Finite volume corrections of non-diagonal form factors

Istvan Vona

REVIEW 3 major objections 4 minor 30 references

Finite volume corrections of non-diagonal form factors

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that mu-term corrections to non-diagonal finite-volume form factors are residues of the F-term integral.

desk verdict A careful and internally consistent residue calculation that cross-checks the F-term formula against mu-terms, conditional on a one-particle regularization that is extended to multiparticle form factors without proof. read the letter →

arxiv 1908.09704 v2 pith:6AHRRNWK submitted 2019-08-26 hep-th

Istvan Vona This is my paper · ORCID

classification hep-th

keywords finitevolumeformfactorsmu-termcorrectionsF-termscalingLee-Yangmodelintegrablequantumfieldtheorybound-statequantizationresiduecalculusdependence

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 thesis establishes that the mu-term corrections to finite-volume non-diagonal form factors in the scaling Lee-Yang model are not an independent effect: they are exactly the residues that the F-term integral picks up when its integration contour is deformed around the S-matrix poles. The proof is explicit for elementary form factors, where the residue sum of the regulated F-term integrand reproduces, term by term, the mu-term formula obtained earlier from bound-state quantization. This agreement matters because it validates the formal F-term formula for general multiparticle states and shows that two apparently different correction schemes are two views of the same analytic structure. If the relation holds generally, it provides a practical dictionary: one contour prescription generates both the leading exponential volume corrections and the bound-state corrections for finite-volume matrix elements.

What carries the argument

The load-bearing object is the regulated form factor $F^r_{M+N+2}(v+i\pi,\{\vartheta+i\pi/2\},v,\{\theta-i\pi/2\})$, defined by subtracting the kinematical pole of the pair of mirror particles symmetrically, together with the contour rule (3.30): the F-term integral is evaluated by averaging two contours that pick residues at $v=\theta_k\pm i\pi/6$, where the two S-matrix poles of the Lee-Yang model sit. The residues of the integrand evaluate to $\pm 2i\,\delta u_{k\mp}$, the same corrections to the fusion angle that bound-state quantization produces from the exponentialized Bethe-Yang equations (3.23). The bootstrap equation $S(\theta)=S(\theta+iu)S(\theta-iu)$ and the S-matrix pole structure at the fusion angle connect these residues to the mu-term formula (4.5).

What would settle it

Compute the F-term integral (4.11) numerically for a two-particle state in the scaling Lee-Yang model without using the residue expansion, and compare it with the mu-term formula (4.5). Any disagreement beyond the stated exponential accuracy would disprove the claimed identity. A direct test is to compare the residue-corrected analytic prediction for $\langle 0|\Phi|\{1,-1\}\rangle_L$ with the exact finite-volume value obtained by the truncated conformal-space numerical method at volumes $mL\sim 6$ to $12$; a systematic gap at the expected order would refute the paper's central claim.

Watch

Extended reading notes

Core claim

The paper's central discovery is a contour-residue identity. For a non-diagonal finite-volume form factor with elementary in-state rapidities $\{\bar\theta^{(0)}\}$, the mu-term correction $\delta^{(\mu)}F_N(\{\bar\theta^{(0)}\})$ of equation (4.5)—derived by representing each physical particle as a pair of constituents with complex rapidities $\bar\theta^{(0)}_k\pm i(u+\delta\bar u_k)$ and expanding the finite-volume normalization—equals the sum of residues $$\frac12\sum_{k,\pm}\pm i\,\mathrm{Res}_{v\to\bar\$theta^{{(0)}}$_k\mp i\pi/6}\left\{F^r_{N+2}(v+i\pi,v,\{\bar\$theta^{{(0)}}$_j-i\pi/2\})e^{-mL\$\cosh$ v}\right\},$$ where $F^r$ is the regulated form factor with the kinematical pole subtracted. The equality is shown by computing the residues of the S-matrix poles in the F-term integrand and using the bootstrap equation to convert the residue terms into the $\delta u_{k\pm}$ quantities of bound-state quantization. This proves the suspected relation between the F-term and mu-term formalisms and thereby underpins the formal derivation of the F-term formula.

Load-bearing premise

The calculation assumes that the regularization of the squared delta function, $2\pi\delta(u-v)=i/(u-v+i\epsilon)-i/(u-v-i\epsilon)$, justified for the one-particle form factor, remains valid when applied to the general multiparticle matrix element; if it does not, the residue sum need not produce the mu-terms.

