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arXiv:1005.3505v3 [hep-ph] 25 Nov 2010
DO-TH 10/09
TTK-10-32
SFP/CPP-10-35
Less space for a new family of fermions
Otto Eberhardt∗,1, Alexander Lenz†,2,1,
and
J¨ urgen Rohrwild‡,3
1Institut f¨ ur Theoretische Physik,
Universit¨ at Regensburg, D–93040 Regensburg, Germany
2Institut f¨ ur Physik,
Technische Universit¨ at Dortmund, D–44221 Dortmund, Germany
3Institut f¨ ur Theoretische Teilchenphysik und Kosmologie,
RWTH Aachen University, D–52056 Aachen, Germany
Abstract
We investigate the experimentally allowed parameter space of an extension of the standard
model (SM3) by one additional family of fermions. Therefore we extend our previous study of
the CKM like mixing constraints of a fourth generation of quarks. In addition to the bounds
from tree-level determinations of the 3×3 CKM elements and FCNC processes (K-, D-, Bd-,
Bs-mixing and the decay b → sγ) we also investigate the electroweak S, T, U parameters, the
angle γ of the unitarity triangle and the rare decay Bs→ µ+µ−. Moreover we improve our
treatment of the QCD corrections compared to our previous analysis. We also take leptonic
contributions into account, but we neglect the mixing among leptons. As a result we find that
typically small mixing with the fourth family is favored, but still some sizeable deviations
from the SM3 results are not yet excluded. The minimal possible value of Vtbis 0.93. Also
very large CP-violating effects in Bsmixing seem to be impossible within an extension of
the SM3 that consists of an additional fermion family alone. We find a delicate interplay
of electroweak and flavor observables, which strongly suggests that a separate treatment of
the two sectors is not feasible. In particular we show that the inclusion of the full CKM
dependence of the S and T parameters in principle allows the existence of a degenerate
fourth generation of quarks.
∗Otto.Eberhardt@physik.uni-regensburg.de
†Alexander.Lenz@physik.uni-regensburg.de
‡rohrwild@physik.rwth-aachen.de
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1 Introduction
Increasing the number of fermion generations (see [1] for a review and [2] for an update)
is probably the most obvious extension of the usual standard model with three generations
(SM3). Although being popular in the 80s, such a possibility was discarded for a long time.
Recently these models (SM4) celebrated a kind of resurrection. Partly, this was due to the fact
that a fourth generation is not necessarily in conflict with electroweak precision observables
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
Besides being a straightforward extension of the SM3, an increase of the number of fermion
generations leads also to several desired effects:
• The authors of [7, 9, 11, 12, 13, 15] have shown that a fourth generation softens the
current low Higgs mass bounds from electroweak precision observables, see e.g. [16], by
allowing considerably higher values for the Higgs mass.
• It might solve problems related to baryogenesis: An additional particle family could
lead to a sizeable increase of the measure of CP-violation, see [17, 18]. Moreover,
such an extension of the SM would increase the strength of the phase transition, see
[19, 20, 21].
• The gauge couplings can in principle be unified without invoking SUSY [22].
• New heavy fermions lead to new interesting effects due to their large Yukawa couplings,
see e.g [23, 24]. Moreover dynamical electroweak symmetry breaking might be triggered
by these heavy new fermions [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. This mechanism
can also be incorporated in models with warped extra dimensions, as done in [36, 37].
There are also some modest experimental deviations that could be explained by the existence
of a fourth generation:
• A new family might cure certain problems in flavor physics (CP-violation in Bs-mixing,
K −π-puzzle, ǫkanomaly,...) see e.g. [38, 39, 40, 41, 42] for some recent work and e.g.
[43, 44] for some early work on 4th generation effects on flavor physics.
• Investigations of lepton universality show a value of the PMNS element Ve4?= 0 at the
2.5 σ level [45].
For more arguments in favor of a fourth generation see e.g. [2] and also [46, 47, 48, 49].
We conclude the list by repeating our statement from [50]: In view of the (re)start of the
LHC, it is important not to exclude any possibility for new physics scenarios simply due to
prejudices. Direct search strategies for heavy quarks at the LHC are worked out e.g. in
[31, 51, 52, 53, 54, 55, 56, 57]. Signatures and consequences for collider physics, such as the
modification of production rates, have been studied e.g. in [58, 31, 59, 60, 61].
In this work we extend our analysis in [50], where we performed an exploratory study
of the allowed parameter range for the CKM like mixing of hypothetical quarks of a fourth
generation. Adding one generation of quarks results in several new parameters. In particular,
we have the new masses mb′ and mt′, and nine parameters (six angles and three phases) in
the 4 × 4 CKM matrix (compared to three angles and one phase in the SM3). Following
our previous strategy, we consecutively add bounds on the CKM structure of the SM4 and
perform a scan though the parameter space of the model to identify the allowed regions;
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while this treatment is insufficient to fit for the central values or standard deviation of the
model parameters, it gives a very reasonable idea of the experimentally possible parameter
space allowing for statements on the size of effects of the model on particularly interesting
flavor observables. Apart from the unitarity of the 4 × 4 matrix and the direct bounds on
the quark masses, the most important input comes from direct measurements of the absolute
values of CKM matrix elements, e.g. from β decay. In [50] the next step was the inclusion
of flavor observables sensitive to FCNC, mediated i.e. by box or penguin diagrams. This led
to some surprising results regarding the possible size of the quark mixing with the fourth
generation quark, as rather large values for the mixing angle s34could not be excluded.
However, in [62] Chanowitz found that the parameter sets, which we gave as an example
for large mixing with the fourth generation, are excluded by electroweak precision constraints,
in particular by the oblique corrections [63]. Moreover, Chanowitz performed the whole
electroweak fit for four different values of the mass of the t′quark. Here, some assumptions
were used: i) the lepton masses are fixed to ml4 = 145 GeV and mν4= 100 GeV, ii)
lepton mixing is not included, iii) the mass difference of the heavy quark doublet is fixed
to mt′ − mb′ = 55 GeV, iv) only mixing between the third and fourth family was included.
Assumptions iii) and iv) were also tested in [62].
Therefore, we supplement the analysis of the flavor sector by the S, T and U parame-
ters; also the lepton masses of the fourth generation have to be taken into account. For the
present work we assume that the neutrinos have Dirac character and neglect the possible
mixing of the fourth neutrino in the lepton sector. Moreover, we extend the set of our FCNC
observables to include also Bs→ µ+µ−and we improve the simplified treatment of the decay
b → sγ by using the full leading logarithmic result. Concerning the tree-level determination
of the CKM elements we include now also the experimental results for the angle γ of the
unitarity triangle, which gives a direct constraint on CKM phases. Similar studies have been
recently performed e.g. in [64, 65, 66, 67]
In Section 2 we present all experimental constraints we use in our analysis. We start with the
parameterization of VCKM4in Section 2.1, next we discuss briefly tree-level determinations
of CKM elements and direct mass limits. The electroweak parameters S, T and U will be
investigated before reviewing the FCNC constraints. We end Section 2 with the allowed
regions for deviations of the SM4 results from the SM3 values.
In Section 3 we determine the bounds on the parameters of the model. After explaining our
general strategy in 3.1, we present the results for the different mixing angles of VCKM4, the
new results for Vcxand Vtx(x = d,s,b), and allowed effects of a fourth generation in neutral
meson mixing.
In Section 4 we give a Wolfenstein-like expansion of the 4 × 4 CKM matrix. With the addi-
tional information from the electroweak sector, tighter constraints on the fourth generation
quark mixing can be utilized leading to a simplified expansion.
We conclude with Section 5.
2 Constraints on VCKM4
2.1 Parameterization of VCKM4
In the SM3 the mixing between quarks is described by the unitary 3×3 CKM matrix [68, 69],
which can be parameterized by three angles, θ12, θ13and θ23(θij describes the strength of
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the mixing between the ith and jth family) and the CP-violating phase δ13. The so-called
standard parameterization of VCKM3reads
VCKM3=
c12c13
s12c13
s13e−iδ13
s23c13
c23c13
−s12c23− c12s23s13eiδ13c12c23− s12s23s13eiδ13
s12s23− c12c23s13eiδ13−c12s23− s12c23s13eiδ13
(2.1)
with
sij:= sin(θij) and cij:= cos(θij).(2.2)
Extending the minimal standard model to include a fourth family of fermions (SM4) intro-
duces 3 additional angles in the CKM matrix θ14,θ24and θ34and 2 additional CP-violating
phases δ14and δ24. To determine the allowed range for these new parameters we use an exact
parameterization of the 4 × 4 CKM matrix. We have chosen the one suggested by Botella
and Chau [70]1, Fritzsch and Plankl [71]2and also by Harari and Leurer [72].
