The neutron is largely spherical and incompressible in atomic nuclei. These two properties are however challenged in the extreme pressure environment of a neutron star. Our variational computation within the Cornell model of Coulomb gauge QCD shows that the neutron (and also the Delta-3/2 baryon) can adopt cubic symmetry at an energy cost of about 150 MeV. Balancing this with the free energy gained by tighter neutron packing, we expose the possible softening of the equation of state of neutron matter.
The atomic nucleus is a close-packed system analogous to a cold quantum liquid. Its hadronic components are very incompressible, as revealed by the nuclear radius scaling with the mass number as r A ≃ r 0 A -1/3 , with unit of length r 0 ≃ 1.2 f m. Nucleons there are quite optimally packed, as the (charge) nucleon radius has been precisely determined [1] in electron scattering, and is approximately 0.88 f m. If we (grossly) think of the nucleon as a solid sphere, 74% of the nuclear volume is occupied, near the Keplerian limit [2] reached by hcp or fcc lattices.
In neutron stars, gravity-compressed nuclear matter finds extreme pressure environments that changes some of its properties. Recently [3] a two-solar mass neutron star has been discovered that takes nuclear physics to the limit (and we have recently proposed [4] that this is so much so as to constrain gravity for strong field intensity). At high pressure and density in the core of the star many exotic phases of hadron matter have been proposed, including a neutron superconductor/superfluid, various meson condensates, quarkyonic or Color-Flavor-Locked condensates, etc. Some of them have already been ruled out by the new, superheavy neutron star [5]. At the crust one expects a thin atomic sheet [6] followed by neutron-rich nuclear matter, and finally almost pure neutron matter (with a small number of protons in βequilibrium) [7].
One intermediate phase that has long been proposed [8] is a neutron crystal. In analogy with condensed matter systems (saliently 3 He, a quantum liquid through T = 0 that solidifies upon compression) neutron matter is assumed to locally adopt a periodic lattice-like arrangement to maximize density. Our new observation is that when the environment looses its isotropy, the nucleon (largely spherical at rest) is subject to directional stresses and may deform to minimize the evacuated volume. Thus, for mass-energy densities ε larger than 140 M eV /f m 3 ≃ 3m 4 π neutrons can no more be considered pointlike objects subject to the rules of local, rotationally-invariant quantum field theory as elementary fields. Their composition and structure should start being taken into account, perhaps revisiting some of the hypothesis underlying the nuclear many body problem approach based on QFT methods (particularly, rotational invariance).
We report an estimate of the deformability of the neutron with the renowned Cornell Hamiltonian,
a field theory model of Quantum Chromodynamics, tractable by many-body methods common in nuclear and condensed matter theory. The relativistic quark fields Ψ interact via the color charge density Ψ † ( x)T a Ψ( x).
The model has been amply exploited to treat the heavy quarkonium spectrum [9,10], as well as both light mesons and baryons [11,12] with the Cornell potential V (r) = σr -4αs 3r . The model ground state ( 3 P 0 quark-antiquark condensed vacuum) is treated in BCS approximation to generate constituent quarks from the current m = 5 M eV quarks in the Hamiltonian, then the second-quantized wavefunction appropriate for 3-quark baryons in terms of Bogoliubov-rotated quark creation operators B † is
(2) To approach lowest-lying baryons by the Rayleigh-Ritz variational principle, we employ the separable wavefunction (ρ, λ being 3-body Jacobi coordinates with Cartesian components ρ x , λ z , etc.)
To choose a convenient variational wavefunction we observe that the graph of (k
, which is depicted in figure 1 for even N , interpolates between spherical symmetry for N = 2 and octahedral (the cube’s) symmetry, reached as N → ∞. Therefore one can take an appropriately normalized ψ N from a precise numerical solution of the two-body problem and use
to study the three-quark system transiting between both symmetry groups.
The Hamiltonian expectation value N |H|N is a function of the two variational parameters α ρ , α λ , and is minimized respect to them to find the best approximation to the neutron mass within this family of functions for N = 2. To compute the matrix elements we perform all relativistic spinor sums numerically, and the three-body nine-dimensional integral by Monte Carlo methods [13]. The model approximation to the nucleon mass is 980 M eV (with a Monte Carlo error of 40 M eV ), the physical mass being 940 M eV . Should higher precision be necessary one should resort to lattice gauge theory [14]. This is not so for our purposes and we proceed to change the wavefunction symmetry, repeating the minimization for varying N .
We find that the neutron mass increases by about 150 M eV between N = 2 and N = 18 (the last already very close to a cubic neutron), as depicted in figure 2.
As a check of the spin-independence of our result, we have repeated the calculation for the ∆(1232) baryon, where all three quark spins are parallel, and obtained an equal excitation energy. The similar outcome is reported in figure 3.
We can offer the following heuristic argument supporting these numerical findings. I
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