Integrability and renormalizability for the fully anisotropic ${\rm SU}(2)$ principal chiral field and its deformations

Abstract

For the class of $1+1$ dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic ${\rm SU}(2)$ Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic ${\rm SU}(2)$ PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

Figures (4)

• Figure 1: The integration contour for the Wilson loop can be moved freely along the cylinder.
• Figure 2: The orientation of the rigid body is uniquely specified by the $3D$ special orthogonal matrix that relates the moving frame $({\bm e}_1,\bm{e}_2,\bm{e}_3)$ to the fixed frame $({\bm E}_1,\bm{E}_2,\bm{E}_3)$. The axes of the moving frame are chosen to coincide with the principal axes of inertia.
• Figure 3: The evolution of $I_1,\, I_2$ and $I_3$ as functions of $\hbar\tau$. The initial conditions at $\tau=0$ were chosen to be $I_1(0)=0.06,\ I_2(0)=0.42,\ I_3(0)=0.10$. The flow remains real and non-singular in the interval $\tau\in (\tau_{\text{min}}, \tau_{\text{max}})$ with $\hbar\tau_{\text{min}}=-0.006$ and $\hbar\tau_{\text{max}}=0.154$ which is marked by the dashed lines.
• Figure 4: The relation between the various models. The Poisson-Lie deformation is represented by an arrow.