VCKM4=
c12c13c14
c13c14s12
c14s13e−iδ13
s14e−iδ14
−c23c24s12− c12c24s13s23eiδ13
−c12c13s14s24ei(δ14−δ24)
c12c23c24− c24s12s13s23eiδ13
−c13s12s14s24ei(δ14−δ24)
c13c24s23
c14s24e−iδ24
−s13s14s24e−i(δ13+δ24−δ14)
−c12c23c34s13eiδ13+ c34s12s23 −c12c34s23− c23c34s12s13eiδ13
−c12c13c24s14s34eiδ14
+c23s12s24s34eiδ24
+c12s13s23s24s34ei(δ13+δ24)
c13c23c34
c14c24s34
−c12c23s24s34eiδ24
−c13c24s12s14s34eiδ14
+s12s13s23s24s34ei(δ13+δ24)
−c13s23s24s34eiδ24
−c24s13s14s34ei(δ14−δ13)
−c12c13c24c34s14eiδ14
+c12c23s13s34eiδ13
+c23c34s12s24eiδ24− s12s23s34
+c12c34s13s23s24ei(δ13+δ24)
−c12c23c34s24eiδ24+ c12s23s34
−c13c24c34s12s14eiδ14
+c23s12s13s34eiδ13
+c34s12s13s23s24ei(δ13+δ24)
−c13c23s34
−c13c34s23s24eiδ24
−c24c34s13s14ei(δ14−δ13)
c14c24c34
(2.3)
For our strategy the explicit form of VCKM4does not matter, it is only important that the
parameterization is exact. Besides the nine parameters of VCKM4we have also the masses of
the fourth generation particles, which we denote as mb′,mt′,ml4and mν4. We do not include
leptonic mixing, yet.
2.2Experimental bounds
In this section we summarize the experimental constraints that have to be fulfilled by the
parameters of the fourth family.
The elements of the 3×3 CKM matrix have been studied intensely for many years and
precision data on most of them is available. In principle there are two different ways to
determine the CKM elements. On the one hand, they enter charged weak decays already at
tree-level and a measurement of e.g. the corresponding decay rate provides direct information
on the CKM elements (see e.g. [73] and references therein). We will refer to such constraints
as tree-level constraints.On the other hand, processes involving flavor-changing neutral
1In the published paper of Botella and Chau there is a typo in the element Vtd: in the last term of Vtd
the factor syhas to be replaced by cy.
2In the published paper of Fritzsch and Plankl there is a typo in the element Vcb: the factor c23has to be
replaced by the factor s23.
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currents (FCNC) are forbidden at tree-level and only come into play at loop level via the
renowned Penguin and Box diagrams. These processes provide strong bounds — referred to
as FCNC constraints — on the structure of the CKM matrix and its elements as well as on
the masses of the heavy virtual particles appearing in the loops.
We will start with the tree-level constraints, since they only depend on the CKM elements and
not on the fermion masses. Next we consider mass constraints on the fourth family members
from direct searches at colliders. Since the oblique electroweak parameters are expected to
reduce the allowed range of masses for a new fermion family notably, we consider them next
and finally we discuss the FCNC constraints.
2.2.1Tree-level constraints for the CKM parameters
Since the (absolute) value of only one CKM element enters the theoretical predictions for
weak tree-level decays, no GIM mechanism or unitarity condition has to be assumed. By
matching theory and experiment the matrix element can be extracted independently of the
number of generations3. Therefore, all tree-level constraints have the same impact on the
4 × 4 matrix as they have on the 3 × 3 one.
We take the PDG values [74] for our analysis:
absolute value
0.97418 ± 0.00027
0.2255 ± 0.0019
0.00393 ± 0.00036
0.230 ± 0.011
1.04 ± 0.06
0.0412 ± 0.0011
> 0.74
relative error
0.028%
0.84%
9.2%
4.8%
5.8%
2.7%
direct measurement from
nuclear beta decay
semi-leptonic K-decay
semi-leptonic B-decay
semi-leptonic D-decay
(semi-)leptonic D-decay
semi-leptonic B-decay
(single) top-production
Vud
Vus
Vub
Vcd
Vcs
Vcb
Vtb
In the following, we denote the absolute values in the table above as |Vi| ± ∆Vi. In addition
to the above tree-level constraints there exists a direct bound on the CKM angle
γ = arg
?
−VudV∗
VcdV∗
ub
cb
?
.
It can be extracted via the decays B → DK,Dπ [75, 76, 77, 78]. In principle the extraction
of γ might be affected by the presence of a fourth generation of fermions [79], but it was
shown in [80] that these effects are negligible. Therefore, γ gives direct information on the
phases of the CKM matrix; with three CP violating phases present, this can provide a useful
piece of information. We use the CKMfitter value from [81] (update of [82, 83])
γ = 73◦−25◦
+22◦ ± 2◦,(2.4)
where the last error accounts for the tiny additional uncertainty due to the additional fermion
generation.
3There is, however, one loop hole: In [45] the possibility of lepton mixing reducing the accuracy of the
determination of e.g. Vudwas discussed. Since we use the more conservative error estimate from the PDG,
our relative error is similar to the final error of Lacker and Menzel, who started with a more ambitious error
for Vudin their analysis [45].
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2.2.2Direct mass limits for the fourth family
The PDG [74] gives from direct searches the following mass limits for a fourth family
mν4> 80.5...101.5GeV,(2.5)
ml4> 100.8GeV,(2.6)
mb′ > 128...268GeV,(2.7)
mt′ > 256GeV. (2.8)
The mass bound on the heavy neutrino depends on the type of neutrino (Dirac or Majorana)
and whether one considers a coupling of the heavy neutrino to e−, µ−or τ−. It is interesting
to note that LEP results in combination with [84] exclude a fourth stable neutrino with m
< 2400 GeV [74]. The quark mass bounds are obtained from direct searches at TeVatron
[85, 86], which were recently updated [87, 88]
mb′ > 338GeV,mt′ > 335GeV. (2.9)
In [89] it was pointed out that in deriving these bounds assumptions about the couplings
of the fourth generation have been made (in [87] it is e.g.
b′is shortlived and that it decays exclusively to tW−, which corresponds to demanding
Vub′ ≈ 0 ≈ Vcb′, mb′ < mt′ and Vtb′ is not extremely small). Without these assumptions the
mass bounds can be weaker, as the extraction of the masses has to be combined with the
extraction of the CKM couplings. The inclusion of this dependence is beyond the scope of
the current work. For some recent papers concerning the mass exctraction of leptons and
quarks, see [90, 91].
In this work we investigate heavy quark masses in the range of 280 GeV to 650 GeV heavy
charged lepton masses in the range of 100 GeV to 650 GeV and heavy neutrino masses in the
range of 90 GeV to 650 GeV. Note that the triviality bound from unitarity of the t′t′S-wave
scattering [92] indicates a maximal t′mass of around 504 GeV [93]. However, this estimate
is based on tree-level expressions and while it seems prudent to treat too high quark masses
with a grain of salt, one should not disregard higher masses based on this estimate alone.
In this context it would be desireable to have e.g. a lattice study of the effect of very heavy
(fourth generation) quarks.
explicitly assumed that the
2.2.3Electroweak constraints
We present here the expressions for the oblique electroweak S,T and U parameters [63] in
the presence of a fourth generation. They were originally defined as
αS = 4e2 d
dq2
?Π33(q2) − Π3Q(q2)?????
xW¯ xWM2
Z
q2=0
,(2.10)
αT =
e2
[Π11(0) − Π33(0)] ,(2.11)
αU = 4e2 d
dq2
?Π11(q2) − Π33(q2)?????
5
q2=0
,(2.12)
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with the electric coupling α and e, Πxydenotes the virtual self-energy contributions to the
weak gauge bosons and with the Weinberg angle expressed as xW= sin2θWand ¯ xW= 1−xW.
In the first paper of Ref. [63] ¯ xWM2
to the famous ρ-parameter [94, 95, 92]
Zwas approximated by M2
W. The T parameter is related
ρ :=
M2
W
¯ xWM2
Z
=: 1 + ∆ρ (2.13)
= 1 + αT . (2.14)
In practice, it turns out to be considerably simpler to reexpress the derivatives in S and U
as differences
d
dq2ΠXY(q2)
q2=0
????
≈ΠXY(M2
Z) − ΠXY(0)
M2
Z
. (2.15)
This approximation works very well for mnew≫ MZand it is used by the PDG [74]. We will
use however the original definitions given in Eqs. (2.10), (2.11), (2.12) with M2
because there are no correction terms and our expressions are exact.
