---
title: Dimension-6 SMEFT Wilson Coefficients
url: https://www.emergentmind.com/topics/dimension-6-smeft-wilson-coefficients
type: topic
---

# Dimension-6 SMEFT Wilson Coefficients

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 \(\Lambda^{-2}\) [2605.18679; 2503.07724]. 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 [2504.02958].

## 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 [2605.18679]. The `wilson` package likewise absorbs \(\Lambda^{-2}\) into the numerical coefficient and expects inputs in units such as \(1/\mathrm{TeV}^2 = 10^{-6}\,\mathrm{GeV}^{-2}\) [1804.05033]. By contrast, CoDEx returns matching coefficients with heavy masses appearing explicitly in denominators, without introducing a separate \(\Lambda\) symbol in the output [1808.04403].

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 \(O^{(6,0)}\) denotes tree-level SM–dimension-6 interference and \(O^{(6,1)}\) one-loop amplitudes with a single dimension-6 insertion [2503.07724]. Many analyses keep only terms linear in the coefficients and neglect \(\mathcal{O}(1/\Lambda^4)\) and dimension-8 contributions; this is explicit in the NLO EWPO and POPxf calculations, and also in the \(U(3)^5\)-symmetric global fit [2503.07724; 2605.18679; 2405.10101]. Other fits nevertheless tabulate quadratic terms,
\[
\sigma_{\alpha,\rm SMEFT}^{\,i}
=
\sigma_{\alpha,\rm SM}^{\,i}
\left(
1+\sum_j A_{\alpha,j}^{\,i}\frac{c_j}{\Lambda^2}
+\sum_{j,k} B_{\alpha,jk}^{\,i}\frac{c_j c_k}{\Lambda^4}
\right),
\]
to quantify the impact of \(\mathcal{O}(1/\Lambda^4)\) effects in one-parameter interpretations [2504.02958].

## 2. Operator bases and flavor structure

The dominant reference basis is the Warsaw basis of Grzadkowski et al., used in the one-loop \(Z\)-decay calculation, the analytic NLO EWPO results, the POPxf NLO decay library, `wilson`, and CoDEx [1808.05948; 2503.07724; 2605.18679; 1804.05033; 1808.04403]. 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 [2503.07724]. The POPxf library also uses the Warsaw basis, but organizes the observable dependence coefficient-by-coefficient in JSON polynomial files [2605.18679].

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 [1808.04403]. A TGC-focused low-energy study instead uses the HISZ bosonic basis and retains only three CP-even bosonic operators,
\[
\mathcal{O}_{WWW},\qquad \mathcal{O}_W,\qquad \mathcal{O}_B,
\]
to parametrize anomalous triple gauge couplings [2203.04673]. 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 \(U(3)^5\) flavor symmetry together with CP conservation reduces this to 41 real coefficients [2405.10101]. 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 [2503.07724]. 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 [2504.02958].

## 3. Renormalization, running, and matching

Dimension-6 Wilson coefficients are renormalized in the \(\overline{\rm MS}\) scheme and obey the standard one-loop SMEFT RGE,
\[
C_i(\mu)=C_{0,i}-\frac{1}{32\pi^2\hat\epsilon}\,\gamma_{ij}C_j,
\qquad
\mu\frac{d}{d\mu}C_i(\mu)=\frac{1}{16\pi^2}\gamma_{ij}C_j(\mu),
\]
with \(\gamma_{ij}\) taken from the one-loop anomalous-dimension matrix of the Warsaw basis computed by Jenkins, Manohar, Trott, Alonso et al., and related work [1808.05948; 2503.07724]. In practical applications the coefficients are interpreted as running quantities evaluated at a definite electroweak scale, often \(\mu=M_Z\) [1808.05948].

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 \(m_W\), all within WCxf conventions [1804.05033]. The flavor-symmetric FCNC analysis computes the complete tree and one-loop matching of \(U(3)^5\)-symmetric dimension-6 SMEFT onto WET operators relevant for down-type FCNC observables, while explicitly including SMEFT corrections to input observables [1903.00500]. CoDEx instead starts from a renormalizable UV Lagrangian with heavy spin-0, spin-\(1/2\), 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 [1808.04403].

Input-parameter schemes are not innocuous bookkeeping devices. The analytic EWPO calculation provides results in five electroweak input schemes,
\[
\{G_F,s_{\rm eff}^2,M_Z\},\quad
\{\alpha(M_Z),s_{\rm eff}^2,M_Z\},\quad
\{G_F,M_W,M_Z\},\quad
\{\alpha(M_Z),M_W,M_Z\},\quad
\{G_F,\alpha(M_Z),M_Z\},
\]
precisely to expose higher-order scheme dependence [2503.07724]. The POPxf NLO library implements \(\{\alpha(0),M_Z,G_F\}\) and \(\{M_W,M_Z,G_F\}\) for \(W\), \(Z\), and EWPO predictions, and explicitly notes that the numerical impact of the choice can be sizable for operators such as \(C_{\phi D}\) and \(C_{\phi WB}\) [2605.18679].

