Dirac sea effects on Heavy Quarkonia decay widths in magnetized matter -- a field theoretic model of composite hadrons

We study the partial decay widths of charmonium (bottomonium) states to ${\rm D\bar D \; (B\bar B)}$ mesons in magnetized (nuclear) matter using a field theoretical model of composite hadrons with quark (and antiquark) constituents. These are computed from the mass modifications of the decaying and produced mesons within a chiral effective model, including the nucleon Dirac sea effects. The mass modifications of the open charm (bottom) mesons are calculated from their interactions with the nucleons and the scalar mesons, whereas the mass shift of the heavy quarkonium state is obtained from the medium change of a scalar dilaton field, $\chi$, which mimics the gluon condensates of QCD. The Dirac sea contributions are observed to lead to a rise (drop) in the quark condensates as the magnetic field is increased, an effect called the (inverse) magnetic catalysis. These effects are observed to be significant and the anomalous magnetic moments (AMMs) of the nucleons are observed to play an important role. For $\rho_B$=0, there is observed to be magnetic catalysis (MC) without and with AMMs, whereas, for $\rho_B=\rho_0$, the inverse magnetic catalysis (IMC) is observed when the AMMs are taken into account, contrary to MC, when the AMMs are ignored. In the presence of a magnetic field, there are also mixings of spin 0 (pseudoscalar) and spin 1 (vector) states (PV mixing) which modify the masses of these mesons. The magnetic field effects on the heavy quarkonium decay widths should have observable consequences on the production the heavy flavour mesons, which are created in the early stage of ultra-relativistic peripheral heavy ion collisions, at RHIC and LHC, when the produced magnetic fields can still be extremely large.

A study of the mass modifications of the charmonium states due to the gluon condensates as well asDD meson loop [59] showed that the dominant contributions are due to the medium modifications of the gluon condensates. In a chiral effective model, the in-medium masses of the heavy quarkonium (charmonium and bottomonium) have been computed from the medium change of a scalar dilaton field [50,51,55], which simulates the gluon condensates of QCD within the effective hadronic model. The chiral effective model, in the original version with three flavours of quarks (SU(3) model) [60][61][62][63], has been used extensively in the literature, for the study of finite nuclei [61], strange hadronic matter [62], light vector mesons [63], strange pseudoscalar mesons, e.g. the kaons and antikaons [64][65][66][67] in isospin asymmetric hadronic matter, as well as for the study of bulk matter of neutron stars [68]. Within the QCD sum rule framework, the light vector mesons [69,70], as well as, the heavy quarkonium states [16][17][18], in (magnetized) hadronic matter have been studied, using the medium changes of the light quark condensates and gluon condensates calculated within the chiral SU(3) model. Using the in-medium masses of the heavy flavour mesons in the (magnetized) hadronic matter, calculated within the chiral effective model, the partial decay widths of the heavy quarkonium states to the open heavy flavour mesons have been studied in (magnetized) hadronic medium [51,71], using a light quark-antiquark pair creation model [72], namely the 3 P 0 model [73][74][75][76] as well as using a field theoretical model for composite hadrons with quark (and antiquark) constituents [77][78][79][80][81]. The effects of magnetic field on the masses of the heavy flavour mesons have been studied in Refs. [82][83][84][85][86][87][88][89], and, it is observed that the spin-magnetic field interaction leads to mixing  [84][85][86][87][88][89]. In the presence of a magnetic field, the studies of the effects of Dirac sea (DS) in the quark matter sector [90][91][92][93] within the Nambu-Jona-Lasinio model [94][95][96], are observed to lead to enhancement of the light quark condensates with increase in the magnetic field, an effect called the magnetic catalysis (MC). The opposite trend of the light quark condensates with magentic field, namely the inverse magnetic catalysis (IMC) is observed in some lattice QCD calculations [97], where the crittical temperature, T c is seen to decrease with increase in the magnetic field. For the nuclear matter, the effects of Dirac sea (DS) have been studied using the Walecka model as well as an extended linear sigma model in Ref. [98]. These are observed to lead to magnetic catalysis (MC) effect for zero temperature and zero density, which is observed as a rise in the effective nucleon mass with the increase in magnetic field. In Ref. [99], the contributions of Dirac sea of the nucleons to the self-energies of the nucleons have been studied in the Walecka model by summing over the scalar (σ) and vector (ω) tadpole diagrams, in a weak magnetic field approximation of the fermion propagator. At zero density, the effects of the Dirac sea are seen to lead to magnetic catalysis (MC) effect at zero temperature [99]. When the anomalous magnetic moments (AMMs) of the nucleons are taken into account, at a finite density and zero temperature, there is observed to be a drop in the effective nucleon mass with increase in the magnetic field. This behaviour with the magnetic field is observed when the temperature is raised from zero to non-zero values, upto the critical temperature, T c , when the nucleon mass has a sudden drop, corresponding to the vacuum to nuclear matter phase transition. The decrease in T c with increase in value of B is identified with the inverse magnetic catalysis (IMC) [99].