Next only the new physics contributions to the S, T and U parameters will be considered,
as the SM values of the oblique parameters are by definition set to zero. Fit results for the
allowed regions of the S, T and U parameters are obtained e.g. by the PDG [74], EWWG
[96], Gfitter [16] and most recent in [14]. Note that the more recent analyses [16, 14] differ
significantly from the old (November 2007) PDG version. Due to more refined experimental
results and an improved theoretical understanding the best fit values shifted significantly
towards higher values of S and T, see Fig. 1 for the Gfitter S-T ellipse [97]; this somewhat
relaxes the previously observed tension with an additional fermion generation.
In the presence of a fourth generation the fermionic contribution to these parameters (before
the necessary subtraction of the SM contribution) reads
W= ¯ xwM2
Z,
S =Nc
6π
4
?
f=1
?
1 −1
3lnm2
uf
m2
df
?
+
1
6π
4
?
f=1
?
1 + lnm2
νf
m2
lf
?
,(2.16)
T =
Nc
16πxW¯ xWM2
Z
?
?
q=u,d,s,...,t′,b′
m2
q−
4
?
f=1
4
?
f′=1
|Vufdf′|2FT(m2
uf,m2
df′)
?
(2.17)
+
1
16πxW¯ xWM2
Z
?
l=νe,e−,...,ν4,l−
4
m2
l−
4
?
f=1
4
?
f′=1
|Vνflf′|2FT(m2
νf,m2
lf′)
, (2.18)
U =Nc
3π
?
4
?
f=1
4
?
f′=1
|Vufdf′|2FU(m2
uf,m2
df′) −5
6
4
?
f=1
1
?
+
1
3π
?
4
?
f=1
4
?
f′=1
|Vνflf′|2FU(m2
νf,m2
lf′) −5
6
4
?
f=1
1
?
.(2.19)
uf denotes the up-type quark of the fth generation, df the down-type quark of the fth
generation, lf the charged lepton of the fth generation and νf the neutrino of the fth
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SS
-0.4-0.4 -0.3-0.3-0.2 -0.2-0.1-0.1000.10.10.2 0.20.30.30.4 0.40.50.5
TT
-0.4 -0.4
-0.3-0.3
-0.2-0.2
-0.1-0.1
00
0.10.1
0.20.2
0.30.3
0.40.4
0.50.5
68%, 95%, 99% CL fit contours
=120 GeV, U=0)
H
(M
[114,1000] GeV
∈
H
M
±
= 173.1
t
m 1.3 GeV
H
M
preliminary
G fitter SM
B
Dec 09
Figure 1: Fit of the electroweak oblique parameters S and T. The plot is taken from [97].
generation. We have used the following functions
FT(m2
1,m2
2) := 2
m2
m2
1m2
1− m2
m2
(m2
2
2
lnm2
m2
1
2
,(2.20)
FU(m2
1,m2
2) := 2
1m2
1− m2
2
2)2+
?
m2
1+ m2
1− m2
2
2(m2
2)−m2
1m2
(m2
2(m2
1− m2
1+ m2
2)3
2)
?
lnm2
m2
1
2
. (2.21)
Both functions are symmetric in their arguments.
The formula for S is very well known - see e.g. [63, 5, 11, 62]. Using instead the PDG
definition we would obtain the following corrections terms to S for heavy quark masses
(m2
q≫ M2
Z)
Scorr.
q
=Nc
6π
?M2
?M2
Z
3m2
b′
?
−1
2+23xW−4
9x2
W
?
+M2
3m2
Z
t′
?
??
−1
2+4 3xW−16
9x2
W
??
,(2.22)
Scorr.
l
= −1
6π
Z
6m2
ν4
+M2
2m2
Z
l4
?
−1
3+13xW− 3x2
W
, (2.23)
which are very small for the allowed mass ranges of the fourth family members. In the param-
eter T no mixing was usually assumed. We give here the full CKM and PMNS dependence.
Our expression for T in Eq. (2.18) agrees with the one quoted in [62], if we make the same
assumptions (only 4 − 3 mixing or 4 − 3 and 4 − 2 mixing is considered).
By defining the SM3 values for S and T as zero, we only need to take the additional con-
tributions due to the fourth generation into account. Keeping the full, previously neglected
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CKM dependences, we obtain
?
S4=
1
3π
2 + lnmb′ mν4
mt′ ml4
?
,(2.24)
T4=
Nc
16πxw¯ xwM2
Z
?
m2
b′ + m2
t′ −
4
?
f=1
4
?
f′=1
|Vufd′
f|2FT
?
m2
uf,m2
df′
?
+ FT
?m2
t,m2
b
??
+
1
16πxw¯ xwM2
Z
?
m2
l4+ m2
ν4−
4
?
f=1
4
?
+ |Vt′b|2lnm2
f′=1
|Vνfl′
f|2FT
?
m2
νf,m2
lf′
??
,(2.25)
U4= −Nc
6π
?
|Vt′d|2lnm2
t′
m2
d
+ |Vt′s|2lnm2
t′
m2
s
t′
m2
b
+ |Vub′|2lnm2
b′
m2
u
+ |Vcb′|2lnm2
b′
m2
c
−2|Vtb′|2FU
?
?m2
ν4
m2
e
t,m2
b′?− 2|Vt′b′|2FU
+ |Vν4µ|2lnm2
?m2
t′,m2
b′??
ν4
m2
−1
6π
|Vν4e|2lnm2
ν4
m2
µ
+ |Vν4τ|2lnm2
τ
+ |Vνel4|2lnm2
l4
m2
νe
+ |Vνµl4|2lnm2
l4
m2
νµ
+|Vµτl4|2lnm2
l4
m2
µτ
− 2|Vν4l4|2FU
?m2
ν4,m2
l4
??
−10
9π+ USM3.
(2.26)
S4has a large positive contribution of about 0.21 which is independent of the parameters
(masses and mixing) of the model. This value can, however, be diminished by the second
logarithmic term that depends on the fermion masses.
In the SM3 the only significant contribution to T reads
Nc
16πxw¯ xwM2
Z
Here safely Vtb= 1 can be assumed. In the SM4, however, Vtbcan in principle differ signifi-
cantly from one, therefore we have the correction term in “+FT(m2
T4. We also have included all previously neglected mixing terms within the SM3 particles.
In principle we also should correct for the charm-strange contribution and for the up-down
contribution with “+FT(m2
below one per mille of the top-bottom contribution, so we do not show these two additional
correction terms in the formula for T4.
With the help of the S and T parameter Chanowitz [62] could exclude the three parameter
sets, which we gave in [50] as an example for a very large mixing between the third and the
fourth generation; these sets have passed all bounds set by precision flavor observables. We
confirm the numbers from table I in [62]. We also tested the approximation of taking only
3 − 4 mixing into account: Comparing with the full CKM dependence the differences are
below 6 % for these three parameter sets.
To simplify the expression for U we approximated
2lnm2
m2
T4=
?m2
t− FT
?m2
t,m2
b
??.(2.27)
t,m2
b)” in the formula for
u,m2
d) + FT(m2
c,m2
s)” , but their numerical effect is considerably
FU(m2
1,m2
2) ≈ −1
1
2
for m2
1≪ m2
2. (2.28)
8
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Moreover we have only shown the contributions of the 4th family explicitly in Eq. (2.26), the
previously neglected rest is denoted by USM3. It will be interesting to see in a future analysis,
whether the large logarithms in the lepton sector will lead to strong constraints on the PMNS-
matrix. In the literature it is typically assumed that U is very small, see [5] for a notable
exception. To our knowledge we incorporate for the first time the full CKM dependence in
U. We find that arbitrary values for mixing and mass parameters could in principle generate
values as large as 7.5 for U4. If one only takes into account mixing parameters that pass the
tree-level flavor constraints still values of O(0.1) seem to be possible; however, in this case
we observe a simultaneous blow up of the T parameter. For T < 0.4, U does not exceed 0.06.
Note that, while still small, this value is larger than the 0.02 effect expected without flavor
mixing [11].
At this point a few comments are appropriate: first, we would like to point out that our
implementation of the S and T parameter is not “exact” from the SM4 point of view. In
principle one would have to perform a full reanalysis of all electroweak data from the SM4
perspective to fit the new values of S and T, as advocated for in [62]. This is, of course,
beyond the scope of the present work and it is generally accepted that a large deviation of
the oblique parameters from their SM values cannot be accommodated in models that do
not introduce new particles coupling to fermions [74]. Secondly, we will henceforth neglect
the effect of lepton mixing due to a non-trivial modification of the PMNS matrix — the off
diagonal elements including the fourth neutrino are in any case required to be small, see [45].
As a first step in our analysis, we only take the tree-level constraints on VCKM4into account
and investigate the parameter ranges that pass the S − T test at the 95% confidence level5.
