Revisiting the minimal universal extra dimensions with mass-dimension-five operators

Our knowledge of the fundamental particles and their interactions in nature is encompassed by the Standard Model (SM) of particle physics. However, the result of several experimental observations that indicates the existence of dark matter (DM) suggests that the SM is not sufficient to support all current data. Through measurements collected by the Planck satellite, we know that approximately 27% of the existing particles contained in the universe are dark [1].

Several experiments are now looking for stronger evidence to prove the existence of DM. One of them is the investigation of the process by which DM is produced from proton-proton collisions in the Large Hadron Collider (LHC). Typical collider experiments are based on the idea that a stable and neutral particle, if produced, would leave the detector without any interaction. In order to identify its production, we would search for energy lost when it leaves the detector. The latter quantity is obtained through energy conservation and we call it missing energy, ${{E\mkern-3.5pt{/}}_T}$ [2].

Hence, the collider strategy is to search for traces of DM presence via the associated production of SM particles plus ${{E\mkern-3.5pt{/}}_T}$ signals. Namely, identification of singular objects within the detector, mono searches, where a single object could be a W or Z boson, top quark, jets, photon or leptons [3–6].

The proposal of a theoretical model that has a suitable DM candidate to explain the experiments remains a challenge for physicists. Models with extra dimensions can be considered promising in the sense they can be tested at colliders and are recurrent in unification theories like superstrings. Therefore, their predictions seem to be part of a viable hypothesis for describing nature [7–11]. We developed our work from an extension of the models with universal extra dimensions initially proposed in [12]. This class of models includes candidates for DM, testable at the LHC. The extra dimensions are universal in the sense that all SM fields are promoted to fields which propagate on the full space-time $\mathcal{M}\times X$, where $\mathcal{M}$ is the flat four-dimensional (4D) Minkowski space and X is a compact space.

In the 4D effective theory, the extra dimensions appear as heavy particle modes, the Kaluza-Klein (KK) towers, identified by an integer n, called the KK number. The lightest mode (zero mode) is identified with the SM particles. The KK mass spectrum is governed by the inverse size of the extra dimensions geometry. In the case of only one extra dimension —minimal universal extra dimension (MUED) model— compactification on the orbifold $X = S^1/Z_2$ allows to have chiral zero modes of fermions.

The MUED is a complete model and has a single free parameter, the radius of the extra dimension, R. However, as a higher-dimensional theory, the model is nonrenormalizable and should be treated as an effective theory valid up to a cutoff scale Λ, usually taken to be around $20R^{-1}$ [8]. Since its proposal, several works have established limits for both parameters: with ATLAS data for dilepton final states and $\sqrt{s} = 8\ \text{TeV}$, a compactification scale up to 900 GeV for $\Lambda R = 40$ is excluded [7]. Reference [10] presents a more stringent limit on the size of the extra dimension. Based on the method of recasting results from ATLAS and CMS Run 02 and considering that the lightest KK particle (LKP) is the U(1)_Y gauge boson $(\gamma_1)$, mass scale up to $R^{-1} = 1500\ \text{GeV}$ for $\Lambda R = 40$ can be excluded at 95% confidence level, according to the study based on dilepton stop search.

On the other hand, taking into account the relic density data, including all coannihilation channels and all resonances, the usual DM candidade of the MUED model, $\gamma_1$, reproduces the observed DM relic density only if $R^{-1} \sim 1.25$ TeV [9].

As one can see additional ingredients are needed in the MUED model, the data must reconcile both cosmological information and collider physics. We explored a minimal extension of the model that includes effective operators in $(3+1)$D that are dimensionless in $(4+1)$D [13–14], keeping the focus on electric and magnetic dipole moments and Higgs portals operators. The new interactions introduce a Dirac fermion χ and its first KK mode $(\lambda_1, \upsilon_1)$ can play the role of DM. With a new fermionic DM candidate, exclusion limits in the free parameter $R^{-1}$ are less restrictive for this minimally modified version of the MUED model. As the new Lagrangian has mass dimension five, the obtained four-dimensional theory is suppressed by the size of the extra dimension without the addition of a new parameter. Therefore, these operators can be portals to prove the size of the extra dimension in the search for DM.

