aainstitutetext: Dipartimento di Fisica, Università di Torino & INFN, Sezione
di Torino, I-10125 Torino, Italybbinstitutetext: Theoretische Physik 1,
Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-
Straße 3, D-57068 Siegen, Germanyccinstitutetext: Albert Einstein Center for
Fundamental Physics, Institut für Theoretische Physik, Universität Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland.ddinstitutetext: CERN, Theoretical
Physics Department, CH-1211 Geneva 23, Switzerlandeeinstitutetext: INFN,
Sezione di Trieste, SISSA, Via Bonomea 265, 34136, Trieste, Italy

# Exploiting dijet resonance searches for flavor physics

Marzia Bordone c,d Admir Greljo e and David Marzocca
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we reinterpret ATLAS and CMS dijet resonance searches to set
robust constraints on all hypothetical tree-level scalar and vector mediators
with masses up to 5 TeV, assuming a diquark or a quark-antiquark coupling with
an arbitrary flavor composition. To illustrate the application of these
general results, we quantify the permissible size of new physics in
$\bar{B}_{q}\to D_{q}^{(*)+}\,\\{\pi,K\\}$ consistent with the absence of
signal in dijet resonance searches. Along the way, we perform a full SMEFT
analysis of the aforementioned non-leptonic $B$ meson decays at leading-order
in $\alpha_{s}$. Our findings uncover a pressing tension between the new
physics explanations of recently reported anomalies in these decays and the
dijet resonant searches. The high-$p_{T}$ constraints are crucial to drain the
parameter space consistent with the low-$p_{T}$ flavor physics data.

††preprint: CERN-TH-2021-031, P3H-21-016, SI-HEP-2021-09††arxiv: 2103.10332

##  1 Introduction

Non-leptonic decays are a challenging playground for the Standard Model (SM),
particularly for non-perturbative approaches to quantum chromodynamics (QCD).
A question which is often neglected in the literature is how much new physics
can hide in these decays? In particular, given the present constraints from
complementary new physics searches at low and high energies, what is the
allowed deviation from the SM predictions? To address this question, we
exploit dijet resonance searches at high-$p_{T}$ colliders as a complementary
probe of the hypothetical new physics entering non-leptonic decays.

We imagine a bosonic mediator $X$ coupled to quarks which, on the one hand,
modifies the low-energy meson decays, while on the other hand, can be directly
produced and (or) decayed at the LHC by the same interactions. Several
challenges are stemming from the lack of knowledge of the underlying
microscopic theory beyond the SM. Firstly, there is a broad range of relevant
masses that needs to be covered. Here we consider $m_{X}\in(50,5000)$ GeV
relying crucially on the latest experimental progress on dijet resonance
searches by ATLAS and CMS collaborations Sirunyan:2018xlo ; Aad:2019hjw ;
Sirunyan:2019vgj ; Aaboud:2018fzt . Secondly, there is a variety of possible
representations and flavor couplings of the mediator $X$, for which the
expected signal rates in $pp$ collisions differ. One of the main outcomes of
our study is the reinterpretation of the existing dijet resonance searches in
a general form applicable to flavor physics. Note that, in the narrow-width
approximation, the presence of several couplings can only add up in the total
$pp\to X\to jj$ rate. This is a very useful feature for flavor physics. Non-
leptonic meson decays depend on the product of two couplings when the
resonance $X$ is integrated out at tree level. The absence of the signal in
the dijet resonance searches sets limits on all $Xq^{i}q^{j}$ couplings
simultaneously, thus also on their product.

In order to exemplify the importance of these results, we investigate
potential new physics (NP) effects in the branching ratio of $\bar{B}_{s}\to
D_{s}^{(*)+}\pi$ and $\bar{B}\to D^{(*)+}K$ decays. These decays are mediated
by the underlying $b\to c\bar{u}d_{i}$ quark-level transitions, where
$d_{i}=d,s$, rendering their SM theory predictions amongst the most reliable
in the sector of non-leptonic decays and are obtained in the framework of QCD
factorisation (QCDF) Beneke:2000ry . Since the quarks entering in these decays
are distinguishable, topologies like penguin contribution or weak annihilation
do not contribute, rendering the description of these decays rather clean. The
most up-to-date predictions for the branching ratios have been presented in
Ref. Bordone:2020gao . Next-to-leading power corrections, arising at order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$, are found to be subleading compared
to the leading-power ones, strengthening the predictive power of QCDF for
these channels.

The possibility of NP effects in four-quarks operators has been already
entertained, with the focus on low-energy inclusive observables, see for
example Refs. Bobeth:2014rda ; Brod:2014bfa ; Jager:2017gal ; Chala:2019fdb ;
Lenz:2019lvd ; Cai:2021mlt . Interestingly, the aforementioned update
uncovered an intriguing tension with the data, not yet thoroughly analyzed in
the NP context. A fit to all the available experimental information concerning
$\bar{B}_{s}\to D_{s}^{(*)+}\pi$ and $\bar{B}\to D^{(*)+}K$ decays is
performed, and the current combination of the experimental measurements for
their branching ratios is extracted Bordone:2020gao . The comparison with the
respective theory predictions shows that the latter always overestimate the
former, with a combined discrepancy of about $4.4\sigma$ Bordone:2020gao .
This trend has already been observed in the literature (see for example Refs.
Beneke:2000ry ; Fleischer:2010ca ; Huber:2016xod ; Cai:2021mlt ) but has
become more apparent due to the updated theory results in Ref. Bordone:2020gao
.

A satisfactory explanation of this puzzle is not yet articulated. On the
theory side, the hypothesis of a big deviation due to the missing subleading
contributions in QCDF seems to be unlikely, since they overshoot the current
estimates by at least one order of magnitude. Hence, it seems motivated to
entertain the possibility of this deviation being due to NP. In this paper, we
try to understand to which extend the NP solution is viable, especially in
connection with bounds from related processes, most notably dijet searches at
high-$p_{T}$. The bounds that we obtain from dijet searches can be applied to
a broader class of four quark operators, beyond the ones mediating $b\to
c\bar{u}d(s)$ transitions Grossman:2006jg ; Ciuchini:2007hx ; Haber:2006ue ;
An:2009zh ; Pirjol:2009vz ; Buras:2010zm ; Feldmann:2012js ;
Altmannshofer:2012ur ; Ciuchini:2012gd ; Franco:2012ck ; Engel:2013lsa ;
Dekens:2014jka ; Inoue:2014nva ; Assad:2017iib ; Bhattacharya:2018msv ;
Dekens:2018bci ; Haba:2018byj ; Chala:2019fdb ; Altmannshofer:2020shb ;
Aebischer:2020mkv ; Alguero:2020xca .

The paper is organised as follows. In Sect. 2 we study dijet searches at the
LHC to set generic constraints on hypothetical new resonances. In Sect. 2.1 we
discuss the constraints on the pair production $pp\to XX\to(jj)(jj)$ from
gauge interactions and subsequent decay into jets, while in Sect. 2.2 we
discuss the single dijet resonance production $pp\to X\to jj$. In Sect. 3 we
present a complete leading-order effective field theory (EFT) analysis of
$b\to c\bar{u}d_{i}$ transitions. In Sect. 4 we consider all simplified tree-
level mediator models able to explain the aforementioned anomaly and confront
the relevant parameter space against the high-$p_{T}$ collider constraints.
The $\Delta F=2$ transitions are typically induced alongside and require a
specific structure to comply with the present bounds while giving sufficient
contribution to non-leptonic decays. For two explicit models, color-sextet
scalar and colorless weak-doublet scalar, we perform a thorough study of the
flavor phenomenology to find the parameter space consistent with the stringent
bounds from $\Delta F=2$ and other observables, while still being capable to
address the anomaly. Nevertheless, we show that the remaining parameter space
is (almost) entirely ruled out by the dijet searches from Sect. 2 pushing the
models towards a strongly coupled regime featuring broad resonances beyond the
validity of perturbation theory. This serves as a practical example of
complementarity between low-energy flavor physics and the high-$p_{T}$
searches at the LHC. Finally, many details of the calculations are left for
Appendices, and we conclude in Sect. 5.

##  2 Dijet searches at high-$p_{T}$

Figure 1:  Representative Feynman diagrams for the pair production $pp\to
XX\to(jj)(jj)$ (left diagram) and the single production $pp\to X\to jj$ (right
diagram) of a dijet resonance $X$ at the LHC. The constraints from the
existing searches are reported in Section 2 for different representations and
flavor interactions.

In this Section we reinterpret the latest ATLAS and CMS dijet searches in
terms of constraints on a generic resonance $X$ coupled to a pair of SM quarks
of an arbitrary flavor, decaying predominantly into jets. There are two
inevitable production mechanisms of $X$ at high-energy hadron colliders.
First, when $X$ is charged under a nontrivial SM gauge representation it can
be pair produced via gauge interactions. Second, the dijet resonance can be
singly produced directly from quark collisions. The representative diagrams of
the two production mechanisms are shown in Fig. 1. The left one corresponds to
QCD pair production, in the case of a colored resonance, which is fairly large
at hadron colliders (other diagrams are not shown for simplicity), while the
diagram on the right represents the single dijet resonance production. In the
most general case, when additional (sizeable) interactions are present, the
resonance decays (promptly) to either dijet, charged leptons, top quark,
electroweak gauge bosons, or exotic charged particles. For comparable rates,
the dijet final state is hardest to detect at hadron colliders due to the
overwhelming QCD background. The constraints in this section are obtained
assuming $\mathcal{B}(X\to jj)=1$. While the rescaling for different
$\mathcal{B}$ is straightforward, if other decay channels are present it might
be worth considering the constraints from the corresponding searches since
they may be stronger than those from dijets.

The total decay width to mass ratio $\Gamma_{X}/m_{X}$ is a crucial parameter
in resonance searches. We will focus mostly on models featuring narrow
resonances in which perturbative calculations are fully under control. By the
optical theorem, the total decay width is related to loop corrections to the
propagator. Collider searches for narrow resonances typically imply the
following condition, $\Gamma_{X}/m_{X}\lesssim 0.1$, due to the limited
detector resolution. In the second part of the study, where we focus on the NP
explanation of $b\to c\bar{u}d_{i}$ anomalies, the narrow width approximation
is valid in a broad mass range assuming the minimal set of couplings.
Nonetheless, we will comment on how much the collider bounds can be relaxed
for a broad resonance, with an increased $\Gamma_{X}/m_{X}$ ratio, while being
cautious about the validity of the calculation.

The resonances from new dynamics may or may not be within the kinematical
reach of the LHC. If the resonance mass $m_{X}$ is above the reach for on-
shell production at the LHC, its effect can be studied in the high-$p_{T}$
dijet tails in terms of four-quark contact interactions. From the experimental
point of view, this requires a qualitatively different approach since it is no
more possible to fit the data with a resonance-like signal over a smooth
background. Other observables, such as the angular distributions of the two
jets, are instead employed. For example, see Ref. Aaboud:2017yvp  for an ATLAS
search and Ref. Alioli:2017jdo  for and EFT analysis in terms of flavor-
universal contact interactions. We leave the analysis of the full set of
flavor-dependent four-quark contact interactions for future work, focusing
here on on-shell narrow resonances. The non-leptonic decays studied in the
second part of the paper focus on the weakly coupled ultraviolet (UV)
completions for which the resonance searches are sufficient, while contact
interactions will be relevant for strongly coupled UV completions.

###  2.1 Pair production of dijet resonances

Even when the couplings to quarks are small, the resonance $X$ is pair-
produced by gauge interactions, as in the left diagram of Fig. 1. The pair
production rate is robustly set by the resonance mass $m_{X}$ and its gauge
representation. We further assume $X$ undergoes a prompt decay to a dijet
final state. The LEP-II bounds rely on QED production in $e^{+}e^{-}\to
X\bar{X}$ and apply for all electrically charged resonances. A narrow scalar
resonance exclusively decaying to $jj$ is ruled out, unless

${\rm(LEP-II)}\;\;\;m_{X^{\pm 1/3}}\gtrsim 80\,{\rm GeV}\,,~{}m_{X^{\pm
1}}\gtrsim 95\,{\rm GeV}~{},$ (1)

see Fig. 9 (c) in Heister:2002jc . Similar limits apply for vector resonances.

Tevatron and LHC bounds require QCD interactions to be effective and thus
apply only to colored resonances. QCD pair production of colored resonances at
hadron colliders is overwhelming. The main challenge in these searches is to
suppress the large multijet background. Nonetheless, the most recent ATLAS and
CMS searches at 13 TeV with about $36\,$fb-1 can robustly exclude pair-
produced colored resonances decaying exclusively to $jj$ Aaboud:2017nmi ;
Sirunyan:2018rlj . In particular, the experimental limits on the complex
scalars, for different color and weak representations under
$(SU(3)_{c},SU(2)_{L})$, are

Scalar | (3,1) | (6,1) | (8,1)
---|---|---|---
$m_{X}>$ |  $410$ GeV (ATLAS) | 820 GeV (ATLAS) |  $1050$ GeV (ATLAS)
$520$ GeV (CMS) | 950 GeV (CMS) | 1000 GeV (CMS)
Scalar | (3,3) | (6,3) | (8,2)
$m_{X}>$ |  $620$ GeV (ATLAS) | 1200 GeV (ATLAS) |  $1200$ GeV (ATLAS)
$750$ GeV (CMS) | 1200 GeV (CMS) | 1200 GeV (CMS)

.

Here we report the upper edge of the exclusion mass window, while the lower
edge extends down to the LEP-II exclusions. In other words, the combination of
all experiments robustly excludes a resonance $X$ with the mass smaller from
what is reported in the table above. The limits on the color triplet and octet
are directly based on the stop and sgluon benchmarks, respectively. Note that
the color octet is a complex field, which doubles the sgluon cross section
used in Aaboud:2017nmi . We neglect small differences in the acceptance times
efficiency for resonances of a different color (and spin), such that
representations not considered by the experimental collaborations are
constrained by comparing the predicted production cross sections with the 95%
confidence level (CL) observed limits from Figure 9 of Ref. Aaboud:2017nmi
and Figure 11 of Ref. Sirunyan:2018rlj . This is validated comparing the
exclusion limits on stop, sgluon, and coloron from Ref. Aaboud:2017nmi . For
the color sextet, we calculate the cross section using MadGraph5_aMC@NLO
Alwall:2014hca  and the UFO model from the FeynRules Alloul:2013bka
repository based on the implementation of Han:2009ya .

The limits on the vector resonances depend on the UV completion. For example,
a vector color triplet can have an additional non-minimal coupling of the type
$\mathcal{L}\supset-ig_{s}\kappa\,X^{\dagger}_{\mu}T^{a}X_{\nu}G^{a\mu\nu}$.
If the resonance is a massive gauge boson left after the breaking of extended
gauge symmetry, the Yang-Mills (YM) case $\kappa=1$ applies. Another example
is the minimal coupling (MC) case $\kappa=0$, which usually leads to
conservative colliders constraints. Reinterpreting the searches Aaboud:2017nmi
; Sirunyan:2018rlj  for these two cases, we find

Vector (3) | YM | MC
---|---|---
$m_{X}>$ |  $1150$ GeV (ATLAS) | 700 GeV (ATLAS)
$1150$ GeV (CMS) | 800 GeV (CMS)

.

The appropriate cross sections are calculated using the UFO model from Ref.
Dorsner:2018ynv . As shown in this example, the limits on vector resonances
are extremely sensitive to the UV completion. Nonetheless, the exclusions are
typically stronger compared to their scalar counterparts.

As a final comment, when a resonance has sizeable couplings to valence quarks,
there is an additional contribution to production $q\bar{q}\to X\bar{X}$ with
$t$-channel quark exchange. Since the overall rate is dominated by the gluon
fusion, the (potential) negative interference with the sub-dominant $q\bar{q}$
diagram has no practical impact on the limits. On the other hand, when the
coupling gets larger, the total cross section is increased, and the previously
quoted exclusions become even more stringent. (For a related study see Fig. 3
in Ref. Dorsner:2014axa .) However, given the limits on the couplings derived
from the single production discussed below, the $t$-channel contribution to
the pair production can be safely neglected.

###  2.2 Dijet resonance

The coupling of the field $X$ to an arbitrary pair of quarks necessarily leads
to a resonant production of $X$ in $pp$ collisions at high enough energies,
followed by a dijet decay signature as shown in Fig. 1 (right). Experiments
have searched for a new dijet resonance, and set competitive constraints over
a wide range of $m_{X}$. We can use these null results to set robust upper
limits on the size of the $X$ coupling to any quark pair, as a function of
$m_{X}$. The idea is the following — the different flavor channels
$q^{i}q^{\prime j}\to X$ add up incoherently in the total cross section — thus
an upper limit on $pp\to X\to jj$ simultaneously bounds the absolute values of
all $Xq^{i}q^{\prime j}$ couplings. We carry out a general analysis of the
latest ATLAS and CMS dijet searches Sirunyan:2018xlo ; Aad:2019hjw ;
Sirunyan:2019vgj ; Aaboud:2018fzt  in the mass range
$m_{W^{\prime}}\in(450,5000)$ GeV for all possible spin-zero and spin-one
mediators considering the most general flavor structure for the couplings.

####  $W^{\prime}$ example

To set up the stage, let us consider a benchmark example. The partial decay
width for a spin-one colorless $W^{\prime}$ resonance with the interaction
Lagrangian

$\mathcal{L}\supset
x_{ij}~{}\bar{u}^{i}_{L}\gamma^{\mu}d^{j}_{L}~{}W^{\prime}_{\mu}+\textrm{h.c.}~{},$
(2)

is given by

$\Gamma_{W^{\prime}\to
u^{i}\bar{d}^{j}}=\frac{m_{W^{\prime}}}{8\pi}|x_{ij}|^{2}~{}.$ (3)

The leading-order cross section for the production of a narrow positively-
charged resonance $W^{\prime}$ in the quark fusion ($u_{i}\bar{d}_{j}\to
W^{\prime}$) at the LHC is determined by the partial decay width of the
inverse process,

$\sigma(pp\to
W^{\prime})=\frac{8\pi^{2}}{3s_{0}}~{}\frac{\Gamma_{W^{\prime}\to
u^{i}\bar{d}^{j}}}{m_{W^{\prime}}}~{}\int_{\tau}^{1}dx\,~{}\frac{1}{x}\,f^{p}_{u^{i}}(x)\,f^{p}_{\bar{d}^{j}}(\tau/x)~{},$
(4)

where $\tau=m^{2}_{W^{\prime}}/s_{0}$ with $\sqrt{s_{0}}$ the collider energy,
and $f^{p}_{q}(x)$ are the parton distribution functions evaluated at the
factorisation scale $\mu_{F}=m_{W^{\prime}}$. In the considered mass range the
higher-order radiative corrections of the inclusive production cross section
are expected to be of $\mathcal{O}(10\%)$. In our numerical calculations we
use MMHT14 NNLO central PDF set Harland-Lang:2014zoa . Analogous expressions
are for the charged-conjugate process.

