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

We investigate the in-medium masses of the pseudoscalar $(D,{\bar D},D_s^{\pm})$, and vector $(D^*,\bar{D}^*, D_s^{*\pm})$, open charm mesons in isospin asymmetric magnetized nuclear matter, accounting for the effects of Dirac sea. The masses are used to study the in-medium partial decay widths of $D^* \rightarrow D \pi$ ($\bar{D}^*\rightarrow \bar{D}\pi$) and $\Psi(3770) \rightarrow D \bar{D}$, using the $^3P_0$ model. The in-medium masses of the open charm mesons are calculated from their interactions with the nucleons and scalar mesons within the generalized chiral effective model, in terms of the scalar and number densities of nucleons and the scalar fields fluctuations. The effects of Landau energy levels of protons and AMMs of the nucleons are also considered in the magnetized nuclear matter. The light quark condensates are modified considerably with magnetic field, leading to (inverse) magnetic catalysis due to the magnetized Dirac sea effects. The magnetic field causes modifications to occur due to the mixing of the pseudoscalar and longitudinal component of the vector mesons, along with the lowest Landau level contribution to the ground state energy of the charged mesons as point particle correction. For the charmonium state $\Psi(3770)$, the effects of the magnetized Dirac sea are incorporated to the mass modifications through the medium modified scalar dilaton field $\chi$ within the chiral model. The in-medium masses and decay widths of the open charm and charmonium mesons thus obtained should have important observable consequences in the production of the open charm mesons and charmonia in peripheral ultra-relativistic heavy ion collision experiments, where huge magnetic fields are expected to be created.


I. INTRODUCTION
The study of the hadron properties under extreme conditions of matter is of great relevance in the high energy heavy ion collision experiments and in some astrophysical objects like the magnetars, neutron stars etc. The heavy flavor mesons, created in the initial stages of these collisions may have impact from the strong magnetic field. As huge magnetic fields are estimated to be generated in the peripheral ultra relativistic heavy ion collision experiments at LHC in CERN and at RHIC in BNL [1][2][3][4][5], in the early phase of the collisions.
However, there is still an open question on the time evolution of the produced magnetic field due to the complexity in the estimation of the electrical conductivity within the medium.
The effects of magnetic field on the in-medium properties of hadrons thus initiate a new area of research with great relevance in the physics of the heavy ion collisions [6].
The experimental observables of the relativistic heavy ion collision experiments are affected by the medium modifications of the hadrons. In the presence of magnetic field, there can be (decrement) enhancement of the light quark condensates with increasing magnetic field, an effect called (inverse) magnetic catalysis [7][8][9][10][11][12]. In the literature, there have been studies on the effect of (inverse) magnetic catalysis on quark matter in the context of NJL model [10,11,[13][14][15]. The effects of Dirac sea at finite magnetic field have been studied on the vacuum to nuclear matter phase transition within the Walecka model and an extended linear sigma model [16]. The (inverse) magnetic catalysis effect has been studied in [17], in the context of nuclear matter, accounting for the finite anomalous magnetic moments of the nucleons. The effects of the anomalous magnetic moments of the nucleons have been taken into account through the weak-field expansion of the fermionic propagators in the calculation of the nucleonic one-loop self energy functions.
