Relating multichannel scattering and production amplitudes in a microscopic OZI-based model
aa r X i v : . [ h e p - ph ] O c t Relating multichannel scattering and productionamplitudes in a microscopic OZI-based model
Eef van Beveren
Centro de F´ısica Te´orica, Departamento de F´ısica, Universidade de CoimbraP-3004-516 Coimbra, Portugal http://cft.fis.uc.pt/eef
George Rupp
Centro de F´ısica das Interac¸c˜oes Fundamentais, Instituto Superior T´ecnicoUniversidade T´ecnica de Lisboa, Edif´ıcio Ciˆencia, P-1049-001 Lisboa, Portugal [email protected] number(s): 11.80.Gw, 11.55.Ds, 13.75.Lb, 12.39.Pn
October 25, 2018
Abstract
Relations between scattering and production amplitudes are studied in a microscopicmultichannel model for meson-meson scattering, with coupling to confined quark-antiquarkchannels. Overlapping resonances and a proper threshold behaviour are treated exactly inthe model. Under the spectator assumption, it is found that the two-particle productionamplitude shares a common denominator with the elastic scattering amplitude, besidesa numerator consisting of a linear combination of all elastic and some inelastic matrixelements. The coefficients in these linear combinations are shown to be generally complex.Finally, the standard operator expressions relating production and scattering amplitudes,viz. A = T /V and ℑ m ( A ) = T ∗ A , are fulfilled, while in the small-coupling limit the usualisobar model is recovered. In a very recent article [1] we have shown that several hadronic three-body decays of
J/ψ , D and D s mesons can be well described, up to moderately high energies, in a model for production pro-cesses derived from the so-called Resonance-Spectrum Expansion (RSE) [2]. The RSE formalismamounts to an effective description of non-exotic meson-meson scattering, based upon quark-antiquark pair creation and annihilation allowing transitions between an infinity of confined q ¯ q states and the meson-meson continuum. An essential feature of the RSE is that it gives riseto closed-form expressions for the S -matrix and even the fully off-shell T -matrix. Hence, exactanalyticity and unitarity properties, as well as a correct (sub)threshold behaviour, are manifestlysatisfied. Moreover, the resulting meson-meson production amplitude can be derived exactly,1oo, by summing the corresponding two-body Born series, the only assumption being that thethird particle acts as a mere spectator [1].In the present paper, we shall further develop the formalism introduced in Ref. [1], so as tocover the most general multichannel case in mesonic 3-body decays. Clearly, at higher energiescompeting inelastic 2-meson channels require that the production amplitude be described bya vector and not a scalar function. The underlying 2-body scattering T -matrix is then a truematrix. Furthermore, also the quark-antiquark sector needs an extension, as there can be mixingof different q ¯ q channels that couple to the same meson-meson channels. This is the case in e.g.the production of I = 0 ππ and K ¯ K pairs, which both couple to the n ¯ n (= ( u ¯ u + d ¯ d ) / √ s ¯ s channels, giving rise to the isoscalar scalar resonances f (600) (alias σ ) and f (980).Finally, having a general and exact — within the model assumptions — expression for productionamplitudes at hand, we may carry out a detailed comparison with the ansatzes employed in otherapproaches, focusing on common features as well as clear differences.Under the spectator approximation, we assume that a pair of mesons is created out of one q ¯ q pair emerging from the original decay, accompanied by a non-interacting spectator meson.In this process, we only consider OZI-allowed [3] strong transitions to pairs of mesons. Theintensities of transitions between the initial q ¯ q pair (its quantum numbers, including flavour, areabbreviated by α ) and the various meson-meson pairs i allowed by quantum numbers are givenby coupling constants g αi determined via the recoupling scheme of Ref. [4]. In Sec. 3, we derivethe matrix elements t ( i → ν, E ) of the scattering amplitude at total CM energy E = √ s , fortransitions between the meson-meson channels i and ν . In the same section, we establish a relationbetween the common denominator D ( E ) of all matrix elements t (anything → anything , E ) andthe numerators of diagonal matrix elements of t ( E ), the latter representing elastic scattering.The production amplitude a ( α → i, E ), which is related to the probability of producing a meson-meson pair i , assuming that a q ¯ q pair emerges in the initial — here not described – stages of thedecay process, is determined in Sec. 4. Note that the process giving rise to the initial q ¯ q pairplus the spectator meson can be either weak or strong yet OZI-suppressed (see Ref. [1] for someexamples).The central result of the present paper is a relation between the production and scatteringamplitudes which can be formulated as a ( α → i, E ) ∝ g αi D ( E ) + i - λ X ν { g αi x