Fully nonlinear excitations of non-Abelian plasma
aa r X i v : . [ h e p - ph ] J un Fully nonlinear excitations of non-Abelian plasma
Vishnu M. Bannur
Department of Physics , University of Calicut, Kerala-673 635, India.
January 8, 2019
Abstract
We investigate fully nonlinear, non-Abelian excitations of quark-antiquark plasma, using relativisticfluid theory in cold plasma approximation. There are mainly three important nonlinearities, coming fromvarious sources such as non-Abelian interactions of Yang-Mills (YM) fields, Wong’s color dynamics andplasma nonlinearity, in our model. By neglecting nonlinearities due to plasma and color dynamics we getback the earlier results of Blaizot et. al. [1]. Similarly, by neglecting YM fields nonlinearity and plasmanonlinearity, it reduces to the model of Gupta et. al. [2]. Thus we have the most general non-Abelianmode of quark-gluon plasma (QGP). Further, our model resembles the problem of propagation of laserbeam through relativistic plasma [4] in the absence of all non-Abelian interactions.
PACS Nos :
Keywords :
Quark-gluon plasma, Non-Abelian, Nonlinearity.1
Introduction :
QGP is a quasi-color-neutral gas of quarks and gluons which exhibits collective behaviour. It is expectedto be formed in relativistic heavy ion collisions (RHICs) experiments, deep inside the neutron star andmight have formed in early universe. A study of collective excitations of QGP are important to diagnosevarious parameters and signatures of QGP. It is also proposed that the chaotic collective modes of QGPgives an estimate of thermalization of QGP in RHICs [3]. From the extensive study of electrodynamicsplasma, we know that there exists various linear and nonlinear excitations in it, governed by electrodynamicinteractions which is an Abelian gauge theory. Here, in QGP, also we expect similar linear and nonlinearmodes, but modified by the non-Abelian interactions, which itself is nonlinear. Therefore, in QGP, thereis two types of nonlinear effects, one coming from usual plasma nonlinearity and other, from non-Abelianeffects. Nonlinear solutions of non-Abelian or Yang-Mills (YM) theory is studied extensively by Matinyan et. al. without plasma, but with Higgs order phase [5]. Later, these studies were extended to QGP byBlaizot et. al. , but without plasma nonlinearity and Wong’s color dynamics [6] which is another non-Abelianeffect in QGP. Various periodic, quasi-periodic, chaotic nonlinear modes and transition from order to chaosby plasma collective effects were studied. A study of stabilization of QCD vacuum instability by plasmacollective modes were studied earlier in [7]. There is another group of work in these lines by Gupta et. al. ,but without the nonlinearity of YM fields and plasma nonlinearity. Nonlinear or Non-Abelian effects comefrom the Wong’s color dynamics. Here, we present a fully nonlinear, non-Abelian excitations, including allnonlinearities: plasma nonlinearity, YM fields nonlinearity and color dynamics nonlinearity.
