rev. 10. 6. 07 BNL-NT-07/18
BI-TP 2007/06
Thermal Hadronization and
Hawking-Unruh Radiation in QCD
P. Castorinaa, D. Kharzeevb and H. Satzc
a) Dipartimento di Fisica, Università di Catania,
and INFN Sezione di Catania, Via Santa Sofia 64 I-95123 Catania, Italia
b) Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA
c) Fakultät für Physik, Universität Bielefeld, D-33501 Bielefeld, Germany
Abstract
We conjecture that because of color confinement, the physical vacuum forms an event
horizon for quarks and gluons which can be crossed only by quantum tunneling, i.e.,
through the QCD counterpart of Hawking radiation by black holes. Since such radiation
cannot transmit information to the outside, it must be thermal, of a temperature de-
termined by the chromodynamic force at the confinement surface, and it must maintain
color neutrality. We explore the possibility that the resulting process provides a com-
mon mechanism for thermal hadron production in high energy interactions, from e+e−
annihilation to heavy ion collisions.
Keywords:
event horizon, color confinement, Hawking-Unruh radiation, multihadron production
PACS: 04.70Dy, 12.38Aw, 12.38Mh, 12.40Ee, 25.75Nq, 97.60Lf
http://arxiv.org/abs/0704.1426v2
1 Introduction
The aim of this paper is to develop a conceptual framework for a universal form of thermal
multihadron production in high energy collisions. Our work is based on two seemingly
disjoint observations:
• Color confinement in QCD does not allow colored constituents to exist in the physi-
cal vacuum, and thus in some sense creates a situation similar to the gravitational
confinement provided by black holes.
• Numerous high energy collision experiments have provided strong evidence for the
thermal nature of multihadron production, indicating a universal hadronization tem-
perature TH ≃ 150− 200 MeV.
We want to suggest that quantum tunnelling through a color event horizon, as QCD
counterpart of Hawking-Unruh radiation from black holes, can relate these observations
in a quite natural way.
The idea that the color confinement of quarks and gluons in hadrons may have a dual
description in terms of a theory in curved space–time is not new. Both gravitational
confinement of matter inside a black hole [1] and the de Sitter solution of the Einstein
equations with a cosmological constant describing a “closed” universe of constant cur-
vature [2] have been proposed as possible descriptions of quark confinement. Soon it
became clear that asymptotic freedom [3] and the scale anomaly [4,5] in QCD completely
determine the structure of low–energy gluodynamics [6]. This effective theory can be con-
veniently formulated in terms of the Einstein–Hilbert action in a curved background. At
shorter distances (inside hadrons), the effective action has the form of classical Yang–Mills
theory in a curved (but conformally flat) metric [7]. The “cosmological constant” present
in this theory corresponds to the non–perturbative energy density of the vacuum, or the
“gluon condensate” [8].
It is worthwhile to also mention here the well–known conjectured holographic correspon-
dence between the large N limit of supersymmetric Yang–Mills theory in 3+1 dimensions
and supergravity in an anti–de Sitter space–time sphere, AdS5 × S5 [9]. This example
illustrates the possible deep relation between Yang–Mills theories and gravity; however,
a conformal theory clearly differs from the examples noted above, in which the scale
anomaly (describing the breaking of conformal invariance by quantum effects) was used
as a guiding principle for constructing an effective curved space–time description.
Let us assume that color confinement indeed allows a dual description in terms of the
gravitational confinement of matter inside black holes. What are the implications of
this hypothesis for hadronic physics? Hawking [10] showed that black holes emit thermal
radiation due to quantum tunneling through the event horizon. Shortly afterwards, Unruh
[11] demonstrated that the presence of an event horizon in accelerating frames also leads
to thermal radiation. It was soon conjectured that the periodic motion of quarks in a
confining potential [12], or the acceleration which accompanies inelastic hadronic collisions
[13–15], are associated with an effective temperature for hadron emission.
Recently, a QCD–based picture of thermal production based on the parton description of
high energy hadronic collisions has been proposed [16, 17]. The effective temperature T
is in this case determined either by the string tension σ, with a relation
, (1)
or, in the gluon saturation regime, by the saturation momentum Qs describing the strength
of the color fields in the colliding hadrons or nuclei, with T ≃ (Qs/2π).
Turning now to the second observation, we recall that over the years, hadron production
studies in a variety of high energy collision experiments have shown a remarkably universal
feature. From e+e− annihilation to p− p and p− p̄ interactions and further to collisions
of heavy nuclei, with energies from a few GeV up to the TeV range, the production
pattern always shows striking thermal aspects, connected to an apparently quite universal
temperature around TH ≃ 150 − 200 MeV [18]. As a specific illustration we recall that
the relative abundance of two hadron species a and b, of masses ma and mb, respectively,
is essentially determined by the ratio of their Boltzmann factors [19],
R(a/b) ∼ exp−{(ma −mb)/TH}. (2)
What is the origin of this thermal behaviour? While high energy heavy ion collisions
involve large numbers of incident partons and thus could allow invoking some “thermal-
isation” scheme through rescattering, in e+e− annihilation the predominant initial state
is one energetic qq̄ pair, and the number of hadronic secondaries per unit rapidity is too
small to consider statistical averages. The case in p− p/p− p̄ collisions is similar.
This enigma has led to the idea that all such collision experiments result in the formation
of a strong color field “disturbing” the physical vacuum. The disturbed vacuum then
recovers by producing hadrons according to a maximum entropy principle: the actually
observed final state is that with the largest phase space volume. While this provides an
intuitive basis for a statistical description, it does not account for a universal temperature.
Why don’t more energetic collisions result in a higher hadronization temperature?
A further piece in this puzzle is the observation that the value of the temperature deter-
mined in the mentioned collision studies is quite similar to the confinement/deconfinement
transition temperature found in lattice studies of strong interaction thermodynamics [20].
While hadronization in high energy collisions deals with a dynamical situation, the ener-
gy loss of fast color charges “traversing” the physical vacuum, lattice QCD addresses
the equilibrium thermodynamics of unbound vs. bound color charges. Why should the
resulting critical temperatures be similar or even identical?
We shall here consider these phenomena as reflections of the QCD counterpart of the
Hawking radiation emitted by black holes [10]. These ultimate stellar states provide a
gravitational form of confinement and hence, as already noted, their physics was quite
soon compared to that of color confinement in QCD [1, 2], where colored constituents
are confined to “white holes” (colorless from the outside, but colored inside). It should
be emphasized from the outset that in contrast to the original black hole physics in
gravitation, where confinement is on a classical level complete, in QCD confinement refers
only to color-carrying constituents; thus, e.g., photons or leptons are not affected.
In black hole physics, as noted above, it was shown that the event horizon for systems
undergoing uniform acceleration leads to quantum tunnelling and hence to thermal radia-
tion [11]. Our aim here is to show that such Hawking-Unruh radiation, as obtained in the
specific situation of QCD, provides a viable account for the thermal behavior observed
in multihadron production by high energy collisions. Furthermore, in the process we also
want to elucidate a bit the common origin of the “limiting temperature” concepts which
have arisen in strong interaction physics over the years.
We begin by reviewing those features of black hole physics and Hawking radiation which
are relevant for our considerations, and then discuss how they can be implemented in
QCD. In particular, we show that modifications of the effective space-time structure, in a
perturbative approach as well as in a non-perturbative treatment based on a large-scale
dilaton field, lead to an event horizon in QCD.
Following this, we present the main conceptual consequences of our conjecture.
• Color confinement and the instability of the physical vacuum under pair produc-
tion form an event horizon for quarks, allowing a transition only through quantum
tunnelling; this leads to thermal radiation of a temperature TQ determined by the
string tension.
• Hadron production in high energy collisions occurs through a succession of such
tunnelling processes. The resulting cascade is a realization of the same partition
process which leads to a limiting temperature in the statistical bootstrap and dual
resonance models.
• The temperature TQ of QCD Hawking-Unruh radiation can depend only on the
baryon number and the angular momentum of the deconfined system. The former
could provide a dependence of TQ on the baryon number density, while the angular
momentum pattern of the radiation allows a centrality-dependence of TQ and elliptic
flow.
• In kinetic thermalization, the initial state information is successively lost through
collisions, converging to a time-independent equilibrium state. In contrast, the
stochastic QCD Hawking radiation is “born in equilibrium”, since quantum tun-
nelling a priori does not allow information transfer.
2 Event Horizons in Gravitation and in QCD
2.1 Black holes
A black hole is formed as the final stage of a neutron star after gravitational collapse [24].
It has a mass M concentrated in such a small volume that the resulting gravitational field
confines all matter and even photons to remain inside the event horizon R of the system:
no causal connection with the outside is possible. As a consequence, black holes have
three (and only three) observable properties: mass M , charge Q and angular momentum
J . This section will address mainly black holes with Q = J = 0; we shall come back to
the more general properties in section 4. We use units of h̄ = c = 1.
The event horizon appears in a study of the gravitational metric, which in flat space has
the form
ds2 = g dt2 − g−1 dr2 − r2[dθ2 + sin2 θdφ2], (3)
using units where c = 1. The field strength of the interaction is contained in the coefficient
g(r),
g(r) =
, (4)
leading back to the Minkowski metric in the large distance limit r → ∞. The vanishing
of g(r) specifies the Schwarzschild radius R as event horizon,
R = 2GM. (5)
It is interesting to note that the mass of a black hole thus grows linearly with R, analogous
to the behaviour of the confining potential in strong interactions: M(R) = (2G)−1 R.
Classically, a black hole would persist forever and remain forever invisible. On a quan-
tum level, however, its constituents (photons, leptons and hadrons) have a non-vanishing
chance to escape by tunnelling through the barrier presented by the event horizon. Equiv-
alently, we can say that the strong force field at the surface of the black hole can bring
vacuum fluctuations on-shell. The resulting Hawking radiation [10] cannot convey any
information about the internal state of the black hole; it must be therefore be thermal.
For a non-rotating black hole of vanishing charge (denoted as Schwarzschild black hole),
the first law of thermodynamics,
dM = TdS (6)
combined with the area law for the black hole entropy [25],
leads to the corresponding radiation temperature
TBH =
. (8)
This temperature is inversely proportional to the mass of the black hole, and since the
radiation reduces the mass, the radiation temperature will increase with time, as the black
hole evaporates. For black holes of stellar size, however, one finds TBH <∼ 2 × 10
−8 ◦K,
which is many orders of magnitude below the 2.7 ◦K cosmic microwave background, and
hence not detectable.
It is instructive to consider the Schwarzschild radius of a typical hadron, assuming a mass
m ∼ 1 GeV:
Rhadg ≃ 1.3× 10−38 GeV−1 ≃ 2.7× 10−39 fm. (9)
To become a gravitational black hole, the mass of the hadron would thus have to be
compressed into a volume more than 10100 times smaller than its actual volume, with a
radius of about 1 fm. On the other hand, if instead we increase the interaction strength
from gravitation to strong interaction [1], we gain in the resulting “strong” Schwarzschild
radius Rhads a factor
, (10)
where αs is the dimensionless strong coupling and Gm
2 the corresponding dimensionless
gravitational coupling for the case in question. This leads to
Rhads ≃
; (11)
with the effective value of αs ∼ O(1) we thus get Rhads ∼ O(1) fm.∗ In other words,
the confinement radius of a hadron is about the size of its “strong” Schwarzschild radius,
so that we could consider quark confinement as the strong interaction version of the
gravitational confinement in black holes [1, 2].
We had seen that the mass of a black hole grows linearly with the event horizon, M =
(1/2G)R, so that in gravitation 1/2G plays the role of the string tension in strong inter-
action physics. The replacement GM2 → αs here leads to
σ ≃ m
≃ 0.16 GeV2, (12)
if one uses the mentioned effective saturation value αs ≃ 3 [26]. The value of αs ∼ 1 thus
gives a reasonable string tension as well as a reasonable radius.
2.2 Quasi-Abelian case
The appearence of an event horizon occurs in general relativity through the modification
of the underlying space-time structure by the gravitational interaction. Such modifica-
tions have also been discussed for other interactions. In particular, it was noted that
in electrodynamics, non-linear in-medium effects can lead to photons propagating along
geodesics which are not null in Minkowski space-time; this can even lead to photon trap-
ping, restricting the motion of photons to a compact region of space [27]. Thus, an
effective Lagrangian L(F ) depending on a one-parameter background field, F = FµνF µν ,
results in a modified metric
gµν = ηµνL′ − 4FαµF αν L′′, (13)
where the primes indicate first and second derivatives with respect to F . Hence
g00 = L′ − 4FL′′ = 0 (14)
defines the radius of the compact region of the theory, i.e., the counterpart of a black
hole [27].
QCD is an inherently non-linear theory, with the physical vacuum playing the role of a
medium [28]. The general structure of the effective Lagrangian in a background field F ,
compatible with gauge invariance, renormalization group results [3] and trace anomaly,
has the unique form [29]
LQCD =
g2(gF )
µνǫ(gF ). (15)
∗In fact, some studies [26] indicate that at large distances, the strong coupling freezes at αs ≃ 3; in
that case the corresponding radius becomes Rhad
≃ 1 fm.