ν ( E ) t ( ν → ν, E ) − g αν x i ( E ) t ( i → ν, E ) } , (1)where the x i stand for momentum distributions that will be specified in Sec. 4 (Eq. 19).We thus obtain the result that the production amplitude is in the first place given by thecommon denominator of the scattering amplitudes. This implies that, within the RSE formalism,resonance poles are identical for production and scattering, at least in the spectator approxima-tion. Secondly, we find that the remainder of the production amplitude to the i -th two-mesonchannel is proportional to the sum of the differences between all possible elastic scattering am-plitudes t ( ν → ν, E ) and the inelastic amplitudes for the i -th channel, t ( i → ν, E ). This does notspoil our conclusion about the resonance poles, since all T -matrix elements share the common denominator. Note, however, that the contribution of the term ν = i vanishes in the expressionbetween braces on the r.h.s. of Eq. (1). Consequently, the amplitude a ( α → i, E ) for the produc-tion of a two-meson pair i does not carry any dependence on the amplitude for elastic scattering i → i . In those cases where the coupling to different meson-meson channels vanishes or can beneglected, implying a 1 × T -matrix, the resulting production amplitude is solely determined bythe common denominator D ( E ) [1]. 2ome words are due about the K -matrix formalism. In general, and so also here, the T -matrixcan be written as T = K/ (1 − iK ), where K is a real symmetric matrix. So, at first sight, itseems that the common denominator of all T -matrix elements is given by 1 − iK . However, thisis not the case. First of all, K is a matrix and not just a real function. But even in the 1 × K can be represented by a real function, it has a denominator itself, the zeros of whichare the K -matrix poles, located at the real energies where some eigenphase shift passes through90 ◦ . The common denominator above is then the sum of the denominator of K plus − i timesthe numerator of K . In general, when K is a matrix, this relation involves the determinant of K . Hence, comparing K -matrix poles, lying on the real axis in the complex energy plane, andresonance poles, which are usually in the second Riemann sheet with respect to the nearest “open”threshold, is far from trivial. Moreover, for some resonances, like the σ and the κ ( K ∗ (800) [5]),the respective K -matrix poles, corresponding to an elastic phase shift δ = 90 ◦ , lie about 350–600MeV higher in energy that the real parts of the respective S -matrix poles, while mixing withother resonances ( f (980) and K ∗ (1430)) further complicates the picture. So rather than making ad hoc assumptions about poles of the production amplitude, we shall straightforwardly derivethe latter, and then see what its properties are.A final remark here concerns Watson’s [6] theorem for production. This theorem implic-itly relies on having a potential which is energy independent or only weakly energy dependent.However, this is not the case here, because the energy dependence of our effective meson-mesonpotential is far stronger than that of the scattering T -matrix. As a consequence, the energydependence of the production experiment does not resemble at all the one of the T -matrix, andall exercises imposing the Watson “theorem” or theorems derived from it are inappropriate here.This issue is analysed in more detail in Ref. [7]. Scattering from a weakly coupled resonant source has been studied in a variety of differentapproaches. For such systems it is observed that resonances occur at energies that are close tothe unperturbed spectrum of the resonant source. Widths and mass shifts can be determined byperturbative methods, and expressed in terms of pole positions of the resonances in the complexenergy plane.Intuitively, however, perturbative methods do not offer the correct strategy for strong interac-tions. Since in the present paper we are interested in obtaining exact relations between scatteringand production amplitudes, which are moreover based on a microscopic description in terms ofquarks, we rather fall back upon an approximate yet exactly solvable theory or model. Such amanifestly unitary and analytic framework is provided by the RSE.The RSE aims at describing the scattering of meson pairs in non-exotic channels, therebyassuming that in the interaction region a meson pair may temporarely transform into a quark-antiquark pair through q ¯ q annihilation and subsequent creation. The transitions of the system,from meson-meson pairs to q ¯ q pairs and vice-versa, are described by an off-diagonal potential V t in the RSE, linking these two sectors to each other. It has a maximum at an interquark distance r which depends on the average effective quark mass and runs from slightly less than 0.2 fm for b ¯ b to about 0.6 fm for light quarks. Furthermore, we assume that this mechanism gives rise tothe dominant meson-meson interaction in non-exotic channels. Here, we limit