Semileptonic Decays of Heavy Baryons in the Relativistic Quark Model

Abstract

Semileptonic and rare semileptonic decays of heavy baryons are studied in the framework of the relativistic quark model based on the quark-diquark picture, quasipotential approach, and quantum chromodynamics (QCD). The form factors parametrizing the matrix elements of the weak transitions are calculated in the whole accessible kinematical range with the comprehensive account of the relativistic effects. The obtained results for the branching ratios and other observables agree well with the available experimental data.

1. Introduction

In recent years significant experimental progress has been achieved in studying decays of heavy baryons. Many decay channels of these baryons were observed and new more precise data are expected in near future, since heavy baryons are copiously produced at the Large Hadron Collider LHC. Weak decays of heavy baryons can serve as an additional source for the determination of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. Such decays can be also used to verify the lepton flavour universality, indications of which violation were reported recently in the semileptonic B meson decays.

Here we review our study of the bottom and charm baryon decays in the framework of the relativistic quark model based on the quark-diquark picture and the quasipotential approach [1,2,3,4,5,6,7]. Heavy baryons are considered to be the heavy-quark–light-diquark bound states. Such approximation reduces very complicated relativistic three-body problem to the subsequent solution of two more simple two-body problems. Firstly, diquark is considered to be a bound state of two quarks. Secondly, baryon is considered to be the bound state of a quark and diquark. It is important to remember that the diquark inside a baryon is composed of two quarks. Thus, it is not a point-like object and its interaction with gluons is smeared by the form factor which can be expressed through the overlap integral of diquark wave functions. It is also important to account for the relativistic nature of light and heavy quarks and diquarks and thus treat them fully relativistically.

To calculate weak decay rates of heavy baryons it is necessary to determine the form factors which parametrize the matrix elements of the weak current between initial and final baryon states. On the basis of the relativistic quark model the explicit expressions for the decay form factors were obtained in terms of the overlap integrals over the baryon wave functions which are known from their mass spectrum calculations [8,9]. All relativistic effects including contributions of the intermediate negative-energy states and wave function transformations from the rest to the moving reference frame were systematically taken into account. The obtained formulas are valid in the whole accessible range of the momentum transfer squared $q^2$ and do not require additional model assumptions. The convenient analytic expressions which accurately reproduce the $q^2$ dependence of form factors are given. The calculated form factors are used for obtaining predictions for the differential and total decay rates and different asymmetry parameters. Good agreement of theoretical results with experimental data is found.

2. Relativistic Quark Model

In the relativistic quark model based on the quark-diquark picture and the quasipotential approach the interaction of two quarks in a diquark and the quark-diquark interaction in a baryon are described by the diquark wave function ${\Psi }_d$ of the bound quark-quark state and by the baryon wave function ${\Psi }_B$ of the bound quark-diquark state, which satisfy the relativistic quasipotential equation of the Schrödinger type [10]

$$\left(\frac{b^2\left(M\right)}{2{\mu }_R}-\frac{{\mathbf{p}}^2}{2{\mu }_R}\right){\Psi }_{d,B}\left(\mathbf{p}\right)=\int \frac{d^{3}q}{{\left(2\pi \right)}^3}V(\mathbf{p},\mathbf{q};M){\Psi }_{d,B}\left(\mathbf{q}\right),$$

(1)

where the relativistic reduced mass and the center-of-mass system relative momentum squared on mass shell are

$${\mu }_R=\frac{M^4-{(m_1^2-m_2^2)}^2}{4M^3},b^2\left(M\right)=\frac{(M^2-{(m_1+m_2)}^2)(M^2-{(m_1-m_2)}^2)}{4M^2},$$

and M is the bound state mass (diquark or baryon), $m_{1,2}$ are the masses of quarks ($q_1$ and $q_2$) which form the diquark or of the diquark (d) and quark (q) which form the baryon (B), and $\mathbf{p}$ is their relative momentum.

The quasipotentials $V(\mathbf{p},\mathbf{q};M)$ of the quark-quark or quark-diquark interaction are constructed with the help of the QCD-motivated off-mass-shell scattering amplitude, projected onto the positive energy states. The effective quark interaction is taken to be the sum of the usual one-gluon exchange term and the mixture of long-range vector and scalar linear confining potentials with the mixing coefficient $\epsilon$. It is also assumed that the vector confining potential contains not only the Dirac term but the Pauli term, thus introducing the anomalous chromomagnetic quark moment $\kappa$. The explicit expressions for the quasipotentials are given in Ref. [1].

In the nonrelativistic limit the usual Cornell-like potential is reproduced

$$V\left(r\right)=-\frac{4}{3}\frac{{\alpha }_s}{r}+Ar+B,$$

(2)

where the QCD running coupling constant with freezing is given by

$${\alpha }_s\left({\mu }^2\right)=\frac{4\pi }{{\displaystyle \left(11-\frac{2}{3}{n}_f\right)ln\frac{{\mu }^2+M_B^2}{{\Lambda }^2}}},\mu =\frac{2m_1{m}_2}{m_1+m_2},$$

(3)

$n_f$ is the number of flavors, and the background mass $M_B=2.24\sqrt{A}=0.95$ GeV, $\Lambda =413$ MeV.

All parameters of the model were fixed previously from calculations of meson and baryon properties [1,10]. The constituent quark masses $m_u=m_d=0.33$ GeV, $m_s=0.5$ GeV, $m_c=1.55$ GeV and the parameters of the linear potential $A=0.18$ GeV${}^2$ and $B=-0.3$ GeV have the usual values of quark models. The value of the mixing coefficient of vector and scalar confining potentials $\epsilon =-1$ and the universal Pauli interaction constant $\kappa =-1$ [10]. Please note that the long-range chromomagnetic contribution to the potential, which is proportional to $(1+\kappa )$, vanishes for the chosen value of $\kappa =-1$.

