Analysis of Λ(1405) production data in the π⁻p → K⁰πΣ reaction at p_π = 1.69 GeV/c

Abstract

We have been studying the (Σπ)⁰ mass distribution based on data from an old Hydrogen Bubble Chamber collected by Thomas et al. concerning the p(π⁻, K⁰)Σπ reaction at p_π = 1.69 GeV/c. In this analysis we investigated the formation of the Λ(1405) state, using the Generalized Optical Theorem and T-matrix operator. We applied a combined transition operator, T_mix, which contains T₂₁ ($ = T_{\pi \Sigma - \bar{K}N}$) and T₂₂ (= T_πΣ−πΣ) due to the initial channel mixing in the Λ(1405) formation. We also considered a possible mixture of I = 0 and 1 in the wave function, as well as some background of the Σ(1385) population.

Contributed by Toshimitsu YAMAZAKI, M.J.A.

I. Introduction

Over the past decade, there has been growing interest in the existence of deeply bound kaonic nuclear states, both theoretically and experimentally. Although a quasi-bound state of Λ(1405) which we call here Λ* has an important role in the production of those states and much research has been conducted,^1)–5) there are still vigorous debates concerning the nature of this Λ* resonance, which was first predicted by Dalitz and Tuan theoretically.⁶⁾ Experimentally, Alston et al. reported the production of Λ* in p(K⁻, Σπ)ππ reaction at 1.15 GeV.⁷⁾

The Particle Data Group (PDG) has adopted the idea that the resonance state of Λ* exists below the $\bar{K}N$ threshold with a mass of $1405.1_{ - 1.0}^{ + 1.3}$ MeV/c² and a width of 50.5 ± 2.0 MeV. It possesses I = 0, S = −1 and decays 100% to Σπ.⁸⁾ Previous studies conducted by the current authors have been adopted by PDG concerning the mass and width of Λ*,^9),10) whereas there are many other theoretical calculations and experimental data, as listed in ref. 8.

In a previous paper,¹¹⁾ our focus was placed on calculating the invariant mass distribution in the K⁻ + p → Σ + π reaction, which was measured by Hemingway.¹²⁾ Also, the role of the K⁻p → Σπ and Σπ → Σπ channels in the spectrum is discussed in detail in ref. 13. Previously, the invariant-mass spectrum had been interpreted theoretically by using one of the transition operators (T matrix elements) without a successful fitting. In this particular transition it was not clear which of the transition operators is responsible and, thus, instead we made use of a combined operator   

\begin{equation} T_{\text{mix}} = (1 - f)T_{21} + fT_{22} \end{equation} [I.1]

where T₂₁ is a transition amplitude for the $T_{\Sigma \pi \leftarrow \bar{K}N}$ process. fT₂₂ is a transition amplitude for the T_Σπ←Σπ process and f is a complex mixing parameter.¹¹⁾ The percentage of T₂₁ is |1 − f|²/(|1 − f|² + |f|²). We then found a good fitting of the Hemingway experimental data with a parameter of f = 0.38 ± 0.02.

In the present work we attempt to analyze some other old experimental data of Thomas et al.¹⁴⁾ concerning the p(π⁻, K⁰) reaction at p_π = 1.69 GeV/c in a Hydrogen Bubble Chamber¹⁴⁾ and the production of a Λ* particle and its decay to the (Σπ)⁰ channel to see whether they can be fitted by a combined operator or not. We also take into account the effect of isospin mixing between I = 0 and I = 1, as well as possible Σ(1385) contamination in the observed spectrum. The framework of the paper is as follows: in section II, the formulation of the problem is discussed. Section III is devoted to analysis and discussion; and the conclusion and summary are described in section IV.

