[Paper Review] Diagonal form factors from non-diagonal ones

Zoltán Bajnok, Chao Wu|arXiv (Cornell University)|Jul 25, 2017

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper proves the asymptotic large-volume form factor formula for diagonal matrix elements in integrable quantum field theories by taking the infinite rapidity limit of a non-diagonal form factor. The key result establishes the Bethe-Yang form of diagonal form factors, completing the proof of the LeClair-Mussardo formula for finite-volume one-point functions and confirming conjectures in AdS/CFT for Heavy-Heavy-Light three-point functions.

ABSTRACT

We prove the asymptotic large volume expression of diagonal form factors in integrable models by evaluating carefully the diagonal limit of a non-diagonal form factor in which we send the rapidity of the extra particle to infinity.

Motivation & Objective

• To prove the conjectured Bethe-Yang (BY) form of diagonal finite-volume form factors in integrable quantum field theories.
• To establish the asymptotic large-volume behavior of diagonal form factors by taking the diagonal limit of a non-diagonal form factor.
• To complete the proof of the LeClair-Mussardo formula for finite-volume one-point functions using the diagonal limit method.
• To provide a rigorous derivation of diagonal form factors that supports their role in the AdS/CFT correspondence, particularly for Heavy-Heavy-Light (HHL) three-point functions.

Proposed method

• The authors analyze the diagonal limit of a non-diagonal form factor by sending one particle's rapidity to infinity, treating the limit carefully to avoid divergences.
• They use the Bethe-Yang quantization condition and the infinite-volume form factor axioms, including the crossing formula, kinematical singularity, and permutation symmetry.
• The method involves expressing the finite-volume form factor as a sum over intermediate states and using the asymptotic behavior of the S-matrix and form factors.
• The diagonal limit is evaluated by isolating the singular terms in the rapidity expansion, particularly focusing on the behavior as the extra particle's rapidity goes to infinity.
• The authors apply the Bethe-ansatz equations to express the coefficients of the polynomial in the volume, showing that the finite-volume form factor becomes a polynomial in the volume and the logarithmic terms.
• Induction is used to generalize the result from the two-particle case to the general n-particle diagonal form factor, verifying consistency with the symmetric form factor at L=0 and the derivative with respect to energy levels.

Experimental results

Research questions

• RQ1 How can the diagonal form factor in the large-volume limit be derived from the known non-diagonal form factor in integrable quantum field theories?
• RQ2 What is the precise asymptotic behavior of diagonal form factors in the Bethe-Yang regime, and how does it relate to the LeClair-Mussardo formula?
• RQ3 Can the diagonal limit of a non-diagonal form factor be taken consistently to recover the conjectured form of diagonal finite-volume form factors?
• RQ4 How does the structure of the diagonal form factor, involving sums over partitions and densities of states, emerge from the non-diagonal limit?
• RQ5 What is the role of the Bethe-ansatz equations in determining the coefficients of the polynomial in the volume for the diagonal form factor?

Key findings

• The diagonal form factor in the large-volume limit is proven to be a polynomial in the volume L, linear in each energy level E_k L, and symmetric in the rapidities of the particles.
• The leading finite-volume correction to the diagonal form factor is fully determined by the Bethe-ansatz equations and the S-matrix, with no additional contributions from the operator structure.
• The diagonal limit of the non-diagonal form factor yields the correct form factor structure, confirming the conjectured Bethe-Yang form for diagonal matrix elements.
• The result confirms the LeClair-Mussardo formula for finite-volume one-point functions, establishing its exactness in the large-volume limit.
• The derivation provides a rigorous foundation for the use of diagonal form factors in the AdS/CFT correspondence, particularly for Heavy-Heavy-Light three-point functions at all coupling strengths.