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Root-$T\bar{T}$ Flows Unify 4D Duality-Invariant Electrodynamics and 2D Integrable Sigma Models

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes that 4D self-dual electrodynamics and 2D integrable sigma models obey one identical first-order PDE, and that a single root-$T\bar T$ flow equation $\partial_\gamma \mathcal{L}=R_\gamma$ governs both.

desk verdict Useful two-parameter constructions, but the claimed universal root-TTbar flow fails for two of the paper's own families; worth refereeing after the overclaim is fixed. read the letter →

arxiv 2507.22808 v1 pith:FZMKS4ZU submitted 2025-07-30 hep-th

H. Babaei-Aghbolagh , Bin Chen , Song He This is my paper
classification hep-th
keywords root-TbarTdeformationduality-invariantelectrodynamicsintegrablesigmamodelsCourant-HilbertmethodauxiliaryfieldformalismmarginaldeformationsBorn-Infeldtheoryclosed-formLagrangians
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

Four-dimensional duality-invariant nonlinear electrodynamics and two-dimensional integrable $\sigma$ models have long been studied in parallel; this paper argues they are two faces of the same equation. The self-duality condition in 4D and the classical integrability condition in 2D reduce to the same first-order PDE, $L_U L_V = -1$ in suitable variables, so a single generating function solves both sectors. Introducing two commuting deformation parameters—an irrelevant $\lambda$ and a marginal $\gamma$—the paper shows that the root-$T\bar T$ flow equation $\partial_\gamma \mathcal{L}=R_\gamma$ holds for every model it constructs, including generalized Born–Infeld, logarithmic, q-deformed, and a new closed-form theory. If the paper is right, exact solvability in two and four dimensions is governed by one common deformation paradigm.

What carries the argument

The load-bearing object is the Courant–Hilbert characteristic ansatz, $\hat L = \ell(\tau) - 2q_1/\ell'(\tau)$ with $\tau = q_2 + q_1/\ell'(\tau)^2$; here $\ell(\tau)$ is an arbitrary generating function and the ansatz converts the shared first-order PDE into an algebraic construction, with the root-$T\bar T$ operator collapsing to $\tau\ell'(\tau)$. A parallel formulation uses an auxiliary field $y=e^\phi$ and a master potential $\Omega(y)$ through $\hat L = -q_1/y + y q_2 - \Omega(y)$, where $R_\gamma = y\Omega'(y)$. The three explicit rescaling rules (16), (18), and (20) then generate marginal flows that preserve the root-$T\bar T$ condition, and these rules, together with the generating function or master potential, carry the classification.

What would settle it

Substitute the new closed-form Lagrangian (45) into the integrability PDE (8) for generic $\lambda$ and $\gamma$ and check whether the equality holds identically; alternatively, construct any duality-invariant Lagrangian satisfying (4) that cannot be written in the characteristic form (9), or exhibit a marginal flow preserving the root-$T\bar T$ condition that is not one of the three listed types.

Watch

Extended reading notes

Core claim

Equations (4) and (8) are the same first-order nonlinear PDE, so four-dimensional self-duality and two-dimensional classical integrability are controlled by one universal characteristic equation. The paper solves this common PDE with the Courant–Hilbert characteristic ansatz $\hat L = \ell(\tau) - 2q_1/\ell'(\tau)$, $\tau = q_2 + q_1/\ell'(\tau)^2$, for an arbitrary generating function $\ell(\tau)$, and equivalently with a new auxiliary-field formulation $\hat L = -q_1/y + y q_2 - \Omega(y)$ with master potential $\Omega(y)$. In these variables the root-$T\bar T$ operator reads $R_\gamma=\tau\ell'(\tau)$ or $R_\gamma=y\Omega'(y)$, and the paper exhibits closed-form Lagrangians for generalized Born–Infeld, logarithmic, q-deformed, and a new theory, all satisfying $\partial_\gamma \mathcal{L}=R_\gamma$. It further classifies the marginal $\gamma$-flows into three transformation rules and gives a general perturbative Lagrangian that reproduces all these models to order $\lambda^3$.

