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In the presence of a magnetic field, the effective hadronic chiral Lagrangian density [91] is given by
$ \begin{aligned}[b]{ \cal L}_{{ \rm{eff}}} =& {\cal L}_{ {\rm{kin}}} + \sum\limits_{{\cal W}={\cal X,Y,A,V,}u}{\cal{L_ {\rm{BW}}}} + \cal{L_ \rm{vec}}+ \cal{L_ {\rm{0}}} \\ &+ \cal{L_ {\rm{scale \;break}}} + \cal{L_ {\rm{SB}}} + \cal{L_ {\rm{mag}}}. \end{aligned} $
(1) In this equation,
$ \cal{L_ \rm{kin}} $ refers to the kinetic energy terms of mesons and baryons.$ \cal{L_ \rm{BW}} $ is the baryon-meson interaction term, where the index$ \cal{ W} $ covers both spin-0 and spin-1 mesons. Here, the baryon masses are generated dynamically through baryon-scalar meson interactions.$ \cal{L_ \rm{vec}} $ involves the dynamical mass generation of vector mesons through couplings with scalar mesons, in addition to bearing the quartic self-interaction terms of these mesons.$ \cal{L_ \rm{0}} $ contains the meson-meson interaction terms that introduce the spontaneous breaking of the chiral symmetry.$ \cal{L_ \rm{scale break}} $ incorporates the scale invariance breaking of QCD through a logarithmic potential given in terms of scalar dilaton field χ.$ \cal{L_ \rm{SB}} $ is the explicit chiral symmetry breaking term, and$ \cal{L_ \rm{mag}} $ is the contribution by the magnetic field. We use the mean-field approximation to simplify the hadronic Lagrangian density under which all the meson fields are considered as classical fields. In this approximation, only the vector fields ($ \omega,\;\rho,\;\phi $ ) and scalar fields (nonstrange scalar field σ, strange scalar field ζ and scalar-isovector field δ) contribute as the expectation value of the other mesons vanishes. The baryon-meson interaction term simplifies to$ {{\cal L}_ {\rm{BW}}}=-\sum\bar\psi_{i}[ m_i^*+ g_{\omega i}\gamma_{0} \omega+ g_{\rho i}\gamma_{0} \rho+ g_{\phi i}\gamma_{0} \phi]\psi_{i}. $
(2) Here, the index i covers the eight lightest baryons n, p, Λ,
$ \Sigma^{-} $ ,$ \Sigma^{0} $ ,$ \Sigma^{+} $ ,$ \Xi^{-} $ ,$ \Xi^{0} $ , and$ g_{\omega i},\; g_{\rho i}, \;g_{\phi i} $ represent the coupling strengths of baryons with the vector mesons ω, ρ, and ϕ, respectively. The effective mass of the baryons denoted as$ m_i^* $ is given by$ \begin{equation} m_i^* = - (g_{\sigma i}\sigma+ g_{\zeta i}\zeta + g_{\delta i}\delta), \end{equation} $
(3) where
$ g_{\sigma i}, \;g_{\zeta i},\; g_{\delta i} $ represent the coupling strengths of baryons with the scalar mesons σ, ζ, and δ respectively. The other terms in the Lagrangian reduce to the following expressions:$ {{\cal L}_ {\rm{vec}}} = \frac{1}{2}( m_\omega^2\omega^2 + m_\rho^2\rho^2+ m_\phi^2\phi^2)\left(\frac{\chi^2}{{\chi_0}^2}\right) + g_4(\omega^4 + 6\rho^2\omega^2 + \rho^4 + 2\phi^4), $
(4) $ \begin{aligned}[b] {{\cal L}_ {\rm{0}}}+{{\cal L}_ {\rm{scale\; break}}}=&-\frac{1}{2}k_0\chi^2(\sigma^2+\zeta^2+\delta^2)+k_1(\sigma^2+\zeta^2+\delta^2)^2+k_2\left(\frac{\sigma^4}{2}+\frac{\delta^4}{2}+3\sigma^2\delta^2+\zeta^4\right)+k_3\chi(\sigma^2-\delta^2)\zeta\\&-k_4\chi^4 -\frac{1}{4}\chi^4\ln{\left(\frac{\chi^4}{\chi_0^4}\right)}+\frac{d}{3}\chi^4\ln\left(\left(\frac{(\sigma^2-\delta^2)\zeta}{{\sigma_0^2}\zeta_0}\right)\left(\frac{\chi}{\chi_0}\right)^3\right), \end{aligned} $ (5) $ {{\cal L}_ {\rm{SB}}}=-\left(\frac{\chi}{\chi_0}\right)^2 \Big[{m_\pi}^2f_\pi \sigma +(\sqrt{2}{m_K}^2f_K-\frac{1}{\sqrt{2}}{m_\pi}^2f_\pi)\zeta\Big]. $
(6) In Eq. (4),
