Initial Singularity of the Little Bang

0.3 MB·English

by  Kenji Fukushima

Initial Singularity of the Little Bang
Kenji Fukushima(1), Franc¸ois Gelis(2), Larry McLerran(1,3)
7
0 February 2, 2008
0
2

n
1. RIKEN BNL Research Center,
a
BrookhavenNational Laboratory,
J
Upton, NY-11973, USA
1
2
2. Service de Physique Th´eorique (URA 2306 du CNRS)

2 CEA/DSM/Saclay, Bˆat. 774
v 91191, Gif-sur-Yvette Cedex, France
6
1 3. Department of Physics, Bldg. 510 A,
4 BrookhavenNational Laboratory,
0
Upton, NY-11973, USA
1
6 Abstract
0
/ The Color Glass Condensate (CGC) predicts the form of the nuclear
h
wavefunction in QCD at very small x. Using this, we compute the wave-
p
functionforthecollision oftwonuclei,infinitesimallyintheforward light
-
p cone. WeshowthattheWignertransformation of thiswavefunction gen-
e eratesrapiditydependentfluctuationsaroundtheboostinvariantclassical
h
solution which describe theGlasma in theforward light cone.
:
v
i RBRC-621
X
BNL-NT-06/41
r SPhT-T06/139
a
1 Introduction
The Color Glass Condensate (CGC) provides a descriptionof the wavefunction
of a hadron at very small values of x [1–5]. The CGC is a high density state of
gluons which is controlled by a weak coupling, due to the high gluon density.
Its properties are computable from first principles in QCD, at leastin the limit
of extremely high density.
The CGC has also been applied to heavy ion collisions in order to generate
the initial conditions for the evolution of matter in the forward light cone [6–
10]. There are boost invariant solutions of the equations of motion supposedly
appropriate for the high energy limit. The matter in the forward light cone,
which we call the Glasma [10], initially has large longitudinal color electric and
1
magneticfields,andaswellasalargevalueoftheChern-Simonschargedensity.
As time evolves, transverse color electric and magnetic fields appear as the
remnants of the decaying longitudinal fields. This occurs as a consequence of
theclassicalequationsofmotion. Eventuallyasthesystembecomesverydilute,
these transverse color electric and magnetic fields may be treated as gluons,
which form a quark-gluon plasma. The solution to the classical equations of
motion is boost invariant and describes a system of expanding classical fields.
It has recently been discovered that the boost invariant solution of the
equations of motion are unstable with respect to rapidity dependent pertur-
bations [11]. This instability has in fact close connections with the Weibel
instabilities encountered in the physics of anisotropic plasmas [12–14], and it
is speculated that such an instability may help the system created after heavy
ion collisions reach a state of local equilibrium [15–21]. Whether or not such
instabilities can grow to sufficient magnitude as to become as large as the clas-
sical solution is the subject of current investigations. One piece of the puzzle
which is not yet understood is the spectrum of the initial fluctuations. It is the
purpose of this paper to derive an expressionfor the probability distribution of
these fluctuations.
If such fluctuations can grow to a magnitude comparable to the classical
boost invariant field, then one has amplified initial quantum fluctuations to
macroscopicchaoticturbulence. Suchturbulencemayultimatelyberesponsible
for producing a thermalized Quark Gluon Plasma. Indeed, it proves difficult to
thermalize an expanding system of quarks and gluons by conventionalmultiple
scattering formulae [22].
The evolution from the initial collision to the final state is shown in Fig. 1.
Because of the similarities between the expansion of the universe in cosmol-
ogy and the expansion of matter in heavy ion collisions, we call the latter the
“little bang”. The initial singularity of cosmology is replaced by the singu-
larity in the classical equations of motion associated with the collision. As in
cosmology, where topological transitions associated with Chern-Simons charge
may be responsible for generating baryon number, during the Glasma phase,
there are also topological helicity flip transitions [10,23]. It is during this time
that instabilities develop, and there may be a Kolmogorovspectrum of density
fluctuations generated [16,19,24]. This spectrum is similar to the spectrum of
density fluctuations generated during inflationary cosmology. After inflation,
thesystemreheatsandthermalizesforminganelectroweakplasma. Inthelittle
bang, thermalization might also be achieved after the Glasma expansion. Of
course such an analogy between the little bang is not perfect, and it has not
been established that the Glasma can in fact become thermalized. Neverthe-
less, in the little bang, all of the physics is in principle understood from QCD
in a small coupling regime, and one should be able ultimately to compute the
evolution of the system.
