Spin–parity identification of newly observed singly charmed baryons in Regge phenomenology

Abstract

In the present article, we employ the Regge phenomenology to describe the mass-spectra of singly charmed baryons. With the assumption of linear Regge trajectories, the relations between Regge slopes, intercepts, and baryon masses are derived. Using these relations we evaluated the Regge slopes ($\alpha ^{'}$) for $\varLambda _{c}^{+}$, $\varSigma _{c}^{++,+,0}$, $\varXi _{c}^{+,0}$, $\varXi _{c}^{'+,0}$, and $\varOmega _{c}^{0}$ baryons and obtain the orbitally excited state masses in the $(J,M^{2})$ plane. Similarly, the Regge parameters ($\beta$ and $\beta _{0}$) are estimated for each Regge line in the $(n,M^{2})$ plane and the radially excited state masses are obtained. Also, the experimental errors are calculated in the obtained results wherever the experimental inputs are taken. Further the Regge trajectories are plotted for our calculated masses in the ($J,M^{2}$) plane. We compared our estimated results with the experimental data and other theoretical predictions. We assign the spin–parity of recently observed experimental states $\varSigma _{c}(2800)$, $\varXi _{c}(2923)$, $\varXi _{c}(3123)$, $\varOmega _{c}(3000)$, $\varOmega _{c}(3050)$, $\varOmega _{c}(3065)$, $\varOmega _{c}(3090)$ and $\varOmega _{c}(3119)$ in this work.

Appendix

The experimental errors are incorporated in the present work. In this section we give the detailed description and calculation of the error analysis. We have calculated the experimental errors wherever the experimental values are taken as input. Since to calculate the error in slope $\alpha _{jjq}$, we have obtained from Eq. (17), firstly we have to obtain the error in $\alpha _{iiq}$. Since we have,

$$\begin{aligned} \alpha _{iiq} = \frac{1}{M^{2}_{J+2} - M^{2}_{J}} \end{aligned}$$

(28)

Equation (28) can be written as,

$$\begin{aligned} \alpha _{iiq} = \frac{1}{k} \end{aligned}$$

(29)

where,

$$\begin{aligned} k = [M^{2}_{J+2} - M^{2}_{J}] \end{aligned}$$

(30)

Hence, error in k can be given as,

$$\begin{aligned} \delta k = \sqrt{\left( 2M_{J+2} \delta M_{J+2}\right) ^{2}+\left( 2M_{J}\delta M_{J}\right) ^{2}} \end{aligned}$$

(31)

Here $M_{J+2}$ and $M_{J}$ are the experimental masses taken as input including their error values $\delta M_{J+2}$ and $\delta M_{J}$ respectively. Now error in slope $\alpha _{iiq}$ from Eq. (29) can be given as,

$$\begin{aligned} \delta \alpha _{iiq} = -\left( \frac{\delta k}{k^{2}}\right) \end{aligned}$$

(32)

Hence with the aid of Eqs. (30), (31), and (32) we can obtain the value of $\delta \alpha _{iiq}$. Now from Eq. (17):

$$\begin{aligned} \dfrac{\alpha ^{'}_{jjq}}{\alpha ^{'}_{iiq}}= & {} \dfrac{1}{2M^{2}_{jjq}}\times [(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}) \nonumber \\{} & {} \pm \sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^2}-4M^{2}_{iiq}M^{2}_{jjq}}],\nonumber \\ \end{aligned}$$

(33)

The above equation can also be written as,

$$\begin{aligned} \alpha ^{'}_{jjq} = \frac{x}{y} \end{aligned}$$

(34)

where,

$$\begin{aligned} \begin{aligned} x&= {\alpha ^{'}_{iiq}}[(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}) \\&\quad +\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^2}-4M^{2}_{iiq}M^{2}_{jjq}}], \end{aligned} \end{aligned}$$

(35)

and

$$\begin{aligned} y = 2M^{2}_{jjq} \end{aligned}$$

(36)

Error in Eq. (34) is given as,

$$\begin{aligned} \delta \alpha ^{'}_{jjq} = \alpha ^{'}_{jjq}\sqrt{\left( \frac{\delta x}{x}\right) ^{2}+\left( \frac{\delta y}{y}\right) ^{2}} \end{aligned}$$

(37)

From Eq. (36) error in y can be given as,

$$\begin{aligned} \delta y = 8M_{jjq}\delta M_{jjq} \end{aligned}$$

(38)

Now to calculate the error in x, we can write Eq. (35) can be written as

$$\begin{aligned} x = \alpha ^{'}_{iiq} \times P \end{aligned}$$

(39)

Hence,

$$\begin{aligned} \delta x = x \sqrt{\left( \frac{\delta \alpha ^{'}_{iiq}}{ \alpha ^{'}_{iiq}}\right) ^{2} + \left( \frac{\delta P}{P}\right) ^{2}} \end{aligned}$$

(40)

where,

$$\begin{aligned} P= & {} \bigg [(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}) \nonumber \\{} & {} +\sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^2}-4M^{2}_{iiq}M^{2}_{jjq}}\bigg ] \end{aligned}$$

