau}_{S_{R}}=0.065\pm 0.027,$ (22)

when the two fits are combined. In order to explain the relation
$R_{K^{(*)}}^{\rm Exp}<R_{K^{(*)}}^{\rm SM}$, the matrix elements $x_{s\mu}$
and $x_{b\mu}^{*}$ have to be opposite in sign. Taking all matrix elements to
be real, and setting both $x_{b\mu}$ and $x_{b\tau}$ positive and $x_{s\mu}$
negative, in order to explain $R_{D^{(*)}}^{\rm Exp}>R_{D^{(*)}}^{\rm SM}$,
the condition $|x_{s\mu}|<(x_{b\mu}+x_{b\tau})V_{cb}/V_{cs}$ has to be
satisfied. Thus in our vector LQ model, both $R_{K^{(*)}}$ and $R_{D^{(*)}}$
anomalies can be explained simultaneously.

LHC constraints:—The LQs can be either singly produced or pair-produced at
hadron colliders Diaz:2017lit ; Dorsner:2018ynv . The singly produced LQ is
accompanied by a lepton in the final state, $qg\to q\to U_{1}\ell$; the pair-
production process can occur via either the gluon–gluon fusion process or the
quark–antiquark annihilation process (mediated by either a t-channel lepton or
an s-channel photon/gluon).

Searches for these direct productions of leptoquarks have been carried out at
the LHC for various final states Aaboud:2019bye ; Aad:2021rrh ; Aad:2020iuy ;
Aad:2020jmj ; CMS:2018 . From these searches, one can derive the collider
exclusion bounds on a given leptoquark mass as a function of its branching
ratio (denoted by $\beta$) into a specific fermion final state.

Using the recent LHC data, a recasting result from Ref. Angelescu:2021lln
shows the current limits for the vector LQs. In particular, for pair-produced
LQs decaying into the final states of $b\bar{b}\tau\bar{\tau}$,
$t\bar{t}\tau\bar{\tau}$, $jj\mu\bar{\mu}$, $b\bar{b}\mu\bar{\mu}$,
$t\bar{t}\mu\bar{\mu}$, $jj\nu\bar{\nu}$, $b\bar{b}\nu\bar{\nu}$ and
$t\bar{t}\nu\bar{\nu}$, the lower limits on its mass are $1.5$ TeV, $2.0$ TeV,
$2.3$ TeV, $2.3$ TeV, $2.0$ TeV, $1.8$ TeV, $1.8$ TeV and $1.8$ TeV,
respectively, assuming $\beta=1$ Angelescu:2021lln . The limits from
$b\bar{b}\tau\bar{\tau}$, $jj\mu\bar{\mu}$, $b\bar{b}\mu\bar{\mu}$,
$jj\nu\bar{\nu}$, and $t\bar{t}\nu\bar{\nu}$ final states can be applied for
the $U_{1}$ model. The most stringent constraints are from $jj\mu\bar{\mu}$
and $b\bar{b}\mu\bar{\mu}$ final states. For the pair of LQs decaying into
different quark-lepton final states case, i.e., $U_{1}\to
b\bar{\tau},t\bar{\nu}$, the lower limit on the vector LQ is about $1.7$ TeV
Sirunyan:2020zbk .

However, these limits are under the assumption that the branching fraction
$\beta=1$ and the interaction between vector leptoquark and gluons is
described by $\kappa g_{s}U_{1\mu}^{\dagger}G^{\mu\nu}U_{1\nu}$ with $g_{s}$
is the strong coupling and $\kappa=1$. Thus if one tunes $\beta$ and $\kappa$
to be small, the limits from LHC direct production searches can be
significantly weakened. Indeed, once switching off the the interaction between
the vector LQ and gluons, i.e., $\kappa=0$, the quark–antiquark annihilation
process becomes dominant; because the quark–antiquark annihilation for the
vector LQ pair production is typically smaller than the gluon fusion, the
limit on the vector LQ mass is reduced to $\sim 1.3$ TeV for the $U_{1}\to
b\bar{\tau},t\bar{\nu}$ searches Sirunyan:2020zbk .

The searches for the singly produced LQs have also been performed at CMS using
data with ${\cal L}=137\leavevmode\nobreak\ {\rm{fb}}^{-1}$ Sirunyan:2020zbk .
The limits on the LQ mass as a function of the LQ-quark-lepton coupling $x$
have been derived; for LQs coupled to the third-generation fermions with
coupling $x=1.5$, the LQ mass $m_{U_{1}}\lesssim 1.2$ TeV is excluded.

LQs can also be searched for in the $qq\to\ell\ell$ process at the LHC in
which the LQ particle serves as a t-channel mediator. Thus LQs are constrained
by the high $p_{T}$ resonance searches via $pp\to\ell\ell^{(\prime)}$ and
$pp\to\ell\nu$ recently performed at the LHC Faroughy:2016osc ;
Schmaltz:2018nls ; Greljo:2017vvb ; Afik:2019htr ; Angelescu:2020uug ;
Angelescu:2021lln ; Greljo:2018tzh ; Marzocca:2020ueu . A recent analysis from
Ref. Angelescu:2021lln  used data from Aad:2020zxo  and CMS:2019tbu  to set
upper bounds on the couplings $x_{b\mu}\lesssim 0.7$, $x_{s\mu}\lesssim 0.5$,
$x_{b\tau}\lesssim 1.0$ and $x_{s\tau}\lesssim 0.7$ for the vector LQ mass
below $1$ TeV in $U_{1}$ model. The limit on $x_{b\mu}$ as a function of
$m_{U_{1}}$ is shown in Fig. 3.

