Production Spectra of $^3$He($\pi$, $K$) Reactions with Continuum Discretized Coupled Channels

We investigate theoretically $\Lambda$ production spectra of $^3$He($\pi$, $K$) reactions at $p_\pi=$ 1.05--1.20 GeV/$c$ in the distorted-wave impulse approximation, using the continuum-discretized coupled-channel method. The production cross section of a $^3_\Lambda$H(1/2$^+$) ground state is also discussed.

In this paper, we focus on the Λ production spectra of 3 He(π, K) reactions at p π = 1.05-1.20 GeV/c in the distorted-wave impulse approximation (DWIA), using the continuumdiscretized coupled-channel (CDCC) method [7] in order to well describe the NN continuum states above the N + N + Λ breakup threshold. We also discuss the production cross section of 3 Λ H(1/2 + ) in 3 He(π − , K 0 ) reactions, considering nuclear medium effects of the πN → ΛK amplitude and recoil effects.

II. CALCULATIONS
Inclusive differential cross sections for nuclear (π, K) reactions in the laboratory frame within the DWIA [7] are given by (in units = c = 1) where [J] = 2J + 1, β is a kinematical factor, and E K , E π , E B and E A are energies of outgoing K, incoming π, hypernuclear states and the target nucleus, respectively; Ψ B and Ψ A are wavefunctions of hypernuclear states and the target nucleus, respectively.F is a strangeness-exchange external operator given bŷ where χ π are distorted waves for outgoing K and incoming π, respectively, which are calculated with the help of the eikonal approximation.Ô j is a baryon operator changing jth nucleon into a Λ hyperon in the nucleus. Figure 1 displays the momentum transfer to the final state, q Λ = |p π − p K |, as a function of the incident pion momentum p π , where p π and p K are the laboratory momenta of π and K in the nuclear reaction, respectively. f πN →ΛK is the πN → ΛK amplitude in nuclear medium, which is obtained by the optimal Fermi-averaging method [8]. It should be noticed that strong E Λ dependence appears in f πN →ΛK because the elementary cross sections for the πN → ΛK reactions depend on the incident pion momentum [9]; we confirm that the optimal Fermi-averaged cross sections of dσ/dΩ = β|f πN →ΛK | 2 at p π = 1.05 and 1.20 GeV/c have strong E Λ dependence [8], as shown Hypernuclear final states are considered as three-body NNΛ systems in the 3 He(π, K) reactions, involving continuum states above the N + N + Λ threshold. Here we employ the CDCC method [10] in order to well describe the NN continuum states as breakup channels.
For the ΛN potential v ΛN , we assume a single Gaussian form which reproduces the scattering length and the effective range in Λp scattering at low energies, fitting into those of NSC97f.
We can also reproduce the experimental value of B Λ = 0.13 MeV for 3 Λ H(1/2 + ) when we slightly modify the strengths of v ΛN by a factor of 0.92. We rewrite a sum over the final states in Eq. (1) as whereĜ(ω) is a complete Green's function for the 2N-Λ systems given by CDCC wavefunctions. Therefore, the inclusive differential cross sections are obtained by the Green's function method [11].

III. RESULTS AND DISCUSSION
In the 3 He(π + , K + ) reactions, we consider the production of the ppΛ states with only NN, 1 S 0 components. Figure 3 displays the calculated inclusive K + spectrum of the 3 He(π + , K + ) reaction at p π = 1.20 GeV/c, θ lab = 3 o , together with partial-wave components of the spectrum. Considering large momentum transfer of q Λ ≃ 360 MeV/c in exothermic (π, K) reactions, we find that many partial waves moderately contribute to the spectrum; there appears an enhancement of the T = 1, J π = 1/2 + (L π = 0 + , S = 1/2) component just above the p + p + Λ threshold. This enhancement may indicate that a pole of the s-wave ppΛ resonance or virtual state resides near the p + p + Λ threshold, as suggested by several three-body calculations.
In the 3 He(π − , K 0 ) reactions, we study the production of the pnΛ states with NN, 3 S 1 and 1 S 0 components, which include the 3 Λ H (T = 0, J π = 1/2 + ) ground state. Figure 4 where a recoil factor of M C /M A is equal to 2/3 for the 3 He target.
displays the calculated inclusive K 0 spectrum of the 3 He(π − , K 0 ) reaction at p π = 1.20 GeV/c, θ lab = 3 o , together with partial-wave components of the spectrum. We find that the integrated production cross section of 3 Λ H amounts to dσ/dΩ ( 3 Λ H) = 3.13 µb/sr. In Table I near the Λ threshold at 1.05 GeV/c are about 1.5 times as large as those at 1.20 GeV/c. This is caused by the fact that the former is larger than the latter in terms of q eff Λ . If the recoil effects are switched off (M C /M A = 2/3 is replaced by 1), the values of dσ/dΩ ( 3 Λ H) are reduced by an order of magnitude or more. Hence we recognize that the recoil effects for production on the light target as 3 He are important in the nuclear (π, K) reactions.

IV. SUMMARY
We have shown the calculated Λ production spectra of the NNΛ systems in the 3 He(π, K) reactions at 1.05-1.20 GeV/c with CDCC which describes the NN continuum states above the N + N + Λ breakup threshold. The production cross section of 3 Λ H(1/2 + ) in the (π − , K 0 ) reaction is evaluated, e.g., dσ/dΩ ( 3 Λ H) ≃ 3 µb/sr at 1.05-1.20 GeV/c, θ lab = 3 o . The recoil effects are very important to the production with the light nuclear target as 3 He, as well as the medium effects of the πN → ΛK amplitudes for nuclear (π, K) reactions. More precise analysis on convergence of the CDCC model space depending on (k max , ℓ max ) should be needed. This investigation is in progress.