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#  Flavor constraints for a vector-like quark of Nelson-Barr type

A. L. Cherchiglia  Centro de Ciências Naturais e Humanas, Universidade Federal
do ABC, 09.210-170, Santo André-SP, Brasil  G. De Conto  Centro de Ciências
Naturais e Humanas, Universidade Federal do ABC, 09.210-170, Santo André-SP,
Brasil  C. C. Nishi  Centro de Matemática, Computação e Cognição, Universidade
Federal do ABC, 09.210-170, Santo André-SP, Brasil

###### Abstract

The Nelson-Barr (NB) mechanism to solve the strong CP problem assumes CP
conservation, arranges vanishing $\bar{\theta}$ at tree-level and requires
vector-like quarks (VLQs) to transmit the CP breaking to the SM. We analyze
the flavor constraints coming from the presence of one such down type VLQ of
NB type by performing a global fit on the relevant flavor observables. A
comparison is made to the case of one generic VLQ. We find that the allowed
parameter space for the VLQ Yukawa couplings and the mixing to the SM are
confined to a region much smaller than in the generic case, making the NB case
falsifiable in principle.

##  I Introduction

The Nelson-Barr (NB) solution nelson ; barr  to the strong CP problem is a
very simple mechanism based on the assumption that CP is a fundamental
symmetry of nature strongCP:CP 111Alternatively, one can assume P is a
fundamental symmetry strongCP:P . We will focus on the solution based on CP.
—hence the QCD vacuum angle is zero at tree level— spontaneously broken so
that the CP violation we observe in the SM is reproduced. The mechanism
proposes a simple recipe to arrange a vanishing contribution to $\bar{\theta}$
at tree level from quark Yukawa couplings, but still allows large CP violation
to arise in the SM. Then the challenge dine ; perez.shalit  is to keep the
calculable radiative corrections to $\bar{\theta}$ to be tiny,
$\bar{\theta}\lesssim 10^{-10}$, conforming to the experimental observations
EDMreview ; nEDM:exp ; EDM:Hg .

A necessary byproduct of the NB mechanism is the presence of vector-like
quarks (VLQ) whose mixing with ordinary quarks after spontaneous CP breaking
is the only source of CP breaking that is transmitted to the SM already at
tree level. 222Large CKM CP violation may be also generated at loop level,
although it requires strong coupling in the mixing term and supersymmetry to
protect $\bar{\theta}$ hiller:01 .  A simple definition of these VLQs of
Nelson-Barr type (NB-VLQs) was given recently for singlet VLQs in Ref. nb-vlq
.333A similar setting was proposed some time ago in Ref. lavoura . The
simplest complete model was proposed 30 years ago by Bento, Branco and Parada
BBP , which contained one down-type VLQ and one complex scalar responsible for
CP breaking. More recently, Ref. NB:CP4  improved on this model by imposing a
non-conventional CP symmetry which required the addition of two VLQs as the
price to protect $\bar{\theta}$ from receiving one-loop corrections. Similarly
to these UV completions, the implementation of the NB idea generically leads
to theories where CP violation arises only in wave-function renormalization
hiller:01 . Flavor alignment may follow if one combines the spontaneous
breaking of CP with the breaking of quark family number meade:SFV . See other
approaches in Refs. scpv:others:recent ; cp-texture .

However, VLQs appear as byproducts in many other theories, many unrelated to
the strong CP problem. Little higgs little.higgs  and composite Higgs
composite.higgs  theories comprise a small subset.444Ref. perez.shalit
improves the quality of the NB solution by making the spontaneous CP breaking
sector composite.  Therefore, if VLQs are ever found at the TeV scale, it is
pressing that we explore the characteristic features of VLQs related to the
strong CP problem and the origin of CP violation in nature.

Here we study in more detail the case of the presence of a NB-VLQ of charge
$-1/3$, focusing on the flavor constraints. We perform a global fit on a set
of flavor observables and compare the effect of a NB-VLQ with a VLQ of generic
type, not necessarily related to the source of CP violation. For one down-type
NB-VLQ, Ref. nb-vlq  has shown that the VLQ couplings to the up type quarks
through the charged current necessarily follow a hierarchical pattern, a
feature that suppresses the flavor changing effects involving lighter quarks.
This nongeneric and hierarchical pattern is interesting if we consider the new
physics scales that we can probe by, e..g, $\Delta F=2$ transitions induced by
new physics contributions to flavor changing operators of the form

$\frac{1}{\Lambda_{ij}^{2}}(\bar{d}_{iL}\gamma^{\mu}d_{j_{L}})^{2}\,.$ (1)

A generic analysis isidori.cern.12  at each sector $(ij)=(sd),(bd),(bs)$ leads
to the rough bound555More precise but still generic bounds can be found in the
same references.

$\frac{1}{\Lambda_{ij}}\lesssim\frac{|V_{ti}^{*}V_{tj}|}{3.4\,\mathrm{TeV}}\,,$
(2)

which follows the hierarchy

$\Lambda_{sd}^{-1}:\Lambda_{bd}^{-1}:\Lambda_{bs}^{-1}\sim 0.009:0.2:1\,.$ (3)

For one NB-VLQ, this kind of hierarchy is automatically induced.666The overall
scale, however, is not determined.  We will obtain quantitative bounds from
the global fit.

We organize this paper as follows: in Sec. II we briefly review the model of
one NB-VLQ and describe the main features. This includes the _seesaw
parametrization_ that was developed previously nb-vlq . In Sec. III we list
the observables that will be used in the global fit. The methods and the
results of the global fit are shown in Sec. IV. We finally conclude in Sec. V.

##  II Review of the model and parametrization

Our model of one down-type VLQ of Nelson-Barr type (NB-VLQ), denoted by
$B_{L,R}$, is defined by the Lagrangian nb-vlq

$\displaystyle-\mathscr{L}$
$\displaystyle=\bar{q}_{iL}\mathscr{Y}^{d}_{ij}Hd_{jR}+\bar{q}_{iL}\mathscr{Y}^{u}_{ij}\tilde{H}u_{jR}$
(4)
$\displaystyle\quad+\bar{B}_{L}\mathscr{M}^{Bd}_{j}d_{jR}+\bar{B}_{L}\mathscr{M}_{B}B_{R}+h.c.,$

where $i,j=1,2,3,$ and with the additional requirement that
$\mathscr{Y}^{u},\mathscr{Y}^{d}$ are real $3\times 3$ matrices,
$\mathscr{M}_{B}$ is a real mass and only $\mathscr{M}^{Bd}$ is a complex row
vector. This structure follows from CP conservation and a $\mathbb{Z}_{2}$
symmetry 777A larger $\mathbb{Z}_{n}$ or $U(1)$ dine , a nonabelian global or
gauge symmetry nelson ; barr  can be also used. Note that the definition based
on $\mathbb{Z}_{2}$ is not sufficient when singlet and doublet VLQs are
simultaneously present.  under which only $B_{L,R}$ are odd, and only
$\mathscr{M}^{Bd}$ breaks CP and $\mathbb{Z}_{2}$ softly (spontaneously)
realizing the Nelson-Barr mechanism that guarantees $\bar{\theta}=0$ at tree-
level nelson ; barr . The SM fields are $q_{iL},d_{iR}$ and $u_{iR}$,
corresponding to the quark doublets, down-type singlets and up-type singlets,
respectively. This Lagrangian contains
$3~{}(\mathscr{Y}^{u})+9~{}(\mathscr{Y}^{d})+2~{}(\mathscr{M}^{Bd})+1~{}(\mathscr{M}_{B})=15$
parameters in total (see explicit parametrization below) using the freedom of
real orthogonal transformations in the $\mathscr{Y}^{u}$ diagonal basis nb-vlq
.

In contrast, a generic VLQ is customarily described by the Lagrangian

$\displaystyle-\mathscr{L}$
$\displaystyle=\bar{q}_{iL}Y^{d}_{ij}Hd_{jR}+\bar{q}_{iL}Y^{u}_{ij}\tilde{H}u_{jR}$
(5) $\displaystyle~{}~{}+\
\bar{q}_{iL}Y^{B}_{i}H\,B_{R}+\bar{B}_{L}M_{B}B_{R}+h.c.,$

where $M_{B}$ is expected to be much larger than the electroweak scale. If
generic, this lagrangian depends on
$3~{}(Y^{u})+7~{}(Y^{d})+5~{}(Y^{B})+1~{}(M_{B})=16$ parameters using now the
freedom of unitary rotations in $Y^{u}$ diagonal basis. We can see that one
more parameter is needed compared to the case of one NB-VLQ. Hence, the NB
case is just a subcase and when the lagrangian (4) is rewritten in the form
(5), the various parameters cannot be independent and correlations necessarily
appear nb-vlq . In special, _only one_ CP violating parameter controls all CP
violation in the NB case while the generic case depends on three CP violating
parameters lavoura.branco ; branco:book ; nir:slac99 . We emphasize this
information in table 1.

$\begin{array}[]{|c|cc|}\hline\cr&\text{\\# of param.}&\text{\\# of CP
odd}\cr\hline\cr\text{SM}&10&1\cr\text{One generic VLQ}&16&3\cr\text{One NB-
VLQ}&15&1\cr\hline\cr\end{array}$ Table 1:  Number of parameters in the flavor
sector of the SM and with the addition of one VLQ.

