Oscillating Magnetized Color Superconducting Quark Stars

Round 1

Reviewer 1 Report

Referee report on "Oscillating magnetized color superconducting quark
stars" by Celi et al.
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The paper studies the structure and oscillations of strange quark star
composed of color superconducting matter in strong magnetic fields.
Its main focus is the oscillation modes of such stars and the
astrophysical constraints on their mass-radius relationship. The paper
is interesting but requires some revision.

1. The assumptions underlying the CFL superconductivity are not
convincing. If there is a mismatch between the Fermi surfaces of
different flavors or colors then one has a suppression of
superconductivity. One sometimes speaks of gapless superconductivity.
The expression used by the authors coincides with the case when the
Fermi surfaces of all u, d, s quarks are equal with \bar\mu being
their common chemical potential. In other words, I am not sure that it
is legitimate to use this expression (see e.g. Alford et al, ApJ 629,
(2005), 969) in the case of a mismatch between the Fermi surfaces.
The author can keep their formula if they insist on it, but it would
be useful to mention the gapless superconductivity and the need for a
more careful study of this issue.

2. There are papers by I. Shovkovy, H. Warringa, J. Noronha around
2007-2008 which deal with CFL matter in strong magnetic fields. The
authors may wish to check their work and its relevance to their work.
Note that strange quark stars in strong fields have been also studied
in Phys. Rev. D 88, 025008 (2013) by Sinha et al.

3. It would be useful to compare the oscillations modes of
non-superconducting (\Delta = 0) and superconducting stars. Does
superconductivity affect the modes significantly? Is it possible to
quantify this effect?

4. How about the r-mode oscillations? See for example, for r-modes of strange quark stars with strong magnetic fields. Can the authors comment on this type of modes?

Author Response

We thank the referee for his/her comments and criticism. We have revised and answered all the items pointed out by the referee.

Changes made in the original version of the manuscript are highlighted in bold letters in the revised version. We also provide a point-by-point response explaining how we have addressed each of the reviewer's comments.

1. The assumptions underlying the CFL superconductivity are not convincing. If there is a mismatch between the Fermi surfaces of different flavors or colors then one has a suppression of superconductivity. One sometimes speaks of gapless superconductivity.

The expression used by the authors coincides with the case when the Fermi surfaces of all u, d, s quarks are equal with \bar\mu being their common chemical potential. In other words, I am not sure that it is legitimate to use this expression (see e.g. Alford et al, ApJ 629, (2005), 969) in the case of a mismatch between the Fermi surfaces. The author can keep their formula if they insist on it, but it would be useful to mention the gapless superconductivity and the need for a more careful study of this issue.

We have added some discussion about this issue in Section 2.2

2. There are papers by I. Shovkovy, H. Warringa, J. Noronha around 2007-2008 which deal with CFL matter in strong magnetic fields. The authors may wish to check their work and its relevance to their work. Note that strange quark stars in strong fields have been also studied in Phys. Rev. D 88, 025008 (2013) by Sinha et al.

We have referenced the articles by I. Shovkovy, H. Warringa and J. Noronha about  CFL matter in strong magnetic fields in Subsection 2.2. The work by Sinha et al. about strange quark stars in strong magnetic fields have been added as a comment in the Introduction among new references and discussion.

3. It would be useful to compare the oscillations modes of non-superconducting (\Delta = 0) and superconducting stars. Does superconductivity affect the modes significantly? Is it  possible to quantify this effect?

We have added a brief discussion on this topic on the manuscript in Section 3.

4. How about the r-mode oscillations? See for example, for r-modes of strange quark stars with strong magnetic fields, Phys. Rev. D 81, 045015 (2010) by Huang et al. Can the authors comment on this type of modes?

The r-modes oscillations appear only in rotating objects; although the magnetic effect on the viscosity of the fluid could be relevant on the r-mode instability, as we do not consider rotation effects in our models, these modes are out of the scope of our work. In Section 3, we have corrected the first sentence to clarify that we study non-rotating QSs.

