Charmed $\Omega_{c}$ weak decays into $\Omega$ in the light-front quark model

The lowest-lying singly charmed baryons include the anti-triplet and sextet states ${\bf B}_{c}=(\Lambda_{c}^{+},\Xi_{c}^{0},\Xi_{c}^{+})$ and ${\bf B}^{\prime}_{c}=(\Sigma_{c}^{(0,+,++)},\Xi_{c}^{{}^{\prime}(0,+)},\Omega_{c}^{0})$, respectively. The ${\bf B}_{c}$ and $\Omega_{c}^{0}$ baryons predominantly decay weakly CroninHennessy:2000bz ; Ammar:2002pf ; Aubert:2007bt ; Yelton:2017uzv ; pdg , whereas the $\Sigma_{c}$ ($\Xi^{\prime}_{c}$) decays are strong (electromagnetic) processes. There have been more accurate observations for the ${\bf B}_{c}$ weak decays in the recent years, which have helped to improve the theoretical understanding of the decay processes Lu:2016ogy ; Geng:2017esc ; Geng:2018plk ; Geng:2018upx ; Hsiao:2019yur ; Zhao:2018mov ; Zou:2019kzq ; Hsiao:2020iwc ; Niu:2020gjw . With the lower production cross section of $\sigma(e^{+}e^{-}\to\Omega_{c}^{0}X)$ Yelton:2017uzv , it is an uneasy task to measure $\Omega_{c}^{0}$ decays. Consequently, most of the $\Omega_{c}^{0}$ decays have not been reanalysized since 1990s AvilaAoki:1989yi ; PerezMarcial:1989yh ; Singleton:1990ye ; Hussain:1990ai ; Korner:1992wi ; Xu:1992sw ; Cheng:1993gf ; Cheng:1996cs ; Ivanov:1997ra , except for those in Pervin:2006ie ; Dhir:2015tja ; Zhao:2018zcb ; Gutsche:2018utw ; Hu:2020nkg ; Geng:2017mxn .

One still manages to measure more than ten $\Omega_{c}^{0}$ decays, such as $\Omega_{c}^{0}\to\Omega^{-}\rho^{+}$, $\Xi^{0}\bar{K}^{(*)0}$ and $\Omega^{-}\ell^{+}\nu_{\ell}$, but with the branching fractions relative to ${\cal B}(\Omega_{c}^{0}\to\Omega^{-}\pi^{+})$ pdg . To extract the absolute branching fractions, the study of $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ is crucial. Fortunately, the $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ decay involves a simple topology, which benefits its theoretical exploration. In Fig. 1a, $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ is depicted to proceed through the $\Omega_{c}^{0}\to\Omega^{-}$ transition, while $\pi^{+}$ is produced from the external $W$-boson emission. Since it is a Cabibbo-allowed process with $V_{cs}^{*}V_{ud}\simeq 1$, a larger branching fraction is promising for measurements. Furthermore, it can be seen that $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}$ has a similar configuration to those of $\Omega_{c}^{0}\to\Omega^{-}\rho^{+}$ and $\Omega_{c}^{0}\to\Omega^{-}\ell^{+}\nu_{\ell}$, as drawn in Fig. 1, indicating that the three $\Omega_{c}^{0}$ decays are all associated with the $\Omega_{c}^{0}\to\Omega^{-}$ transition. While $\Omega$ is a decuplet baryon that consists of the totally symmetric identical quarks $sss$, behaving as a spin-3/2 particle, the form factors of the $\Omega_{c}^{0}\to\Omega^{-}$ transition can be more complicated, which hinders the calculation for the decays. As a result, a careful investigation that relates $\Omega_{c}^{0}\to\Omega^{-}\pi^{+},\Omega^{-}\rho^{+}$ and $\Omega_{c}^{0}\to\Omega^{-}\ell^{+}\nu_{\ell}$ has not been given yet, despite the fact that the topology associates them together.

Based on the quark models, it is possible to study the $\Omega_{c}^{0}$ decays into $\Omega^{-}$ with the $\Omega_{c}^{0}\to\Omega^{-}$ transition form factors. However, the validity of theoretical approach needs to be tested, which depends on if the observations, given by

can be interpreted. Since the light-front quark model has been successfully applied to the heavy hadron decays Zhao:2018zcb ; Bakker:2003up ; Ji:2000rd ; Bakker:2002aw ; Choi:2013ira ; Cheng:2003sm ; Schlumpf:1992vq ; Hsiao:2019wyd ; Jaus:1991cy ; Melosh:1974cu ; Dosch:1988hu ; Zhao:2018mrg ; Geng:2013yfa ; Geng:2000if ; Ke:2012wa ; Ke:2017eqo ; Ke:2019smy ; Hu:2020mxk , in this report we will use it to study the $\Omega_{c}^{0}\to\Omega^{-}$ transition form factors. Accordingly, we will be enabled to calculate the absolute branching fractions of $\Omega_{c}^{0}\to\Omega^{-}\pi^{+}(\rho^{+})$ and $\Omega_{c}^{0}\to\Omega^{-}\ell^{+}\nu_{\ell}$, and check if the two ratios in Eq. (I) can be well explained.