The changing of basis from (4) to (5) is easily described by comparing the
$4\times 4$ mass matrix of the down-type quarks following from (4) and (5),
respectively, after EWSB:

$\text{NB:}\quad\mathscr{M}^{d+B}=\begin{pmatrix}\frac{v}{\sqrt{2}}\mathscr{Y}^{d}&0\cr\mathscr{M}^{Bd}&\mathscr{M}_{B}\end{pmatrix}\,,\quad\text{generic:}\quad
M^{d+B}=\begin{pmatrix}\frac{v}{\sqrt{2}}Y^{d}&\frac{v}{\sqrt{2}}Y^{B}\cr
0&M_{B}\end{pmatrix}\,.$ (6)

Only a unitary transformation from the right is necessary to connect them:

$\mathscr{M}^{d+B}W_{R}=M^{d+B}\,.$ (7)

The relevant exact relations are given by

$\displaystyle M_{B}$
$\displaystyle=\sqrt{\mathscr{M}^{Bd}{\mathscr{M}^{Bd}}^{\dagger}+\mathscr{M}_{B}^{2}}\,,$
(8a) $\displaystyle Y^{B}$
$\displaystyle=\mathscr{Y}^{d}{\mathscr{M}^{Bd}}^{\dagger}/M_{B}\,,$ (8b)
$\displaystyle Y^{d}{Y^{d}}^{\dagger}$
$\displaystyle=\mathscr{Y}^{d}\left({\mathbbm{1}}_{3}-{\mathscr{M}^{Bd}}^{\dagger}\mathscr{M}^{Bd}/M_{B}^{2}\right){\mathscr{Y}^{d}}^{\mbox{\scriptsize$\mathsf{T}$}}\,.$
(8c)

For both NB and generic cases, the form $M^{d+B}$ has a simple interpretation
in the leading quark seesaw approximation888In this approximation, the block
diagonalized form of $M^{d+B}$ is simply given by neglecting the $Y^{B}$ term
off the diagonal. At this order, $Y^{B}$ will only contribute to the VLQ
mixing with the SM but not to the masses.  where $M_{B}\gg v$: $M_{B}$ is the
VLQ mass while $Y^{d}{Y^{d}}^{\dagger}$ is the leading SM down-type Yukawa
matrix squared which contains the CKM CP violation BBP . Therefore, in leading
order, the latter is determined from SM input:

$Y^{d}{Y^{d}}^{\dagger}=\frac{2}{v^{2}}V_{d_{L}}\operatorname{\mathrm{diag}}(m^{2}_{d},m^{2}_{s},m^{2}_{b})V_{d_{L}}^{\dagger}\,,$
(9)

where $V_{d_{L}}$ is the CKM matrix $V_{\rm ckm}^{\rm sm}$, except for
possible phases from the left. If $V_{d_{L}}=V_{\rm ckm}^{\rm sm}$, note that
the righthand side of (9) contains 7 parameters from the SM down sector.

Removing from the counting the three up quark Yukawa couplings in the diagonal
basis, we can now concentrate on the 12 parameters in $\mathscr{M}^{d+B}$ for
the NB-VLQ case or 13 parameters in $M^{d+B}$ for the generic case; cf. (6).
We should now concentrate on how to separate the 7 SM input parameters from
the BSM parameters.

In the generic case, $Y^{d}$ and $Y^{B}$ are independent. Then, in the seesaw
approximation, 7 parameters in $Y^{d}$ can be fixed from (9), with
$V_{d_{L}}=V_{\rm ckm}^{\rm sm}$, and the remaining 5 parameters in $Y^{B}$
and one parameter in $M_{B}$ are free BSM parameters.

In the NB case, $Y^{d}$ and $Y^{B}$ are _not_ independent and a different
parametrization is needed. We now describe the seesaw parametrization devised
in Ref. nb-vlq  for one NB-VLQ.

We first choose (8a) as one of the parameters instead of $\mathscr{M}_{B}$ and
rewrite

$\mathscr{M}^{d+B}=\begin{pmatrix}\frac{v}{\sqrt{2}}\mathscr{Y}^{d}&0\cr
M_{B}w^{\dagger}&M_{B}\sqrt{1-|w|^{2}}\end{pmatrix}\,.$ (10)

Without loss of generality nb-vlq ,

$w={\mathscr{M}^{Bd}}^{\dagger}/M_{B}=\begin{pmatrix}0\cr ib\cr
a\end{pmatrix}\,,$ (11)

with $b<a$ and $a^{2}+b^{2}<1$. So $\\{M_{B},w\\}$ contains three parameters
while $\mathscr{Y}^{d}$, being real and generic, contains 9 parameters.

The combination in the righthand side of (8c) depends on $\mathscr{Y}^{d}$ and
$w$, and is partially fixed by SM input in (9). However, as is clear from Eqs.
(8b) and (8c), both $Y^{d}$ and $Y^{B}$ depends on $\mathscr{Y}^{d}$ and $w$,
and we cannot transfer phases from one to another. We then need to consider
additional phases999In Ref. nb-vlq , these phases were denoted as
$\beta_{2},\beta_{3}$, respectively. in

$V_{d_{L}}=\begin{pmatrix}1&&\cr&e^{i\beta_{1}}&\cr&&e^{i\beta_{2}}\end{pmatrix}V^{\rm
sm}_{\rm ckm}\,,$ (12)

in order to take into account SM input in (9).

Given $\beta_{1},\beta_{2}$, the real matrix $\mathscr{Y}^{d}$ is partially
determined from the inversion formula for (8c); see appendix A and Ref. nb-vlq
for details. The remaining freedom is parametrized by $b$ in $w$ and one
additional angle parameter $\gamma$. In this way, in our seesaw
parametrization, among the 12 parameters in $\mathscr{M}^{d+B}$, the 7
parameters are fixed from the SM input while the following 5 BSM parameters
are free:

$\\{M_{B},b,\gamma,\beta_{1},\beta_{2}\\}\,.$ (13)

So $\mathscr{Y}^{d}$ and $w$ depend on $b,\gamma,\beta_{1},\beta_{2}$, and
$M_{B}$ can be chosen independently. In special, we restrict

$b\in[0.02,1/\sqrt{2}],$ (14)

so that the seesaw approximation is valid within 1% for the mass eigenvalues
and the moduli of the $3\times 3$ block of the CKM matrix nb-vlq . The angle
$\gamma$ and phases $\beta_{1},\beta_{2}$ all vary in the whole range
$[0,2\pi]$.

In the generic as well as NB case, the connection with observables are all
encoded in the rectangular CKM matrix $V\sim 3\times 4$,

$V=U_{u_{L}}^{\dagger}PU_{d_{L}}\approx
V_{d_{L}}\left(\begin{array}[]{c|c}{\mathbbm{1}}_{3}-\mbox{\large$\tfrac{1}{2}$}\delta
X^{d}&\Theta\end{array}\right)\,,$ (15)

where $U_{d_{L}}$ is the matrix that diagonalizes either matrix in (6) from
the left, $P$ is a $3\times 4$ projection matrix, nonvanishing only for
$P_{11}=P_{22}=P_{33}=1$, and

$\Theta\equiv V_{d_{L}}^{\dagger}\theta_{L}\,,\quad\delta
X^{d}\equiv\Theta\Theta^{\dagger}\,.$ (16)

The diagonalizing matrix of the up sector can be ignored,
$U_{u_{L}}={\mathbbm{1}}_{3}$, in the basis where $Y^{u}=\hat{Y}^{u}$. The
expansion parameter in the seesaw is given by

$\theta_{L}=\frac{v}{\sqrt{2}}Y^{B}M_{B}^{-1}\,.$ (17)

The square matrix $X^{d}$ of size $4$ that describes the FCNC coupling to $Z$
is

$X^{d}=V^{\dagger}V\approx\left(\begin{array}[]{c|c}{\mathbbm{1}}_{3}-\delta
X^{d}&\Theta\cr\hline\cr*&\Theta^{\dagger}\Theta\end{array}\right)\,.$ (18)

So we see that all couplings of VLQs to gauge bosons depend solely on the
matrix $\Theta$ within the seesaw approximation.

For definiteness, we employ the phase convention for $V$ such that

$\arg(V)=\begin{pmatrix}0&0&*&*\cr\pi&*&0&*\cr*&*&0&0\end{pmatrix}\,.$ (19)

The stars denote free phases. This convention exhausts all the rephasing
freedom and it is easier to implement than the one coming from the angle-phase
parametrizations. For the $3\times 3$ block, this convention is similar to the
one employed in Ref. branco:book  and differs slightly only by a tiny phase in
the second column. This convention is also very close to the standard
parametrization of the $3\times 3$ CKM or to the angle-phase parametrization
of the $4\times 4$ unitary matrix employed in Ref. nb-vlq . They differ by
tiny phases once SM input is considered.

The phase convention (19) can be chosen directly for the generic case but it
is not automatic for the NB case as the diagonalizing matrix for (6) has
implicit phase relations. We enforce it after the diagonalization process, a
procedure which also redefines $Y^{d}$ and $Y^{B}$ by rephasing from the left.

In Ref. nb-vlq , it was shown that once the SM input was considered, the
mixing of the heavy VLQ with the SM quarks followed a hierarchical structure:

$|V_{uB}|\ll|V_{cB}|\ll|V_{tB}|\,.$ (20)

For a parameter $b$ not so small, the hierarchy roughly follows the hierarchy
of $b$-quark mixing:

$|V_{uB}|:|V_{cB}|:|V_{tB}|\sim|V_{ub}|:|V_{cb}|:|V_{tb}|\approx
0.004:0.04:1\,,$ (21)

a feature that is exact in case the VLQ couples exclusively with the third
family. The hierarchy also applies to $Y^{B}_{i}$ since they are proportional
to $V_{iB}$ at leading order, cf. (17).101010See appendix B for an approximate
function describing the upper boundary of $|Y^{B}_{i}|$ as a function of $b$.
This hierarchy largely renders the model flavor safe as the most restrictive
flavor constraints of flavor changing among the first and second families are
naturally suppressed. Here we will test these constraints more quantitatively
by performing a global fit.

