Newly discovered $\Xi _c^{0}$ resonances and their parameters

Abstract

The aim of the present article is investigation of the newly observed resonances $\Xi _c(2923)^{0}$, $\Xi _c(2939)^{0}$, and $\Xi _c(2965)^{0}$ which are real candidates to charm-strange baryons. To this end, we calculate the mass and pole residue of the ground-state and excited 1P and 2S spin-1/2 flavor-sextet baryons $\Xi _{c}^{\prime 0}$, $\Xi _{c}^{\prime 0}(1/2^{-})$ and $\Xi _{c}^{\prime 0}(1/2^{+})$ with quark content csd, respectively. The masses and pole residues of the ground-state and excited spin-3/2 baryons $\Xi _{c}^{\star 0}$ are found as well. Spectroscopic parameters of these particles are computed in the context of the QCD two-point sum rule method. Widths of the excited baryons are evaluated through their decays to final states $\Lambda _{c}^{+}K^{-}$ and $\Xi _{c}^{\prime 0}\pi$. These processes are explored by means of the full QCD light-cone sum rule method necessary to determine strong couplings at relevant vertices. Obtained predictions for the masses and widths of the four excited baryons, as well as previous results for 1P and 2S flavor-antitriplet spin-1/2 particles $\Xi _c^{0}$ are confronted with available experimental data on $\Xi _{c}^{0}$ resonances to fix their quantum numbers. Our comparison demonstrates that the resonances $\Xi _c(2923)^{0}$ and $\Xi _c(2939)^{0}$ can be considered as 1P excitations of the spin-1/2 flavor-sextet and spin-3/2 baryons, respectively. The resonance $\Xi _{c}(2965)^{0}$ may be interpreted as the excited 2S state of either spin-1/2 flavor-sextet or antitriplet baryon.

Appendix: The invariant amplitudes \(\Pi _{1}^{\mathrm {OPE}}(M^{2},s_{0})\) and \(\Pi _{2}^{\mathrm {OPE}}(M^{2},s_{0})\)

The invariant amplitudes $\Pi _{1(2)}^{\mathrm {OPE}}(p^{2})$ used to calculate the mass and pole residue of the flavor-sextet spin-1/2 baryons after the Borel transformation and subtraction prescriptions take the following form

$$\begin{aligned} \Pi _{1(2)}^{\mathrm {OPE}}(M^{2},s_{0})=\int _{\mathcal {M}^{2}}^{s_{0}}ds\rho _{1(2)}^{\mathrm {OPE}}(s)e^{-s/M^{2}}+\Pi _{1(2)}(M^{2}), \end{aligned}$$

(A.1)

where $\mathcal {M}=m_{c}+m_{s}$. The spectral densities $\rho _{1(2)}^{ \mathrm {OPE}}(s)$ in Eq. (A.1) are found from the imaginary part of the correlation function and encompass essential piece of $\Pi ^{\mathrm {OPE} }(p)$. The Borel transformations of remaining terms in $\Pi ^{\mathrm {OPE} }(p)$ are included into $\Pi _{1(2)}(M^{2})$, and have been calculated directly from the expression of $\Pi ^{\mathrm {OPE}}(p)$.

The functions $\rho _{1(2)}^{\mathrm {OPE}}(s)$ and $\Pi _{1(2)}(M^{2})$ contain components of different dimensions and have the structure

$$\begin{aligned} \rho _{1(2)}^{\mathrm {OPE}}(s)= & {} \rho _{1(2)}^{\mathrm {pert.}}(s)+\sum \rho _{1(2)}^{\mathrm {DimN}}(s),\nonumber \\ \Pi _{1(2)}(M^{2})= & {} \sum \Pi _{1(2)}^{\mathrm { DimN}}(M^{2}). \end{aligned}$$

(A.2)

The $\Pi _{1(2)}^{\mathrm {OPE}}(M^{2},s_{0})$ have been computed by setting $m_{q}=0$ and $m_{s}\ne 0$, and used to perform numerical analyses. The amplitudes $\Pi _{1(2)}^{\mathrm {OPE}}(M^{2},s_{0})$, in general, contain a few hundred terms and exist as Mathematica files. Their explicit expressions are cumbersome, therefore we provide below simplified formulas in which $m_{s}=0$.

