DYNAMICS OF EXCITED INSTANTONS IN THE SYSTEM OF FORCED GURSEY NONLINEAR DIFFERENTIAL EQUATIONS
F. Aydogmus*
Department of Physics, Faculty of Science, Istanbul University 34452, Istanbul, Turkey
Received July 13, 2014
The Gursey model is a 4D conformally invariant pure fermionic model with a nonlinear spinor self-coupled term. Gursey proposed his model as a possible basis for a unitary description of elementary particles following the "Heisenberg dream". In this paper, we consider the system of Gursey nonlinear differential equations (GNDEs) formed by using the Heisenberg ansatz. We use it to understand how the behavior of spinor-type Gursey instantons can be affected by excitations. For this, the regular and chaotic numerical solutions of forced GNDEs are investigated by constructing their Poincare sections in phase space. A hierarchical cluster analysis method for investigating the forced GNDEs is also presented.
DOI: 10.7868/S0044451015020054
1. INTRODUCTION
Solitons wore discovered in the 19th century as nondissipating surface waves 011 water and were later realized to obey nonlinear wave equations [1]. During the past forty years, a rather complete description of solitons has been developed by the productive collaboration of mathematicians and physicists. In mathematical physics, the amount of information 011 nonlinear wave phenomena obtained using solitons is quite high. Today it is known that solitons play an important role in many areas, ranging from condensed matter physics to cosmology.
There are four leading soliton types: instant on, monopole, vortex, and kink ones. Inst ant ons have a finite action with zero energy, and they have been considered as configurations of quantum fields that provide a tunnelling effect between the vacuums that have different topologies in space time. This property of instantons is especially interpreted to overcome the quark confinement problem. Before the instant on solutions were discovered in 1975 by Belavin, Polyakov, Schwarz, and Tyupkin [2] in the Yang Mills theory, this theory of strong interactions appeared to have a symmetry that did not exist in nature; this was known as the axial U(l) problem and was solved by 't Hooft, who realized
E-mail: fatma.aydofimus'ä'gmail.com
that it may even be the most important effect of instant 011 solutions to break the unwanted symmetry [3]. This was the first example of an extended classical solution having a physical consequence in the field theory of particle physics. In recent years, one of the most powerful uses of instantons is in the various topics of both QCD and electroweak theory. Although they play-ail important role in the interface region between par-tonic and hadronic description of strong interactions theoretically, direct experimental evidences for instantons have being lacking until now. However, a careful analysis of Large Hadron Collider (LHC) data at CERN might bring experimental confirmation of such processes [4].
After the success of the Dirac equation in the description of electron dynamics, the first work 011 models including only spinors goes back to Heisenberg [5]. Heisenberg spent his years to formulate a "theory of everything" using only fcrmions. A few decades later, as a possible basis for a unitary description of elementary particles, Gursey proposed a new spinor wave equation that is similar to Heisenberg's nonlinear generalization of the Dirac equation but in addition exhibits ill-variance with respect to conformal transformations [6]. Gursey had to use a nonpolynomial form in order to write a conformally invariant Lagrangian. Gursey s model possesses broader dynamical symmetries compared to Dirac's and Heisenberg et al.'s works. More importantly, Gursey s work is suitable for extensions to other particles with spin [6]. I11 the same year, Ivo-
rtol found sonic classical solutions of Gursey's conformai invariant spinor wave equation via the Hoisonborg ansatz [5,7], which much later wore shown to bo instantons (Gursey instantons) by considering conformai symmetry breaking, which means that (0|^1|0) ^ 0 [8]. The Gursey model is very important because of the similarity of these solutions to solutions of pure Yang Mills theories in four dimensions. As a possible passage to the quantum level, the Poisson bracket structure of this model has also been proposed by the introduction of auxiliary scalar fields and using the Dirac method for constrained systems [9]. In Rof. [10], a Solor-typo soliton solution [11] of the Gursey model with a mass term was given and its phase space behavior was investigated [12].
On the other hand, very recently, the stability behavior of Gursey instantons around their bifurcation points in phase space has been investigated by using the system of Gursey nonlinear differential equations (GNDEs) in a Euclidean configuration with the Hoisonborg ansatz. Moreover, the role of the coupling constant has boon discussed [13,14].
In this paper, we again consider the GNDEs using the Hoisonborg ansatz. Wo use this system to understand how the behavior of Gursey instantons can be affected by excitation. For this, we first look for the stability characterization of Gursey instantons and then investigate the regular and chaotic numerical solutions of forced GNDEs by constructing their Poincaré sections in phase space. We also built the bifurcation diagram of forced GNDEs to find the critical value of the forcing frequency as the control parameter. Besides this, a hierarchical cluster analysis method of investigation is presented to reinforce our conclusions.
2. GURSEY'S CONFORMAL INVARIANT SPINOR WAVE EQUATION AND INSTANTONS
The Gursey wave equation [C] is described by the conformai invariant Lagrangian
L = fî/.f 4< + g(tt )4/3,
(1)
where the fermion field ^ has scale dimension 3/2 and g is a positive dimensionless coupling constant. The conformai invariant spinor wave equation that follows from the above Lagrangian is
hvd^ + ig(M )1/3 0 = 0.
