OU-HEP-230601

Prospects for charged Higgs bosons in natural
SUSY models at the high-luminosity LHC
Howard Baer1,2111Email<EMAIL_ADDRESS>, Vernon Barger2222Email:
<EMAIL_ADDRESS>Xerxes Tata3333Email<EMAIL_ADDRESS>and Kairui
Zhang2333Email<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 and Astronomy, University of Hawaii, Honolulu, HI 53706
USA

We continue our examination of prospects for discovery of heavy Higgs bosons
of natural SUSY (natSUSY) models at the high luminosity LHC (HL-LHC), this
time focussing on charged Higgs bosons. In natSUSY, higgsinos are expected at
the few hundred GeV scale whilst electroweak gauginos inhabit the TeV scale
and the heavy Higgs bosons, $H$, $A$ and $H^{\pm}$ could range up tens of TeV
without jeopardizing naturalness. For TeV-scale heavy SUSY Higgs bosons $H$,
$A$ and $H^{\pm}$, as currently required by LHC searches, SUSY decays into
gaugino plus higgsino can dominate $H^{\pm}$ decays provided these decays are
kinematically accessible. The visible decay products of higgsinos are soft
making them largely invisible, whilst the gauginos decay to $W$, $Z$ or $h$
plus missing transverse energy ($\not\\!\\!\\!{E_{T}}$). Charged Higgs bosons
are dominantly produced at LHC14 via the parton subprocess, $gb\rightarrow
H^{\pm}t$. In this paper, we examine the viability of observing signtures from
$H^{\pm}\rightarrow\tau\nu$, $H^{\pm}\rightarrow tb$ and $H^{\pm}\rightarrow
W,Z,h+\not\\!\\!\\!{E_{T}}$ events produced in association with a top quark at
the HL-LHC over large Standard Model (SM) backgrounds from (mainly)
$t\bar{t}$, $t\bar{t}V$ and $t\bar{t}h$ production (where $V=W,\ Z$). We find
that the greatest reach is found via the SM $H^{\pm}(\rightarrow\tau\nu)+t$
channel with a subdominant contribution from the $H^{\pm}(\rightarrow tb)+t$
channel. Unlike for neutral Higgs searches, the SUSY decay modes appear to be
unimportant for $H^{\pm}$ searches at the HL-LHC. We delineate regions of the
$m_{A}$ vs. $\tan\beta$ plane, mostly around $m_{A}\sim 1-2$ TeV, where
signals from charged Higgs bosons would serve to confirm signals of a heavy,
neutral Higgs boson at the $5\sigma$ level or, alternatively, to exclude heavy
Higgs bosons at the 95% confidence level at the high luminosity LHC.

##  1 Introduction

There are two routes to discovery of supersymmetry (SUSY) at hadron colliders
such as the CERN Large Hadron Collider (LHC): one is via direct pair
production of $R$-parity odd states and the other is via single (or pair)
production of new $R$-parity even states such as the additional heavy Higgs
bosons present in the Minimal Supersymmetric Standard Model (MSSM)[1]. Of
course, strictly speaking, the production of the heavy Higgs boson states is
not necessarily a signal for supersymmetry (unless these are seen via their
decays into supersymmetric particles) since such states are also possible in
non-supersymmetric models with an extended Higgs sector. In the present paper,
we continue our work on the second approach: prospects for SUSY discovery via
the required additional SUSY Higgs bosons. While much work has been done in
this field, our focus is on LHC signals of the heavy Higgs bosons of natural
SUSY models, wherein no large fine-tunings are required in order to gain a
weak scale characterized by $m_{W,Z,h}\simeq 100$ GeV. In previous work, we
examined prospects for SUSY Higgs discovery in natural SUSY via resonance
production of heavy neutral Higgs bosons $H$ and $A$, followed by 1. decays
into Standard Models modes with $H,\ A\rightarrow\tau^{+}\tau^{-}$ being most
promising[2], and 2\. decays into pairs of SUSY particles[3] which offer
qualitatively new channels for SUSY Higgs boson discovery. Within natural
SUSY, once the heavy Higgs boson decay channels to gaugino+higgsino become
open, these may rapidly dominate the branching fractions. This leads to two
effects: 1. the new SUSY decay modes diminish the branching fractions into SM
modes, thus diminishing the expected LHC reach via these decay channels, and
2. the new SUSY decay modes open up new avenues for SUSY Higgs detection,
where these new channels would signal the presence of the expected SUSY
particles. In the present paper, we extend our earlier analyses to include
production of charged SUSY Higgs bosons $H^{\pm}$. Discovery of charged SUSY
Higgs bosons is expected to be more challenging than discovery of the neutral
bosons. This is due to typically smaller production cross sections (for a
given Higgs boson mass) but also to less distinctive discovery signatures. We
investigate here whether this situation still maintains under the rubric of
natural SUSY.

Here, we take the measured value of the $Z$-boson mass as representative of
the magnitude of weak scale, where in the MSSM the $Z$ mass is related to the
weak scale Lagrangian parameters via the electroweak minimization condition

$m_{Z}^{2}/2=\frac{m_{H_{d}}^{2}+\Sigma_{d}^{d}-(m_{H_{u}}^{2}+\Sigma_{u}^{u})\tan^{2}\beta}{\tan^{2}\beta-1}-\mu^{2}$
(1)

where $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are the Higgs soft breaking masses,
$\mu$ is the (SUSY preserving) superpotential higgsino mass parameter and the
$\Sigma_{d}^{d}$ and $\Sigma_{u}^{u}$ terms contain a large assortment of loop
corrections (see Appendices of Ref. [4] and [5] and also [6] for leading two-
loop corrections). We adopt the notion of practical naturalness[7, 8], wherein
the value of an observable ${\cal O}$ is natural if all independent
contributions to ${\cal O}$ are comparable to (within a factor of few), or
smaller than ${\cal O}$. For natural SUSY models, we use the naturalness
measure[9, 4]

$\Delta_{EW}\equiv|{\rm maximal\ term\ on\ the\ right-hand-side\ of\
Eq.~{}(\ref{eq:mzs})}|/(m_{Z}^{2}/2)\;,$ (2)

and take

$\Delta_{EW}\lesssim 30$ (3)

to be natural. For most SUSY benchmark models, the superpotential $\mu$
parameter is tuned to cancel against large contributions to the weak scale
from SUSY breaking. Since the $\mu$ parameter typically arises from very
different physics than SUSY breaking, e.g. from whatever solution to the SUSY
$\mu$ problem that is assumed,111Twenty solutions to the SUSY $\mu$ problem
are recently reviewed in Ref. [10]. then such a “just-so” cancellation is
highly implausible[11] (though logically possible) compared to the case where
all contributions to the weak scale are $\sim m_{weak}$, so that $\mu$ (or
other parameters) need not be tuned.

Several important implications of Eq. (3) for heavy SUSY Higgs searches
include the following.

  * •

The superpotential $\mu$ parameter enters $\Delta_{EW}$ directly, leading to
$|\mu|\lesssim 350$ GeV. This implies that for heavy Higgs searches with
$m_{H^{\pm}}\gtrsim 2|\mu|$, then SUSY decay modes of $H^{\pm}$ should
typically be open. If these additional decay widths to SUSY particles are
large, then the branching fractions to the (usually assumed) SM search modes
would be correspondingly reduced.

  * •

For $|m_{H_{d}}^{2}|\gg|m_{H_{u}}^{2}|$ or $\mu^{2}$, then $|m_{H_{d}}|$ sets
the heavy Higgs mass scale ($m_{A,H,H^{\pm}}\sim|m_{H_{d}}|$) while
$|m_{H_{u}}|$ sets the mass scale for $m_{W,Z,h}$. Then, assuming that
$m_{H^{\pm}}\gg M_{W}$, naturalness requires[12]

$m_{A,H,H^{\pm}}\lesssim m_{Z}\tan\beta\sqrt{\Delta_{EW}}.$ (4)

For $\tan\beta\sim 10$ with $\Delta_{EW}\lesssim 30$, then $m_{H^{\pm}}$ can
range up to $\sim 5$ TeV. For $\tan\beta\sim 40$, then $m_{H^{\pm}}$ stays
natural up to $\sim 20$ TeV (although for large $\tan\beta\gtrsim 20$, then
bottom squark contributions to $\Sigma_{u}^{u}$ become large and provide much
stronger upper limits on natural SUSY spectra[7]).

