A search for microscopic black holes, string balls, and sphalerons in proton-proton collisions at $\sqrt{s}$ = 13 TeV

Using 138 fb⁻¹ of proton-proton collision data at 13 TeV collected by the CMS detector, this study presents model-independent and model-dependent searches that exclude semiclassical black holes and string balls with masses below approximately 8.4–11.4 TeV and 9.0–10.7 TeV, respectively, while setting a 95% confidence level upper limit of 0.0034 on the fraction of quark-quark interactions producing electroweak sphalerons above 9 TeV.

The Gist

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful "smashing machine." It takes two tiny particles (protons) and slams them together at nearly the speed of light. Usually, these collisions just create a shower of ordinary particles, like raindrops after a storm. But physicists are hoping that sometimes, the energy is so intense that it creates something truly exotic and strange.

This paper is a report from the CMS Collaboration, a team of thousands of scientists, who acted like cosmic detectives searching for three specific types of "ghosts" that might appear in these collisions: Microscopic Black Holes, String Balls, and Sphalerons.

Here is the story of their hunt, explained simply.

1. The Three Suspects (What they were looking for)

Think of the Standard Model of physics as the "Rulebook" of the universe. It explains how almost everything works, but it has some holes in it (like why gravity is so weak compared to other forces). These three suspects are theoretical ideas that could patch those holes.

• Microscopic Black Holes (The Tiny Vortex):

  • The Idea: Usually, black holes are massive, like the ones that eat stars. But if our universe has hidden, extra dimensions (like a secret room in a house we can't see), gravity might be much stronger at tiny scales. If you smash particles hard enough, you might create a black hole so small it would vanish instantly.
  • The Analogy: Imagine blowing a soap bubble. Usually, they pop gently. But if you blow with enough force, maybe you could create a "black hole bubble" that sucks everything in and then pops (evaporates) in a flash of light.

• String Balls (The Tangled Yarn):

  • The Idea: String theory suggests particles aren't dots, but tiny vibrating strings. If you heat these strings up enough, they get excited and stretch out into long, messy, tangled balls of energy.
  • The Analogy: Think of a ball of yarn. If you spin it slowly, it's neat. If you spin it wildly fast, it becomes a chaotic, fuzzy ball. A "String Ball" is that fuzzy ball of energy before it settles down into a black hole.

• Sphalerons (The Rule-Breakers):

  • The Idea: In our universe, the number of protons (matter) and antiprotons (antimatter) usually balances out. But the universe is made of matter, not antimatter. Sphalerons are rare, high-energy events that can break the rules, turning matter into antimatter (or vice versa) in a way that explains why we exist.
  • The Analogy: Imagine a strict bouncer at a club who never lets people switch seats. A Sphaleron is like a magical moment where the bouncer falls asleep, and suddenly, everyone swaps seats, breaking the usual order.

2. The Detective Work (How they looked)

The scientists didn't just wait for a black hole to appear; they had to build a trap.

• The Trap (The Event): When a black hole or string ball is created, it doesn't stay hidden. It immediately evaporates (like a snowflake hitting a hot stove) and explodes into a massive shower of particles: jets of energy, electrons, photons, and missing energy.
• The Clues: The team looked for events with high multiplicity. Imagine a normal collision is like a few sparks flying off a hammer. They were looking for a "fireworks explosion" with dozens of sparks flying in all directions.
• The Sphericity Test: Normal collisions (background noise) tend to shoot particles in a flat, pancake-like shape. But a black hole explosion is like a firework going off in the middle of a room—it sprays energy in a perfect sphere. The scientists used a mathematical tool called Sphericity to measure how "round" the explosion was.
• The AI Assistant: Because the data is so complex, they used a machine learning algorithm (a type of AI) trained to recognize the difference between a "normal spark" and a "strange explosion." It's like teaching a dog to sniff out a specific scent in a pile of leaves.

3. The Results (Did they find anything?)

The short answer: No.

After analyzing 138 "inverse femtobarns" of data (which is a fancy way of saying they looked at a huge amount of collision data from 2016 to 2018), they found zero evidence of these exotic objects.

