Renormalization group approach to confinement

Abstract While we have several complementary models of confinement, some of which are phenomenologically appealing, we do not have the ability to calculate analytically even simple aspects of confinement, let alone have a framework to eventually prove confinement. The problem we are facing is to evolve the theory from the perturbative regime to the long distance confining regime. This is generally achieved by renormalization group transformations. With the gradient flow we now have a technique to address the problem from first principles. The primary focus is on the running coupling $$\alpha _S(\mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , from which confinement can be concluded alone. A central point is that the gluon condensate is scale invariant, which reflects its self-similar behavior across different scales. Building on that, we derive $$\alpha _S(\mu ) \simeq \Lambda _S^2/\mu ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≃</mml:mo> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , which evolves to the infrared fixed point $$1/\alpha _S = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in accordance with infrared slavery. The only important factor appears to be the presence of the gluon condensate, which is a universal feature that QCD shares with many other models. The analytical results are supported by numerical simulations.