The matrix element of weak current $J_{\mu }^W={\overline{Q}}^{\prime }{\gamma }_{\mu }(1-{\gamma }_5)Q$ between initial and final baryon states in the considered approach is given by [1]

$$\langle B_{Q^{\prime }}\left(P^{\prime }\right)|J_{\mu }^W|B_Q\left(P\right)\rangle =\int \frac{d^{3}p{d}^{3}q}{{\left(2\pi \right)}^6}{\overline{\Psi }}_{B_{Q^{\prime }}{\mathbf{P}}^{\prime }}\left(\mathbf{p}\right){\Gamma }_{\mu }(\mathbf{p},\mathbf{q}){\Psi }_{B_Q\mathbf{P}}\left(\mathbf{q}\right),$$

(4)

where ${\Gamma }_{\mu }(\mathbf{p},\mathbf{q})={\Gamma }_{\mu }^{\left(1\right)}(\mathbf{p},\mathbf{q})+{\Gamma }_{\mu }^{\left(2\right)}(\mathbf{p},\mathbf{q})$ is the two-particle vertex function that receives relativistic contributions both from the impulse approximation diagram (see Figure 1 of Ref. [1])

$${\Gamma }_{\mu }^{\left(1\right)}(\mathbf{p},\mathbf{q})={\psi }_d^{*}\left(p_d\right){\overline{u}}_{Q^{\prime }}\left(p_{Q^{\prime }}\right){\gamma }_{\mu }(1-{\gamma }^5)u_Q\left(q_Q\right){\psi }_d\left(q_d\right){\left(2\pi \right)}^3\delta ({\mathbf{p}}_d-{\mathbf{q}}_d),$$

(5)

and from the diagrams with the intermediate negative-energy states which are the consequence of the projection onto the positive-energy states in the quasipotential approach (see Figure 2 of Ref. [1])

$$\begin{array}{ccc}\hfill {\Gamma }_{\mu }^{\left(2\right)}(\mathbf{p},\mathbf{q})& =& {\psi }_d^{*}\left(p_d\right){\overline{u}}_{Q^{\prime }}\left(p_{Q^{\prime }}\right)\{{\gamma }_{\mu }(1-{\gamma }^5)\frac{{\Lambda }_Q^{(-)}\left(k\right)}{{ϵ}_Q\left(k\right)+{ϵ}_Q\left(p_{Q^{\prime }}\right)}{\gamma }^0{\mathcal{V}}_{Qd}({\mathbf{p}}_d-{\mathbf{q}}_d)\hfill \\ & & +{\mathcal{V}}_{Qd}({\mathbf{p}}_d-{\mathbf{q}}_d)\frac{{\Lambda }_{Q^{\prime }}^{(-)}\left(k^{\prime }\right)}{{ϵ}_{Q^{\prime }}\left(k^{\prime }\right)+{ϵ}_{Q^{\prime }}\left(q_Q\right)}{\gamma }^0{\gamma }_{\mu }(1-{\gamma }^5)\}{u}_Q\left(q_Q\right){\psi }_d\left(q_d\right),\hfill \end{array}$$

(6)

where

$${\Lambda }^{(-)}\left(p\right)=\frac{ϵ\left(p\right)-\left(m{\gamma }^0+{\gamma }^0\left(\mathbf{\gamma }\mathbf{p}\right)\right)}{2ϵ\left(p\right)}$$

and $\mathbf{k}={\mathbf{p}}_{Q^{\prime }}-\Delta$; ${\mathbf{k}}^{\prime }={\mathbf{q}}_Q+\Delta$; $\Delta ={\mathbf{P}}^{\prime }-\mathbf{P}$; $ϵ\left(p\right)=\sqrt{m^2+{\mathbf{p}}^2}$

The wave function ${\Psi }_{B_{Q^{\prime }}{\mathbf{P}}^{\prime }}$ of the moving baryon is connected with the rest-frame wave function ${\Psi }_{B_{Q^{\prime }}\mathbf{0}}\equiv {\Psi }_{B_{Q^{\prime }}}$ by the transformation

$${\Psi }_{B_{Q^{\prime }}{\mathbf{P}}^{\prime }}\left(\mathbf{p}\right)=D_{Q^{\prime }}^{1/2}\left(R_{L_{{\mathbf{P}}^{\prime }}}^W\right)D_d^{\mathcal{I}}\left(R_{L_{{\mathbf{P}}^{\prime }}}^W\right){\Psi }_{B_{Q^{\prime }}\mathbf{0}}\left(\mathbf{p}\right),\mathcal{I}=0,1,$$

(7)

where $R^W$ – Wigner rotation, and $L_{{\mathbf{P}}^{\prime }}$ is the Lorentz boost from the baryon rest frame to a moving one. The rotation matrix $D_Q^{1/2}\left(R\right)$ of heavy quark spin is given by

$$\left(\genfrac{}{}{0pt}{}{10}{01}\right)D_{Q^{\prime }}^{1/2}\left(R_{L_{{\mathbf{P}}^{\prime }}}^W\right)=S^{-1}\left({\mathbf{p}}_{Q^{\prime }}\right)S\left({\mathbf{P}}^{\prime }\right)S\left(\mathbf{p}\right),S\left(\mathbf{p}\right)=\sqrt{\frac{ϵ\left(p\right)+m}{2m}}\left(1+\frac{\mathbf{\alpha }\mathbf{p}}{ϵ\left(p\right)+m}\right),$$

while the rotation matrices $D_d^{\mathcal{I}}\left(R\right)$ of diquark with spin $\mathcal{I}=0,1$ are as follows: $D_d^0\left(R^W\right)=1$ for scalar diquark and $D_d^1\left(R^W\right)=R^W$ for axial vector diquark.

3. Semileptonic Decay form Factors

The schematic diagram of the semileptonic decay of a heavy baryon $B_Q\to B_{Q^{\prime }}\ell {\nu }_{\ell }$ ($Q=b,c$ and $Q^{\prime }=c,u$) in quark-diquark picture is shown in Figure 1. The heavy quark Q undergoes the weak transition, while the light diquark d acts as a spectator.

