Renormalisation in the curved background

Question

Suppose we have some field theory on a curved background, and the metric tensor $g_{mu nu} (x)$ is a smooth function of the position. For simplicity, let's consider a scalar theory with Lagrangian:
$$
mathcal{L} = -frac{1}{2} g^{mu nu} partial_mu phi  partial_nu phi + V(phi)
$$
In general, the Green function for this operator may look inattractive, and the expressions for loop integrals are unlikely to be treated analytically.
However, renormalisation is a $UV$-effect, and looking at the physical processes at distances, much smaller that the characteristic scale, on which $g_{mu nu} (x)$ changes, it will look approximately constant.
Does it make sense to apply a renormalisation procedure locally, namely:

At each point $x$ - set $g_{mu nu}$ to be a constant
When integrating by parts to get a propagator neglect all terms with derivatives acting on $g_{mu nu}$
Diagonalize the resulting matrix (Green function) in momentum space, which would now have the form $A^{mu nu} (x) k_mu k_nu$ (no summation over $mu, nu$ is assumed)
Apply the Feynman rules in that basic locally

As a result, I expect to have coupling constants to depend on the position $x$ in a certain way.
Or one has to work with the exact Green function to obtain something meaningful?

Dwagg · Answer

Regarding your 4 point procedure: The utility of the momentum-space Feynman rules comes from translation invariance of the action, which is lost in an action with a static metric $g_{munu}(x)$ (not to mention the overall factor $sqrt{-g}$). For instance, we don't have any momentum conserving delta functions. And neglecting all terms with derivatives acting on $g_{munu}$ while computing perturbative corrections to the Green's function seems like an uncontrolled approximation.
However, renormalization is a UV effect and something from the flat-space procedure should survive, as you mentioned. I can't give a complete answer, but I see two possible ways to proceed:

Standard QFT on a curved background (c.f. Carroll for instance). Pick a timelike direction, solve the classical Klein-Gordon equation (for the Gaussian truncated Lagrangian) and obtain a complete set of modes $f_i(x^mu)$ orthonormal under the K-G inner product. The index $i$ can continuous or discrete. Expand the field $phi = sum_i (a_i f_i + a_i^* f_i^*)$ and quantize it as usual. The Green's function is $G(x,y) = sum_i f_i(x) f_i^*(y)$. You can now proceed to do position-space Feynman rules to account for $sqrt{-g} V(phi)$ corrections.
If $g_{munu}approxeta_{munu}$ then you could approximate your Lagrangian as $-frac12 eta^{munu} partial_mu phipartial_nu phi + lambda(x) tilde V(phi,partial phi)$ where $tilde V$ now contains pieces of the kinetic term and $V(phi)$. It seems such theories haven't been studies much (one study). But in principle there is nothing stopping you from proceeding in position-space Feynman rules. If $|lambda(x)|$ is bounded you could even argue that perturbation theory is valid (to whatever extent it typically is). The study I cited works out the 1-loop corrections to the $lambda x^kappa phi^4$ quartic coupling perturbation, where the integrals aren't too difficult and finds an RG fixed point.

Renormalisation in the curved background

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