# Thread: Collins-Perez-Sudarsky-Urrutia Lorentz-invariance naturalness argument, and everything that answers it

**Date:** 2026-09-13
**Thread author:** Argus subagent (collins-naturalness)
**Status:** COMPLETE

Brief asked for: (1) mechanism stated precisely with loop structure and equations; (2) claimed magnitude; (3) experiment compared to and precision; (4) every published rebuttal in full (a: SUSY, b: custodial symmetry/scale separation/regulator evasions, c: scheme-artifact arguments); (5) application to a LATTICE regulator specifically; (6) whether Symanzik improvement changes the loop-induced coefficient.

All quotes below are verbatim from the sources fetched today (2026-09-13). Evidence classes: **Established** = read in the source; **Serious speculation** = argued by authors, not settled; **Anomaly**; **Anecdote**; **Inference** = my own.

---

## 1. THE MECHANISM, STATED PRECISELY [Established — read in full from gr-qc/0403053v4 and hep-th/0603002v1]

**Primary source:** J. Collins, A. Perez, D. Sudarsky, L. Urrutia, H. Vucetich, "Lorentz invariance and quantum gravity: an additional fine-tuning problem?", Phys. Rev. Lett. **93**, 191301 (2004). arXiv:gr-qc/0403053 (v4 is the journal-length version), DOI 10.1103/PhysRevLett.93.191301. Read in full from https://arxiv.org/html/gr-qc/0403053v4. (Note: the task brief called it "an additive violation" — the actual title is "an additional fine-tuning problem?".)

**Companion:** J. Collins, A. Perez, D. Sudarsky, "Lorentz invariance violation and its role in quantum gravity phenomenology", arXiv:hep-th/0603002 (2006), draft chapter for *Towards Quantum Gravity* (ed. D. Oriti, CUP). Read in full from https://arxiv.org/html/hep-th/0603002v1.

**Setup (PRL):** The dispersion law of a particle is obtained by solving their Eq. (2):
E² − p² − m² − Π(E,p) = 0,
where Π is the sum of all self-energy graphs, plus any (small) Lorentz-violating corrections calculated in free-field theory.

**The reasoning, in their words (main text):** "Without a cutoff the graphs have divergences from large momenta/short distances. In the Lagrangian defining the theory, the divergences correspond to terms of dimension 4 (or less) that obey the symmetries of the microscopic theory... Now Planck-scale physics can cut off the divergences, and also modify the formulae to be used when loop momenta are around the Planck scale. But the same power-counting that determines the divergences also determines the natural size of the contributions of Planck-scale virtual momenta: the dominant contributions correspond to operators of dimension 4 (or less) in the Lagrangian... If the microscopic theory violates Lorentz invariance at the Planck scale, then generically we also get Lorentz violation at low energies without any suppression by powers of E/E_P."

**The regulator, exactly:** a preferred-frame cutoff in **Minkowski space** — not a Lorentz-invariant regulator. Their words: "Conventional treatments of effective field theory assume a Lorentz invariant cutoff for ultra-violet divergences; this cutoff is either dimensional regularization or a cutoff on momenta in Euclidean space. In contrast, we examine the effect of an effective cutoff in Minkowski space, as provided by a putative granularity of real space-time."

**The worked loop calculation (their Appendix A, in full):** Yukawa theory: self-energy of a scalar with one fermion loop. They quantify Lorentz violation by
ξ = ∂²Π₁/∂(p⁰)² + ∂²Π₁/∂(p¹)² at p=0 (zero for a Lorentz-invariant self-energy).

Without Planck-scale modifications, their Eq. (A.1):
ξ = −(i g²/π⁴) ∫ d⁴k ([(k⁰)²+(k¹)²] (k²+3M²)) / (k²−M²+iϵ)⁴

Their comment, verbatim: "Were it not that this integral is logarithmically divergent, it could be shown to be zero by continuing k⁰ to imaginary values and then using Euclidean rotation invariance." — This is the crux: the would-be logarithmic divergence is where the Lorentz violation survives; the (E/M_Planck) suppression never appears because the effect sits in the coefficient of the would-be divergence, which is order unity in units of the cutoff.

**Their explicit regulator (Appendix A):** the free-fermion propagator i(γ·k+m)/(k²−m²+iϵ) is modified by a factor of a smooth function f(|k|/Λ) with f(0)=1, f(∞)=0, cutoff parameter Λ (at the Planck scale). Note: a cutoff on **3-momentum |k|**, i.e., explicitly time-space anisotropic in the preferred frame — Lorentz-violating by construction.

Result, their Eq. (A.2):
ξ = (g²/6π²) [ 1 + 2 ∫₀^∞ dx x f′(x)² ]

Their words: "Thus the corresponding Lorentz violation is of order the square of the coupling independently of Λ. The exact value depends on the details of the Planck-scale free propagator, of course. The main point, however, is that the power counting that gives the would-be logarithmic divergence follows from standard arguments in the theory of renormalization, and that it applies to self-energy graphs for all fields."

