Relativistic unitary description of ππ scattering
M. A. Pichowsky, A. Szczepaniak, and J. T. Londergan Department of Physics and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47405
ranging from the two-pion threshold up to 1400 MeV. A unitary framework based on the Bakamjian-Thomas This system provides an excellent test for the framework. construction of relativistic quantum mechanics is used to de- A relativistic treatment is quite important when dealing scribe two-pion scattering from threshold to 1400 MeV. The with particles as light as pions and the interplay between framework properly includes unitarity cuts for one-, two- and strong dynamics and chiral symmetry makes this system three-hadron states and provides an excellent description of quite interesting. The ππ system is somewhat simpler the available data for ππ phase shifts and inelasticities. The role and importance of three-hadron cuts are calculated and than others, in that its study requires only a minimal discussed. complication from the proper implementation of rela- tivistic spins, since both of the two-body states (ππ and KK¯ ) involved are comprised of spin-0 particles. Another I. INTRODUCTION attractive aspect of applying the framework to ππ scat- tering is the relative wealth of experimental data for the isoscalar, S-wave channel. A nonperturbative framework capable of describing the One drawback with using ππ scattering as a touchstone relativistic, coupled-channel scattering of hadrons is pre- may be that inelasticities due to open states of three or sented. The approach is based on a relativistic Hamil- more particles, do not appear to be significant for this tonian formulation with model interactions introduced process; that is, the ρππ and ππππ thresholds seem to into the mass operator, and with few-body states imple- have little impact on the S-andP -wave observables. The mented in a way that maintains the unitarity of the the- effects of the opening of these three- and four-body chan- ory. The elementary degrees of freedom in the framework nels seem to be overwhelmed by the opening of the two- are finite-sized hadrons which provide a natural ultravi- body KK¯ channel. Nonetheless, several important as- olet regularization, ensuring that the scattering ampli- pects of the framework can be explored in an application tudes are finite. to ππ scattering. The Hilbert space is truncated to include only one-, The isoscalar-scalar (I =0,J PC =0++) channel of ππ two- and three-body states. A central and novel fea- scattering has been a subject of numerous and extensive ture of this framework is the explicit inclusion of both studies. The study of meson scattering in this low-energy real and imaginary parts of scattering amplitudes arising region may be an ideal test of our understanding of the from the opening of three-body channels. The proper interplay between bound states in QCD and chiral dy- handling of three-body unitarity cuts is crucial to gain- namics. The region near E = 1000 MeV is perhaps most ing a deeper understanding of several well-known scatter- interesting, as it is dominated by the mixing between ππ ing systems; a good example is πN scattering in the P 11 and KK¯ channels and the isoscalar-scalar f (980) meson channel, which exhibits a significant inelasticity arising 0 resonance. The nature of the f (980) resonance, and the from the intermediate three-body ππN state [1]. 0 question of whether it is comprised of valence quarks or It presents a formidable challenge to develop a general, arises purely from meson scattering dynamics, has been relativistic scattering framework to describe the final- addressed by many authors [2–8]. Above this energy re- state interactions between hadrons, that includes the ef- gion, three additional scalar meson resonances have been fects of three-body unitarity cuts. Nonetheless, a practi- well established. These are referred to as the f (1370), cal framework which can treat hadron reactions beyond 0 f (1500) and the f (1710). It is still unclear which of the lowest-order valence quark picture is clearly desirable. 0 0 these should be considered as quark-antiquark bound For example, the systematic analysis of hadron reactions states, glueballs or possibly resonances arising from dy- in the baryon resonance region currently being conducted namical effects of final-state interactions [9–13]. Thus, in Hall B at the Thomas Jefferson National Accelerator the isoscalar-scalar channel remains a source of great in- Facility (TJNAF) requires that such a framework be used terest and mystery for meson phenomenology. to extract information about baryon resonances in this Although there have been previous studies which em- highly complex dynamical region. The framework devel- ploy a framework similar to the one developed here, there oped herein is an attempt to construct a useful, relativis- are some important differences. Most studies of meson tic framework capable of describing the nonperturbative, scattering dynamics are based on potential models (as is low-momentum transfer final-state interactions between the framework developed here.) However, most other ap- hadrons in a unitary manner. proaches typically include one- and two-particle channels For the first application of the framework developed only; that is, they include s-channel states and several here, a simple model for ππ scattering is introduced and two-particle channels, such as ππ, KK¯ , σσ,etc.They used to described the S and P partial waves for energies
1 either neglect the possibility of open three-particle chan- Sec. II A a proof of the covariance of this approach is nels altogether or only partially implement them. For ex- provided. Furthermore, Betz and Coester [18] show that ample, in the model developed by the Julich group [14], such a framework can satisfy cluster separability. All of the interaction potentials between two-particle channels, these features are desirable for the study of hadron scat- such as ππ-ππ or ππ-KK¯ interactions are obtained us- tering. ing an instantaneous approximation of a meson-exchange In Sec. II A, the Lorentz covariance of the framework model. Such instantaneous approximations generally do is demonstrated and the fully-interacting mass operator not account for absorptive effects due to the opening of is constructed. It is shown that the framework leads three-body channels. Still, there is no question that the toM Lorentz-invariant on-shell T-matrix elements T (E,P) Julich model quite successfully describes the phase shifts for colliding particles with total momentum P and en- and inelasticities of ππ scattering for S, P and D waves. ergy E; that is, one finds T (E,P)=T (√s)where Alternatively, the Krakow group [15] has developed a √s = √E2 P2 is the invariant mass of the system. separable-potential model for ππ scattering. In their cal- The Poincare− generators act on a Hilbert space which, culation, few-body dynamical effects are incorporated by in general, contains an infinite number of states. The including additional, effective two-body channels, such Hilbert space is truncated to include only those states es- as a σσ channel [16]. Their model also obtains excellent sential to describe the scattering system of interest within results for the ππ phase shifts and inelasticities. a particular energy range. Here, only one-, two- and The outline of this article is as follows. In Sec. II, three-particle states are maintained. Following this trun- the relativistic scattering formalism employed herein is cation, the operator form of the Lippmann-Schwinger briefly discussed, beginning with a short proof of the co- equation can be written as a set of coupled integral equa- variance of observables calculated within this framework. tions. The input that determines the dynamics is given Then, the integral equations that relate the one-, two- in terms of the matrix elements of a model potential V . and three-body scattering T -matrices are provided. Once these are provided, the full scattering problem is In Sec. III, the framework is applied to a study of ππ solved in a straightforward manner. scattering. The particle states that are included in the model are discussed, along with the necessary dynam- ical model parameters. The interactions employed in A. Relativistic covariance this study arise from the meson exchanges which cou- ple states of various numbers of particles to each other. A simple realization of the Poincare algebra for an In our framework, these interactions arise from one-, two- interacting system of a finite number of constituents and three-meson intermediate states which may exhibit is given by the Bakamjian-Thomas construction [17]. production thresholds, resulting in absorptive contribu- This approach has the advantage of providing a Lorentz- tions to the kernels and self-energies appearing in the covariant generalization for a large class of noncovari- Lippmann-Schwinger equations. A simple fitting proce- ant microscopic models, such as the constituent quark dure is shown to provide excellent agreement with data model. In principle, a noncovariant microscopic model for ππ scattering phase shifts and inelasticities. Details could be used to obtain matrix elements of the under- of the relevant model dynamics that produce the various lying elementary hadronic potentials V . In this case, features observed in the resulting phase shifts and inelas- one might consider this framework as a means to extend ticities are discussed. Then, it is shown that the numer- the original noncovariant model dynamics, allowing for a ical methods employed herein are sufficient to maintain Lorentz-covariant treatment of scattering phenomena. the unitarity of the framework to better than one part in The explicit construction of the Poincare algebra pro- a million. Finally, in Sec. IV, the article is summarized ceeds as follows. Starting from a system of noninteracting and plans for future studies are presented. particles, described by their coordinates xa,momentapa, spins sa, and masses ma, the Poincare generators are
