Importance of the X ray edge singularity for the detection of relic neutrinos in the PTOLEMY project

Response to the Referee report Zhiyang Tan and Vadim Cheianov

We would like to thank the Referee for the thorough review of our manuscript titled ”Importance of the X ray edge singularity for the detection of relic neutrinos in the PTOLEMY project.” We greatly appreciate the time and effort you have invested in providing constructive feedback to improve the quality of our work. We have carefully considered each of the Referee’s points and we address them below to the best of our ability:

1) The recoil of the Daughter Nucleus (Section 2B): We do not include the impact of the recoil in our model for the sake of simplicity. The impact of the recoil has been addressed in Ref.[17] of our manuscript, and the ways to mitigate the effect of recoil have been addressed in further literature (Refs. [18-24] in the manuscript). We assume that one of the proposed mitigating solutions has been implemented and therefore we focus on the next important harmful effect, which is the shakeup of the Fermi Sea in graphene. If necessary, incorporating the zero-point motion of the nucleus in our model is straightforward - one simply has to compute the convolution of the spectrum obtained in the present manuscript with the Gaussian distribution due to the zero-point motion as described in e.g. Ref. [17].

2) Notation Uniformity (Eq. 8): The Hamiltonian $H_{D−G}$ introduced in Eq. (8) is not equivalent to the Hamiltonian Hg introduced in Eq. (5). The former is the interaction between the ion and the graphene, while the latter is the full Hamiltonian of the graphene system, which also includes the kinetic energy of the electrons of graphene. The definition of $H_{D−G}$ is in Eq. (11) and the definition of $H_g$ is in Eq. (21).

3) Definition of States (Eq. 16): We appreciate the Referee’s observation regarding the undefined states $|... \rangle_{h} $ and $|... \rangle_{t}$ in Eq. (16). To keep the uniformity and consistency, we have changed them to $|... \rangle_{D}$ and $|... \rangle_{M}$. $|0\rangle_{M, D}$ denotes a state where the isotope (mother or daughter) is absent, and $|1\rangle_{M, D} $denotes a state where the isotope is present in the state of zero kinetic energy. We have also addressed them in the main text.

4) Presence of ”i” in Eq. (52): The ”i” appearing in the numerator of Eq. (52) is not a misprint and it indeed represents the imaginary unit. The expression is a way to express the Fourier transform of a sudden external potential, which is also addressed in Ref [52] of our manuscript.

5) Estimation of the Density of Tritium (Eq. 50): We thank the Referee for suggesting a comparison of the estimated Tritium density with the standard graphene density or conversion to mass density for easier experimental comparison. We agree that this would enhance the accessibility of our results and we have included such comparisons in the revised manuscript.

6) Placement of Figures: We acknowledge the Referee’s feedback regarding the placement of figures and agree that it would improve clarity if they were placed on the same page where they are first mentioned. We have done our best to reorganize the figures accordingly, however, some of the figures still appear on a page, which is different from the page of the first mention. Such a placement is not our fault, but it is due to the internal formatting algorithms of LaTeX. 7) Relevant Time Scale for Physical Processes: The referee is, in a sense, correct in saying that all processes happening after beta decay are essentially different ways of redistributing the energy inside the graphene system. However, the statement that these processes are decoupled from the outgoing electron contains a subtle ambiguity, which may be the source of the confusion. We shall illustrate this point with the example of the neutrino capture process, although the arguments also apply to beta-decay.

Firstly, let us remark that the formation time of the beta electron is extremely short (on the order of $10^{−16 }$seconds) and the associated energy scale runs into $KeV$. Clearly, this energy scale has no relevance to the problem of the broadening of the beta-emission line in the neutrino capture process. In fact, the limit in which the formation of the beta electron is treated as instantaneous is mathematically sound and is a good approximation for the treatment of soft excitations created in the process.

Now, following the capture, which we treat as an instantaneous process, the combined graphene + daughter ion + beta-electron system emerges in a quantum superposition of states $|f\rangle= \sum_{k,λ} C_{k,λ}|k\rangle_e|λ\rangle$, where $|λ\rangle$ are Eigenstates of the Hamiltonian of the (graphene + daughter ion) system and $|k\rangle_e $are the eigenstates of the kinetic energy of the beta-electron. The states in the superposition have to obey the energy conservation law $(\hbar^2k^2/2m_e)+E_λ = const$ where the constant on the right-hand side is dictated by the mass defect difference. Due to this constraint, the tensor $C_{k,λ}$ is fundamentally inseparable, and the final state is an entangled state between the outgoing electron and the graphene system in which the kinetic energy of the beta electron is indefinite. Therefore despite being decoupled from graphene and the daughter ion in the dynamical sense, the beta electron remains entangled with this system until the projective measurement of its energy is performed in the calorimeter. This entanglement necessitates the total transition probability to be a convolution of the transition probability function of beta decay and the graphene spectral density function. The influence of the energy redistribution between the graphene system and the daughter ion on the energy spectrum is encapsulated within the spectral density function, defined by Eq. (20).

On a more general note, the effect of the finite lifetime of the products of a reaction on the energy conservation law in that reaction is well-known and described in the literature. The bottom line is that processes leading to the collisional decay of the products of a reaction, even if they happen long after the reaction itself took place, have an effect on the available phase space of the reaction due to the broadening of the ”on-shell” delta-function into a more generous spectral density function, typically a Lorentzian. See, e.g.

L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962/ Perseus Books, Cambridge, MA, 1989).

J. R. Barker, Quantum Theory of high-field transport in semiconductors. Journal of Physics C 6, 2663 (1973)

Once again, we thank you for your valuable feedback, which will undoubtedly strengthen the quality and clarity of our manuscript. We will make the necessary revisions as suggested and look forward to resubmitting the improved version for your consideration.