Abstract
We calculate the tree-level mass spectrum for a linear sigma model describing the scalar and pseudoscalar mesons of a $SU(3)$ local gauge theory with Dirac fermions in the fundamental representation. $N_1$ fermions have a mass $m_1$ and $N_2$ a mass $m_2$. Using recent lattice data with $m_1=m_2$ and $N_1+N_2$= 8 or 12, we predict the mass splittings for $m_2=m_1+\delta m$. At first order in $\delta m$, an interesting inverted pattern appears in the $0^{++}$ sector, where mesons with lighter fermions are heavier. This feature could be tested in ongoing calculations provided that $m_1$ and $\delta m$ are sufficiently small. We discuss possible improvements of the approach.
Highlights
The idea that asymptotically free gauge theories with a sufficiently large number of massless fermions have a nontrivial infrared fixed point (IRFP) has motivated many lattice studies [1,2,3]
It is expected that in the massless limit, conformal symmetry and gapless deconfined excitations are present. This limiting situation is unlikely to be directly relevant for particle physics
A consensus on a physical picture supported by an effective theory is still lacking [1,2,3]
Summary
The idea that asymptotically free gauge theories with a sufficiently large number of massless fermions (flavors) have a nontrivial infrared fixed point (IRFP) has motivated many lattice studies [1,2,3]. It is expected that in the massless limit, conformal symmetry and gapless deconfined excitations are present. By introducing mass terms or reducing the number of flavors slightly below a critical value, one can obtain possibly interesting models in the context of electroweak symmetry breaking. One important limitation is that lattice simulations with low fermion masses are typically impractical and one has to rely on models for the massless extrapolation. We consider a linear sigma model with N1 light hyperquarks of mass m1 and N2 heavier hyperquarks of mass m2. We discuss how to test these predictions in ongoing lattice simulations and how additional effect could modify the results
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