IFT-UAM/CSIC-21-4

Discovery and exclusion prospects for staus
produced by heavy Higgs bosons decays at the LHC

Ernesto<EMAIL_ADDRESS>Victor Mart ́in-
<EMAIL_ADDRESS>Anibal D.
<EMAIL_ADDRESS>Nicolas I.
<EMAIL_ADDRESS>
1Instituto de Física Teórica UAM/CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049, Madrid, Spain

2IFLP, CONICET - Dpto. de Física, Universidad Nacional de La Plata,
C.C. 67, 1900 La Plata, Argentina

3DESY, Notkestraße 85, 22607 Hamburg, Germany

###### Abstract

In a previous work we developed a search strategy for staus produced by the
decay of the heavy CP-even Higgs boson $H$ within the context of the large
$\tan\beta$ regime of the minimal supersymmetric standard model (MSSM) in an
scenario of large stau mixing. Here we study the performance of such search
strategy by confronting it with the complementary mixing pattern in which
decays of both the CP-even and CP-odd heavy Higgs bosons contribute to the
production of $\widetilde{\tau}_{1}\widetilde{\tau}_{2}^{*}\;+\;c.c$ pairs.
Again, we focus on final states with two opposite-sign tau leptons and large
missing transverse energy. We find that our proposed search strategy, although
optimized for the large stau mixing scenario, is still quite sensitive to the
complementary mixing pattern. For instance, with a total integrated luminosity
of only 100 fb-1 we are able to exclude heavy Higgs masses above 850 GeV for
average stau masses higher than 290 GeV. We also extend the results reported
in the preceding work for the large mixing scenario by including now the
exclusion limits at 100 fb-1 and the prospects both for exclusion and
discovery in a potential high luminosity phase of the LHC (1000 fb-1).
Finally, we discuss the possibility to distinguish the two mixing scenarios
when they share the same relevant mass spectrum and both reach the discovery
level with our search strategy.

##  1 Introduction

Among the theories that extend the standard model (SM) of particle physics,
supersymmetry (SUSY) remains as one of the most interesting and promising
candidates (for reviews, see, e.g., [1, 2]). From a phenomenological point of
view, its minimal version with $R$-parity conservation [3], the minimal
supersymmetric standard model (MSSM) [4, 5, 6], has as its main virtues a
solution to the gauge hierarchy problem, a potential unification of SM gauge
couplings at high energies and a viable dark-matter candidate, the lightest
supersymmetric particle (LSP) [7, 8]. The MSSM predicts the existence of
superpartners (sparticles) for each SM particle: squarks/sleptons, gauginos
and higgsinos are the companions of quarks/leptons and gauge and Higgs bosons,
respectively. The MSSM Higgs sector contains two scalar doublets that under
the assumption of a CP-conserving potential leads to a physical spectrum that
includes three neutral Higgs bosons (a light scalar $h$, a heavy scalar $H$,
and a heavy pseudoscalar $A$) and a pair of charged Higgs bosons ($H^{\pm}$),
of which the 125-GeV SM-like Higgs boson [9, 10] can be easily accommodated as
the lightest CP-even Higgs boson $h$ (see for instance [11]). Together with
the phenomenological signals of these additional Higgs bosons, the existence
of sparticles produces a rich phenomenology with characteristic collider
signals that are being searched for at the Large Hadron Collider (LHC).

A particular interesting example are the supersymmetric scalar partners of the
tau leptons, the staus, which are being intensely searched for at the LHC by
the ATLAS and CMS collaborations [12, 13, 14, 15, 16, 17, 18, 19], since in
many scenarios where SM gauged mediators are responsible for the transmission
of SUSY breaking from a hidden sector to the visible sector, staus could be
among the lightest sparticles. In a previous work [20], we have demonstrated
that stau-pair production at the LHC that are originated from the decay of a
heavy scalar Higgs boson, and where the staus subsequently mainly decay into a
tau lepton and the LSP neutralino, can be very promising in the large-
tan$\beta$ regime within MSSM scenarios with large stau mixing [21]. Indeed,
we found that in these regions of the MSSM parameter space, resonant stau-pair
production cross sections are 1-2 orders of magnitude larger than the usually
considered electroweak (EW) production mechanism. The search strategy we
developed was dedicated to scenarios with a stau mixing pattern for which the
only relevant Higgs decay channel into staus was $H\to$
$\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1}$, being $\widetilde{\tau}_{1}$
the lightest stau. This class of stau mixing pattern occurs when the stau
mixing angle is large and the diagonal entries of the stau mass matrix are of
similar value. By means of a set of basic cuts, we obtained signal-to-
background significances at the discovery level for a LHC center-of-mass
energy of 14 TeV and a total integrated luminosity of 100 fb-1. For this new
work we would like to extend our previous analysis and show that our search
strategy also works very well for scenarios with a complementary stau mixing
pattern in which the stau mixing angle is small but the non-diagonal entries
of the stau mass matrix are large, mainly due to a sufficiently large stau
trilinear coupling. In these latter scenarios, contrary to the ones analyzed
in our previous work, not only the CP-even Higgs contributes to the stau pair
production but there is also the CP-odd Higgs contribution which in this case
is non-vanishing, potentially increasing the phenomenological signals.
Furthermore, mixed combinations of heaviest and lightest staus are preferable
produced via the heavy Higgs boson decays and thus we have the decay patterns,
$H/A\to\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{2},\widetilde{\tau}_{2}^{*}\widetilde{\tau}_{1}$,
where $\widetilde{\tau}_{2}$ is the heaviest stau, as the main source of
staus. Since in this mixing pattern $m_{\widetilde{\tau}_{2}}$ can be much
larger than $m_{\widetilde{\tau}_{1}}$, we expect that the the collider
phenomenology of the stau decays products to be quite different from the
patterns analyzed by us before and can potentially help in distinguishing both
mixing patterns from each other. From a more general approach, this search
strategy could be applied to any process at the LHC with an identical
topology, that is, the resonant production of a pair of charged scalars which
decay into a tau lepton and an invisible particle, consisting of final states
with a $\tau$-lepton pair plus a large amount of missing transverse energy
($E_{T}^{\text{miss}}$).

The paper is organized as follows: Section 2 is devoted to review the
theoretical features of the large stau mixing MSSM scenarios we work with,
paying special attention to the main characteristics of the $H/A$ decays into
a stau pair. The collider analysis we develop along this work is presented in
Section 3, together with the description of our search strategy for stau
pairs, originated from heavy Higgs boson resonances and decaying into the
lightest neutralino and a tau lepton. A compendium of the obtained results is
presented in Section 4, for both classes of stau scenarios and with a final
study of the potential discrimination between them, leaving Section 5 for a
discussion of our main conclusions.

##  2 $H/A$ decays into stau leptons within the MSSM

As mentioned in the introduction, we work in the context of the MSSM in the
large $\tan\beta\gg 1$ limit in which the couplings of the heavy Higgs bosons,
$H_{k}=H,A$ to the down-type sfermions of mixed chiralities are given by,

$\displaystyle
g_{H\widetilde{d}_{L}\widetilde{d}_{R}}=-\frac{1}{2}m_{d}[-\mu+A_{d}\tan\beta],$
(1) $\displaystyle
g_{A\widetilde{d}_{L}\widetilde{d}_{R}}=-\frac{1}{2}m_{d}[\mu+A_{d}\tan\beta],$
(2)

where $m_{d}$ is the mass of the down-type fermion, $\mu$ is the Higgsino mass
and $A_{d}$ is the trilinear coupling given in the soft SUSY breaking
Lagrangian. The couplings involving the same chiral states are proportional to
SM fermion and gauge boson masses, do not involve soft SUSY breaking
parameters and therefore cannot be enhanced [22]. In contrast, the couplings
involving different chiral states depend not only on the SM fermion mass but
also on SUSY parameters, namely, the trilinear soft breaking parameter $A_{d}$
and the $\mu$ parameter. In Ref. [21] it was shown that in a regime where
$\tan\beta\gg 1$ and for large values of $A_{d}$, the couplings of Eqs. (1)
and (2) can in fact be enhanced in the case of staus in particular. This is
translated into larger branching fractions to staus, ${\rm
BR}(H_{k}\to\sum_{i,j=1,2}\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j})$,
implying consequentially that the branching ratio to taus, ${\rm
BR}(H_{k}\to\tau^{+}\tau^{-})$, decreases. It is this what allows for the
resurrection of certain regions of the MSSM that seem at first sight to be
excluded from di-tau searches at the LHC. In this regime we can find two
scenarios with sizable branching fraction into staus: in one of them
$\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1}$ is the dominant decay mode
(Scenario I), while in the other the
$\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{2}$ \+ $c.c.$ mode gives the
dominant contribution (Scenario II). Let us take a closer look at these
scenarios in terms of the stau mass matrix and the corresponding mixing
patterns.