Editorial extensions

If this is right

The F-term integral, evaluated with the contour rule (3.30), contains the mu-term corrections; no separate treatment of bound-state constituents is needed to obtain the leading exponential volume dependence.
The equality confirms the formal F-term formula of the companion paper for non-diagonal form factors, since the same residue structure that reproduces the mu-terms is built into the regulated form factor.
The dictionary between the two schemes can be run in reverse: higher-order corrections from bound-state quantization suggest how to construct the corresponding integral terms for higher exponential orders.
Because the derivation uses only the form-factor axioms and the S-matrix pole structure, the same F-term/mu-term equivalence is expected in other diagonal scattering theories with fusion, with the fusion angle $u$ replaced by the model-specific value.
In the diagonal limit, the relation connects the non-diagonal F-term to the known exact finite-volume diagonal form factors, providing an independent consistency check on both approaches.

Reading between the lines

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

The same mechanism suggests a numerical shortcut: for multi-particle states, one could evaluate the F-term correction by the residue sum at the shifted rapidities instead of the full integral, isolating the bound-state contributions directly.
The contour picture suggests that mu-term and F-term corrections are two residues of a single meromorphic integrand; iterating the bound-state expansion to higher orders could yield a constructive route to higher exponential corrections.
A testable extension is to repeat the residue calculation in another integrable theory with a fusion channel, replacing the Lee-Yang fusion angle by the model-specific value; agreement would indicate the relation is generic rather than model-dependent.

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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

This MSc thesis does exactly what it claims: it shows, for elementary finite-volume form factors in the Lee-Yang model, that the residues of the F-term integral reproduce the mu-terms from bound-state quantization. The residue algebra in Appendices B and C is careful and appears internally consistent, and the match with (4.5) is a genuine cross-check between two independent methods. What is new is the explicit N-particle residue calculation; the one-particle vacuum case was already known to Bajnok, and the F-term formula itself comes from the thesis author's own paper [16].

The thesis also does several things well. The organization is clear, the pedagogical introduction to integrable QFT is readable, and the technical core is honest about where the explicit calculation stops. Footnote 1 of Chapter 4 concedes that only elementary form factors are computed and that the general non-diagonal case is argued by non-mixing. That is a real limitation, but it is modest and stated plainly.

The soft spot is the one flagged in the stress test. The derivation starts from the F-term formula (4.8)-(4.10), and that formula depends on regularizing the squared delta function via 2πδ(u−v)=i/(u−v+iϵ)−i/(u−v−iϵ). In [28] this was justified for the one-particle form factor. Here it is extended "everywhere" to multiparticle matrix elements, where the crossed part contains S-matrix prefactors and an N-particle form factor in the disconnected term. The double-pole structure is genuinely more complex, and no independent derivation of the regularization is given. So the agreement with the mu-term is evidence, but not a proof; if the regularization is only valid for N=1, the equality could be accidental. I do not think that kills the paper—the calculation remains a meaningful check—but it is the load-bearing assumption.

Two smaller points. The numerical TCSA comparison is imported from [16] without error bars, so it supports the result but does not independently pin down the analytic regularization. And the self-citation point is real but not a flaw: the F-term formula is not assumed in the mu-term derivation, and the two routes are genuinely different. The circularity burden is moderate, not fatal.

Who is this for? Specialists working on finite-volume form factors in integrable QFTs. It will not change the direction of the field, but it is a useful technical check and a detailed companion to [16]. I would send it to a competent referee rather than desk reject it, with the instruction to focus on the regularization step. As a standalone journal article it is thin; as a thesis chapter or technical note it deserves to be available.

Referee Report

3 major / 4 minor

Summary. This MSc thesis develops the leading exponential finite-volume corrections to non-diagonal form factors in the scaling Lee-Yang model. It first reviews integrable quantum field theory, the bootstrap, and finite-volume methods, then derives the μ-term correction δ^(μ)F_N from bound-state quantization (§4.1, App. B) and the F-term correction from the mirror-model trace (§4.2.1, Eqs. (4.8)–(4.10)). The central result is that evaluating the F-term integral (4.11) with the contour prescription (3.30) and taking the residues at v=θ_k±iπ/6 reproduces the μ-term formula (4.5), thereby establishing the suspected F/μ correspondence for elementary form factors. The appendices contain the residue algebra and the derivation of the μ-term from bound-state quantization.

