Effects of strange molecular partners of $P_{c}$ states in $\gamma p\to K\Sigma$ reactions

The several $P_{c}$ states observed by the LHCb experiment in 2015 and later Aaij et al. (2015, 2019) are the most convincing multiquark candidates, prompting significant interest in investigating their nature Chen et al. (2016); Guo et al. (2018); Liu et al. (2019a). In the hadronic molecular picture, the $P_{c}(4312)$ can be interpreted as a narrow $\overline{D}\Sigma_{c}$ bound state with spin-parity $J^{P}=1/2^{-}$, while the $P_{c}(4440)$ and $P_{c}(4457)$ can be interpreted as two degenerate narrow $\overline{D}^{*}\Sigma_{c}$ bound states with $J^{P}=1/2^{-}\&\ 3/2^{-}$, respectively He and Chen (2019); Chen et al. (2019); Liu et al. (2021). Moreover, a $\overline{D}\Sigma_{c}^{*}$ bound state with $J^{P}=3/2^{-}$ referred to as $P_{c}(4380)$, which is different from the broad one reported by LHCb in 2015, and three $\overline{D}^{*}\Sigma_{c}^{*}$ bound states with $J^{P}=1/2^{-},3/2^{-}\&\ 5/2^{-}$ are also expected to exist, based on heavy quark spin symmetry Du et al. (2020, 2021); Liu et al. (2019b); Yalikun et al. (2021). The successful interpretation of these hidden-charm $P_{c}$ states as the hadronic molecules inspires us to investigate their strange partners.

In the strange sector, S-wave $K\Sigma^{*}$ molecule $N(1875)3/2^{-}$, S-wave $K^{*}\Sigma$ molecules $N(2080)1/2^{-}\&\ 3/2^{-}$ and S-wave $K^{*}\Sigma^{*}$ molecules $N(2270)1/2^{-},3/2^{-}\&\ 5/2^{-}$ are proposed as the strange partners of $P_{c}$ molecular states He (2017); Zou and Dai (2018); Lin et al. (2018); Ben et al. (2023); Wu et al. (2023); Ben and Wu (2024). In Refs. Lin et al. (2018); Ben and Wu (2024), their decay patterns have been calculated using an effective Lagrangian approach. Notably, in the most recent Particle Data Group(PDG) review Navas et al. (2024), the two-star $N(2080)$ listed before the 2012 review has been split into two three-star states: $N(1875)$ and $N(2120)$. For consistency with our previous work, we retain the designation $N(2080)3/2^{-}$ for the possible $K^{*}\Sigma$ molecule, which is not necessarily identified with the $N(2120)$ resonance in the PDG review. Furthermore, the contentious state $N(1535)1/2^{-}$ can also be interpreted as a bound state of $K\Lambda,K\Sigma$ within the molecular picture Zou and Dai (2018); Kaiser et al. (1995); Bruns et al. (2011); Li et al. (2024); Molina et al. (2024).

We have conducted several studies to investigate the effects of these hidden-strange molecules in photoproduction reactions. In Refs. Ben et al. (2023); Wu et al. (2023), the $N(2080)3/2^{-}$ and $N(2270)3/2^{-}$ are introduced in s-channel as the primary contributors to the $\gamma p\to K^{*+}\Sigma^{0}/K^{*0}\Sigma^{+}$ and $\gamma p\to\phi p$ reactions. The theoretical models constructed based on this fit well with the available experimental data for these reactions. Following this, we observe that the differential cross-section data for $\gamma p\to K^{+}\Sigma^{0}$ from CLAS 2010 Dey et al. (2010) exhibits bump structures near the center-of-mass energies $W$ = 1875, 2080 and 2270 MeV, as shown in Fig. 3, corresponding to the Breit-Wigner masses of $N(1875)3/2^{-}$, $N(2080)1/2^{-}\&\ 3/2^{-}$ and $N(2270)1/2^{-},3/2^{-}\&\ 5/2^{-}$. Additionally, the $K\Sigma$ channel is essential in the molecular picture of $N(1535)1/2^{-}$ Molina et al. (2024). These prompt us to focus on the $\gamma p\to K^{+}\Sigma^{0}$ and $\gamma p\to K^{0}\Sigma^{+}$ reactions to test the effects of these seven hidden-strange molecules mentioned above.

