Finding the velocity of a bounded quark of a meson

How would you compute the velocity of one of the quarks of a meson, when we have no reference to a reaction or a collision?

I initially thought we could use the conservation of energy and momentum to deal with it, however, how do we do it without a collision or a decay for example.

Take for example the pions ($$\pi^+$$, $$\pi^-$$, $$\pi^0$$). All three of them have a mass around $$140\text{ MeV/c}^2$$. They consist of one quark ($$u$$ or $$d$$) and one antiquark ($$\bar{u}$$ or $$\bar{d}$$). The rest masses of these quarks are much smaller (only $$2$$ or $$5 \text{ MeV/c}^2$$ respectively). So you see, their masses contribute only very little to the total mass of the pion. Hence the pion mass is mostly from the kinetic energy of the constituent quarks (and may be from the gluons also). That means the quarks are moving within the pion very close to the speed of light.

Another way to find the speed of a quark is to start with the radius of the pion. The article "Charge Radius of Pion" has published a radius of $$0.51 \cdot 10^{-15}$$ m. From Heisenberg's uncertainty relation you can then estimate the momentum $$p$$ of a quark inside the pion as $$p \geq \frac{\hbar}{r} =\frac{6.6\cdot 10^{-16}\text{ eV s}}{0.51\cdot 10^{-15}\text{ m}} =1.3\text{ eV s/m} =390\text{ MeV}/c.$$ Then you can use the definition of relativistic momentum $$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}},$$ resolve this to $$v$$ as a function of momentum $$p$$ and quark mass $$m$$, $$v=c\frac{p}{\sqrt{p^2+m^2c^2}}.$$ Because of $$p\gg mc$$ you will again get a speed $$v$$ very close to $$c$$.

• ah I see. That makes sense. So how exactly would I go about trying to calculate the minimum velocity? Could we use the uncertainty principle? Oct 25, 2021 at 12:03
• @EPICTubeHD I don't know if it is even possible. May be this has been done by numerical methods from Lattice QCD. Oct 25, 2021 at 12:14
• Well I am currently going through a exercise, that's similar to my question above that asks this. The problem is as follows: The J/ψ is a meson consisting of a bound charm-quark anti-charm-quark pair, discovered in 1974 at both SLAC and Brookhaven National Laboratory in the USA. The radius of the J/ψ is ≈ 0.2 fm. The spin of the J/ψ is 1, hence the J/ψ is a boson, albeit not a fundamental boson. Estimate the minimum velocity of the charm-quark in the J/ψ meson. Oct 25, 2021 at 12:21
• @EPICTubeHD Well, that is a different problem. You can answer it by using the uncertainty relation. Oct 25, 2021 at 12:59

I am turning my comments into an answer in order to include a figure:

Quote from Motl's answer to a similar question, about electrons in an atom:

The state of an electron (or electrons) in the atoms isn't an eigenstate of the velocity (or speed) operator, so the speed isn't sharply determined. However, it's very interesting to make an order-of-magnitude estimate of the speed of electrons in the Hydrogen atom (and it's similar for other atoms).

That is why the electrons are in orbitals, not orbits. See this link for details of the measured orbitals of the Hydrogen atom,

There is no consecutive $$Δs$$ to a $$Δt$$ to be able to define a velocity.

On average is a different story, as discussed in the link by Motl

However, it's very interesting to make an order-of-magnitude estimate of the speed of electrons in the Hydrogen atom (and it's similar for other atoms).

In the strong interaction that binds the quarks, a Bohr type model is impossible, all successful calculations have been done with QCD on the lattice, so there is not even a classical approximation to the orbitals of the quarks. The same general principles can be used to get average numbers , as in the link. To ask about a minimum velocity of a quark makes no sense, in my opinion.