# Why is the proton (uud) lighter than the $Δ^0$ (uud) baryon?

Neutron has quark composition udd with spin $$\frac 12$$. $$\Delta^0$$ baryon has quark composition udd with spin $$3 \over 2$$.

On Wikipedia it says that $$\Delta$$ baryons have mass of approximately $$1232 \frac{\mathrm{MeV}}{c^2}$$ while the neutron has mass of approximately $$~939 \frac{\mathrm{MeV}}{c^2}$$.

Why is the neutron lighter than the $$\Delta$$ baryon when they have the same quark composition? Is it because of the difference in spin?

First note that it's the $$\Delta^+$$ particle that is $$uud$$. The $$\Delta^0$$ is $$udd$$ like the neutron.

The $$\Delta^+$$ particle is an excited state of the proton, that is the proton is the ground state of two up and one down quarks and the $$\Delta^+$$ particle is the first excited state. Likewise the $$\Delta^0$$ particle is the first excited state of the neutron.

To create a $$\Delta^+$$ particle from a proton we have to add the energy to excite the proton to a $$\Delta^+$$, and when we add that energy we are adding an equivalent mass related to the energy by Einstein's famous equation $$E=mc^2$$. The mass difference between the proton and $$\Delta^+$$ particle is simply due to this.

This is a general property of bound states. For example when we excite a hydrogen atom from the $$1s$$ to the $$2p$$ state we are adding an energy of $$10.2$$ eV and the mass increases by $$10.2/c^2$$.

• I think what she's asking is why the the one that's the ground state is the ground state, and why the one that's the excited state is the excited state, instead of the other way around. Possible explanation: the Pauli exclusion principle says you can't generate spin 3/2 just by aligning the spin 1/2's of the three quarks. Therefore you need to have some orbital angular momentum in the mix, and this costs energy. – Ben Crowell Mar 9 at 14:47
• Could you explain why the excited state has spin $s=3/2$ and the ground state has spin $s=1/2$? @JohnRennie – SRS Oct 23 at 17:54

In QCD, much as in E&M, there is a chromo-magnetic dipole-dipole contact interaction $$\propto S\cdot {S}'$$, which will obviously differ between spin 1/2 and 3/2 baryons. Bag-modelers have used it to estimate the mass difference between nucleons and deltas with tolerable accuracy.

The quark content is not a good predictor of baryonic mass. Only a small part stems from quark mass, hence from the Higgs mechanism. 98% or so is contributed by the energy of the massless gluons.

• does spin play a part in the baryonic mass? – TaeNyFan Mar 9 at 12:17