UH-511-1324-22

Angular cuts to reduce the $\tau\bar{\tau}j$ background to the higgsino signal
at the LHC
Howard Baer1222Email<EMAIL_ADDRESS>, Vernon Barger2333Email:
<EMAIL_ADDRESS>Dibyashree Sengupta3444Email<EMAIL_ADDRESS>and Xerxes Tata4555Email<EMAIL_ADDRESS>
1Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, OK 73019, USA
2Department of Physics, University of Wisconsin, Madison, WI 53706 USA
3Department of Physics, National Taiwan University, Taipei, Taiwan 10617,
R.O.C.
4Department of Physics and Astronomy, University of Hawaii, Honolulu, HI, USA

We re-examine higgsino pair production in association with a hard QCD jet at
the LHC. We focus on $\ell^{+}\ell^{-}+\not\\!\\!\\!{E_{T}}+j$ events from the
production and subsequent decay,
$\tilde{\chi}_{2}^{0}\rightarrow\tilde{\chi}_{1}^{0}\ell^{+}\ell^{-}$, of the
heavier neutral higgsino. The novel feature of our analysis is that we propose
angular cuts to reduce the irreducible background from
$Z(\rightarrow\tau\bar{\tau})+j$ events more efficiently than the
$m_{\tau\tau}^{2}<0$ cut that has been used by the ATLAS and CMS
collaborations. Additional cuts, needed to reduce backgrounds from $t\bar{t}$,
$WWj$ and $W/Z+\ell\bar{\ell}$ production, are also delineated. We evaluate
the reach of LHC14 for 300 and 3000 fb-1 and stress that the dilepton mass
distribution would serve to characterize the higgsino signal.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

Executive Summary

The ATLAS and CMS collaborations have been searching for light higgsinos with
a compressed spectrum – whose existence is, perhaps, the most robust
prediction of natural SUSY – via monojet events with an additional soft
$e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair, coming mainly from the decay
$\chi_{2}^{0}\rightarrow\chi_{1}^{0}\ell\bar{\ell}$.
$Z(\rightarrow\tau\bar{\tau})+j$ production (where both taus decay
leptonically) is an important irreducible SM background to the higgsino
signal. This background can be considerably reduced by requiring that
$m_{\tau\tau}^{2}<0$, where $m_{\tau\tau}^{2}$ is constructed in the
approximation (valid for relativistic taus from $Z$ decays) that the tau decay
products are collinear with the parent tau direction. We have devised angular
cuts that reduce the tau pair plus jet background much more efficiently than
this di-tau mass cut.

Our results are exhibited in the Table below for three SUSY benchmark points,
BM1, BM2 and BM3 with a higgsino masses around 150, 200 and 300 GeV and
$\Delta m\sim 12,16$ and 4.3 GeV, respectively. Also shown are the important
SM backgrounds. We see that while the di-tau mass cut reduces the tau
background by about a factor of 4 (row 2), the angle cuts reduce this by a
factor of 50 (row 3) with a relatively small loss in signal efficiency. The
last line shows the signal and the background after additional cuts necessary
to reduce the other backgrounds.

cuts/process | $BM1$ | $BM2$ | $BM3$ | $\tau\bar{\tau}j$ | $t\bar{t}$ | $WWj$ | $W\ell\bar{\ell}j$ | $Z\ell\bar{\ell}j$
---|---|---|---|---|---|---|---|---
Basic cuts | 1.2 | 0.19 | 0.07 | 94.2 | 179 | 35.9 | 14.7 | 5.9
Basic+$m_{\tau\tau}^{2}<0$ | 0.92 | 0.13 | 0.043 | 23.1 | 75.6 | 12.8 | 7.7 | 3.2
Basic+angle | 0.68 | 0.12 | 0.04 | 1.8 | 130 | 22 | 11.0 | 4.9
Additional cuts | 0.25 | 0.032 | 0.017 | 0.088 | 0.29 | 0.39 | 0.15 | 0.07
Table: Cross sections (in $fb$) for signal benchmark points and the various SM
backgrounds.

Our projected reach for the signal at the LHC with 300 fb-1 and for 3000 fb-1
is shown by dashed lines in the figure, and compared with corresponding
projections for the HL-LHC obtained by the ATLAS and CMS collaborations.

Figure: Our projections for the LHC reach (dashed lines) along with the
current 95% CL exclusion (ATLAS) and the projected 95% CL exclusions from two
different analyses for the HL-LHC.

##  1 Introduction

It is generally expected that higgsinos cannot be much heavier than the
electroweak scale if weak scale supersymmetry is the new physics that
stabilizes the Higgs sector [1]. The precise upper limit on the higgsino mass
depends on the degree of fine-tuning that is considered acceptable: for no
worse than a part in thirty electro-weak fine-tuning ($\Delta_{\rm EW}<30$)
[2], higgsinos are expected to be lighter than $\sim 350$ GeV, while all other
super-partners could be in the multi-TeV range [2, 3]. For this reason, the
search for light higgsinos has become an important part of the experimental
SUSY program at the LHC [4, 5].

In the generic situation of natural SUSY models where electroweak gauginos are
much heavier than the higgsinos (gauginos with multi-TeV masses do not
destabilize the Higgs vacuum-expectation-value) the mass splitting between the
two lightest neutral higgsinos is very small– $\sim$4-25 GeV– so that the
visible products of higgsino decays to the (higgsino-like) LSP111 We assume
conservation of $R$-parity and also note that the lightest, thermally produced
higgsino comprises only a portion of the observed relic cold dark matter
abundance. are correspondingly soft. As a result, although higgsino pair
production cross sections at LHC14 are substantial ($\sim 100-1000~{}fb$ for
higgsino masses in the 120-350 GeV range), the visible decay products from
higgsino pair production may be hidden under enormous backgrounds from
Standard Model (SM) processes[6]. Because the visible decay products of the
higgsinos are– for practical purposes– invisible, this led to the suggestion
that it might be possible to search for higgsino pairs produced in association
with a hard object $X$, where $X$ is a mono-jet[7], mono-photon[8], mono-gauge
boson[9] or mono-Higgs boson[10]. While it is possible to gain a statistical
significance of the signal exceeding $5\sigma$ by requiring a hard enough cut
on the mono-jet, the signal to background ratio is at the $\sim 1\%$ level, so
that this strategy is viable only if the backgrounds can be controlled to
percent level or below precision.

