QCD sum rule analysis of Heavy Quarkonium states in magnetized matter -- effects of (inverse) magnetic catalysis

The masses of the $1S$ and $1P$ states of heavy quarkonia are investigated in the magnetized, asymmetric nuclear medium, accounting for the Dirac sea effects, using a combined approach of chiral effective model and QCD sum rule method. These are calculated from the in-medium scalar and twist-2 gluon condensates, calculated within the chiral model. The gluon condensate is simulated through the scalar dilaton field, $\chi$ introduced in the model through a scale-invariance breaking logarithmic potential. Considering the scalar fields to be classical, the dilaton field, $\chi$, the non-strange isoscalar, $\sigma (\sim (\langle \bar u u\rangle +\langle \bar d d\rangle ))$, strange isoscalar, $\zeta (\sim \langle \bar s s\rangle)$ and non-strange isovector, $\delta (\sim (\langle\bar u u\rangle-\langle\bar d d\rangle)$) fields, are obtained by solving their coupled equations of motion, as derived from the chiral model Lagrangian. The effects of magnetic field due to the Dirac sea as well as the Landau energy levels of protons, and the non-zero anomalous magnetic moments of the nucleons are considered in the present study. In presence of an external magnetic field, there is also mixing between the longitudinal component of the vector meson and pseudoscalar meson (PV mixing) in both quarkonia sectors, leading to a rise (drop) of the masses of $J/\psi^{||}\ (\eta_c$) and $\Upsilon^{||}(1S)\ (\eta_b$) states. These might show in the experimental observables, e.g., the dilepton spectra in the non-central, ultra-relativistic heavy ion collision experiments at RHIC and LHC, where the produced magnetic field is huge.


I. INTRODUCTION
The study of the in-medium properties of hadrons is an important area of research in the physics of strongly interacting matter. The study of the heavy flavor hadrons [1] has attracted a lot of attention due to its relevance in the ultra-relativistic heavy ion collision experiments. Recently, heavy quarkonia (qq; q = c, b) under extreme conditions of matter i.e., high density and/or high temperature, have been investigated extensively. The medium created in the relativistic, high energy collisions affect the masses and decay widths of the produced particles and have significant observable impacts, e.g., the production and propagation of the particles. In the peripheral heavy ion collisions, strong magnetic fields are expected to be produced, at RHIC in BNL and LHC in CERN [2][3][4][5][6]. However, the time evolution of the produced magnetic field requires detailed knowledge of the electrical conductivity of the medium and proper treatment of the solutions of magneto-hydrodynamic equations [6], which is still an open question. The study of the effects of strong magnetic fields on the in-medium properties of hadrons has initiated a new area of research in the physics of heavy ion collisions. The heavy quarkonia are the bound states of a heavy quark (q = c or, b) and its antiquark. Charmonium (cc) and bottomonium (bb) states have been investigated in the literature using a variety of approaches, i.e., the potential models [7][8][9], the QCD sum rule approach [10][11][12][13][14][15][16][17][18][19][20][21], the coupled channel approach [22], quark-meson coupling model [23,24], a chiral effective model [25][26][27][28], and a field theoretic model for composite hadrons [29,30].
In-medium masses of the ground states of heavy quarkonium in a hadronic medium, have been studied extensively in the literature using the non-perturbative QCD sum rule (QCDSR) approach. In the isospin asymmetric hot nuclear medium, in-medium masses have been studied from the medium modified scalar and twist-2 gluon condensates, calculated within the chiral SU (3) model in terms of the scalar dilaton field χ and other scalar fields, in the absence of a magnetic field [12], and in presence of an external magnetic field [13,14]. At finite magnetic field, Landau energy levels of protons and anomalous magnetic moments (AMMs) of the nucleons contribute to the scalar fields through number (ρ p,n ) and scalar (ρ s p,n ) densities of the nucleons within the magnetized nuclear matter [13][14][15][16]. Thermal modifications of the S-waves bottomonium spectral functions have been investigated using QCDSR with the maximum entropy method [17]. The temperature effects were included through the gluon condensates, which were estimated from the finite temperature lattice QCD data. Investigation of the effects of finite temperature and baryon chemical potential on the mass of the charmonium states were performed by using the QCD perturbative (second-order stark effect) and non-perturbative sum rule methods [18]. The medium effects of temperature and density were incorporated through the gluon condensates calculated in a resonance gas model. In [30][31][32], the magnetically induced mixing between the pseudoscalar and (longitudinal component of) vector charmonium mesons (η c -J/ψ || ) have been investigated using a hadronic effective Lagrangian which leads to a level repulsion between the masses of J/ψ || and η c with increasing magnetic field. In [31], the mass shifts of η c and J/ψ also have been studied using QCDSR framework, considering the mixing effects through the current correlator in the phenomenological side. The OPE side contained the effects of magnetic field in terms of its operator expectation value and the vacuum scalar gluon condensate term up to dimension-4. In the chiral effective model, mass shifts of the heavy quarkonia are obtained through the modifications in the scalar gluon condensates [33], given in terms of the medium modified scalar dilaton field χ, within the chiral SU(3) model [25,27].
