Perturbative calculations of deuteron form factors

We calculate the deuteron charge, electric quadrupole, and magnetic form factors up to next-to-next-to-leading order in chiral effective field theory, treating subleading corrections, especially that of chiral nuclear forces, in perturbation theory. We examine the power counting based on naive dimensional analysis by investigating the ultraviolet cutoff variation of these form factors. We find that the ${\mathrm{N}}^2\mathrm{LO}$ magnetic form factor shows significant cutoff dependence, suggesting the contact current operator responsible be enhanced. After promotion to ${\mathrm{N}}^2\mathrm{LO}$, it indeed renormalizes the magnetic form factor. This is in agreement with a previous work based on renormalization-group analysis. For the charge and quadrupole moments, perturbative calculation allows us to study how they scale with multiple low-energy parameters such as the pion mass, deuteron binding momentum, and momentum transfer.

Figure 1

The LEC values of the ${\mathrm{N}}^2\mathrm{LO}$ contact potentials as functions of $\mathrm{\Lambda }$.

Figure 2

The ratio ${\varepsilon }_r\equiv ({\varepsilon }_{\text{N}{}^2\mathrm{LO}}-{\varepsilon }_{\text{emp}})/({\varepsilon }_{\text{N}{}^2\mathrm{LO}}-{\varepsilon }_{\text{LO}})$ at $k=150.12\mathrm{MeV}$ as a function of $\mathrm{\Lambda }$, where ${\varepsilon }_{\text{LO}}$ and ${\varepsilon }_{\text{N}{}^2\mathrm{LO}}$ are LO and ${\mathrm{N}}^2\mathrm{LO}$ values of the mixing angle ${\varepsilon }_1$, and ${\varepsilon }_{\text{emp}}$ is the empirical value from the Nijmegen partial wave analysis.

Figure 3

The LO deuteron charge, quadrupole, and magnetic form factors at $q=m_{\pi }$ as functions of the UV cutoff value $\mathrm{\Lambda }$.

Figure 4

The ${\mathrm{N}}^2\mathrm{LO}$ deuteron charge, quadrupole, and magnetic form factors as functions of the UV cutoff value $\mathrm{\Lambda }$ for $q=m_{\pi }$.

Figure 5

The ${\mathrm{N}}^2\mathrm{LO}$ correction to $G_M$ as a function of $\mathrm{\Lambda }$ at $q=m_{\pi }$ before renormalization. The legends specify the source driving the correction. See the text for more explanation.

Figure 6

The cutoff variation of ${\mu }_d$ at LO and ${\mathrm{N}}^2\mathrm{LO}$ before renormalization.

Figure 7

The cutoff variation of $G_M$ at ${\mathrm{N}}^2\mathrm{LO}$ after renormalization. The solid circles, squares, and triangles correspond to $q=m_{\pi }$, $2m_{\pi }$, and $3m_{\pi }$.

Figure 8

The deuteron charge, electric quadrupole, and magnetic form factors as a function of $q$. The green (red) bands represent their LO (${\mathrm{N}}^2\mathrm{LO}$) using two cutoff values $\mathrm{\Lambda }=600$ and $2400\mathrm{MeV}$. For the ${\mathrm{N}}^2\mathrm{LO}$ $G_M$, the contact current operator (24) is accounted for. The light blue band represents the ${\mathrm{N}}^2\mathrm{LO}$ $G_M$ before renormalization. The solid circles and error bars represent the experimental data [62, 63]. However, for $G_C$ and $G_M$ the scales are too large for the error bars to be visible.