Open Charm mesons in magnetized nuclear matter
– effects of (inverse) magnetic catalysis

The study of the hadron properties under extreme conditions of density, temperature and magnetic field is of relevance in the ultra-relativistic heavy ion collision experiments and in astrophysical objects like the magnetars, neutron stars etc. The topic of hadron properties in the presence of magnetic fields Hosaka_Prog_Part_Nucl_Phys has attracted a lot of attention in the recent past, due to its relevance in ultra relativistic peripheral heavy ion collision experiments, e.g. at RHIC, BNL and LHC, CERN, where, the magnetic fields produced are estimated to be huge tuchin . The open heavy flavour mesons and the heavy quarkonium states are profusely produced in the early phase of these collisions, when the magnetic field can still be large. However, the time evolution of the magnetic field, which requires solutions of the magnetohydrodynamic equations, along with a proper estimate of the electrical conductivity in the medium, is still an open question. The experimental observables of the relativistic heavy ion collision experiments are affected by the medium modifications of the hadrons. In the presence of a magnetic field, there can be an enhancement (reduction) of the light quark condensate with increase in the magnetic field, an effect called the (inverse) magnetic catalysis kharzeevmc ; kharmc1 ; chernodub ; Preis ; menezes ; Shovkovy ; balicm . The (inverse) magnetic catalysis effect has been studied for the nuclear matter within Walecka model haber ; arghya , which arises from the Dirac sea contributions to the nucleon self energy haber . In the Walecka model, the effective nucleon mass is given as $M^{*}_{N}=M_{N}-g_{\sigma N}\sigma$, where $M_{N}$ is the mass of the nucleon in vacuum (at zero magnetic field) and $g_{\sigma N}$ is the coupling of the nucleon with the scalar field, $\sigma$. The magnetic catalysis (MC) effect is observed as an increase in the nucleon mass with a rise in the magnetic field at zero temperature and zero baryon density, due to the contributions from the Dirac sea to the self energy of the nucleon haber ; arghya . In the no-sea approximation (when the Dirac sea effects are neglected), the mass of the nucleons remains at its vacuum value for zero temperature and zero density. Using the Walecka model, the Dirac sea contributions are calculated through the summation of the scalar and vector tadpole diagrams in Ref. arghya . The self energy of the nucleon is calculated using the weak magnetic field approximation for the nucleon propagator, and, the effects of the anomalous magnetic moments (AMM) of the nucleons are also considered. For zero temperature and zero density, there is observed to be quite dominant contributions from the AMMs, as compared to when these are not taken into account. The AMMs are observed to play an important role when the Dirac sea contributions are taken into account. In Ref. arghya , at finite densities and zero temperature, the AMMs of the nucleons are observed to lead to the opposite trend of the drop in the effective nucleon mass with the magnetic field, and, this trend persists for non-zero temperatures, upto the critical temperature, $T_{c}$. The lowering of $T_{c}$ with increase in the value of the magnetic field, $B$ is identified with the inverse magnetic catalysis (IMC) arghya . This effect is also seen in lattice studies balicm . In the present work, we calculate the self-energy of the nucleon in magnetized isospin asymmetric nuclear matter within the chiral effective model, accounting for the Dirac sea contributions to the nucleon self energy in the weak field approximation for the nucleon propagator arghya , by summing over the scalar ($\sigma$, $\zeta$ and $\delta$) and vector ($\omega$ and $\rho$) tadpole diagrams.

In the presence of a magnetic field, there is mixing of the pseudoscalar meson and the longitudinal component of the vector meson (PV mixing) leading to a drop (increase) in the mass of the pseudoscalar (longitudinal component of the vector) meson. The effects of the PV mixing have been studied for the open and hidden charm mesons Gubler_D_mag_QSR ; charmonium_mag_QSR ; charmonium_mag_lee ; Suzuki_Lee_2017 ; Alford_Strickland_2013 and have been observed to have appreciable modifications to the masses of these mesons. For the charged mesons, there are additional contributions from the Landau levels in the presence of an external magnetic field. The open charm mesons, which are created at the early phase of the heavy ion collision experiments, when the magnetic field is still large, can hence be important tools in probing the effects of the magnetic field, e.g., the mass modifications due to the (inverse) magnetic catalysis, the mixing of pseudoscalar meson and the longitudinal component of the vector meson (PV mixing), additionally, the Landau level contributions (for the charged mesons).

