Some Phenomenological Aspects of Kaon Photoproduction in the Extreme Kinematics

TL;DR

This paper explores the phenomenological aspects of kaon photoproduction in extreme kinematic regions, analyzing amplitude convergence, and extracting coupling constants in forward and backward scattering regions.

Contribution

It provides a detailed analysis of amplitude convergence in the threshold region and methods to extract coupling constants in forward and backward kinematics.

Findings

Amplitude convergence requires expansion up to 12th order at threshold.

Expansion impacts the extraction of coupling constants in different kinematic regions.

Soft Kaon Approximation is not easily applicable in kaon photoproduction.

Abstract

We have investigated phenomenological aspects of kaon photoproduction in three different extreme kinematics. The first kinematics of interest is the threshold region. At the threshold we have investigated the convergence of kaon photoproduction amplitude by expanding the square of the amplitude in terms of the ratio $m_K/m$, where $m_K$ is the mass of kaon and $m$ is the averaged mass of nucleon and $\Lambda$-hyperon. The amplitude is calculated from the appropriate Feynman diagrams by using the pseudovector theory. The contact diagram as a consequence of the PCAC hypothesis is also taken into account in the amplitude. Our finding indicates that the convergence can be only achieved if the amplitude was expanded up to at least $12^{\rm th}$ order. As a consequence, applications of some theoretical calculations based on the expansion the scattering amplitude, such as the Low Energy Theorem or Soft Kaon Approximation, cannot be easily managed in kaon photoproduction. The second kinematics is the forward region, where we could assume only $t$-channel contributes to the process. Here we have investigated the effect of amplitude expansion on the extraction of the coupling constants $g_{K^+\Lambda p}$ and $g^V_{K_1^+\Lambda p}$. The last kinematics is the backward region, where we have also assumed that only $u$-channel survives and we could extract the leading coupling constants $g_{K^+\Lambda p} $ and $g_{K^+\Sigma^0 p}$.

Tables

Table 1. Table 1 : Coupling constants extracted from fittings to experimental data in the previous [ 1 ] and present works and the corresponding χ 2 / N superscript 𝜒 2 𝑁 \chi^{2}/N .

Coupling constant	Previous	Present
$g_{K^{+}\Lambda p}/\sqrt{4\pi }$	$-3.80$	$-3.00$
$g_{K^{+}{\Sigma }^{0}p}/\sqrt{4\pi }$	$1.20$	$1.30$
$G_{K^{\ast }}^V/4\pi$	$-0.79$	$-0.73$
$G_{K^{\ast }}^T/4\pi$	$-0.04$	$0.70$
$G_{K_1}^V/4\pi$	$1.19$	$0.80$
$G_{K_1}^T/4\pi$	$-0.68$	$-1.50$
${\chi }^2/N$	$1.526$	$1.739$

Table 2. Table 2 : Properties of the particles considered in this study [ 17 ] .

Particle	$S$	$J^P$	$I$	$\mu$ (n.m.)	Mass	Width
					(MeV)	(MeV)
$p$	0	${\frac{1}{2}}^{+}$	$\frac{1}{2}$	$2.79284734$	$938.2721$	-
$n$	0	${\frac{1}{2}}^{+}$	$\frac{1}{2}$	$-1.91304273$	$939.5654$	-
$K^{+}$	$1$	$0^{-}$	$\frac{1}{2}$	-	$493.677$	-
${\Sigma }^0$	$-1$	${\frac{1}{2}}^{+}$	$1$	$1.61\pm 0.08$	$1192.642$	-
$\Lambda$	$-1$	${\frac{1}{2}}^{+}$	$0$	$-0.613\pm 0.004$	$1115.683$	-
$K^{\ast +}$	$1$	$1^{-}$	$\frac{1}{2}$	-	$891.76\pm 0.25$	$50.3\pm 0.8$
$K_1$	1	$1^{+}$	$\frac{1}{2}$	-	$1272\pm 7$	$90\pm 20$

Table 3. Table 3 : Contribution of the n 𝑛 n -th order photoproduction amplitude to the cross section at threshold.

