Analysis of Overlapping Resonances with Unitary Breit–Wigner and K-Matrix Approaches

Abstract

We compare two methods for obtaining the parameters of overlapping resonances. The convenience of the Breit–Wigner (BW) approach is based on the fact that it operates with the masses and widths of the states. For several resonances with the same quantum numbers, a sum of BW functions violates the unitarity of the S-matrix. However, unitarity can be maintained by introducing interference phases to a BW implementation of scattering matrix formalism. A background can be added to the BW amplitudes in the standard way by using background phases. The K-matrix method is often used to analyze data related to several resonances with the same quantum numbers. It guarantees the unitarity of the S-matrix, but its parameters can be considered as resonance masses and widths only for well-spaced states. It also does not allow the separation of the resonant and background contributions in scattering amplitudes, which is critically important for determining parameters of wide resonances. To demonstrate the features of these methods, we consider several examples using simulated data.

1. Introduction

The Breit–Wigner function [1] describes partial amplitudes in a form which directly contains the mass and width of resonances. The BW function for one resonance satisfies the unitarity condition; a problem arises, however, when one needs to construct the unitary S-matrix for several resonances with the same quantum numbers.

A scattering operator connecting an initial and a final state, $S_{if}=\langle f\left|S\right|i\rangle$, must be unitary

$$S{S}^{†}=I$$

(1)

and symmetric, $S_{if}=S_{fi}$, to satisfy the time-reversal invariance.

The results of an analysis of any isolated resonance are eventually compared to the BW function (M is the number of channels)

$$S_{ij}={\delta }_{ij}+2i\frac{\sqrt{{\mathsf{\Gamma }}_{ri}{\mathsf{\Gamma }}_{rj}}/2}{E_r-E-i{\mathsf{\Gamma }}_r/2}, {\mathsf{\Gamma }}_r={\displaystyle \sum _i^M{\mathsf{\Gamma }}_{ri}}$$

(2)

(we start with a form without a background), or with the variable $s=E^2$,

$$S_{ij}={\delta }_{ij}+2i\frac{m_r\sqrt{{\mathsf{\Gamma }}_{ri}{\mathsf{\Gamma }}_{rj}}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r}$$

(3)

The unitarity of these expressions, along with a form of the wave function of an unstable state, $\psi (t)=\psi (0)e^{-i\left(E_r-i{\mathsf{\Gamma }}_r/2\right)t/\hslash }$, ${\left|\psi (t)\right|}^2={\left|\psi (0)\right|}^2{e}^{-t/\tau }$, where $\tau =\hslash /{\mathsf{\Gamma }}_r$ is the life-time of a resonance, supports the argument for writing the partial amplitudes in the BW form.

The idea of writing the S-matrix as a sum of resonant terms in a general way, which must satisfy the unitarity constraints, is due to work [2]. We demonstrate that this can be achieved in the form

$$S_{ij}={\delta }_{ij}+2i{\displaystyle \sum _{r=1}^N{e}^{i{\phi }_{ij}^r}}\frac{m_r\sqrt{{\mathsf{\Gamma }}_{ri}{\mathsf{\Gamma }}_{rj}}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r}$$

(4)

where the interference between the states is taken into account by the phases ${\phi }_{ij}^r$. In the original work [2], this expression was written with complex numerators (residues) instead of such phases—both forms are equivalent. This scheme was realized in work [3] for two resonances with constant (energy-independent) widths.

To consider energy-dependent widths, and to take into account channel thresholds, the S-matrix should be given in the form

$$S_{ij}={\delta }_{ij}+2i{F}_{ij}={\delta }_{ij}+2i\sqrt{{\rho }_i}{T}_{ij}\sqrt{{\rho }_j}$$

(5)

where $T_{ij}$ are transition amplitudes and ${\rho }_i$ are phase-space factors. The conditions which the unitarity imposes on the widths ${\mathsf{\Gamma }}_{ri}(s)$ and (constant) phases ${\phi }_{ij}^r$ are formulated in Section 2.

In the K-matrix approach [4], the S-matrix is expressed as

$$S=\frac{I+i\rho K}{I-i\rho K}$$

(6)

From the unitarity and symmetry of the S-matrix, it follows that $K$ is a real and symmetric operator. From Equation (6), it follows that

$$T={(I-i\rho K)}^{-1}K$$

(7)

In the case of one channel, $S=e^{2i\delta }$, where $\delta$ is the scattering phase; thus, $K=\tan \delta$. At the resonance position, $\delta =\pi /2$; thus, $K$ has a pole. A single pole parametrization,

$$K=\frac{m_1{\mathsf{\Gamma }}_1}{m_1^2-s}$$

(8)

gives the standard BW function for the partial amplitude:

$$F=\frac{\rho m_1{\mathsf{\Gamma }}_1}{m_1^2-s-i\rho m_1{\mathsf{\Gamma }}_1}$$

(9)

Thus, $m_1$ and ${\mathsf{\Gamma }}_1$ in Equation (8) are the resonance’s mass and the width in this simple situation.

For several poles and channels, a commonly used parametrization is [5]

$$K_{ij}={\displaystyle \sum _{\alpha }\frac{m_{\alpha }{\mathsf{\Gamma }}_{\alpha }^0{\gamma }_{\alpha i}{\gamma }_{\alpha j}}{m_{\alpha }^2-s}}$$

(10)

For N poles and M channels, the number of free parameters in the K-matrix method is $N(M+1)$ [6]. Parameter $m_{\alpha }$ is often called the “nominal” mass and parameter ${\gamma }_{\alpha i}^{}$ is called the “coupling constant” of the state $\alpha$ to the decay channel $i$. To enhance the similarity with the BW formula, they are usually normalized: ${\displaystyle \sum _i}{\gamma }_{\alpha i}^2$, and ${\gamma }_{\alpha i}^2$ are considered as resonance branching ratios. Notice that all these statements are based on the comparison with the BW expression for an isolated resonance.

