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New physics upper bound on the branching ratio of Bs→l+l−
Physics Letters B 620 (2005) 61–68 www.elsevier.com/locate/physletb
New physics upper bound on the branching ratio of Bs → l +l − Ashutosh Kumar Alok, S. Uma Sankar Department of Physics, Indian Institute of Technology, Bombay, Mumbai 400076, India Received 31 March 2005; received in revised form 31 May 2005; accepted 1 June 2005 Available online 13 June 2005 Editor: G.F. Giudice
Abstract We consider the most general new physics effective Lagrangian for b → sl + l − . We derive the upper limit on the branching ratio for the processes Bs → l + l − where l = e, µ, subject to the current experimental bounds on related processes, B → Kl + l − and B → K ∗ l + l − . If the new physics interactions are of vector/axial-vector form, the present measured rates for B → (K, K ∗ )l + l − constrain B(Bs → l + l − ) to be of the same order of magnitude as their respective Standard Model predictions. On the other hand, if the new physics interactions are of scalar/pseudoscalar form, B → (K, K ∗ )l + l − rates do not impose any constraint on Bs → l + l − and the branching ratios of these decays can be as large as present experimental upper bounds. If future experiments measure B(Bs → l + l − ) to be 10−8 then the new physics giving rise to these decays has to be of the scalar/pseudoscalar form. 2005 Published by Elsevier B.V.
The rare decays of B mesons involving flavour changing neutral interaction (FCNI) b → s has been a topic of great interest for long. Not only will it subject the Standard Model (SM) to accurate tests but will also put strong constraints on several models beyond the SM. In the SM, FCNI occur only via one or more loops. Thus the rare decays of B mesons will provide useful information about the higher-order effects of the SM. Recently, the very high statistics experiments at B-factories have measured non-zero values for the branching ratios for the FCNI processes B → (K, K ∗ )l + l − [1,2], −7 Br B → Kl + l − = 4.8+1.0 −0.9 ± 0.3 ± 0.1 × 10 , −7 Br B → K ∗ l + l − = 11.5+2.6 (1) −2.4 ± 0.8 ± 0.2 × 10 . These branching ratios are close to the values predicted by the SM [3]. However, the SM predictions for them contain about ∼ 15% uncertainty coming from the hadronic form factors. Still, it is worth considering what constraints these measurements impose on other related processes. E-mail address: [email protected] (S.U. Sankar). 0370-2693/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.06.005
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The effective Lagrangian for the four fermion process b → sl + l − gives rise to the exclusive semi-leptonic decays such as B → Kl + l − and B → K ∗ l + l − and also to purely leptonic decays Bs → l + l − , where l = e, µ. (From here onwards, the symbol l represents either e or µ.) Relation between semi-leptonic and purely leptonic B-decays, arising from FCNI generated by heavy Z boson exchange, was briefly considered in [4,5]. The SM predictions for the branching ratios for the decays Bs → e+ e− and Bs → µ+ µ− are (7.58 ± 3.5) × 10−14 and (3.2 ± 1.5) × 10−9 respectively [6]. The large uncertainty in the SM prediction for these branching ratios arises due to the 12% uncertainty in the Bs decay constant and 10% uncertainty in the measurement of Vts . These branching ratios have been calculated in various new physics models. In models with Z -mediated FCNI, one has B(Bs → µ+ µ− ) < 5.8 × 10−8 [7] which is about 20 times larger than the SM prediction. Due to the increased precision in the measurement of B(B → (K, K ∗ )l + l − ), this bound can be improved and the present calculation attempts to do so. B(Bs → l + l − ) are also calculated in multi Higgs doublet models. These models are classified into two types. In the first type, there is natural flavour conservation (NFC) and there are no FCNI at tree level. In such models, there is an additional loop contribution to FCNI, where a charged Higgs boson exchange replaces the SM W -exchange. In a two Higgs doublet model with NFC, branching ratio for Bs → µ+ µ− 10−8 is possible [8]. In the second type, flavour changing processes do occur at tree level, mediated by flavour changing neutral scalars (FCNSs). In such models also a branching ratio of about 10−8 for Bs → µ+ µ− can be achieved [7]. Leptonic and semi-leptonic rare decays of B and Bs mesons have also been calculated in various extensions of SM, such as technicolour [9] and supersymmetry [10]. Their relation to the dark matter problem is considered in [11]. From the experimental side, at present, there exist only upper bounds B(Bs → e+ e− ) < 5.4 × 10−5 [12] and B(Bs → µ+ µ− ) < 5.0 × 10−7 [13]. In this Letter, we consider the most general four fermion effective Lagrangian for b → sl + l − transition due to new physics. We derive upper bounds on the branching ratios for Bs → e+ e− and Bs → µ+ µ− by demanding that the predictions of this new physics Lagrangian for B → K ∗ l + l − and B → Kl + l − should be consistent with the current experimental values. The most general effective Lagrangian for b → sl + l − transitions due to new physics can be written as Leff b → sl + l − = LV A + LSP + LT ,
