Lifetime of Antibaryon Bound in Finite Nuclei

Transcription

1 Commun. Theor. Phys. (Beijing, China) 53 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 1, January 15, 2010 Lifetime of Antibaryon Bound in Finite Nuclei CHEN Xi (í ), LI Ning (ÓÛ), YAO Hai-Bo ( ), QIN Xu-Ming ( Ê ), and WU Shi-Shu ( ) Center for Theoretical Physics and School of Physics, Jilin University, Changchun , China (Received September 7, 2009; revised manuscript received November 9, 2009) Abstract The study of finite nuclei containing antibaryon(s) in addition to nucleons is an interesting topic in nuclear physics. The calculation of the lifetime of an antibaryon embedded in a nucleus was performed in the framework of the standard quantum field theory. It was shown that the annihilation probability of the antibaryon in nuclei is strongly dependent on the effective masses of mesons involved in the annihilation channels. The contribution of the Dirac sea to the annihilation probability makes the lifetime of the antibaryon short. If the Dirac sea effect is neglected, the lifetime of the bound antibaryon tends to be longer with the nuclear density increasing. Particularly, when the nuclear density is larger than a critical value, the antibaryon may exist stably in a nucleus. PACS numbers: t, k, Gv, n Key words: antinucleon in finite nuclei, lifetime of bound antinucleon, annihilation probability of antibaryon 1 Introduction The properties of antibaryon bound in dense nuclear environment are very interesting in nuclear physics. [1 5] Unfortunately the experimental information on the antinucleon effective potential in nuclei is obscured by the strong absorption caused by annihilation. The real part of the antiproton effective potential is as large as MeV and its uncertainty reaches 100% in the deep interior of nucleus. [6 7] As a result it was believed that the baryonium can not survive long enough to be observed. [8] To explore the related experimental phenomena further, the determination of the lifetime of antibaryon embedded in finite nuclei is still a challenging issue. Recently it was proposed that a real antibaryon which is implanted in finite nuclei may survive for a long time by using the phenomenological approach. [9] The lifetime τ of the bound antibaryon was estimated as τ = σ A v rel ρ B, where σ A is a parameter fitted the data of annihilation cross section, v rel is the relative velocity, and ρ B is the nuclear density. Due to the suppressed phase space of the available energy, the annihilation branch ratio could be easily suppressed by a factor of 20 30, and the lifetime was estimated to be fm/c. This amplitude of the lifetime is the minimum to be observed in experiment. Two other methods were also used to estimate the lifetime. The kinetic approach [10] was adopted to calculate the rate of the reaction B N c, where B is baryon (p, n), N is antibaryons ( p, n, Λ), and c is the final meson channel in dense nuclear environment. [11] On the assumptions that the transition probability does not depend sensitively on the particle momentum and that the annihilation branch ratio in dense nuclear matter is proportional to that in vacuum, the partial annihilation width for a given channel (c) was estimated as: Γ c = Γ 0 B c λ c J BN, where Γ 0 is a constant, B c is the branch ratio in vacuum, λ c is available phase space factor, and J BN is the overlap integral appropriate for localized density distributions. By summing up all partial contributions, the total annihilation width is obtained as: Γ total = Γ c. Then the lifetimes was given as fm/c. The dynamical transport model was applied to study the formation and decay of the compressed p-nuclear system. [12] In the Giessen Boltzmann Uehling Uhlenbeck model, [13] the coupled Vlasov equations for nucleon and antiproton were solved. It was demonstrated that the time scale of cold compression process is comparable with the lifetime of antiproton in dense nuclear environment. To estimate the lifetime of the bound antibaryon further, it is desirable to investigate it based on a full quantum field theory. In this paper, the calculation of the antibaryon lifetime in finite nuclei will be performed in the framework of the standard quantum field theory. We show that the annihilation probability of the antibaryon embedded in nuclei is strongly dependent on the effective masses of mesons involved in the annihilation channels and the contribution of the Dirac sea to the annihilation probability makes the lifetime of the antibaryon short. Without the Dirac sea effect, the lifetime of the bound antibaryon tends to be very long near a critical nuclear density and it may exist stably when the density is larger than this critical value. 2 Theoretical Framework In the framework of quantum hadron dynamics, [14 18] the system containing nucleons and antinucleons is usually described by the following Lagrangian density: L = ψ j (iγ µ µ m j )ψ j j=n, B µ σ µ σ 1 2 m2 σσ 2 b 3 σ3 c 4 σ4 Supported by National Natural Science Foundation of China under Grant No and Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No