The following values for the SM fit of the oblique parameters are used [14]6
Sbestfit= 0.03,σS= 0.09,(2.29)
Tbestfit= 0.07,σT= 0.08,(2.30)
ρcorr= 0.867 ,(2.31)
where σxgives the standard deviation of x. The S and T parameters are not independent
quantities; the strength of this correlation is given by ρcorr.
As our work focuses on the flavor aspects of the 4th generation scenario, a short comment
on the famous S-T-ellipses seems to be in order. If the U parameter stays close to zero for
some physics model it is feasible to set U = 0 and work with S and T alone. In this case the
probability distribution of S and T reduces to a two-dimensional Gaussian distribution in
the S-T plane centered on the best fit values. Due to the different σxand due to the strong
correlation the distribution is essentially squeezed and rotated. Hence, the “equiprobability”
lines are no longer circles but ellipses. In fact the two dimensional case is somewhat special
as the problem of finding the ellipse encircling an area corresponding to certain probability
P can be solved analytically. The equation determining the contour for a given confidence
4We did not check whether this is the largest possible value.
5For the final investigation of the allowed parameter range of a fourth fermion family we will use the 99%
confidence level.
6Note that this fit is the most restrictive one currently available. Using instead the results from Gfitter a
little more space is left for a fourth family. We simply decided to use the most recent numbers.
9
Page 11
level CL is then given by
?S − Sbest fit
T − Tbest fit
?T?σSσS σSσTρ
σSσTρ σTσT
?−1?S − Sbest fit
T − Tbest fit
?
= −2ln(1 − CL) .(2.32)
Already at that stage we find some interesting results:
1. S − T test with ‘no leptons’ + ‘no VCKM’
We do not take into account leptons as well as mixing of the quarks. By neglecting the
leptonic contribution one can, of course, not make any conclusions as to how restrictive
the oblique parameters are. However, we still find it instructive to consider the effect
of the various contributions in the S-T plane individually. The scatter plot shows the
accessable region within the 95% CL ellipse of [14]. We find, as expected that the masses
of the fourth quark generation can not be degenerate if they fulfill the constraints from
the oblique parameters. The necessary mass difference is of the order of 50 GeV as
stated in [11].
2. S − T test with ‘with leptons’ + ‘no VCKM’
Next, we include the leptonic contributions (without lepton mixing) and still neglect
CKM mixing. In this case also degenerate values of the quark masses of the fourth
generation are in principle not excluded; however, this would require a significant mass
gap in the lepton doublet to increase T (and preferably reduce S).
10
Page 12
The dark green points indicate values not in conflict with a degeneracy of the quark
masses of the fourth generation.
3. S − T test with ‘no leptons’ + ‘with VCKM’
To study the “unperturbed” effect of a non trivial CKM structure of the fourth gener-
ation, let us discard again the leptons for the moment. The modified CKM structure
results in an increase of the T parameter without changing S, cf. Eqs. (2.24) and (2.25).
In this scenario an increase of T can originate either from a quark mass splitting or
from nonzero mixing of the fourth generation with the SM quarks. However, a mass
splitting must also induce a tiny (logarithmical) contribution to S; so the central area
which could not be reached in scenario ’1’ corresponds to nonzero mixing and tiny mass
splittings.
4. S − T test with ‘with leptons’ + ‘with VCKM’
Here, we use the full expressions for S and T including both, leptons (without mixing)
and CKM mixing. In this scenario we find a maximally allowed mass splitting of
|mt′ − mb′| < 80 GeV for quarks and |ml4− mν4| < 140 GeV for leptons.
Note that this splitting was also observed by the Gfitter group [97]; their fits also show
11
Page 13
a minimal required mass splitting as they do not take the possible effects of a nontrivial
CKM structure into account. Due to the effects of quark mixing, we do not find a lower
bound for the splitting. In fact, a simultaneous degeneracy of quark and lepton masses
is not excluded, even though the S parameter favors larger t′and l4masses.
5. For completeness we also show the S-U plane using the exact expression for U and S.
Summarizing our investigation of the S, T and U parameters we get the following results:
• U is not a priori small; only after the constraints on the quark mixing and the T
parameter are used the maximal value for U is reduced below 0.06.
• The quarks of a 4th generation can be degenerated without violating the 95% CL
constraints from electroweak precision observables7. However, this requires taking into
account the effects of the non trivial flavor sector on T, i.e. mixing of the 4th generation
fermions, or a sufficiently large mass splitting in the lepton sector. Note that, at first
glance, this result seems to be in direct conflict with the standard statement that
a degenerate fourth generation is excluded at the 6 σ level [74] by virtue of the S
7For the 99% CL this statement will even hold stronger.
12
Page 14
parameter. However, this statement always tacitly assumed a trivial CKM structure.
The CKM factors in Eq. (2.25) can lead to T > 0 even if both lepton and quark masses
are degenerate. However, we did not investigate the effect of the Z →¯bb vertex, which
tends to favor small or no mixing. Still one can conclude that the situation for tiny
mass splittings or even degenerate masses drastically improves once mixing is taken
into account.
Finally, we also have to (re)consider the contribution of the Higgs particle, since in the
presence of a fourth family higher values of the Higgs mass may be possible [63].
correction terms to the S, T and U parameters read
The
SH=
1
12πln
M2
H
(117GeV)2, (2.33)
TH= −
3
16π¯ xW
ln
M2
H
(117GeV)2,(2.34)
UH≈ 0.(2.35)
Using that form for the Higgs contributions we implicitly subtract the used value for the
Higgs mass in the fit (117 GeV) and add “our” value.
A heavier Higgs increases S and lowers T. Instead of adding SH and TH to our values
for S and T, we subtracted the Higgs contributions from the fit values to make the diagram
easier to understand. So we shift the ellipse and not our data sets. We investigate the S and
T parameters for three values of the Higgs mass: 117 GeV, 250 GeV and 500 GeV.
The decrease in T is welcome, as it allows even bigger mass splitting (or alternatively larger
mixing); however, the simultaneous increase of S due to the heavy Higgs completely seems
to neutralize or even reverse this effect. Hence, as stated recently in [14] very large values for
the mass of a SM-like Higgs are clearly not favored. However, for a 250 GeV Higgs scenario
the origin (SM3) is outside the ellipse, whereas some SM4 points are inside and thus more
likely.
13
Page 15
Figure 2: Schematic diagrams for the necessary ingredients of the golden plated channel
Bd→ J/ΨKs: B-mixing, Kaon mixing and the decay process itself. The left panel shows the
tree-level and the right panel the penguin mediated decay. The dashed line represents any
current capable of creating a J/Ψ, e.g. two gluons.
2.3Flavor physics constraints — FCNC processes
After addressing the electroweak bounds, we turn to the constraints imposed by precision
observables of flavor physics involving a FCNC. One can hope to impose severe constraints
on the model by utilizing information from such processes as it is well known that the
weak interaction bypasses the Appelquist-Carazzone decoupling theorem [98]; thus, FCNC
processes are very sensitive to contributions of new physics.
However, the selection of flavor physics bounds on a hypothetical fourth family is a non
trivial issue. The reason for this is the fact that some processes known for being theoretically
or experimentally very clean, may in fact specifically require the SM3 setup. Hence, it is
always necessary to check, if a specific feature (of the SM3) crucial e.g. for the data analysis
is preserved in the fourth generation extension. If this is not the case, it may either be
necessary to repeat the analysis without some SM3 simplification — much like the need to
give up 3 × 3 unitarity — or the whole process may not even be feasible anymore.
As an example for how unexpected complications may arise (see also [99] for a more
detailed discussion), we discuss the so-called golden plated channel for the determination of
the Standard Model CKM angle β: Bd→ J/ΨKs[100]. This channel is renowned for being
theoretically very clean (in the SM3). Since the decay process is tree level dominated, it is
usually taken for granted that the contribution of the fourth generation quarks to the decay
is generally small. Therefore the SM4 could, in principle, be an explanation for discrepancies
of the measurement of sin(2β) in the Bd→ J/ΨKsand Bd→ ΦKs, as Bd→ ΦKsis penguin
dominated and as such more sensitive to new physics effects, see [101] for a more detailed
version of this argument. However, it turns out that this elegant picture of the consequences
of the fourth generation is, unfortunately, too simple. The reason for this is the following:
sin(2β) is extracted via time-dependent CP asymmetries. The necessary ingredients are (i) Bd
mixing, (ii) Kaon mixing and the (iii) decay process itself, see Fig. 2 for a schematic picture of
the relevant subprocesses. There are in fact two decay processes, the tree-level decay and the
top mediated penguin decay (c and u penguin are expected to be tiny). However, the beauty
14
Page 16
of this process in the SM3 is that the tree level decay and the t penguin have (to a fantastic
accuracy) the same CKM phase. Hence, it is not necessary to take into account e.g. different
hadronization effects as they will only modify the overall amplitude but not the phase. Adding
an additional generation has two new, separate effects. First of all, the expressions for the box
diagram changes (see formulae below), so that additional CKM factors contribute; therefore,
instead of the CKM angle β a different combination of CKM angles can be extracted from
this process. This is, however, not a problem as one could still use Bd→ J/ΨKsto constrain
the SM4. The real problem is the simultaneous modification of the penguin diagram by the
t′loop. As the t′will introduce some new virtually unconstrained phase, penguin and tree
decay now have different CKM phases; the fourth generation introduces a mismatch between
tree and penguin decay, which makes taking into account hadronization and QCD corrections
mandatory. Therefore, it is not clear what quantity can be extracted from time-dependent
CP asymmetries in Bd→ J/ΨKsin the SM4 scenario. This, of course, limits the usefulness
of this process for constraining the parameters of the model.
