# Muonic Lithium atoms: nuclear structure corrections to the Lamb shift

### Submission summary

 As Contributors: Simone Salvatore Li Muli Arxiv Link: https://arxiv.org/abs/1910.14370v2 (pdf) Date accepted: 2019-12-20 Date submitted: 2019-12-17 01:00 Submitted by: Li Muli, Simone Salvatore Submitted to: SciPost Physics Proceedings Proceedings issue: 24th European Few Body Conference (University of Surrey, U.K.) Discipline: Physics Subject area: Nuclear Physics - Theory Approaches: Theoretical, Computational

### Abstract

In view of the future plans to measure the Lamb shift in muonic Lithium atoms we address the microscopic theory of the $\mu$-$^6$Li$^{2+}$ and $\mu$-$^7$Li$^{2+}$ systems. The goal of the CREMA collaboration is to measure the Lamb shift to extract the charge radius with high precision and compare it to electron scattering data or atomic spectroscopy to see if interesting puzzles, such as the proton and deuteron radius puzzles, arise. For this experiment to be successful, theoretical information on the nuclear structure corrections to the Lamb shift is needed. For $\mu$-$^6$Li$^{2+}$ and $\mu$-$^7$Li$^{2+}$ there exist only estimates of nuclear structure corrections based on experimental data that suffer from very large uncertainties. We present the first steps towards an ab initio computation of these quantities using few-body techniques.

Published as SciPost Phys. Proc. 3, 028 (2020)

### List of changes

We thank the referee for his valuable comments. We would like to address his points below.

1) In our first paragraphs we describe the history of the proton radius puzzle, not only what it is
today. We do not feel that stating that the proton radius puzzle is only a disagreement of the
Paris measurement with the rest is fair, because there are electron scattering experiments from
Mainz that measured a large radius. As nuclear physicists we want to mention them and we
think it is important to understand why they give a large radius. as opposed to the small radius
The PRAD Nature paper was published on Nov 6th, while we submitted the proceedings on
Nov 1st. That is why we did not include it in the figure. Of course, we were aware of these data
(as preliminary data) but we did not want to include unpublished data, especially when they are
from other collaborations. Now that the paper is published, we added the PRAD point to Fig. 1.
We also modified the paragraph:
"New experiments are also being performed. These account for precise measurements of
electron-proton at low momentum transfer, e.g., the Proton Radius (PRad) experiment at JLab
[11] and the muon-proton scattering experiment (MUSE) being commissioned at PSI [12]“
to
"New experiments were performed or are being performed. These account for precise
measurements of electron-proton at low momentum transfer, e.g., muon-proton scattering
experiment (MUSE) being commissioned at PSI [11] and the Proton Radius (PRad) experiment
at JLab [12], that recently measured a small radius, consistent with the muonic atom results.
Furthermore, new ...”

2) We corrected to 2S-2P_1/2.

3) We do not consider the deuteron radius puzzle an independent puzzle, but it has been
coined "deuteron radius puzzle" in Ref.[20], so we think we can call it this way. However, to
take into account the referee's point we substituted the sentence:
" Recent laser spectroscopy experiments in muonic deuterium (μ-2 H ) led to the discovery of
the "deuteron radius puzzle" [20], which is rather similar to the proton radius puzzle."
=>
" Recent laser spectroscopy experiments in muonic deuterium (μ-2 H ) led to the discovery of
the "deuteron radius puzzle" [20], which is rather similar to, but not independent from, the

4) We want to be consistent with the notation of the our recent JPG review (Ref.[22]), so we call
it \delta_TPE. Of course the Coulomb term is not of order (Z\alpha)^5, and the referee is correct
in saying that it is not clearly explained. To amend, right after Eq.(4), we have added the
following sentence:
“ The above terms are all of order $(Z\alpha)^5$, but the Coulomb term (\delta_C^{(0)}) which is
logarithmically enhanced to $(Z\alpha)^6\log(Z\alpha$). We include it in our $\delta_{\text{TPE}}$,
consistently with Ref. [22], where it is also possible to find a full compilation and derivation of these
expressions.”

5) We did not include the expression of \delta_N because it is not the main point of the
proceedings. However, to be more clear we added a footnote:\footnote{Expressions of \delta_N can be found in Eq. (3a),(105) and (106) of Ref.[22].}

6) We corrected for that by moving the definitions of \beta and \lambda after Eq. (14). The squared
neutron charge radius is negative, even though the neutron is neutral, because it can be pictured as
a proton with a negative pion cloud that extends further at larger distance, so that the second
moment of the charge distribution is slightly negative.

"From Eq.(1) it is clear that $\delta_{\text{Z1}}^{(1)}$ and $\delta_{\text{Z3}}^{(1)}$ cancel out
when
considering the total TPE correction, however we still compute them with the purpose of
comparing to Ref. [28], who has estimated only the inelastic part of $\delta_{\text{TPE}}$."

8) We have added the following sentence with 4 new citations, before Section 2:
“Nuclear structure corrections have been studied by various groups, see, e.g., Refs. [23-26].”

We have performed one other change:
* Corrected the value of \eta to 0.33

We hope that now the manuscript can be accepted for publication as a proceedings of the few-
body conference 2019 in Surrey.

With best regards,
Simone, Anna and Sonia