Semi-Linear Neutrino Mass Response

• Semi-linear neutrino mass response is a structural framework where neutrino masses depend linearly on a small symmetry-breaking parameter, with higher-order terms remaining subleading.
• It spans multiple contexts—from low-scale linear seesaw mechanisms and KATRIN beta-decay endpoint analyses to cosmological N-body simulations and SMEFT radiative corrections.
• This approach aids in isolating leading-order effects in neutrino measurements and model predictions, thereby streamlining the interpretation of experimental and theoretical results.

Searching arXiv for relevant papers on semi-linear neutrino mass response and closely related uses of “semi-linear”/linear-response neutrino mass mechanisms. arxiv_search(query="\"semi-linear\" neutrino mass response linear seesaw linear response neutrino mass", max_results=10, sort_by="relevance") arxiv_search(query="\"One line to run them all\" neutrino linear response", max_results=5, sort_by="relevance") arxiv_search(query="(Matute, 2021) OR (Chen et al., 2020) OR (Kleesiek et al., 2018) OR (Nunes et al., 2011) OR (Batra et al., 2023) OR (Huitu et al., 2019) OR (Chakrabarty et al., 2019) OR (Chala et al., 2021)", max_results=10, sort_by="relevance") Semi-linear neutrino mass response denotes a family of constructions in which the neutrino mass sector, or an observable controlled by neutrino mass, is governed at leading order by a linear dependence on a small lepton-number-violating spurion, a background-induced source, or an analytic response kernel, while higher-order terms are subleading, suppressed, or explicitly isolated. In the literature, related “semi-linear” language appears in low-scale linear seesaw model building, endpoint $\beta$-spectroscopy, cosmological massive-neutrino linear response, growing-neutrino quintessence, torsion-induced in-medium masses, and SMEFT radiative feed-down into the Weinberg operator (Matute, 2021, Kleesiek et al., 2018, Chen et al., 2020, Nunes et al., 2011, Chakrabarty et al., 2019, Chala et al., 2021).

1. Taxonomy of semi-linear behavior

The literature uses related terminology in several technically distinct ways. In low-scale seesaw constructions, “linear” or “semi-linear” refers to the fact that the active neutrino mass is proportional to a small lepton-number-violating parameter $\mu$ and only linearly to the Dirac mass, rather than scaling as $m_D^2/M$. In direct mass measurements, the count-rate deformation near the tritium endpoint is linear in $m^2$ over the usual analysis interval. In cosmological simulation, a linear-response kernel $R(k,a)$ maps the cold matter density contrast into the total matter response. In growing-neutrino quintessence, the relevant response is the suppression of the local neutrino mass inside dense lumps relative to the cosmological background. In torsion-based constructions, a semi-linear Dirac equation induces an effective in-medium mass shift. In SMEFT, dimension-six and dimension-seven operators feed semi-linearly into the neutrino mass matrix through one-loop mixing into the Weinberg operator (Matute, 2021, Kleesiek et al., 2018, Chen et al., 2020, Nunes et al., 2011, Chakrabarty et al., 2019, Chala et al., 2021).

Context	Leading semi-linear quantity	Representative relation
Minimal linear seesaw	Light neutrino mass	$$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$
KATRIN endpoint analysis	Folded count rate	$\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$
Cosmological $N$-body linear response	Total-matter correction	$R(k,a)=\delta_m/\delta_{cb}$
Growing-neutrino quintessence	Lump-to-background mass ratio	$r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$
Torsion in matter	Effective in-medium mass shift	$\mu$0
SMEFT feed-down	Loop-corrected mass matrix	$\mu$1

This suggests that “semi-linear neutrino mass response” is best understood as a structural descriptor rather than a single model class.

2. Linear seesaw realizations of neutrino mass

In many of the so-called “low-scale” or “linear-type” seesaws the active neutrino mass is proportional to a small lepton-number-violating parameter $\mu$2 and only linearly to the Dirac mass; one often calls them “linear seesaws.” In the model of Matute, the Standard Model is extended with three right-handed neutrinos and a Dirac pair of extra sterile neutrinos, odd under a discrete $\mu$3 symmetry, while the heavy right-handed neutrinos are pushed to super-heavy scale and essentially decoupled by a conventional Type-I seesaw. After those states are integrated out, the effective neutrino mass matrix in the basis $\mu$4 takes the minimal linear-seesaw form

$\mu$5

with

$\mu$6

In the one-generation or aligned-coupling limit this reduces to $\mu$7. The light-neutrino mass matrix has rank two, so one of the three active neutrinos is massless; the heavy states form a pseudo-Dirac pair with active-sterile mixing angles of order $\mu$8 (Matute, 2021).

