Charmonium decay widths in magnetized matter

TL;DR

This paper investigates how strong magnetic fields and isospin asymmetry in nuclear matter affect the decay widths of charmonium states into D meson pairs, using a combined theoretical approach.

Contribution

It introduces a comprehensive calculation of in-medium charmonium decay widths in magnetized, asymmetric nuclear matter using the $^3P_0$ model and a chiral effective model, considering magnetic field effects and anomalous magnetic moments.

Findings

Magnetic fields significantly alter decay widths of charmonium states.

Isospin asymmetry enhances the medium modifications of decay widths.

Magnetic field effects could influence charmed meson production in heavy ion collisions.

Abstract

We study the partial decay widths of the charmonium states ($J/\psi$, $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$, $\chi_{c2}$) to $D\bar D$ ($D^+D^-$ or $D^0\bar {D^0}$) in isospin asymmetric nuclear matter, in the presence of strong magnetic fields. The in-medium partial decay widths of charmonium states to $D\bar D$ are calculated within a light quark--antiquark pair creation model, namely the $^3P_0$ model, using the in--medium masses of the charmonia as well as $D$ and $\bar D$ mesons in the magnetized nuclear matter obtained within a chiral effective model. The presence of a magnetic field leads to Landau quantization of the energy levels of the proton in the nuclear medium. The effects of magnetic field and isospin asymmetry on the charmonium decay widths to $D\bar D$ are found to be quite prominent. The effects of the anomalous magnetic moments have also been taken into consideration for obtaining the in-medium masses of these heavy flavour mesons, used to calculate the partial decay widths of the charmonium states. The medium modifications of the charmonium decay widths can have observable consequences on the production of the charmed mesons in high energy asymmetric heavy ion collision experiments.

Equations

$$\displaystyle k_{0}\chi^{2}\sigma-4k_{1}\left(\sigma^{2}+\zeta^{2}+\delta^{2}\right)\sigma-2k_{2}\left(\sigma^{3}+3\sigma\delta^{2}\right)-2k_{3}\chi\sigma\zeta-\frac{d}{3}\chi^{4}\bigg{(}\frac{2\sigma}{\sigma^{2}-\delta^{2}}\bigg{)}$$

$$k_0{\chi }^2\zeta -4k_1\left({\sigma }^2+{\zeta }^2+{\delta }^2\right)\zeta -4k_2{\zeta }^3-k_3\chi \left({\sigma }^2-{\delta }^2\right)-\frac{d}{3}\frac{{\chi }^4}{\zeta }$$

$$k_0{\chi }^2\delta -4k_1\left({\sigma }^2+{\zeta }^2+{\delta }^2\right)\delta -2k_2\left({\delta }^3+3{\sigma }^2\delta \right)+2k_3\chi \delta \zeta$$

$$k_0\chi \left({\sigma }^2+{\zeta }^2+{\delta }^2\right)-k_3\left({\sigma }^2-{\delta }^2\right)\zeta +{\chi }^3\left[1+\ln \left(\frac{{\chi }^4}{{\chi }_0^4}\right)\right]+4k_4{\chi }^3$$

$${\Gamma }_{J/\psi \to D\bar{D}}$$

$${\Gamma }_{\psi (3686)\to D\bar{D}}$$

$${\Gamma }_{\psi (3770)\to D\bar{D}}$$

$${\Gamma }_{{\chi }_{c0}\to D\bar{D}}$$

$${\Gamma }_{{\chi }_{c2}\to D\bar{D}}$$

$$E_D=(p_D^2+m_D^2{)}^{1/2},E_{\bar{D}}=(p_D^2+m_{\bar{D}}^2{)}^{1/2},$$

$$\displaystyle p_{D}=\Bigg{(}\frac{M_{\psi}^{2}}{4}-\frac{m_{D}^{2}+m_{\bar{D}}^{2}}{2}+\frac{(m_{D}^{2}-m_{\bar{D}}^{2})^{2}}{4M_{\psi}^{2}}\Bigg{)}^{1/2}.$$

$$\Delta m_{\Psi }=\frac{4}{81}(1-d)\int d{k}^2⟨|\frac{\partial \psi (\vec{k})}{\partial \vec{k}}{|}^2⟩\frac{k}{k^2/m_c+ϵ}\left({\chi }^4-{{\chi }_0}^4\right),$$

$$⟨|\frac{\partial \psi (\vec{k})}{\partial \vec{k}}{|}^2⟩=\frac{1}{4\pi }\int |\frac{\partial \psi (\vec{k})}{\partial \vec{k}}{|}^{2}d\mathrm{\Omega }.$$

Full text

I Introduction

The topic of in-medium properties of hadrons and more recently, of heavy flavour hadrons Hosaka_Prog_Part_Nucl_Phys , is an important and challenging area of research in strong interaction physics, due to its relevance in the ultra relativistic heavy ion collision experiments. The estimated magnetic fields produced in noncentral ultra high energy nuclear collisions are huge ($eB\sim 2m_{\pi}^{2}$ at RHIC, BNL, and, $eB\sim 15m_{\pi}^{2}$ at LHC, CERN) HIC_mag_1 ; HIC_mag_2 ; HIC_mag_3 ; HIC_mag_4 . This has initiated studies of the effects of strong magnetic fields on the in-medium properties of hadrons in the recent years. However, the time evolution of the magnetic field time_evolution_B_HIC_Tuchin_1 ; time_evolution_B_HIC_Tuchin_2 ; time_evolution_B_HIC_Tuchin_3 ; time_evolution_B_HIC_Tuchin_4 ; time_evolution_B_HIC_Ajit is still under debate, which needs the proper estimate of the electrical conductivity of the medium as well as solutions of the magnetohydrodynamic equations.

