Methods of parameterization of amplitudes and extraction of resonances, D-decay amplitudes

Amplitudes used for analyses of two-body interactions very often are not unitary therefore can not guarantee correct results. It is, however, quite easy to construct unitary amplitude or check whether given amplitude fulfills unitarity condition. Only few conditions must be fulfilled to guarantee unitarity. Presently, when in many data analyses very small, overlapping or broad signals are studied, non-unitary effects can significantly influence results and lead to nonphysical interpretation of obtained parameters.


Introduction
Unitarity can be compared to probability that is conserved in nature. Therefore unitarity should play crucial role in analyses of amplitudes in various interactions and decays. Here we concentrate on the simplest case: two-body interactions. The typical easiest way to construct amplitudes containing several resonances is just to add the smallest number of individual amplitudes which sufficiently well describe data. Here we show that amplitudes constructed in such a way may be not unitary and may be not sufficient to describe the data and need additional smooth background. Backgrounds may be interpreted as an effective influence of all omitted singularities of the amplitudes. These singularities usually lie very far from the physical region and are very model dependent, therefore can not be interpreted as real resonances. Resonances which we can find in for example Particle Data Group Tables should be model independent and should play leading role in construction of full amplitudes (phase shifts and inelasticities), both elastic and inelastic. Parameters of resonances (for example mass and width) do, however, depend very much on whether the amplitudes used in analyses were unitary or not.
Generally S -matrix can be expressed as ratio of two Jost functions S (k) = D(−k) D(k) = e 2iδ . To reproduce resonance in our S -matrix let us first assume minimum condition -one zero of the denominator D(k) at k = k j . So D(k) = (k − k j ) and D(−k) = (−k − k j ). One can easily check that in such "one pole" case |S (k)| 1 therefore our amplitude A(k) related with the S (k)-matrix by A(k) = S (k)−1 2ik is not unitary. Adding second-symmetric pole also on the 2nd Riemann sheet but at k = −k * j one gets D(k) = (k−k j )(k + k * j ) and D(−k) = (−k−k j )(−k + k * j ). One can easily check that now |S (k)| = 1 and phase shift δ = (−α − β + γ + ω)/2, where α, β, γ and ω are phases of the poles p, p and zeroes z and z presented on Fig. (1). Full phase shifts and all components are presented on this figure.
All those phase components are given by ArcT an( −Imk j k−Rek j ). It shows that of course only pole p and zero z lying closer to physical region than their mirror pole p and zero z (hereafter "second pole") can produce increase of the phase by π/2 what is characteristic for single resonances. Therefore the role of the second pole decreases with the energy. Nowhere is, however, equal to zero. At the threshold, i.e. right in the middle between p, z p and z influence of the all these poles and zeroes on the amplitude is the same.
. In case of unitary S -matrix defined by Jost functions with two poles and zeroes presented above, corresponding limits are correct, i.e. S (k) −→ −k j * k j −k j * k j so δ(k) −→ 0 and σ(k) −→ 0.

Unitarity for amplitude with more resonance and more channels
For amplitudes with more resonances very popular is isobar model with sum of amplitudes describing, for example, single resonances. For two amplitudes, their sum 2ik what corresponds to S 1 + S 2 = e 2iδ 1 + e 2iδ 2 . Of course |S 1 + S 2 | 1 what means that isobar model violates unitarity. In case of analyses using S -matrix instead of amplitudes one can create product S 1 S 2 which fulfills unitarity (for more resonances described by S i the method is the same). Another popular way of parameterization of amplitudes is to use K-matrix defined by S = (1 + iK)/(1 − iK). Sum of two K-matrices does not violate unitarity.
In case of more that one channel situation becomes more complicated. Because of simple ambiguity k 2 = ± k 2 1 + m 2 1 − m 2 2 every new channel doubles number of Riemann sheets. So in case of n channels one has 2 n Riemann sheets and 2 n−1 poles lying on various Riemann sheets and coming from one single pole appearing in one of the channels in fully decoupled case. All these poles are shifted more or less (it depends on strength of coupling between channels) with respect to position of this original pole. Figure (2) presents Riemann sheets for two channels and schematic positions of poles and zeroes corresponding to one resonance. Names of Riemann sheets are given by signs of imaginary parts of momenta in all channels. For example in two channel case mark (−, +) means that Im(k 1 ) < 0 and Im(k 2 ) > 0  Table 1 presents an example of positions of the S -matrix poles for three resonances found in two channel analysis of scalar-isoscalar ππ interactions below 2 GeV (analysis similar to that in [1]). Underlined are poles which play leading role in the full amplitude. They were found checking distances of the poles from the physical region in complex conformal variable z defined by z = k 1 +k 2 √ The results of such analysis have been confirmed by analysis of phases and squared modules of amplitudes (proportional to cross section) corresponding to each pole. Figures (3) and (4) present phases and squared modules of amplitudes for each pole separately and for pairs of poles related with given resonance. It is seen that in both cases (especially for f 0 (980)) one pole is dominant and the second one plays minor role.
In case of analysis of more than 2 channels one can not define and use similar conformal variable and the simplest and effective method is just analysis of influence of all found poles on the phase shifts and inelasticities as was shown on Figs (3) and (4). Another method relays on presentation of positions of all poles in 3 dimensional combinations of real and/or  Table 1 and their pairs, black lines are for pole 1, blue for 1' and red for both poles together.
imaginary parts of complex momenta in all channels. Reasonable choice of axis and careful analysis of distances of these poles from physical region enables to identify the most prominent poles. Example of results of such 3-channel analysis one can find in [2] (Tables 3-7). Independently on analyzed number of channels and number of found resonances crucial is the use of correct i.e unitary amplitude. In recent analysis of pion electromagnetic form factor [3] authors present various ways of parameterization of e + e − → π + π − cross section and of vector-isoscalar ππ elastic amplitude. In Table 2 compared are parameters of ρ states obtained using Gounaris-Sakurai model and unitary and analytic approach. The latter one gives significantly different results than those from PDG Tables and than obtained using Gounaris-Sakurai model. Mass difference for ρ(770) is about 9 MeV and for ρ and ρ about 170 MeV and 78 MeV respectively. Sign of these differences agrees with what was presented in Section 2 (mass from unitary amplitude is smaller). Small phase produced by second pole (denoted in Section 2 by p ) leads to shift of the main pole p (i.e. shift of the mass) to lower energies  Table 1 and their pairs, black lines are for pole 2, blue for 2' and red for both poles together. in comparison with mass determined by value of phase shift equal to π/2. In case of ρ(770) this shift should be few MeV and for wider states should be bigger what agrees with numbers in Table 2.

Crossing symmetry as additional constraint
It is very advisable to introduce the crossing symmetry requirement to the amplitudes. For identical particles like ππ it quite easy and was proposed by Roy few decades ago [5] and later was developed and applied in number of works in the early 2000s e.g. [6] and [7]. For non-identical mesons like π and K similar analysis was performed recently [8].
Introduction of the crossing symmetry requirement to amplitudes describing scalarisoscalar ππ interactions has led to spectacular successes. One of them was to eliminate the long standing up-down ambiguity in these amplitudes (in favor of the down solution)