Intersecting D6-branes

In this section we consider a D-brane configuration that has a set of features that make it a good starting point for the construction of a string model of particle physics. Since we need fermions, we use a ten-dimensional superstring theory. In this theory, six of the ten dimensions, x4, …, x9, are taken to form a small compact space of finite volume. This is necessary in order to have an effectively four-dimensional spacetime (with coordinates x0, x1, x2, and x3). The compact space is as simple as possible: each dimension is turned into a circle, so that the resulting space is a six-dimensional torus T6.

Duality symmetries and Hamiltonians

Duality symmetries are some of the most interesting symmetries in physics. The term “duality” is generally used by physicists to refer to the relationship between two systems that have very different descriptions but identical physics. The main subject of this chapter is one such situation that arises in closed string theory. You may think that a world where one dimension is curled up into a circle of radius R could easily be distinguished from a world in which the circle has radius α′/R (recall that α′ has units of length-squared), but in closed string theory these two worlds are indistinguishable for any value of R. There is a duality symmetry that relates them to each other. This symmetry is called T-duality, where the T stands for toroidal.

How to examine unphysical sheets of the amplitude

In the previous lecture we have discussed in detail that owing to unitarity, analyticity and crossing symmetry — the general properties of the theory — all the physics of hadron interactions is determined, in principle, by the spectrum of real particles.

We saw that the interaction constant — the measure of the interaction strength — entered only as a residue in the pole of the scattering amplitude. Can it be true that the plethora of phenomena in the hadron world is described by a single quantitative characteristic — the residue?

This situation looks strange and profoundly unsatisfactory from a theoretical point of view. It makes one wonder whether there is not something hidden beyond the mass spectrum that we have introduced.

Philosophy aside, it is important to know how the amplitude behaves in the vicinity of the cut. We cannot say a priori that it changes smoothly there, since the question of smoothness of a multi-valued function is a delicate one.

If we position ourselves near the cut on the physical sheet then, rather close in the energy variable, we have an unphysical sheet about which nothing had been said so far. If there is a singularity (for example, a pole) on the unphysical sheet close to the physical one, the amplitude on the physical sheet would be changing fast.

Quantum Chromodynamics (QCD) was in its infancy when Gribov delivered his lectures on strong interactions. Since then QCD had been established as the true microscopic theory of hadrons.

The main (though not the only) focus of these lectures is to present the ‘old theory’ of hadron interactions (known as reggistics). This theory has realized the ‘Pomeranchuk—Gribov programme’ of describing strong interactions without appealing to the internal structure of hadrons. The old theory was launched in 1958 by the Pomeranchuk theorem and reached a climax in Gribov's prediction of an asymptotic equality of hadron cross sections 15 years later. With the advent of QCD, it was abandoned by the great majority of theorists in the mid-1970s and has been neither taught nor learnt since.

QCD — the ‘new theory’ — is now in its fourth decade. The QCD Lagrangian approach did marvels in describing rare processes. This is the realm of hard interactions that occur at small distances where quarks and gluons interact weakly due to the asymptotic freedom. The domain of expertise of the old theory is complementary: it is about normal size hadron—hadron cross sections, soft interactions that at high energies are dominated by peripheral collisions developing at large distances. QCD only starts to timidly approach this domain, with new generations of researchers borrowing (sometimes improperly) the notions and approaches developed by the ‘old theory’.

We have a serious task ahead: to analyse and sum up the branchings, taking into account how they influence each other, and to learn how to write the amplitudes respecting unitarity.

Reggeon branchings from the s-channel point of view

The s-channel approach

In Lecture 2 we discussed how, given a particle spectrum, we can construct interaction amplitudes with the help of unitarity conditions and analyticity (dispersion relations). The dispersive method is well formulated and straightforward but not very convenient in practice. So instead of iterating the amplitudes through the non-linear unitarity relations we draw series of Feynman diagrams which satisfies (at least term by term) the unitarity condition.

Now we face an opposite situation for the first time. We discovered the branchings and found the specific unitarity conditions they satisfy. Can we construct an effective field theory that would generate the reggeon unitarity conditions?

There are two ways of attacking the problem.

1. One can explicitly construct an effective reggeon QFT directly in the t-channel, identifying reggeons with particles of a non-relativistic (anti-hermitian) field theory in 2 + 1 dimensions.

2. Alternatively, we can solve the problem of constructing the reggeon diagrams which solve the unitarity conditions, working directly in the s-channel.

The two routes yield the same results. We will follow the second one, for a number of reasons.