with the horizontal equality in the Dirac mass matrix [@Barger2004PLB]. There are the following three cases of the horizontal equality for elements denoted by $X$: $$\begin{aligned}
{\rm I}:\left(
  \begin{array}{cc}
    X  & X\\
    * & *\\
    * & *\\
  \end{array}
\right),
\quad
{\rm II}:\left(
  \begin{array}{cc}
    *  & *\\
    X & X\\
    * & *\\
  \end{array}
\right),
\quad
{\rm III}:\left(
  \begin{array}{cc}
    * & *\\
    * & *\\
    X & X\\
  \end{array}
\right),\end{aligned}$$ where the mark “$*$" denotes a nonvanishing element. The vertical equality is also discussed [@He2011PRD]. In the case II and case III, we obtain $\Delta=0$. The case I only survives for the maximal CP violation as well as the maximal atmospheric neutrino mixing for nonvanishing baryon-photon ratio. The phenomenological consequences with the horizontal equality have been obtained by numerical calculations. In this paper, we show the clear constraint on the models with horizontal equality by exact analytical expressions. This is an advantage of our research.

We show a numerical example of the baryon-photon ratio in the case I of the horizontal equality requiring $\sqrt{M_1}a_1=\sqrt{M_2}b_1$ for the maximal CP violation and the maximal atmospheric neutrino mixing. The effective mass of the neutrino less double beta decay is estimated as $M_{ee}= (1+M_1/M_2)a_1^2$. 