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

TL;DR

This paper presents a theoretical study of $ ext{He}^3( ext{pi},K)$ reactions using the continuum-discretized coupled-channel method to analyze $ ext{Lambda}$ production spectra and cross sections.

Contribution

It introduces a novel application of the continuum-discretized coupled-channel method to $ ext{He}^3( ext{pi},K)$ reactions for $ ext{Lambda}$ hypernucleus production analysis.

Findings

Calculated $ ext{Lambda}$ production spectra at specific pion momenta.

Discussed the production cross section of $ ext{He}_ ext{Lambda}$ ground state.

Demonstrated the effectiveness of the continuum-discretized coupled-channel approach.

Abstract

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.

Tables

Table 1. Table 1: Calculated results of the integrated production cross sections of the H Λ 3 ​ ( 1 / 2 + ) superscript subscript H Λ 3 1 superscript 2 {{}^{3}_{\Lambda}{\rm H}(1/2^{+})} ground state, d ​ σ / d ​ Ω 𝑑 𝜎 𝑑 Ω d\sigma/d\Omega  ( H Λ 3 superscript subscript H Λ 3 {{}^{3}_{\Lambda}{\rm H}} ), in the 3 He( π − superscript 𝜋 \pi^{-} , K 0 superscript 𝐾 0 K^{0} ) reactions at p π = subscript 𝑝 𝜋 absent p_{\pi}= 1.05 and 1.20 GeV/ c 𝑐 c .

$p_{\pi }$	${\theta }_{\mathrm{lab}}$		${\left(d\sigma /d\Omega \right)}_{\mathrm{elem}.}^{\pi N\to \Lambda K}$  ($\mu$b/sr)			$q_{\Lambda }$	$q_{\Lambda }^{\mathrm{eff}}$333$q_{\Lambda }^{\mathrm{eff}}\approx \left(M_C/M_A\right)q_{\Lambda }$ where a recoil factor of $M_C/M_A$ is equal to 2/3 for the 3He target.		$d\sigma /d\Omega$ (${}_{\Lambda }{}^{3}H$)  ($\mu$b/sr)
(GeV/$c$)	(degree)		free111$\beta {\left|f_{{\pi }^{-}p\to \Lambda K^0}\right|}^2$ is used.	in medium222$\beta {\left|{\overline{f}}_{{\pi }^{-}p\to \Lambda K^0}\right|}^2$ is used.		(MeV/$c$)	(MeV/$c$)		w/o recoil	full
1.05	3		463	344		354	236		0.15	3.07
	7		459	340		369	246		0.10	2.42
	11		450	332		393	262		0.05	1.59
1.20	3		292	225		326	217		0.22	3.13
	7		287	235		350	233		0.13	2.24
	11		277	236		383	255		0.05	1.33

Equations

$$\frac{d^2\sigma }{d{E}_{K}d{\Omega }_K}=\beta \frac{1}{[J_A]}\sum _{M_A}\sum _B|⟨{\Psi }_B|\hat{F}|{\Psi }_A⟩{|}^2\delta (E_K+E_B-E_{\pi }-E_A),$$

$$\hat{F}=\int d\mathbf{r}{\chi }_K^{(-)\ast }({\mathbf{p}}_K,\mathbf{r}){\chi }_{\pi }^{(+)}({\mathbf{p}}_{\pi },\mathbf{r})\sum _{j=1}^A{\overline{f}}_{\pi N\to \Lambda K}\delta (\mathbf{r}-{\mathbf{r}}_j){\hat{O}}_j,$$

$$\displaystyle{\Psi}_{B}\simeq{\Psi}^{\rm CDCC}_{B}({\bm{r}},{\bm{R}})=\sum_{\alpha=1}^{N_{\rm max}}\sum_{\ell_{2}=0}^{\ell_{\rm max}}\left[\bigl{[}\tilde{\phi}_{\alpha,\ell_{2}}^{(2N)}({\bm{r}})\otimes\varphi^{(\Lambda)}_{\alpha,\ell_{\Lambda}}({\bm{R}})\bigl{]}_{L_{B}}\otimes X^{B}_{I_{\alpha},S_{\alpha}}\right]_{J_{B}}^{M_{B}},$$

$$U_{\alpha\alpha^{\prime}}(R)=\int\!\rho_{\alpha\alpha^{\prime}}({\bm{r}})\Bigl{(}\overline{v}_{\Lambda N}({\bm{R}}-{\bm{r}}/2)+\overline{v}_{\Lambda N}({\bm{R}}+{\bm{r}}/2)\Bigr{)}d{\bm{r}},$$

$$\sum _B|{\Psi }_B⟩⟨{\Psi }_B|\delta (E_B-E_A-\omega )=-\frac{1}{\pi }\mathrm{Im}\hat{G}(\omega ),$$

