Decay properties of $N^*(1895)$

The nature of nucleon resonances is still being debated, while much experimental data are accumulated. In this work, we focus on the negative parity resonance $N^*(1895)$ which is located in the scattering region of various meson-baryon coupled channels, and such dynamics can be crucial in understanding its properties. To test the relevance of such hadron dynamics, we investigate the decay properties of $N^*(1895)$ in detail. We examine how a two pole nature of $N^*(1895)$ is compatible with its observed decay properties. Moreover, we find that the resonance decays into final states involving $\Lambda(1405)$ and $\Sigma(1400)$, where the latter is not yet observed experimentally. Such decay processes can be useful to study the properties of the aforementioned hyperon resonances.


I. INTRODUCTION
The objective of the present work is to obtain the partial decay widths of N * (1895) to light hyperon resonances, which can be useful in unraveling its nature. The state N * (1895) is particularly special as it is the highest mass nucleon known with J π = 1/2 − and the particle data group (PDG) [1] lists all 1/2 − structures found above 1800 MeV together, under the label of N * (1895). Due to this latter fact, it is unclear if one or more states correspond to N * (1895). Indeed, in a previous work [2], we found two poles with overlapping widths associated with N * (1895). The pseudoscalar/vector meson-baryon coupled channel amplitudes obtained in this former work reproduce, for example, the isospin 1/2 and 3/2 πN amplitudes extracted from partial wave analysis [3] of the experimental data and the π − p → ηn and π − p → K 0 Λ cross sections up to a total energy of about 2 GeV.
Having the information on the poles related to N * (1895) as obtained in Ref. [2] using constrains from experimental data, a detailed analysis of its decay properties is important to further reveal its nature. For instance, N * (1895) cannot be described within the naïve quark model [4][5][6][7]. An S 11 resonance, within quark models based on the harmonic oscillator potential, after N * (1535) and N * (1650), is expected to appear with mass > 2100 MeV [6,7]. Hence, coupled channel hadron interactions are expected to play an important role in describing the properties of N * (1895).
In fact, the exchange of N * resonances with masses ≥ 2000 MeV was found to be significant to describe the cross sections of the photoproduction of Λ(1405) near the threshold in Ref. [13]. Given the fact that N * (1895) lies close to the KΛ(1405) and KΣ(1400) thresholds, it should be important to study the contribution of N * (1895) to the photoproduction of Λ(1405) and Σ(1400). The information obtained in this work can also be useful to analyze the process πN → K * πΣ, which is intended to be studied at J-PARC [14].
Having stated the motivation of our work, we would like to dedicate a brief discussion on Σ(1400). There exist evidences for the existence of an isovector resonance with J π = 1/2 − and mass ∼ 1400 MeV, though with less agreement on its properties as obtained from different works [15][16][17][18][19][20][21][22][23]. To bring a consensus on the issue, in a recent work [24], we studied coupled channel meson-baryon scattering for systems with strangeness −1 by determining the unknown parameters of the model using experimental data on the total cross sections of K − p → K − p,K 0 n, ηΛ, π 0 Λ, π 0 Σ 0 , π ± Σ ∓ and the data on the energy level shift and width of the 1s state of the kaonic hydrogen. The work lead to finding an evidence for Σ(1400), besides Λ(1405) and some other higher mass hyperons. In this former work, the coupled channels considered included both pseudoscalar and vector mesons. An advantage of such a treatment is that it allows us to obtain the couplings of the pseudoscalar/vector-baryon channels taken into account to the resonances found in the complex energy plane. In the present work we use the couplings determined in Ref. [24] to study N * (1895) → KΛ (1405) and N * (1895) → KΣ(1400).
In the following section we discuss the formalism of the work where we show that the calculation of the partial widths for N * (1895) → KΛ(1405) and N * (1895) → KΣ(1400) is done by considering different triangle loops involving several meson-baryon channels. In the subsequent section we present and discuss the results obtained which, we hope, are useful for experimental investigations of N * (1895) as well as for the study of the photoproduction of Λ(1405) and Σ(1400).

II. FORMALISM
The main purpose of the present work is to study the decay widths of N * (1895) to different meson-baryon channels and final states involving unstable hyperons, in particular, Λ(1405) and Σ(1400). We take this opportunity to present the results on the branching ratios for N * (1895) decaying to different pseudoscalar/vector-baryon channels and compare them with the available experimental values listed by the PDG. To study these decay processes, we rely on our previous works on the nonstrange [2] and on the strangeness −1 [24] meson-baryon coupled systems, where N * (1895), Λ(1405) and Σ(1400) appear as poles in the complex energy plane of the corresponding amplitudes.
