Spectrum of Nonstrange Singly Charmed Baryons in the Constituent Quark Model

Abstract

We study the masses of radial and orbital excited states of nonstrange singly charmed baryons in the framework of hypercentral Constituent Quark Model (hCQM). To obtain the mass spectra, the Coulomb plus screened potential is employed with the first order correction, which gives a relativistic effect of order $\mathcal {O}$(1/m). The spin-spin, spin-orbit and tensor interactions are included (perturbatively as a spin dependent potential) in order to generate the splitting in mass spectra. We compare our computed mass spectra of nonstrange singly charmed baryons with the other theoretical predictions as well as with the experimental observations. We construct the Regge trajectories of these baryons in the (J, M²) plane. Further, we analyze the strong one pion decay rates for S, P and the D-wave transitions in the framework of Heavy Hadron Chiral Perturbation Theory (HHChPT). Moreover, the electromagnetic properties like magnetic moments, transition magnetic moments and the radiative decay widths are determined for the ground state of these baryons in the constituent quark model.

1 Introduction

The nonstrange singly charmed baryons belong to ${\varLambda }_{c}^{+}$ and Σ_c families, which are classified into SU(3) flavor representation of antisymmetric antitriplet and symmetric sextet group respectively. Their experimental evidences are continuously in progress from the past few years. The latest Review of Particle Physics (RPP) by Particle Data Group (PDG) [1] present the seven states of ${\varLambda }_{c}^{+}$ baryon: Λ_c(2286)⁺, Λ_c(2595)⁺, Λ_c(2625)⁺, Λ_c(2765)⁺, Λ_c(2860)⁺, Λ_c(2880)⁺, Λ_c(2940)⁺ and the three states of isotriplet Σ_c baryons: Σ_c(2455)^++,+,0, Σ_c(2520)^++,+,0, Σ_c(2800)^++,+,0 (see in Table 1). For that the various experimental groups FOCUS, CLEO, BABAR, CDF and Belle have provided the masses as well as other properties of these baryons. Recently, the LHCb Collaboration [2] has given precise measurements of the masses and the strong decay widths of ${\varLambda }_{c}^{+}$ and the isotriplet Σ_c baryons with the statistical and the systematic uncertainties. The more experimental informations are available in review articles [3, 4]. Moreover, the current projects LHCb and Belle II, and the future experimental facilities J-PARC, $\overline {\text {\sffamily {P}}}$ANDA [5, 6] are expected to provide further information regarding singly charmed baryons.

Table 1 Mass, width and J^P value of the nonstrange singly charmed baryons from PDG [1]

The LHCb Collaboration [2] assigned the J^P (J is the total spin and P is parity) value of Λ_c(2880)⁺ and Λ_c(2940)⁺ states as ${\frac {5}{2}}^{+}$ and ${\frac {3}{2}}^{-}$ respectively. They also measured a state Λ_c(2860)⁺ with $J^{P} = {\frac {3}{2}}^{+}$. Except Λ_c(2765)⁺, the spin-parity of all observed ${\varLambda }_{c}^{+}$ baryons have been confirmed as shown in Table 1. On the other hand the J^P value of excited isotriplet Σ_c baryons have not been confirmed yet. The identifications and decay properties of new states of these baryons makes this study challenging. It is interesting to look back to the theory and the phenomenological study to see where the new predictions lie. Till date, the mass spectra of singly charmed baryons have been studied in various potential models using different approaches: a Quantum Chromodynamics (QCD) based quark model [7], relativistic quark potential model [8], relativistic quark-diquark picture [9, 10], QCD sum rule [11], Faddeev method [12] the flux tube model [13], the quasi-linear Regge trajectory ansatz [14], the non-relativistic constituent quark model [15,16,17,18,19], heavy quark limit in the one-boson-exchange potential [20], the heavy quark-light quark cluster picture [21], lattice QCD study [22,23,24] etc..

In order to improve the understanding of quark dynamics and their confinement mechanism inside the baryons, the study of singly charmed baryons containing one heavy quark and two light quarks is an important tool. It can provide some qualitative informations about the chiral symmetry breaking and the heavy quark symmetry. The singly charmed baryons mainly decay via strong interactions and it will be dominant over the electromagnetic or weak decay processes. So far any electromagnetic observation of the singly charmed baryons are not found experimentally. The strong decays and the electromagnetic behavior of the charmed baryons have been studied by several methods and they are: the Heavy Hadron Chiral Perturbation Theory (HHChPT) [25,26,27,28,29,30,31,32,33,34], the chiral structure model [35,36,37], the chiral soliton model [38], the pion mean-field approach [39], the chiral perturbation theory [40, 41], the relativistic constituent three-quark model [42, 43], large N_c limit [44,45,46], the light front quark model [47], the QCD sum rule [48,49,50,51,52], the ³P₀ model [53, 54], a constituent quark model [55, 56], the bag model [57], the non-relativistic approach [58,59,60,61,62,63], the lattice QCD [64,65,66] etc..

In this paper, the masses of radial and orbital excited states of nonstrange singly charmed baryons are calculated. For that, we employ the hypercentral Constituent Quark Model (hCQM) in which Coulomb plus screening potential is used with the first order correction. The obtained mass spectra are presented corresponding to n = 1,2,3,4,5 (n is the principal quantum number) with orbital quantum number L = 0,1,2,3. Such masses are used to draw Regge trajectories in the (J, M²) plane.

This paper is organized as follows: After the introduction, in Section 2 we present details of the hypercentral Constituent Quark Model (hCQM) and discuss the potential model. In Section 3 we analyze the mass spectra and construct the Regge trajectories. In Section 4 the strong one pion decay rates of Λ_c(2765)⁺, Σ_c(2455), Σ_c(2520) and Σ_c(2800) baryons are calculated in HHChPT, and also the electromagnetic properties such as the magnetic moments, transition magnetic moments and the radiative decays for L = 0 are studied in the constituent quark model. At last, we summarize our present work in Section 5.

