Abstract
We investigate the Schmid theorem, which states that if one has a tree level mechanism with a particle decaying to two particles and one of them decaying posteriorly to two other particles, the possible triangle singularity developed by the mechanism of elastic rescattering of two of the three decay particles does not change the cross section provided by the tree level. We investigate the process in terms of the width of the unstable particle produced in the first decay and determine the limits of validity and violation of the theorem. One of the conclusions is that the theorem holds in the strict limit of zero width of that resonance, in which case the strength of the triangle diagram becomes negligible compared to the tree level. Another conclusion, on the practical side, is that for realistic values of the width, the triangle singularity can provide a strength comparable or even bigger than the tree level, which indicates that invoking the Schmid theorem to neglect the triangle diagram stemming from elastic rescattering of the tree level should not be done. Even then, we observe that the realistic case keeps some memory of the Schmid theorem, which is visible in a peculiar interference pattern with the tree level.
Highlights
1, 2 and R as intermediate states
Once the A particle decays into 1 and R, and R decays into 2 and 3, the probability for this process is established and the posterior interaction of 1 and 2 should not modify this probability. This argument is the intuitive statement of the Schmid theorem, which technically reads as follow: Let tt(0) be the S-wave projection of the tree level amplitude, tt, of the diagram of Fig. 1, evaluated in the rest frame of 1+2, referred to the angle between the particles 1 and 3
It is still interesting to note that in the realistic case the reaction studied still has a memory of Schmid theorem, in the sense that the coherent sum of amplitudes gives rise to a width, or cross section, that is even smaller than the incoherent sum of both contributions
Summary
1, 2 and R as intermediate states. This loop function can lead to some singularities (for the limit of zero width of the R particle) when all the three intermediate particles are placed on shell, particle 1 and 3 are antiparallel and a condition is fulfilled which in classical terms can be stated as corresponding to the decay of A at rest into R and 1, assuming R moves forward and 1 backward, R decays into 2 and 3, and letting 3 go forward, 2 eventually goes backward, which means in the same direction of 1. Studies of triangle singularities were done in Ref. There a surprise appeared, known nowadays as Schmid theorem that states that if the rescattering of particles 1 and 2 occurs, going to the same state 1 + 2, the triangle singularity does not lead to any observable effect in magnitudes like cross sections or differential widths. It is reabsorbed by the S-wave of the tree level amplitude (the same mechanism without rescattering, see Fig. 1) modifying only the phase of this partial amplitude. A topic that stimulated the present interest on the subject was the one of isospin violation in the η(1405) decay into π 0 f0(980) versus the isospin allowed decay into π 0a0(980)
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