Analysis of Pseudoscalar and Scalar $D$ Mesons and Charmonium

PHYSICAL REVIEW C 101, 015202 (2020)

Analysis of pseudoscalar and scalar D mesons and charmonium decay width in hot magnetized asymmetric nuclear matter

Rajesh Kumar* and Arvind Kumar† Department of Physics, Dr. B. R. Ambedkar National Institute of Technology Jalandhar, Jalandhar 144011 Punjab, India

(Received 27 August 2019; revised manuscript received 19 November 2019; published 8 January 2020)

In this paper, we calculate the mass shift and decay constant of isospin-averaged pseudoscalar (D+, D0 ) +, 0 and scalar (D0 D0 ) mesons by the magnetic-field-induced quark and gluon condensates at finite density and temperature of asymmetric nuclear matter. We have calculated the in-medium chiral condensates from the chiral SU(3) mean-field model and subsequently used these condensates in QCD sum rules to calculate the effective mass and decay constant of D mesons. Consideration of external magnetic-field effects in hot and dense nuclear matter lead to appreciable modification in the masses and decay constants of D mesons. Furthermore, we also ψ ,ψ ,χ ,χ studied the effective decay width of higher charmonium states [ (3686) (3770) c0(3414) c2(3556)] as a 3 by-product by using the P0 model, which can have an important impact on the yield of J/ψ mesons. The results of the present work will be helpful to understand the experimental observables of heavy-ion colliders which aim to produce matter at finite density and moderate temperature.

DOI: 10.1103/PhysRevC.101.015202

I. INTRODUCTION debateable topic. Many theories suggest that the magnetic field produced in HICs does not die immediately due to the Future heavy-ion colliders (HIC), such as the Japan Proton interaction of itself with the medium. The primary magnetic Accelerator Research Complex (J-PARC Japan), Compressed field induces electric current in the matter and, due to Lenz’s Baryonic Matter (CBM, GSI Germany), Proton AntiProton law, a secondary magnetic field comes into the picture which Annihilation in Darmstadt (PANDA, GSI Germany), and slows down the decay rate of the magnetic field [9–16]. These Nuclotron-based Ion Collider Facility (NICA, Dubna Russia), interactions increase the electric conductivity of the medium will shed light on the nonperturbative regime of the QCD by which further affects the relaxation time of the magnetic exploring the hadronic matter in a high-density and moderate- interaction, and this phenomenon is called the chiral magnetic temperature range [1]. In HICs, two heavy-ion beams are effect [4,6–8]. The presence of the magnetic field affects the smashed against each other and as a by-product quark gluon yield of in-medium/vacuum chiral condensates, and hence the plasma (QGP) comes into existence under extreme conditions location of the critical temperature T is also affected and this of temperature and density, but it lives for a very short interval c process is known as (inverse) magnetic catalysis [6]. of time [2]. Subsequently, with the decrease of temperature, Near the hadron phase transition, it is not possible to detect a phase transition occurs in which the QGP is modified QGP directly due to its short-lived nature, and hence many into hadronic matter by a process called hadronization [2]. other indirect observations are used as a tool to understand Alongside the medium attributes such as isospin asymmetry its existence namely jet quenching [17], strangeness (due to unequal number of protons and neutrons in heavy ion), enhancements [18,19], dilepton enhancements [20–22], and strangeness (due to the presence of strange particles in the J/ψ (ϒ) suppression [23]. In 1986, Matsui and Satz proposed medium), temperature, and density, recently it was found that the idea of J/ψ suppression on the basis of color Debye in HICs, a strong magnetic field is also produced having field 2 2 screening [23]. In this mechanism, when the Debye screening strength of approximately eB = 2–15 mπ (1mπ = 2.818 × 1018 gauss) [3–5]. Since then, physicists have been trying to radius becomes less than the charm quark system’s binding understand how the presence of this magnetic field affects radius, the charm binding force can no longer keep the c and c¯ quarks together. These free charm quarks (antiquarks) form the first- and second-order phase transitions [4,6–9]. The , , time duration for which the magnetic field remains is a very a bound state with free light quarks (u d s) in the medium to form D mesons. The in-medium effects on open charm mesons are more than the quarkonium. This is due to the fact *[email protected] that the in-medium properties of D mesons depend on light †[email protected]; [email protected] quark condensates which varies appreciably with the medium, whereas for charmonia (bottomonia), it depends on gluon Published by the American Physical Society under the terms of the condensates which do not change much with density [9,24]. Creative Commons Attribution 4.0 International license. Further It may also be noted that J/ψ suppression occurs not only distribution of this work must maintain attribution to the author(s) due to QGP formation but also because of density-dependent and the published article’s title, journal citation, and DOI. Funded suppression, comover scattering [25], and nuclear dependence by SCOAP3. of D and B (for ϒ suppression) mesons [26,27]. Higher

2469-9985/2020/101(1)/015202(18) 015202-1 Published by the American Physical Society RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020) bottomonium and charmonium states decay to ϒ and J/ψ of strong magnetic fields is also studied on the properties of mesons, respectively, and hence are considered to be the ρ meson [70], B meson [71,72], charmonium [9,16,73,74], major source of these ground-state mesons [23,28]. Under the and bottomonium states [74,75]. A lot of work has also been effect of different medium conditions, if the mass of the D (B) done without taking into account the effect of the magnetic meson decreases appreciably, then these higher quarkonium field. For example, Tolos et al. investigated the increase in the states will prefer to decay in DD¯ (BB¯) meson pairs rather than mass of D mesons in the nuclear medium using a coupled- in conventional J/ψ (ϒ) mesons. In AA and pA¯ collisions, channels approach [51]. In the QMC model, Tsushima and the decay width of higher quarkonium states and other Khanna observed a negative shift of D mesons in the nuclear experimental observables [29] can be directly measured medium and also discussed the possibility of the formation of experimentally to validate the phenomenological results [30]. D mesic nuclei due to the attractive interaction of D meson The QCD phase diagram is a graphical representation to with the medium constituents [76]. The chiral SU(3) model account for the different QCD regime’s physics with different was generalized to the SU(4) sector to study the in-medium medium parameters. To study this diagram in the hadron mass of pseudoscalar D mesons [77]. In this article, along phase, several potential models are constructed on the basis with the in-medium mass of D meson, authors also studied the of effective-field theory by incorporating basic properties decay width of higher charmonium states into DD¯ pairs using 3 of QCD, notably broken scale invariance and symmetry P0 model. Using QCD sum rules, Wang et al. calculated the breaking [9,31]. Some of these models include the Walecka mass and decay constant of pseudoscalar, scalar, vector, and model [32], the Nambu-Jona-Lasinio (NJL) model [33], the axial vector D mesons by taking the contributions from next- chiral SU(3) model [9,31,34–36], QCD sum rules (QCDSR) to-leading-order terms [63]. Using QCDSR, the contribution [37–42], the quark-meson coupling (QMC) model [43–48], up to the leading-order term has also been used to calculate the and the coupled-channels approach [49–52]. In the above properties of scalar D mesons [38]. By using the unification of approaches, the effect of thermal and quantum fluctuations the chiral SU(3) model and QCDSR, the in-medium mass and are neglected by using mean-field approximations. These decay constants of pseudoscalar, scalar, vector, and axial vec- fluctuations are included by modified potential models such tor D mesons are calculated in the strange hadronic medium as the Polyakov quark meson (PQM) model [53,54], the and a negative (scalar and vector) shift and a positive shift Polyakov loop extended NJL (PNJL) model [55–57], and (pseudoscalar and axial vector) in the mass of D mesons were functional remormalization group (FRG) [58,59] techniques. observed [24,28,65,78]. In these articles, authors have also In addition, the decay width of the heavy mesons has been calculated the in-medium decay width of higher charmonium 3 explained through various models, i.e., the S1 model [60], states [28] and scalar D mesons [24]. The in-medium decay the elementary meson-emission model [61], the flux-tube width of different heavy charmonia is also calculated using 3 3 model [62], and the P0 model [30]. quark antiquark pair P0 model [30] and recently this model Our present work presents three main points. First, we was also used to calculate the decay width of ψ(4260) [79]. calculate the in-medium quark and gluon condensates from The outline of the present paper is as follows: In Secs. II A the chiral SU(3) mean-field model and, second, use them in and II B, we will briefly explain the formalism to calcu- QCDSR to calculate the medium-induced mass and decay late the effective masses and decay constant of pseudoscalar constant of D mesons in the presence of magnetic field. Last, and scalar D mesons under the effect of magnetic field. In 3 by using the P0 model, we study the magnetic-field-induced Sec. II C, we will describe the methodology to calculate the decay width of higher charmonium states. The in-medium decay width of higher charmonium states. In Sec. III,wewill properties of meson under the effect of strong magnetic fields discuss the quantitative results of the present work and in have been studied by various nonperturbative techniques in Sec. IV, we will give a conclusion. the literature [28,63–65]. For example, in Ref. [15], the properties of D meson in strongly magnetized asymmetric II. FORMALISM nuclear matter was studied using the chiral SU(4) model and We use the unification of the chiral SU(3) model and an additional positive mass shift for the charged D meson, QCDSR techniques to study the effective mass and decay due to interaction with the magnetic field, was observed. In constants of scalar and pseudoscalar D mesons. These non- addition, under the effect of magnetic field the mass spectra of perturbative techniques are constructed to understand the low- D mesons and mixing effects between pseudoscalar and vector energy QCD by using renormalization methods [31,36,38,40]. D mesons have been studied with the use of operator product In this section, we gradually discuss the quark and gluon con- expansion technique of QCDSR by Gubler et al. [66]. The densates, in-medium mass and decay constants of D mesons, magnetic-induced decay width of higher charmonium states and the magnetic-field-induced charmonium decay width. into lower charmonium states are calculated with the joint 3 approach of the chiral model and the P0 model [67]. In this work, the author observed appreciable magnetic-field effects A. Quark and gluon condensates from the chiral SU(3) model in cold nuclear matter. Processes such as chiral magnetic We use the nonperturbative chiral SU(3) model, effect and (inverse) magnetic catalysis show great effect on constructed on the basis of effective-field theory. This the physics of deconfinement and chiral symmetry breaking. model incorporates the basic QCD features such as nonlinear The analytic crossover, critical point, and phase transition realization of chiral symmetry and trace anomaly [9,31,36,80– of the QCD phase diagram is studied extensively in the 83]. In this framework, the trace anomaly (broken scale literature [6,68,69]. In addition to these articles, the effect invariance) property of QCD is preserved by the introduction

