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Rare B Knn g decay in two Higgs doublet model (*)

Civil Engineering Department, Near East University - Lefkos¸a, Mersin, Turkey (ricevuto il 6 Giugno 1997; approvato il 14 Luglio 1997)

Summary. — The rare BsK n n g decay is analysed in the context of 2HDM. It is shown that for large values of ctg b the contributions of 2HDM exceeds considerably the SM ones.

PACS 14.80.Bn – Standard-model Higgs bosons.

1. – Introduction

The theoretical and experimental investigations of the rare decays has been a subject of continous interest in the existing literature. The experimental observation of the inclusive b Ksg and exclusive BKK* g [1] decays stimulated the study of rare B-meson decays on a new footing. These decays take place via flavor-changing neutral currents (FCNC) b Ks transitions which are absent in the Standard Model (SM) at tree level and appear only at the loop level. Thus the study of these decays can provide sensitive tests for investigating the structure of SM at the loop level, and they represent the promising objects for establishing “new physics” beyond the standard model [2]. Moreover, the investigation of these rare decays permits precise determination of the fundamental parameters in the SM, which are poorly known at present, such as the Cabibbo-Kobayashi-Maskawa matrix elements [3] and the leptonic decay constants fBs, fBd.

Currently, the main interest of rare B-meson decays is focused on decays for which the SM predicts relatively large branching ratios and can be potentially measurable in the future B-factories. The B Knng is one example of such decays. This process represents an interesting model for the following reasons. Firstly, from the experimental point of view it has a very clear signature, i.e. “missing energy” and isolated photon. Secondly, from the theoretical point of view, the branching ratio is dependent quadratically on the leptonic decay constant fB. Therefore, the investigation

of this decay gives us an alternative way for determining the leptonic decay constant fB.

Besides, an interesting peculiarity of this decay is that the QCD corrections to this decay practically equal zero.

(*) The author of this paper has agreed to not receive the proofs for correction.

As is well known, the SM predicts the branching ratio of the above-mentioned decay of the order A1028_–1029 _{and it will be quite measurable in future B-factories. In}

general, the FCNC is very sensitive to extensions of the SM and provides a unique source of constraints on some new physics scenarios which predict large enhancement of B Knng decay mode (for review see [4] and references therein).

In the present work, we study the process Bs( Bd) Knng in the framework of the

two Higgs doublet model. In sect. 2, the relevant effective Hamiltonian for the BsK

n n g is presented. In sect. 3, using the light-cone QCD sum rules results for estimating

form factors, we calculate the dependence of the branching ratio on the photon energy. Then the branching ratio of Bs( Bd) Knng is estimated and a brief discussion of the

results is given.

2. – Effective Hamiltonian

Firstly, we consider the quark level process b Kqnn ( q4s, d). The relevant Feynman diagrams are displayed in fig. 1. The first three diagrams describe the SM and the last three represent the 2HDM contributions to the b Ksnn decay due to the charged H2 _{boson exchange. In the calculations we will use the so-called model II}

which appears in the two Higgs doublet or minimal supersymmetric version of SM [5]. In this model the interaction Lagrangian of fermions with the charged Higgs fields is:

L4(2k2GF)1 /2[ tg b ULVCKMMDDR1ctg b URMUVCKMDL1tg b NLMEER] H11h.c.

(1)

Here H1 _{is the charged physical Higgs field. U}

L, DR represent left-handed up and

right-handed down quark fields. NL, ER are left-handed neutral and right-handed

charged leptons. MD, MUand ME are the mass matrices for the down, up quarks and

charged leptons, respectively. VCKM is the Cabibbo-Kobayashi-Maskawa matrix. From

eq. (1) it follows that the box diagrams contribution to the process B Ksnn in 2HDM is proportional to the charged lepton mass and therefore gives a negligible contribution. The SM contributions to b Ksnn decay are calculated in [6] and in this work the