Here ǫ(gF ) is the dielectric “constant” of the system in the presence of the background
field; the F -dependence of ǫ(F ) effectively turns the QCD vacuum into a non-linear
medium. On a one-loop pertubative level we have
ǫ(gF ) ≃ 1− β0
, (16)
where β0 = (11Nc − 2Nf)/48π2, with Nc and Nf specifying the number of colors and
flavors, respectively. Using this form in the formalism of [27] leads to gt changing sign
(i.e., gt = 0) at
gF ∗ = Λ2 exp{−4π/β0g2}, (17)
indicating a possible horizon at r∗ ∼ 1/
gF ∗ [30]. It is clear that this line of argument
can at best provide some hints, since we used the lowest order perturbative form of the
beta-function, even though at the horizon perturbation theory will presumably break
down. Nevertheless, we believe that it suggests the possibility of an event horizon for
QCD; the crucial feature is the asymptotic freedom of QCD [3], which leads to ǫ < 1 and
allows g00 to vanish even without the external medium effects required in QED.
2.3 Non-Abelian case
Indeed, a different and more solid suggestion that in QCD there is an event horizon comes
from studying the theory on a curved background. For gluodynamics, such a program is
discussed in [7]. Classical gluodynamics is a scale-invariant theory, but quantum fluctu-
ations break this invariance, with the trace of the energy-momentum tensor introducing
non-perturbative effects, associated with the vacuum energy density ǫvac. It was shown [6]
that low-energy theorems can be used to determine the form of the effective Yang-Mills
Lagrangian in a curved but conformally flat metric
gµν(x) = ηµνe
h(x), (18)
where the dilaton field h(x) is coupled to the trace of the energy momentum tensor, θµµ.
The resulting action has the form
eh(∂µh)
2 − 1
(F aµν)
2 + e2h(ǫvac − 1
θµµ|pert).
; (19)
here ǫvac is the absolute value of the energy density of the vacuum and mG the dilaton
mass; the trace of the energy-momentum tensor has been separated into perturbative and
non-perturbative contributions,
θµµ = θ
µ|pert+ < θµµ >= θµµ|pert − 4ǫvac. (20)
The crucial point for our considerations is that the first term of eq. (19) can be written
eh(∂µh)
2 = R
−g, (21)
defining R as the Ricci scalar of the theory. Hence eq. (19) has the structure of an
Einstein-Hilbert Lagrangian of gluodynamics in the presence of an effective gravitation,
(F aµν)
2 + e2h(ǫvac − 1
θµµ|pert).
; (22)
where G is now given by
. (23)
The relation 1/2G → σ between G and the string tension conjectured above then leads
. (24)
On the other hand, the string tension is just the energy density of the vacuum times the
transverse string area,
σ = ǫvacπr2T . (25)
Combining relations (24) and (25), we have
≃ 0.4 fm, (26)
using mG ≃ 1.5 GeV for the scalar glueball mass. Eq. (26) thus gives us the transverse
extension or horizon of the string.
From eqs. (24) or (25) we can obtain a further consistency check. Given the glueball mass
and the string tension σ ≃ 0.16 GeV2, we find for the vacuum energy density
ǫvac ≃ 3
σ m2G ≃ 0.013 GeV4 ≃ 1.7 GeV/fm
. (27)
This is the value for pure gluodynamics; since the energy density is related to the trace
of the energy–momentum tensor by the relation (20), and
θµµ =
(F aµν)
2 ≃ −b g
(F aµν)
2, (28)
with the coefficient b = 11Nc−2Nf of the β-function, we can estimate that for three-flavor
ǫvacQCD =
11Nc − 2Nf
ǫvac =
ǫvac ≃ 0.01 GeV4, (29)
which is in perfect agreement with the original value of the gluon condensate [8]
≃ 0.012 GeV4; (30)
note that ǫvacQCD = 27/32 〈(αs/π)G2〉).
3 Hyperbolic Motion and Hawking-Unruh Radiation
In general relativity, the event horizon appeared as consequence of the geometrized grav-
itational force, but its occurrence and its role for thermal radiation was soon generalized
by Unruh [11]. A system undergoing uniform acceleration a relative to a stationary ob-
server eventually reaches a classical turning point and thus encounters an event horizon.
Let us recall the resulting hyperbolic motion [34]. A point mass m subject to a constant
force F satisfies the equation of motion
1− v2
= F, (31)
where v(t) = dx/dt is the velocity, normalized to the speed of light c = 1. This equation
is solved by the parametric form through the so-called Rindler coordinates,
x = ξ cosh aτ t = ξ sinh aτ, (32)
where a = F/m denotes the acceleration in the instantaneous rest frame of m, and τ
the proper time, with dτ =
1− v2dt. If we impose the boundary condition that the
velocity at t = 0 vanishes, we have ξ = 1/a and x(t=0) = 1/a. The resulting world line
is shown in Fig. 1. It corresponds to the mass m coming from x = ∞ at t = −∞ at with
a velocity arbitrarily close to that of light, decelerating uniformly until it comes to rest
at the classsical turning point xH = −(1/a), t = 0. Subsequently, it accelerates again and
returns to x = ∞ at t = ∞, approaching the speed of light. For given a, the light cone
originating at a distance xH = 1/a away from the turning point of m defines a space-time
region inaccessible to m: no photon in this region can (classically) ever reach m, in much
the same way as photons cannot escape from a black hole. Here the acceleration is crucial,
of course; if m stops accelerating, it will eventually become visible in the “hidden region”.
mass m
region
hidden
event horizon
Figure 1: Hyberbolic motion
The metric of such a accelerating system becomes in spherical coordinates [31]
ds2 = ξ2a2dτ 2 − dξ2 − ξ2 cosh2 aτ(dθ2 + sin2 θdφ2), (33)
which we want to compare to the black hole metric (3). Making in the latter the coordinate
transformation [32]
, (34)
where the surface gravity κ is given by
, (35)
we obtain for r → R the black hole form
ds2 = η2κ2dt2 − dη2 −R2(dθ2 + sin2θdφ2). (36)
When we compare eqs. (33) and (36), it is evident that the system in uniform acceleration
can be mapped onto a spherical black hole, and vice versa, provided we identify the surface
gravity κ with the acceleration a.
The vacuum through which m travels is, for a stationary observer, empty space. On a
quantum level, however, it contains vacuum fluctuations. The accelerating mass m can
bring these on-shell, using up a (small) part of its energy, so that for m the vacuum
becomes a thermal medium of temperature
. (37)
Consider such a fluctuation into an e+e− pair, flying apart in opposite directions. One
electron is absorbed by the mass m, the other penetrates into the “hidden region” and can
never be detected by m (see Fig. 2. Since thus neither an observer on m nor a stationary
observer in the hidden region can ever obtain access to full information, each will register
the observed radiation as thermal (Einstein-Podolsky-Rosen effect [33]). In other words:
the accelerating mass m sees the vacuum as a physical medium of temperature TU , while
a stationary observer in the hidden region observes thermal radiation of temperature TU
as a consequence of the passing of m.
mass m
Figure 2: Unruh radiation
We here also mention that the entropy in the case of an accelerating system again becomes
1/4 of the event horizon area, as in the black hole case, so that also here the correspondence
remains valid [35].
In the case of gravity, we have the force
F = ma = G
, (38)
on a probe of mass m. With R = 2 GM for the (Schwarzschild) black hole radius, we have
a = 1/(4 GM) for the acceleration at the event horizon and hence the Unruh temperature
(37) leads back to the Hawking temperature of eq. (8).
In summary, we note that constant acceleration leads to an event horizon, which can be
surpassed only by quantum tunnelling and at the expense of complete information loss,
leading to thermal radiation as the only resulting signal.
4 Pair Production and String Breaking
In the previous section, we had considered a classical object, the mass m, undergoing
accelerated motion in the physical vacuum; because of quantum fluctuations, this vacuum
appears to m as a thermal medium of temperature TU . In this section, we shall first
address the modifications which arise if the object undergoing accelerated motion is itself
a quantum system, so that in the presence of a strong field it becomes unstable under
pair production. Next we turn to the specific additional features which come in when
the basic constituents are subject to color confinement and can only exist in color neutral
bound states.
As starting point, we consider two-jet e+e− annihilation at cms energy
e+e− → γ∗ → qq̄ → hadrons. (39)
The initially produced qq̄ pair flies apart, subject to the constant confining force given
by the string tension σ; this results in hyperbolic motion [14] of the type discussed in the
previous section. At t = 0, the q and q̄ separate with an initial velocity v0 = p/
p2 +m2,
where p ≃
s/2 is the momentum of the primary constituents in the overall cms and m
the effective quark mass. We now have to solve Eq. (31) with this situation as boundary
condition; the force
F = σ, (40)
is given by the string tension σ binding the qq̄ system. The solution is
x̃ = [1−
1− v0t̃+ t̃2] (41)
with x̃ = x/x0 and t̃ = t/x0; here the scale factor
1− v20
γ (42)
is the inverse of the acceleration a measured in the overall cms. The velocity becomes
v(t) =
(v0/2)− t̃
1− v0t̃+ t̃2
; (43)
it vanishes for
t̃∗ =
⇒ t∗ = v0
γ, (44)
thus defining
x(t∗) =
1− (v20/4)
as classical turning point and hence as the classical event horizon measured in the overall
cms (see Fig. 3).
xx(t  )*xQ
Figure 3: Classical and quantum horizons in qq̄ separation
Eq. (45) allows the q and the q̄ to separate arbitrarily far, provided the pair has enough
initial energy; this clearly violates color confinement. Our mistake was to consider the
qq̄ system as classical; in quantum field theory, it is not possible to increase the potential
energy of a given qq̄ state beyond the threshold value necessary to bring a virtual qq̄ pair
on-shell. In QED, the corresponding phenomenon was addressed by Schwinger [36], who
showed that in the presence of a constant electric field of strength E the probability of
producing an electron-positron pair is given by
P (M, E) ∼ exp{−πm2e/eE}, (46)
with me denoting the electron mass and e denoting the electric charge. This result is
in fact a specific case of the Hawking-Unruh phenomenon, as shown in [16]. In QCD,
we expect a similar effect when the string tension exceeds the pair production limit, i.e.,
σ x > 2m (47)
where m specifies the effective quark mass. Beyond this point, any further stretching of
the string is expected to produce a qq̄ pair with the probability
P (M,σ) ∼ exp{−πm2/σ}, (48)
with the string tension σ replacing the electric field strength eE . This string breaking
acts like a quantum event horizon xq = 2 m/σ, which becomes operative long before
the classical turning point is ever reached (see Fig. 3). Moreover, the resulting allowed
separation distance for our qq̄ pair, the color confinement radius xQ, does not depend on
the initial energy of the primary quarks.
There are some important differences between QCD and QED. In case of the latter, the
initial electric charges which lead to the field E can exist independently in the physical
vacuum, and the produced pair can be simply ionized into an e+ and an e−. In contrast,
neither the primary quark nor the constituents of the qq̄ pair have an independent ex-
istence, so that in string breaking color neutrality must be preserved. As a result, the
Hawking radiation in QCD must consist of qq̄ pairs, and these can be produced in an in-
finite number of different excitation states of increasing mass and degeneracy. Moreover,
the qq̄ pair spectrum is itself determined by the strength σ of the field, in contrast to the
exponent m2e/E in eq. (46), where the value of E has no relation to the electron mass me.
Hadron production in e+e− annihilation is believed to proceed in the form of a self-similar
cascade [37, 38]. Initially, we have the separating primary qq̄ pair,
γ → [qq̄] (49)
where the square brackets indicate color neutrality. When the energy of the resulting
color flux tube becomes large enough, a further pair q1q̄1 is excited from the vacuum by
two-gluon exchange (see Fig. 4),
γ → [q[q̄1q1]q̄]. (50)
Although the new pair is at rest in the overall cms, each of its constituents has a transverse
momentum kT determined, through the uncertainty relation, by the transverse dimension
rT of the flux tube. The slow q̄1 now screens the fast primary q from its original partner
q̄, with an analoguous effect for the q1 and the primary antiquark. To estimate the qq̄
separation distance at the point of pair production, we recall that the thickness of the
flux tube connecting the qq̄ pair is in string theory given by [41]
r2T =
2k + 1
, (51)
where K is the string length in units of an intrinsic vibration measure. Lattice studies [43]
show that for strings in the range of 1 - 2 fm, the first string excitation dominates, so that
we have
rT = c0
, (52)
with c0 ≃ 1 or slightly larger. Higher excitations lead to a greater thickness and eventually
to a divergence (the “roughening” transition). From the uncertainty relation we then have
. (53)
With this transverse energy is included in eq. (47), we obtain for the pair production
separation xQ
σxq = 2
m2 + k2T ⇒ xq ≃
m2 + (πσ/2 c20) ≃
≃ 1 fm, (54)
with σ = 0.2 GeV2, m2 ≪ σ, and c0 ≃ 1.
Once the new pair is present, we have a color-neutral system qq̄1q1q̄; but since there is a
sequence of connecting string potentials qq̄1, q̄1q1 and q1q̄, the primary string is not yet
broken. To achieve that, the binding of the new pair has to be overcome, i.e., the q1 has to
tunnel through the barrier of the confining potential provided by q̄1, and vice versa. Now
the q excerts a longitudinal force on the q̄1, the q̄ on the q1, resulting in a longitudinal
acceleration and ordering of q1 and q̄1. When (see Fig. 4)
σx(q1q̄1) = 2
m2 + k2T , (55)
Figure 4: String breaking through qq̄ pair production
the q̄1 reaches its q1q̄1 horizon; on the other hand, when
σx(qq̄1) = 2
m2 + k2T , (56)
the new flux tube qq̄1 reaches the energy needed to produce a further pair q2q̄2. The
q̄2 screens the primary q from the q1 and forms a new flux tube qq̄2. At this point, the
original string is broken, and the remaining pair q̄1q2 form a color neutral bound state
which is emitted as Hawking radiation in the form of hadrons, with the relative weights
of the different states governed by the corresponding Unruh temperature. The resulting
pattern is schematically illustrated in Fig. 4.
