c^{}_l c^{}_{\nu} c + s^{}_l s^{}_{\nu} e^{-i\phi}  & c^{}_l s \cr -
s^{}_{\nu} s   & - c^{}_{\nu} s   & c \cr \end{matrix} \right )

accords with the above observation, while the “standard"
parametrization in Eq. (98) does not. That is why the RGEs of
neutrino mixing angles and CP-violating phases in the standard
parametrization are rather complicated.

Here we take the FX form of $U$ to derive the RGEs of neutrino mass
and mixing parameters. Combining Eqs. (135), (136) and the FX form
of $U$, we arrive at

\begin{eqnarray}
\dot{\kappa}^{}_1 = \frac{\kappa^{}_1}{16\pi^2} \left (
\alpha^{}_\kappa + 2 C^{}_\kappa y^2_\tau s^2_\nu s^2 \right ) \; ,
\nonumber \\
\dot{\kappa}^{}_2 = \frac{\kappa^{}_2}{16\pi^2} \left (
\alpha^{}_\kappa + 2 C^{}_\kappa y^2_\tau c^2_\nu s^2 \right ) \; ,
\nonumber \\
\dot{\kappa}^{}_3 = \frac{\kappa^{}_3}{16\pi^2} \left (
\alpha^{}_\kappa + 2 C^{}_\kappa y^2_\tau c^2 \right ) \; , ~~~
%       (137)
\end{eqnarray}

where $\alpha^{}_\kappa \approx -3g^2_2 + 6y^2_t + \lambda$ (SM) or
$\alpha^{}_\kappa \approx -1.2g^2_1 - 6g^2_2 + 6 y^2_t$ (MSSM); and

\begin{eqnarray}
&& \dot{\theta}^{}_l = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} ~
c^{}_\nu s^{}_\nu c \left [ \zeta^{-1}_{13} c^{}_\rho c^{}_{(\rho
-\phi)} + \zeta^{}_{13} s^{}_\rho s^{}_{(\rho - \phi)} -
\zeta^{-1}_{23} c^{}_\sigma c^{}_{(\sigma -\phi)} - \zeta^{}_{23}
s^{}_\sigma s^{}_{(\sigma - \phi)} \right ] \; ,
\nonumber \\
&& \dot{\theta}^{}_\nu = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} ~
c^{}_\nu s^{}_\nu \left [ s^2 \left ( \zeta^{-1}_{12} c^2_{(\sigma
-\rho)} + \zeta^{}_{12} s^2_{(\sigma -\rho)} \right ) + c^2 \left (
\zeta^{-1}_{13} c^2_\rho + \zeta^{}_{13} s^2_\rho \right ) - c^2
\left ( \zeta^{-1}_{23} c^2_\sigma + \zeta^{}_{23} s^2_\sigma \right
) \right ] \; ,
\nonumber \\
&& \dot{\theta} = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} ~ c s \left [
s^2_\nu \left ( \zeta^{-1}_{13} c^2_\rho + \zeta^{}_{13} s^2_\rho
\right ) + c^2_\nu \left ( \zeta^{-1}_{23} c^2_\sigma +
\zeta^{}_{23} s^2_\sigma \right ) \right ] \; ;
%       (138)
\end{eqnarray}

as well as

\begin{eqnarray}
&& \dot{\rho} = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} \left [
\widehat{\zeta}^{}_{12} c^2_\nu s^2 c^{}_{(\sigma -\rho)}
s^{}_{(\sigma -\rho)} + \widehat{\zeta}^{}_{13} \left (s^2_\nu s^2 -
c^2 \right ) c^{}_\rho s^{}_\rho + \widehat{\zeta}^{}_{23} c^2_\nu
s^2 c^{}_\sigma s^{}_\sigma \right ] \; ,
\nonumber \\
&& \dot{\sigma} = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} \left [
\widehat{\zeta}^{}_{12} s^2_\nu s^2 c^{}_{(\sigma -\rho)}
s^{}_{(\sigma -\rho)} + \widehat{\zeta}^{}_{13} s^2_\nu s^2
c^{}_\rho s^{}_\rho + \widehat{\zeta}^{}_{23} \left ( c^2_\nu s^2 -
c^2 \right ) c^{}_\sigma s^{}_\sigma \right ] \; ,
\nonumber \\
&& \dot{\phi} = \frac{C^{}_\kappa y^2_\tau}{16\pi^2} \left [ \left (
c^2_l -s^2_l \right ) c^{-1}_l s^{-1}_l c^{}_\nu s^{}_\nu c \left(
\zeta^{-1}_{13} c^{}_\rho s^{}_{(\rho -\phi)} - \zeta^{}_{13}
s^{}_\rho c^{}_{(\rho -\phi)} - \zeta^{-1}_{23} c^{}_\sigma
s^{}_{(\sigma -\phi)} + \zeta^{}_{23} s^{}_\sigma c^{}_{(\sigma
-\phi)} \right ) \right .
\nonumber \\
&& \left . ~~~~~~~~~~~~~~~~~ + \widehat{\zeta}^{}_{12} s^2
c^{}_{(\sigma -\rho)} s^{}_{(\sigma -\rho)} +
\widehat{\zeta}^{}_{13} \left (s^2_\nu - c^2_\nu c^2 \right )
c^{}_\rho s^{}_\rho + \widehat{\zeta}^{}_{23} \left (c^2_\nu -
s^2_\nu c^2 \right ) c^{}_\sigma s^{}_\sigma \right ] \; ,
%       (139)
\end{eqnarray}

where $\widehat{\zeta}^{}_{ij} \equiv \zeta^{-1}_{ij} -
\zeta^{}_{ij} = 4 \kappa^{}_i \kappa^{}_j/\left ( \kappa^2_i -
\kappa^2_j \right )$, $c^{}_a \equiv \cos a$ and $s^{}_a \equiv \sin
a$ (for $a = \rho$, $\sigma$, $\sigma -\rho$, $\rho -\phi$ or
$\sigma -\phi$).

Some discussions on the basic features of RGEs of three neutrino
masses, three flavor mixing angles and three CP-violating phases are
in order.

(a) The running behaviors of three neutrino masses $m^{}_i$ (or
equivalently $\kappa^{}_i$) are essentially identical and determined
by $\alpha^{}_\kappa$, unless $\tan\beta$ is large enough in the
MSSM to make the $y^2_\tau$-associated term competitive with the
$\alpha^{}_\kappa$ term. In our phase convention,
$\dot{\kappa}^{}_i$ or $\dot{m}^{}_i$ (for $i=1, 2, 3$) are
independent of the CP-violating phase $\phi$.

(b) Among three neutrino mixing angles, only the derivative of
$\theta^{}_\nu$ contains a term proportional to $\zeta^{-1}_{12}$.
Note that $\zeta^{-1}_{ij} = ( m^{}_i + m^{}_j )^2/\Delta m^2_{ij}$
with $\Delta m^2_{ij} \equiv m^2_i - m^2_j$ holds. Current solar and
atmospheric neutrino oscillation data yield $\Delta m^2_{21} \approx
7.7 \times 10^{-5} ~ {\rm eV}^2$ and $\left | \Delta m^2_{32} \right
| \approx \left | \Delta m^2_{31} \right | \approx 2.4 \times
10^{-3} ~ {\rm eV}^2$. So $\theta^{}_\nu$ is in general more
sensitive to radiative corrections than $\theta^{}_l$ and $\theta$.
The evolution of $\theta^{}_\nu$ can be suppressed through the
fine-tuning of $(\sigma -\rho)$. The smallest neutrino mixing angle
$\theta^{}_l$ may get radiative corrections even if its initial
value is zero, and thus it can be radiatively generated from other
neutrino mixing angles and CP-violating phases.

(c) The running behavior of $\phi$ is quite different from those of
$\rho$ and $\sigma$, because it includes a peculiar term
proportional to $s^{-1}_l$. This term, which dominates $\dot{\phi}$
when $\theta^{}_l$ is sufficiently small, becomes divergent in the
limit $\theta^{}_l \rightarrow 0$. Indeed, $\phi$ is not
well-defined if $\theta^{}_l$ is exactly vanishing. But both
$\theta^{}_l$ and $\phi$ can be radiatively generated. We may
require that $\dot{\phi}$ remain finite when $\theta^{}_l$
approaches zero, implying that the following necessary condition can
be extracted from the expression of $\dot{\phi}$ in Eq. (139):

\begin{eqnarray}
\zeta^{-1}_{13} c^{}_\rho s^{}_{(\rho -\phi)} - \zeta^{}_{13}
s^{}_\rho c^{}_{(\rho -\phi)} - \zeta^{-1}_{23} c^{}_\sigma
s^{}_{(\sigma -\phi)} + \zeta^{}_{23} s^{}_\sigma c^{}_{(\sigma
-\phi)} = 0 \; .
%       (140)
\end{eqnarray}

Note that the initial value of $\theta^{}_l$, if it is exactly zero
or extremely small, may immediately drive $\phi$ to its quasi-fixed point. In this case Eq. (140) can be used to understand
the relationship between $\phi$ and two Majorana phases $\rho$ and
$\sigma$ at the quasi-fixed point.

