Magnetic moments of decuplet baryons in asymmetric strange hadronic matter at finite temperature

Abstract

In the present work we study the masses and magnetic moments of decuplet baryons in isospin asymmetric strange hadronic medium at finite temperature using chiral SU(3) quark mean field model. In the strange isospin asymmetric medium, the properties of baryons in chiral SU(3) mean field model are modified through the exchange of scalar fields $\sigma$, $\zeta$ and $\delta$ and the vector fields $\omega$, $\rho$ and $\phi$. The scalar-isovector field $\delta$ and the vector-isovector field $\rho$ signifies the finite isospin asymmetry of the medium. We calculate the in-medium constituent quark masses and masses of decuplet baryons in asymmetric strange matter within the chiral SU(3) quark mean field model and use these as input in the constituent chiral quark model to calculate the in-medium magnetic moments of decuplet baryons for different values of isospin asymmetry and strangeness fraction of hot and dense medium. For calculating the magnetic moments of baryons, contributions of valence quarks, quark sea and orbital angular momentum of quark sea are considered in the calculations.

Appendix

The equations of motion for scalar fields $\sigma$, $\zeta$ the dilaton field, $\chi$, scalar iso-vector field, $\delta$, and, the vector fields $\omega$, $\rho$ and $\phi$ obtained by minimizing the thermodynamic potential are written as (Eq. (22))

$$\begin{aligned} \frac{\partial \Omega }{\partial \sigma }= & {} k_{0}\chi ^{2}\sigma -4k_{1}\left( \sigma ^{2} \, + \, \zeta ^{2} \, + \, \delta ^2\right) \sigma \, - \, 2k_{2}\left( \sigma ^{3} + 3\sigma \delta ^2\right) \nonumber \\ {}{} & {} - 2k_{3}\chi \sigma \zeta \, - \, \frac{\xi }{3}\chi ^{4}\left( \frac{2\sigma }{\sigma ^2-\delta ^2}\right) \nonumber \\ {}{} & {} + \, \frac{\chi ^{2}}{\chi _{0}^{2}}m_{\pi }^{2}f_{\pi } -\left( \frac{\chi }{\chi _{0}}\right) ^{2}m_{\omega }\omega ^{2} \frac{\partial m_{\omega }}{\partial \sigma }\, \nonumber \\ {}{} & {} -\left( \frac{\chi }{\chi _{0}}\right) ^{2}m_{\rho }\rho ^{2} \frac{\partial m_{\rho }}{\partial \sigma }\, + \, \sum _{i} \frac{\partial M_{i}^{*}}{\partial \sigma } \rho _i^s=0,\end{aligned}$$

(34)

$$\begin{aligned} \frac{\partial \Omega }{\partial \zeta }= & {} k_{0}\chi ^{2}\zeta - 4k_{1}\left( \sigma ^{2} \, + \, \zeta ^{2} + \delta ^2\right) \zeta \, - \, 4k_{2}\zeta ^{3} \, - \, k_{3}\chi \sigma ^{2} \ \nonumber \\ {}{} & {} - \, \frac{\xi \chi ^{4}}{3\zeta } \, + \, \frac{\chi ^{2}}{\chi _{0}^{2}} \left( \sqrt{2}m_{K}^{2}f_{K} \, - \, \frac{1}{\sqrt{2}}m_{\pi }^{2}f_{\pi } \right) \nonumber \\ {}{} & {} + \, \sum _{i=\Lambda , \Sigma ^{\pm ,0}, \Xi ^{-,0}} \frac{\partial M_{i}^{*}}{\partial \zeta } \rho _i^s=0,~~~~~ = 0, \end{aligned}$$

(35)

$$\begin{aligned} \frac{\partial \Omega }{\partial \chi }= & {} k_0\chi \left( \sigma ^2+\zeta ^2+\delta ^2\right) -k_3\left( \sigma ^2-\delta ^2\right) \zeta \nonumber \\ {}{} & {} +\frac{2\chi }{\chi _{0}^{2}}\left[ m_{\pi }^{2}f_{\pi }\sigma +\left( \sqrt{2}m_{K}^{2}f_{K} \, - \, \frac{1}{\sqrt{2}}m_{\pi }^{2}f_{\pi } \right) \zeta \right] \nonumber \\{} & {} -\frac{\chi }{\chi _0^2} \left( m_\omega ^2\omega ^2+m_\rho ^2\rho ^2\right) \nonumber \\ {}{} & {} +\left( 4k_4+1+\textrm{ln}\frac{\chi ^4}{\chi _0^4} - \frac{4\xi }{3}\textrm{ln}\left( \left( \frac{\left( \sigma ^2-\delta ^2\right) \zeta }{\sigma _0^2\zeta _0}\right) \right) \right) \chi ^3 =0, \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial \Omega }{\partial \delta }= & {} k_{0}\chi ^{2}\delta -4k_{1}\left( \sigma ^{2} \, + \, \zeta ^{2} \, + \, \delta ^2\right) \delta \, - \, 2k_{2}\left( \delta ^{3} + 3\sigma ^2\delta \right) \nonumber \\ {}{} & {} + \, 2k_{3}\chi \delta \zeta \, - \, \frac{\xi }{3}\chi ^{4}\left( \frac{2\delta }{\sigma ^2-\delta ^2}\right) \, - \, \sum _{i} g_{\delta i} \rho _i^s=0, \end{aligned}$$

(37)

$$\begin{aligned} \frac{\partial \Omega }{\partial \omega }= & {} \left( \frac{\chi ^2}{\chi _0^2}\right) m_\omega ^2\omega +4g_4 \omega ^3+12g_4\omega \rho ^2-\sum _{i} g_{\omega i} \rho _i=0, \end{aligned}$$

(38)

$$\begin{aligned} \frac{\partial \Omega }{\partial \rho }= & {} \left( \frac{\chi ^2}{\chi _0^2}\right) m_\rho ^2\rho +4g_4 \rho ^3+12g_4\omega ^2\rho -\sum _{i} {g_{\rho i}} \rho _i=0, \end{aligned}$$

(39)

and

$$\begin{aligned} \frac{\partial \Omega }{\partial \phi }= \left( \frac{\chi ^2}{\chi _0^2} \right) m_\phi ^2\phi +8g_4\phi ^3- \sum _{i}g_{\phi \,i}\rho _{i}=0, \end{aligned}$$

(40)

respectively. The number (vector) density, $\rho _{i}$, and scalar density, $\rho _{i}^{s}$, of baryons appearing in above equation are given as

$$\begin{aligned} \rho _{i} = \gamma _{i} \int \frac{d^{3}k}{(2\pi )^{3}} \Big (f_i(k)-\bar{f}_i(k) \Big ), \end{aligned}$$

(41)

and

$$\begin{aligned} \rho _{i}^{s} = \gamma _{i} \int \frac{d^{3}k}{(2\pi )^{3}} \frac{m_{i}^{*}}{E^{*}_i(k)} \Big (f_i(k)+\bar{f}_i(k) \Big ), \end{aligned}$$

(42)

respectively, where $f_i(k)$ and $\bar{f}_i(k)$ represent the Fermi distribution functions at finite temperature for fermions and anti-fermions and are expressed as

$$\begin{aligned} f_i(k)= & {} \frac{1}{1+\exp \left[ (E^*_{i}(k) -\nu ^{*}_{i})/k_BT \right] }~~~ \text {and} \nonumber \\ \bar{f}_i(k)= & {} \frac{1}{1+\exp \left[ ( E^*_{i}(k) +\nu ^{*}_{i})/k_BT\right] }~. \end{aligned}$$

(43)