The $K\Sigma$ photoproduction reactions have garnered significant attention both experimentally and theoretically over the past few years, contributing to the study of the light baryon resonance spectrum. In the experimental aspect, various collaborations such as CLAS, SAPHIR, LEPS have contributed large and diverse sets of experimental data on both cross-sections and polarization observables for the reaction $\gamma p\to K^{+}\Sigma^{0}$ Glander et al. (2004); Kohri et al. (2006); Bradford et al. (2006); Dey et al. (2010); Schmieden (2014); Shiu et al. (2018); Jude et al. (2021); Lleres et al. (2007); Paterson et al. (2016); Shiu et al. (2018); Bradford et al. (2007). With the exception of some older measurements, these data generally show no significant discrepancies. For the reaction $\gamma p\to K^{0}\Sigma^{+}$, several collaborations such as SAPHIR, CBELSA, A2 have also provided the experimental data Lawall et al. (2005); Castelijns et al. (2008); Aguar-Bartolome et al. (2013); Ewald et al. (2012); Nepali et al. (2013); Clark et al. (2024), including the latest data of polarization observables from the CLAS Collaboration Clark et al. (2024). However, in comparison to $\gamma p\to K^{+}\Sigma^{0}$, the amount of experimental data available for $\gamma p\to K^{0}\Sigma^{+}$ remains relatively sparse.

Many theoretical works have already been devoted to analyzing the data for $K\Sigma$ photoproduction, based on the effective Lagrangian approaches, isobar models, Regge-plus-resonance models, and so on Wei et al. (2022, 2023); Rönchen et al. (2022); Mart and Kholili (2019); Clymton and Mart (2021); Maxwell (2016); Sarantsev et al. (2005); Steininger and Meissner (1997); Mai et al. (2009); Kaiser et al. (1997); Borasoy et al. (2007); Golli and Širca (2016); Luthfiyah and Mart (2021); Egorov (2020); Lee et al. (2001); Tiator (2018); Corthals et al. (2007). In Refs. Mart and Kholili (2019); Clymton and Mart (2021) and Ref. Rönchen et al. (2022), photoproduction data for $K\Sigma$ have been simultaneously analyzed and effectively described using an isobar model and the Jülich-Bonn dynamical coupled-channel approach, respectively. And the work in Refs. Wei et al. (2022, 2023) provides a comprehensive analysis of the available data for $\gamma n\to K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$ reactions, based on an effective Lagrangian approach in the tree-level Born approximation.

In this work, we employ the methodology used in Refs. Wei et al. (2022, 2023) to simultaneously analyze data for $\gamma p\to K^{+}\Sigma^{0}$ and $\gamma p\to K^{0}\Sigma^{+}$ reactions. Our theoretical model incorporates contributions from $s$-channel exchanges of $N$, $\Delta$, $N^{*}$(including the hadronic molecules with hidden strangeness), and $\Delta^{*}$; $t$-channel exchanges of $K$, $K^{*}(892)$, and $K_{1}(1270)$; $u$-channel exchange of $\Sigma$; and the generalized contact term. We utilize this model to investigate the reaction mechanisms and test the effects of hidden-strange molecules in $\gamma p\to K\Sigma$ reactions.

The article is organized as follows. In Sec. II, we briefly introduce the framework of our theoretical model. Sec. III presents the details of our fitting settings. In Sec. IV, we show the results of our theoretical model along with some discussions. Finally, Sec. V provides the summary and conclusions.

As shown in Fig. 1, the gauge-invariant amplitude of $K\Sigma$ photoproduction reactions in the tree-level effective Lagrangian approach can be expressed as Wei et al. (2022, 2023)

where the terms $M_{s}$, $M_{t}$, $M_{u}$ and $M_{int}$ stand for the amplitudes calculated from the $s$-channel mechanism, $t$-channel mechanism, $u$-channel mechanism and the interaction current, respectively.

Fig. 1(a) presents the $s$-channel with exchanges of $N$, $\Delta$, $N^{*}$, and $\Delta^{*}$. The corresponding resonances are discussed in detail below. First, to investigate the effects of hidden-strange molecular states in $K\Sigma$ photoproduction, we introduce the seven molecules: $N(1535)1/2^{-}$, $N(1875)3/2^{-}$, $N(2080)1/2^{-}\&\ 3/2^{-}$ and $N(2270)1/2^{-},3/2^{-}\&\ 5/2^{-}$. Second, in Ref. Wei et al. (2023), contributions from the $N(1710)1/2^{+}$, $N(1880)1/2^{+}$, $N(1900)3/2^{+}$, $N(1895)1/2^{-}$, $N(2060)5/2^{-}$, $\Delta(1910)1/2^{+}$ and $\Delta(1920)3/2^{+}$ resonances have been taken into account to reproduce the available data for both $\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions. Apart from the $N(1710)1/2^{+}$ which is marked as ”seen” in its decay branching ratio to the $K\Sigma$ channel, all the other considered resonances have sizable branching ratios in PDG Navas et al. (2024). In this work, we disregard the $N(1895)1/2^{-}$ and $N(2060)5/2^{-}$, as their contributions are negligible when considering molecules, and we retain the other five resonances: $N(1710)1/2^{+}$, $N(1880)1/2^{+}$, $N(1900)3/2^{+}$, $\Delta(1910)1/2^{+}$ and $\Delta(1920)3/2^{+}$. Third, to achieve satisfactory numerical results, we refer to the analyses in Refs. Mart and Kholili (2019); Clymton and Mart (2021) and add seven additional resonances that may have significant contributions: $N(1675)5/2^{-}$, $N(1720)3/2^{+}$, $\Delta(1600)3/2^{+}$, $\Delta(1700)3/2^{-}$, $\Delta(1900)1/2^{-}$, $\Delta(1930)5/2^{-}$ and $\Delta(1940)3/2^{-}$. In summary, besides the ground states $N$ and $\Delta$, there are seven molecules, five general $N^{*}$ resonances and seven $\Delta^{*}$ resonances considered in $s$-channel of our theoretical model, which are listed in Table 2.

