aainstitutetext: Dipartimento di Fisica “E. Fermi”, Università di Pisa,
Largo Bruno Pontecorvo 3, I-56127 Pisa, Italybbinstitutetext: INFN, Sezione di
Pisa,
Largo Bruno Pontecorvo 3, I-56127 Pisa, Italyccinstitutetext: Dipartimento di
Fisica e Chimica, Università di L’Aquila,
67100 Coppito, L’Aquila, Italyddinstitutetext: INFN, Laboratori Nazionali del
Gran Sasso,
67010 Assergi, L’Aquila, Italy

# Are the CKM anomalies induced by vector-like quarks? Limits from flavor
changing and Standard Model precision tests

B. Belfatto c,d and Z. Berezhiani<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent high precision determinations of $V_{us}$ and $V_{ud}$ indicate towards
anomalies in the first row of the CKM matrix. Namely, determination of
$V_{ud}$ from superallowed beta decays and of $V_{us}$ from kaon decays imply
a violation of first row unitarity at about $4\sigma$ level. Moreover, there
is tension between determinations of $V_{us}$ obtained from leptonic $K\mu 2$
and semileptonic $K\ell 3$ kaon decays. These discrepancies can be explained
if there exist extra vector-like quarks at the TeV scale, which have large
enough mixings with the lighter quarks. In particular, extra vector-like weak
singlets quarks can be thought as a solution to the CKM unitarity problem and
an extra vector-like weak doublet can in principle resolve all tensions. The
implications of this kind of mixings are examined against the flavour changing
phenomena and SM precision tests. We consider separately the effects of an
extra down-type isosinglet, up-type isosinglet and an isodoublet containing
extra quarks of both up and down type, and determine available parameter
spaces for each case. We find that the experimental constraints on flavor
changing phenomena become more stringent with larger masses, so that the extra
species should have masses no more than few TeV. Moreover, only one type of
extra multiplet cannot entirely explain all the discrepancies, and some their
combination is required, e.g. two species of isodoublet, or one isodoublet and
one (up or down type) isosinglet. We show that these scenarios are testable
with future experiments. Namely, if extra vector-like quarks are responsible
for CKM anomalies, then at least one of them should be found at scale of few
TeV, and anomalous weak isospin violating $Z$-boson couplings with light
quarks should be detected if the experimental precision on $Z$ hadronic decay
rate is improved by a factor of $2$ or so.

##  1 Introduction

The Standard Model (SM) $SU(3)\times SU(2)\times U(1)$ contains three fermion
families with left-handed (LH) components of quarks $q_{Li}=(u_{L},d_{L})_{i}$
and leptons $\ell_{Li}=(\nu_{L},e_{L})_{i}$, $i=1,2,3$, forming doublets and
the right-handed (RH) components $u_{Ri}$, $d_{Ri}$, $e_{Ri}$ being singlets
of isotopic symmetry $SU(2)$ of weak interactions. The charged current weak
interactions in terms of the quark mass eigenstates, up quarks $u,c,t$ and
down quarks $d,s,b$, are described by the coupling

$\displaystyle\frac{g}{2\sqrt{2}}\,W_{\mu}^{+}\begin{array}[]{ccc}\overline{(u\leavevmode\nobreak\
\leavevmode\nobreak\ c\leavevmode\nobreak\ \leavevmode\nobreak\
t)}\end{array}\gamma^{\mu}(1-\gamma^{5})\,V_{\text{CKM}}\left(\begin{array}[]{c}d\\\
s\\\ b\end{array}\right)\leavevmode\nobreak\ +\leavevmode\nobreak\
\text{h.c.}$ (5)

where $V_{\text{CKM}}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix:

$\displaystyle
V_{\text{CKM}}=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{array}\right)$ (9)

In the SM context $V_{\text{CKM}}$ should be unitary. Any deviation from its
unitarity can be a signal of new physics beyond the SM (BSM).

At present, the determinations of $|V_{us}|$ and $|V_{ud}|$ have reached high
enough precision to test with high accuracy the unitarity of the first row in
CKM matrix (9):

$\displaystyle|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1$ (10)

Namely, precision experimental data on kaon decays, in combination with the
latest lattice QCD calculations of the decay constants and form-factors,
provide accurate information about $|V_{us}|$. On the other hand, recent
calculations of short-distance radiative corrections in $\beta$-decays
substantially improved the determination of $|V_{ud}|$. Since the contribution
of $|V_{ub}|\simeq 0.004$ is very small and actually negligible, the test of
the sum rule in eq. (10) is practically equivalent to a Cabibbo universality
check.

In our previous paper ref. Belfatto:2019swo  it was pointed out that there is
a significant (about $4\sigma$) anomaly in the first row unitarity (10), after
using three types of independent determinations of $|V_{us}|$ and $|V_{ud}|$,
which were dubbed as determinations of type A, B and C. Specifically,
determination A corresponds to the direct determination of $|V_{us}|$ from the
kaon semileptonic ($K\ell 3$) decays, B comes from the determination of the
ratio $|V_{us}/V_{ud}|$ obtained from charged kaon leptonic ($K\mu 2$) decays
by comparing them with pion leptonic decays, and C corresponds to the direct
determination of $|V_{ud}|$ from superallowed $0^{+}$–$0^{+}$ nuclear
transitions by employing the value of the Fermi constant obtained from the
muon decay, $G_{F}=G_{\mu}$.

For explaining this anomaly we proposed two possible BSM scenarios. One is
related to a new physics in the lepton sector. Namely, we considered the
horizontal gauge symmetry $SU(3)_{\ell}\times SU(3)_{e}$ between the lepton
families, with $SU(3)_{\ell}$ acting between LH states
$\ell_{Li}=(\nu_{L},e_{L})_{i}$ and $SU(3)_{e}$ acting between RH ones
$e_{Ri}$. We have shown that flavour changing gauge bosons of $SU(3)_{\ell}$
induce an effective operator which contributes to the muon decay in positive
interference with the SM contribution ($W$-boson exchange). In this way, the
muon decay constant $G_{\mu}$ becomes different from the Fermi constant
$G_{F}$: $G_{\mu}=G_{F}(1+\delta_{\mu})$ with
$\delta_{\mu}=(v_{w}/v_{F})^{2}$, where $v_{w}$ and $v_{F}$ respectively are
the electroweak and horizontal symmetry breaking scales. Since the values of
$|V_{us}|$ and $|V_{ud}|$ are normally extracted by assuming $G_{F}=G_{\mu}$,
in this scenario they are shifted by a factor $1+\delta_{\mu}$ while their
ratio is not affected. CKM unitarity is recovered with $\delta_{\mu}\simeq
7\times 10^{-4}$ which corresponds to a horizontal breaking scale of about $6$
TeV. Interesting point is that such a low mass scale for the horizontal gauge
bosons is not in conflict with the stringent experimental limits on the lepton
flavour changing processes as $\mu\rightarrow 3e$, $\tau\to 3\mu$ etc.
Belfatto:2019swo . The breaking scale of $SU(3)_{e}$ symmetry of the RH
leptons can also be as small as few TeV without contradicting experimental
limits B1 .

Another (more straightforward) possibility discussed in ref. Belfatto:2019swo
is to introduce extra vector-like quarks.111After ref. Belfatto:2019swo  the
problem of the CKM unitarity anomaly was addressed with different approaches
in several subsequent papers Grossman:2019bzp ; Pagliara ; Cheung:2020vqm ;
Endo:2020tkb ; Capdevila:2020rrl ; Crivellin:2020ebi ; Kirk:2020wdk ;
Coutinho:2020xhc ; Alok:2020jod ; Crivellin:2021njn .  In particular, with
extra isosinglet quarks of down-type $b^{\prime}$ or up-type $t^{\prime}$ one
can settle the CKM unitarity problem which results from the determination of
$V_{ud}$ from superallowed beta decays (C) and $V_{us}$ from kaon decays (A
and B), whereas by employing the extra quarks forming the weak isodoublet
$(t^{\prime},b^{\prime})$ all the tensions between the independent
determinations A, B and C can in principle be explained.

However, large mixings with SM families induce flavour changing phenomena
which can be in potential conflict with stringent experimental limits. In this
work we give a detailed study of the effects on relevant flavour changing
processes and electroweak observables and constrain the parameter space for
each scenario (extra weak isosinglets of up-type or down-type or weak
isodoublets).

As we will show, there still remains some available parameter space which can
satisfy these stringent constraints but it is very limited and can be excluded
with future experimental data. In particular, it can be excluded if the limits
on masses of extra vector-like species will increase up to $3$ TeV or so or
the limits on some relevant flavour changing phenomena or $Z$ boson physics
will further strengthen. Therefore, all these scenarios can be falsified in
close future.

The paper is organized as follows. Since after ref. Belfatto:2019swo  some new
data appeared, in section 2 we update the analysis of the CKM first row
anomalies. In section 3 we discuss the generalities about the role of
different types of vectorlike quarks in fixing the problem. In sections 4 and
5 we analyze separately the scenarios with extra weak isosinglet quarks of
down-type ($b^{\prime}$) and up-type ($t^{\prime}$), by providing a detailed
study of flavour changing phenomena induced in this scenarios and determining
the available parameter space. In section 6 we perform the analysis in the
case of additional extra weak isodoublet $(t^{\prime},b^{\prime})$. In section
6.5 we discuss some combinations in case more families are introduced. At the
end, in section 7 we give our conclusions.

##  2 Present situation in the determination of $|V_{us}|$ and $|V_{ud}|$

As already stated, the precision of recent determinations of $|V_{us}|$ and
$|V_{ud}|$ allows to test the first row unitarity (10) of CKM matrix.
Deviation from unitarity can be parameterized as

$\displaystyle|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1-\delta_{\text{CKM}}$
(11)

Hence, the value $\delta_{\text{CKM}}$ shows the measure of the unitarity
deficit.

The element $|V_{us}|$ can be directly determined from semileptonic $K\ell 3$
decays ($K_{L}\mu 3$, $K_{L}e3$, $K^{\pm}e3$, etc.) which imply Moulson :

$\displaystyle f_{+}(0)|V_{us}|=0.21654\pm 0.00041$ (12)

where $f_{+}(0)$ is the vector form factor at zero momentum transfer which can
be computed in the lattice QCD simulations. The average of 4-flavor
computations reported by FLAG 2019 is $f_{+}(0)=0.9706(27)$ FLAG2019 . We
combine it with the latest 4-flavor result $f_{+}(0)=0.9696(19)$ Bazavov
(which was not included in FLAG 2019 FLAG2019 ) getting $f_{+}(0)=0.9699(15)$.
In this way, from eq. (12) we obtain the value of $|V_{us}|$ (determination A
in the following) as:

$\displaystyle\text{A}:\qquad|V_{us}|_{\text{A}}=0.22326(55)$ (13)

An independent information (determination B in the following) stems from the
ratio of the kaon and pion leptonic decay rates $K\to\mu\nu(\gamma)$ and
$\pi\to\mu\nu(\gamma)$ which implies PDG18 :

$\displaystyle|V_{us}/V_{ud}|\times(f_{K^{\pm}}/f_{\pi^{\pm}})=0.27599\pm
0.00038$ (14)

Then, by employing the 4-flavour average for the decay constants ratio
$f_{K^{\pm}}/f_{\pi^{\pm}}=1.1932(19)$ reported in FLAG 2019 FLAG2019 , we
obtain:

$\displaystyle\text{B}:\qquad|V_{us}|/|V_{ud}|=0.23130(49)$ (15)

As regards the element $|V_{ud}|$, its most precise determination is obtained
from superallowed $0^{+}$– $0^{+}$ nuclear $\beta$-decays, which are pure
Fermi transitions sensitive only to the vector coupling constant
$G_{V}=G_{F}|V_{ud}|$. The master formula reads Hardy ; Hardy2 :

$\displaystyle|V_{ud}|^{2}=\frac{K}{2G_{F}^{2}\mathcal{F}t\,(1+\Delta_{R})}=\frac{0.97147(20)}{1+\Delta_{R}}$
(16)

where $K=2\pi^{3}\ln 2/m_{e}^{5}=8120.2776(9)\times 10^{-10}$ s/GeV4,
$G_{F}=G_{\mu}=1.1663787(6)\times 10^{-5}$ GeV-2 is the Fermi constant
determined from the muon decay mulan  and $\mathcal{F}t=3072.07(72)\,\text{s}$
is the nucleus independent value derived from $ft$-values of $14$ best
determined superallowed $0^{+}$– $0^{+}$ nuclear transitions by absorbing in
the latter all transition-dependent (so called outer) corrections Hardy2 . The
major uncertainty is related to the transition independent short-distance (so
called inner) radiative correction $\Delta_{R}$ which in 2006 was computed by
Marciano and Sirlin obtaining $\Delta_{R}=0.02361(38)$ Marciano .

However, a recent calculation with improved hadronic uncertainties brought to
a drastically different value $\Delta_{R}=0.02467(22)$ Seng . A more
conservative approach of ref. Marciano2  gives a slightly lower result with
relatively larger uncertainty, $\Delta_{R}=0.02426(32)$. For our analysis we
decided to use a democratic average of these two results,
$\Delta_{R}=0.02454(27)$. This means that we ‘democratically’ take as
uncertainty simply the arithmetical average of the error bars reported in
original refs. Seng  and Marciano2 .222If we would treat the errors as
statistical fluctuations and combine the errors in the standard way, then we
would have to reduce our ‘democratic’ error $(27)$ to $(18)$. However, since
we deal with the results of theoretical calculations, we decided to be
conservative in treating their uncertainties. Then, using this average, from
eq. (16) we get the value of $|V_{ud}|$ (determination C in the following):

$\displaystyle\text{C}:\qquad|V_{ud}|=0.97376(16)$ (17)

The value of $|V_{ud}|$ can be extracted also from the free neutron
$\beta$-decay:

$\displaystyle|V_{ud}|^{2}=\frac{K/\ln
2}{G_{F}^{2}\mathcal{F}_{n}\tau_{n}\,(1+3g_{A}^{2})(1+\Delta_{R})}=\frac{5024.46(30)\leavevmode\nobreak\
{\rm s}}{\tau_{n}(1+3g_{A}^{2})(1+\Delta_{R})}$ (18)

where $\mathcal{F}_{n}=f_{n}(1+\delta^{\prime}_{R})$ is the neutron $f$-value
$f_{n}=1.6887(1)$ corrected by the long-distance QED correction
$\delta^{\prime}_{R}=0.01402(2)$ Hardy3 . However it is less precise due to
limited accuracy in the experimental determination of the neutron lifetime
$\tau_{n}$ and the axial coupling constant $g_{A}=G_{A}/G_{V}$. Moreover,
there is an apparent tension ($4\sigma$) between the neutron lifetime
measurements using the bottle and beam experimental methods, origin of which
requires more profound understanding and perhaps some new physics puzzle .
E.g. by combining the recent determination of axial coupling
$g_{A}=1.27625(50)$ and the “bottle" lifetime $\tau_{n}^{\rm
bottle}=879.4(0.6)$ s as in ref. Belfatto:2019swo , one gets:

$\displaystyle|V_{ud}|=0.97333(33)_{\tau_{n}}(32)_{g_{A}}(11)_{\Delta_{R}}$
(19)

which is compatible with our choice (17) but has about 3 times larger errors.
(Interestingly, by comparing the determinations of $|V_{ud}|$ from free
neutron decays and superallowed $0^{+}$– $0^{+}$ decays, the factor
$1+\Delta_{R}$ cancels out and one obtains an accurate determination of the
neutron lifetime $\tau_{n}=5172.0(1.1)/(1+3g_{A}^{2})=878.7(0.6)$ Czarnecki
which well agrees with $\tau_{n}^{\rm bottle}$ but is in strong tension with
$\tau_{n}^{\rm beam}=888.0(2.0)$ s puzzle .) The measurements of
$\pi^{+}\rightarrow\pi^{0}e^{+}\nu$ branching ratio by PIBETA experiment
Pocanic:2003pf  lead to the independent result $|V_{ud}|=0.9728(30)$ which
however has about 20 times larger uncertainties as compared to (17).
Therefore, we take in consideration only determination C obtained from
superallowed $0^{+}$– $0^{+}$ transitions.

Figure 1:  The purple, blue and red bands correspond respectively to the
values of $|V_{us}|$ from eq. (13), $|V_{us}/V_{ud}|$ from eq. (15) and
$|V_{ud}|=0.97376(16)$ from eq. (17). The best fit point and $1$, $2$ and
$3\sigma$ contours are shown (red cross and red circles) for these values. The
black curve corresponds to the unitarity condition (10). The dashed black
curve corresponds to eq. (11) with the deficit of unitarity
$\delta_{\text{CKM}}=1.48\cdot 10^{-3}$. The green band correspond to the
value $|V_{ud}|=0.97333(47)$ obtained from neutron decay (19). The best fit
point and $1$ and $2\sigma$ contours (green cross and green circles) are
obtained by fitting the latter determination with the other two determinations
$|V_{us}|$ from eq. (13), $|V_{us}/V_{ud}|$ from eq. (15). The dashed gray
curve corresponds to eq. (11) with the deficit of unitarity
$\delta_{\text{CKM}}=2.6\cdot 10^{-3}$.

The tensions between determinations A, B and C are shown in figure 1, which
presents the fit of the values (13), (15), (17), with $V_{us}$ and $V_{ud}$
considered as independent parameters, without imposing unitarity. The
unitarity condition (10) is shown with the black continuous line. The best fit
(minimum $\chi^{2}$) corresponds to

$\displaystyle|V_{us}|=0.22439(36)\qquad|V_{ud}|=0.97373(16)$ (20)

which is $3.6\sigma$ away from the unitarity curve. The $\chi^{2}$ value is
rather large, $\chi^{2}=7.4$, due to the tension between the two
determinations A and B from kaon decays. For the deficit of CKM unitarity the
best fit values (20) imply $\delta_{\text{CKM}}=1.48(36)\cdot 10^{-3}$.

The tensions can be manifested also in another way. We can take
$|V_{us}|_{\text{A}}=0.22326(55)$ from the direct determination A while B and
C can be also translated in $|V_{us}|$ determinations by imposing the
unitarity condition (10). Namely, the value of $|V_{us}|$ obtained in this way
from eq. (15) is:

$\displaystyle|V_{us}|_{\text{B}}=0.22535(45)$ (21)

which is compatible also with a theoretical result $|V_{us}|=0.22567(42)$ from
$K\mu 2$ decays obtained in ref. Martinelli .

From determination C (17) we get instead:

$\displaystyle|V_{us}|_{\text{C}}=0.22756(70)$ (22)

For completeness, in table 1 we show the values of $|V_{ud}|$ and respective
“unitarity” values of $|V_{us}|$ corresponding to the choices of $\Delta_{R}$
as reported in original refs. Marciano , Seng , Marciano2 , indicated as C0,
C1 and C2.

Figure 2 displays the values $|V_{us}|_{\text{A}}$ (13), $|V_{us}|_{\text{B}}$
(21) and $|V_{us}|_{\text{C}}$ (22) with corresponding error bars (shaded
areas). Between determinations obtained from kaon decays, A and B, there is
$2.9\sigma$ tension, which maybe could disappear with more accurate lattice
simulations.333These determinations obtained from 3-flavor lattice
computations FLAG2017  were in fact compatible because of larger error-bars,
see figure 2.  Therefore, we conservatively take a democratic average of
$|V_{us}|_{\text{A}}$ and $|V_{us}|_{\text{B}}$ without reducing error bars
(with the uncertainty taken as arithmetical average of two uncertainties):

$\displaystyle\overline{\text{A+B}}\,:\qquad|V_{us}|=0.22451(50)$ (23)

We see that there is about $5\sigma$ tension between determination A and C and
$3\sigma$ tension between B and C. The discrepancy of the average
$\overline{\text{A+B}}$ from C results in $3.6\sigma$. Let us notice that if
we try to fit the incompatible determinations of $V_{us}$ $A$ (13), $B$ (21),
$C$ (22), we would get the average value $|V_{us}|=0.22511$ but with a ugly
large $\chi^{2}$ value, $\chi^{2}=24$.

