#### Vector-like quarks at the origin of light quark masses and mixing

Eur. Phys. J. C
Vector-like quarks at the origin of light quark masses and mixing
Francisco J. Botella 1
G. C. Branco 0
Miguel Nebot 0
M. N. Rebelo 0
J. I. Silva-Marcos 0
0 Departamento de Física and Centro de Física Teórica de Partículas (CFTP), Instituto Superior Técnico (IST), Universidade de Lisboa (UL) , Av. Rovisco Pais, 1049-001 Lisbon , Portugal
1 Departament de Física Teòrica and IFIC, Universitat de València-CSIC , 46100 Burjassot , Spain
We show how a novel fine-tuning problem present in the Standard Model can be solved through the introduction of a Z6 flavour symmetry, together with three Q = −1/3 quarks, three Q = 2/3 quarks, as well as a complex singlet scalar. The Z6 symmetry is extended to the additional fields and it is an exact symmetry of the Lagrangian, only softly broken in the scalar potential, in order to avoid the domain-wall problem. Specific examples are given and a phenomenological analysis of the main features of the model is presented. It is shown that even for vector-like quarks with masses accessible at the LHC, one can have realistic quark masses and mixing, while respecting the strict constraints on processes arising from flavour changing neutral currents. The vector-like quark decay channels are also described.
1 Introduction
In the framework of the Standard Model (SM), the Brout–
Englert–Higgs mechanism is responsible not only for the
breaking of the gauge symmetry but also for the generation
of fermion masses, through the Yukawa interactions of the
scalar doublet with quarks and leptons. Understanding the
observed pattern of fermion masses and mixing remains a
fundamental open question in Particle Physics. On the
experimental side, there has been great progress and at present the
moduli of the Cabibbo–Kobayashi–Maskawa (CKM) matrix,
VCKM, are reasonably well known [1–3] with clear evidence
that the mixing matrix is non-trivially complex, even if one
allows for the presence of New Physics (NP) beyond the SM
[4–6]. The discovery of the Higgs particle at LHC [7,8]
rena e-mail:
b e-mail:
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ders specially important the measurement of the Higgs
couplings to quarks and charged leptons. The strength of these
couplings is fixed within the SM, but at present we have poor
knowledge of their actual value.
Recently, it has been pointed out [9] that there is a novel
fine-tuning problem in the SM. Contrary to conventional
wisdom, in the SM without extra flavour symmetries, quark
mixing is naturally large, in spite of the large quark mass
hierarchy. In fact, even in the extreme chiral limit, where only
the third family acquires mass, mixing is meaningful, and
in general it is of order one. It is possible to solve this
finetuning problem through the introduction of a simple flavour
symmetry which leads to VCKM = 1. The challenge is then
to achieve quark mixing and masses for the first two
generations.
In this paper we face this challenge and put forward a
framework where a flavour symmetry leads to VCKM = 1
in leading order, with light quark masses and mixing being
generated by the presence of vector-like quarks (VlQ). We
introduce three up-type VlQ and three down-type VlQ and
derive effective mass squared Hermitian mass matrices for
the standard quarks. In this framework one finds a natural
explanation why |V13|2 + |V23|2 is very small, while
allowing for an adequate Cabibbo mixing. In this model there are
Z-mediated and Higgs-mediated Flavour Changing Neutral
Currents (FCNC) which are naturally suppressed. The VlQs
can have masses of the order of one TeV which are at the
reach of the second run of LHC [10–19]. At this stage, it is
worth emphasising that VlQs have been extensively studied
in the literature [20–48] and arise in a variety of frameworks,
including E6 GUTS, models with extra dimensions, models
providing solutions to the strong CP problem without axions,
etc. It has also been pointed out [49] that non-supersymmetric
extensions of the SM with VlQs can achieve unification of
gauge couplings.
The paper is organised as follows: in the next section we
briefly review a novel fine-tuning problem of the Standard
Model and illustrate how it can be solved through the addition
of a flavour symmetry. In Sect. 3, we show how a realistic
quark mass spectrum and pattern of mixing can be generated
through the introduction of vector-like quarks and a complex
scalar singlet. In Sect. 4 we give specific examples examining
the constraints arising from various FCNC processes and
describing the vector-like quark decay channels. Finally, our
conclusions are presented in Sect. 5.
2 Hierarchy and alignment through a flavour symmetry
2.1 A novel fine-tuning problem in the standard model
Recently, it has been pointed out [9] that in the SM there is
a novel fine-tuning problem which stems from the fact that
the natural value of |V13|2 + |V23|2 is of order one in the
SM, to be compared to the experimental value of 1.6 × 10−3.
