Production of a 4He_Lambda hypernucleus in the 4He(pi, K) reactions reexamined

TL;DR

This paper theoretically examines the production cross sections of the $^4_\Lambda$He hypernucleus in the $^4$He($\pi$, $K$) reaction, highlighting the sensitivity to wavefunctions and meson distortions, and providing specific cross section estimates.

Contribution

It reexamines the production of $^4_\Lambda$He hypernucleus using distorted-wave impulse approximation with updated models, emphasizing the effects of wavefunctions and meson distortions.

Findings

Calculated cross section at 1.05 GeV/c is approximately 11 μb/sr.

Production cross sections are sensitive to $\\Lambda$ wavefunctions and meson distortions.

Recoil effects and energy dependence significantly influence the results.

Abstract

We investigate theoretically production cross sections of the $J^\pi=$ 0$^+$ ground state of a $^4_\Lambda$He hypernucleus in the $^4$He($\pi$,~$K$) reaction with a distorted-wave impulse approximation using the optimal Fermi-averaged $\pi N\to K\Lambda$ $t$ matrix. We demonstrate the sensitivity of the production cross sections to the $\Lambda$ wavefunctions obtained from $3N$-$\Lambda$ potentials and to meson distorted waves in eikonal distortions. It is shown that the calculated laboratory cross sections of the 0$^+$ ground state in $^4_\Lambda$He amount to $d\sigma/d\Omega_{\rm lab} \simeq$ 11 $\mu$b/sr at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward direction because of an advantage of the use of the s-shell target nucleus such as $^4$He. The importance of the recoil effects and the energy dependence of the $\pi N\to K\Lambda$ cross sections is also discussed.

Tables

Table 1. Table 1: Kinematical values and the π ​ N → K ​ Λ → 𝜋 𝑁 𝐾 Λ \pi N\to K\Lambda lab cross sections in the nuclear ( π 𝜋 \pi ,  K 𝐾 K ) reactions on a 4 He target at p π = subscript 𝑝 𝜋 absent p_{\pi}= 1.05 GeV/ c 𝑐 c . The non-spin-flip amplitudes in the π ​ N → K ​ Λ → 𝜋 𝑁 𝐾 Λ \pi N\to K\Lambda reaction are used.

${\theta }_{\mathrm{lab}}$	$p_K$	$q$	$q_{\mathrm{eff}}$111 Effective momentum transfer to the $\Lambda$ final state, $q_{\mathrm{eff}}\approx \left(M_C/M_A\right)q$ given in Eq. (32).	$\beta$	$\alpha$	$\alpha {⟨d\sigma /d\Omega ⟩}_{\pi N\to \Lambda K}$
(degree)	(MeV/$c$)	(MeV/$c$)	(MeV/$c$)			free ($\mu$b/sr)	opt ($\mu$b/sr)
0	690.1	359.9	271.1	0.684	0.828	574	464
4	689.5	365.4	275.3	0.686	0.829	561	487
8	687.7	381.3	287.3	0.692	0.834	525	468
12	684.6	406.2	306.1	0.703	0.843	472	416

Table 2. Table 2: Calculated PWIA and DWIA results of the differential lab cross sections of 0 g . s . + subscript superscript 0 formulae-sequence g s 0^{+}_{\rm g.s.} in Λ 4 subscript superscript absent 4 Λ {}^{4}_{\Lambda} He via the 4 He( π + superscript 𝜋 \pi^{+} ,  K + superscript 𝐾 K^{+} ) reaction at p π = subscript 𝑝 𝜋 absent p_{\pi}= 1.05 GeV/ c 𝑐 c . The Isle, SG, and HO ( ℏ ​ ω Λ = Planck-constant-over-2-pi subscript 𝜔 Λ absent \hbar\omega_{\Lambda}= 7.0 MeV) potentials are used with ( σ π , σ K ) = subscript 𝜎 𝜋 subscript 𝜎 𝐾 absent ({\sigma}_{\pi},~{}{\sigma}_{K})= (30 mb, 15 mb). The recoil effects are taken into account.

	${\theta }_{\mathrm{lab}}$	$N_{\mathrm{eff}}^{\mathrm{PW}}$	${\left(d\sigma /d\Omega \right)}_{\mathrm{lab}}^{\mathrm{PW}}$	$D_{\mathrm{dis}}$111 $D_{\mathrm{dis}}=N_{\mathrm{eff}}^{\mathrm{DW}}/N_{\mathrm{eff}}^{\mathrm{PW}}$.	$N_{\mathrm{eff}}^{\mathrm{DW}}$	${\left(d\sigma /d\Omega \right)}_{\mathrm{lab}}^{\mathrm{DW}}$
	(degree)	($\times {10}^{-2}$)	($\mu$b/sr)		($\times {10}^{-2}$)	($\mu$b/sr)
Isle
	0	7.571	35.14	0.334	2.527	11.73
	4	6.951	33.88	0.324	2.254	10.99
	8	5.390	25.25	0.296	1.595	7.47
	12	3.540	14.72	0.248	0.879	3.65
SG
	0	12.68	58.84	0.332	4.211	19.55
	4	11.82	57.62	0.325	3.841	18.72
	8	9.616	45.06	0.303	2.918	13.67
	12	6.876	28.58	0.268	1.846	7.67
HO
	0	5.160	23.95	0.309	1.595	7.40
	4	4.704	22.93	0.298	1.402	6.83
	8	3.574	16.75	0.265	0.948	4.44
	12	2.281	9.481	0.211	0.482	2.00

Table 3. Table 3: Calculated PWIA and DWIA results of the differential lab cross sections of 0 g . s . + subscript superscript 0 formulae-sequence g s 0^{+}_{\rm g.s.} in Λ 4 subscript superscript absent 4 Λ {}^{4}_{\Lambda} He via the 4 He( π + superscript 𝜋 \pi^{+} ,  K + superscript 𝐾 K^{+} ) reaction at p π = subscript 𝑝 𝜋 absent p_{\pi}= 1.05 GeV/ c 𝑐 c . The Isle, SG, and HO ( ℏ ​ ω Λ = Planck-constant-over-2-pi subscript 𝜔 Λ absent \hbar\omega_{\Lambda}= 7.0 MeV) potentials are used with ( σ π , σ K ) = subscript 𝜎 𝜋 subscript 𝜎 𝐾 absent ({\sigma}_{\pi},~{}{\sigma}_{K})= (30 mb, 15 mb). The recoil effects are omitted.

	${\theta }_{\mathrm{lab}}$	$N_{\mathrm{eff}}^{\mathrm{PW}}$	${\left(d\sigma /d\Omega \right)}_{\mathrm{lab}}^{\mathrm{PW}}$	$D_{\mathrm{dis}}$	$N_{\mathrm{eff}}^{\mathrm{DW}}$	${\left(d\sigma /d\Omega \right)}_{\mathrm{lab}}^{\mathrm{DW}}$
	(degree)	($\times {10}^{-2}$)	($\mu$b/sr)		($\times {10}^{-2}$)	($\mu$b/sr)
Isle
	0	0.886	4.113	0.093	0.082	0.382
	4	0.756	3.684	0.077	0.058	0.283
	8	0.466	2.181	0.034	0.016	0.074
	12	0.199	0.829	0.000	0.000	0.000
SG
	0	2.413	11.23	0.162	0.391	1.813
	4	2.154	10.59	0.151	0.325	1.583
	8	1.538	7.572	0.119	0.183	0.857
	12	0.886	3.683	0.071	0.063	0.263
HO
	0	0.542	2.520	0.054	0.029	0.136
	4	0.463	2.256	0.039	0.018	0.092
	8	0.286	1.342	0.010	0.003	0.013
	12	0.128	0.530	0.011	0.002	0.006

Table 4. Table 4: Distortion-parameter dependence of the differential lab cross sections of 0 g . s . + subscript superscript 0 formulae-sequence g s 0^{+}_{\rm g.s.} in Λ 4 subscript superscript absent 4 Λ {}^{4}_{\Lambda} He via the 4 He( π + superscript 𝜋 \pi^{+} ,  K + superscript 𝐾 K^{+} ) reaction at p π = subscript 𝑝 𝜋 absent p_{\pi}= 1.05 GeV/ c 𝑐 c . The Isle potential for Λ Λ \Lambda is used.