II. RELATIVISTIC QUANTUM MECHANICAL H = (m , p )= m2 + p2 , E a a a a FRAMEWORK a a X X p P = p , In this section, a relativistic Hamiltonian framework a a that provides a covariant unitary approach to the study X J = x p + s , of multichannel scattering is described. Lorentz sym- a × a a a metry is maintained by identifying the interactions with X 1 s p the mass operator (that is, the Hamiltonian in the over- K = x , (m , p ) a × a . (2.1) 2{ a E a a }− (m , p )+m all center-of-momentum frame). It is shown in Ref. [17] a a a a that the complete set of Poincare generators can be con- X E structed in a simple way that separates the internal dy- Here, H and P are the total free energy and linear mo- namics from the center-of-momentum (CM) motion. In mentum of the system, J and K are the total angu- lar momentum and boost operators, respectively. The
2 relative coordinates ra, relative momenta ka, center-of- three-meson vertex. The matrix elements of the three- momentum (CM) spins s0 , and the CM coordinates R , meson interaction vertex would be given by a V bc and a cm h | | i total momentum Pcm, and total spin Scm, are introduced would only depend on the internal variables associated via the Gartenhaus-Schwartz transformation which al- with the CM frame where pa = pb + pc = 0. Of course, lows a separation of the internal dynamics and CM mo- these internal variables can be expressed in terms of the tion. In terms of these new variables, the Poincare gen- individual particle momenta in another frame by using erators are given by the boost relations analogous to Eqs. (2.4) and (2.5). Within this framework, the Lorentz covariance of ob- 2 2 H = P + (k1, k2,...), servables may be demonstrated from the following con- M siderations. Construct an invariant -matrix which sat- P = Pcm, p isfies a Lorentz-invariant Lippmann-SchwingerT equation J = R P + S , cm × cm cm (LSE), 1 S P K = R ,H cm × cm , (2.2) 2{ cm }− H + (k , k ,...) = + , (2.8) M 1 2 T V VGT with an invariant interaction , with the constraints, V = H2 H2 m r =0, V I − a a =(P2 + 2) (P2 + 2) a MI − M X = W 2 + HW + WH k =0, a 2 a = V + V + V , (2.9) X M M and an invariant propagator for scattering energy E, Scm ra ka + s0 =0. (2.3) G − × a given by a ! X 2 2 1 =(E H + i)− In Eq. (2.2), the quantity = (k1, k2,...) is referred G 2 − 2 2 1 M M =(E P + i)− . (2.10) to as the free invariant mass in the Schr¨odinger picture. − −M The internal momenta ka are related to the individual Since is independent of the CM momentum P and the particle momenta pa via a free Lorentz transformation scatteringV energy E, one may rewrite the scattering en- to the CM frame, ergy E = √s + P2, in terms of a new variable s, referred to as the invariant mass squared. From Eq. (2.10), one k =Λ(k p )p a a ← a a observes that the propagator (s) is a function of the p P (m , p ) G = p + a · P E a a P (2.4) invariant mass squared only, and one concludes that the a ( + H) − LSE (2.8) depends only on the invariant mass squared M M M s. It follows that the resulting -matrix, (√s) depends T T and the CM frame spins sa0 are related to the individ- only on the invariant mass squared s. ual spins sa via a Wigner rotation corresponding to the It is possible to relate the on-shell matrix elements of product of Lorentz boosts R =Λ(0 pa)Λ(pa this invariant -matrix to the on-shell matrix elements ← ← ka)Λ(ka 0), leading to of a T -matrix thatT is the solution of a non-invariant LSE ← with the interaction potential W , (s) (s) sa0 = D (R)saD (R)∗. (2.5) T (E,P)=W + WG(E) T (E,P), (2.11)
Interactions are incorporated into the Poincare gener- 1 ators by the addition of a term in the free mass operator, where G(E)=(E H + i)− . The relation between the on-shell matrix elements− is given by
(r , k , s0 )= + V. (2.6) M→MI a a a M (√s)=2 s + P2 T (E,P), (2.12) T Thus, transforming the free Hamiltonian H into the in- which can be demonstratedp term by term by expanding teracting Hamiltonian HI , the on-shell matrix elements of in powers of the po- tential V , T H H = H + W, → I = + (E) + O(V 3) W = 2 + P2 2 + P2. (2.7) T V VG V MI − M = W 2 +2EW q p 1 This replacement preserves the canonical commutation +W (E + H) (E + H)W + O(V 3) relations, provided V = V (r , k , s ) is a function of E2 H2+i a a 0a − internal coordinates only and is invariant under rotations 1 =2EW +2EW W + O(V 3) [V,J]=[V,Scm] = 0. For example, consider the case E H +i − for which the elementary interaction is a Yukawa-type =2ET(E,P).
3 In this article, calculations are carried out in the CM drawback of such a truncation is that some symmetries, frame for which E = √s, then the interaction potential such as crossing symmetry, which require the inclusion of W = I = V , and the relevant LSE is many-particle states may be lost. The addition of states M −M with a higher number of particles, such as four-particle T (E)=V + VG(E) T (E), (2.13) states, can in principle be included in a straightforward manner but the resulting set of equations would be far where T (E)=T (E,P = 0). In the CM frame, one finds more complicated than that studied here. (√s)=2√sT(√s). (2.14) The main objective of this work is to develop a frame- T work for handling up to three-body channels in a fully Thus, the on-shell matrix elements of the solution T (E = unitary fashion, by including effects beyond their contri- √s) of the non-invariant LSE in Eq. (2.13) are related by bution to the real part of the effective, two-body poten- Eq. (2.14) to the on-shell matrix elements of the solution tials. The intended application is the description of soft (√s) of the invariant LSE of Eq. (2.8). It follows that final-state interactions in hadron production processes. Tobservables calculated from Eq. (2.13) are equivalent to Such processes are distinguished by their strong cou- those calculated from a Lorentz-invariant theory. plings and low momentum transfers. For this reason, composite hadrons (mesons and/or baryons) are chosen as the fundamental degrees of freedom rather than quarks B. Coupled Lippmann-Schwinger Equations and gluons. The truncation of the Hilbert space to contain only In the above framework, the particle dynamics are one-, two- and three-hadron states may be sufficient given in the center-of-momentum (CM) frame where since, in many applications, states with higher numbers P = 0 by the invariant mass operator, of hadrons contribute very little to two-hadron elastic scattering amplitudes. This suppression arises because
I = + V. (2.15) many-hadron intermediate states typically have a large M M invariant mass, which appears in the denominator of the The quantity , introduced in Eq. (2.2), is the free in- Green function G, tending to weaken its contribution. In- variant mass inM the Schr¨odinger picture and V is the el- terestingly, this suppression of higher-order Hilbert space ementary hadron interaction potential. states is also observed in some quantum field theoreti- The probability amplitude for observing an N-body cal frameworks. In a study of the pion-loop contribtion state βQ with total momentum Q, given an initial N- to the ρ-meson self-energy and charge radius, based on body| statei αP with total momentum P,isgivenbythe a phenomenological application of the Dyson-Schwinger S-matrix element| i βQ S(E) αP .TheT -matrix T (E,P) equations of QCD [19], the covariant, quantum field the- is defined by theh Lippmann-Schwinger| | i equation (LSE) oretic expression for the ρ-meson self-energy was sepa- of Eq. (2.13) and determines the on-shell S-matrix ele- rated into the various time orderings and their relative ments, importance calculated. The time orderings include con- tributions arising from ππ and ρρππ intermediate states, βQ S(E) αP = βQ αP as well as others. In this calculation, it was shown that h | | i h | i 2πi δ( ( β, Q) ( α, P)) βQ T (E,P) αP , (2.16) terms associated with the two-pion intermediate state − E M −E M h | | i contributed more than 95% of the total, while the four- 2 2 where ( α, P)= α + P . hadron states contributed less than 5%. Thus, one ex- TheE potentialM V andM the T -matrix describe all interac- pects that a truncation scheme which neglects states with tions between the variousp channels, including channels four or more hadrons should provide a reasonable descrip- with differing numbers of particles. In general, they tion of the residual strong interactions between mesons do not conserve particle number. Therefore, the LSE and baryons. of Eq. (2.13) represents an countably infinite system of The matrix elements of the potential V describe the coupled-channel equations which couple states of differ- couplings between hadrons that arise from the underly- ent numbers of particles. ing QCD dynamics of quarks and gluons. Color confine- This infinite system of coupled equations may be sim- ment requires that all physical particle thresholds are as- plified by truncating the Hilbert space to include only a sociated with the colorless hadron states. It follows that finite number of states that are expected to contribute the matrix elements of the potential V are real. In this substantially to a given reaction. For the purposes of framework, all of the analytic structure of the T -matrix this study, the Hilbert space is restricted to contain a necessarily arises from the color-singlet hadron poles and finite number of one-, two-, and three-particle states. branch cuts which result from the LSE of Eq. (2.13). Furthermore, the particles are assumed to be of finite Once the Hilbert space has been truncated to include spatial extension, thereby providing an ultraviolet reg- only one-, two- and three-particle states, Eq. (2.13) is ex- ularization to the theory. With these restrictions, the panded and rewritten in a simpler form by labeling each LSE of Eq. (2.13) reduces to a closed system of integral of the Hilbert space operators with subscripts indicating equations which may be solved exactly. Of course, one the numbers of particles they act on. The potential V is
4 0 Σ V K FIG. 2. Schematic diagram of the two-particle self-energy 0 0 Σ, and the two-particle kernel K,asdefinedinEq.(2.21).