II. Theoretical procedure

We start our discussions of the π⁻p → Σ^±π^∓K⁰ reaction by looking at the theoretical procedure applied in the coupled channel $\bar{K}N - \Sigma \pi $ treatment. A tree-level Feynman diagram of the p(π⁻, K⁰) reaction is illustrated in Fig. 1. There are 10 different coupled channels, such as $K^{ - }p,\bar{K}^{0}n,\pi ^{0}\Sigma ^{0},\pi ^{0}\Lambda , \ldots{} $, for this reaction. In the current paper we consider $\bar{K}N$ (Channel 1) and (Σπ)⁰ (Channel 2), since the Λ* of our interest is an I = 0 quasi-bound state in Channel 1 which appears as a Feshbach resonance^15),16) in Channel 2. The T-matrix is given by the following coupled-channel equation;   

\begin{equation} t_{ij} = U_{ij} + \sum_{l = 1,2}U_{il}G_{l}t_{lj}, \end{equation} [II.2]

where i(j) is for Channel 1(2), U_ij is the interaction and G_l is Green’s function with the following loop integration:   

\begin{equation} G_{l} = \frac{2\mu_{l}}{\hbar^{2}}\int d\vec{q}\,g(\vec{q})\frac{1}{k_{l}^{2} - q^{2} + i\epsilon_{0}}g(\vec{q}). \end{equation} [II.3]

Here, μ_l is a reduced mass in the corresponding Channel l, k_l is a relative momentum in the Channel l and $g(\vec{q})$ is a Yukawa-type form factor applied in the separable potentials, v_ij, and the transition matrices, t_ij, of the current calculation:   

\begin{equation} \langle\vec{k_{i}'}|v_{ij}|\vec{k_{j}}\rangle = g(\vec{k_{i}'})U_{ij}g(\vec{k_{j}}), \end{equation} [II.4]

  

\begin{equation} \langle\vec{k_{i}'}|T_{ij}|\vec{k_{j}}\rangle = g(\vec{k_{i}'})t_{ij}g(\vec{k_{j}}), \end{equation} [II.5]

  

\begin{equation} g(\vec{k}) = \frac{\Lambda^{2}}{\Lambda^{2} + \vec{k^{2}}},\quad \Lambda = \frac{m_{B}c}{h}, \end{equation} [II.6]

with Λ = 3.9 fm⁻¹ being a range parameter and m_B is the mass of the exchanged boson (here, ρ meson with m_B = 770 MeV/c² is adopted). The solution of eq. [II.2] in a matrix form is   

\begin{equation} t = [1 - UG]^{-1}U. \end{equation} [II.7]

Fig. 1.

Diagrams for the π⁻p → K⁰(Σπ)⁰ reaction and the internal processes proposed by the present paper: T₂₁ and T₂₂ channels.

In the framework of the model we describe in this article, U is Akaishi-Myint-Yamazaki (AMY) interaction¹⁷⁾ for the I = 0 component of the $\bar{K}N$ interaction:   

\begin{equation} U_{ij} = \frac{1}{\pi^{2}}\frac{\hbar^{2}}{2\sqrt{\mu_{i}\mu_{j}}}\frac{1}{\Lambda}s_{ij}, \end{equation} [II.8]

where μ_i(μ_j) is the reduced mass of Channel i(j) and s_ij is a non-dimensional strength parameter. s_ij is a function of the binding energy and width of the Λ* resonance. We first obtain s₁₁ and s₁₂ from the M_pole and Γ values of the K⁻p state assumed for data fitting. We then calculate the Σπ invariant mass distributions, M(Σπ). In this paper we consider the K⁻p quasi-bound state, $(\bar{K}N)^{I = 0}$, as a Feshbach resonance^15),16) embedded in the Σπ continum using the AMY model.¹⁷⁾ The kinematical variables of the decay rate of Λ(1405) to (Σπ)⁰ are presented in Fig. 2 in the c.m. frame. Using these figures, the decay spectrum, S, of the following two processes   

\begin{align} &K^{-} + p \to \Lambda^{*} \to \Sigma + \pi\quad(a),\quad \\ &\Sigma + \pi \to \Lambda^{*} \to \Sigma + \pi\quad (b), \end{align} [II.9]