The relationship between the K- and the BW- methods and the appearance of relative phases in formula (4), can be demonstrated by considering two states in one channel. With

$$K=\frac{{\gamma }_1^2}{m_1-E}+\frac{{\gamma }_2^2}{m_2-E}$$

(11)

(here, and below, we use variable E and omit $\rho$ only to simplify the expressions illustrating the comparison between the two methods), we obtain the scattering amplitude:

$$F=\frac{{\gamma }_1^2{m}_2+{\gamma }_2^2{m}_1-E\left({\gamma }_1^2+{\gamma }_2^2\right)}{E^2-E\left[m_1+m_2-i\left({\gamma }_1^2+{\gamma }_2^2\right)\right]+m_1{m}_2-i\left({\gamma }_1^2{m}_2+{\gamma }_2^2{m}_1\right)}$$

(12)

Denoting complex roots of the denominator as ${\mu }_1$ and ${\mu }_2$, F can be expressed as a sum of two BW functions:

$$F=\frac{A_1}{{\mu }_1-E}+\frac{A_2}{{\mu }_2-E}$$

(13)

where $A_{1,2}$ are complex quantities:

$$A_1=\frac{{\mu }_1\left({\gamma }_1^2+{\gamma }_2^2\right)-{\gamma }_1^2{m}_2-{\gamma }_2^2{m}_1}{{\mu }_2-{\mu }_1},A_2=\frac{{\mu }_2\left({\gamma }_1^2+{\gamma }_2^2\right)-{\gamma }_1^2{m}_2-{\gamma }_2^2{m}_1}{{\mu }_1-{\mu }_2}.$$

(14)

The real parts of ${\mu }_{1,2}$ give energies (masses) of resonances, the imaginary parts give their widths:

$$E_{1,2}=\mathrm{Re}{\mu }_{1,2}=\frac{m_1+m_2}{2}\pm \frac{1}{2\sqrt{2}}\sqrt{{\left(m_1-m_2\right)}^2-{\left({\gamma }_1^2+{\gamma }_2^2\right)}^2+W}$$

(15a)

$${\mathsf{\Gamma }}_{1,2}=2\mathrm{Im}{\mu }_{1,2}=\frac{{\gamma }_1^2+{\gamma }_2^2}{2}\pm \frac{1}{\sqrt{2}}\frac{\left(m_1-m_2\right)\left({\gamma }_1^2-{\gamma }_2^2\right)}{\sqrt{{\left(m_1-m_2\right)}^2-{\left({\gamma }_1^2+{\gamma }_2^2\right)}^2+W}}$$

(15b)

where

$$W=\sqrt{{\left[{\left(m_1-m_2\right)}^2+{\left({\gamma }_1^2+{\gamma }_2^2\right)}^2\right]}^2+16{\left(m_1-m_2\right)}^2{\gamma }_1^2{\gamma }_2^2}$$

(15c)

Thus, the amplitude F in the K-matrix method is equivalent to the sum of BW functions with complex residues; in other words, with the relative phase, as seen in Equation (4).

The expression for this phase is easy to find. For

$$F={\displaystyle \sum _{n=1}^2\frac{{\alpha }_n/2}{E_n-E-i{\mathsf{\Gamma }}_n/2}}$$

(16)

S is unitary if

$$\frac{{\alpha }_1}{{\alpha }_2}=\frac{{\mathsf{\Gamma }}_1}{{\mathsf{\Gamma }}_2}{e}^{2i\phi }, \mathrm{where} \phi =-\arctan \frac{{\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2}{2\left(E_1-E_2\right)}.$$

(17)

When $m_1$ and $m_2$ are well separated, i.e., $\left|m_1-m_2\right|\gg {\gamma }_1^2+{\gamma }_2^2$, then $E_i=>m_i$, ${\mathsf{\Gamma }}_i=>2{\gamma }_i^2$ and $\phi =>0$. The fact that the amplitude $F$ in the K-method can be presented as a sum of the BW functions is often taken as the justification that the K-matrix pole parameters are close to those of the physical resonances. However, ‘good separation’ is a rather subjective argument; the unitarity of the S-matrix is very crudely violated, even when resonances are well spaced, as shown in Figure 1.

2. Brief Description of the Unitary BW Method

Here, we briefly describe the method of constructing the unitary S-matrix [7]. The regular procedure is presented in Appendix A.

