Probing extended Higgs sector through
We study the constraints on the contribution of new physics in the form of scalar/pseudoscalar operators to the average forward backward asymmetry of muons in and the longitudinal polarization asymmetry of muons in . We find that the maximum possible value of allowed by the present upper bound on is about at C.L. and hence will be very difficult to measure. On the other hand, the present bound on fails to put any constraints on , which can be as high as even if is close to its standard model prediction. The measurement of will be a direct evidence for an extended Higgs sector, and combined with the branching ratio it can even separate the new physics scalar and pseudoscalar contributions.
The quark level flavor changing neutral interaction is forbidden at the tree level in the standard model (SM) and can occur only at the one-loop level. Therefore it can serve as an important probe to test SM at loop level and also constrain many new physics models beyond the SM. This quark level interaction is responsible for the purely leptonic decay and also the semi-leptonic decays . The semi-leptonic decays have been observed by BaBar and Belle babar-03 (); babar-06 (); belle-03 () with the following branching ratios:
These values are close to the SM predictions ali-02 (); lunghi (); kruger-01 (). However there is about uncertainty in these predictions mainly due to the errors in the determination of the hadronic form factors and the CKM matrix element .
The decay is highly suppressed in SM. Its branching ratio is predicted to be blanke (); buchalla (); buras-01 (). This decay is yet to be observed experimentally. Recently the upper bound on its branching ratio has been improved to cdf-07 ()
which is still more than an order of magnitude above its SM prediction. will be one of the important rare decays to be studied at the upcoming Large Hadron Collider (LHC) and we expect that the sensitivity of the level of the SM prediction can be reached with fb of data. schneider (); maria ().
Many new physics models predict an order of magnitude enhancement or more in . These include theories with mediated vector bosons london-97 (), as well as multi-Higgs doublet models that violate london-97 () or obey hewett-89 () natural flavor conservation. In alok-sankar01 (), it was shown that the new physics mediated by vector bosons is highly constrained by the measured values of the branching ratio of . As a result, an order of magnitude enhancement in from new physics vector or axial vector operators is ruled out. On the other hand, such an enhancement from the scalar/pseudoscalar new physics (SPNP) operators is still allowed, since the most stringent bound on the SPNP operators comes from itself. In particular, multi-Higgs doublet models or supersymmetric (SUSY) models with large can give rise to such an enhancement.
Apart from the branching ratios of the purely leptonic and semi-leptonic decays, there are other observables which are sensitive to the SPNP contribution to transitions. These are forward-backward (FB) asymmetry of muons ali-92 () in and longitudinal polarization (LP) asymmetry of muons in handoko-02 (). Both these are predicted to be zero in the SM. Therefore, any nonzero measurement of one of these asymmetries is a signal for new physics. In addition, these asymmetries are almost independent of form factors and CKM matrix element uncertainties, which makes them attractive candidates in searches for new physics. In this paper we investigate what constraints the recently improved upper bound on puts on the possible SPNP contribution to and . Do SPNP operators enhance these observables to sufficiently large values to be measurable in future experiments?
Ii Forward-backward asymmetry in
There are numerous studies in literature of the FB asymmetry of leptons in the SM and its possible extensions ali-00 (); yan-00 (); bobeth-01 (); erkol-02 (); demir-02 (); li-04 (); chen-05 (). In the SM, the FB asymmetry of muons in vanishes (or to be more precise, is negligibly small) because the hadronic current for transition does not have any axial vector contribution. However this asymmetry can be nonzero in multi-Higgs doublet models and supersymmetric models with large , due to the contributions from Higgs bosons. Therefore FB asymmetry in is expected to serve as an important probe to test the existence and importance of an extended Higgs sector erkol-02 (); chen-05 (). Any nonzero measurement of this asymmetry will be a clear signal of new physics.
These measurements are consistent with zero. But on the other hand, they can be as high as within error bars.
ii.1 Calculation of
We consider new physics in the form of scalar/pseudoscalar operators. The effective Lagrangian for the quark level transition can be written as
Here and is the sum of -momenta of and . and are the new physics scalar and pseudoscalar couplings respectively. In our analysis we assume that there are no additional CP phases apart from the single CKM phase. Under this assumption, and are real. Within SM, the Wilson coefficients in eq. (6) have the following values:
The normalized FB asymmetry is defined as
In order to calculate the FB asymmetry, we first need to calculate the differential decay width. The decay amplitude for is given by
where . The relevant matrix elements are
Here, , and . In this paper, we approximate by .