The $D(\bar{D})$ and $D^{*}(\bar{D^{*}})$ mesons are the open heavy flavour mesons, which comprise of a heavy charm quark (antiquark) and a light (u,d) antiquark (quark). Within a chiral effective model, the in-medium masses of the $D$ and $\bar{D}$ mesons have been calculated The in-medium properties of the open heavy flavour (charm and bottom) mesons amd heavy quarkonium states in the absence of a magnetic field have been studied within the model amdmeson ; amarindamprc ; amarvdmesonTprc ; amarvepja ; DP_AM_Ds ; DP_AM_bbar ; DP_AM_Bs ; AM_DP_upsilon . In the presence of a magnetic field, the masses of the open and hidden heavy flavour mesons have been studied within the model in the ‘no sea’ approximation dmeson_mag ; bmeson_mag ; charmonium_mag and their effects on the partial decay widths of the charmonium states to $D\bar{D}$ charmdecay_mag ; charmdw_mag ; open_charm_mag_AM_SPM and bottomonium states to $B\bar{B}$ upslndw_mag .

In the present work, we investigate the masses of the open charm ($D$ and $\bar{D}$) mesons in isospin asymmetric magnetized nuclear matter using a chiral effective model, accounting for the Dirac sea contributions. The effects of the AMMs of the nucleons are also considered in the present work. The PV mixing effect is considered using a phenomenological Lagrangian interaction charmonium_mag_lee ; Iwasaki . The additional contributions from the lowest Landau level (LLL) are taken into account for the charged open charm mesons.

We organize the paper as follows: In section II, we discuss briefly the chiral effective model and the computation of the masses of the $D$ and $\bar{D}$ mesons in the isospin asymmetric magnetized nuclear matter. These are calculated including the contributions of the Dirac sea for the nucleon self-energy, which is observed to a rise (drop) in the magnitude of the light quark condensate with increase in the magnetic field, an effect called the (inverse) magnetic catalysis. as well as, incorporating the effects of the PV mixing in the presence of a magnetic field. In section III, we describe the results for the masses of the open charm mesons due to the effects of the (inverse) magnetic catalysis, PV mixing and for the charged mesons, contributions from the Landau level. In section IV, we summarize the findings of the present work.

We describe the chiral effective model used to study the open charm $D$ and $\bar{D}$ mesons in magnetized nuclear matter. The model is based on a nonlinear realization of chiral symmetry. The breaking of scale invariance of QCD is incorporated into the model through the introduction of a scalar dilaton field, $\chi$. The Lagrangian of the model, in the presence of a magnetic field, has the form

$${\cal L}={\cal L}_{kin}+\sum_{W}{\cal L}_{BW}+{\cal L}_{vec}+{\cal L}_{0}+{\cal L}_{scalebreak}+{\cal L}_{SB}+{\cal L}_{mag}^{B\gamma},$$

(1)

where, ${\cal L}_{kin}$ corresponds to the kinetic energy terms of the baryons and the mesons, ${\cal L}_{BW}$ contains the interactions of the baryons with the meson, $W$ (scalar, pseudoscalar, vector, axialvector meson), ${\cal L}_{vec}$ describes the dynamical mass generation of the vector mesons via couplings to the scalar fields and contains additionally quartic self-interactions of the vector fields, ${\cal L}_{0}$ contains the meson-meson interaction terms, ${\cal L}_{scalebreak}$ is a scale invariance breaking logarithmic potential given in terms of a scalar dilaton field, $\chi$ and ${\cal L}_{SB}$ describes the explicit chiral symmetry breaking. The term ${\cal L}_{mag}^{B\gamma}$, describes the interacion of the baryons with the electromagnetic field, which includes a tensorial interaction $\sim\bar{\psi_{i}}\sigma^{\mu\nu}F_{\mu\nu}\psi_{i}$, whose coefficients account for the anomalous magnetic moments of the baryons dmeson_mag ; bmeson_mag ; charmonium_mag .

