What is a "dynamically generated scale" physically?

Question

A theory like QCD with massless quarks in four dimensions has no explicit mass parameters in its classical Lagrangian. At the quantum level however, instead a mass scale Λ is generated dynamically at the quantum level.

How can I understand physically what a 'dynamical generation of scale' is? Why do we say 'dynamical'? Is it related to the motion of the particle content?

Actually I suppose that it's hard to start with the problem to have intuition about a conformal theory as our own world is obviously non-conformal.

But is there a simple example in physics where I can see (an example where the see is put in quotations is also good. ;) ) how a scale in 'generated'?

Answer 1

Even if a theory is naively (classically) scale invariant (eg: the scalar theory with $\lambda \phi^4$ interaction therm), quantum mechanically, the 4-point scattering amplitude depends on the energy of the scattering particles (as can be shown by a one-loop computation. Tree level computations are the classical approximation). Suppose the scattering amplitude varies monotonically -- then, whatever your coupling at some particular energy, there existsanother energy scale at which the strength of the interaction crosses some predefined number you choose (say $1$ for niceness, or you could even consider $\infty$ which would tell you the "Landau-pole energy"). Thus, the theory has a special scale encoded indirectly in the value of your classically dimensionless coupling $\lambda$.

Similarly, in QCD, even though the gauge coupling is classically dimensionless, but if you account for the energy dependence of the scattering amplitudes, then there is an energy scale ($\Lambda_{\textrm{QCD}} \sim m_{\textrm{proton}}$) at which the gauge theory becomes so strongly coupled that you can never observe independent quarks -- only bound states i.e. "baryons" and "mesons". (This is also known as confinement of colour.)

The scale is said to be "dynamically generated" as it is a consequence of including the dynamical effects of quantum fluctuations. I don't have a better way to motivate the name.