The non-linear Glasma

0.76 MB·English

by  J. Berges

The non-linear Glasma
Ju¨rgen Berges1,2, S¨oren Schlichting1,3
1Institut fu¨r Theoretische Physik, Universita¨t Heidelberg
Philosophenweg 16, 69120 Heidelberg
2ExtreMe Matter Institute EMMI,
GSI Helmholtzzentrum
Planckstr. 1, 64291 Darmstadt
3Theoriezentrum, Institut fu¨r Kernphysik
Technische Universita¨t Darmstadt
Schlossgartenstr. 9, 64289 Darmstadt
We study the evolution of quantum fluctuations in the Glasma created immediately after the
collision of heavy nuclei. It is shown how the presence of instabilities leads to an enhancement
of non-linear interactions among initially small fluctuations. The non-linear dynamics leads to
2
an enhanced growth of fluctuations in a large momentum region exceeding by far the originally
1
unstable band. We investigate the dependence on the coupling constant at weak coupling using
0
classical statistical lattice simulations for SU(2) gauge theory and show how these non-linearities
2
canbeanalyticallyunderstoodwithintheframeworkoftwo-particleirreducible(2PI)effectiveaction

p techniques. Thedependenceonthecouplingconstantisonlylogarithmicinaccordancewithanalytic
e expectations. Concerningtheisotropizationofbulkquantities, ourresultsindicatethatthesystem
S
exhibitsanorder-oneanisotropyonparametricallylargetimescales. Despitethisfact,wefindthat

4 gauge invariant pressure correlation functions seem to exhibit a power law behavior characteristic
for wave turbulence.

]
h
p I. INTRODUCTION fluctuations in the color glass condensate description
- of high energy heavy ion collisions. We work at weak
p
coupling, where the presence of plasma instabilities has
e
h The great interest in relativistic heavy ion collision been established in previous works [16–18], and present
[ experiments is to a large part driven by its possibility results from classical-statistical lattice simulations along

to explore the properties of deconfined strongly inter- with analytic estimates. We investigate in detail the
1
v acting matter described by quantum chromodynamics different dynamical stages of the system undergoing an
7 (QCD). The past decades have revealed remarkable instability and find that in addition to the ’primary’
1 properties of the quark-gluon plasma, probably most Weibel type instability [16–18], ’secondary’ instabilities
8 strikingly its behavior similar to an ideal fluid [1, 2]. emergeduetonon-linearinteractionsofunstablemodes.
0 However these properties are not directly accessible This mechanism is very similar to previous observation
.
9 experimentally as they are encoded in the final particle in non-expanding gauge-theories [19, 20] and cosmo-
0 spectra measured by the detectors at RHIC and the logical models [34] and can be naturally understood in
2 LHC [3–7]. Consequently the extraction of medium the framework of two-particle irreducible (2PI) effective
1
properties crucially depends on theoretical input, such action techniques [35].
:
v as the time when the plasma thermalizes locally, which
i has to be calculated within an ab-initio approach. This paper is organized as follows: In Sec. II we
X
The non-equilibrium dynamics of high energy nuclear present a short review of the dynamics of nuclear col-
r
a collision poses a challenging problem in the underlying lisions in the color glass condensate (CGC) framework
theory of QCD, which in practice can only be addressed [8]. We comment in particular on recent developments
with suitable approximations. In this context a field to include quantum fluctuations within an ab-initio
theoretical framework known as the ’color glass con- approach [9, 10] and show how the discussion in the
densate’ has been developed, which provides a real literature is related to a classical-statistical treatment.
time ab-initio description of nuclear collisions at high In Sec. IV we present results from classical-statistical
energies [8–10]. While several properties of the initial lattice simulations. We focus on non-linear effects and
state right after the collision can be explored within obtain the relevant growth rates and set-in times of
this approach [11–15], present studies have not yet been primary and secondary instabilities. We find that our
able to explain the thermalization mechanism [16, 17]. results are rather insensitive to the value of the strong
Here quantum fluctuations may play an important role coupling constant α as long as α (cid:28) 1. We also
s s
as they break the longitudinal boost invariance of the investigate the impact of instabilities on bulk properties
system and can be strongly amplified in the presence of of the system such as the ratio of longitudinal pressure
plasma instabilities [16–33]. to energy density. Here our results indicate that the
system remains anisotropic on parametrically large time
In this paper we investigate the impact of quantum
2
scales. We summarize our results and conclude with
Sec. V.








II. NON-EQUILIBRIUM DYNAMICS OF
NUCLEAR COLLISIONS 
A. High-energy limit and CGC
From a non-equilibrium point of view an ab-initio ap-
proach to heavy-ion collisions requires to determine the
initial density matrix consisting of two incoming nuclei
in the vacuum and subsequently solving the initial value
problem in quantum chromodynamics. Though this is
beyond the scope of present theoretical methods, one
may apply suitable approximations in the high energy FIG. 1. (color online) Cartoon of the real time evolution of
and weak coupling limit, which make the problem com- a high energy heavy-ion collision.