Upper limits on the coupling as a function of the mass are extracted from the
ATLAS and CMS exclusions on a specific flavor-universal $Z^{\prime}$
benchmarks Aaboud:2018fzt ; Sirunyan:2018xlo ; Aad:2019hjw ; Sirunyan:2019vgj
. These analyses hunt for a resonance in the invariant mass of the two
highest-$p_{T}$ jets consistent with the inclusive $pp\to Z^{\prime}\to jj$
kinematics.111Instead, dedicated searches for the low-mass resonances target
$pp\to Z^{\prime}j\to(jj)j$ process Sirunyan:2019vxa ; Sirunyan:2019pnb .
These probe the parameter space untouchable by previous experiments at lower
energies (see e.g. CDF Aaltonen:2008dn  and UA2 Alitti:1993pn ). However, the
presence of the additional jet complicates the reinterpretation of the bounds
for other spin and flavor cases due to the huge QCD corrections from $gq$ and
$gg$ induced diagrams Rubin:2010xp . In Section 4.2 we recast Sirunyan:2019vxa
to set limits on a $H^{\prime}$ model example. The coupling versus mass limits
are adapted to our cases by equating the total production cross sections for
two models in the fiducial region. Fig. 2 (top panel) shows exclusions on a
$W^{\prime}$ coupled to first-generation quarks, $x_{ud}\neq 0$. The plot
shows 95% CL upper limits on $|x_{ud}|$ in the mass range
$m_{W^{\prime}}\in(450,5000)$ GeV. The vertical axis on the right shows the
corresponding partial decay width $\Gamma_{W^{\prime}}/m_{W^{\prime}}$ in Eq.
(3), justifying the narrow-width approximation.

####  $Xud$ couplings

Let us now reinterpret these bounds for a bosonic complex resonance $X$
(scalar or vector) of any color representation $r\in({\bf 1},{\bf 3},{\bf
6},{\bf 8})$. We define the coupling of $X$ to up and down quarks of arbitrary
flavors $i$ and $j$ as $x_{ij}$ in the interaction Lagrangians in Eq.
(46).222We do not consider non-renormalisable derivative interactions.

$SU(3)_{c}$ | 1 | 3 | 6 | 8
---|---|---|---|---
$\delta_{C}$ | 1 | 2 | 2 | 4/3
$\gamma_{C}$ | 1 | 2/3 | 1/3 | 1/6

spin | 0 | 1
---|---|---
$\delta_{S}$ | 1/2 | 1
$\gamma_{S}$ | 3/2 | 1

Table 1: Production cross sections ($\delta$) and decay widths ($\gamma$)
rescaling factors for resonances $X$ of different color and spin with $Xud$
couplings. For more details see Sect. 2.2 and Fig. 2. The interaction
Lagrangians are defined in Appendix B.

Dijet searches do not discriminate well between different diquark and quark-
antiquark resonances, see the discussion in Ref. Sirunyan:2018xlo . In other
words, the $W^{\prime}$ results can be reinterpreted for other mediators $X$
with different color, spin, and flavor couplings. Comparing the production
cross sections for the same coupling $x_{ij}$, we find

$\frac{\sigma(pp\to X)}{\sigma(pp\to W^{\prime})}=\delta_{C}\delta_{S}~{},$
(5)

where the color and spin $\delta$ factors are reported in Table 1. For
convenience, we also report the color and spin $\gamma$ factors for resonances
$X$, defined as

$\frac{\Gamma_{X\to
u^{i}d^{j}}}{m_{X}}=\gamma_{C}\gamma_{S}\,\frac{\Gamma_{W^{\prime}\to
u^{i}\bar{d}^{j}}}{m_{W^{\prime}}}~{}.$ (6)

The exclusions on the couplings for other cases are obtained by rescaling
those in Fig. 2 (top panel) with the appropriate ratio of parton luminosity
functions. The generic constraints are shown in Fig. 2 (bottom panel) when
combining all experimental searches. Dashed lines are for $qq\to X$ while
solid lines are for $q\bar{q}\to X$. The two are different only for the $ud$
versus $u\bar{d}$ case as expected for valence quarks. The first set should be
used for color triplet and sextet resonances, while the second for color
singlets and octets.

Figure 2:  Experimental limits on a narrow dijet resonance from the LHC
searches at 13 TeV Aaboud:2018fzt ; Sirunyan:2018xlo ; Aad:2019hjw ;
Sirunyan:2019vgj . Top panel is for a spin-1 colorless $W^{\prime}$ coupled to
a single flavor combination of chiral quarks, $x_{ud}\neq 0$. The plot shows
upper limits at 95% CL on the absolute value of the coupling from several CMS
and ATLAS searches. The vertical axis on the right-hand side is the
corresponding partial decay width $\Gamma_{W^{\prime}}/m_{W^{\prime}}$ from
Eq. (3). Bottom panel shows the combined dijet limits on resonances of
different spin and color, as well as, arbitrary flavor couplings $ij$. Dashed
lines are for diquark resonances (color triplets and sextets) while solid
lines are for quark-antiquark resonances (color singlets and octets). The
multiplicative rescaling factors for color ($\delta_{C}$) and spin
($\delta_{S}$) are reported in Table 1. This plot assumes $\mathcal{B}(X\to
jj)=1$ and is valid when the total decay width to mass ratio is
$\Gamma_{X}/m_{X}\lesssim 10\%$.

Figure 3:  Same as Fig. 2 but for a pair of up or a pair of down quarks. The
top figure is for the flavor diagonal couplings while the bottom one is for
the flavor changing. For more details see Sect. 2.2.

####  $Xuu$ and $Xdd$ couplings

Also in this case we define the interaction Lagrangian in Eq. (46), assuming
both quarks are either up-type or down-type. We describe electrically neutral
resonances with real fields, see the footnote below Eq. (46). The constraints
are summarized in Fig. 3 for the flavor diagonal couplings (top plot), as well
as, for the flavor violating (bottom plot). The rescaling factors $\delta_{C}$
and $\delta_{S}$ are the same as in Table 1. The $\delta_{x}$ factor equals 1
with the exception of the diagonal couplings for spin-1 color singlet and
octet $\delta_{x}=1/2$, the off-diagonal couplings for spin-0 color triplet
and sextet $\delta_{x}=4$, and the diagonal couplings for spin-0 color sextet
$\delta_{x}=2$.

#### Discussion

We study the variance of the $pp\to X\to jj$ kinematics for different spins
and flavor couplings and their impact on the reinterpretation of the ATLAS and
CMS bounds derived in a specific model. To this purpose we use
MadGraph5_aMC@NLO Alwall:2014hca  to simulate the process at the partonic
level. Dedicated UFO models for vector and scalar resonances are obtained with
FeynRules Alloul:2013bka . Motivated by the signal acceptance criteria in Ref.
Aaboud:2018fzt , we impose the following partonic-level cuts
$p_{T}(j_{1})>200$ GeV, $p_{T}(j_{2})>100$ GeV, and $|\eta_{1,2}|<2.8$. We
also require the pseudorapidity difference $|\eta_{1}-\eta_{2}|<1.2$ and the
dijet invariant mass $m(jj)>450$ GeV. We study the benchmark point
$m_{X}=0.75$ TeV and $\Gamma_{X}/m_{X}=0.06$. We find that the cut efficiency
for the scalar is about $15$% larger than for the vector. This comes from the
pseudorapidity cuts, in particular, for the scalar, the $2\to 2$ differential
partonic cross section $d\sigma/dt$ does not depend on the Mandelstam
variables $t$ and $u$, while for the vector, it is proportional to $t^{2}$ or
$u^{2}$. Thus, our bounds on the scalar resonances are somewhat conservative
since the collaborations model the signal with the vector resonance. The above
exercise was performed for the coupling with the first generation only. We
repeat the simulation for the scalar resonance for all flavor combinations of
incoming partons: $ud$, $us$, $ub$, $cd$, $cs$, and $cb$. The difference in
the signal acceptance is at most a few percent between different flavors.

Theoretical uncertainties on the production cross section are due to higher-
order radiative corrections, as well as limited knowledge of parton luminosity
functions. The most important NLO QCD corrections depend on the mediator
representation and typically increase the rate by $\mathcal{O}(10\%)$, see
example Ref. Han:2009ya  for the color sextet scalar. In this respect, our
leading-order calculation gives somewhat conservative bounds on the coupling.
Note that the relative error on the upper bound on $|x_{ij}|$ is half of the
relative error on the cross section.

The other source of uncertainties comes from the determination of the parton
luminosity functions. Relative uncertainties on the parton luminosities (the
integral in Eq. (4)) are shown in Fig. 5.10 of Ref. Ball:2017nwa  for three
different PDF collaborations: NNPDF3.1 Ball:2017nwa , CT14 Dulat:2015mca  and
MMHT14 Harland-Lang:2014zoa . These uncertainties are at the level of few to
ten percent across the entire mass range except for the heaviest resonances.
In particular, when $m_{X}\gtrsim 3$ TeV, the relative error from NNPDF3.1
quickly becomes $\mathcal{O}(1)$, while the other two collaborations do not
exhibit this behavior and their relative errors are under control in the
entire mass range of Figs. 2 and 3. We numerically compare MMHT14 and NNPDF3.1
sets breaking down the luminosities by the flavor content. The difference
between sets is particularly prominent for some flavor combinations, for
example, $s\bar{c}$ fusion, and less relevant for others. Hence, this feature
of the NNPDF3.1 set questions the robustness of the limits shown in Figs. 2
and 3 for $m_{X}\gtrsim 3$ TeV which will be resolved by the future updates.

In the analysis above, we always assume $\mathcal{B}(X\to jj)=1$ as expected
in the realistic case for non-leptonic decays given their present sensitivity.
This also provides conservative bounds, as any other final state (e.g.
leptons) would be easier to detect. The most important caveat concerns the
total width of the resonance. The typical detector resolution of the dijet
invariant mass is at the level of $10\%$ Sirunyan:2019vxa ; Sirunyan:2019pnb ;
Aaboud:2018fzt ; Sirunyan:2018xlo ; Aad:2019hjw ; Sirunyan:2019vgj .
Therefore, a narrow resonance is experimentally defined by
$\Gamma_{X}/m_{X}\lesssim 10\%$. This criterion has to be satisfied to apply
the bounds from Fig. 2.

When the resonance $X$ is a bit broader, the dijet limits weaken but do not
disappear completely. Fig. 10 of Ref. Sirunyan:2019vgj  shows how the cross
section times acceptance drops when increasing the total decay width. For
example, assuming $m_{X}=4$ TeV, and increasing the width from $1\%$ to $10\%$
and $40\%$, the limit on the cross section relaxes by a factor $\sim 3$ and
$\sim 15$, respectively. The effect is less pronounced for lighter resonances,
for example, when $m_{X}=2.2$ TeV, these factors are $\sim 1.6$ and $\sim 8$,
respectively. To conclude, the limits on the couplings of a broad resonance
will be relaxed by at most $\mathcal{O}(1)$ factor from those in Fig. 2,
before the control over the calculation is lost and the strongly-coupled
regime is entered.

##  3 General EFT analysis of $b\to cud_{i}$

In this section we study NP effects in non-leptonic $B$ meson decays adopting
a bottom-up approach to keep the discussion as general as possible. To this
purpose, we utilize the methods of effective field theory (EFT). After
presenting the data, we perform the fit in the weak effective Hamiltonian to
identity the preferred parameter space. The results are then interpreted in
the context of the EFT above the electroweak scale (SMEFT).

#### Measurements

The experimental values for the branching fractions for $b\to c\bar{u}d_{i}$
decays are obtained by fitting all available data and are compared with the
most up-to-date SM predictions based on QCDF Bordone:2020gao . The main
difference between the results in Ref. Bordone:2020gao  with respect to
previous analyses (see e.g. Ref. Huber:2016xod ) is the use of updated inputs
for CKM elements, decay constants and form factors for $\bar{B}_{q}\to
D^{(*)+}_{q}$ transitions Bordone:2019guc ; Bordone:2019vic ; Zyla:2020zbs ,
causing shifts in the central values (the largest one for $\bar{B}_{s}\to
D^{*+}_{s}\pi$ decay) and generally a reduction of the uncertainties on the
branching ratios. From the experimental point of view, the non-leptonic
$\bar{B}_{s}\to D_{s}^{(*)+}\pi$ and $\bar{B}\to D^{(*)+}K$ decays are often
measured as part of ratios with other decay channels in order to reduce
experimental errors.

To be conservative we choose to employ the experimental fit in the third
column of Table II of Ref. Bordone:2020gao  (without QCDF inputs but with the
LHCb measurements of $f_{s}/f_{d}$ from semileptonic decays). Let us define
the ratio of the measured branching ratio to the respective SM prediction as

$R(X\to YZ)\equiv\mathcal{B}(X\to YZ)/\mathcal{B}(X\to YZ)_{\text{SM}}~{}.$
(7)

Combining the measurements and SM predictions, including correlations in both
experimental and theoretical uncertainties,333We thank Martin Jung for
providing the associated correlation matrix. we obtain the following result:

$\begin{array}[]{r l}R(\bar{B}_{s}^{0}\to D_{s}^{+}\pi^{-})&=0.704\pm 0.074\\\
R(\bar{B}^{0}\to D^{+}K^{-})&=0.687\pm 0.059\\\ R(\bar{B}_{s}^{0}\to
D_{s}^{*+}\pi^{-})&=0.49\pm 0.24\\\ R(\bar{B}^{0}\to D^{*+}K^{-})&=0.66\pm
0.13\end{array}~{},\qquad\rho=\left(\begin{array}[]{c c c c
}1&0.36&0.16&0.092\\\ 0.36&1&0.072&0.16\\\ 0.16&0.072&1&0.40\\\
0.092&0.16&0.40&1\end{array}\right)~{},$ (8)

where $\rho$ is the correlation matrix and, by definition all $R=1$ in the SM.
The observed branching ratios are consistently smaller than the QCDF
predictions Bordone:2020gao .

$\mathcal{Q}_{V_{LL}}^{ijkl}=(\bar{u}_{L}^{i}\gamma_{\mu}d_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}u_{L}^{l})$ | $\mathcal{Q}_{V_{LL}}^{\prime ijkl}=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}T^{A}u_{L}^{l})$
---|---
$\mathcal{Q}_{V_{RR}}^{ijkl}=(\bar{u}_{R}^{i}\gamma_{\mu}d_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}u_{R}^{l})$ | $\mathcal{Q}_{V_{RR}}^{\prime ijkl}=(\bar{u}_{R}^{i}\gamma_{\mu}T^{A}d_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}u_{R}^{l})$
$\mathcal{Q}_{V_{LR}}^{ijkl}=(\bar{u}_{L}^{i}\gamma_{\mu}d_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}u_{R}^{l})$ | $\mathcal{Q}_{V_{LR}}^{\prime ijkl}=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}u_{R}^{l})$
$\mathcal{Q}_{S_{RL}}^{ijkl}=(\bar{u}_{L}^{i}d_{R}^{j})(\bar{d}_{R}^{k}u_{L}^{l})$ | $\mathcal{Q}_{S_{RL}}^{\prime ijkl}=(\bar{u}_{L}^{i}T^{A}d_{R}^{j})(\bar{d}_{R}^{k}T^{A}u_{L}^{l})$
$\mathcal{Q}_{S_{LR}}^{ijkl}=(\bar{u}_{R}^{i}d_{L}^{j})(\bar{d}_{L}^{k}u_{R}^{l})$ | $\mathcal{Q}_{S_{LR}}^{\prime ijkl}=(\bar{u}_{R}^{i}T^{A}d_{L}^{j})(\bar{d}_{L}^{k}T^{A}u_{R}^{l})$
$\mathcal{Q}_{S_{RR}}^{ijkl}=(\bar{u}_{L}^{i}d_{R}^{j})(\bar{d}_{L}^{k}u_{R}^{l})$ | $\mathcal{Q}_{S_{RR}}^{\prime ijkl}=(\bar{u}_{L}^{i}T^{A}d_{R}^{j})(\bar{d}_{L}^{k}T^{A}u_{R}^{l})$
$\mathcal{Q}_{T_{RR}}^{ijkl}=(\bar{u}_{L}^{i}\sigma_{\mu\nu}d_{R}^{j})(\bar{d}_{L}^{k}\sigma^{\mu\nu}u_{R}^{l})$ | $\mathcal{Q}_{T_{RR}}^{\prime ijkl}=(\bar{u}_{L}^{i}\sigma_{\mu\nu}T^{A}d_{R}^{j})(\bar{d}_{L}^{k}\sigma^{\mu\nu}T^{A}u_{R}^{l})$
Table 2: Low-energy operators relevant for $b\to c\bar{u}d_{i}$ transitions.