Load-bearing premise

The argument assumes that the Courant–Hilbert characteristic ansatz (9) with an arbitrary generating function covers every solution of the shared PDE, and that the three rescaling rules (16), (18), and (20) exhaust all marginal flows preserving the root-$T\bar T$ condition; the paper states this completeness without giving a proof.

Editorial extensions

If this is right

  • Every closed-form model in the paper—generalized Born–Infeld, logarithmic, q-deformed, and the new theory—satisfies the root-$T\bar T$ flow equation $\partial_\gamma \mathcal{L}=R_\gamma$ by construction, so this flow is a universal organizing principle rather than a case-by-case feature.
  • The three $\gamma$-rescaling rules (16), (18), and (20) define three distinct families of marginal deformations, each preserving both duality invariance in 4D and integrability in 2D; the paper asserts these families are exhaustive.
  • Because the 4D and 2D conditions are the same PDE, the $(U,V)\leftrightarrow(q_1,q_2)$ dictionary lets any known duality-invariant electrodynamics be translated into an integrable sigma model and conversely.
  • The new closed-form theory in Section I.C exists simultaneously as a duality-invariant electrodynamics and an integrable sigma model, with a perturbative expansion different from Born–Infeld and q-deformed models.

Reading between the lines

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

  • If the characteristic ansatz (9) is truly general, then every duality-invariant electrodynamics and every integrable sigma model is encoded in a single function $\ell(\tau)$; constructing new $\ell(\tau)$ would automatically produce matched 4D/2D pairs.
  • The paper works classically; an immediate testable extension is to quantize the new closed-form theory and check whether the root-$T\bar T$ flow equation survives as a statement about quantum effective actions.
  • The energy-momentum-independent flow (27) is a departure from previously known flow equations; one could look for a symmetry or conserved quantity behind it, or test whether analogous flows exist for other generating functions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a unified framework connecting four-dimensional duality-invariant nonlinear electrodynamics and two-dimensional integrable sigma models. The connection is based on the observation that the self-duality condition (4) and the integrability PDE (8) have the same form, so both can be solved by a single Courant–Hilbert generating function ℓ(τ) or by an auxiliary-field potential Ω(y). The authors introduce two deformation parameters, an irrelevant λ and a marginal γ, and define three types of γ-rescalings of ℓ(τ). They claim that these generate a universal class of models satisfying the root-TTbar flow ∂γ L = Rγ, construct generalized Born–Infeld, logarithmic, q-deformed, and a new closed-form Lagrangian, and provide a perturbative expansion to O(λ³) with a table of coefficients. The paper is written as a Letter and relies on several computations that are asserted rather than displayed.

Significance. If the central claims were correct, the paper would provide a valuable organizational principle: a single generating function and a single flow equation would control both 4D electromagnetic duality invariance and 2D classical integrability, and the new closed-form models would be concrete nontrivial examples. The perturbative expansion (48)–(50) and Table I are useful summaries of known and new models, and the auxiliary-field formulas (12)–(13) offer a compact alternative presentation. However, the paper's headline universality claim is currently not supported by its own equations, and several of the new closed-form Lagrangians are not printed in a checkable form. The strengths are the explicit model-building strategy and the clear mapping between the CH and auxiliary-field formalisms, but these virtues are undermined by the overbroad flow statements and the absence of derivations for key identities.

