|
|
|
|
 |
 |
 |
 |
 |
 |
 |
 |
 |
|||||||
 |
QCD Made Simple Quantum chromodynamics is conceptually simple. Its realization in nature, however, is usually very complex. But not always. Quantum chromodynamics, familiarly called QCD, is the modern theory of the strong interaction.1 Historically its roots are in nuclear physics and the description of ordinary matter--understanding what protons and neutrons are and how they interact. Nowadays QCD is used to describe most of what goes on at high-energy accelerators. Twenty or even fifteen years ago, this activity was commonly called "testing QCD." Such is the success of the theory, that we now speak instead of "calculating QCD backgrounds" for the investigation of more speculative phenomena. For example, discovery of the heavy W and Z bosons that mediate the weak interaction, or of the top quark, would have been a much more difficult and uncertain affair if one did not have a precise, reliable understanding of the more common processes governed by QCD. With regard to things still to be found, search strategies for the Higgs particle and for manifestations of supersymmetry depend on detailed understanding of production mechanisms and backgrounds calculated by means of QCD. Quantum chromodynamics is a precise and beautiful theory. One reflection of this elegance is that the essence of QCD can be portrayed, without severe distortion, in the few simple pictures at the bottom of the box on the next page. But first, for comparison, let me remind you that the essence of quantum electrodynamics (QED), which is a generation older than QCD, can be portrayed by the single picture at the top of the box, which represents the interaction vertex at which a photon responds to the presence or motion of electric charge.2 This is not just a metaphor. Quite definite and precise algorithms for calculating physical processes are attached to the Feynman graphs of QED, constructed by connecting just such interaction vertices. In the same pictorial language, QCD appears as an expanded version of QED. Whereas in QED there is just one kind of charge, QCD has three different kinds of charge, labeled by "color." Avoiding chauvinism, we might choose red, green, and blue. But, of course, the color charges of QCD have nothing to do with physical colors. Rather, they have properties analogous to electric charge. In particular, the color charges are conserved in all physical processes, and there are photon-like massless particles, called color gluons, that respond in appropriate ways to the presence or motion of color charge, very similar to the way photons respond to electric charge. Quarks and gluons Quarks are spin-1/2 point particles, very much like electrons. But instead of electric charge, they carry color charge. To be more precise, quarks carry fractional electric charge (+ 2e/3 for the u, c, and t quarks, and e/3 for the d, s, and b quarks) in addition to their color charge.
All this would seem to
require 3 ?/font>3
= 9 different color gluons. But one particular combination of
gluons--the color-SU(3) singlet--which responds equally to all charges,
is different from the rest. We must remove it if we are to have a
perfectly color-symmetric theory. Then we are left with only 8 physical
gluon states (forming a color-SU(3) octet). Fortunately, this conclusion
is vindicated by experiment!
The third difference
between QCD and QED, which is the most profound, follows from the
second. Because gluons respond to the presence and motion of color
charge and they carry unbalanced color charge, it follows that
gluons, quite unlike photons, respond directly to one another. Photons,
of course, are electrically neutral. Therefore the laser sword fights
you've seen in Star Wars wouldn't work. But it's a movie about
the future, so maybe they're using color gluon lasers.
U =
UNewton + UGauss + . . . ,
where,
for instance, UNewton = (F ma)2
and UGauss = (?
?/font> E - r)2.
So we can capture all
the laws of physics we know, and all the laws yet to be discovered, in
this one unified equation. But it's a complete cheat, of course, because
there is no useful algorithm for unpacking U, other than to go
back to its component parts. The equations of QCD, displayed in figure
1, are very different from Feynman's satirical unification. Their
complete content is out front, and the algorithms that unpack them flow
from the unambiguous mathematics of symmetry.
A remarkable feature of
QCD, which we see in figure
1, is how few adjustable parameters the theory needs. There is just
one overall coupling constant g and six quark-mass parameters mj
for the six quark flavors. As we shall see, the coupling strength is a
relative concept; and there are many circumstances in which the mass
parameters are not significant. For example, the heavier quarks play
only a tiny role in the structure of ordinary matter. Thus QCD
approximates the theoretical ideal: From a few purely conceptual
elements, it constructs a wealth of physical consequences that describe
nature faithfully.4
Describing reality Besides confinement,
there is another qualitative difference between the observed reality and
the fantasy world of quarks and gluons. This difference is quite a bit
more subtle to describe, but equally fundamental. I will not be able to
do full justice to the phenomenological arguments here, but I can state
the essence of the problem in its final, sanitized theoretical form. The
phenomenology indicates that if QCD is to describe the world, then the u
and d quarks must have very small masses. But if these quarks do have
very small masses, then the equations of QCD possess some additional
symmetries, called chiral symmetries (after chiros, the Greek
word for hand). These symmetries allow separate transformations
among the right-handed quarks (spinning, in relation to their motion,
like ordinary right-handed screws) and the left-handed quarks.