Reviewer 2 Report

The authors study several properties of strongly magnetized superconducting quark stars.  The structure of the magnetized star is obtained imposing the chaotic magnetic field approximation, which corresponds to taking an average of the diagonal space components of the  energy-momentum tensor  defining an effective isotropic pressure and integrating the TOV equations for a spherically symmetric star. For the magnetic field the authors consider a chemical potential dependent parametrization that simulates a magnetic  profile that varies from the surface field of the order of 10^15G to a field as large as 3x10^18 G inside. This parametrization has been used in the literature but meanwhile it has been shown that it is not realistic, see Dexheimer PLB 773,487 (2017) where it was shown that considering a self-consistent calculation with a dipolar magnetic configuration the difference between the surface and central  polar field is below one order of magnitude. Similar conclusions have been drawn by Chatterjee et al MNRAS 447, 3785. In this last work it was also shown that a field weaker that 10^18G in the center was giving rise to a very large deformation. Therefore, it is not clear how significant are the conclusions drawn in the present study. There are two other points the authors did not discuss and that should be addressed: could  the magnetization destroy the quark pairing, and therefore the superconducting phase?; are the equations that determine the oscillation modes not affected by the magnetic field? which is  the limit of validity of the assumption considered?
There are other points that are not very clear
- how strong is the field in the center of the maximum mass configurations of the families shown in fig 5?
- Which EOS was considered for the crust of these stars?
- In Fig 9 the CFL fit curve  cannot be seen. How much do the results obtained for the f mode frequency deviate from the CFL fit?
- Is there a reason why in the present study the universal relations are not so well satisfied as in reference 66?
- in Table 2 could the authors also indicate the baryonic density at the surface?

Author Response

We thank the referee for his/her comments and criticism. We have revised and answered all the items pointed out by the referee.

Changes made in the original version of the manuscript are highlighted in bold letters in the revised version. We also provide a point-by-point response explaining how we have addressed each of the reviewer's comments.

1. The authors study several properties of strongly magnetized superconducting quark stars. The structure of the magnetized star is obtained imposing the chaotic magnetic field approximation, which corresponds to taking an average of the diagonal space components of the  energy-momentum tensor  defining an effective isotropic pressure and integrating the TOV equations for a spherically symmetric star. For the magnetic field the authors consider a chemical potential dependent parametrization that simulates a magnetic  profile that varies from the surface field of the order of 10^15G to a field as large as 3x10^18 G inside. This parametrization has been used in the literature but meanwhile it has been shown that it is not realistic, see Dexheimer PLB 773,487 (2017) where it was shown that considering a self-consistent calculation with a dipolar magnetic configuration the difference between the surface and central  polar field is below one order of magnitude. Similar conclusions have been drawn by Chatterjee et al MNRAS 447, 3785. In this last work it was also shown that a field weaker that 10^18G in the center was giving rise to a very large deformation. Therefore, it is not clear how significant are the conclusions drawn in the present study.

We have enlarged the discussion on this topic in Section 2.1.

2. There are two other points the authors did not discuss and that should be addressed: a) Could the magnetization destroy the quark pairing, and therefore the superconducting phase?

The presence of the magnetic field reinforces color superconductivity. We have added an explanation and a new reference in Section 2.2. An additional comment in Fig. 1 has also been added. 

b) Are the equations that determine the oscillation modes not affected by the magnetic field? Which is the limit of validity of the assumption considered?

Both the structure TOV and the oscillation equations are not directly affected by the magnetic field in the sense that we do not study magnetic Alfven modes driven by the magnetic force. We do include the magnetic field in the construction of the matter EoS and we consider also the pure magnetic pressure term. In this work, we study only the f-mode that is a polar pressure mode, and we are indeed considering the magnetic field contribution in the pressure. The limit to the validity of our approach is related to the fact that we are neglecting the possible deformations and the non-sphericity of the stars, as we work within the chaotic field hypothesis. Within this hypothesis, as the poloidal and toroidal components are considered to be of the same order and they compensate each other, the deformation is negligible independently of the magnetic field strength.

As it can be seen in our results, the effect of the magnetic effects on the f-modes are negligible. This result is coincident with the ones presented in Lander et al. (2010, https://0-academic-oup-com.brum.beds.ac.uk/mnras/article/405/1/318/1022064), where the authors state: “We look at the shift in the frequency of the fundamental mode upon the addition of a magnetic field to the star. This mode is restored by perturbations in the fluid pressure P in the unmagnetized case, so we anticipate that in the magnetic problem the restoring force is perturbations of total (fluid plus magnetic) pressure, P + B2/8π. The magnetic shift in σf , then, should be proportional to B2 – but since magnetic pressure is very modest in magnitude compared with fluid pressure, we expect the frequency shift to be small. For example, using our canonical model star, the magnetic pressure is ∼1 per cent of the fluid pressure at 10^17 G.”

