Effective field theories for van der Waals interactions

Van der Waals interactions between two neutral but polarizable systems at a separation R much larger than the typical size of the systems are at the core of a broad sweep of contemporary problems in settings ranging from atomic, molecular and condensed matter physics to strong interactions and gravity. In this paper, we reexamine the dispersive van der Waals interactions between two hydrogen atoms. The novelty of the analysis resides in the usage of nonrelativistic effective field theories of quantum electrodynamics. In this framework, the van der Waals potential acquires the meaning of a matching coefficient in an effective field theory, dubbed van der Waals effective field theory, suited to describe the low-energy dynamics of an atom pair. It may be computed systematically as a series in R times some typical atomic scale and in the fine-structure constant α. The van der Waals potential gets short-range contributions and radiative corrections, which we compute in dimensional regularization and renormalize here for the first time. Results are given in d space-time dimensions. One can distinguish among different regimes depending on the relative size between 1/R and the typical atomic bound-state energy, which is of order mα2. Each regime is characterized by a specific hierarchy of scales and a corresponding tower of effective field theories. The short-distance regime is characterized by 1/Rmα2 and the leading-order van der Waals potential is the London potential. We also compute next-to-next-to-next-to-leading-order corrections. In the long-distance regime we have 1/R mα2. In this regime, the van der Waals potential contains contact terms, which are parametrically larger than the Casimir-Polder potential that describes the potential at large distances. In the effective field theory, the Casimir-Polder potential counts as a next-to-next-to-next-to-leading-order effect. In the intermediate-distance regime, 1/R∼mα2, a significantly more complex potential is obtained. We compare this exact result with the two previous limiting cases. We conclude by commenting on the van der Waals interactions in the hadronic case.