It is known that the lattice spacing in lattice QCD is not an external parameter and needs to be calculated, also the lattice beta parameter scales the lattice spacing ($a$) and goes as a function of the coupling constant as $$β=\frac{2*N}{g_0^2}$$

Where $N$ is a number of colors.

So the question is, how exactly is the lattice spacing calculated in lattice QCD?

I would also like to know how this lattice spacing is translated into physical units (GeV,fm).


The lattice spacing can be obtained in a procedure usually referred to as scale setting for which one requires a physical quantity computed on the lattice. Take as an example the mass of the proton $am_P$ computed at a given value of the bare gauge coupling $g_0^2$ from the corresponding two-point correlation function. Given knowledge of the experimental value of the proton mass $m_P^{phys}$, the value of the lattice spacing can now be obtained via $$a(g_0^2)=\frac{am_P(g_0^2)}{m_P^{phys}}\,.$$ A result in fm can then be directly derived using $\hbar c=197.3269788(12)$ MeV fm. This scale setting is of course only valid up to lattice artifacts and depends on the quantity used to set the scale.

In practice people usually do not use the proton mass but rather the $\Omega$ or $\Xi$ masses or the decay constants of the pion and/or kaon. Intermediate scales like $r_0$ or $t_0$ which do not have a direct physical counterpart but can be easily and precisely computed on the lattice are also very common.


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