Minimally Extended Standard Model

• The Minimally Extended Standard Model is a family of models that add only the essential fields—such as right-handed neutrinos, singlet scalars, or an extra U(1) gauge factor—to the Standard Model.
• These models target specific phenomenological challenges like neutrino mass generation, anomaly cancellation, and controlled radiative symmetry breaking through minimal additions.
• Different MESM variants serve as benchmark frameworks for studying Dirac, Majorana, and pseudo-Dirac neutrino regimes, as well as for making precise collider and cosmological predictions.

Searching arXiv for the cited works and closely related usages of “minimally extended Standard Model.” The expression “Minimally Extended Standard Model” denotes a family of economical beyond-the-Standard-Model constructions rather than a unique theory. Across the literature, the common principle is structural economy: the Standard Model gauge group is often left unchanged or enlarged by a single factor, the new field content is restricted to the smallest set needed for a targeted purpose, and the added interactions are typically only Yukawa, Majorana, portal, or one extra gauge interaction. In the neutrino literature, this frequently means the Standard Model plus right-handed neutrinos, with either exact lepton-number conservation or completely general Majorana masses; in gauge extensions, it can mean one extra $U(1)$ with anomaly-canceling singlets; in conformal constructions, it means one or two singlet scalars and no explicit mass terms (Perez-Gonzalez et al., 2023).

1. Definitional scope and recurring notion of minimality

A concise way to organize the usages is to distinguish the problem being solved from the minimal ingredients introduced. Taken across the cited works, “minimal” most often means one or more of the following: the gauge group is unchanged or enlarged only by a single Abelian factor; exactly three singlet neutrinos are added, one per active family; one singlet scalar is added when spontaneous breaking of a new symmetry or a seesaw scale is required; and no extra gauge bosons, scalar multiplets, SUSY partners, or higher-dimensional operators are invoked unless they are integral to the construction (Teixeira, 2016).

Context	Minimal ingredients	Primary aim
Dirac neutrino extension	SM + three right-handed neutrinos, conserved lepton number	Accommodate oscillation data
Majorana/pseudo-Dirac neutrino extension	SM + three right-chiral singlets with Majorana masses	Interpolate between Dirac and Majorana limits
Minimal $B-L$ extension	SM + $U(1)_{B-L}$ + three $\nu_R$ + one singlet scalar	Anomaly cancellation and type-I seesaw
Minimal $U(1)_R$ extension	SM + $U(1)_R$ + three $\nu_R$ + one singlet scalar	Chiral gauge extension with seesaw
Conformal/scale-invariant extension	SM + one or two singlet scalars, no mass terms	Radiative symmetry breaking
$SU(3)_C\times SU(N)_L\times U(1)_X$ MESMs	Minimal enlargement of electroweak group	Anomaly-based explanation of three generations

This variety is not merely terminological. It reflects a methodological pattern: minimality is defined relative to a phenomenological target. In one paper the minimal target is neutrino oscillations with no observable charged lepton flavour violation; in another it is a continuous Dirac–Majorana interpolation; elsewhere it is anomaly-free gauging of $B-L$, radiative electroweak symmetry breaking, dark matter, or generation counting. A plausible implication is that the phrase has high local precision within each subfield but low global uniqueness across subfields.

2. Minimal neutrino-sector extensions

One influential usage is the Standard Model minimally extended to accommodate neutrino oscillation data, defined as the Standard Model plus three gauge-singlet right-handed neutrinos while preserving total lepton number, so that neutrinos are purely Dirac particles (Teixeira, 2016). In that case the neutrino Yukawa sector is

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$

and after electroweak symmetry breaking

$B-L$0

There are no Majorana mass terms of the form $B-L$1, no Weinberg operator, and no new scalar multiplets. In this minimal Dirac framework, neutrino oscillations are described by the usual $B-L$2, and the absolute neutrino mass scale remains free.