putationally feasible. This is usually discussed in terms
√
of the light-cone coordinates x± = (t±z)/
2, where at

sufficiently high energies the incoming nuclei travel close and slow degrees of freedom is performed by a renor-
to the light-cone, which is given by x± = 0. The col- malization group procedure prescribed by the JIMWLK
lision takes place around the time when x+ = x− = 0, equations[8]. Inthehighenergylimittheeikonalcurrent
where the center of mass of the nuclei coincides and an Jµ(x) is given by static color sources on the light-cone
a
approximatelyboostinvariantplasmaisformedafterthe and takes the form
collision. Theplasmadynamicsintheforwardlight-cone
(x± > 0) is usually discussed in terms of the co-moving Jaµ(t,x⊥,z)=δµ+(cid:37)(a1)(x⊥)δ(x−)+δµ−(cid:37)(a2)(x⊥)δ(x+),
coordinates (4)
(cid:112)
τ = t2−z2 , η =atanh(z/t), (1) where δµ± is the Kronecker delta in light-cone coordi-
where τ is the proper time in the longitudi- nates and we denote transverse coordinates as x⊥ =
nal direction and η is the longitudinal rapidity. (x1,x2). The color charge densities (cid:37)(a1/2)(x⊥), where
The metric in these coordinates takes the form the superscript (1/2) labels the different nuclei, contain
g (x) = diag(1,−1,−1,−τ2) and we denote the metric all further information about the beam energy, nuclear
µν
determinant as g(x) = det g (x). The dynamics of species and impact parameter dependence. At high col-
µν
the collision and the geometry of the coordinates is lider energies, these have been conjectured to exhibit a
illustrated in Fig. 1. The different colors in the forward universal behavior, which is described by the saturation
light-cone illustrate the dynamics of the longitudinally scale Qs and the value of the strong coupling constant
expanding plasma, which we study in this paper. αs [36, 37]. In this work we simply adapt the McLerran-
Venugopalan(MV)saturationmodel[38],wherethecolor
In the color-glass framework one considers the charge densities of the nuclei are given by uncorrelated
dynamics of the plasma at mid-rapidity (η (cid:28) η ) Gaussian configurations
Beam
and the nuclear partons at high rapidities separately.
In the eikonal approximation the trajectories of the (cid:104)(cid:37)(A)(x )(cid:37)(B)(y )(cid:105)=g2µ2 δAB δ δ(x −y ). (5)
a ⊥ b ⊥ ab ⊥ ⊥
incoming nuclei are unaffected by the collision, while the
dynamics of gluons at mid-rapdity is described by the The model parameter g2µ is proportional to the physi-
classical action calsaturationscaleQs uptologarithmiccorrectionsand
1(cid:90) reflects the properties of the saturated wavefunctions of
S[A]=− Fa (x) gµα(x)gνβ(x) Fa (x), (2) large nuclei [38]. Consequently the current in Eq. (4)
4 µν αβ
x is parametrically large, i.e. formally O(1/g) in powers
where (cid:82) = (cid:82) d4x(cid:112)−g(x) and Fa (x) denotes the non- of the coupling constant. This makes the problem in-
x µν herently non-perturbative and we will come back to this
abelian field strength tensor
aspect when we discuss the impact of quantum fluctua-
Fµaν(x)=∂µAνa(x)−∂νAµa(x)+gfabcAbµ(x)Acν(x).(3) tions. In addition to Eq. (5), we impose a color neutral-
ity constraint on the color charge densities such that the
withLorentzindicesµ,ν andcolorindicesa=1...N2−1
c global color charge vanishes separately for each nucleus,
for SU(N ) gauge theories. In addition, the gauge field
c i.e.
Aa(x) is coupled to an eikonal current Jµ(x), which is
µ a
determined by the properties of the nuclear wavefunc- (cid:90)
d2x (cid:37)(1/2)(x )=0 ∀ a. (6)
tionathighrapidities. Inpractice, theseparationinfast ⊥ a ⊥
3
By specifying the eikonal current according to Eq. (4), thisclassicalevolutionandleadstoaneffectively2+1di-
the longitudinal geometry of the collision has effectively mensionalYangMillstheorycoupledtoanadjointscalar
been reduced to the collision of two-dimensional sheets field [11–14]. In order to study the full 3+1 dimensional
andthereisnolongeralongitudinalscaleinherenttothe Yang Mills theory it is therefore crucial to include quan-
problem. However quantum fluctuations explicitly break tumfluctuations,whichbreaktheboost-invarianceofthe
the longitudinal boost invariance of the system and may system explicitly.
therefore play an important role in the non-equilibrium
dynamics right after the collision [10]. Before we turn
to a detailed discussion of quantum fluctuations, we will C. Quantum fluctuations
briefly review the classical solution to the particle pro-
duction process. We will show later, in Sec. IIC, how Theinclusionofquantumfluctuationshasrecentlyat-
this solution emerges also in the weak-coupling limit of tractedgreatattentionduetotheirexpectedimportance
the quantum field theory, where quantum fluctuations inunderstandingthethermalizationprocess[9,10]. This
can be handled properly. concerns in particular the inclusion of vacuum fluctua-
tions in the initial state, which are quantum in origin
but evolve classically at sufficiently weak coupling and
B. Classical solution short enough times [43]. In contrast to most discussions
in the literature [9, 10], our analysis is based on the
Neglecting quantum fluctuations for the moment, the two particle irreducible (2PI) effective action framework
absence of a longitudinal scale in Eq. (4) leads to boost- [35]. This formalism has been successfully applied to a
invariant solutions of the classical Yang-Mills field equa- variety of similar problems in scalar theories [34, 44] and
tions gauge theories [19, 20, 45] and recently been developed
δS[A] for the problem under consideration here [46]. We give
δAa(x) =−Jaµ(x). (7) a short general introduction to the formalism and show
µ first how the realization proposed in Ref. [10] emerges in
In the classical color glass picture, the strong color-fields this framework. Then we go beyond that discussion and
right after the collision are entirely determined by the identify sub-leading quantum corrections and describe
continuity conditions on the light-cone (x± =0) [39–42]. the non-linear dynamics of instabilities analytically.