ourselves to thecase where V t is considered the only interaction.The intermediate q ¯ q states are supposed to have an unperturbed confinement spectrum de-3ending on the quantum numbers of the system. Its energy eigenvalues are all contained in thetwo-meson scattering matrix, which, consequently, develops corresponding CDD resonance poles.An addtional nice feature of the RSE, which will turn out to be crucial for the constructionof the production amplitude, is the possibility to obtain the closed-form scattering T -matrixboth in configuration and in momentum space. In the former representation, coupled-channelSchr¨odinger equations with the proper boundary conditions directly lead to the full solution,while in the latter picture individual Born terms can be explicitly calculated and then exactlysummed up owing to the general separable nature of the effective meson-meson potential, with noneed to solve the Lippmann-Schwinger [8] integral equations. This allows to verify the correctnessof the momentum-space approach in the scattering case, which is the only method at our disposalto describe production. As we shall see below, a similar Born series can then be written downand summed up. The building blocks of the RSE meson-meson scattering amplitude are the effective meson-mesonpotentials and the free two-meson propagators, which are graphically represented in Fig. 1. It isthe philosophy of the RSE that confinement and decay can be separated. Hence, in the interactionregion, a two-meson system can appear as a permanently confined system consisting of a valencequark and a valence antiquark. Possible intermediate crypto-exotic multiquark states are notconsidered in the RSE. V (effective 2-meson potential) (two-meson propagator)Figure 1: Graphical representation of the building blocks of the RSE two-meson scattering amplitude.The solid lines represent valence quarks and antiquarks as in usual Feynman diagrams. In contrast,the gray areas stand for all possible confining interactions, like gluon exchange, sea-quark loops andtheir higher orders. The effective meson-meson interaction is represented by V . Furthermore, althoughthe mesons in the two-meson propagators are considered pointlike in the RSE, for clarity they are hererepresented by double quark lines connected by confining interactions. The dynamics of the intermediate q ¯ q states is described by a permanently confining Hamil-tonian H c , which has a complete set of eigenstates at eigenvalues representing the confinementspectrum. The other part of the strong interactions, generating transitions between a two-mesonsystem and a q ¯ q state, is given by a transition potential V t . Consequently, the effective meson-meson interaction in Fig. 1 is described by the operator V = V tT [ E − H c ] − V t , (2)where E is the total invariant mass of the coupled-channel system. These interactions and thefree two-meson propagators, both depicted in Fig. 1, can then be used in an ordinary Lippmann-Schwinger [8] approach to scattering. Nevertheless, in constructing the Born series, quark-loopcontributions to all orders are automatically accounted for, as becomes clear from Fig. 2.4 jV + V V + V V V + · · · Figure 2: Graphical representation of the RSE scattering amplitude.The RSE scattering amplitude depicted in this figure has the Born term V ( i → j ) = h i, ~p i | V tT [ E ( p ) − H c ] − V t | j, ~p j i . (3)In Ref. [2] it was shown how, under the RSE assumptions, the integrations can be done analyt-ically. For the present discussion, it is only necessary to mention the generic form of the RSEexpression, given by V ( i → j ) = - λ π ∞ X ℓ =0 (2 ℓ + 1) P ℓ (ˆ p i · ˆ p j ) j ℓ ( p i r ) j ℓ ( p j r ) Z ( ℓ ) ij ( E ) . (4)The overall coupling - λ and the interaction radius r represent the total probability of quark-paircreation/annihilation and the average interquark distance at which such processes take place,respectively; j ℓ stands for the spherical Bessel function for the relative meson-meson angularmomentum ℓ ; ~p i and ~p j are the relative linear momenta in the two-meson channels i and j ,respectively. The matrix Z is a real and symmetric function of the total invariant mass of thesystem.The intermediate q ¯ q systems may have different orbital angular momenta, for the same quan-tum numbers. For example, a meson-meson system with J P C = 1 −− couples to q ¯ q systems ineither an S or a D wave. On the other hand, isosinglet mesons can be mixtures of differentquark flavours, usually n ¯ n and s ¯ s . In such cases, more than one type of q ¯ q states are involved inthe quark loops of the process depicted in Fig. 2. However, since the recurrencies of the perma-nently confined q ¯ q systems are automatically included by the definition of the Born term (4), thenumber of q ¯ q channels that couple to a specific set of two-meson quantum numbers is limited,usually to one or two. Nevertheless, the number of two-meson scattering channels is in principlenot