A weak, attractive, long-range force in Higgs condensates

Due to the peculiar nature of the underlying medium, density fluctuations in a ‘Higgs condensate’ are predicted to propagate for infinitely long wavelengths with a group velocity cs → ∞. On the other hand, for any large but finite cs there is a weak, attractive 1/r potential of strength 1 cs and the energy spectrum deviates from the purely massive form √ p2 +M2 h at momenta smaller than δ ∼ Mh cs . Physically, the length scale δ −1 corresponds to the mean free-path for the elementary constituents in the condensate and would naturally be placed in the millimeter range. 1. In this Letter, I shall discuss some phenomenological aspects of the ground state of spontaneously broken theories: the ‘Higgs condensate’. The name itself (as for the closely related gluon, chiral,..condensates) indicates that, in our view, this represents a kind of medium made up by the physical condensation process of some elementary quanta. If this were true, such a vacuum should support long-wavelength density fluctuations. In fact, the existence of density fluctuations in any known medium is a basic experimental fact depending on the coherent response of the elementary constituents to disturbances whose wavelength is much larger than their mean free path. This leads to an universal description, the ‘hydrodynamical regime’, that does not depend on the details of the underlying molecular dynamics. By accepting this argument, and quite independently of the Goldstone phenomenon, the energy spectrum of a Higgs condensate should terminate with an ‘acoustic branch’, say Ẽ(p) = cs|p| for p → 0, as for the propagation of sound waves in ordinary media. Some arguments suggest that, indeed, the vacuum of a ‘pro forma’ Lorentz-invariant quantum field theory may be such kind of medium. For instance, a fundamental phenomenon as the macroscopic occupation of the same quantum state (say p = 0 in some frame) may represent the operative construction of a ‘quantum aether’ [1, 2]. This would be quite distinct from the aether of classical physics, considered a truly preferred reference frame and whose constituents were assumed to follow definite space-time trajectories. However, it would also be different from the empty space-time of special relativity, assumed at the base of axiomatic quantum field theory to deduce the exact Lorentz-covariance of the energy spectrum. In addition, one should take into account the approximate nature of locality in cutoffdependent quantum field theories. In this picture, the elementary quanta are treated as ‘hard spheres’, as for the molecules of ordinary matter. Thus, the notion of the vacuum as a ‘condensate’ acquires an intuitive physical meaning. For the same reason, however, the simple idea that all deviations from Lorentz-covariance take place at the cutoff scale may be incorrect. In particular, non-perturbative vacuum condensation may give rise to a hierarchy of scales such that the region of Lorentz-covariance is sandwiched both in the highand low-energy region. In fact, in general, an ultraviolet cutoff induces vacuum-dependent reentrant violations of special relativity in the low-energy corner [3]. In the simplest possible case, these extend over a small shell of momenta, say |p| < δ, where the energy spectrum Ẽ(p) deviates from a Lorentz-covariant form. Since Lorentz-covariance becomes an exact symmetry in the local