# How does string theory damp momentum integrals?

Quantum field theory frequently contains $k$-space integrals of rational functions. In this video, Lance Dixon notes that, because string theory doesn't have point particles, a smearing effect damps these integrands, thereby often addressing renormalisation issues such as those for quantum gravity (which has divergent 2-loop diagrams because it multiplies integrands by $k^2$ relative to the case of electromagnetism). What form does this damping take (e.g. is it of the form $e^{-k^2/M^2}$)? Could someone provide or point to a derivation?

An outline of how this mechanism occurs is outlined in Witten's talk, "What Every Physicist Should Know About String Theory". I'll briefly outline it here as well. More or less, the additional symmetry constraints of string theory (i.e. conformal symmetry) prevent the badly behaved UV region from contributing to the integrals in string theory. I'll briefly outline how it works here. In standard relativistic QFT of a spin-$0$ particle (Euclidean signature for simplicity), the amplitude associated to a Feynman diagram of topology $\cal{M}$ is given as $$Z(\cal{M})=\int{exp\bigg(-\int_\cal{M}\bigg[\frac{1}{2e(\tau)}\frac{dx_\mu}{d\tau}\frac{dx^\mu}{d\tau}+\frac{e(\tau)}{2}m^2\bigg]d\tau\bigg)\cal{D}e(\tau)}\cal{D}x^\mu(\tau)$$ where $e(\tau)$ is the 1 dimensional einbein, representing an integral over all metrics. This is the "worldline" formalism, writing a QFT Feynman amplitude as a point particle path integral. Its derivation is rather straightforward, but it is easy to verify that it reproduces the correct amplitudes in simple cases. If we consider the simplest possible case of a single loop, $\cal{M}$ $=S^1$, after gauge fixing the reparameterization invariance of the relativistic single particle action, we get something like $$Z(S^1)=\int_{0}^{\infty}\frac{dT}{T}\int{exp\bigg(-\int_{0}^{T}\bigg[\frac{1}{4}\frac{dx_\mu}{d\tau}\frac{dx^\mu}{d\tau}+m^2\bigg]d\tau\bigg)}\cal{D}x^\mu(\tau)$$ The ordinary path integral is just the standard partition function taken over periodic paths, and is manifestly convergent. The bad UV behavior comes from the short distance ($T\rightarrow0$) behavior of the integral over proper times (the remainder of the integral over metrics after gauge fixing. Here, the proper time is an example of what are more generally in string theory known as moduli). In string theory, the situation is similar, with the 1-dimensional topological manifold that is the Feynman diagram replaced by a 2-dimensional topological manifold, the worldsheet. Once again, you do an integral over all metrics, which after gauge fixing goes to an integral over the moduli. The difference is that in string theory, the action is also conformally invariant, even after gauge fixing, and this restricts the region of integration over the moduli to have some positive nonzero lower bound, rendering the amplitude UV finite.