J/Ψ Mass Shift and J/Ψ-Nuclear Bound State Kazuo Tsushima∗, D. H. Lu†, Gastao Krein∗∗ and Anthony W. Thomas∗ ∗CSSM, School of Chemistry and Physics, University of Adelaide, Adelaide SA 5005, Australia †Department of Physics, Zhejiang University, Hangzhou 310027, P.R.China ∗∗Instituto de Física Teórica, Universidade Estadual Paulista Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, São Paulo, SP, Brazil Abstract. We calculate mass shift of theJ/Ψ meson in nuclear matter arising from the modification of DD,DD∗ andD∗D∗ meson loop contributions to theJ/Ψ self-energy. The estimate includes the in-mediumD andD∗ meson masses consistently. TheJ/Ψ mass shift (scalar potential) calculated is negative (attractive), and is complementary to the attractive potential obtained from the QCD color van der Waals forces. Some results for theJ/Ψ-nuclear bound state energies are also presented. Keywords: J/Ψ-nuclear bound state, in-mediumJ/Ψ self-energy, Quark-meson coupling model. PACS: 21.85.+d,21.65Jk,21.65.Qr,24.85.+P INTRODUCTION The properties of charmonia and charmed mesons in a nuclear medium (nuclei) are still poorly known. However, with the 12 GeV upgrade of the CEBAF accelerator at the Jefferson Lab and with the construction of the FAIR facility, we expect tremendous progress in understanding the properties of these mesons ina nuclear medium. The new facilities will be able to produce low-momenta charmonia and charmed mesons such asJ/Ψ, ψ, D and D∗ in an atomic nucleus. One of the major challenges is to find appropriate kinematical conditions to produce these mesons essentially at rest, or with small momentum relative to the nucleus. Amongst many new possible experimental efforts, searching for theJ/Ψ-nuclear bound states may be one of the most exciting ones. Finding of such bound states would provide us with evidence for the negative mass shift of theJ/Ψ meson discussed here. Studies ofJ/Ψ and ηc binding in nuclei is a relatively long history. The original suggestion [1] that multiple gluon QCD van der Waals forces would be capable of binding a charmonium state, estimated a binding energy as large as 400 MeV in an A= 9 nucleus. However, the same approach but taking into account the nucleon density distributions in the nucleus, Ref. [2] found a maximum of 30 MeV binding energy in a large nucleus. Based on Ref. [3], which showed the mass shift of charmonium in nuclear matter is possible to be expressed similar to the usual second-order Stark effect due to the chromo-electric polarizability of the nucleon, the authors of Ref. [4] obtained a 10 MeV binding forJ/Ψ in nuclear matter, in the limit of an infinitely heavy charm quark mass. Following the same procedure, but keeping the charm quark mass finite and using realistic charmonium bound-state wave-functions, Ref. [5] found 8 MeV binding energy forJ/Ψ in nuclear matter. There are some other studies on theJ/Ψ mass in nuclear medium. QCD sum rules T(R)OPICAL QCD 2010 AIP Conf. Proc. 1354, 39-44 (2011); doi: 10.1063/1.3587583 © 2011 American Institute of Physics 978-0-7354-0912-5/$30.00 39 estimated aJ/Ψ mass decrease in nuclear matter ranging from 4 MeV to 7 MeV [6, 7, 8], while an estimate based on color polarizability [9] gave larger than 21 MeV. Since theJ/Ψ and nucleons have no quarks in common, the quark interchangeor the effective meson exchange potential should be none or negligible to first order in elastic scattering [1]. Furthermore, there is no Pauli blocking even at the quark level. Thus, if the J/Ψ-nuclear bound states are formed, the signal for these states will be sharp and show a clear narrow peak in energy dependence of the cross section. In this contribution we report on our study [10] made for the mass shift ofJ/Ψ. In addition we also report some results obtained for theJ/Ψ-nuclear bound states [11]. J/Ψ MASS SHIFT IN SYMMETRIC NUCLEAR MATTER We study first theJ/Ψ meson mass shift in medium arising from the modification of DD,DD∗ and D∗D∗ meson loop contributions to theJ/Ψ self-energy [10]. As an example, theDD-loop contribution to theJ/Ψ self-energy is shown in Fig. 1. D̄ D J/Ψ J/Ψ FIGURE 1. DD-loop contribution to theJ/Ψ self-energy. To calculate theJ/Ψ self-energy due to theDD,DD∗ andD∗D∗ meson