$({\sigma }_{\pi },{\sigma }_K)$	${\theta }_{\mathrm{lab}}$	$D_{\mathrm{dis}}$	$N_{\mathrm{eff}}^{\mathrm{DW}}$	${\left(d\sigma /d\Omega \right)}_{\mathrm{lab}}^{\mathrm{DW}}$
(mb)	(degree)		($\times {10}^{-2}$)	($\mu$b/sr)
(20, 20)	0	0.387	2.932	13.61
	4	0.378	2.629	12.81
	8	0.351	1.889	8.85
	12	0.304	1.075	4.47
(30, 10)	0	0.372	2.818	13.08
	4	0.363	2.523	12.30
	8	0.335	1.805	8.45
	12	0.287	1.016	4.22
(30, 30)	0	0.243	1.841	8.54
	4	0.234	1.625	7.92
	8	0.206	1.110	5.20
	12	0.160	0.568	2.36

Equations

$$\frac{d^2\sigma }{d{E}_{K}d{\Omega }_K}$$

$$\beta=\biggl{(}1+{E^{(0)}_{K}\over E^{(0)}_{B}}{{p^{(0)}_{K}-p^{(0)}_{\pi}\cos\theta_{\rm lab}}\over p^{(0)}_{K}}\biggr{)}{p_{K}E_{K}\over p^{(0)}_{K}E^{(0)}_{K}},$$

$$\hat{F}$$

$$\omega =E_{\pi }-E_K,\mathbf{q}={\mathbf{p}}_{\pi }-{\mathbf{p}}_K,$$

$$q=(p_{\pi }^2+p_K^2-2p_{\pi }{p}_K\cos {\theta }_{\mathrm{lab}}{)}^{1/2},$$

$$\displaystyle\biggl{(}{d\sigma\over d{\Omega}_{K}}\biggr{)}$$

$$\displaystyle\left({d\sigma\over d\Omega_{K}}\right)_{\rm lab,\theta_{\rm lab}}^{J^{\pi}}=\alpha{1\over{[J_{A}]}}\sum_{m_{A}m_{B}}\biggl{|}\Bigl{\langle}{\Psi}_{B}\Big{|}\,\overline{f}_{\pi N\to K\Lambda}$$

$$\displaystyle\qquad\times\,\chi^{(-)*}_{K}({\bm{p}}_{K},{M_{C}\over M_{B}}{\bm{r}})\chi^{(+)}_{\pi}({\bm{p}}_{\pi},{M_{C}\over M_{A}}{\bm{r}})\Big{|}\Psi_{A}\Bigr{\rangle}\biggr{|}^{2},\qquad$$

$$\alpha =\beta {\left(1+\frac{E_K}{E_B}\frac{p_K-p_{\pi }\cos {\theta }_{\mathrm{lab}}}{p_K}\right)}^{-1}.$$

$$f_{{\pi }^{+}n\to K^{+}\Lambda }=-f_{{\pi }^{-}p\to K^0\Lambda }.$$

$$\displaystyle{\Psi}_{B}=\left[\bigl{[}\phi_{3N}\otimes\varphi^{(\Lambda)}_{\ell_{\Lambda}}\bigr{]}_{L_{B}}\otimes X^{B}_{T_{B},S_{B}}\right]_{J_{B}},$$

$$\displaystyle X^{B}_{T_{B},S_{B}}=\bigl{[}\chi^{(3N)}_{I_{3},S_{3}}\otimes\chi^{(\Lambda)}_{{0},{1/2}}\bigr{]}_{1/2,0},$$

$$\left(-\frac{{\hslash }^2}{2\mu }{\nabla }^2+U_{\Lambda }\right){\phi }_{{\ell }_{\Lambda }}^{(\Lambda )}=-B_{\Lambda }{\phi }_{{\ell }_{\Lambda }}^{(\Lambda )},$$

$$U_{\Lambda }$$

$${\hat{F}}^{\mathrm{ex}}$$

$$U_{\Lambda }(\mathbf{r})$$

$$\displaystyle{\Psi}_{A}={\cal A}\left[\bigl{[}\phi_{3N}\otimes\varphi^{(N)}_{\ell_{N}}\bigr{]}_{L_{A}}\otimes X^{A}_{T_{A},S_{A}}\right]_{J_{A}},$$

$$\displaystyle X^{A}_{T_{A},S_{A}}=\bigl{[}\chi^{(3N)}_{I_{3},S_{3}}\otimes\chi^{(N)}_{{1/2},{1/2}}\bigr{]}_{0,0},$$

$$U_N(\mathbf{r})$$

$$\displaystyle\varphi^{(\Lambda)}_{\ell_{\Lambda}}({\bm{r}})={\biggl{(}\frac{4}{\tilde{b}_{\Lambda}^{3}\sqrt{\pi}}\biggr{)}^{1/2}}\exp{\biggl{(}-\frac{r^{2}}{2\tilde{b}_{\Lambda}^{2}}\biggr{)}}Y_{0}^{0}(\hat{\bm{r}}),$$

$${\tilde{b}}_{\Lambda }=b_{\Lambda }\sqrt{1+m_{\Lambda }/3m_N}=2.634\mathrm{fm},$$

$${\left(\frac{d\sigma }{d{\Omega }_K}\right)}_{\mathrm{lab},{\theta }_{\mathrm{lab}}}^{J^{\pi }}=\alpha {⟨\frac{d\sigma }{d{\Omega }_K}⟩}_{\pi N\to K\Lambda }^{\mathrm{opt}}{N}_{\mathrm{eff}}^{J^{\pi }}({\theta }_{\mathrm{lab}}),$$

$$N_{\mathrm{eff}}^{J^{\pi }}({\theta }_{\mathrm{lab}})$$

$$C_{B}=\Bigl{\langle}X^{B}_{T_{B},S_{B}}\Big{|}\sum_{j=1}^{A}{\hat{O}}_{j}\Big{|}X^{A}_{T_{A},S_{A}}\Bigr{\rangle}.$$

$$F(q)$$

$${\rho }_{{\ell }_{\Lambda }{\ell }_N}^{(tr)}(r)$$

$$N_{\mathrm{eff}}^{0^{+}}({\theta }_{\mathrm{lab}})=2|F(q){|}^2,$$

$${\chi }_K^{(-)\ast }({\mathbf{p}}_K,\mathbf{r}){\chi }_{\pi }^{(+)}({\mathbf{p}}_{\pi },\mathbf{r})=\exp (i\mathbf{q}\cdot \mathbf{r})D(\mathbf{b},z)$$

$$D(\mathbf{b},z)$$

$${\chi }_K^{(-)\ast }({\mathbf{p}}_K,\mathbf{r}){\chi }_{\pi }^{(+)}({\mathbf{p}}_{\pi },\mathbf{r})$$

$${\tilde{j}}_L(q;r)$$

$$D_{\ell }(r)=\frac{2\ell +1}{2}{\int }_{-1}^{1}D(\mathbf{b},z)P_{\ell }(\cos \theta )d(\cos \theta ).$$

