Study of the $\Lambda_{b} \to N^\ast \ell^+ \ell^- $ decay in light cone sum rules

TL;DR

This paper calculates the form factors and branching ratio of the $ ext{Lambda}_b o N^* ext{l}^+ ext{l}^-$ decay using light cone QCD sum rules, predicting a measurable decay rate at LHCb.

Contribution

It introduces a method to eliminate negative parity baryon contributions and provides the first detailed calculation of the decay's form factors and branching ratio.

Findings

Branching ratio is predicted to be large and measurable.

Method successfully isolates the desired decay contributions.

Results support potential observation at LHCb.

Abstract

The form factors of the $\Lambda_{b} \to N^\ast \ell^+ \ell^-$ decay are calculated in the framework of the light cone QCD sum rules. In the calculations the contribution of the negative parity $\Lambda_b^\ast$ baryon is eliminated by constructing the sum rules for different Lorentz structures. Furthermore the branching ratio of the semileptonic $\Lambda_b \to N^\ast \ell^+ \ell^-$ decay is calculated. The numerical study for the branching ratio of the $\Lambda_{b} \to N^\ast \ell^+ \ell^-$ decay indicates that it is quite large and could be measurable at future planned experiments to be conducted at LHCb.

Tables

Table 1. Table 1: The parametrization of the form factors of the Λ b → N ∗ ​ ℓ + ​ ℓ − → subscript Λ 𝑏 superscript 𝑁 ∗ superscript ℓ superscript ℓ \Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-} decay for LQSR–1

Table 2. Table 2: The same as Table 1, but for LQSR–2

Table 3. Table 3: Branching ratios for the Λ b → N ∗ ​ ℓ + ​ ℓ − , ( ℓ = e , μ , τ ) → subscript Λ 𝑏 superscript 𝑁 ∗ superscript ℓ superscript ℓ ℓ 𝑒 𝜇 𝜏 \Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-},~{}(\ell=e,\mu,\tau) decays

Table 4. Table 4: Parameters of the DAs for the N ∗ ​ ( 1535 ) superscript 𝑁 ∗ 1535 N^{\ast}(1535) baryon at μ 2 = 2 ​ G ​ e ​ V 2 superscript 𝜇 2 2 𝐺 𝑒 superscript 𝑉 2 \mu^{2}=2~{}GeV^{2}

Equations

$$⟨{\Lambda }_Q(p-q)$$

$${\bar{u}}_{\Lambda }(p$$

$$⟨{\Lambda }_Q(p-q)$$

$${\bar{u}}_{\Lambda }(p$$

$$⟨\Lambda (p-q)$$

$${\bar{u}}_{\Lambda }(p$$

$${\Pi }_{\mu }^j(p,q)=i\int d^{4}x{e}^{iqx}⟨0\mid T\{{\eta }_{{\Lambda }_b}(0)J_{\mu }^j(x)\}\mid N^{\ast }(p)⟩,$$

$$\displaystyle J_{\mu}^{j}=\left\{\begin{array}[]{ll}\bar{b}\gamma_{\mu}(1-\gamma_{5})d~{},&j=I\\ \bar{b}~{}i\sigma_{\alpha\nu}q^{\nu}(1+\gamma_{5})d~{},&j=II~{}.\end{array}\right.$$

$${\eta }_{{\Lambda }_b}$$

$${\Pi }_{\mu }^j(p,q)=\sum _i\frac{⟨0\mid {\eta }_{{\Lambda }_b}(0)\mid {\Lambda }_b^i(p-q,s)⟩⟨{\Lambda }_b^i(p-q,s)\mid \bar{b}{\Gamma }_{\mu }^{j}d\mid N^{\ast }(p)⟩}{m_i^2-(p-q{)}^2},$$

$$\displaystyle\Gamma_{\mu}^{j}=\left\{\begin{array}[]{ll}\gamma_{\mu}(1-\gamma_{5})~{},&j=I~{},\\ i\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})~{},&j=II~{},\end{array}\right.$$

