Consistent Extensions and Portals

A portal is useful only after the extension on both sides of it is consistent. Start by fixing fields, the faithful gauge group, anomalies, mass generation, and the vacuum; choose a scalar, extended-Higgs, vector, neutral-fermion, or axionlike communication mechanism; then select an EFT, resolved mediator, or larger renormalizable description from the actual momentum transfers. Widths, interference, a prefit validity mask, and versioned likelihood provenance are part of the prediction—not optional details added after a search result is seen.

Enter this chapter

The chapter uses the site’s (+---) metric, natural units, Hermitian generators, and $Q=T_3+Y$. All anomaly sums use left-handed Weyl fields, so a right-handed fermion is represented by its left-handed conjugate. Gauge kinetic terms are made canonical before mass matrices are diagonalized, and the same field transformations act on currents.

For an unstable mediator, the pole convention is

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

An EFT hierarchy refers to every relevant invariant transfer, not only the collider or beam energy. A conclusion about data additionally identifies the observable, likelihood, covariance, response, validity mask, evidence version, and cutoff date. This chapter provides durable constructions and synthetic checks; it contains no current exclusion, ranking, or detector-performance claim.

Choose a route

The nine leaves appear below exactly once in their manifest order.

Route	Use it when you need to	Result you should be able to produce
1. Consistency Checklist for Standard Model Extensions	Decide whether a proposed extension is sufficiently specified to calculate with	The earliest failed gate among representations, anomalies, masses, vacuum, unitarity, flavor, decoupling, observable, and evidence
2. Consistent New Matter and Gauge Sectors	Add fermions, scalars, or a gauge factor	A faithful representation table with allowed masses/Yukawas, exact anomaly sums, positive kinetic terms, symmetry breaking, and decoupling checks
3. Higgs-Singlet Scalar Portals	Couple a gauge singlet through $H^\dagger H$	A complete singlet potential, selected vacuum, scalar masses/mixing, boundedness, widths, and a declared heavy limit
4. Extended Higgs Sectors: Alignment, Custodial Symmetry, and Decoupling	Add a second doublet or another electroweak multiplet	Physical scalar parameters with basis invariance, alignment, custodial/flavor conditions, unitarity, and decoupling separated
5. Vector Portals and Kinetic Mixing	Couple an extra Abelian vector	A canonical kinetic/mass system with induced currents, anomaly-safe charges, mass generation, widths, interference, and the correct switch-off limit
6. Fermion and Neutrino Portals	Connect hidden singlets through neutral-fermion mixing	An exact Takagi mass spectrum, seesaw residual, lepton-number assignment, light-block nonunitarity, and lifetime-domain calculation
7. Axionlike and Pseudoscalar Portals	Organize a shift-symmetric pseudoscalar EFT	A basis translation among derivative, fermionic, and topological couplings with anomaly normalization, amplitude invariance, and QCD-axion distinction
8. Effective, Simplified, and Mediator Descriptions	Decide which degrees of freedom must be resolved	Matched EFT and mediator amplitudes with truncation, pole/width, interference, unitarity, gauge-completion, and double-counting masks
9. Search Validity and Reinterpretation for Portal Models	Apply a released search or likelihood to a portal	A reproducible theory-response package, prefit validity mask, covariance/coverage checks, evidence provenance, and properly bounded negative conclusion

Hard dependencies and suggested order

The hard graph and a useful reading order are not the same:

• Route 1 requires Gauge-Anomaly Cancellation and Quantum Consistency.
• Route 2 requires route 1 and The Global Form of the Standard Model Gauge Group.
• Route 3 requires route 1. Route 4 requires route 1 and Higgs Self-Interactions and the Scalar Potential.
• Route 5 requires route 2 and Gauge-Boson Masses and Electroweak Mixing.
• Route 6 requires route 1 and Neutrino Mass Mechanisms. Route 7 requires route 1 and Strong CP and the Axion Interface.
• Route 8 requires route 1, Effective Field Theory as a Controlled Expansion, and Physical Poles and Tree-Level Factorization.
• Route 9 requires route 8 and Collider Measurements, Fiducial Predictions, and Likelihood Provenance.

