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Generalized Algebra of Currents and Regge Poles*

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Generalized Algebra of Currents and Regge Poles* GENERALIZED ALGEBRA OF CURRENTS AND REGGE POLES* SUSUMU OKUBO Department of Physics and Astronomy, University of Rochester, Rochester, New York (Received 25 January 1967) Abstract Fy using the generalized algebra of currents based upon commutators on the light cone, one can show that one can re-interpret Regge trajectories in terms of our generalized commutators. In this way, the correspondence between our theory and Regge's is one to one, and we can derive many results of Regge poles without in­ voking the concept of complex angular momentum. Besides, many sum rules for the high energy scattering cross-sections are obtained. 1. lotroductioo AT THE moment, there are several successful approaches to problems involving strong inter­ actions. These are the dispersion theory, Regge poles and the algebra pf currents. Of all these three approaches, the dispersion theory is the most universal in its application and hence ex­ tensively utilized even for problems involving Regge poles and the algebra of currents. Apart from this use of the dispersion theory, it looked until now that Regge poles and the algebra of currents have no relation of any kind whatever. Indeed, this was not surprising in view of the fact that the former is essentially connected with high energy phenomena while the latter is so far restricted to giving low-energy ttieorems. However, this is not really the whole storY". In a previous paper [1], it has been shown that an algebra of currents among spatial components of electromagnetic currents may give rise· to a high energy theorem. Unfortunately, this technique is not easily generalizable for more interesting situations. In this note, we shall present another approach where we generalize the concept of the algebra of currents not only on the equal-time but also on more general commutators on the light cone. We shall see that in this way, we are naturally led into investigating the so-called t­ channel and that we may obtain many results of the Regge pole model. For instance, the Barger­ Rubin relation [2] for high energy meson-baryon scattering cross-sections is easily obtainable, together with many others. In fact, it is possible in our theory to re-interpret the Regge tra­ jectories in terms of generalized commutators on the light cone. Although some assumptions must be made for such an interpretation, the whole situation appears to be quite promising and it is hoped that further investigations may clear up these points . • This work is supported in part by the Atomic Energy Commission. 2. Generalized Commutators and Sum Rules We shall consider scattering of an octet of pseudoscalar mesons by some targets which we need not specify at the moment. Now, using the tensor notation [3] of representations of the SU(3) group, one may denote the meson field by a traceless tensor ~ ~(x), (a, b = 1, 2, 3). Then, the standard technique enables us to write down the scattering amplitude T( i .... f) where an incoming meson ~b with 4-momentum k is scattered by a target specified by I a> and an out.- a going meson ~~with 4-momentum k' emerges together with a recoiled residual taget I f3 >. k+k'] T(i .... /) = i(4k k~)-K d4x exp i x x 0 f [ 2 ( 1) where we defined the pseudoscalar current j~(x) by 2 j~(x) = (µ - D xl ~~(x) ( 2) and R~(x) is a sum of equal-time commutators given by (3) 1 oo(xo) + - + _: oo(xo) [a(:) .. c(-:)] 2 'dx 2 1 d 2 0 ~b Actually, the meson mass µ in equation (2) need not be the same for all mesons and may be re­ placed by µb.a Now, it is easy to show that the R~~(x) term does not contribute at all to the imaginary part of the forward scattering amplitude, and hence to the total scattering cross section, by the optical theorem. Thus, it is conceivable that its effects may be negligible in the high energy limit. Indeed, we shall justify this conjecture in the next section. However, we need not assume this in this section. Let us consider the first term on the right hand side (hereafter referred to as R.H.S.) of equation (1). For the high energy limit, we let k0 === k0 .... co with a fixed momentum transfer. Then, the main contribution to the scattering amplitude would come [4] from the most singular parts of the retarded commutator since non-singular parts are expected to give vanishingly small contributions in that limit thanks to the Rieman-Lebesgue lemma on the Fourier transform. Ordi­ narily, we believe that these singular parts are concentrated on the light cone x2 = 0 because