$$⟨0\mid {\eta }_{{\Lambda }_b}\mid {\Lambda }_b(p-q)⟩$$

$$⟨0\mid {\eta }_{{\Lambda }_b^{\ast }}\mid {\Lambda }_b^{\ast }(p-q)⟩$$

$$(/p-m_{N^{\ast }})u_{N^{\ast }}(p)=0,$$

$${\Pi }_{\mu }^I(p,q)$$

$${\Pi }_{\mu }^{II}(p,q)$$

$${ϵ}^{abc}⟨0\mid u_{\alpha }^a(0)d_{\beta }^b(x)d_{\lambda }^c(0)\mid N^{\ast }(p)⟩,$$

$${\Pi }_{\mu }^j$$

$$S_b(x)$$

$${ϵ}^{abc}⟨0\mid u_{\alpha }^a(0)d_{\beta }^b(x)d_{\lambda }^c(0)\mid N^{\ast }(p)⟩,$$

$${\Pi }_{\mu }^j(p,q)=\left[{\Pi }_1^j{p}_{\mu }+{\Pi }_2^j{p}_{\mu }/q+{\Pi }_3^j{\gamma }_{\mu }+{\Pi }_4^j{\gamma }_{\mu }/q+{\Pi }_5^j{q}_{\mu }+{\Pi }_6^j{q}_{\mu }/q\right]{\gamma }_5{u}_{N^{\ast }}.$$

$${\Pi }_i^j\left[(p-q{)}^2,q^2\right]=\sum _{j=1,2,3}{\int }_0^1\frac{{\mathcal{D}}_{ij}(x,q^2)}{{\Delta }^n},$$

$$\Delta =m_b^2-(1-x)q^2+x(1-x)m_{N^{\ast }}^2-x(p-q{)}^2.$$

$${\Pi }_i^j\left[(p-q{)}^2,q^2\right]=\frac{1}{\pi }{\int }_{m_b^2}^{\infty }\frac{ds}{s-(p-q{)}^2}\text{mbox}Im{\Pi }_i^j[(p-q{)}^2,q^2].$$

$$\Delta =x\left[s(x)-(p-q{)}^2\right].$$

$${\int }_{s_0}^{\infty }\frac{ds}{s-(p-q{)}^2}{\rho }_i(q^2)\simeq \frac{1}{\pi }{\int }_{s_0}^{\infty }\frac{ds}{s-(p-q{)}^2}\text{mbox}Im{\Pi }_i^j(q^2),$$

$$\frac{1}{s-(p-q{)}^2}\to e^{-s/M^2}.$$

$$\int dx\frac{{\mathcal{D}}_{ij}^{(\alpha )}(x)}{\Delta }$$

$$\int dx\frac{{\mathcal{D}}_{ij}^{(\alpha )}(x)}{{\Delta }^2}$$

$$\int dx\frac{{\mathcal{D}}_{ij}^{(\alpha )}(x)}{{\Delta }^3}$$

$$s_0=\frac{m_b^2-\bar{x}{q}^2+x\bar{x}{m}_{N^{\ast }}^2}{x}.$$

$$-2{\lambda }_{{\Lambda }_b}{g}_1(q^2)e^{-m_{{\Lambda }_b}^2/M^2}+2{\lambda }_{{\Lambda }_b^{\ast }}{\tilde{g}}_1(q^2)e^{-m_{{\Lambda }_b^{\ast }}^2/M^2}$$

$$2{\lambda }_{{\Lambda }_b}\frac{g_2(q^2)}{m_{{\Lambda }_b}}{e}^{-m_{{\Lambda }_b}^2/M^2}-2{\lambda }_{{\Lambda }_b^{\ast }}\frac{{\tilde{g}}_2(q^2)}{m_{{\Lambda }_b^{\ast }}}{e}^{-m_{{\Lambda }_b^{\ast }}^2/M^2}$$

$$\displaystyle-\lambda_{\Lambda_{b}}e^{-m_{\Lambda_{b}}^{2}/M^{2}}\Big{\{}(m_{\Lambda_{b}}-m_{N^{\ast}})\Big{[}g_{1}(q^{2})-\frac{g_{2}(q^{2})}{m_{\Lambda_{b}}}(m_{\Lambda_{b}}+m_{N^{\ast}})\Big{]}\Big{\}}-$$

$$\displaystyle\lambda_{\Lambda_{b}^{\ast}}e^{-m_{\Lambda_{b}^{\ast}}^{2}/M^{2}}\Big{\{}(m_{\Lambda_{b}^{\ast}}+m_{N^{\ast}})\Big{[}\widetilde{g}_{1}(q^{2})+\frac{\widetilde{g}_{2}(q^{2})}{m_{\Lambda_{b}^{\ast}}}(m_{\Lambda_{b}^{\ast}}-m_{N^{\ast}})\Big{]}\Big{\}}\!\!\!$$