Significance. If the correspondence holds, the thesis provides a nontrivial consistency check between two independent finite-volume schemes: the bound-state quantization of [15] and the mirror-model F-term formula of [16]. The residue calculation in App. C is detailed and internally consistent as far as the text allows one to check, and the matching with (4.5) is not a trivial identity. The pedagogical introduction is also useful. However, the proof is conditional on a regularization of the squared delta function whose multiparticle validity is not established, and the explicit computation covers only elementary form factors; the generalization to arbitrary non-diagonal matrix elements is asserted rather than demonstrated.

major comments (3)

[§4.1–§4.2, footnote 1] The multiparticle F-term formula relies on regularizing the squared delta function as 2πδ(u−v)=i/(u−v+iϵ)−i/(u−v−iϵ) and on the statement that this is correct 'by using this everywhere'. The reference [28] justifies this regularization for the one-particle form factor only. For N>1 the integrand contains S-matrix prefactors and multiparticle form factors in the disconnected terms, and the residue calculation at v=θ_k±iπ/6 depends directly on the resulting double-pole structure. The validity of the regularization for the multiparticle matrix element is therefore load-bearing and is not derived. Please provide a derivation, or an explicit reduction to the one-particle case, or state clearly that the equality between the residues of (4.11) and the μ-term (4.5) is conditional on this regularization.
[Chapter 4, footnote 1] The explicit residue calculation is performed only for elementary form factors ⟨0|O|{n}⟩_L, while the title and abstract claim the result for general non-diagonal form factors ⟨{m}|O|{n}⟩_L. The extension is justified by a non-mixing argument stated in a footnote: poles belonging to the in- or outgoing set do not contribute to the μ-terms of the other set. This assertion is not demonstrated. Because the F-term integrand (4.8) and the residue formula (4.14) contain products over both sets of rapidities, an explicit argument is needed to show that the two sets decouple. Without it, the paper's central claim is established only for elementary form factors, and the general statement should be presented as an extrapolation unless the missing argument is supplied.
[Abstract and §4.2] The F-term formula (4.8)–(4.10) and the contour rule (3.30) are imported from [16], which is co-authored by the thesis author. The comparison with the μ-term from bound-state quantization is therefore a consistency check between two prescriptions rather than an independent derivation of the F-term formula. This is a legitimate and useful result, but the abstract's wording — 'proves the suspected relation' and 'underpins the formal derivation' — overstates the logical status. The paper should state explicitly that the F-term formula is assumed and that the calculation verifies the residue/μ-term correspondence conditional on that assumption and on the delta-function regularization.

minor comments (4)

[Eq. (C.6)] In the displayed formula for δ^(μ)F_N, the first sum appears to contain δ¯u_k both inside and outside the braces, which would give a term quadratic in δ¯u_k and contradict Eq. (4.5). Please check whether this is a typographical error and correct it.
[§2.1.2 and §2.2] There are minor typographical issues: 'model sepciﬁc' should be 'model specific', and 'Schwartz's theorem' should be 'Schwarz's theorem'.
[Figure 5.1] The TCSA comparison shown in Fig. 5.1 is imported from [16] and is presented without error bars. Please state the numerical uncertainty of the TCSA data or refer the reader to the original source for the error estimate.
[§3.2.3] The term 'F-term' is used for both the exponential integral in Eq. (3.15) and the derivative correction to the rapidities. Consider using a consistent notation (e.g., 'F-term integral' and 'F-term rapidity shift') to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the F-term-to-mu-term correspondence is derived as a genuine cross-check, not assumed.

full rationale

The thesis's central claim is that the residue sum of the F-term integral (4.11), evaluated with the contour rule (3.30), reproduces the mu-term expression delta^(mu)F_N in (4.5). Equation (4.5) is derived independently in Sec. 4.1 and App. B from bound-state quantization, while the residue computation in Sec. 4.2.2 and App. C starts from the F-term formula (4.8)-(4.10) and the S-matrix/bootstrap input (3.16)-(3.17). The two calculations share neither the mu-term expression nor the F-term integrand as an input; the equality is a derived identity. The F-term formula itself is imported from [16], a paper coauthored by the thesis author, and the squared-delta regularization is imported from [28] and extended from the one-particle case to the multiparticle case; these are stated assumptions and a minor self-citation, but they are not circular because the cited prior work does not assume the target mu-term equality. The TCSA comparison in Fig. 5.1 is imported from [16] as an external numerical benchmark, not as an input to the derivation. No step reduces by construction to its own input, so no significant circularity is found.

