Model independent analysis of the decay B→K+ℓ⁺+ℓ⁻

Abstract

We demonstrate that the Dalitz-distributions in energies of both unpolarized leptons, produced in the decays B→K+ℓ⁺+ℓ⁻, for ℓ=e, μ, τ, can be a sensitive test for possible new physics beyond the Standard Model. We find measurable characteristics (including the polarization of leptons as well) which are independent of all dirty non-perturbative physics needed in calculating the hadronic form factors. Such characteristics can be used to estimate the presence of possible non-standard mechanisms in a model-independent manner.

Introduction

In this work we will discuss the most general properties of the simplest leptonic decay of B-mesons, namely, B→K+ℓ⁺+ℓ⁻, where $\text{ℓ=e,}\text{μ,}\text{τ}$. This decay, being induced by the flavour changing neutral current (FCNC), is an attractive object for investigation in the future hadronic colliders and B-factories [1]. After the successful detection of the electromagnetic penguins, through the electromagnetic decays of B-mesons, B→K+γ [2]and B→X_s+γ [3], it has been possible to estimate the magnetic moment of the transition b→s+γ, i.e. the magnitude of the Wilson coefficient C₇(m_b). But for this aim it is necessary to know some elements of the CKM matrix [4]. One can address the inverse problem as well. Namely, assuming the validity of the standard formalism of effective Hamiltonians, and a procedure for calculating the coefficient C₇(m_b), it is possible to use the data for the radiative B-decay to estimate |V_ts| [5].

Leptonic rare decays, $\text{B→X}{}_{\mathrm{s}}\text{+ℓ}{}^{\mathrm{+}}\text{+ℓ}{}^{\mathrm{-}}\text{,B→K(K}{}^{\mathrm{\ast }}\text{)+ℓ}{}^{\mathrm{+}}\text{+ℓ}{}^{\mathrm{-}}$ offer better opportunities for studying new physics, than the decays $\text{B→K}{}^{\mathrm{\ast }}\text{+γ}$, and B→K_s+γ. That is, they provide more sensitive search strategies for new physics. For example, the sign of C₇(m_b) which depends on the underlying physics, is not determined by the measurement of $\text{B→K}{}^{\mathrm{\ast }}\text{+γ}$ or B→X+γ. Similarly, in SUSY models [6], both (the negative and positive) signs are allowed, as one scans over the allowed SUSY parameter space. The B→X_s+ℓ⁺+ℓ⁻ amplitude in SM has, in addition to the coefficient C₇, new terms namely, C₉ and C₁₀ 8, 9, 10. So it has been argued that the signs and the magnitudes of the three coefficients C₇,C₉,C₁₀ can, in principle, be determined from the decays B→X_s+γ and B→X_s+ℓ⁺+ℓ⁻ [7]. This strategy of testing SM, and searching for new physics includes also an analysis of polarization phenomena, namely the polarization properties of leptons produced in B→X_s+ℓ⁺+ℓ⁻ 11, 12.

But all these characteristics are also sensitive to the non-perturbative, i.e. long-distance, contributions, where exact calculations face serious problems. This is particularly so, for the exclusive semihadronic B-decays, such as $\text{B→K(K}{}^{\mathrm{\ast }}\text{)+ℓ}{}^{\mathrm{+}}\text{+ℓ}{}^{\mathrm{-}}$. The experimental limits [13]on these decays, while quite close to the SM-based predictions, can only be interpreted in specific models (for computing form factors), which hinders somewhat their transcription in terms of the underlying Wilson coefficients.

So, all these semileptonic FCNC decays will provide precise tests for SM, as they will determine the signs and magnitudes of the three Wilson coefficients, C₇,C₉,C₁₀. Note, that the dilepton mass distribution and the forward -backward asymmetry for the decay B→X_s+ℓ⁺+ℓ⁻ can also be sensitive to the non-SM effects [14].

We will prove here that a simpler decay, namely, B→K+ℓ⁺+ℓ⁻ with definite experimental signature, can particularly be suitable for searching for the manifestations of new physics. As both of the hadrons here have zero spin, the spin structure of the corresponding matrix element and the analysis of the underlying physics is very simple in this case.

The exact calculation of hadronic form factors, using the effective Hamiltonian formalism (in terms of Wilson coefficients), is possible only in the framework of definite versions of the non-perturbative QCD. Therefore, information about the B-meson wave function, and the retardation effects, and a procedure of relativization etc. is needed to obtain some (model dependent) parametrization of the form factors of the axial and vector hadronic currents in the SM-formalism. However, the interval [15]for the predicted branching ratio is wide enough, which is a clear manifestation of the existing difficulties of non-perturbative theory of strong interactions [16].

Under such unsatisfactory theoretical circumstances, we wish to pose the following principal question: How can one find the manifestations of the new physics, in the decay B→K+ℓ⁺+ℓ⁻? In other words, are there any observable characteristics of this decay which are sensitive to new physics? Is it possible to avoid numerous theoretical non-perturbative difficulties in calculating the hadronic form factors, to find measurable decay characteristics, which carry the signatures of new physics independently of all possible un-controllable theoretical uncertainties due to non-perturbative physics?

In this work we will try to obtain answers to these questions. As we will demonstrate below, there exist such observable characteristics, and it is possible to identify definite experimental procedures, for which the exact knowledge of the corresponding form factors does not matter. Thus, the definite reparametrization of the observables introduced in this work, can be used as the new model-independent method for testing SM, and for search of new physics.