Regge theory
In quantum physics, Regge theory (/ˈrɛdʒeɪ/ REJ-ay, Italian: [ˈrɛddʒe]) 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 multiple of ħ but is allowed to take any complex value.
[1] The simplest example of Regge poles is provided by the quantum mechanical treatment of the Coulomb potential
or, phrased differently, by the quantum mechanical treatment of the binding or scattering of an electron of mass
of the binding of the electron to the proton is negative whereas for scattering the energy is positive.
is in quantum mechanics (by solution of the radial Schrödinger equation) found to be given by
Thus in this consideration the orbital momentum can assume complex values.
[2][3][4] Regge trajectories appear as poles of the scattering amplitude or in the related
-matrix is given by the following expression as can be checked by reference to any textbook on quantum mechanics: where
(the gamma function in the numerator) possesses poles at precisely those points which are given by the above expression for the Regge trajectories; hence the name Regge poles.
is the noninteger value of the angular momentum of a would-be bound state with energy
It is determined by solving the radial Schrödinger equation and it smoothly interpolates the energy of wavefunctions with different angular momentum but with the same radial excitation number.
is known as the Regge trajectory function, and when it is an integer, the particles form an actual bound state with this angular momentum.
Shortly afterwards, Stanley Mandelstam noted that in relativity the purely formal limit of
means large energy in the crossed channel, where one of the incoming particles has an energy momentum that makes it an energetic outgoing antiparticle.
This observation turned Regge theory from a mathematical curiosity into a physical theory: it demands that the function that determines the falloff rate of the scattering amplitude for particle-particle scattering at large energies is the same as the function that determines the bound state energies for a particle-antiparticle system as a function of angular momentum.
, which is the squared momentum transfer, which for elastic soft collisions of identical particles is s times one minus the cosine of the scattering angle.
The relation in the crossed channel becomes 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
The range of angles where scattering can be productively described by Regge theory shrinks into a narrow cone around the beam-line at large energies.
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:
The straight-line Regge trajectories were later understood as arising from massless endpoints on rotating relativistic strings.
Since a Regge description implied that the particles were bound states, Chew and Frautschi concluded that none of the strongly interacting particles were elementary.
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.
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.
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, Richard Dolen, David Horn, and Christoph Schmid understood a crucial property that led Gabriele Veneziano to formulate a self-consistent scattering amplitude, the first string theory.
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.
