Electromagnetic Fields in the QCD Vacuum


Professor of Physics, University of Arizona

Among the most far-reaching developments of the recent 25 years of research into the consequences of fundamental interactions is the recognition that the true physical vacuum is a state of considerable complex and physically significant dynamical structure. While the vacuum is empty, i.e. devoid of matter, its quantum wave function can be highly non-trivial, deviating considerably from the non-interacting Fock space wave function to which the perturbative expansion of interactions in quantum field theory refers. Quarks play an active role in shaping the vacuum structure. Being dual carriers of both `color' and `electric' charges they also respond to externally applied electromagnetic fields. Thus, in principle, the presence of the vacuum structure related to strong, Quantum-Chromodynamic (QCD) interactions influences higher order Quantum-Electrodynamic (QED) processes such as is the photon-photon scattering, which is of course completely impossible in classical electrodynamics. I will describe here the current status of the understanding of the vacuum structure as related to strong interactions, and take a fresh look at the possibility of an interference between QCD and QED properties of the vacuum.


Experimental exploration of the vacuum is not a simple task, since in laboratory experiments it is difficult, if not indeed impossible, to perform a comparative study with a vacuum structure free manifold. The search for vacuum melting at high temperature and density of matter as arising in high energy nuclear collisions is to-date the sole widely accepted method to pursue this objective. The vacuum QCD structure is due to attractive glue-glue interaction inherent in the non-Abelian nature of color charges, the naive, i.e. non-interacting product wave function of the vacuum state is known to be unstable and it is generally believed that the QCD-originating structures are the source of the confinement effect which restricts quarks to colorless bound states. Many features of the structured vacuum have been studied in the past 20 years with a variety of methods, but one aspect, the appearance of a glue `condensate' field, i.e vacuum expectation value (VEV) of the gluon field-correlator, in the vacuum state is of particular relevance. Its value obtained from sum-rule methods is today not much different from values first noted nearly 20 years ago.

What is the physical meaning of the vacuum condensate field? Clearly, the vacuum must be field free, so that the appearance of a field correlator has no classical analog, it expresses a Bogoliubov-type rotation away from the trivial Fock space state, induced by the interactions. The effect is often compared to ferro-magnetism since one can prove that one of the QCD instabilities is the magnetic gluon spin-spin interaction. On the other hand, the confinement effect of color charged quarks is best understood invoking anomalous dielectric property. Both these classical analogs are probably applicable. There are many different and equivalent ways to model and understand the glue condensate. Our next interest is in quark fluctuations (condensates as well).

When the quark masses of the two light quarks are on some scale indistinguishable, this implies that the Hamiltonian is symmetric under rotations which mix `u' with `d' quarks; this is an expression of the isospin-SU(2) symmetry of strong interactions. In case that the quark masses would be so small to be irrelevant, this implies that the Hamiltonian would be even more symmetric, specifically it would be invariant under two chiral symmetries denoted by SU(2)_LxSU(2)_R. Here L stands for left, and R for right-handed `polarization'. While m_{q=u,d}=5--15 MeV = 0 to a good approximation on hadronic scale of 1 GeV, there is no sign of the corresponding symmetry, which would be represented in the hadronic spectrum by doublets of hadronic parity states. For example, we find only one isospin doublet of nucleons, not two. Nambu pointed to this symmetry breaking in which the ground state breaks the intrinsic (almost) chiral symmetry of the Hamiltonian. On the other hand, the intrinsic SU(2)xSU(2) symmetry of the Hamiltonian can be deduced from the [Adler-Weisberger] sum rules which relate weak and strong sectors of the Hamiltonian.

By virtue of the Goldstone theorem, in the event that the quark masses vanish exactly, there would be an exactly massless `Goldstone' boson with quantum numbers of the broken symmetry, thus spin zero, negative parity and isospin 1. It is believed that since the chiral symmetry of the Hamiltonian was not really exact, the low mass pion state expresses the massless Goldstone meson of strong interactions. In a way one can then see the parity doublets of all strongly interacting particles as being substituted for by a `direct product' of the Goldstone boson (pion) with the elementary hadron states. This in turn means that many features of the vacuum structure should strongly depend on the small and seemingly irrelevant quark masses. A common illustration of this quantum phenomenon is the proof that when quark masses vanish, the pion mass also vanishes.

We thus see that the Nambu-Goldstone mechanisms assure that the quark vacuum must have a profoundly non-trivial structure which remembers even the values of small quark masses. The Nambu-Jona-Lasinio model of strong interactions incorporates in considerable detail the relevant (spontaneous) symmetry breaking features and has become the most frequently studied model of strong interactions. It also offers an opportunity to explore the interference between electromagnetic and strong forces, which has been considered with an eye for the possible chiral symmetry restoration in fields of extreme strength. However, what interests us here is a small distortion of the vacuum in consequence of a relatively weak Maxwell field being applied, which we shall turn to now. It remains to be seen if this question, also subtly dependent on the exact wave function of the vacuum state, can be studied within the Nambu-Jona-Lasinio approach, or if we need to address these issues in full QCD.