In the present work, the partial decay widths of the charmonium (bottomonium) states to open heavy flavour mesons, DD(BB) are studied in magnetized (nuclear) matter using a field theoretical model of composite hadrons. As the matter created in ultra-relativiistic peripheral heavy ion collisions is dilute, we study the partial decay widths of the lowest quarkonium states in the charm and bottom sectors, ψ(3770) and Υ(4S) (which decay to DD and BB in vacuum). These are investigated for ρ B = 0 as well as for ρ B = ρ 0 , the nuclear matter saturation density, for symmetric as well as asymmetric nuclear matter in the presence of an external magnetic field. The study of effects of temperature on the open charm and charmonium masses (and hence on the charmonium decay widths) [50,51] have been observed to be marginal for small densities (upto ρ 0 ). Within the chiral effective model, the mass shift of the heavy quarkonium states and the open heavy flavour mesons arise from the medium modifications of the dilaton field and the scalar fields, which have marginal modifications due to temperature, and, hence the temperature effects on the quarkonium decay widths (due to mass modification of these mesons) are not taken into account in the present study. The magnetic effects are the most dominant effects for the (dilute) matter resulting from ultra-relativistic peripheral collisons, which include the contributions from the magnetized Dirac sea of nucleons as well as PV mixing, in additon to the Landau level contributions for the charged hadrons. In the chiral effective model, the effects of the Dirac sea are incorporated to the nucleon propagator, through summation of scalar (σ, ζ and δ) and vector (ω and ρ) tadpole diagrams. When the anomalous magnetic moments (AMMs) of the nucleons are not taken into account, for zero density as well as for ρ B = ρ 0 , magnetic catalysis (MC) is observed. However, when the AMMs of nucleons are considered, for ρ B = ρ 0 (both for symmetric and asymmetric nuclear matter), inverse magnetic catalysis (IMC) is observed, i.e., the quark condensate is observed to be reduced with rise in the magnetic field.
The outline of the paper is as follows. In section II, we describe briefly the chiral effective In the presence of a magnetic field, the Lagrangian for SU(3) model has the form [100] where L kin refers to the kinetic energy terms of the baryons and the mesons, L BW is the baryon-meson interaction term, L vec describes the dynamical mass generation of the vector mesons via couplings to the scalar mesons and contain additionally quartic self-interactions of the vector fields, L 0 contains the meson-meson interaction terms, L scalebreak is the scale invariance breaking term and L SB describes the explicit chiral symmetry breaking. The kinetic energy terms are given as where, B is the baryon octet, X is the scalar meson multiplet, Y is the pseudoscalar chiral singlet, χ is the scalar dilaton field, are the field strength tensors of the vector meson multiplet, V µ , the axial vector meson multiplet A µ and the photon field, A µ . In Eq. (2), and the covariant derivative of a field Φ(≡ B, X, Y, χ) reads where u = exp i σ 0 π a λ a γ 5 , with π a and λ a , i = 1, ..8, as the pseudoscalar mesons and the Gell-Mann matrices. The interaction of the baryons with the meson, W (scalar, pesudoscalar, vector, axialvector meson) is given as where, the F -type (antisymmetric) and D-type (symmetric) couplings are defined as T r(BOB)T r(W ). In equation (5), (W, O) ≡ (X, 1), (u, γ 5 ), (V, γ µ ) and (A, γ µ γ 5 ), for the interactions of the baryons with the scalar, the pseudoscalar, the vector and the axialvector mesons respectively.