The phenomenological importance of this definition is sharpened by charged lepton flavour violation. With only light Dirac neutrinos and $B-L$3 loops, processes such as $B-L$4 exist in principle but are suppressed to $B-L$5, far below experimental reach (Teixeira, 2016). Consequently, any observation of cLFV would imply physics beyond this minimal Dirac extension. In that sense, the Dirac MESM functions as a null benchmark.

A different but closely related neutrino MESM is the Standard Model plus three right-handed singlet neutrinos with completely general Majorana masses (Perez-Gonzalez et al., 2023). The neutrino-sector Lagrangian is

$B-L$6

with full $B-L$7 mass matrix

$B-L$8

In the simplifying ansatz used there, $B-L$9 is diagonal and the $U(1)_{B-L}$0 mixing matrix factorizes into three independent $U(1)_{B-L}$1 active–sterile subsystems. Each block has eigenvalues

$U(1)_{B-L}$2

This formulation continuously interpolates between three regimes:

• Seesaw (Majorana) limit: $U(1)_{B-L}$3, with $U(1)_{B-L}$4, $U(1)_{B-L}$5, and $U(1)_{B-L}$6.
• Dirac limit: $U(1)_{B-L}$7, with $U(1)_{B-L}$8 and $U(1)_{B-L}$9.
• Pseudo-Dirac limit: $\nu_R$0, with $\nu_R$1 and $\nu_R$2.

This neutrino MESM is minimal in field content but maximal in neutrino-sector generality. It is technically identical to type-I seesaw field content, yet it does not assume a seesaw hierarchy a priori.

3. Minimal gauge extensions and anomaly structure

A second major class of MESMs enlarges the gauge sector by a single Abelian factor while keeping the rest of the Standard Model intact. The minimal $\nu_R$3 extension is the canonical example: the gauge group becomes

$\nu_R$4

and anomaly cancellation requires three right-handed neutrinos together with one complex singlet scalar $\nu_R$5 or $\nu_R$6 of $\nu_R$7 (Basso, 2011). The Yukawa sector contains

$\nu_R$8

so that $\nu_R$9 generates Majorana masses dynamically and realizes a renormalisable type-I seesaw. In the “pure $U(1)_R$0 limit,” $U(1)_R$1, there is no tree-level $U(1)_R$2 mixing, and

$U(1)_R$3

The model is described as triply minimal: one extra gauge factor, one singlet fermion per generation, and one singlet scalar (Basso, 2011).

A closely related conformal version gauges the same $U(1)_R$4 but imposes classical conformal invariance, so that the scalar potential contains only quartics,

$U(1)_R$5

and $U(1)_R$6 breaking is induced radiatively by the Coleman–Weinberg mechanism (0902.4050). In that setup,

$U(1)_R$7

and the same scalar $U(1)_R$8 both breaks $U(1)_R$9 and provides the seesaw scale.

The minimal $U(1)_R$0 construction instead gauges a chiral right-handed symmetry,

$U(1)_R$1

under which only right-handed fermions carry nonzero charge (Nomura et al., 2017). Here too, three right-handed neutrinos are required for anomaly cancellation and neutrino masses. The Standard Model Higgs itself carries $U(1)_R$2 charge, and a singlet $U(1)_R$3 with charge $U(1)_R$4 breaks the symmetry. The neutral gauge boson mass matrix in the $U(1)_R$5 basis is

$U(1)_R$6

and the precision $U(1)_R$7-mass measurement implies

$U(1)_R$8

That bound is central to the model’s phenomenology: it forces the $U(1)_R$9 breaking scale to $\nu_R$0 or above.

