Predicted Meson Masses: Working Out the Spins

[genloz]

Homework Statement

I'm trying to determine the predicted meson masses but am having a little trouble with working out the spins...

Homework Equations

$$m=(q\overline{q})=m_{1}+m_{2}+A\frac{S_{1} \bullet S_{2}}{m_{1}m_{2}}$$

The Attempt at a Solution

I understand that using a quark up and down mass of 310MeV/c^2 and an A value of 0.06GeV^3 I can simply sub these values into the equation, but how do I determine the spin?

I found an equation:
$$S_{1}\bulletS_{2}=0.5[S(S+1)-S_{1}(S_{1}+1)-S_{2}(S_{2}+1)]$$
but I'm a bit confused about how the quark spins relate to the above formula...

Thanks!

 

[javierR]

A quark has spin s=1/2. If we assume the quarks have a definite z-component of spin (where z is an arbitrary direction we choose), then each can have either sz=+1/2 or
sz=-1/2. The two different quarks can be relatively aligned or anti-aligned, so the dot product s1.s2 gives + or - contribution.

 

[genloz]

Thanks for that! But where does the second equation come into it? What are S, S1 and S2? Is it possible to demonstrate with an example somehow please? Like an up quark (spin +1/2) and a down quark (spin -1/2) how these equations work?

 

[genloz]

Okay, I understand that for spin 0:
$$M=m_{1}+m_{2}-\frac{3A}{4m_{1}m_{2}}$$
and for spin 1:
$$M=m_{1}+m_{2}+\frac{A}{4m_{1}m_{2}}$$

I understand that the $$\pi$$ for example has spin 0 and the $$\rho$$ has spin 1... I know that they both have an up and an antiup quark (or a down and an antidown quark) but I still don't understand how the calculation of spin works...

 

[Ibrajim]

what does A stands for?