Weak Effective Field Theory (WEFT)

• Weak Effective Field Theory (WEFT) is a low-energy framework that integrates out heavy SM particles to capture weak interactions and possible BSM effects.
• It employs a systematic operator expansion with matching and running procedures to connect high-scale physics with precise low-energy predictions.
• WEFT underpins practical analyses in neutrino, flavor, nuclear, and gravitational studies, thereby guiding experimental searches for new physics.

Weak Effective Field Theory (WEFT) is a broad set of low-energy effective field theory frameworks used to parameterize and analyze phenomena involving weak interactions, low-energy manifestations of Standard Model (SM) processes, and possible contributions from heavy new physics via higher-dimensional operators. WEFT generalizes the notion of effective field theory (EFT) below the electroweak scale, targeting the dynamics where heavy SM degrees of freedom (W, Z, Higgs, and the top quark) have been integrated out, and non-renormalizable interactions—arising from physics at higher scales—are systematically captured. These frameworks serve as the foundational language for analyses in flavor physics, neutrino phenomenology, low-energy nuclear dynamics, precision gravity, and BSM searches involving weak-scale new physics.

1. Framework Definition and Hierarchy of Scales

WEFT is constructed by systematically integrating out all heavy states with masses at or above the electroweak scale ($\sim 100$ GeV), matching the full theory onto an effective Lagrangian for the remaining light fields: the light SM fermions, photons, gluons, pions, and composite nuclear degrees of freedom as relevant. The effective Lagrangian admits a tower of operators organized by increasing mass-dimension,

$$\mathcal{L}_\text{WEFT} = \mathcal{L}_\text{SM-light} + \sum_{i} \frac{C^{(d)}_i}{\Lambda^{d-4}}\,Q^{(d)}_i,$$

where the $Q^{(d)}_i$ are local, gauge-invariant operators of dimension $d > 4$, $C^{(d)}_i$ are Wilson coefficients encoding new physics, and $\Lambda$ is the heavy scale. The SMEFT-to-WEFT matching is performed at the electroweak scale, followed by QCD and QED renormalization group running down to the relevant experimental scale (typically $2$–$5$ GeV for neutrino and hadronic observables). This structure enables a consistent incorporation of ultraviolet (UV) physics effects into the low-energy observables, maintaining gauge invariance and a systematic power counting in $E/\Lambda$.

In the context of weak interactions, the charged-current sector typically features effective four-fermion operators (e.g., Fermi theory for beta decay), while the neutral-current sector includes vector and axial-vector couplings, generalized to accommodate BSM interactions. For instance, the general charged-current interaction in WEFT reads (Kopp et al., 25 Sep 2025): $$\mathcal{L}_{\text{WEFT,CC}} \supset -\frac{2V_{jk}}{v^2} \left\{ [\mathbb{1}+\epsilon_L^{(jk)}]_{\alpha\beta}(\bar u^j\gamma^\mu P_L d^k)(\bar e_\alpha \gamma_\mu P_L \nu_\beta) + \cdots \right\} + \text{h.c.},$$ with additional terms for right-handed, scalar, pseudoscalar, and tensor currents. $\epsilon_X^{(jk)}$ are Wilson coefficients parameterizing new physics contributions, $v$ is the Higgs vacuum expectation value, and $V_{jk}$ are CKM matrix elements.

2. Construction and Power Counting

The construction of WEFT follows the philosophy of EFTs: low-energy observables are computed in a series expansion in $E/\Lambda$, with all operators of a given dimension respecting the unbroken gauge and global symmetries. The relevant operators can be derived through a “matching and running” procedure from the full theory (or SMEFT), integrating out heavy fields at each threshold.

Power counting is determined by the dimension of the operator and the ratio of momentum/energy to the cut-off scale. In electroweak processes, the expansion parameter is typically $G_F E^2\ll1$, with $G_F$ the Fermi constant and $E$ the energy scale of the process. In nuclear and hadronic physics, analogous chiral power-counting is employed, as in chiral effective field theory ($\chi$EFT), ordered by $(Q/\Lambda_\chi)^n$ with $Q$ a small external momentum and $\Lambda_\chi$ the chiral symmetry breaking scale.

For processes involving unstable particles (e.g., $W$ and $Z$ bosons, top quark), WEFTs exploit the hierarchy $\Gamma \ll M$ (width much less than mass), expanding amplitudes simultaneously in $\alpha$ and $\delta\equiv \Gamma/M$ (Beneke, 2015). This allows a systematic, gauge-invariant evaluation of resonant and non-resonant contributions.

3. Applications Across Physical Domains

The WEFT formalism underpins a wide range of precision computations and new physics searches:

• Neutrino Physics: WEFT provides the most general, low-energy parameterization of new physics effects in neutrino production (meson and lepton decays), oscillation (modifications to the MSW potential and PMNS matrix), and detection (detection operator structure). The general charged-current and neutral-current four-fermion operators can incorporate right-handed neutrinos and account for lepton-flavor and lepton-number violating processes. Analyses at COHERENT (Bresó-Pla et al., 2 May 2025) and DUNE (Kopp et al., 25 Sep 2025) operate in the $\nu$WEFT framework, which is essential for unified treatments of new physics across production, propagation, and detection.
• Hadronic and Nuclear Weak Interactions: In processes such as $pp$-fusion and beta decay, the nuclear Hamiltonian and weak current operators are constructed in $\chi$EFT up to high order (e.g., N$^3$LO), including EM and weak corrections, with low-energy constants (LECs) fit to few-body observables (Marcucci et al., 2013).
• Precision Gravity and Scattering: The worldline EFT and post-Minkowskian expansions—employed, for example, in gravitational wave physics and Planckian scattering—use WEFT power counting for ultra-relativistic and weak-field interactions (Kol, 2011, Bjerrum-Bohr et al., 2022). The generalized approach facilitates resummation of leading instantaneous transverse exchanges and addresses the computation of classical (and quantum) corrections to scattering angles and tidal Love numbers (Bhattacharyya et al., 4 Aug 2025).
• Electroweak and BSM Phenomenology: Low-energy observables in $B$ and $K$ physics, lepton flavor violation, and searches for new heavy mediators are connected to SMEFT and BSM scenarios via the WEFT operator basis and RG evolution. Bounds from oscillation experiments, low-energy processes, and LHC data are consistently compared (Isidori et al., 2023, Marzocca et al., 2020).

4. Matching, Running, and Operator Basis

Consistent WEFT analyses require precise matching from UV completions or SMEFT at high scales to the low-energy WEFT at the operator level. For example (Kopp et al., 25 Sep 2025), SMEFT Wilson coefficients ($c_i$) at the high scale ($\sim$TeV) are evolved down to the electroweak scale via RGEs and then matched onto the WEFT coefficients ($\epsilon_X$). Further running to hadronic scales incorporates QCD and QED corrections. The connected chain is: $\text{UV/SMEFT} \xrightarrow{\text{matching %%%%37%%%% running}} \text{WEFT @ EW scale} \xrightarrow{\text{QCD running}} \text{WEFT @ GeV}.$ The systematic matching also ensures that physical observables remain manifestly gauge invariant and that cross-experimental analyses (e.g., joint DUNE and LHC fits) are theoretically meaningful.

The operator basis of WEFT is designed to be non-redundant and to respect all relevant symmetries. It allows for complete description of possible BSM contributions—including those violating lepton flavor or parity—and readily incorporates right-handed neutrino and non-standard four-fermion structures as required (Kopp et al., 25 Sep 2025, Bresó-Pla et al., 2 May 2025).

5. Precision Observables and Experimental Probes

WEFT is indispensable for extracting new physics information from and setting precise theoretical predictions for numerous observables:

• Neutrino–nucleus scattering: CE$\nu$NS and quasi-elastic scattering (QES) require WEFT cross-section calculations encompassing the full flavor and chirality structure of new physics operators. Generalized effective “nuclear charges” encode production and detection modifications compactly (Bresó-Pla et al., 2 May 2025).
• Long-baseline oscillation experiments: Probabilities for flavor transitions admit modified production, oscillation, and detection pieces, consistently parameterized within WEFT (Kopp et al., 25 Sep 2025). Simulation toolkits (such as GLoBES-EFT) implement full WEFT predictions, including operator mixing via RGEs.
• Beta decay, muon decay, and nuclear matrix elements: Systematic inclusion of scalar, pseudoscalar, tensor, and right-handed currents captures the leading deviations from the SM (Marcucci et al., 2013).
• Gravitational interactions: Ultra-relativistic and tidal effects are treated using weak-field and worldline WEFT, with observables such as scattering angles, interaction durations, and Love numbers computed from operator coefficients and matching to UV metric solutions (Kol, 2011, Bhattacharyya et al., 4 Aug 2025).

6. Conceptual and Theoretical Advances

Research exploiting the WEFT formalism has yielded significant theoretical insights:

• Classical Versus Quantum Corrections: The formalism demonstrates dominant classical dynamics in shockwave Regge/Planckian scattering limit, with quantum gravitational corrections suppressed by the expansion parameter, supporting a form of “censorship” of quantum gravity (Kol, 2011).
• Positivity and Entropy Constraints: Information—theoretic and thermodynamic arguments (non-negativity of relative entropy) enforce positivity bounds on higher-dimensional operators, reflecting deep connections between analyticity, unitarity, and black hole physics (Weak Gravity Conjecture) (Cao et al., 2022).
• Operator Structure in Near-threshold States: EFT-based generalizations of weak-binding relations clarify the compositeness of hadronic resonances and provide a systematic error methodology in extracting internal state structure from low-energy data (Kamiya et al., 2016, Kamiya et al., 2017).
• Resonance Saturation and Nonlinear Realizations: Matching heavy resonance Lagrangians onto EWET/WEFT allows for direct experimental bounds on resonance masses and couplings in a generic, model-independent manner, crucial for constraining new electroweak physics (Rosell et al., 2020, Rosell et al., 2021).

7. Limitations and Outlook

While WEFT is extremely powerful for parameterizing and interpreting new-physics effects in weak-scale observables at energies below $\sim 100$ GeV, its validity is limited to processes where neglected higher-dimension operators are reliably suppressed and the external kinematics remains far from the integrated-out scales. For precision predictions, theoretical uncertainties from higher-order contributions, operator mixing, and low-energy constants (in chiral and nuclear EFTs) must be carefully controlled. In gravitational contexts, WEFT currently applies predominantly at leading power; higher-order corrections and dynamical finite-size effects require further development (Plestid, 13 May 2024, Bhattacharyya et al., 4 Aug 2025).

As experimental reach extends, the WEFT formalism will underpin the global, model-independent combination of data from neutrino oscillation experiments, flavor physics, astrophysical processes, and gravitational wave observations—providing the standard theoretical backbone for beyond Standard Model searches and precision determinations of SM parameters.