Extended Hqeft Lagrangian and Currents *

From the tree-level heavy quark effective Lagrangian keeping particle-antiparticle mixed sectors we derive the vector current coupling to a hard gluonic field allowing for heavy quark-antiquark pair annihilation and creation. 1. A COMPLETE TREE-LEVEL HQEFT LAGRANGIAN In this work we keep both heavy quark and heavy antiquark coupled sectors in the HQEFT Lagrangian. To our knowledge, until [1] only the paper by Wu [2] in the literature actually dealt with both heavy quark and antiquark fields altogether in the effective Lagrangian. In the present work, based on [3,1], we pursue this line of investigation further, looking for more symmetric expressions and a more transparent physical interpretation. Following the standard reference [4] we introduce the effective fields for a heavy quark bound inside a hadron moving with (four-)velocity v, as h (+) v (x) = e imv·x 1 + v / 2 Q (+) (x) (1) H (+) v (x) = e imv·x 1 − v / 2 Q (+) (x) (2) where Q (+) (x) stands for the (positive energy) fermionic field describing the heavy quark in the full theory; h (+) v (x) and H (+) v (x) represent the " large " and " small " components of a classical spinor field respectively. Similarly for a heavy antiquark, h (−) v (x) = e −imv·x 1 − v / 2 Q (−) (x) (3) H (−) v (x) = e −imv·x 1 + v / 2 Q (−) (x) (4) where Q (−) (x) stands for the (negative energy) anti-fermionic field in the full theory. Introducing the residual momentum q for a heavy quark of total momentum p as p = mv + q, the on-shell condition can be written as v·q = − q 2 2m (5) The underlying idea when introducing the residual momentum is that once removed the large mechanical momentum associated to the heavy-quark mass, only the low-energy modes remain in the effective theory [5]. This is the standard way to handle heavy quarks in singly heavy hadrons according to HQET. However, one may conjecture about the possibility of performing such an energy-momentum shift by introducing a center-of-mass residual momentum for processes involving creation or annihilation of heavy quark-antiquark pairs at tree-level. Hence only low energy modes of the fields (about the heavy quark mass) would be involved likely making meaningful our approach within the framework of an effective theory. In sum, the main …