Very High-Energy Collisions of Hadrons by Richard P. Feynman circa October 1969

1. Very High-Energy Collisions of Hadrons by Richard P. Feynman circa October 1969 Presented by Neil McFadden

2. Purpose • To make suggestions as to how to characterize cross sections and what significant data can be taken • In the true nature of Feynman he says that “he is more sure of the conclusions than of any single argument which suggested them” • So he does not include much of he reasoning about how he got to his results, still though it has been cited 1560 times since publishing

3. Elastic and Inelastic Scattering • 10% of scatterings are Elastic and of those 1/3 of them are interpreted as diffraction dissociation i.e. A+B = A+C both of which approach a constant cross section at very high-energies. • Main focus of this paper are inelastic scatterings because they have not been studied very extensively due to the high amount of data produce from these collisions.

4. Defining Conditions • Assumes transverse momenta are limited in a way independent of the large z-component momenta, where z is the direction of propagation. • If we define the energy of the system to be W and the longitudinal momenta to be Pz,then we can define x = W/Pzsuch that x is in absolute units. • Thus the differential cross section is dxd2Q where dx=f (W,Pz/Q)

5. Two types of experiments • Exclusive experiment: we ask that certain particles, given a x and Q, be formed and no others. For example A+B→ ƩCi+ ƩDiwhere each sum represents particles moving either left or right. • Thus the cross section should vary as (W2)2α(t)-2 where a(t) is the highest Regge trajectory capable of converting the quantum numbers of A into the sum of the C’s, and t is the transverse momentum difference of A and the sum of the C’s • In the case of diffraction dissociation, the cross section should approach the constant ratio of the elastic cross section

6. Inclusive Experiments • An experiment in which we look for special particles with certain x and Q in the final state, but we allow other productions. An example would be measuring the mean number of K+’s in a pp collision • Feynman states that the cross-section approach's a constant as W gets large

7. Inclusive Experiment (continued) • Because the 3-compents of isotopic spin must be exchanged and thus the fields associated with the spins, the particles will radiate(like bremsstrahlung radiation). The faster the spin changes, the greater radiation. This causes the cross section to fall off to a constant value as the energy gets large. • In exclusive experiments (pure two body) this is forbidden because once they start radiating, thus it will no longer be a two body experiment.

8. Small x Limit • Because of the simplicity, we will look at the relations derived from assume a small value of x. • By Lorentz transformation, the fields to be radiated are becoming narrower in the zdirection as W rises. Which in the limit the fields correlates to a delta function in the z direction. • Thus applying Fourier analyze to the field energy, we see that the momentum dPz is uniform.

9. Small x continued • If we assume that each particle of mass μ has energy and the field energy is distributed uniformly. We can then define the mean number of particles to be dPz/E for small x. • Using the differential cross section and the mean number of particles we can define the probability of finding particle i as dxdQ2dPz /E or f(Q,Pz/W)dPz dQ2/

10. Large W • Note that in the limit of large W ,f(Q,Pz/W) is approximately just a function of Q • Also dPz/E approximately becomes dx/x • Because of this dx/x behavior, the mean total number (multiplicity) of any kind of particle will grow as the logarithm of W • If we imagine some independently emitted units, n, the probability that none of them would be emitted would be e-n (as suggested by the Poisson distribution)

11. Final Conclusions • If we include amplitudes with similar n, then the probability that should vary as , where C is moving to the right with almost all the momentum of A. • Again α(t) is the highest Regge trajectory which could carry off the quantum numbers an momentum transfer t needed to change A to C.

12. Regge Theory • l can take on complex numbers and the scattering amplitude A(z) ~ , where l is not necessarily an integer. When l is an integer the particle is bound with that l. • Large z (scattering angle) is equivalent to large t (large energy in cross channels), where one incoming particle has energy momentum that makes it an energetic out going anti particle.

13. More Theory • Thus it is demanded that the function that determines the fall off rate of the scattering amplitude for a particle-particle scattering at large enough energies is the same as the function that determines the bound state energies for particle- antiparticle system, as a function of momentum. • Thus at high energies, first l becomes a function of the squared momentum transfer tand second the cross section grows at a logarithmic rate and finally l(E2)=kE2, which are the Regge lines.