is written as   

\begin{equation} S(x) = \frac{2(2\pi)^{5}}{(\hbar c)^{2}}\frac{E_{\Sigma}E_{\pi}}{E_{\Sigma} + E_{\pi}}|\langle k|T|k_{0}\rangle |^{2}k, \end{equation} [II.10]

where $\vec{k}(\overrightarrow{k_{0}})$ is a relative momentum in the final (initial) channel, T is the transition matrix given in eq. [II.5], and E_Σ(E_π) is the energy of the Σ(π) particle in the final state. The spectrum, S(x), is a function of the invariant-mass variable, x = IM(Σπ), with some fitting parameters. We use a notation in which the already optimized parameters are omitted, then, for example, S(x; M_pole, Γ) means that the parameters M_pole and Γ are to be fitted.

Fig. 2.

Kinematical variables in the center-of-mass frame for the K⁻p → Σπ channel.

The Σ^±π^∓ mass distribution for the π⁻p → Σ^±π^∓K⁰ reaction at 1.69 GeV/c is presented by a histogram in Fig. 3 of ref. 14. The mass and the width of Λ(1405), deduced in this experiment in ref. 14, are M = 1405 MeV/c² and Γ = 45–55 MeV, respectively. The mass range of the data is from x = 1340 MeV/c² to 1430 MeV/c², lying between the Σπ threshold (1332 MeV/c²) and the $\bar{K}N$ threshold (1432 MeV/c²). Therefore, we have 10 points of data, which should be used to perform the χ² analysis, as follows:   

\begin{equation} \chi^{2}(M_{\text{pole}},\Gamma) = \sum\left(\frac{N_{i} - cS_{i}(M_{\text{pole}},\Gamma)}{\sigma_{i}}\right)^{2}, \end{equation} [II.11]

where N_i is the experimental data, σ_i is the statistical errors and S_i(M_pole, Γ) = S(x_i; M_pole, Γ) is the theoretical decay rate calculated with the mass, M_pole, and the width, Γ, as parameters; c is a fixed normalization parameter that makes the theoretical values and the experimental data in the same order and obtained by dividing the maximum value of data by a theoretical one in the same mass.

III. Interpretation of the results

In this section, our theoretical results for the invariant-mass spectrum, M(Σπ), of the mixed transition matrix, T_mix, are investigated in detail. To evaluate the goodness of the performed fitting, the χ² analysis was carried out in each case of combined T₂₁ and T₂₂. Our main concern is to extract the pole position (the best-fit M_pole) and width (the best-fit Γ) of Λ(1405) from the data. By changing T₂₁ and T₂₂, since Λ(1405) is the I = 0 resonance pole on the $\bar{K}N - \Sigma \pi $ (+, −) sheet, we have treated the coupled two major I = 0 channels (as ones having essential roles among 10 channels). Our procedure is a phenomenological one with some fitting parameters, f and (M_pole, Γ), the obtained pole position and width for Λ(1405). Therefore, in subsection A, T_mix with I = 0 is considered and then in subsection B the contribution of the I = 0 and I = 1 components in the spectrum are investigated.

A. Different contributions of T₂₁ and T₂₂ amplitudes.

In this subsection, various combinations of the T₂₁ and T₂₂ components are taken into account. For this purpose, T_mix is defined in eq. [I.1]. The factor f consists of an initial reaction process, b_i, times the channel propagator, G_i(x), as depicted in Fig. 1. The b_i depends on the meson-baryon interaction, which is considered to be on the order of the ρ-meson exchange, ħc/770 MeV. Because this interaction is of short-range nature, the energy dependence of b_i becomes weak in the relevant invariant-mass region between 1332 (Σπ threshold) and 1432 (K⁻p threshold) MeV/c². As for the x-dependence of the propagator, we calculated b_i G_i(x) T_ij(x) in a typical case (Λ(1405) of PDG value) and compared it with f_i T_ij(x), as shown in Fig. 3. Therefore, the factor f is treated as being a constant in our analysis. In this way, in addition to M_pole and Γ, f is considered to be a free parameter in the fitting process, and it is necessary to check the χ² values for various f. The fitting process is performed for three (M_pole, Γ, f) parameters, and χ² values are obtained for each step.