If BSM models are studied in terms of effective field theories and matched
onto the SM or effective theories at lower energies, flavor observables can be
written in terms of SM parameters and the BSM parameters $Y^{B}_{i}$ and the
VLQ mass $M_{B}$. Constraints can then be placed on these BSM parameters. This
study was performed in, e.g., Ref. buras.celis.17 . Information on $Y^{B}$ is
equivalent to the information on the CKM matrix $V$ and at leading order
$Y^{B}$ is proportional to the light-heavy mixing parameters $\theta_{L}$ in
(17). We can extract $Y^{B}$ independently of the basis for right-handed
fields in terms of the mass matrix in (6) as

$\frac{v}{\sqrt{2}}Y^{B}_{i}=\frac{(M^{d+B}{M^{d+B}}^{\dagger})_{i4}}{\sqrt{(M^{d+B}{M^{d+B}}^{\dagger})_{44}}}\,.$
(22)

This relation is also valid if we replace $M^{d+B}$ by $\mathscr{M}^{d+B}$
from the NB case since the two matrices are related by unitary transformation
for right-handed quarks. For the generic case, the mass matrix squared
$(M^{d+B}{M^{d+B}}^{\dagger})$ can be reconstructed from the $4\times 4$
mixing and masses 111111Note that the _mass_ $M_{B}$ here only coincides with
the _parameter_ $M_{B}$ in (6) in the leading seesaw approximation.  as

$M^{d+B}{M^{d+B}}^{\dagger}=U_{d_{L}}\operatorname{\mathrm{diag}}(m_{d}^{2},m_{s}^{2},m_{b}^{2},M_{B}^{2})U_{d_{L}}^{\dagger}\,,$
(23)

while for the NB case $\mathscr{M}^{d+B}$ itself can be used. The $4\times 4$
diagonalizing matrix $U_{d_{L}}$ can be parametrized in terms of the standard
parametrization for the $3\times 3$ CKM matrix and additional angles,
$\theta_{14},\theta_{24},\theta_{34}$, and additional phases
$\delta_{1},\delta_{2}$ nb-vlq ; see also Refs. gouvea ; botella . When a
specific basis is understood, $Y^{B}$ will denote the Yukawa coupling in the
basis where the up-type Yukawas are diagonal $Y^{u}=\hat{Y}^{u}$. However, to
compare with Ref. buras.celis.17 , we will also need

$\tilde{Y}^{B}_{i}\equiv(V^{\dagger}_{d_{L}}Y^{B})_{i}\,,$ (24)

defined in the weak basis where $Y^{d}$ is diagonal when $Y^{B}$ is defined in
the basis where $Y^{u}$ is diagonal.

##  III Phenomenological constraints for $n_{B}=1$

In this section we describe the various observables that will be considered in
the fit. Other observables will be predicted from the fit and these are
discussed as well.

###  III.1 VLQ mass

Because $V_{iB}$ are hierarchical, the $B$ quark couples dominantly with the
top and we can use the current constraint coming from direct searches at the
LHC atlas ; cms :

$M_{B}\gtrsim 1.2\,\mathrm{TeV}\,.$ (25)

Considering the recent analysis cms:20 , we will perform our fit for the
conservative value $M_{B}=1.4\,\mathrm{TeV}$.

Even if $V_{iB}$ are hierarchical and flavor alignment is naturally enforced,
we need to test the structure of $V$ against the various flavor observables.
The various observables that we will use to constrain the model will be
described in the following subsections and is summarized in Table 2.

We reserve the letter $B$ for the heavy quark and the $B$-mesons will be
always denoted with their subscripts $B_{d}^{0}$ and $B_{s}^{0}$.

###  III.2 Constraints from $|V_{ij}|$

Most of the moduli $|V_{ij}|$ of the CKM matrix are extracted from tree level
processes. The exceptions are $|V_{td}|$ and $|V_{ts}|$ which are extracted
from $B_{d}^{0}$ and $B_{s}^{0}$ meson oscillations through box diagrams
involving the top quark. We impose direct constraints on the ones extracted at
tree level while the constraints for $|V_{td}|$ and $|V_{ts}|$ are extracted
directly from $\Delta m_{B_{d}}$ and $\Delta m_{B_{s}}$ by considering the VLQ
contribution; see Sec. III.3. So these entries may deviate from the SM values.

The experimental values for $|V_{ij}|$ are PDG :

$|V_{ij}|_{\rm exp}=\begin{pmatrix}0.97370(14)&0.2245(8)&3.82(0.24)\times
10^{-3}\cr 0.221(4)&0.987(11)&41.0(1.4)\times
10^{-3}\cr\underline{8.0(0.3)\times 10^{-3}}&\underline{38.8(1.1)\times
10^{-3}}&1.013(30)\end{pmatrix}\,.$ (26)

The values for $|V_{td}|$ and $|V_{ts}|$, which are underlined, will not be
considered directly. Note that (26) shows a somewhat lower value for
$|V_{ud}|$ PDG  due to use of updated hadronic uncertainty ramsey:Vud . These
values hint at nonunitarity of the first row of the CKM matrix which can be
interpreted in terms of VLQs berezhiani.19 ; cheung.20 .

For comparison, we can also show the values for the magnitudes of the CKM
elements obtained from the combination of the various experiments and assuming
unitarity. The result reported by PDG 2020 PDG  is

$|V_{ij}|_{\rm exp}^{\rm
fit}=\begin{pmatrix}0.97401(11)&0.22650(48)&0.00361^{+0.00011}_{-0.00009}\cr
0.22636(48)&0.97320(11)&0.04053^{+0.00083}_{-0.00061}\cr\underline{0.00854^{+0.00023}_{-0.00016}}&\underline{0.03978^{+0.00082}_{-0.00060}}&0.999172^{+0.000024}_{-0.000035}\end{pmatrix}\,.$
(27)

The corresponding parameters in the standard parametrization read

$\displaystyle s_{12}$ $\displaystyle=0.22650(48)\,,$ $\displaystyle\quad
s_{13}$ $\displaystyle=0.00361^{+0.00011}_{-0.00009}\,,$ (28) $\displaystyle
s_{23}$ $\displaystyle=0.04053^{+0.00083}_{-0.00061}\,,$
$\displaystyle\quad\delta$ $\displaystyle=1.196^{+0.045}_{-0.043}\,,$

while the Jarlskog invariant is $10^{5}{\times}J=3.00^{+0.15}_{-0.09}$. In the
Wolfenstein parametrization, the same fit leads to

$\displaystyle\lambda$ $\displaystyle=0.22650(48)\,,$ $\displaystyle\quad A$
$\displaystyle=0.790^{+0.017}_{-0.012}\,,$ (29) $\displaystyle\bar{\rho}$
$\displaystyle=0.141^{+0.016}_{-0.017}\,,$ $\displaystyle\quad\bar{\eta}$
$\displaystyle=0.357(11)\,.$

###  III.3 $\Delta m_{B_{d}}$ and $\Delta m_{B_{s}}$

The CKM matrix entries $|V_{td}|$ and $|V_{ts}|$ in the SM are currently
extracted from $B_{d}^{0}$ and $B_{s}^{0}$ meson oscillations through box
diagram contributions. In the presence of VLQs, additional contributions arise
due to the exchange of the VLQs. Or, in an EFT description, from the deviation
from unitarity of the CKM matrix and FCNC through $Z$ exchange.

The $B^{0}_{q}-\bar{B}^{0}_{q}$ mixing for the $q=d,s$ mesons leads to the
mass difference

$\Delta m_{B_{q}}=2|M_{12}^{B_{q}}|\,,$ (30)

as long-distance effects are negligible. This quantity will be sensitive to

$\lambda^{t}_{qb}\equiv V_{tq}^{*}V_{tb}\,.$ (31)

The mixing in the presence of a VLQ can be calculated as saavedra:flavor

$M_{12}^{B_{q}}=\frac{G_{F}^{2}m_{W}^{2}f^{2}_{B_{q}}\hat{B}_{B_{q}}m_{B_{q}^{0}}}{12\pi^{2}}\left[(\lambda^{t}_{qb})^{2}\eta^{B}_{tt}S_{0}(x_{t})+\Delta_{B_{q}}\right]\,,$
(32)

where $x_{t}\equiv m_{t}^{2}/m_{W}^{2}$ and

$\Delta_{B_{q}}=-8X_{qb}\lambda^{t}_{qb}\eta^{B}_{tt}Y_{0}(x_{t})+\frac{4\pi
s^{2}_{w}}{\alpha_{e}}\eta^{B}_{Z}X^{2}_{qb}\,$ (33)

is the VLQ contribution depending on $X_{qb}$. We only retain the dominant top
contribution in the box diagrams. The dominant theoretical error comes from
the combination $f_{B_{q}}\sqrt{B_{B_{q}}}$ of the meson decay constant and
bag parameter. Lattice calculations with three flavors report the average
values $f_{B_{d}}\sqrt{B_{B_{d}}}=225(9)\,\mathrm{MeV}$ and
$f_{B_{s}}\sqrt{B_{B_{s}}}=274(8)\,\mathrm{MeV}$ flag.19 . Other input
parameters are shown in table 3. NLO QCD corrections from running from $m_{W}$
to $m_{b}$ is taken into account by the factors buras:qcd.eta
$\eta^{B}_{tt}=0.55$, $\eta^{B}_{Z}=0.57$, where we neglect the small errors.
$S_{0}$ and $Y_{0}$ are Inami-Lim functions given in appendix C.

###  III.4 $Z\to bb$: $R_{b}$

Given the mixing among the SM $d$-type quarks and the VLQ, not only FCNC will
be generated but also diagonal couplings of SM $d$-type quarks to the $Z$
boson will be modified. These modifications can be strongly constrained by the
ratio $R_{b}$ saavedra:handbook , since this observable shows a good agreement
with the SM prediction; see Table 2.

For definiteness,

$R_{b}=\frac{\Gamma(Z\to b\bar{b})}{\Gamma(Z\to\text{hadrons})}\,.$ (34)

The possible deviations appear in the neutral current with d-type quarks

$\mathcal{L}_{Zqq}=-\frac{g}{2c_{w}}\bar{q}\gamma^{\mu}\left(c_{Lq}P_{L}+c_{Rq}P_{R}\right)bZ_{\mu}\,,$
(35)

where, in our model,

$c_{Lq}=c_{Lq}^{\rm SM}+\delta c_{Lq},\quad c_{Lq}^{\rm
SM}=-1+\frac{2}{3}s_{w}^{2},\quad\delta c_{Lq}=1-X^{d}_{qq},\quad
c_{Rq}=2/3s_{w}^{2}.$ (36)

Since only small deviations from the SM couplings will be allowed, one can use
an approximate formula for $R_{b}$ obtained by considering only first-order
deviations from the tree-level formula

$R_{b}^{0}=\frac{c_{Lb}^{2}+c_{Rb}^{2}}{\sum_{q}(c_{Lq}^{2}+c_{Rq}^{2})},$
(37) $R_{b}=R_{b}^{SM}(1+0.5118\,\delta c_{Ld}+0.5118\,\delta
c_{Ls}-1.8178\,\delta c_{Lb})\,$ (38)

where $R_{b}^{SM}$ is given in Table 2, and $\delta c_{Lq}$ are defined in eq.
(36).