The stau mass matrix is defined as,

$\displaystyle\mathcal{M}^{2}_{\widetilde{\tau}}=\left(\begin{array}[]{cc}m_{\widetilde{\tau}_{11}}&m_{\widetilde{\tau}_{12}}\\\
m^{*}_{\widetilde{\tau}_{12}}&m_{\widetilde{\tau}_{22}}\end{array}\right)=$
(5)
$\displaystyle\left(\begin{array}[]{cc}m_{\tilde{L}_{3}}^{2}+m_{\tau}^{2}+(-1/2+1/3\sin^{2}\theta_{w})m^{2}_{Z}\cos
2\beta&m_{\tau}(A_{\tau}-\mu\tan\beta)\\\
(m_{\tau}(A_{\tau}-\mu\tan\beta))^{*}&m_{\tilde{E}_{3}}^{2}+m_{\tau}^{2}+1/3\sin^{2}\theta_{w}m^{2}_{Z}\cos
2\beta\end{array}\right),$ (8)

where the trilinear coupling, $A_{\tau}$, comes from the soft Lagrangian term
$\mathcal{L}_{\rm soft}\supset
y_{\tau}A_{\tau}\tilde{\bar{e}}_{3}\tilde{L}_{3}H_{d}+c.c.$ The
diagonalization of the mass matrix leads to the following mass eigenvalues and
eigenstates:

$\displaystyle
m_{\widetilde{\tau}_{1},\widetilde{\tau}_{2}}^{2}=\frac{1}{2}(m_{\widetilde{\tau}_{11}}+m_{\widetilde{\tau}_{22}}\pm\Delta_{\widetilde{\tau}}),$
(9)
$\displaystyle\widetilde{\tau}_{1}=\widetilde{\tau}_{L}\cos\theta_{\widetilde{\tau}}+\widetilde{\tau}_{R}\sin\theta_{\widetilde{\tau}},$
(10)
$\displaystyle\widetilde{\tau}_{2}=-\widetilde{\tau}_{L}\sin\theta_{\widetilde{\tau}}+\widetilde{\tau}_{R}\cos\theta_{\widetilde{\tau}}.$
(11)

where
$\Delta_{\widetilde{\tau}}\equiv\sqrt{(m_{\widetilde{\tau}_{11}}-m_{\widetilde{\tau}_{22}})+4m_{\widetilde{\tau}_{12}}}$,
assuming that $m^{*}_{\widetilde{\tau}_{12}}=m_{\widetilde{\tau}_{12}}$, and
the mixing angle can be written as

$\displaystyle\tan
2\theta_{\widetilde{\tau}}=\frac{2m_{\widetilde{\tau}_{12}}}{m_{\widetilde{\tau}_{11}}-m_{\widetilde{\tau}_{22}}}.$
(12)

Now that we have defined the stau sector, let us study the decay of a heavy
Higgs boson, $H_{k}=H,A$, into staus. The decay width is given by

$\displaystyle\Gamma(H_{k}\to\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j})=\frac{G_{F}}{2\sqrt{2}M_{H_{k}}}\lambda^{1/2}_{\widetilde{\tau}_{i}\widetilde{\tau}_{j}H_{k}}g^{2}_{H_{k}\widetilde{\tau}_{i}\widetilde{\tau}_{j}}\quad(i,j=1,2),$
(13)

where

$\displaystyle\lambda_{ijk}=\left(1-\frac{m^{2}_{i}}{m_{k}^{2}}-\frac{m^{2}_{j}}{m_{k}^{2}}\right)^{2}-4\frac{m_{i}^{2}m_{j}^{2}}{m^{4}_{k}},$
(14)

is the kinematic factor in a two-body decay, and
$g_{H_{k}\widetilde{\tau}_{i}\widetilde{\tau}_{j}}$ is the coupling of the
Higgs $H_{k}$ to the staus $\widetilde{\tau}_{i}$ and $\widetilde{\tau}_{j}$.
It is important to note that this coupling is a combination of the chiral
couplings given in Eqs. (1) and (2), and then can be written as

$\displaystyle
g_{H_{k}\widetilde{\tau}_{i}\widetilde{\tau}_{j}}=\sum_{\alpha,\beta=L,R}T_{ij\alpha\beta}g_{H_{k}\widetilde{\tau}_{\alpha}\widetilde{\tau}_{\beta}},$
(15)

where the $T_{ij\alpha\beta}$ are the elements of a $4\times 4$ matrix that
relates the mass eigenstates with the interaction states.

Scenario I is achieved, as commented above, when the mixing angle is maximal,
$\theta_{\widetilde{\tau}}\sim\pi/4$. In this case, according to Eq. (12), it
is not only important that $m_{\widetilde{\tau}_{12}}$ is large but it is also
required that $m_{\widetilde{\tau}_{11}}\sim m_{\widetilde{\tau}_{22}}$. In
such case, there is a cancellation of the contributions that mix chiralities
in the coupling $g_{H\widetilde{\tau}_{1}\widetilde{\tau}_{2}}$ involving
different staus, leaving them only proportional to the chiral diagonal
couplings that as we mentioned before cannot be enhanced. The mass diagonal
couplings $g_{H\widetilde{\tau}_{1}\widetilde{\tau}_{1}}$ and
$g_{H\widetilde{\tau}_{2}\widetilde{\tau}_{2}}$ on the other hand do not
present this cancellation and depend on the parameters that mixes chiralities,
which can be increased by our choice of parameters. Therefore, the Higgs
couplings to $\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{2}$ and
$\widetilde{\tau}_{2}^{*}\widetilde{\tau}_{1}$ suffer a cancellation, while
the couplings to $\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1}$ and
$\widetilde{\tau}_{2}^{*}\widetilde{\tau}_{2}$ are maximal. Since in this
situation the decays into pairs of heavier staus $\widetilde{\tau}_{2}$ are
usually not kinematically available, the decay of $H$ is dominated by the
decays into $\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1}$. Furthermore, as a
consequence of the fact that the mixed-mass couplings are suppressed, the
couplings involving the CP-odd Higgs $A$ to staus are also suppressed, and the
stau production via Higgs boson decays is dominated by the heaviest CP-even
Higgs $H$.

The situation that characterizes Scenario II arises when the mixing angle is
small but $m^{2}_{\widetilde{\tau}_{12}}$ is large due to the $A_{\tau}$ term
[22]. In this case the mixed chiral couplings are maximized, such that the
left-right part of the coupling of $H$ to
$\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{2}$ and the right-left part of the
coupling of $H$ to $\widetilde{\tau}_{2}^{*}\widetilde{\tau}_{1}$ are maximal.
This latter pattern of decay also shows up for the supersymmetric decays of
the CP-odd Higgs $A$ to staus due to CP conservation.

##  3 Collider analysis

In Ref. [20] we developed a search strategy that proved to be very efficient
as a discovery tool within the context of what we call here Scenario I. In
this section we will apply the same analysis which was optimized for Scenario
I to both scenarios in order to test its discovery potential not only for
Scenario I but also for Scenario II. In the two scenarios the staus are
resonantly produced through a heavy Higgs boson: $H$, in the case of Scenario
I, and $H/A$ in the case of Scenario II. As it was mentioned in Ref. [20], the
cross section of the resonant production is significantly larger than that
corresponding to the EW pair production in the mass range
$m_{H/A}\in[800-1200]$ GeV with values of $\tan\beta\in[25-50]$. This fact
relies on the relatively low masses for the scalar and pseudoscalar
resonances, the larger production via bottom fusion for the heavy Higgs bosons
in the large $\tan\beta$ regions compared to the EW-size production via SM
electroweak gauge bosons, and the large values of $A_{\tau}$ that enhance the
decays of the heavy Higgs bosons, allowing non-negligible values of the
branching ratios to staus,
BR$(H\to\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1})\sim 0.1-0.2$ in the case
of Scenario I, and BR$(H/A\to\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j})\sim
0.1-0.4$ ($i,j=1,2$ and $i\neq j$) for Scenario II.

Our analysis focuses on the process $b\bar{b}\to
H/A\to\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j}\to\tau^{+}\widetilde{\chi}_{1}^{0}\tau^{-}\widetilde{\chi}_{1}^{0}$,
with taus decaying hadronically for both scenarios. Regardless on which stau
mass state is produced, the final state involves two opposite-sign tau leptons
and a large amount of missing energy, that comes from the two LSP neutralinos,
$\widetilde{\chi}_{1}^{0}$. In order to test the two stau mixing scenarios, we
have taken the benchmarks points used in Ref. [20] for Scenario I and new ones
produced in such a way that fulfill the requirements of Scenario II, but were
included already in Ref. [21]. A caveat is in order given that the parameter
space considered for our scenarios I and II may be already excluded by the
most recent LHC searches for Higgs bosons decaying into two tau leptons [23,
24], even when considering the additional decays into staus [21]. However, as
mentioned in the conclusion of Ref. [21], points of parameter space that have
just been excluded by the recent di-tau searches can once again be resurrected
considering the supersymmetric decays of the heavy Higgs bosons into staus in
a completely analogous manner. We will therefore continue our analysis with
these older points given that our conclusions regarding the strength of the
search strategies proposed and the possible phenomenological signals will
remain the same as for newly equivalent resurrected parameter points that are
allowed to evade the latest ditau searches.

All the points were computed using SPheno 3.3.8 [25, 26], from which we obtain
all the spectra and phenomenological properties, like the branching ratios. To
test the different points, we produce Monte Carlo events for the $b$-quark
fusion process that dominates the heavy Higgs production in the large
$\tan\beta$ limit, $b\bar{b}\to
H/A\to\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j}\to\tau^{+}\widetilde{\chi}_{1}^{0}\tau^{-}\widetilde{\chi}_{1}^{0}$,
at a center-of-mass energy of $\sqrt{s}=14$ TeV using the tool
MadGraph_aMC@NLO 2.6 [27]. In order to compute the signal cross section we
make use of the tool SusHi [28, 29] that gives the results for Higgs boson
production cross sections at NNLO for the different production modes. The
obtained values confirm that the dominant production mode is $b$-quark
annihilation, with a cross section at least two orders of magnitude larger
than the gluon fusion mode.

