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To be published in Few-Body Systems Suppl.

FewBody Systems

c by Springer-Verlag 2002 To be printed in Austria

Does it make any sense to talk about a ?-isobar?

arXiv:nucl-th/0212008v1 2 Dec 2002

Frieder Kleefeld? Centro de F? ?sica das Interac?oes Fundamentais, Instituto Superior T?cnico, c? e Edif? Ci?ncia, Piso 3, Av. Rovisco Pais, P-1049-001 LISBOA, Portugal ?cio e

Abstract. It is shortly investigated, on what basis an experimentally observed resonance like the ?(1232)-isobar can be embedded into the framework of Quantum Theory (QT), i.e. Quantum Field Theory (QFT) and Quantum Mechanics (QM). After a short discussion of the particle concept in the context of the “bootstrap” idea of G. Chew and S. Mandelstam we will focus on the theoretical formalism being necessary to describe resonances in the Lagrangian or Hamiltonian formulation of QT.

1

Why this title?

During a workshop on the (anticipated) occasion of the 65th birthday of Peter U. Sauer we want to recall the intriguing and revealing question raised by Peter U. Sauer [1] and collegues in the year 1990 on the resonance having excited him most of all: “Does it make any sense to talk about N ? phase shifts?”. In order to be allowed to ask this question we ?rst have to explore the underlying question: “Does it make any sense to talk about a ?-isobar?” or “Do unstable particles in physics have experimentally and theoretically a well de?ned meaning?”. The positive answer to the last question sketched in this short presentation will be based on previous work [2, 3, 4] (and references therein). Certain technical aspects will be displayed in more detail. 2 Particle Concept and Bootstrap

Experimentally it seems to be a not well de?ned problem to classify what is a physical “particle” and to specify most decisively its mass properties in physical data tables. The most accurate theoretical de?nition of a particle has been given by G. Chew (see ref. [5]) who associated “particles” with poles of the scattering matrix. According to this de?nition the Review of Particle Properties [6] contains not only a small number of “stable”, i.e. “elementary” particles

? E-mail

address: kleefeld@c?f.ist.utl.pt

2

(which play the quantum theoretical role of “asymptotic states”) yet lists also a vast majority of “particles”, which are not elementary. Such particles are either considered to be “unstable” or — which is intimately related — “composite” [2, 7] (topical examples are e.g. scalar mesons [8, 9, 10]). As the parametrization of data in terms of (improved) non-relativistic Breit-Wigner ?ts being close to the idea of G. Chew fails in the description of processes of relativistic and strongly interfering unstable “particles” the developement of an adequate consistent relativistic formalism for such “particles” has been imperative. Consider a simple non-relativistic scattering problem at an energy at which the T-matrix can be described to a good approximation by a Breit-Wigner amplitude. In this case the Breit-Wigner amplitude describes the full T-matrix already at “treelevel”. The Breit-Wigner amplitude itself being determined by the solution of a Lippmann-Schwinger equation is perturbatively expanded into a Born series. Correspondingly, in the S-matrix picture of G. Chew the “e?ective action” which may be descibed by “particles” in the sense of G. Chew (which need not all to be stable) describes a scattering process already at “tree-level” while the “e?ective action” is usually perturbatively obtained from the generating functional of an interacting theory of asymptotic ?elds1 . The scenario that G. Chew’s “particles” may be (at least approximately) chosen such that they describe scattering problems already (at least approximately) at “tree-level” may be summarized under the term “bootstrap” introduced by G. Chew and S. Mandelstam [11] (see also ref. [5]). A nice feature of such quasi-bootstrap theories of (local) Chew “particles” is that even for large coupling constants physics is already exhaustively described at “tree-level”2. Yet there emerges the theoretically demanding consequence that not only self-energies, but also coupling constants may be complex valued. In order to handle such theories without getting in con?ict with unitarity, causality, Lorentz covariance, locality, positivity, renormalizability etc. it is not only convenient, but imperative to achieve a consistent non-Hermitian prescription of QT in the context of a Lagrangian or Hamiltonian framework. The solution to this problem has been illuminated to a certain extent in refs. [2, 3, 4]. It consists of the replacement of standard Hermitian QT by an (anti)causal formulation of QT, in which the underlying causal and anticausal Lagrangians (or Hamiltonians) are nonHermitian. As a consequence the non-Hermitian causal and anticausal ?elds being solutions of the causal and anticausal non-Hermitian Lagrange equations of motion represent elements of a generalized (anti)causal, i.e. biorthogonal Fock space. This Fock space contains elementary (“quasi-Hermitian”) and unstable (“non-Hermitian”) ?elds. As stated above we should bear in mind that the calculation of scattering or boundstate problems performed only in the “stable” particle basis is usually highly non-perturbative, while the inclusion of unstable particles might bring us by bootstrapping close to tree-level.