(d) The running behaviors of $\rho$ and $\sigma$ are relatively mild
in comparison with that of $\phi$. A remarkable feature of
$\dot{\rho}$ and $\dot{\sigma}$ is that they will vanish, if both
$\rho$ and $\sigma$ are initially vanishing. This observation
indicates that $\rho$ and $\sigma$ cannot simultaneously be
generated from $\phi$ via the RGEs.

§ HOW TO GENERATE NEUTRINO MASSES?

Neutrinos are assumed or required to be massless in the SM, just
because the structure of the SM itself is too simple to accommodate
massive neutrinos.

* Two fundamentals of the SM are the $SU(2)^{}_{\rm L} \times
U(1)^{}_{\rm Y}$ gauge symmetry and the Lorentz invariance. Both of
them are mandatory to guarantee that the SM is a consistent quantum
field theory.

* The particle content of the SM is rather economical. There
are no right-handed neutrinos in the SM, so a Dirac neutrino mass
term is not allowed. There is only one Higgs doublet, so a
gauge-invariant Majorana mass term is forbidden.

* The SM is a renormalizable quantum field theory. Hence
an effective dimension-5 operator, which may give each neutrino a
Majorana mass, is absent.

In other words, the SM accidently possesses the $(B-L)$ symmetry
which assures three known neutrinos to be exactly massless.

But today's experiments have convincingly indicated the existence of
neutrino oscillations. This quantum phenomenon can appear if and
only if neutrinos are massive and lepton flavors are mixed, and thus
it is a kind of new physics beyond the SM. To generate non-zero but
tiny neutrino masses, one or more of the above-mentioned constraints
on the SM must be abandoned or relaxed. It is intolerable to abandon
the gauge symmetry and Lorentz invariance; otherwise, one would be
led astray. Given the framework of the SM as a consistent field
theory, its particle content can be modified and (or) its
renormalizability can be abandoned to accommodate massive neutrinos.
There are several ways to this goal.

§.§ Relaxing the renormalizability

In 1979, Weinberg extended the SM by introducing some
higher-dimension operators in terms of the fields of the SM itself:

\begin{equation}
{\cal L}^{}_{\rm eff} = {\cal L}^{}_{\rm SM} + \frac{{\cal
L}^{}_{\rm d=5}}{\Lambda} + \frac{{\cal L}^{}_{\rm d=6}}{\Lambda^2}
+ \cdots \; ,
%     (141)
\end{equation}

where $\Lambda$ denotes the cut-off scale of this effective theory.
Within such a framework, the lowest-dimension operator that violates
the lepton number ($L$) is the unique dimension-5 operator
$HHLL/\Lambda$. After spontaneous gauge symmetry breaking, this
Weinberg operator yields $m^{}_i \sim \langle H\rangle^2/\Lambda$
for neutrino masses, which can be sufficiently small ($\leq 1$ eV)
if $\Lambda$ is not far away from the scale of grand unified
theories ($\Lambda \sim 10^{13}$ GeV for $\langle H\rangle \sim
10^2$ GeV). In this sense we argue that neutrino masses can serve as
a low-energy window onto new physics at superhigh energies.

§.§ A pure Dirac neutrino mass term?

Given three right-handed neutrinos, the gauge-invariant and
lepton-number-conserving mass terms of charged leptons and neutrinos

\begin{equation}
-{\cal L}^{}_{\rm lepton} = \overline{\ell^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \overline{\ell^{}_{\rm L}} Y^{}_\nu \tilde{H}
N^{}_{\rm R} + {\rm h.c.} \; ,
%     (142)
\end{equation}

where $\tilde{H} \equiv i\sigma^{~}_2 H^*$ is defined and $\ell_{\rm
L}$ denotes the left-handed lepton doublet. After spontaneous gauge
symmetry breaking, we arrive at the charged-lepton mass matrix
$M^{}_l = Y^{}_l v/\sqrt{2}$ and the Dirac neutrino mass matrix
$M^{}_\nu = Y^{}_\nu v/\sqrt{2}$ with $v \simeq 246 ~ {\rm GeV}$. In
this case, the smallness of three neutrino masses $m^{}_i$ (for
$i=1,2,3$) is attributed to the smallness of three eigenvalues of
$Y^{}_\nu$ (denoted as $y^{}_i$ for $i=1,2,3$). Then we encounter a
transparent hierarchy problem: $y^{}_i/y^{}_e = m^{}_i/m^{}_e \leq
0.5 ~{\rm eV}/0.5 ~{\rm MeV} \sim 10^{-6}$. Why is $y^{}_i$ so
small? There is no explanation at all in this Dirac-mass picture.

A speculative way out is to invoke extra dimensions; namely, the
smallness of Dirac neutrino masses is ascribed to the assumption
that three right-handed neutrinos have access to one or more extra
spatial dimensions. The idea is simply to confine the SM particles
onto a brane and to allow $N^{}_{\rm R}$ to travel in the bulk. For
example, the wave-function of $N^{}_{\rm R}$ spreads out over the
extra dimension $y$, giving rise to a suppressed Yukawa interaction
at $y = 0$ (i.e., the location of the brane):

\begin{equation}
\left[ \overline{\ell^{}_{\rm L}} Y^{}_\nu \tilde{H} N^{}_{\rm R}
\right]^{}_{y=0} \sim \frac{1}{\sqrt{L}} \left[
\overline{\ell^{}_{\rm L}} Y^{}_\nu \tilde{H} N^{}_{\rm R}
\right]^{}_{y=L} \; .
%     (143)
\end{equation}

The magnitude of $1/\sqrt{L}$ is measured by
$\Lambda/\Lambda^{}_{\rm Planck}$, and thus it can naturally be
small for an effective theory far below the Planck scale.

§.§ Seesaw mechanisms

This approach works at the tree level and reflects the essential
spirit of seesaw mechanisms — tiny masses of three known neutrinos
are attributed to the existence of heavy degrees of freedom and
lepton number violation.

* Type-I seesaw — three heavy right-handed neutrinos are
added into the SM and the lepton number is violated by their
Majorana mass term:

\begin{eqnarray}
-{\cal L}^{}_{\rm lepton} = \overline{\ell^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \overline{\ell^{}_{\rm L}} Y^{}_\nu \tilde{H}
N^{}_{\rm R} + \frac{1}{2} \overline{N^{\rm c}_{\rm R}} M^{}_{\rm R}
N^{}_{\rm R} + {\rm h.c.} \; ,
%     (144)
\end{eqnarray}

where $M^{}_{\rm R}$ is the Majorana mass matrix.

* Type-II seesaw — one heavy Higgs triplet is added into the
SM and the lepton number is violated by its interactions with both
the lepton doublet and the Higgs doublet:

\begin{eqnarray}
-{\cal L}^{}_{\rm lepton} = \overline{\ell^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \frac{1}{2} \overline{\ell^{}_{\rm L}} Y^{}_\Delta
\Delta i\sigma^{}_2 \ell^{\rm c}_{\rm L} - \lambda^{}_\Delta
M^{}_\Delta H^T i\sigma^{}_2 \Delta H + {\rm h.c.} \; ,
%     (145)
\end{eqnarray}

\begin{equation} \Delta \equiv \left(\begin{matrix}
\Delta^- & -\sqrt{2} ~ \Delta^0 \cr \sqrt{2} ~ \Delta^{--} &
-\Delta^- \cr \end{matrix} \right)
%     (146)
\end{equation}

denotes the $SU(2)^{}_{\rm L}$ Higgs triplet.

* Type-III seesaw — three heavy triplet fermions are
added into the SM and the lepton number is violated by their
Majorana mass term:

\begin{eqnarray}
-{\cal L}^{}_{\rm lepton} = \overline{\ell^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \overline{\ell^{}_{\rm L}} \sqrt{2} Y^{}_\Sigma
\Sigma^{\rm c} \tilde{H} + \frac{1}{2} {\rm Tr} \left(
\overline{\Sigma} M^{}_\Sigma \Sigma^{\rm c} \right) + {\rm h.c.} \;
%     (147)
\end{eqnarray}

\begin{equation}
\Sigma = \left( \begin{matrix} \Sigma^0/\sqrt{2} & \Sigma^+ \cr
\Sigma^- & -\Sigma^0/\sqrt{2} \cr \end{matrix} \right)
%     (148)
\end{equation}

denotes the $SU(2)^{}_{\rm L}$ fermion triplet.

Of course, there are a number of variations or combinations of these
three typical seesaw mechanisms in the literature.

