Thursday, April 27, 2006

Pomeron

Matt Strassler has just described their fascinating work on
with Richard Brower, Chung-I Tan, and Joe Polchinski. Return 40 years into the past. The research that eventually evolves into string theory is proposed as a theory of strong interactions: something that would be known as a failed theory of strong interactions for the following 30 years. Things only start to slowly change after the 1997 discovery by Juan Maldacena and a steady flow of new insights eventually leads to a nearly full revival of the description of strong interactions using a "dual" string theory, albeit this string theory is more complicated than what was envisioned in the late 1960s. QCD can be equivalently described as the old string theory with some modern updates: higher-dimensional and braney updates.

The basic concepts of the Regge physics included the Regge trajectory, a linear relation between the maximum spin "J" that a particle of squared mass "m^2" can have; the slope - the coefficient "alphaprime" of the linear term "alphaprime times m^2" - is comparable to the inverse squared QCD scale. The dependence of "J" could be given by a general Taylor expansion but both experimentally as well as theoretically, the linear relation was always preferred.

Note that "alphaprime" in "the" string theory that unifies all forces is much much smaller area than the inverse squared QCD scale (the cross section of the proton). We are talking about a different setup in AdS/QCD where the four-dimensional gravity may be forgotten. This picture is not necessarily inconsistent with the full picture of string theory with gravity as long as you appreciate the appropriately warped ten-dimensional geometry.

At this moment, you should refresh your memory about the chapter 1 of the Green-Schwarz-Witten textbook. There is an interesting limit of scattering in string theory (a limit of the Veneziano amplitude) called the Regge limit: the center-of-mass energy "sqrt(s)" is sent to infinity but the other Mandelstam variable "t" - that is negative in the physical scattering - is kept finite. The scattering angle "sqrt(-t/s)" therefore goes to zero.

In this limit, the Veneziano amplitude is dominated by the exchange of intermediate particles of spin "J". Because the indices from the spin must be contracted, the interaction contains "J" derivatives, and it therefore scales like "Energy^J". Because there are two cubic vertices like that in the simple Feynman diagram of the exchange type, the full amplitude goes like "Energy^{2J}=s^J" where the most important value of the spin "J" is the linear function of "t" given by the linear Regge relation above.

The amplitude behaves in the Regge limit like "s^J(t)" where "J(t)" is the appropriate linear Regge relation. You can also write it as "exp(J(t).ln(s))". Because "t=-s.angle^2", you see that the amplitude is Gaussian in the "angle". The width of the Gaussian goes like "1/sqrt(ln(s))" in string units. Correspondingly, the width of the amplitude Fourier-transformed into the transverse position space goes like "sqrt(ln(s))" in string units. That should not be surprising: "sqrt(ln(s))" is exactly the typical transverse size of the string that you obtain by regulating the "integral dsigma x^2" which equals, in terms of the oscillators, "sum (1/n)" whose logarithmic divergence must be regulated. The sum goes like "ln(n_max)" where "n_max" must be chosen proportional to "alphaprime.s" or so.

If you scatter two heavy quarkonia (or 7-7 "flavored" open strings in an AdS/CFT context, think about the Polchinski-Strassler N=1* theory) - which is the example you want to consider - the interaction contains a lot of contributions from various particles running in the channel. But the formula for the amplitude can be written as a continuous function of "s,t". So it seems that you are effectively exchanging an object whose angular momentum "J" is continuous.

Whatever this "object" is, you will call it a pomeron.

In perturbative gauge theory, such pomeron exchange is conveniently and traditionally visualized in terms of Feynman diagrams that are proportional to the minimum power of "alpha_{strong}" that is allowed for a given power of "ln(s)" that these diagrams also contain: you want to maximize the powers of "ln(s)" and minimize the power of the coupling constant and keep the leading terms. When you think for a little while, this pomeron exchange leads to the exchange of DNA-like diagrams: the diagrams look like ladder diagrams or DNA.

There are two vertical strands - gluons - stretched in between two horizontal external quarks in the quarkonia scattering states. And you may insert horizontal sticks in between these two gluons, to keep the diagrams planar. If you do so, every new step in the ladder adds a factor of "alpha_{strong}.ln(s)". You can imagine that "ln(s)" comes from the integrals over the loops.

