Exact finite-volume diagonal matrix elements in the sine-Gordon model reformulated in TBA pseudo-energy language: toward extension to other non-diagonally scattering integrable QFTs

For integrable quantum field theories in 1+1 dimensions, exact finite-volume diagonal matrix elements (expectation values of local operators in arbitrary finite-volume eigenstates) are known for diagonally scattering theories via the LeClair–Mussardo series and, uniquely among non-diagonally scattering theories, for the sine-Gordon model via determinant formulas derived from a nonlinear integral equation (NLIE) specific to its 6-vertex lattice regularization. The source paper states as an open problem to rephrase these determinant formulas in the model-independent language of Thermodynamic Bethe Ansatz (TBA) pseudo-energies and to thereby help develop a general method for computing finite-volume diagonal matrix elements in other non-diagonally scattering integrable QFTs; the O(N) nonlinear sigma model and the principal chiral field are named here as concrete candidates for that longer-term direction. The problem is open: no pseudo-energy reformulation of the sine-Gordon formulas is known, and no exact finite-volume diagonal matrix elements are known for any relativistic non-diagonal model beyond sine-Gordon.

Background

In an integrable quantum field theory (QFT) in 1+1 dimensions, particle scattering preserves particle number and individual momenta, so the two-particle S-matrix determines the theory's interactions. In a diagonally scattering theory the S-matrix is a scalar function of the rapidity difference (the rapidity parametrizes energy and momentum by $m\cosh\theta$ and $m\sinh\theta$), and multi-particle states simply pass through each other. In a non-diagonally scattering theory the S-matrix is matrix-valued: in the sine-Gordon model (the theory of a scalar field with interaction proportional to $\cos\beta\phi$), whose elementary excitations are a soliton and an antisoliton carrying opposite topological charge plus their bound states (breathers), soliton–antisoliton scattering includes backscattering that interchanges the two internal states. Because of this, the standard technology for computing correlation functions at finite volume (equivalently, at finite temperature, by compactifying a spatial direction of length $L$) does not transfer directly. Two approaches exist. First, the LeClair–Mussardo series expresses one-point functions of local operators as an infinite integral series over the connected infinite-volume form factors (matrix elements of local operators between asymptotic multi-particle states, with 'connected' denoting the part remaining in a certain subtraction prescription adapted to coincident rapidities), weighted by filling fractions built from the Thermodynamic Bethe Ansatz; analytic continuation of the source terms to any set of excited-state rapidities gives expectation values in arbitrary finite-volume eigenstates. This series is well established for diagonal scattering. Second, for sine-Gordon one can use an integrable lattice regularization by the 6-vertex model (an exactly solvable two-dimensional vertex model whose continuum limit is sine-Gordon). This yields a nonlinear integral equation (NLIE) for a counting function that encodes the exact finite-volume spectrum, and a 'fermionic basis' of local operators in which the expectation value of any operator in any Hamiltonian eigenstate is given by determinants built from a single function $\omega$ satisfying linear integral equations coupled to the NLIE. Hegedűs (JHEP 01 (2020) 122) completed this construction for arbitrary multi-soliton excited states, checking it in the ultraviolet against Liouville three-point conformal functions (Liouville theory being the exactly solvable conformal field theory governing the sine-Gordon ultraviolet limit) and at large volume against a previously conjectured LeClair–Mussardo-type formula. The limitation, stated explicitly in that paper's conclusions, is twofold. The construction is tied to the sine-Gordon-specific NLIE counting function inherited from the 6-vertex regularization, and no analogue for other non-diagonally scattering theories is known. By contrast, the Thermodynamic Bethe Ansatz (TBA) — the system of nonlinear integral equations for pseudo-energies $\varepsilon_a(\theta)$ whose solutions encode exact finite-volume energies — is available for a wide class of non-diagonally scattering theories: for the O(N) nonlinear sigma model (a relativistic QFT of a field constrained to the unit sphere $S^{N-1}$) and the SU(N) principal chiral field (a relativistic QFT of a field valued in the group SU(N)), both asymptotically free (their effective coupling vanishes logarithmically at short distances), the finite-volume spectrum is described by nested TBA systems, 'nested' because the particles carry internal degrees of freedom that must be resolved level by level through a hierarchy of Bethe-ansatz equations. What is missing is the matrix-element side: a general method to compute finite-volume diagonal matrix elements of local operators in these theories. As a first step toward this, the source proposes to rephrase the sine-Gordon determinant formulas in terms of TBA pseudo-energies, replacing the model-specific counting function by quantities that exist for every theory in the class; this rephrasing is the problem stated below.

Problem Statement

Rephrase the exact finite-volume diagonal matrix elements of the sine-Gordon model — the determinant formulas for the expectation values $\langle\theta_1,\dots,\theta_N|\mathcal{O}|\theta_1,\dots,\theta_N\rangle_L$ of local operators in arbitrary eigenstates of the Hamiltonian on a circle of circumference $L$, built from the function $\omega$ and the NLIE counting function inherited from the 6-vertex lattice regularization — in the language of Thermodynamic Bethe Ansatz pseudo-energies, replacing the model-specific counting function by the rapidity-dependent pseudo-energies $\varepsilon_a(\theta)$ satisfying nonlinear TBA integral equations, quantities that exist for a wide class of non-diagonally scattering integrable QFTs. In the source's words, such a rephrasing "could give a deeper insight into the structure of the formulas and help in finding a generalization to other important models". The hoped-for longer-term direction of that generalization — a motivation for the rephrasing rather than a required deliverable of it — is a method to compute exact finite-volume diagonal matrix elements in other non-diagonally scattering relativistic integrable QFTs, such as the O(N) nonlinear sigma model and the principal chiral field.

References

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