The Qcd Trace Anomaly as a Vacuum Effect

(The vacuum is a medium is the message!) We use arguments taken from the electrodynamics of media to deduce the QCD trace anomaly from the expression for the vacuum energy in the presence of an external color magnetic field. Intuitive discussions of phenomena in quantum field theory frequently take the form of describing them as properties of the vacuum. The vacuum, it is then argued, is just like any other medium, so we can use the intuition developed in other fields of physics to get an understanding of quantum field theory. Perhaps the first example of this was the interpretation of unoccupied negative energy states in Dirac's relativistic electron theory as positrons. Today the obvious analogy is the behavior of electrons in semiconductors, though historically it seems that the concept of holes developed independently in quantum field theory and solid state physics [1]. Another example is the Casimir effect, which is explained in terms of the zero-point fluctuations of the quantized electromagnetic field. A third example is spontaneous symmetry breaking of the vacuum, where the intuition is supplied by the behavior of (anti-)ferromagnets and (for local symmetries) superconductors. As a final example we have the charge-screening properties of the vacuum. Already in the 30s it was realised that the QED vacuum behaves just like an ordinary dielectric medium [2]: Charges placed in the vacuum get screened by polarization effects. Around 1980 Nielsen, [3], and Hughes, [4], showed that asymptotic freedom, the antiscreening property of the QCD vacuum, could be interpreted as (color-) paramagnetism. 1 In this paper we will show that the calculation this result rests upon, the change in vacuum energy due to an external (abelian color-) magnetic field, can also be used to derive the QCD trace anomaly. The point is that the energy is all we need in order to apply some elementary arguments borrowed from the electrodynamics of media to get the stress tensor, and hence the anomaly. The vacuum is indeed just another (color-) electromagnetic medium. Our paper is organized as follows. We first (section 2) review the medium picture of screening (QED) and antiscreening (QCD). In the course of this discussion we quote the the result for the vacuum energy in QCD and introduce the renormalization group equation for charges. In section 3 we then use the expression for the vacuum energy and some " elementary physics " arguments to deduce the the stress …