What is the spin of the particles being exchanged for small values of "t", the so-called intercept (the absolute term in the linear relation)? It is a numerical constant between one and two. Matt essentially confirmed my interpretation that you can imagine QCD to be something in between an open string exchange (whose intercept is one) and a closed string exchange (whose intercept is two). The open string exchange with "J=1" is valid at the weak QCD coupling - it corresponds to a gluon exchange. At strong coupling, you are exchanging closed strings with "J=2".

For large positive values of "t", you are in the deeply unphysical region because the physical scattering requires negative values of "t" (spacelike momentum exchange). But you can still talk about the analytical structure of the scattering amplitude - Mellin-transformed from "(s,t)" to "(s,J)". For large positive "t", you will discover the Regge behavior which agrees with string theory well. Unfortunately, this is the limit of scattering that can't be realized experimentally. Nevertheless, for every value of "t", you find a certain number of effective "particles" that can be exchanged - with spins up to "J" which is linear in "t".

The negative values of "t" can be probed experimentally, and this is where string theory failed drastically in the 1970s: string theory gave much too soft (exponentially decreasing) behavior of the amplitude at high energies even though the experimental data only indicated a much harder (power law) behavior. So now you isolate two different classes of phenomena:
  • the naive string theory is OK for large positive "t"
  • the old string theory description of strong interactions fails for negative "t"; the linear Regge relation must break down here
But the old string theory only fails for negative "t" if you don't take all the important properties of that string theory into account. The most important property that was forgotten 35 years ago was the new, fifth dimension. The spectrum of particles - eigenvalues of "J" - is related to the Laplacian but it is not just a four-dimensional Laplacian; it also includes a term in the additional six dimensions, especially the fifth holographic dimension of the anti de Sitter space. And this term can become - and indeed, does become - important.

What is the spectrum of allowed values of "J" of intermediate states that you can exchange at a given value of "t"? Recall that each allowed value of "J" of the intermediate objects generates a pole in the complex "J" plane - or a cut whenever the spectrum of allowed "J" becomes continuous. For large positive "t", the spectrum contains a few (roughly "alphaprime.t") eigenvectors with positive "J"s, and a continuum with "J" being anything below "J=1". For negative values of "t", you only see the continuum spectrum (a cut) for "J" smaller than one.

Don't forget that the value of "J" appears as the exponent of "s" in the amplitude for the Regge scattering. We are talking about something like "s^{1.08}" or "s^{1.3}" - both of these exponents appear in different kinds of experiments and can't be calculated theoretically at this moment.

Matt argues convincingly that the Regge behavior for large positive "t", with many poles plus the cut below "J=1", is universal. The "empty" behavior at large negative "t" where you only see the continuum below "J=1" is also universal. It is only the crossover region around "t=0" that is model-dependent and where the details of the string-theoretical background enter. And they can calculate the spectrum of "J" as a function of "t" in toy models from string theory. They assume that the string-theoretical scattering in the AdS space takes place locally in ten dimensions, and just multiply the corresponding amplitudes by various kinematical and warp factors - the usual Polchinski-Strassler business.

The spectrum of poles and cuts in the "J" plane reduces to the problem to find the eigenvalues of a Laplacian - essentially to a Schrödinger equation for a particle propagating on a line. You just flip the sign of the energy eigenvalues "E" from the usual quantum mechanical textbooks to obtain the spectrum of possible values of "J". And they can determine a lot of things just from the gravity subsector of string theory - where you exchange particles of spin two (graviton) plus a small epsilon that arises as a string-theoretical correction.

For large positive "t", you obtain a quantum mechanical problem with a locally negative (binding) potential that leads to the discrete states - those that are seen at the Regge trajectory.

When all these things are put together, they can explain a lot about physics observed at HERA. The calculation is not really a calculation from the first principles because they are permanently looking at the HERA experiments to see what they should obtain. But they are not the first physicists who use these dirty tricks: in the past, most physicists were constantly cheating and looking at the experiments most of their time. ;-)

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