Order of the	Cross Section
expansion	($\mu$b)
$0^{\mathrm{th}}$	$33.98$
$1^{\mathrm{st}}$	$19.38$
$2^{\mathrm{nd}}$	$-37.89$
$3^{\mathrm{rd}}$	$-10.58$
$4^{\mathrm{th}}$	$12.56$
$5^{\mathrm{th}}$	$-2.96$
$6^{\mathrm{th}}$	$-0.31$
$7^{\mathrm{th}}$	$1.78$
$8^{\mathrm{th}}$	$0.03$
$9^{\mathrm{th}}$	$0.53$
${10}^{\mathrm{th}}$	$0.66$
${11}^{\mathrm{th}}$	$0.47$
${12}^{\mathrm{th}}$	$0.54$
${13}^{\mathrm{th}}$	$0.55$
${14}^{\mathrm{th}}$	$0.53$
${15}^{\mathrm{th}}$	$0.53$
$\cdots$	$\cdots$
${20}^{\mathrm{th}}$	$0.53$

Table 4. Table 4 : Coupling constants g K + ​ Λ ​ p subscript 𝑔 superscript 𝐾 Λ 𝑝 g_{K^{+}\Lambda p} and g K 1 + ​ Λ ​ p V subscript superscript 𝑔 𝑉 superscript subscript 𝐾 1 Λ 𝑝 g^{V}_{K_{1}^{+}\Lambda p} extracted from the expanded photoproduction amplitude up to n-th order in the forward region.

Order	$K^{+}$ exchange			$K^{+}$ and $K_1$ exchanges
	$g_{K^{+}\Lambda p}$	${\chi }^2/N$		$g_{K^{+}\Lambda p}$	$g_{K_1^{+}\Lambda p}^V$	${\chi }^2/N$
$0^{\mathrm{th}}$	$-5.09$	$1.10$		$-5.09$	$1.49$	$1.11$
$2^{\mathrm{nd}}$	$-5.32$	$0.88$		$-9.55$	$5.93$	$0.72$
$4^{\mathrm{th}}$	$-5.32$	$0.86$		$-8.38$	$5.42$	$0.72$
$6^{\mathrm{th}}$	$-5.32$	$0.86$		$-8.89$	$6.46$	$0.72$
$8^{\mathrm{th}}$	$-5.33$	$0.86$		$-9.07$	$6.79$	$0.72$
${10}^{\mathrm{th}}$	$-5.33$	$0.86$		$-9.13$	$6.89$	$0.72$
${12}^{\mathrm{th}}$	$-5.33$	$0.86$		$-9.15$	$6.93$	$0.72$
${14}^{\mathrm{th}}$	$-5.33$	$0.86$		$-9.08$	$6.79$	$0.72$

Table 5. Table 5 : The leading coupling constants in K + ​ Λ superscript 𝐾 Λ K^{+}\Lambda photoproduction, g K + ​ Λ ​ p subscript 𝑔 superscript 𝐾 Λ 𝑝 g_{K^{+}\Lambda p} and g K + ​ Σ 0 ​ p subscript 𝑔 superscript 𝐾 superscript Σ 0 𝑝 g_{K^{+}\Sigma^{0}p} , extracted from the backward angle data for different orders of expansion.

Order	$g_{K^{+}\Lambda p}$	$g_{K^{+}{\Sigma }^{0}p}$	${\chi }^2/N$
$0^{\mathrm{th}}$	$-172.9\pm 62.5$	$-89.7\pm 34.6$	$26.3$
$2^{\mathrm{nd}}$	$-1.58\pm 1340.3$	$4.80\pm 697.1$	$56.6$
$4^{\mathrm{th}}$	$-457.7\pm 121.9$	$-236.8\pm 66.3$	$27.1$
$6^{\mathrm{th}}$	$-694.0\pm 2160.5$	$-360.1\pm 1152.2$	$36.8$
$8^{\mathrm{th}}$	$-693.3\pm 825.9$	$-360.2\pm 442.9$	$33.2$
${10}^{\mathrm{th}}$	$-695.6\pm 929.4$	$-361.3\pm 498.0$	$33.7$
${12}^{\mathrm{th}}$	$-695.5\pm 926.9$	$-361.3\pm 496.6$	$33.7$
${14}^{\mathrm{th}}$	$-695.5\pm 924.4$	$-361.3\pm 496.3$	$33.7$

