Photon production from gluon splitting and fusion induced by a magnetic field in heavy-ion collisions

This paper investigates the direct photon puzzle in peripheral heavy-ion collisions by calculating the one-loop photon production from gluon splitting and fusion induced by strong magnetic fields during the pre-equilibrium stage, finding that splitting dominates at low energies and that longitudinal anisotropy has a negligible effect on the yield, which is then compared with PHENIX experimental data.

The Gist

The Big Mystery: The "Direct Photon Puzzle"

Imagine two heavy atomic nuclei (like tiny, dense marbles made of protons and neutrons) smashing into each other at nearly the speed of light. This happens in giant particle accelerators. When they crash, they create a super-hot, super-dense soup of particles called a Quark-Gluon Plasma (QGP).

Scientists have been studying this soup for years. They expected to see a certain amount of light (photons) coming out of it. However, they found a mystery:

1. Too much light: There is way more light coming out than their best computer models predicted.
2. Too much "spin": The light isn't just coming out randomly; it's flowing in a specific, oval shape (like a rugby ball) that is surprisingly strong.

This is called the Direct Photon Puzzle. It's like baking a cake and finding it's twice as big as the recipe said it should be, and it's also spinning in a way that shouldn't be possible.

The New Ingredient: The Magnetic Field

The authors of this paper suggest a new ingredient might be the missing piece of the puzzle: Magnetic Fields.

When these heavy nuclei smash together, they don't just create heat; they also create incredibly powerful, short-lived magnetic fields. Think of it like a lightning bolt that exists for a split second but is trillions of times stronger than the strongest magnet on Earth.

The team asked: Could these intense magnetic fields be helping to create that extra light?

The Mechanism: Gluons Splitting and Fusing

Inside the hot soup, the main players are gluons. If protons are like bricks, gluons are the "glue" holding them together. Usually, light is made when quarks (the bricks) interact. But in the very early moments of the crash, there are way more gluons than quarks.

The paper focuses on two specific ways these gluons can turn into light, but only because the magnetic field is there to help:

1. Gluon Fusion (The Merge): Two gluons crash into each other and merge to become a single photon (light).
2. Gluon Splitting (The Breakup): A single gluon breaks apart, and one of the pieces becomes a photon.

The Analogy: Imagine a crowded dance floor (the soup). Usually, people (quarks) pair up to dance. But in this specific scenario, the music (the magnetic field) is so loud and strange that the "glue" guys (gluons) start dancing in a way they never did before. They either merge into a spotlight or break apart to create one.

What the Scientists Did

The team didn't just guess; they did some heavy math to prove it works.

• The Blueprint: First, they had to draw the "blueprint" for how two gluons turn into a photon in the presence of a magnetic field. This is a complex geometric shape in math-land. Previous studies tried to draw this blueprint but had to use shortcuts (approximations) that only worked for weak or very specific magnetic fields.
• The Full Picture: These authors drew the complete blueprint without shortcuts. They calculated exactly how this interaction works for magnetic fields of any strength.
• The Result: They found that Gluon Splitting is the star of the show. At lower energy levels (slower light), the "breakup" process creates way more light than the "merge" process.

Testing Against Reality

They took their new math and compared it to real data from the PHENIX experiment (a real detector at a particle accelerator).

• The Match: When they added the light produced by these gluon processes to their models, the "extra light" problem started to disappear. Their calculations matched the real-world data much better than before.
• The Shape: They also checked if this light would flow in that weird oval shape (elliptic flow). They found that the magnetic field naturally creates this flow without needing to force it.

A Twist: Does the Shape of the Soup Matter?

The scientists wondered: What if the soup isn't a perfect sphere? What if it's stretched out like a cigar because the particles are zooming away fast?

They tried to add this "stretchiness" (anisotropy) into their math. Surprisingly, it didn't change the amount of light much. Whether the soup was a perfect ball or a stretched cigar, the magnetic field was doing the heavy lifting to create the light.

The Takeaway

This paper solves a piece of the "Direct Photon Puzzle" by showing that:

1. The massive magnetic fields created in heavy-ion collisions are real and powerful.
2. These fields allow gluons (which are usually quiet) to turn into light through splitting and fusing.
3. This process explains the "extra light" and its strange flow that scientists have been seeing for years.

In short: The magnetic field is the secret conductor of the orchestra, getting the "glue" particles to play a solo that creates the extra light we've been trying to explain.

Technical Summary

1. Problem Statement: The Direct Photon Puzzle

In relativistic heavy-ion collisions (HIC), experiments (specifically PHENIX) have observed a "direct photon puzzle" characterized by two anomalies:

• Excess Yield: The observed production of direct photons exceeds the predictions of standard thermal hydrodynamical models, particularly at low transverse momentum ($p_T$).
• Large Elliptic Flow ($v_2$): The measured elliptic flow of these photons is surprisingly large, comparable to that of hadrons. Since photons are colorless and do not interact strongly after production, they are expected to be emitted early in the collision when expansion velocities are low, which should result in a small $v_2$.

The authors propose that strong magnetic fields ($B \sim 10^{19}$ G) generated in peripheral collisions during the pre-equilibrium stage may drive additional photon production mechanisms that could explain this excess and the associated anisotropy.