In order to obtain this result, one can examine the extreme
chiral limit where only the third family of quarks has mass
while all other quarks are massless. In this limit, the general
quark mass matrices can be written
Md = U d † diag(0, 0, mb) URd ,
L
Mu = ULu † diag(0, 0, mt ) URu
where ULd,,uR are arbitrary unitary matrices. The quark mixing
matrix is then V 0 = ULu †ULd . Using the freedom to redefine
the quark fields of the massless quarks through two-by-two
unitary matrices one can show that the quark mixing can be
described by an orthogonal two-by-two rotation connecting
only the (c, t ) and (s, b) quarks. This angle is arbitrary and
expected to be of order one.
2.2 A flavour symmetry leading to small mixing
Let us consider the SM and introduce the following Z6
symmetry:
where the Q0L j are left-handed quark doublets, dR0 j and u0R j
are right-handed quark singlets and denotes the Higgs
doublet. The Yukawa interactions are given by
LY =
0 0 0
Yd dR j − Q Li ˜ Yu u R j + h.c.,
and this symmetry leads to the following pattern of texture
zeros for the Yukawa couplings:
which lead to VCKM equal to the identity with only one
nonzero quark mass in each charge sector.
3 Vector-like quarks and generation of realistic quark
masses and mixing
A possible mechanism to generate masses for the light
standard-like quarks is to introduce vector-like quarks. In
our examples we specialise to the case of three down (D0Li ,
D0Ri ,) and three up (UL0i , UR0i ) vector-like isosinglet quarks.
With the introduction of these additional isosinglet quarks
the Yukawa interactions can now be denoted
LY =
0 0 0
(Yd )iα dRα − Q Li ˜ (Yu )iβ u Rβ + h.c.;
here the index i runs from 1 to 3, as in the SM, while the
indices α and β cover all right-handed quark singlets of the
down and up sector, respectively. The following generic bare
mass terms must also be introduced in the Lagrangian:
0 0 0 0
Lb.m. = [− DL j (ηd ) jα dRα − U Lk (ηu )kβ u Rβ ] + h.c.; (6)
here the indices j and k run over all left-handed vector-like
quarks in each sector. As mentioned before, in all examples
that follow i, j and k run from 1 to 3 and therefore α and
β run from 1 to 6 (obviously D0Ri ≡ dRi+3 and UR0i ≡
0
0
u Ri+3). In what follows we extend the Z6 discrete flavour
symmetry introduced in Sect. 2.2 and we introduce a complex
scalar singlet S. In the scalar sector the Z6 symmetry is also
imposed, but we allow for soft breaking terms like e.g., (S2 +
S∗2). These terms are crucial in order to avoid the
domainwall problem. Since the symmetry is discrete, they also avoid
the massless Goldstone boson. This scalar singlet will couple
to the quark singlets in the following way:
0 0
Lg = [− DL j [(gd ) jα S + (gd ) jα S∗] dRα
0 0
− U Lk [(gu )kβ S + (gu )kβ S∗] u Rβ ] + h.c.
We assume that the modulus of the vacuum expectation value
of the field S is of an order of magnitude higher than the
electroweak scale. After spontaneous symmetry breaking the
following mass terms are generated:
LM =
v 0 0 v 0
− √2 d Li (Yd )iα dRα − √2 u0Li (Yu )iα u Rα
0 0 0 0
− D Li (μd )iα dRα − U Li (μu )iα u Rα + h.c.
These terms can be written in a more compact form, as
LM = −
with 6 × 6 mass matrices, Md and Mu , denoted
Md =
Mu =
3.1 Structure of charged and neutral currents
The matrices Md and Mu will be diagonalised through the
following unitary transformations:
≡ URddR
dL ≡ ULd dL
uL ≡ ULu uL
≡ URuu R
where Ad L , Bd L , AuL and BuL are 3 × 6 matrices; the
matrices UL , UR , ULu and U Ru are unitary 6 × 6 matrices and (dL ),
d d
(dR ), (u L ) and (u R ) stand for the components of the six down
and six up mass eigenstate quarks. Unitarity implies that
= Ad† L Ad L + Bd†L Bd L = 1I6×6
with similar relations for the up-type matrices. The charged
currents are given by
with V = Au†L Ad L . The couplings of the Z boson are of the
form
u L W u γ μu L − d L W d γ μu L
with W d = V †V and W u = V V †. After spontaneous
symmetry breaking, Eqs. (5), (6) and (7) give rise to the physical
quark masses, which together with the couplings to the
SMlike Higgs can be denoted
[u L V Dd dR − u R Du V dL ]
d L W d Dd dR − u L W u Du u R
d L W d Dd dR + u L W u Du u R + h.c.
where h is the Higgs field, v is the vacuum expectation value
of the neutral component of the Higgs doublet and G+ and
G0 are the would-be Goldstone bosons, Dd and Du are the
six by six diagonal quark mass matrices.