In the present work, the meson fields are treated as claassical fields. However, the nucleon is treated as a quantum field and the self energy of the nucleon includes the contributions of the Dirac sea through the summation of the tadpole diagrams corresponding to the the non-strange isoscalar ($\sigma$), strange isoscalar ($\zeta$), and non-strange isovector ($\delta$) scalar fields and the vector fields. The masses of the baryons are generated by their interactions with the scalar mesons. Within the chiral effective model, the mass of baryon of species $i$ ($i=p,n$ in the present work of nuclear matter) is given as

$$M_{i}=-g_{\sigma i}\sigma-g_{\zeta i}\zeta-g_{\delta i}\delta,$$

(2)

where the scalar fields $\sigma$, $\zeta$ and $\delta$ are solved along with the dilaton field, $\chi$, from ther coupled equations of motion. The the scalar densities of the proton and neutron, occurring in the equations of the scalar fields ($\sigma$, $\zeta$ and $\delta$) include the contributions from the Dirac sea, calculated in the weak field approximation for the nucleon propagator arghya . The masses of the $D$ and $\bar{D}$ mesons are computed from solution of their dispersion relations, given as dmeson_mag

$$-\omega^{2}+{\vec{k}}^{2}+m_{D(\bar{D})}^{2}-\Pi_{D(\bar{D})}(\omega,|\vec{k}|)=0,$$

(3)

where $\Pi_{D(\bar{D})}$ denotes the self energy of the $D$ ($\bar{D}$) meson in the medium. For the $D$ meson doublet ($D^{0}$,$D^{+}$), and $\bar{D}$ meson doublet (${\bar{D}}^{0}$,$D^{-}$), the self enrgies are given by

	$\displaystyle\Pi(\omega,|\vec{k}|)$	$\displaystyle=$	$\displaystyle\frac{1}{4f_{D}^{2}}\Big{[}3(\rho_{p}+\rho_{n})\pm(\rho_{p}-\rho_{n})\big{)}\Big{]}\omega$		(4)
		$\displaystyle+$	$\displaystyle\frac{m_{D}^{2}}{2f_{D}}(\sigma^{\prime}+\sqrt{2}{\zeta_{c}}^{\prime}\pm\delta^{\prime})$
		$\displaystyle+$	$\displaystyle\Big{[}-\frac{1}{f_{D}}(\sigma^{\prime}+\sqrt{2}{\zeta_{c}}^{\prime}\pm\delta^{\prime})+\frac{d_{1}}{2f_{D}^{2}}(\rho^{s}_{p}+\rho^{s}_{n})$
		$\displaystyle+$	$\displaystyle\frac{d_{2}}{4f_{D}^{2}}\Big{(}({\rho^{s}_{p}}+{\rho^{s}_{n}})\pm({\rho^{s}_{p}}-{\rho^{s}_{n}})\Big{)}\Big{]}(\omega^{2}-{\vec{k}}^{2}),$

and

	$\displaystyle\Pi(\omega,|\vec{k}|)$	$\displaystyle=$	$\displaystyle-\frac{1}{4f_{D}^{2}}\Big{[}3(\rho_{p}+\rho_{n})\pm(\rho_{p}-\rho_{n})\Big{]}\omega$		(5)
		$\displaystyle+$	$\displaystyle\frac{m_{D}^{2}}{2f_{D}}(\sigma^{\prime}+\sqrt{2}{\zeta_{c}}^{\prime}\pm\delta^{\prime})$
		$\displaystyle+$	$\displaystyle\Big{[}-\frac{1}{f_{D}}(\sigma^{\prime}+\sqrt{2}{\zeta_{c}}^{\prime}\pm\delta^{\prime})+\frac{d_{1}}{2f_{D}^{2}}(\rho^{s}_{p}+\rho^{s}_{n})$
		$\displaystyle+$	$\displaystyle\frac{d_{2}}{4f_{D}^{2}}\Big{(}({\rho^{s}_{p}}+{\rho^{s}_{n}})\pm({\rho^{s}_{n}}-{\rho^{s}_{n}})\Big{]}(\omega^{2}-{\vec{k}}^{2}),$

