Chiral Model in Nuclear Medium and Hypernuclear Production

Transcription

1 WDS'10 Proceedings of ontributed Papers, Part III, 2 6, ISBN MATFYZPRESS hiral Model in Nuclear Medium and Hypernuclear Production V. Krejčiřík harles University, Faculty of Mathematics and Physics, Prague, zech Republic. Nuclear Physics Institute, Academy of Sciences, Řež, zech Republic. Abstract. The effective models of strong interaction based on chiral symmetry became an integral part of the current theoretical physics. The models were originally formulated for hadron interactions in vacuum and their extension into the nuclear medium must be handled with a special care. In our contribution, we discuss the impact of the nuclear medium and related effects in case of kaon-nuclear interaction. We use effective separable meson-baryon potentials and coupledchannel techniques to study the properties of antikaon-nucleon interaction. The chirally motivated model is then applied to the hypernuclear production induced by the stopped kaons and the results are compared with those obtained by using alternative approaches. Introduction Due to nonperturbative character of quantum chromodynamics at low and medium energy scales, other approaches how to describe strong interactions have been developed. One of them is the effective theory based on chiral symmetry. However, the perturbative expansion cannot be used in the strange sector due to the presence of the (1405) resonance. In this paper, we present the construction of effective separable meson-baryon potentials and following coupledchannels calculations. Then, Pauli blocking and kaon self-energy are employed into our model in order to describe the effect of nuclear medium. Next, the chiral model is used for the calculation of the hypernuclear production induced by stopped kaons. Theoretical description is based on distorted wave impulse approximation. The chiral model impress at two places here. First, it allows us to calculate directly the branching ratios for elementary kaon-nucleon process. And second, it is used to generate the kaon-nucleus potential which affects the kaon wave function in the initial state. We also study the sensitivity of calculated capture rates to the various inputs. hiral models Following Kaiser et al. [1995] and ieplý and Smejkal [2010], we describe meson-baryon interactions in the formalism of the quantum field theory by the chiral Lagrangian density. The leading order term is given in the language of baryonic (Ψ) nad mesonic (involved in A µ, D µ ) fields by L (1) = Tr ( Ψ B (iγ µ D µ M 0 ) ) + F Tr ( Ψ B γ µ γ 5 [A µ,ψ B ] ) + D Tr ( Ψ B γ µ γ 5 {A µ,ψ B } ). (1) As mentioned above, the straight perturbative expansion is unusable in the energy region of our interest, and therefore we employ Lippmann-Schwinger equation and effective separable potentials, V ij (k,k ) = 1 M i g i (k) ij 2E i ω i f 2 g j(k ) 1 2E j M j ω j, g j (k) = (k/α j ) 2. (2) 2

2 The separable structure of effective potentials leads to the simplification of Lippmann-Schwinger equation to a purely algebraic form t 0 ij = v ij + n v in I n t nj (3) with only one remaining integration I n, I n = ω n d 3 1 l 2π Ω n kn 2 l 2 + iǫ g2 n (l). (4) oupled channels considered in our calculation are: π 0, π 0 Σ 0, π Σ +, π + Σ, K p, K0 n, η, ησ 0, K 0 Ξ 0, K + Ξ. The connection between the two formalisms is achieved by the requirement of equal s-wave scattering lengths calculated up to order q 2. The second order chiral Lagrangian density L (2) contributes to the order q 2 too, but we do not specify it here. The symmetry structure of Lagrangian (given by chiral symmetry) is reflected in the coefficients ij. The fit of remaining free model parameters was done by ieplý and Smejkal [2010]. They used experimental data for low energy scattering of K p into various final state channels, kaonic hydrogen shift and width, and well established threshold branching ratios. They achieved quite successful value χ 2 /N 1.4. Effects of nuclear medium The presented model for meson-baryon interaction is formulated in vacuum. The extension into nuclear medium is performed by two steps, considering the Pauli blocking and the kaon self-energy. Pauli blocking The Pauli principle requires the momentum of intermediate proton or neutron to be greater than the Fermi momentum. Following Waas et al. [1996], this restriction changes the domain of integration in the integral (4) from Ω n = R 3 to Ω n (p F ), Kaon self-energy Ω n (p F ) = { l R 3 ; p + k j l p F }. (5) Lutz [1998] pointed out the necessity of considering the kaon self-energy into the inmedium calculations. Using different words, the mass of kaon is shifted when propagating in nuclear matter compared to the situation in vacuum. Following ieplý et al. [2001], this suggestion leads to the change in the propagator, I n = ω n d 3 1 l 2π Ω n(p F ) kn 2 l 2 Π n + iǫ g2 n(l), (6) where Π n = 2µ KN Vopt+2µ K KN Vopt, N V opt is the optical potential. The K -nucleus potential is assumed to be of t ρ form. Hypernuclear production One of the areas, where the extension of chiral model into the nuclear medium can be tested, is the hypernuclear production induced by stopped kaons. 3