The companion paper (hep-th/0603002) repeats the identical calculation (its Eq. (9), same ξ̃ formula, with the footnote that the cutoff is a modification of the propagators per its Eqs. (6)-(7), f(|p|/Λ) and f̃(|p|/Λ), Λ of order the Planck scale), and states: "Although the exact value depends on the details of the function f, it is bounded below by g²/6π². Lorentz violation is therefore of the order of the square of the coupling, rather than power-suppressed."

**Why dimension-4 operators specifically (companion paper, Eq. (8)):** expansion Π(p) = A + p²B + p^μ p^ν W_μ W_ν ξ̃ + Π^(LI)(p²) + O(p⁴/Λ²), where W_μ is the unit timelike preferred-frame vector. A and B are the usual LI mass and wavefunction renormalization; "the only unsuppressed Lorentz violation is in the third term. Its coefficient ξ̃ is finite and independent of Λ."

## 2. CLAIMED MAGNITUDE [Established]

ξ = (g²/6π²)[1 + 2∫ dx x f′(x)²] ≥ g²/6π². The bracket is ≥ 1 (integrand x f′(x)² ≥ 0). For g ~ 0.3–1 this gives ξ ~ 1.5×10⁻³ to 1.7×10⁻² — hence:

- "effects that are only suppressed by two powers of known standard-model couplings, effects that change the value of c in the dispersion relation... fractional differences around 0.1% to 10%."
- The c-modification factor is 1 + ξ/4 + O(ξ²) (their check via Eq. (2)).
- "some 20 orders of magnitude higher than earlier estimates" (abstract) — earlier estimates being E/E_P or (E/E_P)² dispersion corrections, and m_ew/M_Pl ≃ 10⁻¹⁷ (they quote Kostelecký & Potting, PRD 51, 3923 (1995): Lorentz violation "likely to be suppressed by some power of m_ew/M_Pl ≃ 10⁻¹⁷").
- No log terms appear at this order: the log divergence at one loop in the symmetric limit becomes, under the LV cutoff, a Λ-independent O(1) coefficient; the f′-integral is the only regulator-detail dependence. (PRL Appendix A.)
- Why a single metric renormalization cannot save it: "In a theory with one field it is possible to treat this term as a renormalization of the space-time metric tensor which could remove the observable Lorentz violation. However, there are many fields in the standard model that differ by the sizes of their couplings. Hence renormalization of the metric tensor cannot remove all leading-power Lorentz violation." (Appendix A.)
- Companion paper on how solid the number is: "More exact calculations would use renormalization group methods. But we know from the running of Standard-Model couplings, that this can produce changes of one order of magnitude, not twenty."

## 3. EXPERIMENT COMPARED TO [Established]

PRL: "the observed limits [1,2,3] that c is constant at a fractional level well below 10⁻²⁰" — cited as [1] Coleman & Glashow, PRD 59, 116008 (1999); [2] Colladay & Kostelecký, PRD 58, 116002 (1998); [3] Amelino-Camelia, Ellis, Mavromatos, Nanopoulos, Sarkar, Nature 393, 763 (1998), plus Jacobson, Liberati & Mattingly, Nature 424, 1019 (2003); Sudarsky, Urrutia & Vucetich, PRL 89, 231301 (2002); Alfaro & Palma PRD 67, 083003 (2003); Gleiser & Kozameh PRD 64, 083007 (2001); Bertolami GRG 34, 707 (2002). Predicted 0.1–10% inter-species c differences vs. observed <10⁻²⁰: incompatible by ~17–21 orders ("some 20").

The SUSY rebuttal paper (hep-ph/0505029) uses the sharper, experiment-specific values (all read in the TeX source):
- **Torsion balance / anomalous spin precession:** |b^i| < 10⁻²⁸ GeV (Heckel, Adelberger, Gundlach, Harris, Swanson, "Torsion balance test of spin coupled forces", Orbis Scientiae 1999) → their constraint Eq. (b_limit): (m_s²/(100 GeV)²)·(10¹⁹ GeV/M)·|N_A^i − (3/2)N^i + δ_s N_V^i| < 10⁻¹², i.e. |N_A^i − (3/2)N^i| ≲ 10⁻¹² for m_s ≳ 100 GeV at M = M_Pl.
- **Clock-comparison (Hg magnetometer / Zeeman maser) experiments:** sensitivity to Δω_LV at the 10⁻³² GeV level (Bear, Stoner, Walsworth, Kostelecký, Lane, PRL 85, 5038 (2000); Berglund et al., PRL 75, 1879 (1995)) → (10¹⁹ GeV/M)|N^i| < 10⁻⁹ for nuclear spin precession.
- **Electron-positron g-factor comparison:** |g_e − g_e-bar| < 8×10⁻¹² (Mittleman, Ioannou, Dehmelt, Russell, PRL 83, 2116 (1999)) → (m_e/M)|N_V⁰| < 2×10⁻¹².
- **Astrophysical dispersion-relation searches** (Carroll-Field-Jackiw-type vacuum birefringence, GRB time-of-flight): sensitivity to dim-5 operators at the 10⁻⁵/M_Pl level.
- **Summary number:** dim-5 LV operators constrained at 10⁻¹⁰–10⁻⁵ in units of 1/M_Pl; dim-6 operators suppressed by 1/M_Pl² are still allowed. Soft-breaking-induced dim-4 backgrounds estimated at m_s²/M² ~ 10⁻³² for m_s ~ 1 TeV, M ~ 10¹⁹ GeV — "close to the experimental sensitivity."