The unitarity condition, $S^{†}S=I$, with $S=I+2iF=I+2i\sqrt{\rho }T\sqrt{\rho }$, gives

$$T^{†}-T=i{T}^{†}\rho T$$

(18)

The unitarization procedure becomes technically simpler if, following the idea of works [2,3], to introduce complex vectors of partial widths, ${\overrightarrow{g}}_r$. With $g_{ri}=e^{i{\phi }_{ri}}\left|g_{ri}\right|$, $i=1,\dots ,M$, $r=1,\dots ,N$, the S-matrix is:

$$S_{ij}={\delta }_{ij}+2i\sqrt{{\rho }_i(s)}\left({\displaystyle \sum _{r=1}^N{e}^{i{\phi }_{ij}^{(r)}}}\frac{m_r{\mathsf{\Gamma }}_r^0\left|g_{ri}\right|\cdot \left|g_{rj}\right|}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r(s)}\right)\sqrt{{\rho }_j(s)}$$

(19)

and the T-matrix is:

$$T_{ij}(s)={\displaystyle \sum _{r=1}^N\frac{m_r{\mathsf{\Gamma }}_r^0{g}_{ri}{g}_{rj}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r(s)}}$$

(20)

Here, ${\phi }_{ij}^r={\phi }_{ri}+{\phi }_{rj}$ are energy-independent phases; factors ${\mathsf{\Gamma }}_r^0={\mathsf{\Gamma }}_r(m_r^2)$ are introduced to keep vectors $g_{ri}$ dimensionless. Expression (19) is time-invariant.

The decay branching ratio of a resonance r in channel $i$ is

$$B_{ri}=\frac{{\left|g_{ri}\right|}^2}{{\left|{\overrightarrow{g}}_r\right|}^2}=\frac{{\left|g_{ri}\right|}^2}{{\displaystyle \sum _{k=1}^M{\left|g_{rk}\right|}^2}}$$

(21)

The partial widths are ${\mathsf{\Gamma }}_{ri}={\mathsf{\Gamma }}_r{B}_{ri}$. The total and partial widths can depend on energy, ${\mathsf{\Gamma }}_r(s)={\displaystyle \sum _i^M{\mathsf{\Gamma }}_{ri}(s)}$.

Formulas in the rest of this section become technically simpler if the variable E is used rather than $s=E^2$—all the expressions and algorithms can be rewritten in terms of $s$. For

$$T_{\mathit{ij}}={\displaystyle \sum _{r=1}^N}\frac{{\mathsf{\Gamma }}_r^0{g}_{\mathit{ri}}{g}_{\mathit{rj}}/2}{m_r-E-i{\mathsf{\Gamma }}_r\left(E\right)/2},$$

(22)

let us formulate the conditions that should be imposed on vectors ${\overrightarrow{g}}_r$ to maintain $S_{ij}(E)$ unitarity. From Equation (18) we have

$$\begin{array}{c}i{\displaystyle \sum _{r=1}^N\left[\frac{g_{ri}^{*}{g}_{rj}^{*}}{m_r-E+i{\mathsf{\Gamma }}_r(E)/2}-\frac{g_{ri}{g}_{rj}}{m_r-E+i{\mathsf{\Gamma }}_r(E)/2}\right]}\hfill \\ \hfill +{\displaystyle \sum _{l=1}^N{\displaystyle \sum _{r=1}^N{V}_{lr}}}\frac{g_{ri}{g}_{lj}^{*}}{\left(m_r-E+i{\mathsf{\Gamma }}_r(E)/2\right)\left(m_r-E+i{\mathsf{\Gamma }}_r(E)/2\right)}\equiv 0,\end{array}$$

(23)

where the following notation is used:

$$V_{lr}=\sqrt{{\mathsf{\Gamma }}_l^0{\mathsf{\Gamma }}_r^0}{\displaystyle \sum _{k=1}^M{\rho }_k}(E)g_{lk}^{*}{g}_{rk}$$

(24)

The constraints Equation (23) involves complicated non-linear conditions. To resolve them, we use a method which allows substantial technical difficulties to be overcome; these previously restricted the approach to a maximum of two [3], or three [8] resonances, even when ${\rho }_k\equiv 1$. The method is based on the construction of vectors ${\overrightarrow{g}}_r={\overrightarrow{g}}_r^x+i{\overrightarrow{g}}_r^y$ in such a form that their imaginary parts ${\overrightarrow{g}}_r^y$ are combinations of their real parts ${\overrightarrow{g}}_r^x$: ${\overrightarrow{g}}_r^y=U{\overrightarrow{g}}_r^x$, or

$${\overrightarrow{g}}_r^y=u_{r1}{\overrightarrow{g}}_1^x+u_{r2}{\overrightarrow{g}}_2^x+\dots +u_{rN}{\overrightarrow{g}}_N^x$$

(25)

U is a real anti-symmetric matrix which has a simple form for any particular N and M (see Appendix A). Next, instead of trying to find all components of vectors ${\overrightarrow{g}}_r$, we only find their real parts, and then obtain their imaginary parts using matrix U. Notice that the number of free parameters, $N(M+1)$, is the same as in the K-matrix method.

When $\left|m_r-m_{r^{\prime }}\right|\gg {\mathsf{\Gamma }}_r+{\mathsf{\Gamma }}_{r^{\prime }}$, the matrix elements $u_{rk}\to 0$ and vectors ${\overrightarrow{g}}_r$ become real and orthogonal, $\left({\overrightarrow{g}}_r,{\overrightarrow{g}}_q\right)=0$, and we return to a simple sum of the BW functions without relative phases.