Several studies exist on the in-medium properties of the heavy flavor mesons under the various conditions of matter, for e.g., density, temperature and magnetic fields. Various methodologies are employed to study them as illustrated below. One of them is the chiral effective model [18][19][20][21][22][23][24][25][26][27][28]. Other approaches include the QCD sum rules (QCDSR) [29][30][31][32][33], the coupled channel approach [34], the Quark meson coupling (QMC) model [35,36] quarkonium states have been studied within the chiral effective model framework [18][19][20][21][22][23][24][25], in the absence of magnetic field. In the presence of a magnetic field, the masses of the open and hidden heavy flavour mesons have been studied within the chiral model [26][27][28] and their effects on the in-medium partial decay widths of the charmonium states to DD [37][38][39] and bottomonium states to BB [40]. At finite magnetic field, there is mixing phenomena due to the interaction between the pseudoscalar and the longitudinal component of the vector mesons called the PV mixing, which leads to a decrease (increase) in the masses of the pseudoscalar (longitudinal component of the vector) mesons with increasing magnetic field. The PV mixing effects have been studied for the open heavy flavor mesons and heavy quarkonia in [29][30][31][32][33][38][39][40][41][42][43][44]. For the charged mesons, there are additional contributions from the Landau energy levels in presence of an external magnetic field. The in-medium partial decay widths of the heavy quarkonia (qq; q = c, b) going to open heavy flavor mesons have been studied using a field theoretic model for composite hadrons with quark (and antiquark) constituents [38][39][40], as well as within the 3 P 0 model [37], in presence of an external to the pseudoscalar meson states D(D) and a π meson, are also studied considering the effects of the magnetized Dirac sea on their masses, using the 3 P 0 model [47][48][49][50][51][52][53][54]. In the 3 P 0 model, a light quark-antiquark pair is created in the 3 P 0 state (with quantum numbers similar to vacuum, J P C = 0 ++ ), and this light quark (antiquark) combines with the heavy charm antiquark (quark) of the decaying D * meson at rest, resulting in the production of the open charm D meson and a light flavored π meson [21,54]. The matrix element for the decay of the D * meson state depends on the center-of-mass momentum |p| of the outgoing particles and on the internal structure of the meson states.
We organize the paper as follows: In section II, we discuss in detail the chiral effective model to calculate the in-medium masses of the open charm and charmonium, Ψ(3770) mesons in the magnetized nuclear matter accounting for the Dirac sea effects. The effects of the point particle correction to the ground state mass of the charged mesons are discussed.
The pseudoscalar-vector mesons (PV) mixing in presence of an external magnetic field are obtained using an effective Lagrangian approach and the spin-magnetic field interaction Hamiltonian (only for D s mesons). In section III, using the 3 P 0 model the hadronic decays of the vector mesons D * and Ψ(3770) are described. In section IV, the results of the current investigation are discussed. In section V, we summarize the findings of this work.  [55][56][57] and QCD scale symmetry breaking effect [58,59]. The latter is realized through a logarithmic potential in terms of the scalar dilaton field, χ [60]. General structure of the chiral model Lagrangian density is where, L kin corresponds to the kinetic energy terms of the baryons and the mesons, L BW describes the baryon-meson interactions for W = scalar, pseudoscalar, vector, axial vector mesons, L vec gives the dynamical mass generation of the vector mesons due to the couplings to the scalar fields and contains additional quartic vector mesons self-interactions, L 0 expresses the meson-meson interaction terms, L scale−break is the scale symmetry breaking logarithmic potential in terms of a scalar dilaton field, χ and L SB illustrates the explicit chiral symmetry breaking term. L mag contains the baryon (here nucleons) and magnetic field interactions through the electromagnetic field strength tensor (F µν ) and magnetic vector potential (A µ ), as given in refs. [26][27][28][61][62][63].
The magnetized Dirac sea contributes to the scalar densities of the nucleons, within the chiral model framework . The coupled equations of motion for the scalar fields are derived   from the chiral model Lagrangian density under the classical scalar fields approximation, including the medium effects through the number (ρ p,n ) and scalar (ρ s p,n ) densities of protons and neutrons in the nuclear matter. There are contributions of the Landau energy levels of protons and anomalous magnetic moments (AMMs) of nucleons through ρ p,n and ρ s p,n due to the presence of L mag term in the general Lagrangian density (1), apart from the magnetized Dirac sea contribution described later.
The masses of the baryons are generated by their interactions with the scalar mesons.