An alternative strategy is to detect the soft leptons produced via higgsino
decays in higgsino pair events triggered by a hard monojet (or the $\
\not\\!\\!\\!{E_{T}}$) to further reduce the SM background [11, 12, 13, 14].
Dileptons from $\widetilde{\chi}^{0}_{2}$ decays would necessarily be of
opposite sign (OS) and the same flavour (SF), and further, their invariant
mass is kinematically bounded above by $\Delta m\equiv
m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$. The higgsino mono-
jet plus soft dilepton signal has been studied by several groups, and has been
used by both the ATLAS [4] and CMS [5] collaborations for their searches for
higgsinos with a compressed spectrum. The main purpose of this note is to
point out an analysis strategy involving angular cuts that is more effective
than the currently used method that uses the di-tau invariant mass to beat
down the important SM background to the higgsino signal arising from tau pair
production in association with a hard jet (the leptons come from the decays of
the taus). We also delineate other cuts that reduce the other backgrounds, and
map out the reach of the HL-LHC in the $m_{\widetilde{\chi}^{0}_{2}}$ vs.
$\Delta m$ mass plane. This Snowmass 2021 contribution summarizes and
streamlines the results of Ref. [15] with an emphasis on the angular cuts to
reduce the important irreducible background from $Z\rightarrow\tau\bar{\tau}$
events.

##  2 Higgsino Signal and SM background processes

We focus on the signal from higgsino pair production in association with a QCD
jet at the LHC: $pp\rightarrow\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-},\
\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{2},\
\widetilde{\chi}^{\pm}_{1}\widetilde{\chi}^{0}_{2}+j$, where
$\widetilde{\chi}^{\pm}_{1}\rightarrow\ell\nu\widetilde{\chi}^{0}_{1}$ and
$\widetilde{\chi}^{0}_{2}\rightarrow\ell\bar{\ell}\widetilde{\chi}^{0}_{1}$
($\ell=e,\mu$). Requiring a hard jet boosts the lepton daughters of the
higgsino in addition to providing a trigger for the events. We evaluate SM
backgrounds to the monojet plus OS/SF dilepton higgsino signal from:

  * •

$\tau\bar{\tau}j$ production,

  * •

$t\bar{t}$ production,

  * •

$WWj$ production,

  * •

$W\ell\bar{\ell}j$ production, and

  * •

$Z\ell\bar{\ell}j$ production

in the SM.

For our calculations, we use MadGraph 2.5.5 [16] interfaced to PYTHIA v8 [17]
via the default MadGraph/PYTHIA interface with default parameters for
showering and hadronization to generate $pp$ collision events with
$\sqrt{s}=14$ TeV. Detector simulation is performed by Delphes using the
default Delphes 3.4.2 [18] “ATLAS” parameter card. We adopt the anti-$k_{T}$
jet algorithm [19] with $R=0.6$, the default value in the ATLAS Delphes card,
to form jets. Jet finding in Delphes is implemented via FastJet [20]. We
consider only jets with transverse energy satisfying $E_{T}(jet)>40$ GeV and
pseudorapidity satisfying $|\eta(jet)|<3.0$ in our analysis. We implement the
default Delphes $b$-jet tagger and implement a $b$-tag efficiency of 85% [21].
We identify leptons with $E_{T}>5$ GeV and within $|\eta(\ell)|<2.5$. We label
them as isolated leptons if the sum of the transverse energy of all other
objects (tracks, calorimeter towers, etc.) within $\Delta R=0.5$ of the lepton
candidate is less than $10\%$ of the lepton $E_{T}$.

In order to assess the level of the SUSY signal relative to the SM
backgrounds, we have selected three benchmark points with varying higgsino
masses and varying mass gaps between the heavier higgsinos and the LSP: the
higgsino production rate obviously depends on the higgsino mass, and the
lepton detection efficiency is very sensitive to the higgsino mass gap,
$\Delta m$. We use the computer code Isajet 7.88[22] to generate sparticle
masses. The resulting SUSY Les Houches Accord (SLHA) files are then used as
inputs to Madgraph/Pythia/Delphes for event generation.

The first two benchmark points, BM1 and BM2, are obtained from the two-extra-
parameter non-universal Higgs model (NUHM2) with parameters $m_{0},\ m_{1/2},\
A_{0},\ \tan\beta,\ \mu,\ m_{A}$ and have higgsino masses around 150 GeV and
300 GeV, respectively, and corresponding mass gaps $\Delta m=12$ GeV and 16
GeV. Although not critical for a discussion of the phenomenology of the
signal, we note that both these benchmark points satisfy our naturalness
criterion, $\Delta_{\rm EW}<30$. The third point, labeled as BM3 (GMM′) has
higgsino masses around 200 GeV and a small $\Delta m\simeq 4.3$ GeV. Since
NUHM2 models with $\Delta_{\rm EW}<30$ necessarily have $\Delta m\gtrsim 10$
GeV, we have used the natural generalized mirage mediation model[23], where
$\mu$ is an input, to generate a benchmark point with this small mass gap;
this, of course, makes the signal search a challenge even though the higgsinos
are not particularly heavy. The input parameters and SUSY spectra from the
benchmark points are listed in Table 1, along with some low energy and dark-
matter-related observables along with the degree of electroweak fine-tuning,
which we view as a conservative measure of naturalness.