015202-2 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020) of scalar dilaton field χ [9,31]. Also, the isospin asymmetry [15,36]. The effect of the external magnetic field is taken by of the medium is incorporated by the introduction of scalar adding the Lagrangian density due to magnetic field in the isovector delta field δ and vector-isovector field ρ [36]. The chiral effective Lagrangian density [9,15]. By minimizing the model is built on the assumption of the mean-field potential, thermodynamic potential of the chiral SU(3) model [9,74], in which the mixing of vector and pseudoscalar mesons the coupled equations of motion of the scalar (σ , ζ , δ, χ ), has been neglected, and hence the effect of thermal and and vector (ω, ρ), meson exchange fields are derived and are quantum fluctuations are not studied in the present work given as

σ χ 2 2 2 2 2 3 2 d 4 2 2 s k χ σ − 4k (σ + ζ + δ )σ − 2k (σ + 3σδ ) − 2k χσζ − χ + m fπ = gσ ρ , (1) 0 1 2 3 2 2 π i i 3 σ − δ χ0 d χ 4 χ 2 √ 1 χ 2ζ − σ 2 + ζ 2 + δ2 ζ − ζ 3 − χ σ 2 − δ2 − + 2 − √ 2 = ρs, k0 4k1( ) 4k2 k3 ( ) 2mK fK mπ fπ gζ i i (2) 3 ζ χ0 2 δ 2 2 2 2 3 2 2 4 s k χ δ − 4k (σ + ζ + δ )δ − 2k (δ + 3σ δ) + 2k χδζ + dχ = gδ τ ρ , (3) 0 1 2 3 3 σ 2 − δ2 i 3 i χ 4 4 (σ 2 − δ2 )ζ χ 3 χ σ 2 + ζ 2 + δ2 − σ 2 − δ2 ζ + χ 3 + + − χ 3 − χ 3 k0 ( ) k3( ) 1 ln 4 (4k4 d) d ln 2 χ 3 σ ζ χ0 0 0 0 χ √ χ 2 2 2 1 2 2 2 2 2 + m fπ σ + 2m f − √ m fπ ζ − m ω + m ρ = 0, (4) χ 2 π K K π χ 2 ω ρ 0 2 0 χ 2 2 ω + ω3 + ρ2ω = ρv, mω g4(4 12 ) gωi i (5) χ0 and χ 2 2 ρ + ρ3 + ω2ρ = τ ρv, mρ g4(4 12 ) gρi 3 i (6) χ0 respectively. In the above, the parameters k0, k2, and k4 are fitted to reproduce the vacuum values of scalar meson fields, and the other parameters, such as k1, are constrained to obtain the in-medium mass of nucleon at nuclear saturation density, ρN , and the parameter k3 is selected to generate the masses of η and η mesons. In addition, the parameters fπ , fK and mπ , mK are the decay constants and masses of pions and kaons, respectively. Moreover, the effect of isospin asymmetry is introduced in the τ ρv nuclear matter calculations by the parameter (η =− i 3i i ), where ρs and ρv represent the scalar and vector densities of the 2ρN i i ith nucleon (i = n, p) in the presence of magnetic field which is applied in the z direction [9,84,85] and τ3i is the I3 component of isospin. With the interaction of protons with magnetic field, Landau quantization takes place [9,84]. This circular confined motion disrupts the net momentum k in two parts, i.e., k⊥ (perpendicular to the z axis) and k (parallel to z axis) [9]. The magnetic-field-induced scalar density as well as the vector density of uncharged neutron given in Eqs. (1) and (5)are given as [84,85] ⎡ ⎤ ∞ sμ κ B ∞ m∗ ρs = 1 n n ⎣ − N n ⎦ n n n + ¯n n k⊥dk⊥ 1 dk fk,s fk,s (7) π 2 2 ˜ n 2 =± 0 ∗2 + n 0 Es s 1 mn k⊥ and ∞ ∞ v 1 n n n n n ρ = k⊥dk⊥ dk f , − f¯ , , (8) n 2π 2 k s k s s=±1 0 0 respectively. Similarly, for the charged proton, the scalar and vector densities are given by [84,85] ⎡ ⎤ ν(s=1) ν(s=−1) ∗ max ∞ p max ∞ p |qp|Bm dk dk ρs = p ⎣ f p + f¯p + f p + f¯p ⎦ (9) p 2 k,ν,s k,ν,s k,ν,s k,ν,s 2π p 2 2 p 2 2 ν=0 0 (k ) + (¯mp) ν=1 0 (k ) + (¯mp) and ⎡ ⎤ ν(s=1) ν(s=−1) |q |B max ∞ max ∞ ρv = p ⎣ dkp f p − f¯p + dkp f p − f¯p ⎦, (10) p 2π 2 k,ν,s k,ν,s k,ν,s k,ν,s ν=0 0 ν=1 0

015202-3 RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020) respectively, wherem ¯ p is the induced mass under the effect of scalar fields through the relation [9] magnetic field, which is defined as α χ 2 s a aμν 8 4 2 Gμν G = (1 − d)χ + mπ fπ σ π 9 χ0 = ∗2 + ν| | − μ κ . ρN m¯ p mp 2 qp B s N pB (11) √ 2 1 2 + 2m fK − √ mπ fπ ζ . (17) In the above equations, ν represents the Landau quantized K 2 ∗ levels and m =−(gσ iσ + gζ iζ + gδiτ3iδ) is the effective i The value of d = 2 has been taken from the QCD beta mass of the nucleons. Here gσ , gζ , and gδ represent the 11 i i i β coupling constants of ith nucleons with σ, ζ , and δ fields, function, QCD at the one-loop level [31]. respectively. The effective single-particle energy of a proton √ 2 B. Masses and decay constant of D mesons from QCDSR ˜ p = p 2 + ∗2 + ν| | − μ κ is given by Eν,s (k ) ( mp 2 qp B s N pB) , In this subsection, we discuss the QCDSR to calculate the whereas for a neutron its expression is given as √ 2 in-medium mass shift and decay constant of isospin averaged n n 2 ∗2 n 2 + + E˜ = (k ) + [ m + (k⊥) − sμN κnB] . The constants , , 0 s n pseudoscalar (D D0 ) and scalar (D0 D0 ) mesons. QCDSR ki and s are the anomalous magnetic moment and the spin of is a nonperturbative technique which is based on the Borel n ¯n p the nucleons, respectively. In addition, fk,ν,s, fk,ν,s, fk,s, and transformation and operator product expansion (OPE) method ¯p fk,s represent the finite-temperature distribution functions for [38,40,42,63]. These methods are used to deal with the di- neutrons and protons and their antiparticles and are given as vergence that occurs in the asymptotic perturbative series [38,40]. We will see that the mass and decay constant of these 1 open charm mesons is expressed in terms of scalar and gluon f n = , k,s + β ˜ n − μ∗ 1 exp Es n condensates, which contains the effect of medium parame- 1 ters such as temperature, density, asymmetry, and magnetic f¯n = . (12) field. We start with the two-point current correlation function k,s + β ˜ n + μ∗ 1 exp Es n (q) which represents the Fourier transformation of the time- 1 ordered product of the isospin-averaged meson current, J (x), f p = , k,ν,s + β ˜ p − μ∗ and can be written as [28,63] 1 exp Eν,s p 4 iq.x † p 1 (q) = i d xe T {J (x)J (0)} ρ , , (18) f¯ = . (13) N T k,ν,s + β ˜ p + μ∗ 1 exp Eν,s p where q is the four-momentum and ρN and T represent As we will see, to calculate the effective mass and decay the nucleon density and temperature of the medium. In constant of scalar and pseudoscalar D mesons using QCDSR, Refs. [39,86], the mass splitting of the different oppositely we need the values of quarks and gluon condensates. In the charged mesons are also investigated by dividing the current chiral model, the scalar quark condensates can be related to correlation function in even and odd terms. In this paper, we symmetry breaking via the following relation [9]: have considered the average meson currents of the particle D D¯ and their antiparticle . The average current of scalar and pseudoscalar meson is given by the following mathematical m q¯ q ρ =−L , (14) i i i N SB relations: i c¯(x)q(x) + q¯(x)c(x) J(x) = J†(x) = (19) where LSB is an explicit symmetry-breaking Lagrangian term 2 [9], and by using this equation we formulated the up- and and down-quark condensates, which are expressed as γ + γ = † = c¯(x)i 5q(x) q¯(x)i 5c(x), J5(x) J5 (x) (20) 1 χ 2 1 2 = 2 σ + δ uu¯ ρN mπ fπ ( ) (15) q x u m χ 2 respectively. In the above, the quark operator ( )isfor and u 0 d quarks and c(x) is the charm quark operator. The selection of and q depends on the quark content of the given meson. From the quark composition of D mesons one can easily understand that +, 0 +, 0 χ 2 the (D D ) and (D0 D0 ) form the isospin doublets and they 1 1 2 dd¯ ρ = m fπ (σ − δ) , (16) show mass splitting in the presence of isospin asymmetric N χ π md 0 2 medium [28]. Now, for the nuclear matter in the Fermi gas approximation, we divide the correlation function (q)into respectively. In the above mu and md are the masses of vacuum, static nucleon, and thermal parts as up-quark and down-quark, respectively. Also, by using the ρN broken scale invariance property of QCD [9,31,36], the scalar = + + . . , , α μν (q) 0(q) TN (q) P B (q T ) (21) = s a a 2mN gluon condensate GρN π Gμν G is formulated by the ρN comparison of energy-momentum tensor (EMT) of QCD with where TN (q) is the forward-scattering amplitude and mN the EMT of the chiral model and is expressed in terms of denotes the nucleon mass. The third term, i.e., the pion bath

