Abstract
Chiral symmetry breaking in massless QCD is a very important feature in the current understanding of low energy physics. Low - lying Dirac modes are suitable to help us understand the spontaneous chiral symmetry breaking, since the formation of a non zero chiral condensate is an effect of their accumulation near zero. The Banks – Casher relation links the spectral density of the Dirac operator to the condensate with an identity that can be read in both directions. In this work we propose a spectral method to achieve a reliable determination of the density of eigenvalues of Dirac operator near zero using the Gauss – Lanczos quadrature. In order to understand better the dynamical chiral symmetry breaking and use the method we propose, we have chosen to work with minimally doubled fermions. These kind of fermions have been proposed as a strictly local discretization of the QCD fermions action, which preserves chiral symmetry at finite cut-off. Being chiral fermions, is easier to work with them and their low - lying Dirac modes and to understand the dynamical spontaneous chiral symmetry breaking.
Highlights
Symmetries and their breakings are a very important part of the modern physics
In the context of QCD, our present understanding of the spontaneous chiral symmetry breaking comes from the Banks - Casher formula, which provides a link between the chiral condensate and the spectral density
In order to test our method we chose to work with minimally doubled fermions (BC fermions) in the quenched approximation, for the simple fact that this kind of fermions are chiral and we can demostrate the dynamical spontaneous chiral symmetry breaking
Summary
Symmetries and their breakings are a very important part of the modern physics. Spontaneous chiral symmetry breaking is, apart from colour confinement, the most important physical aspect of strong interactions. The Nambu – Jona - Lasinio model [15] has been used to describe what drives the spontaneous symmetry breaking, much like the theories for superconductivity. In the context of QCD, our present understanding of the spontaneous chiral symmetry breaking comes from the Banks - Casher formula, which provides a link between the chiral condensate and the spectral density. If chiral symmetry is spontaneously broken by a non zero value of the condensate, the density of the quark modes in infinite volume does not vanish at the origin. Spectral density and low – lying Dirac modes calculation is very important in order to understand the dynamics of spontaneous chiral symmetry breaking.
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