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

No new fitted parameters and no new physical entities are introduced. The calculation uses the standard Lee-Yang S-matrix inputs m, Γ and u=π/3 from prior literature, and the small shifts δuk± are functions determined by the quantization conditions, not fitted numbers. The regulated form factors F^[α] are bookkeeping devices for subtracting poles, not new physics. The work rests on the bootstrap and form-factor axioms and on the F-term formula from the author's previous paper.

assumptions (4)

domain assumption The S-matrix has the Lee-Yang bound-state pole structure S(θ) ≈ iΓ²/(θ−2iu) and −iΓ²/(θ−iu) (eq. 3.17), and satisfies the bootstrap equation S(θ)=S(θ+iu)S(θ−iu) (eq. 3.16).
The residue calculation in §4.2.2 and App. C uses these poles and the factorization; the bootstrap equation is introduced as an axiom in §2.3.
domain assumption The form factor axioms (2.25)-(2.29), especially the kinematical and dynamical pole axioms, fix the singularities and residues used in the regulated form factors.
The finite parts F^[α] in App. A and the pole expansions in App. C are constructed from these axioms; they are standard in integrable QFT but not proved in this thesis.
domain assumption The finite-volume form factor formula (3.32) from Pozsgay and Takacs correctly describes the leading polynomial-volume approximation for non-diagonal matrix elements.
The mu-term derivation in §4.1 starts from this formula; the thesis does not rederive it.
domain assumption The F-term formula (4.8)-(4.10), including the delta-function regularization in §4.2.1, is valid for general multiparticle non-diagonal form factors.
The central calculation in §4.2.2 evaluates residues of this formula; it is taken from [16], a paper coauthored by the thesis author, and is exactly the object the thesis aims to support.

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Cite this review

Pith. "Pith review of Finite volume corrections of non-diagonal form factors." pith.science (2026). https://pith.science/paper/6AHRRNWK

@misc{pith2026190809704,
  author       = {Pith},
  title        = {Pith review of: Finite volume corrections of non-diagonal form factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AHRRNWK}},
  note         = {Machine review of arXiv:1908.09704}
}

abstract

This thesis presents L\"uscher's $\mu$- and $F$-term corrections to volume dependence of non-diagonal finite volume form factors in the scaling Lee-Yang model. An explicit calculation proves the suspected relation that the $\mu$-terms known previously from bound state quantization can be obtained from the $F$-term integrals by modifying the contour of integration such that it picks up residues of appropriate poles in the integrand. The fact that these two different approaches for getting the $\mu$-terms give the same result underpins the formal derivation of the $F$-term in arXiv:1904.00492 which was not known until recently. In the meantime, the notions of integrable quantum field theories and those related to their treatment in finite volume are introduced to help understand the topic for readers not familiar with it.

Figures

Figures reproduced from arXiv: 1908.09704 by the authors.

**Figure 2.1.** Visualization of the equality (2.10) in terms of paths. We see from (2.10) that the trace of T(t, λ) is independent of time by the cyclic property, and so q(λ) ≡ Tr(T(t, λ)) is constant. In its expansion q(λ) = P n qnλ n the qn[φ]-s are conserved charges, where we denoted the fields of the integrable model collectively as φ. In a concrete model one could get these charges by solving the PDE (2.9) for the matrix U. S… view at source ↗

**Figure 2.2.** Left: Static soliton and anti-soliton. The shape of their energy density (x) is superimposed on the figure in a scale-less way. Right: Two-soliton and soliton-anti-soliton solutions where the particles are moving with relative velocity v = 0.1c in each; the figure shows the configurations at t = 100m−1 after the scattering events. 2.2. Integrability in quantum field theory One can look at the sinh-Gordon model - ap… view at source ↗

**Figure 2.3.** Graphs for tree-level contributions. another problem: the soliton mass (2.11) is non-perturbative O(β −2 ) in the coupling, and the soliton is not an elementary excitation of the field around a single minimum of the potential which one could treat via standard Feynman perturbation theory.1 It is however possible to circumvent these problems if we assume that those infinitely many charges remain conserved quantities … view at source ↗

Figures from the paper (17 more)