$$\displaystyle-\lambda_{\Lambda_{b}}e^{-m_{\Lambda_{b}}^{2}/M^{2}}\Big{[}g_{1}(q^{2})-\frac{g_{2}(q^{2})}{m_{\Lambda_{b}}}(m_{\Lambda_{b}}+m_{N^{\ast}})\Big{]}+$$

$$\displaystyle\lambda_{\Lambda_{b}^{\ast}}e^{-m_{\Lambda_{b}^{\ast}}^{2}/M^{2}}\Big{[}\widetilde{g}_{1}(q^{2})+\frac{\widetilde{g}_{2}(q^{2})}{m_{\Lambda_{b}^{\ast}}}(m_{\Lambda_{b}^{\ast}}-m_{N^{\ast}})\Big{]}\!\!\!$$

Full text

**Study of the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay in light cone sum rules

**

T. M. Aliev ${}^{a}\!\!$   , T. Barakat ${}^{b}\!\!$   , M. Savcı ${}^{a}\!\!$

a Physics Department, Middle East Technical University, 06531 Ankara, Turkey

b Physics Department, King Saud University

Riyadh 11451, Saudi Arabia e-mail: taliev@metu.edu.tre-mail: tbarakat@ksu.edu.sae-mail: savci@metu.edu.tr

The form factors of the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay are calculated in the framework of the light cone QCD sum rules. In the calculations the contribution of the negative parity $\Lambda_{b}^{\ast}$ baryon is eliminated by constructing the sum rules for different Lorentz structures. Furthermore the branching ratio of the semileptonic $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay is calculated. The numerical study for the branching ratio of the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay indicates that it is quite large and could be measurable at future planned experiments to be conducted at LHCb.

PACS numbers: 11.55.Hx; 13.30.Ce; 14.20.Gk

1 Introduction

Lately, exciting experimental results have been obtained in study of the rare decays of the heavy $\Lambda_{b}$ baryon induced by the flavor changing neutral currents. The rare $\Lambda_{b}\rightarrow\Lambda\ell^{+}\ell^{-}$ decays induced by the $b\to s$ transition were observed by the CDF [1] and LHCb collaborations [2]. Later the detailed analyses of the differential branching ratio and various symmetries of the $\Lambda_{b}\rightarrow\Lambda\ell^{+}\ell^{-}$ decay have been performed performed at LHCb [3]. The LHCb collaboration firstly observed the rare $\Lambda_{b}\rightarrow p\pi^{-}\mu^{+}\mu^{-}$ decay induced by the $b\to d$ transition [4]. This observation motivated the theoretical study of the $\Lambda_{b}\rightarrow N\ell^{+}\ell^{-}$ decay, induced also by the $b\to d$ transition. This decay was studied within the framework of the light cone QCD sum rules method (LCSR) in [5]. The light cone QCD sum rules method (LCSR) [6, 7] is hybrid of the traditional SVZ sum rules [8] and the methods used in hard exclusive processes. The other interesting decays induced by the $b\to d$ transition are the $\Lambda_{b}\to$ nucleon resonance decays. The analysis of these decays can provide complementary information about the properties of the nucleon resonances in principle which could experimentally be studied at LHCb. It should be noted here that the comprehensive study of the nucleon resonance constitutes one of the main research directions of the research program that is planned for future study at Jefferson Laboratory [9]. The properties of the nucleon resonance $N^{\ast}$ in the $\Lambda_{b(c)}\to N^{\ast}\ell\nu$ decay is investigated in framework of the LCSR in [10]. The present work is devoted to the study of the rare $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay in the framework of the LCSR method.

The paper is organized as follows: In section 2 the LCSR for the relevant form factors appearing in the $\Lambda_{b}(\Lambda_{b}^{\ast})\to N^{\ast}$ transitions are obtained. In section 3 present the numerical analysis of the sum rules for the form factors. Using then the obtained results for the form factors we estimate the decay widths of the $\Lambda_{b}(\Lambda_{b}^{\ast})\to N^{\ast}\ell^{+}\ell^{-}$ decays. This section ends with a conclusion.

