stability of classical chromodynamic fields
TRANSCRIPT
Stability of Classical Chromodynamic Fields
Sylwia Bazak1a and Stanis law Mrowczynski1,2b1Institute of Physics, Jan Kochanowski University, ul. Uniwersytecka 7, PL-25-406 Kielce, Poland
2National Centre for Nuclear Research, ul. Pasteura 7, PL-02-093 Warsaw, Poland(Dated: February 11, 2022)
A system of gluon fields generated at the earliest phase of relativistic heavy-ion collisions can bedescribed in terms of classical fields. Numerical simulations show that the system is unstable buta character of the instability is not well understood. With the intention to systematically studythe problem, we analyze a stability of classical chromomagnetic and chromoelectric fields whichare constant and uniform. We consider the Abelian configurations discussed in the past wherethe fields are due to the single-color potentials linearly depending on coordinates. However, wemostly focus on the nonAbelian configurations where the fields are generated by the multi-colornon-commuting constant uniform potentials. We derive a complete spectrum of small fluctuationsaround the background fields which obey the linearized Yang-Mills equations. The spectra of Abelianand nonAbelian configurations are similar but different and they both include unstable modes. Webriefly discuss the relevance of our results for fields which are uniform only in a limited spatialdomain.
I. INTRODUCTION
Soon after the discovery of asymptotic freedom [1, 2], when quantum chromodynamics was recognized as an un-derlying theory of strong interactions, the stability of various configurations of classical chromodynamic fields wasinvestigated [3–8]. These studies, which revealed that numerous configurations are actually unstable, were not per-formed with a specific application in mind, rather it was about better understanding the newborn theory.
Today, classical chromodynamics is often used as an approximation of quantum theory. We are interested in theearly phase of relativistic heavy-ion collisions experimentally studied at RHIC and the LHC. Within the Color GlassCondensate (CGC) approach, see e.g. the review articles [9, 10], color charges of valence quarks of the collidingnuclei act as sources of long wavelength chromodynamic fields which can be treated as classical because of their largeoccupation numbers. The system of non-equilibrium gluon fields created in the nuclear collision is called glasma. Atthe earliest moment of the collision, the glasma is dominated by the chormoelectric and chromomagnetic fields parallelto the beam axis and later on transverse fields show up.
It has been found numerically [11, 12], see also [13], that there is an unstable exponentially growing mode in thecourse of glasma’s evolution. The mode was identified as the Weibel instability [14] which is well known in physics ofelectromagnetic plasma. A presence of the chromodynamic Weibel instability in relativistic heavy-ion collisions wasfirst argued in [15] and further on studied in detail, see the review [16].
The Weibel instability occurs when charged particles with anisotropic momentum distribution interact with themagnetic field generated by the particles. As explained in detail in [16], there is an energy transfer from the particlesto the field which causes its exponential growth. In glasma there are no particles but high-frequency modes of classicalfields are often treated as particles [11, 12].
The problem of unstable glasma was studied in the series of papers [17–20] where a particular attention was paidto strong longitudinal chormoelectric and chromomagnetic fields generated at the earliest phase of nuclear collisions.It was suggested [17–20] that the unstable mode found in the numerical simulation [11, 12] is not the Weibel butrather Nielsen-Olesen instability [21] which occurs when spin 1 charged particles circulate in a uniform magnetic field.There was considered [19] a possible role of the vacuum instability due to strong electric field which according to theSchwinger mechanism [22] causes a spontaneous generation of particle-antiparticle pairs from vacuum. A stability ofoscillatory chromomagnetic fields was also studied [23] in the context of glasma and a coexistence of the Nielsen-Oleseninstability with the phenomenon of parametric resonance was found.
We intend to clarify what are the unstable modes of evolving glasma. Since the simulation [11, 12] was preformedin terms of classical fields we study a stability of classical field configurations. We start with the simplest case ofconstant and uniform chromomagnetic and chromoelectric fields. The fields which are truly constant and uniform areobviously an idealization but our results are relevant for fields which are approximately constant and uniform in alimited space-time domain.
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Here we focus on a specific aspect of nonAbelian theory which has not been explored yet. The constant andhomogeneous chromoelectric and chromomagnetic fields can occur due to the potentials which are of single color andas in electrodynamics linearly depend on coordinates. We call such configurations Abelian. However, the fields can bealso generated by the multi-color non-commuting potentials and then we have genuinely nonAbelian configurations.We note that the Abelian and nonAbelian configurations are physically inequivalent as they cannot be related toeach other by a gauge transformation. It was also proved [24] that there are only these two gauge-inequivalentconfigurations which produce the space-time uniform chromodynamic fields.
Stability of the Abelian configurations of constant and uniform chromoelectric and chromomagnetic fields wasstudied in [5, 6] and later on repeatedly analyzed, see e.g. [18–20, 23]. However, the nonAbelian configurations seemto be more relevant for glasma. The point is that the chromoelectric and chromomagnetic fields Ea, Ba from theearliest phase of the collisions are generated along the beam axis z in a nonAbelian manner [25, 26]. Specifically,the Ea, Ba fields occur due to the transverse pure gauge potentials of initial nuclei Ai1a, A
i2a as Ea = −gfabcAi1bAi2c
and Ba = −gfabcεzijAi1bAj2c, where fabc are the structure constants of the SU(Nc) group and εzij is the totally
asymmetric tensor. We are aware of only one paper [7] where the stability of nonAbelian uniform configuration ofchromomagnetic field was briefly discussed. A presence of an unstable mode was indicated but a complete spectrumof modes was not derived.
