Implications of b →sγ in the Weinberg ThreeHiggsDoublet Models
Darwin CHANG
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ChuanHung CHEN
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ChaoQiang GENG
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Department of Physics, National Tsing Hua University
, Hsinchu,
Taiwan 30043
Using recent experimental measurements on Br(b..... sr) from CLEO, we study the constraints on the charged Higgs sector in various threeHiggsdoublet models. Some phenomenological implications in these models with emphasis on CP violation are presented. In particular, in some of these models; the CP violating muon polarization in Kp3 can be detected using the current KEK experiment E246.

The matrix U+ can be parameterized in a manner similar to the CKM matrix:8)
D. Chang, c..B. Chen and C.Q. Geng
2. Various types of Weinberg threeHiggsdoublet models and their constraints
In SU(2)L X U(I) models with three Higgs doublets, denote the vacuum expecta
tion values (VEVs) of cPi by vie i9;, i=l, 2, 3. The weak eigenstates of charged Higgs
cP/ are related to the mass eigenstates H/ by a unitary matrix U+:
with Vl=ClV, V2=SlC2V, and V3=SlS2V, where Ci=COS()i, si=sin()i (i=I, 2, 3), and 8H are
the charged Higgs mixing angles and CPviolating phase, respectively.
In order to study how the present experimental data constrains the WTHDMs, we
will consider two distinct models (I and II) that naturally avoid treelevel flavor
changing neutral current (FCNC).9) In Model I, one doublet cPl couples to the quark
sector, and cP3 to the lepton sector, but cP2 decouples from the fermions at tree level.
In Model II, (Pt, cP2 and cP3 couple to downtype, uptype quarks and leptons, respective
ly. The Yukawa interactions in terms of the physical charged Higgs bosons are
given by3)
where K is the CKM matrix; and U, D, Nand E are fields for up quarks (e.g., U T
=(u, c, t, down quarks, neutrinos and charged leptons with M u, MD and ME being
the corresponding diagonal mass matrices, respectively. In general, Xi, Yi and Zi are
complex numbers that depend on mixing parameters in the charged Higgs sector.
We assume that H2+ has a much larger mass than Hl + and consider just the light one,
i.e., MH,'}pMH1=MH, since it gives the dominant contribution. We also assume that
the observed CP violation in K > me decays will be accounted for dominantly by the
standard CKM mechanism.
In Model I, from Eqs. (1) and (3), we find that
X=Y=SlC3/Cl,
That is, in this basis, both X and yare purely real numbers, but Z contains a phase
factor. Thus, if we want to get a CPviolating process, it must be semileptonic, such
as the CP violating transverse muon polarization in K+> 7l'0fl+ 1)" decay which can be
Implications of bsr in the Weinberg ThreeHiggsDoublet Models
related to Im(xz*) or Im(yz*).lO)
In fact, it is easy to conclude in Model I that the charged Higgs mediated CP
violation will not contribute to purely nonleptonic or purely leptonic processes. As
has been emphasized repeatedly in the literature for the case of CKM mixing matrix,
the CPviolating combination should be proportional to Im( Vik Vii Vi! Vit) for the case
of three generations. This implies that more than one species of the charged Higgs
bosons must be involved in the process for CP violation to occur. In Model I, only
q1t is involved for the nonleptonic processes and only CP3 is involved for the purely
leptonic processes, and therefore no CP violation from charged Higgs exchange is
possible until very high order in the perturbation series.
In Model II, both CPl and CP2 can be involved in the nonleptonic processes. In
particular, in the basis of Eq. (3), we have
In this model, only one of x, y and z can be made real in general. Hence, nonleptonic
CPviolating quantities, such as the (NEDM), indirect CPviolating parameter and
direct CP violating parameter ', can be induced and should be proportional to
Im(xy*); similarly, CP violating observables in semileptonic processes, such as
K+_7r0f.l+ lip, can also exist and should be related to Im(xz*) or Im(yz*), as before.
The relationships of Im(xy*) with Im(xz*) and Im(yz*) can be written as
y=( ClC2C3+s2s3e i8H )/(SIC2) ,
z= (ClS2C3 + c2s3e i8H) / (SIS2) .
Im(xz*)=( ~: YIm(xy *) ,
Im(yz*)=( ~: YIm(xy *) .
To illustrate the bound on IYI, we choose MH~50 GeV and take mt=174 GeV, MB~5.3
GeV, IVtdl~0.008, IVtbl~l, fBBB1/2~0.18GeV and L1MB=2IMI21~3.36x1013GeV.
With this set of parameters, we obtain
It is interesting to note that the value of IYI is insensitive to the mass of the charged
Higgs boson for mt~174 GeV. The only potential constraint on Ixl comes from D
 fj mixing. However, the current experimental bound with large uncertainties in
the CKM elements is not useful. The value of Izl can be constrained by e  fl
universality in r decays,14),15)
for MH~200GeV.
1X104 < Br(b>sy)<4x104
This new experimental data can give a stringent constraint on new physics beyond
SM.
The branching ratio of b>sy decay in the WTHDMs, including QCD correction,
is given by
D. Chang, G.H. Chen and G.Q. Geng
where a= m/ /MH2 and
if Izl is smaller than the perturbative limit. Thus, from Eqs. (11) and (12), we find
For Model II, the direct bounds on Im(xy*) from the experimental limit on the EDM
of the neutron are given by
F ps is a phase space factor, 7J1 and 172 are QCD corrections factors, xt==mt2 /Mw 2, Yt
== mt2/MH2, and
Implications of b+sy in the Weinberg ThreeHiggsDoublet Models
Fig. 1. The upper bound on Iyl as a function of the
lightest charged scalar mass MH The three
curves correspond to m,=140 (solid), 174 (dot
ted), 190 (dashed) GeV.
