# Phenomenologies in Hypersphere Soliton and String Photon Models
## 2 Baryon Predictions in HSM
The nucleon mass \(M_{N}\) (\(I=1/2\)) and delta baryon mass \(M_{\Delta}\) (\(I=3/2\)) are then given as follows [9]  
\[E=\frac{f_{\pi}}{e}\Bigg{[}2\pi L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left( \frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\bigg{(} \frac{1}{L}\frac{df}{d\mu}+1\bigg{)}^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+ 6\pi^{2}B\Bigg{]}, \tag{3}\]6  
Next, we formulate the magnetic moments of the form [9]  
\[E=\frac{f_{\pi}}{e}\Bigg{[}2\pi L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left( \frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\bigg{(} \frac{1}{L}\frac{df}{d\mu}+1\bigg{)}^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+ 6\pi^{2}B\Bigg{]}, \tag{3}\]7 \[E=\frac{f_{\pi}}{e}\Bigg{[}2\pi L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left( \frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\bigg{(} \frac{1}{L}\frac{df}{d\mu}+1\bigg{)}^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+ 6\pi^{2}B\Bigg{]}, \tag{3}\]8  
where \(M_{N}\) is now given by the nucleon mass with the WOC in (26), given in the first-class formalism. Next, we similarly obtain the axial coupling constant [9]  
\[E=\frac{f_{\pi}}{e}\Bigg{[}2\pi L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left( \frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\bigg{(} \frac{1}{L}\frac{df}{d\mu}+1\bigg{)}^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+ 6\pi^{2}B\Bigg{]}, \tag{3}\]9  
Now, we consider the charge radii. The electric and magnetic isovector mean square charge radii are given in the HSM, respectively [6; 9]  
\[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]0  
where the subscripts \(E\) and \(M\) denote electric and magnetic charge radii, respectively, and \(dV_{B}=\lambda_{B}^{3}\sin^{2}\mu\sin\theta\leavevmode\
obreak\ \mu d \leavevmode\
obreak\ \theta d\phi\) on the hypersphere \(S^{3}\). Note that \(dV_{B}\) is given by the product of three arc lengths: \(\lambda_{B}d\mu\), \(\lambda_{B}\sin\mu d\theta\) and \(\lambda_{B}\sin\mu\sin\theta d\phi\) and \(\lambda_{B}\) is radius of hypersphere soliton. Moreover, we find the charge radii in terms of \(ef_{\pi}\)[6; 9]  
\[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]1 \[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]2  
Shuffling the above baryon and transition magnetic moments, we obtain the model independent sum rules in the HSM [6]  
\[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]3 \[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]4 \[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]5 \[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]6  
Next, we choose \(\langle r^{2}\rangle_{M,p}^{1/2}=0.80\) fm as an input parameter. We then have  
\[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]7  
and exploiting this fixed value of \(ef_{\pi}\) and the phenomenological formulas in (26)-(30), we can proceed to calculate the physical quantities, as shown in \[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]8  
Now, we discuss the predictions in the soliton models. In Table 1, Prediction I and II are given by [PERSON] [9] using the HSM, while Prediction III is given by [PERSON] [3], exploiting the standard Skyrmion model defined in \(R^{3}\) manifold. Here, the input parameters are indicated by \({}^{\star}\). In Prediction I, the two experimental values for \(\langle r^{2}\rangle_{M,p}^{1/2}\) and \(M_{N}\) are used as input parameters. In Predictions II and III, we have exploited the same input parameters associated with \(M_{N}\) and \(M_{\Delta}\) to compare their predictions effectively.