The enhancement of the light quark condensates with increasing magnetic field, is called magnetic catalysis [34][35][36][37]. In the literature, this effect has been studied to a large extent on the quark matter sector using the Nambu-Jona-Lasinio (NJL) model [38][39][40]. In reference [41], magnetic catalysis (MC) have been studied in the context of nuclear matter, through the contributions of magnetized Dirac sea within the Walecka model and an extended linear sigma model. Magnetic catalysis have been observed through the rise in the scalar field σ(∼ ⟨qq⟩) with magnetic field in the vacuum, for zero AMM of the nucleons. As a consequence, the effective nucleon mass, m * N = m N − g σN σ, increases with magnetic field in the vacuum (m N is the vacuum mass of the nucleon and g σN , the σ-nucleon coupling constant in the Lagrangian). In [42], the effects of (inverse) magnetic catalysis have been studied using the weak-field approximation of fermion propagator. The critical temperature of vacuum to nuclear matter phase transition decreases with magnetic field, for the nonzero AMMs of the nucleons, implying an inverse magnetic catalysis (IMC) [43]. The zero value for AMM leads to an opposite behavior, namely to the magnetic catalysis. In the literature there are few works related to the effect of IMC/MC on hadronic properties in the nuclear matter.  [44][45][46], and using the generalized version of the chiral effective model, both in the absence of an external magnetic field [47,48], and in presence of a magnetic field [49,50]. The open heavy flavor mesons have their mass modifications in terms of both the light quark condensates (because of the light quark flavor present in their quark structure) as well as the gluon condensates simulated within the chiral model by the scalar dilaton field χ. The in-medium masses of the light vector mesons (ρ, ω, ϕ) have been studied within the QCD sum rule approach [51]. The medium modifications of the masses obtained from the non-strange (⟨qq⟩; q = u, d for ρ, ω) and strange (⟨ss⟩ for ϕ) light quark condensates and the scalar gluon condensates (∼ ⟨G a µν G aµν ⟩), calculated within the chiral SU (3) model, in the strange asymmetric matter in absence of magnetic field [52], and in the magnetized nuclear medium [53].