The zero mode of the new fermion χ is chosen to have a mass below 1 GeV, and the MUED relic density calculation considered co-annihilations with the n = 1 photon, which is closest in mass to the χ.

We analyze the parameter space which satisfies the DM relic abundance and accommodates the ATLAS data for mono-jet events at $\sqrt{s}= 13\ \text{TeV}$, one of the promising search channels for new physics analysis. We employ machine learning (ML) techniques to classify signal and background data and increase the statistical significance of DM discovery in the mono-jet channel [15].

Our study is motived by the scarce results from new physic searches at the LHC in the context of the current models [3–4], [16]. Our proposal is to concilate the DM search with the extra dimension scenario considering a new fermion on the theory.

The paper is organized as follows. In the next section we present basic aspects of the minimal MUED model, pointing out the interactions terms, fundamental for our study. Following the MUED model, we discuss the mono-jet signature at the LHC, including the possible DM signal through Higgs and photon decay. The fourth section presents the results on relic density in the light of our results and afterward our conclusions.

We consider all SM fields and a new doublet fermion field χ, propagating in one extra dimension, which is compactified on an $S^1/Z_2$ orbifold with the fundamental region $0\leq y \leq \pi R$. We have the action of our model in 5D as follows:

$$\begin{eqnarray} \mathcal{S}&= \int \textrm{d}^4 x \int_0^{\pi R} \textrm{d}y \biggl(\frac{-i\xi(\pi R)^{3/2}}{2}\bar{\chi}\sigma^{M N}(\mathds{1}+\Gamma^5)\chi F_{M N} \nonumber\\ & \quad+\xi(\pi R)^2\bar{\chi}(\mathds{1}+\Gamma^5)\chi H^\dagger H\biggr). \end{eqnarray} \tag{ 1 }$$

The representation for Clifford algebra in 5D is such that

$$\begin{eqnarray*} &&\{\Gamma_{M}, \Gamma_{N}\}=2\eta_{MN} \quad \Gamma_{\mu}=\gamma_{\mu}, \quad \Gamma_{5}=-i\gamma_{5}, \end{eqnarray*}$$

where $N=(x^\mu,y)$. In eq. (1) ξ is the dimensionless constant, H is the Higgs doublet and the 5D spinor $\chi\equiv (\lambda, \upsilon)^T$. The λ and υlon fields are decomposed as

$$\begin{eqnarray} \lambda (x^\mu, y)&= \frac{1}{\sqrt{\pi R}}\left[\lambda_{0}(x^\mu)+\sqrt{2}\sum_{j\geq 1}\left[\lambda_j(x^\mu)\cos\frac{jy}{R}\right]\right], \nonumber\\ \upsilon (x^\mu, y)&= \sqrt{\frac{2}{\pi R}}\sum_{j\geq 1}\left[\upsilon_j(x^\mu)\sin\frac{jy}{R}\right]. \end{eqnarray} \tag{ 2 }$$

The λ field obeys the Neumann boundary condition, whereas υlon the Dirichlet boundary condition. After integrating out the fifth dimension and after eletroweak symmetry breaking, the interaction terms in the $(3+1)$D effective Lagrangian are

$$\begin{eqnarray} & \mathcal{L}_{4D}\supset \kappa[4(h_0+v) h_1 \bar{\lambda}_1\lambda_0+ h_{0}^2 (\bar{\lambda}_1\lambda_1+\bar{\lambda}_0\lambda_0+\bar{\upsilon}_1\upsilon_1)\nonumber\\ & +2 v h_0(\bar{\lambda}_1\lambda_1+\bar{\lambda}_0\lambda_0+\bar{\upsilon}_1\upsilon_1)+\frac{3}{2}h_{1}^2 \bar{\lambda}_1\lambda_1+ h_{1}^2 \bar{\lambda}_0\lambda_0\nonumber\\ & +\frac{1}{2} h_{1}^2 \bar{\upsilon}_1\upsilon_1](\mathds{1}-i\gamma^5)+\frac{1}{2}\kappa \partial_{\nu}A_{\mu}^1 \bar{\lambda}_1\lambda_0[\gamma^{\nu},\gamma^{\mu}](\mathds{1} -\gamma^5)\nonumber\\ & +\frac{1}{2}\kappa\partial_{\nu}A_{\mu}(\bar{\lambda}_1\lambda_1+\bar{\upsilon_1}\upsilon_1+\bar{\lambda}_0\lambda_0)[\gamma^{\nu},\gamma^{\mu}](\mathds{1} -\gamma^5). \end{eqnarray} \tag{ 3 }$$