The relativistic fluid set of equations, in cold plasma limit, is given by [8] m du µ dτ = gI a G µνa u ν , (1)the equation of motion, where m the mass, τ the proper time, g the coupling constant, a the color index, u ν G µνa the field tensor, defined as, G µνa = ∂ µ A νa − ∂ ν A µa + gǫ abc A µb A νc , (2)1n terms of 4-vector potentials A µa . I a is the dynamical color charges which obey Wong’s equation, dI a dτ = − gǫ abc u µ A µb I c . (3)The vector potentials are obtained from the Yang-Mills field equation, ∂ µ G µνa + gǫ abc A µb G µνc = J νa , (4)where J νa is the 4-vector color current produced by various species in plasma with color charges, such asquarks, antiquarks and gluons. For simplicity, here in our analysis, we consider quark-antiquark plasma andthe current density is given by J νa = X species ngI a u ν , (5)where n is the density of each species, determined by the continuity equation, ∂ µ ( nu µ ) = 0 . (6)In general, these are very complicated, coupled, nonlinear equations to be solved and hence one goes forapproximations, such as moving frame ansatz [4], space-homogeneous solutions, so on, to look for specialsolutions. Following Blaizot et. al. [1], let us consider the homogeneous solutions of our set of equations,few of them may be easily solved. The continuity equation for each species may be integrated and we get n ( t ) u ( t ) = constant = n u , (7)where n and u are the density and 0-component of fluid velocity at equilibrium. We also chose a gauge A a = 0 and the spatial part of the equation of motion may be easily integrated to get u j = − gI a A ja m , (8)with the assumption that, at equilibrium, the plasma is at rest. The 0-component fluid velocity is given by u = q u j , (9)and hence u = 1. Similarly, the spatial part of the field equations gives¨ A ia + g h ( A jb A jb ) A ia − ( A jb A ja ) A ib i = g X n I a u i u , (10)and temperal component gives, X n I a = ǫ abc A ib ˙ A ic , (11)2here dot means differentiation with respect to time. Finally, the color dynamics equation reduces to˙ I a = gǫ abc u i u A ib I c . (12)For further simplification, let us use hedgehog ansatz where the color directions are taken to be along thespatial direction and redefine variables as X ≡ gI A x m ; Y ≡ gI A y m ; Z ≡ gI A z m ; , (13)and rescaling time and color charges as t → mI t and I a → I a I , (14)where I is introduced to normalize I a I a = 1, which is one of the constant of motion as can be seen from theequation for color dynamics. Further, from Eq. (11), I a of second species (antiquarks) is opposite to thatof first species (quarks) and hence I a = − I a ≡ − I a . In terms of redefined variables, our simplified set ofequations becomes, ¨ X + ( Y + Z ) X = − ǫ I x X p I x X ) + ( I y Y ) + ( I z Z ) , (15)and ˙ I x = − I y I z ( Y − Z ) p I x X ) + ( I y Y ) + ( I z Z ) , (16)and similar equations for y and z components which may be obtained by cyclic change among x , y and z . The parameter ǫ ≡ ω p I m , where the plasma frequency ω p ≡ n g I m . Above set of equations has twoimmediate constant of motions, namely, I a I a = 1 and( ˙ X + ˙ Y + ˙ Z ) / X Y + Y Z + Z X ) / ǫ q I x X ) + ( I y Y ) + ( I z Z ) = E , (17)the energy. These approximate set of equations retains all important aspects of QGP such as YM nonlinearity,plasma nonlinearity and color dynamics nonlinearity. In the earlier calculations of Blaizot et. al. [1], thecolor dynamics is neglected and in [2], YM nonlinearity is dropped out.In order to extract the results of Blaizot et. al. , let us assume that color charge I a are constant and then,Eq. (15) reduces to, ¨ X + ( Y + Z ) X = − ǫ X p X + Y + Z ) / , (18)3here the square root term is the plasma nonlinearity, coming from the relativistic treatment just like in [4].Further, expanding the plasma nonlinearity term up to 3 rd order in vector potential gives,¨ X + (1 − ǫ
18 )( Y + Z ) X − ǫ X + ǫ X = 0 , (19)which is similar to that of Blaizot et. al. , except with few new terms containing a ( − ǫ ). This new termsmay lead to additional new features like chaotic scattering [9]. This model without these new additionalterm was studied extensively by Matinyan et. al. [5] and Blaizot et. al. [1].Let us look for some other new solutions of our model Eq. (18). For example, a special solution with Z = 0 leads to, ¨ X + Y X = − ǫ X p X + Y ) / , (20)for X and a similar equations for Y with X and Y interchanged. It differs from similar work by Matinyan et. al. , Blaizot et. al. because we have kept the plasma nonlinearity also. Our numerical study shows thatthe plasma nonlinearity enhances the chaos and therefore, increases the order to chaos transition parameterdefined in [5], which will be discussed later.Next let us look at another special solution with X = Y = Z of our general equation Eq. (15) and weget ¨ X + 2 X = − ǫ X √ X , (21)which describes a nonlinear oscillation. It is the more general nonlinear oscillation, including the plasmanonlinearity, than the elliptic functions Cn discussed in [5, 1]. It is easy to see that on neglecting the plasmanonlinearity, we get back Cn or Sn , depending on the strength of non-Abelian parameter compared to plasmafrequency. It is interesting to note that above mode is an exact solution of QGP because for X = Y = Z ,the color dynamics equation shows that color charges are constant. The most general set of equations of our model are nonlinear, coupled equations and may not be easy tosolve. So we have made an approximation, known as hedgehog ansatz and reduced the number of equationsto be solved, but having all non-Abelian and nonlinear features. From these simplified set of equations, wemay get the results of all other earlier works in this field. For example, in Fig. 1, we plotted the Poincaresection of our model with the approximation that the dynamical color charges are constant and Z = 0, Eq.418). Figures 1(a) and 1(b) are for the system without plasma nonlinearity and found that the regular orbitsseen in Fig. 1(a) for ǫ = 5 disappears at the critical value of ǫ = 2, Fig. 1(b) and hence chaotic. Similarfigures with plasma nonlinearity shows changes from ordered orbit islands for ǫ = 8 .
