Effects of Atomic-Scale Electron Density Profile and a Fast and Efficient Iteration Algorithm for Matter Effect of Neutrino Oscillation

Round 1

Reviewer 1 Report

See the attachment.

Comments for author File: Comments.pdf

Author Response

We thank the referee for a thorough review of the manuscript. The answers to the referee’s concerns are below:

In our interpretation, the use of word magnitude in the phrase clearly suggests that we refer to the maximum magnitude of the density spikes. We make this more clearly in the new version of the manuscript. Not clear what effects are we refer to. It is well known that the matter effects are essential for measuring an asymmetry effect by DUNE/LBNF. In addition, the expected asymmetries are small, and therefore a precise treatment of the variation of matter density is important, as claimed in former Ref. [20] (Ref. 1802.06784 makes the claim that given the present parameter of DUNE the density variation might not be as important and its analytical method might suffice, but this should not be view as a limitation to explore better experimental conditions). Regardless, neither of the references listed by the referee discuss the effect of large electron densities near the atomic nuclei, which is one of the issues we analyze in our manuscript. One can never find perfect notations, especially when one plan to reuse them in some published work. As indices, consistent with the standard notations in the literature, we use Greek letters for flavors, and latin letters for the mass eigenstates. We used the same notations in other publications where we take them for previous theoretical work in the field (see e.g. former Ref. [8]). Unfortunately, in the literature one also uses \alpha for the ratio of \Delta m^2_21/\Delta m^3_31, which we continue to use here. The notations might be slightly inconvenient, but they are correct. We included Refs. For the original work of Wolfenstein and Mikheyev in the new version of the manuscript. (5) is relevant for the last part of section 5. We fix this problem in the new version of the manuscript. Most of the energies of interest for beam neutrino physics are larger than 2-5 MeV, and therefore they are the most relevant for our manuscript. We make more clearly in the new version of the manuscript what is the range of Z-number satisfying Eq. (5) (for the above range of energies). Fix it. The units are correct: s is dimensionless (see discussion after Eq. (11)), and N – the number of baseline intervals – can be chosen appropriately. We make this description more clearly in the new manuscript. We included a longer description in the new version of the manuscript. N is the number of intervals on the baseline, the same as in Eq. (11). To avoid further confusions, we changed N to n as a subscript for the neutral current potential (Eq. (6) and after Eq. (27)). We though that’s obvious by comparing Eq. (16) to Eq. (11). An explicit extension of D_1 and D_2 to the 3+1 case was presented in Eq. (27). We make this more clearly in the new version of the manuscript. The iteration method is approximate, while the eigenvalue methods is “exact” (up to numerical errors of finding eigenvalues and eigenvectors in double precision). We already mentioned that increasing the number of intervals decreases the difference. We explain the reasons for describing different baselines in the new version of the manuscript. We did not claim that the direct integration should not work for the “spikey” potential. In reality, we started using it, but found out that for long baseline it requires special attention, and it may provide misleading results. There is no Fig. 8 in the manuscript. For Fig. 5 we mentioned which density profile we used in the last paragraph of section 3 where we describe this figure. We disagree. In our view our section 5 is very important, showing how efficient and convenient is the extension of our iteration method to cases with one or more sterile neutrinos. In fact, we are not even aware of many treatments of matter effects for 3+2 mass eigenstates. The extension of our method to that case is obvious and simple. We emphasize this last aspect in the new version of the manuscript.

We hope that the new version of the manuscript, in addition to our specific answers, would change the referee’s, and the manuscript will be recommended for publication.

Reviewer 2 Report

This paper presents two conclusions: the non-adiabatic transition of neutrinos through condensed matter can be approximated with results assuming the adiabatic constant density model (hereafter as "no-spike approximation"); a new algorithms by iterating the S-matrix <=> Hamiltonian, is faster than the integration method (hereafter as "iteration algorithm"). The reviewer considers these two conclusions, especially the iteration algorithm, interesting to the community and will attract attention related to the analyses of several experiments in the near future. However, the reviewer considers the structure of the paper can be further improved, and the design of the tests and way of presentation to support the conclusions need to be improved. As a result, the reviewer suggests the article to be accepted after major revision.

