Overview

The Strongly Interacting Quark Gluon Plasma and Future Physics

T. D. Lee

Columbia University

and

RIKEN BNL Research Center

At RHIC, recent experiments over the last two years reveal a new state
of nuclear matter, hitherto unknown to us. While the term "quark gluon
plasma (QGP)" has been used in the past, the physical picture was often
modeled after the weakly interacting electron plasma, which is very
different from what has been discovered recently. At RHIC, this new
state of nuclear matter is found to interact very strongly. Since it is
composed of quarks and gluons, we may call this new state of nuclear
matter sQGP, the strongly interacting quark gluon plasma.

The main focus of this workshop is about the present discovery of sQGP.
It is an experimental discovery. The properties of sQGP are defined by
the new findings of experiments at RHIC. While at the same time, there
are many important and impressive theoretical advances, our theoretical
progress remains imperfect. This may seem strange to some of our
experimental colleagues since we do possess quantum chromodynamics (QCD)
as the theory of strong interaction. How come, we still have
difficulties in predicting and understanding the properties of sQGP?

1\) Knowing the equation ≠ having its solutions.

Take the example of the nonrelativistic Schroedinger equation of a
system of charged particles with Coulomb interactions. In principle,
that single N-dimensional second order linear differential equation
should contain in its solutions all informations about

1\. atomic and molecular physics (except for small relativistic
corrections)

2\. condensed matter physics

3\. chemistry

and

4\. biology

It is clear, although we do have the equation for 74 years, we are still
very far from deriving all its solutions.

Just take condensed matter physics, and within that field let us briefly
review the experimental discoveries and theoretical understandings of
superconductivity and superfluidity:

1911 Superconductivity (Kamerlingh Ounes)

1924 λ --transition of He (Kamerlingh Ounes and Boks)

1924 Bose-Einstein Condensation

1938 Two fluid Model of Liquid He (London)

1957 BCS Theory

1986 High T~c~ Superconductivity (Bednorz and Müller)

Superconductivity was discovered in 1911. However, BCS theory was
formulated in 1957, 46 years later after the experimental discovery, and
32 years later after Schroedinger wrote down the basic equation of
quantum mechanics. Yet, even today, we still have difficulties in our
theoretical understanding of the high T~c~ superconductivity.

Superfluidity is another example: λ-transition of He was observed in the
same year (1924) when the theory of Bose-Einstein condensation was
formulated. Still, it took 14 years for F. London to propose the two
fluid model of liquid He. Why should it take so long?

In 1924, the theory of Bose-Einstein condensation was developed for a
system of ideal bosons with no interaction. On the other hand, the
density of liquid helium is quite high, near the close-packing density
of a system of hard spheres. Only much later, we understood that the
critical element of Bose-Einstein condensation is the long range order
of the phase of the Schroedinger wave function; it is insensitive to the
density, since the density is the absolute value squared of the wave
function. Thus, density-wise liquid He is very different from ideal
bosons, but phase-wise these two systems share the same long-range
Bose-Einstein condensation characterization. It took decades for the
best of theorists, including Einstein and others, to be sure of this
conclusion.

Yes, knowing the equation is not the same as having the solutions.
However, having the equation is a prerequisite for a full understanding
of its solutions. This is why a physics discovery is often made by
experimentalists first, then followed by theoretical understanding. For
the progress of our field, a close collaboration between
experimentalists and theorists is of critical importance.

The experimental discovery of sQGP is only the beginning of an era of
new physics. It will lead us to a more complete understanding of quantum
chromodynamics, the basic theory of quarks and gluons. Because sQGP is a
strongly interactive system, once produced its properties will dominate
all subsequent physical phenomena that follow. Thus, sQGP physics may
well be the threshold, through which we will reach LHC physics and
beyond, including perhaps also dark energy and dark matter in our
universe.

2\) sQGP physics and dark energy

Recently, there exist very strong evidence\[1\] that cosmological
constant Λ is not only nonzero, but very large. The energy density of Λ
is negative, with a large magnitude

ρ~Λ~ \~ 3 -- 7 x 10^-6^ GeV/cm^3^, (1)

which is the same order of magnitude as the critical energy density of
our universe, given by

ρ~c~ \~ 1 x 10^-5^ GeV/cm^3^. (2)

What is the origin of this negative energy? Why should ρ~Λ~ and ρ~c~ be
of the same order of magnitude?

In Einstein's first edition of his masterpiece "The Meaning of
Relativity," published in 1922, he wrote\[2\]:

"Matter consists of electrically charged particles. On the basis of
Maxwell's

theory these cannot be conceived of as electromagnetic fields free from
singularities.

