A production representation of partial-wave S matrix is utilized to construct low-energy elastic pion-nucleon scattering amplitudes from cuts and poles on complex Riemann sheets. Among them, the contribution of left-hand cuts is estimated using the O(p 3 ) results obtained in covariant baryon chiral perturbation theory within the extended-on-nass-shell scheme. By fitting to data on partial-wave phase shifts, it is indicated that the existences of hidden poles in S 11 and P 11 channels, as conjectured in our previous paper [Eur. Phys. J. C. 78 (7): 543 (2018)], are firmly established. Specifically, the pole mass of the S 11 hidden resonance is determined to be (895 ± 81) − (164 ± 23)i MeV, whereas, the virtual pole in the P 11 channel locates at (966 ± 18) MeV. It is found that analyses at the O(p 3 ) level improves significantly the fit quality, comparing with the previous O(p 2 ) one. Quantitative studies with cautious physical discussions are also conducted for the other Sand P -wave channels.

The S-and P-wave phase shifts of low-energy pion-nucleon scatterings are analysed using Peking University representation, in which they are decomposed into various terms contributing either from poles or branch cuts. We estimate the left-hand cut contributions with the help of treelevel perturbative amplitudes derived in relativistic baryon chiral perturbation theory up to O( p 2 ). It is found that in S 11 and P 11 channels, contributions from known resonances and cuts are far from enough to saturate experimental phase shift data -strongly indicating contributions from low lying poles undiscovered before, and we fully explore possible physics behind. On the other side, no serious disagreements are observed in the other channels.

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A careful re-analysis is made on e + e − → X(4660) → (Λ cΛc )/(ψ ππ) processes, aiming at resolving the apparent conflicts between Belle and BESIII data above Λ cΛc threshold. We use a model containing a Breit-Wigner resonance and Λ cΛc four-point contact interactions, with which the enhancement right above Λ cΛc threshold is well explained by a virtual pole generated by Λ cΛc attractive final state interaction, located at M V = 4.566 ± 0.007 GeV. Meanwhile, X(4660) remains to be a typical Breit-Wigner resonance, and is hence of confinement nature. Our analysis strongly suggests the existence of the virtual pole with statistical significance of 4.2 standard deviation (σ). Nevertheless, the conclusion crucially depends on the line-shape of cross sections which are of limited statistics, hence we urge new experimental analyses from Belle II, BESIII, and LHCb to settle down the issue.

We give a pedagogical analysis on K-matrix models describing the πN scattering amplitude, in S11 channel at low energies. We show how the correct use of analyticity in the s-channel and crossing symmetry in t- and u-channels leads to a much improved analytic behavior in the negative s region, in agreement with the prediction from chiral perturbation amplitudes in its validity region. The analysis leads again to the conclusion that a genuine N*(890) resonance exists.