Relativistic Atomic Effects of Dark Matter Electron Scattering

Chuan-Yang Xing; Jie Sheng; Shao-Feng Ge

REVIEW 4 major objections 5 minor 1 cited by

Relativistic Atomic Effects of Dark Matter Electron Scattering

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Treating xenon electrons as relativistic bound states lowers dark-matter scattering rates by 30–50%.

desk verdict The QFT reframing is worth taking seriously, but the headline 30–50% relativistic suppression is not established because the two calculations use different effective charges. read the letter →

arxiv 2509.14534 v2 pith:MSMSPBJP submitted 2025-09-18 hep-ph hep-ex

Shao-Feng Ge , Jie Sheng , Chuan-Yang Xing This is my paper · ORCID

classification hep-phhep-ex

keywords darkmatterelectronscatteringatomicformfactorboundelectronsrelativisticeffectsDiracequationxenondirectdetectionsub-GeVK-factor

topics Dark Matter

open problems Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the standard factorization of dark-matter–electron scattering—a free-electron matrix element times an atomic form factor—is not self-consistent, because it imposes free-electron kinematics and phase space on electrons that are bound and off-shell, and in part of parameter space it even yields negative differential cross sections. Starting from quantum field theory, the authors quantize the electron directly in the atomic Coulomb potential, so the initial and final electron states are exact bound and ionized solutions rather than plane waves, and derive a cross section whose atomic content lives in a relativistic K-factor. They evaluate that K-factor for scalar dark-matter interactions in xenon with both Schrödinger and Dirac wave functions. The relativistic treatment reduces the phase-space ratio and differential cross section by 30–50%, an effect that grows with the effective nuclear charge and comes mainly from the amplitude and Coulomb phase of the ionized final-state wave function. A reliable sub-GeV dark-matter search therefore needs both a consistent bound-state formalism and relativistic atomic wave functions.

What carries the argument

The object that carries the calculation is the atomic K-factor $K^S_{n\kappa}(q,\Delta E)$, the sum over final ionized states of the squared inner product of the initial bound-state and final ionized-state electron wave functions with the momentum-transfer plane wave. In the non-relativistic limit this reduces to a sum over overlaps of Schrödinger wave functions, while the relativistic version uses four-component Dirac spinors and is simplified with the Wigner–Eckart theorem into a double sum over $\kappa'$ and $L$, with radial integrals $R_{PP}$ and $R_{QQ}$ over the large and small components. The companion quantity is the phase-space ratio $R(T_r) = \int |q|\, K(q,\Delta E)\,d|q| / (2m_e T_r)$, which measures how far the atomic response is from a free electron at rest and is the quantity that drops by 30–50%. The numerical bottleneck, highly oscillatory radial integrals coming from the confluent hypergeometric function in the ionized wave function, is removed by switching to the integral representation of that function, which turns each radial integral into a compact sum of Gauss hypergeometric functions.

What would settle it

Compute the inner-shell K-factor for xenon using self-consistent many-body Dirac wave functions that include screening and ionic relaxation; if the 30–50% reduction relative to the Schrödinger result shrinks or vanishes, the claimed relativistic effect is an artifact of the single-particle hydrogen-like model rather than a robust prediction.

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Extended reading notes

Core claim

The central claim is that $|M|^2$ for dark-matter scattering off a bound electron does not factorize as $|M_{\rm free}|^2$ times an atomic form factor. Because the initial electron has negative binding energy and a momentum distribution, the free-electron on-shell dispersion and phase space do not apply; imposing them can make $|M|^2$ negative, for example for an N-shell xenon electron with $m_\chi = 10$ keV, $T_\chi = 10$ keV, and recoil energies between 1.5 and 5 keV. A consistent quantum field theory treatment, with electron fields quantized in the bound-state picture and final ionized states normalized by recoil energy, gives a differential cross section integrated over momentum transfer, with the atomic effects encoded in a relativistic K-factor built from Dirac wave functions. For scalar interactions in xenon, replacing the non-relativistic Schrödinger wave functions by Dirac wave functions suppresses the phase-space ratio $R(T_r)$ by 30–50% for the $s$-shell states; the suppression grows with the effective nuclear charge and is driven mainly by the reduced amplitude and altered Coulomb phase of the final ionized wave function, not by a change in the bound-state wave function.

Load-bearing premise

The calculation assumes each xenon electron is a single particle in a hydrogen-like Coulomb potential with a shell-dependent effective charge, ignoring electron–electron screening, correlations, and the rearrangement of the ion left behind; if those many-body effects change the inner-shell wave functions, the size of the claimed relativistic reduction could be different.