**Figure 2.4.** Figure 2.4: Reflection amplitude for the sine-Gordon model where particles are changing their identities. reason why we chose to work with these theories at start: in higher dimensions the existence of a generator with higher rank will render the S-matrix to be trivial, since th…

**Figure 2.5.** Figure 2.5: Left: The complex plane of the invariant s. Physical values come from approaching A, the inverse of the S-matrix is at B, and the physical t-channel amplitude reached at C. Right: The analytic structure after the mapping, on the θ plane. We reach the physical values …

**Figure 2.6.** Figure 2.6: The so-called Coleman-Thun diagram on the Euclidean (E, q) plane. The complex rapidities of the particles are θi = iu¯ j ik, θj = −iu¯ i jk, θk = 0 [PITH_FULL_IMAGE:figures/full_fig_p026_2_6.png]

**Figure 2.7.** Figure 2.7: The simple pole corresponding to an on-shell bound state propagator in the s-channel. and there are such pairs also in Sjk and Sik at θ = iui jk, iu¯ i jk ↔ s, t = m2 i and θ = iuj ik, iu¯ j ik ↔ s, t = m2 j respectively due to bound states i and j, if Γijk 6= 0. The…

**Figure 2.8.** Figure 2.8: The mass triangle. The relevance of the µ k ij height will be explained in subsection 3.2.4 [PITH_FULL_IMAGE:figures/full_fig_p027_2_8.png]

**Figure 2.9.** Figure 2.9: Illustration to the bootstrap equation. applies equation (2.20) Sab(θ) = Saa(θ + iu¯ a ab)Saa(θ − iu¯ a ab) Sbb(θ) = Sab(θ + iu¯ a ab)Sab(θ − iu¯ a ab), where we assumed that a pole of yet unknown origin in the physical strip of Saa is at a fusion angle u b aa for a …

**Figure 2.10.** Figure 2.10: Intuitive pictorial representation of the crossing formula (2.24). The iπ shift is indicated by turning the direction of the particle line. 2. Periodicity axiom14: F O i1i2...iN (θ1 + 2iπ, θ2, . . . , θN ) = F O i2...iN i1 (θ2, . . . , θN , θ1) (2.26) 14By comparing…

**Figure 3.1.** Figure 3.1: Left: Lüscher’s µ-term correction to the mass of a standing particle comes from a virtual process where the particle disintegrates into its virtual (mirror-)particle constituents which fuse into another particle after wrapping the space-time cylinder. Right: The F-…

**Figure 3.2.** Figure 3.2: The energy spectrum of the scaling Lee-Yang model. The discrete points represent the result of the TCSA method mentioned in the introduction of this chapter (i.e. the exact energy values, since the numerical uncertainty is small [16]), while the solid lines are the c…

**Figure 3.3.** Figure 3.3: The original and the mirror model from the finite volume finite temperature Euclidean theory. The arrows show the direction in which the Hamiltonians HL, HR generate the time translation. The one corresponding to the first trace of (3.4) is a zero-temperature QFT i…

**Figure 3.4.** Figure 3.4: A particle with rapidity θj in the R-channel behaves as a defect on which the mirror particles scatter in the L-channel. Source: [16] 3.2.4. TBA for models with fusion and the µ-term In models with bound states the type of singularities of the integrands mentioned in…

**Figure 3.5.** Figure 3.5: The two integration contours whose contributions averaged gives both the µ- and the F-term. Source: [16] On figure (3.6) we compare the different energy corrections dealt with in this section. 44 [PITH_FULL_IMAGE:figures/full_fig_p048_3_5.png]

**Figure 3.6.** Figure 3.6: Lüscher-corrections to the three-particle state with quantum numbers {2, 0, −2} in SLYM. The exact (TCSA) values are subtracted, and so the better an approximation is, the faster it converges to zero. From (3.25) we get “µ1”, the leading order. “F1” is the F-term cor…

**Figure 4.1.** Figure 4.1: Left: Illustration to finite volume form factor hϑM, . . . , ϑ1|O|θ1, . . . , θN iL. Right: The same matrix element in the “thermal” channel, as a trace Tr(e −LHON,M). Source: [16] in particle numbers and momenta, and so one cannot make the boundary conditions perio…

**Figure 4.2.** Figure 4.2: The matrix element hv|ON,M|ui of the non-local operator in the Lchannel with an ingoing mirror particle with u and an outgoing one with v. The directions of the particle lines can be intuitively identified with the imaginary parts of the arguments in 4.7. Source: [1…

