Relativistic spin precession in homogeneous background fields

Alan Kostelecky; Arnaldo J. Vargas; Yunhua Ding

REVIEW 2 major objections 4 minor 65 references

Relativistic spin precession in homogeneous background fields

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read One extended BMT equation now covers spin precession in electromagnetic plus Lorentz-violating backgrounds.

desk verdict A genuinely new covariant BMT extension with SME backgrounds; the completeness of the term enumeration is asserted rather than proven, so Eq. (35) is well-motivated but not rigorously unique. read the letter →

arxiv 2509.05098 v1 pith:4TE2MEYU submitted 2025-09-05 hep-ph

Yunhua Ding , Alan Kostelecky , Arnaldo J. Vargas This is my paper

classification hep-ph

keywords spinprecessionBMTequationLorentzviolationCPTStandard-ModelExtensionelectricdipolemomentstorageringDiracfermion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove that the spin of a charged Dirac particle in a homogeneous electromagnetic field, plus constant Lorentz-violating background fields, precesses according to a single covariant equation, Eq. (35). That equation extends the BMT equation by adding leading-order terms from Lorentz-violating coefficients of mass dimension 3 through 6, including field-dependent combinations for dimensions 5 and 6. The authors establish the result by enumerating independent observer-covariant terms consistent with the spin's orthogonality constraints, fixing the unknown coefficients against a known storage-ring precession frequency, and then promoting b, H, d, g to F-dependent forms via an established field-redefinition protocol. If correct, the equation gives experimenters a model-independent template for searching Lorentz and CPT violation in storage-ring spin-precession experiments, and it reduces to the standard BMT equation when all new coefficients vanish.

What carries the argument

The argument is carried by a term-enumeration algorithm. For each coefficient family, the paper constructs all observer-covariant contractions built from the particle 4-velocity Uμ, spin Sμ, field strength Fμν, the Lorentz-violating coefficients, and the metric or Levi-Civita tensor. Each candidate must be linear in S, F, and the coefficients, have even overall parity (so b and d appear with ε), and satisfy the constraints in Eq. (18), which enforce S·dS/dτ=0 and U·dS/dτ=0 under the assumption that homogeneous coefficients leave dUμ/dτ unchanged. The surviving terms carry unknown constants, fixed by matching the storage-ring result (17), and the d=5,6 pieces follow from the replacement in Eq

What would settle it

Compute the Dirac Hamiltonian for the SME Lagrangian used here, perform a Foldy-Wouthuysen transformation to extract the spin-operator equation of motion at leading order in the coefficients, and compare term-by-term with Eq. (35); a mismatch would show the enumeration is incomplete. Alternatively, measure the storage-ring radial spin-precession frequency (38) at two different magnetic-field values and check that it grows linearly in B with the predicted coefficient combination ˇH33F.

Watch

Extended reading notes

Core claim

The central claim is that Eq. (35) is the complete leading-order relativistic equation for dSμ/dτ for a charged Dirac particle with magnetic and electric dipole moments moving in homogeneous electromagnetic and Lorentz-violating background fields, valid in any inertial frame. It includes the standard BMT terms, then adds contributions proportional to bμ, Hμν, dμν, gμνρ and their dimension-5/6 field-strength-coupled counterparts bF, HF, dF, gF. Each added term is linear in the coefficients, linear in F, and linear in S, and the structure preserves the required velocity-spin orthogonality. The authors fix the overall constants by requiring the laboratory-frame 3-vector form to reproduce the kn

Load-bearing premise

The derivation stands on two linked premises: that the enumerated list of independent covariant terms is complete for each coefficient at leading order, and that homogeneous Lorentz-violating coefficients leave the particle's 4-acceleration unchanged; if either fails, Eq. (35) misses corrections of the same order.