Determination |  $\Delta_{R}$ |  $|V_{ud}|$ | $|V_{us}|$
---|---|---|---
C0 Marciano  |  $0.02361(38)$ |  $0.97420(21)$ | $0.2257(9)$
C1 Seng  |  $0.02467(22)$ |  $0.97370(14)$ | $0.22780(60)$
C2 Marciano2  |  $0.02426(32)$ |  $0.97389(18)$ | $0.22699(77)$
C (our choice) |  $0.02454(27)$ |  $0.97376(16)$ | $0.22756(70)$
Table 1:  Values of $\Delta_{R}$ reported in original references and corresponding values of $|V_{ud}|$ obtained from eq. (16). Values of $|V_{us}|$ are obtained assuming unitarity (10). C represents our ‘democratic’ average (see text).  Figure 2:  Shaded areas show the values of $|V_{us}|$ obtained from determinations A (13), B (15) and C (17) by assuming CKM unitarity (10), while the black line corresponds to the democratic average of A and B (23) (see text). C0, C1, C2 are the values of $|V_{us}|$ obtained from the values of $\Delta_{R}$ reported by original references, as listed in table 1. For comparison we also show the values of $|V_{us}|$ obtained using $3$ flavours lattice QCD simulations as reported in FLAG 17 FLAG2017  and were adopted in Particle Data Group 2018. These determinations have practically no tension with the old determination C0 Marciano . Hence, this picture demonstrate that the CKM tensions in fact emerged due to improved precision of $4$-flavours computations FLAG2019 ; Bazavov  on one side, and due to the changes in inner radiative correction $\Delta_{R}$ Seng ; Marciano2  on the other side.  | A: $|V_{us}|\\!=\\!0.22326(55)$ | B: $|V_{us}|\\!=\\!0.23130(49)|V_{ud}|$ | Average∗
---|---|---|---
C1: $|V_{ud}|\\!=\\!0.97370(14)$ | $2.05(37)\cdot 10^{-3}$ | $1.17(35)\cdot 10^{-3}$ | $1.55(36)\cdot 10^{-3}$
C2: $|V_{ud}|\\!=\\!0.97389(18)$ | $1.68(43)\cdot 10^{-3}$ | $0.78(41)\cdot 10^{-3}$ | $1.17(42)\cdot 10^{-3}$
C: $|V_{ud}|\\!=\\!0.97376(16)$ | $1.94(40)\cdot 10^{-3}$ | $1.05(38)\cdot 10^{-3}$ | $1.43(39)\cdot 10^{-3}$
Table 2:  Values of $\delta_{\text{CKM}}$ obtained for different choices of
the values of $|V_{us}|$ and $|V_{ud}|$.
∗ We average the values of $V_{us}$ given in the columns A and B
conservatively, taking its error bar as arithmetical average of their
uncertainties.

In table 2 we show the landscape of possible values of the unitarity deficit
$\delta_{\text{CKM}}$. We see that depending on the choice of the data this
value spans from about $10^{-3}$ to about $2\cdot 10^{-3}$.

In conclusion, it is clear that the CKM unitarity condition with three
families (10) is inconsistent with the present determinations of $|V_{us}|$
and $|V_{ud}|$. An immediate solution would be to introduce a fourth
sequential family $(t^{\prime},b^{\prime})$, analogous to the three SM
families, with the LH components forming weak isodoublets and RH components
being isosinglets. Then the $3\times 3$ CKM matrix (9) should be extended to a
unitary $4\times 4$ matrix:

$\displaystyle
V_{\text{CKM}}=\left(\begin{array}[]{cccc}V_{ud}&V_{us}&V_{ub}&V_{ub^{\prime}}\\\
V_{cd}&V_{cs}&V_{cb}&V_{cb^{\prime}}\\\
V_{td}&V_{ts}&V_{tb}&V_{tb^{\prime}}\\\
V_{t^{\prime}d}&V_{t^{\prime}s}&V_{t^{\prime}b}&V_{t^{\prime}b^{\prime}}\end{array}\right).$
(28)

and correspondingly the first row unitarity condition would be modified to:

$\displaystyle|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}+|V_{ub^{\prime}}|^{2}=1$
(29)

Comparing this equation with eq. (11) we see that the parameter
$\delta_{\text{CKM}}$ assumes the meaning of the mixing with the fourth
family, $\delta_{\text{CKM}}=|V_{ub^{\prime}}|^{2}$. Therefore for typical
values of $\delta_{\text{CKM}}$ given in table 2 we get
$|V_{ub^{\prime}}|\approx 0.04$, which is comparable with $|V_{cb}|$ and an
order of magnitude larger than $|V_{ub}|\approx 0.004$. It looks not very
natural that the mixing of the first family with fourth family is much
stronger than its mixing with third family, but some models admit this
possibility Bregar . Unfortunately, the existence of a fourth sequential
family is excluded by the limits from electroweak precision data combined with
the LHC data.

However, vector-like quarks can be introduced without any contradiction with
SM precision tests. The LHC limits merely tell that their masses should be
above $1$ TeV or so.

In the rest of this paper we discuss how the anomalies in the first row of CKM
matrix can be solved by introducing extra vector-like fermions. In particular
we will consider the role of weak isosinglets of down-type or up-type and weak
isodoublets.

In fact, two approaches to the problem can be considered. The incompatibility
inside kaon physics may be attributed to some uncertainties which can
disappear maybe soon with more precise determinations, focusing instead on the
average of determinations from kaons. Then the problem consist in solving the
lack of unitarity in the first row of $V_{\text{CKM}}$. The insertion of an
extra vectorlike weak isosinglet, down-type or up-type, is on this line.

Otherwise the discrepancy inside kaon physics can be considered seriously by
looking for a solution addressing the whole situation. However, the $V_{us}$
anomaly (discrepancy between $V_{us}$ determinations from the kaon
semileptonic and leptonic decays) can be considered seriously, 444In fact, a
recent high precision determination of $K\ell 3$ radiative corretions
Seng:2021boy  indicates that the SM electroweak effects are not large enough
to account for $V_{us}$ anomaly. and one has to look for a solution addressing
the whole situation. As it will be shown, a weak isodoublet can in principle
explain all the anomalies.

In these scenarios with extra vector-like families, the unitarity deficit
$\delta_{\text{CKM}}$ will be related to the mixing of the extra quarks with
the SM families.

##  3 The SM with extra vector-like fermions

The Standard Model $SU(3)\times SU(2)\times U(1)$ contains, by definition,
three chiral families of fermions, the LH quarks $q_{Li}=(u_{L},d_{L})_{i}$
and leptons $\ell_{Li}=(\nu_{L},e_{L})_{i}$ transforming as $SU(2)$
isodoublets and the RH components $u_{Ri}$, $d_{Ri}$, $e_{Ri}$ being
isosinglets, with $i=1,2,3$ being the family index. This set of fermions is
free of gauge anomalies. Attractive property of the SM is that these fermions
can acquire masses only via the Yukawa couplings with Higgs doublet $\varphi$:

$\displaystyle{\cal L}_{\rm Yuk}^{\rm
SM}=Y_{u}^{ij}\tilde{\varphi}\,\overline{q_{Li}}u_{Rj}+Y_{d}^{ij}\varphi\,\overline{q_{Li}}d_{Rj}+Y_{e}^{ij}\varphi\,\overline{\ell_{Li}}e_{Rj}\leavevmode\nobreak\
+\leavevmode\nobreak\ {\rm h.c.}$ (30)

where $Y_{u,d,e}^{ij}$ are the Yukawa constant matrices, and
$\tilde{\varphi}=i\tau_{2}\varphi^{\ast}$. The known species of quarks and
leptons are eigenstates of mass matrices $M_{u,d,e}=Y_{u,d,e}v_{w}$ where
$\langle\varphi^{0}\rangle=v_{w}$ is the Higgs VEV. In other words, in the SM
the quark and lepton masses are induced only after the electroweak symmetry
breaking, and their values are proportional to the electroweak scale $v_{w}$.

The quark mass matrices can be diagonalized by bi-unitary transformations

$\displaystyle V_{Lu}^{\dagger}M_{u}V_{Ru}=\widetilde{M}_{u}={\rm
diag}(m_{u},m_{c},m_{t}),\quad
V_{Ld}^{\dagger}M_{d}V_{Rd}=\widetilde{M}_{d}={\rm diag}(m_{d},m_{s},m_{b})$
(31)

and the weak eigenstates in terms of mass eigenstates are:

$\displaystyle\left(\begin{array}[]{c}u_{L1}\\\ u_{L2}\\\
u_{L3}\end{array}\right)=V_{Lu}\left(\begin{array}[]{c}u_{L}\\\ c_{L}\\\
t_{L}\end{array}\right);$ $\displaystyle\left(\begin{array}[]{c}d_{L1}\\\
d_{L2}\\\ d_{L3}\end{array}\right)=V_{Ld}\left(\begin{array}[]{c}d_{L}\\\
s_{L}\\\ b_{L}\end{array}\right)$ (44)

The CKM mixing matrix in $W$ boson charged current couplings (5) emerges as a
combination of the ‘left’ unitary transformations, $V_{\rm
CKM}=V_{Lu}^{\dagger}V_{Ld}$, and thus it should be unitary. As for the
‘right’ matrices $V_{Ru}$ and $V_{Rd}$, in the SM frames they have no physical
significance. Without loss of generality, one can choose a fermion basis in
which one of the Yukawa matrices $Y_{u}$ or $Y_{d}$ is diagonal in which cases
we would have respectively $V_{\rm CKM}=V_{Ld}$ or $V_{\rm
CKM}=V_{Lu}^{\dagger}$.

The SM exhibits a remarkable feature of natural suppression of flavor-changing
neutral currents (FCNC) Glashow ; Paschos : no flavor mixing emerges in
neutral currents coupled to $Z$ boson and Higgs boson. In particular, this
means that $Z$ boson tree level couplings with the fermion mass eigenstates
remain diagonal after rotations (44). On the other hand, the Yukawa matrices
$Y_{u,d,e}$ and mass matrices $M_{u,d,e}$ are proportional and thus by
transformations (31) they are diagonalized simultaneously, so that the Yukawa
couplings of the Higgs boson $H$ with the fermion mass eigenstates are
diagonal. Hence, all FCNC phenomena are suppressed at tree level and emerge
exclusively from radiative corrections. At present, the majority of
experimental data on flavor changing and CP violating processes are in good
agreement with the SM predictions.

Clearly, in the SM framework the unitarity of the CKM matrix as well as the
natural flavor conservation in neutral currents are direct consequences
following from the fact that the three families are in identical
representations of $SU(3)\times SU(2)\times U(1)$.

However, in addition to three chiral families of quarks and leptons, there can
exist extra vector-like species, with the LH and RH in the same
representations of the SM. In particular, one can consider the extra fermion
species in the same representations of $SU(3)\times SU(2)\times U(1)$ as
standard quarks and leptons, namely in the form of weak isosinglets of down
quark type $D_{L,R}$, up quark type $U_{L,R}$ and charged lepton type
$E_{L,R}$, and weak isodoublets $Q_{L,R}=({\cal U},{\cal D})_{L,R}$ and
$L_{L,R}=({\cal N},{\cal E})_{L,R}$ of quark and lepton types.555 Such vector-
like species are predicted in some extensions of the Standard Model. For
example, $D$ and $L$ type species emerge (per each family) in the context of
minimal $E_{6}$ Gursey ; Achiman  or $SU(6)$ Dvali  grand unifications. In
addition, the specifics of the latter model in which Higgs emerges as pseudo-
Goldstone particle, requires at least one copy of $Q$, $U$ and $E$ type
species for inducing the fermion masses and in particular the top quark mass
Barbieri ; su6 .  (Extra vector-like fermions can be introduced also in other
representations as e.g. $SU(2)$ isotriplets which can contain quark or lepton
type fragments but also some fragments with exotic electric charges but here
we do not address these cases.) The mass terms of these species are not
protected by the SM gauge symmetries and hence their masses can be (or must
be) considerably larger than the electroweak scale.

In the following we shall concentrate on the quark sector. Namely, we consider
a theory which, besides the three chiral families of standard quarks $u_{Ri}$,
$d_{Ri}$ and $q_{Li}=(u_{L},d_{L})_{i}$ ($i=1,2,3$), includes some extra
vector-like quark species $U_{L,R}$, $D_{L,R}$ and $Q_{L,R}=({\cal U},{\cal
D})_{L,R}$ which in principle can be introduced in different amounts.
Therefore, along with the standard Yukawa terms for the three chiral
families:666 Hereafter indices $i=1,2,3$ of normal families as well as indices
of extra species are suppressed.

$\displaystyle{\cal L}_{\rm
Yuk}=Y_{u}\tilde{\varphi}\,\overline{q_{L}}u_{R}+Y_{d}\varphi\,\overline{q_{L}}d_{R}\leavevmode\nobreak\
+\leavevmode\nobreak\ {\rm h.c.}$ (45)

the most general Lagrangian of this system must include the mixed Yukawa terms
between the standard and extra species:

$\displaystyle{\cal L}_{\rm Yuk}^{\rm
mix}=h_{U}\tilde{\varphi}\,\overline{q_{L}}U_{R}+h_{D}\varphi\,\overline{q_{L}}D_{R}+h_{\cal
U}\tilde{\varphi}\,\overline{Q_{L}}u_{R}+h_{\cal
D}\varphi\,\overline{Q_{L}}d_{R}\leavevmode\nobreak\ +\leavevmode\nobreak\
{\rm h.c.}$ (46)

and the mass terms

$\displaystyle{\cal L}_{\rm
mass}=M_{U}\overline{{U}_{L}}{U}_{R}+M_{D}\overline{{D}_{L}}{D}_{R}+M_{Q}\overline{Q_{L}}Q_{R}+\mu_{u}\overline{{U}_{L}}u_{R}+\mu_{d}\overline{{D}_{L}}d_{R}+\mu_{q}\overline{q_{L}}Q_{R}\leavevmode\nobreak\
+\leavevmode\nobreak\ {\rm h.c.}$ (47)

where $h_{U,D}$ and $h_{u,d}$ in the Yukawa terms (46) and $M_{U,D,Q}$ and
$\mu_{u,d,q}$ in mass terms (47) are the matrices of proper dimensions
depending on the amounts of extra species. One could introduce also the Yukawa
couplings between extra species:

$\displaystyle{\cal L}_{\rm Yuk}^{\rm
extra}=\lambda_{U}\tilde{\varphi}\,\overline{Q_{L}}U_{R}+\lambda_{\cal
U}\tilde{\varphi}\,\overline{Q_{R}}U_{L}+\lambda_{D}\varphi\,\overline{Q_{L}}D_{R}+\lambda_{\cal
D}\varphi\,\overline{Q_{R}}D_{L}\leavevmode\nobreak\ +\leavevmode\nobreak\
{\rm h.c.}$ (48)

However, they play no relevant role in further discussions and for simplicity
we neglect them.

The vector-like quarks are key players in models with horizontal inter-family
symmetries PLB83 ; Dimopoulos ; PLB85 ; Bagger ; Khlopov1 ; Khlopov2 ; King ;
so10 , in some models of the axion Kim ; Shifman ; Khlopov ; Zphys ; Sakharov
as well as in axionless (Nelson-Barr type) models for solving the strong CP
problem via spontaneous CP or P violation Nelson ; Barr ; Babu ; MPL ;
Senjanovic . In these models the values $M_{U,D,Q}$ and $\mu_{u,d,q}$ in mass
terms (47) are related to the breaking scales of respective symmetries, and in
some models e.g. Babu ; Senjanovic ; Berezhiani:1990  they can be as low as
few TeV.

Interestingly, some of these symmetries (e.g. flavor symmetry or Peccei-Quinn
symmetry) may forbid the direct Yukawa terms (45) but allow the mixed ones
(46) while mass terms $\mu$ and $M$ in (46) can be originated from some
physical scales.777 E.g. in the ‘seesaw’ model of ref. PLB83  the values of
$\mu_{u,d}$ and $M_{U,D}$ are respectively determined by the breaking scales
of left-right symmetry and family symmetry, i.e. by the VEV of the ‘right’
Higgs doublet and VEVs of flavon scalars which break the horizontal symmetry.
Nevertheless, despite that the original constants $Y_{u,d}$ in (45) are
vanishing, the SM Yukawa terms (30) for normal fermions will be induced after
integrating out the heavy states. In particular, provided that mixing mass
terms $\mu_{u,d,q}$ are smaller than $M_{U,D,Q}$, we obtain

$\displaystyle Y_{u}\simeq h_{U}M_{U}^{-1}\mu_{u}+\mu_{q}M_{Q}^{-1}h_{\cal
U},\quad Y_{d}\simeq h_{D}M_{D}^{-1}\mu_{d}+\mu_{q}M_{Q}^{-1}h_{\cal D}$ (49)

In other words, the non-zero quark masses are induced via the mixings with the
extra vector-like species. Such a scenario known as ‘universal’ seesaw
mechanism PLB83 ; Rajpoot ; Davidson ; Rattazzi  is commonly used in
predictive model building for fermion masses and mixings as e.g. PLB83 ;
Rattazzi ; Rattazzi2 ; tanbeta ; Anderson:1993fe ; Berezhiani:1996bv ; Koide1
; Koide2 ; Nesti . In the context of supersymmetric models with flavor
symmetry this mechanism can give a natural realization of the minimal flavor
violation scenario via the alignment of soft supersymmetry breaking terms with
the Yukawa terms MFV ; MFV1 ; MFV2 .

In the following we are not interested in the model details and in possible
dynamical effects of the underlying symmetries broken at higher scales, but
only in the effects of the mixing between the three normal (chiral) and extra
(vector-like) quarks. Therefore, we can conveniently redefine the fermion
basis. Namely, the species $u_{R}$ and $U_{R}$, $d_{R}$ and $D_{R}$, and
$q_{L}$ and $Q_{L}$, are in the identical representations of $SU(3)\times
SU(2)\times U(1)$. Thus, by redefining these species, one can eliminate mixed
mass terms $\mu_{u}$, $\mu_{d}$ and $\mu_{q}$ in (47) by ‘absorbing’ them
respectively in the mass terms $M_{U}$, $M_{D}$ and $M_{Q}$ (this means that
e.g. from $3+n$ RH species with quantum numbers of $d_{R}$ we can always
select $n$ their combinations which ‘marry’ $n$ species of LH fermions ${\cal
D}_{L}$ via mass terms while the remaining $3$ combinations have no mass
terms). In addition, without losing generality, the ‘heavy’ mass matrices
$M_{U}$, $M_{D}$ and $M_{Q}$ can be taken to be diagonal and real.

In this basis the total mass matrices of up type $(u,U,{\cal U})_{L,R}$ and
down type $(d,D,{\cal D})_{L,R}$ quarks, after substituting the Higgs VEV
$\langle\phi\rangle=v_{w}$, read:

$\displaystyle{\cal M}_{\rm
up}=\left(\begin{array}[]{ccc}Y_{u}v_{w}&h_{U}v_{w}&0\\\ 0&M_{U}&0\\\ h_{\cal
U}v_{w}&0&M_{Q}\end{array}\right),\quad\quad{\cal M}_{\rm
down}=\left(\begin{array}[]{ccc}Y_{d}v_{w}&h_{D}v_{w}&0\\\ 0&M_{D}&0\\\
h_{\cal D}v_{w}&0&M_{Q}\end{array}\right)$ (56)

where the blocks $Y_{u,d}$ are matrices of dimensions $3\times 3$. Assuming
that the numbers of extra species $U$, $D$ and $Q$ are respectively $p$, $n$
and $m$, then blocks $M_{U}$, $M_{D}$ and $M_{Q}$ should be correspondingly of
dimensions $p\times p$, $n\times n$ and $m\times m$. Thus, ${\cal M}_{\rm up}$
and ${\cal M}_{\rm down}$ respectively are $(3+p+m)\times(3+p+m)$ and
$(3+n+m)\times(3+n+m)$ matrices.

The mass matrices (56) can be brought to the diagonal forms via bi-unitary
transformations $({\cal V}^{L}_{\rm up})^{\dagger}{\cal M}_{\rm up}{\cal
V}^{R}_{\rm up}=\widetilde{\cal M}_{\rm up}$ and $({\cal V}^{L}_{\rm
down})^{\dagger}{\cal M}_{\rm down}{\cal V}^{R}_{\rm down}=\widetilde{\cal
M}_{\rm down}$. In this way, the initial states of e.g. down-type quarks are
related to their physical states (mass eigenstates) as

$\displaystyle\left(\begin{array}[]{c}d\\\ D\\\ {\cal
D}\end{array}\right)_{\\!\\!L,R}={\cal V}_{\rm
down}^{L,R}\left(\begin{array}[]{c}d^{\prime}\\\ D^{\prime}\\\ {\cal
D}^{\prime}\end{array}\right)_{\\!\\!L,R}=\left(\begin{array}[]{ccc}V_{dd^{\prime}}^{L,R}&V_{dD^{\prime}}^{L,R}&V_{d{\cal
D}^{\prime}}^{L,R}\\\ V_{Dd^{\prime}}^{L,R}&V_{DD^{\prime}}^{L,R}&V_{D{\cal
D}^{\prime}}^{L,R}\\\ V_{{\cal D}d^{\prime}}^{L,R}&V_{{\cal
D}D^{\prime}}^{L,R}&V_{{\cal D}{\cal
D}^{\prime}}^{L,R}\end{array}\right)\left(\begin{array}[]{c}d^{\prime}\\\
D^{\prime}\\\ {\cal D}^{\prime}\end{array}\right)_{\\!\\!L,R}$ (69)

Here $d=(d_{1},d_{2},d_{3})^{T}$ are initial states and
$d^{\prime}=(d,s,b)^{T}$ are the mass eigenstates, and similarly for heavy
species $D$ and ${\cal D}$. Analogously, unitary matrices ${\cal V}^{L,R}_{\rm
up}$ connect the initial up-quark type states $u,U,{\cal U}$ with their mass
eigenstates $u^{\prime},U^{\prime},{\cal U}^{\prime}$, where
$u=(u_{1},u_{2},u_{3})^{T}$ and $u^{\prime}=(u,c,t)^{T}$.