We perform a comparative study of linear stability of Abelian and nonAbelian configurations of constant andhomogeneous chromomagnetic and chromoelectric fields. We are mostly interested in the nonAbelian configurationsbut for a completeness of our study we repeat with minor refinements the stability analyses of Abelian configurationspresented in [5, 6]. Throughout our whole study we use the background gauge while the axial gauges (different for thechromomagnetic and chromoelectric configurations) were applied in [5, 6]. Using one gauge facilitates comparisons ofvarious cases.
We note that comparative analyses of the Abelian and nonAbelian configurations of uniform chromoelectric andchromomagnetic fields can be found in [24] and [27]. A motion of a classical particle was shown to be significantlydifferent in the two cases [24]. Quantum matter fields of spin 0 and 1/2 also behave differently in the background ofAbelian and nonAbelian chromodynamic fields [24, 27]. However, the self-interaction of nonAbelian fields, which isof our main interest, was not studied in [24] and [27].
Our paper is organized as follows. In Sec. II we present the linearized Yang-Mills equations in the background gaugewhich are subsequently used in stability analyses. In Secs. III and IV we discuss, respectively, Abelian and nonAbelianconfigurations of the constant homogeneous chromomagnetic field. Secs. V and VI are devoted analogously to theconstant homogeneous chromoelectric field. Our study is closed in Sec. VII. After summarizing our considerations,we briefly discuss the relevance of our results for fields which are uniform only in a limited spatial domain. Finally,we outline a perspective for further research.
Throughout the paper the indices i, j = x, y, z and µ, ν = 0, 1, 2, 3 label, respectively, the Cartesian spatial coordi-nates and those of Minkowski space. The signature of the metric tensor is (+,−,−,−). The indices a, b = 1, 2, . . . N2
c−1numerate color components in the adjoint representation of SU(Nc) gauge group. We neglect henceforth the prefix‘chromo’ when referring to chromoelectric or chromomagnetic fields. Since we study chromodynamics only, this shouldnot be confusing.
II. LINEARIZED CLASSICAL CHROMODYNAMICS
The Yang-Mills equations written in the adjoint representation of the SU(Nc) gauge group are
Dabµ F
µνb = jνa , (1)
where Dabµ ≡ ∂µδab − gfabcAcµ, jνa is the color current and the strength tensor is
Fµνa = ∂µAνa − ∂νAµa + gfabcAµbA
νc . (2)
The electric and magnetic fields are given as
Eia = F i0, Bia =1
2εijkF kja , (3)
where εijk is the Levi-Civita fully antisymmetric tensor.We assume that the potential Aµa solves the Yang-Mills equation (1) and we consider small fluctuations aµa around
Aµa . So, we define the potential
Aµa(t, r) ≡ Aµa(t, r) + aµa(t, r), (4)
3
such that |A(t, r)| � |a(t, r)|.Assuming that the background potential Aµa satisfies the Lorentz gauge condition ∂µA
µa = 0 while the fluctuation
potential aµa that of the background gauge
Dabµ a
µb = 0, (5)
where Dabµ ≡ ∂µδab − gfabcAcµ, the Yang-Mills equation linearized in aµa can be written as[
gµν(DρDρ)ac + 2gfabcFµνb
]acν = 0. (6)
The background gauge appears particularly convenient for our purposes because different color and space-timecomponents of aaµ are mixed only through the tensor Fµνb which enters Eq. (6). In case of other gauges, e.g. theLorentz gauge ∂µa
µa = 0, the mixing is more severe.
Throughout our analysis, which includes Abelian and nonAbelian configurations of magnetic and electric fields,we use the background gauge which facilitates comparisons of various cases. However, our further considerations arelimited to the SU(2) gauge group when fabc = εabc with a, b = 1, 2, 3.
III. ABELIAN CONFIGURATION OF MAGNETIC FIELD
The constant homogeneous magnetic field along the axis x occurs for a potential Aia which is known from theAbelian theory. Specifically, Aa(t, r) = δa1(0, 0, yB), where B is a constant and r = (x, y, z). Then, using Eqs. (3),one finds Ea(t, r) = 0 and Ba(t, r) = δa1(B, 0, 0). We also note that the only non-vanishing components of thestrength tensor are F zy1 = −F yz1 = B. The Abelian configuration solves the Yang-Mills equations (1) with vanishingcurrent jµa . The nonAbelian terms disappear because there is only one color component. We also note that the chosenpotential satisfies the Lorentz gauge condition.
When Aµa(t, r) = δa1(0, 0, 0, yB), the equation (6) of aµa becomes
�aµa − 2gByεab1∂zaµb − 2gBεab1(δµyazb − δµza
yb )− g2B2y2εac1εcb1aµb = 0. (7)
One sees that the color component aµ1 decouples from the remaining two color components and it satisfies the freeequation of motion. So, the functions aµ1 represent free waves which we do not consider anymore.