The suggested spine is 1 → 2, then one mechanism branch 3–7, followed by 8 → 9 if an observable or search interpretation is needed. A singlet-scalar or neutral-fermion construction can branch directly from route 1; a vector needs the matter/gauge checks in route 2 first. Read all of routes 3–7 only for comparison or a model that genuinely contains several mechanisms.

Readiness diagnostic

This diagnostic is informal and unscored. Use the repair before the associated branch if the ready answer is missing.

Can you do this now?	Ready answer	Direct repair
Build a complete field table	Give spin, representation under the faithful group, all Abelian charges, multiplicities, and mass source	Review The Global Form of the Standard Model Gauge Group, then use routes 1–2
Evaluate gauge consistency	Convert every fermion to a left-handed Weyl field and include spectator dimensions in local anomaly sums; name the separate global test	Repair with Gauge-Anomaly Cancellation and Quantum Consistency, then use routes 1–2
Test a scalar vacuum	Distinguish boundedness, stationarity, positive physical Hessian, global-minimum comparison, and perturbative unitarity	Repair with Higgs Self-Interactions and the Scalar Potential, then use route 3 or 4
Diagonalize mixed kinetic and mass terms	Prove the kinetic matrix is positive, canonicalize it, diagonalize masses second, and rotate currents with both transformations	Repair with Gauge-Boson Masses and Electroweak Mixing, then use route 5
Handle a symmetric neutral-fermion mass	Produce nonnegative Takagi masses and compare an exact answer with its small-mixing residual	Repair with Neutrino Mass Mechanisms, then use route 6
Decide EFT versus a mediator	List every kinematic invariant, locate poles/thresholds, state the retained order, estimate the omitted power, and test widths/unitarity	Repair with Effective Field Theory as a Controlled Expansion and Physical Poles and Tree-Level Factorization, then use route 8
Reuse a search statistically	Identify the exact likelihood/covariance, response, nuisance semantics, overlap, software version, and a mask frozen before fitting	Repair with Collider Measurements, Fiducial Predictions, and Likelihood Provenance, then use route 9

Synthesis: one consistency-first chain

A finite extension can be written schematically as

$$\mathcal L=\mathcal L_{\rm SM} +\mathcal L_{\rm new\ sector} +\mathcal L_{\rm portal}.$$

The split is organizational, not physical: field redefinitions can move mixing among kinetic terms, masses, and currents. The full Lagrangian and its amplitudes are invariant data. The lowest-dimension representative interactions illustrate the mechanism choices:

Mechanism	Representative gauge-invariant interaction	First nonnegotiable checks
Real singlet scalar	$-(\lambda_{hs}/2)H^\dagger H\,S^2$	complete potential, boundedness, vacuum, scalar mixing, invisible/exotic width
Extra Higgs multiplet	$V(H,\Phi)$ plus symmetry-qualified Yukawas	representation and vacuum, basis invariants, custodial/flavor safety, coupled-channel unitarity
Abelian vector	$-(\epsilon/2)B_{\mu\nu}X^{\mu\nu}$	kinetic positivity, current/anomaly consistency, mass generation, longitudinal behavior
Neutral fermion	$-\bar L\widetilde HYN_R+\text{h.c.}$	lepton number, Takagi masses, mixing expansion, matching, total width
ALP	$(a/f_a)c_i\alpha_iF_i\widetilde F_i/(8\pi)$	periodicity, anomaly/global-form normalization, basis translation, cutoff and explicit breaking

Each row still begins with the same field/global-form/anomaly review. Local triangle cancellation does not prove the quotient or global-anomaly conditions Bilal 2008, §§3–4, 7. Likewise, scalar boundedness is not vacuum selection, and a positive mass matrix is not a coupled-channel unitarity test.