In Sec. 2, we first present a natural SUSY benchmark point which then leads to
a natural SUSY Higgs scenario which we previously dubbed $m_{H}^{125}({\rm
nat})$. The $m_{h}^{125}({\rm nat})$ scenario is promoted as a template for
SUSY Higgs searches in that[2] 1\. it leads to a value of $m_{h}\simeq 125$
GeV throughout almost the entire $m_{A}$ vs. $\tan\beta$ search plane and 2\.
the value of $\Delta_{EW}$ is also low (though sometimes exceeding a value of
30 at higher $\tan\beta$ values) thoughout the search plane. In Sec. 3, we
list the dominant charged SUSY Higgs boson production cross sections at LHC14
(LHC with $\sqrt{s}=14$ TeV). It is well-known that the $gb\rightarrow
tH^{\pm}$ subprocesses is the dominant $H^{\pm}$ production mechanism at the
LHC[13, 14, 15]. In Sec. 4, we present charged Higgs boson branching fractions
from the $m_{h}^{125}({\rm nat})$ scenario in the $m_{A}$ vs. $\tan\beta$
plane, and find that indeed SUSY decay modes do become rapidly dominant once
these are kinematic accessible. In Sec. 5, we examine $pp\rightarrow
tH^{\pm}+X$ followed by $H^{\pm}\rightarrow\tau^{\pm}\nu_{\tau}$ and map out
distributions which help obtain signal over SM background levels. In Sec. 6,
we examine $pp\rightarrow tH^{\pm}+X$ followed by $H^{\pm}\rightarrow tb$
decay. In Sec. 7, we study the impact of the SUSY decays of $H^{\pm}$ bosons:
unfortunately, we find that the relevant cross-sections are mostly in the sub-
fb range for mass values where the branching fractions for these modes become
substantial, and (contrary to what we found for $H$ and $A$ bosons) SUSY
decays of $H^{\pm}$ do not offer a viable search strategy. In Sec. 8, we plot
out the reach of high-luminosity LHC for charged Higgs bosons of natural SUSY.
Our summary and conclusions are contained in Sec. 9.

###  1.1 A review of some previous related work

Here, we present a brief (and likely incomplete) review of some related work
on charged Higgs bosons from SUSY. In Ref. [16], it was already emphasized
that detection of a top quark signal in accord with SM expectations would
preclude the decay $t\rightarrow bH^{+}$ and thus require $m_{H^{\pm}}\gtrsim
m_{t}-m_{b}$. In light of present $t\bar{t}$ signal results, this implies
$m_{H^{\pm}}\gtrsim 168$ GeV. This result was already used by Kunszt and
Zwirner in Ref. [17] to form the low $m_{A}$ limit of the proposed $m_{A}$ vs.
$\tan\beta$ heavy SUSY Higgs search plane. Decays of heavy SUSY Higgs boson to
SUSY decay modes were originally explored in Ref. [18, 19, 20, 21] and the
complete set of $H^{\pm}$ decay widths may be found in Appendix C of [1]. In
Ref. [12], these decay modes were examined in the context of natural SUSY. In
that work, it was noted that for $H,\ A,\ H^{\pm}\rightarrow wino+higgsino$
channels, the higgsino decays led to mainly soft, quasi-visible decay debris
whilst the winos decayed dominantly via two-body modes into $W+higgsino$,
$Z+higgsino$ and $h+higgsinos$.

In Ref. [13, 14, 15], it was found that the dominant production process for
charged Higgs bosons at LHC was the reaction $gb\rightarrow tH^{-}+c.c.$. NLO
corrections to this production process were calculated in Ref. [22, 23] and
[24]. Signals from the final state $tH^{\pm}(\rightarrow\tau\nu_{\tau})$ were
examined in Ref. [25] and [26]. The decay channel $H^{+}\rightarrow hW^{+}$
(which is highly suppressed in natSUSY) was examined in Ref. [27]. The use of
three[28] and four[29] $b$-quark tags in $gg\rightarrow tbH^{\pm}$ was
examined. Corrections to the $tbH^{\pm}$ vertex were examined in Ref. [30].
Initial projections of the LHC reach for (charged) SUSY Higgs bosons were
given by Denegri et al.[31] and by Assamagan et al.[32]. Search limits from
LHC for the $H^{\pm}\rightarrow\tau\nu_{\tau}$ decay mode were presented based
on $\sim 36$ fb-1 of integrated luminosity by ATLAS[33] and by CMS[34]. An
ATLAS search for charged Higgs bosons in the $H^{\pm}tb$ mode based on 139
fb-1 was given in Ref. [35]. A guidebook for LHC searches for SUSY and non-
SUSY charged Higgs bosons was provided in Ref. [36] and a review of non-SUSY
charged Higgs is available in Ref. [37].

##  2 A natural SUSY benchmark point and the $m_{h}^{125}({\rm nat})$ scenario

Following our previous work, we here adopt the same natural SUSY benchmark
point as in Ref. [3], which was dubbed $m_{h}^{125}({\rm nat})$ since the
value of $m_{h}$ is very close to its measured value throughout the entire
$m_{A}$ vs. $\tan\beta$ plane. We use the two-extra-parameter non-universal
Higgs model (NUHM2)[38] with parameter space $m_{0},\ m_{1/2},\ A_{0},\
\tan\beta,\ \mu,\ m_{A}$ which is convenient for naturalness studies since
$\mu$ can be set to its natural range of $\mu\sim 100-350$ GeV whilst both
$m_{A}$ and $\tan\beta$ are free parameters.222The NUHM2 framework allows for
independent soft SUSY breaking mass parameters for the scalar fields $H_{u}$
and $H_{d}$ in the Higgs sector, but leaves the matter scalar mass parameters
universal to avoid flavour problems. The parameters $m_{H_{u}}^{2}$ and
$m_{H_{d}}^{2}$ are then traded for $\mu$ and $m_{A}$ in the parameter set
shown in (5). We adopt the following natural SUSY benchmark Higgs search
scenario:

$m_{h}^{125}({\rm nat}):\ m_{0}=5\ {\rm TeV},\ m_{1/2}=1\ {\rm TeV},\
A_{0}=-1.6m_{0},\ \tan\beta,\ \mu=200\ {\rm GeV}\ {\rm and}\ m_{A}.$ (5)

A similar $m_{h}^{125}({\rm nat})$ benchmark model spectrum, but with
$\mu=250$ GeV and $m_{1/2}=1.2$ TeV, was shown in Table 1 of Ref. [2] for
$\tan\beta=10$ and $m_{A}=2$ TeV and so for brevity we do not show the revised
spectrum here. We adopt the computer code Isajet[39] featuring Isasugra[40]
for spectrum generation. The SUSY Higgs boson masses are computed using
renormalization-group (RG) improved third generation fermion/sfermion loop
corrections[41]. The RG improved Yukawa couplings include full threshold
corrections[42] which account for leading two-loop effects[43]. For
$\tan\beta=10$ and $m_{A}=2$ TeV, we note that $\Delta_{EW}=16$ so the model
is indeed EW natural. Also, with $m_{h}=124.6$ GeV, $m_{\tilde{g}}=2.4$ TeV
and $m_{\tilde{t}_{1}}=1.6$ TeV, it is consistent with LHC Run 2 SUSY search
constraints. Most important for our purpose, the two lightest neutralinos,
$\tilde{\chi}_{1}^{0}$ and $\tilde{\chi}_{2}^{0}$, and the lighter chargino,
$\tilde{\chi}_{1}^{\pm}$, are higgsino-like with masses $\sim 200$ GeV while
the neutralino $\tilde{\chi}_{3}^{0}$ is bino-like with a mass of 450 GeV and
the heaviest neutralino and the heavier chargino have masses $\sim 0.86$ TeV.
Thus, the $H,\ A,\ H^{\pm}\rightarrow wino+higgsino$ decay modes turn on for
$m_{H,A,H^{\pm}}\gtrsim 1.1$ TeV (although $H,\ A,\ H^{\pm}\rightarrow
bino+higgsino$ turns on at somewhat lower $m_{A}$ values). It is important to
note that while the value of $\Delta_{EW}$ may change somewhat for small
variations of the parameters that are held fixed in Eq. (5), we expect that
the Higgs sector phenomenology is relatively insensitive to our specific
choice (as long as $\mu\lesssim 300$ GeV to maintain naturalness).

##  3 $H^{\pm}$ production cross sections at LHC14

In Fig. 1, we show leading order total cross sections for production of
charged SUSY Higgs bosons at LHC14, as generated using Pythia[44] for two
values of $\tan\beta=10$ (solid) and 40 (dashed). We show cross sections for
charged Higgs boson production at the LHC via $tH^{\pm}$ production (blue),
resonant $H^{\pm}$ production (light green), $H^{+}H^{-}$ production (light
blue), and $hH^{\pm}$ (magenta), $AH^{\pm}$ (green) and $HH^{\pm}$ (red)
production.