As the above example shows, not all processes can be used to obtain limits on the param-
eters of the SM4 and one has to be careful not to make use of a bound whose experimental
input essentially requires the SM3 setup or some SM3 specific feature.
2.3.1 FCNC constraints with no sensitivity to lepton mixing
Another issue is the sensitivity of some processes to the properties of the leptons. Since
we do not take mixing in the lepton sector into account and in fact assume lepton number
conservation, quantities that are insensitive to the precise structure of the lepton sector are
of course advantageous.
We first consider the mixing of the K-, the D-, the Bd- and the Bs-system.
completeness we repeat the relevant formulae already given in [50]. The virtual part of the
box diagram (here e.g. for Bd-mixing)
For
b
u,c,t(,t’)
W
W
d
d
W
W
b
u,c,t(,t’)
u,c,t(,t’)
u,c,t(,t’)
d
_
b
_
d
_
b
_
is encoded in M12, which is very sensitive to new physics contributions. It is related to the
mass difference of the heavy and light neutral mass eigenstate via
∆M = MBH− MBL= 2|M12|.(2.36)
15
Page 17
In the SM3 one obtains the following relations
MK0
12 ∝ ηcc
?
?
?λBs
λK0
c
?2
?2
?2S0(xt),
S0(xc) + 2ηctλK0
cλK0
t S(xc,xt) + ηtt
?
λK0
t
?2
S0(xt),(2.37)
MBd
12∝ ηtt
λBd
t
S0(xt),(2.38)
MBs
12∝ ηtt
t
(2.39)
with the Inami-Lim functions [102]
S0(x) =4x − 11x2+ x3
4(1 − x)2
?
−3x3ln[x]
2(1 − x)3,
1
1 − y−3
1
1 − x−3
(2.40)
S(x,y) = xy
1
y − x
1
x − y
?1
4+324
1
(1 − y)2
1
(1 − x)2
?
ln[y]
+
?1
4+324
?
ln[x] −3
4
1
1 − x
1
1 − y
?
,(2.41)
where xc,t=
m2
M2
c,t
W, the CKM elements
λK0
x
= VxdV∗
xs,λBd
x = VxdV∗
xb,λBs
x = VxsV∗
xb
(2.42)
and the QCD corrections [103, 104, 105]
ηcc= 1.38 ± 0.3,ηct= 0.47 ± 0.04,ηtt= 0.5765 ± 0.0065.(2.43)
The full expressions for M12can be found e.g. in [103, 106]. In deriving these expressions
the unitarity of the 3 × 3 matrix was explicitly used, i.e.
λX
u+ λX
c+ λX
t= 0. (2.44)
Moreover, in the B-system the CKM elements of the different internal quark contributions
are all roughly of the same size. Only the top contribution, which has by far the largest value
of the Inami-Lim functions, survives. This is not the case in the K-system. Here the top
contribution is CKM suppressed, while the kinematically suppressed charm terms are CKM
favored. Therefore, both have to be taken into account. More information about the mixing
of neutral mesons can be found e.g. in [106, 107].
We define the parameter ∆ as the ratio of the new physics model prediction (in our case
SM4) for a generic observable to the SM3 theory value; thus, it quantifies the deviation from
the standard model [106]. For M12one would then define:
∆ :=MSM4
MSM3
12
12
= |∆|eiφ∆.(2.45)
This representation is convenient as one can effectively map any observable to the complex ∆
plane; this allows a straightforward comparison of the sensitivity of the various observables
to the effect of the model. Experimental data can be mapped analogously by plotting˜∆ :=
16
Page 18
OExp/OSM3in the same complex plane. OExpdenotes the experimental value of a generic
observable and OSM3the theory prediction within the SM3. In the SM4, we obtain
MK0,SM4
12
∝ ηcc
?
λK0
c
?2
cλK0
S0(xc) + 2ηctλK0
cλK0
t S(xc,xt) + ηtt
?
λK0
t
?2
S0(xt) (2.46)
+2ηct′λK0
t′ S(xc,xt′) + 2ηtt′λK0
t λK0
t′ S(xt,xt′) + ηt′t′
?
λK0
t′
?2
S0(xt′),
MBd,SM4
12
∝ ηtt
?
?λBs
λBd
t
?2
?2S0(xt) + ηt′t′?λBs
S0(xt) + ηt′t′
?
λBd
t′
?2
?2S0(xt′) + 2ηtt′λBs
S0(xt′) + 2ηtt′λBd
t λBd
t′ S(xt,xt′), (2.47)
MBs,SM4
12
∝ ηtt
t
t′
tλBs
t′ S(xt,xt′). (2.48)
Note that now also those CKM elements change that describe the mixing within the first
three families! For simplicity we take the new QCD corrections to be (see also [66, 67])
ηt′t′ = ηtt′ = ηtt and ηct′ = ηct. (2.49)
It is interesting to note here that the only information we have currently about the CKM
elements Vtdand Vtscomes from B-and K-mixing plus assuming the unitarity of VCKM3.
The mass difference in the neutral D system
on |Vub′Vcb′|, see [108]8. We redid this analysis in [50] and softened the bound. The mass
difference in the neutral D0-system is typically expressed in terms of the parameter xD:
can be used to infer a very strong bound
xD=∆MD
ΓD
≤2|MD0
12|
ΓD
. (2.50)
For more information on the last inequality see e.g. the discussion in [110, 111]. HFAG [112]
quotes for the experimental value of xD
xD= (0.811 ± 0.334) · 10−2. (2.51)
The main difference compared to the above discussed K- and B-mixing systems is that in
the D-system the theory prediction in the SM3 is theoretically not well under control, see
e.g. [110, 111]. However, the pure contribution of a heavy fourth generation to M12can be
calculated reliably. Using the unitarity of the CKM matrix of the SM4 λD0
0 (with λD0
x
= VcxV∗
d+λD0
s+λD0
b+λD0
b′ =
ux), the full expression for M12reads
MD0
12∝
?
+2λD0
λD0
s
?2
sλD0
S0(xs) + 2λD0
sλD0
bS(xs,xb) +
?
λD0
b
?2
S0(xb) + LD
b′ S(xs,xb′) + 2λD0
bλD0
b′ S(xb,xb′) + LD
+
?
λD0
b′
?2
S0(xb′), (2.52)
8A similar strategy was recently used in [109].
17
Page 19
where the proportionality constant is
G2
FM2
12π2
WMD
f2
DBDη(mc,MW).(2.53)
We use the same numerical values as in [50]. The first line of (2.52) corresponds to the pure
SM3 contribution, the third line is due to contributions of the heavy 4th generation and the
second line is a term arising when SM3- and b′contributions mix:
MD0
12= MD0
12,SM3+ MD0
12,Mix+ MD0
12,b′ .(2.54)
The idea of [108] was to neglect all terms in MD0
with the experimental number for xD, since all perturbative short-distance contributions with
light internal quarks are negligible. Since it is not completely excluded that there might be
large non-perturbative contributions to both MD0
the size of the experimental value of xD, we get the following bound
12, except MD0
12,b′, and to equate this term
12,SM3and MD0
12,Mix(denoted by LD), each of
3MD0,Exp
12
≥ MD0
12,b′ .(2.55)
Allowing this possibility we obtain the following bounds on |Vub′Vcb′|
|Vub′Vcb′| ≤
0.00395 for mb′ = 200GeV,
0.00290 for mb′ = 300GeV,
0.00193 for mb′ = 500GeV.
(2.56)
This bound is still by far the strongest direct constraint on |Vub′Vcb′|.
Next, we consider the b → sγ transition.
of the FCNC decay b → sγ by simply looking at the product of CKM structure and the
corresponding Inami-Lim function D′
In [50] we approximated the treatment
0(xt) [102]9.