015202-4 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020) contribution, represents the thermal effects of the medium and written as [63] is given as [87] + ∗ mN mD/D0 m = 2π ρ a / (28) D/D0 N D D0 4 iq·x † mN mD/D0 P.B.(q, T ) = i d xe T {J (x)J (0)} T . (22) and In the present investigation, we can neglect this thermal 2 2 3 m bρ 4 f / m / mD/D ∗ = c N − D D0 D D0 0 , effects term, as the temperature and magnetic effects of the fD/D 0 2 f / m4 2m m2 medium are incorporated by quark and gluon condensates, D D0 D/D0 N c which are calculated in terms of the meson exchange fields (29) as discussed under Sec. II A [24,78]. By neglecting the third term from Eq. (21), the expression becomes respectively. Hence the effective mass of open charm mesons ρ can be written as = + N . (q) 0(q) TN (q) (23) ∗ ∗ 2mN m = m / + m . (30) D/D0 D D0 D/D0 The forward-scattering amplitude TN (q) can be written as Note that the effective mass (m∗ ) is the total excitation D/D0 4 iq·x † energy of the particle at zero momentum in the medium and TN (ω, q ) = i d xe N(p)|T {J(x)J (0)}|N(p) . (24) mD/D0 denotes a vacuum mass of pseudoscalar and scalar D mesons. The amplitude T (ω, q ) can be related to the DN(D N) N 0 As discussed earlier, the Landau quantization takes place scattering T matrix in the limit of q → 0, with the interaction of charged particle with magnetic field. T , = π + , D/D0 N (mD/D0 0) 8 (mN mD/D0 )aD/D0 (25) This interaction invokes an additional positive shift in the mass of the charged D+, D+ meson and this lead to / 0 where aD D0 is the scattering lengths of DN (D0N) interactions. This scattering matrix can be represented in terms ∗∗ ∗2 m + + = m + +|eB|. D /D D+/D (31) of phenomenological spectral density ρ(ω, 0), which can be 0 0 parametrized in three unknown parameters, a, b, and c [38], On the other hand, due to their uncharged nature the neutral 0 0 2 4 pseudoscalar (D ) and scalar (D0 ) mesons have no modifica- 1 T / (ω, 0) f / m / ρ(ω, 0) =− Im D D0N D D0 D D0 +··· , tion due to magnetic field. π 2 2 2 ω2 − m + iε mc We get the analytic QCDSR in terms of two unknown pa- D/D0 rameters a and b by equating the Borel-transformed forward- d 2 2 2 2 ω, = a δ ω − m / + b δ ω − m / scattering amplitude TN ( q ) in the OPE side with the Borel- dω2 D D0 D D0 transformed forward-scattering amplitude TN (ω, q )inthe 2 + c δ(ω − s0 ). (26) phenomenological side [38]. The parametrized QCDSR are given by equation In above equation, the first term represents the double pole term, which is related to the on-shell effect of T matrices and aCa + bCb = Cf . (32) can be related to the scattering length as The explicit form of Borel-transformed coefficients having 2 amc next-to-leading-order contributions for the pseudoscalar cur- aD/D = . (27) 0 2 4 rent J5(x)is[63] f / m / [−8π(mN + mD/D )] D D0 D D0 0 2 1 mD s0 s0 Furthermore, the second term in Eq. (26) represents the Ca = exp − − exp − , single pole term that relates the off-shell effects of T matrices; M2 M2 m4 M2 D the last term corresponds to the remaining contributions (con- m2 s s C = exp − D − 0 exp − 0 , (33) tinuum), where s0 denotes the continuum threshold parameter. b 2 2 2 M mD M The shift in the above-mentioned masses mD/D0 and decay constants fD/D0 of the open charm mesons can be and

2m (m + m ) f m2 g 2 1 1 m2 C = N H N D D DNH − exp − D f + 2 − 2 2 2 − + 2 2 (mH mN ) mD mc M mD (mH mN ) M 1 (m + m )2 m qq¯ α 4m2 + exp − H N − c N 1 + s 6 − c + 2 − 2 2 π 2 (mH mN ) mD M 2 3M 2 m2 m2 m2 m2 m2 − 1 − c log c − 2 0, c exp c exp − c 3 M2 μ2 M2 M2 M2 1 m2 4m m2 1 α GG m2 + −2 1 − c q†iDq + c 1 − c qiDiDq¯ + s exp − c . (34) 2 N 2 2 N π 2 2 M M 2M 12 N M

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For the scalar current J(x), we have [64,65] 1 m2 s s m2 s s C = exp − D0 − 0 exp − 0 , C = exp − D0 − 0 exp − 0 , (35) a M2 M2 m4 M2 b M2 m2 M2 D0 D0 and 2 2 2 2m (m − m ) fD m gD NH 1 1 m C = N H N 0 D0 0 − exp − D0 f (m − m )2 − m2 m M2 m2 − (m − m )2 M2 H N D0 c D0 H N 1 (m − m )2 m qq¯ m2 + exp − H N + c N exp − c − 2 − 2 2 2 (mH mN ) mD M 2 M 0 1 m2 4m m2 m2 + −2 1 − c q†iDq − c 1 − c qiDiDq¯ exp − c 2 M2 N M2 2M2 N M2 1 α GG 1 m2 m2 1 α GG 1 1 − x m2 + s dx 1 + c exp − c − s dx m4 exp − c . (36) π 2 2 4 π c 2 16 N 0 M M 48M N 0 x M