Full text

Hypernuclei, DWIA, Cross section, Recoil effect

††preprint: J-PARC-TH-0167

I Introduction

Recently, unexpected short lifetimes of a ${{}^{3}_{\Lambda}{\rm H}}$ hypernucleus were measured in hypernuclear production of high-energy heavy-ion collisions HypHI13 ; ALICE16 ; STAR18 ; the world average lifetime of $\tau^{\rm(av)}({{}^{3}_{\Lambda}{\rm H}})=$ $185^{+23}_{-28}$ ps is shorter than the free lifetime $\tau_{\Lambda}=$ $263.2\pm 2.0$ ps by about 30%. However, ALICE Collaboration ALICE18 newly reported the result of $\tau({{}^{3}_{\Lambda}{\rm H}})=$ $237^{+33}_{-36}$ ps which is moderately closer to $\tau_{\Lambda}$. This is one of the most topical issues to study hypernuclear physics Gal19 , and is essential because ${}^{3}_{\Lambda}$H is the lightest hypernucleus which provides valuable information on interacting a $\Lambda$ hyperon with nucleons Kamada98 . In addition, HypHI Collaboration HypHI13 suggested that the lifetime of a ${}^{4}_{\Lambda}$H hypernucleus $\tau(^{4}_{\Lambda}{\rm H})=$ $140^{+48}_{-33}$ ps is shorter than $\tau(^{4}_{\Lambda}{\rm H})=$ $194^{+28}_{-26}$ ps measured in stopped $K^{-}$ experiments at KEK Outa98 , whereas theoretical calculations Motoba92 ; Fuse95 predicted $\tau(^{4}_{\Lambda}{\rm H})=$ 196–264 ps which depend on the $\Lambda$ wavefunctions. To solve the lifetime puzzle, experimental measurements of the ${{}^{3,4}_{\Lambda}{\rm H}}$ lifetime have been planned in ($K^{-}$, $\pi^{0}$) and ($\pi^{-}$, $K^{0}$) reactions on 3,4He targets at J-PARC J-PARC-P73 ; J-PARC-P74 . Therefore, it is important to investigate theoretically production of $A=$ 3, 4 hypernuclei via the ($\bar{K}$, $\pi$) and ($\pi$, $K$) reactions on 3,4He targets Harada19 . Especially, we believe that $\Lambda$ production studies on the 4He target are useful to settle mysterious problems related to the $A=4$ hypernuclei Gal16 , e.g., overbinding/underbinding anomaly Dalitz72 ; Akaishi00 , charge symmetry breaking (CSB) Dalitz64 ; Gal15 , and so on.

Many theoretical studies of hypernuclear spectroscopy have been performed in nuclear ($K^{-}$, $\pi^{-}$), ($\pi^{+}$, $K^{+}$), and ($\gamma$, $K^{+}$) reactions Gal16 . Production cross sections of $\Lambda$ hypernuclear states for each reaction are usually characterized by a specific momentum transfer to a $\Lambda$ hyperon. The ($K^{-}$, $\pi^{-}$) reactions on 4He Harada98 can produce a $\Lambda$ substitutional state in ${}^{4}_{\Lambda}$He under recoilless conditions. On the other hand, the ($\pi^{+}$, $K^{+}$) reactions on 4He enable us to give large momentum transfers of $q=$ 300–500 MeV/$c$ Shinmura86 , populating a $\Lambda$ stretched state which seems to be disadvantageous to the $0s_{N}\to 0s_{\Lambda}$ transition in terms of a momentum matching law Dover80 ; however, the production cross sections of ${}^{4}_{\Lambda}$He via the ($\pi^{+}$, $K^{+}$) reaction on 4He were predicted theoretically Harada06 for the non-mesonic weak decay of ${}^{4}_{\Lambda}$He and ${}^{4}_{\Lambda}$H at J-PARC Ajimura06 . Consequently, it is worth revisiting investigation of ${}^{4}_{\Lambda}$He production via the ($\pi$, $K$) reactions on 4He because we have now been achieved a good understanding of the nuclear ($\pi$, $K$) reactions Harada05 .

In this paper, we investigate theoretically the production cross sections of the $J^{\pi}=$ 0+ ground state ($0^{+}_{\rm g.s.}$) of a ${}^{4}_{\Lambda}$He hypernucleus via the 4He($\pi^{+}$, $K^{+}$) reaction in the distorted-wave impulse approximation. We demonstrate the differential cross sections at $p_{\pi}=$ 1.05 GeV/$c$ in the $K^{+}$ forward direction in the laboratory system. We discuss medium effects of the $\pi N\to K\Lambda$ amplitude in nuclear ($\pi$, $K$) reactions, and recoil effects in the light hypernucleus such as ${}^{4}_{\Lambda}$He. Our results on ${}^{4}_{\Lambda}$He production via 4He($\pi^{+}$, $K^{+}$) reactions are expected to be identical to those on ${}^{4}_{\Lambda}$H mirror production via 4He($\pi^{-}$, $K^{0}$) reactions because charge independence guarantees ${f}_{\pi^{+}n\to K^{+}\Lambda}=-{f}_{\pi^{-}p\to K^{0}\Lambda}$ in the ($\pi$, $K$) reactions on $T=0$ target nuclei.

II Calculations

II.1 Distorted-wave impulse approximation

Let us consider a calculation procedure of hypernuclear production for nuclear ($\pi$, $K$) reactions in the laboratory (lab) frame. The double differential cross sections within the distorted-wave impulse approximation (DWIA) Hufner74 ; Auerbach83 are given by (in units $\hbar=c=1$)

[TABLE]

where $[J]=2J+1$, 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 final states and the initial state of the target nucleus, respectively. The kinematical factor $\beta$ Tadokoro95 ; Koike08 arising from a translation from a two-body meson-nucleon lab system to a meson-nucleus lab system Dover83 is given by

[TABLE]

where $p^{(0)}_{\pi}$ and $p^{(0)}_{K}$ ($E^{(0)}_{K}$ and $E^{(0)}_{B}$) are lab momenta of $\pi$ and $K$ (lab energies of $K$ and $\Lambda$) in the two-body $\pi N\to K\Lambda$ reaction, respectively. Here we considered only the non-spin-flip amplitude because we are interested in the $\Delta S=$ 0 cross sections in the $K$ forward direction. Thus an external operator $\hat{F}$ for the associated production $\pi N\to K\Lambda$ reactions is given by

[TABLE]

where we assume zero-range interaction for the $\pi N\to K\Lambda$ transitions. Distorted waves of $\chi_{K}^{(-)\ast}$ and $\chi_{\pi}^{(+)}$ are obtained with the help of the eikonal approximation Hufner74 . ${\hat{O}}_{j}$ is a baryon operator changing $j$th nucleon into a $\Lambda$ hyperon in the nucleus, and ${\bm{r}}$ is the relative coordinate between the mesons and the center-of-mass (c.m.) of the nucleus; $\overline{f}_{\pi N\to K\Lambda}$ is the Fermi-averaged non-spin-flip amplitude for the $\pi N\to K\Lambda$ reactions in nuclei on the lab frame Harada05 .