∆b→sγ:=|λSM4
t
|2D′
0(xt)2+ 2Re?λSM4
t
λSM4
t′
|λSM3
?D′
0(xt)D′
0(xt)2
0(xt′) + |λSM4
t′
|2D′
0(xt′)2
t
|2D′
,(2.57)
with
D′
0(x) = −−7x + 5x2+ 8x3
12(1 − x)3
x. We assumed that parameters which give a value of ∆b→sγ close to one will
also lead only to small deviations of Γ(b → sγ)SM4/Γ(b → sγ)SM3from one. However, this
crude treatment imposed a too strong bound on the 3 − 4 mixing.
In this work we will use the full leading logarithmic expression for b → sγ, see also [66, 67].
Following [113] we normalize the b → sγ decay rate to the semi leptonic decay rate
R :=Γ(b → sγ)
|Vcb|2
f(z = m2
note that in deriving this formula the unitarity of the 3 × 3 CKM matrix was already used
9The Inami-Lim function D′
0(xt) is proportional to the Wilson-coefficient C7γ(MW).
+x2(2 − 3x)
2(1 − x)4ln[x] (2.58)
and λx≡ λBs
Γ(b → ce¯ ν)=|V∗
tsVtb|2
6α
πf(z)
???C(0)eff
7γ
???
2
, (2.59)
c/m2
b) is a phase space factor, which we will not need later on. It is interesting to
18
Page 20
and the CKM combination λu= V∗
effective Wilson coefficient C(0)eff
C7and C8, which are accompanied by the CKM structure λtand the current-current Wilson
coefficient C2with the corresponding CKM structures λcand λu
usVubwas neglected (in comparison to λcand λt). The
is a linear combination of the penguin Wilson coefficients
7γ
C(0)eff
7γ
(µ)=Ceff1
7
(µ) + Ceff2
7
(µ) + Ceff3
7
(µ), (2.60)
Ceff1
7
(µ) = η
16
23C(0)
7γ(MW),(2.61)
Ceff2
7
(µ) =8
3
?
η
14
23− η
16
23
?
C(0)
8g(MW), (2.62)
Ceff3
7
(µ) =
8
?
i=1
hiηaiC(0)
2(MW),(2.63)
with
η(µ) :=αs(MW)
αs(µ)
. (2.64)
The values for hi and aiare given in Table XXVII of [113]. The initial conditions of the
Wilson coefficients read
C(0)
2(MW) = 1, (2.65)
C(0)
7γ(MW) = −1
2D′
0
?
?
xt=
m2
M2
t
W
?
?
, (2.66)
C(0)
8g(MW) = −1
2E′
0
xt=
m2
M2
t
W
. (2.67)
D′
0(x) is given above in Eq. (2.58), E′
0(x) reads
−1
2E′
0(x) = −2x + 5x2− x3
8(1 − x)3
−
3x2
4(1 − x)4ln[x]. (2.68)
Numerically it turns out that even for large values of mt (up to 1000 GeV) Ceff3
dominant contribution to C(0)eff
7γ
(µ). In [50] we have only taken Ceff1
therefore overestimated the effects of a fourth generation to the branching ratio of the decay
b → sγ. Putting everything together we get
∆b→sγ:=RSM3
RSM4
7
is the
7
into account and
(2.69)
=
????
VSM3
cb
VSM4
cb
????
2?????
λSM4
c
Ceff3
7
− λSM4
λSM3
c
t
(Ceff1
7
Ceff3
7
+ Ceff2
− λSM3
7
) − λSM4
(Ceff1
7
t′
(Ceff1′
7
+ Ceff2′
7
)
t
+ Ceff2
7
)
?????
2
.(2.70)
In [67] it was suggested to use the LO expression for b → sγ at a low scale of µ = 3.22 GeV in
order to reproduce the numerical value of the NLO expression. We have checked that ∆b→sγ
is quite insensitive to a variation of the scale between mband 3 GeV, so we use µ = mb.
19
Page 21
2.3.2 FCNC constraints with sensitivity to lepton mixing
Next we also discuss FCNC processes that are sensitive to lepton mixing. In principle lepton
mixing has to be investigated in the same manner as the quark mixing. For simplicity we have
neglected lepton mixing in this paper. However, we will take into account the conservative
bounds on Veν4, Vµν4and Vτν4given in [45] for the rare decay Bs→ µ+µ−. In the ratio of the
SM4 and SM3 predictions for the branching ratios almost everything cancels out and one is
left with the product of CKM elements and Inami-Lim functions
∆Bs→µµ:=BR(Bs→ µ+µ−)SM4
BR(Bs→ µ+µ−)SM3
??λSM4
=
t
Y0(xt) + λSM4
|λSM3
t′
Y0(xt′)??2
t
Y0(xt)|2
, (2.71)
with the Inami-Lim function
Y0[x] =x
8
?x − 4
x − 1+ 3
x
(x − 1)2ln[x]
?
. (2.72)
Including also the leptonic contributions we have to make the following substitutions [67] in
Eq. (2.71)
Y0(xt) → Y0(xt) − |Uµ4|2S(xt,xν4),
Y0(xt′) → Y0(xt′) − |Uµ4|2S(xt′,xν4),
(2.73)
(2.74)
where S is the box function given in Eq. (2.41), xν4is given by the mass of the fourth
neutrino and Uµ4is the PMNS matrix element describing the mixing between the µ and the
fourth neutrino. In [45] the bound Uµ4< 0.029 was derived. Using this information we find
that leptonic contributions give at most a relative correction of 0.5 per mille, so we can safely
neglect them.
The branching ratio for Bs→ µµ is not measured yet, HFAG quotes [114] (for the current
experimental bound from TeVatron, see also [115])
Br(Bs→ µ+µ−) < 3.6 · 10−8. (2.75)
In the SM3 one expects a value of [67]
Br(Bs→ µ+µ−) = (3.2 ± 0.2) · 10−9.(2.76)
2.3.3 Allowed ranges for the ∆ parameters
Now we come to a crucial point: the fixing of the allowed ranges for the values of the
different ∆s. For our exploratory study - in comparison to a full fit that will be performed
in future - we fix reasonable ranges for the ∆s. Therefore we have to investigate theoretical
and experimental errors. The FCNC quantities ∆Ms, ∆Md and b → sγ are dominated
by theoretical uncertainties. Currently, in particular the hadronic uncertainties are under
20
Page 22
intense discussion, see e.g. [116]. Therefore, we use conservative estimates for the theoretical
errors.
Bound
1 ± 0.3
0 ± 10◦
1 ± 0.3
free
1 ± 0.5
0 ± 0.3
1 ± 0.15
< 15
|∆Bd|
φ∆
Bd
|∆Bs|
φ∆
Bs
Re(∆K)
Im(∆K)
∆b→sγ
∆Bs→µµ
Since we choose for the central values of our ∆s the value one, all resulting allowed parameter
points for VCKM4will be scattered around the SM3 values by definition. For a future fit we
will use ∆s with the central value˜∆ = OExp/OSM3.
This means in other words that we do not take into account some current deviations in flavor
physics in our current analysis, we simply include them in our error band for the ∆s.
3Putting things together—Constraints on the param-
eter space
In order to constrain the mixing with the fourth quark family we perform a scan through
the parameter space of the model. To this end we use the exact parameterization of VCKM4
described in Sec. 2.1, Eq. (2.3). For the tree-level bounds we use the central values and
standard deviations as given in 2.2.1; we allow for a variation at the 2σ level. The restric-
tive Peskin-Takeuchi parameters are allowed to vary at the 99% confidence level of [14],
cp. Sec. 2.2.3. For the quark masses we use a hard lower limit of 280 GeV and allow for
a maximal mass difference of 80 GeV as determined in Sec. 2.2.3. The lepton masses are
chosen to be larger than 100 GeV with a maximal splitting of 140 GeV. For the FCNC we
use the ∆s given in Sec 2.3.3.
Then we generate a large number (O(1011) ) of randomly distributed points in the 13 dimen-
sional parameter space10. For each point we determine the value of CKM matrix elements,
flavor and electroweak observables and check whether the various experimental bounds are
passed (for details see [50]).
3.1Result for the mixing angles
Let us for the moment ignore correlations among the various parameters and focus on the
maximally allowed size of the mixing with the fourth generation. Table 1 shows the limits
on the mixing angles θ14, θ24, and θ34— without and with the electroweak bounds.
As already expected in Sec. 2, the Peskin-Takeuchi parameters impose strong constraints
on the mixing of Standard Model fermions with the fourth generation. These numbers are
comparable with the ones quoted in [62, 67]. The most dramatic effect is observed for the
mixing angle θ34. The maximal size is roughly halfed by the virtue of the T parameter alone.