2. Methodology

The study focuses on photon production via gluon fusion ($gg \to \gamma$) and gluon splitting ($g \to g\gamma$) in the presence of a constant magnetic field. The methodology proceeds in three main stages:

A. Theoretical Framework: The Two-Gluon One-Photon Vertex

• Tensor Structure Analysis: The authors analyze the general tensor structure of the vertex $\Gamma^{\mu\nu\alpha}_{ab}(p_1, p_2, q)$ connecting two gluons and one photon in a magnetic field.
  • They impose constraints from gauge invariance, gluon indistinguishability (symmetry under exchange), and CP invariance.
  • Using the Ritus basis (a set of vectors adapted to the magnetic field direction), they span the 4D Minkowski space.
  • For on-shell gauge bosons, symmetry arguments reduce the possible tensor structures from 27 down to three independent structures.
• One-Loop Calculation:
  • The vertex is computed at the one-loop level involving quark triangles.
  • Unlike previous studies that relied on strong-field approximations (lowest Landau level) or incomplete tensor structures, this work computes the vertex for magnetic fields of arbitrary strength without additional approximations.
  • They utilize Schwinger's proper time representation for the fermion propagator in a magnetic field to handle the breaking of Lorentz symmetry.
  • The calculation includes the full summation of Feynman diagrams (Fig. 2) and Schwinger phases.

B. Photon Yield Calculation

• The invariant momentum distribution is calculated using the derived vertex matrix elements ($\mathcal{M}$).
• The process accounts for both fusion and splitting channels, related by crossing symmetry.
• Initial State Distribution: The authors model the initial gluon distribution in the pre-equilibrium "shattered glasma" phase.
  • Isotropic Case: A Bose-Einstein distribution with a saturation scale $\Lambda_s$ and occupation factor $\eta$.
  • Anisotropic Case: To account for longitudinal expansion and magnetic pressure anisotropy, they introduce an anisotropy parameter $\xi$ in the momentum distribution (both for a simple exponential form and a Color Glass Condensate-inspired form).

C. Comparison with Data

• The theoretical yields are compared against the difference between PHENIX experimental data and state-of-the-art hydrodynamical calculations (which represent the "missing" yield).
• Parameters such as magnetic field strength ($B$), saturation scale ($\Lambda_s$), and occupation factor ($\eta$) are varied to fit the data.

3. Key Contributions

1. Rigorous Vertex Derivation: The paper provides the first complete derivation of the two-gluon one-photon vertex for arbitrary magnetic field strengths, correcting previous works that either used strong-field approximations or lacked the full tensor symmetry properties.
2. Dominance of Splitting: The authors demonstrate that for low photon energies, gluon splitting ($g \to g\gamma$) dominates over gluon fusion ($gg \to \gamma$). This is a crucial finding for modeling the low-$p_T$ spectrum.
3. Anisotropy Analysis: They systematically investigate the impact of longitudinal momentum anisotropy in the initial gluon distribution.
4. Quantitative Fit: The model successfully reproduces the magnitude of the photon excess observed by PHENIX in the 20-30% centrality class without requiring extreme parameters.

4. Key Results

• Low-Energy Dominance: As shown in Figure 3, the gluon splitting contribution (dotted green line) is the primary driver of the photon yield at low energies ($\omega_q < 1.5$ GeV), while fusion becomes less significant. The total yield (solid red line) aligns well with the PHENIX data excess.
• Role of Saturation Scale: Figure 4 illustrates that the gluon saturation scale ($\Lambda_s$) is the critical parameter controlling the curvature of the photon yield spectrum. Adjusting $\Lambda_s$ allows the model to match the spectral shape of the experimental data.
• Magnetic Field Strength: The results hold for magnetic fields of $B = 3m_\pi^2$ and $B = 10m_\pi^2$, suggesting the mechanism is robust across a range of field intensities typical for RHIC energies.
• Anisotropy Impact: Figure 5 reveals that introducing longitudinal anisotropy ($\xi \neq 0$) into the gluon distribution does not significantly alter the total photon yield compared to the isotropic case. The yield remains consistent with the PHENIX data regardless of the anisotropy parameter used.

5. Significance and Conclusion

• Resolution of the Puzzle: The study suggests that magnetic-field-induced gluon splitting and fusion during the pre-equilibrium stage provide a viable mechanism to explain the direct photon excess and potentially contribute to the large elliptic flow ($v_2$).
• Anisotropy and $v_2$: While the yield is insensitive to longitudinal anisotropy, the authors note that the elliptic flow ($v_2$) is likely sensitive to it. The anisotropic pressure and expansion in the pre-equilibrium phase could generate the necessary azimuthal anisotropy in the photon emission to match the observed large $v_2$.
• Future Work: The authors identify the calculation of the elliptic flow coefficient using these anisotropic distributions as a necessary next step to fully resolve the direct photon puzzle.

In summary, this work establishes a rigorous theoretical foundation for magnetic-field-driven photon production in heavy-ion collisions, identifying gluon splitting as the dominant low-energy mechanism and demonstrating that the model can quantitatively reproduce experimental anomalies without relying on unphysical approximations.