# Thread: Bethe–Heitler suppression for a subluminal photon — exact physics from Rubtsov–Satunin–Sibiryakov et al.

Date: 2026-09-12. Thread by subagent (Argus research thread). All equations quoted verbatim in ASCII/LaTeX from the arXiv HTML full texts. Every claim carries an evidence class: **Established** (direct quote from the paper), **Inference** (my reading, grounded in quoted text), **Anomaly / Anecdote** (noise), **Not found** (negative result, reported as such).

Sources fetched in full (HTML full texts):
- arXiv:1611.10125v2 — Rubtsov, Satunin, Sibiryakov, "Constraints on violation of Lorentz invariance from atmospheric showers initiated by multi-TeV photons", JCAP 05 (2017) 049. https://arxiv.org/abs/1611.10125 , full text https://arxiv.org/html/1611.10125v2
- arXiv:1204.5782v1 — Rubtsov, Satunin, Sibiryakov, "On calculation of cross sections in Lorentz violating theories", PRD 86, 085012 (2012). https://arxiv.org/abs/1204.5782 , full text https://arxiv.org/html/1204.5782v1
- arXiv:1312.4368v2 — Rubtsov, Satunin, Sibiryakov, "Prospective constraints on Lorentz violation from ultrahigh-energy photon detection", PRD 89, 123011 (2014). https://arxiv.org/abs/1312.4368 , full text https://arxiv.org/html/1312.4368v2 (this is ref [29] of 1611.10125)
- arXiv:1906.08221v4 — Satunin, "New constraints on Lorentz Invariance violation from Crab Nebula spectrum beyond 100 TeV", Eur.Phys.J.C 79 (2019) 1011. https://arxiv.org/abs/1906.08221 , full text https://arxiv.org/html/1906.08221v4
- arXiv:2312.06307 — Satunin & Sharofeev, "Shower formation constraints on cubic Lorentz Invariance Violation parameters in quantum electrodynamics" (Myers–Pospelov cubic case, LHAASO). https://arxiv.org/abs/2312.06307 , full text https://arxiv.org/html/2312.06307
- Abstract pages: arXiv:2105.07967 (Ma, PRD 104, 063012), arXiv:2105.06647 (Ma, JHEAp 32, 1), arXiv:2106.06393 (Satunin, EPJC 81, 750), arXiv:2106.12350 (LHAASO collab.), arXiv:2510.07234 (Satunin & Troitsky, JETP Lett. 123, 73).

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## (1) What I found

### 1.1 Dispersion-relation convention (arXiv:1611.10125, Sec. 2, eq. (1)) — Established

> "Focusing on the QED sector and keeping up to quartic terms, one writes the dispersion relations for photons and electrons/positrons:
> `E_γ^2 = p_γ^2 + ε_γ p_γ^4 / M_LV,γ^2 ,   E_e^2 = m_e^2 + p_e^2 (1 + δ_e) + ε_e p_e^4 / M_LV,e^2`  (1)
> where ε_γ,e can take values ±1 and we allowed the scales suppressing the quartic contributions for photons and electrons/positrons to be different in general. Note that, without loss of generality, we have set the quadratic correction to the photon dispersion relation to zero, so that the low-energy velocity of photons is normalized to one; this can be always achieved by an appropriate rescaling of the space- or time-coordinates."

So: the convention is (a) a preferred frame (CMB rest frame, stated in the Introduction: "In this framework one postulates existence of a preferred frame, commonly identified with the rest-frame of the CMB"), (b) isotropic quartic-only LV, (c) low-energy photon velocity normalized to 1. The electron *is* given a LV sector in eq. (1) (parameters δ_e, ε_e) but is then **dropped** for the analysis — see 1.4 (quote: "Therefore we will neglect LV in electrons from now on"). For the main shower analysis the electron is luminal. Evidence class: Established.

Also eq. (2) maps the parameters onto the SME: `δ_e = −2 c̊_2^(4) ,   ε_e / M_LV,e^2 = −2 c̊_4^(6) ,   ε_γ / M_LV,γ^2 = −c_(I)00^(6)/√π` (2). Established.

### 1.2 Definition of the effective photon mass (arXiv:1611.10125, Sec. 2C, eq. (8)) — Established

> "The quartic contribution to the dispersion relation can be thought of as an effective momentum-dependent 'photon mass',
> `m_γ,eff^2(p_γ) ≡ E_γ^2 − p_γ^2 = p_γ^4 / M_LV,γ^2` . (8)
> It characterizes the amount of energy that can be transferred from the photon to the decay products."