Since we have $\widetilde{\cal M}_{\rm down}^{2}=({\cal V}^{L}_{\rm
down})^{\dagger}\,{\cal M}_{\rm down}{\cal M}_{\rm down}^{\dagger}\,{\cal
V}^{L}_{\rm down}=({\cal V}^{R}_{\rm down})^{\dagger}\,{\cal M}_{\rm
down}^{\dagger}{\cal M}_{\rm down}\,{\cal V}^{R}_{\rm down}$, unitary matrices
${\cal V}^{L}_{\rm down}$ and ${\cal V}^{R}_{\rm down}$ can be determined by
considering the hermitean squares of ${\cal M}_{\rm down}$:

$\displaystyle{\cal M}_{\rm down}{\cal M}_{\rm
down}^{\dagger}=\left(\begin{array}[]{ccc}v_{w}^{2}Y_{d}Y_{d}^{\dagger}+v_{w}^{2}h_{D}h_{D}^{\dagger}&v_{w}h_{D}M_{D}&v_{w}^{2}Y_{d}h_{\cal
D}^{\dagger}\\\ v_{w}M_{D}h_{D}^{\dagger}&M_{D}^{2}&0\\\ v_{w}^{2}h_{\cal
D}Y_{d}^{\dagger}&0&M_{Q}^{2}+v_{w}^{2}h_{\cal D}h_{\cal
D}^{\dagger}\end{array}\right),$ (73) $\displaystyle{\cal M}_{\rm
down}^{\dagger}{\cal M}_{\rm
down}=\left(\begin{array}[]{ccc}v_{w}^{2}Y_{d}^{\dagger}Y_{d}+v_{w}^{2}h_{\cal
D}^{\dagger}h_{\cal D}&v^{2}_{w}Y_{d}^{\dagger}h_{D}&v_{w}h_{\cal
D}^{\dagger}M_{Q}\\\
v_{w}^{2}h_{D}^{\dagger}Y_{d}&M_{D}^{2}+v_{w}^{2}h_{D}^{\dagger}h_{D}&0\\\
v_{w}M_{Q}h_{\cal D}&0&M_{Q}^{2}\end{array}\right)\,.$ (77)

The off-diagonal entries of these matrices are fixed by the value $v_{w}$, so
that the elements of the off-diagonal blocks $V_{dD^{\prime}}$, $V_{d{\cal
D}^{\prime}}$ etc. in (69) are determined by the ratio of the electroweak
scale to the masses of extra quark species. In the limit when the latter are
very heavy they decouple and their mixings with light quarks become negligibly
small. Thus, in this limit $3\times 3$ block $V_{dd^{\prime}}$ becomes
unitary. The same is true for analogous $V_{uu^{\prime}}$ block in up quark
mixing. However, if the extra quarks are not that heavy and off-diagonal
blocks are not negligible, then $V_{dd^{\prime}}$ and $V_{uu^{\prime}}$ blocks
are no more unitary.

The present experimental limits on the extra quark masses are
$M_{U,D,Q}>(1\div 1.5)$ TeV, depending on their type and decay modes PDG18 .
Therefore, the ratios $\epsilon_{u}=v_{w}/M_{U}$, $\epsilon_{d}=v_{w}/M_{D}$
and $\epsilon_{q}=v_{w}/M_{Q}$ can be considered as small parameters,
$\epsilon_{u,d,q}\leq 0.1$ or so. By inspecting the matrices (73), one can
estimate the elements of the off-diagonal blocks in ${\cal V}_{L,R}^{\rm
down}$ and ${\cal V}_{L,R}^{\rm up}$ as

$\displaystyle|V^{L}_{dD^{\prime}}|\sim\epsilon_{d},\quad|V^{L}_{uU^{\prime}}|\sim\epsilon_{u},$
$\displaystyle\qquad|V^{L}_{d{\cal D}^{\prime}}|,|V^{L}_{u{\cal
U}^{\prime}}|\sim\epsilon_{q}^{2}\qquad$ $\displaystyle\qquad|V^{R}_{d{\cal
D}^{\prime}}|,|V^{R}_{u{\cal U}^{\prime}}|\sim\epsilon_{q},$
$\displaystyle|V^{R}_{dD^{\prime}}|\sim\epsilon_{d}^{2},\quad|V^{R}_{uU^{\prime}}|\sim\epsilon_{u}^{2}\qquad$
(78)

modulo the Yukawa constants which are assumed to be $\leq 1$ for
perturbativity. Therefore, the deviation from unitarity of the “left" matrices
$V_{dd^{\prime}}^{L}$ and $V_{uu^{\prime}}^{L}$ blocks are
$\sim\epsilon^{2}_{u,d}\leq 10^{-2}$. E.g. the first row unitarity of the
matrix ${\cal V}^{L}_{\rm down}$ implies
$|V_{1d}|^{2}+|V_{1s}|^{2}+|V_{1b}|^{2}=1-|V_{1D^{\prime}}|^{2}-|V_{1{\cal
D}^{\prime}}|^{2}=1-\delta_{d}$. Taking into account the above estimations, we
see that the deviation can be as large as $\delta_{d}\sim\epsilon_{d}^{2}\sim
10^{-2}$. Let us recall that the CKM unitarity deficit $\delta_{\rm CKM}$
estimated in previous section is about $(1\div 2)\times 10^{-3}$, see table 2.
Thus, for accounting for the above values of $\delta_{\rm CKM}$, one would
need $\epsilon_{d}=0.03-0.05$. As for $\epsilon^{4}\sim 10^{-4}$
contributions, they are irrelevant and so order $\epsilon^{2}$ mixings as
$V_{1{\cal D}^{\prime}}$ etc. can be safely neglected.

Let us assume, for simplicity, that each of $U$, $D$ and $Q$ type species is
present in one copy, i.e. $p=m=n=1$ (our discussion can be extended in a
straightforward way for arbitrary number of extra species). In this case the
off-diagonal blocks proportional to $v_{w}$ in $5\times 5$ matrices (56)
become columns as e.g. $h_{D}=(h_{1D},h_{2D},h_{3D})^{T}$ or rows as e.g.
$h_{\cal D}=(h_{{\cal D}1},h_{{\cal D}2},h_{{\cal D}3})$. The Yukawa couplings
$Y_{u,d}$ can be presented in the form
$Y_{u}=V_{Lu}\widetilde{Y}_{u}V_{Ru}^{\dagger}$ and
$Y_{d}=V_{Ld}\widetilde{Y}_{d}V_{Rd}^{\dagger}$ where $\widetilde{Y}_{u,d}$
are diagonal $3\times 3$ matrices, $\widetilde{Y}_{u}={\rm
diag}(y_{u},y_{c},y_{t})$ and $\widetilde{Y}_{d}={\rm
diag}(y_{d},y_{s},y_{b})$. Let us denote also $h_{U}=V_{Lu}\widetilde{h}_{U}$,
$h_{D}=V_{Ld}\widetilde{h}_{D}$, $h_{\cal U}=\widetilde{h}_{\cal
U}V_{Ru}^{\dagger}$ and $h_{\cal D}=\widetilde{h}_{\cal D}V_{Rd}^{\dagger}$.

Then for $5\times 5$ unitary matrix of ‘left’ rotations we obtain, with the
precision up to $\epsilon^{2}$ terms:

$\displaystyle{\cal V}_{\rm down}^{L}=\left(\begin{array}[]{ccc}V_{Ld}&0&0\\\
0&1&0\\\
0&0&1\end{array}\right)\\!\left(\begin{array}[]{ccc}\\!\\!\\!\sqrt{1-S_{D}S_{D}^{\dagger}}&S_{D}&0\\\
-S_{D}^{\dagger}&\\!\\!\sqrt{1-S_{D}^{\dagger}S_{D}}&0\\\
0&0&1\end{array}\right)\\!\left(\begin{array}[]{ccc}\\!\\!\\!\sqrt{1-S_{\cal
D}S_{\cal D}^{\dagger}}&0&S_{\cal D}\\\ 0&1&0\\\ -S_{\cal
D}^{\dagger}&0&\sqrt{1-S_{\cal D}^{\dagger}S_{\cal D}}\end{array}\right)$ (88)
$\displaystyle\quad\quad\quad=\left(\begin{array}[]{ccc}V_{Ld}\big{[}1-\frac{1}{2}S_{D}S_{D}^{\dagger}-\frac{1}{2}S_{\cal
D}S_{\cal D}^{\dagger}+\dots\big{]}&V_{Ld}S_{D}&V_{Ld}S_{\cal D}\\\
-S_{D}^{\dagger}&1-\frac{1}{2}S_{D}^{\dagger}S_{D}+\dots&O(\epsilon_{d}\epsilon_{q}^{2})\\\
-S_{\cal D}^{\dagger}&0&1-\frac{1}{2}S_{\cal D}^{\dagger}S_{\cal
D}+\dots\end{array}\right)$ (92)

where the column $S_{D}=\epsilon_{d}\widetilde{h}_{D}$ describes the light
quark ($d,s,b$) mixings with the extra isosinglet species $D$. As for their
mixings with ${\cal D}\subset Q$ from extra isodoublet, $S_{\cal
D}=\epsilon_{q}^{2}\widetilde{Y}_{d}\widetilde{h}_{\cal D}^{\dagger}$, it can
be neglected since, apart of $\epsilon^{2}$ suppression, these are
proportional to the small Yukawa constants $y_{d},y_{s},y_{b}$ in
$\widetilde{Y}_{d}$. Clearly, the unitarity conditions for rows and columns of
this matrix is fulfilled with the precision up to $\sim\epsilon_{d}^{4}$
terms. The matrix ${\cal V}_{\rm up}^{L}$ can be presented in an analogous
form.

Let us discuss now charged current interactions. Considering that $q_{L}$ and
$Q_{L}$ are $SU(2)$ doublets while $U_{L}$ and $D_{L}$ are singlets, the LH
charged current interacting with $W$ boson in terms of initial states and mass
eigenstates reads:

$\displaystyle\frac{g}{\sqrt{2}}\,W^{+}_{\mu}\,\overline{(u_{L}\leavevmode\nobreak\
U_{L}\leavevmode\nobreak\ {\cal
U}_{L})}\,\gamma^{\mu}\,J_{L}\left(\begin{array}[]{c}d_{L}\\\ D_{L}\\\ {\cal
D}_{L}\end{array}\right)=\frac{g}{2\sqrt{2}}\,W^{+}_{\mu}\,\overline{(u^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ U^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
U}^{\prime})}\,\gamma^{\mu}(1-\gamma^{5})\,{\cal V}_{L}^{\rm
mix}\left(\begin{array}[]{c}d^{\prime}\\\ D^{\prime}\\\ {\cal
D}^{\prime}\end{array}\right)$ (99)

where $J_{L}={\rm diag}(1,1,1,0,1)$ and ${\cal V}_{L}^{\rm mix}=({\cal
V}_{L}^{\rm up})^{\dagger}J_{L}{\cal V}_{L}^{\rm down}$, or explicitly

$\displaystyle{\cal V}^{\rm
mix}_{L}=\left(\begin{array}[]{ccc}\\!\\!(1-\frac{1}{2}\Delta_{U})\widetilde{V}(1-\frac{1}{2}\Delta_{D})&(1-\frac{1}{2}\Delta_{U})\widetilde{V}S_{D}&O(\epsilon^{2})\\\
S_{U}^{\dagger}\widetilde{V}(1-\frac{1}{2}\Delta_{D})&S_{U}^{\dagger}\widetilde{V}S_{D}&O(\epsilon^{3})\\\
O(\epsilon^{2})&O(\epsilon^{3})&1\end{array}\right)\\!,\quad\widetilde{V}=V_{Lu}^{\dagger}V_{Ld}$
(103)

where
$\Delta_{U}=S_{U}S_{U}^{\dagger}=\epsilon_{u}^{2}\widetilde{h}_{U}\widetilde{h}_{U}^{\dagger}$
and
$\Delta_{D}=S_{D}S_{D}^{\dagger}=\epsilon_{d}^{2}\widetilde{h}_{D}\widetilde{h}_{D}^{\dagger}$
as far as $O(\epsilon_{q}^{4})$ contributions $S_{\cal U}S_{\cal U}^{\dagger}$
and $S_{\cal D}S_{\cal D}^{\dagger}$ can be neglected.

We are interested in its $3\times 3$ block which describes the transitions
between the quark mass eigenstates $u^{\prime}=(u,c,t)$ and
$d^{\prime}=(d,s,b)$ in charged current:

$\displaystyle
V_{\text{CKM}}=\big{(}1-\frac{1}{2}\Delta_{U}\big{)}\,\widetilde{V}\,\big{(}1-\frac{1}{2}\Delta_{D}\big{)}=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{array}\right)$ (107)

While $\widetilde{V}=V_{Lu}^{\dagger}V_{Ld}$ is unitary $3\times 3$ matrix,
the ‘corrected’ matrix $V_{\rm CKM}$ is not. In particular, deviation from the
unitarity for its rows or columns read, up to order $\epsilon^{2}$ terms,
respectively as888Obviously, the ‘large’ mixing matrix ${\cal V}^{\rm
mix}_{L}$ is not unitary in itself because of the non-unitary factor $J_{L}$
‘sandwiched’ between the unitary matrices ${\cal V}^{\rm up}_{L}$ and ${\cal
V}^{\rm down}_{L}$.

$\displaystyle V_{\rm CKM}V_{\rm
CKM}^{\dagger}=1-\epsilon_{u}^{2}\widetilde{h}_{U}\widetilde{h}_{U}^{\dagger}-\epsilon_{d}^{2}\widetilde{V}\widetilde{h}_{D}\widetilde{h}_{D}^{\dagger}\widetilde{V}^{\dagger},$
$\displaystyle V_{\rm CKM}^{\dagger}V_{\rm
CKM}=1-\epsilon_{d}^{2}\widetilde{h}_{D}\widetilde{h}_{D}^{\dagger}-\epsilon_{u}^{2}\widetilde{V}\widetilde{h}_{U}\widetilde{h}_{U}^{\dagger}\widetilde{V}^{\dagger}$
(108)

In particular, the unitarity deficit for first row (11) we obtain
$\delta_{CKM}=1-|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}\approx\epsilon_{u}^{2}|\widetilde{h}_{U1}|^{2}+\epsilon_{d}^{2}|\widetilde{h}_{D1}|^{2}$,
which can fall in the range of $(1\div 2)\times 10^{-3}$ provided that
$\epsilon_{u}$ and/or $\epsilon_{d}$ are $\sim 0.1$ and the Yukawa constants
$\widetilde{h}_{U1}$ and $\widetilde{h}_{D1}$ are large enough.

Let us discuss the RH sector. Considering that $Q_{R}$ is an $SU(2)$ doublet
while $u_{R}$, $d_{R}$, $U_{R}$ and $D_{R}$ are singlets, the RH charged
current interacting with $W$ boson in terms of initial states and mass
eigenstates reads:

$\displaystyle\frac{g}{\sqrt{2}}\,W^{+}_{\mu}\,\overline{(u_{R}\leavevmode\nobreak\
U_{R}\leavevmode\nobreak\ {\cal
U}_{R})}\,\gamma^{\mu}\,J_{R}\left(\begin{array}[]{c}d_{R}\\\ D_{R}\\\ {\cal
D}_{R}\end{array}\right)=\frac{g}{2\sqrt{2}}\,W^{+}_{\mu}\,\overline{(u^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ U^{\prime}\leavevmode\nobreak\ \leavevmode\nobreak\ {\cal
U}^{\prime})}\,\gamma^{\mu}(1+\gamma^{5})\,{\cal V}_{R}^{\rm
mix}\left(\begin{array}[]{c}d^{\prime}\\\ D^{\prime}\\\ {\cal
D}^{\prime}\end{array}\right)$ (115)

where $J_{R}={\rm diag}(0,0,0,0,1)$ and ${\cal V}_{R}^{\rm mix}=({\cal
V}_{R}^{\rm up})^{\dagger}J_{R}{\cal V}_{R}^{\rm down}$. Thus, presenting
matrices ${\cal V}_{R}^{\rm down}$ which diagonalizes ${\cal M}_{\rm
down}^{\dagger}{\cal M}_{\rm down}$ (73) in the form similar to ${\cal
V}_{L}^{\rm down}$ (88), and analogously doing for and ${\cal V}_{R}^{\rm
up}$, we get up to $O(\epsilon^{2})$ terms:

$\displaystyle{\cal V}^{\rm
mix}_{R}=\left(\begin{array}[]{ccc}S_{u}S_{d}^{\dagger}&0&-S_{u}(1-\frac{1}{2}S_{d}^{\dagger}S_{d})\\\
0&0&0\\\
-(1-\frac{1}{2}S_{u}^{\dagger}S_{u})S_{d}^{\dagger}&0&\;1-\frac{1}{2}S_{d}^{\dagger}S_{d}-\frac{1}{2}S_{u}^{\dagger}S_{u}\end{array}\right)$
(119)

where $S_{d}=\epsilon_{q}\widetilde{h}_{\cal D}^{\dagger}$ and
$S_{u}=\epsilon_{q}\widetilde{h}_{\cal U}^{\dagger}$. Thus, we see that the
mixing with the weak isodoublet $Q$-type species induces RH charged current
interactions between the quark mass eigenstates $u^{\prime}=(u,c,t)$ and
$d^{\prime}=(d,s,b)$, given by the (non-unitary) $3\times 3$ matrix:

$\displaystyle\Delta=\left(\begin{array}[]{ccc}\Delta_{ud}&\Delta_{us}&\Delta_{ub}\\\
\Delta_{cd}&\Delta_{cs}&\Delta_{cb}\\\
\Delta_{td}&\Delta_{ts}&\Delta_{tb}\end{array}\right)=S_{u}S_{d}^{\dagger}=\epsilon_{q}^{2}\widetilde{h}_{u}^{\dagger}\widetilde{h}_{d}=\epsilon_{q}^{2}\left(\begin{array}[]{ccc}h_{u}^{\ast}{h}_{d}&h_{u}^{\ast}{h}_{s}&h_{u}^{\ast}{h}_{b}\\\
h_{c}^{\ast}{h}_{d}&h_{c}^{\ast}{h}_{s}&h_{c}^{\ast}{h}_{b}\\\
h_{t}^{\ast}{h}_{d}&h_{t}^{\ast}{h}_{s}&h_{t}^{\ast}{h}_{b}\end{array}\right)$
(126)

where we denote the elements of row vectors $\widetilde{h}_{\cal U,D}$ in the
light quark mass basis as $\widetilde{h}_{\cal U}=(h_{u},h_{c},h_{t})$ and
$\widetilde{h}_{\cal D}=(h_{d},h_{s},h_{b})$. Therefore, mixing with $Q$-type
fermions violates pure $V-A$ character of the quark interactions with $W$
boson, and vector and axial couplings for each transition are not equal
anymore but have a difference $O(\epsilon_{q}^{2})$.

In fact, instead of purely $V-A$ couplings (5), now we have

$\displaystyle\frac{g}{2\sqrt{2}}\,W^{+}_{\mu}\,\overline{(u\leavevmode\nobreak\
\leavevmode\nobreak\ c\leavevmode\nobreak\ \leavevmode\nobreak\
t)}\,\big{[}\gamma^{\mu}(V_{\rm CKM}+\Delta)-\gamma^{\mu}\gamma^{5}(V_{\rm
CKM}-\Delta)\big{]}\,\left(\begin{array}[]{c}d\\\ s\\\ b\end{array}\right)$
(130)

The presence of RH couplings has a direct implications for our problem. In
particular, determination C from purely Fermi $0^{+}-0^{+}$ transitions now
fixes the vector coupling constant $G_{V}=G_{F}|V_{ud}+\Delta_{ud}|$, instead
of $G_{F}|V_{ud}|$. Analogously, determination A from semileptonic decays
$K\ell 3$, transforms in the determination of the vector coupling
$|V_{us}+\Delta_{us}|$, instead of $|V_{us}|$. On the other hand, since the
leptonic decays $K\mu 2$ and $\pi\mu 2$ are contributed only by axial current,
determination B instead of the ratio $|V_{us}/V_{ud}|$ fixes now the
combination $|V_{us}-\Delta_{us}|/|V_{ud}-\Delta_{ud}|$.

Therefore, instead of (17), (13) and (15), now we have:

$\displaystyle\text{C}\,:\qquad|V_{ud}+\Delta_{ud}|=|V_{ud}|\,|1+\delta_{ud}|=0.97376(16)$
(131)
$\displaystyle\text{A}:\qquad|V_{us}+\Delta_{us}|=|V_{us}|\,|1+\delta_{us}|=0.22326(55)$
(132)
$\displaystyle\text{B}:\qquad\left|\frac{V_{us}-\Delta_{us}}{V_{ud}-\Delta_{ud}}\right|=\left|\frac{V_{us}}{V_{ud}}\right|\,\left|\frac{1-\delta_{us}}{1-\delta_{ud}}\right|=0.23130(49)$
(133)

where
$\delta_{ud}=\Delta_{ud}/V_{ud}=\epsilon_{q}^{2}\,h_{u}^{\ast}h_{d}/V_{ud}$
and
$\delta_{us}=\Delta_{us}/V_{us}=\epsilon_{q}^{2}\,h_{u}^{\ast}h_{s}/V_{us}$
are in general complex numbers. Hence, $Q$-type extra fermion can have
interesting implications and potentially it can resolve all tensions between
A, B and C determinations.999In principle, one can obtain the RH weak currents
also from left-right symmetric models. However, strong limits on $W_{R}$ mass
imply that the mixing $W_{L}$-$W_{R}$ is too small to give the RH
contributions needed to explain the anomalies.

