On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory

Alex Simon; Karol K. Kozlowski

REVIEW 2 major objections 4 minor 21 references

On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Theorem 4.9 proves that truncated k-point correlation functions of the sinh-Gordon model are smooth functions of space-like separated points, given by finite sums of absolutely convergent rapidity integrals.

desk verdict First closed formula for truncated k-point functions in Sinh-Gordon bootstrap, rigorous within a declared class but conditional on a completeness conjecture the paper does not prove. read the letter →

arxiv 2502.03894 v1 pith:UWZBYKCE submitted 2025-02-06 math-ph math.MPnlin.SI

Karol K. Kozlowski , Alex Simon This is my paper · ORCID

classification math-phmath.MPnlin.SI MSC 81T4082B2381U15

keywords sinh-Gordonmodelintegrablequantumfieldtheoryformfactorbootstrapcorrelationfunctionsmulti-pointK-transformS-matrixrapidityintegrals

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

This paper proves that the truncated k-point correlation functions of the integrable 1+1-dimensional $\sinh$-Gordon quantum field theory are smooth functions of the field positions whenever those positions are pairwise space-like separated, and writes each of them explicitly as a finite sum of absolutely convergent multiple integrals over rapidity variables. The integrands are assembled entirely from the model's scattering data: the two-body S-matrix, the form factors of the quantum fields (meromorphic functions encoding the fields' matrix elements between multi-particle states), and plane-wave factors $e^{ip(\gamma)\cdot x}$. Rigorous control in this bootstrap framework previously stopped at two-point functions, with only partial three- and four-point results in the literature, so the paper fills a genuine gap. Theorem 4.9 is, however, conditional: it covers form factors of the K-transform type, and the paper states in Section 2.2 that it is a reasonable but unproven conjecture that every solution of the form-factor bootstrap axioms is of this form.

What carries the argument

The argument rides on four interlocking objects. First, the K-transform representation of form factors, $F^{(O)}_n(\beta_n)=\prod_{a<b} F(\beta_{ab})\, K_n[p^{(O)}_n](\beta_n)$ (equation (2.23)): the discrete summation $K_n$ converts a comparatively simple, $2i\pi$-periodic, holomorphic function $p^{(O)}_n$ into a solution of the bootstrap Riemann–Hilbert axioms, and Lemma 4.2 turns this into a contour-integral bound that controls all derivatives. Second, the recursive definition of the multi-particle kernels $M^{(O)}_{n;m}$ through Axiom V (shifted concatenation and reduction by Dirac masses), which Lemmas 2.4–2.10 resolve into partition sums over subsets of rapidity variables with product-of-$S$-matrix weights. Third, the master combinatorial representation (Proposition B.1) that, after enforcing the Dirac constraints, factorizes the denominator structure into independent chains of rapidity differences (Lemma 4.6), producing denominators $\gamma - \gamma' - i\epsilon$ along ordered chains. Fourth, the analytic engine: Propositions 4.7 and 4.8 deform each integration contour into $R+i\eta^{(ba)}$ with strictly decreasing imaginary parts, so that all denominator singularities are pushed away and the integral converges absolutely; dominated convergence then justifies the $\epsilon\to 0^+$ limits and supplies the smoothness in $x$-space.

What would settle it

To falsify the paper's claim, it would suffice to exhibit a solution of the form-factor bootstrap axioms I–IV that is not of the K-transform form (2.23), or to find a point in the totally space-like region where the finite sum in (4.108) fails to converge absolutely, e.g. by checking numerically a small truncation level for k=3 at a configuration with large rapidity differences.