So, already at this stage, one is able to conclude that a study of the flavor aspects of SM4
10mt′,mb′,ml4,mν4,θ12,θ23,θ13,θ14,θ24,θ34,δ13,δ14,δ24.
21
Page 23
only tree-level
< 0.07
< 0.19
< 0.8
with FCNC bounds
< 0.0535
< 0.145
< 0.67
with electroweak observables
< 0.0535
< 0.121
< 0.35
θ14
θ24
θ34
Table 1: Maximal mixing of SM fermion generations with the fourth generation. The left
column show the effect of tree-level bounds alone, for the central column FCNC bounds where
added and the right-most column gives the limits including electroweak parameters.
Figure 3: Some correlations of the angles and phases.
must not be decoupled from a simultaneous analysis of the electroweak sector.
To illustrate the dependence of T on θ34we also show the scatter plot for T versus θ34:
Next, the correlations between the different angles are examined. The results are depicted
in Fig. 3. Obviously, the maximal mixing angles given in Table 1 cannot be simultaneously
realized; especially θ14 and θ24 show a rather strong correlation and maximal θ24 is only
possible for θ14close to 0.018. Note e.g. the θ14−θ34correlation; simultaneous large mixing
angles θ34 and θ14 are also disfavored. Indeed, this observation is rather natural, as Vtd
22
Page 24
includes a term c12c13c24s14s34eiδ14. Hence, simultaneous large s14and s34would lead to a
large modification of Vtd; Bdmixing would be sensitive to this and indeed proves to be the
most restrictive of the mixing observables. Due to this observation we disfavor a strategy
based on starting from fixed bounds on the mixing angles without taking the correlations
into account.
The large ’voids’ in the δ13− δ14plane can be traced to the effect of the direct limit on the
phases due to the CKM angle γ.
3.2 Results for the CKM elements
Since the 3 × 3 unitarity fixes the values of the second and third row CKM elements rather
precisely in the SM3, it is interesting to see the effect of the lifting of the unitarity constraint
on Vcd, Vcs, Vcb, Vtd, Vtsand Vtb. In Figure 4 we present the possible values for these CKM
matrix elements in the complex plane. Note that a CKM matrix element itself is not a
physical observable as it depends on phase conventions and CKM parameterization; one can,
however, compare the values for the elements once the representation and phase convention
is fixed. The plots correspond to the standard represention, cp. (2.1), which is the limit of
the Botella-Chau representation for zero mixing with a fourth family.
The absolute value of the elements of the second row cannot change much with respect to
the SM3; however, it is interesting to observe that the imaginary part of Vcdand Vcscan be
increased by an order of magnitude. This might be potentially interesting for searches for
CP violation in the charm sector. While Vcbdoes not have an imaginary part in the standard
representation in SM3, a tiny imaginary part can be present in SM4.
The absolute value of both, Vtdand Vts, can be modified (with respect to their SM3 value)
by approximately a factor 2. More important, the imaginary part of Vtscan be an order of
magnitude larger than in the SM. Therefore, one can expect that the weak phases of processes
involving Vts, e.g. Bsmixing, may experience large corrections.
The absolute value of Vtb can be as low as 0.93. Without the constraints coming from
oblique parameters this limit would be much lower — around 0.8. This again shows that the
electroweak sector imposes strong limits on the flavor structure of SM4.
This number is in particular interesting since it can be compared with direct determinations
of Vtbfrom Single-top-production from TeVatron [117, 118, 119]
VTeVatron
tb
= 0.88 ± 0.07. (3.77)
3.3New physics in Bs-mixing
The results for the complex ∆Bsplane is particularly interesting since there might be some
hints on new physics effects in the CP-violating phase of Bsmixing, see [106, 120] and the
web-updates of [82]. In [106] a visualization of the combination of the mixing quantities ∆Ms,
∆Γs, as
of Φsin the complex ∆-plane was suggested. Combining recent measurements [114, 124] for
the phase Φsone obtains a deviation from the tiny SM-prediction [106] in the range of 2 to
3 σ:
sl, which are known to NLO-QCD [103, 121, 122, 123] and of direct determinations
• HFAG: 2.2 σ [114],
• CKMfitter: 2.1...2.5 σ [125, 126],
23
Page 25
Figure 4: Possible modifications of the SM3 CKM matrix elements Vcd, Vcs, VcbVtd, Vtsand
Vtbin the SM4 scenario. Depicted is the real part versus the imaginary part of the CKM
element (in the standard representation). The crossed lines show the SM3 value; Vcbis real
by construction in the SM3.
• UTfit: 2.9 σ [120].
The central values of these deviations cluster around
Φs≈ −51◦. (3.78)
Very recently the D0 collaboration announced a 3.2σ deviation of a linear combination of ad
and as
large negative value of Φs.
As can be read off from the left picture of Figure 5 sizeable values for Φscan also be obtained
in scenarios with additional fermions. Such large values for Φs are not favored, but they
are possible. An enhancement of Φs to large negative values by contributions of a fourth
generation was first predicted in [40], by choosing the parameters of the fourth generations
in such a way that other flavor problems like the B → Kπ puzzle are solved. Once the T
parameter is implemented with full CKM dependence and used as an additional bound on the
CKM elements, a large value of Φsseems to be very unlikely and requires a significant fine
tuning of the parameters, see the right picture of Figure 5. In that respect we differ slightly
from the conclusion of e.g. [40, 66, 67, 128], where very large values for Φsare allowed11.
Here we expect new and considerably more precise data from TeVatron and LHC soon. If the
central value stayed at the current position, the possibility would arise to find new physics
that can not originate from an additional fermion family alone.
It is interesting to study the mass dependence of the phase Φs. Already in [128] it was
noted that large phases clearly favor small t′(and b′) masses and the largest phases require
a value of mt′ close to 300 GeV; this behaviour is also present in our analysis and it seems
sl
sl[127] from the standard model prediction in [106]. This deviation also indicates a
11If we also described the problems in e.g. B → Kπ by a fourth family, we also would exclude the points
with Φsclose to zero and we would predict a sizeable phase - around −20◦- but no points with e.g. −50◦
survive in our analysis. Because of hadronic uncertainties we did not include B → Kπ in our analysis.
24
Page 26
Figure 5: Complex ∆ plane for Bsmixing. The green shaded area corresponds roughly to
the experiment at the 1σ level [114]. The left panel shows the possible phase if one omits the
T parameter, the right panel shows the impact of T on the allowed phase Φs.
that too large quark masses struggle to resolve the tension in the flavor sector should future
experiments confirm e.g. a phase of the order of 30◦. In Fig.6 we show the distribution of the
scatter plot for “light” (< 440 GeV) and “heavy”(> 440 GeV) t′masses12.
As a final remark, we would like to point out that the naively expected huge contributions
to ∆Bsdue to the heavy t′can (still) be veiled by the mechanism described in [50]. The new
contributions in the SM4 fall into two classes. The first class contains the ’direct’ effects
of the new heavy ferminons, i.e., the contributions of diagrams with at least one t′in the
loop. The second class is more subtle as it includes the ’indirect’ effects of the SM4 scenario.
These contributions arise from the breaking of the 3×3 unitarity; the standard model CKM
elements have to be ’thinned’ out to accomodate for the non-zero values of the fourth row
and column matrix elements: this results in a modification of the SM-like contributions.
These two sets of contributions are both sizeable (typically large enough to violate at least
one experimantal bound), but they can cancel to a very large extent. Here is one example
for the case of ∆Bswhich survived all our constraints:
θ12
θ13
θ23
θ14
θ24
θ34
0.22750.0034090.040360.017010.083920.1457
δ13
1.019
δ14
δ24
mt′
mb′
ml4
mν4
0.918251.0787 385 GeV378 GeV602 GeV636 GeV
This parameter set yields
∆Bs= 1 + 0.5379 − 0.9016i
?
???
direct
−0.4196 + 0.4262i
? ???
indirect
= 1.2 e−i23◦. (3.79)
One sees that each contribution is of the size of the Standard model result; however, direct
and indirect contribution have opposite signs and cancel to a large extent. Note that in this
case the 3-4 mixing is strong enough to allow for almost degenerate quark masses.
12Note that the points corresponding to “heavy” t′quarks are placed on top of the one corresponding to
“light” t′s. A light t′and a simultaneous small phase Φsare, of course, still possible.
25
Page 27
Figure 6: Dependence of ∆Bson the t′mass. The dark blue points correspond to masses
below 440 GeV, the yellow (light grey) point correspond to masses heavier than 440 GeV.
The red line indicates a phase Φsof 30◦.
