- Focus will be on so-called "quark-gluon plasma" (QGP)
- Many other interesting phenomena / physics, but we need to narrow our focus because of time constraints
- Fairly easy to think about without knowing great details (e.g., use models that are caricatures of QCD)
- The most interesting physics idea (why RHIC is high priority)
- "new state of matter"
- connection to early universe
- possible connection to astrophysical objects like neutron stars
- phase transitions of very interesting types ("deconfinement" and "chiral symmetry restoration")
- Test our understanding of QCD; qualitatively different physics or conditions or properties we can calculate

- Physics of plasmas
- Rough definition of "ordinary" plasma
- "How Things Work" defines a plasma as a gas-like phase of matter consisting of electrically charged particles, such as positively charged ions and negatively charged electrons.
- Sometimes referred to as the "fourth state" of matter, but ordinarily no sharp transition. That is, there is no phase transition with a discontinuous change in the free energy or specific heat.
- The term "plasma" was introduced by Tonks and Langmuir in 1929, probably as an analogy to particle-like corpuscles in fluid (blood) plasma. In general, refers to a quasi-neutral assembly of charged particles.

- Examples of plasmas and how they are made [see handout]
- The sun (and other stars and out in space). Hydrogen plasma mostly. Temperatures high enough to ionize (or enough radiation). In space, UV radiation is most responsible for ionization.
- Magnetically confined hydrogen plasmas for fusion reactors.
- Checkpoint: Can you find plasmas in this room?

Fluorescent lamps contain argon, neon, and/or krypton gas at low pressure and density (roughly 0.3% atmosphere). One in a thousand atoms is mercury vapor, which is the actual source of light. They are excited by collisions with electrons and emit UV light when they deexcite. This in turn is converted to visible light by a phosphor coating on the tube. The mixture of phosphors is chosen so that they fluoresce over a wide range of visible wavelengths to provide a balanced, white spectrum. The tubes need positively charged mercury ions to maintain neutrality with the electrons -> plasma. They are produced by particularly energetic collisions. - In a neon sign.
- Other examples? (See handout)

- Summary: Characteristics of a plasma
- Because of the "free" charges, the plasma can conduct
an electric current (e.g. in a fluorescent light).
However, it cannot be treated as an ordinary gas that
happens to be electrically
conducting.
- In a neutral gas, the forces between constituents are very strong but short-range, so the dynamics is dominated by two-body interactions (billiard-ball-like or hard sphere for typical systems).
- In a plasma, the forces are coulombic. Therefore they are comparatively weak and long-ranged. This means that there will be collective and coherent effects involving large numbers of particles.
- Both can have screening effects, but from polarization in first case and Debye screening in second case.

- Quasi-neutrality provided by strong electric forces that attract opposite charges. Any separation of charges produces electric fields that tend to compensate.
- Debye length (find the characteristic scales in a problem)
- Think about time scales over which ionic background charges don't move -> uniform charge; only electrons adjust.
- If we put a test charge in the plasma, the observed
electrostatic potential is not \propto 1/r
but \propto e^{-r/l_D}/r
(a "Yukawa"), with characteristic length l_D.

l_D = (T / 4\pi n_0 e^2)^{1/2} , where n_0 is the particle density. [Careful of units here, by the way.]

- Charges are "screened" by other charges in the plasma. [We'll go through the calculation in detail later and give a physical picture of moving charges deflected closer to or further from the test charge.] The effective range of interactions is no longer infinite but the Debye length.
- More loosely, the Debye length is the characteristic length over which positive and negative charges can be separated due to thermal fluctuations. That is, the particle electric potential energy cannot exceed the particle thermal energy kT in order of magnitude. Determine l_D from this idea.
- Think about a slab of thickness l_D in which the charges have been separated so that only one type of charge is in the slab. Electric field is E = 4\pi e n_0 l_D (from Gauss' law), so particle potential energy e\phi is that times l_D. Equating that to T (k=1 here) gives an estimate of l_D. Or, separate charges to either side of the slab and treat like a capacitor, with a constant electric field. The surface charge density of the slab is just the original volume charge density times l_D, so we can find E and \phi as before. The final result is the same for l_D (at least up to factors of 2).

- An ionized gas is a plasma if the Debye length is shorter than characteristic lengths such as size of plasma. Otherwise you wouldn't notice the screening! There are further criteria that we will discuss later.

- Because of the "free" charges, the plasma can conduct
an electric current (e.g. in a fluorescent light).
However, it cannot be treated as an ordinary gas that
happens to be electrically
conducting.
- Summary: How do you make a hot plasma?
- "thermal energy" greater than dissociation energy

- How much of this carries over to QGP?

