In quantum physics, Regge theory is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1959.
Contents
Example
The simplest example of Regge poles is provided by the quantum mechanical treatment of the Coulomb potential
where
Considered as a complex function of
Regge trajectories can be obtained for many other potentials, in particular also for the Yukawa potential.
Regge trajectories appear as poles of the scattering amplitude or in the related
where
History and implications
The main result of the theory is that the scattering amplitude for potential scattering grows as a function of the cosine
where
Shortly afterwards, Stanley Mandelstam noted that in relativity the purely formal limit of
The switch required swapping the Mandelstam variable
...which says that the amplitude has a different power law falloff as a function of energy at different corresponding angles, where corresponding angles are those with the same value of
In 1960 Geoffrey Chew and Steven Frautschi conjectured from limited data that the strongly interacting particles had a very simple dependence of the squared-mass on the angular momentum: the particles fall into families where the Regge trajectory functions were straight lines:
Experimentally, the near-beam behavior of scattering did fall off with angle as explained by Regge theory, leading many to accept that the particles in the strong interactions were composite. Much of the scattering was diffractive, meaning that the particles hardly scatter at all — staying close to the beam line after the collision. Vladimir Gribov noted that the Froissart bound combined with the assumption of maximum possible scattering implied there was a Regge trajectory that would lead to logarithmically rising cross sections, a trajectory nowadays known as the Pomeron. He went on to formulate a quantitative perturbation theory for near beam line scattering dominated by multi-Pomeron exchange.
From the fundamental observation that hadrons are composite, there grew two points of view. Some correctly advocated that there were elementary particles, nowadays called quarks and gluons, which made a quantum field theory in which the hadrons were bound states. Others also correctly believed that it was possible to formulate a theory without elementary particles — where all the particles were bound states lying on Regge trajectories and scatter self-consistently. This was called S-matrix theory.
The most successful S-matrix approach centered on the narrow-resonance approximation, the idea that there is a consistent expansion starting from stable particles on straight-line Regge trajectories. After many false starts, Dolen Horn and Schmidt understood a crucial property that led Gabriele Veneziano to formulate a self-consistent scattering amplitude, the first string theory. Mandelstam noted that the limit where the regge trajectories are straight is also the limit where the lifetime of the states is long.
As a fundamental theory of strong interactions at high energies, Regge theory enjoyed a period of interest in the 1960s, but it was largely succeeded by quantum chromodynamics. As a phenomenological theory, it is still an indispensable tool for understanding near-beam line scattering and scattering at very large energies. Modern research focuses both on the connection to perturbation theory and to string theory.