Dual Unitarization Research Articles

On the Higher Rank Cylinder Kernel

Finite energy corrections are estimated in the dual topological unitarization by introduc­ tion of a planar nonleading Regge trajectory. The cylinder amplitude is controlled by the factorizable kernel of rank two and guarantees automatically the extinction of the planar leading as well as nonleading SU(N) singlet. The idea of pomeronf identity is modified as a natural consequence of the cylindrical mixing mechanism. §I. Introduction The dual topological unitarization (DTU) has provided us with a persuasive approach to hadronic interactions at high energies. 1> The most emblematic con­ sequence of the DTU framework reads the pomeron:f identity, i.e., the bare pome­ ron is identified with the cylindrically renormalized f trajectory.l)-a> Moreover the cylinder amplitude is free of j-plane cuts so long as the planar amplitude obeys the pole-to-pole bootstrap condition. 4> Both of these results have been established in the standard DTU framework in which a cylinder equation is described by the factorizable kernel of rank one. It has not yet been settled, however, whether these results are commonly inherent in the DTU scheme or particularly characteris­ tic of the simplest Fredholm kernel. It will therefore be of physical importance to examine a typical cylinder equation of the higher rank kernel. This is performed in the present note by considering the possible finite energy effect. In § 2 an integral equation is derived for the cylinder by introducing a planar nonleading reggeon. In § 3 the dynamical equation is solved. Conspicuous features of the cylinder are investigated. Finally § 4 is devoted to clarifying the cylindrical mix­ ing mechanism. For the details of the notation and the definition, we refer to Ref. 4) .*>

Flavour and spin structure of linear baryons

Based on the string picture, we construct a phenomenological model for baryons and study their flavour symmetry, exchange degeneracy pattern and spin structure. Baryons on leading trajectories are assumed to have the configuration of two quarks being attached to the ends of a linear string and the third sitting in the middle, called linear baryons. For such linear baryons, a unitarization scheme can be constructed in a manner similar to the dual unitarity scheme for mesons but without recourse to the 1 N expansion. We find that the interchange interaction of the middle quark with one of the other two quarks at the ends of the string can give rise to a large exchange degeneracy breaking of the baryon spectrum. With this non-planar correction, the model of linear baryons can account for the observed pattern of leading baryon states.

Experimental verification of the dual unitarization scheme

The connection between particle production in antiproton-proton annihilation and non-diffractive proton-proton processes is discussed in the framework of the dual unitarization scheme. We calculate relative multiplicity moments for antiproton-proton annihilation using the moments of non-diffractive proton-proton interactions and find good agreement with experimental annihilation data.

Dual unitarization scheme with several trajectories

Consequences of bootstrap with several input Regge trajectories are investigated. We find that in a general (without the duality-diagrams constraints) treatment of bootstrap, consistency requires the intercept of the output Pomeron pole in the one-dimensional case to be larger than one: ${\ensuremath{\alpha}}_{P}(0)>1$, a situation reminiscent of the one in Reggeon field theory. Symmetry breakings of the Pomeron couplings are discussed. These couplings coincide with those of the $f$-dominated Pomeron model of Carlitz, Green, and Zee (CGZ). The case when in the unitarity loops all possible trajectories are exchanged is also considered. Predictions of the dual unitary model for the slopes of differential cross sections for diffractive scattering are made which differ from those of the CGZ model. A comparison with the experimentally available data is performed.

A theoretical approach to low multiplicity diffractive dissociation

We investigate the dynamics of low mass inelastic diffractive production in the framework of the “1/N dual unitarization” scheme. The smallness of inelastic diffractive dissociation is explicitly demonstrated by incorporating a Deck type mechanism with the crucial planar bootstrap equation. Although both inelastic and elastic pomeron couplings are of the same order in 1/N, the origin for their smallness, however, is not identical. Our work further confirms the validity of the iterative procedure, where the elastic amplitude is first generated from only non-diffractive intermediate states (except possibly for central collisions). Using a previous study of the “Cylinder” strength, we present also a semi-quantitative results for the integrated cross-section for low multiplicity diffractive production and compare it with the elastic cross-section at very high energies.

Flavour symmetry breaking and production processes in dual unitarisation

The effects of SU(3) and SU(4) symmetry breaking upon the dual unitarisation scheme are considered, at the planar and cylinder levels. When the effect of cluster threshold masses is considered, it becomes difficult to explain the observed rise in the proton-proton cross section in terms of pomeron flavouring. Estimates are obtained of strange and of charmed production cross sections. The associated production of the phi and psi mesons is also considered.

