Improved action for contact effective field theory

L. Contessi, M. Schäfer, and U. van Kolck

Phys. Rev. A 109, 022814 – Published 22 February 2024

Abstract

We present an improved action for renormalizable effective field theories (EFTs) of systems near the two-body unitarity limit. The ordering of EFT interactions is constrained, but not entirely fixed, by the renormalization group. The remaining freedom can be used to improve the theory's convergence, to simplify its applications, and to connect it to phenomenological models. We exemplify the method on a contact theory applied to systems of up to five ${}^{4}{He}$ atoms. We solve the EFT at leading order including a subleading interaction that accounts for part of the two-body effective range. We show that the effects of such fake range can be compensated in perturbation theory at next-to-leading order, as long as the fake range is smaller or comparable to the experimental effective range. These results open the possibility of using similar improved actions for other many-body systems.

Received 25 October 2023
Accepted 16 January 2024

DOI:https://doi.org/10.1103/PhysRevA.109.022814

Physics Subject Headings (PhySH)

Atomic & molecular clusters Effective field theory Efimov states Few-body systems Nuclear forces

Renormalization

Atomic, Molecular & OpticalNuclear Physics

Authors & Affiliations

L. Contessi ^*

Université Paris-Saclay, CNRS-IN2P3, IJCLab, 91405 Orsay, France

M. Schäfer ^†

Nuclear Physics Institute of the Czech Academy of Sciences, Řež 25068, Czech Republic

U. van Kolck

Université Paris-Saclay, CNRS-IN2P3, IJCLab, 91405 Orsay, France and Department of Physics, University of Arizona, Tucson, Arizona 85721, USA

^*lorenzo@contessi.net
^†m.schafer@ujf.cas.cz

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Vol. 109, Iss. 2 — February 2024

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Images

Figure 1
Calculated dimer binding energy normalized to the dimer binding energy obtained directly from the LM2M2 potential [36], $B_{2} / B_{2}^{LM 2 M 2}$ , as function of the cutoff $Λ$ in units of the three-body scale $Q_{3}$ . Results from the LO and NLO potentials, respectively, $V^{(0)}$ and $V^{(0 + 1)} \equiv V^{(0)} + V^{(1)}$ , are shown by triangles and circles connected with dotted lines. At the two-body level, the improved potentials of type I and II are equivalent. The (blue) hatched and shaded bands show the LO and NLO values from the improved potentials ${\tilde{V}}_{I}^{(0)} / {\tilde{V}}_{I I}^{(0)}$ and ${\tilde{V}}_{I}^{(0 + 1)} / {\tilde{V}}_{I I}^{(0 + 1)}$ , respectively, when the fake range is in the range $6 \leq {(Q_{3} \tilde{R})}^{- 1} \leq 24$ .
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Figure 2
Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and ground tetramer $B_{4}$ (right panel) at LO, normalized to the ground-trimer binding energy $B_{3}$ , as function of the cutoff $Λ$ in units of the three-body scale $Q_{3}$ . Results for the LO potential $V^{(0)}$ are shown by triangles connected with a dotted line. Results for the improved LO potential ${\tilde{V}}_{I}^{(0)}$ are given in red with different symbols for different inverse fake-range values: ${\tilde{R}}^{- 1} = 6 Q_{3}$ (diamonds connected by a dash-dotted line), ${\tilde{R}}^{- 1} = 12 Q_{3}$ (squares, dashed line), and ${\tilde{R}}^{- 1} = 24 Q_{3}$ (circles, solid line). The blue-shaded band shows the LO values from ${\tilde{V}}_{I I}^{(0)}$ when the fake range is in the range $6 \leq {(Q_{3} \tilde{R})}^{- 1} \leq 24$ . The (green) horizontal dashed lines mark the result of the full LM2M2 potential [35].
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Figure 3
Same as Fig. 2 but for the ground-pentamer binding energy $B_{5}$ . The LM2M2 result is from Ref. [31]
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Figure 4
Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and ground pentamer $B_{5}$ (right panel) up to NLO, normalized to the ground-trimer binding energy $B_{3}$ , as function of the cutoff $Λ$ in units of the three-body scale $Q_{3}$ . Results for the LO and NLO potentials $V^{(0)}$ and $V^{(0 + 1)}$ are shown by, respectively, empty triangles connected with a dotted line and empty circles connected with a solid line. Results for the improved NLO potentials ${\tilde{V}}_{I}^{(0 + 1)}$ and ${\tilde{V}}_{I I}^{(0 + 1)}$ at ${\tilde{R}}^{- 1} = 12 Q_{3}$ are given by, respectively, (red) half-filled circles connected by a (red) solid line and (blue) solid circles connected by a (blue) solid line. The (red) hatched area and the (blue) shaded band indicate the spread of $V_{I}^{(0 + 1)}$ and $V_{I I}^{(0 + 1)}$ results, respectively, for fake-range values in the range $6 \leq {(Q_{3} \tilde{R})}^{- 1} \leq 24$ . (For the pentamer, we show $V_{I}^{(0 + 1)}$ results only for one fake-range value, as obtaining accurate results for the entire range is numerically challenging.) The (green) horizontal dashed lines mark the result of the full LM2M2 potential [31, 35].
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Figure 5
Binding energies for the excited trimer $B_{3}^{*}$ (left panel) and ground pentamer $B_{5}$ (right panel) up to NLO, normalized to the ground-trimer binding energy $B_{3}$ , as function of the inverse fake range ${\tilde{R}}^{- 1}$ in units of the three-body scale $Q_{3}$ . Results for the improved NLO potential ${\tilde{V}}_{I I}^{(0 + 1)}$ for cutoff values $Λ \geq 6 Q_{3}$ are given by a (blue) shaded band; extrapolated values for $Λ \to \infty$ at the given ${\tilde{R}}^{- 1}$ are marked by (blue) stars. For comparison, the variation of LO ( $V^{(0)}$ ) and NLO ( $V^{(0 + 1)}$ ) results for cutoff $Λ \geq 6 Q_{3}$ are shown by light and dark (gray) hatched bands, respectively. The (green) horizontal dashed lines mark the result of the full LM2M2 potential [31, 35]. The vertical dotted line indicates the fake range value, ${\tilde{R}}_{c}^{- 1} ≃ 4.8 Q_{3}$ , where the improved LO potential ${\tilde{V}}_{I I}^{(0)}$ exactly reproduces the LM2M2 effective range.
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