# Phenomenologies in Hypersphere Soliton and String Photon Models
## 2 Baryon Predictions in HSM  
Now, we consider the baryon predictions in the first-class Hamiltonian formalism in the HSM. To do this, we introduce the Skyrmion Lagrangian density given by  
\[\mathcal{L}=\frac{f_{\pi}^{2}}{4}\text{tr}(\partial_{\mu}U^{\dagger}\partial^ {\mu}U)+\frac{1}{32e^{2}}\text{tr}[U^{\dagger}\partial_{\mu}U,U^{\dagger} \partial_{\
u}U]^{2} \tag{1}\]  
where \(U\) is an SU(2) chiral field, and \(f_{\pi}\) and \(e\) are a pion decay constant and a dimensionless Skyrme parameter, respectively. In this work, we will treat \(f_{\pi}\) and \(e\) as the model parameters. Here, the quartic term is necessary to stabilize the soliton in the baryon sector. Next, we introduce the hyperspherical three metric on \(S^{3}\) of the form  
\[ds^{2}=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu\ (d\theta^{2}+\sin^{2}\theta\ d \phi^{2}), \tag{2}\]  
where the ranges of the three angles are defined as \(0\leq\mu\leq\pi\), \(0\leq\theta\leq\pi\) and \(0\leq\phi\leq 2\pi\), and \(\lambda\) (\(0\leq\lambda<\infty\)) is a radius parameter of \(S^{3}\). In the HSM, using the Skyrmion Lagrangian density in (1), we obtain the soliton energy \(E\) of the form  
\[E=\frac{f_{\pi}}{e}\Bigg{[}2\pi L\int_{0}^{\pi}d\mu\sin^{2}\mu\left(\left( \frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\right)^{2}+2\bigg{(} \frac{1}{L}\frac{df}{d\mu}+1\bigg{)}^{2}\frac{\sin^{2}f}{\sin^{2}\mu}\right)+ 6\pi^{2}B\Bigg{]}, \tag{3}\]  
where \(L=ef_{\pi}\lambda\) (\(0\leq L<\infty\)) is a dimensionless radius parameter and \(B\) is topological baryon number, which is unity for a single soliton. Here, \(f(\mu)\) is a profile function for hypersphere soliton, and satisfies \(f(0)=\pi\) and \(f(\pi)=0\) for unit topological baryon number. Note that the the profile function \(f\) in the soliton energy \(E\) in (3) satisfies the first-order differential equations  
\[\frac{df}{d\mu}+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}=0,\ \ \ \frac{1}{L}\frac{df}{d\mu}+1=0, \tag{4}\]  
to attain the BPS topological lower bound in the soliton energy [2; 4; 5; 6; 7] given by  
\[E_{B}=\frac{6\pi^{2}f_{\pi}}{e}. \tag{5}\]  
Moreover, in this case, we find the equation of motion for the hypersphere soliton [4; 6]  
\[\bigg{(}L\sin^{2}\mu+\frac{2}{L}\sin^{2}f\bigg{)}\frac{d^{2}f}{d\mu^{2}}+ \bigg{(}L\sin 2\mu+\frac{1}{L}\frac{df}{d\mu}\sin 2f\bigg{)}\frac{df}{d\mu}- \bigg{(}L+\frac{1}{L}\frac{\sin^{2}f}{\sin^{2}\mu}\bigg{)}\sin 2f=0. \tag{6}\]One of the simplest solutions of (6) is the identity map  
\[f(\mu)=\pi-\mu, \tag{7}\]  
in which case the soliton energy in (3) can be rewritten as [4; 6]  
\[E=\frac{3\pi^{2}f_{\pi}}{e}\bigg{(}L+\frac{1}{L}\bigg{)}. \tag{8}\]  
Note that, in order to obtain the BPS topological lower bound \(E_{B}\) in (5) by exploiting the soliton energy \(E\) in (8) associated with the identity map \(f(\mu)=\pi-\mu\) in (7), we use the fixed value \(L=L_{B}\) where  
\[L_{B}\equiv ef_{\pi}\lambda_{B}=1. \tag{9}\]  
Note also that the identity map in (7) is a minimum energy solution for \(L<\sqrt{2}\), while for \(L>\sqrt{2}\), it is a saddle point [20; 21]. Now, we briefly discuss the Dirac quantization of constrained systems [1; 3; 4; 5; 6; 7; 8; 9; 22]. In the HSM, we have the second class constraints for the collective coordinates \(a^{\mu}\) (\(\mu=0,1,2,3\)) and the corresponding canonical momenta \(\pi^{\mu}\) conjugate to \(a^{\mu}\) of the form  
\[\Omega_{1}=a^{\mu}a^{\mu}-1\approx 0,\ \ \ \Omega_{2}=a^{\mu}\pi^{\mu}\approx 0. \tag{10}\]  
Exploiting the Poisson bracket for \(a^{\mu}\) and \(\pi^{\mu}\),  
\[ds^{2}=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu\ (d\theta^{2}+\sin^{2}\theta\ d \phi^{2}), \tag{2}\]0  
we obtain the Poisson algebra for the commutator of \(\Omega_{1}\) and \(\Omega_{2}\)[8; 22]  
\[ds^{2}=\lambda^{2}d\mu^{2}+\lambda^{2}\sin^{2}\mu\ (d\theta^{2}+\sin^{2}\theta\ d \phi^{2}), \tag{2}\]1  
Since this Poisson algebra does not vanish, we call the constraints \(\Omega_{1}\) and \(\Omega_{2}\) the second class.