= Solution
The known active <neutrinos> are too light and too fast to supply the dominant <cold dark matter>. Their relativistic decoupling leaves substantial <free streaming>, which erases rather than seeds sufficiently small-scale structure; their allowed <mass> density is also inadequate for the dominant cold component. Changing a light active <neutrino> from Dirac to Majorana does not remove those problems. What can change the picture is an enlarged neutral-fermion sector containing sterile states with different <masses>, production and decay rates. A sufficiently long-lived <sterile neutrino> in a Majorana state can be a dark-matter candidate; Majorana character alone supplies neither stability nor the correct abundance.
For the <spinor> calculation, put $a=p_z/(E+m)$ and define $A=N(1-a)/2$, $B=N(1+a)/2$. Applying the given <chiral projectors> gives
$$
U_{1L}=A\begin{pmatrix}1\\0\\-1\\0\end{pmatrix},\qquad
U_{1R}=B\begin{pmatrix}1\\0\\1\\0\end{pmatrix},\qquad
V_{2R}=B\begin{pmatrix}1\\0\\1\\0\end{pmatrix}.
$$
Consequently
$$
\overline U_{1R}U_{1L}=2AB
=\frac{E+m}{2}\left(1-\frac{p_z^2}{(E+m)^2}\right)=m,
\qquad
\overline V_{2R}U_{1L}=2AB=m,
$$
where the last equality uses $E^2-p_z^2=m^2$. Same-chirality <Lorentz scalar> products instead vanish: for example $\overline U_{1L}U_{1L}=0$. The nonzero overlaps demonstrate the <Lorentz scalar> structures coupling left and right <chirality>. For $U$ to $U$ this is a particle-number-preserving Dirac coupling; for $U$ to the charge-conjugate $V$ sector it can describe a lepton-number-violating Majorana coupling.
There is a normalization subtlety: both displayed overlaps equal the on-shell <mass> parameter used to define the <spinors>. A <chiral scalar spinor overlap> verifies a coupling structure without fixing an independent Lagrangian coefficient $M_L$. Repeating the on-shell construction with mass parameter $M_L$ would instead give the corresponding overlap $M_L$; this is a different input, rather than a newly derived parameter. The independent coefficients of a <Dirac mass term> and a <Majorana mass term> are defined by
$$
\mathcal L_m=-m(\bar\nu_R\nu_L+\bar\nu_L\nu_R)
-\frac12\left(M_L\overline{\nu_L^c}\nu_L+
M_R\overline{\nu_R^c}\nu_R+\mathrm{h.c.}\right).
$$
The Majorana terms connect a field with its charge conjugate and violate the corresponding continuous <lepton> phase symmetry by two units. With only unbroken <Standard Model> <gauge symmetry>, a left-handed Majorana <mass> also requires <electroweak symmetry breaking> or an appropriate higher-dimensional operator; it is not an arbitrary gauge-invariant bare <mass> for a <lepton> doublet.
Use an all-left-handed Weyl <basis> $\mathbf n_L=(\nu_L,\nu_R^c)^T$. For real parameters the <one-family Dirac-Majorana mass matrix> is
$$
\boxed{\mathcal M=\begin{pmatrix}M_L&m\\m&M_R\end{pmatrix},\qquad
\mathcal L_m=-\frac12\overline{\mathbf n_L^{\,c}}\mathcal M\mathbf n_L+\mathrm{h.c.}}
$$
Writing $(\nu_{1L},\nu_{2L})^T=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}(\nu_L,\nu_R^c)^T$ gives an orthogonal diagonalization, with $\tan2\theta=2m/(M_R-M_L)$ up to the rotation-sign convention. Massive-particle <chirality> and <helicity> are not identical; the familiar left/right <helicity> language becomes an approximation in the relativistic regime.
The <characteristic polynomial> is
$$
\det(\mathcal M-\lambda\mathbf1)
=\lambda^2-(M_L+M_R)\lambda+M_LM_R-m^2.
$$
Its roots are
$$
\boxed{\lambda_\pm=\frac12\left[M_R+M_L\pm
\sqrt{(M_R-M_L)^2+4m^2}\right].}
$$
For Majorana <fermions> the nonnegative physical <masses>, as explained by <physical Majorana mass and signed eigenvalue>, are $|\lambda_\pm|$ after a possible field rephasing; a negative real <eigenvalue> is not a negative physical <rest mass>. When $M_R\gg |m|,|M_L|$,
$$
\lambda_{\rm light}\simeq M_L-\frac{m^2}{M_R},\qquad
\lambda_{\rm heavy}\simeq M_R+\frac{m^2}{M_R}.
$$
The light state is predominantly $\nu_L-(m/M_R)\nu_R^c$, while the heavy state is predominantly $\nu_R^c+(m/M_R)\nu_L$. Each <mass> <eigenstate> combines its Weyl component and charge conjugate to form a Majorana field.
