Theory Calculations for LDMX and LOHENGRIN beyond Coherent Bethe-Heitler Scattering

This paper presents comprehensive theoretical calculations for LDMX, DarkSHINE, and LOHENGRIN experiments that extend beyond standard coherent Bethe-Heitler scattering by including higher-order electromagnetic and kinetic mixing effects, finding that while these contributions have limited impact on signal and background predictions, the LOHENGRIN experiment specifically requires a hadronic calorimeter extension to effectively veto diffractive scattering backgrounds.

The Gist

The Big Picture: Hunting for Invisible Ghosts

Imagine you are trying to find a ghost in a room. You can't see the ghost, but you know it's there because when you throw a ball at a wall, the ball bounces back in a weird way.

This paper is about three new experiments (LDMX, DarkSHINE, and Lohen-GRIN) that are building "ghost hunting" machines. They shoot a beam of electrons (tiny, fast balls) at a heavy metal target (the wall).

• The Goal: They hope to create a "Dark Photon" (the ghost).
• The Clue: If a Dark Photon is created, it flies away invisibly. The only thing the detectors can see is the electron bouncing back. If the electron bounces back with less energy than expected, the scientists say, "Aha! Something invisible took the missing energy!"

The Problem: The "Background Noise"

The problem is that electrons bouncing off a wall is a very common event. Usually, they just bounce off the whole wall smoothly. This is called Coherent Bethe-Heitler scattering. It's like throwing a ball at a solid brick wall; it bounces back predictably.

The scientists in this paper asked: "Is our prediction of how the ball bounces off the wall perfect? Or are we missing some subtle details that could look like a ghost?"

What This Paper Did: Looking Under the Rug

The authors built a much more detailed mathematical map of how these electrons scatter. They realized that previous maps were too simple. They added three new layers of complexity:

1. The Wall isn't just a Wall; it's made of Bricks.

   • Old View: The electron hits the whole nucleus (the wall) as one big smooth object.
   • New View: The electron might actually hit individual protons or neutrons (the bricks) inside the nucleus. Sometimes it bounces off a single brick, causing the wall to rattle. The paper calculates how often this happens and how it changes the electron's path.

2. The "Ghost" can talk to the Bricks, not just the Wall.

   • Old View: The Dark Photon only interacts with the electron.
   • New View: The Dark Photon might also interact with the protons and neutrons inside the target. It's like the ghost can whisper to the bricks, changing how they vibrate.

3. The "Ghost" can be a "Virtual" Guest.

   • Sometimes the Dark Photon doesn't even get created as a real particle. Instead, it pops in and out of existence for a split second (a "virtual" particle) and messes with the math of the collision. The paper calculates how this invisible, fleeting guest changes the final result.

The Tools: A Super-Powered Calculator

To do this, the authors wrote a new computer program called Lohengrin++. Think of this as a super-advanced video game engine.

• Previous engines could only simulate the ball hitting the wall perfectly.
• This new engine can simulate the ball hitting individual bricks, the bricks rattling, and the invisible ghost whispering to them, all at the same time.

The Results: What Did They Find?

After running millions of simulations with their new, detailed map, they found two main things:

1. For Lohen-GRIN (The smaller experiment):
   They found that the "bricks" (individual protons/neutrons) can sometimes get knocked out of the wall and fly into the detector. If the detector isn't big enough to catch these flying bricks, it might mistake them for a ghost signal.

   • The Fix: They recommend that the Lohen-GRIN experiment needs to upgrade its "backstop" (a part of the detector called the HCAL) to catch these stray bricks so they don't fake a ghost signal.

2. For the General Search (LDMX and others):
   Surprisingly, once they accounted for all these new details (hitting bricks, virtual ghosts, etc.), the final prediction for the "Ghost Signal" didn't change much compared to the old, simple predictions.

   • The Takeaway: The old, simple maps were actually pretty good for the main search. The new, complex details mostly just confirm that the background noise is what we thought it was, though they are crucial for understanding specific, tricky parts of the experiment.

Summary

This paper is a "quality control" check for the math behind the ghost hunt.

• They built a better calculator that accounts for the messy reality of atomic nuclei (bricks inside a wall).
• They found that for one specific experiment (Lohen-GRIN), they need a bigger net to catch stray debris.
• They confirmed that for the main search for Dark Matter, the old, simpler math was mostly correct, giving scientists confidence that their "ghost hunting" strategy is solid.

Technical Summary

Technical Summary: Theory Calculations for LDMX and LOHENGRIN beyond Coherent Bethe-Heitler Scattering

Problem Statement
The Light Dark Matter eXperiment (LDMX), DarkSHINE, and LOHENGRIN are proposed fixed-target experiments designed to search for light dark matter (MeV–GeV range) via missing momentum signatures. These experiments rely on electron beams scattering off heavy nuclei to produce dark photons ($A'$) in bremsstrahlung-like processes. Historically, signal and background predictions have relied primarily on coherent Bethe-Heitler (BH) scattering, where the incident electron interacts with the nuclear charge distribution via single-photon exchange.

However, a complete theoretical description requires accounting for contributions beyond the standard coherent BH approximation. Specifically, existing calculations often neglect:

1. Incoherent contributions from diffractive scattering off individual nucleons (protons and neutrons).
2. The coupling of the dark photon to hadronic systems (quarks) as a virtual exchange particle.
3. Virtual Compton scattering (VCS) contributions where the (dark) photon is radiated from the hadronic current.
4. Virtual dark photon corrections to elastic $2 \to 2$ scattering, particularly in the context of leptophilic dark photons.

