$B_{s}\rightarrow K^{\ast 0}$ decay form factors from covariant confined quark model

TL;DR

This paper calculates the $B_{s} ightarrow K^{ ext{*}0}$ transition form factors across all kinematic regions using a covariant confined quark model, aiding in the analysis of rare decay processes.

Contribution

It provides a comprehensive calculation of form factors for $B_{s} ightarrow K^{ ext{*}0}$ decays within a covariant confined quark model, extending previous work.

Findings

Form factors computed across full kinematic range.

Results applicable to $B_{s} ightarrow K^{ ext{*}0}\mu^+\mu^-$ decay analysis.

Supports experimental measurements by LHCb.

Abstract

We evaluate $B_{s}\rightarrow K^{\ast 0}$ transition form factors in the full kinematical region within the covariant confined quark model. The calculated form factors can be used to calculate the $B_{s}\rightarrow K^{\ast 0}\mu^+ \mu^-$ rare decay branching ratio, which was recently measured by LHCb collaboration.

Tables

Table 1. Table 1: Parameters for the approximated form factors in Eq. ( 4 ).

	$A_0$	$A_{+}$	$A_{-}$	$V$	$a_0$	$a_{+}$	$g$
$F\left(0\right)$	0.30	0.21	$-0.23$	0.24	0.21	0.21	0.21
$a$	-0.64	-1.47	-1.55	-1.60	-0.69	-1.48	-1.61
$b$	$-0.28$	0.44	0.52	0.56	$-0.23$	0.45	0.57

Table 2. Table 2: The form factors at maximum recoil q 2 = 0 superscript 𝑞 2 0 q^{2}=0 .

	$V^c\left(0\right)$	$A_0^c\left(0\right)$	$A_1^c\left(0\right)$	$A_2^c\left(0\right)$	$T_1^c\left(0\right)$	$T_3^c\left(0\right)$
CCQM	$0.24\pm 0.02$	$0.18\pm 0.02$	$0.21\pm 0.02$	$0.21\pm 0.02$	$0.21\pm 0.02$	$0.14\pm 0.01$
Ref. Ball:2004rg	0.31	0.36	0.23	0.18	0.26	0.14

Equations

$$\begin{array}[]{ccccc|ccccc}m_{u/d}&m_{s}&m_{c}&m_{b}&\lambda&\Lambda_{B_{s}}&\Lambda_{K^{*0}}&m_{B_{s}}&m_{K^{*0}}\\ \hline\cr\ \ 0.241&\ \ 0.428&\ \ 1.67&\ \ 4.68&\ \ 0.181&\ \ 2.05&\ \ 0.81&\ \ 5.367&\ \ 0.896&\ {\rm GeV}\\ \end{array}$$

$$⟨V(p_2,{ϵ}_2{)}_{[{\bar{q}}_3{q}_2]}|{\bar{q}}_2{O}^{\mu }{q}_1|P_{[{\bar{q}}_3{q}_1]}(p_1)⟩=$$

$$⟨V(p_2,{ϵ}_2{)}_{[{\bar{q}}_3{q}_2]}|{\bar{q}}_2({\sigma }^{\mu \nu }{q}_{\nu }(1+{\gamma }^5))q_1|P_{[{\bar{q}}_3{q}_1]}(p_1)⟩=$$

$$F(q^2)=\frac{F(0)}{1-as+b{s}^2},s=\frac{q^2}{m_1^2},$$

$$A_0$$

$$A_{-}$$

$$a_0$$

$$A_0^c(0)$$

$$2m_2{A}_3^c(q^2)$$

Full text

Last measurements of rare B-decays show deviations with respect to Standart Model predictionsAaij:2013qta ; Aaij:2013iag ; Aaltonen:2011qs ; Aaltonen:2011cn ; Aaij:2013aln . The $b\rightarrow s\ell^{+}\ell^{-}$ and $b\rightarrow d\ell^{+}\ell^{-}$ processes are forbidden at tree-level in Standart Model and sensitive to New Physics contributions in loops.The $b\rightarrow d$ transition is supressed than $b\rightarrow s$ due to CKM matrix elements. However it is interesting to study decays proceeds via flavour-changing neutral-current (FCNC) transition. The $b\rightarrow d$ transition decays observed by LHCb collaboration for $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$LHCb:2012de ; Aaij:2015nea and $\Lambda^{0}\rightarrow\rho\pi^{-}\mu^{+}\mu^{-}$ Aaij:2017ewm decays. Recently the LHCb Collaboration Aaij:2018jhg reported about the measurement of branching ratio of $B_{s}\rightarrow K^{\ast 0}\mu^{+}\mu^{-}$ decay.