restricted.For any number of coupled confinement and scattering channels, the general structure of theamplitude reads ( E = E ( p i ) = E ( p j )) t ( i → j ) = h i, ~p i | t | j, ~p j i = h i, ~p i | ( V + V GV + V GV GV + . . . ) | j, ~p j i (5)= - λ π ∞ X ℓ =0 (2 ℓ + 1) P ℓ (ˆ p i · ˆ p j ) j ℓ ( p i r ) j ℓ ( p j r ) A ( ℓ ) ij ( E ) D ( ℓ ) ( E ) , where A and D are functions of the total invariant mass E satisfying the unitarity condition ℑ m (cid:16) D ( ℓ ) A ( ℓ ) ij ∗ (cid:17) = 2 - λ X ν µ ν p ν j ℓ ( p ν r ) A ( ℓ ) iν A ( ℓ ) jν ∗ . (6)The denominator D contains the full pole structure of the coupled two-meson states. In orderto be a bit more specific, let us consider the scattering of charmed mesons, i.e., D ¯ D , D ∗ ¯ D , D ∗ ¯ D ∗ , D s ¯ D s , D ∗ s ¯ D s and D ∗ s ¯ D ∗ s , all coupled to c ¯ c . For such a process, D has in the RSE the form D ( ℓ ) ( E ) = 1 + 2 i - λ X ν g ν ∞ X n =0 (cid:12)(cid:12)(cid:12) F ( n ) c ¯ c ( r ) (cid:12)(cid:12)(cid:12) E − E n µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) , (7)5here the outer sum runs over all two-meson channels, and the inner sum over all recurrencies n for the operator H c describing confinement in the c ¯ c system. F ( n ) c ¯ c and E n represent the eigenstateand eigenvalue of the n -th recurrency of the H c spectrum, respectively. Furthermore, the g ν standfor the relative couplings of each of the two-meson systems to c ¯ c , while h (1) ℓ is a spherical Hankelfunction of the first kind.The denominator D ( E ) vanishes for E near E n and small overall coupling - λ . In this case,the scattering cross sections in all channels display narrow spikes for values of E in the vicinityof E n ( n = 0, 1, 2, . . . ). Hence, for small - λ , the theoretical cross sections reproduce — up tosmall shifts — the hypothetical c ¯ c confinement spectrum.However, for larger values of - λ the zeros in D are no longer near the eigenvalues of H c , butmove deeper into the complex E plane, farther away from the real axis and with appreciable shiftsfor the real parts as well. Then, the resonance spectrum does no longer reproduce the spectrumof H c : resonances start overlapping and even the number of zeros in D that lie close enough tothe real energy axis to be observed experimentally may change. We believe this describes quiteaccurately the true situation in hadron spectroscopy.Below the lowest threshold, poles, i.e., zeros in D (Eq. 7), come out on the real axis, becausethe expression ij ℓ h (1) ℓ turns real. In that case, expression (5) describes bound c ¯ c states, suchas η c , J/ψ , χ c (1 P ) and ψ (2 S ), yet with an admixture of two-meson components. The energyeigenvalues of these “dressed” states then depend on the value of - λ . It has been observed [9, 10]that charmonium mass shifts with respect to the pure confinement spectrum can be surprisinglylarge in the RSE, as well as in other approaches [11].In the present work, we intend to derive relations among A ( ℓ ) ij , D ( ℓ ) and Z ( ℓ ) ij . In principle,this could be achieved by just performing the calculus outlined in Ref. [2]. However, here weshall allow more general expressions for the Z matrix in the Born term (4). Hence, apart fromthe unitarity condition (6), we must construct a second relation. For that purpose, we write theidentity0 = h i, ~p i | ( T − V − T GV ) | j, ~p j i = - λ π ∞ X ℓ =0 (2 ℓ + 1) P ℓ (ˆ p i · ˆ p j ) j ℓ ( p i r ) j ℓ ( p j r ) × (8) × A ( ℓ ) ij ( E ) D ( ℓ ) ( E ) − Z ( ℓ ) ij ( E ) + 2 i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) A ( ℓ ) iν ( E ) D ( ℓ ) ( E ) Z ( ℓ ) νj ( E ) , which yields the relation D ( ℓ ) Z ( ℓ ) ij = A ( ℓ ) ij + 2 i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) A ( ℓ ) iν Z ( ℓ ) νj . (9)Furthermore, if we assume A ( ℓ ) ij = A ( ℓ )(0) ij + - λ A ( ℓ )(1) ij + - λ A ( ℓ )(2) ij + . . . , (10)then we obtain the following solution to relations (6) and (9):1. The denominator D can be fully expressed in terms of the numerators A , according to D ( ℓ ) = 1 + 2 i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) A ( ℓ ) νν . (11)6. The zeroth- order term of (10) is evidently given by the Born term (4): A ( ℓ )(0) ij = Z ( ℓ ) ij . (12)3. For the higher-order terms of the expansion (10) we obtain the recursion relation A ( ℓ )( n +1) ij = 2 i X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) n A ( ℓ )( n ) νν Z ( ℓ ) ij − A ( ℓ )( n ) iν Z ( ℓ ) νj o . (13)From Eq. (5) we then get a partial-wave scattering amplitude of the form t ℓ ( i → j ) = 2 - λ j ℓ ( p i r ) j ℓ ( p j r ) A ( ℓ ) ij ( E ) D ( ℓ ) ( E ) . (14)For a full definition of this amplitude, satisfying the unitarity conditions for scattering, seeEq. (29). Various opinions exist on how to analyse the final-state interactions of pairs of hadrons emergingfrom a decay process [12–15]. In particular, the production of pion pairs has been studied frommany different angles. Several resonances have been discovered and established in