Limits on an Exotic Higgs Decay From a Recast ATLAS Four-Lepton Analysis

ATLAS uses a variety of trigger streams in selecting events for this analysis. These are the following, where the number(s) in parentheses indicate the minimum $p_{T}$(s) required:

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  single electron (26 GeV),

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  single muon (26 GeV),

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  dielectron symmetric (17 GeV),

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  dimuon symmetric (14 GeV),

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  dimuon asymmetric (22 GeV and 8 GeV),

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  electron (17 GeV) and muon (14 GeV),

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  two electron (12 GeV) and muon (10 GeV), and

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  electron (12 GeV) and two muon (10 GeV).

All electrons must have $|\eta_{e}|<2.47$, while all muons must have $|\eta_{\mu}|<2.7$ ATLAS:2020gty ; ATLAS:2019dpa .

Events are selected using the following cut flow. First, kinematic cuts are imposed on all observed leptons. Electrons must have $p_{T}>4.5$ GeV and $|\eta|<2.47$. Muons must have $p_{T}>3$ GeV and $|\eta|<2.7$. All leptons must be prompt.¹¹1 The longitudinal impact parameter $z_{0}$ relative to the primary vertex must satisfy $|z_{0}\,{\rm{sin}}(\theta)|<0.5$, where $\theta$ is the angle of the lepton track relative to the beampipe. The transverse impact parameter $d_{0}$, measured with uncertainty $\sigma_{d}$, must satisfy $d_{0}/\sigma_{d}<3\ (5)$ for muons (electrons).

Leptons must also be isolated, with loose isolation requirements ATLAS:2016lqx ; ATLAS:2019qmc ; ATLAS:2020auj . Electrons of transverse momentum $p_{T}^{e}$ are required to have $E_{\rm{cone}20}<0.2\,p_{T}$ and $(p_{T})_{\rm{varcone}20}<0.15\,p_{T}$, where

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  $E_{\rm{cone}20}$ is the energy of all particles within a cone of $\Delta R=0.2$ surrounding the electron, and

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  $(p_{T})_{\rm{varcone}20}$ denotes the scalar sum of the transverse momentum (relative to the beam) of all charged particles (with $p^{c}_{T}>1$ GeV and $|\eta|<2.5$) that lie within a cone of radius $\Delta R=$ min(10 GeV/$p_{T}^{e}$, 0.2) around the electron.

Muons are required to satisfy $(p_{T})_{\rm{varcone}30}+0.4\,E_{\rm{neflow}20}<0.16\,p_{T}^{\mu}$, where

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  $E_{\rm{neflow}20}$ is the transverse energy of all neutral particle flow candidates within a cone of $\Delta R=0.2$ surrounding the muon, and

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  $(p_{T})_{\rm{varcone}30}$ denotes the scalar sum of the transverse momentum (relative to the beam) of all charged particles (with $p^{c}_{T}>0.5$ GeV and $|\eta|<2.5$) that lie within a cone of radius $\Delta R=$ min(10 GeV/$p_{T}^{\mu}$, 0.3) around the muon.

If at least four leptons satisfy all of these requirements — we will refer to this as the “ID $\geq 4$” step – the leptons are combined into same-flavor opposite-charge (SFOC) pairs in all allowed combinations. If no combination has at least two SFOC pairs, the event is rejected.

Any two SFOC pairs forms a lepton quadruplet. All possible SFOC quadruplets must have invariant mass $m_{4\ell}<m_{Z}-(5$ GeV), or the event is rejected. Within a quadruplet, the invariant masses of the SFOC pairs are labeled as $m_{12}$ and $m_{34}$, with $m_{12}>m_{34}$. If the SFOC assignment is ambiguous, as in events with 4 leptons of the same flavor, then the pairing is selected that has the smaller value of $m_{12}-m_{34}$. Similarly, if there are multiple quadruplets, the one with the smallest $|m_{12}-m_{34}|$ is chosen.

A cut is then placed on the angular separation between leptons in the quadruplet. Same-flavored leptons must have $\Delta R>0.1$ and different-flavored leptons must have $\Delta R>0.2$.

Since the signal includes two equal-mass lepton pairs, ATLAS requires that $m_{34}/m_{12}>0.85$. Finally, to avoid background from hadronic resonances, all SFOC pairs in the quadruplet (including alternative pairings if the leptons are all of the same flavor) must satisfy

where $m_{\Upsilon(1s)}=9.460$ GeV and $m_{\Upsilon(3s)}=10.355$ GeV.

Since dark photons appear in the ATLAS analysis and in some of the other signals used in this paper, we briefly discuss how we treat dark photons.