(2)
where LV A contains vector and axial-vector couplings, LSP contains scalar and pseudoscalar couplings and LT contains tensor couplings. LT does not contribute to Bs → l + l − because 0|¯s σ µν b|Bs (pB ) = 0. Hence we will drop it from further consideration. First we will assume that the new physics Lagrangian contains only vector and axial-vector couplings. We parametrize it as GF α + − ¯ V + gA s¯ (gV + gA γ5 )γµ bl(g γ5 )γ µ l. LV A b → sl l = √ 2 2 4πsW
(3)
Here the constants g and g are the effective couplings which characterize the new physics. From the above equation, we get Bs → l + l − matrix element to be α GF 0|sγ5 γµ b|Bs l + l − lγ5 γ µ l|0. M Bs → l + l − = (gA gA )√ 2 2 4πsW
(4)
Only the axial vector parts contribute for both the hadronic and leptonic parts of the matrix element. Substituting 0|sγ5 γµ b|Bs = −ifBs pBµ , in Eq. (4) we get
+ −
M Bs → l l
α u(p ¯ l )γ5 v(pl¯). 2 2 4πsW
GF = −i2ml fBs (gA gA )√
(5)
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As we are considering only vector and axial vector currents, helicity suppression is still operative for the Bs → l + l − decay amplitude. The calculation of the decay rate gives 2 G2F fB2s α + − 2 ΓNP Bs → l l = (6) (gA gA ) mBs m2l . 2 8π 4πsW )2 . To estimate the value of (g g )2 , we consider the related Thus the decay rate depends upon the value of (gA gA A A ∗ + − + − decays B → K l l and B → Kl l , which also receive contributions from the effective Lagrangian in Eq. (3). In deriving Eq. (6), we dropped terms proportional to m2l /m2B , as their contribution is negligible. We will make the same approximation in calculating the decay width of semi-leptonic modes also. We first consider the process B → K ∗ l + l − . Here we will have to calculate the following hadronic matrix elements [14]: ∗ K (pK ∗ )sγµ bB(pB ) = iµνλσ ν (pK ∗ )(pB + pK ∗ )λ (pB − pK ∗ )σ V q 2 , ∗ K (pK ∗ )sγ5 γµ bB(pB ) = µ (pK ∗ ) m2 − m2 ∗ A1 q 2 − (.q)(pB + pK ∗ )µ A2 q 2 , (7) B
K
where q = pl + + pl − . In the above equation, a term proportional to qµ is dropped because its contribution to the decay rate is proportional to m2l /m2B . It is assumed that the q 2 dependence of these form factors is well described by a pole fit: V Ai , Ai q 2 = . V q2 = 2 2 ∗ ∗ (mB + mK )(1 − q /mB ) (mB + mK )(1 − q 2 /m2B ) The decay rate is
2 1 G2F m5B 2 α ∗ + − 2 gV + gA IV A , ΓNP B → K l l = 3 2 2 192π 4πsW
(8)
where IV A is the integral over the dilepton invariant mass (z = q 2 /m2B ). Under the assumption that A1 ≈ A2 , IV A is given by IV A = gV2 V 2
zmax dz zmin
where
z 2 2 C1 (z) + gA A1 1−z
zmax dz zmin
z C2 (z), 1−z
(9)
mK ∗ −2 C1 (z) = 2 1 + Φ(z), mB
5m2K ∗ mK ∗ −2 mK ∗ 2 mB 2 1+ z− Φ(z) + C2 (z) = 3 1 − mB 2mK ∗ mB m2B
with Φ(z) = (1 − z)2 + 4z(mK ∗ /mB )2 . The limits of integration for z are given by zmin = (2ml /mB )2 and zmax = )2 can be determined from the measured rate of (1 − mK ∗ /mB )2 . From Eq. (8) we see that, the value of (gA gA 2 2 2 ∗ + − Γ (B → K l l ), provided the value of gV (gV + gA ) is known. For this we consider the decay of B → Kl + l − . The matrix element necessary in this case is [14] + 2 q , K(pK )sγµ bB(pB ) = (pB + pK )µ fKB (10) where again a term proportional to qµ is dropped. The q 2 dependence of the formfactor, again, is approximated by a single pole with mass ≈ mB , f + q2 =
f + (0) . 1 − q 2 /m2B
(11)
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The decay rate is given by 2 + 2 2 G2F m5B α f (0) + − 2 2 . ΓNP B → Kl l = gV gV + gA 2 2 192π 3 4πsW
(12)
We demand that the maximum value of this decay rate is the measured experimental value, i.e., Γexp = ΓNP .
(13)
With this assumption we calculate the upper bound on the decay rate of Bs Eq. (3). Using Eqs. (8), (12) and (13), we get
→ l+l−,
arising due to LV A , given in
BExp (B → Kl + l − ) 2 = × 104 gV2 gV 2 + gA 3.45[f + (0)]2
(14)
2) 2 BExp (B → K ∗ l + l − ) × 104 − 1.58V 2 gV2 (gV 2 + gA 2 2 gV + gA = . gA 8.94A21
(15)
and
In our calculation, we take the formfactors to be [15] f + (0) = 0.319+0.052 −0.041 ,
V = 0.457+0.091 −0.058 ,
A1 =0.337+0.048 −0.043 ,
(16)
and use experimental values of B → (K, K ∗ )l + l − given in [2]. Adding all errors in quadrature, we get 2 2 −2 2 2 −3 gV2 gV 2 + gA = 1.36+0.53 gV + gA = 6.76+4.04 gA −0.44 × 10 , −3.48 × 10 .
