Potential energy in strong- and weak- nuclear interaction

[olgerm]

Schrödinger equation for N-particles is:
$U_{System Potential Energy}(r_1,r_2,r_3,...,r_n,t)-\sum_{n=1}^{n_{max}}(\sum_{d=0}^{d_{max}}(\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dx_n^2})*\frac{ħ^2}{m_n})=i*ħ \frac{dΨ(r_1,r_2,r_3,...,r_n,t)}{dt}$
Where $\sum_{d=0}^{d_{max}}$ is summation over dimensions.
Where $\sum_{n=1}^{n_{max}}$ is summation over particles.

1.Which formula is used to calculate U__{System Potential Energy}(r_1,r_2,r_3,...,r_n,t) energy if we also consider weak nuclear and strong nuclear interaction?
2.Is there any potential between 2 particles with different color charges?
3.Is there any potential between 2 or more particles with same color charges?
4.Which is U__{System Potential Energy}(r_1,r_2,r_3,...,r_n,t) for particle system:
a) One green Up-quark and one blue Down-quark?
b) Two blue Up-quark?
c) One red Down-quark ,one green Up-quark and one blue Down-quark?
d) One blue Up-quark and one green Charm-quark?

 

[fzero]

For the strong force, there isn't really good potential function that is analogous to the Coulomb potential for electrodynamics. Because of confinement, asymptotic states have no net color charge. Basically, if you had some sort of subatomic tweezers that would allow you to pull at a quark inside a proton, instead of pulling out an isolated quark, you would have to do enough work to produce a quark-antiquark pair. You'd end up with another proton and something like a pion.

For heavy enough quarks, one can try to study the bound state problem using an approximate potential, called the Cornell potential

$$ U(r) = - \frac{a}{r} + b r. $$

The first term is like the Coulomb potential, while the 2nd term is a linear potential that mimics the confinement of the quarks. Because of this linear term, we won't find solutions where the average separation between the quarks is arbitrarily large.

At much lower energies, the strong interaction can be effectively described as a theory where mesons are exchanged between hadrons. Then the potential is of Yukawa type

$$U(r) = g^2 \frac{ e^ {-m_\pi r}}{r},$$

which is valid for distances $r\gg 1/m_\pi$.

The weak interaction at low-energies is also of Yukawa type where now $m_W$, the mass of the W-boson appears in the exponent, since it is a theory of exchange of virtual W and Z particles.

 

[olgerm]

For heavy enough quarks, one can try to study the bound state problem using an approximate potential, called the Cornell potential

$$ U(r) = - \frac{a}{r} + b r. $$

Is r distance between two quarks,which have different color charges an U(r) potential between those quarks?
what are values of a and b?

 

[fzero]

Is r distance between two quarks,which have different color charges an U(r) potential between those quarks?
what are values of a and b?

Yes, $r$ is the interquark distance and the color charges will be different because the hadrons are color singlets. The Coulomb term corresponds to a single gluon exchange and the value of $a$ I see quoted in the literature is $a = 4 \alpha_s/3$, where $\alpha_s$ is the running QCD coupling constant. The value of $b$ is called the QCD string tension, which can be computed approximately in a couple of ways. I've found a quoted value of $b\sim 0.87~\text{GeV/fm}$.

I should stress that this is a very approximate description of quark-quark interactions. There are various refinements in the literature, which you can search for under the name NRQCD, for Non-Relativistic QCD or the term quarkonia, which is the name used to refer primarily to bound states of charm-anticharm and bottom-antibottom.