###  III.5 $B_{q}\to\mu^{+}\mu^{-}$

The rare decays $B_{q}\to\mu^{+}\mu^{-}$, $q=d,s$, are extremely suppressed
within the SM because it is mediated by a FCNC and are helicity suppressed.
Because of the purely leptonic final state, it is also very clean
theoretically and then sensitive to new physics.

The experimental value for $B_{s}$ is given in table 2 and it is of the order
of $10^{-9}$. The branching ratio in the presence of a VLQ can be found in the
literature shimizu  and yields

$Br(B_{q}\to\mu^{+}\mu^{-})=\tau_{B_{q}}\frac{G_{F}^{2}}{16\pi}\left(\frac{\alpha_{e}}{\pi
s^{2}_{w}}\right)^{2}f_{B_{q}}^{2}m_{B^{0}_{q}}m_{\mu}^{2}\sqrt{1-\frac{4m_{\mu}^{2}}{m^{2}_{B_{q}}}}\,|\eta_{Y}^{2}|\left|\lambda^{t}_{qb}Y_{0}(x_{t})+\Delta^{B}_{\mu\mu}\right|^{2}\,,$
(39)

where the deviation of the SM is given by

$\Delta^{B}_{\mu\mu}=-\frac{\pi s^{2}_{w}}{\alpha_{e}}X_{qb}\,.$ (40)

The factor $\eta_{Y}=1.0113$ accounts for QCD corrections in a scheme where
NLO electroweak corrections can be neglected buras:Bmumu .

The measurement on $Br(B^{0}_{d}\to\mu^{+}\mu^{-})$ still has a large error
and will not be considered in the fit. We will instead present predictions for
its value based on the fit. Analogously, the constraint coming from
$Br(B^{0}_{s}\to X_{s}\gamma)$ will not be considered because it is weaker
than $Br(B^{0}_{s}\to\mu^{+}\mu^{-})$ shimizu .

###  III.6 $B^{0}_{d}\to J/\psi K_{S}$

The cleanest measurement of the angle $\beta$ of the $db$ unitarity triangle
is given by the $\bar{b}\to c\bar{c}\bar{s}$ transition in $B^{0}_{d}\to
J/\psi K_{S}$. It is dominated by only one weak phase in the dominant tree
level decay and the penguin pollution is also dominated by the same weak
phase. In the presence of the VLQ, additional contributions are induced.

The measurement of the direct CP violation decay amplitude $S_{\psi K_{S}}$
occurs through the time-dependent CP asymmetry of $B^{0}_{d}$ and
$\bar{B}^{0}_{d}$ to the common final state $J/\psi K_{S}$, which is
approximately a CP odd state, and can be written as branco:book

$S_{\psi K_{S}}=\sin(2\beta+2\theta_{B_{d}}-2\theta_{K}),$ (41)

where $\beta$ is the angle

$\beta\equiv\arg\left(-\frac{V_{cd}V_{cb}^{*}}{V_{td}V_{tb}^{*}}\right)=-\arg\left(-\frac{\lambda^{c}_{db}}{\lambda^{t}_{db}}\right)\,,$
(42)

while $\theta_{B}$ and $\theta_{K}$ parametrize the possible deviations from
the SM of the mixings in the $B$ and $K$ system as

$2\theta_{B_{d}}\equiv\arg\frac{M^{B_{d}}_{12}}{(M^{B_{d}}_{12})_{\rm
SM}}\,,\quad 2\theta_{K}\equiv\arg\frac{M^{K}_{12}}{(M^{K}_{12})_{\rm SM}}\,.$
(43)

The small phase
$\epsilon^{\prime}\equiv\arg(-V_{us}V_{cd}V_{ud}^{*}V_{cs}^{*})$ aleksan ,
contributing as $-2\epsilon^{\prime}$ inside the sine in (41), has been
neglected.

###  III.7 $K$ meson system: $\epsilon_{K}$

The $K$ meson system was the first to show measurable CP violation in the SM.
The value of $|\epsilon_{K}|\sim 10^{-3}$ quantifies this indirect CP
violation. The short-distance contribution for the mixing amplitude can be
calculated in the presence of a VLQ as (see Ref. saavedra:flavor  and
references therein)

$M_{12}^{K}=\frac{G_{F}^{2}m_{W}^{2}f^{2}_{K}\hat{B}_{K}m_{K^{0}}}{12\pi^{2}}\left[(\lambda^{c}_{ds})^{2}\eta^{K}_{cc}S_{0}(x_{c})+(\lambda^{t}_{ds})^{2}\eta^{K}_{tt}S_{0}(x_{t})+2\lambda^{c}_{ds}\lambda^{t}_{ds}\eta^{K}_{ct}S_{0}(x_{c},x_{t})+\Delta_{K}\right]\,,$
(44)

where

$\Delta_{K}=-8X_{ds}[\lambda^{c}_{ds}\eta^{K}_{Z}Y_{0}(x_{c})+\lambda^{t}_{ds}\eta^{K}_{tt}Y_{0}(x_{t})]+\frac{4\pi
s^{2}_{w}}{\alpha_{e}}\eta^{K}_{Z}X^{2}_{ds}\,;$ (45)

the last term being the additional contribution from the VLQ. The
$\eta^{K}_{ij}$ are QCD correction factors, $\hat{B}_{K}=0.717(24)$ PDG  is
the bag parameter.

The contribution to the CP violating parameter $\epsilon_{K}$ is given by

$\epsilon_{K}=\kappa_{\epsilon}e^{i\phi_{\epsilon}}\frac{\text{Im}M_{12}^{K}}{\sqrt{2}\Delta
m_{K}}\,,$ (46)

where $\kappa_{\epsilon}\simeq 0.94(2)$ buras:epsK  includes effects of
$\Delta s$ operators and corrections of deviation of
$\phi_{\epsilon}=43.52(5)^{\circ}$ pdg  from $45^{\circ}$. We use
buras.celis.17  $\eta^{K}_{cc}=1.87(76)$, $\eta^{K}_{tt}=0.5765(65)$,
$\eta^{K}_{ct}=0.496(47)$ for the QCD correction factors at NNLO. We also use
$\eta^{K}_{Z}=0.60$ saavedra:flavor . The theoretical prediction for $\Delta
m_{K}$ has large error in long-distance contributions and will not be
considered. The experimental value is considered in (46) instead.

###  III.8 $K_{L}\to\mu\mu$

A strong limit on $\lambda^{t}_{sd}$ and $X_{sd}$ comes from the rare decay
$K_{L}\to\mu^{+}\mu^{-}$. Although clean theoretically, the branching ratio
$Br(K_{L}\to\mu^{+}\mu^{-})\sim 10^{-8}$, containing contributions from a
dispersive part, $\operatorname{\mathrm{Re}}^{2}(A)$, and a absorptive part,
$\operatorname{\mathrm{Im}}^{2}(A)$, is known to be dominated by the latter,
which can be calculated from $Br(K_{L}\to\gamma\gamma)$. The theoretical
prediction for the dispersive part contains large errors from long distance
contributions. So we consider the simple bound on the short distance (SD)
contribution isidori:KLmumu

$Br(K_{L}\to\mu^{+}\mu^{-})_{\rm SD}<2.5\times 10^{-9}\,.$ (47)

A more precise constraint would come by considering the quantity $\chi_{SD}$
isidori:KLmumu  but we limit ourselves to the constraint above.

The short distance part can be calculated by relating it to the decay
$K^{+}\to\mu^{+}\nu$ as saavedra:flavor :

$\frac{Br(K_{L}\to\mu^{+}\mu^{-})_{\rm
SD}}{Br(K^{+}\to\mu^{+}\nu)}=\frac{\tau_{K_{L}}}{\tau_{K^{+}}}\frac{\alpha_{e}^{2}}{\pi^{2}s_{w}^{4}|V_{us}|^{2}}\left[Y_{\rm
NL}\operatorname{\mathrm{Re}}(\lambda^{c}_{sd})+\eta^{Y}_{t}Y_{0}(x_{t})\operatorname{\mathrm{Re}}(\lambda^{t}_{sd})+\Delta_{K_{L}}\right]\,.$
(48)

We use the factor $Y_{\rm NL}=(2.94\pm 0.28)\times 10^{-4}$ at NLO
saavedra:flavor ; buchalla.99  and the QCD correction $\eta^{Y}_{t}=1.012$
saavedra:flavor ; buchalla.93 . Other input values can be found in table 3.
The VLQ contribution is saavedra:flavor

$\Delta_{K_{L}}=C_{U2Z}\operatorname{\mathrm{Re}}X_{sd}\,,$ (49)

where $C_{U2Z}=-\pi s^{2}_{w}/\alpha$ buras:KL .

###  III.9 $K$ meson system: $\epsilon^{\prime}/\epsilon$

The value for $\epsilon^{\prime}/\epsilon$ quantifying direct CP violation in
the SM was until recently predicted to be significantly below the current
experimental world average from NA48 eps':na48  and KTeV eps':ktev
collaborations:

$(\epsilon^{\prime}/\epsilon)_{\rm exp}=(16.6\pm 2.3)\times 10^{-4}\,;$ (50)

see buras:epsp:19  for a summary of the discrepancy before April 2020.
However, the recent 2020 calculation of Ref. buras:epsp:20  reports

$(\epsilon^{\prime}/\epsilon)_{\rm SM}^{(9)}=(13.9\pm 5.2)\times 10^{-4}\,,$
(51)

which is well compatible with the experimental value. This value takes into
account recent lattice calculations of hadronic matrix elements epsp:lattice ,
isospin breaking effects epsp:pich  which include the effects of the nonet of
lowest-lying mesons and NNLO QCD corrections to EW penguin contributions.

The compatibility weakens the possibility for new physics (NP) buras:epsp:19
as the source of deviation but, due to the large theoretical error, there is
still room for a large NP contribution, roughly of the order of buras:epsp:20

$-4\times 10^{-4}\lesssim\left(\frac{\epsilon^{\prime}}{\epsilon}\right)_{\rm
NP}\lesssim+10\times 10^{-4}\,.$ (52)

We take this interval as a $1\sigma$ range in our fits.