FIG. 1. Schematic diagram of the interaction potential ma- trix V of Eq. (2.18). G1 00 G = 0 G 0 , (2.19) 2 00G3 of the form and the T -matrix is V11 V12 V13 V = V21 V22 V23 . (2.17) T T T 11 12 13 V31 V32 V33 T = T21 T22 T23 . (2.20) T31 T32 T33 The part of the potential associated with the coupling of a one-particle state to a two-particle final state is denoted In the CM frame, each submatrix G1, G2 or G3 in V21. The resulting system of integral equations can be Eq. (2.19) is itself diagonal since our hadron states form solved formally in a straightforward manner. a complete, orthogonal set of eigenstates of the free in- It is important to note that each matrix element of variant mass operator . the potential V in Eq. (2.17) is itself a matrix, since it Upon insertion of theseM forms for V , G and T from may contain interactions between any number of different Eqs. (2.18), (2.19) and (2.20) into the LSE (2.13), one particle channels. That is, matrix elements of the form may formally solve this system of integral equations. V21 describe the couplings of any one-particle state with It is convenient to consider the combination of terms any two-particle state. The number of one-, two-, and V22 + V23G3V32, which appears frequently in our formal- three-particle states one wishes to include depends on the ism. These terms play an important role and so are col- specific application. For the application to ππ scattering lected and rewritten as the sum of Σ and K, considered in Sec. III, a further simplification is made by assuming an absence of fundamental interactions in Σ+K V22 + V23G3V32. (2.21) V connecting one-particle states to three-particle states, ≡ and three-particle states to three-particle states. Then, These are referred to as the two-particle self-energy Σ, the potential V takes a simpler form, and the two-particle kernel K. These terms are defined such that the matrix elements of the two-particle self- V11 V12 0 energy Σ contain only terms proportional to a δ-function V = V21 V22 V23 , (2.18) in the relative momentum of the two-particle state. Con- 0 V32 0 sequently, matrix elements of the two-particle kernel K contain all contributions that are not proportional to a and is shown schematically in Fig. 1. δ-function in the relative momentum. The neglected terms V13 = V31† are associated with The two-particle self-energy and kernel are depicted energy-independent transitions between one-particle and schematically in Fig. 2. In the following, it will be- three-particle states. When such terms are neglected the come apparent that Σ and K are the central elements only way in which a one-body state can decay into a of the framework, from which all other quantities are ob- three-body state is through a multiple-step process in- tained. In fact, all effects due to three-particle interme- volving a two-body intermediate state. diate states can be traced back to these two amplitudes. In setting the term V33 = 0, several possible elementary One defines the dressed one- and two-particle Green interactions have been neglected. First, V33 describes functions in the usual manner as direct energy-independent couplings between two three- 1 1 body states, as well as interactions in which two of the G =(G− Π)− , (2.22) 1 1 − particles interact while the third particle is a spectator. 1 1 G2 =(G− Σ)− . (2.23) Such terms may be important. One might argue that it e 2 − is inconsistent to include direct two-body interactions in They are definede in terms of the two-body self-energy Σ V22, but neglect the analogous two-body (plus spectator) and the one-body self-energy Π, where interactions in V33. Nonetheless, in this work such terms are ignored. The significance and role of these interac- Π=V11 + V12G2V21, (2.24) tions will be addressed in future studies. In the truncated Hilbert space, the free Green function V21 = V21 + KG2V21, e 1e is a diagonal matrix =(1 KG2)− V21. (2.25) e − e e e 5 ~ which do not appear in Σ in Eq. (2.30), are part of the Π V21 two-body kernel K.) Upon inserting a complete set of three-body states into ~ ~ Eq. (2.30) and evaluating the resulting expressions in the V21 K V21 overall CM frame with P = 0, one obtains a k ,p FIG. 3. Schematic diagram of the integral equations for (1) ∞ γ13 ( 13 ) Σβα(p, E)= dk13 the one-body self-energy Π and the dressed vertex V21.These 0 2 (mβ ,p) (mα ,p) γ13 Z 1 1 diagrams depict the expressions in (2.24) and (2.25). X E E Vβ1γ13 (k13) Gγ (p, k13,E)pVγ13α1 (k13), (2.31) e × (1) ~ ~ (i) (i) (i) t V21 V12 where Σ = V23 G3V32 . A similar expression is ob- (2) tained for Σβα(p, E). The momentum integration is over t(2) t(2) the relative momentum k13 between the first and third K K particles of the three-body intermediate state, J is the to- FIG. 4. Schematic diagram of the scattering amplitudes tal angular momentum of the system, the sum is over all t(1) and t(2) which enter into the two-particle scattering ma- three-body states γ, and the three-body Green function trix elements. These diagrams depict the expressions in is Eqs. (2.27) and (2.28). 1 G (p, k ,E)= . (2.32) γ 13 E (p, k )+i −Mγ123 13 These quantities are shown schematically in Fig. 3. For brevity the ubiquitous two-body phase space factor, The solution for the two-particle scattering T -matrix can be written as k2 ρ (k ) a (k ,p)= 13 γ13 13 , (2.33) 1 (1) (2) 1 1 γ13 13 3 T = G G (t + t )G G + G G Σ , (2.26) (2π) 2 ( γ (k13),p) 22 2− 2 2 2− 2− 2 E M 13 where and two-body Jacobian e e e (1) t = V21G1V12, (2.27) (k ) ρ (k )= Mγ13 13 , (2.34) t(2) = K + K G t(2), γ13 13 2 (m ,k ) (m ,k ) 2 E γ1 13 E γ3 13 e e e 1 =(1 KG2)− K. (2.28) − e are introduced. The expression in Eq. (2.31) for the two- These two scattering amplitudes are shown schemati- body self-energy Σ(1)(p, E) is depicted in Fig. 5. The e βα cally in Fig. 4. Briefly, the contributions to the two- two-body self-energy Σβα(p, E) is then the sum, body scattering amplitude T22 that proceed through one- body channels are denoted t(1), while contributions that (1) (2) Σβα(p, E)=δΣβα(p)+Σβα(p, E)+Σβα(p, E), (2.35) don’t proceed through one-body channels are denoted t(2);botht(1) and t(2) contain the effects of the two- and where the counter term is chosen to be three-body singularities, but only t(1) contains one-body (1) (2) singularities. δΣα(p)= Σα (p, E)+Σα (p, E) . (2.36) − E= (p) The matrix elements for the dressed Green function G2 Mα12 defined by Eq. (2.23) are given by This is necessary and sufficient to ensure unitarity and 1e − that the stable two-body system α12 is observed asymp- Gβα(p, E)= δαβ(E α12 (p)+i) Σαβ(p, E) . totically with the invariant mass (p). Evaluation −M − Mα12 of the two-body self-energy Σα(p, E) requires calculating e (2.29) the imaginary part and a principal part of the integral in One can collect the terms from Eq. (2.21) contributing Eq. (2.31). For energies E above a three-body thresh- old, these integrals encounter poles in the three-body to the two-body self-energy Σβα, and organize them into the following sum, Green function Gγ (p, k13,E) for values of the relative momentum k13 = k0γ ,wherek0γ satisfies the relation (1) (1) (2) (2) (p, k )=E. Σ=δΣ+V23 G3V32 + V23 G3V32 . (2.30) γ123 0γ MFrom Eq. (2.24), one obtains an expression for the one- Here, the superscript (i) refers to the diagram in which body self-energy in the CM frame, the ith particle in the two-body state emits and subse- quently re-absorbs the particle γ3.ThetermδΣ is iden- 1 ∞ Πα(E)=δΠα + dk aγ (k, 0) tified with the part of the potential V22 that is propor- 2m 12 α1 γ 0 tional to a Dirac δ-function in the relative two-body mo- X12 Z mentum. (All other terms that appear in Eq. (2.21) but V (k) G (k, E) V (k, E) . (2.37) × α1γ12 γ γ12α1 e e 6 Vβ γ γ Vγ α Vγ α β 1 13 1 13 1 α 13 1 α 1 1 β1 1 γ k γ k q 3 13 p q 3 13 p β α 2 2 k23 β2 α 2 FIG. 5. Diagram depicting one of the contributions to the two-body self-energy Σ(1)( ) given in Eq. (2.31). Shown βγ βα p, E V 2 23 here, particle α1 decays into particles γ1 and γ3,andthese FIG. 6. Definitions of the relative momenta and particle subsequently recombine to form particle .Therelative (1) β1 labels for the kernel Kβα (q, p) given in Eq. (2.44). momentum between the intermediate particles γ1 and γ3 is k13, the relative momentum of the incoming state αp1p2 | i is p, and relative momentum of the outgoing state βq1q2 of the two particles in the incoming state emits the ex- | i is q. Solid circles denote matrix elements of the potentials changed particle, the superscript on the third quantity from Eq. (2.18), V32 and V23, evaluated between two- and refers to the direct four-meson interaction. In the first three-body states. two terms, the exchanged particle is subsequently ab- sorbed by the other particle in the outgoing state. Explicit expressions for the most general two-body ker- In Eq. (2.37), m is the mass of the one-body state α1 nels of Eqs. (2.40) and (2.41) in the spherical wave basis αP , V (k) is the vertex function of the