where the $\pm$ signs refer to the $D^{0}$ and $D^{+}$ respectively in equation (4) and to the $\bar{D^{0}}$ and $D^{-}$ respectively in equation (5). In equations (4) and (5), $\sigma^{\prime}(=(\sigma-\sigma_{0}))$, ${\zeta_{c}}^{\prime}(=(\zeta_{c}-{\zeta_{c}}_{0}))$ and $\delta^{\prime}(=(\delta-\delta_{0}))$ are the fluctuations of the scalar-isoscalar fields $\sigma$ and $\zeta_{c}$, and the third component of the scalar-isovector field, $\delta$, from their vacuum expectation values (for zero magnetic field).

The values of the scalar meson fields $\sigma$, $\zeta$, $\delta$ are obtained by solving their coupled equations of motion. In the present work, as has already been mentioned, the contribution of the Dirac sea to the self energy of the nucleon is taken into account through the summation of the tadpole diagrams, which are incorporated into the equations of motion of the fields $\sigma$, $\zeta$ and $\delta$ through the scalar densities of the nucleons. These scalar fields, along with the dilaton field, $\chi$, are calculated for a given baryon density, $\rho_{B}$, given isospin asymmetry, $\eta=(\rho_{n}-\rho_{p})/(2\rho_{B})$, where $\rho_{p,n}$ are the number densities of the proton and neutron respectively. Accounting for the lowest Landau level (LLL) contributions for the charged open charm mesons, the effective mass of $D^{\pm}$ is given as

$$m^{eff}_{D^{\pm}}=\sqrt{{m^{*}_{D^{\pm}}}^{2}+|eB|},$$

(6)

whereas for the neutral ($D^{0}$ and $\bar{D^{0}}$) mesons, the effective masses are given as

$$m^{eff}_{D^{0}(\bar{D^{0}})}=m^{*}_{D^{0}(\bar{D^{0}})}.$$

(7)

In equations (6) and (7), $m^{*}_{D^{\pm},D^{0},\bar{D^{0}}}$ are the masses calculated using the chiral effective model, as the solutions for $\omega$ at $|\vec{k}|=0$, of the dispersion relations for these mesons given by equation (3).

The masses of the charged vector open charm ($D^{*}$ and $\bar{D^{*}}$) mesons, retaining the lowest Landau level contributions ($n$=0), depend on $S_{z}$ (the z-component of the spin vector) and are given as

$$m^{eff}_{{D^{*}}^{\pm}}=\sqrt{{m^{*}}_{{D^{*}}^{\pm}}^{2}+|eB|+gS_{z}|eB|},$$

(8)

whereas for the neutral ${D^{*}}^{0}$ and $\bar{{D^{*}}^{0}}$, the in-medium masses are given as

$$m^{eff}_{D^{*}(\bar{D^{*}})}={m^{*}}_{D^{*}(\bar{D^{*}})}.$$

(9)

It is assumed that the mass shifts of the vector open charm ($D^{*}$ and $\bar{D^{*}}$) mesons (which have same quark-antiquark constituents as $D$ and $\bar{D}$ mesons) are identical to the mass shifts of the pseudoscalar mesons $D$ and $\bar{D}$ mesons, calculated within the chiral effective model open_charm_mag_AM_SPM . This is in line with the QMC model where the masses of the hadrons are obtained from the modification of the scalar density of the light quark (antiquark) constituent of the hadron Hosaka_Prog_Part_Nucl_Phys . The in-medium mass of the vector open charm mesons are thus assumed to be

$$m^{*}_{D^{*}(\bar{D^{*}})}-m_{D^{*}(\bar{D^{*}})}^{vac}={m^{*}}_{D(\bar{D})}-m_{D(\bar{D})}^{vac},$$

(10)

for which there are additional Landau level contributions for the charged ${D^{*}}^{\pm}$ mesons, as given by equation (8). In the presence of an external magnetic field, there is also mixing of the pseudoscalar and the longitudinal component ($S_{z}$=0) of the vector meson. Its effect on the masses of the open charm mesons is also studied in the present work.