$ g_4 $ is the renormalized coupling for the ω field [71, 72]. In the above equations,$ \sigma_0 $ ,$ \zeta_0 $ , and$ \chi_0 $ denote the vacuum values of the scalar fields σ, ζ, and the dilaton field χ, respectively. In Eq. (5), the parameters$ k_0 $ ,$ k_1 $ ,$ k_2 $ ,$ k_3 $ , and$ k_4 $ are phenomenological constants, and their fitting is described in detail in Section V. In Eq. (6),$ f_K $ is the kaon decay constant,$ f_\pi $ is the pion decay constant, and$ m_K $ ,$ m_\pi $ are their respective vacuum masses. Finally, the contribution of the magnetic field incorporated in the Lagrangian term given by$ \begin{array}{*{20}{l}} {{\cal L}_ {\rm{mag}}} =-\bar{\psi_{i}}q_{i}\gamma_{\mu}A^{\mu}\psi_{i}-\dfrac{1}{4}\kappa_{i}\mu_{N}\bar{\psi_{i}}\sigma^{\mu\nu}F_{\mu\nu}\psi_{i}-\dfrac{1}{4}F^{\mu\nu}F_{\mu\nu}. \end{array} $
(7) The second term in Eq. (7), which is a tensorial interaction term, is related to the anomalous magnetic moment (AMM) of the baryons. In this term,
$ \mu_{N} $ is the nuclear Bohr magneton, expressed as$ \mu_{N} $ =$ e/(2m_N) $ , where$ m_N $ is the vacuum mass of the nucleon. Here,$ \kappa_{i} $ is the gyromagnetic ratio corresponding to the anomalous magnetic moment of the baryons. The values of$ \kappa_{i} $ used in our calculations were obtained from Refs. [98–100]. We consider the magnetic field to be uniform and along the z-axis. We set the vector potential as$ A^{\mu} $ = (0, 0, Bx, 0). From the mean-field Lagrangian density, we obtain the coupled equations of motion for the scalar fields σ, ζ, δ, χ, and vector meson fields$ \omega,\; \rho,\; \phi $ . The equations of motion for the scalar fields σ, ζ, and δ are expressed in terms of the scalar densities of baryons [72, 73]. The equations of motion for vector meson fields$ \omega, \rho, \phi $ are expressed in terms of the number densities of baryons [72, 73]. The magnetic field introduces summation over Landau levels in the expressions of the number and scalar densities of charged baryons (i= p,$ \Sigma^{-} $ ,$ \Sigma^{+} $ ,$ \Xi^{-} $ ), which are given by [100–102]$ \;\;\; \rho_i = \frac{eB}{2\pi^2}\Bigg[{\sum\limits_{\nu}^{{\nu^{(S=1)}_{\rm max}}} k_{f, \nu, 1}^i} + {\sum\limits_{\nu}^{{\nu^{(S=-1)}_{\rm max}}} k_{f, \nu, -1}^i}\Bigg], $
(8) $ \begin{aligned}[b] \rho_s^i =& \frac{eBm_i^*}{2\pi^2}\Bigg[{\sum\limits_{\nu}^{\nu^{(S=1)}_{\rm max}} \frac{\sqrt[]{m_i^{*2} + 2eB\nu} + \Delta_i}{\sqrt[]{m_i^{*2} + 2eB\nu}} {\rm ln}\Bigg|\frac{k_{f,\nu,1}^i + E_f^i}{\sqrt[]{m_i^{*2} + 2eB\nu} + \Delta_i}\Bigg|} \\ &+ {\sum\limits_{\nu}^{\nu^{(S=-1)}_{\rm max}} \frac{\sqrt[]{m_i^{*2} + 2eB\nu} - \Delta_i}{\sqrt[]{m_i^{*2} + 2eB\nu}} \ln\Bigg|\frac{k_{f,\nu,-1}^i + E_f^i}{\sqrt[]{m_i^{*2} + 2eB\nu} - \Delta_i}\Bigg|}\Bigg]. \end{aligned} $
(9) Here,
$ k_{f, \nu, s}^i $ is the Fermi momentum of charged baryons,$ E_f^i $ is the Fermi energy, ν is the Landau level, and spin index S = +1($ -1 $ ) corresponds to spin up (spin down) projections for the baryons. The parameter$ \Delta_{i} $ refers to the anomalous magnetic moments of the baryons, expressed as$ \Delta_{i}=-({1}/{2})\kappa_i\mu_N B $ . The Landau levels of charged baryons is enumerated using the expression$ \nu = n+ \dfrac{1}{2}-\dfrac{q_B}{|q_B|}\dfrac{S}{2}, $ where$ q_B $ is the charge of the baryon ($ q_B=e $ for p,$ \Sigma^{+} $ and$ q_B=-e $ for$ \Sigma^{-} $ ,$ \Xi^{-} $ ). The lowest Landau level for a particular spin projection of the charged baryon is obtained by setting n=0 in this expression. The maximum allowed value of Landau level for a charged baryon is determined using the expression$\nu_{\rm max} = \Bigg\lfloor\dfrac{(E_f^i - S \Delta_i)^2 - m_i^{*2}}{2eB}\Bigg\rfloor. $ Here, the floor operator acting on a quantity x, i.e.,$ \lfloor x\rfloor $ is defined as the largest integer less than or equal to x. The Fermi momenta of charged baryons are related to their Fermi energies$ E_f^i $ as$ \begin{array}{*{20}{l}} k_{f, \nu, S}^i = \sqrt[]{(E_f^i)^2 - \Big(\sqrt[]{m_i^{*2} + 2eB\nu} + S\Delta_i\Big)^2}. \end{array} $