2 Form factors of the $\Lambda_{b}(\Lambda_{b}^{\ast})\to\ell^{+}\ell^{-}$

decay in LCSR

In the present section we derive the LCSR for the transition form factors of the $\Lambda_{b}(\Lambda_{b}^{\ast})\to\ell^{+}\ell^{-}$ decay. Before giving the details of the calculations few words about the notation should be mentioned. In all further discussions the negative parity states of the $\Lambda_{b}$ and $N$ baryons are denoted as $\Lambda_{b}^{\ast}$ and $N^{\ast}$, respectively. The $\Lambda_{b}(\Lambda_{b}^{\ast})\to N^{\ast}\ell^{+}\ell^{-}$ decay at the quark level is described by the $b\to d$ transition. At the hadronic level $\Lambda_{b}(\Lambda_{b}^{\ast})\to N^{\ast}\ell^{+}\ell^{-}$ decay is obtained by sandwiching the transition current between the $\Lambda_{b}(\Lambda_{b}^{\ast})$ and $N^{\ast}$ states. The corresponding form factors of the vector, axial vector and tensor currents are defined as,

[TABLE]

The form factors responsible for $\Lambda_{b}^{\ast}\to N^{\ast}$ transition can be obtained from Eqs. (2), (2) and (2) with the help of the following replacements: $f_{i}\to\widetilde{f}_{i}$, $g_{i}\to\widetilde{g}_{i}$, $f_{i}^{T}\to\widetilde{f}_{i}^{T}$, $g_{i}^{T}\to\widetilde{g}_{i}^{T}$, $m_{\Lambda_{b}}\to m_{\Lambda_{b}^{\ast}}$, and $\bar{u}_{\Lambda}(p-q)\to\bar{u}_{\Lambda^{\ast}}(p-q)\gamma_{5}$.

In order to derive the LCSR for these form factors we introduce the correlation function

[TABLE]

where $\eta_{\Lambda_{b}}$ is the interpolating current of the $\Lambda_{b}(\Lambda_{b}^{\ast})$ baryon, $J_{\mu}^{j}(x)$ is the heavy–light transition current which is set to,

[TABLE]

In the calculations we use the following most general form of the interpolating current for the $\Lambda_{b}(\Lambda_{b}^{\ast})$ baryon,

[TABLE]

where $a,~{}b$ and $c$ are the color indices, $C$ is the charge conjugation operator, and $\beta$ is an arbitrary parameter with $\beta=-1$ corresponding to the Ioffe current.

The usual procedure in deriving the LCSR is to calculate the correlation function given in Eq. (4) in two different domains. On one side, insert a complete set of states with the quantum numbers of $\Lambda_{b}$ between the two currents, and isolate the ground state contribution. On the other side use the operator product expansion (OPE) around the light cone where $(p=q)^{2},~{}q^{2}\ll 0$. These two representations of the correlation function are then matched using the dispersion relations and quark–gluon duality ansatz. Finally, applying the Borel transformation in order to kill the the possible subtraction terms which could appear in the dispersion relations, and to suppress the contributions from higher states.

It should be remarked here that the interpolating current $\eta_{\Lambda_{b}}$ has nonzero overlap not only with the $J^{P}=\displaystyle{1\over 2}^{+}$ state but also with the $J^{P}=\displaystyle{1\over 2}^{-}$ state. It is shown in [11] that the mass difference between between the $J^{P}=\displaystyle{1\over 2}^{+}$ and $J^{P}=\displaystyle{1\over 2}^{-}$ states is about $(200-300)~{}MeV$. For this reason the contribution of the negative parity $\Lambda_{b}$ baryon should properly be taken into account.

After having mentioned these cautionary remarks we proceed to calculate the physical part of the correlation function. Saturating Eq. (4) with the ground and first excited $\Lambda_{b}$ baryon we get,

[TABLE]

where

[TABLE]

and summation is performed over the ground and first excited states of the $\Lambda_{b}$ baryon. The decay constants of the positive and negative parity $\Lambda_{b}$ baryons are determined as,

[TABLE]

Using the equation of motion

[TABLE]

and Eqs. (2–2) and (2), for the phenomenological part of the correlation function we get,

[TABLE]

Now we turn our attention to the calculation of the correlation function from the QCD side. At deep Eucledian domain $(p-q)^{2},~{}q^{2}\ll 0$ the product of the two currents can be expanded around the light–cone $x^{2}\simeq 0$. After contracting the heavy quark fields which give the heavy quark propagator, the matrix element

[TABLE]

of the three quarks between the vacuum and the $N^{\ast}$ state is revealed. Decomposition of this matrix element in terms of the distribution amplitudes (DAs) with increasing twist is given in [12] (see Appendix A).

