Shape and soft functions of HQET and SCET in the 't Hooft model

The main application of heavy quark effective theory (HQET) and of soft collinear effective theory (SCET) is in establishing factorization theorems for exclusive and semi-inclusive decays of heavy mesons. However, the calculation of the soft factors from the HQET or SCET factorization relations is, as usual, impeded by the non-perturbative aspect of the strong interactions. In the hope of gaining some insights into some of these quantities we compute them in the 't Hooft model. We find that the B-meson shape function is exactly given by the square of the B-meson light-cone wave-function. The structure of the B-π soft function is more complicated: it is given by the product of wave-functions or by a resonant sum depending on the kinematics. The result simplifies dramatically in the chiral limit, where it can be compared with general arguments based on heavy meson-chiral perturbation theory. No attempt is made to use these results for applications to phenomenology.