Solving 2D QCD with an adjoint fermion analytically

Abstract

We present an analytic approach to solving 1+1 dimensional QCD with an adjoint Majorana fermion. In the UV this theory is described by a trivial CFT containing free fermions. The quasi-primary operators of this CFT lead to a discrete basis of states which is useful for diagonalizing the Hamiltonian of the full strongly interacting theory. Working at large-N , we find that the decoupling of high scaling-dimension quasi-primary operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. This suggests a scheme, whereby, truncating the basis to operators of dimension below ∆ max , one can calculate the low-energy spectrum, parametrically to an accuracy of e −∆max (although the precise accuracy depends on the state). Choosing ∆ max = 9.5 we find very good agreement with the known spectrum obtained earlier by numerical DLCQ methods. Specifically, below the first three-particle threshold, we are able to identify all six single-particle bound-states, as well as several two-particle thresholds.

High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation

“…This light front quantization [9] is also used in numerical solutions of strongly coupled QFTs via a version of HT; some recent work is [10][11][12][13][14][15]. The structure of the unperturbed Hilbert space is different from the equal time case, which leads to important differences in the numerical procedure.…”

“…In figure 4 we test it for the first three energy levels above the vacuum, which should correspond to the operators with dimensions ∆ σ = 1/8, ∆ ǫ = 1, ∆ ∂ 2 σ = 2 + 1/8. The error comes from extrapolating to 10 The central value corresponds to the smallest χ 2 (gc) = N i=1 (yi − f (xi)) 2 /erri 2 . The uncertainty interval was conservatively determined from the condition χ 2 (gc) 3 χ 2 (2.76).…”

A conformal truncation framework for infinite-volume dynamics

“…We are specifically interested in those two-particle states that contribute to the spectral density of the operator φ 2 , which is even under the parity transformation p ⊥ → −p ⊥ . We can therefore restrict our basis to the symmetric, parity-even sector by only including the all minus states F (2) − . Just like the two-particle basis, our set of n-particle states must be symmetric under the exchange of any two momenta.…”

“…Starting with the two-particle case, we see that this mass term leads to a matrix element correction of the form We can then use our new Dirichlet basis functions to rewrite this correction as the integral δM˜ k,˜ k = m 2 dµ 2 µ g (2) k (µ)g (2) k (µ) dp − p − (P − − p − ) F (2) (p) F (2) (p) P − p − + P − P − − p − .…”

“…It is therefore a worthwhile goal to search for non-perturbative methods which may also access dynamical observables. Hamiltonian truncation has recently gained momentum as a means of studying realtime dynamics [1][2][3][4][5][6][7][8][9][10][11][12][13]. The basic idea is to first discretize the QFT in some manner, yielding a Hilbert space consisting of an infinite tower of discrete basis states.…”