18 July 1996
Physics Letters B 381 (1996) 348-352
Cai-Dian Lii, Da-Xin Zhang Physics Department, Technion- Israel Institute of Technology, Haifa 32000, Israel Received 27 February
1996; revised manuscript
Editor: R. Gatto
Abstract We study the loop induced rare decays B,( Bd) + YYVwithin the standard model. Both constituent quark model and pole model are used and the branching ratios turn out to be of the orders of lo-* for B, + yvV and 10m9 for Bd -+ yvti,. These processes can be served to determine the decay constants of B, and Bd.
1. Introduction Rare decays of bottom quark [ l] can be served as tests of the standard model (SM). As the observed process b -+ sy which induced B + K*y  and B + X,y [ 31 at the hadronic level, b -+ sl+l-  are also studied extensively in the past both at inclusive and at exclusive levels. It was recognized that the SM prediction for the branching ratio of B --+ X,vF, which is N 5 x 10d5 , is within one order beyond the present detectivity and this process might be observed soon [ 61. Furthermore, the corresponding exclusive decay B + K* vt is important as an input to determine V& to an accuracy of 10% [ 73. Pure leptonic decays of heavy pseudoscalar mesons into light lepton pairs are helicity suppressed, their branching ratios are [ 81: B(Bs --f ,u+p-) B( B,y4 efe-)
2. Effective Hamiltonian
= 1.8 x 10-9, = 4.2 x 10-14,
which make it difficult to determine f~, from these processes. For Bd the situation is even worse due to the small CKM mixing angles. Although the processes B,s( Bd) + T+T- do not suffer from this suppression 0370-2693/96/$12.00
0 1996 Elsevier Science
mechanism and the branching ratio is about lop7 [ 51 in SM, it is hard to be detected at future B-factory where the efficiency is not better than 10e2. The processes B,(Bd) --f yvp,, whose branching ratios depend quadrically on fs, ( fs), can be taken as a possible alternate to determine the decay constants. For detections of these processes the method of searching for the “missing mass” can be used, which is similar to the case of B + X,vF . In the present work, we study the processes + yet within the SM. We will analyze B,(Bd) B,( Bd) + yvV in the following: In Section 2 the relevant effective Hamiltonian will be given in SM. Constituent quark model and pole model will be used in Section 3 to give the predictions. Finally, Section 4 contains some brief discussion.
Let us start with the quark level process b -+ qvV, with q = s or d. The Feynman diagrams are displayed in Fig. 1. Both box and Z penguin diagrams contribute to this process. The resulting effective Hamiltonian is left handed in SM [9,5] :
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C.-D. Lii, D.-X. Zhang/Physics
Letters B 381 (19%) 348-352
Fig. 1. Feynman diagrams in standard model for b + pp.
7-k= ctqy,PLb>(Fy’+~Lv), with Pt = ( 1 -
ys)/2. The coefficient C is (4)
&%a 7r sin’ 8,
(3) where x = mf/m&. For simplicity, we neglect the QCD correction to this coefficient, whose effects are within 2% if appropriate renormalization point is chosen [ 51. Since the neutrinos are massless in minimal SM, - ’ are forbidden by helicthe processes B4( bq) -+ YY ity. If an additional photon line is attached to any of the charged lines in Fig. 1, the situation will be different: no helicity suppression exists any more. However, when the photon line is attached to one of the internal charged lines, there will be a suppression factor of rnfJrn$, in the Wilson coefficient compared with the ones for b -+ qvP. The reason is that these effective operators are now dimension-8 instead of dimension6. On the other hand, the diagrams in Fig. 1 with photon line connected to one of the external (bottom or strange quark) lines, whose effective Hamiltonian turn out to be ’ We denote mesons with the quark content (bg) as B, for convenience.
will be the dominant contributions to the decay B,(b& + yYF.