Full text

Lambda hypernuclei, production, DWIA, CDCC, recoil effects

††preprint: J-PARC-TH-0156

I Introduction

It has been believed that hypertriton (${{}^{3}_{\Lambda}{\rm H}}$) has lifetime within a few % of the free $\Lambda$ lifetime of $\tau_{\Lambda}=$ 263.2$\pm$2.0 ps because ${{}^{3}_{\Lambda}{\rm H}}$ ($T=$ 0, $J^{\pi}=$ 1/2+) is composed of a deuteron ($I=$ 0, ${}^{3}S_{1}$) and a $\Lambda$ hyperon loosely bound with the small $\Lambda$ binding energy of $B_{\Lambda}=$ 130$\pm$50 keV. It is supported by several theoretical calculations Kamada98 ; see however Gal19 . Recently, unexpected short lifetime of ${{}^{3}_{\Lambda}{\rm H}}$ was measured in hypernuclear production by high-energy heavy-ion collisions HypHI13 ; the world average lifetime of $\tau^{\rm(av)}({{}^{3}_{\Lambda}{\rm H}})=$ 185${}^{+23}_{-28}$ ps is shorter than $\tau_{\Lambda}$ by about 30%. (ALICE Collaboration ALICE18 newly reported the preliminary result of $\tau({{}^{3}_{\Lambda}{\rm H}})=$ 237${}^{+33}_{-36}$ ps which is moderately closer to $\tau_{\Lambda}$.) To solve the hypertriton lifetime puzzle, experimental measurements of the ${{}^{3}_{\Lambda}{\rm H}}$ lifetime are planned by ${}^{3}{\rm He}$($K^{-}$, $\pi^{0}$) and ${}^{3}{\rm He}$($\pi^{-}$, $K^{0}$) reactions at J-PARC. Moreover, HypHI Collaboration Rappold13 found a bound $nn\Lambda$ system, whereas theoretical calculations suggest that no ${}^{3}_{\Lambda}{n}$ ($T=$ 1, $J^{\pi}=$ 1/2+) bound state exists Gal14 . Therefore, it is important to investigate theoretically production of such $NN\Lambda$ systems, e.g., ($\pi$, $K$) reactions on a 3He target, in order to settle the current problems related to three-body hypernuclei.

In this paper, we focus on the $\Lambda$ production spectra of ${}^{3}{\rm He}$($\pi$, $K$) reactions at $p_{\pi}=$ 1.05–1.20 GeV/$c$ in the distorted-wave impulse approximation (DWIA), using the continuum-discretized coupled-channel (CDCC) method Harada15 in order to well describe the $NN$ continuum states above the $N+N+\Lambda$ breakup threshold. We also discuss the production cross section of ${{}^{3}_{\Lambda}{\rm H}(1/2^{+})}$ in ${}^{3}{\rm He}$($\pi^{-}$, $K^{0}$) reactions, considering nuclear medium effects of the $\pi N\to\Lambda K$ amplitude and recoil effects.

II Calculations

Inclusive differential cross sections for nuclear ($\pi$, $K$) reactions in the laboratory frame within the DWIA Harada15 are given by (in units $\hbar=c=1$)

[TABLE]

where $[J]=2J+1$, $\beta$ is a kinematical factor, and $E_{K}$, $E_{\pi}$, $E_{B}$ and $E_{A}$ are energies of outgoing $K$, incoming $\pi$, hypernuclear states and the target nucleus, respectively; $\Psi_{B}$ and $\Psi_{A}$ are wavefunctions of hypernuclear states and the target nucleus, respectively. $\hat{F}$ is a strangeness-exchange external operator given by

