Abstract
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.
Highlights
It has long been realized that deeply inelastic scattering (DIS) off polarized protons probes the physics of the chiral anomaly in QCD [1] though its precise role has been the subject of some debate [2,3,4]
We will focus on the triangle graph [5,6,7,8] whereby the anomaly manifests itself in the coupling of the isosinglet axial vector current to the topological charge density in the polarized proton
As we shall discuss, the offforward matrix element for the polarized g1 structure function contains an infrared pole that appears to diverge in the forward limit [4]
Summary
It has long been realized that deeply inelastic scattering (DIS) off polarized protons probes the physics of the chiral anomaly in QCD [1] though its precise role has been the subject of some debate [2,3,4]. We will focus on the triangle graph [5,6,7,8] whereby the anomaly manifests itself in the coupling of the isosinglet axial vector current to the topological charge density in the polarized proton. While the matrix element of TrðFF Þð0Þ is naively zero in the forward limit, the matrix element as defined above is finite when one combines the contribution from the density matrix jP0ihPj and the triangle operator This is often done in careful perturbative QCD computations by introducing a mass term or like infrared regulator which cancels between the two to give the finite result [67,68,69]. The computation of the triangle graph in this formalism is discussed in Appendix C
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