For each of the above seesaw pictures, one may arrive at the unique
dimension-5 Weinberg operator of neutrino masses after integrating
out the corresponding heavy degrees of freedom:

\frac{{\cal L}^{}_{\rm d=5}}{\Lambda} = \left\{
\begin{array}{lcl}
\displaystyle \frac{1}{2} \left(Y^{}_\nu M^{-1}_{\rm R}
Y^T_\nu\right)^{}_{\alpha\beta} \overline{\ell^{}_{\alpha \rm L}}
\tilde{H} \tilde{H}^T \ell^{\rm c}_{\beta \rm L} + {\rm h.c.} &&
\vspace{0.15cm} \\
\displaystyle -\frac{\lambda^{}_\Delta}{M^{}_\Delta}
\left(Y^{}_\Delta\right)^{}_{\alpha\beta} \overline{\ell^{}_{\alpha
\rm L}} \tilde{H} \tilde{H}^T \ell^{\rm c}_{\beta \rm L} + {\rm
h.c.} &&
\vspace{0.15cm} \\
\displaystyle \frac{1}{2} \left(Y^{}_\Sigma M^{-1}_{\rm \Sigma}
Y^T_\Sigma\right)^{}_{\alpha\beta} \overline{\ell^{}_{\alpha \rm L}}
\tilde{H} \tilde{H}^T \ell^{\rm c}_{\beta \rm L} + {\rm h.c.} &&
\end{array}
\right .

corresponding to type-I, type-II and type-III seesaws. After
spontaneous gauge symmetry breaking, $\tilde{H}$ achieves its vacuum
expectation value $\langle \tilde{H}\rangle = v/\sqrt{2}$ with $v
\simeq 246$ GeV. Then we are left with the effective Majorana
neutrino mass term for three known neutrinos,

\begin{equation}
-{\cal L}^{}_{\rm mass} = \frac{1}{2} \overline{\nu^{}_{\rm L}}
M^{}_\nu \nu^{\rm c}_{\rm L} + {\rm h.c.} \; ,
%     (149)
\end{equation}

where the Majorana mass matrix $M^{}_\nu$ is given by

\begin{equation}
M^{}_\nu = \left\{
\begin{array}{lcl}
\displaystyle -\frac{1}{2} Y^{}_\nu \frac{v^2}{M^{}_{\rm R}} Y^T_\nu
&& ({\rm Type ~I}) \; , \vspace{0.15cm} \\
\displaystyle \lambda^{}_\Delta Y^{}_\Delta \frac{v^2}{M^{}_\Delta }
&& ({\rm Type
~II}) \; , \vspace{0.15cm} \\
\displaystyle -\frac{1}{2} Y^{}_\Sigma \frac{v^2}{M^{}_\Sigma}
Y^T_\Sigma && ({\rm Type ~III}) \; .
\end{array}
\right .
%     (150)
\end{equation}

It becomes obvious that the smallness of $M^{}_\nu$ can be
attributed to the largeness of $M^{}_{\rm R}$, $M^{}_\Delta$ or
$M^{}_\Sigma$ in the seesaw mechanism.

§.§ Radiative origin of neutrino masses

In a seminal paper published in 1972, Weinberg pointed out that “in
theories with spontaneously broken gauge symmetries, various masses
or mass differences may vanish in zeroth order as a consequence of
the representation content of the fields appearing in the
Lagrangian. These masses or mass differences can then be calculated
as finite higher-order effects." Such a mechanism may allow us to
slightly go beyond the SM and radiatively generate tiny neutrino
masses. A typical example is the well-known Zee model,

\begin{eqnarray}
-{\cal L}^{}_{\rm lepton} = \overline{\ell^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \overline{\ell^{}_{\rm L}} Y^{}_S S^- i\sigma^{}_2
l^{\rm c}_{\rm L} + \tilde{\Phi}^T F S^+ i\sigma^{}_2 \tilde{H} +
{\rm h.c.} \; ,
%     (151)
\end{eqnarray}

where $S^{\pm}$ are charged $SU(2)^{}_{\rm L}$ singlet scalars,
$\Phi$ denotes a new $SU(2)^{}_{\rm L}$ doublet scalar which has the
same quantum number as the SM Higgs doublet $H$, $Y^{}_S$ is an
anti-symmetric matrix, and $F$ represents a mass. Without loss of
generality, we choose the basis of $M^{}_l = Y^{}_l \langle H\rangle
= {\rm Diag}\{m^{}_e, m^{}_\mu, m^{}_\tau \}$. In this model
neutrinos are massless at the tree level, but their masses can
radiatively be generated via the one-loop corrections. Given $M^{}_S
\gg M^{}_H \sim M^{}_\Phi \sim F$ and $\langle \Phi\rangle \sim
\langle H\rangle$, the elements of the effective mass matrix of
three light Majorana neutrinos are

\begin{equation}
\left(M^{}_\nu\right)^{}_{\alpha\beta} \; \sim \;
\frac{M^{}_H}{16\pi^2} \cdot \frac{m^2_\alpha - m^2_\beta}{M^2_S}
\left(Y^{}_S\right)^{}_{\alpha\beta} \; ,
%     (152)
\end{equation}

where $\alpha$ and $\beta$ run over $e$, $\mu$ and $\tau$. The
smallness of $M^{}_\nu$ is therefore ascribed to the smallness of
$Y^{}_S$ and $(m^2_\alpha - m^2_\beta)/M^2_S$. Although the original
version of the Zee model is disfavored by current experimental data
on neutrino oscillations, its extensions or variations at the
one-loop or two-loop level can survive.

§ ON THE SCALES OF SEESAW MECHANISMS

As we have seen, the key point of a seesaw mechanism is to ascribe
the smallness of neutrino masses to the existence of some new
degrees of freedom heavier than the Fermi scale $v \simeq 246$ GeV,
such as heavy Majorana neutrinos or heavy Higgs bosons. The energy
scale where a seesaw mechanism works is crucial, because it is
relevant to whether this mechanism is theoretically natural and
experimentally testable. Between Fermi and Planck scales, there
might exist two other fundamental scales: one is the scale of a
grand unified theory (GUT) at which strong, weak and electromagnetic
forces can be unified, and the other is the TeV scale at which the
unnatural gauge hierarchy problem of the SM can be solved or at
least softened by a kind of new physics.

§.§ How about a very low seesaw scale?

In reality, however, there is no direct evidence for a high or
extremely high seesaw scale. Hence eV-, keV-, MeV- and GeV-scale
seesaws are all possible, at least in principle, and they are
technically natural in the sense that their lepton-number-violating
mass terms are naturally small according to 't Hooft's naturalness
— “At any energy scale $\mu$, a set of parameters $\alpha^{}_i
(\mu)$ describing a system can be small, if and only if, in the
limit $\alpha^{}_i (\mu) \rightarrow 0$ for each of these
parameters, the system exhibits an enhanced symmetry." But there are
several potential problems associated with low-scale seesaws: (a) a
low-scale seesaw does not give any obvious connection to a
theoretically well-justified fundamental physical scale (such as the
Fermi scale, the TeV scale, the GUT scale or the Planck scale); (b)
the neutrino Yukawa couplings in a low-scale seesaw model turn out
to be tiny, giving no actual explanation of why the masses of three
known neutrinos are so small; and (c) in general, a very low seesaw
scale does not allow the “canonical" thermal leptogenesis mechanism
to work.

§.§ Seesaw-induced hierarchy problem

Many theorists argue that the conventional seesaw scenarios are
natural because their scales (i.e., the masses of heavy degrees of
freedom) are close to the GUT scale. This argument is reasonable on
the one hand, but it reflects the drawbacks of the conventional
seesaw models on the other hand. In other words, the conventional
seesaw models have no direct experimental testability and involve a
potential hierarchy problem. The latter is usually spoke of when two
largely different energy scales exist in a model, but there is no
symmetry to stabilize the low-scale physics suffering from large
corrections coming from the high-scale physics.

