Two- and Three-Nucleon Contacts in Nuclear EFT

• Two- and three-nucleon contact interactions are short-range operators in EFT that encapsulate low-energy nuclear dynamics.
• They are systematically organized by power counting and symmetry constraints, reducing operator redundancy through Fierz identities.
• Including three-nucleon contacts is crucial for renormalization, stabilizing few-body observables and accurately predicting binding energies.

Two- and three-nucleon contact interactions are the short-range components in effective field theory (EFT) descriptions of low-energy nuclear systems, systematically organized in powers of momenta and constrained by quantum chromodynamics symmetries. They perform central roles in both pionful (chiral) and pionless EFTs, determining renormalizability, spectrum stability, and fit quality for a broad range of observables from two-nucleon scattering to nuclear binding energies and electroweak processes.

1. Operator Structure and Power Counting

Contact interactions, built from nucleon fields $N(x)$, their gradients, and spin/isospin matrices, are ordered by canonical dimension.

Two-Nucleon Contacts

• Leading Order (LO, $Q^0$):

$$\mathcal{L}_{\text{2N}}^{(0)} = -\frac{1}{2}C_S\, (N^\dagger N)^2 - \frac{1}{2}C_T\, (N^\dagger\vec{\sigma} N)\cdot(N^\dagger\vec{\sigma} N)$$

Only two S-wave operators survive, corresponding to spin singlet ($S=0$) and triplet ($S=1$). This minimal set persists in pionless EFT (Kievsky et al., 2016, Girlanda et al., 2011).

• NLO ($Q^2$):

Seven independent operators, quadratic in nucleon momenta (k and q),

$$V_{2N}^{(2)}(q,k) = C_1 q^2 + C_2 k^2 + (C_3 q^2 + C_4 k^2)(\vec{\sigma}_1\cdot\vec{\sigma}_2) + i C_5 (\vec{\sigma}_1+\vec{\sigma}_2)\cdot(q\times k) + C_6 (\vec{\sigma}_1\cdot q)(\vec{\sigma}_2\cdot q) + C_7 (\vec{\sigma}_1\cdot k)(\vec{\sigma}_2\cdot k)$$

No additional two-nucleon contacts at $Q^3$; fifteen further operators enter at N$^3$LO ($Q^4$) (Holt et al., 2011, Bovermann et al., 29 Nov 2024).

Three-Nucleon Contacts

• Leading Contact ($Q^0$ in pionless, N$^2$LO in chiral EFT):

Local S-wave operator, e.g.

$V_{3N}^{\text{CT}} = D_0 \, [N^\dagger N]^3$

or, in regulated form, $W_0 \exp(-r_{ij}^2/r_0^2)\exp(-r_{ik}^2/r_0^2)$ (Kievsky et al., 2016, Schiavilla et al., 2021).

• Subleading Contacts:

After imposing all symmetries and Fierz identities, the minimal basis at N$^4$LO in chiral EFT (N$^2$LO in pionless) consists of 10–13 operators with two derivatives, encoding tensor and spin-orbit structures (Girlanda et al., 2012, Girlanda et al., 2011, Girlanda et al., 2020). The explicit coordinate-space forms include combinations of derivatives, spin, and isospin operators acting among three nucleons.

2. Symmetry Constraints, Fierz Identities, and Unitary Ambiguity

All contact terms respect Galilean/Poincaré invariance, parity, and time-reversal. Systematic reduction via Fierz rearrangements and commutation with the free boost operator yields:

• For three-nucleon contacts: out of an initial $\mathcal{O}(10^2)$ possible two-derivative terms, only 10–13 are independent (Girlanda et al., 2012, Girlanda et al., 2011).
• Poincaré covariance precludes explicit total system momentum at relevant orders, but at N$^3$LO in 2N, $D_{16},D_{17}$ terms with total pair momentum $P$ arise and can be traded for genuine 3N contacts via unitary transformation (Girlanda et al., 2020, Filandri et al., 17 Apr 2024). This redundancy allows spin–orbit and tensor three-body operators to absorb off-shell NN contact ambiguities.

3. Renormalization, Efimov Physics, and Necessity of the 3N Contact

Efimov-Thomas collapse and RG analyses demonstrate that:

• Solely two-nucleon LO contacts are insufficient to renormalize three-body observables. The triton binding energy, for instance, manifests strong cutoff sensitivity and large variation as regulation is removed (Kievsky et al., 2016, Rotureau et al., 2011).
• Inclusion of the three-body contact at LO stabilizes the RG flow, fixes the physical triton binding, and dramatically improves predictions for $^3$He, $^4$He, and other $A=3,4$ observables (Kievsky et al., 2016, Schiavilla et al., 2021, Bovermann et al., 29 Nov 2024).