In the magnetized nuclear matter, the mass modifications of the 1S (J/ψ), 2S (ψ(3686)) and 1D (ψ(3770)) states of charmonium have been studied in terms of the medium modifications of the scalar dilaton field χ, which in turn mimics the gluon condensates of QCD within the chiral effective model [25]. For the charmonium states, there observed to be a mass drop as the density increases beyond the nuclear matter saturation density ρ 0 , for different values of magnetic field, |eB| and isospin asymmetry parameter, η. The dominant effect was coming from the nuclear matter density as compared to the effects from the magnetic field. In [54], the mass shifts of these charmonium states due to the change in the gluon condensate have been calculated using the perturbative QCD approach, to the leading order in density till ρ 0 . The mass shifts obtained in [25], calculated for any baryonic density within the chiral effective model, agrees with the results of [54] in the linear density approximation of −8, −100 and −140 (in MeV) for J/ψ, ψ(3686) and ψ(3770), respectively at ρ B = ρ 0 and |eB| = 0. In-medium masses of the open charm and charmonium mesons have also been studied in the magnetized strange hadronic matter [28]. The PV mixing between the longitudinal component of vector and pseudoscalar open charm (D * || − D) mesons [55] and S-waves of charmonia [30,55] have been studied using a hadronic effective Lagrangian. The in-medium masses thus obtained have been used to study the in-medium hadronic decay widths for D * → Dπ [55] and of ψ(3770) → DD [30,55], accounting for the lowest Landau energy level contributions for the charged D mesons. In [56], the spin-magnetic field interaction between B − B * and η b (4S) − Υ(4S) have been studied using a Hamiltonian approach [57] in the presence of an external magnetic field. The in-medium partial decay widths of Υ(4S) going to BB have been studied using a field theoretic model for composite hadrons with quark (and antiquark) constituents [56]. The in-medium decay widths of charmonium states to DD in the magnetized nuclear matter have also been studied using a light quark-antiquark pair creation model, namely the 3 P 0 model [58]. The spin-magnetic field interaction between the spin-singlet and longitudinal component of the spin-triplet states have been studied using a Hamiltonian formalism of [57] on the 1S states of heavy quarkonia (η c − J/ψ) and (η b − Υ(1S)), which lead to a rise (drop) in the mass of the J/ψ || (η c ) and Υ(1S) || (η b ) with increasing magnetic field [15,16,57]. The effects of (inverse) magnetic catalysis due to the Dirac sea contribution at finite |eB|, have not been considered in the studies mentioned above. The studies of the magnetic field modified hadronic properties, specifically of the heavy flavor mesons have important observable consequences, such as in the formation time of heavy quarkonia, the particle production ratio, etc. in the non-central ultra relativistic heavy ion collision experiments [59,60].
In the present work, we study the in-medium masses of the 1S-wave (vector, J/ψ, pseudoscalar, η c ) and 1P -wave (scalar, χ c0 , axial-vector, χ c1 ) charmonium states as well as the 1S-wave (vector, Υ(1S), pseudoscalar, η b ) and 1P -wave (scalar, χ b0 , axial-vector, χ b1 ) bottomonium states, in a magnetized isospin asymmetric nuclear medium, using the QCD sum rule approach, by incorporating the effects of Dirac sea in presence of an external magnetic field. At finite magnetic field, effects of the pseudoscalar-vector (PV) mixing between the pseudoscalar η c (η b ) and longitudinal component of vector J/ψ || (Υ || (1S)) mesons are studied in both quarkonia sector, accounting for the Dirac sea effects on their in-medium masses.
The outline of the paper is: in section II, the chiral effective model is described briefly to calculate the medium modified gluon condensates. Section III illustrates the QCD Sum Rule framework to calculate the in-medium masses of the lowest lying states of heavy quarkonia.
Mass shifts of the S-wave states due to the pseudoscalar-vector (PV) mesons mixing are introduced in presence of a magnetic field. In section IV, results of the magnetized Dirac sea effects are discussed. Section V summarizes the findings of the present work.
In-medium masses of the quarkonium ground states are computed within the QCD sum rule approach, in terms of the scalar and twist-2 gluon condensates. In the present study, these condensates are calculated within an effective chiral hadronic model [61]. The chiral model is based on the non-linear realization of chiral SU (3) L × SU (3) R symmetry [62][63][64], and the broken scale invariance of QCD [61,65,66]. The QCD scale-invariance breaking is incorporated through a logarithmic potential in the scalar dilaton field χ [67], in the model.
The chiral model Lagrangian density has the following general form [61], L scale−break is the QCD scale symmetry breaking logarithmic potential; the explicit symmetry breaking term is L SB ; finally the magnetic field effects on the charged and neutral baryons in the nuclear medium are given by [25,26,49,50,[68][69][70][71], where, ψ i is the baryon field operator (i = p, n, in case of nuclear matter), the parameter, κ i is related to the anomalous magnetic moment of the i-th baryon, κ p = 3.5856 and κ n = −3.8263, are the gyromagnetic ratio corresponding to the anomalous magnetic moments (AMM) of the proton and the neutron respectively [68][69][70][71][72][73][74][75]. In the magnetized nuclear medium there are contributions from the protons Landau energy levels [72], and the nucleons anomalous magnetic moments [72,75], to the number and scalar densities (ρ i , ρ s i , i = p, n, respectively) of the nucleons [49,50]. In the current study, the Dirac sea contributes to the scalar densities of nucleons at finite magnetic field, including the effects of the anomalous magnetic moments of nucleons within the chiral SU (3) model. One-loop self energy functions of the nucleons are evaluated through summation over the scalars (σ, ζ and δ) and vectors (ρ and ω) tadpole diagrams, using the weak-field expansion of the nucleonic propagator [42], accounting for the AMMs of nucleons, within the chiral effective model.
