Abstract
In [1], we demonstrated that the structure function $g_1(x_B,Q^2)$ measured in polarized deeply inelastic scattering (DIS) is dominated by the triangle anomaly in both the Bjorken limit of large $Q^2$ and the Regge limit of small $x_B$. In the worldline formulation of quantum field theory, the triangle anomaly arises from the imaginary part of the worldline effective action. We show explicitly how a Wess-Zumino-Witten term coupling the topological charge density to a primordial isosinglet ${\bar \eta}$ arises in this framework. We demonstrate the fundamental role played by this contribution both in topological mass generation of the $\eta^\prime$ and in the cancellation of the off-forward pole arising from the triangle anomaly in the proton's helicity $\Sigma(Q^2)$. We recover the striking result by Shore and Veneziano that $\Sigma\propto \sqrt{\chi'(0)}$, where $\chi'$ is the slope of the QCD topological susceptibility in the forward limit. We construct an axion-like effective action for $g_1$ at small $x_B$ that describes the interplay between gluon saturation and the topology of the QCD vacuum. In particular, we outline the role of "over-the-barrier" sphaleron-like transitions in spin diffusion at small $x_B$. Such topological transitions can be measured in polarized DIS at a future Electron-Ion Collider.
Highlights
In our previous paper [1] ( Paper I), we discussed the role of the chiral anomaly in the inclusive polarized deeply inelastic scattering (DIS) process eðkÞ þ NðP; SÞ → eðk0Þ þ X; ð1Þ where k denotes the four-momentum of the lepton (e) which scatters off a polarized target hadron (N) with fourmomentum P 1⁄4 ðPþ; M2=2Pþ; 0⊥Þ and four-spin S via the exchange of a virtual photon γà with fourmomentum q 1⁄4 k − k0
In our previous paper [1], we computed the contribution of the box diagram to the polarized structure function g1ðxB; Q2Þ employing the worldline representation of the fermion determinant in QCD
We demonstrated that the isosinglet triangle anomaly dominates the structure of the box diagram in both the Bjorken and Regge asymptotics of QCD
Summary
In our previous paper [1] ( Paper I), we discussed the role of the chiral anomaly in the inclusive polarized deeply inelastic scattering (DIS) process eðkÞ þ NðP; SÞ → eðk0Þ þ X; ð1Þ where k denotes the four-momentum of the lepton (e) which scatters off a polarized target hadron (N) with fourmomentum P 1⁄4 ðPþ; M2=2Pþ; 0⊥Þ and four-spin S (with S2 1⁄4 −1) via the exchange of a virtual photon γà with fourmomentum q 1⁄4 k − k0. As we noted in Paper I, the coupling of the isosinglet axial vector current to low energy dynamics of gauge fields (represented by the topological charge density) arises from the imaginary part of the worldline effective action. Zumino-Witten (WZW) [9,23] term in the imaginary part of the effective action that couples the isosinglet pseudoscalar field ηto the topological charge density Ω This term plays a fundamental role in the cancellation of the anomaly pole because an identical pole exists in the ηexchange with the proton. The role played by the chiral anomaly in the proton’s spin is deeply tied to the resolution of the UAð1Þ problem While these conclusions, if not the approach, are familiar from the work of Veneziano and collaborators, our framework can be extended to the computation of g1ðxB; Q2Þ in the Bjorken and Regge asymptotics of QCD. The treatment of Ω as an intrinsic low energy degree of freedom, on par with the Goldstone modes of chiral symmetry breaking, is discussed extensively in [3,4,5,15] and will be addressed
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