The energy and momentum transfer to the $\Lambda$ final state is given by

[TABLE]

where $E_{\pi}=(m_{\pi}^{2}+{\bm{p}}_{\pi}^{2})^{1/2}$ and $E_{K}=(m_{K}^{2}+{\bm{p}}_{K}^{2})^{1/2}$ are the lab energies of $\pi$ and $K$ in the nuclear reaction, respectively; $m_{\pi}$ and $m_{K}$ (${\bm{p}}_{\pi}$ and ${\bm{p}}_{K}$) are masses (lab momenta) of $\pi$ and $K$, respectively. Figure 1 displays the momentum transfer to the $0^{+}_{\rm g.s.}$ final state in ${}^{4}_{\Lambda}$He,

[TABLE]

as a function of the incident lab momentum $p_{\pi}$.

The differential lab cross section of a $\Lambda$ hypernuclear state is obtained by the energy integration Morimatsu94

[TABLE]

around a corresponding peak in the inclusive $K$ spectrum. We often adopt the effective number technique into the differential lab cross section within DWIA Hufner74 ; Dover83 ; Koike08 ; Morimatsu94 ; Itonaga94 . Thus the differential lab cross section of the $\Lambda$ bound state with $J^{\pi}$ can be written as

[TABLE]

where ${\bm{r}}$ is the relative coordinate between a $3N$-core nucleus and a nucleon or $\Lambda$ hyperon; the factors of $M_{C}/M_{B}$ and $M_{C}/M_{A}$ take into account the recoil effects where $M_{A}$, $M_{B}$, and $M_{C}$ are masses of the target, the hypernucleus, and the core nucleus, respectively. The kinematical factor $\alpha$ Dover83 is related to $\beta$ in Eq. (2) as

[TABLE]

In Table 1, we show the kinematic values for ${}^{4}_{\Lambda}$He production via the nuclear ($\pi$, $K$) reaction on a 4He target at $p_{\pi}=$ 1.05 GeV/$c$, $\theta_{\rm lab}=$ 0∘, 4∘, 8∘, and 12∘, where we choose $B_{\Lambda}=$ 2.39 MeV as the $\Lambda$ binding energy for $0^{+}_{\rm g.s}$ in ${}^{4}_{\Lambda}$He. Note that the value of $\alpha$ for a 4He target at $p_{\pi}=$ 1.05 GeV/$c$ (0∘) amounts to 0.828 which is 25% larger than that for heavier nuclei; $\alpha\simeq$ 0.66 for a 40Ca target Dover80 .

II.2 Fermi-averaged ($\pi$, $K$) cross sections

It should be noticed that the strong energy dependence of differential lab cross sections appears in the nuclear ($\pi$, $K$) reactions, as discussed in Ref. Harada05 . To consider the $p_{\pi}$ dependence of elementary $\pi N\to K\Lambda$ lab cross sections, we do averaging of the $\pi N\to K\Lambda$ $t$ matrix in the lab frame over a Fermi-momentum distribution, where nuclear effects of a nucleon binding $\varepsilon_{N}$ are naturally taken into account. This procedure is called as the “optimal Fermi-averaging” under the on-energy-shell condition Harada05 . Charge independence guarantees the following relation between the $\pi N\to K\Lambda$ amplitudes:

[TABLE]

Thus the ($\pi^{+}$, $K^{+}$) and ($\pi^{-}$, $K^{0}$) cross sections are identical to each other on the $T=0$ nuclear targets as 4He. Here we employed the elementary $\pi^{-}p\to K^{0}\Lambda$ amplitudes analyzed by Sotona and $\check{\rm Z}$ofka Sotona89 .

Figure 2 shows the optimal Fermi-averaged $\pi N\to K\Lambda$ lab cross sections of $\alpha\langle d\sigma/d\Omega\rangle^{\rm opt}_{\pi N\to K\Lambda}$ including the kinematical factor $\alpha$ in nuclear ($\pi$, $K$) reactions on a 4He target, together with the elementary $\pi N\to K\Lambda$ lab cross sections of $\alpha\langle d\sigma/d\Omega\rangle^{\rm free}_{\pi N\to K\Lambda}$ in free space Sotona89 . The peaks of $\alpha\langle d\sigma/d\Omega\rangle^{\rm free}_{\pi N\to K\Lambda}$ are located at $p_{\pi}\simeq$ 1.05 GeV/$c$ corresponding to $M(\pi N)\simeq$ 1700 MeV/$c^{2}$ in the invariant mass of $\pi$ and $N$ because there exist $N^{*}$ resonances, e.g., $S_{11}$(1680), $P_{11}$(1730), and $P_{13}$(1700). We find that the peaks of $\alpha\langle d\sigma/d\Omega\rangle^{\rm opt}_{\pi N\to K\Lambda}$ are shifted to the position of $p_{\pi}\simeq$ 1.00 GeV/$c$, taken into account the Fermi motion of a struck nucleon under the optimal condition in the nucleus; the shape of $\alpha\langle d\sigma/d\Omega\rangle^{\rm opt}_{\pi N\to K\Lambda}$ is moderately broader than that of $\alpha\langle d\sigma/d\Omega\rangle^{\rm free}_{\pi N\to K\Lambda}$. In Table 1, we also show the values of $\alpha\langle d\sigma/d\Omega\rangle_{\pi N\to\Lambda K}^{\rm opt}$, together with the kinematical factor $\alpha$ at $p_{\pi}=$ 1.05 GeV/$c$.

II.3 Wavefunctions

In our calculations, we have assumed the $3N$ $1/2^{+}$ ground state as a core-nucleus in $A=$ 4 systems Harada98 , i.e., $\phi_{3N}$ is a wavefunction for the $3N$ eigenstates (3He or 3H), and we have neglected the CSB effects in $A=$ 4 hypernuclei. Thus the wavefunction of the $0^{+}$ ground state ($0^{+}_{\rm g.s.}$) in ${}^{4}_{\Lambda}$He ($L_{B}=$ 0, $S_{B}=$ 0, $T_{B}=$ 1/2) is written as

[TABLE]

where $\varphi^{(\Lambda)}_{\ell_{\Lambda}}$ is a relative wavefunction between $3N$ and $\Lambda$, and $X^{B}_{T_{B},S_{B}}$ is the isospin-spin function for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He; $\chi^{(3N)}_{I_{3},S_{3}}$ and $\chi^{(\Lambda)}_{{0},{1/2}}$ are the isospin-spin functions for $3N$ (isospin $I_{3}$, spin $S_{3}$) and $\Lambda$ ($I=$ 0, $S=$ 1/2), respectively. The explicit form of $X^{B}_{T_{B},S_{B}}$ is given in Appendix A. According to Ref. Kurihara82 , we use $\varphi^{(\Lambda)}_{\ell_{\Lambda}}$ which is regarded as a spectroscopic amplitude obtained from a four-body $\Lambda NNN$ wavefunction using realistic central nucleon-nucleon ($NN$) and $\Lambda N$ potentials Dalitz72 . It should be noticed that $\varphi^{(\Lambda)}_{\ell_{\Lambda}}$ includes the contribution of the $\Lambda N$ short-range correlations and also many-body correlations. Thus we consider that $\varphi^{(\Lambda)}_{\ell_{\Lambda}}$ satisfies the Schrödinger equation

[TABLE]

where $\mu$ is the $3N$-$\Lambda$ reduced mass and $B_{\Lambda}$ is the $\Lambda$ binding energy with respect to the ${}^{3}{\rm He}$-$\Lambda$ threshold. The $3N$-$\Lambda$ potential ${U}_{\Lambda}$ is defined as

[TABLE]

where $\bar{V}^{\rm ex}$ is the sum of isospin-spin averaged $\Lambda N$ potentials, and $\hat{F}^{\rm ex}$ is an external operator on the basis of multiple scattering processes Akaishi86 :

[TABLE]

where $Q=1-|\phi_{3N})(\phi_{3N}|$ and $e=E-\hat{H}_{3N}+\hbar^{2}\nabla^{2}/2\mu$. Therefore, ${U}_{\Lambda}$ can be derived from the four-body $\Lambda NNN$ wavefunction, taken into account the $\Lambda N$ short-range correlations Kurihara82 ; Akaishi86 .

