Effective, Simplified, and Mediator Descriptions

Use an EFT when every relevant momentum transfer is well below the lightest omitted mass, a resolved-mediator description when a pole or threshold is kinematically accessible, and a larger renormalizable sector when gauge invariance, anomaly cancellation, mass generation, or longitudinal unitarity requires additional states. “Simplified” describes a deliberately reduced phenomenological model; it is neither automatically gauge complete nor an ultraviolet completion.

Required background. Consistency Checklist for Standard Model Extensions supplies the structural gates. Effective Field Theory as a Controlled Expansion supplies power counting. Physical Poles and Tree-Level Factorization supplies pole residues and factorization.

Helpful background. Forward-Limit Positivity Bounds adds ultraviolet consistency tests when its analyticity and subtraction hypotheses hold.

Four description layers

Layer	Retained information	Appropriate domain	Characteristic failure
Ultraviolet or larger renormalizable sector	all states needed for gauge symmetry, anomalies, masses, and interactions over a declared range	through the included thresholds, below any further cutoff	pretending renormalizability proves ultimate completion
Resolved mediator	propagating mediator, mass, width, currents, interference, and required completion states	near or above its production threshold while its own completion remains controlled	omitting the width, longitudinal sector, or correlated channels
Simplified model	a small phenomenological state/coupling set chosen for a search question	only where its gauge, flavor, unitarity, and width masks pass	treating independent couplings as consistent when symmetry relates them
EFT	local operators through a stated order with matched/running coefficients	all invariants small compared with omitted masses and truncation error acceptable	using the expansion near a pole or mixing operator orders inconsistently

The right layer is process dependent. A mediator can be resolved in one dataset and safely integrated out in another. Conversely, a nominally low center-of-mass energy does not ensure EFT control when a $t$-channel momentum, invariant mass, or boosted tail approaches the omitted scale.

Exact mediator-to-EFT expansion

Strip couplings and widths from a scalar propagator away from its pole:

$$\mathcal A_{\rm med}(s)=\frac{1}{M^2-s}.$$

For $|s|<M^2$,

$$\mathcal A_{\rm med}(s) =\frac{1}{M^2}\left(1+\frac{s}{M^2}+\frac{s^2}{M^4}+\cdots\right).$$

Keeping the first two terms gives

$$\mathcal A_{\rm EFT}^{(8)}(s)=\frac{1}{M^2}+\frac{s}{M^4}.$$

The residual relative to the exact mediator amplitude is exactly

$$\frac{|\mathcal A_{\rm med}-\mathcal A_{\rm EFT}^{(8)}|} {|\mathcal A_{\rm med}|} =\left(\frac{s}{M^2}\right)^2.$$

With $s=E^2$, its log–log slope versus $E/M$ is four. At $M=10$ and $s=25$, the residual is $1/16$. This fixture tests the denominator, retained order, and normalization and can be reproduced directly from the displayed formula.

Restoring couplings gives a leading Wilson coefficient $C^{(6)}\sim g_{\rm in}g_{\rm out}/M^2$. Matching fixes its sign, group factors, Lorentz structure, and scale. RG evolution and operator mixing then transport it to the observable scale. A contact scale $\Lambda$ cannot be identified with $M$ unless the coupling convention is also specified.

Pole, width, and interference

For a resolved unstable mediator, organize the amplitude around its complex pole,

$$\mathcal A(s)=\mathcal A_{\rm nonres}(s) +\frac{R}{s-s_p}, \qquad s_p=M_p^2-iM_p\Gamma_p.$$

The observable contains

$$|\mathcal A_{\rm SM}+\mathcal A_{\rm med}|^2 =|\mathcal A_{\rm SM}|^2+|\mathcal A_{\rm med}|^2 +2\operatorname{Re}(\mathcal A_{\rm SM}^*\mathcal A_{\rm med}).$$

Therefore a signal yield cannot generally be added incoherently to a Standard Model template. A narrow-width approximation requires $\Gamma_p/M_p\ll1$, slowly varying nonresonant amplitudes and response, and isolation from nearby thresholds or poles. When gauge cancellations link resonant and nonresonant diagrams, use a gauge-consistent pole or complex-mass prescription rather than inserting a width into a selected propagator Denner et al. 2005, §§2–3.