Editorial extensions

If this is right

Storage-ring searches for muon or proton electric dipole moments and g−2 can absorb Lorentz-violating shifts by fitting the coefficient combinations in Eq. (38), such as ˇH03, ˇbF, ˇgF, and ˇHF.
In the e→0 limit, the same equation describes neutral spin-1/2 particles in Lorentz-violating backgrounds, since the new terms are independent of the charge.
Each odd-indexed coefficient is CPT-odd, so measurements of the corresponding precession terms separate CPT violation from CPT-even Lorentz violation.
The storage-ring application shows that some coefficient combinations, e.g. ˇb11F and ˇb22F in the idealized ring, appear only as a sum and cannot be measured independently in that geometry.
When all new coefficients vanish, Eq. (35) reduces to the BMT equation, so the extension is backward-compatible with existing precision tests.

Reading between the lines

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

Beyond the paper: the same enumeration could be extended to subleading operators with d≥7, where couplings with two powers of F or derivative-dependent coefficients would enter; nothing in the method prevents that extension.
Beyond the paper: because Earth's rotation modulates laboratory-frame coefficients at sidereal frequencies, averaging storage-ring data over days could break degeneracies such as ˇb11F+ˇb22F that remain in the idealized instantaneous measurement.
Beyond the paper: extending Eq. (35) to weakly inhomogeneous fields, e.g. quadrupole focusing fields in storage rings, is a natural testable next step, since field gradients could induce precession terms proportional to ∂F.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a solid, useful extension of the BMT equation to SME-type Lorentz-violating backgrounds, and Eq. (35) is genuinely new. The main gap is that the enumeration of allowed terms is asserted, not proven, and the calibration anchors are prior results from the same group, so completeness is the fragile point.

What's new: the covariant equation (35) covering d=3 through d=6 operators, including field-dependent d=5/6 pieces, in arbitrary inertial frames. The storage-ring application (38)-(39) shows how it feeds into experiments. The paper does what it claims, and it's clearly written. The reduction to standard BMT is clean, and the authors are upfront about the scope (no gravity, no nonconcordant frames, no subleading terms).

I disagree with a hard-line reading that the completeness gap is fatal. The method of enumerating covariant contractions and fixing constants by matching to a known limit is standard in SME work, and the four enumerated bases (1 b-term, 2 H-terms, 3 d-terms, 6 g-terms) look plausible. The constraints (18) kill many candidates, and the final equation passes the BMT limit. But the stress-test note is right: there's no counting argument, no Young-tableau or invariant-theory proof, and the matching to Eq. (17) only constrains combinations that survive in the storage-ring geometry. A missing contraction that vanishes there would be invisible. That's a genuine soft spot, and a short appendix with a systematic count would settle it. It doesn't make the paper wrong, but it means the uniqueness claim in Eq. (35) is not fully established.

The second load-bearing premise—that homogeneous LV coefficients produce no correction to dU/dτ—is imported from the Finsler/Berwald literature with several references. That's better supported, but still a premise.

The orbit-averaging in the application is sketched; it checks out. No code or data, but the algebra is explicit enough to reproduce.

Bottom line: this is a paper for people doing storage-ring searches for Lorentz and CPT violation and for anyone who wants a frame-independent BMT extension with SME coefficients. It deserves a serious referee, and it got one. If I were the editor, I'd send it out. I'd cite it if I worked in this area.

Referee Report

2 major / 4 minor

Summary. This paper extends the Bargmann-Michel-Telegdi (BMT) equation for the spin precession of a charged Dirac particle to include Lorentz-violating background fields in the Standard-Model Extension (SME). The treatment covers spin-dependent operators of mass dimension d=3 and 4 in the propagation sector (coefficients b_μ, H_μν, d_μν, g_μνρ) and d=5 and 6 operators involving the electromagnetic field strength (coefficients b_F, H_F, d_F, g_F). The method is to construct observer-covariant terms linear in the spin, the coefficients, and F_μν, impose the constraints (2) through the Lorentz-violating restrictions (18), fix the overall constants by matching the known storage-ring spin-precession frequency (17), and thereby assemble the extended relativistic BMT equation, Eq. (35). The paper then applies this equation to a model storage ring and derives the radial spin-precession frequency, Eq. (38).

