Abstract
Schr\"{o}dinger field theory with an attractive self-interaction possess non-topological extended solutions with a finite energy in both finite and infinite-volume cases, namely, bright solitons. The analytical form of the solution itself is well-known, though analytical investigation of the quantum fluctuations in this background still requires more thorough investigation, for instance, analytical computation of quantum corrections to this background within the saddle-point approximation. In the present work this gap is filled. Both 2-point Green's function and quantum corrections to the background are analytically computed and properly renormalized by means of momentum cut-off procedure. It is deduced that quantum corrections are indeed small provided that particle number is large. Also, we see that perturbation modes of continuum spectrum at bright soliton background generate a gap in the energy spectrum. Moreover, it turns out that the whole spectrum is continuous modulo zero-modes, which is similar to Sine-Gordon solitons.
Highlights
In quantum field theory two types of solitons are usually considered
The situation is more subtle with nontopological solitons, which can be stabilized by means of some conserved global current and correspond to fixed global charge [14]
The procedure of quantizing nontopological solitons in the relativistic case was established in [15], but no actual computation according to this procedure was ever done,2 in spite of attempts given for instance in works like [16], where integration along symmetry direction was carried out
Summary
In quantum field theory two types of solitons are usually considered. The first type is topological solitons [1,2,3,4],1 which exist thanks to some nontrivial mapping of internal field space on coordinate space or space-time. There are some works that address questions of fluctuating modes [17,18,19], neither analytical nor numerical computations of the Green’s function or quantum corrections to the background of nontopological soliton like Qball by means of semiclassical methods was carried out. The rest of the article is organized as follows: In Sec. II I present classical properties of bright solitons and connection to the relativistic theory, in Sec. III I introduce the inverse Green’s function of quantum fluctuations in the background of the bright soliton and invert this operator in Sec. IV, where I discuss backreactions on the background and their compliance with the demand of fixed particle number, in Sec. V quantum correction to the classical energy is computed
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