Hadronic Light By Light ScatteringEdit
Hadronic light-by-light scattering (HLbL) is a subtle, nonperturbative piece of the Standard Model puzzle that contributes to the muon's anomalous magnetic moment. In quantum field theory language, it is a class of processes in which a muon interacts with three photons that couple through a hadronic intermediate state. Because the strong interaction binds quarks into hadrons and operates in a regime where perturbation theory is unreliable, HLbL is one of the most challenging contributions to compute with controlled uncertainties. It sits alongside other hadronic effects, such as hadronic vacuum polarization, as a key source of theoretical error in the prediction for the muon anomalous magnetic moment, often denoted a_mu.
HLbL is central to precision tests of the Standard Model because the muon is heavier than the electron and therefore more sensitive to higher-mass scales and new physics. The experimental measurement of a_mu has achieved remarkable precision, and the comparison to theory hinges on reliably estimating HLbL as part of the Standard Model prediction. The status of HLbL calculations has evolved substantially in the 21st century, moving from model-centric estimates to first-principles lattice calculations and dispersive approaches that aim to reduce model dependence while maintaining rigorous control of uncertainties. Detailed discussions of these efforts can be found in the rundowns of hadronic contributions to the muon g-2 and in surveys of nonperturbative QCD techniques.
Overview
Hadronic light-by-light scattering is a four-point process in which the muon interacts with a hadronic blob, mediated by virtual photons. The complexity arises because the hadronic blob encompasses all possible strongly interacting intermediate states, from light pseudoscalar mesons to multi-hadron continua. The calculation thus must address a wide range of energies and resonant structures. The HLbL contribution is typically separated into distinct pieces that are dominated by recognizable hadronic phenomena, each with its own theoretical treatment and set of uncertainties. The largest piece often comes from the exchange of light pseudoscalar mesons, but other contributions—such as charged pion loops and short-distance quark-loop-like structures—play important roles too. See for example discussions of the pi0, eta, and eta' pole contributions, as well as nonpole and loop effects in various hadronic channels.
Linking to the broader framework, HLbL sits within the broader category of hadronic effects in the Standard Model, alongside hadronic vacuum polarization and other nonperturbative dynamics described by Quantum Chromodynamics Quantum Chromodynamics. The muon’s anomalous magnetic moment is an observable that accumulates discrepancies between experiment and theory, and HLbL is one of the components where the theory community has focused intense efforts to sharpen the prediction. For context, the muon and its magnetic moment are discussed in resources on muon and anomalous magnetic moment.
Theoretical foundations
The HLbL amplitude is constrained by general principles such as gauge invariance, Lorentz symmetry, and the analytic structure of quantum field theory. Because it probes a nonperturbative regime of QCD, most practical calculations rely on a combination of theoretical frameworks and phenomenology:
Model-based, large-Nc QCD approaches and resonance saturation. In these methods, the hadronic blob is approximated by dominant resonances and guided by the organizing principle that, in the limit of a large number of colors (Nc), QCD reduces to a theory of mesons and their interactions. Vector meson dominance (VMD) and related ideas are often invoked to model the couplings of photons to hadrons. See discussions of Vector meson dominance and Large-Nc QCD as organizing ideas.
Chiral effective theories and pseudoscalar poles. At very low energies, chiral perturbation theory provides a controlled expansion in momenta and light quark masses. The exchange of light pseudoscalar mesons, especially the pi0 via its transition form factor to two photons, dominates the HLbL landscape in many models.
Dispersive and data-driven approaches. In recent years, dispersive methods have been developed to connect HLbL to experimental data on processes like gamma gamma* -> hadrons, using dispersion relations to constrain the amplitude from measured cross sections and decay rates. This program aims to reduce model dependence and improve the reliability of uncertainties.
Lattice QCD calculations. First-principles lattice simulations compute HLbL amplitudes from the underlying theory of QCD+QED on a discretized spacetime grid. Lattice approaches are progressing toward controlled continuum and infinite-volume limits and are increasingly able to provide independent cross-checks of model-based estimates.
Hybrid strategies. Many contemporary analyses combine elements from several frameworks, using data inputs where possible and calibrating model pieces with lattice or dispersive results to constrain unknowns.
The overarching goal across these approaches is to isolate the parts of HLbL that can be anchored to experimental information or fundamental theory, while keeping a transparent accounting of uncertainties and correlations among contributions.
Dominant contributions to HLbL
Pseudoscalar meson poles. The lightest pseudoscalar mesons—pi0, eta, and eta'—couple to photons through the axial-vector anomaly, producing pole-like contributions when two photons couple to these mesons and then connect to the muon line. Among these, the pi0 is the most important due to its strong coupling and relatively light mass. The accuracy of these pieces rests on the pion transition form factor, describing how the pi0 couples to photons with different virtualities.
