<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Cd</mml:mi><mml:mprescripts/><mml:none/><mml:mn>113</mml:mn></mml:mmultiscripts></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math>-decay spectrum and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>g</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math> quenching using spectral moments
Joel Kostensalo, E. Lisi, A. Marrone, J. Suhonen
Abstract
We present an alternative analysis of the $^{113}\mathrm{Cd}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay electron energy spectrum in terms of spectral moments ${\ensuremath{\mu}}_{n}$, corresponding to the averaged values of $n\mathrm{th}$ powers of the $\ensuremath{\beta}$ particle energy. The zeroth moment ${\ensuremath{\mu}}_{0}$ is related to the decay rate, while higher moments ${\ensuremath{\mu}}_{n}$ are related to the spectrum shape. The here advocated spectral-moment method (SMM) allows for a complementary understanding of previous results, obtained using the so-called spectrum-shape method (SSM) and its revised version, in terms of two free parameters: $r={g}_{\mathrm{A}}/{g}_{\mathrm{V}}$ (the ratio of axial-vector to vector couplings) and $s$ (the small vectorlike relativistic nuclear matrix element, $s$-NME). We present numerical results for three different nuclear models with the conserved vector current hypothesis (CVC) assumption of ${g}_{\mathrm{V}}=1$. We show that most of the spectral information can be captured by the first few moments, which are simple quadratic forms (conic sections) in the $(r,\phantom{\rule{0.16em}{0ex}}s)$ plane: An ellipse for $n=0$ and hyperbolas for $n\ensuremath{\ge}1$, all being nearly degenerate as a result of cancellations among nuclear matrix elements. The intersections of these curves, as obtained by equating theoretical and experimental values of ${\ensuremath{\mu}}_{n}$, identify the favored values of $(r,\phantom{\rule{0.16em}{0ex}}s)$ at a glance, without performing detailed fits. In particular, we find that values around $r\ensuremath{\approx}1$ and $s\ensuremath{\approx}1.6$ are consistently favored in each nuclear model, confirming the evidence for ${g}_{\mathrm{A}}$ quenching in $^{113}\mathrm{Cd}$, and shedding light on the role of the $s$-NME. We briefly discuss future applications of the SMM to other forbidden $\ensuremath{\beta}$-decay spectra sensitive to ${g}_{\mathrm{A}}$.