this channel.However, there still are many open questions, of which the most intriguing one probably is theformation of the f (980) resonance [16–28]. As such, this resonance seems to be one of the keyissues for understanding strong interactions. It lies close the K ¯ K threshold, couples relativelyweakly to pions, comes on top of a much broader structure, namely the f (600), and is furthermorenot very distant from a broad resonance around 1.35 GeV, viz. the f (1370) [15].It is our understanding that mesonic resonances, like the f (600) and the f (980), forman integral part of the whole meson family. Therefore, we have developed a model for all q ¯ q phenomena, including those involving charm and bottom. Here, we wish to develop a new toolfor data analysis, which is an amplitude for the description of final-state interactions in two-mesonsubsystems emerging in decay processes involving other particles. This production amplitude isbased on the two-meson scattering amplitude given in Eq. (5). vq ¯ q MM + q ¯ q MMv V + q ¯ q MMv V V + · · · Figure 3:
Graphical representation of the RSE production amplitude. The transition q ¯ q → M M ,denoted by V t in the text, is here represented by v ; the resulting effective M M interaction is denotedby V . For the description of the final-state interactions of meson pairs in production processes, itis common practice to make the spectator assumption, according to which the other emerginghadrons do not interact strongly with the pair. Evidently, this is an approximation, which isjustified by the observation that in most production processes resonances involving the third (orfourth, . . . ) hadron are much higher in mass than the energies considered for the pair. Here,7e moreover assume that the meson pair is generated from an initially produced q ¯ q pair. Ouramplitude for the production of a meson pair, including all higher-order contributions from final-state interactions, is depicted in Fig. 3. Also using expression (5) for the scattering amplitude,we are led to define for the production amplitude a ( α → i ) = h i, ~p i | (1 + T G ) V t | ( q ¯ q ) α , E i = (15)= h i, ~p i | V t | ( q ¯ q ) α , E i + X ν Z d k ν D i, ~p i | T | ν, ~k ν E G (cid:16) ~k ν (cid:17) D ν, ~k ν | V t | ( q ¯ q ) α , E E = - λ √ π X ℓ,m ( − i ) ℓ j ℓ ( p i r ) Y ( ℓ ) m ( ˆ p i ) Q ( α ) ℓ q ¯ q ( E ) ×× g αi − i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) g αν A ( ℓ ) iν ( E ) D ( ℓ ) ( E ) . Here, Q ( α ) ℓ q ¯ q represents the overlap with the initial q ¯ q distribution, having quantum numbers α andrelative interquark angular momentum ℓ q ¯ q . Notice that the latter quantum number is related- though unequal - to the relative two-meson angular momentum ℓ , because of total-angular-momentum and parity conservation. Below, we shall discuss the properties of production ampli-tude (15) for pairs of interacting mesons. P i = X ν c ν T νi ? The result (15) agrees to some extent with the expression proposed in Refs. [29, 30]. Like here,the authors of Ref. [30] based their ansatz on the OZI rule [3] and the spectator picture, soas to find that the production amplitude can be written as a linear combination of the elastic t ℓ ( i → i ) and inelastic t ℓ ( i → ν = i ) scattering amplitudes, with coefficients that do not carryany singularities, but are rather supposed to depend smoothly on the total CM energy of thesystem.Indeed, if we carry out the substitution (14), we find for our production amplitude the ex-pression a ( α → i ) = - λ √ π X ℓ,m ( − i ) ℓ Y ( ℓ ) m ( ˆ p i ) Q ( α ) ℓ q ¯ q ( E ) ( g αi j ℓ ( p i r ) − i X ν µ ν p ν h (1) ℓ ( p ν r ) g αν t ℓ ( i → ν ) ) , (16)which contains a linear combination of elements of the scattering amplitude, with coefficientssmooth in E .However, Ref. [30] concluded from the relation ℑ m ( A ) = T ∗ A (17)that the production amplitude must be given by a real linear combination of the elements of thetransition matrix. A similar conclusion, based on a K -matrix parametrisation, can be found inRef. [31]. In contrast, we arrive at a different conclusion, namely that, as the Hankel functionof the first kind is a complex function for real arguments, the coefficients must be complex , inagreement with experimental analyses [13, 32, 33] as well as with the theoretical work of theIshidas [34, 35]. 