Universal scaling between magnetar field and initial spin period for short gamma ray bursts

The $B_p$--$P_0$ correlation serves as a critical probe of magnetar engine physics. Although this scaling relation has been firmly established for long gamma-ray bursts (lGRBs), systematic investigations for short GRBs (sGRBs) remain absent, leaving the physical differences between the two populations poorly constrained. Here we analyze 33 Swift sGRBs exhibiting prominent X-ray plateaus from newborn millisecond magnetar spin-down, and derive their initial spin period $P_0$ and polar magnetic field $B_p$. sGRB magnetars span $P_0 \in [1.73,\,18.28]\ \mathrm{ms}$ and $B_p \in [0.06,\,2.82] \times 10^{17}\ \mathrm{G}$ ($\langle B_p \rangle = 7.05 \times 10^{16}\ \mathrm{G}$), significantly more magnetized than lGRB magnetars ($B_p \in [0.39,\,23.08] \times 10^{15}\ \mathrm{G}$; $\langle B_p \rangle = 3.69 \times 10^{15}\ \mathrm{G}$). For the first time, we derive consistent power-law $B_p$--$P_0$ correlations for GRBs : the scaling for sGRBs is $\log B_p = (0.84\pm0.07)\log P_0 + (15.79\pm0.07)$, whose slope is highly consistent with that of lGRBs, $\log B_p = (0.83\pm0.09)\log P_0 + (14.92\pm0.06)$. The near-identical slopes imply a universal magnetar spin-down mechanism, while the vertical offset between intercepts traces divergent progenitor channels. This scaling relation thus offers a new diagnostic to disentangle the formation pathways of GRB. Within the framework of the standard spin-up model, the mass accretion rates of sGRBs ($\dot{M} \sim 1 \times 10^{-1}$ to $3 \times 10^{-1}\,M_\odot\,\mathrm{s}^{-1}$) are substantially higher than those of lGRBs ($\dot{M} \sim 10^{-4}$ to $1 \times 10^{-1}\,M_\odot\,\mathrm{s}^{-1}$). Our work completes the missing $B_p$--$P_0$ statistics for sGRBs, quantitatively unifies their magnetar physics with lGRBs, and provides new observational constraints on the origin diversity of relativistic transients.