Equations

$$\gamma (k)+p(p_p)⟶K^{+}(q)+\Lambda (p_{\Lambda }),$$

$$\mathcal{M}(m_K\to 0,q=0)={\bar{u}}_{\Lambda }(p_{\Lambda })\left[-\frac{g_{K^{+}\Lambda p}}{m_p+m_{\Lambda }}{\gamma }_5/ϵ\right]u_p(p_p).$$

$$\mathcal{M}$$

$$M_1$$

$$M_2$$

$$M_3$$

$$M_4$$

$$\displaystyle\frac{|\mbox{\boldmath${k}$}|}{|\mbox{\boldmath${q}$}|}\sigma_{\rm tot}$$

$${|\mathcal{M}|}^2$$

$$-10.6\le g_{K^{+}\Lambda p}\le -15.6.$$

$$m$$

$$x$$

$$y$$

$$\kappa$$

$$g_{\Lambda }$$

$$g_{\Sigma }$$

$$G_{\Lambda }$$

$$G_{\Sigma }$$

$$G_{K^{\ast }}$$

$$G_{K_1}$$

$$|\mathcal{M}{|}_{(0)}^2$$

$$|\mathcal{M}{|}_{(1)}^2$$

$$|\mathcal{M}{|}_{(2)}^2$$

$$|\mathcal{M}{|}_{(3)}^2$$

$$|\mathcal{M}{|}_{(4)}^2$$

$$|\mathcal{M}{|}_{(5)}^2$$

$$|\mathcal{M}{|}_{(6)}^2$$

$$|\mathcal{M}{|}_{(7)}^2$$

$$|\mathcal{M}{|}_{(8)}^2$$

$$|\mathcal{M}{|}_{(9)}^2$$

Full text

\PACS

13.60.Le, 13.60.Rj, 13.75.Jz

1 Introduction

In the previous works we have studied kaon photo- and electroproduction near their production thresholds by using the pseudo-scalar (PS) and pseudo-vector (PV) theories [1, 2]. Extended studies to cover not only the threshold energy but also the resonance region have been also performed [3, 4, 5]. These studies indicate that compared to the PV coupling the PS one leads to a better agreement with experimental data. This is the reason that the PS coupling is commonly used in the phenomenological investigations of kaon photoproduction. We note, however, that this is not the case in pion production, where the PV coupling can be and mostly used in the production formalism. The reason is that the cross section of kaon production obtained in experiments is two order of magnitude smaller than that of pion production, whereas the Feynman diagrams and the coupling constants used in the theoretical formulation of both processes are similar. Since in photoproductions the pion thresholds are the lowest, pion photoproduction has been used as the main reaction for investigating the Low Energy Theorem (LET) [6, 7]. LET is derived by using the PV coupling and expanding the pion photoproduction amplitude in terms of the ratio of pion and nucleon masses. Since this ratio is considerably small and the expansion can quickly converge, terms with higher orders can be obviously neglected. We also note that the successful Chiral Perturbation Theory formulation assumes that pion is a pseudovector particle.

On the other hand, kaon mass is much heavier than pion mass. Therefore, for kaon photoproduction LET has not been seriously considered since the convergence cannot be easily achieved. However, an attempt to derive LET for the radiative decay width of charged kaon $K\rightarrow l+\nu+\gamma$ within the so-called Soft Kaon Approximation (SKA) was performed more than five decades ago [9]. Comparison with the result obtained from the Pole Dominance Approximation [11] reveals the fact that the result of the SKA is shifted upward by approximately 20%. Such a significant difference indicates that further studies of SKA are strongly required. Presumably, the problem originates from the intrinsic properties of kaon as compared to the pion, e.g., the heavier mass of kaon. A systematic and careful investigation of the convergence of kaon scattering amplitude is therefore very important to this end. As a first step, we might investigate the convergence of kaon photoproduction amplitude, for which experimental data are abundant nowadays.