Following the theory framework of Gopalakrishna:2008dv , we introduce a $U(1)_{D}$ coupled to the SM through kinetic mixing between the new gauge boson $X_{\mu}$ and the hypercharge gauge boson $B_{\mu}$ (all quantities with a caret are the bare ones before diagonalization). The relevant terms in the Lagrangian are

where $\hat{X}_{\mu\nu}$ $(B_{\mu\nu})$ is the field strength tensor of $\hat{X}_{\mu}$ ($B_{\mu}$), and $\xi$ is a small constant. We can diagonalize the kinetic terms and calculate the mass eigenstates of the massless photon, $Z$, and $X$. The branching ratio of the dark photon to various SM final states, calculated with DarkCast Baruch:2022esd , are shown in Fig. 3.

The ATLAS MC study includes the simulation of five points in parameter space, all with $m_{h_{D}}=50$ GeV. We study the three cases $m_{A^{\prime}}=8,\ 15,\ 20$ GeV; the other two have off-shell dark photons ($m_{A^{\prime}}>\frac{1}{2}m_{h_{D}}$) and lie outside our focus.

For the MC study, a simulation is done at 13 TeV center of mass, combining MadGraph 5_aMC@NLO Alwall:2014hca for the partonic process and PYTHIA 8.230 Sjostrand:2014zea for showering and hadronization, using the A14 parton-shower tune TheATLAScollaboration:2014rfk . The NNPDF3.0nlo parton distribution function is used NNPDF:2014otw . Pile-up is not included. The hard process $pp\rightarrow Z\rightarrow A^{\prime}h_{D}$, $h_{D}\rightarrow A^{\prime}A^{\prime}$ was generated in MadGraph using the HAHM model Schabinger:2005ei ; Curtin:2013fra ; Curtin:2014cca with $\epsilon=10^{-2}$, $\kappa=10^{-10}$, $m_{h_{D}}=50$ GeV, and 6 GeV $\leq m_{A^{\prime}}\leq$ 24 GeV. Branching ratios were computed automatically in MadGraph, and Madspin was used to allow the three $A^{\prime}$s to decay to SM fermion-antifermion pairs.

The ATLAS MC study takes a shortcut that ensures that all events have at least four signal leptons (here lepton means electron or muon.) This is done by letting one of the $A^{\prime}$ decay into any kinetically allowed fermion pair, including quarks and neutrinos, while restricting the decay of the other two $A^{\prime}$ to $ee$ or $\mu\mu$. The contribution of decays to $\tau\tau$ was ignored, as the analysis cuts remove most such events.²²2A matching procedure is adopted if $m_{A^{\prime}}\geq\frac{1}{2}m_{h_{D}}$ or $m_{A^{\prime}}+m_{h_{D}}\geq m_{Z}$. We ignore this issue here, since we only focus on the part of parameter space where $h_{D}$ and all $A^{\prime}$s are on-shell.

This shortcut only approximately reproduces the count of 6 lepton events versus 4, as the combinatorics of leptonic decays is not strictly correct. This results in a small error in the ATLAS MC study simulation relative to a full simulation of the model. In using this ATLAS MC study to estimate efficiencies, we use the same, slightly erroneous, method. However, in reproducing the full analysis and in imposing limits on other models, we do complete simulations, avoiding this approximation.

One additional detail is that the ATLAS Monte Carlo study imposes the trigger after the cut flow. This unrealistic ordering means that, after having matched the ATLAS MC study, we have to estimate the uncertainties in the efficiencies arising from the reversed order. Fortunately these appear small; see Section 3.1.2.

The cut flow of the MC study mostly follows the flow in the analysis as described above in Section 2.2, although it begins by imposing a “Monte Carlo (MC)” cut that removes all simulated events except those that have at least four leptons with $p_{T}>2$ GeV and $\eta<3$. Cuts are then imposed in the order described above in Section 2.2.

In this section, we will elaborate the procedures to obtain our limits on the signal strength $\mu_{\rm sig}\equiv{\cal L}\ \sigma(pp\to Z\to A^{\prime}h_{D}\to 4\ell+X)$, where ${\cal L}$ is the integrated luminosity of 139 fb^-1. We produce truth-level samples as for the MC study (except that we simulate the full model rather than just four-lepton events.) We apply the trigger pathways as cuts, along with the recalibration factor $r_{{\rm trig}}$ on events with four or more leptons, to set our initial event sample. We then apply the cuts of the ATLAS analysis to those that remain, keeping only a fraction $r_{\rm lep}$ of the leptons that pass the ID $\geq 4$ cut.