#### Low-energy effective field theory

The most general theoretical framework for short-distance NP effects in $b\to
c\bar{u}d_{i}$ ($i=1,2$) transitions is the low-energy effective field theory
(LEFT) Jenkins:2017jig . Here we perform a NP analysis including the full set
of relevant operators $\mathcal{O}_{i}$ in the basis of Ref. Jenkins:2017jig
(for the list see Eq. (34) of Appendix A). These operators, however, are not
in a convenient form to evaluate the hadronic matrix elements we are
interested in. Therefore, we Fierz them to the $\mathcal{Q}_{i}^{(\prime)}$
listed in Table 2, where

$\mathcal{L}_{\text{NP}}=\sum_{i=1}^{7}(a_{i}\,\mathcal{Q}_{i}+a^{\prime}_{i}\,\mathcal{Q}^{\prime}_{i})+{\rm
h.c.}~{}.$ (9)

The corresponding matching relations between the two bases are reported in Eq.
(36). The operators $\mathcal{Q}^{cbiu}_{V_{LL}}$ and $\mathcal{Q}^{\prime
cbiu}_{V_{LL}}$ correspond to the SM color-allowed and color-suppressed
operators $Q_{2}$ and $Q_{1}$ of the CMM basis Chetyrkin:1997gb ,
respectively. In these conventions, the SM Wilson coefficients are

$(a_{V_{LL}}^{cbiu})^{\rm
SM}=-C_{2}\frac{4G_{F}V_{cb}V_{ui}^{*}}{\sqrt{2}}~{},\quad(a_{V_{LL}}^{\prime
cbiu})^{\rm SM}=-C_{1}\frac{4G_{F}V_{cb}V_{ui}^{*}}{\sqrt{2}}~{},$ (10)

where $C_{2}=+1.010$ and $C_{1}=-0.291$ Gorbahn:2004my . The hadronic matrix
elements for the NP operators are evaluated at leading order in $\alpha_{s}$
and leading power in $1/m_{b}$. As in the SM, the NP operators can be grouped
in color-allowed and color-suppressed ones. As a consequence of color algebra,
we have

$\langle
D_{q}^{+(*)}P^{-}|\mathcal{Q}^{\prime}_{i}|\bar{B}_{q}\rangle=0+\mathcal{O}(\alpha_{s}/N_{c})~{},$
(11)

regardless of the chirality structure of $\mathcal{Q}_{i}^{\prime}$ operators.
Introducing $\alpha_{s}$ corrections generates contributions from the color-
suppressed operators proportional to $\alpha_{s}/N_{c}$. These contributions
in the SM are small compared to the leading ones since they are further
suppressed by the Wilson coefficient $C_{1}\ll C_{2}$. A recent computation of
color-suppressed topologies Cai:2021mlt  showed that they are even more
subleading than what naively expected, compared to color-allowed ones. This
strengthens our hypothesis of disregarding color-suppressed
$\mathcal{Q}_{i}^{\prime}$ operators for this NP analysis. Furthermore, we
stress that this choice does not affect the constraining power of dijet
resonance searches.

The non-zero matrix elements give rise to the following decay amplitudes for
$\bar{B}_{q}\to D_{q}^{+(*)}P^{-}$ (see Appendix A for details):

$\begin{split}\mathcal{A}(\bar{B}_{q}\to
D_{q}^{+}P^{-})=&\,\mathcal{A}(\bar{B}_{q}\to
D_{q}^{+}P^{-})_{\text{SM}}\times\\\
&\left\\{1+\frac{1}{2\sqrt{2}G_{F}V_{cb}V^{*}_{ui}C_{2}}\bigg{[}\big{(}-a_{V_{LL}}^{cbiu}+a_{V_{RR}}^{cbiu}+a_{V_{LR}}^{cbiu}-a_{V_{LR}}^{uibc}\big{)}\right.\\\
&\left.+\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}-m_{c})}\big{(}a_{S_{RL}}^{cbiu}-a_{S_{LR}}^{cbiu}-a_{S_{RR}}^{cbiu}+a_{S_{RR}}^{uibc}\big{)}\bigg{]}\right\\}\,,\\\
\end{split}$ (12) $\begin{split}\mathcal{A}(\bar{B}_{q}\to
D_{q}^{*+}P^{-})=&\,\mathcal{A}(\bar{B}_{q}\to
D_{q}^{*+}P^{-})_{\text{SM}}\times\\\
&\left\\{1+\frac{1}{2\sqrt{2}G_{F}V_{cb}V^{*}_{ui}C_{2}}\bigg{[}\big{(}-a_{V_{LL}}^{cbiu}-a_{V_{RR}}^{cbiu}+a_{V_{LR}}^{cbiu}+a_{V_{LR}}^{uibc}\big{)}\right.\\\
&+\left.\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}+m_{c})}\big{(}a_{S_{RL}}^{cbiu}+a_{S_{LR}}^{cbiu}-a_{S_{RR}}^{cbiu}-a_{S_{RR}}^{uibc}\big{)}\bigg{]}\right\\}\,,\end{split}$
(13)

where $i=d,s$ corresponds to $P^{-}=\pi^{-},K^{-}$, respectively.

#### Fit to the data

These decay amplitudes are used to calculate the ratio of the branching
fractions to the respective SM prediction in Eq. (7) as

$R(X\to YZ)=\frac{|\mathcal{A}(X\to YZ)|^{2}}{|\mathcal{A}(X\to YZ)_{\rm
SM}|^{2}}~{},$ (14)

from which we perform a fit using Eq. (8) and Eqs. (12)–(13). Only half of the
NP coefficients contributing to the amplitudes in Eq. (12) and Eq. (13) can
simultaneously explain the observed suppression in both final states with a
$D_{q}^{+}$ or a $D_{q}^{+*}$ meson. These are $a^{cbiu}_{V_{LL}}$,
$a^{cbiu}_{V_{LR}}$, $a^{cbiu}_{S_{RR}}$ and $a^{cbiu}_{S_{RL}}$. The other
half can not fit the data well. Regarding the flavor structure, new
interactions with both strange and down quarks are needed.

Figure 4:  Low-energy EFT fit to $\bar{B}_{q}\to D_{q}^{+(*)}P^{-}$ decays.
Dashed and solid lines show $68\%$ and $95\%$ CL regions for vector operators
(left panel) and scalar operators (right panel). The gray dotted line is
consistent with the relative size following the CKM ratio $V_{us}/V_{ud}$.

The results of the fits are shown in Fig. 4. The left (right) panel is for new
vector (scalar) interactions. The horizontal and the vertical axes are for the
couplings to down and to strange quarks, respectively. The dashed and solid
green lines describe the boundaries of the $68\%$ and $95\%$ CL regions. The
discrepancy between SM predictions and measurements manifests as a shift of
the preferred region from the origin. The best-fit point in the left (right)
plot improves the fit to data with respect to the SM by $\chi^{2}_{\rm
SM}-\chi^{2}_{\rm best-fit}\approx 36\,(35)$ and corresponds to

$\begin{split}\text{vector:}~{}\Big{\\{}&a_{V_{LL}}^{cbdu}-a_{V_{LR}}^{cbdu}\approx
0.23V_{ud}~{}\text{TeV}^{-2}~{},\quad
a_{V_{LL}}^{cbsu}-a_{V_{LR}}^{cbsu}\approx 0.24V_{us}~{}\text{TeV}^{-2}~{},\\\
\text{scalar:}~{}\Big{\\{}&a_{S_{RR}}^{cbdu}-a_{S_{RL}}^{cbdu}\approx
0.26V_{ud}~{}\text{TeV}^{-2}~{},\quad
a_{S_{RR}}^{cbsu}-a_{S_{RL}}^{cbsu}\approx
0.31V_{us}~{}\text{TeV}^{-2}~{}.\end{split}$ (15)

The preferred size of the effective operators suggests ultraviolet completion
not far above the TeV scale. Furthermore, we observe that in both cases the
fits are compatible with a CKM-like flavor structure, with the operators
involving the strange quark being Cabibbo-suppressed with respect to those
with the down quark, as shown with the gray dotted lines in Fig. 4. This is a
desirable trait from the flavor model building perspective.

#### Standard model effective field theory

The EFT coefficients in Fig. 4 are reported at scale $\mu_{R}=m_{b}$. To
establish connections with possible UV completions, these results have to be
appropriately extrapolated to high energies. The low-energy EFT coefficients
are evolved up to the EW scale and then matched at tree-level to the SMEFT.
These are finally evolved to the UV scale (see App. A for details). In the
SMEFT, the theory is supplement with a series of gauge-invariant irrelevant
operators of increasing canonical dimension. Among all possible dimension-six
SMEFT coefficients, we focus on the dimension-six four-fermion operators that
either contribute directly at tree-level to $b\to c\bar{u}d_{i}$ or strongly
mix with such operators. In Table 3, we list all these operators. Other tree-
level effects in the SMEFT, such as $W$-vertex corrections, are better
constrained elsewhere, and can not give sizable effect to $\bar{B}_{q}\to
D_{q}^{(*)+}\\{\pi,K\\}$ decays.

$[\mathcal{O}_{qq}^{(1)}]_{ijkl}=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{q}_{L}^{k}\gamma_{\mu}q_{L}^{l})$ | $[\mathcal{O}_{qq}^{(3)}]_{ijkl}=(\bar{q}_{L}^{i}\sigma^{a}\gamma_{\mu}q_{L}^{j})(\bar{q}_{L}^{k}\sigma^{a}\gamma_{\mu}q_{L}^{l})$
---|---
$[\mathcal{O}_{ud}^{(1)}]_{ijkl}=(\bar{u}_{R}^{i}\gamma_{\mu}u_{R}^{j})(\bar{d}_{R}^{k}\gamma_{\mu}d_{R}^{l})$ | $[\mathcal{O}_{ud}^{(8)}]_{ijkl}=(\bar{u}_{R}^{i}T^{A}\gamma_{\mu}u_{R}^{j})(\bar{d}_{R}^{k}T^{A}\gamma_{\mu}d_{R}^{l})$
$[\mathcal{O}_{qd}^{(1)}]_{ijkl}=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{d}_{R}^{k}\gamma_{\mu}d_{R}^{l})$ | $[\mathcal{O}_{qd}^{(8)}]_{ijkl}=(\bar{q}_{L}^{i}T^{A}\gamma_{\mu}q_{L}^{j})(\bar{d}_{R}^{k}T^{A}\gamma_{\mu}d_{R}^{l})$
$[\mathcal{O}_{qu}^{(1)}]_{ijkl}=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{u}_{R}^{k}\gamma_{\mu}u_{R}^{l})$ | $[\mathcal{O}_{qu}^{(8)}]_{ijkl}=(\bar{q}_{L}^{i}T^{A}\gamma_{\mu}q_{L}^{j})(\bar{u}_{R}^{k}T^{A}\gamma_{\mu}u_{R}^{l})$
$[\mathcal{O}_{quqd}^{(1)}]_{ijkl}=(\bar{q}_{L}^{i}u_{R}^{j})(i\sigma^{2})(\bar{q}_{L}^{k}d_{R}^{l})$ | $[\mathcal{O}_{quqd}^{(8)}]_{ijkl}=(\bar{q}_{L}^{i}T^{A}u_{R}^{j})(i\sigma^{2})(\bar{q}_{L}^{k}T^{A}d_{R}^{l})$
Table 3: SMEFT operators relevant for $b\to c\bar{u}d_{i}$ transitions.

##  4 Simplified models

The SMEFT operators identified in the previous section can be generated
already at tree-level by integrating out a new bosonic field $X$ coupled to
quark currents. Here we list the complete set of new scalar and vector
mediators which generate the relevant operators shown in Fig. 4 at tree-level
with renormalisable interactions deBlas:2017xtg , without also necessarily
inducing dangerous $\Delta F=2$ transitions at tree-level

$\begin{split}\\!\\!\\!\\!\text{spin-0:}&\quad\left\\{\begin{array}[]{lll}\Phi_{1}=({\bf
1},{\bf 2},1/2),{}{}&\Phi_{8}=({\bf 8},{\bf 2},1/2),{}{}&\\\
\Phi_{3}=({\bf\bar{3}},{\bf 1},1/3),{}{}&\Psi_{3}=({\bf\bar{3}},{\bf
3},1/3),{}{}&\Phi_{6}=({\bf 6},{\bf 1},1/3),\end{array}\right.\\\
\\!\\!\\!\\!\text{spin-1:}&\quad\left\\{\mathcal{Q}_{3}=({\bf 3},{\bf
2},1/6),\quad\mathcal{Q}_{6}=({\bf\bar{6}},{\bf 2},1/6)\right.~{}.\end{split}$
(16)

Here, the SM gauge representations are reported in the format
$(SU(3)_{c},SU(2)_{L},U(1)_{Y})$. Among other mediators that generate at tree-
level the effective operators listed in Eq. (LABEL:eq:SMEFT_basis), colored
vectors $({\bf 3},{\bf 2},-5/6)$ and $({\bf\bar{6}},{\bf 2},-5/6)$ are not
viable since the coefficients $a_{S_{LR}}^{cbiu}$ do not fit the anomaly. On
the other hand, the vector triplet $W^{\prime}=({\bf 1},{\bf 3},0)$, vectors
$({\bf 8},{\bf 1},0)$ and $({\bf 8},{\bf 3},0)$, and the scalar $({\bf 6},{\bf
3},1/3)$ mediate a neutral meson mixing at tree-level even with the minimal
set of couplings required to fit the anomaly. Hence, we do not consider them
further given the stringent constraints on $\Delta F=2$ transitions. We refer
to Ref. Iguro:2020ndk  for a more detailed discussion of the $W^{\prime}$
case.

The main goal of this section is to show how the high-$p_{T}$ searches at the
LHC, specifically those from dijet signatures, can test the solutions of the
anomaly for all viable mediators. We separate the discussion into two
subsections based on the pair production dijet resonance searches from Sect.
2.1. In particular, in Sect. 4.1 we focus on colored resonances which receive
important constraints from the pair production at the LHC, while the colorless
doublet $\Phi_{1}$ is studied in isolation in Sect. 4.2. The colorless
mediator can in principle be much lighter since the relevant bound comes only
from the LEP-II collider.

The single dijet resonance searches derived in Section 2.2 can be used in both
cases. Non-leptonic meson decays depend on the product of two couplings when
the resonance is integrated out at tree level. In particular, the product of
the couplings entering those decays satisfies

$|x_{q^{i}q^{j}}\,x^{*}_{q^{k}q^{l}}|=|x_{q^{i}q^{j}}|\times|x_{q^{k}q^{l}}|~{},$
(17)

where both terms on the right-hand side are simultaneously constrained from
non-observation of $\sigma(pp\to X\to jj)$ at high-$p_{T}$. Using this
inequality, we can limit NP contributions in $\bar{B}_{q}\to
D_{q}^{(*)+}\,\\{\pi,K\\}$ decays.444We only consider scenarios in which the
anomaly is attributed to a single new mediator. One could explore a
possibility of several mediators, each passing the dijet bounds, while adding
up in $\bar{B}_{q}\to D_{q}^{(*)+}\,\\{\pi,K\\}$.

###  4.1 Colored mediators

Figure 5:  High-$p_{T}$ constraints from the single dijet resonance production
(gray) and the QCD-induced pair production of dijet resonances (brown)
compared with the best-fit region from non-leptonic $B$ decays. The
constraints are imposed on the product of the two relevant couplings as a
function of the mass for colored mediators listed in Eq. (16). Note that the
constraints from Sect. 2.2 are strictly applicable for
$\Gamma_{X}/m_{X}\lesssim 10\%$ which is not necessarily the case in the upper
parts of the plot, depending on the relative sizes of the couplings. The
anomaly in $\bar{B}_{q}\to D_{q}^{+(*)}P^{-}$ selects the best-fit region at
68% CL (green) and 95% CL (yellow). Shown with the red dashed lines in the
top-left plot ($\Phi_{6}$) are the limits from the meson mixing for two
representative choices of $y_{13}^{R}$ coupling. For more details see Sect.
4.1.

As discussed in Sect. 2.1, the QCD-induced pair production at the LHC sets
robust lower limits on the masses of the colored mediators in the range 0.5
TeV to 1.15 TeV, depending on the representation. Note that, complementary to
the pair production, the single mechanism is effective for heavy resonances.
The combination of single and pair production dismisses all these mediators as
the explanation of the anomaly, see Fig. 5.

In the following we show the interplay between the dijet bounds and the fit to
the anomaly for each mediator, leaving the details on the models and their EFT
matching to App. B.

#### Color-sextet diquark $\Phi_{6}$

The SM is extended with the singlet sextet diquark scalar $\Phi_{6}=({\bf
6},{\bf 1},1/3)$. The relevant interaction Lagragian is:

$\mathcal{L}_{\Phi_{6}}\supset
y^{L}_{ij}{\bf\Phi}_{6}^{\alpha\beta\dagger}\bar{q^{c}}^{(\alpha|}_{Li}(i\sigma_{2})q^{|\beta)}_{Lj}+y^{R}_{ij}{\bf\Phi}_{6}^{\alpha\beta\dagger}\bar{u^{c}}^{(\alpha|}_{Ri}d^{|\beta)}_{Rj}+{\rm
h.c.}~{},$ (18)

where
$\psi_{i}^{(\alpha|}\psi_{j}^{|\beta)}=\frac{1}{2}(\psi_{i}^{\alpha}\psi_{j}^{\beta}+\psi_{i}^{\beta}\psi_{j}^{\alpha})$,
and $y^{L}$ is an antisymmetric matrix. The components of the sextet
representations are given as ${\bf\Phi}_{6}^{\alpha\beta}\equiv
S^{i}_{\alpha\beta}\Phi_{6}^{i}$, where $i=1,\ldots,6$ and the symmetric color
matrices $S^{i}_{\alpha\beta}$ are given in Eq. (47). The anomaly can be
addressed by switching on only two couplings:

$y^{L}=\left(\begin{array}[]{ccc}0&y^{L}_{12}&0\\\ -y^{L}_{12}&0&0\\\
0&0&0\end{array}\right)~{},\qquad
y^{R}=\left(\begin{array}[]{ccc}0&0&y^{R}_{13}\\\ 0&0&0\\\
0&0&0\end{array}\right)~{}.$ (19)

It is worth noticing that the structure of the left-handed couplings $y^{L}$
of Eq. (19) is compatible with the approximate $U(2)_{q}$ symmetry of the SM
Lagrangian, where the first two families transform as a doublet while the
third as a singlet Barbieri:2011ci . Indeed, since the antisymmetric
combination of the $q_{L}^{i=1,2}$ doublets transforms as a singlet, the
$U(2)_{q}$ symmetry would predict $y^{L}_{12}\sim\mathcal{O}(1)$ while
$y^{L}_{13},y^{L}_{23}\ll 1$. Regarding the right-handed couplings, the
$U(2)_{u}\times U(2)_{d}$ symmetry would predict $y^{R}_{33}\sim 1$ while all
other terms should be suppressed. By introducing a spurion $V_{u}=(a_{u},0)$,
with $a_{u}\ll 1$, transforming as a doublet of $U(2)_{u}$ it is possible to
generate a small value of $y^{R}_{13}\sim a_{u}\ll 1$. This spurion is not
required by the minimal breaking of the symmetry necessary to generate the SM
Yukawas. We thus conclude that this setup could be compatible with a non-
minimally broken $U(2)^{5}$ flavor symmetry if $y^{L}_{12}\sim 1$,
$y^{R}_{33}\sim 1$ and $y^{R}_{13}\ll 1$.

