[1]$V_{#1}^{*}$

# Diquark Explanation of $b\to s\ell^{+}\ell^{-}$

Andreas Crivellin<EMAIL_ADDRESS>Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer
Institut, CH–5232 Villigen PSI, Switzerland  Matthew Kirk<EMAIL_ADDRESS>Departament de Física Quàntica i Astrofísica (FQA), Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona (UB), Spain

###### Abstract

The discrepancies between $b\to s\ell^{+}\ell^{-}$ data and the corresponding
Standard Model predictions point to the existence of new physics with a
significance at the $5\sigma$ level. While previously a lepton flavour
universality violating effect was preferred, the new $R(K^{(*)})$ and
$B_{s}\to\mu^{+}\mu^{-}$ measurements are now compatible with the Standard
Model, favouring a lepton flavour universal beyond the Standard Model
contribution to $C_{9}$. Since heavy new physics is generally chiral, and
because of the stringent constraints from charged lepton flavour violation,
this poses a challenge for model building. In this article, we point out a
novel possibility: a diquark, i.e. a coloured scalar, induces the Wilson
coefficient of the $(\bar{s}\gamma^{\mu}P_{L}b)(\bar{c}\gamma_{\mu}P_{L}c)$
operator at tree-level, which then mixes into $O_{9}$ via an off-shell photon
penguin. This setup allows for a lepton flavour universal effect of
$C_{9}\approx-0.5$, without violating bounds from $\Delta M_{s}$,
$\Delta\Gamma$, $B\to X_{s}\gamma$ and $D^{0}-\bar{D}^{0}$ mixing. This
scenario predicts a small and negative $C_{9}^{\prime}$ and a light diquark,
preferably with a mass around $500\,$GeV, as compatible with the CMS di-di-jet
analysis, and a deficit in the inclusive $b\to c\bar{c}s$ rate.

††preprint: PSI-PR-23-35, ZU-TH 54/23

##  I Introduction

While the Standard Model (SM) Cabibbo-Kobayashi-Maskawa (CKM) mechanism
Kobayashi and Maskawa (1973) of quark flavour violation was established by the
$B$ factories Belle Abashian  _et al._ (2002) and BaBar Aubert  _et al._
(2002), there is still room for new physics (NP) at the order of $10\%$ in
flavour changing neutral current (FCNC) processes. Such FCNC observables are
loop suppressed and thus particularly sensitive to beyond-the-SM
contributions. In fact, there are long-lasting hints for NP in $b\to
s\ell^{+}\ell^{+}$ observables. However, the picture changed radically with
the release of the latest LHCb results for the ratios $R(K^{(*)})={\rm
Br}[B\to K^{(*)}\mu^{+}\mu^{-}]/{\rm Br}[B\to K^{(*)}e^{+}e^{-}]$ Aaij  _et
al._ (2023a, b), superseding their previous measurements Aaij  _et al._
(2014a, 2017, 2022a). While previously all global fits Altmannshofer and
Stangl (2021); Geng  _et al._ (2021); Algueró  _et al._ (2022a); Hurth  _et
al._ (2022); Kowalska  _et al._ (2019); Ciuchini  _et al._ (2023a); D’Amico
_et al._ (2017) preferred lepton flavour universality (LFU) violating NP
Crivellin and Hoferichter (2021); Crivellin and Matias (2022), now data is not
only consistent with LFU but even stringently limits deviations from it.

Nonetheless, the case for physics beyond the SM in $b\to s\ell^{+}\ell^{+}$
transitions remains very strong (see Ref. Capdevila  _et al._ (2023) for a
recent review). The main tensions with the SM predictions are within the
angular $B\to K^{*}\mu^{+}\mu^{+}$ observable $P_{5}^{\prime}$ Descotes-Genon
_et al._ (2013a); Aaij  _et al._ (2016a, 2020a, 2021a), the total branching
ratio and angular observables in $B_{s}\to\phi\mu^{+}\mu^{-}$ Aaij  _et al._
(2015, 2021b, 2021c) as well as in Br$[B\to K\mu^{+}\mu^{-}]$ Aaij  _et al._
(2014b, 2016b)111Measurements of these decays were also performed by the
ATLAS, CMS and Belle collaborations Chatrchyan  _et al._ (2013); Khachatryan
_et al._ (2016); Sirunyan  _et al._ (2018); Aaboud  _et al._ (2018a); Sirunyan
_et al._ (2021) but with less precise results., with tensions at the
$2-4\sigma$ level in each of these modes. In fact, while SM predictions are
challenging, due to the hadronic form factors involved Ball and Zwicky (2005);
Horgan  _et al._ (2014); Bouchard  _et al._ (2013); Bharucha  _et al._ (2016);
Gubernari  _et al._ (2019) (including non-local charm-loop contributions
Khodjamirian  _et al._ (2010, 2013); Gubernari  _et al._ (2021)), the first
lattice calculation over the full $q^{2}$ range of Br$[B\to K\mu^{+}\mu^{-}]$
leads to a stronger tension of $4.7\,\sigma$ Parrott  _et al._ (2023).
Furthermore, $P_{5}^{\prime}$, being an optimised angular observable Matias
_et al._ (2012); Descotes-Genon  _et al._ (2013b, c) possess a reduced
sensitivity to the form factors and semi-inclusive decays at high $q^{2}$,
that are independent of hadronic form factors, are fully compatible with the
other observables Isidori  _et al._ (2023). Finally, dispersive methods based
on analyticity confirm previous error estimates for the form factors Bobeth
_et al._ (2018); Gubernari  _et al._ (2023) (including their non-local parts).