To determine the temperature of the hadronic Hawking radiation, we return to the original
pair excitation process. To produce a quark of momentum kT , we have to bring it on-shell
and change its velocity from zero to v = kT/(m
2 + k2T )
1/2 ≃ 1. This has to be achieved in
the time of the fluctuation determined by the virtuality of the pair, ∆τ = 1/∆E ≃ 1/2kT .
The resulting acceleration thus becomes
≃ 2 kT ≃
2πσ/c0 ≃ 1 GeV, (57)
which leads to
≃ 160− 180 MeV (58)
for the hadronic Unruh temperature. It governs the momentum distribution and the
relative species abundances of the emitted hadrons.
A given step in the evolution of the hadronization cascade of a primary quark or antiquark
produced in e+e− annihilation thus involves several distinct phenomena. The color field
created by the separating q and q̄ produces a further pair q1q̄1 and then provides an
acceleration of the q1, increasing its longitudinal momentum. When it reaches the q1q̄1
confinement horizon, still another pair q2q̄2 is excited; the state q̄1q2 is emitted as a
hadron, the q̄2 forms together with the primary q a new flux tube. This pattern thus
step by step increases the longitudinal momentum of the “accompanying” q̄i as well as
of the emitted hadron. This, together with the energy of the produced pairs, causes a
corresponding deceleration of the primary quarks q and q̄, in order to maintain overall
energy conservation. In Fig. 5, we show the world lines given by the acceleration q̄i → q̄i+1
(qi → qi+1) and that of formation threshold of the hadrons q̄iqi+1 and the corresponding
antiparticles.
hadron radiation
quark acceleration
Figure 5: Quark acceleration and hadronization world lines
The energy loss and deceleration of the primary quark q in this self-similar cascade,
together with the acceleration of the accompanying partner q̄i from the successive pairs
brings q and q̄i closer and closer to each other in momentum, from an initial separation
qq̄1 of
s/2, until they finally are combined into a hadron and the cascade is ended. The
resulting pattern is shown in Fig. 6.
hadrons
Figure 6: Hadronization in e+e− annihilation
The number of emitted hadrons, the multiplicity ν(s), follows quite naturally from the
picture presented here. The classical string length, in the absence of quantum pair for-
mation, is given by the classical turning point determined in eq. (45). The thickness of a
flux tube of such an “overstretched” string is known [41]; from eq. (51) we get
R2T =
2k + 1
ln 2K, (59)
where K is the string length. From eq. (45) we thus get
R2T ≃
s (60)
for the flux tube thickness in the case of the classical string length. In parton language, the
logarithmic growth of the transverse hadron size is due to parton random walk (”Gribov
diffusion” [42]); this phenomenon is responsible for diffraction cone shrinkage in high–
energy hadron scattering.
Because of pair production, the string breaks whenever it is stretched to the length xq
given in eq. (54); its thickness rT at this point is given by eq. (51). The multiplicity can
thus be estimated by the ratio of the corresponding classical to quantum transverse flux
tube areas,
ν(s) ∼ R
s, (61)
and is found to grow logarithmically with the e+e− annihilation energy, as is observed
experimentally over a considerable range.
We note here that in our argumentation we have neglected parton evolution, which would
cause the emitted radiation (e.g., q̄1q2 in Fig. 4) to start another cascade of the same
type. Such evolution effects result eventually in a stronger increase of the multiplicity.
The formation of a white hole does not affect the production of hard processes at early
times (e.g., multiple jet production), which is responsible for an additional growth of the
measured multiplicity.
A further effect we have not taken into account here is parton saturation. At sufficiently
high energy, stronger color fields can lead to gluon saturation and thus to a higher temper-
ature determined by the saturation momentum [16]. The resulting system then expands
and hadronizes at the universal temperature determined by the string tension.
It interesting to compare the separation of two energetic light quarks, as we have consid-
ered here, with that of two static heavy quarks Q and Q̄. From quarkonium studies it is
known that
2(MD −mc) ≃ 2(MB −mb) ≃ 1.2 GeV, (62)
where MD(MB) and mc(mb) are the masses of open charm (beauty) mesons and of the
corresponding charm (beauty) quarks, respectively. The energy needed to separate a
heavy QQ̄ pair thus is independent of the mass of the heavy quarks, indicating that
the string breaking involved here is really a consequence of the vacuum, through qq̄ pair
excitation. With
σxQ ≃ 1.2 GeV ⇒ xQ ≃ 1.2 fm (63)
we find that the resulting separation threshold for pair excitation agrees well with that
found above in eq. (54). Lattice QCD studies lead to similar results.
Up to now, we have considered hadron production in e+e− annihilation, in which the
virtual photon produces a confined colored qq̄ pair as a “white hole”. Turning now to
hadron-hadron collisions, we note that here two incident white holes combine to form a
new system of the same kind, as schematically illustrated in Fig. 7. Again the resulting
string or strong color field produces a sequence of qq̄ pairs of increasing cms momentum,
leading to the well-known multiperipheral hadroproduction cascade shown in Fig. 8. We
recall here the comments made above concerning parton evolution and saturation; in
hadronic collisions as well, these phenomena will affect the multiplicity, but not the relative
abundances.
In the case of heavy ion collisions, two new elements enter. The resulting systems could
now have an overall baryon number, up to B = 400 or more. To take that into account,
we need to consider the counterpart of charged black holes. Furthermore, in heavy ion
collisions the resulting hadron production can be studied as function of centrality, and
peripheral collisions could lead to an interaction region with an effective overall angular
momentum. Hence we will also consider rotating black holes. In the next section, we then
summarize the relevant features of black holes with Q 6= 0, J 6= 0.
γ h h
(a) (b)
Figure 7: “White hole” structure in e+e− annihilation (a) and hadronic collisions (b)
hadrons
Figure 8: Hadronization in hadron-hadron collisions
5 Charged and Rotating Black Holes
As mentioned, an outside observer the only characteristics of a black hole are its mass M ,
its electric charge Q, and its spin or angular momentum J . Hence any further observables,
such as the event horizon or the Hawking temperature, must be expressable in terms of
these three quantities.
The event horizon of a black hole is created by the strong gravitational attraction, which
leads to a diverging Schwarzschild metric at a certain value of the spatial extension R.
Specifically, the invariant space-time length element ds2 is at the equator given by
ds2 = (1− 2GM/R) dt2 −
1− 2GM/R
dr2, (64)
with r and t for flat space and time coordinates; it is seen to diverge at the Schwarzschild
radius RS = 2GM . If the black hole has a net electric charge Q, the resulting Coulomb
repulsion will oppose and hence weaken the gravitational attraction; this will in turn
modify the event horizon. As a result, the corresponding form (denoted as Reissner-
Nordström metric) becomes
ds2 = (1− 2GM/R +GQ2/R2) dt2 − 1
1− 2GM/R +GQ2/R2
dr2. (65)
For this, the divergence leads to the smaller Reissner-Nordström radius
RRN = GM (1 +
1−Q2/GM2), (66)
which reduces to the Schwarzschild radius RS for Q = 0. The temperature of the Hawking
radiation now becomes [24, 39]
TBH(M,Q) = TBH(M, 0)
1−Q2/GM2
1−Q2/GM2 ) 2
; (67)
its functional form is illustrated in Fig. 9. We note that with increasing charge, the
Coulomb repulsion weakens the gravitational field at the event horizon and hence de-
creases the temperature of the corresponding quantum excitations. As Q2 → GM2, the
gravitational force is fully compensated and there is no more Hawking radiation.
0.5 1.0
Q / GM
T   (M,Q) / T   (M,0)
BH BH
Figure 9: Radiation temperature for a charged black hole
In a similar way, the effect of the angular momentum of a rotating black hole can be
incorporated. It is now the centripetal force which counteracts the gravitational attraction
and hence reduces its strength. The resulting Kerr metric must take into account that in
this case the rotational symmetry is reduced to an axial symmetry, and with θ denoting
the angle relative to the polar axis θ = 0, it is (at fixed longitude) given by
ds2 =
1− 2GMR
R2 + j2 cos2 θ
dt2 − R
2 + j2 cos2 θ
R2 − 2GMR + j2
dr2 − (R2 + j2 cos2 θ) dθ2. (68)
The angular momentum of the black hole is here specified by the parameter j = J/M ; for
a = 0, we again recover the Schwarzschild case. The general situation is now somewhat
more complex, since eq. (68) leads to two different divergence points. The solution
RK = GM (1 +
1− j2/(GM)2 ) (69)
defines the actual event horizon, corresponding to absolute confinement. But the resulting
black hole is now embedded in a larger ellipsoid
RE = GM (1 +
1− [j2/(GM)2] cos2 θ), (70)
as illustrated in Fig. 10. The two surfaces touch at the poles, and the region between them
is denoted as the ergosphere. Unlike the black hole proper, communication between the
ergosphere and the outside world is possible. Any object in the ergosphere will, however,
suffer from the rotational drag of the rotating black hole and thereby gain momentum. We
shall return to this shortly; first, however, we note that the temperature of the Hawking
radiation from a rotating black hole becomes
TBH(M,J) = TBH(M, 0)
1− j2/(GM)2
1− j2/(GM)2
. (71)
For a non-rotating black hole, with j = 0, this also reduces to the Hawking temperature
for the Schwarzschild case.
ergosphere
black hole
Figure 10: Geometry of a rotating black hole
To illustrate the effect of the ergosphere, imagine radiation from a Schwarzschild black hole
emitted radially outward from the event horizon. In the case of a Kerr black hole, such an
emission is possible only along the polar axis; for all other values of θ, the momentum of the
emitted radiation (even light) will increase due to the rotational drag in the ergosphere.
This effect ceases only once the radiation leaves the ergosphere. Since the amount of
drag depends on θ, the momentum of the radiation emitted from a rotating black hole,
as measured at large distances, will depend on the latitude at which it is emitted and
increase from pole to equator.
Finally, for completeness, we note that for black holes with both spin and charge (denoted
as Kerr-Newman), the event horizon is given by
RKN = GM (1 +
1− [Q2/GM2]− [j2/(GM)2]), (72)
and the radiation temperature becomes [24, 39]
TBH(M,Q, J) = TBH(M, 0, 0)
1− (GQ2 + j2)/(GM)2
1− (GQ2 + j2)/(GM)2 ) 2 + j2/(GM)2
. (73)
The decrease of TBH for Q 6= 0, J 6= 0 expresses the fact that both the Coulomb repulsion
and the rotational force counteract the gravitational attraction, and if they win, the black
hole is dissolved.
The dependence of a black hole on its basic properties M,Q, J is very similar to the
dependence of a thermodynamic system on a set of thermodynamic observables. The first
law of thermodynamics can be written as
dE = TdS + φdQ+ ωdJ, (74)
expressing the variation of the energy with entropy S, charge Q and spin J ; here φ denotes
the electrostatic potential per charge and ω the rotational velocity. The corresponding
relation in black hole thermodynamics becomes
dM = TBHdSBH + ΦdQ + ΩdJ, (75)
where the entropy SBH is defined as the area of the event horizon,
SBH =
π(R2KN + j
. (76)
The temperature is given by eq. (73), and
4πQRRN
, Ω =
specify the electrostatic potential Φ and the rotational velocity Ω.
The considerations of this section were for spherical black holes. As seen above, such
objects are in fact equivalent to uniformly accelerating systems. An application to actual
high energy collisions involves a further assumption. Thermal Hawking-Unruh radiation
arises already from a single QQ̄ system, as seen above in the discussion of e+e− anni-
hilation. If we treat the systems produced in heavy ion collisions as black holes of an
overall baryon number and/or an overall spin, we are assuming that the collision leads to
a large-scale collective system, in which each accelerating parton is affected by totality of
the other accelerating partons. This assumption clearly goes beyond our event horizon
conjecture and, in particular, it need not be correct in order to obtain thermal hadron
production.
6 Baryon Density and Angular Momentum
6.1 Vacuum Pressure and Baryon Repulsion
We now want to consider the extension of charged black hole physics to color confinement
in the case of collective systems with a net baryon number. In eq. (67) we had seen that
the reduction of the gravitational attraction by Coulomb repulsion in a charged black hole
modifies the event horizon and hence in turn also the temperature of Hawking radiation.
The crucial quantity here is the ratio Q2/GM2 of the repulsive overall Coulomb force,
Q2/R2, to the attractive overall gravitational force, GM2/R2, at the horizon.
In QCD, we have a “white” hole containing colored quarks, confined by chromodynamic
forces or, equivalently, by the pressure of the physical vacuum. If the system has a non-
vanishing overall baryon number, the baryon-number dependent interaction will also affect
the forces at the event horizon. The simplest instance of such a force the repulsion between
quarks due to Fermi statistics, but more generally, there will be repulsive effects of the
type present in cold dense baryonic matter, such as neutron stars. The resulting pressure
will modify the confinement horizon and hence lead to a corresponding modification of
the Hawking-Unruh temperature of hadronization.
By using the conjectured correspondence between black hole thermodynamics and the
thermodynamics of confined color charges, we translate black hole mass, charge and grav-
itational constant into white hole energy, net baryon number and string tension,
{M,Q,G} ↔ {E,B, 1/2σ}. (78)
Hence eq. (67) leads us to the relation
TQ(B) = TQ(B = 0)
1− 2σ B2/E2
1− 2σB2/E2 ) 2
; (79)
for the dependence of the hadronization temperature on the ratio of net baryon number
B and energy E, with TQ(B = 0) given by eq. (58). Its functional form is the same as
that illustrated in Fig. 9.