potential V , | i α1γ12 12 are complicated and not particularly enlightening. How- Vγ12α1 (k, E) is the vertex function for the dressed vertex ever, in the model application to ππ scattering consid- V21, Gγ (k, E) is the dressed two-body Green function, ered in Sec. III, the resulting kernel is relatively sim- ande aγ12 (k, 0) is a factor from Eq. (2.33) associated with ple. The only matrix elements of V23 that are of interest thee phasee space of the two-body system γ12.Theone- in this application are those associated with the transi- body mass counter term δΠ is fixed by demanding that tions of the forms ππ ππρ, ππ ππf , ππ KKρ¯ α → → 0 → the elements of the one-body self-energies be identically and ππ KKf¯ , ππ πKK¯ ∗.Ineachofthese → 0 → zero when the driving energy E = (mα1 , P). In the CM hadron states, at least two of the three particles are spin- frame, the mass renormalization conditionE is 0 mesons. For these interactions, the plane-wave matrix (1) elements of the potentials V23 are of the form Re Πα(E) =0. (2.38) E mα1 → γ Kkk V (1) α Pp =(2π)3δ(K P)δ 123 13 32 12 γ2,α2 In this framework, the finite size of the hadrons involved h | | i − 2 (mα , p) results in vertex form factors, such as V (k), which fall × E 2 − β1γ12 γ kk V α p , (2.42) off sufficiently rapidly with k to ensure the convergence ×h 13 13| 21| 1 i of all integrals. Therefore, the counter terms δΠandδΣ where the matrix elements of V12 are of the form are both finite. s λ Having obtained expressions for the dressed one- and ( 1) γ3 − γ kk V α p =(2π)3δ(k p) − two-boby Green functions, one next considers the two- 13 13 21 1 h | | i − 2sγ3 +1 body scattering amplitudes t(1) and t(2). These scatter- sγ3 sγ3 D ( k , p) k Y ∗ (kˆ ) V (k ). (2.43) ing amplitudes depend on the one- and two-body Green λγ ,λ 13 13 sγ , λ 13p γ13α1 13 3 − 3 − functions, as well as the two-body kernel K.Thetwo- Xλ body kernel is comprised of three contributions (1) The vertex for V32 appears in Figs. 5 and 6 as the right- K = K(1) + K(2) + K(4pt), (2.39) most interaction vetrtex. In the partial-wave basis, the kernel K(1) isshowninFig.6andisgivenby where +1 P (x) 1 K(1)J (q, p)=2π dx J (1) (2) (1) βα K = V G V , (2.40) 1 4π 2 (mγ3 , p q) 23 3 32 γ123 Z− E − (1) (2) X s K(2) = V G V , (2.41) V (k ) γ3 (q, p) V (k ) 23 3 32 β2γ23 23 S γ13α1 13 , (2.44) × E γ (q, p, p q)+i (4pt) 123 and K is the part of the potential V22 that is not pro- −M − − portional to a Dirac δ-function in the relative momentum. where This latter term is the direct four-point coupling of four s =0 S γ3 (q, p)=1, mesons, and it is depicted as four meson lines converging m2 g + k k on a single point in Fig. 1. In Eq. (2.39), the parentheti- s =1 µ γ3 µν γ3µ γ3ν ν S γ3 (q, p)=q − p , (2.45) cal superscripts on the first two quantities refer to which β1 (k ) (k ) α2 Mγ23 23 Mγ13 13 !
7 ∞ (2)J=sα for scalar exchanges (sγ3 = 0) and vector exchanges 1 Vβ12α1 (q)= dk aγ12 (k, 0) Ωβγ (q, k) Vγ12α1 (k). (sγ3 = 1), respectively, and PJ (x) are the usual Legen- 0 γ12Z dre polynomials in x = p q/pq. The three four-momenta X e appearing in Eq. (2.44) are· (2.50)
µ Finally, the two-body scattering amplitude t(1) is q =( (mβ , q), q) , β1 E 1 µ J J p =( (m , p), p) , V (q)Gγ0 γ1 (E)Vγ α (p) α2 α2 (1)J β12γ10 1 1 12 µ E − − tβα (q, p)= (2.51) kγ =( (mγ3 , p q), p q) . (2.46) 2 mγ mγ1 3 γ γ √ 10 E − − X10 1 e e e Expressions corresponding to the matrix elements of The complete expression for the two-body scattering am- (2) (1) Kβα(q, p) can be obtained in a similar manner. plitude T22 is obtained by adding this expression for t Once specific forms of the model vertex form factors to t(2) according to Eq. (2.26).
Vγ13α1 (k13)andVβ2γ23 (k23) are provided and substituted In the previous sections, it was demonstrated that the into Eq. (2.44), the matrix elements of the kernel K are explicit solution to the scattering problem involving one- computed numerically. One can proceed to solve the in- , two-, and three-body states can be obtained by per- tegral equation in Eq. (2.28) for the scattering ampli- forming several integrations and one matrix inversion. tude t(2). In the CM frame, the integral equation for the The matrix inversion is necessary to obtain the two-body partial-wave scattering amplitude t(2)J (q, p) has the form (2)J βα Moller amplitude Ωβα (q, p).
t(2)J (q, p)=KJ (q, p)+ ∞dk a (k, 0) βα βα γ12 III. APPLICATION TO ππ SCATTERING γ 0 X12 Z KJ (q, k) G (k, E) t(2)J (k, p), (2.47) × βγ γ γα In this section, the framework is applied to ππ scatter- where a (k, 0) is the usual two-body phase space fac- ing. Simple model forms of the elementary vertex form γ12 e factors V (q) are introduced, and solutions for the tor. Obtaining the solution of this integral equation is β12α1 complicated by the presence of the two-body pole in the self-energies Πβα(E)andΣβα(p, E) and scattering am- plitudes t(1)J (q, p), t(2)J (q, p), and V (q) are obtained two-body Green function Gγ (k), and possibly the appear- βα βα β12α1 numerically. Several interesting aspects of the obtained ance of three-body unitarity cuts in both G (k)andthe γ solutions are discussed. It should be emphasized that two-body kernel KJ (q, pe). The method used to solve e βα the model introduced in Sec. III A is preliminary and the this integral equation is adapted from Ref.e [20]. It in- manner in which the model parameters are fit to the data volves obtaining a two-body Moller operator Ω(2),whose th may be overly simplistic, as it focuses on reproducing J partial-wave matrix element satisfies, only a few observables and therefore does not represent an exhaustive or complete study of the dynamics of ππ (2)J ∞ (2)J tβα (q, p)= dk aγ12 (k, 0) Ωβγ (q, k) Kγα(k, p). scattering. The motivation is to provide a demonstration γ 0 X12Z of the framework and exhibit the features of the model, (2.48) and its ability to describe the scattering of a system of strongly-coupled particles with emphasis on the multi- (2)J particle channel aspect. More complete studies of meson After a solution for tβα (q, p) is obtained, one proceeds to obtain an explicit expression for the two-body scatter- scattering within the present framework will be the sub- ing amplitude t(1), which is given by Eq. (2.27). Since ject of future articles. the solution of the intermediate dressed one-body Green In Sec. III A, the dynamical assumptions are discussed along with the model parameters. A detailed list of the function G1 is obtained from the one-body self-energy from Eq. (2.37), all that remains is to determine the form states included in the Hilbert space is provided. The model parameters are determined using a simple method of the dressede vertices V and V . The integral equa- 12 21 to fit experimental data for the ππ isoscalar-scalar phase tion for the dressed vertex is obtained for the transpose shift and ρ-meson decay width, using the S-wave phase of the dressed vertex from Eq. (2.25), e e shifts from Ref. [21]. In Sec. III B the resulting phase
∞ shifts, inelasticities and cross sections are provided and Vβ12α1 (q)=Vβ12α1 (q)+ dk aγ12 (k, 0) compared to the data, and some aspects of the f0(980) γ 0 ¯ X12 Z scalar meson are discussed in terms of a KK bound state. J=s e K α1 (q, k) G (k, E) V (k) (2.49) × βγ γ γ12α1 A. Dynamical model for ππ scattering The similarity between this integrale equatione and the in- tegral equation of Eq. (2.47) with J = sα1 which de- termines t(2) is clear. It follows that the solution to The model is intended to describe the scattering in a Eq. (2.49) is just range of center-of-momentum (CM) energies from thresh-
8 TABLE I. Masses and widths of mesons. Masses that are old (E =2mπ 280 MeV) to about E = 1400 MeV. Above 1400 MeV,≈ it is important to include in more de- underlined have been fixed to reproduce the accepted values. tail the effects of the three scalar mesons observed in this All other values are obtained from the model calculation. In region. For this preliminary study, however, it is possi- the present study, the width of the K∗ meson was not calcu- lated. All values are given in units of MeV. ble to avoid making strong assumptions concerning these scalar mesons, hence the model will not be accurate in πKρK∗ f0(1350) f0(980) this energy region. In the following, only the isoscalar- mass 140 500 770 890 1350 996 scalar I =0,J = 0) and isovector-vector (I =1,J =1) width 0 0 150 — 805 46 channels are considered. The motivation is to explore some of the interesting physical aspects of the present TABLE II. Isospin coupling constants for kernel ( ) framework and to estimate the importance of including Kβα q, p for isoscalar, S-wave (I =0,J PC =0++) and isovector, three-body states in such a model of hadron scattering. PC P -wave (I =1,J =1−−) scattering. The assumptions of the dynamical model are summarized below. Channel Exchange I =0,J =0 I =1,J =1 Two-body states: For the channels and energies ex- ππ ππ ππf0 11 ¯ ↔ ¯ ¯ plored herein, it is assumed that ππ scattering is primar- KK