Combining the processes discussed above in a global fit together with all
other available data on $b\to s\ell^{+}\ell^{+}$ transitions leads to a
coherent picture. In fact, while before the $R(K^{(*)})$ update, the most
strongly favoured scenarios were at least two-dimensional Algueró  _et al._
(2019), now a single one-dimensional scenario is clearly favoured: the
$C_{9}^{U}$ scenario with a significance around $5\sigma$ Gubernari  _et al._
(2022); Ciuchini  _et al._ (2023b); Algueró  _et al._ (2023); Wen and Xu
(2023). This means that a left-handed $b-s$ current and a vectorial flavour-
universal lepton current ($B_{s}\to\mu^{+}\mu^{-}$ Aaij  _et al._ (2022b, c);
Aaboud  _et al._ (2019); Tumasyan  _et al._ (2023a) constrains an axial
current) is needed.

This poses a challenge for model building since both tree-level leptoquark
effects Hiller and Schmaltz (2014); Alonso  _et al._ (2015) as well as loop
contributions of new scalars and fermions Gripaios  _et al._ (2016); Arnan
_et al._ (2017); Grinstein  _et al._ (2018); Arnan  _et al._ (2019), in
general, give a chiral lepton current and have difficulties respecting the
stringent bounds from $\mu\to e$ flavour violating Crivellin  _et al._ (2018a)
unless multiple generations are involved Crivellin  _et al._ (2022). This
leaves $Z^{\prime}$ models Buras and Girrbach (2013); Gauld  _et al._ (2014);
Algueró  _et al._ (2022b) as well as leptoquarks which generate a LFU effect
in $C_{9}^{U}$ via a tau-loop with an off-shell photon Crivellin  _et al._
(2019a, 2020a); Fuentes-Mart´$\mathrm{i}$n  _et al._ (2020); Algueró  _et al._
(2022b); Bordone and Fernández Navarro (2023); Fernández Navarro and King
(2023) as the remaining (simple) options. However, also in these cases
$B_{s}-\bar{B}_{s}$ mixing Di Luzio  _et al._ (2018), LEP and LHC constraints
Greljo  _et al._ (2023); Aebischer  _et al._ (2023) make a full explanation
challenging.

An alternative scenario that can naturally generate $C_{9}^{U}$ is a NP
contribution to the Wilson coefficient of the
$(\bar{s}\gamma^{\mu}P_{L}b)(\bar{c}\gamma_{\mu}P_{L,R}c)$ operator Jäger  _et
al._ (2018) that mix into $C_{9}$ via an off-shell photon penguin Bobeth  _et
al._ (2014). As a tree-level effect in $\bar{s}b\bar{c}c$ operators is
necessary, only $Z^{\prime}$ bosons, heavy gluons, Higgses Iguro and Omura
(2018); Crivellin  _et al._ (2019b); Kumar (2023); Iguro (2023) or diquarks
(DQs) come into mind de Blas  _et al._ (2018). For the the first two options
the possible effect is stringently limited by $B_{s}-\bar{B}_{s}$ mixing (to
which these particles also contribute at tree-level Aebischer  _et al._
(2020a)) and there is only a small region in parameter space left that works
for 2HDMs Iguro (2023). Therefore, we will consider the DQs in this article
which have not been studied so far. Out of the 8 different scalar DQs Giudice
_et al._ (2011); Pascual-Dias  _et al._ (2020), there is only a single
representation that couples to left-handed down-type quarks, can have flavour-
diagonal couplings to up-type quarks and does not lead to tree-level effects
in $\Delta F=2$: the scalar $SU(3)_{c}$ triplet, $SU(2)_{L}$ singlet with
hypercharge $-1/3$ which we call $\phi$.222This field is also known as the
$S_{1}$ leptoquark when it couples to quarks and leptons instead of two
quarks. However, to avoid proton decay, we will assume that it couples only to
quarks which can be achieved by e.g. assuming that lepton number is conserved.

DQs are not only theoretically well motivated, e.g. by $E_{6}$ models Hewett
and Rizzo (1989) or the R-parity violating MSSM Barbier  _et al._ (2005), but
also lead to interesting LHC signatures Ma  _et al._ (1999); Kilic  _et al._
(2008); Cheung and Cho (2003); Arnold  _et al._ (2010); Shu  _et al._ (2010);
Dorsner  _et al._ (2010a, b); Gresham  _et al._ (2011); Patel and Sharma
(2011); Grinstein  _et al._ (2011); Ligeti  _et al._ (2011): They are
candidates from an explanation Crivellin  _et al._ (2023) of the ATLAS di-jet
Aad  _et al._ (2021) and the CMS di-di-jet Tumasyan  _et al._ (2023b)
excesses. In fact, the non-resonant CMS analysis shows a weaker-than-expected
limit for a diquark $SU(3)_{c}$ triplet with a mass around $500\,$GeV.

In the next section, we will define our model and perform the matching on the
effective theory before we continue with the phenomenological analysis in Sec.
III can conclude in Sec. IV.