In the Wolfenstein parameterization, the CKM matrix elements are adopted as $V_{cs}=V_{ud}=1-\lambda^{2}/2$ with $\lambda=0.22453\pm 0.00044$ pdg . We take the lifetime and mass of the $\Omega_{c}^{0}$ baryon and the decay constants $(f_{\pi},f_{\rho})=(132,216)$ MeV from the PDG pdg . With $(c_{1},c_{2})=(1.26,-0.51)$ at the $m_{c}$ scale Buchalla:1995vs , we determine $a_{1}$. In the generalized factorization, $N_{c}$ is taken as an effective color number with $N_{c}=(2,3,\infty)$ Hu:2020nkg ; Gutsche:2018utw ; Hsiao:2019wyd ; Hsiao:2019ann , in order to estimate the non-factorizable effects. For the $\Omega_{c}^{+}(css)\to\Omega^{-}(sss)$ transition form factors, the theoretical inputs of the quark masses and parameter $\beta$ in Eq. (II.2) are given by Geng:2013yfa ; Geng:2000if

where $\beta_{c(s)}$ is to determine $\phi^{(\prime)}(x^{(\prime)},k_{\perp}^{(\prime)})$ for $\Omega_{c}^{0}$ $(\Omega^{-})$. We hence extract $F^{V}_{i}$ and $F^{A}_{i}$ in Table 1. For the momentum dependence, we have used the double-pole parameterization:

with $m_{F}=1.86$ GeV.

Using the theoretical inputs, we calculate the branching fractions, whose results are given in Table 2.

In Table 2, we present ${\cal B}_{\pi}$ and ${\cal B}_{\rho}$ with $N_{c}=(2,3,\infty)$. The errors come from the form factors in Table 1, of which the uncertainties are correlated with the charm quark mass. By comparison, ${\cal B}_{\pi}$ and ${\cal B}_{\rho}$ are compatible with the values in Ref. Gutsche:2018utw ; however, an order of magnitude smaller than those in Refs. Xu:1992sw ; Cheng:1996cs , whose values are obtained with the total decay widths $\Gamma_{\pi(\rho)}=2.09a_{1}^{2}(11.34a_{1}^{2})\times 10^{11}$ s^-1 and $\Gamma_{\pi(\rho)}=1.33a_{1}^{2}(4.68a_{1}^{2})\times 10^{11}$ s^-1, respectively. We also predict ${\cal B}_{e}=(5.4\pm 0.2)\times 10^{-3}$ as well as ${\cal B}_{\mu}\simeq{\cal B}_{e}$, which is much smaller than the value of $127\times 10^{-3}$ in Pervin:2006ie . Only the ratios ${\cal R}_{\rho/\pi}$ and ${\cal R}_{e/\pi}$ have been actually observed so far. In our work, ${\cal R}_{\rho/\pi}=2.8\pm 0.4$ is able to alleviate the inconsistency between the previous value and the most recent observation. We obtain ${\cal R}_{e/\pi}=1.1\pm 0.2$ with $N_{c}=2$ to be consistent with the data, which indicates that $({\cal B}_{\pi},{\cal B}_{\rho})=(5.1\pm 0.7,14.4\pm 0.4)\times 10^{-3}$ with $N_{c}=2$ are more favorable.

The helicity amplitudes can be used to better understand how the form factors contribute to the branching fractions. With the identity $H^{V(A)}_{-\lambda_{\Omega}-\lambda_{f}}=\mp H^{V(A)}_{\lambda_{\Omega}\lambda_{f}}$ for the ${\bf B}^{\prime}_{c}(J^{P}=1/2^{+})$ to ${\bf B}^{\prime}(J^{P}=3/2^{+})$ transition Gutsche:2018utw , $H_{\pi}^{2}$ in Eq. (II.3) can be rewritten as $H_{\pi}^{2}=2(|H_{\frac{1}{2}{\bar{0}}}^{V}|^{2}+|H_{\frac{1}{2}{\bar{0}}}^{A}|^{2})$. From the pre-factors in Eq. (17), we estimate the ratio of $|H_{\frac{1}{2}{\bar{0}}}^{V}|^{2}/|H_{\frac{1}{2}{\bar{0}}}^{A}|^{2}\simeq 0.05$, which shows that $H_{\frac{1}{2}{\bar{0}}}^{A}$ dominates ${\cal B}_{\pi}$, instead of $H_{\frac{1}{2}{\bar{0}}}^{V}$. More specifically, it is the $F_{4}^{A}$ term in $H_{\frac{1}{2}{\bar{0}}}^{A}$ that gives the main contribution to the branching fraction. By contrast, the $F_{1,3}^{A}$ terms in $H_{\frac{1}{2}{\bar{0}}}^{A}$ largely cancel each other, which is caused by $F_{1}^{A}M_{-}\simeq F_{3}^{A}\bar{M}^{\prime}_{-}$ and a minus sign between $F_{1}^{A}$ and $F_{3}^{A}$ (see Table 1); besides, the $F_{2}^{A}$ term with a small $F_{2}^{A}(0)$ is ignorable.