# On the phase structure of vector-matrix scalar model in four dimensions

## Abstract

The leading-order equations of the \(1/N\) – expansion for a vector-matrix model with interaction \(g\phi _a^*\phi _b\chi _{ab}\) in four dimensions are investigated. This investigation shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. For small values of the coupling the phion propagator behaves as free. In the strong-coupling region the asymptotic behavior drastically changes – the propagator in the deep Euclidean region tend to some constant limit. The phion propagator in the coordinate space has a characteristic shell structure. At the critical value of coupling that separates the weak and strong coupling regions, the asymptotic behavior of the phion propagator is a medium among the free behavior and the constant-type behavior in strong-coupling region. The equation for a vertex with zero transfer is also investigated. The asymptotic behavior of the solutions shows the finiteness of the charge renormalization constant. In the strong-coupling region, the solution for the vertex has the same shell structure in coordinate space as the phion propagator. An analogy between the phase transition in this model and the re-arrangement of the physical vacuum in the supercritical external field due to the “fall-on-the-center” phenomenon is discussed.

## 1 Introduction

In the present paper we consider a vector-matrix model of the complex scalar field \(\phi _a\) (phion) and real scalar mass-less field \(\chi _{ab}\) (chion) with interaction \(g\phi _a^*\phi _b\chi _{ab}\) in four dimensions (\(a, b=1,\ldots , N\)). This model, known as well as a scalar Yukawa 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 (squarks). Despite its well-known imperfection associated with its instability (or more precisely, the metastability [1]), this model, as the simplest model of the interaction of fields, often used as a prototype of more substantive theories to elaborate the various non-perturbative approaches in the quantum field theory (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8]).

The solution of the equation for the phion propagator in the leading order of the 1 / *N* – expansion shows a change of the asymptotic behavior in the deep Euclidean region in a vicinity of a certain critical value of the coupling constant. For small values of the coupling the propagator behaves as free, which is consistent with the wide-spread opinion about the dominance of perturbation theory for this super-renormalizable model. In the strong-coupling region, however, the asymptotic behavior changes dramatically – the propagator in the deep Euclidean region tend to some constant limit. At the critical value of coupling that separates the weak and strong coupling regions, the asymptotic behavior of the propagator (1 / *p*) is a medium among the free behavior and the constant-type behavior in strong-coupling region

The similar change of asymptotic behavior was found also with solution of the system of Schwinger–Dyson equations in some approximations [9, 10].

In the present paper we obtain solutions of the nonlinear equation for the phion propagator in the linearized approximation, correctly describing both the asymptotic behavior and the behavior at small momenta, and also investigate the asymptotic behavior of the vertex for zero transfer momentum.

The existence of a critical coupling in the scalar Yukawa model was noticed by practically all authors who have investigated this model using different methods (see, e.g., [3, 4, 5] and references therein). This critical constant is generally regarded as a limit on the coupling constant for a self-consistent description of the model by some method. In our approach, however, the self-consistent positive solutions in Euclidean region exist also for the strong coupling, and the existence of the critical coupling looks more like as a phase transition in accordance with the general definition of the phase transition as a sharp change of properties of the model with a smooth change of parameters.

The structure of the paper is as follows. In Sect. 2, equations of the leading-order of the 1 / *N* – expansion for this model and its renormalization are given. In Sect. 3, equation for phion propagator is investigated. It is shown that there exists a critical value of the coupling at which the asymptotic behavior changes. Solutions of the equation in linearized approximation are obtained in three regions: in the weak-coupling region, at the critical value of the coupling, and in the strong-coupling region. In the strong coupling region, the propagator in the coordinate space has a characteristic shell structure. In Sect. 4, an equation for a vertex with zero transfer is investigated. The asymptotic behavior of the solutions in these regions shows the finiteness of the charge renormalization constant. In the strong-coupling region, the solution for the vertex in coordinate space has the same shell structure as the propagator. Discussion and conclusion are contained in Sect. 5. An analogy is made between the phase transition in the model under consideration and the re-arrangement of the physical vacuum in the supercritical external field.

## 2 Preliminaries

*N*– expansion for such models is well-known (see [11, 12]). In this paper we consider only the leading order of this expansion.

*D*is the Dyson equation

*V*is the three-point function. Since \(V=O(\frac{1}{\sqrt{N}})\), we see, that \(D=D_c+O(1/N)\), i.e., the chion propagator in the leading-order of 1 /

*N*–expansion is the free propagator.

^{1}

To renormalize the leading order one should to add three counter-terms: \(\delta m^2\) (renormalization of the phion mass), *z* (the phion-field renormalization) and \(z_g\) (the coupling renormalization), and in this super-renormalizable model does the only counterterm \(\delta m^2\) be infinite

This zero-momentum normalization condition plays a very important role in the construction of the analytic solutions obtained below. When normalizing at another point, the equations become more complicated, and it is hardly possible to solve them without using numerical methods.

