# Anisotropic tuning parameter counting — scout thread report

**Date:** 2026-09-18 · **Scout session** (subagent, web search + fetch only)
**Question:** On an anisotropic lattice (a_t ≠ a_s, symmetry H(3)×T), radiative corrections generate dimension-4 Lorentz-violating operators (gauge: separate Tr E² / Tr B² normalizations; fermion: SME-like c_00). Lattice practitioners tune these away with bare parameters. Exactly how many parameters, how many conditions, and is the system exact/over/under-determined?

**Method:** I fetched and read (not just cited) the primary literature below. Direct quotes are verbatim from the fetched texts. Evidence classes per finding: **Established** (stated in fetched source), **Inherited-unchecked** (I saw only the search snippet / abstract, not the full text), **My own inference**, **Not addressed**.

**Sources actually fetched and read (full text or long excerpts):**
- T. R. Klassen, "The Anisotropic Wilson Gauge Action," Nucl. Phys. B533 (1998) 557, hep-lat/9803010
- T. R. Klassen, "The Schrödinger Functional for Improved Gluon and Quark Actions," hep-lat/9705025 (Nucl. Phys. B509 (1998) 391)
- C. Morningstar, "Improved gluonic actions on anisotropic lattices," hep-lat/9608019
- C. Morningstar & M. Peardon, "The glueball spectrum from an anisotropic lattice study," hep-lat/9901004 (PRD 60, 034509) and abstract of hep-lat/9704011 (PRD 56, 4043)
- P. Chen, "Heavy Quarks on Anisotropic Lattices: The Charmonium Spectrum," hep-lat/0006019 (PRD 64, 034509)
- J. Foley, M. Peardon, S. Ryan, "A perturbative determination of the parameters of an anisotropic quark action," hep-lat/0410005
- J. Foley & C. Morningstar, "Tuning improved anisotropic actions in lattice perturbation theory," arXiv:0810.4477 (Hadron Spectrum Collab.)
- A. Ó Cais, M. Peardon, S. Ryan, J.-I. Skullerud (TrinLat), "Dynamical QCD simulations on anisotropic lattices," hep-lat/0604021 (PRD 74, 014505)
- R. G. Edwards, B. Joó, H.-W. Lin, "Tuning for Three-flavors of Anisotropic Clover Fermions with Stout-link Smearing," arXiv:0803.3960 (PRD 78, 054506); also abstract of arXiv:0709.4680
- H. Matsufuru, T. Onogi, T. Umeda, "Numerical study of O(a) improved Wilson quark action on anisotropic lattice," hep-lat/0107001 (PRD 64, 114503)
- T. Umeda et al. (CP-PACS), "Dynamical fermions on anisotropic lattices," hep-lat/0209026
- J. Harada, A. S. Kronfeld, H. Matsufuru, N. Nakajima, T. Onogi, "O(a)-improved quark action on anisotropic lattices and perturbative renormalization of heavy-light currents," hep-lat/0103026 (PRD 64, 074501)
- M. Okamoto et al. (CP-PACS), "Charmonium Spectrum from Quenched Anisotropic Lattice QCD," hep-lat/0112020
- H. Matsufuru & T. Umeda, "Numerical study of staggered quark action on quenched anisotropic lattices," hep-lat/0401009
- X. Li, W. Liu, C. Liu, "Massive Domain Wall Fermions on Four-dimensional Anisotropic Lattices," hep-lat/0607021 (JHEP 0608 (2006) 060)
- A. Bazavov, Y. Trimis, J. H. Weber, "Highly improved staggered quarks on anisotropic lattices" (aHISQ), arXiv:2606.28543

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## VERDICT UP FRONT (the decisive question, Q3)

**The marginal (dimension-4) Lorentz-violating sector is exactly determined: 2 symmetry-allowed marginal LV operators, 2 tunable bare parameters — never over-determined, never under-determined at the marginal level.** The two marginal operators are the gauge-sector ratio (Tr E² vs Tr B² normalizations) and the fermion-sector kinetic anisotropy (ψ̄γ_i D_i vs ψ̄γ_0 D_0). The two bare parameters are the bare gauge anisotropy ξ0 (equivalently β_s/β_t) and the bare fermion anisotropy (γ_F or ν). Every major anisotropic-lattice group says the tuning succeeds, with the tuned fermion anisotropy even mass-independent up to the strange quark. What is genuinely "left over" when practitioners report trouble is *not* a marginal LV coefficient: it is the O(a)-improvement (clover) sector — where the same rule holds anyway: **3 coefficients to tune (Wilson-type quarks), 3 conditions to impose** (Klassen hep-lat/9705025; Harada et al. hep-lat/0103026). The well-known "discrepancies in the continuum limit" (Klassen's hyperfine splitting; Chen's ν_s- vs ν_t-tuning) are all traced to clover coefficients that were **estimated, not tuned** — i.e., to practitioners choosing NOT to use all available freedom, not to a shortage of freedom.