The same structural dependence appears in the simplest linear seesaw reviewed by Cepedello, Hirsch and Madge. There the field content includes two $\mu$9-singlet fermions per family, $m_D^2/M$0 and $m_D^2/M$1, together with a second Higgs doublet $m_D^2/M$2 carrying lepton number $m_D^2/M$3. After electroweak symmetry breaking,

$m_D^2/M$4

and in the basis $m_D^2/M$5 the full $m_D^2/M$6 mass matrix is

$m_D^2/M$7

For $m_D^2/M$8, block diagonalization yields

$m_D^2/M$9

Since $m^2$0, neutrino masses vanish linearly as $m^2$1. The six heavy Majorana states pair into three quasi-Dirac fermions with mass $m^2$2, and the light-heavy mixing is characterized at leading order by $m^2$3. Because $m^2$4 may be $m^2$5 while $m^2$6 is small, the model permits sizeable $m^2$7 and collider processes such as $m^2$8, $m^2$9, and $R(k,a)$0, followed by $R(k,a)$1 and $R(k,a)$2 (Batra et al., 2023).

A third realization is the $R(k,a)$3-model with Froggatt-Nielsen suppression. In that framework the neutral-fermion mass matrix contains blocks $R(k,a)$4, $R(k,a)$5, $R(k,a)$6, and $R(k,a)$7, and the effective light-neutrino mass is

$R(k,a)$8

The first contribution is quadratic in the small lepton-number-violating spurion $R(k,a)$9, whereas the second is linear and constitutes the genuine linear-seesaw piece. For $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$0, $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$1, and $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$2, the model reproduces $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$3, $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$4, and $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$5 inside the $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$6 global-fit range (Huitu et al., 2019).

These constructions establish the central particle-physics meaning of semi-linear response: the neutrino mass matrix depends linearly on a symmetry-breaking quantity that can be naturally small.

3. Semi-linear dependence in direct neutrino-mass measurements

In tritium $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$7-decay spectroscopy, the semi-linear structure concerns the observable rather than the origin of the mass. The KATRIN experiment aims to determine the effective electron neutrino mass $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$8 with a sensitivity of $$m_\nu \simeq -(m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T)$$9 at $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$0 C.L. The differential spectrum near the endpoint is

$\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$1

where $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$2 in the quasi-degenerate approximation and $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$3. Expanding near the endpoint for $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$4 and $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$5,

$\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$6

To first order in $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$7, the spectrum is therefore linear in $\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$8,

$\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)$9

After folding with the apparatus response function $N$0, the integral count rate becomes

$N$1

This semi-linear approximation is highly accurate in the usual analysis window of a few eV below the endpoint, and the leading neglected contribution is proportional to $N$2 (Kleesiek et al., 2018).

The KATRIN response function incorporates MAC-E transmission, scattering in the WGTS, energy loss, cyclotron-radiation loss, source geometry, detector efficiency, and background. Because $N$3 is independent of $N$4, all instrumental effects enter through the zero-mass rate $N$5 and the slope function $N$6, without spoiling the linearity in $N$7 over the usual interval. In the analysis window $N$8 eV, a profile-likelihood, Neyman/Feldman-Cousins, or Bayesian treatment may be used, and the linearized model is particularly convenient because $N$9 becomes the single shape parameter of interest (Kleesiek et al., 2018).

A recurrent misconception is to identify this semi-linear behavior with a fundamental neutrino-mass model. Here the linearity refers only to the endpoint response of the measured rate to $R(k,a)=\delta_m/\delta_{cb}$0, not to the microscopic origin of neutrino mass.

4. Cosmological linear-response formulations for massive neutrinos

In cosmological $R(k,a)=\delta_m/\delta_{cb}$1-body simulation, the relevant semi-linear structure is a response closure for the neutrino density contrast. Chen et al. treat the neutrino phase-space distribution as collisionless in a Newtonian subhorizon patch and start from the linearized Boltzmann solution

$R(k,a)=\delta_m/\delta_{cb}$2

In the free-streaming limit $R(k,a)=\delta_m/\delta_{cb}$3,

$R(k,a)=\delta_m/\delta_{cb}$4

whereas in the clustering limit $R(k,a)=\delta_m/\delta_{cb}$5,

$R(k,a)=\delta_m/\delta_{cb}$6

These two limits are connected by the rational interpolator

$R(k,a)=\delta_m/\delta_{cb}$7

or, in terms of the cold component,

$R(k,a)=\delta_m/\delta_{cb}$8

The PM-grid Poisson solve is then modified by a one-line replacement,

$R(k,a)=\delta_m/\delta_{cb}$9

No integrals, look-up tables, or interpolation are required; the only cost is two extra real arithmetic operations per $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$0-cell (Chen et al., 2020).