(17)
)2 can have, is Thus the maximum value (gA gA 2 −3 ) = 6.76+4.04 (gA gA −3.48 × 10 .
(18)
→ l+l−,
The branching ratio for Bs due to LV A , to be 2 2 ) , B Bs → µ+ µ− = 4.54 × 10−6 fB2s (gA gA ) . B Bs → e+ e− = 1.06 × 10−10 fB2s (gA gA
(19)
)2 from Eq. (18), we get Substituting fBs = 240 ± 30 MeV [16] and the maximum value for (gA gA −14 −9 , B Bs → µ+ µ− = 1.74+1.13 B Bs → e+ e− = 4.06+2.65 −2.34 × 10 −1.00 × 10 .
(20)
Therefore the upper bounds on the branching ratios are B Bs → e+ e− < 6.71 × 10−14 , B Bs → µ+ µ− < 2.87 × 10−9
(21)
at 1σ and B Bs → e+ e− < 1.20 × 10−13 ,
(22)
B Bs → µ+ µ− < 5.13 × 10−9
at 3σ . These rates are close to the SM predictions. The reason for this is quite simple. The decay rate for an exclusive semi-leptonic process can be written as Γ = (c.c.)2 (f.f.)2
phase space,
(23)
where c.c. is the coupling constant and f.f. is the form factor. The measured rates for the exclusive semi-leptonic decays are close to the SM predictions. And we assumed that the new physics predictions for these processes are equal to their corresponding experimental values. Also, the same set of form factors are used in both SM and new physics calculations. Thus the assumption that new physics predictions for semi-leptonic branching ratios are
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equal to their experimental values (which in turn are equal to their SM predictions) implies that the couplings of new physics are very close to the couplings of the SM. This is why our new physics prediction for the purely leptonic mode is also close to the SM prediction. Therefore, new physics, whose effective Lagrangian for b → sl + l − consists of only vector and axial vector currents, cannot boost up the rate of Bs → l + l − due to the present experimental constraints coming from the decays B → Kl + l − and B → K ∗ l + l − . For the reasons explained above, using a different set of form factors, as for example those given in [17], will not change the upper bound on Bs → l + l − significantly. In fact, we find that the change is less than 10%. )2 by the following procedure. We equate the new physics We can obtain a more stringent upper bound on (gA gA ∗ + − contribution for Γ (B → (K, K )l l ) to the difference between the experimental value and the SM contribution. This, in turn, leads to a much more stringent upper bound on contribution of LV A to Bs → l + l − . In fact, at 1σ , this bound is consistent with 0. At 3σ we get B Bs → e+ e− < 7.89 × 10−14 , (24) B Bs → µ+ µ− < 3.37 × 10−9 , which are again comparable to the SM predictions. Comparing these results with the ones obtained by previous assumption, we see that there is not much difference in the branching ratios. This occurs due to the relatively large errors in both the experimental measurements and SM predictions for Γ (B → (K, K ∗ )l + l − ). Thus we conclude that the presently measured values of B → (K, K ∗ )l + l − do not allow any large boost in the contribution of LV A to Bs → l + l − . We now consider the new physics effective Lagrangian to consist of scalar/pseudoscalar couplings GF α + − ¯ S + gP γ5 )l. LSP b → sl l = √ (25) s¯ (gS + gP γ5 )bl(g 2 2 4πsW The matrix element for the decay Bs → l + l − is given by GF α + − gP 0|sγ5 b|Bs gS u(p ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) . M Bs → l l = √ 2 2 4πsW
(26)
On substituting 0|¯s γ5 b|Bs = −i
fBs m2Bs mb + ms
,
(27)
we get
+ −
M Bs → l l
fBs m2Bs α GF ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) , = −igP √ gS u(p 2 m + m 2 4πsW b s
(28)
where mb and ms are the masses of bottom and strange quark, respectively. Here we take the quark masses from Particle Data Group obtained under MS scheme [18]. We see that in this case there is no helicity suppression, i.e., the rates for the decays Bs → e+ e− and Bs → µ+ µ− will be the same provided gP and gS for both electrons and muons are the same. The calculation of the decay rate gives G2F ΓNP Bs → l + l − = gP2 gS 2 + gP 2 16π
α 2 4πsW
2
fB2s m5Bs (mb + ms )2
.
