Dilaton in scalar QFT: a no-go theorem

1- The paper gives a self-contained proof of an elementary fact.

2- The discussion is quite readable and pedagogical.

1- The focus of the paper appears to be misplaced.

In this manuscript, the authors prove that for four-dimensional scalar theories with quartic interactions a massless dilaton that may exist at the classical level does not exist in the quantum theory (already at one loop). The authors elucidate this assertion in three nicely presented examples, and they subsequently provide a general proof which is mathematically correct.

While this manuscript is quite readable and contains a result that appears to be new, my main concern with this work is that the focus on the dilaton being massless is unjustified and in fact misplaced. What the authors actually show is that there is no quantum theory, within the set of examples considered, that has a renormalized tree-level potential that permits spontaneous breaking of scale symmetry. Any such potential of the classical theory gets modified at the quantum level to eliminate the possibility of scale symmetry breaking vacua. Therefore, the question of a dilaton being massless in the quantum theory does not even make sense, since a dilaton can arise in the first place only when we have spontaneous breaking of scale symmetry.

In light of these comments, I would like to invite the authors to reconsider the focus of their work and amend their manuscript accordingly. The authors should emphasize that a dilaton does not even arise in the quantum theory. It is misleading to suggest that it acquires a mass in going from the classical to the quantum regime, because that assumes that it exists in the standard interpretation in both regimes.

The authors should also elaborate on why they choose to simply consider the renormalized tree-level potential and not include Coleman-Weinberg radiative corrections, which in fact may allow for spontaneous breaking of scale symmetry in the effective potential. In that case the dilaton interpretation would remain, but then away from a fixed point the dilaton would presumably acquire a mass due to the explicit breaking by the scale anomaly (beta function). In a more complicated setting, this approach has been pursued, for example, in arXiv:1105.2370 (Ref. [7] in the manuscript).

As it stands, I do not recommend this manuscript for publication. However, I am willing to consider it further if the authors choose to resubmit a revised version.

This paper addresses and interesting topic as to whether massless dilatons can exist in non supersymmetric theories. The arguments given at best render this plausible rather than being robust. In section 3 some detailed arguments are presented in what is usually called the biconical model. This has been the subject of some recent interest, see for instance in 2005.03676. This has a very detailed discussion.

The authors discussion in the paper under review appears to be closely related to the question as to whether there exist exact marginal operators in non supersymmetric theories. Is this possible for very large N which is then broken by finite N effects? Could massless dilatons exist in some large N limit? How would fermions affect the argument.

Some of the discussion seems rather weak. In massless QCD chiral symmetry breaking with massless fermions is what allows massive fermions to be present. The authors might also have commented on the conformal anomalies where dilatons might play a role in saturating them as in the Komargodski Schwimmer discussion of the a theorem.

I note that in the text Goldstone has become Gondstone in one place.

Finally there are old papers such as

Spontaneous breakdown of conformal symmetry #51 C.J. Isham(ICTP, Trieste),
Abdus Salam(ICTP, Trieste), J.A. Strathdee(ICTP, Trieste) (1970) Published
in: Phys.Lett.B 31 (1970) 300-302 This has 81 c itations which should be followed up

Phenomenological actions for spontaneously-broken conformal symmetry #77 John R. Ellis(CERN) (1971)
Published in: Nucl.Phys.B 26 (1971) 536-546

see above, arguments are rather weak.

I think this could be published in this journal subject to changes.

see above