SMEFT and HEFT in Standard Model Observables

SMEFT and HEFT are different controlled descriptions of deviations from Standard Model observables. SMEFT expands in canonical dimension with the Higgs in a linear $SU(2)_L$ doublet; HEFT realizes electroweak symmetry nonlinearly and uses a chiral power counting in which the physical Higgs is a singlet. A valid prediction must declare the framework, basis, input scheme, perturbative order, and truncation before a coefficient is interpreted.

Required background. Use the parameter coordinates of the Standard Model Lagrangian, the framework distinction in SMEFT and HEFT, and the observable input translations of electroweak renormalization schemes.

Helpful background. Basis translation and reproducibility supplies the invariant round-trip test for field redefinitions and equation-of-motion relations.

Two expansions, two hypotheses

SMEFT assumes a mass gap to additional degrees of freedom and a linearly transforming Higgs doublet. Its Lagrangian is

$$\mathcal L_{\mathrm{SMEFT}} =\mathcal L_{\mathrm{SM}} +\sum_{d>4}\sum_i\frac{C_i^{(d)}(\mu)}{\Lambda^{d-4}} \mathcal O_i^{(d)}(\mu).$$

Canonical dimension supplies the primary expansion in $E/\Lambda$ and $v/\Lambda$, while loops supply a second expansion in couplings divided by $16\pi^2$. Redundant operators are removed using integration by parts, algebraic identities, and leading equations of motion; the Warsaw basis is one complete baryon-number-conserving dimension-six choice Grzadkowski et al. 2010, §§2–3.

HEFT instead packages the Goldstone modes in

$$U(x)=\exp\!\left(\frac{i\pi^I(x)\sigma^I}{v}\right), \qquad U\mapsto g_LUg_R^\dagger,$$

and treats the physical scalar $h$ as a singlet. A leading bosonic term is

$$\mathcal L_{\mathrm{HEFT}} \supset \frac{v^2}{4}F_C(h/v) \operatorname{Tr}\!\left[(D_\mu U)^\dagger D^\mu U\right] +\frac12(\partial h)^2-V(h),$$

with an arbitrary analytic function $F_C$ inside the EFT domain. The ordering is chiral: derivatives, weak couplings, fermion masses, and loops carry assigned weights rather than being sorted solely by canonical dimension. A systematic light-Higgs electroweak chiral construction is given in Buchalla, Catà, and Krause 2014, §§2–4.

SMEFT occupies a constrained region of HEFT parameter space when a smooth doublet coordinate exists and the coefficient functions satisfy the corresponding relations. HEFT is not simply “SMEFT with more operators,” and a HEFT coefficient cannot be inserted into a dimension-six SMEFT likelihood without a declared matching map.

Worked SMEFT deformation: the neutral mass relation

Consider the Warsaw-basis operator

$$\mathcal O_{HD} =\left(H^\dagger D_\mu H\right)^* \left(H^\dagger D^\mu H\right), \qquad \Delta\mathcal L=\frac{C_{HD}}{\Lambda^2}\mathcal O_{HD}.$$

In unitary gauge at $h=0$, $H=(0,v/\sqrt2)^T$, so

$$H^\dagger D_\mu H =\frac{iv^2}{4}(gW_\mu^3-g'B_\mu),$$

and therefore

$$\mathcal L_{\mathrm{mass}}^{\mathrm{neutral}} =\frac{v^2}{8}\left(1+\frac{C_{HD}v^2}{2\Lambda^2}\right) (gW_\mu^3-g'B_\mu)^2+O(\Lambda^{-4}).$$

At fixed Lagrangian coordinates $(g,g',v)$ this produces the fractional shift $\delta m_Z^2/m_Z^2=C_{HD}v^2/(2\Lambda^2)$ and leaves the charged mass term unchanged at tree level. That is not yet an experimental prediction. If $m_Z$ is itself an input, the same equation must be inverted, shifting derived parameters and every other observable. For input coordinates $p_a$ and observable $X$,

$$\delta X =\delta X_{\mathrm{direct}} +\sum_a\frac{\partial X_{\mathrm{SM}}}{\partial p_a}\,\delta p_a,$$

where $\delta p_a$ is found by holding the chosen input observables fixed. Omitting the second term makes the result input-scheme dependent for the wrong reason. The complete dimension-six prediction workflow, including parameter shifts, is reviewed in Brivio and Trott 2019, §§4–6.