major comments (3)
  1. [§II.B (Courant–Hilbert approach), Eqs. (17), (19), (21); Conclusions, Eq. (47)] The universal root-TTbar flow ∂γ L = Rγ is not satisfied by the Type-II and Type-III families introduced in this section. For Type-II, Eq. (19) gives ∂γ L = ℓ(λ,γ)(τ), and for Type-III, Eq. (21) gives ∂γ L = ∓(ℓ - τℓ′)/d; neither equals Rγ = τℓ′(τ). A concrete check for the Type-II rescaled Born–Infeld model at q1 = 0 gives ℓ(τ) = e^γ/λ(√(1+2λτ)-1), so ∂γ L = ℓ(q2) = e^γ(q2 - λq2²/2 + ...), while Rγ = τℓ′(τ)|_{τ=q2} = e^γ(q2 - λq2² + ...); these differ at order λ. Therefore the unqualified statement in the Conclusions that ∂γL = Rγ “holds across all models” (Eq. (47)) is false as written. The claim must be restricted to the Type-I rescaling and to the auxiliary-field constructions, or the terminology must be changed so that Type-II and Type-III are described as γ-flows that are not root-TTbar flows.
  2. [§II.B, after Eq. (21)] The assertion that the three rescalings (16), (18), and (20) “exhaust all marginal flows compatible with the integrability condition” is a completeness claim that is not proved. The perturbative analysis (48)–(49) only determines the γ-dependence of the coefficients within the assumed ansatz ℓ(λ,γ)(τ) = Σ n_i λ^i f_i(γ) τ^{i+1}; it does not rule out other one-parameter families that preserve the PDE (8). Since the claimed “universal class” and the phrase “systematically spans” in the abstract depend on this exhaustiveness, a proof or a precise statement of the subclass covered by the three transformations is needed.
  3. [§I.C and Eqs. (39), (41), (45)] The new closed-form Lagrangians are not well-formed as printed, which prevents verification of the central new results. Equation (39) contains the ambiguous expression “Σ3λ” and unbalanced parentheses; Eq. (41) has an unmatched parenthesis structure around the terms involving Λ^{1/3}; and Eq. (45) is printed as a ratio of sums without clear bracketing. In addition, the text repeatedly states that “one can explicitly demonstrate” the root-TTbar flow (e.g., for Eq. (26), Eq. (30), Eq. (35), and Eq. (45)) without showing the calculation. Given that the flow identity is the load-bearing claim of the paper, these verifications must be supplied or referenced to a detailed appendix.
minor comments (5)
  1. [Eq. (28)] The trace formula for T^μ_μ in the Type-III generalized Born–Infeld sector appears to have an unbalanced parenthesis or a missing bracket in the denominator; please re-typeset the expression.
  2. [Eq. (30)] The logarithmic Lagrangian is printed with an ambiguous radical “√(1+4λe^{-γ}q1-4λ²q1q2+1)” and the log argument is not clearly delimited; please rewrite with explicit parentheses.
  3. [Eq. (36)] The definition of Δ contains an extra closing parenthesis “))” after the radicand; please correct.
  4. [Footnote 1 and Conclusions] Footnote 1 contains the informal sentence “Using this dictionary, you can get:” and should be rewritten in a formal style; also, “electricmagentic” in the Conclusions is a typo for “electromagnetic.”
  5. [§I.A.1] The sentence “The energy-momentum tensor derived for the Lagrangian L̂^ST_GBI, sector yields the trace:” is missing a verb or punctuation; please rephrase.

Circularity Check

3 steps flagged · score 5.0 of 10

The gamma-flow equations are imposed by the chosen rescalings of ℓ and Ω; the explicit Lagrangians remain nontrivial.

  1. self definitional [Courant–Hilbert approach, Eqs. (16)–(21)]
    "The marginal γ-flow corresponds to the universal flow equations in (11) through rescalings of ℓλ(τ) and the coupling λ, as: ℓ(λ,γ)(τ)=eγℓ¯λ(τ), ¯λ=eγλ, defining a universal class of integrable theories where the deformation ∂γˆL(λ,γ)=τℓ′(τ), preserves integrability via consistent transformations of the Lax structure while maintaining the root-T¯T operator structure."

    Under the CH ansatz (9), ∂γL equals the partial γ-derivative of ℓ at fixed τ because the implicit τ-dependence cancels, giving Lγ=ℓγ. The three rescalings are engineered so that Type-I has ℓγ=τℓ′ term-by-term in the weak-field expansion (15), Type-II has ℓγ=ℓ, and Type-III has ℓγ=∓(ℓ−τℓ′). Hence Eqs. (17), (19), and (21) are identities enforced by the chosen γ-dependence, not flow equations derived from an independent principle. The subsequent claim that these 'exhaust all marginal flows' merely enumerates the three rescalings just defined.