Defining the functions
T± = a02 ± ia03, X± = ax2 ± iax3 ,
Y ± = ay2 ± iay3, Z± = az2 ± iaz3,
(8)
the equation (7) provides (�± 2igBy∂z + g2B2y2
)T± = 0, (9)(
�± 2igBy∂z + g2B2y2)X± = 0, (10)(
�± 2igBy∂z + g2B2y2)Y ± ± 2igBZ± = 0, (11)(
�± 2igBy∂z + g2B2y2)Z± ∓ 2igBY ± = 0. (12)
The equations of T± and X± have the diagonal form. To diagonalize the equations of Y ± and Z± one defines thefunctions
U± ≡ Y + ± iZ+, W± ≡ Y − ± iZ−, (13)
which allow one to change the equations (11) - (12) into(� + 2igBy∂z ± 2gB + g2B2y2
)U± = 0, (14)(
�− 2igBy∂z ∓ 2gB + g2B2y2)W± = 0. (15)
4
We assume that the functions aµa depend on t, x, z as e−i(ωt−kxx−kzz). Since the functions should be real, only theirreal parts are of physical meaning. Now, the equations (9), (10) and (14), (15) read(
− ω2 + k2x + (kz ∓ gBy)2 − d2
dy2
)T±(y) = 0, (16)
(− ω2 + k2x + (kz ∓ gyB)2 − d2
dy2
)X±(y) = 0, (17)
(− ω2 ± 2gB + k2x + (kz − gyB)2 − d2
dy2
)U±(y) = 0, (18)
(− ω2 ∓ 2gB + k2x + (kz + gyB)2 − d2
dy2
)W±(y) = 0. (19)
We note that one obtains exactly the same equations (17), (18) and (19) using the temporal axial gauge a0a = 0 whichwas applied in Refs. [5, 6].
Since the eigenenergy Schrodinger equation of harmonic oscillator can be written as(− 2mE +m2ω2(y − y0)2 − d2
dy2
)ϕ(y) = 0, (20)
where m is the oscillator mass, E its energy and ω is the frequency of the corresponding classical oscillator, oneobserves that Eqs. (16) - (19) coincide with Eq. (20) under the following replacements
ω2 + d− k2x → 2mE , gB → mω, ± kzgB
→ y0, (21)
where d = 0 for Eqs. (16), (17) and d = ∓2gB for Eqs. (18), (19).Since E = ω(n+ 1/2) with n = 0, 1, 2, . . . , the frequency squared ω2 is
ω20 = 2gB
(n+
1
2
)+ k2x, n = 0, 1, 2, . . . (22)
for Eqs. (16), (17) and
ω2± = 2gB
(n+
1
2
)± 2gB + k2x, n = 0, 1, 2, . . . (23)
for Eqs. (18), (19). It should be stressed that although we refer to the Schrodinger equation to find the frequencies(22), (23) the solutions are purely classical - the Planck constant ~ does not show up in the final formulas. Thefrequencies squared (22), (23) are ‘quantized’ because the fluctuation field aµa is assumed to be limited everywhere.This is analogous to the requirement that a wave function, which solves the Schrodinger equation, is normalizable.
One sees that ω20 ≥ 0 and ω2
+ ≥ 0 for any n but ω2− = −gB + k2x for n = 0 and consequently, it is negative for
k2x < gB. Then, there are unstable modes of U− and W+ which grow as eγt with γ ≡√gB − k2x. This is the well-
known Nielsen-Olesen instability [21]. The unstable modes are paired with the overdamped modes which decrease intime as e−γt.
Let us now discuss a character of the solutions of Eqs. (16) - (19). The potentials aµa are assumed to depend ontime as eiωt but to see how a given combination of aµa evolves in time, one should consider only the real parts of aµa .
Modes T± and X±
The modes T± and X± are stable. Assuming that T+ 6= 0 while T− = X± = U± = W± = 0, one finds that T+
represents the wave which rotates in the two-dimensional color space spanned by the colors 2 and 3. The mode T−
is similar but it rotates in the opposite direction than T+. The modes X± behave as T±.
Modes U± and W±
The modes U+ and W− are always stable. When U+ 6= 0 and U− = X± = W± = 0, one finds that U+ representsthe wave which rotates in both two-dimensional 2-3 color and y-z coordinate spaces. There is analogous situationwith the stable modes W−. The modes U− and W+ can be stable or unstable. If the modes are stable, they aresimilar to U+ and W−. In case of unstable and overdamped modes U− and W+, which depend on time as eγt ande−γt, the small field wave does not rotate neither in color nor in coordinate space.
5
IV. NONABELIAN CONFIGURATION OF MAGNETIC FIELD
A nonAbelian configuration of Aia which produces a constant homogeneous magnetic field Ba = δa1(B, 0, 0) can bechosen as
Aµa =
0 0 0 0
0 0 0√B/g
0 0√B/g 0
, (24)
where the Lorentz index µ numerates the columns and the color index a numerates the rows. The potential (24),which obviously satisfies the Lorentz gauge condition, does not solve the Yang-Mills equation (1) with jµa = 0. Insteadone gets 0 0 0 0
0 0 0 g1/2B3/2
0 0 g1/2B3/2 0
= jµa . (25)
Following [7], we assume that the current, which enters the Yang-Mills equation, equals the left-hand side of Eq. (25).Then, the potential (24) solves the Yang-Mills equations (1).