(10) For neutral baryons (i= n, Λ,
$ \Sigma^{0} $ ,$ \Xi^{0} $ ), there is no contribution of Landau quantization in the presence of an external magnetic field. The number and scalar densities are given by$ \begin{aligned}[b] \rho_i =& \frac{1}{4\pi^2}\sum\limits_{S=\pm1} \Bigg(\frac{2}{3}(k_{f,S}^i)^3 + S\Delta_i\Bigg[(m_i^* + S\Delta_i)k_{f, S}^i \\&+ (E_f^i)^2\Bigg\{{\rm arcsin}\Bigg(\frac{m_i^* + S\Delta_i}{E_f^i}\Bigg) - \frac{\pi}{2}\Bigg\}\Bigg]\Bigg), \end{aligned} $
(11) $ \rho_s^i = \frac{m_i^*}{4\pi^2}\sum\limits_{S=\pm1}\Bigg[k_{f,S}^iE_f^i - (m_i^* + S\Delta_i)^2\ln\Bigg|\frac{k_{f,S}^i + E_f^i}{m_i^* + S\Delta_i}\Bigg|\Bigg]. $
(12) The Fermi momenta of neutral baryons
$ k_{f,S}^i $ are related to their Fermi energies$ E_f^i $ as$ \begin{array}{*{20}{l}} k_{f, S}^i = \sqrt[]{(E_f^i)^2 - (m_i^* + S\Delta_i)^2}. \end{array} $
(13) We consider the hyperonic matter in equilibrium, resulting in five chemical equilibrium equations of baryons. They are given by
$ \begin{array}{*{20}{l}} \Lambda + \Lambda\rightleftharpoons p+{\Xi^-}, \end{array} $
(14) $ \begin{array}{*{20}{l}} \Lambda + \Lambda\rightleftharpoons n+\Xi^0, \end{array} $
(15) $ \begin{array}{*{20}{l}} \Sigma^-+p \rightleftharpoons \Lambda + n, \end{array} $
(16) $ \begin{array}{*{20}{l}} \Sigma^++n \rightleftharpoons \Lambda+p, \end{array} $
(17) $ \begin{array}{*{20}{l}} \Sigma^0+\Sigma^-\rightleftharpoons \Xi^-+n.\end{array} $
(18) The above equations constrain the chemical potentials
$ \mu_i $ of all the baryons in the medium. Because the particles on the left- and right-hand sides are in chemical equilibrium, their chemical potentials can be equated and expressed as$ \begin{array}{*{20}{l}} 2\mu_\Lambda=\mu_p+\mu_{\Xi^-}, \quad \end{array} $
(19) $ \begin{array}{*{20}{l}} 2\mu_\Lambda=\mu_n+\mu_{\Xi^0}, \quad \end{array} $
(20) $ \begin{array}{*{20}{l}} \mu_{\Sigma^-}+\mu_p=\mu_n+\mu_{\Lambda}, \end{array} $
(21) $ \begin{array}{*{20}{l}} \mu_{\Sigma^+}+\mu_n=\mu_p+\mu_{\Lambda}, \end{array} $
(22) $ \begin{array}{*{20}{l}} \mu_{\Sigma^-}+\mu_{\Sigma^0}=\mu_n+\mu_{\Xi^-}. \end{array} $
(23) These equations can be further rewritten in terms of the effective chemical potential of baryons
$ \mu_i^* $ through the relation$ \mu_i^* =\mu_i-(g_{\rho i}\rho+g_{\omega i}\omega+g_{\phi i}\phi) $ . The effective chemical potential$ \mu_i^* $ is numerically equal to the Fermi energy of baryons at temperature T=0. These chemical equilibrium equations provide the necessary equations of constraint in studying the strange hadronic matter under external magnetic fields. The equations of motion of scalar fields are then solved self-consistently at different magnetic fields for given values of the total baryon density$ \rho_B= \sum_i\rho_i $ , isospin asymmetry parameter$ \eta= ({-\sum_i{I_{3i}\rho_i}})/ {\rho_B} $ , and strangeness fraction$f_s= ({\sum_i|{S_{ i}|\,\rho_i})/}{\rho_B}$ . Here,$ I_{3i} $ is the third component of the isospin, and$ S_ i $ is the strangeness quantum number for the$ i^{\,\rm th} $ baryon. The strangeness fraction is a measure of the relative population of hyperons (with appropriate weight factors owing to the number of strange quarks in the hyperons) among all the baryons present in the medium.In the following section, we describe the interaction of open charm mesons with strongly magnetized strange hadronic matter and their medium mass modifications.