The hadronic matrix elements for the semileptonic decays of baryons with scalar diquark (${\Lambda }_Q$) are parameterized in terms of six invariant form factors

$$\begin{array}{l}\langle {\Lambda }_q(p^{\prime },s^{\prime })|V^{\mu }|{\Lambda }_Q(p,s)\rangle ={\overline{u}}_{{\Lambda }_q}(p^{\prime },s^{\prime })\left(f_1^V\left(q^2\right){\gamma }^{\mu }-f_2^V\left(q^2\right)i{\sigma }^{\mu \nu }\frac{q_{\nu }}{M_{{\Lambda }_Q}}+f_3^V\left(q^2\right)\frac{q^{\mu }}{M_{{\Lambda }_Q}}\right)u_{{\Lambda }_Q}(p,s),\\ \langle {\Lambda }_q(p^{\prime },s^{\prime })|A^{\mu }|{\Lambda }_Q(p,s)\rangle ={\overline{u}}_{{\Lambda }_q}(p^{\prime },s^{\prime })\left(f_1^A\left(q^2\right){\gamma }^{\mu }-f_2^A\left(q^2\right)i{\sigma }^{\mu \nu }\frac{q_{\nu }}{M_{{\Lambda }_Q}}+f_3^A\left(q^2\right)\frac{q^{\mu }}{M_{{\Lambda }_Q}}\right){\gamma }_5{u}_{{\Lambda }_Q}(p,s),\end{array}$$

(8)

where M and $u(p,s)$ are masses and Dirac spinors of the baryons.

Using Equation (4) for the decay matrix element of the weak current the form factors are expressed as overlap integrals of baryon wave functions. (The spin-flavour structure of the wave functions of the members of the light baryon octet in the quark-diquark model is given in Ref. [11].) Contribution both of leading ${\Gamma }^{\left(1\right)}$ and subleading order ${\Gamma }^{\left(2\right)}$ vertex functions as well as relativistic transformation of the baryon wave function form the rest to moving reference frame are taken into account. The explicit expressions are given in Ref. [1]. Substituting in these expressions the baryon wave functions, found in calculation of their mass spectra, we obtain the values of the decay form factors and determine their dependence on $q^2$ in the whole kinematical region. As an example in Figure 2 we plot form factors for the ${\Xi }_b\to {\Xi }_c$ transitions. The values of form factors at $q^2=0$ and $q^2=q_{\max }^2$ as well as accurate analytic approximation of numerical results for form factors in the accessible $q^2$ range are given in Refs. [1,2,3,4].

4. Semileptonic Decays

Using the obtained form factors and helicity formalism we calculate the total and differential semileptonic decays of heavy baryons as well as other important observables.

4.1. Bottom Baryon Decays

The semileptonic differential decay rates of the ${\Lambda }_b$ baryon are plotted in Figure 3. In

Table 1. Comparison of theoretical predictions for baryon semileptonic decays with experimental data.

Parameter	Our	Ivanov	Pervin	Dutta	Ke	Azizi	Detmold	Experiment
	RQM	CCQM	SRQM	ELA	LFQM	QCDSR	Lattice	PDG
	[1,2]	[12,13,14,15]	[16]	[17]	[18]	[20]	[21]	[22]
${\Lambda }_b\to {\Lambda }_{c}l\nu$
$\Gamma$ (ns${}^{-1}$)	44.2		53.9
$\Gamma /|V_{cb}{|}^2$ (ps${}^{-1}$)	29.1						$21.5\pm 0.8\pm 1.1$
$Br$ (%)	6.48	6.9		4.83	6.3	$6.04\pm 1.70$		$6.2_{-1.3}^{+1.4}$
${\Lambda }_b\to {\Lambda }_c\tau \nu$
$\Gamma$ (ns${}^{-1}$)	13.9		20.9
$\Gamma /|V_{cb}{|}^2$ (ps${}^{-1}$)	9.11						$7.15\pm 0.15\pm 0.27$
$Br$ (%)	2.03	2.0		1.63		$1.87\pm 0.52$
${\Lambda }_b\to pl\nu$
$\Gamma /|V_{ub}{|}^2$ (ps${}^{-1}$)	18.7	13.3	7.55				$25.7\pm 2.6\pm 4.6$
$Br$ (%)	0.045	0.029		0.0389	0.0254
${\Lambda }_b\to p\tau \nu$
$\Gamma /|V_{ub}{|}^2$ (ps${}^{-1}$)	12.1	9.6	6.55				$17.7\pm 1.3\pm 1.6$
$Br$ (%)	0.029	0.021		0.0275
${\Xi }_b\to {\Xi }_{c}l\nu$
$\Gamma /|V_{cb}{|}^2$ (ps${}^{-1}$)	25.7
$Br$ (%)	6.15			9.22
${\Xi }_b\to {\Xi }_c\tau \nu$
$\Gamma /|V_{cb}{|}^2$ (ps${}^{-1}$)	8.4
$Br$ (%)	2.00			2.35
${\Xi }_b\to \Lambda l\nu$
$\Gamma /|V_{ub}{|}^2$ (ps${}^{-1}$)	10.0
$Br$ (%)	0.026

we compare our predictions in the framework of the relativistic quark model (RQM) for the total semileptonic decay rates of bottom baryons with the results of other theoretical approaches [12,13,14,15,16,17,18,19,20,21] and available experimental data from Particle Data Group (PDG) [22]. The most comprehensive results for different decay parameters were previously obtained in the covariant confined quark model (CCQM) [12,13,14,15], with which we find the general agreement. The semirelativistic quark model (SRQM) is used in Ref. [16], while effective Lagrangian approach (ELA) with form factors calculated on the lattice [21] is employed in Ref. [17]. The authors of Refs. [18,19] made calculations in the light-front quark model (LFQM) and in QCD light-cone sum (LCSR) rules, respectively. Calculations in the framework of QCD sum rules (QCDSR) in full theory are given in Ref. [20]. The only experimental data are available for the branching ratio of the ${\Lambda }_b\to {\Lambda }_c^{+}{l}^{-}{\overline{\nu }}_l$ decay ($l=e,\mu$). All theoretical predictions agree well with data within error bars. However, note that lattice calculations [21] give somewhat lower predictions for the branching ratios normalized by the square of the corresponding CKM matrix element for ${\Lambda }_b\to {\Lambda }_c$ transitions but give higher results for ${\Lambda }_b\to p$ transitions than other approaches.