Data on the energy level shift and width of the 1s state of the kaonic hydrogen were also considered in Ref. [24]. As a result, two sets of fits of similar quality were found, denoted as "Fit I" and "Fit II" in Ref. [24]. In case of Fit I, two close lying poles appeared around 1400 MeV in the isovector amplitudes, while in Fit II one pole was found with isospin 1 around 1400 MeV. The state related to these poles was represented as Σ(1400). Thus, both fits implied the presence of Σ(1400), one indicating a possible double pole nature of the state while the other relating a single pole to it. However, only one of the two poles of Fit I was found to be stable under changes in the lowest order amplitudes used in the model, such as the consideration (or not) of the contributions originating from the u-channel interaction (see Ref. [24] for more details). This latter pole is very similar to the single pole found in Fit II. In the present work, we, thus, use the pole position found in Fit II of Ref. [24] for describing the properties of Σ(1400). In the two sets of fits obtained in Ref. [24], a double pole associated with Λ(1405) was found, in agreement with the analysis [23,25,26] of the data on the electroproduction and photoproduction of Λ(1405). Since the quality of Fit I and II of Ref. [24] was similar, and, as mentioned above, we are going to use the results of Fit II for Σ(1400), for consistency, we use the results of the same fit for describing the properties of Λ(1405). For convenience of the reader, the aforementioned pole positions of Λ(1405) and Σ(1400) are given in Table I of the present manuscript.  [2,24]. Notice that two poles are associated with N * (1895) and Λ(1405).

State
Pole position (MeV) The findings of Refs. [2,24] allowed us to consider that the transition amplitudes among the different meson-baryon channels in the vicinity of a pole can be expressed in terms of a scattering matrix T ij as where z R corresponds to the pole position associated with the resonance in the complex plane and g i g j is the product of the couplings of the resonance to channels i and j, and can be determined by calculating the residue of T ij . In Refs. [2,24], we obtained the couplings of N * (1895), Λ(1405) and Σ(1400) to the different related coupled channels. Using these couplings, the partial decay widths of N * (1895) to different pseudoscalar/baryon channels can be calculated in a straightforward way. The calculation of the amplitudes, and, consequently, the decay widths, for the processes N * + (1895) → K + Λ(1405) and N * + (1895) → K + Σ 0 (1400) is more complex, as we discuss in the following section. Based on the properties found in Refs. [2,24] for N * (1895), Λ(1405) and Σ(1400), the decay processes N * (1895) → KΛ(1405), KΣ(1400) proceed through the diagrams shown in Fig. 1.
To obtain the amplitudes for the diagrams in Fig. 1, we use the following Lagrangians for the vertices involving mesons [27,28]: where the couplings are related to the pion decay constant and the vector meson mass as and the matrices for the mesons are For the vertices involving baryons, we set effective Lagrangians which are compatible with the conventions followed in Refs. [2,24] such that we can use the couplings of the resonances to meson-baryon channels obtained in these former works, The field H * in Eqs. (4) represents Σ(1400) or Λ(1405), and the couplings g P BN * , g V BN * , g P BH * , g V BH * are taken from Refs. [2,24]. The factor √ 3 in the Lagrangians for the vertices involving a vector meson is due to the fact that the Breit-Wigner amplitudes in Refs. [2,24], for spin 1/2 of the V B system, are written in terms of the g V BB * couplings as Note that Eq. (4) leads to a spin dependent VB → VB amplitude such that, when projected on spin 1/2, it becomes in agreement with Eq. (5).
Having discussed the Lagrangians for the different vertices necessary to describe the decay of N * (1895) to K + H * 0 , we can now start calculating the amplitudes for the different diagrams shown in Fig. 1. We begin by writing the amplitude for the diagram in Fig. 1 where we have followed the four momentum attribution shown in Fig. 2. The summation over the index j, in Eq. (8), refers to considering different three hadron channels in the triangle loop which can contribute to the diagram in Fig. 2a. The list of such three-hadrons channels Four-momentum labels for the particles involved in the N * → K + Σ * process.
is given in Table IV Table IV in the Appendix A for each three-hadron loop present in the diagram of Fig. 1a.