Flavor symmetry breaking in DTU

The implications of the breaking of SU (N) flavor symmetry are studied at the planar and cylinder levels of the Dual Topological Unitarization scheme. It is shown that the ρ intercept is constrained to lie between 0.51 and 0.54, and that SU (4) symmetry is necessarily more strongly broken that SU (3). The matrix structure of the cylinder bootstrap equations is shown to suppress many of the problems of the “cylinder extinction of the planar poles”, and to lead to a sensible singularity spectrum.

Elastic peak and hadron size from at-channel viewpoint

A consistent approach for understanding hadron size and the elastic diffraction peak from at-channel viewpoint is presented. Our approach is orthogonal to that of ans-channel additive quark model. We consider the problem of determining the elastic amplitude as the shadow of multiparticle production in the framework of dual topological unitarization (DTU), when pion exchanges are included in addition to the usual vector-tensor exchanges which generate the Pomeron. The inclusion of pion exchange does not affect the Pomeron trajectory substantially, but plays an important role in the computation of Pomeron residues to external particles. Good agreement with elastic data as well as total cross sections is obtained.

Low mass diffractive dissociation in a simple t-dependent dual bootstrap model

The smallness of inelastic diffractive dissociation is explicitly demonstrated, in the framework of the “ 1 N dual unitarization” scheme, by incorporating a Deck type mechanism with the crucial planar bootstrap equation. Although both inelastic and elastic pomeron couplings are of the same order in 1 N , the origin for their smallness, however, is not identical.

Study of a second-order hadronic process: Theπ−p→φnreaction

We present a semiquantitative analysis of the energy and momentum-transfer dependence of the truly ''second-order'' hadronic reaction ..pi../sup -/p ..-->.. phin, where the input is provided by certain ''first-order'' hadronic processes. Unitarity, analyticity and crossing determine both the imaginary and real parts of the amplitudes without the explicit use of dispersion relations. Dual unitarization naturally explains the marked difference between unnatural- and natural-parity exchanges, the former being responsible for the anomolous s and t behavior of ..pi../sup -/p ..-->.. phin.

Implication of charm and strangeness for the convergence of the “1/N dual unitarization” scheme

Due to the substantialSU4-symmetry breaking, the convergence of the “1/N topological expansion” is considerably improved in processes involving charmed quarks. We define an appropriate effective quark flavour numberN1 for each type of quark i=u, s, c, and a simple model suggests thatNu≈2.8,Ns≈4.7 andNc≈14, thus exhibiting the improved convergence, especially in the charm sector. One also obtainsσel/σ T≈ 1/N2 withN2 being replaced by the correspondingNi's, depending on the constituents of the colliding particles.

The bare parameters of Gribov's lagrangian in a simple t-dependent dual bootstrap model

In the context of the “ 1 N Dual Unitarization” scheme, we present an explicit dynamical study of the triple bare pomeron mechanism which governs the interaction term in Gribov's lagrangian. Together with the previously established bare pomeron slope and intercept, controlling respectively, the kinetic and mass terms in Gribov's lagrangian, this work demonstrates the viability of the “ 1 N Dual Unitarization” approach for a field theory of interaction bare pomerons.

S-channel discontinuities of the torus insertion in the dual unitarisation scheme

The refined analysis of the s-channel discontinuities of the torus insertion in the topological expansion is performed. Assuming the dependence on reggeon masses of the fixed-pole residue similar to that given by the Veneziano model we obtain almost exact cancellations of the discontinuities which in the one-dimensional approximation were argued to be responsible for the exchange degeneracy breaking between ϱ and A 2 trajectories. Implications on different f-IP schemes are discussed.

Dual unitarization approach to current-induced reactions

We establish an ansatz which incorporates the photon into the structure of duality diagrams. This then allows us to apply the Mueller-Regge formalism to deep-inelastic electron scattering. We derive Bjorken scaling, in particular the behavior (1 - x)/sup ..beta../ near x = 1 for the structure function, with ..beta.. compatible with the value deduced from the data. We also give results on the relative sizes of sea and valence quark distributions.

Baryons in dual unitarization and dynamical thresholds

Baryon exchange and baryon resonance production is introduced in the dual unitarisation scheme. The dynamical threshold for the production of meson and baryon resonances is incorporated. It is shown that the intercepts of the ω and the f, which are generated by baryons, are suppressed by the above dynamical threshold effects to α ω(0) ⋍ 0 and α f(0) < 0. The pomeron is shifted slightly upwards by baryon production. An upper limit is determined for the ratio of the crossed and uncrossed produced baryon lines. The breaking of the Freund-Rosner-Walz rule is discussed.