The constraints on vectors ${\overrightarrow{g}}_r$ are the following:

$${\displaystyle \sum _{k=1}^M{\mathsf{\Gamma }}_r^0{\rho }_k}(E)|g_{rk}{|}^2=-\frac{1}{S}\left[S+2Q_r\right]{\mathsf{\Gamma }}_r,$$

(26)

$${\displaystyle \sum _{k=1}^M{\mathsf{\Gamma }}_r^0{\rho }_k}(E)\mathrm{Re}\left(g_{qk}^{*}{g}_{rk}\right)=-\frac{1}{S}\left[2F_{qr}\left(m_q-m_r\right)-i{G}_{qr}\left({\mathsf{\Gamma }}_q+{\mathsf{\Gamma }}_r\right)\right],$$

(27)

$${\displaystyle \sum _{k=1}^M{\mathsf{\Gamma }}_r^0{\rho }_k}(E)\mathrm{Im}\left(g_{qk}^{*}{g}_{rk}\right)=-\frac{1}{S}\left[2G_{qr}\left(m_q-m_r\right)+i{F}_{qr}\left({\mathsf{\Gamma }}_q+{\mathsf{\Gamma }}_r\right)\right],$$

(28)

$$r=1,\dots ,N;q=r+1,\dots ,N.$$

Constant coefficients $S, Q_r, F_{qr}, G_{qr}$ are determined via the elements of matrix U.

The widths are given by Equation (26):

$${\mathsf{\Gamma }}_r(E)=-\frac{S}{S+2Q_r}{\mathsf{\Gamma }}_r^0{\displaystyle \sum _{k=1}^M{\rho }_k}(E){\left|g_{rk}\right|}^2$$

(29)

Formulas (25)–(29) provide the algorithm for the method. For any particular case, $N=2$, $N=3$, etc.; the coefficients $S, Q_r, F_{qr}, G_{qr}$ are given by simple expressions, presented in Appendix A.

In the case of one resonance, this approach reduces to the traditional BW function:

$$S_{ij}={\delta }_{ij}+2i\sqrt{{\rho }_i{\rho }_j}\frac{{\mathsf{\Gamma }}_1^0{g}_{1i}{g}_{1j}/2}{m_1-E-i{\mathsf{\Gamma }}_1(E)/2},i,j=1,\dots ,M.$$

(30)

Vector ${\overrightarrow{g}}_1={\overrightarrow{g}}_1^x$ is real, and ${\mathsf{\Gamma }}_1^0{g}_{1i}{g}_{1j}=\sqrt{{\mathsf{\Gamma }}_{ri}{\mathsf{\Gamma }}_{rj}}$. With Equation (29), it gives the analogue of Flatte’s formula:

$$T_{ij}=\frac{{\mathsf{\Gamma }}_1^0{g}_{i1}{g}_{1j}/2}{m_1-E-\frac{i}{2}{\mathsf{\Gamma }}_1^0{\displaystyle \sum _{k=1}^M{\rho }_k(E)}{\left(g_{1k}^x\right)}^2}$$

(31)

or in variable s,

$$T_{ij}=\frac{m_1{\mathsf{\Gamma }}_1^0{g}_{i1}{g}_{1j}}{m_1^2-s-i{m}_1{\mathsf{\Gamma }}_1^0{\displaystyle \sum _{k=1}^M{\rho }_k(s)}{\left(g_{1k}^x\right)}^2}$$

(32)

In a fitting procedure, the free parameters are: mass $m_1$ and the components of vector ${\overrightarrow{g}}_1={\overrightarrow{g}}_1^x$.

If a resonance is lying above all the thresholds, its mass is simply $m_1$ and its width is

$${\mathsf{\Gamma }}_1(E)={\mathsf{\Gamma }}_1^0{\displaystyle \sum _{k=1}^M{\rho }_k}(E){\left(g_{1k}^x\right)}^2.$$

(33)

If a resonance is lying between the thresholds L and $(L+1)$, there are two options: to set ${\rho }_k(E)=0$ in the energy region below the corresponding threshold, $E<E_k$, or to continue it as ${\rho }_k=>i{\sigma }_k\left|{\rho }_k\right|$ (${\sigma }_k$ can take the values $\pm 1$ to consider different Riemann sheets). The effective mass is then

$${\tilde{m}}_1(E)=m_1+\frac{{\mathsf{\Gamma }}_1^0}{2}{\displaystyle \sum _{k=L+1}^M{\sigma }_k\left|{\rho }_k(E)\right|{\left(g_{1k}^x\right)}^2}$$

(34)

and the width is

$${\mathsf{\Gamma }}_1(E)={\mathsf{\Gamma }}_1^0{\displaystyle \sum _{k=1}^L{\rho }_k}(E){\left(g_{1k}^x\right)}^2.$$

(35)

For the case of two resonances and one channel, this algorithm leads to the same expression for the amplitude T as in the K-matrix method (as in Section 1).

Let us illustrate the method with the example of two resonances and two channels, $N=2$, $M=2$. The goal is to find vectors $\overrightarrow{g_r}$.

For two channels, matrix U is

$$U=\left(\begin{array}{cc}0& -\alpha \\ \alpha & 0\end{array}\right)$$

(36)

($\alpha$ is the real parameter, $0\le \alpha <1$), i.e.,

$${\overrightarrow{g}}_1^y=-\alpha {\overrightarrow{g}}_2^X, {\overrightarrow{g}}_2^y=\alpha {\overrightarrow{g}}_1^X.$$

(37)

The coefficients in the unitary constraint Equations (26)–(28) are:

$$S=1-{\alpha }^2,Q_1=Q_2={\alpha }^2,F_{12}=-\alpha ,G_{12}=0.$$

(38)

Six real quantities, for example, masses $m_1$ and $m_2$, parameter $\alpha$, and $g_{11}^x$, $g_{12}^x$, $g_{22}^x$ can be taken as independent parameters in data fitting. The remaining real component $g_{21}^x$ is determined from Equations (27) and (28).