parameter | $BM1$ | $BM2$ | $BM3\ (GMM^{\prime})$
---|---|---|---
$m_{0}$ | 5000 | 5000 | $\textendash$
$m_{1/2}$ | 1001 | 1000 | $\textendash$
$A_{0}$ | -8000 | -8000 | $\textendash$
$\tan\beta$ | 10 | 10 | 10
$m_{3/2}$ | $\textendash$ | $\textendash$ | 75000
$\alpha$ | $\textendash$ | $\textendash$ | 4
$c_{m}$ | $\textendash$ | $\textendash$ | 6.9
$c_{m3}$ | $\textendash$ | $\textendash$ | 6.9
$a_{3}$ | $\textendash$ | $\textendash$ | 5.1
$\mu$ | 150 | 300 | 200
$m_{A}$ | 2000 | 2000 | 2000
$m_{\tilde{g}}$ | 2425.4 | 2422.6 | 2837.3
$m_{\tilde{u}_{L}}$ | 5295.9 | 5295.1 | 5244.6
$m_{\tilde{u}_{R}}$ | 5427.8 | 5426.5 | 5378.0
$m_{\tilde{e}_{R}}$ | 4823.7 | 4824.5 | 4813.2
$m_{\tilde{t}_{1}}$ | 1571.7 | 1578.4 | 1386.9
$m_{\tilde{t}_{2}}$ | 3772.0 | 3773.0 | 3716.7
$m_{\tilde{b}_{1}}$ | 3806.7 | 3807.6 | 3757.8
$m_{\tilde{b}_{2}}$ | 5161.2 | 5160.2 | 5107.7
$m_{\tilde{\tau}_{1}}$ | 4746.8 | 4747.5 | 4729.8
$m_{\tilde{\tau}_{2}}$ | 5088.6 | 5088.2 | 5075.7
$m_{\tilde{\nu}_{\tau}}$ | 5095.4 | 5095.0 | 5084.8
$m_{\widetilde{\chi}^{\pm}_{2}}$ | -857.1 | -857.6 | -1801.9
$m_{\widetilde{\chi}^{\pm}_{1}}$ | -156.6 | -311.6 | -211.1
$m_{\widetilde{\chi}^{0}_{4}}$ | -869.0 | -869.8 | -1809.3
$m_{\widetilde{\chi}^{0}_{3}}$ | -451.3 | -454.7 | -1554.4
$m_{\widetilde{\chi}^{0}_{2}}$ | 157.6 | 310.1 | 207.0
$m_{\widetilde{\chi}^{0}_{1}}$ | -145.4 | -293.7 | -202.7
$m_{h}$ | 124.5 | 124.6 | 125.4
$\Omega_{\widetilde{\chi}^{0}_{1}}^{std}h^{2}$ | 0.007 | 0.023 | 0.009
$BF(b\rightarrow s\gamma)\times 10^{4}$ | 3.1 | 3.1 | 3.1
$BF(B_{s}\rightarrow\mu^{+}\mu^{-})\times 10^{9}$ | 3.8 | 3.8 | 3.8
$\Delta_{\rm EW}$ | 13.9 | 21.7 | 26.0
Table 1: Input parameters and sparticle masses in GeV units for the two NUHM2
model benchmark points (BM1 and BM2) and one natural mirage mediation SUSY
benchmark point (BM3 (GMM’)) introduced in the text. We take $m_{t}=173.2$
GeV. The input parameters for the natural(generalized) mirage mediation model
such as $\alpha$ and $c_{m}$ have been calculated from $m_{0}^{MM}$ and
$m_{1/2}^{MM}$ which are taken equal to the corresponding NUHM2 model values
of $m_{0}$ and $m_{1/2}$, respectively. The $c_{m}$ and $c_{m3}$ have been
taken equal to each other so that masses of first/second and third generation
sfermions are equal at the GUT scale so as to also match the NUHM2 models in
the second and third columns of the table.

##  3 Higgsino Signal Analysis

For the SUSY signal from higgsinos, we generate events from the reactions
$pp\rightarrow\widetilde{\chi}_{1}^{\pm}\widetilde{\chi}^{0}_{2}j$,
$\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{2}j$ and
$\widetilde{\chi}_{1}^{+}\widetilde{\chi}_{1}^{-}j$, where
$\widetilde{\chi}^{0}_{2}\rightarrow\widetilde{\chi}^{0}_{1}\ell^{+}\ell^{-}$
and $\widetilde{\chi}^{\pm}_{1}\rightarrow\ell\nu\widetilde{\chi}^{0}_{1}$.
The dilepton plus jet signal together with the cross sections from the various
backgrounds listed in Sec. 2, after a series of cuts described below, is shown
in Table 2. The first entry labeled $BC$ (for before cuts) actually has parton
level cuts implemented at the Madgraph level. These cuts serve to regularaize
subprocesses that are otherwise divergent. Also, for the backgrounds with a
hard QCD ISR (labeled as $j$ in the table header), we require $p_{T}(j)>80$
GeV to efficiently generate events with a hard jet. For the backgrounds
including $\gamma^{*},Z^{*}\rightarrow\ell\bar{\ell}$ ($\ell=e$ or $\mu$), we
require $m(\ell\bar{\ell})>1$ GeV. We also require $p_{T}(\ell)>1$ GeV and
$\Delta R(\ell\bar{\ell})>0.01$, again at the parton level. The $W$ daughters
of top quarks in $t\bar{t}$ events are forced to decay leptonically (into $e$,
$\mu$ or $\tau$), but not so the $W$-bosons in first entry of the $WWj$
column. These parton events are then fed into PYTHIA and analysed using the
DELPHES detector simulation. The leading order cross sections (in $fb$), for
both the signal as well as for the backgrounds, are listed in row 1, labelled
$BC$. At this stage, the signal is dwarfed by SM backgrounds.

