# Thread: KM sign convention for the SME photon-sector dispersion relation (c_(I)jm^(d))

**Date:** 2026-09-11
**Thread task:** Establish, with exact quotes and equation numbers, the sign convention of the SME photon-sector vacuum dispersion relation for the CPT-even nonbirefringent coefficients c_(I)jm^(d).

**Sources used (all fetched and read this session):**
- V.A. Kostelecký, M. Mewes, "Electrodynamics with Lorentz-violating operators of arbitrary dimension," Phys. Rev. D 80, 015020 (2009), arXiv:0905.0031 — read as full ar5iv HTML (https://ar5iv.labs.arxiv.org/html/0905.0031)
- V.A. Kostelecký, M. Mewes, "Constraints on relativity violations from gamma-ray bursts," Phys. Rev. Lett. 110, 201601 (2013), arXiv:1301.5367 — read as full ar5iv HTML (https://ar5iv.labs.arxiv.org/html/1301.5367)
- V.A. Kostelecký, N. Russell, "Data Tables for Lorentz and CPT Violation," Rev. Mod. Phys. 83, 11 (2011), arXiv:0801.0287v19 — read as full arXiv HTML (https://arxiv.org/html/0801.0287v19)

**Evidence class:** All quotes below are **Established** — verbatim from the published/arXiv papers, with equation numbers verified against the ar5iv/arXiv HTML sources. Equation TeX is reproduced exactly as the source `alttext` gives it.

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## 1. The vacuum photon dispersion relation as KM write it

**KM 2009, Eq. (75)** (end of Sec. IV.3, "Vacuum models"; verbatim, TeX exact):

> p⁰ ≃ (1 − ς⁰ ± √((ς¹)² + (ς²)² + (ς³)²)) p ,    (75)

i.e. `p^{0}\simeq\big(1-\varsigma^{0}\pm\sqrt{(\varsigma^{1})^{2}+(\varsigma^{2})^{2}+(\varsigma^{3})^{2}}\,\big)p,` — the two leading-order solutions of the vacuum dispersion relation. The sign in front of the root is the birefringence (±) splitting; the ς⁰ term enters with a **minus** sign. Source: https://ar5iv.labs.arxiv.org/html/0905.0031 (Eq. S4.E75).

Preceding it, Eq. (74) is the vacuum scalar dispersion relation:

> (p^μ p_μ − (ĉ_F)^μν p_μ p_ν)² − 2(χ̂_w)^αβγδ (χ̂_w)_{αμγν} p_β p_δ p^μ p^ν − 4(p^μ(k̂_AF)_μ)² ≃ 0 .    (74)

**KM 2013 restates the same relation as its Eq. (1)** (verbatim):

> The dispersion relations connecting the photon energy E and momentum p for the two modes can be written in the compact but implicit form
> E = (1 − ς⁰ ± √((ς¹)² + (ς²)² + (ς³)²)) p ,    (1)
> where the dimensionless quantities ς^a = ς^a(E, θ, φ) depend both on E and on the photon direction of propagation …

`E=\big(1-\varsigma^{0}\pm\sqrt{(\varsigma^{1})^{2}+(\varsigma^{2})^{2}+(\varsigma^{3})^{2}}\,\big)p,` — same sign structure as KM 2009 Eq. (75).

## 2. Definition of ς⁰ in terms of c_(I)jm^(d)

**Step 1 — KM 2009, Eq. (76)** (verbatim, TeX exact):

> ς⁰ = ½ (ĉ_F)^μν p_μ p_ν / ω² ,
> (ς¹)² + (ς²)² = ½ (χ̂_w)^αβγδ (χ̂_w)_{αμγν} p_β p_δ p^μ p^ν / ω⁴ ,
> ς³ = − p^μ (k̂_AF)_μ / ω² ,    (76)
> "contain coefficients for nonbirefringent CPT-even, birefringent CPT-even, and birefringent CPT-odd effects, respectively."

**Step 2 — KM 2009, Eq. (81)**: the spin-weighted-spherical-harmonic expansion of the Stokes-parameter combinations, quoted verbatim (TeX exact):

> ς⁰ = Σ_{djm} ω^{d−4} (−1)^j ₀Y_jm(p̂) c^{(d)}_{(I)jm} ,
> ς¹ ± iς² = Σ_{djm} ω^{d−4} (−1)^j ±₂Y_jm(p̂) ( k^{(d)}_{(E)jm} ∓ i k^{(d)}_{(B)jm} ) ,
> ς³ = Σ_{djm} ω^{d−4} (−1)^j ₀Y_jm(p̂) k^{(d)}_{(V)jm} .    (81)

TeX exact for the first line: `\varsigma^{0}=\sum_{djm}\omega^{d-4}(-1)^{j}\,\phantom{}_{0}Y_{jm}(\mbox{\boldmath$\hat{p}$})\,c^{(d)}_{(I)jm},`. Note: ₀Y_{jm} is the spin-weight-0 spherical harmonic; for the isotropic case j = m = 0, ₀Y₀₀ = 1/√4π. The (−1)^j factor is a definitional convenience, per the text following (81): "the factors of (−1)^j in Eq. (81) have been introduced for convenience and to match the definitions in Ref. [km_apjl]". It is irrelevant for j=0.