**Figure 5.1.** Figure 5.1: Comparison of the same kind of volume corrections to the form factor h0|Φ|{1, −1}iL as explained below the figure 3.6 for the energy (here Φ is the deformation field of the scaling Lee-Yang model, whose form factor was “measured”). The figure shows the difference be…

**Figure 5.2.** Figure 5.2: Interpretation of the formula (4.9). A member of a virtual particle pair wraps the cylinder and both of them reaches the operator. Source: [16] The universal contour shown in Fig. (3.5) for the integrals - which gave both the µ-terms and the F-term correction in case…

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Works this paper leans on

30 extracted references · 17 canonical work pages

[16]

Leading exponential finite size corrections for non-diagonal form factors

Z. Bajnok, M. Lajer, B. Szepfalvi, and I. Vona, `` Leading exponential finite size corrections for non-diagonal form factors ,'' http://dx.doi.org/10.1007/JHEP07(2019)173 JHEP 07 (2019) 173 , http://arxiv.org/abs/1904.00492 arXiv:1904.00492 [hep-th]

work page Pith review arXiv 2019
[28]

Field theoretical derivation of L\"uscher's formula and calculation of finite volume form factors

Z. Bajnok, J. Balog, M. Lajer, and C. Wu, `` Field theoretical derivation of Luscher's formula and calculation of finite volume form factors ,'' http://dx.doi.org/10.1007/JHEP07(2018)174 JHEP 07 (2018) 174 , http://arxiv.org/abs/1802.04021 arXiv:1802.04021 [hep-th]

work page Pith review arXiv 2018
[15]

Luscher's mu-term and finite volume bootstrap principle for scattering states and form factors

B. Pozsgay, `` Luscher's mu-term and finite volume bootstrap principle for scattering states and form factors ,'' http://dx.doi.org/10.1016/j.nuclphysb.2008.04.021 Nucl. Phys. B802 (2008) 435--457 , http://arxiv.org/abs/0803.4445 arXiv:0803.4445 [hep-th]

work page Pith review arXiv 2008
[1]

Bajnok, `` How Integrability Works (for AdS/CFT) ,'' http://dx.doi.org/10.5506/APhysPolB.47.2451 Acta Phys

Z. Bajnok, `` How Integrability Works (for AdS/CFT) ,'' http://dx.doi.org/10.5506/APhysPolB.47.2451 Acta Phys. Polon. B47 (2016) 2451--2477

work page doi:10.5506/aphyspolb.47.2451 2016
[2]

Mussardo, Statistical field theory

G. Mussardo, Statistical field theory . Oxford Univ. Press, New York, NY, 2010

work page 2010
[3]

Bajnok and L

Z. Bajnok and L. S amaj, ``Introduction to integrable many-body systems III ,'' Acta Physica Slovaca. Reviews and Tutorials 61 no. 2, (2011) 129--271

work page 2011
[4]

Dorey, `` Exact S-matrices ,'' in Conformal field theories and integrable models

P. Dorey, `` Exact S-matrices ,'' in Conformal field theories and integrable models. Proceedings, Eotvos Graduate Course, Budapest, Hungary, August 13-18, 1996 , pp. 85--125. 1996. http://arxiv.org/abs/hep-th/9810026 arXiv:hep-th/9810026 [hep-th]

arXiv 1996
[5]

Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory

R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory . North-Holland Publishing Company, 1982

work page 1982

Show all 30 references

[6]

R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The analytic S-matrix . Cambridge Univ. Press, Cambridge, 1966

1966
[7]

A. B. Zamolodchikov, `` Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-Yang Models ,'' http://dx.doi.org/10.1016/0550-3213(90)90333-9 Nucl. Phys. B342 (1990) 695--720

1990 doi
[8]

Mitra, `` Elasticity, Factorization and S-matrices in (1+1)-Dimensions ,'' http://dx.doi.org/10.1016/0370-2693(77)90063-6 Phys

P. Mitra, `` Elasticity, Factorization and S-matrices in (1+1)-Dimensions ,'' http://dx.doi.org/10.1016/0370-2693(77)90063-6 Phys. Lett. 72B (1977) 62--64

1977 doi
[9]