Such a seesaw-induced fine-tuning problem means that the SM Higgs
mass is very sensitive to quantum corrections from the heavy degrees
of freedom in a seesaw mechanism. For example,

\delta M^2_H = \left\{
\begin{array}{lcl}
\displaystyle -\frac{y^2_i}{8\pi^2} \left(\Lambda^2 + M^2_i
\ln\frac{M^2_i}{\Lambda^2} \right) &\hspace{-0.3cm}& ({\rm I})
\vspace{0.2cm} \\
\displaystyle \frac{3}{16\pi^2} \left[ \lambda^{}_3 \left(\Lambda^2
+ M^2_\Delta \ln\frac{M^2_\Delta}{\Lambda^2}\right) + 4
\lambda^2_\Delta M^2_\Delta \ln\frac{M^2_\Delta}{\Lambda^2} \right]
&\hspace{-0.28cm}& ({\rm II})
\vspace{0.2cm} \\
\displaystyle -\frac{3y^2_i}{8\pi^2} \left(\Lambda^2 + M^2_i
\ln\frac{M^2_i}{\Lambda^2} \right) &\hspace{-0.3cm}& ({\rm III})
\end{array}
\right .

in three typical seesaw scenarios, where $\Lambda$ is the regulator
cut-off, $y^{}_i$ and $M^{}_i$ (for $i=1,2,3$) stand respectively
for the eigenvalues of $Y^{}_\nu$ (or $Y^{}_\Sigma$) and $M^{}_{\rm
R}$ (or $M^{}_\Sigma$), and the contributions proportional to $v^2$
and $M^2_H$ have been omitted. The above results show a quadratic
sensitivity to the new scale which is characteristic of the seesaw
model, implying that a high degree of fine-tuning would be necessary
to accommodate the experimental data on $M^{}_H$ if the seesaw scale
is much larger than $v$ (or the Yukawa couplings are not extremely
fine-tuned in type-I and type-III seesaws). Taking the type-I seesaw
scenario for illustration, we assume $\Lambda \sim M^{}_i$ and
require $|\delta M^2_H| \leq 0.1 ~{\rm TeV}^2$. Then the above
equation leads us to the following rough estimate:

\begin{eqnarray}
M^{}_i \sim \left[\frac{(2\pi v)^2 |\delta
M^2_H|}{m^{}_i}\right]^{1/3} \leq 10^7 {\rm GeV} \left[\frac{0.2 ~
{\rm eV}}{m^{}_i}\right]^{1/3} \left[\frac{|\delta M^2_H|}{0.1 ~
{\rm TeV}^2}\right]^{1/3} \; .
%     (153)
\end{eqnarray}

This naive result indicates that a hierarchy problem will arise if
the masses of heavy Majorana neutrinos are larger than about $10^7$
GeV in the type-I seesaw scheme. Because of $m^{}_i \sim y^2_i
v^2/(2M^{}_i)$, the bound $M^{}_i \leq 10^7$ GeV implies $y^{}_i
\sim \sqrt{2m^{}_i M^{}_i}/v \leq 2.6 \times 10^{-4}$ for $m^{}_i
\sim 0.2$ eV. Such a small magnitude of $y^{}_i$ seems to be a bit
unnatural in the sense that the conventional seesaw idea attributes
the smallness of $m^{}_i$ to the largeness of $M^{}_i$ other than
the smallness of $y^{}_i$.

There are two possible ways out of this impasse: one is to appeal
for the supersymmetry, and the other is to lower the seesaw scale.
We shall follow the second way to discuss the TeV seesaw mechanisms
which do not suffer from the above-mentioned hierarchy problem.

§.§ Why are the TeV seesaws interesting?

There are several reasons for people to expect some new physics at
the TeV scale. This kind of new physics should be able to stabilize
the Higgs-boson mass and hence the electroweak scale; in other
words, it should be able to solve or soften the unnatural gauge
hierarchy problem. It has also been argued that the
weakly-interacting particle candidates for dark matter should weigh
about one TeV or less. If the TeV scale is really a fundamental
scale, may we argue that the TeV seesaws are natural? Indeed, we are
reasonably motivated to speculate that possible new physics existing
at the TeV scale and responsible for the electroweak symmetry
breaking might also be responsible for the origin of neutrino
masses. It is interesting and meaningful in this sense to
investigate and balance the “naturalness" and “testability" of TeV
seesaws at the energy frontier set by the LHC.

As a big bonus of the conventional (type-I) seesaw mechanism, the
thermal leptogenesis mechanism provides us with an elegant dynamic
picture to interpret the cosmological matter-antimatter asymmetry
characterized by the observed ratio of baryon number density to
photon number density, $\eta^{}_B \equiv n^{}_B/n^{}_\gamma = (6.1
\pm 0.2) \times 10^{10}$. When heavy Majorana neutrino masses are
down to the TeV scale, the Yukawa couplings should be reduced by
more than six orders of magnitude so as to generate tiny masses for
three known neutrinos via the type-I seesaw and satisfy the
out-of-equilibrium condition, but the CP-violating asymmetries of
heavy Majorana neutrino decays can still be enhanced by the resonant
effects in order to account for $\eta^{}_B$. This “resonant
leptogenesis" scenario might work in a specific TeV seesaw model.

Is there a TeV Noah's Ark which can naturally and simultaneously
accommodate the seesaw idea, the leptogenesis picture and the
collider signatures? We are most likely not so lucky and should not
be too ambitious at present. In the following we shall concentrate
on the TeV seesaws themselves and their possible collider signatures
and low-energy consequences.

§ TEV SEESAWS: NATURAL AND TESTABLE?

The neutrino mass terms in three typical seesaw mechanisms have been
given before. Without loss of generality, we choose the basis in
which the mass eigenstates of three charged leptons are identified
with their flavor eigenstates.

§.§ Type-I seesaw

Given $M^{}_{\rm D} = Y^{}_\nu v/\sqrt{2}~$, the approximate type-I
seesaw formula in Eq. (150) can be rewritten as $M^{}_\nu = -
M^{}_{\rm D} M^{-1}_{\rm R} M^T_{\rm D}$. Note that the $3\times 3$
light neutrino mixing matrix $V$ is not exactly unitary in this
seesaw scheme, and its deviation from unitarity is of ${\cal
O}(M^2_{\rm D}/M^2_{\rm R})$. Let us consider two interesting
possibilities. (1) $M^{}_{\rm D} \sim {\cal O}(10^2)$ GeV and
$M^{}_{\rm R} \sim {\cal O}(10^{15})$ GeV to get $M^{}_\nu \sim
{\cal O}(10^{-2})$ eV. In this conventional and natural case,
$M^{}_{\rm D}/M^{}_{\rm R} \sim {\cal O}(10^{-13})$ holds. Hence the
non-unitarity of $V$ is only at the ${\cal O}(10^{-26})$ level, too
small to be observed. (2) $M^{}_{\rm D} \sim {\cal O}(10^2)$ GeV and
$M^{}_{\rm R} \sim {\cal O}(10^{3})$ GeV to get $M^{}_\nu \sim {\cal
O}(10^{-2})$ eV. In this unnatural case, a significant
“structural cancellation" has to be imposed on the textures of
$M^{}_{\rm D}$ and $M^{}_{\rm R}$. Because of $M^{}_{\rm
D}/M^{}_{\rm R} \sim {\cal O}(0.1)$, the non-unitarity of $V$ can
reach the percent level and may lead to observable effects.

Now we discuss how to realize the above “structural cancellation"
for the type-I seesaw mechanism at the TeV scale. For the sake of
simplicity, we take the basis of $M^{}_{\rm R} = {\rm Diag}\{M^{}_1,
M^{}_2, M^{}_3\}$ for three heavy Majorana neutrinos ($N^{}_1,
N^{}_2, N^{}_3$). It is well known that $M^{}_\nu$ vanishes if

\begin{eqnarray}
M^{}_{\rm D} = m \left( \begin{matrix} y^{}_1 & y^{}_2 & y^{}_3 \cr
\alpha y^{}_1 & \alpha y^{}_2 & \alpha y^{}_3 \cr \beta y^{}_1 &
\beta y^{}_2 & \beta y^{}_3 \cr \end{matrix} \right) \; , ~~~~
\sum^3_{i=1} \frac{y^2_i}{M^{}_i} = 0
%     (154)
\end{eqnarray}

simultaneously hold. Tiny neutrino masses can be generated from tiny
corrections to the texture of $M^{}_{\rm D}$ in Eq. (154). For
example, $M^\prime_{\rm D} = M^{}_{\rm D} - \epsilon X^{}_{\rm D}$
with $M^{}_{\rm D}$ given above and $\epsilon$ being a small
dimensionless parameter (i.e., $|\epsilon| \ll 1$) yields

\begin{eqnarray}
M^\prime_\nu = -M^\prime_{\rm D} M^{-1}_{\rm R} M^{\prime T}_{\rm D}
\simeq \epsilon \left( M^{}_{\rm D} M^{-1}_{\rm R} X^T_{\rm D} +
X^{}_{\rm D} M^{-1}_{\rm R} M^T_{\rm D} \right) \; ,
%     (155)
\end{eqnarray}

from which $M^\prime_\nu \sim {\cal O}(10^{-2})$ eV can be obtained
by adjusting the size of $\epsilon$.

A lot of attention has recently been paid to a viable type-I seesaw
model and its collider signatures at the TeV scale. At least the
following lessons can be learnt:

* Two necessary conditions must be satisfied in order to
test a type-I seesaw model at the LHC: (a) $M^{}_i$ are of ${\cal
O}(1)$ TeV or smaller; and (b) the strength of light-heavy neutrino
mixing (i.e., $M^{}_{\rm D}/M^{}_{\rm R}$) is large enough.
Otherwise, it would be impossible to produce and detect $N^{}_i$ at
the LHC.

* The collider signatures of $N^{}_i$ are essentially
decoupled from the mass and mixing parameters of three light
neutrinos $\nu^{}_i$. For instance, the small parameter $\epsilon$
in Eq. (155) has nothing to do with the ratio $M^{}_{\rm
D}/M^{}_{\rm R}$.