After contracting the heavy b–quark fields, the correlation takes the form,

[TABLE]

[TABLE]

where $K_{i}$ are the modified Bessel functions of the second kind, and $G_{\alpha\beta}$ is the gluon field strength tensor.

Using the expression of the heavy quark propagator and definition of the matrix

[TABLE]

in terms of the DAs of the $N^{\ast}$ baryon, we can calculate the correlation function from the QCD side. Note that, using the equation of motion $(\not\!{p}-m_{N^{\ast}})u_{N^{\ast}}(p)=0$, the correlation function can be decomposed into six independent functions as follows,

[TABLE]

After performing Borel transformation, these invariant functions in the correlation function can in general be written as,

[TABLE]

where

[TABLE]

The explicit forms of ${\cal D}_{ij}$ are quite lengthy and for this reason we do not present them here. We can write (17) as a dispersion integral in $(p-q)^{2}$ as follows,

[TABLE]

Making the replacement $s(x)=m_{b}^{2}-(1-x)q^{2}+x(1-x)m_{N^{\ast}}^{2}-x(p-q)^{2}$, the denominator of Eq. (17) takes the form,

[TABLE]

Using the quark–hadron ansatz the contribution of the hadronic states can be represented as,

[TABLE]

where $s_{0}$ is the continuum contribution. Finally Borel transformation can be performed on the hadronic and physical sides with the help of the replacement

[TABLE]

In implementing the Borel transformation and the continuum subtraction we use the following relations,

[TABLE]

where $x_{0}$ is the solution of the equation

[TABLE]

Equating the coefficients of the structures $p_{\mu}\gamma_{5}$, $p_{\mu}\hbox to0.0pt{/\hss}{q}\gamma_{5}$, $\gamma_{\mu}\gamma_{5}$, $\gamma_{\mu}\hbox to0.0pt{/\hss}{q}\gamma_{5}$, $q_{\mu}\gamma_{5}$, and $q_{\mu}\hbox to0.0pt{/\hss}{q}\gamma_{5}$ we get the following sum rules for the invariant functions of the transition current ($\bar{b}\gamma_{\mu}d$),

[TABLE]

The results for the form factors induced by the $\bar{b}\gamma_{\mu}\gamma_{5}d$ current can be obtained from Eq. (2) by making the following replacements: $g_{i}\rightarrow-f_{i}$, $\widetilde{g}_{i}\rightarrow-\widetilde{f}_{i}$, $m_{N^{\ast}}\rightarrow-m_{N^{\ast}}$, and $\Pi_{i}^{I}\rightarrow\Pi_{i}^{I\prime}$.

The sum rules of the ($\bar{b}i\sigma_{\mu\nu}q^{\nu}d$) transition current can be obtained in similar manner, which are given below,

[TABLE]

where $\Pi_{1}^{II}$, $\Pi_{2}^{II}$, $\Pi_{3}^{II}$, and $\Pi_{4}^{II}$ are the invariant functions for the structures $p_{\mu}\hbox to0.0pt{/\hss}{q}\gamma_{5}$, $q_{\mu}\hbox to0.0pt{/\hss}{q}\gamma_{5}$, $q_{\mu}\gamma_{5}$, and $\gamma_{\mu}\gamma_{5}$, respectively. The sum rules for the ($\bar{b}i\sigma_{\mu\nu}q^{\nu}\gamma_{5}d$) transition current can be obtained from Eq. (2) by making the replacements $g_{i}^{T}\rightarrow f_{i}^{T}$, $\widetilde{g}_{i}^{T}\rightarrow\widetilde{f}_{i}^{T}$, $m_{N^{\ast}}\rightarrow-m_{N^{\ast}}$, and $\Pi_{i}^{II}\rightarrow\Pi_{i}^{II\prime}$. The explicit form of these invariant functions are quite lengthy, and for this reason we do not present them in this work.

