Researchers use the S-matrix bootstrap to examine whether string theory is the only consistent theory of quantum gravity

String theory encompasses all of the quantum gravity theory space that is allowed. Credit: Guerrieri. Penedones & Vieira.
The S-matrix boottrap is a numerical method which can be used to constrain or determine the scattering amplitudes for particles in quantum field theory. It uses simple principles. This technique has been used by some physicists to study various physics theories and phenomena over the past several decades.

Researchers from Tel Aviv University and cole Polytechnique Fdrale de Lausanne have further developed the S-matrix boottrap and are trying to use it in different areas of Physics. The team published a paper in Physical Review Letters that attempted to use the S-matrix bootstrap to study string theory, a well-known physics theory which describes the fundamental elements of the universe as one-dimensional (1D),'strings' instead of point-like particles.

Andrea Guerrieri, one the researchers behind the study, explained to that the S-matrix boottrap was an old idea. It became less popular after the emergence and use of quantum chromodynamics for the description of the strong nuclear force. The S-matrix boottrap's goal is to constrain (or determine) the scattering intensities of particles in quantum theory using only basic principles such as causality and the fact the probabilities can not exceed 1. We brought a new topic to the table by proposing a systematic numerical algorithm that would implement all of these principles.

Although there are a few dozen papers that have been published on physical systems, they focus mainly on large particles. Guerrieri and colleagues were able to use the bootstrap to study scattering by massless particles such as transverse fluctuations in a color flux tube, in quantum chromodynamics, or in masslesspions.

Joo Pendones, another researcher who was involved in the study, said that once we discovered that the method worked for massless particles, we knew that we should test it for gravitons. These are the mediators for the gravitational force. To reduce technical difficulties, we decided to impose supersymmetry within 10 spacetime dimensions.

Guerrieri and Pedro Vieira's recent work can be applied to any theory or method of quantum gravity. However, their study focused on the S-matrix boottrap to examine string theory, and to test whether it is the only consistent theory of quantum gravity.

Penedones stated that Einstein's Theory of General Relativity (GR), describes gravitational interactions at long distances and (equivalently), low energy. It can be used, for example, to calculate how gravitons with large wavelengths (much greater than the Planck lengths) scatter from one another. We cannot trust the prediction by GR if the wavelength is decreased at any point. In reality, GR's prediction will have probabilities higher than 1.

The theory must be corrected at short wavelengths/wavelengths or at high energy to improve the reliability and predictability of GR's predictions. This correction is commonly referred to as an "ultraviolet(UV) completion".

Diagram of graviton scattering. Credit: Guerrieri. Penedones & Vieira.

Guerrieri stated that a theory about quantum gravity is often a UV completion for GR. "We investigated the first correction of the GR predictions at the low energy limit in our latest paper. This is what parameter alpha stands for in our paper.

String theory is the most well-known theory of quantum gravity. The researchers have shown that the alpha parameter in string theory can be given a particular set of values.

Vieira explained that alpha in string theory is related to tension of the fundamental strings (in Planck units). "We set out to investigate in this paper what theory of quantum gravity describes the real world. We wanted to be agnostic about it and just ask what alpha values such a theory could produce.

Surprisingly, researchers discovered that the alpha values that a theory describing gravity could have according to S-matrix bootstrap numerical calculations were exactly the same as those produced by string theory. These findings may have some interesting implications. However, there were some limitations to their study that could affect its validity.

Vieira stated that "The first caveat to our work is that it is numerical. It used computer clusters in order to scan over large sets of scattering ampltudes and extrapolate to all consistent scattering ampltudes." This extrapolation procedure is not perfect. We find that alpha is 0.13 0.02 and an estimated error range. String theory would allow any alpha > 0.1389, which is (within error bars) perfectly on our bound.

The researchers plan to decrease the numerical error caused by the S-matrix boottrap used to determine if the results are unchanged in future studies. Their study also had a second problem. It was limited in that they used a simplified set-up, since it was only in 10 dimensions and with supersymmetry. However, the real world is only four-dimensional, and supersymmetry has yet to be experimentally observed.

Guerrieri stated that "Nevertheless, we believe that 10d with Supersymmetry is a good starting place for such explorations." "This is because 10d superstring theory has the simplest form and we have sharp predictions against which to compare our numerics. A second reason is that supersymmetry allows us to relate gravitons, which are difficult objects to mathematically tame due to their spinning nature, to simpler supersymmetric counterparts that do not spin.

In their latest study, the researchers focused on the main research question: Is spring theory the only consistent theory for quantum gravity? Although they couldn't answer the question with absolute certainty, their findings showed that the S-matrix boottrap might be able to help. Guerrieri and Penedones would like to continue their calculations in the future without supersymmetry and with lower spacetime dimensions.

Vieira stated that so far, the values of Alpha allowed by general principles are consistent with string theory's. This is compatible with the positive answer to the above question, but it is not strong evidence. We can however study other parameters (let's call them beta and gamma), which are sub-leading corrections for the predictions of GR low energies. String Theory predicts that there are an infinite number of them. The evidence will be strong if we can prove that the permitted range of parameters in the S-matrix Bootstrap principles coincides with String Theory.

Continue reading: Explain gravity without string theory

Additional information: Andrea Guerrieri and colleagues, Where Is String Theory within the Space of Scattering Amplitudes (Physical Review Letters 2021). Andrea Guerrieri and colleagues, Where is String Theory in Space of Scattering Amplitudes (2021). DOI: 10.1103/PhysRevLett.127.081601 Miguel F. Paulos et al, The S-matrix bootstrap. Part I: QFT In AdS, Journal of High Energy Physics 2017 DOI: 10.1007/JHEP11(2017)133 Joan Elias Mir et al, Flux Tube S -Matrix Bootstrap, Physical Review Letters (2019). DOI: 10.1103/PhysRevLett.123.221602 Andrea L. Guerrieri et al, S-matrix bootstrap for effective field theories: massless pions, Journal of High Energy Physics (2021). DOI: 10.1007/JHEP06(2021)088 Journal information: Physical Review Letters

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