A Model for Dark Moments of the W Boson

This paper extends the concept of "dark moment" couplings to the Standard Model W boson within a specific portal matter model, demonstrating that while direct W boson interactions with dark photons remain difficult to detect at the LHC due to small rates and large backgrounds, the direct production of the underlying portal matter scalars offers a more promising discovery channel already constrained by existing searches.

The Gist

Here is an explanation of the paper "A Model for Dark Moments of the W Boson" using simple language and creative analogies.

The Big Picture: The Invisible Neighbor

Imagine our universe is a house (the Standard Model) where we know all the furniture and people. But we know there's a "Dark Sector" next door (Dark Matter) that we can't see or touch directly. We suspect they are there because of gravity, but we don't know how they interact with our house.

Usually, scientists think these two worlds talk to each other through a "leaky pipe" called Kinetic Mixing. It's like a tiny, almost invisible crack in the wall where a little bit of light (photons) from our house leaks into the dark house, and vice versa. This allows us to detect the dark side indirectly.

This paper asks a new question: What if there isn't just a leaky pipe? What if there are secret, hidden tunnels inside the walls that allow the "Dark Neighbor" to shake hands with specific, heavy furniture in our house, like the W Boson (a heavy particle that helps hold atoms together)?

The Cast of Characters

To build these secret tunnels, the author introduces a new type of matter called Portal Matter (PM). Think of these as "dual-citizens."

• They live in our world (they have Standard Model properties).
• They also live in the Dark world (they have Dark charges).
• In this specific story, these dual-citizens are ghostly ghosts (scalar particles) that are very light and can float around.

The "Dark Moment" Analogy

In the old story (Kinetic Mixing), the dark neighbor just bumps into the light, and the light gets a tiny bit of "darkness" on it.

In this new story, the author suggests something more complex. Imagine the W Boson is a heavy, spinning top. Usually, it spins perfectly. But because these "dual-citizen" ghosts are running in circles around the top, they create a tiny, invisible magnetic field around it. This is called a "Dark Moment."

• The Metaphor: Think of the W Boson as a dancer. The "Dark Moment" is like the dancer suddenly picking up a tiny, invisible fan (the Dark Photon) that spins with them. The dancer doesn't own the fan, but the fan is attached to them because of the ghosts running around them.
• The Catch: This only works if the ghosts running around aren't all the same weight. If they are all identical, their effects cancel out (like two people pushing a car from opposite sides with equal force). But if one ghost is slightly heavier than the other, the balance tips, and the "fan" (Dark Photon) gets attached to the dancer.

The Experiment: Can We See It?

The author calculated what happens if we smash particles together at the Large Hadron Collider (LHC), the world's biggest particle accelerator.

1. The Goal: We want to see a W Boson fly out and immediately drop off a Dark Photon. Since the Dark Photon is invisible, we would see the W Boson fly off, and then suddenly, the whole system would have "Missing Energy" (MET) because the Dark Photon vanished into the dark sector.
2. The Result: The author found that this "Dark Moment" interaction is indeed stronger than the old "leaky pipe" (Kinetic Mixing) idea. It's like finding a wider tunnel.
3. The Problem: Even though the tunnel is wider, it's still too narrow to see through the noise. The LHC produces billions of collisions. The "signal" (the W Boson dropping the Dark Photon) is like trying to hear a whisper in a stadium full of screaming fans (the Standard Model background). The author concludes that even with the future "High-Luminosity" LHC, we probably won't be able to hear that whisper.

The Twist: The "Real" Treasure

Here is the most exciting part of the paper. While the "Dark Moment" signal is too quiet to hear, the dual-citizen ghosts themselves (the Portal Matter scalars) might be loud enough to see!

• The Analogy: Imagine you are trying to hear a whisper (the Dark Moment). You can't. But, the people whispering (the Portal Matter scalars) are actually wearing bright neon jackets. If you look for the people in the neon jackets, you can see them easily, even if you can't hear the whisper.
• The Discovery: The author shows that we can produce these "neon jacket" particles directly at the LHC. When they decay, they create a very specific signature: W Bosons + Missing Energy.
• The Good News: This signal is much louder than the whisper. It's loud enough that the LHC might already be able to see it, or will definitely see it soon.
• The Bad News: Because these particles are so easy to make, they create a lot of "noise" that actually makes it even harder to hear the original "Dark Moment" whisper.