KK KKf0 11 ↔ ππ ππ ππρ -1 -1 ily determined by the dynamics arising from the coupling ↔ ¯ KK¯ KK¯ KKρ¯ -1 -1 of the ππ and KK two-body channels. Hence, ππ and ↔ ¯ | i ππ KK¯ πKK∗ √2-1 KK are the only two-body channels included in the ↔ − Hilbert| i space. One-body states: It is assumed that the coupling of the KK¯ system is strong enough to result in the appear- the 1300–1700 MeV region, is in the width of this reso- ance of a narrow resonance in the scalar-isoscalar chan- nance. In order to fit the model parameters to the ππ nel at E 980 MeV. This state is identified with the phase shifts requires a single effective resonance with a J PC =0++≈ f (980) meson. Since this scalar meson is 0 very large width. It is found that the model resonance presumed to arise from final-state interactions as a quasi- has a decay width of 805 MeV, which is approximately bound KK¯ state, it is not part of the free Hilbert space, the sum of the widths of the three observed resonances and there is no bare mass associated with it. Rather, it in this region. appears as a pole in the analytically-continued T -matrix. ¯ Three-body states: Only three-body states are included Furthermore, in the limit that the two-body ππ and KK that can couple to the ππ or KK¯ states through the ab- channels decouple, this pole moves to the real-energy PC sorption or emission of the isovector J =1−− ρ(770), axis below the two-kaon threshold; that is, it becomes isodoublet J P =1 K (892), and J PC =0++ f mesons. a KK¯ bound state in this limit. The identification of − ∗ 0 ¯ Thus, the three-body states included in this study are the f0(980) meson as a KK molecule is controversial. ππρ , ππf , KKρ¯ , KKf¯ , πKK¯ and πKK . Although it appears as a molecular state in this model, 0 0 ∗ ∗ | Toi summarize,| i | thei | hadronici | statesi included| in thisi the “true” nature of the f0(980) meson remains an open model of ππ scattering are question. In contrast to the f (980) meson, it is assumed that at 0 f0 , ρ , least one of the scalar resonances observed in the mass | i | i ππ , KK¯ , region between 1300 and 1700 MeV will be a QCD bound | i | ¯i ¯ ¯ state; that is, a state which arises as a bound state whose ππρ , KKρ , πKK∗ , ππf0 , KKf0 , (3.1) | i | i | i | i | i constituents are quarks, antiquarks and gluons. Such states do not arise from the meson final-state interac- where the f0 meson refers to the f0(1350) meson. (The tions, they are not bound states of mesons, and hence f0(980) meson is expected to appear in the model as a ¯ must be included in the model as bare states with bare KK resonance.) The values of the bare masses of these masses. particles are provided in Table I and are underlined to Experiments reveal the presence of several reso- indicate that they are input parameters. As discussed nances in the scalar-isoscalar channel between 1300 and in Sec. II, one- and two-body counter terms are included 1700 MeV. A complete study of the ππ scattering sys- in the elementary interaction potentials V11 and V22,re- tem in this energy range requires the inclusion of each spectively, such that the bare masses given in Table I of these resonances into the model. However, to simplify coincide with the dressed masses of the mesons. the present study, all of these resonances are modeled Model vertex form factors: The vertices in the model in terms of a single scalar resonance. The resonance is are assumed to be finite-sized and hence require the ap- assumed to have a mass of 1350 MeV, which gives it propriate form factors for the relative three-momentum a mass similar to the lightest of the resonances above q. They are given by the universal form: PC ++ kaon threshold, referred to as the J =0 f0(1370) 16π q2/Λ2 meson. One ramification of choosing a single scalar res- V (q)=a e− β1β2α1 , (3.2) β12α1 β1β2α1 2s +1 onance to model the effect of all observed resonances in r max
9 TABLE III. Coupling strengths and momentum scales for where smax =maxsβ1 ,sβ2 ,sα1 is the largest spin of the particles involved.{ In the present} study s =0for the vertex form factors. The values of the momentum scales are given in MeV. vertices involving the f0 meson, and s =1forver- ¯ ¯ tices involving the ρ or K∗ mesons. The vertex cou- β1β2α1 ππf0 KKf0 ππρ KKρ KπK∗ pling constants aβ1β2α1 and form factor momentum scales aβ1β2α1 12.4 5.08 20.0 20.0 20.0
Λβ1β2α1 are chosen to provide a good fit to the data Λβ1β2α1 875 875 875 875 875 for the isoscalar-scalar ππ phase shift δππ and the ρ- meson decay width Γρ ππ = 150 MeV. The parameter search was limited in a→ number of ways. First, the var- TABLE IV. Coupling strengths and momentum scales for ious meson-exchange form factor scales Λ in Ta- the four-point meson interactions. The values of the momen- β1β2α1 tum scales are given in MeV. ble III were all constrained to be the same value and less than 1 GeV. The direct four-meson couplings in Ta- ππX KKX¯ 4π 4K ble IV were chosen to be one of two scales, the first a 19.8 11.0 17.2 0.0 was taken to be 125 MeV larger, and the second to be Λ 1000 1000 750 750 125 MeV smaller than the meson-exchange scales in Ta- ble III. In the following, it is shown that the vector- exchange interactions contribute little to the observables of three-body intermediate states to the study of meson considered. To reduce the number of parameters, the scattering and final state interactions, rather than to test strength of the vector-exchange vertices were taken to be a particular interaction model for ππ scattering. identically equal a = a ¯ = a . The isospin ππρ KKρ πKK∗ Once the forms of the model vertices are fixed, the factors that arise in a calculation of the meson-exchange only observables used in the fit are the scalar-isoscalar kernels KJ (q, p), such as in Eq. (2.44), are given in Ta- βα ππ phase shift δ , the existence of the KK¯ resonance ble II. ππ (referred to as the f (980)), and the decay width of the Direct interactions: In addition to meson exchanges, 0 ρ(770) vector meson. All measurable reaction channels it is important to include real-valued potentials that are not used in the fitting procedure, since this paper directly couple two pseudoscalar mesons to two pseu- represents more a proof of principle of the framework doscalar mesons, as a part of the K(4pt) kernel in rather than a complete phenomenological analysis of ππ Eq. (2.39). Such interaction potentials could arise from scattering. The resulting values of the coupling constants the direct coupling of four mesons to a virtual-quark loop. are provided in Tables III and IV. Here, two direct four-point interactions are considered. The first is intended as a way to mimic some of the ef- fects of dynamical chiral symmetry breaking. This inter- B. Phase shifts and inelasticities action is taken to be of the form given by the elementary potential V22 and is referred to as the direct 4π (or 4K) interaction. In a partial-wave basis, the form of the four- Below all three-body thresholds, the non-trivial part of the S-matrix in Eq. (2.16) can be written as a 2 2 pion interaction is given by × unitary matix Sβα(E)withα, β denoting the only two (4π)J J 2 q2/Λ2 p2/Λ2 open channels ππ and KK¯ .TheS-matrix in the J th K (q, p)=16π (qp) a e− 4π e− 4π , (3.3) ππ,ππ 4π partial wave can be parametrized in terms of two phase and the four-kaon term K(4K)J (q, p) = 0. The second shifts δππ and δKK¯ , and one inelasticity ηππ, four-point interaction is a short-ranged attraction mod- J i Sαα(E) eled as a t-channel exchange of a heavy scalar-isoscalar δα(E)=− ln , (3.5) meson. Its form is given by the scalar-exchange kernel of 2 ηα(E) J Eq. (2.44) and the two-body self-energy Σ of Eq. (2.31). ηα(E)= Sαα(E) , (3.6) For simplicity it is treated exactly as if two additional | | three-body states, for α = ππ, KK¯ . Below all three-body thresholds there are only two stable channels, ππ and KK¯ . Hence, there is ¯ ππX , KKX , (3.4) only one inelasticity parameter ηππ = ηKK¯ . For energies | i | i E above the lowest stable three-body threshold, one must with mX = 1500 MeV, were added to the Hilbert space. augment the S-matrix by including all stable three-body Clearly, modelled in this manner, for energies E>mX + states. Consequently, its parametrization requires more 2mπ = 1780 MeV, the state ππX can go on-energy- than two phase shifts and one inelasticity. Nonetheless, | i shell. However, the calculations described herein are for one may still use Eqs. (3.5) and (3.6) to define the phase energies less than 1400 MeV, so that ππX is never on- shifts δ (E), and inelasticities η (E) for the two channels | i α α energy-shell. α = ππ and KK¯ . Of course, above the threshold of a Again, the objective is to study the framework devel- stable three-body state ηππ = ηKK¯ . oped in this paper. That is, it is interesting to assess The ππ phase shift, as defined6 by Eq. (3.5) and ob- the importance and possibility of including the dynamics tained from our model, is shown as a solid curve in Fig. 7.
