---
title: Semi-Linear Neutrino Mass Response
url: https://www.emergentmind.com/topics/semi-linear-neutrino-mass-response
type: topic
---

# Semi-Linear Neutrino Mass Response

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="2107.14726 OR 2011.12504 OR 1806.00369 OR 1102.1664 OR 2305.00994 OR 1903.12126 OR 1904.06036 OR 2104.08248", 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 [2107.14726] [1806.00369] [2011.12504] [1102.1664] [1904.06036] [2104.08248].

## 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 [2107.14726] [1806.00369] [2011.12504] [1102.1664] [1904.06036] [2104.08248].

| 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 | \(m_\alpha^{\rm eff}=(3\kappa/8)\sum_f \lambda_\alpha\lambda_f n_f\) |
| SMEFT feed-down | Loop-corrected mass matrix | \(\delta m_\nu \propto \ln(\Lambda/\mu)\,[\gamma_{55}C_5+\cdots]\) |

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 \(\mu\) 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 \(Z_2\) 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 \((\nu_L,N_R^c,N_L)\) takes the minimal linear-seesaw form
\[
\mathcal M=
\begin{pmatrix}
0 & m_D & \mu\\
m_D^T & 0 & M_N\\
\mu^T & M_N^T & 0
\end{pmatrix},
\]
with
\[
m_\nu \simeq -[m_D M_N^{-1}\mu^T+\mu M_N^{-1}m_D^T]\sim m_D\mu/M_N.
\]
In the one-generation or aligned-coupling limit this reduces to \(m_\nu \simeq 2m_D\mu/M_N\). 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 \(\theta\simeq m_D/M_N\) [2107.14726].

The same structural dependence appears in the simplest linear seesaw reviewed by Cepedello, Hirsch and Madge. There the field content includes two \(SU(2)_L\)-singlet fermions per family, \(\nu^c\) and \(S\), together with a second Higgs doublet \(\chi_L\) carrying lepton number \(L[\chi_L]=-2\). After electroweak symmetry breaking,
\[
m_D^{ij}=\frac{Y_\nu^{ij}v_\Phi}{\sqrt2},\qquad
M_L^{ij}=\frac{Y_S^{ij}v_\chi}{\sqrt2},
\]
and in the basis \((\nu_L,\nu^c,S)\) the full \(9\times 9\) mass matrix is
\[
\mathcal M_\nu=
\begin{pmatrix}
0 & m_D & M_L\\
m_D^T & 0 & M_R\\
M_L^T & M_R^T & 0
\end{pmatrix}.
\]
For \(M_R\gg m_D\gg M_L\), block diagonalization yields
\[
m_\nu \approx m_D(M_R^{-1}M_L^T)+(M_LM_R^{-1})m_D^T.
\]
Since \(M_L\propto v_\chi\), neutrino masses vanish linearly as \(v_\chi\to 0\). The six heavy Majorana states pair into three quasi-Dirac fermions with mass \(\sim M_R\), and the light-heavy mixing is characterized at leading order by \(V=m_D M_R^{-1}\). Because \(Y_S\) may be \(\mathcal O(1)\) while \(v_\chi\) is small, the model permits sizeable \(\ell_i\to \ell_j\gamma\) and collider processes such as \(e^+e^-\to N_iN_i\), \(e^-\gamma\to N_iH^-\), and \(e^+e^-\to H^+H^-\), followed by \(H^\pm\to \ell_i^\pm N\) and \(N\to \ell_j^\pm W^\mp\) [2305.00994].

A third realization is the \(331\)-model with Froggatt-Nielsen suppression. In that framework the neutral-fermion mass matrix contains blocks \(m^D\), \(m^N\), \(m'^N\), and \(M\), and the effective light-neutrino mass is
\[
m_\nu =4\,m^{D\dagger}(m'^{N\dagger})^{-1}M^*(m'^{N*})^{-1}m^{D*}
-\bigl[m^{N*}(m'^{N*})^{-1}2m^{D*}+2m^{D\dagger}(m'^{N\dagger})^{-1}m^{N\dagger}\bigr].
\]
The first contribution is quadratic in the small lepton-number-violating spurion \(m^D\), whereas the second is linear and constitutes the genuine linear-seesaw piece. For \(u\simeq v_2\simeq 50\,{\rm TeV}\), \(v_1\simeq v'\simeq 10^2\,{\rm GeV}\), and \(\epsilon=0.23\), the model reproduces \(\Delta m^2_{21}\simeq 7.5\times 10^{-5}\,{\rm eV}^2\), \(\Delta m^2_{31}\simeq 2.5\times 10^{-3}\,{\rm eV}^2\), and \(|U_{\rm PMNS}|\) inside the \(3\sigma\) global-fit range [1903.12126].

