Contribution of vector resonances to the $\bar{B}_{d}^{0}\to\bar {K}^{*0}\mu^{+}\mu^{-}$ decay

Abstract

The fully differential angular distribution for the rare flavor-changing neutral current decay $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) \mu^{+}\mu^{-}$ is studied. The emphasis is placed on accurate treatment of the contribution from the processes $\bar{B}_{d}^{0} \to\bar{K}^{*0} (\to K^{-} \pi^{+}) V$ with intermediate vector resonances V=ρ(770),ω(782),ϕ(1020),J/ψ,ψ(2S),… decaying into the μ ⁺ μ ⁻ pair. The dilepton invariant-mass dependence of the branching ratio, longitudinal polarization fraction f _L of the $\bar{K}^{*0}$ meson, and forward–backward asymmetry A _FB is calculated and compared with data from Belle, CDF and LHCb. It is shown that inclusion of the resonance contribution may considerably modify the branching ratio, calculated in the SM without resonances, even in the invariant-mass region far from the so-called charmonia cuts applied in the experimental analyses. This conclusion crucially depends on values of the unknown phases of the B ⁰→K ^∗0 J/ψ and B ⁰→K ^∗0 ψ(2S) decay amplitudes with zero helicity.

Appendix: Amplitudes of B→K ∗ V decays

An important ingredient of the resonant contribution is amplitude of the decay of B meson into two vector mesons, B(p)→V ₁(q,ϵ ₁)+V ₂(k,ϵ ₂), with on-mass-shell meson V ₂ ($k^{2} = m_{2}^{2}$) and off-mass-shell meson V ₁ ($q^{2} \ne m_{1}^{2}$). For the case of two on-mass-shell final mesons one can write the amplitude in the form [42]

(30)

in terms of three invariant amplitudes S ₁, S ₂ and S ₃, V _CKM is a CKM factor. The quantities S ₁, S ₂ and S ₃ may be complex and involve two types of phase, CP-conserving strong phases and CP-violating weak phases. In general, the invariant amplitudes are a sum of several interfering amplitudes, S _1j , S _2j and S _3j , respectively. Then the phase structure of S ₁, S ₂ and S ₃ is

$$ S_k=\sum_{j}|S_{kj}| e^{i \varphi_{kj}}e^{i \delta_{kj}} \quad (k=1,2,3 ) , $$

(31)

where φ _1j , φ _2j , and φ _3j are the CP-violating weak phases and δ _1j , δ _2j , and δ _3j are the CP-conserving strong phases.

Using CPT invariance, we can represent the matrix element for the charge-conjugate decay $\bar{B}(p) \to\bar{V}_{1}(q,\epsilon_{1})\* \bar{V}_{2}(k,\epsilon_{2})$ as

(32)

where $\bar{S}_{1}$, $\bar{S}_{2}$, and $\bar{S}_{3}$ can be derived from S ₁, S ₂, and S ₃ by reversing the sign of the CP-violating phase. Note that if the B→V ₁ V ₂ decay is invariant under the CP symmetry, then $\bar{S}_{1}=S_{1}$, $\bar{S}_{2}=S_{2}$, and $\bar{S}_{3}=S_{3}$. On the other hand, if all CP-conserving phases of invariant amplitudes are equal to zero, then $\bar{S}_{1}=S^{*}_{1}$, $\bar{S}_{2}=S^{*}_{2}$, and $\bar{S}_{3}=S^{*}_{3}$.

The helicity amplitudes in terms of three invariant amplitudes, S ₁, S ₂, and S ₃ are

(33)

From the decomposition Eq. (33) one finds the following relations between the helicity amplitudes and the invariant amplitudes S ₁, S ₂, S ₃:

$$ \begin{aligned} \lefteqn{H_0 = - \frac{1}{2 \hat{m}_1 \hat{m}_2} \biggl( \bigl(1- \hat{m}_1^2-\hat{m}_2^2\bigr) S_1 + \frac{S_2}{2} \lambda\bigl(1,\hat{m}_1^2, \hat{m}_2^2\bigr) \biggr), } \\ \lefteqn{H_\pm= S_1 \pm\frac{S_3}{2} \sqrt {\lambda\bigl(1,\hat{m}_1^2, \hat{m}_2^2\bigr)} ,} \end{aligned} $$

(34)

with $\lambda(1,\hat{m}_{1}^{2},\hat{m}_{2}^{2}) \equiv(1-\hat{m}_{1}^{2})^{2} - 2\hat{m}_{2}^{2}(1+\hat{m}_{1}^{2}) +\hat{m}_{2}^{4}$ and $\hat{m}_{1(2)} \equiv m_{1(2)}/m_{B}$.