[TABLE]

where $\chi_{K}^{(-)\ast}$ and $\chi_{\pi}^{(+)}$ are distorted waves for outgoing $K$ and incoming $\pi$, respectively, which are calculated with the help of the eikonal approximation. ${\hat{O}}_{j}$ is a baryon operator changing $j$th nucleon into a $\Lambda$ hyperon in the nucleus. Figure 1 displays the momentum transfer to the final state, $q_{\Lambda}=|{\bm{p}}_{\pi}-{\bm{p}}_{K}|$, as a function of the incident pion momentum $p_{\pi}$, where ${\bm{p}}_{\pi}$ and ${\bm{p}}_{K}$ are the laboratory momenta of $\pi$ and $K$ in the nuclear reaction, respectively. $\overline{f}_{\pi N\to\Lambda K}$ is the $\pi N\to\Lambda K$ amplitude in nuclear medium, which is obtained by the optimal Fermi-averaging method Harada04 . It should be noticed that strong $E_{\Lambda}$ dependence appears in $\overline{f}_{\pi N\to\Lambda K}$ because the elementary cross sections for the $\pi N\to\Lambda K$ reactions depend on the incident pion momentum Sotona89 ; we confirm that the optimal Fermi-averaged cross sections of ${{d}{\sigma}/{d}\Omega}=\beta|\overline{f}_{\pi N\to\Lambda K}|^{2}$ at $p_{\pi}=$ 1.05 and 1.20 GeV/$c$ have strong $E_{\Lambda}$ dependence Harada04 , as shown in Fig. 2.

Hypernuclear final states are considered as three-body $NN\Lambda$ systems in the 3He($\pi$, $K$) reactions, involving continuum states above the $N+N+\Lambda$ threshold. Here we employ the CDCC method Kamimura86 in order to well describe the $NN$ continuum states as breakup channels. Thus the wavefunctions ${\Psi}_{B}$ with $J^{\pi}$ in the $LS$-coupling scheme can be written as

[TABLE]

where $\tilde{\phi}_{\alpha,\ell_{2}}^{(2N)}({\bm{r}})$ is the $NN$ wavefunction having bound and continuum-discretized states for angular momentum $\ell_{2}$, spin ${}^{1}S_{0}$ or ${}^{3}S_{1}$ in channel $\alpha$, $\varphi^{(\Lambda)}_{\alpha,\ell_{\Lambda}}({\bm{R}})$ is the relative wavefunction between $NN$ and $\Lambda$ with $\ell_{\Lambda}$, and $X^{B}_{I_{\alpha},S_{\alpha}}$ is the isospin-spin function for $NN\Lambda$. Because we omit spin-flip processes in the ($\pi$, $K$) reactions on the 3He (1/2+) target, the final states on $J^{\pi}=|L^{\pi}\pm 1/2|$ with $L^{\pi}=$ 0+, 1-, 2+, $\cdots$, and $S=$ 1/2 can be populated. We obtain $\varphi^{(\Lambda)}_{\alpha,\ell_{\Lambda}}$, solving a coupled-channel equation with the potential $U_{\alpha\alpha^{\prime}}$ given by the microscopic $2N$-$\Lambda$ folding model;

[TABLE]

where $\rho_{\alpha\alpha^{\prime}}({\bm{r}})$ is the nucleon or transition density, and $\overline{v}_{\Lambda N}$ is the spin-averaged $\Lambda N$ potential. For the $\Lambda N$ potential $v_{\Lambda N}$, we assume a single Gaussian form which reproduces the scattering length and the effective range in $\Lambda p$ scattering at low energies, fitting into those of NSC97f. We can also reproduce the experimental value of $B_{\Lambda}=$ 0.13 MeV for ${}^{3}_{\Lambda}$H(1/2+) when we slightly modify the strengths of $v_{\Lambda N}$ by a factor of 0.92. We rewrite a sum over the final states in Eq. (1) as

[TABLE]

where $\hat{G}(\omega)$ is a complete Green’s function for the $2N$-$\Lambda$ systems given by CDCC wavefunctions. Therefore, the inclusive differential cross sections are obtained by the Green’s function method Morimatsu85 .

