Two-particle Form Factors Research Articles

A study of form factors in relativistic mixed-flux AdS3

We study the two-particle form-factors for the relativistic limit of the integrable S-matrix of the mixed-flux AdS3 × S3 × T4 string theory. The S-matrix theory was formally constructed in two distinct ways by two different teams. We focus on the massive theory built up by Frolov, Polvara and Sfondrini, and derive expressions for the minimal solutions to the axioms, in both integral and manifestly meromorphic form, and then proceed to apply the off-shell Bethe ansatz method of Babujian et al. We obtain the integral formulas for the two-particle complete form-factors and check the axioms at this particle number.

Quantum Energy Inequalities in Integrable Models with Several Particle Species and Bound States

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough–Dodd, the Federbush, and the O(n)-nonlinear sigma models.

Entanglement entropy from form factors in textrm{T}overline{textrm{T}} -deformed integrable quantum field theories

In two recent papers [1, 2] we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised textrm{T}overline{textrm{T}} -deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to textrm{T}overline{textrm{T}} -perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for textrm{T}overline{textrm{T}} -perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of the recent publication [3] where the same problem has been addressed.

Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data

We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at s > 0 in the 2d ϕ4 theory. We use the form factor of the stress tensor at s ≤ 0 and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. We independently construct the “pure” S-matrix bootstrap bounds (bootstrap without including matrix elements of local operators), and find that the sinh-Gordon model and its analytic continuation the “staircase model” saturate these bounds. Surprisingly, the ϕ4 two-particle scattering amplitude also very nearly saturates these bounds, and moreover is extremely close to that of the sinh-Gordon/staircase model.

Two-point functions and bootstrap applications in quantum field theories

We study two-point functions of local operators and their spectral representation in UV complete quantum field theories in generic dimensions focusing on conserved currents and the stress-tensor. We establish the connection with the central charges of the UV and IR fixed points. We re-derive “c-theorems” in 2d and show the absence of their direct analogs in higher dimensions. We conclude by focusing on quantum field theories with a mass gap. We study the stress tensor two-particle form factor, derive implications of unitarity and define concrete bootstrap problems in generic dimensions.

An Operator Product Expansion for Form Factors II. Born level

Form factors in planar mathcal{N} = 4 Super-Yang-Mills theory admit a type of non-perturbative operator product expansion (OPE), as we have recently shown in [1]. This expansion is based on a decomposition of the dual periodic Wilson loop into elementary building blocks: the known pentagon transitions and a new object that we call form factor transition, which encodes the information about the local operator. In this paper, we compute the two-particle form factor transitions for the chiral part of the stress-tensor supermultiplet at Born level; they yield the leading contribution to the OPE. To achieve this, we explicitly construct the Gubser-Klebanov-Polyakov two-particle singlet states. The resulting transitions are then used to test the OPE against known perturbative data and to make higher-loop predictions.

Consistency checks for two-body finite-volume matrix elements. II. Perturbative systems

Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. Working in a finite cubic volume with periodicity $L$, we derive a $1/L$ expansion of the matrix element through $\mathcal O(1/L^5)$ and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the $1/L$ expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, $1/L^3$, in contradiction to earlier work, which found a $1/L^2$ contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.

Form factors of the spin-1 analogue of the eight-vertex model

The twenty-one-vertex model, the spin $1$ analogue of the eight-vertex model is considered on the basis of free field representations of vertex operators in the $2\times 2$-fold fusion SOS model and vertex-face transformation. The tail operators, which translate corner transfer matrices of the twenty-one-vertex model into those of the fusion SOS model, are constructed by using free bosons and fermions for both diagonal and off-diagonal matrix elements with respect to the ground state sectors. Form factors of any local operators are therefore obtained in terms of multiple integral formulae, in principle. As the simplest example, the two-particle form factor of the spin operator is calculated explicitly.

Comments on world-sheet form factors in AdS/CFT

We study form factors in the light-cone gauge world-sheet theory for strings in AdS5 ×S5. We perturbatively calculate the two-particle form factor in a closed sector to one-loop in the near-plane-wave limit and to two-loops in the Maldacena–Swanson limit. We also perturbatively solve the functional equation which follows from the form factor axioms for the world-sheet theory and show that the ‘minimal’ solution correctly reproduces the discontinuities of the perturbative calculations. Finally we propose a prescription, valid for polynomial orders of the inverse world-sheet length, for extracting the finite-volume world-sheet matrix element from the form factors and show that the two-excitation matrix element matches with the thermodynamic limit of the spin–chain description of certain tree-level SYM structure constants.

Form factors in the Bullough-Dodd-related models: The Ising model in a magnetic field

We consider a certain modification of the free-field representation of the form factors in the Bullough-Dodd model. The two-particle minimal form factors are eliminated from the construction. We consequently obtain a convenient representation for the multiparticle form factors, establish recurrence relations between them, and study their properties. We use the proposed construction to obtain the free-field representation of form factors for the lightest particles in the Φ 1,2 -perturbed minimal models. As an important example, we consider the Ising model in a magnetic field. We verify that the results obtained in the framework of the proposed free-field representation agree with the corresponding results obtained by solving the bootstrap equations.

Single-ion anisotropy in Haldane chains and the form factor of theO(3)nonlinear sigma model

We consider spin-1 Haldane chains with single-ion anisotropy, which exists in known Haldane chain materials. We develop a perturbation theory in terms of anisotropy, where magnon-magnon interaction is important even in the low temperature limit. The exact two-particle form factor in the O(3) nonlinear sigma model leads to quantitative predictions on several dynamical properties including dynamical structure factor and electron spin resonance frequency shift. These agree very well with numerical results, and with experimental data on the Haldane chain material Ni(C$_5$H$_{14}$N$_2$)$_2$N$_3$(PF$_6$).