(41)

Let, $P = a+b$ where,

$$\begin{aligned} a = (4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}) \end{aligned}$$

(42)

and

$$\begin{aligned} b = \sqrt{{{(4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq}})^2}-4M^{2}_{iiq}M^{2}_{jjq}} \end{aligned}$$

(43)

Now error in P can be given as,

$$\begin{aligned} \delta P = \sqrt{(\delta a)^{2} + (\delta b)^{2}} \end{aligned}$$

(44)

From Eq. (42) we can have,

$$\begin{aligned} \delta a = \sqrt{(32M_{ijq} \delta M_{ijq} )^{2} + (2M_{iiq} \delta M_{iiq})^{2} + (2M_{jjq} \delta M_{jjq})^{2}} \end{aligned}$$

(45)

Here $M_{ijq}$, $M_{jjq}$, and $M_{iiq}$ are the experimental masses which are taken as input and $\delta M_{ijq}$, $\delta M_{jjq}$, and $\delta M_{iiq}$ respectively, represents their error values. Now to calculate error in b, Eq. (43) is written as,

$$\begin{aligned} b = \sqrt{F} \end{aligned}$$

(46)

Since, error in b is given as,

$$\begin{aligned} \delta b = \frac{1}{2} \left( \frac{\delta F}{\sqrt{F}}\right) \end{aligned}$$

(47)

where,

$$\begin{aligned} F = (4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})^{2}-4M^{2}_{iiq}M^{2}_{jjq} \end{aligned}$$

(48)

We can write Eq. (48) as,

$$\begin{aligned} F = H - G \end{aligned}$$

(49)

where,

$$\begin{aligned} H = (4M^{2}_{ijq}-M^{2}_{iiq}-M^{2}_{jjq})^{2} \end{aligned}$$

(50)

and,

$$\begin{aligned} G = 4M^{2}_{iiq}M^{2}_{jjq} \end{aligned}$$

(51)

Hence,

$$\begin{aligned} \delta F = \sqrt{(\delta H)^{2}+(\delta G)^{2}} \end{aligned}$$

(52)

Now, form Eqs. (45) and (50) we can write the error in H as,

$$\begin{aligned} \delta H = 2H^{1/2} \delta a \end{aligned}$$

(53)

Putting the value of $\delta a$ and H from Eqs. (45) and (50) into Eq. (53), we can obtain the value of $\delta H$. Now, from Eq. (51) we can have,

$$\begin{aligned} \delta G = 16 M^{2}_{iiq}M^{2}_{jjq}\times \sqrt{\left( \frac{2\delta M_{iiq}}{ M_{iiq}}\right) ^{2}+\left( \frac{2\delta M_{jjq}}{ M_{jjq}}\right) ^{2}} \end{aligned}$$

(54)

Inserting the values of $\delta H$ and $\delta G$ into Eq. (52), we get the value of $\delta F$. Using the value of $\delta F$ and form Eq. (48) we can find $\delta b$. Hence, from Eqs. (32), (40), (41), and (44) we can calculate the value of $\delta x$. Now putting the values of $\delta x$ and $\delta y$ from Eqs. (40) and (38) respectively, in Eq. (37) we can finally get the error in slope value i.e. $\delta \alpha ^{'}_{jjq}$ which are shown in Table 1.

In the similar manner, we can evaluate the error in slope ($\alpha _{ijq}$) obtained from Eq. (18). Once we get error in slope values, now we have obtained the error values in excited state masses, obtained from Eq. (24). Now Eq. (24) can also be written as,

$$\begin{aligned} M_{J+1} = \sqrt{Q} \end{aligned}$$

(55)

where,

$$\begin{aligned} Q = M^{2}_{J}+\frac{1}{\alpha ^{'}} \end{aligned}$$

(56)

Error in Q from Eq. (56) can be given as,

$$\begin{aligned} \delta Q = \sqrt{(2M_{J}\delta M_{J})^{2}+\left( \frac{\delta \alpha ^{'}}{\alpha ^{'}}\right) ^{2}} \end{aligned}$$

(57)

where $\alpha ^{'}$ and $\delta \alpha ^{'}$ represents the value of slopes we have determined in the present work for singly charmed baryons and their error values we have obtained from above calculation. Also error in excited state masses $M_{J+1}$ is represented by $\delta M_{J+1}$. Hence from Eq. (55) we can have,

$$\begin{aligned} \delta M_{J+1} = \frac{1}{2}\left( \frac{\delta Q}{\sqrt{Q}}\right) \end{aligned}$$

(58)

Hence from Eqs. (56), (57), and (58) can obtained the error values in excited state masses in the ($J,M^{2}$) plane which are shown in Tables 2, 3, 4, 5 and 6.

Similarly, we can determine the error values for Regge slopes and intercept calculated in the ($n,M^{2}$) plane which are shown in Table 7 and also error in the radially excited state masses evaluated in the ($n,M^{2}$) plane for singly charmed baryons.