The structural comments are as follows:
1. The title of the article emphasize this article would be about the iteration algorithm, while most of the abstract is addressing the no-spike approximation. This could confuses the readers what is the content of the article and shadows the genesis of the paper. The reviewer suggest to modify the title and the abstract to include both. For example
- change title to "Effects of atomic-scale electron density profile and a fast and efficient iteration algorithm for matter effect of neutrino oscillation".
- line 8, "probabilities, but ... " -> "probabilities. During the study, because the integration in the computation is extremely time-consuming, we explored a new iteration-based algorithm for computing the neutrino survival/appearance probability in matter. The algorithm is shown to take only 1/X of the time used by the integration algorithm under the same accuracy. ..." Here X should be tested and shown as the concrete evidence of "fast and efficient".

2. In figure 3, the evidence of no-spike approximation should be that two assumptions gave consistent results, while the results showing inconsistent results are shown, and they are not explained in the figure caption. The author should replace the figure 3 with the cases where tolerance is set to 10^{-12} and two curves align. The author can put two curves overlapped and show in another pad the ratio of two curves. In the caption the tolerance should also be mentioned.

3. The accuracy of iteration algorithm is shown in figure 4 and figure 5, while they are contradictory. According to figure 4, the iteration algorithm can reach 0.1% accuracy in 15 iterations, while in figure 5 the drift is around 5% (0.005/~0.1). The author should address the source of discrepancy or eliminate it.

4. The more important conclusion of the article, that the new iteration algorithm is fast, is not proven. A sub-section or one paragraph close to line 81 should be added reporting the actual time consumed using iteration algorithm and integration algorithm reaching the same accuracy.

5. Although "non-adiabatic" is mentioned several time, it seems from Equation (9) the author use the adiabatic assumption. Besides, line 60 mentions 2--5 MeV threshold. The day-night asymmetry of B-8 solar neutrinos is due to matter effect of the earth, and interested neutrino energies is 6~10 MeV, just around the threshold. The author could mention this in line 60 and maybe address the question carefully for solar neutrinos in future studies.
Here is the end of the structural comments.

The detailed comments are as follows:
1. Line 1, remove citation in the abstract. The reference is not visible to readers reading only abstracts.

2. Line 21, should include references, such as
- Lisi, E., & Montanino, D. (1997). Earth regeneration effect in solar neutrino oscillations: An analytic approach. Physical Review D - Particles, Fields, Gravitation and Cosmology, 56(3), 1792–1803. https://doi.org/10.1103/PhysRevD.56.1792
- Blennow, M., Ohlsson, T., & Snellman, H. (2004). Day-night effect in solar neutrino oscillations with three flavors. Physical Review D, 69(7), 073006. https://doi.org/10.1103/PhysRevD.69.073006
- Long, H. W., Li, Y. F., & Giunti, C. (2013). Day-night asymmetries in active-sterile solar neutrino oscillations. Journal of High Energy Physics, 2013(8), 56. https://doi.org/10.1007/JHEP08(2013)056

3. Line 23. The matter effect need to be important with significant amount of matter. Xe-136 will only be present in the detector in ton scale and will introduce negligible effects. More proper examples would be Fe, O, Is, Mg, if they qualify "medium-Z" elements, which are major elements of the earth.

4. Line 30. It is unclear and the author should extend the argument about "there are no local mass eigenstates in matter".

5. Line 30. ".. in matter, but ..." The "And" is more suitable than "but" here.

6. Line 37. "This approach" is not clear. Do you mean the iteration approach described in section 3? If so, "this approach" should be briefly explained here. Besides, because the author did not justify the application of this method to rapidly varying density cases, the author should not mention the spike-effect in describing the algorithm. One sentence should also

7. Line 36-37. "We are investigating" Change to "In this article, we investigated .."

8. Equation (4), line 41 etc. units (cm, pm etc.) should be in straight fonts, not Italic.

9. Equation (6), maybe "Vn(x)"?

10. Line 53. The potential of neutrons and protons will be narrower spikes compared with electrons. It is not trivial that they can cancel out. The author should address this issue.