The properties of the heavy flavour mesons, e.g., the open charm and open bottom mesons Gubler_D_mag_QSR ; machado_1 ; B_mag_QSR ; dmeson_mag ; bmeson_mag as well as the charmonium states charmonium_mag_QSR ; charmonium_mag_lee ; charmonium_mag ; jpsi_etac_mag have been studied in the literature in the presence of strong magnetic fields. In the presence of a strong magnetic field, there can be mixing of the longitudinal components of the vector charmonium states, e.g, $J/\psi$, $\psi^{\prime}(\equiv\psi(3686))$, with their pseudoscalar partners $\eta_{c}$, $\eta^{\prime}_{c}$ Suzuki_Lee_2017 ; Alford_Strickland_2013 which might show in the dilepton spectra as anolamous $\eta_{c}$ and $\eta^{\prime}_{c}$ peaks, in addition to the $J/\psi$ and $\psi^{\prime}$ peaks. In Ref. Suzuki_Lee_2017 , the formation times for the quarkonia have been studied and it is observed that with increasing magnetic field strength, the formation times of the vector quarkonia become larger, whereas the formation time of their pesudoscalar partners become faster Suzuki_Lee_2017 . With a larger formation time, the $J/\psi$ could survive through the initial thermal effects, thus enhancing its survivability, whereas the $\eta_{c}$, $\eta^{\prime}_{c}$ peaks in the dilepton spectra could probe an early magnetic field. The heavy ion collision experiments involve heavy nuclei which have much larger number of neutrons than protons, and hence the isospin asymmetry effects on the hadron properties are important to study. In the present work, we study in-medium decay widths of the charmonium states ($J/\psi$, $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$ and $\chi_{c2}$) to $D\bar{D}$, in isospin asymmetric nuclear matter in the presence of strong magnetic fields. These decay widths are computed using the ${}^{3}P_{0}$ model 3p0_1 ; 3p0_2 ; 3p0_3 ; 3p0_4 ; friman ; amarvepja , from the mass modifications of the charmonium states as well as $D$ and $\bar{D}$ mesons calculated using a chiral effective model.

The heavy quarkonium states, e.g., the charmonium and bottomonium states have been studied extensively in the literature, using potential models eichten_1 ; eichten_2 ; satz_1 ; satz_2 ; satz_3 ; satz_4 ; satz_5 ; repko . The effects of the gluonic fluctutations on the quarkonium states have been studied in Refs. pes1 ; pes2 ; voloshin , with a color Coulomb potential for the interaction of the heavy quark, $Q$ and heavy antiquark, $\bar{Q}$, within the quarkonium state. Assuming the separation of $Q$ and $\bar{Q}$, to be small compared to the characteristic scale of the gluonic fluctuations, the leading order contribution of a multipole expansion of the quarkonium state to the gluonic field leads to the mass of the quarkonium state to be proportional to the gluon condensate. In Ref. leeko , the in-medium masses of the charmonium states have been studied using leading order QCD formula pes1 and the linear density approximation for the gluon condensate in the nuclear medium. Within the QCD sum rule approach, the mass modifications of the charmonium states are due to the medium changes of the gluon condensate kimlee ; klingl ; amarvjpsi_qsr ; jpsi_etac_mag ; moritalee_1 ; moritalee_2 ; moritalee_3 ; moritalee_4 , whereas, the open heavy flavour mesons are modified due to the interaction with the light quark condensates in the hadronic medium open_heavy_flavour_qsr_1 ; open_heavy_flavour_qsr_2 ; open_heavy_flavour_qsr_3 ; open_heavy_flavour_qsr_4 ; Wang_heavy_mesons_1 ; Wang_heavy_mesons_2 ; arvind_heavy_mesons_QSR_1 ; arvind_heavy_mesons_QSR_2 ; arvind_heavy_mesons_QSR_3 . The heavy flavour mesons have also been studied in the literature, using quark meson coupling model open_heavy_flavour_qmc_1 ; open_heavy_flavour_qmc_2 ; open_heavy_flavour_qmc_3 ; qmc_1 ; qmc_2 ; qmc_3 ; qmc_4 ; krein_jpsi ; krein_17 , using heavy quark symmetry and interaction of these mesons with nucleons via pion exchange Yasui_Sudoh_pion , using heavy meson effective theory Yasui_Sudoh_heavy_meson_Eff_th , studying the heavy flavour meson as an impurity in nuclear matter Yasui_Sudoh_heavy_particle_impurity as well as using the coupled channel approach ltolos ; ljhs ; mizutani_1 ; mizutani_2 ; HL ; tolos_heavy_mesons_1 ; tolos_heavy_mesons_2 . Within the chiral effective model Schechter ; paper3 ; kristof1 , the in-medium charmonium masses are obtained from the medium changes of a scalar dilaton field, which mimicks the gluon condensates of QCD amarindamprc ; amarvdmesonTprc ; amarvepja and the mass modifications of the $D$ and $\bar{D}$ mesons arise due to their interactions with the baryons and scalar mesons in the hadronic medium amarindamprc ; amarvdmesonTprc ; amarvepja ; amdmeson ; DP_AM_Ds . The chiral effective model has been used extensively in the literature, for the study of finite nuclei paper3 , strange hadronic matter kristof1 , light vector mesons hartree , strange pseudoscalar mesons, e.g. the kaons and antikaons kaon_antikaon ; isoamss ; isoamss1 ; isoamss2 in isospin asymmetric hadronic matter, as well as for the study of bulk matter of neutron stars pneutronstar . The light vector mesons ($\omega$, $\rho$ and $\phi$) are modified in the hadronic medium, predominantly due to the medium changes in the light quark condensates hatsuda within the QCD sum rule approach. These vector mesons in (magnetized) hadronic matter have been studied using QCD sum rule calculations from the medium changes of the light quark condensates and gluon condensates calculated within the chiral SU(3) model am_vecmeson_qsr ; vecqsr_mag . The kaons and antikaons have been recently studied in the presence of strong magnetic fields using this model kmeson_mag . The model has been used to study the open heavy flavour (charm and bottom) mesons amarindamprc ; amarvdmesonTprc ; amarvepja ; amdmeson ; DP_AM_Ds ; DP_AM_bbar ; DP_AM_Bs , the heavy quarkonium states amarvdmesonTprc ; amarvepja ; AM_DP_upsilon , the partial decay widths of the heavy quarkonium states to the open heavy flavour mesons, in the hadronic medium amarvepja using a light quark creation model friman , namely the ${}^{3}P_{0}$ model 3p0_1 ; 3p0_2 ; 3p0_3 ; 3p0_4 as well as using a field theoretic model for composite hadrons amspmwg ; amspm_upsilon . Recently, the effects of magnetic fields on these open and hidden heavy flavour mesons have been investigated dmeson_mag ; bmeson_mag ; charmonium_mag ; upsilon_mag using the chiral effective model.