(29)
The branching ratio is given by fB2 gP2 (gS 2 + gP 2 ) B Bs → l + l − = 0.17 s . (mb + ms )2
(30)
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To estimate the value of gP2 (gS 2 + gP 2 ), we again consider the related decay B → K ∗ l + l − . Its matrix element, due to LSP is given by GF α M B → K ∗l+l− = √ (31) ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) gP K ∗ sγ5 b|B gS u(p 2 2 4πsW as K ∗ |sb|B = 0. The pseudoscalar hadronic matrix element is given by [19] ∗ 2mK ∗ A0 q 2 (q · ). K sγ5 b|B = −i mb − ms
(32)
The q 2 dependence of the formfactor is described by a pole fit A0 q 2 =
A0 (0) . (1 − q 2 /m2B )
(33)
The full calculation gives ΓNP B → K ∗ l + l − =
G2F m5B 256π 3
α 2 4πsW
2
2mK ∗ mb − ms
2
mB 2 2 2 2 2 A0 (0) gP gS + gP ISP , 2mK ∗
(34)
where ISP
zmax = dz zmin
z (1 − z)2
1+
m2K ∗ m2B
2 −z
−
4m2k ∗ m2B
3/2 .
(35)
The limits of integration for the dilepton invariant mass (z = q 2 /m2B ) are once again given by zmin = (2ml /mB )2 and zmax = (1 − mK ∗ /mB )2 . Now we assume that the maximum value of this decay rate is the measured experimental value. Thus from Eq. (34), we get (mb − ms )2 BExp (B → K ∗ l + l − ) × 103 . gP2 gS 2 + gP 2 = 2.16[A0 (0)]2
(36)
Taking the value of A0 (0) to be 0.471+0.127 −0.059 [15], we get −2 gP2 gS 2 + gP 2 = 4.02+2.41 −1.41 × 10 .
(37)
Substituting the value of gP2 (gS 2 + gP 2 ) in Eq. (30) we get, −5 B Bs → l + l − = 2.10+1.38 −0.93 × 10 .
(38)
The upper bound on B(Bs → µ+ µ− ) from the above equation is much higher than the present experimental upper bound [13]. Thus we see that the measured values of B(B → (K, K ∗ )l + l − ) do not provide any useful constraint on LSP contribution to B(Bs → µ+ µ− ). The significance of this result is that if a future experiment, such as LHC-b [20] observes B(B → µ+ µ− ) 10−8 , one can confidently assert that the new physics giving rise to this large a branching ratio must necessarily be of scalar/pseudoscalar type. Comparing the expression in Eq. (30) to the experimental upper bound in [13], we obtain the bound gP2 gS 2 + gP 2 10−3 . (39) In deriving the limits on the scalar/pseudoscalar couplings of leptons in Eq. (39), all terms proportional to lepton masses have been neglected. This is valid in general multi-Higgs models. However, in some models, such as two Higgs doublet model, the scalar and pseudoscalar couplings of fermions are proportional to
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their masses [21]. In such models, the SM and the new physics amplitudes are both proportional to the lepton mass and the interference between them should be taken into account [22]. For these models, the ratios RK = B(B → Kµ+ µ− )/B(B → Ke+ e− ) and RK ∗ = B(B → K ∗ µ+ µ− )/B(B → K ∗ e+ e− ) provide stronger limits on new physics scalar/pseudoscalar couplings of muons than B(Bs → µ+ µ− ) [23]. The measured values of semi-leptonic branching ratios quoted in Eq. (1) are obtained using SM Monte Carlo. The model dependence in obtaining these branching ratios is included in the error. One must exercise care in obtaining constraints on new physics from the above measurements. Usually, differential decay distributions are very sensitive to the form of the interaction. Hence one must include the decay distribution due to new physics in the Monte Carlo before obtaining constraint on the new physics, if differential distributions are studied. However, here we are concerned only with the total decay rates. The decay rate due to new physics is a simple addition to the SM rate. Thus, even if new physics decay rate is included in the Monte Carlo, the measured values of branching ratios will not be very different. The experimental values contain a 20% statistical error. A change of similar magnitude, due to a different Monte Carlo, will not change our conclusions. Conclusions. We considered the most general effective Lagrangian for the flavour changing neutral process b → sl + l − , arising due to new physics. We showed that the present experimental values of B(B → (K, K ∗ )l + l − ) set strong bounds on B(Bs → l + l − ) if the effective Lagrangian is product of vectors/axial-vectors. Given that the above semi-leptonic decay rates of B-mesons are comparable to their SM predicted values, we showed that the rate for purely leptonic decays of Bs cannot be much above the their SM predicted values. We have also derived a 3σ upper bound on B(Bs → µ+ µ− ) < 5 × 10−9 arising from Z -mediated flavour changing neutral currents. If the effective Lagrangian for b → sl + l − is product of scalars/pseudoscalars then present experimental values of B(B → (K, K ∗ )l + l − ) do not lead to any useful bound on B(Bs → l + l − ). This leads us to the very important conclusion that, if a future experiment observes Bs → l + l − with a branching ratio greater than 10−8 , then the new physics responsible for this decay must of be scalar/pseudoscalar type.
Acknowledgements This work grew out of the discussions with Prof. Roger Forty of CERN, during WHEPP-8. We thank Prof. Forty and other participants of WHEPP-8 for discussions on rare B-decays.