We begin this paper by a discussion of the evolution of an unstable sys-
tem in quantum mechanics. We discuss a Gaussian initial wavefunction in the
path-integralrepresentationinwhichadescriptionequivalenttotheSchwinger-
Keldyshformalism[25]naturallyarises. Weshowthatthedistributionofinitial
2
t
z = -t z = t
Perfect Fluid
Quark Gluon Plasma Hadronization
τ~ 1 - 10 fm/c
Topological Excitations
Glasma Density Fluctuations, Thermalization
τ~ 0.1 - 1 fm/c
Event Horizon
Initial Singularity Quantum Fluctuations
z τ~ 0 - 0.1 fm/c
Initial Nuclei as CGC Coherent, High-density Gluons
Figure 1: A light cone diagramillustrating the evolution ofmatter produced in
heavy ion collisions
fluctuationsisgivenbytheWignertransformofthequantummechanicalwave-
function [26]. These initial fluctuations are further evolved by solving classical
equations of motion. We then proceed to show that the same procedure works
in quantum field theory. We then compute the quantum field theoretical wave-
function which describes the collision of two hadrons, at a time infinitesimally
in the forward light-cone, after the collision. This should provide a practical
algorithmforthe computationofthe evolutionofmatterproducedinveryhigh
energy heavy ion collisions.
The necessity of summing over the initial quantum fluctuations when solv-
ing the classical Yang-Mills equations also appears when one goes beyond the
classical approximation for the description of the fields in the forward light-
cone. The formalism necessary for calculating loop corrections in a regime of
strong sources and fields has been developed in Ref. [27] (and applied to the
computation of the production of quark pairs – the simplest among the NLO
contributions – in Ref. [28]). In this systematic approach, the instability of the
boostinvariantclassicalsolutionmanifestsitselfasadivergenceintheone-loop
correction to the inclusive gluon spectrum, and it seems that the resummation
of these divergent terms leads directly to an average over fluctuations of the
classical initial conditions [29].
The description we advocate here has also some philosophical similarities
with the work of Kharzeev et al., who make an analogy to Hawking radia-
tion[30]. WeareunabletoidentifyaHawkingtemperatureinourcomputation,
but the idea that there is an initial spectrum of fluctuations generated at the
initial singularity due to a quantum treatment of this singularity is similar.
3
2 The Case of Quantum Mechanics
2.1 General Formulation
We shall consider the time evolution of a generic quantum mechanical system
from the initial time t to the final time t . If the system is as simple as an
I F
harmonic oscillator, the quantum fluctuations at the final time t can be ex-
F
pressedintermsofthe initialquantumfluctuationatt whichevolveaccording
I
to the classical equations of motion. In more complicated situations beyond
the harmonic oscillator problem, the formulation we develop in what follows
still holds unchanged, as long as we can make a Gaussianapproximationin the
weak coupling regime.
Let us consider a simple example [31]. Supposing that we have a convex
parabolic potential as sketched in Fig. 2 until the initial time t , we know that
I
the ground state has a Gaussian distribution of quantum fluctuations around
the minimum of the potential. Let us now assume that the potential suddenly
becomes inverted at t : then the system is subject to an instability. If the
I
problemwaspurelyclassical,theunstablestatewouldnotdecayatallprovided
it lies exactly on the top of the potential curve. The decay into stable states
is howeverunavoidablein quantum mechanics because ofquantum fluctuations
aroundtheminimum. Aslongashigherorderquantumfluctuationsarenegligi-
ble, itshouldbe acceptable to approximatethe time evolutionclassicallyunder
such a potential causing the instability. As a result, we can anticipate that
the convolutionof the initial dispersionandthe classicalevolutionprovides the
later distribution of quantum fluctuations at t , and we shall verify this in the
F
forthcoming discussion. We would like to emphasize here that our formulation
is not limited to this sortof specific instability problem,but applicable to more
generic problems.
t=t t=t >t
I F I
t <t
I
Classical
Gaussian Evolution
Fluctuation
Figure 2: Schematic picture of the time evolution of quantum fluctuations.
