Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data

Abstract

We both review and augment the lightcone conformal truncation (LCT) method. LCT is a Hamiltonian truncation method for calculating dynamical quantities in QFT in infinite volume. This document is a self-contained, pedagogical introduction and "how-to" manual for LCT. We focus on 2D QFTs which have UV descriptions as free CFTs containing scalars, fermions, and gauge fields, providing a rich starting arena for LCT applications. Along our way, we develop several new techniques and innovations that greatly enhance the efficiency and applicability of LCT. These include the development of CFT radial quantization methods for computing Hamiltonian matrix elements and a new SUSY-inspired way of avoiding state-dependent counterterms and maintaining chiral symmetry. We walk readers through the construction of their own basic LCT code, sufficient for small truncation cutoffs. We also provide a more sophisticated and comprehensive set of Mathematica packages and demonstrations that can be used to study a variety of 2D models. We guide the reader through these packages with several examples and illustrate how to obtain QFT observables, such as spectral densities and the Zamolodchikov C-function. Specific models considered are finite N c QCD, scalar φ 4 theory, and Yukawa theory.

Chiral Limit of 2d QCD Revisited with Lightcone Conformal Truncation

“…At finite-N c , we encounter matrix elements with generic particle numbers. Following the procedure of [7], we arrive at an expression (B.18) Directly evaluating (B.18) by brute force for non-integer powers of x 2 , x 3 can be computationally expensive. However, notice that the Jacobi polynomials x α (1−x) α P (2s,2s) (1− 2x) form a complete basis of polynomials on x ∈ [0, 1] with boundary condition x α (1 − x) α .…”

“…A model which remained out of reach, however, was a 2d version of real 4d QCD, where the strongly bound quarks have nonvanishing masses much lighter than the strong coupling scale (set in 2d by the gauge coupling g). As we will review below, though it is possible to accommodate either strictly massless quarks or heavy quarks [1][2][3][4][5][6][7][8], for light quarks, bound state wavefunctions develop features which can be difficult to approximate. Consequently, DLCQ as well a generic Lightcone Conformal Truncation (LCT) approach will lead to slow convergence.…”

Spinors and conformal correlators

“…It is obvious that the factorization observed in the Minkowski momentum space OPE present important advantages, even though it only applies to Wightman functions. The second direction of research is Hamiltonian truncation in infinite volume [44,45], where one of the computational bottlenecks is the evaluation of matrix elements that correspond to a particular limit of the Wightman 3-point functions [46]. Finally, the last area in which this work can have an impact is quantum field theory in de Sitter space-time.…”

Tadpoles and vacuum bubbles in light-front quantization

“…Recent calculations of the critical coupling in two-dimensional φ 4 theory [1][2][3][4][5][6][7][8][9][10][11][12][13] have shown that there is a discrepancy in the nonperturbative equivalence of equal-time and light-front quantization. Although this discrepancy can be explained with a computed shift in the renormalized mass [5,6,8,14], this explanation is a correction to the light-front calculation, rather than a direct calculation.…”