But there is no such
symmetry among the observed strongly interacting particles; they do not
come in opposite-parity pairs. So if QCD is to describe the real world,
the chiral symmetry must be spontaneously broken, much as rotational
symmetry is spontaneously broken in a ferromagnet.
Clearly, it's a big
challenge to relate the beautifully simple concepts that underlie QCD to
the world of observed phenomena. There have been three basic approaches
to meeting this challenge:
While these and
other massive numerical calculations give impressive and useful
results, they are not the end of all desire. There are many
physically interesting questions about QCD for which the known
numerical techniques become impractical. Also, it is not entirely
satisfying to have our computers acting as oracles, delivering
answers without explanations.
Compare those
multiparticle hadronic events to collisions in which leptons, say muons,
are produced. In that case, about 99% of the time one observes simply a
muon and an antimuon, emerging in opposite directions. But
occasionally--in about 1% of the muonic final states--a photon is
emitted as well.
If history had happened
in a different order, the observation of jet-like hadronic final states
would surely have led physicists to propose that they manifest
underlying phenomena like those displayed on the right-hand side of figure
3. Their resemblance to leptonic scattering and QED would be too
striking to ignore.
Eventually, by studying
the details of how energy was apportioned among the jets, and the
relative probabilities of different angles between them, the physicists
would have deduced directly from experimental data that there are light
spin-1/2 and massless spin-1 objects lurking beneath the appearances,
and how these covert objects couple to one another. By studying the rare
4-jet events, they could even have learned about the coupling of the
spin-1 particles to each other. So all the basic couplings we know in
QCD might have been inferred, more or less directly, from experiment.
But there would still be one big puzzle: Why are there jets, rather than
simply particles?
The answer is profound,
and rich in consequences. It is that the strength with which gluons
couple depends radically on their energy and momentum. "Hard''
gluons, which carry a lot of energy and momentum, couple weakly; whereas
the less energetic "soft'' gluons, couple strongly. Thus, only
rarely will a fast-moving colored quark or gluon emit
"radiation" (a gluon) that significantly redirects the flow of
energy and momentum. That explains the collimated flows one sees in
jets. On the other hand, there can be a great deal of soft radiation,
which explains the abundant particle content of the jets. So, in a
rigorous and very tangible sense, we really do get to see the quarks and
gluons--but as flows of energy, not individual particles.
The actual history was
different. The need for asymptotic freedom in describing the strong
interaction was deduced from much more indirect clues, and QCD was
originally proposed as the theory of the strong interaction because it
is essentially the unique quantum field theory having the property of
asymptotic freedom.9 From these ideas, the existence of jets,
and their main properties, were predicted before their experimental
discovery.5
High temperature QCD
Asymptotic freedom
implies that QCD physics gets simpler at very high temperature.
That would seem implausible if you tried to build up the
high-temperature phase by accounting for the production and interaction
of all the different mesons and baryon resonances that are energetically
accessible at high temperature. Hoping to bypass this forbidding mess,
we invoke a procedure that is often useful in theoretical physics. I
call it the Jesuit Stratagem, inspired by what I'm told is a credal
tenet of the Order: "It is more blessed to ask forgiveness than
permission.'' The stratagem tells you to make clear-cut simplifying
assumptions, work out their consequences, and check to see that you
don't run into contradictions.
In this spirit we
tentatively assume that we can describe high-temperature QCD starting
with free quarks and gluons. In an ideal (noninteracting) gas of quarks,
antiquarks, and gluons at high temperature, most of the energy and
pressure will be contributed by particles with large energy and
momentum. How do interactions affect these particles? Well,
significantly deflecting such a particle requires an interaction with
large momentum transfer. But such interactions are rare because, as
asymptotic freedom tells us, they are governed by rather weak coupling.
