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The Standard Model effective operator expansion

State the SMEFT field and symmetry assumptions, count operator orders, and connect a normalized dimension-five interaction to an illustrative neutrino scale.

Before you begin
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The question for today

What you’ll understand

  • Identify the assumptions defining a linear-Higgs SMEFT.
  • Distinguish the dimension-five neutrino operator from dimension-six corrections.
  • Keep observable truncation and input-parameter shifts consistent.

The Standard Model effective field theory parameterizes heavy new interactions using the Standard Model fields and gauge symmetry. Its usefulness depends on those assumptions: new particles must not need to propagate explicitly at the energies being analyzed, and the Higgs is treated as a linearly transforming doublet. The expansion is not an assertion that every possible low-energy extension has this form. It provides a disciplined framework in which different experiments can constrain shared Wilson coefficients, provided bases, inputs, and retained orders are made compatible.

Field content and symmetry precede an operator list

Take the gauge group SU(3) color times SU(2) weak times U(1) hypercharge, with the ordinary Standard Model matter multiplets and a Higgs doublet H. Operators must be gauge and Lorentz invariant before electroweak symmetry breaking. Imposing baryon or lepton number conservation is an additional choice, not a consequence of gauge invariance alone. Flavor indices, Hermitian conjugation, and reality conditions must also be stated before quoting a count of independent interactions.

With no additional light fields, the leading dimension-five interaction has the schematic two-component form (L⋅H)(L⋅H)(L\cdot H)(L\cdot H), where weak indices and the two spinor indices are contracted to singlets. It violates lepton number by two units and generates Majorana neutrino masses after the Higgs acquires its expectation value. If lepton number is imposed as exact, this term is absent. At dimension six, many baryon-conserving and potentially baryon-violating structures occur; a complete basis removes algebraic, integration-by-parts, and equation-of-motion redundancy.

LSMEFT=LSM−c52Λ(L⋅H)(L⋅H)+h.c.+∑iCi(6)Λ2Qi(6)+⋯ ,mν=c5v22Λ\mathcal L_{\mathrm{SMEFT}}=\mathcal L_{\mathrm{SM}}-\frac{c_5}{2\Lambda}(L\cdot H)(L\cdot H)+\mathrm{h.c.}+\sum_i\frac{C_i^{(6)}}{\Lambda^2}Q_i^{(6)}+\cdots,\qquad m_\nu=\frac{c_5v^2}{2\Lambda}
The neutrino relation is for one illustrative flavor with H0=(v+h)/√2 and Weyl mass convention −mννν/2+h.c.; take positive c5 for the example. Fermions have dimension3/2 and H dimension1, so the displayed neutrino operator has dimension5. General flavor coefficients form a symmetric matrix.↔ Scroll wide equations horizontally.

A gauge-invariant expansion has several low scales

The heavy scale must be large compared with the energies and momentum transfers used in the analysis, and a conventional electroweak expansion also treats v over Lambda as small. Near a new narrow resonance, integrating out that state fails even if one inclusive average energy looks modest. If an additional light scalar, sterile neutrino, or other degree of freedom is physically relevant, the EFT field content must be enlarged. A nonlinear Higgs realization is a distinct organization with different operator relationships.

Dimension-six interactions can shift the quantities used to define Standard Model parameters, including measured electroweak inputs. Predictions require a specified input scheme and a consistent parameter translation before comparing direct vertex effects across observables. Coefficients depend on the basis and renormalization scale, while suitable physical combinations do not. Treating every experimental constraint as a measurement of one isolated coefficient can hide correlations, flat directions, or assumptions about all the coefficients set to zero.

δA6∼C6E2Λ2,∣A∣2∣Λ−4⊃∣δA6∣2+2Re⁡(ASM∗δA8)\delta\mathcal A_6\sim C_6\frac{E^2}{\Lambda^2},\qquad |\mathcal A|^2\big|_{\Lambda^{-4}}\supset|\delta\mathcal A_6|^2+2\operatorname{Re}(\mathcal A_{\mathrm{SM}}^*\delta\mathcal A_8)
The amplitude estimate applies to a specified dimension-six matrix element after factoring its couplings and kinematics. Additional v/Λ factors can enter other interactions. At order Λ−4, dimension-six squares do not by themselves comprise the complete observable correction.↔ Scroll wide equations horizontally.

Coefficients constrain combinations, not unique ultraviolet stories

For a chosen coefficient normalization, a measured neutrino mass can be converted to an illustrative scale only after assuming the coefficient size. A small coefficient could arise from weak couplings, loop factors, flavor structure, or a symmetry-breaking parameter. The same low-energy mass can therefore correspond to many heavy scales. Likewise, a pattern of dimension-six coefficients may disfavor a particular ultraviolet model without identifying a unique replacement.

