Tachyon condensation and quark mass in modified Sakai-Sugimoto model
  This paper continues the investigation of the modified Sakai-Sugimoto model proposed in arXiv:0708.3233. Here we discuss in detail numerical solutions to the classical equations for the brane profile and the tachyon condensate. An ultraviolet cut-off turns out to be essential because the numerical solutions tend to rapidly diverge from the desired asymptotic solutions, beyond a sufficiently large value of the holographic coordinate. The required cut-off is determined by the non-normalizable part of the tachyon and is parametrically far smaller than that dictated by consistency of a description in terms of 10-dimensional bulk gravity. In arXiv:0708.3233 we had argued that the solution in which the tachyon field goes to infinity at the point where the brane and antibrane meet has only one free parameter, which may be taken to be the asymptotic brane-antibrane separation. Here we present numerical evidence in favour of this observation. We also present evidence that the non-normalizable part of the asymptotic tachyon solution, which is identified with quark mass in the QCD-like boundary theory, is determined by this parameter. We show that the normalizable part of the asymptotic tachyon solution determines the quark condensate, but this requires holographic renormalization of the on-shell boundary brane action because of the presence of infinite cut-off dependent terms. Our renormalization scheme gives an exponential dependence on the cut-off to the quark mass. We also discuss meson spectra in detail and show that the pion mass is nonzero and satisfies the Gell-Mann-Oakes-Renner relation when a small quark mass is switched on. 
Avinash Dhar and Partha Nag