Watch

Extended reading notes

Core claim

The central claim, Theorem 4.9, is stated on the model's own terms: for smooth compactly supported test functions $g_1,\dots,g_k$ whose supports sit in the totally space-like region $\mathbb{D}_{\rm space;+}$ (defined by $x_{ab}^2<0$ and $x_{a;1}>x_{b;1}$ for all $a<b$), the distribution induced by the truncated $k$-point function $(f_{\rm vac}, O_1[g_1] O_2^{(r_1)}[g_2]\cdots O_k^{(r_{k-1})}[g_k] f_{\rm vac})$ is given by a smooth function $W_{\bf r}(x_1,\dots,x_k)$. The paper computes this function in closed form, equation (4.108): it is the finite sum over integer vectors ${\bf n}\in\mathbb{N}_{\bf r}$ of integrals over rapidities $\gamma^{(ba)}$, each carried on a horizontal contour $R+i\eta^{(ba)}$ with strictly decreasing heights, of the product of the S-matrix factor $S(\gamma)$, one form factor $F^{(O_p)}$ per operator evaluated at mixed rapidity arguments with a $+i\pi$ shift, and the oscillatory factor $\prod_{b>a} e^{ip(\gamma^{(ba)})\cdot x_{ba}}$, all divided by ${\bf n}!(2\pi)^{|{\bf n}|}$ times a phase $e^{-2i\pi n_{ba}\omega_{ba}}$. Each of these integrals converges absolutely, so the representation is meaningful pointwise, not only in the distributional sense.

Load-bearing premise

The whole theorem rests on the assumption that every form factor of the sinh-Gordon model is of the K-transform type (2.23) with p_n satisfying conditions (a)–(d), a conjecture the paper explicitly states is unproven.

Editorial extensions

If this is right

For any fixed truncation ${\bf r}$, the $k$-point function on space-like configurations is now an explicitly computable finite sum of absolutely convergent integrals, so it can be evaluated pointwise and, in principle, numerically to any desired accuracy.
The mixed representation of Proposition 4.11 is constructed precisely so that the paper (in a planned companion work) can verify the local commutativity property of the fields; combined with the present rigorous construction, that would establish the Wightman axioms for the truncated correlators whenever the remaining convergence assumption holds.
The same bootstrap-plus-reduction scheme is expected by the authors to yield closed multipoint formulas for other integrable models with diagonal scalar S-matrices, such as the sine-Gordon model, whose form factors are of the same general type.
The full (untruncated) $k$-point function is conjectured to be the sum over ${\bf r}$ of the newly constructed $W_{\bf r}$ (Conjecture 4.10); what is missing is only a proof of absolute convergence of that series, which the paper leaves open.
For $k=2$ the framework reproduces the structure whose convergence had already been settled, so the result upgrades the known rigorous two-point theory to arbitrary $k$ at space-like separation.

Reading between the lines

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

Reading Lemma 4.6 structurally, each summand of (4.108) corresponds to wiring the rapidity variables $\gamma^{(ba)}$ into ordered chains $a_0<a_1<\dots<a_\ell$ with denominators $\gamma-\gamma'-i\epsilon$ along the chain; the paper does not pursue this diagrammatic reading, but it suggests graph-organized resummations and sampling schemes that could be developed independently.
A natural next step the paper does not attempt: use the uniform estimates behind Proposition 4.8 to attack the ${\bf r}$-series of Conjecture 4.10 directly, e.g. by proving a summability bound over ${\bf n}\in\mathbb{N}_{\bf r}$; success would promote the conjectured full $k$-point formula to a theorem.
If the completeness conjecture on the K-transform class were to fail, Theorem 4.9 would still hold for every operator whose form factors fall in the class, making the paper's results conditional but not empty; the practical question is therefore the scope of the class, not the validity of the formula.
A successful verification of the Wightman axioms from these representations would make the paper's construction a complete constructive definition of the sinh-Gordon model, a long-standing goal for interacting quantum field theories; this consequence is an inference from the announced program, not something the paper claims to have achieved.

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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

Dear colleague,

This is the first paper to give a closed, rigorous formula for truncated k-point functions with k>=3 in the Sinh-Gordon model. That part is real. Theorem 4.9 is a finite sum of absolutely convergent rapidity integrals that defines a smooth function on the totally space-like region, and the machinery needed to get there—especially the combinatorial master representation and the contour-shift estimates—is substantial and looks sound. No fitted parameters, no normalization games. The paper earns its technical reputation.

The soft spots are not hidden, which I respect. The main one is the completeness of the K-transform class: the paper only proves the theorem for form factors of the form (2.23), and Section 2.2 explicitly says the converse is not established. So the result is conditional on a reasonable but open conjecture about the full operator content of the model. The abstract oversells this by saying "the Sinh-Gordon quantum field theory" without the qualifier. That is a presentation issue; the body is honest. A revised version should fix the abstract and state the theorem as conditional on the K-transform completeness, or prove that completeness.

