Formation of Deeply Bound Pionic Atoms in Sn Isotopes

We study the formation of deeply bound pionic atoms in the (d,3He) reaction theoretically. At different scattering angles, we find that the different subcomponents dominate the formation spectra because of the matching condition of the reaction. We also find that the pionic 1s state which is free from the residual interaction effects appears clearly in 117Sn(d, 3He) spectra. We conclude that the observation of the (d,3He) reaction for these new cases will provide more systematic and accurate information on the pionic bound states, and it will help to develop the study of the pion properties and the partial restoration of chiral symmetry in nuclei.


Introduction
Deeply bound pionic atom is one of the best systems to deduce pion properties at finite density and to obtain precise information on partial restoration of chiral symmetry in nuclei [1]. The deeply bound states have been experimentally produced in the forward (d, 3 He) reactions with Pb and Sn isotope targets [2,3] by following theoretical predictions [4,5]. In Ref. [3], the binding energy and width of the 1s states have been precisely measured in three Sn isotopes and isospin-density dependence of the s-wave pion-nucleus potential has been deduced. From these observations, the reduction of the chiral order parameter qq in nucleus was concluded.
To develop the studies of pion properties and symmetry restoration in nuclei further, we need to obtain more precise and systematic information on deeply bound pionic states. The information is, for example, necessary for the unique determination of the pion-nucleus interaction, which is required to fix the potential strength related to chiral symmetry [6].
In this paper, we consider theoretically two new studies of pionic atom formation in the (d, 3 He) reaction. One is the pionic atom formation in the (d, 3 He) reaction at finite angles [7][8][9], where we can expect to have the manifestation of different subcomponents of pion and neutron hole states due to the matching condition with different momentum transfer, and expect to determine the binding energies and widths of various pionic states simultaneously in each nucleus.
The other is the pionic atom formation on the odd nuclear target, which has not been investigated so far. For example, the odd nucleus 117 Sn has a spin of 1 2 . After the proton pick-up (d, 3 He) reaction, we have the contributions for the pionic atom formation on the ground state of the even-even daughter nucleus 116 Sn with the quantum number of 0 + . This pionic state does not have the additional shift due to the residual interaction effect [10,11]. For the even nuclear target cases, since the final pionic states are the pion-particle plus neutron-hole [π ⊗ n −1 ] states, the residual interaction effects may shift the binding energies and widths.
We use the effective number approach to calculate the pionic atom formation cross sections [9]. We refine the theoretical model used in Refs. [4,5,12,13] to study the angular dependence of the (d, 3 He) spectra by including the kinematical correction factors K in Eq. (1) as explained below. The (d, 3 He) reaction cross section in the laboratory frame is expressed as, where dσ dΩ He lab ele indicates the elementary cross section for the d + n → 3 He + π − reaction in lab frame, which is extracted from the experimental data [14].
The effective number N eff is defined as, where φ π and ψ j n indicate the wave functions of the pion bound state in the daughter nucleus and the neutron bound state in the target nucleus, respectively. For the neutron wave function ψ j n , we use the calculated wave function using the neutron potential reported in Ref. [15]. The wave functions of the projectile (d) and the ejectile ( 3 He) are denoted by χ * He and χ d . The kinematical correction factor K is defined as [9], where the superscript 'A' indicates the momentum and energy which should be evaluated in the kinematics of the nuclear target case. The superscript 'lab' indicates that all kinematical variables are evaluated in the laboratory frame.

Numerical Results and Discussions
In Fig. 1, we show the calculated spectra at finite angles for the bound pionic states formation in the 122 Sn(d, 3 He) and 117 Sn(d, 3 He) reactions. We find that the both spectra have a strong angular dependence and the shape of the spectra are much different at finite angles from that at 0 • . In the 122 Sn(d, 3 He) spectra, the largest peak structure at Q = −137.8 MeV in the forward spectra is strongly suppressed at finite angles and the spectra show the structure of three peaks at θ lab dHe ≥ 2 • . In the 117 Sn(d, 3 He) spectra, we find that we can see clearly the peak structure of the pionic 1s state formation with the ground state of the even-even nucleus 116 Sn at Q = −135.8 MeV. This state does not have the additional shift due to the residual interaction effect. Therefore, we can expect that we obtain more precise information than that of the even nuclear target case [16].

Summary
We study the formation of deeply bound pionic atoms in the (d, 3 He) reactions theoretically. We develop the formula to include the kinematical correction factors to the effective number approach to obtain more realistic angular dependence of the (d, 3 He) spectra. We show the angular dependence of the 122 Sn(d, 3 He) and 117 Sn(d, 3 He) spectra at T d = 500 MeV. We find that the 122 Sn(d, 3 He) spectra are dominated by the subcomponents including the (2 p) π state at larger scattering angles θ lab dHe ≥ 2 • , while they are dominated by the (1s) π and (2s) π states at forward angles. Thus, we can conclude that we can obtain information on deeply bound pionic 2 p state in addition to 1s and 2s states by observing the spectra at finite angles. As indicated in Ref. [6], the observation of several deeply pionic bound states in a certain nucleus will help to deduce precise information of pion properties and chiral dynamics at finite density [3]. We also find that the pionic 1s state which is free from the residual interaction effects appears clearly in 117 Sn(d, 3 He) spectra. We believe that our results provide a good evaluation for further experimental studies of the states reported here, which should contribute to the development of the field.