The equation of motion of the small field aµa (6) is found to be
�aµa + 2gA(εa3b∂y + εa2b∂z)aµb − g
2A2(εa2eεe2b + εa3eεe3b)aµb + 2g2A2εa1b(δµyazb − δµzayb ) = 0, (26)
where A ≡√B/g.
Assuming that aµa(t, x, y, z) = e−i(ωt−k·r)aµa , where k = (kx, ky, kz) and r = (x, y, z), Eqs. (26) are changed into thefollowing set of algebraic equations
M tB~at = 0, (27)
MxB~ax = 0, (28)
MyzB
~ayz = 0, (29)
where
M tB = Mx
B =
−ω2 + k2 + 2g2A2 −2igAky 2igAkz
2igAky −ω2 + k2 + g2A2 0
−2igAkz 0 −ω2 + k2 + g2A2
, (30)
MyzB =
−ω2 + k2 + 2g2A2 −2igAky 2igAkz 0 0 0
2igAky −ω2 + k2 + g2A2 0 0 0 −2g2A2
−2igAkz 0 −ω2 + k2 + g2A2 0 2g2A2 0
0 0 0 −ω2 + k2 + 2g2A2 −2igAky 2igAkz
0 0 2g2A2 2igAky −ω2 + k2 + g2A2 0
0 −2g2A2 0 −2igAkz 0 −ω2 + k2 + g2A2
,
(31)
and
~at =
at1
at2
at3
, ~ax =
ax1
ax2
ax3
, ~ayz =
ay1
ay2
ay3
az1
az2
az3
. (32)
Since the equations (27), (28) and (29) are all homogeneous, they have solutions if
detM tB = 0, detMx
B = 0, detMyzB = 0, (33)
which are the dispersion equations.
6
FIG. 1. ω2n(k) as a function of k2 for Θ = π/2. FIG. 2. ω2
n(k) as a function of Θ for k2 = gB.
A. Equations detM tB = 0 and detMx
B = 0
Computing the determinant of the matrix M tB , the dispersion equation detM t
B = 0 becomes
(−ω2 + k2 + gB)(ω4 − ω2(2k2 + 3gB) + k4 + gB(3k2 − 4k2T ) + 2g2B2
)= 0, (34)
where k ≡ |k| and kT ≡√k2y + k2z . The solutions are
ω2±(k) =
1
2
(2k2 + 3gB ±
√g2B2 + 16gBk2T
), ω2
0(k) = gB + k2. (35)
One observes that ω2±(k) and ω2
0(k) are always positive. Consequently the modes ±ω+(k), ±ω−(k) and ±ω0(k) are
real and stable. The solutions of the equation detMxB = 0 are obviously the same as those of detM t
B = 0.
The waves represented by the solutions of the equations detMxB = 0 and detM t
B = 0 rotate not in the two-dimensional color subspace, as the analogous solutions of the Abelian configuration, but in the three-dimensionalspace.
B. Equation detMyzB = 0
Computing the determinant of the matrix MyzB as
detMyzB =
[− 6g6A6 + (k2 − ω2)3 + g4A4(k2 − 4k2T − ω2) + 4g2A2(k2 − ω2)(k2 − k2T − ω2)
]2, (36)
one finds that the dispersion equation detMyzB = 0 is cubic in x ≡ ω2 and it reads
x3 + a2x2 + a1x+ a0 = 0, (37)
with
a2 ≡ −4g2A2 − 3k2, (38)
a1 ≡ g4A4 + 8g2A2k2 − 4g2A2k2T + 3k4, (39)
a0 ≡ 6g6A6 − g4A4k2 + 4g4A4k2T − 4g2A2k2(k2 − k2T )− k6. (40)
We note that because of the square in the determinant (36) each solution of the cubic equation is doubled.As well known, see e.g. [28], all three roots of a cubic equation can be found algebraically. Since the coefficients
a0, a1, a2 are real, the character of the roots depends on a value of the discriminant
∆ = 18 a0a1a2 − 4 a32a0 + a21a22 − 4 a31 − 27 a20. (41)
One distinguishes three cases:
7
2
3 ( )
gB
k
2
gB
k
2 ( )
gB
k
2
gB
k
FIG. 3. Unstable modes in the nonAbelian (left panel) and Abelian (right panel) configurations as a function of k2 and Θ.
• if ∆ > 0, the roots are real and distinct;
• if ∆ = 0, the roots are real and at least two of them coincide;
• if ∆ < 0, one root is real and the remaining two are complex.
The discriminant (41) with the coefficients (38), (39), (40) is computed as
∆
16g3B3= 9g3B3 + 68g2B2k2T + 49gBk4T + 16k6T . (42)
As seen, ∆ > 0 and there are three distinct real solutions of the equation detMyzB = 0.