When both resonances lie above the 2nd threshold, $E_2<m_1<m_2$,

$$g_{21}^x=\frac{1}{{\rho }_1\left(m_2\right)g_{11}^x}[\frac{2\alpha }{{\mathsf{\Gamma }}_1^0{(1-{\alpha }^2)}^2}(m_1-m_2)-{\rho }_2(m_2)g_{12}^x{g}_{22}^x]$$

(39)

After this, the components ${\overrightarrow{g}}_r^y$ can be calculated with Equation (37). This completes the construction of the amplitudes

$$T_{\mathit{ij}}=\frac{{\mathsf{\Gamma }}_1^0{g}_{1i}{g}_{1j}/2}{m_1-E-i{\mathsf{\Gamma }}_1\left(E\right)/2}+\frac{{\mathsf{\Gamma }}_2^0{g}_{2i}{g}_{2j}/2}{m_2-E-i{\mathsf{\Gamma }}_2\left(E\right)/2}$$

(40)

The functions ${\mathsf{\Gamma }}_r(E)$ are given by expression (29), their values at $E=m_r$,

$${\mathsf{\Gamma }}_r(m_r)=\frac{1-{\alpha }^2}{1+{\alpha }^2}{\mathsf{\Gamma }}_r^0{\displaystyle \sum _{k=1}^2{\rho }_k(m_r){\left|g_{rk}\right|}^2}$$

(41)

can be considered as resonance “widths”.

If the 2nd threshold is located between $m_1$ and $m_2$, i.e., $m_1<E_2<m_2$, the mass of the 1st resonance, ${\tilde{m}}_1$, is determined by the zero of the real part of the denominator in the first term in Equation (40), i.e., by the root of the equation

$$E-{\tilde{m}}_1-\frac{1-{\alpha }^2}{2(1+{\alpha }^2)}{\mathsf{\Gamma }}_1^0\left|{\rho }_2(E)\right|\cdot {\left|g_{12}\right|}^2=0$$

(42)

so this resonance width is

$${\tilde{\mathsf{\Gamma }}}_1\left({\tilde{m}}_1\right)=\frac{1-{\alpha }^2}{1+{\alpha }^2}{\mathsf{\Gamma }}_1^0{\rho }_2\left({\tilde{m}}_1\right){\left|g_{12}\right|}^2$$

(43)

The component $g_{21}^x$ can be found from Equation (39) with ${\tilde{m}}_1$ instead of $m_1$.

3. Examples: K-Matrix vs. Unitary BW Approach

Let us consider two resonances in two channels, $S=I+2iF=I+2i\sqrt{\rho }T\sqrt{\rho }$, where $\rho$ is a diagonal $2\times 2$ matrix. In the unitary BW expression,

$$S_{ij}={\delta }_{ij}+2i\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^2\frac{m_r{\mathsf{\Gamma }}_r(m_r)g_{ri}{g}_{rj}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r(s)}},$$

(44)

there are six independent parameters: $m_1,m_2,\alpha ,g_{11}^x,g_{12}^x,g_{22}^x$. (The product $g_{\mathit{ri}}{g}_{\mathit{rj}}=e^{i{\phi }_{ij}^{\left(r\right)}}\left|g_{\mathit{ri}}\right|\left|g_{\mathit{rj}}\right|$).

The K-matrix pole terms in expression (10),

$$K_{ij}={\gamma }_{1i}{\gamma }_{1j}\frac{m_1{\mathsf{\Gamma }}_1^0}{m_1^2-s}+{\gamma }_{2i}{\gamma }_{2j}\frac{m_2{\mathsf{\Gamma }}_2^0}{m_2^2-s},$$

(45)

have the same number of independent parameters: $m_r$, ${\mathsf{\Gamma }}_r^0$, and two ${\gamma }_{ri}$ (with $\gamma$ being normalized, ${\displaystyle {\sum }_i{\gamma }_{ri}^2}=1$). The K-matrix amplitudes are

$$F=\frac{\sqrt{\rho }K\sqrt{\rho }}{1-i\rho K}=\frac{1}{1-{\rho }_1{\rho }_{2}D-i\left({\rho }_1{K}_{11}+{\rho }_2{K}_{22}\right)}\left(\begin{array}{cc}{\rho }_1\left(K_{11}-i{\rho }_{2}D\right)& \sqrt{{\rho }_1{\rho }_2}{K}_{12}\\ \sqrt{{\rho }_1{\rho }_2}{K}_{21}& {\rho }_2\left(K_{22}-i{\rho }_{1}D\right)\end{array}\right),$$

(46)

where $D=K_{11}{K}_{22}-K_{12}^2$.

Let us start with the important observation that the K-matrix amplitudes $T_{ij}$ become zero (or are very close to zero values) between the $m_1$ and $m_2$ locations. This feature directly follows from expression (7), it is, therefore, retained in any modification of the K-matrix approach. In the case of two resonances, equation $T_{12}(E)=0$ (i.e., $K_{12}(E)=0$) is a linear equation; in the case of three states, it is a quadratic equation—these results directly follow from expressions (10) and (46). The zeros in amplitudes $T_{11}$ and $T_{22}$ are slightly shifted from the zero position in $T_{12}$. Thus, resonances in scattering amplitudes are, in fact, well spaced (a polynomial added to expression (10) for $K_{ij}$ does not change this observation, see Appendix B). As a result, this method can work well only when describing experimental data with sufficiently well-separated states.

Since the purpose of this article is to demonstrate the features of the two methods, it is more useful to consider simple examples than discuss a real physical problem in which there are always ambiguous questions, such as the choice of the most significant channels, the number of resonances, etc. Particular physical problems are beyond the scope of this paper and will be studied separately.