To select out signal events, we require:

  * •

two opposite sign, same flavour (OS/SF) isolated leptons with $p_{T}(\ell)>5$
GeV, $|\eta(\ell)|<2.5$,

  * •

there be at least one jet in the event, i.e., $n_{j}\geq 1$ with
$p_{T}(j_{1})>100$ GeV for identified calorimeter jets,

  * •

$\Delta R(\ell\bar{\ell})>0.05$ (for $\ell=e$ or $\mu$),

  * •

$\not\\!\\!\\!{E_{T}}>100$ GeV, and

  * •

a veto of tagged $b$-jets, $n$($b$-jet)=0.

The cross sections after this set of cuts, labeled as C1 are shown in the next
row of Table 2.

cuts/process | $BM1$ | $BM2$ | $BM3(GMM^{\prime})$ | $\tau\bar{\tau}j$ | $t\bar{t}$ | $WWj$ | $W\ell\bar{\ell}j$ | $Z\ell\bar{\ell}j$
---|---|---|---|---|---|---|---|---
$BC$ | 83.1 | 9.3 | 31.3 | 43800.0 | 41400 | 9860 | 1150.0 | 311
$C1$ | 1.2 | 0.19 | 0.07 | 94.2 | 179 | 35.9 | 14.7 | 5.9
$C1+m_{\tau\tau}^{2}<0$ | 0.92 | 0.13 | 0.043 | 23.1 | 75.6 | 12.8 | 7.7 | 3.2
$C1+angle$ | 0.68 | 0.12 | 0.04 | 1.8 | 130 | 22 | 11.0 | 4.9
$C2$ | 0.29 | 0.049 | 0.019 | 0.088 | 0.99 | 0.49 | 0.18 | 0.14
$C3$ | 0.25 | 0.032 | 0.017 | 0.088 | 0.29 | 0.39 | 0.15 | 0.07
Table 2: Cross sections (in $fb$) for signal benchmark points and the various
SM backgrounds listed in the text after various cuts. The row labelled BC
denotes parton level cross sections after the requirement $p_{T}(j)>80$ GeV,
along with minimal cuts implemented to regulate divergences, and also includes
the leptonic branching fractions for decays of both the top quarks in the
$t\bar{t}$ column. The remaining rows list the cross sections after a series
of analysis cuts detailed in the text.

At this stage, the leading irreducible physics background from
$\tau\bar{\tau}j$ events is two orders of magnitude larger than the signal,
while the background from top pair production (which nominally contains two
untagged $b$-jets) is a factor of two larger than this.

###  3.1 Reducing the di-tau plus jet background

####  3.1.1 $m_{\tau\tau}^{2}$ cut

After C1 cuts, the decay $Z\rightarrow\tau\bar{\tau}$ of an on-shell high
$p_{T}$ $Z$ boson recoiling against a hard QCD jet is the leading source of
the $\tau\bar{\tau}j$ events. This means that this background can be greatly
reduced if it would be possible to reconstruct the di-tau invariant mass
despite the presence of the neutrinos that are present when the taus decay
leptonically. This is possible because the taus from $Z$-boson decays are
typically very relativistic. In the approximation that the leptons and
neutrinos from the decay of a relativistic tau are all exactly collimated
along the parent tau direction, we can write the momentum carried off by the
two neutrinos from the decay
$\tau_{1}\rightarrow\ell_{1}\bar{\nu}_{\ell_{1}}\nu_{\tau_{1}}$ of the first
tau as $\xi_{1}\vec{p}(\ell_{1})$ and, similarly, as
$\xi_{2}\vec{p}(\ell_{2})$ for the second tau. Momentum conservation in the
plane transverse to the beams then requires that

$-\sum_{jets}\vec{p}_{T}(j)=(1+\xi_{1})\vec{p}_{T}(\ell_{1})+(1+\xi_{2})\vec{p}_{T}(\ell_{2}).$
(1)

These two equations can be solved for $\xi_{1}$ and $\xi_{2}$ given that
$\vec{p}_{T}(j)$ and $\vec{p}_{T}(\ell_{1,2})$ are all measured, and used to
evaluate the momenta of the individual taus. This then allows us to evaluate
the invariant mass squared of the di-tau system which (within the collinear
approximation for tau decays) is given by,

$m_{\tau\tau}^{2}=(1+\xi_{1})(1+\xi_{2})m_{\ell\ell}^{2}.$ (2)

We show the distribution of $m_{\tau\tau}^{2}$ constructed using Eq. (2) for
both signal events as well as for the various backgrounds in Fig. 1 after the
cut set C1 and also requiring $n_{j}=1$ to further reduce the top background.
As expected, this peaks sharply around $m_{Z}^{2}$ for the $\tau\bar{\tau}j$
events (red histogram). In contrast, for signal and other SM background
sources, where the isolated lepton and $\vec{\not\\!\\!\\!{E_{T}}}$ directions
are uncorrelated, the $m_{\tau\tau}^{2}$ distributions are very broad and peak
at even negative values of $m_{\tau\tau}^{2}$. The $m_{\tau\tau}^{2}$ variable
provides a very good discriminator between $\tau\bar{\tau}j$ background and
signal. The cut $m_{\tau\tau}^{2}<0$ [13] has, in fact, been used in ATLAS [4]
and CMS [5] for their analyses. We see from the third row of Table 2 that the
$\tau\bar{\tau}j$ background is reduced by a factor $\sim 4$ while the signal
is reduced by 25-40%. The efficiency with which the di-tau background is
reduced is limited because the tail of the $\tau\bar{\tau}j$ background
extends out to negative values of $m_{\tau\tau}^{2}$; this happens because of
tau pair production from virtual photons, the breakdown of the collinear
approximation for asymmetric $Z$ decays and finally hadronic energy
mismeasurements which skew the direction of both $\vec{p}_{T}(j)$ and of $\
\vec{\not\\!\\!\\!{E_{T}}}$.