The meson fields of the chiral model Lagrangian are treated as the classical fields, whereas the nucleons as the quantum fields in the evaluation of the Dirac sea contribution to the scalar fields. The scalar dilaton field, χ simulates the scalar gluon condensate ⟨ αs π G a µν G aµν ⟩, as well as the twist-2 gluon operator ⟨ αs π G a µσ G aσ ν ⟩, within the model. The energy momentum tensor, T µν derived from the χ-dependent terms in the chiral model Lagrangian density is thus [12] The QCD energy momentum tensor, in the limit of finite current quark mass contains a symmetric trace-less part and a trace part [76], with the leading order QCD β function [12], β QCD (g) = − g 3 (4π) 2 (11 − 2 3 N f ), for three color quantum numbers of QCD, and N f = 3 number of quark flavors. Here, m i 's (i = u, d, s) are the current quark masses. The medium expectation value of the twist-2 gluon operator is, where u µ is the 4-velocity of the nuclear medium taken to be at rest in the present investigation, namely, u µ = (1, 0, 0, 0). The energy momentum tensor of QCD is thus [12] T Comparing the expressions of energy momentum tensor from equations (6) and (3), the expressions for G 2 (the twist-2 component) and the scalar gluon condensate are given by multiplying both sides with u µ u ν − gµν 4 and g µν respectively. These are given by and [77], The expectation values of the scalar and the twist-2 gluon condensates in magnetized nuclear medium, depend on the in-medium values of the non-strange scalar-isoscalar field σ, the strange scalar-isoscalar field ζ, the non-strange scalar-isovector field δ and the scalar dilaton  i of the i-type of quarkonium ground state (i= vector, pseudoscalar, scalar, and axial-vector) is given as [78], Where M i n is the n-th moment of the i-type meson state and m q (q = c, b) is the running heavy quark mass dependent on the renormalization scale ξ. Using the operator product expansion technique [OPE], the moment can be written as [10,78], Where A i n , a i n , b i n , and c i n are the Wilson coefficients. The coefficients, A i n result from the bareloop diagram of perturbative QCD, a i n are the contributions of the perturbative radiative corrections, and b i n are related to the scalar gluon condensate through By substituting the expression for the scalar gluon condensate from equation (8), The coefficients c i n are associated with the twist-2 gluon condensates as which using equation (7), reduces to The ξ-dependent parameters m q (q = c, b) and the running coupling constant α s are [12,78] [12,79], and with , and n f = 4, α s (4m 2 c ) ≃ 0.23 in the charm quark sector, and n f = 5, α s (4m 2 b ) ≃ 0.15 in the bottom quark sector [79]. The Wilson coefficients, A i n , a i n and b i n are given in [78] for different J P C quantum numbers of states, for e.g., the pseudoscalar, vector, scalar, axial-vector channels. The c i n s' are listed for the vector and pseudoscalar (1S states) channels in [10], for the 1P -wave states (scalar and axial-vector) c i n s' are calculated using a background field technique [20]. At finite magnetic field, mixing of the pseudoscalar (P ≡ η c (1S)) and vector (V ≡ J/ψ) charmonium states are considered through the interaction Lagrangian [30-32, 46, 55] where m av = (m V + m P )/2, m P and m V are the masses for the pseudoscalar and vector charmonium states,F µν is the dual electromagnetic field strength tensor. In equation (17), the coupling parameter g P V is fitted from the observed value of the radiative decay width, Where The effective Lagrangian given by equation (17) is observed to lead to the mass modifications of the longitudinal component of J/ψ and η c states in the presence of magnetic field. In equation (19), the effects of the magnetized Dirac sea, the Landau quantization of protons and the AMMs of the nucleons are incorporated through the in-medium values of m P,V . The in-medium masses of m P , m V are calculated using the QCDSR approach. Effects of the spin-magnetic field interaction have been studied for the S-wave states of heavy quarkonia [15,16,56,57,80]. This leads to a level repulsion between the mass eigenstates of the spin-0 and longitudinal component of the spin-1 states. The interaction leads to a mixing effect between (Υ(1S)−η b ). Thus, the effective masses of Υ || (1S) and η b , accounting for the mass shifts due to the spin-magnetic interaction Hamiltonian (−⃗ µ. ⃗ B) [57], In the above equation, m * Υ(1S)/η b denotes the in-medium masses of the 1S-wave bottomonium states calculated within QCDSR, accounting for the Dirac sea effects. ∆m sB is the mass shift due to the spin-magnetic field interaction, given by , and g is chosen to be 2 (ignoring the anomalous magnetic moments of the bottom quark (anti-quark)). The Hamiltonian approach is taken to study the spin-mixing effects in the bottomonium sector (unlikecc), due to the lack of experimental data on the bottomonium radiative decay width, Υ(1S) → η b γ.