The mass of baryon of species i (i = p, n in the present work of nuclear matter) is given as ( In the generalized SU(4) version of the chiral SU(3) model the interaction term involving the pseudoscalar open charm D andD mesons, [19][20][21]26] In equation (3) The sea contribution for the spin-1/2, electrically charged fermions is given by the free energy of the magnetized vacuum [16] where, ν denotes the Landau level and sum over this quantity is implied. The single particle energies are ϵ k,ν = k 2 z + 2ν|qB| + M 2 when AMM of the fermion is taken to be zero, and α ν (= 2 − δ ν0 ) is the spin degeneracy of each Landau level which is 1 for the lowest level with ν = 0. For non-zero AMM of the charged fermions, the single fermion energies turn into ϵ k,ν,s = k 2 z + ( 2ν|qB| + M 2 + sκ f B) 2 , with an extra summation over the spin-projection s = ±1 in eq.(4) with α ν = 1 for all ν. In the mean-field approximation, Ω N,sea takes the form of free fermions by including all interactions to the in-medium mass term as in eq.(2), is not a pure vacuum term. Equation (4) involves divergent integral. The various regularization schemes of dimensional regularization or, proper time method have been employed in the literature to regularize the divergent integral [11,16,64] and renormalizing the remaining term by proper scaling of the physical quantities like charge and magnetic field etc. [16].
The free energy of the Dirac sea contribution is minimized to give the scalar fields equations of motion, with the impact of the sea term on their solutions in magnetic field.
The dispersion relation for the D mesons are obtained from the Fourier transforms of the equations of motions of the mesons obtained from the chiral effective Lagrangian, given by where Π D(D) denotes the self energy and m D(D) is the vacuum mass of the D (D) meson.
The self energies of D meson (D 0 , D + ) andD meson doublet (D 0 , D − ) are given in terms of the scalar fields fluctuations, the number and the scalar densities of the nucleons, as given in ref. [26]. For the charmed, strange meson D s , the corresponding mass is m Ds , Π Ds is the D s meson self energy, which is same for D + s and D − s mesons in nuclear matter [22] Π(ω, | ⃗ k|) In equation (6), ζ ′ (= ζ − ζ 0 ) is the fluctuation of the strange scalar-isoscalar field ζ(∼ ⟨ss⟩). The value of f Ds is determined from the particle data group (PDG) as 250 MeV [65]. The solution of equation (5) for variable ω at | ⃗ k| = 0 gives the in-medium mass of the corresponding pseudoscalar open charm meson. To obtain this, the coupled equations of motion of the scalar fields (σ, ζ, δ and χ), as derived from the chiral model Lagrangian by taking classical scalar fields assumption, are solved at given baryon density ρ B , isospin asymmetry η = (ρ n − ρ p )/(2ρ B ), and magnetic field |eB|.
In the presence of an external magnetic field, for e.g. ⃗ B along the z direction, the energy level of an electrically charged point particle with spin S, mass m is given by E n,Sz (p z ) = p 2 z + (2n + 1 + 2S z )|eB| + m 2 ; with p z as the continuous momentum along the z-axis of Bẑ, S z = −S, −S + 1, ..S, is the spin-component along the magnetic field direction and n is the Landau level [29]. As mentioned in refs. [66,67], the minimal effective mass corresponding to the lowest energy state with p z = 0 can be considered as the ground state mass of the charged point-like particle. For spin-0, charged open charm mesons, It is referred to as the point particle correction ignoring the internal structure of the mesons.
However, in our work, the medium effects are incorporated mainly due to the interactions of the D mesons with the nucleons and scalar mesons within the generalized chiral effective model. The contributions of Dirac sea and the Fermi sea in the magnetic matter are obtained in this way apart from the correction due to the point particle assumption. Although, at weak field region, higher Landau levels (n ≥ 0) are situated infinitesimally close to the ground state n = 0. In ref. [29], the sum over all Landau levels for the charged D mesons have been taken into account using the formalism of QCD Borel sum rule. In equations (7), m * D ± denote the masses of the corresponding D mesons calculated from the solutions of eq.(5) for ω at | ⃗ k| = 0. For spin-1 mesons the ground state mass due to the point particle correction is therefore given by It is assumed that the mass shifts of the vector open charm (D * andD * ) mesons (which have the same quark-antiquark constituents as D andD mesons) are identical to the mass shifts of the pseudoscalar mesons D andD mesons, calculated within the chiral effective model [39]. 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 [6]. The in-medium mass of the vector open charm mesons are thus given by [40] Using these values of m * D * in eq.(8), the point particle corrections for the vector, charged D * mesons are obtained for the spin projections of S z = −1, 0, 1 along Bẑ direction.