Adapting the Fock-Schwinger gauge condition (A =0),
τ
where the classical Yang-Mills action takes the form The quantum evolution equations can be formu-
S[A]=(cid:90) τdτ dη d2x (cid:20) 1 (∂ Aa)2+ 1(∂ Aa)2 lfiaetleddopinertaetromrssAˆoaf(xth)edeenxopteecdtabtyion values of the gauge
⊥ 2τ2 τ η 2 τ i µ
1 1 (cid:21) Aa(x)=(cid:104)Aˆa(x)(cid:105), (11)
− FaFa − FaFa , (8) µ µ
2τ2 ηi ηi 4 ij ij
and the time ordered two-point correlation function
(i = 1,2) the initial state right after the collision can
be specified at τ = 0+, where the chromo magnetic and Gaµbν(x,y)=(cid:104)TAˆaµ(x)Aˆbν(y)(cid:105), (12)
electric fields are given by [39–42]
The two independent parts of the propagator Gab(x,y)
µν
A (x )=α(1)(x )+α(2)(x ), A =0, (9) can be expressed in terms of the spectral and statistical
i ⊥ i ⊥ i ⊥ η
two-point correlation functions
E =0, E (x )=ig[α(1)(x ),α(2)(x )].
i η ⊥ i ⊥ i ⊥
i
Here αi(1/2)(x⊥) are pure gauge configurations which de- Gaµbν(x,y)=Fµaνb(x,y)− 2sgn(x0−y0)ρaµbν(x,y)(13)
scribe the Yang-Mills field outside the light-cone. They
which are associated to the commutator and anti-
are related to the nuclear color charge densities by [39–
commutator
42]
(cid:68)(cid:104) (cid:105)(cid:69)
−i ρab(x,y)=i Aˆa(x),Aˆb(y) , (14)
αi(N)(x⊥)= g eigΛ(N)(x⊥)∂ie−igΛ(N)(x⊥) , µν µ ν
1(cid:68)(cid:110) (cid:111)(cid:69)
Fab(x,y)= Aˆa(x),Aˆb(y) −Aa(x)Ab(y).(15)
∂i∂iΛ(N)(x⊥)=(cid:37)(N)(x⊥), (10) µν 2 µ ν µ ν
anddependontransversecoordinatesonly. Therelations Here expectation values are given by the trace over the
(9) and (10) specify the ’Glasma’ initial state at τ =0+ initial vacuum density matrix in the presence of the
right after the collision. The time evolution in the for- eikonalcurrents. Theinitialdensitymatrixisspecifiedin
wardlight-conecanbestudiednumericallybysolvingthe the remote past (t →−∞), where the background field
0
lattice analogue of the classical evolution equations and Aa vanishes and the statistical fluctuations Fab take the
µ µν
has been studied extensively [11–14]. However the lon- vacuum form (see e.g. Ref. [46]). In contrast, the initial
gitudinal boost invariance of the system is preserved in valuesofthespectralfunctionareentirelydeterminedby
4
equations for connected one and two-point correlation
functions follow from the stationarity of the two particle
= +
irreducible (2PI) effective action [47, 48]
Γ [A,G]=S[A]+ itr(cid:2)lnG−1(cid:3)+ itr(cid:2)G−1[A]G(cid:3)
FIG.2. Verticesinnon-abeliangaugetheoryinthepresence 2PI 2 2 0
of background gauge fields.
+Γ [A,G]+const. (17)
2
andformaclosedsetofcoupledevolutionequations. The
theequaltimecommutationrelations,whichintemporal
set of equations is given by the evolution equation of the
gauge (A =0) read
0
macroscopic field
ρab(x,y)(cid:12)(cid:12) =0,
∂x0ρaµµbνν(x,y)(cid:12)(cid:12)xx00==yy00 =−δab(cid:112)−gµgν(x)δ((cid:126)x−(cid:126)y), δδASaµ[A(x]) =−Jaµ(x)− 2itr(cid:20)δδGA−0aµ1([xA)]G(cid:21)− δΓδA2[aµA(,xG)]
∂x0∂y0ρaµbν(x,y)(cid:12)(cid:12)x0=y0 =0, (16) (18)
and are valid at all times.1 The gauge field expectation and the evolution equations for spectral and statistical
values in Eq. (11) correspond to the Glasma background two point correlation functions, which can be written as
fields, while the spectral and statistical two-point func- [35]
tions contain the quantum fluctuations. The evolution
(cid:104) (cid:105) (cid:90) x0
iG−1,µγ[x;A]+Π(0),µγ(x) ρcb(x,y)=− dz Π(ρ),µγ(x,z)ρcb(z,y), (19)
0,ac ac γν ac γν
y0
(cid:104) (cid:105) (cid:90) x0 (cid:90) y0
iG−1,µγ[x;A]+Π(0),µγ(x) Fcb(x,y)=− dz Π(ρ),µγ(x,z)Fcb(z,y)+ dz Π(F),µγ(x,z)ρcb(z,y). (20)
0,ac ac γν ac γν ac γν
−∞ −∞
Here we denote (cid:82)bdz = (cid:82)bdz0(cid:82) ddz(cid:112)−g(z) and hand side of Eq. (18) and the self-energy contributions
a a
iG−1,µν[x;A] denotes the free inverse propagator entering Eqns. (19) and (20) have been calculated to
0,ab three loop order (g6) in Ref. [45] and the corresponding
iG−1,µν[x;A]= γ−1(x) Dac[A] γ(x) gγαgµν Dcb[A] expressions in co-moving (τ,η) coordinates can be found
0,ab γ α
inRef.[46]. Beforeweturntoamoredetaileddiscussion
−γ−1(x) Dac[A] γ(x) gγνgµα Dcb[A]
γ α of the right hand side contributions of Eqns. (18), (19)
−g fabc Fµν(x)[A], (21) and(20), itisinsightfultoconsiderfirsttheleadingpart
c
in a weak coupling expansion. We will see shortly how
(cid:112)
with γ(x) = −g(x) and we introduced the (back- this recovers the classical solution for the background
ground) covariant derivative field, as discussed in Sec. IIB, while initial state vac-
uum fluctuations are already included to leading order
Dab[A]=∂ δab−gfabcAc . (22)
µ µ µ intermsoftheconnectedtwo-pointcorrelationfunctions.