1 Here 2A

we assume the existence of a Lagrangian and ?elds which is in general not guaranteed. typical example for a bootstrap theory is the “quark-level linear sigma model” (see e.g. ref. [10, 12] and references therein). Even for Hermitian ?elds nature is well described at tree-level and loop-corrections are estimated to be very small [13].

3

3

The (anti)causal Klein-Gordon (KG) ?eld (“Nakanishi-?eld”)

In order to tackle a spin-3/2-isospin-3/2-resonance like the ?(1232) we ?rst have to understand the properties of underlying (iso)spin 0, 1/2, 1 resonances. The (anti)causal spin 0 ?eld has been studied for the ?rst time3 by N. Nakanishi [14, 15] in the context of his “Complex-Ghost Relativistic Field Theory” de?ning a complex (resonance) mass M = m ? i (Γ/2) and denoting the “free” Lagrangian for the respective causal (anticausal) KG ?eld φr (x) (φ+ (x)) with r isospin (N ? 1)/2 as (r = 1, . . . , N ): L(x) =

r

1 (?φr (x))2 ? M 2 (φr (x))2 + (?φ+ (x))2 ? M ? 2 (φ+ (x))2 r r 2

By variation of the action one obtains the respective causal and anticausal KGequations ( ? 2 + M 2 ) φr (x) = 0 and ( ? 2 + M ? 2 ) φ+ (x) = 0, respectively. As r Im [M ] = ?Γ/2 = 0 they are solved by a Laplace transform 4,5 : φr (x) = = φ+ (x) r =

! !

d4 p “ δ(p2 ? M 2 ) ” ar (p) e?ipx (2π)3 d3 p ar (p) e?ipx + c+ (p) e+ipx 0 r (2π)3 2 ω(p) p =ω(p) 3 ? ? d p cr (p) e?ip x + a+ (p) e+ip x r (2π)3 2 ω ? (p) p0 =ω(p)

The Canonical conjugate momenta to φr (x) and φ+ (x) are: r Πr (x) := δ L(x) ! = ?0 φr (x) , δ (?0 φr (x))

+ Πr (x) :=

δ L(x) ! = ?0 φ+ (x) r + δ (?0 φr (x))

The consistent (anti)causal quantization requires even for resonance ?elds equal-time commutation relations. The respective non-vanishing relations are:

+ [ φr (x, t), Πs (y, t) ] = i δ 3 (x ? y ) δrs , [ φ+ (x, t), Πs (y, t) ] = i δ 3 (x ? y ) δrs r

Hence the non-vanishing commutation relations in momentum space are: [ ar (p), c+ (p ′ ) ] s [ cr (p), a+ (p ′ ) ] s = (2π)3 2ω (p) δ 3 (p ? p ′ ) δrs = (2π)3 2ω ? (p) δ 3 (p ? p ′ ) δrs