Fig. 3.

Difference between the x-dependence b_iG_i(x) and a constant f_i, where v₂ is the final-state phase volume.

We took the same M_pole and Γ for both channels, and regarded them as being free parameters varied in the “Λ* mass region”, since both of the peak structures (seen in Fig. 3, for example) in T₂₁ spectrum and T₂₂ spectrum come from a single pole lying on the $\bar{K}N - \Sigma \pi $ (+, −) sheet.¹⁸⁾ Currently, the chiral dynamics models claim double poles, a higher-mass pole (1st pole) and a lower-mass pole (2nd pole), for Λ(1405). The peak structure in the T₂₁ spectrum comes from the 1st pole. However, some chiral papers, like Magas et al.,²⁰⁾ insist that the peak structure in the T₂₂ spectrum comes from the 2nd pole. This statement concerning the role of the 2nd pole seems to be incorrect. Both of the peak structures in the T₂₁ spectrum and the T₂₂ spectrum come from the 1st pole, as discussed in refs. 13 and 18. This means that, even if the chiral double-pole picture is adopted, the same (M_pole, Γ) parameter set should be used for both channels, as is done in this paper.

In the following, for each value of f, a set of (M_pole and Γ) that gives the best fit are obtained, as presented in Fig. 4. It can be seen from Figs. 8 and 9 explained in the next section that $\chi _{\text{min}}^{2}$ = 3.9 is attained for f = 0.5, M_pole = 1370 MeV/c² and Γ = 67 MeV, and the contribution of each channel is about 50%.

Fig. 4.

Spectra of the best fit for mixed channels with each fixed f parameter: f = 0.0, 0.2, …, 1.0. The M_pole, Γ and χ² values of each curve are shown in addition to the experimental data.

To see how sensitively the pole mass is determined by χ² fitting, diagrams of χ² versus the pole mass for various f parameters are depicted in Fig. 5. In this figure, it is clear that f = 0.5 gives the minimum χ² when χ² = 3.9. In the next figure (M_pole, Γ) are fixed to PDG value and the optimum f is obtained (see Fig. 6). In this figure $\chi _{min}^{2}$ = 29 is attained for f = 0.2, which means a 94% contribution of the T₂₁ channel and 6% T₂₂ ones.

Fig. 5.

Diagrams of χ² versus the pole mass for different values of f: 0.0, 0.2, …, 1.0. f = 0.5 gives the $\chi _{min}^{2}$ = 3.9.

Fig. 6.

Spectra for the optimum f parameters; for each curve (M_pole, Γ) is fixed to PDG values ($1405.1_{ - 1.0}^{ + 1.3}$, 50.5 ± 2.0) in MeV/c².

B. The contribution of the I = 0 and I = 1 components in the spectrum.

In this section we investigate the importance of the isospin I = 0 and I = 1 components as well as their mixing in the calculated spectrum. The charge-basis T-matrices are related to isospin-basis T-matrices as   

\begin{equation} |T_{\Sigma^{+}\pi^{-}}|^{2} \approx \frac{1}{3}|T_{I = 0}|^{2} + \frac{1}{2}|T_{I = 1}|^{2} + \sqrt{\frac{2}{3}} \,{\text{Re}}|T_{I = 0}^{*}T_{I = 1}|, \end{equation} [III.12]

  

\begin{equation} |T_{\Sigma^{-}\pi^{+}}|^{2} \approx \frac{1}{3}|T_{I = 0}|^{2} + \frac{1}{2}|T_{I = 1}|^{2} - \sqrt{\frac{2}{3}} \,{\text{Re}}|T_{I = 0}^{*}T_{I = 1}|. \end{equation} [III.13]

The absolute values of the I = 0 and I = 1 components for the T₂₁ and T₂₂ channels are depicted in Fig. 7, where the I = 1 component is calculated using Shevchenko’s interaction.¹⁹⁾ This figure shows that the I = 1 component has a small contribution in the total spectrum, and for the smaller region of the invariant mass M(Σπ) it has a monotonic decrease, whereas the I = 0 part has a large portion and includes the resonance state.