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New physics upper bound on the branching ratio of Bs → l +l − Ashutosh Kumar Alok, S. Uma Sankar Department of Physics, Indian Institute of Technology, Bombay, Mumbai 400076, India Received 31 March 2005; received in revised form 31 May 2005; accepted 1 June 2005 Available online 13 June 2005 Editor: G.F. Giudice
Abstract We consider the most general new physics effective Lagrangian for b → sl + l − . We derive the upper limit on the branching ratio for the processes Bs → l + l − where l = e, µ, subject to the current experimental bounds on related processes, B → Kl + l − and B → K ∗ l + l − . If the new physics interactions are of vector/axial-vector form, the present measured rates for B → (K, K ∗ )l + l − constrain B(Bs → l + l − ) to be of the same order of magnitude as their respective Standard Model predictions. On the other hand, if the new physics interactions are of scalar/pseudoscalar form, B → (K, K ∗ )l + l − rates do not impose any constraint on Bs → l + l − and the branching ratios of these decays can be as large as present experimental upper bounds. If future experiments measure B(Bs → l + l − ) to be 10−8 then the new physics giving rise to these decays has to be of the scalar/pseudoscalar form. 2005 Published by Elsevier B.V.
The rare decays of B mesons involving flavour changing neutral interaction (FCNI) b → s has been a topic of great interest for long. Not only will it subject the Standard Model (SM) to accurate tests but will also put strong constraints on several models beyond the SM. In the SM, FCNI occur only via one or more loops. Thus the rare decays of B mesons will provide useful information about the higher-order effects of the SM. Recently, the very high statistics experiments at B-factories have measured non-zero values for the branching ratios for the FCNI processes B → (K, K ∗ )l + l − [1,2], −7 Br B → Kl + l − = 4.8+1.0 −0.9 ± 0.3 ± 0.1 × 10 , −7 Br B → K ∗ l + l − = 11.5+2.6 (1) −2.4 ± 0.8 ± 0.2 × 10 . These branching ratios are close to the values predicted by the SM [3]. However, the SM predictions for them contain about ∼ 15% uncertainty coming from the hadronic form factors. Still, it is worth considering what constraints these measurements impose on other related processes. E-mail address: [email protected] (S.U. Sankar). 0370-2693/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physletb.2005.06.005
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The effective Lagrangian for the four fermion process b → sl + l − gives rise to the exclusive semi-leptonic decays such as B → Kl + l − and B → K ∗ l + l − and also to purely leptonic decays Bs → l + l − , where l = e, µ. (From here onwards, the symbol l represents either e or µ.) Relation between semi-leptonic and purely leptonic B-decays, arising from FCNI generated by heavy Z boson exchange, was briefly considered in [4,5]. The SM predictions for the branching ratios for the decays Bs → e+ e− and Bs → µ+ µ− are (7.58 ± 3.5) × 10−14 and (3.2 ± 1.5) × 10−9 respectively [6]. The large uncertainty in the SM prediction for these branching ratios arises due to the 12% uncertainty in the Bs decay constant and 10% uncertainty in the measurement of Vts . These branching ratios have been calculated in various new physics models. In models with Z -mediated FCNI, one has B(Bs → µ+ µ− ) < 5.8 × 10−8 [7] which is about 20 times larger than the SM prediction. Due to the increased precision in the measurement of B(B → (K, K ∗ )l + l − ), this bound can be improved and the present calculation attempts to do so. B(Bs → l + l − ) are also calculated in multi Higgs doublet models. These models are classified into two types. In the first type, there is natural flavour conservation (NFC) and there are no FCNI at tree level. In such models, there is an additional loop contribution to FCNI, where a charged Higgs boson exchange replaces the SM W -exchange. In a two Higgs doublet model with NFC, branching ratio for Bs → µ+ µ− 10−8 is possible [8]. In the second type, flavour changing processes do occur at tree level, mediated by flavour changing neutral scalars (FCNSs). In such models also a branching ratio of about 10−8 for Bs → µ+ µ− can be achieved [7]. Leptonic and semi-leptonic rare decays of B and Bs mesons have also been calculated in various extensions of SM, such as technicolour [9] and supersymmetry [10]. Their relation to the dark matter problem is considered in [11]. From the experimental side, at present, there exist only upper bounds B(Bs → e+ e− ) < 5.4 × 10−5 [12] and B(Bs → µ+ µ− ) < 5.0 × 10−7 [13]. In this Letter, we consider the most general four fermion effective Lagrangian for b → sl + l − transition due to new physics. We derive upper bounds on the branching ratios for Bs → e+ e− and Bs → µ+ µ− by demanding that the predictions of this new physics Lagrangian for B → K ∗ l + l − and B → Kl + l − should be consistent with the current experimental values. The most general effective Lagrangian for b → sl + l − transitions due to new physics can be written as Leff b → sl + l − = LV A + LSP + LT ,
