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An Upper Bound on Atomic-Structure Corrections to the Germanium X-Ray Bound on CSL

R. E. W. Kho

Published Sep 5, 2026• 0 citations

Abstract

The strongest experimental bound on white, mass-proportional continuous spontaneous localization (CSL) at the canonical correlation length rC = 10-7 m is the Majorana Demonstrator's 2022 search for collapse-induced X-ray emission from germanium (Phys Rev Lett 129, 080401). That bound was derived assuming the emission rate scales as 1/E with the coherent-nucleus-plus-electron amplitude S = Np2 + Ne. In 2024, Piscicchia and colleagues showed the true atomic-system emission spectrum departs from 1/E in exactly the band the Majorana search used, and called for the Majorana analysis to be reconsidered. This paper bounds that departure two ways. First, an assumption-free bound: the two k-dependent terms of the emission strength are bounded separately by the true extrema of sinc(x) (minimum -0.2172, not -1), giving 0 ≤ S(k) ≤ Np2+Ne2+2|sincmin|NpNe for every photon momentum and every atomic configuration. For germanium (Np = Ne = 32) this caps any atomic-structure correction at a factor 2.361 (not the looser 3.879 obtained from the naive triangle inequality, which is not attainable): the tightest conceivable corrected ceiling is 2.09 × 10-15 s-1, and the GRW parameter point survives it by a further factor 20.9. Second, a Hartree-Fock argument, given hypotheses stated explicitly in the paper, that within that structure the correction can only weaken the bound, not tighten it. Under the best currently published spectrum (Manti, Bortolotti, Diósi, Piscicchia and Curceanu 2026, arXiv:2608.07205), GRW survives by 79-87 times, against a published margin of 49.4 times. All arithmetic is executed in VERIFY.py and reproduced in VERIFY-OUTPUT.txt. Combined with a scoped 34-source census of the field's other constraint classes, the result is that the white-CSL window at rC = 10-7 m remains open and is bounded against future spectrum corrections by a proved ceiling, strengthened by a hypothesis-bearing argument for its sign.

DOI: 10.5281/zenodo.22314149

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The strongest experimental bound on white, mass-proportional continuous spontaneous localization (CSL) at the canonical correlation length rC = 10-7 m is the Majorana Demonstrator's 2022 search for collapse-induced X-ray emission from germanium (Phys Rev Lett 129, 080401). That bound was derived assuming the emission rate scales as 1/E with the coherent-nucleus-plus-electron amplitude S = Np2 + Ne. In 2024, Piscicchia and colleagues showed the true atomic-system emission spectrum departs from 1/E in exactly the band the Majorana search used, and called for the Majorana analysis to be reconsidered. This paper bounds that departure two ways. First, an assumption-free bound: the two k-dependent terms of the emission strength are bounded separately by the true extrema of sinc(x) (minimum -0.2172, not -1), giving 0 ≤ S(k) ≤ Np2+Ne2+2|sincmin|NpNe for every photon momentum and every atomic configuration. For germanium (Np = Ne = 32) this caps any atomic-structure correction at a factor 2.361 (not the looser 3.879 obtained from the naive triangle inequality, which is not attainable): the tightest conceivable corrected ceiling is 2.09 × 10-15 s-1, and the GRW parameter point survives it by a further factor 20.9. Second, a Hartree-Fock argument, given hypotheses stated explicitly in the paper, that within that structure the correction can only weaken the bound, not tighten it. Under the best currently published spectrum (Manti, Bortolotti, Diósi, Piscicchia and Curceanu 2026, arXiv:2608.07205), GRW survives by 79-87 times, against a published margin of 49.4 times. All arithmetic is executed in VERIFY.py and reproduced in VERIFY-OUTPUT.txt. Combined with a scoped 34-source census of the field's other constraint classes, the result is that the white-CSL window at rC = 10-7 m remains open and is bounded against future spectrum corrections by a proved ceiling, strengthened by a hypothesis-bearing argument for its sign.

The Majorana Demonstrator neutrinoless double-beta decay experiment comprises a 44 kg (30 kg enriched in ^{76}Ge) array of p-type, point-contact germanium detectors. With its unprecedented energy resolution and ultralow backgrounds, Majorana also searches for rare event signatures from beyond standard model physics in the low energy region below 100 keV. In this Letter, we test the continuous spontaneous localization (CSL) model, one of the mathematically well-motivated wave function collapse models aimed at solving the long-standing unresolved quantum mechanical measurement problem. While the CSL predicts the existence of a detectable radiation signature in the x-ray domain, we find no evidence of such radiation in the 19-100 keV range in a 37.5 kg-y enriched germanium exposure collected between December 31, 2015, and November 27, 2019, with the Demonstrator. We explored both the non-mass-proportional (n-m-p) and the mass-proportional (m-p) versions of the CSL with two different assumptions: that only the quasifree electrons can emit the x-ray radiation and that the nucleus can coherently emit an amplified radiation. In all cases, we set the most stringent upper limit to date for the white CSL model on the collapse rate, λ, providing a factor of 40-100 improvement in sensitivity over comparable searches. Our limit is the most stringent for large parts of the allowed parameter space. If the result is interpreted in terms of the Diòsi-Penrose gravitational wave function collapse model, the lower bound with a 95% confidence level is almost an order of magnitude improvement over the previous best limit.