Second, Proposition 2.1 from Babujian-Karowski is imported without proof. That is standard, but here the whole representation theory rests on it, so the referee should ask for a proof or a very precise citation with the result stated verbatim.

Third, the theorem covers only the totally space-like ordered region D_{space;+}. The paper says as much, but the abstract skips it. The extension to other space-like chambers is sketched via Proposition B.1 and is not fully carried out. Time-like separations remain open.

The stress-test note is correct but it does not describe a flaw the paper tries to hide. It is a declared limitation. So my verdict is conditional, not negative. The central argument holds up for the class it claims.

Who should read this: anyone working on the bootstrap program, integrable QFT, or Wightman-axiom checks for non-perturbative models. The introduction and Theorem 4.9 give a good entry point; Section 4 is for the dedicated referee.

I would send this to a serious referee. The result is important enough to warrant careful checking, even if the completeness conjecture means it is not the fully general statement the abstract implies.

Referee Report

2 major / 4 minor

Summary. This paper develops a bootstrap derivation of r-truncated k-point Wightman functions for the 1+1-dimensional Sinh-Gordon model. After setting up the Fock space, S-matrix and form-factor axioms, the authors recall the K-transform representation (2.23) of form factors, prove several equivalent combinatorial representations of the multi-particle kernels M_{n;m}^{(O)}, and use them to convert the smeared correlation function into a finite sum over rapidity integrals. The main technical work is in Section 4: Lemmas 4.2–4.6 supply bounds and a chain-factorization that justify the ε→0 limits, and Propositions 4.7–4.8 establish well-definedness. Theorem 4.9 then gives the explicit smooth density W_r(x_1,...,x_k) on the totally space-like region D_{space;+}, Eq. (4.108), as a finite sum of absolutely convergent integrals; the untruncated k-point function is stated as Conjecture 4.10. The proof is explicitly restricted to form factors of K-transform type, and the paper states in Section 2.2 that the completeness of this class is not established.

Significance. If the result is read as a theorem about the K-transform class, it is a substantial and credible advance: it gives the first closed rigorous expressions for multipoint (k≥3) truncated correlators in the Sinh-Gordon bootstrap program, with explicit estimates, no fitted parameters, and a clear falsifiable formula. The paper is also honest about its limitations, notably the conjectural nature of the full series and the explicit restriction to space-like separated supports. However, because the completeness of the K-transform representation is open, the unqualified claim that the paper supplies the truncated k-point function of "the Sinh-Gordon quantum field theory" is not established. This is a load-bearing qualification rather than a presentation issue.

major comments (2)

[§2.2, Eq. (2.23); Theorem 4.9, Eq. (4.108)] Theorem 4.9 is proved only for operators whose form factors are of K-transform type (2.23) with p_n satisfying (a)–(d). The paper itself states in Section 2.2 that it has not been established that every solution of the form-factor bootstrap axioms is of this form. Since Lemma 4.2 and the estimates in §4.1 use the representation (2.23) in an essential way, Theorem 4.9 cannot be read as a theorem about arbitrary admissible Sinh-Gordon fields unless that completeness conjecture is true. The abstract and the final sentence of the Introduction overstate the result. I ask the authors to either prove the completeness of the K-transform class or explicitly state Theorem 4.9 and the abstract-level claim with the class restriction, moving the unrestricted statement to a conjecture.
[§2.2, Proposition 2.1] Proposition 2.1 is quoted without proof from [3] and is the only bridge between the bootstrap axioms and the explicit representation used throughout. Because the present paper's goal is rigorous derivation, the dependence on this external result should be made fully precise: if [3] contains a proof, give the exact statement and location; otherwise, mark Proposition 2.1 as an assumption. This is not a mere formality: without it, the derivation does not start from the axioms alone.

minor comments (4)