Solving these equations we can eliminate the $\Lambda^{\ast}$ pole from the sum rules. As the result we obtain the desired sum rules responsible for the $\Lambda_{b}\to N^{\ast}$ transition. In the next section we present our numerical results on these form factors.

3 Numerical analysis

In this section we present our numerical results on the form factors that describe the $\Lambda_{b}\to N^{\ast}$ transition. First let us specify the input parameters which are needed in performing the numerical calculations. The masses of the $\Lambda_{b}$ and $\Lambda_{b}^{\ast}$ baryons which we use in our calculations are $\Lambda_{b}=5.62~{}GeV$ and $\Lambda_{b}^{\ast}=5.85~{}GeV$, and the mass of the nucleon is $m_{N^{\ast}}=1.52~{}GeV$ [13]. The residues $\lambda_{\Lambda_{b}}$ and $\lambda_{\Lambda_{b}^{\ast}}$ of the relevant baryons are taken from [5] having the values $\lambda_{\Lambda_{b}}=(6.5\pm 1.5)\times 10^{-2}~{}GeV^{3}$ and $\lambda_{\Lambda_{b}^{\ast}}=(7.5\pm 2.0)\times 10^{-2}~{}GeV^{3}$. The mass of the $b$ quark is assigned to its $\overline{MS}$ given as $m_{b}=(4.16\pm 0.03)~{}GeV$ [13]. The values of the quark condensates of the light quarks are taken as, $\langle\bar{u}u\rangle(1~{}GeV)=\langle\bar{d}d\rangle(1~{}GeV)=-(246_{-19}^{+28}~{}MeV)^{3}$. As has already been noted the main nonperturbative parameters are DAs of the $N^{\ast}$ baryon. The expressions of the $N^{\ast}$ DAs, as well as the coefficients $\phi_{i}^{(\pm,0)}$, $\psi_{i}^{(\pm,0)}$, and $\xi_{i}^{(\pm,0)}$, appearing in the DAs are obtained in [12] (also in [14, 15, 16, 17, 18]), and for completeness they are presented in Appendix A.

The sum rules for the transition form factors contain three auxiliary parameters: The Borel mass parameter $M^{2}$, the continuum threshold $s_{0}$ and the arbitrary parameter $\beta$. The Borel mass parameter and the continuum threshold $s_{0}$ are determined from the criteria that the sum rule dictates, i.e., the suppression of the contributions coming from the continuum states and the higher twist contributions should be satisfied. Our analysis shows that the working regions of $M^{2}$ and $s_{0}$ lie in the region $M^{2}=(10\pm 5)~{}GeV^{2}$, $s_{0}=(40\pm 1)~{}GeV^{2}$, when aforementioned conditions are fulfilled, and hence sum rules predictions are reliable. The final step of our analysis is is the determination of the working region of the parameter $\beta$. Our numerical study shows that when $-1\leq\cos\theta\leq-0.5$, where $\tan\theta=\beta$ the results for the residues and masses are rather stable with respect to the variation of $\beta$, and we choose $\beta=-1$.

The LCSR predictions are reliable up to the range $q^{2}\leq q_{max}^{2}=(m_{\Lambda_{b}}-m_{N^{\ast}})^{2}$. In order to calculate the decay width the LCSR predictions for the form factors need to be extrapolated to the whole physical region. For this purpose we use the z–series parametrization that is proposed in [19],

[TABLE]

where $t_{0}=(m_{\Lambda_{b}}-m_{N^{\ast}})^{2}$, $t_{+}=(m_{B}+m_{\pi})^{2}$. The parametrization that best reproduces the form factors predicted by the LCSR in the region $q^{2}\leq 11~{}GeV^{2}$, is given as

[TABLE]

For the pole masses we use,

[TABLE]

In Tables 1 and 2 we present the fit parameters $a_{0}$, $a_{1}$ and $a_{2}$ that results from our numerical analysis.

Using the definition of the form factors the differential decay width is calculated in the standard manner whose result is given below:

[TABLE]

where $v_{\ell}=\sqrt{1-4m_{\ell}^{2}/q^{2}}$ is the lepton velocity, $\lambda(1,r,s)=1+r^{2}+s^{2}-2r-2s-2rs$, $s=q^{2}/m_{\Lambda_{b}}$, and $r=m_{N^{\ast}}^{2}/m_{\Lambda_{b}}^{2}$, $\alpha$ is the fine structural constant, and the expressions of $\Gamma_{1}(s)$ and $\Gamma_{2}(s)$ are given in the Appendix–B.