A. Charmonium states
In this subsection, the results for the in-medium masses of the lowest S-wave: J/ψ ( 3 S 1 ), η c ( 1 S 0 ) and P -wave: χ c0 ( 3 P 0 ), χ c1 ( 3 P 1 ), states of charmonium are discussed in the presence of magnetized asymmetric nuclear matter, accounting for the effects of the magnetized Dirac sea (denoted as DS). In the sum rule approach, masses are obtained by calculating the moments (M i n ) for all the four meson currents: vector ( 3 S 1 ), pseudoscalar ( 1 S 0 ), scalar ( 3 P 0 ) and axial-vector ( 3 P 1 ). The moments M i n , are given in terms of the perturbative Wilson coefficients and the non-perturbative gluon condensates of QCD, as given by equation (10).
Wilson coefficients are calculated for different J P C quantum numbers of the currents and are independent of the medium effects [10,20,78]. The mass formula in the sum rule framework [equation (9)], depends on the running charm quark mass, m c (ξ), and the running coupling constant, α s (ξ), which are functions of the renormalization scale ξ, given by equations (15) and (16) respectively. The scalar gluon condensate ⟨ αs π G a µν G aµν ⟩ is connected to the ϕ b term, whereas, the twist-2 gluon condensate G 2 to the ϕ c term, which incorporate the medium effects in the mass calculation. The gluon condensates are obtained from the chiral effective model [equations (7) and (8) with increasing magnetic field, accounting for the Dirac sea contribution, as shown in figures (2)-(3). In figure 2, ⟨ αs π G a µν G aµν ⟩ 1/4 , corresponding to the scalar gluon condensate, in units of MeV is plotted as a function of magnetic field |eB| in units of m 2 π , at ρ B = 0, ρ 0 for symmetric and asymmetric nuclear matter, including the effects of the Dirac sea at finite of G 2 [equation (7)] are different for S waves with ξ = 1 [10] and P waves with ξ = 2.5 [20] states of charmonium. In figure (3), the variation in the twist-2 gluon condensate G 2 is coming from that of the scalar dilaton field χ and other scalar fields with respect to |eB|, following equation (7) Lagrangian approach, accounting for the additional effects of Dirac sea to the masses. The effective masses of J/ψ || and η c thus given by equation (19). The nuclear matter saturation density, ρ 0 is taken to be 0.15 fm −3 in the present work [61].
The value of the renormalization scale, ξ = 1 is chosen for the S-wave [10] charmonium states and ξ = 2.5 [20] for the P -wave charmonium states, to study their respective in-medium values of the twist-2 gluon condensate, G 2 is zero, and of ⟨ αs π G a µν G aµν ⟩ is (373.02 MeV) 4 at vacuum. The vacuum masses of χ c0 and χ c1 are obtained as 3720.95 MeV and 3878.55 MeV, respectively. The mass shifts of J/ψ at ρ B = ρ 0 and |eB| = 0 of −4.21 MeV calculated in this work can be compared with −8 MeV of mass shifts obtained using the leading order QCD formula (similar to second-order Stark effect), in the linear density approximation [54].
