Resonance parameters from K-matrix and T-matrix poles
aa r X i v : . [ nu c l - t h ] F e b Resonance parameters from K matrix and T matrix poles R. L. Workman, R. A. Arndt and M. W. Paris
Center for Nuclear Studies, Department of PhysicsThe George Washington University, Washington, D.C. 20052 (Dated: October 31, 2018)
Abstract
We extract K matrix poles from our fits to elastic pion-nucleon scattering and eta-nucleonproduction data in order to test a recently proposed method for the determination of resonanceproperties, based on the trace of the K matrix. We have considered issues associated with theseparation of background and resonance contributions, the correspondence between K matrix and T matrix poles, and the complicated behavior of eigenphases. PACS numbers: PACS numbers: 11.55.Bq, 11.80.Et, 11.80.Gw
1n a study by Ceci and collaborators [1], a method for resonance parameter extractionwas proposed, based on properties of the trace of the T matrix and the associated K matrix,from a multi-channel fit to scattering data. The relevant relations are [1],Tr( K ) = ˜Γ R / E R − E + N X j = R tan δ j , (1)and Tr( T ) = ˜Γ R / E R − E − i ˜Γ R / N X j = R e iδ j sin δ j , (2)where ˜Γ R / R / E R − E ) tan δ B , (3) E R is the resonance energy, Γ R is the full width, and δ B is a background phase, and N isthe number of included channels. The index R labels the j = R element of the diagonal K matrix, and δ j is an eigenphase. These expressions follow directly from the expressions forthe K and T matrices in terms of the eigenphasesTr( K ) = N X j =1 tan δ j , Tr( T ) = N X j =1 e iδ j sin δ j , (4)assuming a single diagonal element of the K matrix has the resonant formtan δ R = Γ R / E R − E + tan δ B , (5)with a K matrix pole at E = E R and the non-pole behavior collectively described by thebackground phase. The quantities Γ R , E R and δ B are considered as functions of the energy, E . The importance of Eqs.(1) and (2) for Ref.[1] is that while the position of the T matrixpole of the first term in Eq.(2), E R − i ˜Γ R /
2, and its residue, ˜Γ R /
2, depend on δ B , theposition of the K matrix pole in Eq.(1), E R , and its residue, Γ R , do not. This is the modelindependence[2] cited in Ref.[1]. In light of this, the authors of Ref.[1] suggest the use of K matrix pole positions and residues to give a model-independent characterization of resonancestructure. Their method involves the determination of the K matrix, from a given T matrix,from which the pole positions, E R , and their residues, Γ R , are extracted.We have explicitly tested this method with a set of amplitudes determined in recent fitsto pion-nucleon scattering and eta-nucleon production data. Using the T matrix in a given2artial wave, determined in fits to the observed data [3], we determine the K matrix, fromwhich we extract the pole positions and residues for real energies. This analysis yields atleast two results which undermine the utility of using the positions and residues of K matrixpoles as model-independent characterizations of resonance structure. First, we show thatassuming a different form for Eq.(5) alters the finding of Ref.[1] that the K matrix pole andresidue are independent of the background, tan δ B . Second, using the T matrix determinedin Ref.[3], we numerically calculate the related K matrix (see Eq.(6)) and find that thereare poles in the T matrix which have no nearby poles in the K matrix for real energies.This would seem to obviate the use of K matrix poles in characterizing resonance structuresobserved in scattering experiments, since the structures present in the T matrix do notnecessarily appear in the K matrix.Prior to describing our numerical results, we revisit the derivation of Eqs.(1) and (2) inrelation to the assumptions of Eq.(5). We then compare the result of Ref.[1] with the resultobtained with a different assumption (following Dalitz[4]) for the parameterization[5] of theresonant eigenphase of Eq.