Charged pion loops. Loops formed by charged pions interacting with photons provide a substantial nonpole contribution. These loops probe intermediate-energy dynamics and are sensitive to how pions couple to photons and to each other.
Quark-loop and higher-resonance contributions. Short-distance components resemble a quark loop but are sensitive to nonperturbative QCD effects. Heavier resonances beyond the light pseudoscalars also contribute, though their impact is typically smaller and increasingly model-dependent.
Multi-hadron and more complex intermediate states. Beyond the simplest pieces, the HLbL amplitude includes a spectrum of multi-hadron states, rescattering effects, and continuum contributions that require careful treatment to avoid double counting or omissions.
Each piece has its own history of calculation, with model assumptions, lattice determinations, and dispersive constraints contributing to a composite estimate.
Methods and calculations
Lattice QCD. Lattice studies aim to compute HLbL directly from the QCD Lagrangian with controlled systematic errors. Progress includes simulations that address finite-volume effects, discretization errors, and the treatment of the QED sector. The lattice program provides a valuable, increasingly competitive cross-check of model-based estimates and dispersive analyses.
Dispersive approaches. Dispersive methods express HLbL in terms of measurable amplitudes and cross sections, using dispersion relations to relate the hadronic light-by-light tensor to physical processes such as gamma gamma* -> hadrons. This pathway emphasizes data constraints and can help reduce model dependence.
Phenomenological and resonance-based models. Model builders employ large-Nc-inspired frameworks, chiral Lagrangians, and resonance saturation to capture the dominant physics, particularly for the pi0, eta, and eta' pole contributions and the low-energy interactions of photons with hadrons. These approaches are valuable for intuition and for cross-checking more rigorous methods.
Cross-checks and error budgets. A central challenge is the careful accounting of uncertainties, including correlations among different contributions, the treatment of transition form factors, and the matching between low-energy effective theories and short-distance QCD.
Relevance to physics
Muon g-2 and the Standard Model test. The muon’s anomalous magnetic moment, a_mu, encapsulates quantum corrections from all sectors of the Standard Model. HLbL is a key part of the hadronic sector that currently limits the precision of the Standard Model prediction. Ongoing improvements in HLbL calculations directly affect the comparison between theory and experiment, such as results from the Muon g-2 experiments and their successors. See muon g-2 and anomalous magnetic moment for context.
Interplay with experimental data. Dispersive and data-driven strategies emphasize experimental inputs, while lattice results strive for independent, first-principles validation. The convergence of different methods strengthens confidence in the HLbL estimate, while persistent discrepancies would point to either methodological gaps or, more speculatively, new physics scenarios.
Broader implications for nonperturbative QCD. HLbL studies illuminate how hadrons couple to photons in regimes where quark and gluon degrees of freedom are confining and strongly interacting. The same physics touches on form factors, resonance structures, and the transition between low-energy effective theories and fundamental QCD.
Controversies and debates
Model dependence versus first principles. A long-standing tension exists between results obtained from phenomenological models (with resonance saturation and VMD inputs) and those obtained from lattice QCD and dispersive analyses. Advocates of data-driven and lattice methods argue that model reliance can underestimate systematic uncertainties, while model builders stress the interpretability and tractability of their frameworks. The field continues to refine how best to quantify and propagate uncertainties.
Transition form factors and short-distance behavior. The pi0 transition form factor, which governs the pi0 pole contribution, is central to HLbL. Different parametrizations and experimental inputs can lead to noticeable variations in the estimated HLbL piece. The debate centers on how to incorporate high-virtuality behavior consistently with QCD constraints while remaining faithful to data.
Finite-volume and discretization issues in lattice QCD. Lattice computations must contend with finite-volume effects, discretization errors, and the challenges of simulating QED in a finite box. Teams pursue increasingly sophisticated techniques to control these systematics, and results from different lattice groups are now beginning to show coherent trends even as they are not yet in exact agreement.
Data quality and dispersion relations. Dispersive approaches depend on the quality and completeness of experimental data for related processes. Critics emphasize the need for comprehensive measurements across energy ranges, while proponents highlight the elegance of a framework that ties HLbL to observable quantities rather than model assumptions.
Implications for new physics interpretations. Because HLbL contributes significantly to the theoretical error bar in a_mu, the degree of control over HLbL shapes how skeptics and proponents interpret any residual discrepancy between experiment and the Standard Model. As methods improve, the field expects a sharper, more robust assessment of whether the muon g-2 anomaly hints at new physics or rests on uncertain hadronic inputs.