2 Methodology

The spectroscopy of light and heavy flavor baryons are usually studied in relativistic (or non-relativistic) approach of quantum mechanics. In this section, we introduce the non-relativistic treatment in the framework of hypercentral Constituent Quark Model (hCQM). Such a model is well established and have been used to determine the properties of light, heavy-light and heavy-heavy flavored baryons (see in Refs. [17, 18, 62, 67, 68]). It has a spin-independent term with confining potential, gives a confining effect at a long range quark separations. The relative Jacobi coordinates (ρ and λ) are employed to see the dynamics of three quarks system. We express the Hamiltonian as [69, 70],

$$ H=\frac{P{_{x}^{2}}}{2m} + V(x) $$

(1)

where x is the six-dimensional hypercentral coordinate and $m = \frac {2m_{\rho } m_{\lambda }}{m_{\rho }+m_{\lambda }}$ gives the reduced mass of the baryonic system. Here, the coordinates ρ and λ are,

$$ \boldsymbol{\rho}=\frac{1}{\sqrt2} (\mathbf{r_{1}}-\mathbf{r_{2}}) and \boldsymbol{\lambda}=\frac{\mathbf{r_{1}}+\mathbf{r_{2}}-2\mathbf{r_{3}}} {\sqrt {6}}; $$

(2)

the relative Jacobi coordinates, that describe the relevant degrees of freedom for the dynamics of three constituent quarks. Here, r_i (i = 1,2,3) represents the i^th coordinate of the constituent quarks. The reduced mass of these coordinates are [71],

$$ m_{\rho} = \frac{2m_{1}m_{2}}{(m_{1}+m_{2})} and m_{\lambda}=\frac{2m_{3}(m{_{1}^{2}}+m{_{1}^{2}}+m_{1}m_{2})}{(m_{1}+m_{2})(m_{1}+m_{2}+m_{3})}. $$

(3)

For the $\varLambda _{c}^{+}$ baryon the constituent quarks are u, d and c and the Σ_c baryon has three isospin states with a different quark constitutions: ${\varSigma }_{c}^{++}$ (uuc), ${\varSigma }_{c}^{+}$ (udc) and ${\varSigma }_{c}^{0}$ (ddc). To calculate the masses of these isotriplet Σ_c baryons separately, we consider different constituent quark masses (taken from Ref. [18]) of the light quarks (u and d), which are m_u = 0.338 GeV, m_d = 0.350 GeV and for the charm quark (c), m_c = 1.275 GeV.

The hyperspherical coordinates, hyperradias (x) and hyperangle (ξ), are defined in the form of Jacobi coordinates (ρ and λ) as, $x=\sqrt {\rho ^{2}+\lambda ^{2}}$ and $\xi = arctan \left (\frac {\rho ^{2}}{\lambda ^{2}} \right )$. In the center-of-mass frame, an expression of kinetic energy operator $\frac {P{_{x}^{2}}}{2m}$ (appear in the (1)) for a three quarks system is,

$$ \frac{P{_{x}^{2}}}{2m}=-\frac{\hbar^{2}}{2m}(\triangle_{\rho} + \triangle_{\lambda})=-\frac{\hbar^{2}}{2m} \left( \frac{\partial^{2}}{\partial x^{2}}+\frac{5}{x}\frac{\partial}{\partial x}+\frac{L^{2}({\varOmega})}{x^{2}}\right) $$

(4)

where $L^{2}({\varOmega })=L^{2}({\varOmega }_{\rho }, {\varOmega }_{\lambda }, \xi )$ represents the quadratic Casimir operator in the six-dimensional rotational group O(6) and its eigenfunctions are the hyperspherical harmonics Y_[γ]l_ρl_λ(Ω_ρ, Ω_λ, ξ) satisfying the eigenvalue relation,

$$ L^{2} Y_{[\gamma]}l_{\rho}l_{\lambda} ({\varOmega}_{\rho}, {\varOmega}_{\lambda}, \xi) = -\gamma(\gamma+4)Y_{[\gamma]}l_{\rho}l_{\lambda} ({\varOmega}_{\rho}, {\varOmega}_{\lambda}, \xi). $$

(5)

Here, Ω_ρ and Ω_λ are the angles of the hyperspherical coordinates. The total angular momentum is L = L_ρ + L_λ, and the angular momentum associated with the Jacobi coordinates ρ and λ are, l_ρ and l_λ, respectively. The eigenvalues of L² are given by − γ(γ + 4), where γ = 2n + ρ + λ represents the grand angular momentum quantum number with the non-negative integer value (n).

2.1 The Potential Model

The non-relativistic interaction potential V (x) classified into spin-independent V_SI(x) and spin-dependent V_SD(x) part of the potential. The effective spin-independent static potential V_SI(x) is simply the sum of Lorentz vector (Coulomb) V_V(x) and Lorentz scalar (confining) V_S(x) terms. In the present study we used the first order correction (say V_I(x)) in V_SI(x) [72], i.e.,

$$ V_{SI}(x)=V_{V}(x)+V_{S}(x)+V_{I}(x). $$

(6)

Here, V_V(x) is the non-relativistic QCD potential between two quarks q₁ and q₂ in a baryon (q₁q₂q₃), which is written in a Coulombic form by considering color wavefunction as,

$$ V_{V}(x) = -\frac{2}{3}\frac{\alpha_{s}}{x}, $$

(7)

where x is the inter-quark separation and the parameter α_s represents the strong running coupling constant

$$ \alpha_{s} = \frac{\alpha_{s}(\mu_{0})}{1+\left( \frac{33-2n_{f}}{12\pi}\right)\alpha_{s}(\mu_{0})ln\left( \frac{m_{1}+m_{2}+m_{3}}{\mu_{0}}\right)}. $$