Fig. 7.

Absolute values for the T₂₁ and T₂₂ channels with the I = 1 and I = 0 components.

To check the effects of the I = 1 contribution we showed T_mix = (1 − f)T₂₁ + fT₂₂ (f = 0.5) with various percentages of I = 1 mixing the range of which is illustrated in Figs. 8(a) and (b). Figure 8(a) represents 0%, 25% and 50% I = 1 mixing and the χ² values for each curve. It is obvious from this figure that the best fit belongs to the I = 0 contribution, and that adding I = 1 does not change the results very much, and even makes it worse. Finally, Fig. 8(b) helps to clarify this situation, and shows that considering the I = 1 component does not change the χ² values, and that the shift of the pole position becomes negligible.

Fig. 8.

(a) I = 1 contributions (from 0% to 50%) for T_mix with a fixed M_pole = 1370 MeV/c², Γ = 67 MeV and (b) I = 1 contributions (from 0% to 80%) for T_mix channels (f = 0.5): χ² and M_pole versus the percentages of I = 1 are depicted.

Figure 9 shows the I = 1 contributions (from 0% to 100%) for T_mix = (1 − f)T₂₁ + fT₂₂ with various f values (from 0.0 to 0.8). From this figure it is found that as f increases from 0.0 to 0.5, χ² decreases and as f increases from 0.5 to 0.8, χ² increases (except 100% I = 1 mixing). The best result with χ²/NDF = 0.4 is obtained for 0% I = 1 mixing at f = 0.5. It should be noticed that this best fit to the data for 0% I = 1 mixing at f = 0.5 is the consequence of an equal footing treatment of the I = 0 and I = 1 channels, as explained below. In the χ² fitting process for the curves in Fig. 9, the parameters f and implicitly M_pole, change the I = 0 spectrum through the entrance-channel weight and pole position, and the I = 1 mixing parameter, m_I=1, controls the I = 1 spectrum. When we consider the χ² contour map on the (f, m_I=1) plane, which is an equal-footing treatment of the I = 0 and I = 1 components, the lowest χ² value is realized at f = 0.5 and the m_I=1 = 0% point, as can be clearly seen from Fig. 9. Thus, it is justified to neglect the I = 1 component and to apply the I = 0 two-channel model in the present analysis of the Thomas et al. data.¹⁴⁾

Fig. 9.

I = 1 contributions (from 0% to 100%) for T_mix = (1 − f)T₂₁ + fT₂₂ channel: χ² versus the percentages of I = 1 for various f (from 0.0 to 0.8) is presented. The minimum value of χ² = 3.9 is obtained at 0% I = 1 mixing for f = 0.5.

The T_mix spectrum is decomposed into the T₂₁ and T₂₂ channels, shown in Fig. 10, to verify the dependence of the mixed channel on its components. It is found that T₂₁ has the dominant part of the mixed spectrum, but at the same time adding T₂₂ leads to a shift in the peak position for the total spectrum and a large cancellation due to the interference part. Therefore, the behaviors of the T_mix and T₂₁ channels are different in the presence of T₂₂ mixing.

Fig. 10.

Decomposition of the T_mix best-fit spectrum (f = 0.5) (solid black curve) to its components: T₂₁ (dashed red curve) and T₂₂ (dotted blue curve) parts. The large cancellation due to the interference part and the dominance of the T₂₁ channel is clear.

C. Effects of the population of the Σ(1385) resonance.

In this subsection, the possible background contribution of the production and decay of Σ⁰(1385) in the invariant mass spectrum with a branching ratio of 12% is considered.

