[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:

- Start with n=0
- Loop through the A nucleons
- 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.
- 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).

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).

- quarks are spin-1/2 fermions => fermi statistics
- 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.)

- How can three 10 MeV quarks make a 1 GeV baryon?
- "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.

- 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)

- 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)

- 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)

- A
- Very simple picture; neglects or downplays important physics
- chiral symmetry breaking in particular
- but good for intuition and determining approximate scales

- 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!]

- What about QGP?
- The simple picture of inward bag pressure of magnitude B balanced by outward pressure of quarks implies new phases possible
- If pressure inside increased, then at some point it is greater than inward bag pressure and quarks are not contained => new phase of matter
- large pressure of quark matter when
- temperature of matter is high
- baryon number density is large

- Consider each in turn next time.

Return to 880.05 home page
syllabus

Copyright © 1997,1998 Richard Furnstahl and James Steele.