3.2 Extension of the symmetry to the full Lagrangian
In the fermion sector, as mentioned above, we introduce three
down-type and three up-type vector-like quarks. In the scalar
sector, in addition to the standard Higgs, we have introduced
a complex scalar S. We extend the symmetry to the full
Lagrangian, with the new fields transforming in the following
way under the family symmetry:
UR01 → UR01 UR02 → e−iτ UR02 UR03 → e2iτ UR03;
S → eiτ S; τ = 26π (16)
together with the transformations for the standard-like quarks
specified in Eq. (2). The singlet scalar S is introduced in order
to be able to obtain realistic quark masses and mixing,
without breaking the symmetry at the Lagrangian level, except
for the soft breaking in the scalar potential. An alternative
option would be to softly break the symmetry also in the
fermion sector, through the introduction of bare mass terms
for the heavy fermions. Since the symmetry would be only
softly broken, the model would maintain its
renormalisability. We find it more appealing to have the symmetry exact
in the fermion sector. We assume that the scale of
spontaneous symmetry breaking of the S fields is higher than the
electroweak scale.
Tables 1 and 2 summarise the information on the
combination of the different fermionic charges and allow one to infer
what is the pattern of the mass matrices. The first three rows
come from Yukawa terms of the form given by Eq. (5) and
therefore are only allowed when the fermionic charge
cancels the one coming from the scalar doublet. In these cases
we write this charge explicitly. In the forbidden terms we
Table 1 Down sector, summary
of transformation properties. In
the forbidden terms we put a
bullet sign. We denote by 1 the
entries corresponding to allowed
bare mass terms. The fermionic
charges are given for those
terms that are allowed through
couplings to scalar fields: , S
or S∗, to which we assign
appropriate charges
Table 2 Up sector, summary of
transformation properties. In the
forbidden terms we put a bullet
sign. We denote by 1 the entries
corresponding to allowed bare
mass terms. The fermionic
charges are given for those
terms that are allowed through
couplings to scalar fields: , S
or S∗, to which we assign
appropriate charges
put a bullet sign. The last three rows come from bare mass
terms of the form given by Eq. (6) or else from couplings
to the field S. We denote with 1 the entries corresponding to
allowed bare mass terms and by the fermionic charges those
terms that allow coupling to either S or S∗. Again we use
bullets for the forbidden terms. The introduction of these singlet
scalar field provides a rationale for the choice of terms that
would otherwise softly break the symmetry and would look
arbitrary.
3.3 Effective Hermitian squared mass matrix
The 6 × 6 mass matrices Md , Mu are diagonalised through
the bi-unitary transformations:
d† d
UL Md UR = Dd ≡ diag(dd , Dd )
where dd ≡ diag (md , ms , mb), Dd ≡ diag (MD1, MD2,
MD3) and with MDi standing for the heavy Q = −1/3 quark
masses. An analogous equation holds for Mu . In our
examples the diagonalisation of Md and Mu is done through
an exact numerical calculation. However, in order to have
an idea of the main physical features involved, it is useful
to perform an approximate evaluation of UL , ULu and of the
d
quark mass eigenvalues. For this purpose, it is useful to write
UL , ULu in block form:
d
UL =
where K , R, S, T are 3 × 3 matrices. For simplicity, we
drop the indices d, and u. In Appendix A we show that the
deviations of the unitarity of the matrix K are naturally small:
K K † = 1 − R R†
R ≈
≈ (m/M )
S ≈
K † K = 1 − S† S
X X † + M M †
K −1Heff K = d2
The matrices Kd , Ku can be evaluated from an effective
Hermitian squared matrix Heff through:
The derivation of Heff is given in Appendix A.
3.4 Qualitative analysis of quark mixing
We analyse here the qualitative features of quark mixing
among standard quarks. As we have seen in the previous
section, in this framework the 3 × 3 VCKM matrix is
generated through the mixing of standard quarks with
vectorlike quarks, leading to an effective Hermitian squared mass
matrix given by Eq. (24). For simplicity, we will assume that
the dominant contribution to VCKM arises from the
diagonalisation of the down quark sector.
