SM EFT Operators: Structure & Phenomenology

• SM EFT Operators are higher-dimensional terms added to the SM Lagrangian that encapsulate heavy new physics via suppression by the energy scale Λ.
• They are classified into types, such as four-scalar and scalar–fermion operators, that can generate significant one-loop corrections to the Higgs mass, addressing naturalness.
• Their effects are probed through precision electroweak tests and collider signatures like enhanced Higgs pair production, linking low-energy observables to UV physics.

The Standard Model Effective Field Theory (SM EFT) operator formalism provides a systematic, model-independent way to parameterize potential effects of heavy new physics on low-energy observables by augmenting the Standard Model Lagrangian with higher-dimensional local operators constructed from SM fields and their derivatives. These operators are suppressed by inverse powers of a large energy scale $\Lambda$, associated with the masses of heavy degrees of freedom that have been integrated out. The resulting expansion encodes the virtual effects of such particles and organizes new interactions by increasing operator dimension, SM symmetry structure, and field content.

1. Structure and Classification of SM EFT Operators

The effective Lagrangian in the SM EFT framework is expressed as a power series in $1/\Lambda$: $$\mathcal{L}_\mathrm{eff} = \mathcal{L}_\mathrm{SM} + \sum_{n=5}^{\infty}\frac{1}{\Lambda^{n-4}}\sum_i f_i^{(n)} \mathcal{O}_i^{(n)}$$ where each operator $\mathcal{O}_i^{(n)}$ must be Lorentz invariant, SM gauge invariant, and local with mass dimension $n>4$. The Wilson coefficients $f_i^{(n)}$ encapsulate information about the couplings and masses of the heavy UV states.

Within this vast operator space, only a very restricted subset is relevant for one-loop quadratic divergences in the Higgs sector—i.e., those that can address the “little hierarchy problem” by generating contributions to $m_h^2$ canceled against SM loops up to scales $\Lambda \gg \text{TeV}$ (Bar-Shalom, 2014).

The relevant classes are:

• Type I (Four-Scalar Operators):

Operators involving four Higgs doublets (with possibly derivatives), e.g.

$$\begin{aligned} \mathcal{O}_S^{(2k+4)} &= |\phi|^2 \Box^k |\phi|^2\ \mathcal{O}^{(2k+4)} &= (\phi^\dagger \tau_I \phi) D^{2k} (\phi^\dagger \tau_I \phi)\ \mathcal{O}_{\tilde{s}}^{(2k+4)} &= \frac{1}{4}(\phi^\dagger \tau_I \tilde{\phi}) D^{2k} (\tilde{\phi}^\dagger \tau_I \phi) \end{aligned}$$

Here $\phi$ denotes the SM Higgs doublet and $\tau_I$ are SU(2) generators.

• Type II (Fermion-Scalar Operators):

Operators that contain two scalar fields and two SM fermions (possibly with derivatives), constructed when heavy fermions couple to $\phi$ and a light SM fermion $\psi$:

$\mathcal{O}_{(\Psi-\psi)}^{(2k+4)} = |\phi|^2 \ [\bar{\psi} (i \slashed{D})^{2k-1} \psi]$

These are generated from tree-level exchange of heavy fermions $\Psi$.

These two classes exhaust the possibilities for operators that contribute at one loop to $O(\Lambda^2)$ corrections to $m_h^2$. Operators not tree-generated by heavy physics (e.g., loop-generated dimension-six four-fermion operators) are subleading in this respect.

2. Operator Matching, Loop Corrections, and EFT Naturalness

The principle of “EFT Naturalness” is the imposition that the extra one-loop corrections to $m_h^2$ from dimension-$n\geq5$ operators must cancel (or reduce) the quadratically-sensitive SM contributions: $$\delta m_h^2 = \delta m_h^2(\text{SM}) + \delta m_h^2(\text{EFT}) \lesssim m_h^2$$ where

$$\delta m_h^2(\text{SM}) = \frac{\Lambda^2}{16 \pi^2}[24 x_t^2 - 6(2x_W^2+x_Z^2+x_h^2)],\qquad x_i = \frac{m_i}{v}$$

and

$$\delta m_h^2(\text{EFT}) = -\frac{\Lambda^2}{16\pi^2} F_\text{eff}$$

with $F_\text{eff}$ a specific linear combination of operator coefficients: $$F_\text{eff} = \sum_{k=0}^\infty \frac{1}{k+1} \sum_\Phi f_\Phi^{(2k+4)} - \sum_{k=0}^\infty \frac{1}{k+2} \sum_X f_X^{(2k+6)} - \sum_{k=1}^\infty \frac{(-1)^k}{k+1} \sum_{(\Psi,\psi)} f_{(\Psi-\psi)}^{(2k+4)}$$ The fine-tuning measure

$$\Delta_h \equiv \frac{|\delta m_h^2|}{m_h^2} = \frac{\Lambda^2}{16\pi^2 m_h^2} |F_\text{eff} - 8.2|$$

quantifies the degree of cancellation required. For “EFT Naturalness,” one seeks $\Delta_h \sim O(1)$.