Figure 1: Distribution in $m_{\tau\tau}^{2}$ for the three SUSY BM models with
$\mu=150,\ 200$ and 300 GeV introduced in the text, along with SM backgrounds
after $C1$ cuts augmented by $n_{j}=1$.

In Sec.3.1.2, we describe angle cuts which can reduce the di-tau background
very efficiently with only moderate loss of the higgsino signal, and suggest
that these replace the $m_{\tau\tau}^{2}<0$ cut that has been used by ATLAS
and CMS for their compressed higgsino search.

####  3.1.2  Angle cuts

In the transverse plane, the di-tau pair must recoil against the hard QCD
radiation with the transverse plane opening angle between the taus
significantly smaller than $\pi$. The central idea behind our proposal is
that, in the transverse plane, the $\not\\!\\!\\!{E_{T}}$ vector must lie
between the directions of the two taus. For relativistic taus, the tau
direction is, of course, essentially the same as the observable direction of
its charged lepton daughter as illustrated in Fig. 2. We require the azimuthal
angles $\phi_{\ell}$ and $\phi_{\bar{\ell}}$ for each lepton to lie between
$0$ and $2\pi$, and define $\phi_{max}=max(\phi_{\ell},\phi_{\bar{\ell}})$ and
$\phi_{min}=min(\phi_{\ell},\phi_{\bar{\ell}})$. Then for
$\vec{\not\\!\\!\\!{E_{T}}}$ to lie in between the tau daughter lepton
directions we must have, 222This works as long as
$|\phi_{\ell}-\phi_{\bar{\ell}}|<\pi$. If
$|\phi_{\ell}-\phi_{\bar{\ell}}|>\pi$, define
$\phi_{\ell}^{\prime}=\phi_{\ell}+\pi$,
$\phi_{\bar{\ell}}^{\prime}=\phi_{\bar{\ell}}+\pi$ and $\phi_{\hskip
2.84526pt\not\\!\\!\\!{E_{T}}}^{\prime}=\phi_{\hskip
2.84526pt\not\\!\\!\\!{E_{T}}}+\pi$, (all modulo $2\pi$) along with
$\phi_{max}=max(\phi_{\ell}^{\prime},\phi_{\bar{\ell}}^{\prime})$, and
likewise, $\phi_{min}=min(\phi_{\ell}^{\prime},\phi_{\bar{\ell}}^{\prime})$,
and then require, $\phi_{min}<\phi_{\hskip
2.84526pt\not\\!\\!\\!{E_{T}}}<\phi_{max}$.

$\phi_{min}<\phi_{\hskip 2.84526pt\not\\!\\!\\!{E_{T}}}<\phi_{max}.$

Notice that, by definition, $\phi_{max}-\phi_{min}<\pi$, and for a boosted tau
pair, often significantly smaller than $\pi$.

Figure 2: Sketch of a ditau background event to the di-lepton plus jet plus
$\not\\!\\!\\!{E_{T}}$ signature in the transverse plane of the event. Here
$\ell_{1}$ and $\not\\!\\!\\!{E_{T}}_{1}$ denote the transverse momentum of
the lepton and of the vector sum of the neutrinos from the decay of the first
tau, and likewise $\ell_{2}$ and $\not\\!\\!\\!{E_{T}}_{2}$.
$\not\\!\\!\\!{E_{T}}$(tot) is the resultant $\not\\!\\!\\!{E_{T}}$ in the
event. Notice that because the taus are expected to be relativistic,
$\ell_{i}$ and $\not\\!\\!\\!{E_{T}}_{i}$ vectors are nearly collimated along
the direction of the $i^{th}$ tau ($i=1,2$).

To characterize the $Z(\rightarrow\tau\bar{\tau})+j$ background, we show in
Fig. 3 a scatter plot of these events in the
$\phi_{1}\equiv\phi_{max}-\phi_{\hskip 2.84526pt\not\\!\\!\\!{E_{T}}}$ vs.
$\phi_{2}\equiv\phi_{\hskip 2.84526pt\not\\!\\!\\!{E_{T}}}-\phi_{min}$ plane.
If the collinear approximation for tau decays holds, we would expect that the
$\tau\bar{\tau}j$ background selectively populates the top right quadrant with
$\phi_{1}>0$ and $\phi_{2}>0$ with
$\phi_{1}+\phi_{2}=\phi_{max}-\phi_{min}<\pi$, and significantly smaller than
$\pi$ when the tau pair emerges with a small opening angle in the transverse
plane. We see from the figure that there is a small, but significant, “spill-
over” into the region where $\phi_{1}$ or $\phi_{2}$ assumes small negative
values; i.e. where $\vec{\not\\!\\!\\!{E_{T}}}$ lies just outside the cone
formed by $\vec{\ell_{1}}$ and $\vec{\ell_{2}}$. This spill-over arises from
asymmetric decays of the $Z$ where one of the taus (the one emitted backwards
from the $Z$ direction) is not as relativistic so that the collinear
approximation works rather poorly, or because hadronic energy mismeasurements
skew the direction of $\vec{\not\\!\\!\\!{E_{T}}}$. Indeed we see from the top
frame of Fig. 3 that the $\tau\bar{\tau}j$ background mostly populates the
triangle in the top-right quadrant of the $\phi_{1}$ vs. $\phi_{2}$ plane, and
$\phi_{1}+\phi_{2}<f\pi$ where the fraction $0<f<1$, with a spill-over into
the strips where one of $\phi_{1,2}$ is slightly negative. For signal events
and for the other backgrounds, $\phi_{\hskip 2.84526pt\not\\!\\!\\!{E_{T}}}$
will be uncorrelated with $\phi_{min}$ and $\phi_{max}$, and so their scatter
plots will extend to the other quadrants. This is illustrated for the
$t\bar{t}$ background in the middle frame of Fig. 3 and for signal point BM1
in bottom frame. In these cases, we indeed see a wide spread in $\phi_{1}$ and
$\phi_{2}$ between $\pm 2\pi$.