4 Taylor expansion of VCKM4
In [50] we gave a Wolfenstein-like expansion of the CKM matrix for four fermion generations,
which allows for a first estimate of possible effects of the fourth generation. Since the inclusion
of the electroweak oblique parameters led to tighter constraints for the mixing, one can now
further improve this expansion.
In the SM3 the hierarchy of the mixing between the three quark families can conveniently
be visualized via a Taylor expansion in the small CKM element Vus ≈ 0.2255 = λ: the
Wolfenstein parameterization [129].
Following [130]13we define
Vub= s13e−iδ13=: Aλ4(˜ ρ + i˜ η), (4.1)
Vus= s12(1 + O(λ8)) =: λ,
Vcb= s23(1 + O(λ8)) =: Aλ2.
(4.2)
(4.3)
For the case of 4 generations one also needs the possible size, i.e. the power in λ of the new
CKM matrix elements. Using the results of the previous section we find:
|Vub′| ≤ 0.0535 ≈ 1.05λ2
|Vcb′| ≤ 0.123 ≈ 0.54λ1≈ 2.38λ2
|Vtb′| ≤ 0.35 ≈ 1.55λ1
13Note that due to historical reasons the element Vubis typically defined to be of order λ3, while it turned
out that it is numerically of order λ4.
(1.05λ2),
(2.8λ2),
(3.0λ1).
26
Page 28
In brackets we show the results of our previous analysis in [50]. The biggest effect of the
inclusion of the electroweak precision constrains was the reduction of the allowed mixing
between the third and fourth family. The mixing between the first and the fourth family
can still be considerably larger than the mixing between the first and the third family. This
bound is still dominated by D-mixing.
• Defining the Vub′ as
Vub′ =s14e−iδ14=: λ2(x14− iy14),
one obtains
⇒s14= λ2?
⇒c14= 1 − λ4x2
x2
14+ y2
14,
14+ y2
2
14
+ O?λ8?
. (4.4)
The parameters x14and y14are effectively smaller than or equal to 1 for all cases.
• Let us further define the matrix element Vcb′ via
Vcb′ = c14s24e−iδ24=: (x24− iy24)λ1.(4.5)
Comparison with Eq. (2.3) then gives
⇒ s24e−iδ24= (x24− iy24)λ + O?λ5?,
⇒ c24= 1 +1
2
?−x2
24− y2
24
?λ2+ O?λ5?. (4.6)
• the last ingredient is the element Vtb′:
Vtb′ = c14c24s34=: Bλ (4.7)
and therefore
sin(θ34) = Bλ +1
2λ3?Bx2
+1
24+ By2
24
?+ O?λ5?,
cos(θ34) = 1 −B2λ2
28λ4?−B4− 4B2x2
24− 4B2y2
24
?+ O?λ5?. (4.8)
Expanding the CKM4 matrix up to and including order λ4, the matrix elements take the
form :
Vud= 1 −λ2
2+18λ4?−4x2
14− 4y2
14− 1?+ O?λ5?,Vus= λ,
Vub= A(˜ ρ − i˜ η)λ4,Vub′ = (x14− iy14)λ2,
(4.9)
27
Page 29
Vcd= − λ +1
2λ3?x2
2λ2?−x2
8λ4?−4A2− 2x2
Vcb=Aλ2,
24+ y2
24
?− (x14+ iy14)(x24− iy24)λ4+ O?λ5?,
24− 1?+
?y2
Vcs=1 +1
24− y2
+1
24 24− 1?− x4
24− y4
24+ 2y2
24− 1?+ O?λ5?,
Vcb′ =λ2(x24− iy24) , (4.10)
Vtd=λ3(A − Bx14− iBy14) + (−iAη − Aρ + Bx24+ iBy24)λ4+ O?λ5?,
Vts= − Aλ2− Bλ3(x24+ iy24) +1
2λ4?AB2− Ax2
24− Ay2
24+ A − 2Bx14− 2iBy14
?+ O?λ5?,
Vtb=1 −B2λ2
2
+1
8λ4?−4A2− B4− 4B2x2
24− 4B2y2
24
?+ O?λ5?,
Vtb′ =Bλ,(4.11)
Vt′d=λ2(−x14− iy14) + λ3(x24+ iy24)+
+1
2λ4?−2AB + (x14+ iy14)?B2+ x2
Vt′s=λ2(−x24− iy24) + λ3(AB − x14− iy14) +1
24+ y2
24
?+ x14+ iy14
?B2+ 1?λ4(x24+ iy24) + O?λ5?,
?+ O?λ5?,
2
Vt′b= − Bλ −1
2λ3?B?x2
2λ2?−B2− x2
8λ4?−B4− 2B2x2
24+ y2
24
??− Aλ4(x24+ iy24) + O?λ5?,
?+
24− 2B2y2
Vt′b′ =1 +1
24− y2
24
+1
24− 2x2
24y2
24− x4
24− 4x2
14− y4
24− 4y2
14
?+ O?λ5?.
(4.12)
Naturally, this expansion cannot take into account correlations among the various CKM
elements. However, the expansion is quite useful if one wants a rough estimate of the maximal
size of a product of CKM elements determining the impact of the fourth generation on a
certain process.
5 Conclusion
We have investigated the experimentally allowed parameter range for a hypothetical 4×4
quark mixing matrix. Therefore we extended our previous study [50] of the CKM like mixing
28
Page 30
constraints of a fourth generation of quarks.
Besides the tree-level determinations of the 3×3 CKM elements we also included the angle
γ of the unitarity triangle, which turned out to be a rather severe bound for the phases of
VCKM4. Next we included the electroweak S, T, U parameters. Here we reproduced some
of the results from [62], in particular we also excluded the three examples of very large
mixing presented in [50]. In this paper we have included for the first time the full CKM
dependence of the T and the U parameter. Doing so, we found that a mass degenerate
fourth family of quarks is not excluded in that respect we differ e.g. from [11, 74]. While
degenerate quark masses can also arise if the lepton masses are adjusted accordingly, see
Sect. 2.2.3 and [14], including the full CKM dependence allows for a greater ’flexibility’ in
the parameter space: e.g. only then a simultaneous separate degeneracy of leptons and quarks
of the fourth generation would not be excluded — this may be of interest if one wants to
invoke a symmetry to motivate the tiny mass splittings. In addition we found also that large
values of the parameter U are not excluded a priori; only after applying the T parameter
constraint we are left with small values of U.
Concerning the FCNC constraints we studied K-, D-, Bd-, Bs-mixing and the decay b → sγ.
In contrast to [50] we also included bounds to the rare decay Bs→ µ+µ−and we improved
our treatment of the QCD corrections to b → sγ. It turned out that the naive bound for
b → sγ used in [50] was too restrictive.
Performing a scan over the whole parameter space of the SM4 we found that typically small
mixing with the fourth family is favored, but still some sizeable deviations from the SM3
results are not yet excluded. We demonstrated explicitly that e.g. effects of O(100%) in Bs
mixing are not excluded, yet. Concerning CP-violation in Bsmixing, we could have an almost
arbitrarily large phase Φswithout violating the tree-level constraints. After switching on the
T constraint (99 % CL of [14]) we could exclude values of the O(−50◦) for the weak mixing
phase Φs, while values of O(−20◦) can easily be obtained - this is still about two orders of
magnitudes larger than the SM3 prediction Φs= 0.24◦±0.08◦[106]. In that respect we differ
slightly from [40, 66, 67, 128]. If the real value of Φswas −50%, this could not be achieved
by an extension of the SM3, which consists only of an additional fermion family14. Here new
results for the B-mixing observables from TeVatron and LHCb are very desireable.
We found a minimal possible value for Vtbof 0.93 within the framework of the SM4, which
can be compared with the result from single top production at the TeVatron [117, 118, 119]:
VTeVatron
tb
= 0.88±0.07. If in nature a central value of Vtb= 0.88 was realized, this could not
be explained by the SM4 alone. Here also more precise experimental data are desireable.
In general we found a delicate interplay of electroweak and flavor observables, which strongly
suggests that a separate treatment of the two sectors is not feasible.
In our opinion the next steps to determine the allowed parameter space of the SM4 consist
of i) performing a combined electroweak and CKM fit, ii) including lepton mixing and iii)
including even more precision observables like e.g. Rbor B → K∗ll.
Acknowledgements
We thank Heiko Lacker, Jens Erler and M. Vysotsky for clarifying discussions and Johann
Riedl for providing us the computer code from our previous project. Moreover, we thank
Thorsten Feldmann for clarifying discussions concerning Ref. [67]. This work was supported
14This statement holds only if the current allowed range for the S and T parameters will not change.
29
Page 31
in part by the DFG Sonderforschungsbereich/Transregio 9 “Computergest¨ utzte Theoretische
Teilchenphysik”.
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