9.4 Instantons and the θ-vacuum in QCD

Instantons are mind-bending solutions in QCD that connect different vacuum states. They're like secret tunnels between universes, helping explain weird stuff like why some particles have mass when they shouldn't. It's trippy, but super important.

The θ-vacuum is like a cosmic merry-go-round of these different states. It's described by a number θ, which messes with the laws of physics. This leads to the strong CP problem - a huge mystery about why the universe isn't more messed up than it is.

Instantons in QCD

Concept and role of instantons in non-perturbative QCD

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• Instantons are classical solutions to the equations of motion in non-Abelian gauge theories (QCD) that describe tunneling transitions between topologically distinct vacuum states
• These topologically distinct vacuum states are characterized by different winding numbers or Chern-Simons numbers related to the non-trivial topology of the gauge group
• Instantons play a crucial role in understanding non-perturbative aspects of QCD by contributing to processes that cannot be described by perturbation theory (U(1)A anomaly, strong CP problem)
• The presence of instantons in QCD leads to the violation of classical symmetries (U(1)A symmetry) and the emergence of new physical phenomena (η' meson mass, θ-vacuum)

Visualization and action of instantons

• Instantons can be visualized as localized, finite-action field configurations in Euclidean spacetime that interpolate between different vacuum states
• The instanton action is proportional to $8π^2/g^2$, where $g$ is the coupling constant
• Instanton effects become more significant at strong coupling or low energies due to the inverse relationship between the action and the coupling constant
• Example: In the semi-classical approximation, the tunneling probability between vacuum states is proportional to $e^{-S_I}$, where $S_I$ is the instanton action, demonstrating the non-perturbative nature of instanton effects

Properties of Instantons

Self-duality equations and solutions

• Instantons are obtained by solving the self-duality or anti-self-duality equations for the gauge field strength tensor: $F_{\mu\nu} = \pm^*F_{\mu\nu}$, where $^*F_{\mu\nu}$ is the dual field strength tensor
• The self-duality equations ensure that the instantons are local minima of the Euclidean action and satisfy the classical equations of motion
• The BPST (Belavin-Polyakov-Schwarz-Tyupkin) instanton is the simplest and most well-known instanton solution in SU(2) gauge theory, given by $A_\mu(x) = i\sigma_{\mu\nu}(x-x_0)_\nu / [(x-x_0)^2 + \rho^2]$, where $\sigma_{\mu\nu}$ are the 't Hooft symbols, $x_0$ is the instanton center, and $\rho$ is the instanton size
• Instantons in SU(N) gauge theories can be constructed using the 't Hooft ansatz, which involves embedding the SU(2) BPST instanton into the larger gauge group

Instanton action, number, and moduli space

• The instanton action is quantized in units of $8π^2/g^2$, and the instanton number (topological charge) is an integer that counts the number of times the gauge field winds around the group manifold
• Example: In SU(2) gauge theory, the instanton number is given by $Q = \frac{1}{32\pi^2} \int d^4x \, F^a_{\mu\nu} {^*}F^{a\mu\nu}$, where $a$ is the color index
• Instanton solutions have a moduli space that describes the collective coordinates of the instanton (position, size, orientation in group space)
• The integration over these collective coordinates is crucial for calculating instanton contributions to physical observables
• Example: The one-instanton contribution to the partition function in SU(N) gauge theory is proportional to $\int d^4x_0 \, d\rho \, \rho^{-5} \, (8\pi^2/g^2)^{2N} \, e^{-8\pi^2/g^2}$, where $x_0$ and $\rho$ are the instanton position and size, respectively

The θ-Vacuum in QCD

Structure and parametrization of the θ-vacuum

• The θ-vacuum is a superposition of topologically distinct vacuum states in QCD, characterized by different winding numbers or Chern-Simons numbers
• The θ-vacuum is parametrized by an angular parameter $\theta$, which appears in the QCD Lagrangian as a term proportional to $\theta$ times the topological charge density: $\mathcal{L}_\theta = (\theta g^2/32\pi^2) F_{\mu\nu}^*F^{\mu\nu}$
• The presence of the $\theta$-term in the QCD Lagrangian has profound implications for the structure of the vacuum and the symmetries of the theory, particularly breaking the CP symmetry and leading to the strong CP problem

Topological structure and instanton transitions

• The θ-vacuum has a non-trivial topological structure, with a periodic potential that exhibits multiple degenerate minima separated by potential barriers
• Instantons mediate transitions between different θ-vacua, and their contributions to the path integral lead to physical effects (resolution of the U(1)A anomaly, appearance of a non-zero η' meson mass)
• Example: The transition probability between adjacent θ-vacua is proportional to $e^{-S_I}$, where $S_I$ is the instanton action, demonstrating the role of instantons in connecting different topological sectors
• The value of the $\theta$ parameter is experimentally constrained to be very small ($\theta < 10^{-10}$) by measurements of the neutron electric dipole moment, which is a signature of CP violation, constituting the strong CP problem

Instantons vs Strong CP Problem

Witten-Veneziano mechanism and the U(1)A problem

• Instantons provide a non-perturbative mechanism for understanding the structure and properties of the θ-vacuum in QCD
• The presence of instantons leads to the breakdown of the U(1)A symmetry and the appearance of a non-zero mass for the η' meson, which would otherwise be a Goldstone boson in the chiral limit
• The resolution of the U(1)A problem through instantons is known as the Witten-Veneziano mechanism, which relates the η' mass to the topological susceptibility of the QCD vacuum
• Example: The Witten-Veneziano formula for the η' mass is given by $m_{\eta'}^2 = \frac{4N_f}{f_\pi^2} \chi_t$, where $N_f$ is the number of flavors, $f_\pi$ is the pion decay constant, and $\chi_t$ is the topological susceptibility

Strong CP problem and proposed solutions

• Instantons play a crucial role in the strong CP problem by inducing a non-zero value for the $\theta$ parameter in the QCD Lagrangian, which breaks the CP symmetry
• The experimental upper bound on the neutron electric dipole moment constrains the value of $\theta$ to be extremely small, which is unnatural from a theoretical perspective and constitutes the strong CP problem
• Proposed solutions to the strong CP problem (Peccei-Quinn mechanism, axion hypothesis) involve modifying the QCD Lagrangian to dynamically relax the $\theta$ parameter to zero
• Instanton-induced effects (axion potential, topological susceptibility) are crucial for understanding the dynamics of the proposed solutions to the strong CP problem
• Example: In the Peccei-Quinn mechanism, the axion field $a(x)$ couples to the topological charge density as $\mathcal{L}_a = -(\xi/32\pi^2) a(x) F_{\mu\nu}^*F^{\mu\nu}$, where $\xi$ is a constant, allowing the axion to dynamically cancel the $\theta$-term and solve the strong CP problem