In order to be consistent with the facts, it is necessary to introduce
energy terms, not

contained in Maxwell's theory, so that the single electric particles may
hold

together in spite of the mutual repulsions between their elements,
charged with

electricity of one sign. For the sake of consistency with this fact,
Poincaré

has assumed a pressure to exist inside these particles which balances
the electro-

static repulsion. It cannot, however, be asserted that this pressure
vanishes outside

> the particles. We shall be consistent with this circumstance if, in
> our phenomenological presentation, we add a pressure term. This must
> not, however,
>
> be confused with a hydrodynamical pressure, as it serves only for the
>
> energetic presentation of the dynamical relations inside matter."

Immediately after this paragraph, Einstien inserted a negative scalar
pressure term

\- p

into his equation.

This negative pressure term is, of course, identical to the cosmological
constant Λ. Only later, in the second and later editions of the same
book, Einstein added an appendix, in which he said\[3\]

" The objection to this solution is that one has to introduce a negative
pressure,

for which there exists no physical justification."

Now, there does exist strong experimental evidence for its existence.
However, is there any theoretical justification or understanding of this
large negative pressure term?

Actually, negative pressure is a familiar concept for a nuclear
physicist. It has been used extensively in the MIT bag model\[4\]. A
similar idea was also developed by Dirac\[5\] in his theory of muon. In
the following, we give a general theoretical framework for the existence
of negative pressure; it is the same original reasoning that led Wick
and myself to propose the use of relativistic heavy ion collisions for
possible creation of new states of nuclear matter \[6\].

Let us pre-suppose the existence of a scalar field φ, which can be
either a fundamental field or a composite one, made of other fields,
such as the σ-model and Higgs field. Since by using the products of any
field, one can always construct a scalar component, the physical basis
for the existence of such a scalar component φ is quite general. The
vacuum is a scalar, hence, φ is coupled to the vacuum and therefore, to
the inertia of any physical particle. In the physical vacuum state, the
field φ has an expectation value designated by φ ~vac~. Consider the
state of any single i th particle. Define its coupling g~i~ to φ  by

> m~i~ = g~i~ φ~vac~ (3)

where m~i~ is its physical mass. Next, consider the transformation

> φ → φ + c (4)

where c is a constant; correspondingly the vacuum state is changed to an
excited state, in which the expectation value of φ changes as follows:

> φ~vac~ → φ~vac~ + c. (5)

Correspondingly the mass m~i~ of any i th particle (except φ itself)
will also be altered, with

> m~i~ → m~i~ + g~i~c (6)

Choose the constant c to be given by

> c = - φ~vac~ (7)

Accordingly, (5) and (6) become

> φ~vac~ → 0 (8)

and

> m~i~ → 0 (9)

Since in any field theory, one can always construct a scalar component
φ, this means for any single physical i th particle, there exists an
excited single particle state in which its mass is zero. The generality
of this argument is the origin of negative energy.

The transformation (4) transforms the vacuum state to an excited state,
with a different expectation value from φ ~vac~. Its excited energy is
proportional to the energy density function U(φ) and the volume Ω of the
excitation. An example of U(φ) is given in Figure 1, with the abscissa φ
denoting the value φ~vac~ + c in (5).

For a single particle state in an infinite volume, transformations (8)
and (9) lead to excited states with an infinite excitation energy. But
for a system of particles within a finite volume, such changes may occur
spontaneously, as we shall see.

Figure 2 gives the energy balance of the MIT bag model of a single
nucleon, consisting of three quarks. One assumes the expectation value
of φ to be φ~vac~ outside the bag, but zero inside. Thus, the quark mass
is zero inside the bag, yielding a quark matter energy E~Q~ proportional
to 1/R, where R is the bag radius. Since the magnitude E~p~ of the
energy due to the negative pressure --p is proportional to R^3^, from
equipartition of energy, it follows that

> E~Q~ = 3 E~p~ (10)

The total energy E of the system is determined by

> E - E~p~ = E~Q~. (11)

From (10), it follows then

> E = 4 E~p~. (12)

There is an alternative version, the SLAC bag model\[7\], in which one
assumes the function U(φ) to have two degenerate minima, with U(0) = p =
0. What replaces the negative pressure is a surface tension s, giving
instead of (11), the following energy balance for the total energy E:

> E -- E~s~ = E~Q~ (13)

where E~Q~ is the quark energy similar to the MIT bag, proportional to
1/R. But instead of E~p~, one has

> E~s~ = 4πR^2^s (14)