Possible restriction on internal symmetry from unitarity and the strength of hadronic cross-sections

We study the role of the self-consistent, nonlinear unitarity equation in the « 1/N dual unitarization » approach with at-dependent factorizable model, without and with inelastic diffraction. In the impact parameter representation of the unitarity equation, the input is the « cylinder » (nondiffractive) mechanism,i.e. the bare pomeron, which is strongly constrained by the planar bootstrap. A nontrivial lower bound is obtained for the (effective) quark flavour numberN in terms of the space-time properties of the model. We also investigate the dependence of the strength of hadronic cross-sections, and especially the ratioσel/σT, on bothN and the energy. Satisfactory numerical estimates are derived for the above physical quantities, in the NAL-ISR regime, by using our previous quantitative study of the « cylinder ».

Baryonium states in multiquark spectroscopy

The spectrum of qq qq states, examined in a quark-gluon model combined with dual unitarization, yields two types of baryoniums both of which have narrow widths into mesons. One type has normal hadronic widths ∼ 100 MeV into B B . The other is narrow, ∼ 10 MeV, and goes into B B only reluctantly, preferring, if possible, to decay by cascade, and, being a consequence only of hidden colour, is an important object to verify experimentally.

Baryons and dual unitarization

Processes involving baryons are discussed in the scheme of dual unitarization. In particular, the topological expansion is generalized to any hadronic S-matrix elements involving baryons and/or mesons. Our expansion is based on a model for the baryon propagator, which is a set of three planar Feynman diagrams joined at a junction line. The resulting expansion is a double expansion in 1/ N ( N= the number of quark flavours) and in the number of baryon loops. Based on this, several new observations are made in phenomenological problems, and a unifying point of view is stressed. The scheme is evidently crossing invariant, and unitarity constraints are imposed order by order in 1/ N and in the baryon loop number.

Baryon and Baryonium States with String-Junctions in the Dual Unitarization Scheme

\'Ve consider a baryon and a family of baryonium states m the dual unitarization scheme taking account of topological structures; a baryon has a Y shaped structure with a string­ junction at the center and baryonium states denoted by iJ,, M, and }IJ, have a pair of junction and antijunction. Coupled channel bootstrap equations are given for the baryonic and meeoonic reggeons and for the pomeron. The breaking of the exchange degeneracy between the iJ and /3 trajectories of baryons is given through a non-planar diagram with a twisted meson bridge between two membranes swept by strings; we have a.;,-aNp~O.l. The intercepts of the family of baryonium states are that aM.(±)<aM,(±)<a.,,,( +)<aM,(-), where (±) denote charge conjugation parity, and the triple-reggeon couplings iJnNI1\Il are shown to be g' (iJ,<±l 1\IIM) <g'(fVI,<±l N£1\4.) <o' (.A/f,<+J },11\ll) <o' (fVJ,<-l j'vJ.f'vf), where lvf denotes an ordinary natural parity meson: The ratio g (iVJ, <-l 1\111\11) : (} (lVJ, H BE) =1: 2 is encouraging to the idea that the difficulty of the 'o extinction can be remedied by the existence of the ]\1, (-).

Renormalization of trajectories with zero quantum numbers in dual unitarization

The renormalization of the $f$, ${f}^{\ensuremath{'}}$, $\ensuremath{\omega}$, and $\ensuremath{\varphi}$ trajectories through the crossed-loop (or cylinder) insertion is exhaustively studied in the context of dual unitarization. The trajectory functions $\ensuremath{\alpha}(t)$, mixing angles, and couplings are obtained for the range $\ensuremath{-}1.0\ensuremath{\lesssim}t\ensuremath{\lesssim}2.0$ ${\mathrm{GeV}}^{2}$. In particular, it is found that the $f$ is renormalized upwards by a considerable amount, acquiring an intercept ${\ensuremath{\alpha}}_{f}(0)\ensuremath{\simeq}0.93$ and slope ${\ensuremath{\alpha}}_{f}^{\ensuremath{'}}(0)\ensuremath{\simeq}0.4$. It becomes still flatter for $t&lt;0$. These characteristics, together with its symmetry properties, strongly support its identification with the experimental Pomeron. At the same time, the mass splitting of $f$ and ${A}_{2}$ is correctly given as about 0.1 ${\mathrm{GeV}}^{2}$. The $\ensuremath{\omega}$ trajectory is renormalized downwards, and acquires a considerable admixture of the strange component, in agreement with the phenomenological estimates of $\frac{{\ensuremath{\gamma}}_{\mathrm{KK}}^{\ensuremath{\omega}}}{{\ensuremath{\gamma}}_{\mathrm{KK}}^{\ensuremath{\rho}}}\ensuremath{\simeq}1.4$. Furthermore, the $t$ dependence of the calculated quantities in general verifies the prediction that renormalization effects decrease as $t$ increases.