Electromagnetic properties of ground-state and excited-state pseudoscalar mesons

The axial-vector Ward-Takahashi identity places constraints on particular properties of every pseudoscalar meson. For example, in the chiral limit all pseudoscalar mesons, except the Goldstone mode, decouple from the axial-vector current. Nevertheless, all neutral pseudoscalar mesons couple to two photons. The strength of the ${\pi }_n^0\gamma \gamma$ coupling, where $n=0$ denotes the Goldstone mode, is affected by the Abelian anomaly's continuum contribution. The effect is material for $n\ne 0$. The ${\gamma }^{*}{\pi }_n{\gamma }^{*}$ transition form factor, ${\mathcal{T}}_{{\pi }_n}(Q^2)$, is nonzero $\forall n$, and ${\mathcal{T}}_{{\pi }_n}(Q^2)\approx (4{\pi }^2/3)(f_{{\pi }_n}/Q^2)$ at large $Q^2$. For all pseudoscalars but the Goldstone mode, this leading contribution vanishes in the chiral limit. In this instance the ultraviolet power-law behavior is $1/Q^4$ for $n\ne 0$, and we find numerically ${\mathcal{T}}_{{\pi }_1}(Q^2)\simeq (4{\pi }^2/3)(-\langle \overline{q}q\rangle /Q^4)$. This subleading power-law behavior is always present. In general its coefficient is not simply related to $f_{{\pi }_n}$. The properties of $n\ne 0$ pseudoscalar mesons are sensitive to the pointwise behavior of the long-range piece of the interaction between light quarks.

Figure 1

This axial-vector Ward-Takahashi identity is an analogue of Eq. (37). It is valid if, and only if: the dressed-quark propagator, S, is obtained from Eq. (34); the axial-vector vertex, ${\Gamma }_{5\mu }$, is obtained from Eq. (13) with the kernel constructed from S and Eq. (35); the pseudoscalar vertex is constructed analogously; and the unamputated renormalized quark-antiquark scattering matrix, $G=(\mathit{SS})+(\mathit{SS})K(\mathit{SS})+(\mathit{SS})K(\mathit{SS})K(\mathit{SS})+[\dots ]$, is constructed from the elements just described.

Figure 2

Small-$Q^2$ behavior of the ${\gamma }^{*}(Q){\pi }_n(P){\gamma }^{*}(Q)$ transition form factor, defined in Eq. (48), calculated with the current-quark mass in Eq. (67). The ground state's two-photon coupling suggested by Eq. (50) is marked by “×.”

Figure 3

Calculated large-$Q^2$ behavior of the ${\gamma }^{*}(Q){\pi }_n(P){\gamma }^{*}(Q)$ transition form factor, Eq. (48): diamonds—ground state, $n=0$; and circles—first excited state, $n=1$. The solid lines are Eq. (51) with either $f_{{\pi }_0}$ or $f_{{\pi }_1}$ from Table , as appropriate.

Figure 4

Large-$Q^2$ behavior of the ${\gamma }^{*}(Q){\pi }_1(P){\gamma }^{*}(Q)$ transition form factor, Eq. (48): Diamonds—the result obtained with $\stackrel{\wedge }{m}$ in Eq. (67); Circles—our chiral limit calculation ($\stackrel{\wedge }{m}=0$); Solid line—the curve $\frac{4{\pi }^2}{3}(0.22\mathrm{GeV}){}^3/Q^4$.

Figure 5

Evolution of ground and first excited state pseudoscalar mesons’ electromagnetic charge radii with the model's scale parameter ω. Dotted line: $r_{\pi }=0.66$fm, which indicates the experimental value of the ground state's radius. We must estimate the derivative in Eq. (76) numerically. That is the primary source of the numerical error depicted in the figure, which corresponds to a relative error $\lesssim 1$% for $n=0$ and $\lesssim 3$% for $n=1$.