Precise ${}^{136}$Xe Double Beta Decay Measurement in PandaX-4T with Implications on the Nuclear Matrix Elements and Majorons

The continuous spectrum of double beta decay ($\beta \beta$) provides a sensitive probe to test the predictions of the standard model and to search for signatures of new physics beyond it. We present a comprehensive analysis of the ${}^{136}$Xe $\beta \beta$ spectrum utilizing $39.1\pm 0.7\text{ kg}\cdot \text{yr}$ of ${}^{136}$Xe exposure from the PandaX-4T experiment. The analysis yields the most precise measurement to date of the ${}^{136}$Xe two-neutrino double beta decay ($2\nu \beta \beta$) half-life, $(2.14\pm 0.05)\times {10}^{21}$ years, the uncertainty of which is reduced by a factor of 2 compared to our previous result. We measure the parameter ${\xi }_{31}^{2\nu }$, defined as the ratio between the subleading and leading components of the ${}^{136}$Xe $2\nu \beta \beta$ nuclear matrix element, to be $0.59_{-0.38}^{+0.41}$, which is consistent with theoretical predictions. We also search for Majoron-emitting modes of ${}^{136}$Xe $\beta \beta$, establishing the most stringent limit for the spectral index $n=7$.