The outline of the paper is as follows : In section II, we describe briefly the quark pair creation model, namely the ${}^{3}P_{0}$ model 3p0_1 ; 3p0_2 ; 3p0_3 ; 3p0_4 ; friman used to compute the in-medium partial decay widths of the charmonium states to $D\bar{D}$ in magnetized nuclear matter. The medium modifications of these decay widths are computed from the mass modifications of the charmonium states, $D$ and $\bar{D}$ mesons calculated within a chiral effective model. In section III, we discuss the results obtained in the present investigation of these in-medium charmonium decay widths in asymmetric nuclear matter in presence of strong magnetic fields. In section IV, we summarize the findings of the present study.

II Decay widths of Charmonia to $D\bar{D}$ within ${}^{3}P_{0}$ model

In the present work, we compute the in-medium partial decay widths of the charmonium states ($J/\psi$, $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$, $\chi_{c2}$) to $D\bar{D}$ ($D^{+}D^{-}$ or $D^{0}\bar{D^{0}}$) in magnetized nuclear matter, using the ${}^{3}P_{0}$ model. In this model, a light quark-antiquark pair is created in the ${}^{3}P_{0}$ state, and this light quark (antiquark) combines with the heavy charm antiquark (charm quark) of the decaying charmonium state at rest, resulting in the production of the open charm $D$ and $\bar{D}$ mesons.

The in-medium partial decay widths of charmonium states $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$, $\chi_{c2}$ to $D\bar{D}$ were studied using ${}^{3}P_{0}$ model in Ref. friman , where the masses of the $D$ and $\bar{D}$ mesons were assumed to have same medium modification, in addition to the degeneracy of the masses of the $D^{+}$ and $D^{0}$ within the $D$ doublet, and of $D^{-}$ and $\bar{D^{0}}$ within the $\bar{D}$ doublet in symmetric nuclear matter. In the asymmetric strange hadronic matter, the partial decay widths of the charmonium states, $J/\psi$, $\psi(3686)$ and $\psi(3770)$ to $D\bar{D}$ pair were calculated within the ${}^{3}P_{0}$ model, using the mass modifications of these charmonium states, $D$ and $\bar{D}$ mesons calculated within the chiral effective model amarvepja .

The mass modifications of the open charm meson are calculated from their interactions with the isoscalar scalar mesons ($\sigma$ and $\zeta$), isovector scalar meson, $\delta$, and the nucleons in the magnetized asymmetric nuclear matter dmeson_mag , whereas the masses of the charmonium states are modified due to the medium change of the gluon condensates, simulated by the scalar dilation field, $\chi$ charmonium_mag within the chiral effective model. The in-medium masses of the $D$ and $\bar{D}$ mesons in the magnetied asymmetric nuclear matter were calculated in Ref. dmeson_mag , within the frozen glueball approximation, i.e., neglecting the medium modifications of the dilaton field, $\chi$. In the present work, we take into account the medium modification of the dilaton field, $\chi$, by solving the Euler Lagrange equations of motions for the scalar fields, $\sigma$, $\zeta$ and $\delta$, as well as $\chi$ self consistently, for given values of baryon density, isospin asymmetry parameter and magnetic field. Accounting for the medium modifications of the $\chi$ field, however, are observed to give rise to marginal modifications to the masses of the $D$ and $\bar{D}$ meson masses as compared to the case of frozen glueball approximation. The medium modifications of the charmonium decay widths to $D\bar{D}$ in the magnetized nuclear matter arise due to the mass modifications of the open charm and charmonium states, which depend on the medium changes of the scalar fields and the dilaton field. To undertand the in-medium behaviour of these fields, which modify the masses of the charmonium and open charm mesons, and hence the charmonium decay widths to $D\bar{D}$, we write explicitly the coupled equations of motion of the scalar fields ($\sigma$, $\zeta$, $\delta$) and the dilaton field, $\chi$. These are given as upsilon_mag

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the above, $\rho_{i}^{s}(i=p,n)$ are the scalar densities for the nucleons, and, $\sigma_{0}$, $\zeta_{0}$ and $\chi_{0}$ are the vacuum values for these scalar fields.