It would be interesting to test the prediction (79) against experimental data; one could
identify B with the net baryon number per unit rapidity dNB/dy and E with the total
transverse energy per unit rapidity dET/dy. The reduction of the hadronization temper-
ature with baryon number could thus occur in two ways. A sufficient decrease of the
collision energy, e.g. from peak SPS to AGS energy, will strongly reduce dET/dy, while
dNB/dy is not affected as much. This leads to the known decrease of T (µB) with increas-
ing µB [44], and it will be interesting to see if the form (79) agrees with the observed
behaviour. A second, novel possibility would be to consider hadrochemistry as a func-
tion of rapidity. At peak SPS energy, dNB/dy remains essentially constant out to about
y = 2, while dET/dy drops by more than a factor of two from y = 0 to y = 2 [45]. A
similar behaviour occurs at still lower collision energies. Hence it would seem worthwhile
to check if an abundance analysis at large y indeed shows the expected decrease of the
hadronization temperature.
6.2 Angular Momentum and Non-Central Collisions
The dependence of Hawking radiation on the angular momentum of the emitting sys-
tem introduces another interesting aspect for the “white hole evaporation” we have been
considering. Consider a nucleus-nucleus collision at non-zero impact parameter b. If the
interaction is of collective nature, the resulting interaction system may have some angular
momentum orthogonal to the reaction plane (see Fig. 11). In central collisions, this will
not be the case, nor for extremely peripheral ones, where one expects essentially just
individual nucleon-nucleon collisions without any collective effects.
If it possible to consider a kinematic region in which the interacting system does have an
overall spin, then the resulting Hawking radiation temperature should be correspondingly
reduced, as seen in eq. (71). The effect is not so easily quantified, but simply a reduction
Figure 11: Rotating interaction region in non-central AA collision
of the hadronization temperature for non-central collisions would quite indicative. Such a
reduction could appear only in the temperature determined by the relative abundances,
since, as we shall see shortly, the transverse momentum spectra should show modifications
due to the role of the ergosphere.
We next turn to the momentum spectrum of the Hawking radiation emitted from a ro-
tating white hole. As discussed in section 4, such radiation will exhibit an azimuthal
asymmetry due to the presence of the ergosphere, which by its rotation will affect the
momentum spectrum of any passing object. At the event horizon, the momentum of all
radiation is determined by the corresponding Hawking temperature (71); but the passage
of the ergosphere adds rotational motion to the emerging radiation and hence increases
its momentum. As a result, only radiation emitted directly along the polar axis will have
momenta as specified by the Hawking temperature; with increasing latitude θ (see Fig.
12)a, the rotation will increase the radiation momentum up to a maximum value in the
equatorial plane.
spectators spectators
ergosphere
Figure 12: Transverse plane view of a non-central AA collision
Hawking radiation from a rotating source thus leads for nuclear collisions quite naturally
to what in hydrodynamic studies is denoted as elliptic flow. It is interesting to note
that both scenarios involve collective effects: while in hydrodynamics, it is assumed that
non-central collisions lead to an azimuthally anisotropic pressure gradient, we have here
assumed that such collisions lead to an overall angular momentum of the emitting system.
Concluding this section we emphasize that the results obtained here for the Hawking
temperatures of systems with finite baryon density or with an effective overall spin depend
crucially on the assumption of collectivity. If the different nucleon-nucleon interactions
in a heavy ion collision do not result in sufficiently collective behavior, the corresponding
modifications of Tq do not apply. In the case of black holes with spin, we moreover have
no way to relate in a quantitative way centrality and overall spin. Both cases do show,
however, that such extensions lead to qualitatively reasonable modifications.
7 Temperature and Acceleration Limits
We had seen that the underlying confinement dynamics of high energy hadron collisions
and e+e− annihilation led to a limit on the acceleration (or the corresponding deceleration)
in the self-similar hadronization cascade - a limit which can be specified in terms of the
string tension. In turn, this led to a limiting Unruh hadronization temperature
. (80)
We emphasize that a Hawking-Unruh temperature as such can a priori have any value;
it is the universal limit on the acceleration that leads to a universal temperature for the
emitted hadron radiation.
In the study of strongly interacting matter, temperature limits are well-known and arise
for an ideal gas of different composite constituents (“resonances” or “fireballs” of varying
mass M), if the composition law provides a sufficiently fast increase of the degeneracy
ρ(M) with M . If the number of states of a constituent of mass M grows exponentially,
ρ(M) ∼ M−a exp{bM}, (81)
with constants a and b, then the grand canonical partition function for an ideal gas in a
volume V
Z(T, V ) =
(2π)3
dMρ(M)
d3p exp{−
p2 +M2 /T}
, (82)
diverges for
T > TH ≡ 1/b, (83)
so that the Hagedorn temperature TH [18] constitutes an upper limit for the temperature
of hadronic matter.
In the dual resonance model [46], the resonance composition pattern is governed by linearly
rising Regge trajectories,
α′M2n = n+ α0, n = 1, 2, ..., (84)
in terms of the universal Regge slope α′ ≃ 1 GeV−2 and a constant (of order unity)
specifying the family (π, ρ, ...). For an ideal resonance gas in D − 1 space and one time
dimension, one then obtains [47]
(D + 1), b = 2π
, (85)
leading to the temperature limit
. (86)
In string theory, the Regge resonance pattern is replaced by string excitation modes,
retaining the same underlying partition structure, with α′ = 1/2πσ relating Regge slope
and string tension. Hence we get
for the corresponding limiting temperature. For D = 4, this coincides with eq. (1)
from [16] and agrees within 20 % with the Unruh temperature (80) determined by the
lowest string excitation alone (c0 = 1 in eq. (58)).
Prior to the dual resonance model, Hagedorn had determined the level density ρ(M) of
fireballs composed of fireballs, requiring the same composition pattern at each level [18].
The resulting bootstrap condition leads to [48]
a = 3 b = r0
(2 ln 2− 1)
]−1/3
≃ r0, (88)
where r0 measures the range of the strong interaction. With r0 ≃ 1 fm, we thus get
TH ≃ 0.2 GeV for the limiting temperature of hadronic matter. If we identify r0 with the
pair production separation xq obtained in eq. (54), we get
and hence again agreement with the hadronic Unruh temperature (80).
Hadronic matter as an ideal gas of constituents with self-similar composition spectra
(“resonances of resonances” or “fireballs of fireballs”) thus leads to an upper limit of
the temperature, because the level density of such constituents increases exponentially.
What does this have to do with the limiting acceleration found in the qq̄ cascade of e+e−
annihilation?
To address this problem, it is useful to recall the underlying reason for the exponential
increase of the level density in the dual resonance model and the bootstrap model. The
common origin in both cases is a classical partition problem, which in its simplest form
[40] asks: how many ways ρ(M) are there to partition a given integer M into ordered
combinations of integers? As example, we have for M = 4 the partitions 4, 3+1, 1+3,
2+2, 2+1+1, 1+2+1, 1+1+2, 1+1+1+1; thus here ρ(M = 4) = 8 = 2M−1. It can be
shown that this is generally valid, so that
ρ(M) =
exp{M ln 2}. (90)
For a “gas of integers”, T0 = 1/ ln 2 would thus become the limiting temperature; the
crucial feature in thermodynamics is the exponential increase in the level density due to
the equal a priori weights given to all possible partitions.
Returning now to the quark cascade in e+e− annihilation, we note that the form we have
discussed above is a particular limiting case. We assumed that the color field of the
separating qq̄ excites in the first step one new pair from the vacuum; in principle, though
with much smaller probability, it can also excite two or more. The same is true at the
next step, when the tunnelling produces one further pair: here also, there can be two
or more. Thus the e+e− cascade indeed provides a partition problem of the same kind.
What remains to be shown are the two specific features of our case: that the dominant
decay chain is one where in each step one hadron is produced, which provides the constant
deceleration of the primary quark and antiquark.
The statistical bootstrap model as well as the dual resonance model lead to self-similar
decay cascades, starting from a massive fireball (or resonance), which decays into further
fireballs, and so on, until at the end one has light hadrons. In Fig. 13a we illustrate such
a cascade for the case where the average number k̄ of constituents per step in the decay
(or composition) partition pattern
M → M11 +M12 + ... +M1k; M11 → M21 +M22 + ...+M2k; ... (91)
is k = 3. In the statistical bootstrap model, k̄ can be can be determined [49]; it is found
that the crucial feature here is the power term multiplying the exponential increase in eq.
(81). For a < 5/2, the distribution in k is given by
F (k) =
(ln 2)k−1
(k − 1)!
, (92)
so that the average becomes
k̄ = 1 + 2 ln 2 ≃ 2.4. (93)
The dominant decay (∼ 70%) is thus into two constituents, with 24 % three-body and 6
% four-body decays. While in general the fireball mass M could decrease in each step by
M/k, i.e., by an amount depending on M , the case a < 5/2 is found to be dominated by
one heavy and one soft light hadron,
M → M1 + h1; M1 → M2 + h2; ... (94)
where hi denotes final hadrons; the pattern is shown in Fig. 13b. Moreover, the three-
and four-body decays also lead to one heavy state plus soft light hadrons. The decay thus
provides a uniform decrease of the fireball mass by the average hadron mass or transverse
energy.
(a) (b)
Figure 13: Fireball decay patterns
We therefore conclude that the hadronization pattern we had obtained for e+e− anni-
hilation is indeed also connected to the same partition problem as the one leading to
exponential level densities.
8 Stochastic vs. Kinetic Thermalization
In statistical mechanics, a basic topic is the evolution of a system of many degrees of
freedom from non-equilibrium to equilibrium. Starting from a non-equilibrium initial
state of low entropy, the system is assumed to evolve as a function of time through
collisions to a time-independent equlibrium state of maximum entropy. In other words,
the system loses the information about its initial state through a sequence of collisions
and thus becomes thermalized. In this sense, thermalization in heavy ion collisions was
studied as the transition from an initial state of two colliding beams of “parallel” partons
to a final state in which these partons have locally isostropic distributions. This “kinetic”
thermalization requires a sufficient density of constituents, sufficiently large interaction
cross sections, and a certain amount of time.
From such a point of view, the observation of thermal hadron production in high energy
collisions, in particular in e+e− and pp interactions, is a puzzle: how could these systems
ever “have reached” thermalization? Already Hagedorn [50] had therefore concluded
that the emitted hadrons were “born in equilibrium”. Given an exponentially increasing
resonance mass spectrum, it remained unclear why collisions should result in a thermal
system.
Hawking radiation provides a stochastic rather than kinetic approach to equilibrium, with
a randomization essentially provided by the quantum physics of the Einstein-Podolsky-
Rosen effect. The barrier to information transfer due the event horizon requires that
the resulting radiation states excited from the vacuum are distributed according to max-
imum entropy, with a temperature determined by the strength of the “confining” field.
The ensemble of all produced hadrons, averaged over all events, then leads to the same
equilibrium distribution as obtained in hadronic matter by kinetic equilibration. In the
case of a very high energy collision with a high average multiplicity already one event can
provide such equilibrium; because of the interruption of information transfer at each of
the successive quantum color horizons, there is no phase relation between two successive
production steps in a given event. The destruction of memory, which in kinetic equili-
bration is achieved through sufficiently many successive collisions, is here automatically
provided by the tunnelling process.
So the thermal hadronic final state in high energy collisions is not reached through a
kinetic process; it is rather provided by successively throwing dice.
9 Conclusions
We have shown that quantum tunnelling through the color confinement horizon leads to
thermal hadron production in the form of Hawking-Unruh radiation. In particular, this
implies:
• The radiation temperature TQ is determined by the transverse extension of the color
flux tube, giving
, (95)
in terms of the string tension σ.
• The multiplicity ν(s) of the produced hadrons is approximately given by the increase
of the flux tube thickness with string length, leading to
ν(s) ≃ ln
s, (96)
where
s denotes the cms collision energy. Parton evolution and gluon saturation
will, however, increase this, as will early hard production. The universality of the
resulting abundances is, however, not affected.
• The temperature of Hawking radiation can in general depend on the charge and
the angular momentum of the emitting system. The former here provides a baryon-
number dependence of the hadronization temperature and predicts a decrease of TQ
for sufficiently high baryon density. The latter provides the basis for the possibility
of elliptic flow and of a dependence of TQ on the centrality of AA collisions.
• The limiting temperature obtained in the statistical bootstrap and the dual reso-
nance or string model arises from a self-similar composition pattern leading to an
exponentially growing level density. We find that the underlying partition problem
also leads to the cascade form obtained for hadron emission in high energy collisions,
so that the dynamic and the thermodynamic limits have the same origin.
• In statistical QCD, thermal equilibrium is reached kinetically from an initial non-
equilibrium state, with memory destruction through successive interactions of the
constituents. In high energy collisions, tunnelling prohibits information transfer and
hence leads to stochastic production, so that we have a thermal distribution from
the outset.
We close with a general comment. In astrophysics, Hawking-Unruh radiation has so far
never been observed. The thermal hadron spectra in high energy collisions may thus
indeed be the first experimental instance of such radiation, though in strong interaction
instead of gravitation.
Acknowledgements
D. K. is grateful to E. Levin and K. Tuchin for useful and stimulating discussions. H. S.
thanks M. Kozlov for helpful discussions and a critical reading to the text. The work of
D. K. was supported by the U.S. Department of Energy underContract No. DE-AC02-
98CH10886.