S. R. Coleman and H. J. Thun, `` On the Prosaic Origin of the Double Poles in the Sine-Gordon S-matrix ,'' http://dx.doi.org/10.1007/BF01609466 Commun. Math. Phys. 61 (1978) 31

1978 doi
[10]

J. L. Cardy and G. Mussardo, `` S-matrix of the Yang-Lee Edge Singularity in Two-Dimensions ,'' http://dx.doi.org/10.1016/0370-2693(89)90818-6 Phys. Lett. B225 (1989) 275--278

1989 doi
[11]

H. M. Babujian, A. Fring, M. Karowski, and A. Zapletal, `` Exact form-factors in integrable quantum field theories: The Sine-Gordon model ,'' http://dx.doi.org/10.1016/S0550-3213(98)00737-8 Nucl. Phys. B538 (1999) 535--586 , http://arxiv.org/abs/hep-th/9805185 arXiv:hep-th/980...

1999 arXiv
[12]

Mussardo, `` Off critical statistical models: Factorized scattering theories and bootstrap program ,'' http://dx.doi.org/10.1016/0370-1573(92)90047-4 Phys

G. Mussardo, `` Off critical statistical models: Factorized scattering theories and bootstrap program ,'' http://dx.doi.org/10.1016/0370-1573(92)90047-4 Phys. Rept. 218 (1992) 215--379

1992 doi
[13]

Karowski and P

M. Karowski and P. Weisz, `` Exact Form-Factors in (1+1)-Dimensional Field Theoretic Models with Soliton Behavior ,'' http://dx.doi.org/10.1016/0550-3213(78)90362-0 Nucl. Phys. B139 (1978) 455--476

1978 doi
[14]

Luscher, `` Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories

M. Luscher, `` Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States ,'' http://dx.doi.org/10.1007/BF01211589 Commun. Math. Phys. 104 (1986) 177

1986 doi
[17]

S. J. van Tongeren, `` Introduction to the thermodynamic Bethe ansatz ,'' http://arxiv.org/abs/1606.02951 arXiv:1606.02951 [hep-th] . [J. Phys.A49,no.32,323005(2016)]

2016 arXiv
[18]

Bajnok, ``Integr \'a lhat \'o modellek \'e s az A d S / CFT megfeleltet \'e s,''

Z. Bajnok, ``Integr \'a lhat \'o modellek \'e s az A d S / CFT megfeleltet \'e s,''
[19]

Teschner, `` On the spectrum of the Sinh-Gordon model in finite volume ,'' http://dx.doi.org/10.1016/j.nuclphysb.2008.01.021 Nucl

J. Teschner, `` On the spectrum of the Sinh-Gordon model in finite volume ,'' http://dx.doi.org/10.1016/j.nuclphysb.2008.01.021 Nucl. Phys. B799 (2008) 403--429 , http://arxiv.org/abs/hep-th/0702214 arXiv:hep-th/0702214 [hep-th]

2008 arXiv
[20]

Bajnok and O

Z. Bajnok and O. el Deeb, `` Form factors in the presence of integrable defects ,'' http://dx.doi.org/10.1016/j.nuclphysb.2010.02.004 Nucl. Phys. B832 (2010) 500--519 , http://arxiv.org/abs/0909.3200 arXiv:0909.3200 [hep-th]

2010 arXiv
[21]

Pozsgay, I

B. Pozsgay, I. M. Szecsenyi, and G. Takacs, `` Exact finite volume expectation values of local operators in excited states ,'' http://dx.doi.org/10.1007/JHEP04(2015)023 JHEP 04 (2015) 023 , http://arxiv.org/abs/1412.8436 arXiv:1412.8436 [hep-th]

2015 arXiv
[22]

Dorey and R

P. Dorey and R. Tateo, `` Excited states by analytic continuation of TBA equations ,'' http://dx.doi.org/10.1016/S0550-3213(96)00516-0 Nucl. Phys. B482 (1996) 639--659 , http://arxiv.org/abs/hep-th/9607167 arXiv:hep-th/9607167 [hep-th]

1996 arXiv
[23]

Pozsgay and G

B. Pozsgay and G. Takacs, `` Form-factors in finite volume I: Form-factor bootstrap and truncated conformal space ,'' http://dx.doi.org/10.1016/j.nuclphysb.2007.06.027 Nucl. Phys. B788 (2008) 167--208 , http://arxiv.org/abs/0706.1445 arXiv:0706.1445 [hep-th]

