In several publications (e.g. http://arxiv.org/abs/1506.06864) a "colour Coulomb interaction" between quarks was mentioned. What kind of interaction is that, is it electromagnetic or strong?


The colour Coulomb interaction is another name for the one-gluon exchange potential between (typically heavy) quarks or other sources having colour. It is a straightforward generalisation of the Abelian (i.e. standard) Coulomb potential stemming from one-photon exchange,

$$ V(r) = -\alpha/r \; ,$$

where $r$ is the distance between the sources taken to be elementary charges $\pm e$ for simplicity. In this case the potential strength is just the fine structure constant, $\alpha = e^2/4\pi = 1/137$ (in natural units, $\hbar = 1 = c$). Note that $1/r$ is basically the Fourier transform of the photon propagator

$$ \left. \frac{1}{q^2} \right|_{q^0 = 0} = - \frac{1}{\mathbf{q}^2} \; , $$

at zero energy transfer, $q^0 = 0$, corresponding to the static approximation.

To obtain the non-Abelian generalisation one just replaces

$$ \alpha \to C_2(R) \, \alpha_s \; , $$

where $\alpha_s$ is the strong interaction coupling and $C_2(R) = T^a(R) T^a(R)$ the quadratic Casimir in the representation $R$ of colour SU(3). For quarks, $R = F = [3]$, the fundamental representation and $C_2(F) = 4/3$. For octet sources such as gluons, $R = A = [8]$, the adjoint representation, and $C_2(A) = 3$.

NB: For SU(N), the Casimir operators are $C_2([N]) = (N^2-1)/2N$ and $C_2([N^2-1]) = N$, see e.g. Peskin and Schroeder, Ch. 15.4.

The fact that the non-Abelian potential is proportional to $C_2(R)$ is widely believed to hold beyond perturbation theory, i.e. also for the string tensions in different representations. This conjecture goes under the name of Casimir scaling, see e.g. Jeff Greensite's review on confinement.


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