10 360
315 1.2 270 1 225 0.8 (deg) 180 ππ η
ππ 0.6
δ 135 0.4 90
45 0.2
0 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.4 0.6 0.8 1 1.2 1.4 E (GeV) E (GeV)
FIG. 7. The ππ scattering phase δππ in the scalar, isoscalar FIG. 8. The ππ scattering inelasticity ηππ in the scalar, channel as a function of the CM driving energy E. The data isoscalar channel as a function of the CM driving energy E. are from Refs. [22] (open squares), [23] (closed circles), [24] The data are from Ref. [22]. (up triangles), and [25] (down triangles).
the phase motion tends to increase, until finally, in the The model provides an excellent description of the ππ limit that the coupling between the two channels goes to phase shift data depicted in Fig. 7 [22–25], and the in- zero, the phase motion becomes a step-function of mag- elasticity ηππ, as shown in Fig. 8. The overall trend of nitude 180 degrees. Such a phase motion is completely the pion scattering phase shift δππ is positive and in- unobservable, and could therefore be ignored altogether creases slowly with energy E. This is indicative of a weak (although it is relevant to Levinson’s theorem, which re- and attractive effective ππ scattering potential. At the lates overall changes in the phase shifts from threshold kaon threshold, a rapid phase motion is apparent. It re- to infinite energy to the number of bound states in a sults from the presence of a narrow, f0(980) scalar meson. system). Above the two-kaon threshold, the phase shift continues The importance of the coupling between the two-body to increase slowly, at a rate similar to the increase in the channels ππ and KK¯ can be estimated quantitatively by phase shift below the threshold. recalculating the pion phase shift δππ after removing the Below all other thresholds, the ππ channel is the only kaon state KK¯ from the Hilbert space. The result is open channel, and unitarity requires that the inelasticity shown as the| dottedi curve in Fig. 9. However, from the ηππ = 1 here. This is clearly observed in Fig. 8, where above argument, perhaps a better indication of the im- the calculated inelasticity ηππ has a value consistent with portance of the KK¯ channel is obtained by letting the unity below the threshold of the KK¯ channel at 1 GeV. couplings that lead to a mixing between the KK¯ and ππ Had four-body states been included into this framework, channels go smoothly to zero. (In practice, this is done by one might have expected to see a decrease in the inelas- not allowing the one-body scalar f0(1350) to have a bare ticity η due to the opening of the four-pion state, which ¯ ππ coupling to the KK state, and setting aK∗πK = 0. This has a threshold of E =4mπ 0.560 GeV. However, the has a minimal effect on the dynamics but prevents mixing data in Fig. 8 [22] seem to suggest≈ that the contribution the ππ and KK¯ states.) In this limiting case, the two | i ¯ | i of the four-pion state to ππ scattering is negligible. This states KK and f0(980) are coupled to the ππ state, can be seen by noting the lack of any systematic deviation but the| contributionsi | theyi make to ππ scattering| i go to from ηππ = 1 for the range of energies 4mπ
11 360 1.000
315 0.999 270 0.998 225 (deg) 180 ππ 0.997 η ππ
δ 135 0.996 90 0.995 45
0 0.994 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.1 1.2 1.3 1.4 1.5 E (GeV) E (GeV)
FIG. 9. The ππ scattering phase δππ in the scalar, isoscalar FIG. 10. The ππ inelasticity ηππ in the isoscalar-scalar channel versus the CM energy E. The full calculation (solid channel with no kaons. curve) is compared to the calculation (dashed and dotted ¯ curves) in the limit the KK and ππ states become decou- p 1 pled. The data are from Ref. [22]. = 0 z −(4π)2 2E ππ t(1)J (p ,p )+t(2)J (p ,p ) , (3.8) can contribute more than half of the total phase shift × ππ,ππ 0 0 ππ,ππ 0 0 δππ. At energies above the two-kaon threshold, its im- 2 2 portance quickly dimishes and vanishes altogether above where p0 = E /4 mππ is the magnitude of the on- energy-shell three-momentum− of the pions, z is the E = 1150 MeV. p ππ The importance of the mixing between the ππ and KK¯ wave function renormalization of the two-pion state ππ , (1)J (2)J | i channels can also be observed in the pion inelasticity ηππ and tβα (q, p)andtβα (q, p) are the two-body scattering shown in Fig. 8. Just above the two kaon threshold, the amplitudes, obtained from Eqs. (2.47) and (2.51), respec- inelasticity plummets to a minimum value ηππ 0.31. In tively. The two-body scattering cross section σβ α(E) the limit that the coupling to the kaon channel≈ goes to can be written in terms of the partial-wave scattering← zero, as described above, the inelasticity takes on a very cross sections according to different appearance.
In Fig. 10, the two pion inelasticity is plotted above the ∞ J ¯ σβ α(E)= σβ α(E). (3.9) KK threshold. Below the threshold its value is unity. ← ← The lowest multi-particle state to which two-pion flux JX=0 can be lost is the three-particle channel ππρ . The pro- | i In terms of the scattering amplitudes, these partial-wave duction threshold of this state is 1050 MeV. It is clear cross sections are given by that one observes a slow decrease in the inelasticity ηππ above 1050 MeV, due to the opening of the ππρ channel. J σβ α(E)= The effect of this channel on the inelasticity is very small, ← 2J +1 z z 2 with a minimum value η 0.994. β α t(1)J (q ,p )+t(2)J (q ,p ) , (3.10) ππ ≈ 4π 64π2E2 βα 0 0 βα 0 0 The energy dependence of the S-wave pion scatter- ing inelasticity η and phase shift δ are conveniently ππ ππ where p0 and q0 are the on-energy-shell solutions to plotted together in an Argand diagram. In Fig. 11, the α(p0)=E and β(q0)=E, respectively. The re- J=0 M M partial-wave scattering amplitude aππ (E) is plotted in sulting cross section for elastic, S-wave ππ scattering is the complex plane as a parametric function of the CM shown as a solid curve in Fig. 12. It is finite at thresh- energy E. This figure clearly shows the rapid rise in the old, exhibits a maximum value of 43 millibarns at E elastic ππ phase shift just below 1 GeV, and the resulting 600 MeV, and a sharp decrease at the position of the≈ dramatic loss of flux from the elastic channel as soon as f0(980) scalar meson resonance. This sudden drop oc- the KK¯ channel opens up. curs just below the KK¯ threshold, as can be seen upon For spinless particles, the amplitude is given by examination of the inset plot in Fig. 12 which depicts a closeup of the KK¯ threshold region. η e2iδππ 1 aJ (E)= ππ − , (3.7) The dot-dashed curve is the resulting cross section ππ 2i J=0 ¯ σKK¯ ππ for the two-kaon production process ππ KK. This← cross section is considerably smaller than that→ of the
12 1.0 0.9 0.8 150 0.7 8
125 6 1.3 1.2 4 1.4 1.1 0.6 100 2
(mb) 0 75 0.95 1.00 1.05
1.5 J=0 0.5 σ 50
25 1.6 0.4 1.0 0 1.7 0.4 0.6 0.8 1.0 1.2 1.4 0.3 -0.5 0.0 0.5 E (GeV) FIG. 12. S-wave cross section σJ=0(E) in millibarns for FIG. 11. Argand diagram for ππ scattering at all energies. ππ ππ (solid curve), KK¯ KK¯ (dashed curve), and → → Plotted is the energy dependence of the real and imaginary ππ KK¯ (dot-dashed curve). The inset shows a detail of J=0 → ¯ parts of the partial-wave scattering amplitude aππ (E)for the region around the KK threshold at E =1GeV. the J = 0 partial wave from Eq. (3.7) or Eq. (3.8). The curve is calculated for all energies E>2mπ, and annotated with the corresponding energies E in GeV, from threshold to above of theoretical models, or theoretical assumptions, in or- E =1.7GeV. der to fit the experimental observables. In the worst-case scenario, the resulting ππ phase shift data may be more representative of the extraction methods employed than elastic ππ scattering cross section, reaching its maximum of the actual ππ scattering process. value of 4.6 millibarns just above the two-kaon threshold The extraction of the ππ phase shifts from experi- at E = 1 GeV. Its small size is a result of the weak cou- ment is a difficult and long-standing problem of hadron pling between the two-pion and two-kaon channels. As physics. At present, it is impossible to construct an ex- discussed above, a weak coupling of these channels is nec- periment in which a pion beam is scattered from a pion essary to ensure a narrow f0(980) meson. If the mixing target. Hence, other techniques are required to extract between the pion and kaon channels were stronger, the the ππ phase shift from experimental observables. One f0(980) meson would more easily decay into two pions, possibility is to use decays that produce two pions in the tending to increase its width significantly. final state, and attempt to extract the ππ phase shifts The dashed curve in Fig. 12 is the S-wave cross section ¯ from the final-state interactions. for elastic KK scattering. This cross section is compara- The procedure employed by Ref. [24] is extract the bly huge, having its maximum at the two-kaon threshold phase shifts from the electroweak kaon decay K+ energy. Its size can be compared to that of the two-pion + 0 + → π π e νe. The results are shown as open squares in elastic cross section, Figs. 9 and 13. Extraction of the phase shift using decays J=0 with more than two particles in the final state requires σππ ππ(E =2mπ) 18.3mb, J=0 ← ≈ some knowledge of transition form factors for the cou- σKK¯ KK¯ (E =2mK) 734 mb. ← ≈ pling of a kaon, two pions and the W boson to determine + + + The very large KK¯ cross section arises from the scalar K π π−W . A nice feature of employing the elec- ¯ troweak→ decay K+ π+π0e+ν is that the two pions are f0(980) meson which lies just below the KK. The pres- → e ence of a bound state just below the two-body scattering the only strongly-interacting particles in the final state. threshold will generally tend to increase the size of the Hence, one would expect that the ππ interactions would cross section dramatically. be the dominant contribution to the dressing of the final As was discussed earlier in Sec. III A, the model pa- state. However, this approach is hampered by two ex- rameters were chosen to provide a good fit to the S-wave perimental difficulties. The first is the lack of statistics. pion phase shift δ data from Ref. [21]. In Fig. 7, the This particular kaon decay represents only a small frac- ππ 5 + tion (4 10− ) of the total K -meson decay width, which resulting phase shifts are also in excellent agreement with × the data analyses of Ref. [22], [24], and [25]. However, is already extremely small. The second is the fact that it is important to realize that direct comparison of these the energies and angles of the outgoing leptons provide a model results with these ππ scattering data must be done small lever arm with which to vary the CM energy E of with caution. Extraction of these data requires the use the two final-state pions.
