Photons, jets and missing momentum from a two-vector dark sector

This paper investigates the LHC phenomenology of a two-vector dark sector model, demonstrating that a binned analysis of the $\gamma+\text{jets}+E_T^{\text{miss}}$ signature significantly improves sensitivity to parameter space regions consistent with the observed dark matter relic abundance compared to inclusive missing-transverse-momentum searches.

The Gist

Imagine the universe is like a giant, bustling party. We know most of the guests (the "Standard Model" particles like electrons and quarks), but we suspect there's a huge crowd of invisible guests (Dark Matter) making up about 26% of the party. We can't see them, but we know they're there because they're pulling on the visible guests with gravity.

This paper is a detective story about how we might finally spot these invisible guests at the Large Hadron Collider (LHC), the world's biggest particle accelerator.

The Cast of Characters: The "Dark Duo"

Usually, scientists imagine Dark Matter as a single, shy character hiding in the shadows. This paper proposes a different story: a two-person team from a "Dark Sector."

1. The Stable One (X1): This is the Dark Matter candidate. It's the "good guy" who never leaves the party and never changes. It's invisible to our detectors.
2. The Unstable One (X2): This is the heavier partner. It's like a messenger that doesn't last long. It shows up, does something, and then quickly transforms.

The Rulebook (Dark Parity):
There's a special rule in this Dark Sector called "Dark Parity." It's like a bouncer at the club who says, "The stable one (X1) is allowed to stay forever, but the unstable one (X2) has to leave eventually." This rule ensures X1 is the Dark Matter we are looking for.

The Connection: The "Magic Door"

How do these invisible Dark guests interact with our visible world? They don't have a direct handshake (like a normal force). Instead, they use a "Magic Door" made of Dimension-Six Operators.

Think of this as a very weak, high-tech radio signal. It's so faint that you need a very loud shout (high energy) to hear it. The paper suggests that the only way these Dark guests talk to us is through a specific type of signal involving the hypercharge field (a fundamental force in our universe).

Because of the "Dark Parity" rule, they can't talk to us one-on-one. They need two of them to interact with us at the same time. It's like trying to open a heavy door that requires two people to push it simultaneously.

The Detective Work: What Happens at the LHC?

The scientists at the LHC smash protons together to create energy. Sometimes, this energy is enough to create a pair of these Dark guests (X1 and X2) along with a few jets of regular matter (quarks/gluons).

Here is the sequence of events the paper predicts:

1. The Creation: A collision creates the Dark Duo (X1 + X2) and a jet of regular particles.
2. The Transformation: The unstable partner (X2) is too heavy to stay. It immediately decays (transforms) into the stable partner (X1) and a photon (a particle of light).
3. The Escape: The stable partner (X1) is invisible. It flies away, taking its energy with it.
4. The Clue: Since energy must be conserved, if we see a bright flash of light (the photon) and a jet of particles, but the total energy doesn't add up, we know something invisible ran away. This missing energy is called "Missing Transverse Momentum."

The Signature: The paper looks for a specific "fingerprint" at the detector:

• One bright photon (the light from the transformation).
• One or more jets (the debris from the collision).
• Missing energy (the invisible Dark Matter running away).

The Detective's Strategy: The "Three-Bin" Trick

The authors compared two ways to look for this signal:

1. The "Inclusive" Approach (The Net): This is like casting a wide net and catching everything with missing energy above a certain level. It's simple, but it catches a lot of "noise" (background events that look like the signal but aren't).
2. The "Three-Bin" Approach (The Sieve): This is the paper's main innovation. Instead of just looking for any missing energy, they split the data into three buckets based on how much energy is missing:
   • Bucket 1: Low missing energy.
   • Bucket 2: Medium missing energy.
   • Bucket 3: High missing energy.

Why does this help?
Imagine you are looking for a rare bird. If you just look at the whole forest, you might miss it because there are too many other birds. But if you know the rare bird only flies at high altitudes, you can ignore the low branches and focus your search on the high canopy.

Similarly, the "Dark Duo" signal tends to produce higher missing energy than the background noise. By splitting the data into bins, the scientists can see the "shape" of the energy distribution. They found that this "Three-Bin" strategy is much better at spotting the signal because it ignores the noisy low-energy background and focuses on the high-energy tail where the signal hides.

