Renormalization and Universality of Van der Waals forces

Abstract

Abstract. Renormalization ideas can profitably be exploited in conjunction with the superposition principle of boundary conditions in the description of model independent and universal scaling features of the singular and long range Van der Waals force between neutral atoms. The dominance of the leading power law is highlighted both in the scattering as well as in the bound state problem. The role of off-shell two-body unitarity and causality within the Effective Field Theory framework on the light of universality and scaling at low energies is analyzed.

Coarse graining $$\pi \pi $$ π π scattering

“…Singular potentials are commonplace within EFT and finite solutions exist in a renormalization sense within well specified conditions, as discussed at length in the NN scattering case [78][79][80]. Applications for αα-scattering [81,82] and atom-atom scattering [83,84] are well documented by now (see e.g. [85] for a sucint and pedagogical presentation).…”

“…In fact, phase shifts in any partial wave δ l ( p) are convergent when the short distance cut-off goes to zero, provided the scattering length α l is fixed. This corresponds to a renormalization program already developed in previous studies [78][79][80][81][82][83][84][85] that will not be pursued any further here. The upshot of these considerations in the ππ case is that generally one needs a cut-off r c > 0.5 fm to prevent the appearance of unphysical bound states generated by the TPE potential.…”

Van der Waals Forces and Photon-Less Effective Field Theory

“…( 2), and dismissing any explicit reference to the underlying photon exchange. However, even at very low energies, causality arguments provide the shortest possible value for r c , which for vdW forces yields r c > 0.6R [2]. Using for illustration a square well (SW) potential with range r c and depth V 0 , V eff (r) = −V 0 θ (r c − r), one obtains…”

“…Numerically we get MC 0 /R ∼ −15 and MC 2 /R 3 ∼ 2 for Λ ∼ π/(2R), in agreement with the previous SW and DS analysis. The values of Λ where the EFT low energy parameters diverge correspond to an upper bound above which C 0 and C 2 become complex, violating the self-adjointness of the potential [2] and the Lagrangian, L (x) = L † (x). Thus, off-shell two-body unitarity and hence three-body unitarity are jeopardized for Λ R ≥ 4, despite the phase shift being real and on-shell unitarity being fulfilled.…”

“…where C n are the dispersion coefficients which are accurately known for many diatomic systems (see e.g. a compilation in [2]). The vdW length R = (MC 6 /h 2 ) 1 4 characterizes the size of the forces.…”