For any particular state (i = vector, pseudoscalar, scalar and axial-vector) of heavy quarkonia, the quantity m * i , in equation (9), in terms of the ratio of two consecutive moments and parameters involving the renormalization scale, ξ, can be varied as a function of n. The minimum value of m * i corresponds to the physical mass of the state. In our present study, the ratio of two consecutive moments (M i n−1 (ξ)/M i n (ξ)) in the moment sum rule approach is adopted to obtain the mass of the lowest lying resonance [78]. For higher values of n, the effects of the higher lying resonances and continuum can be neglected. However, for large values of n, the incorporation of the higher dimensional operators in the OPE side do not let the first order perturbation theory to hold. To minimize the contributions of the higher dimensional operators, the value of Q 2 0 and hence of ξ must be chosen nonzero [79]. Proper choice of ξ sets a range of values for n, which saturates the phenomenological side with the lowest lying resonance [79]. This range is called the stability region in n which changes with ξ. For small values of n, breakdown occurs in the stability region due to the contribution of the higher lying resonances. In our calculations, the stability region of n, for the 1S and 1P states of charmonium and bottomonium are in accordance with the findings of [10,20,78,79]. In figure 4, the m * i are plotted as functions of n for the four lowest lying states of charmonium, at ρ B = 0, ρ 0 (η = 0) for eB = 0. In this figure, the physical mass of the 1S-waves vector (J/ψ) and pseudoscalar (η c ) states of charmonium are obtained at n = 8 and n = 9, respectively for ξ = 1. For both the 1P -waves scalar (χ c0 ) and axial-vector (χ c1 ) states, it is obtained at n = 9.
At zero density and finite magnetic field, only Dirac sea effect is there with no protons' Landau level contribution, in the absence of matter part. The masses are calculated by considering the nonzero anomalous magnetic moments (AMMs) of the nucleons and compared to the case when AMM is taken to be zero. At finite density matter, the scalar fields are and 1P wave states of charmonium, taking into account the PV mixing effect of (J/ψ || − η c ) for the 1S states in (6). The parameter g P V ≡ g ηcJ/ψ in the phenomenological Lagrangian (17), is evaluated to be 2.094 from the observed radiative decay width of Γ(J/ψ → η c γ) in vacuum, 92.9 keV [81], using equation (18). This effect leads to the rise (drop) in the masses of J/ψ || (η c ) states with magnetic field as shown in figures (5) and (6)  η c is identified as a level repulsion between the states with increasing magnetic field. The level repulsion occurred due to the magnetically induced PV mixing between J/ψ || and η c using an effective hadronic interaction Lagrangian [30][31][32]. In the QCD sum rule approach, the magnetically induced PV mixing effects have been incorporated through the current correlator in the phenomenological side and the magnetic field effects through the operator expectation value in the OPE side, leading to the level repulsion between J/ψ || − η c with increasing magnetic field [31,32]. The level repulsion from the two approaches of QCDSR and the hadronic effective Lagrangian [in the 2 nd order of |eB| and leading order in ( m V −m P 2mav ) of equation (19)], have been observed to be in good agreement in the weak-field region (below |eB| ∼ 0.1 GeV 2 ) with a slight deviation as |eB| increases further [31,32]. The vacuum masses of J/ψ and η c were found to be 3.092 GeV and 3.025 GeV, respectively, for ⟨ αs π G 2 ⟩ of (0.35 GeV ) 4 , using the Borel transform of sum rule where the Borel curves indicated the level repulsion between J/ψ and η c at |eB| = 0 and 5m 2 π , accounting for the phenomenologically incorporated mixing effects in the spectral ansatz of QCDSR [31,32]. In the present work, the vacuum masses for J/ψ and η c are obtained as 3.196 GeV and 3.067 GeV by using ⟨ αs π G 2 ⟩ of (0.37 GeV ) 4 from the chiral SU (3) model. An increase (decrease) in the energy levels of the longitudinal component of J/ψ (η c ) with magnetic field has been studied due to the spin-magnetic field interaction Hamiltonian within a Cornell potential model [57]. In an external magnetic field, the quarkonia have a conserved pseudomomentum instead of a conserved center-of-mass momentum. The spin-mixing also lead to the suppression of J/ψ decays to lepton pairs and turn on decays of η c state which should experimentally be realized as a reduction in the dilepton yields of J/ψ and the appearance of a peak at m ηc in the dilepton spectrum. In [57], an approximate suppression of 11% of the J/ψ decays have been predicted.
