How does string theory damp momentum integrals?

Question

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?

Answer 1

In field theory, graviton scattering amplitudes become UV divergent in position space as the asymptotic states become coincident in position space. Since quantum gravity is not expected to have a serendipitous UV fixed point, the only way to salvage UV finiteness is by "smearing the interaction out". In QFT, and even more so in a theory of gravity, there simply aren't any good ways to do this consistently in a manner that preserves particles and modifies the interaction, since naïve methods to do this obviously violate the fundamental principles of global and local Lorentz symmetry respectively. String theory offers a clear, perturbative solution to this problem.

1. Modular Invariance

Note that the relationship between perturbative integrals for calculating string and ordinary particle amplitudes (or indeed, defining them, in the case of string theory) is formally tenuous (though as I will demonstrate later there is a hidden link that will illustrate the general principles behind the UV regulation). The reason is that string theory differs from ordinary QFT on account of the topological properties of the 2D worldsheet while summing over them in the path integral - this behaviour is simply absent for 1D "spacetime" Feynman diagrams. In other words, string theory introduces interactions as characterised by global, topological properties, while still retaining the same local properties of the worldsheet. From the partition function viewpoint, when one refers to "momentum damping", it is really a novel method of computing S-matrix amplitudes that are UV-finite, rather than affixing an exponentially decaying "regulator" in existing QFT amplitudes. In a sense, the "minimal" string length scale acts as a physical UV cutoff. This is made rigorous by the following fact:

Consider the closed string partition function on the torus (you may find the derivation in any string theory textbook). On account of modular invariance, at no point do we have to integrate over the UV region of the moduli space of inequivalent tori. Equivalently, this restriction to the fundamental region is necessitated by the decoupling of the ghosts. Modular invariance in turn comes from the sum over metrics and topologies in the Polyakov path integral and the Diff$\times$Weyl invariance of the theory.

The string partition function seems opaque, so an interesting way to see the contrast is through the following: consider the partition function for an ordinary, free bosonic particle: $$ Z\sim\int\mathcal D\phi\ \exp\left(- \int\mathrm{d}^Dx\ \partial_\mu\phi\partial^\mu\phi-\frac12m^2\phi^2 \right) \\=\int\mathcal D\phi\ \exp\left(-\frac12 \int\mathrm{d}^Dx\ \phi(-\partial^2+m^2)\phi\right) \\\sim\exp\left(\frac12 \int\mathrm{d}^Dp\ \ln(p^2+m^2)\right) \\\overset{\text{1-loop}}{=}\int\mathrm{d}^Dp\ \int_0^\infty\frac{\mathrm dl}{2l}\ e^{-(p^2+m^2)l} \\=\int_0^\infty\frac{\mathrm dl}{l^{1+D/2}}\ e^{-m^2l} $$ Generalising to the infinite particle spectrum of string theory and implementing level matching, we end up with $$ Z=\int\frac{\mathrm d^2\tau}{\mathrm{Im}(\tau)^2}\ \frac{1}{\mathrm{Im}(\tau)^{12}}\frac{1}{|\eta(\tau)|^{48}} $$ with $\tau$ a modular parameter, and this looks eerily similar to the one-loop string partition function. The only difference is modular invariance, which forces us to restrict the string partition function to the "fundamental domain" contrary to the QFT version that runs into the UV region where $\mathrm{Im}(\tau)$ becomes small.

String Theory	Ordinary QFT
$\mathrm{Re}(\tau)\in\left[-\frac12,\frac12\right]$	$\mathrm{Re}(\tau)\in\left[-\frac12,\frac12\right]$
$\mathrm{Im}(\tau)~:~|\tau|\ge1$	$\mathrm{Im}(\tau)\in[0,\infty)$

^Source

The blue region is the fundamental region on $\mathbb H$, and the red region indicates the deep UV. The reluctance of string theory to set foot in the latter is a manifestation of the physical cutoff (or smearing) effected by the string length scale, as alluded to previously.

For more general worldsheets (and for open string scattering), there emerge other divergences, but all of these are IR (long-distance) effects, and have analogues in QFT where we know how to tame or reinterpret them. Indeed, the lightest states of the string end up regulating the behaviour in the UV - in modern parlance, this is called the "UV-IR correspondence".