Parameter | Scenario I | Scenario II
---|---|---
$m_{A}$ [GeV] | 947.5 | 1148.9
$\tan\beta$ | 33.8 | 45.33
$M_{1}$ [GeV] | 100 | 100
$M_{2},\,M_{3}$ [GeV] | 2200 | 2200
$\mu$ [GeV] | -327.2 | -273.13
$A_{\tau}$ [GeV] | -859.4 | 1125
$m_{\tilde{L}_{3}}$ [GeV] | 412.9 | 591.9
$m_{\tilde{E}_{3}}$ [GeV] | 393.8 | 363.1
$m_{H}$ [GeV] | 947.6 | 1149
$m_{\widetilde{\tau}_{1}}$ [GeV] | 367.5 | 350.7
$m_{\widetilde{\tau}_{2}}$ [GeV] | 408.4 | 583.9
$m_{\widetilde{\chi}_{1}^{0}}$ [GeV] | 99 | 98.2
Table 1: Benchmark points of Scenario I and Scenario II that are used to
develop the collider analysis.

In Table 1 we show two particular benchmark points that are representative of
each scenario and that we use to prove our search strategy. In Scenario I we
use a point with a heavy Higgs mass $m_{H}=947.6$ GeV, $\tan\beta=33.8$, and a
lightest stau mass $m_{\widetilde{\tau}_{1}}=367.5$ GeV. With these values the
production cross section for the heavy Higgs boson $H$ is $\sigma_{bbH}=194.2$
fb for $b$-quark annihilation and $\sigma_{ggH}=3.2$ fb for gluon fusion. As
we remarked before, the large $\tan\beta$ value makes the $b$-quark
annihilation cross section larger than gluon fusion. The values of the
branching fraction of the heavy Higgs boson $H$ decaying into staus and tau
leptons are 0.17 and 0.09, respectively. As we can see here, the enhancement
of the decay into staus leads to a decrease in the branching ratio to tau
leptons. The branching fraction of the lightest stau into a tau lepton and the
lightest neutralino is ${\rm
BR}(\widetilde{\tau}_{1}\to\tau\widetilde{\chi}_{1}^{0})=0.98$. Thus the total
cross section for the process $pp\to
H\to\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1}\to\tau\widetilde{\chi}_{1}^{0}\tau\widetilde{\chi}_{1}^{0}$
at $\sqrt{s}=14$ TeV is $\sigma^{\rm total}_{\rm S-I}=31.7$ fb. In the case of
Scenario II, we have $m_{H}=1149$ GeV, $m_{A}=1148.9$ GeV, and
$\tan\beta=45.33$. In this benchmark point the production cross section is
roughly $\sigma_{bbH}\sim\sigma_{bbA}\sim 120$ fb. The relatively large
production cross section despite the value of the masses is due to the large
value of $\tan\beta$ that enhances the $b$-quark annihilation production for
both CP-even and CP-odd Higgs bosons. For this scenario the stau masses are
$m_{\widetilde{\tau}_{1}}=350.7$ GeV and $m_{\widetilde{\tau}_{2}}=583.9$ GeV,
and the branching fraction of the heavy neutral Higgs bosons are ${\rm
BR}(A\to\sum_{i,j=1,2}^{i\neq
j}\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j})=0.25$ and ${\rm
BR}(H\to\sum_{i,j=1,2}^{i\neq
j}\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j})=0.22$. We can notice that for
this benchmark point the decay of the heavy Higgs bosons to a lightest stau
pair is zero in the case of the CP-odd Higgs boson and ${\rm
BR}(H\to\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{1})=0.01$ for the CP-even
one. This is just a realization of the properties of this scenario where there
is an enhancement of the chiral couplings and the non-mixed states of the
staus. The branching ratios for both stau states are ${\rm
BR}(\widetilde{\tau}_{2}\to\tau\widetilde{\chi}_{1}^{0})=0.28$ and ${\rm
BR}(\widetilde{\tau}_{1}\to\tau\widetilde{\chi}_{1}^{0})=0.82$ respectively.
Therefore, the total cross section of the process $pp\to
H/A\to\sum_{i,j=1,2}\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j}\to\tau\widetilde{\chi}_{1}^{0}\tau\widetilde{\chi}_{1}^{0}$
for this Scenario-II benchmark point at $\sqrt{s}=14$ TeV is $\sigma^{\rm
total}_{\rm S-II}=13.2$ fb. This cross section is smaller than the one
obtained in Scenario I, even when in Scenario II there are two resonant states
contributing to the production of the staus. However, this was expected since
in Scenario I $m_{H/A}$ is lighter than it is in Scenario II and, on top of
that, the branching ratio of staus is almost saturated by the
$\tau\widetilde{\chi}^{0}_{1}$ channel, which is not the case in Scenario II.

Background | Cross section [fb]
---|---
$t\bar{t}$ | 10125
$W$+jets | 6.257$\times 10^{6}$
$Z$+jets | 4.254$\times 10^{6}$
$WW$ | 1188.6
$ZZ$ | 183.3
Table 2: List of backgrounds and their cross sections to the process
$b\bar{b}\to
H/A\to\widetilde{\tau}_{i}^{*}\widetilde{\tau}_{j}\to\tau^{+}\widetilde{\chi}_{1}^{0}\tau^{-}\widetilde{\chi}_{1}^{0}$
at the LHC at a center-of-mass energy of $\sqrt{s}=14$ TeV.

The main backgrounds of the considered process are $t\bar{t}$, $W$+jets,
$Z$+jets, $WW$ and $ZZ$, and they are listed in Table 2 with their cross
sections at $\sqrt{s}=14$ TeV. In principle, one has to include the QCD
multijet background as well, however this is highly suppressed once the cuts
involving large amounts of $E_{T}^{\rm miss}$ are applied, as shown in Ref.
[20]. Although all the events corresponding to the background processes have
been generated at leading order, the cross sections for $t\bar{t}$, $WW$ and
$ZZ$ have been rescaled with K-factors of 1.5, 1.4, and 1.3, respectively,
extracted from Ref. [27]. In addition, the cross sections for the $W$+jets and
$Z$+jets backgrounds have been estimated by considering up to two light jets.
It is important to note that for the $t\bar{t}$, $W$+jets, and $WW$
backgrounds we have included only the decay of the $W$ boson into
$\tau\nu_{\tau}$, while in the case of the $ZZ$ and $Z$+jets backgrounds, we
have considered the decays $ZZ\to\tau^{+}\tau^{-}\nu\bar{\nu}$ and
$Z\to\tau^{+}\tau^{-}$, respectively. Both the signal and the different
backgrounds have been generated with MadGraph_aMC@NLO 2.6 [27] and showered
with PYTHIA 8 [30], while the detector response has been simulated with
Delphes 3 [31]. The implementation of the different cuts of the search
strategy that we present below have been carried out with MadAnalysis 5 [32]
in the expert mode.

###  3.1 Search strategy

We will describe here the search strategy that we follow which was first
proposed in Ref. [20]. First of all, we apply some basic selection cuts that
define the final state that we are searching for. We require that both signal
and background events exhibit two opposite-sign tau leptons and we also demand
that they have the following properties:

$\displaystyle p_{T}^{\tau_{1}}>50\,{\rm GeV}\,,\,p_{T}^{\tau_{2}}>40\,{\rm
GeV}\,,\,|\eta^{\tau}|<2.47\,.$ (16)

Here we define $\tau_{1}$ and $\tau_{2}$ as the leading and subleading tau
leptons, respectively, $p_{T}$ is the transverse momentum of the corresponding
tau lepton and $\eta^{\tau}$ is its pseudorapidity. Given the topology of the
signal process, one must rely on the large amount of transverse missing
energy, $E_{T}^{\rm miss}$, coming from the two LSP neutralinos escaping the
detector in order to discriminate it from the background. For such reason, in
this analysis we take into account kinematic variables that depend directly on
$E_{T}^{\rm miss}$:

  1. 1.

The transverse mass $m_{T}$, defined as

$\displaystyle m_{T}(\vec{p}_{T}^{\,\,i},\,\vec{p}_{T}^{\,\,\rm
inv})=\sqrt{m^{2}_{i}+2\left(\sqrt{m^{2}_{i}+|\vec{p}_{T}^{\,\,i}|^{2}}E_{T}^{\rm
miss}-\vec{p}_{T}^{\,\,i}\cdot\vec{p}_{T}^{\,\,\rm inv}\right)},$ (17)

where $i$ denotes the detected particle with transverse momentum
$\vec{p}^{\,\,i}_{T}$ and mass $m_{i}$, and $\vec{p}_{T}^{\,\,\rm inv}$ is the
total missing transverse momentum.

  2. 2.

The stransverse mass $m_{T2}$, which is designed to target events with two
sources of missing transverse momentum:

$\displaystyle
m_{T2}=\min\limits_{\not{\vec{p}}_{1}+\not{\vec{p}}_{2}=\vec{p}_{T}^{\,\,\rm
inv}}\left\\{\max\left[m_{T}(\vec{p}^{i}_{T},\not{\vec{p}}_{1}),m_{T}(\vec{p}^{j}_{T},\not{\vec{p}}_{2})\right]\right\\},$
(18)

where $i$ and $j$ are the two visible states from the parent decays, and
$\not{\vec{p}}_{1}$ and $\not{\vec{p}}_{2}$ are the corresponding missing
transverse momenta. The power of the $m_{T2}$ variable comes from the fact
that its distribution presents an endpoint around the mass of the parent
decaying particle. This feature makes this variable quite efficient to
discriminate between the signal and the $t\bar{t}$ and $WW$ backgrounds.

|
---|---
|
Figure 1: Four different distributions of kinematic variables after the
selection cuts are applied to the signal and background events. On each case
we show the distribution of the signal for the two scenarios along with those
corresponding to the backgrounds listed in Table 2. Top left panel: Transverse
mass of the leading tau lepton, $m_{T}^{\tau_{1}}$. Top right panel:
Transverse mass of the subleading tau lepton, $m_{T}^{\tau_{2}}$. Bottom left
panel: Invariant mass of the two tau leptons, $m_{\tau\tau}$. Bottom right
panel: Stransverse mass $m_{T2}$.