The method was implemented in GADGET-2 and validated for $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$1, PM$r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$2, box $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$3 Mpc/$r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$4. For $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$5 eV, the total matter power spectrum agrees with full integral linear-response runs to $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$6 for $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$7, and the CDM+baryon power spectrum to $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$8. Even up to $r_m=\langle m_\nu\rangle/\bar m_\nu\ll 1$9 eV, the CDM+baryon power is reproduced to $\mu$00 and the total to $\mu$01. Memory is identical to a pure $\mu$02CDM run of the same PM-grid size, and the runtime overhead is essentially zero, in practice $\mu$03 of the total runtime (Chen et al., 2020).

This use of linear response does not generate neutrino mass. Instead, it parameterizes the gravitational effect of already massive neutrinos on structure formation through an algebraic response operator.

5. Nonlinear environments: quintessence, torsion, and effective mass shifts

Growing-neutrino quintessence provides a different notion of neutrino mass response, in which the mass itself depends on a scalar field,

$\mu$04

The linearized perturbation of the cosmon satisfies

$\mu$05

implying a linear neutrino-mass perturbation

$\mu$06

However, once $\mu$07, the linear approximation breaks down. In a static spherical lump, the full field equation becomes

$\mu$08

with $\mu$09. The mass-weighted average inside the lump,

$\mu$10

defines the ratio

$\mu$11

For typical lumps containing $\mu$12–$\mu$13 of all cosmic neutrinos, one finds $\mu$14–$\mu$15. The associated Newtonian potential is reduced by exactly this factor relative to the linear extrapolation. The paper’s fiducial example uses $\mu$16, $\mu$17, $\mu$18 eV, and finds a suppression of local $\mu$19 by $\mu$20 inside realistic lumps (Nunes et al., 2011).

A second environment-induced realization appears in Einstein-Cartan-Kibble-Sciama gravity. There the spin connection is decomposed as

$\mu$21

and the contorsion $\mu$22 is algebraically determined by fermion axial currents. The resulting neutrino equation of motion is semi-linear: $\mu$23 In a homogeneous, unpolarized background of electrons, protons, and neutrons, the mean-field reduction yields an effective mass shift

$\mu$24

so that the matter Hamiltonian can be written with

$\mu$25

This modifies the in-medium mixing angle exactly as an ordinary matter potential would do (Chakrabarty et al., 2019).

These cases show that semi-linear response can either fail because nonlinear backreaction freezes the mass, or emerge because an auxiliary geometric field induces an effectively linear medium correction.

6. SMEFT feed-down, renormalisation structure, and conceptual boundaries

Within SMEFT, the tree-level Majorana mass arises from the Weinberg operator

$\mu$26

Chala and Titov compute the leading-logarithmic correction to the neutrino mass matrix to dimension seven and show that the dimension-five operator mixes at one loop with eleven independent dimension-six operators and five dimension-seven operators. The correction can be written as

$\mu$27

Among the explicit entries are

$\mu$28

The analysis identifies dimension-seven operators which, despite violating lepton number, do not renormalise neutrino masses at one loop, and exhibits the specific combination

$\mu$29

with vanishing anomalous dimension into the Weinberg operator. For $\mu$30 TeV and $\mu$31, the relative shift $\mu$32 is $\mu$33 (Chala et al., 2021).

The conceptual boundary is therefore sharp. In linear seesaw models, the semi-linear dependence is part of the tree-level mass-generation mechanism. In KATRIN, it is an endpoint expansion of the response of the measured rate to $\mu$34. In cosmology, it is a closure scheme for neutrino perturbations. In growing-neutrino quintessence and torsion-based models, it is a medium- or field-dependent effective-mass response. In SMEFT, it is a radiative feed-down of higher-dimensional operators into the neutrino mass matrix. A plausible implication is that the phrase “semi-linear neutrino mass response” is best reserved for the leading-order linear dependence together with a controlled account of the terms that ultimately limit that linearization.