Furthermore, kaon photoproduction at forward and backward angles is also of interest, since at the two extreme directions contributions of $t$ and $u$ channels, would be dominant. As a consequence, theoretical formulation of kaon photoproduction in both cases could be extremely simplified. This is valid not only in kaon photoproduction, but also in the electromagnetic production of meson in general. For instance, in Ref. [12] it is shown that it is possible to extract the pion electromagnetic form-factor by merely using the $t$-channel diagram, provided that only the pion electroproduction data obtained in the forward direction are used in the analysis. It is also important to mention here that forward angles are very decisive for hypernuclear production, since only in this kinematics the nuclear cross section is sufficiently large. The nuclear form factor suppresses this cross section strongly as the kaon scattering angle increases. Since the elementary operator for this purpose is constructed from the kaon photoproduction amplitude, an accurate description for the forward angles production is inevitable [13]. To check whether or not the $u$-channel contribution dominates the kaon photoproduction amplitude, measurement of kaon photoproduction at backward angles has been also performed at SPring8 more than a decade ago [14].

In this paper we investigate the convergence of kaon photoproduction amplitude by expanding the squared amplitude, which is proportional to the cross section, in terms of the ratio $m_{K}/m$, where $m_{K}$ is the mass of kaon and $m$ is the averaged mass of nucleon and $\Lambda$-hyperon. For this purpose, we construct the minimal model that can explain experimental data very close to the threshold. We use the suitable Feynman diagrams with PV coupling based on our previous studies [1, 2]. The analytic calculation of the amplitude expansion was performed with the help of Mathematica software. The same method is also used to derive the reaction amplitudes in the forward and backward angles.

This paper is organized as follows. In Section 2 we present the kinematics of the photoproduction process. Section 3 discusses the expansion of the photoproduction amplitude in general. The corresponding numerical result is given in Section 4. Section 5 is devoted for the discussion of kaon photoproduction in the forward region. The case of backward angles photoproduction is given in Section 6. We will summarize our finding in Section 7. The expansion of the photoproduction amplitude up to $9^{\rm th}$ order is given in Appendix A.

2 Kinematics

Let us consider the kaon photoproduction process

[TABLE]

where the four-momenta of the photon $\gamma$, proton $p$, kaon $K^{+}$ and hyperon $\Lambda$ are explicitly indicated.

To derive the photoproduction amplitude we use the first order Feynman diagrams shown in Fig. 1. Following the previous works on pion photoproduction, here we consider kaon as a pseudovector particle, instead of pseudoscalar one. As will be explained in the next Section, the choice is also supported by the fact that the expansion of the PS amplitude has a serious problem since the PS amplitude contains the large ${\cal O}(m/m_{K})$ terms, instead of the small and expandable ${\cal O}(m_{K}/m)$ terms. The conservation of axial-vector current requires the kaon to be massless. Therefore, at $m_{K}\rightarrow 0$ and $q=0$ (at threshold), the amplitude to the first order of electromagnetic coupling can be written as

[TABLE]

This term is known as the contact (seagull) term, which contributes in the zeroth order of amplitude and it is considered as a consequence of the partially conserved axial current (PCAC) hypothesis [8]. Besides the Born and contact terms we also include two more intermediate states that have been proven to be important to achieve a better agreement with experimental data. They are the $K^{*}$ and $K_{1}$ vector meson resonances. The total amplitude reads

[TABLE]

where $\epsilon$ is the photon polarization, $\mu_{p}$ and $\mu_{\Lambda}$ represent the magnetic moments of proton and $\Lambda$-hyperon, respectively, while $s$, $t$ and $u$ indicate the Mandelstam variables. In Eq. (3) we have introduced $M=1$ GeV to make the transition strengths $g_{K^{*}K^{+}\gamma}$ and $g_{K_{1}K^{+}\gamma}$ dimensionless. Note that since in the phenomenological studies of kaon photoproduction the two transition strengths cannot be explicitly separated, we define $G^{V,T}_{K^{*}}=g_{K^{*}K^{+}\gamma}\,g_{K^{*}\Lambda p}^{V,T}$ and $G^{V,T}_{K_{1}}=g_{K_{1}K^{+}\gamma}\,g_{K_{1}\Lambda p}^{V,T}$.

