In page 9 of the introductory chaper of *Renormalization and Effective Field Theory* (the introductory chapter is available free [here](http://bookstore.ams.org/surv-170)), Kevin Costello defines a propagator $P$ for the Laplace operator $D$ in a Riemannian spacetime $M$. 


In order to define a "length-scale version of the renormalization group flow", he provides a formula for this propagator in terms of the heat kernel $K\_\tau$ of the Laplace operator: 
$$P(x,y):=\int\_0^\infty d\tau\; K\_\tau(x,y).\tag{1}$$
Using the interpretation of the heat kernel as a transition probability, one may then rigorously write (expanding upon equation 1)
$$P(x,y)=\int\_0^\infty d\tau \int\_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp\left(-\int\_0^\tau \|d\gamma\|^2\right).\tag{2}$$
My question regards the interpretation of this "path integral" above involving $\gamma$, as something is confusing to me. Since Costello defines $\tau$ to be the proper time of the worldline $\gamma$ as measured by the Riemannian metric, it occurred to me that the integrand in equation (2) vanishes for $\tau < \text{dist}(x,y)$, as there will not exist paths with lengths smaller than that of the geodesic.


Therefore, we can write, more subtly,
$$P(x,y)=\int\_{\text{dist}(x,y)}^\infty d\tau \int\_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp\left(-\int\_0^\tau \|d\gamma\|^2\right).\tag{2}$$
Look how, naturally, a length-scale cutoff emerges. Clearly, I have made a serious conceptual error. Which is it?



Spurious length-scale cutoff emerges in propagator defined in Costello's "Renormalization and EFT"
There is no space cutoff because it is $d(x,y)$ which depends on the two points. If you put a constant $c$ in your integral $\int\_c^\infty d\tau$ then yes you would have introduced a spurrious cutoff in position space.
If you are in $\mathbb{R}^d$ you can write the free massless propagator as
$$
(-\Delta)^{-1}(x,y)=\int\_0^{\infty}\frac{dl}{l} l^{-(d-2)}
u\left(\frac{x-y}{l}\right)
$$
for some nonnegative, positive semidefinite smooth function $u$ with **compact support** say in the unit Euclidean ball around the origin.
See for example my article ["A complete renormalization group trajectory between two fixed points"](https://link.springer.com/article/10.1007%2Fs00220-007-0352-x). Comm. Math. Phys. 276 (2007), no. 3, 727-772.
This can be very useful for a rigorous renormalization group analysis (see [this MO answer](https://mathoverflow.net/questions/62770/what-mathematical-treatment-is-there-on-the-renormalization-group-flow-in-a-spac/63089#63089)).


Clearly, there is no cutoff above because the integral over length scales $l$ is from $0$ to $\infty$. However, because of the support property of the infinitesimal multiscale slice $u((x-y)/l)$, the previous expression is equal to
$$
\int\_{|x-y|}^{\infty}\frac{dl}{l} l^{-(d-2)}
u\left(\frac{x-y}{l}\right)\ .
$$
There is a theory for this type of compactly supported multiscale decompositions developed by Brydges, Guadagni, Mitter, Bauerschmidt, Talarczyk, and others. See the JSP article by Mitter ["On a finite range decomposition of the resolvent of a fractional power of the Laplacian"](https://link.springer.com/article/10.1007%2Fs10955-016-1507-y) (and its [erratum](https://link.springer.com/article/10.1007/s10955-016-1687-5)) for a recent account.