(8)

In the above expression, we consider α_s(μ₀ = 1GeV ) = 0.6 and n_f is the number of active quark flavors contributing effectively in quark-gloun loops. The quantity (33 − 2n_f) must be greater than zero. Therefore, n_f will never larger than six. In the present calculation we take n_f = 3. So we have an approximately α_s = 0.46 for both $\varLambda _{c}^{+}$ and Σ_c baryons. We choose screened potential as a scalar potential,

$$ V_{S}(x)= a \left( \frac{1-e^{-\mu x}}{\mu}\right), $$

(9)

where a is the string tension and the constant μ occurring in screened potential is the screening factor. For $x \ll \frac {1}{\mu }$, the screened potential is behaving like a linear potential ax and for $x \gg \frac {1}{\mu }$ it becomes a constant $\frac {a}{\mu }$. So it is interesting to study the mass spectroscopy with screened potential which gives the masses of the higher excited states lesser compared to the linear potential [73, 74]. Here, we set μ = 0.04 GeV. Figure 1 shows the behavior of spin-independent Coulomb plus screened potential (V_CP+SP) for ${\varLambda }_{c}^{+}$ and ${\varSigma }_{c}^{+}$ baryons.

Fig. 1

The nature of non-relativistic spin independent part of QCD potential (V_SI(x)), ${\varLambda }_{c}^{+}$ baryon (left) and ${\varSigma }_{c}^{+}$ baryon (right). V_SI(x) is changing with respect to inter-quark separation (x) for α_s = 0.46. The blue line indicates the short range Coulomb interaction (V_CP) and the added long range interaction potential say Coulomb plus screened potential (V_CP+SP) is presented by orange line

The first order correction V_I(x) can be written in the form of Casimir charges of the fundamental and the adjoint representation such as $C_{F} = \frac {2}{3}$ and C_A = 3 respectively,

$$ V_{I}(x) = - C_{F} C_{A} \frac{\alpha{_{S}^{2}}}{4x^{2}}. $$

(10)

The spin-dependent potential V_SD(x) determine the mass difference between degenerate baryonic states given by [75,76,77],

$$ V_{SD}(x) = V_{SS}(x)(\mathbf{S_{\rho}}\cdot \mathbf{S_{\lambda}}) + V_{\gamma S}(x)(\mathbf{L} \cdot \mathbf{S}) + V_{T}(x) \left[S^{2} - \frac{3(\mathbf{S} \cdot \mathbf{x})(\mathbf{S} \cdot \mathbf{x})}{x^{2}}\right], $$

(11)

where V_SS(x), V_γS(x) and V_T(x) are the spin-spin, spin-orbit interaction and the spin-tensor interaction terms (for details see Ref. [18]).

The six-dimensional hyperradial Schrödinger equation is solved numerically using Mathematica notebook [78],

$$ \left[\frac{1}{2m}\left( -\frac{d^{2}}{dx^{2}}+\frac{\frac{15}{4}+\gamma(\gamma+4)}{x^{2}}\right)+V(x)\right]\phi_{\gamma}(x) = E_{B} \phi_{\gamma}(x). $$

(12)

Here, m is the reduced mass (see in (1)) and E_B gives the binding energy of the baryonic states, and V (x) = V_SI(x) + V_SS(x) + V_LS(x) + V_T(x), is the total potential of the baryonic system.

The spin average masses are determined by taking a summation of model quark masses with its binding energy,

$$ M_{SA} = E_{B} + m_{q_{1}} + m_{q_{2}} + m_{q_{3}}. $$

(13)

Therefore, the total mass is,

$$ M_{total} = M_{SA} + V(x) - V_{SI}(x). $$

(14)

In this way, we calculate the excited state masses of the nonstrange singly charmed baryons. We analyze them in the next section.

3 Mass Spectra and Regge Trajectories

The masses of the radial and the orbital excited states of nonstrange singly charmed baryons are calculated in the non-relativistic framework of hypercentral Constituent Quark Model (hCQM). Here, the screened potential is used as a confining potential with first order correction. Our calculated masses are presented in Tables 2, 3, 4, 5 and 6 with other theoretical predictions and known experimental observations. Our results are used to draw Regge trajectories in the (J, M²) plane (see in Figs. 2 and 3). Regge trajectories can be used to determine the possible quantum number of the particular hadronic state [14].

Fig. 2

The $(M^{2} \rightarrow J)$ Regge trajectory of ${\varLambda }_{c}^{+}$ baryon with natural parity. An experimental available measurements are presented by cross sign

Fig. 3

The $(M^{2} \rightarrow J)$ Regge trajectories of ${\varSigma }_{c}^{+}$ baryon with natural parity (left) and unnatural parity (right)

Table 2 Predicted masses of the radial and the orbital excited states of ${\varLambda }_{c}^{+}$ baryon (in GeV)

Table 3 Predicted masses of the radial excited states of isotriplet Σ_c baryons (in GeV)

Table 4 Predicted masses of the orbital excited states of ${\varSigma }_{c}^{++}$ baryon (in GeV)

Table 5 Predicted masses of the orbital excited states of ${\varSigma }_{c}^{+}$ baryon (in GeV)

Table 6 Predicted masses of the orbital excited states of ${\varSigma }_{c}^{0}$ baryon (in GeV)

The Regge trajectories are plotted in (J, M²) plane with natural and unnatural parities given by $P= (-1)^{J-\frac {1}{2}}$ and $P= (-1)^{J+\frac {1}{2}}$ respectively. Hence, the states $J^{P}={\frac {1}{2}}^{+}$, ${\frac {3}{2}}^{-}$, ${\frac {5}{2}}^{+}$ and ${\frac {7}{2}}^{-}$ are available with the natural parity and the states $J^{P}={\frac {3}{2}}^{+}$, ${\frac {5}{2}}^{-}$, ${\frac {7}{2}}^{+}$ and ${\frac {9}{2}}^{-}$ are for the unnatural parity. Using the equation,

$$ J=\alpha M^{2}+\alpha_{0}, $$

(15)

we construct the Regge trajectories in the (J, M²) plane. Here, α and α₀ are the fitted slopes and the intercepts respectively. The straight lines are obtained by the linear fitting. For the indication our calculated masses are presented by various mathematical symbols and the experimental observations by cross sign with a particle name.