Figure 3:  Anomolous flavour non-diagonal couplings of $Z$-boson with SM
families, contributing to flavour changing processes at tree level.

Figure 4:  Box diagrams induced by the presence of vector-like quarks.

However, mixing with extra vector-like species (with non-standard isosipin
content) affects the natural flavor conservation of the SM. Namely, by
integrating out the heavy isosinglet states $D$ and $U$ one induces the
following effective operators for the quark couplings with $Z$-boson

$\displaystyle\frac{g}{\cos\theta_{W}}\overline{q_{Li}}\gamma^{\mu}\left[(T_{3}-\text{Q}\sin\theta_{W}^{2})\delta_{ij}-T_{3}\frac{h_{Di}h_{Dj}^{\dagger}\varphi^{\dagger}\varphi}{M_{D}^{2}}\right]q_{Lj}Z_{\mu}\,,$
$\displaystyle\frac{g}{\cos\theta_{W}}\overline{q_{Li}}\gamma^{\mu}\left[(T_{3}-\text{Q}\sin\theta_{W}^{2})\delta_{ij}-T_{3}\frac{h_{Ui}h_{Uj}^{\dagger}\tilde{\varphi}^{\dagger}\tilde{\varphi}}{M_{U}^{2}}\right]q_{Lj}Z_{\mu},$
(134)

as it is shown in figure 3, where $T_{3}$ is the weak isospin and Q the
electric charge, while mixing with the extra doublet $Q$ induces anomalous
isospin violating couplings with the RH states

$\displaystyle\frac{g}{\cos\theta_{W}}\left[-\frac{1}{2}\frac{h_{di}^{\dagger}h_{dj}\varphi^{\dagger}\varphi}{M_{Q}^{2}}+\frac{1}{3}\sin\theta_{W}^{2}\delta_{ij}\right]\overline{d_{Ri}}\gamma^{\mu}d_{Rj}Z_{\mu},$
$\displaystyle\frac{g}{\cos\theta_{W}}\left[\frac{1}{2}\frac{h_{ui}^{\dagger}h_{uj}\tilde{\varphi}^{\dagger}\tilde{\varphi}}{M_{Q}^{2}}-\frac{2}{3}\sin\theta_{W}^{2}\delta_{ij}\right]\overline{u_{Ri}}\gamma^{\mu}u_{Rj}Z_{\mu},$
(135)

Thus, such anomalous couplings, flavor-non-diagonal between the mass
eigenstates, after substitution of the VEV $\langle\phi\rangle=v_{w}$,
contribute at tree level in the flavor changing phenomena as
$K^{0}-\overline{K}^{0}$ mixing etc. inducing four-fermion effective operators

$\displaystyle-\frac{h_{Di}h_{Dj}^{\dagger}h_{Dk}h_{Dm}^{\dagger}v_{w}^{2}}{4\,M^{4}_{D}}(\overline{d_{Li}}\gamma^{\mu}d_{Lj})(\overline{d_{Lk}}\gamma_{\mu}d_{Lm}),\quad\quad-\frac{h_{Ui}h_{Uj}^{\dagger}h_{Uk}h_{Um}^{\dagger}v_{w}^{2}}{4\,M^{4}_{U}}\overline{u_{Li}}\gamma^{\mu}u_{Lj}\overline{u_{Lk}}\gamma^{\mu}u_{Lm}$
(136)

and analogously for RH states. These operators parametrically are order
$G_{F}(v_{w}/M)^{4}\sim v_{w}^{2}/M^{4}$. On the other hand, box diagrams
shown in the upper-left part of figure 4 (and analogously for up-type quarks)
induce operators which parametrically are order $1/M^{2}$:

$\displaystyle\frac{h_{Di}h_{Dj}^{\dagger}h_{Dk}h_{Dm}^{\dagger}}{128\pi^{2}\,M^{2}_{D}}(\overline{d_{Li}}\gamma^{\mu}d_{Lj})(\overline{d_{Lk}}\gamma_{\mu}d_{Lm}),\quad\quad\frac{h_{Ui}h_{Uj}^{\dagger}h_{Uk}h_{Um}^{\dagger}}{128\pi^{2}\,M^{4}_{U}}\overline{u_{Li}}\gamma^{\mu}u_{Lj}\overline{u_{Lk}}\gamma^{\mu}u_{Lm}$
(137)

but are suppressed by a loop factor. Regarding the RH sector, as in the upper-
right part of figure 4 we have:

$\displaystyle\frac{h_{di}^{\dagger}h_{dj}h_{dk}^{\dagger}h_{dm}}{64\pi^{2}\,M^{2}_{D}}(\overline{d_{Ri}}\gamma^{\mu}d_{Rj})(\overline{d_{Rk}}\gamma_{\mu}d_{Rm}),\quad\quad\frac{h_{ui}^{\dagger}h_{uj}h_{uk}^{\dagger}h_{um}}{64\pi^{2}\,M^{4}_{U}}\overline{u_{Ri}}\gamma^{\mu}u_{Rj}\overline{u_{Rk}}\gamma^{\mu}u_{Rm}$
(138)

In the presence of all type of fermions, also left-right operators can be
induced, as shown in figure 4 (bottom).

In addition, extra vector-like leptons induce analogous contributions in
lepton sector. However, in the following, for brevity, we shall concentrate
only on quark sector, and study one by one implications of $D$-type, $U$-type
and $Q$-type extra fermions.

Namely, in each of these cases we put limits that emerge from flavour changing
phenomena ($K^{0}$-$\bar{K}^{0}$, $D^{0}$-$\bar{D}^{0}$,
$B^{0}$-$\bar{B}^{0}$, flavour changing leptonic and semileptonic meson
decays) and flavour conserving observables ($Z$-boson physics, low energy
electroweak observables). For down-type weak singlets these limits were first
discussed in ref. Lavoura . However in that work contributions of box diagrams
involving heavy states were not discussed, whereas, as we will show, these
diagrams become dominant if heavy vector-like quarks have masses larger than
$\sim 3$ TeV.

##  4 Extra down-type isosinglet

Let us examine in details the implications of the addition of a down-type
vector-like weak isosinglet (D-type) couple of quarks $D_{L}=d_{L4}$ and
$D_{R}=d_{R4}$ involved in the mixing with the SM three families
$q_{Li}=(u_{Li},d_{Li})^{T}$, $u_{Ri}$ and $d_{Ri}$, $i=1,2,3$. New Yukawa
terms and Dirac mass terms should be added to the Lagrangian density besides
the standard Yukawa terms:
$h_{dj}^{\prime}\varphi\overline{q_{Lj}}d_{Ri}^{\prime}+m_{i}\overline{d_{L4}}d_{Ri}^{\prime}+\text{h.c.}$.
Since the four species of right-handed singlets $d_{Ri}^{\prime}$ have
identical quantum numbers, a unitary transformation can be applied on the four
components $d_{Ri}^{\prime}$ so that $m_{j}=0$ for $j=1,2,3$ and $d_{R4}$ is
identified with the combination making the Dirac mass term with the left
handed singlet $d_{L4}$. Thus the Yukawa Lagrangian of this system can be
written as:

$\displaystyle
y^{u}_{ij}\tilde{\varphi}\overline{q_{Li}}u_{Rj}+y^{d}_{ij}\varphi\overline{q_{Li}}d_{Rj}+h_{dj}\varphi\overline{q_{Lj}}d_{R4}+M_{4}\overline{d_{L4}}d_{R4}+\text{h.c.}$
(139)

where, without losing generality, the mass term $M_{4}$ can be taken real and
positive. Then the down-type quarks mass matrix looks like:

$\displaystyle\overline{d_{Li}}\mathbf{m}^{(d)}_{ij}d_{Rj}+\text{h.c.}=$
$\displaystyle=$
$\displaystyle(\overline{d_{L1}},\overline{d_{L2}},\overline{d_{L3}},\overline{d_{L4}})\left(\begin{array}[]{ccc|c}&&&h_{d1}v_{w}\\\
&\mathbf{y}^{(d)}_{3\times 3}v_{w}&&h_{d2}v_{w}\\\ &&&h_{d3}v_{w}\\\ \hline\cr
0&0&0&M_{4}\end{array}\right)\left(\begin{array}[]{c}d_{R1}\\\ d_{R2}\\\
d_{R3}\\\ d_{R4}\end{array}\right)+\text{h.c.}$ (148)

where $v_{w}=174$ GeV is the SM Higgs vacuum expectation value (VEV) (for a
convenience, we use this normalization of the Higgs VEV instead of the
“standard" normalization $\langle\phi\rangle=v/\sqrt{2}$, i.e.
$v=\sqrt{2}v_{\rm w}$) and $\mathbf{y}^{(d)}_{3\times 3}$ is the $3\times 3$
matrix of Yukawa couplings. The mass matrix $\mathbf{m}^{(d)}$ can be
diagonalized with positive eigenvalues by a biunitary transformation:

$\displaystyle
V_{L}^{(d)\dagger}\mathbf{m}^{(d)}V_{R}^{(d)}=\mathbf{m}^{(d)}_{\text{diag}}=\text{diag}(y_{d}v_{w},y_{s}v_{w},y_{b}v_{w},M_{b^{\prime}})$
(149)

where $V^{(d)}_{L,R}$ are two unitary $4\times 4$ matrices.
$\mathbf{m}^{(d)}_{\text{diag}}$ is the diagonal matrix of mass eigenvalues
$m_{d,s,b}=y_{d,s,b}v_{w}$ and $M_{b^{\prime}}\approx M_{4}$. Weak eigenstates
in terms of mass eigenstates are:

$\displaystyle\left(\begin{array}[]{c}d_{L1}\\\ d_{L2}\\\ d_{L3}\\\
d_{L4}\end{array}\right)=V_{L}^{(d)}\left(\begin{array}[]{c}d_{L}\\\ s_{L}\\\
b_{L}\\\ b^{\prime}_{L}\end{array}\right)\,,\qquad
V^{(d)}_{L}=\left(\begin{array}[]{cccc}V_{L1d}&V_{L1s}&V_{L1b}&V_{L1b^{\prime}}\\\
V_{L2d}&V_{L2s}&V_{L2b}&V_{L2b^{\prime}}\\\
V_{L3d}&V_{L3s}&V_{L3b}&V_{L3b^{\prime}}\\\
V_{L4d}&V_{L4s}&V_{L4b}&V_{L4b^{\prime}}\end{array}\right)$ (162)

As for up-type quarks, the up-quark Yukawa matrix can be taken diagonal,
$\mathbf{y}^{u}=\text{diag}(y_{u},y_{c},y_{t})$, so that also the mass matrix
$\mathbf{m}_{u}=\text{diag}(m_{u},m_{c},m_{t})=\text{diag}(y_{u},y_{c},y_{t})\times
v_{w}$ is diagonal, and $u_{1},u_{2},u_{3}$ correspond to the mass eigenstates
$u,c,t$. Since only the three down-type quarks $d_{L1},d_{L2},d_{L3}$ couple
with $W$-bosons, the Lagrangian for the charged weak interactions expressed in
terms of mass eigenstates become:

$\displaystyle\mathcal{L}_{\text{cc}}$
$\displaystyle=\frac{g}{\sqrt{2}}\left(\begin{array}[]{ccc}\overline{u_{L1}}&\overline{u_{L2}}&\overline{u_{L3}}\end{array}\right)\gamma^{\mu}\left(\begin{array}[]{c}d_{L1}\\\
d_{L2}\\\ d_{L3}\end{array}\right)W_{\mu}^{+}+\text{h.c.}=$ (167)
$\displaystyle=\frac{g}{\sqrt{2}}\left(\begin{array}[]{ccc}\overline{u_{L}}&\overline{c_{L}}&\overline{t_{L}}\end{array}\right)\gamma^{\mu}\tilde{V}_{\text{CKM}}\left(\begin{array}[]{c}d_{L}\\\
s_{L}\\\ b_{L}\\\ b^{\prime}_{L}\end{array}\right)W_{\mu}^{+}+\text{h.c.}$
(173)

where

$\displaystyle\tilde{V}_{\text{CKM}}=$
$\displaystyle\tilde{V}_{L}^{(d)}=\left(\begin{array}[]{cccc}V_{L1d}&V_{L1s}&V_{L1b}&V_{L1b^{\prime}}\\\
V_{L2d}&V_{L2s}&V_{L2b}&V_{L2b^{\prime}}\\\
V_{L3d}&V_{L3s}&V_{L3b}&V_{L3b^{\prime}}\end{array}\right)=\left(\begin{array}[]{ccc|c}&&&V_{ub^{\prime}}\\\
&V_{\text{CKM}}&&V_{cb^{\prime}}\\\
&&&V_{tb^{\prime}}\end{array}\right)=\left(\begin{array}[]{cccc}V_{ud}&V_{us}&V_{ub}&V_{ub^{\prime}}\\\
V_{cd}&V_{cs}&V_{cb}&V_{cb^{\prime}}\\\
V_{td}&V_{ts}&V_{tb}&V_{tb^{\prime}}\end{array}\right)$ (183)

is the $3\times 4$ submatrix of $V_{L}^{(d)}$ in eq. (162), obtained by
cutting the last row. Obviously, $\tilde{V}_{\text{CKM}}$ is not unitary
anymore. More precisely, although the unitarity condition is violated for
columns, it still holds for rows:
$\tilde{V}^{(d)}_{L}\tilde{V}^{(d){\dagger}}_{L}=1_{3\times 3}$, with
$1_{3\times 3}$ being the $3\times 3$ identity matrix. In particular, the
first row unitarity condition of CKM matrix is modified to

$\displaystyle|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}+|V_{ub^{\prime}}|^{2}=1$
(184)

which is the extended unitarity condition for the first row. The elements of
the fourth column of $\tilde{V}_{\text{CKM}}$ determine the strength of the
violation of SM CKM unitarity.

Figure 5:  Determinations of $|V_{us}|$ obtained using eq. (184) with $|V_{ub^{\prime}}|=0.038$, with the dataset in eqs. (13), (15), (22), to be compared with figure 2.  Determination |  $|V_{us}|$ value | $|V_{ub^{\prime}}|=0.038$
---|---|---
A | $0.22326(55)$ | $0.22326(55)$
B | $0.22535(45)$ | $0.22518(45)$
C | $0.22756(70)$ | $0.22429(71)$
A+B | $0.22451(35)$ | $0.22441(35)$
A+B+C | $0.22511(31)$ | $0.22439(31)$
Table 3:  Values of $V_{us}$ obtained from the dataset in eqs. (13), (15),
(22). In the first column the SM unitarity of CKM matrix is used, while in the
second column the extended unitarity (184) is used with
$|V_{ub^{\prime}}|=0.038$.

The dataset A, B, C from eqs. (13), (15), (17) can be fitted in this scenario
by using the extended unitarity (184). The best fit point
($\chi^{2}_{\text{dof}}=7.4$) is obtained in $|V_{us}|=0.22439(36)$, with:

$\displaystyle|V_{ub^{\prime}}|^{2}=1.48(36)\times
10^{-3}\;,\qquad\quad|V_{ub^{\prime}}|=0.038(5)$ (185)

At $95$% C.L. the needed additional mixing is
$|V_{ub^{\prime}}|=0.038^{+0.008}_{-0.011}$. In figure 5 it is shown how the
present situation would look like by choosing $|V_{ub^{\prime}}|=0.038$. The
data are listed in table 3. Determinations B and C are shifted with respect to
the ones in figure 2, while A remains unchanged. Determination C, which is
obtained from $V_{ud}$ from superallowed beta decays, becomes perfectly
aligned with the average of the determinations of $V_{us}$ obtained from kaon
decays, because of the extended unitarity relation (184). The $\chi^{2}$ is
rather large $\chi^{2}_{\text{dof}}=7.4$, because of the remaining tension
between the determinations of $V_{us}$ extracted from kaon physics (A and B).

However, the mixing $V^{(d)}_{L}$ induces non-standard couplings of $Z$-boson
with the LH down quarks, since the normal families $d_{Li}$, $i=1,2,3$ reside
in doublets while $d_{L4}$ is a weak singlet (the mixing $V^{(d)}_{R}$ of the
RH quarks does not give the same effect since all RH states $d_{Ri}$ are in
identical representations of the SM). In fact, $Z$-boson couples to a fermion
species $f$ (LH or RH) as $Z_{\mu}\overline{f}(T_{3}+Q\sin^{2}\theta_{W})f$,
where $T_{3}$ is the weak isospin projection and $Q$ the electric charge.
Therefore, $Q$-dependent couplings remain diagonal between the mass
eigenstates $d,s,b,b^{\prime}$, while the $T_{3}$ dependent part gets non-
diagonal couplings. The weak neutral current Lagrangian for down quarks reads:

$\displaystyle\mathcal{L}_{\text{nc}}=$
$\displaystyle\frac{g}{\cos\theta_{W}}\left[-\frac{1}{2}\left(\begin{array}[]{ccc}\overline{d_{L1}}&\overline{d_{L2}}&\overline{d_{L3}}\end{array}\right)\gamma^{\mu}\left(\begin{array}[]{c}d_{L1}\\\
d_{L2}\\\
d_{L3}\end{array}\right)+\frac{1}{3}\sin^{2}\theta_{W}\left(\overline{\mathbf{d}_{L}}\gamma^{\mu}\mathbf{d}_{L}+\overline{\mathbf{d}_{R}}\gamma^{\mu}\mathbf{d}_{R}\right)\right]Z_{\mu}=$
(190) $\displaystyle=$
$\displaystyle\frac{g}{\cos\theta_{W}}\left[-\frac{1}{2}\left(\begin{array}[]{cccc}\overline{d_{L}}&\overline{s_{L}}&\overline{b_{L}}&\overline{b^{\prime}_{L}}\end{array}\right)\gamma^{\mu}V^{(d)}_{\text{nc}}\left(\begin{array}[]{c}d_{L}\\\
s_{L}\\\ b_{L}\\\
b^{\prime}_{L}\end{array}\right)+\frac{1}{3}\sin^{2}\theta_{W}\left(\overline{\mathbf{d}_{L}}\gamma^{\mu}\mathbf{d}_{L}+\overline{\mathbf{d}_{R}}\gamma^{\mu}\mathbf{d}_{R}\right)\right]Z_{\mu}$
(196) $\displaystyle V^{(d)}_{\text{nc}}=$ $\displaystyle
V_{L}^{(d){\dagger}}\text{diag}(1,1,1,0)V_{L}^{(d)}=\tilde{V}_{L}^{(d){\dagger}}\tilde{V}_{L}^{(d)}$
(197)

where $\mathbf{d}$ is the column vector of the four down-type quarks
$d,s,b,b^{\prime}$. As comes out from eq. (190), the non-unitarity of
$\tilde{V}^{(d)}_{L}$ is at the origin of non-diagonal couplings with $Z$
boson, which means flavor changing neutral currents (FCNC) at tree level.
Explicitly, the weak isospin dependent part of the $Z$ coupling is given by:

$\displaystyle
V_{\text{nc}}^{(d)}=V_{L}^{(d){\dagger}}\text{diag}(1,1,1,0)V_{L}^{(d)}=$
$\displaystyle=\\!\left(\begin{array}[]{c@{\hspace{1\tabcolsep}}c@{\hspace{1\tabcolsep}}c@{\hspace{1\tabcolsep}}c}1\hfil\hskip
6.0pt&0\hfil\hskip 6.0pt&0\hfil\hskip 6.0pt&0\\\ 0\hfil\hskip
6.0pt&1\hfil\hskip 6.0pt&0\hfil\hskip 6.0pt&0\\\ 0\hfil\hskip
6.0pt&0\hfil\hskip 6.0pt&1\hfil\hskip 6.0pt&0\\\ 0\hfil\hskip
6.0pt&0\hfil\hskip 6.0pt&0\hfil\hskip
6.0pt&0\end{array}\right)\\!+\\!\left(\begin{array}[]{c@{\hspace{1\tabcolsep}}c<EMAIL_ADDRESS>6.0pt&-V_{L4d}^{*}V_{L4s}\hfil\hskip 6.0pt&-V_{L4d}^{*}V_{L4b}\hfil\hskip
9.0pt&-V_{L4d}^{*}V_{L4b^{\prime}}\\\ -V_{L4s}^{*}V_{L4d}\hfil\hskip
6.0pt&-|V_{L4s}|^{2}\hfil\hskip 6.0pt&-V_{L4s}^{*}V_{L4b}\hfil\hskip
9.0pt&-V_{L4s}^{*}V_{L4b^{\prime}}\\\ -V_{L4b}^{*}V_{L4d}\hfil\hskip
6.0pt&-V_{L4b}^{*}V_{L4s}\hfil\hskip 6.0pt&-|V_{L4b}|^{2}\hfil\hskip
9.0pt&-V_{L4b}^{*}V_{L4b^{\prime}}\\\ -V_{L4b^{\prime}}^{*}V_{L4d}\hfil\hskip
6.0pt&-V_{L4b^{\prime}}^{*}V_{L4s}\hfil\hskip
6.0pt&-V_{L4b^{\prime}}^{*}V_{L4b}\hfil\hskip
9.0pt&|V_{L1b^{\prime}}|^{2}\\!+\\!|V_{L2b^{\prime}}|^{2}\\!+\\!|V_{L3b^{\prime}}|^{2}\end{array}\right)$
(206)