- Rough definition of "ordinary" plasma
- Overview: Quantum Chromodynamics (QCD) [just intro here]
- More details later; just the basics here to introduce vocabulary and basic pictures (cartoons)
*Underlying*degrees of freedom: quarks and gluons- Show pictures of atoms, nuclei, and quarks.
- quarks come in 3 colors and six flavors
- up, down, strange, and charm quarks will enter our discussion at some point.

- Basic vocabulary:
**asymptotic freedom:**interaction becomes weak at high energies (short distances), so perturbation theory is useful.**color confinement:**at low energies the observed strongly interacting particles ("hadrons") are colorless. No free quarks or gluons. The dynamics is nonperturbative; hard to calculate!**hadrons:**bound states of quarks and gluons. Examples are protons, neutrons, pions, and kaons. Since they are the building blocks of nuclei, QCD in turn is the underlying theory of nuclei!

- Analogy to quantum electrodynamics (QED)
- Start with lagrangian of (classical) QCD. Looks very similar to QED lagrangian (more later). [See handout with copy of transparency.]
- Compare QED and QCD side by side [transparency and handout]. Interaction mediated by exchange of massless gauge bosons (photon and gluon, respectively). Color charge in QCD in analog to electric charge in QED. But comes in 3 rather than 1 type (with red/anti-red analogous to +/-). Photons are uncharged, gluons carry color charge (eight possibilities). The latter has important consequences when considering electromagnetic vs. color force.
- Consider vacuum polarization in QED vs. QCD.
- Draw perturbative diagrams for vacuum fluctuation correction to force between two electrons
- The physics is that vacuum fluctuations (e+e- pairs that appear virtually even in the absence of matter -- uncertainty principle!) are polarized by a charge source, which leads to screening [draw picture]. Long wavelength photons see the screened charge; ordinary electron charge is defined in this limit. Short wavelength photons probe "bare" charge, so the effective charge increases. We say that it "runs" with the scale being probed.
- The consequence is that the QED vacuum (lowest energy state) acts like a dielectric medium with \epsilon_{QED} > 1, and \epsilon depends on the resolution (four-momentum or distance). Since relativity tells us that \mu x \epsilon =1, this means we have a diamagnetic medium. In principle, all charged fermions participate in the screening.
- In QCD, we have screening by q\qbar pairs just like e+e- in QED. Now they are screening color charge, not electric charge. So we'd expect the same running of the coupling.
- However, we have a new effect since gluons also carry color charge. A color charge radiates virtual gluons that disperse the color charge through a gluon cloud around the charge. The full charge is only seen at large distances (low momentum probes) while at short distances there is less color charge effectively. (Think of probing inside of a ball of charge instead of a point charge. When you get inside the force decreases. There is a "form factor" that descreases with increasing momentum q.) [Note that the picture here is gauge dependent.]
- These two effects compete and in QCD the second wins. The result is "asymptotic freedom". The corresponding \epsilon is less than 1 ("antiscreens") and \mu is thus greater than 1 ("paramagnetic").

- One-loop running couplings for QED and QCD

QED: \alpha_{QED} = \alpha / (1 - (\alpha/3\Pi)\log(Q^2/m_e^2))

QCD: \alpha_{QCD} = \alpha(mu^2) / (1 + \alpha(\mu^2)(\beta_0/4\Pi)\log(Q^2/\mu^2))

= {12\pi \over (33-2N_f) \ln(Q^2/\Lambda^2)}

where \beta_0 = 11 - 2/3 N_f.- Q is a typical momentum scale (such as the momentum transfer) of the process in question. Large Q^2 means high resolution, so short distances are probed. Low Q^2 <-> long distances.
- So plotted against Q^2, the QED coupling starts small (1/137) and grows logarithmically. The QCD coupling starts small as Q^2 -> infinity, and grows with lower Q^2 until the one-loop formula becomes invalid.
- Note that if the number of flavors N_f was large enough, we'd have a situation like QED!
- At high temperatures, \mu=T is relevant, which implies asymptotic freedom and weak interactions. [However, we still expect nonperturbative physics at the highest temperatures we will reach.]
- At high densities, the chemical potential is relevant, with the same conclusion.
- This whole discussion is only quantitative if the coupling is small. At longer distances (lower energy), the coupling grows until perturbation theory is no longer valid.