We see from Fig. 1 that the dominant production mechanism at LHC14 is via the
$gb\rightarrow tH^{\pm}$ subprocess [13, 14, 15]. For $\tan\beta=40$, the
cross section varies from $\sim 10^{3}$ fb at low $m_{H^{\pm}}$ to $\sim
10^{-2}$ fb at $m_{H^{\pm}}\sim 3$ TeV. These dominant cross sections are
enhanced by large $\tan\beta$. If we compare $tH^{\pm}$ production to the rate
for $b\bar{b},gg\rightarrow A,\ H$ production[2] for $\tan\beta=10$ and
$m_{A}\sim 1.5$ TeV, we find that charged Higgs production rates are
suppressed compared to resonance production of $A$ by a factor $\sim 5$. And
since $\sigma(b\bar{b},gg\rightarrow A)\sim\sigma(b\bar{b},gg\rightarrow H)$,
charged Higgs production compared to resonance $H,\ A$ production is
suppressed by about an order of magnitude. This seems reasonable since charged
Higgs production occurs in association with a (spectator) top quark in
contrast to the resonantly produced neutral $H$ or $A$ boson.

Figure 1: The total cross section for $pp\rightarrow H^{\pm}+X$ via various
production mechanisms at LHC14 for $\tan\beta=10$ (solid) and $\tan\beta=40$
(dashed).

The next largest cross section is direct resonance production of $H^{\pm}$ via
(Yukawa suppressed) $u\bar{d}$ fusion or via (parton distribution function
suppressed) $c\bar{s}$ fusion. Note that this is suppressed relative to
neutral $H/A$ resonance production from $b\bar{b}$ fusion by the smaller
Yukawa couplings (of the first two generation of quarks) and/or small
Kobayashi Maskawa mixing elements. These light green curves show a $\tan\beta$
enhancement due to the Yukawa couplings involved in the production mechanism.
The resonance production cross section is typically about two orders of
magnitude below $tH^{\pm}$ associated production.

The next largest production cross sections are $HH^{\pm}$ and $AH^{\pm}$ which
are produced via $W^{*}$ exchange followed closely by $H^{+}H^{-}$ pair
production which takes place via $s$-channel $\gamma^{*}$ and $Z^{*}$
exchange. All these production vertices involve gauge interactions and so are
$\tan\beta$ independent. As a result, the $\tan\beta=10$ and $40$ curves lie
on top of one another. These three cross sections are kinematically suppressed
because they involve the production of a pair of heavy bosons, and are $\sim
1.5-3$ orders of magnitude below the dominant $tH^{\pm}$ cross sections.

In magenta, we show $hH^{\pm}$ associated production which occurs dominantly
via $s$-channel $W^{*}$ exchange where the production vertex includes a factor
$g\cos(\alpha+\beta)$ (see Eq. (8.110) of Ref. [1])333The convention for Higgs
mixing angle $\alpha$ in Ref. [1] differs from often-used conventions which
result in the mixing angle factor $\cos(\alpha-\beta)$. which vanishes in the
decoupling limit. As a result, these cross sections are suppressed from the
dominant $tH^{\pm}$ cross section by $\sim 3-5$ orders of magnitude depending
on $m_{H^{\pm}}$ and $\tan\beta$. They also feature a dip at certain values of
$m_{H^{\pm}}$ which occurs when the Higgs mixing angle $\alpha$ is such that
$\cos(\alpha+\beta)\rightarrow 0$. An additional cross section
$\sigma(pp\rightarrow W^{\pm}H^{\mp}+X)$ occurs at the loop level, and so is
highly suppressed and we do not include it here: see Ref. [45].

In light of our discussion of the various production cross sections, for our
HL-LHC SUSY charged Higgs reach analysis we will restrict ourselves to the
dominant $tH^{\pm}$ production process. In Fig. 2 we show the values of
$\sigma(pp\rightarrow tH^{\pm}+X)$ at LHC14 in the $m_{A}$ vs. $\tan\beta$
plane for our $m_{h}^{125}({\rm nat})$ scenario. The largest cross sections
$\sim 10^{3}$ fb are denoted by dark red whilst the lowest cross sections
$\sim 10^{-3}$ fb are denoted dark blue. We also show the latest ATLAS 95% CL
exclusion limit from their search for $H,\ A\rightarrow\tau^{+}\tau^{-}$
events using LHC13 with 139 fb-1 in several non-natural Higgs scenarios which
nonetheless maintain $m_{h}\sim 125$ GeV[46]. Thus, to uncover new physics at
HL-LHC in the SUSY Higgs sector, we will mainly focus our attention on
$m_{H^{\pm}}$ values in the TeV range.

Figure 2: The dominant $pp\rightarrow tH^{\pm}+X$ cross section at
$\sqrt{s}=14$ TeV in the $m_{A}$ vs. $\tan\beta$ plane. The region to the left
of the dashed blue line, is currently excluded at the 95% confidence level by
limits from ATLAS searches for $pp\rightarrow H,\
A\rightarrow\tau^{+}\tau^{-}$ [46].

##  4 $H^{\pm}$ branching fractions in natSUSY

The LHC signal from charged Higgs boson production will clearly depend on how
$H^{\pm}$ decays. For TeV scale values of $m_{H^{\pm}}$, the dominant SM
decays are via $H^{\pm}\rightarrow tb$ and $H^{\pm}\rightarrow\tau\nu_{\tau}$.
Decays to $h$, $W$ and $Z$ bosons are dynamically suppressed. Charged Higgs
boson decays via the gaugino plus higgsino modes can also be important if
these are not kinematically suppressed. With this in mind, in Fig. 3, we show
some select $H^{\pm}$ branching fractions (BFs) in the $m_{A}$ vs. $\tan\beta$
plane for the model plane (Eq. 5). The branching fractions are color-coded,
with the larger ones denoted by red whilst the smallest ones are denoted by
dark blue. The branching fractions are extracted from the Isasugra code[39].

In Fig. 3a) we show the BF for $H^{+}\rightarrow t\bar{b}$. This decay mode to
SM particles is indeed dominant for $m_{A}\lesssim 1$ TeV and for larger
values of $\tan\beta\gtrsim 20-30$. In frame b), we show the
BF($H^{+}\rightarrow\tau^{+}\nu_{\tau}$). Like $H^{+}\rightarrow t\bar{b}$,
this mode is enhanced at large $\tan\beta$ and has provided the best avenue
for SUSY charged Higgs discovery/exclusion plots so far.

While SUSY decay modes of $H^{\pm}$ to higgsino pairs are also open in these
regions, these decay modes are suppressed by mixing angles for reasons
discussed below. Supersymmetry requires that there is a direct gauge
coupling[1]

${\cal L}\ni-\sqrt{2}\sum_{i,A}{\cal
S}_{i}^{\dagger}gt_{A}\bar{\lambda}_{A}\psi_{i}+H.c.$ (6)

where ${\cal S}_{i}$ labels various matter and Higgs scalar fields of the
MSSM, $\psi_{i}$ is the fermionic superpartner of ${\cal S}_{i}$ and
$\lambda_{A}$ is the gaugino with gauge index $A$. Also, $g$ is the
corresponding gauge coupling for the gauge group in question and the $t_{A}$
are the corresponding gauge group matrices. Letting ${\cal S}_{i}$ be the
Higgs scalar fields, we see there is an unsuppressed coupling of the Higgs
scalars to a gaugino and a higgsino as mentioned earlier. This coupling can
lead to dominant SUSY Higgs boson decays to SUSY particles when the gaugino-
plus-higgsino decay channel is kinematically unsuppressed. But it also shows
why the heavy Higgs decay to higgsino pairs is suppressed by mixing angles for
$|\mu|\ll|M_{1,2}|$, once we recognize that a Higgs boson-higgsino-higgsino
coupling is forbidden by gauge invariance.