##  II Setup and Observables

There are 8 representations of scalar DQs with couplings to quarks $SU(2)_{L}$
singlets or doublets which can have either symmetric or anti-symmetric
couplings in flavour space. In order to get a sizable effect in $b\to
s\ell^{+}\ell^{-}$ via the operator
$(\bar{s}\gamma^{\mu}P_{L}b)(\bar{c}\gamma_{\mu}P_{L,R}c)$, we need 1)
simultaneous couplings to up-type and down-type quarks 2) flavour diagonal
couplings to up-quarks (i.e. not anti-symmetric ones) 3) left-handed down-
quarks must be involved 4) no tree-level effect in $B_{s}-\bar{B}_{s}$ mixing.
These requirements only leave a single representation, i.e. IV in the
conventions of Ref. Giudice  _et al._ (2011) whose mass we label $M_{\phi}$.

The couplings to quarks are given by

$\mathcal{L}=\left(\dfrac{1}{2}\tilde{\lambda}_{ij}^{L}(\bar{Q}_{i}^{I,\alpha})^{c}Q_{j}^{J,\beta}\epsilon_{IJ}+\tilde{\lambda}_{ij}^{R}\bar{u}^{\alpha
c}_{i}d^{\beta}_{j}\right)\phi^{\gamma}\epsilon_{\alpha\beta\gamma}+\text{h.c.}\,.$
(1)

Here, $\alpha,\beta,\gamma$ are color indices, $I,J$ ($i,j$) $SU(2)_{L}$
(flavour) indices and $c$ stands for charge conjugation. Note that
$\tilde{\lambda}^{L}$ can be chosen to be symmetric in flavour space, i.e.
$\tilde{\lambda}_{ij}^{L}=\tilde{\lambda}^{L}_{ji}$, without loss of
generality. After electroweak symmetry breaking, the quark doublets decompose
into their $SU(2)_{L}$ components and in the mass eigenbasis we have

$\mathcal{L}={\varepsilon_{\alpha\beta\gamma}}\left({{\lambda_{ij}^{L}}\bar{u}_{i}^{\alpha
c}{P_{L}}d_{j}^{\beta}+{\lambda_{ij}^{R}}{{\bar{u}}_{i}^{\alpha
c}}{P_{R}}{d_{j}^{\beta}}}\right){\phi^{\gamma}}+\text{h.c.}\,,$ (2)

where we absorbed the rotation matrices into the definition of $\lambda^{R}$
and, working in the down basis, defined
$V_{ii^{\prime}}^{*}\tilde{\lambda}_{i^{\prime}j}^{L}=\lambda^{L}_{ij}$.

###  II.1 Tree-level Matching

The Lagrangian

$\mathcal{L}=-N_{ij}\sum\limits_{X=1}^{10}{\left({C_{X}^{{i}{j}}Q_{X}^{{i}{j}}+C_{X}^{\prime{i}{j}}Q_{X}^{\prime{i}{j}}}\right)}$
(3)

with $N_{ij}=4G_{F}V_{ib}V_{*}{js}/{{\sqrt{2}}}$ and the operators

$\displaystyle Q_{1}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{\gamma_{\mu}}{P_{L}}b^{\beta}}\right)\left({\bar{s}^{\beta}{\gamma^{\mu}}{P_{L}}u_{j}^{\alpha}}\right),$
(4) $\displaystyle Q_{2}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{\gamma_{\mu}}{P_{L}}b_{L}^{\alpha}}\right)\left({\bar{s}_{L}^{\beta}{\gamma^{\mu}}{P_{L}}u_{j}^{\beta}}\right),$
$\displaystyle Q_{7}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{P_{R}}b^{\beta}}\right)\left({\bar{s}^{\beta}{P_{R}}u_{j}^{\alpha}}\right),$
$\displaystyle Q_{8}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{P_{R}}b^{\alpha}}\right)\left({\bar{s}_{L}^{\beta}{P_{R}}u_{j}^{\beta}}\right),$
$\displaystyle Q_{9}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{\sigma_{\mu\nu}}{P_{R}}b^{\beta}}\right)\left({\bar{s}^{\beta}{\sigma^{\mu\nu}}{P_{R}}u_{j}^{\alpha}}\right),$
$\displaystyle Q_{10}^{{i}{j}}$
$\displaystyle=\left({\bar{u}_{i}^{\alpha}{\sigma_{\mu\nu}}{P_{R}}b^{\alpha}}\right)\left({\bar{s}_{L}^{\beta}{\sigma^{\mu\nu}}{P_{R}}u_{j}^{\beta}}\right),$

defines the charged current interactions. Our diquark, integrated out at tree
level at the electroweak scale, leads to the coefficients

