The role of the chiral anomaly in polarized deeply inelastic scattering I: Finding the triangle graph inside the box diagram in Bjorken and Regge asymptotics

We revisit the role of the chiral "triangle" anomaly in deeply inelastic scattering (DIS) of electrons off polarized protons employing a powerful worldline formalism. We demonstrate how the triangle anomaly appears at high energies in the DIS box diagram for the polarized proton structure function $g_1(x_B,Q^2)$ in both the Bjorken limit of large $Q^2$ and in the Regge limit of small $x_B$. We show that the operator product expansion is not required to extract the anomaly in either asymptotics though it is sufficient in the Bjorken limit. Likewise, the infrared pole in the anomaly arises in both limits. The leading contribution to $g_1$, in both Bjorken and Regge asymptotics, is therefore given by the expectation value of the topological charge density, generalizing a result previously argued by Jaffe and Manohar to hold for the first moment of $g_1$. In follow-up work, we will show how our results motivate the derivation of a helicity-dependent effective action incorporating the physics of the anomaly at small $x_B$ and shall discuss the QCD evolution of $g_1(x_B,Q^2)$ in this framework.