- Confinement
- Since the color forces become stronger as we separate
charges, the perturbative picture becomes inadequate.
We must deal with
*nonperturbative*physics. - We can draw cartoons about what happens [see transparencies] but the only quantitative calculations to date are done with lattice gauge theory (see below).
- Bottom line is that color charges are confined in hadrons.
- A useful caricature of a hadron is the bag model. We think of hadrons as bags of quarks (draw pictures). The bag separates the vacuum outside from inside (think of two different mediums, with magnetic permeability \mu infinite outside). The original picture was of free quarks inside with a trivial vacuum (\mu=1), but this is not consistent with other known aspects of hadrons. In any case, the quarks are confined in the bag volume.
- The vacuum inside is higher energy though, so there is pressure (labeled B) to collapse the bag. The quarks provide a corresponding pressure inside to balance (because they are at nonzero momentum since confined --- think uncertainty principle!).

- Since the color forces become stronger as we separate
charges, the perturbative picture becomes inadequate.
We must deal with
- QCD vacuum: spontaneous symmetry breaking ("hidden symmetry")
- ferromagnetic analogy
- We'll do the detailed thermodynamics later (at least in the mean-field approximation).
- Nearest-neighbor spin-spin interaction for atoms
on a lattice:

H_{jk} = - 2J \vec S_j \cdot \vec S_k --> -2J S_j_z S_k_z (for simplicity)

The total magnetization is the sum of the moments from each atom, which are in turn proportional to the spin at that site. (Aside: H actually from identical fermions--electrons-- being unable to occupy the same state, which leads to an exchange force; magnetic dipole interaction is much weaker. This drops rapidly with distance, motivating the nearest-neighbor limitation.) - Consider the situation with no external magnetic field.
- With the usual interaction considered (J>0), the interaction energy is lowest when spins are aligned. So at zero temperature the spins are highly correlated. (Note that we're not being very careful in this discussion!) This means there is a net magnetization (per unit volume). But this magnetization picks out a single direction, which means that the underlying 3d rotational symmetry is hidden. (Imagine doing experiments within a ferromagnet to look for that symmetry.)
- This is an example of "spontaneous symmetry breaking," because a symmetry of the hamiltonian, in this case rotational symmetry, is not manifest in the ground state. (Note that it real life you will get "domains"; cf. disordered chiral condensates later.)
*So how come this correlated state is not the lowest energy state as the temperature is increased?*

Trick question: It*is*the lowest energy state, but it is not energy that is minimized at nonzero temperature. Rather for this problem it is the free energy F = E - TS, where S is the entropy. Since the entropy is lowest (zero!) for the lowest energy state (only one configuration), it becomes "advantageous" to go to an ensemble that has higher energy states but with larger entropy as the temperature increases.- At high temperatures (above the Curie temperature) the rotational symmetry is restored: all directions are equivalent again.
- See transparency with picture of magnetization
from Reif (or draw it). The magnetization
is an "order parameter" for the phase transition.

- ferromagnetic analogy
- In QCD we also have spontaneous symmetry breaking. A good way to discuss it is in terms of lattice gauge theory, which has some analogs to a lattice of spins.

- Lattice gauge calculations
- Exploits Feynman path integral representation of field theories. One sums over all possible things that can happen (i.e. all possible configurations of the quark and gluon fields), with a weighting factor. (Cf., extremizing the action in classical mechanics.) This gives the partition function for QCD (cf., the partition function for the ferromagnet). Once we have the partition function we can calculate thermodynamic quantities. [If we can deal with infinite dimensional integrals plus nasty infinities!]
- Just like one can make an ordinary integral doable on a computer
by making a continuous variable discrete, one can make a path
integral doable by making space and time discrete.
At the same time, it serves to
*regularize*the ultraviolet divergences in QCD by providing a momentum cutoff. - Quark fields are given values on the lattice points and color fields are represented by "link variables" that join the lattice points. This incorporates the gauge symmetry of QCD.
- These link variables can become correlated in color with their neighbors, in analogy to the ferromagnet example. This leads to a low-temperature vacuum with a color (gluon) condensate (cf. superconductivity).
- There is also a quark condensate, which signifies the spontaneous breaking of chiral symmetry. What is the order parameter?
- Many more details later! One of our goals is to understand how this works.
- As the temperature is raised, thermal motion tends to disrupt the correlations, just as in the ferromagnet case. The QCD symmetries are restored in the ground state and quarks and gluons become deconfined.
- Indicate a phase transition at temperatures around 150 MeV with energy densities of a few GeV/fm^3. This is exhibited by a sharp rise in the energy density and a smoother rise in the pressure near this temperature. [see figure on page 13 of LRP handout.] Signals rapid rise in effective number of degrees of freedom.
- There are still many open questions even about this aspect, since details of the transition (such as its order) depend on parameters (such as the quark masses) that are not solidly simulated yet. There are also questions about lattice artefacts.