The non-zero spectral and statistical parts of the self-
energy Π(ρ/F)[A,G] on the right hand side and the local In order to isolate the leading contributions one
part Π(0)[G] on the left hand side make the evolution has to take into account the strong external currents
equations non-linear in the fluctuations. In general Jµ ∼ O(1/g), which induce non-perturbatively large
a
they contain contributions from the vertices depicted background fields Aa(x) ∼ O(1/g). In contrast, the
µ
in Fig. 2, where in addition to the classical three gluon statistical fluctuations Fab(x,y) originate from initial
µν
vertex there is a three gluon vertex associated with state vacuum and are therefore initally O(1). The
the presence of a non-vanishing background field. The spectral function ρab(x,y) has to comply with the
µν
explicit expressions for the derivatives on the right equal time commutation relations (16) and is therefore
parametrically O(1) at any time. Considering only the
leading contributions in a weak coupling expansion, the
evolution equation (18) reduces to its classical form (c.f.
1 Note that Eq. (16) is valid also for (τ,η) coordinates, when re-
placing x0 = τ and x3 = η and imposing the Fock-Schwinger
(Aτ =0)gaugecondition.
5
Eq. (7)) Dynamical power counting
Π(F): one loop
δS[A]
=−Jµ(x), (23)
δAa(x) a
µ
and the evolution equations for the spectral and statisti-
g2 F2 , β =1
cal two-point correlation functions at leading order read
Π(F): two loop
iG−1,µγ[x;A] ρcb(x,y)=0, (24)
0,ac γν
iG−1,µγ[x;A] Fcb(x,y)=0, (25)
0,ac γν
where sub-leading contributions are suppressed by at
leastafactorofg2 relativetotheleadingcontribution. It
is important to realize that at this order the evolution of
theGlasmabackgroundfieldsdecouplesfromthatofthe g4 F3(ρ), β =4/3
fluctuations, i.e. there is no back-reaction from the fluc- Π(0) (local)
tuationsonthebackgroundfields. Thereforethedynam-
ics of the background fields remains unchanged and one
recoverstheclassicalfieldsolutionsdiscussedinSec.IIB.
In addition the evolution of vacuum fluctuations of the g2 F , β =2
initial state is taken into account by Eqns. (24) and (25)
tolinearorderinthefluctuations. Tothisorderthequan- TABLE I. Self energy diagrams to two loop order (g4). The
tum field theory is known to agree with the classical sta- value β corresponds to the classification in the dynamical
tisticaltheory[43]andEqns.(24,25)canequivalentlybe power-counting scheme. The one-loop diagram in the top
obtainedbyconsideringthelinearizedclassicalevolution panelyieldsthefirstrelevantcorrection,whilediagramswith
equationsforsmallfluctuations2(seee.g. Ref.[10]). The higher values of β become important at later times.
linear approximation in Eqns. (24) and (25) yields a ma-
jorsimplificationtoEqns.(19)and(20),asonecansolve
Eqns. (24) and (25) independently from the evolution of higherorderself-energycorrections,whichnaturallycure
thebackgroundfield. ThishasbeenexploitedinRef.[10] the divergencies. The associated power counting is dis-
to obtain the spectrum of initial fluctuations right after cussed in Sec. IID, where we identify the diagrammatic
the collision. In turn, the range of validity of the ap- contributions,whichcontainthefirstrelevantcorrections
proximation is limited to the domain where fluctuations andanalyzetheirimpactonthedynamicsoftheinstabil-
remain parametrically small. This is, however, not the ity. A systematic way to include self-energy corrections
case in the forward light-cone (τ >0), where Eqns. (24) toallorderswasoutlinedinRef.[10]andwillbediscussed
and (25) exhibit plasma instabilities associated to expo- in more detail in Sec. III on the classical-statistical ap-
nential growth of statistical fluctuations [16, 17] proximation.
(cid:112) (cid:112)
Fab(τ,τ(cid:48),x ,y ,ν)∝exp[Γ(ν)( g2µτ + g2µτ(cid:48))],
µν T T
(26)
D. Dynamical power counting and non-linear
dynamics
forcharacteristicmodes,whereνistheFouriercoefficient
with respect to relative rapidity according to
Inordertogobeyondafixed-ordercouplingexpansion
Fab(x,y)=(cid:90) dνFab(x,y,ν)eiν(ηx−ηy) . (27) of the 2PI effective action, one has to develop a power
µν 2π µν counting scheme which takes into account not only the
suppression by the small coupling constant but also the
In this regime fluctuations can become parametrically
enhancement due to parametrically large fluctuations in
large and strongly modify the naive power counting.
the presence of non-equilibrium instabilities. This has
Since the approximation underlying Eqns. (24) and (25)
been worked out in detail for scalar theories [34, 44] and
is not energy conserving, one encounters exponential di-
applies in a similar way to gauge theories [19, 20]. In
vergences when stressing Eqns. (24, 25) beyond their
this power counting scheme self-energy corrections are
rangeofvalidity[49]. Inthisregimeitiscrucialtoinclude
classified according to powers of the coupling constant
g as well as powers of the background field Aa(x) and
µ
the statistical fluctuations Fab(x,y). For a generic
µν
self-energy contribution containing powers gnFmAlρk,
2 This can be seen immediately by solving Eq. (25) in terms
of mode functions F(x,y) = δA(x)δA(y), which then in- the integers n,m and l yield the suppression factor from
divdually satisfy the linearized classical evolution equations the coupling constant (n) as well as the enhancement
iG−01[x;A]δA(x)=0. due to a parametrically large background field (l) and
6
parametrically large fluctuations (m). The ’weight’ of This hierarchy of diagrams has important conse-
thespectralfunction(k)remainsparametricallyoforder quences for the dynamics of systems undergoing an
one at all times as encoded in the equal-time commu- instability, where initially small fluctuations grow expo-
tation relations. For parametrically large macroscopic nentially in time. At early times statistical fluctuations
fields Aa(x) ∼ 1/g one expects a sizable self-energy are small and the system is accurately described by
µ
correction once fluctuations have grown as large as the set of equations (23, 24, 25), which give rise to the
Fab(x,y) ∼ 1/g(n−l)/m for characteristic modes. The linear instability regime [16, 17]. At later times, when
µν
hierarchy emerging from a classification of diagrams in statistical fluctuations have grown larger, self-energy
terms of β =(n−l)/m is shown in Tab. I. The one loop corrections become important and alter the dynamics
diagram shown in the upper panel contains two three of the system. In this regime self-energy contributions
gluon vertices, which give rise to a suppression factor g2 with smaller values of β become important at an earlier
((n−l)=2). Ontheotherhandthediagramisenhanced stage as compared to contributions with higher values
by two statistical propagators in the loop (m = 2) and of β, as each diagram requires Fab(x,y) ∼ g−β to yield
µν
we can classify the overall contribution as O(g2F2). a significant contribution. The one-loop diagram in the
Similarly one can analyze the two loop diagrams and top panel of Tab. I is of order O(g2F2) in the dynamical
the tadpole diagram also shown in Tab. I. The tadpole power counting. The contribution of this diagram
diagram contains a suppression factor g2 from the four becomes of O(1) as soon as statistical fluctuations
gluon vertex (n =2, l =0) and one statistical propaga- have grown as large as O(1/g), while all higher order
tor (m = 1), such that the overall contribution can be self-energy corrections are still suppressed by at least a
classifiedasO(g2F). Thetwoloopdiagramsareoforder fractional power of the coupling constant. Accordingly
g4 (n−l = 4) in the coupling constant and contain at thereisaregimewheretheoneloopdiagramdisplayedin
most three statistical propagators (m = 3). The overall thetoppanelofTab.Iyieldstheonlyrelevantcorrection.