3.1 ${{\varLambda }_{c}^{+}}$ States

We fix the ground state (1S) mass of ${\varLambda }_{c}^{+}$ baryon with the experimentally known value 2.286 GeV [1] and then calculate its excited state mass spectra. For 2S state with $J^{P} = {\frac {1}{2}}^{+}$, our result is 2.785 GeV, close to PDG [1] by a mass difference of 19 MeV and also in agreement with the theoretical predictions [9,10,11, 16, 20, 21]. For the higher radial excitation (3S, 4S and 5S states), our results are in accordance with the predictions of D. Ebert et al. [10] (see in Table 2).

In 2011, CDF [79] measured the masses, 2592.25 ± 0.24(stat) ± 0.14(syst) MeV/c² and 2628.11 ± 0.13(stat) ± 0.14(syst) MeV/c², from their decay into ${{\varLambda }_{c}^{+}} \pi ^{+} \pi ^{-}$. These states are assigned with the quantum numbers $J^{P} = {\frac {1}{2}}^{-}$ and ${\frac {3}{2}}^{-}$ respectively. For the 1P state with $J^{P} = {\frac {1}{2}}^{-}$ and ${\frac {3}{2}}^{-}$, our predictions are 2.573 GeV and 2.568 GeV respectively, which are smaller than Ref. [1] and the theoretical predictions [9,10,11, 16, 20, 21]. Here, 2.573 GeV with $J^{P} = {\frac {1}{2}}^{-}$ is in good agreement with the lattice QCD result 2.578 GeV [24]. For 2P state with $J^{P} = {\frac {1}{2}}^{-}$ and ${\frac {3}{2}}^{-}$, our predicted masses are 2.978 GeV and 2.970 GeV respectively, which are close to the results of Refs. [9,10,11, 18, 21] as well as the result of ${\frac {3}{2}}^{-}$ is reasonably close to the experimental observation 2939.6 MeV [1]. Our predictions of 3P, 4P and 5P states are larger than Ref. [10].

In 2007, the BABAR Collaboration [80] measured a state with mass 2881.9 ± 0.1(stat) ± 0.5(syst) MeV/c² and then in the same year Belle Collaboration [85] identified its spin-parity as $J^{P} = {\frac {5}{2}}^{+}$. At latest, the LHCb Collaboration [2] observed the mass and the quantum number of Λ_c(2880)⁺ baryon as: 2881.75 ± 0.29(stat) ± 0.07(syst) MeV, $J^{P}={\frac {5}{2}}^{+}$. They also found a new resonance state Λ_c(2860)⁺ having a mass 2856.1${^{+2.0}_{-1.7}}$(stat) ± 0.5(syst) MeV with $J^{P}={\frac {3}{2}}^{+}$. For these states Λ_c(2880)⁺ and Λ_c(2860)⁺, our results are 2.876 GeV and 2.865 GeV having mass difference of 20 MeV and 16 MeV with Ref. [1]. For the 2D-state, our predictions are 3.256 GeV and 3.244 GeV for spin-parity ${\frac {3}{2}}^{+}$ and ${\frac {5}{2}}^{+}$ respectively, which are in accordance with the results of Ref. [9] and larger than the other theoretical predictions [10, 16, 18, 20]. Still there are no experimental evidences available for the F-states. Our predictions of the 1F states are 3.152 GeV and 3.136 GeV for the spin-parity ${\frac {5}{2}}^{-}$ and ${\frac {7}{2}}^{-}$ respectively, which are compatible with predictions of Refs. [9,10,11, 20, 21]. For 3D, 4D, 2F and 4F states, our results are overestimated from Refs. [10, 16]. By taking a comparison with Ref. [10] the mass differences are increasing with n (principle quantum number).

Figure 2 shows that the Regge trajectories of the ${\varLambda }_{c}^{+}$ baryon and they are almost parallel and equidistant. The square masses of the ${\varLambda }_{c}^{+}$ baryon fit nicely to the linear trajectories in (J, M²) plane. The available experimental data are well matched with the corresponding Regge trajectories obtained in our model. Still the quantum state of Λ_c(2765)⁺ baryon is not confirmed experimentally. On the Regge line such a state Λ_c(2765)⁺ is following Λ_c(2940)⁺ state in (J, M²) plane. So it may have 2S state with spin $\frac {1}{2}$.

3.2 Σ_c States

Just after the discovery of ${\varLambda }_{c}^{+}$ baryon at Brookhaven National Laboratory (BNL) in 1975 [81], the Fermi National Accelerator Laboratory (FNAL) found the evidence of Σ_c baryons [82]. It has three isospin states with different quarks constitutions: ${\varSigma }_{c}^{++}$ (uuc), ${\varSigma }_{c}^{+}$ (udc) and ${\varSigma }_{c}^{0}$ (ddc). In the present study, their masses are calculated separately by considering unequal light quarks (u and d) masses. The Refs. [8, 9, 16, 20, 21] consider the same light quarks masses, so here our predictions are compared only with ${\varSigma }_{c}^{+}$ states.