The Hamilton operator is derived in the standard way from the Lagrangian6,7 : H

3 The

=

r

d 3p

1 1 ω(p) c+ (p), ar (p) + ω ? (p) a+ (p), cr (p) r r 2 2

considerations in the 2nd article of ref. [2] have been done without knowing refs. [14, 15]. 4 Let ω(p) := p2 + M 2 (ω(0) = M ) and ar (p) := ar (p)|p0 =ω(p) , c+ (p) := ar (?p)|p0 =ω(p) . r 5 For details with respect to the Laplace transform involving a symbolic δ-distribution denoted as “δ(p2 ? M 2 )” for the complex resonance mass M see refs. [16, 15, 2]! + + 6H = d 3x { r (Πr (x) (?0 φr (x)) + (?0 φr (x)) Πr (x) ) ? L(x)} 7 In 1-dim. QM we have H = 1 ω [c+ , a ] + 1 ω ? [a+ , c ] ± ± (± for Bosons/Fermions with 2 2 [c, a+ ]? = 1). This (anti)causal Harmonic Oscillator is diagonalized by the (normalized) 1 (c+ )n (a+ )m |0 with Enm = ω (n ± 1 ) + ω ? (m ± 2 ). right eigenstates |n, m = √ 1 2

n! m!

4

Standard real-time ordering leads to the causal “Nakanishi” propagator: 0| T ( φr (x) φs (y) ) |0 = i

!

d 4 p e?i p(x?y) δrs (2 π)4 p2 ? M 2

The anticausal propagator is obtained by Hermitian conjugation or by a vacuum expectation value of an anti-real-time ordered product of two anticausal ?elds. 4 Lorentz boost, spinors and polarization vectors

The de?nition of spinors and polarization vectors for spin 1/2 and 1 resonances requires the introduction of the respective concept of a Lorentz boost of unstable ?elds (see e.g. refs. [2]). For any symmetric metric g?ν Lorentz transformations (LTs) Λ?ν are de?ned by Λ?ρ g?ν Λνσ = gρσ . Let n? be a timelike unit 4-vector (n2 = 1) and ξ ? an arbitrary complex 4-vector with ξ 2 = 0. The 4 independent LTs8 relating ξ ? with its “restframe” (i.e. ξ ? = Λ ?ν (ξ) n ν ξ 2 and n ν ξ 2 = ξ ? Λ ?ν (ξ)) are (P ? := 2 n ? n ν ? g ?ν = re?ection matrix): ν Λ ?ν (ξ) Λ ?ν (ξ) = ± = ± g ?ρ ? g ?ν ? ξ2 ξ2 ξ2 ?ξ·n ξ2 ?ξ·n n? ? n? ? ξ? ξ2 ξ? ξ2 nρ ? nν ? ξρ ξ2 ξν ξ2 P ρν

Let ξ ? be e.g.9 the 4-momentum p? of a“force-free”10 unstable particle with complex mass M in a (+, ?, ?, ?) metric. Its restframe is de?ned by n? = (1, 0)? 11 . For such a spin 1/2 Fermion we de?ne spinors u(p) ≡ v(?p) and their transpose (i.e. uc (p) := uT (p) C with C = i γ 2 γ 0 ) by the generalized Dirac equation ( p? p2 )u(p) = 0 12 . Lorentz covariance of this Dirac equation requires S ?1 (Λ(p)) γ ? S(Λ(p)) = Λ ?ν (p) γ ν for u(p, s) = S(Λ(p)) u( p2 n, s) resulting for the metric (+, ?, ?, ?) and the proper orthochronous LTs in u(p, s)|Re[p0 ]>0 = p+ p2 p2 ) p+ p2 p2 ) u( p2 n, s) = p+ p0 + = χ+ , 0 s p2 p2 p+ p0 + χs 0 p2 p2 .