Amplitude order and the quadratic-term question

For a dimension-six amplitude,

$$\mathcal A =\mathcal A_{\mathrm{SM}} +\frac1{\Lambda^2}\sum_iC_i\mathcal A_i^{(6)} +O(\Lambda^{-4}).$$

Squaring gives

$$\begin{aligned} |\mathcal A|^2={}&|\mathcal A_{\mathrm{SM}}|^2 +\frac{2}{\Lambda^2}\operatorname{Re}\!\sum_i C_i\mathcal A_{\mathrm{SM}}^*\mathcal A_i^{(6)}\\ &+\frac1{\Lambda^4}\sum_{ij}C_iC_j^* \mathcal A_i^{(6)}\mathcal A_j^{(6)*} +\frac{2}{\Lambda^4}\operatorname{Re} \sum_k C_k^{(8)}\mathcal A_{\mathrm{SM}}^*\mathcal A_k^{(8)}+\cdots. \end{aligned}$$

Keeping the dimension-six square while dropping dimension-eight interference is not a complete $O(\Lambda^{-4})$ calculation. It may still be reported as a sensitivity or positivity diagnostic if labeled explicitly and accompanied by a validity estimate. If the linear interference vanishes because of helicity, CP, or phase-space selection, the missing terms become especially important rather than automatically negligible.

Basis translation is an invariant round trip

Let two nonredundant bases be related at the retained order by $C'=R^{-T}C$ and matrix elements by $\mathcal A'=R\mathcal A$. Then

$$C'^T\mathcal A'=C^T\mathcal A.$$

This contraction—not the individual coefficient—is the invariant. A valid translation must also transform input shifts, anomalous dimensions, flavor assumptions, and any covariance or prior defined in coefficient space. The minimum round trip is:

1. transform the Lagrangian coefficients and all declared conventions to the target basis;
2. recompute or transform the observable at the same EFT and loop order;
3. apply the inverse map;
4. recover the original amplitude and input observables through the retained order.

A failure can signal a transposed anomalous-dimension convention, an omitted equation-of-motion term, or a prior/likelihood that was treated as basis invariant when it was not.

Framework and validity decision

Question	SMEFT answer	HEFT answer
Higgs transformation	component of a linear doublet	singlet accompanying nonlinear Goldstones
Leading ordering	canonical dimension plus loops	chiral dimension plus loops
Higgs-coupling relations	correlated by doublet analyticity	independent coefficient functions unless constrained
Natural regime	decoupling heavy physics with $E,v\ll\Lambda$	nonlinear electroweak dynamics with a controlled chiral scale
Invalid shortcut	arbitrary resummation of selected dimension-six terms	assigning SMEFT dimension counting to chiral operators

Before fitting, record the process energy, coefficient normalization, flavor hypothesis, RG scale, input scheme, loop order, retained linear/quadratic terms, and an event-level validity mask. A bin whose characteristic invariants approach the declared cutoff cannot be rescued by a small best-fit coefficient after the fact.

The typed handoff to a precision analysis is

$$\left\{\text{SMEFT or HEFT},\ \text{basis and flavor},\ C_i(\mu),\ \text{input shifts},\ \text{amplitude order},\ \text{validity mask},\ \text{theory covariance}\right\}.$$

It contains no current limits. Those require a released likelihood and the observable semantics developed in Precision Standard Model.

References

• Brivio, Ilaria, and Michael Trott. “The Standard Model as an Effective Field Theory.” Physics Reports 793 (2019): 1–98, §§4–6. DOI.
• Buchalla, Gerhard, Oscar Catà, and Claudius Krause. “Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO.” Nuclear Physics B 880 (2014): 552–573, §§2–4. DOI.
• Grzadkowski, Bohdan, Michał Iskrzyński, Mikołaj Misiak, and Janusz Rosiek. “Dimension-Six Terms in the Standard Model Lagrangian.” Journal of High Energy Physics 2010, no. 10 (2010): 085, §§2–3. DOI.