Spectral Function Approach in Photon Emission

• Spectral Function Approach is a rigorous framework that expresses thermal photon emission rates in hot hadronic matter through medium-induced correlators and vacuum spectral functions.
• It uses chiral reduction formulas and pion density expansions to incorporate key processes such as pion bremsstrahlung and resonance decays, ensuring parameter-free calculations.
• Integrating with hydrodynamic models, this approach enables precise comparisons with SPS, RHIC, and LHC data, offering robust insights into QCD matter dynamics.

The spectral function approach constitutes a precise and systematically improvable framework to describe photon emission in hot and dense hadronic matter produced in heavy ion collisions, as formulated in (0911.2426). By relating medium-induced photon rates to experimentally accessible vacuum correlation functions and using chiral reduction techniques and density expansions in pion number, the method circumvents model dependencies inherent in kinetic calculations, provides comprehensive coverage of relevant processes, and establishes robust connections with phenomenological data.

1. Spectral Function Methodology

The core premise is to relate the differential photon production rate in a thermalized hadronic gas to the finite-temperature electromagnetic current–current correlator: $$q^0\, \frac{dN}{d^3q} = -\frac{\alpha_{\text{em}}}{4\pi^2}\, W(q),$$ where

$$W(q) = \int d^4x\, e^{-iq\cdot x} \, \text{Tr}\left(e^{-(H-F)/T} J^\mu(x) J_\mu(0)\right).$$

For real photons ($q^2 = 0$), one employs the spectral representation,

$$W(q) = \frac{2}{1 + e^{q^0/T}}\, \text{Im}\, W^F(q),$$

with $W^F(q)$ the Fourier transform of the time-ordered thermal correlator. The imaginary part of this correlator—i.e., the spectral function—encodes the absorptive processes responsible for photon emission. The remarkable aspect of this approach is that, using well-established relations, $W(q)$ may be fully recast in terms of vacuum spectral functions, making direct contact with quantities measured in $e^+e^-$ annihilation and $\tau$ decays. This mapping eliminates the need to catalog each kinetic channel individually.

2. Chiral Reduction Formulas

A pivotal technical step is the application of chiral reduction formulas, which enable the systematic re-expression of thermal correlation functions, order by order in the pion density, as functionals of vacuum (i.e., T=0) spectral functions. For instance, the one-pion contribution to the time-ordered correlator is written as: $$W^F_\pi(q,k) = \frac{12}{q_\pi^2} [q^2 \operatorname{Im} \Pi_V(q^2)] - \frac{6}{f_\pi^2} [(k+q)^2 \operatorname{Im} \Pi_A((k+q)^2)] + (q \to -q) + \text{mixing terms}.$$ Here, $\Pi_V$ and $\Pi_A$ are the vector and axial-vector current correlators, respectively, and $f_\pi$ is the pion decay constant. The resulting formulation allows all processes with specified pion number in the density expansion to be generated automatically—no specification of each reaction type (e.g., $X \to \pi\gamma$) is necessary beyond this formalism.

3. Density Expansion and Inclusion of Kinetic Processes

The spectral function approach organizes the full rate as a virial-like expansion: $$W^F(q) = W_0 + \int d\pi_1\, W_\pi + \frac{1}{2}\int d\pi_1 d\pi_2\, W_{\pi\pi} + \cdots,$$ with $d\pi$ the appropriately normalized pion phase-space measure. The $W_0$ (vacuum) term vanishes for on-shell photons due to electromagnetic gauge invariance. The $W_\pi$ (one-pion) term addresses channels like $X\to\pi\gamma$, while the $W_{\pi\pi}$ (two-pion) term encodes all processes with two final-state pions, notably including pion bremsstrahlung ($\pi\pi \to \pi\pi\gamma$), $\rho$ radiative decays, and related mechanisms. Carrying the expansion to second order in pion density is essential: two-pion terms dominate the low-energy photon spectrum and must be included for quantitative accuracy.

In the soft-pion limit, the two-pion contribution further reduces to the difference of axial and vector spectral functions, precisely matching sum rule expectations.

4. Relation to Vacuum Correlation Functions

Crucially, the mapping to vacuum spectral functions, $\operatorname{Im} \Pi_V$ and $\operatorname{Im} \Pi_A$, ensures the rates are anchored to experimental data, minimizing uncertainties from modeling. For one-pion contributions, the dominant term is

$$\frac{12}{f_\pi^2} q^2 \operatorname{Im} \Pi_V(q^2),$$

and for two-pion processes, spectral structures corresponding to vector meson exchange and resonance effects emerge. This strategy provides a seamless, parameter-free input for the calculation. Moreover, the spectral function approach allows for a unified treatment of both real and virtual photon (dilepton) emission, an essential feature given the experimental overlap between photons and low-mass dileptons.

5. Quantitative Comparison with Measurements

The rates derived via the spectral function approach, combined with hydrodynamic modeling, show strong agreement with direct photon and dilepton data.

• SPS: Inclusion of two-pion processes enhances the soft photon yield at low $q_t$, explaining the direct photon upper limits detected by WA98. Pion bremsstrahlung, a second order process in the pion density, is necessary to reproduce the observed spectrum for $q_t \lesssim 0.8$ GeV.
• RHIC: Using two distinct hydrodynamic evolution schemes (RHIC1, RHIC2), the computed spectra, combined with prompt photons, reproduce PHENIX observations. The sensitivity to the hydrodynamic initialization is explored; in the "RHIC2" scenario, a higher QGP yield is predicted due to earlier hydrodynamic expansion. The methodology is also checked against low-mass dilepton measurements, with process-dependent systematics ($\sim$15%) accounted for by a correction factor ($\mathcal{S}$).
• LHC: Predictions indicate a crossing of the hadronic and QGP photon yields at $q_t \sim 1$ GeV, and the domination of prompt production above 2.5–3 GeV $q_t$.

6. Hydrodynamic Integration and Space-Time Evolution

Realistic comparison with experiments necessitates folding the static (equilibrium) rate calculations over the evolving space–time profile of the heavy ion collision. This is accomplished using a (2+1)D hydrodynamic simulation tuned to reproduce hadronic multiplicities, $p_t$ spectra, and elliptic flow, initialized with entropy and baryon density profiles from a Glauber model and evolved over a cooling timescale. The integrated photon and dilepton yields are hence directly comparable to measured spectra, allowing theory-experiment comparisons over the full range from SPS to LHC energies.

7. Summary and Significance

The spectral function approach for thermal photon emission in heavy ion collisions provides a rigorous, data-driven, and systematically improvable framework:

• All relevant processes (up to two final-state pions), including bremsstrahlung and resonance contributions, are incorporated via chiral reduction and density expansion.
• Photon emission rates are written entirely in terms of vacuum spectral functions, offering parameter-free and experimentally-constrained calculations.
• Integration with hydrodynamic models enables detailed phenomenological comparison and predictions.
• The observed agreement with direct photon and dilepton measurements at SPS and RHIC, and predictions for LHC, validate the framework.

The principal advances are the reduction in model dependency, the ability to systematically include relevant hadronic processes, and the robust link to experimentally measured spectral functions. This approach underpins contemporary theoretical analyses of electromagnetic emission in hot QCD matter and provides a benchmark for future developments and experimental strategies.