For the calculation of cross section with the full amplitude it is customary to decompose the transition amplitude in Eq. (3) into the gauge and Lorentz invariant matrices $M_{j}$, i.e., [15]

[TABLE]

where

[TABLE]

with $P=(p_{p}+p_{\Lambda})/2$ and $\epsilon_{\mu\nu\rho\delta}$ the Levi-Civita tensor. The form functions $A_{i}$ given in Eq. (4) can be used to calculate the cross section.

The transition amplitude ${\cal M}$ in Eq. (3) can be also written within the PS theory by replacing ${g_{K^{+}\Lambda p}}\gamma_{5}\not{q}/(m_{\Lambda}+m_{p})$ in the $s$ and $u$ channels and ${g_{K^{+}\Lambda p}}\gamma_{5}(\not{q}-\not{k})/(m_{\Lambda}+m_{p})$ in the $t$ channel with $g_{K^{+}\Lambda p}\gamma_{5}$ [1]. However, it is found that the expansion of the PS amplitude is difficult because the expansion contains the terms that are proportional to $x^{-1}$, where $x=m_{K}/m$. Thus, to achieve the convergence is a daunting task in this case. The problem originates from the fact that in the PS theory the amplitude is not proportional to the kaon momentum $q$, as in the case of PV theory [see Eq. (3)].

3 Expansion of the Amplitude

The transition amplitude of kaon photoproduction obtained from the Feynman diagrams shown in Fig. 1 is given in Eq. (3). The form functions $A_{i}$ are intended for the calculation of the cross section in the data fitting as well as for obtaining the result of full calculation which will be compared with the approximations made in the present work [1]. Since we limit our study to energies very close to threshold, we do not use nucleon resonances in our calculation. The latter is also important to significantly simplify the amplitude formulation.

There are very few experimental data points near the threshold, as will be shown in the next Section. The closest ones to the threshold are obtained from the SAPHIR data [16]. Since we exclude the nucleon resonances the calculated full amplitude in the present calculation differs from our previous model [1]. As a consequence, we must refit the coupling constants in order to obtain a reasonable amplitude. The result is shown in Table 1, where we compare our present result with that of the previous PV calculation [1]. From Table 1 it appears that the coupling constants obtained from refitting the experimental data differ, although not dramatically, from those of previous work. The larger value of $\chi^{2}/N$ obtained in the present work is understandable because the number of free parameters decreases due to the exclusion of nucleon and hyperon resonances.

The square of photoproduction amplitude is expanded in terms of the ratio between kaon and baryon masses in the reaction. Different from the pion photoproduction, in the kaon photoproduction the baryon in the initial and final states are proton and $\Lambda$-hyperon, respectively. Therefore, we define the baryon mass $m$ as the average of the proton mass $m_{p}$ and $\Lambda$-hyperon mass $m_{\Lambda}$ and the ratio becomes $x=m_{K}/m=2m_{K}/(m_{p}+m_{\Lambda})$. Furthermore, to simplify the formalism, we also make the assumption that $m_{p}\approx m_{\Lambda}$ in the present work.

In this work the expansion of the square of amplitude has been performed up to $n=20$, in order to reach the convergence. The expansion has been done analytically with the help from the Mathematica software. We calculate the expansion of the squared amplitude instead of the amplitude itself because we find it is more simple and practical to use and to compare with experimental data as well as the result of other calculations. We aware that there exist a number of studies which compare the results with multipoles amplitudes. Nevertheless, for this exploratory study we limit our calculation to the total cross section. At the threshold the (reduced) total cross section of kaon photoproduction can be written as

[TABLE]

with

[TABLE]

where the subscript indicates the order of expansion. Therefore, $\mathcal{|M|}_{(n)}^{2}\propto x^{n}$. The formulas of $\mathcal{|M|}_{(n)}^{2}$ with $n$ up to 9 are given in Appendix A. For higher order amplitudes the formulas are very complicated and too long to be written in this paper, although it is produced by Mathematica and relatively simple for numerical computation. Nevertheless, for the sake of completeness in the present work we still calculate the cross section with $n$ up to 20, in spite of the fact that to reach the convergence the number of order is significantly less than 20, as will be discussed in the next Section.