In Fig. 1 we depict the distributions of several variables after applying the
selection cuts defined above to the signal and background events. The two top
panels represent the transverse mass of the leading and the subleading tau
leptons. From these two distributions we can see that the background events
concentrate in the low transverse mass region, $m_{T}^{\tau}\lesssim 200$ GeV
for the leading tau lepton and $m_{T}^{\tau}\lesssim 120$ GeV in the case of
the subleading tau lepton. On the other hand, the distribution of the signal
events reaches heavier transverse masses due to the fact that there is more
missing transverse energy coming from the neutralinos. For that reason we
require the transverse mass of the two tau leptons to be greater than 120 GeV.
The bottom left panel shows the invariant mass of the tau lepton pair. We see
that it is easy to discriminate the events coming from the $ZZ$ and $Z+$jets
backgrounds since they peak at $m_{\tau\tau}\sim m_{Z}$. Therefore, we set a
cut on the invariant mass of the two tau leptons of $m_{\tau\tau}>100$ GeV.
The bottom right panel depicts the $m_{T2}$ variable. From the shape of the
distribution, it is clear that this variable is crucial to discriminate
between the signal and the background events. The aim of the $m_{T2}$ variable
is to select the processes in which there is a large amount of missing
transverse energy, $E_{T}^{\rm miss}$, coming from at least two sources.
Moreover, recall that this variable exhibit an endpoint around the mass of the
parent decaying particle. These features explain the quick decrease for the
background distributions, which are mostly concentrated at $m_{T2}<150$ GeV.
On the other hand, the signal distribution extends towards higher values of
$m_{T2}$. Based on this we impose the cut $m_{T2}>180$ GeV that has a
significant impact on the background events. In addition to the cuts already
explained, we have also included a cut in the angular separation of the two
tau leptons, $\Delta R(\tau_{1},\,\tau_{2})$, and imposed a $b$-jet
veto111Given that in our scenarios the main production mechanism of the
resonances is the $b$-quark annihilation, it could be interesting to study the
signal process without imposing the $b$-jet veto. In principle, one could
define different signal regions according to the number of $b$-tagged jets,
allowing for a better discovery rate and efficiency. However, the proper
identification of $b$-jets arising from the initial state is a complex task
and would require a detailed study on its own, which is out of the scope of
this paper. along with the requirement that the number of light jets is
smaller than 2. All the cuts are summarized in Table 3, where the left column
contains the selection cuts and the right column includes the cuts that define
our signal region. It is important to emphasize here that the fact that our
search strategy for stau pairs is based on their production through heavy
resonances ($H$ and $A$ Higgs bosons) allows us to impose a much more
restrictive cut on the $m_{T2}$ variable than in strategies based on the usual
electroweak stau production (see for example Table 1 of [19]). In the latter
case, the background is much less suppressed and unfortunately mimics the
signal better.

Selection Cuts | Signal Region Cuts
---|---
2 OS taus |  $N_{b}=0$ & $N_{j}<2$
$p_{T}^{\tau_{1}}>50\,{\rm GeV}$ | $\Delta R(\tau_{1},\tau_{2})<3.5$
$p_{T}^{\tau_{2}}>40\,{\rm GeV}$ |  $m_{T}^{\tau_{1}},m_{T}^{\tau_{2}}>120$ GeV
$|\eta^{\tau}|<2.47$ |  $m_{\tau\tau}>100$ GeV
|  $m_{T2}>180$ GeV
Table 3: List of cuts performed in the collider analysis. The first column
shows the selection cuts that define the process. In the second column we
depict the selected cuts in order to discriminate the signal from the
background.

We can now test the efficiency of the search strategy by applying it to the
benchmark points of Scenarios I and II given in Table 1. In order to simulate
the background, we have followed the same procedure as in Ref. [20],
generating the same number of events as it is indicated in Table 4 for every
background source. Following similar searches [12, 14] we assume a systematic
uncertainty of 30% on the estimated sum of all backgrounds. In order to
compute the significance of the signal events, $S$, with respect to the
background events, $B$, including the potential systematic uncertainties we
use [33]

$\displaystyle\mathcal{S}_{\rm
dis}=\sqrt{2\left((B+S)\log\left(\frac{(S+B)(B+\sigma_{B}^{2})}{B^{2}+(S+B)\sigma_{B}^{2}}\right)-\frac{B^{2}}{\sigma_{B}^{2}}\log\left(1+\frac{\sigma_{B}^{2}S}{B(B+\sigma_{B}^{2})}\right)\right)},$
(19)

where $\sigma_{B}=(\Delta B)B$, with $\Delta B$ being the relative systematic
uncertainty, in our case $\Delta B=$ 30%. In Table 4 we show the number of
events for every source of background and the signal events of both scenarios
at a LHC center-of-mass energy of $\sqrt{s}=14$ TeV and for a total integrated
luminosity of $\mathcal{L}=100$ fb-1. For each scenario we show a column with
the number of events if no cut is applied and a second column with the number
of events after applying the cuts. We see that with a 30% of systematic
uncertainties, a signal significance of 6.62$\sigma$ for Scenario I and
5.24$\sigma$ for Scenario II are obtained for a luminosity of 100 fb-1. In
spite of the differences between the two scenarios in terms of the nature of
the stau states, the decaying resonance and the Higgs-stau coupling, the
analysis appears to be efficient in both cases.

| Scenario I | Scenario II
---|---|---
| No Cuts | SR | No Cuts | SR
Signal | 3171 | 28.78 | 1317 | 21.16
$t\bar{t}$ | 1012500 | 2.03 | 1012500 | 2.03
$W$+jets | $6.257\times 10^{8}$ | 0.65 | $6.257\times 10^{8}$ | 0.65
$Z$+jets | $4.254\times 10^{8}$ | 1.01 | $4.254\times 10^{8}$ | 1.01
$WW$ | 118860 | 0 | 118860 | 0
$ZZ$ | 18330 | 0.37 | 18330 | 0.37
$\mathcal{S}_{\rm dis}$ | $\mathcal{O}(10^{-5})$ | 6.62 | $\mathcal{O}(10^{-5})$ | 5.24
Table 4: Number of signal and background events at $\sqrt{s}=14$ TeV for an
integrated luminosity of $\mathcal{L}=100$ fb-1 before and after applying
cuts. The last line represents the signal significance obtained using Eq.
(19).

One can also think the other way around and imagine that no significant signal
events are found for a given luminosity. In that situation, one can set 95%
C.L. exclusion limits by using the exclusion significance as follows [33]:

$\displaystyle\mathcal{S}_{\rm
exc}=\sqrt{2\left(B\log\left(\frac{B}{S+B}\right)+S\right)}\leq 1.64,$ (20)

where $B$ is the total number of background events and $S$ is the number of
signal events at a given luminosity $\mathcal{L}$. In the next section we will
analyze a set of points for each scenario in terms of exclusion and discovery
significances using this search strategy.

##  4 Results

In this section we use the search strategy described above and test it against
several benchmark points from both Scenario I and Scenario II. For each
benchmark point we study the efficiency of the analysis in terms of potential
exclusion at 95% C.L. and discovery signal significance by considering two
values of integrated luminosity at $\sqrt{s}=14$ TeV, $\mathcal{L}=100$ fb-1,
to be easily reached at the next run of the LHC, and $\mathcal{L}=1000$ fb-1,
corresponding to the high-luminosity LHC (HL-LHC). With these two values of
the luminosity we can explore the future prospects and the reach of this
search strategy in terms of physical parameters such as the mass of a new
heavy (pseudo) scalar.

###  4.1 Scenario I:
$H\to\widetilde{\tau}_{1}\widetilde{\tau}_{1}^{*}\to\tau^{+}\widetilde{\chi}_{1}^{0}\tau^{-}\widetilde{\chi}_{1}^{0}$

From Scenario I we take 27 benchmark points that were described in Ref. [20].
These points are characterized by different values of $m_{H}$, $\tan\beta$,
$A_{\tau}$, and $m_{\widetilde{\tau}_{1}}$. We have applied the search
strategy described in Section 3.1 and we have studied the exclusion power of
the analysis as well as the signal significance of discovery. The results are
shown in Fig. 2, where the orange points correspond to excluded benchmarks and
the blue ones to the benchmarks that cannot be ruled out at 95% C.L. by our
analysis.

|
---|---
|
Figure 2: Potential exclusion at 95$\%$ C.L. obtained from Eq. (20) in the
[$m_{H}$, $A_{\tau}$] plane (top) and [$\tan\beta$, $A_{\tau}$] plane
(bottom), within Scenario I, for a center-of-mass energy of $\sqrt{s}=$ 14 TeV
and total integrated luminosities of 100 fb-1 (left panel) and 1000 fb-1
(right panel). We display in orange the benchmark points that are excluded at
95% C.L. and in blue those that are allowed.

For $\mathcal{L}=100$ fb-1 23 of the 27 benchmarks points are excluded. All
the points with heavy Higgs boson mass smaller than 1000 GeV are ruled out.
Above this value, $m_{H}>$ 1000 GeV, there is still one benchmark with
$m_{H}$= 1075 GeV that can be excluded, while the remaining 4 benchmarks in
this mass region are allowed. This is due to the fact that the benchmark point
with $m_{H}=$ 1075 GeV has also a value of $A_{\tau}$ large enough to enhance
the coupling $g_{Hdd}$ and then the branching ratio into staus, which is
$\sim$16%. Moreover, this point has a negative value of $\mu$ and then its
contribution to the coupling in Eq. (1) adds to that corresponding to the
$A_{\tau}$. Finally, as can be seen in the plane [$\tan\beta$, $A_{\tau}$],
this benchmark point includes a large value of $\tan\beta$ which increases the
production cross section compensating the suppression due to the large value
of $m_{H}$. Furthermore, a large value of $\tan\beta$ also enhances $g_{Hdd}$.