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 \(\beta\)-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 \(m(\nu_e)\) with a sensitivity of \(0.2\,\mathrm{eV}\) at \(90\%\) C.L. The differential spectrum near the endpoint is
\[
\frac{d\Gamma}{dE}
=\frac{G_F^2|V_{ud}|^2}{2\pi^3}(g_V^2+3g_A^2)F(Z,E)\,p\,(E+m_e)\,
\sum_f P_f\,\epsilon_f\sqrt{\epsilon_f^2-m^2}\,\Theta(\epsilon_f-m),
\]
where \(m\equiv m_{\rm eff}\) in the quasi-degenerate approximation and \(\epsilon_f=E_0-V_f-E\). Expanding near the endpoint for \(\Delta\equiv E_0-E\ll E_0\) and \(\Delta\gtrsim m\),
\[
\epsilon_f\sqrt{\epsilon_f^2-m^2}
=\Delta_f^2-\frac12 m^2-\frac18\,\frac{m^4}{\Delta_f^2}+\cdots.
\]
To first order in \(m^2\), the spectrum is therefore linear in \(m^2\),
\[
\frac{d\Gamma}{dE}\biggr|_{\Delta\ll E_0}\approx K(E)\Bigl[\sum_f P_f\Delta_f^2-\frac12 m^2\Bigr].
\]
After folding with the apparatus response function \(R(E,qU)\), the integral count rate becomes
\[
\dot N(qU;m^2)\simeq N_0(qU)-m^2 S_1(qU)+\mathcal O(m^4).
\]
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 \(m^4/\Delta^2\) [1806.00369].

The KATRIN response function incorporates MAC-E transmission, scattering in the WGTS, energy loss, cyclotron-radiation loss, source geometry, detector efficiency, and background. Because \(R\) is independent of \(m\), all instrumental effects enter through the zero-mass rate \(N_0(qU)\) and the slope function \(S_1(qU)\), without spoiling the linearity in \(m^2\) over the usual interval. In the analysis window \(\Delta\in[1,30]\) eV, a profile-likelihood, Neyman/Feldman-Cousins, or Bayesian treatment may be used, and the linearized model is particularly convenient because \(m^2\) becomes the single shape parameter of interest [1806.00369].

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 \(m^2\), not to the microscopic origin of neutrino mass.

## 4. Cosmological linear-response formulations for massive neutrinos

In cosmological \(N\)-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
\[
\delta_\nu(k,s)= -k^2 \int_{s_i}^{s} ds'\,a^2(s')\,\phi(k,s')\,(s-s')\,F[q(s,s')].
\]
In the free-streaming limit \(k\gg k_{\rm FS}\),
\[
\delta_\nu^{\rm FS}(k,a)\simeq \frac{k_{\rm FS}^2(a)}{k^2}\,\delta_m(k,a),
\]
whereas in the clustering limit \(k\ll k_{\rm FS}\),
\[
\delta_\nu^{C}(k,a)\simeq \delta_{cb}(k,a)\simeq \delta_m(k,a).
\]
These two limits are connected by the rational interpolator
\[
\delta_\nu(k,a)\approx
\frac{k_{\rm FS}^2(a)}{[k+k_{\rm FS}(a)]^2}\,\delta_m(k,a),
\]
or, in terms of the cold component,
\[
R(k,a)\equiv \frac{\delta_m(k,a)}{\delta_{cb}(k,a)}
=\frac{[k+k_{\rm FS}(a)]^2(1-f_\nu)}
{[k+k_{\rm FS}(a)]^2-k_{\rm FS}^2(a)f_\nu}.
\]
The PM-grid Poisson solve is then modified by a one-line replacement,
\[
k^2\Phi(\mathbf k,a)=
-\frac32\,\mathcal H^2(a)\,\Omega_m(a)\,R(k,a)\,\delta_c(\mathbf k,a).
\]
No integrals, look-up tables, or interpolation are required; the only cost is two extra real arithmetic operations per \(k\)-cell [2011.12504].

The method was implemented in GADGET-2 and validated for \(N_p=1024^3\), PM\(=1024^3\), box \(=512\) Mpc/\(h\). For \(\Sigma m_\nu\lesssim 0.5\) eV, the total matter power spectrum agrees with full integral linear-response runs to \(\lesssim 1\%\) for \(0.01\lesssim k\lesssim 5\,h/{\rm Mpc}\), and the CDM+baryon power spectrum to \(\lesssim 0.1\%\). Even up to \(\Sigma m_\nu\approx 0.93\) eV, the CDM+baryon power is reproduced to \(0.1\%\) and the total to \(\sim 1.2\%\). Memory is identical to a pure \(\Lambda\)CDM run of the same PM-grid size, and the runtime overhead is essentially zero, in practice \(\lesssim 1\%\) of the total runtime [2011.12504].