(2)
where LV A contains vector and axial-vector couplings, LSP contains scalar and pseudoscalar couplings and LT contains tensor couplings. LT does not contribute to Bs → l + l − because 0|¯s σ µν b|Bs (pB ) = 0. Hence we will drop it from further consideration. First we will assume that the new physics Lagrangian contains only vector and axial-vector couplings. We parametrize it as GF α + − ¯ V + gA s¯ (gV + gA γ5 )γµ bl(g γ5 )γ µ l. LV A b → sl l = √ 2 2 4πsW
(3)
Here the constants g and g are the effective couplings which characterize the new physics. From the above equation, we get Bs → l + l − matrix element to be α GF 0|sγ5 γµ b|Bs l + l − lγ5 γ µ l|0. M Bs → l + l − = (gA gA )√ 2 2 4πsW
(4)
Only the axial vector parts contribute for both the hadronic and leptonic parts of the matrix element. Substituting 0|sγ5 γµ b|Bs = −ifBs pBµ , in Eq. (4) we get
+ −
M Bs → l l
α u(p ¯ l )γ5 v(pl¯). 2 2 4πsW
GF = −i2ml fBs (gA gA )√
(5)
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As we are considering only vector and axial vector currents, helicity suppression is still operative for the Bs → l + l − decay amplitude. The calculation of the decay rate gives 2 G2F fB2s α + − 2 ΓNP Bs → l l = (6) (gA gA ) mBs m2l . 2 8π 4πsW )2 . To estimate the value of (g g )2 , we consider the related Thus the decay rate depends upon the value of (gA gA A A ∗ + − + − decays B → K l l and B → Kl l , which also receive contributions from the effective Lagrangian in Eq. (3). In deriving Eq. (6), we dropped terms proportional to m2l /m2B , as their contribution is negligible. We will make the same approximation in calculating the decay width of semi-leptonic modes also. We first consider the process B → K ∗ l + l − . Here we will have to calculate the following hadronic matrix elements [14]: ∗ K (pK ∗ )sγµ bB(pB ) = iµνλσ ν (pK ∗ )(pB + pK ∗ )λ (pB − pK ∗ )σ V q 2 , ∗ K (pK ∗ )sγ5 γµ bB(pB ) = µ (pK ∗ ) m2 − m2 ∗ A1 q 2 − (.q)(pB + pK ∗ )µ A2 q 2 , (7) B
K
where q = pl + + pl − . In the above equation, a term proportional to qµ is dropped because its contribution to the decay rate is proportional to m2l /m2B . It is assumed that the q 2 dependence of these form factors is well described by a pole fit: V Ai , Ai q 2 = . V q2 = 2 2 ∗ ∗ (mB + mK )(1 − q /mB ) (mB + mK )(1 − q 2 /m2B ) The decay rate is
2 1 G2F m5B 2 α ∗ + − 2 gV + gA IV A , ΓNP B → K l l = 3 2 2 192π 4πsW
(8)
where IV A is the integral over the dilepton invariant mass (z = q 2 /m2B ). Under the assumption that A1 ≈ A2 , IV A is given by IV A = gV2 V 2
zmax dz zmin
where
z 2 2 C1 (z) + gA A1 1−z
zmax dz zmin
z C2 (z), 1−z
(9)
mK ∗ −2 C1 (z) = 2 1 + Φ(z), mB
5m2K ∗ mK ∗ −2 mK ∗ 2 mB 2 1+ z− Φ(z) + C2 (z) = 3 1 − mB 2mK ∗ mB m2B
with Φ(z) = (1 − z)2 + 4z(mK ∗ /mB )2 . The limits of integration for z are given by zmin = (2ml /mB )2 and zmax = )2 can be determined from the measured rate of (1 − mK ∗ /mB )2 . From Eq. (8) we see that, the value of (gA gA 2 2 2 ∗ + − Γ (B → K l l ), provided the value of gV (gV + gA ) is known. For this we consider the decay of B → Kl + l − . The matrix element necessary in this case is [14] + 2 q , K(pK )sγµ bB(pB ) = (pB + pK )µ fKB (10) where again a term proportional to qµ is dropped. The q 2 dependence of the formfactor, again, is approximated by a single pole with mass ≈ mB , f + q2 =
f + (0) . 1 − q 2 /m2B
(11)
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The decay rate is given by 2 + 2 2 G2F m5B α f (0) + − 2 2 . ΓNP B → Kl l = gV gV + gA 2 2 192π 3 4πsW
(12)
We demand that the maximum value of this decay rate is the measured experimental value, i.e., Γexp = ΓNP .
(13)
With this assumption we calculate the upper bound on the decay rate of Bs Eq. (3). Using Eqs. (8), (12) and (13), we get
→ l+l−,
arising due to LV A , given in
BExp (B → Kl + l − ) 2 = × 104 gV2 gV 2 + gA 3.45[f + (0)]2
(14)
2) 2 BExp (B → K ∗ l + l − ) × 104 − 1.58V 2 gV2 (gV 2 + gA 2 2 gV + gA = . gA 8.94A21
(15)
and
In our calculation, we take the formfactors to be [15] f + (0) = 0.319+0.052 −0.041 ,
V = 0.457+0.091 −0.058 ,
A1 =0.337+0.048 −0.043 ,
(16)
and use experimental values of B → (K, K ∗ )l + l − given in [2]. Adding all errors in quadrature, we get 2 2 −2 2 2 −3 gV2 gV 2 + gA = 1.36+0.53 gV + gA = 6.76+4.04 gA −0.44 × 10 , −3.48 × 10 .