Empirical values of the singlet ($a_0 \sim-24$ fm) and triplet ($a_1 \sim +5.4$ fm) scattering lengths, both much larger than the interaction range, create near-unitarity conditions and necessitate the three-nucleon counterterm for consistency (Kievsky et al., 2016).

4. Fitting Low-Energy Constants and Regulator Choices

LECs in the two- and three-nucleon sectors are fixed by a hierarchy of observables:

• Two-nucleon sector: np/pp phase shifts, scattering lengths, and deuteron binding energy at LO/NLO/N$^3$LO (with $\chi^2$/datum $\lesssim$ 1.7 for N$^3$LO fits up to 25 MeV) (Schiavilla et al., 2021, Bovermann et al., 29 Nov 2024).
• Three-nucleon sector: The three-body contact is universally fit to triton binding, with subleading contacts (c_D, $E_i$) constrained by $\beta$-decay rates, $^3$He binding, nd scattering lengths, and analyzing powers (Baroni et al., 2018, Bovermann et al., 29 Nov 2024, Witała et al., 2021, Schiavilla et al., 2021).

Contact terms are regulated by Gaussian or nonlocal momentum-space cutoffs, typical $\Lambda=$ 320-550 MeV; in coordinate space, $R_0 = 0.8-2.5$ fm. Regulator artifacts can affect short-range operator equivalence at finite cutoff, but predictions for light nuclei are robust under reasonable cutoff variation (Schiavilla et al., 2021, Bovermann et al., 29 Nov 2024).

5. Three-Nucleon Contacts: Subleading Structures and Phenomenological Impact

The two-derivative subleading 3N operator basis (10–13 terms) includes central, tensor, spin-orbit, and projective operators, essential for:

• Achieving full renormalization and RG invariance in $A=3$ systems at N$^2$LO/N$^3$LO in both pionless and chiral EFT (Girlanda et al., 2012, Girlanda et al., 2011).
• Absorbing ambiguities from off-shell 2N contacts (Girlanda et al., 2020, Filandri et al., 17 Apr 2024).
• Describing polarization observables and solving longstanding puzzles (e.g., proton–deuteron $A_y$) only when promoted from N$^4$LO to lower order (Kievsky et al., 2016, Filandri et al., 17 Apr 2024).

Table: Minimal subleading three-nucleon contact operator basis (Girlanda et al., 2011, Girlanda et al., 2012)

Operator type	Structurally independent terms	Generic symbol(s)
Central (no spin)	1–4	$[N^\dagger N]^3,\dots$
Tensor	2–4	$S_{ij},\dots$
Spin–Orbit	2–4	$\mathbf{L}_{ij}\cdot \mathbf{S}_{ij},\dots$
Isospin	1–3	$\boldsymbol\tau_i\cdot \boldsymbol\tau_j,\dots$

Full expressions, including all spatial derivatives and isospin structures, are detailed in (Girlanda et al., 2011, Girlanda et al., 2012).

6. Practical Applications: Lattice, Monte Carlo, and Nuclear Structure Calculations

Contact-interaction Hamiltonians are crucial in various computational frameworks:

• Few-body and lattice EFT: Non-perturbative lattice EFT simulations fix 2N and 3N LECs to NN and triton data; smearing regularizations and improved discretizations deliver $A=3$ energies, radii, and weak decay rates at the percent level (Bovermann et al., 29 Nov 2024).
• AFDMC/GFMC methods: Variational Quantum Monte Carlo methods faithfully incorporate local/regulator-modulated contact terms, producing accurate $A=2-4$ binding energies and densities (Lynn et al., 2017, Baroni et al., 2018, Schiavilla et al., 2021).
• Open-shell nuclei and shell model: Pure contact ("pionless") Hamiltonians yield systematic underprediction of excitation gaps and wrong level orderings in $p$- and $sd$-shell nuclei, demonstrating the necessity of pion-exchange (tensor and spin–orbit) components for realistic shell evolution (Lyu et al., 22 Nov 2025).

7. Outlook and Theoretical Considerations

• Inclusion of three-nucleon contacts at LO is compulsory for RG-invariant predictions and improved convergence for light nuclei.
• Subleading short-range operators (tensor, spin–orbit) must be promoted by up to two orders in the chiral expansion; their fit parameters absorb redundancies originating from off-shell 2N contact structures (Girlanda et al., 2020, Kievsky et al., 2016, Girlanda et al., 2012).
• A unified description from few-body ($A=2-4$) to medium- and heavy-mass systems hinges on accurately fitted contact LECs, regulator consistency, and proper operator basis choices (Hüther et al., 2019).

A plausible implication is that future EFT fits will require systematic treatment of all promoted three-nucleon contact terms, possible large-scale lattice or Monte Carlo refits as higher-precision experimental and theoretical constraints emerge, and rigorous exploration of operator equivalence classes as dictated by symmetry and unitary ambiguity.