The NP contribution coming from the VLQ can be computed from the simplified
formula buras.celis.17

$\left(\frac{\epsilon^{\prime}}{\epsilon}\right)_{\rm
NP}=P_{7}\operatorname{\mathrm{Im}}(C^{sd}_{7})\,,$ (53)

where we consider only the dominant contribution coming from the operator
$Q_{7}$. The Wilson coefficient at the electroweak scale

$C^{sd}_{7}=-\frac{\alpha}{6}\frac{\pi}{G_{F}^{2}m_{W}^{2}M_{B}^{2}}\frac{\tilde{Y}^{B}_{s}{\tilde{Y}^{B^{*}}_{d}}}{\lambda^{u}_{sd}}\,,$
(54)

depends on the Yukawa coefficients defined in eq. (24). The coefficient in
front is

$P_{7}=-102.02-1.32B_{6}^{(1/2)}+2040.38B_{8}^{(3/2)}\,,$ (55)

where we use $B_{6}^{(1/2)}=1.36\pm 0.23$ and $B_{8}^{(3/2)}=0.79\pm 0.05$
buras:epsp:20  without considering the errors.

##  IV Numerical results

###  IV.1 Methods

Here we describe the fit procedure. We use the same set of observables to
perform the fit in three cases: (a) SM, (b) addition of one NB-VLQ and (c)
addition of one generic VLQ. Therefore, these three cases can be compared
directly. The fit is performed for $M_{B}=1.4\,\mathrm{TeV}$, as allowed by
direct search constraints (25).

The observables considered in the fit are listed in table 2. For most of the
observables, we assume gaussian likelihood and consider

$\chi^{2}=-2\ln\mathcal{L}=\sum_{i}\frac{\left(\mathcal{O}_{i\,\rm
theo}-\mathcal{O}_{i\,\rm exp}\right)^{2}}{\sigma_{i}^{2}}\,,$ (56)

where $\mathcal{O}_{i\,\rm theo}$ is the functional form predicted by the
model, $\mathcal{O}_{i\,\rm exp}$ is the central experimental value with error
$\sigma_{i}$. The exception is the observable $Br(K_{L}\to\mu^{+}\mu^{-})$,
listed in the end of table, which is treated differently from (56). We include
it through the function

$f(x;a,b,r)=\begin{cases}\left(\frac{a-x}{r(b-a)}\right)^{2},&x\leq a,\cr
0,&a<x<b\cr\left(\frac{x-b}{r(b-a)}\right)^{2},&x\geq b,\cr\end{cases}\,,$
(57)

which is flat in the range $x\in[a,b]$ but grows quadratically outside that
range. The rate of growth is controlled by $r$, chosen to be $r=0.01$. For $x$
being $Br(K_{L}\to\mu^{+}\mu^{-})$, this function is added to the $\chi^{2}$
in (56) with the ranges $[a,b]$ listed in table 2. The reason for the
different treatment of this observable is that it is still subjected to large
theoretical uncertainties but, nevertheless, strongly constrain certain BSM
parameters. Therefore, we choose a function which does not contribute to the
$\chi^{2}$ if the observable lies within the allowed range but quickly
disfavors values which are outside that range.

Regarding the absolute values of the CKM elements in (26), we do not consider
the experimental values for $|V_{td}|$ and $|V_{ts}|$, and instead we use the
measured values for $\Delta m_{B_{q}}$, $q=d,s$, which receive contributions
from the VLQ. We also symmetrize the errors and inflate the $1\sigma$ ranges
of (26) by 50% because the high precision of the measurements of the first row
of the CKM matrix hints to unitarity violation if taken at face value
ramsey:Vud ; berezhiani.19 ; cheung.20 .

The input parameters are listed in table 3. The parameters
$f_{B_{d}}{\hat{B}_{B_{d}}}^{\nicefrac{{1}}{{2}}}$,
$f_{B_{s}}{\hat{B}_{B_{s}}}^{\nicefrac{{1}}{{2}}}$, $f_{B_{s}}$ listed in this
table, and $\eta^{K}_{cc}$ described in Sec. III.7, have large theoretical
errors and thus are considered as nuisance parameters which are also included
in the fit as in (56). We also consider the correlation between the first two
of these nuisance parameters.

$\begin{array}[]{|c|c|c|c|}\hline\cr\text{Observable}~{}~{}\mathcal{O}_{i}&\text{Experimental}&\text{SM
(c.v [95\%] CL)}\\\
\hline\cr|V_{ij}|&\text{Eq.\,}\eqref{|V|:exp}^{a}&\text{--}\\\ \hline\cr\Delta
m_{B_{d}}\;[\mathrm{ps^{-1}}]&0.5065(19)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&0.5060\;[0.5031,0.5104]\\\
\hline\cr\Delta
m_{B_{s}}\;[\mathrm{ps^{-1}}]&17.749(21)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&17.747\;[17.711,17.787]\\\
\hline\cr Br(B_{s}\to\mu^{+}\mu^{-})&3.0(4)\times
10^{-9}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&(3.40\;[3.13,3.67])\times
10^{-9}\\\ \hline\cr|\epsilon_{K}|&2.228(11)\times
10^{-3}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&(2.227\;[2.206,2.248])\times
10^{-3}\\\ \hline\cr S_{\psi K_{S}}~{}[\sin
2\beta_{d}]&0.699(17)\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&0.71\;[0.68,0.74]\\\
\hline\cr R_{b}&0.21629(66)\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Rb:Exp}{\@@citephrase{(}}{\@@citephrase{)}}}}&0.21582(11)\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Gfitter:Rb}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr(\epsilon^{\prime}/\epsilon)_{\rm NP}&[-4,10]\times
10^{-4}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{buras:epsp:20}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{--}\\\
\hline\cr\hline\cr Br(K_{L}\to\mu^{+}\mu^{-})_{\rm SD}&<2.5\times
10^{-9}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{isidori:KLmumu}{\@@citephrase{(}}{\@@citephrase{)}}}}&(0.86\;[0.76,0.95])\times
10^{-9}\\\ \hline\cr\end{array}$ Table 2:  Observables considered in the fit.
a We exclude $|V_{td}|,|V_{ts}|$, symmetrize the errors and inflate them in
50%; see text. The last column is taken from our fit, except for $R_{b}$,
which is not considered in the SM fit.  $\begin{array}[]{|c|c|}\hline\cr
G_{F}\;[\mathrm{GeV^{-2}}]&1.1663787(6)\times
10^{-5}\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
s_{w}^{2}(m_{Z})~{}(\overline{\text{MS}})&0.23121(4)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
m_{W}\;[\mathrm{GeV}]&80.379(12)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
m_{Z}\;[\mathrm{GeV}]&91.1876(21)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\alpha(m_{Z})^{-1}~{}{(\overline{\text{MS}})}&127.952(9)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\alpha_{s}(m_{Z})&0.1179(10)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
m_{t}(m_{t})\;[\mathrm{GeV}]&161.9\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{marquard}{\@@citephrase{(}}{\@@citephrase{)}}}}^{a}\\\
\hline\cr
m_{c}(m_{c})\;[\mathrm{GeV}]&1.27\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
y_{d}(m_{Z})&1.58\times 10^{-5}\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{antusch}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr y_{s}(m_{Z})&3.13\times 10^{-4}\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{antusch}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr y_{b}(m_{Z})&1.639\times
10^{-2}\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{antusch}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
<EMAIL_ADDRESS>fit}}}\hfil\lx<EMAIL_ADDRESS>\hline\cr
A&0.820^{+0.012}_{-0.014}\\\ \hline\cr\lambda&0.22655^{+0.00060}_{-0.00066}\\\
\hline\cr 10^{5}\times J&3.18^{+0.10}_{-0.12}\\\
\hline\cr\end{array}\begin{array}[]{|c|c|}\hline\cr
m_{B_{d}}\;[\mathrm{MeV}]&5279.58(17)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
m_{B_{s}}\;[\mathrm{MeV}]&5366.88(16)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
f_{B_{d}}{\hat{B}_{B_{d}}}^{\nicefrac{{1}}{{2}}}\;[\mathrm{MeV}]&225(9)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
f_{B_{s}}{\hat{B}_{B_{s}}}^{\nicefrac{{1}}{{2}}}\;[\mathrm{MeV}]&274(8)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\rho(f_{B_{d}}{\hat{B}_{B_{d}}}^{\nicefrac{{1}}{{2}}},f_{B_{s}}{\hat{B}_{B_{s}}}^{\nicefrac{{1}}{{2}}})&0.951\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{corr:fBds,buras.celis.17}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\tau_{B_{s}}\;[\mathrm{ps}]&1.515(4)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
f_{B_{s}}\;[\mathrm{MeV}]&228.4(3.7)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{flag.19}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr
m_{K^{0}}\;[\mathrm{MeV}]&497.611(13)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\Delta m_{K}\;[\mathrm{MeV}]&3.484(6)\times
10^{-12}\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
f_{\pi}\;[\mathrm{MeV}]&130.2(1.2)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\ \hline\cr
f_{K}/f_{\pi}&1.192(2)\,\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{flag.19}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
<EMAIL_ADDRESS>fit}}}\hfil\lx<EMAIL_ADDRESS>\hline\cr\overline{\rho}&0.154^{+0.009}_{-0.012}\\\
\hline\cr\overline{\eta}&0.351^{+0.010}_{-0.008}\\\ \hline\cr&\\\
\hline\cr\end{array}$ Table 3:  Input parameters. Some parameters for specific
observables are listed in the text. a We consider QCD corrections up to four
loops with $m_{t}^{\rm pole}=172.4(7)\,\mathrm{GeV}$ PDG  and
$\alpha_{s}(m_{t}^{\rm pole})=0.1077$. The last two rows list the Wolfenstein
parameters from the SM fit.

To perform the minimization of the $\chi^{2}$ function and sampling the nearby
region we use MultiNest multinest  accessed through the Python wrapper
PyMultiNest py.multi .