3. Model calculations The effective Hamiltonian given in (4) is not enough to analyze the processes B, -+ yvP. In addition, models are needed to do the calculations at the hadronic level. First we use a simple constituent quark model (see for example [ 111) . In this model both of the (anti) quarks are treated non-relativistically with the same velocity of the hadron. Thus (anti)quarks masses are now constituent quark masses of the order of several hundred MeV. We use further the interpolating field technique [ 121 which relates all the hadronic matrix elements of the present case to the decay constants of the mesons. The amplitude for B, --+yvP decay is:
C-D. Lii, D.-X. Zhung / Physics Letters B 381 (1996) 348-352
The first term inside the bracket comes from the CPodd part of B,, while the second term results from the CP-even part of B,. Since mb >> m4 (q = d, s), we can safely neglect the term of I/Q. Then after squaring the amplitude, both CP-odd and CP-even parts give the same contribution to the decay rate. Performing the phase space integration over one of the two Dalitz variables, and summing over the three generation of neutrinos, we get the differential decay width versus the vv invariant mass: Fig. 2. Differential decay rate of ES --+ [email protected]
versus the VP invariant mass m$, with the solid line denoting results calculated from constituent quark model, and the dashed line corresponding to that of pole model.
The decay width is:
(7) Using LY= l/132, m, = 0.51 GeV, m, = 176 GeV and 1vbv: 1= 0.04, we get
lY(B,? + yvv)
= 9.5 x lo-=’
(8) If the lifetime is taken as r( B,) = 1.34 x lo-t2s [ lo], the branching ratio is found to be 1.9 x lo-*. The differential decay rate versus rnZi.is displayed in Fig. 2 (solid line). For Bd meson decay, we take I&,~~/ = 0.01 and md = 0.35 GeV. The decay width is then 2 r(B,
1.2 X lo-=’
which corresponds to the branching ratio 2.6 x 10e9, if uses of r(Bd) = 1.50 X 10-‘2s [lo] and fs, = 0.2 GeV are made. We also apply the pole model to calculate the B, meson decay. The main contribution here is the radiative transition of the B, meson into an intermediate (virtual) BG. The Feynman diagram is shown in Fig. 3. The subsequent process Bz --f vD is deter-
Fig. 3. Feynman diagram in pole model for B,, + yvP, in SM by the effective Hamiltonian while the B,B,*y transition is taken as mined
(IO) This corresponds to the first part of Eq. (5)) which is for the CP-odd part of B,. Hereafter, we will concentrate only on the CP-odd combination of B, and B4. The effective coupling constant is usually explained as a function of the invariant mass of the intermediate B;. Here we simply derive its value from the constituent quark model as a constant [ 111,
(11) The 1/Mb term can also be neglected compared to the 1/m, term. Note that the potential model [ 131 gives
C.-D. L.ii, D.-X. Zhmg / Physics Letters B 381(19%)
i\d = 0.59 GeV, A, =
& = 4.93 GeV
(14) Here again we give the differential cross section for the purpose of being used experimentally:
After integrating is obtained as
- mZc)3m$ ‘I (m& - m$)* ’ 4
over the phase space, the decay width
The branching ratio is obtained as 2.4 x 10w9, also quite close to that obtained in constituent quark model.
4. Conclusion We predict the branching ratios in SM for B, --f yvfi to be 10e8 and for Bd -+ yvV to be 10e9. With these branching ratios, they are hopeful to be detected at future B factories or LHC. They can provide alternate channels for measuring f B, ( fB, ) .
We thank G. Eilam and M. Gronau for helpful discussions. The research of D.X. Zhang is supported in part by Grant 542 l-3-96 from the Ministry of Science and the Arts of Israel.
S. Playfer and S. Stone, HEPSY 95-01.
[21 R. Ammar, et al., CLEO Collaboration. The formulas ( 15>, ( 16) are similar to the constituent quark model case Eqs. (6), (7). It is easy to see that if mB; = m&, fB; = f&, are taken in Eqs. (15>, (16), they will reduce exactly to Eqs. (6), (7). Now we use mBf = 5.42 GeV and get:
---f yvF) = 8.8 x lo-*’
r11 For review see, A.J. Buras, M.K. Harlander, Heavy flavours, pp. 58-201, eds. A.J. Buras and M. Lindner (World Scientific, Singapore) ; A. Ali, Nucl. Phys. B (Proc. Suppl.) 39BC ( 1995) 408;
+ 6y2 + 12y - 12 -y)*ln(l
.f(Y) = -5y3
= 1.1 X
which is quite close to ( 11) . Now the amplitude the decay B, -+ yvF is:
For Bd meson, we get r(&
It is only slightly different from Eq. (8)) because mB; is not quite different from mB,. The branching ratio is 1.8 x 10e8, if fBT = fB, = 0.2 GeV is used. The differential decay rate calculated in pole model is also displayed in Fig. 2 (dashed line) as function of m&.
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