In the companion paper (arXiv:1204.5782, Discussion, unnumbered): `m_γ^2(k) ≡ E_γ^2 − k^2 = ξ k^4 / M^2` and `m_e^2(p) ≡ E_e^2 − p^2 = m^2 + 2ϰ p^2 + 2g p^4 / M^2` — the two-sector generalization ("LV introduces effective momentum-dependent masses for the photon and electron"). Established. Note 1906.08221 uses the *unsquared* effective mass: `m_γ,eff ≡ √(E_γ^2 − k_γ^2) = E_γ^2 / M_LV` (unnumbered, "Photon decay" section). Established.

### 1.3 The suppression condition for Bethe–Heitler — THE criterion |m_eff²| vs 4m_e² — Established

arXiv:1611.10125, Sec. 3.1 "Suppression of the Bethe-Heitler process". LI cross section, eq. (14):

> `σ_BH = 28 Z² α³ / (9 m_e²) · ( log(183/Z^{1/3}) − 1/42 )` (14), σ_BH ≈ 0.51 b on nitrogen (Z=7), ⟨X_0⟩ = m_at/σ_BH ≃ 57 g cm⁻².

Then, verbatim:

> "As pointed out in [28,26], LV changes the cross section of the Bethe–Heitler process. Qualitatively this can be understood as follows. The electron mass in the expression (14) characterizes the momentum transfer between the photon and nucleus required to produce the e⁺e⁻ pair. In the LV case the momentum transfer is shifted due to the presence of the effective photon mass (8). Thus, up to a factor of order one, the modified Bethe–Heitler cross section can be estimated as (14) with the replacement
> `m_e^2 ↦ |m_e^2 − m_γ,eff^2(p_γ)/4|` . (15)
> This modification is not relevant for superluminal photons as the cross section essentially remains close to its value in the LI theory as long as 0 < m_γ,eff²(p_γ) < 4m_e², i.e. as long as the photon decay is forbidden; for higher values of m_γ,eff² photon decay provides the dominant signature of LV. However, for subluminal photons the modification of the Bethe–Heitler cross section can be important. If
> `m_γ,eff²(p_γ) < 0 ,  |m_γ,eff²(p_γ)| ≫ 4m_e²` (16)
> the cross section gets strongly suppressed."

**The criterion is therefore: photon effective mass squared *negative* (subluminal), and its magnitude large against 4m_e².** Note the factor 1/4 in (15): suppression threshold is |m_γ,eff²| ≫ 4m_e², equivalent to |m_γ,eff| ≫ 2m_e. Established.

### 1.4 The modified cross-section formula — Established

arXiv:1611.10125, eq. (17)–(18), citing [26] = arXiv:1204.5782:

> "These qualitative arguments are supported by an explicit calculation in LV QED. Under the conditions (16) the modified cross section reads [26],
> `σ_BH^LV = (16 Z² α³)/(3 |m_γ,eff²(p_γ)|) · log(1/(α Z^{1/3})) · log( |m_γ,eff²(p_γ)| / (2 m_e²) )` . (17)
> The suppression factor
> `σ_BH^LV / σ_BH ≃ (12 m_e² M_LV,γ²)/(7 E_γ⁴) · log( E_γ⁴ / (2 m_e² M_LV,γ²) )` (18)
> quickly decreases with energy."

The underlying exact two-sector result (arXiv:1204.5782, eq. (40), with screening; the model has photon LV ξ/M² and electron LV ϰ, g/M² in the Lagrangian, its eq. (1); dispersion relations its eqs. (2),(3)):

> `σ_γZ→Ze⁺e⁻ = (4 Z² α³)/(3 k |ω_LV|) · [ 2 log(1/(α Z^{1/3})) + (1/2) log(k|ω_LV|/m²) ] · log(k|ω_LV|/m²)` (40),
> valid under `|k ω_LV| ≫ m²` (their eq. (37)); ω_LV taken at x=±1. The LI result to compare: `σ_γZ→Ze⁺e⁻^LI = (28 Z² α³)/(9 m²) × { log(2k/m) − 109/42 (no screening) ; log(183/Z^{1/3}) − 1/42 (with screening) }`.

Verbatim from 1204.5782 §3.4: "We see that in the regime (37) LV strongly suppresses the cross section of pair production on nuclei. Besides, the LV cross section is dominated, according to (34), by the configurations when one of the produced fermions carries most of the energy (|x|≈1). This is in contrast to the standard QED where the energy distribution of the pair is smooth over the whole range −1 ≤ x ≤ 1."