So interactions do not really invalidate the overall picture. To put it
another way, if we treat the hadron jets generated by quarks, antiquarks,
or gluons as quasiparticles "dressed" in hadronic garb, then
we have a nearly ideal gas of quasiparticles. So it seems that ignoring
the interactions was a valid starting point. The stratagem has
succeeded.
What about real
experiments? Unfortunately our only access to the quarkgluon plasma
is through the production of tiny, short-lived nuclear fireballs, of
which we detect only the debris. Interpreting the data requires
complicated modeling. In the quest for evidence of the quarkgluon
plasma, there are two levels to which one might aspire. At the first
level, one might hope to observe phenomena that are very difficult to
interpret from a hadronic perspective but have a simple qualitative
explanation based on quarks and gluons. Several such effects have been
observed by the CERN heavy-ion program in recent years.11 But
there is a second, more rigorous level that remains a challenge for the
future. Using fundamental aspects of QCD theory, similar to those I
discussed in connection with jets, one can make quantitative predictions
for the emission of various kinds of "hard" radiation from a
quarkgluon plasma. We will not have done justice to the concept of
a weakly interacting plasma of quarks and gluons until some of these
predictions are confirmed by experiment.
High density QCD Why might we hope that
QCD simplifies in the limit of large density? Again we use the Jesuit
Stratagem. Assume we can neglect interactions. Then, to start with,
we'll have large Fermi surfaces for all the quarks. (The Fermi surface
bounds the smallest momentum-space volume into which you can pack all
those fermions, even at zero temperature.) This means that the active
degrees of freedom--the excitations of quarks near the Fermi
surface--have large energy and momentum. And so we might be tempted to
make essentially the same argument we used for the high-temperature,
low-density regime and declare victory once again.
On further reflection,
however, we find this argument too facile. For one thing, it doesn't
touch the gluons, which are, after all, spin-1 bosons. So they
are in no way constrained by the Pauli exclusion principle, which blocks
the excitation of low-momentum quarks. The low-momentum gluons interact
strongly, and because they were the main problem all along, it is not
obvious that going to high density really simplifies things very much.
A second difficulty
appears when we recall that the Fermi surfaces of many condensed-matter
systems at low temperature are susceptible to a pairing instability that
drastically changes their physical properties. This phenomenon underlies
both superconductivity and the superfluidity of helium-3. It arises
whenever there is an effective attraction between particles on opposite
sides of the Fermi surface. As elucidated by John Bardeen, Leon Cooper,
and Robert Schrieffer, even an arbitrarily weak attraction can, in
principle, cause a drastic restructuring of the ground state.
A nominally small
perturbation can have such a big effect because we're in the realm of
degenerate perturbation theory. Low-energy excitation of pairs of
particles on opposite sides of the Fermi surface, with total momentum
zero, can be scattered into one another. By orchestrating a coherent
mixture of such excitations, all pulling in the same direction, the
system gains an energy advantage.
In condensed-matter
physics, the occurrence of superconductivity is a difficult and subtle
affair. That's because the fundamental interaction between electrons is
Coulomb repulsion. In the classic metallic superconductors, an effective
attraction arises from subtle retardation effects involving phonons. In
the cuprate superconductors, the cause is still obscure.
In QCD, by contrast,
the occurrence of what we might call "color superconductivity"
is a relatively straightforward phenomenon.12 That's because
the fundamental interaction between two quarks, unlike that between two
electrons, is already attractive! One can see this by a
group-theoretical argument: Quarks form triplet representations of color
SU(3). A pair of quarks, in the antisymmetric color state, form an
antitriplet. So when two quarks are brought together, the effective
color charge is reduced by a factor of two compared to when they were
separated. The color flux emerging from them is reduced, lessening the
energy in the color field. That implies an attractive force. So we
should consider carefully what color superconductivity can do for us.
Two of the central
phenomena of ordinary superconductivity are the Meissner effect and the
energy gap. The Meissner effect is the inability of magnetic fields to
penetrate far into the body of a superconductor. Supercurrents arise to
cancel them out. Electric fields are, of course, also screened by the
motion of charges. Thus electromagnetic fields in general become
short-range. Effectively it appears as if the photon has acquired a
mass. Indeed that is just what emerges from the equations. We can
therefore anticipate that in a color superconductor, gluons will acquire
mass. That's very good news, because it removes our problem with the low
energymomentum gluons.