1 and In the above equations, M2 is the Borel mass operator and α † sGG qq¯ N , q iDq N , qiDiDq¯ N , and π N are the nucleon α s a aμν expectation values of different quark and gluon condensates. Gμν G ∞ π , = −x 1 −t 2 = 2/ N Also, (0 x) e 0 dt t+x e and mc mc x. We will see qq α α m later that as compared to scalar quark condensates ¯ N , = s a aμν − s a aμν 2 N . † Gμν G Gμν G (43) the impact of other quark condensates (OQC) q iDq N , π ρ π ρN N vac qiDiDq¯ N is very small on the observables of the D mesons. The nucleon expectation values of the chiral condensates The condensate qiDiDq¯ ρN appearing in Eq. (41) can be can be calculated by using calculated in terms of light quark condensates using equations [28,89] d3 p Oρ = Ovac + 4 nF N(p)|O|N(p) 1 2 N 3 ρ + σ ρ = . ρ (2π ) 2Ep qiDiDq¯ N 8 qg¯ s Gq N 0 3GeV N (44) ρN = Ovac + ON , (37) and 2mN 2 2 qg¯ σ Gq ρ = λ qq¯ ρ + 3.0GeVρ . (45) where O denotes an operator. Now, by taking the expectation s N N N values at both sides of above equation, In this article, we have used the condensate value † = . 2 ρ ρ q iDq N 0 18 GeV N from the linear density approxi- O =O + N O , ρN vac N mations results [89]. Now, in Eq. (32), to calculate the values 2mN of two unknowns a and b, we need one more equation, which 2mN = 1 O = O −O , can be obtained by differentiation of Eq. (32) with z M2 , N ( ρN vac ) (38) ρN i.e., O O d d d where vac and N denote the vacuum operator and a C + b C = C . (46) nuclear matter-induced operator in the Fermi gas model, dz a dz b dz f respectively [88]. Following this, the nucleon expectation values of light By solving Eqs. (32) and (46) simultaneously, we get the quark and gluon condensates are expressed as following mathematical formulas to ﬁnd a and b, 2m − d − − d = − N , Cf dz Cb Cb dz Cf uu¯ N [ uu¯ ρ uu¯ vac] (39) = , N ρ a d d N Ca − Cb − Cb − Ca dz dz 2mN d d dd¯ = [dd¯ ρ −dd¯ ] , (40) − − − N N vac Cf dz Ca Ca dz Cf ρN = . b d d (47) Cb − Ca − Ca − Cb 2mN dz dz qiDiDq¯ = [qiDiDq¯ ρ −qiDiDq¯ ] , (41) N N vac ρ N The obtained values of a and b are used to calculate the mass 2m † = † − † N , shift and decay constant of D mesons given by Eqs. (28) and q iDq N [ q iDq ρN q iDq vac] (42) ρN (29), respectively.

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C. In-medium decay width of higher charmonium 92], which is a quark-antiquark pair creation model. In this 3 3 states using P0 model model, a light quark pair is generated in the P0 state (vac- In the present work, one objective is to calculate the de- uum), and one of the quarks (antiquarks) is combined with cay width of higher charmonium states [ψ(3686),ψ(3770), the heavy charm quark from the decaying charmonium at zero χ ,χ ¯ momentum. The matrix element for the decay C → D + D¯ c0(3414) c2(3556)] to pseudoscalar DD mesons. In order 3 to calculate this observable, we rely on the P0 model [30,90– (where C is charmonia) is given as [30] ∝ 3 φ − φ − φ − v . MC→DD¯ d kc C (2kc 2kD ) D(2kc kD ) D¯ (2kc kD ) u¯kc,s −kc,s (48)

In the above, kc − kD and kD − kc represent the momentum of the charm quark and the charm antiquark of charmonia C. Since | |=| | the decaying particle is assumed to be at rest, the magnitude of the momentum of D and D¯ meson is the same, i.e., kD kD¯ . v The term [¯ukc,s −kc,s] denotes the wave function of the quark-antiquark pair in the vacuum and for the charmonium. We start with the harmonic oscillator wave function [24,30], − 2 2 1 n L L+ 3 2n! R k L+ 2 2 ψ = (−1) (−ι) R 2 exp L 2 (R k )Y (k), (49) nLML + + 3 2 n lm n L 2 + 1 L 2 2 2 where Ln (R k ) denotes associate Laguerre polynomial, Ylm(k) represents the spherical harmonics, and R is the radius of the charmonia. Further, the decay rate of the charmonium state decaying into a DD¯ pair can be represented as [24,93]

pDEDED¯ 2 (C → D + D¯ ) = 2π |MLS| . (50) mC = 2 ∗ + 2 = 2 ∗ + 2 Here ED mD PD, ED¯ mD¯ PD, and the center-of-mass momentum ∗ ∗ ∗ ∗ 2 / m2 m2 + m2 m2 − m2 1 2 = C − D D¯ + D D¯ . pD 2 (51) 4 2 4mC

In the above, mC is the mass of charmonia and MLS is the invariant matrix amplitude. Using Eq. (50), the decay rate of different higher charmonium states can be represented as [28,30,93] / 2 π 1 2 7 + 2 2 − 2 2 2 + 2 y2 EDED¯ 2 (3 2r ) (1 3r ) 2r (1 r ) − = γ 2 3 + 2 2(1+2r2 ) , ψ(3686)→DD¯ y 1 y e (52) mψ (3686) 32(1 + 2r2 )7 (1 + 2r2 )(3 + 2r2 )(1 − 3r2 ) / 7 2 π 1 2 11 + 2 y2 EDED¯ 2 5 r 1 r − = γ 2 3 − 2 2(1+2r2 ) , ψ(3770)→DD¯ y 1 y e (53) mψ (3770) 32 1 + 2r2 5(1 + 2r2 ) 5 2 2 2 E E r 1 + r − y 1/2 D D¯ 2 9 2 2 χ → ¯ = π γ 2 3 y 1 − y e 2(1+2r ) , (54) c0 (3414) DD + 2 + 2 2mχc0 (3414) 1 2r 3(1 2r ) and III. NUMERICAL RESULTS AND DISCUSSIONS / π 1 2 10 5 + 2 2 y2 EDED¯ 2 r (1 r ) − In this section, we will discuss our observations on the = γ 2 5 2(1+2r2 ) . χ (3556)→DD¯ y e c2 mχ (3556) 15(1 + 2r2 )7 effect of the magnetic field on masses and decay con- c2 +, 0 +, 0 (55) stants of pseudoscalar (D D ) and scalar (D0 D0 ) mesons and in-medium decay widths of higher charmonium states β ψ ,ψ ,χ χ In the above equations, the variables r and D incorporate [ (3686) (3770) c0(3414) and c2(3556)] in asymmetric the modification of the wave function due to the nodal struc- nuclear matter at finite temperature. As discussed earlier, the ture of the initial- and final-state mesons [94,95] and their light quark condensates and gluon condensates have been values are fitted with the help of the experimental partial decay calculated by using the chiral SU(3) model and the different width of ψ(4040) to DD¯ mesons [30]. The parameter y is parameters used in the model are shown in Table I. In addition, expressed as, y = pD and the parameter γ denotes the strength the value of charm quark mass m , running coupling constant βD c of the vertex and fitted using the experimental decay width of αs, coupling constant gDNH, and constant λ are approximated [ψ(3770) → DD¯ ][28,30]. The decay width of different to be 1.3, 0.45, 6.74, and 0.5 GeV, respectively [28,63]. The higher charmonia can be calculated by the above equations vacuum masses of D mesons are taken as 1.869, 1.864, 2.355, + 0 + 0 by using the effective mass of D meson obtained from the and 2.350 GeV for D , D , D0 , and D0 mesons, respectively. QCDSR calculations. The vacuum values of the decay constant for pseudoscalar and

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TABLE I. Values of different parameters. N= 0 N=4 0 -2 -2 k0 k1 k2 k3 k4 ) 3 -4 -4 − − − 2.53 1.35 4.77 2.77 0.218 GeV -6 -6 −3 σ0 (MeV) ζ0 (MeV) χ0 (MeV) d ρ0 (fm ) -3 -8 -8 −93.29 −106.8 409.8 0.064 0.15 (10 N mπ (MeV) mK (MeV) fπ (MeV) fK (MeV) g4 -10 -10 u> 139 498 93.29 122.14 79.91 -12 -12 gσ N gζ N gδN gωN gρN -8 -8 ∗ ∗

2 ) 7–9 GeV , respectively. Moreover, the values of parameters γ , 4 1.95 1.95 3 βD, and r used in the P0 model are taken as 0.281, 0.30, and GeV 1.9 1.9

