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| CMS-EXO-24-028 ; CERN-EP-2026-048 | ||
| A search for microscopic black holes, string balls, and sphalerons in proton-proton collisions at $ \sqrt{s} = $ 13 TeV | ||
| CMS Collaboration | ||
| 12 April 2026 | ||
| Submitted to the Journal of High Energy Physics | ||
| Abstract: A search for microscopic black holes, string balls, and electroweak sphalerons using proton-proton collisions at $ \sqrt{s} = $ 13 TeV recorded with the CMS detector at the CERN LHC during the 2016--2018 data taking, and corresponding to an integrated luminosity of 138 fb$ ^{-1} $, is presented. Two search strategies based on control samples in data are used. Model-independent limits on the cross section of physics phenomena with multiple energetic jets, leptons, and photons are set using a method that relies on the shape invariance of the scalar sum of the transverse momenta of all objects in the event. Model-dependent limits on black hole and sphaleron production are set using a newly introduced method that has been developed for the identification of collider events with distinct kinematic features by separating them into classes based on phase space proximity. In the context of models with large extra dimensions, semiclassical black holes and string balls with masses below 8.4--11.4 TeV and 9.0--10.7 TeV, respectively, are excluded at 95% confidence level, significantly extending the reach beyond previous searches. Results of a dedicated search for electroweak sphalerons are used to derive an upper limit of 0.0034 at 95% confidence level on the fraction of quark-quark interactions above the nominal sphaleron transition energy threshold of 9 TeV. | ||
| Links: e-print arXiv:2604.10732 [hep-ex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; | ||
| Figures | |
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Figure 1:
The $ S_\text{T} $ (left) and sphericity (right) distributions for various BH (with $ n = $ 2) and sphaleron signal models are plotted along with the corresponding distributions for simulated QCD multijet background events. The distributions are normalized to unit area. |
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Figure 1-a:
The $ S_\text{T} $ (left) and sphericity (right) distributions for various BH (with $ n = $ 2) and sphaleron signal models are plotted along with the corresponding distributions for simulated QCD multijet background events. The distributions are normalized to unit area. |
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Figure 1-b:
The $ S_\text{T} $ (left) and sphericity (right) distributions for various BH (with $ n = $ 2) and sphaleron signal models are plotted along with the corresponding distributions for simulated QCD multijet background events. The distributions are normalized to unit area. |
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Figure 2:
The SVM score distributions for simulated QCD multijets, and selected BH (with $ n = $ 2) and sphaleron models, before (left) and after (right) the sphericity requirement. The distributions are normalized to unit area. |
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Figure 2-a:
The SVM score distributions for simulated QCD multijets, and selected BH (with $ n = $ 2) and sphaleron models, before (left) and after (right) the sphericity requirement. The distributions are normalized to unit area. |
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Figure 2-b:
The SVM score distributions for simulated QCD multijets, and selected BH (with $ n = $ 2) and sphaleron models, before (left) and after (right) the sphericity requirement. The distributions are normalized to unit area. |
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Figure 3:
The SVM score vs. the $ S_\text{T} $ distributions for simulated QCD multijet background (left) and the BH signal model B1 with $ M_\text{D} = $ 2 TeV, $ M_\text{BH} = $ 10 TeV, and $ n = $ 2 (right), after the $ S > $ 0.1 selection. |
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Figure 3-a:
The SVM score vs. the $ S_\text{T} $ distributions for simulated QCD multijet background (left) and the BH signal model B1 with $ M_\text{D} = $ 2 TeV, $ M_\text{BH} = $ 10 TeV, and $ n = $ 2 (right), after the $ S > $ 0.1 selection. |
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Figure 3-b:
The SVM score vs. the $ S_\text{T} $ distributions for simulated QCD multijet background (left) and the BH signal model B1 with $ M_\text{D} = $ 2 TeV, $ M_\text{BH} = $ 10 TeV, and $ n = $ 2 (right), after the $ S > $ 0.1 selection. |
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Figure 4:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) in the SI-VR in data is indicated by the black dots. The background prediction is represented by the red line, and the gray band corresponds to the background modeling uncertainty. The lower panels show the difference between observed data and the background prediction, normalized by the total uncertainty. |