In frame c), we show
BF($H^{+}\rightarrow\tilde{\chi}_{1}^{0}\tilde{\chi}_{2}^{+}$), where
$\tilde{\chi}_{1}^{0}$ is dominantly higgsino-like and $\tilde{\chi}_{2}^{+}$
is dominantly wino-like for natural SUSY models like the $m_{h}^{125}({\rm
nat})$ scenario. Here, we see that for larger values of $m_{A}\simeq
m_{H^{+}}\gtrsim 1.2$ TeV, then this mode turns on, and at least for moderate
$\tan\beta\sim 10-20$ (which is favored by naturalness[12]), rapidly comes to
dominate the $H^{+}$ decay modes along with the neutral wino+higgsino channels
$H\rightarrow\tilde{\chi}_{2}^{+}\tilde{\chi}_{2}^{0}$ (frame d)) and
$H^{+}\rightarrow\tilde{\chi}_{1}^{+}\tilde{\chi}_{4}^{0}$ (frame e)). In our
analysis we have assumed that the gaugino mass parameters unify at the high
scale, so that at the weak scale $M_{1}\simeq{1\over 2}M_{2}$: as a result,
$\chi_{3}^{0}$ is dominantly bino-like, $\chi_{4}^{0}$ and $\chi_{2}^{\pm}$
are dominantly wino-like, while $\tilde{\chi}_{1,2}^{0}$ and $\chi_{1}^{\pm}$
are mainly higgsino-like. The sum of these three wino+higgsino decay channels
thus dominate the $H^{+}$ decay branching fractions for $m_{H,A}\gtrsim 1.2$
TeV and low-to-moderate values of $\tan\beta$. For high values of $\tan\beta$,
the $b$ and $\tau$ Yukawa couplings become large, and SM decays to fermions
once again dominate SUSY decays. Decays of $H^{+}$ to gauge boson pairs and to
$h$ are unimportant in the decoupling limit as mentioned above. For
completeness, we also show in frame f) the decay mode
$H\rightarrow\tilde{\chi}_{1}^{+}\tilde{\chi}_{3}^{0}$ which is to
higgsino+bino. This mode is large only in a small region of $m_{H}\sim 1$ TeV
and modest $\tan\beta$ where the mode $H\rightarrow bino+higgsino$ decay has
turned on, but where $H\rightarrow wino+higgsino$ has yet to become
kinematically open. Decays to winos dominate decays to binos because the
$SU(2)$ gauge coupling $g$ is larger than the hypercharge gauge coupling
$g^{\prime}$.

Figure 3: Branching fractions in the $m_{A}$ vs. $\tan\beta$ plane for $H^{+}$
to a) $t\bar{b}$, b) $\tau^{+}\nu_{\tau}$, c)
$\tilde{\chi}_{2}^{+}\tilde{\chi}_{1}^{0}$, d)
$\tilde{\chi}_{2}^{+}\tilde{\chi}_{2}^{0}$, e͡)
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{4}^{0}$ and f)
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{3}^{0}$ from Isajet 7.88[39] for the model
line introduced in the text.

##  5 Search for $H^{\pm}\rightarrow\tau\nu_{\tau}$

Next, we turn to the examination of the prospects for discovering the charged
Higgs boson produced at the HL-LHC via $pp\rightarrow H^{\pm}t+X$, followed by
$H^{\pm}\rightarrow\tau\nu_{\tau}$ decay. For signal and $2\rightarrow 2$
background processes listed below, we use the Pythia event generator[44]. For
$2\rightarrow 3$ BG processes such as $Wb\bar{b}$, $Zb\bar{b}$ and $ht\bar{t}$
production, we adopt Madgraph[47] for the subprocess calculation but then
interface with Pythia for parton showers, hadronization and underlying event.
Our final state particles are then fed into the Delphes[48] detector
simulation program which includes a jet-finding algorithm and routines for
identifying both $b$-jets and hadronic tau jets (labelled as $\tau_{h}$).

In Delphes, a jet object is reconstructed using an anti-$k$ algorithm with
$p_{T}(min)>25$ GeV and $\Delta R<0.4$. For a baseline jet we require:

  * •

$|\eta(j)|<4.7$.

For a baseline $b$-jet, besides the requirement for a baseline jet, we further
require

  * •

the jet to be tagged as a $b$-jet by Delphes.

For a signal $\tau_{h}$-jet, besides the requirement of baseline jet, we
further require

  * •

the jet must be a tagged hadronic tau-jet $\tau_{h}$ by Delphes with

  * •

$|\eta(\tau_{h})|<2.5$.

For the baseline lepton isolation requirement, we require

  * •

$p_{T}(min)(e/\mu)>5$ GeV.

For a signal lepton, besides the requirement for baseline lepton isolation, we
further require

  * •

$|\eta_{e,(\mu)}|<2.47\ (2.5)$ and

  * •

$p_{T}(e(\mu))>20\ (25)$ GeV.

###  5.1
$H^{\pm}\rightarrow\tau\nu_{\tau}\rightarrow\tau_{h}+\not\\!\\!\\!{E_{T}}$
channel

In this channel, we search for
$H^{\pm}\rightarrow\tau_{h}+\not\\!\\!\\!{E_{T}}$ along with the presence of a
spectator $t$-jet which is signalled by the presence of a tagged $b$-jet. We
include SM BGs from $t\bar{t}$, single top, $Wb\bar{b}$, $Zb\bar{b}$, $WW$,
$WZ$, $ZZ$, $Zh$, $Wh$ and $ht\bar{t}$ production. We first require:

  * •

exactly one signal $\tau_{h}$-jet with no baseline leptons; the no baseline
lepton requirement targets events where the spectator top decays hadronically,
though of course events with a semileptonic decaying top could contribute if
the lepton evades detection;

  * •

$n(b-{\rm jet})\geq 1$, where $b$ here (and in the rest of Sec. 5) refers to
baseline $b$-jets, and

  * •

a neutrino reconstruction method444In $t\bar{t}$ events where one of the tops
decays hadronically, and the other leptonically, so that the
$\not\\!\\!\\!{E_{T}}$ comes only from the (massless) neutrino; i.e.
$\not\\!\\!\\!{E_{T}}_{x,y}=p(\nu)_{x,y}$, one can construct $p(\nu)_{z}$
assuming that the $W$ boson from top decay is on-shell. This vetoes about half
the potentially enormous $t\bar{t}$ background with a loss of less than 10% of
the signal. is employed here. If the invariant mass of the reconstructed
neutrino, the signal $\tau_{h}$-jet and any of the tagged baseline $b$-jets in
the event reconstructs to $m=m_{t}\pm 50$ GeV, then the event is vetoed.

This latter requirement is imposed to veto a portion of the very large
$pp\rightarrow t\bar{t}$ background.

With these remaining events, an examination of various distributions of signal
and background (that we do not show for brevity) leads us to impose the
following analysis cuts:

  * •

$\not\\!\\!\\!{E_{T}}>350$ GeV,

  * •

$\Delta\phi(\tau_{h},\vec{\not\\!\\!\\!{E_{T}}})>30^{\circ}$,

  * •

$\Delta\phi(b_{1},\vec{\not\\!\\!\\!{E_{T}}})>50^{\circ}$, where $b_{1}$ is
the leading baseline $b$-jet, and

  * •

$min(\Delta R(b,\tau_{h}))>0.9$, where the $b$ loops over all tagged baseline
$b$-jets in the event.

After these cuts, the resulting transverse mass distribution
$m_{T}(\tau_{h},\not\\!\\!\\!{E_{T}})$ is shown in Fig. 4. As expected, the
signal histograms peak around the value of $m_{H^{\pm}}$, while the
backgrounds yield falling distributions. Our goal in each signal channel is to
look for an excess above the SM backgrounds in the largest transverse mass
bins which are most sensitive to TeV-scale charged Higgs decay. From the
distribution, the solid colored histograms represent the various BGs, of which
the dominant is light yellow: $t\bar{t}$. The signal distributions, labelled
as dotted curves for several benchmark scenarios as listed, can emerge from BG
at large values of $m_{T}$, provided the signal is large enough.

Figure 4: Distribution in $m_{T}(\tau_{h},\not\\!\\!\\!{E_{T}})$ from
$pp\rightarrow H^{\pm}(\rightarrow\tau\nu)+t+X$ followed by the decay
$\tau\rightarrow hadrons+\not\\!\\!\\!{E_{T}}$. We also show dominant SM
backgrounds.

###  5.2
$H^{\pm}\rightarrow\tau\nu_{\tau}\rightarrow\ell+\not\\!\\!\\!{E_{T}}$ channel

In this subsection, we examine the $pp\rightarrow tH^{\pm}+X$ production
reaction where $H^{\pm}\rightarrow\tau\nu_{\tau}$ followed by
$\tau\rightarrow\nu_{\tau}+(\ell\nu_{\ell})$ channel, where $\ell=e$ or $\mu$.
For this signal channel, we require:

  * •

exactly one signal lepton and no other baseline leptons,

  * •

no jets have been $\tau$-tagged ($\tau_{h}$-veto); here, we are again
targeting events where the top decays hadronically, and the tau decays
leptonically.

  * •

$n(b-{\rm jet})\geq 1$ and

  * •

the neutrino reconstruction method described above is employed here. If the
invariant mass of the reconstructed neutrino, the signal lepton plus any of
the baseline $b$-jets in the event is within $m=m_{t}\pm 50$ GeV, then event
is vetoed.

