$K^0\Lambda$ Photoproduction with Nucleon Resonances

TL;DR

This paper models the $K^0 \Lambda$ photoproduction process off neutrons using an effective Lagrangian and Regge approach, highlighting the importance of the $N(1685)$ resonance in matching experimental data.

Contribution

It introduces a combined effective Lagrangian and Regge model to analyze $K^0 \Lambda$ photoproduction, emphasizing the role of the $N(1685)$ resonance.

Findings

The $N(1685)$ resonance significantly improves data fit.

The model reproduces total and differential cross sections.

Resonance parameters are extracted from experimental data.

Abstract

We investigate the reaction mechanism of $K^0 \Lambda$ photoproduction off the neutron target, i.e., $\gamma n \to K^0 \Lambda$, in the range of $W\approx 1.6-2.2$ GeV. We employ an effective Lagrangian method at the tree-level Born approximation combining with a Regge approach. As a background, the $K^*$-Reggeon trajectory is taken into account in the $t$ channel and $\Lambda$ and $\Sigma$ hyperons in the $u$-channel Feynman diagram. In addition, the role of various nucleon resonances listed in the Particle Data Group (PDG) is carefully scrutinized in the $s$ channel where the resonance parameters are extracted from the experimental data and constituent quark model. We present our numerical results of the total and differential cross sections and compare them with the recent CLAS data. The effect of the narrow nucleon resonance $N(1685,1/2^+)$ on cross sections is studied in detail and it turns out that its existence is essential in $K^0 \Lambda$ photoproduction to reproduce the CLAS data.

Tables

Table 1. Table 1: Nucleon resonances listed in the PDG  Tanabashi:2018oca and N ​ ( 1685 , 1 / 2 + ) 𝑁 1685 1 superscript 2 N(1685,1/2^{+}) and their strong coupling constants g K ​ Λ ​ N ∗ subscript 𝑔 𝐾 Λ superscript 𝑁 g_{K\Lambda N^{*}} . The decay amplitudes G ​ ( ℓ ) 𝐺 ℓ G(\ell) are obtained from Ref.  Capstick:1998uh and the branching ratio Br N ∗ → K ​ Λ subscript Br → superscript 𝑁 𝐾 Λ \mathrm{Br}_{N^{*}\to K\Lambda} from the PDG  Tanabashi:2018oca .

State	Rating	$G\left(\ell \right)\left[{\mathrm{MeV}}^{1/2}\right]$	$g_{K\Lambda N^{\ast }}$	${\mathrm{Br}}_{N^{\ast }\to K\Lambda }[\%]$	$\left|g_{K\Lambda N^{\ast }}\right|$	$g_{K\Lambda N^{\ast }}$(final)
$N(1650,1/2^{-})$	****	$-3.3\pm 1.0$	$-0.78$	$5-15$	$0.59-1.02$	$-0.78$
$N(1675,5/2^{-})$	****	$0.4\pm 0.3$	$1.23$			$1.23$
$N(1680,5/2^{+})$	****	$0.1\pm 0.1$	$-2.84$			$-2.84$
$N(1700,3/2^{-})$	***	$-0.4\pm 0.3$	$2.34$			$2.34$
$N(1710,1/2^{+})$	****	$4.7\pm 3.7$	$-7.49$	$5-25$	$4.2-9.4$	$-4.2$
$N(1720,3/2^{+})$	****	$-3.2\pm 1.8$	$-1.80$	$4-5$	$1.8-2.0$	$-1.1$
$N(1860,5/2^{+})$	**	$-0.5\pm 0.3$	$1.40$	seen		$1.40$
$N(1875,3/2^{-})$	***	$1.7\pm 1.0$	$-2.47$	seen		$-2.47$
$N(1880,1/2^{+})$	***			$12-28$	$4.5-6.4$	$3.0$
$N(1895,1/2^{-})$	****	$2.3\pm 2.7$	$0.34$	$13-23$	$0.58-0.77$	$0.34$
$N(1900,3/2^{+})$	****			$2-20$	$0.53-1.7$	$0.6$
$N(1990,7/2^{+})$	**	$1.5\pm 2.4$	$0.61$			$0.61$
$N(2000,5/2^{+})$	**	$-0.5\pm 0.3$	$0.61$			$0.61$
$N(2060,5/2^{-})$	***	$-2.2\pm 1.0$	$-0.52$	seen		$-0.52$
$N(2120,3/2^{-})$	***	$1.7\pm 1.0$	$-1.05$			$-1.05$
$N(2190,7/2^{-})$	****	$-1.1$	$0.67$			$0.67$
$N(1685,1/2^{+})$					$-0.9$

Equations

$${\mathcal{L}}_{K\Lambda N^{\ast }}^{1/2^{\pm }}$$

$${\mathcal{L}}_{K\Lambda N^{\ast }}^{5/2^{\pm }}$$

$$⟨K(\vec{q})\Lambda (-\vec{q},m_f)|-i{\mathcal{H}}_{\mathrm{int}}|N^{\ast }(\vec{0},m_j)⟩=4\pi M_{N^{\ast }}\sqrt{\frac{2}{|\vec{q}|}}\sum _{\ell ,m_{\ell }}⟨\ell m_{\ell }\frac{1}{2}{m}_f|j{m}_j⟩Y_{\ell ,m_{\ell }}(\hat{q})G(\ell ),$$

