⚛️ phenomenology

Pion bremsstrahlung in the splitting function formalism and the dark photon production

This paper investigates the production of dark photons with masses between 0.4 and 3.5 GeV in negatively charged pion-proton collisions via inelastic pion bremsstrahlung and QCD Drell-Yan-like processes, estimating total cross sections and energy distributions relevant for the NA64h experiment and secondary pion beams in T2K, DUNE, and SHiP.

Original authors: Dmitry Gorbunov, Ekaterina Kriukova

Published 2026-03-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dmitry Gorbunov, Ekaterina Kriukova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Hunting for a "Ghost" Particle

Imagine the Standard Model of physics as a very well-organized library. We know almost all the books (particles) in it. But astronomers and physicists have noticed that the library seems to be missing some shelves. There is "dark matter" out there, and we don't know what it's made of.

To find it, scientists propose a "Vector Portal." Think of this as a secret door connecting our known world (the library) to a hidden, dark room (the dark sector). The key to this door is a hypothetical particle called the Dark Photon ( $\gamma'$ ). It's like a cousin to the regular photon (light), but it's heavy and invisible to our eyes. It can only be seen if it briefly "mixes" with normal light.

The goal of this paper is to figure out how to create these Dark Photons in a lab and how to spot them.

The Setup: The Particle Cannon

The authors are looking at experiments where a high-speed beam of negative pions (unstable, short-lived particles) smashes into a block of iron (a target).

The Analogy: Imagine firing a stream of fast-moving tennis balls (pions) at a brick wall (the iron target).
The Goal: When these tennis balls hit the wall, they might shed a piece of themselves or interact in a way that creates a new, invisible "ghost" ball (the Dark Photon) that flies off.

The paper focuses on two specific ways this "ghost" can be created:

The "Braking" Method (Pion Bremsstrahlung): The pion hits the wall, gets jolted, and emits a Dark Photon as it slows down or changes direction.
The "Annihilation" Method (Drell-Yan): Inside the pion and the proton, tiny particles called quarks meet, annihilate, and turn directly into a Dark Photon.

The Problem: The Old Map Was Wrong

For a long time, scientists used a specific map (called Chiral Perturbation Theory or ChPT) to predict how often these Dark Photons would be made. This map works great for slow-moving particles, like a car driving through a quiet neighborhood.

However, in experiments like NA64h (at CERN), the pions are moving at nearly the speed of light (50 GeV). This is like driving a Formula 1 car at 200 mph through that same neighborhood.

The Authors' Discovery: They realized the old map (ChPT) breaks down at these speeds. It's like trying to use a map of a walking path to navigate a supersonic jet. The math predicts things that don't actually happen because it ignores the complex "traffic" (resonances and heavy particles) that appears at high speeds.
The Fix: They threw out the old map for high speeds and built a new one using a technique called Factorization.

The New Method: The "Splitting" Strategy

Instead of trying to calculate the entire messy collision from scratch, the authors split the problem into two simpler parts:

The Split: Imagine the incoming pion is a runner carrying a backpack. As it runs, it "splits" off a Dark Photon. The authors calculated the probability of this "splitting" happening using a new formula (the Splitting Function).
The Crash: Once the Dark Photon is split off, the remaining pion crashes into the target. They used real-world experimental data to see how often these crashes happen, rather than relying on theoretical guesses.

The Result: This new method is much more accurate for high-speed collisions. It shows that for Dark Photons with masses between 0.4 and 1.3 GeV, the "Braking" method (Bremsstrahlung) is actually the most common way to make them.

The Rival: The "Annihilation" Method

The paper also looked at the second method (Drell-Yan), where quarks inside the particles annihilate to create the Dark Photon.

The Analogy: If the "Braking" method is like a runner dropping a coin, the "Annihilation" method is like two runners colliding and turning into a coin.
The Finding: For heavier Dark Photons (masses 1.3 to 3.5 GeV), this "Annihilation" method becomes the dominant way they are produced.

Why This Matters: The "Energy Signature"

The most exciting part of the paper is that these two methods produce Dark Photons with different energy signatures.

The Braking Method: Produces Dark Photons that keep most of the original speed. They are like a runner who drops a coin but keeps running at full speed. These high-energy particles are easier to detect in experiments like NA64h because they travel far and hit detectors with significant force.
The Annihilation Method: Produces Dark Photons that are "slower" (lower energy relative to the beam).