Standard Model backgrounds from $t\bar{t}$, single top, $Wb\bar{b}$, $WW$,
$WZ$, $Wh$, $ht\bar{t}$ and $Zh$ can also lead to the same event topology as
the signal.

Examination of various distributions leads us to impose the following analysis
cuts:

  * •

$\not\\!\\!\\!{E_{T}}>350$ GeV,

  * •

$\not\\!\\!\\!{E_{T}}_{,rel}:=$
$\not\\!\\!\\!{E_{T}}\cdot\sin{(min(\Delta\phi,\frac{\pi}{2}))}>150$ GeV,
where $\Delta\phi$ is the azimuthal angle between the
$\vec{\not\\!\\!\\!{E_{T}}}$ and the closest lepton or jet with $p_{T}>25$
GeV.

  * •

$\Delta\phi(\ell,\vec{\not\\!\\!\\!{E_{T}}})>30^{\circ}$,

  * •

$\Delta\phi(b_{1},\vec{\not\\!\\!\\!{E_{T}}})>70^{\circ}$, where $b_{1}$ is
the leading baseline $b$-jet and

  * •

$min(\Delta R(b,\ell))>1.2$, where the $b$ loops over all tagged baseline
$b$-jets in the event.

The resultant $m_{T}(\ell,\not\\!\\!\\!{E_{T}})$ distribution is displayed in
Fig. 5. The dominant BG at low $m_{T}$ comes from $t\bar{t}$ production
(light-yellow histogram) while the dominant BG at high $m_{T}$ comes from $WW$
production. We also show several signal benchmark distributions (dotted
curves) which may cause an excess of events over background expectations at
high $m_{T}(\ell,\not\\!\\!\\!{E_{T}})$. In this case, the signal
distributions shown will only cause a slight excess above background at high
$m_{T}$. But combined with the other channels, this signal channel can
slightly increase the overall significance.

Figure 5: Distribution in $m_{T}(\ell,\not\\!\\!\\!{E_{T}})$ from
$pp\rightarrow H^{\pm}(\tau\nu)+t+X$ followed by
$\tau\rightarrow\ell+\not\\!\\!\\!{E_{T}}$ decay. We also show dominant SM
backgrounds.

###  5.3 $H^{\pm}\rightarrow\tau\nu_{\tau}$ with $t\rightarrow
b\ell\nu_{\ell}$ channel

In this channel, we attempt to extract the signal from $pp\rightarrow
tH^{\pm}$ production followed by
$H^{\pm}\rightarrow\tau_{h}+\not\\!\\!\\!{E_{T}}$ but where the spectator
$t$-quark decays semi-leptonically: $t\rightarrow b\ell\nu_{\ell}$. SM
backgrounds from $t\bar{t}$, single top, $WW$, $WZ$, $Wh$, $Zh$ and
$ht\bar{t}$ production are included in our analysis.

We require:

  * •

exactly one signal lepton and no other baseline leptons,

  * •

exactly one signal $\tau_{h}$-jet,

  * •

the charges of the signal lepton and the $\tau_{h}$-jet must be OS (opposite-
sign) and

  * •

$n(b-{\rm jet})\geq 1$.

Examination of the resultant signal and background distributions leads us to
the following additional analysis cuts:

  * •

$\not\\!\\!\\!{E_{T}}>350$ GeV,

  * •

$\not\\!\\!\\!{E_{T}}_{rel}>150$ GeV,

  * •

$\Delta\phi(\ell,\vec{\not\\!\\!\\!{E_{T}}})>20^{\circ}$,

  * •

$\Delta\phi(\tau_{h},\vec{\not\\!\\!\\!{E_{T}}})>80^{\circ}$,

  * •

$\Delta\phi(b_{1},\vec{\not\\!\\!\\!{E_{T}}})>50^{\circ}$, where $b_{1}$ is
the leading baseline $b$-jet and

  * •

$min(\Delta R(b,\tau_{h}))>1.2$, where the $b$ loops over all baseline
$b$-jets in the event.

The resultant $m_{T}(\tau_{h},\not\\!\\!\\!{E_{T}})$ distribution is shown in
Fig. 6. The dominant BG at low $m_{T}$ again comes from $t\bar{t}$ production.
At high $m_{T}$, then the various signal histograms can cause a noticeable
increase in the expected $m_{T}$ distribution beyond SM expectations, albeit
with a rather low event rate.

Figure 6: Distribution in $m_{T}(\tau_{h},\not\\!\\!\\!{E_{T}})$ from
$pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow\tau_{h}+\not\\!\\!\\!{E_{T}}$ decay and also $t\rightarrow
b\ell\nu_{\ell}$ decay. We also show dominant SM backgrounds.

###  5.4 LHC reach in $H^{\pm}\rightarrow\tau\nu_{\tau}$ channel

Using the analysis cuts for the various signal channels delineated above, we
can now create reach plots to show the LHC14 discovery sensitivity or
exclusion limits for $pp\rightarrow H^{\pm}t$ production in the $m_{A}$ vs.
$\tan\beta$ plane. We use the $5\sigma$ level to claim discovery of a charged
Higgs boson and assume the true distribution one observes experimentally
corresponds to signal-plus-background. We then test this against the
background-only distribution in order to see if the background-only hypothesis
can be rejected at the $5\sigma$ level. Specifically, we use the binned
transverse mass distributions (bin width of 25 GeV) from each signal channel
as displayed above to obtain the discovery/exclusion limits.

In the case of the exclusion plane, the upper limits for exclusion of a signal
are set at 95% CL; one assumes the true distribution one observes in
experiment corresponds to background-only. The limits are then computed using
a modified frequentist $CL_{s}$ method[49] where the profile likelihood ratio
is the test statistic. In both the exclusion and discovery planes, the
asymptotic approximation for obtaining the median significance is
employed[50].

In Fig. 7, we plot our result for the discovery/exclusion regions via the
$H^{\pm}\rightarrow\tau\nu$ channel for the HL-LHC with $\sqrt{s}=14$ TeV and
3000 fb-1 of integrated luminosity in the $m_{A}$ vs. $\tan\beta$ plane using
our $m_{h}^{125}({\rm nat})$ benchmark scenario (which is quite typical for
natural SUSY models[51]). In frame a), we plot the $5\sigma$ discovery reach
using the combined three channels discussed previously. Above the dashed black
line, experiments at the HL-LHC should be able to discover $H^{\pm}$ at the
LHC operating at $\sqrt{s}=14$ TeV, assuming an integrated luminosity of 3000
fb-1. The green and yellow bands display the $\pm 1\sigma$ and $\pm 2\sigma$
uncertainties in our mapping of the discovery region. The region above the
dashed blue line is excluded by ATLAS searches for $H/A\rightarrow\tau\tau$
events, albeit in a scenario with decoupled superpartners [46]. From the plot,
we see that a discovery region does indeed emerge, starting around $m_{A}\sim
500$ GeV and $\tan\beta\sim 18$ and extends out to $m_{A}\sim 3$ TeV for
$\tan\beta\sim 50$ where both the $\sigma(pp\rightarrow tH^{\pm}+X)$ and the
branching fraction for $H^{+}\rightarrow\tau\nu_{\tau}$ decays are both
enhanced. The discovery region pinches off below $\tan\beta\sim 15$ where the
$H^{\pm}\rightarrow\tau\nu_{\tau}$ branching ratio becomes too small.

In frame b), we plot the 95% CL exclusion limit for HL-LHC for our combined
three signal channels. The exclusion limit now extends out to beyond
$m_{A}\sim 3$ TeV for large $\tan\beta\sim 50$. We also see that the exclusion
contour extends to about $\tan\beta\sim 10$ for relatively light $m_{H^{+}}$;
however, this part of the plane is already excluded by ATLAS searches.

Figure 7: In a), we show the $5\sigma$ discovery region of the $m_{A}$ vs.
$\tan\beta$ plane for $pp\rightarrow H^{\pm}t+X$ followed by
$\tau\rightarrow\ell+\not\\!\\!\\!{E_{T}}$ decay for HL-LHC with 3000 fb-1. In
b), we plot the corresponding 95% CL exclusion region. The region above the
dashed blue curve is excluded by ATLAS searches for $H/A\rightarrow\tau\tau$
events.

##  6 Search for $H^{\pm}\rightarrow tb$

In this section, we examine the capability of HL-LHC to discover charged Higgs
bosons in the $H^{\pm}\rightarrow tb$ decay channel. Recent limits have been
placed within this search channel by the ATLAS collaboration using 139 fb-1 of
data[35]. Our analysis proceeds similarly to Sec. 5 except now we place an
emphasis on the presence of high-$p_{T}$ top-jets in the final state.