*g*is the renormalized coupling.

## 3 Phion propagator

*(i) The weak coupling:*\(\lambda <1\). In the weak-coupling region the asymptotic solution at large \(p^2\) is

- (a)
\(y_0\) has the right asymptotic behavior and \(y_1=o(y_0)\) at \(t\rightarrow \infty \);

- (b)
the initial conditions (17) at \(t=0\) are fulfilled.

*(ii) The critical coupling:* \(\lambda =1\).

*F*is the Gauss hypergeometrical function [14].

*(iii) The strong coupling:* \(\lambda >1\).

This super-critical case requires some comment. Zero-momentum normalization condition (7) implies that the phion-field renormalization is \(z=1-\Sigma '(0)\). Using definition (9) and formula (12), it is easy to see that \(\Sigma '(0)=\lambda \). Consequently, the phion-field renormalization constant is \(z=1-\lambda \). In the strong-coupling region *z* is negative that looks like a pathology.

*on the mass shell*, the negativity of the field renormalization constant \(z_m=1-\Sigma '(-m^2)\) indicates the existence of problems with unitarity. To solve the problem of the sign of \(z_m\), it is necessary to solve the equation for the propagator at an arbitrary normalization point \(p^2 = \mu ^2\) and then to continue this solution in the point \(\mu ^2 = -m^2\). The normalization of the renormalized propagator \(\Delta (p^2) \) at point \(\mu ^2\) leads to the renormalized Eq. (8) where the renormalized mass operator \(\Sigma _r\) will be

This reasoning, of course, is not evidence, but rather a suggestive consideration, because it implies a positivity of the propagator in the pseudo-Euclidean region up to the point \(p^2=-m^2\). The detailed study of the equation for the propagator at an arbitrary normalization point, involving the use of numerical methods, is beyond the scope of this paper. We only note that the positivity of the propagator in the Euclidean region can be regarded as a physical principle, and solutions that violate this principle should be rejected. The change of the sign in the pseudo-Euclidean region in the interval from zero to \(m^2\) would mean the singularity of the propagator at some point \(m_0^2<m^2\), which can be interpreted as dynamical generation of a new light particle – a fact in itself is quite interesting. In any case, this point requires further study.

*m*is a renormalized mass that does not coincide with the physical mass of the phion \(m_\phi \), since we use zero-momentum normalization. To determine the physical mass of a phion, it is necessary go to the pseudo-Euclidean Minkowski space and determine the position of poles of the propagator, i.e. zeros of the inverse propagator. The inverse propagator in Minkowski space for the strong coupling region is

^{2}

In the context of the linearized equation considered here it is not clear whether this shell structure an immanent property of the model or is an artifact of linearization. An argument in favor of the first statement is the study of the vertex function carried out in the next section.

In concluding this section, we note that the main effect associated with the non-linearity of the renormalized equation for the propagator is a sharp change in the asymptotic behavior in the deep Euclidean region at the critical point and in the super-critical strong-coupling region. This phenomenon, of course, does not depend on the linearization procedure, which is used to obtain analytic expressions suitable in the whole range of momenta.

## 4 Vertex at zero transfer momentum

In this section, we consider the equation for the vertex function (11) with zero transfer momentum \(k = 0\). Our goal will be to prove the finiteness of the charge renormalization constant \(z_g\), which is determined from the Eq. (10). It is clear that for this proof, it suffices to prove the convergence of the integral in (10).

*(i) The weak coupling:*\(\lambda <1\). In the weak coupling region approximating the propagator by its asymptotics

*(ii) The critical coupling:* \(\lambda =1\).

*(iii) The strong coupling*\(\lambda >1\). In the strong coupling region the approximation of the propagator by its asymptotics (29) leads to equation for \(\Gamma \), which can be reduced to Eq. (31), and the solution of Eq. (37) with propagator \(\Delta _s\) has the form

*x*-space we obtain from (41):

## 5 Discussion

The equations of the leading order of 1 / *N*-expansion in the scalar vector-matrix model have self-consistent positive solutions in the Euclidean region not only in the weak-coupling region, (where a dominance of the perturbation theory in this model is obvious), but also in the strong-coupling region.