We assume that the 6 × 6 down quark mass matrix
has a Froggatt–Nielsen structure [50], with its elements
parametrised in terms of a small parameter λ, with all other
parameters of order one:
where μ ≈ mb. Using Eq. (24) one can derive that Heff has
the following structure:
Heff ∼ ⎝
r = |a3|2 + |c3|2
r = |a2|2 + |b2|2 + |c2|2
and (ii) the upper right off-diagonal blocks of the quark mass
matrices that read
The matrices in Eqs. (29) and (30) are proportional to the
Higgs vacuum expectation value; therefore its matrix
elements should be of the same order of magnitude as the
corresponding bottom or top mass, or smaller. The heavy
vectorlike quarks mass sectors Md and Mu are
Using, Eq. (24) and taking into account that He f f =
VCKMdd2VC†KM with dd = diag (m2d , ms2, m2b), one obtains
2
where Ki j are order one, namely K23 ≡ (c2 y), K13 ≡ (c3z).
Identifying λ with the Cabibbo parameter, one sees that the
experimental values of |Vub|, |Vcb| can be obtained. A more
detailed analysis can show that the full VCKM matrix can be
obtained, in agreement with experiment. In this section, we
illustrate in a qualitative way how one can obtain a correct
VCKM at low energies starting from a 6×6 quark mass matrix
parametrised by powers of a small parameter λ and order one
parameters. In the next section, we give a realistic example
with an exact diagonalisation of the 6 × 6 quark mass
matrices.
4 Realistic examples
Following the notation in Eq. (10) we present one full
realistic example; mass matrices are given at the MZ scale in
units of GeV. Among the matrices coming from electroweak
symmetry breaking we have: (i) the mass matrices that
connect light (ordinary) quarks among themselves md and mu ;
they should contain the dominant contributions to the b and
t quarks masses, allowed by the symmetry in Eq. (2),
Md = ⎝
Mu = ⎝
⎛ 767.538
0
0
⎛ 1295.99
0
0
It is clear that the light masses agree with the light masses
of the SM [51,52]. From the diagonalisation of Md and Mu
we also obtain the non-unitary 6 × 6 CKM matrix V . Its
moduli are given by
These matrices fix approximately the masses of the new
heavy vector-like singlet quarks. Finally, the other matrices
that connect the heavy sector with the light one are
⎛ 0 6.4 · 10−4
⎜ π 0
⎜⎜ −0.393 π + 0.0188
arg (V ) = ⎜⎜⎜⎝ 11..5673 −−11..1506
−1.09 1.55
−1.197 −8.2 · 10−3 −3.109 −2.3 · 10−3 ⎞
00 −11.2.923 −0.26.6584 −0.26.6163 ⎟⎟
2.02 0.091 −0.516 −0.107 ⎟⎟⎟ .
−1.56 −0.345 −0.895 −0.896 ⎟⎠
1.57 2.79 2.25 2.17
and the arguments of its matrix elements are given in leading
order by
The scale of the matrices in Eqs. (31), (32) and (33) is, for
most entries similar or higher than the electroweak scale: the
off-diagonal matrices being always smaller than the heavy
mass matrix sectors in order to be the responsible of giving
mass to the light quark masses and generating the CKM
mixing. Following the standard diagonalisation of Md and Mu
the quark mass spectrum (in GeV at the MZ scale) is
This generalised non-unitary CKM matrix V deserves several
comments:
1. The first upper left 3×3 block reproduces to a great extent
the SM CKM mixing matrix, including the phases.
2. It is remarkable that all the moduli—except |Vtb|—of the
3 × 3 light sector of V agree with the SM fitted values
within 1.5σ , in fact most are within 1σ .
3. A very important difference is in the element |Vtb|, which
is incompatible with the SM value. Disentangling this
value from the SM one is certainly an experimental
challenge for single top production.
4. Looking at the four independent phases that can be
defined in the 3×3 light sector β, γ , βs and χ —related to
the phases of Vtd , Vub, Vts and Vus , see Refs. [35,53]—
we cannot find any relevant difference among the phases
of this example and the phases of the SM. We get in our
model
5. Since the new quark singlets do not couple directly to the
SU(2)L gauge bosons, in the limit of no mixing among
chiral and vector-like singlet quarks only the upper left
3 × 3 sector of V is different from zero. This explains
the smallness of the entries in the other sectors of V
mass mU1 = 1313 GeV: U1 → t Z and U1 → t h. The
moduli of the matrix elements that control the FCNC in the down
sector are given by W d = V †V :
and indicates that the elements VUi d , VUi s and VUi b can
induce new physics effects that could appear for
example in b → d, b → s or s → d transitions. Transitions
induced by Vu Di , VcDi and Vt Di should be smaller than
the latter.