2008 arXiv
[24]

Pozsgay and G

B. Pozsgay and G. Takacs, `` Form factors in finite volume. II. Disconnected terms and finite temperature correlators ,'' http://dx.doi.org/10.1016/j.nuclphysb.2007.07.008 Nucl. Phys. B788 (2008) 209--251 , http://arxiv.org/abs/0706.3605 arXiv:0706.3605 [hep-th]

2008 arXiv
[25]

Bajnok and C

Z. Bajnok and C. Wu, `` Diagonal form factors from non-diagonal ones ,'' http://arxiv.org/abs/1707.08027 arXiv:1707.08027 [hep-th]

arXiv
[26]

Leclair and G

A. Leclair and G. Mussardo, `` Finite temperature correlation functions in integrable QFT ,'' http://dx.doi.org/10.1016/S0550-3213(99)00280-1 Nucl. Phys. B552 (1999) 624--642 , http://arxiv.org/abs/hep-th/9902075 arXiv:hep-th/9902075 [hep-th]

1999 arXiv
[27]

B. Pozsgay, `` Form factor approach to diagonal finite volume matrix elements in Integrable QFT ,'' http://dx.doi.org/10.1007/JHEP07(2013)157 JHEP 07 (2013) 157 , http://arxiv.org/abs/1305.3373 arXiv:1305.3373 [hep-th]

2013 arXiv
[29]

P. D. Powell, `` Calculating Determinants of Block Matrices ,'' http://arxiv.org/abs/1112.4379 arXiv:1112.4379 [math.NA]

arXiv
[30]

write newline

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [16]

Leading exponential finite size corrections for non-diagonal form factors

Z. Bajnok, M. Lajer, B. Szepfalvi, and I. Vona, `` Leading exponential finite size corrections for non-diagonal form factors ,'' http://dx.doi.org/10.1007/JHEP07(2019)173 JHEP 07 (2019) 173 , http://arxiv.org/abs/1904.00492 arXiv:1904.00492 [hep-th]

work page Pith review arXiv 2019

[2] [28]

Field theoretical derivation of L\"uscher's formula and calculation of finite volume form factors

Z. Bajnok, J. Balog, M. Lajer, and C. Wu, `` Field theoretical derivation of Luscher's formula and calculation of finite volume form factors ,'' http://dx.doi.org/10.1007/JHEP07(2018)174 JHEP 07 (2018) 174 , http://arxiv.org/abs/1802.04021 arXiv:1802.04021 [hep-th]

work page Pith review arXiv 2018

[3] [15]

Luscher's mu-term and finite volume bootstrap principle for scattering states and form factors

B. Pozsgay, `` Luscher's mu-term and finite volume bootstrap principle for scattering states and form factors ,'' http://dx.doi.org/10.1016/j.nuclphysb.2008.04.021 Nucl. Phys. B802 (2008) 435--457 , http://arxiv.org/abs/0803.4445 arXiv:0803.4445 [hep-th]

work page Pith review arXiv 2008

[4] [1]

Bajnok, `` How Integrability Works (for AdS/CFT) ,'' http://dx.doi.org/10.5506/APhysPolB.47.2451 Acta Phys

Z. Bajnok, `` How Integrability Works (for AdS/CFT) ,'' http://dx.doi.org/10.5506/APhysPolB.47.2451 Acta Phys. Polon. B47 (2016) 2451--2477

work page doi:10.5506/aphyspolb.47.2451 2016

[5] [2]

Mussardo, Statistical field theory

G. Mussardo, Statistical field theory . Oxford Univ. Press, New York, NY, 2010

work page 2010

[6] [3]

Bajnok and L

Z. Bajnok and L. S amaj, ``Introduction to integrable many-body systems III ,'' Acta Physica Slovaca. Reviews and Tutorials 61 no. 2, (2011) 129--271

work page 2011

[7] [4]

Dorey, `` Exact S-matrices ,'' in Conformal field theories and integrable models

P. Dorey, `` Exact S-matrices ,'' in Conformal field theories and integrable models. Proceedings, Eotvos Graduate Course, Budapest, Hungary, August 13-18, 1996 , pp. 85--125. 1996. http://arxiv.org/abs/hep-th/9810026 arXiv:hep-th/9810026 [hep-th]

arXiv 1996

[8] [5]

Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory

R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory . North-Holland Publishing Company, 1982

work page 1982