13 Another method is to extract the final state interac- tions of the two-pion production process πp ππn, employed by Ref. [25] (down triangles), Ref. [22]→ (open 360 squares), and Ref. [23] (closed circles), shown in Fig. 7. 315 These studies require some theoretical input in order to 270 perform the extraction of the ππ scattering phase shift 225 δππ and inelasticity ηππ; hence, they are not direct mea- surements of the ππ scattering phase shift. 180 (deg) Our model parameters were originally fit to the ππ 135 phase shifts obtained by an analysis [23] of an exper- ππ + δ 90 imentatCERNinvolvingπ−p π π−n at 17.2 GeV. Recently, this same data was re-examined→ by Kaminski et 45 al. [22], with weaker model assumptions than were used 0 in Ref. [23]. The work of Ref. [22] provides an exhaus- -45 tive and nearly complete study of the ππ phase shift. In 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 particular, there is no assumption that pion exchange is + E (GeV) the dominant mechanism for the process π−p π π−n. → Consequently, this analysis seems to be more general FIG. 13. Importance of the different model contributions than the others. A relative phase ambiguity in the anal- to ππ scattering phase shift. The solid curve is the full calcula- ysis of Ref. [22] provides four possible, distinct solutions tion. The dashed curve is obtained by removing the one-body for the ππ phase shifts and inelasticities. Two of these so- state f0(1350) from the Hilbert space. The dot-dashed curve lutions seem to have an unphysical inelasticity ηππ below is obtained by completely removing the scalar f0(1350) from the two-kaon threshold and can be discarded. The other the theory, both in one-body and three-body states. The two solutions, denoted the “up-flat” and “down-flat” so- dotted curve is same as the dot-dashed curve, but all scalar lutions, are very similar in appearance and neither can couplings (the “X”-exchanges and contact terms) are also set be dismissed on qualitative grounds. The phase shifts to zero. This is the effect of just the vector-meson exchanges. Above threshold it is negative with a minimum of δππ -2.0 from the “down-flat” solution of Ref. [22] are shown as ≈ degrees at E 1.2 GeV. The dot-dot-dashed curve is ob- solid circles in Figs. 9 and 13. ≈ tained by weakening the coupling of the one-body f0(1350) The general behavior of the ππ phase shift δππ is pos- itive, which is indicative of an attractive ππ scattering state to the ππ state by 10 percent. The data points are from Fig. 7. amplitude. An abrupt increase in the pion phase shift is evident at 1 GeV, which is due to the combined effects ¯ of the opening of the KK threshold and the crossing of Second, they provide the most important contribution to the scalar f0(980) resonance. off-diagonal matrix elements of the two-body scattering Apart from this feature, which in this particular model kernel K. That is, they provide the strongest source is a result of the delicate mixing between the ππ and of mixing for the two-pion and two-kaon states in this ¯ KK channels, the calculated pion phase shift δππ ex- model. hibits a steady, gentle increase from the threshold at Both of these effects tend to produce an attraction for 280 MeV to above 1400 MeV! In the model, this slowly the two pions. In particular, the strong attraction nec- increasing behavior arises from subtle cancellations be- essary to bind the kaons to form the f0(980) resonance tween the attractive potentials of the heavy scalar-meson results in a strong attraction in the two-pion channel as exchanges and the repulsive, direct four-pion interaction well. The amount of mixing between the two-pion and from Eq. (3.3) which is intended to model the effect of two-kaon states dictates the attraction felt by the pions. dynamical chiral symmetry breaking. The different form Hence, when the coupling between ππ and f0(1350) factor scales involved in these interactions (see Tables III states is artificially reduced by as little| i as 10%,| the re-i and IV) are chosen to provide this slowly increasing, weak sult is significant, as shown by the dot-dot-dashed curve phase shift observed in the two-pion channel. Typically, in Fig. 13. When the one-body state f0(1350) is re- scalar potentials by themselves provide a strong attrac- moved entirely from the Hilbert space,| the resulti is the tion in the S-wave ππ channel, that leads to a rapidly dashed curve. The resulting pion phase shift is negative rising phase shift just above the ππ threshold, which then and close to zero below the two-kaon threshold, and pos- quickly falls away. This behavior is not seen in the ππ itive above the threshold. The absence of the one-body phase shift δ . ππ s-channel state f0(1350) reduces the mixing between the The importance and role of the scalar resonances in two-pion and two-kaon states, which results in a nearly ππ scattering can be appreciated by a close examination stable (and very narrow) kaon bound state f0(980). In of Fig. 13. The effect of the heavy s-channel resonances this case, one observes a f0(980) bound state with a width which are collectively modeled by the single f0(1350) in that has been reduced from 46 MeV to 0.28 MeV! This is the model is two fold. First, their presence leads to a a result of the fact that the f0 ππ decay must proceed strong attractive potential for energies below 1400 MeV. →
14 through K∗ exchange in the kernel K, which provides only a weak mixing of the ππ and KK¯ states. When all scalar mesons are removed from the the- 0 ory entirely; that is, when the couplings that lead to 10 the existence of one-body states f0(1350) , and three- two-body f ππ f KK¯ | i a body states 0 and 0 are set to zero ππf0 = -2 | i | i 10 aKKf¯ 0 = 0, the resulting phase shift is shown in Fig. 13 by the dot-dashed curve. The slightly repulsive behavior is a result of the combined effect of the attractive scalar- -4 X and repulsive four-point interactions associated with 10 three-body chiral symmetry, given by the parameters in Table IV. To minimize the number of free parameters all of the -6 couplings to vector mesons are chosen to be equal to each 10 other aππρ = aKKρ¯ = aπKK∗ . These coupling strengths unitarity violation were then determined by solving P -wave ππ scattering 012345 6 7 8 9 10 11 12 13 14 15 at the ρ meson mass E = mρ = 770 MeV, and requir- a ing that the ρ-meson width reproduced the experimental value Γρ = 150 MeV, as given in Table I. It is found FIG. 14. Fraction of incoming two-pion flux at energy that the resulting coupling strength leads to a vector- E = 1400 MeV in the isoscalar-scalar channel, appearing as meson exchange interaction kernel K which provides a “two-body” or “three-body” final states, or lost to unitarity very weak repulsion for ππ scattering. This is illustrated violations. The fractions are shown versus the exponent a of a by the dotted curve in Fig. 13, where all of the cou- the adiabatic scale =10− GeV. plings except those involving the vector mesons (aρππ, a ¯ , a ¯ ) are set to zero. The resulting phase shift ρKK K∗πK labeled “two-body” and “three-body” represent the frac- is negative (repulsive) and very small; its largest absolute tions of the missing two-pion flux that appear as outgoing value is about 2 degrees. Thus, in this model, ρ-meson two-body (ππ or KK¯ ) and three-body (ππρ) states, re- exchange is negligible in the S wave. This model differs spectively. Since the Hilbert space is restricted to one-, from the analysis of Ref. [14], in which they report that two- and three-body states, this should account for all the attractive potential (which leads to the binding of the flux. However, in practice, the numerical methods the KK¯ into the f (980) resonance) is primarily due to 0 employed introduce violations to the unitarity condition ρ-meson exchange, which is strong and attractive in their of Eq. (3.11). The region below the solid curve in Fig. 14 model. In the framework, the exchange of a spin-1 meson represents the fraction of flux that completely disappears in the kernel K results in a very weak and mildly repul- from the theory; such a loss of flux violates unitarity. One sive interaction. This difference is a result of how the observes that for values of 10 7 GeV, the violation spin couplings and energy denominators of the meson- − is less than 10 6 of the outgoing≤ flux and is therefore exchange propagators are implemented in the kernels of − negligible. In the application to ππ scattering described the two frameworks. above, the value of =10 12 GeV (or a = 12) was em- Before ending this section, a final comment concern- − ployed. ing the accuracy of the numerical methods employed is provided. The accuracy is measured using the unitarity condition (or optical theorem), IV. SUMMARY AND FUTURE DIRECTIONS