Fig. 2. The upper bound on IIm(xY*)1 as a function
of the lightest charged scalar mass MH The
three curves correspond to m,=140 (solid), 174
(dotted), 190 (dashed) GeV.
12(I=x)4 [(75x8x2)(Ix)+x(1218x)lnx] ,
which increases for larger M H Therefore, we see that, in Model I, the bound in Eq.
(19) derived from the limits b+sr is compatible with that from the B 13 mixing given
in Eq. (11).
In Model II, using Eqs. (15)~(18), the upper bound on IIm(xy*)1 from the branch
ing ratio of b+sr decay can be derived to be
The results are presented in Fig. 2. Explicitly we have, with mt ~ 174 GeV,
IIm(xy*)1 < 1.7 for MH ~ 50 GeV ,
<3.3 for MH~200 GeV .
From these results, we can see that the measumement of b+sr decay gives a stronger
constraint on IIm(xy*)1 than that from the direct contribution of the neutron EDM in
Eq. (14).
Now, one may conclude that the measurement of the inclusive branching ratio of
the b+sr decay in CLEO gives a more strict constraint on Model II than that on
ModelL
D. Chang, c.R. Chen and C.Q. Geng
3. Consequences in semileptonic processes
We now utilize the constrained parameters presented in the previous section to
investigate the contributions to the semileptonic processes due to the various types of
charged Higgs WTHDMs. To illustrate our results, in particular on Model I, we will
concentrate on quantities involving semileptonic decays such as the T violating
transverse muon polarization in K+> 7(0f..L+ v" and the branching ratio of B> X rv,.
The first one is of particular interest because of the KEK experiment E246 in progress.
Since the CLEO measumement on b>s/, decay gives different constraints on the
parameters in Models I and II, we will discuss the effects separately.
For the muon polarization in K+> 7(0f..L+ v", we use10)
where BrSM(B>Xrv,)=(2.300.25)%, R=(mbm,xz*)/MH 2, and I and J are phase
space factors, defined by Ref. 12)
Br(B> X rv,) = BrSM(B> X rv,)
P/(K+> 7(0f..L+ v,,) < 2.6 x 103 ,
and the branching ratio of the semileptonic B decay
Implications of bsr in the Weinberg ThreeBiggsDoublet Models
respectively. Basically, the charged Higgs contribution to Br(B X rll,.) is not only
related to Im(xz*) but to the mass of charged Higgs. According to the result of Eq.
(23), the contribution of the charged Higgs is proportional to (Im(xz*)/MH2)2. Unfor
tunately, even with the current lower charged Higgs mass limit of 45 GeV, the
contribution of the charged Higgs boson to Br(B X rll,.) is already suppressed.
Model II: In order to obtain predictions in Model II, we will encounter the
undetermined ratios of the VEVs. However, there exist some plausible patterns,
referred to as scenario I and scenario II, discussed in the literature. 19),20) In scenario
I the VEVs are roughly comparable in magnitude, i.e., VI ~ V2 ~ V3, and in scenario II
the VEVs of the three Higgs bosons are proportional to the masses of the fermions to
which they couple, respectively, i.e., Vl:V2:V3~ ms:mc:mp..*)
As indicated in Eq. (7), the bounds on Im(xz*), Im(yz*) are related to those on
Im(xy*) given by Eq. (21). So, we have
Im(xz*) ~ Im(xy*) (scenario I) ,
Im(xz*) ~ 196 Im(xy*) (scenario II) .
Combining Eqs. (21) and (28), we find
Pf(K+Jr0.u+ lip.) < 3.5 X 105 (scenario I) ,
Pf(K+Jr.u+Ilp.)<7.0xl03 (scenario II)
Br(BXrllr)~BrSM(BXrllr) (scenario I),
Br(BXrllr)<3.0 x 102 (scenario II) ,
taking MH =50 GeV. It is interesting to note that, by choosing different scenarios, the
results in Model II, as shown in Eq. (29) for the muon polarization of K+Jr.u+llp., can
be significantly different. The charged Higgs contribution to Br(B X rllr) shown in
Eq. (30) is small in both scenarios I and II. The main reason is due to the strict bsr
constraint on Im(xz*).
Note that in both models, the potential value of Pf(K+Jr.u+Ilp.) could be larger
than the value 5 x 104 that E246 of KEK proposes to measure. Even if the measure
ment results only in a new upper bound of the polarization, it can already put new,
interesting limits on parameters of the charged Higgs sector of most of the THDM's.
4. Concluding remarks
In this paper we have studied the constraints on some three Higgs doublet models
from the measurement of bsr. We have found that this measurement gives a
stronger constraint on IIm(xy*)1 in Model II than that from the neutron EDM as shown
in Eq. (21), but in Model I it gives a weaker constraint on Iyl. This is compatible with
*) Usually, one uses the mass ratios of the third generation as the ratios of the VEVs in scenario II to
get large values, but here we take the ratios among the second generation since the consequences of the large
third generation ratios due to the topquark mass are ruled out by experiments.
the constraint from other processes such as that from the B  B mixing shown in Eq.
(11).
We have shown that although there is an enhancement, the range of Br(BXn/r)
in both models could always be within the predicted range for the standard model due
to the current lower charged Higgs mass limit and the bsr constraint on Im(xy*).
However, the value for T violating muon polarization in K+ 1(0f1.+ 1/1' could be
large in both Models I and II and it is accessible to the ongoing T violation experi
ment of E246 at KEK.2l) This experiment could tell us which variant of Weinberg's
threeRiggsdoublet models can be accepted and what kind of the ratios of the VEVs
we should choose.
Acknowledgements
This work is supported in part by the National Science Council of the Republic
of China under grants NSC842112M007016 and NSC842112M007041.