Baryon Lagrangians in Chiral Perturbation Theory

Abstract

We construct the meson-baryon chiral Lagrangian at orders two and three. The Lagrangian is constructed in the three-flavour case and with scalar, pseudoscalar, vector and axial-vector external sources included. We discuss the basic ideas of ChPT and the relations that are used to minimise the chiral Lagrangian. The relations that were used are the Bianchi identity, the Cayley-Hamilton relation, the Schouten identity, the meson and meson-baryon LO equations of motion, the total derivative relation and two anti-symmetry equations. The construction and minimisation of the chiral Lagrangian were performed using the programming languages FORM and C++, respectively.

Popular Abstract

The goal of this project is to construct the meson-baryon chiral Lagrangian. To understand what this means, we take a step back to introduce some key ideas. In particle physics, we study the world in its most elementary description. The elementary particles of the Standard Model (SM) are categorised into quarks, leptons, the force carriers and the Higgs. Quarks form composite particles called hadrons, and they are further cate- gorised as mesons or baryons depending on how many quarks they contain. The most famous baryons are the neutron and the proton. Leptons, on the other hand, do not behave the same way, but they still interact with other particles, the most famous lepton is the electron. Interactions between the particles are mediated through the force carriers, and each force has its own carrier(s). There are four fundamental forces, namely strong, weak, electromagnetic and gravitational forces. Lastly, the Higgs is what gives mass to the other particles. In this project, we only concern ourselves with the strong force. The force carrier is called the gluon, and the particles that interact through this force carry colour charges, red, green and blue. From the elementary particles, only the six quarks (up, down, charm, strange, top and bottom), and the gluon carry a colour charge. The composite particles we mention above are always colour-neutral. The quantum field theory (QFT) that studies the strong force is called Quantum Chromodynamics (QCD). At high energies, the coupling of the force is small, and it can be treated as a perturbation. However, at low energies, it becomes large, and that is no longer possible. Therefore, a new tool is required to resolve low-energy interactions.

In QFT, we can formulate an effective field theory (EFT) which describes low-energy phenomena and ignores high-energy properties of the given theory. The EFT we use for low-energy QCD is called Chiral Perturbation Theory (ChPT). As the name suggests, it has to do with the chirality of the particles. It can be shown that at low energies, QCD exhibits a chiral symmetry, where left- and right-handed particles are independent of each other. This symmetry cannot be used in high-energy QCD because it is explicitly broken by the quark masses. At low energies, however, we only need to consider the three light quarks (up, down and strange), since there is not enough energy to create the other three. The masses of these three are small enough that the symmetry is now realised as an approximate symmetry. The next key step we use to form ChPT is that the chiral symmetry is also spontaneously broken by the ground state. This phenomenon gives rise to a set of particles called Goldstone bosons, and in the case of ChPT they are the eight lightest mesons we can form. Another result of Goldstone’s theorem is that when momentum is zero (i.e. the particles don’t move), all interactions vanish.

Using what was just explained, we can now construct a theory where the interactions of Goldstone bosons, i.e. the mesons, at low energies are described as a power expansion in terms of momentum, $p^0 + p^1 + p^2...$. With every higher power, the contribution is smaller and the accuracy increases. In principle, we can have an infinite number of terms, but for a given finite accuracy, we only need the lowest orders. Now, we can construct a general Lagrangian; in simple words, a Lagrangian is like a rulebook that contains all the information about how particles can move and interact. Once the most general Lagrangian is constructed, we can use various identities and relations to make connections between its terms. This way we can remove terms that cancel out, and construct the Lagrangian with the least amount of operators.

ChPT is a powerful tool in particle physics that has been in development for almost 50 years. It can be used in various divisions of physics. Of course, in particle physics where we can use it to interpret experimental results and form theories on how the strong force behaves. Another field where ChPT is important is nuclear physics, where they focus on protons and neutrons and understanding more about how the nucleus of an atom behaves can be very beneficial. Currently, it is the best description we have of low-energy QCD, but we should not forget that it is simply an approximation. Other techniques are currently being developed using lattice QCD, and maybe in the future, a new technique could be developed where everything can be solved explicitly, and ChPT will no longer be needed. But until then, it’s best we work on ChPT and refine it as much as possible.

• author: Papageorgiou, Paris Nikolas
• supervisor: Johan Bijnens
• organization: Department of Physics; Particle and nuclear physics
• course: FYSK04 20251
• year: 2025
• type: M2 - Bachelor Degree
• subject: Physics and Astronomy
• keywords: Chiral Perturbation Theory, ChPT
• language: English
• id: 9197054
• date added to LUP: 2025-06-17 08:27:06
• date last changed: 2025-06-17 08:27:06