11. Line 125 and line 128. The two conclusions are not supported by the text.
- For the first conclusion, in the text the author only presented the proper way to apply the algorithm without comparing it with a robust algorithm. Change "the new algirhtm can be extended to the case ..." To "how to extend the new algorithm to the case ..."
- For the second conclusion, only one sentence in line 108 addresses this conclusion. It is weak and described as "preliminary", so it should not appear in the conclusion section.
Here is the end of detailed comments.

Author Response

We would like to thank the referee for her/his careful reading of our manuscript and for the extremely constructive suggestions and comments. We essentially adopted and/or addressed all of the referee’s recommendations. Our specific answers are below:

Structural comments: We changed the title to the form the referee suggested.

We changed the part of the abstract to better describe the content of the manuscript, as the referee suggested.

The main reason of including Fig. 3 in its present form was that it could have actually been related to a realistic experiment. As one can see from the figure, there is little effect at 1300 km that corresponds to DUNE/LBNF experiment baseline, but it would have been significant for 7300 km, which is the distance between Fermilab and Gran Sasso or that from CERN to Sanford South Dakota. Clearly, we did not explain that in the manuscript. Regarding the potential "bug" in ZVODE routine, we also extensively check that using Julia's DifferentiaEquations package, which has a much richer set of routines that can integrate stiff and non-stiff systems of complex differential equations, and the behavior was similar to that we experience with ZVODE. We should have mention that in the original version of the manuscript, but we thought that mentioning ZVODE would suffice. These two shortcomings are addressed in the new version of the manuscript. We hope that these changes will better serve the message of the manuscript, rather than showing two overlapping curves. The shift of the two curves in Fig. 4 were not intended to indicate the error of the calculations, but rather to make them visible. For this calculation we actually used 150 iterations, that reduce the actual difference between the two calculations to 10^(-5), the same as the error used for direct integration. We make this aspect more clearly in the new version of the manuscript. Although a complete direct comparison with other algorithms could be hardly comprehensive because of large number of choices for cases and alternative algorithms, we estimated a factor of at least 20 between our iterations algorithm and that used in (former) Ref. [20]. We make more clearly in the new version of the manuscript. In the neutrino oscillations literature, the (non)adiabatic behavior refers to evolution of the in-medium mass eigenstates. Eq. (9) is only used to obtain the approximate Eq. (11) from the more complicated Eq. (5). Eq. (5) can be used directly to obtain the evolution of flavor amplitudes, or it can be re-written in terms of a complicated evolution of in-medium mass-eigenstates (as discussed in former Ref. [1]), which avoid here. We actually only used adiabatic vs non-adiabatic distinction in the introduction to emphasize the difficulties of using the traditional local in-medium mass eigenstates. In (former) Ref. [1] we showed that for steeply varying electron densities the known results for the emission and detection are standing. In this paper we actually developed methods that directly integrate the vacuum mass eigenstates propagating through matter described by Eq. (11). We make this more clearly in the new version of the Introduction.

Detailed comments: We removed the Ref. from the abstract. We added all references as requested. We changed the example of enhancement of electron density to that of 28Si. We make this aspect more clearly in the new version of the manuscript. We made the change. We made appropriate changes to the last paragraph of the introduction to answer the referee’s question, and also addressing A.5. We made the change. We made the change. We made the change. We addressed this issue in section 5. We made appropriate changes in the last paragraph of conclusions.

Reviewer 3 Report

Really interesting work!  A couple of issues with referencing need to be addressed.

Has ref [1] been submitted for peer review?  Please either provide a journal reference for this or explain the status of its publication?

In section 5 conclusions are presented but no results - you mention another paper but don't reference it.  If it has not been published yet please either remove the conclusions (line 106 onwards in this section) or include your results and methodology in this paper?  The general description of the extension to steriles is fine and can stay in any case.