  2. self definitional [New auxiliary field formalism, Eqs. (22)–(24)]
    "Transformations introduced in Eq. (22) result in the formation of the rootT ¯T -flow equation within the theory’s Lagrangian, expressed as: ∂γˆL(λ,γ)=˜y˜Ω′(˜y)."

    With Ω̃(ỹ)=eγΩ(e−γỹ) and ỹ=eγy as used in Eq. (37), the stationarity condition Ω̃′=q1/ỹ²+q2 cancels the ỹγ terms and leaves ∂γL̃=ỹΩ̃′(ỹ) by the chain rule alone. The root-T¯T form is therefore built into the transformation, and the resulting Lagrangians (35), (39), and (45) are Legendre transforms of these engineered potentials. Satisfying Eq. (24) is a consistency condition of the ansatz, not an independent universal prediction.

1 more flagged steps
  1. other [Courant–Hilbert approach, after Eq. (21)]
    "These exhaust all marginal flows compatible with the integrability condition and encompass perturbative and exact closed-form integrable theories."

    No independent characterization of 'marginal flows' is supplied; the only flows considered are the three transformations (16), (18), and (20), so the exhaustiveness assertion is true by stipulation. If 'marginal' were instead defined by preservation of the root-T¯T condition, Types II and III would not qualify, as they obey the ℓ-flow and the trace flow rather than Rγ=τℓ′; the classification is therefore circular rather than proven.

full rationale

The core construction is not empirically circular: the Courant–Hilbert ansatz (9) and the auxiliary-field ansatz (12) are standard, and the new closed-form Lagrangians (26), (30), (39), (41), and (45) are genuine solutions with independent content. However, the claimed universal flow equations are manufactured by the γ-rescalings (16), (18), (20), and (22). For the CH ansatz one has Lγ=ℓγ, and each rescaling is chosen so that ℓγ equals τℓ′, ℓ, or ∓(ℓ−τℓ′), respectively, so Eqs. (17), (19), (21), and (24) are definitional consequences rather than discovered predictions. The exhaustiveness statement after Eq. (21) is likewise a stipulation. Separately, and this is a correctness issue rather than a circularity, the abstract's claim that all γ-coupled theories preserve the root-T¯T condition is not satisfied by the Type-II and Type-III families: for Type-II GBI at q1=0, ∂γL=eγ(q2−λq2²/2+⋯) while τℓ′=eγ(q2−λq2²+⋯), which differ at O(λ). Because the explicit solutions remain nontrivial, the circularity is partial rather than total.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The ledger focuses on load-bearing inputs: the characteristic ansatz and the auxiliary-field representation are imported from prior work, while the weak-field expansion and gamma-flow rules are introduced here without completeness proofs. The generating function and expansion coefficients act as free functional inputs. No new physical particles, forces, or dimensions are introduced.

free parameters (2)
  • generating function ℓ(τ) or potential Ω(y) = model-dependent; new theory uses Ω(y)=1/λ(2/3 y+1/3 y^2-1)
    The entire classification is parameterized by this free functional input, chosen per model rather than derived from independent data.
  • coefficients n_i in the perturbative CH expansion = new theory: n1=1/4, n2=1/12, n3=1/48; other values in Table I
    These constants are introduced by hand in equation (48) to assign each model; they are not fixed by an independent principle.
assumptions (4)
  • domain assumption The self-duality condition L_U L_V = -1 and the integrability condition L_q1 L_q2 = -1 have identical solution structure, and the Courant-Hilbert ansatz (9) solves both.
    This is the core unification step, relying on the PDE arguments in the cited literature and the method from Courant and Hilbert; the paper gives no general derivation that (9) is exhaustive.
  • ad hoc to paper The weak-field expansion ℓ_λ(τ)=τ+λO(τ^2)+... covers the CH functions of all integrable theories considered, with finite coefficients n_i and prescribed gamma-functions.
    The paper postulates this expansion to define the generalized perturbative theory; its convergence and generality are not established.
  • domain assumption The auxiliary-field formalism with Lagrangian (12) and constraint Ω'(y)=q1/y^2+q2 represents all relevant self-dual or integrable models.
    The paper imports this as a prior result from the cited literature, and the new theory in section I.C is built on it.
  • ad hoc to paper The gamma-rescalings (16), (18), and (20) define flows that preserve the root-T Tbar condition and exhaust all marginal flows.
    These rules are introduced by hand as field transformations; exhaustiveness is asserted without proof.