Other gauge-generalized MESMs broaden the idea of minimality further. The “super-weak” extension introduces a single $\nu_R$1, three right-handed neutrinos, and one complex scalar singlet $\nu_R$2, with neutrino masses from a type-I seesaw and a light sterile neutrino dark matter candidate (Trocsanyi, 2023). The flipped-$\nu_R$3 model uses

$\nu_R$4

and realizes a minimal type-I seesaw with only two right-handed neutrinos coupling to the active sector, while the third serves as a $\nu_R$5-odd Majorana dark matter state (Nam, 2020). The $\nu_R$6 construction is called minimal because it adds only one extra $\nu_R$7 gauge boson beyond the Standard Model, though anomaly cancellation then requires one exotic $\nu_R$8-dimensional fermion multiplet in addition to the Standard-Model-like one (Herrera et al., 2017).

4. Conformal and scale-invariant minimal extensions

A distinct lineage of MESMs imposes classical scale invariance or classical conformal invariance. In these models, “minimal” refers not primarily to neutrino masses or anomaly cancellation but to the smallest scalar extension capable of generating electroweak symmetry breaking radiatively and remaining perturbatively viable at high scales (Boyle et al., 2011).

The Minimal Dimensionless Standard Model is defined as the minimal renormalizable extension of the Standard Model with purely dimensionless couplings, successful electroweak symmetry breaking via the Coleman–Weinberg mechanism, and a see-saw mechanism for neutrino mass (Boyle et al., 2011). Three variants are identified: MDSM$\nu_R$9, MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$0, and MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$1. MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$2 adds one real singlet scalar, MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$3 adds one complex singlet scalar, and MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$4 adds one complex singlet scalar together with $SU(3)_C\times SU(N)_L\times U(1)_X$5. The favored MDSM$SU(3)_C\times SU(N)_L\times U(1)_X$6 combines a dynamical seesaw scale, a $SU(3)_C\times SU(N)_L\times U(1)_X$7, and the possibility of inflation, dark matter, and leptogenesis.

A related construction, the Minimal Scale Invariant extension of the Standard Model, keeps the Standard Model gauge group fixed and adds a single complex scalar singlet $SU(3)_C\times SU(N)_L\times U(1)_X$8, with tree-level scalar potential

$SU(3)_C\times SU(N)_L\times U(1)_X$9

The model admits Type I, Type II, and Type III flat directions under the Gildener–Weinberg analysis, and a notable Type-II $B-L$0-symmetric realization with $B-L$1 yields maximal spontaneous CP violation while remaining perturbative up to the Planck scale (Alexander-Nunneley et al., 2010). In that case the heavy scalar spectrum includes

$B-L$2

and the pseudo-Goldstone boson of broken scale invariance receives its mass at one loop.

A still more restrictive conformal survey finds that the minimal successful conformal extension of the Standard Model Higgs sector is not a one-singlet model but the Standard Model Higgs doublet plus two real gauge singlet scalars, one of which acquires a VEV and mixes with the Higgs, while the other remains $B-L$3-odd and can serve as a dark matter candidate (Helmboldt et al., 2016). The scalar potential is

$B-L$4

In the viable region, the model predicts sizable Higgs–singlet mixing, a light pseudo-Goldstone boson, and a heavier stable scalar $B-L$5 with mass in the range $B-L$6 (Helmboldt et al., 2016).

These conformal constructions are minimal in a stronger structural sense than many gauge extensions: they attempt to eliminate explicit mass parameters altogether. A plausible implication is that “minimality” here concerns the parameter basis as much as the field content.

5. Cosmology, relic neutrinos, and the Dirac–Majorana interpolation

The neutrino MESM of three right-chiral singlets with Majorana masses has acquired a specific cosmological role through relic-neutrino phenomenology. In this framework, the cosmic neutrino background can probe the Dirac, pseudo-Dirac, and Majorana regimes continuously, because the capture rate and related observables depend smoothly on the lepton-number-violating scale $B-L$7 (Perez-Gonzalez et al., 2024).

