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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.

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 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 at one loop. 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. The lightest states of the string end up regulating the behaviour in the UV - in modern parlance, this is called the "UV-IR correspondence".

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 (whose derivation may be found in any string theory book worth its weight). The only difference is modular invariance, which in turn comes from the sum over topologies in the Polyakov path integral and Diff$\times$Weyl invariance - this forces us to restrict to the "fundamental domain" contrary to the QFT version that runs into the UV region where $\mathrm{Im}(\tau)$ becomes small.

 ^Source

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.

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, and 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?.

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 the infinite particle spectrum of string theory is the factor that contributes to this - the infinite sum over the individually divergent exchanges of string resonances actually brings 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.

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.

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.

^Source

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".

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?.

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.

The blue region is the fundamental region, 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.

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.

Clearly 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 tenets of global and local Lorentz symmetry respectively. String theory offers a clear, perturbative solution to this problem.

For more general worldsheets (and for open string scattering), there emerge other divergences, but all of these are IR or long-distance effects, and have analogues in QFT where we know how to tame or reinterpret them.

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, and I will take a very modern outlook.

One essential tenet of the bootstrap program is the correspondence between poles in scattering amplitudes and the particle spectrum, and 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.

Take the tree-level contribution to graviton-graviton scattering, with the string theory result dimensionally reduced to 4 dimensions, as compared to the amplitude derived from the non-renormalisable field theory:

with poles corresponding to the massive tower of states in string theory. Now take a look at the high-energy limit, with $s\to\infty$ and $s/t$ constant (I don't remember the exact factors, but this is roughly correct): $$ \mathcal A^{(h)}_{\text{string}}\to\ \ \sim\exp\left(-\alpha'(s\ln s+t\ln t+u\ln u)\right) $$

due to the properties of the gamma function. Thus at high energies, the amplitude is indeed exponentially damped, a world away from the power-law behaviour of field theory. Here the infinite particle spectrum is the factor that contributes to this - the infinite sum over the individually divergent string resonance exchanges actually brings about this damped UV behaviour.

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.

^Source

The blue region is the fundamental region, 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.

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, and 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:

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 the infinite particle spectrum of string theory is the factor that contributes to this - the infinite sum over the individually divergent exchanges of string resonances actually brings 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.