Non-abelian magnetic field interaction

Question

Consider the magnetic part of a non-Abelian field. I want to know if we can define Hamiltonian for the interaction of this sector with the spin (something such as Hamiltonian of interaction of magnetic moment and electromagnetic field.)

Answer 1

As Lorentz invariance is inherently followed in a quantum field theory, we ususally encounter quantities that are 4-vectors. But in some cases like NRQCD, it is more convenient to resolve the Yang-Mills field strength tensor into terms like what we have in classical EM field theory.

When we integrated out the hard scale of mass, $m$ from the full QCD lagrangian, we get the effective field theory of NRQCD. The heavy quark part of the NRQCD lagrangian is given as:

\begin{equation} L_\psi = \psi^\dagger \Biggl\{ iD^0 + c_k^{(1)} \frac{{\textbf D}^2}{2m_1} + c_4^{(1)} \frac{{\textbf D}^4}{8m_1^3} + c_F^{(1)} g \frac{\sigma \cdot {\textbf B}}{2m_1} \end{equation} \begin{equation} + c_D^{(1)} g\frac{{\textbf D} \cdot {\textbf E}- {\textbf E} \cdot {\textbf D}}{8m_1^2} + ic_S^{(1)}g\frac{\sigma \cdot({\textbf D} \times {\textbf E}- {\textbf E} \times {\textbf D})}{8m_1^2} \Biggl\} \psi \end{equation} \begin{equation} + (\mathrm{terms \; for \; heavy \;light \; quark \; local \; interaction}) \end{equation}

where,
$\quad$ $\sigma$ are the Pauli matrices,
$\quad$ $\textbf{D}=\nabla-ig\mathbf{A}$,
$\quad$ $\textbf{E}^i=F^{i0}$ is the chromoelectric field
$\quad$ $\textbf{B}^i=-\varepsilon_{ijk} F^{jk}/2$ is the chromomagnetic field.
$\quad$ $F^{jk}$ is the spacial part of the full QCD field strength tensor, $F^{\mu\nu}$.
$\quad$ $c$ terms are the Wilson coefficents of the EFT.

The bracket term is the free field part plus the coupling to soft gluons. In this EFT heavy quark-antiquark pairs cannot be created anymore, so it is convenient to use non-relativistic Pauli spinors ($\psi$) instead of Dirac spinors for the heavy quark spin states.

The term with $\sigma \cdot {\textbf B}$ is remenicent of quark spin interaction with chromomagnetic field. So, it is possible to invoke chromoelectric and chromomagnetic field description for non-abelian gauge theory, but it is suitable only for some non-relativistic systems.

Reference:
page 26, Effective Field Theories of QCD for Heavy Quarkonia at Finite Temperature, Jacopo Ghiglieri,arXiv:1201.2920 [hep-ph]