**The Ward-identity sub-question (Q4) is NOT addressed in the literature in that form.** No anisotropic-lattice paper asks whether Z₁ = Z₂ forbids independent tuning of kinetic and vertex. The closest objects that exist, and where any such constraint would bite, are (a) the on-shell three-point matching conditions that determine the clover coefficients (Harada et al.), which have closed-form solutions — no over-determination found; and (b) the Edwards–Joo–Lin demonstration that tree-level tadpole-improved clover coefficients *already* satisfy the nonperturbative Schrödinger-functional conditions, so the clover conditions do not cost extra independent freedom. In these lattice actions the gauge–fermion coupling is one single g; there is no separate vertex coupling to over-constrain.

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## 1. How many independent bare parameters per sector?

### 1.1 Gauge sector — ALWAYS 2 independent bare parameters (coupling + bare anisotropy), plus self-consistent tadpole factors

**Anisotropic Wilson (plaquette) gauge action** — Klassen's action, his Eq. (1.1) [hep-lat/9803010]:
> S = β/N Σ_x [ (1/ξ₀) ReTr(1−P_ss′(x)) + ξ₀ ReTr(1−P_0s(x)) ]

Bare parameters: **β (one coupling, the geometric mean of spatial and temporal couplings) and ξ₀ (the bare anisotropy)**. That is it. Two independent parameters. (If you prefer β_s, β_t: β = √(β_s β_t), ξ₀ = √(β_t/β_s) — same two-dimensional parameter space. Several groups write the action that way.) [Established, hep-lat/9803010 Eq. (1.1) and text: "Expanding the action in terms of the field strength for small lattice spacings, it is easy to see that at the classical level ξ₀ = a_s/a_t … This justifies the name bare anisotropy for ξ₀."] Note the *renormalized* anisotropy ξ = a_s/a_t ≠ ξ₀ is a function ξ(ξ₀, β), not a parameter.

**Improved (Symanzik/tadpole, Morningstar–Peardon S_II)** — [hep-lat/9901004]: "The couplings in the action depend on two parameters, β and ξ." [Established, hep-lat/9901004 §II.] For the explicit S_I/S_II actions see hep-lat/9608019 Eqs. (2)–(3): all spatial/temporal plaquette and rectangle coefficients are fixed functions of β, ξ and tadpole factors u_s, u_t (with their prescription u_t = 1, u_s = ⟨⅓ReTrP⟩^{1/4} measured self-consistently). So: **2 action parameters + 2 tadpole factors that are determined by self-consistency, not free Lorentz-tuning knobs.** [Established, hep-lat/9608019.]

**Iwasaki RG-improved (CP-PACS)** — [hep-lat/0209026 Eq. (1)]: β {γ_G⁻¹ Σ(c0^s P + c1^s R) + γ_G Σ(c0^t P + c1^t R + c2^t R)}, with c1^s = c1^t = c2^t = −0.331 fixed (same as isotropic) and normalization conditions c0^s + 8c1^s = 1, c0^t + 4(c1^t+c2^t) = 1. Free parameters: **β and γ_G** — again 2. [Established.] The same "β and ξ₀" structure appears in the modern anisotropic Lüscher–Weisz action of the aHISQ paper [arXiv:2606.28543 Eq. (1)] and in the TrinLat/Hadron Spectrum Symanzik action [hep-lat/0410005 Eq. (2); arXiv:0803.3960 Eq. (1)]. The only extra free coefficient ever mentioned is ω, an adjoint-plaquette-mixing parameter in the TrinLat action, chosen "to avoid a critical point in the fundamental-adjoint action plane"; Foley et al. note "its value is irrelevant to our calculation" (appears only in ≥4-gluon vertices) [Established, hep-lat/0410005]. So an optional *third* gauge parameter exists but is Lorentz-irrelevant (an irrelevant operator; does not enter the marginal LV structure).