$\displaystyle C_{1}^{{i}{j}}$
$\displaystyle=-C_{2}^{{i}{j}}=-\dfrac{{\lambda_{i2}^{L*}\lambda_{j3}^{L}}}{{2N_{ij}M_{\phi}^{2}}},$
(5) $\displaystyle C_{7}^{{i}{j}}$
$\displaystyle=-C_{8}^{{i}{j}}=-4C_{9}^{{i}{j}}=4C_{10}^{{i}{j}}=\dfrac{{\lambda_{i2}^{L*}\lambda_{j3}^{R}}}{{2N_{ij}M_{\phi}^{2}}},$
(6)

with the primed coefficients obtained by
${\lambda^{L}\leftrightarrow\lambda^{R}}$.333It can be seen that $C_{X}^{uc}$
(which give $b\to u\bar{c}s$ transitions) are enhanced relative to the SM by
the large ratio $V_{us}/V_{ub}$. However, the effects of these operators in
meson mixing-related observables is too small Lenz  _et al._ (2022).
Furthermore, $C^{cu}_{X}$ (which give $b\to c\bar{u}s$ transitions) have been
considered as part of a potential explanation of the discrepancy between the
SM prediction with QCD factorization and experiment for $\bar{B}^{0}_{(s)}\to
D^{*+}_{(s)}\\{\pi^{-},K^{-}\\}$ Huber  _et al._ (2016); Bordone  _et al._
(2020); Cai  _et al._ (2021); Endo  _et al._ (2022); Beneke  _et al._ (2021);
Fleischer and Malami (2022, 2023). However as no analysis has been done with
just NP in $\bar{s}b\bar{c}u$, this interesting future direction is beyond the
scope of this article.

###  II.2 $b\to s\ell^{+}\ell^{-}$

The effective Lagrangian governing $b\to s\ell\ell$ transitions is given by

$\mathcal{L}\supset N_{33}\frac{\alpha}{4\pi}\begin{aligned}
&\left[C_{9}(\bar{s}\gamma^{\mu}P_{L}b)(\bar{\ell}\gamma_{\mu}\ell)+C_{10}(\bar{s}\gamma^{\mu}P_{L}b)(\bar{\ell}\gamma_{\mu}\gamma^{5}\ell)\right],\end{aligned}$
(7)

where the primed operators and coefficients are again obtained by exchanging
$L$ and $R$. The tree-level induced operators $Q_{1,2}^{cc}$ generate via
mixing

$\displaystyle C_{9}(m_{b})=$ $\displaystyle
8.5C_{1}^{cc}(M_{W})+2.1C_{2}^{cc}(M_{W})$ (8) $\displaystyle+$
$\displaystyle\left[3.1C_{1}^{cc}(M_{W})+0.32C_{2}^{cc}(M_{W})\right]\times
h(q^{2}),$

at the $B$ meson scale (and similarly for $C_{9}^{\prime}$) with

$h(q^{2},m_{c})=-\frac{4}{9}\left[\ln\frac{m_{c}^{2}}{m_{b}^{2}}-\frac{2}{3}+(2+z)a(z)-z\right],$
(9)

$a(z)=\sqrt{|z-1|}\arctan\frac{1}{\sqrt{z-1}}$, and $z=4m_{c}^{2}/q^{2}$. Note
that $h$ also includes finite subleading $q^{2}$-dependent effects (which we
evaluate at $q^{2}=\qty{5}{{\rm GeV}^{2}}$) Jäger  _et al._ (2018).

The threshold effects from top quark loops induce at the EW scale

$\displaystyle\frac{{-C_{9}}}{1-4s_{W}^{2}}={C_{10}}$
$\displaystyle=-\frac{{\lambda_{32}^{L*}\lambda_{33}^{L}}}{{2{e^{2}}V_{tb}V_{ts}^{*}}}f\left({\frac{{m_{t}^{2}}}{{{M_{\phi}^{2}}}}}\right),$
(10)

with $f(x)=x(x-\log x-1)/(x-1)^{2}$ and $C_{9,10}^{\prime}$ obtained from
$C_{9,10}$ by exchanging $L$ and $R$.

###  II.3 $B\to X_{s}\gamma$

The relevant effective operators are defined by

$\mathcal{L}\supset
N_{33}C_{7\gamma(8g)}\frac{e(g_{s})m_{b}}{16\pi^{2}}\bar{s}\sigma^{\mu\nu}(t^{a})P_{R}bF_{\mu\nu}(G^{a}_{\mu\nu}),$
(11)

with primed operators obtained through $P_{R}\to P_{L}$. We have two NP
contributions to $C_{7\gamma}$. The mixing of the $\bar{s}b\bar{c}c$ operators
into $C_{7\gamma}$ leads to Jäger  _et al._ (2018, 2020)

$\displaystyle C_{7\gamma}(m_{b})\begin{aligned}
=&0.02C_{1}^{cc}(M_{W})-0.19C_{2}^{cc}(M_{W})-1.0C_{7}^{cc}(M_{W})\\\
-&0.47C_{8}^{cc}(M_{W})+4.0C_{9}^{cc}(M_{W})+0.47C_{10}^{cc}(M_{W})\\\
-&\frac{m_{c}}{m_{b}}(2.5C_{7}^{cc}+1.3C_{8}^{cc}-10C_{9}^{cc}+0.89C_{10}^{cc})\times
y,\end{aligned}$ (12)

where, similarly to the case of $C_{9}$, we have included sub-leading $q^{2}$
dependent terms via $y=-(1+2\log(m_{c}^{2}/m_{b}^{2}))/6$. The equivalent
result for the primed operators is again obtained by an exchange of
chiralities.444In principle, there are potential RGE contributions from
$\bar{s}b\bar{c}c$ operators in $C_{8g}$ which are however expected to be very
small (see appendix of Ref. Jäger  _et al._ (2018)).