The two-sector "mass" that enters is the combination ω_LV (defined in 1204.5782 eq. (14) and 1312.4368 eq. (6)): `ω_LV(x) = −ϰ k + ξ k³/(2M²) − g k³/(4M²) (1+3x²)` — it mixes photon-sector ξ and electron-sector ϰ, g. In 1312.4368, eq. (4) gives the general estimate `σ_BH^LV ∼ Z² α³ / m_eff²(k)`, with the explicit statement: "if the effective mass evaluated at the energy of the primary photon significantly exceeds m, the BH process will be strongly suppressed", and eq. (7) `σ_BH^LV ≃ (8 Z² α³)/(3 k |ω_LV(1)|) · log(1/(α Z^{1/3})) · log(k|ω_LV(1)|/m²)`, "upon identifying k|ω_LV(1)| as the precise expression for m_eff²(k)". Established.

### 1.5 Numerical bounds and at what energies — Established

arXiv:1611.10125 (Crab nebula, HEGRA to 75 TeV, H.E.S.S. flare 2013 to ~40 TeV):
- HEGRA: `M_LV,γ > 2.1×10¹¹ GeV (ε_γ = −1) at 95% CL` (26a); SME bound `c_(I)00^(6) < 4×10⁻²³ GeV⁻²` (26b).
- H.E.S.S.: `M_LV,γ > 1.3×10¹¹ GeV (ε_γ = −1) at 95% CL` (27a); `c_(I)00^(6) < 10⁻²² GeV⁻²` (27b).
- Same paper, other channels (review): photon decay bound `M_LV,γ > 2.8×10¹² GeV (ε_γ=+1)` (10); EBL absorption `M_LV,γ ≳ 3×10¹¹ GeV` (11), `> 7.5×10¹¹ GeV` (12) (ε_γ=−1); UHE-CR non-observation `M_LV,γ ≳ 1.2×10²² GeV` (13) (ε_γ=−1). Time-of-flight: `6.4×10¹⁰ GeV` AGN (6), `1.3×10¹¹ GeV` GRB (7).
- Projections: CTA 100 TeV `M_LV,γ > 1.7×10¹² GeV` (28); ~400 TeV arrays `≳ 3×10¹³ GeV` (30); a few 10¹⁹ eV photon events → `≳ 4×10²³ GeV` (following [29]=1312.4368, 2014).
- 1906.08221 (Tibet ASγ bin at E = 140 TeV, and HAWC): subluminal shower-formation bound `M_LV > 1.4×10¹² GeV, 95% CL` (14); superluminal splitting `> 4.1×10¹⁴ GeV` (13); photon decay `> 1.9×10¹³ GeV, 5σ` (15); HAWC decay bounds `M_LV > 1.0 (1.4)×10¹³ GeV` (16).
- 2312.06307 (cubic, Myers–Pospelov, LHAASO 100 TeV–PeV): `E_LIV ≳ O(10²⁰ GeV)` (95% CL), stated to be "significantly weaker than existing birefringence constraints but is independent", with ratio `σ_LIV/σ_LI ≃ 1.7 (m² E_LIV / E_γ³) log( E_γ³ / (2 m² E_LIV) )` (9).

### 1.6 The 2014 UHE-photon predecessor (arXiv:1312.4368) — Established

This is the paper ref [29] of 1611.10125. It is where the *general two-sector* framework and the effective-mass criterion live, and it contains the double-sided prospective bounds from 10¹⁹–10²⁰ eV photon detection. Key verbatim: dispersion relations (2) `E_γ² = k² + Σ_{n≥3} a_n kⁿ/M^{n−2}`, `E_e∓² = m² + p² + Σ_{n≥2} b_n∓ pⁿ/M^{n−2}`; effective masses (3) `m_γ,eff²(k) = Σ_{n≥3} a_n kⁿ/M^{n−2}`, `m_e∓,eff²(p) = m² + Σ_{n≥2} b_n∓ pⁿ/M^{n−2}`. General condition: "For negative ω_LV(x) < −m²/k LV significantly suppresses the cross section of the BH process" and the prospective constraints (8) `ω_LV(x) ≲ 2m²/(k(1−x²))`, `−10m²/k ≲ ω_LV(1)` leading to "double-sided limits" ("a detection of photon-induced air showers with energies above 10¹⁹ eV... will put tight double-sided limits on Lorentz violation in the sector of quantum electrodynamics" — abstract).