The Lagrangian for the vector meson interaction is written as The masses of ω, ρ and ϕ are fitted from m V , µ and λ V . The Lagrangian describing the interaction for the scalar mesons, X, and pseudoscalar singlet, Y , is given as [61] with I 2 = Tr(X + iY ) 2 , I 3 = det(X + iY ) and I 4 = Tr(X + iY ) 4 . In the above, χ is the scalar dilaton field which is introduced in order to mimic the QCD trace anomaly, i.e. the non-vanishing energy-momentum tensor where G a µν is the gluon field tensor, and, the second term in the trace accounts for the finite quark masses, with m i as the current quark mass for the quark of flavor, i = u, d, s. The scale breaking and the explicit chiral symmetry breaking terms are given as [60,61] π fπ − 1), here m π and m K are the masses of the pion and K-meson, and, f π and f K , their decay widths.
In the present investigation, we use the mean field approximation, where all the meson fields are treated as classical fields. In this approximation, only the scalar and the vector fields contribute to the baryon-meson interaction, L BW since for all the other mesons, the expectation values are zero. The various terms of the Lagrangian density in the mean field approximation are given as L vec = 1 2 The baryon-scalar meson interactions generate the baryon masses and the parameters corresponding to these interactions are adjusted so as to obtain the baryon masses as their experimentally measured vacuum values. In equation (11), the effective mass of the baryon of type i (i = p, n, Λ, Σ ±,0 , Ξ 0,− ) is given as which is calculated from the values of the scalar fields in the magnetized medium, and, the masses with the vacuum values of the scalar fields correspond to the experimentally measured vacuum values of the baryons.
The explicit chiral symmetry breaking term is given as In the above, the matrix, whose trace gives the Lagrangian density corresponding to the explicit chiral symmetry breaking in the chiral SU(3) model, has been explicitly written down. Comparing the above term with the explicit chiral symmetry breaking term of the Lagrangian density in QCD given as one obtains the nonstrange quark condensates (⟨ūu⟩ and ⟨dd⟩) and the strange quark condensate (⟨ss⟩) to be related to the the scalar fields, σ, δ and ζ as It might be noted here that with the choice for A p in the explicit symmetry breaking term as given by equation (10), together with the constraints σ 0 = −f π , ζ 0 = − 1 √ 2 (2f K − f π ) assure that the PCAC-relations of the pion and kaon are fulfilled. Using one loop QCD β 11Nc , with N c = 3, the number of colors and N f as the number of quark flavor, in the trace of energy momentum tensor in QCD given by equation (8) and equating with θ µ µ of the chiral model the scalar gluon condensate gets related to the dilaton fleld as [51] in the limiting situation of massless quarks in the energy momentum tensor of QCD given by equation (8).
The term L Bγ mag in the Lagrangian given by equation (1), describes the interacion of the baryons with the electromagnetic field, and, is given as [101][102][103] where, ψ i corresponds to the i-th baryon. The tensorial interaction of baryons with the electromagnetic field given by the second term in the above equation is related to the anomalous magnetic moments of the baryons. We choose the magnetic field to be uniform and along the z-axis, and take the vector potential to be A µ = (0, 0, Bx, 0). The number and scalar densities of the proton have contributions from the Landau energy levels and the neutrons have contributions to their number and scalar densities due to the anomalous magnetic moment, in the presence of a magnetic field [101,102]. The expresssions for the number and scalar densities of the proton in the presence of a uniform magnetic field (chosen to be along z-direction) and accounting for the anomalous magnetic moments for the nucleons are given as [104][105][106] and where, k The number density and the scalar density of neutrons are given as The Fermi momentum, k (n) f,S for the neutron with spin projection, S (S = ±1 for the up (down) spin projection), is related to the Fermi energy for the neutron, E In the equations (22)- (27), the parameter ∆ i is related to the anomalous magnetic moment for the nucleon, i (i = p, n) as ∆ i = − 1 2 κ i µ N B, where, κ i , occurring in the second term in the Lagrangian density given by Eq. (21), is the value of the gyromagnetic ratio of the nucleon corresponding to the anomalous magnetic moment of the nucleon. In the present study of magnetized (nuclear) matter, the meson fields are treated as classical in the mean field approximation, and nucleons as quantum fields and the self energies of the nucleons include the contributions from the Dirac sea. In addition