Large width can mean large couplings, many open channels, overlapping states, or a poor particle description. Each possibility changes matching and the observable differently. The ratio $\Gamma/M$ is thus a validity diagnostic, not a nuisance to be freely scanned independently of the couplings that generate it.

Gauge completion and longitudinal unitarity

A simplified interaction such as

$$X_\mu\left(g_q\bar q\gamma^\mu q+g_\chi\bar\chi\gamma^\mu\chi\right)$$

must answer four questions: Is the current invariant under the unbroken Standard Model group? Are the $X$ charges anomaly free? What generates $m_X$? What cancels high-energy longitudinal growth? If the answers require a hidden Higgs, spectator fermions, or electroweak-related couplings, those ingredients enter before the energy reaches their threshold or the partial-wave bound. Simplified spin-1 and pseudoscalar constructions and their common gauge-completion failures are reviewed in Kahlhoefer 2017, §§3.2–3.3.

The same logic applies to scalar mediators: electroweak gauge invariance can require Higgs mixing or an additional doublet, and large trilinear couplings can destabilize the potential. A phenomenological vertex is useful only with its domain and omitted consistency conditions visible.

Construct the validity mask before inference

For each parameter point and event/bin, define a Boolean or weighted mask from independent tests:

1. Resolution: all relevant $|s|$, $|t|$, $|u|$, invariant masses, and virtualities satisfy the retained hierarchy.
2. Truncation: the first omitted operator contribution is below a predeclared tolerance under the chosen power counting.
3. Pole/width: the EFT excludes a resolved pole; a mediator model includes every open width and an appropriate line shape.
4. Perturbativity/unitarity: running couplings and partial-wave eigenvalues remain controlled up to the largest retained scale.
5. Gauge/flavor completion: Ward identities, anomaly sums, mass generation, and required flavor relations pass.
6. No double counting: a mediator exchange and the local operator generated by that same exchange are not both added unless a subtraction/matching scheme is declared.
7. Observable support: the theoretical response and interpolation were validated over every retained bin.

Freeze this mask before fitting or generating coverage ensembles. Choosing it after seeing which region improves the likelihood changes the statistical procedure. EFT truncation prescriptions for collider tails must be part of the theory prediction, not an unreported post-fit edit Busoni et al. 2014, §§2–3.

Independent checks and failure modes

• Reproduce the exact $E^4/M^4$ relative residual away from the pole and show its breakdown as $s\to M^2$.
• Match one full-theory amplitude, including its sign and interference, at two kinematic points rather than only one total rate.
• Verify the optical/pole width from the same couplings used in production and decay.
• Test a longitudinal amplitude or Ward identity and every local/global anomaly before accepting a spin-1 simplified model.
• Vary the matching scale and operator basis consistently; physical predictions should change only beyond the retained order.

Generic matching remains with Matching, Decoupling, and Thresholds. Applying a frozen mask and released likelihood belongs to Search Validity and Reinterpretation for Portal Models; live exclusions and model comparisons belong to Effective Field Theory and Tests of the Standard Model.

References

• Busoni, Giorgio, Andrea De Simone, Enrico Morgante, and Antonio Riotto. “On the Validity of the Effective Field Theory for Dark Matter Searches at the LHC.” Journal of Cosmology and Astroparticle Physics 2014, no. 6 (2014): 060. DOI.
• Denner, Ansgar, Stefan Dittmaier, Michael Roth, and Lars H. Wieders. “Electroweak Corrections to Charged-Current $e^+e^-\to4f$ Processes: Technical Details and Further Results.” Nuclear Physics B 724 (2005): 247–294. DOI.
• Kahlhoefer, Felix. “Review of LHC Dark Matter Searches.” International Journal of Modern Physics A 32 (2017): 1730006. DOI.