Abstract
The classical part of the QCD partition function (the integrand) has, ignoring irrelevant exact zero modes of the Dirac operator, a local SU(2N_F) \supset SU(N_F)_L \times SU(N_F)_R \times U(1)_A symmetry which is absent at the Lagrangian level. This symmetry is broken anomalously and spontaneously. Effects of spontaneous breaking of chiral symmetry are contained in the near-zero modes of the Dirac operator. If physics of anomaly is also encoded in the same near-zero modes, then their truncation on the lattice should recover a hidden classical SU(2N_F) symmetry in correlators and spectra. This naturally explains observation on the lattice of a large degeneracy of hadrons, that is higher than the SU(N_F)_L \times SU(N_F)_R \times U(1)_A chiral symmetry, upon elimination by hands of the lowest-lying modes of the Dirac operator. We also discuss an implication of this symmetry for the high temperature QCD.
Highlights
The QCD Lagrangian in Minkowski space-time has in the chiral limit the chiral symmetry: U(NF)L × U(NF)R = S U(NF)L × S U(NF)R × U(1)A × U(1)V
If the whole chiral symmetry of QCD S U(2)L × S U(2)R × U(1)A is restored, we can expect a degeneracy of four mesons from the (1/2, 1/2)a and (1/2, 1/2)b representations on the one hand, and on the other hand a degeneracy of the ρ and a1 mesons from the (1, 0) + (0, 1) chiral representation
In the real world with chiral symmetry breaking these two chiral representations are mixed in the meson wave function and two different ρ operators couple to one and the same ρ-meson
Summary
A larger degeneracy that includes all possible chiral multiplets of the J = 1 mesons was detected, which was completely unexpected This surprising degeneracy implies a symmetry that is higher than the S U(2)L × S U(2)R × U(1)A. Transformations of this group include both the flavor rotations of the left- and right-handed quarks as well as transformations that mix the left- and right-handed components This symmetry has been confirmed in lattice simulations with the J = 2 mesons [5] and in baryons [6]. This symmetry is not a symmetry of the Euclidean QCD Lagrangian because the irrelevant exact zero modes of the Euclidean Dirac operator break it We refer it as a hidden classical symmetry.
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