Remarks on pole trajectories for resonances

We discuss in general terms pole trajectories of resonances coupling to a continuum channel as some strength parameter is varied. It is demonstrated that, regardless the underlying dynamics, the trajectories of poles that couple to the continuum in a partial wave higher than s-wave are qualitatively the same, while in case of s-waves the pole trajectory can reveal important information on the internal structure of the resonance. In addition we show that only molecular (or extraordinary) states appear near thresholds naturally, while more compact structures need a significant fine tuning in the parameters. This study is of current relevance especially in strong interaction physics, since lattice QCD may be employed to deduce the pole trajectories for hadronic resonances as a function of the quark mass thus providing additional, new access to the structure of s-wave resonances.


Introduction
If all mesons wereqq states then there would be no natural reason for poles in scattering amplitudes to occur very close to thresholds. At large values of N c , the number of colors in QCD, allqq mesons become narrow with (nearly) unchanged mass [1]. Thus, their masses have no relation to the masses of the mesons to which they couple 1 . Accordingly, the ρ mass is not related to 2m π , nor is the K * mass related to m K + m π . On the contrary, there is good reason for "extraordinary" hadrons, often called hadronic molecules, to have masses close to thresholds [6]. So the mere fact that the f 0 (980) and a 0 (980) appear very near KK threshold is already a reason to be suspicious that they may not be simpleqq states. The same applies to unusual charmonium states that have been found near charm-anticharm meson thresholds like the famous X(3872) located very close to the D 0D0 * thresholdfor a recent review see [7].
In this paper we look carefully at the way that the manifestations of poles in scattering amplitudes change as the poles approach thresholds as some strength parameter is varied -here one may think of varying the quark masses in lattice QCD calculations. This has acquired a renewed interest after the trajectory of the σ or f 0 (500) resonance pole as a function of the quark mass was predicted by us within unitarized Chiral Perturbation Theory [8]. A similar trajectory as that of the σ was soon shown to be followed by the controversial κ or K(800) resonance in the isospin 1/2 scalar πK scattering partial wave, including the appearance of a virtual states at sufficiently large pion masses [9]. Recently the existence of such a virtual bound state in πK scattering at high pion masses has been confirmed by lattice calculations [10]. The subtleties of the extraction of resonance parameters from lattice QCD simulations performed at a finite volume are outlined in detail in Refs. [11,12] and will not be discussed further here.
While finishing this work, we became aware of a theoretical study [13] of the scaling of hadron masses near an s-wave threshold, showing that the bound state energy is not continuously connected to the real part of the resonance energy. In this paper we have another look at this issue which allows us to provide various additional, non-trivial insights. In particular, we demonstrate that there is a qualitative difference between the pole trajectories of resonances that couple to the relevant continuum channel is an s-wave or in a higher partial wave: As a consequence of analyticity a resonance is characterized by two poles on the second sheet, one located at s = s R and one located at s = s * R . For narrow resonances only one of them is close to the physical region. As some strength parameter is increased, the two poles start to approach each other. We will show on general grounds that while for higher partial waves the poles meet at the corresponding two meson threshold, for s-waves the poles can still be located inside the complex plane even for the real part of the pole position at or below threshold. As a consequence, s-wave-trajectories are controlled by an additional dimensionful parameter, namely the value of s where the two poles meet below threshold which may be related to the structure of the state. In other words, generic trajectories of s-wave resonances do lead to poles whose real part of the position is below threshold, but whose imaginary part of the position does not vanish, before giving rise to virtual bound states, and then bound states, as some strength parameter is varied. While this observation is in line with the findings of Refs. [8,14], it is in vast conflict with "common wisdom" that the imaginary part of a pole has to be identified with one half of its decaying width, for this implies that, if the "resonance mass" -identified with the real part of the pole position -lies below threshold, the pole necessarily has to lie on the real axis. Of course, such identification is a fair approximation for narrow resonances far from thresholds or other singularities, but very inappropriate in the cases we will show below.
The paper is organized as follows: in the next Section we discuss general properties of the poles that appear in the S-matrix, paying especial attention to poles that occur in partial waves with angular momenta higher than 0, and in particular to the role of the centrifugal barrier which is absent in the scalar partial waves. Next we consider the trajectories of resonance poles in the complex plane as a function of some strength parameter, and how they can become bound states. In the next section we briefly review Weinberg's compositeness criterion and reformulate it in terms of the parameters introduced in the previous section. The possible behaviors are then illustrated with two models of scattering in separable potentials within non-relativistic scattering theory, one with a single channel and another one in a two-channel system. In Section 4 we analyze the realistic examples of the pole trajectories of the σ or f 0 (500) scalar meson and the ρ(770) as functions of the quark masses, obtained from the combination of Chiral Perturbation Theory and a single channel dispersion relation obtained in [8]. We show how the generic features discussed in this paper show up in these two cases. In particular, we can conclude that the f 0 (500) or sigma meson would have a predominantly molecular nature, if the pion mass were of the order of 450 MeV or higher. In the final Section we summarize our results.

