A comparative analysis of a model of complex scalar field φ and real scalar field χ with interaction for the real and purely imaginary values of coupling g in perturbative and non-perturbative regions is provided. In contrast to the usual Hermitian version (real g), which is asymptotically free and energetically unstable, the non-Hermitian PT-symmetric theory (imaginary g) is energetically stable and not asymptotically free. The non-perturbative approach based on Schwinger-Dyson equations reveals new interesting feature of the non-Hermitian model. While in the Hermitian version of theory the phion propagator has the non-physical non-isolated singularity in the Euclidean region of momenta, the non-Hermitian theory substantially free of this drawback, as the singularity moves in the pseudo-Euclidean region.
Non-Hermitian PT-symmetric quantum models, open at the end of the last century [
In works of Bender et al. [
In this paper we study the scalar Yukawa model, i.e., a model of a complex scalar field
in a zero-dimensional space becomes the usual improper integral
which converges for
In the coupling-constant perturbation theory, this model also has a very similar to the
Section 3 presents an attempt to go beyond the coupling-constant perturbation theory. The formalism of bilocal source is used to build a non-perturbative expansion of the system of the Schwinger-Dyson equations, and equation for the phion propagator in the leading approximation of this expansion is investigated. A remarkable property is established: for the Hermitian theory the phion propagator has a non-isolated singularity in the Euclidean region of momenta while for the Hermitian theory this singularity (an origin of a cut) moves in a pseudo- Euclidean region, i.e., from the point of view of the analytic properties the non-Hermitian theory is preferable.
We consider the model of interaction of a complex scalar field
in a d-dimensional Euclidean space
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
In the dimensional regularization (
Here
and by adopting the MS scheme [
The independence of initial (bare) quantities and unrenormalized Green functions from the ’t Hooft scale
Here
Counter-terms (5) allow us to calculate renormalization-group coefficients1
These renormalization-group coefficients quite similar to corresponding coefficients of
At
with the boundary condition
For
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
etc.
The situation in this case is also similar to
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
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
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
The differentiation of (19) over
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
which only contains the derivatives over the bilocal source
Since
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 [
Equation (23) can be used for the construction of the mean-field expansion (see [
In this paper we consider the expansion, based on the Equation (24) (see also [
For logarithm
Equation
which determines the phion propagator can be regarded as an equation that determines implicitly
Assuming the unique solvability of the Equation (26), we can move to a new function variable
From definitions (26) and (27) it follows that
and SDE (25) takes the form
In this equation, it is assumed that
which follows from the relation
SDE (29) tells us a non-perturbative expansion of the generating functional
Next-to-the-leading-order equation is
where
At the source being switched off, Equation (31) is the equation for the leading-order phion propagator:
A differentiation of equation (31) on
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
leads to the renormalized equation in momentum space
where
Below we consider the case of massless chion:
we obtain
Introducing dimension-less function
where
which is reduced to the non-linear fourth-order differential equation
This differential equation enables us to calculate the asymptotics of
where
i.e., for Hermitian theory with
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
Our results demonstrate that the non-Hermitian PT-symmetric scalar Yukawa model has interesting properties
both perturbative and non-perturbative. In the perturbation region of small momenta,
fixed point, a non-trivial fixed point of Wilson-Fisher type. As expected, the properties of the scalar Yukawa model in the perturbative region completely analogous to the corresponding properties of
For a complete description of the leading-order ladder expansion, including its renormalization group analysis, it is necessary to solve Equation (34) for the three-point function. This is a very difficult task, since this equation contains a nontrivial phion propagator, described by Equation (33). Perhaps for the renormalization-group analysis, clarifying the nature of the behavior of couplings in the asymptotic region is sufficient to solve a more limited problem, namely the calculation of the vertex function at zero momentum (which is, however, also very difficult). We can assume that in the Hermitian case the theory retains the property of asymptotic freedom, and everything will return to own. For the non-Hermitian PT-symmetric theory a prediction of the answer is harder. In any case, the results indicate that the non-Hermitian scalar Yukawa model has, compared with the Hermitian version, a number of attractive features, which make it a very interesting object of study.
Author is grateful to the participants of IHEP Theory Division Seminar for useful discussion.
Vladimir E. Rochev, (2016) Hermitian vs PT-Symmetric Scalar Yukawa Model. Journal of Modern Physics,07,899-907. doi: 10.4236/jmp.2016.79081