The random free field scalar theory

1) Clarity
2) Relevance to the theory of disordered systems as well as CFT descriptions

1) Lack of consideration of previous literature

In their manuscript Piazza, Serone and Tevisani consider a free scalar field theory coupled to a Gaussian quenched random source and derive two manifestly conformally invariant descriptions of the theory. One involves treating the external random source as a quantum field and the other one relies on Cardy’s parametrization of the replica fields after making use of the replica trick to handle the quenched disorder. The description of the theory greatly simplifies in these two formalisms, which allows the authors to describe operator product expansion, normal ordering of composite operators, and other properties of more standard conformal field theories while providing exact maps between the correlation functions in the original and in the new descriptions. The new description of the free random-source scalar field theory also allows the authors to unveil new, exotic continuous symmetries, in particular nilpotent ones that in Cardy’s parametrization bear some strong resemblance to the supersymmetries of the Parisi-Sourlas construction for the random-field Ising model.

Despite being focused only on a free theory, the paper clarifies a number of properties of random-field systems and casts them in the CFT language. It is interesting and clearly written. I therefore recommend publication in SciPost Physics.

I would nonetheless like the authors to consider the following comments that touch upon comparison with previous literature on random field models:

- In the Introduction, the others state that “basic notions of ordinary QFTs such as how to properly define an RG flow (…) have started to be analyzed for quenched disorder QFTs only very recently”. Here and in several places of the manuscript, the authors overlook the many studies based on the functional RG that has been developed to precisely define RG flows in the presence of quenched disorder. To cite a few: D. Fisher, Phys. Rev. B 31, 7233 (1985), P. Le Doussal et al., Phys. Rev. E 69, 026112 (2004), P. Le Doussal, Ann. Phys. 325, 49 (2010), K. Wiese, J. Phys. A 17, S1889 (2005), M. Baczyk et al., JSTAT P06010 (2014), M. Tissier et al., Phys. Rev. B 85, (2012), etc.

- In a similar vein, accommodating the existence of additional correlators (page 3) and addressing the issue of assigning scaling dimensions in random-field models (pages 4, 6, 8) is precisely what has been done with the notion of a zero-temperature fixed point and the introduction of an additional scaling dimension for a running temperature: see the articles cited above.

Clearly this literature is not framed in the same CFT language and has not addressed several points considered in the present paper but a few words of acknowledgement or, better, of comparison would be welcome.

- Long-range random-field models have been considered, still in a functional RG framework, in
M. Baczyk et al., Phys. Rev. B 88, 014204 (2013) and Balog et al., JSTAT P1017 (2014); the former in particular investigates the Parisi-Sourlas supersymmetry with long-range interactions and disorder correlations and its breakdown.

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1. a simple exactly solvable model
2. a pedagogical paper.

1. difficult to reconcile with other methods
2. the model is well defined only in d>4

There has been growing interest in reconsidering random field spin models, particularly the Random Field Ising Model (RFIM), using a representation that dates back to the linear transformations of fields proposed by J. Cardy in 1985. This is a complex problem that has remained unresolved for many years. Significant progress was achieved about a decade ago through the use of functional renormalization group methods, especially in their non-perturbative form. Recent advances in conformal field theory methods have provided an alternative approach to studying this problem; however, to my knowledge, the perspectives offered by the two approaches have yet to be reconciled. This is difficult to realize because, despite the fact that both methods use the renormalization group language, the ways in which they attack this difficult problem are very different.

In the present manuscript the authors apply the conformal field theory (CFT) approach to a much simpler system, the random free field theory (RFFT). This is an exactly solvable model without interactions but with quenched random field. They show that the RFFT after changing variables can be viewed as a simple non-interacting CFT whose primary operators are related to the original elementary fields in a peculiar way.
In contrast to ordinary field theories, conformal behavior is observed only when considering specific linear combinations of correlators of local operators.
The authors provide two different effective conformal field theories with and without replicas and derive the explicit maps of correlation functions between these theories and the original theory with quenched random field.

The random field system with local interactions can then be studied using the RG flow considering the above effective conformal theories as a starting UV fixed point.

The problem under consideration is interesting, and the presentation is clear and pedagogically sound, making the manuscript suitable for publication. However, I am uncertain whether it meets the high standards of novelty and breakthrough required for general acceptance in SciPost Physics.
I encourage the authors to address several points that could enhance the manuscript and add significant value.

As mentioned by the authors, the RFFT model does not satisfy cluster decomposition for $d<4$. A natural question that arises for the reader, therefore, is: what are the consequences of this for the scaling behavior in physically relevant dimensions below $d=4$?

Is the RFFT model in the same universality class as the critical RFIM above $d=6$?

Since the the RFFT model is exactly solvable, could one compute the exact distribution of the order parameter at criticality?

The authors consider an extension of their model to the case where the action $S_0$ is replaced by a generalized free theory. Could the calculations be further extended to the case of correlated random fields, such as power-law correlated random fields?

In the seminal RG framework for the critical behavior of spin systems with random fields, one typically refers to a zero-temperature or infinite-disorder fixed point that governs the criticality of these disordered systems. It would be useful to discuss the connection between this picture and the approach developed by the authors.

The authors identify the $O(-2)$ symmetry in their CFT. Another known example of disordered systems that exhibit this hidden symmetry is Charge Density Waves (CDW) pinned by disorder, which is also described by a zero-temperature fixed point and can be mapped onto a pure system with $O(-2)$ symmetry (see K.J. Wiese and A.A. Fedorenko, Phys. Rev. Lett. 123, 197601 (2019)). Is this a common property of systems controlled by a zero-temperature fixed point?

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