We have added comments on this issue in Section 3 and 4.

3. There are other points that are not very clear

a) how strong is the field in the center of the maximum mass configurations of the families shown in fig 5?

We have added the central magnetic field in Table 2.

b) Which EOS was considered for the crust of these stars?

We do not consider any crust in our model, we are modeling the so-called bare quark stars. We have added a comment to clarify this in Section 1.

4. In Fig 9 the CFL fit curve  cannot be seen. How much do the results obtained for the f mode frequency deviate from the CFL fit?

We have re-calculated the CFL fit for our results in order to compare with the work by Flores and Lugones, Phys. Rev. C 2017, 95, 025808; we also have added a discussion on this issue in Section 3.

5. Is there a reason why in the present study the universal relations are not so well satisfied as in reference 66?

We have added a comment about this difference and we have added a new fit in Section 3. 

6. In Table 2 could the authors also indicate the baryonic density at the surface?

We have added the surface baryonic density in Table 2.

Reviewer 3 Report

In manuscript ``Oscillating magnetized color superconducting quark stars’’ the authors, Marcos Celi, Mauro Mariani, Milva Gabriela Orsaria , Lucas Tonetto, consider interesting problem. Problem  itself is suitable to be considered in Special Issue: ``Superfluidity and Superconductivity in Neutron Stars.’’ However in my opinion at least in the present form the manuscript cannot be published. The problem is not clearly explained. The authors not critically apply  results obtained in various  publications for neutron stars (NS) and hybrid stars (HS),  to the case of quark stars (QS), for which they   might be not applicable.

It is not clear why QS should satisfy the data on M_max, which are well explained by NS or HS? What to do with other numerous  data on pulsars (?), with cooling of NS (?),  r-mode damping (?),  etc.

Origin of presence of strong magnetic fields in (QSs) quark stars is not explained  and model for its distribution is not presented.

Especially unclear is why magnetic field  should be so different at surface and in interior, as it is assumed in Tabl.1 and as follows from Eq. (14), which was introduced for neutron stars and hybrid stars having  extended surface layer.

Line 34-35. The statement should be corrected since there are neutron stars with low magnetic fields.

Eq. (1): how many Landau levels do you consider? How strong field should be in order one could retain one level?  Do you consider interior?  How do you model   the surface layer, where magnetic field decreases several orders of magnitude following your Table 1?

Why do you employ simple Bag model? Other more extended models of QSs have been used in  the literature (light bag,  heavy bag -heavy quark quasiparticles, color superconductivity has been  included in them, even fit to lattice data has been considered, etc). Extra references and explanations why do you focus on a oversimplified model are required.

Table 1 and Eq. (14):  why do you think that B_max is 10^15 G if B_min  is 10^13 and B_max is 10^18 G if B_min  is 10^15?

Why Eq. (14) uses (not introduced) notations B_sup and B_cen rather than B_min and B_max?

Line 86-87 you write that B_max is field in the center. From where does it follow?  What is physics behind relation Eq. (14), does it work for quark stars? What is the surface layer of quark stars (density, thickness, origin of the field)?

Eq. (15): What Gap profile do you use? If you assume step-function density profile what is the origin of  the difference between B_min   and B_max?

Fig. 2 is not clearly explained.

Line 118-119: what does mean slowly? Next, you consider \nu up to  \sim 2000 Hz. Is it slow rotation? What is value of  Kepler frequency? Do not you think that superconductivity can crucially change consideration of rotation?

Line 126-127: why 2.01 M_sol is the constraint for quark stars?  Is not it more reliable that this object is the neutron star or hybrid star?  Important things are  explained in the manuscript.

Why should QSs oscillate as neutron stars and why QSs should obey the same description of modes as NSs?

I could continue to numerate unclear statements.

Author Response

We thank the referee for his/her comments and criticism. We have revised and answered all the items pointed out by the referee.

Changes made in the original version of the manuscript are highlighted in bold letters in the revised version. We also provide a point-by-point response explaining how we have addressed each of the reviewer's comments.

