Effective Field Theory for Ultrasoft Momenta in NRQCD and NRQED

Two particle non-relativistic bound states in QCD (QED) have, at least, three well separated scales: the mass m (hard scale), the typical relative momentum |~ p| (soft scale) and the typical bound state energy E (ultrasoft scale). This allows to introduce a hierarchy of effective field theories when sequentially integrating out each of these scales. After integrating out the hard scale m QCD (QED) becomes a non-relativistic theory, the so called NRQCD (NRQED) [1]. This effective theory is local in space and time and it is still naturally written in terms of quark fields. Integrating out the soft scale leads to a so far elusive effective field theory which we call potential NRQCD (pNRQCD). We claim that this effective theory is local in time but it is non-local in space and it is naturally written in terms of wave function fields. We present some original results for the above mentioned effective field theories. In section 2 we give the matching coefficients for the four-quark operators at one loop for NRQED and also for NRQCD in the case of different quark masses. In section 3 we put forward our proposal for pNRQCD and give some results for pNRQED.