Three quarks for you!

Let the mass of up quark be $m_u$ and that of down quark be $m_d$ and their charges be $q_u$ and $q_d$ . Now we need to calculate $m_d$ - $m_u$ . For that, taking into consideration a necessary equivalence (i.e the conservation of energy with mass effect taken in account), the total energy of a neutron( $E_n$ ) $-$ total energy of a proton( $E_p$ ) = $($ Mass difference between Neutron and a proton $)\times c^{2}$ .

We are given that mass difference as $2.306\times10^{-30}$ . For now let us denote it by $d$ . This implies $E_n - E_p = dc^{2} (1)$ . Now for calculating $E_n$ , $E_n$ = Energy due to quark masses $+$ electrostatic potential energy. The energy due to the quark mass $=$ Total mass $\times c^{2}$ $= (2m_d + m_u)\times c^{2}$ . And the electrostatic potential energy of the three quark system $= \frac{Kq_u q_d}{r} + \frac{Kq_d q_u}{r} + \frac{Kq_d q_d}{r}$ where $r$ denotes the distance between quarks. This implies, $E_n = (2m_d + m_u)c^{2} + \frac{2Kq_u q_d}{r} + \frac{Kq_d q_d}{r}$ . Similarly for a proton having two up quarks and one down quark $E_p = (m_d + 2m_u)c^{2} + \frac{2Kq_u q_d}{r} + \frac{Kq_u q_u}{r}$ .

Substituting these values in Equation $1$ we get, $E_n - E_p = (m_d - m_u)c^{2} + \frac{Kq_d q_d}{r} - \frac{Kq_u q_u}{r} = dc^{2}$ . Solving for $m_d - m_u$ we get $m_d - m_u = d + \frac{K(q_uq_u - q_dq_d)}{r\times c^{2}}$ .

Substituting $q_u = -2e/3 = 1.068\times 10^{-19} , q_d = e/3 = -5.34\times 10^{-20}$ $K = 9\times 10^{9}, r = 10^{-15} , c^{2} = 9\times 10^{16}, d =2.306\times 10^{-30}$ we get $m_d-m_u = 3.161468\times 10^{-30}$ .

The difference of masses is due to two different factors. The first is the electrostatic energy of proton and neutron, given by the charges of the quarks. The second is the energy due to the masses.

The total charge of the proton is $-e$ , and the total charge of the neutron is zero. Both values may be constructed by using three quarks for each particle: for the proton: up, up, down ; for the neutron: up, down, down (if you want to see more information about quarks, you may click here ).

As the quarks have the same distance between then, we may consider that they are located at the vertices of an equilateral triangle of edge $r$ , and then calculate the electrostatic energy of their interactions ( up-down , up-up or down-down ) to find the energy values we want.

For the proton $\left(U_p\right)$ :

$U_p = 2U_{u-d} + U_{u-u} = \frac {2K}{r} \left(\frac{-2e}{3}\right) \left(\frac{e}{3}\right) + \frac {K}{r} \left(\frac{-2e}{3}\right) \left(\frac{-2e}{3}\right) = 0$

For the neutron $\left( U_n \right)$ :

$U_n = 2U_{u-d} + U_{d-d} = \frac {2K}{r} \left(\frac{-2e}{3}\right) \left(\frac{e}{3}\right) + \frac {K}{r} \left(\frac{e}{3}\right) \left(\frac{e}{3}\right) = -\frac{Ke^2}{3r}$

Let's assume that the mass of the quark up is responsible for the energy $E_u = m_u c^2$ , and the quark down , for the energy $E_d = m_p c^2$ .

Given the composition of proton and neutron, their total energies are:

$E_p = 2E_u + E_d$

$E_n = 2E_d + E_u -\frac{Ke^2}{3r}$

The difference between $E_n$ and $E_p$ , following the enunciate, has to give our difference of masses, $\Delta m = 2.306 \times 10^{-30} kg$ , times $c^2$ :

$E_n - E_p = E_d - E_u - \frac{Ke^2}{3r}$

For the masses:

$\Delta mc^2 = (m_d - m_u)c^2 - \frac{Ke^2}{3r}$

Solving for $\left(m_d - m_u\right)$ , we may obtain that:

$m_d - m_u = \Delta m + \frac{Ke^2}{3rc^2}$

Using the numerical values given in the enunciate:

$m_d - m_u = 3.16 \times 10^{-30} kg$ .