Within the high-luminosity phase of the LHC, it could be possible to exclude
benchmark points with heavy Higgs boson masses above 1 TeV. However, as can be
seen from the right panel of Fig. 2, it seems that for trilinear couplings
smaller than 1 TeV, our analysis cannot probe heavy Higgs boson masses above
1.1 TeV, even with values of $\tan\beta$ as large as 42.

|
---|---
|
Figure 3: Signal significance in the [$m_{H}$, $A_{\tau}$] plane (top) and
[$\tan\beta$, $A_{\tau}$] plane (bottom), within Scenario I for a center-of-
mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities of 100
fb-1 (left panel) and 1000 fb-1 (right panel). Red circles correspond to
significances below the evidence level ($\mathcal{S}<3\sigma$), blue circles
to significances between the evidence level and the discovery one
($3\sigma<\mathcal{S}<5\sigma$), and green circles to significances larger
than the discovery level ($\mathcal{S}>5\sigma$).

In Fig. 3 we show the discovery prospects for each of the 27 benchmarks in the
[$m_{H}$, $A_{\tau}$] and [$m_{H}$, $\tan\beta$] planes. Signal significances
in the ranges $\mathcal{S}<3\sigma$, $3\sigma<\mathcal{S}<5\sigma$ (evidence
level) and $\mathcal{S}>5\sigma$ (discovery level) are displayed in red, blue
and green, respectively. We see that most of the benchmarks with $m_{H}$ below
1 TeV lie in the evidence or the discovery level. Only those around $m_{H}=$
910 GeV with a trilinear coupling $A_{\tau}=$ 720 GeV are below the evidence
level due to the small branching ratio of $H$ into staus, which is almost 10%.
Among these benchmark points, solely one could be tested by increasing the
luminosity to 1000 fb-1. The benchmarks in the mass region $m_{H}>$ 1 TeV are
difficult to probe even at $\mathcal{L}=1000$ fb-1. Again the exception is the
point with $m_{H}=$ 1075 GeV, due to the combination of a large trilinear
coupling ($A_{\tau}=$ 1 TeV) that leads to a branching ratio of 16%, and a
value of $\tan\beta=$ 43 that is large enough to increase the production cross
section despite the large heavy Higgs boson mass. From the low panels of Fig.
3, we see that there is a region with $\tan\beta$ $\in$ (37-41) and
$m_{H}\geq$ 1 TeV in which the search strategy is not efficient. This is due
to the fact that the production cross section decreases as $m_{H}$ grows, and
the benchmarks in the mass region above 1 TeV correspond to $\tan\beta$ values
that are not large enough to compensate this trough their impact on the
production cross section and the decay rate. It seems that values of
$\tan\beta$ above 41 are required in order to test the region $m_{H}>$ 1 TeV
with our search strategy.

Figure 4: Potential areas of exclusion at 95$\%$ C.L. within Scenario I for a
center-of-mass energy of $\sqrt{s}=$ 14 TeV. The area above the full line,
here in dark gray, represents the one that could be possibly excluded by this
analysis at 100 fb-1 of integrated luminosity if no evidence of signal is
found. The area below it, here in light gray, that is defined by the dashed
line shows the potentially excluded range at 1000 fb-1 of integrated
luminosity. However, the dashed line here is not visible since it goes below
masses of the lightest stau of $m_{\widetilde{\tau}_{1}}<200$ GeV. The shaded
red area is forbidden because the decay mode
$H\to\tilde{\tau}_{1}\tilde{\tau}_{1}^{\ast}$ is kinematically closed.

By performing an interpolation based on the 27 benchmark points studied above,
we can now interpret the obtained results in the [$m_{H}$,
$m_{\widetilde{\tau}_{1}}$] plane. This is shown in Fig. 4, where we display
the 95% C.L. exclusion limits for $\mathcal{L}=100$ fb-1 (dark gray) and
$\mathcal{L}=1000$ fb-1 (light gray). The red area on the left top corner is
kinematically forbidden. We see that the search strategy is able to probe most
of the [$m_{H}$, $m_{\widetilde{\tau}_{1}}$] plane with a luminosity of
$\mathcal{L}=100$ fb-1. On the other hand, a higher luminosity is required to
gain sensitivity in the region at $m_{H}>$ 930 GeV and
$m_{\widetilde{\tau}_{1}}<$ 260 GeV because the large values of $m_{H}$
reduces the production cross section and also the small values of
$m_{\widetilde{\tau}_{1}}$ lead to a substantial decrease in the amount of
$E_{T}^{\rm miss}$, which makes the $m_{T2}$ cut less powerful. The last
explains the fact that for a given heavy Higgs boson mass above 930 GeV we can
move from the allowed to the excluded region by increasing the stau mass. With
$\mathcal{L}=1000$ fb-1, the search strategy becomes sensitive to the whole
area comprised by heavy Higgs boson masses between 750 GeV and 1100 GeV and
stau masses between 200 GeV and 450 GeV.

|
---|---
Figure 5: Signal significance in the [$m_{H}$, $m_{\tilde{\tau}_{1}}$] plane,
within Scenario I, for a center-of-mass energy of $\sqrt{s}=$ 14 TeV and total
integrated luminosities of 100 fb-1 (left panel) and 1000 fb-1 (right panel).
The dark gray area above the dashed line is the discovery level region ($\geq$
5 standard deviations), while the light gray area above the solid line is the
evidence level region ($\geq$ 3 standard deviations). Finally, the white area
below the solid line corresponds to signal significances smaller than
3$\sigma$ and the red area is kinematically forbidden.

In Fig. 5 we present the same contour plot in the [$m_{H}$,
$m_{\widetilde{\tau}_{1}}$] plane as in Fig. 2 but for the discovery prospects
of the signal, for a total integrated luminosity of $\mathcal{L}=100$ fb-1
(left panel ) and 1000 fb-1 (right panel). We depict the evidence level
(3$\sigma$) as a light green area limited by a solid black line whereas the
discovery level (5$\sigma$) is shown as a darker green area limited by a
dashed black line. From the left panel we see that for $\mathcal{L}=100$ fb-1
the search strategy is sensitive to $m_{H}<$ 850 GeV regardless the value of
the stau mass (within the considered range). For $m_{H}>$ 850 GeV the
sensitivity is lost for stau masses below 300 GeV due to the same two reasons
discussed before in the case of the exclusion plot: on the one hand, the
signal cross section decreases considerably for large values of $m_{H}$, and
on the other one, small stau masses produce a final state with less energetic
tau leptons and lower $E_{T}^{\rm miss}$, which in turn reduces the
discrimination power of crucial kinematic variables as $m_{T2}$ or $m_{T}$.
This high $m_{H}$ range can still be probed if larger values of the stau mass
are considered. In particular, the discovery level is reached for
$m_{\widetilde{\tau}_{1}}>$ 350-370 GeV. For an integrated luminosity of 1000
fb-1, our analysis cover most of the considered area in the [$m_{H}$,
$m_{\widetilde{\tau}_{1}}$] plane. However, its sensitivity is not enough to
reach the region with $m_{H}>$ 900 GeV and $m_{\widetilde{\tau}_{1}}<$ 260
GeV. We see that this region of high $m_{H}$ and low
$m_{\widetilde{\tau}_{1}}$ is very challenging even within the context of the
HL-LHC.

###  4.2 Scenario II
$H/A\to\widetilde{\tau}_{1,2}\widetilde{\tau}_{2,1}^{*}\to\tau^{+}\widetilde{\chi}_{1}^{0}\tau^{-}\widetilde{\chi}_{1}^{0}$

The parameters involved in Scenario II are $m_{H}$, $m_{A}$, $\tan\beta$,
$A_{\tau}$, $m_{\widetilde{\tau}_{1}}$, and $m_{\widetilde{\tau}_{2}}$. Thus,
we have in this case an additional parameter arising from the stau sector,
namely, $m_{\widetilde{\tau}_{2}}$. We select in this case 228 benchmark
points. We explore the results obtained for each of them in terms of the
parameters $A_{\tau}$, $\tan\beta$, and $m_{A}$ first and then in the stau
sector, that we characterize by using the average of the two stau masses and
their difference

$\displaystyle{\overline{m}_{\widetilde{\tau}_{12}}}=\frac{m_{\widetilde{\tau}_{1}}+m_{\widetilde{\tau}_{2}}}{2},\quad\quad\Delta
m=m_{\widetilde{\tau}_{2}}-m_{\widetilde{\tau}_{1}}.$ (21)

We choose $m_{A}$ instead of $m_{H}$ because it is a natural parameter in the
MSSM and also one can obtain $m_{H}$ making use of $m_{A}$. Furthermore, as we
move in the decoupling limit we find usually that $m_{A}$ $\sim$ $m_{H}$.

|
---|---
|
Figure 6: Potential exclusion at 95$\%$ C.L. in the [$m_{A}$, $A_{\tau}$]
plane (top plots) and [$\tan\beta$, $A_{\tau}$] plane (bottom plots), within
Scenario II, for a center-of-mass energy of $\sqrt{s}=$ 14 TeV and total
integrated luminosities of 100 fb-1 (left panels) and 1000 fb-1 (right
panels). Benchmarks excluded by the analysis are shown in orange, while the
allowed ones are shown in blue.