1. In manuscript ``Oscillating magnetized color superconducting quark stars’’ the authors, Marcos Celi, Mauro Mariani, Milva Gabriela Orsaria , Lucas Tonetto, consider interesting problem. Problem  itself is suitable to be considered in Special Issue: ``Superfluidity and Superconductivity in Neutron Stars.’’ However in my opinion at least in the present form the manuscript cannot be published. The problem is not clearly explained. The authors not critically apply  results obtained in various  publications for neutron stars (NS) and hybrid stars (HS),  to the case of quark stars (QS), for which they might be not applicable.

We have added some paragraphs and references to the revised version of the  manuscript.

1. It is not clear why QS should satisfy the data on M_max, which are well explained by NS or HS? What to do with other numerous  data on pulsars (?), with cooling of NS (?),  r-mode damping (?),  etc.

It is not yet totally conclusive what kind of objects are the observed and constrained ones known as Neutron Stars; up to now, there exist hadron, hybrid and quark star models that are not discarded (see for example the recent review on the MIT bag model and quark stars by Lopes et al., 2021, https://0-doi-org.brum.beds.ac.uk/10.1088/1402-4896/abef34). The structure equations do not differ among hadron, hybrid or quarks stars; the different kinds of the composition should be considered in the EoS models as we do in this work. The possibility to predict observable quantities for these different models results is of great importance to determine the actual structure and composition of these compact objects.

There exist works such as Bombaci et al. (2021, 10.1103/physrevlett.126.162702), Tangphati et al. (2022, 10.1140/epjc/s10052-022-10024-6) that consider some of the most recent observed neutron stars to be quark stars.

1. Origin of presence of strong magnetic fields in (QSs) quark stars is not explained and model for its distribution is not presented.

Regarding the magnetic field, we have enlarged the discussion and added references about its distribution and strength in Subsection 2.1.

1. Especially unclear is why magnetic field should be so different at surface and in interior, as it is assumed in Tabl.1 and as follows from Eq. (14), which was introduced for neutron stars and hybrid stars having  extended surface layer.

The possibility for compact objects to reach central magnetic fields of about 10^17-10^18 Gauss is not discarded, and there also exist some MHD simulations that suggest these central magnetic field values. In particular, some works suggest that even a magnetic field as large as 10^20 Gauss could be reached in the core of Quark stars [Ferrer et al. (2010, http://0-dx-doi-org.brum.beds.ac.uk/10.1103/PhysRevC.82.065802), Chu et al. (2018, https://0-doi-org.brum.beds.ac.uk/10.1016/j.physletb.2018.01.064), Chu et al. (2021, https://ui.adsabs.harvard.edu/link_gateway/2021EPJC...81...93C/doi:10.1140/epjc/s10052-020-08800-3)]. A discussion on this issue and the correspondent references have been added in Subsection 2.1.

1. Line 34-35. The statement should be corrected since there are neutron stars with low magnetic fields. 

We have corrected this statement.

1. Eq. (1): how many Landau levels do you consider? How strong field should be in order one could retain one level?  Do you consider interior?  How do you model   the surface layer, where magnetic field decreases several orders of magnitude following your Table 1?

The model we adopt evaluates the maximum Landau level that should be considered for each magnetic field value automatically and without any approximation or truncation in the sum over levels. In this sense, we treat the high and low MF regimes within the same formalism and code.

In particular, our model goes up to the following Landau levels for the different particles and MF scenarios:

• Low MF scenario

• B=B_min

• quark u \nu_max = 6778

• quark d \nu_max = 13556

• quark s \nu_max = 13556

• B=B_max

• quark u \nu_max = 14

• quark d \nu_max = 29

• quark s \nu_max = 29

• Magnetar scenario

• B=B_min

• quark u \nu_max = 667195

• quark d \nu_max = 1334391

• quark s \nu_max = 1334391

• B=B_max

• quark u \nu_max = 40419

• quark d \nu_max = 80838

• quark s \nu_max = 80838

Sotani et al (2015, https://0-doi-org.brum.beds.ac.uk/10.1093/mnras/stu2677) found that B should be greater than \sim 10^19 Gauss in order for the quark matter to settle only in the lowest Landau level.

1. Why do you employ simple Bag model? Other more extended models of QSs have been used in  the literature (light bag,  heavy bag -heavy quark quasiparticles, color superconductivity has been  included in them, even fit to lattice data has been considered, etc). Extra references and explanations why do you focus on a oversimplified model are required.

We have added some discussion and references about this issue in the Introduction.

1. Table 1 and Eq. (14):  why do you think that B_max is 10^15 G if B_min  is 10^13 and B_max is 10^18 G if B_min  is 10^15?

