Model-independent resonance parameter extraction using the trace of K and T matrices

A model-independent method for the determination of Breit-Wigner resonance parameters is presented. The method is based on eliminating the dependence on the choice of channel basis by analyzing the trace of the K and T matrices in the coupled-channel formalism, rather than individual matrix elements of the multichannel scattering matrix.


Introduction
A general problem in theoretical baryon physics is to make a connection between resonances that are predicted by various quark models and experiment. A reasonable way to proceed is by identifying the poles of analytic functions that are able to describe simultaneously all experimental data in a multiplicity of existing channels with theoretically predicted resonant states. Therefore, properly and uniquely extracting resonance parameters from experiment is a task of primary importance. We emphasize the problem of uniqueness. The work described here is motivated by the need to extract Breit-Wigner resonance parameters from multichannel partialwave analyses (PWAs) in a model-independent way. Many PWAs of similar experimental data produce similar partial waves, while the extracted Breit-Wigner resonance parameters are often quite different. This fact can easily be seen in the Review method for extracting Breit-Wigner parameters from any unitary multichannel analysis able to provide the full T matrix, using the trace of the corresponding K matrix. Since all Breit-Wigner parameterizations are equivalent at the energy of the K-matrix pole, the parameters obtained using this method should be directly compared to those from quark models and lattice QCD. In order to connect the results of a model-independent K matrix extraction with those of a model-dependent analysis, e.g. based on the T matrix, we shall keep the relations defining multichannel T and K matrices as general as possible. It turns out that the T -matrix trace simplifies the formalism without loss of generality, and shows resonant behavior more prominently than any T matrix element does. To illustrate this, we shall take the T matrix from an earlier analysis [7] and recalculate the resonance parameters. The T -matrix trace happens to show resonant behavior at energies matching those of the K-matrix poles.

Multichannel scattering
The essence of any multichannel theory is the fact that the evolution of a system is no longer described by scalars, but by operators acting in an orthonormal wave-function space, and the transition probabilities for physical (measurable) processes are given by the matrix elements of their representation in the chosen basis. Once this basis is specified, the evolution of the system is described by solving equations which are matrices in the multichannel space, rather than scalar equations.
All equations given here are considered to be matrix relations, unless matrix indices are explicitly stated. The transition probability P a→b that a twobody system from initial channel |a; q ends up in the final two-body (or quasi-two-body) channel |b; q is given by the absolute square of the scattering S qmatrix element P a→b = | b; q|Ŝ q |a; q | 2 , where q designates all quantum numbers conserved in the scattering reaction, and a and b are channels. In the case of πN scattering we have conserved spin, parity, and almost conserved isospin (charge symmetry is only slightly violated). Conservation of probability is ensured if the S matrix (for simplicity, we drop q henceforth) is unitary. Therefore, the S matrix can be written as S = e 2iδ , where δ is some matrix Hermitian in the channel indices. Hermitian matrices have real eigenvalues and are diagonalized by uni-tary matrices. The δ matrix is related to a real, diagonal matrix δ D by a unitary transformation δ = U † δ D U , where U is a unitary matrix. The S matrix is evidently diagonalized by the same transformation, so S = U † e 2iδD U .
The K matrix [6,9] is defined as K = i(I −S)/(I + S), where I is the unit matrix. The K matrix can, in the eigenstate basis, be written using the diagonal matrix δ D as K = U † tan δ D U . The K matrix is Hermitian because S is unitary, and symmetric because of time-reversal invariance, so K is, in fact, a real matrix. Thus, U is a real orthogonal matrix that we henceforth designate as O.
Every diagonal N × N matrix can be spanned in the ortho-normal vector basis {E 1 , . . . , E N }: so, in our case, we have where δ i is the ith diagonal element of δ D , also known as the eigenphase shift, and N is the number of channels. We define the coupling matrices χ i to be and these matrices turn out to be ortho-normal projectors: where δ ij is the Kronecker δ symbol. The trace of a matrix is, by definition, a sum of its diagonal elements. A trace has two particularly important properties: i) the trace of a product of matrices is invariant with respect to cyclic permutations, Tr [ABC] = Tr [BCA]; and ii) the trace is a distributive function with respect to scalars α and β, The orthogonal transformation in definition (3) conserves the trace of a matrix, so It follows that the matrices K and T can be written as the sums χ j e iδ j sin δ j , (6) where the connection between the K and T matrices is given by the relation