In Fig. 6 we show the results of applying the 95% C.L. exclusion condition of
Eq. (20) to the Scenario-II benchmarks in the [$m_{A}$, $A_{\tau}$] (top) and
[$\tan\beta$, $A_{\tau}$] (bottom) planes for total integrated luminosities of
100 fb-1 (left) and 1000 fb-1 (right). From the plots on the top panel we see
that the exclusion power of the search strategy extends to masses up to 1200
GeV. Again, for a given mass, the sensitivity increases for higher values of
$|A_{\tau}|$. For $\mathcal{L}=100$ fb-1 all the points with $m_{A}\lesssim
840$ GeV are excluded even for the lowest values of $A_{\tau}$ considered here
($|A_{\tau}|$ $\sim$ 500 GeV). For $\mathcal{L}=1000$ fb-1 this conclusion is
valid for masses below 860 GeV. Above these masses the sensitivity depends on
the specific values of $A_{\tau}$ and $\tan\beta$. Regarding this last
parameter, we see from the bottom panels that all the points with
$\tan\beta\geq$ 47 ($\tan\beta\geq$ 42) are excluded for $\mathcal{L}=100$
fb-1 (1000 fb-1). The main reason for this behavior is the fact that larger
values of $\tan\beta$ enhance the $b$-quark annihilation production cross
section and the coupling with the staus at the same time, so that the search
strategy is quite efficient even for benchmark points with large $m_{A}$ and
$m_{H}$ and relatively low values of $A_{\tau}$ ($|A_{\tau}|$ $\sim$ 500 GeV).

|
---|---
|
Figure 7: Signal significance in the [$m_{A}$, $A_{\tau}$] plane (top plots)
and [$\tan\beta$, $A_{\tau}$] plane (bottom plots), within Scenario II for a
center-of-mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities
of 100 fb-1 (left panels) and 1000 fb-1 (right panels). Benchmarks with
significances below the evidence level ($\mathcal{S}<3\sigma$), between the
evidence level and the discovery level ($3\sigma<\mathcal{S}<5\sigma$) and
above the discovery level ($\mathcal{S}>5\sigma$) are shown in red, blue and
green, respectively.

In Fig. 7 we depict, in the same parameters planes as in Fig. 6, the results
corresponding to the signal significance prospects for $\mathcal{L}=100$ fb-1
(left) and 1000 fb-1 (right). Similarly to the results discussed above for the
exclusion limits, benchmarks with higher values of $|A_{\tau}|$ are more
likely to be detected by the search strategy. Specifically, all the benchmarks
with $|A_{\tau}|$ $\geq$ 1125 GeV have signal significances above 3$\sigma$,
with most of them reaching the discovery level. We note that for
$\mathcal{L}=1000$ fb-1 some of these benchmarks lie in the high mass region
with $m_{A}$ $\geq$ 1110 GeV. On the other hand, for both luminosities the
points with masses below 840 GeV exhibit significances at the discovery level
in spite of the relatively small value of $A_{\tau}$ ($\sim$ 500 GeV). The
same conclusions about the impact of the value of $A_{\tau}$ in the
significance can be read off on the lower panels. In addition, we also see
that for the case of $\mathcal{L}=1000$ fb-1 all the points with $\tan\beta$
$\geq$ 46 reach significances above 3$\sigma$. In fact, more than half of the
benchmarks lying in that region correspond to significances at the discovery
level.

|
---|---
|
Figure 8: Potential exclusion at 95$\%$ C.L. in the [$m_{A}$,
$\overline{m}_{\tilde{\tau}_{12}}$] plane (top) and [$\Delta m$,
$\overline{m}_{\tilde{\tau}_{12}}$] (bottom), within Scenario II, for a
center-of-mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities
of 100 fb-1 (left panels) and 1000 fb-1 (right panels). Benchmarks excluded by
the analysis are shown in orange, while the allowed ones are shown in blue.

Let us turn now to the results in terms of the stau variables defined in Eq.
(21). These are shown in Fig. 8 in the planes [$m_{A}$,
$\overline{m}_{\widetilde{\tau}_{12}}$] (top panels) and [$\Delta m$,
$\overline{m}_{\widetilde{\tau}_{12}}$] (bottom panels) for $\mathcal{L}=100$
fb-1 (left) and $\mathcal{L}=1000$ fb-1 (right). We see that all the points
with $\overline{m}_{\widetilde{\tau}_{12}}\geq$ 300 GeV are excluded in the
case of $\mathcal{L}=100$ fb-1, while this value decreases to
$\overline{m}_{\widetilde{\tau}_{12}}\geq$ 270 GeV for $\mathcal{L}=1000$
fb-1. This conclusion is also visible in the [$\Delta m$,
$\overline{m}_{\widetilde{\tau}_{12}}$] plane, from which we also note that in
the case of $\mathcal{L}=100$ fb-1 the points with
$\overline{m}_{\widetilde{\tau}_{12}}\geq$ 300 GeV appear to be easily tested
when the value of $\Delta m$ is smaller. The same conclusion can be drawn from
the plots corresponding to $\mathcal{L}=1000$ fb-1 for points with average
stau masses below 270 GeV. This behaviour is due to the fact that the $m_{T2}$
cut is more efficient for smaller values of $\Delta m$ since this kinematic
variable was originally designed to tag a pair of decaying particles with
equal mass.

|
---|---
|
Figure 9: Signal significance in the [$m_{A}$,
$\overline{m}_{\tilde{\tau}_{12}}$] plane (top) and [$\Delta m$,
$\overline{m}_{\tilde{\tau}_{12}}$] (bottom), within Scenario II for a center-
of-mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities of 100
fb-1 (left panels) and 1000 fb-1 (right panels). Benchmarks with significances
below the evidence level ($\mathcal{S}<3\sigma$), between the evidence level
and the discovery level ($3\sigma<\mathcal{S}<5\sigma$) and above the
discovery level ($\mathcal{S}>5\sigma$) are shown in red, blue and green,
respectively.

The results of the signal significance in terms of stau variables are shown in
Fig. 9. In this case most of the points with
$\overline{m}_{\widetilde{\tau}_{12}}\geq$ 300 GeV reach the discovery level.
In contrast, all the points with stau masses below this value cannot be probed
with the analysis at $\mathcal{L}=100$ fb-1. This situation improves only a
bit at $\mathcal{L}=1000$ fb-1, since in this case some points with
$\overline{m}_{\widetilde{\tau}_{12}}\leq$ 300 GeV show evidence level.
However, there are no points reaching the discovery level in this region. The
increase in luminosity from 100 fb-1 to 1000 fb-1 also makes that a
considerable number of points in the mass region above 300 GeV with large
values of $m_{A}$ or $\Delta m$ become accessible. In the case of the
parameter $\Delta m$, the behaviour is the same as in the exclusion plots. For
a given $\overline{m}_{\widetilde{\tau}_{12}}$ value, the efficiency of the
search strategy increases for smaller $\Delta m$ values.

Figure 10: Exclusion limits at 95% C.L. in the [$m_{A}$,
$\overline{m}_{\tilde{\tau}_{12}}$] plane, within Scenario II, for a center-
of-mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities of 100
fb-1 (dark gray) and 1000 fb-1 (light gray).

As we did in Section 4.1, we can interpolate the obtained results and show
them in contour-line plots. In Fig. 10 the contour-line plot of the exclusion
potential at 95% C.L. in the [$m_{A}$, $\overline{m}_{\tilde{\tau}_{12}}$]
plane is depicted. In this figure the dark gray area represents the exclusion
region for $\mathcal{L}=100$ fb-1, while the light gray area corresponds to
$\mathcal{L}=1000$ fb-1. We can observe that the search strategy is able to
exclude the region with $m_{A}\leq 850$ GeV. This region is slightly increased
to $m_{A}\leq 870$ GeV for $\mathcal{L}=1000$ fb-1. Above these masses, the
search strategy excludes in general average stau masses that are greater than
275-290 GeV and 250-275 GeV for $\mathcal{L}=100$ fb-1 and $\mathcal{L}=1000$
fb-1, respectively. As in the case of Scenario I, the proposed search strategy
is not sensitive to the region of low values of stau masses due to the
specific kinematic variables that drive its discrimination power.

|
---|---
Figure 11: Signal significance in the [$m_{A}$,
$\overline{m}_{\tilde{\tau}_{12}}$] plane, within Scenario II, for a center-
of-mass energy of $\sqrt{s}=$ 14 TeV and total integrated luminosities of 100
fb-1 (left panel) and 1000 fb-1 (right panel). The dark green and light green
areas correspond to significances at the evidence level (3$\sigma$) and at the
discovery level (5$\sigma$).