General properties of S-matrix poles
In this work we only consider one continuum channel. This implies that the S-matrix has one right hand cut, starting at s = (2m) 2 -the so called unitarity cut 2 . As a consequence there are two sheets and, as usual, we call first or physical sheet the one corresponding to a momentum with a positive imaginary part. The S matrix evaluated on sheet I (II) is written as S I (s) (S II (s)). If no subscript is given, the expression holds for both sheets. It follows directly from unitarity and analyticity that [15] S I (s) = 1/S II (s) and [S(s)] * = S(s * ) . (1) As a consequence a pole on the second sheet immediately implies a zero on the first and vice-versa. In addition, if there is a pole at s = s 0 , there must also be a pole at s = s * 0 , i.e., poles outside the real axis occur in conjugate pairs. Furthermore, it can be shown that the only poles allowed on the physical sheet are bound state poles, namely, those located on the real axis below threshold.
A different, but equivalent, way to discuss the pole structure of the S-matrix is to use the k-plane: instead of the Mandelstam variable s, the center of mass momentum k is used to characterize the energy of the system. The two quantities are related via The obvious advantage is that there is no right hand cut with respect to k and correspondingly there is only one sheet. It follows directly from the definition that the upper (lower) half plane of the complex k-plane, defined by positive (negative) values of the imaginary part of k, maps onto the first (second) sheet of the s-plane. The conditions derived above from Eq. (1) translate into the k-plane as follows: the only poles allowed in the upper half plane are on the imaginary axis and in the lower half plane appear as mirror images with respect to the imaginary axis. The relation between the different planes is illustrated in Fig. 1. On the one hand, it becomes clear from the figure that the resonance pole located at r is the one closest to the physical axis and therefore physically more relevant than the Now, assuming that there is at least one resonance pole, and that it is not too far away from threshold, we are now in the position of writing down the most general expression for the S-matrix in the vicinity of that pole or its conjugate partner. For the derivation it is easier to use the k plane, and thus we assume that there is a resonance pole at k = k p − iγ with γ > 0. For a resonance, k p is a real number and we choose k p > 0, for, as commented above, it corresponds to the pole closest to the physical axis. Then, from the above considerations it follows that there is in addition a pole at k = −k p − iγ and zeros at k = ±k p + iγ. We may therefore, dropping terms of higher order in k, and for a particular partial wave ℓ, write the following general expression for the S-matrix element in the vicinity of the pole [15]: where φ(k) is a smooth function, real valued for real, positive values of k. For simplicity this phase factor will be dropped in what follows. Using the definition of the T matrix, S = 1 + 2ikT , we may write For elastic scattering unitarity provides a stringent link between the real and the imaginary part of T , that actually allows the T -matrix elements to be described in terms of its phase δ as Of course, it is straightforward to recast the above expressions in terms of s instead of k. One finds for example with s 0 = 4 k 2 p + γ 2 + m 2 .

ℓ > 0 partial wave threshold behavior and poles
In general the centrifugal barrier demands, for momenta much smaller than some typical scale µ, that the scattering amplitude behaves as T ℓ ∝ k 2ℓ . If we are only interested in this low energy region, the constraint translates into the replacement Of course, this amplitude should only be used for k much smaller than the typical scale µ, not beyond. Note that ℓ = 0 waves are unaffected by this change, but, for example, ℓ = 1 waves now have poles whenever iγk 2 + k ± k p = 0, namely at: These are four poles in conjugated pairs, but of course, they are only meaningful if they lie within the low momentum region of validity of our amplitude. We can ensure that we have only one conjugated pair within this region if we requireγk p ≪ 1, (which is nothing but assuming that both parameters are natural, i.e.,γ ≪ 1/µ and k p ≪ µ). In such case we can expand so that the four poles lie at: One should not worry about the unphysical conjugated pair of poles lying on the first sheet, since our amplitude has been constructed for 1/γ ≫ k p and thus these spurious poles are deep in the complex plane, beyond the range of applicability of our approach, which is however valid for the two poles not too far from threshold. A similar pattern emerges for even higher partial waves, with a physical pair for small k and additional unphysical pairs of poles beyond the applicability region of our amplitude.