In summary, the CKM sector of our mixing matrix V
reproduces very well the SM case except for a minor but
definite deviation in Vtb at the 1% level.
4.1 The FCNC structure
In spite of the great similarity between the 3 × 3 sector of
our mixing matrix V and the SM CKM matrix, we know that
in this model we will have FCNC at tree level both in the up
and in the down sectors. Of course we expect to have these
FCNC highly suppressed, but to check these expectations
we present the matrices that control these FCNC both in the
couplings to the Z and to the Higgs h bosons. The moduli of
the matrix elements that control the FCNC in the up sector
are given by W u = V V †:
Owing to the fact that W d can induce K 0–K 0, Bd0 − Bd and
0
Bs0 − Bs0 mixing at tree level through Z exchange, the
offdiagonal elements Wdds , Wddb and Wsdb cannot be too large.
But as we can see they are well below (two to three orders of
magnitude) the values from previous analyses [32,33] that
avoid conflicting with meson mixing constraints. The
bottom line is that the 3 × 3 light sector of |W d | is very well
approximated by 13×3.
But this is not the end of the story. We have now an
enlarged 6 × 6 CKM matrix V with elements connecting
light and heavy quarks of order 10−3 to 10−4. These matrix
elements enter into the loops that generate FCNC with heavy
quarks running inside the loop and with several Inami–Lim
(IL) functions [54] growing with the square of the heavy
quark masses. So, a priori, one has to check that the
product of these heavy masses with these suppressed couplings
does not spoil the great success of the SM in FCNC
processes. At this point it is worthwhile recalling that we have
fixed our model with vector-like quarks by demanding that
a particular texture mass structure, imposed by symmetries,
should reproduce the light quark masses and the dominant
In the light sector Wuuc = 1.1 · 10−8 is too small to be
in conflict with D0–D0 mixing or D0 → μμ. Values like
Wqut ∼ 10−17 make t → Z u, Z c extremely suppressed.
The reminiscent of Vtb = 1 is here Wtut = 0.977 another
challenging deviation from the SM to be checked more likely
in loops. It is important to realise that the light 3 × 3
sector resembles very much the identity matrix 1I3×3 except for
Wtut . Values like W u
U1t = 0.149 will dictate the neutral
current dominant decay channels of the heavy quark U1 of
CKM mixing matrix. On the other hand, the symmetry does
not impose any constraint on the product of heavy masses
square and mixing between light and heavy fermions.
Therefore we should check the relevant constraints.
4.2 Loop FCNC constraints
In general the structure of FCNC at one loop level in this
model can be quite involved. But, as we have seen, the tree
level flavour changing coupling are very much suppressed. So
i=1
∼ 1 +
i=1
= 1.6 × 10−2,
= 0.13.
Including the charm and the leading QCD corrections, and
defining the corrections with respect the SM,
arg (M12)
− 1,
− 1,
we get for the K 0, Bd and Bs mixings
let us comment on the different processes where, as we will
see, tree level FCNC contributions can be safety neglected:
1. The contributions that go with Wiuj and Widj in FCNC tree
level contributions to F = 1 processes like Bq → μμ.
These NP pieces are proportional to Wbdq to be compared
with αem Vt∗b Vtq times an IL function of order one
coming from the SM piece. The transitions where these NP
contributions are the largest ones are rare kaon decays
where we have Wdds /αem Vt∗d Vts 10−2. So for example
K + → π +νν or K L → μμ are not affected by these
tree level contributions. The corresponding decays in the
B meson systems are even less affected.
2. The contributions that go with Wiuj 2 and Widj 2 in
FCNC tree level contributions to F = 2 processes like
D0–D0 or K 0–K 0, Bd0 − Bd0 and Bs0–Bs0. In these cases,
we are neglecting Wdds with respect to √αem Vt∗d Vts and
similarly for the Bd , Bs and D neutral meson systems.
The largest NP correction appears in the kaon case and
is of order 10−6 times the SM contribution.
3. When the tree level FCNC are small, Barenboim and
Botella [55] showed that the leading NP contribution to
meson mixing, induced by this FCNC enters at order
αem Wdds Vt∗d Vts , to be compared with αem Vt∗d Vts 2. So,
in our example, these contributions are at most of order
10−4. Again, we can neglect these contributions.
Taking into account the previous considerations, we can
analyse all the F = 1, 2 processes neglecting the tree level
FCNC effects.