Figure 3 shows ${U}_{\Lambda}$ as a function of the distance between $3N$ and $\Lambda$. We recognize that this potential has a central repulsion and an attractive tail, so we call it as “Isle” potential Fuse95 ; Kurihara82 ; Akaishi86 ; Kurihara85 ; the central repulsion originates predominantly from the $\Lambda N$ short-range correlations due to the repulsive core of the $\Lambda N$ potentials, and it plays an important role in describing the lifetime of ${}^{4}_{\Lambda}$He in precise experimental studies on the mesonic weak decay of $\Lambda\to p\pi^{-}$ Outa98 ; Fuse95 . For convenience of use, we parameterize $U_{\Lambda}$ into a two-range Gaussian form as

[TABLE]

where $V_{C}=$ 91.61 MeV, $V_{A}=$ $-$80.88 MeV, $b_{C}=$ 1.14 fm, and $b_{A}=$ 1.69 fm, reproducing the experimental data of $B_{\Lambda}^{\rm exp}=$ $2.39\pm 0.03$ MeV. To clearly see the effect of the central repulsion, we also introduce a single-range Gaussian (SG) potential of $U_{\Lambda}({\bm{r}})=-38.5\exp\{-(r/b_{A})^{2}\}$ with $b_{A}=$ 1.70 fm, adjusting to $B_{\Lambda}=$ 2.39 MeV. The SG potential is often used as a phenomenological one.

The wavefunction of the $0^{+}$ ground state in 4He ($L_{A}=$ 0, $S_{A}=$ 0, $T_{A}=$ 0) is written as

[TABLE]

where ${\cal A}$ is the anti-symmetrized operator for nucleons, and $\varphi^{(N)}_{\ell_{N}}$ is a relative wavefunction between $3N$ and $N$; $X^{A}_{T_{A},S_{A}}$ and $\chi^{(N)}_{{1/2},{1/2}}$ are the isospin-spin functions for 4He and $N$ ($I=$ 1/2, $S=$ 1/2), respectively. Here we used $\varphi^{(N)}_{\ell_{N}}$ obtained from the $3N$-$N$ potential ${U}_{N}$ which was derived from a microscopic four-body calculation with a central $NN$ potential of Tamagaki’s C3G Akaishi86 . Therefore, $\varphi^{(N)}_{\ell_{N}}$ includes the contribution of the $NN$ short-range correlations and also many-body correlations Kurihara82 . For convenience of use, this potential ${U}_{N}$ is parametrized into useful Gaussian forms as

[TABLE]

with $V_{1}=$ 156.28 MeV, $V_{2}=$ $-$185.66 MeV, $V_{3}=$ $-$9.56 MeV, $b_{1}=$ 1.21 fm, $b_{2}=$ 1.58 fm, and $b_{3}=$ 2.82 fm Harada90 , making a fit to the experimental data of the binding energy $B_{N}=$ 20.6 MeV and the nuclear root-mean-square (r.m.s.) distance of $\langle{r}^{2}_{N}\rangle^{1/2}=$ 1.87 fm between ${}^{3}{\rm He}$ and $n$.

Figure 4 shows that $3N$-$\Lambda$ density distributions $\rho_{\Lambda}(r)$ between $3N$ and $\Lambda$ for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He, as a function of the relative distance, together with $3N$-$N$ density distributions $\rho_{N}(r)$ between $3N$ and $N$ for $0^{+}_{\rm g.s.}$ in 4He. We find that the $3N$-$\Lambda$ distribution obtained from the Isle potential differs fairly from that obtained from the SG potential around the nuclear inside; the $3N$-$\Lambda$ distribution is considerably suppressed at the nuclear center, and it is pushed outside, as discussed in Refs. Kurihara82 ; Akaishi86 ; Kurihara85 ; Harada90 . The relative r.m.s. distance between $3N$ and $\Lambda$ becomes $\langle{r}^{2}_{\Lambda}\rangle^{1/2}=$ 3.57 fm for Isle, which is 8% larger than 3.31 fm for SG.

To see properties of our $\varphi^{(\Lambda)}_{\ell_{\Lambda}}$ obtained by $U_{\Lambda}$ in Fig. 3, we also consider a single-particle (s.p.) harmonic oscillator (HO) wavefunction with the size parameter $b_{\Lambda}=(\hbar/m_{\Lambda}\omega_{\Lambda})^{1/2}$, which is used as simple model calculations Dover80 ; Cieply01 . Here we choose $b_{\Lambda}=$ 2.233 fm ($\hbar\omega_{\Lambda}=$ 7.0 MeV), simulating the value of the relative r.m.s. distance of $\langle r^{2}_{\Lambda}\rangle^{1/2}=$ 3.3–3.6 fm for the $3N$-$\Lambda$ distribution obtained by four-body $\Lambda NNN$ calculations using the $NN$ and $\Lambda N$ potentials Dalitz72 ; Bodmer85 ; Hiyama01 ; we have $\langle r^{2}_{\Lambda}\rangle^{1/2}=$ 3.23 fm for HO. The relative $3N$-$\Lambda$ wavefunction with ${\ell_{\Lambda}}=0$ for HO is written as

[TABLE]

where the size parameter with the c.m. correction denotes

[TABLE]

where $m_{\Lambda}$ and $m_{N}$ are masses of a $\Lambda$ hyperon and a nucleon, respectively. We confirm $\langle r^{2}_{\Lambda}\rangle^{1/2}=\tilde{b}_{\Lambda}\sqrt{3/2}=$ 3.23 fm for HO. Figure 4 also shows the $3N$-$\Lambda$ distribution for HO taken into account the c.m. correction which is absolutely essential in light nuclei. We expect that the difference among the $3N$-$\Lambda$ distributions obtained from several models in Fig. 4 is clearly observed in production cross sections of nuclear ($\pi$, $K$) reactions giving a large momentum transfer to the $\Lambda$ final state. On the other hand, we confirm that the $3N$-$N$ distribution for HO using the size parameter $b_{N}=$ 1.329 fm ($\hbar\omega_{\Lambda}=$ 23.5 MeV) is in good agreement with that obtained from the C3G potential; the relative r.m.s. distance between $3N$ and $N$ has $\langle r^{2}_{N}\rangle^{1/2}=\tilde{b}_{N}\sqrt{3/2}=$ 1.88 fm which is quite close to 1.87 fm for C3G, where $\tilde{b}_{N}=b_{N}\sqrt{4/3}=$ 1.535 fm.