We fix the ground state (1S) of isotriplet Σ_c baryons with experimental value from Ref. [1], and calculate their respective radial and orbital excited states. From the quark mass hierarchy, the down (d) quark is heavier than up (u) quark, so the masses of the ${\varSigma }_{c}^{0}$ baryons are expected to be higher than the masses of ${\varSigma }_{c}^{++}$ baryons. Many experimental observations [79, 83, 84, 86] gives negative mass splittings of M(Σ_c(2455)⁰ −Σ_c(2455)⁺⁺) and M(Σ_c(2520)⁰ −Σ_c(2520)⁺⁺). Expected by the models (for details see Ref. [87]), in our case both these isospin mass splittings are positive. ${\varSigma }_{c}^{+}$ has been observed in a decay mode $\varLambda _{c}^{+} \pi ^{0}$ by single experiment [88], because the detectors are inefficient for the π⁰ identification as a decay product. The isotriplet Σ_c baryons have two spin states: $\frac {1}{2}$ and $\frac {3}{2}$. For the 2S state with $J^{P} = {\frac {1}{2}}^{+}$ and $J^{P} = {\frac {3}{2}}^{+}$, our results are 2.912 GeV and 2.951 GeV, consistent with the results 2.901 GeV and 2.936 GeV of D. Ebert et al. [10]. For the 3S state, the mass corresponding to $J^{P} = {\frac {1}{2}}^{+}$ is 3.266 GeV, which is nearer to 3.271 GeV of [10], and for $J^{P} = {\frac {3}{2}}^{+}$ it is 3.288 GeV, overestimated from Refs. [8, 9, 16, 20, 21] (see in Table 3).

Experimentally, only the first orbital excited states of isotriplet Σ_c baryons are observed [84, 92] and still their J^P values are not known. Our calculated orbital excited state masses of the isotriplet Σ_c baryons are presented in Tables 4, 5 and 6. For the 1P, 2P and 1D states of ${\varSigma }_{c}^{+}$ baryon our predictions are in agreement with the results of Refs. [8,9,10, 21] and, for the 3P,4P, 2D and 1F states our results are smaller than [10] (see in Table 5). That means, the screening effect comes into the picture that gives the lower mass of the higher excited states.

Figure 3 shows the Regge trajectories of ${\varSigma }_{c}^{+}$ baryon with natural and unnatural parities in the (J, M²) plane. For the massive states of ${\varSigma }_{c}^{+}$ baryon the trajectories are to become horizontal, which leads to smaller slopes for heavy quarks (for more study see Refs. [14, 89,90,91]). From the spectroscopy and the Regge trajectories here we are unable to predict the possible J^P value of the isotriplet Σ_c(2800) baryons.

4 Properties

4.1 Strong Decays

We are using our calculated masses for the determination of the decay properties, that is important to identify the J^P value of newly observed experimental states. As stated in the introduction, singly charmed baryons have one charm quark and two light quarks. So it provides an excellent base for rectifying the heavy quark symmetry of the heavy quark and the chiral symmetry of the light quarks in the low energy regime. A Heavy Hadron Chiral Perturbation Theory (HHChPT) represents the chiral Lagrangian in which the heavy quark symmetry and the chiral symmetry are incorporated (see Refs. [26,27,28,29]). Such a Lagrangian describes the strong interactions of heavy baryons with the emission of light pseudoscalar mesons. It contains strong coupling constants: g₁ and g₂ for the P-wave transitions, h₂ to h₇ for the S-wave transitions and h₈ to h₁₅ used for the D-wave transitions. The expressions of partial decay widths are taken from Refs. [29,30,31,32,33] and the pion decay constant is, f_π = 130.2 MeV [1].

4.1.1 P-wave Transitions

The P-wave couplings take place among the s-wave baryons. For the decay ${\varSigma }_{c}(2520) \rightarrow {\varSigma }_{c}(2455) \pi$, experimentally the mass difference between these two singly charmed baryons is around 65 MeV [1]. A single pion do not have such amount of phase space for this decay. Therefore, such a decay is kinematically prohibited and we cannot extract coupling constant g₁ here. The g₂ can be obtained from the allowed decay channels: ${\varSigma }_{c}(2455)^{++} \rightarrow {\varLambda }_{c}^{+} \pi ^{+}$, ${\varSigma }_{c}(2455)^{0} \rightarrow {\varLambda }_{c}^{+} \pi ^{-}$, ${\varSigma }_{c}(2520)^{++} \rightarrow {\varLambda }_{c}^{+} \pi ^{+}$ and ${\varSigma }_{c}(2520)^{0} \rightarrow {\varLambda }_{c}^{+} \pi ^{-}$ as

$$ \mid g_{2} \mid = 0.550{_{-0.027}^{+0.013}}, 0.544{_{-0.018}^{+ 0.010}}, 0.561{_{-0.007}^{+ 0.005}}, 0.570{_{-0.009}^{+ 0.007}} $$

(16)

respectively, using their masses and respective decay widths from PDG-2018 [1]. Therefore, an average ∣g₂∣₂₀₁₈ = $0.566{_{-0.027}^{+0.013}}$ (using PDG-2018 [1]), which is nearer to ∣g₂∣₂₀₁₅ = $0.565{_{-0.024}^{+0.011}}$ of Ref. [31] and ∣g₂∣₂₀₀₆ = 0.591 ± 0.023 of Ref. [30]. The non-relativistic quark model determined the coupling constants g₁ and g₂ in the form of an axial-vector coupling $g{_{A}^{q}}$ in a single light quark transition $u \rightarrow d$, written as [29]

$$ g_{1}=\frac{4}{3}g{_{A}^{q}}, g_{2}=\sqrt{\frac{2}{3}}g{_{A}^{q}}. $$

(17)