$${\Gamma }_{N^{\ast }\to K\Lambda }=\sum _{\ell }|G(\ell ){|}^2.$$

Full text

photoproduction, nucleon resonances, effective Lagrangian

††preprint: INHA-NTG-03/2019

I Introduction

The investigation of kaon photoproduction at low energies plays an important role in understanding the features of baryon resonances which weakly couple to $\pi N$ and $\eta N$ channels. There have been a lot of theoretical and experimental works on the charged $K^{+}\Lambda$ photoproduction in the literature, e.g., MAID Drechsel:2007if and SAID Arndt:2006bf , and the contributions of various nucleon resonances are examined. In the neutral $K^{0}\Lambda$ channel, the background contribution is much reduced, compared to its charged channel due to the absence of charged particle exchanges. Moreover,it is related to the narrow nucleon resonance $N(1685,1/2^{+})$, of which the existence is previously supported in the $\gamma n\to\eta n$ reaction revealing a clear bump structure on the cross section near $W=1.68$ GeV Kuznetsov:2006kt . This phenomena is called the neutron anomaly in $\eta$ photoproduction. It is expected that the $N(1685,1/2^{+})$ could be observed also in $K^{0}\Lambda$ photoproduction, since its threshold is below the mass of this narrow resonance and the $s\bar{s}$ component is contained in the $N(1685,1/2^{+})$. In the present talk, we summarize a recent work Kim:2018qfu , where $K^{0}\Lambda$ photoproduction was studied with emphasis on the $N(1685,1/2^{+})$.

II Theoretical Formalism

We employ an effective Lagrangian method at the tree-level Born approximation to study the reaction mechanism of $\gamma n\to K^{o}\Lambda$. The relevant Feynman diagrams are drawn in Fig. 1.

We refer to Ref. Kim:2018qfu for detailed explanation of the theoretical formalism of the background term and the relevant form factors. In this talk, we concentrate on the structures of effective Lagrangians and coupling constants for the resonance term as shown in Fig. 1(b). The effective Lagrangians for $K\Lambda N^{*}$ interactions are written as

[TABLE]

where the $\Gamma^{+}(\Gamma^{-})=\gamma^{5}(1)$ is determined according to the parity of the nucleon resonances. $N^{*}$, $N^{*}_{\mu}$, $N^{*}_{\mu\nu}$, and $N^{*}_{\mu\nu\alpha}$ are the Rarita-Schwinger spin $J=$ $\frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$, and $\frac{7}{2}$ fields, respectively Rarita:1941mf .

We derive the strong coupling constants, $g_{K\Lambda N^{*}}$, in Eq. (1) from the quark model predictions where information on the partial decay amplitude for the $N^{*}\to K\Lambda$ channel is provided Capstick:1998uh . They have the relation Kim:2017nxg :

[TABLE]

where $\langle\ell\,m_{\ell}\,\frac{1}{2}\,m_{f}|j\,m_{j}\rangle$ and $Y_{\ell,m_{\ell}}({\hat{q}})$ are the Clebsch-Gordan coefficients and spherical harmonics, respectively. Then, the decay width is calculated from the partial decay amplitude $G(\ell)$

[TABLE]

We introduce sixteen nucleon resonances from the PDG Tanabashi:2018oca that are likely to couple strongly to the $\gamma N$ and $K\Lambda$ vertices and the $N(1685,1/2^{+})$. All the relevant parameters are listed in Table 1. Here we compare the results from the quark model predictions with those from the experimental data. The values of the coupling constants are mostly close to each other, although the signs can be determined only from the quark model. In the case of the $N(1880,1/2^{+})$ and $N(1900,3/2^{+})$, there exist only the experimental data. Thus their strong coupling constants are determined by using the PDG data and their signs are determined phenomenologically.

III Results and Discussions

We present the results of our calculation for the total cross section in Fig. 2. It is shown that the nucleon resonances play a crucial role in reproducing the CLAS data Compton:2017xkt . At the threshold region (W $\leq$ 1.85 GeV), the contribution of $K^{*}$-Reggeon exchange (dotted line) reaches only the level of around $25\,\%$ compared with CLAS data. The main contributions of the nucleon resonances are plotted in the right panel of Fig. 2. The $N(1650,1/2^{-})$ and $N(1720,3/2^{+})$ are mostly responsible and the magnitude from the narrow resonance $N(1685,1/2^{+})$ reaches around 0.25 $\mathrm{\mu b}$.

Figure 3 draws the numerical results of the differential cross sections. The dotted, dot-dashed, and solid curves indicate the contribution from $K^{*}$-Reggeon exchange, that from the $N^{*}$ exchanges, and the total contribution, respectively. The dashed one stands for the total contribution except for the narrow resonance $N(1685,1/2^{+})$. The inclusion of the $N(1685,1/2^{+})$ enhances the results in the backward angle and reduces them in the forward angle by the interference effect. It is also interesting that the dip structures begin to appear as the angle $\cos\theta$ increases, which is consistent with the present results. This indicates strongly the existence of the $N(1685,1/2^{+})$ in the $\gamma n\to K^{0}\Lambda$.

IV Summary and conclusion

We investigated the mechanism of the $K^{0}\Lambda$ photoproduction with the help of the effective Lagrangian method. We have found that the $N(1650,1/2^{-})$ and $N(1720,3/2^{+})$ give the most dominant contributions to the cross section. While the effect of the $N(1685,1/2^{+})$ on the total cross section is very small, it is essential to describe the differential cross sections near the threshold region (1.6 $\leq$ W $\leq$ 1.8 GeV).

Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. 2018R1A5A1025563).