contribution is thus O(g4F3) in the dynamical power
counting. The classification in terms of β = (n−l)/m In order to investigate the impact of the one-loop
shows that the only diagram yielding a contribution to correction in more detail, we will in the following
β = 1 is the one-loop diagram in the upper panel of neglect all contributions to the memory integrals in
Tab. I. The leading contribution of higher order loop Eqns. (19, 20) which originate from outside the for-
diagrams can be classified as β = 2L/(L + 1), where ward light cone. This assumption is justified at weak
L ≥ 1 is the number of loops. The tadpole diagram coupling, where fluctuations are sufficiently small until
yields β = 2. Finally there are self-energy contributions the time when they exhibit exponential growth due to
which contain powers of the spectral function instead the presence of the instability. We can then switch to
of statistical propagators. These give rise to even a discussion in (τ,η) coordinates and employ the Fock-
higher values of β ≥ 2 and are therefore suppressed at Schwinger (A = 0) gauge condition in the following.
τ
sufficiently weak coupling. The self-energy contributions from the one-loop diagram
in Tab. I (top panel) then take the form [46]
Π(F),µµ(cid:48)(x,y,ν)=−g2 (cid:90) →−V µνγ (cid:20)Fbb(cid:48)(x,y,ν(cid:48))Fcc(cid:48)(x,y,ν−ν(cid:48))− 1ρbb(cid:48)(x,y,ν(cid:48))ρcc(cid:48) (x,y,ν−ν(cid:48))(cid:21)←V−µ(cid:48)ν(cid:48)γ(cid:48) ,(28)
-o-,aa(cid:48) 2 x,abc νν(cid:48) γγ(cid:48) 4 νν(cid:48) γγ(cid:48) y,a(cid:48)b(cid:48)c(cid:48)
ν(cid:48)
Π(ρ),µµ(cid:48)(x,y,ν)=−g2 (cid:90) →−V µνγ (cid:104)Fbb(cid:48)(x,y,ν(cid:48))ρcc(cid:48) (x,y,ν−ν(cid:48))+ρbb(cid:48)(x,y,ν(cid:48))Fcc(cid:48)(x,y,ν−ν(cid:48))(cid:105)←V−µ(cid:48)ν(cid:48)γ(cid:48) , (29)
-o-,aa(cid:48) 2 x,abc νν(cid:48) γγ(cid:48) νν(cid:48) γγ(cid:48) y,a(cid:48)b(cid:48)c(cid:48)
ν(cid:48)
wherex=(τ ,x )collectivelylabelsthepropertimeand (no summation over b and c), where the derivative oper-
x ⊥
transverse coordinates, ν is the Fourier coefficient with ator ∂˜µ,x =(∂ ,−∂ ,−iτ−2ν) only acts on the propa-
a τx x⊥ x
respect to relative rapidity and Lorentz indices take the gator with color index a, and the contribution from the
values µ=1,2,η. The three gluon vertices in Eqns. (28) background field is given by
and (29) take the form
VA,µνγ =−g[ gµν(x)C Aγ(x) (32)
Vµνγ =V0,µνγ +VA,µνγ , (30) x,abc ab,cd d
x,abc x,abc x,abc + gνγC Aµ(x)+gµγC Aν(x)]
whereinadditiontotheclassicalthreegluonvertexthere bc,ad d ca,bd d
isacontributionfromthebackgroundfield. Theclassical with
three gluon vertex can be written as
(cid:104) C =facefbde+fadefbce. (33)
V0,µνγ =fabc gµν(x)(−∂˜γ,x−2∂˜γ,x) (31) ab,cd
x,abc c b
(cid:105) Inordertoinvestigatetheimpactoftheaboveself-energy
+gνγ(x)(∂˜bµ,x−∂˜cµ,x)+gµγ(x)(2∂˜cν,x+∂˜bν,x) corrections,wefirstnotethattheρρterminEq.(28)isa
7
genuine quantum correction, which is absent in the clas- of statistical fluctuations in the initial state. The prior
(cid:112)
sicalstatisticaltheory[43]. Ontheotherhandstatistical give rise to a constant contribution g2µτ , while
Primary
fluctuationsgrowexponentiallyaccordingtoEq.(26)for the latter enter only logarithmically in this estimate.
unstable modes, such that the dominant contribution in
Eq.(28)arisesfromtheclassical(FF)contributionsand The emergence of secondary instabilities again modifies
one can safely neglect the sub-leading quantum correc- the power-counting, and one has to take into account
tions. Therighthandsideoftheevolutionequations(20) also the contributions which originate from modes
then receive exponentially enhanced contributions from which exhibit secondary instabilities. Also higher order
theself-energies(28)and(29), whichgrowexponentially self-energy corrections become increasingly important as
in (proper) time. This behavior can be verified explic- time proceeds and non-linear amplification can repeat
itly by performing a ’memory expansion’ of Eq. (20), i.e. itself, until at some point the growth of instabilities
evaluating the memory integrals on the right hand side saturates and occupancies become as large as O(1/g2).