At present, the tension between predictions of the Standard Model and experimental data in the B meson sector is observed for the ratio of branching fractions of semileptonic B decays to $D^{(*)}$ mesons involving $\tau$ and muon or electron. Therefore it is very important to search for the similar decays in baryon sector. We can define the following ratios of the ${\Lambda }_b$ baryon branching fractions

$$R_{{\Lambda }_c}=\frac{Br({\Lambda }_b\to {\Lambda }_c\tau \nu )}{Br({\Lambda }_b\to {\Lambda }_{c}l\nu )},R_p=\frac{Br({\Lambda }_b\to p\tau \nu )}{Br({\Lambda }_b\to pl\nu )},R_{{\Lambda }_{c}p}=\frac{{\int }_{15{\mathrm{GeV}}^2}^{q_{max}^2}\frac{d\Gamma ({\Lambda }_b\to p\mu \nu )}{d{q}^2}d{q}^2}{{\int }_{7{\mathrm{GeV}}^2}^{q_{max}^2}\frac{d\Gamma ({\Lambda }_b\to {\Lambda }_c\mu \nu )}{d{q}^2}d{q}^2}.$$

(9)

Our predictions for these ratios are given in

Table 2. Comparison of theoretical predictions for the ratios of the decay rates with experimental data.

Ratio	Our	Dutta	Lattice	Experiment
	[1]	[17]	[21]	LHCb [23]
$R_{{\Lambda }_c}$	0.313	0.3379	$0.3318\pm 0.0074\pm 0.0070$
$R_p$	0.649	0.7071
$R_{{\Lambda }_{c}p}$	$(0.78\pm 0.08)\frac{|V_{ub}{|}^2}{|V_{cb}{|}^2}$	0.0101	$(1.471\pm 0.095\pm 0.109)\frac{|V_{ub}{|}^2}{|V_{cb}{|}^2}$	$(1.00\pm 0.04\pm 0.08)\times {10}^{-2}$

in comparison with calculations [17] based on lattice values of weak decay form factors [21]. Results of predictions for $R_{{\Lambda }_c}$ are in good agreement, while our $R_p$ value is slightly lower than the estimate in Ref. [17]. Please note that lattice determination of form factors is done in the region of small recoils of the final baryon $q^2\sim q_{\max }^2$ and then their values are extrapolated to the whole kinematical region, which is broad especially for the heavy-to-light ${\Lambda }_b\to pl{\nu }_l$ decay. In our model we explicitly determine the form factor $q^2$ dependence in the whole kinematical range without extrapolations.

The LHCb collaboration [23] measured the ratio of the heavy-to-heavy and heavy-to-light semileptonic ${\Lambda }_b$ decays in the limited interval of $q^2$. Such measurement is very important since it allows for the first time to extract the ratio of CKM matrix elements $|V_{ub}|/|V_{cb}|$ from the ${\Lambda }_b$ baryon decays and compare it to the corresponding ratio determined from B and $B_s$ meson decays. Our prediction for the ratio $R_{{\Lambda }_{c}p}$ in comparison with lattice result [21] and experimental value is given in Table 2. From this table we see that our value of the coefficient in front of $|V_{ub}{|}^2/{|V_{cb}|}^2$ is significantly lower than lattice one. This is the result of the above mentioned deviation of our (and other quark model calculations) from lattice predictions for normalized by the square of CKM matrix element value for heavy-to-heavy and heavy-to-light semileptonic ${\Lambda }_b$ decays. This deviation even increases in the ratio.

Comparing our result for $R_{{\Lambda }_{c}p}$ with experimental data we find

$$\frac{|V_{ub}|}{|V_{cb}|}=0.113\pm 0.011{{|}_{\mathrm{theor}}\pm 0.006|}_{\exp },$$

in good agreement with the experimental ratio of these matrix elements extracted from inclusive decays

$$\frac{|V_{ub}{|}_{\mathrm{incl}}}{|V_{cb}{|}_{\mathrm{incl}}}=0.105\pm 0.006,$$

and with the corresponding ratio found in our previous analysis of exclusive semileptonic B and $B_s$ meson decays [$|V_{cb}|=(3.90\pm 0.15)\times {10}^{-2}$, $|V_{ub}|=(4.05\pm 0.20)\times {10}^{-3}$]

$$\frac{|V_{ub}|}{|V_{cb}|}=0.104\pm 0.012.$$

4.2. Charm Baryon Decays

In Figure 4 we plot the differential decay rates for ${\Lambda }_c\to \Lambda l{\nu }_l$ and ${\Lambda }_c\to nl{\nu }_l$ ($l=e,\mu$) semileptonic decays. In

Table 3. Theoretical predictions for the ${\Lambda }_c$ baryon semileptonic decays and experimental data.

Parameter	Our	Ivanov	Pervin	Liu	Azizi	Experiment
	RQM	CCQM	SRQM	LCSR	LCSR	PDG
	[3]	[13,14,15]	[16]	[24]	[25]	[22]
${\Lambda }_c^{+}\to \Lambda e^{+}{\nu }_e$
$\Gamma$ (ns${}^{-1}$)	162	139	236
$Br$ (%)	3.25	2.78	4.72	$3.0\pm 0.3$		$3.6\pm 0.4$
${\Lambda }_c^{+}\to \Lambda {\mu }^{+}{\nu }_{\mu }$
$\Gamma$ (ns${}^{-1}$)	157	135	236
$Br$ (%)	3.14	2.69	4.72	$3.0\pm 0.3$		$3.5\pm 0.5$
${\Lambda }_c^{+}\to n{e}^{+}{\nu }_e$
$\Gamma$ (ns${}^{-1}$)	13.4		13.5
$\Gamma /|V_{cd}{|}^2$ (ps${}^{-1}$)	0.265	0.20			$8.21\pm 2.80$
$Br$ (%)	0.268	0.207	0.27		$8.69\pm 2.89$
${\Lambda }_c^{+}\to n{\mu }^{+}{\nu }_{\mu }$
$\Gamma /|V_{cd}{|}^2$ (ps${}^{-1}$)	0.260	0.19			$8.3\pm 2.85$
$Br$ (%)	0.262	0.202			$8.78\pm 2.89$

we compare our results for various ${\Lambda }_c$ semileptonic decay observables with the recent predictions of other theoretical approaches [13,14,15,16,24,25] and available experimental data [22]. The most detailed predictions are given in the framework of the covariant confined quark model [13,14,15]. The authors present the values not only of decay rates and branching fractions but also of different asymmetries and polarization parameters. We find good agreement with their results. The semirelativistic quark model is used in Ref. [16]. Different versions of QCD light-cone sum rules (LCSR) are employed in Refs. [24,25]. We compare theoretical predictions with experimental values given in PDG [22], which are available for the branching fractions of ${\Lambda }_c\to \Lambda l{\nu }_l$ decays. All theoretical predictions reasonably agree with data. For the ${\Lambda }_c\to nl{\nu }_l$ decay no data are available at present. Most of the theoretical predictions [13,14,15,16,24], except the light cone QCD sum rule approach [25], give close values for the branching fractions $Br({\Lambda }_c\to nl{\nu }_l)=0.2–0.3\%$.

