How Long Does the Hydrogen Atom Live?

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In my opinion, this is a simple relevant paper that deserves to be published. The references need to be improved, in the sense that many appear as ? probably due to some LaTeX bug. Similarly, LaTeX bugs that produce wrong symbols such as "CEfl" in eq. (2) and line 117 etc have to be be fixed.

Author Response

We thank the referee for their report and will fix the latex formatting errors.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript is mainly on the exotic decays of Hydrogen induced by  dark particles, and relevant exotic neutron decays. This is very interesting for the beyond standard model community, in light of the theoretical and experimental efforts to look for the signatures of baryon number violation and proton decays. I would recommend publication at Universe, if the authors could address the following minor issues:

1. I could not understand how the charge-violating decay of electron e-> \gamma + missing energy can be used to set limit on Hydrogen lifetime [cf. line 53-55]. The authors should explain it more clearly.

2. The subscript for \Gamma in Eq.(2) below line 48 and “H->OEfl” in line 117 might be a typo. The authors should correct it.

3. The authors should fix the question marks for the references.

Author Response

We thank the referee for their report. Below are our responses to their points.

1. The signature at Borexino from the H decay that we study is simply the injection of a photon with a spread in energy determined by the mass of the final state particle(s). This is similar to the final charge-violating electron decay searched for in Ref. [5]: a monochromatic photon with an energy determined by the final state that the electron decays into. Since Ref. [5] provided the observed spectrum along with the expected signal, we could perform a conservative recasting of their data to search for the same final state with a slightly different spectrum (conservative because we could set limits by assuming all the photons seen were due to our signal).

2-3. We will fix the typos mentioned.

Reviewer 3 Report

Comments and Suggestions for Authors

This paper explores the possibility that the atomic hydrogen may be unstable and could decay into other particles, such as photons, dark matter candidates, and etc. This process can be described by some EFT models, where the corresponding coupling parameters can be constrained by the Borexino data. The claims made by the authors are novel and interesting. This paper is also clearly and logically organized. In my opinion, this paper is suitable for publication in Universe.

Although the paper is suitable for publication in this journal, I would suggest that the authors consider or address the following concerns: 

(1)

On page 6, in the bottom paragraph, the authors argue that the Borexino data can probe Lambda as large as “~100–1000 PeV”, which, I think, is a very high energy. On page 4, Lambda is defined at the hadron-level in Eq. (8). If you analyze the process at the quark level, Eq.(8) needs to be replaced by a dimension-9 operator rather than the dimension-6 operator. Then, the corresponding energy scale at the quark level would be very different from the coupling parameter (i.e. Lambda) presented by the authors at the hadron level. Therefore, the derived bounds on the magnitude of the coupling parameter (e.g. Lambda) depend on whether you evaluate them at the quark level or at the hadron level. It might be helpful for the reader if the authors explain what the physical meaning of Lambda is. Is it corresponding to an energy scale of new physics e.g. a mass of a new boson, or just a meaningless parameter? 

(2)

In this paper, the limit on the atomic hydrogen lifetime is obtained indirectly based on the parameter f_mol defined in Eq. (12) on page 5. If I understand correctly, the initial limit is actually imposed on the proton and electron as a whole contained in some compounds such as the water molecule, and etc. Then, such an initial limit is translated to the limit on the atomic hydrogen using the parameter f_mol. It would be helpful if the authors explain more about the reason why the atomic hydrogen is so special that the limit imposed on the proton and electron as a whole contained in the compounds need to be translated to the limit on the atomic hydrogen. In principle, the derived limit on the lifetime can be translated to any other molecules that contain the proton and electron by defining an appropriate f_mol. Similarly, I also wonder whether or not such a limit can be interpreted as the limit on the lifetime of the compounds used in the experiments e.g. the lifetime of the water molecule? 

(3)

On page 5, the parameter f_mol defined in Eq. (12) is called “the reduction in the probability of finding the electron at the location of the proton in the molecular state than that in atomic hydrogen” and the authors assume that it takes the value 0.5. However, to my knowledge, for some molecules containing heavy elements, the probability of finding the electron at the location of the proton or nucleus might be higher than that in atomic hydrogen due to a stronger attractive force from the highly charged nucleus of the heavy element. Therefore, for heavy molecules, the derived limit on the lifetimes could be weaker. Would it give rise to the instability of matter that might contradict the present experimental observation?

