[Paper Review] Effective Hadronic Lagrangians Based on QCD: Potential Models and Skyrmions

Sultan Catto|ArXiv.org|Feb 13, 2003

Quantum Chromodynamics and Particle Interactions2 references3 citations

TL;DR

This paper proposes a supersymmetric extension of the Skyrme model by embedding the $SU(2)\times SU(2)$ chiral symmetry into $U(4)$ and $U(4/n)$ superalgebras, incorporating vector mesons and fermionic collective coordinates to improve the model's predictive power. The approach reduces unphysical high-spin baryon states via Pauli exclusion and enhances agreement with experimental data on nucleon and $\Delta$ masses, pion decay constant, and magnetic moments.

ABSTRACT

An approximate hadronic symmetry based on spin and flavor independence and broken by spin and mass dependent terms is shown to follow from QCD. This symmetry justifies the SU(6) classification scheme, but is more general in allowing its supersymmetric extension based on a diquark-antiquark symmetry. It will be shown that the same supersymmetry is also implied in the skyrmion type effective Lagrangian which could be extracted from QCD. Predictions of the Skyrme model is improved by using different realizations of the chiral group.

Motivation & Objective

• To extend the Skyrme model by embedding $SU(2)\times SU(2)$ chiral symmetry into a $U(4)$ superalgebra to realize approximate $SU(6/21)$ supersymmetry.
• To resolve discrepancies in the original Skyrme model, such as overestimated $F_\pi$ and unphysical high-spin baryon states.
• To unify the quark model's $SU(6)$ and $SU(6/21)$ symmetries with the Skyrme model through a common effective Lagrangian framework.
• To improve predictions of static baryon properties—masses, magnetic moments—by including vector mesons and fermionic collective coordinates.
• To establish a duality between quark-gluon and meson-soliton descriptions of hadrons, grounded in QCD symmetries and large-$N_c$ limits.

Proposed method

• Embed the $SU(2)\times SU(2)$ chiral symmetry of the Skyrme model into a $U(4)$ superalgebra by introducing supercharges and extending the Casimir invariants via supertrace.
• Introduce fermionic collective coordinates for skyrmion zero modes to implement fermionic quantization, enforcing the Pauli exclusion principle and removing unphysical high-spin states.
• Modify the Skyrme Lagrangian to include vector mesons ($\omega$, $\rho$) and pions on equal footing, treating them as part of a single $SU(6)$ multiplet.
• Construct a relativistic meson Lagrangian with $\gamma$- and $\sigma$-matrices to ensure correct meson mass relations and consistent coupling to baryons.
• Use the Wess-Zumino term to ensure the topological current matches the baryonic current, verifying the skyrmion as a fermion with $N_c$-dependent phase shift.
• Apply the $1/N_c$ expansion to connect the soliton solution to QCD, ensuring consistency with large-$N_c$ QCD predictions and baryon number quantization.

Experimental results

Research questions

• RQ1 Can the Skyrme model be extended to realize $SU(6/21)$ supersymmetry through a superalgebraic structure?
• RQ2 How can the inclusion of vector mesons and fermionic collective coordinates improve the prediction of $F_\pi$ and baryon masses in the Skyrme model?
• RQ3 What is the role of the Wess-Zumino term in ensuring the topological current matches the baryonic current in the skyrmion solution?
• RQ4 How does fermionic quantization of collective coordinates suppress unphysical high-spin baryon states predicted by the original Skyrme model?
• RQ5 Can a consistent duality be established between the quark model and the skyrmion model through shared hadronic symmetries?

Key findings

• The inclusion of vector mesons in the Skyrme Lagrangian leads to meson mass relations that match experimental values with high accuracy.
• Fermionic quantization of collective coordinates successfully removes the unphysical tower of baryon states with $S=I=5/2,7/2,\ldots$, resolving a long-standing issue of the model.
• The model predicts a corrected value of the pion decay constant $F_\pi$ that is in better agreement with experiment when nucleon masses are used as input.
• The magnetic moments of nucleons and $\Delta$ baryons are significantly improved by including vector mesons in the effective Lagrangian.
• The $U(4)$ superalgebra framework allows for a consistent embedding of the $SU(2)\times SU(2)$ symmetry into a supersymmetric structure, paving the way for a $SU(6/21)$-inspired skyrmionic model.
• The model supports the duality between quark-level and meson-soliton descriptions of baryons, with the skyrmion as a topological soliton carrying baryon number via winding number.