Dimension-6 SMEFT Wilson Coefficients

• Dimension-6 SMEFT Wilson coefficients are coupling parameters multiplying gauge-invariant operators of dimension six, encoding new physics effects in low-energy observables.
• They are computed using various normalization conventions and power counting methods, with analyses often including both linear and quadratic contributions.
• These coefficients critically impact electroweak precision tests and global fits, and are evolved via renormalization-group equations in different operator bases.

Dimension-6 SMEFT Wilson coefficients are the couplings multiplying gauge-invariant operators of canonical dimension six in the Standard Model Effective Field Theory. In the conventions used across current phenomenology, the effective Lagrangian is written either as

$$\mathcal{L}=\mathcal{L}_{\rm SM}+\sum_i \frac{\hat C_i^{(6)}}{\Lambda^2}\,\mathcal{O}_i^{(6)}$$

or as

$L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$

with the two notations differing only by normalization conventions for the coefficients and the explicit appearance of $\Lambda^{-2}$ (Bellafronte et al., 18 May 2026, Biekötter et al., 10 Mar 2025). These coefficients encode the low-energy imprint of heavy or weakly coupled new physics, run with the renormalization scale, and enter observables through tree-level interference, loop corrections with a single dimension-6 insertion, and, in some analyses, quadratic $\mathcal{O}(1/\Lambda^4)$ terms (Collaboration, 3 Apr 2025).

1. Definition, normalization, and power counting

The modern literature uses several closely related normalizations for dimension-6 Wilson coefficients. In one common convention,

$C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$

so that the coefficient itself has mass dimension $-2$; this is the convention adopted in the POPxf implementation of NLO SMEFT predictions for Higgs and electroweak observables (Bellafronte et al., 18 May 2026). The wilson package likewise absorbs $\Lambda^{-2}$ into the numerical coefficient and expects inputs in units such as $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$ (Aebischer et al., 2018). By contrast, CoDEx returns matching coefficients with heavy masses appearing explicitly in denominators, without introducing a separate $\Lambda$ symbol in the output (Bakshi et al., 2018).

Power counting is usually organized simultaneously in operator dimension and loop order. In the NLO electroweak-precision treatment,

$$O_s^{\rm LO}=O_s^{(4,0)}+v_s^2\,O_s^{(6,0)},\qquad O_s^{\rm NLO}=O_s^{\rm LO}+\frac{1}{v_s^2}O_s^{(4,1)}+O_s^{(6,1)},$$

where $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$0 denotes tree-level SM–dimension-6 interference and $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$1 one-loop amplitudes with a single dimension-6 insertion (Biekötter et al., 10 Mar 2025). Many analyses keep only terms linear in the coefficients and neglect $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$2 and dimension-8 contributions; this is explicit in the NLO EWPO and POPxf calculations, and also in the $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$3-symmetric global fit (Biekötter et al., 10 Mar 2025, Bellafronte et al., 18 May 2026, Bartocci, 2024). Other fits nevertheless tabulate quadratic terms,

$L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$4

to quantify the impact of $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$5 effects in one-parameter interpretations (Collaboration, 3 Apr 2025).

2. Operator bases and flavor structure

The dominant reference basis is the Warsaw basis of Grzadkowski et al., used in the one-loop $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$6-decay calculation, the analytic NLO EWPO results, the POPxf NLO decay library, wilson, and CoDEx (Dawson et al., 2018, Biekötter et al., 10 Mar 2025, Bellafronte et al., 18 May 2026, Aebischer et al., 2018, Bakshi et al., 2018). The analytic EWPO calculation works with the 59 CP-even/odd dimension-6 operators of the Warsaw basis, grouped into eight classes, and retains completely general flavor indices (Biekötter et al., 10 Mar 2025). The POPxf library also uses the Warsaw basis, but organizes the observable dependence coefficient-by-coefficient in JSON polynomial files (Bellafronte et al., 18 May 2026).

Alternative bases remain important in restricted sectors. CoDEx can output Wilson coefficients in either the Warsaw or SILH basis, with renormalization-group evolution implemented only in Warsaw because the anomalous-dimension matrix is known there (Bakshi et al., 2018). A TGC-focused low-energy study instead uses the HISZ bosonic basis and retains only three CP-even bosonic operators,

$L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$7

to parametrize anomalous triple gauge couplings (Choudhury et al., 2022). The coexistence of Warsaw, SILH, and HISZ parameterizations suggests that numerical coefficient values are basis-dependent objects, even when the corresponding physical predictions are translated consistently.

