$\Delta$-baryon axialvector and pseudoscalar form factors, and associated PCAC relations

Abstract

A quark+diquark Faddeev equation treatment of the baryon bound state problem in Poincaré-invariant quantum field theory is used to deliver parameter-free predictions for all six $\Delta$-baryon elastic weak form factors. Amongst the results, it is worth highlighting that there are two distinct classes of such $\Delta$-baryon form factors, $(g_1,g_3,G_{\pi \Delta \Delta })$, $(h_1,h_3,H_{\pi \Delta \Delta })$, the functions within each of which are separately linked via partial conservation of axial current (PCAC) and Goldberger-Treiman (GT) relations. Respectively within each class, the listed form factors possess qualitatively the same structural features as the nucleon axial, induced pseudoscalar, and pion-nucleon coupling form factors. For instance, the $\Delta$-baryon $g_1$ axial form factor can reliably be approximated by a dipole function, characterised by an axial charge $g_A^{{\Delta }^{+}}=0.71(9)$ and mass-scale $m_A^{\Delta }=0.95(2)m_{\Delta }$. Moreover, the two distinct $\Delta$-baryon PCAC form factor relations are satisfied to a high degree of accuracy on a large range of $Q^2$; the associated GT relations present good approximations only on $Q^2/m_{\Delta }^2\simeq 0$; and pion pole dominance approximations are reliable within both classes. There are two $\pi \Delta \Delta$ couplings: $g_{\pi \Delta \Delta }=10.46(1.88)$; $h_{\pi \Delta \Delta }=35.73(3.75)$; and the associated form factors are soft. Such couplings commonly arise in phenomenology, which may therefore benefit from our analyses. A flavour decomposition of the axial charges reveals that quarks carry $71$\% of the $\Delta$-baryon spin. The analogous result for the proton is $\approx 65$\%.

Comments: 19 pages, 9 figures, 2 tables