$V^{(d)}_{L}$ can be parameterized by $6$ angles and $10$ phases. However,
four phases can be eliminated by phase transformations of $d,s,b,b^{\prime}$
states, so that $V^{(d)}_{L}$ can be presented as:

$\displaystyle
V^{(d)}_{L}=\left(\begin{array}[]{cccc}V_{L1d}&V_{L1s}&V_{L1b}&V_{L1b^{\prime}}\\\
V_{L2d}&V_{L2s}&V_{L2b}&V_{L2b^{\prime}}\\\
V_{L3d}&V_{L3s}&V_{L3b}&V_{L3b^{\prime}}\\\
V_{L4d}&V_{L4s}&V_{L4b}&V_{L4b^{\prime}}\end{array}\right)\simeq
V_{3L}^{(d)}L^{(d)}=$ (211)
$\displaystyle=\left(\begin{array}[]{c@{\hspace{0.5\tabcolsep}}c@{\hspace{0.5\tabcolsep}}c@{\hspace{1\tabcolsep}}c}\hfil\hskip
3.0pt&\hfil\hskip 3.0pt&\hfil\hskip 6.0pt&0\\\ \hfil\hskip 3.0pt&V_{3\times
3}^{(d)}\hfil\hskip 3.0pt&\hfil\hskip 6.0pt&0\\\ \hfil\hskip 3.0pt&\hfil\hskip
3.0pt&\hfil\hskip 6.0pt&0\\\ 0\hfil\hskip 3.0pt&0\hfil\hskip
3.0pt&0\hfil\hskip
6.0pt&1\end{array}\right)\\!\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&c^{d}_{L3}&-\tilde{s}^{d}_{L3}\\\
0&0&\tilde{s}^{d*}_{L3}&c^{d}_{L3}\end{array}\right)\\!\left(\begin{array}[]{cccc}1&0&0&0\\\
0&c^{d}_{L2}&0&-\tilde{s}^{d}_{L2}\\\ 0&0&1&0\\\
0&\tilde{s}^{d*}_{L2}&0&c^{d}_{L2}\end{array}\right)\\!\left(\begin{array}[]{cccc}c^{d}_{L1}&0&0&-\tilde{s}^{d}_{L1}\\\
0&1&0&0\\\ 0&0&1&0\\\
\tilde{s}^{d*}_{L1}&0&0&c^{d}_{L1}\end{array}\right)\\!\approx$ (228)
$\displaystyle\approx\left(\begin{array}[]{c@{\hspace{0.5\tabcolsep}}c@{\hspace{0.5\tabcolsep}}c@{\hspace{1\tabcolsep}}c}\hfil\hskip
3.0pt&\hfil\hskip 3.0pt&\hfil\hskip 6.0pt&0\\\ \hfil\hskip 3.0pt&V_{3\times
3}^{(d)}\hfil\hskip 3.0pt&\hfil\hskip 6.0pt&0\\\ \hfil\hskip 3.0pt&\hfil\hskip
3.0pt&\hfil\hskip 6.0pt&0\\\ 0\hfil\hskip 3.0pt&0\hfil\hskip
3.0pt&0\hfil\hskip
6.0pt&1\end{array}\right)\left(\begin{array}[]{cccc}c^{d}_{L1}&0&0&-\tilde{s}^{d}_{L1}\\\
-\tilde{s}^{d}_{L2}\tilde{s}^{d*}_{L1}&c^{d}_{L2}&0&-\tilde{s}^{d}_{L2}\\\
-\tilde{s}^{d}_{L3}\tilde{s}^{d*}_{L1}&-\tilde{s}^{d*}_{L2}\tilde{s}^{d}_{L3}&c^{d}_{L3}&-\tilde{s}^{d}_{L3}\\\
\tilde{s}^{d*}_{L1}&\tilde{s}^{d*}_{L2}&\tilde{s}^{d*}_{L3}&c^{d}_{L1}c^{d}_{L2}c^{d}_{L3}\end{array}\right)$
(237)

$c^{d}_{Li}$ are cosines and $\tilde{s}^{d}_{Li}$ are complex sines of angles
in the $1\,4$, $2\,4$, $3\,4$ family planes parameterizing the mixing of the
first three families with the vector-like quark:

$\displaystyle\tilde{s}^{d}_{Li}=\sin\theta^{d}_{Li4}e^{i\delta^{d}_{Li}}=s^{d}_{Li}e^{i\delta^{d}_{Li}}\;,\qquad
c^{d}_{Li}=\cos\theta^{d}_{Li4}$
$\displaystyle\delta^{d}_{Lij}=\delta^{d}_{Li}-\delta^{d}_{Lj}$ (238)

and corresponding to the elements of the last row:

$\displaystyle V_{L4d}\approx\tilde{s}_{L1}^{d*}\,,$ $\displaystyle
V_{L4s}\approx\tilde{s}_{L2}^{d*}\,,$ $\displaystyle
V_{L4b}=\tilde{s}_{L3}^{d*}$ (239)

Since it is the relative phase of the elements which will come into play, we
also defined the relative phase of the elements in eq. (4). $V_{3L}^{(d)}$
contains $3$ angles and $3$ phases; it diagonalizes the $3\times 3$ Yukawa
matrix $\mathbf{y}^{(d)}_{3\times 3}$ in eq. (4). After applying this $3\times
3$ diagonalization, since the elements in $V_{3L}^{(d)}$ are small, the mixing
of the three SM families with the vector-like species is still described by
the couplings: $h_{di}=V^{(d)}_{Lij}\tilde{h}_{dj}\approx\tilde{h}_{di}$.
Equation (237) is true at order
$O(|h_{di}|(y_{i})^{2}\frac{v_{w}^{3}}{M_{b^{\prime}}^{3}}+|h_{dj}|^{2}|h_{di}|\frac{v_{w}^{3}}{M_{b^{\prime}}^{3}})$,
$i,j=1,2,3$, $y_{i}=y_{d,s,b}$, with:

$\displaystyle\frac{1}{2}\tan(2\theta^{d}_{Li4})\approx\frac{|h_{di}|v_{w}}{M_{b^{\prime}}}\;,\qquad\quad\delta^{d}_{Li}=\arg(h_{di})-\pi$
(240)

that is:

$\displaystyle\tilde{s}^{d}_{Li}\approx-\frac{h_{di}v_{w}}{M_{b^{\prime}}}$
(241)

Because of small mixing angles, $V_{3\times 3}^{(d)}$ is practically equal to
the $3\times 3$ submatrix of $V^{(d)}_{L}$ in (162). In fact, for example in
the chosen parameterization (237), the main corrections regard the elements:
$V_{L3d}\simeq V_{3\times 3\,3d}-\tilde{s}^{d}_{L1}\tilde{s}^{d}_{L3}$,
$V_{L2d}\simeq V_{3\times 3\,2d}-\tilde{s}^{d}_{L1}\tilde{s}^{d}_{L2}$,
$V_{L3s}\simeq V_{3\times 3\,3s}-\tilde{s}^{d}_{L2}\tilde{s}^{d}_{L3}$,
$V_{L1d}\simeq V_{3\times 3\,1d}\,c_{L1}^{d}$. However, it will be shown that,
in order to have $s^{d}_{L1}\approx|V_{L4d}|\approx|V_{ub^{\prime}}|\approx
0.03$, it should be that at most $s^{d}_{L3}=|V_{L4b}|<7.4\times 10^{-3}$ and
$s^{d}_{L2}\approx|V_{L4s}|<5.8\times 10^{-4}$. Then:

$\displaystyle[V_{L}^{(d)}]_{i\beta}\simeq[V^{(d)}_{3\times 3}]_{i\beta}$
(242)

As regards charged currents, $\tilde{V}_{\text{CKM}}$ in (183) can be
described by $6$ moduli and $9$ phases, $6$ of which can be absorbed into the
quark fields. For the submatrix $V_{\text{CKM}}$ in (183), it holds that:

$\displaystyle[V_{\text{CKM}}]_{\alpha\beta}=\sum_{i=1}^{3}V^{(u)*}_{L\,i\alpha}V_{L\,i\beta}^{(d)}\simeq\sum_{i=1}^{3}V^{(u)*}_{L\,i\alpha}V_{3\times
3\,i\beta}^{(d)}$ (243)

Then, after rephasing the quark fields, $V_{\text{CKM}}$ can be in the usual
parameterization with $3$ angles and one phase. Also another phase can be
absorbed, so we can always choose $\delta^{d}_{L1}=0$ without loss of
generality. From (243), for the elements of the fourth column of
$\tilde{V}_{\text{CKM}}$ in (183) it holds that:

$\displaystyle V_{ub^{\prime}}\approx-
V_{L4d}^{*}V_{ud}-V_{L4s}^{*}V_{us}-V_{L4b}^{*}V_{ub}\approx-V_{L4d}^{*}$
(244) $\displaystyle V_{cb^{\prime}}\approx-
V_{L4d}^{*}V_{cd}-V_{L4s}^{*}V_{cs}-V_{L4b}^{*}V_{cb}$ (245) $\displaystyle
V_{tb^{\prime}}\approx-
V_{L4d}^{*}V_{td}-V_{L4s}^{*}V_{ts}-V_{L4b}^{*}V_{tb}\simeq-V_{L4b}^{*}$ (246)

where the last approximations come from the mentioned constraints on the
mixings.

It should be noted that also the couplings of quarks with the real Higgs are
not diagonal if the vector-like species are added. In fact, the matrix of
Yukawa couplings and the mass matrix (4) are not proportional anymore, and
then they are not diagonalized by the same transformation. In particular,
left-handed light quarks are coupled with $b^{\prime}_{R}$ with coupling
constants which can be in principle of order $O(1)$. In fact, the Higgs
couplings with quarks are:

$\displaystyle(\overline{d_{L1}},\overline{d_{L2}},\overline{d_{L3}},\overline{d_{L4}})\left(\begin{array}[]{ccc|c}&&&\tilde{h}_{d1}\\\
&\mathbf{y}^{(d)}_{3\times 3}&&\tilde{h}_{d2}\\\ &&&\tilde{h}_{d3}\\\
\hline\cr 0&0&0&0\end{array}\right)\left(\begin{array}[]{c}d_{R1}\\\ d_{R2}\\\
d_{R3}\\\ d_{R4}\end{array}\right)\frac{H^{0}}{\sqrt{2}}+\text{h.c.}\approx$
(255) $\displaystyle\approx$
$\displaystyle(\overline{d_{L}},\overline{s_{L}},\overline{b_{L}},\overline{b^{\prime}_{L}})L^{(d){\dagger}}\left(\begin{array}[]{ccc
c}y_{d}&0&0&h_{d1}\\\ 0&y_{s}&0&h_{d2}\\\ 0&0&y_{b}&h_{d3}\\\
0&0&0&0\end{array}\right)R^{(d)}\left(\begin{array}[]{c}d_{R}\\\ s_{R}\\\
b_{R}\\\
b^{\prime}_{R}\end{array}\right)\frac{H^{0}}{\sqrt{2}}+\text{h.c.}\approx$
(264) $\displaystyle\approx$
$\displaystyle(\overline{d_{L}},\overline{s_{L}},\overline{b_{L}},\overline{b^{\prime}_{L}})\left(\begin{array}[]{ccc
c}c_{L1}^{d}y_{d}&\tilde{s}_{R2}^{d*}h_{d1}&\tilde{s}_{R3}^{d*}h_{d1}&h_{d1}\\\
\tilde{s}_{R1}^{d*}h_{d2}&y_{s}&\tilde{s}_{R3}^{d*}h_{d2}&h_{d2}\\\
\tilde{s}_{R1}^{d*}h_{d3}&\tilde{s}_{R2}^{d*}h_{d3}&y_{b}&h_{d3}\\\
-\tilde{s}_{L1}^{d*}y_{d}&-\tilde{s}_{L2}^{d*}y_{s}&-\tilde{s}_{L3}^{d*}y_{b}&\sum_{i=1}^{3}(-\tilde{s}_{Li}^{d*}h_{di})\end{array}\right)\left(\begin{array}[]{c}d_{R}\\\
s_{R}\\\ b_{R}\\\
b^{\prime}_{R}\end{array}\right)\frac{H^{0}}{\sqrt{2}}+\text{h.c.}$ (273)

where $L^{(d)}$ is defined in (237) and $R^{(d)}$ is analogously defined. Then
in principle FC couplings between light quarks emerge at tree level. However
the mixing angles of the SM right-handed quarks with $b^{\prime}_{R}$ are much
smaller than angles in the left-handed sector,
$s_{Ri}^{d}\approx\frac{y_{i}|h_{di}|v_{w}^{2}}{M^{2}_{b^{\prime}}}$, where
$s_{Ri}^{d}$ is defined in the same way as $s_{Li}^{d}$ in eqs. (237), (4).

It should also be noticed that, because of the large mixing with the first
family, the extra quark $b^{\prime}$ would mainly decay into $u$ or $d$ quark
via the couplings with $W$, $Z$, $H$. The CMS experiment put lower limits on
the mass of vector-like quarks coupling to light quarks, which in our scenario
imply $M_{b^{\prime}}\gtrsim 700$ GeV CMS . It should be noticed that, with
this constraint, $|V_{ub^{\prime}}|\simeq 0.03$ can be obtained if
$|h_{d1}|\gtrsim 0.1$, much larger than the Yukawa constant of the bottom
quark $y_{b}$. In turn, by taking $|V_{ub^{\prime}}|>0.03$ in
$M_{b^{\prime}}=|h_{d1}|v_{w}/|V_{ub^{\prime}}|$, and assuming (for the
perturbativity) $|h_{d1}|<1$, there is an upper limit on the extra quark mass,
$M_{b^{\prime}}<5.8$ TeV.

In the following sections experimental limits from FCNC and electroweak
observables are examined. The results are summarized in section 4.5, in table
6 and figures 13, 14.

Table 4 displays the values used in computations. Since the effects of new
physics on effective operators generated at the tree-level within the SM can
in general be neglected, the experimental determinations of the CKM matrix
quantities derived by tree level processes can be used in our BSM scenarios.

Quantity | Value | Quantity | Value
---|---|---|---
$|V_{cd}|$ |  $0.221(4)$ |  $M_{t}$[GeV] | $172.76(30)$
$|V_{cb}|$ |  $0.0410(14)$ |  $m_{c}(m_{c})$[GeV] | $1.27(2)$
$|V_{ts}|$ |  $0.0388(11)$ |  $m_{s}(2\text{GeV})$[MeV] |  $93.12(69)$ FLAG2019
$|V_{td}|$ |  $0.0080(3)$ |  $m_{b}(m_{b})$[GeV] | $4.18(3)$
$|V_{ub}|$ |  $0.00382(24)$ |  $m_{K^{0}}$[MeV] | $497.611(13)$
$|V_{us}|$ |  $0.2245(8)$ |  $\tau_{K_{L}}$[s] | $5.116(21)\cdot 10^{-8}$
$|V_{cs}|$ |  $0.987(11)$ |  $\tau_{K_{S}}$[s] | $8.954(4)\cdot 10^{-11}$
$\lambda$ |  $0.22653(48)$ |  $M_{K^{+}}$[MeV] | $493.677(16)$
$\bar{\rho}$ |  $0.123(32)$ |  $\tau_{K^{+}}$[s] | $1.2380(20)\cdot 10^{-8}$
$A$ |  $0.799(28)$ |  $m_{\mu}$[MeV] | $105.6583745(24)$
$\bar{\eta}$ |  $0.382(29)$ |  $\tau_{D^{+}}$[s] | $1.040(7)\cdot 10^{-12}$
$\alpha_{s}(M_{Z})$ |  $0.1185(16)$ |  $M_{D^{+}}$[MeV] | $1869.65(5)$
$\alpha(M_{Z})^{-1}$ |  $127.952(9)$ |  $\tau_{D^{0}}$[s] | $4.101(15)\cdot 10^{-13}$
$\sin^{2}\theta_{W}(M_{Z})$ |  $0.23121(4)$ |  $m_{D^{0}}$[MeV] | $1864.83(5)$
$G_{F}[\text{GeV}^{-2}]$ |  $1.1663787(6)\cdot 10^{-5}$ |  $\tau_{B^{0}_{d}}$[s] | $1.519(4)\cdot 10^{-12}$
$m_{W}$[GeV] |  $80.379(12)$ |  $M_{B^{0}_{d}}$[MeV] | $5279.65(12)$
$m_{Z}$[GeV] |  $91.1876(21)$ |  $\tau_{B^{0}_{sH}}$[s] | $1.620(7)\cdot 10^{-12}$
$m_{H}$[GeV] |  $125.10(14)$ |  $M_{B^{0}_{s}}$[MeV] | $5366.88(14)$
Table 4:  The central values are employed in the computations. All values are
taken from ref. PDG18 , except $m_{s}(2\text{GeV})=93.12(69)$ MeV from ref.
FLAG2019 . The adopted values of $\lambda$, $A$, $\bar{\rho}$ and $\bar{\eta}$
are those received using only tree-level inputs in the global fit PDG18 .

###  4.1 Limits from rare kaon decays

####  4.1.1 $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$

The decay $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ is one of the golden modes for
testing the SM, since long-distance contributions are negligibly small. The
effective Lagrangian comes from a combination of $Z$-penguin and box-diagram
and it is given by Buchalla:1995vs :

$\displaystyle\mathcal{L}_{\text{SM}}$
$\displaystyle=-\frac{4G_{F}}{\sqrt{2}}\frac{\alpha(M_{Z})}{2\pi\sin^{2}\theta_{\text{W}}}\sum_{\ell=e,\mu,\tau}[V_{cs}^{*}V_{cd}X^{\ell}(x_{c})+V_{ts}^{*}V_{td}X^{\ell}(x_{t})](\overline{s_{L}}\gamma^{\mu}d_{L})(\overline{\nu_{\ell
L}}\gamma_{\mu}\nu_{\ell L})+\text{h.c.}$ (274)

$X(x_{a})$ are the Inami-Lim function including QCD and electroweak
corrections, with $x_{a}=m^{2}_{a}/M^{2}_{W}$, $a=c,t$. The index $\ell$
denotes the lepton flavor. The dependence on the charged lepton mass is
negligible for the top contribution due to $m_{t}\gg m_{\ell}$,
$X^{\ell}(x_{t})=X(x_{t})$, whereas, being $m_{\tau}$ comparable with $m_{c}$,
for the c-quark contribution the box with $\ell=\tau$ gives somewhat different
contribution from $\ell=e,\mu$ Buchalla:1995vs ,
$X^{e}(x_{c})=X^{e}(x_{c})\neq X^{\tau}(x_{c})$. Therefore, it is used an
averaged value
$\bar{X}=\frac{1}{3}\left(X^{e}(x_{c})+X^{\mu}(x_{c})+X^{\tau}(x_{c})\right)$.
Then the effective Lagrangian (274) can also be written as:

$\displaystyle\mathcal{L}_{\text{SM}}$
$\displaystyle=-\frac{4G_{F}}{\sqrt{2}}\mathcal{F}_{K}(\overline{s_{L}}\gamma^{\mu}d_{L})\sum_{e,\mu,\tau}(\overline{\nu_{\ell
L}}\gamma_{\mu}\nu_{\ell L})$ $\displaystyle\mathcal{F}_{K}$
$\displaystyle=\frac{\alpha(M_{Z})}{2\pi\sin^{2}\theta_{\text{W}}}\Big{(}V_{ts}^{*}V_{td}X(x_{t})+(V_{cs}^{*}V_{cd})\bar{X}(x_{c})\Big{)}=$
$\displaystyle=-\frac{\alpha(M_{Z})}{2\pi\sin^{2}\theta_{\text{W}}}\lambda^{5}\Big{(}A^{2}(1-\rho-i\eta)X(x_{t})+(1-\frac{\lambda^{2}}{2})P_{c}(X)\Big{)}$
(275)

where $P_{c}(X)=\frac{1}{\lambda^{4}}\bar{X}$, and we used the Wolfenstein
parameterization:

$\displaystyle\text{Re}(V_{cs}^{*}V_{cd})=-\lambda(1-\frac{\lambda^{2}}{2})\,,$
$\displaystyle\text{Re}(V_{ts}^{*}V_{td})=-(1-\frac{\lambda^{2}}{2})A^{2}\lambda^{5}(1-\rho)\,,$
$\displaystyle\text{Im}(V_{ts}^{*}V_{td})=A^{2}\lambda^{5}\eta\,.$ (276)

The top contribution gives $X(x_{t})=1.481$ Buras15  (central value). For the
charm contribution, using the values in table 4, from the formula in ref.
Brod1  we obtain $P_{c}(X)\approx 0.351$. By using $\rho=0.126$, $\eta=0.392$
and the central values in table 4 we have
$\mathcal{F}_{K}\approx(-3.7+i\,1.2)\times 10^{-6}$, $|\mathcal{F}_{K}|\approx
3.9\times 10^{-6}$. The predicted SM contribution for the branching ratio can
be written as:

$\displaystyle\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}\approx
k_{K^{+}}|\mathcal{F}_{K}|^{2}\>,$ $\displaystyle
k_{K^{+}}=\frac{G_{F}^{2}\tau_{K^{+}}M^{5}_{K^{+}}f^{2}_{+}(0)\mathcal{I}_{\nu}^{+}}{64\pi^{3}}\approx
5.43$ (277)

where $f_{+}(0)=0.9699(15)$ is the form factor,
$\mathcal{I}_{\nu}^{+}=0.15269$ is the phase space integral Mescia:2007kn  and
we have used the values in table 4 for the kaon mass and mean life,. In this
way, we obtain our benchmark value
$\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}\approx 0.82\cdot
10^{-10}$, which is within the range of the estimate reported by PDG
Br$(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}=(0.85\pm 0.05)\cdot
10^{-10}$ PDG18 . The SM expectation is compatible with the experimental
branching ratio PDG18 :

$\displaystyle\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{exp}}=(1.7\pm
1.1)\cdot 10^{-10}$ (278)

With future experimental precision, any deviation from the SM prediction of
this golden mode branching ratio would indicate towards new physics.