1.04, respectively [28]. Our further discussion of this section -2

is divided into three subsections. (10 1.85 1.85 N

G 1.8 1.8 A. Magnetic-field-induced condensates and mass of nucleons (e) (f) 1.75 1.75 In this subsection, we show the results for the medium- 1234567 1234567 induced light quark and gluon condensates, which are cal- eB/m 2 eB/m 2 culated using the chiral model described in Sec. II A.From T=0 MeV T=100 MeV the expressions given in Eqs. (15)–(17), one can see that T=50 MeV T=150 MeV σ ζ δ the condensates depend on the scalar fields , , , and ¯ FIG. 1. The light quark condensates ( uu¯ ρN and dd ρN )and χ, which are solved under different conditions of medium α s a aμν gluon condensate π Gμν G ρ (denoted by GρN ) is plotted for such as density, magnetic field, temperature, and asymmetry η = N [9,74]. In Figs. 1 and 2, we have plotted the scalar up-quark symmetric nuclear matter ( 0) as a function of magnetic field eB under different conditions of medium. condensate uu¯ ρ , down-quark condensate dd¯ ρ , and gluon α N μν N s a a condensate π Gμν G with respect to magnetic field at ρN isospin symmetry parameters η = 0 and 0.5, respectively. We interaction of charged protons with the magnetic field. This have shown the results at nucleon densities ρN = ρ0 and 4ρ0 interaction disturbs the equality between the scalar density and temperatures T = 0, 50, 100, and 150 MeV. In Fig. 1, of protons and neutrons and hence the magnitude of δ field at η = 0, we can see that the magnitude of the up- and down- become nonzero [74]. quark condensates decreases with the increase in the magnetic In Figs. 1(e) and 1(f), we also show the results for scalar field. One can also conclude that the density effects are also gluon condensate and observed that the magnitude of the appreciable as the magnitude of quark condensates decrease gluon condensate changes much less as compared to light with the increase in the density. Moreover, inclusion of tem- quark condensates. This is why open charm mesons expe- perature effects increase the magnitude of quark condensates rience a larger mass shift than the ground-state charmonia but the trend concerning the magnetic field remains the same. [9,28]. The value of gluon condensate at nuclear saturation 2 For example, in symmetric nuclear medium for eB = 4mπ ,the density decreases as a function of the magnetic field and this ¯ ρ = ρ − . − . ρ = ρ values of uu¯ ρN ( dd ρN ), at N 0,is 10 28 ( 7 42), trend becomes more appreciable at N 4 0. Furthermore, −11.16 (−8.02), and −11.41 (−8.18) times 10−3 GeV3 for we observe that the temperature effects on gluon condensate T = 0, 100, and 150 MeV, respectively, and for ρN = 4ρ0 are opposite those of quark condensates. This is because the it modifies to −4.81 (−3.61), −5.40 (−4.10), and −5.79 gluon condensate depends on the fourth power of the χ field (−4.27) times 10−3 GeV3. It may be noted that despite η = 0, along with the σ and ζ fields [see Eq. (17), whereas quark the value of up- and down-quark condensates are different, condensates [see Eq. (15) and (16)] hae only σ - and δ-field which is contradictory to the previous work (at zero magnetic dependence [74]. field) as up- and down-quark are isospin partners and hence In Fig. 2,atη = 0.5, we observe similar behavior of indistinguishable in symmetric nuclear matter [78]. This is condensates as a function of the magnetic field as was at 2 because of Landau quantization, which occurs due to the η = 0, except at low temperature. For example, at eB = 4mπ ,

015202-8 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020)

N= 0 N=4 0 linear densitya pprox chiral SU(3) model -2 -2

) 1.0 1.05 3 -4 -4 (a) (b) 0.8 0 GeV -6 -6 1.0

-3 0.6 q>

-8 -8 0 0.95 (10 0.4 /G /< q N N N -10 -10 0.2

G 0.9 u> -12 -12 q> 0.0

(a) (b)

d> -8 -8

4 1.95 1.95 to the generation of additional fermion antifermion condensates in the presence of magnetic ﬁeld [6,69,74].

GeV 1.9 1.9

-2 Furthermore, in order to understand the validity of our calculations of quark and gluon condensates at higher density (10 1.85 1.85

N of nuclear matter, in Fig. 3 we compare the quark and gluon G 1.8 1.8 condensate using the chiral SU(3) model with that of linear (e) (f) 1.75 1.75 density approximation. In literature, the quark condensate 1234567 1234567 in linear density approximation has been calculated by the 2 2 σ ρ eB/m eB/m ρ = + N N σ expression qq¯ N qq¯ 0 mu+md , where N is taken as T=0 MeV T=100 MeV 45 MeV [63] and qq¯ 0 is the vacuum value of quark con- T=50 MeV T=150 MeV densate. In addition, in the same approximation gluon con- = − ¯ densates have been calculated by the expression GρN G0 FIG. 2. The light quark condensates ( uu¯ ρN and dd ρN )and α 8 ρ s a aμν mN N [96], where G0 is vacuum value of gluon condensate gluon condensate π Gμν G (denoted by GρN ) is plotted for 9 ρN asymmetric nuclear matter (η = 0) as a function of magnetic field and mN is vacuum value of nucleon mass. On the other side, eB under different conditions of medium. the quark and gluon condensate are evaluated from the chiral model using Eqs. (15)–(17). From Fig. 3, we observe that in linear density approximation the quark and gluon condensate ¯ ρ = ρ − . − . the values of uu¯ ρN ( dd ρN )at N 0 are 11 49 ( 7 57), decreases linearly with density, whereas in chiral-model cal- −11.90 (−7.93), and −11.34 (−7.69) times 10−3 GeV3 for culations, the condensates shows slow decrease (nonlinear) T = 0, 100, and 150 MeV, respectively, and for ρN = 4ρ0 with respect to density. A similar response of quark and gluon it modifies to −6.76 (−3.77), −6.84 (−3.95), and −6.24 condensate in the medium has been observed in Refs. [96–99]. (−3.86) times 10−3 GeV3. We also observed a crossover Therefore, the quark and gluon condensates evaluated in the behavior in the plot of gluon condensate. In this case, the value chiral model validate us to calculate observables in high of scalar gluon condensate increases for low temperature, density and these calculations are not exactly in linear density whereas it decreases for high temperature. This is because in approximation. a medium having a large number of neutrons, the behavior of In Fig. 4, we have shown the magnetic-field-induced mass ∗ the neutron scalar density in low-temperature modifies, hence (mi ) of proton and neutron under the effect of different the δ field also modifies [15,74]. Therefore, from Eqs. (15) values of temperature, magnetic field, density, and isospin and (16), one can see that in symmetric nuclear matter, these asymmetry. In this figure, for symmetric matter (η = 0), we quark condensates are directly proportional to scalar fields σ observe the mass of proton/neutron decrease with the increase δ and , and hence the behavior of qq¯ ρN with different medium in magnetic field. We also observed the mass of nucleons parameters is same as that of the σ field (as the δ field has decreases heavily with the increase in density and follows much less variation with the magnetic field for η = 0) [74]. the same trend as a function of magnetic field. Inclusion of On the other hand, for asymmetric matter, the condensates temperature effects shows increase in mass at particular value have mixed contributions of σ and δ fields as the δ field varies of magnetic field. This is because in chiral model, the mass appreciably with the increase in the magnetic field. Moreover, of nucleons are calculated on the basis of scalar fields σ , ζ , σ δ ∗ =− σ + ζ + τ δ the gluon condensate has the dependence on scalar fields , and by the relation mi (gσ i gζ i gδi 3i ), which ζ , and χ. The scalar fields as a function of magnetic field are solved under the influence of scalar density of nucleons.

015202-9 RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020)

N= 0 N=4 0 N= 0 N=4 0 800 800 1880 1880 =0 (a) =0 (b) =0 (a) =0 (b) 700 700 1860 1860

600 600 1840 1840

500 500 (MeV) 1820 1820 (MeV) p ** + *

D 1800 1800 m 400 400 m 300 300 1780 1780

200 200 1760 1760 1234567 1234567 1234567 1234567 800 800 1880 1880 =0.5 (c) =0.5 (d) =0.5 (c) =0.5 (d) 700 700 1860 1860

600 600 1840 1840

500 500 (MeV) 1820 1820 (MeV) ** p + *

D 1800 1800

m 400 400 m 300 300 1780 1780

200 200 1760 1760 1234567 1234567 1234567 1234567 800 800 1820 1820 =0 =0 =0 (e) =0 (f) (e) (f) 1800 1800 700 700 1780 1780 600 600 1760 1760

500 500 (MeV) (MeV) *

0 1740 1740 n * D

m 400 400 m 1720 1720 300 300 1700 1700

200 200 1680 1680 1234567 1234567 1234567 1234567 800 800 1820 1820 =0.5 =0.5 =0.5 (g) =0.5 (h) (g) (h) 1800 1800 700 700 1780 1780 600 600 1760 1760

500 500 (MeV) * (MeV)

0 1740 1740 n * D

m 400 400 m 1720 1720 300 300 1700 1700

200 200 1680 1680 1234567 1234567 1234567 1234567 2 2 eB/m 2 eB/m 2 eB/m eB/m T=0 MeV T=100 MeV T=0 MeV T=100 MeV T=50 MeV T=150 MeV T=50 MeV T=150 MeV

+ 0 FIG. 4. The effective masses of protons and neutrons are plotted FIG. 5. The effective mass of pseudoscalar D (charged) and D for symmetric and asymmetric nuclear matter as a function of (uncharged) mesons is plotted as a function of magnetic ﬁeld eB magnetic ﬁeld eB under different conditions of medium. under different conditions of medium.

splitting is larger as in case of symmetric matter. For example, One can see in spite of symmetric nuclear matter, the masses 2 ∗ ∗ at eB = 4mπ and η = 0.5, the values of m (m )atρN = ρ0 is of protons and neutrons are slightly different. For example, p n 2 ∗ ∗ 614.96 (602.46), 669.44 (628.35), and 614.96 (602.46) MeV at eB = 4mπ and η = 0, the values of m (m )atρN = ρ0 is p n for T = 0, 100, and 150 MeV, respectively, and for ρN = 630.03 (630.76), 616.44 (617.35), and 568.24 (569.78) MeV 4ρ it modifies to 331.52 (310.45), 339.12 (320.58), and = , ρ = ρ 0 for T 0 100, and 150 MeV, respectively, and for N 4 0 315.63 (303.74) MeV. In asymmetric matter the δ field varies it modifies to 257.3 (260.80), 289.98 (293.06), and 311.45 appreciably and due to Landau effects the scalar densities of (314.07) MeV. As discussed earlier, this is because of nonzero proton and neutron generate this crossover behavior. value of δ field, which arises due to the Landau quantization. On the other hand, for asymmetric nuclear matter, the mass of nucleons decreases as a function of density (similar in sym- B. Mass and shift in decay constant of pseudoscalar metric matter). Furthermore, the mass of nucleon decreases and scalar D mesons for large values of temperature but increases for lower values Here we will discuss how mass and shift in decay con- of temperature as a function of magnetic field. Here the mass stants of isospin-averaged pseudoscalar (D+, D0 ) and scalar

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+ 0 0 + TABLE II. The values of the magnetic-ﬁeld-induced masses of D , D , D0,andD0 mesons (in units of MeV).