A nucleon axial coupling $g{_{A}^{N}}$ = 1.25 can be reproduced by taking $g{_{A}^{q}}$ = 0.75 [26, 27, 29]. Therefore, the above equations are,

$$ g_{1}=1, g_{2}=0.61. $$

(18)

Hence, the quark model predictions of coupling constants are in accordance with the results obtained in the framework of HHChPT (see in (16)). The lattice QCD studies [64] give g₁ and g₂ slightly different as, 0.56 ± 0.13 and 0.41 ± 0.08, respectively. Refs. [44, 45] used leading order approximation of large N_c limit and calculated g₂ = $g{_{A}^{q}}/\sqrt {2}$ = 0.88 and, recently the chiral structure model predict, $g_{2} = 0.688{_{-0.035}^{+0.013}}$ [37], which are larger than the experimental values and the quark model predictions (see in (18)). Taking an experimental average value of ∣g₂∣₂₀₁₈ = $0.566{_{-0.027}^{+0.013}}$ and assuming the validity of the quark model relations among different coupling constants (see in (17)) implies $g{_{A}^{q}}$ = $0.694{_{-0.025}^{+0.013}}$ and ∣g₁∣₂₀₁₈ = $0.925{_{-0.063}^{+0.017}}$.

The decay widths for the P-wave transitions are listed in the Table 8. The first column represents the decay widths corresponding to the masses obtained in the screened potential with ∣g₂∣ = $0.550{_{-0.027}^{+0.013}}$ and the second column used the masses from PDG-2018 [1] with the same g₂. The third and the fourth column used the masses from PDG-2014 and PDG-2006 in the framework of HHChPT as in Ref. [31] and Ref. [30] respectively. We compare our results with other theoretical predictions and the experimental measurements. Moreover, the ratio of the decay widths of Σ_c(2520) to Σ_c(2455) is $\sim 7$ and it will be same in the limit of heavy quark symmetry.

4.1.2 S and D-wave Transitions

The S and D-wave couplings take place between p-wave and s-wave baryons. The couplings h₂,..., h₇ are dimensionless employed for the S-wave couplings and h₈,..., h₁₅ used for the D-wave couplings has a dimension E^− 1. In HHChPT, the coupling constants h₂ and h₈ can be extracted from the decays $\varLambda _{c}(2595)^{+} \rightarrow {\varSigma }_{c}(2455) \pi$ and $\varLambda _{c}(2625)^{+} \rightarrow {\varSigma }_{c}(2455) \pi$ respectively. Experimentally, the mass difference M(Λ_c(2595)⁺) − M(Σ_c(2455)) ≈ 139 MeV, therefore, such a decay is kinematically barely allowed. The CDF [79] measured the decay width 2.59 ± 0.30 ± 0.47 MeV/c² of Λ_c(2595)⁺ into ${\varLambda }_{c}^{+} \pi ^{+} \pi ^{-}$ decay mode, having $h{_{2}^{2}}$ = 0.36 ± 0.04 ± 0.07, that implies h₂ = 0.60 ± 0.07, which is close to h₂ = 0.57${_{-0.197}^{+0.322}}$ of Ref. [29] and 0.63 ± 0.07 of Ref. [31]. The decays of isotriplet Σ_c(2800) into ${\varLambda }_{c}^{+} \pi$ are governed by the coupling h₃. For the S-wave transition, the quark model relations of the couplings are

$$ \mid h_{4} \mid = 2 \mid h_{2} \mid, \mid h_{3} \mid = \frac{\sqrt{3}}{2} \mid h_{4} \mid; $$

(19)

and for the D-wave transitions [29,30,31]

$$ \mid h_{8} \mid = \mid h_{9} \mid = \mid h_{10} \mid, \mid h_{11} \mid = \sqrt{2} \mid h_{10} \mid. $$

(20)

The coupling h₁₀ can be extracted from the decay of isotriplet ${\varSigma }_{c}({\frac {3}{2}}^{-})^{++,+,0}$ into ${\varLambda }_{c}^{+} \pi$ end particles. Using the world average masses of isotriplet Σ_c(2800)^++,+,0 and their respective decay widths from Ref. [1], we obtained h₁₀ as,

$$ \mid h_{10} \mid = 0.999{_{-0.161}^{+0.162}} \times 10^{-3}, 0.942{_{-0.819}^{+ 0.436}} \times 10^{-3}, 0.956{_{-0.055}^{+ 0.129}} \times 10^{-3} $$

(21)

in MeV^− 1 respectively, which are larger than the estimation of naive dimensional analysis 0.4 × 10^− 3 MeV^− 1 and compatible with $0.85{_{-0.08}^{+0.11}} \times 10^{-3}$ of Ref. [31]. Using h₁₀, the couplings h₈ and h₁₁ can be determined from the quark model relations and they are involved in the decays of ${\varLambda }_{c}(2625)^{+} \rightarrow {\varSigma }_{c}(2455) \pi$ and ${\varSigma }_{c}({\frac {3}{2}}^{-})^{++,+,0} \rightarrow {\varSigma }_{c}(2455) \pi / {\varSigma }_{c}(2520) \pi$ respectively. Using the couplings h₃ = 1.049 ± 0.107, $h_{10} = 0.999{_{-0.161}^{+0.162}} \times 10^{-3}$ MeV^− 1 = h₈ and $h_{11} = 1.413{_{-0.227}^{+0.229}} \times 10^{-3}$ MeV^− 1, we calculate the decay rates for the S and D-wave transitions (see in Table 8). Our results are compatible with Refs. [30, 31] and other theoretical predictions.