Let us start by considering two overlapping states (peaks) in the amplitudes ${\left|F_{\mathit{ij}}\right(E\left)\right|}^2$ located near $1.3$ and $1.6$ GeV, with widths about $0.15÷0.3$ GeV. We also consider different positions of the 2^nd channel threshold: below both states, at $E_2=1.22$, and between the states, at $E_2=1.38$ (the position of the 1^st threshold is fixed at $E_1=0.5$; all quantities here and below are in GeV). Next, we generate “data” which qualitatively correspond to these situations—the details about such data generation are given in Appendix B. (Note that these data are similar to the data on scattering amplitudes in the case of vector ${\omega }^{\prime }$ resonances.) To do this, we draw smooth curves (different peak heights and widths can be considered), then discretize and randomize these curves and introduce dispersions (error bars at each point are generated randomly with the upper limit of 0.05). Examples of such data are presented in Figure 2 for $E_2=1.22$ and Figure 3 for $E_2=1.38$. The peaks overlap and, at least in one channel, are not well resolved, as often occurs in real situations. We then fit these data with the unitary BW formulas (44)—red lines, and K-matrix (46)—blue lines.

In calculations, we use the phase factors ${\rho }_i(s)=\sqrt{(s-s_i)/s}$ (these can be modified by including the barrier factors). The continuation ${\rho }_2(s)=>\pm i\left|{\rho }_2(s)\right|$ below the $E_2$ threshold can be used in expression (32) for the amplitudes $T_{ij}$.

In the unitary BW formula (44) for the threshold at $E_2=1.22$, the values of the six independent parameters are:

$$m_1=1.35,m_2=1.65,\alpha =0.07,g_{11}^x=-0.43,g_{12}^x=0.41,g_{22}^x=0.49$$

the fit gives ${\chi }^2/d=0.49$ ($d$ is degrees of freedom number).

For the threshold at $E_2=1.38$, the values of these parameters are:

$$m_1=1.36,m_2=1.65,\alpha =0.08,g_{11}^x=-0.42,g_{12}^x=0.38,g_{22}^x=0.36$$

the fit gives ${\chi }^2/d=0.53$. The remaining parameters can be calculated using the six independent parameters. (After the $g_{ri}$ values are found, the value of a ‘technical’ parameter $\alpha$ is no longer required.)

Table 1. BW parameters for two threshold positions.

$\mathbf{Threshold} {\mathit{E}}_2=1.22$						$\mathbf{Threshold} {\mathit{E}}_2=1.38$
$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$	$g_{r1}$	$g_{r2}$	$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$	$g_{r1}$	$g_{r2}$
1.32	0.26	53	47	−0.43 − i0.03	0.41 − i0.04	1.36	0.19	55	45	−0.43 − i0.02	−0.38 − i0.03
1.65	0.318	44	56	0.43 − i0.03	0.49 + i0.03	1.65	0.14	42	58	0.30 − i0.03	0.36 − i0.03

shows the BW resonance parameters—the branching ratios $B_{ri}$ are calculated with formula (21), the phases ${\phi }_{ij}^r$ can be obtained from the $g_{ri}$ values, and ${\mathsf{\Gamma }}_r={\mathsf{\Gamma }}_r(m_r)$. By the direct substitution of these $g_{ri}$ in formula (44), it can be confirmed that matrix $S(E)$ is unitary for any E.

The fit based on the K-matrix formula (46) for the threshold $E_2=1.22$, gives the following values for the six independent parameters:

$$m_1=1.36,m_2=1.63,{\mathsf{\Gamma }}_1=0.27,{\mathsf{\Gamma }}_2=0.37,{\gamma }_{11}=0.77,{\gamma }_{21}=0.68$$

resulting in ${\chi }^2/d=8.72$.

For the threshold $E_2=1.38$, the parameters are:

$$m_1=1.37,m_2=1.63,{\mathsf{\Gamma }}_1=0.32,{\mathsf{\Gamma }}_2=0.19,{\gamma }_{11}=0.73,{\gamma }_{21}=0.63$$

resulting in ${\chi }^2/d=4.75$.

Table 2. K-matrix parameters for two threshold positions.

$\mathbf{Threshold} {\mathit{E}}_2=1.22$				$\mathbf{Threshold} {\mathit{E}}_2=1.38$
$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$	$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$
1.36	0.27	59	41	1.37	0.32	54	46
1.63	0.37	47	53	1.63	0.20	40	60

contains the K-matrix parameters. The branching ratios are $B_{ri}={\gamma }_{ri}^2$.

Large values of ${\chi }^2$ reflect the fact that the K-matrix method cannot adequately describe the data in situations when wide resonances are not well spaced, such as in the channel $1\to 2$, in Figure 2 and Figure 3. However, this method works perfectly well when resonances are well resolved, such as in Figure 4, where the data have substantial dips between the resonances in all three channels (here, for brevity, we consider only one threshold location, $E_2=1.22$). The quality of the fits in both methods is practically the same, ${\chi }^2/d\approx 0.5$. The resonance parameters for Figure 4 are shown in

Table 3. BW and K-matrix parameters for data in Figure 4.

BW						K
$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$	$g_{r1}$	$g_{r2}$	$m_r$	${\mathsf{\Gamma }}_r$	$B_{r1}$	$B_{r2}$
1.30	0.19	49	51	−0.39 − i0.02	−0.37 − i0.03	1.30	0.28	45	55
1.60	0.09	18	82	−0.11 − i0.05	−0.25 + i0.06	1.60	0.12	27	73

.

