Effective operators and new-physics searches

In four spacetime dimensions, a Lagrangian density has mass dimension four. If an operator $\mathcal O_i^{(d)}$ has dimension $d$, writing a dimensionless Wilson coefficient $C_i^{(d)}$ requires a factor $1/\Lambda^{d-4}$. Gauge invariance must hold under the chosen low-energy symmetry realization. For the Standard Model effective field theory with a linearly transforming Higgs doublet, use the full Standard Model gauge structure rather than adding an arbitrary electrically neutral interaction after selecting a vacuum.

At dimension five, the familiar lepton-number-violating operator built from two lepton doublets and two Higgs fields can generate Majorana neutrino masses after symmetry breaking. At dimension six, many operators alter scattering, decays, and precision relations. Equations of motion and integration by parts relate apparently different terms, so a complete independent basis is required before counting coefficients. The basis labels are bookkeeping choices; physical predictions agree after parameters and coefficients are transformed consistently.

For a process without a relevant dimension-five contribution, write the amplitude schematically as $\mathcal A_{\mathrm{SM}}+\mathcal A_6/\Lambda^2+\mathcal A_8/\Lambda^4+\cdots$. Squaring gives a leading interference term at order $1/\Lambda^2$. At order $1/\Lambda^4$, the squared dimension-six contribution appears together with interference between the Standard Model and dimension-eight amplitudes. Keeping only one of these terms is an additional assumption about power counting, selection rules, or a restricted model.

An interference term can vanish or be suppressed by helicity, CP, or another selection rule. That does not automatically authorize ignoring all higher-dimensional effects. A truncated linear prediction may even become negative in a bin for large coefficients, signaling use outside its reliable domain rather than a negative physical probability. State the retained terms, input-parameter convention, loop order, and estimate of missing orders before interpreting a fitted coefficient as a controlled deformation of the Standard Model.

A heavy mediator with coupling factors $g_1g_2$ can generate a contact coefficient of order $g_1g_2/M^2$, with signs and group factors set by the model. A bound on that combination does not give a model-independent lower bound on $M$ without assumptions about the couplings. The experimentally probed energies must also stay safely below the heavy scale for the local expansion to apply. Strong couplings or resonances can change which hierarchy controls the omitted terms.

Measurements usually constrain correlated combinations of operators. Setting every coefficient except one to zero can produce a much tighter interval than a simultaneous fit allowing cancellations or flat directions. Matching at the heavy scale and running to measurement scales can mix coefficients, so combining low-energy decays with collider data requires consistent evolution. A useful search result publishes likelihood information and validity assumptions, allowing multiple ultraviolet models to be tested without pretending that one contact-interaction plot identifies a unique microscopic explanation.