Gauge Symmetries, Exact Symmetries and Conserved Charges in Minimal Massive Gravity

Kang Liu; Xiao-Mei Kuang

REVIEW 4 major objections 4 minor 111 references

Gauge Symmetries, Exact Symmetries and Conserved Charges in Minimal Massive Gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that in Minimal Massive Gravity a third gauge symmetry carried by the auxiliary field forces a replacement of the Kosmann derivative, and constructs the associated conserved charge.

desk verdict A careful MMG analysis with a real new auxiliary-field symmetry, but the claimed new exact symmetry is undone by the paper's own condition (68), which forces the nontrivial Lorentz vector to zero. read the letter →

arxiv 2507.17635 v1 pith:RGMGMMYG submitted 2025-07-23 hep-th

Kang Liu , Xiao-Mei Kuang This is my paper

classification hep-th PACS 04.60.-m04.20.Fy04.70.-s11.15.-q

keywords MinimalMassiveGravitycovariantphasespacegaugesymmetriesexactKosmannderivativeconservedchargesauxiliaryfieldBTZblackhole

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

Minimal Massive Gravity is a three-dimensional theory whose fields are a triad, a spin connection, and an auxiliary one-form. The paper shows that the auxiliary field carries a third gauge symmetry beyond Lorentz rotations and spacetime translations, and that this new symmetry prevents the standard symmetry algebra from closing. To recover exact symmetries in the sense of the covariant phase-space Noether-charge method, the paper argues that the familiar Kosmann derivative must be extended to the transformation $\delta_{(\xi,\beta_\xi,\bar\beta_\xi)}=L_\xi+\delta^j_{\beta_\xi}+\delta^m_{\bar\beta_\xi}$. Under two conditions on the Lorentz parameters and one integrability condition, this transformation leaves all three fields invariant along a Killing vector, producing a conserved charge that combines the diffeomorphism charge with two gauge charges and becomes gauge invariant in the limit $1+\sigma\alpha\to 0$. If correct, the construction is a general mechanism for building exact symmetries and conserved charges in theories with auxiliary fields.

What carries the argument

The load-bearing object is the new transformation $\delta_{(\xi,\beta_\xi,\bar\beta_\xi)}=L_\xi+\delta^j_{\beta_\xi}+\delta^m_{\bar\beta_\xi}$ (Eq. 45), a linear combination of a Lie derivative, a Lorentz transformation with parameter $\beta_\xi$, and the new auxiliary translation with parameter $\bar\beta_\xi$. The auxiliary translation $\delta^m_\chi$ is defined by its action on $e$, $\omega$, and $h$ in Eqs. (19)--(21); it originates from the auxiliary field $h$ that MMG adds to the triad and spin connection. Conditions (46a)--(46c) together with the integrability condition (48) fix $\beta_\xi$ and $\bar\beta_\xi$ so that $\delta$ annihilates all fields along a Killing vector. The Kosmann derivative $K_\xi=L_\xi+\delta^j_{\lambda_\xi}$, the Lorentz-covariant completion of the Lie derivative used in the Noether-charge method, is the object being replaced.

What would settle it

Take a non-maximally-symmetric MMG solution, such as one from the logarithmic branch the paper cites, solve for $h$ from (12), and check whether $e^\mu_I h^\nu_I$ is symmetric. If it is not, compute the difference in (B2) for a generic transformation and see whether it is still a total derivative; failing that would show the new symmetry and charge depend on a condition that is not generally satisfied.

