Pion form factor in the chiral limit of a hard-wall AdS/QCD model

We describe a formalism to calculate form factor and charge density distribution of the pion in the chiral limit using the holographic dual model of QCD with hard-wall cutoff. We introduce two conjugate pion wave functions and present analytic expressions for these functions and for the pion form factor. They allow us to relate such observables as the pion decay constant and the pion charge electric radius to the values of chiral condensate and hard-wall cutoff scale. The evolution of the pion form factor to large values of the momentum transfer is discussed, and results are compared to existing experimental data.

Figure 1

Left: Pion decay constant $f_{\pi }$ as a function of $a$ for fixed ${\alpha }^{1/3}=424\mathrm{MeV}$. Right: Function $n(a)$

Figure 2

Functions $\phi (\zeta ,a)$ (left) and $\psi (\zeta ,a)$ (right) for several values of $a$: $a=0$ (uppermost lines), $a=1$, $a=2.26$, $a=5$, $a=10$ (lowermost lines).

Figure 3

Left: Function $\rho (\zeta ,a)$ for $a=0$, $a=1$, $a=2.26$, $a=5$, $a=10$. Middle: Densities $\rho (\zeta ,2.26)$ for pion and ${\rho }_{\rho }(\zeta )$ for $\rho$-meson in the hard-wall model. Right: Same for densities multiplied by $\zeta$.

Figure 4

$⟨r_{\pi }^2⟩$ in ${\mathrm{fm}}^2$ for $z_0=z_0^{\rho }$ as a function of $a$.

Figure 5

Left: Contributions to pion form factor $F_{\pi }(Q^2)$ from ${\Psi }^2$-term (lower curve), from ${\Phi }^2$-term (middle curve) and total contribution (upper curve). Right: Same for $Q^2{F}_{\pi }(Q^2)$.

Figure 6

Left: Pion form factor $F_{\pi }(Q^2)$ from the holographic model (upper curve) in comparison with the monopole interpolation $F_{\pi }^{\mathrm{mono}}(Q^2)$ (lower curve). Right: Ratio $F_{\pi }(Q^2)/F_{\pi }^{\mathrm{mono}}(Q^2)$.

Figure 7

Left: Model density $z{\rho }^{\mathrm{mod}}(z)$ (measured in ${\mathrm{fm}}^{-1}$) is larger than the density $z|\Phi (z,{\kappa }_{\pi }){|}^2$ for large $z$ (displayed in fm). Right: Ratio $F_{\pi }^{\mathrm{mod}}(Q^2)/F_{\pi }^{\mathrm{mono}}(Q^2)$ for $B=1/4$.