Second, the direct matching at the new physics scale (to be taken the weak
scale) is

$\displaystyle{C_{7\gamma}}(M_{W})$
$\displaystyle=\frac{{-1}}{{M_{\phi}^{2}}}\frac{2}{N_{33}}\left({\lambda_{j2}^{L*}\lambda_{j3}^{R}\frac{{{m_{{u_{j}}}}}}{{{m_{b}}}}\left[\frac{1}{3}{{f_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)-\frac{2}{3}{g_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)}\right]+\lambda_{j2}^{L*}\lambda_{j3}^{L}\left[\frac{1}{3}{{{\tilde{f}}_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)-\frac{2}{3}{{\tilde{g}}_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)}\right]}\right),$
$\displaystyle C_{8g}(M_{W})$
$\displaystyle=\frac{-1}{{M_{\phi}^{2}}}\frac{2}{N_{33}}\left({\lambda_{j2}^{L*}\lambda_{j3}^{R}\frac{{{m_{{u_{j}}}}}}{{{m_{b}}}}\left[f_{\Phi}\left(\frac{m_{u_{j}}^{2}}{M_{\phi}^{2}}\right)+{g_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)\right]+\lambda_{j2}^{L*}\lambda_{j3}^{L}\left[\tilde{f}_{\Phi}\left(\frac{m_{u_{j}}^{2}}{M_{\phi}^{2}}\right)+{{\tilde{g}}_{\Phi}}\left({\frac{{m_{{u_{j}}}^{2}}}{{M_{\phi}^{2}}}}\right)\right]}\right),$
(13)

with the loop functions defined in Ref. Crivellin  _et al._ (2018b), and
primed operators obtained as usual via $L\leftrightarrow R$. These
coefficients are then evolved to the $B$ meson scale resulting in

$C_{7{\gamma}}^{(\prime)}(m_{b})=0.65C_{7\gamma}^{(\prime)}(M_{W})+0.10C_{8g}^{(\prime)}(M_{W})\,.$
(14)

The latest SM prediction for the inclusive radiative decay $B\to X_{s}\gamma$
Misiak  _et al._ (2020)

$\text{Br}[B\to X_{s}\gamma]^{\rm SM}=$3.40(17)\text{\times}{10}^{-4}$\,,$
(15)

is in good agreement with the experimental average Amhis  _et al._ (2023)

$\text{Br}[B\to X_{s}\gamma]^{\rm EXP}=$3.49(19)\text{\times}{10}^{-4}$\,.$
(16)

This measurement stringently constrains a BSM contribution to $C_{7\gamma}$.
In addition, asymmetries in $B\to K^{*}e^{+}e^{-}$ Aaij  _et al._ (2020b) are
very sensitive to $C_{7\gamma}^{\prime}$. We performed a fit to $C_{7\gamma}$
and $C_{7\gamma}^{\prime}$ using smelli Stangl (2021); Aebischer  _et al._
(2019); Straub (2018).

###  II.4 $B_{s}-\bar{B}_{s}$ Mixing

In the SM, EW box diagrams give rise to $B_{s}-\bar{B}_{s}$ mixing, which can
be measured through observables including $\Delta M_{s}$, $\Delta\Gamma_{s}$,
and $a_{\text{sl}}^{s}$ that can in principle constrain our NP model.

The mass difference $\Delta M_{s}=2|M_{12}^{s}|$ in the SM is

$\Delta M_{s}^{\text{SM}}=(18.2^{+0.6}_{-0.8})\,\text{ps}^{-1}\,,$ (17)

using the non-perturbative bag parameter combination from Ref. Greljo  _et
al._ (2023) (based on results in Refs. Bazavov  _et al._ (2016); Dowdall  _et
al._ (2019); King  _et al._ (2019)), the FLAG 2023 average for $f_{B_{s}}$
from FLAG 2023 Aoki  _et al._ (2022) (based on results in Refs. Dowdall  _et
al._ (2013); Bussone  _et al._ (2016); Hughes  _et al._ (2018)), $m_{t}$ from
PDG 2022 Workman  _et al._ (2022), and the Spring 2021 CKMfitter collaboration
results Charles  _et al._ (2005); CKM , which should be compared to the latest
HFLAV average of Amhis  _et al._ (2023)

$\Delta M_{s}^{\rm EXP}=\qty{17.765\pm 0.006}{\per}\,.$ (18)