Figure 3: The $2\sigma$ favored regions for the muon $g-2$ (blue band),
$R_{K^{(*)}}$ (pink band) and $R_{D^{(*)}}$ (gray band) on the
$(m_{U_{1}},x_{b\mu})$ plane. Here we set $x_{s\mu}=-0.005$ and
$x_{b\tau}=5x_{b\mu}$. The black solid line indicates the LHC high $p_{T}$
searches, and the vertical purple solid line represents the leptoquark
searches via pair-production at the LHC Angelescu:2021lln .

Numerical results:—Here we present our numerical results. We choose
$x_{b_{\tau}}=5x_{b\mu}$ so that the new physics contributions to the $b\to
c\tau\nu$ process are larger than those to the $b\to c\mu\nu$ process, leading
to $R_{D^{(*)}}>R_{D^{(*)}}^{\rm SM}$. Note that varying the $x_{b_{\tau}}$
coupling does not affect the $R_{K^{(*)}}$ and $\Delta a_{\mu}$ computations.
We choose $x_{s\mu}=-0.005$ which has an opposite sign to $x_{b\mu}$ in order
to satisfy the $R_{K^{(*)}}$ anomaly which is required to obtain
$R_{K^{(*)}}^{\rm Exp}<R_{K^{(*)}}^{\rm SM}$. Such a small value of $x_{s\mu}$
also satisfies the condition $|x_{s\mu}|<(x_{b\mu}+x_{b\tau})V_{cb}/V_{cs}$
needed for a positive contribution to the $R_{D^{(*)}}$.

Fig. 3 shows the 2$\sigma$ regions of the parameter space of our vector LQ
model, for the muon $g-2$, $R_{K^{(*)}}$, and $R_{D^{(*)}}$ anomalies. It is
remarkable that there exists a large parameter space ranging from 1 TeV to
above 4 TeV, in which one can simultaneously explain the muon $g-2$,
$R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies. The LHC constraints including the
high $p_{T}$ di-lepton searches and the pair-produced LQ searches are also
displayed on Fig. 3; both LHC constraints are adopted from Ref.
Angelescu:2021lln . Because we have $x_{b\tau}=5x_{b\mu}$, the branching
fraction of ($U_{1}\to b\tau$) dominates, namely BR$(U_{1}\to b\tau)\sim 1$.
Thus we adopt the limits in the $b\bar{b}\tau\bar{\tau}$ final state searches
for pair-produced LQs at the LHC Aaboud:2019bye ; Angelescu:2021lln , which
rule out the $U_{1}$ boson with a mass below $1.5$ TeV. The LHC high $p_{T}$
data Aaboud:2019bye  exclude the $x_{b\mu}>1$ region which have already put
constraints on the 2 $\sigma$ region for $R_{K^{(*)}}$ interpretation when
$m_{U_{1}}>3.5$ TeV. We note that the large parameter space in our model, from
1.5 TeV to 4 TeV, in which both the muon g-2 anomaly and the various B
anomalies can be explained are allowed by the current LHC searches on LQs.

Conclusions:–We have shown that the vector LQ $U_{1}=({\bf 3},{\bf 1},2/3)$
can simultaneously explain the muon $g-2$ anomaly and the $B$ decay anomalies,
including the $R_{K^{(*)}}$ and $R_{D^{(*)}}$ anomalies, while satisfying the
experimental constraints from the large hadron collider, including searches on
singly produced and pair-produced LQs, and the high $p_{T}$ dilepton searches.

We found that in order to provide a sizeable positive new physics contribution
to the muon g-2 anomaly while satisfying various LHC constraints, the vector
LQ $U_{1}$ has to couple to left-handed and right-handed fermions equally,
namely $x_{L}=x_{R}$, so that only vector couplings are present in our model.
Unlike the scalar LQ model, the new physics contributions from the vector LQ
to $R_{K^{(*)}}$ and $R_{D^{(*)}}$ can be somewhat adjusted independently in
the parameter space of our model. Thus conditions $R_{K^{(*)}}^{\rm
Exp}<R_{K^{(*)}}^{\rm SM}$ and $R_{D^{(*)}}^{\rm Exp}>R_{D^{(*)}}^{\rm SM}$
can be satisfied with ease. The LHC searches for LQ decays set stringent
constraints on the LQ mass, $m_{U_{1}}\gtrsim{\cal O}(1)$ TeV. The couplings
of LQs to quark and lepton are strongly constrained to be $x\lesssim{\cal
O}(1)$ in the LHC high-$p_{T}$ searches for LQ mass extending to several TeV.
Taking into account all the LHC constraints, there still exists a large
parameter space in which the muon $g-2$ anomaly recently reported by the
Fermilab g-2 experiment and the various $B$ decay anomalies including
$R_{K^{(*)}}$ and $R_{D^{(*)}}$ can be simultaneously explained in the vector
leptoquark model in the mass range from 1.5 TeV to several TeV.

Acknowledgement:– The work is supported in part by the National Natural
Science Foundation of China under Grant No. 11775109. Van Que Tran would like
to thank the Institute of Physics, Academia Sinica, Taiwan for its hospitality
during this work.

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