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## (2) THE ANSWER: relative vs absolute — subluminal *relative to the electron*, evaluated in the preferred frame

**Short answer: the suppression is a *relative-velocity* (photon-vs-electron) effect. In the papers' convention the electron is luminal (v_e = 1 in the preferred frame), and the criterion m_γ,eff² < 0 with |m_γ,eff²| ≫ 4m_e² is exactly "the photon is subluminal relative to the electron."** When electron LV is switched on, the same physics is carried by the combination ω_LV, which is precisely the photon-vs-pair energy deficit (i.e., the relative velocity difference including the electron sector), not by the absolute photon velocity. Evidence class: Established (the criterion and convention) + Inference (the "relative" reading, supported by the papers' own statements below).

The quotes that settle it:

1. Electron set luminal (arXiv:1611.10125, Sec. 2, right after eq. (5)): **Established**
> "We are going to see that the constraints on LV in the photon dispersion relation that can be obtained from the current data are significantly weaker than for electrons. Therefore we will neglect LV in electrons from now on."
Then the pair-production threshold section carries the footnote: "Recall that we neglect LV in electrons" (footnote 5). And the dispersion convention fixes the zero of velocity: "we have set the quadratic correction to the photon dispersion relation to zero, so that the low-energy velocity of photons is normalized to one" (see 1.1). With δ_e = ε_e = 0, electron velocity ≡ 1 ≡ low-energy photon velocity; m_γ,eff² < 0 (ε_γ = −1) means the photon's phase velocity drops below 1 at high energy — **subluminal relative to the electron.** Established.

2. The physical reading of (15) is the momentum transfer between photon and nucleus set by the *effective masses*: "The electron mass in the expression (14) characterizes the momentum transfer... In the LV case the momentum transfer is shifted due to the presence of the effective photon mass (8)" (1611.10125, quoted in full in 1.3). Established.

3. That only the *combination* of photon and electron velocities matters — the two-sector generalization — is explicit in 1204.5782: **Established**
- ω_LV(x) (eq. 14) contains ϰ (electron dim-4), g (electron dim-6) AND ξ (photon dim-6): `ω_LV(x) = −ϰk + ξk³/(2M²) − gk³/(4M²)(1+3x²)`. All BH physics depends only on this combination, not on ξ alone. The energy-conservation formula (eq. 30) shows its meaning: `E_γ − E_1 − E_2 ≈ ω_LV(x) − 2(p_y²+p_z²)/(k(1−x²)) − q_y²/(2k) − q_x`, so ω_LV is the photon-vs-pair energy excess/deficit — i.e. the relative velocity of photon and electron sectors projected into the process. Established.
- Footnote 4 (on Ref. [17], which assumes electron/positron at unit velocity and photon velocity modified): "This is related to our setup by a rescaling of the space coordinates and therefore is physically equivalent." — i.e., a global rescaling that changes *both* velocities is unphysical; the physical content is the *relative* velocity. Established (statement of equivalence), and it directly backs the "relative" answer.
- The dimensional-analysis rule in the Discussion of 1204.5782: "the momentum transfer between the nucleus and the photon is of order 2m_e(k)" with `m_e²(k) ≡ E_e² − k² = m² + 2ϰk² + 2g k⁴/M²` and `σ_γZ→Ze⁺e⁻ ~ Z²α³/m_e²(k)` — the relevant scale is the *effective mass of the produced pair* (which includes electron-sector LV), evaluated at the photon momentum. Established.
- 1312.4368 states it most generally: "We focus on the opposite situation, m_e∓,eff ≳ m_γ,eff", and "if the effective mass evaluated at the energy of the primary photon significantly exceeds m, the BH process will be strongly suppressed" (see 1.6). Established.

4. A global-absolute-frame reading is **not** what the formulas express. Note carefully: the papers do write everything in the preferred (CMB) frame — "one postulates existence of a preferred frame, commonly identified with the rest-frame of the CMB" (1611.10125 intro) — but the *observable* they compute depends on the relative photon–electron velocity through m_γ,eff² (electron luminal) or ω_LV (two sectors). If an *absolute* criterion were operative, electron-sector LV would be irrelevant; it is not — ϰ and g enter ω_LV with the same weight class as ξ (all three terms of ω_LV are k, k³, k³ suppressed as ϰ, gk²/M², ξk²/M², and the constraint paper 1312.4368 explicitly writes double-sided bounds on the *combination*). Inference from equations quoted above (the papers do not spell out "relative not absolute" in so many words; the closest they come is footnote 4 of 1204.5782 and the ω_LV structure).

**Caveat (Inference):** the equivalence "subluminal relative to electron" ⟺ "m_γ,eff² < 0" holds *in the papers' normalization* (electron luminal). If one allowed a non-luminal electron, the condition generalizes to k|ω_LV| ≫ m² with ω_LV < 0, i.e. the sign of the photon–electron velocity difference, still evaluated in the preferred frame. No paper in this lineage states the fully covariant "relative velocity" phrasing explicitly; the relative interpretation is forced by the structure of ω_LV and by footnote 4.

