Lightcone Effective Hamiltonians and RG Flows

TL;DR

The paper develops a general prescription to incorporate lightcone zero-mode effects into an effective LC Hamiltonian $H_{ m eff}$ that reproduces ET results for CFTs deformed by relevant operators. It introduces a Dyson-series-based matching and a momentum-space diagnostic to identify when zero-mode contributions generate nonlocal or delta-function terms in LC time, and shows how many theories reduce to vacuum-energy shifts or bare-parameter renormalizations, while in certain large-$N$ or holographic contexts they encode bulk dynamics via a background profile $ abla ext{φ}_{ m cl}$. The method is tested across several examples (vacuum energy, Ising model, $oldsymbol{\oldsymbol{oldsymbol{\phi^4}}}$ theory, holographic models, and the $oldsymbol{O}(N)$ model), recovering known results like Burkardt's mass counterterm, clarifying discrepancies between LC and ET quantizations, and highlighting its perturbative validity and nonperturbative caveats. The results point toward a practical route for LC Hamiltonian truncation in large-$N$ QFTs and potentially for RG flows in theories with AdS duals.

Abstract

We present a prescription for an effective lightcone (LC) Hamiltonian that includes the effects of zero modes, focusing on the case of Conformal Field Theories (CFTs) deformed by relevant operators. We show how the prescription resolves a number of issues with LC quantization, including i) the apparent non-renormalization of the vacuum, ii) discrepancies in critical values of bare parameters in equal-time vs LC quantization, and iii) an inconsistency at large N in CFTs with simple AdS duals. We describe how LC quantization can drastically simplify Hamiltonian truncation methods applied to some large N CFTs, and discuss how the prescription identifies theories where these simplifications occur. We demonstrate and check our prescription in a number of examples.

Figures (12)

• Figure 1: Triangle diagram associated with the matrix element $\langle{\mathcal{O}}|V|{\mathcal{O}}'\rangle$, demonstrating the flow of momentum. Each leg of the triangle can be thought of as the momentum space two-point function of a fictitious "building block" operator. Lightcone zero modes are defined to be contributions where one of the legs has vanishing lightcone momentum.
• Figure 2: Numeric solution of the bulk profile $\phi(z)$ in the toy model. ( Black, solid): exact numeric solution; ( red, dashed): asymptotic value at $\phi = \sqrt{\frac{-m^2}{g_4}}$; ( blue, dotted): free theory behavior $\phi = \lambda z^{d-\Delta_R}$. Parameters are $\Delta_R = 1.7$, $\lambda =0.1$, $g_4 = 0.5$, $z_{\rm UV} = 10^{-4}$, $d=2$, all in units of $\ell_{\rm AdS}=1$.
• Figure 3: General structure of "plant" diagrams which lead to effective Hamiltonian contributions in large $N$ theories. The zero modes (dashed lines) created by the relevant deformation ${\mathcal{O}}_R$ must only connect to the physical states (solid lines) via a single contact interaction.
• Figure 4: Using our prescription, the Ising four-point function $\langle T\varepsilon\varepsilon T\rangle$ gives rise to a Hamiltonian matrix element involving the effective interaction $\psi \frac{1}{\partial} \psi$. This contribution arises due to the factor of $\delta(x^+)$ in the $\chi$ propagator (dashed line).
• Figure 5: General structure of plant diagrams which contribute to the effective Hamiltonian in $\phi^4$ theory. These higher-point correlation functions correspond to Feynman diagrams which contribute to the one-particle mass in equal-time quantization but vanish in lightcone quantization. By including these terms in $H_\textrm{eff}$, our prescription eliminates the naive discrepancy between the two quantization schemes.