For $M_R=10^{12}\,\mathrm{GeV}$, \b[the heavy <mass> is approximately $10^{12}\,\mathrm{GeV}$]. In the type-I limit $M_L\simeq0$,
$$
\boxed{m_{\rm light}\simeq10^{-3}
\left(\frac{m}{1\,\mathrm{GeV}}\right)^2\,\mathrm{eV}.}
$$
The <seesaw light-mass estimate needs a Dirac scale>: no unique numerical light <mass> follows because the Dirac <mass> and $M_L$ were not supplied. As an explicitly conditional illustration, choosing $m=100\,\mathrm{GeV}$ and $M_L=0$ gives $10\,\mathrm{eV}$; this is an assumed Dirac scale, not additional exam data or a prediction from $M_R$ alone. A heavy seesaw state is also not automatically stable enough to be <dark matter>.
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2016/paper-1-seesaw-scales.png]
{title=A large right-handed <mass> produces reciprocal light and heavy scales}
{description=Original exact one-family <eigenvalues> for $M_L=0$, expressed relative to an unspecified fixed Dirac <mass>. Negative signed <eigenvalues> are shown by their positive physical <masses>; no numerical Dirac <mass> is assumed.}
The conventional minimal <Pati-Salam model> <gauge group> is
$$
\boxed{SU(4)_C\times SU(2)_L\times SU(2)_R.}
$$
The group printed in this paper is different: it contains two $SU(4)$ factors and only one $SU(2)$. This is a discrepancy in the original PDF, not just in the transcription. The standard-model discussion below uses the conventional intended <Pati-Salam model> model. A literal theory with the printed product would require additional <fermion> representations and symmetry-breaking data that are not specified.
The <Pati-Salam model> unifies the three <quark> <color charges> with <lepton number> as a fourth <color charge>. A family forms $(4,2,1)$ left-handed and $(4,1,2)$ right-handed multiplets, including a <right-handed neutrino>. The extended <color charge> group contains quark-lepton <gauge bosons>, while the right-handed weak factor supplies right-handed charged currents after symmetry breaking. The <Pati-Salam charge embedding> is
$$
Q=T_{3L}+T_{3R}+\frac{B-L}{2}.
$$
The original paper explicitly develops the smaller local subgroup $SU(2)_L\times SU(2)_R\times SU(4)_C$, as well as a larger underlying global structure; neither distinction licenses silently replacing the printed group.<Lepton number as the fourth color by Jogesh Pati and Abdus Salam>[§II.B, equation (2), printed p. 276; §II.C, charge assignments]{ref} The product group is a quark-lepton unification scheme rather than a simple <gauge group> with one automatically common coupling. Breaking the right-handed and $B-L$ symmetries can generate a large $M_R$ and light/heavy <Majorana neutrinos> through the seesaw. The <gauge group> alone does not uniquely fix their <masses> or guarantee a dark-matter lifetime.
The most important laboratory test of Majorana character is <neutrinoless double beta decay>,
$$
(A,Z)\longrightarrow(A,Z+2)+e^-+e^-,
$$
with no outgoing <neutrinos>. Under light-neutrino exchange, the amplitude involves
$$
m_{\beta\beta}=\left|\sum_iU_{ei}^{\,2}m_i\right|,\qquad
[T_{1/2}^{0\nu}]^{-1}=G_{0\nu}|\mathcal M_{0\nu}|^2
\left|\frac{m_{\beta\beta}}{m_e}\right|^2.
$$
The coherent squared mixing elements allow phase cancellations. A null result constrains this combination within the assumed mechanism, rather than proving that all <neutrinos> are Dirac.
The observable is a peak in summed electron <energy> at the decay <Q value>, unlike the continuous two-neutrino spectrum. Enriched source material, large exposure, good <energy> resolution, underground operation and radiopure construction suppress backgrounds. <germanium> arrays use the source as a semiconductor detector; <xenon> time-projection chambers combine <ionization>, light and event-position information; <bolometers> measure deposited <energy> through a <temperature> rise. The GERDA collaboration's Phase-I study is a concrete example of an enriched-germanium source-detector search with background suppression.<Results on neutrinoless double beta decay of 76Ge from GERDA Phase I by the GERDA Collaboration>[Abstract and Phase-I source-detector method, arXiv:1307.4720]{ref} <energy> calibration, resolution tails, radon and material backgrounds, cosmic-ray vetoes, signal-like event <topology> and sideband fits all need independent checks before a Q-value excess is convincing. Collider same-sign dilepton processes can test heavier Majorana states, but require a specified production and decay mechanism. \b[Lepton-number violation, absolute-mass measurements and oscillation data provide complementary information; oscillations alone do not settle Majorana character.]
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