Photo-production of lowest $\Sigma^*_{1/2^-}$ state within the Regge-effective Lagrangian approach

Since the lowest $\Sigma^{*}$ state, with quantum numbers spin-parity $J^{P} =1/2^{-}$, is far from established experimentally and theoretically, we have performed a theoretical study on the $\Sigma^*_{1/2^-}$ photo-production within the Regge-effective Lagrangian approach. Taking into account that the $\Sigma^*_{1/2^-}$ couples to the $\bar{K}N$ channel, we have considered the contributions from the $t$-channel $K$ exchange diagram. Moreover, these contributions from $t$-channel $K^*$ exchange, $s$-channel nucleon pole, $u$-channel $\Sigma$ exchange, and the contact term, are considered. The differential and total cross sections of the process $\gamma n \to K^{+}\Sigma^{*-}_{1/2^-}$ are predicted with our model parameters. The results should be helpful to search for the $\Sigma^*_{1/2^-}$ state experimentally in future.

The analyses of the relevant data of the process K − p → Λπ + π − suggest that there may exist a Σ * 1/2 − resonance with mass about 1380 MeV [16,17], which is consistent with the predictions of the unquenched quark models [20]. The analyses of the K * Σ photoproduction also indicate that the Σ * 1/2 − is possibly buried under the Σ * (1385) peak with mass of 1380 MeV [21], and it is proposed to search for the Σ * 1/2 − in the process Λ c → ηπ + Λ [22]. A more delicate analysis of the CLAS data on the process γp → KΣπ [23] suggests that the Σ * 1/2 − peak should be around 1430 MeV [13]. In Refs. [24,25], we suggest to search for such state in the processes of χ c0 (1P ) →ΣΣπ and χ c0 (1P ) →ΛΣπ. In addition, Ref. [26] has found one Σ * 1/2 − state with mass around 1400 MeV by solving coupled channel scattering equations, and Ref. [27] suggests to search for this state in the photo-production process γp → K + Σ * 0 1/2 − . It's worth mentioning that a Σ * (1480) resonance with J P = 1/2 − has been listed on the previous version of RPP [28]. As early as 1970, the Σ * (1480) resonance was reported in the Λπ + , Σπ, and pK 0 channels of the π + p scattering in the Princeton-Pennsylvania Accelerator 15-in.∼hydrogen bubble chamber [29,30]. In 2004, a bump structure around 1480 MeV was observed in the K 0 S p(p) invariant mass spectrum of the inclusive deep inelastic ep scattering by the ZEUS Collaboration [31]. Furthermore, a signal for a resonance at 1480 ± 15 MeV with width of 60 ± 15 MeV was observed in the process pp → K + pY * 0 [32]. Theoretically, the Σ * (1480) was investigated within different models [33][34][35][36]. In Ref. [36], the S-wave meson-baryon interactions with strangeness S = −1 were studied within the unitary chiral approach, and one narrow pole with pole position of 1468−i 13 MeV was found in the second Riemann sheet, which could be associated with the Σ * (1480) resonance. However, the Σ * (1480) signals are insignificant, and the existence of this state still needs to be confirmed within more precise experimental measurements.
As we known, the photo-production reactions have been used to study the excited hyperon states Σ * and Λ * , and the Crystal Ball [37][38][39], LEPS [40], and CLAS [23] Collaborations have accumulated lots of relevant experimental data. For instance, with these data, we have analyzed the process γp → KΛ * (1405) to deepen the understanding of the Λ * (1405) nature in Ref. [41]. In order to confirm the existence of the Σ * (1480), we propose to investigate the process γN → KΣ * (1480) 1 within the Regge-effective Lagrange approach.
Considering the Σ * (1480) signal was first observed in the π + Λ invariant mass distribution of the process π + p → π + K + Λ, and the significance is about 3 ∼ 4σ [30], we search for the charged Σ * (1480) in the process γn → K + Σ * − 1/2 − , which could also avoid the contributions of possible excited Λ * states. We will consider the t-, s-, u-channels diagrams in the Born approximation by employing the effective Lagrangian approach, and the t-channel K/K * exchanges terms within Regge model. Then we will calculate the differential and total cross sections of the process γn → K + Σ * − 1/2 − reaction, which are helpful to search for Σ * 1/2 − experimentally. This paper is organized as follows. In Sec. II, the theoretical formalism for studying the γn → K + Σ * − (1480) reactions are presented. The numerical results of total and differential cross sections and discussion are shown in Sec. III. Finally, a brief summary is given in the last section.