In the following, we will use lower case letter such as $j$ ($b$) to denote
the small radius jets (tagged $b$-jets) while upper case letter such as $J$
($T$) denote large radius jets (top-jets).

The parameters for baseline reconstructed particles are, for charged leptons
$l$:

  * •

$|\eta(e,\mu)|<4$,

  * •

$p_{T}(e)>20$ GeV, $p_{T}(\mu)>25$ GeV.

For small radius jets $j$:

  * •

jet reconstruction with the anti-$k_{T}$ algorithm,

  * •

cone size $\Delta R<0.4$,

  * •

$p_{T}(j)>25$ GeV,

  * •

$|\eta(j)|<4.7$.

For a tagged baseline $b$-jet:

  * •

satisfy the above small radius jet requirements,

  * •

tagged $b$-jet by Delphes.

For a large radius jet $J$:

  * •

jet reconstruction using the Cambridge/Aachen algorithm (CA)[52, 53],

  * •

cone size $\Delta R<1.5$,

  * •

$p_{T}(J)>300$ GeV and

  * •

$|\eta(J)|<2$.

For a tagged top-jet $T$:

  * •

satisfy the above large radius jet ($J$) requirements,

  * •

$T$ is tagged by the HEPTopTagger2 algorithm[54, 53].

Also, for candidate events with an isolated lepton or a signal $b$-jet, we
further require

  * •

$|\eta(e)|<2.47$, $|\eta(\mu)|<2.5$ and $|\eta(b)|<2.4$.

The analysis is then separated into four orthogonal channels depending on
whether or not the final state does or does not contain a tagged $T$-jet, and
whether or not it contains an isolated lepton.

In all cases, the small radius $b$-jet (denoted as $b_{1}$ below) arising
directly from the $H^{\pm}$ decay is determined by the following procedure.

  * •

we require $p_{T}(b_{1})>350$ GeV,

  * •

$|\eta(b_{1})|<1.5$,

  * •

$R(b_{1},J_{1})>1.5$ (so this means $b_{1}$ must be outside the cone of the
fat jet top candidate).

  * •

$m(j,j^{\prime},b_{1})$ cannot be in the top mass range $[125,225]$ GeV, where
$j$, $j^{\prime}$ are any small radius jet pairs in the event.

  * •

If multiple candidates satisfying these conditions are found, the one with the
hardest $p_{T}$ is taken as $b_{1}$.

  * •

Events are vetoed if no $b$-jets satisfy these conditions.

Then, $m(T_{1},b_{1})$ is used to reconstruct the mass of the $H^{\pm}$ in all
cases. Note that in the channels where the HEPTopTagger2 has positively tagged
a top-jet, i.e. the $1t,1t1l$ channels, it is the four vector reconstructed by
the algorithm that is taken as $T_{1}$. But in the channels where
HEPTopTagger2 fails to identify a top-jet, then it is the fat jet itself that
is taken as the $T_{1}$. The $m(T_{1},b_{1})$ distributions are then shown as
the final results for each signal channel.

The background samples being considered for the hadronic channels $[1t,1t(no\
tag)]$ are $t\bar{t}$ (with $\geq 3$ truth $b$’s removed to avoid double
counting with $t\bar{t}b\bar{b}$ events that are separately simulated), single
top, $t\bar{t}b\bar{b}$ and $tttt$. The backgrounds for the semileptonic
channels $[1t1l,1t(no\ tag)1l]$ are $t\bar{t}$ (again with $\geq 3$ truth $b$s
removed), $t\bar{t}b\bar{b}$ and $t\bar{t}t\bar{t}$. We have not simulated
$t\bar{t}V$ events as we require at least three $b$-jets which has been shown
to be very small [55].

###  6.1 Single tagged top channel without signal leptons

In this channel, we search for $pp\rightarrow tH^{\pm}+X$ with
$H^{\pm}\rightarrow tb$ decay. The primary $t$-quark in the $tH^{\pm}$ final
state tends to be non-central, and so rarely produces a tagged top-jet, but
does more often produce a tagged $b$-jet. So for this channel, we focus on
reconstructing the decay-produced top jet from $H^{\pm}\rightarrow tb$, where
TeV-scale charged Higgs decay gives rise to a well-collimated $T$-jet. Thus,

  * •

The HEPTopTagger2 algorithm that we use has tagged exactly one top from the
large radius boosted ($R<1.5$, $p_{T}(J)>300$ GeV) jet $n_{T}=1$. (This fat
jet is denoted as $J_{1}$ below).

The top-jet four vector reconstructed by the HEPTopTagger2 is used as $T_{1}$.
The four vector for the subjet $b$ reconstructed by the tagger is denoted as
$b_{2}$. We further require the following.

  * •

At least 3 $b$-jets: $n_{b}\geq 3$, of which at least two of them must satisfy
the signal $b$-jet requirements listed above.

  * •

At least 6 jets: $n_{j}\geq 6$.

  * •

No isolated leptons: $n_{l}=0$.

Based on examination of various signal/background distributions, we also
require

  * •

$H_{T}>1200$ GeV,

  * •

$m(b,b^{\prime})>215$ GeV, where the $b$ and $b^{\prime}$ are the $b$-jet pair
with the max $p_{T}$ in the events,

  * •

$max(R(b^{\prime},H^{\pm}))>1.5$, where the $b^{\prime}$ are any $b$-jets in
events that are not $b_{1}$ and $b_{2}$. The $H^{\pm}$ is reconstructed from
$T_{1}$, $b_{1}$,

  * •

$min(R(b^{\prime},b_{1}))<2.8$, where $b^{\prime}$ are any $b$-jets in the
events.

At this point, we are able to construct the distribution $m(tb)\equiv
m(T_{1},b_{1})$ shown in Fig. 8. The dominant background distributions are
shown as solid colored histograms whilst several signal benchmark models are
shown as dashed histograms. From the plot, we see that the signal
distributions roughly reconstruct $m(H^{\pm})$ while background is dominated
by $t\bar{t}b\bar{b}$ production at low mass, and by $t\bar{t}$ at high
invariant mass. The goal then is to search for resonant signal bumps against
the continuum of expected backgrounds. If this bump is buried under the SM
background, this channel will make a negligible contribution to the
significance when combined with other channels.

Figure 8: Distribution in $m(tb)$ from $pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow tb$ decay. These events include a top-jet tagged by
HEPTopTagger2. We also show dominant SM backgrounds.

###  6.2 Single top (no tag) channel without signal leptons

In this channel, the HEPTopTagger2 fails to tag any tops. However,

  * •

there is at least one large radius boosted ($R<1.5$, $p_{T}>300$ GeV) jet
$n_{J}>=1$ with trimmed mass[56] $115\ {\rm GeV}<m_{J}<190$ GeV, and at least
one small radius ($R<0.4$) $b$-jet within the cone of the large radius jet
$J$. Then, the fat Jet with the hardest $p_{T}$ is taken as the top candidate
arising directly from the charged Higgs decay (denoted as $J_{1}/T_{1}$). The
hardest $b$ within $J_{1}/T_{1}$ cone is taken as $b_{2}$.

  * •

At least 3 $b$ jets: $n_{b}\geq 3$, of which at least two of them must satisfy
the signal $b$-jet requirements listed above.

  * •

At least 6 jets: $n_{j}\geq 6$ and

  * •

No isolated leptons: $n_{l}=0$.

We then require:

  * •

$H_{T}>1200$ GeV,

  * •

$m(b,b^{\prime})>215$ GeV, where the $b$ $b^{\prime}$ are the $b$-jet pairs
with the max $p_{T}$ in the events,

  * •

$max(R(b^{\prime},H^{\pm}))>2.4$, where the $b^{\prime}$ are any $b$-jets in
events that are not $b_{1}$ and $b_{2}$. The $H^{\pm}$ is reconstructed from
$T_{1}$, $b_{1}$,

  * •

$0.8<min(R(b^{\prime},b_{1}))<2.8$, where the $b^{\prime}$ are any $b$\- jets
in the events.

The distribution $m(tb)$, reconstructed from $T_{1}$ and $b_{1}$, is shown in
Fig. 9 where again the $m(tb)$ roughly reconstructs $m(H^{\pm})$ and where
again the $t\bar{t}$ and $t\bar{t}b\bar{b}$ are the dominant backgrounds.

Figure 9: Distribution in $m(tb)$ from $pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow tb$ decay, but no fat jet tagged as top by HEPTopTagger2.
We also show dominant SM backgrounds.

###  6.3 Single top (tagged) plus lepton channel

In this signal channel, we again require tagged top-jet, but now also require
the presence of an isolated lepton arising from semileptonic decay of one of
the tops.