## 4. EVERY PUBLISHED REBUTTAL

### (a) SUSY protection [Established]

**Bolokhov, Groot Nibbelink, Pospelov, "Lorentz violating supersymmetric quantum electrodynamics", PRD 72, 015013 (2005), arXiv:hep-ph/0505029** (Pavel A. Bolokhov¹,³, Stefan Groot Nibbelink², Maxim Pospelov — confirmed from the TeX source). 37 pages; read in full from the arXiv TeX source.

**The problem they start from (their Eq. "transmute"):** dimensional transmutation of dim-5 LV into dim-3:
[LV]_dim3 ~ (loop factor) Λ_UV² × [LV]_dim5.
"If the UV cutoff scale Λ_UV is of the order of M, huge dimension three operators are generated." They note Myers & Pospelov (hep-ph/0301124, PRL 90, 211601) showed dim-5 LV operators coupled to fully symmetric traceless 3-index tensors are protected against quadratic divergences — "but this solves the naturalness problem only partially, as this does not provide an argument as to why dimension three and four operators cannot be induced at tree level."

**What SUSY buys, exactly (their Sec. 1, verbatim):** "A recent paper [GrootNibbelink:2004za] proposed that supersymmetry (SUSY) could provide a powerful selection rule on admissible forms of LV interactions. In particular, it has been shown that in the Minimal Supersymmetric Standard Model (MSSM) the requirements of SUSY and gauge invariance restrict LV operators to be of dimension five or higher. Therefore SUSY solves the naturalness problem of LV physics. Once SUSY is softly broken, the quadratic UV divergences are effectively stabilized at the supersymmetric threshold."

The underlying PRL is **Groot Nibbelink & Pospelov, "Lorentz violation in supersymmetric field theories", PRL 94, 081601 (2005), arXiv:hep-ph/0404271**, whose abstract states: "We show that in the supersymmetric Standard Model the lowest possible dimension for such operators is five, and therefore they are suppressed by at least one power of an ultra-violet energy scale, providing a possible explanation for the smallness of Lorentz violation and its stability against radiative corrections. Supersymmetric Lorentz noninvariant operators do not lead to modifications of dispersion relations at high energies thereby escaping constraints from astrophysical searches."

Mechanism: with exact N=1 SUSY, the LV couplings must be superfields and gauge-invariant; the only available structures are dim-5 and dim-6 operators built from vector superfields (photon+photino) and chiral superfields ((s)electrons), parametrized by three vector backgrounds N^μ, N_+^μ, N_-^μ and one irreducible rank-3 tensor T^{μνλ} (CPT-violating, dim 5) plus dim-6 CPT-even two-index tensor backgrounds. **There are no superselected (i.e., gauge+SUSY invariant) dim-3 or dim-4 LV operators at all** — that is the protection.

Quantum results (proved in hep-ph/0505029): (i) no destabilizing quadratically divergent D-terms arise; (ii) gauge anomalies unaffected by LV operators, so **no Chern–Simons term is generated even at loop level**; (iii) in exact SUSY only logarithmic divergences run — they solve the one-loop RG equations for dim-5 operators; (iv) EOM reduction relates dim-5 to dim-3: [LV]_dim3 ~ m_e² [LV]_dim5, controlled by the electron mass.

**The residual suppression — soft breaking [Established, key numbers]:** With soft SUSY breaking (scalar electron masses m_s):
[LV]_dim3 ~ m_soft² [LV]_dim5 (dimensional transmutation at one loop).
"Although a loop effect, this constitutes a dramatic enhancement compared to the case with unbroken SUSY, as m_soft²/m_e² > 10¹⁰." They explicitly show no CS term is generated even with soft breaking. Induced dim-4 LV backgrounds: m_s²/M² ~ 10⁻³² for m_s ~ 1 TeV, M ~ 10¹⁹ GeV — near current sensitivity but not excluded. **Net: the SUSY answer is that dim-4 LV is forbidden (not just suppressed) while SUSY is exact; after soft breaking the induced coefficients are down by (m_soft/M_Pl)² ~ 10⁻³² rather than O(g²) ~ 10⁻².** (Established from source; whether this "solves" naturalness vs. "postpones" it to the soft-breaking scale is a judgment the authors themselves flag via the m_soft²/m_e² > 10¹⁰ enhancement.)

