Models, measurements, and effective field theory: Proton capture on ${}^7\mathrm{Be}$ at next-to-leading order

We employ an effective field theory (EFT) that exploits the separation of scales in the $p$-wave halo nucleus ${}^8\mathrm{B}$ to describe the process ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ up to a center-of-mass energy of 500 keV. The key leading-order (LO) and next-to-leading-order (NLO) results appeared in our earlier papers. Here we first present full details of the EFT calculation. We develop the lagrangian and power counting in terms of velocity scaling, thereby making manifest that the Coulomb force between ${}^7\mathrm{Be}$ and proton plays a major role in both scattering and radiative capture at these energies: Coulomb interactions must be included to all orders in ${\alpha }_{\mathrm{em}}$. The EFT calculation of the capture reaction is then carried out using Feynman diagrams computed in time-ordered perturbation theory, so we recover existing quantum-mechanical technology such as the Lippmann-Schwinger equation and the two-potential formalism for the treatment of the Coulomb-nuclear interference. Meanwhile, the strong interactions and the E1 operator are dealt with via EFT expansions in powers of momenta, with a breakdown scale set by the size of the ${}^7\mathrm{Be}$ core, $\mathrm{\Lambda }\approx 70$ MeV/c. This is worked out up to NLO in the EFT expansion; at this order the relevant physics in the different channels that enter the radiative capture reaction is encoded in ten different EFT couplings. The result is a model-independent parametrization for the reaction amplitude in the energy regime of interest. In the second part of the paper we consider other approaches that have been used to describe ${}^7\mathrm{Be}{(p,\gamma )}^8\mathrm{B}$ in this energy range. We discuss the relationship of EFT to each of these approaches in qualitative terms and then make the connection quantitative by determining what the ten NLO EFT coefficients are in five different calculations that we consider representative. The EFT parameters are of natural size in all five cases, which shows that each of these earlier calculations corresponds to a particular point in the EFT parameter space. This understanding of the relationship between EFT and other ways of computing ${}^7\mathrm{Be}{(p,\gamma )}^8\mathrm{B}$ allows us to update earlier results for the dependence of $S\left(0\right)$ on asymptotic normalization coefficients and scattering lengths, since EFT separates dependence on these asymptotic quantities from dependence on shorter-distance contributions to the matrix element. We also summarize the fit to experimental capture data presented in our earlier work and explain why we obtain an extrapolated $S\left(0\right)$ with a markedly smaller error bar than that of the previous standard evaluation. Finally, we demonstrate that the only ${\mathrm{N}}^2\mathrm{LO}$ corrections in ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ come from an inelasticity that is practically of $\mathrm{N}{}^3\mathrm{LO}$ size in the energy range of interest, and so the truncation error in our calculation is effectively $\mathrm{N}{}^3\mathrm{LO}$.

Figure 1

The top equation shows the leading-order fully dressed propagator for $\varphi , D_{\varphi }^{\mathrm{LO}}$ (LHS), which results from the resummation of the bubble diagrams (RHS). The propagators in these diagrams are the free ones $D_{\varphi }^{\left(0\right)}$, as discussed in the main text. The lower equation expands the filled oval as a series of 0, 1, 2,...Coulomb photon exchanges (double wavy lines) between the charged proton (solid line) and core (long dashed line). This line labeling will be used throughout the entire paper.

Figure 2

The NLO $\varphi$ propagator $D_{\varphi }^{\mathrm{NLO}}$ as a sum of zero and one insertions of the dimer kinetic energy term that encodes the effective range. The bracketed diagrams containing two, three,… insertions are strictly higher order, but resumming them allows us to exactly match the ERE.

Figure 3

The NLO $\pi$ propagator, $D_{\pi }^{\mathrm{NLO}}$, is a sum of shaded SE (${\mathrm{\Sigma }}_{\pi }$) insertions, by using the free propagator as the LO one, $D_{\pi }^{\mathrm{LO}}$. Again shading the bubble diagram means a sum of diagrams with 0 to $\infty$ number of Coulomb photon exchanges. To differentiate the $\pi$ propagator from the $\varphi$ one, here, and in the following discussion, we use a filled rectangle to label $D_{\pi }$.

Figure 4

A LO diagram (first) and a NLO diagram (second) for $s$-wave to $p$-wave bound state radiative capture. The LO diagram is due to the gauged $n-c-\pi$ coupling. In the second diagram, the filled box is for the contact coupling term discussed in Eq. (6). Comparing the two gives the size of the coupling $L_{E1}$.

Figure 5

Diagrams for $nc$ scattering due to transverse photon exchange. The LHS is the Feynman diagram. The corresponding time-ordered perturbation theory diagrams are on the RHS.

Figure 6

The general scattering $T$-matrix caused by the Coulomb potential and strong interaction in $s$-wave channel. The bracketed diagrams are the pure Coulomb scattering. The last two diagrams are the strong interaction LO and NLO $T$-matrix.

Figure 7

The LO diagrams for radiative capture to a shallow $p$-wave bound state. The open elongated box denotes the LO propagators for the $s$-wave ${\varphi }_{\left(1\right)}$ and ${\varphi }_{\left(2\right)}$ dimers. The black filled box is the ${}^8\mathrm{B}$ state. The particle spins are not shown explicitly here, but discussed in the main text. These four diagrams $\sim V^{\frac{1}{2}}{\left(\frac{V}{V_{\mathrm{\Lambda }}}\right)}^{\frac{1}{2}}$. The corresponding diagrams with the photon coupled to the proton line are not shown here for simplicity.

Figure 8

The NLO diagrams for radiative capture to a shallow $p$-wave bound state. They scale as $V^{\frac{1}{2}}{\left(\frac{V}{V_{\mathrm{\Lambda }}}\right)}^{\frac{3}{2}}$. The filled boxes in the (V) and (VI) diagrams are the effective range correction contact coupling, as used in Fig. 2; those in (VII) and (VIII) are the $h_{\left({}^3{P}_2^{*}\right)}$ couplings (the dotted line in the bubbles labels the ${}^7{\mathrm{Be}}^{*}$ field); the one in diagram (IX) denotes the $L_1$ and $L_2$ E1 contact couplings defined in Eq. (55).

Figure 9

Upper panel: $S$ factors from the EFT fits (continuous) and the original potential [3, 24] and cluster [4] models (discrete circles). From top to bottom, the first three curves are the Davids and Typel potential models, with scattering lengths in both channels increasing from top to bottom. Next comes the cluster model, then at the bottom the Navratil potential model. Lower panel: Absolute values of $S$-factor residuals between the original models and fitted EFTs as fractions of the total. Symbols correspond to models in the same way as in the top panel.

Figure 10

The matrix element of the EM vertex $L_{EM}$—defined in Eq. (A3)—between initial free $n-c$ state and final free $n-c$ plus one transverse photon state.