At \(\lambda =1\) the asymptotic behavior of the propagator (1 / *p*) is a medium among the free behavior \(1/p^2\) at \(\lambda <1\) and the constant-type behavior in strong-coupling region \(\lambda >1\). The phion propagator in the strong-coupling region asymptotically approaches to a constant. It is not something unexpected, if we remember the well-known conception of the static ultra-local approximation, or “static ultralocal model” (see [15] and references therein). In this approximation, all the Green functions are combinations of \(\delta \)-functions in the coordinate space that are constants in momentum space. Of course, this approximation is physically trivial. In contrast to the ultra-local approximation, our solution of linearized approximation has the standard pole behavior for the small momenta.

*Ze*is the nuclear charge) has some of specific features. The Dirac equation with a potential corresponding to a point charge

*Ze*is not correct for \(Z>137\): here the ”fall on the center“ known from quantum mechanics occurs (see, for example, [20]). This fall on the center is related with the term \(1/r^2\) in a potential of the relativistic Coulomb problem and is the main reason for this re-arrangement of the vacuum. The potential

*U*, which corresponds to the propagator, is defined by comparing the Born approximation of the non-relativistic quantum theory with the lower approximation of the relativistic theory (see, e.g., Pauli [21], Bethe and Morrison [22], Gross [23]):

## Footnotes

## Notes

### Acknowledgements

Author is grateful to the participants of IHEP Theory Division Seminar for useful discussion and the anonymous referee for their helpful comments.

## References

- 1.F. Gross, C. Savkli, J. Tjon, Phys. Rev. D
**64**, 076008 (2001)ADSCrossRefGoogle Scholar - 2.C. Savkli, F. Gross, J. Tjon, Phys. Atom. Nucl.
**68**, 842 (2005)ADSCrossRefGoogle Scholar - 3.S. Ahlig, R. Alkofer, Ann. Phys.
**275**, 113 (1999)ADSCrossRefGoogle Scholar - 4.R. Rosenfelder, A.W. Schreiber, Eur. Phys. J. C
**25**, 139 (2002)ADSCrossRefGoogle Scholar - 5.V. Sauli, J. Phys. A Math. Theor.
**36**, 8703 (2003)ADSMathSciNetGoogle Scholar - 6.T. Nieuwenhuis, J.A. Tjon, Few Body Syst.
**21**, 167 (1996)ADSCrossRefGoogle Scholar - 7.V. Sauli, J. Adam Jr., Phys. Rev. D
**67**, 085007 (2003)ADSCrossRefGoogle Scholar - 8.L.M. Abreu, A.P.C. Malbouisson, J.P.C. Malbouisson, E.S. Nery, R. Rodrigues da Silva, Nucl. Phys. B
**881**, 327 (2014)ADSCrossRefGoogle Scholar - 9.V.E. Rochev, J. Phys. A Math. Theor.
**46**, 185401 (2013)ADSMathSciNetCrossRefGoogle Scholar - 10.V.E. Rochev, Phys. Atom. Nucl.
**78**, 443 (2015)ADSCrossRefGoogle Scholar - 11.G. ’t Hooft, Nucl. Phys. B
**75**, 461 (1974)ADSCrossRefGoogle Scholar - 12.A.A. Slavnov, Theor. Math. Phys.
**51**, 307 (1982)Google Scholar - 13.T.V. Lysova, Differ. Equ.
**39**, 857 (2016)Google Scholar - 14.A. Erdelyi,
*Higher Transcendental Functions*, vol. 1, 2 (McGraw-Hill, New York, 1953)zbMATHGoogle Scholar - 15.R.J. Rivers,
*Path Integral Methods in Quantum Field Theory*(Cambridge Univ. Press, Cambridge, 1987)CrossRefGoogle Scholar - 16.C.M. Bender, J. Phys. Conf. Ser.
**631**, 012002 (2015)CrossRefGoogle Scholar - 17.V.S. Popov, Zh. Eksp. Teor. Fiz.
**59**, 965 (1970)Google Scholar - 18.W. Greiner, B. Müller, J. Rafelski,
*Quantum Electrodynamics of Strong Fields*(Springer, Berlin, 1985)CrossRefGoogle Scholar - 19.V.M. Kuleshov, V.D. Mur, N.B. Narozhny, A.M. Fedotov, YuE Lozovik, V.S. Popov, Phys. Uspekhi
**58**, 785 (2015)ADSCrossRefGoogle Scholar - 20.L.D. Landau, E.M. Lifshitz,
*Quantum mechanics*(Pergamon Press, New York, 1989)zbMATHGoogle Scholar - 21.W. Pauli,
*Meson Theory of Nuclear Forces*(Interscience, New York, 1948)zbMATHGoogle Scholar - 22.H. Bethe, P. Morrison,
*Elementary Nuclear Theory*(Wiley, New York, 1956)CrossRefGoogle Scholar - 23.F. Gross,
*Relativistic Quantum Mechanics and Field Theory*(Wiley, New York, 1999)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}