Strong force

The strong force (or strong interaction) is one of the fundamental forces in nature. It is responsible for holding protons and neutrons together in atomic nuclei and quarks inside protons and neutrons, and is the strongest fundamental force. It is carried by gluons, massless particles travelling at the speed of light.

Lagrangian
To find the Lagrangian in the strong field, in terms of pilitron theory, we start with 4 assumptions:


 * Everything that has color charge is affected by the strong force.
 * Gluons, which transmit this force, have color charge.
 * Therefore, as you move away from a color-charged particle, you experience denser and denser strong field (gluon density), until eventually it gets so dense that not even gluons can escape.
 * If 2 color charges get too close to each other, they start repelling.

If we have two constants, the strong attraction constant, $$S$$ and the strong repulsion constant, $$g_R$$, we can say that the repulsive kinetic energy is:


 * $$E = -Sd^2 + g_R$$

Additionally, the force must be proportional to the product of the absolute value (magnitude) of the 2 color charges:


 * $$E = -Sd^2|\chi_1||\chi_2| + |\chi_1||\chi_2|g_R$$

Finally, to get the Lagrangian, we multiply by the cosine of the angle between them:


 * $$\mathcal{L} = -Sd^2|\chi_1||\chi_2|\cos \theta + |\chi_1||\chi_2|g_R\cos \theta$$

The constants
To find a way to calculate the related constants, let's first assume that $$\cos \theta = 1$$. This leaves us with the Lagrangian:


 * $$\mathcal{L} = -Sd^2|\chi_1||\chi_2| + |\chi_1||\chi_2|g_R$$

Let's now divide both sides by the product of color charges:


 * $$-Sd^2 + g_R = \frac{\mathcal{L}}{|\chi_1||\chi_2|}$$

Now divide both sides by $$d^2$$:


 * $$-S + \frac{g_R}{d^2} = \frac{\mathcal{L}}{|\chi_1||\chi_2|d^2}$$

Rearrange this:


 * $$\frac{g_R}{d^2} - S = \frac{\mathcal{L}}{|\chi_1||\chi_2|d^2}$$

Now, let's say we have some distance, $$D_C$$, for which the attractive and repulsive part is balanced. We know that this is possible because protons and neutrons are stable. In this case, $$\mathcal{L} = 0$$, which leaves us with:


 * $$\frac{g_R}{D_C^2} - S = 0$$

Which shows that:


 * $$\frac{g_R}{D_C^2} = S$$

This then leads to:


 * $$D_C^2 = \frac{g_R}{S}$$

Which means that this critical distance has to be:


 * $$D_C = \sqrt{\frac{g_R}{S}}$$

This is not enough to evaluate the constants. Now, if we pick a distance, $$D_1$$, such that when 2 quarks separated by that distance give each other a Lagrangian of $$1 \mathrm{J}$$, and the product of the color charges is $$1$$. This gives us:


 * $$g_R - SD_1^2 = 1 \mathrm{J}$$

Again, dividing both sides by $$D_1^2$$:


 * $$\frac{g_R}{D_1^2} - S = \frac{1 \mathrm{J}}{D_1^2}$$