Flavor assumptions strongly affect the counting of independent Wilson coefficients. The fully general SMEFT contains 2499 real dimension-6 coefficients for three generations in the Warsaw basis, while imposing exact $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$8 flavor symmetry together with CP conservation reduces this to 41 real coefficients (Bartocci, 2024). The 2025 analytic EWPO calculation imposes no flavor symmetry at all, keeps explicit flavor indices, takes CKM equal to the identity, and finds that only flavor-diagonal combinations enter EWPO at linear order (Biekötter et al., 10 Mar 2025). The CMS combined EFT interpretation instead adopts the topU3l flavor symmetry of SMEFTsim v3, which reduces the theory space to 129 CP-even operators, of which 64 are retained in the final fit (Collaboration, 3 Apr 2025).

3. Renormalization, running, and matching

Dimension-6 Wilson coefficients are renormalized in the $L=L^{(4)}+\sum_i C_i(\mu)\,Q_i(\mu),$9 scheme and obey the standard one-loop SMEFT RGE,

$\Lambda^{-2}$0

with $\Lambda^{-2}$1 taken from the one-loop anomalous-dimension matrix of the Warsaw basis computed by Jenkins, Manohar, Trott, Alonso et al., and related work (Dawson et al., 2018, Biekötter et al., 10 Mar 2025). In practical applications the coefficients are interpreted as running quantities evaluated at a definite electroweak scale, often $\Lambda^{-2}$2 (Dawson et al., 2018).

Matching between EFTs is equally central. The wilson package automates one-loop running in the complete dimension-6 SMEFT, tree-level matching onto WET at the electroweak scale, and QCD/QED running below $\Lambda^{-2}$3, all within WCxf conventions (Aebischer et al., 2018). The flavor-symmetric FCNC analysis computes the complete tree and one-loop matching of $\Lambda^{-2}$4-symmetric dimension-6 SMEFT onto WET operators relevant for down-type FCNC observables, while explicitly including SMEFT corrections to input observables (Hurth et al., 2019). CoDEx instead starts from a renormalizable UV Lagrangian with heavy spin-0, spin-$\Lambda^{-2}$5, or spin-1 fields, integrates them out at tree level and one loop, and outputs the induced dimension-6 SMEFT coefficients in Warsaw or SILH form (Bakshi et al., 2018).

Input-parameter schemes are not innocuous bookkeeping devices. The analytic EWPO calculation provides results in five electroweak input schemes,

$\Lambda^{-2}$6

precisely to expose higher-order scheme dependence (Biekötter et al., 10 Mar 2025). The POPxf NLO library implements $\Lambda^{-2}$7 and $\Lambda^{-2}$8 for $\Lambda^{-2}$9, $\mathcal{O}(1/\Lambda^4)$0, and EWPO predictions, and explicitly notes that the numerical impact of the choice can be sizable for operators such as $\mathcal{O}(1/\Lambda^4)$1 and $\mathcal{O}(1/\Lambda^4)$2 (Bellafronte et al., 18 May 2026).

A notable extension of the usual homogeneous SMEFT running arises in the presence of a light axion-like particle. In the ALP+SMEFT EFT, one-loop ALP exchange generates inhomogeneous source terms,

$\mathcal{O}(1/\Lambda^4)$3

so that non-zero dimension-6 SMEFT coefficients are induced even if the heavy-state matching contribution vanishes at $\mathcal{O}(1/\Lambda^4)$4 (Galda et al., 2021).

4. How the coefficients enter observables

The observable content of dimension-6 Wilson coefficients is now available well beyond tree level. A particularly explicit example is provided by the one-loop SMEFT treatment of $\mathcal{O}(1/\Lambda^4)$5-boson decays. Restricting to the Warsaw-basis subset

$\mathcal{O}(1/\Lambda^4)$6

the weak-mixing-angle shift is

$\mathcal{O}(1/\Lambda^4)$7

which in turn induces universal tree-level shifts

$\mathcal{O}(1/\Lambda^4)$8

The same coefficient maps into anomalous TGCs through

$\mathcal{O}(1/\Lambda^4)$9

while $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$0 controls

$C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$1

In that calculation, $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$2 enters $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$3 already at tree level, whereas $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$4, $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$5, and $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$6 contribute only at one loop within the chosen operator subset (Dawson et al., 2018).

The 2025 analytic EWPO calculation generalizes this logic to a fully flavor-general NLO treatment of $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$7 and $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$8 widths, asymmetries, ratios, and derived electroweak observables. At LO the $C_i \equiv \frac{\hat C_i^{(6)}}{\Lambda^2},$9 partial widths depend on oblique-type coefficients $-2$0 and $-2$1, plus vertex corrections $-2$2, $-2$3, $-2$4, $-2$5, and $-2$6; at NLO, four-fermion operators and dipoles enter through one-loop insertions (Biekötter et al., 10 Mar 2025). The NLO POPxf library extends the same degree of control to all 2- and 4-body Higgs decays, all $-2$7 and $-2$8 decays, electroweak precision observables, $-2$9 at $\Lambda^{-2}$0 GeV, and differential $\Lambda^{-2}$1 spectra in $\Lambda^{-2}$2 (Bellafronte et al., 18 May 2026).