The ground state of a steady system for t < t has a Gaussian distribution
I
of quantum fluctuations. The distribution at a later time t is given by the
F
classical evolution from the initial distribution at t=t .
I
Inthe languageofquantumphysicsthegroundstatewavefunctionatt em-
I
bodiesthezero-pointoscillation. Thepositionandmomentumfluctuateaccord-
ing to the uncertainty principle. The expectation value of physical observables
4
is defined as an average weighted by the wavefunction. Let us consider an ob-
servable (x ) which is a function of the position x at the final time t . [We
O F F F
specifically denote the position variable at t and t as x and x respectively
I F I F
inordertodistinguishthemfromthepositionatothertimes.] Bydefinitionthe
expectation value at t is given by
F
= dx ψ(x ,t )2 (x ). (1)
hOitF F | F F | O F
Z
One can easily generalize the above expression to a situation involving observ-
ables that depend not only on the position x but also on the momentum p .
F F
Using p = i∂/∂x and (x , i∂/∂x ) acting on ψ(x ,t ), one can reach
F − F O F − F F F
the same conclusion as in Eq. (10) in what follows, but for simplicity we only
perform the derivation for the case of Eq. (1).
Thepath-integralformulationnaturallyprovidesustherelationbetweenthe
initial and final wavefunctions,
t
F
ψ(x ,t )= dx x(t) p(t) ψ(x ,t ) exp i dt px˙ H(p,x) , (2)
F F I D D I I −
Z (cid:26) ZtI (cid:27)
(cid:2) (cid:3) (cid:2) (cid:3)
where the integrals over the paths x(t) and p(t) extend from t to t , with the
I F
boundary condition x(t ) = x and x(t ) = x . The Hamiltonian is denoted
I I F F
by H(p,x). We will perform the path-integral approximately in order to find
an analytically manageable expression.
Weexpandthephaseintheexponentialaroundthestationarypoint,inorder
to reduce Eq. (2) to Gaussian functional integrals. The stationary condition
leads to the classical path x = xc(t) and p = pc(t) determined by Hamilton’s
equations of motion,
∂H(pc,xc) ∂H(pc,xc)
x˙c = , p˙c = (3)
∂pc − ∂xc
with the initial conditions xc(t ) = x and pc(t ) = p . We note that the final
I I I I
position xc(x ,p ) xc(t ) is unique with x and p given, which means that
F I I ≡ F I I
wemustconsidertheinitialmomentump asafunctionofx andx . Naturally,
I I F
the quantum fluctuation around the classical path must be restricted at t and
I
t to satisfy the boundary condition δx(t )=δx(t )=0. Up to the quadratic
F I F
order in quantum fluctuations, therefore, we have
t
ψ(x ,t )= dx ψ(x ,t ) exp i Fdt pcx˙c H(pc,xc)
F F I I I −
Z (cid:26) ZtI (cid:27)
(cid:2) (cid:3)
t
F
δx(t) δp(t) exp i dt δpδx˙
(4)
× D D
δx(tI)=δZx(tF(cid:2))=0 (cid:3) (cid:26) ZtI h
∂2H ∂2H ∂2H
δp2 δx2 2 δpδx .
− ∂p2 − ∂x2 − ∂p∂x
i(cid:27)
5
In the expansion of the argument of the exponential, the linear order terms in
the fluctuations vanish because of the classical equations of motion. The first
integralin the above expressionis the classicalpartcorrespondingto the WKB
approximation,wherethex dependence is implicitthroughthe factthatxc(t)
F
is the classicaltrajectorythat starts at x and ends at x . The secondintegral
I F
represents the one-loop quantum corrections leading to the determinant of the
propagatorwith the Dirichletboundarycondition. It is obviousfromthe above
expression that, if H is the Hamiltonian of an harmonic oscillator (therefore
consistingonlyofquadratictermsinxandp),e.g.,H = 1p2+1ω2x2,theWKB
2 2
approximation is exact. It is because none of ∂2H/∂p2=1, ∂2H/∂x2=ω2, and
∂2H/∂p∂x=0 depends on x nor p , and thus the integration with respect to
I I
fluctuations merely produces an irrelevant purely numerical factor.