Figure 3: Scatter plots in the $\phi_{1}$ vs. $\phi_{2}$ plane for
$\tau\bar{\tau}j$ and $t\bar{t}$ backgrounds and for the signal point BM1
after $C1$ cuts, requiring also that $n_{j}=1$. Also shown are the lines where
$\phi_{1}+\phi_{2}<\pi/2,\pi$ along with the strip cuts in Eq. (4). In the
strips in the top left and bottom right of each frame and along the positive
$\phi_{1}$ and $\phi_{2}$ axes, $\ \vec{\not\\!\\!\\!{E_{T}}}$ is close to the
boundary of the lepton cone.

To efficiently veto the $\tau\bar{\tau}j$ background, we examined nine cases
of angular cuts [15]. Specifically, we allowed $f=1,2/3$ and 1/2 to optimize
the effect of the boost of the di-tau on the transverse plane opening angle,
and also allowed for the width of the spill-over strip (to negative values of
$\phi_{1}$ or $\phi_{2}$) equal to 0, $\pm\pi/20$ and $\pm\pi/10$. We found
that $S/\sqrt{BG(\tau\bar{\tau}j)}$ was optimized for LHC14 with 3000 fb-1
when we,

${\rm veto\ the\ triangle}\ \ \ \phi_{1},\ \phi_{2}>0\ \ {\rm with}\ \
\phi_{1}+\phi_{2}<\pi/2,$ (3)

along with an additional veto to further reduce background from the spill-over
of $\ \vec{\not\\!\\!\\!{E_{T}}}$ outside of the cone defined by the charged
leptons:

${\rm strip\ cuts:\ veto}\ |\phi_{1,2}|<\pi/10,\ {\rm and}\ {\rm veto}\
\phi_{1,2}\ {\rm in\ the\ two\ corner\ strips.}$ (4)

In the horizontal (vertical) strip in the upper left (lower right) quadrant of
each frame of Fig. 3, $\phi_{2}$ ($\phi_{1}$) is within $\pi/10$ of $2\pi$
which, of course, is the same $-\pi/10<\phi_{1,2}<0$ (modulo $2\pi$), and
hence just outside the lepton cone.

We list signal and background rates after C1 cuts together with the cuts (3)
and (4) – collectively referred to as angle cuts from this point on – in row 4
of Table 2. In this case, we find that $\tau\bar{\tau}j$ background is reduced
from cut set C1 by a factor $\sim 52$ (compared to a factor $\sim 4$ for the
$m_{\tau\tau}^{2}<0$ cut) whilst signal efficiency for the point BM1 is almost
60% (compared to $\sim 75$% for the $m_{\tau\tau}^{2}<0$ cut). We also see
that signal efficiency for the other two benchmark points is nearly the same
for the angular and for the $m_{\tau\tau}^{2}<0$ cuts. We regard the angular
cuts as a significantly improved method for almost completely removing the
irreducible $\tau\bar{\tau}j$ background. The other SM backgrounds are not as
efficiently reduced by the angular cut as by the $m_{\tau\tau}^{2}<0$ cut, and
need other cuts to reduce them to manageable levels.

###  3.2 Further cuts to remove top and $W$-pair backgrounds

We have just seen that while the angle cuts greatly reduce the irreducible
$\tau\bar{\tau}j$ background, SM backgrounds from $t\bar{t}$ and $WWj$ events
(followed by leptonic decays of the top and $W$-bosons) and also from
$W/Z\ell\bar{\ell}+j$ production still completely dwarf the signal. Since the
top pair background typically has a higher jet multiplicity, requiring
$n_{j}=1$ increases the signal-to-background ratio (as already mentioned
earlier). The hard jet and $\not\\!\\!\\!{E_{T}}$ distributions are both
backed up against their cut value, and given the already small signal cross
section, it is not helpful to require a harder cut on these variables.

In contrast, the transverse momenta of the leptons from on-shell top and $W$
decays then have broader distributions than the signal leptons. We found that
requiring upper limits on $|p_{T}(\ell_{2})|$ and
$H_{T}(\ell\bar{\ell})\equiv|p_{T}(\ell_{1})|+|p_{T}(\ell_{2})|$ indeed
enhances the signal over the background [15]. The transverse momenta of
leptons and neutrinos from $W$ and top decays have comparable magnitudes; in
contrast the momentum scale for signal leptons is set by $\Delta m$ while that
for signal $\ \not\\!\\!\\!{E_{T}}$ is set by the higgsino mass. As a result,
the ratio $\ \not\\!\\!\\!{E_{T}}/H_{T}(\ell\bar{\ell})$ is expected to be
considerably harder for the higgsino signal than for the SM background as
first noted by the ATLAS collaboration [24]. Finally, we note that the
dilepton mass distribution for leptons from
$\widetilde{\chi}^{0}_{2}\rightarrow\widetilde{\chi}^{0}_{1}\ell\bar{\ell}$
decays is kinematically bounded by
$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}\lesssim
20-25$ GeV for the compressed higgsino search, and further, that the invariant
mass of dileptons from $\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}$ production
events will also tend to be smaller than for background events, simply because
leptons from the chargino decays are soft. Again, we refer the interested
reader to Ref.[15] where these distributions and others are explicitly shown.

