Title: Renormalisation Procedure for IR Divergences in Tree-Level in-in Late-Time de

We formulate a renormalisation procedure for IR divergences of tree-level
in-in late-time de Sitter correlators. These divergences are due to the
infinite volume of spacetime and are analogous to the divergences that appear
in AdS dealt with by holographic renormalisation. Regulating the theory using
dimensional regularisation, we show that one can remove all infinities by
adding local counterterms at the future boundary of dS in the Schwinger-Keldysh
path integral. The counterterms amount to renormalising the late-time bulk
field. We frame the discussion in terms of bulk scalar fields in dS, using
tree-level correlators of massless and conformal scalars for illustration. The
relation to AdS via analytic continuation is discussed, and we show that
different versions of the analytic continuation appearing in the literature are
equivalent to each other. In AdS, one needs to add counterterms that are
related to conformal anomalies, and also to renormalise the source part of the
bulk field. The analytic continuation to dS projects out the traditional AdS
counterterms, and links the renormalisation of the sources to the
renormalisation of the late-time bulk field. We use these results to establish
holographic formulae that relate tree-level dS in-in correlators to CFT
correlators at up to four points, and we provide two proofs: one using the
connection between the dS wavefunction and the partition function of the dual
CFT, and a second by direct evaluation of the in-in correlators using the
Schwinger-Keldysh formalism. The renormalisation of the bulk IR divergences is
mapped by these formulae to UV renormalisation of the dual CFT via local
counterterms, providing structural support for a possible duality. We also
recast the regulated holographic formulae in terms of the AdS amplitudes of
shadow fields, but show that this relation breaks down when renormalisation is
required.

The conclusions of the text are as follows:

• A renormalisation procedure for IR divergences of tree-level in-in late-time de Sitter correlators has been formulated.
• These divergences are due to the infinite volume of spacetime and are similar to the divergences found in AdS.
• By using dimensional regularisation, all infinities can be removed by adding local counterterms at the future boundary of dS in the Schwinger-Keldysh path integral.
• The discussion is focused on bulk scalar fields in dS, specifically using tree-level correlators of massless and conformal scalars as examples.
• The relationship between AdS and dS, via analytic continuation, is discussed.
• In AdS, counterterms related to conformal anomalies need to be added, as well as renormalisation of the source part of the bulk field.
• The analytic continuation to dS projects out traditional AdS counterterms and links the renormalisation of sources to the renormalisation of the late-time bulk field.
• Using these results, holographic formulae are established that relate tree-level dS in-in correlators with CFT correlators.
• Two proofs are provided: one using the connection between the dS wavefunction and the partition function of the dual CFT, and another by direct evaluation of in-in correlators using the Schwinger-Keldysh formalism.
• The renormalisation of bulk IR divergences is mapped to UV renormalisation of the dual CFT via local counterterms, supporting a possible duality.
• Regulated holographic formulae are also reinterpreted in terms of the AdS amplitudes of shadow fields, but this relation breaks down when renormalisation is required.

Future Roadmap

Based on the conclusions of the text, there are several potential challenges and opportunities on the horizon:

Potential Challenges:

1. Further development of the renormalisation procedure for IR divergences in tree-level in-in late-time de Sitter correlators, including exploring its applicability to other types of fields and correlators.
2. Understanding the nature and implications of infinite volume of spacetime in dS and its relation to the divergences.
3. Investigating the connection between AdS and dS via analytic continuation in more detail, particularly exploring different versions of the analytic continuation and their equivalence.
4. Addressing the breakdown of the relation between regulated holographic formulae in terms of AdS amplitudes of shadow fields when renormalisation is required. Identifying the limitations and potential alternative approaches.

Potential Opportunities:

1. Exploring the implications of the established holographic formulae that relate tree-level dS in-in correlators to CFT correlators. Investigating their generalization to higher-point correlations and other types of fields.
2. Further investigating the structural support provided by the mapping of bulk IR divergences to UV renormalisation of the dual CFT via local counterterms. Exploring the potential insights and applications of this duality.
3. Expanding the knowledge on the connection between the dS wavefunction and the partition function of the dual CFT, potentially uncovering new links and implications.

Overall, the conclusions of the text provide a foundation for future research and exploration in the field of renormalisation in tree-level in-in late-time de Sitter correlators. The challenges and opportunities outlined above can guide researchers in their efforts to further understand the concepts and implications discussed in the text.

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