Mass difference between two baryons

Question

I have been trying to make sense out of this (unsuccessfully for days). It's an exercise on Particle Physics. Exercise asks to calculate the mass difference between baryons ($cuu$) with \begin{equation*} J^p = (3/2)^+ and (1/2)^+ \end{equation*} respectively, by using the operator: \begin{equation*} H_{ss} = K \sum_{i<j} \frac{\vec{S_i}\vec{S_j}}{m_i m_j} \end{equation*}

Where $K$ is a constant, $S$ is the spin operator, and $m$ are the masses.

I know beforehand the solution for the operator must be: \begin{equation*} H_{ss} =\frac{K}{2 m_u^2}(s(s+1) - \frac{3}{2}) + \frac{K}{2 m_u m_c} (J(J+1) - s(s+1) - \frac{3}{4}) \end{equation*}

But I can't derive this exact result.

My process has been:

-Express the product of spin operators as:

\begin{equation*} \vec{S_i}\vec{S_j} = \frac{1}{2} [(S_{ij})^2 - S_i^2 - S_j^2] \end{equation*}

Use that: \begin{equation*} S_{ii}^2 = s(s+1) (eigenvalue) \end{equation*} \begin{equation*} S_{i}^2 = \frac{3}{4} (eigenvalue) \end{equation*} \begin{equation*} J^2 = S_{uuc}^2 = S_{uc}^2 + S_c^2 + 2 S_{uc}S_u \end{equation*}

\begin{equation*} J^2 = J(J+1) (eigenvalue) \end{equation*}

-And getting:

\begin{equation*} H_{ss} =\frac{K}{2 m_u^2}(s(s+1) - \frac{3}{2}) + \frac{K}{2 m_u m_c} (J(J+1) - s(s+1) - 2S_{uc}S_u - \frac{3}{2}) \end{equation*}

So the problem seems to be the term: \begin{equation*} - 2S_{uc}S_u \end{equation*}

How do I compute it? Am I doing something wrong? (other than just substituting eigenvalues as if they were equal to their operators, I hope it's understandable).

Answer 1

It's hard to understand what you are doing, really.

Assuming you map cuu to i =1,2,3, and inserting eigenvalues where appropriate, you have $$ j(j+1)=\vec J ^2= (\vec S_1+\vec S_2+\vec S_3)^2= 9/4+ 2(\vec S_1\cdot \vec S_2+ \vec S_1\cdot \vec S_3+\vec S_2\cdot \vec S_3). $$ You also appear to be defining s as the spin of the uu diquark, $$ 2\vec S_2\cdot \vec S_3= s(s+1) -3/2, $$ so that $$ 2(\vec S_1\cdot \vec S_2+ \vec S_1\cdot \vec S_3)= j(j+1) -9/4 +3/2-s(s+1). $$

Substituting these two into your degeneracy-lifting hamiltonian, $$ H_{ss} =\frac{K}{2 m_u^2}\left ( s(s+1) - \frac{3}{2}\right ) + \frac{K}{2 m_u m_c} \left ( j(j+1) - s(s+1) - \frac{3}{4}\right )~. $$

This is all elementary addition of spins; little to do with the baryons in your title, at heart.