-
We examine the medium modifications of the masses of open charm mesons in asymmetric magnetized strange hadronic matter. These mesons interact with light quark condensates, which are modified significantly in the hadronic medium. Here, the chiral SU(3) has been generalized to the chiral SU(4) to include the charmed mesons and their interactions with the light hadronic sector [64–66, 74]. The interaction Lagrangian density of D and
$ \bar{D} $ mesons with the strange hadronic medium in the chiral effective model is given by [76]$ \begin{aligned}[b] {\cal L}_{\rm int} = & -\frac {\rm i}{8 f_D^2} \Big [3\Big (\bar p \gamma^\mu p +\bar n \gamma ^\mu n \Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) - (\partial_\mu {{D^0}}) {\bar D}^0 \Big ) +\Big(D^+ (\partial_\mu D^-) - (\partial_\mu {D^+}) D^- \Big )\Big ) + \Big (\bar p \gamma^\mu p -\bar n \gamma ^\mu n \Big) \Big( \Big({D^0} (\partial_\mu \bar D^0) - (\partial_\mu {{D^0}}) {\bar D}^0 \Big ) \\ &- \Big( D^+ (\partial_\mu D^-) - (\partial_\mu {D^+}) D^- \Big )\Big ) + 2\Big((\bar{\Lambda}^{0}\gamma^{\mu}\Lambda^{0}) \Big( \Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) + \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big) \Big)\\ &+ 2 \Big(\Big(\bar{\Sigma}^+\gamma^{\mu}\Sigma^+ + \bar{\Sigma}^-\gamma^{\mu}\Sigma^-\Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) + \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big)\Big)\\ &+ \Big(\bar{\Sigma}^+\gamma^{\mu}\Sigma^+ - \bar{\Sigma}^-\gamma^{\mu}\Sigma^-\Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) - \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big)\Big)\Big)\\ &+2\Big(\bar{\Sigma}^{0}\gamma^{\mu}\Sigma^{0}\Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) + \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big) \Big)\\ &+ \Big(\bar{\Xi}^{0}\gamma^{\mu}\Xi^{0} + \bar{\Xi}^-\gamma^{\mu}\Xi^-\Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) + \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big)\Big)\\ &+\Big(\bar{\Xi}^{0}\gamma^{\mu}\Xi^{0} - \bar{\Xi}^-\gamma^{\mu}\Xi^-\Big) \Big(\Big({D^0} (\partial_\mu \bar D^0) -(\partial_\mu {{D^0}}) {\bar D}^0 \Big) - \Big(D^+ (\partial_\mu D^-) - (\partial_\mu D^+) D^- \Big)\Big)\Big ]\\ &+ \frac{m_D^2}{2f_D} \Big [ (\sigma +\sqrt 2 \zeta_c)\big (\bar D^0 { D^0}+(D^- D^+) \big ) +\delta \big (\bar D^0 { D^0})-(D^- D^+) \big ) \Big ] - \frac {1}{f_D}\Big [ (\sigma +\sqrt 2 \zeta_c ) \Big ((\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) +(\partial _\mu {D^-})(\partial ^\mu {D^+}) \Big )\\ &+ \delta \Big ((\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) -(\partial _\mu {D^-})(\partial ^\mu {D^+}) \Big ) \Big ] + \frac {d_1}{2 f_D^2}(\bar p p +\bar n n +\bar{\Lambda}^{0}\Lambda^{0}+\bar{\Sigma}^+\Sigma^++\bar{\Sigma}^{0}\Sigma^{0} +\bar{\Sigma}^-\Sigma^-+\bar{\Xi}^{0}\Xi^{0}+\bar{\Xi}^-\Xi^-) \big ( (\partial _\mu {D^-})(\partial ^\mu {D^+}) \\ &+(\partial _\mu {{\bar D}^0})(\partial ^\mu {D^0}) \big ) + \frac {d_2}{2 f_D^2} \Big [ \Big(\bar p p+\frac{1}{6}\bar{\Lambda}^{0}\Lambda^{0} +\bar{\Sigma}^+\Sigma^++\frac{1}{2}\bar{\Sigma}^{0}\Sigma^{0}\Big) (\partial_\mu {\bar D}^0)(\partial^\mu {D^0}) \\ &+\Big(\bar n n+\frac{1}{6}\bar{\Lambda}^{0}\Lambda^{0} +\bar{\Sigma}^-\Sigma^-+\frac{1}{2}\bar{\Sigma}^{0}\Sigma^{0}\Big) (\partial_\mu D^-)(\partial^\mu D^+)\Big ]. \\[-8pt] \end{aligned} $ (24) In Eq. (24), the first term (with coefficient