Author Response

The results reported in Ref. [1] were published in abbreviated form in a conference proceedings: AIP Conference Proceedings 2165, 020012 (2019); https://doi.org/10.1063/1.5130973 . A slightly extended version of Ref. [1] was accepted for publication in the European Physical Journal A (EPJA). I was expected to have it published (at least online) by this time, but apparently there were some delays, probably due to the Holidays season. I'll add in the revised version of the manuscript a reference to the AIP Proceedings and the EPJA.

The statements regarded the treatment of the neutrino spikes around the atomic nuclei for lower neutrino energies are accurate and truthful. I performed some numerical calculation using Eq. (5) under some artificially conditions, as described, and I plan report this analysis in the near future. At this point I don't see any other option of justifying the use of average neutron densities for neutrino energies lower than 1.5 GeV. I don't know if anyone else, including the reviewers, has a solution to this problem, but at the very least I believe that the community should be aware of the problem. Therefore, we are standing on our statements from (former) lines 106-111.

Reviewer 4 Report

My comments can be found in the attached file.

Comments for author File: Comments.pdf

Author Response

We thank the referee for highlighting the need of having better analyses tools for varying electron density effects in neutrino oscillations experiments, and for her/his comments that may help to improve the manuscript. Below are our answers.

The main reason of including Fig. 3 was that it could have actually been related to a realistic experiment. As one can see from the figure, there is little effect at 1300 km that corresponds to DUNE/LBNF experiment baseline, but it would have been significant for 7300 km, which is the distance between Fermilab and Gran Sasso or that from CERN to Sanford South Dakota. Clearly, we did not explain that in the manuscript. Regarding the potential "bug" in ZVODE routine, we also extensively check that using Julia's DifferentiaEquations package, which has a much richer set of routines that can integrate stiff and non-stiff systems of complex differential equations, and the behavior was similar to that we experience with ZVODE. We should have mention that in the original version of the manuscript, but we thought that mentioning ZVODE would suffice. These two shortcomings are addressed in the new version of the manuscript.

The referee asserts that “People approximately infer the electron density from the matter density. So it is good to discuss the difference between the two cases with and without the electron spikes based on the same varying-density profile. However, the authors did not answer this question.” To our understanding, we actually addressed the referee’s two cases in sections 3 and 4.

We are very much aware of the standard procedure of using piecewise the S-matrix for constant density regions when integrating through the Earth crust. Our understanding is that this procedure is mainly justified by the PREM model of the Earth density (Dziewonski, Adam M.; Anderson, Don L. (June 1981). "Preliminary reference Earth model" (PDF). Physics of the Earth and Planetary Interiors. 25 (4): 297–356, doi:10.1016/0031-9201(81)90046-7). However, we believe that one would like to be able to investigate more accurate models of varying Earth crust density. In addition, the constant matter S-matrix for 3 and high number of neutrino mass eigenstates are quite complicated (especially for 4 and higher mass eigenstates), and it seems to lack a simple mechanism of improving their accuracy. In contrast, our iterations algorithm is very simple, can be easily implemented by pre-calculating the product of the last three matrices in Eq. (25), and can be easily extended to any number of mass eigenstates. Moreover, as shown in the manuscript, the new algorithm can be used to justify the use of average electron densities (rather than considering the spikes) and in some cases the use of average neutron densities when sterile neutrinos are considered. Therefore, we respectfully ask the referee to support the publication of the manuscript in journal Universe.

