Abstract
We present the saddle-point approximation for the effective Hamiltonian of the quantum kink in two-dimensional linear sigma models to all orders in the time-derivative expansion. We show how the effective Hamiltonian can be used to obtain semiclassical soliton form factors, valid at momentum transfers of order the soliton mass. Explicit results, however, hinge on finding an explicit solution to a new wave-like partial differential equation, with a time-dependent velocity and a forcing term that depend on the solution. In the limit of small momentum transfer, the effective Hamiltonian reduces to the expected form, namely H = (P^2 + M^2)^(1/2), where M is the one-loop corrected soliton mass, and soliton form factors are given in terms of Fourier transforms of the corresponding classical profiles.
Highlights
AND MOTIVATIONThe description of soliton states in quantum field theory—the foundations of which were laid out in the mid 1970s—is a beautiful subject where basic notions of quantum field theory operate in the background of exact solutions to nonlinear differential equations; for popular reviews see [1,2,3]
We show how the effective Hamiltonian can be used to obtain semiclassical soliton form factors, valid at momentum transfers of order the soliton mass
A standard renormalization prescription can be made for the class of linear sigma models discussed above where the effect of the counterterms is such that the renormalized potential, VðφÞ 1⁄4 V0ðφÞ þ Vc:t:ðφÞ, has the same form as V0, but with m20 replaced by m2 þ Δm2, where m2 is a finite renormalized mass, and the condition fixing Δm2 is that the tadpole for the fluctuation field around the vacuum vanishes to all orders in perturbation theory
Summary
The description of soliton states in quantum field theory—the foundations of which were laid out in the mid 1970s—is a beautiful subject where basic notions of quantum field theory operate in the background of exact solutions to nonlinear differential equations; for popular reviews see [1,2,3]. In this paper we demonstrate that one can access the high momentum transfer regime of solitons in two-dimensional linear sigma models by working directly with the exact field-theoretic soliton-sector Hamiltonian obtained in [10,11]. Quantum fluctuations around a solution to Eq (1.1) can be treated in the usual perturbative manner.5 Integrating out these degrees of freedom results in the one-loop and higherorder contributions to the soliton effective Hamiltonian, viewed as an expansion in g. A summary of the results presented here appears in [24]
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