Precision calculation of the quartet-channel $p-d$ scattering length

We present a fully perturbative calculation of the quartet-channel proton-deuteron scattering length up to next-to-next-to-leading order in pionless effective field theory. We use a framework that consistently extracts the Coulomb-modified effective-range function for a screened Coulomb potential in momentum space and allows for a clear linear extrapolation back to the physical limit without screening. Our result of ${{}^{4}a}_{p\text{−}d}=(10.9\pm 0.4)$ fm agrees with older experimental determinations of this quantity but deviates from potential-model calculations and a more recent result from Black et al., which find larger values around 14 fm. As a possible resolution to this discrepancy, we discuss the scheme dependence of Coulomb subtractions in a three-body system.

Figure 1

Leading $O\left(\alpha \right)$ diagrams involving Coulomb photons.

Figure 2

Integral equation for the full (i.e., strong + Coulomb) scattering quartet-channel amplitude ${\mathcal{T}}_{\mathrm{full}}$.

Figure 3

Photon-mass dependence and extrapolation of ${{}^{4}a}_{p\text{−}d}$. Dotted lines: LO result; dashed lines: NLO result; solid lines: $\mathrm{N}{}^2\mathrm{LO}$ result. Each calculation was performed at three different cutoffs: $\Lambda =140$ MeV (thick lines), $\Lambda =280$ MeV (medium lines), and $\Lambda =560$ MeV (thin lines). RK and $O\left(\alpha \right)$ indicate the different Coulomb-counting schemes (see text). The bands (shown for the RK results only) reflect the expected EFT expansion uncertainty.

Figure 4

$S$-wave $p-d$ quartet channel scattering phase shifts as functions of the center-of-mass momentum $k$. The curves and bands are as in Fig. 3. Experimental $p-d$ phase-shift data are shown from Refs. [6] (diamonds) and [34] (circles). The crosses are the results from the AV18 potential-model calculation reported in Ref. [35].

Figure 5

$p-d$ effective-range function with screened Gamow factor and pure-Coulomb phase shifts from T-matrix based on the EFT calculation. $\mathrm{N}{}^2\mathrm{LO}$ calculation with cutoff $\Lambda =560$ MeV.

Figure 6

Same as Fig. 5, but now with ${\mathcal{T}}_{\mathrm{c}}$ calculated from a simple two-body Yukawa potential and for a larger cutoff $\Lambda =56000$ MeV.

Figure 7

Photon-mass and cutoff dependence of ${{}^{4}a}_{p\text{−}d}$ with a simple two-body Yukawa subtraction at $\mathrm{N}{}^2\mathrm{LO}$ in the $O\left(\alpha \right)$ counting scheme.