The real solutions of the cubic equation (37) can be written down in the Viete’s trigonometric form [28]
xn = 2
√−p3
cos
[1
3arccos
(3q
2p
√−3
p
)− 2π(n− 1)
3
]− a2
3, (43)
where n = 1, 2, 3 and
p ≡ 3a1 − a223
, q ≡ 2a32 − 9a2a1 + 27a027
. (44)
These formulas assume that p < 0 and that the argument of the arccosine belongs to [−1, 1]. These conditions areguaranteed as long as ∆ > 0 which is the case under consideration.
We show ω2n(k) with n = 1, 2, 3 as a function of k2 for Θ = π/2 in Fig. 1 and ω2
n(k) as a function of Θ for k2 = gBin Fig. 2. The angle Θ defines the orientation of the wave vector k with respect to the magnetic field along the axisx. Therefore, kT = k sin Θ. One observes that ω2
1(k) and ω22(k) are everywhere positive and the corresponding modes
±ω1(k) and ±ω2(k) are stable. There is a domain shown in the left panel of Fig. 3 where ω23(k) is negative. For
comparison we show in the right panel of Fig. 3 the domain of instability of the Abelian configuration discussed inSec. III. The Abelian mode depends only on kx = k cos Θ. One observes that the domain of instability of the Abelianconfiguration extends to infinity for a wave vector which is perpendicular to the magnetic field. In case of nonAbelianconfiguration, the domain of instability is limited for any orientation of the wave vector. We note that with the pureimaginary unstable modes which exponentially grow in time there are paired overdamped modes which exponentiallydecay in time.
The waves represented by the solutions of the equation detMyzB = 0 rotate in the y-z plane, as the analogous
solutions of the Abelian configuration, but the rotation in the color space is not in the two-dimensional subspace butin the three-dimensional space.
8
V. ABELIAN CONFIGURATION OF ELECTRIC FIELD
The constant homogeneous electric field along the axis x occurs for a potential Aia which is known from the Abeliantheory. Specifically, Aµa(t, r) = δa1(−xE, 0, 0, 0), where E is a constant and r = (x, y, z). Then, using Eqs. (3),one finds Ea(t, r) = δa1(E, 0, 0) and Ba(t, r) = 0. We also note that the only non-vanishing elements of Fµνa areF x01 = −F 0x
1 = E. The chosen potential solves the Yang-Mills equations (1) with vanishing current. The nonAbelianterms disappear because there is only one color component. We also note that the chosen potential satisfies theLorentz gauge condition.
When Aµa(t, r) = δa1(−xE, 0, 0, 0), the equation (6) of aµa becomes
�aµa − 2gExεa1b∂0aµb + 2gEεa1b(δµ0axb + δµxa0b) + g2E2x2εa1dεd1baµb = 0. (45)
One sees that the color component aµ1 decouples from the remaining two color components and it satisfies the freeequation of motion. So, the functions aµ1 represent free waves which we do not consider any more.
Defining the functions
T±(x) = a02(x)± ia03(x), X±(x) = ax2(x)± iax3(x),
Y ±(x) = ay2(x)± iay3(x), Z±(x) = az2(x)± iaz3(x),(46)
Eqs. (45) provides the equations (�∓ 2igEx∂0 − g2E2x2
)T± ± 2igEX± = 0, (47)(
�∓ 2igEx∂0 − g2E2x2)X± ± 2igET± = 0, (48)(
�∓ 2igEx∂0 − g2E2x2)Y ± = 0, (49)(
�∓ 2igEx∂0 − g2E2x2)Z± = 0. (50)
The equations of Y ± and Z± have a diagonal form while the equations of T± and X± are diagonalized using
G± ≡ T+ ±X+, H± ≡ T− ±X−. (51)
Then, Eqs. (47) and (48) provide (�− 2igEx∂0 ± 2igE − g2E2x2
)G± = 0, (52)(
� + 2igEx∂0 ∓ 2igE − g2E2x2)H± = 0. (53)
Assuming that the functions G±, H±, Y ±, Z± depend on t, y, z as e−i(ωt−kyy−kzz) we find(k2y + k2z ± 2igE − (ω + gEx)2 − d2
dx2
)G±(x) = 0, (54)
(k2y + k2z ∓ 2igE − (ω − gEx)2 − d2
dx2
)H±(x) = 0, (55)
(k2y + k2z − (ω ± gEx)2 − d2
dx2
)Y ±(x) = 0, (56)
(k2y + k2z − (ω ∓ gEx)2 − d2
dx2
)Z±(x) = 0. (57)
We note that one obtains exactly the same equations (54), (55) and (56) using the axial gauge aza = 0 which wasapplied in Ref. [5].
Since the eigenenergy Schrodinger equation of inverted harmonic oscillator can be written as(− 2mE −m2ω2(x− x0)2 − d2
dx2
)ϕ(x) = 0, (58)
one sees that Eqs. (54) - (57) coincide with Eq. (58) under the following replacements
k2y + k2z + d → − 2mE , gE → mω, ± ω
gE→ y0, (59)
9
FIG. 4. ω2n(k) as a function of k2 for Θ = 0. FIG. 5. ω2
n(k) as a function of k2 for Θ = 0.05.
where d = 0 for Eqs. (56) and (57) and d = ±2igE for Eqs. (54) and (55). In the latter case we deal with theSchrodinger equation of non-Hermitian Hamiltonian.