As stated by the problem, a neutron consists of 2 down quarks and an up quark. The energy of a neutron is therefore equal to the total energy of the quarks. The quarks energy itself is contribution from 2 component, mass of the quarks and electrostatic potential energy of the quarks.

The former can be calculated using the mass-energy equivalence $E=mc^2$ and the latter can be calculated using $E=\frac {kq_1q_2}{r}$ . The quarks are located at corners of an equilateral triangle of sides $r$ .

From the explanation above, we can start the quantitative calculation.
$E_n=E_m+E_e$ where $E_n, E_m, and E_e$ stands for energy of neutron, mass energy, and electrostatic energy subsequently.

$E_m=2m_dc^2+m_uc^2$ and $E_e= 2\frac {kq_uq_d}{r}+\frac {kq_d^2}{r}$ . Hence, $E_n=2m_dc^2+m_uc^2+ 2\frac {kq_uq_d}{r}+\frac {kq_d^2}{r}$ .

Doing the same for proton, we have
$E_p=m_dc^2+2m_uc^2+ 2\frac {kq_uq_d}{r}+\frac {kq_u^2}{r}$

Noticing that $E_p=m_pc^2$ and $E_n=m_nc^2$ and substituting to the aforementioned equations for each proton and neutron, and finally subtracting them, we have:
$(m_n-m_p)c^2=(m_d-m_u)c^2+\frac{k}{r}(q_d^2-q_u^2)$
Using the given quantities, we finally have $m_d-m_u=3.16 \times 10^{-30} kg$

The net electrostatic energy (relative to charges at infinity) in a proton is 0, since $q_u^2 + 2q_u q_d=0$ . Likewise, the neutron's energy is $(k/r) (q_d^2 + 2q_u q_d) = -ke^2/3r = - 7.68 \times 10^{-14}$ J, so the neutron has less electrostatic potential energy than the proton. This energy converts into $E/c^2 = 8.53 \times 10^{-31}$ kg.

Since the neutron already weighs more than the proton but contains less electrostatic energy, the $2.31 \times 10^{-30}$ kg difference would be even larger without the influence of charge. The "chargeless" mass of the neutron is thus $3.16 \times 10^{-30}$ kg higher than that of the proton. Counting components (ddu - uud), this is also equal to the difference in quark masses.

First, let's define $q_u=Ue$ and $q_d=De$ for the charge of the up and down quark, respectively.

Since we can treat the quarks as point charges, the electrostatic energy is just the usual electrostatic potential energy, for example between two point charges $E=Kq_1q_2/r$ . For the three charges in the proton the total electrostatic energy is

$E_p=Ke^2(U^2+2UD)/r=0$

while for the neutron it's $E_n=Ke^2(D^2+2UD)/r=-Ke^2/3r=-7.7 \times 10^{-14}~J$ .

The difference between the neutron and proton mass will give us the difference between the down quark and up quark mass with a little linear algebra and $E=mc^2$ to convert energy into mass. We have

$m_n-m_p=((2m_d+m_u)+E_n/c^2)-((2m_u+m_d)+E_p/c^2)$ .

This implies that $m_d-m_u=(m_n-m_p)-(E_n-E_p)/c^2=3.16 \times 10^{-30}~kg$ .

Now, the astute problem solver might ask why in the world we'd ask for the difference in quark masses. Wouldn't a more interesting problem be to actually find the quark masses themselves from the neutron and proton mass? We could have done that, but while the difference between the masses would remain correct, the actual masses calculated for the quarks would have been way off. This is because the picture of a proton or neutron being made up of just three quarks is, well, a little incorrect. There is actually a whole lot more to a proton and neutron. But, by asking about the difference in the masses we can construct a problem that gets pretty close to the right value in the real world as the "whole lot more" roughly cancels out.