Divergences in the hadronic light-by-light amplitude of the holographic soft-wall model

TL;DR

This work reveals fundamental divergences in holographic soft-wall QCD models when assessing CS-term–driven observables, such as the VVA correlator and hadronic light-by-light scattering. Through a WKB analysis, it shows excited-state transition form factors grow exponentially, causing non-convergent mode sums for both sum rules and HLbL, while a full 5D calculation cures the VVA divergence but leaves HLbL divergent, underscoring a serious limitation of the standard SW setup for muon g-2. The authors demonstrate that background modifications alone are insufficient to fix these pathologies and show that scalar-extended CS terms or moving to V-QCD-type constructions can tame the divergences and satisfy short-distance constraints. Hard-wall models naturally avoid these issues and align better with dispersive/experimental data, suggesting a direction for more reliable holographic inputs to HLbL. The results place strong constraints on holographic QCD model-building and point toward viable paths (e.g., V-QCD) to achieve finite HLbL amplitudes and consistent MV-SDC behavior.

Abstract

We use the WKB approximation to uncover divergences and instances of non-commuting limits in a large class of holographic soft-wall models. We show that the infinite sum over single resonance contributions for a variety of observables involving the Chern-Simons term, such as the VVA correlator or the hadronic light-by-light tensor, does not converge. These divergences can in some cases (such as the VVA correlator) be traced back to non-commuting limits and avoided by working directly in the 5-dimensional setup with bulk-to-boundary propagators. However, the hadronic light-by-light scattering tensor also diverges in the 5-dimensional formulation with corresponding Green functions, preventing a correct implementation of the Melnikov-Vainshtein short-distance constraint and even leading to an infinite contribution to the muon $g-2$. We also discuss modifications of the standard soft-wall model that are able to resolve this issue.

Figures (8)

• Figure 1: The pseudoscalar potential in units of GeV$^2$ and the 4 regions into which the whole range of the $z$ coordinate (in units of GeV$^{-1}$) is divided for $E_n\sim 40$ GeV$^2$, which is indicated by the dashed line. The two locations where the dashed line intersects $V_P(z)$ are the points $z_l,z_r$ where $k^2=0$. The three lines separating the regions are located at $z_1,z_2,z_3$. In region II and IV the oscillating and exponentially decaying WKB solutions respectively are appropriate. In region I we use a Bessel function, while in region III, an Airy function is used.
• Figure 2: A comparison of the WKB mode function (blue) and the full numerical solution (dashed black). The vertical black lines indicate the boundaries $z_i$ of the four regions, and the dotted gray curve indicates the WKB solution \ref{['eq:WKBeqtext']}. Clearly the points $z_l,z_r$ marked by red lines where $\frac{1}{\sqrt{k}}$ diverges are outside region II.
• Figure 3: The ratio of \ref{['eq:TFFscaling']} with $C$ omitted and \ref{['eq:TFFlargen']} for $Q_i^2=0$ GeV$^2$ (black), $Q_i^2=1.5$ GeV$^2$ (red), and $Q_i^2=2.5$ GeV$^2$ (blue) for the first 70 modes. The convergence to the analytic expression \ref{['eq:TFFscaling']} is faster for smaller $Q_i^2$.
• Figure 4: Plot of the SW results for the $a_\mu$ contributions of excited pseudoscalars and excited axials, Table \ref{['tab:g-2modesnew-mq']} (blue) and \ref{['tab:g-2modesnew-mq-axial']} (orange)
• Figure 5: The Witten diagrams needed in the calculation of the HLbL tensor. Single lines denote a vector bulk-to-boundary propagator, while double lines denote a bulk-to-bulk propagator of the axial gauge field $A_{\mu}(x,z)$.