Pole trajectories as a function of a strength parameter
In the construction presented in the previous section it was assumed that k p is a real number -then the equations describe a resonance. We will now generalize this investigation by considering the movement of the poles as some strength parameter is varied. Therefore we will study how the resonance properties change when varying k p . In particular, it is interesting to observe the trajectories of the poles for k p → 0 especially very close to threshold. Of course, as long as two conjugate poles exist, their trajectories have to be symmetric with respect to the imaginary k axis. Let us first follow the trajectories followed by conjugate poles for ℓ > 0 partial waves. As a consequence of Eq. (7), they will come infinitesimally close to k pole = 0. However, as commented above there can be no poles of the S-matrix on the physical cut. This property is automatically implemented in Eq. (3) for when k p = 0 and γ = 0 simultaneously, the zeros in the numerator and denominator cancel to yield S(k = 0) = 1. The resulting pole trajectories in the s plane are illustrated for ℓ = 1 in Fig. 2.
In contrast, for s-waves the point where the two conjugate poles meet each other on the imaginary k axis is not fixed except for the condition that no poles in the physical axis should exist in the first Riemann sheet, and in particular not at k = 0. But that leaves the whole negative axis for s-wave poles to meet when k p decreases and the point where the two trajectories meet, −iγ, is a non-trivial parameter of the underlying dynamics. This is one of the central messages of this paper. In order to extend our discussion to poles below threshold, we need to continue analytically k p to complex values k p = iκ p , with κ p real and positive so that k 2 p is real and negative. Of course, for our amplitude to make sense we still keep the conditionγκ p ≪ 1. Now, for the ℓ > 0 case, we find two physical poles at k pole = iκ p (±1 +γκ p ). In the s-plane, these are two poles below threshold but one in the first and another one in the second Riemann sheet. The resonance has become a bound state. Sinceγκ p ≪ 1 they lie almost symmetrically with respect to the threshold. As seen from the s-plane, this is the typical structure of subthreshold poles in the first and second Riemann sheets.
In contrast, for scalar waves, we find two poles in the imaginary axis at k pole = i(±κ p − γ). Note that, as κ p grows, the two poles start separating from each other and move apart from the "meeting point", −iγ. In the k-plane both move along the imaginary k-axis, the physically more relevant one is located at −i(γ − κ p ) while the other one is located at −i(γ + κ p ). Correspondingly in the s-plane the two poles move along the real axis below threshold, but both of them lying on the second sheet until κ p = γ. Eventually, when κ p > γ, the first pole moves to the physical sheet -the virtual state turns into a bound state. The corresponding motion of the poles is illustrated in Fig. 3.
One may define the mass M of a particle as the real part of the corresponding pole position in the complex plane. It is therefore interesting to follow this point as γ and k p vary. For s-waves, in general one finds a striking non-analytic behavior in M at the point where k p = 0. On the other hand, for partial waves higher than s-waves the behavior much smoother 3 . This non-analyticity of the hadron mass when the conjugate poles reach the real axis as k p → 0 has been recently studied in detail within the general formalism of Jost functions in [13]. The conclusion is a similar warning to the one we raised in Ref. [8] about the naive mass extrapolation formulas for states which appear near thresholds on the lattice, although within a more general framework. In this work we will illustrate this nonanalyticity in passing, when explaining the different possible pole trajectories on the basis of various examples below, whereas for the analytic formalism of those mass singularities we simply refer the reader to [13].

