In a previous publication, it is pointed out that in the L_2I,2J = S₁₁ channel partial wave πN scattering amplitude, there exist two virtual poles. [1] One lies below but close to ${c}_{L}=\tfrac{{\left({m}_{N}^{2}-{m}_{\pi }^{2}\right)}^{2}}{{m}_{N}^{2}}$, and another lies above but close to ${c}_{R}\,={m}_{N}^{2}\,+2{m}_{\pi }^{2}$. Here [c_L, c_R] defines the cut of partial wave amplitude caused by the u channel nucleon exchange. [2] Meanwhile, it is found that the dispersion representation for the phase shift has to be modified, and the net effect of the new contributions from virtual poles and the additional cut is vanishingly small, leaving previous analyses on the S₁₁ partial wave unchanged. In this note, we will extend our previous analysis to all partial waves L_2I,2J. We find that the existence of virtual poles are quite universal, but different channels behave rather differently.

The on-shell T-matrix elements for elastic πN scatterings depend on three Mandelstam variables:obeying the constraintwith m_N and m_π the physical masses of the nucleon and the pion, respectively. In the isospin limit, the isospin amplitude can be decomposed as [3]where τ^a is Pauli matrices, and χ_i (χ_f) corresponds to the isospin wave function of the initial (final) nucleon state. The amplitudes with isospins I = 1/2, 3/2 can be written asAs for the Lorentz structure, for an isospin index I = 1/2, 3/2,where the superscripts (${s}^{{\prime} }$), (s) denote the spins of the Dirac spinors, and q(p) and ${q}^{{\prime} }({p}^{{\prime} })$ are the 4-momenta of initial and final pions (nucleons), respectively. Functions A and B are scalar functions of s and t. Helicity amplitudes can be obtained by substituting the nucleon spinors by helicity eigenstates in the centre of mass frameThe first and second subscripts of T refer to the helicity of the initial and final nucleon, respectively, and ‘ + , − ’ are shorthands for helicity + 1/2 and − 1/2, respectively; ${z}_{s}=\cos \theta $ with θ the scattering angle. Partial wave amplitudes for total angular momentum J can be written aswhere d^J is the standard Wigner d-function. Substituting equation (6) into equation (7), then using the relation between the Wigner d-function and the Legendre polynomialone getswith where F stands for A or B.

The parity eigenstates can be constructed by the linear combinations of helicity amplitudeswhich satisfy the optical theorem for partial waves,The kinematic factor ρ(s) is defined aswith ${s}_{R}={\left({m}_{N}+{m}_{\pi }\right)}^{2}$ and ${s}_{L}={\left({m}_{N}-{m}_{\pi }\right)}^{2}$. In fact, amplitudes ${T}_{\pm }^{I,J}$ stand for scattering states with orbital angular momentum l = J ∓ 1/2 and parity $P={\left(-1\right)}^{l-1}$, respectively. The partial wave S-matrix is defined bywhich satisfies S^†S = 1. The T-matrix in the second Riemann sheet is written as

The scalar function A^I and B^I may be calculated by chiral perturbation theory at low energies.⁴(In the region [s_L, s_R] chiral expansions are expected to work well.) Here, we present the results at the ${ \mathcal O }({p}^{1})$ level,where F and g denote the pion decay constant and the axial vector coupling constant, respectively. Here we however only need to concern the $1/(u-{m}_{N}^{2})$ term, which coefficient is immune to any chiral corrections, and its sign determines the existence of the zeros (poles) of the S-matrix.

