Consider a 2d scalar field theory with quartic interaction
$$S[\phi]=\int d^2x \left((\nabla\phi)^2+m^2[\phi^2+g\phi^4]\right)$$
I want to compute the partition function
$$ Z[m,g]=\int\mathcal{D}\phi\,e^{-S[\phi]}$$
say as a function of $m,g$. I want to do this in $m^2\to\infty$ limit keeping $g$ finite. Note this is the regime where the quartic coupling and mass both go to infinity while their ratio is fixed.
Short statement of the question.
When $m^2$ is large the saddle point method seems appropriate. I expect that the leading term is given by the the one-loop partition function $\log\det (-\Delta+m^2)$ of the quadratic action and that the $\phi^4$ term will give further $1/m^2$ corrections. However, the naive perturbation theory leads to all loop diagrams being of the same order in $m^2$. Is there a way to organize the perturbative expansion such that it gives meaningful $1/m^2$ corrections? In principle I am interested in a theory defined on a curved space and the answer should in an expansion in terms of $R/m^2$ where $R$ is the scalar curvature.
My attempt to do a naive perturbative expansion and why it fails.
Since there is a large parameter in the action, I try to use the saddle point expansion. The saddle point configuration is just $\phi=0$ so the action already is written for the fluctuations about the saddle point. Next, one expects quadratic term to dominate while the quartic term to produce corrections in the form of $1/m^2$ expansion. However if I try to do naive perturbation theory this turns out to be false.
Consider a simplest diagram without self-contractions which turns out to be three-loop and write it in coordinate space
$$\left<\left(gm^2\int d^2x\phi^4\right)^2\right>\simeq g^2m^4 \int d^2x'\int d^2x G^4(x-x')\simeq g^2 m^4 V \int d^2x\,\, G^4(r)$$
Here $V$ is formally the volume of space $V=\int d^2x $. If flat space it is infinite so we could put the theory in a finite box or on a closed surface but I think these details are irrelevant.
Now, naively the propagator of a heavy field should behave as $G\propto m^{-2}$ so that $G^4\propto m^{-8}$ and the whole diagram is proportional to $m^{-4}$. However, the actual propagator for the massive field in two dimensions is up to a constant
$$G(r)=K_0(mr),\qquad\qquad(-\Delta+m^2)G(r)=\delta^{(2)}(r)$$
Here $K_0(r)$ is the zeroth order modified Bessel function. It has a logarithmic singularity at $r=0$ and decays exponentially at $r\to\infty$. So actually $\int d^2x G^4(r)=\int d^2x K^4_0(mr)\propto m^{-2}$ and not $m^{-8}$. As a result the whole diagram is proportional to $m^2$. It is easy to see by similar arguments that suppressing factors $m^{-2}$ are not associated with propagators but rather with vertices in the diagrams. However as each vertex carries a factor $m^2$ coming from the action all diagrams in the perturbation theory have the same order $m^2$. It is not hard to trace these problems to the fact that the kinetic term in the action is not multiplied by $m^2$ so the power of $m^2$ is not the real loop-counting parameter.
I the quartic couplig $g$ can be treated as small, then this naive perturbative expansion is sensible as expansion in powers of $g$. However if I insist on keeping $g$ of order one is there a way to reorgonanize the expansion to get $m^{-2}$ corrections described by a finite amount of diagrams?
I should perhaps note that I have very little experience with perturbative expansions of this kind. The solution may be simple, for example to use an improved propagator, or much more complex. Pointers to the literature are also very welcome.
Q&A (4831)