of Eq. (20) around the latest (proper) times of interest In this regime every truncation at a fixed loop order
[34, 44]. We find that the dominant contribution origi- breaks down and the problem has to be addressed in a
natesfromthestatisticalself-energyinEq.(28),whereas fully non-perturbative way. While in scalar quantum
thecontributionfromthespectralself-energyinEq.(29) field theories there are different ways to address this
iseffectivelyβ =2intheaboveclassificationschemeand problem, involving e.g. large N resummation techniques
thus becomes important only at later times. The modi- [50], the most frequently employed approach in gauge
fiedevolutionequationsinthisregimethentaketheform theories is the classical-statistical approximation.
δ2
iG−1,µγ[x,ν;A]Fcb(x,y,ν)= τΠ(F),µγ(x,y,ν)g (y),
0,ac γν 2 -o-,ab γν
(34)
where δ is the extent in time for which the memory
τ III. CLASSICAL-STATISTICAL
integrals are evaluated. To obtain Eq. (34), we per-
APPROXIMATION
formedaleadingorderTaylorexpansionoftheintegrand
inEq.(20)aroundτ =τ andmadeuseoftheequaltime
z y
commutation relations in Eq. (16) to estimate the spec- In the classical statistical approximation all fluc-
tralfunction. Theone-loopintegralΠ(F), whichappears tuations evolve classically and one neglects genuine
on the right hand side of Eq. (34), is dominated by the quantum fluctuations, such as the (ρρ) quantum term
contributions from unstable modes and is proportional in Eq. (28). When transforming to the language of
(cid:112)
to exp[2Γ g2µτ] at equal times. The contribution on expectation values, as employed in the previous section,
0
the right hand side acts as a source term in the evolu- it can be shown that the prescription resums an infi-
tionequationofstatisticalfluctuations. Asiswellknown nite subset of diagrams [43], such that the dynamics
from various examples of self-interacting quantum field of fluctuations is treated on equal footing with the
theories, this term leads to a non-linear amplification of background fields. The set of diagrams included in the
instabilities,where’secondary’instabilitieswithstrongly classical-statistical treatment can be identified as the
enhanced growth rates emerge over a large range of mo- self-energycorrections,whichcontainthemostpowersof
menta. This has been observed in scalar field-theories the statistical propagator as compared to powers of the
[34, 44] as well as in non-abelian gauge theories [19, 20] spectral function for each topology [43]. Accordingly,
andisarathergenericfeatureofself-interactingtheories this corresponds to resumming the leading effects of
undergoing an instability. We will show in Sec. IV that the instability to all orders in the coupling constant
thephenomenonofnon-linearamplificationalsoemerges [10]. In contrast to expansions at fixed loop orders,
in numerical simulations of the unstable Glasma and the classical statistical approximation thus provides a
plays a crucial role in understanding gauge-invariant ob- robust approximation scheme, which is particularly well
servables. The characteristic time scale for non-linear suited for problems involving large statistical fluctua-
amplification to take place can be infered by comparing tions. However there are problems associated with the
themagnitudeofthenon-linearcontributionsinEq.(34) Rayleigh-Jeans divergence, which concern the handling
tothecontributionsofthebackgroundfield. Intheweak of ultra-violet divergences and the approach to thermal
coupling limit this time scale is parametrically given by equilibrium at late times, which are discussed in more
detail in the literature [43].
(cid:112)g2µτ g(cid:28)∼1 1 ln g−2 , (35)
Sec 2Γ
0 In the classical-statistical theory observables (cid:104)O(x)(cid:105)
cs
where Γ is the characteristic growth rate of primary are calculated as an ensemble average of classical
0
instabilities and we assumed Fab ∼ O(1) at τ = 0+ for field solutions A [A ,E ], which individually satisfy
µν cl τ0 τ0
characteristic modes. In addition to Eq. (35) there are the Yang-Mills evolution equations. The canonical
sub-leading contributions associated to the delayed set field variables A and E at initial time τ are dis-
τ0 τ0 0
in of primary instabilities and the spectral distribution tributed according to a phase space density functional
8
W[A ,E ], such that [43] initial conditions at τ = 0+, while the classical evolu-
τ0 τ0
tion equations become independent of the coupling con-
(cid:90)
(cid:104)O(x)(cid:105) = DA DE W[A ,E ] O [A ,E ]. stant. For the Glasma background fields this can be
cs τ0 τ0 τ0 τ0 cl τ0 τ0 achieved most efficiently by replacing gAa(x) → A˜a(x)
µ µ
(36) and g(cid:37)(1/2)(x ) → (cid:37)˜(1/2)(x ) for all expressions in
⊥ ⊥
Sec. IIB, where in the MV model prescription
HereO [A ,E ]denotesthefactthattheobservableis
cl τ0 τ0
evaluated as a functional of the classical field solution (cid:104)(cid:37)˜(A)(x )(cid:37)˜(B) (y )(cid:105)=g4µ2δ(x −y ). (39)
a ⊥ b ⊥ ⊥ ⊥
(cid:90)
Here the model parameter g2µ is directly related to the
O [A ,E ]= DA O[A] δ(A−A [A ,E ]),
cl τ0 τ0 cl τ0 τ0 saturation scale Q without further powers of the cou-
s
(37) pling constant appearing in the expression [38]. Accord-
ingly the defining equation (39) is indeed independent of
where A [A ,E ] is the classical Yang-Mills field so-
cl τ0 τ0 the value of the coupling constant. In contrast, the cou-
lution with initial conditions A = A and E = E
cl τ0 cl τ0 plingconstantgappearsexplicitlyintheinitialspectrum
at initial time τ . In practice, Eqns. (36) and (37) state
0 of fluctuations, given by
that vacuum fluctuations of the initial state are added
othnetsoupbseoqfutehnet cblaacsksigcraoluenvdolufiteilodnakteepinsittiraalcktimofea,llwnhoinle- F˜µaνb(x,y)(cid:12)(cid:12)(cid:12)τ=τ(cid:48)=0+ = g2 Fµaνb(x,y)(cid:12)(cid:12)τ=τ(cid:48)=0+ (40)
linearities. Thesetofequations(36)and(37)isprecisely and similarly for derivatives at initial time. Here it is
the same as in Ref. [10], where it has been obtained as important to note that the magnitude of the vacuum
a partial resummation scheme of the perturbative cor- fluctuations Fab(x,y) on the right hand side is indepen-
µν
rections due to vacuum fluctuations of the initial state. dent of the value of the coupling constant. Thus the
TheprocedureoutlinedinRef.[10]consistofahybridap- initial suppression of vacuum fluctuations compared to
proach,whichemploysthelinearizedevolutionequations the boost invariant background fields is the only mea-
(23, 24, 25) outside the forward light-cone, where fluctu- sure of the strong coupling constant, present in the
ations are small, while switching to a classical-statistical classical-statistical field theory. We will exploit this fact
description in the forward light-cone. This has the ad- in Sec. IVB, where we vary the amplitude of initial fluc-
vantage that the evolution of the background field and tuations in our simulations to study the coupling depen-
thespectrumoffluctuations,whichenterthephase-space dence of our results.