2 p2 (p0 + v c (p, s)|Re[p0 ]>0

8 They 9 In

=

v c ( p2 n, s)

2 p2 (p0 +

are well known as ortho-chronous/non-ortho-chronous proper/improper LTs. “Thermal Field Theory” the inverse temperature β = 1/T may be treated as an imaginary time. Hence ξ ? = (t ? i β, x )? and ξ 2 n? = ( (t ? i β)2 ? x 2 , 0)? are related by a LT! 10 A remarkable “bootstrap” feature of (anti)causal QT is that a boost of such resonances does not produce particles — in contrast to the “traditional” boost of interacting particles! 11 The space-time trajectories of such a particle are determined by the condition p · x = const, or — in 1 dimension — by x1 = (ω(p)/p1 ) t + const. If the movement proceeds along the real time axis, the quantity ω(p)/p1 has the interpretation of a complex velocity. 12 Note that uc (p) = u(? p? ) = v(p? ). The spin projections are introduced such that for Re[p0 ] = 0 there holds the spinor normalization condition sgn(Re[p0 ]) s u(p, s) vc (p, s) = p + p2 . The restframe 2-spinors used above obey χ+ χs′ = δss′ and s χs χ+ = 12 . s s

5

The spin-projection vector s ? (p) = Λ? (p) s ν ( p2 , 0) = Λ? (p) (0, s) ν with ν ν |s|2 = 1 obeys s ? (p)s? (p) = ?1 and p? s? (p) = 0. For Re[p0 ] > 0 there holds γ5 s(p) u(p, s) = 2s u(p, s) and γ5 s(p? ) v c (p, s) = 2s v c (p, s) (with s = ±1/2). Similarly, de?ning 3 orthonormal Cartesian unitvectors e(i) (i = x, y, z) by e(i) · e(j) = δ ij we may introduce polarization vectors by ε ?(i) (p) := (j) Λ? (p) ε ν (i) ( p2 , 0) = Λ? (p) (0, e(i) ) ν ful?lling ε ? (i) (p) ε? (p) = ? δ ij , ν ν (i) p? ε? (p) = 0 and i ε ? (i) (p) ε ν (i) (p) = ?g ?ν + (p? pν )/p2 . All these identities can be projected on the complex mass shell p2 = M 2 , e.g. for p0 = ω(p). 5 The (anti)causal Dirac and Proca ?eld

As shown in refs. [2, 3, 4] the (anti)causal Lagrangian for an isospin (N ? 1)/2 ? spin 1/2 Fermionic resonance is given by (r = 1, . . . , N )(M := γ0 M + γ0 ): L(x) =

r

1 ? ? c 1 ? 1 c ψr (x) ( i ? ?M ) ψr (x) + ψ r (x) ( i ? ?M ) ψr (x) 2 2 2

As a whole we have four Lagrange equations of motion, i.e. (i ? ? M ) ψr (x) = 0 c ? c and (i ? ? M ) ψr (x) = 0 and the two transposed equations for ψr (x) and ψr (x). As for the KG ?eld their solution is obtained by a Laplace transform yielding13 : ψr (x)

c ψr (x)

=

s

d3 p [e?iqx b r (p, s) u(p, s) + eiqx d+ (p, s) v(p, s)] r (2π)3 2ω(p)

? ? d3 p [eiq x b+ (p, s) uc (p, s) + e?iq ·x dr (p, s) v c (p, s)] r 3 2ω ? (p) (2π) ? ? d3 p [eiq x b+ (p, s) u(p, s) + e?iq x dr (p, s) v (p, s)] ? ? r (2π)3 2ω ? (p)

=

s

ψr (x)

c ψr (x)

=

s

=

s

d3 p [e?iqx br (p, s) uc (p, s) + eiqx d+ (p, s) v c (p, s)] r (2π)3 2ω(p)

Canonical (“real-equal time”) quantization leads — due to Fermi-statistics — to the following non-vanishing momentum-space anticommutation relations: { br (p, s), d+ (p ′ , s′ ) } r′ + { dr (p, s), br′ (p ′ , s′ ) } = = (2π)3 2 ω (p) δ 3 (p ? p ′ ) δrr′ δss′ (2π)3 2 ω ? (p) δ 3 (p ? p ′ ) δrr′ δss′