The practical importance of these corrections is explicit. The one-loop $\Lambda^{-2}$3-decay study states that the SMEFT effects under discussion are “of order a few percent, of the same size as Standard Model electroweak corrections” (Dawson et al., 2018). The POPxf analysis likewise finds that NLO corrections are often at the level of a few percent of the LO SMEFT contribution, but can be larger for particular operator–observable combinations; scheme dependence is especially relevant for $\Lambda^{-2}$4 and $\Lambda^{-2}$5 in precision fits (Bellafronte et al., 18 May 2026).

5. Global fits and public infrastructures

Dimension-6 Wilson coefficients are increasingly constrained only in large correlated fits. Under $\Lambda^{-2}$6 flavor symmetry and CP conservation, a global analysis of 41 dimension-6 coefficients combines parity-violating experiments, EWPO, Higgs data, top interactions, flavor observables, dijet production, and lepton scattering (Bartocci, 2024). In that fit, NLO SMEFT contributions improve the bounds on $\Lambda^{-2}$7, $\Lambda^{-2}$8, and $\Lambda^{-2}$9 by about two orders of magnitude relative to LO, while the 10 coefficients entering EWPO already at tree level remain comparatively stable; all 41 coefficients are compatible with the SM within $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$0, and almost all satisfy

$1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$1

with $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$2 and $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$3 as the notable weaker directions (Bartocci, 2024).

A complementary large-scale fit is provided by CMS, which combines seven sets of Run-2 measurements probing Higgs boson, electroweak vector boson, top quark, and multi-jet production together with LEP/SLC electroweak precision observables (Collaboration, 3 Apr 2025). That analysis determines constraints on 64 individual Wilson coefficients and on 42 principal-component-like linear combinations, with the 42 directions varied simultaneously. The CMS likelihood mixes full experimental likelihoods and Gaussian simplified likelihoods, and its linear SMEFT parameterization is built directly from the matrices of coefficients $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$4 and $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$5 extracted from process-by-process simulations (Collaboration, 3 Apr 2025).

Several public infrastructures now support such fits. POPxf stores NLO SMEFT predictions as JSON polynomials with explicit metadata on basis, scale, and input scheme, and is designed for use in HEPfit, SMEFiT, and custom likelihood codes (Bellafronte et al., 18 May 2026). wilson provides automated running and matching between SMEFT and WET using the WCxf interchange format (Aebischer et al., 2018). CoDEx serves the opposite direction, namely matching from renormalizable UV models to Warsaw or SILH dimension-6 coefficients and then evolving them to the electroweak scale (Bakshi et al., 2018).

6. UV interpretation and theoretical constraints

Dimension-6 SMEFT Wilson coefficients need not arise exclusively from integrating out heavy states in a simple tree-level way. CoDEx exhibits explicit UV completions in which real singlet or triplet scalars, heavy fermions, and heavy vectors generate coefficients such as $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$6, $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$7, $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$8, and $1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}$9 at tree level or one loop, with the corresponding mass and coupling dependence displayed analytically (Bakshi et al., 2018). The ALP+SMEFT analysis shows that even if no heavy state has been integrated out, a light ALP with dimension-5 interactions can radiatively source dimension-6 SMEFT coefficients such as $\Lambda$0, $\Lambda$1, $\Lambda$2, $\Lambda$3, dipoles, and a wide set of four-fermion operators through one-loop running (Galda et al., 2021).

The sign and allowed size of dimension-6 coefficients are not universally fixed by positivity arguments. For four-fermion operators, dispersion relations lead to sum rules of the form

$\Lambda$4

so the relevant dispersive object is a difference of cross sections plus a possible contribution from infinity. Weakly coupled UV completions can generate either sign for the corresponding coefficients, and the paper explicitly demonstrates both possibilities (Azatov et al., 2021). In that sector there is therefore no simple positivity bound analogous to the familiar dimension-8 story.

The purely gluonic sector behaves differently. For the dimension-6 operators

$\Lambda$5

causality and unitarity imply that they can exist only in the presence of certain dimension-8 four-gluon operators, leading to inequalities such as

$\Lambda$6

together with mixed determinant-type bounds (Ghosh et al., 2022). This does not contradict the four-fermion analysis: it reflects the distinct structure of the forward amplitudes in the gluonic sector.

These results collectively indicate that dimension-6 SMEFT Wilson coefficients are not merely bookkeeping devices for contact interactions. They are renormalized, basis-dependent parameters whose interpretation depends on flavor assumptions, input schemes, loop order, and matching conditions; they are tied simultaneously to electroweak precision observables, Higgs and diboson measurements, top and jet spectra, low-energy flavor data, and ultraviolet consistency requirements (Dawson et al., 2018, Biekötter et al., 10 Mar 2025, Bellafronte et al., 18 May 2026).