The in-medium masses of the J/ψ, η c , χ c0 , and χ c1 states of charmonium (decrease) increase with increasing magnetic field at ρ 0 , due to the (inverse) magnetic catalysis effect (without PV mixing effect of the 1S waves), for the case of (nonzero AMMs) zero AMM of the nucleons. In figure (6), the masses of J/ψ (η c ) increase (decrease) with magnetic field, both with and without AMMs of the nucleons in the DS contribution, when the PV mixing effect is considered between (J/ψ-η c ). Although the rate at which mass rises (drops) is  [14], at ρ B = ρ 0 , T = 0, in the symmetric nuclear matter, using the Borel sum rule. The study of the mass shifts of charmonia due to the medium modifications of the gluon condensates in hot and dense hadronic matter, lead to the downward mass shifts of J/ψ within the QCD perturbative (second-order stark effect) approach [18]. The mass shifts calculated using the Borel transformed QCD sum rule lead to approximately twice as large mass shifts for the χ c1 state than for the mass shifts of J/ψ [18], which can be compared with the mass shifts of −9.14 MeV for χ c1 which is nearly twice as much as −4.21 MeV for the J/ψ calculated at ρ B = ρ 0 , T = 0 and |eB| = 0, in the present work. The gluon condensates were calculated using a hadronic resonance gas model as compared to the chiral effective model used in the present study.
In figure 8, the in-medium masses of (a) η c , (b) J/ψ, (c) χ c0 and (d) χ c1 charmonium states are plotted as functions of the magnetic field, at ρ B = 0, ρ 0 for η = 0, 0.5 to show the density effects on the masses, accounting for the effects of Dirac sea. In ref. [9], the charmonia spectra (in MeV) of 2980.3 (η c ), 3097.36 (J/ψ), 3415.7 (χ c0 ) and 3508.2 (χ c1 ) were obtained perturbatively, in a potential model. The values of the QCD gluon condensates, ⟨ αs π G a µν G aµν ⟩ and G 2 (for 1S and 1P states) at ρ 0 , η = 0 and with AMM, are given respectively, as: (371. π , for ⟨ αs π G a µν G aµν ⟩ and G 2 of 1S and 1P states, respectively. The masses plotted incorporating the effects of magnetized Dirac sea, are denoted as "with DS, with AMM" when the anomalous magnetic moments of the nucleons are considered and "with DS, w/o AMM" for zero AMM. The different behavior of the pseudoscalar meson mass with magnetic field in the absence of nuclear matter, can be attributed to the variation in the Wilson coefficients for different channels.

B. Bottomonium states
In this section, the results for the in-medium masses of the bottomonium ground states, namely the 1S-wave: Υ(1S) ( 3 S 1 ), η b ( 1 S 0 ) and 1P -wave: χ b0 ( 3 P 0 ), χ b1 ( 3 P 1 ), are illustrated in the magnetic asymmetric nuclear matter with the additional contribution from the magnetized Dirac sea. The masses are calculated using the QCD sum rule approach in a similar fashion as described above for the corresponding 1S and 1P waves states of charmonium.
The running bottom quark mass, m b (ξ) and the running coupling constant, α s (ξ), as given by equations (15) and (16) respectively, are different from the charmonium states, as the number of current quark flavors is n f = 5 for bottom quark, which is n f = 4 for the charm quark, also the ξ-independent values of the quark mass and couplings (in equations (15) and (16), respectively) are different in the two sectors of charm and bottom quarks. The scalar gluon condensate, ⟨ αs π G a µν G aµν ⟩ through the ϕ b term and the twist-2 gluon condensate, G 2 in ϕ c term, incorporate the effects of density, magnetic fields and isospin asymmetry of the nuclear medium on the bottomonium masses. In the present investigation, we have studied the values with increasing magnetic field are found to be similar in both sectors. Therefore, using the values of ϕ b and ϕ c , the in-medium masses of the bottomonium ground states are calculated using equation (9). In presence of an external magnetic field, the spin-magnetic field interaction are studied for the 1S states of bottomonium using a Hamiltonian approach [57]. The effective masses of Υ || (1S) and η b , taking into account the spin-mixing effect, are computed using equation (20).   [17]. The peak positions are at higher energies than the vacuum masses of these states. However, it is due to the contribution of the excited states along with the ground state in the spectral density function [17], which can also be the case in "pole+continuum" assumption of the usual sum rules where the mass of the ground state are obtained at higher values of energy than the actual vacuum mass of the state [17]. In  [14], by incorporating the magnetic field effects due to the Landau quantization of protons and AMMs of the nucleons. In [14], the Borel sum rule has been employed to study the in-medium mass of the lowest lying heavy quarkonia in hot magnetized nuclear matter.