The description layer is fixed by resolution. Away from a pole,

$$\frac{g_{\rm in}g_{\rm out}}{M^2-s} =\frac{g_{\rm in}g_{\rm out}}{M^2} \left(1+\frac{s}{M^2}+\cdots\right),$$

while near it the complex pole and all interfering amplitudes are resolved. This transition must retain matching signs, operator order, running, width, and the states required by gauge symmetry. A simplified model occupies only the domain where those omitted ingredients are demonstrably irrelevant; it is not promoted to a complete theory by fitting data.

Finally, the model prediction is folded into a pole, pseudo-observable, or fiducial response and a released likelihood. The reproducibility record contains dataset and DOI/version/checksum, observable definition, nuisance covariance, model/width convention, software identity, overlap, frozen validity mask, corrections, evidence cutoff, and supersession. Reinterpretation standards emphasize that reusable statistical models and response information determine what conclusions a release can support Abdallah et al. 2020, §§2–5.

Informal synthesis review

These prompts are work-product checks, not registered assessment. A satisfactory response states conventions, shows a reproducible calculation, and stops at the first failed domain.

1. Define the extension. Construct a field/interaction table for one new sector. Criteria: faithful representations, all allowed renormalizable terms, mass sources, exact local anomaly sums, a named global test, and remnant symmetries. Repair: routes 1–2.
2. Select and solve a portal. Choose one mechanism and diagonalize its kinetic/mass system. Criteria: positivity, physical eigenvalues, rotated currents, switch-off limit, and one independent trace/determinant or amplitude check. Repair: route 3, 4, 5, 6, or 7 according to the fields.
3. Test the vacuum and high-energy domain. Criteria: boundedness, stationary/global comparison, physical Hessian, RG/threshold range, and every relevant $J=0$ partial-wave eigenvalue. Repair: routes 1, 3, and 4.
4. Choose a description. Compare an exact mediator amplitude with its local expansion. Criteria: reproduce the first omitted power, locate the pole, derive the width from the same couplings, retain Standard Model interference, and freeze a validity mask. Repair: return to route 8 and recompute the exact-to-local residual.
5. Construct a reinterpretation package. Criteria: exact observable and dataset identity, released likelihood/covariance, nuisance and overlap semantics, response/code checksums, prefit mask, interpolation closure, coverage test, corrections, and evidence cutoff. Repair: return to route 9 and validate the package on the synthetic covariance and coverage fixtures.
6. Diagnose an overclaim. A valid portal point is described as experimentally favored because it improves a private recast. Criteria: separate internal consistency, fit quality, calibration, look-elsewhere scope, detector provenance, and official evidence; state the narrower supportable conclusion. Repair: routes 1 and 9.

Purpose-keyed exits

• Reproduce exact group factors, anomaly sums, common-center phases, and the vectorlike fixture with exact arithmetic.
• Reproduce vector/scalar mixing, Takagi masses, and the mediator-to-EFT residual with declared conventions.
• Check covariance, nuisance profiling, coverage, and prefit masks on the synthetic fixture before using released inputs.
• For general matching and running, return to Matching, Decoupling, and Thresholds; for global-form or anomaly formalism, return to Gauge Structure, Global Form, and Observables.
• For light-neutrino mass and mixing rather than hidden-sector communication, continue to Neutrino and Lepton Physics.
• For a structured particle/nuclear sequence, use QFT for particle and nuclear physics; its capstone remains unauthored, so the checks here are informal.
• For dated searches, exclusions, or model comparison, continue to Effective Field Theory and Tests of the Standard Model.
• To choose another chapter, return to Gauge Theories and the Standard Model.

References

• Abdallah, Waleed, et al. “Reinterpretation of LHC Results for New Physics: Status and Recommendations after Run 2.” SciPost Physics 9 (2020): 022. DOI.
• Bilal, Adel. “Lectures on Anomalies.” arXiv:0802.0634 [hep-th] (2008). arXiv.