(5).The K matrix we use is real for energies above all thresholds considered in the problem,and is related to the T matrix by T = K (1 − iK ) − . (6)The real symmetric K matrix is diagonalized by an orthogonal transformation, U as K D = U T K U. (7)This matrix also diagonalizes the T matrix, and therefore the S matrix, defined as S =1 + 2 iT . Since S is a unitary matrix,( S D ) ij = U T SU = δ ij e iδ i (8)where the δ i are eigenphases. Using the relation between K and T matrices above, we have( K D ) ij = i (1 − S D )(1 + S D ) − = δ ij tan δ i . (9)Having determined the eigenphases, we can reconstruct the physical T matrix T if = ( U T D U T ) if = X α U iα U fα e iδ α sin δ α . (10)3aking the trace of Eqs. (9) and (10) gives the result in Eq. (4).We first examine the use of these relations in a simple scenario including a single resonanteigenphase and neglect any background effects. Assuming only one eigenphase ( δ R ) passesthrough π/ E R and neglecting others, the resonant eigenphase must have theform tan δ R = Γ R / E R − E , (11)which leads to T if = 12 Γ / i Γ / f E R − E − i Γ R / . (12)with Γ i = U iR Γ R and P i U iR = 1 (orthogonality) giving P i Γ i = Γ R . The result, Eq.(12)is consistent with Eqs.(1) and (2) for δ B = 0. Next we consider how background can beadded, and whether a single dominant eigenphase is appropriate. These questions have beenaddressed in the works of Dalitz [4], Goebel and McVoy [6], and Weidenm¨uller [7].Consider first the addition of a background phase to Eq.(11). One way of doing thisis the ans¨atz of Eq.(5) employed in Ref.[1] (see also Ref.[8]) and used to obtain Eqs.(1)and (2). This leads to the model independence of Ref.[1] described above. An alternativeparameterization of the resonant eigenphase is considered in Refs.[4, 6, 7]. The ans¨atz usedthere also assumes a single dominant eigenphase, which rises through π/
2, but posits aphase-addition rule: the resonant eigenphase, ˜ δ R has the form˜ δ R = ˜ δ B + δ P , (13)where ˜ δ B is the background phase which determines the eigenphase far from the resonanceenergy, and δ P is the resonant (pole) contribution. This form of resonance and backgroundseparation modifies the above conclusion of model independence. We consider the phase-addition rule in some detail to clarify this point.As a function with a simple pole, the resonant contribution, δ P may be written in generalas tan δ P = γ ( E ) / E ∗ ( E ) − E , (14)where the position of the pole is given by E ∗ P ( E ∗ ( E ∗ P ) − E ∗ P = 0) and the function γ ( E )goes to a non-zero constant at the pole. Note that, far from the pole, the eigenphase shift ˜ δ R reduces to the non-pole part, ˜ δ B . Using Eqs.(13) and (14) we compute the resonant element4f the diagonal K matrix as tan ˜ δ R = γ + ( E ∗ − E ) tan ˜ δ B ( E ∗ − E ) − γ tan ˜ δ B , (15)which leads to a K matrix with the traceTr( K ) = Γ( E ) / E ∗ P ( E ) − E + N X j = R tan δ j , (16)where Γ( E ) / γ/ E ∗ − E ) tan ˜ δ B and the location of the pole in Tr( K ) is E ∗ P , where[( E ∗ ( E ) − E ) − γ ( E )2 tan ˜ δ B ( E )] (cid:12)(cid:12)(cid:12)(cid:12) E = E ∗ P = 0 , (17)In general, E ∗ P = E ∗ P and the pole position of the K matrix, E ∗ P depends on the background,tan ˜ δ B . The residue also depends on ˜ δ B since Γ( E ∗ P ) = γ/ cos ˜ δ B .We could anticipate this result by comparing the forms Eqs.(5) and (13). In Eq.(5),used in Ref.[1], the location of the K matrix pole, E R is independent of δ B . The resonantstructure, Γ / [2( E R − E )], and the non-resonant contribution are assumed to be decoupled.That is, if tan δ B is a bounded function of the energy, its value cannot affect the energy wherethe resonant eigenphase, δ R is π/
2. In the “Dalitz form,” Eq.(13), the location of the polein the K matrix, determined by the energy E R where the phase δ R → π/
2, is affected by the“background phase,” δ B . Since the true form of the K matrix is unknown, the existence ofalternative forms complicates the extraction of pole positions. In fact, in dynamical modelsof scattering amplitudes, the location of the K matrix pole is expected to depend, perhapsstrongly, on the non-resonant (or background) contribution to the amplitude [9].Turning to the T matrix, in place of Eq.(12), the result is T if = 12 e i ˜ δ B Γ ′ / i Γ ′ / j E ′ R − E − i Γ ′ / U iR U jR e i ˜ δ B sin ˜ δ B , (18)for the corresponding T matrix element with resonance ‘mass’ and ‘width’ shifted fromthe K matrix pole parameters. Thus, a different scheme for the addition of resonanceand background contributions can alter the relationship between K matrix and T matrixresonance masses.As another example of the model dependence of K and T matrix poles, and to address thequestion of whether a single resonant eigenphase is appropriate, we consider the following5