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Pith. "Pith review of Root-$T\bar{T}$ Flows Unify 4D Duality-Invariant Electrodynamics and 2D Integrable Sigma Models." pith.science (2026). https://pith.science/paper/FZMKS4ZU

@misc{pith2026250722808,
  author       = {Pith},
  title        = {Pith review of: Root-$T\barT$ Flows Unify 4D Duality-Invariant Electrodynamics and 2D Integrable Sigma Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZMKS4ZU}},
  note         = {Machine review of arXiv:2507.22808}
}
abstract

We present a unified framework that connects four-dimensional duality-invariant nonlinear electrodynamics and two-dimensional integrable sigma models via the Courant-Hilbert and new auxiliary field formulations, both governed by a common generating function and a generating potential, respectively. Introducing two commuting deformation parameters, $\lambda$ (irrelevant) and $\gamma$ (marginal), we identify a universal class of $\gamma$-flows, including the root-$T\bar{T}$ deformation and its rescaled variants. Our approach generalizes conventional single-coupling structures via novel field transformations that extend to a two-parameter space ($\lambda$,$\gamma$) while preserving the root-$T\bar{T}$ flow condition for all $\gamma$-coupled theories. We construct several integrable models, including generalized Born-Infeld, logarithmic, q-deformed, and a new closed-form theory applicable to both electrodynamics and integrable systems. This unified framework, based on the unique form of the root-$T\bar{T}$ flow, systematically spans duality-invariant nonlinear electrodynamics in 4D and their exact 2D integrable counterparts.

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Forward citations

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Reference graph

Works this paper leans on

43 extracted references · 15 canonical work pages · cited by 4 Pith papers

  1. [1]

    Extension from a single coupling constant λ to a two- parameter space (λ,γ )

    work page

  2. [2]

    Preservation of the root-T ¯T flow condition for all γ- coupled theories 3 The fundamental transformation rule takes the general form: L(λ)→ ˜L(λ,γ ) =Lseed +λL1(γ) +λ2L2(γ) +..., (14) whereLseed is the Lagrangian for the seed theories in 2D or 4D andLn(γ) is carefully constrained to maintain compatibil- ity with the root-T ¯T operator. The Lagrangian tra...

    work page

  3. [3]

    ℓ(τ) = 1 λ ( 1− ( 1− 1 qλτ )q)

    q-Deformed Theories A large family of integrable q-deformed theories can be constructed for the CH-function ℓ(τ), subject to the initial conditionℓ(λ = 0,τ ) =τ. ℓ(τ) = 1 λ ( 1− ( 1− 1 qλτ )q) . (32) These theories are deformed by the root-T ¯T flow, as governed by the operatorRγ in the CH-function framework. For various values of the parameter q, the PDE...

    work page

  4. [4]

    This systematic framework yields novel families of closed-form integrable models, all of which re- main consistent with the universal root T ¯T -flow equations introduced in the previous section. A. Theories from Courant–Hilbert approach

    work page

  5. [5]

    This function is defined by the following expres- sion: ℓλ(τ) =− 1 λ ( 1− √ 1 + 2λτ )

    Generalized Born-Infeld theories We examine a canonical example of integrable models and invariant electromagnetic theories possessing Born-Infeld du- ality, characterized by the CH-functionℓ(λ)(τ) with coupling constantλ. This function is defined by the following expres- sion: ℓλ(τ) =− 1 λ ( 1− √ 1 + 2λτ ) . (25) By applying the transformations in (16) t...

    work page

  6. [6]

    These models can be systematically de- rived through appropriate transformations of the fundamental equation (16) when applied to the CH-function ℓ(τ) =− 1 λ log(1−λτ)