Editorial extensions

If this is right

Existing XENON-style exclusion limits on sub-GeV dark matter that use non-relativistic Schrödinger form factors would loosen by roughly the same 30–50%, because the true bound-electron cross section is smaller than assumed.
The unphysical negative differential cross sections that appear for fast incoming dark matter, such as cosmic-ray-boosted dark matter, disappear in the consistent formalism, so the full parameter space becomes usable.
Because the suppression grows with effective nuclear charge, inner-shell ionization in high-$Z$ targets like xenon is affected most, while outer shells and lighter atoms remain closer to the non-relativistic prediction.
The phase-space ratio returns to unity at high recoil energy, so relativistic atomic corrections matter mainly in the low-energy window relevant to sub-GeV dark matter rather than for very energetic electron recoils.
The same bound-state formalism applies to other neutral projectiles that ionize atomic electrons, notably neutrinos, so predicted solar-neutrino and supernova-neutrino electron-recoil rates in xenon inherit the same relativistic correction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the growth of the suppression with effective nuclear charge suggests that even heavier targets such as gold or lead would show a larger relativistic reduction in inner-shell atomic factors; a cross-element scan with the same formalism would test this scaling.
Beyond the paper, only scalar interactions are demonstrated; for vector or axial-vector couplings the electron bilinear does not reduce to a single scalar overlap, so the relativistic correction could differ in size or sign, and applying the same Wigner–Eckart machinery to those couplings is a natural next step.
Beyond the paper, a self-consistent many-body treatment of xenon could shift the quoted 30–50% number, since screening and ionic relaxation affect exactly the inner-shell wave functions where the relativistic effect is largest.
Beyond the paper, the same phase-shift mechanism should be visible in neutrino–electron scattering in liquid xenon, where the predicted recoil spectrum near threshold would fall below the Schrödinger-based prediction by a measurable factor.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the formal part is the real contribution. The claim that the usual free-electron factorization can produce negative cross section is correct and worth having; the Furry-picture quantization and the Wigner-Eckart reduction of the K-factor are worked out carefully. I would send this to a referee. But I would not let the abstract keep the 30–50% number until the authors isolate relativity from model recalibration.

The negative-cross-section pathology in Sec. 2 is real. They take the free-electron matrix element, impose bound-electron kinematics, and show a large region where |M|^2 goes negative. Anyone using the standard factorization should know this. The new formalism in Secs. 3–4 is also mostly sound: quantize the electron in the atomic potential, keep only the energy-conserving delta, use the ionized-state phase space, and recover the usual atomic factor in the non-relativistic limit. The analytic trick for the oscillatory radial integrals using the integral representation of 1F1 is useful and seems correct as far as I can check.

The relativistic suppression itself is not entirely new—Roberts et al. and Bloch et al. already reported reduced rates with Dirac wave functions—but the explicit demonstration of the pathology and the systematic reduction of the integrals are new elements. Credit where it is due.

Now the soft spot. The headline number is computed as a ratio of K-factors, not from the full differential cross section, and the relativistic and non-relativistic wave functions are not evaluated in the same potential. For 5s they quote Z_eff = 6.87 for Dirac and 6.5 for Schrödinger, with correspondingly different binding energies. They choose effective charges to match atomic data, so the comparison mixes a real relativistic phase shift with a 5–6% change in the potential. The paper itself notes that the suppression grows with Z_eff. That means the “relativistic effect” is not cleanly separated from recalibration. A clean calculation would use identical Z_eff for both wave functions, or at least show the R(Tr) comparison at fixed Z_eff. Until that is done, 30–50% is a model-dependent estimate, not a demonstrated relativistic correction.

There is also a mismatch in Fig. 4: the caption says n=3 for the right panel while the text describes 1s and 2s. That is minor, but it makes reconstruction harder. There is no code or data, and no uncertainty estimate, which is consistent with the paper being a formalism paper, but it does mean the reader cannot independently check the numerical integrals.

Who is this for? Direct detection theorists, especially those who use QEdark/DarkARC-type atomic factors for sub-GeV DM. They should read Sec. 2 and the formalism; they should be careful with the suppression estimate. I would accept for peer review, but insist on the fixed-Z_eff comparison and the full differential cross section before publication.