2 No. 1 Lifetime of Antibaryon Bound in Finite Nuclei ωµν ω µν m2 ω ωµ ω µ + d 4 (ωµ ω µ ) ρ µν ρ µν m2 ρ ρ µ ρ µ + j=n, B ψ j ( g σj σ g ωj ω µ γ µ g ρj ρ µ γ µ τ j e j A µ1 + τ 3 γ µ )ψ j F µνf µν, (1) where ψ represents the baryon field, N and B are nucleons (p, n) and antibaryons ( p, n), respectively. The meson degrees of freedom include σ (J π = 0 + ), ω (J π = 1 ), and ρ (J π = 1 ). A µ is the Coulomb field. e j is the charge number of baryon j, τ is the isospin vector, and the vector field tensor is defined as F µν = µ F ν ν F µ. The g mj (m = σ, ω, ρ) is the coupling parameter between meson and baryon. From the above Lagrangian density L, the Hamiltonian density H can be obtained. By using the S-matrix, the partial annihilation probability W c for a certain final meson channel is written as W c = d 3 V S fi 2 p f 2π 3 (2) T f with S fi = δ fi + ( i) n d 4 x 1 n! n d 4 x n f T[H 1 (x 1 ) H n (x n )] i, (3) where S fi is the S-matrix for a given Feynman diagram. i is the initial baryon and antibaryon bound state, and f is the final meson state. The T in Eq. (2) is the time factor, and the T in Eq. (3) is the time-ordered product. For instance, the lowest order S-matrix of p + p π + π + channel is obtained as: S = igπ 2 2π ωk1 ω k2 M 2V δ(e p + E p ω k1 ω k2 ) (4) with M = η and µ k1 ( p α p η )µ k2 (p η p β ) E β ω k2 E η + iε + µ k1( p α p η )µ k2 ( p η p β ) E β ω k2 Ēη iε + µ k2( p α p η )µ k1 (p η p β ) E β ω k1 E η + iε + µ k2( p α p η )µ k1 ( p η p β ) E β ω k1 Ēη iε, (5) µ k ( p α p η ) = g π d 3 x ψ α (x)γ 5 ψ η (x)e ikx, (6) where µ k has been expanded to the bound state, and k is the momentum of π meson. The subscripts (α, β) represents the initial state of baryons, and η is the medium state of baryon propagator. The meson energy is given by ω k = m 2 + k 2, where m is the meson effective mass. The meson effective mass is sensitive to the nuclear density. So the meson generating threshold, the final meson momentum, and the partial annihilation probability W c are strongly affected by the nuclear density. The total annihilation probability W total = c W c can be calculated and the lifetime τ is the reciprocal of the total annihilation probability, τ = 1/W total. Therefore the kinds and amounts of the partial annihilation channels are crucial to the lifetime of the antibaryon in the finite nuclei environment. The annihilation of the antibaryon with one of the nucleons in 16 O + N produces many kind of particles with different possible states. It leads the contributions of many channels to the annihilation probability. The available energy and the final particle generating threshold determine the possible kind of final particles and their momentum and energy. In our picture, the annihilation occurs in finite nuclei environment, we take account into the following process. The survival nucleons of the finite nuclei absorb the bounce momentum to fulfill the conservation of 4- momentum, and the single meson final state was reckoned in the summation of W total. But the contribution of the single final state diagram is zero in vacuum. Since we focus on the annihilation process, the following channels were ignored, such as the transition p + p p + p, the hyperon production p + p Y + Ȳ, and the transition through the medium doorway state p + p M i + M j c, [8] where Y and Ȳ are the hyperons, M i and M j are the medium states of meson. Only the final states composed of mesons are taken account into here. In the multi-meson final state calculation, the lowest order S-matrix for each Feynman diagram was considered. In our treatment, we have used the G-parity. The G-parity is an internal symmetry of the exchange meson in strong interaction. It is the combination of the isospin symmetry and the C-conjugation rule, and is usually defined as G = Ce iπi2, where I 2 is the second axis in isospin