Round 2

Reviewer 1 Report

Comments:
\begin{enumerate}
\item Notation: for the mass index it is common to use mid-alphabet Latin characters, $i,j,k$, specifically to avoid how the indices on $U$ appear in e.g.~eq.~2.
\item References: reference 3 (in v2) after the term MSW effect is mentioned is still incorrect, as is the term in this context. The matter effect was pointed out by Wolfenstein, Mikheyev and Smirnov pointed out a particular application of the matter effect that enhances a small mixing angle to a large one. This effect does not seem to apply here in this paper.
\item Errors: there is still a missing square after eq.~11 during the discussion of the mass ordering.
\end{enumerate}

Conclusion: I still do not recommend the paper for publication. Beyond the above mentioned superfluous problems, I still believe that this paper does not meet the publication criteria. The paper compares two integration techniques of a numerically challenging problem, and points out that one is better than the other. While both return the correct result in theory, one is significantly more computationally expensive and the authors apparently fail to converge to the correct solution in one case. If the focus of the paper was on numerical integration techniques the paper could be completely rewritten with that in mind; that said there is a body of literature on this topic already with many more sophisticated techniques. In the context of neutrino oscillation physics, their main result is that the highly varying electron number density is extremely well approximated by a uniform electron number density. This is the assumption that has been used since the matter effect has been identified so the result is of marginal interest at best. If that were the main presentation of the paper it would probably be suitable for publication, however highly misleading results (e.g.~fig.~3) make this paper a potential detriment to the community with little to no gain.

Author Response

We change the index of mass eigenstates from “a” to “j”, as requested by the referee. We use “i” as the index for the intervals on the dimensionless baseline s.

We removed reference 3, as requested by the referee. We thought that was justified in the context of the first sentence of the introduction.

We fixed the \Delta m^2 (we thought we did in the first revision, but it appears that it fell through the cracks – apologies).

We changed the caption to Fig. 3, directly acknowledging that the black curve is incorrect, emphasizing the relevance given in the text. We hope that this would remove any contention that Fig. 3 is misleading.

Reviewer 2 Report

The conclusion of this article is neutrino oscillations in density with spikes are equivalent to that in constant density. This is proven in Section 4.

With the word "However" in the abstract (line 8), which I suggest to remove but the author insist to keep it, I expect the author prove that, different from the integration method, the iteration method should not be subject to this numerical problem. However the author only proves the equivalence mathematically. I think probably the iteration is also subject to numerical problems in density with spikes.

The reviewer suggest to modify the abstract

line 8 remove "We found little evidence that these spikes affect the standard oscillations probabilities.". Because the author considers "the numerical integration under these conditions could be treacherous, and could be leading to erroneous results."(line 147).  Remove "However" in line 8. Because the iteration method is also subject to numerical problems. (if not, please prove it with a figure like figure 3 using iteration method). Line 10. change "seems to be" to "is". and add proof for "20 times faster".  Add to the end of line 10. "With this method, we found little evidence that these spikes affect the standard oscillations probabilities.".

Please also modify the caption of figure 3. It is very confusing.

   5.  Add a sentence to figure 3 caption: The difference is incorrect and due to numerical problems. From Section 4 we can see they should be equivalent.

The other conclusion, that the iteration method is fast and efficient is not proven. I have mentioned this in the last comments but the author has ignored it. 

   6.  A sub-section or one paragraph close to line 81 should be added reporting the actual time consumed using iteration algorithm and integration algorithm reaching the same accuracy. The author should show how many seconds are consumed in both methods.

Author Response

We implemented all changes of the abstract suggested by the referee.

We changed the caption to Fig. 3 indicating that the black curved is incorrect.

Regarding the speed of the algorithm, we do not believe that including running times would be beneficial to the manuscript. That could expose the manuscript to criticism regarding the conditions under which the timing was performed, such as processor type, compilers used, cache memory, etc. We rather give a justification based on the number of iterations needed to obtain the same accuracy. In addition, we provide a qualitative explanation of why our algorithm gets such a gain relative to the direct integration of Eq. (11). All this information is now included in the extended last paragraph of section 3. We also added a phrase about the speed up in the Conclusions (we forgot, not ignored, the advice in the 1^st response).

One should keep in mind that we do not claim that our algorithm is the fastest in all situations, but rather faster than the direct integration method in the cases where analytic/approximate methods are not accurate enough.

Reviewer 4 Report

The authors have answered my previous concerns, I recommend the publication of this article. 

Author Response

We thank the referee.