F = 2 pure loop constraints
The neutral meson mixing is therefore dominated by the box
diagrams that generalise the SM one, with all species of heavy
quarks plus the top and the charm quarks—or the bottom and
the strange quarks for D0–D0 mixing—running inside the
loop.1 If we define as usual λqaq = Va∗q Vaq —for mesons
with down quarks—, the dominant corrections to the
mixing (M12)qq of the meson with quark content qq , with
respect to the SM box diagram with internal top quarks, can
be written as
1 Note that in the SM, the Inami–Lim function that appears in the box
runs for c and t quarks after using unitarity 3 × 3. In our case we have
for the kaon system, for example, 6 V †V ds = Wdds ,
so there is an additional contributioni=—1 tVoidwVhi∗sat=we are considering
in the main text—proportional to the quark u contribution and to
Wdds , and therefore negligible. So we can extend the sum from c, t to
c, t, U1, U2, U3.
where S(xt ), S(xUi , xt ) are the IL functions defined as in
[53] and xt , xUi = (mt /MW )2, mUi /MW 2 . In our case we
have for the first and second corrections in the kaon system
We conclude that these kinds of models are compatible
with the actual analysis beyond the SM, but they can have
sizeable effects. For example this model does not modify
MBd / MBs but can give 12% corrections to MBd and
MBs . Furthermore, K can have 12% corrections.
F = 1 pure loop constraints
In these processes—for example q → q μμ—the
corrections to the top dominated SM amplitudes (when necessary,
charm has to be taken into account) are given by
A (q → q μμ)SM ∼ 1 +
i, j=c,t,U1,U2,U3
The IL functions Y xUi grow with the square of the quark
mass Ui —also N xUi , xUi . This fact, apparently, will make
the second term more relevant than the corresponding one
in F = 2 processes: here there is only a λUqqi /λqtq
sup2
suppression as in Eq. (40).
But the WUui Ui are very small—the new quarks are singlets
under SU(2)L —, therefore the last term has a piece
enforcing decoupling and cancelling partially the second piece.2
For more details one can see Refs. [56–59]. A similar
structure appears in q → q νν. If we define, as in the F = 2
processes, the deviation from the SM model
r ( A → B) =
− 1,
r B+ → K +νν = 0.20.
r K + → π +νν = 0.20, r B+ → π +νν = 0.21,
Some of these predictions can be definitely excluded or
verified very soon by the LHC experiments. One example is the
effect predicted for Bs → μμ, which deviates significantly
from the SM value [60,61]. If the RK and P5 anomalies [62]
have a common origin and therefore are both related to lepton
universality breaking, it is clear that the scheme developed
here cannot explain them. In this context only with the
introduction of vector-like fermions in the leptonic sector can one
produce lepton universality breaking.
We also have analysed other loop mediated processes and,
for example, we get for the oblique corrections [63,64]
T = 0.22 ,
S = 0.06,
U = 0.003.
4.3 The heavy vector-like quark decay channels
In our model the new heavy quarks decay mainly
throughout charged or neutral currents. The decays can be
characterised by Q j → qi B. where Q j = U j , D j , is the new
heavy fermion, B = Z , h, W ± and qi the final state fermion,
namely u, c, t, d, s or b. The different partial decay widths
can be written as
2 Note that this term is usually overlooked in the literature.
r (K L → μμ)S D = 0.32, r (Bd → μμ) = 0.31,
Furthermore, under the same conditions, we also have
⎧ Vi j for B = W −,
and XiBj,Q j = ⎨ V ji for B = W +,
⎩ Wi Qj for B = h, Z ,
fW ± (x , r ) = f Z (x , r ) ,
fh (x , r ) ∼ f Z (x , r ) ∼ 1, (49)
where the last two relations are valid when mqi , MZ , MW , Mh
m Q j . Detailed formulae are included in Appendix B. In
this regime, to a very good approximation [25],
for qi and qi in the same generation, as suggested by the
subindex i : for example c and s. It turns out that, under
reasonable conditions, as explained in Appendix B, it is quite
common to have V ji 2 ∼ Wiuj 2 and Vi j 2 ∼ Widj 2.
Therefore we also have
Although many searches for new heavy quarks assume
Q j → qi Z : Q j → qi h : Q j → qi W =
1 : 1 : 2 and Br Q j → qi Z + Br Q j → qi h +
Br Q j → qi W 1, the latter is unjustified and the total
Q j decay width can be distributed among the decay
channels to different generations, while the 1 : 1 : 2 pattern of
branching ratios “ per generation” is maintained [30]. This
fact happens in the present model and we show in Tables 3
and 4 the dominant decay channels of the heavy quarks.