The Results: What Did They Find?

• The "Net" (Inclusive) failed to find much: It could only see very light Dark Matter, and even then, it was in a region that cosmologists think is unlikely (because it would create too much Dark Matter for the universe to handle).
• The "Sieve" (Three-Bin) succeeded: By using the three buckets, they could see much heavier Dark Matter. Crucially, this method allowed them to probe a region of the universe that is compatible with what we actually observe. It found a "sweet spot" where the Dark Matter exists in just the right amount to match our universe's history.

The Caveat: The "Map" Limitation

The authors are honest about a limitation. Their "Magic Door" (the interaction) is described by a mathematical theory called an Effective Field Theory (EFT). This theory works well at low energies, like a map that works great for walking around a town but breaks down if you try to drive a car at 200 mph.

If the Dark Matter particles are extremely heavy (very high energy), the "map" might not be accurate anymore. The paper acknowledges that their results for the heaviest particles are "benchmarks"—best guesses based on the current map—but a more complete theory (a "UV completion") would be needed to be 100% sure about the heaviest scenarios.

Summary

In simple terms, this paper says:
"We have a new theory about Dark Matter being a pair of particles. If we smash particles together at the LHC, we might see a flash of light and a jet, with some energy mysteriously disappearing. By carefully sorting the data into three groups based on how much energy is missing, we can find this signal much better than before. This method allows us to look for Dark Matter in a range that actually makes sense for our universe, whereas the old, simpler methods would have missed it."

Technical Summary

Technical Summary: Photons, jets and missing momentum from a two-vector dark sector

Problem and Motivation
The microscopic nature of dark matter (DM) remains unknown despite extensive experimental efforts. While many models introduce a dark sector, a specific class of scenarios involves a "dark parity" symmetry that forbids renormalizable kinetic mixing between the Standard Model (SM) and the dark sector. In such cases, interactions arise only through higher-dimensional operators. This paper investigates the phenomenology of a minimal effective field theory (EFT) containing two neutral massive vector states, $X_1$ and $X_2$, both odd under a dark-parity symmetry. The lightest state, $X_1$, is stable and serves as a DM candidate, while the heavier state, $X_2$, is unstable. Unlike the canonical dark-photon scenario, the leading interactions with the SM in this model arise from dimension-six operators involving the hypercharge field strength tensor. The authors aim to determine the detectability of this specific topology at the Large Hadron Collider (LHC), specifically the signature of a photon, jets, and missing transverse momentum ($\gamma + \text{jets} + E_T^{\text{miss}}$), arising from the radiative decay $X_2 \to X_1 \gamma$.

Methodology
The study employs a comprehensive Monte Carlo simulation and statistical analysis framework:

1. Theoretical Framework: The model is defined by an effective Lagrangian containing dimension-six operators coupling the dark vectors to the SM hypercharge field strength ($B_{\mu\nu}$). The operators are antisymmetric in dark-flavor indices, necessitating at least two distinct vector states. The analysis assumes a benchmark EFT scale $\Lambda = 1$ TeV and Wilson coefficients $c_B = \tilde{c}_B = 1$.
2. Cosmological Context: The paper evaluates the relic abundance of $X_1$ under two production mechanisms:
   • Freeze-out: Assuming thermal equilibrium, the annihilation cross-section (scaling as $\Lambda^{-8}$) determines the relic density.
   • Freeze-in: Assuming the dark sector never reaches equilibrium, production occurs via rare scattering processes. The analysis focuses on the hierarchy $m_{X_1} < T_{rh} < m_{X_2}$, where $T_{rh}$ is the reheating temperature.
3. Collider Simulation:
   • Signal Generation: Events for $pp \to X_1 X_2 + j$ followed by $X_2 \to X_1 \gamma$ are generated at $\sqrt{s} = 13.6$ TeV using MadGraph5_aMC@NLO at leading order (LO). Parton showering and hadronization are handled by Pythia 8, and detector simulation is performed with Delphes 3 (ATLAS configuration).
   • Backgrounds: Major SM backgrounds include $Z(\to \nu\bar{\nu})\gamma j$, $W(\to \ell\nu)\gamma j$, and $t\bar{t}\gamma$. The $t\bar{t}\gamma$ yield is conservatively rescaled by a factor of 10 to account for potential statistical fluctuations and missing higher-order effects.
   • Event Selection: A cut-based strategy targets a hard isolated photon ($p_T^\gamma > 150$ GeV), a recoil jet ($p_T^j > 100$ GeV), and significant missing transverse momentum ($E_T^{\text{miss}} \ge 200$ GeV). Angular separations and lepton/b-jet vetoes are applied to suppress reducible backgrounds.
4. Statistical Analysis: Two strategies are compared:
   • Inclusive Single-SR: A single signal region with $E_T^{\text{miss}} \ge 200$ GeV.
   • Binned Three-SR: The same events are partitioned into three exclusive $E_T^{\text{miss}}$ bins (200–350 GeV, 350–450 GeV, and $\ge 450$ GeV) to retain coarse shape information.
   • Limits are derived using a profile-likelihood ratio with the $CL_s$ method, considering systematic uncertainties on background normalizations.