Note that string theory has not been proven rigorously to be UV finite to all orders - the method above only works at one-loop and two-loop in bosonic string theory once vertex operator additions are factored in. It is however widely expected that the general principle of modular invariance on the worldsheet illustrated above will regulate UV divergences and leave string theory both finite and unitary, as advertised.

2. Amplitude Structure

The topological properties of the string worldsheet also conspire to deliver an equivalent viewpoint on the UV damping of string theory, via the structure of amplitudes. Historically, it was the other way around, with Regge theory and the S-matrix bootstrap program being interpreted later on primarily by Schwarz and Scherk as the foundations of the string theory that we know today.

One essential tenet of the bootstrap program is the correspondence between poles in scattering amplitudes and the particle spectrum. In field theory, as in string theory, scattering amplitudes (say $2\to 2$ for simplicity) depend on the coupling constant and the kinematical Mandelstam variables $s, t, u$. In field theory, which contains several Feynman diagrams at each loop order, each contributing to only one of the s-, t- or u-channels. But clearly there is only one string worldsheet diagram at each topological order, which means that the corresponding amplitude will have poles in all channels. Furthermore, crossing symmetry, part of the dual resonance model in pre-string theory times, and manifest in the modern calculation of amplitudes, dictates that the string amplitude corresponds to an infinite sum over field theory amplitudes in exactly one of the channels. These features, combined with unitarity, analyticity and Lorentz invariance essentially allow you to determine (the leading contributions to) the amplitude structure directly, to "bootstrap" it, as it were. This allowed, for instance, Veneziano to determine the open-string tachyon scattering amplitude without the Polyakov action. Also see: What are bootstraps?.

Similarly, we can bootstrap the tree-level contribution to graviton-graviton scattering, dimensionally reduce it to 4 dimensions, and compare it to the amplitude derived from the non-renormalisable field theory:

$$ \mathcal A^{(h)}_{\text{string}}\sim\frac{\Gamma{(1-\frac{\alpha's}4)}\Gamma{(1-\frac{\alpha't}4)}\Gamma{(1-\frac{\alpha'u}4)}}{\Gamma{(1+\frac{\alpha's}4)}\Gamma{(1+\frac{\alpha't}4)}\Gamma{(1+\frac{\alpha'u}4)}}\ \mathcal A^{(h)}_{\text{field}} $$

with poles from the numerator corresponding to the massive tower of states in string theory. Thus there is "gamma function damping" with respect to the field theory amplitude (this prefactor also becomes unity in the $\alpha'\to0$ limit). Now take a look at the high-energy limit, with $s\to\infty$ and $s/t$ constant: $$ \mathcal A^{(h)}_{\text{string}}\to\ \ \sim\exp\left(-\alpha'(s\ln \alpha's+t\ln\alpha't+u\ln\alpha'u)\right) $$

due to the properties of the gamma function. Thus at high energies, the amplitude does indeed have exponential falloff, far better than the power-law behaviour of field theory. Here it is clear that the infinite particle spectrum of string theory is another factor that contributes to this - the infinite sum over the individually divergent exchanges of string resonances actually converges to an exponential decay to bring about this damped UV behaviour. Again, this is not a universal proof for UV finiteness, but it is once again expected that this general principle holds for all string scattering amplitudes.

Answer 2

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.

Answer 3

Another answer is based on the explicit representation of loop momentum amplitudes in string theory. It's for instance described in D'Hoker and Phong 1988 (the review) and is at the heart of the "chiral splitting" formalism. Long story short, when you're given a $g$-loop string integrand, you can undo a Gaussian integral so as to make appear explicitly the loop momentum. This will give you expression that possess a term like: $$\exp(-\sum_{I,J=1}^g \ell_I \Im \Omega_{IJ}\ell_j)$$ where $\ell_I$ are the $g$ loop momenta and $\Omega_{IJ}$ is the so-called period matrix, of size $g\times g$, which generalises the complex structure $\tau$ at one-loop (which is a one-by-one matrix in this case).

And now you may use modular invariance to see that $\tau$ is always non-zero and larger than some finite constant ($\Im \tau>\sqrt{3}/2$) (same sort of relations on $\Omega$), so that large $\ell$ produce an exponential suppression of the loop integrand.