What is the non-local interaction in HQET?

[Einj]

Hi everyone. I have been studying the Heavy Quark Effective Theory and at a certain point we have a Lagrangian like:
$$
\mathcal{L}=\bar h_v iD\cdot v h_v+\bar h_vi\gamma_\mu D^\mu_\perp\frac{1}{iD\cdot v+2m_Q}i\gamma_\nu D_\perp^\nu h_v.
$$
$h_v$ is the field representing the heavy quark, $v$ is the velocity of the heavy quark and $D_\mu$ is the usual covariant derivative.

I read that this Lagrangian is non-local but I can't understand why. Do you have any idea?

 

[Hepth]

Is it because you're choosing what the momentum "v" is? Therefor things are no longer technically lorentz invariant, as the theory only holds in the limit that v is "stationary". Basically you're choosing a specific POV to choose the problem.

 

[Bill_K]

$$
\mathcal{L}=\bar h_v iD\cdot v h_v+\bar h_vi\gamma_\mu D^\mu_\perp\frac{1}{iD\cdot v+2m_Q}i\gamma_\nu D_\perp^\nu h_v.
$$

I read that this Lagrangian is non-local but I can't understand why. Do you have any idea?

It's because of the operator $\frac{1}{iD\cdot v+2m_Q}$, which implies an integration over all x. Or you can expand it in a power series and get derivatives of all orders.

 

[Einj]

Great, thanks!