weight W[A ,E ], can be obtained analytically. The
τ0 τ0
phase-spaceaverageinEq.(36)canthenbetakenonthe
Cauchy surface τ =0+, such that only the dynamics in B. Initial conditions at τ =0+
0
the forward light-cone has to be studied within classical-
statistical lattice simulations. We follow this approach Within the classical-statistical framework, the initial
but employ a simpler spectrum of initial fluctuations, as conditions for the time-evolution in the forward light-
specified in Sec. IIIB, instead. This simplification is jus- cone are given at τ = 0+ by the set of equations (9)
tified at sufficiently weak coupling, where the spectrum and (10), complemented by the spectrum of initial fluc-
offluctuationsisquicklydominatedbythegrowthofpri- tuations. While in general it is necessary to implement
mary instabilities rather than the initial spectrum. the spectrum of fluctuations as specified in Ref. [10], it
is also clear that at sufficiently weak coupling many de-
tails of the spectral shape of the initial fluctuations be-
A. Coupling dependence comeirrelevantduetothepresenceofinstabilities,which
quickly dominate the spectrum. Since implementing the
An important property of the classical-statistical de- spectrum of fluctuations in Ref. [10] is numerically chal-
scription is the independence of the gauge coupling con- lenging,wewillthereforesticktoasimplerchoice,where
stantg,inthesensethattheclassical-evolutionequations in accordance with previous works [16, 17] the statistical
are invariant under a change of variables fluctuations initially take the form
Aaµ(x)→g Aaµ(x), Fµaνb(x,y)→g2Fµaνb(x,y), (38) Fµaνb(x,y)(cid:12)(cid:12)τ=τ(cid:48)=0 =0, (41)
whereas the spectral function remains unaffected, as en- ∂τFµaνb(x,y)(cid:12)(cid:12)τ=τ(cid:48)=0 =0,
coded in the equal time commutation relations.3 With ∂τ∂τ(cid:48)Fµaνb(x,y)(cid:12)(cid:12)τ=τ(cid:48)=0 =(cid:104)δEµa(x⊥,ηx)δEνb(y⊥,ηy)(cid:105).
the rescaling (38) the entire coupling dependence in the
with
classical-statistical evolution can be absorbed into the
δEa(x ,η)=∂ f(η)ea(x ) (42)
i ⊥ η i ⊥
δE (x ,η)=−f(η)D ea(x ). (43)
η ⊥ i i ⊥
3 Intheclassicaltheory−itimesthecommutatoristobereplaced suchthatδEa(x)isanadditivecontributiontotheback-
µ
bythePoissonbrackets[43]. ground field Ea(x) at initial time. The advantage of this
µ
9
-2
10
ν=4
4 10-4 ν=12
-µ] ν=43
2g ν=71
-6
[ 10 ν=94

ν) ν=200
,
τ
Π( 10-8
2
g

]
τ -10
µ 10
2
g
[
-12
10
0 10 20 30 40 50
[g2µτ]1/2
FIG. 3. (color online) Time evolution of the pressure-pressure correlator Π (τ,ν) for different rapidity wave numbers ν.
L
Once the initial fluctuations have grown larger one observes the emergence of the secondary instabilities, with growth rates.
Subsequentlytheinstabilitypropagatestowardshighermomentauntilsaturationoccursandthesystemexhibitsamuchslower
dynamics. TheresultsareobtainedfortheMVmodelparameterg2µL =22.6onlatticeswithN =16andN =1024sites.
⊥ ⊥ η
The initial fluctuations are parametrized by ∆=10−10 and b=0.01 according to Eq. (44).
construction is that the Gauss constraint is satisfied ex- of boost non-invariant fluctuations is irrelevant for the
plicitly for arbitrary functions f(η) and ea(x ). For the dynamics of unstable modes, it is clear that for the
i ⊥
numerical simulations we choose non-linear regime the size of the initial fluctuations
matters. In view of the spectrum of initial fluctuations
(cid:104)eai(pT)ebj(qT)(cid:105)=∆2 δijδabδ(pT +qT), (44) obtainedinRef.[10], theratiooftheinitialamplitudeof
(cid:104)f(ν)f(ν(cid:48))(cid:105)=e−2b|ν|δ(ν+ν(cid:48)), (45) fluctuations compared to the amplitude of the (squared)
background field is parametrically of the order of the
where p and ν are the Fourier coefficients with respect strong coupling constant g2. We study this dependence
T
to relative transverse coordinates and relative rapidity. in Sec. IVB by considering different amplitudes of the
Here b is a (small) number, which regulates the ultra- initialfluctuations,ascharacterizedbythedimensionless
violetdivergenceandthedimensionlessparameter∆con- parameter ∆2 ∼ g2 (see Sec. IIIB). We restrict our
trols the initial amplitude of small wave-number fluctu- analysis to weak coupling (g2 (cid:28) 1), where classical-
ations. While this construction does not respect the de- statistical methods are expected to provide an accurate
tails of the spectral composition, the parameter ∆ pro- description of the quantum dynamics on large time
vides a measure of the coupling constant ∆2 ∼ g2 (see scales.