* The non-unitarity of $V$ might lead to some
observable effects in neutrino oscillations and other
lepton-flavor-violating or lepton-number-violating processes, if
$M^{}_{\rm D}/M^{}_{\rm R} \leq {\cal O}(0.1)$ holds.

* The clean LHC signatures of heavy Majorana neutrinos are
the $\Delta L =2$ like-sign dilepton events, such as $pp \to
W^{*\pm} W^{*\pm} \to \mu^\pm \mu^\pm jj$ and $pp \to W^{*\pm} \to
\mu^\pm N_i \to \mu^\pm \mu^\pm jj$ (a dominant channel due to the
resonant production of $N^{}_i$).

Some instructive and comprehensive analyses of possible LHC events
for a single heavy Majorana neutrino have recently been done, but
they only serve for illustration because such a simplified type-I
seesaw scenario is actually unrealistic.

§.§ Type-II seesaw

The type-II seesaw formula $M^{}_\nu = Y^{}_\Delta v^{}_\Delta =
\lambda^{}_\Delta Y^{}_\Delta v^2/M^{}_\Delta$ has been given in Eq.
(150). Note that the last term of Eq. (145) violates both $L$ and
$B-L$, and thus the smallness of $\lambda^{}_\Delta$ is naturally
allowed according to 't Hooft's naturalness criterion (i.e., setting
$\lambda^{}_\Delta =0$ will increase the symmetry of ${\cal
L}^{}_{\rm lepton}$). Given $M^{}_\Delta \sim {\cal O}(1)$ TeV, for
example, this seesaw mechanism works to generate $M^{}_\nu \sim
{\cal O}(10^{-2})$ eV provided $\lambda^{}_\Delta Y^{}_\Delta \sim
{\cal O}(10^{-12})$ holds. The neutrino mixing matrix $V$ is exactly
unitary in the type-II seesaw mechanism, simply because the heavy
degrees of freedom do not mix with the light ones.

There are totally seven physical Higgs bosons in the type-II seesaw
scheme: doubly-charged $H^{++}$ and $H^{--}$, singly-charged $H^+$
and $H^-$, neutral $A^0$ (CP-odd), and neutral $h^0$ and $H^0$
(CP-even), where $h^0$ is the SM-like Higgs boson. Except for
$M^2_{h^0}$, we get a quasi-degenerate mass spectrum for other
scalars: $M^2_{H^{\pm \pm}} = M^2_\Delta \approx M^2_{H^0} \approx
M^2_{H^\pm} \approx M^2_{A^0}$. As a consequence, the decay channels
$H^{\pm \pm} \to W^\pm H^\pm$ and $H^{\pm \pm} \to H^\pm H^\pm$ are
kinematically forbidden. The production of $H^{\pm\pm}$ at the LHC
is mainly through $q\bar{q} \to \gamma^*, Z^* \to H^{++}H^{--}$ and
$q\bar{q}^\prime \to W^* \to H^{\pm\pm}H^\mp$ processes, which do
not rely on the small Yukawa couplings.

The typical collider signatures in this seesaw scenario are the
lepton-number-violating $H^{\pm\pm} \to l^\pm_\alpha l^\pm_\beta$
decays as well as $H^+ \to l^+_\alpha \overline{\nu}$ and $H^- \to
l^-_\alpha \nu$ decays. Their branching ratios

\begin{eqnarray}
&& {\cal B}(H^{\pm\pm} \to l^\pm_\alpha l^\pm_\beta) = \frac{
|(M^{}_\nu)^{}_{\alpha\beta}|^2 \left(2 - \delta^{}_{\alpha\beta}
\right)}{\displaystyle \sum_{\rho,\sigma}
|(M^{}_\nu)^{}_{\rho\sigma}|^2} \; ,
\nonumber \\
&& {\cal B}(H^+ \to l^+_\alpha \overline{\nu}) = \frac{\displaystyle
\sum_\beta |(M^{}_\nu)^{}_{\alpha\beta}|^2}{\displaystyle
\sum_{\rho,\sigma} |(M^{}_\nu)^{}_{\rho\sigma}|^2} \;
%     (156)
\end{eqnarray}

are closely related to the masses, flavor mixing angles and
CP-violating phases of three light neutrinos, because $M^{}_\nu = V
\widehat{M}^{}_\nu V^T$ with $\widehat{M}^{}_\nu = {\rm
Diag}\{m^{}_1, m^{}_2, m^{}_3\}$ holds. Some detailed analyses of
such decay modes together with the LHC signatures of $H^{\pm\pm}$
and $H^{\pm}$ bosons have been done in the literature.

It is worth pointing out that the following dimension-6 operator can
easily be derived from the type-II seesaw mechanism,

\begin{eqnarray}
\frac{{\cal L}^{}_{\rm d=6}}{\Lambda^2} =
\left(Y^{}_\Delta\right)^\dagger_{\rho\sigma}}{4 M^2_\Delta}
(\overline{\ell^{}_{\alpha \rm L}} \gamma^\mu \ell^{}_{\sigma \rm
L}) (\overline{\ell^{}_{\beta \rm L}} \gamma^{}_\mu \ell^{}_{\rho
\rm L}) \; , \;\;
%     (157)
\end{eqnarray}

which has two immediate low-energy effects: the non-standard
interactions of neutrinos and the lepton-flavor-violating
interactions of charged leptons. An analysis of such effects
provides us with some preliminary information:

* The magnitudes of non-standard interactions of
neutrinos and the widths of lepton-flavor-violating tree-level
decays of charged leptons are both dependent on neutrino masses
$m^{}_i$ and flavor-mixing and CP-violating parameters of $V$.

* For a long-baseline neutrino oscillation experiment, the
neutrino beam encounters the earth matter and the electron-type
non-standard interaction contributes to the matter potential.

* At a neutrino factory, the lepton-flavor-violating
processes $\mu^-\rightarrow e^-\nu^{}_e\overline{\nu}^{}_\mu$ and
$\mu^+\rightarrow e^+\overline{\nu}_e\nu^{}_\mu$ could cause some
wrong-sign muons at a near detector.

Current experimental constraints tell us that such low-energy
effects are very small, but they might be experimentally accessible
in the future precision measurements.

§.§ Type-(I+II) seesaw

The type-(I+II) seesaw mechanism can be achieved by combining the
neutrino mass terms in Eqs. (144) and (145). After spontaneous gauge
symmetry breaking, we are left with the overall neutrino mass term

\begin{eqnarray}
-{\cal L}^{}_{\rm mass} = \frac{1}{2} ~ \overline{\left( \nu^{}_{\rm
L} N^{\rm c}_{\rm R}\right)} \left( \begin{matrix} M^{}_{\rm L} &
M^{}_{\rm D} \cr M^T_{\rm D} & M^{}_{\rm R} \end{matrix} \right)
\left(
\begin{matrix} \nu^{\rm c}_{\rm L} \cr N^{}_{\rm R} \end{matrix}
\right) + {\rm h.c.} \; ,
%     (158)
\end{eqnarray}

where $M^{}_{\rm D} = Y^{}_\nu v/\sqrt{2}$ and $M^{}_{\rm L} =
Y^{}_\Delta v^{}_\Delta$ with $\langle H \rangle \equiv v/\sqrt{2}$
and $\langle \Delta \rangle \equiv v^{}_\Delta$ corresponding to the
vacuum expectation values of the neutral components of the Higgs
doublet $H$ and the Higgs triplet $\Delta$. The $6\times 6$ mass
matrix in Eq. (158) is symmetric and can be diagonalized by the
unitary transformation done in Eq. (28); i.e.,

\begin{eqnarray}
\left( \begin{matrix} V & R \cr S & U \end{matrix} \right)^\dagger
\left( \begin{matrix} M^{}_{\rm L} & M^{}_{\rm D} \cr M^T_{\rm D} &
M^{}_{\rm R} \end{matrix} \right) \left(\begin{matrix} V & R \cr S &
U \end{matrix} \right)^* = \left( \begin{matrix} \widehat{M}^{}_\nu
& {\bf 0} \cr {\bf 0} & \widehat{M}^{}_N
\end{matrix} \right) \; ,
%     (159)
\end{eqnarray}

where $\widehat{M}^{}_\nu = {\rm Diag}\{m^{}_1, m^{}_2, m^{}_3\}$
and $\widehat{M}^{}_N = {\rm Diag}\{M^{}_1, M^{}_2, M^{}_3\}$.
Needless to say, $V^\dagger V + S^\dagger S = VV^\dagger +
RR^\dagger = {\bf 1}$ holds as a consequence of the unitarity of
this transformation. Hence $V$, the flavor mixing matrix of light
Majorana neutrinos, must be non-unitary if $R$ and $S$ are non-zero.

