Comment on the article by D. Borah and B. Karmakar"Linear seesaw for Dirac neutrinos with A4 flavour symmetry", Phys. Lett. B789 (2019) 59-70, arXiv: 1806.10685

D. Borah and B. Karmakar in Phys. Lett. B789 (2019) have proposed an A4 flavoured linear seesaw model to realise light Dirac neutrinos. In this comment article, we show that some neutrino Yukawa interactions were missed in the model, thus implying that a different formula would be needed to determine the effective neutrino mass matrix, with significantly different results. Our result shows that, unlike stated in Phys. Lett. B789 (2019), that the inverted neutrino mass spectrum is not ruled out.

y a 3 φ + S ) andS L S R (y ξ 4 ξ † + y s 4 φ † S + y a 4 φ † S ) must exist in the neutrino sector since they are invariant under all symmetries of the model. Therefore, the two corresponding contributions have to be added in Eq. (22) of Ref. [1] and the mixing between the heavy neutrinos N L N R and S L S R are generated at renormalizable level, with the corresponding mass matrices M N N , M SS given by: where In this case, the full neutrino mass matrix has the following form which implies that the light Dirac neutrinos mass matrix can be written as [3]: where the different entries are given by: Here m ν = aI, m ′ ν = a ′ I, M νS = bI, M ′ νS = b ′ I and x 3,4 , a 3,4 , s 3,4 are given by Eq. (2) and all the other parameters (e.g, x 1,2 , a 1,2 , s 1,2 ,...) are the same as in Ref. [1].
To diagonalise the light active neutrino mass matrix m ν in Eq. (5), we define a Hermitian matrix M, given by where with A, A 1,2,3,4 and B 12 are defined in Eqs. (6) and (7).
The mass matrix M is diagonalized by the rotation matrix U 13 given by Eq. (21) of Ref. [1] and the light active neutrino masses m 2 1,2,3 are given by We see that the leptonic mixing matrix in this case is the same as in Ref. [1], however, the expressions of neutrino masses are different from each other. For instance, in the case of normal neutrino mass hierarchy, by taking the best fit values on neutrino mass squared differences given in Ref. [2], ∆m 2 21 = m 2 2 − m 2 1 = 7.53 × 10 −5 eV 2 and ∆m 2 32 = m 2 3 − m 2 2 = 2.51 × 10 −3 eV 2 , we find the solution: A = 0.00121735 eV + m 22 , B = −0.00129265 eV.
The absolute values of neutrino masses as well as the neutrino mass ordering is still unknown, however, we can use the neutrino oscillation experimental data for normal hierarchy given in Ref.
On the other hand, in the inverted neutrino mass spectrum [2], Ref. [1] where the inverted neutrino mass hierarchy is not allowed.

Conclusion
We have shown that some neutrino Yukawa terms must be added in the A 4 flavored linear seesaw model of Ref. [1]. When these terms are included, the obtained results are significantly different than the ones reported in Ref. [1]. Namely, in the normal neutrino mass spectrum, the leptonic mixing matrix in the case where the missed Yukawa terms are added is the same as in [1], however, the expressions for neutrino masses are different from each other. On the other hand, in the inverted neutrino mass spectrum, our result is completely different from the one obtained in Ref. [1] and the sum of neutrino masses satisfies the current cosmological constraints, thus implying, unlike stated in Ref. [1] that this scenario is not ruled out.