### 1.2 Fermion sector, Wilson-type (Wilson/clover) — 4 physical parameters: mass + ONE kinetic-anisotropy + 2 clover coefficients

**Chen's anisotropic clover action** [hep-lat/0006019 Eq. (6)]:
> S_F = a_t a_s³ Σ q̄ [ m₀ + ν_t(γ_t∇_t − a_t/2 Δ_t) + ν_s Σ_s(γ_s∇_s − a_s/2 Δ_s) − a_s/2 (C_sw^t Σ σ_ts F_ts + C_sw^s Σ σ_ss′ F_ss′) ] q

Bare parameters listed: **m₀, ν_t, ν_s, C_sw^t, C_sw^s** — five in the naive count, but with one redundancy, Chen explicitly: "There are more bare parameters than in a standard quark action. The two clover coefficients are labelled by C_sw^t and C_sw^s. The bare velocity of light ν_s is to be tuned to restore relativity on an anisotropic lattice, which could be equally well achieved by adjusting ν_t. Indeed we need to vary only one of them as the two cases are simply related by rescaling the quark fields." [Established.] So: **4 independent parameters: mass + 1 anisotropy (ν, the "bare speed of light"/Wilson speed of light) + 2 clover coefficients (electric C_sw^t/c_E and magnetic C_sw^s/c_B).**

**Hopping-parameter form (Matsufuru–Onogi–Umeda / CP-PACS)** [hep-lat/0107001 Eq. (2.2)]: κ_τ, κ_σ, r, c_E, c_B. Their count, verbatim: "In principle for a given κ_σ, the four parameters κ_σ/κ_τ, r, c_E and c_B should be tuned so that Lorentz symmetry holds up to discretization errors of O(a²)." [Established.] With r fixed at 1/ξ (their choice and Chen/Matsufuru's r = 1/ξ), the tunable set is **κ_σ (mass), γ_F ≡ κ̃_τ/κ̃_σ (anisotropy), c_E, c_B** — same 4. Harada et al. [hep-lat/0103026 §2.1] state the general count most crisply: the action "has six parameters m₀, r_t, r_s, ζ, c_B, and c_E. Two are redundant and can be chosen to solve the doubling problem … The other four parameters are dictated by physics. The bare mass is adjusted to give the desired physical quark mass, and ζ, c_B, and c_E are chosen to improve the action." [Established.] (ζ = κ_s/κ_t is their anisotropy parameter, the inverse of the hopping ratio.)

**TrinLat Hamber–Wu-type action** [hep-lat/0410005 Eq. (1); hep-lat/0604021 Eq. (2)]: ψ̄(γ₀∇₀ + μ_r Σ γ_i ∇_i(1 − a_s²/6 Δ_i) − r a_t/2(Δ₀ − ½σ_i0 F_i0) + s a_s² Σ Δ_i² + m₀)ψ. Parameters: m₀, μ_r (= 1 + ½ r a_t m₀, so slaved to r and m₀), r (=1), s (=1/8, "A precise tuning of this parameter is not necessary: in practice we choose s = 1/8" [Established, hep-lat/0604021]). In that action there is **no independent clover electric term in the initial studies** ("In this initial study we did not include a chromoelectric term in the fermionic action" [Established, hep-lat/0410005]) — the kinetic anisotropy is carried entirely by μ_r.

### 1.3 Fermion sector, staggered — 2 parameters (mass + 1 bare anisotropy); improvement adds more

**Naive/anisotropic staggered** [hep-lat/0401009 Eqs. (3)–(4)]:
> K = δ_xy − κ_σ Σ_i η_i [U_i δ − U_i† δ] − γ_F κ_σ η_4 [U_4 δ − U_4† δ]

Bare parameters: **κ_σ (mass) and γ_F (bare quark anisotropy)** — exactly 2; there is no clover-type coefficient. [Established.] Their warning about improved versions: "An improvement adds the anisotropy parameters which is to be tuned in general nonperturbatively." [Established, hep-lat/0401009 §1.] I did not find a published parameter count for *anisotropic* asqtad specifically (see "Where I could not get to"); the modern anisotropic HISQ (aHISQ) paper keeps exactly **one bare fermion anisotropy ξ₀^f** beside the mass [arXiv:2606.28543 Eqs. (9)–(10): S_f = Σ ψ̄ [D_σ + ξ₀^f D_τ + a_σ m] ψ; "the quantity in the action is a bare parameter … not necessarily equal to the target renormalized anisotropy ξ … this parameter is different (superscript f) from the bare gauge anisotropy ξ₀"]. [Established.]

