Forward doubly-virtual Compton scattering off an unpolarized deuteron in pionless effective field theory

We calculate the forward unpolarized doubly-virtual Compton scattering (VVCS) off the deuteron in the framework of pionless effective field theory, up to next-to-next-to-next-to-leading order (N3LO) for the longitudinal and next-to-leading order for the transverse amplitude. The charge elastic form factor of the deuteron, obtained from the residue of the longitudinal VVCS amplitude, is used to extract the value of the single unknown two-nucleon one-photon contact coupling that enters the longitudinal amplitude at N3LO. We also study the lowest spin-independent generalized polarizabilities of the deuteron. The calculated unpolarized VVCS amplitude provides a high-precision model-independent input for a future calculation of the two-photon-exchange correction to the Lamb shift of muonic deuterium.

Figure 1

The LO $NNT$ matrix for the spin-triplet channel. Here and below, big disk (square) vertices denote insertions of a $NNT$ matrix ($NN$ potential), with the number inside a vertex showing the order of the vertex in the $\cancel{\pi }\mathrm{EFT}$ counting. The LO $T$ matrix in the spin-singlet ${}^{1}S_0$ channel is obtained analogously.

Figure 2

Examples of diagrams that have an external loop attached via an insertion of the LO triplet $NNT$ matrix. Such diagrams are not included in the four-point function. The crossed vertex denotes the leading-order $O\left(P^0\right)NNd$ coupling.

Figure 3

Correction to the $NNT$-matrix vertex at order $k\ge 0$, defined in terms of the dressed potential at that order (denoted by the square with the primed index $k^{\prime }$). Only the triplet channel is shown. The dressed potentials at the orders we are considering are shown in Fig. 4.

Figure 4

Dressed potential at $O\left(P^0\right)$, $O\left(P\right)$, and $O\left(P^2\right)$. Only the triplet channel is shown.

Figure 5

Corrections to the $NNd$ vertex at $O\left(P\right)$, $O\left(P^2\right)$, and $O\left(P^3\right)$, with the numbers denoting the order of the correction. Not to be confused with the $NNT$ matrices defined in Figs. 1 and 3, which are denoted by bigger discs and have four external legs.

Figure 6

Graphical expression for $i\mathrm{\Sigma }\left(E\right)$ up to N3LO.

Figure 7

Diagrams contributing to $\mathcal{M}$ at LO, i.e., $O\left(P^{-3}\right)$ for ${\mathcal{M}}_L$ and $O\left(P^{-1}\right)$ for ${\mathcal{M}}_T$. Dotted vertices are the sum of the charge and the magnetic moment couplings from the single-nucleon Lagrangian; the big gray disk denotes the LO $NNT$ matrix in the singlet channel. Crossed graphs are not shown.

Figure 8

Contributions to $\mathcal{M}$ at NLO, i.e., $O\left(P^{-2}\right)$ for ${\mathcal{M}}_L$ and $O\left(P^0\right)$ for ${\mathcal{M}}_T$, due to NLO corrections in the $NN$ interaction. The notation is as in Fig. 7. Crossed graphs are not shown.

Figure 9

Contributions to $\mathcal{M}$ at NLO due to the magnetic contact terms $L_1^{M{1}_V}$ and $L_2^{M{1}_S}$ (shown as diamonds). The rest of the notation is as in Fig. 7. These graphs give an $O\left(P^0\right)$ contribution to ${\mathcal{M}}_T$ only. Crossed graphs are not shown.

Figure 10

Diagrams that contribute to ${\mathcal{M}}_L$ at NNLO, i.e., $O\left(P^{-1}\right)$, due to NNLO terms in the $NN$ interaction. The nucleon-photon vertex is exclusively the minimal coupling. In addition to diagrams that actually contribute to $f_L$, we also show those that are necessary in order to keep the electromagnetic gauge invariance. The vertical double dashed lines indicate possible insertions of a LO $NNT$ matrix in the spin-triplet channel. Crossed graphs are not shown.

Figure 11

Correction to ${\mathcal{M}}_L$ due to the photon coupling proportional to ${\widehat{r}}_E^2$, denoted by the black cross vertex, contributing at $O\left(P^{-1}\right)$. The rest of the notation is as in Fig. 10. Crossed graphs are not shown.

Figure 12

Diagrams that contribute to ${\mathcal{M}}_L$ at N3LO, i.e., $O\left(P^0\right)$, due to N3LO terms in the $NN$ interaction. In addition to diagrams that actually contribute to $f_L$, we also show those that are necessary in order to keep the electromagnetic gauge invariance. Gray squares marked by “P” show insertions of the $P$-wave $NN$ interactions. The rest of the notation is as in Fig. 10. Crossed graphs are not shown.

Figure 13

Contributions to ${\mathcal{M}}_L$ at N3LO due to the electric contact terms (shown as crossed diamonds). The rest of the notation is as in Fig. 10. Crossed graphs are not shown.

Figure 14

Correction to ${\mathcal{M}}_L$ due to the photon coupling proportional to ${\widehat{r}}_E^2$, contributing at $O\left(P^0\right)$. The rest of the notation is as in Figs. 10 and 11. Crossed graphs are not shown.

Figure 15

Deuteron charge form factor at LO (dash-dot-dotted black), NLO (dash-dotted blue), NNLO (dashed green), and N3LO (solid red, with the band showing the uncertainty due to higher-order terms). The result of the $\chi \mathrm{EFT}$ fit [46] is shown by the purple dotted curve.

Figure 16

Generalized deuteron polarizabilities: (a) ${\alpha }_{E1}\left(Q^2\right)$ and (b) ${\beta }_{M1}\left(Q^2\right)$. The LO, NLO, NNLO, and N3LO results for ${\alpha }_{E1}\left(Q^2\right)$ in the left panel are coded as in Fig. 15. In the right panel, the LO and NLO results for ${\beta }_{M1}\left(Q^2\right)$ are shown, respectively, by the black dashed and the red solid curve, with the band showing the estimate of higher-order contributions.

Figure 17

Generalized deuteron polarizabilities: (a) ${\alpha }_L\left(Q^2\right)$ and (b) $[{\alpha }_{E1}+{\beta }_{M1}]\left(Q^2\right)$. The curves are coded as in the left and right panel of Fig. 16, respectively. The curves for $[{\alpha }_{E1}+{\beta }_{M1}]\left(Q^2\right)$ are obtained using the exact expression $\left|\mathit{q}\right|=\sqrt{{\nu }^2+Q^2}$ and reproduce at $Q=0$ the static values of ${\alpha }_{E1}+{\beta }_{M1}$ at the respective order (see the text for details).

Figure 18

Fourth-order generalized Baldin sum rule. The curves are coded as in the right panel of Fig. 16.