Referee Report

4 major / 5 minor

Summary. This paper argues that the standard factorization of DM-bound-electron scattering into a free-electron matrix element times an atomic form factor is inconsistent, because imposing free-electron kinematics on bound electrons can produce negative squared matrix elements (Sec. 2). It then develops a QFT formalism in the Furry picture, quantizing electrons in the atomic Coulomb potential (Sec. 3), derives the cross section, recovers the non-relativistic factorization in the appropriate limit (Sec. 4), and computes scalar-interaction atomic K-factors for xenon using Dirac and Schrödinger hydrogenic wave functions (Sec. 5). The authors report a 30%–50% reduction in the phase-space ratio R(T_r) when relativistic wave functions are used, and conclude that a relativistic treatment is necessary.

Significance. If the quantitative claim is established, the paper provides a cleaner theoretical basis for DM-electron scattering calculations and a practical method for evaluating relativistic atomic form factors; the formal QFT derivation, the Wigner-Eckart reduction of the K-factor, and the analytic transformation of the radial integrals are useful contributions. The paper is also explicit about the negative-cross-section pathology of the factorized treatment, which is an important caution for the field. However, the headline 30%–50% reduction currently rests on a model-dependent comparison that is not fully controlled, so the significance is somewhat tempered until the quantitative claim is isolated and validated.

major comments (4)

[Secs. 5.3–5.4, Eq. (5.30), Fig. 4] The comparison supporting the 30%–50% reduction is not controlled with respect to the effective charge. The Dirac and Schrödinger bound states are computed with different Z_eff values (e.g., Z_nκ=6.87 vs Z_nl=6.5 for 5s, quoted in Sec. 5.3), and Fig. 3 (right) shows that the relativistic phase shift grows with Z_eff. If both Z values are calibrated to the same empirical binding energy, the comparison uses different Coulomb potentials and therefore conflates the relativistic treatment of the electron with the different effective charge required by the hydrogenic model. Please provide a control calculation at identical Z_eff for the two wave-function sets, or explicitly quantify how much of the reported 30%–50% reduction is attributable to the Z_eff difference rather than to Dirac vs Schrödinger dynamics.
[Sec. 5.4, Eq. (5.2), definition of R(T_r)] The headline reduction is extracted from the K-factor-only ratio R(T_r)=∫|q|K(ΔE,q)d|q|/(2m_e T_r), not from the full differential cross section in Eq. (5.2), which contains |M_χ^{SS}|^2 D_SS^2 inside the q integral. Since the DM spinor factor and the mediator propagator depend on q, the reduction in dσ/dT_r is not necessarily 30%–50% unless those factors are effectively q-independent over the integration range. The abstract and conclusion generalize beyond what is actually computed. Please compute the full cross-section ratio for a representative mediator mass, or state explicitly the conditions under which it reduces to the K-factor ratio.
[Fig. 4 and Fig. 2] The central quantitative figures are internally inconsistent. The Fig. 4 right-panel caption says n=3, l=0–3, while the text describes the panel as showing 1s and 2s sub-shells with varying Z_eff; the Fig. 2 caption quotes T_r=24 keV while the text says T_r=50 keV. Because these figures carry the paper's main quantitative claim, the captions and the text must be reconciled and the exact states and Z_eff values used in each panel must be specified unambiguously.
[Secs. 4.2 and 5.2, hydrogenic approximation] The calculation assumes that each xenon electron is a single electron in a hydrogen-like Coulomb potential with a shell-dependent Z_eff, and it does not include electron-electron screening beyond the effective charge, final-state relaxation, or many-body correlations. If the 30%–50% reduction is intended as a xenon-specific prediction, the paper should justify this approximation for inner shells and compare with existing many-body or RPA calculations (e.g., Refs. [38,68]); if the result is meant as an illustration of the formalism, that limitation should be stated prominently in the abstract and conclusion.

minor comments (5)

[Eq. (5.29a)] The spinor trace is written as Tr[\bar u(me)u(me)\bar u(me)u(me)], while the subsequent text switches to the projection u(me)\bar u(me) and γ^0. Please align the notation and define the spinor normalization convention explicitly.
[Sec. 5.2 heading] There is a typo: 'Comloub' should be 'Coulomb' in the heading of Sec. 5.2.
[Sec. 5.4] The phrase 'our precious work [58]' should read 'our previous work [58]'. There are also typos such as 'qunatum' near Eq. (5.30b).
[Fig. 2 caption] The color assignment for the ionized wave functions is inconsistent between the caption (T_r=24 keV) and the main text (T_r=50 keV); please unify.
[General] No numerical tables or code are provided for the K-factors or the phase-space ratios. Given the nontrivial radial integrations, a small table of R(T_r) values or a code repository would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QFT formalism and K-factor comparison are self-contained; the 30-50% reduction is a direct numerical output, not a fitted or self-cited input.