4 Numerical Results

The momentum and energy distributions of the calculated reduced total cross section are shown in Fig. 2, where we also compare the result of previous calculations obtained by using PS and PV couplings [1] along with the available experimental data [19]. As discussed above as well as in Ref. [1] the PS model yields a better agreement with experimental data. This is clearly shown in Fig. 2. Furthermore, we can also see that the PV model without hyperon and nucleon resonances produces a slightly different cross section compared to the full calculation. This result indicates that without hyperon and nucleon resonances the PV model can still nicely work and provide a good framework for the purpose of comparison with the present work. Furthermore, from Fig. 2 we find that the calculated reduced total cross section at threshold is around 0.65 $\mu$b. This is clearly smaller than the experimental data data near the threshold. However, we also observe that the experimental error bar in this case is significantly large. Thus, we still believe that for the sake of comparison made in the present work Fig. 2 justifies that we can use the simplified model, i.e., the model constructed from Born terms along with the $K^{*}$ and $K_{1}$ vector mesons.

By using the PV model with parameters given in the third column of Table 1 and the particle properties taken from the Particle Data Group (PDG) listed in Table 2 we calculate the reduced total cross section at threshold for each of the expansion order. The result is listed in Table 3 and graphically displayed as a function of $n$ (order of expansion) in Fig. 3, where the bold green line indicates the full order calculation as our reference value. From Table 3 and Fig. 3 it is apparent that the lowest order calculation yields very large discrepancy with the full calculation. Figure 3 shows that the expansion starts to converge at the $9^{\rm th}$ order. Although at this order a small discrepancy still exists (about 6%), we have also to admit that we have made an assumption for baryonic mass $m$, which could also create sizable error.

The result discussed above indicates that compared to the case of pion photoproduction the LET for kaon photoproduction would require higher order terms. Actually, by excluding the $K^{*}$ and $K_{1}$ contributions we can achieve the convergence relatively faster, i.e., it converges at the $n=7$ order. However, it is well known that the Born terms alone cannot reproduce the kaon photoproduction data, even near the threshold. This is in contrast to the pion photoproduction case. Note that in this work we expand the square of amplitude, instead of the amplitude itself. As a consequence, it is natural if the order of expansion required in the first case is twice larger than that of the second case. Nevertheless, as we explained above, the convergence of this squared amplitude is reached with $n=9$. Therefore, the convergence of linear amplitude would be reached with $n>4$. This is still not comparable to the case of pion photoproduction.

The expansion of squared amplitude given in Appendix A reveals the fact that there are zeroth order and first order squared amplitudes that depend solely on the coupling constant $g_{K^{+}\Lambda p}$ [see Eqs. (22) and (23)]. Should we propose a LET by using only these term, as in the case of pion photoproduction, then we would find that the value of $g_{K^{+}\Lambda p}$ must be much smaller than the prediction of SU(3) symmetry if the calculated cross section must be comparable to experimental data. The reason is that the cross section of kaon photoproduction is much smaller than that of the pion one. To sum up, we can safely say that the theories which are based on low order expansion of kaon mass, such as SKA and Chiral Perturbation Theory, should include higher order terms in order to produce the experimental data.

Should we try to create a LET by using higher order terms in our formalism, then the main problem would be the number of unknown parameters. We can also see that the expansion described above reduces the number of unknown parameters, i.e., the coupling constants of $K_{1}$ and $K^{*}$ intermediate states. From Eq. (3) we see that there are two coupling constants for each of them. From our assumption to generalize the baryonic masses, i.e., $m_{p}\equiv m_{\Lambda}\equiv m$, the two couplings reduces to only one coupling [see Eqs. (20) and (21)]. As a result, in this formalism we have only four unknown coupling constants. Nevertheless, it is by no means easy to derive LET from these 4 parameters. We have to find other mechanisms to reduce them. This is the interest of our future study, i.e., to create higher order LET for kaon photoproduction. Furthermore, in the future we will consider the expansion of amplitude, instead of the squared one described in the present work.

