Probing torsion field with Einstein-Cartan theory at the HL-LHC: an angular distribution case study

This study utilizes simulated High Luminosity LHC data to investigate the angular distribution of high-mass dimuon pairs in the Collins-Soper frame, employing a simplified Einstein-Cartan model to establish 95% confidence level upper limits on the masses of a spin-2 dark neutral gauge boson and the torsion field.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle smasher. It fires two beams of protons at each other at nearly the speed of light, creating a chaotic explosion of tiny particles. Usually, scientists look for specific "new" particles in this debris. But sometimes, the new physics isn't a single heavy particle; it's a subtle change in how the debris flies apart.

This paper is like a detective story where the author, S. Elgammal, is trying to find a hidden "twist" in the fabric of space-time using data from the future version of the LHC, called the HL-LHC (High-Luminosity LHC).

Here is the breakdown of the investigation in simple terms:

1. The Mystery: Is Space-Time "Twisted"?

In our everyday world, we think of space as a smooth stage where particles perform. However, a theory called Einstein-Cartan theory suggests that space-time might actually have a "twist" or "torsion" to it, kind of like a screw thread instead of a smooth cylinder.

The author is looking for evidence of this "torsion field." If it exists, it would act like a heavy, invisible bridge that allows particles to turn into Dark Matter (the invisible stuff that holds galaxies together) and a new, invisible "dark" particle.

2. The Clue: The "Angle" of the Debris

When the LHC smashes protons, it often creates pairs of muons (heavy cousins of electrons). In the standard "textbook" physics (the Standard Model), these muon pairs fly out in a predictable pattern, like water spraying from a hose.

The author focuses on a specific angle called $\cos\theta_{CS}$.

• The Analogy: Imagine throwing a ball. In the standard model, the ball tends to fly forward or backward in a specific way. But if the "torsion field" exists, it would act like a magical wind that makes the ball fly in a perfectly symmetrical, round pattern instead.
• The author uses computer simulations to see if the "twisted" space-time model creates a different angle pattern than the standard model.

3. The Setup: A Future Simulation

Since the HL-LHC (which will run at 14 TeV of energy) hasn't fully started collecting this specific data yet, the author used a computer simulation.

• Think of this as a "flight simulator" for particle physics.
• They programmed the computer to crash protons together 3,000 times more than previous experiments (3000 "fb" of data).
• They created a "signal" (the torsion theory) and mixed it with "background noise" (standard particle collisions like Drell-Yan, top quarks, etc.).

4. The Filter: Cleaning the Noise

The problem is that the "signal" (the torsion effect) is very quiet and gets drowned out by the "noise" (standard collisions).

• The Analogy: Imagine trying to hear a whisper (the torsion signal) in a stadium full of cheering fans (the background noise).
• To solve this, the author applied a set of strict filters (cuts). They looked for events where:
  • The muons and the missing energy (the dark matter running away) were perfectly opposite each other (back-to-back).
  • There were very few other "junk" particles (jets) flying around.
  • The energy matched specific predictions.
• These filters acted like a noise-canceling headphone, silencing the fans so the whisper could be heard.

5. The Findings: What the Detective Found

After applying the filters, the author found two main things:

A. The Shape is Different
The "twisted" model produced a distinct, symmetrical shape in the angle data (a spin-2 signature), while the standard model looked different. This proves that if we see this specific shape in real data, it would be a smoking gun for this new theory.

B. The "Exclusion Limits" (The "Safe Zones")
The author didn't find the torsion field yet (because they were using simulated data, not real data). Instead, they calculated upper limits.

• The Analogy: Imagine searching for a lost dog in a forest. You don't find the dog, but you can say, "If the dog were this big, we would have seen it by now. So, the dog must be smaller than X, or in a part of the forest we haven't checked."
• The paper calculates exactly which masses of the "torsion field" and the "dark gauge boson" (the new particle) are ruled out at a 95% confidence level.
  • For example, if the dark boson weighs 200 GeV, the torsion field cannot weigh between 1,396 and 5,545 GeV. If it did, we would have seen it.

Summary

This paper is a "proof of concept" for a future experiment. It says:

1. Theory: If space-time has a twist (torsion), it changes the angle at which particles fly.
2. Method: We can spot this by looking at high-energy muon pairs at the future HL-LHC and using strict filters to ignore background noise.
3. Result: We haven't found it yet, but we have mapped out exactly which "weights" of these new particles are impossible to exist based on our current theoretical understanding. If the HL-LHC runs and finds a signal in the "allowed" zones, it could rewrite our understanding of gravity and dark matter.

Important Note: The paper strictly deals with theoretical physics simulations. It does not claim to have found dark matter, nor does it suggest any immediate medical or technological applications. It is purely about testing the laws of the universe at the smallest scales.

Technical Summary

Technical Summary: Probing Torsion Field with Einstein-Cartan Theory at the HL-LHC

Problem Statement
The primary objective of this study is to investigate potential signatures of physics beyond the Standard Model (SM) at the High-Luminosity Large Hadron Collider (HL-LHC). Specifically, the analysis targets the production of high-mass dimuon pairs ($\mu^+\mu^-$) accompanied by significant missing transverse energy ($E_T^{miss}$), a signature indicative of dark matter production. The study focuses on distinguishing these signals from dominant SM backgrounds, particularly the Drell-Yan process, by analyzing the angular distribution of the dimuon pairs. The theoretical framework employed is a simplified model based on Einstein-Cartan (EC) theory, which posits the existence of a torsion field ($TS$) mediating interactions between quarks and dark matter, leading to the production of a dark neutral gauge boson ($A'$) and dark matter particles ($\chi$).