The Argand diagrams corresponding to the amplitude $F_{11}$ in Figure 2 and Figure 4 are shown in Figure 5.

4. Background

Taking background into account in a rigorous way is especially important when several resonances are not well resolved, or their existence is not well established. Even a small background can hide a resonance [9,10], as in the situation with a vector $\rho (1250)$ meson overlapping with other wide vector states, $\rho (1450)$ and $\rho (1600)$.

Usually, to account for the background in the K-matrix description, polynomials are added to the pole terms in $K_{ij}$:

$$K_{ij}={\displaystyle \sum _a\frac{{\gamma }_{ai}{\gamma }_{aj}{m}_a{\mathsf{\Gamma }}_a^0}{m_a^2-s}+}\left(A_{ij}+B_{ij}s\right)$$

(47)

The form (47) does not allow the background to be presented in terms of background phases, $S_{ij}^B=e^{i({\beta }_i+{\beta }_j)}$; in this form, here, even the number of parameters is different to that in (47): for example, for two channels and an energy-independent background, there are two parameters ${\beta }_1$ and ${\beta }_2$, versus three $A_{ij}$ in $K_{ij}$. Thus, polynomials in Equation (47) only allow the use of additional fitting parameters besides the pole parameters.

Another important fact is that the K-matrix resonances remain well spaced even when polynomials in $K_{ij}$ are taken into account; for details see Appendix B.

In the BW scheme, the S-matrix can be separated in a sum of the resonant and background terms from the very beginning:

$$S_{ij}=B_{ij}+2i\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^N\frac{m_r{\mathsf{\Gamma }}_r^0{A}_{ri}{A}_{rj}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r(s)}}$$

(48)

The unitary and symmetric background matrix B can be constructed as described below.

If matrix B is diagonal, i.e., it is non-zero only in channels with $i=j$,

$$B_{ij}={\delta }_{ij}{e}^{i{\beta }_i} (i,j=1,\dots ,M),$$

vectors ${\overrightarrow{A}}_r$ in (48) can be presented as

$$A_{rk}=e^{i{\beta }_k}{g}_{rk}=\left(\cos {\beta }_k+i\sin {\beta }_k\right)\left(g_{rk}^x+i{g}_{rk}^y\right),$$

where $g_{ri}$ can be found with the procedure described in the previous sections. Matrix S, in such a situation, is

$$S_{ij}=e^{i\left({\beta }_i+{\beta }_j\right)}\left[{\delta }_{ij}+2i\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^N{e}^{i\left({\phi }_{ri}+{\phi }_{rj}\right)}}\frac{m_r{\mathsf{\Gamma }}_r^0\left|g_{ri}\right|\cdot \left|g_{rj}\right|}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r}\right]$$

(49)

or

$$S_{ij}=e^{i\left({\beta }_i+{\beta }_j\right)}\left[{\delta }_{ij}+2i\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^N{e}^{i{\phi }_{ij}^r}}\frac{m_r\sqrt{{\mathsf{\Gamma }}_{ri}{\mathsf{\Gamma }}_{rj}}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r}\right]$$

(50)

If matrix B is non-diagonal, i.e., a background potentially is not zero in all channels, it is helpful to present it in form [3], $B=W{e}^{2i\overrightarrow{\beta }}{W}^T$, where W is a real orthogonal matrix, $W^{-1}=W^T$ and $e^{2i\overrightarrow{\beta }}$ is a diagonal matrix ($W=I$ when matrix B is diagonal). Writing matrix B as $B=b{b}^T$ with $b=W{e}^{i\overrightarrow{\beta }},$ we define ${\overrightarrow{g}}_r=b^T{\overrightarrow{A}}_r$; thus, ${\overrightarrow{A}}_r=b{\overrightarrow{g}}_r$ and matrix S can be expressed as $S=b\tilde{S}{b}^T$, where $\tilde{S}$ does not contain a background:

$${\tilde{S}}_{ij}={\delta }_{ij}+2i\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^N\frac{m_r{\mathsf{\Gamma }}_r^0{g}_{ri}{g}_{rj}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r}}$$

(51)

Obviously, $S$ is unitary if matrix $\tilde{S}$ is unitary. Therefore, we can independently find vectors $g_{ri}$ in $\tilde{S}$, and then return to the matrix $S$.

Matrix W can be constructed as a sequence of $M(M-1)/2$ rotations:

$$W=R_1\cdot R_2\cdot \dots \cdot R_{M(M-1)/2}$$

(52)

Matrix $R_k=\left\{r_{ij}^{(k)}\right\}$ $(i,j=1,\dots ,M)$ is convenient to write in the form:

$$r_{pp}^{(k)}=r_{qq}^{(k)}=\cos {\psi }_k, r_{pq}^{(k)}=-r_{qp}^{(k)}=\sin {\psi }_k,$$

(53)

$p=1,\dots ,M,$ $q=p+1,\dots ,M$; here ${\psi }_k\in [0,2\pi )$ are the rotation angles.

