A Tachyonic Gluon Mass: between Infrared and Ultraviolet

The gluon spin coupling to a Gaussian correlated background gauge field induces an effective tachyonic gluon mass. It is momentum dependent and vanishes in the UV only like 1/p2. In the IR, we obtain stabilization through a positive mconf(p 2) related to confinement. Recently a purely phenomenological tachyonic gluon mass was used to explain the linear rise in the qq̄ static potential at small distances and also some long standing discrepancies found in QCD sum rules. We show that the stochastic vacuum model of QCD predicts a gluon mass with the desired properties. Recently it was argued [1, 2] that an effective negative gluon (mass) produces the linear rise of the static quark-antiquark potential observed in lattice calculations [3] at rather short distances below the nonperturbative IR scale of QCD. This can also be related to an unconventional Q piece in the running QCD coupling αs(Q ) [4, 5] indicated again by lattice data on the gluon condensate and the 3-gluon vertex [6]. Later in ref. [7], it was shown that a linear term in the potential for small distances appears in the evaluation of the Wilson loop integral in a Gaussian correlated background gauge field if the paramagnetic (spin related) gluon coupling is taken into account. A gluon propagator with a nonperturbative tachyonic mass in the evaluation of the operator product expansion for QCD sum rules also solves old discrepancies in the pion and scalar gluonium channels [2]. In ref. [2], the view was taken that a gluon mass is not just a convenient way to parametrize nonperturbative infrared (IR) effects, but that a tachyonic gluon mass also enters into the basic short distance operator product expansions. In this letter, we propose less radically that a tachyonic gluon mass does appear at momenta above the QCD scale, although at very high momenta/small distances it goes to zero. Thus, in the real ultraviolet regime (UV), the basic perturbative operator product expansion should be valid. Of course, if one discusses an operator product expansion in an effective theory including a tachyonic gluon mass in which very high momenta are already integrated out, this mass shows up in the expansion. However, at very small distances, the expansion loses its basis. In the IR, a tachyonic gluon mass would cause divergences. Thus there should be an overcompensating positive effective gluon (mass). In the case of the hot electroweak theory close to the symmetric phase, we have argued earlier [9] for the following picture: a tachyonic gauge boson mass related to spin-spin forces is effective at intermediate momenta and a positive (mass) related to the area law of confinement is dominant in the gauge boson propagator in the IR. Our equations in ref. [9] were written for a general dimension d and gauge group SU(N), but we had restricted our discussion to the case d = 3 and N = 2, which is the relevant one for the discussion of the electroweak phase transition. In this letter, we analyze the QCD case d = 4 and N = 3. Concerning a 1/Q behavior consider the following simple function: (αQ + βQ). If β ≫ α is large, it behaves like 1 βQ for Q < β/α, i.e. for rather large Q!