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RG flows in de Sitter: cfunctions and sum rules
by Manuel Loparco
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Submission summary
Authors (as registered SciPost users):  Manuel Loparco 
Submission information  

Preprint Link:  https://arxiv.org/abs/2404.03739v2 (pdf) 
Date accepted:  20240903 
Date submitted:  20240731 12:11 
Submitted by:  Loparco, Manuel 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the renormalization group flow of unitary Quantum Field Theories on twodimensional de Sitter (dS) spacetime. We prove the existence of two functions of the radius of dS that interpolate between the central charges of the UV and IR fixed points of the flow when tuning the radius $R$ while keeping the mass scales fixed. The first is constructed from certain components of the twopoint function of the stress tensor evaluated at antipodal separation. The second is the spectral weight of the stress tensor in the $\Delta=2$ discrete series. This last fact implies that the stress tensor of any unitary QFT in dS$_2$ must interpolate between the vacuum and states in the $\Delta=2$ discrete series irrep. We verify that the cfunctions are monotonic for intermediate radii in the free massive boson and free massive fermion theories, but we lack a general proof of said monotonicity. We derive a variety of sum rules that relate the central charges and the cfunctions to integrals of the twopoint function of the trace of the stress tensor and to integrals of its spectral densities. The positivity of these formulas implies $c^{\text{UV}}\geq c^{\text{IR}}$. In the infinite radius limit the sum rules reduce to the well known formulas in flat space. Throughout the paper, we prove some general properties of the spectral decomposition of the stress tensor in dS$_{d+1}$.
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Author comments upon resubmission
List of changes
Abstract
 Shortened abstract, removed figure from abstract
Introduction
 Corrected minor typos and sentences
 Restated "..the radius of the sphere is a useful tunable parameter which can be used to follow RG flows in any QFT of interest.." rather than "..the radius of the sphere is a valuable IR regulator.."
Section 2
 Corrected minor typos
 Added a broad outline of the general logic of this section at its beginning
 In 2.1 emphasized that although the words "de Sitter" are used, the derivations throughout the paper are signature agnostic.
 In 2.2 emphasized that the conservation equations are $E_i=0$ and emphasized more the logic involved in solving (2.25).
 In 2.3 added a discussion after equation (2.37) expliciting what is rigorously known and what is a conjecture regarding the presence of discrete series contributions in the KallénLehmann decomposition of scalar operators in de Sitter
 In 2.4 added a paragraph outlining a concrete criterion to establish whether $\lim_{R\to0}c_1(R)=c^{\text{UV}}$ in a given theory.
Section 3
 Around equation (3.13), emphasized that assumption (2.45) is used in this derivation. Then, in (3.22) we weaken that assumption. This concretely explains why $\lim_{R\to0}c_2(R)=c^{\text{UV}}$ in a broader class of theories than $c_1(R)$.
Section 4
 Fixed minor typos
 In 4.1, restructured the discussion on the free massive boson theory.
 Fixed typo in equation (4.7) and absorbed the Heaviside theta function in the spectral density to be consistent with the notation in section 3. Added some explanatory sentences.
Appendix
 In (A.4), added explicit forms of $q_i(\sigma)$ functions.
 Swapped orders of appendices B and C.
Published as SciPost Phys. 17, 079 (2024)
Reports on this Submission
Anonymous Report 2 on 2024824 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2404.03739v2, delivered 20240824, doi: 10.21468/SciPost.Report.9643
Strengths
1 Clear exposition in the main body
2 Technicalities in the appendices
Report
The manuscript presents the study of RG of QFTs on $dS_2$ and $S^2$. In particular, it is focused on the definition of two functions $c_{1,2}$ of the radius $R$ of $dS_2/S^2$ that interpolate between the central charges of the CFTs in the UV and IR at the extremes of the RG flow. Such extremes are formally achieved with the two limits $R\to 0$ (UV) and $R\to \infty$ (IR). Both functions are built from the stressenergy tensor.
The paper is well written, especially in the main body. All the technical details are proven and the mathematical structure consists of a good level of accuracy and rigor.
Overall, the study presented is a relevant contribution to the knowledge of QFTs on dS. The comprehensive analysis and rigorous methodology employed in the research underscore its relevance and potential impact on further studies in this domain.
Requested changes
The requested changes in the first version of the manuscript have been implemented in this second version.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report 1 by Edoardo Lauria on 2024823 (Invited Report)
Report
I am happy with the changes made by the author. I recommend this paper paper for publication.
Recommendation
Publish (meets expectations and criteria for this Journal)