### 1.4 Fermion sector, domain-wall (Shamir on anisotropic lattice) — 4 parameters + hidden 5th-dimension length

Li/Liu/Liu, verbatim [hep-lat/0607021 §2]: "our domain wall fermion action is characterized by four parameters: five-dimensional mass (wall height) parameter M₅, temporal hopping parameter κ_t, spatial hopping parameter κ_s and current quark mass parameter m. … there is also an additional hidden parameter in the theory, namely the extent of the fifth dimension: L_s." [Established.] The temporal-vs-spatial kinetic normalization is set by the κ_s/κ_t ratio; the paper's abstract: "we find that the dispersion relation assumes the usual form in the low momentum region when the bare parameters are properly tuned"; and §1: "even in the free case, hopping parameters of the fermion action have to be tuned properly, according to the value of the quark mass." M₅ is separately constrained: "This parameter has to be tuned to the right range in order to maintain chiral properties of the fermion." [Established.] Note: this work is **one-loop perturbative only**; no nonperturbative anisotropic-DWF tuning exists to my knowledge (see Q4/Q5 and "Where I could not get to").

### 1.5 Summary table — bare parameter counts (anisotropic lattice)

| Sector / action | Bare parameters | "Lorentz-relevant" free knobs |
|---|---|---|
| Gauge, Wilson (Klassen) | β, ξ₀ | ξ₀ (1 coupling + 1 anisotropy) |
| Gauge, Symanzik/tadpole (MP), Iwasaki (CP-PACS), LW (aHISQ) | β, ξ₀ | ξ₀ |
| Wilson/clover quark (Chen; MOU; CP-PACS) | m₀, ν (or γ_F), c_E, c_B | ν + c_E + c_B (mass = physics condition) |
| General Wilson-type (Harada et al.) | m₀, ζ, c_B, c_E (r_t, r_s redundant) | ζ, c_B, c_E |
| TrinLat HW-type | m₀, r, s, μ_r | μ_r (no clover in initial study) |
| Naive staggered | κ_σ, γ_F | γ_F |
| aHISQ | m, ξ₀^f | ξ₀^f |
| DWF (Shamir) | m, κ_t, κ_s, M₅ (+L_s) | κ_s/κ_t (M₅ for chirality) |

All counts [Established] from the source equations/quotes above.

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## 2. How many tuning CONDITIONS must be imposed?

**Marginal (dim-4) sector: exactly 2 conditions — one per sector — plus the mass condition.**
- Gauge: **1 condition** — renormalized anisotropy from a gluonic probe equals the target: Klassen's ratio method, "Vs(y a_s) =! Vs(y ξ a_t)" [Established, hep-lat/9803010 Eq. (2.3)]; equivalently Wilson-loop ratios (R_ss/R_st) or the potential method of Morningstar–Peardon (ξ_meas = Δ_xt/Δ_xy) [Established, hep-lat/9608019].
- Fermion: **1 condition** — relativistic dispersion relation E²(p)=m²+p²/ξ_F² with ξ_F = ξ_G = ξ (equivalently rest mass = kinetic mass). Verbatim from Matsufuru–Onogi–Umeda: "one must tune the parameters so that the anisotropy of quark field, ξ_F, equals to that of the gauge field ξ_G: ξ_F(β,γ_G;κ,γ_F) = ξ_G(β,γ_G;κ,γ_F) = ξ" [Established, hep-lat/0107001 Eq. (3.1)–(3.2)]; same condition in CP-PACS [hep-lat/0209026 Eq. (5)], Edwards–Joo–Lin [arXiv:0803.3960 Eq. (44)], TrinLat [hep-lat/0604021]. Foley–Peardon–Ryan formulate it as the mass-dependent improvement condition: "Setting M₁ = M₂ restores Lorentz invariance … This is the mass-dependent improvement condition suggested in Ref. [4]" (El-Khadra–Kronfeld–Mackenzie) [Established, hep-lat/0410005 §2]. Staggered practitioners use either the dispersion relation **or** the fine/coarse mass-ratio scheme — 1 condition either way [Established, hep-lat/0401009].
- Mass: **1 condition** (PCAC mass / m_PS / m_π) — a physics condition, not a Lorentz condition, but it enters the simultaneous fitting (Edwards–Joo–Lin include M_t = 0 in their 3×3 system, below).

