Carroll Fermions

Dear Editor, dear Referees,

We thank the anonymous referee for reading our reply to their first report, and for suggesting new improvements of our manuscript in this second report. We added some further clarifications that we detail in the following list of changes. In particular, we tried to stress that the halving of degrees of freedom in the electric limit noted by the referee is an artifact induced by our way of rescaling the fields in eq. (2.13). One could as well rescale all spinors with the same factor of $\tilde{c}$ and obtain an electric action containing the same number of spinor components as in the original action. Our choice somehow stresses that in order to obtain a magnetic limit it is not sufficient to use the rescalings that we introduced in eq. (2.8) while discussing the Carrollian transformation rules of spinors. One crucially has to implement these rescalings in a `twisted' way as in (2.19), where we rescaled the two projected components in the opposite way in the two Dirac spinors entering the Lagrangian (2.10). Otherwise one gets an electric limit as detailed after (2.13).
We hope that these additional clarifications will make our paper suitable for publication.

- Our electric and magnetic Carroll Lagrangian are invariant under the Carroll symmetries. In particular, the boost symmetry is obviously realised in the electric Lagrangian, however a non-trivial cancellation happens when checking that also the magnetic Lagrangian is boost invariant. Since our magnetic action in first-order formalism precisely matches the one written in the Lagrangian formalism, we expect that the same non-trivial cancellation happens when computing the Poisson brackets of its energy density. Since in our paper we mainly focus on the Lagrangian formalism, we decided to just comment about this below (2.40) rather than explicitly showing it.

- We slightly modified the text in the paragraph containing eq. (2.11) and we added eq. (2.12) to make the starting point of the following discussions more explicit. Consistently with the comments in our resubmission cover letter, we also stressed again above (2.13) that the rescalings proposed in that equation are not the only way to obtain an electric limit (the same rescalings were already mentioned at the beginning of the preceding paragraph, but we agree that it is more appropriate to stress the point in the paragraph dedicated to the Electric limit). We also added an explicit comment below (2.16) about the fact that even if we started from a ``magnetic'' rescaling of the relativistic fields we ended with ``electric'' Carrollian transformations thanks to the disappearing of $\psi_-$ and $\chi_-$ in the limit. We also modified footnote 10 to explain that this is not to be considered as a general feature of the electric limit, but only as a consequence of our way to define a particular electric limit, eventually leading to the minimal electric Lagrangian (2.18).

- We modified the sentence about the unification of the limits below eq. (2.18) and we made more explicit that we provided a framework in which, differently from Dirac's action, we can perform either an electric or a magnetic limit. On the other hand, we are not able identify both limits at the same time as was done for bosonic theories via the c-expansion of the relativistic action in arXiv: 2110.02319. Motivated by the referee's comment we explained in footnote 12 at pag. 8 why the c-expansion of relativistic bosonic theories does not seem to provide a promising setup to unify the limits in the fermionic case. Indeed, in the bosonic case, to show that the electric (magnetic) case occurs in leading (subleading) order requires the application of a so-called Hubbard-Stratonovic transformation, which can only be performed if the leading divergence can be written as a square. This is true in bosonic actions where the kinetic term is a square, but this is no longer the case in fermionic actions. In fact, this difference with the bosonic case is precisely the reason that it took some effort to define a Carroll limit for fermions. Nevertheless, the c-expansion method applied to fermionic theories remains an interesting issue to explore.

- We added a small remark about the potential applications of Carrollian fermionic field theories in the study of flat space holography at the beginning of the last paragraph of pag. 2.