In the leading-order approximation, the type-(I+II) seesaw formula
reads as

\begin{equation}
M^{}_\nu \approx M^{}_{\rm L} - M^{}_{\rm D} M^{-1}_{\rm R} M^T_{\rm
D} \; .
%     (160)
\end{equation}

Hence type-I and type-II seesaws can be regarded as two extreme
cases of the type-(I+II) seesaw. Note that two mass terms in Eq.
(160) are possibly comparable in magnitude. If both of them are
small, their contributions to $M^{}_\nu$ may have significant
interference effects which make it practically impossible to
distinguish between type-II and type-(I+II) seesaws; but if both of
them are large, their contributions to $M^{}_\nu$ must be
destructive. The latter case unnaturally requires a significant
cancellation between two big quantities in order to obtain a small
quantity, but it is interesting in the sense that it may give rise
to possibly observable collider signatures of heavy Majorana

Let me briefly describe a particular type-(I+II) seesaw model and
comment on its possible LHC signatures. First, we assume that both
$M^{}_i$ and $M^{}_\Delta$ are of ${\cal O}(1)$ TeV. Then the
production of $H^{\pm\pm}$ and $H^\pm$ bosons at the LHC is
guaranteed, and their lepton-number-violating signatures will probe
the Higgs triplet sector of the type-(I+II) seesaw mechanism. On the
other hand, ${\cal O}(M^{}_{\rm D}/M^{}_{\rm R}) \leq {\cal O}(0.1)$
is possible as a result of ${\cal O}( M^{}_{\rm R}) \sim {\cal
O}(1)$ TeV and ${\cal O}(M^{}_{\rm D}) \leq {\cal O}(v)$, such that
appreciable signatures of $N^{}_i$ can be achieved at the LHC.
Second, the small mass scale of $M^{}_\nu$ implies that the relation
${\cal O}(M^{}_{\rm L}) \sim {\cal O}(M^{}_{\rm D} M^{-1}_{\rm R}
M^T_{\rm D})$ must hold. In other words, it is the significant but
incomplete cancellation between $M^{}_{\rm L}$ and $M^{}_{\rm D}
M^{-1}_{\rm R} M^T_{\rm D}$ terms that results in the non-vanishing
but tiny masses for three light neutrinos. We admit that dangerous
radiative corrections to two mass terms of $M^{}_\nu$ require a
delicate fine-tuning of the cancellation at the loop level. But this
scenario allows us to reconstruct $M^{}_{\rm L}$ via the excellent
approximation $M^{}_{\rm L} = V \widehat{M}^{}_\nu V^T + R
\widehat{M}^{}_N R^T \approx R \widehat{M}^{}_N R^T$, such that the
elements of the Yukawa coupling matrix $Y^{}_\Delta$ read as

\begin{equation}
\left(Y^{}_\Delta\right)^{}_{\alpha \beta} \; = \;
\frac{\left(M^{}_{\rm L}\right)^{}_{\alpha \beta}}{v^{}_\Delta}
\approx \sum^3_{i=1} \frac{R^{}_{\alpha i} R^{}_{\beta i}
M^{}_i}{v^{}_\Delta} \; ,
%     (161)
\end{equation}

where the subscripts $\alpha$ and $\beta$ run over $e$, $\mu$ and
$\tau$. This result implies that the leptonic decays of $H^{\pm
\pm}$ and $H^\pm$ bosons depend on both $R$ and $M^{}_i$, which
actually determine the production and decays of $N^{}_i$. Thus we
have established an interesting correlation between the singly- or
doubly-charged Higgs bosons and the heavy Majorana neutrinos. To
observe the correlative signatures of $H^\pm$, $H^{\pm\pm}$ and
$N^{}_i$ at the LHC will serve for a direct test of this type-(I+II)
seesaw model.

§.§ Type-III seesaw

The lepton mass terms in the type-III seesaw scheme have already
been given in Eq. (147). After spontaneous gauge symmetry breaking,
we are left with

\begin{eqnarray}
&& -{\cal L}^{}_{\rm mass} = \frac{1}{2} ~\overline{\left(
\nu^{}_{\rm L} ~ \Sigma^0 \right)} \left( \begin{matrix} {\bf 0} &
M^{}_{\rm D} \cr M^T_{\rm D} & M^{}_\Sigma \cr \end{matrix} \right)
\left(
\begin{matrix} \nu^{\rm c}_{\rm L} \cr {\Sigma^{0}}^{\rm c} \cr
\end{matrix} \right) + {\rm h.c.} \; ,
\nonumber \\
&& -{\cal L}^\prime_{\rm mass} = \overline{\left( e^{}_{\rm L} ~
\Psi^{}_{\rm L} \right)} \left( \begin{matrix} M^{}_l & \sqrt{2}
M^{}_{\rm D} \cr {\bf 0} & M^{}_\Sigma \cr \end{matrix} \right)
\left( \begin{matrix} E^{}_{\rm R} \cr \Psi^{}_{\rm R} \cr
\end{matrix} \right) + {\rm h.c.} \; , ~~~~~
%     (162)
\end{eqnarray}

respectively, for neutral and charged fermions, where $M^{}_l =
Y^{}_l v/\sqrt{2}~$, $M^{}_{\rm D} = Y^{}_\Sigma v/\sqrt{2}~$, and
$\Psi = \Sigma^- + {\Sigma^+}^{\rm c}$. The symmetric $6\times 6$
neutrino mass matrix can be diagonalized by the following unitary

\begin{eqnarray}
\left( \begin{matrix} V & R \cr S & U \cr \end{matrix}
\right)^\dagger \left( \begin{matrix} {\bf 0} & M^{}_{\rm D} \cr
M^T_{\rm D} & M^{}_\Sigma \cr \end{matrix} \right) \left(
\begin{matrix} V & R \cr S & U \cr \end{matrix} \right)^*
= \left( \begin{matrix} \widehat{M}^{}_\nu & {\bf 0} \cr {\bf 0} &
\widehat{M}^{}_\Sigma \cr \end{matrix} \right) \; ,
%     (163)
\end{eqnarray}

where $\widehat{M}^{}_\nu = {\rm Diag}\{m^{}_1, m^{}_2, m^{}_3 \}$
and $\widehat{M}^{}_\Sigma = {\rm Diag}\{M^{}_1, M^{}_2, M^{}_3 \}$.
In the leading-order approximation, this diagonalization yields the
type-III seesaw formula $M^{}_\nu = -M^{}_{\rm D} M^{-1}_\Sigma
M^T_{\rm D}$, which is equivalent to the one derived from the
effective dimension-5 operator in Eq. (150). Let us use one sentence
to comment on the similarities and differences between type-I and
type-III seesaw mechanisms: the non-unitarity of the $3\times 3$
neutrino mixing matrix $V$ has appeared in both cases, although the
modified couplings between the $Z^0$ boson and three light neutrinos
differ and the non-unitary flavor mixing is also present in the
couplings between the $Z^0$ boson and three charged leptons in the
type-III seesaw scenario.

At the LHC, the typical lepton-number-violating signatures of the
type-III seesaw mechanism can be $pp \to \Sigma^+ \Sigma^0 \to
l^+_\alpha l^+_\beta + Z^0W^-(\to 4j)$ and $pp \to \Sigma^- \Sigma^0
\to l^-_\alpha l^-_\beta + Z^0W^+(\to 4j)$ processes. A detailed
analysis of such collider signatures have been done in the
literature. As for the low-energy phenomenology, a consequence of
this seesaw scenario is the non-unitarity of the $3\times 3$ flavor
mixing matrix $N$ ($\approx V$) in both charged- and neutral-current
interactions. Current experimental bounds on the deviation of
$NN^\dagger$ from the identity matrix are at the $0.1\%$ level, much
stronger than those obtained in the type-I seesaw scheme, just
because the flavor-changing processes with charged leptons are
allowed at the tree level in the type-III seesaw mechanism.