(4)

Although the title of the paper is attractive and interesting, it only provides very limited information on the content of the paper.

 

 

 

 

Author Response

We thank the referee for their report. Below are our responses to their points.

1. This is a helpful point. If one calls the scale of the dimension-9 operator at the quark level M, it is related to \Lambda via M^5~(0.01 GeV^3 \Lambda^2) so that \Lambda~100 PeV corresponds to M~250 GeV. We will mention this in an updated version of the draft.

2/3. The probability for the (exotic) electron capture decay depends on the value of the electron wavefunction at the position of the proton as is the case for all electron capture decays. In atomic hydrogen this is a standard undergraduate calculation. Molecular H or H in an organic compound has a different value of the (hybridized) electron wavefunction at the position of the proton. The proportional change in this relative to atomic hydrogen is what we call f_mol. This is in general something that can be calculated in quantum chemistry. In hydrocarbons this is generally a number of O(1) since they're covalently bonded. Our ref. [17] contains computations of this in a simple hydrocarbon, finding a value larger than what we normalize on (0.5). Since this only enters the limit on the mixing angle as the 1/2 power we don't expend a large effort on determining the precise value of this in pseudocumene but we show the dependence of the result on this value that can be computed by quantum chemists.

The limits on the atomic hydrogen lifetime derived from the limits on the lifetime of a hydrogen-containing compound would depend on the value of f_mol for that compound. Some heavier hydrocarbons than pseudocumene considered here could have similar (or larger) values of f_mol. However, of course one would have to instrument an experimental volume with said compound as well as Borexino did with p-c. As for the stability of other matter, the strongest limits come from the (relatively weakly bound) Be-9 atom. We have shown limits that come from the stability of Be-9 in Figs. 3 and 4. Because of the nuclear binding energy the reach in mass is less than from hydrogen. The nuclear binding energy of other stable matter makes their mass reach even less than that of Be-9.

Reviewer 4 Report

Comments and Suggestions for Authors

Comments for author File: Comments.pdf

Author Response

We thank the referee for their report. Below are our responses to their points.

1. The parameter \lambda could be a complex number. However this is a CPT invariant QFT so that anti-H has the same total lifetime as H in the models we consider. The intrinsic lifetime of the  anti-H made and stored at ALPHA can be used to set a limit on the parameters we consider. In practice this limit is much weaker than those inferred from the Borexino search we recast.

2. We will add the +h.c. to make the hermiticity manifest.

3. We have made a notational choice to set the Wilson coefficient to 1 in Eq. (8) in our definition of \Lambda. We note that if we write the operator in (8) at the quark level as a dim-9 operator with scale 1/M^5 (which modulo couplings corresponds to the exchange of a particle of mass M) then M is related to \Lambda via M^5~(0.01 GeV^3 \Lambda^2) so that \Lambda~100 PeV corresponds to M~250 GeV. We will mention this in an updated version of the draft.

4. We obtained this quenching factor and FWHM by using the values taken from Ref. [5], where they show the effect of a monochromatic photon of 256 keV in Fig. 1; the peak is shifted lower by the factor 0.86 and is well fit by a gaussian with FWHM that we quote with a rate modulated by the efficiency we use. We will mention this in the draft. It is reasonable to assume that these numbers are uncertain at the less than 10% level by fitting the injected signal in Ref. [5]'s Fig. 1--the constraints that we derive are not extremely sensitive to varying these parameters in this range. As for f_mol, this carries a larger uncertainty. We used a value that is representative of a typical organic compound. These have been computed in the quantum chemistry community for many years and a more refined estimate for pseudocumene could be obtained. That is beyond the scope of our work and we note that the lifetime limit is linearly sensitive to this quantity. In the interesting region of parameter space for Borexino, its limits are ~50x stronger than the direct search in neutron decay and so changing the value of f_mol by a factor of several (which is larger than reasonable) does not change the situation qualitatively.

5. In the case of an operator of the form we consider but with the electron replaced with a muon, unfortunately muonic H would not provide very strong constraints. This is because with the larger muon rest mass, normal muon capture decays on H or other nuclei can occur (since the muon mass is much larger than the proton-neutron mass difference). Thus, although one could probe larger \chi masses, if we are looking for changes in the rate of muon capture processes where a \chi is emitted instead of a neutron, the shifts would be at the level of \theta^2 or less than a part in 10^18, well below the sensitivity of muon capture experiments.

6-8. We will fix the typos that have been pointed out here.