Theoretical predictions [26,27,28,29] for the semileptonic ${\Xi }_c$ decay branching fractions are compared in

Table 4. Comparison of theoretical predictions for the ${\Xi }_c$ semileptonic decay branching fractions (in %) with available experimental data.

Decay	Our	Zhao	Geng	Geng	Azizi	Experiment
	RQM	LFQM	$\mathbf{SU}\left(3\right)$	$\mathbf{SU}\left(3\right)$	LCSR
	[4]	[26]	[27]	[28]	[29]
$Br({\Xi }_c^0\to {\Xi }^{-}{e}^{+}{\nu }_e)$	2.38	1.35	$4.87\pm 1.74$	$2.4\pm 0.3$	$7.26\pm 2.54$	$5.58\pm 2.62$
$Br({\Xi }_c^0\to {\Xi }^{-}{\mu }^{+}{\nu }_{\mu })$	2.31			$2.4\pm 0.3$	$7.15\pm 2.50$
$Br({\Xi }_c^{+}\to {\Xi }^0{e}^{+}{\nu }_e)$	9.40	5.39	$3.{38}_{-2.26}^{+2.19}$	$9.8\pm 1.1$	$28.6\pm 10.0$	$6.58\pm 3.85$
$Br({\Xi }_c^{+}\to {\Xi }^0{\mu }^{+}{\nu }_{\mu })$	9.11			$9.8\pm 1.1$	$28.2\pm 9.9$
$Br({\Xi }_c^{+}\to \Lambda e^{+}{\nu }_e)$	0.127	0.082		$0.166\pm 0.018$
$Br({\Xi }_c^{+}\to \Lambda {\mu }^{+}{\nu }_{\mu })$	0.124

. The light-front quark model is used for the weak decay form factor and branching fraction calculations in Ref. [26]. The predictions of Refs. [27,28] are based on the application of the SU(3) flavor symmetry, while the light-cone QCD sum rules are employed in Ref. [29]. The experimental branching fractions are obtained by multiplying values [22] for the ratios $\Gamma ({\Xi }_c^0\to {\Xi }^{-}{e}^{+}{\nu }_e)/\Gamma ({\Xi }_c^0\to {\Xi }^{-}{\pi }^{+})=3.1\pm 1.1$ and $\Gamma ({\Xi }_c^{+}\to {\Xi }^0{e}^{+}{\nu }_e)/\Gamma ({\Xi }_c^{+}\to {\Xi }^{-}{\pi }^{+}{\pi }^{+})=2.3_{-0.8}^{+0.7}$ by the recently measured by the Belle Collaboration branching fractions $Br({\Xi }_c^0\to {\Xi }^{-}{\pi }^{+})=(1.80\pm 0.50\pm 01.14)\%$ [30] and $Br({\Xi }_c^{+}\to {\Xi }^{-}{\pi }^{+}{\pi }^{+})=(2.86\pm 1.21\pm 0.38)\%$ [31]. We find reasonable agreement of our predictions for the CKM favored ${\Xi }_c\to \Xi \ell {\nu }_{\ell }$ decays with the results of Refs. [26,27,28] and experimental data, while the light-cone QCD sum rule values for branching fractions are substantially higher and disagree with data by more than a factor of 2 for the ${\Xi }_c^{+}\to {\Xi }^0{e}^{+}{\nu }_e$ decay. Please note that the ratio $\Gamma ({\Xi }_c^0\to {\Xi }^{-}{e}^{+}\mathrm{anything})/\Gamma ({\Xi }_c^0\to {\Xi }^{-}{\pi }^{+})=1.0\pm 0.5$ [22] combined with the Belle data [30,31] leads to the semi-inclusive branching fraction $Br({\Xi }_c^0\to {\Xi }^{-}{e}^{+}\mathrm{anything})=(1.80\pm 1.07)\%$ in good agreement with our result. There is no data yet for the CKM suppressed ${\Xi }_c^{+}\to \Lambda {\ell }^{+}{\nu }_{\ell }$ decays and theoretical evaluations differ substantially. Our prediction is about a factor of 6 larger than the light-cone quark model value [26] and a factor of 3 larger than the SU(3) flavor symmetry result [28]. The difference for the $Br({\Xi }_c^{+}\to \Lambda {\ell }^{+}{\nu }_{\ell })$ with Ref. [26] originates from larger values of form factors predicted by our model.

We can combine our predictions for the semileptonic ${\Xi }_c$ baryon decays with our analysis of the semileptonic ${\Lambda }_c$ decays to test the flavor $SU\left(3\right)$ symmetry. Under the exact $SU\left(3\right)$ symmetry the following relations should hold [26,27]

$$\frac{\Gamma ({\Lambda }_c\to ne{\nu }_e)}{|V_{cd}{|}^2}=\frac{3\Gamma ({\Lambda }_c\to \Lambda e{\nu }_e)}{2|V_{cs}{|}^2}=\frac{6\Gamma ({\Xi }_c\to \Lambda e{\nu }_e)}{|V_{cd}{|}^2}=\frac{\Gamma ({\Xi }_c\to \Xi e{\nu }_e)}{|V_{cs}{|}^2}.$$

(10)

In

Table 5. Predictions for the ratios $\Gamma /|V_{cq}{|}^2$ in ps${}^{-1}$ ($q=s,d$)