(2)
In Rof. [15], ■ip'tp for spinor-type instant on solutions are also related to spontaneous symmetry breaking of
the full conformal group and tfnp is then characterized by being invariant under the transformations of a special subgroup [16], which in turn reflects the final symmetry properties of the ground state of the system as
R^W) = ~
u
u —x
d,1 + {xd+2d)x,1
where
Rfi —
uP„
(00) = 0, (3)
(4)
and u is a parameter with the dimensions of length, PIÀ is the momentum operator, and Dtl is a conformal scale-invariant operator in the four-dimensional
Euclidean space time. We then find that = ±
g (a2 + x2 )
for a solution related to the special case (instantem) [15] of a Euclidean configuration of the Heisenberg ansatz [5]
t = [i.t:,n,iX(») + (5)
where c is an arbitrary spinor constant and ;\ (.s) and tp(s) are real functions of s = x2 = r2 + t2 (xi = x, ./>2 = y, x3 = z, .1:4 = t) in the Euclidean space time, i. e., r2 = x'l + x'2 + £3. Substituting Eq. (5) in Eq. (2) with
imp = i
■lit = h) - 2s
ds
■ 2Ù-V7,1
dç(s)
ds
cc.
(6)
and
(tt)1/3= (sxisf + çisf)^)1/^ (7) we obtain the system of nonlinear differential equations
+ - .g(Tr)1/3 x
CIS
x [«v(«)2 + v{s)*]1/a <p(s) = 0, (8a) 2d^s± +g(Mi/s [n(s)2 + ^(s)2]V3 v(s) = ^ (gb)
CIS
where we write a. = g(CC)1/3 for brevity. Substituting \ = .4.s_<7F(«) and tp = Bn~TG(u), with u = ln.s and a = r + 1/2, r = 3/4, and .42 = B2 [7], we obtain the dimensionless form of the system of nonlinear ordinary coupled differential equations (8) as
2^ + -F(u)-<*(AB)V3 x du 2
x [F(t
■ G(uf]1/?'G(u) = 0, (9a)
4 >K9T<£>, libiii. 2
241
dG(u)
du
-G(u) + a(AB)1/3 x x [F(m)2 + G(m)2]1/3
F(ti) = 0, (9b)
whoro F and G arc dinionsionloss functions of«, and ,4 and B arc constants [7]. Wo call those equations the Gursey nonlinear differential equations (GNDEs) and the solutions of GNDEs with a(AB)1/3 = 1 are the "Gursey instant oris" given in [8]. It is difficult to obtain these exact solutions directly, and therefore numerical simulations were performed [13]. Moreover, the role of the coupling constant in the evolution of 4D spinor-type Gursey instantons in phase space has boon investigated elsewhere via the Heisenberg ansatz [13,14].
For the stability characterization of Gursey instantons, we find the fixed points of GNDEs as functions of a(AB)1/3. They are
I 8[n'(,4Z?)/3]3/2 ' 8[n<(.4Z?)/3]3/2 J ' eigenvalues belonging to these fixed
(10)
The eigenvalues belonging to these fixed points are
1 QajABf^'FG (F2 + G2 )2/3
80
n2(ABf?3(F2 + G2)2/3
1/2
(ID
Substituting the above fix points gives purely imaginary eigenvalues for all a(AB)1/3 > 0. Hence, the equilibrium points are elliptic in character. An elliptic fixed point has a closed orbit around it [13,14]. As can be seen from Fig. 1 (plotted for a(AB)1/3 = 1), the phase-space dynamics of Gursey instantons has an undamped Duffing-type stability characteristic. This behavior does not depend on the values of the coupling constant [13,141.
3. REGULAR AND CHAOTIC SOLUTIONS OF FORCED GNDEs
The main aim of this paper is to investigate the characteristics of forced GNDEs by reporting the Poincare sections on the dinionsionloss phase space (F(tt), G(u)) and the bifurcation diagram to see how the stable behavior of Gursey instantons can be affected by external forcing.
We redefine forced GNDEs by using a new constant ft = a(AB)1/3 as
tZF(u) du
¡F(u)
■ ft [F(uf + G(uf]1/?' G(u) = 0, (12a)
G
Fig. 1. Undamped Duffing-type stability characterization of Gursey instantons for a(AB)l/i = 1; the equilibrium points are ; ^^^T" ] anc'
(3\/3 3\/3 \ i ' i
dG(-u) du
-G(u)
ft[F(,
G(u)
,1/3
F(u) = ucos(uiu). (12b)
In the forced system, we can consider two main parameters: the amplitude and the frequency of excitation. Hero, a is the amplitude of the external forcing and ui is its frequency. Forced GNDEs can be converted to
tZF(u) du
ilG(u) du
^ft[F(,
G(u)
,1/3
G(«) = 0, (13a)
ft [F(uf+G(uf]1/?' F(u) = ucos[ojH (</
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