Breit-Wigner parameterization
Elements of tan δ D , as well as the χ j , are functions of energy or a corresponding kinematical variable, and their description requires modeling of the energy dependence of numerous functions. We see resonances in scattering reactions as real poles of the K matrix. The rth element of the diagonal matrix tan δ D can be written as [9] tan where the selected pole term is parametrized in Breit-Wigner form, and it is singled out from other contributions, designated collectively as the background term at resonance tan δ r B . The Breit-Wigner mass (M r ) and total width (Γ r ) parameters are allowed to be functions of the center-of-mass total energy W . The reported Breit-Wigner parameters M R r and Γ R r are given by the values of M r (W ) and Γ r (W ) evaluated at an energy equal to the corresponding resonance mass M R r : where we have explicitly written M r and Γ r from Eq. (8) as functions of energy W . The corresponding K and T matrices are given by the equations where the second term in each equation is the coupled-channel background contribution, and Γ ′ r /2 represents Γ r /2 + (M r − W ) tan δ r B . When W equals the mass of the resonance, Γ ′ r is manifestly equal to Γ r . Although these relations are in general a sum over several resonances r, here they are written for one resonance for simplicity.
If there is a pole in the K matrix at some energy M R r , then the matrix element χ r ab at that energy gives the coupling strength of the resonance with mass M R r and total decay width Γ R r from channel a to channel b. The diagonal element of the matrix χ r is the branching ratio x r a of a given resonance to the channel a x r a = χ r aa . (12)

Extraction procedure
The channel dependence of resonance parameters can be reduced significantly by using only diagonal elements of the T and K matrices. In practice, these matrices can be obtained either by unitary coupledchannel partial-wave analyses, or by using partialwave T matrices obtained in diverse single-channel PWAs as input to a unitary coupled-channel formalism, and refitting them to obtain a unitary set of all coupled-channel T matrix elements.
Channel dependence is completely removed from the sums because the traces of the K and T matrices are the same as the traces of their similar diagonal partners tan δ D and e iδD sin δ D , respectively. The same is also evident from Eq. (5). Consequently, Eqs. (10) and (11) are simplified by taking the traces The last relation, i.e. the T -matrix trace, would be a good starting point for model-dependent extraction methods. However, instead of putting considerable effort into modeling the background and energyand channel-dependent resonance parameters, we use the following procedure: (i) The parameter extraction procedure starts when a full T matrix has been obtained from an energy-dependent partial-wave analysis of experimental data. (ii) Contrary to the usual prescription, where Eq. (11) is used to obtain resonance parameters from the T matrix in a model-dependent way, we use Eq. (7) to obtain the full K matrix from the known T matrix.
(iii) Poles of Tr K are found to obtain a set of resonance masses M R 1 , · · · M R NR , where N R is the number of resonances. (iv) Multiplying both sides of Eq. (14) by (M R k − W ) and setting the energy W to the value of the kth resonance mass (M R k ), the corresponding resonance width is isolated: All other contributions to the K matrix trace, i.e. background, other resonances, and channel-couplings, are removed in this limiting process (this relation turns out to be similar to Eq.(16) in Ref. [10] for the case of the various πN isospin channels). (v) The branching ratio of a resonance to a given channel can be obtained in similar manner, but this time using the diagonal K-matrix element, K aa from Eq. (10) and definition (12) where, as before, all undesired contributions vanish. (vi) Steps (iv) and (v) are then repeated for all resonances found in (iii).