The Takeaway:
If an experiment sees a Dark Photon, scientists can look at how much energy it has to figure out how it was made.

High energy? It was likely made by the "Braking" method.
Lower energy? It was likely made by the "Annihilation" method.

Conclusion: A New Roadmap for Hunters

The authors conclude that previous studies might have underestimated the number of Dark Photons produced by the "Braking" method because they used the wrong map (ChPT) for high-speed collisions.

By using their new "Splitting" map, they show that:

NA64h (using a 50 GeV pion beam) is a very promising place to find Dark Photons in the 0.4–1.3 GeV range.
Other major experiments like T2K, DUNE, and SHiP (which use proton beams that create secondary pions) also have the potential to find these particles, but the "energy signature" will help them distinguish the signal from the background noise.

In short, the paper provides a better, more accurate guide for scientists hunting for the "ghost" particles that could explain the dark side of our universe.

1. Problem Statement

The paper addresses the production of dark photons ( $\gamma'$ or $A'_\mu$ ) in negatively charged pion-proton collisions ( $\pi^- p \to \gamma' X$ ). Dark photons are hypothetical vector portal mediators that couple to the Standard Model (SM) electromagnetic current via kinetic mixing ( $\epsilon$ ).

The authors focus on two primary production mechanisms in fixed-target experiments (specifically NA64h, SpinQuest, T2K, DUNE, and SHiP):

Inelastic Pion Bremsstrahlung: The emission of a dark photon from an incident pion.
Drell–Yan-like Process: The annihilation of a quark from the pion and an antiquark from the proton ( $q\bar{q} \to \gamma'$ ).

The Core Issue: A previous study (Ref. [7]) claimed that pion bremsstrahlung dominates dark photon production in experiments like SpinQuest. However, that study relied on Leading Order (LO) Chiral Perturbation Theory (ChPT) to calculate the cross-section. The authors of this paper argue that LO ChPT is invalid for the high-energy regimes (pion momenta $P \sim 50$ GeV) relevant to these experiments because the expansion parameter exceeds the theory's applicability limits ( $Q \gtrsim 4\pi F_\pi$ ).

2. Methodology

A. Critique of ChPT Applicability

The authors first demonstrate that LO ChPT fails for elastic $\pi^- p$ scattering at energies typical of the NA64h experiment ( $P=50$ GeV, $\sqrt{s} \approx 10$ GeV).

They show that the center-of-mass energy ( $E_{cm} \approx 5$ GeV) is far outside the low-energy regime where ChPT is valid.
They prove that imposing cuts on the momentum transfer ( $t$ ) or scattering angle to force "validity" does not resolve the discrepancy between LO ChPT predictions and experimental data (e.g., Fermilab Single Arm Spectrometer data).
Conclusion: LO ChPT cannot be used to estimate the cross-section for high-energy pion bremsstrahlung; a new approach is required.

B. New Approach: Splitting Function Formalism

To estimate the inelastic pion bremsstrahlung cross-section ( $\pi^- p \to \gamma' X$ ), the authors propose a factorization approach analogous to the Altarelli-Parisi splitting functions used in QCD.

Factorization: The process is factorized into a hard subprocess ( $\pi^- \to \gamma' \pi^-$ ) and the inclusive inelastic scattering cross-section ( $\pi^- p \to X$ ).
Splitting Function ( $w_\pi$ ): They derive a new splitting function for the charged pion, incorporating:
- The pion electromagnetic form factor ( $F_{em}^\pi$ ).
- A phenomenological off-shell hadronic form factor ( $F_{virt}$ ) to handle the virtuality of the intermediate pion.
- Dawson Correction: To preserve unitarity and handle the longitudinal polarization of the massive dark photon, they apply a correction to subtract the Goldstone mode contribution, ensuring the calculation remains consistent in the quasi-real approximation.
Kinematics: The calculation is performed in the laboratory frame, integrating over the dark photon energy ( $E_{\gamma'}$ ) and polar angle, subject to constraints ensuring the validity of the quasi-real approximation.

C. Drell–Yan-like Process in Perturbative QCD

For the competing mechanism, the authors calculate the cross-section for the Drell–Yan-like process ( $q\bar{q} \to \gamma'$ ) within perturbative QCD.