Vacuum polarization, photon-photon scattering, and pair production

The quantum vacuum state will respond in a subtle way to less dramatic perturbations. The study of this subtle response requires high precision experiments. The well known example is the vacuum polarization (VP) effect that on distance scale of the electron-Compton-wavelength alters the 1/r-nature of Coulomb's law by a 0.1%-relative-strength correction. The VP effect is based in the response of the vacuum to highly inhomogeneous field-strength, and is in general interpreted as a dielectric particle-hole (electron-positron) photon polarizibility. Because of its range, the VP-potential can only be detected in the vicinity of the atomic nucleus. VP-effect does not violate the superposition of electromagnetic fields, thus there is no possibility here for an effective new interaction, an `anomaly' such as photon-photon scattering, or photon-electromagnetic field interaction.

Nonlinear QED vacuum effects were in depth explored in the limit that gradients of the electromagnetic fields are insignificant on the scale of the electron Compton wave length. The observation of this so called Euler-Heisenberg (EH) effective action has provided a major challenge to the experimental community seeking to verify this interesting prediction of quantum-electrodynamics (QED). Considerable effort is being mounted to obtain a direct measurement of the birefringence effect arising from anisotropic index of refraction of the vacuum exposed to strong external magnetic field, and traversed by a laser beam. As the smallness of pure QED effects testifies, the symmetry principle of gauge invariance related to charge conservation, and the process of charge renormalization combine to `protect' the electromagnetic interactions in strength and shape, these effects are not present in the nonlinear effective interaction, an anomaly, which allows photon-photon scattering.

The smallness of the usual vacuum polarization effects in QED, and thus of the still smaller nonlinear EH effect arises because of gauge invariance related to charge conservation, and the process of charge renormalization. These effects combine to `protect' the electromagnetic interactions in strength and shape. However, this does not explain why other interactions do not contribute to the nonlinear anomalies of the Maxwell field. This is the reason to expect that macroscopic Maxwell-EM-fields could provide a better environment to look for QCD related QED-vacuum effects. Namely, the infrared gluonic instabilities impact the dual-charged quarks and should thus influence via their Maxwell charges the QED vacuum sector. Were this indeed strong enough, this would not only facilitate the measurement of the photon-photon scattering, but more importantly, such an effect would present an opportunity to explore the elusive structure of the QCD-vacuum.

The nonlinear effective action of Heisenberg and Euler has an analytic structure that comprises an imaginary component akin to the situation one finds for unstable decaying states when interactions are turned on. Schwinger in 1951 identified this feature with the pair production instability of the vacuum, a process in principle possible when potentials are present that can rise more than 2mc^2, the gap energy of the vacuum state, and this is of course the case in the presence of even arbitrarily weak, but constant electrical fields of infinite range.

Open questions

The glue condensate can be thought to generate a highly variable background field for the quarks, which respond to it in the way that we see when studying quark condensation effects in the vacuum. Will, in a comparison, a weak Maxwell-magnetic field have any measurable impact on this situation? It is hard to tell, there are different arguments and approximations that lead to quite contradictory answers. The difficulty is that we really do not have the wavefuntion of the quark condensate, though many of the matrix elements are understood. Without the wave function it is difficult to argue what a symmetry breaking small perturbation can do, since the properties of the vacuum with respect to the quark structure are completely dominated by these subtle symmetry effects we described above.

We cannot at present predict the magnitude of the QCD-vacuum deformation by an applied Maxwell-magnetic field. However, a perturbative analysis suggests that the up-quark fluctuations in the structured QCD vacuum dominate, and are much greater than the pure QED Euler-Heisenberg effect. We have also shown, assuming that the glue condensate field is constant on scale of the quark Compton wavelength, that full re-summation of all possible diagrams including n-loop series is required for the proper evaluation of the effective action. We also realize that non-perturbative `constant' condensate results led some to believe that the QED-QCD vacuum interference effect is tiny. The premise of derivation of this result is, however, intrinsically inconsistent, in that one assumes a constant glue condensate but latest QCD-lattice simulations show that glue condensate fluctuations are occurring on a scale well below hadronic size. This means that the variation of the condensate is an essential and yet unaccounted factor in non-perturbative evaluation of the QED-QCD effective action. Does that matter? Indeed since it is easy to show that the effective action we have in hand is sensitively dependent on the assumption that the fields are constant. Can we than completely ignore the popular assumption that the glue condensate provides the scale in the QCD based light-light scattering effect? Frankly, I am persuaded that this is the case.

An interesting fact to keep in mind is that the QCD-glue condensate field is nearly 100 times stronger than the so called critical QED field strength, but unlike the externally applied QED field it is believed to be a stochastic, fluctuating vacuum field. Therefore a complete treatment of the problem we described here requires a totally novel description in which an ultra strong, but stochastic short range gauge field is superposed with a weak Maxwell `drift' field in driving the effective quark action.

In summary, the theoretical understanding of the QED-QCD vacuum structure is not in any way settled at this time. The experimental and theoretical work must proceed, as the opportunity is there to study QCD vacuum structure using precision optical QED probes.

This work was supported in part by a grant from the U.S. Department of Energy, DE-FG03-95ER40937.

for more details see: hep-ph/9806389 [abs, src, ps, other]

``Electromagnetic Fields in the QCD Vacuum'' (J. Rafelski and H.Th Elze)
in Proceedings of: Workshop on Quantum Chromodynamics, Paris, France 1--6 June 1998, ;
H. Fried and B. Muller, edts World Scientific, Singapore

and Publications after1991
Conference Proceedings after1991

October 23, 1998