Note: hep-ph/0505029 does NOT cite gr-qc/0403053 directly (checked the .bbl); it frames the problem through Myers-Pospelov and GrootNibbelink-Pospelov. Polchinski (below) does identify SUSY as "the one known example" evading the Collins et al. argument: "Supersymmetry is the one known example, in which both the space and time derivative kinetic terms arise from a single superfield. If supersymmetry is exact at the Planck scale and broken only at a much lower scale, then the observed Lorentz breaking may be suppressed to a sufficient degree." He also cites **Jain & Ralston, PLB 621, 213 (2005), arXiv:hep-ph/0502106** as the second reference on the SUSY escape.

### (b) Custodial symmetry / separation of scales / regulator-specific evasions [Established, with open disputes]

- **Scale separation (Belenchia, Gambassi, Liberati, "Lorentz violation naturalness revisited", JHEP 1606 (2016) 049, arXiv:1601.06700, read in full):** They redo the Collins et al. Yukawa model, confirm the scalar one-loop percolation, and find (their key formula, Eq. (49)): for an MDR with leading LIV terms of order (E/M)^{2n}, the percolation scales as **Δc ∝ (Λ/M)^{2n}** where Λ is a Lorentz-invariant EFT cutoff and M the LIV scale, M ≫ Λ. "This implies that if any such scenario could be successfully applied to the SM, values of Λ below 10¹⁰ GeV would be sufficient to reconcile the most interesting (CPT invariant) case n=1 with current observational constraints." They stress (their words): "We are not arguing here that the one discussed below is a protection mechanism which works for the entire Standard Model... What we want to emphasise, via a toy-model computation, is that the separation of scales could be one, or part of a, solution." They also find: fermion percolation vanishes for m_φ = m_ψ with common LIV (Δc = 0, Eq. (38)) but reappears as −g²/48π² for f̃ = 1; dissipation does not percolate (no imaginary contributions), dispersion does; scale separation protects both. Important nuance they add: "note that neglecting Δ and Δ̃, i.e. the MDR, is possible only at the price of identifying the EFT and the MDR scales. This is tantamount to assuming no new physics between the Standard Model scale and the Planck one" — i.e., they say Collins et al.'s single-scale treatment is itself a modeling assumption.
- **Gravitational-sector confinement (Pospelov & Shang, "On Lorentz violation in Horava-Lifshitz type theories", PRD 85, 105001 (2012), arXiv:1010.5249):** LIV confined to the gravity sector, matter tree-level LI; Planck-suppressed matter-gravity vertices tame percolation; exploits that gravitational LIV is less stringently tested.
- **Gravitational shielding / strong coupling (Afshordi, "Why is High Energy Physics Lorentz Invariant?", arXiv:1511.07879):** perturbative unitarity/weak coupling of LIV EFTs with gravity constrains observable LIV: violation ≲ 10⁻¹⁰ E(eV)⁻⁴ or the theory is strongly coupled beyond meV; conjecture that high-energy theories are "shielded" from LIV like color confinement. (Serious speculation — self-described conjecture.)
- **Emergent Lorentz invariance via strong dynamics (Bednik, Pujolàs, Sibiryakov, "Emergent Lorentz invariance from Strong Dynamics: Holographic examples", JHEP 11 (2013) 064, arXiv:1305.0011):** RG flow toward LI accelerated by strong coupling near the Planck scale — the Nielsen–Ninomiya/Picek line (Nielsen & Ninomiya, Nucl. Phys. B 141, 153 (1978); Nielsen & Picek, PLB 114, 141 (1982)): "the standard logarithmic running towards a LI theory in the infrared is generically not fast enough... a strong coupling close to the Planck scale can sufficiently enhance the running."
- **Physical LIV regulator with fine-tuning quantified (Cortés & López-Sarrión, "Fine-tuning problems in quantum field theory and Lorentz invariance: a scalar-fermion model with a physical momentum cutoff", IJMPA 32, 1750084 (2017), arXiv:1607.00744, read in full):** A Lifshitz-type finite theory (physical momentum cutoff Λ as LIV regulator). They quantify exactly the fine-tuning needed: (i) scalar mass: λ̃/c̃_s ∫ dx x/K_s(x²) = ẏ²/c̃_f ∫ dx x/K_f(x²) (their Eq. (22)) — a SUSY-like boson/fermion cancellation of Λ² contributions, but with no symmetry enforcing it ("the cancellation is due to a choice of the parameters of the model which should be modified order by order in the perturbative expansion"); (ii) velocity equality for power-suppressed LIV at one loop: c̃_f² − c̃_s² = (ẏ²/6π²c̃_f)∫dx [1−K_f+2(1−K_s)]/[x K_s(K_s+K_f)²] − (ẏ²/4π²c̃_f)∫dx (1−K_f²)/(x K_f³) (their Eq. (26)). "At each order in perturbation theory the required fine-tuning changes and the choice of functions K_s, K_f has to be modified." They also flag that gauge invariance is incompatible with the momentum-scale regulator ("This would add a new fine-tuning problem"). Their framing: "Instead of considering the absence of fine-tunings as a consistency requirement, we propose to use it as a guide" — i.e., they accept the fine-tuning.
- **Random/causal-set discreteness without a preferred frame:** Dowker, Henson, Sorkin, MPLA 19, 1829 (2004) (random causal sets; cited approvingly by both sides — Collins et al. explicitly: "In [28] Dowker et al. show how, by the use of a random causal set of points, space-time can be made discrete while Lorentz invariance is preserved in a suitable sense"); Rovelli & Speziale, PRD 67, 064019 (2003) (minimum length ≠ LIV; angular momentum eigenvalue analogy).