For the low MF case, we consider surface MF values according to the observed values for regular low MF pulsars; for the magnetar case, we consider surface MF values according to the observed values for magnetars. For the central MF values, we adopt, both for low MF and for magnetar case, values suggested in the literature expected for compact objects. References and discussion have been added on this topic on Subsection 2.1.

1. Why Eq. (14) uses (not introduced) notations B_sup and B_cen rather than B_min and B_max?

We have corrected this inaccurate notation throughout the manuscript.

1. Line 86-87 you write that B_max is field in the center. From where does it follow?  What is physics behind relation Eq. (14), does it work for quark stars? What is the surface layer of quark stars (density, thickness, origin of the field)?

We have enlarged the discussion and increased the number of references on the magnetic field distribution and strength on Subsection 2.1. Regarding the surface layer, we do not consider any crust in our model, we are modeling the so-called bare quark stars. We have added a comment and a reference to clarify this in Section 1.

1. Eq. (15): What Gap profile do you use? If you assume step-function density profile what is the origin of  the difference between B_min   and B_max?

The gap we mentioned in equation (15) does not represent the total energy density of the EoS, but only the term from the CFL contribution. In this sense, there exists no step-function density profile due to this gap. This gap, within our CFL model, has no profile or dependence with any other quantity, as it is a free parameter. In particular, it is not related with the MF values; hence, the values for B_min and B_max do not come from the gap value or vice versa.

1. Fig. 2 is not clearly explained.

We have added a more detailed explanation in the caption of this Figure.

1. Line 118-119: what does mean slowly? Next, you consider \nu up to  \sim 2000 Hz. Is it slow rotation? What is value of  Kepler frequency? Do not you think that superconductivity can crucially change consideration of rotation?

We do not consider rotation effects in our models and the use of ‘slowly’ was an inaccuracy. In Section 3, we have corrected the first sentence to clarify that we study non-rotating QSs. The value of the frequency $\nu$ is not related to rotation but to the frequency of the non-radial polar pressure fundamental mode. This oscillation mode does not imply or is a consequence of any rotation and so the Kepler frequency is not relevant in our study.

1. Line 126-127: why 2.01 M_sol is the constraint for quark stars?  Is not it more reliable that this object is the neutron star or hybrid star?  Important things are  explained in the manuscript

As we mentioned in a previous point, it is not yet totally conclusive what kind of objects are the observed and constrained ones known as Neutron Stars; up to now, there exist hadron, hybrid and quark star  models that are not discarded. 

1. Why should QSs oscillate as neutron stars and why QSs should obey the same description of modes as NSs?

In general, the structure equations do not differ among hadron, hybrid or quarks stars; the different kinds of composition should be considered in the EoS models, as we do in this work. In particular, the oscillation modes are given by the equations proposed by Detweiler and Lindblom (1985) [59]. In that work, they assume a Schwarschild’s metric to describe the perturbations under the notation presented in Lindblom and Detweiler (1983) Ap. J. Suppl, 53,73. They present their results considering stellar models within barotropic and zero-temperature EoSs, which are coincident conditions with those we are taking into account in our paper. For the metric perturbations, they use the Thorne, K.S.; Campolattaro (1967) equations, which are obtained for small, adiabatic, non-radial perturbations from the TOV equations, under the general relativity framework. Since the hypothesis taken in account to obtain the oscillation equations are coincident with the ones considered in this paper, and with the non-radial perturbation of relativistics compact objects in general, there is no reason to think this description would not suit the QS studies.

Round 2

Reviewer 2 Report

The authors have answered to all my comments and modified the manuscript accordingly.  Taking the  magnetic field dependence just as hypothetical and not realistic the study still seems to show the magnetic field has almost no influence on the f-modes. I suggest that lines from 103 to 115 are written more carefully and the information in references 53 and 54 should be taken as valid and give hints how the intensity of the magnetic field changes. Also, for instance,  reference 41 gives some information: in fig 5 a  poloidal field configuration is represented and the field changes from 0.6 x 10**18G in the center to approximately 0.1 to  0.2x 10**18G on the surface; the toroidal field represented in fig 1 has a similar variation except that it is completely distributed inside the star as expected from a toroidal configuration. The authors take an ad-hoc parametrization when they should have taken into consideration the recent developments. An exponential increase of B does seems to be not realistic if the B increases very fast, i.e. much more than an order of magnitude from the surface to the center. The authors have, however, shown, that the effect of the magnetic field is negligible. I guess if they had considered a polynomial change of B the same conclusions would be valid. Could they stress their parametrization of B is just hypothetical, and  complete their discussion referring to the expected results if a  dependence as the one in refs 53 and 54 or 41, would have been used?