When an analysis retains terms linear in dimension-six coefficients, its error assessment must address dimension-eight interference and dimension-six squares, along with missing loops and input uncertainties. In channels where Standard Model interference is suppressed, a quadratic contribution can be phenomenologically important, but that does not automatically justify ignoring every term of equal inverse-scale order. A transparent result states its assumptions, energy range, coefficient correlations, and truncation rather than presenting a fit as a model-independent discovery scale.

Work it through

A neutrino scale with its coefficient assumption exposed

Use the one-flavor convention above with c5=1, v=246 GeV, and an illustrative neutrino mass mν=0.050 eV. Infer Λ and repeat for c5=0.010 while keeping the mass fixed. Use 1 eV=10−9 GeV.

  1. The mass is 0.050×10−9=5.0×10−11 GeV0.050\times10^{-9}=5.0\times10^{-11}\,\mathrm{GeV}. The relation requires Λ=c5v2/(2mν)\Lambda=c_5v^2/(2m_\nu) in this explicit one-half convention.

  2. For c5=1, the result is 2462/(1.0×10−10)=6.0516×1014 GeV246^2/(1.0\times10^{-10})=6.0516\times10^{14}\,\mathrm{GeV}. The factor of two is fixed by the displayed operator and mass normalizations.

  3. For c5=0.010 the scale is a hundred times smaller, 6.0516×1012 GeV6.0516\times10^{12}\,\mathrm{GeV}. Both parameter choices yield the same illustrative mass, so the mass alone has not determined a unique heavy scale.

The result

The inferred scales are approximately 6.05×10¹⁴ GeV and 6.05×10¹² GeV for the two stated coefficient assumptions.

The living notebook

Explore the model

Interactive lab

Interactive lab · scattering

A squared weight is different from an interference term

For each fresh comparison, Reset and set BOTH Scattering angle θ=90° and Coupling g. Compare g=0.50 and g=1.00, predicting schematic weights 1/16 and 1, a sixteenfold change. The exact toy weight is g⁴(1+cos²θ); it contains no Standard Model reference amplitude or EFT interference. It is not a physical cross section, a SMEFT fit, or a particle trajectory.

A squared weight is different from an interference termA schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Current readings: Coupling factor g⁴: 0.24; Angular factor 1 + cos²θ: 1.329; Illustrative weight: 0.319. All parameters are adjustable with the labeled controls below.Incoming termsOutgoing termsθSchematic interaction term · no particle trajectories
Incoming termsOutgoing termsAngle θ
Coupling factor g⁴
0.24
Angular factor 1 + cos²θ
1.329
Illustrative weight
0.319
t = 0 s

A schematic term diagram with dimensionless weight g⁴(1 + cos²θ). Lines denote terms, not particle paths; this is not a process cross-section or loop calculation. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA squared contribution can scale differently from a linear interference correction; this toy weight cannot supply the missing amplitude information.

Pause & reason

Check your understanding

Under the ordinary linear-Higgs SMEFT assumptions, what does the dimension-five (L·H)(L·H) interaction imply?

Your turn

Try it yourself

In a declared toy SMEFT channel, let real ASM=1 and δA6=C6(E/Λ)² with C6=0.50, E=0.20 TeV, and Λ=1.0 TeV. Compute the rate through linear interference and the extra dimension-six square. Explain why adding that square alone is not generally a complete Λ−4 calculation.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First find the amplitude shift. Keep track of the inverse-scale order of each contribution after squaring, not only whether it is numerically small.

Compare with the solution

The amplitude shift is 0.50(0.20)2=0.0200.50(0.20)^2=0.020. The leading squared amplitude is one, and linear interference adds 0.040, so the linear-order result is 1.040. The dimension-six square adds 0.0202=0.00040.020^2=0.0004. Dimension-eight interference with ASM also enters at order Λ−4 and is not specified by these inputs. Thus 1.0404 is the square of the displayed partial amplitude but is not automatically a complete prediction at that higher EFT order. Additional assumptions could justify emphasizing it, but those assumptions would need to be stated and checked for this channel.

Keep the idea

Take it with you

  • SMEFT requires specified fields, gauge symmetry, and a controlled hierarchy above electroweak scales.
  • Operator normalization and coefficient assumptions determine any inferred neutrino scale.
  • Dimension-six fits need input-scheme, correlation, running, and truncation information.

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