In Fig. 11 we show a similar contour-line plot as in Fig. 10 but for the
signal significances at $\mathcal{L}=100$ fb-1 (left) and $\mathcal{L}=1000$
fb-1 (right). The dark (light) gray area corresponds to significances at the
discovery (evidence) level. For $\mathcal{L}=100$ fb-1 the evidence level is
reached for masses $m_{A}\leq$ 825 GeV regardless the value of
$\overline{m}_{\widetilde{\tau}_{12}}$, while for masses above 825 GeV the
average stau mass needs to be larger than 300 GeV. For values of $m_{A}$ below
$\sim 780$ GeV significances at the discovery level are obtained within all
the considered $\overline{m}_{\widetilde{\tau}_{12}}$ range. It is interesting
to note that the discovery contour line drastically grows towards large values
of $\overline{m}_{\widetilde{\tau}_{12}}$ for $m_{A}$ $>$ 1090 GeV. This is
because for such high values of $m_{A}$, stau masses above 450 GeV are
required in order to reach 5$\sigma$ significances (see the upper left panel
of Fig. 9). For $\mathcal{L}=1000$ fb-1 the 3$\sigma$ region extends to
$m_{A}$ $\sim$ 850 GeV regardless the value of
$\overline{m}_{\widetilde{\tau}_{12}}$, and for $m_{A}$ $\geq$ 850 GeV
significances at the evidence level are obtained for
$\overline{m}_{\widetilde{\tau}_{12}}$ above 275-290 GeV. By looking at the
5$\sigma$ contour line, we conclude that significances at the discovery level
can be obtained for $m_{A}$ $<$ 815 GeV for any
$\overline{m}_{\widetilde{\tau}_{12}}$ within the range under study. Moreover,
larger masses can still reach significances at the discovery level if
$\overline{m}_{\widetilde{\tau}_{12}}$ is approximately above 300-320 GeV.
This is in contrast to the case of $\mathcal{L}=100$ fb-1, where the region in
which $m_{A}$ $>$ 1090 GeV is particularly challenging and requires quite
larger values of $\overline{m}_{\widetilde{\tau}_{12}}$ in order to reach the
discovery level.

###  4.3 Potential discrimination between stau mixing scenarios

We explore now the possibility to distinguish the two mixing scenarios once
they reach the discovery level with our search strategy and in the specific
case in which they exhibit similar relevant mass spectra
($m_{H},m_{\widetilde{\tau}_{1}}$, and $m_{\widetilde{\tau}_{2}}$). As stated
above, in Scenario I the tau leptons in the final state arise from the decay
of a pair of $\widetilde{\tau}_{1}$, while in Scenario II they originate from
the decay of the pair $\widetilde{\tau}_{1}\widetilde{\tau}^{*}_{2}$ or its
conjugate ($\widetilde{\tau}_{1}^{*}\widetilde{\tau}_{2}$). Thus, the main
difference between the signals associated to these scenarios relies on the
difference between the stau masses. In this sense, one may expect that
kinematic variables such as $m^{\tau_{1}}_{T}$, $m^{\tau_{2}}_{T}$, and
$m_{T2}$ will be sensitive to the mass splitting and therefore be well suited
to discriminate between scenarios.

Benchmark | $m_{H}$ | $m_{\widetilde{\tau}_{1}}$ | $m_{\widetilde{\tau}_{2}}$
---|---|---|---
SI-7 | 951 GeV | 367 GeV | 409 GeV
SI-20 | 1075 GeV | 320 GeV | 388 GeV
SII-47 | 951 GeV | 367 GeV | 409 GeV
SII-82 | 1099 GeV | 352 GeV | 583 GeV
Table 5: List of the relevant parameters of the four benchmarks used for the
comparison between Scenarios I and II.

In order to establish the extent of the above statement, we will compare the
two stau mixing scenarios by considering two benchmarks belonging to Scenario
I and two ones corresponding to Scenario II. The relevant parameters of these
four benchmarks are listed in Table 5. Note that the benchmarks SI-7 and
SII-47 have the same relevant mass spectrum whereas this is not the case for
SI-20 and SII-82. However, we can justify the use of this pair for the sake of
comparison as follows. First of all, for Scenario I, the considerable
difference in the value of $m_{\widetilde{\tau}_{2}}$ is not relevant since
only the light stau contributes to the process, and then a benchmark belonging
to it with exactly the same value of $m_{\widetilde{\tau}_{2}}$ than the
benchmark SII-82 would have exactly the same distributions as the SI-20.
Second, the differences in $m_{H}$ (24 GeV) and $m_{\widetilde{\tau}_{1}}$ (32
GeV) are significantly smaller than the mass splitting present in SII-82 (231
GeV) and then will not affect the main conclusions arising from the comparison
between the distributions.

In Fig. 12 we show the distributions corresponding to $m^{\tau_{1}}_{T}$,
$m^{\tau_{2}}_{T}$, and $m_{T2}$ after applying the cuts of our search
strategy (see Table 3). On the left panels (right panels) we compare the
distributions of the benchmarks SI-7 and SII-47 (SI-20 and SII-82). In the
case of benchmarks SI-7 and SII-47, we see that out of the three considered
variables222For the two comparisons between scenarios presented in this
section we have also explored many other distributions of variables such as
$E_{T}^{\text{miss}},|p_{T}^{\tau_{2}}/p_{T}^{\tau_{1}}|,m_{\tau\tau}$ or
$\Delta R(\tau_{1},\tau_{2})$. Since none of these distributions has proven to
be useful to discriminate between the stau mixing scenarios we do not include
any results in this regard. only the $m^{\tau_{1}}_{T}$ exhibits some
sensitivity to the mixing pattern, with the peaks of the distributions of SI-7
and SII-47 shifted by approximately 80 GeV. The difficulty to distinguish
these two benchmarks comes from the fact that the splitting between the stau
masses in SII-47 ($\Delta m=42$ GeV) is too small to produce traceable changes
in distributions based on the tau leptons in the final state. The case of the
benchmarks SI-20 and SII-82 is more promising since now the mass splitting is
significantly higher ($\Delta m=231$ GeV). In fact, as we can see from the
right panels of Fig. 12, not only the $m^{\tau_{1}}_{T}$ distributions are
shifted but also both the $m^{\tau_{2}}_{T}$ and $m_{T2}$ distributions
present different endpoints according to the benchmark. The $m^{\tau_{2}}_{T}$
distribution for SI-20 has an endpoint in $\sim 400$ GeV, while for SII-82 the
distribution extends until $\sim 600$ GeV. Thus, a cut such as
$m^{\tau_{2}}_{T}>350$ GeV rejects the majority of SI-20 events while
retaining a significant number of SII-82 events. The same behaviour occurs in
the $m_{T2}$ distributions, with the endpoint being $\sim 275$ GeV for SI-20
and around $475$ GeV for SII-82. Again, we see that by means of requiring
$m^{\tau_{2}}_{T}$ to be above 275 GeV, we are able to get rid off all the
SI-20 events while still keeping a substantial amount of SII-82 events. As
expected, we see that the higher the stau mass splitting the better the chance
of distinguishing between mixing patterns through the inspection of kinematic
distributions in the proposed signal region.

|
---|---
|
|
Figure 12: Distributions of the kinematic variables
$m^{\tau_{1}}_{T},m^{\tau_{2}}_{T}$ and $m_{T2}$ (from top to bottom) after
applying the cuts listed in Table 3 for benchmarks SI-7 and SII-47 (left
panels) and SI-20 and SII-82 (right panels).

##  5 Conclusions

In this paper we have proven that our stau pair search strategy, developed in
[20] and applied to a type of MSSM scenario in the large-$\tan\beta$ regime
with large stau mixing, dominated by decays of the heavy CP-even Higgs $H$ to
a pair of lightest staus, $\widetilde{\tau}_{1}\widetilde{\tau}_{1}^{*}$
(Scenario I), is also very efficient in the complementary scenario in which
decays of both the CP-even and CP-odd heavy Higgs contribute mainly to the
production of $\widetilde{\tau}_{1}\widetilde{\tau}_{2}^{*}\;+\;c.c$ pairs
(Scenario II), and focusing also on the stau decays that drive to final states
made up of a $\tau$-lepton pair and a large amount of missing transverse
energy.

This search strategy, with a luminosity of $\mathcal{L}=100$ fb-1, allows us
to set exclusion limits at the 95% C.L. for most of the [$m_{H}$,
$m_{\widetilde{\tau}_{1}}$] parameter space of Scenario I, if $m_{H}<$ 930 GeV
and $m_{\widetilde{\tau}_{1}}>$ 260 GeV. With a HL-LHC luminosity of 1000 fb-1
the search strategy is able to exclude the whole Scenario-I area comprised by
heavy-Higgs masses between 750 GeV and 1100 GeV and stau masses between 200
GeV and 450 GeV. On the other hand, for $\mathcal{L}=100$ fb-1, we can reach
signal significances at the evidence level if $m_{H}<$ 850 GeV regardless the
value of the stau mass (within the considered range). If one requires
discovery level significances, stau masses above 350 GeV are needed. For
$\mathcal{L}=1000$ fb-1, our analysis is sensitive to most of the considered
area in the [$m_{H}$, $m_{\widetilde{\tau}_{1}}$] plane, although not enough
to reach the region with $m_{H}>$ 900 GeV and $m_{\widetilde{\tau}_{1}}<$ 260
GeV.

With regard to Scenario II, the search strategy excludes the region with
$m_{A}\leq 850$ GeV at the 95% C.L with $\mathcal{L}=100$ fb-1. This region
extends slightly to $m_{A}\leq 870$ GeV for $\mathcal{L}=1000$ fb-1. Above
these masses, the search strategy sets 95% C.L exclusion limits for average
stau masses that are greater than 275-290 GeV (250-275 GeV) for
$\mathcal{L}=100$ fb-1 ($\mathcal{L}=1000$ fb-1). Considering a luminosity of
100 fb-1, significances at the discovery level are obtained for masses
$m_{A}\leq$ 780 GeV regardless the value of
$\overline{m}_{\widetilde{\tau}_{12}}$, while for masses above 780 GeV the
average stau mass needs to be larger than 300-320 GeV or even higher ($\sim
450$ GeV) when $m_{A}$ is above 1090 GeV. In the case of the HL-LHC with
$\mathcal{L}=1000$ fb-1, 5$\sigma$ significances can be reached for $m_{A}$
$<$ 815 GeV for any $\overline{m}_{\widetilde{\tau}_{12}}$ within the range
under study. In addition, larger masses can still give rise to discovery-level
significances if $\overline{m}_{\widetilde{\tau}_{12}}$ is approximately above
300-320 GeV.