The expressions for the decay widths of the charmonium states in the asymmetric nuclear matter, considered in the present investigation, are given as friman

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In the above, $M_{\Psi}$ is the in-medium mass of the corresponding charmonium state ($\Psi=J/\psi,\psi(3686),\psi(3770),\chi_{c0},\chi_{c2}$), $E_{D}$, $E_{\bar{D}}$ are energies of the outgoing $D$ and ${\bar{D}}$ mesons given as

[TABLE]

with $m_{D}$ and $m_{\bar{D}}$ as the in-medium masses of the $D$ and $\bar{D}$ mesons, and $p_{D}$ is the 3-momentum of the produced open charm mesons in the centre of mass frame given as

[TABLE]

In the expressions for the partial decay widths of the charmonium states to $D\bar{D}$, $\gamma$ is the coupling strength related to the strength of the ${}^{3}P_{0}$ vertex. It signifies the probability for creating a light quark-antiquark pair. The wave functions of the charmonium states as well as $D(\bar{D})$ mesons are assumed to be the wave functions with harmonic oscillator potential. The ratio $r=\beta/\beta_{D}$, where $\beta$ is the strength of the harmonic potential of the parent chamonium state and $\beta_{D}$ is the strength of harmonic potential of the daughter $D(\bar{D})$-meson. The scaled momentum $x$ is defined as $x=p_{D}/\beta_{D}$. In this study the value of $\gamma$ is chosen to be 0.33 which reproduces the observed decay widths of $\psi(3770)$ to $D^{+}D^{-}$ as well as to $D^{0}\bar{D^{0}}$ in vacuum amarvepja . The value of $\beta_{D}$ is taken as 0.31 GeV, consistent with the decay widths of $\psi(4040)$ to $D\bar{D}$, $D^{*}\bar{D}$, $D\bar{D^{*}}$ and $D^{*}\bar{D^{*}}$ in vaccum leeko . As we might see from the expressions of the decay widths of the charmonium states ($J/\psi,\psi(3686),\psi(3770),\chi_{c0},\chi_{c2}$) to $D\bar{D}$ given by the equations (5)–(9), these decay widths depend on the variable, $x$, which is the center of mass momentum, $p_{D}$ in units of $\beta_{D}$, as a polynomial multiplied by an exponential term. The dependence of these decay widths on the density at given values of isospin asymmetry and magnetic field are thus determined by their dependence on $p_{D}$, given by equation (11), in terms of the in-medium masses of the charmonium state as well as $D$ and $\bar{D}$ mesons. Depending on the form of the polynomial, the in-medium decay widths of these charmonium states, are observed to have very different behaviour.

We use an effective chiral $SU(3)$ model paper3 to study the in-medium masses of the charmonium states in asymmetric nuclear matter in the presence of an external magnetic field charmonium_mag . The model is based on the nonlinear realization of chiral symmetry weinberg ; coleman ; bardeen and broken scale invariance paper3 ; kristof1 ; hartree . The scale invariance breaking is through a logarithmic potential given in terms of a scalar dilaton field heide1 , and the medium modification of the dilaton field gives the medium modification of the gluon condensate, used to calculate the charmonium mass in the nuclear medium in the presence of an external magnetic field. The contribution of the magnetic field is incorporated broderick1 ; broderick2 ; Wei ; mao into the chiral effective model to study the mass modifications of the charmonium states charmonium_mag as well as open charm mesons dmeson_mag in the magnetized nuclear matter, including the effects of anomalous magnetic moments (AMM) of the nucleons broderick1 ; broderick2 ; Wei ; mao ; amm ; VD_SS_1 ; VD_SS_2 ; aguirre_fermion . The AMM effects are taken into consideration in the present study of the partial charmonium decay widths to $D\bar{D}$ in the magnetized nuclear matter, through the in-medium masses of the charmonium state as well as the open charm mesons. In the presence of the AMM effects, there are contributions to the nucleon fields, due to the tensorial interaction of the nucleons with the electromagnetic field $(-\frac{1}{4}\kappa_{i}\mu_{N}{\bar{\psi_{i}}}\sigma^{\mu\nu}F_{\mu\nu}\psi_{i})$, in addition to the vectorial interaction term $(-{\bar{\psi_{i}}}q_{i}\gamma_{\mu}A^{\mu}\psi_{i})$, where, $\psi_{i}$ corresponds to the $i$-th baryon (proton and neutron for nuclear matter, as considered in the present work) dmeson_mag ; bmeson_mag ; charmonium_mag ; broderick1 ; broderick2 ; Wei ; mao ; amm ; VD_SS_1 ; VD_SS_2 ; aguirre_fermion . The values of $\kappa_{p}$ and $\kappa_{n}$ are given as $3.5856$ and $-3.8263$ respectively, which are the values of the gyromagnetic ratio corresponding to the anomalous magnetic moments of the proton and neutron respectively. The scalar densities and number densities of the nucleons are modified with the AMM effects due to the tensorial interaction term. With inclusion of the AMM effects, the values for the scalar fields and dilaton field, which are obtained by solving the coupled equations of motion (1)–(4), are modified, as compared to when the AMM effects are not taken into consideration. Hence, the mass shifts of the charmonium states, as well as the $D$ and $\bar{D}$ meson are modified, which in turn modify the charmonium decay widths to $D\bar{D}$.

As has already been mentioned, the mass shift in the charmonium state arises due to the medium modification of the scalar gluon condensate, and hence due to the change in the value of the dilaton field in the chiral effective model, and is given as leeko ; amarvepja ; charmonium_mag

[TABLE]

where

[TABLE]

In the above, $m_{c}$ is the mass of the charm quark, taken as 1.95 GeV, $m_{\Psi}$ is the vacuum mass of the charmonium state and $\epsilon=2m_{c}-m_{\Psi}$ is the binding energy. $\psi({\bf k})$ is the wave function of the charmonium state in the momentum space, which has been assumed to be Gaussian friman ; amarvepja ; leeko .

