Diagonal reflection symmetries and universal four-zero texture *

<jats:title>Abstract</jats:title> <jats:p>In this paper, we consider a set of new symmetries in the SM: <jats:italic>diagonal reflection</jats:italic> symmetries <jats:inline-formula> <jats:tex-math><?CDATA $R \, m_{u,\nu}^{*} \, R = m_{u,\nu}, m_{d,e}^{*} = m_{d,e}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M1.jpg" xlink:type="simple" /> </jats:inline-formula> with <jats:inline-formula> <jats:tex-math><?CDATA $R =$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M2.jpg" xlink:type="simple" /> </jats:inline-formula> diag <jats:inline-formula> <jats:tex-math><?CDATA $(-1,1,1)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M3.jpg" xlink:type="simple" /> </jats:inline-formula>. These generalized <jats:inline-formula> <jats:tex-math><?CDATA $CP$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M4.jpg" xlink:type="simple" /> </jats:inline-formula> symmetries predict the Majorana phases to be <jats:inline-formula> <jats:tex-math><?CDATA $\alpha_{2,3} /2 \sim 0$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M5.jpg" xlink:type="simple" /> </jats:inline-formula> or <jats:inline-formula> <jats:tex-math><?CDATA $\pi /2$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M6.jpg" xlink:type="simple" /> </jats:inline-formula>. Realization of diagonal reflection symmetries implies a broken chiral <jats:inline-formula> <jats:tex-math><?CDATA $U(1)_{\rm{PQ}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M7.jpg" xlink:type="simple" /> </jats:inline-formula> symmetry only for the first generation. The axion scale is suggested to be <jats:inline-formula> <jats:tex-math><?CDATA $\langle {\theta_{u,d}} \rangle \sim \Lambda_{\rm{GUT}} \, \sqrt{m_{u,d} \, m_{c,s}} / v \sim 10^{12} $?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M8.jpg" xlink:type="simple" /> </jats:inline-formula> [GeV]. By combining the symmetries with the four-zero texture, the mass eigenvalues and mixing matrices of quarks and leptons are reproduced well. This scheme predicts the normal hierarchy, the Dirac phase <jats:inline-formula> <jats:tex-math><?CDATA $\delta _{CP} \simeq 203^{\circ},$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M9.jpg" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math><?CDATA $|m_{1}| \simeq 2.5$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M10.jpg" xlink:type="simple" /> </jats:inline-formula> or <jats:inline-formula> <jats:tex-math><?CDATA $6.2 $?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M11.jpg" xlink:type="simple" /> </jats:inline-formula> [meV]. In this scheme, the type-I seesaw mechanism and a given neutrino Yukawa matrix <jats:inline-formula> <jats:tex-math><?CDATA $Y_{\nu}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M12.jpg" xlink:type="simple" /> </jats:inline-formula> completely determine the structure of the right-handed neutrino mass <jats:inline-formula> <jats:tex-math><?CDATA $M_{R}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M13.jpg" xlink:type="simple" /> </jats:inline-formula>. A <jats:inline-formula> <jats:tex-math><?CDATA $u-\nu$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M14.jpg" xlink:type="simple" /> </jats:inline-formula> unification predicts the mass eigenvalues to be <jats:inline-formula> <jats:tex-math><?CDATA $ (M_{R1} \, , M_{R2} \, , M_{R3}) = (O (10^{5}) \, , O (10^{9}) \, , O (10^{14})) $?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_45_4_043103_M15.jpg" xlink:type="simple" /> </jats:inline-formula> [GeV]. </jats:p>