η = 0 η = 0.5 T = 0 T = 100 T = 0 T = 100

2 eB/mπ ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ∗∗ mD+ 4 1826 1786 1835 1792 1828 1788 1833 1791 6 1834 1792 1843 1798 1839 1801 1844 1801 ∗ mD0 4 1770 1704 1784 1713 1779 1734 1795 1735 6 1766 1694 1780 1703 1789 1737 1794 1731 ∗∗ m + 4 2448 2495 2441 2490 2446 2493 2442 2491 D 0 6 2458 2505 2450 2501 2454 2494 2450 2499 m∗ 4 2451 2516 2440 2509 2436 2493 2431 2492 D0 0 6 2459 2524 2443 2517 2435 2491 2431 2495

+, 0 of mass change from 1826 to 1786 MeV. The inclusion of (D0 D0 ) mesons modifies with magnetic field and other medium properties. As discussed earlier in Sec. II B, the above temperature effects increases the mass at particular values of 2 observables are calculated in the QCDSR by using quark and density. For example, at η = 0, eB = 4mπ , and ρN = 4ρ0, gluon condensates. In Figs. 5 and 7, we have plotted the when we move from T = 0 to 100 MeV, the values of mass masses of pseudoscalar and scalar D mesons, respectively, as change from 1786 to 1792 MeV. a function of external magnetic field in nuclear matter. The For neutral pseudoscalar D0 meson there will not be any values of the effective mass of pseudoscalar and scalar D additional positive mass shift [see Eq. (31)] and therefore the mesons in the presence of magnetic field and other medium effective mass in this case does not increase but decreases properties are shown in Table II for better comparison. In with the increase in magnetic field (see Table II). However, Fig. 5, for symmetric nuclear matter, we observe that the the temperature and density effects remain the same as for + + 2 effective mass of charged D meson increases with the in- D meson. For instance, at eB = 4mπ , η = 0, and T = 0 crease in magnetic field and hence the magnitude of negative (100) MeV, the mass of D0 meson is 1770 (1784) and 1704 + mass shift of D meson decreases. For example, at η = 0, (1713) MeV for ρN = ρ0 and 4ρ0, respectively, and for eB = 2 ρN = ρ0, and T = 0, the value of mass increase from 1826 6mπ , the value changes to 1766 (1780) and 1694 (1703) MeV. 2 2 to 1834 MeV, when we move from eB = 4mπ to 6mπ .This It is observed that for both pseudoscalar mesons, if we go is because the D+ meson is a charged meson and with the from symmetric matter to highly asymmetric matter, then interaction of a magnetic field additional positive mass shift the effect of temperature become less appreciable except for [see Eq. (31)] comes into the picture. If we do not consider the T = 150 MeV. The isospin asymmetry effects should be quite + additional mass shift, then we observe a negative in-medium visible for D (cd¯) and D0 (cu¯) mesons as they are isospin mass shift which increases with the increase in magnetic field. partner of each other. For nonmagnetic nuclear matter, appre- The density effect on the mass shift of D+ meson is also very ciable isospin effects are observed [28]. In Ref. [28], the mass + prominent as the mass shift of D+ meson increase with the of the D0 meson decreased, whereas the mass of D increases 2 increase in density. For instance, at eB = 4mπ , η = 0, and as we go from symmetric to asymmetric nuclear matter. But in + T = 0 MeV, when we move from ρN = ρ0 to 4ρ0, the values the present case, the asymmetry effects on the D meson are

+ 0 2 TABLE III. Magnetic-ﬁeld-induced mass shift of D and D mesons (in MeV) at eB = 4mπ are compared with the mass shift obtained without magnetic ﬁeld [28]. We have also considered and compared the contribution of other quark condensates (OQC).

η = 0 η = 0.5 T = 0 T = 100 T = 0 T = 100

ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 All condensates m∗∗ −43 −83 −34 −77 −41 −81 −36 −78 D+ m(B = 0) −64 −110 −55 −103 −68 −112 −60 −108 OQC = 0 m∗∗ −60 −92 −51 −85 58 −89 −52 −86 m(B = 0) −62 −101 −53 −94 −66 −104 −58 −99 All condensates m∗ −94 −160 −80 −151 −85 −130 −69 −129 D0 m(B = 0) −92 −163 −79 −153 −81 −141 −72 −137 OQC = 0 m∗ −89 −146 −76 −137 −71 −116 −65 −115 m(B = 0) −90 −154 −77 −144 −79 −133 −69 −128

015202-11 RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020)

+ 0 = 2 TABLE IV. Magnetic-ﬁeld-induced mass shift of D0 and D0 mesons (in MeV) at eB 4mπ are compared with the mass shift obtained without magnetic ﬁeld [24]. We have also considered and compared the contribution of additional condensates.

η = 0 η = 0.5 T = 0 T = 100 T = 0 T = 100

ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 All condensates m∗∗ 93 140 86 135 91 114 87 136 + = D0 m(B 0) 64 125 58 120 68 127 62 123 OQC = 0 m∗∗ 87 133 80 128 85 131 81 129 m(B = 0) 66 138 59 132 70 140 63 137 All condensates m∗ 101 166 90 159 86 143 81 142 0 = D0 m(B 0) 87 162 76 156 78 148 72 143 OQC = 0 m∗ 110 174 99 168 95 156 90 151 m(B = 0) 89 180 79 173 81 164 73 160 compensated for by the additional positive mass shift due to with the increase in temperature while follows the same trend Landau interaction, and hence we see fewer crossover temper- as a function of magnetic field. The asymmetric effects on 0 + ature effects. On the other hand, the effective mass of the D scalar D0 meson are also compensated for by the magnetic meson modifies appreciably in asymmetric nuclear matter and field as for the pseudoscalar case. † shows crossover behavior. It shows a slight increase for high As we pointed out, condensates q iDq N and qiDiDq¯ N temperature, but for low temperature it decreases appreciably. have less effect as compared to qq¯ N condensates on in- 2 For example, at eB = 6mπ , η = 0.5, and T = 0 (100) MeV, medium properties of charmed mesons (see Tables III and themassoftheD0 meson is 1789 (1794) and 1737 (1731) IV for pseudoscalar and scalar mesons, respectively). In these MeV for ρN = ρ0 and 4ρ0, respectively. This crossover is a tables, we have also compared the mass shift with and without reflection of the behavior of quark and gluon condensates in magnetic effect. The results of mass shift in the absence of asymmetric magnetized nuclear matter as shown in Fig. 2. magnetic field have been taken from Ref. [28] (pseudoscalar) As can be seen from Fig. 7, contrary to pseudoscalar and Ref. [24] (scalar). In this comparison, we observed that D mesons, we observed a positive mass shift for scalar the presence of magnetic field affects the mass shift of charged + 0 D0 and D0 mesons. The fact that the effective mass will pseudoscalar and scalar D meson significantly, whereas for decrease or increase depends on the sign of the scattering neutral D meson the effects are good but not prominent length [see Eq. (27)]. In heavy meson-nucleon bound states, as compared to charged meson. This is due to additional the negative or positive sign of scattering length determines positive mass shift in the medium which occurs via Landau whether the DN interactions are attractive or repulsive [63]. + quantization as discussed earlier also. The enhancement in mass of the D0 meson is more than In our knowledge, the in-medium masses of D mesons at 0 the D0 meson as a function of magnetic field. As discussed, finite temperature and density of nuclear matter considering the charged meson experiences additional positive shift due external magnetic fields have not been evaluated yet within to induced Landau levels. For scalar D mesons, we observed any model. In Ref. [15] the mass of pseudoscalar D meson in an opposite behavior of masses as a function of temperature strongly magnetized asymmetric cold nuclear matter has been comparative to the pseudoscalar case. In symmetric nuclear calculated solely in the effective chiral SU(4) model. In this matter, the effective mass of the scalar D meson decreases article, the mass splitting between D+ (D0 ) and D− (D¯ 0 ) has

+ 0 0 + TABLE V. The values of magnetic-ﬁeld-induced shift in decay constants of D , D , D0,andD0 mesons (in units of MeV).

