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

The study of the low-lying excited $ \Lambda^* $ and $ \Sigma^* $ hyperon resonances is one of the most important issues in hadron physics. In particular, since $ \Lambda(1405) $ was experimentally discovered [1, 2], its nature has garnered significant attention [3–8], and one explanation for $ \Lambda(1405) $ is the $ \bar{K}N $ hadronic molecular state [9–15]. In addition, for the isospin $ I=1 $ partner of $ \Lambda(1405) $, the lowest $ \Sigma^{*}_{1/2^{-}} $ is crucial to understand light baryon spectra. At present, $ \Sigma^*(1620) $ with $ J^P = 1/2^{-} $ is listed in the latest version of the Review of Particle Physics (RPP) [16]. It should be emphasized that the $ \Sigma^*(1620) $ state is a one-star baryon resonance. Many studies indicate that the lowest $ \Sigma^*_{1/2^-} $resonance is still far from being established, and its mass has been predicted to lie in the range $ 1380 \sim 1500 $ MeV [13, 17–20]. Thus, searching for the lowest $ \Sigma^*_{1/2^-} $ is helpful for understanding low-lying excited baryons with $ J^P=1/2^- $ and light flavor baryon spectra.

The analyses of relevant data on the process $ K^- p \to \Lambda \pi^+\pi^- $ suggest that a $ \Sigma^*_{1/2^{-}} $ resonance may exist with a mass of approximately 1380 MeV [17, 18], which is consistent with the predictions of unquenched quark model [21]. The analyses of $ K^{*}\Sigma $ photo-production also indicate that $ \Sigma^*_{1/2^{-}} $ is possibly buried under the $ \Sigma^*(1385) $ peak with a mass of 1380 MeV [22], and the search for $ \Sigma^*_{1/2^{-}} $ in the process $ \Lambda^+_c \to \eta\pi^+\Lambda $ has been proposed [23]. A more delicate analysis of CLAS data on the process $ \gamma p \to K \Sigma \pi $ [24] suggests that the $ \Sigma^*_{1/2^{-}} $ peak should be around 1430 MeV [13]. In Refs. [25, 26], we suggest searching for such a state in the processes $ \chi_{c0}(1P) \to \bar{\Sigma} \Sigma\pi $ and $ \chi_{c0}(1P)\to \bar{\Lambda}\Sigma\pi $. In addition, in Ref. [27], one $ \Sigma^*_{1/2^{-}} $ state was found with a mass of approximately 1400 MeV by solving coupled channel scattering equations, and Ref. [28] suggests to search for this state in the photo-production process $ \gamma p \to K^{+} \Sigma^{*0}_{1/2^-} $.

It is worth mentioning that a $ \Sigma^*(1480) $ resonance with $ J^P =1/2^- $ has been listed on the previous version of the RPP [29]. As early as 1970, the $ \Sigma^*(1480) $ resonance was reported in the $ \Lambda \pi^{+} $, $ \Sigma \pi $, and $ p\bar{K}^{0} $ channels of $ \pi^+ p $ scattering in the Princeton-Pennsylvania Accelerator 15-in. hydrogen bubble chamber [30, 31]. In 2004, a bump structure around 1480 MeV was observed in the $ K^0_S p(\bar{p}) $ invariant mass spectrum of inclusive deep inelastic $ ep $ scattering by the ZEUS Collaboration [32]. Furthermore, a signal for a resonance at $ 1480 \pm 15 $ MeV with a width of $ 60 \pm 15 $ MeV was observed in the process $ pp \to K^{+}pY^{*0} $ [33]. $ \Sigma^*(1480) $ has been investigated theoretically within different models [34–37]. In Ref. [37], S-wave meson-baryon interactions with strangeness $ S=-1 $ were studied within the unitary chiral approach, and one narrow pole with a pole position of $1468-{\rm i} \ 13 \ \rm MeV$ was found in the second Riemann sheet, which may be associated with the $ \Sigma^*(1480) $ resonance. However, $ \Sigma^*(1480) $ signals are insignificant, and the existence of this state still needs to be confirmed within more precise experimental measurements.

Photo-production reactions have been used to study the excited hyperon states $ \Sigma^* $ and $ \Lambda^* $, and the LEPS [38] and CLAS [24] Collaborations have accumulated considerable relevant experimental data. For instance, with these data, we analyzed the process $ \gamma p\to K\Lambda^*(1405) $ to deepen our understanding of the nature of $ \Lambda^*(1405) $ in Ref. [39]. To confirm the existence of $ \Sigma^*(1480) $, we propose to investigate the process $ \gamma N \to K\Sigma^*(1480) $^① within the Regge-effective Lagrangian approach.

