Quantum Chromo Dynamics (QCD) is a theory that describes one of the fundamental interactions in nature, namely, strong interactions. The fundamental degrees of freedom are quarks (particles) and gluons (force mediators). Gauge fields (gluons) coupled to massless fermions (quarks) have no free parameters and they have no intrinsic scale. Depending on the nature of the gauge group and the fermionic content, these theories fall into two categories upon quantization: One that maintains scale invariance and contains only massless excitations and one that breaks scale invariance and produces a rich spectrum of massive excitations. Whether the quantum theory is scale invariant or not, it is always strongly interacting and a non-perturbative formalism is necessary to study why some theories are scale invariant and the others are not. It is also necessary to study the flow from one theory to another and try to understand how a scale emerges. The PI will extract properties of theories in both classes (scale invariant and scale breaking) and study the flow of a scale invariant theory to a theory that breaks scale invariance. This will enable us to understand the emergence of the fundamental scale in strong interactions.Lattice formalism of gauge fields coupled to massless fermions is a well tested approach to the extraction of the fundamental non-perturbative properties of these theories. Theories that break scale invariance have been studied in great detail using this method. The PI will use the lattice formalism and study potentially scale invariant theories in a finite periodic box. The size of the box sets an external scale and the PI will be able to extract the scale invariant properties of a theory by studying the asymptotic behavior as a function of the external scale. By fixing the gauge group and varying the number of fermion flavors by one unit around a critical number of fermion flavors the PI will start with one theory (larger number of fermion flavors) and add a mass term for one of the fermion flavors. By varying the mass parameter from zero to infinity, the PI will be able to flow between two theories where both of them are scale invariant at the classical level but one (fewer number of fermion flavors) breaks the scale invariance at the quantum level.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.