II. FORMALISM
The reaction mechanisms of the Σ * (1480) (≡ Σ * ) photo-production process are depicted in the Fig. 1, where we have taken into account the contributions from the t-channel K and K * exchange term, s-channel nucleon pole term, u-channel Σ exchange term, and the contact term, respectively.
To compute the scattering amplitudes of the Feynman diagrams shown in Fig. 1 within the effective Lagrangian approach, we use the Lagrangian densities for the electromagnetic and strong interaction vertices as used in Refs. [27,[42][43][44][45][46] where e(= √ 4πα) is the elementary charge unit, A µ is the photon filed, andê ≡ (1+τ 3 )/2 denotes the charge operator acting on the nucleon field.κ N ≡ κ pê +κ n (1−ê) is the anomalous magnetic moment, and we take κ n = −1.913 for neutron [15]. M N and M Σ denote the masses of nucleon and the ground-state of Σ hyperon, respectively. The strong coupling g KN Σ is taken to be 4.09 from Ref. [47]. The g γKK * = 0.254 GeV −1 is determined from the experimental data of Γ K * →K+γ [15] and the value of g K * N Σ * = −3.26 − i0.06 is taken from Ref [26]. In addition, the coupling g KN Σ * = 8.74 GeV is taken from Ref. [36], and the transition magnetic moment µ ΣΣ * = 1.28 is taken from Ref. [27] With the effective interaction Lagrangian densities given above, the invariant scattering amplitudes are defined as where u Σ * and u N stand for the Dirac spinors, respectively, while ǫ µ (k 1 , λ) is the photon polarization vector and the sub-indice h corresponds to different diagrams of Fig. 1. The reduced amplitudes M µ h are written as In order to keep the full photoproduction amplitudes considered here gauge invariant, we adopt the amplitude of the contact term It is known that the Reggeon exchange mechanism plays a crucial role at high energies and forward angles [48][49][50][51], thus we will adopt Regge model for modeling the t-channel K and K * contributions by replacing the usual pole-like Feynman propagator with the corresponding Regge propagators as follows, with α K (t) = 0.7 GeV −2 × (t − M 2 K ) and α K * (t) = 1 + 0.83 Gev −2 × (t − M 2 K * ) the linear Reggeon trajectory. The constants s K 0 and s K * 0 are determined to be 3.0 GeV 2 and 1.5 GeV 2 , respectively [52]. Here, the α ′ K and α ′ K * are the Regge-slopes. Then, the full photo-production amplitudes for γn → K + Σ * − 1/2 − reaction can be expressed as While F Regge with M i and q i being the masses and four-momenta of the intermediate baryons, and the Λ i is the cut-off values for baryon exchange diagrams. In this work, we take Λ s = Λ u = 1.5 GeV, and will discuss the results with different cut-off. Finally, the unpolarized differential cross section in the center of mass (c.m.) frame for the γn → KΣ * − 1/2 − reaction reads where s denotes the invariant mass square of the center of mass (c.m.) frame for Σ * 1/2 − photo-production. Here k c.m.

III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we show our numerical results of the differential and total cross sections for the γn → K + Σ * − 1/2 − reaction. The masses of the mesons and baryons are taken from RPP [15], as given in Table I. In addition, the mass and width of the Σ(1480) are M = 1480 ± 15 GeV and Γ = 60 ± 15 GeV, respectively [28]. First we show the angle dependence of the differential cross sections for the γn → K + Σ * − 1/2 − reaction in Fig. 2, where the the center-of-mass energies W = √ s varies from 2.0 to 2.8 GeV. The black curves labeled as 'Total' show the results of all the contributions from the t-, s-, u-channels, and contact term. The blue-dot curves and red-dashed curves stand for the contributions from the u-channel Σ exchange and t-channel K exchange mechanism, respectively. The magenta-dot-dashed curves and the green-dot curves correspond to the contributions from the s-channel and t-channel K * exchange diagrams, respectively, while the cyan-dot-dashed curves represent the contribution from the contact term. According to the differential cross sections, one can find that the t-channel K meson exchange term plays an important role at forward angles for the process γn → K + Σ * − 1/2 − , mainly due to the Regge effects of the t-change K exchange. The K-Reggeon exchange shows steadily increasing behavior with cosθ c.m. and falls off drastically at very forward angles. In addition, the u-channel Σ exchange term mainly contribute to the backward angles for both processes. It should be stressed that the contribution from the tchannel K * exchange term is very small and could be safely neglected for the process γn → K + Σ * − 1/2 − , which is consistent with the results of Ref. [27].
In addition to the the differential cross sections, we have also calculated the total cross section of the γn → K + Σ * − 1/2 − reaction as a function of the initial photon energy. The results are shown in Fig. 3. The black curve labeled as 'Total' shows the results of all the contributions, including t-, s-, u-channels and contact term. The blue-dot and red-dashed curves stand for the contributions from the u-channel Σ exchange and t-channel K exchange mechanism, respectively. The magenta-dotdashed and the green-dot curves show the contribution of s-channel and t-channel K * exchange diagrams, respectively, while the cyan-dot-dashed curve represents the Finally, we also show the total cross section for γn → K + Σ * − 1/2 − with the cut-off Λ s/u = 1.2, 1.5, and 1.8 GeV in Fig. 4, where one can find the total cross sections are weakly dependence on the value of the cut-off. Since the precise couplings of the Σ(1480) are still unknown, the (Color online) Total cross section for γn → K + Σ * 1/2 − is plotted as a function of the lab energy Eγ. The black curve labeled as 'Total' shows the results of all the contributions, including t-,s-,u-channels and contact term. The blue-dot and red-dashed curves stand for the contributions from the effective Lagrangian approach u-channel Σ exchange and t-channel K exchange mechanism, respectively. The magenta-dot-dashed and the green-dot curves show the contribution of s-channel and t-channel K * exchange diagrams, respectively, while the cyan-dot-dashed curve represents the contribution of the contact term. future experiment would be helpful to constrain these couplings if the state Σ(1480) is confirmed.

IV. SUMMARY
The lowest Σ * − 1/2 − is far from established, and its existence is important to understand the low-lying excited baryon with J P = 1/2 − . There are many experimental hints of the Σ * (1480), which has been listed in the previous version of the Review of Particle Physics. We propose to search for this state in the photoproduction process to confirm its existence.
Assuming that the J P = 1/2 − low lying state Σ * (1480) has a sizeable coupling to theKN according the study of Ref. [36], we have phenomenologically investigated the γn → K + Σ * − 1/2 − reaction by considering the contributions from the t-channel K/K * exchange term, s-channel nucleon term, u-channel Σ exchange term, and contact term within the Regge-effective Lagrange approach. The differential cross sections and total cross sections for these processes are calculated with our model parameters. The total cross section of γn → K + Σ * − 1/2 − is about 4.3 µb around E γ = 2.3 GeV. We encourage our experimental colleagues to measure γn → K + Σ * − 1/2 − process.