Nuclear and Few-Body EFT

Use chiral EFT when pion exchange is resolved and pionless EFT when every external momentum is well below the pion mass. In either case, a result is controlled only after the degrees of freedom, promoted nonperturbative interactions, regulator and matching data, current operators, breakdown scale, and correlated uncertainties are declared. This chapter builds that complete two- or three-body calculation contract and separates regulator-dependent potentials and couplings from observable amplitudes and matrix elements.

Enter this chapter

The chapter uses natural units $\hbar=c=1$, the global (+---) metric, and Hermitian internal-symmetry generators. For nonrelativistic scattering, $M$ is the particle mass, $k$ the center-of-mass momentum, $a$ the scattering length, and $r_e$ the effective range. The regulator $\Lambda$ is not the breakdown scale $\Lambda_b$; controlled calculations use resolved momenta $Q\ll\Lambda_b$ and test an appropriate window in $\Lambda$.

The partial-wave amplitude convention is

$$T(k)=\frac{4\pi}{M} \frac{1}{k\cot\delta(k)-ik}, \qquad \operatorname{Im}T^{-1}=-\frac{Mk}{4\pi}.$$

This fixes the unitarity sign and the physical-sheet bound-state pole at $k=i/a$ for positive $a$ in the zero-range limit. State and multipole normalizations must still be declared for current matrix elements.

Choose a route

The six leaves appear below in their intended inventory order. A “method” page constructs a calculation; the final “reference” page supplies a comparison and evidence procedure.

Route	Use it when you need to	Result you should be able to produce
1. Chiral EFT with Pions and Nucleons	Resolve pion exchange and organize pion–nucleon operators	Nonlinear pion/nucleon fields, a chiral index, and a declared heavy-baryon or covariant matching scheme
2. Nuclear Forces in the Chiral Expansion	Construct and iterate two- or three-nucleon interactions	A force hierarchy with reducibility, promotions, regulator, fitted data, and cutoff/order tests separated
3. Pionless EFT and Shallow Two-Body Systems	Treat a large S-wave scattering length below the pion scale	A resummed, regulator-independent amplitude with pole sheet, effective-range corrections, and exact elastic unitarity
4. Three-Body Renormalization and Universality	Add a third particle in a resonant attractive channel	A running three-body counterterm, one declared datum, and properly delimited universal correlations
5. Electroweak Currents in Few-Body Systems	Predict a charge, transition, rate, or response	A consistently regulated one- plus many-body current between matched nuclear states, with normalization and covariance
6. Nuclear Predictions, Uncertainties, and Evidence Across Methods	Compare calculations or attach validated uncertainties	An observable identity, joint covariance, calibration/validation split, and dated-evidence boundary

Hard dependencies and suggested order

The hard dependency graph is smaller than a suggested reading sequence:

• Route 1 requires Chiral Lagrangians and Low-Energy QCD.
• Route 2 requires route 1 and Nuclear and Few-Body EFT Architecture.
• Route 3 requires that same architecture; Partial-Wave Unitarity is helpful but not hard.
• Route 4 requires route 3. Route 5 requires route 2.
• Route 6 requires route 5 and Validation and Theory Uncertainties; route 4 is helpful when the comparison includes three-body observables.

For a chiral force or current calculation, the suggested sequence is 1 → 2 → 5 → 6. For a large-scattering-length problem, use 3 → 4 and add 6 when reporting correlated predictions. A pionless electroweak application still benefits from route 2 before route 5 because the force–current consistency and potential-versus-observable distinction are shared even though its explicit pions and chiral force formulas are not.

Readiness diagnostic

This check is informal and unscored. If an answer is incomplete, use the repair before attempting the associated branch.

Can you do this now?	Ready answer	Repair
Identify $Q$, $\Lambda_b$, and the active fields	Name the external momenta, shallow poles or pion mass counted as low, the first omitted excitation, and whether pions are resolved	Review the scale-and-degree-of-freedom workflow in Nuclear and Few-Body EFT Architecture, then start at route 1 or 3 above
Distinguish a perturbative insertion from promoted iteration	Explain why a small energy denominator or $\lvert a\rvert\gg\Lambda_b^{-1}$ can make a nominal interaction order unity	Repair with the promoted-interaction analysis in Nuclear and Few-Body EFT Architecture, then use route 2 or 3
Fix the sign of the imaginary part and classify a pole sheet	Derive $\operatorname{Im}T^{-1}=-Mk/(4\pi)$ and distinguish bound, virtual, and resonance poles	Repair with Partial-Wave Unitarity, then use route 3
Separate calibration from validation and combine correlated inputs	Name which data fit each coefficient, reserve a held-out test, and retain shared off-diagonal covariance	Repair with Validation and Theory Uncertainties, then use route 6

