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Turning the nuclear energy density functional method into a proper effective field theory: reflections

  • Regular Article -Theoretical Physics
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Abstract

Nuclear energy density functionals (EDFs) have a long history of success in reproducing properties of nuclei across the table of the nuclides. They capture quantitatively the emergent features of bound nuclei, such as nuclear saturation and pairing, yet greater accuracy and improved uncertainty quantification are actively sought. Implementations of phenomenological EDFs are suggestive of effective field-theory (EFT) formulations and there are hints of an underlying power counting. Multiple paths are possible in trying to turn the nuclear EDF method into a proper EFT. I comment on the current situation and speculate on how to proceed using an effective action formulation.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This article is a commentary for the EPJA topical issue: “The tower of effective (field) theories and the emergence of nuclear phenomena” and therefore does not include data.]

Notes

  1. The term “effective field theory” is often restricted to mean a local Lagrangian formulation of a low-energy theory, with “effective theory” a more general designation. We mostly have in mind strict EFTs but for convenience will use that designation even for the more general cases.

  2. The important distinction between DFT as formalized for the Coulomb many-body problem and the nuclear EDF approach has been stressed by Duguet and collaborators [26]. We will not address this issue explictly until we consider zero modes in Sect. 3.5. Until then we will generally use DFT and EDF interchangeably.

  3. Because \(v_{\mathrm{ext}}\) is typically given rather than eliminated, for a closer analogy we would also define \(\varOmega _\mu (N) \equiv F(N) - \mu N\), which depends explicitly on both N and \(\mu \). This gives the grand potential when minimized with respect to N [86].

  4. A Minkowski-space formulation of the effective action with time-dependent sources leads naturally to an RPA-like generalization of DFT that can be used to calculate properties of collective excitations.

  5. The functionals will change with resolution or field redefinitions; only stationary points are observables. This can be seen from Eq. (18), where \(\varGamma [\rho ]\) is not the expectation value of \({\widehat{H}}\) in an eigenstate unless \(J = J[\rho _\mathrm{gs}]\).

  6. For the Minkowski-space version of this discussion, see Ref. [90].

  7. There are important formal details [94], such as that we need E[v] to be concave to carry out the transform.

  8. In other contexts, such densities are called “intrinsic”, but this has a different meaning in the context of symmetry breaking, so “internal” is typically used instead.

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Acknowledgements

I acknowledge many illuminating discussions over many years with my colleagues on effective field theory and energy density functionals that have contributed to my reflections here. However, all misunderstandings, misstatements, and misinterpretations are my own. Supported in part by the US National Science Foundation under Grant no. PHY-1614460 and the NUCLEI SciDAC Collaboration under US Department of Energy MSU subcontract RC107839-OSU.

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Furnstahl, R.J. Turning the nuclear energy density functional method into a proper effective field theory: reflections. Eur. Phys. J. A 56, 85 (2020). https://doi.org/10.1140/epja/s10050-020-00095-y

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