(17)
)2 can have, is Thus the maximum value (gA gA 2 −3 ) = 6.76+4.04 (gA gA −3.48 × 10 .
(18)
→ l+l−,
The branching ratio for Bs due to LV A , to be 2 2 ) , B Bs → µ+ µ− = 4.54 × 10−6 fB2s (gA gA ) . B Bs → e+ e− = 1.06 × 10−10 fB2s (gA gA
(19)
)2 from Eq. (18), we get Substituting fBs = 240 ± 30 MeV [16] and the maximum value for (gA gA −14 −9 , B Bs → µ+ µ− = 1.74+1.13 B Bs → e+ e− = 4.06+2.65 −2.34 × 10 −1.00 × 10 .
(20)
Therefore the upper bounds on the branching ratios are B Bs → e+ e− < 6.71 × 10−14 , B Bs → µ+ µ− < 2.87 × 10−9
(21)
at 1σ and B Bs → e+ e− < 1.20 × 10−13 ,
(22)
B Bs → µ+ µ− < 5.13 × 10−9
at 3σ . These rates are close to the SM predictions. The reason for this is quite simple. The decay rate for an exclusive semi-leptonic process can be written as Γ = (c.c.)2 (f.f.)2
phase space,
(23)
where c.c. is the coupling constant and f.f. is the form factor. The measured rates for the exclusive semi-leptonic decays are close to the SM predictions. And we assumed that the new physics predictions for these processes are equal to their corresponding experimental values. Also, the same set of form factors are used in both SM and new physics calculations. Thus the assumption that new physics predictions for semi-leptonic branching ratios are
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equal to their experimental values (which in turn are equal to their SM predictions) implies that the couplings of new physics are very close to the couplings of the SM. This is why our new physics prediction for the purely leptonic mode is also close to the SM prediction. Therefore, new physics, whose effective Lagrangian for b → sl + l − consists of only vector and axial vector currents, cannot boost up the rate of Bs → l + l − due to the present experimental constraints coming from the decays B → Kl + l − and B → K ∗ l + l − . For the reasons explained above, using a different set of form factors, as for example those given in [17], will not change the upper bound on Bs → l + l − significantly. In fact, we find that the change is less than 10%. )2 by the following procedure. We equate the new physics We can obtain a more stringent upper bound on (gA gA ∗ + − contribution for Γ (B → (K, K )l l ) to the difference between the experimental value and the SM contribution. This, in turn, leads to a much more stringent upper bound on contribution of LV A to Bs → l + l − . In fact, at 1σ , this bound is consistent with 0. At 3σ we get B Bs → e+ e− < 7.89 × 10−14 , (24) B Bs → µ+ µ− < 3.37 × 10−9 , which are again comparable to the SM predictions. Comparing these results with the ones obtained by previous assumption, we see that there is not much difference in the branching ratios. This occurs due to the relatively large errors in both the experimental measurements and SM predictions for Γ (B → (K, K ∗ )l + l − ). Thus we conclude that the presently measured values of B → (K, K ∗ )l + l − do not allow any large boost in the contribution of LV A to Bs → l + l − . We now consider the new physics effective Lagrangian to consist of scalar/pseudoscalar couplings GF α + − ¯ S + gP γ5 )l. LSP b → sl l = √ (25) s¯ (gS + gP γ5 )bl(g 2 2 4πsW The matrix element for the decay Bs → l + l − is given by GF α + − gP 0|sγ5 b|Bs gS u(p ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) . M Bs → l l = √ 2 2 4πsW
(26)
On substituting 0|¯s γ5 b|Bs = −i
fBs m2Bs mb + ms
,
(27)
we get
+ −
M Bs → l l
fBs m2Bs α GF ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) , = −igP √ gS u(p 2 m + m 2 4πsW b s
(28)
where mb and ms are the masses of bottom and strange quark, respectively. Here we take the quark masses from Particle Data Group obtained under MS scheme [18]. We see that in this case there is no helicity suppression, i.e., the rates for the decays Bs → e+ e− and Bs → µ+ µ− will be the same provided gP and gS for both electrons and muons are the same. The calculation of the decay rate gives G2F ΓNP Bs → l + l − = gP2 gS 2 + gP 2 16π
α 2 4πsW
2
fB2s m5Bs (mb + ms )2
.