Summary of Weinberg's Criterion
In Refs. [17] Weinberg developed a criterion for compositeness for bound states that occur in the s-wave of scattering amplitudes (under which circumstances this formalism can be generalized to resonances is described in Ref. [18]). The starting point is the scattering amplitude near threshold that may be expressed in terms of the scattering length, a, and the effective range, r as 4 Weinberg derived relations between the scattering length, a, the effective range, r, and the wave function renormalization constant for the particle described by the S-matrix pole, Z, Here β is the typical momentum scale of the binding interactions -in our case either β ∼ m π or larger, depending on whether single pion exchange is important in the process -and R is the inverse of the imaginary momentum corresponding to the S-matrix pole when the pole in S occurs at s = 4(m 2 − κ 2 ) = 4m 2 − 4mB. For a bound state κ > 0 and for a virtual state κ < 0 (in both cases B > 0). On general grounds one can show that in leading order in an expansion in 1/(Rβ) Z can be interpreted as the probability to find the ordinary, compact component in the wave function of the physical state; especially 0 ≤ Z ≤ 1.
Assuming that the effective range approximation is valid for all momenta of interest, from Eq. (11) evaluated at the pole follows a single kinematic relation among a, r, and R Weinberg found that a predominantly composite state (or hadronic molecule or extraordinary hadron) has Z ≈ 0. In the case of a weakly bound particle, where R ≫ 1/β, this criterion reduces to a ∼ R and r ∼ 1/β, with the range term in Eq. (14) just providing a small correction. Note that within potential scattering one can show that the terms of order 1/β are typically positive. On the other hand, a predominantly elementary state has Z ≈ 1 and therefore a ∼ 0, or more accurately a ∼ 1/β, and and |r| ≫ R. As a result, in order to get a bound state near threshold for a predominantly elementary state a fine tuning between the range term and the scattering length term is necessary in Eq. (14) 5 . This clearly demonstrates, that it is way more natural to find composite states near thresholds than elementary states.
The argument can be easily expressed in terms of the parameters introduced in the previous section. From Eq. (4) one finds Using Eq. (12) this can be translated to As explained in the previous section, for a shallow bound state one has κ p − γ = κ ≪ β. Thus, we again recover that the most natural situation for a near threshold pole is Z ≃ 0, since only if both κ p and γ are individually much smaller than β and at the same time γ ≪ κ p , then Z ≃ 1, referring to an elementary state. Clearly, for this to be realized fine tuning is necessary. In the following we will illustrate the patterns described above on two simple models. (Some pole trajectories within a specific coupled channel model were already shown in [19]). Both of these models are based on non-relativistic scattering in a separable potential. Model A is a single channel separable potential. If the potential is attractive and strong enough, it can generate an S-matrix pole. In [6] it was argued that this is a "toy model" for an extraordinary hadron which would vanish as N c → ∞. Model B is a two channel model where there is no diagonal interaction in the open channel, but there is a bound state in the closed channel (a Feshbach resonance). In [6] it was argued that this is a model for an "ordinary hadron", whose width would go to zero as N c → ∞.

Model A
Model A has a separable potential that only couples to a single partial wave with angular momentum l. The scattering amplitude in all other partial waves is zero. For the partial wave with angular momentum l, the Schrödinger equation is To make things simple v(r) is chosen such that the integrals can be done analytically: Although for r ∼ 1/µ the behavior of the system depends on the form chosen for v(r), the behavior for r ≪ 1/µ is genuine. Then one finds for the scattering amplitude, f l (k), where ξ l (k) = ∞ 0 rj l (kr)v(r), and in particular Here units are chosen such that µ = 1 -accordingly the model should reproduce the genuine behavior discussed above for k ≪ 1. One can compute N(k) and D(k) explicitly; for the s-wave (ℓ = 0), For s-wave poles located near a threshold one may use Weinberg's criterion to pin down the degree of compositeness of the corresponding physical state. Here closeness to the threshold translates into k ≪ 1. Then we may read off from the expressions given above A bound state is present only if the interactions are attractive and a > 0, which translates into λ > 1 (λ < 0 refers to a repulsive interaction). In addition, r is always positive -which means that in Eq. (12) for the range the 1/β term dominates. Thus, the pole is located very near threshold only for r A /(2a A ) ≪ 1, as follows straightforwardly from Eq. (14) -within the model this ratio does not exceed 0.3 showing that Model A produces extraordinary hadrons in the whole parameter range where bound states are produced. Equivalently one may also directly calculate the wave function renormalization constant for the s-wave -one finds it consistent with 0 within the uncertainties. Thus, the single partial wave separable potential generates an S-matrix pole dynamically that mimics an extraordinary hadron: a hadronic molecule.

Model B
This model is designed to show the scattering effects of a confined state when it can be treated non-relativistically using the Schroedinger equation. It is described in detail in [6], Section II.C, and will not be repeated here. Basically it maps onto a separable potential model but with λ → λ/(E − E 0 ), where E 0 is the energy of the confined channel bound state. (Feshbach showed that near such a state the two channel Schrödinger Equation collapses to a single channel equation with a separable potential.) Here are the numerator and denominator of the s-wave scattering amplitude for this model: The only difference between the two models is that the 1 in D 0 (k) in model A is replaced by k 2 − k 2 0 in model B -said differently: model A is recovered from model B in the limit k 2 0 → ∞ whileλ = −λ/k 2 0 is kept finite. In particular, this implies that in some areas of parameter space model B describes, as model A, composite states.
However, because of the new extra parameter, k 0 , the location of a near threshold pole in the scattering amplitude is no longer directly linked to the scattering length and as a consequence Weinberg's criterion for compositeness can be evaded. This is intuitively clear: If the coupling to the confined state is very weak or repulsive, it should appear like an elementary particle in the scattering channel. This is the case for very small λ and negative k 2 0 . If one takes, for instance,λ = 0.1 and k 0 = 0.3i one obtains a bound state with Z ≃ 1, which is an elementary state.
In summary, for certain parameters, that need to be fine-tuned considerably, the "Feshbach" resonance model can describe a bound state in a confined channel that couples to scattering in an open channel that does not satisfy Weinberg' criterion for compositeness and thus should be interpreted as a genuine ("ordinary") state.

