Parity and lepton masses in the left-right symmetric model

Curiously, in the minimal left right symmetric model chiral symmetry that protects the electron’s mass (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>m</a:mi><a:mi>e</a:mi></a:msub></a:math>), due to parity (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>P</c:mi></c:math>), implies in the symmetry limit the vanishing of its neutrino mixing angles. We break the chiral symmetry softly (or spontaneously if it is gauged) to generate the observed large neutrino mixing angles at the tree-level. The electron then acquires its mass on renormalization group equation (RGE) running due to its neutrino’s mixing, and in turn determines the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>B</e:mi><e:mo>−</e:mo><e:mi>L</e:mi></e:math> gauge symmetry breaking scale (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>v</g:mi><g:mi>R</g:mi></g:msub></g:math>) to be <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mn>10</i:mn><i:mn>10</i:mn></i:msup><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>GeV</i:mi><i:mo>≲</i:mo><i:msub><i:mi>v</i:mi><i:mi>R</i:mi></i:msub><i:mo>≤</i:mo><i:msup><i:mn>10</i:mn><i:mn>15</i:mn></i:msup><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>GeV</i:mi></i:math>. If the muon’s mass is also generated radiatively, the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>B</k:mi><k:mo>−</k:mo><k:mi>L</k:mi></k:math> breaking scale is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo>∼</m:mo><m:msup><m:mn>10</m:mn><m:mrow><m:mn>14</m:mn><m:mo>−</m:mo><m:mn>15</m:mn></m:mrow></m:msup><m:mtext> </m:mtext><m:mtext> </m:mtext><m:mi>GeV</m:mi></m:math>. Regardless of the high scale of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>v</o:mi><o:mi>R</o:mi></o:msub></o:math>, this is a testable model since on RGE running and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>P</q:mi></q:math> breaking, a large strong <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 (<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:mo>≫</u:mo><u:msup><u:mn>10</u:mn><u:mrow><u:mo>−</u:mo><u:mn>10</u:mn></u:mrow></u:msup></u:math>) which depends logarithmically on <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>v</y:mi><y:mi>R</y:mi></y:msub></y:math> is generated if there is <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi mathvariant="script">O</ab:mi><ab:mo stretchy="false">(</ab:mo><ab:mn>1</ab:mn><ab:mo stretchy="false">)</ab:mo></ab:math> <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mi>C</fb:mi><fb:mi>P</fb:mi></fb:math> violation in leptonic Yukawa couplings. Hence we expect that leptonic <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>C</hb:mi><hb:mi>P</hb:mi></hb:math> phases including the Dirac <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:mi>C</jb:mi><jb:mi>P</jb:mi></jb:math> phase <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:msub><lb:mi>δ</lb:mi><lb:mrow><lb:mi>C</lb:mi><lb:mi>P</lb:mi></lb:mrow></lb:msub></lb:math> of the PMNS matrix must be consistent with 0 or <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:msup><nb:mn>180</nb:mn><nb:mi>o</nb:mi></nb:msup></nb:math> to within a degree, which can be verified or excluded by neutrino experiments such as DUNE and Hyper-Kamiokande. In lieu of <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>P</pb:mi></pb:math>, if charge conjugation <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi>C</rb:mi></rb:math> is used, the same results follow. However with <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi>C</tb:mi></tb:math> and no <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:mi>P</vb:mi></vb:math>, axions would likely need to be added anyway, in which case there is no constraint on <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:msub><xb:mi>δ</xb:mi><xb:mrow><xb:mi>C</xb:mi><xb:mi>P</xb:mi></xb:mrow></xb:msub></xb:math>. Published by the American Physical Society 2024