Significance. If Eq. (35) is correct and complete, this is a useful and significant result: it provides a ready-to-use relativistic spin-precession equation valid in arbitrary inertial frames for the dominant SME coefficients of dimensions 3 through 6, and it reduces to the standard BMT equation when all Lorentz-violating coefficients vanish. The construction is parameter-free in the sense that no new free parameters are introduced: the constants are fixed by matching the known limit (17). The concrete storage-ring prediction (38) is falsifiable and directly relevant to ongoing and proposed EDM and g-2 searches. However, the central claim of completeness of the enumerated term basis is not proven in the manuscript, and this is the main barrier to accepting the equation as the full leading-order extension.

major comments (2)

[Eqs. (19), (23), (27), (31) and assembly of Eq. (35)] The central claim that Eq. (35) is the complete leading-order BMT extension rests on four enumeration assertions ('we find', 'examination reveals') with no counting argument, no table of contractions, and no invariant-theory proof. The constraints (18) are necessary but not sufficient; for example, for d_μν and g_μνρ additional epsilon/metric contractions that vanish in the particular storage-ring geometry are not shown to reduce to the displayed basis. Because the constants are calibrated to a single component, Eq. (17), in a single geometry, any independent structure that vanishes in that geometry is invisible to the calibration. Please supply a systematic enumeration—listing all parity-even Lorentz contractions built from U, S, F, the coefficients, η, and ε, then imposing (18) and index symmetries—or an independent Foldy-Wouthuysen derivation, to prove completeness.
[Before Eq. (18)] The statement that homogeneous Lorentz-violating coefficients produce no correction to dU/dτ is load-bearing: if dU/dτ had a Lorentz-violating piece, the constraints (2) would mix it with the Lorentz-violating spin terms and alter the term list and the constants in Eq. (35). The manuscript cites Refs. [55-65] rather than proving this for the present charged-particle system coupled to F_μν. Please state the precise theorem from the Finsler/Berwald literature and either prove or explicitly verify its applicability when the particle is charged and an electromagnetic field is present; even a short argument from the wave-packet limit of the Dirac equation would settle this point.

minor comments (4)

[After Eq. (23)] The text says 'k_H1 and k_H1 are constants'; the second should be k_H2.
[After Eq. (21)] The matching for k_b says averaging β over the circular motion is required, but for the 3-component of Eq. (21) the β-dependent terms vanish and b'_3 = b3 in the ring geometry. Clarify whether an average is actually being performed or whether the relevant terms simply drop out.
[Eqs. (12)-(15)] The paper does not explicitly state why other spin-independent d=3,4 coefficients (such as a_μ and c_μν) are absent from the term base. A sentence explaining that they do not contribute to spin precession at leading order would make the restriction clearer.
[Eq. (33)] The claimed equivalence between Eq. (32) and the compact g_μνρ form (33) is not demonstrated. In particular, the cancellation of the trace part g(T) is not shown. If this is intended as an equivalent expression, the cancellation should be stated or referenced explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: constants are calibrated to a prior storage-ring result, and the general covariant equation has independent content.

full rationale

The derivation is an observer-covariant construction rather than a circular fit. The method section specifies the building blocks (U^mu, S^mu, F^mu nu, LV coefficients, metric, Levi-Civita tensor), imposes linearity in S^mu and in the small coefficients, and enforces the constraints (18). The premise that homogeneous LV coefficients produce no correction to dU/dtau is imported from Refs [55-65] (Finsler/Berwald geometry), which is external literature and not an assumption of the target result. The term lists in Eqs (19), (23), (27), and (31) are asserted by 'we find' without a counting proof; this is a legitimate rigor/completeness concern, but it is not a circular reduction because the final equation is not defined in terms of those lists. The constants are fixed by matching the 3-component of the storage-ring spin-precession frequency, Eq (17), from Refs [42,43]. This is transparent calibration: that component is an input, not claimed as a prediction. The resulting full tensor equation (35) and the radial application (38) go beyond the calibration component and are derived consequences of the covariant ansatz, constraints, and Lorentz transformations. The cited inputs are published, parameter-free derivations from the same SME Lagrangian that do not assume Eq (35) and are experimentally falsifiable, so under the review rules they count as independent evidence rather than circularity. No equation is both an input and a predicted output in a way that would constitute a circular step.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The derivation is calibrated rather than fully self-contained. The term bases are asserted complete, all constants are fixed by matching to Eq. (17) from Refs [42,43] (author overlap: Kostelecky, Vargas), and the d=5,6 content is imported via replacement (16) from Ref [28] (author overlap: Ding, Kostelecky). No new entities are invented: the check coefficients in Eq. (39) are linear combinations of existing SME coefficients. The central equation is therefore only as secure as the completeness claim and the correctness of the anchor results.

