Electromagnetic currents and magnetic moments in chiral effective field theory $(\chi \mathrm{EFT})$

A two-nucleon potential and consistent electromagnetic currents are derived in chiral effective field theory $(\chi \mathrm{EFT})$ at, respectively, $Q^2$ (or ${\mathrm{N}}^2\mathrm{LO}$) and $eQ$ (or ${\mathrm{N}}^3\mathrm{LO}$), where $Q$ generically denotes the low-momentum scale and $e$ is the electric charge. Dimensional regularization is used to renormalize the pion-loop corrections. A simple expression is derived for the magnetic dipole $(M1)$ operator associated with pion loops, consisting of two terms, one of which is determined, uniquely, by the isospin-dependent part of the two-pion-exchange potential. This decomposition is also carried out for the $M1$ operator arising from contact currents, in which the unique term is determined by the contact potential. Finally, the low-energy constants entering the ${\mathrm{N}}^2\mathrm{LO}$ potential are fixed by fits to the $\mathit{np}$ $S$- and $P$-wave phase shifts up to 100 MeV laboratory energies.

Figure 1

Diagrams illustrating contributions to the $\mathit{NN}$ potential entering at LO ($Q^0$), panels (a) and (b), and ${\mathrm{N}}^2\mathrm{LO}$ ($Q^2$), panels (c)–(i). Nucleons and pions are denoted by solid and dashed lines, respectively. The filled circle in panel (c) represents the vertex from contact Hamiltonians containing two gradients of the nucleons’ fields. Only one among the possible time orderings is shown for each contribution with more than one vertex.

Figure 2

Diagrams illustrating one- and two-body currents entering at LO ($e{Q}^{-2}$), panel (a); NLO ($e{Q}^{-1}$), panels (b) and (c); and ${\mathrm{N}}^2\mathrm{LO}$ ($e{Q}^0$), panel (d). Nucleons, pions, and photons are denoted by solid, dashed, and wavy lines, respectively. The square represents the $(Q/m_N){}^2$ relativistic correction to the LO one-body current. Only one among the possible time orderings is shown for the NLO currents.

Figure 3

Diagrams illustrating one-loop two-body currents entering at ${\mathrm{N}}^3\mathrm{LO}$ ($eQ$); notation as in Fig. 2. Only one among the possible time orderings is shown for each contribution.

Figure 4

Diagrams illustrating loop corrections to tree-level two-body currents; notation as in Fig. 2. Only one among the possible time orderings is shown for each contribution.

Figure 5

Same as in Fig. 4.

Figure 6

Diagrams illustrating ${\mathrm{N}}^4\mathrm{LO}$ ($e{Q}^2$) loop corrections to tree-level currents not included in the present study; notation as in Fig. 2.

Figure 7

The $S$-wave $\mathit{np}$ phase shifts, obtained with cutoff parameters $\Lambda =500$, 600, and 700 MeV, are denoted by dash (red), dot-dash (green), and solid (blue) lines, respectively. The filled circles represent the phase-shift analysis of Ref. [16].

Figure 8

Same as in Fig. 7 but for $P$-wave phase shifts.

Figure 9

Same as in Fig. 7 but for $D$-wave phase shifts. The dash-double-dot (orange) line is obtained in first-order perturbation theory for the $T$ matrix by including only the one- and two-pion-exchange parts of the ${\mathrm{N}}^2\mathrm{LO}$ potential.

Figure 10

Same as in Fig. 9 but for $F$-wave phase shifts.

Figure 11

Same as in Fig. 9 but for $G$-wave phase shifts.

Figure 12

Same as in Fig. 9 but for the mixing angles ${\epsilon }_J$.

Figure 13

The $S$-wave and $D$-wave components of the deuteron, obtained with cutoff parameters $\Lambda =500,600$, and 700 MeV and denoted by dash (red), dot-dash (green), and solid (blue) lines, respectively, are compared with those calculated from the Argonne $v_{18}$ potential (dash-double-dot black lines).

Figure 14

Set of time-ordered diagrams for the contribution illustrated by the single diagram (i) in Fig. 4. Notation as in Fig. 2.

Figure 15

Subset of time-ordered diagrams for the contribution illustrated by the single diagram (j) in Fig. 4. See text for discussion. Notation as in Fig. 2.