Decay	Our	Exact $\mathbf{SU}\left(3\right)$	Difference
${\Lambda }_c\to ne{\nu }_e$	0.265	0.265
${\Lambda }_c\to \Lambda e{\nu }_e$	0.167	0.177	6%
${\Xi }_c\to \Lambda e{\nu }_e$	0.059	0.044	34%
${\Xi }_c\to \Xi e{\nu }_e$	0.215	0.265	19%

we test these relations for the ratios $\Gamma /|V_{cq}{|}^2$, where the corresponding CKM matrix element is used. In the first column we give the semileptonic decay process. In the second column we present predictions of our model. The third column contains the flavor $SU\left(3\right)$ symmetry result using relations (10) and $\Gamma ({\Lambda }_c\to ne{\nu }_e)/{|V_{cd}|}^2$ as an input, while the last column gives the relative difference in %. From this table we see that the flavor $SU\left(3\right)$ is broken for the charmed baryon semileptonic decays especially for ${\Xi }_c$ where its breaking is about 20–35%. This is the consequence of the larger mass of the s quark in comparison with the $u,d$ quarks and the employed quark-diquark picture of baryons.

5. Rare Semileptonic ${\Lambda }_{\mathbf{b}}$ Baryon Decays

The effective Hamiltonian for the rare $b\to s$ transitions is given by

$${\mathcal{H}}_{\mathrm{eff}}=-\frac{4G_F}{\sqrt{2}}{V}_{ts}^{*}{V}_{tb}\sum _{i=1}^{10}{c}_i{\mathcal{O}}_i,$$

(11)

where $G_F$ is the Fermi constant, $V_{tj}$ are CKM matrix elements, $c_i$ are the Wilson coefficients. ${\mathcal{O}}_i$ are the standard model operators:

$${\mathcal{O}}_i\sim \left(\overline{s}b\right)\left(\overline{c}c\right),\mathrm{for}i=1\dots 6;{\mathcal{O}}_8\sim m_b\overline{s}(\sigma ·G)b$$

and the only operators with a tree-level non-vanishing matrix element in $b\to s{\ell }^{+}{\ell }^{-}$:

$$\begin{array}{l}{\mathcal{O}}_7=\frac{e}{4{\pi }^2}{\overline{s}}_L{\sigma }_{\mu \nu }{m}_b{b}_R{F}^{\mu \nu },\\ {\mathcal{O}}_9=\frac{e^2}{4{\pi }^2}{\overline{s}}_L{\gamma }^{\mu }{b}_L\overline{\ell }{\gamma }_{\mu }\ell ,\\ {\mathcal{O}}_{10}=\frac{e^2}{4{\pi }^2}{\overline{s}}_L{\gamma }^{\mu }{b}_L\overline{\ell }{\gamma }_{\mu }{\gamma }_5\ell .\end{array}$$

(12)

Then the matrix element of the $b\to s{l}^{+}{l}^{-}$ transition amplitude between baryon states has the following form

$$\mathcal{M}({\Lambda }_b\to \Lambda l^{+}{l}^{-})=\frac{G_F\alpha }{2\sqrt{2}\pi }|V_{ts}^{*}{V}_{tb}|\left(T_{\mu }^{\left(1\right)}\left(\overline{l}{\gamma }^{\mu }l\right)+T_{\mu }^{\left(2\right)}\left(\overline{l}{\gamma }^{\mu }{\gamma }_{5}l\right)\right),$$

(13)

where

$$\begin{array}{ccc}\hfill T_{\mu }^{\left(1\right)}& =& c_9^{eff}\langle \Lambda |\overline{s}{\gamma }^{\mu }(1-{\gamma }^5)b|{\Lambda }_b\rangle -\frac{2m_b}{q^2}{c}_7^{eff}\langle \Lambda |\overline{s}i{\sigma }^{\mu \nu }{q}_{\nu }(1+{\gamma }^5)b|{\Lambda }_b\rangle ,\hfill \\ \hfill T_{\mu }^{\left(2\right)}& =& c_{10}\langle \Lambda |\overline{s}{\gamma }^{\mu }(1-{\gamma }^5)b|{\Lambda }_b\rangle \hfill \end{array}$$

(14)

$T^{\left(m\right)}$ ($m=1,2$) are expressed through the form factors and the Wilson coefficients.

The matrix elements of the flavour changing neutral current for the rare ${\Lambda }_b\to \Lambda l^{+}{l}^{-}$ baryon decays are parametrized by the following set of invariant form factors

$$\begin{array}{ccc}\hfill \langle \Lambda (p^{\prime },s^{\prime })|\overline{s}{\gamma }^{\mu }b|{\Lambda }_b(p,s)\rangle & =& {\overline{u}}_{\Lambda }(p^{\prime },s^{\prime })\left(f_1^V\left(q^2\right){\gamma }^{\mu }-f_2^V\left(q^2\right)i{\sigma }^{\mu \nu }\frac{q_{\nu }}{M_{{\Lambda }_b}}+f_3^V\left(q^2\right)\frac{q^{\mu }}{M_{{\Lambda }_b}}\right)u_{{\Lambda }_b}(p,s),\hfill \\ \hfill \langle \Lambda (p^{\prime },s^{\prime })|\overline{s}{\gamma }^{\mu }{\gamma }_{5}b|{\Lambda }_b(p,s)\rangle & =& {\overline{u}}_{\Lambda }(p^{\prime },s^{\prime })\left(f_1^A\left(q^2\right){\gamma }^{\mu }-f_2^A\left(q^2\right)i{\sigma }^{\mu \nu }\frac{q_{\nu }}{M_{{\Lambda }_b}}+f_3^A\left(q^2\right)\frac{q^{\mu }}{M_{{\Lambda }_b}}\right){\gamma }_5{u}_{{\Lambda }_b}(p,s),\hfill \\ \hfill \langle \Lambda (p^{\prime },s^{\prime })|\overline{s}i{\sigma }^{\mu \nu }{q}_{\nu }b|{\Lambda }_b(p,s)\rangle & =& {\overline{u}}_{\Lambda }(p^{\prime },s^{\prime })\left(\frac{f_1^{TV}\left(q^2\right)}{M_{{\Lambda }_b}}\left({\gamma }^{\mu }{q}^2-q^{\mu }\cancel{q}\right)-f_2^{TV}\left(q^2\right)i{\sigma }^{\mu \nu }{q}_{\nu }\right)u_{{\Lambda }_b}(p,s),\hfill \\ \hfill \langle \Lambda (p^{\prime },s^{\prime })|\overline{s}i{\sigma }^{\mu \nu }{q}_{\nu }{\gamma }_{5}b|{\Lambda }_b(p,s)\rangle & =& {\overline{u}}_{\Lambda }(p^{\prime },s^{\prime })\left(\frac{f_1^{TA}\left(q^2\right)}{M_{{\Lambda }_b}}\left({\gamma }^{\mu }{q}^2-q^{\mu }\cancel{q}\right)-f_2^{TA}\left(q^2\right)i{\sigma }^{\mu \nu }{q}_{\nu }\right){\gamma }_5{u}_{{\Lambda }_b}(p,s),\hfill \end{array}$$