III Results and Discussion

In the 3He($\pi^{+}$, $K^{+}$) reactions, we consider the production of the $pp\Lambda$ states with only $NN$, ${}^{1}S_{0}$ components. Figure 3 displays the calculated inclusive $K^{+}$ spectrum of the 3He($\pi^{+}$, $K^{+}$) reaction at $p_{\pi}=$ 1.20 GeV/$c$, $\theta_{\rm lab}=$ 3o, together with partial-wave components of the spectrum. Considering large momentum transfer of $q_{\Lambda}\simeq$ 360 MeV/$c$ in exothermic ($\pi$, $K$) reactions, we find that many partial waves moderately contribute to the spectrum; there appears an enhancement of the $T=$ 1, $J^{\pi}=$ 1/2+ ($L^{\pi}=$ 0+, $S=$ 1/2) component just above the $p+p+\Lambda$ threshold. This enhancement may indicate that a pole of the $s$-wave $pp\Lambda$ resonance or virtual state resides near the $p+p+\Lambda$ threshold, as suggested by several three-body calculations.

In the 3He($\pi^{-}$, $K^{0}$) reactions, we study the production of the $pn\Lambda$ states with $NN$, ${}^{3}S_{1}$ and ${}^{1}S_{0}$ components, which include the ${}^{3}_{\Lambda}{\rm H}$ ($T=$ 0, $J^{\pi}=$ 1/2+) ground state. Figure 4 displays the calculated inclusive $K^{0}$ spectrum of the 3He($\pi^{-}$, $K^{0}$) reaction at $p_{\pi}=$ 1.20 GeV/$c$, $\theta_{\rm lab}=$ 3o, together with partial-wave components of the spectrum. We find that the integrated production cross section of ${}^{3}_{\Lambda}{\rm H}$ amounts to $d\sigma/d\Omega\,(^{3}_{\Lambda}{\rm H})=$ 3.13 $\mu$b/sr. In Table 1, we show the values of $d\sigma/d\Omega\,(^{3}_{\Lambda}{\rm H})$ at $p_{\pi}=$ 1.05 and 1.20 GeV/$c$ with $\theta_{\rm lab}=$ 3o, 7o, and 11o in order to see the sensitivity to the in-medium $\pi N\to\Lambda K$ amplitudes and to momentum transfers. We find that the values of $d\sigma/d\Omega\,(^{3}_{\Lambda}{\rm H})$ at 1.05 and 1.20 GeV/$c$ are similar, although the values of $(d\sigma/d\Omega)^{{\small\pi N\to\Lambda K}}_{\rm elem.}$ near the $\Lambda$ 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 $q^{\rm eff}_{\Lambda}$. If the recoil effects are switched off ($M_{C}/M_{A}=2/3$ is replaced by 1), the values of $d\sigma/d\Omega\,(^{3}_{\Lambda}{\rm H})$ are reduced by an order of magnitude or more. Hence we recognize that the recoil effects for production on the light target as ${}^{3}{\rm He}$ are important in the nuclear ($\pi$, $K$) reactions.

IV Summary

We have shown the calculated $\Lambda$ production spectra of the $NN\Lambda$ systems in the ${}^{3}{\rm He}$($\pi$, $K$) reactions at 1.05–1.20 GeV/$c$ with CDCC which describes the $NN$ continuum states above the $N+N+\Lambda$ breakup threshold. The production cross section of ${{}^{3}_{\Lambda}{\rm H}(1/2^{+})}$ in the ($\pi^{-}$, $K^{0}$) reaction is evaluated, e.g., $d\sigma/d\Omega\,({{}^{3}_{\Lambda}{\rm H}})\simeq$ 3 $\mu$b/sr at 1.05–1.20 GeV/$c$, $\theta_{\rm lab}=$ 3o. The recoil effects are very important to the production with the light nuclear target as ${}^{3}{\rm He}$, as well as the medium effects of the $\pi N\to\Lambda K$ amplitudes for nuclear ($\pi$, $K$) reactions. More precise analysis on convergence of the CDCC model space depending on ($k_{\rm max}$, $\ell_{\rm max}$) should be needed. This investigation is in progress.

The authors would like to thank H. Tamura and A. Gal for valuable comments. This work was supported by JSPS KAKENHI Grant Numbers JP16K05363.