For example, D1—the lightest of the heavy vector-like
quarks in this model—has around 20% of its branching ratio
to decay channels in the third generation and some 80%
decaying to the second generation (Z s, hs, W c). These are
not the most common ways of searching for these down-type
vector-like quarks, although the different results by ATLAS
and CMS can be adapted. Note that neither the decay to the
third family is the dominant one nor there is a unique
family entering in the dominant decay products. Similar patterns
Table 3 Decays of new
down-type quarks
Branching ratio to channel (%)
1.3 × 10−2
8.6 × 10−6
Branching ratio to channel (%)
appear in D2 decays. As far as the up vector-like quarks are
concerned the lightest U1 decays dominantly to the third
family while the other (heavier) two decay to the second family.
4.4 Other examples
We have presented a paradigmatic example to show that it
is possible and even relatively easy to accomplish our goal.
Indeed, in our example the light quark masses and CKM
mixing are generated from the couplings of ordinary quarks
to the new heavy vector-like quarks, keeping the extended
symmetry that explains why to first order the CKM matrix is
the identity. In addition, we predict in the flavour electroweak
phenomenology many deviations from the SM, some of them
at 2σ level or more. Note that we have also a definite
deviation of 3 × 3 unitarity, albeit very difficult to disentangle
experimentally: |Vtb| ∼ 0.9877. Also we have six new heavy
quarks that can be seen at LHC, some of them with very
peculiar characteristic decays. From the present example, one can
get the wrong impression that if these new quarks are not
discovered at the LHC and if the different deviations predicted
in the flavour and electroweak sectors are not established,
the main ideas of this paper are no longer relevant. This is
not the case. On the contrary, we will show that there are
many additional solutions which, without spoiling the main
characteristics of the low mass sector, have the feature that
New Physics effects smoothly decouple in the low energy
phenomenology. In order to see how this decoupling arises,
let us consider Eq. (10) and label our explicit example in
Eqs. (29)–(33) with a superscript “(1)”, i.e., the matrices in
Eqs. (29)–(33) are
Let us construct a second solution, labelled “(2)” that differs
by the real numbers ρd and ρu :
Now, for ρd , ρu > 1 we obtain another solution which
reproduces “essentially” the same light quark masses and
mixings, while deviations from the SM decouple as ρd and ρu
are increased. The basic equations are Eqs. (19)–(24).
Equau
tion (24) shows that Heff does not change at leading order
in going from M(u1) to M(u2) and so Ku(2) ∼ Ku(1). At the
same time Eqs. (20)–(22) tell us that Ru(1) and Su(1) scale as
Ru(2) ∼ ρu−1 Ru(1) and Su(2) ∼ ρu−1 Su(1), which leads to a small
change in Ku(2) taking it closer to unitarity. The invariance of
Heuf(f1) → Heuf(f2) under the scaling in Eq. (54) is at the origin
of the fact that, once we have a solution of the type presented
here, we have a continuum of solutions with essentially the
same light quark masses and mixing and with heavy quarks
much heavier and more decoupled. It remains to check that
the effects from heavy quarks running inside the loops also
decouple. This can easily be seen by realizing that our CKM
matrix can also be written
V =
†
From this it is evident that the submatrix Ru Kd scales to
†
ρ−1 Ru Kd and consequently we have a very simple scaling
u
law from the up heavy sector scaling:
VUi d j → ρu−1VUi d j , λUdiid j → ρu−2λUdiid j ,
∼
as ρu−2. The second correction scales as ρu−2 ln ρu according
to the behaviour of the IL function S xUi , xt . Behaviour
similar to the latter one appears in the F = 1 processes
following the cancellations explained in Eq. ( 44).
To be more specific we present briefly the results for a
(2) constructed with ρu = 2 and
second example with Mu
keeping the rest of the model unchanged. For F = 2
processes we get, instead of the results in Eq. (43), the following:
mU2i and therefore it decreases
B0 ⎠
s
= ⎜⎜ 0.07 ⎟⎟ , ⎜⎜ δφ Bd0 ⎟⎟
⎝ 0.06 ⎠ ⎝ δφ Bs0 ⎠
On average, the corrections to the SM mixing values get
reduced from a 12% to some 6%. In the case of F = 1
processes instead of Eq. (45) we get
r (K L → μμ)S D = 0.16, r (Bd → μμ) = 0.16,
r (Bs → μμ) = 0.16,
r K + → π +νν = 0.11, r B+ → π +νν = 0.11,
r B+ → K +νν = 0.10.
In this sector, the deviation from the SM model gets reduced
by a factor of 2. In this model “(2)” the deviations in these
processes are at the 11–16% level, showing a smooth
decoupling of the effects while at the same time the heavy up-type
quarks get heavier masses: mU1 , mU2 , mU 3 ∼ (2.6, 3, 4.5)
TeV. As for the oblique corrections, they have the following
values:
T = 0.11,
S = 0.05,
U = 0.004.