III Results and Discussions

We investigate the in-medium partial decay widths of the charmonium state $\psi$ ($J/\psi$, $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$ and $\chi_{c2}$) to $D\bar{D}$ pair in the magnetized nuclear matter. The medium modifications of these decay widths are calculated from the medium modifications of their masses. The in-medium masses of the charmonium states, $J/\psi$, $\psi(3686)$ and $\psi(3770)$ in asymmetric nuclear matter in the presence of strong magnetic fields, have already been studied using a chiral effective model in Ref. charmonium_mag . These masses were calculated from the medium changes of a scalar dilaton field, which simulates the gluon condensates of QCD, within the effective hadronic model through broken scale invariance. In the present work, the mass modifications of the 1P charmonium states, $\chi_{c0}$ as well as $\chi_{c2}$ have been studied, which are obtained from the medium modifications of the scalar dilaton field, $\chi$, from the equation (12), using the wave function for the charmonium state to be of harmonic oscillator wave function for 1P state. The values of the strength of the harmonic oscillator potential, $\beta$ for $J/\psi$, $\psi(3686)$ and $\psi(3770)$ (assuming these states to be 1S, 2S and 1D states) are calculated from their rms radii $\langle r^{2}\rangle$ of 0.472 fm2, 0.962 fm2 and 1 fm2 respectively leeko ; amarvepja . For the 1P states, $\chi_{c0}$ and $\chi_{c2}$, we extract the values of $\beta$ from a linear extrapolation of the vacuum mass versus $\beta$ graph of these 1P states as well as the $\psi(3686)$ and $\psi(3770)$. The values of $\beta$ for $\chi_{c0}$ (vacuum mass of 3414.7 MeV) and $\chi_{c2}$ (vacuum mass of 3556.17 MeV) are obtained as 0.44 GeV and 0.41 GeV respectively.

The masses of the $D$ and $\bar{D}$ mesons are obtained in the present work, from their interactions with the nucleons and the scalar mesons, $\sigma$, $\zeta$ and $\delta$ in the isospin asymmetric magnetized nuclear matter. The values of these meson fields are solved from the coupled equations of these fields as well as of the scalar dilaton field in the mean field approximation. The study of $D(\bar{D})$ mesons in asymmetric nuclear matter in the presence of a strong magnetic field in Ref. dmeson_mag , used the frozen glueball aprroximation, i.e., $\chi$ to be fixed at the vacuum value of $\chi$ as 409.77 MeV. The masses of the $D$ and $\bar{D}$ mesons in the magnetized nuclear matter in the present work are observed to have modifications (though small) as compared to the masses obtained in Ref. dmeson_mag , due to the medium dependence of the dilaton field $\chi$ considered in the present investigation.

The masses of the $D(D^{+},D^{0})$ and $\bar{D}(D^{-},\bar{D^{0}})$ mesons are shown in figures 1 amd 2, as functions of the baryon density (in units of nuclear matter saturation density) for the values of magnetic field as $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$, with the values of the isospin asymmetry parameter as $\eta$=0, 0.3 and 0.5, with the anomalous magnetic moments (AMM) of the nucleons taken into consideration. These results are compared with the case when the AMM of the nucleons are neglected. In the isospin symmetric nuclear matter, the masses (in MeV) of the $D^{+}$ and $D^{0}$ are observed to be 1813.6 (1811.8) and 1787 (1785.5) at the nuclear matter saturation density for magnetic field (in units of 1/e) of $4m_{\pi}^{2}$ with (without) AMM effects. With the larger density pf 4$\rho_{0}$, these are modified to 1557 (1542.6) and 1526.5 (1510.4). These values may be compared to 1809.7 (1807.8) and 1782.9 (1781.4) at $\rho_{B}=\rho_{0}$ and 1539.9 (1524.9) and 1508.5 (1492.7) at $\rho_{B}=4\rho_{0}$, in the frozen glueball approximation dmeson_mag . The modifications of the masses of the $D$ mesons are thus observed to be rather small, when the medium modifications of the dilaton field are taken into account, as compared to the frozen glueball approximation. For the $\bar{D}$ mesons, in isospin symmetric nuclear matter, the masses of $D^{-}$ and $\bar{D^{0}}$ mesons are obtained as 1864 (1862.2) and 1837.8 (1836.4) at $\rho_{0}$ and 1740.2 (1722.7) and 1711.5 (1692.6) at 4$\rho_{0}$ with (without) AMM effects. The isospin asymmetry as well as AMM effects are observed to be larger at the higher magnetic field of $eB=8m_{\pi}^{2}$ for $D^{0}$ meson, whereas $D^{+}$ is rather insensitive to these effects, as can be seen from figure 1. For $D^{-}$ as well as $\bar{D^{0}}$ mesons, the asymmetry as well as AMM effects are appreciable for the larger value of the magnetic field ($eB=8m_{\pi}^{2}$), as can be seen from figure 2.

The in-medium masses of the charmonium states, $J/\psi$, $\psi(3686)$ and $\psi(3770)$ in the magnetized asymmetric nuclear matter have been studied within the chiral effective model charmonium_mag . In the present work, we study the mass modifications of the P-wave charmonium states, $\chi_{c0}$ and $\chi_{c2}$, and investigate the partial decay widths of the charmonium states ($J/\psi,\psi(3686),\psi(3770),\chi_{c0},\chi_{c2}$) to $D\bar{D}$ in the magnetized nuclear matter.