3. UV Origin: Heavy Field Classification and Operator Coefficient Structure

The higher-dimensional operators arise from integrating out heavy renormalizable extensions of the SM at the scale $M > \Lambda$. Their UV origins are:

• Heavy Scalars/Vectors (Type I):
  • Scalar singlets ($S\sim(1,1)_0$), triplets ($\vec{\Delta}\sim(1,3)_0$, $\tilde{\Delta}\sim(1,3)_1$)
  • Vector extensions also contribute (e.g., heavy triplet vectors)
  • Corresponding operators: $|\phi|^2\Box^k|\phi|^2$, $$(\phi^\dagger\tau_I\phi)D^{2k}(\phi^\dagger\tau_I\phi)$$
• Heavy Fermions (Type II):
  • Fields coupling via $u_\Psi\ \bar{\Psi}\phi\psi$ or $\bar{\Psi}\tilde{\phi}\psi$ (for $\Psi$ sharing SM quantum numbers with $\phi\psi$ or $\tilde\phi\psi$)
  • Tree integration out of $\Psi$ leads to $|\phi|^2 \ \bar{\psi}(i\slashed{D})^{2k-1}\psi$ operators

The Wilson coefficients for these operators are functions of the heavy mass $M$ and coupling $u_\Phi$ (scalars) or $y_\Psi$ (fermions): $$f_\Phi^{(2k+4)}(\Lambda) = |u_\Phi/M_\Phi|^2 (-\Lambda^2/M_\Phi^2)^k$$ These correlations restrict $f_i^{(n)}$ to a subspace in the EFT parameter space, determined by the UV spectrum and coupling structure.

4. Experimental Constraints and Collider Phenomenology

While these operators typically do not maximize their impact on low-energy precision Higgs observables or production/decay rates, several experimental probes are relevant:

• Electroweak Precision Observables: Contributions to the $\rho$-parameter from operators involving heavy triplets (scalar/vector) or $SU(2)$-current structures imply

$\Lambda \lesssim 10~\mathrm{TeV}$

as an upper bound from agreement with electroweak precision tests.

• Collider Signatures:
  • Enhanced Higgs pair production via $s$- or $t$-channel exchange of heavy states: $VV\rightarrow hh$ (possible rates enhanced relative to SM prediction)
  • Processes like $q\bar{q}$ or $\ell^+\ell^-\rightarrow hh$ and associated production via heavy-fermion $t$-channel exchange
  • For heavy singlet extensions, small Higgs-singlet mixing leads to suppressed LHC-visible signatures, but may become relevant at future colliders
• Perturbativity Constraints: Strong-coupling regimes for the heavy sector are limited by partial-wave unitarity, imposing further constraints on allowed $u_\Phi$ (or $y_\Psi$).

5. Implications for the Structure and Phenomenology of SMEFT Operators

Only a highly restricted subset of SMEFT operators, namely those generated at tree level by integrating out heavy physics and those capable of generating $O(\Lambda^2)$ corrections to $m_h^2$, are relevant for addressing the little hierarchy problem (Bar-Shalom, 2014). The key implications are as follows:

• Phenomenological Focus: Constraints on/evidence for these few operator structures directly test classes of UV models that can soften Higgs-sector naturalness.
• Correlated Coefficient Structures: Matching relates heavy spectrum and couplings to operator coefficients, meaning observed patterns would signal the nature of the UV completion.
• EFT Utility: The SMEFT framework enables a bottom-up link from low-energy observables (Higgs mass stability, deviations in double Higgs production, $\rho$-parameter shifts) to UV parameter regions (masses/couplings of heavy states).
• Operator Set Restriction: Broad classes of higher-dimensional operators—such as those only generated at loop-level, or not correlated with the Higgs sector as described—do not contribute to ameliorating the little hierarchy problem at one-loop.

By classifying the tree-level-generated operators, determining their UV origins, and relating their coefficients to underlying model parameters, the paper shows that only types I and II operators with precisely specified SM gauge and Lorentz structures admit one-loop quadratic corrections substantial enough to impact Higgs naturalness. The set of such operators is much smaller than the full SMEFT basis at a given mass dimension and is tightly constrained by the structure of UV models compatible with naturalness requirements.

6. Summary Table: Key Operator Classes and Their Physical Roles

Operator Class	Generic Structure	UV Origin	Observables Affected	Collider Signature
Type I: Four-Scalar	$|\phi|^2\,\Box^k\,|\phi|^2$	Heavy scalar/vector	Higgs mass correction, $\rho$-param	Higgs pair production, VBF
Type II: Scalar–Fermion	$|\phi|^2\,\bar{\psi}(i\slashed{D})^{2k-1}\psi$	Heavy fermion (Yukawa-linked)	Higgs mass correction	$q\bar{q}\to hh$, $pp\to hh\ell$

Trade-offs are set by experimental constraints (precision EW, LHC), model-building consistency (perturbativity, matching), and the need for correlated operator structures (operator coefficients are UV-not arbitrary). Thus, the EFT approach “softens” the Higgs naturalness problem by transparent criteria on SMEFT operator structure and UV origin, providing a focused direction for both model-building and precision experimental investigation.