Quarks and Nuclear Matter

Why does nuclear matter exist?

Only two numbers: (!)

EB~ 8 MeV/nucleon

ρ0 ~ .17 N fm-3


Conventional Assumptions:

  1. Ignore Coulomb interactions
  2. Nucleons form Fermi Gas
  3. Forces given by meson exchange potential
But potentials have ~ 10 separate operators, many parameters derived from a mixture of fits to low energy scattering and theoretical prejudices

Note also Fermi energy ~ ρ02/3
Conventional π exchange will give E ~ -1/ρ0 at large densities: need to introduce hard-core forces to stop nuclei collapsing.

"A suitable nuclear Hamiltonian has not yet been found"

Also: evidence that nuclear degrees of freedom are not enough: Hence maybe we should look at quarks as a model for nuclear matter Several attempts to describe nuclear matter in terms of constituent quarks.

Quark Model: lightning introduction:

QCD is non-perturbative: only rigorous method for handling multi-quark systems to date is lattice QCD. This predicts that the interaction between two massive quarks at large distance takes the form of a string or tube, with energy proportional to its length.

This gives the

Flux Tube Model

Potential between quark and anti-quark for mesons.



For baryons, flux tubes arrange to minimise the total tube length.
N

Warm-up exercise:

ignore 1-gluon (i.e. Coulomb and hyperfine interactions), variational calculation with
with 2 params α,β. Very simple:
Efree = T+V

This can be minimised:
. Minimisation wrt α has to be done numerically, giving α = 1.74
. Use Natural to extend it to many quark systems and hence

Nuclear Matter

A good phenomenological model should be able to at least reproduce qualitatively all the overall bulk properties of nuclear matter. In particular,

Many body calculation:

P.J.S. Watson, Nucl. Phys. A494 (1989) 543.
Have to make a variational calculation: need some new ingredients
  1. SU(3) is too hard: use SU(2) colour so that nucleons are qq.
  2. need to fix α = 1.74.
  3. need to fix colour on a given quark.
  4. Need an infinite number of quarks: to mimic continuous medium, use cyclic boundary conditions. Number of pairs N varied from 2 to 7: results vary little for N = 3 to 7.
  5. Many body Schrodinger equation
    needs Metropolis algorithm to solve it.
  6. Quarks are fermions: must antisymmetrise wave-functions
  7. Many-body potential has to be minimised: V(r) = σ r becomes
    . Can only be done by brute force (NP hard problem)
Metropolis Algorithm: N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E.Teller, J. Chem. Phys. 21 (1953) 1087.
  1. Choose quarks in random positions
  2. Calculate ψ
  3. Move one quark at random
  4. Calculate ψnew
  5. Find random number P, 0<P <1
  6. If P >|ψnew/ψ|2 reject the move, go to 2.
  7. Otherwise accept the move, find T and V

    and go back to 1.
Gives an unbiased estimate of the energy, after enough moves (typically 105). Variance is small if the wave function is good, however, it is very time consuming. see D.Ceperley, G.V. Chester, and M.H. Kalos, Phys. Rev. B16 (1977) 3081.
We can check the low-density limit using the analytic result above. We can also find a high energy limit (when the quarks form an interacting Fermi-gas):
A useful trick: After fixing α, there are two parameters : ρ and ρ, where ρ = N/ L3. Can use scaling to reduce this to one parameter:

θ = arctan (ρ1/3/β) ,
ζ = ( β2 + ρ2/3)1/2
In terms of these,the binding energy for an interacting system becomes Ebind(β,ρ) = Ebind(θ,ζ) = ζ2 T(θ) + V(θ)/ζ- Eo
Allows ζmin to be found immediately: reduces complexity enormously. Results: for β
and for energy:

So what did we forget?

This model appears to be promising, but various effects have been ignored, such as Boyce did SU(3) calculation (M.M. Boyce and P.J.S. Watson, Nuc. Phys. A580 (1994) 500.): about 1000 hours of computer time. (also did calculation with a quadratic potential V(r) = σ' r2 and a corresponding energy:
No qualitative change. Need a quick and dirty model to see if any of the other effects are important.Hence a toy:

The mesonic molecule

Two heavy quarks Q and two relatively light antiquarks q, which can move without disturbing Q's:

i.e. the Born-Oppenheimer approximation is valid. By varying the distance, R, between the heavy quarks a mesonic-molecular potential, U(R) can be computed. . Expect that if this system can bind, so can nuclear matter.

The Flux-Bubble Model


combines both the non-perturbative (flux-tubes) and perturbative (one-gluon exchange) aspects of QCD in a consistent fashion. However, to avoid Van der Waals forces these regimes must be separated. Flux-Bubble Models and Mesonic Molecules, M. Boyce, J. Treurniet, and P.J.S. Watson, Carleton preprint 1997) Note that

Results:

for the linear potential and its various variants, including the flux-bubble potential, we get a "pseudo-Morse" potential between the heavy quarks. None of the extensions to the string-flip potential will give rise to nuclear binding.

 Maybe our wave-function is wrong. Note the similarity between the mesonic-molecular system and H2 molecular system. Key reason for molecular binding is screening effect of electrons which tend to be localized in between the protons. Chemists use "bond-centred" wave functions. Siegfried Flugge, Practical Quantum Mechanics I,
Instead of we use

Old Wave function
New Wave function
Results:
Note huge increase in depth of the pseudo-Morse potential from ~5 MeV to ~ 80 MeV. Now sufficient to bind quark molecules with mQ ~ 660 MeV. All of the subtle effects seen with the old wave function are overwhelmed by the depth of the potential. More important, one cn estimate the mean potential in nuclear matter. The above curve can be fitted with . This gives ~ 49 MeV, can be compared with the value of ~50 MeV, which is usual value for a Fermi gas model of nucleus (we got lucky!) Should be able to use this wave function with just a string-flip potential, with α fixed,
Conclusion: maybe (just maybe) we have a working model for nuclear matter. Now all I need is a good graduate student!