    Logarithmic theories We consider a class of two-dimensional integrable logarith- mic field theories characterized by two independent coupling constants,γ andλ. These models can be systematically de- rived through appropriate transformations of the fundamental equation (16) when applied to the CH-function ℓ(τ) =− 1 λ log(1−λτ). (29) The transformation proc...

    work page

  7. [7]

    To achieve this, we consider the following potential: Ω(y) = 1 4λ ( (y−3 + 3y)− 4 ) , (33) wherey is determined by: y = √ 2λq1 + √ 9 + 4λ(λq2 1− 3q2) 3− 4λq2

    q-deformed theories We demonstrate that an appropriate potential Ω(y) new aux- iliary field formalism reproduces theq-deformed theory in the CH approach forq = 3/4 and yields the corresponding flow equations∂γ ˆL(λ,γ) =Rγ in (13). To achieve this, we consider the following potential: Ω(y) = 1 4λ ( (y−3 + 3y)− 4 ) , (33) wherey is determined by: y = √ 2λq1...

    work page

  8. [8]

    Born and L

    M. Born and L. Infeld, Proc. Roy. Soc. Lond. A144, 425 (1934)

    1934

Show all 43 references
  1. [9]

    E. S. Fradkin and A. A. Tseytlin, Phys. Lett. B 163, 123 (1985)

    1985

  2. [10]

    M. R. Garousi, Phys. Rept. 702, 1 (2017), arXiv:1702.00191 [hep-th]

    2017 arXiv

  3. [11]

    L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987)

    1987

  4. [12]

    Bialynicki-Birula, (1984)

    I. Bialynicki-Birula, (1984)

    1984

  5. [13]

    M. K. Gaillard and B. Zumino, inA Newton Institute Euroconfer- ence on Duality and Supersymmetric Theories (1997) pp. 33–48, arXiv:hep-th/9712103

    1997 arXiv

  6. [14]

    G. W. Gibbons and D. A. Rasheed, Nucl. Phys. B 454, 185 (1995), hep-th/9506035

    1995 arXiv

  7. [15]

    G. W. Gibbons and D. A. Rasheed, Phys. Lett. B 365, 46 (1996), arXiv:hep-th/9509141

    1996 arXiv

  8. [16]

    Ferko, S

    C. Ferko, S. M. Kuzenko, L. Smith, and G. Tartaglino- Mazzucchelli, Phys. Rev. D 108, 106021 (2023), 2309.04253

    2023 arXiv

  9. [17]

    Á. J. Murcia, (2025), arXiv:2507.16502 [hep-th]

    2025

  10. [18]

    F. A. Smirnov and A. B. Zamolodchikov, Nucl. Phys. B 915, 363 (2017), 1608.05499

    2017 arXiv

  11. [19]

    Cavaglià, S

    A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, JHEP 10, 112 (2016), 1608.05534

    2016 arXiv

  12. [20]

    Conti, L

    R. Conti, L. Iannella, S. Negro, and R. Tateo, JHEP 11, 007 (2018), arXiv:1806.11515 [hep-th]

    2018 arXiv

  13. [21]

    Babaei-Aghbolagh, K

    H. Babaei-Aghbolagh, K. Babaei Velni, S. He, and Z. Pezhman, (2025), arXiv preprint, 2504.10361

    2025 arXiv

  14. [22]

    B. Chen, S. He, and J. Hou, (2025), arXiv:2506.03129 [hep-th]

    2025

  15. [23]

    Babaei-Aghbolagh, K

    H. Babaei-Aghbolagh, K. Babaei Velni, D. M. Yekta, and H. Mo- hammadzadeh, JHEP 04, 187 (2021), arXiv:2012.13636 [hep- th]

    2021 arXiv

  16. [24]

    Ferko, H

    C. Ferko, H. Jiang, S. Sethi, and G. Tartaglino-Mazzucchelli, JHEP 02, 016 (2020), arXiv:1910.01599 [hep-th]

    2020 arXiv

  17. [25]

