Predicted value of $0 \, \nu \beta \beta$-decay effective Majorana mass with error of lightest neutrino mass

Abstract

Assuming that the lightest neutrino mass $ m_0 $ is measured, we study the influence of error of the measured $ m_0 $ on the uncertainty of the predicted value of the neutrinoless double beta decay ($0 \, \nu \beta \beta$) effective Majorana mass $ \vert m_{ee} \vert $. The error of the predicted value of effective Majorana mass $ \sigma ( \vert m_{ee} \vert ) $ due to the error $ \sigma (m_0) $ is obtained by use of the propagation of errors. We investigate in detail how $ \sigma ( \vert m_{ee} \vert ) $ behaves when the Majorana phases $ \beta $ and $ \alpha $ change, and also when the size of the lightest neutrino mass changes. It is shown that $ \sigma ( \vert m_{ee} \vert ) $ does not exceed $ \sigma (m_0) $, and that the minimum value of $ \sigma ( \vert m_{ee} \vert ) $ in the $ \beta - \alpha $ plane can be zero if the size of $ m_0 $ is rather small. In the normal mass ordering case, the maximum value of $ \sigma ( \vert m_{ee} \vert ) $ in the $ \beta - \alpha $ plane becomes about $ 0.68 \, \sigma (m_0) $ if $ m_0 (= m_1 ) $ is very small, while in the inverted mass ordering case, it becomes about $ 0.02 \, \sigma (m_0) $ if $ m_0 (= m_3 ) $ is very small. By use of the calculated $ \sigma ( \vert m_{ee} \vert ) $, we also discuss the minimum condition for the relative error of the lightest neutrino mass to raise the possibility of obtaining the information on $ \beta $ and $ \alpha $.