Considerations on the Schmid theorem for triangle singularities

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. arXiv:1809.06890v1 [hep-ph]

Threshold cusps and triangle singularities in hadronic reactions

“…This is the so-called Schmid theorem. But for the reactions involving inelastic rescattering processes [129,285,286], the situation will be quite different from the single-channel case discussed in Ref. [74].…”

“…Then, the TS condition of cos θ = 1 in the 2 + 3 c.m. frame [286] constrains the region of TS on the upper-half arc of the phase space boundary in Fig. 21; the condition that the particles 1 and B are parallel in the particle A rest frame restricts the region to the upper-left arc of the phase space boundary in Fig.…”

“…The validity and limitation of Schmid theorem have been studied in Refs. [75,129,285,286,299]. Let us take a look at the triangle-loop amplitude in Fig.…”

“…(66) with a dispersion integral [74,129], or evaluating the q integral of Eq. (46) explicitly [286], as given in, e.g., Ref. [301] with the nonrelativistic reduction, the most singular part of T (loop) is given by (note the sign difference from Ref.…”

“…[301] with the nonrelativistic reduction, the most singular part of T (loop) is given by (note the sign difference from Ref. [286] due to the choice of convention)…”