space. By using G-parity transformation, the wave function of anti-nucleon and nucleon are connected. The positive-energy anti-nucleons solutions of Dirac Equation and the nuclear potential for antibaryons are obtained by this way. [21] Also we apply the relativistic mean field (RMF) approximation. [14 18] The nucleons and antinucleons are supposed to obey the Dirac equation coupled to the mean meson field. The wave function of baryons determines the nuclear density, and the nuclear density gives scalar (S) and vector (V ) potentials. S and V provide attractive and repulsive effect to baryons. The σ and ω meson fields affect the distribution of baryons reversely. In this self-consistent scheme, the nuclear density distribution, the baryon and antibaryon wave function, and the energy state of 16 O + B can be obtained. 3 Results and Discussions In the present work, two well-known parameter sets, i.e., NL3 [19] and NLZ2 [20] were used. The method of Ref. [21] is adopted to obtain the wave function of the

3 130 CHEN Xi, LI Ning, YAO Hai-Bo, QIN Xu-Ming, and WU Shi-Shu Vol. 53 antibaryon, and the Feynman amplitude is expanded to the bound states. In Fig. 1, the density distributions of the normal nucleus 16 O and the system 16 O + B are displayed. Because of the strong attractive effect, the nucleons tend to concentrate around the antibaryon. The distribution of antibaryon concentrates in the range of about 1.5 fm. The protons and neutrons are localized in the range of about 3.0 fm. The central density of 16 O is about ρ fm 3, but the central density of 16 O + B is about ρ 0. It should be mentioned that the central density of 16 O + B could be even larger when other parameter sets such as TM1 is used. [9] Such a high nuclear density is an important characteristic of the system 16 O + B, and it plays a crucial role in both the nucleon and the meson effective mass. Fig. 1 The baryon and antiproton densities in the system 16 O+ p. The results from the two parameter sets NLZ2 and NL3 are compared. For the normal nucleus 16 O, the two parameter sets give essentially the same baryon density, thus only the result from NLZ2 is shown. In the present RMF calculations, the scalar potential S 350 MeV and the vector potential V 300 MeV. For the nucleons, the depth of the net potential is V S 50 MeV. Following the assumption that the behavior of the antiproton in dense nuclear environment could be described validly by G-parity, [21] the scalar potential for an antiproton is the same as a proton, but the sign of the vector potential changes. Therefore the net potential for antiproton is much deeper, V S 650 MeV. The attractive effect for antiproton is enhanced by the G- transformed vector potential. Considering the high density in the 16 O + B system, the binding energy of antibaryon and surrounding nucleons are much lower than that of normal 16 O. In other word, the cold compression [9] of the nuclear system manifests as the high nucleon density and the deep bound states. In Table 1, we show the binding energy of occupied single particle states in 16 O + p. The available energy for the annihilation is Q = M B + M B E B E B, where M and E B mean the mass of baryon and the binding energy, respectively. The subscript B ( B) represents baryon (antibaryon). The deep binding energy leads to a small energy available for the annihilation. For example, the average available energy for p+ p c in 16 O + p is about MeV. If there are enough available energies, the multi-π channels are the main part for the annihilation, and the channels containing heavy mesons such as ρ, ω contribute less than 15% according to the experiment. [8] But in the phase space of small average available energy, the probability of generating mesons with larger masses is suppressed [e.g., for a 0 (880)], or even be closed [e.g., for b 1 (1235)]. [8] Therefore we focus on the channels composed of ρ, π, and ω mesons. In our calculation, the contributions of ω and ρ mesons are less than 5%. In the phase space of the average available energy of the 16 O + p system, the contribution of the 3-π channel is only 1/1000 of those of 1, 2-π meson channels. But with the available energy increasing, the contributions of 3 or more multi-π channel increase fast, and become to dominate when the available energy is higher than about 1400 MeV. However, the channels of 3 or more mesons are almost closed in the ground state of the system 16 O + p. Therefore only 1 and 2-meson