The proton–deuteron system in pionless EFT revisited

Abstract

We provide a detailed discussion of the low-energy proton-deuteron system in pionless effective field theory, considering both the spin-quartet and doublet S-wave channels. Extending and amending our previous work on the subject, we calculate the 3 He-3 H binding energy difference both perturbatively (using properly normalized trinucleon wave functions) and non-perturbatively by resumming all O(α) Coulomb diagrams in the doublet channel. Our nonperturbative result agrees well with a calculation that involves the full off-shell Coulomb T-matrix. Carefully examining the cutoff-dependence in the doublet channel, we present numerical evidence for a new three-nucleon counterterm being necessary at next-to-leading order if Coulomb effects are included. Indeed, such a term has recently been identified analytically. We furthermore make a case for a simplified Coulomb power counting that is consistent throughout the bound-state and scattering regimes. Finally, using a "partially screened" full off-shell Coulomb T-matrix, we investigate the importance of higher-order Coulomb corrections in low-energy quartet-channel scattering.Effective field theories are powerful tools that can be used to carry out calculations in a formalism involving directly the relevant degrees of freedom for the physical system under consideration. In particular, in nuclear systems at very low energies and momenta, pionexchange effects cannot be resolved and one can hence use the so-called pionless effective field theory. This approach only includes short-range contact interactions between nucleons [1-3] and is constructed to reproduce the effective range expansion [4] in the two-body system.The applicability of this approach, which recovers Efimov's universal approach to the three-nucleon problem [5,6], stems from the experimental fact that the S-wave nucleonnucleon scattering lengths in both the 3 S 1 (isospin 0) channel, a d ≈ 5.42 fm, and in the 1 S 0 (isospin 1) channel, a t ≈ −23.71 fm, are significantly larger than the range of interaction of about 1.4 fm set by the inverse pion mass. The corresponding effective ranges, on the other hand, have the values 1.75 and 2.73 fm, respectively, and are thus of the expected natural order of magnitude [7,8]. In Refs. [9][10][11][12][13][14], the formalism has been extended to the three-nucleon sector. The situation there is particularly interesting because the triton can be interpreted as an approximate Efimov state. Recently, a fully perturbative calculation of neutron-deuteron scattering up to next-to-next-to-leading order has been carried out by J. Vanasse [15], using a novel technique that circumvents the (numerically expensive) calculation of full off-shell quantities.Since most low-energy nuclear experiments involve more than one charged particle (proton), it is very important to discuss the treatment of Coulomb effects. Although this interaction can be treated as a perturbative correction for intermediate and higher energies, it becomes strong close to threshold and has to be treated nonperturbati...

Triton charge radius to next-to-next-to-leading order in pionless effective field theory

“…The "physical" triton vertex function is calculated using Eq. (29). Using the definition of B 0 and the triton wavefunction renormalization, the LO renormalization for the triton vertex function G 0 (B, p) is…”

“…[7] for a review) has been used with great success in the two-body sector calculating deuteron electromagnetic form factors [8,9], NN scattering [8,10,11], neutron-proton (np) capture [8,9,12] to (<∼1%) [13], proton-proton fusion [14][15][16], and neutrino deuteron scattering [17]. Progress has also been made in the three-body sector with calculations of nd scattering [6,[18][19][20][21][22][23], pd scattering [24][25][26], nd capture [27][28][29], and the energy difference between 3 H and 3 He [25,30,31]. Previous three-body calculations of nd scattering in EFT(/ π) made use of the partial resummation technique [21].…”

Power counting and Wilsonian renormalization in nuclear effective field theory

“…Trivially this conclusion applies too for the 3 He nucleus and doublet proton-deuteron scattering at LO, though here we have the additional complication of Coulomb. 60 Recently it has been discovered that Coulomb is able to modify the counting at NLO, 61,62 but only if it is treated non-perturbatively. 63 On the contrary for the spin-3/2 configurations (the quartet) of the three-nucleon system the size of the three-nucleon contact is the one expected in NDA.…”

The proton–deuteron scattering length in pionless EFT

“…More details of the formalism can be found, for example, in Ref. [13] and earlier references therein. The full leading-order dibaryon propagator Δ d (p 0 , p) is obtained by dressing the bare expression with nucleon bubbles to all orders and matching −y [7,8].…”

“…We present here results in this "RK" scheme as well as for an alternative "O(α) counting that includes this diagram already at LO, i.e., iterating it together with the other diagrams shown in Fig. 1(d) [13].…”