A realistic example: Pole trajectories of the σ and ρ mesons as a function of quark masses
In order to illustrate with a realistic example what was described in the previous sections, we now show the results for the pole trajectories of the ρ-meson and the σ-meson calculated within the inverse amplitude method (IAM) [22]. The approach uses Chiral Perturbation Theory (ChPT) predictions to a given order to fix the subtraction constants of an elastic partial wave dispersion relation. This leads to an amplitude consistent with elastic unitarity that by construction matches the ChPT expansion when re-expanded at low energies and at the same time generates the poles associated to the σ and ρ resonances in pion-pion scattering [23]. Note that the numerical values of the low energy constants obtained when fitting the IAM to scattering data might slightly differ from those of ChPT since they absorb higher order effects. Since the whole QCD quark mass dependence is included up to the desired order in terms of the ChPT expansion of the pion mass and decay constant, one can study the quark mass dependence of both the σ and ρ resonances [8]. We will now discuss the resulting pole trajectories is some more detail.
In Fig. 4 we show the pole movement in the second sheet for both σ and ρ. The pole movement of the σ in the k-plane is shown in the right panel of Fig. 5. Not only provides us the k-plane with a different look at the positions and movements of S-matrix poles, it    , respectively. We see that for both k p and γ very simple two parameter fitting functions provide a reasonable representation of the full results. We start with the physical, non-vanishing values for both γ and k p for the σ as well as the ρ. As the pion mass gets increased k p decreases significantly and eventually vanishes while γ changes relatively little. At the point where k p = 0 the two poles meet at the real axis below threshold for the σ and at exactly at threshold for the ρ, as explained above. When the quark masses are increased further, one σ pole moves towards the ππ threshold, while the other one moves away from the threshold along the real s (imaginary k) axis.
We can now come back to the discussion of Sec. 3 and apply the formalism to the σ as derived from the IAM. In case of the σ the range of forces is set by m ρ . The σ becomes a bound state at m π = 450 MeV. At this point we have Thus we conclude from this analysis that at least for m π > 450 MeV the σ is predominantly of molecular nature. Note that, both for simplicity and in order to be conservative, we have shown calculations for the IAM to one-loop from [8], although the two-loop calculation has also been performed [24]. In that case a similar behavior is found, including the appearance of a virtual pole, although for pion masses m π > 300 MeV.
Given the large similarity of the pole trajectory of the σ meson and that found for the controversial K(800) scalar resonance (or κ) with the IAM using SU(3) ChPT [9], a similar conclusion seems unavoidable for the K(800), especially since the virtual pole predicted as the pion mass increases was recently confirmed in a lattice-QCD calculation [10].

Summary
In this paper we discussed on general grounds the properties of pole trajectories as some strength parameter is varied for resonances coupling to the continuum in different partial waves. There is a qualitatively different behavior for states that couple in an s-wave compared to all higher partial waves: only for s-wave states the two, complex conjugate resonance poles on the second sheet meet at some value of the strength parameter below the threshold -for all other partial waves this meeting point is located exactly at threshold. Using Weinberg's compositeness criterion we were able to show that there is a connection between the value of the mentioned subthreshold meeting point and the composition of the wave function of the physical state. To illustrate the mentioned properties we investigated two models: Model A gives hadronic molecules, which one might also call extraordinary hadrons, for all values of the coupling that lead to a pole, while the more general Model B allows for a near threshold state with a prominent elementary component, however, this requires a significant amount of fine tuning.
In lattice QCD resonance poles move as quark masses are varied. Since most simulations at present are still performed at such values of the quark quark masses/lattice spacings that the resonances can not decay to the continuum, so called chiral extrapolations are necessary to relate the lattice results to the real world parameters. For extraordinary swaves those need to contain striking non-analyticities. This is illustrated in this paper by employing the quark mass dependence of the σ pole as predicted by the inverse amplitude method in combination with one loop chiral perturbation theory. On the basis of this study we were also able to provide a simple parameterization for the pole trajectories that contains the mentioned non-analyticity and should proof useful in future lattice studies.