Ising Correlations and Elliptic Determinants

Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors - matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.

Free field construction for the eight-vertex model: representation for form factors

The free field realization of the eight-vertex model is extended to form factors. It is achieved by constructing off-diagonal with respect to the ground state sectors matrix elements of the Λ operator which establishes a relation between corner transfer matrices of the eight-vertex model and of the SOS model. As an example, the two-particle form factor of the σ z operator is evaluated explicitly.

Exact zero-temperature correlation functions for two-leg Hubbard ladders and carbon nanotubes

Motivated by recent work of Lin, Balents, and Fisher [Phys. Rev. B 58, 1794 (1998)], we compute correlation functions at zero temperature for weakly coupled two-leg Hubbard ladders and $(N,N)$ armchair carbon nanotubes. In this paper it was argued that such systems renormalize towards the SO(8) Gross-Neveu model, an integrable theory. We exploit this integrability to perform the computation at the SO(8) invariant point. Any terms breaking the SO(8) symmetry can be treated systematically in perturbation theory, leading to a model with the same qualitative features as the integrable theory. Using said correlators, we determine the optical conductivity, the single-particle spectral function, and the $I\ensuremath{-}V$ curve for tunneling into the system from an external metallic lead. The frequency, \ensuremath{\omega}, dependent optical conductivity is determined exactly for $\ensuremath{\omega}<3m$ $(m$ being the fermion particle mass in the SO(8) Gross-Neveu model). It is characterized by a sharp ``exciton'' peak at $\ensuremath{\omega}=\sqrt{3}m,$ followed by the onset of the particle-hole continuum beginning at $\ensuremath{\omega}=2m.$ Interactions modify this onset to $\ensuremath{\sigma}(\ensuremath{\omega}+2m)\ensuremath{\sim}{\ensuremath{\omega}}^{1/2}$ and not the ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ one would expect from the van Hove singularity in the density of states. Similarly, we obtain the exact single-particle spectral function for energies less than $3m.$ The latter possesses a $\ensuremath{\delta}$ function peak arising from single-particle excitations, together with a two-particle continuum for $\ensuremath{\omega}>~2m.$ The final quantity we compute is the tunneling $I\ensuremath{-}V$ curve to lowest nonvanishing order in the tunneling matrix elements. For this quantity, we present exact results for voltages, $V<(1+\sqrt{3})m.$ The resulting differential conductance is marked by a finite jump at $\ensuremath{\omega}=2m,$ the energy of the onset of tunneling into the continuum of two-particle states. Through integrability, we are able to characterize this jump exactly. All calculations are done through form-factor expansions of correlation functions. These give exact closed form expressions for spectral functions because the SO(8) Gross-Neveu model is massive: each term in the expansion has an energy threshold below which it does not contribute. Thus, we obtain exact results below certain thresholds by computing a finite number of terms in this series. Previous to this paper, the only computed form factor of SO(8) Gross-Neveu was the two-particle form factor of an SO(8) current with two fundamental fermions. In this paper we compute the set of all one- and two-particle form factors for all relevant fields, the currents as well as the kinks and fermions.

The field theory of the q→4 + Potts model

The q-state Potts model in two dimensions exhibits a first-order transition for q>4. As q→4 + the correlation length at this transition diverges. We argue that this limit defines a massive integrable quantum field theory whose lowest excitations are kinks connecting 4+1 degenerate ground states. We construct the S-matrix of this theory and the two-particle form factors, and hence estimate a number of universal amplitude ratios. These are in very good agreement with the results of extrapolated series in q −1/2 as well as Monte Carlo results for q=5.

Wave function renormalization constants and one-particle form factors in Dl(1) Toda field theories

We apply the method of angular quantization to the calculation of the wave function renormalization constants in D 1 (1) affine Toda quantum field theories. A general formula for the wave function renormalization constants in ADE Toda field theories is proposed. We also calculate all one-particle form factors and some of the two-particle form factors of an exponential field.

Off-critical correlations in the Ashkin-Teller model

We use the exact scattering description of the scaling Ashkin-Teller model in two dimensions to compute the two-particle form factors of the relevant operators. These provide an approximation for the correlation functions whose accuracy is tested against exact sum rules.

Exact two-particle matrix elements in S-matrix preserving deformation of integrable QFTs

In a recent paper it was shown that the response of an integrable QFT under variation of the Unruh temperature can be computed from a S-matrix preserving deformation of the form factor approach. We give explicit expressions for the deformed two-particle form factors for various integrable models: The Sine-Gordon and SU(2) Thirring model, several perturbed minimal CFTs and the real coupling affine Toda series. A uniform pattern is found to emerge when both the S-matrix and the deformed form factors are expressed in terms Barnes' multi-periodic functions.

Correlation functions in the two-dimensional Ising model in a magnetic field at T = Tc

The one- and two-particle form factors of the energy operator in the two-dimensional Ising model in a magnetic field at T = T c are exactly computed within the form factor bootstrap approach. Together with the matrix elements of the magnetisation operator already computed by G. Delfino and G. Mussardo [Nucl. Phys. B 455 (1995) 724], they are used to write down the large distance expansion for the correlators of the two relevant fields of the model.

Functional equations of form factors for diagonal scattering theories

The form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two-particle form factors. For A 2 (1) affine Toda field theory, which is the simplest non-self-conjugate theory, form factors are derived up to four-body ones and identification of the operator is done. Generalizing this identification to the AN N (1) affine Toda cases, we find the two-particle form factors. We also determine the additional pole structure of form factors which comes from the double pole of the S-matrices of the A N (1) theory. For A N theories, the existence of the conserved Z N+1 charge leads to the division of the set of form factors into N + 1 decoupled sectors.