T T † = T †(G G†)T, (3.11) − − Herein, a framework suitable for the description of non- which is derived from the fact that V in Eq. (2.18) is Her- perturbative hadron scattering, based on the Bakamjian- mitian. Evaluating Eq. (3.11) between two-pion states, Thomas formulation of relativistic quantum mechanics one obtains an equation that relates the two-pion flux is introduced. In Sec. II, it is shown that by including missing from the forward direction to the one-, two-, and the interactions into the free mass operator ,onecan three-body outgoing flux observed leaving the scattering ensure that observables calculated in the frameworkM are center. This relation provides a sensitive check of the Lorentz coviariant. When the Hilbert space is truncated numerical methods employed. to contain only one-, two- and three-body states, the The fraction of lost two-pion flux observed as outgoing resulting Lippmann-Schwinger equations (LSEs) form a two- or three-particle states is shown versus the expo- closed set of coupled integral equations. The solution of a nent a of the adiabatic scale =10− GeV in Fig. 14 these integral equations is obtained numerically. for CM energy E = 1400 GeV. For any energy E greater A significant improvement of this framework over ear- than the two-pion scattering threshold, there can be no lier work is that the full effect of three-body states have stable one-particle state. It follows that no one-particle been included. The three-body Green functions appear outgoing flux can be observed. In Fig. 14, the regions only in the two-body scattering kernel K and the two-
15 body self-energy Σ, and for energies E above the three- hybrid meson decays have so far ignored effects of final body thresholds, unitary branch cuts associated with state interactions. The best candidate for the exotic me- these thresholds appear both in K and Σ. It follows son, the π1(1600) [26] was found in the ρπ decay channel, that the kernel K and two-body self-energy Σ are com- which is predicted to be suppressed with respect to other plex functions of the energy E. The appearance of such two body decay channels, in particular the b1π [27,28]. three-body branch cuts provides the means for important This shift of strength from b1π to ρπ could be explained dynamical effects, such as three-hadron production and by mixing with the three-body, ωππ intermediate state decays into three-hadron states, which are automatically which is believed to have a strong coupling to these two– accounted for in our framework. Such effects have hith- meson channels. erto been ignored for the most part in previous studies. The inclusion of three-body cuts into the integral equa- tions requires appropriate numerical methods to be em- ACKNOWLEDGMENTS ployed. These methods, necessary to solve the coupled set of LSEs are shown to maintain the unitarity of the This work is supported by the U.S. Department of En- theory to better than one part in one million. To demon- ergy under contract DE-FG02-87ER40365 and the Na- strate the utility of the framework, a preliminary study of tional Science Foundation under contract PHY0070368. ππ scattering is carried out in Sec. III. The simple model, introduced in Sec. III A, is able to provide an excellent description of the ππ phase shifts and inelasticities. The main purpose for developing this framework is to provide a means of incorporating the dynamics of [1] J. A. Johnstone and T.-S. H. Lee, Phys. Rev. C34, 243 low-momentum transfer, final-state interactions into the (1986). study of hadronic processes for energies up to a couple of [2] R. Jaffe, Phys. Rev. D 15, 267 (1977). GeV. In this energy region, the comparison between ex- [3] J. Weinstein and N. Isgur, Phys. Rev. D 27, 588 (1983). [4] T. Barnes, Phys. Lett. B165, 434 (1985). perimental data and theorectical predictions from models [5] D. Morgan and M. R. Pennington, Phys. Lett. B258, 444 of QCD for quark and gluon dynamics, are often made (1991). difficult due to the presence of final-state interactions. [6] J. A. Oller, E. Oset, and J. R. Pelaez, Phys. Rev. Lett. The soft rescattering of final-state hadrons tends to mask 80, 3452 (1998). the QCD dynamics of interest. The framework developed [7]E.Klempt,B.C.Metsch,C.R.Munz,andH.R.Petry, herein provides a tractable means to incorporate the ef- Phys. Lett. B361, 160 (1995). fects of final-state interactions into studies of hadronic [8] N. N. Achasov and G. N. Shestakov, Usp. Fiz. Nauk. 161, phenomena. 53 (1991). Towards this end, the framework is constructed to be [9] Particle Data Group, Eur. Phys. J. C 15, 1 (2000). a consistent extension for the constituent quark model. [10] C. J. Morningstar and M. J. Peardon, Phys. Rev. D 60, It provides a means to unquench the quark model by pro- 034509 (1999); G. S. Bali et al. [SESAM Collaboration], viding for hadron loops, multi-particle thresholds and the Nucl. Phys. Proc. Suppl. 53, 239 (1997) unitarity branch cuts associated with these. The result is [11] S. Narison, Nucl. Phys. B509, 312 (1998) the generation of complex-valued scattering amplitudes. [12] P. Minkowski and W. Ochs, Eur. Phys. J. C9, 283 (1999) It is an extension of the quark model, in that the quark [13] R. Kaminski, L. Lesniak, and B. Loiseau, Eur. Phys. J. model may be used to provide the elementary couplings C9, 141 (1999) and form factors for the hadronic interactions in V .The [14] G. Janssen, B. C. Pearce, K. Holinde, and J. Speth, Phys. framework uses this real potential V to generate the full Rev. D52, 2690 (1995). scattering solution. [15] R. Kaminski, L. Lesniak, and J.-P. Maillet, Phys. Rev. Future applications of the framework will focus on D50, 3145 (1994). the dynamics of nucleon resonances and exotic, hybrid [16] R. Kaminski, L. Lesniak, B. Loiseau, Phys. Lett. B413, mesons that are the subject of current and proposed ex- 130 (1997). [17] B. Bakamjian and L. H. Thomas, Phys. Rev. 92, 1300 periments at TJNAF. In the baryon case, for energies up (1953). to about 2 GeV, there are some well-known and strik- [18] M. Betz and F. Coester, Phys. Rev. C21, 2505 (1980). ing examples in πN scattering for which three-body ef- [19] M. A. Pichowsky, S. Walawalkar, and S. Capstick, Phys. fects are crucial in understanding the experimental ob- Rev. D60, 054030 (1999). servables. For example, in the L2I,2J = P11 channel, the [20] M. I. Haftel and F. Tabakin, Nucl. Phys. 158, 1 (1970). πN inelasticity arising from the three-body ππN state is [21] B. R. Martin, D. Morgan, and G. Shaw, in “Pion-Pion very large [1]. It is likely that a complete understand- Interactions in Particle Physics” (Academic, London, ing of the P11 πN scattering channel and the mysterious 1976), pp. 87-101, and references therein. N ∗(1440) resonance requires the full implementation of [22] R. Kaminski, L. Lesniak, and K. Rybicki, Acta. Phys. three-body unitary cuts that this framework provides. Polon. B31, 895 (2000). The effect of three-body cuts in exotic partial waves [23] G. Grayer, et al., Nucl. Phys. B75, 189 (1974). can also be very important. Most theoretical studies of [24] L. Rosselet, et al., Phys. Rev. D15, 574 (1977).
16 [25] V. Shrinivasan et al., Phys. Rev. D12, 681 (1975). [26] G. S. Adams et al. [E852 Collaboration], Phys. Rev. Lett. 81, 5760 (1998). [27] N. Isgur, R. Kokoski and J. Paton, Phys. Rev. Lett. 54, 869 (1985). [28] P. Page, E. S. Swanson, and A. P. Szczepaniak, Phys. Rev. D59, 034016 (1999).
17