to using the mean field approximation, where the meson fields are replaced by their expectation values, we also use the approximations where, ρ s i and ρ i are the scalar and number density of fermion of species, i (neutron and proton in the present investigation). Using the scalar densities of the nucleons in the presence of magnetic field, the values of the scalar fields, σ, ζ and δ are obtained by solving their coupled equations of motion, for given values of the baryon density, isospin asymmetry parameter and magnetic field. The last terms in equations (23) and (26) correspond to the contributions of the Dirac sea for the scalar densities of proton and neutron. The magnetized Dirac sea contribution to the nucleon self-energy has been calculated by summing over the tadpole diagrams arising due to the interaction of the nucleons with the scalar field σ within the Walecka model in the weak magnetic field approximation [99]. Generalizing to include the interactions of the nucleons to the strange ζ and the non-strange isovector δ scalar fields as well, in addition to the interaction with the non-strange σ field, for the chiral effective model used in the present investigation, the contribution due to the magnetized Dirac sea to the self-energy of the i-th nucleon (i = p, n) is given as where, q i is the charge and ∆ i = − 1 2 κ i µ N B is related to the anomalous magnetic moment of the baryon i (p and n in the present investigation). The baryon-pseudoscalar meson interaction term (the Weinberg-Tomozawa term) is then written as where the first term is the Weinberg-Tomozawa term, L SM E is the scalar exchange term, and L 1strange , L d 1 and L d 2 are the range terms. The scalar meson exchange term is obtained from explicit symmetry breaking term given by (10), with the generalizations: π fπ − 1), and the scalar meson multiplet for the SU(4) and SU(5) cases, is given as The range terms are obtained from the interaction terms [51] and, In the above equations, u occurring in in the expressions of u µ and Γ µ given by equations (3) and (4), is given as, u = exp( i σ 0 λ a π a γ 5 ) where, λ a , are the 4 × 4 (5 × 5) Gell-Mann matrices with a = 1, ...15 (a = 1, ...24) for the generalization to the case of SU(4) (SU (5) These are given as where Π F (F ) , denotes the self energy of the meson F (≡ D, B),F (≡D,B) in the medium.
Explicitly, the self energies for the D andD are given as [101] Π and where the ± signs refer to the D 0 and D + respectively in equation (35) and to theD 0 and D − respectively in equation (36). For the B meson doublet (B + ,B 0 ), andB meson doublet ( B − ,B 0 ), the self energies are given by [102] and where the ± signs refer to the B + and B 0 respectively in equation (37) and to the B − andB 0 mesons respectively in equation (38). The terms in the self-energies refer to the leading Weinberg-Tomozawa term and the sub-leading terms (the scalar exchange term and the range terms) in chiral perturbation expansion. The parameters d 1 and d 2 are fitted from the KN scattering lengths [51]. In equations (35)- (38), where m * F (F ) is the mass of the open charm (bottom) meson obtained as solution of the dispersion relation given by equation (34). The mass shift of the heavy quarkonium states arises from the medium modification of the scalar gluon condensate in the leading order and is given as [56][57][58][59] ∆m Ψ(Υ) = 1 18 which, using equation (20) gives the mass shift of the heavy quarkonium state as [50,51] ∆m where ⟨| ∂ψ(k) ∂k In equation (41), d is a parameter introduced in the scale breaking term in the Lagrangian, χ and χ 0 are the values of the dilaton field in the magnetized medium and in vacuum respectively. The wave functions of the quarkonium states, ψ(k) are assumed to be harmonic is the binding energy of the charmonium (bottomonium) state of mass, m ψ(Υ) . It might be noted here that the leading order mass formula (given by equation (40)) was derived using the binding of the heavy quark and antiquark in the heavy quarkonium state to be Coulombic. This is a good approximation for the ground state, but not realistic for the excited states [59], as the mass shift formula contains derivatives of the wave function, which measure the dipole size of the system. The wave functions for the charmonium and bottomonium states are assumed to be harmonic oscillator type, with the strengths of the potential determined from the rms radii of the quarkonium states. The mass shifts of the heavy quarkonium states are thus obtained from the values of the dilaton field, χ (using equation (41)). The for the heavy quarkonia [78,81,[85][86][87], the open charm mesons [79] and the strange (K andK) mesons [107]. In equation (43), 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.