Watch

Extended reading notes

Core claim

The paper's central discovery is that once the auxiliary field's new 'auxiliary translation' $\delta^m_\chi$ is included, the Kosmann derivative is no longer the correct generator of exact symmetries in MMG. The replacement is $\delta_{(\xi,\beta_\xi,\bar\beta_\xi)}=L_\xi+\delta^j_{\beta_\xi}+\delta^m_{\bar\beta_\xi}$ (Eq. 45), with $\beta_\xi$ and $\bar\beta_\xi$ determined by the three conditions (46) and by the integrability condition $\Lambda_0=(1-\ell^2\sigma^2\mu^2(1+\sigma\alpha)^2)/(\alpha\ell^2)$ (Eq. 48). Acting along a Killing vector, this transformation leaves the triad, spin connection, and auxiliary field invariant to the relevant order, so the covariant phase-space method applies. The resulting conserved charge is $D^N_\xi=D_\xi+Q^j_{\beta_\xi}+Q^m_{\bar\beta_\xi}$ (Eq. 50), where $Q^m$ is a new 'auxiliary charge'; in the small-$(1+\sigma\alpha)$ limit the charge is approximately gauge invariant. The rotational BTZ solution verifies the construction and reproduces the first law $dE=2T_H\,dS+\Omega_H\,dJ$ with the expected MMG entropy.

Load-bearing premise

The derivation assumes that the contraction $e^\mu_I h^\nu_I$ is symmetric, so $h_{\mu\nu}=h_{\nu\mu}$; if general MMG solutions admit non-symmetric $h$, the Lagrangian-variation proof and the conserved-charge construction do not go through.

Editorial extensions

If this is right

The new charge $D^N_\xi$ reproduces the BTZ first law $dE=2T_H\,dS+\Omega_H\,dJ$, with entropy $S=A_h\left((\sigma+\alpha C)+\frac{r_-}{\mu\ell r_+}\right)$ matching earlier MMG results.
When the Killing vector's Lorentz parameter vanishes ($\varepsilon=0$), the two gauge-charge terms cancel and $D^N_\xi$ reduces to the ordinary diffeomorphism charge $D_\xi$.
In the limit $1+\sigma\alpha\to 0$ with $\mu\to\infty$, the total charge becomes approximately gauge invariant, which gives a handle on symmetries in the regime where the naive translation algebra fails.
The complete symmetry algebra of MMG is parameter-dependent: Lorentz plus translation closes on the redefined fields $(\tilde\omega,\tilde e)$ at generic parameters, while Lorentz plus auxiliary translation closes only at $1+\sigma\alpha=0$ on the enlarged space $(\hat\omega,\tilde e,h)$.
The same field-redefinition and charge-construction scheme is expected to apply to other three-dimensional Chern-Simons-like gravity theories and to non-maximally-symmetric solutions, with extra care when $h$ is not proportional to the triad.

Reading between the lines

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

Beyond the paper: if the auxiliary translation is generic in first-order theories with an auxiliary one-form, the same 'replace the Kosmann derivative' recipe could produce conserved charges for other massive 3D gravity models and for boundary-condition studies.
Beyond the paper: the parameter-dependent closure of the algebra suggests the symmetry structure of MMG changes character at $1+\sigma\alpha=0$; checking whether $D^N_\xi$ changes discontinuously there would be a direct test.
Beyond the paper: applying the construction to a non-maximally-symmetric solution with $h$ not proportional to the triad would test the paper's own caveat about the symmetry of $e^\mu_I h^\nu_I$; a failure at that point would delimit exactly where the new charge exists.

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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

Have you had a chance to look at Liu and Kuang's MMG paper? It is a mixed bag. The genuinely new thing is the identification of an auxiliary-field translation symmetry δm in MMG, with its action on the fields and the associated charge Qm. The covariant phase space analysis is careful, and the BTZ thermodynamics (E, J, S) correctly reproduces known results. That part is useful and will be a reference for whoever works on Chern-Simons-like theories. The field-redefinition trick to get closed algebras (31)–(32) is also neat.

But the central claim—that the Kosmann derivative should be replaced by a new transformation δ(ξ,β,β̄) generating exact symmetries—does not survive contact with the paper's own equations. The only explicit nonzero solution for β̄, Eq. (67), is annihilated by condition (68), which sets exactly its defining expression to zero. The zero branch is always a solution; the paper acknowledges this, and so the 'new' charge Qm reduces to zero in the BTZ example. No independent existence proof is given; Appendix C2 basically assumes the wanted closure. This is more damaging than the symmetric-h restriction, which at least is stated in a footnote. There is also an inconsistency at Eq. (26c): it sets B3 = Eh∧φ = 0, while Appendix B1 gives a nonzero boundary term for the c=1 case, contradicting the claim that δm acts without using equations of motion. These are not minor typos—they undercut the main result.

