We consider the model of interaction of a complex scalar field (phion) and a real scalar field (chion) with the Lagrangian

in a d-dimensional Euclidean space near. At the coupling g is dimensionless, and the theory contains ultraviolet divergences which can be eliminated with a standard recipe by the renormalization of fields and vacuum expectations (Green functions).

The perturbation theory on the renormalized coupling constant g gives us the following expressions for the renormalized 1PI functions:

Propagators of the phion

and of the chion

A vertex:

Here are the renormalized masses of the phion and the chion. are counter-terms of the renormalization of the masses and fields of the phion and the chion correspondingly, is a counter-term of the renormalizarion of coupling, and

In the dimensional regularization ():

Here is a ’t Hooft scale. We define the dimensionless coupling as

and by adopting the MS scheme [6] , we get the counter-terms:

The independence of initial (bare) quantities and unrenormalized Green functions from the ’t Hooft scale leads to the renormalization group equation:

Here is the one-particle-irreducible function with n phion and l chion tails.

Counter-terms (5) allow us to calculate renormalization-group coefficients¹

These renormalization-group coefficients quite similar to corresponding coefficients of -theory (see [3] [6] ). As for -theory the scalar Yukawa model near possesses only a Gaussian fixed point, and near this point the couplings scale according their scaling dimension.

At the scalar Yukawa model is asymptotically free as the -theory. The running coupling (invariant charge) is a solution of equation

with the boundary condition Here.

For -function (7) the solution of this equation at is

i.e. the model possesses the typical asymptotically-free behavior at high momenta with all consequences.

For the non-hermirian PT-symmetric theory one should make the substitution

in formulae of above Subsections. Thus, the expression for -function takes the form

etc.

The situation in this case is also similar to -theory (see [3] ). -function vanishes, except of the Gaussian point, at the fixed point of Wilson-Fisher type:

Near the Gaussian point couplings are still defined by their canonical dimensions. Near the non-Gaussian fixed point (14) the scale behavior is modified in accordance with the linearized renormalization group equations. At fixed points merge into one Gauss point.

The running coupling in this case is

i.e., the theory at large momenta has the trivial-type behavior, and the perturbation theory in this asymptotic region cannot be applicable.

To construct the non-perturbative approximation we will use the formalism of Schwinger-Dyson equations (SDE).

The generating functional of Green functions (vacuum averages) of the model with Lagrangian (1) is the functional integral

Here is the bilocal source of phions², j is the single source of chions.

The translational invariance of the functional integration measure leads to relations

and

which can be rewritten as the functional-differential SDE for generating functional G:

and

Here and are bare phion and chion masses. Equation (18) allows us to express all Green functions with chion legs in terms of functions that contain phions only. For logarithm this equation has the form

The differentiation of (19) over gives us the three-point chion-phion function

where

is the two-particle phion function. The differentiation of (19) over j with taking into account Equation (20) gives us the chion propagator:

etc. Thus, for a complete description of the model we need to know phion Green function only.

Excluding with the help of the SDE (18) a differentiation over j in SDE (17), we obtain at the equation

which only contains the derivatives over the bilocal source.

Since, then Bose-symmetry entails the relation

reflecting crossing symmetry of the two-particle function, and, accordingly, the Equation (23) can be written as

Both equations give the same coupling-constant perturbation series, and are completely equivalent from the point of view of some visionary exact solutions of Schwinger-Dyson equations. However, these equations give different non-perturbative expansion. This is due to the incomplete structure of the leading-order multi-particle functions of such expansions in terms of crossing symmetry. It is a peculiar feature of some non-perturbative approximations. In order to restore crossing symmetry lost in the leading-order approximation, it is necessary to consider the next-to-leading-order approximation. (A more detailed discussion of this issue see in the papers [11] [12] and references therein).

Equation (23) can be used for the construction of the mean-field expansion (see [11] ). In the language of Feynman diagrams the leading order of this expansion corresponds to the summation of the chains and its structure actually reproduce the renormalization-group summation of the previous section.

In this paper we consider the expansion, based on the Equation (24) (see also [12] ). In the language of Feynman diagrams the leading order of this expansion corresponds to the summation of ladder graphs, so we'll call it the ladder expansion.

For logarithm this equation has the form

Equation

which determines the phion propagator can be regarded as an equation that determines implicitly as a functional of:

Assuming the unique solvability of the Equation (26), we can move to a new function variable and define the generating functional of Legendre transform

From definitions (26) and (27) it follows that

and SDE (25) takes the form

In this equation, it is assumed that is a functional of new functional variable, what can be done, using the condition of connection

which follows from the relation

SDE (29) tells us a non-perturbative expansion of the generating functional, which based on the following leading approximation

Next-to-the-leading-order equation is

where is a functional of, defined by condition of connection (30).

At the source being switched off, Equation (31) is the equation for the leading-order phion propagator:

A differentiation of equation (31) on and taking into account connection condition (30) together with equation (20) gives us the equation for the three-point function:

Lets go to the Equation (33) for the phion propagator. To eliminate ultraviolet divergences in Equation (33) is sufficient to introduce counter-terms of phion-field renormalization and mass. The normalization of the renormalized propagator at zero momentum

leads to the renormalized equation in momentum space

where is the renormalized mass operator, and

Below we consider the case of massless chion:. In this case nonlinear integral Equation (35) can be reduced to an integral Volterra-type equation, which, in turn, is reduced to a differential equation. Using the formula of massless integration in six-dimensional space

we obtain

Introducing dimension-less function

where we obtain the integral equation

which is reduced to the non-linear fourth-order differential equation

This differential equation enables us to calculate the asymptotics of for large t:

where

i.e., for Hermitian theory with the asymptotic behavior becomes purely imaginary. In order to understand what is happening with the propagator in Euclidean region, consider a simplified model with the same UV behavior. This model is based on the following approximation of mass operator (37) in a high-momentum region:

The equation for the inverse propagator u takes the form:

The cutoff at the lower limit of integration is introduced in order to avoid mass singularities (in the case insignificant).

The exact solution of Equation (42) is

i.e., an asymptotic behavior at large momentum given by the same formula (40).

Thus, we can conclude that for the usual Hermitian theory with the propagator in the ladder approximation has the non-physical non-isolated singularity in the Euclidean region, while for the non-Hermitian theory with, this singularity moves in a pseudo-Euclidean region, i.e., the non-Hermitian theory is more preferable from the standpoint of the analytical properties of the propagator.