Use (2.92) to study the relative masses of the Λ, Σ, Σ*, , , and ^* baryons of Figs. 2.8 and 2.1

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Use (2.92) to study the relative masses of the Λ, Σ, Σ*, , ,...

Use (2.92) to study the relative masses of the $\Lambda, \Sigma, \Sigma *$, $\Lambda_c, \Sigma_c$, and $\Sigma_c^*$ baryons of Figs. 2.8 and 2.14. Each baryon is a qqQ composite, where $\mathrm{q}=\mathrm{u}$ or $\mathrm{d}$ and $\mathrm{Q}=\mathrm{s}$ or $\mathrm{c}$. For the $\Lambda$ and $\Sigma$ baryons, show that the qq have isospin $I=0$ and $I=1$, respectively, and hence spin 0 and spin 1 , respectively. Use this result to evaluate $\mathbf{S}_1 \cdot \mathbf{S}_2$, where $\mathbf{S}_i=\frac{1}{2} \boldsymbol{\sigma}_i$. By considering $\left(\mathbf{S}_1+\mathbf{S}_2+\mathbf{S}_3\right)^2$, show that $$ \left(\mathbf{S}_1+\mathbf{S}_2\right) \cdot \mathbf{S}_3=0,-1 \text {, and }+\frac{1}{2} \quad \text { for } \Lambda, \Sigma \text {, and } \Sigma^* \text {, respectively. } $$ Thus, confirm that (2.92) gives $$ \begin{aligned} & m\left(\Lambda_{\mathrm{Q}}\right)=m_0-\frac{3 a^{\prime}}{2 m_u^2} \\ & m\left(\Sigma_{\mathrm{Q}}\right)=m_0+\frac{2 a^{\prime}}{m_u^2}\left(\frac{1}{4}-\frac{m_u}{m_Q}\right) \\ & m\left(\Sigma_{\mathrm{Q}}^*\right)=m_0+\frac{a^{\prime}}{m_u^2}\left(\frac{1}{2}+\frac{m_u}{m_Q}\right) \end{aligned} $$ where $m_0=2 m_u+m_Q$. Verify that $[\sec (2.84)]$ $$ \left[m\left(\Sigma_c\right)-m\left(\Lambda_c\right)\right]=\frac{m_s}{m_c} \frac{\left(m_{c_c}-m_u\right)}{\left(m_s-m_u\right)}[m(\Sigma)-m(\Lambda)] \simeq 0.16 \mathrm{GeV}, $$ The masses of the other $\frac{1}{2}^{+}$and $\frac{3}{2}^{+}$baryons can also be calculated from (2.92) in terms of $a^{\prime}$ and the quark masses. Considering the crude nature of the model, the quantitative agreement between the predictions and the observed masses is impressive. Indeed, all the observed features are reproduced. It is straightforward to enlarge the calculation to include hadrons containing $b$ quarks.

Use (2.92) to study the relative masses of the $\Lambda, \Sigma, \Sigma *$, $\Lambda_c, \Sigma_c$, and $\Sigma_c^*$ baryons of Figs. 2.8 and 2.14. Each baryon is a qqQ composite, where $\mathrm{q}=\mathrm{u}$ or $\mathrm{d}$ and $\mathrm{Q}=\mathrm{s}$ or $\mathrm{c}$. For the $\Lambda$ and $\Sigma$ baryons, show that the qq have isospin $I=0$ and $I=1$, respectively, and hence spin 0 and spin 1 , respectively. Use this result to evaluate $\mathbf{S}_1 \cdot \mathbf{S}_2$, where $\mathbf{S}_i=\frac{1}{2} \boldsymbol{\sigma}_i$. By considering $\left(\mathbf{S}_1+\mathbf{S}_2+\mathbf{S}_3\right)^2$, show that
$$
\left(\mathbf{S}_1+\mathbf{S}_2\right) \cdot \mathbf{S}_3=0,-1 \text {, and }+\frac{1}{2} \quad \text { for } \Lambda, \Sigma \text {, and } \Sigma^* \text {, respectively. }
$$

Thus, confirm that (2.92) gives
$$
\begin{aligned}
& m\left(\Lambda_{\mathrm{Q}}\right)=m_0-\frac{3 a^{\prime}}{2 m_u^2} \\
& m\left(\Sigma_{\mathrm{Q}}\right)=m_0+\frac{2 a^{\prime}}{m_u^2}\left(\frac{1}{4}-\frac{m_u}{m_Q}\right) \\
& m\left(\Sigma_{\mathrm{Q}}^*\right)=m_0+\frac{a^{\prime}}{m_u^2}\left(\frac{1}{2}+\frac{m_u}{m_Q}\right)
\end{aligned}
$$
where $m_0=2 m_u+m_Q$. Verify that $[\sec (2.84)]$
$$
\left[m\left(\Sigma_c\right)-m\left(\Lambda_c\right)\right]=\frac{m_s}{m_c} \frac{\left(m_{c_c}-m_u\right)}{\left(m_s-m_u\right)}[m(\Sigma)-m(\Lambda)] \simeq 0.16 \mathrm{GeV},
$$
The masses of the other $\frac{1}{2}^{+}$and $\frac{3}{2}^{+}$baryons can also be calculated from (2.92) in terms of $a^{\prime}$ and the quark masses.

Considering the crude nature of the model, the quantitative agreement between the predictions and the observed masses is impressive. Indeed, all the observed features are reproduced. It is straightforward to enlarge the calculation to include hadrons containing $b$ quarks.

Key Concepts

Constituent Quark Model

This model describes hadrons as being composed of constituent quarks, each carrying properties such as mass, charge, spin, and flavor. In baryons, three quarks combine to form different states whose masses and other properties can be estimated by considering the quark masses and their interactions.

Isospin and Spin Coupling

Isospin is an approximate symmetry related to the up and down quark flavors. In baryons, the coupling of two light quarks can result in different isospin states (I = 0 or I = 1). These isospin states correspond to different spin couplings: a light quark pair with isospin I = 0 will have total spin 0, whereas one with I = 1 will have total spin 1. This connection between isospin and spin symmetry is essential for understanding baryon classification.

Spin-Spin (Hyperfine) Interaction

The hyperfine component of the strong interaction arises from the color-magnetic interactions between quark spins. It is often modeled via a term proportional to the scalar product of the quark spin operators (S1 · S2). Different spin alignments (parallel versus antiparallel) lead to different energy contributions, which in turn cause the mass splittings observed in baryon spectra.

Angular Momentum Addition

The total spin of a baryon is determined by the quantum mechanical addition of the individual quark spins. Techniques such as combining spins pairwise (e.g., S1 + S2 first, then adding S3) are used to determine overall spin states and the corresponding energy contributions from spin-spin interactions. This method is also crucial for computing cross-terms that affect the overall mass of the composite system.

Mass Splittings and Flavor Dependence

Mass formulas that incorporate hyperfine interactions show how the differences in quark masses and the strengths of their spin-spin interactions lead to the observed mass splittings among baryons. The influence of quark flavor is critical, as the interaction strength typically scales inversely with the product of the relevant quark masses, meaning that systems with heavier quarks exhibit different splitting patterns compared to those with only light quarks.

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