$ -{\rm i}/8{f_D}^2 $ ) is the vectorial Weinberg-Tomozawa interaction term, which is obtained from the kinetic energy term$ \cal{L_ \rm{kin}} $ in Eq. (1) [64–66]. The general structure of the Weinberg-Tomozawa term for the SU(4) case is given in Ref. [65]. The second term (with coefficient$ m^2_D/2f_D $ ) is the scalar meson exchange term that is obtained from the explicit symmetry-breaking term$ \cal{L_ \rm{SB}} $ in Eq. (1) [64–66]. The next three terms in the above Lagrangian density ($ \sim(\partial_\mu {\bar D} $ )($ \partial^\mu {D} $ )) are known as the range terms. The first range term (with the coefficient ($ - 1/ f_D $ )) is obtained from the kinetic energy term of the pseudoscalar mesons [65]. Here,$ f_D $ is the decay constant of D mesons. The terms with coefficients ($ d_1/2f_D^2 $ ) and ($ d_2/2f_D^2 $ ) are the$ d_1 $ and$ d_2 $ range terms, respectively [65]. The parameters$ d_1 $ and$ d_2 $ are determined by a fitting of the empirical values of the Kaon-Nucleon scattering lengths [103–105] for I = 0 and I = 1 channels [76, 77].The interaction Lagrangian density of
$ D_s $ mesons in the strange hadronic medium is given by [70]$ \begin{aligned}[b] {\cal L}_{\rm int}=&-\frac{\rm i}{4 f_{D_{S}}^{2}}\big[(2(\bar{\Xi}^{0} \gamma^{\mu} \Xi^{0}+\bar{\Xi}^- \gamma^{\mu} \Xi^-)+\bar{\Lambda}^{0} \gamma^{\mu} \Lambda^{0}+\bar{\Sigma}^+ \gamma^{\mu} \Sigma^+ +\bar{\Sigma}^{0} \gamma^{\mu} \Sigma^{0}+\bar{\Sigma}^- \gamma^{\mu} \Sigma^-)(D_{S}^+(\partial_{\mu} D_{S}^-)-(\partial_{\mu} D_{S}^+) D_{S}^-)\big] \\&+\frac{m_{D_{S}}^{2}}{\sqrt{2} f_{D_{S}}}\big[(\zeta^{\prime}+\zeta_{c}^{\prime})(D_{S}^+ D_{S}^-)\big] -\frac{\sqrt{2}}{f_{D_{S}}}\big[(\zeta^{\prime}+\zeta_{c}^{\prime})((\partial_{\mu} D_{S}^+)(\partial^{\mu} D_{S}^-))\big] +\frac{d_{1}}{2 f_{D_{S}}^{2}}\big[(\bar{p} p+\bar{n} n+\bar{\Lambda}^{0} \Lambda^{0}+\bar{\Sigma}^+ \Sigma^++\bar{\Sigma}^{0} \Sigma^{0}\\&+\bar{\Sigma}^- \Sigma^-+\bar{\Xi}^{0} \Xi^{0} +\bar{\Xi}^- \Xi^-)((\partial_{\mu} D_{S}^+)(\partial^{\mu} D_{S}^-))\big] +\frac{d_{2}}{2 f_{D_{S}}^{2}}\big[(2(\bar{\Xi}^{0} \Xi^{0}+\bar{\Xi}^- \Xi^-)+\bar{\Lambda}^{0} \Lambda^{0}+\bar{\Sigma}^+ \Sigma^++\bar{\Sigma}^{0} \Sigma^{0}+\bar{\Sigma}^- \Sigma^-)((\partial_{\mu} D_{S}^+)(\partial^{\mu} D_{S}^-))\big]. \end{aligned} $ (25) In Eq. (25), the first term (with coefficient
$ -{\rm i}/4{f_{D_s}}^2 $ ) is the Weinberg-Tomozawa interaction term [70]. The second term (with coefficient$ m^2_{D_S}/\sqrt{2}f_{D_s} $ ) in Eq. (25) is the scalar meson exchange term. The third term (with the coefficient ($ -\sqrt{2}/ f_{D_s} $ )) is the first range term. The fourth and fifth terms with coefficients ($ d_1/2f_{D_s}^2 $ ) and ($ d_2/2f_{D_s}^2 $ ) are the$ d_1 $ and$ d_2 $ range terms, respectively. The interaction Lagrangian density given in Eqs. (24) and (25) lead to equations of motion for D,$ \bar{D} $ , and$ D_s $ mesons, respectively, and their Fourier transforms lead to the dispersion relations given by$ \begin{array}{*{20}{l}} -\omega^2 + \overrightarrow{k}^2 + m_{j}^2 - \Pi_{j}(\omega,|\overrightarrow{k}|) = 0. \end{array} $
(26) Here, the index j denotes the various open charm mesons D,