Various incoherent contributions of the Σ⁰(1385) resonance from 0 to 100% are expressed by $|T_{\text{tot}}|^{2} = |T_{\text{mix}}|^{2} + A_{\Sigma ^{*}}^{2}|BW(\Sigma ^{*})|^{2}$. BW(Σ*) is a Breit-Wigner amplitude for the Σ(1385) resonance with the mass, 1384 MeV/c², and width, 36 MeV, presented in PDG, added to the previous spectra; |T_mix|². The peak position moves by changing $A_{\varSigma ^{*}}^{2}$.

To show the very complicated behavior of the parameters (f, χ², $A_{\Sigma ^{*}}$, Γ and M_pole) with respect to each other, it is better to illustrate them in some individual figures. Figure 11 (upper) depicts the χ² versus f that shows a minimum of around f = 0.5. To make this subject clear, the mass and width of the resonance are shown in Fig. 11 (lower) at the same values of f. It is deduced from this figure that the minimum value of χ² is obtained for Γ = 67 MeV and M_pole = 1370 MeV/c². To clarify this matter we illustrate the variations of χ² versus f for different portions of Σ(1385) in Fig. 12(a). For larger values of Σ(1385), moving of the mass occurs rapidly while the width shows a broad peak at around f = 0.5, which gives the minimum value of χ² (see the (b) and (c) parts of Fig. 12).

Fig. 11.

(upper) Diagram of χ² and (lower) M and Γ of Λ* versus f (effects of various contributions of the T₂₁ and T₂₂ channels).

Fig. 12.

Diagrams of (a) χ², (b) M and (c) Γ of Λ* versus f (various contributions of T₂₁ and T₂₂ channels) for different mixtures of Σ(1385).

Finally, the best fit for T_mix with 0% Σ(1385) is shown in Fig. 13. This figure has two curves: one for PDG values of the mass and width (M_pole = 1405.1 MeV/c² and Γ = 50.5 MeV) with f = 0.2 and χ² = 29 together with the curve of the best fit for M_pole = 1370 MeV/c² and Γ = 67 MeV, f = 0.5 and χ² = 3.9.

Fig. 13.

The best fit for T_mix with 0% Σ(1385): The solid line shows the curve of fixed M_pole = $1405.1_{ - 1.0}^{ + 1.3}$ MeV/c² and Γ = 50.5 ± 2.0 MeV (PDG values), f = 0.2 and χ² = 29. The dashed line is for M_pole = $1370_{ - 5.1}^{ + 6.2}$ MeV/c², Γ = 67 ± 5 MeV, f = 0.5 and χ² = 3.9.

IV. Conclusion

Within the framework of the $\bar{K}N$-$\Sigma \pi $ coupled channels, we have investigated the mass distribution, $M_{\Sigma ^{ \pm }\pi ^{ \mp }}$, in the π⁻p → Σ^±π^∓K⁰ reaction at $p_{\pi ^{ - }}$ = 1.69 GeV/c provided by Thomas et al.¹⁴⁾

In this analysis the mixture of the two transition operators as T_mix = (1 − f)T₂₁ + fT₂₂ was applied, and showed that the mixture of f = 0.5 gives a better fit to the Thomas et al. spectrum. We then considered a mixture of the two isospin states I = 0 and 1. The experimental data are better accounted for by a small admixture of the I = 1 state. Finally, we considered a possible incoherent mixture of the Σ(1385) resonance component in the observed spectrum. The best-fit mass and width after considering the Σ(1385) mixture yield M_pole = $1370_{ - 5.1}^{ + 6.2}$ MeV/c² and Γ = 67 ± 5 MeV (0% Σ(1385) gives the best result), as shown in Fig. 13. The present results contradicts Magas et al.’s two-pole postulation²⁰⁾ in which the Thomas et al.’s peak structure corresponds to the 2nd-pole effect that appears in the T₂₂ spectrum. Finally, for a better conclusion concerning the mass and width of the Λ* resonance we look forward to near-future experiments with high statistics.

Acknowledgment

This work is supported by a Grant-in-Aid for Scientific Research of MEXT of Japan.

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