Sec. IIIA) and we will vary its size in Sec. IVB to study
the coupling dependence. The discussion of our results is organized as fol-
lows: In Sec. IVA we investigate the dynamics of the
instability at weak coupling and show how deviations
IV. LATTICE RESULTS from the linear regime emerge in terms of secondary
instabilities. To further analyze this behavior, we obtain
the relevant growth rates for primary and secondary
In this section we present results from classical-
instabilities as well as the corresponding set-in times.
statistical lattice simulations of the Glasma evolution in
Subsequently, in Sec. IVB, we investigate the depen-
the presence of boost non-invariant fluctuations. While
dence on the coupling constant by varying the size of
the existence of a non-equilibrium instability has been
initial fluctuations.
established in previous simulations [16, 17], we focus
on the non-linear regime where unstable modes have
If not stated otherwise we perform simulations on
grown large enough to significantly alter the dynamics.
N = 16, N = 1024 and N = 32, N = 128 lattices
In contrast to the linear regime, where the initial size ⊥ η ⊥ η
10
and we employ the set of parameters g2µ N a = 22.6
⊥ ⊥ 2500
and N a = 1.6 in accordance with Ref. [16]. We study
η η
the time evolution of the gauge-invariant pressure-
2000
pressure correlation function
Π2(τ,η,η(cid:48))=(cid:104)P (τ,x ,η)P (τ,y ,η(cid:48))(cid:105) , (46)
L L ⊥ L ⊥ T 1500
τ
µ
where P (x) is the longitudinal pressure as a function 2
L g
of space and time arguments and < .,. > denotes av- 1000 Primary (∆=10-10)
T
erageovertransversecoordinatesandclassical-statistical Primary (∆=10-15)
ensembleaverage. WewillfrequentlyemploytheFourier 500 Secondary (∆=10-10)
transforms of Π2L(τ,η,η(cid:48)) with respect to relative rapid- Saturation of growth (∆=10-10)
ity, i.e. we consider
0
(cid:90) 0 50 100 150 200
Π2(τ,ν)=(N a )−1 dη dη(cid:48) Π (τ,η,η(cid:48)) e−iν(η−η(cid:48)) , ν
L η η L
(47) 2
Fluctuations: ∆=10-10
andusuallyshowresultsforΠ (τ,ν),i.e. thesquareroot
L
of the above expression. The details of our lattice setup 1.5 Primary
are described in more detail in the appendix. 1/2 x Secondary
ν) 1
Γ(
A. The unstable Glasma
0.5
In a first step we study the time evolution of the
gauge-invariant pressure-pressure correlator Π (τ,ν)
L
for a fixed value of the amplitude of initial fluctuations
∆=10−10 and b=0.01. The results are shown in Fig. 3 0
0 20 40 60 80 100 120 140
for different rapidity wave numbers ν as a function of
ν
time. From Fig. 3 one observes a sequence of differ-
entdynamicalregimeswhicharecharacterizedasfollows:
FIG. 4. (color online) (top) Set in times of primary and
secondary instabilities as a function of rapidity wave number
At very early times (cid:112)g2µτ (cid:46) 2 one observes a pe- ν. Once secondary instabilities set in, the growth quickly
riod of rapid initial growth, which is presumably caused extendstohigherrapiditywavenumbers. (bottom)Growth
by the dephasing dynamics of the strong background ratesΓ(ν)asafunctionofrapiditywavenumberνforprimary
fields, that takes place roughly on the same time scale and secondary instabilities. The results are obtained for the
[11–14, 17, 51]. However at weak coupling, i.e. for small MVmodelparameterg2µL⊥ =22.6onlatticeswithN⊥ =16
fluctuations, this constitutes a rather small effect as the and Nη =1024 sites.
unstable modes exhibit their dominant growth at later
times.
wave number exhibit smaller growth rates as compared
The rapid initial period is followed by a regime to modes with higher rapidity wave number, while at
where the Glasma instability [16, 17] is operative and large ν the primary growth rates become approximately
modes with non-zero rapidity wave number exhibit constant. The numerical values are compatible with the
exponential amplification. The instability sets in with a results reported in Ref. [16], where characteristic growth
delay for higher momentum modes and the functional rates were obtained from a convolution of the spectrum.
form is well described by an exponential of the form
(cid:112)
exp[Γ(ν) g2µτ],withthemomentumdependentgrowth While the primary instability continues to set-in
rate Γ(ν), as seen for ν = 4,12 in Fig. 3. To further for higher momentum modes, one observes from Fig. 3
investigate this behavior we fit a set of continuous thatatlatertimesmodeswithintermediate(ν =43, 71)
piecewise linear functions to the modes displayed in and small (ν = 4) rapidity wave number suddenly
Fig. 3 in order to obtain the relevant growth rates and exhibit much higher growth rates than previously
set-in times. The results of these fits are shown in Fig. 4 observed. This change in the dynamics becomes evident
asafunctionofrapiditywavenumberν. Fromtheupper when shortly after modes with even higher rapidity
panel of Fig. 4 one observes that the primary set-in wave numbers (ν = 94, 200) exhibit even stronger
times follow a linear behavior, as reported in Ref. [16]. growth rates, such that the spectrum extends quickly to
The primary growth rates are shown in the lower panel the ultra-violet and the instability propagates towards
of Fig. 4. One observes that modes with small rapidity higher momenta. This is precisely the signature of