• The Verdict: The universe, at the energy levels they tested, seems to follow the standard rules. No tiny black holes, no string balls, and no rule-breaking sphalerons were found.
• The Silver Lining: Just because they didn't find the "ghosts" doesn't mean the search was a failure. It's like searching for a needle in a haystack and not finding it. You now know for sure that the needle isn't in that part of the haystack.

4. Why This Matters (The "So What?")

Even though they didn't find the new particles, this paper is a huge success for science because it pushed the boundaries of what we know.

• Setting the Limits: They can now say with 95% confidence: "If microscopic black holes exist, they must be heavier than 8.4 to 11.4 TeV." (Think of this as saying, "If there is a ghost, it must be bigger than a house.")
• Improving the Search: They used new, smarter methods (like the AI and the "shape invariance" technique) to filter out background noise better than ever before. This means future searches will be even more sensitive.
• Ruling Out Theories: By not finding these objects, they are forcing theorists to rethink their ideas. If the black holes are heavier than we thought, maybe the "extra dimensions" of the universe are smaller or different than we guessed.

Summary

The CMS team acted like the ultimate cosmic hunters. They built a sophisticated net (the detector and AI), cast it into the deepest ocean of energy (the LHC collisions), and pulled it up. While the net came up empty of the specific "monsters" they were hunting, they proved that the ocean is deeper and more mysterious than we thought, and they have drawn a new, more precise map of where those monsters might be hiding next time.

Technical Summary

1. Problem Statement and Physics Motivation

The paper addresses the search for three theoretical Beyond the Standard Model (BSM) phenomena predicted to exist at the TeV energy scale:

• Microscopic Black Holes (BHs): Predicted in models with large extra dimensions (ADD and RS models) where the fundamental Planck scale ($M_D$) is lowered to the TeV range. If $M_D$ is low enough, high-energy proton-proton collisions could create microscopic BHs that decay via Hawking radiation into high-multiplicity final states.
• String Balls (SBs): Highly excited, long, jagged string states predicted by string theory, existing in the mass region between the string scale ($M_S$) and the threshold for BH formation. They also decay into many Standard Model (SM) particles.
• Electroweak Sphalerons: Unstable solutions in the electroweak sector that violate baryon ($B$) and lepton ($L$) numbers ($\Delta B = \Delta L = 3N_{CS}$). They are crucial for explaining the matter-antimatter asymmetry in the universe but require energies above a transition threshold ($E_{sph} \approx 9$ TeV) to occur.

The Challenge: These signals manifest as events with high multiplicity of energetic jets, leptons, and photons, which are difficult to distinguish from the dominant Standard Model background: Quantum Chromodynamics (QCD) multijet production. Previous searches were limited by integrated luminosity and background modeling techniques.

2. Methodology

The analysis utilizes the full Run 2 dataset collected by the CMS detector at the LHC, corresponding to an integrated luminosity of 138 fb$^{-1}$ at $\sqrt{s} = 13$ TeV. The search employs two distinct strategies to estimate backgrounds and set limits.

A. Event Reconstruction and Selection

• Trigger: Events are selected based on the scalar sum of transverse momenta ($H_T$) of jets, with thresholds of 900 GeV (2016) and 1050 GeV (2017–2018).
• Preselection: A requirement of $S_T > 2$ TeV (scalar sum of $p_T$ of all objects) and $N \ge 3$ or $4$ objects (depending on the method).
• Object Definition: Jets, electrons, photons, and muons are reconstructed using the Particle-Flow (PF) algorithm.
• Sphericity Cut: A cut of $S > 0.1$ is applied. Signal events (isotropic decays) are expected to be more spherical than QCD background events (planar jets). This cut improves sensitivity by ~70% on average.

B. Background Estimation Methods

The paper introduces two complementary approaches:

1. Model-Independent Approach (Shape Invariance - SI):

   • Assumption: The shape of the $S_T$ distribution for QCD background is invariant with respect to the object multiplicity ($N$).
   • Method: A control region (CR) with low multiplicity ($N=3$) is used to fit the $S_T$ spectrum using 16 analytic functions. The best-fit function is extrapolated to higher multiplicities ($N \ge 4$) to predict the background in the Signal Region (SR).
   • Validation: The prediction is validated in a region ($2.5 < S_T < 5.2$ TeV) before the signal region ($S_T > 5.2$ TeV).