- Nuclear matter phase diagram
- Have transparencies with the
three different versions
- Also on the web
- Which is the most "realistic"? Look at the various features, such as the axis labels, the pictures of quarks confined and deconfined, and the numerical values for transition points.
- Note the units in the wallchart figure versus the others. Natural units with temperature measured in MeV make much more sense (note: k=1).

- Thermodynamics variables
- T on y-axis. What should be on x-axis?
- "density": energy density, baryon density, matter density,
chemical potential? (Note that it is not the same in all
the sample diagrams!) How would the picture
differ and does it make a difference?

- Uniform (infinite) system in thermal and chemical equilibrium
- Static thermodynamic quantities only here (like energy density and pressure)
- In "real life" we have finite systems
- May be far from equilibrated!

- Checkpoint: If you created a free nucleon in empty space, what region would it see? Where would the interior of a large nucleus sit? Followup: Why are nuclei under "ordinary" conditions considered to be close to zero temperature?
- Checkpoint: What would the analogous diagram for water look like? Try plotting T vs. V first (at fixed N).
- Low T and \rho: Liquid-gas phase transition
- Nuclear matter saturation curve.
- What do you expect at low densities and temperatures? Coexistence of low-density nucleon gas and clumps of matter near nuclear saturation density.

- What is the signal for a phase transition? (E.g., is there an "order parameter" like the net magnetization in the ferromagnet case?)

- Have transparencies with the
three different versions
- Estimates of critical temperature and density for transition
- Much more complete discussion later. Here estimate the magnitude of transition temperature and density based on simple considerations.
- Consider bag picture of hadrons. Quarks and gluons confined within a certain radius. If bags overlap over large distances we can expect deconfinement, so we use this to estimate critical parameters. The bags are somewhere from 0.6 to 1.0 fm in radius.
- Two ways to fill space with hadrons [pictures!]:
- As we increase the temperature at fixed baryon density the density of mesons (and baryon-antibaryon pairs) will increase.
- As we increase the density, the nucleon bags get closer together.

- Start with zero (or very low) baryon density.
*How does the baryon density change as we heat up the system? What about the density of other hadrons?*The dominant meson will be the pion.- To a reasonable approximation we'll have a weakly interacting gas of thermal pions [except near T_c!].
- Pions are the lightest
*hadron*, so they are the only mesons for which there is any appreciable density at temperatures below a couple hundred MeV. Pion masses are 139.6 MeV for charged and 135.0 MeV for neutral pions. [Aside to class: How do you account for the difference? Possibilities: electromagnetism, current quark mass differences, condensate differences. All sources of isospin symmetry violation. Mass of proton is 938 MeV, mass of neutron is 940 MeV; are these consistent (think about electromagnetic energy; how could you estimate this and what sign for the mass?).] - Chiral symmetry tells us they will be weakly interacting at low temperature (therefore low energy). So assume we can treat the heated vacuum as a noninteracting gas of pions.
- Pions are bosons, so they will be distributed in momentum according to a Bose-Einstein distribution f(p,T), where f(p,T) = 1/(e^{(E_\pi-\mu_\pi)/T} - 1), with E_\pi the relativistic pion energy. This is the occupation number for a pion mode with momentum \vec p. Note the "-1" since bosons. If massless, E_\pi = p, so much simpler.
- We assume chemical equilibrium; what does this imply about the chemical potential? The important thing about chemical potentials is that they are associated with conserved quantities. The free energy F is minimized with respect to thermodynamic variables like T and V. If the number of particles is also determined by thermodynamics (i.e., it is not conserved), then F is minimized wrt N, with T and V constant. That is, (\partial F/\partial N)_{T,V} = 0. But this is just the definition of the chemical potential \mu, so \mu=0. For ordinary nonrelativistic gases, the number of atoms is conserved. For pions, there are processes that change the number of pions; it is not a conserved quantity. So \mu_\pi=0.
- The total density of pions is found by summing over all possible momenta weighted by f(p,T). That is, we integrate over all momenta to find n_\pi(T). Don't forget the multiplicity of 3 for the three charge states of the pion.
- We can do this integral numerically with the empirical pion mass or else analytically with zero mass (it turns out not to make much difference; students will show this later). In either case, changing the integral to dimensionless form shows that the density scales like T^3. So the number density of pions increases rapidly with temperature. Results for massless pions are derived in Chapter 9.
- When the density is as large as one pion in the volume occupied by the pion, they are packed and we can expect a phase transition to a QGP. So equate 1/(4/3 \pi R_\pi^3) to n_\pi(T_c) and find T_c. For a reasonable range of R_\pi this puts T_c between 150 and 300 MeV.
- For massless pions, the density at 200 MeV is 0.38 pions/fm^3. This temperature corresponds to R_\pi = 0.86 fm with m_\pi = 0.
- We'll look at a variety of other estimates later in the quarter.
- What about the energy density?

