Role of the chiral anomaly in polarized deeply inelastic scattering. III. Wess-Zumino-Witten contributions and chiral Ward identities for finite quark mass

We extend our prior results on the worldline computation of the axial vector-vector-vector (AVV) triangle anomaly in polarized deeply inelastic scattering (DIS) to the finite mass case by computing in addition the pseudoscalar-vector-vector (PVV) triangle graph. For the well-studied QED case, we show explicitly how the off-forward AVV pole exactly cancels an identical PVV pole. We then demonstrate the dramatic difference in QCD due to the chiral condensate, which qualitatively modifies anomalous Ward identities. As in the massless case, the anomaly pole in QCD is canceled by the dynamics of a primordial isosinglet pseudoscalar <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>η</a:mi><a:mo stretchy="false">¯</a:mo></a:mover></a:math>-meson, whose Wess-Zumino-Witten coupling to the topological charge density shifts the pole to the physical <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>η</e:mi><e:mo>′</e:mo></e:msup></e:math> mass, with the finite quark mass contribution differing by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>O</g:mi><g:mo stretchy="false">(</g:mo><g:mn>10</g:mn><g:mo>%</g:mo><g:mo stretchy="false">)</g:mo></g:math> from the Witten-Veneziano formula. We obtain a compact analytic expression for the finite mass corrections to Shore and Veneziano’s result that the proton’s net quark helicity <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="normal">Δ</k:mi><k:mi mathvariant="normal">Σ</k:mi><k:mo>∝</k:mo><k:msqrt><k:mrow><k:msubsup><k:mi>χ</k:mi><k:mi>QCD</k:mi><k:mo>′</k:mo></k:msubsup><k:msub><k:mo stretchy="false">|</k:mo><k:mrow><k:mi>m</k:mi><k:mo>=</k:mo><k:mn>0</k:mn></k:mrow></k:msub><k:mo stretchy="false">(</k:mo><k:mn>0</k:mn><k:mo stretchy="false">)</k:mo></k:mrow></k:msqrt></k:math>, the forward slope of the topological susceptibility in the chiral limit, and show they are of the order of a few percentages. Our prior prediction that the polarized DIS structure function <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi>g</r:mi><r:mn>1</r:mn></r:msub></r:math> is quenched by sphaleronlike topological transitions at small <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>x</t:mi></t:math> is unaffected by quark mass effects. Our results illustrate how worldline computations of anomalous processes, in synergy with lattice computations and nonet chiral perturbation theory, can uncover novel nonperturbative features of QCD at the Electron-Ion Collider.