# Mass of mesons and baryons

In my introduction to nucleons and particles (basic) course, we were given the following formula to compute the mass of mesons and baryons : \begin{eqnarray} M(\text{meson}) & = & m_1 + m_2 + A \frac{\vec{S}_1 \vec{S}_2}{m_1 m_2} \\ m(\text{baryon}) & = & m_1 + m_2 + m_3 + A \left( \frac{\vec{S}_1 \vec{S}_2}{m_1 m_2} + \frac{\vec{S}_1 \vec{S}_3}{m_1 m_3} \frac{\vec{S}_3 \vec{S}_2}{m_3 m_2}\right) \end{eqnarray}

$\vec{S}_1 \vec{S}_2 = \frac{1}{2} ( \vec{S}^2 - \vec{S}^2_1 - \vec{S}^2_2)$

$\vec{S}_1 \vec{S}_2 = \frac{1}{4} \hbar ^2$ for vector mesons $(s=1)$

$\vec{S}_1 \vec{S}_2 = - \frac{3}{4} \hbar ^2$ for pseudo-scalar mesons $(s=0)$

$A=160 \left( \frac{2 m_u}{\hbar} \right) Mev/c^2$.

I guess the small m's stand for the masses of the composing quarks.

Now what we got for an explanation isn't very clear to me. I understood that quarks are only like, $0.1 \%$ of the mass, and that the rest comes from the kinetic energy of the quarks and their confinement inside the nucleon.

What I don't get is:

1) how do we compute this ? (i.e., how do we know the spin of a meson ? And why are the values of $\vec{S}_1 \vec{S}_2$ such ?)

2) Where do these formula come from ?

I have no background in QFT.

• first time I see this, sounds like numerology. Do you have a link ? a rererence? – anna v Jun 1 '18 at 15:24
• i know :'( i know it sounds like bad word from me, but my professor is the worst. We have a "syllabus" made of slides, which are made up from one or two formulas without any definitions of the variables, and he doesn't explain anything in the lectures. thus i have no reference, and no link. And it's the only slide on that subject. – Reflets de Lune Jun 1 '18 at 15:53
• This is likely a version of the "component quark model", in which case the $m$s are the "component quark masses" which are very different from (much larger than) the so called "bare masses". It's worth checking just so you'll know if the results you get are reasonable or not. – dmckee Jun 1 '18 at 17:18
• These formulas are in griffiths particle physics i believe – Triatticus Jun 1 '18 at 17:18
• you must mean constituent quark masses en.wikipedia.org/wiki/Constituent_quark ~300mev for up and down – anna v Jun 1 '18 at 18:20