II.4 Differential cross sections

We consider the differential lab cross sections of the hypernuclear bound state with $J^{\pi}$ on a closed-shell target nucleus with $J_{A}^{\pi}=$ 0${}^{+}_{\rm g.s.}$ like 4He, adapting the effective number technique into the DWIA Hufner74 ; Dover83 ; Koike08 ; Morimatsu94 ; Itonaga94 . Substituting Eqs. (15), (10), and (29) into Eq. (7), we obtain

[TABLE]

where $\alpha\langle d\sigma/d\Omega_{K}\rangle_{\pi N\to K\Lambda}^{\rm opt}$ is the optimal Fermi-averaged $\pi N\to K\Lambda$ lab cross section, as discussed in Sect. II.2. The effective number of nucleons $N^{J^{\pi}}_{\rm eff}$ for $\Lambda$ production of the $J^{\pi}$ final state in the $LS$-coupling scheme is written as

[TABLE]

where $\ell_{\Lambda}+L+\ell_{N}$ must be even due to the non-spin-flip reaction. Thus, only natural parity states with $J^{\pi}=$ $0^{+}$, $1^{-}$, $2^{+}$, $3^{-}$, $\cdots$ for the $3N+\Lambda$ systems can be populated. $C_{B}$ is the isospin-spin spectroscopic amplitude between the $\Lambda$ final state of ${}^{4}_{\Lambda}$He and the initial state of ${}^{4}{\rm He}$, which is given by

[TABLE]

The form factor $F(q)$ in Eq. (22) is given as

[TABLE]

with the $N\to\Lambda$ transition densities

[TABLE]

and the distorted waves $\widetilde{j}_{L}(q;z)$ considering the nuclear distortions by mesons in the DW approximation, as we will express in Eq. (30); ${M_{C}/M_{A}}$ is the so-called recoil factor for the momentum transfer $q$. Here we approximated to $M_{C}/M_{A}\approx M_{C}/M_{B}$ in Eq. (7). For $J^{\pi}=$ $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He, we take $\ell_{\Lambda}=\ell_{N}=L=0$ and $C_{B}=\sqrt{2}$ in Eq. (22), then we obtain

[TABLE]

which is expected to observe the hypernuclear fine structure because $F(q)$ is generally sensitive to the nature of the distribution of $\rho^{(tr)}_{00}(r)$, as a function of $q$.

II.5 Meson distorted waves

Full distorted waves of the $\pi$-nucleus and the $K$-nucleus are important to reproduce absolute values of the cross sections. Because the ($\pi$, $K$) reaction requires a large momentum transfer with a high angular momentum, we simplify the computational procedure in the eikonal approximation to the distorted waves of the meson-nucleus states Hufner74 ; Dover80 ; Dover83 ; Koike08 :

[TABLE]

with

[TABLE]

where $\sigma_{\pi}$ ($\sigma_{K}$) is the averaged total cross section in $\pi N$ ($KN$) elastic scatterings, and $\alpha_{\pi}$ ($\alpha_{K}$) is the ratio of the real and imaginary part of the corresponding forward scattering amplitudes; ${\bm{b}}$ is the impact parameter. $\rho({\bm{r}})\equiv\rho({\bm{b}},z)$ is a matter-density distribution fitting to the data on the nuclear charge density Vries86 . We assume $\alpha_{\pi}=\alpha_{K}=$ 0 which affect hardly the following results, and we study $\sigma_{\pi}$= 20–30 mb and $\sigma_{K}$= 10–30 mb Hufner74 ; Dover80 ; ($\sigma_{\pi}$, $\sigma_{K}$)= (30 mb, 15 mb) is chosen as a standard value Harada05 . Reducing the r.h.s. in Eq. (27) by partial-wave expansion, we obtain

[TABLE]

with

[TABLE]

where $D_{\ell}(r)$ is a distortion function Harada05 defined as

[TABLE]

Here $z=r\cos{\theta}$, ${\bm{r}}^{2}={\bm{b}}^{2}+z^{2}$, and $P_{\ell}(x)$ is a Legendre polynomial. If the distortion is switched off, $\widetilde{j}_{L}(q;r)$ is equal to ${j}_{L}(qr)$ which is a spherical Bessel function with $L$.

The production probability for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He is expected to be only $10^{-2}$, which is roughly estimated as $\exp{(-(bq)^{2}/2)}$ with $bq\simeq 3$ Dover80 , due to the $0s_{N}\to 0s_{\Lambda}$ transition with $\Delta L=$ 0 in nuclear ($\pi$, $K$) reactions because the $\Lambda$ continuum states can be populated predominately by the large angular momentum transfer. Figure 5 displays the distorted waves $\widetilde{j_{0}}(q;r)$ for $\pi$ and $K$ in the 4He($\pi$, $K$) reactions at $p_{\pi}=$ 1.05 GeV/$c$ ($\theta_{\rm lab}=$ 0∘) which leads to $q=$ 360 MeV/$c$. We find that the values of $\widetilde{j_{0}}(q;r)$ are reduced near the center of the nucleus due to the nuclear absorption in the distorted wave, in comparison with the plane waves which are obtained with ($\sigma_{\pi}$, $\sigma_{K}$) = (0 mb, 0 mb). We also find that $\widetilde{j_{0}}(q;r)$ spread outside by taking into account the recoil effects which bring us to use the effective momentum transfer

[TABLE]

in the $A=4$ hypernuclei, as seen in Table 1. We recognize a node in $\widetilde{j_{0}}(q;r)$ at $r=r_{n}$ satisfied as

[TABLE]

so we have the point of $r_{1}=(M_{A}/M_{C})\pi/q=$ 2.29 fm located on the outside of the nucleus having $\langle r^{2}_{N}\rangle^{1/2}=$ 1.88 fm for 4He. If we omit the recoil effects ($M_{C}/M_{A}\to 1$), we have $r_{1}\to\pi/q=$ 1.72 fm which is located on the inside of the nucleus. Therefore, the recoil effects must be taken into account for the light nuclear target in the ($\pi$, $K$) reactions providing the large momentum transfer to the $\Lambda$ final state, as we will discuss in Sect. III.3.

III Results and Discussion

Let us consider the $\Lambda$ production for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He via the ($\pi$, $K$) reactions on the 4He target at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward direction. We will discuss the meson distortion effects, comparing between the differential lab cross sections in PWIA and DWIA, and we will study the sensitivity of the cross sections to the $\Lambda$ wavefunctions using the effective number technique of Eq. (19).

III.1 Differential cross sections for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He

III.1.1 PWIA v.s. DWIA

In Table 2, we show the calculated PWIA and DWIA results of the differential lab cross sections for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward-direction angles of $\theta_{\rm lab}=$ 0∘–12∘. The differential lab cross section of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He accounts for

[TABLE]

at $p_{\pi}=$ 1.05 GeV/$c$ ($\theta_{\rm lab}=$ $0^{\circ}$), using the Isle potential. When we take into account the distortion with $({\sigma}_{\pi},~{}{\sigma}_{K})=$ (30 mb, 15 mb), we obtain the differential lab cross section of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He as

[TABLE]

We confirm that the differential lab cross sections in DWIA are relatively reduced by a distortion factor which is defined as

[TABLE]

as shown in Table 2. We find that the absolute values of the differential lab cross sections for SG (HO) are larger (smaller) than those for Isle. This is because the overlaps between wavefunctions between $N$ and $\Lambda$ are larger inside the nucleus in the order of SG, Isle, and HO, as seen in Fig. 4.