The perturbative contribution and nonperturbative terms with dimensions 3, 4, 5 and 7 in the case of the spectral density $\rho _{1}^{\mathrm {OPE} }(s)$ are given by the expressions:

$$\begin{aligned} \rho _{1}^{\mathrm {pert.}}(s)= & {} \frac{(5+2\beta +5\beta ^{2})}{2048\pi ^{4}s^{2}}\left[ m_{c}^{6}(8s-m_{c}^{2})+s^{3}(s-8m_{c}^{2})+12m_{c}^{4}s^{2}\ln \left( \frac{ s}{m_{c}^{2}}\right) \right] ,\\ \rho _{1}^{\mathrm {Dim3}}(s)= & {} \frac{\langle \overline{s}s\rangle +\langle \overline{d}d\rangle }{192\pi ^{2}s^{2}}m_{c}\left( m_{c}^{2}-s\right) ^{2}\left( 1+4\beta -5\beta ^{2}\right) ,\\ \rho _{1}^{\mathrm {Dim4}}(s)= & {} \frac{\langle g_{s}^{2}G^{2}\rangle }{3072\pi ^{4}s^{2}}\left( s-m_{c}^{2}\right) \left[ 8m_{c}^{2}\left( 1+\beta +\beta ^{2}\right) +s\left( 5+2\beta +5\beta ^{2}\right) \right] ,\\ \rho _{1}^{\mathrm {Dim5}}(s)= & {} \frac{\langle \overline{d}g_{s}\sigma Gd\rangle +\langle \overline{s}g_{s}\sigma Gs\rangle }{768\pi ^{2}s^{2}}m_{c}(\beta -1) \left[ m_{c}^{2}(7+11\beta )-6s(1+\beta )\right] ,\\ \rho _{1}^{\mathrm {Dim7}}(s)= & {} \frac{\langle g_{s}^{2}G^{2}\rangle \left[ \langle \overline{s}s\rangle +\langle \overline{d}d\rangle \right] }{384\pi ^{2}s^{2}}m_{c}(1-\beta ^{2}). \end{aligned}$$

The function $\Pi _{1}(M^{2})$ is composed of the following components:

$$\begin{aligned} \Pi _{1}^{\mathrm {Dim6}}(M^{2},s_{0})= & {} \frac{\langle \overline{s}s\rangle \langle \overline{d}d\rangle }{72}\left( 11\beta ^{2}+2\beta -13\right) e^{-m_{c}^{2}/M^{2}},\\ \Pi _{1}^{\mathrm {Dim7}}(M^{2},s_{0})= & {} \frac{\langle g_{s}^{2}G^{2}\rangle \left[ \langle \overline{s}s\rangle +\langle \overline{d}d\rangle \right] }{ 864\pi ^{2}m_{c}}\left( \beta ^{2}+\beta -2\right) e^{-m_{c}^{2}/M^{2}},\\ \Pi _{1}^{\mathrm {Dim8}}(M^{2},s_{0})= & {} -\frac{\langle g_{s}^{2}G^{2}\rangle ^{2}}{27\cdot 2^{13}\pi ^{4}M^{2}}\left( 13\beta ^{2}+10\beta +13\right) e^{-m_{c}^{2}/M^{2}}\\&+\frac{\langle \overline{s}g_{s}\sigma Gs\rangle \langle \overline{d}d\rangle }{288M^{4}}(1-\beta ) \\&\times \left[ m_{c}^{2}(26+22\beta )+M^{2}\left( 25+23\beta \right) \right] e^{-m_{c}^{2}/M^{2}},\\ \Pi _{1}^{\mathrm {Dim9}}(M^{2},s_{0})= & {} \frac{\langle \overline{s} g_{s}\sigma Gs\rangle \langle g_{s}^{2}G^{2}\rangle }{27\cdot 2^{11}\pi ^{2}m_{c}M^{4}}(1-\beta )\\&\times \left[ m_{c}^{2}(31+11\beta )-2M^{2}(1+\beta ) \right] e^{-m_{c}^{2}/M^{2}}, \\ \Pi _{1}^{\mathrm {Dim10}}(M^{2},s_{0})= & {} 0. \end{aligned}$$