15, Fig. 1(c), into chaoticmotion for ǫ = 6. Therefore, the critical value of ǫ for the order-to-chaos transition is higher with plasmanonlinearity. The chaos seen with ǫ = 6 . ǫ ( ǫ = 2) to have chaos. Hence the plasma nonlinearityenhances the chaos which is an additional new feature compared to the results of Blaizot et. al. [1]. Anotherspecial solution of our model with I a = constant is X = Y = Z , Eq. (21), which is not an elliptic functionsas in [1], but little more general nonlinear oscillation.Next, in Fig. 2, we plotted the general solutions of our model, as an example u x (Eq. (8)), with hedgehogansatz for different values of ǫ with the same initial conditions and we see that as the ǫ decreases the systembecomes more and more chaotic, which is, qualitatively, similar to the results of Gupta et. al. . For a large ǫ , say, ǫ = 100 (Fig.1 (a)), the amplitude of oscillations is small and the YM nonlinearity and plasmanonlinearity may be negligible and hence it is just the Abelian oscillations, modulated by color dynamics.As ǫ decreases, amplitude increases and all nonlinearities due to YM fields, color dynamics and plasmanonlinearity come into play and drive the system to chaotic motion as can be seen from Figs. 2(b) to 2(c)with intermittent oscillations. For ǫ = 0, Fig. 2(d), the chaotic oscillations is mainly due to YM nonlinearity.Similar behaviour is also seen in the other components of velocity. We have studied fully nonlinear, non-Abelian excitations of quark-antiquark plasma using relativistic fluidtheory. It exhibits new features like a special nonlinear oscillation, different from elliptic functions, andenhancement of chaos. Further, we have found that on neglecting color dynamics and plasma nonlinearity,we get back the results of Blaizot et. al. [1] and by neglecting YM fields nonlinearity and plasma nonlinearity,we obtain the results of Gupta et. al. [2]. Hence we have the most general nonlinear, non-Abelian modesof QGP. In general, all three nonlinearities are always there in the system. For small amplitude excitations( | X | , | Y | and | Z | are ≪ References [1] J. P. Blaizot and E. Iancu, Phys. Rev. Lett. , 3317 (1994).[2] S. S. Gupta, P. K. Kaw and J. C. Parikh, Phys. Lett. B498 , 223 (2005).[3] B. Muller and A. Trayanov, Phys. Rev. Lett. , 3387 (1992); C. Gong, Phys. Lett. B298 , 257 (1993);V. M. Bannur, Phys. Rev.
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A25 , 3945 (1992).6igure 1: Poincare sections of our model (with Z = 0 and I a = constant) without plasma nonlinearity (Figs.(a) for ǫ = 5 and (b) for ǫ = 2), and with plasma nonlinearity (Figs. (c) for ǫ = 8 .
15 and (d) for ǫ = 6).Figure 2: Exact numerical solutions of our model, as an example u x , for different values of ǫ with the sameinitial conditions (Figs. (a) for ǫ = 100, (b) for ǫ = 20, (c) for ǫ = 2 and (d) for ǫ = 0). Figure Caption:7