8elation (17), which can be also found in Ref. [36] basically stems from the operator relations AV = (1 + T G ) V = V + T GV = T , the symmetry of T , the realness of V and the unitarityof 1 + 2 iT , which gives ℑ m ( A ) V = ℑ m ( AV ) = ℑ m ( T ) = T ∗ T = T ∗ AV . This leads, for non-singular potentials V , to relation (17). In Appendix A, we show that notwithstanding the complex coefficients in Eq. (16), relation (17) is satisfied for the scattering and production amplitudes ofEqs. (5) and (15), respectively. Consequently, relation (17) does not impose a realness conditionon the coefficients in Eq. (16). Besides the sum over transition matrix elements, our prodution amplitude (16) also containsan extra term ∝ g αi j ℓ ( p i r ). Such a term was not considered in Refs. [29–31]. However, inthe works of Graves-Morris [37] and Aitchison & collaborators [38–40], the possible existenceof an additional real contribution was anticipated. Here, it follows straightforwardly from thereasonable assumption that the produced meson pair originates from an initial q ¯ q pair.It is generally agreed that production and scattering have the same singularity structure inthe complex energy plane. At first sight, this is not obvious from expressions (15) and (16).However, the second term between braces in Eq. (15) can, using Eq. (11), be rewritten as follows: g αi − i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) g αν A ( ℓ ) iν D ( ℓ ) = (18)= 1 D ( ℓ ) ( g αi + 2 i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) h g αi A ( ℓ ) νν − g αν A ( ℓ ) iν i) = g αi D ( ℓ ) + 2 i - λ X ν = i µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) g αi A ( ℓ ) νν D ( ℓ ) − g αν A ( ℓ ) iν D ( ℓ ) . From this equation it is obvious that, in our approach, scattering and production have exactlythe same poles in the complex energy plane, as they share the global denominator D . The pole structure of our production amplitude is exhibited very explicitly in formula (18), andshows that it is completely given by D , the very same denominator that determines the pole struc-ture for elastic scattering. The conclusion is that resonance shapes are different for productionand scattering because they are largely determined by the respective numerators. Moreover, pre-cisely the numerator A ii describing elastic scattering in the i -th two-meson channel has droppedout of expression (18). Hence, when restricted to a one-channel model, our production amplitudeis completely determined by just the denominator D .The result (18) may be substituted into relation (15). Moreover, using expression (14) for thepartial-wave amplitudes, we arrive at a ( α → i ) = - λ √ π X ℓ,m ( − i ) ℓ j ℓ ( p i r ) Y ( ℓ ) m ( ˆ p i ) Q ( α ) ℓ q ¯ q ( E ) ×× g αi D ( ℓ ) + i X ν = i µ ν p ν h (1) ℓ ( p ν r ) " g αi t ℓ ( ν → ν ) j ℓ ( p ν r ) − g αν t ℓ ( i → ν ) j ℓ ( p i r ) . (19)9quation (19) is the central result of our paper. It explicitly relates the ingredients of elasticscattering to the amplitude for production in the spectator approximation. We were able toachieve this because in the RSE one can determine in an analytically closed form all terms ofthe perturbation expansions (5) [2] and (15). Hence, relations (11), (12) and (13) can be derivedand explicitly verified. We may thus conclude that at least for a nonrelativistic (NR) microscopicmodel, i.e., at low energies, production and scattering are related to one another through Eq. (19). P = T /V
Expression (18) takes an extremely simple form in the case that all inelasticity is either absentor neglected. For the ℓ -th partial wave of the production amplitude (15), we then obtain a ( ℓ ) ∝ - λ j ℓ ( pr ) Q ( α ) ℓ q ¯ q ( E ) 1 D ( ℓ ) . (20)This is exactly the generic form of the production amplitude used in a paper by Roca, Palomar,Oset and Chiang [7], when at the f (600) resonance the inelastic contribution KK → ππ isneglected, resulting in a ∝ T /V . Here, we get in the 1-channel case from Eq. (12) that A = Z ,which then precisely yields T /V = 1 / D . In the one-channel approximation, we obtain from the scattering amplitude (5) for the cotangentof the scattering phase shift δ ( ℓ ) ( E ) the expressioncotg (cid:16) δ ( ℓ ) ( E ) (cid:17) = n ℓ ( pr ) j ℓ ( pr ) − - λ µp j ℓ ( pr ) A ( ℓ ) , (21)where the spherical Neuman function is represented by n ℓ .Now, D in formule (20) is related to A in formula (21) through Eq. (11). After some algebra,we get a ( ℓ ) ∝ - λ j ℓ ( pr ) Q ( α ) ℓ q ¯ q ( E ) − tan (cid:16) δ ( ℓ ) ( E ) (cid:17) j ℓ ( pr ) /n ℓ ( pr ) cos (cid:16) δ ( ℓ ) ( E ) (cid:17) eiδ ( ℓ ) ( E ) . (22)For S -waves ( ℓ = 0) this becomes a (0) ∝ - λ j ( pr ) Q ( α ) ℓ q ¯ q ( E ) (cid:16) δ (0) ( E ) (cid:17) tan ( pr ) cos (cid:16) δ (0) ( E ) (cid:17) eiδ (0) ( E ) . (23)With respect to the dependence on the phase δ (0) ( E ), this expression has exactly the same formas the S -wave production amplitude given by Boito and Robilotta in Ref. [41], which is based onWatson’s formalism [6] via the work of Pennington [42]. For the meson-loop phase ω ( s ) definedin Ref. [41], we obtain here pr . However, our resonance poles are determined in quite a differentmanner than in Ref. [41]. Whereas in the RSE the resonance poles are all contained in A inexpression (21) for the cotangent of the phase shift, in the formalism employed in Ref. [41] eachof the resonance poles for S -wave production has to be put into the corresponding expression byhand, one by one. 10 .6 Breit-Wigner resonances Again in the one-channel case, one deduces from Eq. (7) for D the form D ( ℓ ) ( E ) = 1 + 2 i - λ ∞ X n =0 (cid:12)(cid:12)(cid:12) F ( n ) ( r ) (cid:12)(cid:12)(cid:12) E − E n µp j ℓ ( pr ) h (1) ℓ ( pr ) . (24)For small - λ one finds a zero of D in the vicinity of E n , say at E n + ∆ E n , where∆ E n ≈ - λ (cid:12)(cid:12)(cid:12) F ( n ) ( r ) (cid:12)(cid:12)(cid:12) µ n p n n j ℓ ( p n r ) n ℓ ( p n r ) − i j ℓ ( p n r ) o . (25)Here, µ n and p n are the reduced mass and relative linear momentum of the two-meson systemat E = E n , respectively. Note that the imaginary part of ∆ E n is negative, as it should be forresonance poles in the second Riemann sheet. Below threshold we obtain poles on the real energyaxis, since ij ℓ h (1) ℓ becomes real for purely imaginary arguments. The latter poles represent two-meson bound states, as argued above. For the following discussion we shall only consider polesabove threshold.For D we obtain D ( ℓ ) ( E ) ∝ Y n ( E − E n − ∆ E n ) . (26)Consequently, denoting the residue at the n -th pole by α n , we get1 D ( ℓ ) ( E ) ∝ X n α n ( E − E n − ∆ E n ) , (27)which is nothing but a Breit-Wigner [43] expansion over a series of resonances, as employed inthe isobar formalism [40, 44–47]. Of course, things become more involved than in Eq. (27) when - λ is not small and resonancesstart to overlap. Overlapping resonances have been studied extensively in the past [48]. Here, itis no longer possible then to deduce simple approximations for expression (24).Besides extending the formalism of Ref. [6] to coupled channels and overlapping resonances,our work also seems to interpolate between the results of Ref. [42] and Ref. [7]. K -matrix The K -matrix, which is related to the tangent(s) of the scattering phase shift(s), is defined by K = T [ 1 + iT ] − . (28)As follows from the unitarity condition, K is a real (symmetric) matrix for real CM energy E .In the one-channel approximation and in a particular partial wave, K is given by the inverseof expression (21) for the cotangent of the scattering phase shift. For more channels, relationslike Eq. (21) become very complicated expressions in terms of A and D . The reason is that theinverse of the expression (5) has to be determined. Numerically this is no problem, of course, butanalytically it is extremely tedious in the general multichannel case. In particular, for a relationbetween the common denominator D and K , which is needed for the leading term in expression(19), nothing simpel follows. Moreover, the pole positions for both scattering and productionstem from D , and not from K . Hence, the excercise to express the production amplitude in termsof the K -matrix seems pointless. 11 Summary and Concluding Remarks
The two-meson production amplitude (19) has been rigorously calculated, to all orders, froma relatively general expression for a two-meson scattering amplitude (Eq. (5)) dominated by s -channel resonances. The latter had already been succesfully tested for c ¯ c and b ¯ b states, mesonswith open charm and bottom, and also in the light-quark sector.One might object that a model with no t -channel exchanges is too restricted for drawinggeneral conclusions. However, one should be aware of the — quoting T¨ornqvist [49] — “well-known dual-model result for ¯ qq resonances that a sum of s -channel resonances also describes t - and u -channel phenomena.” In the context of duality, Harari [50] formulated a necessarycondition for an s -channel description to reproduce certain t -channel effects, namely the existenceof “strong correlations between the different s -channel resonances.” Well, this is exactly whatour infinite RSE sum over confinement states guarantees. Further proof showing the RSE modelto be realistic is its correct threshold behaviour in elastic ππ scattering [51].Another possible critique of our method could be its NR nature. Nevertheless, in practicalphenomenological applications to spectroscopy and elastic scattering, relative momenta and re-duced masses in the two-meson channels have been consistently defined in a relativistic way, thusensuring proper kinematics at much higher energies than the underlying NR formalism seems tosupport. Such a minimal treatment of relativity is indeed common practice in many relativisedquark models. Our successful description of the spectroscopy and scattering properties of thelight scalar mesons [52] provides additional evidence that this approach to relativity is reasonable.This is also supported by our very recent first application of the present production formalismin the single-channel case [1].It thus seems fair to conclude that production amplitudes can in general contain terms whichare not proportional to scattering T -matrix elements and, moreover, that the proportionalitycoefficients are complex. Acknowledgements
We wish to thank I. J. R. Aitchison, D. V. Bugg and C. Hanhart for useful discussions. Thiswork was supported in part by the
Funda¸c˜ao para a Ciˆencia e a Tecnologia of the
Minist´erio daCiˆencia, Tecnologia e Ensino Superior of Portugal, under contract PDCT/FP/63907/2005.