Figure 6:  Tree level contribution to the rare kaon decays
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K^{0}\rightarrow\pi^{0}\nu\bar{\nu}$
arising from the mixing of the SM families with the extra downtype vectorlike
quark.

In our BSM scenario with extra vectorlike $b^{\prime}$ quark, the non-diagonal
couplings of $Z$-boson with light quarks in eqs. (190) induce at tree level
the operator with the same structure as the SM one (274), as shown in the
diagram in figure 6:

$\displaystyle\mathcal{L}_{\text{new}}=$
$\displaystyle-\frac{g^{2}}{M^{2}_{Z}\cos^{2}\theta_{W}}\left(\frac{1}{2}V_{L4s}^{*}V_{L4d}\right)(\overline{s_{L}}\gamma^{\mu}d_{L})\frac{1}{2}\sum_{e,\mu,\tau}(\overline{\nu_{\ell\text{L}}}\gamma_{\mu}\nu_{\ell\text{L}})=$
$\displaystyle=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}\frac{1}{2}V_{L4s}^{*}V_{L4d}(\overline{s_{L}}\gamma^{\mu}d_{L})\sum_{e,\mu,\tau}(\overline{\nu_{\ell\text{L}}}\gamma_{\mu}\nu_{\ell\text{L}})$
(279)

Thus, the new operator (4.1.1) contributes to the decay
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, in interference with the SM (274). The
total branching ratio becomes:

$\displaystyle\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{tot}}\approx\,k_{K^{+}}\left|\frac{1}{2}V_{L4s}^{*}V_{L4d}+\mathcal{F}_{K}\right|^{2}=$
$\displaystyle=\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}\left|\frac{\frac{1}{2}V_{L4s}^{*}V_{L4d}}{\mathcal{F}_{K}}+1\right|^{2}$
(280)

The experimental result (278) implies, at $95\%$ C.L., the upper limit:

$\displaystyle\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{tot}}<3.9\cdot
10^{-10}$ (281)

Then we have:

$\displaystyle\left|\frac{\frac{1}{2}V_{L4s}^{*}V_{L4d}}{\mathcal{F}_{K}}+1\right|<2.2$
(282)

figure 13 shows the constraint (282) in terms of the modulus and phase of the
elements $V_{L4d}$, $V_{L4s}$, using
$V_{L4s}^{*}V_{L4d}=|V_{L4s}^{*}V_{L4d}|e^{i\delta^{d}_{L21}}$,
$\delta^{d}_{L21}=\delta^{d}_{L2}-\delta^{d}_{L1}$, from eqs. (4) and (239).
Depending on the unknown relative phase, the limit results in the constraint
on the modulus:

$\displaystyle|V_{L4s}^{*}V_{L4d}|<(0.9\div 2.5)\cdot 10^{-5}$ (283)

As can be seen, in the most conservative case of destructive interference, the
new amplitude can be up to three times the SM amplitude:

$\displaystyle\frac{|V_{L4s}^{*}V_{L4d}|}{2|\mathcal{F}_{K}|}$
$\displaystyle\lesssim\Delta_{K^{+}}=3.2$ (284)

In this case we can express the experimental limit in terms of the ratio
$\Delta_{K^{+}}$:

$\displaystyle|V_{L4s}^{*}V_{L4d}|$ $\displaystyle\lesssim 2.5\cdot
10^{-5}\left[\frac{\Delta_{K^{+}}}{3.2}\right]\left[\frac{|\mathcal{F}_{K}|}{3.9\cdot
10^{-6}}\right]$ (285)

####  4.1.2 $K_{\text{L}}\rightarrow\pi^{0}\nu\bar{\nu}$

The second golden mode is the decay
$K_{\text{L}}\rightarrow\pi^{0}\nu\bar{\nu}$. In the Standard Model it is
described by the same Lagrangian as in (4.1.1). However this decay proceeds
almost entirely through direct CP violation, hence it is completely dominated
by short-distance loop diagrams with top quark exchanges and the charm
contribution can be neglected Buras:1998raa . In fact, with the phase
convention $C|K^{0}\rangle=|\bar{K}^{0}\rangle$ (or
$CP|K^{0}\rangle=-|\bar{K}^{0}\rangle$) we have:

$\displaystyle\langle\pi^{0}|(\bar{d}s)_{\text{V$-$A}}|\bar{K}^{0}\rangle=-\langle\pi^{0}|(\bar{s}d)_{\text{V$-$A}}|K^{0}\rangle$
(286) $\displaystyle
K_{\text{L}}=\frac{1}{\sqrt{2}}[(1+\epsilon)K^{0}+(1-\epsilon)\bar{K}^{0}]$
(287)

However, the terms of indirect ($\Delta S=2$) CP violation (proportional to
$\epsilon$) can be neglected in this decay and the dominant contribution comes
from the imaginary part of the operator (4.1.1):

$\displaystyle\langle\pi^{0}\nu\bar{\nu}|\mathcal{L}_{\text{SM}}|\frac{K^{0}+\bar{K}^{0}}{\sqrt{2}}\rangle$
$\displaystyle=-\frac{4G_{F}}{\sqrt{2}}\frac{i\,2\text{Im}\mathcal{F}_{K}}{\sqrt{2}}\langle\pi^{0}|(\overline{s_{L}}\gamma^{\mu}d_{L})|K^{0}\rangle\sum_{\ell=e,\mu,\tau}\langle\overline{\nu_{\ell\text{L}}}\gamma_{\mu}\nu_{\ell\text{L}}\rangle$
(288)

The SM predicted branching ratio can be written as:

$\displaystyle\text{Br}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})_{\text{SM}}\approx
k_{K_{L}}|\mathcal{F}_{K}|^{2}\>,$ $\displaystyle
k_{K_{L}}=\frac{G_{F}^{2}\tau_{K_{L}}M^{5}_{K^{0}}f^{2}_{+}(0)\mathcal{I}_{\nu}^{L}}{64\pi^{3}}$
(289)

where $\mathcal{I}_{\nu}^{+}=0.16043(31)$ is the phase space integral
Mescia:2007kn . Putting the values of kaon mass and lifetime (see table 4) we
obtain $\text{Br}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}\approx
3.5\times 10^{-11}$, which practically agrees with the estimate reported by
PDG $\text{Br}(K_{\text{L}}\rightarrow\pi^{0}\nu\bar{\nu})_{\text{SM}}=(3.0\pm
0.2)\times 10^{-11}$ PDG18 .

On the other hand, the experimental limit on this decay is PDG18 :

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\pi^{0}\nu\bar{\nu})_{\text{exp}}<3.0\times
10^{-9}\quad(90\%\,\text{C.L.})$ (290)

which is two orders of magnitude larger than the SM expectation. Therefore,
there is still much room for new physics.

In our BSM scenario with extra vectorlike $b^{\prime}$ quark, the new
contribution comes from the imaginary part of the Lagrangian in eq. (4.1.1)
Since the interference term with the SM contribution can be neglected, the
experimental limit can be directly applied to the new contribution:

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\pi^{0}\nu\bar{\nu})_{\text{new}}\approx
k_{K_{L}}\frac{1}{4}\big{(}\text{Im}(V_{L4s}^{*}V_{L4d})\big{)}^{2}<\text{Br}_{\text{exp}}$
(291)

So in the given parameterization (4), (239) we have:

$\displaystyle|\text{Im}(V_{L4s}^{*}V_{L4d})|=|V_{L4s}^{*}V_{L4d}||\sin(\delta^{d}_{L21})|<2.2\times
10^{-5}\left[\frac{\text{Br}_{\text{exp}}}{3.0\times
10^{-9}}\right]^{\frac{1}{2}}$ (292)

This condition is shown in figure 13. Then, the present experimental limit
allows the new contribution to be one order of magnitude larger than the SM
contribution:

$\displaystyle\frac{|\text{Im}(V_{L4s}^{*}V_{L4d})|}{2|\text{Im}\mathcal{F}_{K}|}<9.3\left[\frac{\text{Br}_{\text{exp}}}{3.0\cdot
10^{-9}}\right]^{\frac{1}{2}}$ (293)

So the discovery of the decay $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ with
branching ratio larger than the SM expectation can be a signal for BSM
physics.

####  4.1.3 $K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-}$

The decay $K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-}$ contains a direct CP
violating contribution, indirect CP violating contribution, interference
between them, and also a small CP conserving contribution PDG18 . The CP
conserving contribution to the amplitude is dominated by a two photon exchange
$K_{L}\rightarrow\pi^{0}\gamma\gamma\rightarrow\pi^{0}e^{+}e^{-}$, with both
an absorptive and a dispersive part. Using the the decay
$K_{L}\rightarrow\pi^{0}\gamma\gamma$, it is estimated that the CP-conserving
branching ratio is of order $\sim O(10^{-13})$ PDG18 . The indirectly CP
violating amplitude also derives from the coupling of leptons to photons, it
is given by the long-distance dominated $K_{S}\rightarrow\pi^{0}e^{+}e^{-}$
amplitude times the CP parameter $\epsilon_{K}$ Buchalla:1995vs . The complete
CP-violating contribution to the rate assuming a positive sign for the
interference term is estimated to be
$\text{Br}(K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-})_{\text{CPV}}\approx(3.1\pm
0.9)\times 10^{-11}$, where the three contributions from indirect,
interference and direct CP violation are $(1.76,\,0.9,\,0.45)\times 10^{-11}$
respectively PDG18 .

Only the direct CP-violating amplitude can be calculated in detail within the
SM, since it is short distance dominated. The relevant operator contains the
vector part of the hadronic current and both axial and vector component of the
leptonic current, and it is given by:

$\displaystyle\mathcal{L}_{\text{SM,SD}}$
$\displaystyle=-\frac{G_{F}\alpha(M_{Z})}{\sqrt{2}\,2\pi}V_{ts}^{*}V_{td}(\overline{s}\gamma_{\mu}d)\left(\tilde{y}_{V}(\overline{e}\gamma^{\mu}e)+\tilde{y}_{A}(\overline{e}\gamma^{\mu}\gamma^{5}e)\right)+\text{h.c.}$
(294)

where $\tilde{y}_{V,A}$ are a combination of Inami-Lim functions of box and
$Z$ and $\gamma$ penguin diagrams, as defined in ref. Buras:1997fb . The
direct CP-violating branching ratio in the SM can be written as:

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-})_{\text{SM,SD}}=k_{Ke}\left(\frac{G_{F}\alpha(M_{Z})}{\sqrt{2}\,2\pi}\right)^{2}\left(\frac{2\text{Im}V_{ts}^{*}V_{td}}{\sqrt{2}}\right)^{2}(\tilde{y}_{A}^{2}+\tilde{y}_{V}^{2})$
(295) $\displaystyle
k_{Ke}=\frac{1}{2}\frac{M^{5}_{K^{0}}\tau_{K_{L}}f^{2}_{+}(0)\mathcal{I}_{e}}{192\pi^{3}}$
(296)

where $\mathcal{I}_{e}=0.16043(31)$ Mescia:2007kn  is the phase space
integral.

Figure 7:  Tree level contribution to the rare kaon decays
$K^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-}$, $K^{0}\rightarrow\mu^{+}\mu^{-}$
($\ell=e,\mu$) arising from the mixing of the SM families with the extra
downtype vector-like quark.

The new effective Lagrangian contributing to this decay (as shown in figure 7)
is:

$\displaystyle\mathcal{L}_{\text{new}}=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}V_{L4s}^{*}V_{L4d}(\overline{s}\gamma^{\mu}d)\left[g_{V}\overline{e}\gamma_{\mu}e+g_{A}\overline{e}\gamma_{\mu}\gamma_{5}e\right]+\text{h.c.}$
(297)

where $g_{A}=\frac{1}{2}$,
$g_{V}=-\frac{1}{2}+2\sin^{2}\theta_{\text{W}}\approx-0.038$. The experimental
limit on this decay is PDG18 :

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-})_{\text{exp}}<2.8\times
10^{-10}\quad(90\%\,\text{C.L.})$ (298)

which is one order of magnitude bigger than the SM expectation. Then, we can
consider the SM contribution as negligible and let the new contribution to be
the dominant one, imposing a limit directly on the new contribution:

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\pi^{0}e^{+}e^{-})_{\text{new}}=k_{Ke}G_{F}^{2}\,[\text{Im}(V_{L4s}V_{L4d}^{*})]^{2}\left[g_{V}^{2}+g_{A}^{2}\right]<\text{Br}_{\text{exp}}$
(299)

In the given parameterization (4), (239) then:

$\displaystyle|\text{Im}(V_{L4s}V_{L4d}^{*})|=|V_{L4s}V_{L4d}^{*}||\sin(\delta^{d}_{L12})|<1.7\times
10^{-5}\left[\frac{\text{Br}_{\text{exp}}}{2.8\cdot
10^{-10}}\right]^{\frac{1}{2}}$ (300)

where the central values of all the quantities have been used. Also the decay
$K_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ gives a comparable constraint, being
the experimental limit
$\text{Br}(K_{\text{L}}\rightarrow\pi^{0}\mu^{+}\mu^{-})_{\text{exp}}<3.8\times
10^{-10}\quad(90\%\,\text{C.L.})$, to be compared with the SM expectation
$\text{Br}(K_{\text{L}}\rightarrow\pi^{0}\mu^{+}\mu^{-})=(1.5\pm 0.3)\times
10^{-11}$ (assuming positive interference between the direct- and indirect-CP
violating components) PDG18 .

####  4.1.4 $K_{\text{L}}\rightarrow\mu^{+}\mu^{-}$

The rare decay $K_{\text{L}}\rightarrow\mu^{+}\mu^{-}$ is a CP conserving
decay. Its short-distance part is given by $Z$-penguins and box diagrams.
However, this decay is dominated by a long-distance contribution from a two-
photon intermediate state. In fact, the full branching ratio can be written as
Buchalla:1995vs :

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})=|\text{Re}A|^{2}+|\text{Im}A|^{2}\,,$
$\displaystyle\text{Re}A=A_{\text{SD}}+A_{\text{LD}}$ (301)

with Re$A$ and Im$A$ denoting the dispersive and absorptive contributions,
respectively. The absorptive (imaginary) part of the long-distance component
is determined by the measured rate for $K_{L}\rightarrow\gamma\gamma$ to be
$\text{Br}_{\text{abs}}(K_{L}\rightarrow\mu^{+}\mu^{-})=(6.64\pm 0.07)\cdot
10^{-9}$ and it almost completely saturates the observed rate
Br$(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})_{\text{exp}}=(6.84\pm 0.11)\cdot
10^{-9}$ PDG18 . The real part of the long-distance amplitude cannot be
derived directly from experiment. However in ref. Isidori  it is estimated an
upper bound on the short distance contribution:

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})_{\text{SD}}<2.5\cdot
10^{-9}$ (302)

As shown below, the SM prediction for the short distance contribution results:
$\text{Br}(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})_{\text{SD,SM}}\approx
0.9\cdot 10^{-9}$ Buras:1997fb . Then the condition (302) provides a
constraint on new physics scenarios.

Since only the axial component of both hadronic and lepton current contributes
to the decay, the effective Lagrangian describing the short distance
contribution to the decay in the SM can be written as Buchalla:1995vs :

$\displaystyle\mathcal{L}_{\text{SM,SD}}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\frac{\alpha(M_{Z})}{2\pi\sin^{2}\theta_{\text{W}}}\big{(}V_{cs}^{*}V_{cd}Y(x_{c})+V_{ts}^{*}V_{td}Y(x_{t})\big{)}(\overline{s}\gamma^{\mu}\gamma_{5}d)(\overline{\mu}\gamma_{\mu}\gamma_{5}\mu)+\text{h.c.}=$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\mathcal{F}_{L2}(\overline{s}\gamma^{\mu}\gamma_{5}d)(\overline{\mu}\gamma_{\mu}\gamma_{5}\mu)+\text{h.c.}$
(303)

$Y(x_{a})$, $x_{a}=m^{2}_{a}/M^{2}_{W}$, are the Inami-Lim functions including
QCD and electroweak corrections, whose leading order term is a linear
combination of the axial components of $Z$-penguins and box-diagrams.
Numerically $Y(x_{c})\approx 3.3\times 10^{-4}$, $Y(x_{t})\approx 0.97$
Buchalla:1995vs . In eq. (303) we defined the constant
$\mathcal{F}_{L2}\approx(-2.1+i\,0.78)\times 10^{-6}$.

In the scenario with extra isosinglet quark, the non-diagonal couplings of
$Z$-boson with SM families due to the mixing of light quarks with the
$b^{\prime}$-quark leads to the Lagrangian analogous to (297), contributing to
the decay $K_{\text{L}}\rightarrow\mu^{+}\mu^{-}$ at tree level (figure 7):

$\displaystyle\mathcal{L}_{\text{new}}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{L4s}^{*}V_{L4d}(\overline{s}\gamma_{\mu}\gamma_{5}d)\frac{1}{2}(\overline{\mu}\gamma^{\mu}\gamma_{5}\mu)+\text{h.c.}$
(304)

By using again the phase convention $C|K^{0}\rangle=+|\bar{K}^{0}\rangle$ we
have:

$\displaystyle\langle
0|\bar{d}\gamma_{\mu}\gamma_{5}s|\bar{K}^{0}\rangle=+\langle
0|\bar{s}\gamma_{\mu}\gamma_{5}d|K^{0}\rangle$ (306)

and neglecting indirect CP violation:

$\displaystyle K_{L}\approx K_{2}=\frac{1}{\sqrt{2}}(K^{0}+\bar{K}^{0})$ (307)

Then we get:

$\displaystyle\langle\mu^{+}\mu^{-}|\mathcal{L}_{\text{SM,SD}}|K_{L}\rangle=\frac{G_{F}}{\sqrt{2}}\frac{2\text{Re}(\mathcal{F}_{L2})}{\sqrt{2}}\langle
0|\bar{s}\gamma_{\mu}\gamma_{5}d|K^{0}\rangle[\bar{u}(p^{(\mu)})\gamma^{\mu}\gamma_{5}v(p^{(\bar{\mu})})]$
(308)
$\displaystyle\langle\mu^{+}\mu^{-}|\mathcal{L}_{\text{new}}|K_{L}\rangle=\frac{G_{F}}{\sqrt{2}}\frac{\text{Re}(V_{L4s}^{*}V_{L4d})}{\sqrt{2}}\langle
0|\bar{s}\gamma_{\mu}\gamma_{5}d|K^{0}\rangle[\bar{u}(p^{(\mu)})\gamma^{\mu}\gamma_{5}v(p^{(\bar{\mu})})]$
(309)

Then we can define the branching ratio given by the amplitude of the short
distance contribution:

$\displaystyle\text{Br}(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})_{\text{SD}}\\!=k\left[\text{Re}(2\mathcal{F}_{L2}+V_{L4s}^{*}V_{L4d})\right]^{2},$
$\displaystyle
k\\!=\\!\frac{\text{Br}(K^{+}\rightarrow\mu^{+}\nu_{\mu})\tau_{K_{L}}M_{K^{0}}\sqrt{1\\!-\\!4\frac{m^{2}_{\mu}}{M^{2}_{K^{0}}}}}{\tau(K^{+})M_{K^{+}}\left(1\\!-\\!\frac{m^{2}_{\mu}}{M^{2}_{K^{+}}}\right)^{2}|V_{us}|^{2}}$
(310)

where we have used $\langle
0|\bar{s}\gamma_{\mu}\gamma_{5}d|K^{0}\rangle=ip_{\mu}f_{K}$,
$\text{Br}(K^{+}\rightarrow\mu^{+}\nu_{\mu})=0.6356(11)$ and
$|V_{us}|=0.2252$. In absence of new physics, the short distance contribution
coincides with the SM expectation
$\text{Br}(K_{\text{L}}\rightarrow\mu^{+}\mu^{-})_{\text{SD,SM}}$.