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Figure 4-a:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) in the SI-VR in data is indicated by the black dots. The background prediction is represented by the red line, and the gray band corresponds to the background modeling uncertainty. The lower panels show the difference between observed data and the background prediction, normalized by the total uncertainty. |
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Figure 4-b:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) in the SI-VR in data is indicated by the black dots. The background prediction is represented by the red line, and the gray band corresponds to the background modeling uncertainty. The lower panels show the difference between observed data and the background prediction, normalized by the total uncertainty. |
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Figure 5:
Post-fit $ S_\text{T} $ distributions in VR-FAIL (left) and VR-PASS (right) regions in data. The gray hatched areas include both statistical and systematic uncertainties in the background prediction (yellow histogram). The red line corresponds to the signal model B1, with $ M_\text{D} = $ 2 TeV, $ M_\text{BH} = $ 10 TeV, and $ n = $ 2. The lower panels show the difference between observed data and the background prediction, normalized by the total uncertainty. |
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Figure 6:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) SI-SR in data, indicated by the black dots, along with the background prediction and its uncertainty represented by the red line and gray band, respectively. Lower panel as in Fig. 4. |
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Figure 6-a:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) SI-SR in data, indicated by the black dots, along with the background prediction and its uncertainty represented by the red line and gray band, respectively. Lower panel as in Fig. 4. |
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Figure 6-b:
The $ S_\text{T} $ distribution in the $ N \geq $ 4 (left) and $ N \geq $ 7 (right) SI-SR in data, indicated by the black dots, along with the background prediction and its uncertainty represented by the red line and gray band, respectively. Lower panel as in Fig. 4. |
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Figure 7:
Expected and observed model-independent 95% CL upper limits on the cross section times acceptance for multiplicity $ N \geq $ 4, where the inner (outer) band represents the 68% (95%) quantile of the expected limit (left), and the observed limits with different minimum object multiplicity requirements (right). |
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Figure 7-a:
Expected and observed model-independent 95% CL upper limits on the cross section times acceptance for multiplicity $ N \geq $ 4, where the inner (outer) band represents the 68% (95%) quantile of the expected limit (left), and the observed limits with different minimum object multiplicity requirements (right). |
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Figure 7-b:
Expected and observed model-independent 95% CL upper limits on the cross section times acceptance for multiplicity $ N \geq $ 4, where the inner (outer) band represents the 68% (95%) quantile of the expected limit (left), and the observed limits with different minimum object multiplicity requirements (right). |
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Figure 8:
Post-fit $ S_\text{T} $ distributions in the FAIL (left) and PASS (right) regions in data. The gray shaded area includes both statistical and systematic uncertainties in the background prediction (yellow histogram) while the red and blue lines are B1 signal examples, as noted in the legends. Lower panel as in Fig. 5. |
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Figure 9:
Expected and observed 95% CL upper limits on the cross section for a semiclassical nonrotating BH model (B1) with $ n = $ 2 and $ M_\text{D} = $ 2 TeV (left) or $ M_\text{D} = $ 4 TeV (right), as a function of $ M_\text{BH} $. The blue curves represent the theoretical cross section values. The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
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Figure 9-a:
Expected and observed 95% CL upper limits on the cross section for a semiclassical nonrotating BH model (B1) with $ n = $ 2 and $ M_\text{D} = $ 2 TeV (left) or $ M_\text{D} = $ 4 TeV (right), as a function of $ M_\text{BH} $. The blue curves represent the theoretical cross section values. The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
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Figure 9-b:
Expected and observed 95% CL upper limits on the cross section for a semiclassical nonrotating BH model (B1) with $ n = $ 2 and $ M_\text{D} = $ 2 TeV (left) or $ M_\text{D} = $ 4 TeV (right), as a function of $ M_\text{BH} $. The blue curves represent the theoretical cross section values. The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
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Figure 10:
Excluded $ M_{\text{BH}}^{\text{min}} $ values as functions of $ M_\text{D} $ and $ n $ for a variety of BLACKMAX (left) and CHARYBDIS2 (right) BH models. |
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Figure 10-a:
Excluded $ M_{\text{BH}}^{\text{min}} $ values as functions of $ M_\text{D} $ and $ n $ for a variety of BLACKMAX (left) and CHARYBDIS2 (right) BH models. |
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Figure 10-b:
Excluded $ M_{\text{BH}}^{\text{min}} $ values as functions of $ M_\text{D} $ and $ n $ for a variety of BLACKMAX (left) and CHARYBDIS2 (right) BH models. |
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Figure 11:
Expected and observed 95% CL upper limits for SB models with $ M_{\text{S}} = $ 3.5 TeV and $ g_{\text{S}} = $ 0.2 (left) and excluded SB mass values as functions of $ M_{\text{S}} $ at $ g_{\text{S}}=0.2, $ 0.3, and 0.4 (right). The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
png pdf |
Figure 11-a:
Expected and observed 95% CL upper limits for SB models with $ M_{\text{S}} = $ 3.5 TeV and $ g_{\text{S}} = $ 0.2 (left) and excluded SB mass values as functions of $ M_{\text{S}} $ at $ g_{\text{S}}=0.2, $ 0.3, and 0.4 (right). The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
png pdf |
Figure 11-b:
Expected and observed 95% CL upper limits for SB models with $ M_{\text{S}} = $ 3.5 TeV and $ g_{\text{S}} = $ 0.2 (left) and excluded SB mass values as functions of $ M_{\text{S}} $ at $ g_{\text{S}}=0.2, $ 0.3, and 0.4 (right). The inner (outer) band represents the 68% (95%) quantile of the expected limit. |
png pdf |
Figure 12:
Expected and observed 95% CL upper limits on the pre-exponential factor for the sphaleron model with $ p({N_\text{CS}})= $ 0.5 (left), and observed limits with $ p({N_\text{CS}})= $ 0, 0.5, and 1 (right). The inner (outer) band (left) represents the 68% (95%) quantiles of the expected limit. |
png pdf |
Figure 12-a:
Expected and observed 95% CL upper limits on the pre-exponential factor for the sphaleron model with $ p({N_\text{CS}})= $ 0.5 (left), and observed limits with $ p({N_\text{CS}})= $ 0, 0.5, and 1 (right). The inner (outer) band (left) represents the 68% (95%) quantiles of the expected limit. |
png pdf |
Figure 12-b:
Expected and observed 95% CL upper limits on the pre-exponential factor for the sphaleron model with $ p({N_\text{CS}})= $ 0.5 (left), and observed limits with $ p({N_\text{CS}})= $ 0, 0.5, and 1 (right). The inner (outer) band (left) represents the 68% (95%) quantiles of the expected limit. |
| Tables | |
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Table 1:
Parameter combinations of the fundamental Planck scale $ M_\text{D} $ and minimum black hole mass $ M_{\text{BH}}^{\text{min}} $ (both in TeV) that exclude a given number of extra dimensions $ n^{\text{max}} $, for BH models generated with BLACKMAX and CHARYBDIS2. Each row collects all ($ M_\text{D} $, $ M_{\text{BH}}^{\text{min}} $) pairs yielding the same exclusion level. Combinations with $ M_{\text{BH}}^{\text{min}} < M_\text{D} $ are unphysical and not considered. The first row indicates the pairs that are excluded for any number of extra dimensions in the ADD model, while the last row shows the parameters where no exclusion could be made in this analysis. |
| Summary |
| A dedicated search for black holes, string balls, and sphalerons produced in proton-proton collisions at $ \sqrt{s} = $ 13 TeV using data collected with the CMS detector has been presented. No excesses above the standard model background predictions are observed. We set 95% confidence level (CL) model-independent limits on cross section of hypothetical signals characterized by a large multiplicity of energetic jets, leptons, and photons. The model-independent results demonstrate approximately a factor of four improvement in the cross section limit compared to the previous CMS analysis. The model-dependent results exclude at 95% CL semiclassical black holes and string balls with masses below 8.4--11.4 TeV and 9.0--10.7 TeV, respectively, depending on the model and the number of extra dimensions. This extends the exclusion reach by 1--1.6 TeV and 1.3--1.9 TeV, respectively. The observed (expected) upper limit on the sphaleron pre-exponential factor for the nominal electroweak sphaleron transition energy of 9 TeV is 0.0034 (0.0035) at 95% CL, which is strengthened by a factor of 6.2 (3.4) compared to the previous best limit of 0.021 (0.012) from CMS [21]. These are the most stringent limits on the sphaleron pre-exponential factor to date. A significant improvement in the model-dependent study over previous results comes from an improved understanding of parton distribution functions. Additional significant gains can be traced to both the increased integrated luminosity, and the enhanced background rejection provided by the sphericity and phase space distance event selection requirements. |
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Compact Muon Solenoid LHC, CERN |
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