200 1400 1600 1800 2000W [MeV]0306090120150180 δ [ d e g ] FIG. 1: The eigenphases in a four-channel fit to the S partial wave from the SP06 solution fromSAID. simple S matrix from Refs.[6, 7] S ij = e i ( φ i + φ j ) " δ ij + i Γ i / Γ j / E R − E − i Γ / , (19)to show the effect of background on eigenphases. This S matrix is symmetric and far from theresonance energy is diagonal (the elastic background approximation) with eigenphases φ i [10,11]. Applying the unitary transformation diagonalizing Eq.(19) and taking the determinant,yields e i P i δ i = e i P i φ i E R − E + i Γ / E R − E − i Γ / , (20)where δ i is an eigenphase and the last factor has the phase behavior of an elastic resonance at E R . From Eq.(20) we see the above phase-addition rule, Eq.(13), if only a single eigenphaseis significant. In general, however, it is the sum of eigenphases that displays resonancebehavior.A few further comments on the parameterization of eigenphases may be useful. Wei-denm¨uller [7] has shown that individual eigenphases have an energy dependence determinedlargely by the background. Through an application of Wigner’s no-crossing theorem, he findsno single eigenphase increasing by π , except for special values of the background phases.6 JT T poles K poles S (1500 ,
50) (1650 ,
40) 1535 1675 P (1360 ,
80) (1390 , † − − P (1665 , − D (1515 , − D (1655 ,
70) 1760 F (1675 ,
60) (1780 , − − TABLE I: Pole positions in complex energy plane of T and K matrix for the πN → πN reactionfrom SAID [3] for isospin T = partial waves. Each T pole position is expressed in terms ofits real and imaginary parts ( M R , − Γ R /
2) in MeV. Only K matrix pole positions which satisfy1 . < W < . † This pole is located on the second Riemann sheet.
As a result, the eigenphases ‘repel’ rather than crossing, the N eigenphases individuallyincreasing only by an average of π/N over the width of the resonance in some cases. Anexample of this behavior is given in Fig.(1), which shows the eigenphases calculated from SAID [3] for the S partial wave, containing two resonances.Goebel and McVoy have applied the eigenvalue method to resonant d − α scattering [6]data to explicitly study this behavior. Eigenvalues for this two-channel scattering matrixwere also given, showing the appearance of square-root branch points which complicate theenergy dependence [12]. There is a cancellation occurring when the sum of eigenvaluesor eigenphases is taken, and this supports the basic idea of using a trace, as proposed inRef. [1]. A direct relation between resonance energy and the sum of eigenphases is given bythe equation [13, 14] tr Q = 2¯ h X i dδ i dE , (21)relating the trace of Smith’s time-delay matrix to the energy derivative of the sum of eigen-phases, δ i . One diagonal element of the Q matrix has recently been shown to correlateprecisely with the T-matrix pole positions of resonances [14].To explicitly test the method of Ref. [1], we have taken amplitudes determined from ourfits to pion-nucleon elastic scattering data [3], and the reaction πN → ηN . The parame-terization we use is based on the Chew-Mandelstam (CM) K matrix, which builds in cutsassociated with the opening of ηN , π ∆, and ρN channels. The CM form is analytic and7enerates a T matrix which is unitary and can be continued into the complex plane to findpoles on the various sheets associated with the ηN , π ∆, and ρN channels. The fits can,in principle, include couplings to any of the channels, though the π ∆ and ρN channels arenot constrained by data. However, the amplitude associated with each channel has, by con-struction, the proper threshold behavior, cuts, and pole positions. Amplitudes in the elasticchannel are further constrained by forward and fixed- t dispersion relations.Thus far, we have implicitly assumed that there is a direct correspondence between K matrix and T matrix poles. It is known [15] that this is not true in general. For example,in the CM approach it is possible to generate T matrix poles for the resonances withoutexplicitly adding a pole to the CM K matrix [16]. If, however, a pure CM K matrix polerepresentation is used K ij = γ i γ j ( ρ i ρ j ) / E K − E , (22)the resulting T matrix is T ij = γ i γ j ( ρ i ρ j ) / E K − E − P n γ n C n − i P n γ n ρ n (23)where ρ i is the phase space factor for the i th channel, and C i is the real part of the Chew-Mandelstam function, obtained by integrating phase space factors over appropriate unitaritycuts.The fit under consideration uses a parameterization of a CM K matrix[16], from whichthe unitary T matrix is calculated. The K matrix, defined by Eq.(6), is computed fromthe calculated T matrix as K = T (1 + iT ) − . The resulting K matrix was checked forconsistency by reproduction of the T matrix from Eq.(6), and checked for unitarity at each ℓ JT T poles K poles S (1595 ,
70) 1660 P (1770 , − P (1210 ,