In figures 3 amd 4, the masses of the 1P states are plotted as functions of the baryon density (in units of nuclear matter saturation density) for the values of magnetic field as $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$, with the values of the isospin asymmetry parameter as $\eta$=0, 0.3 and 0.5, with the anomalous magnetic moments (AMM) of the nucleons taken into consideration. These results are compared with the case when the AMM of the nucleons are neglected. The mass of $\chi_{c0}$ (in MeV) is observed to be modified from its vacuum value of 3414.7 to 3385.57 (3309) at a density of $\rho_{0}(4\rho_{0})$ for $eB=4m_{\pi}^{2}$ for symmetric nuclear matter, with AMM effects, and 3384.8 (3303.7) when AMM of nucleons are not taken into consideration. The mass shifts are oberved to be smaller with isospin asymmetry effects, as well as with increase in the magnetic field. The mass shifts are observed to be larger when the AMM of the nucleons are not considered. The mass of $\chi_{c2}$ is plotted in figure 4. For symmetric nuclear matter, at a density of $\rho_{0}(4\rho_{0})$ for $eB=4m_{\pi}^{2}$, the mass of $\chi_{c2}$ (in MeV) is observed to be modified from its vacuum value of 3556.17 to 3511 (3393) with AMM effects, and 3509.9 (3382.9) when AMM of nucleons are not taken into consideration. The effects of isospin asymmetry as well as magnetic fields on the mass of $\chi_{c2}$ plotted in figure 4 are observed to be similar to that of $\chi_{c0}$ in the present work as well as to the mass modifications of the charmonium states $J/\psi$, $\psi(3686)$ and $\psi(3770)$ already studied in Ref. charmonium_mag . This is due to the fact that the mass modifications of the charmonium states are determined by the medium change of the scalar dilaton field in the magnetized nuclear matter. The medium change of the mass of the charmonium state is observed to be quite insensitive to the isospin asymmetry in the medium. This can be understood in the following manner. The mass shift of the charmonium state is obtained from the shift of the dilaton field, $\chi$ and is proportional to the shift in the fourth power of $\chi$ from the vacuum value (as can be seen from equation (12)), in the limit of massless quarks in the trace anomaly charmonium_mag as considered in the present investigation. In the mean field approximation, the meson fields are replaced by their expectation values and as has already been mentioned, the value of $\chi$ is obtained through the solution of coupled equations of the scalar fields, $\sigma$, $\zeta$, $\delta$ and $\chi$ given by equations (1)–(4). In the presence of the isospin asymmetry in the medium, i.e. for nonzero values of the isospin asymmetry parameter, $\eta=(\rho_{n}-\rho_{p})/(2\rho_{B})$, the number density of the neutron is larger than the proton density and the scalar densities of the proton and neutron become different, with the value of $\delta$ being proportional to their difference. The magnitudes of $\sigma$ and $\zeta$, become larger in the asymmetric nuclear matter than in the symmetric nuclear matter case, due to the contributions from $\delta^{2}$ term, as can be seen from the equations for $\sigma$ and $\zeta$ fields given by (1) and (2). However, due to the much smaller magntidue of $\delta$ as compared to $\sigma$ and $\zeta$ (e.g., $\delta\sim$ 2 MeV, $\sigma\sim$ 45 MeV, $\zeta\sim$ 94 MeV at $\rho_{B}=2\rho_{0}$ for $eB=8m_{\pi}^{2}$ and $\eta$=0.3), the modifications for $\sigma$ and $\zeta$ are very small in the asymmetric nuclear matter as compared to symmetric nuclear matter. It might be noted here that in the absence of a magnetic field, the scalar densities for the proton and neutron remain the same and the value of $\delta$ is zero, whereas, in the presence of a magnetic field, even for the case of isospin symmetry in the medium ($\rho_{p}=\rho_{n}$), the scalar densities are different for the proton and neutron (due to contribution of Landau energy levels for proton in the presence of a magnetic field), which gives rise to a nonzero value (though small, with a maximum value of about 0.5 MeV for $eB=8m_{\pi}^{2}$) for the isovector scalar meson, $\delta$. The dependence of $\chi$ on $\sigma$ and $\zeta$ (as can be seen from equation (4)) in the leading order, is proportional to \big{(}(\sigma^{2}-\delta^{2})\zeta\big{)}/($\sigma^{2}+\zeta^{2}+\delta^{2}$). The magnitude of the isovector scalar field $\delta$, remains small ($\sim$ few MeV) as compared to the magnitudes of $\sigma$ or $\zeta$. This is the reason why the modification of the dilton field $\chi$ due to isospin asymmetry remains small, and subsequently the mass shifts of the charmonium states (which are proportional to ($\chi^{4}-\chi_{0}^{4}$)) remain insensitive to isospin asymmetry of the medium.

We investigate the in-medium decay widths of the charmonium states to $D\bar{D}$ in the magnetized nuclear matter, for the values of the isospin asymmetry parameter $\eta$ as 0, 0.3 and 0.5 and magnetic fields $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$. For $eB=8m_{\pi}^{2}$ and in the case when the AMM effects of the nucleons are neglected, the decay width of $J/\psi\rightarrow D^{0}\bar{D^{0}}$ is observed to be possible above a density of around 5$\rho_{0}$ in symmetric nuclear matter. There is observed to be an increase in this decay width with density, reaching a value of around 27.9 MeV at 6$\rho_{0}$. The decay $J/\psi\rightarrow D\bar{D}$ is not observed in any other case considered in the present work.

The decay widths of $\psi(3686)$ to $D\bar{D}$ in magnetized isospin asymmetric nuclear matter are plotted as functions of the baryon density in units of nuclear matter saturation density, in figure 5 for different values of the magnetic field and isospin asymmetry parameter. These are plotted for the channels (i) $D^{+}D^{-}$, (ii) $D^{0}\bar{D^{0}}$ as well as (iii) the sum of these two channels. The decay of $\psi(3686)$ is not possible in vacuum since the mass of this charmonium state is smaller than the combined mass of $D$ and $\bar{D}$ in vacuum. However, at certain densities, these decays become possible, when the center of mass momentum, $p_{D}$ becomes non-zero with the medium modifications of the masses of the charmonium as well as the outgoing $D$ and $\bar{D}$ mesons friman ; amarvepja . In symmetric nuclear matter ($\eta$=0), in the presence of magnetic field, $eB=4m_{\pi}^{2}$, shown in panel (a) of figure 5, the threshold densities above which the decays $\psi(3866)$ to $D^{+}D^{-}$ and $\psi(3866)$ to $D^{0}\bar{D^{0}}$ become possible are 4.5$\rho_{0}$ and 2.54$\rho_{0}$ respectively when the AMM of the nucleons are taken into account, and 4.1$\rho_{0}$ and 2.37$\rho_{0}$ when the AMM of nucleons are not considered. The higher value for the threshold density for the decay width of $\psi(3866)$ to $D^{+}D^{-}$, as compared to the channel $\psi(3866)$ to $D^{0}\bar{D^{0}}$ is due to the reason that the masses of the charged $D$ as well as charged $\bar{D}$ mesons have positive shifts due to the contributions from the lowest Landau levels dmeson_mag , which makes the mass of $D^{+}D^{-}$ to be larger than $D^{0}\bar{D^{0}}$. For the magnetic field $eB=8m_{\pi}^{2}$ in symmetric nuclear matter, as shown in panel (b), the decay of $\psi(3866)$ to $D^{0}\bar{D^{0}}$ becomes possible at a density of 2.4$\rho_{0}$ (2.2$\rho_{0}$) for the cases of with (without) AMM effects. The decay of $\psi(3866)$ to $D^{+}{D^{-}}$ is not observed even upto a density of 6$\rho_{0}$.