As discussed in detail in [29], there are no normalizable solutions of the Schrodinger equation of inverted harmonicoscillator which reflects the fact that the solutions run away either to plus or minus infinite. In this sense theconfiguration of constant electric field is genuinely unstable.
VI. NONABELIAN CONFIGURATION OF ELECTRIC FIELD
A nonAbelian configuration of Aia, which produces a constant homogeneous electric field Ea = δa1(E, 0, 0), can bechosen as
Aµa =
0 0 0 0√E/g 0 0 0
0√E/g 0 0
, (60)
where the Lorentz index µ numerates the columns and the color index a numerates the rows. The potential (60),which obviously satisfies the Lorentz gauge condition, does not solve the Yang-Mills equation (1) with jµa = 0. Insteadone gets 0 0 0 0
g1/2E3/2 0 0 00 −g1/2E3/2 0 0
= jµa . (61)
Following [7], we assume that the current, which enters the Yang-Mills equation, equals the left-hand side of Eq. (61).Then, the potential (60) solves the Yang-Mills equations (1).
The equation of motion of the small field aµa (6) is found to be
�aµa + 2gA(εa2b∂0 + εa3b∂x)aµb + g2A2(εa2eεe2b − εa3eεe3b)aµb + 2g2A2εa1b(δµ0axb + δµxa0b) = 0. (62)
where A ≡√E/g.
Assuming that aµa(t, x, y, z) = e−i(ωt−k·r)aµa , where k = (kx, ky, kz) and r = (x, y, z), Eqs. (62) are changed into thefollowing set of algebraic equations
M txE
~atx = 0, (63)
MyE~ay = 0, (64)
MzE~az = 0, (65)
10
FIG. 6. ω2n(k) as a function of k2 for Θ = π/2. FIG. 7. ω2
n(k) as a function of Θ for k2 = 0.5 gE.
where
M txE =
−ω2 + k2 −2igAkx −2igAω 0 0 0
2igAkx −ω2 + k2 + g2A2 0 0 0 −2g2A2
2igAω 0 −ω2 + k2 − g2A2 0 2g2A2 0
0 0 0 −ω2 + k2 −2igAkx −2igAω
0 0 −2g2A2 2igAkx −ω2 + k2 + g2A2 0
0 2g2A2 0 2igAω 0 −ω2 + k2 − g2A2
, (66)
MyE = Mz
E =
−ω2 + k2 −2igAkx −2igAω
2igAkx −ω2 + k2 + g2A2 0
2igAω 0 −ω2 + k2 − g2A2
(67)
and
~atx =
a01
a02
a03
ax1
ax2
ax3
, ~ay =
ay1
ay2
ay3
, ~az =
az1
az2
az3
. (68)
Since the equations (63), (64) and (65) are all homogeneous, they have solutions if
detM txE = 0, detMy
E = 0, detMzE = 0, (69)
which are the dispersion equations.
A. Equations detMyE = 0 and detMz
E = 0
Let us start with the equation detMyE = 0. Computing the determinant of the matrix (67) as
detMyE = −ω6 + (4g2A2 + 3k2)ω4 −
(3g4A4 + 4g2A2(k2 − k2x) + 3k4
)ω2 + k6 − g4A4k2 + 4g4A4k2x − 4g2A2k2k2x, (70)
11
2
3 ( )
gE
k
2
gE
k
2
3 ( )
gE
k
2
gE
k
FIG. 8. ω23(k) (left panel) and negative part of ω2
3(k) (right panel) as functions of Θ and k2.
the dispersion equation detMyE = 0 is again the cubic equation (37) but the coefficients are
a2 ≡ −4g2A2 − 3k2, (71)
a1 ≡ 3g4A4 + 4g2A2(k2 − k2x) + 3k4, (72)
a0 ≡ −k6 + g4A4k2 − 4g4A4k2x + 4g2A2k2k2x. (73)
The discriminant equals
1
4g6A6∆ = 9g6A6 + 4g4A4(13k2 − 7k2x) + 4g2A2(25k4 + 14k2k2x − 119k4x) + 64(k2 + k2x)3, (74)
and it is positive. Consequently, the solutions, which are real, can be written, as previously, in the Viete’s trigonometricform (43).
The solutions are shown in Figs. 4 - 8. The x component of the wave vector k is expressed as kx = k cos Θ and thesolutions are shown as functions of k2 or Θ. The spectrum of modes of the equation detMy
E = 0 is rather complex. InFigs. 4 and 5 one observes the mode coupling of ω2
2(k) and ω23(k). One could think that the curves of ω2
2(k) and ω23(k)
computed for Θ = 0 and shown in Fig. 4 cross each other. However, when the curves are computed for Θ = 0.05 andshown in Fig. 5 one sees that the curves instead only approach each other. For bigger values of Θ the curves ω2
2(k)and ω2
3(k) are well separated. The phenomenon of mode coupling is discussed in detail and explained in §64 of thetextbook [30].