2HDM contributions are taken into account (see also [7]). The resulting effective Hamiltonian is Heff4 Ga 2k2 p sin2_u w VtbV *tss gm( 1 2g5) b n gm( 1 2g5) n Q (2) Q x 8

{

y

z

}

tOmW2, y 4mt2OmH2 and sin2uW is the Weinberg angle. tg b is the ratio of

the vacuum expectation values of the two Higgs doublets in 2HDM, and it is a free parameter of the model. The constraints on the tg b are usually obtained from B0_{- B}0_,

K0- K0_{mixings, b Ksg decay width, semileptonic bKcn}tt decay and given by [8] as

0.7 Gtg bG0.6

g

1

1 GeV

h

Fig. 2. – The dependence of the branching ratio on the charged Higgs boson mass at different values of ctg b.

The next step is, starting from this effective Hamiltonian, to calculate the Bs( d )K

n n g decay at the hadronic level. Note that the pure leptonic decay B Knn is forbidden

by the helicity conservation. The process b Ksnng is described by the diagrams in which the photon is attached to any charged lines in fig. 1. In this case no helicity suppression exists any longer. However, it is not necessary to calculate the contributions of all diagrams. Indeed, when the photon is radiated from internal charged lines, the contributions of such diagrams are suppressed by factor

(

bOmW2(mH2)

)

instead of dimension 6. So, it is enough to consider the contributions only of the diagrams, where the photon is emitted by initial b or s quarks.

In order to calculate the process BsK n n g we need the following matrix element:

agNsgm( 1 2g5) bNBb .

(4)

This matrix element can be parameterized in the following way: ag(q) Ns gm( 1 2g5) bNBb 4

(5)

4 k4 pa

mB2

[emnrse *nprqsf (p2) 1i

(

)

where a is the fine-structure constant, em and q are the polarization vector and

four-momenta of the photon, p is the transfer momentum, and f (p2), g(p2) are the parity-conserving and the parity violation form factors. These form factors were calculated in the light-cone QCD sum rules [9] and the results are:

f (p2) 4 f ( 0 ) ( 1 2p2_/m2 1)2 , (6) g(p2) 4 g( 0 ) ( 1 2p2_/m2 2)2 , (7)

where f ( 0 ) 41 GeV, m14 5 , 6 GeV, g( 0 ) 4 0.8 GeV, m24 6.5 GeV.

After standard calculations, and using the above values it is not difficult to show that the decay width for B Knng is

dG dEg 4 G 2_a3 NVtbVtsN2 ( 2 p)4 mB3sin4uw Eg3(mB22 2 mBEg)[N fN21 NgN2] Q (8) Q

{

y

g

h

z

}

GF4 1.17 Q 1025 GeV22, a 41/137, mb4 4.7 GeV, mB4 5.28 GeV, NVtbV *tsN 4 0.04 and

t(Bd) 41.56Q10212 s21. NVtbV *tdN C 0.01 and t(Bs) 41.34Q10212s21[10].

In fig. 2 we present the dependence of the branching ratio of BsK n n g on mH for

different values of ctg b. From this figure we see that for large values of ctg b (for example ctg b 450) 2HDM contribution to the branching ratio exceeds by the order of two to three the SM one.

However, using the experimental restrictions on tg b it is clear from this that ctg b cannot be a larger one, and the maximum possible value of ctg b C1.5. Using this value of ctg b one can see that the 2HDM contribution in practice is smaller than SM.

In fig. 3 we present the dependence of the branching ratio on the photon energy Eg,

for mH4 100 GeV in a), mH4 300 GeV in b) at tg b 4 1 .

We see that the spectrum is slightly asymmetric as a result of a balance between a highly asymmetric resonance-type behavior given by the non-perturbative contribu-tions and perturbative photon emission.

Finally, note that the results for BdK n n g can be easily obtained from BsK n n g

replacing VtbV *ts by VtbV *td. It is obvious that the branching ratio of BdK n n g is one

order smaller than those of BsK n n g.

* * *

The author thanks T. M. ALIEV for many useful discussions and encouragement during this work.

R E F E R E N C E S

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