I've been working on a calculation to generalize the form factor (Eq. 26 of attached paper) to include particles from wider lateral distances from the cascade axis. Formulae that were single terms now become sums, as I choose to model the contribution from wide-ranging particles as a sum of exponentials rather than a single exponential distribution. See attached version of the paper (you can clone it using git: kingbee.mps.ohio-state.edu:/home/hanson.369/AskaryanPaper). As an example of what I'm describing, see also the two attached plots. The first is an original figure from the paper, where I model the lateral charge in the cascade with a single exponential distribution in rho'. The other figuire is a sum of exponentials.
Finally, I've copied below the latest reviewer comments below in bold font.
Reviewer #3: Reviewer Comments for Complex Analysis of Askaryan Radiation: A Fully Analytic Treatment including the LPM effect and Cascade Form Factor.
This study presents a code based on analytical calculations of Askaryan pulses. The authors have provided new form factors, essential for these calculations, derived from first principles. The release of a publicly available code is a welcome addition and the reviewer applauds the authors for doing this. The work done for this paper merits publication.
However, there is one major issue that needs to be addressed before publication. Given that the authors are presenting an open source code it seems that another section comparing their results to previous results in the existing literature is necessary. For example, Equation 39 is an improved and updated version of equation 34. The reader is left wondering what the impact of using Equation 34 vs. 39 is in their simulations. They should be compared explicitly in this paper. Some comparisons to prior results are done in Section 7.2 but it should be brought out of the appendices into a main section. Someone looking to apply this code will want to know in what cases it gives the same results as previously published results and in which cases it deviates significantly, if at all.
There are also some issues with the summary section. In the first paragraph of Section 5 states "The fully analytic calculations and associated code require no a priori MC analysis, making them computationally efficient and accurate." This is simply not true. Using a a parameterization based on MC analysis is just as efficient once it has been obtained. In any case, the analytic calculations have to be compared to MC analysis for validation. It is worth mentioning that the treatment of the LPM effect will be less accurate since it is treated analytically as an elongation. Simulations presented in Alvarez-Muniz el al., PRD, 84, 103003 (2011) show that the LPM effect for ultra-high energy shower tends to produce longitudinal profiles with random clumps of particles rather than a smooth elongation. Without an analytical way to accurately model the stochastic behavior of these clumps the authors cannot claim this approach is more computationally efficient and accurate. The
accuracy and efficiency of the calculations, particularly in comparison to previous work, has not been treated in the main body of the text. Therefore this conclusion is not supported by the body of the paper.
The last paragraph of this section states "Rejecting the thermal noise in favour of neutrino signals is an exercise in the mathematical analysis of thermal fluctuations [51]. Armed with a firm theoretical understanding of the Askaryan effect, this challenge is made easier." The use of a parametric approach is just as valid and leads to the same conclusion. The authors should focus on whether their treatment, whether it be analytical, parametric or MC based, provides a more accurate and efficient model rather than lauding the fact that it was derived analytically.
The following issue is mainly about style and presentation. I will not require the authors do this as it is not the reviewers job to rewrite the paper for the authors, but I strongly recommend it. It is very difficult to follow what the original contributions to the calculations of Askaryan signals are. The paper spends way too much space presenting known results that can simply be referenced. Sections 3.1, 3.2, 3.3, 3.4.1 can be almost be cut altogether. The contents of these sections should be summarized into one short section referencing the material as appropriate rather than reproducing the previously published results. The results and treatment in Section 3.4.2 seems out of place and should be the starting point of Section 4. This is apparent since equation 27 and 34 are the same and it really only needs to be presented once. As far as the reviewer can tell, equation 39 is the new result for the field. The paper should focus on presenting this derivation as succinctly
as possible, with enough commentary for an expert to reproduce it, and move on to discussing the implications to simulations.
Reviewer #1:
I cannot accept the paper. The implementation of the LPM effect (one of the two new additions in the paper w.r.t. older literature, as written in the title of it) is NOT physical.
I repeat my argument:
At low frequency the field is proportional to the total tracklength, i.e. proportional to the total area = integral[N(X) dX] under the longitudinal development, with N(X) the number of particles at depth X. The tracklength depends linearly with energy, or in other words it is practically constant at a fixed shower energy (shower-to-shower fluctuations of the tracklength are small). Even this can be seen in Fig. 9 right panel of Cillis et al. to use the same reference as the authors used in their arguments. The shower tracklength at a fixed energy is due to low-energy physics at a the few MeV scale, and it is unaffected by the LPM effect.
As a consequence:
The conclusion: "The LPM effect is found to modify the low-frequency emission" is NOT correct.
Showing an enhacement of a factor larger than 5 at frequencies between 1 and 100 MHz in the field at the Cherenkov angle of a 10 PeV shower with LPM effect, with respect to that in a 10 PeV
shower without the LPM effect (left panel of Fig. 9 of the current version of the paper) is NOT correct.
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