Note that the polarized decay amplitudes can be expressed in several different but equivalent bases. For example, the helicity amplitudes can be related to the spin amplitudes in the transversity basis (A ₀,A _∥,A _⊥) defined in terms of the linear polarization of the vector mesons via:

$$A_0 = H_0 ,\qquad A_\parallel= \frac{H_+ + H_-}{\sqrt{2}} , \qquad A_\perp= \frac{H_+ - H_-}{\sqrt{2}} , $$

A ₀, A _∥, A _⊥ are related to S ₁, S ₂ and S ₃ of Eq. (30) via

$$ \begin{aligned} \lefteqn{A_0 = -\frac{1}{2 \hat{m}_1 \hat{m}_2} \biggl( \bigl(1- \hat{m}_1^2-\hat{m}_2^2\bigr) S_1 + \frac{S_2}{2} \lambda\bigl(1,\hat{m}_1^2, \hat{m}_2^2\bigr) \biggr),} \\ \lefteqn{A_\parallel= \sqrt{2} S_1 ,\qquad A_\perp= \sqrt{\frac{\lambda(1,\hat{m}_1^2,\hat{m}_2^2)}{2}} S_3 .} \end{aligned} $$

(35)

The amplitude $\bar{A}_{\lambda}$ (λ=0,∥,⊥) are related to the invariant amplitudes of the $\bar{B} \to\bar{V}_{1} \bar{V}_{2}$ decay by the formulas

$$ \begin{aligned} \lefteqn{\bar{A}_0 = - \frac{1}{2 \hat{m}_1 \hat{m}_2} \biggl( \bigl(1- \hat{m}_1^2-\hat{m}_2^2\bigr) \bar{S}_1 + \frac{\bar{S_2}}{2} \lambda\bigl(1,\hat{m}_1^2, \hat{m}_2^2\bigr) \biggr) ,} \\ \lefteqn{\bar{A}_\parallel= \sqrt{2} \bar{S}_1 , \qquad\bar{A}_\perp=- \sqrt{\frac{\lambda(1,\hat{m}_1^2,\hat{m}_2^2)}{2}} \bar{S}_3 .} \end{aligned} $$

(36)

If the B→V ₁ V ₂ decay is invariant under CP transformation, then $\bar{A}_{0}=A_{0}$, $\bar{A}_{\|}=A_{\|}$, and $\bar{A}_{\perp}=-A_{\perp}$.

The decay width is expresses as follows:

(37)

The matrix element for the $B_{d}^{0}\to K^{*0} V$ decay, where V=ρ ⁰,ω,ϕ,J/ψ(1S),ψ(2S),… mesons, we can represent as

(38)

Next, we define the normalized amplitudes:

$$ \begin{aligned} \lefteqn{h_\lambda\equiv \frac{A_\lambda}{\sqrt{\sum_{\lambda^\prime} |A_{\lambda^\prime}|^2}} ,} \\ \lefteqn{\sum_\lambda |h_\lambda|^2 = 1 \quad\bigl(\lambda, \lambda^\prime= 0, \parallel, \perp\bigr) .} \end{aligned} $$

(39)

By putting m ₁=m _V , $m_{2} = m_{K^{*}}$ and using (37), (39) we obtain the relation between the amplitudes h _λ and A _λ of the process under study $B_{d}^{0} \to K^{*0} V$ for any vector meson V=ρ ⁰,ω,ϕ,J/ψ(1S),ψ(2S),…:

(40)

where BR(…) is the branching ratio of $B_{d}^{0} \to K^{*0} V$ decay and τ _B is the lifetime of a B meson.

Solving Eqs. (35) we find the scalars S ₁,S ₂ and S ₃, and then extend the helicity amplitudes $A_{\lambda}^{V}$ off the mass shell of the meson V, i.e. for $q^{2} \ne m_{V}^{2}$. We introduce the phases $\delta_{\lambda}^{V} \equiv\mathrm{arg}(h_{\lambda}^{V})$, $\delta_{i}^{V} \equiv\mathrm{arg}(S_{i}^{V})$, where i=1,2,3. Then we have

(41)

The branching ratio and decay amplitudes are given in Table 5.

Table 5 Branching ratio [55], and decay amplitudes for ${ B}_{d}^{0}\to{K}^{*0} \rho^{0}$ [56], ${B}_{d}^{0}\to{ K}^{*0} \omega$ [56] and ${ B}_{d}^{0}\to{ K}^{*0} \phi$, ${ B}_{d}^{0}\to{ K}^{*0} J/\psi$, ${ B}_{d}^{0}\to{ K}^{*0} \psi(2S)$ [55]