4 Λ π , which are detected by experiments. Thus we con- sider that K Σ( 1 2 $^{−}$) photoproduction may also contribute to the measured cross sections. We constrain our study to processes with s and p wave hadronic vertices, since the contributions from higher waves are relatively sup- pressed. We also neglect the contributions from some res- onances either for lack of information on the couplings or the couplings are small. Thus the main contributions to K Σ( 1 2 $^{−}$) photoproduction are from the t-channel K me- son exchange, the s-channel N exchange, the u-channel Σ( 1 2 $^{−}$) exchange (and Λ exchange for γp → K $^{+}$Σ$^{0}$( 1 2 $^{−}$)) and the contact term. The Feynman diagrams are the same as Fig. 1. The effective Lagrangian for KN Σ( 1 2 $^{−}$) coupling can be expressed as [21] Since previous evidence for the new Σ( 1 2 $^{−}$) indicates its mass to be around Σ $^{∗}$(1385) [17], here we assume its mass be the same as Σ $^{∗}$(1385). The coupling constant g$_{KN}$$_{Σ}$ and the relevant cut-off parameter Λ$_{M}$ are unknown pa- rameters, and will be tuned to fit the data. where the coupling constant g$_{KN}$$_{Σ}$ is to be fitted to exper- iments; The isospin structure is the same as Eq.(3). The effective Lagrangians for γKK and γNN are described in Eq.(1) and Eq.(4), respectively. For γ Σ( 1 2 $^{−}$)Σ( 1 2 $^{−}$) vertex, the effective Lagrangian can be expressed as [21] The effective Lagrangian methods employ hadronic fields as basic degrees of freedom for the interactions, which introduce many parameters such as the coupling constants and cutoff parameters. Many of the parame- ters are not well constrained, and approximations such as flavor SU(3) symmetry relation or model predictions are used. These may bring some uncertainty in the calculation. For example, in this paper, we consider 3 PDG resonances in s-channel as in Ref. [12]. Their masses, widths and coupling constants are not well con- strained by experiments. Our choices of the parameters from the allowed range of PDG [19] and model calcu- lations [20, 22] certainly have big uncertainties, similar as discussed in Ref. [24] for the study of γn → K $^{+}$Σ $^{−}$. However, for the cross sections and beam asymmetry of the reaction γn → K $^{+}$Σ $^{∗−}$(1385) → K $^{+}$Λ π − at E$_{γ}$ = 1 . 5 − 2 . 4 GeV reported by the LEPS collabora- tion [9], these 3 resonances mainly contribute to cross sections around E$_{γ}$ = 1 . 8 GeV, and their contributions are small for E$_{γ}$ > 2.1 GeV. Even without constraint from other sources, adjusting these parameters cannot repro- duce the measured beam asymmetry qualitatively. We need to find other ingredients to describe the cross sec- tions and beam asymmetry data simultaneously. The measurement by the LEPS Collaboration is lim- ited to forward angles with cos θ$_{c}$$_{.}$$_{m}$$_{.}$ ≥ 0 . 6, where θ$_{c}$$_{.}$$_{m}$$_{.}$ is the forward angle of the produced K meson in the c.m. frame. The theoretical predictions by the original set of parameters ( h = 1 for the contact term) [12] deviate from the LEPS results for the linear beam asymmetry [9], which can be interpreted as where Q$_{Σ}$ is the electric charge (in unite of e ), and κ$_{Σ}$ de- notes the anomalous magnetic moment for Σ( 1 2 $^{−}$), and we take the predicted values κ$_{Σ}$ 0 = − 0 . 43 and κ$_{Σ}$ − = − 1 . 74 from the diquark model [16]. For the γ ΛΣ $^{0}$( 1 2 $^{−}$) coupling in the γp → K $^{+}$Σ0 process, the effective Lagrangian is expressed as [21] where we take g$_{γ}$$_{ΛΣ}$ = 1 . 16. Note that although this coupling has some uncertainty, it is largely suppressed and has very small effect in this process. For each vertex in the Feynman diagrams, a form factor is attached. Similar to the K Σ $^{∗}$( 3 2 $^{+}$) photoproduction processes, we adopt the form factor as in Eq.(15) for t- channel K meson exchange, and Eq.(16) for s-channel and u-channel processes. The contact term for K Σ( 1 2 $^{−}$) photoproduction is re- lated to the Lagrangian of Eq.(26). Following Refs. [12, 18], we adopt the contact current for γp → K $^{+}$Σ$^{0}$( 1 2 $^{−}$), where the Lorenz index ν couples to that of the photon, and C ν is expressed as Eq.(19), where h = 1 is taken. For γn → K $^{+}$Σ $^{−}$( 1 2 $^{−}$) process, we adopt the contact current: where C ν is expressed as Eq.(21), where h = 1 is taken. With these contact currents, one can check that the total amplitudes are gauge invariant. where σ$_{⊥}$ and σ$_{‖}$ denote the cross sections for beam po- larization vertical and parallel to the reaction plane, re- spectively. In this work, we find two possible ingredients to explain the observed beam asymmetry. One is by including the possible contribution of K Σ( 1 2 $^{−}$) photoproduction, and another is to assume a different h parameter for the γn reaction from the γp reaction. To incorporate the new ingredients to fit both cross sections and beam asymme- try, the minimum set of parameters to be adjusted from those used in Ref. [9, 12] are listed in Table I. For both schemes, the narrower widths for the three N ∗ and Δ ∗ resonances are used, but still within the PDG uncertain- ties [19]. The reached χ 2 of the two schemes are also A$_{beam}$ = σ$_{⊥}$ − σ$_{‖}$ σ$_{⊥}$ + σ$_{‖}$ , (31) M ν c = ieg$_{KN}$$_{Σ}$C ν (29) M ν c = $_{ie}$√$_{2}$$_{g}$$_{KN}$$_{Σ}$$_{C}$ ν , (30) L γ ΛΣ = eg$_{γ}$$_{ΛΣ}$ 4( M$_{Λ}$ + M$_{Σ}$ ) Σ γ$_{5}$σ$_{µν}$ Λ F νµ + H . c . , (28) L γ ΣΣ = − e Σ( γ $^{µ}$A$_{µ}$ Q$_{Σ}$ − κ$_{Σ}$ 2 M$_{N}$ σ $^{µν}$∂$_{ν}$ A$_{µ}$ )Σ , (27) L KN Σ = − ig$_{KN}$$_{Σ}$K Σ · τN + H . c . , (26) III. RESULTS AND DISCUSSIONS