Furthermore, the validity of the Weizsäcker-Williams (WW) approximation, often used to simplify these calculations, has been questioned in phase space regions relevant to these experiments (specifically where the recoil electron carries a small fraction of the beam energy).

Methodology
The authors present a comprehensive calculation framework for differential cross sections up to third order in the electromagnetic fine-structure constant ($\alpha^3$) and fourth order in the kinetic mixing parameter ($\varepsilon^4$, though the leading signal terms are $\mathcal{O}(\alpha^3 \varepsilon^2)$).

• Theoretical Framework: The study extends the Standard Model (SM) with a dark sector containing a massive vector boson ($A'$) and dark matter particles. The $A'$ couples to SM fermions via kinetic mixing. The scattering process $e^- + H \to e^- + X$ is analyzed in two regimes:
  • Coherent Regime: Interaction with the entire nucleus (treated as a scalar field for Tungsten isotopes). The hadronic current is parameterized by a nuclear form factor $F(q^2)$.
  • Diffractive Regime: Incoherent interaction with individual nucleons. The hadronic current is parameterized by Dirac and Pauli form factors ($F_1, F_2$) for protons and neutrons, utilizing the Kelly parameterization for spacelike momentum transfers and Vector Meson Dominance (VMD) for timelike transfers.
• Amplitude Construction: The calculation includes:
  • $2 \to 3$ Processes: Real emission of a dark photon or QED photon. The amplitude is decomposed into Bethe-Heitler (photon emitted by the electron) and Virtual Compton Scattering (photon emitted by the hadronic target) terms.
  • $2 \to 2$ Processes: Elastic scattering including virtual dark photon exchange. For leptophilic dark photons, one-loop corrections to the leptonic current are included.
• Implementation: The authors developed a C++ numerical framework, Lohengrin++, utilizing the Cuba library (Vegas algorithm) for phase space integration. The code implements fully differential phase space measures and analytic expressions for squared amplitudes, allowing for arbitrary experimental cuts.
• Experimental Constraints: The study incorporates realistic kinematic selections for LDMX (8 GeV beam) and LOHENGRIN (3.2 GeV beam), including momentum and angular acceptance cuts for the recoil electron, as well as vetoes for hadronic activity and photons based on detector capabilities (ECAL/HCAL).

Key Contributions

1. Generalization of Scattering Formalism: The paper generalizes the hadronic current in Bethe-Heitler scattering to include both the nuclear charge distribution and the magnetic moments/charges of individual nucleons. It also incorporates the dark photon as a virtual exchange particle between leptonic and hadronic currents.
2. Inclusion of VCS and Diffractive Terms: The authors explicitly calculate contributions from Virtual Compton Scattering (radiation from the hadronic target) and diffractive scattering off nucleons, which were previously often omitted or approximated.
3. Virtual Corrections: The work includes one-loop corrections for leptophilic dark photons in $2 \to 2$ scattering, ensuring a consistent $\mathcal{O}(\alpha^3 \varepsilon^2)$ prediction when combined with real emission.
4. Phase Space Analysis: A detailed analysis of the kinematically available phase space for $2 \to 2$ and $2 \to 3$ processes is provided, contrasting the elastic line with the missing momentum signal region.
5. Numerical Framework: The release of Lohengrin++ allows for the integration of these complex amplitudes with realistic experimental cuts, moving beyond the Weizsäcker-Williams approximation.

Results

• Signal and Background: Numerical results for integrated signal yields and differential distributions (electron energy fraction $\xi_e$ and scattering angle $\theta_e$) are presented for both LDMX and LOHENGRIN scenarios.
• Impact of Beyond-Coherent Terms: Within the relevant dark photon mass range (MeV–GeV) and under realistic experimental selections (which typically require low-energy recoil electrons and veto hadronic activity), the contributions from diffractive scattering and VCS beyond coherent Bethe-Heitler scattering have only a limited effect on the predicted signal and dominant background.
• LOHENGRIN Specifics: The study finds that the LOHENGRIN experiment requires an extension of its Hadronic Calorimeter (HCAL) to effectively veto background processes originating from diffractive scattering, where quasi-free nucleons may be ejected.
• Virtual Corrections: For leptophilic dark photons, the interference between one-loop virtual corrections and tree-level QED graphs is shown to be of the same order as real emission contributions. However, in the inclusive signal regions, these virtual effects are necessary for a consistent cross-section prediction but do not drastically alter the exclusion limits compared to the tree-level real emission.
• Elastic Line Sensitivity: The paper notes that detecting the interference of a "vanilla" dark photon on the elastic line ($2 \to 2$) would require precision measurements at the $\mathcal{O}(10^{-5})$ level to distinguish the BSM signal from higher-order QED corrections, which is currently beyond the scope of standard LDMX-like strategies.

Significance
The paper claims to provide a more consistent and complete theoretical treatment of dark photon production in fixed-target experiments at $\mathcal{O}(\alpha^3 \varepsilon^2)$. By moving beyond the coherent Bethe-Heitler approximation and the Weizsäcker-Williams method, the authors establish that while the inclusion of diffractive and VCS terms is theoretically necessary for a rigorous description, their numerical impact on the primary missing momentum search strategy is modest for the proposed experimental setups. The work serves to validate the existing search strategies while highlighting specific detector requirements (such as the HCAL extension for LOHENGRIN) needed to control backgrounds from incoherent scattering. The development of Lohengrin++ offers a tool for future studies to incorporate these higher-order effects and complex kinematic selections without relying on potentially inaccurate approximations.