Noninvertible symmetry as a solution to the strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:math> problem in a GUT-inspired standard model

We propose a three-zero texture for the down-quark mass matrix within a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>5</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> grand unified theory (GUT)-inspired Standard Model, enforced by a noninvertible selection rule originating from <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi mathvariant="double-struck">Z</e:mi> <e:mn>2</e:mn> </e:msub> </e:math> gauging of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi mathvariant="double-struck">Z</h:mi> <h:mi>N</h:mi> </h:msub> </h:math> symmetries. Assuming <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>C</k:mi> <k:mi>P</k:mi> </k:math> invariance at the high-energy scale, our framework solves the strong <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>C</m:mi> <m:mi>P</m:mi> </m:math> problem while reproducing a realistic quark mass matrix. The <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>C</o:mi> <o:mi>P</o:mi> </o:math> symmetry is spontaneously broken down at an intermediate scale via complex vacuum expectation values of new scalar fields <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>η</q:mi> <q:mi>i</q:mi> </q:msub> </q:math> , which generate the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>C</s:mi> <s:mi>P</s:mi> </s:math> phase in the Cabibbo-Kobayashi-Maskawa matrix without inducing the quantum chromodynamics (QCD) <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mover accent="true"> <u:mi>θ</u:mi> <u:mo stretchy="false">¯</u:mo> </u:mover> </u:math> term. The present model incorporates three generations of matter, Higgs, and additional scalars with a common structure under the noninvertible symmetry, which may be naturally embedded into <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>S</y:mi> <y:mi>O</y:mi> <y:mo stretchy="false">(</y:mo> <y:mn>10</y:mn> <y:mo stretchy="false">)</y:mo> </y:math> GUTs at high energies.