In an external magnetic field there is another important effect to be considered due to the mixing between the pseudoscalar (P) and the longitudinal component (S z =0) of the vector (V ) mesons. The PV mixing effect is observed to lead to an appreciable increase (decrease) in the mass of the longitudinal component of vector (pseudoscalar) mesons with increasing magnetic field [29][30][31][38][39][40][41][42][43]. In the present study, the PV mixing effect is considered for the neutral (D 0 − D * 0 ,D 0 −D * 0 ) and charged (D ± − D * ± ) open charm mesons, accounting for the magnetized Dirac sea contribution on the masses. The PV mixing effect arise due to an effective hadronic Lagrangian accounting for the P V γ interaction vertices [30,31,38,39], where P and V µ represent the pseudoscalar and the vector meson fields, respectively,F µν is the dual electromagnetic field strength tensor and m av is the average of the masses of the pseudoscalar and vector mesons, m av = (m P + m V )/2. The masses of the P and V || mesons due to the PV mixing effects can thus be written as [31,38] where, M 2 − = m 2 V − m 2 P and c P V = g P V |eB|; with m P,V being the effective masses of the pseudoscalar and the vector mesons, given by equations (7), (8), as calculated in the magnetized nuclear matter within the chiral effective model.
The coupling parameter of the effective interaction Lagrangian or the PV mixing strength, g P V is fitted from the experimentally known radiative decay width as follows where p cm = is the center of mass momentum of the final state particles. For the charged D * ± mesons the measured radiative decay width of Γ [D * ± → D ± + γ] = 1.33 keV, is used to calculate the mixing strength between D * ± and D ± from eq.(12). The total decay width for the neutral D * 0 meson has not yet been measured, although the decay branching ratio of Γ(D * 0 → D 0 π 0 ) : Γ(D * 0 → D 0 γ) = 64.7 : 35.3 is known [65]. From isospin considerations, the decay constant for the D * 0 → D 0 π 0 can be obtained. Thus the decay width for the neutral D * 0 → D 0 π 0 can be determined. From the above mentioned decay branching ratio, the value for g D 0 D * 0 can be estimated [29], The hadronic effective Lagrangian of L D * Dπ = gπ(∂ µ D)D * µ is formulated to calculate the hadronic decay widths of D * → Dπ modes from the imaginary part of the D − π loop.
The decay widths corresponding to the charged D * ± mesons are fitted to their measured values [65] to obtain the charged mesons decay constants. By equations (13)- (14), the decay constant corresponding to D * 0 → D 0 π 0 mode is obtained which is further used to calculate the decay width of D * 0 → D 0 π 0 . Therefore, the decay branching ratio for the neutral mesons give the value of the PV mixing strength for the neutral mesons g D * 0 D 0 . The effective Lagrangian L D * Dπ consists of only the leading order term in derivative, ignoring the higher derivative contributions which may also give impact to the calculated decay width.
Due to this uncertainty, a method based on the constituent quark model have been proposed to determine the mixing strength between D ± and D * ± mesons [29], in which the u and d quarks are taken to be non-relativistic in the heavy charm quark limit. The mixing-strength between pseudoscalar and vector open charm mesons in the constituent quark model depends on the constituent quark masses and the quark charge, and is dominated by its light quark component. In ref. [29], the mixing strength corresponding to the neutral mesons is obtained from the ratio of | g D 0 D * 0 g D ± D * ± | = 3.5, to be 3.278 which is very close to our estimate of 3.44 determined using the previous method of effective Lagrangian. However, in the constituent quark model, the values of the constituent light quark u, d masses can be changed with magnetic field as has been estimated in the context of NJL model in various quark matter studies at finite magnetic field [11,15]. These modifications may lead to a change in the values of the mixing strengths in the effective Lagrangian eq.(10), with the variation in magnetic field. In the present work, we rely on the framework of chiral effective Lagrangian to compute the magnetic field dependence of chiral order parameters, with the baryons and mesons as the effective degrees of freedom. To consider such effect on the change of constituent quark mass with magnetic field, is not within the scope of the present framework.