### (c) "The whole thing is a scheme artifact" arguments [Established — claims and counter-claims]

- **Burgess, Cline, Filotas, Matias, Moore, JHEP 03 (2002) 043** (higher-dimensional LV scenario). Collins et al.'s reading: "Loop corrections involving gravitons have Lorentz-violating power-law divergences. These authors set the power-law divergences to zero, one justification for which is the use of dimensional regularization... This appears to give only power-suppressed Lorentz violation, in contradiction with our estimates. However... Corresponding to the power-law divergences in the effective theory are finite contributions to coefficients, including those of Lorentz-violating operators of dimension 4. The natural size of the Lorentz violation in their scenario is therefore the same as ours." — I.e., the "disappearance" in dim reg is claimed to be an artifact of a scheme that discards power-law pieces; the natural-size argument survives. (Dispute unresolved in the papers I have read; no published exchange between the two groups located.)
- **Myers & Pospelov, PRL 90, 211601 (2003)** stated the same fine-tuning problem in EFT language, then "claim[ed] to find a condition in which the Lorentz violation disappears at leading power" (per Collins et al.). **Perez & Sudarsky, PRL 91, 179101 (2003)** — a Comment — counter: "their argument does not work" (Collins et al.: "as explained by two of us [21]"). Myers & Pospelov, arXiv:gr-qc/0402028 (added note in PRL v4): they deduce the low-energy theory must have a cutoff far below Planck — "However, they do not explain the problems of implementing such a cutoff Lorentz-invariantly in Minkowski space" (Collins et al.'s Note added).
- **Alfaro's schemes (PRL 94, 221302 (2005); PRD 72, 024027 (2005))** — naturally small LIV via LV-in-the-measure regulators (LV cutoff factor R(k) = −Λ²/(k²−Λ²+ak₀²+iϵ), and LV dim reg with g_μν = η_μν + αϵ W_μ W_ν). Collins et al.'s critique (companion paper Sec. 7, read in full): first scheme "suffers from a routing dependence and is therefore not well-defined... the regulator factor has a pole at k² = Λ²... similar to Pauli-Villars... cannot be considered a normal quantum theory"; second scheme "does not correspond to any normal definition of a QFT, and no rationale is given."
- **Nonlocal regulator (Evens, Moffat, Kleppe, Woodward (1991))** — nonlocal regularization; **Jain & Joglekar, IJMPA 19, 3409 (2004)** argue it violates causality and is physically unacceptable (as reported in hep-th/0603002 Sec. 8).

**Polchinski's decisive frame (J. Polchinski, Comment on 1106.1417, CQG 29, 088001 (2012), arXiv:1106.6346, read in full):** "The question investigated by CPSUV was 'O(1) Lorentz violations at the Planck scale: do they lead to unacceptably large effects?' The question addressed by the models of GRP is quite different: 'Small Lorentz violations at all scales: do they lead to unacceptably large effects?'" And: "almost all models of observable high energy Lorentz violation, and proposed Lorentz-violating theories of quantum gravity, are ruled out by low energy tests; the only known exceptions are based on supersymmetry." He also notes the general effective-field-theory basis: "A local operator of dimension Δ is induced with a coefficient of order M^{4−Δ}... operators of dimension ≤ 4 provide a direct window onto the symmetries of the high energy theory, no matter how high the scale of symmetry breaking," and the Standard Model "admits a large number of dimension 4 operators that are gauge invariant but not Lorentz invariant, for example the spatial gradient terms for each of the 19 gauge multiplets."

## 5. APPLICATION TO A LATTICE REGULATOR — YES, BY GAMBINI, RASTGOO & PULLIN [Established; dispute in the literature]

**R. Gambini, S. Rastgoo, J. Pullin, "Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects?", Class. Quantum Grav. 28, 155005 (2011), arXiv:1106.1417, subj: gr-qc + hep-lat + hep-th.** Read in full. This is the one paper that runs the Collins-type calculation with a genuine lattice regulator:

- Model: Yukawa scalar+fermion on a spacetime lattice, lattice spacing a ~ Planck scale, NOT taken to zero. Fermion propagator their Eq. (1): S(k) with sin(ak_j)/a and sin(bak_0)/(ba) dispersion (b = time/space spacing ratio, b→1 continuum), Euclidean. They compute the lattice analog of ξ̃ — their Eq. (5): a 4D integral over the Brillouin zone (−π/a, π/a)³ × (−π/(ba), π/(ba)).
- **Result (their Eq. (6) and text):** "If one has a finite lattice spacing and computes the integral (7), if one takes the same spacing in space and in time, the integral vanishes by symmetry reasons. If one takes different spacings in space and time (but such that the limit is isotropic, for instance b = 1 + μa...) one gets a finite result, that in the limit lim_{a→0} ξ_a = 0, and if one expands the finite result in powers of a, one gets that the leading contribution is of the order of aμ." — i.e., on the spacetime lattice the loop-induced LIV scales as (lattice spacing)×(anisotropy parameter) and vanishes in the continuum limit, NOT the O(g²) percent effect of Collins et al.
- They are explicit about scope: "we have not done a LQG treatment of the model, but only mimicked the situation using a lattice." They also state the key limit-order caveat: "It is important to notice the order in which limits are taken. If one takes a → 0 before computing the integral one is back to the continuum Lorentz-invariant calculation."
- **Spatial lattice + continuum time** (the canonical/LQG situation): "The integral in the p⁰ variable is now computed on an unbound domain and leads again to the types of effects Collins et al. pointed out." They then argue the LQG "problem of time" + physical-clock smearing may cure the p⁰ divergence (their Eqs. (7)-(8), width σ ~ Planck), calling it "just a sketch, with many implications yet to be fleshed out."
- They also give a Lorentz-violating **Pauli–Villars** example (their Eq. (10)) with induced ξ ~ m⁴/M⁴.

**Polchinski's rebuttal of the lattice claim (1106.6346, read in full — direct exchange):** "The first model is a Euclidean lattice theory, where the ratios of the time and space lattice steps are taken equal in the continuum limit. The issues discussed in this paragraph were already raised in CPSUV. A Euclidean lattice with equal steps along different axes has discrete rotational symmetries, which forbids the dimension 4 terms that would violate the Euclidean Lorentz (i.e. rotational) invariance; indeed, this is essential to the success of lattice gauge theory. However, we are interested in a Lorentzian world, and a Lorentzian lattice with equal time and space steps has no such enhanced symmetry. In terms of symmetry this is better modeled by a Euclidean lattice with unequal steps (heuristically the ratio of steps is the imaginary i), in which case the calculation of GRP confirms the large effect seen in CPSUV." And on analytic continuation from Euclidean: "a lattice propagator has an analytic structure not consistent with unitarity." On the Pauli–Villars model: "the Lorentz-violating term, in the second part of the propagator, is of relative order m⁴/M⁴ at all scales, so the small breaking is put in by hand. For the gravitational models investigated in CPSUV, the breaking is of order one at the Planck scale. If the Lorentz-breaking term in the propagator (1) is enhanced to order one at the Planck scale, then so is the induced low-energy breaking."

**Collins, Perez, Sudarsky on ordinary lattices (hep-th/0603002, Sec. 8, read in full — this is their own on-the-record position):** "if the cutoff theory is defined on an ordinary spatial lattice, boost invariance is completely broken by the rest frame of the lattice. Therefore all the issues discussed in this paper apply to the construction of the renormalized continuum limit, and fine-tuning is needed to get Lorentz invariance. This is acceptable for a mathematical definition of a QFT, but not in a theory that has a claim on being a fundamental theory." They note the Euclidean lattice functional integral only enjoys discrete axis-exchange symmetries, "enough to restrict counterterms to those that give SO(4) invariance in the continuum limit," and pose: "So one is left with a spatial lattice, or some variant, as the only obvious physical regulator of a QFT" — and therefore the search for a naturally LI physical regulator is connected to the search for QG.

**Bottom line for item 5:** The only published application of the argument to an actual lattice regulator is Gambini–Rastgoo–Pullin (1106.1417), who find the spacetime-lattice loop-induced LIV vanishes as a·μ in the equal-spacing/continuum limit — and Polchinski's comment (1106.6346) shows the effect vanishes precisely because equal Euclidean steps have the hypercubic discrete rotational symmetry that forbids dim-4 terms, which is the Euclidean analog of Lorentz invariance and does not exist for a Lorentzian lattice or for generic anisotropy; GRP themselves show a spatial lattice with continuum time reproduces the Collins-type effect. The dispute is live: GRP's central message is that non-perturbative LQG treatments may evade the argument; Polchinski counters that the EFT/Wilsonian argument is not perturbative and applies to any background-independent theory in the near-flat regime. I did not find any further published reply from GRP to Polchinski.

Lattice-QCD-side context (not a response to Collins et al., but the standard lore): the hypercubic lattice breaks Euclidean rotational invariance down to the hypercubic group (PDG Lattice QCD review, e.g. pdg.lbl.gov/2022/reviews/rpp2022-rev-lattice-qcd.pdf); rotational invariance is restored in the continuum limit for sufficiently symmetric discretizations; and there is a literature on restoring rotational invariance at the observable level (e.g., de Soto, "Restoring rotational invariance for lattice QCD propagators", JHEP 10 (2022) 069, arXiv:2204.12189; Morningstar et al., PRD 86, 054505 (2012)). This literature is about extracting rotationally invariant results from hypercubic data or about H(4)-symmetric observables — it does not discuss the Collins-type loop-induced coefficient.