In equation (43), the coupling parameter g P V is fitted from the observed value of the radiative decay width, Γ(V → P + γ). Assuming the spatial momenta of the heavy quarkonia to be zero, there is observed to be mixing between the pseudoscalar and the longitudinal component of the vector field from their equations of motion obtained with the phenomenological P V γ interaction given by equation (43). The physical masses of the pseudoscalar  where M 2 The effective Lagrangian term given by equation (43) has been observed to lead to the mass modifications of the longitudinal J/ψ and η c due to the presence of the magnetic field, which agree extermely well with a study of these charmonium states using a QCD sum rule approach incorporating the mixing effects [85,86]. The PV mixing effects for the open charm mesons (due to D − D * andD −D * mixings) [79], in addition to the mixing of the charmonium states (due to J/ψ − η c , ψ ′ − η ′ c and ψ(3770) − η ′ c mixings) [78,79], as calculated using the phenomenological Lagrangian given by equation (43) have been observed to lead to appreciable drop (rise) in the mass of the mixing effects have been estimated from the mixing of spin with the external magnetic field [81], using the Hamiltonian [84,88].
which decribes the interaction of the magnetic moments of the quark (antiquark) with the external magnetic field. In the above, µ i = g|e|q i S i /(2m i ) is the magnetic moment of the i-th particle, g is the Lande g-factor (taken to be 2 (−2) for the quark (antiquark)), q i , S i , m i are the electric charge (in units of the magnitude of the electronic charge, |e|), spin and mass of the i-th particle [86,88]. This interaction leads to a drop (increase) of the mass of the pseudoscalar (longitudinal component of the vector meson) given as [84] ∆M where ∆ = 2g|eB|((q 1 /m 1 ) − (q 2 /m 2 ))/∆E, ∆E = m V − m P is the difference in the masses of the pseudoscalar and vector mesons. It was observed in Ref. [81] that the partial decay where, M q is the constituent mass of the light quark (antiquark). The subscript q of the field operators in equation (47)  Assuming the initial and final state mesons to be bound by a harmonic oscillator potential, the explict constructions for the vector quarkonium states ψ(3770) (corresponding to 1D state) and Υ(4S), at rest (with spin projection m) are given as [77,80,112] |ψ m (3770 with u ψ(1D) (k) = 16 15 and, with, In equations (48) and (50) D, B,F ≡D,B), with finite momenta are constructed in terms of the constituent quark field operators, obtained from the quark field operators of these mesons at rest through a Lorentz boosting [110]. These are given as andQ(Q), with Q = (c, b) inF (F ) meson are then given as [109], where, the expression for A M (|p|) is written in Appendix A. The decay width is calculated to be with p 0 F (F ) (|p|) = m 2 F (F ) +|p| 2 1/2 , and, |p|, the magnitude of the momentum of the outgoing F (F ) meson is given as, In the above, the masses of the When we include the PV mixing effect, the expression for the decay width is modified to In the above, the first term corresponds to the transverse polarizations for the quarkonium state, M , whose masses remain unaffected by the mixing of the pseudoscalar and vector charmonium states. The second term in (58)  There is observed to be enhancement of the quark condensates (calculated from the scalar fields σ and ζ using equation (18) Including the effects of the Dirac sea of the nucleons, the masses of the open charm [113], the bottom meson mesons [114], and the heavy quarkonia states [115] have been studied in magnetized (nuclear) matter. The inclusion (exclusion) of the AMMs of nucleons give rise to the IMC (MC) for ρ B = ρ 0 , which lead to very different behaviours for the masses of the quarkonium states ψ(1D) and Υ(4S), with a drop (increase) in the mass, with increase in the magnetic field, when the PV effects are not taken into account [115]. For the open heavy flavour mesons, there is observed to be a monotonic increase with magnetic field when the AMMs are not taken into account, whereas, there is obserevd to be an intitial increase followed by a drop in these masses when the magnetic field is further increased, and the behaviour remains similar when the PV mixing effects are also taken into account [113].