The Wolfenstein parameters obtained from the SM fit are also listed in table
3. We can see they are similar to the fit from PDG in (29), although the
values for $A,\bar{\rho}$ and $J$ are larger.

The 95% CL interval for the observables considered in the fit are also shown
in table 2. We can see that the agreement with the experimental values are
excellent. Moreover, our result for $B_{s}^{0}\to\mu^{+}\mu^{-}$ is compatible
with $Br(B_{s}^{0}\to\mu^{+}\mu^{-})=3.23(27)\times 10^{-9}$ of Ref.
Bs->mumu:theo  as well as with
$Br(B_{s}^{0}\to\mu^{+}\mu^{-})=(3.41\;[3.01,3.81])\times 10^{-9}$ from the
fit in Ref. buras.celis.17 . Our intervals for $\Delta m_{B_{d}}$ and $\Delta
m_{B_{s}}$ are also compatible with Refs. buras.celis.17 ; deltaM . The value
of $\epsilon_{K}$ agrees with eps.K .

###  IV.2 Results

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\text{Observable}~{}~{}\mathcal{O}_{i}&\text{Experimental}&\text{SM
(c.v [95\% CL])}&\text{NB-VLQ}&\text{gen-VLQ }\\\
\hline\cr\rule{0.0pt}{20.00003pt}|V_{tb}|&\parbox{40.00006pt}{1.013(30)\\\
\text{[eq.\,\eqref{|V|:exp}]}}&\parbox{93.00017pt}{0.99910 \\\ [0.99905,
0.99916]}&[0.9982,0.9992]&[0.9981,0.9991]\\\ \hline\cr
10^{10}{\times}Br(B_{d}^{0}\to\mu^{+}\mu^{-}){}^{a}&1.1^{+1.4}_{-1.3}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PDG}{\@@citephrase{(}}{\@@citephrase{)}}}}&1.08\;[0.99,1.16]&[0.75,2.15]&[0.0,8.9]\\\
\hline\cr 10^{10}{\times}Br(K_{L}\to\mu^{+}\mu^{-})_{\rm
SD}&<25~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{isidori:KLmumu}{\@@citephrase{(}}{\@@citephrase{)}}}}&8.6\;[7.6,9.5]&[4.4,13.7]&[0,25]\\\
\hline\cr 10^{4}{\times}(\epsilon^{\prime}/\epsilon)_{\rm
NP}&[-4,10]~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{buras:epsp:20}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{--}&[-6.0,6.6]&[-11,17]\\\
\hline\cr\end{array}$ Table 4:  Observables predicted from the fit. The last
two columns show the intervals at 95% CL. a We use the central value for
$f_{B_{d}}=192.0(4.3)$ flag.19  without the error.

In table 4 we show observables that are predicted from the fit. We extract
only the interesting observables and compare the case of SM, one NB-VLQ and
one generic VLQ. For all these cases we show the $95\%$ CL interval for
$M_{B}=1.4\,\mathrm{TeV}$.

We can see that the value for $|V_{tb}|$ can indeed be lowered with the
presence of one VLQ saavedra:flavor  but probing it will require a precision
which is hard to attain.

For the rare decay $B_{d}^{0}\to\mu^{+}\mu^{-}$, our value obtained for the SM
is very close to the one obtained in Ref. Bs->mumu:theo ,
$Br(B_{d}^{0}\to\mu^{+}\mu^{-})=1.06(9)\times 10^{-10}$. In the presence of
one generic VLQ, a large interval is still allowed. For the NB case, the
allowed interval is larger than the SM but narrower than the generic case.

For the kaon decay $K_{L}\to\mu^{+}\mu^{-}$, all the values up to the upper
limit can be reached for the generic VLQ case but for the NB-VLQ the interval
shrinks to a narrower interval. Both these intervals are expectedly wider than
for the SM.

A similar conclusion also applies to the VLQ contribution to
$\epsilon^{\prime}/\epsilon$: a large contribution is still possible for the
generic case but a much smaller contribution is possible for the NB case. In
fact, in the NB case, the predicted maximal contribution at 95% CL is smaller
than allowed experimentally. The obtained intervals differ from Ref.
buras.celis.17  because of the updated calculation reported in Ref.
buras:epsp:20 .

We show in Figs. 1, 2 and 3 the allowed regions for the combination
$\tilde{Y}^{B}_{i}\tilde{Y}^{B*}_{j}$, cf. (24), in the complex plane for the
sectors $(ij)=(sd),(bd),(bs)$, respectively. These quantities were denoted as
$\Lambda_{ij}$ in Ref. buras.celis.17  and can be compared here. The region in
yellow (orange) corresponds to the generic VLQ case at 95% CL (68% CL) while
the region in blue corresponds to the NB-VLQ case at $95\%$ CL. In all sectors
the allowed regions for the NB-VLQ case are much smaller than the generic case
due to the correlations of the model. In the sectors $(sd)$ and $(bd)$, the
95% CL region for the NB case is even smaller than the 65% CL region of the
generic case. For comparison, we also show in a dashed contour the approximate
95% CL regions from the global fit in Ref. buras.celis.17  which used
$M_{B}=1\,\mathrm{TeV}$ (our value is $M_{B}=\,\mathrm{1.4}\,\mathrm{TeV}$).
Some comments are in order.

  * •

For the $(sd)$ sector, we use the most constraining observables as in Ref.
buras.celis.17 : $Br(K_{L}\to\mu\bar{\mu})_{\rm SD}$ and
$(\epsilon^{\prime}/\epsilon)_{\rm NP}$. The region is constrained
horizontally and vertically, respectively, by these observables. Our
implementation of $Br(K_{L}\to\mu\bar{\mu})_{\rm SD}$, however, is different
and we obtain a region wider horizontally. Vertically, we consider an updated
(2020) constraint for $(\epsilon^{\prime}/\epsilon)_{\rm NP}$ buras:epsp:20
which differs from the constraint in Ref. buras.celis.17  of 2016. The net
result is an upward shift in the region. The region is possibly wider due to
the change in the VLQ mass from 1 TeV to 1.4 TeV.

  * •

For the $(bd)$ sector, our region for the generic case is much larger than the
region found in Ref. buras.celis.17  because we do not include the observable
$Br(B^{+}\to\pi^{+}\mu\bar{\mu})$ which is the most constraining for that mass
range. Nevertheless, the regions are roughly compatible and our region for NB
case is entirely contained.

  * •

For the $(bs)$ sector, we use the most constraining observables as in Ref.
buras.celis.17 : $Br(B_{s}\to\mu\bar{\mu})$. The obtained region for the
generic case is largely compatible, with possible enlargement due to the
increase in $M_{B}$. For the NB case, the region is much narrower in the
imaginary part.

Figure 1: Allowed regions for $\tilde{Y}^{B}_{s}\tilde{Y}^{B*}_{d}$ for one
generic VLQ (95% CL in yellow, 68% CL in orange) and one NB-VLQ (95% CL in
blue) for $M_{B}=1.4\,\mathrm{TeV}$. The approximate 95% CL region extracted
from the global fit in Ref. buras.celis.17  for $M_{B}=1\,\mathrm{TeV}$ is
also shown (dashed contour).  Figure 2:  Allowed regions for
$\tilde{Y}^{B}_{b}\tilde{Y}^{B*}_{d}$ for one generic VLQ (95% CL in yellow,
68% CL in orange) and one NB-VLQ (95% CL in blue) for
$M_{B}=1.4\,\mathrm{TeV}$. The approximate 95% CL region extracted from the
global fit in Ref. buras.celis.17  for $M_{B}=1\,\mathrm{TeV}$ is also shown
(dashed contour).  Figure 3:  Allowed regions for
$\tilde{Y}^{B}_{b}\tilde{Y}^{B*}_{s}$ for one generic VLQ (95% CL in yellow,
68% CL in orange) and one NB-VLQ (95% CL in blue) for
$M_{B}=1.4\,\mathrm{TeV}$. The approximate 95% CL region extracted from the
global fit in Ref. buras.celis.17  for $M_{B}=1\,\mathrm{TeV}$ is also shown
(dashed contour).

Note that, after we factor $V_{d_{L}}$, the mixing $\Theta_{i}$ in (15) of up-
quarks with the heavy $B$ quark through $W$ exchange is given by

$\Theta_{i}=\frac{v}{\sqrt{2}}\tilde{Y}^{B}_{i}/M_{B},$ (58)

i.e., it is proportional to $\tilde{Y}^{B}_{i}$ in (24); see eqs. (16) and
(17). So the constraints in Figs. 1, 2 and 3 can be interpreted as constraints
on $\Theta_{i}\Theta_{j}^{*}$. Similarly, flavor changing neutral currents
couple to the $Z$ boson through $X^{d}$ in (18), which is proportional to

$X^{d}_{ij}\approx-\Theta_{i}\Theta_{j}^{*}=-\frac{v^{2}}{2M_{B}^{2}}(\tilde{Y}^{B}_{i}\tilde{Y}^{B*}_{j})\,,\quad
i\neq j\,.$ (59)

The approximation refers to the leading seesaw approximation. The factor
inside the parentheses is exactly the quantity appearing in Figs. 1, 2 and 3.
Indeed, if we link the flavor changing physics scale in (1) to
$1/\Lambda_{ij}^{2}\sim|\tilde{Y}^{B}_{i}\tilde{Y}^{B*}_{j}|/M_{B}^{2}$, apart
from a global factor, we can extract from the figures the milder hierarchies
for the NB case,

$\Lambda_{sd}^{-1}:\Lambda_{bd}^{-1}:\Lambda_{bs}^{-1}\sim 0.18:0.4:1\,.$ (60)

In part, this is due to a larger set of observables we consider, which is not
restricted to $\Delta F=2$ observables. It is important to emphasize that the
NB scheme leads to a _hierarchy_ of $\tilde{Y}^{B}_{i}$ inherited from the SM
Yukawa couplings but the overall scale is not determined nb-vlq . In special,
such a scale is not suppressed by the bottom Yukawa, such as in the MFV
setting, and is only limited by perturbativity from the theoretical side while
flavor observables in the $(bs)$ sector, cf. Fig. 3, allows
$|\tilde{Y}^{B}_{b}|\sim 0.3$ for $M_{B}=1.4\,\mathrm{TeV}$, considering that
$|\tilde{Y}^{B}_{b}\tilde{Y}^{B*}_{s}|\sim|V_{ts}(\tilde{Y}^{B}_{b})^{2}|$.
For larger $M_{B}$, larger values are allowed. See appendix B for approximate
formulas for $\tilde{Y}^{B}$.