Considering the $ \Sigma^*(1480) $ signal was first observed in the $ \pi^+\Lambda $ invariant mass distribution of the process $\pi^+ p\to \pi^+K^+\Lambda$, and the significance is approximately $ 3 \sim 4 \sigma $ [31], we search for charged $ \Sigma^*(1480) $ in the process $\gamma n \to K^{+}\Sigma^{*-}_{1/2^-}$, which may also avoid the contributions of possible excited $ \Lambda^* $ states. We consider the t-, s-, and u-channel diagrams in the Born approximation by employing the effective Lagrangian approach, and the t-channel K/$ K^{*} $ exchanges terms within the Regge model. Then, we calculate the differential and total cross sections of the process $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $, which helps the experimental search for $ \Sigma^*_{1/2^-} $.

This paper is organized as follows. In Sec. II, the theoretical formalism for studying the $ \gamma n \to K^+ \Sigma^{*-}(1480) $ reaction is presented. The numerical results of the total and differential cross sections and discussion are shown in Sec. III. Finally, a brief summary is given in the last section.

The reaction mechanisms of the $ \Sigma^*(1480) (\equiv \Sigma^*) $ photo-production process are depicted in Fig. 1, where we consider the contributions from the t-channel K and $ K^* $ exchange terms, s-channel nucleon pole term, u-channel Σ exchange term, and contact term.

Figure 1.  Mechanisms of the $ \gamma n \to K^+\Sigma^{*-}_{1/2^-} $ process. (a) t-channel K/$ K^* $ exchange terms, (b) s-channel nuclear term, (c) u-channel Σ exchange term, and (d) contact term. $ k_{1} $, $ k_{2} $, $ p_{1} $, and $ p_{2} $ represent the four-momenta of the initial photon, kaon, neutron, and $ \Sigma^*(1480) $, 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 in Refs. [28, 40–45],

where $ e(=\sqrt{4\pi\alpha}) $ is the elementary charge unit, $ A^{\mu} $ is the photon field, and $ \hat{e}\equiv (1+\tau_{3})/2 $ denotes the charge operator acting on the nucleon field. $ \hat{\kappa}_{N} \equiv \kappa_{p} \hat{e}+\kappa_{n}(1-\hat{e}) $ is the anomalous magnetic moment, and we take $ \kappa_{n}= -1.913 $ for the neutron [16]. $ M_N $ and $ M_{\Sigma} $ denote the masses of the nucleon and the ground-state of the Σ hyperon, respectively. The strong coupling $ g_{KN\Sigma} $ is taken to be 4.09 from Refs. [46–48]. $ g_{\gamma K K^{*}}=0.254\ \rm GeV^{-1} $ is determined from the experimental data of $\Gamma_{K^{*} \to K^+ \gamma}$ [16], and the value of $ g_{K^{*}N\Sigma^{*}}=-3.26-{\rm i}~0.06 $ is taken from Ref. [27]. In addition, the coupling $ g_{KN\Sigma^{*}}=8.74 $ GeV is taken from Ref. [37], and the transition magnetic moment $ \mu_{\Sigma\Sigma^*}=1.28 $ is taken from Ref. [28]^②.

In addition to the pseudoscalar coupling of Eq. (5), the vertex of $ KN\Sigma $ may be described with the Lagrangian density of axial-vector coupling as follows [49, 50]

We discuss the difference between the two schemes in the next section. In this work, we perform the calculations with the Lagrangian density of Eq. (5) for the vertex of $ KN\Sigma $ in the following.

With the effective interaction Lagrangian densities given above, the invariant scattering amplitudes are defined as

where $ u_{\Sigma^{*}} $ and $ u_{N} $ represent the Dirac spinors, $ \epsilon_{\mu}(k_{1},\lambda) $ is the photon polarization vector, and the sub-index h corresponds to the different diagrams of Fig. 1. The reduced amplitudes $ \mathcal{M}^{\mu}_{h} $ are written as

To maintain gauge invariance in the full photoproduction amplitudes considered here, we adopt the amplitude of the contact term

for $ \gamma n \to K^{+} \Sigma^{*-}_{1/2^-} $.

