Abstract
Polarizable atoms interacting with a charged wire do so through an inverse-square potential, V = −g/r2. This system is known to realize scale invariance in a nontrivial way and to be subject to ambiguities associated with the choice of boundary condition at the origin, often termed the problem of ‘fall to the center’. Point-particle effective field theory (PPEFT) provides a systematic framework for determining the boundary condition in terms of the properties of the source residing at the origin. We apply this formalism to the charged-wire/polarizable-atom problem, finding a result that is not a self-adjoint extension because of absorption of atoms by the wire. We explore the RG flow of the complex coupling constant for the dominant low-energy effective interactions, finding flows whose character is qualitatively different when g is above or below a critical value, gc. Unlike the self-adjoint case, (complex) fixed points exist when g > gc, which we show correspond to perfect absorber (or perfect emitter) boundary conditions. We describe experimental consequences for wire-atom interactions and the possibility of observing the anomalous breaking of scale invariance.
Highlights
The quantum mechanics of the attractive inverse-square potential poses conceptual challenges that are not encountered for quantum motion in potentials that are less singular at the origin [1]
We demonstrate how to apply Point-particle effective field theory (PPEFT) to non-Hermitian sources by analyzing an explicit physical system that can be realized in a laboratory
Just like in the self-adjoint case, we find that the interactions between bulk fields and microscopic sources can be efficiently parameterized in terms of an action localized on a source located at the origin
Summary
The quantum mechanics of the attractive inverse-square potential poses conceptual challenges that are not encountered for quantum motion in potentials that are less singular at the origin [1]. The default boundary condition cannot be used because of competition between the inverse-square potential and the centrifugal barrier that means that for some values of system parameters both linearly independent solutions to the radial Schrodinger equation are singular at r = 0 Because both solutions are singular the condition of boundedness at the origin does not provide a useful criterion for distinguishing amongst them, and so some other choice is needed. The resulting construction has a definite effective-Lagrangian flavour [17] (for a review, see for instance [18]), with the only difference being that the relevant Lagrangian is first-quantized from the point of view of the source, which could be a single domain wall, string-like defect or point particle For this reason this boundary condition proposal is known as ‘pointparticle effective field theory’ (PPEFT).
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