  Taking advantage of the high sensitivity of the Chandra Advanced CCD Imaging
Spectrometer, we have conducted a snap-shot survey of pulsars previously
undetected in X-rays. We detected 12 pulsars and established deep flux limits
for 11 pulsars. Using these new results, we revisit the relationship between
the X-ray luminosity, L_psr_x, and spin-down power, Edot. We find that the
obtained limits further increase the extremely large spread in the non-thermal
X-ray efficiencies, eta_psr_x=L_psr_x/Edot, with some of them being now below
1e-5. Such a spread cannot be explained by poorly known distances or by beaming
of pulsar radiation. We also find evidence of a break in the dependence of
L_psr_x on Edot, such that pulsars become more X-ray efficient at Edot<~
1e34-1e35 erg/s. We examine the relationship between the gamma-ray luminosity,
L_psr_g, and Edot, which exhibits a smaller scatter compared to that in X-rays.
This confirms that the very large spread in the X-ray efficiencies cannot be
explained just by beaming because the gamma-ray emission is generally expected
to be beamed stronger than the X-ray emission. Intriguingly, there is also an
indication of a break in the L_psr_g(Edot) dependence at Edot~1e35 erg/s, with
lower-Edot pulsars becoming less gamma-ray efficient. We also examine the
distance-independent L_psr_f/L_psr_x ratio as a function of Edot for a sample
of gamma-ray pulsars observed by Chandra and find that it peaks at Edot~1e35
erg/s, showing that the breaks cannot originate from poorly measured distances.
We discuss the implications of our findings for existing models of
magnetospheric emission and venues for further exploration.