Results and discussions
To illustrate the usefulness of our method, resonance parameters from a unitary, multi-resonance, coupled-channel analysis [7] have been extracted. The channels used in the analysis were πN , ηN , and an effective two-body channel designated as π 2 N . Extracted parameters are given in Table 1. The proposed model gives resonance parameters very close to the values obtained by a complicated method of diagonalizing the matrix of the generalized Breit-Wigner function denominator, with minimal calculation.
We have also compared the K-matrix trace to that of the T matrix. It can be seen in Fig. 1 that the Breit-Wigner resonance positions obtained by looking for the poles in Tr K (indicated by gray vertical lines) directly correspond to the positions of peaks in Im(Tr T ), and of zeros in Re(Tr T ). The peaks of the T -matrix elements corresponding to individual channels, however, show a certain deviation from that behavior. This suggests that fitting individual channels in order to obtain resonance parameters in-  Table 1 Resonance parameters extracted using the K-matrix procedure given in this paper are listed in bold face. The original T matrix was taken from Ref. [7] where the channels used were πN , ηN , and an effective two-body channel π 2 N . For comparison, Breit-Wigner resonance parameters from the original reference are shown below.
troduces an uncontrolled error, which is avoided if the trace of the T matrix is used. Unexpectedly, and contrary to previous findings, the resonance parameters produced by the K-matrix extraction method presented here, are in accordance with values obtained by the original analysis as well as with the T -matrix trace. The procedure involves no fitting, diagonalizing, nor modeling of the energy dependence of the resonance parameters and background. Furthermore, a model-independent procedure cannot be given with the T -matrix formalism, because background makes a substantial contribution to the T matrix, even at an energy equal to the resonance mass, M R . The T -matrix background is removed at a complex energy equal to the T -matrix pole position. This might be the reason why extractions of T -matrix poles work much better than Tmatrix extractions of Breit-Wigner parameters. By using the trace of the K matrix, background has been completely removed from consideration at the resonance energies. With regard to the differences between the two approaches listed in Table 1, it is rather striking that all of them can be explained by arguments presented in the original analysis. Since an effective π 2 N channel was introduced in [7] to parametrize the first inelasticity in each partial wave, the parameters of lowlying resonances should be much better determined than those of heavier ones (especially the third resonances in S 11 and D 13 ). A better quality of parameters is also expected for resonances that couple more strongly to the measured channels considered here. Therefore, N (1720) P 13 and the resonance(s) in G 17 have unrealistic parameters since they are completely driven by the effective channel, as can be clearly seen from Fig. 1. These problems should be removed by the explicit inclusion of additional channels in the partial-wave analysis.
The parameters of the two lowest resonances in the S 11 and D 13 partial waves, as well as those of the D 15 , second P 11 , and F 15 resonances, are in rough accordance with quark-model expectations for their masses and partial widths [11], with the exception of the mass of the second D 13 , which is predicted to be roughly degenerate with the second S 11 and the D 15 resonance. This disagreement could be explained by the large coupling of this state to the effective channel. The large width and the somewhat larger mass of the first P 11 (Roper) resonance extracted using the K-matrix procedure bring these parameters closer to those of the class of quarkmodel calculations based on one-gluon exchange potentials and pair creation for strong decays.

Conclusions
We have presented a model-independent method for resonance parameter extraction using the Kmatrix formalism. It is shown that real poles of the K matrix are related to the resonant behavior of the trace of the T matrix. Our resonance parameter extraction procedure is simple and straightforward once the full T matrix is known. Unrealistic extracted parameters for some higher mass resonances point to the need to include additional channels in partial-wave analyses.
At the energies of the K-matrix poles, the influence of background and channel mixing is eliminated, so only parameter values obtained at this particular energy should be compared directly to the predictions of quark model and lattice QCD calculations.
This model-independent procedure cannot be extended to the T -matrix formalism because background makes a substantial contribution to the T matrix, even at the resonance energies M R . This might be the reason why methods that extract Tmatrix poles work much better than those which extract Breit-Wigner parameters from the T matrix. By using the trace of the K matrix, the background has been completely removed from consideration at resonance energies.