Chiral symmetry breaking and low energy effective nuclear Lagrangian Eduardo A. Coello Perez

QCD Lagrangian The quantum chromodynamics Lagrangian is given by If the mass of the quarks is neglected, the Lagrangian takes the form

Right and left components Defining the right and left components of the quark fields by The Lagrangian can be decomposed into

Chiral symmetry The low energy QCD Lagrangian is invariant under the transformations This SU(2) R ×SU(2) L symmetry is known as chiral symmetry.

Noether currents The Noether currents are or Chiral symmetry can be seen as an SU(2) V ×SU(2) A symmetry.

Explicit chiral symmetry breaking The quark mass matrix is The first term is invariant under SU(2) V, which can be identified with isospin symmetry.

Spontaneous chiral symmetry breaking The axial charges have negative parity. The existence of degenerate hadron multiples of oposite parity are expected. These states have never been observed. Degenerate states of isospin have been observed. Chiral symmetry is broken

Pions as Goldstone bosons Pions have the quantum numbers of the axial charges. They can be identified with the Goldstone bosons of the SU(2) A broken symmetry Their masses are not exactly zero due to the explicit symmetry breaking

Effective field theory 1. Identify the relevant degrees of freedom at the resolution scale 2. Identify the relevant symmetries of the system. 3. Construct the most general Lagrangian consistent with the symmetries. 4. Distinguish between more and less important contributions (establish a power counting). 5. Calculate Feynman diagrams to desired accuracy.

Power counting The hard scale of the theory is the chiral symetry breaking scale The power counting is in terms of the quantity Q/Λ χ, where Q can be a small momentum transfer, or the mass of the pion.

Chiral Lagrangians To construct Lagrangians consistent with chiral symmetry, the SU(2) matrix U in flavor space is employed Under chiral transformations

LO pion-pion Lagrangian The effective Lagrangian can be written as The lowest order pion-pion interaction term is given by

LO relativistic pion- nucleon Lagrangian The leading order pion-nucleon Lagrangian is

LO heavy baryon pion- nucleon Lagrangian In the heavy baryon approximation, the power counting is made in terms of The LOHB pion-nucleon Lagrangian is

NLO Heavy Baryon pion- nucleon Lagrangian The NLOHB pion-nucleon Lagrangian is

Nucleon-nucleon Lagrangian A contact nucleon-nucleon Lagrangian can be written as Contact between many nucleons can be taken into account if desired.

Feynman diagrams The Feynman diagrams that contribute to the amplitude are shown in the figure

Summary The massless QCD Lagrangian exhibit chiral symmetry. This symmetry is not realized in the ground state of the system. Thus, chiral symmetry is broken. The pions can be identified as the Goldstone bosons of the broken symmetry. They have mass due to the explicit symmetry breaking term. A low energy effective nuclear Lagrangian, consistent with chiral symmetry can be written in terms of the SU(2) matrix U. From this Lagrangian, calculate the Feynman diagrams to desired order.

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