The product of the spinors, gamma matrices and the numerator of the expression within the curly brackets in Eq. (8) can be worked out as with M H * denoting the mass of H * . The integration on dq 0 in Eq. (8) can be done analytically by using Cauchy's theorem. It is then convenient to rewrite Eq. (9) showing its explicit dependence on q 0 . By doing so Eq. (8) becomes where χ † , χ correspond to the two-component spinors of H * and N * , respectively. The factors N H * , N N * in Eq. (10) are related to the normalization of the Dirac spinors for H * and N * where, although, N N * is unity in the centre of mass frame we still keep it in the equations for completeness. The definitions of A i,j 's are as given below. The subscript i on A i,j refers to the power of q 0 multiplied to A i,j and the index j indicates the three-hadron channel in the loop. Defining the four-momenta in the centre of mass frame as: and The integration on the q 0 variable can be done analytically, to obtain an expression like A cut-off Λ 600 − 700 MeV is used in the integration on the three-momentum to be consistent with the work in Refs. [2,24]. The variation of Λ in this range allows us to estimate the uncertainties of our results. The analytical expressions for N i,j and D j are given in the Appendix B.
To proceed further, we recall that the decay N * → K + H * occurs in p-wave and we, thus, need to write the final state projected on the partial wave l=1. Following Ref. [29], we write a state of two particles with spins S 1 , S 2 , with the centre of mass momentum k, projected on a partial wave l as where S, J and M, µ represent the total spin, total angular momentum and their zcomponents, respectively. Using Eq. (19) and denoting the spins of H * and N * as S H * and S N * and their third components as m H * and m N * , we can write the amplitude for diagram in Fig. 1a, for m N * = 1/2 (the amplitude for m N * = −1/2 can be obtained analogously) as which, from Eq. (17), can be explicitly written as Note that the dependence on the spin projections of H * and N * appearing in Eq. (20) is shown as subscripts for the spinors χ † and χ in Eq. (21).
We can now write the amplitudes for the diagram in Fig. 1b t and for the diagram in Fig. 1c t where the constants D j and F j come from the trace in Eq.  Tables V and VI, respectively, of the Appendix A. As in the case of the amplitude t a , we can write the amplitudes t b and t c as a polynomial of q 0 and integrate on the q 0 variable to be able to write where N i,j and D j are as given in Eqs. (B2)-(B7) of the Appendix B. The expressions for B i,j and C i,j can also be found in the Appendix B.
Having the amplitudes t b and t c , we need to project them on p-wave, as done for t a [see Eq. (20)] and the final amplitude for the transition N * → K + Σ * is the coherent sum of the amplitudes for the three diagrams in Fig. 1 t

III. RESULTS AND DISCUSSIONS
Having obtained the amplitudes for the diagrams shown in Fig. 1 for the processes N * + → K + Σ * 0 and N * + → K + Λ * , we calculate the corresponding partial decay widths as where H * denotes the hyperon resonance, Σ * or Λ * .
For the sake of clarity in the presentations of the results, we represent the two poles of N * (1895) found in Ref. [2] as N * 1 (1895) (for the lower pole at 1801 − i96 MeV) and N * 2 (1895) (for the higher pole at 1912 − i54 MeV). Similarly, we shall refer to the lower and upper mass poles of Λ(1405) (see Table I) as Λ 1 (1405) and Λ 2 (1405), respectively.
Before discussing the results, it is important to mention that although the central mass value of N * 1 (1895) is below the H * -kaon threshold(s), the decay width N * 1 → K + H * is finite, due to the width of N * 1 (1895) (see Table I), which can be taken into account through the convolution of the width [given by Eq. (27)] over the varying mass of N * as In Eq. (28), Γ N * →KH * (m) is calculated using Eq. (27), with the mass of N * varying in the range ± 2Γ N * , and Decay process Partial width (MeV) is a normalization factor. As a result we obtain the widths which are summarised in Table. II. The uncertainty in the results is determined by allowing the cut-off, Λ, on the three-momentum integration to vary in the range 600 − 700 MeV. We refer the reader to Eq. (17) to look for the dependence on Λ in the formalism. We would like to add here that the H * 's also have finite decay widths, which we considered analogously to the way we take into account the width of N * . We find that the widths of H * 's do not practically change the results in Table II.