Note that the J^P values of Σ_c(2800) are not yet confirmed experimentally. It may have either ${\frac {1}{2}}^{-}$ or ${\frac {3}{2}}^{-}$ total spin. So here in the framework of HHChPT we are using the same masses from PDG-2018 [1] in both the cases of total spin. By considering $J^{P} = {\frac {1}{2}}^{-}$, calculated decay widths of isotriplet Σ_c(2800) are close to PDG-2018 [1] rather than the $J^{P} = {\frac {3}{2}}^{-}$. Hence, the Σ_c(2800) are more likely to be Σ_c(1/2⁻). Using h₈ = $0.999{_{-0.161}^{+0.162}} \times 10^{-3}$ MeV^− 1 the S-wave transitions of Λ_c(2625)⁺ into isotriplet Σ_c(2455) are calculated. For these decays our present calculated masses doesn^′t provide such amount of phase space and the masses from PDG-2018 [1] gives the decay rates which are overestimated from Refs. [30, 32, 33, 92]. Such differences are because of the selection of coupling h₈, HHChPT-2007 [30] and HHChPT-2015 [31] are used h₈ ≤ $0.85{_{-0.08}^{+0.11}}$ × 10^− 3 MeV^− 1, and the Ref. [33] used h₈ = 0.4 × 10^− 3 MeV^− 1 from [29].

4.2 Electromagnetic Properties

To probe the electromagnetic properties (magnetic moments, transition magnetic moments, form factors etc.) of baryons, the study of radiative decay processes are important. Such properties are used to expose the inner structures of the baryons. In this section, we study the magnetic moments and transition magnetic moments of the ground state nonstrange singly charmed baryons in the constituent quark model. For radiative decays, the transitions are taking place among the participating baryons by an exchange of massless photons. So it doesn^′t contain phase space restrictions and that^′s why some of the radiative decay modes are contributed significantly to the total decay rate.

4.2.1 Magnetic Moments

The magnetic moment of the baryon (μ_B) is purely the function of masses and spin of their internal quarks constitutions. It can be expressed in the form of expectation value [19, 33, 61],

$$ \mu_{B} = {\sum}_{q} \left\langle {\varPhi}_{sf} \middle| \hat{\mu}{_{q}}_{z} \middle| {\varPhi}_{sf} \right\rangle; q=u,d,c. $$

(22)

Here, Φ_sf is the spin-flavor wave function of the participating baryon and $\hat {\mu }{_{q}}_{z}$ is the z-component of the magnetic moment of the individual quark given by,

$$ \hat{\mu}{_{q}}_{z} = \frac{e_{q}}{2 m{_{q}^{eff}}} \cdot \hat{\sigma}_{q_{z}}, $$

(23)

where e_q is the charge and $\hat {\sigma }_{q_{z}}$ is the z-component of the spin of the constituent quark. The effective quark mass $(m{_{q}^{eff}})$ gives the mass of the bound quark inside the baryon by taking into account its binding interactions with other two quarks and is defined as,

$$ m{_{q}^{eff}} = m{_{q}} \left( {1 + \frac{\left\langle H \right\rangle}{\sum\limits_{q} m{_{q}}}}\right), $$

(24)

where $\sum \limits _{q} m_{q}$ is the sum of constituting quark mass and the Hamiltonian ${\left \langle H \right \rangle } = M - {\sum \limits _{q} m{_{q}}}$, where M is the measured or predicted baryon mass.

Using these equations and taking the constituent quark masses from section 2 and the baryon masses from the spectrum, we determine the ground state magnetic moments of the nonstrange singly charmed baryons in the unit of nuclear magnetons $\left (\mu _{N} = \frac {e\hbar }{2m_{p}}\right )$. We present our results with the predictions obtained from various approaches in Table 9.

4.2.2 Radiative Decays

An expression of electromagnetic decay width is written in the form of radiative transition magnetic moments $(\mu _{B_{C}^{\prime }\rightarrow B_{c}})$ [33, 63],

$$ {\Gamma}_{\gamma} = \frac{k^{3}}{4\pi}\frac{2}{2J+1}\frac{e}{m{_{p}^{2}}}\mu{_{{B{_{c}}} \rightarrow {B{_{c}^{\prime}}}}^{2}}. $$

(25)

Here, k is the photon momentum,

$$ k = \frac{M{_{B_{c}}^{2}} - M{_{B_{c}^{\prime}}^{2}}}{2M_{B}{_{c}}}, $$

(26)

m_p is the mass of proton and J represents the total angular momentum of the initial baryon (B_c). $M{_{B_{c}}}$ and $M{_{B_{c}^{\prime }}}$ are the mass of the initial and final state baryon respectively.

For transition magnetic moments $(\mu _{B_{c} \rightarrow B{_{c}^{\prime }}})$, repeating the same procedure as we discussed in the above subsection by sandwiching the magnetic moment operator (23) between the appropriate initial $({\varPhi }_{sf_{B{_{c}}}})$ and final $({\varPhi }_{sf_{B{_{c}^{\prime }}}})$ state spin-flavour wave functions of nonstrange singly charmed baryons,

$$ \mu{_{{B_{c}} \rightarrow B{_{c}^{\prime}}}} = \langle{\varPhi}_{sf_{B{_{c}}}} \vert \hat{\mu}{_{q}}_{z} \vert {\varPhi}_{sf_{B{_{c}^{\prime}}}} \rangle $$

(27)

For example: in order to determine the radiative decay of Σ_c(2520)⁺⁺ into Σ_c(2455)⁺⁺, first we need to calculate the transition magnetic moment as,

$$ \mu_{{\varSigma}_{c}(2520)^{++} \rightarrow {\varSigma}_{c}(2455)^{++}} = \left \langle{\varPhi}_{sf_{{\varSigma}_{c}(2520)^{++}}} \right \vert \hat{\mu}{_{q}}_{z} \left\vert {\varPhi}_{sf_{{\varSigma}_{c}(2455)^{++}}}\right\rangle $$