full rationale

The paper's derivation is self-contained. Section 3 constructs the cross section from the QFT overlap definition, with bound and ionized electron fields quantized in the Coulomb potential; Eqs. (5.2)-(5.3) define the relativistic K-factor directly from the spatial wave functions. The non-relativistic comparison in Eqs. (4.16)-(4.18) and (5.29a)-(5.30a) is independently re-derived, not assumed from the cited literature. The 30-50% reduction in R(Tr) shown in Fig. 4 is a numerical output of the Dirac versus Schrodinger radial integrals; it is not used to set any effective charge or other parameter. Effective charges are empirical inputs calibrated to binding energies (Eq. (5.13), WebElements), not to the target ratio, so the central claim is not a fitted result renamed as a prediction. Self-citations such as [58, 73, 95] supply context and the phase-space-ratio diagnostic, but the load-bearing equations appear in this paper; no uniqueness theorem or ansatz is imported from the authors' prior work. The separate Z_eff calibration for Dirac and Schrodinger hydrogen-like orbitals is a modeling-accuracy concern, not a circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a single-particle hydrogen-like atomic model with empirical effective charges, and on standard angular-momentum algebra. No new particles or forces are introduced.

free parameters (1)

Effective nuclear charge Z_nκ (or Z_eff) = Varies per shell, e.g., 6.87 for Xe 5s (relativistic) and 6.5 (non-relativistic); other shells from WebElements
Input atomic parameter used to define the hydrogen-like Coulomb potential and wave functions in Sec. 5.2. It determines binding energies and hence the K-factor curves in Figs. 2-4.

assumptions (4)

domain assumption The atomic electron is described by a single-particle Dirac (or Schrödinger) equation in a hydrogen-like Coulomb potential with an effective charge Z_nκ; electron-electron interactions and many-body screening are neglected.
Invoked in Sec. 5.2 (Eqs. 4.8-4.11) and used for all numeric results. The paper labels this the hydrogen-like approximation [105]. Real xenon has 54 electrons, so this is a strong simplification.
domain assumption The ionized final-state electron experiences the same effective charge as the initial bound state (Z_niκi), with no relaxation of the remaining ion.
Stated in Sec. 5.2.2: 'For an electron that is ionized from the initial state |n_i,κ_i>, it should share the same effective charge Z_niκi as the initial state...'. This ignores final-state screening changes.
standard math Standard quantum mechanics theorems: Wigner-Eckart theorem, Wigner-3j orthogonality, spherical harmonic addition, and the analytic integral formula for ∫ |r|^α e^{-βr} j_L(qr) dr expressed as a Gauss hypergeometric function.
Used in Secs. 5.1 and 5.2.3 to reduce the K-factor; these are textbook results [93,106,107].
domain assumption The DM-electron interaction is assumed to be scalar (ΓS = 1) for the concrete K-factor calculation, while the factorization critique in Sec. 2 uses a vector interaction example.
The quantitative reduction is computed only for scalar interactions; extension to other Lorentz structures is asserted but not demonstrated.

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Cite this review

Pith. "Pith review of Relativistic Atomic Effects of Dark Matter Electron Scattering." pith.science (2026). https://pith.science/paper/MSMSPBJP

@misc{pith2026250914534,
  author       = {Pith},
  title        = {Pith review of: Relativistic Atomic Effects of Dark Matter Electron Scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSMSPBJP}},
  note         = {Machine review of arXiv:2509.14534}
}

abstract

The dark matter scattering with atomic bound electrons is a crucial avenue for exploring the sub-GeV mass range. The commonly used factorization, where atomic effects are encoded in an overall form factor multiplying the free-electron scattering matrix element, is not necessarily true. Especially, the free-electron kinematics and phase space cannot consistently apply for off-shell bound electrons. Starting from the first principles of quantum field theory, we establish a theoretically consistent formalism to account for the atomic effects. By taking the scalar-type interaction as an example, we investigate the difference between the non-relativistic and relativistic calculations to show that the relativistic effects can lead to a $30\% \sim 50\%$ reduction in the scattering phase space and differential cross section. In other words, not just a theoretically consistent formalism for the atomic effects but also relativistic calculation with Dirac equation are necessary.

Figures

Figures reproduced from arXiv: 2509.14534 by the authors.

**Figure 2.** The bound (left panel) and ionized (right panel) electron wave functions for typical quantum states. [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** (Left) Ionized electron wave functions with the same quantum numbers ( [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The comparison of the phase space ratios [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]

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