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	Introduction
	Event Horizons in Gravitation and in QCD
	Black holes
	Quasi-Abelian case
	Non-Abelian case
	Hyperbolic Motion and Hawking-Unruh Radiation
	Pair Production and String Breaking
	Charged and Rotating Black Holes
	Baryon Density and Angular Momentum
	Vacuum Pressure and Baryon Repulsion
	Angular Momentum and Non-Central Collisions
	Temperature and Acceleration Limits
	Stochastic vs. Kinetic Thermalization
	Conclusions

  We conjecture that because of color confinement, the physical vacuum forms an
event horizon for quarks and gluons which can be crossed only by quantum
tunneling, i.e., through the QCD counterpart of Hawking radiation by black
holes. Since such radiation cannot transmit information to the outside, it must
be thermal, of a temperature determined by the chromodynamic force at the
confinement surface, and it must maintain color neutrality. We explore the
possibility that the resulting process provides a common mechanism for thermal
hadron production in high energy interactions, from $e^+e^-$ annihilation to
heavy ion collisions.

Introduction
The aim of this paper is to develop a conceptual framework for a universal form of thermal
multihadron production in high energy collisions. Our work is based on two seemingly
disjoint observations:
• Color confinement in QCD does not allow colored constituents to exist in the physi-
cal vacuum, and thus in some sense creates a situation similar to the gravitational
confinement provided by black holes.
• Numerous high energy collision experiments have provided strong evidence for the
thermal nature of multihadron production, indicating a universal hadronization tem-
perature TH ≃ 150− 200 MeV.
We want to suggest that quantum tunnelling through a color event horizon, as QCD
counterpart of Hawking-Unruh radiation from black holes, can relate these observations
in a quite natural way.
The idea that the color confinement of quarks and gluons in hadrons may have a dual
description in terms of a theory in curved space–time is not new. Both gravitational
confinement of matter inside a black hole [1] and the de Sitter solution of the Einstein
equations with a cosmological constant describing a “closed” universe of constant cur-
vature [2] have been proposed as possible descriptions of quark confinement. Soon it
became clear that asymptotic freedom [3] and the scale anomaly [4,5] in QCD completely
determine the structure of low–energy gluodynamics [6]. This effective theory can be con-
veniently formulated in terms of the Einstein–Hilbert action in a curved background. At
shorter distances (inside hadrons), the effective action has the form of classical Yang–Mills
theory in a curved (but conformally flat) metric [7]. The “cosmological constant” present
in this theory corresponds to the non–perturbative energy density of the vacuum, or the
“gluon condensate” [8].
It is worthwhile to also mention here the well–known conjectured holographic correspon-
dence between the large N limit of supersymmetric Yang–Mills theory in 3+1 dimensions
and supergravity in an anti–de Sitter space–time sphere, AdS5 × S5 [9]. This example
illustrates the possible deep relation between Yang–Mills theories and gravity; however,
a conformal theory clearly differs from the examples noted above, in which the scale
anomaly (describing the breaking of conformal invariance by quantum effects) was used
as a guiding principle for constructing an effective curved space–time description.
Let us assume that color confinement indeed allows a dual description in terms of the
gravitational confinement of matter inside black holes. What are the implications of
this hypothesis for hadronic physics? Hawking [10] showed that black holes emit thermal
radiation due to quantum tunneling through the event horizon. Shortly afterwards, Unruh
[11] demonstrated that the presence of an event horizon in accelerating frames also leads
to thermal radiation. It was soon conjectured that the periodic motion of quarks in a
confining potential [12], or the acceleration which accompanies inelastic hadronic collisions
[13–15], are associated with an effective temperature for hadron emission.
Recently, a QCD–based picture of thermal production based on the parton description of
high energy hadronic collisions has been proposed [16, 17]. The effective temperature T
is in this case determined either by the string tension σ, with a relation
, (1)
or, in the gluon saturation regime, by the saturation momentum Qs describing the strength
of the color fields in the colliding hadrons or nuclei, with T ≃ (Qs/2π).
Turning now to the second observation, we recall that over the years, hadron production
studies in a variety of high energy collision experiments have shown a remarkably universal
feature. From e+e− annihilation to p− p and p− p̄ interactions and further to collisions
of heavy nuclei, with energies from a few GeV up to the TeV range, the production
pattern always shows striking thermal aspects, connected to an apparently quite universal
temperature around TH ≃ 150 − 200 MeV [18]. As a specific illustration we recall that
the relative abundance of two hadron species a and b, of masses ma and mb, respectively,
is essentially determined by the ratio of their Boltzmann factors [19],
R(a/b) ∼ exp−{(ma −mb)/TH}. (2)
What is the origin of this thermal behaviour? While high energy heavy ion collisions
involve large numbers of incident partons and thus could allow invoking some “thermal-
isation” scheme through rescattering, in e+e− annihilation the predominant initial state
is one energetic qq̄ pair, and the number of hadronic secondaries per unit rapidity is too
small to consider statistical averages. The case in p− p/p− p̄ collisions is similar.
This enigma has led to the idea that all such collision experiments result in the formation
of a strong color field “disturbing” the physical vacuum. The disturbed vacuum then
recovers by producing hadrons according to a maximum entropy principle: the actually
observed final state is that with the largest phase space volume. While this provides an
intuitive basis for a statistical description, it does not account for a universal temperature.
Why don’t more energetic collisions result in a higher hadronization temperature?
A further piece in this puzzle is the observation that the value of the temperature deter-
mined in the mentioned collision studies is quite similar to the confinement/deconfinement
transition temperature found in lattice studies of strong interaction thermodynamics [20].
While hadronization in high energy collisions deals with a dynamical situation, the ener-
gy loss of fast color charges “traversing” the physical vacuum, lattice QCD addresses
the equilibrium thermodynamics of unbound vs. bound color charges. Why should the
resulting critical temperatures be similar or even identical?
We shall here consider these phenomena as reflections of the QCD counterpart of the
Hawking radiation emitted by black holes [10]. These ultimate stellar states provide a
gravitational form of confinement and hence, as already noted, their physics was quite
soon compared to that of color confinement in QCD [1, 2], where colored constituents
are confined to “white holes” (colorless from the outside, but colored inside). It should
be emphasized from the outset that in contrast to the original black hole physics in
gravitation, where confinement is on a classical level complete, in QCD confinement refers
only to color-carrying constituents; thus, e.g., photons or leptons are not affected.
In black hole physics, as noted above, it was shown that the event horizon for systems
undergoing uniform acceleration leads to quantum tunnelling and hence to thermal radia-
tion [11]. Our aim here is to show that such Hawking-Unruh radiation, as obtained in the
specific situation of QCD, provides a viable account for the thermal behavior observed
in multihadron production by high energy collisions. Furthermore, in the process we also
want to elucidate a bit the common origin of the “limiting temperature” concepts which
have arisen in strong interaction physics over the years.
We begin by reviewing those features of black hole physics and Hawking radiation which
are relevant for our considerations, and then discuss how they can be implemented in
QCD. In particular, we show that modifications of the effective space-time structure, in a
perturbative approach as well as in a non-perturbative treatment based on a large-scale
dilaton field, lead to an event horizon in QCD.
Following this, we present the main conceptual consequences of our conjecture.
• Color confinement and the instability of the physical vacuum under pair produc-
tion form an event horizon for quarks, allowing a transition only through quantum
tunnelling; this leads to thermal radiation of a temperature TQ determined by the
string tension.
• Hadron production in high energy collisions occurs through a succession of such
tunnelling processes. The resulting cascade is a realization of the same partition
process which leads to a limiting temperature in the statistical bootstrap and dual
resonance models.
• The temperature TQ of QCD Hawking-Unruh radiation can depend only on the
baryon number and the angular momentum of the deconfined system. The former
could provide a dependence of TQ on the baryon number density, while the angular
momentum pattern of the radiation allows a centrality-dependence of TQ and elliptic
flow.
• In kinetic thermalization, the initial state information is successively lost through
collisions, converging to a time-independent equilibrium state. In contrast, the
stochastic QCD Hawking radiation is “born in equilibrium”, since quantum tun-
nelling a priori does not allow information transfer.
2 Event Horizons in Gravitation and in QCD
2.1 Black holes
A black hole is formed as the final stage of a neutron star after gravitational collapse [24].
It has a mass M concentrated in such a small volume that the resulting gravitational field
confines all matter and even photons to remain inside the event horizon R of the system:
no causal connection with the outside is possible. As a consequence, black holes have
three (and only three) observable properties: mass M , charge Q and angular momentum
J . This section will address mainly black holes with Q = J = 0; we shall come back to
the more general properties in section 4. We use units of h̄ = c = 1.
The event horizon appears in a study of the gravitational metric, which in flat space has
the form
ds2 = g dt2 − g−1 dr2 − r2[dθ2 + sin2 θdφ2], (3)
using units where c = 1. The field strength of the interaction is contained in the coefficient
g(r),
g(r) =
, (4)
leading back to the Minkowski metric in the large distance limit r → ∞. The vanishing
of g(r) specifies the Schwarzschild radius R as event horizon,
R = 2GM. (5)
It is interesting to note that the mass of a black hole thus grows linearly with R, analogous
to the behaviour of the confining potential in strong interactions: M(R) = (2G)−1 R.
Classically, a black hole would persist forever and remain forever invisible. On a quan-
tum level, however, its constituents (photons, leptons and hadrons) have a non-vanishing
chance to escape by tunnelling through the barrier presented by the event horizon. Equiv-
alently, we can say that the strong force field at the surface of the black hole can bring
vacuum fluctuations on-shell. The resulting Hawking radiation [10] cannot convey any
information about the internal state of the black hole; it must be therefore be thermal.
For a non-rotating black hole of vanishing charge (denoted as Schwarzschild black hole),
the first law of thermodynamics,
dM = TdS (6)
combined with the area law for the black hole entropy [25],
leads to the corresponding radiation temperature
TBH =
. (8)
This temperature is inversely proportional to the mass of the black hole, and since the
radiation reduces the mass, the radiation temperature will increase with time, as the black
hole evaporates. For black holes of stellar size, however, one finds TBH <∼ 2 × 10
−8 ◦K,
which is many orders of magnitude below the 2.7 ◦K cosmic microwave background, and
hence not detectable.
It is instructive to consider the Schwarzschild radius of a typical hadron, assuming a mass
m ∼ 1 GeV:
Rhadg ≃ 1.3× 10−38 GeV−1 ≃ 2.7× 10−39 fm. (9)
To become a gravitational black hole, the mass of the hadron would thus have to be
compressed into a volume more than 10100 times smaller than its actual volume, with a
radius of about 1 fm. On the other hand, if instead we increase the interaction strength
from gravitation to strong interaction [1], we gain in the resulting “strong” Schwarzschild
radius Rhads a factor
, (10)
where αs is the dimensionless strong coupling and Gm
2 the corresponding dimensionless
gravitational coupling for the case in question. This leads to
Rhads ≃
; (11)
with the effective value of αs ∼ O(1) we thus get Rhads ∼ O(1) fm.∗ In other words,
the confinement radius of a hadron is about the size of its “strong” Schwarzschild radius,
so that we could consider quark confinement as the strong interaction version of the
gravitational confinement in black holes [1, 2].
We had seen that the mass of a black hole grows linearly with the event horizon, M =
(1/2G)R, so that in gravitation 1/2G plays the role of the string tension in strong inter-
action physics. The replacement GM2 → αs here leads to
σ ≃ m
≃ 0.16 GeV2, (12)
if one uses the mentioned effective saturation value αs ≃ 3 [26]. The value of αs ∼ 1 thus
gives a reasonable string tension as well as a reasonable radius.
2.2 Quasi-Abelian case
The appearence of an event horizon occurs in general relativity through the modification
of the underlying space-time structure by the gravitational interaction. Such modifica-
tions have also been discussed for other interactions. In particular, it was noted that
in electrodynamics, non-linear in-medium effects can lead to photons propagating along
geodesics which are not null in Minkowski space-time; this can even lead to photon trap-
ping, restricting the motion of photons to a compact region of space [27]. Thus, an
effective Lagrangian L(F ) depending on a one-parameter background field, F = FµνF µν ,
results in a modified metric
gµν = ηµνL′ − 4FαµF αν L′′, (13)
where the primes indicate first and second derivatives with respect to F . Hence
g00 = L′ − 4FL′′ = 0 (14)
defines the radius of the compact region of the theory, i.e., the counterpart of a black
hole [27].
QCD is an inherently non-linear theory, with the physical vacuum playing the role of a
medium [28]. The general structure of the effective Lagrangian in a background field F ,
compatible with gauge invariance, renormalization group results [3] and trace anomaly,
has the unique form [29]
LQCD =
g2(gF )
µνǫ(gF ). (15)
∗In fact, some studies [26] indicate that at large distances, the strong coupling freezes at αs ≃ 3; in
that case the corresponding radius becomes Rhad
≃ 1 fm.
Here ǫ(gF ) is the dielectric “constant” of the system in the presence of the background
field; the F -dependence of ǫ(F ) effectively turns the QCD vacuum into a non-linear
medium. On a one-loop pertubative level we have
ǫ(gF ) ≃ 1− β0
, (16)
where β0 = (11Nc − 2Nf)/48π2, with Nc and Nf specifying the number of colors and
flavors, respectively. Using this form in the formalism of [27] leads to gt changing sign
(i.e., gt = 0) at
gF ∗ = Λ2 exp{−4π/β0g2}, (17)
indicating a possible horizon at r∗ ∼ 1/
gF ∗ [30]. It is clear that this line of argument
can at best provide some hints, since we used the lowest order perturbative form of the
beta-function, even though at the horizon perturbation theory will presumably break
down. Nevertheless, we believe that it suggests the possibility of an event horizon for
QCD; the crucial feature is the asymptotic freedom of QCD [3], which leads to ǫ < 1 and
allows g00 to vanish even without the external medium effects required in QED.