loops, we use the effective Lagrangian densities (in the following we denote byψ the field representing J/Ψ), LψDD = igψDD ψµ [D ( ∂µD ) − ( ∂µD ) D ] , (1) LψDD∗ = gψDD∗ mψ εαβ µν ( ∂ αψβ )[( ∂ µD∗ ν ) D+D(∂ µD∗ν) ] , (2) LψD∗D∗ = igψD∗D∗ { ψµ [( ∂µD∗ ν ) D∗ ν −D∗ ν (∂µD∗ ν ) ] + [( ∂µψν)D∗ν −ψν (∂µD∗ν )] D∗µ + D∗ µ [ψν (∂µD∗ ν ) − ( ∂µψν)D∗ ν ]} , (3) with the coupling constant value,gΨDD = gΨDD∗ = gΨD∗D∗ = 7.64. Only the scalar part contributes to theJ/Ψ self-energy, and the scalar potential for theJ/Ψ meson is the difference of the in-medium,m∗ ψ , and vacuum,mψ , masses ofJ/Ψ, ∆m= m∗ ψ −mψ , (4) with the masses obtained from m2 ψ = (m0 ψ) 2+Σ(k2 = m2 ψ) . (5) 40 Herem0 ψ is the bare mass andΣ(k2) is the totalJ/Ψ self-energy obtained from the sum of the contributions from theDD, DD∗ and D∗D∗ loops. The in-medium mass,m∗ ψ , is obtained likewise, with the self-energy calculated withmedium-modifiedD andD∗ meson masses. In the calculation we use phenomenological form factors to regularize the self-energy loop integrals following a similar calculation for theρ self-energy [12]: uD,D∗(q2) = ( Λ2 D,D∗ +m2 ψ Λ2 D,D∗ +4ω2 D,D∗(q2) )2 . (6) For the verticesJ/ΨDD,J/ΨDD∗ andJ/ΨD∗D∗, we use the form factorsFDD(q2) = u2 D(q 2), FDD∗(q2) = uD(q2)uD∗(q2), andFD∗D∗(q2) = u2 D∗(q2), respectively, withΛD andΛD∗ being the cutoff masses, and the common value,ΛD = ΛD∗ is used. 0 0.5 1 1.5 2 2.5 3 ρ B /ρ0 (ρ0 =0.15 fm -3 ) 1800 1900 2000 m * D , D * ( M eV ) m* D* m* D 0 0.5 1 1.5 2 2.5 3 ρ B /ρ0 (ρ0 =0.15 fm -3 ) -50 -40 -30 -20 -10 0 m * Ψ − m Ψ ( M eV ) cutoff 1000 MeV cutoff 1500 MeV cutoff 2000 MeV cutoff 3000 MeV DD+DD*+D*D* loops FIGURE 2. In-medium masses of theD∗ (the solid line) andD (the dashed line) mesons calculated in the QMC model (left panel), and theJ/Ψ potential in symmetric nuclear matter (right panel). In theright panel the solid, dotted, dashed, and dash-dotted curves correspond to the common cut-off values in the dipole form factors in Eq. (6), 1000, 1500, 2000, and 3000 MeV, respectively. To calculate the in-mediumJ/Ψ self-energy arising from theDD,DD∗ and D∗D∗ meson loops, we need to include the in-medium masses ofD and D∗ mesons con- sistently [10, 13]. For this purpose, we rely on the quark-meson coupling (QMC) model [14, 15]. The QMC model is a quark-based, relativisticmean field model of nu- clear matter and nuclei [14, 15]. Relativistically moving confined light quarks in the nu- cleon bags self-consistently interact directly with the scalar-isoscalarσ , vector-isoscalar ω, and vector-isovectorρ mean fields generated by the light quarks in the (other) nu- cleons. These meson mean fields are responsible for the nuclear binding. The direct interaction between the light quarks and the scalarσ field is the key of the model, which induces thescalar polarizabilityat the nucleon level, and generates the nonlinear scalar potential (effective nucleon mass), or the density (σ -field) dependentσ -nucleon cou- pling. This gives a novel, new saturation mechanism for nuclear matter. The model has opened tremendous opportunities for the studies of finite nuclei and hadron properties in a nuclear medium (nuclei), based on the quark degrees of freedom. Many successful applications of the model can be found in Ref. [15]. 41 TABLE 1. In-mediumJ/Ψ massm∗ J/Ψ and the individual loop contributions to the mass difference ∆mat nuclear matter density, for different values of the cutoffΛD(= ΛD∗). All quantities are in MeV. ΛD m∗ J/Ψ DD DD∗ D∗D∗ ∆m 1000 3081 −3 −2 −11 −16 1500 3079 −3.5 −2.5 −12 −18 2000 3077 −4 −3 −13 −20 3000 3072 −6.5 −5 −12.5 −24 In Fig. 2 we show the in-medium masses of theD∗ and D mesons (left panel) calculated in the QMC model [10, 13], and theJ/Ψ potential in symmetric nuclear matter (right panel). To see the ambiguity due to the cut-offvalues in the form factors, we calculate the potential with the cut-off values,ΛD = ΛD∗ = 1000,1500,2000, and 3000 MeV. In addition we list each meson loop contribution at nuclear matter densityρ0 (= 0.15 fm−3) in Table 1. We regard the results with the cut-off values 1500 and 2000 MeV as our predictions. At normal nuclear matter density these correspond to about 18 and 20 MeV attractions, respectively. J/Ψ-NUCLEAR BOUND STATE First, we show theJ/ψ-nuclear potentials calculated [11] based on the method described in the previous section. As examples, we show in Fig. 3 the potentials felt by theJ/Ψ meson in4He (left panel) and208Pb (right panel) nuclei. 