These considerations lead us to include the analysis cut set C2 to enhance the
higgsino signal over the top and $WWj$ backgrounds that dominate after the
angle cuts:

  * •

the cut set C1 together with the ${\rm angle\ cuts}$,

  * •

$n(jets)=1$,

  * •

$p_{T}(\ell_{2}):5-15$ GeV,

  * •

$H_{T}(\ell\bar{\ell})<60$ GeV

  * •

$\not\\!\\!\\!{E_{T}}/H_{T}(\ell\bar{\ell})>4$, and

  * •

$m(\ell\bar{\ell})<50$ GeV.

We have checked that the requirement $m(\ell\bar{\ell})<50$ GeV, efficiently
reduces much of the background while retaining most of the higgsino signal.

We see from the penultimate row of Table 2 that after C2 cuts, the leading
$t\bar{t}$ background has dropped by a factor $\sim 130$, and the total SM
background has dropped to $\sim 1.1$%, while the signal is retained with an
efficiency of 40-60%. At this point, the total background is just below 2 fb.
Clearly, the signal cross section is small, and the large integrated
luminosities expected at the HL-LHC will be necessary for the detection of the
signal if the higgsino mass is close to its naturalness bound of 300-350 GeV,
or if the higgsino spectrum is maximally compressed to the $4-5$ GeV level,
consistent with electroweak naturalness.

To further enhance the signal relative to (particularly the top) background,
we note that for the signal, where the SUSY particles recoil strongly against
the ISR jet, we expect nearly back-to-back $\vec{p}_{T}(jet)$ and
$\vec{\not\\!\\!\\!{E_{T}}}$ vectors. This correlation is expected to be
somewhat weaker from $t\bar{t}$ background events and also for
$W\ell\bar{\ell}j$ events because these intrinsically contain additional
activity from decay products that do not form jets or identified leptons. The
dilepton-plus-$\not\\!\\!\\!{E_{T}}$ cluster transverse mass
$m_{cT}(\ell\bar{\ell},\not\\!\\!\\!{E_{T}})$ and the relative values of
$E_{T}(jet)$ and $\not\\!\\!\\!{E_{T}}$ (which tend to be more balanced for
the signal than for the backgrounds) serve to give added distinction between
the signal and backgrounds and provide additional discriminators. These
considerations led us to impose the additional cut set C3 that includes:

  * •

all C2 cuts,

  * •

$\Delta\phi(j_{1},\not\\!\\!\\!{E_{T}})>2.0$

  * •

$m_{cT}(\ell\bar{\ell},\not\\!\\!\\!{E_{T}})<100$ GeV

  * •

$p_{T}(j_{1})/\not\\!\\!\\!{E_{T}}<1.5$

  * •

$|p_{T}(j_{1})-\not\\!\\!\\!{E_{T}}|<100$ GeV.

We show the OS/SF dilepton invariant mass after these C3 cuts in Fig. 4. The
total background, shown in gray, is essentially flat, whereas signal-plus-
background is shown by the colored histogram, and correspond to a) BM1 with
$\Delta m=12$ GeV, b) BM2 with $\Delta m=16$ GeV and c) BM3 with $\Delta
m=4.3$ GeV. The idea here is to look for systematic deviations from SM
background predictions in the lowest $m(\ell\bar{\ell})$ bins. Those bins with
a notable excess could determine the kinematic limit
$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$.
By taking only the bins with a notable excess, i.e.
$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$,
then it is possible to compute the cut-and-count excess above expected
background to determine a discovery limit or exclusion bound. The shape of the
distribution of the excess below the
$\widetilde{\chi}^{0}_{2}\rightarrow\widetilde{\chi}^{0}_{1}\ell\bar{\ell}$
end point depends on the relative sign of the lighter neutralino eigenvalues
(these have opposite signs for higginos) and so could serve to check the
consistency of higgsinos as the origin of the signal[25]. Of the three cases
shown, this would be possible at the HL-LHC only for the point BM1, since the
tiny signal to background ratio precludes the possibility of determining the
signal shape in the other two cases. It may be worthwhile to examine whether
sophisticated machine learning methods could lead to a better signal and
background discrimination that allows us to obtain the neutral higgsino mass
gap and also the signal shape, particularly for the two difficult cases in
Fig. 4

Figure 4: Distribution of $m(\ell^{+}\ell^{-})$ for the three SUSY BM models
with $\mu=150,\ 300$ and 200 GeV, and for the SM backgrounds after $C3$ cuts.

##  4 LHC reach for higgsinos with 300-3000 fb-1

The distributions in Fig. 4 suggest that our final analysis cut set C4
include,

  * •

C3 cuts, along with

  * •

$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$.

The location of the mass gap is clearly visible for BM1, but is obscured by
the background for the other two cases. To implement the last
$m(\ell\bar{\ell})$ cut, we suggest examining the cross section with
$m_{\ell\ell}<m_{\ell\ell}^{\rm cut}$ for varying the value of
$m_{\ell\ell}^{\rm cut}$ and looking for a rise in the event rate where events
from
$\widetilde{\chi}^{0}_{2}\rightarrow\widetilde{\chi}^{0}_{1}\ell\bar{\ell}$
would be expected to accumulate. We recognize that with the cut-and-count
technique this may be not be possible for the difficult cases BM2 and BM3, but
in the following, we will optimistically assume that once we have the data,
the region where the higgsino signal is beginning to accumulate will be self-
evident.

(a)

(b)

(c)

(d)

Figure 5: The projected $5\sigma$ reach and 95% CL exclusion of the HL-LHC
with 3000 fb-1 in $\mu$ for four different NUHM2 model lines with a) $\Delta
m=4$ GeV, b) $\Delta m=8$ GeV, c) $\Delta m=12$ GeV and d) $\Delta m=16$ GeV
after $C3$ \+
$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$
cuts.