The mixing between the pseudoscalar and the longitudinal component of the vector (PV mixing) mesons in the presence of a magnetic field, is observed to lead to appreciable modifications to their masses charmonium_mag_QSR ; Gubler_D_mag_QSR ; charmonium_mag_lee ; Suzuki_Lee_2017 ; Alford_Strickland_2013 ; charmdw_mag ; open_charm_mag_AM_SPM ; upslndw_mag ; Iwasaki . In the present work of the study of the in-medium masses of the open charm mesons, the PV mixing effects for the neutral open charm mesons ($D^{0}-{{D^{*}}^{0}}^{||}$ and $\bar{D^{0}}-\bar{{D^{*}}^{0}}^{||}$ mixings) as well as the charged mesons ($D^{+}-{{D^{*}}^{+}}^{||}$ and ${D^{-}}-{{D^{*}}^{-}}^{||}$ mixings) are considered. The PV mixing is taken into account through a phenomenological Lagrangian which was used to study the mixing of the charmonium states charmonium_mag_lee ; Suzuki_Lee_2017 . The effects due to the PV ($J/\psi-\eta_{c}$, $\psi(2S)-\eta_{c}(2S)$ and $\psi(1D)-\eta_{c}(2S)$) mixings on the masses of the charmonium states in magnetized nuclear matter calculated within a chiral effective model, have been studied in Ref. charmdw_mag . The effects of the mass modifications due to PV mixing on the decay width $\psi(3770)\rightarrow D\bar{D}$ were observed to be quite appreciable due to the $\psi(1D)-\eta_{c}(2S)$ mixing charmdw_mag , as well as due to the PV mixing of the open charm ($D(\bar{D})-D^{*}(\bar{D^{*}})$ mesons open_charm_mag_AM_SPM . The $PV\gamma$ interaction Lagrangian is given as,

$$\mathcal{L}_{PV\gamma}=\frac{g_{PV\gamma}}{m_{av}}e\tilde{F}_{\mu\nu}(\partial^{\mu}P)V^{\nu},$$

(11)

where P and $V^{\mu}$ represent the pseudoscalar and the vector fields, respectively, $\tilde{F}_{\mu\nu}$ is the dual field strength tensor of the external magnetic field and and $m_{av}$ is the average of the masses of the pseudoscalar and vector mesons, $(m_{av}=(m_{P}+m_{V})/2)$. $g_{PV}$ is the coupling constant for the radiative decay, $V\rightarrow P\gamma$. The value of the coupling parameter, $g_{PV\gamma}$ is fixed from the observed decay width of $V\rightarrow P\gamma$. The masses of the pesudoscalar and the longitudinal component of the vector meson, due to the PV mixing are given as

$$m_{V^{||},P}^{2}=M_{+}^{2}+\frac{c_{PV}^{2}}{m_{av}^{2}}\pm\sqrt{M_{-}^{4}+\frac{2c_{PV}^{2}M_{+}^{2}}{m_{av}^{2}}+\frac{c_{PV}^{4}}{m_{av}^{4}}}$$

(12)

with $M_{\pm}^{2}=m_{V}^{{eff}^{2}}\pm m_{P}^{{eff}^{2}}$ and $c_{PV}=g_{PV\gamma}eB$; with $m_{V,P}^{eff}$ are the effective masses of the vector and pseudoscalar mesons.These effective masses, given by (6), (7), (8) and (9), include the effects of the Dirac sea calculated in the chiral effective model, with additional contributions due to the lowest Landau level contributions for the charged mesons.