The Conclusion

1. The Theory: It is possible for the W Boson to have a "Dark Moment" interaction with the Dark Sector, created by a specific mix of new particles.
2. The Reality Check: Trying to detect this interaction directly (the whisper) is likely impossible with current or near-future technology because the background noise is too loud.
3. The Opportunity: However, the particles that create this interaction (the neon jackets) are light enough and common enough that we might be able to find them directly at the LHC right now.
4. The Future: If we find these new particles, we will have proven that the "Dark Moment" mechanism exists, even if we can't see the moment itself. It's a case of "seeing the smoke to know there's a fire," even if we can't see the flame.

In short: The paper suggests that while the "ghostly handshake" between the W Boson and Dark Matter is too faint to catch, the "ghosts" doing the shaking might be caught in the act, giving us a new way to solve the mystery of Dark Matter.

Technical Summary

Here is a detailed technical summary of the paper "A Model for Dark Moments of the W Boson" by Thomas G. Rizzo.

1. Problem Statement

The paper addresses the search for interactions between the Standard Model (SM) and the Dark Sector, specifically focusing on the Dark Photon (DP) ($V$) and its potential coupling to the SM $W$ boson.

• Standard Scenario: The most common portal is Kinetic Mixing (KM), where loops of "Portal Matter" (PM) fields induce a mixing between the SM photon and the DP. This gives SM charged particles an effective small charge ($e\epsilon Q_{em}$) under the DP.
• The Gap: Recent work suggested that in specific UV-complete models, PM loops could generate "dark moment" couplings (dipole moments or form factors) for SM fermions, even if they are electrically neutral. However, it was previously argued that SM gauge bosons (like the $W$) could not acquire such couplings if the PM consisted of Vector-Like Fermions (VLF) due to Bose symmetry and charge conjugation invariance forcing loop cancellations.
• The Question: Can scalar Portal Matter generate non-vanishing "dark moment" couplings for the $W$ boson ($WWV$ vertex), and if so, is the resulting production rate ($WV \to W + \text{MET}$) observable at the LHC? Furthermore, how does this compare to the direct production of the PM scalars themselves?

2. Methodology and Model Framework

The author employs a specific scalar extension of the SM to investigate this phenomenon.

• Model Construction:

  • Gauge Group: $SU(2)_L \times U(1)_Y \times U(1)_D$.
  • Scalar Sector: In addition to the SM Higgs doublet ($\Phi$, $Q_D=0$), the model introduces:
    1. A complex $SU(2)_L$ triplet ($\Sigma$, $Y=0$, $Q_D=1$).
    2. A complex singlet ($S$, $Q_D=1$).
  • Symmetry Breaking: Both $\Sigma$ and $S$ acquire vacuum expectation values (vevs), $v_t$ and $v_s$ (order of a few GeV), which break $U(1)_D$ and generate the DP mass ($m_V$). The SM doublet vev $v_d \approx 246$ GeV generates the $W$ mass.
  • Parameter Space: The analysis focuses on the regime where $v_t < v_s$ (parameter $t = v_t/v_s < 1$) to suppress tree-level contributions to the $\rho$ parameter (T-parameter), ensuring consistency with precision electroweak data.

• Mechanism for Dark Moments:

  • Unlike VLFs, the scalar components of the triplet ($\Sigma^{\pm}_{1,2}$) and the singlet/triplet neutral states ($H, h_D$) can have non-degenerate masses due to mixing with the SM Higgs and the specific structure of the scalar potential.
  • The paper calculates 1-loop triangle diagrams involving charged scalars ($\Sigma^\pm$) and neutral scalars ($H, h_D$).
  • Key Insight: Because the masses of the scalars in the loop are non-degenerate ($m_1 \neq m_2$), the Bose symmetry cancellation that suppresses fermionic loops does not occur. This generates a finite, loop-induced effective vertex $\Gamma_{\alpha\beta\mu}$ for $W^* \to W V$.

• Calculations:

  • The effective vertex is parameterized by form factors ($a_0, a_1, a_2, a_3$) derived from dimension-4 and dimension-6 operators.
  • The differential cross-section for $q\bar{q}' \to W^* \to WV$ is calculated at Leading Order (LO).
  • Benchmark Points: Two specific mass configurations (BM1 and BM2) are defined with scalar masses in the 300–600 GeV range to demonstrate the phenomenology.