$ \bar{D} $ ,$ D_s $ , and$ m_{j} $ is the vacuum mass of the corresponding open charm meson. Here,$ \Pi_{j}(\omega,|\vec{k}|) $ denotes the self-energy of the open charm mesons in the medium. For D mesons, the self-energy is given by$ \begin{aligned}[b] \Pi_{{D}} (\omega, |\vec k|) = & \frac {1}{4 f_D^2}\Big [3 (\rho_p +\rho_n) \pm (\rho_p -\rho_n) +2\big(\left( \rho_{\Sigma^+}+ \rho_{\Sigma^-}\right) \pm \left(\rho_{\Sigma^+}- \rho_{\Sigma^-}\right) \big)+2(\rho_{\Lambda^{0}}+\rho_{\Sigma^{0}}) +( \left( \rho_{\Xi^{0}}+ \rho_{\Xi^-}\right) \pm \left(\rho_{\Xi^{0}}- \rho_{\Xi^-}\right)) \Big ] \omega \\ &+\frac {m_D^2}{2 f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') + \Big [- \frac {1}{f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') +\frac {d_1}{2 f_D ^2} (\rho_p ^s +\rho_n ^s\\ &+\rho_{\Lambda^{0}}^s+\rho_{\Sigma^+}^s+\rho_{\Sigma^{0}}^s +\rho_{\Sigma^-}^s +\rho_{\Xi^{0}}^s+\rho_{\Xi^-}^s) +\frac {d_2}{4 f_D ^2} \Big ((\rho _p^s +\rho_n^s) \pm ({\rho} _p^s -{\rho}_n^s)+\frac{1}{3}{\rho} _{\Lambda^0}^s\\ &+({\rho}_{\Sigma^+}^s +{\rho} _{\Sigma^-}^s) \pm ({\rho} _{\Sigma^+}^s -{\rho}_{\Sigma^-}^s) +{\rho} _{\Sigma^{0}}^s \Big ) \Big ] (\omega ^2 - {\vec k}^2), \end{aligned} $ (27) where
$ \pm $ refers to$ D^0 $ and$ D^+ $ , respectively. For$ \bar{D} $ mesons, the self-energy is given by$ \begin{aligned}[b] \Pi_{\bar{D}}(\omega, |\vec k|) = & -\frac {1}{4 f_D^2}\Big [3 (\rho_p +\rho_n) \pm (\rho_p -\rho_n) +2\big(\left( \rho_{\Sigma^+}+ \rho_{\Sigma^-}\right)\pm \left(\rho_{\Sigma^+}- \rho_{\Sigma^-}\right) \big)+2(\rho_{\Lambda^{0}}+\rho_{\Sigma^{0}}) +( \left( \rho_{\Xi^{0}}+ \rho_{\Xi^-}\right) \pm \left(\rho_{\Xi^{0}}- \rho_{\Xi^-}\right))\Big ] \omega \\ &+\frac {m_D^2}{2 f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') + \Big [- \frac {1}{f_D} (\sigma ' +\sqrt 2 {\zeta_c} ' \pm \delta ') +\frac {d_1}{2 f_D ^2} ({\rho}_p^s +{\rho}_n^s\\ &+{\rho}_{\Lambda^{0}}^s+{\rho}_{\Sigma^+}^s +{\rho}_{\Sigma^{0}}^s+{\rho}_{\Sigma^-}^s +{\rho}_{\Xi^{0}}^s+{\rho}_{\Xi^-}^s) +\frac {d_2}{4 f_D ^2} \Big (({\rho}_p^s +{\rho}_n^s) \pm ({\rho}_p^s -{\rho}_n^s)+\frac{1}{3}{\rho}_{\Lambda^{0}}^s \\ &+({\rho} _{\Sigma^+}^s +{\rho} _{\Sigma^-}^s) \pm ({\rho}_{\Sigma^+}^s -{\rho}_{\Sigma^-}^s) +{\rho} _{\Sigma^{0}}^s \Big ) \Big ] (\omega ^2 - {\vec k}^2), \end{aligned} $
(28) where
$ \pm $ refers to$ \bar{D^0} $ and$ D^- $ , respectively. The self-energy for strange-charmed mesons is expressed as$ \begin{aligned}[b] \Pi_{D_s}(\omega,|\vec{k}|)=&\left[\left(\frac{d_{1}}{2 f_{D_{S}}^{2}}\left(\rho_{p}^{s}+\rho_{n}^{s}+\rho_{\Lambda}^{s}+\rho_{\Sigma^+}^{s}+\rho_{\Sigma^{0}}^{s}+\rho_{\Sigma^-}^{s}+\rho_{\Xi^{0}}^{s}+\rho_{\Xi^-}^{s}\right)\right)\right. +\left(\frac{d_{2}}{2 f_{D_{S}}^{2}}\left(2\left(\rho_{\Xi^{0}}^{s}+\rho_{\Xi^-}^{s}\right)+\rho_{\Lambda}^{s}+\rho_{\Sigma^+}^{s}+\rho_{\Sigma^{0}}^{s}+\rho_{\Sigma^-}^{s}\right)\right)\\& \left.-\left(\frac{\sqrt{2}}{f_{D_{S}}}\left(\zeta^{\prime}+\zeta_{c}^{\prime}\right)\right)\right]\left(\omega^{2}-\vec{k}^{2}\right) \pm\left[\frac{1}{2 f_{D_{S}}^{2}}\left(2\left(\rho_{\Xi^{0}}+\rho_{\Xi^-}\right)+\rho_{\Lambda}+\rho_{\Sigma^+}+\rho_{\Sigma^{0}}+\rho_{\Sigma^-}\right)\right] \omega +\left[\frac{m_{D_{S}}^{2}}{\sqrt{2} f_{D_{S}}}\left(\zeta^{\prime}+\zeta_{c}^{\prime}\right)\right]. \end{aligned} $
(29) Here, the
$ \pm $ signs in the co-efficient of ω refer to$ {D_S}^+ $ and$ {D_S}^- $ mesons, respectively. In Eqs. (27), (28), and (29),$ \sigma^\prime $ = ($ \sigma - \sigma_0 $ ),$ \zeta^\prime_c $ = ($ \zeta_c - \zeta_{c0} $ ),$ \delta^\prime $ = ($ \delta - \delta_0 $ ), and$ \zeta^\prime $ = ($ \zeta - \zeta_0 $ ) denote the fluctuations of scalar fields from their vacuum values. The fluctuation$ \zeta^\prime_c $ has been observed to be negligible [106], and its contribution to the in-medium masses of open charm mesons is neglected in this investigation. The charged open charm mesons (j=$ D^+ $ ,$ D^- $ ,$ {D_S}^+ $ ,$ {D_S}^- $ ) have an additional positive mass modification in magnetic fields, which, retaining only the lowest Landau level, is given by$ \begin{array}{*{20}{l}} m^{\rm eff}_{j} = \sqrt[]{m_{j}^{*2} + |eB|}. \end{array} $