We then follow the approach in the ATLAS search ATLAS:2023jyp , dividing the signal region into multiple bins of $\bar{m}_{\ell\ell}\equiv(m_{\ell_{1}\ell_{2}}+m_{\ell_{3}\ell_{4}})/2$ with 1 GeV width. Each relevant $\bar{m}_{\ell\ell}$ bin is treated as an independent channel. In particular, for the $i$-th signal bin of a certain $\{m_{h_{D}},m_{A^{\prime}}\}$ benchmark, its average signal yield $S_{i}(m_{h_{D}},m_{A^{\prime}})$ is determined by

with

Here $\epsilon_{\rm sig}(m_{h_{D}},m_{A^{\prime}})<1$ is the overall signal efficiency, $\kappa_{i}(m_{h_{D}},m_{A^{\prime}})$ is the normalized shape factor of the dilepton resonance peak, and $N_{d}$ is the total number of relevant $\bar{m}_{\ell\ell}$ bins. For brevity, we will omit the dependence of $S_{i}$, $\epsilon_{\rm sig}$ and $\kappa_{i}$ on the model parameters $m_{h_{D}},m_{A^{\prime}}$ in all following equations, and (unless otherwise specified) expressions should be understood to implicitly account for these parameters. Meanwhile, the corresponding background yield $B_{i}$ for each channel is directly provided in Ref. ATLAS:2023jyp . Following the $CL_{S}$ method Junk:1999kv ; Read:2002hq ; Cowan:2010js widely accepted by the LHC community, the observed and expected limits on $\mu_{\rm sig}$ are then obtained once $\epsilon_{\rm sig}$ and $\{\kappa_{i}\}$ are known.

As shown in Sec. 3.1, the $\epsilon_{\rm sig}$ of the three benchmark points in Table 2 match with the results of the ATLAS’s MC study once the necessary recalibration factors $r_{\rm lep},~{}r_{\rm trig}$ are applied. With the same set of recalibration factors, we calculate $\epsilon_{\rm sig}$ for all other mass values.

The signal shapes $\left\{\kappa\right\}$ are provided at multiple values of $m_{h_{D}},m_{A^{\prime}}$ in the supplementary material of ATLAS:2023jyp . We interpolate between these shapes to obtain the signal shape at other parameter points. Thanks to the excellent detector resolution on lepton momenta, the reconstructed $\bar{m}_{\ell\ell}$ peaks are narrow, allowing us to take the four leading $\bar{m}_{\ell\ell}$ bins to calculate $\kappa_{i}$ by normalizing their sum to unity. However, the signal region is always dominated by at most two $\bar{m}_{\ell\ell}$ bins close to $m_{A^{\prime}}$. We keep only the two leading $\bar{m}_{\ell\ell}$ bins in the following analysis to reduce computational cost.

Our methods rest on poorly known and crudely defined recalibration factors, so the systematic uncertainties on our estimates of $\epsilon_{\rm sig}$ are large. We also share ATLAS’s systematic uncertainty on the backgrounds $B_{i}$. It is clear that systematic uncertainties were important for ATLAS,⁴⁴4Were we to account only for statistical uncertainties, our limits would be stronger than those obtained by ATLAS. and we must account for them as well.

We introduce the systematic effects by promoting the overall signal efficiency $\epsilon_{\rm sig}$ and the expected background yield $B_{i}$ in the $i$-th bin to nuisance parameters instead of constants. The true distributions are unknown, but here we will approximate them by log-normal distributions. We also assume that, for each model benchmark $\{m_{h_{D}},m_{A^{\prime}}\}$, the uncertainties in the two $\bar{m}_{\ell\ell}$ bins that we retain are strongly correlated, so that they can be approximately described by the same parameter.

We then introduce parameters $\delta_{\epsilon}$ and $\delta_{B}$ for each $\{m_{h_{D}},m_{A^{\prime}}\}$ benchmark, defined as follows:

and

where $\bar{\epsilon}_{\rm sig}$ and $\bar{B}_{i}$ are the central values obtained from the previous procedure. Even when $\sigma_{\epsilon}$ or $\sigma_{B}$ is $\mathcal{O}(1)$, the distribution in Eq. (5) ensures the expected event yields are positive.

