Resolve the scales in a weakly coupled plasma

For a relativistic equilibrium plasma at weak gauge coupling g, typical thermal particles have momenta of order T. Electric collective momenta are of order gT. In a non-Abelian plasma, a further magnetostatic scale of order $g^2T$ becomes nonperturbative. Numerical constants and logarithms refine these parametric estimates, so naming the scales does not supply an exact magnetic mass. The hierarchy is useful only when coupling and other masses or chemical potentials leave those regions sufficiently separated.

A hard thermal loop has hard internal momentum and soft external energy or momentum. Its induced response includes denominators associated with propagation of charged particles along their velocities. The resulting soft effective description is generally nonlocal in space and time. Gauge invariance links its self-energy and vertices through identities, so resumming one propagator while neglecting equally important vertices can be inconsistent. A scalar tadpole resummation is therefore an instructive starting analogy rather than a replacement for the gauge-theory construction.

A weak, static electric source polarizes the plasma. At leading order, the static longitudinal response has a denominator $k^2+m_D^2$, whose three-dimensional Fourier transform gives an exponentially screened Coulomb potential. This definition sets frequency to zero before studying long spatial scales. A spatially uniform oscillating perturbation instead probes a dynamical limit, including plasma oscillations. Frequency and momentum limits cannot be interchanged merely because both are called soft; the measured response and its tensor component must be held fixed.

For high-temperature QCD with massless quark flavors and zero chemical potentials, the leading Debye coefficient includes both gluons and quarks. Its particle-content dependence differs from the simple QED formula. Static magnetic fields are not screened by this same leading electric mass. In non-Abelian theory, sufficiently long magnetic scales require additional nonperturbative physics. A Yukawa fit to an electric response does not establish a complete solution of the magnetic sector or a mass for all physical polarizations.

The dynamical hard-loop response can absorb energy from a spacelike disturbance through Landau damping, involving hard particles whose velocity matches its phase variation. This collisionless mechanism does not require assigning a constant decay width to every mode. A timelike collective pole, a spacelike continuum, and a static screening length describe distinct structures of the same medium response. The word damping should therefore be accompanied by the excitation and frequency-momentum regime, rather than used as a single universal thermal rate.

At still longer times, actual collisions and sometimes repeated collinear processes control transport and equilibration. A collisionless hard-loop effective action alone does not calculate a finite viscosity or every ultrasoft relaxation rate. Matching to kinetic and long-wavelength theories requires their additional power counting and conservation constraints. A quantitatively useful approximation states the external scales, the temperature and particle content, the coupling regime, and the order retained. Extending a leading screening formula to a strongly coupled plasma without evidence would be an uncontrolled extrapolation.