§.§ Inverse and multiple seesaws

Given the naturalness and testability as two prerequisites, the
double or inverse seesaw mechanism is another interesting
possibility of generating tiny neutrino masses at the TeV scale. The
idea of this seesaw picture is to add three heavy right-handed
neutrinos $N^{}_{\rm R}$, three SM gauge-singlet neutrinos
$S^{}_{\rm R}$ and one Higgs singlet $\Phi$ into the SM, such that
the gauge-invariant lepton mass terms can be written as

\begin{eqnarray}
-{\cal L}^{}_{\rm lepton} = \overline{l^{}_{\rm L}} Y^{}_l H
E^{}_{\rm R} + \overline{l^{}_{\rm L}} Y^{}_\nu \tilde{H} N^{}_{\rm
R} + \overline{N^{\rm c}_{\rm R}} Y^{}_S \Phi S^{}_{\rm R} +
\frac{1}{2} \overline{S^{\rm c}_{\rm R}} \mu S^{}_{\rm R} + {\rm
h.c.} \; ,
%     (164)
\end{eqnarray}

where the $\mu$-term is naturally small according to 't Hooft's
naturalness criterion, because it violates the lepton number. After
spontaneous gauge symmetry breaking, the overall neutrino mass term
turns out to be

\begin{eqnarray}
-{\cal L}^{}_{\rm mass} \; =\; \frac{1}{2}
~\overline{\left(\nu^{}_{\rm L} ~N^{\rm c}_{\rm R} ~S^{\rm c}_{\rm
R}\right)} \left( \begin{matrix} {\bf 0} & M^{}_{\rm D} & {\bf 0}
\cr M^T_{\rm D} & {\bf 0} & M^{}_S \cr {\bf 0} & M^T_S & \mu \cr
\end{matrix} \right) \left( \begin{matrix} \nu^{\rm c}_{\rm L} \cr N^{}_{\rm
R} \cr S^{}_{\rm R} \cr \end{matrix} \right) \; ,
%     (165)
\end{eqnarray}

where $M^{}_{\rm D} = Y^{}_\nu \langle H\rangle$ and $M^{}_S =
Y^{}_S \langle \Phi\rangle$. A diagonalization of the symmetric
$9\times 9$ matrix $\cal M$ leads us to the effective light neutrino
mass matrix

\begin{equation}
M^{}_\nu \approx M^{}_{\rm D} \frac{1}{M^T_S} \mu \frac{1}{M^{}_S}
M^T_{\rm D} \;
%     (166)
\end{equation}

in the leading-order approximation. Hence the smallness of
$M^{}_\nu$ can be attributed to both the smallness of $\mu$ itself
and the doubly-suppressed $M^{}_{\rm D}/M^{}_S$ term for $M^{}_{\rm
D} \ll M^{}_S$. For example, $\mu \sim {\cal O}(1)$ keV and
$M^{}_{\rm D}/M^{}_S \sim {\cal O}(10^{-2})$ naturally give rise to
a sub-eV $M^{}_\nu$. One has $M^{}_\nu = {\bf 0}$ in the limit $\mu
\rightarrow {\bf 0}$, which reflects the restoration of the
slightly-broken lepton number. The heavy sector consists of three
pairs of pseudo-Dirac neutrinos whose CP-conjugated Majorana
components have a tiny mass splitting characterized by the order of

Going beyond the canonical (type-I) and inverse seesaw mechanisms,
one may build the so-called “multiple" seesaw mechanisms to further
lower the seesaw scales.

§ NON-UNITARY NEUTRINO MIXING

It is worth remarking that the charged-current interactions of light
and heavy Majorana neutrinos are not completely independent in
either the type-I seesaw or the type-(I+II) seesaw. The standard
charged-current interactions of $\nu^{}_i$ and $N^{}_i$ are already
given in Eq. (34), where $V$ is just the light neutrino mixing
matrix responsible for neutrino oscillations, and $R$ describes the
strength of charged-current interactions between $(e, \mu, \tau)$
and $(N^{}_1, N^{}_2, N^{}_3)$. Since $V$ and $R$ belong to the same
unitary transformation done in Eq. (28) or Eq. (159), they must be
correlated with each other and their correlation signifies an
important relationship between neutrino physics and collider

It can be shown that $V$ and $R$ share nine rotation angles
($\theta^{}_{i4}$, $\theta^{}_{i5}$ and $\theta^{}_{i6}$ for $i=1$,
$2$ and $3$) and nine phase angles ($\delta^{}_{i4}$,
$\delta^{}_{i5}$ and $\delta^{}_{i6}$ for $i=1$, $2$ and $3$). To
see this point clearly, let us decompose $V$ into $V = A V^{}_0$,
where $V^{}_0$ is the standard (unitary) parametrization of the
$3\times 3$ PMNS matrix in which three CP-violating phases
$\delta^{}_{ij}$ (for $ij=12, 13, 23$) are associated with
$s^{}_{ij}$ (i.e., $c^{}_{ij} \equiv \cos\theta^{}_{ij}$ and
$\hat{s}^{}_{ij} \equiv e^{i\delta^{}_{ij}} \sin\theta^{}_{ij}$).
Because of $VV^\dagger = AA^\dagger = {\bf 1} - RR^\dagger$, it is
obvious that $V \rightarrow V^{}_0$ in the limit of $A \rightarrow
{\bf 1}$ (or equivalently, $R \rightarrow {\bf 0}$). Considering the
fact that the non-unitarity of $V$ must be a small effect (at most
at the percent level as constrained by current neutrino oscillation
data and precision electroweak data), we expect $s^{}_{ij} \leq
{\cal O}(0.1)$ (for $i=1,2,3$ and $j=4,5,6$) to hold. Then we obtain

\begin{eqnarray}
R = \left( \begin{matrix} \hat{s}^*_{14} & \hat{s}^*_{15} &
\hat{s}^*_{16} \cr \hat{s}^*_{24} & \hat{s}^*_{25} & \hat{s}^*_{26}
\cr \hat{s}^*_{34} & \hat{s}^*_{35} & \hat{s}^*_{36} \cr
\end{matrix} \right)
%     (167)
\end{eqnarray}

as an excellent approximations. A striking consequence of the
non-unitarity of $V$ is the loss of universality for the Jarlskog
invariants of CP violation, $J^{ij}_{\alpha\beta} \equiv {\rm
Im}(V^{}_{\alpha i} V^{}_{\beta j} V^*_{\alpha j} V^*_{\beta i})$,
where the Greek indices run over $(e, \mu, \tau)$ and the Latin
indices run over $(1,2,3$). For example, the extra CP-violating
phases of $V$ are possible to give rise to a significant asymmetry
between $\nu^{}_\mu \rightarrow \nu^{}_\tau$ and
$\overline{\nu}^{}_\mu \rightarrow \overline{\nu}^{}_\tau$

The probability of $\nu^{}_\alpha \rightarrow \nu^{}_\beta$
oscillations in vacuum, defined as $P^{}_{\alpha\beta}$, is given by

\begin{eqnarray}
P^{}_{\alpha\beta} = \frac{\displaystyle \sum^{}_i |V^{}_{\alpha
i}|^2 |V^{}_{\beta i}|^2 + 2 \sum^{}_{i<j} {\rm Re} \left(
V^{}_{\alpha i} V^{}_{\beta j} V^*_{\alpha j} V^*_{\beta i} \right)
\cos \Delta^{}_{ij} - \sum^{}_{i<j} J^{ij}_{\alpha\beta}
\sin\Delta^{}_{ij}} {\displaystyle \left(
VV^\dagger\right)^{}_{\alpha\alpha} \left(
VV^\dagger\right)^{}_{\beta\beta}} \; ,
%     (168)
\end{eqnarray}

where $\Delta^{}_{ij} \equiv \Delta m^2_{ij} L/(2E)$ with $\Delta
m^2_{ij} \equiv m^2_i - m^2_j$, $E$ being the neutrino beam energy
and $L$ being the baseline length. If $V$ is exactly unitary (i.e.,
$A = {\bf 1}$ and $V = V^{}_0$), the denominator of Eq. (168) will
become unity and the conventional formula of $P^{}_{\alpha\beta}$
will be reproduced. Note that $\nu^{}_\mu \rightarrow \nu^{}_\tau$
and $\overline{\nu}^{}_\mu \rightarrow \overline{\nu}^{}_\tau$
oscillations may serve as a good tool to probe possible signatures
of non-unitary CP violation. To illustrate this point, we consider a
short- or medium-baseline neutrino oscillation experiment with
$|\sin\Delta^{}_{13}| \sim |\sin\Delta^{}_{23}| \gg
|\sin\Delta^{}_{12}|$, in which the terrestrial matter effects are
expected to be insignificant or negligibly small. Then the dominant
CP-conserving and CP-violating terms of $P(\nu^{}_\mu \rightarrow
\nu^{}_\tau)$ and $P(\overline{\nu}^{}_\mu \rightarrow
\overline{\nu}^{}_\tau)$ are

\begin{eqnarray}
P(\nu^{}_\mu \rightarrow \nu^{}_\tau) \approx \sin^2 2\theta^{}_{23}
\sin^2 \frac{\Delta^{}_{23}}{2} - 2 \left( J^{23}_{\mu\tau} +
J^{13}_{\mu\tau} \right) \sin\Delta^{}_{23} \; ,
\nonumber \\
P(\overline{\nu}^{}_\mu \rightarrow \overline{\nu}^{}_\tau) \approx
\sin^2 2\theta^{}_{23} \sin^2 \frac{\Delta^{}_{23}}{2} + 2 \left(
J^{23}_{\mu\tau} + J^{13}_{\mu\tau} \right) \sin\Delta^{}_{23} \; ,
%     (169)
\end{eqnarray}

where the good approximation $\Delta^{}_{13} \approx \Delta^{}_{23}$
has been used in view of the experimental fact $|\Delta m^2_{13}|
\approx |\Delta m^2_{23}| \gg |\Delta m^2_{12}|$, and the
sub-leading and CP-conserving “zero-distance" effect has been
omitted. For simplicity, I take $V^{}_0$ to be the exactly
tri-bimaximal mixing pattern (i.e., $\theta^{}_{12} =
\arctan(1/\sqrt{2})$, $\theta^{}_{13} =0$ and $\theta^{}_{23}
=\pi/4$ as well as $\delta^{}_{12} = \delta^{}_{13} = \delta^{}_{23}
=0$) and then arrive at

\begin{equation}
2\left( J^{23}_{\mu\tau} + J^{13}_{\mu\tau} \right) \; \approx \;
\sum^6_{l=4} s^{}_{2l} s^{}_{3l} \sin \left( \delta^{}_{2l} -
\delta^{}_{3l} \right) \; .
%     (170)
\end{equation}