The radiative decay for D * s has been observed but not measured experimentally. We use the Hamiltonian formulation to calculate the spin-mixing between spin-0 (D s ) and longitudinal component of spin-1 (D * s ) mesons. In the presence of an external magnetic field, the Hamiltonian accounting for the spin-magnetic field interaction is given by [40,42,43] where, is the mass difference between the vector and pseudoscalar (D * s and D s ) mesons calculated within the chiral effective model. The effective masses due to spin-mixing are obtained from eq.(16), accounting for the DS effects in the magnetized nuclear matter.
The mass shifts of the charmonium states have been evaluated from the modifications of the gluon condensate in nuclear medium by the QCD second order Stark effect, in the limit of large charm quark mass [68]. In [20,21,28], the mass shift is proportional to the medium modifications of the scalar gluon condensate ⟨ αs π G a µν G aµν ⟩, which is simulated by a scale-invariance breaking logarithmic potential in the scalar dilaton field χ within the chiral effective model. In-medium effects of density, isospin asymmetry, and magnetic field, are incorporated through the number and scalar densities of nucleons in the solutions of the scalar fields σ, ζ, δ and χ. As described in section II, the additional effects of the magnetized Dirac sea are investigated in the present study and is considered to calculate the in-medium partial decay widths of charmonium going to open charm mesons, which may give rise to important phenomenological impacts in the production of these mesons as well as in the suppression of J/ψ. In chiral model, the scalar gluon condensate is expressed in terms of the fourth power of the scalar dilaton field in the limit of zero current quark masses. Thus introducing the mass shifts from the medium modifications of the scalar dilaton field χ, in the magnetized nuclear matter including the Dirac sea contribution, as with ∂ψ( ⃗ k) Here m c is the charm quark mass of 1.95 GeV [28]. m Ψ is the vacuum mass of Ψ(3770), ϵ = 2m c − m Ψ is its binding energy. The bound state wave function of Ψ(3770) is obtained using the harmonic oscillator potential model to calculate the mass shifts. The partial decay widths of Ψ(3770) → DD are calculated in the 3 P 0 model [21,54]. In the harmonic potential, the wave function of Ψ(1D) state is given for N = 1 and l = 2, as [54] N ′ is the normalization constant. L k p (z) is the associated Laguerre polynomial. β 2 = M ω/ℏ characterizes the strength of the harmonic potential. M = m c /2 is the reduced mass of charm quark-antiquark bound state. In eq.(17), ψ( ⃗ k) is the Fourier transform of the wave function in coordinate space, eq. (19) for N = 1 and l = 2. Thus, the Gaussian function multiplied by a polynomial is generated using the harmonic potential in the bound state problem of charm quark and antiquark, and used in the context of 3 P 0 model [21,54] which considers the internal structure of mesons as a quark-antiquark bound state to calculate the decay widths. The value of β for Ψ(3770) is taken to be 0.37 GeV by fitting from its root mean squared radius ⟨r 2 ⟩ = 1 f m 2 [68]. The sizes of the charmonium states are related to the strength of the harmonic potential β. Variation of β, hence of r rms with |eB| can be obtained from the mass shifts of the corresponding state, for Ψ(3770) it is given by [21,54]. The r.m.s radius of Ψ(3770) as a function of magnetic field is plotted in section (IV), from its mass shifts. The variation of r.m.s radii for the different states of charmonium and bottomonium have been studied in hadronic matter [21,25,54].