## 6. DOES SYMANZIK / TREE-LEVEL IMPROVEMENT CHANGE THE LOOP-INDUCED COEFFICIENT? — NOT ADDRESSED ANYWHERE I COULD FIND [Null result, stated plainly; with analysis]

**I searched:** lattice-QCD literature on "Lorentz/rotational symmetry restoration" (de Soto 2204.12189; Morningstar et al. PRD 86 054505; PDG lattice reviews 2012/2020/2022); "Symanzik improvement at one loop" (Lüscher–Weisz; García Pérez & van Baal hep-lat/0002029; "improved staggered quarks" hep-lat/0302014; "O(a²) corrections to propagator and bilinears of Wilson/clover fermions" 0811.4499; "1-loop matching coefficients for improved staggered bilinears" hep-lat/0208018; "Testing improved actions" hep-lat/9607007; "QCD on 2+2 anisotropic lattices" hep-lat/0209013); LIV phenomenology literature citing gr-qc/0403053 (Belenchia et al. 1601.06700; Cortés–López-Sarrión 1607.00744; the Springer-cited set); a direct Brave query joining "Collins, Perez, Sudarsky" with "Brillouin"/"lattice spacing"/"improved action" (0 results); a query on Symanzik + "rotational invariance" + one-loop cancellation (0 results). 

**Finding: nobody has published the calculation "does Symanzik/tree-level improvement of a lattice action change the loop-induced Lorentz-violating coefficient?"** The question does not appear in the LIV-naturalness literature (all sides use continuum regulators or uninproved lattice actions), and the lattice-QCD improvement literature quantifies improvement only in terms of the Symanzik effective action's O(a²) terms at LOW momenta, not in terms of a Collins-type coefficient from a Brillouin-zone-wide loop integral.

**What the pieces assembled here imply (Inference, labeled as such — the closest the literature comes):**
- Symanzik improvement is defined by the Symanzik EFT: the lattice action is matched to a continuum effective action so that O(a²) terms cancel to a given order in the low-momentum expansion. At tree level it cancels O(b²) artifacts in on-shell quantities at low momentum (b = lattice spacing); at one loop, improvement coefficients are tuned to cancel O(g²a²) artifacts (this is exactly what "one-loop Symanzik improvement" means, e.g., García Pérez & van Baal; Lüscher–Weisz).
- The Collins-type coefficient ξ comes from the would-be log-divergent part of a loop integral whose dominant contribution is at loop momenta ~ the cutoff (the whole Brillouin zone for a lattice). Symanzik improvement adjusts operators that differ from the naive action by O(a²)/O(g²a²) terms — i.e., the improvement counterterms change the integrand only at O(a² p²) relative order, and the Brillouin-zone-scale contribution to the divergent part is not suppressed by p² (loop momenta are ~ π/a, not ≪ a⁻¹). Nothing in the improvement program removes or refines the leading logarithm/power piece that produces the LV coefficient; improvement modifies the low-momentum behavior (which is where the tree-level artifacts live), not the cutoff-scale behavior (where the LV coefficient lives).
- The one concrete lattice computation that exists (Gambini–Rastgoo–Pullin, above) used the NAIVE lattice action, not an improved one. Under Polchinski's analysis the vanishing of their ξ_a comes from the discrete rotational symmetry of the equal-step Euclidean lattice — a property that holds for the naive action and is destroyed by anisotropy, and I see no mechanism by which Symanzik improvement (which is about a²-suppressed terms) would restore the vanishing that is already symmetry-driven for equal steps; conversely, improvement cannot create that symmetry for a Lorentzian or anisotropic lattice where it is absent.
- Caveats (open items, not resolved anywhere I found): (i) fixed-point/classically-perfect actions have continuum-like dispersion over the whole Brillouin zone (poles at (p+2πl)² for all l — mentioned in recent fixed-point-action work, e.g. arXiv:2502.03315 discussion) — whether a "perfect" action changes the LV-loop story is untested; (ii) no one has computed a one-loop-improved lattice action's ξ̃ directly; (iii) whether the whole Collins argument applies unchanged to a Euclidean-continued lattice theory (where the true symmetry is Euclidean) versus a genuinely Lorentzian lattice is precisely the GRP–Polchinski dispute.

