Hermitian vs PT-Symmetric Scalar Yukawa Model

A comparative analysis of a model of complex scalar field $\phi$ and real scalar field $\chi$ with interaction $g\phi^*\phi\chi$ for the real and purely imaginary values of coupling $g$ in perturbative and non-perturbative regions. 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 singulatity moves in the pseudo-Euclidean region.


Introduction
Non-Hermitian PT-symmetric quantum models, open at the end of the last century [1], currently have a wide use in various fields of physics (see review [2] and references therein). For a quantum field theory the introduction into circulation the non-Hermitian PT-symmetric models is interesting as an extension of a narrow class of Hermitian models with acceptable physical and mathematical properties (such as stability, unitary, and renormalisability) and opens new possibilities for describing the properties of high-energy particles.
In works of Bender et al. [3] [4] the PT-symmetric model of a scalar field with the interaction 3 φ has been investigated. As has long been known (see [5]), the Hermitian version of this model is asymptotically free, but the unstability of the cubic interaction leads to the fact that models of this type previously considered exclusively as a methodical examples (see, for example [6]). For the non-Hermitian version of this model with imaginary coupling, however, the main argument of the unstability-the cubic potential is unbounded below-becomes invalid since the set of complex numbers is not an ordered set. Moreover, in work [3], the arguments are given in favor of energy stability of the non-Hermitian model with cubic interaction. The analysis of Bender et al. (see [3] [4]) indicates that the 3 6 igφ theory is like a 4 4 gφ theory: it is energetically stable, renormalized and has the trivial-type ultraviolet behavior, i.e., compared with the conventional 3 6 φ model, the PT-symmetric 3 6 igφ theory exhibits new interesting properties.
In this paper we study the scalar Yukawa model, i.e., a model of a complex scalar field φ (phion) and a real field χ (chion) with the interaction gφ φχ * . This model is used in nuclear physics as a simplified version of the Yukawa model without spin degrees of freedom, as well as an effective model of the interaction of scalar quarks [7] [8]. If the coupling constant g takes purely imaginary values and the field χ is a pseudoscalar, such a model is PT-symmetric. As expected, this model is a very similar to the 3 φ theory. All arguments of Bender et al. (see [3]) concerning the unstability of the Hermitian theory and stability of non-Hermitian PT-symmetric theory fully extended to the scalar Yukawa model. An additional argument is the consideration (in the spirit of [9]) a zero-dimensional version of the theory. The partition function which converges for 2 0 g < (non-Hermitian case) and diverges for 2 0 g > (Hermitian case). In the coupling-constant perturbation theory, this model also has a very similar to the 3 φ theory. Section 2 briefly presents the results of the coupling-constant perturbation theory and based on the perturbation theory renormalization-group analysis for this model. As well as the 3 6 φ theory the Hermitian scalar Yukawa model in a six-dimensional space is asymptotically free. The non-Hermitian scalar Yukawa model in the 6 d = −  has, besides the Gaussian fixed point, also the non-Gaussian fixed point of Wilson-Fisher type. At 6 d = the non-Hermitian scalar Yukawa model, as well as 3 6 φ theory is ultraviolet unstable, and to describe the ultraviolet region we need to go beyond the perturbation theory.
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.

Perturbation Theory
We consider the model of interaction of a complex scalar field φ (phion) and a real scalar field χ (chion) with the Lagrangian ( ) At 6 d = 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 , .
x y x y Here , m µ are the renormalized masses of the phion and the chion.
Here κ is a 't Hooft scale. We define the dimensionless coupling as and by adopting the MS scheme [6], we get the counter-terms:

Renormalization Group. Hermitian Theory
The independence of initial (bare) quantities and unrenormalized Green functions from the 't Hooft scale κ leads to the renormalization group equation: Here nl Γ is the one-particle-irreducible function with n phion and l chion tails. Counter-terms (5) allow us to calculate renormalization-group coefficients 1 ( ) These renormalization-group coefficients quite similar to corresponding coefficients of 3 6 φ -theory (see [3]  For β -function (7) the solution of this equation at i.e. the model possesses the typical asymptotically-free behavior at high momenta with all consequences.

Renormalization Group. Non-Hermitian Theory
For the non-hermirian PT-symmetric theory one should make the substitution , g ig g i g δ δ → → in formulae of above Subsections. Thus, the expression for β -function takes the form ( ) etc.
The situation in this case is also similar to 3 φ -theory (see [3]). β -function vanishes, except of the Gaussian point * 0 g = , 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 6 d = 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.

Shcwinger-Dyson Equations
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 η is the bilocal source of phions 2 , j is the single source of chions. The translational invariance of the functional integration measure leads to relations which can be rewritten as the functional-differential SDE for generating functional G: , , and Here 0 m and 0 µ are bare phion and chion masses. Equation (18) The differentiation of (19) over η gives us the three-point chion-phion function is the two-particle phion function. The differentiation of (19) over j with taking into account Equation (20) gives us the chion propagator: 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 . , x y x y x y y x δη δ δ δη Next-to-the-leading-order equation is

Ladder Expansion
, , δη 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: we obtain 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 3 6 igφ theory (see [3]). The non-perturbative ladder expansion of Section 3 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 singulatity moves to the pseudo-Euclidean region.
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.