Ingeneral,theone-loopintegralgives[detG−1(xc)]−1/2 whereGistheprop-
agator of a fluctuation in the presence of the classical background xc. Usually,
this determinant has no explicit pc dependence since usual quantum theories
have at most quadratic terms with respect to the canonical momenta. These
quantum corrections alter H(pc,xc) to be H(pc,xc) i lnG−1(xc) and can be
− 2
seen as an effective potential in the classical Hamiltonian.
Fromnowonweassumethatthexc dependenceinH(pc,xc)issostrongthat
we can drop the xc dependence from the quantum corrections i lnG−1(xc).
−2
This assumption should be checked case by case. Generally speaking, this is
acceptable in instability problems like the one illustrated in Fig. 2. The quan-
tum corrections modify the tree-level potential, which would eventually affect
the equations of motion. But if the tree-level potential is steep enough, the al-
terationduetoquantumcorrectionsshouldbe negligible. Hence,ourtreatment
isequivalenttoassumingthatthetime evolutionisdominantlygovernedbythe
tree-level potential.
The probability distribution for the positions at final time is therefore cal-
culated with the classical parts alone as
ψ(x ,t )2 = dx′ dx ψ∗(x′,t′)ψ(x ,t )
| F F | I I I I I I
Z (5)
t
exp i F dt pcx˙c H(pc,xc) pc′x˙c′ H(pc′,xc′) .
× − − −
(cid:26) ZtI h(cid:0) (cid:1) (cid:0) (cid:1)i(cid:27)
The phase difference induced by the different initial conditions, x and x′, can
I I
be simplified thanks to Hamilton’s equations of motion (3) as
t
F dt pcx˙c H(pc,xc) pc′x˙c′ H(pc′,xc′)
− − −
ZtI = h(cid:0)tF dt {pc(t),xc(t(cid:1))} (cid:0)dpcx˙ +pcdx˙c ∂H(cid:1)idpc ∂Hdxc
ZtI Z{pc′(t),xc′(t)}(cid:26) − ∂pc − ∂xc (cid:27)
x
I
= dx˜ p (x˜,x ). (6)
−Zx′I I I I F
6
From the middle to the last line only the surface term of the time integration
remains. There is no finite contribution from the t = t side because x is
F F
common. It should be noted that p (x ,x ) in the last line is a function which
I I F
gives the value of the initial momentum p as a function of the initial and
I
final positions, i.e. p (x ,x )=p and p (x′,x )=p′. Here, in order to get a
I I F I I I F I
convenientexpression,weshalldefineP astheinitialmomentumcorresponding
I
to the average of the two initial positions (and the same final position x ) by
F
P p (X ,x ), (7)
I ≡ I I F
where X 1(x +x′). Then we can write the last line of Eq. (6) as follows,
I ≡ 2 I I
x 1δx
I 2 I
dx˜ p (x˜,x )= d∆x˜ p (X +∆x˜,x )
−Zx′I I I I F −Z−12δxI I I I I F
1 ∂2p (X ,x )
P δx I I F δx3+ (8)
≃− I I − 24 ∂X2 I ···
I
where we defined δx x x′. Within the Gaussian approximation that we
I ≡ I − I
will employ later, it is enough for us to keep only the first term in the above
expansion.
The expectation value at the final time is now transformed in the WKB
approximationfrom Eq. (1) into
hOitF = dxF dx′I dxI ψ∗(x′I,tI)ψ(xI,tI)e−iPIδxI O(xF). (9)
Z
Here we can replace the x integration by an integration over P . Indeed, x ,
F I F
P and X are related by the fact that the classical path that starts at the
I I
initial position X and with the initial momentum P ends at the position x ,
I I F
i.e. x = xc(X ,P ). Such a replacement is accompanied by the Jacobian
F F I I
∂xc/∂P as a function of X , P , and t . One can, in principle, obtain the
| F I| I I F
Jacobian once one solves the classical path xc(x ,p ). This is, however, only
I I
a prefactor of the exponential for the phase of which we made the stationary-
point approximation. It should be, therefore, consistent to drop the Jacobian
prefactor off within the WKB approximation.