The decay width of the quarkonium state ψ(1D) (Υ(4S)) (decaying at rest) to DD (BB) depends on the magnitude of the momentum of the outgoing open heavy flavour mesons, |p|, given by equation (57) [113,115]. However, there is increase in the masses of the charged D ± mesons due to the lowest Landau level (LLL) contributions, which leads to a drop in the decay width for the charged open charm meson pair final state in the presence of a magnetic field, whereas, the decay width of charmonium to neutral DD is observed to drop marginally with increase in the magnetic field, in the absence of PV mixing, as can be seen from panel (a) in figure 1. The contributions due to PV mixing have been observed to be significant in Ref. [78,79]. The Dirac sea contributions are taken into account using the summation of the tadpole diagrams, using weak field approximation for the nucleon propagator [99], and,  (58)) on the magnitude of the momentum of the outgoing B(B) meson (given by equation (57)) as a polynomial term multiplied by a gaussian contribution, and the node occurs when the polynomial part becomes zero. The nodes arise from taking into consideration the internal structure of the mesons in terms of the quark and antiquark constituents [51,72,78,81].
On the other hand, a phenomenological interaction, L int ∼ Υ µ (B(∂ µ B) − (∂ µB )B), without accounting for the internal structure of the mesons, leads to the decay widths, which increase monotonically with increase in |p|.
In figure 5 there is observed to be a non-smooth behaviour of the decay widths in both charged and neutral DD channels at around eB ∼ 7m 2 π (as can be seen in panel (d) of figure 5). This behaviour of the decay widths (which depend on |p|) arises from dependence of the mass of the bottomonium state, Ψ(4S) (hence of |p|) with the magnetic field, which is observed to be non-smooth at around this value of eB [115].
In figure 6, the decay widths are shown accounting for the B(B)−B * (B * ) mixings. In the absence of DS effects, there is observed to be appreciable effect due to these mixings which are observed to lead to only marginal modifications, when the Υ(4S) − η b (4S) mixing is also considered (see panels (b) and (d) as compared to panels (a) and (c)). There is observed to be a node in the decay width for the B + B − final state at around eB ∼ 7m 2 π in the absence of DS effects, without and with the PV mixing effects taken into account, as can be seen from panels (a) and (c). In the presence of DS effects, the initial rise is followed by a drop leading to vanishing of the decay width and again an increase as the magnetic field is further where λ i is the fraction of the energy (mass) of the hadron carried by the quark (antiquark), with i λ i = 1. For a hadron in motion with four momentum p, the field operators for quark annihilation and antiquark creation, for t=0, are obtained by Lorentz boosting the field operator of the hadron at rest, and are given as [110] Q (p) (x, t) = dk (2π) 3/2 S(L(p))U (k)Q(k + λp) exp[i(k + λp) · x − iλp 0 t] (A. 6) and,Q In the above, λ is the fraction of the energy of the hadron, carried by the constituent quark (antiquark). In equations (A.6) and (A.7), L(p) is the Lorentz transformation matrix, which yields the hadron at finite four-momentum p from the hadron at rest, and is given as [109] L µ0 = L 0µ = p µ m H ; L ij = δ ij + p i p j m H (p 0 + m H ) , k(−k) [110]. This is similar to the quasipotential approach, where the Lorentz transformation plays the role of a translation [116]. Using the composite model picture with Lorentz transformations as considered in the present work, the various properties of hadrons, e.g., charge radii of the proton and pion, the nucleon magnetic moments [108,109] have been studied. S(L(p')) to be unity. We shall also take the approximate forms (with a small momentum expansion) for the functions f (|k|) and g(|k|) of the field operator as given by g(|k|) = 1/ (2k 0 (k 0 + M )) 1/2 ≃ 1/(2M ), and f (|k|) = (1 − g 2 k 2 ) 1/2 ≈ 1 − ((g 2 k 2 )/2) [77].
The expression for the decay width of M → FF is obtained as given by equation (56).
The expression for A M (|p|) in the decay width is given as