The new physics contribution at the electroweak scale is given by

$\displaystyle{C_{1}}$
$\displaystyle=\dfrac{{-1}}{{64{\pi^{2}}}}\sum\limits_{j,k=1}^{3}{\lambda_{j2}^{L*}}\lambda_{j3}^{L}\lambda_{k2}^{L*}\lambda_{k3}^{L}{D_{2}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right),$
(19) $\displaystyle{C_{2}}$
$\displaystyle={C_{3}}=\dfrac{{-1}}{{32{\pi^{2}}}}\sum\limits_{j,k=1}^{3}{{m_{{u_{j}}}}}{m_{{u_{k}}}}\lambda_{j2}^{R*}\lambda_{j3}^{L}\lambda_{k2}^{R*}\lambda_{k3}^{L}{D_{0}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right),$
(20) $\displaystyle{C_{4}}$
$\displaystyle=\dfrac{{-1}}{{16{\pi^{2}}}}\sum\limits_{j,k=1}^{3}\begin{aligned}
&\left({m_{{u_{j}}}}{m_{{u_{k}}}}\lambda_{j2}^{R*}\lambda_{j3}^{L}\lambda_{k2}^{L*}\lambda_{k3}^{R}{D_{0}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right)-\frac{1}{2}\lambda_{j2}^{R*}\lambda_{j3}^{R}\lambda_{k2}^{L*}\lambda_{k3}^{L}{D_{2}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right)\right.\\\
&\left.\;+m_{u_{j}}m_{u_{k}}V_{j3}V_{*}{k2}\lambda^{R}_{j3}\lambda^{R*}_{k2}\left[g^{2}D_{0}\left(m_{u_{j}}^{2},m_{u_{k}}^{2},M_{W}^{2},M_{\phi}^{2}\right)-\sqrt{2}G_{F}D_{2}\left(m_{u_{j}}^{2},m_{u_{k}}^{2},M_{W}^{2},M_{\phi}^{2}\right)\right]\right),\end{aligned}$
(21) $\displaystyle{C_{5}}$
$\displaystyle=\dfrac{{-1}}{{16{\pi^{2}}}}\sum\limits_{j,k=1}^{3}\begin{aligned}
&\left({m_{{u_{j}}}}{m_{{u_{k}}}}\lambda_{j2}^{R*}\lambda_{j3}^{L}\lambda_{k2}^{L*}\lambda_{k3}^{R}{D_{0}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right)-\frac{1}{2}\lambda_{j2}^{R*}\lambda_{j3}^{R}\lambda_{k2}^{L*}\lambda_{k3}^{L}{D_{2}}\left({m_{{u_{j}}}^{2},m_{{u_{k}}}^{2},M_{\phi}^{2},M_{\phi}^{2}}\right)\right.\\\
&\left.\;-m_{u_{j}}m_{u_{k}}V_{j3}V_{*}{k2}\lambda^{R}_{j3}\lambda^{R*}_{k2}\left[g^{2}D_{0}\left(m_{u_{j}}^{2},m_{u_{k}}^{2},M_{W}^{2},M_{\phi}^{2}\right)-\sqrt{2}G_{F}D_{2}\left(m_{u_{j}}^{2},m_{u_{k}}^{2},M_{W}^{2},M_{\phi}^{2}\right)\right]\right).\end{aligned}$
(22)

$C_{1,2,3}^{\prime}$ are obtained by exchanging
$\lambda^{L}\leftrightarrow\lambda^{R}$ in $C_{1,2,3}$, and the effective
Hamiltonian and the loop functions are defined in Appendix A and C of Ref.
Crivellin  _et al._ (2013).

The width difference for $B_{s}$ mesons can be calculated as
$\Delta\Gamma_{s}=2|\Gamma_{12}^{s}|\cos(\arg(-M_{12}^{s}/\Gamma_{12}^{s}))$,
where the current SM prediction is Lenz

$\Delta\Gamma_{s}=\qty{0.0895\pm 0.0131}{\per}\,,$ (23)

and the latest HFLAV average555There is some tension between the ATLAS, CMS,
and LHCb measurements, such that this HLFAV average applied a scale factor of
1.8 (according to the PDG prescription) to the total uncertainty. is Amhis
_et al._ (2023)

$\Delta\Gamma_{s}=\qty{0.083\pm 0.005}{\per}\,.$ (24)

Our model alters this quantity through the $\bar{s}b\bar{c}c$ operators, where
the full NP contributions were calculated in Refs. Jäger  _et al._ (2018,
2020).

The semi-leptonic asymmetry for $B_{s}$ mesons can be calculated as
$a_{\text{sl}}^{s}=\Im(\Gamma_{12}^{s}/M_{12}^{s})$, where the current SM
prediction is Lenz
$a_{\text{sl}}^{s,\text{SM}}=$2.2(3)\text{\times}{10}^{-5}$$ and the latest
HFLAV average is Amhis  _et al._ (2023)
$a_{\text{sl}}^{s,\text{exp}}=$-60(280)\text{\times}{10}^{-5}$$. As our NP
does not have the severe GIM cancellation of the SM contribution to
$\Gamma_{12}^{s}$ (see the discussion in e.g. Ref. Artuso  _et al._ (2016)),
we will predict an asymmetry that is twice the size, but with opposite sign,
as the expected SM value (i.e.
$a_{\text{sl}}^{s,\text{NP}}\approx$-4\text{\times}{10}^{-5}$$), even without
CP violating couplings. However, differentiating this from the SM value would
require a huge improvement in the experimental precision.

###  II.5 $D^{0}-\bar{D}^{0}$ mixing

The $\Delta C=2$ coefficients which give a short-distance contribution to
$\Delta M_{D}$ are

$\displaystyle{C_{1}}$
$\displaystyle=\frac{1}{64\pi^{2}M_{\phi}^{2}}\left[\left(\lambda^{L}\lambda^{L\dagger}\right)_{12}\right]^{2},$
(25) $\displaystyle{C_{2}}$ $\displaystyle={C_{3}}=0,$ (26)
$\displaystyle{C_{4}}$
$\displaystyle=C_{5}=-\frac{1}{32\pi^{2}M_{\phi}^{2}}\left(\lambda^{L}\lambda^{L\dagger}\right)_{12}\left(\lambda^{R}\lambda^{R\dagger}\right)_{12}\,.$
(27)

$C_{1,2,3}^{\prime}$ are found from $C_{1,2,3}$ by exchaning
$\lambda^{L}\leftrightarrow\lambda^{R}$.