We require

  * •

The HEPTopTagger2 has tagged exactly one top from the large radius boosted
($R<1.5$, $p_{T}>300$ GeV) jet $n_{T}=1$.

As before the top four vector reconstructed by the HEPTopTagger2 is denoted as
$T_{1}$. The four vector for the subjet $b$ reconstructed by the tagger is
denoted as $b_{2}$.

  * •

At least 3 $b$-jets: $n_{b}\geq 3$, of which at least two of them must satisfy
the signal $b$-jet requirements listed above.

  * •

At least 4 jets: $n_{j}\geq 4$.

  * •

Exactly one signal isolated leptons: $n_{l}=1$.

We further require

  * •

$H_{T}>1200$ GeV,

  * •

$m(b,b^{\prime})>215$ GeV, where the $b$, $b^{\prime}$ are the $b$ pairs with
the max $p_{T}$ in the events,

  * •

$max(R(b^{\prime},H^{\pm}))>1.5$, where $b^{\prime}$ are any $b$-jets in the
events that are not $b_{1}$ and $b_{2}$. The $H^{\pm}$ is reconstructed from
$T_{1}$ and $b_{1}$.

  * •

$min(R(b^{\prime},b_{1}))>1$, where the $b^{\prime}$ are any $b$-jets in the
events, and

  * •

$R(b_{1},l)>0.9$.

The $m(tb)$ invariant mass distribution from the signal BM models and
backgrounds, again constructed by combining $T_{1}$ and $b_{1}$, are shown in
Fig. 10. In this case, the $t\bar{t}b\bar{b}$ background is dominant in the
range where $m(tb)$ reconstructs $m(H^{\pm})$.

Figure 10: Distribution in $m(tb)$ from $pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow tb$ decay, a fat jat tagged as a top-jet by HEPTopTagger2,
and an isolated lepton from the decay of one of the top quarks. We also show
dominant SM backgrounds.

###  6.4 Single top (no tag) plus lepton channel

In this channel, we examine events where the HEPTopTagger2 fails to tag any
top jets but there is a lepton from the decay of one of the tops. We require,

  * •

there is at least one large radius boosted ($R<1.5$, $p_{T}>300$ GeV) jet
$n_{J}\geq 1$ with trimmed mass [56] $115\ {\rm GeV}<m_{J}<190$ GeV, and at
least one small radius ($R<0.4$) $b$-jet within the cone of the large radius
jet $J$. Then, the fat Jet is taken as the hardest $p_{T}$ top candidate
directly from the charged Higgs decay (denoted as $J_{1}/T_{1}$). The hardest
$b$-jet within $J_{1}/T_{1}$ is taken as $b_{2}$.

  * •

At least 3 $b$-jets: $n_{b}\geq 3$, of which at least two of them must satisfy
the signal $b$-jet requirements listed above.

  * •

At least 6 jets: $n_{j}\geq 6$.

  * •

Exactly one signal isolated lepton: $n_{l}=1$.

We also require

  * •

$H_{T}>1200$ GeV,

  * •

$m(b,b^{\prime})>215$ GeV, where the $b$, $b^{\prime}$ are the $b$ pairs with
the max $p_{T}$ in the events,

  * •

$max(R(b^{\prime},H^{\pm}))>1.1$, where $b^{\prime}$ are any $b$-jets in the
events that are not $b_{1}$ and $b_{2}$. $H^{\pm}$ is reconstructed from
$T_{1}$ and $b_{1}$,

  * •

$min(R(b^{\prime},b_{1}))>1$, where $b^{\prime}$ are any $b$-jets in the
events and

  * •

$R(b_{1},l)>0.9$.

The $m(tb)$ distribution (again constructed by combining $T_{1}$ and $b_{1}$)
for signal and background events with non-tagged top-jets and an isolated
lepton is shown in Fig. 11.

Figure 11: Distribution in $m(tb)$ from $pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow tb$ decay. We also require that there is no fat jet tagged
as a top-jet by HEPTopTagger2 but that there is an isolated lepton from the
decay of one of the top quarks. We also show dominant SM backgrounds.

###  6.5 LHC reach in $H^{\pm}\rightarrow tb$ channel

As in the $H^{\pm}\rightarrow\tau\nu_{\tau}$ analysis, after adopting the
above cuts for the various signal channels, we can now create reach plots in
terms of discovery sensitivity or exclusion limits for $pp\rightarrow
H^{\pm}t+X$ followed by $H^{\pm}\rightarrow tb$ in the $m_{A}$ vs. $\tan\beta$
plane. In this case, we use the binned $m(tb)$ distributions (bin width of 25
GeV) from each signal channel as displayed above to obtain the
discovery/exclusion limits.

In Fig. 12, we show our results for the discovery/exclusion regions via the
$H^{\pm}\rightarrow tb$ channel for the HL-LHC with $\sqrt{s}=14$ TeV and 3000
fb-1 of integrated luminosity in the $m_{A}$ vs. $\tan\beta$ plane using our
$m_{h}^{125}({\rm nat})$ benchmark scenario. In frame a), we plot the
$5\sigma$ discovery reach using the combined four $H^{\pm}\rightarrow tb$
signal channels listed above. The dashed black line denotes the computed reach
while the green and yellow bands display the $\pm 1\sigma$ and $\pm 2\sigma$
uncertainty. From the plot, we see that a discovery region is indeed found,
starting around $m_{A}\sim 500$ GeV and $\tan\beta\sim 18$. For these combined
$H^{\pm}\rightarrow tb$ signal channels, the discovery region extends out to
$m_{A}\sim 1.6$ TeV for $\tan\beta\sim 50$. The discovery region pinches off
below $\tan\beta\sim 25$ where the signal, after analysis cuts, becomes too
small relative to the standard model background. Unfortunately, the entire
discovery region lies within the portion of the plane that already appears to
be excluded by the ATLAS search for $H/A\rightarrow\tau\tau$ decays [46].

In frame b), we plot the 95% CL exclusion limit for HL-LHC for our combined
four signal channels. The exclusion limit now extends out to $m_{A}\sim 2.1$
TeV for large $\tan\beta\sim 50$, and well outside the ATLAS excluded region
denoted by the dashed blue line. We also see that the exclusion contour
extends somewhat below $\tan\beta\sim 20$ for lighter $m_{H^{+}}\sim 1$ TeV.
Also, a small exclusion region has now appeared at low $\tan\beta\sim 3$,
which has also appeared in the ATLAS analysis[35].

Figure 12: In a), we plot the $5\sigma$ discovery region of the $m_{A}$ vs.
$\tan\beta$ plane for $pp\rightarrow H^{\pm}t+X$ followed by
$H^{\pm}\rightarrow tb$ decay for HL-LHC with 3000 fb-1. In b), we plot the
corresponding 95% CL exclusion region. The dashed blue line shows the boundary
of the region excluded at the 95% confidence level in Ref.[46].

##  7 Search for $H^{\pm}\rightarrow$ SUSY at HL-LHC

In Ref. [3], we examined $s$-channel production of heavy neutral SUSY Higgs
bosons followed by decays to SUSY particles, where in natural SUSY $H,\
A\rightarrow gaugino+higgsino$ was the dominant decay mode (when kinematically
open) except where $\tan\beta$ was very large. The heavier higgsinos decay to
soft visible particles plus $\not\\!\\!\\!{E_{T}}$ whilst the gauginos decay
via $W,\ Z,\ h+\not\\!\\!\\!{E_{T}}$. This led to discovery channels of $H,\
A\rightarrow W,\ Z,\ h+\not\\!\\!\\!{E_{T}}$ at HL-LHC, with accessible
parameter regions mapped out in the $m_{A}$ vs. $\tan\beta$ plane for natSUSY
in Ref.[3]. The number of signal events after cuts were typically in the range
of tens of events at best at HL-LHC with 3000 fb-1 of integrated luminosity.

The question here is then: are there lucrative charged Higgs
$H^{\pm}\rightarrow SUSY$ decay channels available for HL-LHC? We saw in Fig.
3 that for moderate $\tan\beta$ and $m_{H^{\pm}}>m(gaugino)+m(higgsino)$ that
the SUSY decay modes also become the dominant decay channels for charged Higgs
bosons. And like their neutral Higgs counterparts, the final state
configurations for $H^{\pm}\rightarrow SUSY$ end up being $H^{\pm}\rightarrow
W,\ Z,\ h+\not\\!\\!\\!{E_{T}}$ according to the various charged Higgs and
sparticle branching fractions from Isajet[39].