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## (3) What I could NOT find, and where I looked

1. **A paper that writes the |m_eff²| ≫ 4m_e²-style criterion explicitly with BOTH photon and electron LV turned on.** The two-sector papers (1204.5782, 1312.4368) do include electron-sector coefficients (ϰ, g), but they express everything through ω_LV(x), not through a "|m²_γ,eff − m²_e,eff|" comparison. Eq. (15) of 1611.10125 (the m_e² ↦ |m_e² − m_γ,eff²/4| replacement) is stated only for the luminal-electron case. I did not find the explicit two-sector generalization of eq. (15) / (16) in any of these papers. Not found — searched: full texts of 1611.10125, 1204.5782, 1312.4368, 1906.08221, 2312.06307.

2. **A treatment of BH suppression for a photon that is subluminal in the preferred frame but superluminal *relative to an even faster electron*.** No paper addresses the case where electron-sector LV reverses the relative velocity. From the structure of ω_LV one can see it would move the suppression (ω_LV < 0) across parameter space, but nothing in the literature I found states this. Not found — searched: the five full texts above plus web searches.

3. **The claimed follow-ups "Satunin 2019 arXiv:1906.08221" and "Satunin arXiv:2109.xxxxx".** 1906.08221 exists and is covered (see 1.5). The 2021 Tibet/LHAASO follow-up is arXiv:2106.06393 (Satunin, EPJC 81, 750) — I verified the abstract only; the two-sided constraints are photon-splitting (superluminal) and shower suppression (subluminal), i.e., photon-sector only; I did not find any electron-sector coefficients in it (abstract-level verification only). Not found at full-text level.

4. **Any paper in this lineage treating electron-sector LV in the *shower-suppression* bounds numerically.** 1906.08221 explicitly declines: footnote 1: "LV in the electron sector is not considered here since those constraints are more stronger than in the photon sector [20], see also discussion in [16]." The 2021 LHAASO-collaboration analysis (arXiv:2106.12350) and the Ma et al. papers (2105.07967, 2105.06647) treat photon-sector LV on EBL/CMB pair production (threshold shifts), not two-sector BH suppression. 2510.07234 (Satunin & Troitsky, GRB 221009A) fits LIV vs ALPs against the Carpet-3/LHAASO joint spectrum — photon-sector scenario; abstract-level only in this thread. Not found.

5. **Where I did not go:** I did not read the HEGRA/H.E.S.S./Tibet/HAWC/LHAASO experimental papers themselves, nor the SME data tables of Kostelecký–Russell, nor the EBL-model papers (refs 51–58, 78 of 1611.10125). The Vankov–Stanev paper (Phys. Lett. B 538 (2002) 251, astro-ph/0202388 = ref [28] of 1611.10125) — the original proposal that LV suppresses BH — was identified from the reference list but not fetched; its exact equations are quoted through 1611.10125 only. If precise credit for the first "decrease of photon velocity suppresses shower formation" statement is needed, fetch astro-ph/0202388 next.

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## One-line summary for the parent session

Bethe–Heitler (γ + nucleus → e⁺e⁻ + nucleus) is suppressed when the photon's effective mass squared is negative and huge: `m_γ,eff² ≡ E_γ² − p_γ² < 0, |m_γ,eff²| ≫ 4m_e²` (1611.10125 eq. (16)); cross section replaced by `σ_BH^LV = (16Z²α³)/(3|m_γ,eff²|) log(1/(αZ^{1/3})) log(|m_γ,eff²|/2m_e²)` (eq. 17), suppression factor `~ (12 m_e² M_LV²)/(7 E_γ⁴) log(E_γ⁴/(2m_e²M_LV²))` (eq. 18). Electron is taken luminal; the criterion is **subluminal relative to the electron** — the papers' convention normalizes the low-energy photon velocity to 1 and the LV is written in the preferred (CMB) frame; with electron LV included the physics goes through the relative-velocity combination ω_LV(x) = −ϰk + ξk³/(2M²) − gk³/(4M²)(1+3x²) (1204.5782 eq. 14), and footnote 4 of 1204.5782 shows only the relative velocity matters. Bounds: M_LV,γ > 2.1×10¹¹ GeV (HEGRA, 75 TeV), 1.3×10¹¹ GeV (H.E.S.S., 40 TeV), 1.4×10¹² GeV (Tibet, 140 TeV) at 95% CL — all for the subluminal quartic case.