The spectral shapes of the fluxes of electrons and positrons and the average residence time of cosmic rays in the Galaxy

The cosmic ray energy spectra encode very important information about the mechanisms that generate relativistic particles in the Milky Way, and about the properties of the Galaxy that control their propagation. Relativistic electrons and positrons traveling in interstellar space lose energy much more rapidly than more massive particles such as protons and nuclei, with a rate that grows quadratically with the particle energy $E$. One therefore expects that the effects of energy loss should leave observable signatures in the $e^\mp$ spectra, in the form of softenings centered at the critical energy $E^*$. This quantity is determined by the condition that the total energy loss suffered by particles during their residence time in the Galaxy is of the same order of the initial energy. If the electrons and positrons are accelerated in discrete (quasi) point--like astrophysical objects, such as Supernova explosions or Pulsars, the stochastic nature of the sources should also leave observable signatures in the $e^\mp$ energy spectra and angular distributions above a second (higher) critical energy $E^\dagger$, determined by the condition that particles with $E \gtrsim E^\dagger$ can propagate only for a maximum distance shorter than the average separation between sources. In this work we discuss the theoretically expectations for the signatures of the energy loss effects on the electron and positron spectra, and compare these predictions with the existing observations. Recent measurements of the $(e^- + e^+$) flux have discovered the existence of a prominent spectral break at $E \simeq 1$~TeV. This spectral feature can perhaps be identified with the critical energy $E^*$. An alternative hypothesis is to assmume that $E^*$ has a much lower value of order few GeV. Resolving this ambiguity is of great importance for our understanding of Galactic cosmic rays.