**Step 3 — KM 2013, Eq. (2)–(3)** (same convention, verbatim):

> ς^a(E, θ, φ) = Σ_d E^{d−4} ς^{(d)a}(θ, φ),  (a = 0,1,2,3),    (2)
> ς^{(d)0}(θ, φ) = Σ_jm Y_jm(θ, φ) c^{(d)}_{(I)jm} ,    (3)
> ς^{(d)3}(θ, φ) = Σ_jm Y_jm(θ, φ) k^{(d)}_{(V)jm} .    (3)

## 3. THE PLAIN ANSWER: what does c_(I)00^(6) > 0 mean?

**c_(I)00^(6) > 0 corresponds to SUBLUMINAL photon propagation: v_phase < 1 and E < |p| at high energy.**

One-line algebra. Isotropically, ς⁰ = E^{d−4} ₀Y₀₀ c^{(d)}_{(I)00} = E² c^{(6)}_{(I)00}/√4π for d = 6 (E ≃ p at leading order). Insert into Eq. (75):

E = (1 − ς⁰) p  ⟹  E/p = 1 − E²c^{(6)}_{(I)00}/√4π  < 1  for c^{(6)}_{(I)00} > 0  ⟹  **subluminal, E < |p|**.

Same conclusion from the velocity defect — KM 2009 Eq. (200) (verbatim):

> δv ≃ −ς⁰ = − Σ_{djm} E^{d−4} ₀Y_jm(n̂) c^{(d)}_{(I)jm} .    (200)

and in the isotropic limit, KM 2009 Eq. (199) (verbatim):

> δv ≃ (1/√4π) Σ_d E^{d−4} ( −c^{(d)}_{(I)00} ± k^{(d)}_{(V)00} ) ,    (199)

so for the CPT-even isotropic d=6 term, δv = −E²c^{(6)}_{(I)00}/√4π < 0 ⟺ c^{(6)}_{(I)00} > 0 ⟹ v < 1. **The minus sign on c_(I) is explicit in the paper's own equations (199)–(200).**

Consistency cross-check, KM 2009 (text preceding Table 13, verbatim): "An isotropic higher-order correction of the form δv = ξ₂E² has also been considered [xi2]. This case corresponds to the d = 6 coefficient c^{(6)}_{(I)00} = −√4π ξ₂." I.e., a positive (superluminal) δv = ξ₂E² requires **negative** c^{(6)}_{(I)00} — confirming positive c = subluminal.

## 4. Where the sign convention is restated / any sign footnotes

- **KM 2009, Eq. (200)** is the operative statement of the convention (quoted above): "the velocity defect including anisotropies is given by δv ≃ −ς⁰ = −Σ_{djm} E^{d−4} ₀Y_jm(n̂) c^{(d)}_{(I)jm}." This is the paper's explicit, unambiguous definition: positive c_(I) ⟹ negative velocity defect.
- **KM 2013** restates the identical convention with no change: Eq. (1) (E = (1 − ς⁰ ± …)p), Eq. (3) (ς^{(d)0} = Σ Y_jm c^{(d)}_{(I)jm}), and the group-velocity defect (text after Table 2, verbatim): "δv_gr = (d−3)E^{d−4}(−ς^{(d)0} ± |ς^{(d)+}|) in the CPT-even case … but in the isotropic limit one obtains instead the simpler expressions **δv_gr = −(d−3)E^{d−4} c^{(d)}_{(I)00}/√4π** in the CPT-even case …". Same minus sign, now on the group velocity.
- **No footnote warning about sign ambiguity specific to c_(I) exists in KM 2009** (the ar5iv HTML of the paper contains no footnote elements at all; none found by search of the full text).
- **The Data Tables (0801.0287v19)** contain only the *general* metric-signature caveat, not a c_(I)-specific warning (verbatim, Sec. IV opening, after the Sun-centered-frame conventions): "Note that some of the literature on the SME in Minkowski spacetime adopts a metric η_{μν} of opposite sign, following the common present usage in quantum physics instead of the one in relativity. Under this alternative convention, terms in the Lagrange density with an odd number of index contractions have opposite signs to those appearing in this work. The numerical results for the SME coefficients in the tables are unaffected by the convention." Nothing in it changes the c_(I) vacuum dispersion sign, which enters through Eq. (75)/(199)/(200) as quoted.
- A "sign ambiguity" risk would only apply to the *birefringent* combinations (the ± root and ς³'s sign), which the papers flag via the helicity-sensitive ± in δv = ±ξ₁E / ±ξ₂E² (KM 2009 text before Eq. (199): "δv = ±ξ₂E², where the sign indicates helicity"). The nonbirefringent c_(I) sign is fixed by Eqs. (199)–(201).