[§3.1, heading] The heading "A premilinary expression" contains a typo; it should read "A preliminary expression".
[§4.1, proof of Lemma 4.4] In the proof of Lemma 4.4, the sentence "one infers (4.30)" appears to refer to the Lemma's own bound (4.35); please fix the cross-reference.
[Theorem 4.9, Eq. (4.108)] Theorem 4.9 uses the notation η^{(ba)} in (4.108) without repeating the hierarchy (4.91) from Proposition 4.8; please add a sentence specifying that the same order is assumed or that the integral is independent of the chosen sequence.
[Throughout] The text contains numerous grammatical and typographical slips, for example "Taken the L2-structure", "an operator O(x) on hShG as an integral operator", and "in a strip of fixed with" in §4.1. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the main theorem is conditional on an explicitly disclosed K-transform completeness conjecture, not on a circular fit or self-citation chain.

full rationale

After walking the derivation chain, I find no circular step. The paper starts from the S-matrix (2.2) and the form-factor bootstrap axioms I-IV, solves Axiom V recursively in Propositions 2.2 and Lemmas 2.4-2.10, rewrites the smeared truncated k-point function as a finite sum over rapidity integrals in Propositions 3.1-3.2, then establishes uniform bounds and contour-deformation arguments in Lemmas 4.2-4.6, leading to the explicit formula W_r(x_1,...,x_k) in Theorem 4.9, equation (4.108). No fitted parameter or normalization is involved: the final expression is built only from the S-matrix, form factors, combinatorial coefficients, and exponentials e^{ip(gamma^{(ba)}) . x_{ba}}, with no quantity normalized to the target correlation function. The one structural assumption is the K-transform representation (2.23) imported from Babujian-Karowski [3], where form factors are built from functions p_n satisfying conditions (a)-(d). The paper explicitly states this limitation in Section 2.2: 'To the best of our knowledge, it has not been established yet that every solution of the bootstrap axioms is given by (2.23) for some solution p(O)n(beta_n|ell_n) to a)-d) above. This seems however a reasonable conjecture, and, in the following, we shall only focus on this kind of solutions.' This is a disclosed domain restriction, not an equation that defines the result in terms of itself; if the completeness conjecture fails, Theorem 4.9 would remain valid for the stated class but would not cover every bootstrap solution. The self-citations [14] and [15] appear as contextual references to prior convergence results and a survey, and they are not used to justify Theorem 4.9. No uniqueness theorem is imported from the authors' own work to forbid alternative constructions; the paper instead flags the open completeness question. Therefore the derivation is self-contained within its explicitly stated assumptions, and the only caveat is a correctness/scope limitation rather than circular reasoning.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data. The only constants are the coupling b and the operator parameters omega_O, s_O, w_O, all of which are inputs from the bootstrap axioms. No new physical entities are postulated; the r-truncation is a bookkeeping device. The main imported ingredient is the K-transform representation from [3], whose completeness is explicitly conjectural.

assumptions (6)

domain assumption The Fock space hShG with translation and boost actions (eqs (2.6)-(2.12)) is the state space of the model.
Defines the Hilbert space and relativistic kinematics used in all matrix elements.
domain assumption The Sinh-Gordon S-matrix is the diagonal scalar S(beta) in (2.2), with no bound states.
Input from [10]; the bootstrap axioms and all kernels are built from this S-matrix.
domain assumption Form factor bootstrap axioms I-V (Sections 2.2 and 2.3) are the definition of the quantum fields.
The paper works within the bootstrap program and assumes these axioms rather than constructing fields from a Lagrangian.
ad hoc to paper Every form factor has the K-transform representation (2.23) with p_n satisfying a)-d), and this class is complete.
Explicitly stated after Proposition 2.1: completeness is a reasonable conjecture but unproven; all later results depend on this form.
ad hoc to paper The functions p_n obey 2i pi-periodicity, symmetry, and cosh growth bounds as in Proposition 2.1.
Used in Lemma 4.2 to justify the contour integral representation and bounds for form factors.
standard math Barnes G-function asymptotic expansions (A.16)-(A.17) hold uniformly with differentiable remainders.
Needed in Lemma 4.2 to control the two-particle form factor F at infinity.