One observes in Figs. 4 - 7 that ω23(k) can be negative. Then, there is a pair of pure imaginary modes, one is
unstable and one is overdamped. We show ω23(k) as a function of k2 and Θ in the left panel of Fig. 8. In the right
panel of Fig. 8 one sees the domain of k2 and Θ where ω23(k) is negative. The solutions of the equation detMz
E = 0
are obviously the same as those of detMyE = 0.
B. Equation detM txE = 0
Let us now discuss the equation detM txE = 0. We start with the simple special k = 0 when the determinant of the
matrix (66) is computed as
detM txE = ω4(ω4 − 4g2A2ω2 + 7g4A4)2. (75)
The solutions of the equation detM txE = 0 are: the double solution ω2 = 0 and double solutions
ω2± = (2± i
√3 )g2A2, (76)
12
3 34g E
2
gE
k
FIG. 9. ∆4g3E3 as a function of Θ and k2.
3 34g E
2
gE
k
FIG. 10. Negative part of ∆4g3E3 as a function of Θ and k2.
which give the mode frequencies
ω(+,±) = ±71/4gA(
cos(φ/2) + i sin(φ/2)), ω(−,±) = ±71/4gA
(cos(φ/2)− i sin(φ/2)
), (77)
where
φ = arctg(√3
2
). (78)
One observes that the modes ω(+,+) and ω(−,−) are unstable as their imaginary parts are positive.The general case of k 6= 0 is much more complicated than the k = 0 case, but there is an important simplification.
The determinant of the matrix (66), which is the polynomial of ω2 of order 6, appears to be the square of thepolynomial of order 3 that is
detM txE (79)
=[− ω6 + (4g2A2 + 3k2)ω4 − (7g4A4 + 4g2A2(k2 − k2x) + 3k4)ω2 + 3g4A4k2 + 4g4A4k2x − 4g2A2k2k2x + k6
]2.
So, we have again the cubic equation (37) in ω2 with the coefficients
a2 ≡ −4g2A2 − 3k2, (80)
a1 ≡ 7g4A4 + 4g2A2(k2 − k2x) + 3k4, (81)
a0 ≡ −3g4A4k2 − 4g4A4k2x + 4g2A2k2k2x − k6. (82)
We note that because of the square in the determinant (79) each solution of the cubic equation is doubled.The discriminant ∆ with the coefficients (80), (81) and (82) equals
∆
4g6A6= −147g6A6 + 4g4A4(29k2 + 153k2x)− 4g2A2(23k4 + 82k2k2x + 167k4x) + 64(k2 + k2x)3. (83)
It can be either positive or negative as shown in Figs. 9 and 10.When ∆ > 0 the solutions of the cubic equation are real and can be expressed in the Viete’s trigonometric form
(43). When ∆ < 0 there are one real and two complex solutions which can be found using the Cardano formula [28].The solutions are
x1 = −1
2(u+ v) +
i√
3
2(u− v)− 1
3a2, (84)
x2 = −1
2(u+ v)− i
√3
2(u− v)− 1
3a2, (85)
x3 = u+ v − 1
3a2, (86)
13
FIG. 11. ω2n(k) as a function of k2 for Θ = 0.05. FIG. 12. ω2
n(k) as a function of k2 for Θ = π/4.
where
u ≡3
√−q
2+
√q2
4+p3
27, v ≡
3
√−q
2−√q2
4+p3
27. (87)
The solutions in both regions ∆ < 0 and ∆ > 0 are shown in Figs. 11-14. Figs. 11-13 present the real solutions fromthe domain of ∆ > 0 combined with the real parts of the complex solutions from ∆ < 0. The solutions are shownas functions of k2 for three values of Θ = 0.05, π/4, π/2. In Fig. 14 we show the imaginary parts of the complex
solutions from the domain ∆ < 0. The whole spectrum of the solutions of the equation detM txE = 0 is far not simple.
We first note that the real parts of ω21(k) and ω2
2(k) are equal to each other (Reω21(k) = Reω2
2(k)) in the region∆ < 0 while the imaginary parts shown in Fig. 14 are of the opposite sign (Imω2
1(k) = −Imω22(k)).
One sees in Figs. 11-13 that the real part of low momentum solution ω21(k) or ω2
2(k) from the region ∆ < 0 bifurcatesinto two real solutions ω2
1(k) and ω22(k) from the region ∆ > 0. The bifurcation occurs at ∆ = 0.
The solution ω23(k) is real in both regions ∆ < 0 and ∆ > 0 and it is smooth at ∆ = 0, as seen in Figs. 11-13.
Fig. 11 shows that, as in case of the equation detMyE = 0 discussed in Sec. VI A, the solution ω2
2(k) is coupled toω23(k).There are two unstable modes which occur as square roots of ω2
1(k) and ω22(k) from the region ∆ < 0. These are
the modes with positive imaginary parts which at k = 0 are equal to ω(+,+) and ω(−,−) from Eq. (77). Fig. 14 showsthat the imaginary parts are maximal at k = 0 and equal to Imω(+,+) = Imω(−,−). So, the fastest modes grow as eγt
with γ = Imω(+,+) ≈ 0.57√gE.
