Searching for missing direct photons in heavy-ion collisions with P and CP violation

Technical Summary: Searching for missing direct photons in heavy-ion collisions with P and CP violation

Problem Statement
The production of direct photons in relativistic heavy-ion collisions is traditionally modeled using a combination of perturbative QCD for short-distance processes and hydrodynamics for the collective behavior of the quark-gluon plasma (QGP). While this framework has been largely successful, it faces a persistent challenge known as the "missing direct photons puzzle." Specifically, conventional models fail to simultaneously describe the soft part of the photon spectrum and its azimuthal anisotropy (elliptic flow, $v_2$). Standard approaches either underpredict the photon yield or predict an azimuthal variation that is too mild compared to experimental data.

Previous attempts to resolve this have focused on synchrotron radiation emitted by the QGP due to strong magnetic fields. While this mechanism provides additional photons and enhances $v_2$ due to the anisotropic emission perpendicular to the magnetic field, the resulting elliptic flow often exceeds experimental observations. Furthermore, the inclusion of plasma rotation tends to amplify this magnetic field effect, exacerbating the discrepancy. This paper posits that the elliptic flow $v_2$ is strongly dependent on the chiral properties of the QGP, specifically the transport coefficients associated with chiral magnetic and anomalous Hall currents, which violate P and CP symmetries.

Methodology
The authors develop a theoretical framework to compute photon radiation in the presence of anomalous currents by solving the modified Dirac equation for fermions in a constant magnetic field $\mathbf{B}$ and the presence of P- and CP-violating parameters. These parameters are introduced via an effective field theory term coupling the chiral vector current $j^\mu_5$ to a background pseudoscalar field $\theta$, defined by the vector $b_\mu = \partial_\mu \theta$. The vector $b_\mu$ has two non-vanishing components in this study: $b_0$ (the chiral chemical potential, $\mu_5$) and $b_3$ (the chiral gradient along the magnetic field).

The core methodological contribution is the derivation of exact wavefunctions (spinors) for fermions satisfying the equation:
$$(i \not{D} - \gamma_5 \not{b} - m)\psi = 0$$
where $D$ is the covariant derivative. The authors:

1. Decouple the Dirac equation: They reduce the first-order Dirac equation to a second-order form to solve for the spinor components in a constant magnetic field using the symmetric Landau gauge.
2. Derive Exact Spinors: They construct exact solutions for both the $n=0$ (lowest Landau level) and $n>0$ cases, expressing the spinors in terms of harmonic oscillator eigenfunctions. These solutions account for the mixing of chiralities induced by $b_\mu$.
3. Compute Radiation Rates: Using these exact wavefunctions, they calculate the S-matrix element for synchrotron radiation. They derive the differential emission rate for a single fermion and, subsequently, for a non-interacting plasma by applying Fermi-Dirac statistical weights.
4. Numerical Simulation: The authors perform numerical calculations for the radiation intensity of a single particle and the integrated photon yield ($v_0$) and elliptic flow ($v_2$) for a plasma of up and down quarks. They utilize parameters relevant to the QGP, including a thermal quark mass of $m \approx 300$ MeV and magnetic fields typical of heavy-ion collisions ($|e|B \approx 18000$ MeV$^2$).

Key Contributions

• Exact Wavefunctions: The paper provides the first exact solutions to the Dirac equation in the presence of a constant magnetic field, a chiral chemical potential ($b_0$), and a longitudinal chiral gradient ($b_3$). This extends previous work that was often restricted to the chiral limit or ignored the longitudinal gradient.
• Chiral Gradient ($b_3$) Inclusion: Unlike similar recent calculations (e.g., Shovkovy and Wang) that focused primarily on $b_0$, this work explicitly incorporates $b_3$, associated with the longitudinal component of the anomalous Hall current, and demonstrates its distinct impact on the dispersion relations and spinor structure.
• Derivation of Chiral Magnetic Effect: The derived wavefunctions are used to independently derive the chiral magnetic current, verifying that the computed current is independent of $b_3$, as theoretically expected.
• Resolution of the $v_2$ Discrepancy: The study demonstrates that while synchrotron radiation alone predicts an elliptic flow that is too large, the inclusion of P- and CP-violating parameters ($b_0$ and $b_3$) significantly reduces $v_2$.

Results
The numerical results for a QGP plasma with temperature $T=300$ MeV and magnetic field $|e|B = m_\pi^2$ yield the following:

• Photon Yield ($v_0$): The presence of non-zero $b_0$ or $b_3$ increases the total number of emitted photons by a factor of approximately 4 compared to the standard case. This enhancement saturates when the parameters reach an order of magnitude of $10^{-2}m$ (approx. 3 MeV).
• Elliptic Flow ($v_2$): The most significant finding is the reduction of the elliptic flow coefficient. In the absence of chiral parameters, synchrotron radiation yields a $v_2$ that exceeds experimental data. However, turning on either $b_0$ or $b_3$ significantly suppresses $v_2$.
• Comparison with Data: The reduced $v_2$ values, when combined with the enhanced photon yield, bring the theoretical predictions into a range comparable to experimental data from the PHENIX collaboration (20-40% centrality). The authors note that the result for $v_2$ is robust as it does not depend on the arbitrary length and time scales used to normalize the total yield.
• Helicity Splitting: The study observes a splitting in radiation intensity based on the $s$- and $u$-helicity of the initial particles, a feature enhanced by the presence of $b_0$ and $b_3$.

Significance and Claims
The paper claims that the interplay between synchrotron radiation and chiral transport coefficients offers a viable solution to the missing direct photons puzzle. Specifically, the authors argue that the chiral magnetic current, flowing parallel to the magnetic field, causes relativistic quarks to emit photons primarily in the forward direction, thereby counteracting the perpendicular emission characteristic of standard synchrotron radiation. This mechanism naturally reduces the elliptic flow $v_2$ to levels consistent with observations while simultaneously boosting the photon yield.

The authors modestly frame their work as a consistent theoretical framework that bridges the gap between hydrodynamic models and experimental data. They suggest that a rotating plasma with chiral imbalance (a realistic scenario in heavy-ion collisions) could fully resolve the puzzle, noting that adding rotation to their Hamiltonian is a straightforward extension. They acknowledge that their inclusion of a finite fermion mass (thermal mass) is a novel aspect compared to massless fermion calculations, as it prevents the decoupling of the Dirac equation and allows for chirality-changing kinematics. The paper concludes by identifying the extension to transverse chiral parameters ($b_1, b_2$) as a future goal requiring a dedicated study.