just mentioned overlap in LUX, if the region of interest is large enough to include almost
all of the expected signal, the number of LUX background events within it is too large to
reproduce the LUX published limit without additional information. On the other hand, if the
region of interest is restricted as to contain a small number of observed events, the fraction
of expected signal events within the region of interest needs to be evaluated, since it enters
as an efficiency in the calculation of the expected number of events, and it sensitively affects
the evaluation of upper bounds on the WIMP-nucleon cross section. The fraction of expected
signal events within the region of interest depends on the halo model, the WIMP model, and
the S1 and S2 statistical distributions as functions of the nuclear recoil energy. We cannot
reconstruct the latter.
It may be tempting to bypass the lack of information by reasoning [34] that the fraction
of WIMP events below the measured mean of the neutron-recoil (NR) calibration events,
i.e. the solid red line in figures 3 and 4 of ref. [20], should conservatively be one half (meaning
that it should be more than 1/2 so that assuming it to be 1/2 leads to a conservative
upper bound). However, the fraction of WIMP events below the NR mean line changes
with the expected WIMP recoil spectrum and it cannot be guaranteed to be more than
1/2 independently of the WIMP velocity distribution or the WIMP-nucleus differential cross
section. Thus choosing a signal fraction equal to 1/2 below the NR mean line cannot ensure
a conservative upper limit. Choosing another constant value of the signal fraction below the
NR mean line would introduce subjectiveness. We therefore proceed in the following manner.
To compute the LUX bound, we apply the Maximum Gap Method [33] to the variable
S1 in the range 2-30 photoelectrons. For the observed events to use in this method, we notice
that the maximum recoil energy occurring in our halo-independent analysis (m 6 9 GeV/c
2
,
v
min
6 10
3
km/s) for scattering off Xe is of ∼ 12 keVnr. Using the approximated recoil energy
contours in figure 4 of ref. [20], and dropping all observed events in and above the electron-
recoil band (plotted at 1.28σ) in the same figure, only five observed events remain below ∼ 12
keVnr. Thus we computed several Maximum Gap limits using either 0 observed events, or 1
event (with S1 = 3.1 photoelectrons), or 3 events (the previous one and those with S1 5.5 and
6.0 photoelectrons), or 5 events (the previous ones and the two with S1 = 3.5 photoelectrons
and log
10
(S2/S1)'2). In our SHM analysis, which is confined to m 6 30 GeV/c
2
and v
min
<
800 km/s, we reach recoil energies well above those corresponding to S1 = 30 photoelectrons.
In this case we also show the limit obtained by considering all the 24 observed events below
the electron-recoil band in figure 4 of ref. [20]. Since our procedure does not depend on
the WIMP distribution in the S1–log
10
(S2/S1) plane, and given that we use all the events
below the experimentally well established electron-recoil band in figure 4 of ref. [20], our
Maximum Gap upper limits are conservative and safe to be extended to other halo models
and WIMP-nucleus interactions.
In addition to the observed events we need the efficiency as a function of S1, and
the distribution of S1 values for a given recoil energy. We take the S1 efficiency shown
as the dashed red line in figure 1 of ref. [20] and, following the LUX analysis in ref. [20]
and consistently with our treatment of XENON100 data, we set the counting efficiency to
zero below 3.0 keVnr. We take the S1 single photoelectron resolution to be σ
PMT
= 0.37
photoelectrons [35]. We reconstruct the quenching factor using the recoil energy contour
lines in figure 4 of ref. [20], assigning to each recoil energy the S1 value at the NR mean line.
This procedure is approximately correct where the recoil energy contours in the figure are
approximately vertical, i.e. for the low S1 values on which our light WIMP limits depend the
most. Even for the largest WIMP masses we reach in figure 2, for which the recoil energies
– 3 –