0 1 2 3 fm -30 -25 -20 -15 -10 -5 0 5 J/ Ψ po te nt ia l ( M eV ) cutoff 1500 MeV cutoff 2000 MeV 4 He 0 5 10 fm -25 -20 -15 -10 -5 0 5 J/ Ψ po te nt ia l ( M eV ) cutoff 1500 MeV cutoff 2000 MeV 208 Pb FIGURE 3. J/Ψ potentials in4He (left panel) and208Pb (right panel) nuclei for two values of the cutoff in the form factors, 1500 (the solid line) and 2000 (the dashed line) MeV. The nucleon density distributions for nuclei (208Pb nucleus in this report) are also calculated within the QMC model [16]. For a4He nucleus, we use the parametrization for the density distribution obtained in Ref. [17]. Using a local density approximation we have calculated theJ/Ψ potentials in nuclei. With the potentials obtained in this manner, 42 we then calculate theJ/Ψ-nuclear bound state energies. The detail and complete results will be reported elsewhere [11]. We follow the procedure applied in the earlier work [18] for the η- andω-nuclear bound states. In this study we consider the situation that the J/Ψ meson is produced nearly at rest (recoilless kinematics in experiments). Then, it should be a very good approximation to neglect the possible energy difference between the longitudinal and transverse components [19] of theJ/Ψ wave function. In addition we consider theJ/Ψ meson is produced in medium and heavy mass nuclei,40Ca, 90Zr and 208Pb, where possible center-of-mass corrections should be small. Thus, after imposing the Lorentz condition, to solve the Proca equation, aside from a possible width, is equivalent to solve the Klein-Gordon equation, [ ∇2+E2 Ψ −m∗2 Ψ (r) ] φΨ(r) = 0, (7) whereEΨ is the total energy of theJ/Ψ meson, andm∗ Ψ(r) is the in-medium mass of the J/Ψ in a nucleus. Since in free space the width ofJ/Ψ meson is∼ 93 keV [20], we can ignore this tiny natural width. Thus, we may simply solve theKlein-Gordon equation Eq. (7) without worrying about the width, under the situation we consider now. We list in table 2 the bound state energies obtained. The results show that theJ/Ψ meson is expected to form nuclear bound states. TABLE 2. Bound state energies obtained for theJ/Ψ me- son. With the situation that theJ/Ψ meson is produced in recoilless kinematics, the width due to the strong interactions is all set to zero, as well as its natural width of∼ 93 keV. ΛD,D∗ = 1500 MeV ΛD,D∗ = 2000 MeV E (MeV) E (MeV) 40 Ψ Ca 1s -15.15 -17.42 1p -11.20 -13.32 1d -6.91 -8.85 90 Ψ Zr 1s -16.40 -18.69 1p -13.92 -16.14 1d -11.08 -13.21 208 Ψ Pb 1s -16.80 -19.06 1p -15.34 -17.57 1d -13.62 -15.81 SUMMARY AND CONCLUSION We have estimated mass shift of theJ/Ψ meson in nuclear matter arising from the modification ofDD,DD∗ andD∗D∗ meson loop contributions to theJ/Ψ self-energy, consistently including the in-medium masses of theD and D∗ mesons calculated in the quark-meson coupling model. Then, we have calculated the J/Ψ-nuclear potentials using a local density approximation, where the nuclear density distributions are also calculated within the quark-meson coupling model. We emphasize that, in the model, all the coupling constants between the applied meson mean fields and the light quarks 43 in the nucleon, as well as those in theD andD∗ mesons, are all equal and calibrated by the nuclear matter saturation properties. Using theJ/Ψ-nuclear potentials obtained in this manner, we have solved the Klein-Gordon equation which is reduced from the Proca equation, and obtained theJ/Ψ-nuclear bound state energies for40Ca, 90Zr and 208Pb nuclei, where possible center-of-mass corrections and in-medium width, which are expected to be small, are all neglected. Our results show that theJ/Ψ meson is expected to form nuclear bound states. In the future we need to study the case that theJ/Ψ meson is produced with a finite momentum. In this case theJ/Ψ self-energy in nuclear medium, applied to calculate the J/Ψ potential, will have an imaginary part. Then,J/Ψ meson will naturally get a momentum or energy dependent finite width. 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