Using the above matrix elements, the double differential decay width can be calculated as
Also, and is the angle between the momenta of meson and in the dilepton centre of mass frame. The kinematical variables are bounded as
The form factors can be calculated in the light cone QCD approach. Their dependence is given by ali-00 ()
The calculation of FB asymmetry gives
From eq. (17), it is clear that is proportional to , and to the scalar new physics coupling . In the minimal supersymmetric standard model (MSSM) and two Higgs doublet models, itself is proportional to and . Hence a large FB asymmetry is possible only for exceptionally large values of .
The average FB asymmetry is obtained by integrating the numerator and denominator of eq. (17) separately over dilepton invariant mass, which leads to
where is the total branching ratio of . The numerator in eq. (21) can be calculated to be
whereas the total branching ratio, including the contribution of SPNP operators, is given by bobeth-01 ()
In the SM calculation of , two vector form factors, and , as well as the tensor form factor appear. The SPNP contribution, on the other hand, is only through . We have made the assumption that the fractional uncertainties in all the form factors are the same. The dependence in the numerator and denominator of eq. (21) cancels completely, whereas the errors due to the form factors uncertainties cancel partially. We conservatively take the net error in to be , leading to
ii.2 Constraints on from
We now want to see what constraints the present upper bound on puts on the maximum possible value of . The present experimental upper limit on is an order of magnitude larger than the SM prediction. In such a situation, the SM amplitude for this decay will be much smaller than the new physics amplitude and hence can be neglected in determining the constraints on new physics couplings, and . In other words, we will assume that SPNP operators saturate the present upper limit. Therefore we need to consider only the contribution of to the decay rate of .
The decay amplitude for is given by
where and are the masses of bottom and strange quark, respectively. The calculation of the branching ratio gives
Here we have neglected terms of order and approximated by . Taking , we get
where we have taken the lower bound for the coefficient in eq. (29). Thus, the allowed region in the – parameter space is the interior of a circle of radius centered at the origin.
In alok_amol_uma (), it was shown that the SPNP operators cannot lower below its SM prediction. Therefore from eq. (24), the maximum value of with the current upper bound on is at . If is bounded to , the maximum value of will be .
A naive estimation suggests that the measurement of an asymmetry of a decay with the branching ratio at C.L. with only statistical errors require
number of events. For , if is at C.L., then the required number of events will be as high as ! Therefore it is very difficult to observe such a low value of FB asymmetry in experiments. Hence FB asymmetry of muons in will play no role in testing SPNP.
Iii Longitudinal polarization asymmetry in
The longitudinal polarization asymmetry of muons in is a clean observable that depends only on SPNP operators. It vanishes in the SM, whereas its value is nonzero if and only if the new physics contribution is in the form of scalar operator. Therefore any nonzero measurement of this observable will confirm the existence of an extended Higgs sector. The observable was introduced in ref. handoko-02 (), though the corresponding analysis in the context of had been carried out earlier botella (); kll (); geng (); ecker (). In this section, we will determine the allowed values of consistent with the present upper bound on , and explore the correlation between these two quantities.
where and are the strengths of the scalar, pseudoscalar and axial vector operators respectively. Note that the effective Lagrangian in eq. (32) is essentially the same as the effective Lagrangian given in eq. (5). Here we have dropped and terms which do not contribute to . In addition, the in eq. (32) is the sum of SM and new physics contributions.
In SM, the scalar and pseudoscalar couplings and receive contributions from the penguin diagrams with physical and unphysical neutral scalar exchange and are highly suppressed:
Also, , where is the Inami-Lim function inamilim ()
with . Thus, .
Here is the lifetime of . Eq. (35) represents the most general expression for the branching ratio of .
We now derive an expression for the lepton polarization. In the rest frame of , we can define only one direction , the three momentum of . The unit longitudinal polarization 4-vectors along that direction are
Transformation of unit vectors from the rest frame of to the center of mass frame of leptons (which is also the rest frame of meson) can be accomplished by the Lorentz boost. After the boost, we get
where is the muon energy.
The longitudinal polarization asymmetry of muons in is defined as
Thus we get handoko-02 ()
with . It is clear from eq. (40) that can be nonzero if and only if , i.e. for to be nonzero, we must have contribution from SPNP operators. Within the SM, and hence .
Eq. (41) represents a general relation between the longitudinal polarization asymmetry and the branching ratio of .
We now explore the correlation between and . It is quite obvious that when , we can neglect the SM contribution in obtaining the bounds on and . However if is of the order of the SM prediction, then we will have to take into account the SM contribution as well. Therefore it is reasonable to consider both the cases separately.
We first consider the constraints on coming from the present upper bound on . Fig. 1 shows the plot between and for three different values of