In the isospin asymmetric nuclear medium with $\eta$=0.3, the densities above which the decays of $\psi(3866)$ to $D^{+}D^{-}$ and $\psi(3866)$ to $D^{0}\bar{D^{0}}$ become possible are observed to be 3.3$\rho_{0}$ and 3.7$\rho_{0}$ respectively for $eB=4m_{\pi}^{2}$ as can be seen from panel (c) of figure 5 when the AMM of the nucleons are taken into account. These densities are modified to 3.12$\rho_{0}$ and 3.5$\rho_{0}$, when the AMM effects are not taken into consideration. The contrasting behaviour of the threshold density for $\psi(3686)\rightarrow D^{+}D^{-}$ to be smaller than the channel $\psi(3686)\rightarrow D^{0}\bar{D^{0}}$ for the isospin asymmetric case with $\eta$=0.3, as compared to the isospin symmetric case for $eB=4m_{\pi}^{2}$, is due to the reason that the center of mass momentum of the outgoing $D$ and $\bar{D}$ mesons, $p_{D}$, has larger contribution from the last term of equation (11) for the case of outgoing mesons as $D^{+}D^{-}$ as compared to $D^{0}\bar{D^{0}}$, since the difference in the masses of $D^{+}$ and $D^{-}$ is much bigger than the difference in $D^{0}$ and $\bar{D^{0}}$ masses. When the magnetic field is increased to $eB=8m_{\pi}^{2}$, for asymmetric nuclear matter with $\eta$=0.3, as can be seen from panel (d) of figure 5, the values for the threhold densities for decay to $D^{+}D^{-}$ and $D^{0}\bar{D^{0}}$ are 4.6$\rho_{0}$ and 3.6$\rho_{0}$ respectively for the case without AMM effects. The higher threhold density for the charged $D\bar{D}$ decay as compared to neutral $D\bar{D}$ decay is similar to the case of the isospin symmetric case. This is because the masses of the charged mesons have larger positive mass shifts with increase in the magnetic field to $eB=8m_{\pi}^{2}$. When the AMM effects are taken into account, the threshold dnesity for $\psi(3866)$ to $D^{0}\bar{D^{0}}$ is about 4$\rho_{0}$ and the decay of $\psi(3866)$ to $D^{+}D^{-}$ does not become possible even upto a density of 6$\rho_{0}$.

With further increase in the isospin asmmetry, with $\eta$=0.5, only the decay channel of $\psi(3686)$ to $D^{+}D^{-}$ is possible, above a density of 2.9$\rho_{0}$ and 5$\rho_{0}$ for $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$ respectively, when AMM of the nucleons are considered. In the case when the AMM effects are not taken into account, there is no dependence on the magnetic field, for the decay to $D^{0}\bar{D^{0}}$, as the system consists of only neutrons for $\eta$=0.5 and the only effect of magnetic field can be through the AMM of neutron. However, in the presence of a magnetic field, the masses of the charged $D^{\pm}$ mesons are modified due to the contribution from the lowest Landau level. For the case of without AMM effects and for $\eta$=0.5, the decay of $\psi(3686)$ to $D^{+}D^{-}$ is possible above a density of about 2.7 (4)$\rho_{0}$ for $eB=4(8)m_{\pi}^{2}$, and the decay to the neutral $D\bar{D}$ pair is not observed even upto a density of 6$\rho_{0}$.

In figure 6, the partial decay widths of $\psi(3770)$ to $D\bar{D}$ are shown for different values of the isospin asmmetry parameter and magnetic fields, accounting for the effects of AMMs of nucleons. These results are compared with the case when the AMMs of nucleons are not taken into consideration. For symmetric nuclear matter ($\eta$=0), there is observed to be an initial decrease of the decay width upto a density of around 1.5$\rho_{0}$ followed by an increase at high densities both for the cases of $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$, as can be seen from panels (a) and (b) of figure 6. For $eB=4m_{\pi}^{2}$, the decay is solely through the channel $\psi(3770)\rightarrow D^{0}\bar{D^{0}}$ upto a density of around 4.8$\rho_{0}$ (5.3$\rho_{0}$), above which the decay to the charged $D\bar{D}$ mesons also becomes possible. For the higher magnetic field of $eB=8m_{\pi}^{2}$, as can be seen from panel (b) of figure 6, the decay to the charged $D\bar{D}$ mesons does not become possible even upto a density of 6$\rho_{0}$. For isospin asymmetric nuclear matter with $\eta$=0.3, there is observed to be an intial drop in the partial decay width of $\psi(3770)\rightarrow D\bar{D}$ (solely due to contribution from the channel $\psi(3770)\rightarrow D^{0}\bar{D^{0}}$) with density, which becomes very small ($\sim$ 0.28 (0.11) MeV) at around 1.05$\rho_{0}$ (0.9$\rho_{0}$) with (without) the AMM effects of the nucleons. The value of the decay width for $\psi(3770)\rightarrow D^{0}\bar{D^{0}}$ remains similar upto a density of around 3.75$\rho_{0}$ (3.6$\rho_{0}$), with (without) AMM effects, above which there is observed to be an increase in this decay width with density. The decay channel $\psi(3770)\rightarrow D^{+}{D^{-}}$ is observed to become possible above a density of around 3.1$\rho_{0}$ (2.9$\rho_{0}$) when the AMM effects are considered (neglected). For the magnetic field $eB=8m_{\pi}^{2}$, the behaviour is similar to case of $eB=4m_{\pi}^{2}$, for the case when the AMMs of the nucleons are not taken into account. On the other hand, for the case when AMM effects are considered, there is no decay to charged $D\bar{D}$ pair observed even upto a density of 6$\rho_{0}$. For the isospin asymmetry parameter, $\eta$=0.5, the decay is only through the channel $\psi(3770)\rightarrow D^{+}D^{-}$, for both $eB=4m_{\pi}^{2}$ and $eB=8m_{\pi}^{2}$.