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,
\[
\frac{d}{d\ln\mu}C_i^{\rm SMEFT}
-
\gamma_{ji}^{\rm SMEFT} C_j^{\rm SMEFT}
=
\frac{S_i}{(4\pi f)^2},
\]
so that non-zero dimension-6 SMEFT coefficients are induced even if the heavy-state matching contribution vanishes at \(\Lambda=4\pi f\) [2105.01078].

## 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 \(Z\)-boson decays. Restricting to the Warsaw-basis subset
\[
\mathcal{O}_{HWB},\quad \mathcal{O}_{HW},\quad \mathcal{O}_{HB},\quad \mathcal{O}_W,
\]
the weak-mixing-angle shift is
\[
\delta s_W^2
=
-\frac{s_W c_W}{c_W^2-s_W^2}\,
\frac{v^2}{\Lambda^2} C_{HWB},
\]
which in turn induces universal tree-level shifts
\[
\delta g_{L,R}^{Zf}=Q_f\,\delta s_W^2.
\]
The same coefficient maps into anomalous TGCs through
\[
\delta g_1^Z=-\frac{\delta s_W^2}{c_W^2},\qquad
\delta \kappa^Z=-2\,\delta s_W^2,
\]
while \(C_W\) controls
\[
\lambda^V = 3\,\frac{v}{\Lambda^2}M_W\,C_W,\qquad V=Z,\gamma.
\]
In that calculation, \(C_{HWB}\) enters \(Z\to f\bar f\) already at tree level, whereas \(C_W\), \(C_{HW}\), and \(C_{HB}\) contribute only at one loop within the chosen operator subset [1808.05948].

The 2025 analytic EWPO calculation generalizes this logic to a fully flavor-general NLO treatment of \(W\) and \(Z\) widths, asymmetries, ratios, and derived electroweak observables. At LO the \(Z\) partial widths depend on oblique-type coefficients \(C_{HD}\) and \(C_{HWB}\), plus vertex corrections \(C_{Hl}^{(1,3)}\), \(C_{He}\), \(C_{Hq}^{(1,3)}\), \(C_{Hu}\), and \(C_{Hd}\); at NLO, four-fermion operators and dipoles enter through one-loop insertions [2503.07724]. The NLO POPxf library extends the same degree of control to all 2- and 4-body Higgs decays, all \(W\) and \(Z\) decays, electroweak precision observables, \(e^+e^-\to ZH\) at \(\sqrt{s}=240,365,500\) GeV, and differential \(d\Gamma/dM_{Z^*}\) spectra in \(H\to4\ell\) [2605.18679].

The practical importance of these corrections is explicit. The one-loop \(Z\)-decay study states that the SMEFT effects under discussion are “of order a few percent, of the same size as Standard Model electroweak corrections” [1808.05948]. 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 \(C_{\phi D}\) and \(C_{\phi WB}\) in precision fits [2605.18679].

## 5. Global fits and public infrastructures

Dimension-6 Wilson coefficients are increasingly constrained only in large correlated fits. Under \(U(3)^5\) 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 [2405.10101]. In that fit, NLO SMEFT contributions improve the bounds on \(C_{qd}^{(1)}\), \(C_{qu}^{(1)}\), and \(C_{ud}^{(1)}\) 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 \(2\sigma\), and almost all satisfy
\[
\left|\frac{C_i}{\Lambda^2}\right|<\frac{10}{\mathrm{TeV}^2}
\quad\text{at 95\% CL},
\]
with \(C_{dd}\) and \(C_{dd}'\) as the notable weaker directions [2405.10101].

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 [2504.02958]. 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 \(A_{\alpha,j}^i\) and \(B_{\alpha,jk}^i\) extracted from process-by-process simulations [2504.02958].

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 [2605.18679]. `wilson` provides automated running and matching between SMEFT and WET using the WCxf interchange format [1804.05033]. 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 [1808.04403].

## 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 \(C_H\), \(C_{HD}\), \(C_{HW}\), and \(C_W\) at tree level or one loop, with the corresponding mass and coupling dependence displayed analytically [1808.04403]. 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 \(C_{HG}\), \(C_{HWB}\), \(C_{Hl}^{(1,3)}\), \(C_{Hq}^{(1,3)}\), dipoles, and a wide set of four-fermion operators through one-loop running [2105.01078].

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
\[
\left.\frac{dA_{ab}}{ds}\right|_{s=0}
=
\int\frac{ds}{\pi s}\big[\sigma_{ab}(s)-\sigma_{a\bar b}(s)\big]
+
C_\infty,
\]
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 [2112.02302]. 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
\[
Q_{G^3}^{(1)}=f^{abc}G_\mu^{a\nu}G_\nu^{b\rho}G_\rho^{c\mu},
\qquad
Q_{G^3}^{(2)}=f^{abc}\widetilde G_\mu^{a\nu}G_\nu^{b\rho}G_\rho^{c\mu},
\]
causality and unitarity imply that they can exist only in the presence of certain dimension-8 four-gluon operators, leading to inequalities such as
\[
c_8^{(3)}>\frac{9}{4}c_6^2,\qquad
c_8^{(4)}>\frac{9}{4}c_6'^2,
\]
together with mixed determinant-type bounds [2211.01322]. 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 [1808.05948; 2503.07724; 2605.18679].

Source: https://www.emergentmind.com/topics/dimension-6-smeft-wilson-coefficients