VII. SUMMARY, DISCUSSION AND OUTLOOK
Linear stability of classical magnetic and electric fields, which are constant and uniform, have been studied. Wehave considered the Abelian configurations where the fields are generated by a single-color potential linearly dependingon coordinates and the nonAbelian configurations where the fields occur due to constant non-commuting potentialsof different colors. While the potentials of Abelian configurations solve the sourceless Yang-Mills equations thenonAbelian configurations require non-vanishing currents to satisfy the equations. All four cases have been analyzedusing the same background gauge condition. The Abelian and nonAbelian configurations are physically inequivalentand indeed the spectra of eigenmodes of small fluctuations around the background fields are different though similar.The spectra include unstable modes in all cases.
One asks how our analysis changes when the fields are uniform not in the infinite coordinate space but only in alimited domain of a size L? Assuming that the domain is a cube centered at r = 0 and demanding that the realpotentials aµa vanish at the edge of the cube, the wave vectors kx, ky, kz should be replaced as
(kx, ky, kz) −→ (2lx + 1, 2ly + 1, 2lz + 1)π
L, (88)
14
FIG. 13. ω2n(k) as a function of k2 for Θ = π/2. FIG. 14. Imaginary parts of ω2
1(k) and ω22(k) as functions of
k2 for Θ = 0, π/4, π/2.
where lx, ly, lz are integer numbers. Consequently, a spectrum of eigenmodes becomes discrete and some unstablemodes can disappear. As an illustrative example we consider the unstable mode of the Abelian configuration ofmagnetic field which is ω2 = −gB+k2x. After the replacement (88) it becomes ω2 = −gB+π2(2lx+1)2/L2. One seesthat ω2 < 0 at least for lx = 0 if gBL2 > π2. So, the instability exists if the field is sufficiently strong and uniformover a sufficiently big domain. Since all unstable modes we found occur at small wave vectors there are analogousconditions for existence of instabilities of the fields which are uniform only in a limited spatial domain.
We intend to extend our analysis to the case of parallel electric and magnetic fields which are both present atthe same time. Such a situation is expected at the earliest phase of relativistic heavy-ion collisions described in theframework of Color Glass Condensate. To make our considerations more relevant for relativistic heavy-ion collisionswe plan to analyze stability not of the fields which are constant and uniform but rather the fields which are invariantunder Lorentz boosts along the collision axis.
ACKNOWLEDGMENTS
This work was partially supported by the National Science Centre, Poland under grant 2018/29/B/ST2/00646.
[1] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).[2] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).[3] J. E. Mandula, Phys. Lett. B 67, 175 (1977).[4] J. E. Mandula, Phys. Rev. D 14, 3497 (1976).[5] S. J. Chang and N. Weiss, Phys. Rev. D 20, 869 (1979).[6] P. Sikivie, Phys. Rev. D 20, 877 (1979).[7] T. N. Tudron, Phys. Rev. D 22, 2566 (1980).[8] G. Passarino, Phys. Lett. B 176, 135 (1986).[9] E. Iancu and R. Venugopalan, in Quark–Gluon Plasma 3, eds. R.C. Hwa and X.-N. Wang (World-Scientific, Singapore,
2004).[10] F. Gelis, Int. J. Mod. Phys. A 28, 1330001 (2013).[11] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96, 062302 (2006).[12] P. Romatschke and R. Venugopalan, Phys. Rev. D 74, 045011 (2006).[13] K. Fukushima, Phys. Rev. C 76, 021902 (2007).[14] E.S. Weibel, Phys. Rev. Lett. 2, 83 (1959).[15] St. Mrowczynski, Phys. Lett. B 314, 118 (1993).[16] St. Mrowczynski, B. Schenke and M. Strickland, Phys. Rept. 682, 1 (2017).[17] A. Iwazaki, Phys. Rev. C 77, 034907 (2008).
15
[18] A. Iwazaki, Prog. Theor. Phys. 121, 809 (2009).[19] H. Fujii and K. Itakura, Nucl. Phys. A 809, 88 (2008).[20] H. Fujii, K. Itakura and A. Iwazaki, Nucl. Phys. A 828, 178 (2009).[21] N. K. Nielsen and P. Olesen, Nucl. Phys. B 144, 376 (1978).[22] J. S. Schwinger, Phys. Rev. 82, 664 (1951).[23] J. Berges, S. Scheffler, S. Schlichting and D. Sexty, Phys. Rev. D 85, 034507 (2012).[24] L. S. Brown and W. I. Weisberger, Nucl. Phys. B 157, 285 (1979) [erratum: Nucl. Phys. B 172, 544 (1980)].[25] G. Chen, R. J. Fries, J. I. Kapusta and Y. Li, Phys. Rev. C 92, 064912 (2015).[26] T. Lappi and L. McLerran, Nucl. Phys. A 772, 200 (2006).[27] M. H. Friedman, Y. Srivastava and A. Widom, J. Phys. G 23, 1061 (1997).[28] I. N. Bronshtein and K. A. Semendyayev, Handbook of Mathematics (Van Nostrand Reinhold, New York, 1985).[29] G. Barton, Annals Phys. 166, 322 (1986).[30] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).