(30) In the above equation,
$ m_{j}^{*} $ are solutions of the dispersion relations given by Eq. (26) for ω at$ |\overrightarrow{k}| $ = 0. For the neutral open charm mesons (j=$ D^0 $ ,$ \bar{D^0} $ ), there is no contribution from the Landau quantization effects, and their effective mass in the medium is given by$ \begin{array}{*{20}{l}} m^{\rm eff}_{j} = {m_{j}}^{*}. \end{array} $
(31) -
In this section, we describe the mass shifts of charmonia
$ (c\bar{c}) $ such as$ J/\psi $ ,$ \psi(3686) $ ,$ \psi(3770) $ ,$ \chi_{c0} $ ,$ \chi_{c2} $ , which are the 1S, 2S, 1D,$ 1^3P_0 $ , and$ 1^3P_2 $ states, respectively, in strange hadronic matter in the presence of strong magnetic fields. The heavy quarkonium states are modified in a hadronic environment due to modifications of gluon condensates [79–83]. The trace anomaly in QCD indicates that the trace of the energy-momentum tensor is non-zero when the scale symmetry is broken. A non-zero trace of the energy-momentum tensor in QCD originates from the gluon condensates and finite quark mass contributions. This scale invariance breaking is simulated in the chiral effective Lagrangian given by Eq. (1) at the tree level through the scale breaking term:$ \begin{aligned}[b] {\cal L}_{\rm scalebreak}=& -\frac{1}{4} \chi^{4} {\rm ln} \frac{\chi^{4}}{\chi_{0}^{4}} + \frac{d}{3} \chi^{4} \\&\times{\rm ln} \Bigg( \frac{\left( \sigma^{2} - \delta^{2}\right)\zeta } {\sigma_{0}^{2} \zeta_{0}} \left( \frac{\chi}{\chi_{0}}\right) ^{3}\Bigg). \end{aligned} $
(32) Comparing the expressions for the trace of energy-momentum tensor from QCD, and from the scale-breaking Lagrangian given by Eq. (32), we obtain [107, 108]
$ \begin{equation} \theta_{\mu}^{\mu} = \langle \frac{\beta_{\rm QCD}}{2g} G_{\mu\nu}^{a} G^{\mu\nu a} \rangle + \sum\limits_i m_i \bar {q_i} q_i \equiv -(1 - d)\chi^{4}. \end{equation} $
(33) The second term in the trace accounts for the finite quark masses, with
$ m_i $ being the current quark mass for the quark of flavor,$ i=u,d,s $ . Here, parameter d originates from the second term in$ {\cal L}_{\rm scalebreak} $ given by Eq. (32). The QCD β function at one-loop level is given by$ \begin{equation} \beta_{\mathrm{QCD}}(g)=-\frac{11 N_{c} g^{3}}{48 \pi^{2}}\left(1-\frac{2 N_{f}}{11 N_{c}}\right). \end{equation} $
(34) Here,
$ N_c=3 $ is the number of colors, and$ N_f $ is the number of quark flavors. In the above equation, the first term in the parentheses results from the antiscreening contribution of the gluons, and the second term results from the screening contribution of quark pairs. Using Eqs. (33) and (34), we obtain the scalar gluon condensate that is related to the dilaton field as$ \begin{equation} \langle\frac{\alpha_{s}}{\pi} G_{\mu \nu}^{a} G^{\mu \nu a}\rangle=\frac{24}{\left(33-2 N_{f}\right)}\left[(1-d) \chi^{4}+\sum\limits_{i} m_{i} \bar{q}_{i} q_{i}\right]. \end{equation} $
(35) The second term,
$ \sum_i m_i \bar q_i q_i $ , which is related to the explicit chiral symmetry breaking term$ {\cal L}_{\rm SB} $ in Eq. (1), is given by [27]$ \sum_i m_i \bar {q_i} q_i = \Big[ 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 \Big]. $
(36) In the chiral effective model, the leading order mass shift formula of the charmonium states is given by [65, 66]
