Two-component Duality and Strings

The Veneziano model [1], the starting point of string theory, addressed the at that point much studied and phenomenologically successful idea of two-component duality. Here I would like to recall this idea and give its meaning in modern terms. At the risk of letting the cat out of the bag at too early a stage, let me right away say that the two components in question will turn out to be the open and the closed hadronic strings.

The argument for two-component duality is the following. Unlike quarks, hadrons (mesons and baryons) are obviously not elementary objects. Yet the particles appearing in the initial, final and intermediate states of the hadronic S-matrix are precisely these composite objects and not their constituent quarks. With elementary particles it is obvious how to calculate the S-matrix, just use the Feynman rules. For instance when studying tree-level Bhabha-scattering in QED, we are instructed to add the s-and t-channel photon-pole diagrams, as in Fig. 1. But there we have a lagrangian and the photon is an “elementary” particle whose field appears in this lagrangian. When dealing with composite states, these are represented by an infinite sum + . . . of Feynman diagrams in the quantum field theory theory of the elementary fields out of which the composite particles are built. In the simplest case we can think of these diagrams as Bethe-Salpeter ladder-diagrams. If only one kind of line (field) )is involved in these ladder-diagrams, as for instance with a gΦ 3 lagrangian, then adding the diagrams in which the composite particle pole appears in the s-and in the t-channels, would lead to the doublecounting of the one-box diagram as in Fig. 2. If instead of Bethe-Salpeter ladder-diagrams, we would be summing over all planar “fishnet”-diagrams of say a scalar QFT with cubic and quartic self interaction theory (an example of such a planar fishnet diagram is given in Fig. 3), then not only the one-box diagram, but each and every diagram would be counted twice. This partial, or total double-counting is the essential difference between a theory of elementary and of composite particles.

In their seminal paper, Dolen, Horn and Schmid [2] have shown that for the composite hadrons studied in the laboratory a full such double counting would be involved. I distinctly remember the excitement with which we learned of their paper from Murray Gell-Mann, when he visited us in 1967 before the circulation of the preprint. Dolen, Horn and Schmid carefully showed that in hadronic scattering processes such as πN-charge-exchange, ? ? ? ?

Figure 3: Full double-counting when adding s-and t-channel planar fishnet diagrams.

both the smooth t-channel Regge exchange and the bumpy s-channel resonances account for the full amplitude. This is possible because the imaginary part of the t-channel (Regge) exchange, averages over the contributions of the absorptive parts of the direct s-channel resonances. Therefore the two should not be added, to avoid double-counting. Rather they are dual to each other.

But even for elastic scattering amplitudes for which the s-channel is devoid of resonances such as π + π + , pp, K + p, etc… there are Regge-exchanges and for them there is also diffraction with the corresponding Pomeron-exchange. How can this be? The imaginary part of the Regge-exchange comes from the term e -iπα(t)) in the Regge signature factor. But this term has opposite signs for even-and odd-signature Regge-exchanges. If these are to average to zero, to match the absence of resonances in the s-channel, then the even and oddsignature hadronic Regge poles must come in degenerate pairs with matching residue-functions. This exchange degeneracy is observed experimentally. For instance, extrapolating the rectilinear ρ Regge trajectory to spin 2, it does indeed go through the f -meson point, as required by exchange degeneracy.

But all this still does not take the Pomeron into account. As a consequence of unitarity, diffraction should correspond not to tree-level, but to higher order processes and it was conjeectured by Freund [3] and by Harari [4] that, unlike the other Regge poles, the Pomeron is dual not to s-channel resonances, but to s-channel non-resonant background. With this FH-conjecture a two-component picture has thus emerged, in which besides the the mesonic and baryonic Regge trajectories dual to resonances, there is a second component, the Pomeron, dictated by unitarity as a largely t-channel flavor singlet trajectory, dual to non-resonant s-channel background. This two-component picture accounted for a vast body of data. The remaining question was how to account for the crucial features of the Pomeron this way.

Mesonic Regge poles and their dualities were modeled by Rosner [5] Harari [6] with what were called “duality diagrams” -such as the one in Fig. 4 -and what are in retrospect clearly open string diagrams, Chan-Paton [7] rules and all. For the Pomeron loop diagrams are dictated by unitarity. The simp

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