Synthesis: one calculation contract

The two regimes share one logic but not one expansion. With explicit pions, a connected irreducible diagram has chiral index

$$\nu=-2+2A-2C+2L+\sum_v\Delta_v, \qquad \Delta_v=d_v+\frac{n_v}{2}-2.$$

For a connected irreducible two-nucleon potential this reduces to $\nu=2L+\sum_v\Delta_v$. Reducible propagation then enhances selected diagrams and motivates solving a scattering or bound-state equation; this iteration is an additional infrared statement, not contained in $\nu$ alone. Regulator dependence of the potential is allowed, while matched on-shell observables must be stable to the claimed order. The force hierarchy and its limitations are reviewed in Epelbaum, Hammer, and Meißner 2009, §§ 3–4.

With pions integrated out and $1/|a|\sim Q$, the leading contact is instead promoted so that

$$T_{\rm LO}(k)=\frac{4\pi}{M} \frac{1}{-1/a-ik}.$$

Natural range corrections remain perturbative unless a separate tuning is declared. In an attractive resonant three-body channel, this two-body input is insufficient: one three-body datum fixes the phase of a log-periodically running counterterm. The combined pionless and three-body construction is reviewed in Hammer, König, and van Kolck 2020, § II.

Whichever strong branch is used, an electroweak prediction has the form

$$\langle\Psi_f|J^\mu_{\rm 1b}+J^\mu_{\rm 2b}+\cdots|\Psi_i\rangle.$$

The states and current must share their EFT order, regulator, fitted coefficients, and unitary convention Krebs 2020, §§ 4–5. Finally, parameter, truncation, regulator, numerical, current, radiative, lattice, and experimental effects enter a joint covariance. Cutoff variation is a renormalization diagnostic; it becomes a probability statement only through an explicit, validated statistical model Furnstahl et al. 2015, §§ II–IV.

Informal synthesis review

These prompts are work-product checks, not registered assessment. A satisfactory answer states assumptions, performs the indicated check, and distinguishes fitted inputs from predictions.

1. Choose a theory. Given momenta, pion mass, shallow poles, and an intended observable, choose chiral or pionless fields and state $Q/\Lambda_b$. Criteria: identify every retained field, the first omitted mode, and a failure condition. Repair: return to routes 1 and 3 in the route table.
2. Renormalize an amplitude. Show which two-body interaction is iterated, what data fix it, and how an observable changes after the regulator is varied and the data are refit. Criteria: reproduce the elastic imaginary part, track dimensions, and separate residual cutoff artifacts from truncation probability. Repair: return to routes 2 and 3, then repeat the cutoff scan with the same fitted input.
3. Test the three-body sector. Decide from the ultraviolet kernel whether a leading three-body counterterm is required. Criteria: if it is, supply exactly one three-body datum, verify discrete cutoff scaling, and delimit finite-range or inelastic failure. Repair: return to route 4.
4. Build a current prediction. Specify wave functions, current rank, multipole/normalization convention, shared coefficients, and radiative factors. Criteria: test the vector identity or appropriate axial relation, common-regulator stability, final-state unitarity, and one held-out observable. Repair: return to routes 5 and 6.
5. Compare two methods. Construct the covariance of their difference. Criteria: align observable identities, include cross-covariance from shared data, state the evidence date and versions, and reject unsupported extrapolation. Repair: return to route 6 and EFT Inference, Power Counting, and Truncation.

Purpose-keyed exits

• Reproduce the chiral-counting, sharp-cutoff two-body-amplitude, and limit-cycle checks with declared inputs and tolerances.
• For generic matching, power counting, and EFT architecture before nuclear specialization, return to Effective Theories in Practice.
• For pion and symmetry-breaking inputs from QCD, continue within Confinement, Chiral Symmetry, and Low-Energy QCD.
• For finite-volume spectra and current matrix elements, continue to Finite-Volume Spectra and Amplitudes.
• For a research-grade correlated truncation analysis, continue to EFT Inference, Power Counting, and Truncation.

References

• Epelbaum, Evgeny, Hans-Werner Hammer, and Ulf-G. Meißner. “Modern Theory of Nuclear Forces.” Reviews of Modern Physics 81 (2009): 1773–1825. DOI.
• Furnstahl, R. J., N. Klco, D. R. Phillips, and S. Wesolowski. “Quantifying Truncation Errors in Effective Field Theory.” Physical Review C 92 (2015): 024005. DOI.
• Hammer, Hans-Werner, Sebastian König, and U. van Kolck. “Nuclear Effective Field Theory: Status and Perspectives.” Reviews of Modern Physics 92 (2020): 025004. DOI.
• Krebs, Hermann. “Nuclear Currents in Chiral Effective Field Theory.” European Physical Journal A 56 (2020): 234. DOI.