After the introduction of this discussion, I  recommend the present work for publication in Universe.

Author Response

The authors have answered to all my comments and modified the manuscript accordingly.  Taking the  magnetic field dependence just as hypothetical and not realistic the study still seems to show the magnetic field has almost no influence on the f-modes. I suggest that lines from 103 to 115 are written more carefully and the information in references 53 and 54 should be taken as valid and give hints how the intensity of the magnetic field changes. Also, for instance,  reference 41 gives some information: in fig 5 a  poloidal field configuration is represented and the field changes from 0.6 x 10**18G in the center to approximately 0.1 to  0.2x 10**18G on the surface; the toroidal field represented in fig 1 has a similar variation except that it is completely distributed inside the star as expected from a toroidal configuration. The authors take an ad-hoc parametrization when they should have taken into consideration the recent developments. An exponential increase of B does seems to be not realistic if the B increases very fast, i.e. much more than an order of magnitude from the surface to the center. The authors have, however, shown, that the effect of the magnetic field is negligible. I guess if they had considered a polynomial change of B the same conclusions would be valid. Could they stress their parametrization of B is just hypothetical, and  complete their discussion referring to the expected results if a  dependence as the one in refs 53 and 54 or 41, would have been used?

After the introduction of this discussion, I  recommend the present work for publication in Universe.

We have rewritten more carefully the final paragraphs from subsection 2.1 regarding the MF parametrizations. We have also added a new discussion regarding our results and the parametrization dependence in the ‘Summary and discussion’ section.

We thank the referee for his/her careful reading of the manuscript. We have added a comment in the Acknowledgements.

Reviewer 3 Report

The authors accepted my criticism and essentially reworked the manuscript. The authors clarified their  point of view on problems I mentioned and did corrections in text and added three dozen of  new references. I think that in the present form the manuscript can be published after a minor revision.

Few statements should be corrected:

• Lines 35-36: 10^{13} G are ordinary NS, not magnetars. Correct to 10^{14}G.

(2) Lines 135-142: statement is not quite correct.

For 1s_0 state pairing, in superconductors of the second order (e.g. for protons in NS)  for H>H_{c2} and for H>H_c in superconductors of the first order the  pairing is destroyed.   For H<H_{c2} magnetic field may exist in vortices, cf. 1709.10340. In case of NS, in all mentioned cases magnetic field is expelling during a long time period, cf. Nature 224 (1969) 872. In cases of pairing in P-wave states superfluidity/superconductivity may remain also for H>H_{c2}, as in ferromagnetic superconductors and in some phases of color superconductors, cf. PRD 101 (2020) 056011. So, statement, Lines 135-142, should be somehow corrected.

Author Response

The authors accepted my criticism and essentially reworked the manuscript. The authors clarified their  point of view on problems I mentioned and did corrections in text and added three dozen of  new references. I think that in the present form the manuscript can be published after a minor revision.

Few statements should be corrected:

(1) Lines 35-36: 10^{13} G are ordinary NS, not magnetars. Correct to 10^{14}G.

We have corrected this statement.

(2) Lines 135-142: statement is not quite correct.

For 1s_0 state pairing, in superconductors of the second order (e.g. for protons in NS)  for H>H_{c2} and for H>H_c in superconductors of the first order the  pairing is destroyed.   For H<H_{c2} magnetic field may exist in vortices, cf. 1709.10340. In case of NS, in all mentioned cases magnetic field is expelling during a long time period, cf. Nature 224 (1969) 872. In cases of pairing in P-wave states superfluidity/superconductivity may remain also for H>H_{c2}, as in ferromagnetic superconductors and in some phases of color superconductors, cf. PRD 101 (2020) 056011. So, statement, Lines 135-142, should be somehow corrected.

We have corrected the statement pointed out by the referee by adding a brief discussion about superconductors of first and second order. We have added the references suggested by the referee and two more references in Section 2.2.

We thank the referee for his/her careful reading of the manuscript. We have added a comment in the Acknowledgements.