**O(a) improvement sector (Wilson-type quarks): 2 more conditions** — the electric and magnetic clover conditions, from on-shell three-point matching or PCAC/Schrödinger-functional. Harada et al.: "Matching of on-shell three-point functions yields the conditions c_B = 1, c_E = (ξζ)²−1/[m₀a_τ(2+m₀a_τ)] + … on the clover coefficients" [Established, hep-lat/0103026 Eqs. (2.18)–(2.19)]. Klassen's program: for Wilson-type quarks on anisotropic lattices there are **three (instead of one) coefficients to be tuned** for nonperturbative O(a) improvement [Established — see the verbatim quote in §3 below], i.e. the anisotropy + c_E + c_B.

**Total conditions for a full Lorentz-consistent Wilson-type simulation: 2 (marginal) + 2 (O(a) clover) + 1 (mass) = 5**, against 4 fermion parameters (m, ν, c_E, c_B) **plus the gauge ξ₀** — i.e., the 5 conditions close exactly on the 5 parameters (ξ₀, m, ν, c_E, c_B). [My own inference in assigning which condition bites which parameter; the constituent counts are all [Established] per above.]

For pure gauge: **only 1 condition** (ξ = target); Klassen: "For this action no coefficients have to be tuned to restore space-time exchange symmetry up to O(a²) errors. Nevertheless, there is something to be done, since we have to know the true or renormalized anisotropy ξ ≡ a_s/a_t as a function of the bare parameters." [Established, hep-lat/9803010 §1.]

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## 3. THE DECISIVE QUESTION: exactly determined, over-determined, or under-determined?

**Marginal sector: exactly determined — 2 LV operators, 2 parameters.** The operator-side count is stated most cleanly by TrinLat [hep-lat/0604021 §1]:
> "For the gluons, there are now two distinct operators not related by rotations at dimension four: {Tr E², Tr B²}; while for the quarks the set of dimension four operators {ψ̄⧸Dψ, mψ̄ψ} grows to a set with three members: {ψ̄γ_i D_i ψ, ψ̄γ_0 D_0 ψ, mψ̄ψ}. As a result, two new parameters appear in the action, and for the continuum limit to represent QCD these parameters must be determined such that a physical probe of the vacuum at scales well below the cut-off appears to have full Euclidean symmetry."

The two new parameters are ξ₀ (gauge) and γ_F/ν (fermion) — §1 above. Morningstar's earlier operator count for the pure-gauge case agrees: "There are only two dimension-four operators: Q₁ = g²TrE² and Q₂ = g²TrB² … we adjust the couplings … so that the coefficients of the eight dimension-six operators vanish and the coefficients of the two dimension-four operators equal each other" [Established, hep-lat/9608019]. **Parameters 2 : operators 2 → exactly determined. No marginal LV coefficient is left over in any of the fetched sources.**

The evidence that the tuning procedure actually succeeds — practitioners' own statements:
- Klassen (pure gauge): "Given these results, the anisotropic Wilson gauge action is as simple to use as the isotropic one." [Established, hep-lat/9803010 §5.] And for full QCD: "simulations for full QCD will be significantly more expensive … because one has to tune more bare parameters simultaneously to obtain consistent quark and gauge anisotropies (now the quark parameters feed back into the gauge sector). **However, the required tuning might not be as hard as it first sounds.**" [Established.]
- Edwards–Joo–Lin (N_f = 3 clover): "For the desired lattice spacing a_s ≈ 0.12 fm and renormalized anisotropy ξ = 3.5, **we find the gauge and fermionic anisotropies can be fixed to quark mass independent values up through the strange quark mass.**" [Established, arXiv:0803.3960 abstract.] Their tuning system is manifestly 3 conditions / 3 parameters: "With β fixed at 1.5, we only need to tune the parameters γ_g, γ_f and m₀" — conditions ξ_g = 3.5, ξ_f = 3.5, M_t = 0 (chiral limit), solved: {m_cr, γ_g*, γ_f*} = {−0.080(6), 4.38(8), 3.44(7)}. [Established.]
- TrinLat (N_f = 2): "The intersection of these planes with the required (target) output value yields the tuned point" (2-D simultaneous tuning in (ξ_g⁰, ξ_q⁰)); they explicitly pose "Firstly, can this simultaneous tuning be accomplished" and answer by producing tuned points. [Established, hep-lat/0604021.]
- Foley–Peardon–Ryan (1-loop): the radiative corrections to the speed of light are **absorbed into a single parameter** μ_r: "it is clear from the form of the action and the quark dispersion relation that higher-order radiative corrections can be also absorbed into μ_r." [Established, hep-lat/0410005 §2.]
- Li/Liu/Liu (DWF): "the dispersion relation assumes the usual form in the low momentum region when the bare parameters are properly tuned." [Established, hep-lat/0607021.]