The dependence of the differential branching ratios on $q^{2}$ for the $\Lambda_{b}\to N^{\ast}\mu^{+}\mu^{-}$ and $\Lambda_{b}\to N^{\ast}\tau^{+}\tau^{-}$ decays, at $s_{0}=40~{}GeV^{2}$, $M^{2}=25~{}GeV^{2}$ and $\beta=-1$ are presented in Figs. (1) and (2), respectively. In these figures the graphical results predicted by the LCSR–1 and LCSR–2 cases are shown together.

Performing integration over $s$ in the range $4m_{\ell}^{2}/m_{\Lambda_{b}}^{2}\leq s\leq(1-\sqrt{r})^{2}$, we obtain the branching ratios for the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}~{}(\ell=e,\mu,\tau)$ decays in the case when long distance effects are due to the $J/\psi$ family, and these results are given in Table 3.

It follows from these results that, especially for the $e$ and $\mu$ channels, the branching ratios are quite large and could potentially be measurable at LHCb. The discovery of these decays would provide useful information about the inner structure of the $N^{\ast}$ baryon.

Finally, a comparison of the $\Lambda_{b}\to N\ell^{+}\ell^{-}$ and $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decays shows that the central values of the branching ratio of the former is approximately two (eight) times larger than the $\Lambda_{b}\to N^{\ast}e^{+}e^{-}$ ($\Lambda_{b}\to N^{\ast}\tau^{+}\tau^{-}$) decays.

Conclusion

In this work the transition form factors of the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decay are estimated, which is an alternative approach to extract information about the inner properties of the $N^{\ast}$ baryon. The contribution coming from the $\Lambda_{b}^{\ast}$ baryon is eliminated by constructing the sum rules with the choice of several different Lorentz structures. Using this result, we also calculate the branching ratio of the $\Lambda_{b}\to N\ell^{+}\ell^{-}$ and decays. We see that the branching ratios of the $\Lambda_{b}\to N^{\ast}e^{+}e^{-}$ and $\Lambda_{b}\to N^{\ast}\mu^{+}\mu^{-}$ seem to be large enough to be detected at LHCb.

4 Acknowledgment

We sincerely thank Dr. M. Emmerich for providing us explicit forms of the DAs of the $N^{\ast}$ baryon. One of the authors, T. Barakat, thanks to the International Scientific Partnership Program ISPP at the King Saud University for funding his research work through ISPP No: 0038.

Appendix A: $N^{\ast}$ distribution amplitudes

In this Appendix, we present the $N^{\ast}$ DAs, which are necessary to calculate the $\Lambda\to N^{\ast}$ transition form factors. The DAs of the $N^{\ast}$ baryon are defined from the matrix element $\langle 0\left|\epsilon^{abc}u_{\alpha}^{a}(a_{1}x)d_{\beta}^{b}(a_{2}x)d_{\gamma}^{c}(a_{3}x)\right|N^{\ast}(p)\rangle$. The general decomposition of this matrix in terms of the DAs of the $N^{\ast}$ baryon is given below. (see [12]),

[TABLE]

The functions labeled with calligraphic letters in the above expression do not possess definite twists but they can be written in terms of the $N^{\ast}$ distribution amplitudes (DAs) with definite and increasing twists via the scalar product $p\!\cdot\!x$ and the parameters $a_{i}$, $i=1,2,3$. The relations between the two sets of DAs for the $N^{\ast}$, and for the scalar, pseudo-scalar, vector, axial vector and tensor DAs for nucleons are:

[TABLE]

where the terms in $x^{2}$, $\mathcal{V}_{1}^{M},\mathcal{A}_{1}^{M}$ and $\mathcal{T}_{1}^{M}$ are left aside.

The distribution amplitudes $F[a_{i}(p\!\cdot\!x)]$= $S_{i}$, $P_{i}$, $V_{i}$, $A_{i}$, $T_{i}$ can be represented as:

[TABLE]

where, $x_{i}$ with $i=1,~{}2$ and $3$ are longitudinal momentum fractions carried by the participating quarks.

