Dimension-6 EFT Operators in SMEFT

• Dimension-6 EFT operators are higher-dimensional, gauge-invariant terms that parametrically extend the Standard Model by incorporating new heavy physics effects.
• They are constructed using minimal bases like the Warsaw basis and modify Higgs, electroweak, and flavor observables through distinct Lorentz structures and energy-growing terms.
• Collider analyses leverage these operators in VBF/VBS processes to extract limits on Wilson coefficients and probe new physics by studying altered kinematics and anomalous couplings.

Dimension-6 operators in effective field theory (EFT) provide a systematic, gauge-invariant framework for parametrizing extensions to the Standard Model (SM) at energies below a new physics scale $\Lambda$, where new heavy degrees of freedom have been integrated out. These operators, of canonical mass dimension 6, encapsulate the dominant, unsuppressed effects of new physics in observables accessible at the LHC and other current collider experiments. They play a central role in precision Higgs, electroweak, and flavor physics, and are directly relevant for interpreting constraints from vector boson fusion (VBF) and vector boson scattering (VBS) at high energies.

1. Theoretical Framework and Construction

Dimension-6 operators in EFT arise in the low-energy expansion of the full theory Lagrangian, where the heavy fields are absent and their effects appear as higher-dimensional, nonrenormalizable terms organized by operator dimension. The general form of the SMEFT (SM Effective Field Theory) Lagrangian is

$$\mathcal{L}_{\mathrm{SMEFT}} = \mathcal{L}_{\mathrm{SM}} + \sum_{i} \frac{c_{i}}{\Lambda^{2}} O_{i}^{(6)} + \cdots$$

where $O_i^{(6)}$ are a complete set of gauge- and Lorentz-invariant dimension-6 operators built from SM fields, $c_i$ are Wilson coefficients, and $\Lambda$ is the heavy new physics scale. Redundancies among operators are eliminated using equations of motion and field redefinitions, leading to minimal bases such as the "Warsaw basis" for SMEFT or the "Warsaw-up" for scenarios including right-handed neutrinos.

The construction covers all bilinear, trilinear, and quadrilinear combinations of SM field strengths and covariant derivatives consistent with the $SU(3)_c \times SU(2)_L \times U(1)_Y$ gauge symmetry and baryon/lepton number conservation (if imposed).

Dimension-6 operator classes include:

• Higgs-gauge operators: Modify Higgs and vector boson couplings and tensor structures.
• Triple/Quartic gauge operators: Induce anomalous TGCs and QGCs, e.g., $O_{WWW}$, $O_W$, $O_B$.
• Four-fermion operators: Modify contact terms, e.g., neutral- and charged-current processes.
• Fermionic dipole and Yukawa corrections: Affect flavor and CP-violating processes.

2. Explicit Operator Examples and Their Roles

Operators of direct relevance to electroweak and Higgs-sector phenomenology in VBF/VBS processes include:

Bosonic operators (Warsaw basis, see (Rauch, 2016)): $$\begin{aligned} O_{WWW} &= \mathrm{Tr}\left[ \widehat{W}_{\mu\nu} \widehat{W}^{\nu\rho} \widehat{W}_{\rho}^{~\mu} \right], \ O_{W} &= (D_{\mu}\Phi)^{\dagger} \widehat{W}^{\mu\nu} (D_{\nu}\Phi), \ O_{B} &= (D_{\mu}\Phi)^{\dagger} \widehat{B}^{\mu\nu} (D_{\nu}\Phi), \ O_{\phi W} &= \Phi^{\dagger} \Phi\, W^a_{\mu\nu} W^{a, \mu\nu}, \ O_{\phi B} &= \Phi^{\dagger} \Phi\, B_{\mu\nu} B^{\mu\nu}, \ \end{aligned}$$ where $\Phi$ is the SM Higgs doublet, $W^a_{\mu\nu}$ and $B_{\mu\nu}$ are the $SU(2)$ and $U(1)$ field strengths, and $D_\mu$ is the covariant derivative.

Fermion current-Higgs operators relevant for high-energy VBF (Han et al., 2023): $$\begin{aligned} O_Q^{(3)} &= (\bar Q\,\sigma^a\gamma^\mu Q)\,(i\Phi^\dagger\sigma^a\overleftrightarrow{D}_\mu\Phi), \ O_Q &= (\bar Q\,\gamma^\mu Q)\,(i\Phi^\dagger\overleftrightarrow{D}_\mu\Phi), \ O^u_R &= (\bar u_R\,\gamma^\mu u_R)\,(i\Phi^\dagger\overleftrightarrow{D}_\mu\Phi), \ O^d_R &= (\bar d_R\,\gamma^\mu d_R)\,(i\Phi^\dagger\overleftrightarrow{D}_\mu\Phi), \end{aligned}$$ which modify the $qqV$ currents, lead to modified VBF/VBS kinematics with amplitudes growing as $\hat{s}/\Lambda^2$.