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Cite this review

Pith. "Pith review of On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory." pith.science (2026). https://pith.science/paper/UWZBYKCE

@misc{pith2026250203894,
  author       = {Pith},
  title        = {Pith review of: On multipoint correlation functions in the Sinh-Gordon 1+1 dimensional quantum field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWZBYKCE}},
  note         = {Machine review of arXiv:2502.03894}
}

abstract

This work provides a closed, explicit and rigorous expression for the appropriately truncated $k$-point function of the integrable 1+1 dimensional Sinh-Gordon quantum field theory. The results are obtained within the bootstrap program setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[3]

Sine-Gordon breather form factors and quantum ﬁeld equati ons

H. Babujian and M. Karowski, "Sine-Gordon breather form factors and quantum ﬁeld equati ons.", J.Phys.A 35 (2002), 9081–9104

work page 2002
[1]

Quantum S-matrix of the (1 +1) dimensional Toda chain

A.E. Arinshtein, V .A. Fateev, and A.B. Zamolodchikov, "Quantum S-matrix of the (1 +1) dimensional Toda chain.", Phys. Lett. B 87 (1979), 389–392

work page 1979
[2]

Exact form factors in integrable quantum ﬁeld theo- ries: the sine-Gordon model

H. Babujian, A. Fring, M. Karowski, and A. Zapletal, "Exact form factors in integrable quantum ﬁeld theo- ries: the sine-Gordon model." , Nucl. Phys. B 538 (1999), 535–586

work page 1999
[4]

Multipoint Green’s functions in 1 + 1 dimensional inte- grable quantum ﬁeld theories

H. M. Babujian, M. Karowski, and A.M. Tsvelik, "Multipoint Green’s functions in 1 + 1 dimensional inte- grable quantum ﬁeld theories." , Nucl.Phys. B 917 (2017), 122–153

work page 2017
[5]

The intrinsic coupling in integrable quantum ﬁeld theories

J. Balog, M. Niedermaier, F. Niedermayer, A. Patrascioi u, E. Seiler, and P . Weisz,"The intrinsic coupling in integrable quantum ﬁeld theories." , Nucl.Phys. B 583 (2000), 614–670

work page 2000
[6]

Angular quantization and form factors in massive integrab le models

V . Brazhnikov and S. Lukyanov, "Angular quantization and form factors in massive integrab le models." , Nucl. Phys. B 512 (1998), 616–636

work page 1998
[7]

Potts correlators and the static three-quark potential

M. Caselle, G. Delﬁno, P . Grinza, O. Jahn, and N. Magnoli, "Potts correlators and the static three-quark potential.", J. Stat.Mech. 0603 (2006), P03008

work page 2006
[8]

A multivariate Faa-di-Bruno formula with applications

G.M. Constantine and T.H. Savits, "A multivariate Faa-di-Bruno formula with applications." , Trans. Amer. Math. Soc. 348 (1996), 503–520

work page 1996

Show all 21 references

[9]

An Asymptotic Expansion of the Double Gamma Function

C. Ferreira and J.L. López, "An Asymptotic Expansion of the Double Gamma Function." , J. Approx. Theor. 111 (2001), 298–314. 69

2001
[10]

Two-dimensional quantum ﬁeld theories having exact solut ions

V .M. Gryanik and S.N. V ergeles,"Two-dimensional quantum ﬁeld theories having exact solut ions.", J. Nucl. Phys. 23 (1976), 1324–1334

1976
[11]

Zür F ormulierung quantisierter Feldtheorien

K. Symanzik H. Lehmann and W. Zimmerman, "Zür F ormulierung quantisierter Feldtheorien.", Nuov. Cim. 1 (1955), 205–225

1955
[12]

Complete S-matrix of the massive Thirring model

M. Karowski and H.J. Thun, "Complete S-matrix of the massive Thirring model." , Nucl. Phys. B 130 (1978), 295–308

1978
[13]

F orm-factors in the SU(2)-invariant Thirring model

A.N. Kirillov and F.A. Smirnov, "F orm-factors in the SU(2)-invariant Thirring model.", J. Soviet. Math. 47 (1989), 2423–2450

1989
[14]

Bootstrap approach to 1+1 dimensional integrable quantum ﬁeld theories: the case of the Sinh-Gordon model

K.K. Kozlowski, "Bootstrap approach to 1+1 dimensional integrable quantum ﬁeld theories: the case of the Sinh-Gordon model .", Proc. Int. Cong. Math. 2022 5 (2022), 4096–4118

2022
[15]