Methodology
The analysis utilizes simulated proton-proton collision data at a center-of-mass energy of $\sqrt{s} = 14$ TeV, corresponding to an integrated luminosity of $3000 \text{ fb}^{-1}$, representing the HL-LHC configuration.

• Theoretical Model: The signal model is derived from Einstein-Cartan theory. It involves the annihilation of quark-antiquark pairs mediated by a heavy torsion field ($TS$) into a pair of dark matter particles ($\chi\bar{\chi}$). One dark matter particle radiates a dark gauge boson ($A'$) via bremsstrahlung, which subsequently decays into a muon pair ($A' \to \mu^+\mu^-$). The model assumes fixed parameters: dark matter mass $M_\chi = 500$ GeV, coupling constants $g_\eta = 0.125$ (torsion-fermion) and $g_D = 1.0$ (dark gauge boson-dark matter), while treating the torsion mass ($M_{TS}$) and dark gauge boson mass ($M_{A'}$) as free variables.
• Simulation: Signal and SM background events were generated using MadGraph5 aMC@NLO (at NLO), hadronized with Pythia 8, and processed through a fast detector simulation (DELPHES) configured for the HL-LHC.
• Kinematic Variable: The core discriminant is the angular distribution of the dimuon system in the Collins-Soper (CS) frame, specifically the variable $\cos\theta_{CS}$. This frame is chosen to minimize distortions arising from the unknown transverse momenta of the incoming partons.
• Event Selection: A two-stage selection strategy was employed:
  1. Pre-selection: Requires two oppositely charged muons with $p_T > 30$ GeV, $|\eta| < 2.5$, and isolation criteria.
  2. Tight Selection: To suppress SM backgrounds (Drell-Yan, $t\bar{t}$, $tW$, dibosons), additional cuts were applied: $\Delta\phi(\mu^+\mu^-, E_T^{miss}) > 2.5$, relative transverse energy difference $|E_T^{\mu\mu} - E_T^{miss}|/E_T^{\mu\mu} < 0.4$, back-to-back orientation $\cos(\text{angle}_{3D}) < -0.75$, jet multiplicity $N_{jets} < 1$, and an invariant mass window around the hypothesized $M_{A'}$.

Key Contributions

1. Angular Distribution Analysis: The study demonstrates that the simplified EC model predicts a distinct angular distribution for the signal compared to the SM Drell-Yan background. The signal exhibits a symmetric distribution around zero in $\cos\theta_{CS}$, characteristic of a spin-2 boson decay, whereas the SM background shows a different shape.
2. Optimization of Selection Cuts: The paper details the optimization of kinematic cuts to maximize the signal-to-background ratio. It quantifies the "N-1 efficiency" for each cut, showing that the proposed criteria effectively suppress Drell-Yan and ZZ backgrounds while maintaining high efficiency for the signal at high muon transverse momenta.
3. Statistical Interpretation: Using the profile likelihood method and the modified frequentist $CL_s$ approach, the analysis establishes exclusion limits and discovery potential.

Results

• Discovery Potential: The analysis calculates the integrated luminosity required to achieve a $5\sigma$ discovery significance. For a torsion mass $M_{TS} = 4000$ GeV and $M_{A'} = 200$ GeV, a $5\sigma$ discovery is achievable with $500 \text{ fb}^{-1}$. For a heavier torsion mass of $M_{TS} = 5000$ GeV, the required luminosity increases to over $2000 \text{ fb}^{-1}$. Similarly, for $M_{A'} = 400$ GeV, $160 \text{ fb}^{-1}$ is sufficient for $M_{TS} = 4000$ GeV, while $600 \text{ fb}^{-1}$ is needed for $M_{TS} = 5000$ GeV.
• Exclusion Limits: The study sets 95% confidence level (CL) upper limits on the product of the production cross-section and the branching ratio ($\sigma \times \text{Br}$) as a function of the mediator mass ($M_{TS}$).
  • For $M_{A'} = 200$ GeV, the excluded range for $M_{TS}$ is 1396–5545 GeV.
  • For $M_{A'} = 300$ GeV, the range is 1402–6310 GeV.
  • For $M_{A'} = 400$ GeV, the range is 1537–7026 GeV.
  • For $M_{A'} = 500$ GeV, the range is 1677–6927 GeV.
• Background Suppression: The application of the tight selection criteria successfully eliminates Drell-Yan and ZZ backgrounds and significantly reduces contributions from $t\bar{t}$, $tW$, WW, and WZ processes, rendering the signal distinguishable in the $\cos\theta_{CS}$ distribution.

Significance
The paper claims that analyzing the angular distribution ($\cos\theta_{CS}$) of high-mass dimuon pairs is an effective strategy for differentiating between Standard Model backgrounds and new physics scenarios involving torsion fields and dark matter. By utilizing the specific spin-2 characteristics predicted by the Einstein-Cartan model, the study provides a method to set stringent limits on the masses of the torsion field and the dark gauge boson. The results suggest that the HL-LHC, with its high integrated luminosity, has the potential to either discover such torsion-mediated dark matter production or exclude significant portions of the parameter space for these models. The work serves as a case study for how angular variables can complement mass spectrum analyses in the search for new physics.