channels are important here. Table 1 The binding energy of occupied p and p states in the nucleus 16 O + p. Nucleus κ p = 1 κ p = 2 κ p = 1 κ p = 1 NL MeV 49.7 MeV 21.2 MeV MeV NLZ MeV 49.8 MeV 21.2 MeV MeV The annihilation probabilities are calculated for the channel p + p π + π + with different effective mass for π. The results are shown in Fig. 2 (the Dirac sea effect is considered in the calcutions) and Fig. 3 (the Dirac sea effect is neglected), respectively. Similar features could be found in both figures. The Dirac sea effect has no influence to the probability for the annihilation process p + p π + π + in free space. Therefore the curves at nude mass in both figures are the same. The higher is the nuclear density, the lower is the annihilation probability. For each curve there is a peak at a certain available energy. The probability decreases at high energy values and it disappears rapidly. The Dirac sea effect on the annihilation probability can be clearly found by comparing Figs. 2 and 3. In the two cases in which the Dirac sea effect is considered or ne-

4 No. 1 Lifetime of Antibaryon Bound in Finite Nuclei 131 glected, the density dependences of the effective masses of the ρ, π, and ω mesons in the nuclear matter are very different in the density range ρ 0 < ρ < 2ρ 0. [22 24] Generally speaking, the effective mass of each meson decreases with the nuclear density if the Dirac sea effect is included, but it increases if the Dirac sea effect is neglected. The different meson effective masses result in different threshold energies for a specific channel. In Fig. 3, it is seen that the threshold energy increases with the density increasing in the case of no Dirac sea effect. For the same nuclear density, the contribution of Dirac sea makes the annihilation probability increase. With the nuclear density increasing, the annihilation probability obtained from the calculation with the Dirac sea effect considered decreases slowly, but that without Dirac sea effect drops quickly. from the data available in the range 0 < ρ < 2ρ 0. In our calculation, we have used the extrapolated meson effective mass. The total annihilation probability curves versus the nuclear density are plotted in Fig. 4. It is obvious that the total annihilation probabilities for the two systems, 16 O + p and 16 O+ n, increase with the nuclear density when the Dirac sea effect is taken into account, but they decrease with the nuclear density when the Dirac sea effect is not included. The annihilation probability of each channel decreases at the range of the high nuclear density for the two modes (i.e., the Dirac sea effect is considered or not) as is seen in Figs. 2 and 3. But the different density dependence of the meson effective mass in these two modes determines the different amount of open channels. The high meson effective mass in the no sea case limits the number of open channels strongly. In high nuclear density, only π-channels can be open. Oppositely, the low meson effective mass with the Dirac sea effect enhances the amount of open channels. The splitting of p and n annihilation probability roots in the different binding energy of p and n and the splitting of the effective mass of the meson with different isospin, especially the splitting of m π +, m π 0, and m π. In the curves corresponding to the no-sea approximation, there is a minimum value at about 1.5ρ 0. This value comes from the competition between the one π and two π channels. In the curves with the Dirac sea effect, there is a small peak at about 0.5ρ 0, which is due to the appearance of the channel p + p ω. Fig. 2 The annihilation probability of the channel p + p π + π + versus the available energy at different nuclear density with the Dirac sea effect. Fig. 4 The total annihilation probability of p and n versus the nuclear density ratio ρ/ρ 0. Fig. 3 The annihilation probability of the channel p + p π + π + versus the available energy at different nuclear density without the Dirac sea effect. It should be mentioned that the density dependence of the meson effective mass is uncertain at higher nuclear densities. [22 24] But the meson effective mass in high density nuclear matter can be estimated by extrapolating From the total annihilation probability, we can estimate the lifetime of an antiproton in 16 O + p, which is shown as the function of the density in Fig. 5. Since the annihilation process is strongly suppressed in high nuclear density when the Dirac sea effect is not considered, the corresponding lifetime of the antiproton in 16 O + p becomes longer with the density increasing except in the range around the density ρ 1.5ρ 0. At a critical density ρ c = 2.685ρ 0, the lifetime tends to be infinite. The