In the following, we considering the minimal set of couplings introduced in
Eq. (19). The non-vanishing $a_{X}$ coefficients for $\bar{B}\to
D_{q}^{(*)}P^{-}$ decays are

$a_{S_{RR}}^{cbdu}\approx\frac{2}{3}\kappa_{\rm
RGE}^{S}V_{cs}\frac{y^{L*}_{12}y^{R}_{13}}{M_{\Phi_{6}}^{2}}\approx\frac{0.26V_{cs}}{~{}\text{TeV}^{2}}~{},\qquad
a_{S_{RR}}^{cbsu}\approx-\frac{2}{3}\kappa_{\rm
RGE}^{S}V_{cd}\frac{y^{L*}_{12}y^{R}_{13}}{M_{\Phi_{6}}^{2}}\approx\frac{-0.31V_{cd}}{~{}\text{TeV}^{2}}~{},$
(20)

where $\kappa_{\rm RGE}^{S}\approx 1.65\,(1.85)$ for
$M_{\Phi_{6}}=1\,(5)~{}\text{TeV}$. The $1\sigma$ and $2\sigma$ regions from
the anomaly fit in the plane of the product of the two couplings and the
mediator mass is shown as a green and yellow band in the top-left panel of
Fig. 5, respectively. This state contributes to precisely measured flavor-
violation processes at one-loop level. We study the relevant constraints in
Appendix C, which place an upper limit on $y^{L}_{12}$ as function of the
mass. The stronger bound comes from $D^{0}$ and $K^{0}$ mixing, and the limit
is shown in Fig. 5 (top-left) as dashed and dotted red lines, for two
different assumptions on the right-handed coupling $y^{R}_{13}$. As shown in
the plot, these complementary constraints from low-energy measurements are not
able to probe the interesting parameter space.

The dijet resonance limits become less effective when the resonance is
broader,

$\frac{\Gamma_{\Phi_{6}}}{M_{\Phi_{6}}}=\frac{8|y^{L}_{12}|^{2}+|y^{R}_{13}|^{2}}{16\pi}\gtrsim
0.1~{},$ (21)

but the perturbativity of the model comes into question. Fixing the best-fit
value for the product of the two couplings, the condition above is always
violated for $M_{\Phi_{6}}\gtrsim 2$ TeV. The strongest dijet constraints on
the two couplings $y^{L}_{12}$ and $y^{R}_{13}$ entering Eq. (20), arise from
the processes $us\to\Phi_{6}$ and $ub\to\Phi_{6}$, respectively. The product
of the two couplings, that contributes directly to the anomaly, is bounded
since $|y^{L}_{12}y^{R}_{13}|<|y^{L}_{12}|^{\rm max}|y^{R}_{13}|^{\rm max}$.
Compared to the generic Lagrangian in Eq. (46) the couplings are given by
$x_{ij}=2y^{L}_{ij}$ for the left-handed quarks and $x_{ij}=y^{R}_{ij}$ for
the right-handed ones. As shown in Fig. 5 (top-left), the region preferred by
the anomaly is excluded by the dijet searches for all masses where the theory
is perturbative.

### Color-triplet diquark $\Phi_{3}$

The scalar triplet $\Phi_{3}=({\bf\bar{3}},{\bf 1},1/3)$ couples to the SM
quarks as

$\mathcal{L}_{\Phi_{3}}\supset
y^{qq}_{ij}\epsilon_{\alpha\beta\gamma}\Phi_{3}^{\alpha}\bar{q}^{\beta}_{Li}(i\sigma_{2})q^{c\,\gamma}_{Lj}+y^{du}_{ij}\epsilon_{\alpha\beta\gamma}\Phi_{3}^{\alpha}\bar{d}^{\beta}_{Ri}u^{c\,\gamma}_{Rj}+{\rm
h.c.}~{},$ (22)

where $y^{qq}_{ij}$ is a symmetric matrix. Baryon number conservation must be
imposed to suppress the couplings to quark and leptons, that would otherwise
mediate proton decay. The coupling structure that allows fitting the anomaly
with least suppression demands three non-vanishing couplings $y^{du*}_{31}$,
$y^{qq}_{12}$, and $y^{qq}_{22}$, such that

$a_{S_{RR}}^{cbdu}=-2.6\frac{y^{qq}_{12}y^{du*}_{31}}{M_{\Phi_{3}}^{2}}\approx\frac{0.26V_{ud}}{~{}\text{TeV}^{2}},\quad
a_{S_{RR}}^{cbsu}=\frac{(-2.6y^{qq}_{22}+0.60y^{qq}_{12})y^{du*}_{31}}{M_{\Phi_{3}}^{2}}\approx\frac{0.31V_{us}}{~{}\text{TeV}^{2}}.$
(23)

A good fit requires the relation $y^{qq}_{22}\approx 0.50y^{qq}_{12}$.

The partonic processes that give the strongest constraints on the couplings
relevant to this model are the same as in the scalar sextet case, as well as
the the relations between the $y^{qq/du}_{ij}$ and $x_{ij}$ couplings. As
shown in Fig. 5 (top-right), the dijet searches firmly exclude the parameter
space relevant for the anomaly in all the perturbative range of the model.

Potentially strong limits from loop-induced flavor-violating processes might
require a particular coupling structure. We do not discuss them further since
the dijet searches are already quite restrictive. The case of the scalar
$\Psi_{3}=({\bf\bar{3}},{\bf 3},1/3)$ is discussed in the appendix and shares
analogous features as the scenario where $\Phi_{3}$ only couples to LH quarks.
This scenario is not so advantageous for the anomaly since it involves a
sizable coupling to the top quark, which implies stronger collider
constraints. For this reason, we do not consider it separately, referring to
App. B for more details.

### Color-octet scalar $\Phi_{8}=(8,2,1/2)$

The scalar octet $\Phi_{8}$ couples to quarks with the Lagrangian

$\mathcal{L}_{\Phi_{8}}\supset
y^{qu}_{ij}\Phi_{8}^{\alpha\dagger}i\sigma_{2}\bar{q}^{T}_{Li}T^{\alpha}u_{Rj}+y^{dq}_{ij}\Phi_{8}^{\alpha\dagger}\bar{d}_{Ri}T^{\alpha}q_{Lj}+{\rm
h.c.}~{}.$ (24)

In order to fit the anomaly with minimal CKM suppression and least possible
effect in dijet searches we consider the following non-vanishing couplings:
$y^{dq}_{31}$, $y^{dq}_{32}$, and $y^{qu}_{21}$. The low-energy coefficients
induced by these couplings are:

$a_{S_{RR}}^{cbiu}\approx
0.44V_{cs}\frac{y^{dq*}_{3i}y^{qu}_{21}}{M_{\Phi_{8}}^{2}}~{},$ (25)

where $i=1,2$. We take $y^{dq*}_{32}=V_{us}/V_{ud}~{}y^{dq*}_{31}$ in order to
be consistent with the relative effect observed in $K$ and $\pi$ channels. In
the case of the color-octet representation, the limits from QCD pair
production are particularly strong, forbidding masses below $\sim 1$ TeV.
Furthermore, the relatively small numerical factor in the low-energy
coefficients $a_{S_{RR}}^{cbiu}$ requires larger couplings to fit the anomaly
compared to the previous models. These facts, combined, exclude a successful
explanation of the observed deviation with this setup, as shown in Fig. 5
(bottom-left).

#### Vectors $\mathcal{Q}_{3}$ and $\mathcal{Q}_{6}$

The triplet and sextet vectors $\mathcal{Q}_{3}=({\bf 3},{\bf 2},1/6)$ and
$\mathcal{Q}_{6}=(\bar{\bf 6},{\bf 2},1/6)$ interact with SM quarks as

$\mathcal{L}_{\mathcal{Q}}\supset
g^{\mathcal{Q}_{3}}_{ij}\mathcal{Q}_{3}^{\alpha\mu\dagger}\epsilon_{\alpha\beta\gamma}\bar{d}^{\beta}_{Ri}\gamma_{\mu}(i\sigma_{2})q^{c\gamma}_{Lj}+\frac{1}{2}g^{\mathcal{Q}_{6}}_{ij}\mathcal{Q}_{6}^{\alpha\beta\mu\dagger}\bar{d}^{(\alpha|}_{Ri}\gamma_{\mu}(i\sigma_{2})q^{c|\beta)}_{Lj}+{\rm
h.c.}~{}.$ (26)

The tree-level contribution to the low-energy EFT coefficients relevant for
the anomalies is given by

$a_{S_{RL}}^{cbiu}=\frac{4}{3}\kappa_{\rm
RGE}^{S}V^{*}_{ui}V_{cj}\left(\frac{g^{\mathcal{Q}_{3}*}_{3i}g^{\mathcal{Q}_{3}}_{ij}}{M_{\mathcal{Q}_{3}}^{2}}-\frac{g^{\mathcal{Q}_{6}*}_{3i}g^{\mathcal{Q}_{6}}_{ij}}{M_{\mathcal{Q}_{6}}^{2}}\right)~{},$
(27)

where $\kappa_{\rm RGE}^{S}\approx 2.23$ for a scale of 1 TeV. The two states
give the same contribution, up to a sign change for one coupling. The
combination which has a weaker CKM suppression is obtained with these three
couplings only: $g^{\mathcal{Q},dq}_{31}$, $g^{\mathcal{Q},dq}_{12}$, and
$g^{\mathcal{Q},dq}_{22}$. In particular,

$a_{S_{RL}}^{cbdu}\approx\frac{3.0V_{cs}V_{ud}^{*}}{M_{\mathcal{Q}}^{2}}g^{\mathcal{Q}*}_{31}g^{\mathcal{Q}}_{12}\approx-\frac{0.26V_{ud}}{~{}\text{TeV}^{2}}~{},\quad
a_{S_{RL}}^{cbsu}\approx\frac{3.0V_{cs}V_{ud}^{*}}{M_{\mathcal{Q}}^{2}}g^{\mathcal{Q}*}_{31}g^{\mathcal{Q}}_{22}\approx-\frac{0.31V_{us}}{~{}\text{TeV}^{2}}~{}.$
(28)

To fit the anomaly, we impose the relation
$g^{\mathcal{Q}}_{22}=g^{\mathcal{Q}}_{12}V_{us}^{*}/V_{ud}^{*}$.

In this scenario the leading partonic processes for dijet production are
$dc\to\mathcal{Q}_{3}^{-1/3*}$, $du\to\mathcal{Q}_{3}^{-1/3*}$ (Cabibbo-
suppressed), and $ds\to\mathcal{Q}_{3}^{2/3*}$, induced by
$g^{\mathcal{Q}}_{12}$, and $bu\to\mathcal{Q}_{3}^{-1/3*}$ and
$bd\to\mathcal{Q}_{3}^{2/3*}$, induced by $g^{\mathcal{Q}}_{31}$. The
strongest limits are from the $ds$ ($du$ in the high-mass region) and $bu$
induced ones. Also, in this case, the dijet constraints exclude a weakly-
coupled solution of the anomaly, see Fig. 5 (bottom-right).

###  4.2 Colorless scalar doublet model

The scalar doublet $\Phi_{1}=({\bf 1},{\bf 2},1/2)$ is among the possible
tree-level mediators capable to fit the anomaly, Eq. (16). It is, however, a
unique mediator in the list since it is not charged under QCD and therefore
not sufficiently constrained by the pair production at the LHC. For generic
Yukawa couplings, its neutral component will mediate $\Delta F=2$ transitions
at tree-level. This can be avoided by a suitable alignment in flavor space.
While such alignment is theoretically rather unappealing, it is still
motivated to consider this option as it opens up a qualitatively different
region of parameter space where the mediator mass is comparable to the
heaviest particles in the SM. In the case of colored mediators discussed
above, $pp\to XX\to(jj)(jj)$ searches imply a mass gap from the SM states, and
to fit the anomaly, this means larger couplings, such that $pp\to X\to jj$
becomes important. For this reason, we study the simplified $\Phi_{1}$ model
in details.

Having the same quantum numbers as the SM Higgs boson, the two states will mix
in general. This would disrupt the precise flavor alignment required to pass
the meson mixing constraints and must be forbidden. For the sake of this
simplified analysis, we just assume that $\Phi_{1}$ is the mass eigenstate
corresponding to the doublet which does not take a vacuum expectation value
and that no mixing is present at tree-level. Regarding its Yukawa couplings,
we consider two different benchmark scenarios, designed ad-hoc to avoid tree-
level contributions to meson mixing:

Benchmark I — The couplings of the extra scalar $\Phi_{1}$ are exclusively to
the right-handed down quarks and are diagonal in the down-quark mass basis,

$\mathcal{L}_{\Phi_{1}}^{\rm
Yuk}=y^{d}_{i}\,\Phi_{1}^{\dagger}\bar{d}^{i}_{R}q^{i}_{L}\,+{\rm h.c.},$ (29)

where $q^{i}_{L}=(V^{*}_{ji}u^{j}_{L},d^{i}_{L})^{T}$. A possible mechanism
behind this alignment is discussed in Egana-Ugrinovic:2018znw ; Egana-
Ugrinovic:2019dqu ; Egana-Ugrinovic:2021uew . Integrating out the scalar
$\Phi_{1}$, the LEFT operators $L_{ud}^{V1(8),LR}$ are generated at low
energies, which contribute to the $a_{S_{RL}}^{ijkl}$ coefficients as

$a_{S_{RL}}^{cbiu}=\kappa_{\rm
RGE}V_{cb}V^{*}_{ui}\frac{y^{d*}_{3}y^{d}_{i}}{M^{2}_{\Phi_{1}}}~{},$ (30)

where $\kappa_{\rm RGE}\approx 2.0$ for $M_{\Phi_{1}}=200\,\text{GeV}$,
derived using DsixTools 2.0 Fuentes-Martin:2020zaz . This structure is
compatible with the fit in the right panel of Fig. 4, where the relation
$y^{d}_{1}=y^{d}_{2}$ allows to simplify the analysis.

Benchmark II — The couplings of $\Phi_{1}$ are aligned to the right-handed
bottom quark and to the right-handed up quark:

$\mathcal{L}_{\Phi_{1}}^{\rm
Yuk}=y^{d}_{3}\,\Phi_{1}^{\dagger}\bar{b}_{R}q^{3}_{L}\,+y^{u}_{1}\,\bar{\tilde{q}}^{1}_{L}u_{R}\,\tilde{\Phi}_{1}+{\rm
h.c.},$ (31)

where $q^{3}_{L}=(V^{*}_{jb}u^{j}_{L},b_{L})^{T}$,
${\tilde{q}}^{1}_{L}=(u_{L},V_{uj}d^{j}_{L})^{T}$, and $y^{u}_{1}$ and
$y^{d}_{3}$ are complex numbers. In this setup the relevant LEFT coefficients
generated at low energies are $L_{uddu}^{S1,RR}$, which contribute as

$a_{S_{RR}}^{cbiu}=\kappa_{\rm
RGE}V_{cb}V_{ui}\frac{y^{d*}_{3}y^{u}_{1}}{M^{2}_{\Phi_{1}}}~{},$ (32)

where $\kappa_{\rm RGE}\approx 2.07$ for $M_{\Phi_{1}}=200$ GeV. Also this
benchmark fits very well the excess in the hadronic $B$ decays.

Figure 6:  Radiative corrections to the SM Higgs Yukawa couplings from the
desired $\Phi_{1}$ interactions.

Let us now briefly discuss the stability of this alignment under radiative
corrections. The couplings of $\Phi_{1}$ with fermions induce at one loop
mixing with the SM Higgs. With the coupling structures outlined above, the
largest contribution in both benchmarks arise from the bottom-quark loop,
giving a mixing angle $\sim
y^{d}_{3}Y_{b}/(16\pi^{2})\sim\mathcal{O}(10^{-4})$. In Benchmark I (II) this
induces a correction to the down (up) quark Yukawa, as in Fig. 6, with an
estimated size of

$\delta
Y_{d(u)}\sim\frac{y^{d}_{1}(y^{u}_{1})y^{d}_{3}Y_{b}}{16\pi^{2}}\frac{m_{H}^{2}}{M_{\Phi_{1}}^{2}}\sim~{}0.3Y_{d}(0.6Y_{u}),$
(33)

where we used the best-fit values from the fits to the anomaly. Due to the
small expected size of this effect and the $\Phi_{1}$ couplings being flavor-
diagonal, no fine tuning against radiative corrections is required to keep
this alignment and to avoid tree-level FCNC. Nonetheless, the odd flavor
alignments still remain to be justified.

#### Collider constraints

Despite different flavor structures, the two benchmarks share a similar
collider phenomenology. We focus on one of them for simplicity. The leading
high-$p_{T}$ constraints on the first benchmark are complied in Fig. 7 for the
entire valid range of $m_{\Phi_{1}}$. The limits from single dijet resonance
searches, compared with the region preferred by the non-leptonic $B$ decays,
firmly exclude masses above $m_{\Phi_{1}}\gtrsim 450$ GeV as the explanation
of the anomaly. These are obtained by applying the general results from Fig. 3
using Eq. (17) for the charge-neutral component of the $SU(2)_{L}$ doublet.
Note that this component exclusively decays to a dijet final state for all
$m_{\Phi_{1}}$, unlike the charged one, which has a significant branching
ratio to a top quark and a jet when $m_{\Phi_{1}}>m_{t}$. The introduction of
large mass splitting between the two $SU(2)_{L}$ components potentially helps
to avoid bounds on the neutral component. However, this is very well
constrained by the electroweak precision tests, see for example
Faroughy:2016osc . While the single dijet searches at the LHC are as effective
as for other mediators, on the contrary, we find the pair-produced dijet
searches do not probe the leftover parameter space of interest. The reason for
this is that the pair production cross section for uncolored particles at the
LHC is not large enough to match the current sensitivity. The lower limit on
the mass comes only from LEP-II as discussed in Section 2.1, in particular,
$m_{\Phi_{1}}\gtrsim 95$ GeV.