2.3 Non-Abelian case
Indeed, a different and more solid suggestion that in QCD there is an event horizon comes
from studying the theory on a curved background. For gluodynamics, such a program is
discussed in [7]. Classical gluodynamics is a scale-invariant theory, but quantum fluctu-
ations break this invariance, with the trace of the energy-momentum tensor introducing
non-perturbative effects, associated with the vacuum energy density ǫvac. It was shown [6]
that low-energy theorems can be used to determine the form of the effective Yang-Mills
Lagrangian in a curved but conformally flat metric
gµν(x) = ηµνe
h(x), (18)
where the dilaton field h(x) is coupled to the trace of the energy momentum tensor, θµµ.
The resulting action has the form
eh(∂µh)
2 − 1
(F aµν)
2 + e2h(ǫvac − 1
θµµ|pert).
; (19)
here ǫvac is the absolute value of the energy density of the vacuum and mG the dilaton
mass; the trace of the energy-momentum tensor has been separated into perturbative and
non-perturbative contributions,
θµµ = θ
µ|pert+ < θµµ >= θµµ|pert − 4ǫvac. (20)
The crucial point for our considerations is that the first term of eq. (19) can be written
eh(∂µh)
2 = R
−g, (21)
defining R as the Ricci scalar of the theory. Hence eq. (19) has the structure of an
Einstein-Hilbert Lagrangian of gluodynamics in the presence of an effective gravitation,
(F aµν)
2 + e2h(ǫvac − 1
θµµ|pert).
; (22)
where G is now given by
. (23)
The relation 1/2G → σ between G and the string tension conjectured above then leads
. (24)
On the other hand, the string tension is just the energy density of the vacuum times the
transverse string area,
σ = ǫvacπr2T . (25)
Combining relations (24) and (25), we have
≃ 0.4 fm, (26)
using mG ≃ 1.5 GeV for the scalar glueball mass. Eq. (26) thus gives us the transverse
extension or horizon of the string.
From eqs. (24) or (25) we can obtain a further consistency check. Given the glueball mass
and the string tension σ ≃ 0.16 GeV2, we find for the vacuum energy density
ǫvac ≃ 3
σ m2G ≃ 0.013 GeV4 ≃ 1.7 GeV/fm
. (27)
This is the value for pure gluodynamics; since the energy density is related to the trace
of the energy–momentum tensor by the relation (20), and
θµµ =
(F aµν)
2 ≃ −b g
(F aµν)
2, (28)
with the coefficient b = 11Nc−2Nf of the β-function, we can estimate that for three-flavor
ǫvacQCD =
11Nc − 2Nf
ǫvac =
ǫvac ≃ 0.01 GeV4, (29)
which is in perfect agreement with the original value of the gluon condensate [8]
≃ 0.012 GeV4; (30)
note that ǫvacQCD = 27/32 〈(αs/π)G2〉).
3 Hyperbolic Motion and Hawking-Unruh Radiation
In general relativity, the event horizon appeared as consequence of the geometrized grav-
itational force, but its occurrence and its role for thermal radiation was soon generalized
by Unruh [11]. A system undergoing uniform acceleration a relative to a stationary ob-
server eventually reaches a classical turning point and thus encounters an event horizon.
Let us recall the resulting hyperbolic motion [34]. A point mass m subject to a constant
force F satisfies the equation of motion
1− v2
= F, (31)
where v(t) = dx/dt is the velocity, normalized to the speed of light c = 1. This equation
is solved by the parametric form through the so-called Rindler coordinates,
x = ξ cosh aτ t = ξ sinh aτ, (32)
where a = F/m denotes the acceleration in the instantaneous rest frame of m, and τ
the proper time, with dτ =
1− v2dt. If we impose the boundary condition that the
velocity at t = 0 vanishes, we have ξ = 1/a and x(t=0) = 1/a. The resulting world line
is shown in Fig. 1. It corresponds to the mass m coming from x = ∞ at t = −∞ at with
a velocity arbitrarily close to that of light, decelerating uniformly until it comes to rest
at the classsical turning point xH = −(1/a), t = 0. Subsequently, it accelerates again and
returns to x = ∞ at t = ∞, approaching the speed of light. For given a, the light cone
originating at a distance xH = 1/a away from the turning point of m defines a space-time
region inaccessible to m: no photon in this region can (classically) ever reach m, in much
the same way as photons cannot escape from a black hole. Here the acceleration is crucial,
of course; if m stops accelerating, it will eventually become visible in the “hidden region”.
mass m
region
hidden
event horizon
Figure 1: Hyberbolic motion
The metric of such a accelerating system becomes in spherical coordinates [31]
ds2 = ξ2a2dτ 2 − dξ2 − ξ2 cosh2 aτ(dθ2 + sin2 θdφ2), (33)
which we want to compare to the black hole metric (3). Making in the latter the coordinate
transformation [32]
, (34)
where the surface gravity κ is given by
, (35)
we obtain for r → R the black hole form
ds2 = η2κ2dt2 − dη2 −R2(dθ2 + sin2θdφ2). (36)
When we compare eqs. (33) and (36), it is evident that the system in uniform acceleration
can be mapped onto a spherical black hole, and vice versa, provided we identify the surface
gravity κ with the acceleration a.
The vacuum through which m travels is, for a stationary observer, empty space. On a
quantum level, however, it contains vacuum fluctuations. The accelerating mass m can
bring these on-shell, using up a (small) part of its energy, so that for m the vacuum
becomes a thermal medium of temperature
. (37)
Consider such a fluctuation into an e+e− pair, flying apart in opposite directions. One
electron is absorbed by the mass m, the other penetrates into the “hidden region” and can
never be detected by m (see Fig. 2. Since thus neither an observer on m nor a stationary
observer in the hidden region can ever obtain access to full information, each will register
the observed radiation as thermal (Einstein-Podolsky-Rosen effect [33]). In other words:
the accelerating mass m sees the vacuum as a physical medium of temperature TU , while
a stationary observer in the hidden region observes thermal radiation of temperature TU
as a consequence of the passing of m.
mass m
Figure 2: Unruh radiation
We here also mention that the entropy in the case of an accelerating system again becomes
1/4 of the event horizon area, as in the black hole case, so that also here the correspondence
remains valid [35].
In the case of gravity, we have the force
F = ma = G
, (38)
on a probe of mass m. With R = 2 GM for the (Schwarzschild) black hole radius, we have
a = 1/(4 GM) for the acceleration at the event horizon and hence the Unruh temperature
(37) leads back to the Hawking temperature of eq. (8).
In summary, we note that constant acceleration leads to an event horizon, which can be
surpassed only by quantum tunnelling and at the expense of complete information loss,
leading to thermal radiation as the only resulting signal.
4 Pair Production and String Breaking
In the previous section, we had considered a classical object, the mass m, undergoing
accelerated motion in the physical vacuum; because of quantum fluctuations, this vacuum
appears to m as a thermal medium of temperature TU . In this section, we shall first
address the modifications which arise if the object undergoing accelerated motion is itself
a quantum system, so that in the presence of a strong field it becomes unstable under
pair production. Next we turn to the specific additional features which come in when
the basic constituents are subject to color confinement and can only exist in color neutral
bound states.
As starting point, we consider two-jet e+e− annihilation at cms energy
e+e− → γ∗ → qq̄ → hadrons. (39)
The initially produced qq̄ pair flies apart, subject to the constant confining force given
by the string tension σ; this results in hyperbolic motion [14] of the type discussed in the
previous section. At t = 0, the q and q̄ separate with an initial velocity v0 = p/
p2 +m2,
where p ≃
s/2 is the momentum of the primary constituents in the overall cms and m
the effective quark mass. We now have to solve Eq. (31) with this situation as boundary
condition; the force
F = σ, (40)
is given by the string tension σ binding the qq̄ system. The solution is
x̃ = [1−
1− v0t̃+ t̃2] (41)
with x̃ = x/x0 and t̃ = t/x0; here the scale factor
1− v20
γ (42)
is the inverse of the acceleration a measured in the overall cms. The velocity becomes
v(t) =
(v0/2)− t̃
1− v0t̃+ t̃2
; (43)
it vanishes for
t̃∗ =
⇒ t∗ = v0
γ, (44)
thus defining
x(t∗) =
1− (v20/4)
as classical turning point and hence as the classical event horizon measured in the overall
cms (see Fig. 3).
xx(t  )*xQ
Figure 3: Classical and quantum horizons in qq̄ separation
Eq. (45) allows the q and the q̄ to separate arbitrarily far, provided the pair has enough
initial energy; this clearly violates color confinement. Our mistake was to consider the
qq̄ system as classical; in quantum field theory, it is not possible to increase the potential
energy of a given qq̄ state beyond the threshold value necessary to bring a virtual qq̄ pair
on-shell. In QED, the corresponding phenomenon was addressed by Schwinger [36], who
showed that in the presence of a constant electric field of strength E the probability of
producing an electron-positron pair is given by
P (M, E) ∼ exp{−πm2e/eE}, (46)
with me denoting the electron mass and e denoting the electric charge. This result is
in fact a specific case of the Hawking-Unruh phenomenon, as shown in [16]. In QCD,
we expect a similar effect when the string tension exceeds the pair production limit, i.e.,
σ x > 2m (47)
where m specifies the effective quark mass. Beyond this point, any further stretching of
the string is expected to produce a qq̄ pair with the probability
P (M,σ) ∼ exp{−πm2/σ}, (48)
with the string tension σ replacing the electric field strength eE . This string breaking
acts like a quantum event horizon xq = 2 m/σ, which becomes operative long before
the classical turning point is ever reached (see Fig. 3). Moreover, the resulting allowed
separation distance for our qq̄ pair, the color confinement radius xQ, does not depend on
the initial energy of the primary quarks.
There are some important differences between QCD and QED. In case of the latter, the
initial electric charges which lead to the field E can exist independently in the physical
vacuum, and the produced pair can be simply ionized into an e+ and an e−. In contrast,
neither the primary quark nor the constituents of the qq̄ pair have an independent ex-
istence, so that in string breaking color neutrality must be preserved. As a result, the
Hawking radiation in QCD must consist of qq̄ pairs, and these can be produced in an in-
finite number of different excitation states of increasing mass and degeneracy. Moreover,
the qq̄ pair spectrum is itself determined by the strength σ of the field, in contrast to the
exponent m2e/E in eq. (46), where the value of E has no relation to the electron mass me.
Hadron production in e+e− annihilation is believed to proceed in the form of a self-similar
cascade [37, 38]. Initially, we have the separating primary qq̄ pair,
γ → [qq̄] (49)
where the square brackets indicate color neutrality. When the energy of the resulting
color flux tube becomes large enough, a further pair q1q̄1 is excited from the vacuum by
two-gluon exchange (see Fig. 4),
γ → [q[q̄1q1]q̄]. (50)
Although the new pair is at rest in the overall cms, each of its constituents has a transverse
momentum kT determined, through the uncertainty relation, by the transverse dimension
rT of the flux tube. The slow q̄1 now screens the fast primary q from its original partner
q̄, with an analoguous effect for the q1 and the primary antiquark. To estimate the qq̄
separation distance at the point of pair production, we recall that the thickness of the
flux tube connecting the qq̄ pair is in string theory given by [41]
r2T =
2k + 1
, (51)
where K is the string length in units of an intrinsic vibration measure. Lattice studies [43]
show that for strings in the range of 1 - 2 fm, the first string excitation dominates, so that
we have
rT = c0
, (52)
with c0 ≃ 1 or slightly larger. Higher excitations lead to a greater thickness and eventually
to a divergence (the “roughening” transition). From the uncertainty relation we then have
. (53)
With this transverse energy is included in eq. (47), we obtain for the pair production
separation xQ
σxq = 2
m2 + k2T ⇒ xq ≃
m2 + (πσ/2 c20) ≃
≃ 1 fm, (54)
with σ = 0.2 GeV2, m2 ≪ σ, and c0 ≃ 1.
Once the new pair is present, we have a color-neutral system qq̄1q1q̄; but since there is a
sequence of connecting string potentials qq̄1, q̄1q1 and q1q̄, the primary string is not yet
broken. To achieve that, the binding of the new pair has to be overcome, i.e., the q1 has to
tunnel through the barrier of the confining potential provided by q̄1, and vice versa. Now
the q excerts a longitudinal force on the q̄1, the q̄ on the q1, resulting in a longitudinal
acceleration and ordering of q1 and q̄1. When (see Fig. 4)
σx(q1q̄1) = 2
m2 + k2T , (55)
Figure 4: String breaking through qq̄ pair production
the q̄1 reaches its q1q̄1 horizon; on the other hand, when
σx(qq̄1) = 2
m2 + k2T , (56)
the new flux tube qq̄1 reaches the energy needed to produce a further pair q2q̄2. The
q̄2 screens the primary q from the q1 and forms a new flux tube qq̄2. At this point, the
original string is broken, and the remaining pair q̄1q2 form a color neutral bound state
which is emitted as Hawking radiation in the form of hadrons, with the relative weights
of the different states governed by the corresponding Unruh temperature. The resulting
pattern is schematically illustrated in Fig. 4.
To determine the temperature of the hadronic Hawking radiation, we return to the original
pair excitation process. To produce a quark of momentum kT , we have to bring it on-shell
and change its velocity from zero to v = kT/(m
2 + k2T )
1/2 ≃ 1. This has to be achieved in
the time of the fluctuation determined by the virtuality of the pair, ∆τ = 1/∆E ≃ 1/2kT .