This memo addresses a question: In the “substrate field–soliton” model (v2 paper), canthe intuition “observation collapse = irreversible physical process” be realized as a concretedynamical collapse model? The answer is yes, but with a decisive number attached. Weprove that: (i) The master equation driven by the noise naturally derived from the masscoupling m(ϕ) is structurally isomorphic to the CSL (Continuous Spontaneous Localization) model, and the two ad hoc assumptions of CSL—mass-weighted coupling and positionbasis—are no longer assumptions in this model, but corollaries of the equivalence principleand the mass density operator; (ii) The collapse rate is proportional to Newton’s constant,Γ ∝ Gm2, realizing the motivation of the Di´osi–Penrose program that “collapse strengthis given by gravity,” though the mechanism is different (substrate noise rather than spacetime superposition instability); (iii) Decisive verdict: If the noise is taken as the quantumvacuum fluctuation of ϕ, the single-nucleon collapse rate is ∼ 10−28 s−1, which is 12 orders of magnitude smaller than the value required by GRW—the pure quantum fluctuationversion cannot solve the measurement problem. To achieve true collapse, “intrinsic probability” must be interpreted as a classical true-random component of the substrate condensate,whose power D becomes a new free parameter. In this case, the model becomes a CSL witha specific mediating field, and its heating rate directly interfaces with existing experimentalconstraints (X-ray spontaneous emission, cantilever noise, LISA Pathfinder). We presenta complete parameter space analysis, including the “localization-heating squeeze” identityP/λ = ℏ2/(2m0rC2), which explains why models with rC ∼ 1 fm are strictly ruled out, whilethose with rC ∼ 10−7 m survive.

In 2021, a low-background experiment at the Gran Sasso National Laboratory searched for spontaneous radiation predicted by the standard Markovian Diósi-Penrose (DP) dynamics and obtained the new bound

R_0 > 5.4 \times 10^{-11}\,\mathrm{m}

at

95\%

probability for the short-length cutoff parameter. For germanium at the experimental temperature, the parameter-free prescription identifying

R_0

with the spatial quantum spread of a nucleus gives

R_0 \simeq 5 \times 10^{-12}\,\mathrm{m}

. That particular combination, Diósi's stochastic dynamics and the Penrose-motivated mass-density resolution, was therefore excluded. Subsequent searches strengthened the cutoff bound. The Majorana Demonstrator improved it in 2022, and XENONnT has now obtained the world-leading limits

R_0 > 4.9 \times 10^{-10}\,\mathrm{m}

at

90\%

confidence level and

R_0 > 4.5 \times 10^{-10}\,\mathrm{m}

at

95\%

confidence level. These advances reinforce the 2021 exclusion but do not alter its logical scope: they \emph{parametrically constrain} the Markovian, effectively white-noise DP dynamics, not Penrose's (and Diósi's) lifetime criterion independently of a dynamics, and not all colored-noise, dissipative, or otherwise modified gravity-related spontaneous collapse theories. We reconstruct what Gran Sasso did and did not rule out and place that conclusion in the present experimental context.

We study lower and upper bounds on the parameters for stochastic state vector reduction, focusing on the mass-proportional continuous spontaneous localization (CSL) model. We show that the assumption that the state vector is reduced whan a latent image is formed, in photography or etched track detection, requires a CSL rate parameter

λ

that is larger than conventionally assumed by a factor of roughly

2 \times 10^{9\pm 2}

, for a correlation length

r_C

of

10^{-5}{\rm cm}

. We reanalyze existing upper bounds on the reduction rate and conclude that all are compatible with such an increase in

λ

. The best bounds that we have obtained come from a consideration of heating of the intergalactic medium (IGM), which shows that

λ

can be at most

\sim 10^{8\pm 1}

times as large as the standard CSL value, again for

r_C=10^{-5} {\rm cm}

. (For both the lower and upper bounds, quoted errors are not purely statistical errors, but rather are estimates reflecting modeling uncertainties.) We discuss modifications in our analysis corresponding to a larger value of

r_C

. With a substantially enlarged rate parameter, CSL effects may be within range of experimental detection (or refutation) with current technologies. ADDED NOTE: A careful calculation of radiation from atomic systems in the CSL model (Adler and Ramazanoglu, arXiv:0707.3134) has reinstated the bound obtained by Fu as the best upper bound on lambda, with significant implications for CSL model phenomenology.

In this paper new upper limits on the parameters of the Continuous Spontaneous Localization (CSL) collapse model are extracted. To this end the X-ray emission data collected by the IGEX collaboration are analyzed and compared with the spectrum of the spontaneous photon emission process predicted by collapse models. This study allows to obtain the most stringent limits within a relevant range of the CSL model parameters, with respect to any other method. The collapse rate

\lambda

and the correlation length

r_C

are mapped, thus allowing to exclude a broad range of the parameter space.

We calculate the photon emission rate from a general atomic system in the mass-proportional CSL model. For an isolated charged particle emitting kilovolt gamma rays, our results agree with those obtained by Fu. For a neutral atomic system, photon emission is strongly suppressed for photon wavelengths much larger than the atomic radius. However, for kilovolt gamma rays, Fu’s result is modified by a structure factor that is of order unity, giving no rate suppression. Our calculation is readily generalized to the case of non-white noise, noise couplings that are not mass-proportional, and general (non-Gaussian) spatial correlation functions, and corresponding results are given. We briefly discuss the implications of our calculation for upper bounds on the CSL model parameters. 1