By using the upper bound in eq. (302) on the branching ratio (310) we have:

$\displaystyle\left|\text{Re}(2\mathcal{F}_{L2}+V_{L4s}^{*}V_{L4d})\right|$
$\displaystyle<6.9\times 10^{-6}$ (311)

This constraint is shown in figure 13 in terms of the modulus
$|V_{L4s}^{*}V_{L4d}|$ and relative phase $\delta_{L21}^{d}$, as parameterized
in (4), (239). In terms of the ratio between the new amplitude (304) and the
SM amplitude (303), eq. (311) implies:

$\displaystyle\left|1+\frac{\text{Re}(V_{L4s}^{*}V_{L4d})}{2\text{Re}(\mathcal{F}_{L2})}\right|<1.7\,,\qquad
0.7<\frac{|\text{Re}(V_{L4s}^{*}V_{L4d})|}{2|\text{Re}\mathcal{F}_{L2}|}<2.7$
(312)

or, by using
$\text{Re}(V_{L4s}^{*}V_{L4d})=|V_{L4s}^{*}V_{L4d}|\cos\delta^{d}_{L21}$ we
have:

$\displaystyle-0.28\times
10^{-5}<|V_{L4s}^{*}V_{L4d}|\cos\delta^{d}_{L21}<1.11\times 10^{-5}\,.$ (313)

where we used the Wolfenstein parameterization (276).

####  4.1.5 $K_{\text{S}}\rightarrow\mu^{+}\mu^{-}$

The effective Lagrangian (303) gives the short distance contribution to the
decay $K_{\text{S}}\rightarrow\mu^{+}\mu^{-}$:

$\displaystyle\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{SD,SM}}=4k_{K_{S}}\left[\text{Im}(\mathcal{F}_{L2})\right]^{2}$
(314)

where $k_{K_{S}}$ is the same as $k$ in eq. (310) with the change
$\tau_{K_{L}}\rightarrow\tau_{K_{S}}$. It is obtained
$\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{SD}}\approx 0.2\cdot
10^{-12}$. There are also long distance contributions arising from the two
photon intermediate state which result in the rate
$\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{LD}}\approx 5.1\cdot
10^{-12}$ Cirigliano . The experimental upper limit for this decay was found
by the LHCb collaboration LHCbmumupub :

$\displaystyle\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{exp}}<1.0\cdot
10^{-9}\quad(95\%\,\text{C.L.})$ (315)

Then, an extra contribution is allowed to be higher than both the short and
long-distance contributions to the decay, so we can impose an upper limit on
the new contribution arising in the scenario with the extra isosinglet. The
new decay channel is described by the effective Lagrangian (304) and gives the
rate:

$\displaystyle\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{new}}=k_{K_{S}}\left[\text{Im}(V_{L4s}^{*}V_{L4d})\right]^{2}<\text{Br}_{\text{exp}}$
(316)

We have:

$\displaystyle\text{Br}(K_{\text{S}}\rightarrow\mu^{+}\mu^{-})_{\text{new}}<1.0\cdot
10^{-9}$ (317)
$\displaystyle|\text{Im}(V_{L4s}^{*}V_{L4d})|=|V_{L4s}^{*}V_{L4d}||\sin\delta^{d}_{L21}|<1.0\times
10^{-4}\left[\frac{\text{Br}_{\text{exp}}}{1.0\cdot 10^{-9}}\right]^{1/2}$
(318)

where we used the parameterization in eqs. (4), (239).

###  4.2 Limits from neutral mesons systems

####  4.2.1 $K^{0}$-$\bar{K}^{0}$ mixing

Figure 8:  Box diagram for the transition $K^{0}\rightarrow\bar{K}^{0}$ in the
SM, $q_{\alpha},q_{\beta}=u,c,t$.

In the SM the short-distance contribution to the transition
$K^{0}(d\bar{s})\leftrightarrow\bar{K}^{0}(\bar{d}s)$ arises from weak box
diagrams (figure 8). The effective Lagrangian describing this contribution is
given by:

$\displaystyle\mathcal{L}^{\text{SM}}_{\Delta S=2}$
$\displaystyle=-\frac{G^{2}_{F}m^{2}_{W}}{4\pi^{2}}\left[\lambda^{2}_{c}S_{0}(x_{c})+\lambda^{2}_{t}S_{0}(x_{t})+2\lambda_{c}\lambda_{t}S_{0}(x_{c},x_{t})\right](\overline{s_{L}}\gamma^{\mu}d_{L})^{2}+\text{h.c.}$
(319)

where $\lambda_{a}=V_{as}^{*}V_{ad}$, $x_{a}=\frac{m_{a}^{2}}{m_{W}^{2}}$ and
$S_{0}(x_{i})$ are the Inami-Lim functions Inami:1980fz :

$\displaystyle S_{0}(x)=x$
$\displaystyle\left(\frac{4-11x+x^{2}}{4(1-x)^{2}}-\frac{3x^{2}\ln
x}{2(1-x)^{3}}\right)$ (320) $\displaystyle S_{0}(x_{j},x_{k})=x_{j}x_{k}$
$\displaystyle\left[\left(\frac{1}{4}-\frac{3}{2(x_{j}-1)}-\frac{3}{4(x_{j}-1)^{2}}\right)\frac{\log
x_{j}}{x_{j}-x_{k}}+\right.$
$\displaystyle\left.\left(\frac{1}{4}-\frac{3}{2(x_{k}-1)}-\frac{3}{4(x_{k}-1)^{2}}\right)\frac{\log
x_{k}}{x_{k}-x_{j}}\right.$
$\displaystyle\left.-\frac{3}{4(x_{j}-1)(x_{k}-1)}\right]$ (321)

which, being $x_{c}=m_{c}^{2}/m_{W}^{2}\ll 1$, for the c-quark contribution
reduce to:

$\displaystyle S_{0}(x_{c})\approx x_{c}$ (322) $\displaystyle
S_{0}(x_{c},x_{t})\approx
x_{c}\left[\ln\frac{x_{t}}{x_{c}}-\frac{3x_{t}}{4(1-x_{t})}-\frac{3x_{t}^{2}\ln
x_{t}}{4(1-x_{t})^{2}}\right]$ (323)

keeping only linear terms in $x_{c}$ Buchalla:1995vs .

The weak short-distance contribution to the mass splitting $\Delta
m_{K}=m_{K_{L}}-m_{K_{S}}$ and the CP-violating parameter $\epsilon_{K}$ are
described by the off-diagonal term $M_{12}$ of the mass matrix of neutral
kaons, which is given by:

$\displaystyle M_{12}=-\frac{1}{2m_{K^{0}}}\langle K^{0}|\mathcal{L}_{\Delta
S=2}|\bar{K}^{0}\rangle$ (324)

The hadronic matrix element in vacuum insertion approximation (VIA) is written
as:

$\displaystyle\langle
K^{0}|(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}|\bar{K}^{0}\rangle=\frac{2}{3}f^{2}_{K}m^{2}_{K^{0}}$
(325)

with the phase choice $CP|K^{0}\rangle=-|\bar{K}^{0}\rangle$ Branco . $f_{K}$
is the kaon decay constant, which can be estimated in lattice QCD to be
$f_{K}=155.7(0.7)$ MeV FLAG2019 , and $m_{K^{0}}$ is the neutral kaons mass.
Then, the SM Lagrangian gives:

$\displaystyle M_{12}^{\text{SM}}$
$\displaystyle=\frac{G^{2}_{F}m^{2}_{W}}{12\pi^{2}}(\eta_{1}\lambda^{*2}_{c}S_{0}(x_{c})+\eta_{2}\lambda^{*2}_{t}S_{0}(x_{t})+2\eta_{3}\lambda^{*}_{c}\lambda^{*}_{t}S_{0}(x_{c},x_{t}))f^{2}_{K}m_{K^{0}}B_{K}$
(326)

where we also added the factors $\eta_{1}=1.38\pm 0.20$, $\eta_{2}=0.57\pm
0.01$, $\eta_{3}=0.47\pm 0.04$ which describe short-distance QCD effects
Buchalla:1995vs . The factor $B_{K}$ is the correction to the VIA which is
estimated from lattice QCD calculations, giving $B_{K}=0.7625(97)$ FLAG2019 .

The modulus $|M_{12}|$ and the imaginary part $\text{Im}M_{12}$ respectively
describe short-distance contributions in the mass splitting and CP-violation
in $\bar{K}^{0}\rightarrow K^{0}$ transition. In the SM, in the standard
parameterization of $V_{\text{CKM}}$ we have:

$\displaystyle|M_{12}^{\text{SM}}|\,\approx\,|\text{Re}M_{12}^{\text{SM}}|\,\approx\frac{G^{2}_{F}m^{2}_{W}}{12\pi^{2}}\eta_{1}$
$\displaystyle|\lambda^{2}_{c}|S_{0}(x_{c})\approx\frac{G^{2}_{F}}{12\pi^{2}}\eta_{1}|\lambda^{2}_{c}|m^{2}_{c}f^{2}_{K}m_{K^{0}}B_{K}$
(327)
$\displaystyle\text{Im}M_{12}^{\text{SM}}=-\frac{G_{F}^{2}m_{W}^{2}}{6\pi^{2}}f^{2}_{K}m_{K^{0}}B_{K}$
$\displaystyle\left[\eta_{1}\text{Re}(\lambda_{c})\text{Im}(\lambda_{c})S_{0}(x_{c})+\right.$
$\displaystyle\left.+\big{(}\eta_{2}\text{Re}(\lambda_{t})S_{0}(x_{t})+\eta_{3}\text{Re}(\lambda_{c})S_{0}(x_{c},x_{t})\big{)}\text{Im}(\lambda_{t})\right]$
(328)

The mass difference between mass eigenstates is given by Buchalla:1995vs :

$\displaystyle\Delta m_{K}\approx 2|M_{12}|+\Delta m_{\text{LD}}$ (329)

where $\Delta m_{\text{LD}}$ is the long-distance contribution, which is
difficult to evaluate. Nevertheless, the short distance contribution $\Delta
m_{K}\approx 2|M_{12}^{\text{SM}}|\simeq 2.2\cdot 10^{-15}\,\text{GeV}$ gives
a dominant contribution to the experimentally measured value PDG18 :

$\displaystyle\Delta M_{K}=(3.484\pm 0.006)\times 10^{-15}\,\text{GeV}$ (330)

The CP violation is parameterized by $\epsilon_{K}$, which, with the phase
choice $CP|K^{0}\rangle=-|\bar{K}^{0}\rangle$, is almost determined by short
distance physics Branco :

$\displaystyle|\epsilon_{K}|\approx\frac{|\text{Im}[M_{12}]|}{\sqrt{2}\Delta
m_{K}}$ (331)

in the standard parameterization of $V_{\text{CKM}}$. From experimental data
it is obtained PDG18 :

$\displaystyle|\epsilon_{K}|=(2.228\pm 0.011)\times 10^{-3}$ (332)

Figure 9:  Contributions to $\bar{K}^{0}$-$K^{0}$ transition due to the
insertion of the additional vectorlike down-type weak isosinglet couple
$b^{\prime}_{L},b^{\prime}_{R}$ in quark mixing.

In our BSM scenario with extra vectorlike $b^{\prime}$ quark, the non-diagonal
couplings of $Z$ and Higgs bosons with light quarks in eqs. (190), (255)
induce the same operator as in (319), both at tree level and loop level,
interfering with the SM in the transition $\bar{K}^{0}\leftrightarrow K^{0}$,
as shown in figure 9.

The new tree level contribution translates into the effective Lagrangian:

$\displaystyle\mathcal{L}^{\text{tree}}_{\Delta S=2}=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}(V_{L4d}^{*}V_{L4s})^{2}(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}$
(333)

However if $M_{b^{\prime}}$ is of order few TeV, the loop contribution to
$\bar{K}^{0}$-$K^{0}$ mixing becomes important. In particular, for
$M_{b^{\prime}}=3.1$ TeV the effective operator for the box diagram with Higgs
and $Z$ bosons exchange in figure 9 gives the same contribution as the
effective operator describing the tree level diagram. Moreover the box diagram
contribution grows as $\propto M^{2}_{b^{\prime}}$. In fact, the effective
Lagrangian describing the loop contribution in figure 9 is:

$\displaystyle\mathcal{L}^{\text{box}}_{\Delta S=2}\approx$
$\displaystyle-\frac{(h_{d1}h_{d2}^{*})^{2}}{128\pi^{2}M^{2}_{b^{\prime}}}(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}$
(334)

where $h_{di}$ are the Yukawa couplings defined in eq. (139). However, we are
interested in fixing a value for the elements of the mixing matrix rather than
setting the Yukawas $h_{di}$. Then, it is convenient to express the Lagrangian
explicitly in terms of the mixing elements. From eqs. (239), (241), we have:

$\displaystyle h_{d1}\approx-V_{L4d}^{*}\frac{M_{b^{\prime}}}{v_{w}}\,,\qquad
h_{d2}\approx-V_{L4s}^{*}\frac{M_{b^{\prime}}}{v_{w}}\,.$ (335)

Then, using $G_{F}/\sqrt{2}=1/4v_{w}^{2}$, the Lagrangian can be written as:

$\displaystyle\mathcal{L}^{\text{box}}_{\Delta
S=2}\approx-\frac{G^{2}_{F}\,M^{2}_{b^{\prime}}}{16\pi^{2}}(V_{L4d}^{*}V_{L4s})^{2}(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}$
(336)

Hence, the complete new contribution is:

$\displaystyle\mathcal{L}^{\text{new}}_{\Delta S=2}$
$\displaystyle=\mathcal{L}^{\text{tree}}_{\Delta
S=2}+\mathcal{L}^{\text{box}}_{\Delta S=2}\approx$
$\displaystyle\approx-\frac{G_{F}}{\sqrt{2}}(V_{L4d}^{*}V_{L4s})^{2}\left(1+\frac{G_{F}M^{2}_{b^{\prime}}}{8\sqrt{2}\pi^{2}}\right)(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}=$
$\displaystyle=-\frac{G_{F}}{\sqrt{2}}(V_{L4d}^{*}V_{L4s})^{2}f(M_{b^{\prime}})(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}$
(337) $\displaystyle f(M_{b^{\prime}})$
$\displaystyle=1+\left(\frac{M_{b^{\prime}}}{3.1\,\text{TeV}}\right)^{2}$
(338)

where we defined the function $f(M_{b^{\prime}})$ of the mass of the extra
quark $b^{\prime}$. From the Lagrangian (4.2.1), there is an additive
contribution $M_{12}^{\text{new}}$ to the mixing mass $M_{12}$ (see eq.
(324)), participating both in the mass splitting and CP-violating effects.
Then, the new contribution is constrained by both CP-conserving and CP-
violating observables.

Regarding the CP-conserving part, which originates the mass splitting $\Delta
m_{K}$, we can confront the moduli of the two components of the mixing mass
and impose that the new contribution $|M_{12}^{\text{new}}|$ is less than a
fraction $\Delta_{K}$ of the short-distance SM contribution
$|M_{12}^{\text{SM}}|$:

$\displaystyle|M_{12}^{\text{new}}|<|M_{12}^{\text{SM}}|\,\Delta_{K}$ (339)

where we defined real and positive $\Delta_{K}$. This is analogous to
comparing the modulus of the coefficients of the effective operators in eqs.
(319) and (4.2.1). We evaluate the constraint at leading order of both SM and
new physics contributions, neglecting QCD corrections. Hence, eq. (339)
corresponds to:

$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})\left|V_{L4d}^{*}V_{L4s}\right|^{2}<\,\frac{G^{2}_{F}m^{2}_{c}}{4\pi^{2}}|\lambda^{2}_{c}|\,\Delta_{K}$
(340)

eq. (339) can also be translated into a constraint on the scale of the new
contribution:

$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})\left|(V_{L4d}^{*}V_{L4s})^{2}\right|<\frac{1}{\>\Lambda_{sd}^{2}}\,,\qquad\Delta_{K}=\left(\frac{1.9\cdot
10^{3}\,\text{TeV}}{\Lambda_{ds}}\right)^{2}$ (341)

where we defined the minimum allowed equivalent scale $\Lambda_{sd}^{2}$ of
the new contribution. We estimate the constraint with $\Delta_{K}=1$, that is
by imposing the condition $|M_{12}^{\text{new}}|<|M_{12}^{\text{SM}}|$,
meaning that the additional processes contribute to $\Delta m_{K}$ at most as
the SM short-distance contribution. Anyway, we leave $\Delta_{K}$ as free
parameter in next equations so to allow to reevaluate the results with a
different constraint. Then, using eqs. (4), (239), from the condition (340) we
have:

$\displaystyle|V_{L4d}^{*}V_{L4s}|<1.7\times
10^{-4}\left[\frac{f(1\,\text{TeV})}{f(M_{b^{\prime}})}\right]^{1/2}\left[\Delta_{K}\right]^{1/2}$
(342)

where we evaluated the constraint for the benchmark value $M_{b^{\prime}}=1$
TeV.

Regarding the contribution to the CP-violating parameter $\epsilon_{K}$, the
imaginary part of the new contribution to the mixing mass
$\text{Im}M_{12}^{\text{new}}$ can be constrained to be a fraction
$\Delta_{\epsilon_{K}}$ of the SM contribution:

$\displaystyle|\text{Im}M_{12}^{\text{new}}|<|\text{Im}M_{12}^{\text{SM}}|\,\Delta_{\epsilon_{K}}\,,$
(343)

where we defined real and positive $\Delta_{\epsilon_{K}}$. At leading order
of the new physics contribution, eq. (343) is equivalent to comparing the
magnitude of the imaginary part of operators (319) and (4.2.1):

$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})|\text{Im}[(V_{L4d}^{*}V_{L4s})^{2}]|<$
$\displaystyle\,\frac{G_{F}^{2}m^{2}_{W}}{4\pi^{2}}2\left|\eta_{1}\text{Re}(\lambda_{c})\text{Im}(\lambda_{c})S_{0}(x_{c})+\right.$
$\displaystyle\left.+\big{(}\eta_{2}\text{Re}(\lambda_{t})S_{0}(x_{t})+\eta_{3}\text{Re}(\lambda_{c})S_{0}(x_{c},x_{t})\big{)}\text{Im}(\lambda_{t})\right|\,\Delta_{\epsilon_{K}}$
(344)

eq. (344) can also be translated into a constraint on the scale of the new
contribution:

$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})\left|\text{Im}[(V_{L4d}^{*}V_{L4s})^{2}]\right|<\frac{1}{\>\Lambda_{sd,\text{Im}}^{2}}\,,\qquad\Delta_{\epsilon_{K}}=\left(\frac{1.6\cdot
10^{4}\,\text{TeV}}{\Lambda_{ds,\text{Im}}}\right)^{2}$ (345)

where we defined the minimum allowed equivalent scale
$\Lambda_{sd,\text{Im}}^{2}$ of the new contribution. The numerical value is
obtained by using the Wolfenstein parameterization as in eq. (276), with
$\text{Im}(V_{cs}^{*}V_{cd})=-A^{2}\lambda^{5}\eta$. We make an estimation
choosing $\Delta_{\epsilon_{K}}=0.4$, corresponding to
$\Lambda_{ds,\text{Im}}=2.5\cdot 10^{4}\,\text{TeV}$. Using eqs. (4) and
(239), from the condition (344) we have:

$\displaystyle|V_{L4d}^{*}V_{L4s}|\sqrt{|\sin[2(\delta^{d}_{L12})]|}<1.3\cdot
10^{-5}\left[\frac{f(1\,\text{TeV})}{f(M_{b^{\prime}})}\right]^{1/2}\left[\frac{\Delta_{\epsilon_{K}}}{0.4}\right]^{1/2}$
(346)

where we evaluated the constraint for the benchmark value $M_{b^{\prime}}=1$
TeV. The constraints are also shown in figure 13 for $M_{b^{\prime}}=1$ TeV,
in terms of the modulus and relative phase of the elements
$V_{L4d}^{*}V_{L4s}$.

Let us elaborate a little more about the two contributions to
$\mathcal{L}^{\text{new}}_{\Delta S=2}$ from the tree and box diagrams. The
tree level operator (4.2.1) can be written in terms of the Yukawa constants
$h_{di}$, in order to be compared with the operator (334). Using
$G_{F}/\sqrt{2}=1/4v_{w}^{2}$ and eq. (335), we have:

$\displaystyle\mathcal{L}^{\text{tree}}_{\Delta
S=2}=-\frac{(h_{d1}h_{d2}^{*})^{2}}{4}\frac{v_{w}^{2}}{M_{b^{\prime}}^{4}}(\overline{d_{L}}\gamma^{\mu}s_{L})^{2}+\text{h.c.}$
(347)

which parametrically is a factor $v_{w}^{2}/M_{b^{\prime}}^{2}$ less than the
box operator (334). That is why, after compensating the loop factor, for
$M_{b^{\prime}}>3.1$ TeV the box contribution becomes comparable or more
important than the tree level contribution.