The resulting acceleration thus becomes
≃ 2 kT ≃
2πσ/c0 ≃ 1 GeV, (57)
which leads to
≃ 160− 180 MeV (58)
for the hadronic Unruh temperature. It governs the momentum distribution and the
relative species abundances of the emitted hadrons.
A given step in the evolution of the hadronization cascade of a primary quark or antiquark
produced in e+e− annihilation thus involves several distinct phenomena. The color field
created by the separating q and q̄ produces a further pair q1q̄1 and then provides an
acceleration of the q1, increasing its longitudinal momentum. When it reaches the q1q̄1
confinement horizon, still another pair q2q̄2 is excited; the state q̄1q2 is emitted as a
hadron, the q̄2 forms together with the primary q a new flux tube. This pattern thus
step by step increases the longitudinal momentum of the “accompanying” q̄i as well as
of the emitted hadron. This, together with the energy of the produced pairs, causes a
corresponding deceleration of the primary quarks q and q̄, in order to maintain overall
energy conservation. In Fig. 5, we show the world lines given by the acceleration q̄i → q̄i+1
(qi → qi+1) and that of formation threshold of the hadrons q̄iqi+1 and the corresponding
antiparticles.
hadron radiation
quark acceleration
Figure 5: Quark acceleration and hadronization world lines
The energy loss and deceleration of the primary quark q in this self-similar cascade,
together with the acceleration of the accompanying partner q̄i from the successive pairs
brings q and q̄i closer and closer to each other in momentum, from an initial separation
qq̄1 of
s/2, until they finally are combined into a hadron and the cascade is ended. The
resulting pattern is shown in Fig. 6.
hadrons
Figure 6: Hadronization in e+e− annihilation
The number of emitted hadrons, the multiplicity ν(s), follows quite naturally from the
picture presented here. The classical string length, in the absence of quantum pair for-
mation, is given by the classical turning point determined in eq. (45). The thickness of a
flux tube of such an “overstretched” string is known [41]; from eq. (51) we get
R2T =
2k + 1
ln 2K, (59)
where K is the string length. From eq. (45) we thus get
R2T ≃
s (60)
for the flux tube thickness in the case of the classical string length. In parton language, the
logarithmic growth of the transverse hadron size is due to parton random walk (”Gribov
diffusion” [42]); this phenomenon is responsible for diffraction cone shrinkage in high–
energy hadron scattering.
Because of pair production, the string breaks whenever it is stretched to the length xq
given in eq. (54); its thickness rT at this point is given by eq. (51). The multiplicity can
thus be estimated by the ratio of the corresponding classical to quantum transverse flux
tube areas,
ν(s) ∼ R
s, (61)
and is found to grow logarithmically with the e+e− annihilation energy, as is observed
experimentally over a considerable range.
We note here that in our argumentation we have neglected parton evolution, which would
cause the emitted radiation (e.g., q̄1q2 in Fig. 4) to start another cascade of the same
type. Such evolution effects result eventually in a stronger increase of the multiplicity.
The formation of a white hole does not affect the production of hard processes at early
times (e.g., multiple jet production), which is responsible for an additional growth of the
measured multiplicity.
A further effect we have not taken into account here is parton saturation. At sufficiently
high energy, stronger color fields can lead to gluon saturation and thus to a higher temper-
ature determined by the saturation momentum [16]. The resulting system then expands
and hadronizes at the universal temperature determined by the string tension.
It interesting to compare the separation of two energetic light quarks, as we have consid-
ered here, with that of two static heavy quarks Q and Q̄. From quarkonium studies it is
known that
2(MD −mc) ≃ 2(MB −mb) ≃ 1.2 GeV, (62)
where MD(MB) and mc(mb) are the masses of open charm (beauty) mesons and of the
corresponding charm (beauty) quarks, respectively. The energy needed to separate a
heavy QQ̄ pair thus is independent of the mass of the heavy quarks, indicating that
the string breaking involved here is really a consequence of the vacuum, through qq̄ pair
excitation. With
σxQ ≃ 1.2 GeV ⇒ xQ ≃ 1.2 fm (63)
we find that the resulting separation threshold for pair excitation agrees well with that
found above in eq. (54). Lattice QCD studies lead to similar results.
Up to now, we have considered hadron production in e+e− annihilation, in which the
virtual photon produces a confined colored qq̄ pair as a “white hole”. Turning now to
hadron-hadron collisions, we note that here two incident white holes combine to form a
new system of the same kind, as schematically illustrated in Fig. 7. Again the resulting
string or strong color field produces a sequence of qq̄ pairs of increasing cms momentum,
leading to the well-known multiperipheral hadroproduction cascade shown in Fig. 8. We
recall here the comments made above concerning parton evolution and saturation; in
hadronic collisions as well, these phenomena will affect the multiplicity, but not the relative
abundances.
In the case of heavy ion collisions, two new elements enter. The resulting systems could
now have an overall baryon number, up to B = 400 or more. To take that into account,
we need to consider the counterpart of charged black holes. Furthermore, in heavy ion
collisions the resulting hadron production can be studied as function of centrality, and
peripheral collisions could lead to an interaction region with an effective overall angular
momentum. Hence we will also consider rotating black holes. In the next section, we then
summarize the relevant features of black holes with Q 6= 0, J 6= 0.
γ h h
(a) (b)
Figure 7: “White hole” structure in e+e− annihilation (a) and hadronic collisions (b)
hadrons
Figure 8: Hadronization in hadron-hadron collisions
5 Charged and Rotating Black Holes
As mentioned, an outside observer the only characteristics of a black hole are its mass M ,
its electric charge Q, and its spin or angular momentum J . Hence any further observables,
such as the event horizon or the Hawking temperature, must be expressable in terms of
these three quantities.
The event horizon of a black hole is created by the strong gravitational attraction, which
leads to a diverging Schwarzschild metric at a certain value of the spatial extension R.
Specifically, the invariant space-time length element ds2 is at the equator given by
ds2 = (1− 2GM/R) dt2 −
1− 2GM/R
dr2, (64)
with r and t for flat space and time coordinates; it is seen to diverge at the Schwarzschild
radius RS = 2GM . If the black hole has a net electric charge Q, the resulting Coulomb
repulsion will oppose and hence weaken the gravitational attraction; this will in turn
modify the event horizon. As a result, the corresponding form (denoted as Reissner-
Nordström metric) becomes
ds2 = (1− 2GM/R +GQ2/R2) dt2 − 1
1− 2GM/R +GQ2/R2
dr2. (65)
For this, the divergence leads to the smaller Reissner-Nordström radius
RRN = GM (1 +
1−Q2/GM2), (66)
which reduces to the Schwarzschild radius RS for Q = 0. The temperature of the Hawking
radiation now becomes [24, 39]
TBH(M,Q) = TBH(M, 0)
1−Q2/GM2
1−Q2/GM2 ) 2
; (67)
its functional form is illustrated in Fig. 9. We note that with increasing charge, the
Coulomb repulsion weakens the gravitational field at the event horizon and hence de-
creases the temperature of the corresponding quantum excitations. As Q2 → GM2, the
gravitational force is fully compensated and there is no more Hawking radiation.
0.5 1.0
Q / GM
T   (M,Q) / T   (M,0)
BH BH
Figure 9: Radiation temperature for a charged black hole
In a similar way, the effect of the angular momentum of a rotating black hole can be
incorporated. It is now the centripetal force which counteracts the gravitational attraction
and hence reduces its strength. The resulting Kerr metric must take into account that in
this case the rotational symmetry is reduced to an axial symmetry, and with θ denoting
the angle relative to the polar axis θ = 0, it is (at fixed longitude) given by
ds2 =
1− 2GMR
R2 + j2 cos2 θ
dt2 − R
2 + j2 cos2 θ
R2 − 2GMR + j2
dr2 − (R2 + j2 cos2 θ) dθ2. (68)
The angular momentum of the black hole is here specified by the parameter j = J/M ; for
a = 0, we again recover the Schwarzschild case. The general situation is now somewhat
more complex, since eq. (68) leads to two different divergence points. The solution
RK = GM (1 +
1− j2/(GM)2 ) (69)
defines the actual event horizon, corresponding to absolute confinement. But the resulting
black hole is now embedded in a larger ellipsoid
RE = GM (1 +
1− [j2/(GM)2] cos2 θ), (70)
as illustrated in Fig. 10. The two surfaces touch at the poles, and the region between them
is denoted as the ergosphere. Unlike the black hole proper, communication between the
ergosphere and the outside world is possible. Any object in the ergosphere will, however,
suffer from the rotational drag of the rotating black hole and thereby gain momentum. We
shall return to this shortly; first, however, we note that the temperature of the Hawking
radiation from a rotating black hole becomes
TBH(M,J) = TBH(M, 0)
1− j2/(GM)2
1− j2/(GM)2
. (71)
For a non-rotating black hole, with j = 0, this also reduces to the Hawking temperature
for the Schwarzschild case.
ergosphere
black hole
Figure 10: Geometry of a rotating black hole
To illustrate the effect of the ergosphere, imagine radiation from a Schwarzschild black hole
emitted radially outward from the event horizon. In the case of a Kerr black hole, such an
emission is possible only along the polar axis; for all other values of θ, the momentum of the
emitted radiation (even light) will increase due to the rotational drag in the ergosphere.
This effect ceases only once the radiation leaves the ergosphere. Since the amount of
drag depends on θ, the momentum of the radiation emitted from a rotating black hole,
as measured at large distances, will depend on the latitude at which it is emitted and
increase from pole to equator.
Finally, for completeness, we note that for black holes with both spin and charge (denoted
as Kerr-Newman), the event horizon is given by
RKN = GM (1 +
1− [Q2/GM2]− [j2/(GM)2]), (72)
and the radiation temperature becomes [24, 39]
TBH(M,Q, J) = TBH(M, 0, 0)
1− (GQ2 + j2)/(GM)2
1− (GQ2 + j2)/(GM)2 ) 2 + j2/(GM)2
. (73)
The decrease of TBH for Q 6= 0, J 6= 0 expresses the fact that both the Coulomb repulsion
and the rotational force counteract the gravitational attraction, and if they win, the black
hole is dissolved.
The dependence of a black hole on its basic properties M,Q, J is very similar to the
dependence of a thermodynamic system on a set of thermodynamic observables. The first
law of thermodynamics can be written as
dE = TdS + φdQ+ ωdJ, (74)
expressing the variation of the energy with entropy S, charge Q and spin J ; here φ denotes
the electrostatic potential per charge and ω the rotational velocity. The corresponding
relation in black hole thermodynamics becomes
dM = TBHdSBH + ΦdQ + ΩdJ, (75)
where the entropy SBH is defined as the area of the event horizon,
SBH =
π(R2KN + j
. (76)
The temperature is given by eq. (73), and
4πQRRN
, Ω =
specify the electrostatic potential Φ and the rotational velocity Ω.
The considerations of this section were for spherical black holes. As seen above, such
objects are in fact equivalent to uniformly accelerating systems. An application to actual
high energy collisions involves a further assumption. Thermal Hawking-Unruh radiation
arises already from a single QQ̄ system, as seen above in the discussion of e+e− anni-
hilation. If we treat the systems produced in heavy ion collisions as black holes of an
overall baryon number and/or an overall spin, we are assuming that the collision leads to
a large-scale collective system, in which each accelerating parton is affected by totality of
the other accelerating partons. This assumption clearly goes beyond our event horizon
conjecture and, in particular, it need not be correct in order to obtain thermal hadron
production.
6 Baryon Density and Angular Momentum
6.1 Vacuum Pressure and Baryon Repulsion
We now want to consider the extension of charged black hole physics to color confinement
in the case of collective systems with a net baryon number. In eq. (67) we had seen that
the reduction of the gravitational attraction by Coulomb repulsion in a charged black hole
modifies the event horizon and hence in turn also the temperature of Hawking radiation.
The crucial quantity here is the ratio Q2/GM2 of the repulsive overall Coulomb force,
Q2/R2, to the attractive overall gravitational force, GM2/R2, at the horizon.
In QCD, we have a “white” hole containing colored quarks, confined by chromodynamic
forces or, equivalently, by the pressure of the physical vacuum. If the system has a non-
vanishing overall baryon number, the baryon-number dependent interaction will also affect
the forces at the event horizon. The simplest instance of such a force the repulsion between
quarks due to Fermi statistics, but more generally, there will be repulsive effects of the
type present in cold dense baryonic matter, such as neutron stars. The resulting pressure
will modify the confinement horizon and hence lead to a corresponding modification of
the Hawking-Unruh temperature of hadronization.
By using the conjectured correspondence between black hole thermodynamics and the
thermodynamics of confined color charges, we translate black hole mass, charge and grav-
itational constant into white hole energy, net baryon number and string tension,
{M,Q,G} ↔ {E,B, 1/2σ}. (78)
Hence eq. (67) leads us to the relation
TQ(B) = TQ(B = 0)
1− 2σ B2/E2
1− 2σB2/E2 ) 2
; (79)
for the dependence of the hadronization temperature on the ratio of net baryon number
B and energy E, with TQ(B = 0) given by eq. (58). Its functional form is the same as
that illustrated in Fig. 9.