In eq. (3) we keep the zeroth and first KK modes of the particles in the Lagrangian; $v = 246\ \text{GeV}$ is the vacuum expectation value of the Higgs field; the coupling constants in the $(3+1)$D theory are naturally dependent on the size of the extra dimension, $\kappa\equiv \xi\pi R$. h₁ and $A_{\mu}^1$ are the first KK mode for Higgs and photon fields, respectively.

In this section our candidate to DM, the lightest χ mode, is studied in the light of the LHC data run at $\sqrt{s} = 13\ \text{TeV}$. In particular we will focus on mono-jet final states [3], arising from $pp \rightarrow \lambda_1 \lambda_1 + g(q)$ and $pp \rightarrow \upsilon_1 \upsilon_1 + g(q)$ processes ¹ . At leading order, the main Feynman diagrams for our signal production in t and s channel are shown in fig. 1.

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The fermionic DM $\chi\equiv(\lambda_1, \upsilon_1)$ is produced via the decay of a Higgs boson or a photon, both produced from interactions with quarks, or through effective couplings to gluons (gluon-gluon fusion). The effective vertex shown in fig. 1 with the contributions of the colored KK partners, was implemented in Madgraph5 [17]. The interaction vertices between the DM and the bosons can be easily obtained by deriving functionally the Lagrangian in eq. (3) with respect to the fields. Thus,

$$\begin{eqnarray} \Gamma_{\chi_1 \chi_1 h_0} &= 2\kappa v(\gamma_5+\mathds{1}),\\ \Gamma_{\chi_1 \chi_1\gamma} &= \displaystyle\frac{1}{2}\kappa[{p\mkern-3pt{/}}, \gamma^{\mu}]+\displaystyle\frac{1}{2}\kappa[\gamma^{\mu},{p\mkern-3pt{/}}]\gamma^5, \end{eqnarray} \tag{ 4 }$$

where p is the photon momentum.

In this channel, one looks for events with one high-p_T jet, higher than $100\text{--}200\ \text{GeV}$ in the central region of the detector, with pseudorapidity $\lvert\eta\rvert < 2.4$, and ${{E\mkern-3.5pt{/}}_T}$ above roughly 200 GeV in the 13 TeV analyses for the ATLAS and CMS detectors [3,18]. The dominant irreducible SM background for this final state comes from $Z + j$, with the Z boson subsequently decaying invisibly $Z \rightarrow \nu\nu$. There is also a subdominant irreducible background from $W + j$, with $W \rightarrow \tau \nu$, where the τ decays hadronically. In addition, there are backgrounds from $W + j$ with $W \rightarrow \mu\nu$ or e ν, where the lepton is either missed or misidentified as a jet. However, $Z + j$ constitutes approximately 60% of events.

In order to simulate the mono-jet events at the LHC, we implemented the model in Madgraph5 [19] using the FeynRules [20] package. The CheckMATE 2 Collaboration tools [21–24] was used to verify, for a given set of $R^{-1}$ and DM masses, whether the model is excluded or not at 95% CL by comparing the result with the experimental data analysis [3]. For each signal region, CheckMATE 2 computes the expected number of signal events S after cuts, and comparing it to the 95% CL upper limit $S^{95}_{\textit{exp}}$, given a signal error $\Delta S$. For the sake of comparison, we consider the results from the ATLAS Collaboration data analysis presented in ref. [3].