3. Key Contributions

1. Extension to Gauge Bosons: The paper successfully extends the concept of "dark moments" from fermions to the $W$ boson, demonstrating that scalar Portal Matter can generate these couplings without requiring an extension of the $U(1)_D$ gauge group to a non-abelian structure.
2. Quantitative Cross-Section Analysis: It provides the first detailed calculation of the $WV$ production cross-section at the LHC (13/14 TeV) arising specifically from these loop-induced dark moment couplings, comparing it directly to the standard Kinetic Mixing scenario.
3. Direct vs. Indirect Production: The paper shifts the focus from the indirect $WV$ signal to the direct production of the Portal Matter scalars ($\Sigma_1, \Sigma_2, H$), showing that these processes dominate the phenomenology and offer better discovery prospects.
4. Phenomenological Constraints: It re-evaluates existing LHC search limits (ATLAS/CMS) for $W + \text{MET}$ and $WW + \text{MET}$ channels in the context of this specific scalar model, identifying regions of parameter space already constrained or accessible at the High-Luminosity LHC (HL-LHC).

4. Results

• $WV$ Production Rate:

  • The dark moment-induced cross-section for $WV$ production is found to be $\sim 10-20$ times larger than the rate predicted by pure Kinetic Mixing (KM) for comparable parameters.
  • However, the absolute rate remains small (order of 100–200 attobarns).
  • Conclusion: The signal is completely swamped by Standard Model backgrounds (primarily $WZ$ with $Z \to \nu\bar{\nu}$) and is unobservable at the LHC, even at the HL-LHC, via the $W + \text{MET}$ channel alone.

• Direct Scalar Production:

  • Pair Production ($\Sigma_1^+ \Sigma_1^-$): Via $q\bar{q} \to \gamma^*/Z^* \to \Sigma_1^+ \Sigma_1^-$. The cross-section is significant (order of fb) for masses up to $\sim 450-500$ GeV.
  • Associated Production ($\Sigma_1 h_D$): Via $q\bar{q}' \to W^* \to \Sigma_1 h_D^\dagger$. This process is also sizable and leads to a $W + \text{MET}$ final state.
  • Decay Chains:
    • $\Sigma_1^\pm$ decays dominantly to $W^\pm h_D$ (where $h_D$ is a light dark Higgs decaying to DM/MET).
    • This results in final states of $W^+W^- + \text{MET}$ (from pair production) or $W + \text{MET}$ (from associated production).

• Comparison with Backgrounds:

  • The direct production of PM scalars yields rates 2 orders of magnitude larger than the indirect $WV$ signal.
  • Consequently, the "background" from direct scalar production actually overwhelms the $WV$ signal, making the $WV$ search irrelevant compared to the direct search for the scalars.

• Experimental Constraints:

  • ATLAS/CMS Searches: Existing searches for chargino pair production (decaying to $W$ + LSP) and $W + \text{MET}$ (high purity jets) are reinterpreted.
  • Limits: Current data excludes $\Sigma_1$ masses roughly below 230–370 GeV depending on the channel and analysis assumptions.
  • Future Prospects: The HL-LHC is predicted to probe a significant fraction of the remaining parameter space (masses up to $\sim 450-500$ GeV) using the $W + \text{MET}$ and $WW + \text{MET}$ channels.

5. Significance

• Theoretical: The work demonstrates that scalar Portal Matter can generate unique "dark moment" interactions for gauge bosons without requiring complex non-abelian dark sectors, offering a distinct signature from fermionic models.
• Phenomenological: It highlights a crucial lesson for Dark Sector searches: Indirect loop-induced signals (like $WV$) may be sub-dominant to direct production of the loop particles. In this model, the "new physics" is not the $WV$ vertex itself, but the scalar particles running in the loop.
• Experimental Strategy: The paper argues that the most viable path to discovering this specific class of dark sector models is not searching for anomalous $W$ decays, but rather searching for direct production of light scalar triplets in $W + \text{MET}$ and $WW + \text{MET}$ channels. It provides a concrete roadmap for how the HL-LHC can test these models, potentially ruling out or discovering the scalar Portal Matter within the next decade.