Relaxation time approximation revisited and non-analytical structure in retarded correlators

Dear Editors and Author,

In the present manuscript, the author presents arguments to derive the relaxation time approximation, a widely used phenomenological approximation, from the linearized Boltzmann equation.

Before I can recommend it to be published, I would like the following comments/questions to be addressed.

1. Comment: in the introduction, second paragraph, the author mentions some results on the holographic approach to the thermalization/hydrodynamization. It is also worth noting more recent work such as https://arxiv.org/pdf/2406.06685, where the authors suggest the existence of initial states that take arbitrary long times to thermalize, just as in the manuscripts references 13, 14 or 36.

2. Also in the introduction, second paragraph of page 2: the author mentions hard sphere potential, but, as far as I have seen does not mention https://arxiv.org/abs/2309.09335, which deals with the linearized collision term spectrum for this interaction.

3. Is there a physical intuition for employing the variable g (introduced in the first paragraph of page 3) instead of the usual Mandelstam variables, such as 's'? Or is it just convenient so that the results in ref. 33 (displayed in page 7, at the end of sec. III) are employed?

4. In Eq. (9), the author in the paragraph containing "This can be achieved by redefining the weight function within the inner product definition as (...)", it should also be mentioned that the corresponding integrals should converge. This may exclude very large values for \alpha.

5. In Sec. IIIB, last paragraph. Couldn't it be that the eigenvalue \gamma_6 also grows very large, since it also possesses an \alpha dependence? From ref. 36, the interplay between E_p and the linearized collision operator can lead to a very different behavior regarding the spectrum.

6. In the end of Sec IIIC: the gapless continuous eigenspectrum of what the author calls \mathcal{L}{0} was also confirmed in Ref. 36 and generalized in Ref. 23. Then, the interplay of E_p with what the author calls \mathcal{L} (whose spectrum is known for phi4 theory) is what is crucial, but I would regard their argument as an equivalent justification.

7. In Eq. 22, I would recommend using \Tilde{\chi}, for the sake of clarity

8. In theorem 2: should v_0 be a \nu_0? Is it a typo?

9. In page 8, is cos \theta, the angle between "boltzmann" momentum 3-vector p and "fourier" momentum 3-vector k? If so, I wouldn't expect it to commute with the linearized collision term, since E_p = u.p seen as an operator in Hilbert space also doesn't. Is there any physical intuition behind, or is an assumption to simplify the analysis?

10. The derivation made in the end of Sec. V seems to be quite similar to the one in ref 19, but ref 19 uses \mathcal{L}_{1} in the current authors notation, whereas I the current author, as far as I understood, wants to justify a constant relaxation time RTA, which becomes that of Anderson-Witting for Landau matching conditions. Anyways, ref 19 should be mentioned in this section, even so if the author means something different.

Best regards.

Ask for minor revision

I believe this is a very interesting contribution but before I can recommend acceptance I need clarification of the following points.

• When generalizing the RTA as in this paper it is necessary to make sure that the right hand side of the kinetic equation remains a symmetric operator and that relevant conservation laws are enforced. It is not clear to me that the generalization discussed in this paper matches these criteria. For further discussion see

Peralta-Ramos, J.; Calzetta, E. Macroscopic approximation to relativistic kinetic theory from a nonlinear closure. Phys. Rev. D 87, 034003 (2013).

Rocha, G.; Denicol, G.; Noronha, J. Novel Relaxation Time Approximation to the Relativistic Boltzmann Equation. Phys. Rev. Lett. 127, 042301 (2021, ). (ref. 28)

Rocha, G.; Ferreira, M.; Denicol, G.; Noronha, J. Transport coefficients of quasiparticle models within a new relaxation time approximation of the Boltzmann equation. Phys. Rev. D 106, 036022 (2022).

Kandus, A.; Calzetta, E. Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation. Entropy 26, 927 (2024).

** If I understand correctly, the point is that in generalizations of the RTA other than Anderson - Witting, after diagonalizing the collision operator, there remains a momentum dependence of the mode mean life coming from the momentum dependence of the relaxation time. However this problem may be solved by diagonalizing the whole right hand side of the kinetic equation as a single operator. The authors should justify why their approach is the right one.

I believe it would be fair to refer to Marle's work together with Anderson-Witting's. See

Marle, C. Sur l’etabissement des équations de l’hydrodynamique des fluids relativistes dissipatifs. I.—L’équation de Boltzmann relativiste. Ann. Inst. Henri Poincaré (A) 10, 67–126 ( 1969). Marle, C. Sur l’etabissement des équations de l’hydrodynamique des fluids relativistes dissipatifs. II.—Méthodes de résolution approchée de l’equation de Boltzmann relativiste. Ann. Inst. Henri Poincaré (A) 10, 127–194 (1969).

Ask for minor revision