    Babaei-Aghbolagh, S

    H. Babaei-Aghbolagh, S. He, and H. Ouyang, JHEP 09, 137 (2024), arXiv:2407.03698 [hep-th]

    2024 arXiv

  18. [26]

    Courant and D

    R. Courant and D. Hilbert, Methods of Mathematical Physics, V ol. II(Wiley-Interscience, New York, 1962)

    1962

  19. [27]

    Lechner, P

    K. Lechner, P. Marchetti, A. Sainaghi, and D. P. Sorokin, Phys. Rev. D 106, 016009 (2022), arXiv:2206.04657 [hep-th]

    2022 arXiv

  20. [28]

    J. G. Russo and P. K. Townsend, (2025), arXiv:2505.08869 [hep-th]

    2025

  21. [29]

    Babaei-Aghbolagh, K

    H. Babaei-Aghbolagh, K. B. Velni, D. M. Yekta, and H. Moham- madzadeh, Phys. Lett. B 829, 137079 (2022), arXiv:2202.11156 [hep-th]

    2022 arXiv

  22. [30]

    Bandos, K

    I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, Phys. Rev. D 102, 121703 (2020), arXiv:2007.09092 [hep-th]

    2020 arXiv

  23. [31]

    J. G. Russo and P. K. Townsend, Phys. Rev. D 109, 105023 (2024), 2401.06707

    2024 arXiv

  24. [32]

    J. G. Russo and P. K. Townsend, (2024), arXiv preprint, 2407.02577

    2024 arXiv

  25. [33]

    J. G. Russo and P. K. Townsend, JHEP 06, 191 (2024), arXiv:2404.09994 [hep-th]

    2024 arXiv

  26. [34]

    Borsato, C

    R. Borsato, C. Ferko, and A. Sfondrini, Phys. Rev. D 107, 086011 (2023), arXiv:2209.14274 [hep-th]

    2023 arXiv

  27. [35]

    Babaei-Aghbolagh, K

    H. Babaei-Aghbolagh, K. Babaei Velni, D. Mahdavian Yekta, and H. Mohammadzadeh, Phys. Rev. D 106, 086022 (2022), arXiv:2206.12677 [hep-th]

    2022 arXiv

  28. [36]

    Ferko, A

    C. Ferko, A. Sfondrini, L. Smith, and G. Tartaglino- Mazzucchelli, Phys. Rev. Lett. 129, 201604 (2022), arXiv:2206.10515 [hep-th]

    2022 arXiv

  29. [37]

    Conti, J

    R. Conti, J. Romano, and R. Tateo, JHEP 09, 085 (2022), arXiv:2206.03415 [hep-th]

    2022 arXiv

  30. [38]

    S. He, Y . Li, H. Ouyang, and Y . Sun, (2025), arXiv:2503.09997 [hep-th]

    2025 arXiv

  31. [39]

    E. A. Ivanov and B. M. Zupnik, in 4th International Workshop on Supersymmetry and Quantum Symmetries: 16th Max Born Symposium (2002) pp. 235–250, arXiv:hep-th/0202203

    2002 arXiv

  32. [40]

    E. A. Ivanov and B. M. Zupnik, Phys. Atom. Nucl. 67, 2188 7 (2004), arXiv:hep-th/0303192

    2004 arXiv

  33. [41]

    Ferko and L

    C. Ferko and L. Smith, Phys. Rev. Lett. 133, 131602 (2024), arXiv:2405.05899 [hep-th]

    2024 arXiv

  34. [42]

    Bielli, C

    D. Bielli, C. Ferko, L. Smith, and G. Tartaglino-Mazzucchelli, Phys. Rev. Lett.134, 101601 (2025), arXiv:2407.11636 [hep-th]

    2025 arXiv

  35. [43]

    Bandos, K

    I. Bandos, K. Lechner, D. Sorokin, and P. K. Townsend, JHEP 03, 022 (2021), arXiv:2012.09286 [hep-th]. SUPPLEMENTAL MATERIAL S.4. Generalized perturbative theory For general theories, the CH function can incorporate higher- order terms inτ with coupling toλ. A notable property...

    2021 arXiv

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