3 Figure 1. Elementary branching ratios as a function of nucleon density. Formalism The distorted wave impulse approximation (DWIA) formalism was developed by Gal and Klieb [1986]. The capture rate per one stopped kaon can be written as a product of three terms, R if = KF R(K N πy ) R if /Y, (7) where KF is the kinematic factor, R(K N πy ) represents the branching ratio for the elementary process, and R if /Y, which is loosely called capture rate per hyperon, contains the overlay of kaon, pion, nucleon, and hyperon wave functions. Inputs The chiral model presented in previous chapter (including Pauli blocking and kaon selfenergy) was used to calculate the elementary branching ratios R(K N πy ). The dependence of BR on nucleon density is shown in the Figure 1. We see that the curve is relatively flat in the pertinent region (around ρ 0 /2). onsequently, we used the values corresponding to the nucleon density ρ = ρ 0 /2 in our calculation (BR1). Additionally, we used values obtained in the vacuum (BR2) and results of emulsion experiments (BR3) for comparison. Branching ratios we used are summarized in Table 1. We see that the differences between various approaches are smaller than 30 %. onsidering that this difference is smaller compared to other effects involved in hypernuclear production (mainly pion distortion, kaon w.f., etc.), we used only BR1 in our next calculations. Table 1. Elementary branching ratios used in our calculations. branching BR1 BR2 BR3 ratio ρ = ρ 0 /2 ρ = 0 16 O R(K n π )[10 2 ] R(K p π 0 )[10 2 ] The kaon atomic wave function was generated by solving the Klein-Gordon equation with a potential consisting of two parts, the finite-size oulomb potential plus first order vacuum polarization corrections, and the strong-interaction optical potential parametrized phenomenologically by Friedman et al. [1994]: V K opt(r) = 4π 2µ K ( 1 + µ )[ K b + B M N 4 ( ) ρ(r) ν ] ρ(r). (8) ρ(0)

4 Figure 2. The sensitivity of the capture rates to the kaon wave function. We used three different parameter sets, which are specified in Table 2. The choice [K χ ] represents the chiral model presented in previous chapters, [K eff ] and [K DD ] denote phenomenological potentials taken from Friedman et al. [1994]. For a reference, we also show the respective potential depths in the last column of the table. Moreover, we performed the calculation with a pure electromagnetic potential ([K coul ]) to check the impact of the strong interaction. Table 2. Parameters of the kaonic optical potential. set b [fm] B [fm] ν Vopt K 0) [MeV] [K χ ] i [K eff ] i [K DD ] i i The sensitivity of the capture rates to the choice of K wave functions is demonstrated in Figure 2. It appears that the capture rate is a decreasing function of the K -nucleus potential depth. Results We performed a calculation of -hypernuclear production for target nuclei from lithium to oxygen. The results for the production of hyperon in the 1s state are summarized in Table 3. Table 3. alculated capture rates (in units of 10 3 ) for the summed 1s production (1 transitions). K potential 7 Li 9 Be O [K χ ] [K DD ] We show results obtained with kaon-nucleus potentials [K χ ] and [K DD ], which represent the two main directions for how the K -nucleus interaction is treated at present. It appears that the capture rate is a decreasing function of A. The ratio of 1s capture rate in 7 Li to that in 16 O is 2.66 for [K χ ] and 5.27 for [K DD ]. Put differently, the ratio of rates related to [K χ ] potential to rates related to [K DD ] potential increases from approximately 2 for lithium up to about 4 for oxygen. This trend probably owes to the node structure of the kaon atomic wave functions. 5

5 Our calculation for both 1s (1 transition) and 1p (0 + and 2 + transition) production for target nuclei B,, and 16 O are summarized in Table 4. The experimental data for boron comes from Ahmed et al. [1996] and data for carbon and oxygen from Tamura et al. [1994]. Table 4. alculated capture rates per stopped K (in units of 10 3 ) for production of 1s states (1 transition) and 1p states (0 + and 2 + transitions) and selected experimental rates. transition input B 16 O 1 exp. rates 0.28 ± ± ± 0.06 [K χ ] [K DD ] [K χ ] [K DD ] [K χ ] [K DD ] [K χ ] [K DD ] exp. rates 0.35 ± ± ± 0.16 We see that the agreement with experiment is still not fully satisfactory and capture rates are generally smaller than the measured values. Since the absolute normalization of capture at rest experimental rates is a delicate issue, we suggest to focus on the A dependence of the measured rates and various ratios between them in the following analysis. onclusion We presented the model of meson-baryon interaction based on chiral symmetry. Our model employs the effective separable potentials and desired quantities are obtained by coupled-channel calculations employing the Lippmann-Schwinger equation. The Pauli blocking and kaon selfenergy are involved into our calculation in order to consider the effects of nuclear medium. The production of hypernuclei induced by stopped kaons was calculated. The capture rates appear to be the decreasing function of K -nucleus optical potential depth. We also observed that capture rates are a decreasing function of atomic number. Additionally, the decrease is faster for the deeper K -nucleus potential, and just this effect provides the most significant and testable difference between chiral and phenomenological potentials. The absolute agreement with experiment is still unsatisfactory. More details of our work on hypernuclear production can be found in Krejčiřík et al. [2010]. Acknowledgments. The present work was supported by the GAUK grant no and the GAR grant no. 202/09/1441. I also want to thank to my supervisor Aleš ieplý. References Kaiser N., Siegel P. B., Weise W.: Nucl.Phys.A, 594, , ieplý A., Smejkal J., Eur.Phys.J.A, 43, , Waas T., Kaiser N., Weise W., Phys. Lett.B, 365, 16, Lutz M., Phys.Lett.B, 426, 20, ieplý A., Friedman E., Gal A., Mareš J., Nucl.Phys. A, 696, , Gal A., Klieb L., Phys.Rev., 34, , Friedman E., Gal A., Batty. J.: Nucl.Phys. A, 579, , Ahmed M. W., ui X., Empl A., et al., Phys. Rev., 68, , Tamura H., Hayano R. S., Outa H., Yamazaki T., Prog.Theor.Phys. Suppl., 117, 1 15, Krejčiřík V., ieplý A., Gal A., Phys.Rev., 82, 1 7,