In the present work, we study the effects of magnetic field on the pseudoscalar ($D$ and $\bar{D}$) and vector ($D^{*}$ and $\bar{D^{*}}$) open charm mesons in magnetized (nuclear) matter. The in-medium masses are calculated using a chiral effective model for the $D$ and $\bar{D}$ mesons due to their interactions with the nucleons and the scalar mesons, including the Dirac sea contributions, with additional Landau level contributions for the charged $D^{\pm}$ mesons. It might be noted here the neutron, which is electrically charge neutral, interacts with the magnetic field only due to its non-zero value of the anomalous magnetic moment. The effects of the magnetic field on the masses are computed for zero baryon density as well as, for $\rho_{B}=\rho_{0}$, the nuclear matter saturation density, The effects of isospin asymmetry as well as anomalous magnetic moments (AMMs) of the nucleons are also studied in the present work. The Dirac sea contributions lead to increase in the values of the scalar fields, $\sigma(\sim\langle\bar{u}u\rangle+\langle\bar{d}d\rangle)$ and $\zeta(\sim\langle\bar{s}s\rangle)$, leading to decrease in the nucleon self energy, i.e., enhancement of the light quark condensates (an effect called magnetic catalysis), as compared to when this effect is not considered. The values (in MeV) of $\sigma$ and $\zeta$ are modified from the zero baryon density and zero magnetic fields values of $-93.3$ and $-106.6$ to $-94.537(-102.018)$ and $-107.19(-109.9)$ for $eB=3m_{\pi}^{2}$, without (with) accounting for the AMMs of the nucleons. These lead to the effective nucleon masses to be 948 (1026) MeV in the absence (presence) of the AMMs of the nucleons. Hence there is a dominant rise in the nucleon mass due to the AMMs of the nucleons, as compared to when AMMs are not taken into account, which was also observed in Ref. arghya .

In figure 1, at zero baryon density, the masses of the charged and neutral pseudoscalar and vector mesons are plotted accounting for the Dirac sea contributions to the nucleon self-energy (observed to lead to magnetic catalysis) as well as PV mixing effects. The $D-{D^{*}}$ and $\bar{D}-\bar{D^{*}}$) mixings result in a drop (increase) in the mass of the $D({D^{*}}^{||})$ and $\bar{D}(\bar{D^{*}}^{||})$ meson. The open charm masses are plotted accounting for the AMMs of the nucleons and compared to the cases when the AMMs are not taken into account (shown as dotted lines). While accounting for AMMs of the nucleons, at zero density, the solutions for the scalar fields do not exist for $eB$ higher than around $4m_{\pi}^{2}$ in the weak field approximations as used here. The masses for $D^{\pm}$ are observed to be identical to each other (so also for the $D^{0}$ and $\bar{D^{0}}$ mesons). This can be understood from the disperion relations of the $D$ and $\bar{D}$ mesons given by (3), (4) and (5). For $\rho_{B}=0$ (hence $\rho_{p}$ and $\rho_{n}$ are both zero), implying that the Weinberg-Tomozawa contribution is zero. The dispersion relation satisfied by the $D^{+}(D^{0})$ meson is then same as that of $D^{-}(\bar{D^{0}})$ meson, and hence, the masses are identical within the charged and the neutral sectors. The effective masses of ${D^{*}}(\bar{D^{*}})$ mesons, which are calculated, using equation (10) are also identical within the charged as well as neutral sectors. The effects of the AMMs on the $D$ and $\bar{D}$ mesons are through the Dirac sea (DS) contributions to the scalar densities of proton and neutron, $\rho_{p}^{s}$ and $\rho_{n}^{s}$, occurring in the $d_{1}$ and $d_{2}$ range terms in their self-energies, as well as, due to the values of the scalar fields, which are calculated accounting for contributions from Dirac sea through $\rho_{p}^{s}$ and $\rho_{n}^{n}$. The modifications due to AMMs are observed to be negligible upto $eB\sim 2m_{\pi}^{2}$ and remains small for higher values of the magnetic field, upto $eB\sim 4m_{\pi}^{2}$, beyond which the solutions of the scalar fields do not exist for $\rho_{B}=0$, when AMMs of nucleons are taken into account, in the weak magnetic field approximation arghya as used in the present work.

