Towards Quantum Simulations of Sphaleron Dynamics at Colliders

TL;DR

The paper tackles the nonperturbative, real-time dynamics of sphalerons in high-energy collisions by leveraging a tractable 1+1D $O(3)$ nonlinear sigma model as a toy model. It develops a lattice formulation to identify and stabilize the sphaleron configuration, and demonstrates convergence of the lattice sphaleron to the continuum solution as $a o0$ and $L$ grows. A quantum-simulation framework is then proposed, with Hamiltonian encoding, qubit mappings, and trotterized time evolution, illustrating the substantial circuit-depth challenges and outlining strategies to initialize and extract dynamical observables such as decay processes. The work lays the groundwork for quantum nonperturbative studies of topological field configurations and their interactions with particles, potentially enabling first-principles insights into baryon-number–violating processes in the electroweak theory. Future directions include computing production rates, tracking Chern-Simons number evolution, and incorporating fermions.

Abstract

Sphaleron dynamics in the Standard Model at high-energy particle collisions remains experimentally unobserved, with theoretical predictions hindered by its nonperturbative real-time nature. In this work, we investigate a quantum simulation approach to this challenge. Taking the $1+1$D $O(3)$ model as a protocol towards studying dynamics of sphaleron in the electroweak theory, we identify the sphaleron configuration and establish lattice parameters that reproduce continuum sphaleron energies with controlled precision. We then develop quantum algorithms to simulate sphaleron evolutions where quantum effects can be included. This work lays the ground to establish quantum simulations for studying the interaction between classical topological objects and particles in the quantum field theory that are usually inaccessible to classical methods and computations.

Figures (5)

• Figure 1: ${\mathbf n}(x)$ configurations in the $x-z$ plane at different $x$ for the sphaleron solution.
• Figure 2: Same as FIG. Fig. \ref{['fig:true_sphaleron']} but for the solution in Eq.(\ref{['eq:wrongsphaleron']}).
• Figure 3: Relative errors of the sphaleron energy on the lattice to its continuum value $E_{\rm sph}$, with $\delta E\equiv E_{\mathrm{sph}}(L,a)-E_{\mathrm{sph}}$ for different lattice volume $L$ and lattice spacing $a$ with $\omega=1/2$.
• Figure 4: The sphaleron configuration on the lattice across different number of lattice sites. The parameters are chosen to be $L=12$, $\omega=1/2$, and $g=1$. The thin black line is the sphaleron solution in the continuum.
• Figure 5: The energy density of the sphaleron configuration on the lattice across different number of lattice sites $2N+1$. The parameters are chosen to be $L=12$, $\omega=1/2$, and $g=1$. The thin black line is the sphaleron solution in the continuous limit. To get a symmetric distribution, we modify the kinetic energy part of the $k$th site in Eq. (\ref{['eq:latt-energy']}) to be $[1-\frac{1}{2}\cos (\xi_k - \xi_{k + 1})-\frac{1}{2}\cos (\xi_{k-1} - \xi_{k })]$.