Convince yourself that — as for the causal KG ?eld — the propagator of a causal spin 1/2 Fermion is obtained by standard Fermionic real-time ordering 14 :

c 0| T ( (ψr (x))α ( ψs (y))β ) |0 = i

13 Here 14 The

!

d 4 p e?i p(x?y) ( p + M )αβ δrs (2 π)4 p2 ? M 2

we have de?ned q ? := (ω(p), p)? = p? |p0 =ω(p) and br (p, s) := br (p, s)|p0 =ω(p) ,

:= br (?p, s)|p0 =ω(p) and u(p, s) := u(p, s)|p0 =ω(p) , v(p, s) := v(p, s)|p0 =ω(p) . anticausal propagator is obtained by Hermitian conjugation or by a vacuum expectation value of a anti-real-time ordered product of two anticausal ?elds.

d+ (p, s) r

6

As discussed in ref. [2] we may use the formalism of Jun-Chen Su [17] to introduce — with the polarization vectors derived above — a renormalizable and unitary Lagrangian for a massive (anti)causal vector ?eld. In combining our results for a spin 0, 1/2 and 1 resonance the door of (anti)causal QT is opened widely for an (anti)causal ?-isobar. Acknowledgement. This work has been supported by the Funda?ao para c? a Ci?ncia e a Tecnologia (FCT) of the Minist?rio da Ci?ncia e da Tece e e nologia (e do Ensinio Superior) of Portugal, under Grant no. PRAXIS XXI/BPD/20186/99 and SFRH/BDP/9480/2002. References 1. H. Garcilazo et al.: Phys. Rev. C 42, 2315 (1990) 2. F. Kleefeld: hep-ph/0211460; F. Kleefeld, E. van Beveren, G. Rupp: Nucl. Phys. A 694, 470 (2001) 3. F. Kleefeld: Doctoral Thesis (University of Erlangen-N¨rnberg, Germany, u 1999); F. Kleefeld: Acta Physica Polonica B 30, 981 (1999) 4. F. Kleefeld: pp. 69-77 in Vol. I of Proc. XIV Int. Seminar on High En. Phys. Probl. (ISHEPP 98), 17.-22.8.1998, JINR, Dubna, Russia, nucl-th/9811032 5. F. Capra: “Bootstrap physics: a conversation with Geo?rey Chew”, in Berkeley Chew Jubilee 1984:247 (LBL-18372), kek-library 198412075; B. Diu: “Qu’est-ce qu’une particule ?l?mentaire?”, collection de monograee phies de physique publi?e sous la direction de P. Aigrain, A. Blanc-Lapierre, e ? J. Friedel, M. L?vy, c 1965 par Masson et C ie – Editeurs, Paris e 6. K. Hagiwara et al. (Particle Data Group): Phys. Phys. D66, 010001 (2002) 7. N.D. Hari Dass and V. Soni: hep-th/0204177 8. E. van Beveren et al.: hep-ph/0211411 9. F. Kleefeld et al.: Phys. Rev. D 66, 034007 (2002) 10. E. van Beveren et al.: Mod. Phys. Lett. A 17, 1673 (2002) 11. G.F. Chew and S. Mandelstam: Nuovo Cim. 19, 752 (1961) 12. M.D. Scadron et al.: hep-ph/0211275 13. L.-H. Chan, R. W. Haymaker: Phys. Rev. D 7, 402 (1973) 14. N. Nakanishi: Phys. Rev. D 5, 1968 (1972) 15. N. Nakanishi: Prog. Theor. Phys. Suppl. 51, 1 (1972) 16. N. Nakanishi: Prog. Theor. Phys. 19, 607 (1958) 17. Jun-Chen Su: hep-th/9805195, 9805192, 9805193, 9805194.

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