Fig. 1(b) illustrates the $t$-channel, which includes exchanges of $K$ and $K^{*}(892)$ as considered in Refs. Wei et al. (2022, 2023), along with the $K_{1}(1270)$, which may also contribute. Fig. 1(c) depicts the $u$-channel with only the exchange of the bound state $\Sigma$. As noted in Ref. Maxwell (2016), adding more resonances in the $u$-channel did not materially improve the result. Therefore, we neglect other baryon exchanges in the $u$-channel to reduce the number of fit parameters in our theoretical model, providing a cleaner background for testing the effects of molecules. Additionally, Fig. 1(d) presents the interaction current, which is modeled by a generalized contact current to ensure the gauge invariance of the full photoproduction amplitudes Wei et al. (2023).

Most parts of the formalism, including the Lagrangians, propagators, form factors attached to hadronic vertices, the gauge-invariance preserving term, and the interaction coupling constants, are detailed in Refs. Wei et al. (2022, 2023). For brevity, we do not repeat them here but present only the additional content relevant to the theoretical model in this work.

First, this work focuses on the $\gamma p\to K^{+}\Sigma^{0}$ and $\gamma p\to K^{0}\Sigma^{+}$ reactions, so the coupling constant $g_{\gamma K^{0}K^{*0}}=-0.631$ referred to Ref. Wang et al. (2018), and different isospin factors will be used in the specific calculations. Second, the Lagrangians Maxwell (2016) and the propagator used for $K_{1}(1270)$ are presented below:

Here, $M_{K}$ and $M_{N}$ denote the masses of $K$ and $N$. The $g_{\gamma KK_{1}}$, $g_{\Sigma NK_{1}}^{(1)}$ and $g_{\Sigma NK_{1}}^{(2)}$ are the electromagnetic and hadronic coupling constants treated as fit parameters. $M_{K_{1}}$ and $\Gamma_{K_{1}}$ are the mass and width for $K_{1}(1270)$ with four-momentum $p$.

Finally, we briefly explain the treatment of the molecules in the $s$-channel. Take $N(2080)3/2^{-}$ as an example, which is assumed to be a pure $S$-wave molecular state of $K^{*}$ and $\Sigma$. In principle, in the hadronic molecular picture, both the electromagnetic and hadronic couplings of it are dedicated by the loop diagrams illustrated in Fig. 2 Lin et al. (2018). Here for simplicity, we just follow Ref. Ben et al. (2023) to calculate the tree-level approximation by introducing the effective Lagrangians Wei et al. (2022) of $N^{*}$ with spin-parity $J^{P}=3/2^{-}$ for $N(2080)3/2^{-}$:

In addition, we attach a phase factor Exp[$i\phi_{R}$] in front of the tree-level amplitude, to partially mimic the loop contributions as illustrated in Fig. 2. Similarly, all the other hidden-strange molecules are treated in the same manner, and the Lagrangians introduced for them are referred to Ref. Wei et al. (2022), corresponding via the spin-parity. The masses of $N(1535)1/2^{-}$, $N(1875)3/2^{-}$, $N(2080)1/2^{-}\&\ 3/2^{-}$, $N(2270)1/2^{-},3/2^{-}\&\ 5/2^{-}$ are taken as 1535, 1875, 2080 and 2270 MeV, respectively. Furthermore, the width $\Gamma_{R}$ and coupling constants $g_{\gamma NR}^{(1)}$, $g_{\gamma NR}^{(2)}$ and $g_{K\Sigma R}$—which depend on the choice of cutoff parameters in Refs. Lin et al. (2018); Ben and Wu (2024)—along with the phase $\phi_{R}$ of molecules, are treated as fit parameters.

The fit parameters of this theoretical model are adjusted to match the experimental data in a $\chi^{2}$ minimization using MINUIT James and Roos (1975); Dembinski and et al. (2020). Below, we present our selected settings for the experimental data and fit parameters.