The energy gap means
that it costs a finite amount of energy to excite electrons from their
superconducting ground state. That's quite unlike what we had for the
free Fermi surface. So the original pairing instability, having run its
course, is no longer present.
Now with both the
sensitivity to small perturbations (pairing instability) and the bad
actors (soft gluons) under control, the remaining effects of
interactions really are small and under good theoretical control. Once
again, the Jesuit Stratagem has served us well.
Colorflavor locking Here we discover the
remarkable phenomenon of colorflavor locking.13
Ordinarily the perfect symmetry among different quark colors is quite
distinct and separate from the imperfect symmetry among different quark
flavors. But in the
imagined colorflavor locked state they become correlated. Both
color symmetry and flavor symmetry, as separate entities, are
spontaneously broken, and only a certain mixture of them survives
unscathed.
Colorflavor
locking in high-density QCD drastically affects the properties of quarks
and gluons. As we have already seen, the gluons become massive. Due to
the commingling of color and flavor, the electric charges of particles,
which originally depended only on their flavor, are modified.
Specifically, some of the gluons become electrically charged, and the
quark charges are shifted. The electric charges of these particles all
become integral multiples of the electron's charge!
Thus the most striking
features of confinement--the absence of long-range color forces, and
integer electric charge for all physical excitations--emerge as simple,
rigorous consequences of color superconductivity. Also, because both
left- and right-handed flavor symmetries are locked to color, they are
also effectively locked to each other. Thus chiral symmetry, which
required independent transformations among the left- and the
right-handed components of the quarks, is spontaneously broken.
Altogether, there is a
striking resemblance between the calculated properties of the
low-energy excitations in the high-density limit of QCD and the expected
properties--based on phenomenological experience and models--of hadronic
matter at moderate density. This suggests the conjecture that there is
no phase transition separating them.
Unfortunately both
numerical and direct experimental tests of this conjecture seem out of
reach at the moment. So it is not certain that the mechanisms of
confinement and chiral-symmetry breaking we find in the calculable,
high-density limit are the same as those that operate at moderate or low
density. Still, I think it astonishing that these properties, which have
long been regarded as mysterious and intractable, have been simply--yet
rigorously--demonstrated to occur in a physically interesting limit of
QCD.
I have tried to
convince you of two things: first, that the fundamentals of QCD are
simple and elegant, and second, that these fundamentals come into their
own, and directly describe the physical behavior of matter, under
various extreme conditions.
References 2. R. Feynman, QED: The Strange Theory of Light and Matter
Princeton University Press, Princeton, N. J. (1985).
3. R. Feynman, The Feynman Lectures on Physics,
Addison-Wesley, New York (1964) vol. II, p. 25-10.
4. F. Wilczek, Nature 397, 303 (1999).
5. S. Aoki et al., http://xxx.lanl.gov/abs/hep-lat/9904012
6. G. Hanson et al., Phys. Rev. Lett. 35, 1609 (1975).
7. D. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). H.
Politzer, Phys. Rev. Lett. 30, 1346 (1973).
8. For a recent review, see I. Hinchcliffe, A. Manohar, http://xxx.lanl.gov/abs/hep-ph/0004186.
9. D. Gross, F. Wilczek, Phys. Rev. D 9, 980 (1974).
10. For a recent review, see F. Karsch, http://xxx.lanl.gov/abs/hep-lat/9909006.
11. U. Heinz, M. Jacob, CERN preprint nucl-th/0002042
(2000).
12. For a recent review, including extensive references, see T. Schäfer,
http://xxx.lanl.gov/abs/hep-ph/9909574.
13. M. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B 537,
443 (1999).
14. M. Schmelling, http://xxx.lanl.gov/abs/hep-ex/9701002.
15. S. Gottlieb et al., Phys.
Rev. D 47, 3619 (1993). |
150
MeV or 1.5 ?1012
K), the only important particles are the spinless pi mesons: p+,
p,
and p0.
They represent 3 degrees of freedom. But from a quarkgluon
description we come to expect many more degrees of freedom, because
there are 3 flavors of light spin-1/2 quarks, each of which comes in 3
colors. If you then include 2 spin orientations, antiquarks, and 8
gluons, each with 2 polarization states, you end up with 52 degrees of
freedom in place of the 3 for pions. So we predict a vast increase in
the energy density, at a given temperature, as you go from a hadron gas
to a quarkgluon plasma. And that is what the calculations displayed
in