(15)

where $u_{{\Lambda }_b}(p,s)$ and $u_{\Lambda }(p^{\prime },s^{\prime })$ are Dirac spinors, $q=p^{\prime }-p$. We need to calculate four additional form factors $f_{1,2}^{TV}\left(q^2\right)$ and $f_{1,2}^{TA}\left(q^2\right)$ which parametrize matrix element of tensor and axial tensor current matrix elements. The form factors of the rare weak ${\Lambda }_b\to \Lambda$ transition are plotted in Figure 5 [5].

The effective Wilson coefficient $c_9^{\mathrm{eff}}$ contains additional pertubative and long-distance contributions

$$c_9^{\mathrm{eff}}=c_9+{\mathcal{Y}}_{\mathrm{pert}}\left(q^2\right)+{\mathcal{Y}}_{\mathrm{BW}}\left(q^2\right).$$

(16)

The perturbative part is equal to

$$\begin{array}{c}{\mathcal{Y}}_{\mathrm{pert}}\left(q^2\right)\hspace{1em}=\hspace{1em}h\left(\frac{m_c}{m_b},\frac{q^2}{m_b^2}\right)(3c_1+c_2+3c_3+c_4+3c_5+c_6)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{2}h\left(1,\frac{q^2}{m_b^2}\right)(4c_3+4c_4+3c_5+c_6)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{2}h\left(0,\frac{q^2}{m_b^2}\right)(c_3+3c_4)+\frac{2}{9}(3c_3+c_4+3c_5+c_6),\end{array}$$

(17)

where

$$\begin{array}{ccc}\hfill h\left(\frac{m_c}{m_b},\frac{q^2}{m_b}\right)& =& -\frac{8}{9}ln\frac{m_c}{m_b}+\frac{8}{27}+\frac{4}{9}x-\frac{2}{9}(2+x){|1-x|}^{1/2}\left\{\begin{array}{cc}ln\left|\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}\right|-i\pi ,\hfill & x\equiv \frac{4m_c^2}{q^2}<1,\hfill \\ 2arctan\frac{1}{\sqrt{x-1}},\hfill & x\equiv \frac{4m_c^2}{q^2}>1,\hfill \end{array}\right.\hfill \\ \hfill h\left(0,\frac{q^2}{m_b}\right)& =& \frac{8}{27}-\frac{4}{9}ln\frac{q^2}{m_b}+\frac{4}{9}i\pi .\hfill \end{array}$$

The long-distance (nonperturbative) contributions are assumed to originate from the $c\overline{c}$ resonances ($J/\psi ,{\psi }^{\prime }\cdots$) and have a usual Breit-Wigner structure:

$${\mathcal{Y}}_{\mathrm{BW}}\left(q^2\right)=\frac{3\pi }{{\alpha }^2}\sum _{V_i=J/\psi ,\psi \left(2S\right)\cdots }\frac{\Gamma (V_i\to l^{+}{l}^{-})M_{V_i}}{M_{V_i}^2-q^2-i{M}_{V_i}{\Gamma }_{V_i}}.$$

(18)

We include contributions of the vector $V_i\left(1^{--}\right)$ charmonium states: $J/\psi$, $\psi \left(2S\right)$, $\psi \left(3770\right)$, $\psi \left(4040\right)$, $\psi \left(4160\right)$ and $\psi \left(4415\right)$, with their masses ($M_{V_i}$), leptonic [$\Gamma (V_i\to l^{+}{l}^{-})$] and total (${\Gamma }_{V_i}$) decay widths taken from PDG [22].

The lepton angle differential decay distribution is given by

$$\frac{d^2\Gamma ({\Lambda }_b\to \Lambda l^{+}{l}^{-})}{d{q}^{2}dcos\theta }=\frac{d\Gamma ({\Lambda }_b\to \Lambda l^{+}{l}^{-})}{d{q}^2}\left(\frac{3}{8}(1+{cos}^2\theta )(1-F_L)+A_{FB}^{\ell }cos\theta +\frac{3}{4}{F}_L{sin}^2\theta \right),$$

(19)

where $\theta$ is the angle between the ${\Lambda }_b$ baryon and the positively charged lepton in the dilepton rest frame. The lepton forward-backward asymmetry is defined by [13,14,15]

$$A_{FB}^{\ell }\left(q^2\right)=\frac{\frac{d\Gamma }{d{q}^2}\left(\mathrm{forward}\right)-\frac{d\Gamma }{d{q}^2}\left(\mathrm{backward}\right)}{\frac{d\Gamma }{d{q}^2}},$$

(20)

and $F_L\left(q^2\right)$ is the fraction of longitudinally polarized dileptons.

The hadron angle differential distribution of the decay ${\Lambda }_b\to \Lambda (\to p{\pi }^{-})l^{+}{l}^{-}$ is given by

$$\frac{d^2\Gamma ({\Lambda }_b\to \Lambda l^{+}{l}^{-})}{d{q}^{2}dcos{\theta }_h}=Br(\Lambda \to p{\pi }^{-})\frac{d\Gamma ({\Lambda }_b\to \Lambda l^{+}{l}^{-})}{d{q}^2}\frac{1}{2}\left(1+2A_{FB}^{h}cos{\theta }_h\right),$$

(21)

where ${\theta }_h$ is the angle between the proton and the $\Lambda$ baryon in the ${\Lambda }_b$ rest frame. $A_{FB}^h\left(q^2\right)$ is the hadron forward-backward asymmetry. The other useful observable is the combined hadron-lepton forward-backward asymmetry $A_{FB}^{h\ell }$. It is proportional to the coefficient in front of the term $cos\theta cos{\theta }_h$ in the threefold joint angular decay distribution for the decay of the unpolarized ${\Lambda }_b$ [13,14,15].