####  4.2.2 $B^{0}_{d,s}$-$\bar{B}_{d,s}^{0}$ mixing

Figure 10:  Box diagram for the transition $B^{0}\rightarrow\bar{B}^{0}$ in
the SM.

In the SM the dominant contribution to
$B^{0}_{d}(\bar{b}d)$-$\bar{B}_{d}^{0}(\bar{d}b)$ and
$B^{0}_{s}(\bar{b}s)$-$\bar{B}_{s}^{0}(\bar{s}b)$ mixings comes from box-
diagrams with internal top-quark, as shown in figure 10, while the charm quark
and the mixed top-charm contributions are entirely negligible. The effective
Lagrangians for $B^{0}_{d,s}$-$\bar{B}_{d,s}^{0}$ mixings are Buchalla:1995vs
:

$\displaystyle\mathcal{L}^{\text{SM}}_{\Delta
B(d)=2}=-\frac{G_{F}^{2}}{4\pi^{2}}m^{2}_{W}S_{0}(x_{t})(V_{tb}^{*}V_{td})^{2}(\overline{b_{L}}\gamma_{\mu}d_{L})^{2}+\text{h.c.}$
(348) $\displaystyle\mathcal{L}^{\text{SM}}_{\Delta
B(s)=2}=-\frac{G_{F}^{2}}{4\pi^{2}}m^{2}_{W}S_{0}(x_{t})(V_{tb}^{*}V_{ts})^{2}(\overline{b_{L}}\gamma_{\mu}s_{L})^{2}+\text{h.c.}$
(349)

where $S_{0}(x)$ is the function in eq. (320). Then, analogously to the kaons
system, in VIA:

$\displaystyle\langle\bar{B}_{d,s}^{0}|(\overline{b_{L}}\gamma^{\mu}q_{L})^{2}|B^{0}_{d,s}\rangle=\frac{2}{3}f^{2}_{B_{d,s}}M^{2}_{B^{0}_{d,s}}$
(350)

and

$\displaystyle
2M_{B^{0}_{d,s}}M_{12}^{(d,s)*}=\langle\bar{B}_{d,s}^{0}|-\mathcal{L}_{\Delta
B(d,s)=2}|B^{0}_{d,s}\rangle$ (351) $\displaystyle\Delta
M_{d,s}=2|M_{12}^{(d,s)}|$ (352)

Putting together (350), (351), (352), in the SM:

$\displaystyle\Delta
M_{d,s}^{\text{SM}}=\frac{G_{F}^{2}m_{W}^{2}}{6\pi^{2}}f^{2}_{B_{d,s}}m_{B_{d,s}}B_{B_{d,s}}|(V_{tb}V_{td/s}^{*})^{2}|\eta_{B}S_{0}(x_{t})$
(353)

where $\eta_{B}$ is the QCD factor, $\eta_{B}=0.551$ Buchalla:1995vs , and the
factors $B_{B_{d,s}}$ are correcting the VIA. From lattice QCD calculations:
$f_{B_{d}}\sqrt{B_{B_{d}}}=225(9)$ MeV, $f_{B_{s}}\sqrt{B_{B_{s}}}=274(8)$ MeV
FLAG2019 . In contrast to $\Delta m_{K}$, long distance contributions are
estimated to be very small in neutral B meson systems. and $\Delta M_{d,s}$ is
very well approximated by the relevant box diagrams Buchalla:1995vs . The
experimental results are PDG18 :

$\displaystyle\Delta M_{d\,\text{exp}}=(3.334\pm 0.013)\cdot
10^{-13}\,\text{GeV}$ (354) $\displaystyle\Delta M_{s\,\text{exp}}=(1.1683\pm
0.0013)\cdot 10^{-11}\,\text{GeV}$ (355)

Figure 11:  Contributions to $B^{0}$-$\bar{B}^{0}$ transition due to the
insertion of the additional vectorlike down-type weak isosinglet couple
$b^{\prime}_{L},b^{\prime}_{R}$ in quark mixing.

In our BSM scenario with extra vectorlike $b^{\prime}$ quark, the non-diagonal
couplings of $Z$-boson and Higgs boson with light quarks in eqs. (190), (255)
induce the operator with the same structure as in (348), interfering with the
SM in the transition $\bar{B}^{0}_{d,s}\leftrightarrow B^{0}_{d,s}$. The new
contributions at tree level and loop level are analogous to the operators
giving $K^{0}$ mixing (333), (336), as shown in diagrams in figure 11. Then,
analogously to the neutral kaons system, the new contribution to neutral
B-mesons mixing is:

$\displaystyle\mathcal{L}^{\text{new}}_{\Delta
B(d)=2}\approx-\frac{G_{F}}{\sqrt{2}}(V_{L4d}^{*}V_{L4b})^{2}f(M_{b^{\prime}})(\overline{d_{L}}\gamma^{\mu}b_{L})^{2}+\text{h.c.}$
(356) $\displaystyle\mathcal{L}^{\text{new}}_{\Delta
B(s)=2}\approx-\frac{G_{F}}{\sqrt{2}}(V_{L4s}^{*}V_{L4b})^{2}f(M_{b^{\prime}})(\overline{s_{L}}\gamma^{\mu}b_{L})^{2}+\text{h.c.}$
(357) $\displaystyle
f(M_{b^{\prime}})=1+\left(\frac{M_{b^{\prime}}}{3.1\,\text{TeV}}\right)^{2}$
(358)

From the Lagrangians (356), (357), there is an additive contribution
$M_{12}^{\text{new}}$ to the mixing mass $M_{12}^{(d,s)}$ (351), and thus to
the mass difference $\Delta M_{d,s}$ (352). We can constrain the new
contribution to be less than a fraction $\Delta_{B_{(d,s)}}$ of the SM
contribution $\Delta M_{d,s}^{\text{SM}}=2|M_{12}^{(d,s)\text{SM}}|$ given in
eq. (353). This is analogous to comparing the coefficients of the effective
operators in eqs. (356), (357) with the SM ones (348), (349):

$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})|V_{L4d}^{*}V_{L4b}|^{2}<\frac{G_{F}^{2}}{4\pi^{2}}m^{2}_{W}|V_{td}|^{2}S(x_{t})\cdot\Delta_{B_{d}}=\frac{1}{\Lambda_{bd}^{2}}$
(359)
$\displaystyle\frac{G_{F}}{\sqrt{2}}f(M_{b^{\prime}})|V_{L4s}^{*}V_{L4b}|^{2}<\frac{G_{F}^{2}}{4\pi^{2}}m^{2}_{W}|V_{ts}|^{2}S(x_{t})\cdot\Delta_{B_{s}}=\frac{1}{\Lambda_{bs}^{2}}$
(360)

where we evaluate the constraint at leading order of both SM and new physics
contributions, neglecting QCD corrections. We use $\Delta_{B_{(d,s)}}=0.3$ as
a benchmark value, corresponding to the scales $\Lambda_{bd}=1.0\cdot
10^{3}\,\text{TeV}$, $\Lambda_{bs}=2.1\cdot 10^{2}\,\text{TeV}$. Then we
obtain (with $m_{t}(m_{t})=163$ GeV):

$\displaystyle|V_{L4d}^{*}V_{L4b}|<3.3\times
10^{-4}\left[\frac{f(1\,\text{TeV})}{f(M_{b^{\prime}})}\right]^{1/2}\left[\frac{\Delta_{B_{d}}}{0.3}\right]^{1/2}$
(361) $\displaystyle|V_{L4s}^{*}V_{L4b}|<1.6\times
10^{-3}\left[\frac{f(1\,\text{TeV})}{f(M_{b^{\prime}})}\right]^{1/2}\left[\frac{\Delta_{B_{s}}}{0.3}\right]^{1/2}$
(362)

####  4.2.3 $D^{0}$-$\bar{D}^{0}$ mixing

In the SM $D^{0}(c\bar{u})$-$\bar{D}^{0}(\bar{c}u)$ mixing receives
contributions from box diagrams, dipenguin diagrams and from long-distance
effects Branco . In box diagrams only the internal strange and down quarks
contribute effectively. Differently from the case of kaons and B-mesons, the
masses of $s$ and $d$ are small compared to the mass of the external $c$
quarks, which then cannot be neglected and an extra operator appears in the
effective Lagrangian. In VIA approximation, the box contribution results
$|M_{12}|\sim 10^{-17}\div 10^{-16}$ GeV Branco . Also dipenguin diagrams can
give a contribution not much smaller than box diagrams Petrov . Long-distance
effects are expected to be large Branco . However, since their contribution is
non-perturbative and cannot be computed reliably, there is room for new
physics, which in principle can be the dominant contribution to the mass
difference $\Delta m_{D}$ in $D^{0}$ system. In fact, the experimental value
of the mass difference in the $D^{0}$ mesons system is PDG18 :

$\displaystyle\Delta m_{D\,\text{exp}}=(6.25^{+2.70}_{-2.90})\times
10^{-15}\text{GeV}$ (363)

which at $95\%$ C.L. allows values two orders of magnitude bigger than the SM
short-distance expectation.

Figure 12: New contribution to $D^{0}-\bar{D^{0}}$ mixing,
$q=d,s,b,b^{\prime}$.

In the scenario with extra down-type quark, the additional box diagrams with
internal $b^{\prime}$ quarks contribute to $D$ mesons mixing, as shown in
figure (12). The corresponding effective Lagrangian is:

$\displaystyle\mathcal{L}^{\text{new}}_{\Delta C=2}\approx$
$\displaystyle-\frac{G^{2}_{F}m^{2}_{W}}{4\pi^{2}}(V_{ub^{\prime}}V_{cb^{\prime}}^{*})^{2}S_{0}(x_{b^{\prime}})(\overline{u_{L}}\gamma^{\mu}c_{L})^{2}+\text{h.c.}$
(364)

with $x_{b^{\prime}}=\frac{M_{b^{\prime}}^{2}}{m_{W}^{2}}$ (taking
$M_{b^{\prime}}\simeq 1$ TeV, the two mixed contributions would be competitive
only if $|V_{ub^{\prime}}V_{cb^{\prime}}^{*}|<2\cdot 10^{-7}$, but we are
considering scenarios with larger mixings). Since the new effective operator
(364) originates from charged currents, this process involves directly the
elements $V_{ub^{\prime}}$, $V_{cb^{\prime}}$ of the enlarged CKM matrix
$\tilde{V}_{\text{CKM}}$ (183). This operator gives an additional contribution
$M^{D}_{12\,\text{new}}$ to the mixing mass and thus contributes to the mass
difference $\Delta m_{D}$:

$\displaystyle\Delta
m_{D\text{new}}=2|M^{D}_{12\,\text{new}}|\simeq\frac{G^{2}_{F}m^{2}_{W}}{6\pi^{2}}(V_{ub^{\prime}}V_{cb^{\prime}}^{*})^{2}S_{0}(x_{b^{\prime}})f^{2}_{D}m_{D^{0}}$
(365)

where $f_{D}=212.0\pm 0.7$ MeV FLAG2019  is the decay constant. Since the new
contribution can be regarded as the dominant one, we can think that the new
operator can account up to the entire mass difference in D-mesons system. Then
we can set a constraint directly on the new contribution. We can constrain
$\Delta m_{D\,\text{new}}$ by using use the $95\%$ C.L. experimental limit
obtained from eq. (363):

$\displaystyle\Delta m_{D\text{new}}<\Delta m_{D\,\text{exp}}=1.2\times
10^{-14}\,\text{GeV}$ (366)
$\displaystyle|V_{ub^{\prime}}V_{cb^{\prime}}^{*}|<4.6\times
10^{-4}\left[\frac{\Delta m_{D\,\text{exp}}}{1.2\cdot
10^{-14}}\right]^{1/2}\left[\frac{1\,\text{TeV}}{M_{b^{\prime}}}\right]$ (367)

The last scaling in eq. (367) holds since
$S_{0}(x_{b^{\prime}})\sim\frac{1}{4}x_{b^{\prime}}$ for
$M_{b^{\prime}}\gtrsim 2$ TeV.

###  4.3 Limits from rare $B$ mesons decays

####  4.3.1 Rare semileptonic B decays

Results on rare B-decays can constrain mixings of the extra vector-like quark
with SM families because of the new contributions to FCNC processes involving
$b\rightarrow s(d)$ originated at tree level by non-diagonal couplings of
$Z$-boson with SM quarks.

The new effective Lagrangian contributing to $b\rightarrow
s(d)\ell^{+}\ell^{-}$ decays are:

$\displaystyle\mathcal{L}_{\text{new}}=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}V_{L4d}V_{L4b}^{*}(\overline{b_{L}}\gamma^{\mu}d_{L})\left[(-\frac{1}{2}+\sin^{2}\theta_{W})(\overline{\ell_{L}}\gamma_{\mu}\ell_{L})+\sin^{2}\theta_{W}(\overline{\ell_{R}}\gamma_{\mu}\ell_{R})\right]$
(368) $\displaystyle\mathcal{L}_{\text{new}}=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}V_{L4b}^{*}V_{L4s}(\overline{b_{L}}\gamma^{\mu}s_{L})\left[(-\frac{1}{2}+\sin^{2}\theta_{W})(\overline{\ell_{L}}\gamma_{\mu}\ell_{L})+\sin^{2}\theta_{W}(\overline{\ell_{R}}\gamma_{\mu}\ell_{R})\right]$
(369)

giving both exclusive decays, such as $B\rightarrow\pi\ell^{+}\ell^{-}$ and
$B\rightarrow K\ell^{+}\ell^{-}$, and inclusive decays $B\rightarrow
X_{s(d)}\ell^{+}\ell^{-}$, where $X_{s(d)}$ stands for anything with a $s(d)$
quark.

As regards exclusive decays, experimental branching fractions are below SM
predictions PDG18 . The decays $B^{0}\rightarrow K^{0*}\ell^{+}\ell^{-}$ and
$B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ are the best studied and tensions
with SM were found related to the quantity $P^{\prime}_{5}$ and to lepton
universality test PDG18  (and references therein).

Inclusive decays are also studied in order to avoid hadronic form factors. As
regards $B\rightarrow X_{s}\ell^{+}\ell^{-}$, branching fractions can be
analyzed both from a theoretical and experimental point of view in low and
high dilepton invariant mass regions, that is
$1\,\text{GeV}^{2}<q^{2}<6\,\text{GeV}^{2}$ and $q^{2}>14.4\text{GeV}^{2}$
regions (the intermediate region is dominated by charmonia resonances).
Experimental measurements of the inclusive decay were made by both Belle Belle
and BaBar BaBar . Their data are consistent with SM expectations at $95\%$
C.L. Huber:2015sra . In order to have a rough estimate on the constraint for
the mixings $|V_{L4d}V_{L4b}^{*}|$, $|V_{L4s}V_{L4b}^{*}|$, we can consider
the total branching fraction averaged between electrons and muons, which
experimentally results PDG18 :

$\displaystyle\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{exp}}=(5.8\pm 1.3)\times 10^{-6}$ (370)

SM calculations predict $\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{SM}}=(4.6\pm 0.8)\times 10^{-6}$
Ghinculov:2003qd . Let us define the branching ratio which would be given only
by the new contribution:

$\displaystyle|A_{\text{new}}|^{2}=\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{new}}\simeq$
$\displaystyle\simeq\text{Br}(B\rightarrow
X_{c}\ell^{+}\nu_{\ell})\frac{|V_{L4b}^{*}V_{L4s}|^{2}[(-\frac{1}{2}+\sin^{2}\theta_{W})^{2}+\sin^{4}\theta_{W}]}{|V_{cb}|^{2}}<\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{max}}$ (371)

$\text{Br}(B\rightarrow X_{s}\ell^{+}\ell^{-})_{\text{max}}$ is defined here
as the maximal allowed value for the branching ratio generated by the
amplitude of the new contribution alone and we use the experimental branching
ratio $\text{Br}(B\rightarrow X_{c}\ell^{+}\nu_{\ell})=0.1065(16)$ PDG18 .
Similarly $|A_{\text{SM}}|^{2}=\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{SM}}$. Then, at $95\%$ C.L. of the experimental
result (370), we should have:

$\displaystyle 3.25\times
10^{-6}<|A_{\text{new}}+A_{\text{SM}}|^{2}<8.35\times 10^{-6}$ (372)

Then, in order to set the least stringent constraint, we can consider the case
of destructive interference:

$\displaystyle|A_{\text{new}}|<\sqrt{8.35\times
10^{-6}}\left(1+\frac{|A_{\text{SM}}|}{\sqrt{8.35\times 10^{-6}}}\right)$
(373)

By using the $95\%$ C.L. interval also for the SM branching ratio,
$|A_{\text{SM}}|^{2}=6.2\times 10^{-6}$, the resulting constraint is:

$\displaystyle|V_{L4b}^{*}V_{L4s}|<1.9\times
10^{-3}\left[\frac{\text{Br}(B\rightarrow
X_{s}\ell^{+}\ell^{-})_{\text{max}}}{2.9\times 10^{-5}}\right]^{\frac{1}{2}}$
(374)

and, for this case of destructive interference, $|V_{L4b}^{*}V_{L4s}|>1.5\cdot
10^{-3}$.

As regards $b\rightarrow d\ell^{+}\ell^{-}$ transitions, the LHCb
collaboration measured the branching ratio for the decay
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ PDG18 , LHCbpi0 :

$\displaystyle\text{Br}(B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})_{\text{exp}}=(1.75\pm
0.22)\cdot 10^{-8}$ (375)

which agrees with SM predictions, for example
$\text{Br}(B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})_{\text{SM}}=1.88^{+0.32}_{-0.21}\cdot
10^{-8}$ Ali:2013zfa
$\text{Br}(B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})_{\text{SM}}=(2.04\pm
0.21)\cdot 10^{-8}$ Bailey:2015nbd . Other than that, upper limits on
exclusive decays include Belle constraint
$\text{Br}(B^{+}\rightarrow\pi^{+}\ell^{+}\ell^{-})<4.9\cdot 10^{-8}$ ($90\%$
C.L.) Belle3  and BaBar result BaBar3 :

$\displaystyle\text{Br}(B^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-})_{\text{exp}}<5.3\cdot
10^{-8}\qquad 90\%\,\text{C.L.}$ (376)

$\ell=e$ or $\mu$. The SM expectation for the $B^{0}$ decay can be obtained
from the predictions of $B^{+}$ branching ratios by multiplying by the factor
$(\tau_{B^{0}}/\tau_{B^{+}})/2$, where $\tau_{B^{0}}/\tau_{B^{+}}=1.076(4)$
PDG18 . Let us define as before the decay rate which would arise only from the
new contribution:

$\displaystyle|A^{\text{new}}_{B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-}}|^{2}=\text{Br}(B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-})_{\text{new}}\simeq
k_{B^{\pm}}|V_{L4d}V_{L4b}^{*}|^{2}$ (377)
$\displaystyle|A^{\text{new}}_{B^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-}}|^{2}=\text{Br}(B^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-})_{\text{new}}\simeq
k_{B^{0}}|V_{L4d}V_{L4b}^{*}|^{2}$ (378)

with

$\displaystyle
k_{B^{\pm}}=2\frac{\text{Br}(B^{\pm}\rightarrow\pi^{0}\ell^{+}\nu_{\ell})[(-\frac{1}{2}+\sin^{2}\theta_{W})^{2}+\sin^{4}\theta_{W}]}{|V_{ub}|^{2}}$
(379) $\displaystyle
k_{B^{0}}=\frac{1}{2}\frac{\text{Br}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})[(-\frac{1}{2}+\sin^{2}\theta_{W})^{2}+\sin^{4}\theta_{W}]}{|V_{ub}|^{2}}$
(380)

The factors of $2$ take into account the isospin symmetry relation. We also
take advantage of the results in PDG18
$\text{Br}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu_{\ell})=(1.50\pm 0.06)\cdot
10^{-4}$, $\text{Br}(B^{\pm}\rightarrow\pi^{0}\ell^{+}\nu_{\ell})=(7.80\pm
0.27)\cdot 10^{-5}$, $\ell=e$ or $\mu$. Then we should have:

$\displaystyle 1.3\times 10^{-8}<\,$
$\displaystyle|A_{B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-}}^{\text{new}}+A_{B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-}}^{\text{SM}}|^{2}<2.2\times
10^{-8}$ (381)
$\displaystyle|A_{B^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-}}^{\text{new}}+A_{B^{0}\rightarrow\pi^{0}\ell^{+}\ell^{-}}^{\text{SM}}|^{2}<5.3\cdot
10^{-8}$ (382)

where we defined $|A_{\text{SM}}|^{2}=\text{Br}_{\text{SM}}$. We used the
$95\%$ C.L. of the experimental result (375) and the upper limit (376). In
order to set the constraint, we can consider the case of destructive
interference between the new contribution and the SM one:

$\displaystyle|V_{L4d}V_{L4b}^{*}|<\frac{|A_{B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-}}^{\text{SM}}|}{\sqrt{k_{B^{\pm}}}}\left(1+\frac{\sqrt{2.2\times