It is known that the Reggeon exchange mechanism plays a crucial role at high energies and forward angles [51–54]; thus, we adopt the Regge model when modeling the t-channel K and $ K^{*} $ contributions by replacing the usual pole-like Feynman propagator with the corresponding Regge propagators as follows

where $ \alpha_{K}(t)=0.7\; {\rm GeV}^{-2}\times(t-M_{K}^{2}) $ and $\alpha_{K^{*}}(t)=1+ 0.83 {\rm GeV}^{-2}\times(t-M_{K^{*}}^{2})$ are the linear Reggeon trajectories. The constants $ s^{K}_{0} $ and $ s^{K^{*}}_{0} $ are determined to be 3.0 GeV$ ^{2} $ and 1.5 GeV$ ^{2} $, respectively [55]. Here, $ \alpha'_K $ and $ \alpha'_{K^*} $ are the Regge-slopes.

Then, the full photo-production amplitudes for the $ \gamma n \to K^+ \Sigma^{*-}_{1/2^-} $ reaction can be expressed as

where $ \mathcal{F} ^{\rm Regge}_{K} $ and $ \mathcal{F}^{\rm Regge}_{K*} $ represent the Regge propagators. The form factors $ f_{s} $ and $ f_{u} $ are included to suppress the large momentum transfer of the intermediate particles and describe their off-shell behavior because the intermediate hadrons are not point-like particles. For s-channel and u-channel baryon exchanges, we use the following form factors [40, 56]

where $ M_{i} $ and $ q_{i} $ are the masses and four-momenta of the intermediate baryons, and $ \Lambda_{i} $ represents the cut-off values for the baryon exchange diagrams. In this study, we take $\Lambda_s= \Lambda_u=1.5$ GeV and discuss the results with different cut-offs.

Finally, the unpolarized differential cross section in the center of mass (c.m.) frame for the $ \gamma n \to K\Sigma^{*-}_{1/2^{-}} $ reaction reads as

where s denotes the invariant mass square of the center of mass (c.m.) frame for $ \Sigma^{*}_{1/2^{-}} $ photo-production, and ${\rm d}\Omega = 2\pi {\rm d}{{\rm cos\theta} _{\rm c.m.}}$, with $ \theta_{\rm c.m.} $ as the polar outgoing K scattering angle. Here, $ \vec{k}_1^{\rm c.m.} $ and $ \vec{k}_{2} ^{\rm c.m.} $ are the three-momenta of the photon and K meson in the c.m. frame,

In this section, we present our numerical results on the differential and total cross sections for the $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ reaction. The masses of the mesons and baryons are taken from the RPP [16], as given in Table 1. In addition, the mass and width of $\Sigma^*(1480)$ are $ M=1480\pm15 $ MeV and $ \Gamma=60\pm15 $ MeV, respectively [29].

Particle	Mass$/\rm MeV$
n	939.565
$ \Sigma^{-} $	1197.449
$ K^{+} $	493.677
$ K^{-} $	493.677
$ K^{*} $	891.66

Table 1.  Particle masses used in this study.

First, we show the angle dependence of the differential cross sections for the $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ reaction in Fig. 2, where the center-of-mass energies $ W=\sqrt{s} $ varies from $ 2.0 $ to $ 2.8 $GeV. The black curves labeled as "Total" are the results of all the contributions from the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves correspond to the contributions from the s-channel and t-channel $ K^{*} $ exchange diagrams, respectively, whereas the cyan-dot-dashed curves represent the contributions from the contact term. According to the differential cross sections, we find that the t-channel K meson exchange term plays an important role at forward angles for the process $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $, mainly because of the Regge effects of t-channel K exchange. The K-Reggeon exchange exhibits a steadily increasing behavior with $ {\rm cos}\theta_{\rm c.m.} $ and falls off drastically at very forward angles, which is consistent with the results of Ref. [28]. In the appendix, we show that the contribution from the t-channel K exchange is zero in the forward angle ($ \theta_{\rm c.m.}=0 $) and the backward angle ($ \theta_{\rm c.m.}=\pi $). In addition, the u-channel Σ exchange term mainly contributes to the backward angles. It should be emphasized that the contribution from the t-channel $ K^* $ exchange term is small and can be safely neglected for the process $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $, which is consistent with the results of Ref. [28].

