# $R_\xi$ gauges and the EM-field

$$R_\xi$$-gauges are said to be a generalization of the Lorenz gauge. I dont quite get why we add the term $$\mathcal L_{GF} = - \frac{(\partial_\mu A ^\mu)^2}{2\xi}\tag{1}$$ to the Lagrangian. If i calculate the EOM for the electromagnetic field $$A^\mu$$ adding the "gauge-breaking"-term (1) so for the Lagrangian $$\mathcal L = - \frac 1 4 F_{\mu\nu}F^{\mu\nu} + j_\mu A^\mu - \frac{(\partial_\mu A^\mu)^2}{2\xi}\tag{2}$$ i get EOM $$\partial_{\nu}(\partial^\mu A^\nu + \left(\frac 1 \xi - 1\right)\partial^\nu A^\mu) = j^\mu\tag{3}$$ which for $$\xi = 1$$ just gives the Lorenz gauge. Why is the $$\xi = 1$$ gauge now called Feynman-'t Hooft gauge? And what exactly is the use of this extra term since there is no "real" use for it?

In the Lagrangian of the path integral, a popular class of gauge-fixing terms is of the form$$^1$$ $${\cal L}_{GF}~=~-\frac{\chi^2}{2\xi}.$$ The word "gauge" is here confusingly used in 2 ways:

1. The gauge-fixing function $$\chi$$, e.g. Lorenz gauge $$\chi=\partial_{\mu}A^{\mu}$$, Coulomb gauge $$\chi=\vec{\nabla}\cdot \vec{A}$$, etc.

2. The gauge parameter $$\xi>0$$, e.g. Feynman – 't Hooft gauge $$\xi=1$$, Landau gauge $$\xi=0^+$$, etc.

Both gauge-fixing choices $$(\chi,\xi)$$ should be made. So e.g. OP is properly speaking considering Lorenz gauge $$\chi=\partial_{\mu}A^{\mu}$$ in Feynman – 't Hooft gauge $$\xi=1$$.

Different gauges are useful in answering different questions. It should be stressed that the gauge-fixing condition is in generical only imposed in a quantum average sense in the path integral, cf. e.g. my Phys.SE answers here & here.

Why the path integral needs gauge-fixing is e.g. discussed in this, this, this & this Phys.SE posts.

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$$^1$$ Some gauges may require non-trivial Faddeev-Popov (FP) terms. For a BRST formulation, see e.g. this Phys.SE post. More general types of gauge-fixing is possible in the Batalin-Vilkovisky (BV) formulation.