III.1.2 $N^{0^{+}}_{\rm eff}$ v.s. $q$

To see the features of the $\Lambda$ production of the nuclear ($\pi$, $K$) reactions, we study the effective number of nucleons $N_{\rm eff}^{J^{\pi}}$, as a function of the momentum transfer $q$ which is determined by the incident lab momentum $p_{\pi}$ and the $K$ forward-direction angles of $\theta_{\rm lab}$.

Figure 6(a) displays the calculated results of $N_{\rm eff}^{0^{+}}$, using the $\Lambda$ wavefunctions obtained from the Isle, SG and HO potentials. The essential physical features of the endothermic nuclear ($\pi$, $K$) reactions can be well understood in terms of $N_{\rm eff}^{0^{+}}$ in PWIA, as suggested by Dover et al. Dover80 . We confirm that the magnitudes of $N_{\rm eff}^{0^{+}}$ obtained from the SG, Isle, and HO potentials in PWIA are large in this order. Note that the nuclear ($\pi$, $K$) reactions at $p_{\pi}=$ 1.05 GeV/$c$ ($\theta_{\rm lab}=$ $0^{\circ}$–$12^{\circ}$) provide $q=$ 360–410 MeV/$c$ corresponding to the region of the halftones drawn in Fig. 6, where the recoil effects are very important owing to $q_{\rm eff}\simeq 0.75q$ in ${}^{4}_{\Lambda}$He. On the other hand, exothermic nuclear ($\bar{K}$, $\pi$) reactions with $p_{\pi}=$ 0.79 GeV/$c$ ($\theta_{\rm lab}=$ $0^{\circ}$–$12^{\circ}$) have $q=$ 68–126 MeV/$c$ which denote small momentum transfers, thus the recoil effects do not so affect the differential lab cross sections. In the region of $q>$ 600 MeV/$c$, we find that the values of $N_{\rm eff}^{0^{+}}$ obtained from the Isle potential fall off, and their slopes are steeper than those with the SG potential; a dip appears in the region of $q=$ 600–800 MeV/$c$. This behavior is well known to come from high momentum components in the $\Lambda N$ and $NN$ wavefunctions due to short-range correlations, as discussed in Ref. Shinmura86 . We also find a dip in $N^{0^{+}}_{\rm eff}$ for HO which includes only the $NN$ correlations.

Figure 6(b) shows the calculated results of $N_{\rm eff}^{0^{+}}$ with DWIA. The distortions reduce the magnitudes of $N_{\rm eff}^{0^{+}}$ by about three times (see Table 2), changing slightly the shapes of $N_{\rm eff}^{0^{+}}$ in the region of $q<$ 400 MeV/$c$. The slopes of $N_{\rm eff}^{0^{+}}$ become gradually steeper as increasing $q$ as in the case of $q>$ 400 MeV/$c$, and they grow a dip at the region of $q=$ 520–620 MeV/$c$ which can be achieved by $\theta_{\rm lab}=$ 24∘–34∘. These behaviors may originate from high momentum components generated by meson distortions, because $D_{0}(r)$ significantly modifies $j_{0}(qr)$ inside the nucleus. Therefore, the distortion effects as well as the recoil effects are very important for large momentum transfer processes which can be realized in the ($\pi$, $K$) reactions.

III.2 Comparison with eikonal-oscillator approximation

We also consider the differential lab cross sections using the single-particle (s.p.) harmonic oscillator (HO) wavefunctions and the eikonal distortions by mesons, referring to it as the “eikonal-oscillator” approximation Dover80 . This is often employed as nuclear model calculations for several reactions Dover83 ; Cieply01 . When we use the HO wavefunctions for both nucleon and $\Lambda$, we can express the eikonal distorted waves as

[TABLE]

at the $K$ forward direction angle of $\theta_{\rm lab}=$ 0∘; the nuclear thickness function for the $A=Z+N$ target nucleus is defined as

[TABLE]

with the averaged total cross section $\bar{\sigma}=(\sigma_{\pi}+\sigma_{K})/2$ for the $\pi N$ and $KN$ elastic scatterings. Thus we have the effective number of nucleons $N^{J^{\pi}}_{\rm eff}$ for $0^{+}_{\rm g.s.}$ in Eq. (19), which is rewritten as Dover80 ; Cieply01

[TABLE]

where the distorted-wave integral Dover80 ; Cieply01 is defined by

[TABLE]

Here the mean HO size parameter denotes

[TABLE]

These formulas give us good insight for the nuclear ($\pi$, $K$) reactions. The total effective number of nucleon $N_{\rm eff}^{\rm tot}$ at $\theta_{\rm lab}=$ $0^{\circ}$ is defined by the sum of all contributions of the $\Lambda$ final states. In the closure approximation, it can be easily written as

[TABLE]

so that the values of $N_{\rm eff}^{\rm tot}(0^{\circ})$ amount to 2.00, 1.22, and 0.99 for $\bar{\sigma}=$ 0 mb (PW), 20 mb, and 30 mb, respectively. Figure 6 also displays the values of $N_{\rm eff}^{0^{+}}$ calculated in the eikonal-oscillator approximation of Eq. (39). In the case of PWIA, the magnitude of $N^{0^{+}}_{\rm eff}$ is as large as that obtained from the HO potential, and it is smaller than that obtained from the Isle potential. In the case of DWIA ($\bar{\sigma}=$ 30 mb), the magnitude of $N^{0^{+}}_{\rm eff}$ is as large as that for Isle, and these slopes are similar to each other in the region of $q<$ 400 MeV/$c$. Thus we obtain $(d\sigma/d\Omega)^{\rm DW}_{\rm lab}=$ 10.41, 9.83, 6.90, and 3.64 $\mu$b/sr at $p_{\pi}=$ 1.05 GeV/$c$ in $\theta_{\rm lab}=$ 0∘, 4∘, 8∘, and 12∘, respectively; they are comparable to $(d\sigma/d\Omega)^{\rm DW}_{\rm lab}=$ 11.73, 10.99, 7.47, and 3.65 $\mu$b/sr for Isle. Because high momentum components of neither the $N$ and $\Lambda$ wavefunctions nor the distorted waves for mesons are included in the eikonal-oscillator approximation, we confirm that the shape of $N^{0^{+}}_{\rm eff}$ has no dip as increasing $q$.

III.3

Recoil effects

The recoil effects should be needed in the ($\pi$, $K$) reactions on a very light nuclear target such as 4He; the quantity of the recoil factor $M_{C}/M_{A}\simeq$ 3/4 = 0.75 characterizes the importance of the recoil effects in the nuclear systems. To see the sensitivity to the recoil effects quantitatively, we demonstrate the differential lab cross sections when we omit the recoil effects ($M_{C}/M_{A}\to 1$) in Eq. (7) using the Isle, SG, and HO potentials.

In Table 3, we show the calculated DWIA (PWIA) results of the differential lab cross sections omitting the recoil effects via the 4He($\pi$, $K$) reaction at $p_{\pi}=$ 1.05 GeV/$c$. We find $d\sigma/d\Omega_{\rm lab}=$ 0.382 $\mu$b/sr (4.113 $\mu$b/sr) at $\theta_{\rm lab}=$ 0∘. Surprisingly, this value is an order of magnitude smaller than $d\sigma/d\Omega_{\rm lab}=$ 11.73 $\mu$b/sr (35.14 $\mu$b/sr) which is already shown in Table 2. The recoil effects have a great influence on $d\sigma/d\Omega_{\rm lab}$ depending on the radial behavior of the distorted waves for mesons. Here we consider the overlap function defined as

[TABLE]

which corresponds to the integrand in Eq. (24). Figure 7 displays the behaviors of ${\cal Y}(q;r)$ for various types choosing the $\Lambda$ wavefunctions and the distortion parameters, as a function of the radial distance. When we omit the recoil effects ($M_{C}/M_{A}\to 1$), we find that the node point at $r_{1}$ where ${\cal Y}(q;r_{1})=0$ must be shifted toward the nuclear inside. As a result, the integral value of $\int_{0}^{\infty}{\cal Y}(q;r)dr=F(q)$ is significantly reduced by cancellation between the positive and negative values over integration regions in ${\cal Y}(q;r)$, as illustrated in Fig. 7.