5 Kaon Photoproduction in the Forward Direction

In the forward direction it is well known that the denominator of the $t$-channel propagator ($t-m_{K}^{2}$) is small. As a consequence, the corresponding propagator becomes large and the $t$-channel contribution dominates the production amplitude. The total amplitude given by Eq. (3) can be reduced and the unknown parameters can be limited to the coupling constants $g_{K^{+}\Lambda p}$ and $g^{V}_{K_{1}^{+}\Lambda p}$ only. Since very close to the threshold there exist no experimental data in the forward angles, we extend the model to cover the resonance region, where |\mbox{\boldmath{q}}|\neq 0.

It is important to note that the $t$-channel exchange is not individually gauge invariant. In the PV coupling theory, the nucleon exchange in the $s$-channel and the contact term must be included to restore the gauge invariance. In the PS coupling the electric part of the $s$-channel alone is sufficient to restore the gauge invariance. This is in contrast to the anomalous magnetic moment part, which is already gauge invariant by design, but not included in this work [22].

In the present analysis, we only have the coupling $g_{K^{+}\Lambda p}$ as the unknown parameter, which can be directly extracted by fitting the model prediction to the experimental cross section obtained from the CLAS and LEPS collaborations. To this end the closest available data are within $18^{\circ}<\theta<25^{\circ}$. The result of extraction is listed in Table.4, where we can see that the coupling constant is relatively consistent to all orders. As expected, the $\chi^{2}/N$ decreases as we go to higher orders. Actually, the sign of the coupling $g_{K^{+}\Lambda p}$ cannot be determined from the fitting process because the amplitude is calculated from the squared of the coupling. Therefore, we fix the sign by using the SU(3) prediction [23], although Table 4 exhibits that the extracted value is smaller than the SU(3) prediction by almost 50%. Nevertheless, we also note that such values were commonly obtained in the kaon photoproduction analyses without including hadronic form factors [24, 25]. To remind the reader, within 20% symmetry breaking the SU(3) predicts [23]

[TABLE]

The vector meson coupling $g^{V}_{K_{1}^{+}\Lambda p}$ can be also estimated by utilizing the SU(3) symmetry. However , since the extracted coupling from fitting process is the product $g_{K_{1}^{+}K^{+}\gamma}\,g^{V}_{K_{1}^{+}\Lambda p}$, there is a further uncertainties coming from the determination of the electromagnetic coupling $g_{K_{1}^{+}K^{+}\gamma}$. Thus, we do not pursue to compare our present result with the SU(3) prediction in this case.

By adding the $K_{1}(1270)$ exchange to the existing $K^{+}$ one, the considered coupling constants become the $g_{K^{+}\Lambda p}$ and $g^{V}_{K_{1}^{+}\Lambda p}$. The extracted values along with the corresponding $\chi^{2}$ are given in the last three columns of Table 4. It is apparent from this Table that the $K_{1}$ exchange is very important in the forward angle kaon photoproduction, since by adding the $K_{1}$ exchange the $\chi^{2}$ decreases and the $g_{K^{+}\Lambda p}$ increases approaching the SU(3) value. To our knowledge, previous calculation has shown that the $K_{1}$ exchange is very important to bring the extracted $g_{K^{+}\Lambda p}$ closer to the SU(3) prediction [31]. Therefore, our present finding corroborates this result.

As stated before, the sign of coupling constants cannot be determined from the numerical calculation. Different from the $K_{1}$ exchange, the $K^{*}$ intermediate state has much smaller contribution in our calculation. The corresponding couplings, $g^{V}_{K^{*}\Lambda N}$ and $g^{T}_{K^{*}\Lambda N}$, should have very large values in order to produce a small effect in the calculated cross section. Thus, we do not show the corresponding values and discuss them here.

Compared to the result obtained from the single $t$-channel intermediate state, adding the $K_{1}$ exchange seems to slow down the convergence rate as can be seen by comparing the second and fourth columns of Table 4. In the single intermediate state the convergence is already achieved by calculating the amplitude up to second order. On the contrary, the result obtained from using both $K^{+}$ and $K_{1}^{+}$ mesons seems to be not convergent even at the 14th order calculation.