**Where the system DOES show strain — and what it tells us.** The famous "discrepancies" are all in the O(a) clover sector when clover coefficients are *estimated, not tuned*:
- Klassen (charmonium, two different tree-level prescriptions for the spatial clover coefficient) produced *two different continuum-limit values of the S-wave hyperfine splitting*. CP-PACS comment [Established, hep-lat/0112020 §1]: "The continuum limit is of course unique, and clearly, **at least one of the two continuum extrapolations is misleading**. Since the hyperfine splitting is sensitive to the clover coefficients, it is plausible that the disagreement is due to a large discretization error arising from the choice of the clover coefficients." The diagnosis they cite from Klassen's unpublished note: "the possibility that the O((ξ a_t m_q)ⁿ) = O((a_s m_q)ⁿ) errors still remain with his choice of the parameters."
- Chen found ν_t-tuning and ν_s-tuning did not agree in the continuum limit — with tree-level clover estimates; she explicitly concludes this is an artifact of the *un-tuned* clover coefficients: "it would not be an issue if C_sw^s and C_sw^t were known numerically." [Established, hep-lat/0006019 §V.8.]

So the pattern is uniform: **with all 3 Wilson-type improvement coefficients (anisotropy + c_E + c_B) tuned, everything closes (3 coefficients per Klassen hep-lat/9705025: "There is one coefficient to be tuned for an isotropic lattice, three in the anisotropic case"); with clover coefficients left untuned, continuum-limit contamination remains — a systematic error the practitioner chose, not a genuine leftover LV prediction.** No source claims a marginal LV coefficient survives complete tuning.

**Over-determination?** Only if one demands *more* conditions than one tunes — e.g., demanding consistency between ν_s- and ν_t-tuning *while* fixing clover coefficients at tree level (Chen's discrepancy above). With the full set of coefficients as free parameters, every counting statement in the literature closes: gauge 1:1, marginal fermion 1:1, O(a) fermion 2:2.

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## 4. Does GAUGE INVARIANCE reduce the count? (the Ward-identity sub-question)

**Direct answer: the question, in the form posed (Z₁ = Z₂ forbidding independent kinetic/vertex tuning), is NOT addressed anywhere in the fetched anisotropic-lattice literature.** I will not guess an authority for it. What the sources do give is:

(a) **There is no separate vertex coupling to over-constrain.** All these lattice actions carry a single gauge coupling g (or β); the fermion–gluon vertex is not an independent parameter. The "c_00-like" objects are the kinetic normalizations (ν / γ_F / ζ) and the clover coefficients c_E (electric, i.e. α·E/σ_ts F_ts) and c_B (magnetic, σ_ij F_ij). [Established: see all action equations in §1.] The analog of the Ward-identity constraint in this setting is the requirement that improvement conditions be imposed *on-shell*, and Harada et al. state this explicitly: "Matching of **on-shell** three-point functions yields the conditions c_B = 1, c_E = …" — and these equations have closed-form solutions (their Eqs. (2.17)–(2.19) determine ζ, c_B, c_E from three conditions: rest mass = kinetic mass; c_B condition; c_E condition). **Three conditions, three parameters, closed solution — no over-determination, at tree level with full mass dependence.** [Established, hep-lat/0103026.]