The same equipartition of energy principle gives, instead of (10) and
(12),

> E~Q~ = 2 E~s~, (15)

and

> E = 3 E~s~. (16)

We now take our universe to be a large spherical bag\[8\] of volume Ω
and radius R, related by

> Ω = 4πR^3^/3. (17)

The total energy E of the universe is given by an approximate formula,
identical to that in the MIT bag, but with (11) replaced by

> E -- E~Λ~ = E~M~, (18)

in which E~Λ~ is the negative energy due to the cosmological constant Λ,
and E~M~ the matter energy in our universe, with

> E~Λ~ = Ωρ~Λ~ ∝ R^3^ (19)

and

> E~M~ = constant/R, (20)

Just as E~p~ and E~Q~ in the MIT bag model. We assume that in the early
period of the universe, the matter energy is dominated by its long wave
length components and that leads to (20). Setting the derivative of E
with respect to R to be zero, we have from equipartition of energy

> E~Λ~/E~M~ = 1/3, (21)

as in (10).

If we assume the universe to be like a SLAC bag, then E~Λ~ is replaced
by a surface energy term E~s~, with

> E~s~/E~M~ = 1/2, (22)

as in (15). In either case, the amount of dark energy (i.e., negative
energy) in the universe is of the same order of magnitude as the total
amount of matter energy as indicated by (1) and (2). Considering our
over simplification of neglecting the non-Euclidean geometrical effect,
the shorter wave length contribution to the matter energy and the
dynamical effect of the expansion of our universe, the above order of
magnitude agreement is very encouraging.

So, a nucleon is a small bag. A nucleus is a slightly bigger bag, and
our universe is a much much bigger bag. The pressure of a MIT bag is
exerted by the physical space outside the bag. Similarly, the negative
pressure described by the cosmological constant may also be due to the
physical space outside our universe. Through RHIC and the study of sQGP,
we can change the bag pressure p and the quark energy E~Q~; likewise by
examining carefully the change of matter energy E~M~ within our
universe, we may gain insight to the dynamical properties of the
cosmological constant Λ and its possible space-time variation. With
that, we may also derive some information about the physical space
outside the radius of our universe.

3\) sQGP Physics at LHC and Beyond.

The strongly interactive quark gluon plasma can be produced by Au -- Au
collisions at RHIC. Therefore it will be produced at LHC. Since sQGP is
produced by deutron-Au collisions at RHIC (in the forward direction), it
will also be produced through pp collisions at LHC. Because its strong
interactions, sQGP once produced will dominate other physics phenomena
that follow. This begins a new era of physics.

At a small distance, r, the QCD coupling g(r) also becomes small. But
QCD is nonlinear, there could be unstable soliton-like solutions in
which the gluon amplitude is proportional to 1/g(r). Its excitation
energy should be about 1/r. For r about 1/100 of a fermi or smaller, the
excitation energy would be about 100 GeV or larger. It is quite possible
that these soliton-like solutions may be described by a phenomenological
super-symmetrical field. However, the super-symmetry is at best only
approximate. Consequently, these soliton-like solutions are unstable.
Such super-symmetrical resonances, if exist, could be an exciting mix
between two fields: the traditional particle physics and the newly
emerging sQGP physics.

The complexity of sQGP physics clearly signals the need of intensity,
and the importance to have RHIC II. Likewise, e-RHIC can provide the
unique opportunity of using e as a probe for the analysis of sQGP, using
the much weaker QED to analyze the very strong QCD.

The discovery of the strongly interacting quark gluon plasma is a
historical event. For this occasion, I wish to dedicate the verse shown
in Figure 3.

REFERENCES

1\. P.J.E. Peebles and Bharat Ratra, Rev. Mod. Phys. 75 (2003) 559.

2\. A. Einstein, The Meaning of Relativity, 1922, Princeton University
Press, p. 106.

3\. A. Einstein, The Meaning of Relativity, 1950, Princeton University
Press, p. 111.

4\. A. Chodos, R. J. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weiskopf,
Phys. Rev. D (1974) 3471.

5\. P. A. M. Dirac, Proc.Roy. Soc. A 268 (1962) 57.

6\. T.D. Lee and G. C. Wick, Phys. Rev. D 9 (1974) 2291.

7\. W.A. Bardeen, M.S. Chanowitz, S. D. Drell, M. Weinstein and T. M.
Yan,

> Phys. Rev D 11 (1975) 1094.

8\. T. D. Lee, Chinese Phys. Lett. (2004) 1187.

FIGURE CAPTIONS

Figure 1. A phenomenological potential function U(φ)

e 2. MIT bag model

Figure 3 One Plasma, a verse in the style of J.R.R. Tolkien