Order: Calculations are performed at both Leading Order (LO) and Next-to-Leading Order (NLO) in the strong coupling constant $\alpha_s$ .
Inputs: They utilize Parton Distribution Functions (PDFs) for the pion (JAM21PionPDFnlo) and the proton (CT14).
Narrow-Width Approximation (NWA): The cross-section is derived assuming the dark photon decays into SM particles ( $e^+e^-$ , $\mu^+\mu^-$ , etc.) with a total width $\Gamma_{tot}$ .
Transverse Momentum: They incorporate experimental fits for the average transverse momentum $\langle k_\perp \rangle$ of the lepton pair to model the energy distribution.

3. Key Contributions

Refutation of LO ChPT: The paper definitively establishes that LO ChPT is inapplicable for calculating pion bremsstrahlung cross-sections at energies $P \gtrsim 50$ GeV, invalidating previous claims of dominance based on that framework.
Derivation of New Splitting Function: The authors provide a rigorous derivation of the pion splitting function $w_\pi(z, k_\perp^2)$ for dark photon emission, including necessary off-shell form factors and unitarity corrections.
Comprehensive Cross-Section Comparison: They provide the first simultaneous, consistent calculation of both inelastic pion bremsstrahlung and the Drell–Yan-like process for the same kinematic conditions.
Energy Spectra Analysis: The paper calculates the differential energy spectra and mean energies of dark photons for various experimental setups (NA64h, T2K, DUNE, SHiP), highlighting distinct kinematic signatures for the two production mechanisms.

4. Results

Mass Dependence of Dominance:
- $m_{\gamma'} \in [0.4, 1.3]$ GeV: Inelastic pion bremsstrahlung is the dominant production mechanism. This is largely due to the enhancement from the pion electromagnetic form factor (specifically the $\rho(770)$ resonance contribution).
- $m_{\gamma'} \in [1.3, 3.5]$ GeV: The Drell–Yan-like process becomes the dominant mechanism.
NA64h Experiment ( $P=50$ GeV):
- The inelastic bremsstrahlung spectrum shows a significant contribution from high-energy dark photons ( $E_{\gamma'} \approx P$ ), making them highly detectable via missing energy signatures.
- The Drell–Yan spectrum peaks at lower energies ( $E_{\gamma'} < P/2$ ).
Secondary Pions (T2K, DUNE, SHiP):
- The authors estimate the mean energy of dark photons produced by secondary pions in proton beam-dump experiments.
- For T2K ( $P=30$ GeV), the quasi-real approximation breaks down for $m_{\gamma'} > 1.1$ GeV.
- For DUNE ( $P=120$ GeV) and SHiP ( $P=400$ GeV), the mean energy of produced dark photons increases with the beam momentum, but the relative contribution of bremsstrahlung vs. Drell–Yan follows the mass-dependent dominance established above.
Re-evaluation of SpinQuest/DarkQuest: The authors suggest that previous conclusions regarding the dominance of pion bremsstrahlung in the SpinQuest experiment need re-evaluation, as the Drell–Yan process (previously neglected or underestimated in that context) may play a more significant role than previously thought, especially for higher mass ranges.

5. Significance

Experimental Strategy: The paper provides crucial guidance for current and future dark photon searches. It suggests that experiments should look for different kinematic signatures depending on the target mass range:
- Low Mass ( $<1.3$ GeV): Focus on high-energy forward particles (bremsstrahlung signature).
- High Mass ( $>1.3$ GeV): Focus on lower-energy distributions characteristic of Drell–Yan production.
Theoretical Robustness: By replacing the invalid ChPT approach with a factorization formalism and perturbative QCD, the paper offers a more reliable theoretical framework for interpreting data from fixed-target experiments.
Discrimination Power: The distinct energy distributions ( $E_{\gamma'}$ ) of the two mechanisms allow experimentalists to potentially distinguish between the production channels, aiding in the identification of the underlying physics if a signal is observed.

In summary, this work corrects a significant theoretical error in the field of dark photon phenomenology and provides a robust, dual-mechanism framework for calculating production rates and energy spectra across a wide mass range, directly impacting the sensitivity projections for major experiments like NA64h, DUNE, and SHiP.