Key Contributions and Results

• Decay Properties: The authors demonstrate that for the benchmark parameters, the heavier vector $X_2$ decays promptly ($c\tau \sim \text{nm}$ to fm) across the relevant mass range, justifying the prompt-photon assumption. The branching ratio $BR(X_2 \to X_1 \gamma)$ is unity below the $Z$-boson threshold and remains dominant ($\sim 77\%$) at large mass splittings due to the electroweak mixing of the hypercharge field.
• Cosmological Constraints:
  • In the freeze-out scenario, the region of parameter space that reproduces the observed relic density ($\Omega_{DM}h^2 \approx 0.12$) is identified. Regions with lower masses or smaller mass splittings tend to overproduce DM.
  • In the freeze-in scenario, the required EFT scale $\Lambda$ depends strongly on the reheating temperature $T_{rh}$. Lower $T_{rh}$ (e.g., 7 GeV) allows for $\Lambda$ values ($\sim 1-2.5$ TeV) that are potentially accessible at the LHC, whereas higher $T_{rh}$ pushes $\Lambda$ to scales ($\sim 5-12$ TeV) where collider rates are negligible.
• Collider Sensitivity:
  • Inclusive Analysis: The single-SR strategy has limited reach, excluding masses only up to $m_{X_1} \approx 300$ GeV and $m_{X_2} \approx 850$ GeV for $139 \text{ fb}^{-1}$. Crucially, this reach lies entirely in the region where the model would overproduce dark matter in a standard freeze-out scenario, rendering it cosmologically disfavored if $X_1$ constitutes all DM.
  • Binned Analysis: The three-SR strategy substantially improves sensitivity by exploiting the harder $E_T^{\text{miss}}$ spectrum of the signal compared to the steeply falling SM background. This approach extends the exclusion reach to $m_{X_1} \approx 850-900$ GeV and $m_{X_2} \approx 1.8-2.0$ TeV for $139 \text{ fb}^{-1}$.
  • Significance of Binning: The binned analysis successfully probes regions of parameter space compatible with the observed relic abundance in the standard freeze-out scenario, which the inclusive analysis misses.
  • HL-LHC Projections: For $3000 \text{ fb}^{-1}$, the three-SR reach extends to $m_{X_1} \approx 950$ GeV and $m_{X_2} \approx 2.2$ TeV.

Significance and Claims
The paper claims that searches for $\gamma + \text{jets} + E_T^{\text{miss}}$ are a sensitive probe for dimension-six portals connecting the SM to vector dark sectors. A primary finding is the critical importance of utilizing differential information (even coarse binning) in the missing transverse momentum spectrum. The authors assert that while an inclusive counting strategy fails to probe cosmologically viable regions (those consistent with the observed relic density), a simple three-bin strategy can access parameter space compatible with standard thermal freeze-out.

The authors explicitly note the limitations of their study: the results are presented as benchmark sensitivities within an EFT framework. For high-mass points where the kinematic scales approach or exceed the EFT cutoff ($\Lambda = 1$ TeV), the validity of the contact-operator description is questionable, and a fully conservative interpretation would require a UV completion or event-by-event truncation. The study does not claim to discover new physics but rather establishes the expected reach and the methodological advantage of shape-based analyses for this specific class of dark sector models.