After all, rewriting x = X + 1δx and x′ = X 1δx and integrating
I I 2 I I I − 2 I
with respect to δx , we formally reach the following expression,
I
t
hOiF
= dX dδx dP ψ∗(X 1δx ,t )ψ(X + 1δx ,t )e−iPIδxI xc(X ,P )
I I I I − 2 I I I 2 I I O F I I
Z
(cid:0) (cid:1)
= dX dP W(X ,P ) xc(X ,P ) , (10)
I I I I O F I I
Z
(cid:0) (cid:1)
where W(X ,P ) is the Wigner transform of the product of the two initial
I I
wavefunctions, defined as
W(X ,P ) dδx ψ∗(X 1δx ,t )ψ(X + 1δx ,t )e−iPIδxI . (11)
I I ≡ I I − 2 I I I 2 I I
Z
7
This is our formula and the purpose of our work is to calculate the Wigner
function analytically for the problem of heavy ion collisions, which should be
a necessary input for numerical studies provided that the classical equations of
motion are solved.
2.2 Simple Example
It would be instructive to demonstrate how the formula (10) works for an har-
monic oscillator for which explicit calculations are feasible. Let us estimate
the average of the observable x2 in a direct computation and by means
O ≡ F
of Eq. (10) and then compare the results. As we have already mentioned, the
WKB approximation is exact in the case of the harmonic oscillator problem,
which means that the two results should coincide. This is a trivial check of the
validity of our formula (10).
We shall characterize the initial state as a Gaussian [31];
ψ(x ,t =0)= π12b −12 exp x2/2b2 . (12)
I I − I
The Green’s function in the inverte(cid:0)d har(cid:1)monic o(cid:0)scillator(cid:1)with the Hamiltonian
H = 1p2 1ω2x2, which is defined by
2 − 2
i∂ +∂2+ω2x2 G(x,x′;t)=δ(t)δ(x x′),
t x −
lim G(x,x′;t)=δ(x x′), (13)
(cid:0) (cid:1)
t→0+ −
is known [31] to be
G(x,x′;t)= 2πisinhωt −21 exp i (x2+x′2)coshωt 2xx′ , (14)
2sinhωt −
(cid:0) (cid:1) (cid:26) h i(cid:27)
from which we can directly calculate the wavefunctions at later time by
ψ(x ,t )= dx G(x ,x ;t )ψ(x ,0). (15)
F F I F I F I
Z
What we are calculating is the expectation value of the position dispersion
when the potential is inverted at the initial time t = 0. After some algebraic
I
procedures we find,
ψ(x ,t )2 = π12B2(t ) −1exp x2/B2(t ) , (16)
F F | F − F F
where B2(t ) (cid:12)(cid:12) b2cosh2ωt (cid:2)+b−2sinh2(cid:3)ωt . (cid:2)Then, it is st(cid:3)raightforward to
F ≡ F F
compute the average value of =x2,
O F
= dx ψ(x ,t ) 2x2 = 1B2(t ). (17)
hOitF F F F F 2 F
Z
(cid:12) (cid:12)
Thisresultindicatesthatthedispe(cid:12)rsionofth(cid:12)epositiongrowsexponentiallywith
increasing time under the inverted potential just as sketched in Fig. 2.
8
Let us next consider the same problem according to the formula (10). The
Wigner function obtained from Eq. (12) is
1
W(X ,P )= exp X2/b2 b2P2 , (18)
I I π − I − I
(cid:0) (cid:1)
where we have adjusted the normalization factor, though it is only an unim-
portant prefactor, so that dX dP W(X ,P ) = 1. The classical path is
I I I I
fixed by Hamilton’s equations of motion with the initial inputs xc(0)=X and
R I
pc(0)=P , leading to the final point at t=t as a function of X and P ,
I F I I
P
xc(X ,P )=X coshωt + I sinhωt . (19)
F I I I F ω F
AfterperformingGaussianintegrationswecanreadilyarriveattheexpectation
value,
1
= dX dP W(X ,P )xc2(X ,P )= B2(t ). (20)
hOitF I I I I F I I 2 F
Z
We see that our formula perfectly reproduces the result from the direct evalua-
tion in this simple case of the harmonic oscillator. The merit of our method is
that we do not have to calculate the Green’s function and the wavefunction at
the final time directly.
