Non-perturbative renormalization of the energy momentum tensor in the 2d O(3) nonlinear sigma model

Abstract

The two-dimensional O(3) nonlinear sigma model is a well known toy model for studying non-perturbative phenomena in quantum field theory. A central challenge is the renormalization of the energy-momentum tensor, which is complicated by the nonlinear realization of the $O(3)$ symmetry leading to non-trivial operator mixing patterns, and by large discretization artifacts affecting the determination of renormalization constants. We present results for the renormalization constants in the non-singlet sector, employing a modified lattice action with shifted boundary conditions and defining the renormalized coupling through the gradient flow. While we obtain a precise determination of the relative mixing constant $z_T$, the overall normalization $Z_T$ remains inaccessible due to large discretization artifacts. We discuss the origins of these difficulties and outline possible paths forward.

Figures (5)

• Figure 1: Gradient flow coupling $g^2_{\mathrm{GF}}$ as a function of bare coupling $g^2_0$ for three different actions at $N_0 = 12$ (a) and $N_0 = 18$ (b). For renormalized couplings below $g^2_{\mathrm{GF}} \approx 0.08$, the modified constraint action reaches a given value of $g^2_{\mathrm{GF}}$ at larger bare coupling.
• Figure 2: $g_0 \to 0$ approach for the action $\langle S \rangle - S_{g^2_0 \to 0}$ (a) and the EMT one-point functions $\langle T_{0\nu} \rangle - \langle T_{0\nu} \rangle_{g^2_0 \to 0}$ (b) as a function of $g^2_{\mathrm{GF}}$ at $N_0 = 12$ and $\xi = 1/2$. The action density shows qualitatively different behaviour from the EMT one-point functions, while the latter show similar deviations from the free-theory limit across all three actions.
• Figure 3: Systematic effects from flowtime discretization (a) and tuning of the bare coupling (b) for $N_0 = 64$. In (a), we show that discretization artifacts are well below $0.01\%$ for our chosen step size. In (b), we fit a second order polynomial to determine the bare coupling at $g^2_{\mathrm{GF}} = 0.06$, with errors estimated via bootstrap.
• Figure 4: Results for $z_T$ as a function of $g^2_0$ (a) and the individual EMT one-point functions entering its determination (b). The tree-level subtraction worsens results for $N_0 \leq 18$ and makes no difference for $N_0 \geq 32$. The similar deviations of $\langle T_{00}\rangle_{1/2}$ and $\langle T_{01}\rangle_{1/2}$ from their free-theory values lead to cancellations in the ratio defining $z_T$.
• Figure 5: Results for $Z_T$ as a function of $g^2_0$ (a) and the individual observables entering its determination (b). In (a), due to large statistical errors, $Z_{T,2p}$ is only shown up to $N_0 = 32$. Both methods, $Z_{T,\mathrm{log}}$ from equation \ref{['eq:ZT_log_lat']} and $Z_{T,2p}$ from equation \ref{['eq:ZT_2p_lat']}, show large deviations from the tree-level expectation while remaining mutually compatible, indicating that the dominant discretization artifacts are common to both.