On convergence of form factor expansions in the inﬁnite vol ume quantum Sinh-Gordon model in 1+1 dimensions

, "On convergence of form factor expansions in the inﬁnite vol ume quantum Sinh-Gordon model in 1+1 dimensions.", Inventiones mathematicae 233 (2023), 725–827

2023
[16]

Free ﬁeld representation for massive integrable models

S. Lukyanov, "Free ﬁeld representation for massive integrable models." , Comm. Math. Phys. 167 (1995), 183–226

1995
[17]

Algebraic Bethe ansatz and correlation functions

N.A. Slavnov, "Algebraic Bethe ansatz and correlation functions." , World Scientiﬁc, 2022

2022
[18]

F orm factors in completely integrable models of quantum ﬁe ld theory

F.A. Smirnov, "F orm factors in completely integrable models of quantum ﬁe ld theory.", Advanced Series in Mathematical Physics, vol. 14, World Scientiﬁc, 1992

1992
[19]

PCT, Spin and statistics, and all that

R. Streater and A. Wightman, "PCT, Spin and statistics, and all that." , W.A. Benjamin inc., New Y ork New Y ork, vol. 1, The mathematical physics monograph series, 1964

1964
[20]

Factorized S-matrices in two dimensions as the exact so- lutions of certain relativistic quantum ﬁeld theory models

A.B. Zamolodchikov and Al.B. Zamolodchikov, "Factorized S-matrices in two dimensions as the exact so- lutions of certain relativistic quantum ﬁeld theory models .", Ann. of Phys. 120 (1979), 253–291

1979
[21]

Exact Two-Particle S-Matrix of Quantum Sine-Gordon Solit ons

Al.B. Zamolodchikov, "Exact Two-Particle S-Matrix of Quantum Sine-Gordon Solit ons.", Comm. math. Phys. 55 (1977), 183–186. 70

1977

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [3]

Sine-Gordon breather form factors and quantum ﬁeld equati ons

H. Babujian and M. Karowski, "Sine-Gordon breather form factors and quantum ﬁeld equati ons.", J.Phys.A 35 (2002), 9081–9104

work page 2002

[2] [1]

Quantum S-matrix of the (1 +1) dimensional Toda chain

A.E. Arinshtein, V .A. Fateev, and A.B. Zamolodchikov, "Quantum S-matrix of the (1 +1) dimensional Toda chain.", Phys. Lett. B 87 (1979), 389–392

work page 1979

[3] [2]

Exact form factors in integrable quantum ﬁeld theo- ries: the sine-Gordon model

H. Babujian, A. Fring, M. Karowski, and A. Zapletal, "Exact form factors in integrable quantum ﬁeld theo- ries: the sine-Gordon model." , Nucl. Phys. B 538 (1999), 535–586

work page 1999

[4] [4]

Multipoint Green’s functions in 1 + 1 dimensional inte- grable quantum ﬁeld theories

H. M. Babujian, M. Karowski, and A.M. Tsvelik, "Multipoint Green’s functions in 1 + 1 dimensional inte- grable quantum ﬁeld theories." , Nucl.Phys. B 917 (2017), 122–153

work page 2017

[5] [5]

The intrinsic coupling in integrable quantum ﬁeld theories

J. Balog, M. Niedermaier, F. Niedermayer, A. Patrascioi u, E. Seiler, and P . Weisz,"The intrinsic coupling in integrable quantum ﬁeld theories." , Nucl.Phys. B 583 (2000), 614–670

work page 2000

[6] [6]

Angular quantization and form factors in massive integrab le models

V . Brazhnikov and S. Lukyanov, "Angular quantization and form factors in massive integrab le models." , Nucl. Phys. B 512 (1998), 616–636

work page 1998

[7] [7]

Potts correlators and the static three-quark potential

M. Caselle, G. Delﬁno, P . Grinza, O. Jahn, and N. Magnoli, "Potts correlators and the static three-quark potential.", J. Stat.Mech. 0603 (2006), P03008

work page 2006

[8] [8]

A multivariate Faa-di-Bruno formula with applications

G.M. Constantine and T.H. Savits, "A multivariate Faa-di-Bruno formula with applications." , Trans. Amer. Math. Soc. 348 (1996), 503–520

work page 1996