It is our purpose to discuss in this note, for this kind of problem, a general method which allows in principle the determination of certain cross sections which are out of reach of direct experimental measurement.The method is based on the observation of threshold anomalies of the kind of a cusp or rounded step in cross sections for reactions with three particles in the final state. 2 ' 3 The necessary requirements are that we must have (at least) two different reactions with three particles in the final state, that one of the particles is common to both the considered reactions, and that the phenomenon is analyzed by keeping the total energy fixed. In processes of this kind it is very important to have a good energy resolution for the incoming beam; a spread of the initial energy can in fact destroy the threshold effect which usually involves a small portion of the final spectrum. In this respect an ideal situation presents itself in the three-particle decays of an unstable particle whose mass is well determined. As an example we will discuss the decay of the K* meson into three pions, for which all the requirements for the appearance of the effect are satisfied*:#+^+ + ^0+7^ (b) 8 H 0 tJberall, Phys. Rev 0 119, 365 (1960). See also D. Yovanovitch, Phys 0 Rev. J17, 1580 (1960), for experimental confirmation. 9 A. Fujii and H. Primakoff, Nuovo cimento 12, 327 (1959); L. Wolfenstein, Nuovo cimento 13, 319 (1959); M 0 Morita and A. Fujii, Phys. Rev. 118, 606 (1960); G. Flammand and K. W. Ford, Phys e Rev. 116, 1591 (1959).If we fix our attention on the ir + meson coming from the decay (b) and we vary its kinetic energy E x starting from its maximum value, we see that at first there can be no corresponding TT + from (a) with the same energy, the decay (a) being there energetically forbidden due to the heavier mass of the charged with respect to the neutral pions. By decreasing E± and therefore by increasing the relative energy E 2° of the (7T°, 7T°) pair, we get to the point E^ = 2 m* -2m° = 9.2 Mev where a (TT", TT + ) pair can be created at rest (in its center-of-mass system) in a (7T°, 7r°) collision. By increasing E 2 b over that value it is by now clear that the decay (a) will also be energetically possible so that£ 2 > or the corresponding E 19 acts essentially as the threshold energy for the reaction 7r° + 7r°-*7r~ + 7r+. At E x a threshold effect of the kind of a cusp or rounded step will be observed in the spectrum of the 7r + coming from decay (b) and the size of such an effect will depend critically on the lowenergy value of the pion-pion charge exchange cross section. Since at threshold the (ir" 9 ir + ) pair is in a relative S wave, from conservation of parity (here applicable) and total angular momentum, it follows that the threshold effect will appear in the relative S wave of the (ir°, TT°) pair. Further, due to the small relative energy of the {if 0 9 ff°) P ai r at the 7r~7r + threshold, the relative waves with l> 0 will be negligible, so that the 7T+ common to both decays will also be in the S wa...
A relativistic wave equation for the description of a composite system with nonrelativistic internal motion is deduced from the Bethe-Salpeter equation. It is shown that when the internal motion can be described by algebraic methods, as in the case of the hydrogen atom, the proposed equation for the motion of the system as a whole (motion of the cm.) is equivalent to a Majorana-type equation, free from the well-known difficulties such as a spacelike solution. The hydrogen-atom example is discussed in some detail. It is shown how its 6*0(4,1) spectrum-generating algebra contains the spin part of the generators of the Lorentz group. The solutions found to the Majorana-type equation, which is able to describe both discrete and continuum eigenstates, span a representation of an 50(4,2) algebra.
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