The $B_{s}\rightarrow K^{\ast 0}$ transition form factors were studied in light-cone sum rule Ball:2004rg ; Straub:2015ica and lattice QCD Horgan:2015vla techniques. In view of these development, we calculate $B_{s}\rightarrow K^{\ast 0}$ form factors within the covariant confined quark model(CCQM).

The covariant confined quark modelEfimov:1988yd is an effective quantum field approach to hadronic interactions based on an interaction Lagrangian of hadrons interacting with their constituent quarks. The value of the coupling constant follows form the compositeness condition $Z_{H}=0$, where $Z_{H}$ is the wave function renormalization constant of the hadron. Matrix elements of the physical processes are generated by a set of quark loop diagrams according to the $1/N_{c}$ expansion. The ultraviolet divergences of the quark loops are regularized by including vertex functions for the hadron-quark vertices. These function also describe finite size effects related to the non-pointlike hadrons. The quark confinement Branz:2009cd is built-in through an infrared cutoff on the upper limit of the scale integration to avoid the appearance of singularities in matrix elements. The infrared cutoff parameter $\lambda$ is universal for all processes. The covariant confined quark model has limited number of parameters: the light and heavy constituent quark masses, the size parameters which describe the size of the distribution of the constituent quarks inside the hadron and the infrared cutoff parameter $\lambda$. They are determined by a fit to available experimental data.

In calculations we used next values of the model parameters which are shown in Eq. (1).

[TABLE]

Below, we list the definitions of the dimensionless invariant transition form factors together with the covariant quark model expressions that allow one to calculate them. We closely follow the notation used in our papers Dubnicka:2016nyy ; Dubnicka:2015iwg .

[TABLE]

[TABLE]

We use $P=p_{1}+p_{2}$ and $q=p_{1}-p_{2}$ and the on–shell conditions $\epsilon_{2}^{\dagger}\cdot p_{2}=0$, $p_{i}^{2}=m_{i}^{2}$. Since there are three quark species involved in the transition, we have introduced a two–subscript notation $w_{ij}=m_{q_{j}}/(m_{q_{i}}+m_{q_{j}})$ $(i,j=1,2,3)$ such that $w_{ij}+w_{ji}=1$. The form factors defined in Eq. (3) satisfy the physical requirement $a_{0}(0)=a_{+}(0)$, which ensures that no kinematic singularity appears in the matrix element at $q^{2}=0$.

The form factors are calculated in the full kinematical region of momentum transfer squared and results of our numerical calculations are with high accuracy approximated by the parametrization

[TABLE]

the relative error is less than 1$\%$. The values of $F(0)$, $a$, and $b$ are listed in Table 1.

The curves are depicted in Fig. 1.

The obtained errors of the fitted parameters were of the order of $10\%$. Indeed it follows that form factors at $q^{2}=0$ were calculated with $10\%$ uncertainties. This implies at least $10\%$ uncertainty in form factors in the full kinematical region of momentum transfer squared.

For reference it is useful to relate the above form factors to those used, e.g., in Ref. Ball:2004rg (we denote them by the superscript c). The relations read

[TABLE]

We note in addition that the form factors (5) satisfy the constraints

[TABLE]

Since $a_{0}(0)=a_{+}(0)=g(0)$ we display in Table 2 the form factors $A_{0}^{c}(0)=(m_{1}-m_{2})[A_{0}(0)-A_{+}(0)]/(2m_{2})$, $A_{1}^{c}(0)=A_{0}(0)(m_{1}-m_{2})/(m_{1}+m_{2})$, $A_{2}^{c}(0)=A_{+}(0)$, $T_{1}^{c}(0)=g(0)$ and $T_{3}^{c}(0)=\lim_{\,q^{2}\to 0}(m_{1}^{2}-m_{2}^{2})(a_{+}-a_{0})/q^{2}$ obtained in our model and compare them with those from light-cone sum rule Ball:2004rg .

Acknowledgment

We thank Prof. Mikhail A. Ivanov for the continuous support through out this work and for useful discussions of some aspects.

The work has been carried out under financial support of the Program of the Ministry of Education and Science of the Republic of Kazakhstan IRN number AP05132978.

Author A. Issadykov is grateful for the support by the JINR, grant number 18-302-03.