III. THE 3 P 0 MODEL
In this section we discuss the in-medium partial decay widths of (D * → Dπ,D * →Dπ), and (Ψ(3770) → DD), by taking into account the internal structures of the parent and the outgoing mesons using the 3 P 0 model [54]. into a D and π mesons is given by [37,51,54] In this expression, m D * is the mass of the parent D * meson. E D and E π are the energies of the outgoing D and π mesons, respectively, with m D , the in-medium mass of D meson and m π is the mass of π meson. p is the magnitude of the 3-momentum in the center of mass (c.o.m) frame.
In equation (20), γ is the coupling constant related to the strength of the 3 P 0 vertex [51,52].  [21,48,68]. Decay widths depend on the variable, x, which is the center of mass momentum, p in units of β avg , as a polynomial multiplied by an exponential term.
The in-medium partial decay widths of the charmed, strange mesons, D * ± s → D ± s π 0 can be determined on the same footing as D * mesons, from eq.(20) by using the corresponding masses of D s and D * s mesons, with the coupling parameter and the harmonic oscillator strengths taken to be the same as the D * mesons decays.

B. Decay of Ψ(3770) meson to DD
The hadronic decay width for the charmonium state, Ψ(3770) going to DD mesons is calculated in this subsection, using the 3 P 0 model. The in-medium masses of the parent and daughter particles are obtained within the chiral effective model, accounting for the effects of Dirac sea in the magnetized nuclear matter. The decay width is given by [37,54] Similar to the previous decay channel (D * → Dπ), γ Ψ signifies the probability for creating the light quark-antiquark pair, is chosen to be 0.33 [21], thus reproducing the observed decay widths  [11,16]. However in line with the argument given in [16], the correction due to the sea contribution in absence of magnetic field is very small as compared to the magnetized DS part, hence it is not considered here. The magnetic field-dependent sea part is responsible for the phenomenon of magnetic catalysis in vacuum and in specific case for nuclear matter, which is of concern in the present study. The field-independent sea contribution is of main concern in theory where it induces the chiral symmetry breaking effect in vacuum for coupling strength larger than a critical coupling for e.g., in the NJL model, which in our case is already incorporated by the nucleonic mass term through baryon-scalar meson interactions.
Finally the renormalized magnetized sea is independent of any renormalization scale (either magnetic field or nucleon mass) after minimization of the free energy. In this work, the AMM of the charged fermions is considered with the contribution of lowest Landau level in the single particle energies. The single particle energies of charged fermions incorporate the summation over all Landau levels for zero AMM while calculating the renormalized magnetized sea contribution. For non-zero AMM, however to avoid the complexity of the renormalization of the sea term and given the fact that the lowest Landau level is a perfectly valid approximation in the strong field limit, we restrict our calculations to the lowest level protons along the field axis. There is only ν = 0 level contribution for |eB| > 2m 2 π . Hence, the LLL approximation at finite AMM in the magnetized Dirac sea calculation can be adopted at finite density as well as in vacuum. It is shown in figure (1) by the variation of σ with |eB|, that the solutions corresponding to (with AMM+LLL) and (without AMM+sum over all Landau levels) coincide in the lower magnetic field region. Hence, our approach is justified in the low-field regime by the similarity between the solutions of the two cases. In [43,45,46], the two-particle Hamiltonian of the constituent quark model in a magnetic field has been constructed in the non-relativistic approach. The center-of-mass momentum of the system is not conserved due to the breaking of translational invariance by vector potential, instead one obtains a conserved quantity called pseudomomentum K. Particular choice of the potential, either harmonic or, Cornell potential form is considered in solving the eigenvalue equation for the relative Hamiltonian of the particle-antiparticle pair, with the corresponding wave functions and energy eigenvalues. Solving the Schrödinger equation for such system of particles with non-zero K lead to the quantization of the energy eigenvalues.
