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Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe
by Paolo Benincasa, William J. Torres Bobadilla
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Submission summary
Authors (as registered SciPost users):  William J. Torres Bobadilla 
Submission information  

Preprint Link:  scipost_202201_00007v1 (pdf) 
Date accepted:  20220517 
Date submitted:  20220111 16:55 
Submitted by:  Torres Bobadilla, William J. 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The way we organise perturbation theory is of fundamental importance both for computing the observables of relevance and for extracting fundamental physics out of them. If on one hand the different ways in which the perturbative observables can be written make manifest different features ({\it e.g.} symmetries as well as principles such as unitarity, causality and locality), on the other hand precisely demanding that some concrete features are manifest lead to different ways of organising perturbation theory. In the context of flatspace scattering amplitudes, a number of them are already known and exploited, while much less is known for cosmological observables. In the present work, we show how to systematically write down both the wavefunction of the universe and the flatspace scattering amplitudes, in such a way that they manifestly show physical poles only. We make use of the invariant definition of such observables in terms of {\it cosmological polytopes} and their {\it scattering facet}. In particular, we show that such representations correspond to triangulations of such objects through hyperplanes identified by the intersection of their facets outside of them. All possible triangulations of this type generate the different representations. This allows us to provide a general proof for the conjectured allloop causal representation of scattering amplitudes. Importantly, all such representations can be viewed as making explicit a subset of compatible singularities, and our construction provides a way to extend Steinmann relations to higher codimension singularities for both the flatspace scattering amplitudes and the cosmological wavefunction.
Published as SciPost Phys. 12, 192 (2022)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202248 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202201_00007v1, delivered 20220408, doi: 10.21468/SciPost.Report.4893
Strengths
1. original
2. it can give rise to new developments
Report
In the submitted paper the authors presented a systematic way of generating different representations for both the wavefunction of the universe in cosmology and flatspace scattering amplitudes, by making use of their invariant definition in terms of cosmological polytopes. The different representations arise from the triangulations of the cosmological polytopes and its scattering facet. Their characteristic feature is to have physical poles only. The related combinatorial description allows to consider all these representations on the same footing, recovering both the oldfashioned perturbation theory and the causal representation as well as find new ones. Moreover the authors derive new Steinmann like relations for the dS wavefunction and give a proof for the allloop causal representation of scattering amplitudes. The paper is well presented and I recommend it for publication, in its present form.
Report #1 by Anonymous (Referee 1) on 202231 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202201_00007v1, delivered 20220301, doi: 10.21468/SciPost.Report.4598
Report
This work extends and gives new applications for the cosmological polytope program. The authors derive several interesting results on the analytic structure of the cosmological wavefunction and in addition use the flatspace limit to translate these results into new statements about scattering amplitudes in Minkowski space. In particular, they derive new Steinmann like relations for the dS wavefunction and give a proof for the allloop causal representation of scattering amplitudes.
There are a few minor presentation issues the authors that may be worth considering:
1) Below equation 9 and above the graphical equation there is a sentence " ... which identifies those vertices $\mathcal{Z}_i$ of $ \mathcal{P}_{g}$ such that $\mathcal{Z}_i\cdot\mathcal{W}^{(g)}$, i.e. that are not on the facet".
I believe the authors are missing the condition $\mathcal{Z}_i\cdot\mathcal{W}^{(g)}>0$ in that sentence.
2) The authors introduce the small circle marking above eqn 12 but define it later in figure 4. It may be useful to define this notation earlier.
3) It would also be useful for referencing to number the graphical equations, i.e. the graphical equation above eqn 12.
Modulo these minor issues, the paper is wellwritten. Although the analysis can be very technical, the authors do give a succinct review of the polytope technology. Once the above presentation issues are taken into account, I am happy to recommend this paper for publication.
Requested changes
See (1)(3) listed above.