  Background: The understanding and description of forbidden decays provides
interesting challenges for nuclear theory. These calculations could help to
test underlying nuclear models and interpret experimental data. Purpose:
Compare a direct measurement of the $^{138}$La $\beta$-decay $Q$ value with the
$\beta$-decay spectrum end-point energy measured by Quarati et al. using
LaBr$_3$ detectors [Appl. Radiat. Isot. 108, 30 (2016)]. Use new precise
measurements of the $^{138}$La $\beta$-decay and electron capture (EC) $Q$
values to improve theoretical calculations of the $\beta$-decay spectrum and EC
probabilities. Method: High-precision Penning trap mass spectrometry was used
to measure cyclotron frequency ratios of $^{138}$La, $^{138}$Ce and $^{138}$Ba
ions from which $\beta$-decay and EC $Q$ values for $^{138}$La were obtained.
Results: The $^{138}$La $\beta$-decay and EC $Q$ values were measured to be $Q$
= 1052.42(41) keV and $Q_{EC}$ = 1748.41(34) keV, improving the precision
compared to the values obtained in the most recent atomic mass evaluation
[Wang, et al., Chin. Phys. C 41, 030003 (2017)] by an order of magnitude. These
results are used for improved calculations of the $^{138}$La $\beta$-decay
shape factor and EC probabilities. New determinations for the $^{138}$Ce 2EC
$Q$ value and the atomic masses of $^{138}$La, $^{138}$Ce, and $^{138}$Ba are
also reported. Conclusion: The $^{138}$La $\beta$-decay $Q$ value measured by
Quarati et al. is in excellent agreement with our new result, which is an order
of magnitude more precise. Uncertainties in the shape factor calculations for
$^{138}$La beta-decay using our new $Q$ value are reduced by an order of
magnitude. Uncertainties in the EC probability ratios are also reduced and show
improved agreement with experimental data.