free parameters (7)

k_b = -2
Overall constant of the b-type term, Eq. (19); fixed by matching the boosted rest-frame result (21) to the storage-ring frequency (17) after orbit averaging.
k_H1 = -k_H2 = 2
Constants of the two H-type terms, Eq. (23); constraint (18) forces k_H1 = -k_H2 and matching to (17) sets the value.
k_d1 = -k_d2 = -m
Constants of the d_[mu nu] terms, Eq. (27); set by constraints (18) and matching to (17).
k_d3 = 2m
Constant of the d_(mu nu) term, Eq. (27); set by matching to (17) after rest-frame evaluation and boost.
k_g1, k_g4 = -2m
Constants of the axial and mixed-symmetry g terms, Eq. (31); fixed by matching to (17).
k_g6 = 2m
Constant of the U^mu g^(M) term, Eq. (31); constraint (18) gives k_g4 + k_g6 = 0, and matching sets k_g6.
k_g2, k_g3, k_g5 = 0
Constants of the trace and mixed-symmetry terms, Eq. (31); forced to zero by constraints (18).

assumptions (7)

ad hoc to paper The enumerated covariant term sets for each coefficient class are complete at leading order
Asserted without proof at four points in the paper; the final equation depends on no missing independent contractions existing.
domain assumption Homogeneous Lorentz-violating coefficients do not correct the 4-acceleration dU^mu/dtau
Needed to enforce constraints (18) on the LV terms alone; supported only by citations to Finsler geometry and Berwald-space results (Refs [55-65]).
domain assumption The background fields F^mu nu and the SME coefficients are homogeneous and small
The derivation linearizes in F and in the coefficients and neglects their derivatives; stated in the methodology paragraph.
domain assumption The storage-ring result (17) from Refs [42,43] is correct
All constants of the general equation are fixed by matching to this expression; it was derived by Foldy-Wouthuysen in prior same-group papers and adopted by experiments.
domain assumption The field-redefinition replacement (16) from Ref [28] is valid
Used to convert d=3,4 contributions into d=5,6 contributions; Ref [28] is a same-group paper and the protocol is not re-derived here.
domain assumption Only the listed d=3,4 operators and d=5,6 F-dependent operators contribute at leading order
Based on the SME operator enumeration of Refs [18,27,29]; subleading and Lorentz-invariant corrections are explicitly set aside.
domain assumption Spin precession is linear in the spin 4-vector S^mu (Uhlenbeck-Goudsmit)
Explicitly adopted with citation to Ref [52]; standard in spin-dynamics treatments.

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Cite this review

Pith. "Pith review of Relativistic spin precession in homogeneous background fields." pith.science (2026). https://pith.science/paper/4TE2MEYU

@misc{pith2026250905098,
  author       = {Pith},
  title        = {Pith review of: Relativistic spin precession in homogeneous background fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TE2MEYU}},
  note         = {Machine review of arXiv:2509.05098}
}

read the original abstract

The Bargmann-Michel-Telegdi equation, which describes the precession of the spin of a charged Dirac particle moving in a homogeneous electromagnetic field, is generalized to include also other homogeneous background fields. The treatment incorporates observable coefficients that govern operators of mass dimensions three through six in the underlying Dirac effective field theory. A relativistic formulation valid in arbitrary inertial frames is obtained. The results are applicable to searches for new physics beyond the Standard Model, including searches for Lorentz and CPT violation.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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