Figure 7:  The compilation of the high-$p_{T}$ collider constraints on the
$\Phi_{1}$ model (Benchmark I) together with the best-fit region from non-
leptonic $B$ decays. See Section 4.2 for details.

We are left with the mass range $m_{\Phi_{1}}\in[95-450]$ GeV. To attack this
parameter space, let us consider the CMS search for a light dijet resonance
reported in Ref. Sirunyan:2019vxa . The resonance has a large $p_{T}$ as it
recoils from another jet in the production, i.e. $pp\to Xj$ process. The $X$
decay products are collimated leading to a fat jet topology. The signal model
used in this analysis is a leptophobic $Z^{\prime}$ with flavor universal
vector-like couplings to quarks. The collaboration performed a simulation of
$Z^{\prime}+$jet(s) with the parton-level filter $H_{T}>400$ GeV (for details
see Sirunyan:2019vxa ). The events are selected by asking for at least one AK8
jet (anti-$k_{T}$ with $R=0.8$) with $p_{T}>525$ GeV for $m_{X}<175$ GeV or
CA15 jet (Cambridge-Aachen with $R=1.5$) with $p_{T}>575$ GeV for $m_{X}>175$
GeV. In both cases, the pseudorapidity is $|\eta|<2.5$. The exclusion limits
are reported on the coupling $g^{\prime}_{q}$ as a function of the
$Z^{\prime}$ mass, see Fig. 4 in Sirunyan:2019vxa . To translate these limits
to our case, we perform the MadGraph5_aMC@NLO Alwall:2014hca  partonic level
simulation of $pp\to H^{\prime}j\to(jj)j$ imposing the same $H_{T}$, $\eta$,
and $p_{T}$ cuts, as well as, $\Delta R<0.8\,(1.5)$ for the resonance decay
products. We repeat the same procedure for the $Z^{\prime}$ model used by the
CMS collaboration. The translation is done by comparing the two fiducial cross
sections for several benchmark masses.555This procedure will be followed to
derive bounds for mediators with different spins and representations in the
future study. Also, the partonic-level simulation gives only an estimate of
the difference in the signal acceptance between $H^{\prime}$ and $Z^{\prime}$
models and will be superseded by a full-fledged detector-level simulation.

In addition to this, important constraints come from the top sector. In
particular, searches for $\Phi_{1}\to tb$, when $m_{\Phi_{1}}>m_{t}$, and for
$t\to b\Phi_{1}$, when $m_{\Phi_{1}}<m_{t}$, exclude most of the remaining
parameter space except for a small window around $m_{\Phi_{1}}\approx m_{t}$,
where the two processes are kinematically suppressed (see Fig. 7). The bounds
on $y_{3}^{d}$ from top decays are extracted from the dedicated ATLAS search
Sirunyan:2020aln . The exclusion shown in red color in Fig. 7 is the product
of this bound with the dijet bound from Sirunyan:2019vxa . Similarly, the
exclusion limits shown in blue color are a combination of the ATLAS search in
the $t\bar{t}b$ final state ATLAS:2020jqj , the CDF search for a resonance in
$pp\to tb$ Aaltonen:2009qu ; Bianchi:2011xsw  and the dijet search
Sirunyan:2019vxa .

In conclusion, there is a blind spot around $m_{\Phi_{1}}\approx m_{t}$ which
needs a dedicated collider search to be covered. We used MadGraph5_aMC@NLO
Alwall:2014hca  to calculate processes with off-shell $\Phi_{1}$ and/or top
quark. We identify the following signatures at the LHC which could further
squeeze the interesting parameter space: $t\bar{t}\Phi_{1}$, single top, and
$V\Phi_{1}$ where $V=W,Z$. Further improvements of the light dijet resonance
searches such as Sirunyan:2019vxa  would also be beneficial.

#### Flavour Constraints

Concerning constraints from low-energy observables, we note that in Benchmark
II flavor changing processes from a $b$ to lighter down-type quarks are always
proportional to $y^{u}_{1}$ and are then suppressed by the up-quark mass.
Similar arguments can be applied to charm physics, where only the coupling
$y^{d}_{3}$ enters. However, in this case, the strong suppression comes from
the CKM. Therefore we conclude that Benchmark II is insensitive to flavor
constraints. In the case of Benchmark I, it is not straightforward to draw
analogous conclusions. In Appendix D we investigate a specific parameter-space
point for $m_{\phi_{1}}\approx m_{t}$ and $y^{d}_{i}$, finding that also
Benchmark I cannot be excluded by low-energy flavour constraints. Small
variations around this point do not change our conclusions.

##  5 Conclusions

The size of potential new physics effects in non-leptonic $B$ meson decays is
restricted by complementary processes. A typical example is neutral meson
mixing, which is predicted to accompany non-leptonic decays in many extensions
of the SM. $\Delta F=2$ transitions are usually the leading constraints on
such models, implying unobservable effects in non-leptonic decays. It is,
however, not difficult to find exceptions to this prevalent expectation. In
this work, we provide examples of untuned models owning specific flavor
structure and dynamical content, that are safe on all complementary processes
in flavor physics. Nevertheless, we also find that such cases are severely
constrained by the dijet resonance searches at high-$p_{T}$ colliders.
Intuitively, new mediators have to compete with the tree-level $W$ boson
exchange, and to have an appreciable effect, their masses and couplings turned
out to be perfectly suited for high-$p_{T}$ searches at the LHC. The
connection is even more profound since the minimal set of couplings needed to
generate the desired effect in non-leptonic decays can not be neglected in
high-$p_{T}$ dijet searches, unlike for example in $\Delta F=2$ where the
effect can be loop-induced and flavor-suppressed.

Recognising the importance of dijet searches for non-leptonic decays, we
reinterpret the existing ATLAS and CMS searches as a constraint on a generic
narrow spin-0 and spin-1 resonance, which can be a color singlet, triplet,
sextet, or octet, coupled to a diquark or a quark-antiquark pair with an
arbitrary flavor composition. The pair production of colored resonances $pp\to
XX\to(jj)(jj)$ decaying dominantly to jets sets a robust lower limit on their
mass as explained in Sect. 2.1. The single dijet resonance production $pp\to
X\to jj$ is discussed in Sect. 2.2 and the main results are summarised in
Figs. 2 and 3. Since different flavor channels do not interfere, the absence
of the signal imposes limits on all of them simultaneously. The results
derived in this section go beyond $\bar{B}_{q}\to D_{q}^{(*)+}\,\\{\pi,K\\}$
decays and can be used to limit new physics contributions in other hadronic
decays.

Regarding $\bar{B}_{q}\to D_{q}^{(*)+}\,\\{\pi,K\\}$ decays, we perform a full
EFT analysis for arbitrary new physics at leading-order in $\alpha_{s}$, see
Sect. 3. The main results are shown in Fig. 4. The discrepancy between the SM
theory prediction and measurements observed in Ref. Bordone:2020gao  leads to
the best-fit region favoring new physics. We find this as an excellent
opportunity to test the importance of dijet constraints. Based on the EFT
analysis, we write an exhaustive list of simplified mediator models which can
be matched to the best-fit region at low energies. These are reported in Eq.
(16).

Assuming pragmatically the minimal set of couplings needed to fit the anomaly,
we study the collider limits for every viable tree-level mediator. For colored
mediators, in Sect. 4.1, we show that the combination of pair and single
production of dijet resonances excludes weakly-coupled new physics
explanations of the anomalies, as illustrated in Fig. 5. The pair production
is effective for light resonances, while the single production excludes
heavier resonances. Besides, we also carry out a thorough study of flavor
phenomenology for the color-sextet scalar model to show that other low-energy
bounds such as $\Delta F=2$ can naturally be avoided, see Appendix C. Thus,
the high-$p_{T}$ dijet constraints are crucial to dismiss this model as the
origin of the discrepancy.

Finally, we study the colorless weak-doublet scalar $\Phi_{1}$ in Sect. 4.2
which is not sufficiently constrained by the pair production at the LHC.
However, we show that the consistency of flavor data requires a specific
flavor alignment, predicting sizeable couplings with the top quark. As a
consequence, top-quark physics rules out the interesting parameter space for
all hypothetical $m_{\Phi_{1}}$ apart from a narrow window around the top
mass, see Fig. 7. This blind point is quite resilient on flavor bounds which
are carefully checked in Appendix D, and will likely be covered by the future
collider studies involving the top quark, as well as by further progress in
dijet resonance searches.

###### Acknowledgements.

We thank Javier Fuentes-Martín for collaboration in the early stages of this
work. We also thank Martin Jung for useful communications. The work of MB is
supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under grant 396021762 - TRR 257 “Particle Physics Phenomenology after the
Higgs Discovery” and by the Italian Ministry of Research (MIUR) under grant
PRIN 20172LNEEZ. DM acknowledges support by the INFN grant SESAMO and MIUR
grant PRIN 2017L5W2PT. The work of AG has received funding from the Swiss
National Science Foundation (SNF) through the Eccellenza Professorial
Fellowship “Flavor Physics at the High Energy Frontier” project number 186866.
This work is also partially supported by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme,
grant agreement 833280 (FLAY).

##  Appendix A Details of the EFT analysis

### Hadronic matrix elements for the NP operators

The set of operators in the LEFT that we consider is:

$\displaystyle[\mathcal{O}_{ud}^{V1,LL}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}u_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}d_{L}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{V8,LL}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}u_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}T^{A}d_{L}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{V1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}\gamma_{\mu}u_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{V8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}\gamma_{\mu}T^{A}u_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{V1,LR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}u_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{V8,LR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}u_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{du}^{V1,LR}]_{ijkl}$
$\displaystyle=(\bar{d}_{L}^{i}\gamma_{\mu}d_{L}^{j})(\bar{u}_{R}^{k}\gamma^{\mu}u_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{du}^{V8,LR}]_{ijkl}$
$\displaystyle=(\bar{d}_{L}^{i}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{u}_{R}^{k}\gamma^{\mu}T^{A}u_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{uddu}^{V1,LR}]_{ijkl}$
$\displaystyle=(\bar{u}^{i}_{L}\gamma_{\mu}d_{L}^{j})(\bar{d}^{k}_{R}\gamma^{\mu}u_{R}^{l})\,\,,$
$\displaystyle[\mathcal{O}_{uddu}^{V8,LR}]_{ijkl}$
$\displaystyle=(\bar{u}^{i}_{L}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{d}^{k}_{R}\gamma^{\mu}T^{A}u_{R}^{l})\,\,,$
$\displaystyle[\mathcal{O}_{ud}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}u_{R}^{j})(\bar{d}_{L}^{k}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}T^{A}u_{R}^{j})(\bar{d}_{L}^{k}T^{A}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{uddu}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}d_{R}^{j})(\bar{d}_{L}^{k}u_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{uddu}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}T^{A}d_{R}^{j})(\bar{d}_{L}^{k}T^{A}u_{R}^{l})$
(34)

In order to evaluate the necessary matrix elements, apply Fierz transformation
to get the following set of operators:

$\displaystyle\mathcal{Q}_{V_{LL}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}T^{A}u_{L}^{l})\,,$
$\displaystyle\mathcal{Q}_{V_{LL}}^{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}d_{L}^{j})(\bar{d}_{L}^{k}\gamma^{\mu}u_{L}^{l})\,,$
$\displaystyle\mathcal{Q}_{V_{RR}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}\gamma_{\mu}T^{A}d_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{V_{RR}}^{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}\gamma_{\mu}d_{R}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{V_{LR}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}T^{A}d_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}T^{A}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{V_{LR}}^{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\gamma_{\mu}d_{L}^{j})(\bar{d}_{R}^{k}\gamma^{\mu}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{RL}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}T^{A}d_{R}^{j})(\bar{d}_{R}^{k}T^{A}u_{L}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{RL}}^{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}d_{R}^{j})(\bar{d}_{R}^{k}u_{L}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{LR}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}T^{A}d_{L}^{j})(\bar{d}_{L}^{k}T^{A}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{LR}}^{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}d_{L}^{j})(\bar{d}_{L}^{k}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{RR}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}T^{A}d_{R}^{j})(\bar{d}_{L}^{k}T^{A}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{S_{RR}}^{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}d_{R}^{j})(\bar{d}_{L}^{k}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{T_{RR}}^{\prime ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\sigma_{\mu\nu}T^{A}d_{R}^{j})(\bar{d}_{L}^{k}\sigma^{\mu\nu}T^{A}u_{R}^{l})\,,$
$\displaystyle\mathcal{Q}_{T_{RR}}^{ijkl}$
$\displaystyle=(\bar{u}_{L}^{i}\sigma_{\mu\nu}d_{R}^{j})(\bar{d}_{L}^{k}\sigma^{\mu\nu}u_{R}^{l})\,,$
(35)

The relation between the two sets of operators is given by

$\displaystyle a_{V_{LL}}^{\prime ijkl}$
$\displaystyle=2\,[L_{ud}^{V1,LL}]_{ilkj}-\frac{1}{3}\,[L_{ud}^{V1,LL}]_{ilkj}\,,$
$\displaystyle a_{V_{LL}}^{ijkl}$
$\displaystyle=\frac{1}{3}\,[L_{ud}^{V1,LL}]_{ilkj}+\frac{4}{9}\,[L_{ud}^{V1,LL}]_{ilkj}\,,$
$\displaystyle a_{V_{RR}}^{\prime ijkl}$
$\displaystyle=2\,[L_{ud}^{V1,RR}]_{ilkj}-\frac{1}{3}\,[L_{ud}^{V1,RR}]_{ilkj}\,,$
$\displaystyle a_{V_{RR}}^{ijkl}$
$\displaystyle=\frac{1}{3}\,[L_{ud}^{V1,RR}]_{ilkj}+\frac{4}{9}\,[L_{ud}^{V1,RR}]_{ilkj}\,,$
$\displaystyle a_{V_{LR}}^{\prime ijkl}$
$\displaystyle=[L_{uddu}^{V1,LR}]_{ijkl}\,,$ $\displaystyle a_{V_{LR}}^{ijkl}$
$\displaystyle=[L_{uddu}^{V1,LR}]_{ijkl}\,,$ $\displaystyle a_{S_{RL}}^{\prime
ijkl}$
$\displaystyle=-4\,[L_{ud}^{V1,LR}]_{ilkj}+\frac{2}{3}\,[L_{ud}^{V1,LR}]_{ilkj}\,,$
$\displaystyle a_{S_{RL}}^{ijkl}$
$\displaystyle=-\frac{2}{3}\,[L_{ud}^{V1,LR}]_{ilkj}-\frac{8}{9}\,[L_{ud}^{V1,LR}]_{ilkj}\,,$
$\displaystyle a_{S_{LR}}^{\prime ijkl}$
$\displaystyle=-4\,[L_{du}^{V1,LR}]_{kjil}+\frac{2}{3}\,[L_{du}^{V1,LR}]_{kjil}\,,$
$\displaystyle a_{S_{LR}}^{ijkl}$
$\displaystyle=-\frac{2}{3}\,[L_{du}^{V1,LR}]_{kjil}-\frac{8}{9}\,[L_{du}^{V1,LR}]_{kjil}\,,$
$\displaystyle a_{S_{RR}}^{\prime ijkl}$
$\displaystyle=-[L_{ud}^{S1,RR}]_{ilkj}+\frac{1}{6}\,[L_{ud}^{S8,RR}]_{ilkj}+[L_{uddu}^{S8,RR}]_{ijkl}\,,$
$\displaystyle a_{S_{RR}}^{ijkl}$
$\displaystyle=-\frac{1}{6}\,[L_{ud}^{S1,RR}]_{ilkj}-\frac{2}{9}\,[L_{ud}^{S8,RR}]_{ilkj}+[L_{uddu}^{S1,RR}]_{ijkl}\,,$
$\displaystyle a_{T_{RR}}^{\prime ijkl}$
$\displaystyle=-\frac{1}{4}\,[L_{ud}^{S1,RR}]_{ilkj}+\frac{1}{24}\,[L_{ud}^{S8,RR}]_{ilkj}\,,$
$\displaystyle a_{T_{RR}}^{ijkl}$
$\displaystyle=-\frac{1}{24}\,[L_{ud}^{S1,RR}]_{ilkj}-\frac{1}{18}\,[L_{ud}^{S8,RR}]_{ilkj}\,$
(36)

We can now evaluate the hadronic matrix elements for the operators
$\mathcal{Q}_{i}$ at leading power in $1/m_{b}$ and leading order in
$\alpha_{s}$. We note that, due to color algebra, the matrix elements
$\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{i}^{\prime}\ket{\bar{B}_{q}}=0$ at leading
order in $\alpha_{s}$. We further notice that
$\bra{P^{-}D_{q}^{+}}\mathcal{Q}^{(\prime)}_{T_{RR}}\ket{\bar{B}_{q}}=0$ at
any order.
For a pseudoscalar $D_{q}^{+}$ in the final state we obtain:

$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{V_{LL}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D^{*}_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{V_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{V_{LR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{V_{LR}}^{uibc}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{S_{RL}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}-m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{S_{LR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}-m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{S_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}-m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{S_{RR}}^{uibc}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}(m_{B_{q}}^{2}-m_{D_{q}}^{2})F_{0}^{\bar{B}_{q}\to
D_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}-m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{T_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=\,0\,,$
$\displaystyle\bra{P^{-}D_{q}^{+}}\mathcal{Q}_{T_{RR}}\ket{\bar{B}_{q}}$
$\displaystyle=\,0\,,$ (37)

where $f_{P}$ is the decay constant of the $P^{-}$ meson and $F_{0}$ is the
scalar form factor for $\bar{B}_{q}\to D_{q}^{+}$ decay as in Ref.
Bordone:2019guc . For a vector $D_{q}^{*+}$ in the final state we obtain:

$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{V_{LL}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{V_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{V_{LR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{V_{LR}}^{uibc}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{S_{RL}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}+m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{S_{LR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=-\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}+m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{S_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}+m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{S_{RR}}^{uibc}\ket{\bar{B}_{q}}$
$\displaystyle=+\frac{i}{4}f_{P}\sqrt{\lambda_{P}}\,A_{0}^{\bar{B}_{q}\to
D^{*}_{q}}\frac{m_{P}^{2}}{(m_{u}+m_{d_{i}})(m_{b}+m_{c})}\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{T_{RR}}^{cbiu}\ket{\bar{B}_{q}}$
$\displaystyle=\,0\,,$
$\displaystyle\bra{P^{-}D_{q}^{*+}}\mathcal{Q}_{T_{RR}}^{uibc}\ket{\bar{B}_{q}}$
$\displaystyle=\,0\,,$ (38)

where $A_{0}$ is the scalar form factor for $\bar{B}_{q}\to D_{q}^{+*}$ decay
as in Ref. Bordone:2019guc , and
$\lambda_{P}=m_{B_{q}}^{4}+m_{D^{*}_{q}}^{4}+m_{P}^{4}-2m_{P}^{2}(m_{B_{q}}^{2}+m_{D^{*}_{q}}^{2})-2m_{B_{q}}^{2}m_{D^{*}_{q}}^{2}$.