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,
\[
m_\nu(\phi)=\hat m\,\exp\!\bigl[-\beta\,\phi/M\bigr],\qquad
V(\phi)=M^4\exp\!\bigl[-\alpha\,\phi/M\bigr].
\]
The linearized perturbation of the cosmon satisfies
\[
\delta\phi_{\rm lin}(k)= -\frac{\beta}{M}\,\frac{a^2}{k^2+a^2m_\phi^2}\,\delta\rho_\nu(k),
\]
implying a linear neutrino-mass perturbation
\[
\delta m_{\nu,\rm lin}(k)=
-\frac{\beta^2}{M^2}m_\nu(\bar\phi)\frac{a^2}{k^2+a^2m_\phi^2}\delta\rho_\nu(k).
\]
However, once \(|\delta m_{\nu,\rm lin}|\sim m_\nu\), the linear approximation breaks down. In a static spherical lump, the full field equation becomes
\[
\phi''(r)+\frac{2}{r}\phi'(r)=\frac{dV}{d\phi}(\phi)-\frac{\beta}{M}m_\nu(\phi)n_\nu(r),
\]
with \(m_\nu(r)=\hat m e^{-\beta\phi(r)/M}\). The mass-weighted average inside the lump,
\[
\langle m_\nu\rangle=\frac{1}{N_\nu}\,4\pi\!\int_0^{r_{\rm vir}} r^2 n_\nu(r)m_\nu(r)\,dr,
\]
defines the ratio
\[
r_m(N_\nu)\equiv \frac{\langle m_\nu\rangle}{\bar m_\nu}\ll 1.
\]
For typical lumps containing \(N_\nu/N_{\rm tot}=10^{-6}\)–\(10^{-2}\) of all cosmic neutrinos, one finds \(r_m\sim 10^{-1}\)–\(10^{-3}\). The associated Newtonian potential is reduced by exactly this factor relative to the linear extrapolation. The paper’s fiducial example uses \(\alpha=10\), \(\beta=-52\), \(\bar m_{\nu 0}=2.3\) eV, and finds a suppression of local \(\Delta\alpha/\alpha\) by \(f\sim 10^{-3}\) inside realistic lumps [1102.1664].

A second environment-induced realization appears in Einstein-Cartan-Kibble-Sciama gravity. There the spin connection is decomposed as
\[
A_\mu^{ab}=\omega_\mu^{ab}(e)+K_\mu^{ab},
\]
and the contorsion \(K_\mu^{ab}\) is algebraically determined by fermion axial currents. The resulting neutrino equation of motion is semi-linear:
\[
i\gamma^\mu\nabla_\mu\psi_\alpha-m_{0\alpha}\psi_\alpha
-\frac{3i\kappa}{8}\sum_\beta \lambda_\beta (\bar\psi_\beta\gamma^I\psi_\beta)\gamma_I\psi_\alpha=0.
\]
In a homogeneous, unpolarized background of electrons, protons, and neutrons, the mean-field reduction yields an effective mass shift
\[
m_\alpha^{\rm eff}\equiv \frac{3\kappa}{8}\sum_{f=e,p,n}\lambda_\alpha\lambda_f n_f,
\]
so that the matter Hamiltonian can be written with
\[
V_\alpha = m_\alpha^{\rm eff}\simeq \frac{3\kappa}{8}\lambda_\alpha\sum_f \lambda_f n_f.
\]
This modifies the in-medium mixing angle exactly as an ordinary matter potential would do [1904.06036].

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
\[
O_5\equiv O_{LH}^{(5)}=\epsilon_{ij}\epsilon_{mn}(L_p^i C L_q^m)H^jH^n,
\qquad
(m_\nu^{(0)})_{pq}=-\,\frac{v^2}{\Lambda}C_5^{pq}.
\]
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
\[
\delta m_\nu^{pq}
= -\,\frac{v^2}{16\pi^2\Lambda}\ln(\Lambda/\mu)
\Bigl[
\gamma_{55}C_5^{pq}
+\frac{v^2}{\Lambda^2}\sum_i \gamma_{5,i}C_i
+\frac{v^2}{\Lambda^2}\sum_j \gamma_{5,j}C_j
\Bigr].
\]
Among the explicit entries are
\[
\gamma_{5,5}=3g_2^2-2\lambda_H-6\,{\rm Tr}(Y_u^2),\qquad
\gamma_{5,H}=6,\qquad
\gamma_{5,HL^{(1)}}=0.
\]
The analysis identifies dimension-seven operators which, despite violating lepton number, do not renormalise neutrino masses at one loop, and exhibits the specific combination
\[
O_* = g_2 O_{LHW}-4 O_{LHD1}
\]
with vanishing anomalous dimension into the Weinberg operator. For \(\Lambda=1\) TeV and \(C_H\simeq 5\), the relative shift \(|\delta m_\nu(C_H)-\delta m_\nu({\rm SM})|/|\delta m_\nu({\rm SM})|\) is \(\simeq 50\%\) [2104.08248].

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 \(m^2\). 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.

Source: https://www.emergentmind.com/topics/semi-linear-neutrino-mass-response