Integrability in QCD and beyond

TL;DR

This review argues that four-dimensional Yang–Mills dynamics harbor hidden integrability in several important limits, each realized through an effective spin-chain description related to Heisenberg-type magnets. It connects the renormalization-group evolution of light-cone operators in QCD/SYM, the high-energy Regge behavior, and the Seiberg–Witten low-energy actions in ${\mathcal N}=2$ theories to integrable spin chains with groups such as $SL(2,\mathbb{R})$ and its supersymmetric extensions $SL(2|\mathcal{N})$. The work also discusses the gauge/string duality, showing how gauge-theory integrability maps to classical integrable sigma-models on curved backgrounds, and illuminates the use of Baxter equations, finite-gap methods, and Baxter Q-operators in solving these systems. While strong evidence of integrability emerges in perturbative regimes and certain subsectors, the origin and fate of integrability at higher loops and nonperturbative effects remain open questions, with the gauge/string correspondence offering a promising unifying perspective.

Abstract

Yang--Mills theories in four space-time dimensions possess a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. It turns out that the effective Yang--Mills dynamics in several important limits is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations. In this review we explain the general phenomenon of complete integrability and its realization in several different situations. As a prime example, we consider in some detail the scale dependence of composite (Wilson) operators in QCD and super-Yang--Mills (SYM) theories. High-energy (Regge) behavior of scattering amplitudes in QCD is also discussed and provides one with another realization of the same phenomenon that differs, however, from the first example in essential details. As the third example, we address the low-energy effective action in a N=2 SYM theory which, contrary to the previous two cases, corresponds to a classical integrable model. Finally, we include a short overview of recent attempts to use gauge/string duality in order to relate integrability of Yang--Mills dynamics with the hidden symmetry of a string theory on a curved background.

Figures (6)

• Figure 1: Examples of a "vertex" correction (a), "exchange" diagram (b) and self-energy insertion (c) contributing to the renormalization of three-quark operators in Feynman gauge. Path-ordered gauge factors are shown by the dashed lines. The set of all diagrams includes possible permutations.
• Figure 2: The spectrum of eigenvalues for the conserved charge, $q$ in ($a$) and for the helicity-3/2 Hamiltonian, $\mathcal{E}_{N,q}$ in ($b$).
• Figure 3: The flow of energy eigenvalues for the Hamiltonian ${\mathcal{H}}(\epsilon)$ for $N=30$ (see text). The solid and the dash-dotted curves show the parity-even and parity-odd levels, respectively. The two vertical dashed lines indicate ${\mathcal{H}}_{3/2}\equiv{\mathcal{H}}(\epsilon=0)$ and ${\mathcal{H}}_{1/2}\equiv{\mathcal{H}}(\epsilon=1)$, respectively. The horizontal dotted line shows position of the "ground state" that corresponds to $q=0$.
• Figure 4: Feynman diagrams contributing to the one-loop dilatation operator for the two chiral superfields on the light-cone.
• Figure 5: The dependence of the energy $E_3$ and the conserved charges $q_3$ on the total spin $h=1/2+i\nu_h$ for $n_h=0$. Three curves correspond to the trajectories with $(\ell_1,\ell_2)=(0,2)\,, (2,2)$ and $(4,2)$.