### LEFT RGE

Here we give the one-loop renormalization group equations for the LEFT
coefficients. We run from the bottom mass scale up to the EW scale with 5
quark flavors. The solutions to the one-loop RGE is
$\vec{L}(m_{b})=U(m_{b},m_{Z})\vec{L}(m_{Z})$, where the vector of EFT
coefficients is

$\vec{L}=\left(\begin{array}[]{c}{[}L_{ud}^{V1,LL}{]}_{21i3}\\\
{[}L_{ud}^{V8,LL}{]}_{21i3}\\\ {[}L_{ud}^{V1,RR}{]}_{21i3}\\\
{[}L_{ud}^{V8,RR}{]}_{21i3}\\\ {[}L_{uddu}^{V1,LR}{]}_{23i1}\\\
{[}L_{uddu}^{V8,LR}{]}_{23i1}\\\ {[}L_{uddu}^{V1,LR}{]}_{1i32}\\\
{[}L_{uddu}^{V8,LR}{]}_{1i32}\\\ {[}L_{ud}^{V1,LR}{]}_{21i3}\\\
{[}L_{ud}^{V8,LR}{]}_{21i3}\\\ {[}L_{du}^{V1,LR}{]}_{i321}\\\
{[}L_{du}^{V8,LR}{]}_{i321}\\\ {[}L_{uddu}^{S1,RR}{]}_{23i1}\\\
{[}L_{uddu}^{S8,RR}{]}_{23i1}\\\ {[}L_{ud}^{S1,RR}{]}_{21i3}\\\
{[}L_{ud}^{S8,RR}{]}_{21i3}\\\ {[}L_{uddu}^{S1,RR}{]}_{1i32}\\\
{[}L_{uddu}^{S8,RR}{]}_{1i32}\\\ {[}L_{ud}^{S1,RR}{]}_{123i}\\\
{[}L_{ud}^{S8,RR}{]}_{123i}\end{array}\right)$ (39)

and, using one-loop anomalous dimensions of LEFT operators Jenkins:2017dyc
implemented in DsixTools 2.0 Celis:2017hod ; Fuentes-Martin:2020zaz  we get

$\begin{split}&U(m_{b},m_{Z})=\\\
&{\tiny\left(\begin{array}[]{cccccccccccccccccccc}1.03&-0.1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
-0.47&1.18&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&1.03&-0.1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&-0.47&1.18&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1.03&0.13&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0.57&1.7&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1.03&0.13&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0.57&1.7&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&1.02&0.13&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0.57&1.68&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&1.02&0.13&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0.57&1.68&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1.82&-0.04&-0.44&-0.17&0&0&0&0.\\\
0&0&0&0&0&0&0&0&0&0&0&0&-0.42&0.82&0.34&-0.13&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&-0.44&-0.17&1.84&-0.04&0&0&0&0.\\\
0&0&0&0&0&0&0&0&0&0&0&0&0.32&-0.13&-0.42&0.84&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1.82&-0.04&-0.44&-0.17\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-0.42&0.82&0.34&-0.13\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&-0.44&-0.17&1.84&-0.04\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0.32&-0.13&-0.42&0.84\\\
\end{array}\right)}\end{split}$ (40)

### SMEFT operators and matching to LEFT

The SMEFT operators relevant to this process are

$\mathcal{L}_{\rm SMEFT}=\sum_{i}C_{i}\mathcal{O}_{i}~{},$ (41)

with

$\displaystyle[\mathcal{O}_{qq}^{(1)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{q}_{L}^{k}\gamma_{\mu}q_{L}^{l})\,,$
$\displaystyle[\mathcal{O}_{qq}^{(3)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}\sigma^{a}\gamma_{\mu}q_{L}^{j})(\bar{q}_{L}^{k}\sigma^{a}\gamma_{\mu}q_{L}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{(1)}]_{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}\gamma_{\mu}u_{R}^{j})(\bar{d}_{R}^{k}\gamma_{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{ud}^{(8)}]_{ijkl}$
$\displaystyle=(\bar{u}_{R}^{i}T^{A}\gamma_{\mu}u_{R}^{j})(\bar{d}_{R}^{k}T^{A}\gamma_{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{qd}^{(1)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{d}_{R}^{k}\gamma_{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{qd}^{(8)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}T^{A}\gamma_{\mu}q_{L}^{j})(\bar{d}_{R}^{k}T^{A}\gamma_{\mu}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{qu}^{(1)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}\gamma_{\mu}q_{L}^{j})(\bar{u}_{R}^{k}\gamma_{\mu}u_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{qu}^{(8)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}T^{A}\gamma_{\mu}q_{L}^{j})(\bar{u}_{R}^{k}T^{A}\gamma_{\mu}u_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{quqd}^{(1)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}u_{R}^{j})(i\sigma^{2})(\bar{q}_{L}^{k}d_{R}^{l})\,,$
$\displaystyle[\mathcal{O}_{quqd}^{(8)}]_{ijkl}$
$\displaystyle=(\bar{q}_{L}^{i}T^{A}u_{R}^{j})(i\sigma^{2})(\bar{q}_{L}^{k}T^{A}d_{R}^{l})\,.$

Tree-level matching between the LEFT and the SMEFT, in the down-quark mass
basis:

$\displaystyle[L_{ud}^{V1,LL}]_{prst}$
$\displaystyle=V_{pi}V^{*}_{rj}\left([C_{qq}^{(1)}]_{ijst}+[C_{qq}^{(1)}]_{stij}-[C_{qq}^{(3)}]_{ijst}-[C_{qq}^{(3)}]_{stij}+\frac{2}{N_{c}}([C_{qq}^{(3)}]_{itsj}+[C_{qq}^{(3)}]_{sjit})\right)$
$\displaystyle[L_{ud}^{V8,LL}]_{prst}$
$\displaystyle=4V_{pi}V^{*}_{rj}\left([C_{qq}^{(3)}]_{itsj}+[C_{qq}^{(3)}]_{sjit}\right)$
$\displaystyle[L_{ud}^{V1,RR}]_{prst}$ $\displaystyle=[C_{ud}^{(1)}]_{prst}$
$\displaystyle[L_{ud}^{V8,RR}]_{prst}$ $\displaystyle=[C_{ud}^{(8)}]_{prst}$
$\displaystyle[L_{ud}^{V1,LR}]_{prst}$
$\displaystyle=4V_{pi}V^{*}_{rj}[C_{qd}^{(1)}]_{ijst}$
$\displaystyle[L_{ud}^{V8,LR}]_{prst}$
$\displaystyle=4V_{pi}V^{*}_{rj}[C_{qd}^{(8)}]_{ijst}$
$\displaystyle[L_{du}^{V1,LR}]_{prst}$ $\displaystyle=[C_{qu}^{(1)}]_{prst}$
$\displaystyle[L_{du}^{V8,LR}]_{prst}$ $\displaystyle=[C_{qu}^{(8)}]_{prst}$
(43) $\displaystyle[L_{uddu}^{V1,LR}]_{prst}$ $\displaystyle=0$
$\displaystyle[L_{uddu}^{V8,LR}]_{prst}$ $\displaystyle=0$
$\displaystyle[L_{ud}^{S1,RR}]_{prst}$
$\displaystyle=V_{pi}[C_{quqd}^{(1)}]_{irst}$
$\displaystyle[L_{ud}^{S8,RR}]_{prst}$
$\displaystyle=V_{pi}[C_{quqd}^{(8)}]_{irst}$
$\displaystyle[L_{uddu}^{S1,RR}]_{prst}$
$\displaystyle=-V_{pi}[C_{quqd}^{(1)}]_{stir}$
$\displaystyle[L_{uddu}^{S8,RR}]_{prst}$
$\displaystyle=-V_{pi}[C_{quqd}^{(8)}]_{stir}$

The SMEFT coefficients must then be RG evolved from the EW scale up to the
scale at which the heavy states are integrated out. To this aim we use
DsixTools Celis:2017hod ; Fuentes-Martin:2020zaz .

As example we show the case of the diquark benchmark model, for which the only
relevant operators are $[\mathcal{O}_{quqd}^{(1)}]$ and
$[\mathcal{O}_{quqd}^{(8)}]$. The four coefficients contributing to the
process from the matching in Eq. (50) and only non-vanishing couplings
$y^{L}_{12}$ $y^{R}_{13}$ are

$\vec{C}=\left([C_{quqd}^{(1)}]_{1123},[C_{quqd}^{(8)}]_{1123},[C_{quqd}^{(1)}]_{2113},[C_{quqd}^{(8)}]_{2113}\right)^{t}~{}.$
(44)

They evolve from $m_{Z}$ up to $m_{\Phi_{6}}$ as
$\vec{C}(m_{Z})=\mathcal{U}(m_{Z},m_{\Phi_{6}})\vec{C}(\Phi_{6})$, with

${\tiny\mathcal{U}(m_{Z},M_{\Phi_{6}})=\left(\begin{array}[]{cccc}1.39&-0.03&0.18&0.08\\\
-0.2&0.92&-0.18&0.08\\\ 0.18&0.08&1.39&-0.03\\\ -0.18&0.08&-0.2&0.92\\\
\end{array}\right)_{1~{}\text{TeV}}~{},\quad\left(\begin{array}[]{cccc}1.67&-0.04&0.33&0.12\\\
-0.35&0.88&-0.33&0.12\\\ 0.33&0.12&1.67&-0.04\\\ -0.33&0.12&-0.35&0.88\\\
\end{array}\right)_{5~{}\text{TeV}}~{}}$ (45)

where we show the evolution for two values of the diquark mass. In the case of
the scalar doublet more operators are generated and the RG evolution has been
performed using DsixTools.

##  Appendix B Details on the tree-level mediators

### Conventions for dijet limits

Let us define here the couplings $x_{ij}$ of a bosonic resonance $X$ with
quark bilinears used in the dijet analysis. For a given representation of the
resonance (spin, SU(3)c) and a coupling to $q_{i}q^{\prime}_{j}$, with
$q^{(\prime)}=u,d$ of arbitrary flavors $i$, $j$, we define the interaction
Lagrangians as666Flavor matrices $x_{ij}$ are arbitrary complex matrices
unless $q=q^{\prime}$ when $x_{ij}$ is symmetric for scalar sextet, anti-
symmetric for scalar triplet, and Hermitian for real vector singlet and octet
where $+\text{h.c.}$ is removed from the lagrangian.

$\begin{split}(0,{\bf 1}):\qquad&\mathcal{L}\supset
x_{ij}\,X\,\bar{q}_{i}P_{X}q^{\prime}_{j}\;+\text{h.c.}~{},\\\ (0,{\bf
3}):\qquad&\mathcal{L}\supset
x_{ij}\,X^{\alpha}\,\epsilon_{\alpha\beta\gamma}\,\bar{q^{c}}^{\beta}_{i}P_{X}q^{\prime\gamma}_{j}+\text{h.c.}~{},\\\
(0,{\bf 6}):\qquad&\mathcal{L}\supset
x_{ij}\,X^{m}\,S^{m}_{\alpha\beta}\,\bar{q^{c}}^{(\alpha|}_{i}P_{X}q^{\prime|\beta)}_{j}+\text{h.c.}~{},\\\
(0,{\bf 8}):\qquad&\mathcal{L}\supset
x_{ij}\,X^{A}\,\bar{q}_{i}T^{A}P_{X}q^{\prime}_{j}\;+\text{h.c.}~{},\\\
(1,{\bf 1}):\qquad&\mathcal{L}\supset
x_{ij}\,X_{\mu}\,\bar{q}_{i}\gamma^{\mu}P_{X}q^{\prime}_{j}\;(+\text{h.c.})~{},\\\
(1,{\bf 3}):\qquad&\mathcal{L}\supset
x_{ij}\,X_{\mu}^{\alpha}\,\epsilon_{\alpha\beta\gamma}\,\bar{q^{c}}^{\beta}_{i}\gamma^{\mu}P_{X}q^{\prime\gamma}_{j}+\text{h.c.}~{},\\\
(1,{\bf 6}):\qquad&\mathcal{L}\supset
x_{ij}\,X_{\mu}^{m}\,S^{m}_{\alpha\beta}\,\bar{q^{c}}^{(\alpha|}_{i}\gamma^{\mu}P_{X}q^{\prime|\beta)}_{j}+\text{h.c.}~{},\\\
(1,{\bf 8}):\qquad&\mathcal{L}\supset
x_{ij}\,X_{\mu}^{A}\,\bar{q}_{i}T^{A}\gamma^{\mu}P_{X}q^{\prime}_{j}\;(+\text{h.c.})~{},\end{split}$
(46)

where the chirality projector $P_{X}$ can be either $P_{L/R}$ for left/right
spinors. Also, $T^{A}$ are the generators of $SU(3)_{c}$,
$\psi_{i}^{(\alpha|}\psi_{j}^{|\beta)}=\frac{1}{2}(\psi_{i}^{\alpha}\psi_{j}^{\beta}+\psi_{i}^{\beta}\psi_{j}^{\alpha})$,
and $S^{m}_{\alpha\beta}$ are the symmetric color matrices

$\begin{split}&S^{1}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)~{},\qquad
S^{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&0\end{array}\right)~{},\qquad S^{3}=\left(\begin{array}[]{ccc}0&0&0\\\
0&1&0\\\ 0&0&0\end{array}\right)~{},\\\
&S^{4}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)~{},\qquad S^{5}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&1\end{array}\right)~{},\qquad
S^{6}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
1&0&0\end{array}\right)~{},\end{split}$ (47)

which satisfy the matrices satisfy

$\text{Tr}S^{m}\bar{S}_{n}=\delta^{m}_{\,n}~{},\qquad\sum_{m}S^{m}_{\alpha\beta}\bar{S}^{\gamma\delta}_{m}=\frac{1}{2}(\delta_{\alpha}^{\delta}\delta_{\beta}^{\gamma}+\delta_{\alpha}^{\gamma}\delta_{\beta}^{\delta})~{}.$
(48)

where the conjugate matrices are given by
$\bar{S}^{\alpha\beta}_{m}=S^{m}_{\alpha\beta}$.

### Scalar color-sextet $\Phi_{6}=(6,1,1/3)$

Let us start the study of the tree-level mediators listed in Eq. (16) with the
singlet sextet diquark $\Phi_{6}=({\bf 6},{\bf 1},1/3)$. This state has also
been studied in Han:2009ya ; Chen:2018stt . The relevant interaction Lagragian
is:

$\mathcal{L}\supset
y^{L}_{ij}{\bf\Phi}_{6}^{\alpha\beta\dagger}\bar{q}^{c,(\alpha|}_{Li}(i\sigma_{2})q^{|\beta)}_{Lj}+y^{R}_{ij}{\bf\Phi}_{6}^{\alpha\beta\dagger}\bar{u}^{c(\alpha|}_{Ri}d^{|\beta)}_{Rj}+{\rm
h.c.}~{},$ (49)

where $y^{L}$ is an antisymmetric matrix. The components of the sextet
representations are given as ${\bf\Phi}_{6}^{\alpha\beta}\equiv
S^{m}_{\alpha\beta}\Phi_{6}^{m}$, where $m=1,\ldots,6$ and the symmetric color
matrices $S^{m}_{\alpha\beta}$ are given in Eq. (47). The conjugate
representation is given by
${\bf\Phi}_{6}^{\alpha\beta\dagger}=\bar{S}^{\alpha\beta}_{m}\Phi_{6}^{m*}=S_{\alpha\beta}^{m}\Phi_{6}^{m}$.
Matching $\Phi_{6}$ to the SMEFT at tree-level one has deBlas:2017xtg :

$\begin{split}[C_{qq}^{(1)}]_{ijkl}&=[C_{qq}^{(3)}]_{ilkj}=\frac{y^{L*}_{ik}y^{L}_{jl}}{4M_{\Phi_{6}}^{2}}~{},\\\
[C_{ud}^{(1)}]_{ijkl}&=\frac{2}{3}[C_{ud}^{(8)}]_{ijkl}=\frac{y^{R}_{jl}y^{R*}_{ik}}{3M_{\Phi_{6}}^{2}}~{},\\\
[C_{quqd}^{(1)}]_{ijkl}&=\frac{2}{3}[C_{quqd}^{(8)}]_{ijkl}=4\frac{y^{L}_{ki}y^{R*}_{jl}}{3M_{\Phi_{6}}^{2}}~{}.\end{split}$
(50)

In terms of the $a$ EFT coefficients at the $m_{b}$ scale:

$\begin{split}a_{V_{LL}}^{cb\alpha u}&=-\frac{4}{3}\kappa_{\rm
RGE}^{V}\sum_{i\neq\alpha;\,j=1,2}V_{ci}V^{*}_{uj}\frac{(y^{L*}_{\alpha
i}y^{L}_{j3})}{M_{\Phi_{6}}^{2}}~{},\\\ a_{V_{RR}}^{cb\alpha
u}&=\frac{1}{3}\kappa_{\rm
RGE}^{V}\frac{(y^{R*}_{2\alpha}y^{R}_{13})}{M_{\Phi_{6}}^{2}}~{},\\\
a_{S_{RR}}^{cb\alpha u}&=\frac{2}{3}\kappa_{\rm
RGE}^{S}\sum_{i\neq\alpha}V_{ci}\frac{y^{L*}_{\alpha
i}y^{R}_{13}}{M_{\Phi_{6}}^{2}}~{},\\\ a_{S_{RR}}^{u\alpha
bc}&=-\frac{2}{3}\kappa_{\rm
RGE}^{S}\sum_{i=2,3}V_{ui}\frac{y^{L*}_{i3}y^{R}_{2\alpha}}{M_{\Phi_{6}}^{2}}~{},\end{split}$
(51)

where we already imposed that $y^{L}$ is antisymmetric and $\kappa_{\rm
RGE}^{V,S}$ describe the effect of RGE from $m_{\Phi_{6}}$ to $m_{b}$. For
instance $\kappa_{\rm RGE}^{S}\approx 1.65\,(1.85)$ for
$m_{\Phi_{6}}=1\,(5)~{}\text{TeV}$.