Still, the paper deserves referee time. The algebra machinery is serious, the writing is honest about the difficulties (Appendices B2 and C1 discuss failure modes), and the identified inconsistencies are concrete enough for an author to fix. The gauge-symmetry classification and the auxiliary charge are worth salvaging. If the exact-symmetry claim is retracted or properly solved, the paper could be solid.

For the record: I would not cite it in its current form, but I might bring it to a reading group if a student wants to see a cautionary tale about path-ordered exponentials. Verdict: accept for peer review, with major revision expected.

Referee Report

4 major / 4 minor

Summary. The paper studies Minimal Massive Gravity (MMG) in the first-order formalism using covariant phase space methods. It identifies three gauge symmetries: Lorentz transformations, spacetime translations, and a new translation associated with the auxiliary field h. It constructs field redefinitions that close the algebra in two different regimes, derives gauge charges, and proposes a new exact-symmetry transformation delta_(ξ,βξ,β̄ξ) = L_ξ + delta^j_βξ + delta^m_β̄ξ that is claimed to replace the Kosmann derivative when the new translation symmetry is included. The associated conserved charge D^N_ξ = D_ξ + Q^j_βξ + Q^m_β̄ξ is asserted to be gauge-invariant in the small (1+σα) limit. The construction is applied to the rotational BTZ black hole, and the first law is reproduced. The central claim is that the new transformation generates exact symmetries and that the new charge contains a nontrivial auxiliary-field contribution.

Significance. If the construction were correct, it would offer a novel mechanism for generating exact symmetries in three-dimensional gravity theories with auxiliary fields, and would extend the covariant phase space approach beyond the usual Kosmann derivative. The paper correctly identifies the existence of three independent gauge transformations in MMG and provides explicit computations of the closed algebras and of the BTZ first law. However, the central exact-symmetry construction has serious gaps that undermine the main claim, so the significance rests on whether these gaps can be repaired.

major comments (4)

[Sec. IV.B, Eqs. (67)-(68)] The claimed nontrivial solution for β̄ξ is annihilated by the very condition meant to define it. Equation (67) defines β̄ξK as a path-ordered exponential times a Lorentz vector dψ, while equation (68) imposes that exactly the same expression (with the background forms) vanishes. For a first-order linear equation such as (66), the homogeneous equation has only the zero solution in a simply connected region; the paper provides no argument for a nontrivial kernel. Consequently β̄ξ = 0, Q^m_β̄ξ in (50) and (70)-(71) vanishes, and D^N_ξ reduces to the Kosmann-type charge. The new exact symmetry and the auxiliary charge are therefore not established. A separate existence proof for a nonzero β̄ξ is required.
[Eq. (26c) and Appendix B.1] The statement B^3_φ = E_h ∧ φ = 0 in Eq. (26c) is inconsistent with the derivation in Appendix B.1. Setting a=0, b=0, c=1 in (B5) gives B_φ = T(e) ∧ φ + α [e∧h] ∧ φ = E_h ∧ φ, which is not identically zero off shell. Either the '=0' is an error, or the calculation in (24c) is being evaluated on shell. Since the text claims that the gauge symmetries are derived without using equations of motion, this inconsistency is load-bearing.
[Footnote to Eq. (B2)] The derivation of the gauge transformations and their action on the Lagrangian assumes that the contraction e^μ_I h^ν_I is symmetric, i.e., h_μν = h_νμ. The footnote acknowledges that this condition relies on the specific on-shell form of h. This contradicts the claim in Sec. II.B that the transformations (19)-(21) generate gauge symmetries without using equations of motion. The analysis is therefore restricted to a subset of solutions (e.g., h proportional to the triad), and the generality of the gauge-symmetry result is not established.
[Appendix C.2, Eqs. (C7)-(C9)] The derivation of the condition (46c) and the resulting integrability condition (48) rests on assumptions that have not been derived. The text 'directly supposes' δ(...)h ≈ 0 in (C7) and 'directly assumes' λξK can be determined by K_ξK h ≈ 0 in (C8). The subsequent equation (C9) for β̄ξ therefore follows from the very property that the new transformation is supposed to enforce. This circularity, together with the vanishing of β̄ξ noted above, means the existence of the proposed exact symmetry is not proven.