The SM prediction is currently unclear, as a naive calculation gives a result
four or five orders of magnitude too small, while other estimates (albeit not
from first principles) suggest the SM alone could give a result
$x^{\text{SM}}\sim\qty{0.1}{}$ Wolfenstein (1985); Donoghue  _et al._ (1986);
Falk  _et al._ (2002, 2004); Cheng and Chiang (2010); Jiang  _et al._ (2018);
Lenz  _et al._ (2020a), where

$x\equiv\frac{2\Delta M_{D}}{\Gamma_{D}},$ (28)

is the commonly reported observable in $D^{0}-\bar{D}^{0}$-mixing. Comparing
this to the current HLFAV average Amhis  _et al._ (2023)

$x^{\text{\rm EXP}}=\qty{0.407\pm 0.044}{},$ (29)

we take a conservative approach and allow the short-distance NP contribution
to be up to twice the size of the experimental value (i.e. we impose $\Delta
M_{D}^{\text{NP}}\leq 2\Delta M_{D}^{\text{exp}}$, allowing in principle for a
100 cancellation between BSM and SM).

###  II.6 $B_{s}/B_{d}$ lifetime ratio

The ratio of the $B_{s}$ to $B_{d}$ lifetimes is a theoretically clean
observable, which benefits from many cancellations of uncertainties. The
latest SM prediction is Lenz  _et al._ (2023)

$\frac{\tau(B_{s})^{\rm SM}}{\tau(B_{d})^{\rm SM}}=$1.003(6)$\,,$ (30)

while on the experimental side, the 2023 HFLAV average is Amhis  _et al._
(2023)

$\frac{\tau(B_{s})^{\rm EXP}}{\tau(B_{d})^{\rm EXP}}=$1.002(4)$\,.$ (31)

Readers should note that this combination has, similarly to the
$\Delta\Gamma_{s}$ case, a tension between ATLAS, LHCb and CMS measurements of
the $B_{s}$ lifetime. With regards to the theoretical prediction, the
uncertainty of the experimental extraction of the non-hadronic Darwin term
matrix element leads to an alternative SM prediction of $1.028(11)$, which
shows a much larger tension with experiment (see Ref. Lenz  _et al._ (2023)
for a detailed discussion). In this case, there are potentially sizable, but
currently unknown, contributions from $C_{7-10}^{cc}$ to the Darwin operator
coefficient which renders the theoretical prediction incomplete.666The
contribution from the SM operators $C_{1,2}^{cc}$ has only recently been
calculated in Refs. Lenz  _et al._ (2020b); Mannel  _et al._ (2020); Moreno
(2021).

##  III Phenomenology

We now turn to the phenomenology of our model. First of all, we set
$M_{\phi}=500\,$GeV which is compatible with the non-resonant paired di-jet
search of CMS due to the weaker-than-expected limit in this mass region
Tumasyan  _et al._ (2023b). Note that using a light DQ helps to reduce the
relative effect in $\Delta F=2$ processes since here the DQ contribution is
proportional to $\lambda^{4}/M_{\phi}^{2}$ while for all other flavour
processes the leading DQ effect has a $\lambda^{2}/M_{\phi}^{2}$ scaling.

The product of $\tilde{\lambda}^{L}_{23}$ and $\tilde{\lambda}^{L}_{22}$ is
necessary to give the effect in $b\to s\ell^{+}\ell^{-}$ via $C_{9}$ while the
product of $\tilde{\lambda}^{R}_{22}$ and $\tilde{\lambda}^{R}_{23}$
($\lambda^{R}_{12}$) helps to weaken the bound from $B_{s}-\bar{B}_{s}$ mixing
($D^{0}-\bar{D}^{0}$). To avoid a chirally enhanced effect in $b\to s\gamma$
$\lambda^{R}_{33}\approx 0$ is helpful. Furthermore, to avoid bounds from di-
jet resonance searches Aaboud  _et al._ (2018b); Bordone  _et al._ (2021) and
Kaon mixing, we assume the left-handed coupling involving the first generation
to be approximately zero. Thus we consider the following structure for the DQ
quark couplings

$\tilde{\lambda}^{L}=\begin{pmatrix}0&0&0\\\
0&\tilde{\lambda}^{L}_{22}&\tilde{\lambda}^{L}_{23}\\\
0&\tilde{\lambda}^{L}_{23}&0\end{pmatrix}\,,\qquad\lambda^{R}=\begin{pmatrix}0&\lambda^{R}_{12}&0\\\
0&\lambda^{R}_{22}&\lambda^{R}_{23}\\\ 0&0&0\end{pmatrix}\,.$ (32)

Note that we input ${\tilde{\lambda}}^{L}$ in the down quark basis (not
including CKM rotations).

Since $\tilde{\lambda}^{L*}_{22}\tilde{\lambda}^{L}_{23}$ must be positive to
give the preferred sign in $C_{9}$ we set for simplicity
$\tilde{\lambda}^{L}_{22}=\tilde{\lambda}^{L}_{23}$ and
$\lambda^{R}_{22}=\lambda^{R}_{23}$, assume real couplings and show the
preferred regions of the various observables in Fig. 1. We see that a
reasonably sizable LFU $C_{9}$, of the order of $-0.5$, can be generated in
our model while still respecting the other experimental constraints. For
$D^{0}-\bar{D}^{0}$-mixing, we show the regions compatible with our fine-
tuning argument for $\lambda^{R}_{12}=0$ and floating it within two different
(small) ranges which are compatible with LHC di-jet searches. Interestingly,
our model predicts $|C_{9}^{\prime}|\ll|C_{9}|$ but with the same sign as
(slightly) preferred by the current global fit Algueró  _et al._ (2023). Note
that generating the sizeable negative $C_{9}$ leads to a positive shift in
$\Delta\Gamma_{s}$. This is more in line with the latest experimental result
from CMS, which is in slight tension with LHCb and ATLAS measurements Amhis
_et al._ (2023).