It would be interesting to test the prediction (79) against experimental data; one could
identify B with the net baryon number per unit rapidity dNB/dy and E with the total
transverse energy per unit rapidity dET/dy. The reduction of the hadronization temper-
ature with baryon number could thus occur in two ways. A sufficient decrease of the
collision energy, e.g. from peak SPS to AGS energy, will strongly reduce dET/dy, while
dNB/dy is not affected as much. This leads to the known decrease of T (µB) with increas-
ing µB [44], and it will be interesting to see if the form (79) agrees with the observed
behaviour. A second, novel possibility would be to consider hadrochemistry as a func-
tion of rapidity. At peak SPS energy, dNB/dy remains essentially constant out to about
y = 2, while dET/dy drops by more than a factor of two from y = 0 to y = 2 [45]. A
similar behaviour occurs at still lower collision energies. Hence it would seem worthwhile
to check if an abundance analysis at large y indeed shows the expected decrease of the
hadronization temperature.
6.2 Angular Momentum and Non-Central Collisions
The dependence of Hawking radiation on the angular momentum of the emitting sys-
tem introduces another interesting aspect for the “white hole evaporation” we have been
considering. Consider a nucleus-nucleus collision at non-zero impact parameter b. If the
interaction is of collective nature, the resulting interaction system may have some angular
momentum orthogonal to the reaction plane (see Fig. 11). In central collisions, this will
not be the case, nor for extremely peripheral ones, where one expects essentially just
individual nucleon-nucleon collisions without any collective effects.
If it possible to consider a kinematic region in which the interacting system does have an
overall spin, then the resulting Hawking radiation temperature should be correspondingly
reduced, as seen in eq. (71). The effect is not so easily quantified, but simply a reduction
Figure 11: Rotating interaction region in non-central AA collision
of the hadronization temperature for non-central collisions would quite indicative. Such a
reduction could appear only in the temperature determined by the relative abundances,
since, as we shall see shortly, the transverse momentum spectra should show modifications
due to the role of the ergosphere.
We next turn to the momentum spectrum of the Hawking radiation emitted from a ro-
tating white hole. As discussed in section 4, such radiation will exhibit an azimuthal
asymmetry due to the presence of the ergosphere, which by its rotation will affect the
momentum spectrum of any passing object. At the event horizon, the momentum of all
radiation is determined by the corresponding Hawking temperature (71); but the passage
of the ergosphere adds rotational motion to the emerging radiation and hence increases
its momentum. As a result, only radiation emitted directly along the polar axis will have
momenta as specified by the Hawking temperature; with increasing latitude θ (see Fig.
12)a, the rotation will increase the radiation momentum up to a maximum value in the
equatorial plane.
spectators spectators
ergosphere
Figure 12: Transverse plane view of a non-central AA collision
Hawking radiation from a rotating source thus leads for nuclear collisions quite naturally
to what in hydrodynamic studies is denoted as elliptic flow. It is interesting to note
that both scenarios involve collective effects: while in hydrodynamics, it is assumed that
non-central collisions lead to an azimuthally anisotropic pressure gradient, we have here
assumed that such collisions lead to an overall angular momentum of the emitting system.
Concluding this section we emphasize that the results obtained here for the Hawking
temperatures of systems with finite baryon density or with an effective overall spin depend
crucially on the assumption of collectivity. If the different nucleon-nucleon interactions
in a heavy ion collision do not result in sufficiently collective behavior, the corresponding
modifications of Tq do not apply. In the case of black holes with spin, we moreover have
no way to relate in a quantitative way centrality and overall spin. Both cases do show,
however, that such extensions lead to qualitatively reasonable modifications.
7 Temperature and Acceleration Limits
We had seen that the underlying confinement dynamics of high energy hadron collisions
and e+e− annihilation led to a limit on the acceleration (or the corresponding deceleration)
in the self-similar hadronization cascade - a limit which can be specified in terms of the
string tension. In turn, this led to a limiting Unruh hadronization temperature
. (80)
We emphasize that a Hawking-Unruh temperature as such can a priori have any value;
it is the universal limit on the acceleration that leads to a universal temperature for the
emitted hadron radiation.
In the study of strongly interacting matter, temperature limits are well-known and arise
for an ideal gas of different composite constituents (“resonances” or “fireballs” of varying
mass M), if the composition law provides a sufficiently fast increase of the degeneracy
ρ(M) with M . If the number of states of a constituent of mass M grows exponentially,
ρ(M) ∼ M−a exp{bM}, (81)
with constants a and b, then the grand canonical partition function for an ideal gas in a
volume V
Z(T, V ) =
(2π)3
dMρ(M)
d3p exp{−
p2 +M2 /T}
, (82)
diverges for
T > TH ≡ 1/b, (83)
so that the Hagedorn temperature TH [18] constitutes an upper limit for the temperature
of hadronic matter.
In the dual resonance model [46], the resonance composition pattern is governed by linearly
rising Regge trajectories,
α′M2n = n+ α0, n = 1, 2, ..., (84)
in terms of the universal Regge slope α′ ≃ 1 GeV−2 and a constant (of order unity)
specifying the family (π, ρ, ...). For an ideal resonance gas in D − 1 space and one time
dimension, one then obtains [47]
(D + 1), b = 2π
, (85)
leading to the temperature limit
. (86)
In string theory, the Regge resonance pattern is replaced by string excitation modes,
retaining the same underlying partition structure, with α′ = 1/2πσ relating Regge slope
and string tension. Hence we get
for the corresponding limiting temperature. For D = 4, this coincides with eq. (1)
from [16] and agrees within 20 % with the Unruh temperature (80) determined by the
lowest string excitation alone (c0 = 1 in eq. (58)).
Prior to the dual resonance model, Hagedorn had determined the level density ρ(M) of
fireballs composed of fireballs, requiring the same composition pattern at each level [18].
The resulting bootstrap condition leads to [48]
a = 3 b = r0
(2 ln 2− 1)
]−1/3
≃ r0, (88)
where r0 measures the range of the strong interaction. With r0 ≃ 1 fm, we thus get
TH ≃ 0.2 GeV for the limiting temperature of hadronic matter. If we identify r0 with the
pair production separation xq obtained in eq. (54), we get
and hence again agreement with the hadronic Unruh temperature (80).
Hadronic matter as an ideal gas of constituents with self-similar composition spectra
(“resonances of resonances” or “fireballs of fireballs”) thus leads to an upper limit of
the temperature, because the level density of such constituents increases exponentially.
What does this have to do with the limiting acceleration found in the qq̄ cascade of e+e−
annihilation?
To address this problem, it is useful to recall the underlying reason for the exponential
increase of the level density in the dual resonance model and the bootstrap model. The
common origin in both cases is a classical partition problem, which in its simplest form
[40] asks: how many ways ρ(M) are there to partition a given integer M into ordered
combinations of integers? As example, we have for M = 4 the partitions 4, 3+1, 1+3,
2+2, 2+1+1, 1+2+1, 1+1+2, 1+1+1+1; thus here ρ(M = 4) = 8 = 2M−1. It can be
shown that this is generally valid, so that
ρ(M) =
exp{M ln 2}. (90)
For a “gas of integers”, T0 = 1/ ln 2 would thus become the limiting temperature; the
crucial feature in thermodynamics is the exponential increase in the level density due to
the equal a priori weights given to all possible partitions.
Returning now to the quark cascade in e+e− annihilation, we note that the form we have
discussed above is a particular limiting case. We assumed that the color field of the
separating qq̄ excites in the first step one new pair from the vacuum; in principle, though
with much smaller probability, it can also excite two or more. The same is true at the
next step, when the tunnelling produces one further pair: here also, there can be two
or more. Thus the e+e− cascade indeed provides a partition problem of the same kind.
What remains to be shown are the two specific features of our case: that the dominant
decay chain is one where in each step one hadron is produced, which provides the constant
deceleration of the primary quark and antiquark.
The statistical bootstrap model as well as the dual resonance model lead to self-similar
decay cascades, starting from a massive fireball (or resonance), which decays into further
fireballs, and so on, until at the end one has light hadrons. In Fig. 13a we illustrate such
a cascade for the case where the average number k̄ of constituents per step in the decay
(or composition) partition pattern
M → M11 +M12 + ... +M1k; M11 → M21 +M22 + ...+M2k; ... (91)
is k = 3. In the statistical bootstrap model, k̄ can be can be determined [49]; it is found
that the crucial feature here is the power term multiplying the exponential increase in eq.
(81). For a < 5/2, the distribution in k is given by
F (k) =
(ln 2)k−1
(k − 1)!
, (92)
so that the average becomes
k̄ = 1 + 2 ln 2 ≃ 2.4. (93)
The dominant decay (∼ 70%) is thus into two constituents, with 24 % three-body and 6
% four-body decays. While in general the fireball mass M could decrease in each step by
M/k, i.e., by an amount depending on M , the case a < 5/2 is found to be dominated by
one heavy and one soft light hadron,
M → M1 + h1; M1 → M2 + h2; ... (94)
where hi denotes final hadrons; the pattern is shown in Fig. 13b. Moreover, the three-
and four-body decays also lead to one heavy state plus soft light hadrons. The decay thus
provides a uniform decrease of the fireball mass by the average hadron mass or transverse
energy.
(a) (b)
Figure 13: Fireball decay patterns
We therefore conclude that the hadronization pattern we had obtained for e+e− anni-
hilation is indeed also connected to the same partition problem as the one leading to
exponential level densities.
8 Stochastic vs. Kinetic Thermalization
In statistical mechanics, a basic topic is the evolution of a system of many degrees of
freedom from non-equilibrium to equilibrium. Starting from a non-equilibrium initial
state of low entropy, the system is assumed to evolve as a function of time through
collisions to a time-independent equlibrium state of maximum entropy. In other words,
the system loses the information about its initial state through a sequence of collisions
and thus becomes thermalized. In this sense, thermalization in heavy ion collisions was
studied as the transition from an initial state of two colliding beams of “parallel” partons
to a final state in which these partons have locally isostropic distributions. This “kinetic”
thermalization requires a sufficient density of constituents, sufficiently large interaction
cross sections, and a certain amount of time.
From such a point of view, the observation of thermal hadron production in high energy
collisions, in particular in e+e− and pp interactions, is a puzzle: how could these systems
ever “have reached” thermalization? Already Hagedorn [50] had therefore concluded
that the emitted hadrons were “born in equilibrium”. Given an exponentially increasing
resonance mass spectrum, it remained unclear why collisions should result in a thermal
system.
Hawking radiation provides a stochastic rather than kinetic approach to equilibrium, with
a randomization essentially provided by the quantum physics of the Einstein-Podolsky-
Rosen effect. The barrier to information transfer due the event horizon requires that
the resulting radiation states excited from the vacuum are distributed according to max-
imum entropy, with a temperature determined by the strength of the “confining” field.
The ensemble of all produced hadrons, averaged over all events, then leads to the same
equilibrium distribution as obtained in hadronic matter by kinetic equilibration. In the
case of a very high energy collision with a high average multiplicity already one event can
provide such equilibrium; because of the interruption of information transfer at each of
the successive quantum color horizons, there is no phase relation between two successive
production steps in a given event. The destruction of memory, which in kinetic equili-
bration is achieved through sufficiently many successive collisions, is here automatically
provided by the tunnelling process.
So the thermal hadronic final state in high energy collisions is not reached through a
kinetic process; it is rather provided by successively throwing dice.
9 Conclusions
We have shown that quantum tunnelling through the color confinement horizon leads to
thermal hadron production in the form of Hawking-Unruh radiation. In particular, this
implies:
• The radiation temperature TQ is determined by the transverse extension of the color
flux tube, giving
, (95)
in terms of the string tension σ.
• The multiplicity ν(s) of the produced hadrons is approximately given by the increase
of the flux tube thickness with string length, leading to
ν(s) ≃ ln
s, (96)
where
s denotes the cms collision energy. Parton evolution and gluon saturation
will, however, increase this, as will early hard production. The universality of the
resulting abundances is, however, not affected.
• The temperature of Hawking radiation can in general depend on the charge and
the angular momentum of the emitting system. The former here provides a baryon-
number dependence of the hadronization temperature and predicts a decrease of TQ
for sufficiently high baryon density. The latter provides the basis for the possibility
of elliptic flow and of a dependence of TQ on the centrality of AA collisions.
• The limiting temperature obtained in the statistical bootstrap and the dual reso-
nance or string model arises from a self-similar composition pattern leading to an
exponentially growing level density. We find that the underlying partition problem
also leads to the cascade form obtained for hadron emission in high energy collisions,
so that the dynamic and the thermodynamic limits have the same origin.
• In statistical QCD, thermal equilibrium is reached kinetically from an initial non-
equilibrium state, with memory destruction through successive interactions of the
constituents. In high energy collisions, tunnelling prohibits information transfer and
hence leads to stochastic production, so that we have a thermal distribution from
the outset.
We close with a general comment. In astrophysics, Hawking-Unruh radiation has so far
never been observed. The thermal hadron spectra in high energy collisions may thus
indeed be the first experimental instance of such radiation, though in strong interaction
instead of gravitation.
Acknowledgements
D. K. is grateful to E. Levin and K. Tuchin for useful and stimulating discussions. H. S.
thanks M. Kozlov for helpful discussions and a critical reading to the text. The work of
D. K. was supported by the U.S. Department of Energy underContract No. DE-AC02-
98CH10886.
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	Introduction
	Event Horizons in Gravitation and in QCD
	Black holes
	Quasi-Abelian case
	Non-Abelian case
	Hyperbolic Motion and Hawking-Unruh Radiation
	Pair Production and String Breaking
	Charged and Rotating Black Holes
	Baryon Density and Angular Momentum
	Vacuum Pressure and Baryon Repulsion
	Angular Momentum and Non-Central Collisions
	Temperature and Acceleration Limits
	Stochastic vs. Kinetic Thermalization
	Conclusions