In some constituent quark model studies at finite magnetic field involving neutral heavy-light mesons and states of heavy-quarkonia, the pseudomomentum is also taken to be zero to solve the eigenvalue problem for mesons at rest [45,46]. In this case the Hamiltonian preserves by taking only the lowest Landau level corresponding to its point particle correction to the ground state energy [66,67]. The quantization of the energy eigenvalues for non-zero K, as expressed in [43] corresponding to harmonic potential for a bound state of quark-antiquark pair, for e.g., J/ψ, η c , involve the contributions of magnetic field through the minimal substitution (p → (p − eA)) and spin-magnetic field interaction. The spin-magnetic field interaction for the system of particle-antiparticle pair lead to the mixing between the spin-0 (pseudoscalar) and longitudinal component of spin-1 (vector) mesons [43,45,46] as well as non-zero shift in the energy of the transverse components (only for heavy-light system) [44,45]. mesons. However, the fluctuation in the heavy charm quark condensate ζ ′ c , is neglected in our present calculation [26]. The in-medium masses of the vector open charm and charmed, strange mesons, D * (D * ) and D * ± s are obtained by calculating their mass shifts from the mass shifts of the corresponding pseudoscalar mesons using equation (9). The additional effects of magnetic field due to the particle-like correction to the ground state mass, are considered on the masses of the electrically charged mesons, as given by equation (7) for the pseudoscalar meson (D ± , D ± s ), and by (8) (20). Due to the isospin considerations, the coupling constant γ, related to the 3 P 0 vertex for the channels D * ± → D ± π 0 , and D * + → D 0 π + (with its charge conjugate mode, D * − →D 0 π − ) is chosen to be 0.265 and 0.368, respectively. The value of γ for the D * ± s → D ± s π 0 channel is taken to be the same as for the D * ± → D ± π 0 channel. For D * 0 → D 0 π 0 it is 0.264 (including the charge conjugate channel). The decay of Ψ(3770) to field due to the point particle correction with LLL contribution in the ground state energy.
Around 4m 2 π , the effects of PV mixing becomes prominent for the charged mesons, unlike the neutral D mesons, where the PV mixing effect is significant at non-zero magnetic field.
For D ± mesons, the contribution of PV mixing prevents the monotonous increase in mass with magnetic field at around 10m 2 π where the mass plateaus. In figure 4, the masses of the pseudoscalar D + mesons and the vector D * + mesons are plotted with increasing magnetic field at nuclear matter saturation density ρ B = ρ 0 for symmetric matter (η = 0). The masses of these mesons are a consequence of the combined effects of the Dirac sea contribution, nuclear matter contribution in external magnetic field and the PV mixing effects. In 4a and 4c, the DS effects have not been considered. The D + and D * + meson masses increase with increasing magnetic field due to the point particle correction. Due to (D + − D * +|| ) mixing, the masses of pseudoscalar meson D + are lower as compared to the case when PV mixing is not considered. Similarly the masses for the vector D * + mesons are larger when PV mixing is considered as compared to when it is not considered. In 4b and 4d, the DS effects have been considered. There are significant changes in mass spectrum when the AMM of the nucleons are considered. Without considering the AMM of the nucleons, the masses of the mesons increase with increasing magnetic field due to magnetic catalysis (MC) along with the LLL contribution to the masses. However for non-zero nucleonic AMM at finite density matter, the mass of the mesons gradually increase (instead of the steep increase as mentioned earlier) till 10m 2 π and then decreases, as there is a competing effect due to inverse magnetic catalysis (IMC) against the point particle correction in this case. In both the above mentioned cases, PV mixing results in an overall decrease in mass for the pseudoscalar D + meson, and increase for the vector D * + meson.  