For the lifetime ratio observables discussed in the previous section, we find,
on the one hand, that it does not provide a useful bound on our model when
relying on the SM prediction in Eq. (30). On the other hand, the tension with
experiment of the alternative SM prediction cannot be resolved in our model
without inducing sizable discrepancies elsewhere. Therefore, we do not show
these observables in Fig. 1, awaiting better future theoretical control.

Figure 1: Experimental constraints given our assumed UV couplings matrices.
The black and red vertical and horizontal lines show the generated LFU
contributions to $C_{9}$ and $C_{9}^{\prime}$, respectively.

##  IV Conclusions and Outlook

There are persistent and significant tensions in $b\to s\ell^{+}\ell^{-}$
observables. They are most pronounced in Br[$B\to K\mu^{+}\mu^{-}$], in the
angular observable $P_{5}^{\prime}$ (in $B\to K^{*}\mu^{+}\mu^{-}$) and the
total branching ratio as well as angular observables in
$B_{s}\to\phi\mu^{+}\mu^{-}$. In combination with the constraints from
$R(K^{(*)})$ and $B_{s}\to\mu^{+}\mu^{-}$ they point towards lepton flavour
universal NP in $C_{9}$ with a significance at the $5\sigma$ level. This poses
a challenge for model building because heavy NP is generally chiral, while for
generating a dominant effect in $C_{9}$, a vectorial lepton current is needed.
Furthermore, bounds from charged lepton flavour violation require a separation
of the electron and muon sectors, which is difficult to achieve in many
models.

In this article, we proposed a novel model explaining the $b\to
s\ell^{+}\ell^{-}$ anomalies: an $SU(3)_{c}$ triplet scalar DQ with
hypercharge $Y=-1/3$. This field can generate a LFU effect in $C_{9}$ via the
mixing of the $(\bar{s}\gamma^{\mu}P_{L}b)(\bar{c}\gamma_{\mu}P_{L}c)$
operator into $O_{9}$. Since only quarks are directly involved in this model,
charged lepton flavour violation is automatically respected. Furthermore, due
to the weaker-than-expected LHC limit, DQs can still be relatively light,
around 500 GeV.

We show that in this setup one can obtain $C_{9}\approx-0.5$ while respecting
the bounds from $B_{s}-\bar{B}_{s}$ and $D^{0}-\bar{D}^{0}$ mixing as well as
LHC searches. We predict a small negative value of $C_{9}^{\prime}$ as well as
a positive shift in $\Delta\Gamma_{s}$ of around 20 w.r.t. the SM prediction.
For $b\to s\gamma$ our model predicts NP shifts in
$|C_{7\gamma}^{(\prime)}|\approx 0.05$. Furthermore, DQs are candidates for
sizable effects in hadronic decays of mesons, like
$\epsilon^{\prime}/\epsilon$ Aebischer  _et al._ (2020b); Buras (2021) and
could explain several anomalies in non-leptonic $B$ decays like the
longitudinal polarization in $B_{q}\to K^{*}\bar{K}^{*}$ Algueró  _et al._
(2021), the $B\to K\pi$ puzzle Hofer  _et al._ (2011); Fleischer  _et al._
(2018); Crivellin  _et al._ (2020b), the $B_{(s)}\to D_{(s)}\\{K,\pi\\}$ Huber
_et al._ (2016); Bordone  _et al._ (2020); Cai  _et al._ (2021); Endo  _et
al._ (2022); Beneke  _et al._ (2021); Fleischer and Malami (2022, 2023)
discrepancy within QCD factorisation or the LHC di-di-jet excess Crivellin
_et al._ (2023).

Finally, our NP contributions to the $C_{1,2}^{cc}$ coefficients have opposite
sign than $C_{1}^{cc}$ in the SM. This reduces the theory prediction for the
inclusive branching ratio $b\to c\bar{c}s$ where in the SM we have Br$[b\to
c\bar{c}s]_{\rm SM}=23\pm 2$% Krinner  _et al._ (2013). However, while the
situation for the experimental determination of this fully inclusive quantity
is currently quite unclear, and $O(30\%)$ effects in Br$[b\to c\bar{c}s]$ are
possible, our model prediction is in line with the so-called “missing charm
puzzle” Lenz (2000). A clarification of the experimental situation would
therefore be of great interest.

###### Acknowledgements.

The work of A.C. is supported by a Professorship Grant (PP00P2_211002) of the
Swiss National Science Foundation. M.K. acknowledges support from a Maria
Zambrano fellowship, from the State Agency for Research of the Spanish
Ministry of Science and Innovation through the “Unit of Excellence
Mar$́\mathrm{i}$a de Maeztu 2020-2023” award to the Institute of Cosmos
Sciences (CEX2019-000918-M), and from PID2019-105614GB-C21 and 2017-SGR-929
grants.

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