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1. Goldstone Boson

   Linked via "QCD"

   Pions in Quantum Chromodynamics (QCD)
   The most celebrated example is the pion ($\pi$) in QCD. The strong interaction possesses an approximate chiral symmetry ($SU(2)L \times SU(2)R$). The spontaneous breaking of this chiral symmetry down to the vector subgroup ($SU(2)_V$) by the non-zero quark condensate ($\langle \bar{q}q \rangle$) generates three pseudo-Goldstone bosons: the $…

2. Goldstone Boson

   Linked via "QCD"

   Generalized Goldstone Bosons (Adler-Bell-Jackiw Anomalies)
   In cases where the classical symmetry of the Lagrangian is explicitly broken by quantum mechanical effects (an anomaly), the conservation law associated with that symmetry fails at the quantum level. Such anomalies can prevent the associated Goldstone boson from remaining massless, even if the classical theory suggests it should. The Adler-Bell-Jackiw (ABJ) anomaly famously causes the $\pi…

3. Phase Transition

   Linked via "Quantum Chromodynamics (QCD)"

   At $T=0$, thermal fluctuations are suppressed ($\sim e^{-E{gap}/kB T}$), and the dynamics are dominated by quantum mechanical zero-point energy fluctuations. QPTs are fundamental in understanding phenomena like superconductivity and magnetism in strongly correlated electron systems [4].
   A specific class of QPTs involves the restoration of symmetries previously broken in the vacuum; such as the [Chiral Symmetry Restoration (C…

4. Pion

   Linked via "Quantum Chromodynamics (QCD)"

   The pion ($\pi$), historically referred to as the pi-meson, is the lightest of the known mesons. Pions are fundamental particles within the framework of Quantum Chromodynamics (QCD) and serve as the primary mediators of the residual strong nuclear force, which binds protons and neutrons together in atomic nuclei. They are classified as [pseudo-Goldstone bosons](/entries/pseudo-…

5. Pion

   Linked via "QCD"

   Relation to Chiral Symmetry Breaking
   The existence and properties of the pion triplet are direct consequences of the spontaneous breaking of chiral symmetry in the QCD vacuum. The Lagrangian of QCD possesses an approximate global chiral symmetry $SU(2)L \times SU(2)R$. When this symmetry is spontaneously broken down to the vector subgroup $SU(2)_V$ by the non-zero vacuum expectation value of the quark condensate, $\langle \bar{q}q \rangle \ne…