of the micro-causality. Then, we shall show in the next section that indeed one may obtain a diffraction shrinkage from such assumptions. Further, we assume that on the light cone x 2 = 0, the commutator will contain only unitary singlet and octet components, i.e. it is assumed that at x2 = 0, we have (4) where Qo ( x). ( 5a) (5b) ac a -c 2 Ba -c + E/ Qd( x) Q"ix) - Dbd(x) = BdQb(x) b - 3 b (5c) -f 2 C· -a + ~ Baoc - 3 Bd Qb (x) 9 b d Q/x). ac ac ac . Note that Sbi x), Fbi x) and Dbi x) correspond to the unitary singlet, an .f-type octet, and a d-type octet, respectively. In general, we should have three more terms on the R.H.S. of equa­ tion (4), corresponding to 10, TO and 27 dimensional representations of the SU(3) group. It is worth emphasizing that we did not take Q~(x) and Q~(x) in equation (5) to be traceless for a reason to be e~plained later, although we can always normalize them to be traceless by suitable redefinitions. In this section, we do not assume any explicit form for Q0 (x), Q~(x) and Q~(x). Now, we shall present a motivation for our assumption equation (4). We know that for an equal-time commutator, we shall have an analogue of equation (4) for a commutator of two octet currents. For instance, equal-time commutation relations among spatial components of vector or axial-vector octet currents in the ordinary algebra of currents [5] have the form of equation (4). As we shall see shortly, this assumption appears to be somehow intimately related to the Regge pole model where contributions from Q0 (x), Q~(x) and Q~(x) ma.y correspond to exchanges of the Pomeranchukon, the 1- vector nonet and the 2+ tensor nonet trajectories, respectively. We may also remark that the importance of the t-channel in the high energy phenomena is natu­ rally incorporated in our theory. So far, we have used the terminology of the SU(3) group, thus implicitly assuming the validity of the SU(3) symmetry. However, ih reality we need.not assume its exact validity. In­ deed, one may assume equation (4) independently of the SU(3) symmetry at x 2 =0. Especially, it ma.y hold valid even if the SU(3) is badly broken. This situation is again quite analogous to the ordinary algebra of currents [5]. In fact, as we shall see soon, this may explain why the Barger-Rubin relation [2] and others are well satisfied experimentally in spite ~f possible large SU(3) breaking. In view of this, we shall not assume the exact validity of the SU(3) symmetry in this paper unless otherwise so stated. When we have to take account of the contribution from the R~~(x) term of equation (3), then we once again assume that R~~(x) is a sum of unitary singlet and octets as in equation (4). In this case, this assumption is more plausible since we are considering equal-time commutators. However, for simplicity, we shall omit to mention the contribution from this term since most conclusions presented in this section are really unaffected by its inclusion. Only when its in­ clusion may change our results shall we refer to it, although its effects are actually negli­ gible as we shall show in the next section. Now, it is easy to calculate the scattering amplitude in the high energy limit from equations (1), (4) and (5). For instance, one may get (6) - ... k 013) where for simplicity we omitted the integration symbol together. with other multiplicative factors. From equation (6), one immediately finds the following: (7) Restricting ourselves to the discussion of the forward elastic scattering (a = 13> and taking the imaginary parts of both sides, one finds the following relation among total scattering cross-sections because of the optical theorem: (8) Further, as a special case of equation (8), one gets the familiar Barger-Rubin relation [2] (9) where we used the charge symmetry for the baryon-pion system. Note that in the ordinary deriva­ tion of this equation, we assume the validity of the exact SU(3) symmetry together with ex­ changes of Reggeized unitary singlet and octets particles in the t-channel. However, in our derivation, the exact validity of the SU(3) is not necessarily assumed, although the validity of equation (9) is maintained only in the high energy limit where one can neglect the mass differences of incoming pion and kaons in comparison with the incident energy. This may account for the satisfactory agreement of equation (9) with experiment [2] in spite of possible large violations of the SU(3) symmetry. One may remark also that' equation (8) must be valid for any state I a>. Hence, one may re­ place the proton and the neutron in equation (9) by He 3 and H3 nuclei, respectively.
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