^{3}.

The propagators of unstable particles are considered in framework of the convolution representation. Spectral function is found for a special case when the propagator of scalar unstable particle has Breit-Wigner form. The expressions for the dressed propagators of unstable vector and spinor fields are derived in an analytical way for this case. We obtain the propagators in modified Breit-Wigner forms which correspond to the complex-mass definition.

Two standard definitions of the mass and width of unstable particles (UP), which are usually considered in the literature, have different nature. The on-mass-shell (OMS) scheme defines the mass

Traditional way to construct the dressed propagator of UP is the Dyson summation which introduces the width and redefines the mass of UP. This procedure runs into some problems which are widely discussed in the literature. One of such problems follows from the d’Alembert convergence criterion

The peculiarities of Dyson summation lead to the lack of uniqueness in constructing the propagators of unstable particles. There are several different expressions for the numerator of vector-boson propagator

All abovementioned definitions are connected with the structure of dressed propagators which follows from Dyson summation. As was noted above, this procedure runs into some problems which are widely discussed in literature. An alternative approach is based on the spectral representation of the propagator of UP. It has a long history [

In this work, we consider the structure of the propagators in the framework of the spectral-representation approach and on account of the Dyson procedure. As was noted early, Dyson summation is not well-defined at peak range, while the spectral approach cannot be applied far from the peak. So, we have used the information which follows from both approaches in the domains of their validity. We suppose that the propagator of scalar UP in spectral representation coincides with the BW one in the intersection of their domains of definition. Using this assumption we define the spectral function of boson UP and apply it for the case of the vector UP’s propagator. We show that this strategy strictly leads to the propagators which have the structure of the modified Breit-Wigner ones (

The paper is organized as follows. In Section

The structure of propagator for the case of scalar UP can be represented in the following convolution form:

The principal problem of the approach under consideration is to define the spectral function

Here, we consider the special case of the spectral function for scalar UP in the assumption that the scalar propagator has a conventional BW form:

To define

Let us consider the theoretical status of the result and possible consequences of the presence of negative mass parameter

Now we evaluate the contribution of the negative component. The spectral function

In this section, the result (

Thus, UP can be described in the framework of two different hierarchical levels—“fundamental” level, by the integral representations (

To define the structure of vector propagator, we assume that the spectral function

We should note that both the scalar and vector propagators of UP can be represented in the form with universal complex mass squared:

The propagator of a free fermion can be represented in two equivalent forms:

The main result of the previous two sections is the modified BW expressions (

We have derived just the same expressions for the case of dressed propagators in the momentum representation, that is, for complex mass

The problems of renormalization procedure in the effective theory arise at next-to-leading order. To date there is no fully established treatment of UP within perturbation theory, although many solutions have been proposed [

The definitions of the mass and width of UP, as a rule, are closely connected with the construction of the dressed propagators. It was underlined in this work that traditional approaches, which are based on the Dyson procedure and spectral representation, have formally crucial peculiarities. We have considered the structure of the propagators of UP in the phenomenological approach which is based on the spectral representation. The spectral function describes the distribution of continuous (indefinite, smeared) mass parameter, contains a principal information concerning UP, and defines a spectral structure of the propagators.

In this work, we have analyzed a special case of the spectral function which follows from matching the model and standard scalar BW propagator. This function contains the parameters

The author declares that there is no conflict of interests regarding the publication of this paper.

The work has been supported by Southern Federal University Grant no. 213.01-2014/013-VG.

^{0}resonance