Figure 2.  (color online) $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ differential cross sections as a function of $ \rm{cos \theta_{\rm c.m.}} $ are plotted for $ \gamma n $-invariant mass intervals $ \left(\rm in\ \rm GeV \ units \right) $. The black curve labeled as "Total" represent the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves denote the contributions from the effective Lagrangian approach u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot-dashed curves represent the contributions of the s-channel and t-channel $ K^{*} $ exchange diagrams, respectively, whereas the cyan-dot-dashed curves represent the contributions of the contact term.

As mentioned in Sec. II, in addition to the pseudoscalar coupling of Eq. (5), the vertex of $ KN\Sigma $ may also be described with the Lagrangian density of axial-vector coupling [Eq. (8)]. We also present the contribution from the u-channel Σ exchange with the Lagrangian densities of Eqs. (5) and (8) in Fig. 3; we find that both of them contribute to the backward angles. Because the contribution from the u-channel is small, it is expected that either pseudoscalar coupling or axial-vector coupling for $ KN\Sigma $ does not affect our results significantly.

Figure 3.  (color online) $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ differential cross sections with only the contribution from u-channel Σ exchange. The magenta-dot-dashed curves represent the results obtained with the axial-vector coupling of Eq. (8), and the blue-dotted curves represent the results obtained with the pseudoscalar coupling of Eq. (5).

In addition to the differential cross sections, we also calculate the total cross section of the $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ reaction as a function of the initial photon energy. The results are shown in Fig. 4. The black curve labeled as "Total" represents the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves represent the contributions of the s-channel and t-channel $ K^{*} $ exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term. For the $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ reaction, its total cross section attains a maximum value of approximately $ 4.2 ~\mu $b at $ E_{\gamma}=2.4 \ \rm GeV $. It is expected that $ \Sigma^*(1480) $ can be observed by future experiments in the process $ \gamma n \to K^{+} \Sigma^{*-}\left(1480\right) \to \Sigma^{-}\pi^{0}/\Sigma^{0}\pi^{-}/\Sigma^{-}\gamma $.

Figure 4.  (color online) Total cross section for $ \gamma n \to K^{+}\Sigma^{*}_{1/2^-} $ is plotted as a function of the lab energy $ E_{\gamma} $. The black curve labeled as "Total" represents the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the effective Lagrangian approach u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves represent the contributions of the s-channel nucleon term and t-channel $ K^{*} $ exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term.

Finally, we show the total cross section for $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ with the cut-off $ \Lambda_{s/u}=1.2 $, $ 1.5 $, and $ 1.8 $ GeV in Fig. 5. We find that the total cross sections are weakly dependent on the value of the cut-off. Because the precise couplings of $\Sigma^*(1480)$ are still unknown, future experiments would benefit from constraining these couplings if the state $\Sigma^*(1480)$ is confirmed.

Figure 5.  (color online) Total cross section for $ \gamma n \to K^{+}\Sigma^{*}_{1/2^-} $ with the cut-off $ \Lambda_{s/u}=1.2 $, $ 1.5 $, and $ 1.8 $ GeV.

The lowest $ \Sigma^{*-}_{1/2^-} $ is far from established, and its existence is important to understand low-lying excited baryons with $ J^P=1/2^- $. There are many experimental hints of $ \Sigma^*(1480) $, as listed in the previous version of the RPP. 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 $ \Sigma^*\left(1480\right) $ has a sizeable coupling to $ \bar{K}N $ according to the study of Ref. [37], we phenomenologically investigate the $ \gamma n \to K^{+} \Sigma^{*-}_{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 Lagrangian approach. The differential and total cross sections for these processes are calculated with our model parameters. The total cross section of $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ is approximately $4.2 ~ \mu$b around $ E_{\gamma}=2.4 \ \rm GeV $. We encourage our experimental colleagues to further measure the $ \gamma n \to K^{+}\Sigma^{*-}_{1/2^-} $ process.

In this appendix, we show that the contributions from the t-channel K exchange in the forward angle ($ \theta=0 $) and backward angle ($ \theta=\pi $) are zero. In the c.m frame, the four-momenta of the outgoing K are

where $ \vec{k}_2^{\rm c.m.} $ is the three-momentum of K [Eq. (21)]. The polarization vectors of the photon with momentum $ \vec{k}_1^{\rm c.m.} $ in the helicity basis are

We can easily find that $ k_2^\mu \epsilon_\mu(\vec{k}_1^{\rm c.m.},\lambda=0,\pm 1)=0 $ for $\theta= 0,\pi$, which implies that the amplitude of Eq. (11) for t-channel K exchange will be zero in the forward angle ($ \theta=0 $) and backward angle ($ \theta=\pi $).