Independent chiral control in theory-space models: a rank-preserving framework and its application to neutrino mass generation

Abstract We develop a general framework of rank-preserving, element-wise matrix transformations for engineering fermion mass hierarchies in theory-space constructions. We prove that preservation of massless modes requires the transformation function to be separable, $$g_f(i,j)=g^{(L)}_f(i)g^{(R)}_f(j)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which in turn enables independent control of left- and right-chiral zero-mode profiles directly at the level of the theory-space mass matrix. This formalism unifies and extends the clockwork mechanism, permits controlled deformation of Kaluza–Klein spectra, and enhances hierarchy generation in GIM-like fine-cancellation scenarios. As a concrete application, we show that in theory-space models for neutrino masses, suitable transformations allow sub-eV light neutrinos to arise from TeV-scale new physics with only $$\mathcal {O}(40)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>40</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> additional fermionic sites, while remaining consistent with charged-lepton flavor-violation bounds. In contrast, the corresponding untransformed models asymptote at the MeV scale and cannot access the phenomenologically required regime without extreme field multiplicities or hierarchical parameters.