In Figure 6, Figure 7, Figure 8 and Figure 9 we plot our predictions [5] for the differential branching ratios $dBr/d{q}^2$, lepton $A_{FB}^{\ell }\left(q^2\right)$ and hadron $A_{FB}^h\left(q^2\right)$ forward-backward asymmetries as well as the fraction of longitudinally polarized dileptons $F_L\left(q^2\right)$ for rare decays ${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$ and ${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$ in comparison with available experimental data [32,33]. By solid (dashed) lines we plot theoretical results obtained without (with) inclusion of the long-distance contributions to the Wilson coefficients coming from the charmonium resonances. Experimental data for the ${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$ decay from the LHCb [32] and CDF [33] Collaborations are plotted by dots with solid and dashed error bars, respectively.

In

Table 6. Comparison of theoretical predictions for baryon rare decay branching fractions ($\times {10}^{-6}$) with available experimental data.

Decay	Our	Ivanov	Aliev	Wang	Liu	Mott	Gan	Experiment
	RQM	CCQM	LCSR	LCSR	RQM	NRQN	LCSR	PDG
	[5,6]	[13,14,15]	[34]	[35]	[36]	[37]	[38]	[22]
${\Lambda }_b\to \Lambda e^{+}{e}^{-}$	1.07	1.0	4.6(1.6)		$1.21\sim 2.32$		$2.03\left({}_9^{26}\right)$
${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$	1.05	1.0	4.0(1.2)	$6.1\left({}_{1.7}^{5.8}\right)$	$0.53\sim 0.89$	0.70		$1.08\left(28\right)$
${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$	0.26	0.2	0.8(3)	$2.1\left({}_{0.6}^{2.3}\right)$	$0.037\sim 0.083$	0.22
${\Lambda }_b\to n{e}^{+}{e}^{-}$	0.0381
${\Lambda }_b\to n{\mu }^{+}{\mu }^{-}$	0.0375
${\Lambda }_b\to n{\tau }^{+}{\tau }^{-}$	0.0121

we compare different theoretical predictions [13,14,15,34,35,36,37,38] for the total branching fractions of rare semileptonic ${\Lambda }_b$ decays with available experimental data [22]. The presented values include results of the relativistic and nonrelativistic quark model calculations [13,14,15,36,37] as well as evaluations based on various versions of the light-cone QCD sum rules [34,35,38]. At present, experimental data are available for ${\Lambda }_b\to \Lambda {\mu }^{+}{\mu }^{-}$ decay only. The values obtained in our model and Refs. [13,14,15,36,37] agree well with data, while other results are significantly larger. From Table 6 we also see that predictions for the rare ${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$ decay vary significantly. Our results [5,6,7] are close to those from quark models Refs. [13,14,15,37]. On the other hand, the light-cone QCD sum rules [34,35] predict significantly larger values while the quark model [36] gives a significantly lower value. Thus experimental measurement of the rare ${\Lambda }_b\to \Lambda {\tau }^{+}{\tau }^{-}$ decay branching fraction can help to discriminate between theoretical approaches. In this table we also include our predictions for the rare semileptonic ${\Lambda }_b\to n{l}^{+}{l}^{-}$ decay rates.

6. Rare Radiative ${\Lambda }_b$ Baryon Decay

The exclusive rare radiative decay rate ${\Lambda }_b\to \Lambda \gamma$ for the emission of a real photon ($k^2=0$) is given by

$$\Gamma ({\Lambda }_b\to \Lambda \gamma )=\frac{\alpha }{64{\pi }^4}{G}_F^2{m}_b^2{M}_{{\Lambda }_b}^3|V_{tb}{V}_{ts}{|}^2|c_7^{\mathrm{eff}}\left(m_b\right){|}^2(|f_2^{TV}\left(0\right){|}^2+|f_2^{TA}\left(0\right){|}^2){\left(1-\frac{M_{\Lambda }^2}{M_{{\Lambda }_b}^2}\right)}^3.$$

(22)

Our predictions for the branching fraction are given in

Table 7. Comparison of theoretical predictions for the rare radiative decay branching fraction ($\times {10}^{-5}$) with available experimental data.

Decay	Our	Mannel	Ivanov	Wang	Gan	Colangelo	Experiment
	RQM	HQS	CCQM	LCSR	LCSR	QCDSR	LHCb
	[5,6]	[39]	[13,14,15]	[35]	[38]	[40]	[41]
${\Lambda }_b\to \Lambda \gamma$	1.0	$0.77{(}_{19}^{22})$	0.4	$0.73\left(15\right)$	$0.061{(}_{13}^{14})$	3.1(6)	$0.75\left(15\right)\left(6\right)\left(7\right)$
${\Lambda }_b\to n\gamma$	0.037

in comparison with other theoretical values [13,14,15,35,38,39,40] and the experimental data [41]. Our result [5,6] is consistent with the values from Refs. [35,39], but about a factor of 2 larger than the prediction of the covariant constituent quark model [13,14,15]. The result of the light-cone QCD sum rule study [38] is about an order of magnitude lower, while the value obtained within three-point QCD sum rules in the heavy quark limit [40] is more than a factor 3 larger than other theoretical predictions. Very recently the LHCb Collaboration [41] measured the decay branching fraction $Br({\Lambda }_b\to \Lambda \gamma )$. It is in nice agreement with our prediction and the results of [13,14,15,35,39], but is significantly different form [38,40].

7. Conclusions

We studied the semileptonic decays of heavy baryons in the framework of the relativistic quark-diquark picture. All calculations were done with the comprehensive account of the relativistic effects without application heavy quark expansion. All parameters of the model were fixed in previous considerations of meson and baryon properties. The wave functions of baryons were taken from their mass spectra calculations. It was found that obtained results agree well with available experimental data and some theoretical predictions.