A Generic relation between production and scattering
In order to arrive at a relation equivalent to Eq. (17) for the here proposed scattering andproduction amplitudes, we define T ( ℓ ) ij = − √ µ i p i µ j p j t ℓ ( i → j ) = − - λ √ µ i p i µ j p j j ℓ ( p i r ) j ℓ ( p j r ) A ( ℓ ) ij D ( ℓ ) . (29)For this object, also using relations (6), one easily finds X ν T ( ℓ ) ∗ iν T ( ℓ ) νj = 4 - λ √ µ i p i µ j p j j ℓ ( p i r ) j ℓ ( p j r ) X ν µ ν p ν j ℓ ( p ν r ) A ( ℓ ) ∗ iν A ( ℓ ) νj (cid:12)(cid:12)(cid:12) D ( ℓ ) (cid:12)(cid:12)(cid:12) (30)12 - λ i √ µ i p i µ j p j j ℓ ( p i r ) j ℓ ( p j r ) A ( ℓ ) ∗ ij D ( ℓ ) ∗ − A ( ℓ ) ij D ( ℓ ) = 12 i n T ( ℓ ) ij − T ( ℓ ) ∗ ij o = ℑ m (cid:16) T ( ℓ ) ij (cid:17) . Furthermore, we define A ( ℓ ) αi = √ µ i p i j ℓ ( p i r ) g αi − i - λ X ν µ ν p ν j ℓ ( p ν r ) h (1) ℓ ( p ν r ) g αν A ( ℓ ) iν D ( ℓ ) , (31)for which, by substituting definition (29), we may also write A ( ℓ ) αi = g αi j ℓ ( p i r ) √ µ i p i + i X ν g αν √ µ ν p ν h (1) ℓ ( p ν r ) T ( ℓ ) iν . (32)For this object we study, in accordance with relation (17), the imaginary part ℑ m (cid:16) A ( ℓ ) αi (cid:17) = X ν g αν √ µ ν p ν i n ih (1) ℓ ( p ν r ) T ( ℓ ) iν + ih (2) ℓ ( p ν r ) T ( ℓ ) iν ∗ o (33)= 12 X ν g αν √ µ ν p ν n j ℓ ( p ν r ) (cid:16) T ( ℓ ) iν + T ( ℓ ) iν ∗ (cid:17) + in ℓ ( p ν r ) (cid:16) T ( ℓ ) iν − T ( ℓ ) iν ∗ (cid:17)o = X ν g αν √ µ ν p ν n j ℓ ( p ν r ) ℜ e (cid:16) T ( ℓ ) iν (cid:17) − n ℓ ( p ν r ) ℑ m (cid:16) T ( ℓ ) iν (cid:17)o , where we denote the spherical Hankel function of the second kind by h (2) ℓ = h (1) ∗ ℓ = j ℓ − in ℓ .Next, we use the fact that ℜ e ( T ) = T ∗ + i ℑ m ( T ), and, moreover, substitute subsequentlyrelations (30) and (32): ℑ m (cid:16) A ( ℓ ) αi (cid:17) = X ν g αν √ µ ν p ν n j ℓ ( p ν r ) T ( ℓ ) iν ∗ + i h (1) ℓ ( p ν r ) ℑ m (cid:16) T ( ℓ ) iν (cid:17)o (34)= X ν g αν √ µ ν p ν j ℓ ( p ν r ) T ( ℓ ) iν ∗ + i X ν ′ X ν g αν √ µ ν p ν h (1) ℓ ( p ν r ) T ( ℓ ) ν ′ ν T ( ℓ ) iν ′ ∗ = X ν T ( ℓ ) iν ∗ ( g αν j ℓ ( p ν r ) √ µ ν p ν + i X ν ′ g αν ′ √ µ ν ′ p ν ′ h (1) ℓ ( p ν ′ r ) T ( ℓ ) νν ′ ) = X ν T ( ℓ ) iν ∗ A ( ℓ ) αν . This demonstrates that for our amplitudes a relation exists which is equivalent to the one shownin Eq. (17).
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