These operators induce new Lorentz structures and/or momentum dependences at the $hVV$ or $VVVV$ vertices (see (Djouadi et al., 2013)) and are tightly constrained by precision and high-energy LHC data.

3. Phenomenological Impact in VBF and VBS

Dimension-6 operators are particularly impactful in high-momentum-transfer regions of VBF and VBS. Modifications at the $hVV$ or $VVVV$ vertices alter the kinematic distributions of the associated jets and bosons—most notably the rapidity separation $\Delta y_{jj}$, dijet invariant mass $m_{jj}$, and transverse momenta $p_T^j$—and can also change the efficiency of selection cuts designed to isolate the VBF topology (Djouadi et al., 2013, Han et al., 2023).

For example, in $qq \to qqh$ (VBF) or $qq \to qqVV$ (VBS), the presence of higher-dimensional $qqVH$, $qqVVV$, or $hVV$ operators leads to:

• Enhancement of the tails of $m_{jj}$ and $p_{T,j}$ distributions for constructive interference, or suppression depending on signs of Wilson coefficients.
• Altered selection acceptances due to shifted rapidity and $p_T$ spectra.
• Distinctive energy-growing deviations: interference terms scale as $E^2/\Lambda^2$, squared BSM terms as $E^4/\Lambda^4$.

These kinematic modifications allow high-scale new physics to be probed even when $\Lambda$ is significantly above the collider energy.

4. Operator-Induced Anomalous Couplings and Their Relation to Observables

Dimension-6 operators provide a gauge-invariant origin for traditional anomalous coupling parametrizations encountered in phenomenological Lagrangians:

• The $O_{WWW}$ operator generates the anomalous trilinear gauge coupling $\lambda_V$.
• $O_W$ and $O_B$ modify $\Delta\kappa_V$ and $\Delta g_1^Z$ for triple and quartic gauge vertices.

Contact terms such as $O_{Q}^{(3)}$ and $O_{Q}$ also appear in effective analyses of VBF $h \to b\bar{b}$, where they dominate the high-$p_T$ phase space, and are essential to global SMEFT fits incorporating LHC data from multiple production and decay modes (Han et al., 2023).

The mapping of Wilson coefficients $c_i$ to experimental observables is performed through the computation of interference and squared terms in the relevant kinematic bins, allowing bounds on $\Lambda$ and individual or combinations of $c_i$ to be extracted.

5. Experimental Constraints and Analysis Strategies

High-energy VBF studies, especially in $h \to b\bar{b}$, $h \to \gamma\gamma$, or leptonic diboson decay channels, are highly sensitive to dimension-6 operators due to the growth of BSM contributions with energy (Han et al., 2023, Djouadi et al., 2013). Experimental strategies include:

• Lifting the upper cut on tagging-jet $p_T$ to probe large $|t|$ (momentum transfer).
• Binning observables such as $p_T^h$, $m_{jj}$, or $p_{T, j_1}$ to maximize sensitivity to energy-growing terms.
• Employing profile-likelihood or binned counting analyses to extract limits on individual Wilson coefficients or new physics scales, accounting for both statistical and systematic uncertainties.

For instance, in (Han et al., 2023), the one-operator-at-a-time analysis yields 95% C.L. bounds on the new physics scale in the $0.5$–$1.8$ TeV range, depending on the operator, using HL-LHC projections.

6. Operator Interplay, EFT Validity, and Unitarity

The predictive power of dimension-6 EFT is tied to the assumption of a mass gap and a convergent expansion. As BSM contributions grow rapidly with energy, amplitudes may violate unitarity at partonic energies $E \sim \mathcal{O}(\Lambda)$ unless new physics is introduced to restore consistency (Rauch, 2016, Djouadi et al., 2013). Profiles in observables must therefore be restricted to regions where $E \lesssim \Lambda$ and kinematic distributions are dominated by interference rather than quadratic BSM terms.

A related issue is the mixing of operators under renormalization and the running of Wilson coefficients from the new physics scale down to collider energies, which must be systematically included in precision fits for robust bounds.

7. Context and Outlook in Collider Physics

Dimension-6 operators serve as the central focus of global fits to LHC and future collider data, providing a model-independent parametrization of new physics effects in Higgs, electroweak, and flavor observables. Their impact in VBF/VBS processes is especially pronounced due to the clean experimental signatures and characteristic energy growth. The synergy between VBF high-scale probes, precision Higgs coupling measurements, and complementary channels enables a comprehensive exploration of the TeV-scale landscape.

Across multiple analyses (Han et al., 2023, Djouadi et al., 2013, Rauch, 2016), dimension-6 EFT operators underpin the quantitative interpretation of deviations in jet kinematics, diboson rates, and angular observables, enabling the formulation of robust constraints on BSM theories in the absence of direct new particle discoveries.