The gray region in fig. 2 represents the excluded region by the mono-jet events at $\sqrt{s}=13\ \text{TeV}$, obtained with CheckMATE 2. Our DM candidate $\chi_1$ with mass smaller than $\sim820\ \text{GeV}$ is excluded according to the ${{E\mkern-3.5pt{/}}_T}$ + jet signature in the ATLAS experiment. Hence, a compactification scale up to $\sim820\ \text{GeV}$ is forbidden.

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The cuts implemented on CheckMATE 2 to select the signals with ${{E\mkern-3.5pt{/}}_T} +$jet in the yellow band in fig. 2 were established from the ATLAS detector results, and are described below:

• –  
  the leading jet has $p_T > 250\ \text{GeV}$ and $\eta < 2.4$;
• –  
  the separation in the azimuthal plane of $\Delta\phi(\textit{jet}, p_{T}^{\textit{miss}}) > 0.4$ between the missing transverse momentum direction and each selected jet;
• –  
  a minimum of $1000\ \text{GeV}$ in the ${{E\mkern-3.5pt{/}}_T}$ thresholds.

Now, we turn to investigate the mass range allowed for the channel, the statistical significance of discovering DM in the collider at $\sqrt{s}=13\ \text{TeV}$. We consider a particular code including the boosted decision trees (BDTs) method [15] to eliminate background events and characterize DM. The following kinematic variables were used to train the events: ${{E\mkern-3.5pt{/}}_T}$, $p_{T_{j}}$, $\eta_{j}$, $\Delta R(j, {{E\mkern-3.5pt{/}}_T})$, $\Delta \phi(j, {{E\mkern-3.5pt{/}}_T})$ and the fractional p_T difference, $\frac{{{E\mkern-3.5pt{/}}_T}-p_{T_{j}}}{p_{T_{j}}}$. We reconstruct them using the Les Houches Event (lhe) files generated by Madgraph5 in a Python routine. The backgrounds $W +$ jet and $Z +$ jet were considered in the analysis. In fig. 3 are displayed the distribution of events according to their kinematic variables.

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The main message from the distribution of data in fig. 3 is the separation of classes concerning the signal and background events. It shows that signal and background distributions can be well classified because the arrangement points related to each event are not overlapping, so they have separable variables. Indeed, we compute in fig. 4 the confusion matrix to evaluate the accuracy of a classification using the Python module scikit-learn learn [25]. We see that $\sim90$% of the signal events are classified correctly.

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The best cuts found in order to increase the signal to background ratio, require the events to satisfy $p_{T_{j}}>550\ \text{GeV}$. Analyzing the scattering plot of fig. 3, we verify that this cut eliminates most of the background events. In table 1 are shown the number of signal events, S, for the analyzed mass range and the number of background events, B, after the cut.

Table 1:. Number of signal events in the $p p \rightarrow \chi_1 \chi_1 +$ jet channel after demanding the cut $p_{T_j}>550\ \text{GeV}$. For each signal and background dataset, we initially generate 50000 events.

Background B after the cut

$m_{\chi_1}$ (GeV)	S after the cut
1400	18
1250	46
1100	93
1000	172
900	291
820	474
$Z+$jet	4550
$W+$jet	5023

Considering this cut, the signal/back $\sqrt{2((S+B)\ln(1+S/B)-S)}$, S and B the number of signal and backgrounds events, respectively, as the statistical test and an integrated luminosity of $30\ \text{fb}^{-1}$, the statistical significance of discovery DM are in fig. 2.

The result of our analysis establishes that DM masses in the range $\sim [820\text{--}900]\ \text{GeV}$ can lead to an evidence signal of $[3\sigma\text{--}4\sigma]$ at the LHC $13\ \text{TeV}$ with mono-jet events. By increasing the integrated luminosity, the possibility of a more detailed study for particles with masses greater than $900\ \text{GeV}$ is opened, since in this case we would have a significant number of events in this region, improving the statistical significance of DM discovery.

