# Advanced : HQET Derivative manipulation

1. ### Hepth

529
This is a semi-advanced question for anyone with HQET experience.

When you expand out the Lagrangian, as well as the wave functions, you get things like the equation of motion that are expressed in terms of the covariant derivative and the heavy quark velocity :
$$\def\lts#1{\kern+0.1em /\kern-0.65em #1} i (v\cdot D) h_v = \left(\frac{1}{2 m_Q} \lts{D}^2_{\perp} - \frac{i}{4 m_Q^2} \lts{D}_{\perp} (v \cdot D) \lts{D}_{\perp} + ...\right)h_v$$

So when constructing currents you have some really long products of these D operators. Is there a standard strategy to simplification? Such as moving all (v.D)'s to the left or right. I can come up with the commutation relations so that I can write a simple mathematica script that can rearrange them however I see fit, also getting me the sigma.G term.

I'm just wondering if anyone out there has done anything where they could say "oh yeah, the fastest way is to just reqwrite it all in terms of etc etc then do etc.)

As it stands now depending on the order I expand to its quite a long expression.

(Also any direction to books or articles may be useful, though in articles it always seem they avoid including any code or tricks, and the only real books are like HQP Manohar/Wise.

2. ### Einj

448
In my opinion there no easy way to simplify currents. The HQET is a fairly complicated theory, but I can be wrong. Have you ever tried Neubert review? I don't know if it contains what you are looking for but it's a fairly good reference. Sometimes it can be complementary to Wise's book. The full title is: Matthias Neubert - Heavy-quark symmetry