## Summary of the exchange (who said what)

1. **Collins, Perez, Sudarsky, Urrutia, Vucetich (2004, gr-qc/0403053)** + Collins, Perez, Sudarsky (2006, hep-th/0603002): Planck-scale preferred-frame cutoff → O(g²) percent-level dim-4 LIV via would-be-log-divergent self-energies; ruled out by c-constancy < 10⁻²⁰; fine-tuning needed unless a custodial symmetry exists (none known to them). SUSY is not discussed by them as the solution (they say no such custodial symmetry "appears to be known" in the PRL; the companion acknowledges SUSY-type protection only implicitly via the "custodial symmetry" category).
2. **Myers & Pospelov (2003, hep-ph/0301124)**: same problem in EFT language; claimed a leading-power cancellation condition. **Perez & Sudarsky (2003, PRL 91, 179101)**: that condition fails.
3. **Groot Nibbelink & Pospelov (2005, hep-ph/0404271); Bolokhov, Groot Nibbelink & Pospelov (2005, hep-ph/0505029)**: exact SUSY forbids dim-3/4 LV entirely; soft breaking re-introduces them at (m_soft/M_Pl)² ~ 10⁻³²; no CS term; dim-5 constraints 10⁻⁵–10⁻¹⁰/M_Pl.
4. **Burgess, Cline, Filotas, Matias, Moore (2002)**: dim-reg-based power-suppressed LV — Collins et al. call the cancellation a scheme artifact.
5. **Gambini, Rastgoo, Pullin (2011, 1106.1417)**: genuine lattice regulator; equal-step Euclidean lattice kills ξ̃; anisotropy gives O(aμ); spatial-only lattice reproduces Collins-type effects; clock smearing sketch. **Polchinski (2012, 1106.6346)**: the equal-step vanishing is the Euclidean discrete rotational symmetry, not available in Lorentzian physics; small-at-all-scales ≠ O(1)-at-Planck; SUSY is the only known exception.
6. **Belenchia, Gambassi, Liberati (2016, 1601.06700)**: scale separation Λ ≪ M suppresses percolation as (Λ/M)^{2n}; Λ ≲ 10¹⁰ GeV suffices for n=1; dissipation doesn't percolate. **Cortés & López-Sarrión (2017, 1607.00744)**: quantifies the fine-tunings (Λ² mass cancellation + velocity equality) that a physical LIV regulator requires at one loop; fine-tuning must be re-tuned order by order.
7. **Pospelov & Shang (2012, 1010.5249)**: LIV confined to gravity sector tames percolation. **Afshordi (2015, 1511.07879)**: unitarity/strong-coupling shielding conjecture. **Bednik, Pujolàs, Sibiryakov (2013, 1305.0011)**: emergent LI via strong coupling.
8. **Dowker, Henson, Sorkin (2004); Rovelli & Speziale (2003)**: discreteness need not imply LIV (causal sets; minimum-length analogy) — cited by Collins et al. themselves.

## Where I did NOT look (honest scope statement)

- Full text of Burgess et al. JHEP 03 (2002) 043, Myers & Pospelov PRL 90 211601 and gr-qc/0402028, Pérez & Sudarsky PRL 91 179101, Jain & Ralston hep-ph/0502106, Pospelov & Shang 1010.5249, Bednik et al. 1305.0011, Sindoni 0712.3518, Alfaro PRL 94 221302, Dowker et al. 2004, Rovelli & Speziale 2003 — I have their abstracts and/or the exact characterizations given by both sides in the papers I read in full. I did not use INSPIRE citation lists exhaustively (the API call returned irrelevant recent hits on a malformed query; I did not retry with the correct recid form).
- I could not open the PDF of hep-ph/0505029 via the pdf tool (document-extract plugin disabled) but read the complete TeX source instead, which is the same content.
- No attempt was made to read the LQG-specific papers (Gambini–Pullin 1999; Alfaro et al.) in full; they are context, not the target.

## Sources with URLs/arXiv numbers (all fetched 2026-09-13)

- gr-qc/0403053 (full HTML v4): https://arxiv.org/html/gr-qc/0403053v4 ; DOI 10.1103/PhysRevLett.93.191301
- hep-th/0603002 (full HTML v1): https://arxiv.org/html/hep-th/0603002v1 ; /pdf/hep-th/0603002
- hep-ph/0505029 (full TeX): https://arxiv.org/e-print/hep-ph/0505029 ; DOI 10.1103/PhysRevD.72.015013
- hep-ph/0404271 (abstract + journal refs): https://arxiv.org/abs/hep-ph/0404271 ; DOI 10.1103/PhysRevLett.94.081601
- 1106.1417 (full HTML): https://arxiv.org/html/1106.1417v1 ; DOI 10.1088/0264-9381/28/15/155005
- 1106.6346 (full HTML): https://arxiv.org/html/1106.6346v2 ; DOI 10.1088/0264-9381/29/8/088001
- 1601.06700 (full HTML v2): https://arxiv.org/html/1601.06700v2 ; DOI 10.1007/JHEP06(2016)049
- 1607.00744 (full HTML v2): https://arxiv.org/html/1607.00744v2 ; DOI 10.1142/S0217751X17500841
- 1511.07879 (abstract): https://arxiv.org/abs/1511.07879
- 1010.5249 (as cited in 1601.06700); 1305.0011 (abstract + journal info); hep-ph/0502106 (as cited)