In fig. 5 we present the production cross-section for the ${{E\mkern-3.5pt{/}}_T}$ + jet process at LHC $\sqrt{s} = 13\ \text{TeV}$. We only show the range of allowed values, according to the data from the exclusion analysis performed. For DM masses in the range $\sim [820\text{--}1000]\ \text{GeV}$, we have the cross-sections $\sim [100\text{--}200]\ \text{fb}$, which could be within LHC sensitivity.

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We show the total cross-section, with both Higgs boson and photon-mediated processes. However, we emphasize that the dominant contribution comes from photon-mediated events, since the interaction vertex is dependent on the photon moment. Besides that, in our model the mass of the DM particle is larger than half the Higgs mass, $m_{\chi_1} \geq \frac{1}{2} M_{H}$, hence there is no invisible two-body Higgs decay and the detection of the DM particles in collider experiments becomes much more difficult, since the cross-sections are rather modest through Higgs off shell in this portal [26].

In this section, we assume that the n = 1 KK-χ fermion is the DM candidate, and present some of the Feynman diagrams relevant to determine its cosmological relic density. Besides the standard annihilation into quarks, leptons and gauge bosons, mediated by the SM-like Higgs boson or photon, coannihilations with $\gamma_1$ are also possible. These can be mediated by the new fermions χ. These channels can lead to both Higgs boson or photon in the final state, and involve the contribution of the interaction vertices in eq. (4). The main Feynman diagrams are shown in fig. 6.

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As the DM is heavy, all the annihilation channels into SM particles are open. We also take into account the co-annihilation effects with $\gamma_1$, since we assume that the $n=0 \chi$ mode has mass less than $1\ \text{GeV}$.

We present in fig. 7 the expected fermionic DM relic abundance as a function of coupling constant in 4D, $k=\xi \pi R$, computed with the MadDM v.3.0 package [27]. According to the LHC discovery analysis in fig. 2, the best value found for $R^{-1}$ was $\sim850\ \text{GeV}$, so we fixed this value in the relic computation.

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For solutions falling exactly in the black band the totality of DM can be explained by $\chi_1$. Hence, the correct amount of DM is obtained for couplings κ between $\sim [10^{-4}, 10^{-3}]\ \text{GeV}^{-1}$. When we increase κ, we also increase $\sigma_{\textit{eff}}$ and as a consequence $\Omega h^2$ decreases.

We have investigated in this letter the phenomenology of a fermionic DM established in the context of this minimally modified version of the MUED model.

We focused on the electric and magnetic dipole moments and Higgs portals operators. As they are effective operators from a 4D point of view, the cutoff scale is a remnant of the compacting scale $R^{-1}$.

We have examined the collider signatures associated to the mono-jet +${{E\mkern-3.5pt{/}}_T}$ channel at the LHC. In particular, using the XGboost library we classify the signal and background events, identifying the most efficient cut. In this case, processes in which $p_{T_j} < 550\ \text{GeV}$ were discarded. Hence, we have found a viable DM mass range in the region $820\text{--}900\ \text{GeV}$, in which the statistical significance for its discovery is up to $4 \sigma$ if the integred luminosity is $30\ \text{fb}^{-1}$.

Using the Run 2 LHC mono-jet+${{E\mkern-3.5pt{/}}_T}$ analysis incorporated on CheckMATE 2, we have derived a lower bound on $R^{-1}$ of $820\ \text{GeV}$, which is less restrictive than the current limit for this parameter established in the context of the original version of the MUED model.

We have also presented the $\chi_1$ relic abundance for $R^{-1}=850\ \text{GeV}$. In this scenario, the 4D couplings, κ, in the range $[10^{-3}, 10^{-4}]\ \text{GeV}^{-1}$ are sufficient for DM of our model encompass all relic abundance currently measured.

Hence, we have identified the regions of parameters where DM predictions are in agreement with experimental constraints, such as those coming from DM relic abundance and LHC search.

LCD thanks "Coordenação de Aperfeioamento de Pessoal de Nível Superior - Brasil" (CAPES - Finance Code 001) for the financial support. The research was supported by resources supplied by the Center for Scientific Computing (NCC/Grid Unesp) of the São Paulo state University (UNESP).