The recoil effects are often omitted in several model calculations for nuclear reactions with large momentum transfers, e.g., ($\pi$, $K$) Dover80 , ($\bar{K}$, $K$) Dover83 , ($\bar{K}$, $p$) Cieply01 , and (stopped $\bar{K}$, $\pi$) Matsuyama88 reactions on 12C in which the recoil effects are not so important because $M_{C}/M_{A}\simeq(A-1)/A$ = 11/12 = 0.917 for $A=$ 12. But we must pay attention to the recoil effects when applying it to light nuclear systems such as 4He ($M_{C}/M_{A}\simeq$ 3/4 = 0.75). We believe that the calculated cross sections Dover83 or calculated production probabilities Matsuyama88 of ${}^{4}_{\Lambda}$He are perhaps underestimated by an order of magnitude for lack of the recoil effects Koike07 .

III.4 Dependence on distortion parameters

Due to strong absorptions of mesons in nuclei, e.g., $\pi^{+}$ at $p_{\pi}=$ 1.0–1.5 GeV/$c$ in the $N^{*}$ resonance region, the magnitude of the cross section may be also affected by meson distortions. To understand the distortion effects quantitatively, we demonstrate the differential lab cross sections of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He, considering various eikonal distortions with parameters of ($\sigma_{\pi}$, $\sigma_{K}$). In Table 4, we show the calculated results of $d\sigma/d\Omega_{\rm lab}$ at $p_{\pi}=$ 1.05 GeV/$c$. The magnitudes of the cross sections are reduced as increasing these parameters in DWIA. Thus the differential lab cross sections of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He at $\theta_{\rm lab}=$ 0∘ amount to $(d\sigma/d\Omega)_{\rm lab}\simeq$ 8–14 $\mu$b/sr which depend on $\pi$ and $K$ distorted waves for $\sigma_{\pi}=$ 20–30 mb and $\sigma_{K}=$ 10–30 mb. It should be noticed that the parameter dependence gives an indication of the accuracy of our results within the eikonal distortion. Fully realistic distorted waves obtained from meson-nucleus optical potentials would be needed to make a more quantitative discussion.

III.5 Dependence on the incident momentum

In Fig. 8, we display the differential lab cross sections of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He via the 4He($\pi$, $K$) reactions at $\theta_{\rm lab}=$ 0∘, 4∘, 8∘, and 12∘, as a function of the incident lab momentum $p_{\pi}$. Here we used the $\Lambda$ wavefunctions obtained from the Isle potential and the eikonal distortions with $({\sigma}_{\pi},~{}{\sigma}_{K})=$ (30 mb, 15 mb). We find that the differential lab cross sections slightly increase, as increasing $p_{\pi}$. This trend seems to be opposite to that of $\alpha\langle d\sigma/d\Omega_{\rm lab}\rangle^{\rm opt}_{\pi N\to K\Lambda}$, as seen in Fig. 2. This comes from the fact that the momentum transfers in this region decrease as increasing $p_{\pi}$ (see Fig. 1), together with the nature of $N_{\rm eff}^{0^{+}}$ which must be taken into account the recoil effects.

IV Summary and Conclusion

We have investigated theoretically the production cross sections of the $0^{+}$ ground state of a ${}^{4}_{\Lambda}$He hypernucleus in the 4He($\pi$, $K$) reaction with a distorted-wave impulse approximation using the optimal Fermi-averaged $\pi N\to K\Lambda$ $t$ matrix. We have demonstrated the sensitivity of the production cross section to the $3N$-$\Lambda$ potentials and to the eikonal distorted waves for mesons. We have calculated the differential lab cross sections of $d\sigma/d\Omega_{\rm lab}$ at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward-direction angles of $\theta_{\rm lab}=$ 0∘–12∘. The results can be summarized as follows:

• (1)

The calculated differential lab cross section of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He amounts to $d\sigma/d\Omega_{\rm lab}\simeq$ 11 $\mu$b/sr at $p_{\pi}=$ 1.05 GeV/$c$, $\theta_{\rm lab}=$ 0∘–4∘, as in the case of the Isle potential.

• (2)

The recoil effects enlarge the cross section of ${}^{4}_{\Lambda}$He via the 4He($\pi$, $K$) reactions by an order of magnitude, whereas the meson distortions reduce the cross section by 30%.

• (3)

It is important to take into account the energy dependence of the $\pi N\to K\Lambda$ cross sections for a good description of the nuclear ($\pi$, $K$) reactions.

• (4)

The differential lab cross sections of ${}^{4}_{\Lambda}$H via 4He($\pi^{-}$, $K^{0}$) reactions are the same as those of ${}^{4}_{\Lambda}$He via 4He($\pi^{+}$, $K^{+}$) ones owing to charge independence in nuclear physics.

In conclusion, we have shown that the differential lab cross sections of $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He amount to $d\sigma/d\Omega_{\rm lab}\simeq$ 11 $\mu$b/sr at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward direction because of a major advantage of the use of the s-shell nuclear targets such as 4He. It would be appropriate to study the $\Lambda$ production from the ($\pi$, $K$) reactions on the $s$-shell 3,4He targets in order to study the lifetime measurements of a ${}^{3,4}_{\Lambda}$H hypernucleus in production followed by mesonic decay processes. This investigation is now in progress Harada19 .

Acknowledgements.

The authors would like to thank Professor A. Sakaguchi, Professor H. Tamura, and Dr. A. Feliciello for many valuable discussions. This work was supported by Japan Society for the Promotion of Science (JSPS), KAKENHI Grant Numbers JP16K05363.

Appendix A Explicit forms of isospin-spin functions for $0^{+}_{\rm g.s.}$ in 4He and ${}^{4}_{\Lambda}$He

The isospin-spin function $X^{A}_{T_{A},S_{A}}$ for $0^{+}_{\rm g.s.}$ in 4He ($T_{A}=0$, $m_{T_{A}}=0$; $S_{A}=0$, $m_{S_{A}}=0$) in Eq. (15) is explicitly written as

[TABLE]

where $p_{\uparrow}$ and $n_{\uparrow}$ ($p_{\downarrow}$ and $n_{\downarrow}$) denote spin states with $m_{z}={+1/2}$ (${-1/2}$) for a proton and a neutron, respectively. The isospin-spin function $X^{B}_{T_{B},S_{B}}$ for $0^{+}_{\rm g.s.}$ in ${}^{4}_{\Lambda}$He ($T_{B}=1/2$, $m_{T_{B}}=+1/2$; $S_{B}=0$, $m_{S_{B}}=0$) in Eq. (10) is explicitly written as

[TABLE]

where $\Lambda_{\uparrow}$ ($\Lambda_{\downarrow}$) denotes a spin state with $m_{z}={+1/2}$ ($-1/2$) for a $\Lambda$ hyperon.