- Now consider zero temperature but high density.
- We make the same type of argument, only now simply finding the density corresponding to closely packed nucleons.
- \rho_c = 1/(4/3 \pi R_N^3) turns out to be about 3 times nuclear saturation density (which is the density inside heavy nuclei like lead, about 1/6 fm^{-3}). This is probably an underestimate by a factor of 2 or so based on more sophisticated considerations.
- We can estimate the corresponding energy density by noting that a lower bound will be the nucleon mass times the density. This is about 1 GeV/fm^3.

- We hope to create this phase transition in the laboratory. It is believed to have taken place in the early universe at about 10^-5 seconds after the big bang and may exist in the deep interior of neutron stars.
- The use of "phase transition" here may not be strictly correct, in that the transition may not necessarily be associated with a rapid change in an order parameter.

- Course overview
- We've divided the course according to the three basic questions we will address about the quark-gluon plasma.
- How do we make a QGP?
- The answer is: by colliding very high energy ions. Our goal is to understand why we expect this to work and how one describes the physics of such a collisions.
- Characteristically, high-energy nucleus-nucleus collisions involve large amounts of energy, of which a large fraction is deposited in a small region of space in a short time -> large energy densities.
- RHIC is designed to accelerate ions to about 100
GeV/nucleon.
Gold on gold (Au on Au) means 100 x 197 GeV for each nucleus
and com (center-of-mass) energy \sqrt{s} = 2 x 19.7 TeV = 39.4
TeV.

Checkpoints: Is this more mass energy or kinetic energy? What is each beam momentum? Estimate \gamma. - LHC proposal is to generate 3 TeV/nucleon in the com, so 1262 TeV for Pb on Pb!
- Ok, so we have a lot of energy available. If it just comes
in and goes out again, nothing interesting happens. The phase
diagram is for an
*equilibrated*system. So what happens to the energy in a collision? - Suppose nuclei collide head-on. If they are very
transparent, they pass through and not much energy is
deposited. Claim: This is not what happens!
- inelastic nuclear collisions have large cross sections and are highly inelastic
- nucleon-nucleon cross section at high energies is dominated by inelastic part
- colliding nucleons lose about half their energies, which is deposited in the com and carried away by pions and other mesons
- nuclei-nuclei collisions have many inelastic NN collisions, which are roughly additive
- furthermore, the nuclei are Lorentz contracted into thin disks (draw picture). This means that the NN collisions will nearly all be at the same time and space point.
- The bottom line is a large amount of energy deposited
in a small region of space in a short duration of time.
The claim is an energy density of a few GeV/fm^3 can be
reached.

Checkpoint: compare to energy density of nuclear matter. Energy density is due to mass of nucleon primarily, so 1 GeV x 1/6 nucleons/fm^3. Therefore an order of magnitude greater in the collision.

- There are two ways in which high energy densities are
reached:
- "stopping" or "baryon-rich QGP" region. COM energy \sqrt{s} about 5 to 10 GeV/nucleon. The nuclear matter is nearly stopped.
- "baryon-free QGP" region. COM energy \sqrt{s} greater than 100 GeV/nucleon. The matter is slowed down but not stopped. The energy deposited in the "central rapidity" region (next time understand vocabulary; for now draw picture). This has special astrophysics interest, since net baryon content of early universe is very small.
- Compare scenarios on phase diagrams.

- The starting point is what happens when single nucleons collide at ultra high energies. Read chapters 2 and 3 for next time.

- What is a QGP like?
- Here we'll focus on the thermodynamics of a quark-gluon plasma and the transition from hadronic matter.
- We'll also understand what lattice gauge calculations are all about, since they are our most reliable tool at present to tell us about nonperturbative QCD at high temperatures.

- How do we measure a QGP?
- What are the signatures that we have created a QGP? This is a nontrivial problem!
- How do we measure details about the QGP?

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Copyright © 1997,1998 Richard Furnstahl and James Steele.