minor comments (4)

[Sec. IV.A] There is a typo: 'Appendxi C 2' should read 'Appendix C 2'.
[Sec. IV.B, around Eq. (55)] The term 'Gibbon-Hawking term' should be 'Gibbons-Hawking term' (reference [44]).
[Sec. II.B, Eq. (27)] The phrase 'anti-commutation relations' in the text preceding Eq. (27) is misleading; the operations are ordinary commutators of transformations, and the terminology should be adjusted for clarity.
[Sec. IV.B, Eq. (71)] The expression for Q^j_βξ + Q^m_β̄ξ contains an index mismatch: the first term is written as ε1(σe1 - (1/μ)ω1), but the notation for the Lorentz vector components is not introduced; using explicit component indices consistently would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised nontrivial Lorentz vector β̄ξ is defined to vanish by Eq. (68), so the new auxiliary charge Q^m and the claimed new exact symmetry reduce by construction to the standard Kosmann charge.

self definitional [Section IV.B, Eqs. (66)-(68), with charge in Eqs. (70)-(72)]
"Under the integrable condition (48), the solution of ¯βξK can vanish, or take the form in the path-ordered exponential P exp: ¯βξK = [P exp(− ∫_C ωnew + σµ(1+ σα)enew) ∧ dψ], where C is a three-dimensional path and ψ is an Lorentz vector satisfying [P exp(− ∫_C ω + σµ(1+ σα)e) ∧ dψ] = 0."

Equation (67) defines the candidate nonzero β̄ξ as exactly the bracket [P exp(...)∧dψ]; Eq. (68) then requires that same bracket to vanish. Hence the only solution exhibited is β̄ξ = 0. Since the auxiliary charge Q^m_β̄ in (70)-(71) is linear in β̄ξ, and βξ reduces to the Kosmann parameter λξ when β̄ξ vanishes, the 'new' conserved charge D^N collapses to the standard diffeomorphism/Kosmann charge. The paper later invokes 'different solutions of ¯βξ' to motivate the extended field space, but no nonzero solution is ever exhibited; the claimed auxiliary-field contribution is zero by construction.

full rationale

The Section II/III derivations of the three gauge symmetries, their algebras, and the covariant-phase-space charges are self-contained algebraic rewritings of the Lagrangian and symplectic potential; the BTZ entropy/energy/first-law results are checked against earlier independent results ([19,66,79,81]) rather than fitted, so those parts are not circular. The circularity is confined to the central Section IV construction: the nonzero Lorentz vector β̄ξ that would make Q^m new is introduced via a path-ordered exponential, but the paper's own condition (68) sets that exponential bracket to zero, forcing β̄ξ=0. Consequently the 'new exact symmetry' and 'new conserved charge' (50)/(70) reduce by construction to the Kosmann/diffeomorphism charge, and the claimed auxiliary contribution is a self-definitional zero. This is a load-bearing collapse of the paper's central claim, though it does not affect the auxiliary-field gauge-symmetry algebra results. The symmetric-h restriction noted in the footnote to Eq. (B2) is an honest scope limitation, not a circular step; Appendix C2 also explicitly assumes δ(...)h≈0 rather than deriving it, which further explains why no independent nonzero β̄ξ is provided.