The explicit expressions for the $\Lambda$ DAs up to twist 6 are given as:

Twist–$3$ DAs:

[TABLE]

Twist–$4$ DAs:

[TABLE]

Twist–$5$ DAs:

[TABLE]

Twist-6:

[TABLE]

Finally the $x^{2}$ corrections to the corresponding expressions ${\cal V}_{1}^{M}$, ${\cal A}_{1}^{M}$, ${\cal T}_{1}^{M}$ for the leading twist DAs $V_{1}$, $A_{1}$ and $T_{1}$ in the momentum fraction space are given as:

[TABLE]

where

[TABLE]

The expression for the axial–vector function ${\cal A}_{1}^{M(u)}(x_{2})$ is given as:

[TABLE]

with

[TABLE]

Similarly, we get for the function ${\cal T}_{1}^{M(u)}(x_{2})$:

[TABLE]

where

[TABLE]

The following functions are encountered to the above amplitudes and they can be defined in terms of the 8 independent parameters, namely $f_{N^{\ast}}$, $\lambda_{1}$, $\lambda_{2}$ and $f_{1}^{u},f_{1}^{d},f_{2}^{d},A_{1}^{u},V_{1}^{d}$:

[TABLE]

where the parameters $A_{1}^{u},~{}V_{1}^{d},~{}f_{1}^{d},~{}f_{1}^{u}$ and $f_{2}^{d}$ are defined as [16],

[TABLE]

The numerical values of the parameters $\varphi_{10},~{}\varphi_{11},~{}\varphi_{20},~{}\varphi_{21},~{}\varphi_{22},~{}\eta_{10},~{}\eta_{11}$ and $f_{N^{\ast}}/\lambda_{1}^{N^{\ast}}$, and $\lambda_{1}^{N^{\ast}}/\lambda_{1}^{N}$ are presented in Table 4 (this Table is taken from [10]).

Appendix B: Differential widths for the

$\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ decays

In this Appendix we present the differential widths for the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-},~{}(\ell=e,\mu,\tau)$ decays. After lengthy, but straightforward calculations, for the differential rate of the $\Lambda_{b}\to N^{\ast}\ell^{+}\ell^{-}$ we get,

[TABLE]

where $s=q^{2}/m^{2}_{\Lambda_{b}}$, $r=m_{N^{\ast}}^{2}/m^{2}_{\Lambda_{b}}$, $G_{F}=1.17\times 10^{-5}$ GeV*-2* is the Fermi coupling constant, $v=\sqrt{1-4m_{\ell}^{2}/q^{2}}$ is the lepton velocity, and $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$ is the usual triangle function. For the element of the CKM matrix $\left|V_{tb}V_{td}^{\ast}\right|=(8.2\pm 0.6)\times 10^{-3}$ has been used [13]. The functions $T_{1}(s)$ and $T_{2}(s)$ are given as:

[TABLE]

[TABLE]

The differential decay width for the $\Lambda_{b}^{\ast}\rightarrow N^{\ast}\ell^{+}\ell^{-}$ transition cam be obtained from the differential decay width for the $\Lambda_{b}\rightarrow N^{\ast}\ell^{+}\ell^{-}$ by making the following replacements: $F_{i}\rightarrow\widetilde{G}_{i}$, $G_{i}\rightarrow\widetilde{F}_{i}$, $m_{N}\rightarrow-m_{N}$, and by changing the sign in front of the terms $\mbox{\rm Re}[F_{4}^{\ast}F_{5}]$, $\mbox{\rm Re}[F_{4}^{\ast}F_{6}]$, and $\mbox{\rm Re}[G_{4}^{\ast}G_{5}]$, as well as $m_{\Lambda_{b}}\rightarrow m_{\Lambda_{b}^{\ast}}$, $s\rightarrow s^{\prime}=q^{2}/m_{\Lambda_{b}^{\ast}}^{2}$, and $r\rightarrow r^{\prime}=m_{N^{\ast}}^{2}/m_{\Lambda_{b}^{\ast}}^{2}$. where

[TABLE]

Figure captions

Fig. (1) The dependence of the differential branching ratio for the $\Lambda_{b}\rightarrow N^{\ast}\mu^{+}\mu^{-}$ transition on $s$, at $s_{0}=40~{}GeV^{2}$, and $M^{2}=25~{}GeV^{2}$.

Fig. (2) The same as in Fig. (1), but for the $\Lambda_{b}\rightarrow N^{\ast}\tau^{+}\tau^{-}$ transition.