There is one frequently considered simplified scenario where the combinations
appearing in Figs. 1, 2 and 3 all vanish and no constraint can be extracted.
That is the case where the heavy quark $B$ only couples with the third SM
family saavedra:handbook  and

$\tilde{Y}^{B}=(0,0,y_{B})^{\mbox{\scriptsize$\mathsf{T}$}}\,.$ (61)

Similar conclusions apply to scenarios where $B$ couples only to the first
family or only to the second family. For these cases, it is more interesting
to show directly the constraint on $|V_{iB}|\approx|(V_{d_{L}}\Theta)_{i}|$,
which also quantifies the deviation from unitarity for the $3\times 3$ sub-
block. We show the allowed regions in Fig. 4. The color code is the same as in
Fig. 1. We can see that for the generic case (yellow/orange) all three
$|V_{iB}|$ can reach values of the same order of magnitude (not
simultaneously) but for the NB case (blue), the allowed values are
hierarchical for $i=t,c,u$. The latter pattern is compatible with the
hierarchy (20) but it follows (21) only roughly. Therefore, it is clear that
$|V_{tB}|$ (similarly $|Y^{B}_{3}|$) can reach values as large as in the
generic case but the other mixings (Yukawas) are automatically suppressed.

Figure 4: Allowed regions of $|V_{iB}|$ for one generic VLQ (95% in yellow,
68% in orange) and one NB-VLQ (95% CL in blue).

The constraint for the mixing $|V_{iB}|$ has another advantage: it is more
applicable to different VLQ masses. The reason is that our flavor observables
are essentially sensitive to $X^{d}$ which depends on the CKM matrix $V$. The
mixing $V_{iB}$, however, scales approximately as (17) and a larger mass
implies a smaller mixing for the same Yukawa $Y^{B}$. So the constraints in
Figs. 1, 2 and 3 can be approximately adapted to other VLQ masses after
rescaling while the constraints on Fig. 4 can be approximately applied to any
mass. 121212This scaling breaks down for masses larger than tens of TeV due to
the one-loop box contribution that starts to dominate the mixings of neutral
mesons ligeti.wise .

Concerning CP violation, we show in Fig. 5 the 95% CL intervals for some
Jarlskog invariants involving only the $3\times 3$ sub-block of CKM. The
definition we use is nb-vlq

$J_{ijkl}=\operatorname{\mathrm{Im}}[V_{ij}V^{\dagger}_{jk}V_{kl}V^{\dagger}_{li}]\,.$
(62)

The usual Jarlskog invariant for the SM is $J_{uscb}=J_{1223}$ and we show in
the gray band the allowed 95% interval obtained from the fit. With the
presence of one VLQ, the CKM matrix becomes $3\times 4$ and the Jarlskog
invariants are not all the same. We also show the intervals for $J_{1223}$ and
$J_{2233}$ for a generic VLQ and for a NB-VLQ. The invariant $J_{2233}$ was
shown in Ref. nb-vlq  to present a larger variation for one NB-VLQ. We can
indeed see in Fig. 5 that the interval for $J_{2233}$ with one VLQ is larger
than in the SM for both the generic and the NB case. The deviation of
$J_{2233}$ from the SM value we see in Fig. 5 could be tested in the future by
a more precise determination of $\phi_{s}$ proportional to the angle between
$V_{cs}V_{cb}^{*}$ and $V_{ts}V_{tb^{*}}$ which enters precisely in
$J_{2233}=\operatorname{\mathrm{Im}}[V_{cs}V^{*}_{ts}V_{tb}V^{*}_{cb}]$.
Currently, the errors are no smaller than 40% phi_s  but the precision at LHCb
at High-luminosity LHC with 300 fb-1 is expected to be around 10%. LHCb.high .

Figure 5: 95% CL intervals for Jarlskog invariants, cf. (62), for the case of
one generic VLQ and one NB-VLQ. The gray band is the 95% CL interval for the
SM.

Concerning the 5 BSM parameters (13) of our seesaw parametrization for one NB-
VLQ, there are essentially no constraints on the parameters
$b,\beta_{1},\beta_{2},\gamma$. The mass $M_{B}$ can be constrained in direct
searches in colliders but are not directly constrained by flavor observables
which are dependent on the mixing between the SM quarks and the VLQ.

For a qualitative comparison, we briefly discuss the case of one up-type NB-
VLQ in appendix D. The salient features are the same: the Yukawa couplings
$Y^{T}_{i}$ of the heavy up-type VLQ $T_{L,R}$ with the SM quarks are
_hierarchical_ reflecting the hierarchy in the up-type Yukawas and in the CKM
mixing. For this reason, for $M_{T}\sim 1.3\,\mathrm{TeV}$, obeying flavor
diagonal electroweak precision constraints automatically suppresses flavor
changing observables involving lighter families.

##  V Conclusions

By performing a global fit on interesting low energy flavor observables we
constrain the possibilities for the presence of one VLQ of charge $-1/3$. We
compare the case of one generic VLQ and one VLQ of Nelson-Barr type. Only the
latter would be responsible for transmitting the spontaneous CP violation from
a CP breaking sector to the SM in the Nelson-Barr setting. We see that, using
flavor observables, distinguishing the case of one NB-VLQ from a generic VLQ
of down type will be a challenging task. Nevertheless the presence of one NB-
VLQ is definitely falsifiable once a VLQ is found by other means and its
flavor signature lies in a region of parameter space outside of what is
allowed for one NB-VLQ but within the region allowed for one generic VLQ. The
results presented in table 4 and Figs. 1, 2 and 3 show that the parameter
space constrained by flavor observables is significantly smaller for the NB
case compared to the generic case due to the inherent correlations that appear
in the model, a feature which was uncovered in Ref. nb-vlq .

In the process, we update the allowed parameter space for the presence of one
generic VLQ, which can be directly compared to the regions obtained in Ref.
buras.celis.17 . These constraints should apply to all limiting cases where
one VLQ of down type is present. For example, the often considered case of a
VLQ coupling only to the third family is just a subcase and must have the
allowed parameter space contained in the generic region. On the other hand,
the NB case is a specific example of a scenario where the VLQ couples
dominantly with the third family but also couples to the lighter families with
hierarchically smaller couplings.

In summary, scenarios involving VLQs of Nelson-Barr type are simple scenarios
where the presence of VLQs are motivated to solve the strong CP problem.
Distinguishing a VLQ of NB type from a VLQ unrelated to the origins of CP
breaking in nature is a very difficult but interesting task that warrants
further investigation.

###### Acknowledgements.

C.C.N. acknowledges partial support by Brazilian Fapesp, grant 2014/19164-6,
and CNPq, grant 304262/2019-6. G.D.C. acknowledges financial support by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001. We thank Andre Lessa for helpful comments.

##  Appendix A Inversion formula

Here we briefly review the inversion formula for (8c) developed in Ref. nb-vlq
for the seesaw parametrization of one NB-VLQ. The Yukawa couplings $Y^{d}$ in
(8c) and $Y^{B}$ in (8b) depend on the quantities $\mathscr{Y}^{d}$ and $w$
but $Y^{d}$ should be in accord with SM input (9). The inversion formula
allows us to take the latter into account automatically, leaving
$\mathscr{Y}^{d}$ and $w$ dependent on $\\{b,\gamma,\beta_{1},\beta_{2}\\}$.
The VLQ mass $M_{B}$ completes the five free BSM parameters.

Given $\beta_{1},\beta_{2}$ in (12), $\mathscr{Y}^{d}$ in (8c) can be
determined as

$\mathscr{Y}^{d}=A\mathcal{O}B^{-1}$ (63)

where

$\displaystyle A$
$\displaystyle=\Big{(}\operatorname{\mathrm{Re}}(Y^{d}{Y^{d}}^{\dagger})\Big{)}^{1/2}\,,$
(64) $\displaystyle B$
$\displaystyle=\operatorname{\mathrm{diag}}(1,\sqrt{1-b^{2}},\sqrt{1-a^{2}})\,,$

and the orthogonal matrix $\mathcal{O}$ will be defined below. The matrix $B$
contains one free parameter, chosen to be $b$, while $a$ is fixed from

$\frac{a}{\sqrt{1-a^{2}}}\frac{b}{\sqrt{1-b^{2}}}=\mu\,.$ (65)

The value of $0<\mu<1$ comes from the real antisymmetric matrix

$C\equiv A^{-1}\operatorname{\mathrm{Im}}(Y^{d}{Y^{d}}^{\dagger})A^{-1}\,,$
(66)

with eigenvalues $(0,i\mu,-i\mu)$. The real orthogonal matrix $\mathcal{O}$ in
the formula (63) transforms $C$ to the canonical form

$\mathcal{O}^{\mbox{\scriptsize$\mathsf{T}$}}C\mathcal{O}=\mu\begin{pmatrix}1&&\cr&0&-1\cr&1&0\end{pmatrix}\,.$
(67)

This matrix $\mathcal{O}$ is not unique as (67) is invariant by additional
rotation on the 2-3 plane. Then, we can write

$\mathcal{O}=\mathcal{O}_{0}\begin{pmatrix}1&&\cr&\cos\gamma&\sin\gamma\cr&-\sin\gamma&\cos\gamma\end{pmatrix}\,,$
(68)

where $\mathcal{O}_{0}=(e_{1}|e_{2}|e_{3})$ is a representative matrix
satisfying (67). The first column vector $e_{1}$ is fixed as the eigenvector
of $C$ with eigenvalue zero. The sign is determined so that
$\mathscr{Y}^{d}_{11}>0$. For our numerical code, we choose $e_{2}$ and
$e_{3}$ so that $u_{-}=e_{2}+ie_{3}$ is the eigenvector of $C$ with eigenvalue
$-i\mu$. We choose the convention that $u_{-}$ has real positive first
component so that $e_{3}$ has vanishing first component.