η = 0 η = 0.5 T = 0 T = 100 T = 0 T = 100

2 eB/mπ ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ∗ − − − − − − − − fD+ 4 6.07 8.05 5.18 7.38 5.85 8.05 5.08 7.38 6 −6.31 −8.53 −5.43 −7.89 5.79 −8.53 −5.33 −7.89 f ∗ 4 −9.15 −13.27 −7.71 −12.80 −7.17 −10.57 −6.52 −10.45 D0 0 6 −9.51 −14.81 −8.14 −13.86 −7.14 −10.26 −6.55 −10.86 ∗ f + 4 −9.26 −15.25 −8.29 −14.65 −9.03 −15.04 −8.45 −14.77 D 0 6 −9.49 −15.65 −8.53 −15.09 −9.01 −14.84 −8.45 −14.81 f ∗ 4 −12.37 −20.74 −10.97 −19.88 −10.43 −17.81 −9.73 −17.71 D0 0 6 −12.78 −21.66 −11.39 −20.83 −10.40 −17.53 −9.82 −18.09

015202-12 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020)

N= 0 N=4 0 N= 0 N=4 0 0 0 2540 2540 =0 (a) =0 (b) =0 (a) =0 (b) 2520 2520 -2 -2 2500 2500 -4 -4 (MeV)

(MeV) 2480 2480 * ** + 0

-6 -6 + D D f 2460 2460 m -8 -8 2440 2440

-10 -10 2420 2420 1234567 1234567 1234567 1234567 2540 2540 0 0 =0.5 =0.5 =0.5 =0.5 (c) (d) (c) (d) 2520 2520 -2 -2 2500 2500 -4 -4 (MeV) 2480 2480 (MeV) ** 0 * + +

-6 -6 D D 2460 2460 f m -8 -8 2440 2440 2420 2420 -10 -10 1234567 1234567 1234567 1234567 2560 2560 -4 -4 =0 (e) =0 (f) =0 (e) =0 (f) 2540 2540 -6 -6 2520 2520 -8 -8 2500 2500 (MeV) * (MeV) 0 2480 2480

-10 -10 0 * D 0 D

m 2460 2460 f -12 -12 2440 2440 -14 -14 2420 2420 -16 -16 1234567 1234567 1234567 1234567 2560 2560 =0.5 =0.5 -4 -4 (g) (h) =0.5 (g) =0.5 (h) 2540 2540 -6 -6 2520 2520 -8 -8 2500 2500 (MeV) *

0 2480 2480 (MeV) -10 -10 0 D * 0

m 2460 2460 D f -12 -12 2440 2440 -14 -14 2420 2420 1234567 1234567 -16 -16 2 2 1234567 1234567 eB/m eB/m 2 2 eB/m eB/m T=0 MeV T=100 MeV T=50 MeV T=150 MeV T=0 MeV T=100 MeV T=50 MeV T=150 MeV + 0 FIG. 7. The effective mass of scalar D0 (charged) and D0 (un- FIG. 6. The shift in decay constant of pseudoscalar D+ (charged) charged) mesons is plotted as a function of magnetic ﬁeld eB under and D0 (uncharged) mesons is plotted as a function of magnetic ﬁeld different conditions of medium. eB under different conditions of medium.

the magnetic field. In our future work, we will also include the mixing effects of D mesons in the presence of magnetic field. been calculated using the self-energy of D mesons. In cold and In Fig. 6 (Fig. 8), for a given value of density, temperature, ∗ asymmetric nuclear matter, the mass shift of D mesons has and isospin asymmetry the shift in decay constant fD of been compared with and without taking contributions from pseudoscalar (scalar) D meson is plotted as a function of mag- anomalous magnetic moment (AMM). In Ref. [66], authors netic field. The values of in-medium shift in decay constant of used hadronic QCD sum rules in which the magnetic-field ef- pseudoscalar and scalar D meson in the presence of magnetic fects are introduced on both phenomenological as well as the field and other medium properties are shown in Table V.In ∗ OPE side to evaluate the effect of magnetic field on D meson symmetric nuclear matter, the magnitude of fD increases properties (at zero density and temperature). The additional as a function of magnetic field, whereas it decreases with mixing effects are examined on the phenomenological side by the increase in temperature. But for η = 0.5, the crossover adding a spectral ansatz term. Besides the mixing effects, an behavior is observed as was the case for effective masses. This additional perturbative positive mass shift is also found due to is because the shift in decay constant is calculated using the

015202-13 RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020)

N= 0 N=4 0 N= 0 N=4 0 -6 -6 120 120 =0 (a) =0 (b) =0 (a) =0 (b) -8 -8 100 100 80 80 -10 -10 (MeV)

* 60 60 0 + -12 -12 (3686) (MeV) D f 40 40 -

-14 -14 D + 20 20 D -16 -16 0 0 1234567 1234567 1234567 1234567 -6 -6 =0.5 =0.5 120 120 (c) (d) =0.5 (c) =0.5 (d) -8 -8 100 100

-10 -10 80 80 (MeV)

* 60 60 0 + -12 -12 D (3686) (MeV) f 40 40 -

-14 -14 D + 20 20 D -16 -16 0 0 1234567 1234567 1234567 1234567 -8 -8 =0 =0 120 120 -10 -10 (f) =0 (e) =0 (f) -12 -12 100 100 -14 -14 80 80 (MeV) -16 -16 * 0 0

D -18 -18 (3686) (MeV)

f 60 60 -20 -20 0

D 40 40 -22 -22 0

(e) D -24 -24 1234567 1234567 20 20 1234567 1234567 -8 -8 =0.5 =0.5 (h) 120 120 -10 -10 =0.5 (g) =0.5 (h) -12 -12 100 100 -14 -14

(MeV) -16 -16 80 80 * 0 0

D -18 -18 f (3686) (MeV) 60 60 -20 -20 0

D 40 40 -22 -22 0

(g) D -24 -24 1234567 1234567 20 20 1234567 1234567 eB/m 2 eB/m 2 eB/m 2 eB/m 2 T=0 MeV T=100 MeV T=50 MeV T=150 MeV T=0 MeV T=100 MeV T=50 MeV T=150 MeV + FIG. 8. The shift in decay constant of scalar D0 (charged) and D0 (uncharged) mesons is plotted as a function of magnetic field eB FIG. 9. The effective decay width of the charmonium state 0 ψ ¯ under different conditions of medium. (3686) decaying into DD pairs is plotted as a function of magnetic field eB under different conditions of medium. shift in effective mass [see Eq. (29)] in the QCDSR. It may decay width of charmonia ψ(3686) and ψ(3770), respec- be noted that for scalar mesons, the shift in decay constant is tively, decaying into DD¯ mesons as a function of magnetic negative despite the positive mass shift. field in the nuclear matter. In Table VI, we have listed the observed values of in-medium decay width of charmonia decaying to D+D− and D0D¯ 0 pairs. As one can see from C. In-medium decay width of higher charmonium states Fig. 9, a nonzero value of decay width of ψ(3686) to DD¯ Now we will see how the obtained magnetically induced pair is observed because the threshold value of DD¯ pair masses of pseudoscalar D meson affect the in-medium decay generation is less than the mass of parent meson. For decay ψ ,ψ ,χ , ψ → + − width of higher charmonia [ (3686) (3770) c0(3414) channel (3686) D D , the value of partial decay width χ ¯ ψ c2(3556)] decaying to DD pairs. We have neglected the of (3686) decreases with the increase in magnetic field and medium modifications of the parent charmonia in the present it becomes zero for high values of magnetic field. The zero work. In Figs. 9 and 10, we have shown the variation of partial decay width arises when the threshold value of the D+D− pair

015202-14 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020)

TABLE VI. The values of in-medium decay widths (in MeV) of higher charmonium states ψ(3686), ψ(3770) to D+D− and D0D¯ 0 pairs for different conditions of the medium.

