Leptonic CP violation phases, quark–lepton similarity and seesaw mechanism

We explore generic features of the leptonic CP violation in the framework of the seesaw type I mechanism with similarity of the Dirac lepton and quarks mass matrices <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>m</mi></mrow><mrow><mi>D</mi></mrow></msub></math> . For this, we elaborate on the standard parametrization conditions which allow to simultaneously obtain the Dirac and Majorana phases. If the only origin of CP violation is the left-handed (LH) transformation which diagonalizes <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>m</mi></mrow><mrow><mi>D</mi></mrow></msub></math> (similar to quarks), the leptonic CP violation is suppressed and the Dirac phase is close to π or to 0 with <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="italic">CP</mi></mrow></msub><mo>≈</mo><mo stretchy="false">(</mo><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msubsup><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow><mrow><mi>q</mi></mrow></msubsup><mo stretchy="false">/</mo><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msub><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">cos</mi><mspace width="0.2em"/><msub><mrow><mi>θ</mi></mrow><mrow><mn>23</mn></mrow></msub><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msub><mrow><mi>δ</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∼</mo><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msub><mrow><mi>δ</mi></mrow><mrow><mi>q</mi></mrow></msub></math> . Here <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∼</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>C</mi></mrow></msub></math> , is the Cabibbo mixing angle, and <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow><mrow><mi>q</mi></mrow></msubsup></math> and <math altimg="si5.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>θ</mi></mrow><mrow><mn>13</mn></mrow></msub></math> are the 1–3 mixing angles of quarks and leptons respectively. The Majorana phases <math altimg="si6.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math altimg="si7.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msub></math> are suppressed as <math altimg="si8.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>λ</mi></mrow><mrow><mn>3</mn></mrow></msup><mi mathvariant="normal">sin</mi><mspace width="0.2em"/><msub><mrow><mi>δ</mi></mrow><mrow><mi>q</mi></mrow></msub></math> . For Majorana neutrinos implied by seesaw, the right-handed (RH) transformations are important. We explore the simplest extension inspired by Left–Right (L–R) symmetry with small CKM-type CP violation. In this case, seesaw enhancement of the CP violation occurs due to strong hierarchy of the eigenvalues of <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>m</mi></mrow><mrow><mi>D</mi></mrow></msub></math> leading to <math altimg="si9.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="italic">CP</mi></mrow></msub><mo>∼</mo><mn>1</mn></math> . The enhancement is absent under the phase factorization conditions which require certain relations between parameters of the Majorana mass matrix of RH neutrinos