As for the parameter ranges, the angular and phase variables,
$\gamma,\beta_{1},\beta_{2}$, should all be confined to $[0,2\pi]$. The $b$
parameter is confined to the range (14) so that the quality of the seesaw
approximation is guaranteed within 1%.

##  Appendix B Approximate formulas for the Yukawas

The inversion formula reviewed in appendix A solves for $\mathscr{Y}^{d}$ in
(8c) in terms of the SM Yukawa $Y^{d}$ and this relation is exact in the
leading seesaw approximation. We can obtain approximate relations for $Y^{B}$
which makes explicit the hierarchy of these couplings.

Substituting the solution (63) in (8b) we obtain

$Y^{B}=\left[\operatorname{\mathrm{Re}}(Y^{d}{Y^{d}}^{{\dagger}})\right]^{1/2}\mathcal{O}\begin{pmatrix}0\cr
ix_{b}\cr\displaystyle\frac{\mu}{x_{b}}\end{pmatrix}\,,$ (69)

where $x_{b}\equiv\frac{b}{\sqrt{1-b^{2}}}$ and we have used (65). Considering
$Y^{d}$ is hierarchical and ignoring the real part, we obtain for $b\leq
1/\sqrt{2}$ the approximate bound

$|Y^{B}_{i}|\lesssim
y_{b}|V_{ib}|\frac{\mu}{x_{b}}\sim\frac{\mu}{x_{b}}\times(10^{-4},10^{-3},0.024)\,.$
(70)

Since $|V_{iB}|\approx|(\theta_{L})_{i}|=\frac{v}{\sqrt{2}}|Y^{B}_{i}|/M_{B}$,
this expression describes to a good approximation the upper boundary of Fig. 3
in Ref. nb-vlq  as a function of $b$ (typically $\mu\sim 1$). As the CKM
matrix is hierarchical, we can also write
$|\tilde{Y}^{B}_{i}|\sim|Y^{B}_{i}|$. Equation (70) clearly shows the
hierarchy among the couplings $Y^{B}_{i}$. It also shows that _the absolute
scale_ of the Yukawa couplings of the NB-VLQ are _not_ necessarily suppressed
by the bottom Yukawa for $x_{b}\ll 1$. Indeed, our fit showed that it can
reach $|Y^{B}_{3}|\sim y_{b}/x_{b}\sim 0.3$ for $M_{B}=1.4\,\mathrm{TeV}$ in
conformity with other fits of the _generic_ VLQ case buras.celis.17 .

For completeness, we can quantify the quality of the leading seesaw
approximation by considering the deviation of the real heaviest mass
eigenvalue of (6) compared to the input mass $M_{B}$:

$\frac{\delta
M_{B}}{M_{B}}\approx\mbox{\large$\tfrac{1}{2}$}|\theta_{L}|^{2}\lesssim\frac{m_{b}^{2}}{2M^{2}_{B}}\frac{\mu^{2}}{x^{2}_{b}}\,.$
(71)

The mixing $\theta_{L}$ is related to $Y^{B}$ by (17). With this relation, one
can limit the maximum deviation allowed by setting a lower limit for $b$.

##  Appendix C Auxiliary functions

The Inami-Lim functions inami.lim  are

$\displaystyle S_{0}(x,y)$
$\displaystyle=\frac{x\left(x^{2}-8x+4\right)y\log(x)}{4(x-1)^{2}(x-y)}+\frac{x\left(y^{2}-8y+4\right)y\log(y)}{4(y-1)^{2}(y-x)}-\frac{3xy}{4(x-1)(y-1)}$
(72) $\displaystyle S_{0}(x)$
$\displaystyle=\frac{x^{3}-11x^{2}+4x}{4(1-x)^{2}}-\frac{3x^{3}\log(x)}{2(1-x)^{3}}\,,$
$\displaystyle Y_{0}(x)$
$\displaystyle=\frac{3x^{2}\log(x)}{8(x-1)^{2}}-\frac{3x}{8(x-1)}+\frac{x}{8}\,.$

We follow the convention of Ref. saavedra:flavor . Note that
$S_{0}(x)=\lim_{y\to x}S_{0}(x,y)$.

##  Appendix D One up-type NB-VLQ

In this case, the down-type VLQs $B_{L,R}$ are replaced by up-type VLQs
$T_{L,R}$ and the Lagrangians (4) (real basis) and (5) (generic basis) can be
adapted accordingly. The Yukawa couplings that will be constrained by flavor
observables are

$-\mathscr{L}\subset\tilde{Y}^{T}_{i}\bar{q}_{iL}\tilde{H}T_{R}\,,$ (73)

in the basis where $Y^{u}$ is diagonal and $Y^{d}$ carries the CKM of the SM.
Note that $T$ in $\tilde{Y}^{T}$ is a label and not the transpose operation.
To avoid confusion, we will omit this label in the following discussion.
Similarly, $Y=Y^{T}$ without the tilde is the same Yukawa in the basis where
$Y^{d}$ is diagonal. Because of CKM hierarchy, $|\tilde{Y}_{i}|\sim|Y_{i}|$.

The full CKM matrix will be $4\times 3$ of the form

$V\approx\left(\begin{array}[]{c}{\mathbbm{1}}_{3}-\mbox{\large$\tfrac{1}{2}$}\Theta\Theta^{\dagger}\\\
\hline\cr\Theta^{\dagger}\end{array}\right)V_{u_{L}}^{\dagger}\,,$ (74)

where $V_{u_{L}}^{\dagger}=V_{ckm}$ is the $3\times 3$ CKM matrix within the
SM, except for possible two additional phases. The coupling $\tilde{Y}$
controls the mixing of the VLQ with the SM quarks as, in leading order,

$\Theta_{i}=\frac{v}{\sqrt{2}}\tilde{Y}_{i}M_{T}^{-1}\,.$ (75)

The FCNC coupling to the $Z$ now depends on $X^{u}=VV^{\dagger}$.

In order to reproduce the SM up-type Yukawa couplings, a relation analogous to
(8c) needs to be satisfied and the up-type NB-VLQ will also couple to SM
quarks hierarchically.131313Note that due to $y_{t}\gg y_{b}$, the leading
seesaw approximation is much less precise for an up-type NB-VLQ than for a
down-type NB-VLQ, cf. (71). But once we consider (76), the approximation is
better than $1\%$.  Owing to this hierarchy in the VLQ mixing, we can use the
constraints from the case where the VLQ couples only to the third family. For
example, the oblique parameters $S,T$ constrain saavedra:handbook

$|V_{Tb}|\approx|\Theta_{3}|\lesssim 0.12\,,$ (76)

for $M_{T}>1.3\,\mathrm{TeV}$. This leads to

$|\tilde{Y}_{3}|\lesssim
0.9\times\left(\frac{M_{T}}{1.3\,\mathrm{TeV}}\right)\,.$ (77)

This constraint is stronger than other electroweak observables that deviate
due to modified $Zbb$ coupling.

Adapting the approximate relation (70) to an up-type NB-VLQ, we obtain

$|Y_{i}|\lesssim
y_{t}|V_{ti}^{*}|\frac{\mu}{x_{b}}\sim\frac{\mu}{x_{b}}\times(0.009,0.04,1)\,.$
(78)

Therefore, if we keep the hierarchy in (78),141414Strictly speaking, this
hierarchy may be subjected to an order of magnitude variation as the Yukawas
do not need to saturate the upper limits in (78); see Fig. 3 in Ref. nb-vlq
for the down-type NB-VLQ. In fact, to respect (77), $Y_{3}$ will be far from
saturating the upper limit.  the constraint (77) naturally translates into a
much stronger constraint on the other Yukawas:

$|\tilde{Y}_{1}|\lesssim
0.008\times\left(\frac{M_{T}}{1.3\,\mathrm{TeV}}\right)\,,\quad|\tilde{Y}_{2}|\lesssim
0.04\times\left(\frac{M_{T}}{1.3\,\mathrm{TeV}}\right)\,.$ (79)

So due to the hierarchy of $\tilde{Y}$, analogously to the down-type NB-VLQ,
the Yukawa couplings of the up-type NB-VLQ with lighter quarks are
automatically suppressed. As a result, flavor changing observables involving
lighter quarks may not be as relevant as for a generic up-type VLQ without the
hierarchy in the couplings.

Ref. ligeti.wise  analyzed various flavor observables for all VLQ
representations coupling to the SM. Due to the hierarchy in (78), the
constraint (77) coming from flavor diagonal electroweak precision observables
is more important than flavor violating ones. For example, the decay
$D^{0}\to\mu^{+}\mu^{-}$ constrains the combination ligeti.wise

$|\tilde{Y}_{1}\tilde{Y}_{2}^{*}|<0.11\times\left(\frac{M_{T}}{1.3\,\mathrm{TeV}}\right)^{2}\,.$
(80)

But the combination of the limits in (79) is much stronger. Analogous
considerations apply to other limits.

Although not very important for $M_{T}\sim\mathrm{TeV}$, the one-loop
contributions to neutral meson mixings may become important for larger masses
because the amplitudes scale as $Y^{4}/M_{T}^{2}$ (box diagram) instead of
$Y^{2}/M_{T}^{2}$. Let us collect the results of Ref. ligeti.wise :

$\displaystyle|\operatorname{\mathrm{Im}}Y_{1}Y^{*}_{2}|$
$\displaystyle<0.002\times\frac{M_{T}}{1.3\,\mathrm{TeV}}\,,$ (81)
$\displaystyle|\operatorname{\mathrm{Re}}Y_{1}Y^{*}_{2}|$
$\displaystyle<0.03\times\frac{M_{T}}{1.3\,\mathrm{TeV}}\,,$
$\displaystyle|Y_{1}Y^{*}_{3}|$
$\displaystyle<0.05\times\frac{M_{T}}{1.3\,\mathrm{TeV}}\,,$
$\displaystyle|Y_{2}Y^{*}_{3}|$
$\displaystyle<0.2\times\frac{M_{T}}{1.3\,\mathrm{TeV}}\,.$

Notice the different scaling in $M_{T}$. The last two constraints become more
important than (77), combined with (78), for $M_{T}\sim 7\,\mathrm{TeV}$.

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