5 Conclusions
We have presented a simple solution to a novel fine-tuning
problem present in the SM. The solution involves the
introduction of a Z6 flavour symmetry, together with vector-like
quarks of Q = −1/3 and Q = 2/3 charges, as well as a
complex singlet scalar. In the absence of vector-like quarks only
the bottom and the top quarks acquire masses and VCKM = 1.
We have shown that in the presence of the vector-like quarks
which mix with the standard quarks, a realistic quark mass
spectrum can be obtained and a correct CKM matrix can be
generated. It is remarkable that these results are obtained in a
framework where Z6 is an exact symmetry of the Lagrangian,
only softly broken in the scalar potential. This breaking is
essential in order to avoid the domain-wall problem. In the
literature, there are a large number of works addressing the
flavour problem, [65–69] in various frameworks, including
supersymmetric models or in scenarios including also mirror
fermions or vector-like quarks. The distinctive feature of the
present work is the fact that the full VCKM matrix is entirely
generated through the mixing with vector-like quarks. We
have presented specific realistic examples and have analysed
various FCNC processes as well as the decay channels of the
vector-like quarks. It is also remarkable that in the framework
of fully realistic models, some of these vector-like quarks are
within the reach of the second LHC run.
Acknowledgements The authors acknowledge financial support from
the Spanish MINECO under Grant FPA2015-68318-R, by the Severo
Ochoa Excellence Center Project SEV-2014-0398, by Generalitat
Valenciana under Grant PROMETEOII/ 2014/049 and by Fundação
para a Ciência e a Tecnologia (FCT, Portugal) through the projects
CERN/FIS-NUC/0010/2015, and CFTP-FCT Unit 777 (UID/FIS/00777
/2013) which are partially funded through POCTI (FEDER),
COMPETE, QREN and EU. M.N. acknowledges support from FCT through
the postdoctoral Grant SFRH/BPD/112999/2015. The authors also
acknowledge the hospitality of Universidad de Valencia, IFIC, and
CFTP at IST Lisboa during visits for scientific collaboration.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Derivation of effective Hermitian squared
mass matrix
In order to derive the expression given for Heff in Eq. (24)
we start from
using the notation of Eq. (17), where, for simplicity, we have
dropped the indices d and u for down and up. We can rewrite
this equation explicitly, in terms of each one of the four 3 × 3
blocks, using Eq. (10) for the matrix M:
(mm† + ωω†) K + (m X † + ω M †) S = K d2,
(mm† + ωω†) R + (m X † + ω M †) T = R D2,
(X m† + M ω†) K + (X X † + M M †) S = S d2,
(X m† + M ω†) R + (X X † + M M †) T = T D2.
Since the term S d2 is much smaller than the other two,
Eq. (63) can be approximated by
−(X X † + M M †)−1(X m† + M ω†) K .
Replacing S in Eq. (61) we obtain to a good approximation
[(mm† + ωω†) − (m X † + ω M †)
×(X X † + M M †)−1(X m† + M ω†)] K = K d2,
K −1Heff K = d
with Heff given by Eq. (24).
Appendix B: Vector-like quark decays
VU d = Ru† Kd ,
WUu u = Ru† Ku ,
so we can write
VU d = WUu u Ku−1 Kd .
At leading order Ku−1 ∼ Ku† and we get, to a high accuracy
(and similarly for Vu D and WddD)
VU d ∼ WUu u Vud , Vu D ∼ Vud WddD.
From these expressions it is easy to prove that the relations in
Eq. (71 ) hold, at least for the dominant decay channel of the
corresponding heavy up or down quark. The key ingredient
is that Vud ∼ 1I, in Eq. (72), with corrections at most of order
λ = 0.22.
where each submatrix connects the corresponding types of
quarks u, U and d, D (in the present model, all four
submatrices are 3 × 3). Similarly, we also introduce
Following again the notation in Sect. 3.3 for the up and down
sectors, we have
fh (x , y) = (1 + y − x ) f (x , y),
fV (x , y) =
V = Z , W ±,
(1 − y2) + x (1 + y) − 2x 2 f (x , y),
f (x , y) =
1 −
√y + √x 2
1 −
√y − √x 2.
Let us explain the origin and validity of the relations
|V ji |2 ∼ |(W u )i j |2,
|Vi j |2 ∼ |(W d )i j |2,
where the prime means that i and i correspond to the same
generation number, i = i . For that purpose we introduce, in
self-explanatory matrix notation,
The functions that appear in Sect. 4.3 that are needed for the
heavy quark decays are
V =
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