Substituting equation (17) into equation (9) and using the Neumann equationwhere Q_l is the Legendre function of the second type, one can getin which the neglected terms ⋯ denote the parts regular at c_L and c_R (see discussions below), which receive chiral corrections. The definition of y(s) isParticularly, y(c_R) = 1, y(c_L) = −1. Using the explicit expressions of Q_l(s),withone finds that the function ${B}_{C,S}^{I,J}(s)$ contains the termThis term leads to the existence of three branch points at s = 0, c_L, c_R in function ${B}_{C,S}^{I,J}$, and then in parity eigenstate amplitudes ${T}_{\pm }^{I,J}$. Further, when s tends to c_R, it is easy to find thatThe sign of the infinity is independent of the angular momentum J. Things will be different, however, when s tends to c_L. It turns out that ${T}_{\pm }^{1/2,J}$ tends to $\mp {\left(-1\right)}^{J+1/2}\infty $, and ${T}_{\pm }^{3/2,J}$ tends to $\pm {\left(-1\right)}^{J+1/2}\infty $. Taking the definition of S-matrix equation (15), and the fact that 2iρ(s) is negative at s = c_R, c_L, conclusions can be drawn that:⁵(The emergence of S matrix zeros due to kinematical reasons is known, and χPT amplitudes can be consistently embedded into a unitary S matrix element. See for example the discussion in [4].) Taking a few examples, in S₃₁ channel there is no virtual pole, while in p waves,A modification of the dispersion representation for function f defined in [1] in each channel may be needed accordingly, similar to what has been discussed in S₁₁ channel [1]. Conclusions are similar in all channels: the contribution from the virtual pole to the phase shift is always exactly canceled by the additional contribution from the dispersion integral in f, when v_L = c_L (v_R = c_R). Taking for example the case that there exists only one virtual pole v_L on the left. In this situation,where ${S}^{{v}_{L}}$ represents the ‘S matrix’ of the newly found virtual pole v_L, while S_i depicts all other known poles. ⁷(For more details about the production representation, one can refer to [6–8].) Such an S^cut is real and negative when s ∈ (s_L, c_L),⁸(S^phy(s_L) = +1 according to equation (15). This is because s_L is the branch point caused by the u channel continuous absorptive singularities, T(s_L) itself is regular. For resonances Sⁱ = +1, for bound state or virtual state Sⁱ = −1, when s = s_L. Particularly S^v_L is negative when s ∈ (s_L, v_L) and positive when s ∈ (v_L, c_L).) so the ‘spectral function’ $f=\mathrm{ln}{S}^{\mathrm{cut}}/2{\rm{i}}\rho $ maintains a cut ∈ (s_L, c_L) and its contribution $\tfrac{s}{\pi }{\int }_{{s}_{L}}^{{c}_{L}}\tfrac{\mathrm{Im}[\mathrm{ln}{S}^{\mathrm{cut}}(s^{\prime} )/2{\rm{i}}\rho (s^{\prime} )]}{s^{\prime} (s^{\prime} -s)}{\rm{d}}s^{\prime} $ exactly cancels the contribution from the virtual pole when v_L = c_L. In another situation when there exists a virtual pole on the right, i.e. v_R ∈ (c_R, s_R), the ‘spectral function’ f is defined the same as in [1]:⁹(Except P₁₁ channel, where there exists a nucleon bound state, the contribution of the virtual state remains.) where the function $\bar{\rho }(s)$ is the ‘deformed’ ρ(s) with its cut ∈ [s_L, s_R], while the cut of the latter is defined on ( −∞ , s_L] ∪ [s_R, + ∞ ). Further, function $\bar{f}(s)$ is identical to f(s) when s lies in the physical region. Now −S^cut > 0 when s ∈ (s_L, c_L) and − S^cut < 0 when s ∈ (c_R, s_R). So $\mathrm{ln}-{S}^{\mathrm{cut}}/2{\rm{i}}\rho (s)$ contains a cut (c_R, s_R), on which its imaginary part is canceled by that from the second term on the r. h. s. of equation (26). Hence $\bar{f}$ contains an additional cut (s_L, c_R) coming from $-\pi /2\bar{\rho }(s)$, and its contribution on (s_L, c_R) is exactly canceled by that of the virtual pole when v_R = c_R. As a consequence all calculations made in [9] remain correct with very high accuracy.

The following discussions are dedicated to determining the location of those zeros. Let us focus on I = 1/2 for the moment. The explicit expressions of S-matrix elements readwithwhere ⋯ refers to possible but irrelevant regular terms, andwhere $W\equiv \sqrt{s}$ denotes the center-of-mass frame energy and ${E}_{N}=(s+{m}_{N}^{2}-{m}_{\pi }^{2})/2\sqrt{s}$ is the nucleon energy. It is worth stressing that the function ${{ \mathcal B }}_{\pm }^{1/2,J}(s)$ is immune of χPT corrections.

Let ${v}_{R\pm }^{J}\in ({c}_{R},{s}_{R})$ denotes the position of the zero of ${S}_{\pm }^{1/2,J}(s)$, which givesSuppose ${v}_{R\pm }^{J}$ approaches c_R for sufficiently large J, as will be verified a short while later, the solution of equation (30) may be obtained by iteration methodUsing equation (29) and the fact that P_l[y(c_R)] = P_l(1) = 1, one finds that ${{ \mathcal B }}_{\pm }^{1/2,J}({c}_{R})$ is negative and independent of J. Further, when J tends to ∞ , ${{ \mathcal A }}_{\pm }^{1/2,J}({c}_{R})$ goes to infinity. To prove this, notice that the function O_l−1 defined in equation (22) behaves as (taking l = 2k + 1 for example)Hence, according to equation (28), as J → ∞ , the leading contribution of ${{ \mathcal A }}_{\pm }^{1/2,J}({c}_{R})$ is¹⁰(The contribution from chiral corrections will not diverge, since the contribution itself is regular when s ∈ (s_L, s_R), and notice that ∣P_l(x)∣ < 1 when x ∈ [ −1, +1] for arbitrary l—which is proved using the Laplace formula of Legendre function.) From above, one concludes that ${v}_{R\pm }^{J}$ gets closer and closer to c_R as J increases, according to equation (31). Hence, c_R is the accumulation point of poles on the second sheet. What we have discussed is actually an example of a general mathematical theorem: as the sequence is bounded, it must have an accumulation, i.e. it has one convergent subsequence. Similarly, analyses can be made on the situation in the line segment (s_L, c_L). One finds that both ${v}_{L+}^{1/2,J}$ and ${v}_{L-}^{1/2,J}$ approaches c_L when J → ∞ . Therefore, following the steps of [10], we prove that s = c_L, c_R are two accumulations of singularities of T^1/2(s, t), on the second sheet of the complex s-plane. However, the proof in [10] on the statement that s = 0 is an accumulation of singularities on the second sheet of s plane for ππ scattering amplitudes is fully non-perturbative. Our proof given here is valid for all orders of perturbation chiral expansions.

The analyses can also be applied to the isospin 3/2 case, and only the final results are presented here:One further proves that the accumulation of zeros at c_L occurs and that s = c_L is an accumulation of singularities of T^3/2(s, t) on the second sheet of the s plane.