Figures 2 and 3 show the effects of the Dirac sea (DS) as well as the PV mixing on the masses of $D^{+}$ and ${D^{*}}^{+}$ mesons for the isospin symmetric and asymmetric (with $\eta$=0) nuclear matter in (b) and (d) and compared with the case when the Dirac sea contributions are not taken into account (shown in (a) and (c) respectively). In the ‘no sea’ approximation (Dirac sea neglected), the values of the scalar fields, and hence the open charm meson masses calculated within the chiral effective model is observed to be insensitive to the change in magnetic field in the absence of the Landau level contributions dmeson_mag . The increase in the masses of $D^{+}$ and ${{D^{*}}^{+}}^{||}$ ($S_{z}$=0) are due to the positive Landau contributions in (a) for the case of ’no sea’ approximation. There is observed to be a drop (increase) in the mass of the $D^{+}$ (${{D^{*}}^{+}}^{||}$) meson due to PV mixing. The effect of inclusion of the AMMs of the nucleons is observed to be quite crucial on the masses of these mesons, due to the fact that there is inverse magnetic catalysis, as opposed to the magnetic catalysis observed when AMMs are ignored. In the absence of AMMs of the nucleons, there is observed to be moderate increase in the masses $D^{+}$ and ${D^{+}}^{*}$ with magnetic field, whereas, for the case of accounting for the AMMs, there is observed to be large drop in the masses at high values of the magnetic field.

In figures 4 and 5, the effects of the Dirac sea (DS) contributions and PV mixing are shown on the masses are plotted for $D^{0}$ and ${D^{*}}^{0}$ mesons for $\rho_{B}=\rho_{0}$ and for $\eta=0$ and $\eta=0.5$ respectively. Similar to the case of $D^{+}$ and ${D^{*}}^{+}$, the Dirac sea contributions have a significant effect on the masses of these neutral open charm mesons.

The masses of the $\bar{D}$ and $\bar{D^{*}}$ mesons are plotted for the charged sector in figures 6 and 7, and for the neutral sector in figures 8 and 9, for $\eta=0$ and $\eta=0.5$ respectively. It is observed that the most dominant contribution due to the magnetic field on the masses of the open charm mesons arise from the effects of the Dirac sea.

To summarize, we have investigated the effects of magnetic field on the masses of the open charm pseudoscalar ($D$ and $\bar{D}$) and vector (${D^{*}}$ and $\bar{D^{*}}$) in magnetized (nuclear) matter, accounting for the effects from the Dirac sea (DS), PV mixing and additional Landau level contribtions for the charged mesons. The effects of isospin asymmetry as well as AMMs of the nucleons are also studied in the present work. The effects of the Dirac sea is observed to lead to an enhancement (reduction) of the quark condensates, an effect called (inverse) magnetic catalysis, which is observed as an increase (drop) in the magnitude of the scalar fields, $\sigma$ and $\zeta$. At zero density, there is observed to be magnetic catalysis (MC) effect, which is observed to be enhanced when the AMMs of the nucleons are also taken into account. At $\rho_{B}=\rho_{0}$, both for symmetric ($\eta$=0) and asymmetric nuclear matter (with $\eta$=0.5), there is still observed to be magnetic catalysis when the AMMs of the nucleons are neglected, whereas, the trend becomes opposite with the magnetic field, leading to inverse magnetic catalysis (IMC) in the presence of the AMMs of the nucleons. The effects of the Dirac sea contribution on the masses of the open charm mesons is observed to be the most dominant effect due to the magnetic field, the effect being much larger than the PV mixing. The effect should modify the yields pf the open and hidden charm mesons arising from ultra-relativistic peripheral heavy ion collision experiments, where the created magnetic field is huge.