(28)

the spin-flavour wave functions (Φ_sf) of Σ_c(2520)⁺⁺ and Σ_c(2455)⁺⁺ baryons are expressed as,

$$ \left\vert {\varPhi}_{sf_{{\varSigma}_{c}(2520)^{++}}} \right\rangle = (uuc) \cdot \left( \frac{1}{\sqrt{3}}(\uparrow\uparrow\downarrow+\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow)\right) $$

(29)

$$ \left\vert {\varPhi}_{sf_{{\varSigma}_{c}(2455)^{++}}} \right\rangle = (uuc) \cdot \left( \frac{1}{\sqrt{6}}(2\uparrow\uparrow\downarrow-\uparrow\downarrow\uparrow-\downarrow\uparrow\uparrow)\right) $$

(30)

Following the orthogonal condition of quark flavor and spin states, for example $\left \langle u\uparrow u \uparrow c\downarrow \vert u \uparrow u\downarrow c\uparrow \right \rangle = 0$, we get an expression of transition magnetic moment as

$$ \mu_{{\varSigma}_{c}(2520)^{++} \rightarrow {\varSigma}_{c}(2455)^{++}} = \frac{2\sqrt{2}}{3} \left (\mu_{u} - \mu_{c} \right). $$

(31)

In this way, we determine an expressions of transition magnetic moments of other nonstrange singly charmed baryons (see in Table 10). Our calculated transition magnetic moments and their corresponding radiative decay widths are listed in Tables 10 and 11 respectively. Our results are compared with other theoretical predictions and are in accordance with A. Majethiya et al. [63] and smaller than the predictions of Ref. [61].

5 Summary

In this work, the excited state mass spectra of nonstrange singly charmed baryons are calculated in the framework of hypercentral Constituent Quark Model (hCQM). Here, we have used screened potential as a confining potential. In order to see the relativistic effect in the heavy light baryonic systems we added the first order correction. Our calculated masses are listed in Tables 2, 3, 4, 5 and 6. The predicted mass spectra are in agreement with the other theoretical predictions and the experimental measurements where available. For the ${{\varLambda }_{c}^{+}}$ baryon, our results are in consistent with the results obtained by Z. Shah et al. [18] using the linear confinement potential as a scalar potential. We have seen the screening effect in isotriplet Σ_c baryons, which gives lowered masses for higher excited states compared to the masses obtained from the linear potential. The calculations of excited states masses allow us for plotting the data on Regge lines according to their quantum number with natural and unnatural parities in (J, M²) plane, which helps to assign the J^P value of an experimental unknown states. Regge trajectories identified the Λ_c(2765)⁺ baryon with 2S state and having a spin $\frac {1}{2}$ in our case (see in Fig. 2). Because of the screening effect in isotriplet ${{\varSigma }_{c}^{+}}$ baryon the Regge trajectories approaches a straight horizontal line for the higher excited states as shown in Fig. 3. That are in accordance with the argument of decreasing the Regge slopes with increasing the quark masses (see Refs. [14, 89,90,91]). From the Fig. 3 we can not predict exactly the spin-parity of the first orbital excited state of isotriplet Σ_c baryons.

The strong decay rates of nonstrange singly charmed baryons are analyzed in the framework of Heavy Hadron Chiral Perturbation Theory (HHChPT). Using the masses and the decay widths of these baryons from PDG-2018 [1], first we have extracted couplings constants g₂, h₂ (from CDF [79]) and h₁₀ from their respective decay channels. And the couplings h₃, h₈, h₉ and h₁₁ are obtained from the quark model relations. Such couplings are in accordance with the quark model expectation and other theoretical predictions. These coupling control the strong one pion decay rates in HHChPT. Using the masses from PDG-2018 [1] and from the screened potential spectrum the strong decay rates for the S, P and D-wave transitions are calculated separately. Our results are listed in Tables 7 and 8, and compared with other predictions. The strong decay rates of isotriplet Σ_c(2800) baryons are calculated with the total spin ${\frac {1}{2}}^{-}$ and ${\frac {3}{2}}^{-}$. Their decay rates corresponding to spin ${\frac {1}{2}}^{-}$ are close to the experimental observations rather than the spin ${\frac {3}{2}}^{-}$. Hence, we assign the J^P quantum number of isotriplet Σ_c(2800) as ${\frac {1}{2}}^{-}$. The magnetic moments, transition magnetic moments and the radiative decay rates are calculated for the ground state nonstrange singly charmed baryons in the constituent quark model, which are presented in Tables 9, 10 and 11 with other theoretical predictions.

Table 7 The strong P-wave transitions (in MeV) of the s-wave baryons

Table 8 The strong S and D-wave transitions (in MeV) between p and s-wave baryons

Table 9 Magnetic moments of the nonstrange singly charmed baryons (in μ_N)

Table 10 The transition magnetic moments $\left \vert \mu {_{{B_{c}} \rightarrow B{_{c}^{\prime }}}} \right \vert$ of nonstrange singly charmed baryons (in μ_N)

Table 11 The radiative decay widths (Γ_γ) of nonstrange singly charmed baryons (in keV)

In the present study, our aim is satisfied for the determination of J^P value of experimentally measured unknown states of nonstrange singly charmed baryons: Λ_c(2765)⁺ and isotriplet Σ_c(2800). The spectroscopy and the Regge trajectories predict the Λ_c(2765)⁺ as a 2S state with J^P = ${\frac {1}{2}}^{+}$. And the strong decays analysis of isotriplet Σ_c(2800) baryons predict J^P = ${\frac {1}{2}}^{-}$ as a first orbital excitation. This model is successful for the study of nonstrange singly charmed baryons. Our predictions will help the experimentalists as well as theoreticians towards the understanding of their dynamics. So we would like to extend this scheme for the study of singly bottom baryons.