Formal Developments for Lorentz-Violating Dirac Fermions and Neutrinos

Round 1

Reviewer 1 Report

In this paper the authors consider Lorentz violation for fermions within the non-minimal Standard-Model Extension (SME). They obtain two results. The first one is the full propagator for the single-fermion Dirac theory modified by Lorentz violation. The second one is the dispersion relation for a theory of N fermion flavors with the most general possible Lorentz-violating (LV) SME-type operator bilinear in the fermion field.

These results are new and I think they are eventually worthy of publication.

Nevertheless, there are a few issues I would like to point out.

In the paragraph above Eq. (2a) the authors define the LV operators that appear in eq. (1b) and below. However, they do not explain that they include operators of arbitrary rank constructed with powers of the momentum. This needs to be pointed out, with a reference to the literature.

The authors follow the formalism developed in [11] for spinor fields of N flavors. In order to represent the most general bilinear in fermion fields, allowing as well for the possibility of Majorana couplings, the N fields are joined with their charge conjugates in a 2N-dimensional multiplet. This is not explained in the paper, forcing the reader to read the relevant parts of [11] first. In order that the paper be reasonably self-contained, a paragraph should be included summarizing this construction.

More seriously, the authors make rather confusing statements as to what their construction amounts to. In the abstract (and also in the introduction) they state that it gives "the dispersion equation for a theory of N Dirac fermions and N Majorana neutrino flavors". From which it would seem that it is a framework for 2N fermions/neutrinos. In fact, the construction of [11] allows to write down the most general fermion bilinear for N (not 2N) spinor fields, while allowing for the presence of Dirac as well as Majorana couplings. So it would seem to me that the result the authors obtain in eqs. (24) is a general result in this sense, and I don't understand why they refer in the text below to "N Dirac neutrino flavors and correspondingly N Majorana neutrino flavors". It is important that this be cleared up, and these misleading statements be corrected.

A section 6.2 is included in which the case N = 3/2 is considered. Indeed, the authors announce in the introduction that N is allowed to be half-integer, describing in such a case either "2N neutrinos of Dirac type or 2N neutrinos of Majorana type". I do not see how taking N to be half-integer makes any sense. It certainly is not consistent with the construction in [11]. Either the authors have to explain what is meant exactly, or they should eliminate the half-integer case. In fact, the case in section 6.2 does not even include Lorentz violation, so it is not clear what relevance it has in the context of this paper.

The authors describe an interesting recursive algorithm in section 4 to compute the determinant of the (complicated) matrix D_\nu. I tried to apply it to the case N = 1, but it didn't quite work out, presumably because some of the index ranges are wrongly indicated. If I am not mistaken, the text above eq. (12) should start with "The base is to define 2N sets of .... with k \in {0,...,2N-1}.". Moreover, the phrase above eq. (13) should read: "Hence, for k \in {0,...,2N-2} fixed, ....". While the construction then seems to work at least for N=1, a proof that it does work in general is not provided. It would be nice if the authors could include one or, at least, a reference to the literature.

In view of the above, I consider this paper is suitable for publication in Symmetry provided the authors address the issues outlined above.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript presented two different results: (1) Lorentz-violating Dirac Fermion propagator, (2) arbitrary flavor number Dirac and Majorana neutrino dispersion relations. Their main outcome is that the structure of neutrino dispersion relations is the same as Lorentz-violating operators, which could be applied to high energy physics phenomenology.

I have a minor comment. Eqs. (29a) and (29b) should be changed to Eq. (29). I am not sure why two equation numbers are used, while there is only one equation.

Author Response

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Author Response File: Author Response.pdf