$ \begin{aligned}[b] \Delta m_{\psi}=& \frac{1}{18} \int {\rm d}k^{2} \langle \vert \frac{\partial \psi (\vec k)}{\partial {\vec k}} \vert^{2} \rangle \frac{k}{k^{2} / m_{c} + \epsilon} \\&\times \bigg ( \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle - \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle _{0} \bigg ), \end{aligned} $
(37) where
$ \begin{equation} \langle \vert \frac{\partial \psi (\vec k)}{\partial {\vec k}} \vert^{2} \rangle =\frac {1}{4\pi}\int \vert \frac{\partial \psi (\vec k)}{\partial {\vec k}} \vert^{2} {\rm d}\Omega. \end{equation} $
(38) In Eq. (37),
$ m_{c} $ is the mass of the corresponding heavy quark, and$ \epsilon $ =$ 2m_{c}- m_{\Psi} $ represents the binding energy of the corresponding charmonium state. Here,$ \langle \dfrac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle $ and$ \langle \dfrac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle_{0} $ are the expectation values of the scalar gluon condensates in the magnetized medium and vacuum, respectively.$ \psi(k) $ is the wave function in the momentum space normalized as$ \int \dfrac{{\rm d}^{3} k}{(2 \pi)^{3}}|\psi(k)|^{2}=1 $ . The wave functions for these heavy quarkonium states are considered to be harmonic oscillator wave functions and are given by [90]$ \psi_{N,l} =N {Y_{l}}^m ( \theta, \phi)(\beta ^2r^2)^{l/2} {\rm e}^{-({1}/{2}) \beta ^2r^2} L_{N-1}^{l+({1}/{2})} (\beta ^2r^2). $
(39) Here,
$ {L_{p}}^{k}(z) $ is the associated Laguerre polynomial, and$ \beta^2 $ =Mω/h characterizes the strength of the harmonic potential, where$ M = m_{c}/2 $ . The β values for$ J/\psi $ ,$ \psi(3686) $ ,$ \psi(3770) $ are obtained by fitting their root mean squared radii [83, 84]. The β values for$ \chi_{c0} $ and$ \chi_{c2} $ are obtained using the linear extrapolation of the β vs. vacuum mass of the charmonium states [94]. For$ N_f $ =3, the difference in the value of scalar gluon condensate in the medium and vacuum is given by$ \begin{aligned}[b]& \bigg ( \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle - \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle _{0} \bigg )\\=&\frac{8}{9}\left[(1-d)\left(\chi^{4}-\chi_{0}^{4}\right)+m_{\pi}^{2} f_{\pi} \sigma^{\prime}+\left(\sqrt{2} m_{K}^{2} f_{K}-\frac{1}{\sqrt{2}} m_{\pi}^{2} f_{\pi}\right) \zeta^{\prime}\right]. \end{aligned} $
(40) In the above equation, the terms proportional to
$ \sigma ' $ and$ \zeta ' $ originate from the finite quark mass term$ \sum_i m_i \bar {q_i} q_i $ . When the mass of light quarks is neglected, the terms proportional to$ \sigma ' $ and$ \zeta ' $ vanish. The change in the gluon condensates in the limit of massless light quarks is given by$ \bigg ( \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle - \langle \frac{\alpha_{s}}{\pi} G_{\mu\nu}^{a} G^{\mu\nu a}\rangle _{0} \bigg )=\frac{8}{9}\left[(1-d)\left(\chi^{4}-\chi_{0}^{4}\right)\right]. $
(41) Under this limit, the mass shift of charmonia given by Eq. (37) reduces to [93]
$ \begin{equation} \Delta m_{\psi}^{(m_i=0)}= \frac{4}{81} (1 - d) \int {\rm d}k^{2} \langle \vert \frac{\partial \psi (\vec k)}{\partial {\vec k}} \vert^{2} \rangle \frac{k}{k^{2} / m_{c} + \epsilon} \left( \chi^{4} - {\chi_0}^{4}\right). \end{equation} $
(42) Hence, when the masses of light quarks are neglected, the mass shift for the charmonium states is observed to be proportional to the modification of the dilaton field.
Open charm mesons and charmonia in magnetized strange hadronic matter
- Received Date: 2022-02-07
- Available Online: 2022-08-15
Abstract: We investigate the in-medium masses of open charm mesons (D(