η = 0 η = 0.5 T = 0 T = 100 T = 0 T = 100

2 eB/mπ ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0 ρ0 4ρ0

D+D− [ψ(3686)] 0 75 97 59 99.9 82 96 69 98 4 3397129328961593 6 14930 86482082 D+D− [ψ(3770)] 0 71 35 74 41 69 32 73 36 4 7458716374617163 6 7163646768696369 ψ D0D¯ 0 [ (3686)] 0 100 47 94 58 95 71 86 76 4 101 45 98 55 95 78 90 79 6 100 35 100 44 95 81 90 75 ψ D0D¯ 0 [ (3770)] 0 51 1.3 62 4.6 60 10 67 14 4 451 574 61156515 6 310 531 61176513

becomes more than the parent meson’s mass. The decrement However, for η = 0.5, it increases for lower values of in decay width is due to the calculation of partial decay temperature and decreases for higher values of temperature. width [see Eq. (52)] which depends on the energy ED+/D− that These observations are contradictory to the explanation ∗∗ further depends on the in-medium mass mD+/D− . As discussed, given in the previous paragraph as the high decrease in the in the present work, we have taken averaged meson current mass causes a decrease in decay width, too. In Eq. (52), for D+ and D− mesons [see Eq. (20)] and hence the effective the decay width is the product of Gaussian and polynomial masses of D+ and D− mesons will be the same (and similarly expression. In high density, the polynomial part of the decay for scalar D0 and D¯ 0 )[24,28,63]. Since the effective mass of width dominates the Gaussian part. In Table VI,wehavealso charged mesons increases in magnetic field, this will lead to compared the results with and without magnetic effects. We an increase of threshold value of D+ D− pairs and hence a see that the increase in magnetic field causes more decrease drop in dcay width. From Fig. 5, we can see that the mass of in the value of decay width in low density as compared to D+ mesons (threshold value of D+ D− pairs) decreases with high density for the D+D− decay channel, and for a neutral the increase in density, whereas it increases with the increase D meson pair, the decay width increases for low density and in temperature. This explains why the partial decay width of decreases for high density. ψ(3686) increases with the increase in density, whereas it In Fig. 10, we plot the results of decay width for de- decreases as a function of temperature. cay channels ψ(3770) → D+D− and ψ(3770) → D0D¯ 0 for When we move from isospin-symmetric medium to - the same parameters. For the former case, at lower density asymmetric medium, we observed appreciable effects for and particular value of temperature, the decay probability density and magnetic field, whereas the least effects are decreases slowly with the increase in magnetic field. The shown as a function of temperature. This is the reflection of temperature and asymmetric effects are appreciable in the the observation of the in-medium mass of daughter meson high magnetic-field regime. At high density, the trend of in the respective medium, which occurs due to the unbal- decay width is exactly opposite as a function of magnetic anced behavior of the scalar density of neutrons and protons field and temperature. The values of decay width increase under magnetic effects [74]. In the decay channel ψ(3686) → with the increase in magnetic field which is due to the higher D0D¯ 0,inFig.5(c) [Fig. 5(g)]atη = 0(0.5), we observe that mass of parent meson ψ(3770). The mass of parent meson the decay width remains almost the same (decrease) for low rectifies the center-of-mass momentum pD, which results in values of temperature, whereas for higher values of temper- the modification of the Gaussian expression. The interplay ature it increase as a function of magnetic field. Note that between Gaussian and polynomial expression leads to the the trend of decay width of decay channel ψ(3686) → D0D¯ 0 above observations. In the same figure, for the decay channel ψ → 0 ¯ 0 η = with magnetic field is opposite to decay channel ψ(3686) → (3770) D D ,at 0, the decay width decreases with D+D−. This is because as D0 and D¯ 0 are neutral charged increasing magnetic field for high as well as low density. mesons, they do not exhibit Landau quantization and hence However, for high density the decay probability is much − y2 mass does not increase but decreases as a function of magnetic lower due to the presence of the Gaussian term [e 2(1+2r2 ) ] field as discussed in the last subsection in detail. in Eq. (53) and the larger drop in mass of the neutral D0 In the high-density regime, at η = 0, the partial decay meson. In highly asymmetric matter, the value of the decay width of the parent meson decreases with the increase in probability increases for low temperature and decreases for magnetic field and increases as a function of temperature. higher temperature as a function of magnetic field. This result

015202-15 RAJESH KUMAR AND ARVIND KUMAR PHYSICAL REVIEW C 101, 015202 (2020)

N= 0 N=4 0 appreciably and the threshold value become less than the 100 100 mass of the parent meson, then decay is possible. We have =0 (a) =0 (b) observed zero decay probability for the χ(3414) mesons in 80 80 D+D− pairs at all conditions of the medium. But for decay 0 ¯ 0 60 60 product D D , we have observed ﬁnite decay probability in the high-density regime for the nonmagnetic case. For

(3770) (MeV) 40 40 example, in nonmagnetic cold nuclear matter at η = 0 and

- ρ = ρ ρ χ

D N 0 (4 0 ), the value of the decay width of (3414) is

+ 20 20

D 0 (5.6) MeV. Moreover, for decay channel χ(3556) → DD¯ , 0 0 we have observed a small finite decay width for low values 1234567 1234567 2 of magnetic field up to eB/mπ = 2 at high density only. For 100 100 =0.5 =0.5 η = = ρ (c) (d) instance, at 0, T 0, and 4 0, we observed the values 2 80 80 of decay width of χ(3556) to be 124 (158) MeV at eB/mπ = 4 (6). The values of decay width decrease with the increase in 60 60 temperature but the trend with respect to the magnetic field remains same, whereas for asymmetric matter it decreases (3770) (MeV) 40 40

- for lower temperature and increases for higher temperature D

+ 20 20 with respect to the magnetic field. For example, at η = 0.5, D ρ = ρ = 0 0 N 4 0, and T 0 MeV, the value of the decay width is 2 1234567 1234567 41 (35) MeV at eB/mπ = 4 (6), whereas at T = 100 MeV 100 100 =0 =0 the value of the decay width changes to 39 (47) MeV. The (e) (f) value of the decay width of both χ mesons increases in 80 80 the symmetric magnetic nuclear matter as compared to zero 60 60 magnetic-field data, whereas for asymmetric nuclear matter it shows asymmetric variation due to δ-field corrections. (3770) (MeV) 40 40 In Ref. [67], the magnetic-field-induced decay width of 0

D 20 20 these four charmonium states has been calculated by consid- 0 D ering the in-medium masses of the charmonia as well as D 0 0 and D¯ mesons in the combined approach of the chiral SU(3) 1234567 1234567 3 100 100 model and the P0 model at zero temperature only, whereas as =0.5 (g) =0.5 (h) discussed above in detail, our calculations are done at ﬁnite 80 80 temperature which is important for heavy-ion collisions.

60 60 IV. CONCLUSIONS

(3770) (MeV) 40 40

0 In the present investigation, we calculated the modiﬁcation D 20 20 0

D in the in-medium masses and decay constants of pseudoscalar 0 0 and scalar D mesons under the effect of external magnetic 1234567 1234567 ﬁeld at ﬁnite temperature, asymmetry, and density of the 2 2 eB/m eB/m nuclear medium. To calculate the in-medium mass, we used

T=0 MeV T=100 MeV the combined approach of QCD sum rules and the chiral T=50 MeV T=150 MeV SU(3) model. In nuclear matter, the external magnetic field interacts with charged proton and uncharged neutron (due to FIG. 10. The effective decay width of the charmonium state nonzero anomalous magnetic moment) which results in the ψ(3770) decaying into DD¯ pairs is plotted as a function of magnetic field eB under different conditions of medium. modification of scalar and vector densities of nucleons. This magnetic-field-induced density is used to evaluate the coupled scalar fields of a chiral model which are further used to calcu- is a reflection of the variation of in-medium mass in asym- late the medium modified scalar quark and gluon condensates metric magnetized nuclear matter. With the comparison of [74]. We found prominent effects of magnetic field on the ψ → + − + + decay width of (3770) D D in the absence of magnetic charged D and D0 mesons, whereas for neutral D mesons, field, one can see that the value of decay width at a particular the effects are smaller. We found a negative (positive) mass combination of medium parameters increases in the magnetic shift for pseudoscalar (scalar) D mesons. The temperature and field due to Landau levels and Gaussian interaction, whereas density effects are also quite appreciable. The intervention of it decreases for the D0D¯ 0 case. magnetic field depletes the effect of isospin asymmetry effects We have also calculated the decay widths of excited char- for the charged one but for uncharged mesons it was quite χ χ χ χ monium states c0 and c2 . The vacuum mass of c0 and c2 appreciable with crossover effects. As an application part, we is less than the threshold value for the decay products (DD¯ ) calculated the in-medium decay width of higher charmonia 3 and therefore the decay χ(3414) → DD¯ and χ(3556) → DD¯ in the P0 model. In this calculation, we neglected the mass is not possible. However, if the mass of the D meson drops modification of parent charmonia. We observed appreciable

015202-16 ANALYSIS OF PSEUDOSCALAR AND SCALAR D … PHYSICAL REVIEW C 101, 015202 (2020) changes in the decay widths of ψ(3686) and ψ(3770) but ACKNOWLEDGMENT less modiﬁcation in χ(3414) and χ(3556). The calculated R.K. sincerely acknowledges support from Ministry of decay width may suppress the J/ψ production and hence may Science and Human Resources Development (MHRD), Gov- decrease its yield. The experimental veriﬁcation of obtained ernment of India, via Institute fellowship under the National results can be done in facilities such as CBM, PANDA, Institute of Technology Jalandhar. J-PARC, and NICA.

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