3 Generalization to Quantum Field Theories
It is not difficult to extend the formula (10) described in Quantum Mechanics
to the case of Quantum Field Theories. Let us consider in this section the case
of a real scalar field theory.
3.1 Introduction
For a generic scalar field theory defined by a Hamiltonian density [φ,π] with
H
field φ and canonicalmomentum π, we expect that the counterpart of Eq. (10)
is
= dφ (x) dπ (x) W[φ ,π ] φc[φ ,π ] , (21)
hOitF I I I I O F I I
Z
where φc[φ ,π ] is the(cid:2)solution(cid:3)(cid:2)of the (cid:3)classical equa(cid:2)tions of m(cid:3)otion at time
t withFtheIiniItial conditions φc(t ,x) = φ (x) and πc(t ,x) = π (x). This
F I I I I
classical field is a solution of Hamilton’s equations of motion,
δH[πc,φc] δH[πc,φc]
∂ φc = , ∂ πc = . (22)
t δπc t − δφc
HereH denotestheintegratedHamiltonianH d3x . TheWignerfunction
≡ H
is defined in the same way as in Quantum Mechanics by
R
W[φ ,π ] dδφ (x) Ψ∗[φ 1δφ ,t ]Ψ[φ + 1δφ ,t ]e−iRd3xπI(x)δφI(x)
I I ≡ I I − 2 I I I 2 I I
Z
(cid:2) (cid:3) (23)
9
interms ofthe initialwavefunctionΨ[φ ,t ]. Similarexpressionscouldbewrit-
I I
ten for gauge fields after unphysical redundant degrees of freedom are removed
byfixingthegauge. Inthenextsubsectionwewillexplainhowonecanconstruct
the initial wavefunction Ψ[φ ,t ] in practice.
I I
3.2 Wavefunction from the Annihilation Operator
Let us consider the wavefunction and the associated Wigner function for the
free realscalarfieldtheory,which is the simplest extensionto a QuantumField
Theory of the harmonic oscillator that we considered previously in Quantum
Mechanics. In the Hamiltonian formulation, one can express the field operator
φˆ(x) and its canonical momentum operator πˆ(x) in terms of the creation and
annihilation operators as
d3k 1
φˆ(x)= (2π)3√2ω aˆkeik·x+aˆ†ke−ik·x , (24)
Z k
d3k ω(cid:0) (cid:1)
πˆ(x)=−i (2π)3 2k aˆkeik·x−aˆ†ke−ik·x , (25)
Z r
(cid:0) (cid:1)
in the Schroedinger representation in which operators are time independent.
The dispersion relation for the free theory is ω = k2+m2. The canonical
k
quantizationconditionis[φˆ(x),πˆ(x′)]=iδ(3)(x−x′)porequivalently[aˆk,aˆ†k′]=
(2π)3δ(3)(k k′). In momentum space, we have
−
1
φˆ(k)≡ d3xφˆ(x)e−ik·x = √2ω aˆk+aˆ†−k , (26)
Z k
(cid:0) (cid:1)
ω
πˆ(k)≡ d3xπˆ(x)e−ik·x =−i 2k aˆk−aˆ†−k , (27)
Z r
(cid:0) (cid:1)
for which the canonical commutation relation is deduced as [φˆ(k),πˆ(k′)] =
i(2π)3δ(3)(k k′). Let us consider an initial state which is an eigenstate of
−
the operator φˆ(k),
φˆ(k)Ψ[φ ,t ]=φ (k)Ψ[φ ,t ]. (28)
I I I I I
From the commutator between φˆ(k) and πˆ(k′), we see that πˆ(k) can be repre-
sented as the differential operator with respect to the eigenvalue φ ( k), that
I −
is,
δ
πˆ(k)Ψ[φ ,t ]= i Ψ[φ ,t ]. (29)
I I − δφ ( k) I I
I −
Here the functional derivative in momentum space is understood in our con-
vention as δf(k)/δf(k′) = (2π)3δ(3)(k k′). The ground state is the vacuum
−
defined by the annihilation operator, i.e.
aˆkΨ[φ ,t ]=0. (30)
I I
10