5 132 CHEN Xi, LI Ning, YAO Hai-Bo, QIN Xu-Ming, and WU Shi-Shu Vol. 53 antiproton in 16 O + p has no possibility to be annihilated and exists stably when ρ ρ c. The reason is the following. When some meson mass thresholds are larger than the available energy, the related channels are closed. For the 16 O + p system, the maximum available energy is about MeV. When ρ ρ c, the effective masses of π, ω, or ρ exceed MeV. Thus all possible meson channels are closed and the lifetime of the antiproton becomes infinite at and beyond the critical density. According to these theoretical results, the related phenomena should be experimentally observed near and beyond the critical nuclear density under the condition of no Dirac sea effect. Fig. 5 The lifetime of p in 16 O + p without the Dirac sea effect. It should be noted that the critical density for the 16 O + n system is about 2.94ρ 0, a little bit higher than that for the 16 O + p system. This higher critical density comes from the different binding energy of single particle state and the splitting of the effective mass of the meson in different isospin states. At the density well below the critical density ρ c, the annihilation probability from the present calculation is quantitatively smaller by 2 3 orders than that obtained by using the phenomenological approach, [9] the kinetic approach, [11] and the dynamical transport model. [12] We note that our calculation is based on a full quantum field theory from which more reasonable results are expected. In order to make more accurate predictions, the reliable density dependence of the effective mass of mesons in nuclear medium are needed. 4 Summary We calculated the lifetime of an antibaryon embedded in a finite nucleus based on the standard quantum field theory. It was found that the annihilation probability of the antibaryon in nuclei is strongly dependent on the meson effective mass and the contribution of the Dirac sea to the annihilation probability makes the antibaryon lifetime short. Without the Dirac sea effect, the lifetime of the bound antibaryon tends to be very long at a critical nuclear density and it may exist stably when the density is larger than the critical nuclear density. Acknowledgment CHEN Xi is very grateful to Prof. Shan-Gui Zhou for useful discussions. References [1] C.Y. Wong, A.K. Kerman, G.R. Satchler, and A.D. Mackellar, Phys. Rev. C 29 (1984) 574. [2] E.H. Auerbach, C.B. Dover, and S.H. Kahana, Phys. Rev. Lett. 46 (1981) 702. [3] H. Heiselberg, A.S. Jensen, A. Miranda, and G.C. Oades, Phys. Lett. B 132 (1983) 279. [4] A.J. Baltz, C.B. Dover, M.E. Sainio, A. Gal, and G. Toker, Phys. Rev. C 32 (1985) [5] S.G. Zhou, J. Meng, and P. Ring, Phys. Rev. Lett. 91 (2003) [6] C.Y. Wang, A.K. Kerman, G.R. Satchler, and A.D. Mackellar, Phys. Rev. C 29 (1984) 574. [7] C.J. Batty, E. Friedman, and A. Gal, Phys. Rep. 287 (1997) 385. [8] C.B. Dover, T. Gutsche, M. Maruyama, and A. Faessler, Prog. Part. Nucl. Phys. 29 (1992) 87. [9] T. Bürvenich, I.N. Mishustin, L.M. Satarov, J.A. Maruhn, H. Stöcker, and W. Greiner, Phys. Lett. B 542 (2002) 261. [10] W. Cassing, Nucl. Phys. A 700 (2002) 618. [11] I.N. Mishustin, L.M. Satarov, T.J. Bürvenich, H. Stöcker, and W. Greiner, Phys. Rev. C 71 (2005) [12] A.B. Larionov, I.N. Mishustin, L.M. Satarov, and W. Greiner, Phys. Rev. C 78 (2008) [13] A.B. Larionov, O. Buss, K. Gallmeister, and U. Mosel, Phys. Rev. C 76 (2007) [14] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1985) 1. [15] P.G. Reinhard, Rep. Prog. Phys. 52 (1989) 439. [16] P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193. [17] D. Vretenar, A. Afanasjev, G. Lalazissis, and P. Ring, Phys. Rep. 409 (2005) 101. [18] J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, and L.S. Geng, Prog. Part. Nucl. Phys. 57 (2006) 470. [19] G. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55 (1997) 540. [20] M. Bender, K. Rutz, P.G. Reinhard, J.A. Maruhn, and W. Greiner, Phys. Rev. C 60 (1999) [21] G. Mao, H. Stöker, and W. Greiner, Int. J. Mod. Phys. E 8 (1999) 389. [22] G. Song, Y.J. Zhang, and R.K. Su, Chin. Phys. Lett. 13 (1996) 89. [23] S. Biswas and A.K. Dutt-Mazumder, Phys. Rev. C 77 (2008) [24] H.B. Yao and S.S. Wu, Chin. Phys. C 10 (2008) 842.