The SM background distributions ${\bar{B}}_{i}$ and systematic uncertainties $\sigma_{B,i}$ are provided in the ATLAS analysis ATLAS:2023jyp . The quantity $\sigma_{\epsilon}$ for each mass benchmark can be decomposed into several independent terms:

where $\sigma_{\rm trig}$, $\sigma_{\rm lep}$ are the uncertainties of the corresponding recalibration factors $r_{\rm trig}$, $r_{\rm lep}$ obtained in Sec. 3.1, and $\sigma_{\rm theo}$ is a theoretical uncertainty.⁵⁵5Note that the lepton identification uncertainties for the four leptons are strongly correlated and thus appear in Eq. (6) collectively. We now explain how we estimate these uncertainties.

The two recalibration factors characterize the discrepancy between our simplified simulation, which does not include detector efficiencies and trigger thresholds, and a more sophisticated simulation. But our success in matching the MC study suggests a method for estimating the uncertainties in these two factors. Neither would be expected to be close to one, and of course neither can exceed one. This motivates us to assign the 1$\sigma$ fluctuation of $\epsilon_{\rm sig}$ to be $\frac{1}{2}(1-r_{\rm sig})=0.11$, which puts the unreasonable circumstance of $r_{\rm sig}>1$ a full 2$\sigma$ away from the central value.⁶⁶6 This uncertainty is comparable to the difference between our $r_{\rm lep}$ value and ATLAS’s average lepton efficiency in this $p_{T}$ range, as one can infer from ATLAS technical reports ATLAS:2016lqx ; ATLAS:2019qmc . The corresponding $\sigma_{\rm lep}$ is therefore $(1-r_{\rm lep})/2r_{\rm lep}=0.14$. Similarly, we take $\sigma_{\rm trigger}$ to be $(1-r_{\rm trig})/2r_{\rm trig}=0.12$.

The term $\sigma_{\rm theo}$ denotes theoretical uncertainties induced by perturbative calculations, hadronization, and parton distribution functions. Excluding any detector effects, the simulations described in Sec. 3.1 are almost identical to the ones in the ATLAS $Z\to 6f$ study. We therefore set this term to be $\sigma_{{\rm theo}}=0.14$, the same as in the ATLAS analysis.

Taking this into account, the overall $\sigma_{\epsilon}$ is 0.59, conservative but not unreasonable for a theoretical study. Its size makes clear why we have chosen log-normal distributions rather than Gaussian distributions for our analysis.

There are other contributors to $\sigma_{\epsilon}$ not included above, such as effects from pile-up and from detector energy resolution. However, according to ATLAS:2023jyp , the size of such “experimental uncertainties” (in which pile-up is included) on signal efficiency is at most $7\%$, too small to impact the above value of $\sigma_{\epsilon}$. We thus do not include these terms for simplicity.

With systematic effects modeled in this way, we obtain 95% C.L. limits on $\sigma(pp\to Z\to A^{\prime}h_{D}\to 4\ell+X)$ for various $\{m_{h_{D}},m_{A^{\prime}}\}$ benchmarks where the $h_{D}\to A^{\prime}A^{\prime}$ decay is on-shell. (Note that this limit is on four-lepton final states, not on three-$A^{\prime}$ final states.) For $m_{h_{D}}=20,30,50$ GeV, we plot our expected and observed limits in Fig. 4, together with the those of ATLAS.⁷⁷7We do not place limits for $m_{A^{\prime}}=8,12$. These are adjacent to the $\Upsilon$ mass cut, making it difficult for us to determine the signal shape $\left\{\kappa_{i}\right\}$.

Our results match the shape of ATLAS’s limits, reproducing the $m_{h_{D}}$ and $m_{A^{\prime}}$ dependence in both expected and observed limits. Our bounds are weaker by a factor that varies mainly between 1.5 and 1.9, except at the smallest mass $m_{A^{\prime}}=6$ GeV where it tends higher. This is not surprising given the crude nature of our event simulation, as captured in the large systematic errors on our recalibration factors.

A minor issue is that our recast observed limits tend to fluctuate more than the ATLAS ones, which is most obvious in the $\{m_{h_{D}},m_{A^{\prime}}\}=\{50,19\}$ GeV benchmark point. Since the expected limits do not experience similar fluctuations, the cause probably lies in the likelihood distribution of observed events. The statistical interpretation we adopt takes the background event count per $\bar{m}_{\ell\ell}$ bin and only accounts for the resolution via the signal shape. The potential correlation of background event counts across nearby $\bar{m}_{\ell\ell}$ bins, due to detector resolution, cannot be extracted from the public data, and this may leave us more sensitive to bin-to-bin fluctuations than is the case for ATLAS.

This largely successful reproduction of the results of the ATLAS analysis now gives us confidence to proceed to recasting the analysis for other BSM signatures, in particular for the $H\to 8f$ decay mode.