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).


1 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.

  • $\begingroup$ Thank you! I'm terrible trying to explain myself, but your answer totally solved my problem. It was indeed the spin addition where I had problems. $\endgroup$ Jan 14, 2021 at 15:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.