The effects of baryon density as well as of magnetic fields are shown to the masses of (a) . The rise (drop) in the Υ || (1S)(η b ) mass with increasing magnetic field should manifests in the suppression of dilepton decays of Υ(1S) state and a profuse production of lepton pairs from η b state due to the states' spin-mixing. In [57], approximately 2.8% suppression of dilepton decays of Υ(1S) have been predicted which should instead appear as a peak in the dilepton spectrum at the invariant mass of m η b . However, due to the given finite resolution of the detector, it may be experimentally difficult to resolve this properties of 1S bottomonium states in the dilepton invariant mass spectrum [57]. The spectra of heavy quarkonia have been investigated using a potential model [9], consisting of a relativistic kinetic energy term, a linear confining potential with scalar and vector relativistic corrections as well as the perturbative one-loop QCD short distance potential. The masses are calculated using a variational technique. The bottomonia mass spectra (in MeV) are given as 9413.7 (η b ), 9460.69 (Υ(1S)), 9861.12 (χ b0 ) and 9891.33 (χ b1 ) [9].
The general pattern in the mass variation are of similar kind in the charmonium and bottomonium sectors with change in the density and as well as in the magnetic fields, accounting for the (inverse) magnetic catalysis effect, with (without) the PV mixing for the 1S states. The effects of magnetic field to the in-medium masses of the heavy quarkonia within the magnetized nuclear medium, are observed to be much more prominent when taken through the Dirac sea contribution for zero and finite baryon density. The magnetic field effects, as seen from the figures, are almost negligible (without PV mixing of S waves) when Dirac sea contribution is not taken into account. The in-medium masses accounting for the PV mixing effect in the 1S-wave states ofcc andbb mesons, get modified considerably at finite density matter due to AMMs of the nucleons through the Dirac sea. Incorporation

V. SUMMARY
In the summary, the in-medium masses of the 1S and 1P waves states of charmonium and bottomonium are studied using the QCD sum rule approach, in the magnetized nuclear medium, accounting for the Dirac sea effects. The medium effects are incorporated through the scalar and twist-2 gluon condensates, calculated in terms of the medium modified scalar dilaton field, χ, and other scalar fields (σ, ζ, δ), within the chiral effective model. The effects of magnetic field come from the Landau energy levels of protons and the non-zero anomalous magnetic moments of the nucleons, in the magnetized nuclear matter. In the current work, the contribution of the magnetized Dirac sea is incorporated through the scalar densities of nucleons within the chiral effective model. Appreciable modifications in the condensates are obtained due to the Dirac sea effect along with the Landau level contribution of protons, in comparison to the case when Dirac sea effect is not considered. At zero density, the contribution of magnetic field is realized only through the magnetized Dirac sea, with no effect from the Landau energy levels of protons. The nonzero anomalous magnetic moments (AMMs) of protons and neutrons have noticeable effects on the in-medium masses, through the magnetized Dirac sea. In-medium masses of the charmonium and bottomonium ground states, accounting for the Dirac sea effects, are observed to decrease with increasing magnetic field, at finite density matter (ρ B = ρ 0 ), when AMMs are non-zero, but show opposite behavior (increasing mass with |eB|) for zero AMMs of the nucleons. The magnetic fields thus have significant contribution on the in-medium properties (masses and hence on the decay widths) of heavy quarkonia due to the effects of (inverse) magnetic catalysis. This