GLYPH<Minus> 7 GLYPH<Minus> 6 GLYPH<Minus> 5 GLYPH<Minus> 4 GLYPH<Minus> 3 GLYPH<Minus> 2 GLYPH<Minus> 1 20 40 60 80 100 120 Viscosity for repulsive bosons PSfrag replacements η/s µ/T FIG. 23: The ratio η/s as a function of µ/T for repulsive bosons. XII. APPENDIX A: THE BETA FUNCTION Consider a single free boson with action eq. (2). Setting the external energy and momentum to zero, the 4-point vertex at tree-level plus 1-loop is The ω integral can be done by closing the contour in the upper half plane picking up the pole at ω = − k $^{2}$/ 2 m + iϵ : Introducing an ultraviolet cut-off Λ: Γ (4) = − ig − ( − ig ) 2 2 ∫ dωd $^{d}$k (2 π ) d +1 ( i w − k $^{2}$/ 2 m + iϵ ) ( i − ω − k $^{2}$/ 2 m + iϵ ) (107) 1 − loop = − i mg 2 2 ∫ d $^{d}$k (2 π ) d 1 k 2 (108) Γ (4) = − i ( g − mg $^{2}$Λ d − 2 π d/ $^{2}$Γ( d/ 2)2 $^{d}$( d − 2) ) (109) 43