Assumptions & free parameters 4 free parameters · 5 assumptions · 4 invented entities

The construction rests on standard Cartan and covariant phase space mathematics, on the MMG action and equations from prior literature, and on several restrictions introduced by the authors: symmetric h, the small (1 + sigma alpha) limit, the integrability condition (48), and hand-chosen field-redefinition coefficients. No new physical entity with independent external evidence is introduced; the new objects are mathematical operators and a charge defined inside the construction.

free parameters (4)

Parameter limit 1 + sigma alpha approx 0 = 1 + sigma alpha -> 0
The closed algebra (32) for delta^m holds only at 1 + sigma alpha = 0, and the exact-symmetry construction is verified only to linear order in (1 + sigma alpha). This is a chosen parameter regime, not a datum.
Integrability constraint Lambda_0 = Lambda_0 = [1 - l^2 sigma^2 mu^2 (1 + sigma alpha)^2] / (alpha l^2)
Derived in Appendix C2 from requiring the first-order equation for bar-beta_xi to be integrable via the Frobenius criterion. It selects a special parameter surface on which the construction works, so it is a restrictive condition rather than an independent input.
BTZ solution parameter C in h = C mu e = C = (1 - alpha l^2 Lambda_0) / (2 l^2 mu^2 (1 + sigma alpha)^2)
In the worked example the auxiliary field is assumed proportional to the triad. The value of C is fixed by the maximally symmetric solution and is used to derive the integrability condition.
Field-redefinition coefficients (a2, b3, c1, c2) = Satisfying relations (30)
These coefficients are chosen by hand so that the algebras (31) close on the redefined fields (tilde-omega, tilde-e). They are not fitted to data, but they are additional structure introduced by the authors to make the symmetry algebra work.

assumptions (5)

domain assumption The MMG Lagrangian (4) and equations of motion (7) are correct as given in refs [13,17].
The entire construction starts from this action and these equations of motion; the paper does not re-derive the model.
standard math Cartan calculus, interior products, bracket identities (A4)-(A8), and covariant phase space identities are standard.
Used throughout Sections II-IV; no formal proof is supplied for these background identities.
ad hoc to paper The contraction e^mu_I h^nu_I is symmetric, i.e. h_mu_nu = h_nu_mu.
Footnote 7 states that the derivation of the gauge-symmetry variation relies on this condition and that only solutions satisfying it are analyzed.
domain assumption For maximally symmetric MMG solutions, the Lie derivative of h along a Killing vector satisfies L_xiK h + [h, lambda_xiK] approx 0 (Eq. C8).
This is needed to derive Eq. (C9) and hence the parameter constraint (48); it is asserted rather than proven for general solutions.
standard math The first-order equation (66) for bar-beta_xi is integrable under the Frobenius criterion, yielding the constraint (48).
Appendix C2 uses d_omega d_omega = [R(omega), dot] to impose compatibility; this is an application of the Frobenius theorem, but the calculation is not independently verified here.

invented entities (4)

Auxiliary translation symmetry delta^m_chi
purpose: Gauge symmetry generated by the auxiliary field h, in addition to Lorentz and translation symmetries.
Defined in Eqs. (19)-(21); it is a mathematical operator within the construction and carries no observable prediction outside the paper.
Modified exact-symmetry transformation delta_(xi,beta_xi,bar-beta_xi)
purpose: Replaces the Kosmann derivative to generate exact symmetries on (e, omega, h) in MMG.
Proposed in Eq. (45); the existence of beta_xi and bar-beta_xi is assumed through conditions (46) and checked only on a BTZ example.
Lorentz vector bar-beta_xi
purpose: Measures the approximate incompleteness of translation symmetry and contributes to the new conserved charge.
Introduced in Eq. (45) and (64); no independent external handle is provided beyond the internal consistency of the construction.
Conserved charge D^N_xi
purpose: Proposed conserved charge for MMG exact symmetries, combining diffeomorphism and gauge charges.
Defined in Eq. (50). It reproduces the known first law in the BTZ example, but that is an internal consistency check rather than a falsifiable prediction.

