# Course Outline for Physics 880.05

## III. A. Thermodynamics and Phase Transitions (Chs. 9,13) Part 1

### Comment on Generating Binomial Distribution

[Thanks to Randy Wells and Rick Mohr.] To generate a binomial distribution of n collisions out of A total with the probability of each being P:

2. Loop through the A nucleons
3. For each one, generate a random number between 0 and 1. If the number is less than P, add 1 to n (it's a hit!). Otherwise continue.
4. After the loop, n is number you want!

You can check that this works by generating a large number of n's and calculating the mean and variance. You should find A*P and A*P*(1-P).

### Recap of last time: Initial Energy Density

We'll quickly summarize the inputs to Bjorken's estimate of the initial energy density in a high-energy heavy-ion collision. We'll first plug in numbers using the empirical result for dN/dy and then review how the result is consistent with our Glauber picture (see notes from last period).

### Quantum Chromodynamics: Things you should know

1. quarks are spin-1/2 fermions => fermi statistics
2. quarks come in six flavors
• up (u), down (d), charm (c), strange (s), bottom (b), top (t)
• they can be grouped into three "generations":
(u,d) + (\nu_e,e)
(c,s) + (\nu_mu,mu)
(t,b) + (\nu_tau,tau)
with masses covering a large range, with u,d less than 10 MeV and top over 100 GeV.
• Each flavor is uniquely specified by its combination of six quantum numbers: Q, I_z, C, S, T, B. (The last four are zero unless the quark is the matching flavor.)
3. How can three 10 MeV quarks make a 1 GeV baryon?
4. "Current" quark mass vs. effective "constituent" quark mass
• A particle in a medium can have an effective mass due to its interactions with the medium. E.g., it can drag around a cloud of medium particles, becoming a quasiparticle with a different inertia (mass).
• Here the medium is the complicated QCD vacuum (ground state). A quark picks up an effective mass, referred to as the constituent quark mass. (From spontaneously broken chiral symmetry.)
• Constituent up and down quarks have mass roughly M_nucleon/3.
5. QCD is a gauge field theory
• gluons arise from the principle of local gauge invariance, which allows different gauge choices at each point; more later when we talk about lattice QCD
• gluons are spin-1 bosons
• in principle 9 types of gluons, but one is color singlet => doesn't couple to color. Implies long-ranged force between color singlets?
• SU(3)_c. Requiring color symmetry greatly restricts possible interactions (terms in the lagrangian)
• Non-Abelian => asymptotic freedom (recall that screening of charges is overcome by dispersion of color by gluons)
6. Deep-inelastic scattering
• probe quarks and gluons on short distance scales
• quarks at large momentum transfer behave as if almost free (with current quark mass)
• reflected in phenomenon of Bjorken scaling
• "running coupling" implies that perturbation theory is useful at large energies, momenta (simple Feynman diagrams are much more important than complicated ones)

### Bag Model

1. Idea of color confinement motivates bag picture of hadrons
• A caricature of the true physics
• quarks treated as massless particles inside bag and infinitely massive outside (so they never go there).
• quarks apply outward pressure since confined (think of the uncertainty principle \Delta p \sim 1/\Delta x => force on walls of bag)
• Bag pressure B balances
• phenomenological summary of nonperturbative physics
• two types of vacuum (ground state) inside and out, with different energy densities
• quark color sources hollow out a bubble of the higher-energy vacuum => bag!
• higher-energy vacuum inside implies vacuum pressure tends to make bag collapse
• Net color charge zero inside (so no flux escapes)
2. Very simple picture; neglects or downplays important physics
• chiral symmetry breaking in particular
• but good for intuition and determining approximate scales
3. Details:
• inside, quark wave functions satisfy Dirac equation for massless particles. Wave functions are four-component spinors.
• look for solutions with definite energy, yields simple equation for spatial dependence.
• boundary condition on surface of bag follows from condition that there is no probability flux out
• wave function doesn't vanish on surface, but \bar\psi \psi does
• implies an eigenvalue condition involving the radius of the bag: discrete states
• Lowest energy with 3 quarks in s1/2 state
• p^0*R = 2.04 (consistent with uncertainty relation)
• so total kinetic energy of N quarks is \propto 1/R
• So energy lowered by R increasing (outward pressure)
• Inward pressure (or energy density) is B, where B^(1/4) is typically taken to be from 145 MeV to 235 MeV.
• The total energy is:
E(R) = 2.04N/R + 4\pi/3 R^3*B
• Determine R by minimizing E => equilibrium
• dE/dR=0 => B^{1/4} = (2.04 N/4\pi)^1/4 1/R
• With R_0 = 0.8 fm => B^{1/4} = 206 MeV
• What is E_0? [problem set!]