### Scalar color-triplet $\Phi_{3}=({\bf\bar{3}},{\bf 1},{\bf 1/3})$

The interaction Lagragian to SM quarks is:777This state could also potentially
couple to quarks and leptons, as a leptoquark deBlas:2017xtg . Allowing for
such couplings, together with the diquark ones, would induce proton decay. To
avoid this we must thus assign baryon number $B(\Phi_{3})=2/3$ and impose at
least $B$ conservation.

$\mathcal{L}_{\Phi_{3}}\supset
y^{qq}_{ij}\epsilon_{\alpha\beta\gamma}\Phi_{3}^{\alpha}\bar{q}^{\beta}_{Li}(i\sigma_{2})q^{c\,\gamma}_{Lj}+y^{du}_{ij}\epsilon_{\alpha\beta\gamma}\Phi_{3}^{\alpha}\bar{d}^{\beta}_{Ri}u^{c\,\gamma}_{Rj}+{\rm
h.c.}~{},$ (52)

where $y^{qq}_{ij}$ is a symmetric matrix. Matching to the $\Phi_{3}$ to the
SMEFT at tree-level one has deBlas:2017xtg :

$\begin{split}[C_{qq}^{(1)}]_{ijkl}&=-[C_{qq}^{(3)}]_{ilkj}=\frac{y^{qq}_{ik}y^{qq*}_{lj}}{2M_{\Phi_{3}}^{2}}~{},\\\
[C_{ud}^{(1)}]_{ijkl}&=-\frac{1}{3}[C_{ud}^{(8)}]_{ijkl}=\frac{y^{du}_{ki}y^{du*}_{lj}}{3M_{\Phi_{3}}^{2}}~{},\\\
[C_{quqd}^{(1)}]_{ijkl}&=-\frac{1}{3}[C_{quqd}^{(8)}]_{ijkl}=4\frac{y^{qq}_{ki}y^{du*}_{lj}}{3M_{\Phi_{3}}^{2}}~{}.\end{split}$
(53)

No $\Delta F=2$ processes are generated at tree-level. In terms of the $a$’s
coefficients, one has (keeping into account the symmetricity of $y^{qq}$):

$\begin{split}a_{V_{LL}}^{cb\alpha u}&=-\frac{4}{3}\kappa_{\rm
RGE}^{V_{LL}}V_{ci}V^{*}_{uj}\frac{y^{qq*}_{j3}y^{qq}_{i\alpha}}{M_{\Phi_{3}}^{2}}~{},\\\
a_{S_{RR}}^{cb\alpha u}&=-\frac{2}{3}\kappa_{\rm
RGE}^{S}V_{ci}\frac{y^{du*}_{31}y^{qq}_{i\alpha}}{M_{\Phi_{3}}^{2}}~{},\\\
a_{S_{LL}}^{cb\alpha u}&=-\frac{2}{3}\kappa_{\rm
RGE}^{S}V_{ui}\frac{y^{du*}_{\alpha
2}y^{qq}_{i3}}{M_{\Phi_{3}}^{2}}~{},\end{split}$ (54)

where $\kappa_{\rm RGE}^{V_{LL}}\approx 1.56$ and $\kappa_{\rm RGE}^{S}\approx
4.02$ for $M_{\Phi_{3}}=1~{}\text{TeV}$. We consider two possible coupling
structures to fit the anomaly.

1) Benchmark $V_{LL}$ \- Setting $y^{du}=0$, we only generate the $a_{V_{LL}}$
coefficients. To minimise the impact of dijet bounds we must avoid a strong
CKM suppression in the low-energy coefficients. This can be achieved with
three non-vanishing couplings:

$a_{V_{LL}}^{cbdu}=-1.98\frac{y^{qq*}_{13}y^{qq}_{12}}{M_{\Phi_{3}}^{2}}\approx\frac{0.23V_{ud}}{~{}\text{TeV}^{2}},\quad
a_{V_{LL}}^{cbsu}=\frac{-1.98y^{qq*}_{13}y^{qq}_{22}+0.46y^{qq*}_{13}y^{qq}_{21}}{M_{\Phi_{3}}^{2}}\approx\frac{0.24V_{us}}{~{}\text{TeV}^{2}}.$
(55)

Note that the coupling to third generation $y^{qq}_{13}$ induces a decay of
$\Phi_{3}$ to tops, which will put a constraint on the model even stronger
than the dijet one.

2) Benchmark $S_{RR}$ \- If the only non-zero RH coupling is $y^{du}_{31}$,
then $a_{S_{LL}}^{cb\alpha u}=0$ and we could get a good fit via the
$a_{S_{RR}}$ coefficients. A strong CKM suppression is avoided with the
couplings:

$a_{S_{RR}}^{cbdu}=-2.6\frac{y^{qq}_{12}y^{du*}_{31}}{M_{\Phi_{3}}^{2}}\approx\frac{0.26V_{ud}}{~{}\text{TeV}^{2}},\quad
a_{S_{RR}}^{cbsu}=\frac{(-2.6y^{qq}_{22}+0.60y^{qq}_{12})y^{du*}_{31}}{M_{\Phi_{3}}^{2}}\approx\frac{0.31V_{us}}{~{}\text{TeV}^{2}}.$
(56)

The best-fit for the anomalies is obtained for $y^{qq}_{22}\approx
0.50y^{qq}_{12}$. In this scenario the coupling to the top quark is suppressed
by $V_{ts}$, strongly reducing the relative branching ratio and thus the
corresponding constraints. It is thus more favorable than the $V_{LL}$
benchmark.

### Scalar color-triplet $\Psi_{3}=({\bf\bar{3}},{\bf 3},{\bf 1/3})$

This scalar couples to quarks as

$\mathcal{L}_{\Psi_{3}}\supset
y^{qq}_{ij}\epsilon_{\alpha\beta\gamma}\Psi_{3}^{A\alpha}\bar{q}^{\beta}_{Li}\sigma^{A}(i\sigma_{2})q^{c\,\gamma}_{Lj}+{\rm
h.c.}~{},$ (57)

where $y^{qq}_{ij}$ is an antisymmetric matrix. The SMEFT coefficients
generated integrating out this state at tree-level are deBlas:2017xtg :

$[C_{qq}^{(1)}]_{ijkl}=3[C_{qq}^{(3)}]_{ijkl}=3\frac{y^{qq}_{ki}y^{qq*}_{lj}}{2M_{\Psi_{3}}^{2}}~{}.$
(58)

No $\Delta F=2$ processes are generated at tree-level, since $\Phi_{3}$
couples up to down quarks only. A potentially good benchmark to fit the
anomaly is with two non-vanishing couplings $y^{qq}_{12}$ and $y^{qq}_{23}$,
giving:

$a_{V_{LL}}^{cbdu}=\frac{4}{3}\kappa_{\rm
RGE}^{V_{LL}}V_{us}^{*}V_{cs}\frac{y^{qq*}_{23}y^{qq}_{12}}{M_{\Psi_{3}}^{2}}~{},\quad
a_{V_{LL}}^{cbsu}=\frac{4}{3}\kappa_{\rm
RGE}^{V_{LL}}V_{us}^{*}\frac{y^{qq*}_{23}(-V_{cd}y^{qq}_{12}+V_{cb}y^{qq}_{23})}{M_{\Psi_{3}}^{2}}~{}.$
(59)

This setup is analogous to the $V_{LL}$ benchmark of $\Phi_{3}$, including a
decay to top quarks induced by $y^{qq}_{23}$. They share a similar
phenomenology, except for the fact that the weak-triplet has a stronger
constraint from pair-production of dijet resonances, see Sect. 2.1. For these
reasons we do not consider this model further.

### Scalar color-singlet $\Phi_{1}=(1,2,1/2)$

This scalar has the same quantum numbers as the SM Higgs. To avoid potentially
very strong constraints while at the same time being able to fit the observed
low-energy anomaly we must avoid that this state mixes too strongly with the
SM Higgs and that it takes a non-zero vacuum expectation value. This can be
achieved by appropriately tuning the scalar potential.

The Yukawa couplings to quarks are

$\mathcal{L}_{\Phi_{1}}^{\rm
Yuk}=y^{qd}_{ij}\,\Phi_{1}^{\dagger}\bar{d}^{i}_{R}q^{j}_{L}\,+y^{qu}_{ij}\,\Phi_{1}^{\dagger}\epsilon\bar{q}^{i}_{L}u_{R}^{j}+{\rm
h.c.},$ (60)

Matching at tree-level to the SMEFT, the Yukawa couplings induce the following
coefficients deBlas:2017xtg :

$\begin{split}[C_{quqd}^{(1)}]_{ijkl}&=-\frac{y^{qu}_{ij}y^{dq*}_{lk}}{M_{\Phi_{1}}^{2}}~{},\\\
[C_{qd}^{(1)}]_{ijkl}&=\frac{1}{6}[C_{qd}^{(8)}]_{ijkl}=-\frac{1}{6}\frac{y^{dq*}_{li}y^{dq}_{kj}}{M_{\Phi_{1}}^{2}}~{},\\\
[C_{qu}^{(1)}]_{ijkl}&=\frac{1}{6}[C_{qu}^{(8)}]_{ijkl}=-\frac{1}{6}\frac{y^{qu*}_{jk}y^{qu}_{il}}{M_{\Phi_{1}}^{2}}~{}.\\\
\end{split}$ (61)

While in general the neutral component can induce meson mixing at tree-level,
this can be avoided by a suitable alignment of the couplings. The two
benchmarks we consider, and the corresponding matching to the low-energy EFT,
are discussed in Section 4.2.

### Scalar color-octet $\Phi_{8}=(8,2,1/2)$

This state has the same electroweak quantum numbers as the SM Higgs, but it is
in the adjoint representation of $SU(3)_{c}$. It contains a neutral component
that can potentially mediate meson mixing at tree-level. The interaction
Lagrangian is

$\mathcal{L}_{\Phi_{8}}\supset
y^{qu}_{ij}\Phi_{8}^{A\dagger}i\sigma_{2}\bar{q}^{T}_{Li}T^{A}u_{Rj}+y^{dq}_{ij}\Phi_{8}^{A\dagger}\bar{d}_{Ri}T^{A}q_{Lj}+{\rm
h.c.}~{},$ (62)

Matching to the SMEFT at tree-level one has deBlas:2017xtg :

$\begin{split}[C_{quqd}^{(8)}]_{ijkl}&=-\frac{y^{dq*}_{lk}y^{qu}_{ij}}{M_{\Phi_{8}}^{2}}~{},\\\
[C_{qd}^{(1)}]_{ijkl}&=-\frac{4}{3}[C_{qd}^{(8)}]_{ijkl}=-\frac{2}{9}\frac{y^{dq*}_{li}y^{dq}_{kj}}{M_{\Phi_{8}}^{2}}~{},\\\
[C_{qu}^{(1)}]_{ijkl}&=-\frac{4}{3}[C_{qu}^{(8)}]_{ijkl}=-\frac{2}{9}\frac{y^{qu*}_{jk}y^{qu}_{il}}{M_{\Phi_{8}}^{2}}~{}.\\\
\end{split}$ (63)

The low-energy EFT coefficients relevant for a successful fit to the anomaly
are given by

$a_{S_{RR}}^{cb\alpha u}=\frac{2}{9}\kappa_{\rm
RGE}^{S}\frac{y^{dq*}_{b\alpha}V_{ci}y^{qu}_{iu}}{M_{\Phi_{8}}^{2}}~{},$ (64)

where $\alpha=1,2$ and $\kappa_{\rm RGE}^{S}=1.98(2.27)$ for a mass of 1 TeV
(5 TeV). To minimise CKM suppression and the effect of the dijet constraints
one can consider the following non-vanishing couplings: $y^{dq}_{31}$,
$y^{dq}_{32}$, and $y^{qu}_{21}$. With this choice there is no contribution to
meson mixing at tree level.

### Vectors $\mathcal{Q}_{3}=(3,2,1/6)$ and $\mathcal{Q}_{6}=(\bar{6},2,1/6)$

The colored vectors $\mathcal{Q}_{3}$ and $\mathcal{Q}_{6}$ couple to SM
quarks as

$\mathcal{L}_{\mathcal{Q}}\supset
g^{\mathcal{Q}_{3}}_{ij}\mathcal{Q}_{3}^{\mu\dagger}\epsilon_{ABC}\bar{d}^{B}_{Ri}\gamma_{\mu}(i\sigma_{2})q^{cC)}_{Lj}+\frac{1}{2}g^{\mathcal{Q}_{6}}_{ij}\mathcal{Q}_{6}^{AB\mu\dagger}\bar{d}^{(A|}_{Ri}\gamma_{\mu}(i\sigma_{2})q^{c|B)}_{Lj}+h.c.$
(65)

Matching to the SMEFT at tree-level one has deBlas:2017xtg , for
$\mathcal{Q}_{3}$

$\begin{split}[C_{qd}^{(1)}]_{ijkl}&=-\frac{1}{3}[C_{qd}^{(8)}]_{ijkl}=\frac{2}{3}\frac{g^{\mathcal{Q}_{3}}_{ki}g^{\mathcal{Q}_{3}*}_{lj}}{M_{\mathcal{Q}_{3}}^{2}}~{},\end{split}$
(66)

and for the $\mathcal{Q}_{6}$

$\begin{split}[C_{qd}^{(1)}]_{ijkl}&=\frac{2}{3}[C_{qd}^{(8)}]_{ijkl}=\frac{2}{3}\frac{g^{\mathcal{Q}_{6}}_{ki}g^{\mathcal{Q}_{6}*}_{lj}}{M_{\mathcal{Q}_{6}}^{2}}~{},\end{split}$
(67)

In terms of the low-energy EFT coefficients we have

$a_{S_{RL}}^{cb\alpha u}=\frac{4}{3}\kappa_{\rm
RGE}^{S}V^{*}_{ui}V_{cj}\left(\frac{g^{\mathcal{Q}_{3}*}_{3i}g^{\mathcal{Q}_{3}}_{\alpha
j}}{M_{\mathcal{Q}_{3}}^{2}}-\frac{g^{\mathcal{Q}_{6}*}_{3i}g^{\mathcal{Q}_{6}}_{\alpha
j}}{M_{\mathcal{Q}_{6}}^{2}}\right)~{},$ (68)

where $\kappa_{\rm RGE}^{S}\approx 2.23$ for a UV mass scale of 1 TeV. The two
states give the same contribution, up to a change of sign for one coupling, so
we can focus on $\mathcal{Q_{3}}$ since it has weaker bounds from QCD pair
production. The combination with less CKM suppression is obtained with these
three couplings only: $g^{\mathcal{Q}_{3},dq}_{31}$,
$g^{\mathcal{Q}_{3},dq}_{12}$, and $g^{\mathcal{Q}_{3},dq}_{22}$:

$a_{S_{RL}}^{cbdu}=\frac{2.8V_{cs}V_{ud}^{*}}{M_{\mathcal{Q}_{3}}^{2}}g^{\mathcal{Q}_{3}*}_{31}g^{\mathcal{Q}_{3}}_{12}\approx-\frac{0.26V_{ud}}{~{}\text{TeV}^{2}}~{},\quad
a_{S_{RL}}^{cbsu}=\frac{2.8V_{cs}V_{ud}^{*}}{M_{\mathcal{Q}_{3}}^{2}}g^{\mathcal{Q}_{3}*}_{31}g^{\mathcal{Q}_{3}}_{22}\approx-\frac{0.31V_{us}}{~{}\text{TeV}^{2}}~{}.$
(69)

To fit the anomaly we take
$g^{\mathcal{Q}_{3}}_{22}=g^{\mathcal{Q}_{3}}_{12}V_{us}^{*}/V_{ud}^{*}$.

##  Appendix C Flavor constraints on the scalar sextet $\Phi_{6}$

Let us now discuss possible constraints from low-energy processes on the
solution to the anomaly obtained in the main text.

### $\Delta F=2$

Figure 8:  Diagrams inducing $\Delta F=2$ processes in the sextet diquark
model. The dashed line represents the $\Phi_{6}$ propagator.