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Cite this review

Pith. "Pith review of Gauge Symmetries, Exact Symmetries and Conserved Charges in Minimal Massive Gravity." pith.science (2026). https://pith.science/paper/RGMGMMYG

@misc{pith2026250717635,
  author       = {Pith},
  title        = {Pith review of: Gauge Symmetries, Exact Symmetries and Conserved Charges in Minimal Massive Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGMGMMYG}},
  note         = {Machine review of arXiv:2507.17635}
}

read the original abstract

In this paper, we investigate a three-dimensional gravitational model known as Minimal Massive Gravity (MMG), which includes an auxiliary field, using the covariant phase space method. Our analysis reveals the presence of three gauge symmetries whose algebras close via field recombination and parameter classification within this framework. Upon incorporating these additional symmetries within a specific limit of parameters, we find that the Kosmann derivative should be replaced by a novel transformation compatible with Wald's approach, which establishes a new mechanism for generating exact symmetries and constructing their corresponding conserved charge in theories with auxiliary fields, extending beyond standard methods. However, this transformation does not yield closed algebras on the space of fundamental fields. We find that this corresponds to a Lorentz vector that characterizes the approximate completeness of translation symmetry. As a result, we obtain a gauge invariant charge at a certain limit of parameters, which emerges as a nontrivial combination of the diffeomorphism charge and integrable gauge charges.

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Works this paper leans on

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INTRODUCTION Developing a quantum gravity theory remains a persistent challenge that combines techni- cal complexity with unresolved conceptual gaps

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Reviewed August 6, 2026 · model on record in the stance chip above.

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Derivation of (24) 24

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Discussions on (31) 25

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The proof of exact symmetries generated by δ(ξ,βξ, ¯βξ) 27

Derivation of (32) 27 C. The proof of exact symmetries generated by δ(ξ,βξ, ¯βξ) 27

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Failure of expressing ¯βξK in term of βξK by (46a) and (46b) 27

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INTRODUCTION Developing a quantum gravity theory remains a persistent challenge that combines techni- cal complexity with unresolved conceptual gaps

Seeking for the condition (46c) and constraint (48) 29 References 30 I. INTRODUCTION Developing a quantum gravity theory remains a persistent challenge that combines techni- cal complexity with unresolved conceptual gaps. Notably, standard Einstein-Hilbert gravity in three-dimensional spacetime is topological in the sense that it does not possess local de...

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Derivation of (24) We elucidate the derivation of Equation (24) in this part. We begin with a general transformation δtol φ consisting of δj φ, δt φ and δm φ with constants a, b and c δtol φ = aδj φ+ bδt φ+ cδm φ , (B1) which operates on Lagrangian L, (see (6) and (7)), yielding δtotal φ L− d(aI j φθ+ bI t φθ+ cI m φ θ)= Eh∧{a[h, φ]+ b(σµ(1+ σα)[h, φ]+ Λ0...

[7] [7]

Discussions on (31) In this part, we clarify the properties of the closed algebras (31) formed by the transforma- tions in the new space fields (28), and show the failure to incorporate the new transformation δm φ into a closed algebra. The operation of δm χ on ˜ω and ˜e gives δm χ ˜ω = αa1[e, χ]+ µ(1+ σα)[h, χ]+ c1(dωχ+ α[h, χ]+ σµ(1+ σα)[e, χ]) , (B7a) ...

[8] [8]

Derivation of (32) The double operations of δm φ on ˜ω, ˜e, and h are δm φ1δm φ2 ˜ω = (c1ασµ(1+ σα)+ σµ2(1+ σα)2+ c1σαµ(1+ σα))[[e, φ1] , φ2] +(c1α2+ µα(1+ σα)+ c1µ(1+ σα))[[h, φ1] , φ2] +(c1α+ µ(1+ σα))[ dωφ1, φ2] , (B13a) δm φ1δm φ2 ˜e= (a2α2+ σµα(1+ σα)+ c2σµα(1+ σα))[[e, φ1] , φ2]+ c2α[dωφ1, φ2] +(c2µ(1+ σα)+ c2α2)[[h, φ1] , φ2] , (B13b) δm φ1δm φ2h= ...