Thus, we have also examined the prospects for $pp\rightarrow
tH^{\pm}\rightarrow t+SUSY\rightarrow t+(W,\ Z,\ h)+\not\\!\\!\\!{E_{T}}$ at
HL-LHC. An essential difference of $pp\rightarrow tH^{\pm}+X$ compared to
$pp\rightarrow H,\ A+X$ at LHC is that for a given heavy Higgs mass, the top-
quark plus charged Higgs cross section is typically suppressed by an order of
magnitude or more from $s$-channel neutral heavy Higgs production. A plot of
charged Higgs production cross section times $BF(H^{\pm}\rightarrow SUSY)$ in
fb in the $m_{A}$ vs. $\tan\beta$ plane is shown in Fig. 13 for the
$m_{h}^{125}({\rm nat})$ scenario. From the plot, for $m_{A}\gtrsim 1TeV$
where decays to SUSY particles begin to become important, the cross sections
lie in the sub-fb regime. In addition, one typically tries to tag the
spectator $t$\- or $b$\- jet in $tH^{\pm}$ production which also leads to a
reduction in signal level. Then one must factor in further SUSY decay
branching fractions to $W,\ Z,\ h+\not\\!\\!\\!{E_{T}}$ along with $W,\ Z,\ h$
branching fractions into observable final states. The upshot is: the reduced
signal channels compared to SM backgrounds $t\bar{t}$, $t\bar{t}b\bar{b}$,
$t\bar{t}W$, $t\bar{t}Z$, $t\bar{t}h$ etc. did not lead to any compelling
discovery channels that we could find.

Figure 13: Plot of charged Higgs production cross section times
$BF(H^{\pm}\rightarrow SUSY)$ in fb in the $m_{A}$ vs. $\tan\beta$ plane for
the $m_{h}^{125}({\rm nat})$ scenario.

##  8 Regions of the $m_{A}$ vs. $\tan\beta$ plane accessible to HL-LHC via
charged and neutral Higgs boson searches in natSUSY

We have found so far that a charged Higgs boson signal should be accessible to
the HL-LHC in two different channels: $H^{\pm}\rightarrow\tau\nu_{\tau}$ and,
to a lesser degree, via $H^{\pm}\rightarrow tb$. The regions of the $m_{A}$
vs. $\tan\beta$ plane which are available to HL-LHC via $5\sigma$ discovery
and $95\%$ CL exclusion have been mapped out. At this point, it is worthwhile
to compare the reach of HL-LHC via charged Higgs boson searches to the reach
which can be achieved via $s$-channel $H$ and $A$ signals as delineated for
natSUSY in Ref. [2] and [3].

In Fig. 14, we plot out the $5\sigma$ reach of HL-LHC with 3000 fb-1 in the
$m_{A}$ vs. $\tan\beta$ plane for heavy charged and neutral Higgs boson
signals in the natSUSY scenario. The red dashed contour shows the computed
$5\sigma$ discovery reach via the $H,\ A\rightarrow\tau\bar{\tau}$ channel. Of
all channels assessed so far, this provides the maximal discovery reach due to
higher production cross sections $pp\rightarrow H,\ A+X$, lower backgrounds in
the ditau decay channel and the capability to reconstruct the ditau invariant
mass $m(\tau\bar{\tau})$ using the ability to roughly reconstruct the missing
neutrino momentum. The discovery region lies above the contour whch extends
from $\tan\beta\sim 8$ for $m_{A}=1$ TeV to $\tan\beta\sim 35$ for $m_{A}=2.4$
TeV. The decay branching fraction for $A\rightarrow\tau\bar{\tau}$ is of
course enhanced by the well-known factor $\tan^{2}\beta$.

Figure 14: Plot of $5\sigma$ discovery projections for heavy SUSY Higgs boson
searches at HL-LHC in the $m_{A}$ vs. $\tan\beta$ plane for the
$m_{h}^{125}({\rm nat})$ scenario.

Next most important is the yellow-dashed contour for $pp\rightarrow H,\ A$
with $H,\ A\rightarrow SUSY\rightarrow(W,Z,\ h)+\not\\!\\!\\!{E_{T}}$. These
combined channels determine the ultimate reach and only turn on for
$m_{A}\gtrsim 1$ TeV where $H,\ A\rightarrow SUSY$ becomes kinematically
accessible in the natSUSY scenario. The reach via SUSY decays is comparable
with the $H,\ A\rightarrow\tau\bar{\tau}$ reach for $m_{A}\sim 1.2$ TeV, but
for $m_{A}\gtrsim 1.5$ TeV, the reach via SUSY decays drops off more quickly
for larger $\tan\beta$ values mainly because the $H$ and $A$ decays to SM
fermions are enhanced by large Yukawa couplings, suppressing the branching
fractions for decays to SUSY particles. The green-dashed contour denotes the
HL-LHC $5\sigma$ discovery reach via $pp\rightarrow tH^{\pm}+X$ followed by
$H^{\pm}\rightarrow\tau\nu_{\tau}$ decay. For a given $m_{A}$, the
$pp\rightarrow tH^{\pm}+X$ cross section is well below the resonantly enhanced
$pp\rightarrow H,\ A$ and furthermore one cannot reconstruct a charged Higgs
invariant mass via the $H^{\pm}\rightarrow\tau\nu$ channel: as a result, the
reach via the charged Higgs channel is substantially less than the the reach
via $pp\rightarrow H,\ A\rightarrow\tau\bar{\tau}$. The blue dashed contour
denotes the $5\sigma$ discovery reach for $pp\rightarrow tH^{\pm}+X$ with
$H^{\pm}\rightarrow tb$. This discovery region is mainly applicable at large
$\tan\beta$. Within the model, a substantial region of parameter space is
accessible to HL-LHC via several discovery channels, but we should keep in
mind that portions of this plane is excluded by the ATLAS search, albeit in a
model with decoupled superpartners[46].

In Fig. 15, we show all four contours, but now as 95% CL exclusion limits,
should no signal appear at the HL-LHC. For $m_{A}\gtrsim 1.6$ TeV, the main
exclusion would come from not discovering $H,\ A\rightarrow\tau\bar{\tau}$
while for lower $m_{A}\lesssim 1.6$ TeV the main exclusion come from not
discovering $H,\ A\rightarrow SUSY\rightarrow(W,\ Z,\
h)+\not\\!\\!\\!{E_{T}}$, where the exclusion contour dips to very low
$\tan\beta\sim 3$ (where $m_{h}$ becomes lighter than 125 GeV). The charged
Higgs exclusion contours are contained within the $H,\
A\rightarrow\tau\bar{\tau}$ exclusion contour.

Figure 15: Plot of 95% CL exclusion projections for heavy SUSY Higgs boson
searches at HL-LHC in the $m_{A}$ vs. $\tan\beta$ plane for the
$m_{h}^{125}({\rm nat})$ scenario.

##  9 Conclusions

In this paper we have investigated the ability of HL-LHC to discover the
charged Higgs bosons of supersymmetric theories in the natural SUSY scenario.
We believe that the natSUSY scenario is strongly motivated in that it
naturally explains the measured magnitude of the weak scale which arises from
a conspiracy of the weak scale soft terms: $\mu\sim 100-400$ GeV while
$m_{H_{u}}^{2}$ is radiatively driven to rather small negative values at the
weak scale (EW symmetry is barely broken). Other sparticle masses can be much
larger, lying in the TeV or beyond region since their contributions to the
weak scale are suppressed (at least) by loop factors. Unnatural SUSY models
which predict much higher values for $m_{weak}$ are regarded as rather
implausible since they will require an unnatural conspiracy/finetuning of
parameters in order to gain $m_{weak}\sim 100$ GeV.

The charged Higgs boson masses can range from their present lower limits from
LHC searches up into the multi-TeV range (depending on $\tan\beta$) with
little cost to EW naturalness. Furthermore, once their masses exceed
$m(higgsino)+m(gaugino)$, then the decay to SUSY particles can become
dominant. This reduces heavy Higgs decay to SM particle signals as expected in
unnatural scenarios such as 2HDMs, but also opens up possible new avenues for
heavy Higgs discovery. In this paper, we have delineated search strategies for
charged Higgs bosons in both the $H^{\pm}\rightarrow\tau\nu_{\tau}$ and
$H^{\pm}\rightarrow tb$ channels and have also computed the regions of $m_{A}$
vs. $\tan\beta$ parameter space which are accessible to HL-LHC as the
$5\sigma$ and 95% CL contours. The HL-LHC reach for charged Higgs bosons is
typically contained within the stronger reach via $s$-channel production of
heavy neutrals $H$ and $A$. For searches at the HL-LHC, decays of the charged
Higgs boson to superpartners appear to be unimportant. Nonetheless, there do
exist regions where all of $H^{\pm}$, $H$ and $A$ may be discovered.

Acknowledgements:

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics under Award Number DE-
SC-0009956 and DE-SC-0017647.

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