contribution, lead to this slight increase in the masses. For the case of zero AMM, one obtains magnetic catalysis in magnetized nuclear matter, hence there obtained a sharp rise as compared to finite AMM case. The behavior of the masses of D s and D * s mesons due to spin-mixing are similar as that described in vacuum. Therefore, the observed behavior is a combined effect of (inverse) magnetic catalysis, point-particle correction and level repulsion due to spin-mixing of D s − D * || s mesons. Similar variation with slight change in the values are obtained for the asymmetric matter case. Clearly channel II has a greater decay width than channel I. This is due to the fact that channel I has D + as its daughter particle which gets mass contribution from the LLL and PV mixing. Channel II has D 0 as daughter particle and receives negative mass shift due to the PV mixing. Hence channel II has a greater decay width than channel I.   meson. These particles receive additional effect due to their point particle correction to the ground state energy in an external magnetic field. Besides, their vacuum mass is greater than the daughter particles D 0D0 corresponding to channel II. As such, the decay width for channel I is lower than channel II. At around 4m 2 π the sum of the product particles' masses in channel I becomes larger than the parent meson. Hence, at higher magnetic field, the decay width for this channel becomes zero. The total decay width (I+II) is plotted. Plot (b) shows the decay width for I and II when PV mixing is taken into account. For channel I, the decay width first decreases around 4m 2 π and then increases with magnetic field. This is due to the fact that initially at low magnetic field, the LLL contribution dominates the mass of D + D − . At higher magnetic field, the PV mixing dominates and the decay width increases. For channel II, the decay width increases till 4m 2 π , then decreases. At around 10m 2 π it increases again. In figure 15 the decay widths of Ψ(3770) → D + D − (I) and Ψ(3770) → D 0D0 (II) and the total decay width encompassing the two channels are plotted as function of |eB|/m 2 π at ρ B = ρ 0 at η = 0, 0.5 incorporating the Dirac sea effects. Figures 15a and 15c show the decay widths when PV mixing has not been considered, figures 15b and 15d show the π when AMM is not considered. Thus when AMM is considered, the masses of the neutral D undergo slight decrease in mass as compared to when AMM is not considered (discussed previously) and have slightly higher value of decay widths. The total decay width is obtained by adding the values of decay widths for individual channels. Fig.15c shows that the decay widths for channel I is greater than channel II around 2eB/m 2 π . For channel I, the decay widths   However, in non-central ultra-relativistic heavy ion collision experiments at RHIC, LHC, a strong magnetic field have been estimated to be produced at the early stages of collisions [1,2], which eventually decrease with time depending on the electrical conductivity of the produced medium and needs solutions of the magnetohydrodynamic equations [71]. The study of the in-medium hadronic properties under a static magnetic field can be executed if the distance over which the magnetic field varies considerably, is much bigger than the size of the hadrons. The effect of magnetic field on the dissociation rate of produced quarkonium in the central rapidity region of relativistic heavy ion collisions have been studied and it has been observed that the quarkonium dissociation energy increases with increasing magnetic field [71] and with quarkonium momentum. In recent heavy flavor measurements from ALICE in pp, p-Pb, Pb-Pb collision systems, production of charmed mesons are described by QCD-inspired model within uncertainties, in which ratio of different particle species, for e.g., of D s /D 0 is sensitive to the hadronisation mechanism in vacuum and in medium. In the present work, we have studied the Dirac sea contribution over a range of magnetic field from |eB| = 0 to 12m 2 π at ρ B = 0 and ρ 0 , which should cover the dynamics at the freeze-out hypersurface (FOHS) where the estimated magnetic field is low, around 0.07 GeV 2 [72], hence may give rise to observable impacts as illustrated here. For the D meson spectra in LHC, a larger suppression is observed at low to intermediate transverse momentum and is larger in the most centrality class, as compared to the semi-peripheral centrality class, due to the final state effects as measured by the nuclear modification factor in p−P b collisions. These results are compatible with model considering cold nuclear matter effects [70], hence the effects of magnetized matter on the properties of heavy flavor hadrons specifically on the charm mesons are of considerable phenomenological importance. The medium modified masses of the open charm mesons D 0 , D s as studies in our present work in the magnetic matter may thus have important observable impacts in the particle production ratio of D s /D 0 , from the recent measurements of relative abundances of such particle species in ALICE measurements [70].