Next, in Table III we provide the branching ratios for each of the two poles of N * (1895) to different PB and VB channels in the isospin base and compare them with the experimental values, whenever possible. We calculate the N * → PB, VB decay widths as and convolute over the width of N * by using Eq. (35) in Eq. (28). As can be seen, we obtain compatible results. Notice that the last column of Table III is a compilation of findings from the PDG [1], which shows that the partial widths to the different pseudoscalar-and vector-baryon channels are of the same order in spite of the larger phase space available in the former case. Such findings from experimental data cannot be easily described within the quark model. In fact, the couplings obtained in Ref. [2] show that N * (1895) couples more strongly to the vector-baryon channels, which clearly indicates that the hadron dynamics plays an important role in describing the properties of N * (1895).
To finalize the discussions on the decay widths, it is important to consider another possible source of uncertainty present in the model which is the relative phases in the Lagrangians.
The relative phases among the Lagrangians in Eq. (4) are set as in Refs. [2,24] where the couplings of the N * /H * to the PB/VB channels were determined. However, there may exist an ambiguity in the relative phase among the Lagrangians used for the meson vertices [Eqs. (2) and (3)]. It is then important to discuss the sensitivity of our results on the ambiguity in the relative phase of the PPV and VVP Lagrangians. In case of the N * (1895) decay to KΛ(1405), we find that the amplitude for the diagram in Fig. 1b gives the dominant contribution such that the results are basically insensitive to the relative phase among the PPV and VVP vertices. For the N * (1895) decay to KΣ(1400) the contribution of Fig. 1c is such that there exists a large cancellation between the amplitudes of N * 1 (1895) → KΣ(1400) and N * 2 (1895) → KΣ(1400). As a consequence the decay width of the superposed N * (1895) to KΣ(1400) depends weakly on the relative phase of the PPV and VVP vertices. For example, if we consider g V V P → −g V V P in Eq. (3) and fix the cut-off Λ = 700 MeV to regularize the triangular loops, we obtain the following decay widths which should be compared with Eqs. (31)  cross sections of such a process requires the energy dependent N * + (1895) → K + H * 0 vertex.
Having this in mind, we show in Fig. 5 the real (solid lines) and imaginary parts (dashed lines) of the amplitudes for the processes N * + 1,2 → K + Λ 1,2 and K + Σ 0 (1400) in the energy region of interest.

IV. SUMMARY
In this work we have studied the decay process of N * (1895) to channels involving light hyperon resonances, which are KΛ(1405) and KΣ(1400). We also provide the information on the decays of N * (1895) to various pseudoscalar-and vector-baryon channels. The formalism should be relevant to describe a process, like, γp → KΛ(1405), on which data already exists [30,31]. The results obtained in our work can also be useful in the analyses of other processes producing light hyperons through the exchange of N * (1895) in the intermediate state, for example, πN → K * πΣ, which is intended to be studied at JPARC [14].  Fig. 1a and the value of the C j factors in Eq. (8), together with the couplings g H * V B and g N * P B in the charge base in terms of those given in Refs. [2,24] in the isospin base.  Fig. 1b and the value of the D j factors in Eq. (22), together with the couplings g H * P B and g N * V B in the charge base in terms of those given in Refs. [2,24] in the isospin base.  Fig. 1c and the value of the F j factors in Eq. (23), together with the couplings g H * V B and g N * V B in the charge base in terms of those given in Refs. [2,24] in the isospin base.
g Λ * ρΣ 2 3 g N * K * Σ K * + (892)Σ 0 ω 1 √ 2 g Σ * ωΣ 1 3 g N * K * Σ K * + (892)Λω 1 √ 2 g Λ * ωΛ g N * K * Λ K * + (892)Σ 0 φ 1 g Σ * φΣ 1 3 g N * K * Σ K * + (892)Λφ 1 g Λ * φΛ g N * K * Λ K * 0 (892)Σ + ρ + 1 Appendix B: Expressions for N i,j , D j , B i,j and C i,j As mentioned in section II, the amplitudes for the different diagrams in Fig. 1, as given by Eqs. (17), (24) and (25) are proportional to N i,j ( q )/D j ( q ). The index i indicates that N i,j is the numerator resulting from the q 0 integration on terms proportional to (q 0 ) i . The index j signifies that N i,j ( q )/D j ( q ) is the result of the q 0 integration for the jth channel in the loop. To facilitate writing the expressions of N i,j and D j , we label the energies (masses) of the particles in the triangle loop with four-momentum k − q, q and P − k + q as, E 1j (m 1j ), E 2j (m 2j ) and E Bj (m Bj ), respectively, such that, in the center of mass frame: Using the above definitions, we can write the numerators N i,j ( q) as Finally, the terms C i,j , in Eq. (25), are