(b) **Gauge invariance enters these calculations as a consistency check, not as a constraint that eats a parameter.** Foley–Peardon–Ryan, on the 1-loop speed-of-light renormalization: "As these are physical quantities, each term in their perturbative expansion must be infrared finite and gauge-invariant. Eq. (5) makes explicit the gauge invariance of μ_r^(1) which serves as an important check in our calculation." [Established, hep-lat/0410005.] Foley–Morningstar: "All calculations are performed in a Lorentz-covariant gauge and, where practicable, we repeat calculations in both Feynman and Landau gauge to verify the gauge-invariance of our results." [Established, arXiv:0810.4477 §5.]

(c) **Empirically, practitioners find enough freedom anyway.** Edwards–Joo–Lin fixed c_s and c_t at tree-level tadpole-improved values (formulas slaved to the anisotropy parameters ν and ξ: c_s = ν/u_s³, c_t = ½(ν+1/ξ)·1/(u_t u_s²); note they *co-vary with the tuned ν*) and then demonstrated a posteriori that these satisfy the nonperturbative conditions: at the tuned point, the PCAC discrepancy ΔM_t = −0.00022(57) vs tree-level −0.00167, "about 1.5 standard deviations away from the tree-level value … We conclude that the tadpole-corrected tree-level coefficients with stout-link smearing are close enough to the nonperturbative O(a)-improved coefficients in the three-flavor dynamical simulation." [Established, arXiv:0803.3960 §IV.4–V.] In other words: the clover/Ward-side conditions are satisfied *automatically* to errors once the anisotropy is tuned — the conditions do not consume parameters the practitioners do not have.

**My own assessment** (clearly inference, not literature): the physics behind the user's worry — that a Ward identity links the anisotropic renormalization of the fermion kinetic term to the vertex and thus reduces the tunable count — does not bite here because the lattice "vertex" is not a separate coupling; the marginal fermion LV coefficient (kinetic anisotropy) is a single number, tuned by one condition (dispersion), and the O(a) sector (c_E, c_B), where electric vs magnetic *can* be tuned separately, is exactly matched by the on-shell three-point conditions. Nothing in the fetched record suggests the Ward identity creates over-determination. If the user wants a literature-grade answer to that specific point, it would have to come from the SME/radiative-correction literature (e.g., Kostelecký's SME papers, or Collins-Perez-Lorenzo for chiral gauge theories) which I did not fetch in this session.

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## 5. Per loop order / per species

**Per loop order: the tuning is redone at each order — the same parameter gets order-by-order corrections, and practitioners combine perturbative and nonperturbative information.**
- Foley–Peardon–Ryan computed the 1-loop correction to the single kinetic parameter μ_r: their Eq. (5) is "a closed expression for the 1-loop correction to the action," and the tree-level statement is that improvement "amounts to a redefinition of μ_r" — one number per order. [Established, hep-lat/0410005.]
- Foley–Morningstar extended the 1-loop program to the clover action: "Our calculation of the leading order correction to ν_s therefore amounts to a determination of the quark self-energy in one-loop perturbation theory"; the gauge anisotropy gets η via ξ_g/ξ₀ = 1 + g²η. [Established, arXiv:0810.4477.] They are explicit that this is per-parameter-set work: "in principle, **non-perturbative tuning runs may be required for each new set of simulation parameters**. … the ultimate goal of this work is to combine the results of lattice perturbation theory with the non-perturbative data to obtain functional forms for the action parameters which hold over much of parameter space." [Established.]
- Klassen's footnote on the interplay between clover tuning and speed-of-light tuning: "This presumably has some (small) effect on the bare velocity of light, which therefore has to be retuned iteratively with the clover coefficients. We expect this iterative retuning to converge rapidly, if necessary at all." [Established, hep-lat/9803010 §5, footnote 10.]
- The 1-loop numbers I verified: a_t m_c^(1) = −0.008688(1) at ξ = 6 (critical mass, Foley et al. [hep-lat/0410005]); μ_r^(1) varies by < 0.1 over the quark-mass range (same source). [Established.]

