Dilaton in scalar QFT: a no-go theorem in 4-epsilon dimensions

The arguments given are elementary and are only applied at one loop. I dare say the results about the absence of a dilaton in purely scalar theories in the epsilon expansion are correct but are hardly conclusive about the absence more generally in non supersymmetric theories.

The main argument in section 4 is almost trivial and rather transcends the examples given in previous sections.

This paper makes a rather minor point but perhaps this has not been made explicitly before and so to this extent the paper has originality.

Are there examples in the literature where dilatons are known in non unitary theories? Are such examples easy to construct? Perhaps a few comments in this direction might be added.

This manuscript proves that conformal fixed points in scalar models in $d=4-\varepsilon$ have potentials that do not allow for spontaneous scale symmetry breaking (SSSB). The authors begin with a discussion of classical potentials that display SSSB and proceed to show that when quantum mechanical corrections are taken into account, the generated couplings under renormalization are such that SSSB is not possible. The proof provided is simple and mathematically correct.

This manuscript deserves publication as it adds to the knowledge we have about the behavior of conformal field theories in the $\varepsilon$ expansion below $d=4$. However, one could argue that the result presented may not be enough on its own to warrant publication.

My suggestion in order to ameliorate this concern is for the authors to also discuss the case of scalar models in $d=3-\varepsilon$, where the scalar potential involves the sixth power of the scalar field. In that case, the authors' argument in section 4 does not appear to go through in its current form, although there might be refinements. The case of $d=6-\varepsilon$ could also be discussed, where the scalar potential is cubic in the field and again the authors' argument does not appear sufficient to prove a statement similar to $d=4-\varepsilon$ in its current form. These cases could be discussed briefly in the conclusion and outlook section.

1- Discuss $d=3-\varepsilon$ case 2- Discuss $d=6-\varepsilon$ case