Since the electric charge of $\Phi_{6}$ is $1/3$, no $\Delta F=2$ processes
are generated at tree-level. However, meson mixing can arise at the loop level
via the diagrams shown schematically in Fig. 8. The boxes with two $\Phi_{6}$
propagators are proportional to the structure $(y^{L}y^{L\dagger})$ or
$(y^{R}y^{R\dagger})$. Given the effective Hamiltonian for down-quark $\Delta
F=2$ processes,

$\begin{split}\mathcal{H}_{\Delta
F=2}\supset&C_{VLL}^{q_{i}q_{j}}\left(\bar{q}^{i}_{L}\gamma_{\mu}q^{j}_{L}\right)^{2}+C_{VRR}^{q_{i}q_{j}}\left(\bar{q}^{i}_{R}\gamma_{\mu}q^{j}_{R}\right)^{2}+\\\
&+C_{LR1}^{q_{i}q_{j}}\left(\bar{q}^{i}_{L}\gamma_{\mu}q^{j}_{L}\right)\left(\bar{q}^{i}_{R}\gamma^{\mu}q^{j}_{R}\right)+C_{LR2}^{q_{i}q_{j}}\left(\bar{q}^{i}_{R}q^{j}_{L}\right)\left(\bar{q}^{i}_{L}q^{j}_{R}\right),\end{split}$
(70)

where $q=u,d$ and color is always contracted within the same current.
Integrating out the diquark at one-loop level at the UV mass scale one has
Chen:2018stt :

$\begin{split}C_{VLL}^{d_{i}d_{j}}&\approx\frac{3(y_{L}^{\dagger}y_{L})_{ij}^{2}}{16\pi^{2}M_{\Phi_{6}}^{2}}-\frac{(Vy_{L}^{*})_{3i}(V^{*}y_{L})_{3j}(V^{*}_{ti}V_{tj})}{8\pi^{2}M_{\Phi_{6}}^{2}}\left(2\frac{m_{W}^{2}}{v^{2}}0.56+\frac{m_{t}^{4}}{v^{4}}1.48\right),\\\
C_{VRR}^{d_{i}d_{j}}&=\frac{3(y_{R}^{\dagger}y_{R})_{ij}^{2}}{256\pi^{2}M_{\Phi_{6}}^{2}},\\\
C_{LR1}^{d_{i}d_{j}}&\approx\frac{(y_{L}^{\dagger}y_{L})_{ij}(y_{R}^{\dagger}y_{R})_{ij}}{64\pi^{2}M_{\Phi_{6}}^{2}}-\frac{(y_{R}^{\dagger}y_{R})_{ij}(V^{*}_{ti}V_{tj})}{32\pi^{2}M_{\Phi_{6}}^{2}}\left(\frac{m_{W}^{2}m_{t}^{2}}{v^{4}}1.48+\frac{m_{t}^{2}}{v^{2}}1.67\right),\\\
C_{LR2}^{d_{i}d_{j}}&\approx-\frac{10(y_{L}^{\dagger}y_{L})_{ij}(y_{R}^{\dagger}y_{R})_{ij}}{64\pi^{2}M_{\Phi_{6}}^{2}}+\frac{(y_{R}^{\dagger}y_{R})_{ij}(V^{*}_{ti}V_{tj})}{32\pi^{2}M_{\Phi_{6}}^{2}}\left(\frac{m_{W}^{2}m_{t}^{2}}{v^{4}}1.48+\frac{m_{t}^{2}}{v^{2}}1.67\right),\end{split}$
(71)
$\begin{split}C_{VLL}^{u_{i}u_{j}}&\approx\frac{3(Vy_{L}^{*}y_{L}^{T}V^{\dagger})_{ij}^{2}}{16\pi^{2}M_{\Phi_{6}}^{2}},\\\
C_{VRR}^{u_{i}u_{j}}&=\frac{3(y_{R}^{*}y_{R}^{T})_{ij}^{2}}{256\pi^{2}M_{\Phi_{6}}^{2}},\\\
C_{LR1}^{u_{i}u_{j}}&\approx\frac{(Vy_{L}^{*}y_{L}^{T}V^{\dagger})_{ij}(y_{R}^{*}y_{R}^{T})_{ij}}{64\pi^{2}M_{\Phi_{6}}^{2}},\\\
C_{LR2}^{u_{i}u_{j}}&\approx-\frac{10(Vy_{L}^{*}y_{L}^{T}V^{\dagger})_{ij}(y_{R}^{*}y_{R}^{T})_{ij}}{64\pi^{2}M_{\Phi_{6}}^{2}}~{}.\end{split}$
(72)

Concerning the Wilson coefficient $C_{VLL}^{d_{i}d_{j}}$, the loop functions
for the mixed $W-\Phi_{6}$ contributions have been computed explicitly,
yielding

$\displaystyle C_{VLL}^{d_{i}d_{j}}\bigg{|}_{W-\Phi_{6}}=$
$\displaystyle-(N_{c}-1)\frac{(Vy_{L}^{*})_{3i}(V^{*}y_{L})_{3j}(V^{*}_{ti}V_{tj})}{16\pi^{2}}\frac{m_{W}^{2}}{v^{2}}m_{t}^{2}\times$
(73)
$\displaystyle\bigg{[}\frac{1}{(m_{t}^{2}-M^{2})(m_{t}^{2}-m_{W}^{2})}+\frac{m_{t}^{2}}{(m_{t}^{2}-M^{2})^{2}(M^{2}-m_{W}^{2})}\log\left(\frac{M^{2}}{m_{t}^{2}}\right)$
$\displaystyle+\frac{m_{t}^{2}}{(m_{t}^{2}-m_{W}^{2})^{2}(M^{2}-m_{W}^{2})}\log\left(\frac{m_{t}^{2}}{m_{W}^{2}}\right)\bigg{]}$

The effect of the RGE from the diquark to the electroweak scale is mainly due
to the QCD anomalous dimension. The $C_{VLL}^{q_{i}q_{j}}$ operators are
rescaled by the factor $\eta^{6/21}$, where
$\eta=\alpha_{s}(M_{\Phi_{6}})/\alpha_{s}(m_{t})$. The other anomalous
dimensions are well known and can be seen also in Chen:2018stt .

Particularly dangerous for a possible solution to the anomaly are the
contributions from the left-handed coupling $y_{L}$. A way to avoid them is to
require that $(y^{L}y^{L\dagger})={\rm diag}(x,x,y)$, which is indeed realised
by the coupling structure fixed in Eq. (19), which gives
$(y^{L}y^{L\dagger})={\rm diag}(|y^{L}_{12}|^{2},|y^{L}_{12}|^{2},0)$.

Figure 9:  95% CL limit on $y^{L}_{12}$ from meson mixing (red),
$K^{+}\to\pi^{+}\nu\nu$ (purple), and $(\epsilon^{\prime}/\epsilon)$ (blue),
as a function of the diquark mass.

With this choice, the two diquark boxes in the top row of Fig. 8 give zero
contribution to Kaon and $B_{d/s}$-mixing, while a strongly CKM-suppressed
contribution to $D$-mixing. The box with the $W$ boson shown in the bottom row
of Fig. 8 gives a non-vanishing contribution only to Kaon mixing, which is
also strongly CKM suppressed:

$\begin{split}C_{VLL}^{sd}[M_{\Phi_{6}}]&=\frac{|y^{L}_{12}|^{2}(V^{*}_{ts}V_{td})^{2}}{8\pi^{2}M_{\Phi_{6}}^{2}}\left(2\frac{m_{W}^{2}}{v^{2}}0.56+\frac{m_{t}^{4}}{v^{4}}1.48\right),\\\
C_{VLL}^{cu}[M_{\Phi_{6}}]&=-\frac{3|y^{L}_{12}|^{4}(V_{cb}V^{*}_{ub})^{2}}{16\pi^{2}M_{\Phi_{6}}^{2}}.\\\
\end{split}$ (74)

Taking the low-energy matrix elements from Aebischer:2020dsw  and the
constraints from UTFit Bona:2007vi ,888We use the updated results shown by L.
Silvestrini at the La Thuile conference in 2018. we get the limit on
$y^{L}_{12}$ as a function of the diquark mass as shown as a red line in Fig
9. This bound is reported as an upper limit on the product
$y^{L*}_{12}y^{R}_{13}$ as a function of $m_{\Phi_{6}}$ of Fig. 5 by setting
$y^{R}_{13}$ to either its maximal possible value from perturbativity or to be
equal to $y^{L}_{12}$.

### $K^{+}\to\pi^{+}\nu\nu$

The golden-channel Kaon decay can be described by the effective Hamiltonian
Buras:1998raa

$\mathcal{H}_{\rm
eff}\supset\frac{G_{F}V^{*}_{ts}V_{td}}{\sqrt{2}}\frac{\alpha}{\pi}C_{\nu_{\alpha}}^{L}(\bar{s}_{L}\gamma_{\mu}d_{L})(\bar{\nu}_{\alpha}\gamma^{\mu}(1-\gamma_{5})\nu_{\alpha})+h.c.~{},$
(75)

where $\alpha=1,2,3$ is the neutrino flavor and we do not consider lepton
flavor violation. The SM contribution is given by

$C_{\nu_{\alpha}}^{L,SM}=-\frac{1}{s_{W}^{2}}\left(X_{t}+\frac{V_{cs}^{*}V_{cd}}{V^{*}_{ts}V_{td}}X_{c}^{\alpha}\right)~{},$
(76)

where $X_{t}\approx 1.48$, $X_{c}^{e}=X_{c}^{\mu}\approx 1.053\times 10^{-3}$,
and $X_{c}^{\tau}\approx 0.711\times 10^{-3}$. The diquark contribution is
lepton flavor universal and is given by Chen:2018stt

$C_{\nu}^{L,\Phi_{6}}=\frac{1}{2s_{W}^{2}}\frac{v^{2}(Vy_{L}^{*})_{32}(V^{*}y_{L})_{31}}{m_{W}^{2}V^{*}_{ts}V_{td}}I_{Z}\left(\frac{m_{t}^{2}}{M_{\Phi_{6}}^{2}}\right)~{},$
(77)

where the loop function is

$I_{Z}(y_{t})=-\frac{y_{t}}{1-y_{t}}-\frac{y_{t}\log
y_{t}}{(1-y_{t})^{2}}~{}.$ (78)

The branching ratio is given by

$\text{Br}(K^{+}\to\pi^{+}\nu\nu)=2\text{Br}(K^{+}\to\pi^{+}\nu_{e}\nu_{e})_{\rm
SM}\left|1+\frac{C_{\nu}^{L,\Phi_{6}}}{C_{\nu_{e}}^{L,SM}}\right|^{2}+\text{Br}(K^{+}\to\pi^{+}\nu_{\tau}\nu_{\tau})_{\rm
SM}\left|1+\frac{C_{\nu}^{L,\Phi_{6}}}{C_{\nu_{\tau}}^{L,SM}}\right|^{2}~{},$
(79)

where $\text{Br}(K^{+}\to\pi^{+}\nu_{e}\nu_{e})_{\rm SM}=3.06\times 10^{-11}$
and $\text{Br}(K^{+}\to\pi^{+}\nu_{\tau}\nu_{\tau})_{\rm SM}=2.52\times
10^{-11}$. The most recent measurement from NA62 is CortinaGil:2020vlo 999NA62
results on $K^{+}\to\pi^{+}\nu\nu$ have been updated in a presentation at the
ICHEP2020 conference.

$\text{Br}(K^{+}\to\pi^{+}\nu\nu)=(11.0~{}^{+4.0}_{-3.5})\times 10^{-11}$ (80)

The limit on $y_{12}^{L}$ as a function of the diquark mass is shown as a
purple line in Fig. 9.

### $\epsilon^{\prime}/\epsilon$

The decay amplitude $s\to du\bar{u}$ is induced at tree-level in our setup,
the imaginary part is strongly constrained by the direct CP violation effects
in $\epsilon^{\prime}/\epsilon$. A general master formula of
$\epsilon^{\prime}/\epsilon$ in terms of EFT coefficients evaluated at the EW
scale has been obtained in Ref. Aebischer:2018quc . With the active couplings
$y^{L}_{12}$ and $y^{R}_{13}$, the relevant operators are the purely left-
handed vector-vector ones. In the notation of Aebischer:2018quc  one has

$(\epsilon^{\prime}/\epsilon)_{\rm
BSM}=\sum_{u_{i}=u,c}\left(P^{u_{i}}_{VLL}\text{Im}\left[C^{u_{i}}_{VLL}\right]+\widetilde{P}^{q}_{VLL}\text{Im}\left[\widetilde{C}^{q}_{VLL}\right]\right)~{},$
(81)

where the numerical coefficients are $P^{u}_{VLL}\approx-4.3$,
$\widetilde{P}^{u}_{VLL}\approx 1.5$, $P^{c}_{VLL}\approx 0.7$, and
$\widetilde{P}^{c}_{VLL}\approx 0.2$. The rotation to the LEFT operator basis
in Eq. (34) is given by

$\begin{split}\mathcal{N}C^{u_{i}}_{VLL}&=[L^{V1LL}_{ud}]_{ii21}-\frac{1}{6}[L^{V8LL}_{ud}]_{ii21}~{},\\\
\mathcal{N}\widetilde{C}^{u_{i}}_{VLL}&=\frac{1}{2}[L^{V8LL}_{ud}]_{ii21}~{},\end{split}$
(82)

where $\mathcal{N}=(1~{}\text{TeV})^{-2}$. Matching to the SMEFT, Eq. (43),
and then to the diquark model using Eq. (50) and keeping only $y^{L}_{12}$ we
get:

$(\epsilon^{\prime}/\epsilon)_{\rm BSM}\approx 1.3\times
10^{-4}\frac{|y^{L}_{12}|^{2}}{M_{\Phi_{6}}^{2}/~{}\text{TeV}^{2}}~{},$ (83)

where the phase is only due to the CKM. Using the approximate upper bound for
the BSM contribution of $(\epsilon^{\prime}/\epsilon)_{\rm BSM}\lesssim
10\times 10^{-4}$ we get the constraint shown in Fig. 9 as a blue line, which
is weaker than those discussed above.

##  Appendix D Flavor constraints on the colorless scalar $\Phi_{1}$

We discuss possible constraints from low-energy processes for the $\Phi_{1}$.
In particular, we focus on a specific benchmark point for the solution of the
anomaly: $M_{\Phi_{1}}\sim m_{t}$, $y^{d}_{3}\sim 0.6$ and
$y^{d}_{2}=y^{d}_{1}\sim 0.17$.

##### $\Delta F=2$

The scalar $\Phi_{1}$ generates contributions to neutral meson mixing at loop
level through box diagrams. In the notation of Eq. (70), only the Wilson
coefficients $C_{VRR}^{q_{i}q_{j}}$ and $C_{LR_{2}}^{q_{i}q_{j}}$ are non-
zero. Using the results in Refs. Grossman:1994jb ; Jung:2010ik  we get

$\displaystyle C_{VRR}^{q_{i}q_{j}}=\,$
$\displaystyle\frac{1}{128\pi^{2}M_{\Phi_{1}}^{2}}(V_{ti}V^{*}_{tj})^{2}(y_{j}^{d*})^{2}(y_{i}^{d})^{2}y_{t}I_{1}(y_{t})\,,$
(84) $\displaystyle C_{LR_{2}}^{q_{i}q_{j}}=\,$
$\displaystyle\frac{1}{16\sqrt{2}\pi^{2}M_{\Phi_{1}}^{2}}G_{F}m_{W}^{2}(V_{ti}V^{*}_{tj})^{2}(y_{j}^{d*})(y_{i}^{d})F(y_{t},x_{t})\,,$
(85)

where $y_{q}=m_{q}^{2}/M_{\Phi_{1}}^{2}$, $x_{q}=m_{q}^{2}/m_{w}^{2}$. Using
our benchmark point and the expression of the loop functions in Refs.
Grossman:1994jb ; Jung:2010ik , we have $I_{1}(1)=1/3$ and $F(1,x_{t})=8.09$.
We use the low-energy matrix elements from Aebischer:2020dsw , finding that
our results for this scenario are compatible with the current limits by UTFit
Bona:2007vi  in both the $B_{d}$ and $B_{s}$ cases. Similar conclusions can be
drawn in the case of neutral $K$ and $D$ mixings.

##### $Z\to b\bar{b}$

The contributions to $Z\to b\bar{b}$ decays come from penguin type diagrams.
In particular, the $\Phi_{1}$ generates right-handed vector interactions,
which yield

$\displaystyle g_{b}^{R}\equiv$
$\displaystyle\,(g_{b}^{R})_{\text{SM}}-\frac{(y_{3}^{d})^{2}}{32\pi^{2}}\left[f_{1}(y_{t})+\frac{\alpha_{s}}{3\pi}f_{2}(y_{t})\right]\,.$
(86)

The loop functions $f_{1}(y_{t})$ and $f_{2}(y_{t})$ are reported in Refs.
Jung:2010ik ; Degrassi:2010ne  and in our benchmark point they assume the
values $f_{1}(1)=1/2$ and $f_{2}(1)=-13/6$. We compare with the current value
extracted by electroweak fits. In Ref. ALEPH:2005ab , the fitted value of
$g_{b}^{R}=0.0962\pm 0.0063$ is obtained. In this scenario, the contribution
due to NP to $g_{b}^{R}$ is well below the uncertainty reported in Ref.
ALEPH:2005ab .

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

The contributions to $b\to s\ell^{+}\ell^{-}$ decays come from penguin type
diagrams. We follow the conventions for the effective low-energy Hamiltonian
in Ref. Blake:2016olu . The $\Phi_{1}$ generates the NP Wilson coefficients
$\mathcal{C}_{7^{\prime}}$, $\mathcal{C}_{9^{\prime}}$ and
$\mathcal{C}_{10^{\prime}}$, which are lepton-flavour universal. Concerning
$\mathcal{C}_{9^{\prime}}$ and $\mathcal{C}_{10^{\prime}}$, using the results
in Grinstein:1988me , we get

$\mathcal{C}_{9^{\prime}}(\mu_{W})=-\mathcal{C}_{10^{\prime}}(\mu_{W})=-\frac{1}{g^{2}\sin^{2}\theta_{w}}y_{2}^{d}y_{3}^{d^{*}}B(y_{t})\,,$
(87)

where $B(1)=-1/8$ and $\mu_{W}=m_{W}$. At the low scale $\mu_{b}=m_{b}$ we get

$\mathcal{C}_{9^{\prime}}(\mu_{b})=-\mathcal{C}_{10^{\prime}}(\mu_{b})\sim
0.13\,.$ (88)

The best sensitivity to these Wilson coefficients is achieved in $b\to
s\mu^{+}\mu^{-}$ data, which show interesting deviations w.r.t. the SM
expectations. The current status is found in Alguero:2019ptt ;
Aebischer:2019mlg ; Ciuchini:2020gvn ; Alok:2019ufo , where several NP
scenarios are analysed. Our predictions for $\mathcal{C}_{9^{\prime}}$ and
$\mathcal{C}_{10^{\prime}}$ are not excluded, but are also not able to explain
the tensions in $b\to s\mu^{+}\mu^{-}$ data.

The $\Phi_{1}$ generates also dipole operators and in particular
$\mathcal{O}_{7^{\prime}}$ receives $m_{b}$ enhanced contributions. Following
Besmer:2001cj , we get

$\mathcal{C}_{7^{\prime}}(\mu_{W})=-\frac{1}{2}y_{2}^{d}y_{3}^{d}\frac{v^{2}}{M_{\Phi_{1}}^{2}}\left[\frac{2}{3}F_{1}(y_{t})+F_{2}(y_{t})\right]\,,$
(89)

where $F_{1}(y_{t})=F_{2}(y_{t})\to 1/24$. The RGE evolution of
$\mathcal{C}_{7^{\prime}}$ is the same as for $\mathcal{C}_{7}$. At the low
scale $\mu_{b}=m_{b}$ and at leading order in QCD, we have Buchalla:1995vs

$\frac{\mathcal{C}_{7^{\prime}}(\mu_{b})}{\mathcal{C}_{7}(\mu_{b})}=1.6\%\,,$
(90)

which is below the current bound in Ref. Aaij:2020umj .

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