(29)
The branching ratio is given by fB2 gP2 (gS 2 + gP 2 ) B Bs → l + l − = 0.17 s . (mb + ms )2
(30)
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To estimate the value of gP2 (gS 2 + gP 2 ), we again consider the related decay B → K ∗ l + l − . Its matrix element, due to LSP is given by GF α M B → K ∗l+l− = √ (31) ¯ l )v(pl¯) + gP u(p ¯ l )γ5 v(pl¯) gP K ∗ sγ5 b|B gS u(p 2 2 4πsW as K ∗ |sb|B = 0. The pseudoscalar hadronic matrix element is given by [19] ∗ 2mK ∗ A0 q 2 (q · ). K sγ5 b|B = −i mb − ms
(32)
The q 2 dependence of the formfactor is described by a pole fit A0 q 2 =
A0 (0) . (1 − q 2 /m2B )
(33)
The full calculation gives ΓNP B → K ∗ l + l − =
G2F m5B 256π 3
α 2 4πsW
2
2mK ∗ mb − ms
2
mB 2 2 2 2 2 A0 (0) gP gS + gP ISP , 2mK ∗
(34)
where ISP
zmax = dz zmin
z (1 − z)2
1+
m2K ∗ m2B
2 −z
−
4m2k ∗ m2B
3/2 .
(35)
The limits of integration for the dilepton invariant mass (z = q 2 /m2B ) are once again given by zmin = (2ml /mB )2 and zmax = (1 − mK ∗ /mB )2 . Now we assume that the maximum value of this decay rate is the measured experimental value. Thus from Eq. (34), we get (mb − ms )2 BExp (B → K ∗ l + l − ) × 103 . gP2 gS 2 + gP 2 = 2.16[A0 (0)]2
(36)
Taking the value of A0 (0) to be 0.471+0.127 −0.059 [15], we get −2 gP2 gS 2 + gP 2 = 4.02+2.41 −1.41 × 10 .
(37)
Substituting the value of gP2 (gS 2 + gP 2 ) in Eq. (30) we get, −5 B Bs → l + l − = 2.10+1.38 −0.93 × 10 .
(38)
The upper bound on B(Bs → µ+ µ− ) from the above equation is much higher than the present experimental upper bound [13]. Thus we see that the measured values of B(B → (K, K ∗ )l + l − ) do not provide any useful constraint on LSP contribution to B(Bs → µ+ µ− ). The significance of this result is that if a future experiment, such as LHC-b [20] observes B(B → µ+ µ− ) 10−8 , one can confidently assert that the new physics giving rise to this large a branching ratio must necessarily be of scalar/pseudoscalar type. Comparing the expression in Eq. (30) to the experimental upper bound in [13], we obtain the bound gP2 gS 2 + gP 2 10−3 . (39) In deriving the limits on the scalar/pseudoscalar couplings of leptons in Eq. (39), all terms proportional to lepton masses have been neglected. This is valid in general multi-Higgs models. However, in some models, such as two Higgs doublet model, the scalar and pseudoscalar couplings of fermions are proportional to
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their masses [21]. In such models, the SM and the new physics amplitudes are both proportional to the lepton mass and the interference between them should be taken into account [22]. For these models, the ratios RK = B(B → Kµ+ µ− )/B(B → Ke+ e− ) and RK ∗ = B(B → K ∗ µ+ µ− )/B(B → K ∗ e+ e− ) provide stronger limits on new physics scalar/pseudoscalar couplings of muons than B(Bs → µ+ µ− ) [23]. The measured values of semi-leptonic branching ratios quoted in Eq. (1) are obtained using SM Monte Carlo. The model dependence in obtaining these branching ratios is included in the error. One must exercise care in obtaining constraints on new physics from the above measurements. Usually, differential decay distributions are very sensitive to the form of the interaction. Hence one must include the decay distribution due to new physics in the Monte Carlo before obtaining constraint on the new physics, if differential distributions are studied. However, here we are concerned only with the total decay rates. The decay rate due to new physics is a simple addition to the SM rate. Thus, even if new physics decay rate is included in the Monte Carlo, the measured values of branching ratios will not be very different. The experimental values contain a 20% statistical error. A change of similar magnitude, due to a different Monte Carlo, will not change our conclusions. Conclusions. We considered the most general effective Lagrangian for the flavour changing neutral process b → sl + l − , arising due to new physics. We showed that the present experimental values of B(B → (K, K ∗ )l + l − ) set strong bounds on B(Bs → l + l − ) if the effective Lagrangian is product of vectors/axial-vectors. Given that the above semi-leptonic decay rates of B-mesons are comparable to their SM predicted values, we showed that the rate for purely leptonic decays of Bs cannot be much above the their SM predicted values. We have also derived a 3σ upper bound on B(Bs → µ+ µ− ) < 5 × 10−9 arising from Z -mediated flavour changing neutral currents. If the effective Lagrangian for b → sl + l − is product of scalars/pseudoscalars then present experimental values of B(B → (K, K ∗ )l + l − ) do not lead to any useful bound on B(Bs → l + l − ). This leads us to the very important conclusion that, if a future experiment observes Bs → l + l − with a branching ratio greater than 10−8 , then the new physics responsible for this decay must of be scalar/pseudoscalar type.
Acknowledgements This work grew out of the discussions with Prof. Roger Forty of CERN, during WHEPP-8. We thank Prof. Forty and other participants of WHEPP-8 for discussions on rare B-decays.
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