For instance, for three channels

$$b=W\left(\begin{array}{ccc}{e}^{i{\beta }_1}& 0& 0\\ 0& e^{i{\beta }_2}& 0\\ 0& 0& e^{i{\beta }_3}\end{array}\right)$$

where

$$W=R_1{R}_2{R}_3=\left(\begin{array}{ccc}\cos {\psi }_1& sin{\psi }_1& 0\\ -sin{\psi }_1& \cos {\psi }_1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos {\psi }_2& sin{\psi }_2\\ 0& -sin{\psi }_2& \cos {\psi }_2\end{array}\right)\left(\begin{array}{ccc}\cos {\psi }_3& 0& sin{\psi }_3\\ 0& 1& 0\\ -sin{\psi }_3& 0& \cos {\psi }_3\end{array}\right)$$

The quantities ${\beta }_k$, ${\psi }_k$ can be taken as fitting parameters (they can be functions of energy).

We, then, have

$$F_{ij}=(S_j-I)/2i={(B_F)}_{ij}+\sqrt{{\rho }_i{\rho }_j}{\displaystyle \sum _{r=1}^N\frac{m_r{\mathsf{\Gamma }}_r^0{A}_{ri}{A}_{rj}}{m_r^2-s-i{m}_r{\mathsf{\Gamma }}_r(s)}}$$

(54)

with the matrix

$$B_F=W\left(\begin{array}{ccc}\sin {\beta }_1{e}^{i{\beta }_1}& \dots & 0\\ \dots & \dots & \dots \\ 0& \dots & \sin {\beta }_M{e}^{i{\beta }_M}\end{array}\right)W^T$$

(55)

Thus, the background in the unitary BW method can be described by formulas (54) and (55).

To demonstrate this scheme, let us take the case of two resonances and three channels using the resonance parameters in

Table 4. BW pole functions.

r	${\mathit{m}}_{\mathit{r}}$	${\mathbf{\Gamma }}_{\mathit{r}}$	${\mathit{B}}_{\mathit{r}1} (\%)$	${\mathit{B}}_{\mathit{r}2} (\%)$	${\mathit{B}}_{\mathit{r}3} (\%)$
Resonance 1	1.30	0.10	40	40	20
Resonance 2	1.70	0.30	20.8	59.4	19.8

.

Next, we include a background in these resonant unitary BW amplitudes to see how it modifies the scattering amplitudes. In Figure 6, graphs of ${\left|F_{11}(E)\right|}^2$ and ${\left|F_{12}(E)\right|}^2$ (chosen as examples) are shown for the cases of diagonal and non-diagonal backgrounds. The assigned background parameters (the choice of these values is determined only for the purpose of illustrating the complex role of the background in the scattering amplitudes) are listed in the Figure 6 caption. For brevity, we use an energy-independent background.

It is seen that even a small ${10}^{°}$ diagonal background modifies the resonance shapes. The non-diagonal background affects even the peak locations. For example, the second peak in ${\left|F_{11}(E)\right|}^2$ is shifted by about 0.15 from $m_2$, the “correct” one. Since a background contribution may be very different to the intuitively expected smooth plateau in each of the channels, an adequate treatment of the background is absolutely necessary when describing overlapping multichannel states.

Because a priori there are no criteria for choosing the background parametrization, an additional uncertainty to resonance parameters is added. If some particular parametrization fits the data substantially better than the others, it can be considered as a practical criterion. If this is not the case, then a simplified approach may be the following: first, evaluate the background by smoothing (“smearing”) the data [11,12]—this procedure will remove all the peaks; then, subtract the background obtained in this way from the data and analyze the remainder using methods that preserve unitarity—a similar approach was used in work [13] to analyze overlapping ${\omega }^{\prime }(1400)$ and ${\omega }^{\prime }(1600)$ resonances (in the approximation of constant widths).

5. Conclusions

The parameters of broad inelastic resonances depend on the way the scattering data are analyzed. Obviously, the interference between resonances is the central part of an analysis and interpretation. In this respect, the usual Breit–Wigner parametrization which does not satisfy unitarity, is very unreliable (as shown in Figure 1); for example, it can possibly lead to deviations in mass of more than 100 MeV in the case of ${\rho }^{\prime }$ mesons [14].

There are different unitarization techniques to describe resonant states, both theoretical and phenomenological. Many analyses describe multi-channel resonances as poles on unphysical Riemann sheets for scattering amplitudes, as in works [10,15]. Resonances can be given by poles and corresponding zeros on an uniformization plane, which allows the description of broad multi-channel resonances [16]. The inverse amplitude method is another unitarization technique to present resonances and to enlarge the energy applicability region of effective Lagrangians [17].

We compare the results of the unitary BW approach and the standard widely used K-matrix method [5]. Both methods lead to similar results when resonances in all channels are well resolved, with a better ability of the BW method to resolve wide overlapping resonances. Another advantage of the BW method is that scattering amplitudes can be separated in the resonant and background contributions. A background, in terms of the phase shifts, does not necessarily form a smooth plateau in scattering amplitudes, and its adequate treatment is critically important for describing overlapping multichannel states.

Papers on overlapping resonances often include statements that because a sum of the BW functions violates the unitarity, a BW description cannot be used by default. Here, we draw attention to the fact that the unitary BW form for several states is possible when the BW functions are taken with the proper phase factors. From a theoretical point of view, it worth knowing that the problem formulated a long time ago [2], to present the unitary S-matrix as a sum of BW terms, has a regular solution. This approach can be used in fitting procedures, along with the K-matrix and other methods. Despite the K-matrix description being perfectly adequate as unitary parametrization for relatively narrow resonances, the names of its parameters, e.g., “masses”, “widths”, “and branching ratios” are only borrowed from the BW description through the comparison with a single resonance situation. A unitary BW model-independent method to describe overlapping resonances, with the parameters having direct physical meaning and the background being treated in a regular quantum mechanics form [18], can be a good addition to other unitarization techniques.