My question concerns the delta++ baryon. I understand that the rest mass of quarks makes up a very small percentage of the total mass of a baryon, but I don't understand exactly what it is about delta ++ (uuu) which makes it have greater rest mass than a proton (uud). If anyone can explain this I would be very grateful! Thanks
The rest mass of the aggregate particle is due to: (1) the rest mass of the constituent parts (maybe 2% for a proton) (2) the kinetic energy of the constituent parts ... this adds to the rest mass because it is confined to the particle, and does not appear as external motion. So why is the delta++ baryon more massive? I cannot answer that directly, but we do know that the confined kinetic energy is greater than for a proton; we also know that the delta++ is unstable. To go further would require a chromodynamics analysis of the particle ...
They have spin 3/2 and an antisymmetric color wavefunction. Due to some QCD stuff I don't understand, this influences their mass significantly.
This is certainly an interesting subject. Note that p (uud) and Δ^{+} (uud) have different masses, and similarly n (udd) and Δ^{0} (udd) have different masses. Interestingly, there is a Ʃ^{++} particle (uuc). http://en.wikipedia.org/wiki/Charmed_sigma_baryon http://pdg.lbl.gov/2008/listings/b104.pdf http://pdg.lbl.gov/2012/tables/rpp2012-tab-baryons-Charm.pdf Regarding baryon mass, there is this: Spontaneous violation of chiral symmetry in QCD vacuum is the origin of baryon masses and determines baryon magnetic moments and their other static properties B. L. Ioffe http://link.springer.com/article/10.1134/S1063778809070151# Abstract: "A short review is presented of the spontaneous violation of chiral symmetry in QCD vacuum. It is demonstrated that this phenomenon is the origin of baryon masses in QCD. . . ." and The Origin of Mass in QCD - http://www.slac.stanford.edu/econf/C040802/papers/L010.PDF
There is a spin-spin interaction between quarks that is repulsive in the spin 1 state of two quarks, and attractive in the spin zero state.
The ^{++} has a different symmetry than the proton. Keep in mind that quarks are fermions and must obey Pauli's exclusion principle. What that means is that a Baryons wavefunction must be anti-symmetric under the exchange of two quarks. There is more than one way to do that. The baryon's wavefunction may be though as a product of three factors. The first is the spacial factor which is a function of the coordinates - Those are the usual solutions you get from the Schroendiger equations where the quarks spins (each has a spin 1/2) may combine either in alignment to form a spin 3/2 baryon or it may combine in a partially misaligned way forming a spin 1/2 baryon. The second factor of the wavefunction is the factor where the isospins combine in a way very similar to the way plain spin combines. The up and down quarks are the up isospin and down isospin members of a isospin half multiplet (hence their names). They may combine in alignment forming a isospin 3/2 multiplet which has four members - Those are the Δs. Or they may combine in a partially misaligned form leading to a isospin 1/2 multiplet which has two members - Those are the nucleons (proton and neutron). The third factor of the wavefunction is the color factor where the color quantum numbers of the quarks combine. *Color confinement requires that the color combine forming a color singlet (white). that combination is antisymmetric (as required by Pauli's principle) *That means that the product of the other two factors must be symmetric so that the overall wavefunction is antisymmetric. *There are two ways to accomplish that. Either you have a spin 1/2 isospin 1/2 particle (nucleons) or you have a spin 3/2 isospin 3/2 particle (The Δs). *The last thing to keep in mind is the fact that the spin 3/2 wavefunction has a higher energy because all three quarks cannot be in their lowest energy state at the same time (again, because of Pauli's exclusion principle)
It's their QCD spin-spin interaction: "chromomagnetism". The quarks' spins are parallel in delta baryons and mixed in nucleons. One finds similar splittings in other baryons, and one concludes from them that the strange quarks' chromomagnetic moment is somewhat less than those of the up and down quarks. This is a result of strange quarks' greater rest mass, about 100 MeV instead of about 2 to 5 MeV.