For tritium capture,

$B-L$8

the capture rate in the simplified pseudo-Dirac framework can be written as

$B-L$9

(Perez-Gonzalez et al., 2023). In the Majorana limit,

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$0

whereas in the Dirac limit,

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$1

If the lightest neutrino is still relativistic today, the familiar Dirac-to-Majorana ratio is modified. For normal ordering with a massless lightest Dirac neutrino,

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$2

(Perez-Gonzalez et al., 2023).

The later analysis generalizes this to several relic-neutrino observables—tritium capture, the Stodolsky effect, coherent scattering, and an accelerator proposal—and shows that neutral-current interactions induce transitions between $$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$3 and $$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$4, so a density matrix formalism is required (Perez-Gonzalez et al., 2024). The cosmological phase difference satisfies, in the ultra-relativistic limit,

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$5

and the oldest neutrinos in the universe can in principle probe

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$6

below which neutrinos behave as Dirac fermions for all practical purposes (Perez-Gonzalez et al., 2024). In the earlier capture-only analysis, a characteristic sensitivity around

$$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$7

appears when the lightest active neutrino remains relativistic today (Perez-Gonzalez et al., 2023).

This cosmological application is notable because it uses the same minimal field content as the type-I seesaw, but explores the opposite corner of parameter space: not a large $$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$8, but an extraordinarily small one. The resulting pseudo-Dirac phenomenology is not an additional model; it is a limiting regime of the same MESM.

6. Phenomenological role, misconceptions, and broader significance

A recurring misconception is that “minimally extended Standard Model” names a single canonical Lagrangian. The literature surveyed here shows otherwise. In one common usage it means SM + three Dirac right-handed neutrinos with conserved lepton number (Teixeira, 2016); in another it means SM + three right-handed neutrinos with arbitrary Majorana masses (Perez-Gonzalez et al., 2023); elsewhere it means SM + $$\mathcal{L}_\nu = -\,\overline{L}_\alpha\,Y_{\nu\,\alpha\beta}\,\tilde H\,\nu_{R\beta} + \text{h.c.},$$9 + three $B-L$00 + one singlet scalar (Basso, 2011); in classically conformal work it may mean SM + one or two singlet scalars with no tree-level mass terms (Helmboldt et al., 2016). These are not mutually contradictory; they are distinct minimal completions of distinct phenomenological tasks.

Another misconception is that minimality automatically implies a weak phenomenology. The evidence is mixed. In the Dirac-neutrino MESM, cLFV is effectively absent, with $B-L$01 (Teixeira, 2016). By contrast, minimal $B-L$02 or $B-L$03 extensions predict $B-L$04 bosons, singlet Higgs states, and heavy neutrinos with explicit collider signatures (Basso, 2011). Conformal minimal models can imply light pseudo-Goldstone scalars, sizeable Higgs mixing, and stable scalar dark matter (Helmboldt et al., 2016). The neutrino MESM with general $B-L$05 yields observable modifications of C$B-L$06B detection rates, including pronounced pseudo-Dirac dips when $B-L$07 (Perez-Gonzalez et al., 2023).

The broader significance of the MESM idea lies in its methodological role. These models are often used as baseline theories: they solve one sharply defined deficit of the Standard Model with the smallest plausible set of new ingredients and thereby provide a controlled reference point for interpreting null results or excesses. In cLFV, the minimal Dirac extension is a no-signal benchmark; in neutrino cosmology, the SM plus $B-L$08 with general $B-L$09 is the most economical framework in which Dirac, pseudo-Dirac, and Majorana behaviour can be compared continuously; in gauge extensions, minimal $B-L$10 and $B-L$11 models isolate the consequences of anomaly cancellation and seesaw dynamics without larger hidden sectors.

Taken together, these works suggest that the MESM should be understood not as a single model but as a family of economy principles applied to different open problems of the Standard Model. The phrase retains its force precisely because it is comparative: each MESM asks how much of the Standard Model can be kept unchanged once one demands neutrino masses, anomaly-free gauging, radiative symmetry breaking, dark matter, leptogenesis, or relic-neutrino observables.