50) (1460 , − D (1630 , − D (2000 , − F (1820 , − TABLE II: Pole positions in complex energy plane as in Table I for isospin T = partial waves. K matrix was then searched for poles at energies associated with well-knownresonances. When poles did appear in a given amplitude, we confirmed that they appearedin each associated amplitude at the same energy. However, we did not generally find polesclosely associated with ( T matrix) resonance energies, nor did we find that each resonanceproduced a K matrix pole, as shown in Tables I and II. If an explicit pole was insertedinto the CM K matrix, then this approach generated a corresponding K matrix pole. Thiswas the case for the ∆(1232) resonance, where we found a K matrix pole at 1232 MeV. K matrix poles also appeared near the N (1535), N (1650), and ∆(1620) resonance masses,in the πN S and S partial waves, though no explicit CM K matrix poles were used inthe fit. For the P and D partial waves, however, no CM K matrix poles were used inthe fit, and no K matrix poles were found near the resonance masses. In all of these cases,resonance poles did appear in the corresponding T matrices.In conclusion, we have not found a simple association between K matrix and T matrixpoles for use in the extraction of resonance properties. We have argued that: i) K matrixpoles are not generally independent of background contributions, ii) a pole in the T matrixdoes not necessarily imply a pole in the K matrix. Therefore, K matrix poles do not appearto be useful candidates for characterizing resonance parameters obtained from scatteringamplitudes. Applied to a particular S matrix obtained from a fit to pion-nucleon and eta-nucleon scattering data [3] we find no one-to-one association between K matrix and T matrixpoles. We have also noted that the separation of background and resonance contributions isnot unique and that eigenphase behavior may be more complicated than the form chosen inRef. [1]. We have noted an explicit counterexample for the parameterization of the resonanteigenphase, specifically Eq.(13), which violates the model independence of Ref.[1]. We arecurrently exploring the behavior of eigenphases using S matrices from scattering amplitudesin order to determine whether eigenphase repulsion is as common as suggested in Ref. [6]. Acknowledgments
This work was supported in part by the U.S. Department of Energy Grant DE-FG02-99ER41110 and funding provided by Jefferson Lab.9
1] S. Ceci, A. Svarc, B. Zauner, D.M. Manley, and S. Capstick, Phys. Lett.
B 659 , 228 (2008).[2] “Model independence” here is used in a sense similar to that considered in R. M. Davidsonand N. C. Mukhopadhyay, Phys. Rev. D , 20 (1990). We note that the analysis there isspecific to the elastic resonance, P (1232).[3] R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Workman, Phys. Rev. C, , 045205(2006)[4] R. Dalitz, Ann. Rev. Nucl. Sci. , 339 (1963). We refer to Eqs. 3.23 to 3.26 of this review.[5] R. M. Davidson and N. C. Mukhopadhyay, Phys. Rev. D , 71 (1991). In this work, threepossible parameterizations of tan δ are considered.[6] C.J. Goebel and K.W. McVoy, Phys. Rev. 164, 1932 (1967).[7] H.A. Weidenm¨uller, Phys. Lett. , 441 (1967).[8] See A.D. Martin and T.D. Spearman, Elementary Particle Theory, North Holland, Amster-dam, 1970, p. 410, and Ref. [15], Ch. 18. Signs differ but appear to be correct in Ref. [15].[9] M. W. Paris and R. Workman, in preparation[10] Note that a product form of background and resonant S matrices S = S B S R , generally yieldsa non-symmetric S matrix. This violates time-reversal invariance, which is required to obtaina real K matrix.[11] The 2 × S matrix in Eq.(19) is sufficiently simple to allow an explicit calculation of eigen-phases [6], σ ± , using σ ± = 12 ( S + S ) ± h ( S − S ) + 4 S i / . (24)[12] Though individual eigenphases have a complicated behavior, the sum is much better behaved.See, for example, the discussion by M.L. Goldberger and C.E. Jones, Quanta: Essays intheoretical physics dedicated to Gregor Wentzel, Ed. by P.G.O. Freund, C.J. Goebel, and Y.Nambu, U. of Chicago Press, Chicago, 1970, p. 209.[13] F.T. Smith, Phys. Rev. , 349 (1960).[14] R.L. Workman and R.A. Arndt, ‘Resonance signals in pion-nucleon scattering from speedplots and time delay’, arXiv:0808.0342.[15] R. Levi Setti and T. Lasinski, Strongly Interacting Particles (U. of Chicago Press, Chicago,1973), Ch. 18.
16] The Chew-Mandelstam formulation has been discussed in R.A. Arndt, J.M. Ford, andL.D. Roper, Phys. Rev. D , 1085 (1985)., 1085 (1985).