Bosonic string massive scalar amplitudes and absence of “chaos”

Abstract We compute directly in covariant formalism various four point massive scalar amplitudes in bosonic string with scalars up to level 11. The most “difficult” amplitude we consider is four identical scalars at level 4, i.e. the first non trivial amplitude with four identical massive scalars. All other amplitudes have at least one tachyon. All computed amplitudes are perfectly regular and show no sign of erratic behavior. While this could be naively expected it is not completely trivial because of higher spins exchanged. We give a naive argument based on a very simple model of why this can reasonably be expected and we check on the explicit examples that the intuition derived from the model is sensible. Nevertheless our computations involve states up to $$N\sim 10$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> while the erratic amplitudes have $$N \gtrsim 50$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≳</mml:mo> <mml:mn>50</mml:mn> </mml:mrow> </mml:math> so there is still the possibility of an onset of erratic behaviour at higher N . Furthermore we discuss the complications due to null states (BRST exact states) in actually computing the covariant bosonic string spectrum and how these can be overcome by using a “ $$\Pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Π</mml:mi> </mml:math> gauge” which makes manifest that physical states at any level can be derived as the only solutions of a set of equations without further work. Moreover when an irrep is allowed at certain level and in a certain dimension its degeneracy is generically the same in all higher dimensions as shown by Curtright et al at the end of 80s.