Finally, under the assumption that the LHC will be able to discover staus by
means of our search strategy, we have outlined the potential for
discriminating between the two possible scenarios of stau mixing within the
large-$\tan\beta$ regime, when they share the same relevant mass spectrum and
both reach 5$\sigma$ significances with our search strategy. We have shown
that kinematic cuts in variables sensitive to the stau mass splitting, such as
$m^{\tau_{1}}_{T}$, $m^{\tau_{2}}_{T}$, and $m_{T2}$, may be useful to discern
which of these two types of MSSM scenario is realized in nature. By comparing
two pairs of benchmarks, one with $\Delta m=42$ GeV and the other with $\Delta
m=231$ GeV, we have illustrated the fact that the discrimination power of
these variables depends essentially on how large the mass splitting is.

As a main conclusion, we can say that our search strategy is really efficient
for the discovery or for the exclusion of heavy (scalar or pseudoscalar) Higgs
bosons decaying into a stau pair. From a more general point of view, our
collider analysis could be applied to any process at the LHC with the resonant
production of a pair of charged scalars which decay into a tau lepton and a
dark-matter candidate, resulting in final states with a $\tau$-lepton pair
plus a large amount of $E_{T}^{\text{miss}}$.

## Acknowledgments

The work of EA is supported by the “Atracción de Talento” program (Modalidad
1) of the Comunidad de Madrid (Spain) under the grant number
2019-T1/TIC-14019. This work has been also partially supported by CONICET and
ANPCyT under projects PICT 2016-0164 (EA, AM, NM), PICT 2017-2751 (EA, AM,
NM), PICT 2017-2765 (EA), and PICT 2017-0802 (AM). VML acknowledges support by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany‘s Excellence Strategy – EXC 2121“Quantum Universe” – 390833306. VML
thanks warmly IFLP in La Plata for kind hospitality hosting him during the
completion of this work.

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Forward Proton Measurements with ATLAS

M. Schmidt1$\star$ on behalf of the ATLAS Collaboration111Copyright 2022 CERN
for the benefit of the ATLAS Collaboration. Reproduction of this article or
parts of it is allowed as specified in the CC-BY-4.0 license.

1 University of Wuppertal, Germany

<EMAIL_ADDRESS>
51st International Symposium on Multiparticle Dynamics (ISMD2022)
Pitlochry, Scottish Highlands, 1-5 August 2022 10.21468/SciPostPhysProc.?

## Abstract

The ALFA subdetector is designed to measure elastic proton-proton scattering
from which the total cross section and $\rho$-parameter are determined. In
2016, special runs at a center-of-mass energy of $\sqrt{s}=13$ TeV and with
$\beta^{\star}=2.5$ km were recorded.

###### Contents

  1. 1 Introduction
  2. 2 Experimental Setup
  3. 3 Kinematics
  4. 4 Reconstruction

##  1 Introduction

The goal of this measurement campaign [1] is to study elastic proton-proton
scattering, especially within the very interesting Coulomb-Nuclear-
Interference (CNI) region. In order to study the energy evolution, for these
runs special optics with large $\beta^{\star}$ values of 2.5 km in combination
with high center-of-mass energy of $\sqrt{s}=13$ TeV have been chosen which
makes it possible to reach down to very small momentum-transfers as
illustrated in Figure 1.

Figure 1: Model prediction of the differential elastic cross-section [2]

An important quantity to be measured is the $\rho$-parameter which is defined
as the fraction of the real part of the elastic scattering amplitude and the
imaginary part for small $t$:

$\rho=\left.\frac{\mathrm{Re}[f_{\mathrm{el}}(t)]}{\mathrm{Im}[f_{\mathrm{el}}(t)]}\right|_{t\rightarrow
0}$ (1)

The total cross-section can be calculated from the differential elastic cross-
section with the help of the optical theorem:

$\sigma_{\mathrm{tot}}=4\pi\,\mathrm{Im}\,[f_{\mathrm{el}}(t\rightarrow 0)]$
(2)

The description of this analysis is divided into 2 parts. This document covers
the experimental setup and theoretical background.

##  2 Experimental Setup

The ALFA detector [3] contains 4 roman pot (RP) detector stations that are
placed around 240 m away from the interaction point (IP) of the ATLAS
detector. The inner stations on the A- and C-side have a distance of 237 m to
the IP, whereas the outer stations are positioned 245 m from the IP. A sketch
of the experimental setup on the left side of Figure 2 illustrates the
arrangement of all stations and detectors.

Figure 2: Experimental Setup of ALFA and a sketch of 2 roman pot stations on
one side of the detector in the outgoing beam direction [1]

Due to the back-to-back scattering of both elastically interacting protons,
the 8 detectors can be divided into 2 groups. The upper detectors at side A
and the lower detectors at side C are combined to Arm 1, whereas Arm 2
contains the other 4 detectors. Each station contains 2 main detectors (upper
and lower MDs) and 2 overlap detectors (OD+ and OD-) that contain an upper and
a lower part. The MDs contain $2\times 10$ layers of 64 scintillating fibers
with a side length of 0.5 mm attached to 64-channel Multi-Anode
Photomultipliers (MaPMTs). If a proton hits these fibers, the produced photons
are internally reflected in the fibers and kick out a photoelectron when
reaching the photocathode of the related MaPMT. The resulting avalanche is
then detected on the segmented anode. For digitization, an analog Multi-Anode
ReadOut Chip (MAROC) in combination with a Field-Programmable Gate Array
(FPGA) is used. From the channel number related to the hit fiber, the proton
position in each layer can be reconstructed and converted into $x$ and
$y$-coordinates.

The right side of Figure 2 shows a sketch of 2 RP stations including both MDs
and ODs on the A- or C-side in the beam direction of the outgoing protons from
the IP. The upper and lower parts are colored in blue and green respectively.

##  3 Kinematics

During the elastic scattering process of the 2 interacting protons, the total
energy and momentum of both particles are conserved. Hence, the momentum
transfer $t$, which is defined as the square of the difference between the
momenta of the incoming and outgoing proton, can be calculated according to

$t=q^{2}=(p_{1}-p_{3})^{2}=4p^{2}\sin^{2}\left(\frac{\theta}{2}\right)$ (3)

as shown in Figure 3. For small scattering angles, the sine function can be
approximated according to $\sin\theta\approx\theta$ which leads to the
following relation:

$t\approx(p\theta)^{2}$ (4)

The momentum transfer is therefore simply given as the squared product of the
beam momentum and the scattering angle. Since the beam momentum is known, the
scattering angle of every single event has to be determined with ALFA, in
order to reconstruct the momentum transfer for every scattered proton pair.

$p_{3}$$p_{4}$$p_{1}$$p_{2}$$\theta$$p_{1}$$p_{3}$$q=p_{3}-p_{1}$$\theta/2$
Figure 3: Kinematics of elastic proton-proton scattering

##  4 Reconstruction

Before the $t$-spectrum can be reconstructed, an alignment procedure has to be
performed which mainly consists of an iterative procedure and a
$\chi^{2}$-optimization [4]. For improving the results, a physics-driven
alignment [1] is additionally required for this analysis. After the elastic
scattering process, the outgoing protons are deflected by LHC quadrupole
magnets. The measured angle $\theta$ is therefore not identical to the
scattering angle $\theta^{\star}$ at the IP. In order to calculate
$\theta^{\star}$, the beam optics transport matrix elements have to be taken
into account. Several reconstruction methods with different levels of
precision have been developed throughout the past years. For the $t$-spectrum
reconstruction of this measurement campaign, the so-called subtraction method
delivers the best performance.

This method contains the computation of the difference between the measured
$x$ and $y$-coordinate of each detector side. The result then has to be
divided by the sum of the matrix elements $M_{12}$ for both sides which
directly results in $\theta^{\star}$:

$\theta^{*}=\frac{\\{x,y\\}_{A}-\\{x,y\\}_{C}}{M_{12,A}+M_{12,C}}$ (5)

Figure 4: Selecting elastic events from correlation between $y$-values [1]

With the help of the determined momentum transfer of each event and the number
of events within the covered $t$-range, the elastic cross-section can be
calculated. The full analysis contains several steps such as the selection of
elastic events as shown in Figure 4 for the example of the correlation between
$y$-values of both detector sides. Additionally, background subtraction,
acceptance, unfolding corrections, reconstruction efficiency, and
normalization with the luminosity, measured independently, are required.

The total cross-section is then given by

$\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{1}{16\pi}\left|-8\pi\alpha\hbar
c\frac{G^{2}(t)}{|t|}e^{i\alpha\phi(t)}+(\rho+i)\frac{\sigma_{\mathrm{tot}}}{\hbar
c}e^{\frac{-B|t|-Ct^{2}-D|t|^{3}}{2}}\right|^{2}$ (6)

The left addend represents the Coulomb amplitude, whereas the right one stands
for the nuclear term. Therefore, $G$ denotes the electric form factor of the
proton and $B$ the nuclear slope parameter. This formula is only valid for
small momentum transfers $t\rightarrow 0$. All parameters including $\rho$ and
$\sigma_{\mathrm{tot}}$ can be determined with a fit to the reconstructed
$t$-spectrum. The inelastic cross-section is then simply given by the
difference between the total cross-section and total elastic cross-section:

$\sigma_{\mathrm{inel}}=\sigma_{\mathrm{tot}}-\sigma_{\mathrm{el}}$ (7)

The total elastic cross-section is determined by integrating the nuclear part
of the differential elastic cross-section. The inelastic cross-section is
therefore not an independent measurement, but rather a derived quantity based
on the main fit result.

## References

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