Using these C4 cuts, then we computed the signal cross section for four model
lines in the NUHM2 model for values of $\mu:100-400$ GeV and with $m_{1/2}$
values adjusted such that the
$m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$ mass gap is fixed
at 4, 8, 12 and 16 GeV. While $\mu$ and $m_{1/2}$ are variable, the values of
$m_{0}=5$ TeV, $A_{0}=-1.6m_{0}$, $\tan\beta=10$ and $m_{A}=2$ TeV are fixed
for all four model lines. In Fig. 5, we show the signal cross section after C4
cuts, along with the $5\sigma$ reach and the 95% CL exclusion for LHC14 with
300 and 3000 fb-1. We see from Fig. 5 that the HL-LHC typically increases the
higgsino reach by about 70-100 GeV (50-60 GeV for the lowest values of $\Delta
m=4$ GeV) as compared to the reach that can be obtained with an integrated
luminosity of 300 fb-1.

Figure 6: The projected $5\sigma$ reach and 95% CL exclusion contours for
LHC14 with 300 and 3000 fb-1 in the $m_{\widetilde{\chi}^{0}_{2}}$ vs. $\Delta
m$ plane after $C4$ cuts. Also shown is the current 95% CL exclusion (ATLAS)
and the projected 95% CL exclusions from two different analyses for the HL-LHC
[26].

In Fig. 6, we translate the results of Fig. 5 into contours in the
$m_{\widetilde{\chi}^{0}_{2}}$ vs. $\Delta m$ plane. The shaded region is
excluded by LEP2 chargino searches. The region left of the contour labelled
ATLAS is currently excluded by LHC searches at the 95%CL. We also show
projections for what future searches at the HL-LHC would probe at the 95%CL
[26]: ATLAS (soft-lepton A) and CMS (soft-lepton B). The various dashed
contours show our projections for the signal reach. Our focus here has been on
higgsino mass gaps $\lesssim 20-25$ GeV, that are generically expected in
natural SUSY models. For larger mass gaps, it may be best to search for
higgsinos via the hard multilepton events, without any requirement of a QCD
jet. The reach projections in Fig. 6 may be compared to theoretical
expectations, both from bottom-up naturalness considerations and from the
string landscape [27].

Before closing this section, we note that we have only considered physics
backgrounds in our analysis. The ATLAS collaboration [4] has, however,
reported that a significant portion of the background comes from fake leptons,
both $e$ and $\mu$. Accounting for these detector-dependent backgrounds
require data driven methods which are beyond the scope of our analysis. We
remark, however, that for any specified value of the fake rate, it should be
possible to make a rough estimate of the impact of the fakes on the reach
contours in Fig. 6 using the cross sections in Fig. 5 since if the fakes
increase the background by a factor $f$, the cross section necessary to
maintain the same significance for the signal would have to increase by
$\sqrt{f}$.

##  5 Summary

Light higgsinos of mass $\sim 100-400$ GeV with a compressed spectrum are the
most robust prediction of natural supersymmetry, and could be the only
directly accessible superpartners at the LHC. Here, we have re-examined the
prospects for a search for soft opposite-sign/same flavor dilepton plus
$\not\\!\\!\\!{E_{T}}$ from higgsino pair production in association with a
hard monojet at the LHC. The dileptons originating from
$\widetilde{\chi}^{0}_{2}\rightarrow\ell\bar{\ell}\widetilde{\chi}^{0}_{1}$
would exhibit a distinctive kinematic edge with
$m(\ell\bar{\ell})<m_{\widetilde{\chi}^{0}_{2}}-m_{\widetilde{\chi}^{0}_{1}}$,
while the monojet and the accompanying $\not\\!\\!\\!{E_{T}}$ serve as event
triggers.

Our emphasis has been on the reduction of the main irreducible background to
the higgsino signal from $\tau\bar{\tau}j$ production in the SM, where the
soft leptons come from the decays of the taus. To this end, we have proposed
angular cuts (see Sec. 3.1.2) which appear to be more efficient in reducing
this background than the $m_{\tau\tau}^{2}$ cut that has been used by the
ATLAS and CMS collaborations in their analysis of the higgsino signal. Of
course, additional cuts are needed to further reduce the reducible background
from $t\bar{t}$ production, and subdominant backgrounds from $WWj$,
$W\ell\bar{\ell}j$ and $Z\ell\bar{\ell}j$ production. Table 2 shows a
comparison between the new angular cuts and the currently used di-tau mass
cut, and also provides a cut flow after various other analysis cuts discussed
in the text.

Our final result is summarized by the dashed contours in Fig. 6. Our analysis
works best for $\Delta m$ values $\sim 15-20$ GeV but drops off for smaller
and much larger mass gaps. We mention that mass gaps smaller than about 4 GeV
occur only for very heavy gauginos that fail to satisfy naturalness
expectations, while higgsinos with an increasingly uncompressed spectrum can
be more effectively searched for via multilepton channels. We see from Fig. 6
that the HL-LHC with 3000 fb-1 gives a $5\sigma$ discovery reach to
$m_{\widetilde{\chi}^{0}_{2}}\sim 240$ GeV, with the 95% CL exclusion limit
extending to $\sim 325$ GeV for $\Delta m\sim 16$ GeV. The signal reach would
be even further enhanced if it becomes possible to extend the lepton
acceptance to lower $p_{T}$ values, or reliably increase $b$-quark rejection
even beyond 80-85% that has already been achieved.

Acknowledgements:

This research is based upon work supported in part by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier
Research Centers program under Award Number DE-SC-0009956 and U.S. Department
of Energy Grant DE-SC-0017647. The work of DS was supported by the Ministry of
Science and Technology (MOST) of Taiwan under Grant No. 110-2811-M-002-574.

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