It is suggested that the long-wavelength excitation spectrum of superfluid He4 may consist of a branch of quasiparticle excitations which has a gap at zero momenta in addition to a branch of hydrodynamic phonons. A phenomenological two-branch spectrum is constructed. The model is suggested by the structure of a theory of boson pairing similar to the BCS theory of fermions. The low-temperature specific heat of He4 is calculated within the model and compared with experimental data. It is found that a two-branch-model excitation spectrum can describe the low-temperature anomaly in the specific heat without contradicting existing direct measurement of the excitation spectrum. Some features of a possible microscopic theory of the two-branch spectrum are discussed, and the possibilities of making direct measurements testing the idea of a two-branch excitation spectrum are considered.