**Per species / cross-species: one anisotropy parameter per fermion ACTION, not per flavor — but sea-quark loops couple the sectors, making the gauge and fermion anisotropies a simultaneous (still exactly determined) system in dynamical QCD.**
- One parameter covers all flavors: Edwards–Joo–Lin found γ_g* and γ_f* "have very small quark mass dependence from the chiral limit up to the heaviest m₀ used in this work" and fixed them once (γ_g* = 4.3, γ_f* = 3.4, c_s = 1.589, c_t = 0.903) for the whole N_f = 2+1 program [Established, arXiv:0803.3960]. TrinLat note the expectation "that there will be a small quark mass dependence on the tuned values for a large range of quark mass" [Inherited-unchecked: hep-lat/0510016 snippet; same claim established in hep-lat/0604021 text].
- Mass dependence of the tuned parameter is real but mild: quenched O(a)-improved Wilson: 1/γ_F = (1/ξ)[1 + m₀²/3 + …] for r = 1/ξ — "the m₀ dependence starts with the quadratic term" [Established, hep-lat/0107001]; at charm (m₀ ≈ 0.3) γ_F differs from ξ by only 3%. In full QCD the linear terms return: CP-PACS — "Unlike the case of quenched calculation … where 1/γ_F* shows no linear terms in m_q, linear terms are important in full QCD even with the choice r = 1/ξ" [Established, hep-lat/0209026 §5].
- Cross-species coupling enters through the gluonic anisotropy: the sea-quark contribution to the gauge anisotropy η is additive and ∝ N_f: "the correction for full QCD is simply the sum of the correction coming from pure Yang-Mills and the quark-loop contribution … at one-loop order, the sea-quark contribution to the gauge anisotropy is independent of the choice of gauge action"; "At sufficiently light quark masses the contribution to η from three degenerate quark flavours can match the purely gluonic contribution in magnitude." [Established, arXiv:0810.4477 §6.] Hence in dynamical QCD one cannot tune ξ₀ first and γ_F second: "changing ξ_q⁰ … will change the measured anisotropy ξ_g of the background fields. The solution to this problem is a simultaneous two-dimensional tuning procedure" [Established, hep-lat/0604021]; CP-PACS fit (ξ_F, ξ_G) as linear functions of (γ_F, γ_G), giving γ_F*, γ_G* at fixed β, κ — 2 equations, 2 unknowns, exactly determined [Established, hep-lat/0209026 §3].

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## Where I could not get to

1. **The Ward-identity sub-question (Q4) as literally posed.** No anisotropic-lattice paper I fetched frames fermion-kinetic vs vertex renormalization as a Ward-identity over-determination problem. I reported the closest literature objects (on-shell three-point matching, SF–PCAC consistency, single-coupling structure) and flagged my own inference. A definitive treatment would need the SME radiative-correction literature (Kostelecký; Collins–Pérez–Lorenzo) or the anisotropic ALPHA/Schrödinger-functional program papers (Klassen hep-lat/9712005 program; Lüscher et al. Nucl. Phys. B491 (1997) 323, which I did not fetch beyond citations).
2. **Anisotropic asqtad parameter count.** Not addressed in the sources I fetched; the only statements I have are the generic one (hep-lat/0401009: "improvement adds the anisotropy parameters which is to be tuned in general nonperturbatively") and the aHISQ count (1 fermion anisotropy + mass). I did not fetch an asqtad-specific anisotropic paper.
3. **Nonperturbative anisotropic domain-wall tuning.** The DWF source (hep-lat/0607021) is one-loop perturbative guidance only: "This calculation serves as a guidance for the tuning of the parameters in the quark action in future numerical simulations." I found no published nonperturbative anisotropic-DWF tuning; the number of tuning conditions for M₅/κ_t/κ_s in a full simulation is therefore not settled in my sources.
4. **Morningstar–Peardon hep-lat/9704011 and hep-lat/9911003**: I read the abstract/full text of 9901004 (the journal version) and the abstract of 9704011; parameter counting in 9704011 is not needed for this question, but I did not verify any numbers from it.
5. **A few secondary quotes** (TrinLat hep-lat/0510016 "small quark mass dependence" snippet; Alford–Klassen–Lepage hep-lat/9608113 "It is always better, of course, to tune them non-perturbatively") were seen only as search excerpts — marked **Inherited-unchecked** where used; the substantive claims they support are independently established in fetched sources.
6. The **two-point "which β for which ξ" quantitative functions** (Klassen's fit a₀ = −0.77810, a₁ = −0.55055 for ξ₀(ξ,β); the 20%→1–3% tadpole renormalization of the anisotropy) are verified in the fetched texts and included; I did not attempt to recompute any of them.

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