For the spectral density $\rho _{2}^{\mathrm {OPE}}(s)$, we get

$$\begin{aligned} \rho _{2}^{\mathrm {pert.}}(s)= & {} \frac{m_{c}(13-2\beta -11\beta ^{2})}{1536\pi ^{4}s}\\&\times \left[ m_{c}^{6}+9sm_{c}^{4}-9m_{c}^{2}s^{2}-s^{3}+6m_{c}^{2}s(s+m_{c}^{2})\ln \left( \frac{s}{m_{c}^{2}}\right) \right] ,\\ \rho _{2}^{\mathrm {Dim3}}(s)= & {} \frac{\langle \overline{s}s\rangle +\langle \overline{d}d\rangle }{192\pi ^{2}s}\left( m_{c}^{2}-s\right) ^{2}\left( 1+4\beta -5\beta ^{2}\right) ,\\ \rho _{2}^{\mathrm {Dim4}}(s)= & {} \frac{\langle g_{s}^{2}G^{2}\rangle }{9216\pi ^{4}m_{c}s}\left( 1-\beta \right) \\&\times \left[ (m_{c}^{2}-s)\left( s(13+11\beta )+m_{c}^{2}(53+67\beta )\right) \right. \\&\left. +3m_{c}^{2}s(11+13\beta )\ln \left( \frac{s}{m_{c}^{2}}\right) \right] ,\\ \rho _{2}^{\mathrm {Dim5}}(s)= & {} \frac{\langle \overline{d}g_{s}\sigma Gd\rangle +\langle \overline{s}g_{s}\sigma Gs\rangle }{768\pi ^{2}s}(1-\beta )\\&\times \left[ m_{c}^{2}(5+\beta )-6s(1+\beta )\right] . \end{aligned}$$

The function $\Pi _{2}(M^{2})$ is determined by the components

$$\begin{aligned} \Pi _{2}^{\mathrm {Dim6}}(M^{2},s_{0})= & {} \frac{\langle \overline{s}s\rangle \langle \overline{d}d\rangle }{24}m_{c}\left( 5\beta ^{2}+2\beta +5\right) e^{-m_{c}^{2}/M^{2}},\\ \Pi _{2}^{\mathrm {Dim7}}(M^{2},s_{0})= & {} \frac{\langle g_{s}^{2}G^{2}\rangle \left[ \langle \overline{s}s\rangle +\langle \overline{d}d\rangle \right] }{ 3456\pi ^{2}}\left( \beta ^{2}-8\beta +7\right) e^{-m_{c}^{2}/M^{2}},\\ \Pi _{2}^{\mathrm {Dim8}}(M^{2},s_{0})= & {} \frac{\langle g_{s}^{2}G^{2}\rangle ^{2}}{27\cdot 2^{13}\pi ^{4}m_{c}M^{2}}(m_{c}^{2}-2M^{2})\left( 11+2\beta -13\beta ^{2}\right) e^{-m_{c}^{2}/M^{2}}\\&+\frac{\langle \overline{s} g_{s}\sigma Gs\rangle \langle \overline{d}d\rangle }{144M^{4}}m_{c} \\&\times \left[ M^{2}\left( \beta -1\right) ^{2}-3m_{c}^{2}(5+2\beta +5\beta ^{2})\right] e^{-m_{c}^{2}/M^{2}},\\ \Pi _{2}^{\mathrm {Dim9}}(M^{2},s_{0})= & {} \frac{\langle \overline{s} g_{s}\sigma Gs\rangle \langle g_{s}^{2}G^{2}\rangle }{27\cdot 2^{11}\pi ^{2}M^{4}}m_{c}^{2}(\beta ^{2}+28\beta -29)e^{-m_{c}^{2}/M^{2}}, \\ \Pi _{2}^{\mathrm {Dim10}}(M^{2},s_{0})= & {} 0. \end{aligned}$$