The sensitivity of the $K_{1}$ exchange is depicted in the top panel of Fig. 4. It is apparent from this panel that the contribution of this state is significant near the threshold region, where its effect is shifting the calculated cross section closer to experimental data. The convergence of the squared amplitude expansion is displayed in the bottom panel of Fig. 4. Obviously, after the second order expansion the expansion is practically convergent. This is also supported by the fact that the error bars of the present data cannot resolve the the differences produced by different order calculations (see the bottom panel of Fig. 4).

To conclude this Section we might safely say that as in the case of the determination of pion electromagnetic form factor from the $t$-channel pion electroproduction [12], the extraction of the leading kaon coupling constant $g_{K^{+}\Lambda p}$ can be approximated by only using the $t$-channel intermediate states $K$ and $K_{1}$.

6 Kaon Photoproduction at Backward Angles

In general, the differential cross section obtained from the $u$-channel contribution is relatively smaller than that obtained from the $K^{+}$ exchange in the $t$-channel. In the backward direction, however, the magnitude of Mandelstam variable $|u|$ is very small and, as a consequence, the corresponding contribution becomes significant and can be expected to dominate all contributions [28]. Note that different from the $t$-channel term, the $u$-channel amplitude is individually gauge invariant, since the exchanged particle $\Lambda$ is neutral and, therefore, the $u$-channel amplitude contributes only the magnetic moment term.

Since we have two exchange particles in the $u$-channel, i.e., $\Lambda$ and $\Sigma^{0}$, the unknown parameters in this case are $g_{K^{+}\Lambda p}$ and $g_{K^{+}\Sigma^{0}p}$ couplings. These coupling constants were extracted from fitting the kaon photoproduction data with $152^{\circ}<\theta<161^{\circ}$. The result is shown in Fig. 5, where we compare the calculated cross section obtained from the expanded squared amplitude with 0th up to 14th orders with the CLAS 2010 and LEPS 2007 data. From Fig. 5 we can see that the cross sections obtained from expansions with different orders show a large variance only at low energies, where unfortunately no data are available in this kinematics. Therefore, experimental measurement to this end is strongly required in order to check whether or not the high order expansions are decisive in the backward direction.

The extracted coupling constants along with the corresponding $\chi^{2}/N$ obtained from expansions with different orders are listed in Table 5. In contrast to the case of forward angles, in the backward direction both the $\chi^{2}/N$ and the extracted coupling constants are very large. The latter are even much larger than the prediction of SU(3). This result indicates that photoproduction of kaon at the backward angles cannot be explained by merely using the $u$-channel. We observe that the coupling constants can be greatly decreased if we add the $s$-channel contribution. Previous analyses have shown that including hyperon resonances could also alleviate this problem [32, 33].

7 Summary and Conclusions

We have investigated the convergence of the expansion of squared PV amplitude for kaon photoproduction at threshold. We found that the expanded PV amplitude starts to converge at $9^{\rm th}$ order. Should we use solely the zeroth and first order expansions, then the main coupling constant $g_{K\Lambda p}$ must be much smaller than the SU(3) prediction, if the calculated cross section is fitted to experimental data close to threshold. Therefore, our present investigation concludes that LET using the expansion of lowest order $m_{K}/m$ would be difficult to derive. We have also extracted the leading coupling constants in kaon photoproduction by the expanding the squared PV amplitude at forward and backward directions by solely utilizing the $t$- and $u$-channel amplitudes, respectively. It is found that, unless the $K_{1}$ vector meson was included, the extracted coupling constants in the forward direction are smaller than the SU(3) prediction. Adding the $K_{1}$ significantly improves the agreement of our calculation with the SU(3) prediction. In the backward direction we found a different result, the extracted coupling constants are much larger than the SU(3) value.

8 Acknowledgments

This work has been supported by the PITTA A Grant, Universitas Indonesia, under contract No. NKB-0449/UN2.R3.1/HKP.05.00/2019.

Appendix A Expansion of the Transition Amplitude

As described in Section 3 we expand the square of the amplitude according to

$\mathcal{|M|}^{2}=\sum_{n=0}^{20}\mathcal{|M|}_{(n)}^{2}$ where $\mathcal{|M|}_{(n)}^{2}\propto x^{n}$. To simplify the formulas we define the following quantities.

[TABLE]

The $n$-th order of the expanded amplitude is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]