In figure 7, the effects of isospin asymetry, magnetic field and density on the partial decay widths of $\chi_{c0}$ to $D\bar{D}$ are shown. There is observed to be a threshold density above which the decay is observed to be possible for both the charged and neutral $D\bar{D}$ channels. There is observed to be an initial increase with density, followed by a drop when the density is further increased. The AMM effects are observed to be large for higher value of magnetic field.

In figure 8, the partial decay widths of $\chi_{c2}\rightarrow D\bar{D}$ is plotted as functions of the baryon density for different values of isospin asymmetry parameter and magnetic fields, both with and without AMM effects of the nucleons. There is observed to be a threshold density for both the charged and neutral $D\bar{D}$ pair channels, and the decay width is observed to increase monotonically with density. This behaviour can be understood looking at the polynomial part (in $p_{D}$) of the partial decay width which increases with $p_{D}$, which in turn is an increasing function of the baryon density. This behaviour of monotonic increase in the decay width of $\chi_{c2}$ was also observed in Ref. friman , as a function of the mass drop in $D$ and $\bar{D}$ mesons in the nuclear medium.

The partial decay widths of the charmonium to $D\bar{D}$ depend on the center of mass momentum of the $D(\bar{D})$ meson, $p_{D}$ (given by equation (11), through a polynomial part multiplied by an exponential part, as can be seen from the equations (5)–(9). The stark difference of the $\chi_{c0}$ and $\chi_{c2}$ decay widths is due to the difference in the forms of the polynomial part, which has a monotonic increase with density in the case of $\chi_{c2}$ above a threshold density when the decay to $D\bar{D}$ becomes possible. On the other hand, the decay width for $\chi_{c0}$ (above the threshold density when the decay becomes possible) shows an initial increase followed by a drop with further increase in the density. Similar behaviour for the decay wdiths of $\chi_{c0}$ and $\chi_{c2}$ to $D\bar{D}$ were also observed in Ref. friman . The medium modifications of these decay widths were calculated from the mass modifications of the $D$ and $\bar{D}$ mesons, which were assumed to be same in the symmetric nuclear matter in Ref. friman . The $\psi(3686)$ decay width shows a monotonic increase, whereas $\psi(3770)\rightarrow D\bar{D}$ decay width has an initial drop followed by a rise with further increase in the density. As has already been mentioned, the dependence of the charmonium decay width to $D\bar{D}$ on density is determined by the form of the polynomial and the exponential parts in terms of the center of mass momentum, $p_{D}$.

IV Summary

We have studied the effects of magnetic field on the partial decay widths of the charmonium states, $J/\psi(3097)$, $\psi(3686)$, $\psi(3770)$, $\chi_{c0}$ and $\chi_{c2}$ to $D\bar{D}$ in isospin asymmetric nuclear matter. These are obtained using the mass modifications of the $D$ and $\bar{D}$ mesons as well as the charmonium states, calculated within a chiral effective model. The mass modifications of the open charm mesons, $D$ ($D^{0}$,$D^{+}$) and $\bar{D}$ ($\bar{D^{0}}$, $D^{-}$) arise due to their interactions with the nucleons and the scalar mesons, where as the in-medium masses of the charmonium states are calculated from the medium changes of the scalar dilaton field, which simulates the gluon condesates of QCD within the chiral effective model. In the presence of the magnetic field, the proton has contributions from the Landau energy levels. There is observed to be unequal masses for the $D^{0}$ and $D^{+}$ within the $D$ meson doublet, as well as for $\bar{D^{0}}$ and $D^{-}$ within the $\bar{D}$ doublet, even in isospin symmetric nuclear matter, due to difference in the interactions with the proton and neutron, as well as due to contributions from the Landau energy levels for the charged $D$ and $\bar{D}$ mesons, in the presence of magnetic field. In symmetric nuclear matter, the threshold densities for which the decay widths of the charmonium state to $D\bar{D}$ turn out to be smaller for the $D^{0}\bar{D^{0}}$ channel as compared to for $D^{+}D^{-}$ channel, as the masses of the $D^{+}$ as well as $D^{-}$ have positive shifts in the presence of magnetic field. The effects of isospin asymmetry, magnetic fields are observed to be quite appreciable on the partial decay widths for the charged and neutral $D\bar{D}$ pair channels, which will affect the production of these mesons in high energy asymmetric heavy ion collision experiments.

Acknowledgements.

One of the authors (AM) is grateful to ITP, University of Frankfurt, for warm hospitality and acknowledges financial support from Alexander von Humboldt Stiftung when this work was initiated. Amal Jahan CS acknowledges the support towards this work from Department of Science and Technology, Govt of India, via INSPIRE fellowship scheme (INSPIRE Code IF170745).