## 5. Group velocity / time-of-flight for c_(I)00^(6) > 0

**c_(I)00^(6) > 0 makes high-energy photons arrive LATER than low-energy ones** (they are slower).

Formulae, verbatim:

- KM 2013 (isotropic, CPT-even): **δv_gr = −(d−3)E^{d−4} c^{(d)}_{(I)00}/√4π**; for d=6: δv_gr = −3E²c^{(6)}_{(I)00}/√4π < 0 when c > 0 (quoted above; text following Table 2).
- KM 2009 arrival-time formula (Table 13 caption, verbatim): "The difference in velocity between photons of different energies leads to an arrival-time difference given by [km_apjl]
  t₂ − t₁ ≈ ∫₀ᶻ (v₁ − v₂)/H_z dz ≈ (E₂^{d−4} − E₁^{d−4}) ∫₀ᶻ (1+z)^{d−4}/H_z dz Σ_jm ₀Y_jm c^{(d)}_{(I)jm} ,   **(201)**
  where the source redshift is z and t₁, t₂ are the propagation times for photons with observed energies E₁, E₂ and velocities v₁, v₂."

With δv = −ς⁰, v₁ − v₂ = δv(E₁) − δv(E₂) = (E₂^{d−4} − E₁^{d−4})·(coefficient combination). For c > 0 and E₂ > E₁: t₂ − t₁ > 0 — the higher-energy photon arrives **later**. (For c < 0 it arrives earlier.)
- KM 2009 text also gives the flat-space estimate (Sec. VI.1, verbatim): "In a vacuum dispersion study involving a source at baseline distance L, the quantity of interest is the change δt ≃ δvL in arrival time of the signal."

## 6. Energy-scale formula

- KM 2009 does **not** define a symbol such as "E_LV" or "M_LV" in terms of c^{(d)}_{(I)00}. The energy dependence enters as the power E^{d−4} through Eq. (81)/(199) (ς⁰ ∝ ω^{d−4}); for d=6 the defect grows as E².
- The closest thing to a scale statement is the model-size estimate, KM 2009 Sec. II.1 (verbatim): "one simple estimate has coefficients varying as 𝒦_(d) ∼ ζ M_Planck^{4−d}, where ζ is of order 1" — an estimate for the cartesian coefficients from Planck-scale physics, not a dispersion-relation energy scale. (The `𝒦_(d) ∼ ζ M_Planck^{4−d}` relation is in the paragraph immediately preceding Eq. (4) region; the arXiv HTML does not number it as a displayed equation.)
- Limits are quoted dimensionally, e.g. KM 2009 Table 13: |c^{(6)}_{(I)00}| < 4×10⁻¹⁶ GeV⁻² (GRB 021206, vacuum isotropic). If one defines a "Lorentz-violation scale" via δv = (E/E_LV)², then with δv = −c^{(6)}_{(I)00}E²/√4π one gets E_LV = (√4π/|c^{(6)}_{(I)00}|)^{1/2}, but **that is our inference, not a KM formula** — KM/Datatables never write E_LV for the SME photon sector.

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## Bottom line

1. KM 2009 Eq. (75): p⁰ ≃ (1 − ς⁰ ± √((ς¹)²+(ς²)²+(ς³)²)) p. The ς⁰ term enters with **minus** sign.
2. KM 2009 Eq. (81): ς⁰ = Σ_{djm} ω^{d−4}(−1)^j ₀Y_jm(p̂) c^{(d)}_{(I)jm}. (KM 2013 Eq. (3) identical without (−1)^j.)
3. **c_(I)00^(6) > 0 ⟹ SUBLUMINAL (v_ph < 1, E < |p|)**: E = (1−ς⁰)p, ς⁰ = E²c/√4π > 0 ⟹ E/p < 1; δv = −ς⁰ < 0 (KM 2009 Eq. (199)–(200), KM 2013 δv_gr formula).
4. Restatements: KM 2009 Eq. (200) and KM 2013 (Eqs. (1),(3), δv_gr after Table 2) restate the same convention explicitly. No c_(I)-specific sign footnote exists; Data Tables carry only the general metric-signature note (numerical results unaffected).
5. c_(I)00^(6) > 0 ⟹ high-energy photons arrive **later** (δv_gr = −3E²c/√4π < 0; t₂−t₁ ∝ (E₂²−E₁²)c > 0, KM 2009 Eq. (201)).
6. No E_LV formula in KM; energy enters as E^{d−4}; the only scale remark is 𝒦_(d) ∼ ζ M_Planck^{4−d} (KM 2009 Sec. II.1).

**URLs:** KM 2009: https://arxiv.org/abs/0905.0031 ; KM 2013: https://arxiv.org/abs/1301.5367 ; Data Tables v19: https://arxiv.org/abs/0801.0287 and https://arxiv.org/html/0801.0287v19 ; ar5iv HTML of KM 2009: https://ar5iv.labs.arxiv.org/html/0905.0031