2. Model-Dependent Approach (Phase Space Distance - PS):

   • Novelty: Uses a new metric based on the intrinsic geometry of the phase space manifold to distinguish signal from background.
   • Metric: The distance between two events is calculated using global coordinates on a simplex and hypersphere derived from the event's 4-vectors.
   • Machine Learning: A Support Vector Machine (SVM) is trained using pairwise phase space distances as input. The SVM score separates signal-like events from background.
   • Alphabet Method: Background is estimated using a "PASS/FAIL" region technique. Events with SVM score $< 0.63$ (FAIL) are used to predict the background in the PASS region (score $\ge 0.63$) via a transfer function.

C. Signal Simulation

• BHs: Simulated using BLACKMAX and CHARYBDIS2 generators with various assumptions (rotating/non-rotating, graviton emission, remnant models).
• SBs: Generated with CHARYBDIS2.
• Sphalerons: Generated with BARYOGEN.
• Parameters: Scanned over fundamental Planck scale ($M_D$), string scale ($M_S$), extra dimensions ($n$), and sphaleron energy ($E_{sph}$).

3. Key Contributions

• Novel Phase Space Metric: The introduction of a phase space distance metric combined with SVM classification represents a significant advancement in identifying collider events with distinct kinematic features, moving beyond traditional cut-based or simple multivariate analyses.
• Improved Background Modeling: The Shape Invariance method provides a robust, data-driven background estimate that relies on the shape of the $S_T$ distribution rather than Monte Carlo simulation for the dominant QCD background.
• Data Volume: This is the most comprehensive search to date using the full Run 2 dataset (138 fb$^{-1}$), significantly increasing statistical power compared to previous CMS analyses (e.g., the 35.9 fb$^{-1}$ analysis in Ref. [21]).
• Updated Parton Distribution Functions (PDFs): The analysis utilizes the NNPDF3.1 set, which significantly impacts the predicted cross-sections for high-mass states compared to older PDF sets used in previous searches.

4. Results

No significant excess of events was observed over the Standard Model background predictions in either the model-independent or model-dependent searches.

• Model-Independent Limits:

  • Set 95% Confidence Level (CL) upper limits on the cross-section times acceptance for generic signals with high multiplicity.
  • Achieved a factor of ~4 improvement in cross-section limits compared to previous CMS results, driven by increased luminosity and the sphericity selection.

• Model-Dependent Limits (Black Holes & String Balls):

  • Black Holes: Excluded semiclassical BHs with masses below 8.4–11.4 TeV (depending on the model and $n$). This extends the exclusion reach by 1.0–1.6 TeV compared to previous searches.
  • String Balls: Excluded masses below 9.0–10.7 TeV, extending the reach by 1.3–1.9 TeV.
  • Extra Dimensions: For specific models (e.g., B1 with $M_D=4$ TeV, $n=4$), the exclusion of extra dimensions was extended, with the analysis excluding $n \ge 2$ for a wide range of $M_D$ and $M_{min}^{BH}$.

• Sphaleron Limits:

  • Derived an upper limit on the pre-exponential factor ($p_{sph}$), representing the fraction of quark-quark interactions above the sphaleron threshold that undergo the transition.
  • For the nominal threshold $E_{sph} = 9$ TeV, the observed limit is 0.0034 (expected 0.0035) at 95% CL.
  • This is a factor of 6.2 (observed) and 3.4 (expected) improvement over the previous best limit from CMS.

5. Significance

This paper represents the state-of-the-art in the search for TeV-scale gravity and topological transitions in high-energy physics.

1. Stringent Constraints: It provides the most stringent limits to date on the existence of microscopic black holes, string balls, and electroweak sphalerons, effectively ruling out these phenomena in the mass ranges probed by the LHC Run 2.
2. Methodological Advancement: The successful application of phase space geometry and machine learning (SVM) to background estimation demonstrates a powerful new toolkit for future BSM searches, particularly for signals with complex, high-multiplicity final states.
3. Theoretical Impact: The results constrain the parameter space of Large Extra Dimension models and string theory scenarios, pushing the lower bounds on the fundamental Planck scale and string scale higher than ever before. The lack of observation suggests that if these phenomena exist, they occur at energy scales beyond the current LHC reach or require different production mechanisms.