Given $s^{}_{2l} \sim s^{}_{3l} \sim {\cal O}(0.1)$ and
$(\delta^{}_{2l} - \delta^{}_{3l}) \sim {\cal O}(1)$ (for
$l=4,5,6$), this non-trivial CP-violating quantity can reach the
percent level. When a long-baseline neutrino oscillation experiment
is concerned, however, the terrestrial matter effects must be taken
into account because they might fake the genuine CP-violating
signals. As for $\nu^{}_\mu \rightarrow \nu^{}_\tau$ and
$\overline{\nu}^{}_\mu \rightarrow \overline{\nu}^{}_\tau$
oscillations under discussion, the dominant matter effect results
from the neutral-current interactions and modifies the CP-violating
quantity of Eq. (170) in the following way:

\begin{eqnarray}
2\left( J^{23}_{\mu\tau} + J^{13}_{\mu\tau} \right) \Longrightarrow
\sum^6_{l=4} s^{}_{2l} s^{}_{3l} \left[ \sin \left( \delta^{}_{2l} -
\delta^{}_{3l} \right) + A^{}_{\rm NC} L \cos \left( \delta^{}_{2l}
- \delta^{}_{3l} \right) \right] \; ,
%     (171)
\end{eqnarray}

where $A^{}_{\rm NC} = G^{}_{\rm F} N^{}_n /\sqrt{2}~$ with $N^{}_n$
being the background density of neutrons, and $L$ is the baseline
length. It is easy to find $A^{}_{\rm NC} L \sim {\cal O}(1)$ for $L
\sim 4 \times 10^3$ km.

§ CONCLUDING REMARKS

I have briefly described some basic properties of massive neutrinos
in an essentially model-independent way in these lectures, which are
largely based on the book by Dr. Shun Zhou and myself <cit.> and
on a few review articles or lectures [2]—[6].
It is difficult to cite all the relevant references. I apologize for
missing other people's works due to the tight page limit of these
proceedings. For the same reason I am unable to write in the
cosmological matter-antimatter asymmetry and the leptogenesis
mechanism, although they were discussed in my lectures. Here let me
just give a few remarks on the naturalness and testability of TeV
seesaw mechanisms.

Although the seesaw ideas are elegant, they have to appeal for some
or many new degrees of freedom in order to interpret the observed
neutrino mass hierarchy and lepton flavor mixing. According to
Weinberg's third law of progress in theoretical physics, “you
may use any degrees of freedom you like to describe a physical
system, but if you use the wrong ones, you will be sorry." What
could be better?

Anyway, we hope that the LHC might open a new window for us to
understand the origin of neutrino masses and the dynamics of lepton
number violation. A TeV seesaw might work (naturalness?) and
its heavy degrees of freedom might show up at the LHC (testability?). A bridge between collider physics and neutrino
physics is highly anticipated and, if it exists, will lead to rich

I am indebted to the organizers of AEPSHEP 2012 for their invitation
and hospitality. This work is supported in part by the National
Natural Science Foundation of China under grant No. 11135009.

[1] Z.Z. Xing and S. Zhou, Neutrinos in Particle Physics,
Astronomy and Cosmology, Zhejiang University Press and
Springer-Verlag (2011).

[2] Z.Z. Xing, plenary talk given at ICHEP2008; Int.
J. Mod. Phys. A 23, 4255 (2008).

[3] Z.Z. Xing, Prog. Theor. Phys. Suppl. 180, 112 (2009).

[4] H. Fritzsch and Z.Z. Xing, Prog. Part. Nucl. Phys.
45, 1 (2000);

[5] Z.Z. Xing, Int. J. Mod. Phys. A 19, 1

[6] Z.Z. Xing, Lectures given at the 2010 Schladming Winter
School on Masses and Constants, Austria, 2010; published in
Nucl. Phys. Proc. Suppl. 203-204, 82 (2010).


# Preparation in Bohmian Mechanics

Carlo Rovelli  Aix-Marseille University, Université de Toulon, CPT-CNRS,
F-13288 Marseille, France.  Perimeter Institute, 31 Caroline Street N,
Waterloo ON, N2L2Y5, Canada  The Rotman Institute of Philosophy, 1151 Richmond
St. N London N6A5B7, Canada

###### Abstract

According to Bohmian mechanics, we see the particle, not the pilot wave. But
to make predictions we need to know the wave. How do we learn about the wave
to make predictions, if we only see the particle? I show that the puzzle can
be solved, but only thanks to decoherence.

## The problem

In Bohmiam mechanics Goldstein (2017), the state of a system of $N$ degrees of
freedom (or “particles”) is given by a couple $(\Psi(x_{n}),X_{n})$, namely a
wave function $\Psi(x_{n})$, or “pilot wave”, and the positions $X_{n}$ of the
particles. Here $n=1,...,N$. The dynamics is given by the two equations

$i\hbar\;\partial_{t}\Psi=H\Psi,\ \ \ \
\partial_{t}X_{n}=\frac{\hbar}{m_{n}}\left.\frac{\partial_{x_{n}}\Psi}{|\Psi|}\right|_{x_{n}=X_{n}},$
(1)

where $H$ is the Hamiltonian operator and $m_{n}$ the mass of the $n$th
particle. The first is the Schrödinger equation, the second is Bohm’s guidance
equation.

A theorem states that if at some time we do not know the position of the
particles but we know their probability distribution and this satisfies
$\rho({X_{n}})=|\Psi(X_{n})|^{2}$, then the dynamics conserves this equation.
The value of this equation “initially” is either assumed or said to be
attained via a suitable equilibration process Valentini (2010), depending on
the versions of the Bohmiam theory considered. In either case, if we know the
wave function, we can give probabilistic predictions about where the particles
will go.

According to the theory, what we see, are the actual positions $X_{n}$ of the
variables (“particles”). Hence, in a sense, the particle positions are
“manifest” variables and the wave is “hidden”. In a different sense, the
particle positions are called the “hidden variables”, because they are
variables added to conventional wave mechanics to make it (non local and)
deterministic.

There is one aspect of this theory that I have never understood. In order to
make predictions, we have to know $\Psi$. But how can we know $\Psi$, if what
we see is only the particle positions $X_{n}$? The theory gives us a
probability distribution for the particle position, given the wave; but it
does not give us a recipe for guessing the wave, given the particle positions
that we see. (On the problem, see Bohm and Hiley (1984); Bohm  _et al._
(1987).)

In textbook quantum mechanics, the state is determined by a _preparation_ ,
which is essentially a measurement. A measurement, in conventional quantum
theory, _collapses_ the wave function on the eigenstate of the observed
eigenvalue. Hence we know the state because we made an observation, and we the
theory instructs us to _assume_ that the state is then the corresponding
eigenstate. If we see spin up, we then say the state is $|+\rangle$.

But nothing of the sort is directly true in Bohmiam mechanics: if we see the
particle in the position $X$, the wave can still be virtually anything. Hence
we cannot infer much about the wave, from observing particles. Hence we cannot
measure the wave. But then how can we make predictions, given that the wave is
needed for predictions?

Here, I give an answer to this question. Maybe this answer is known, but I
have not found it in the literature. The answer is more subtle than what I
expected, requires thinking about decoherence and does not require any
hypothesis about special states in the past.

## An insufficient solution

In Bohmian literature, one reads that after a standard measurement the state
of a System (with variable $s$) and Apparatus (with pointer variable $a$) has
the form

$\Psi(s,a)=\sum_{n}c_{v}\ \psi_{n}(s)\phi_{n}(a),$ (2)

where $n$ labels the eigenvalues of measured quantity, $\psi_{n}$ are the
corresponding eigenstates, and $\phi_{n}$ are states of the pointer variable
that are peaked on well disjoint regions $R_{n}$ of its configuration space.
If the Bohmian pointer variable $a=A$ is in one of these regions, then the
_effective_ state of the system $\psi(s)=\Psi(s,A)$ is in fact in the
eigenstate of the measured quantity.

All this is fine, of course, but steps are missing. The key one is that system
and apparatus do _not_ end up in a state of the form (2) if they are in a
generic state before the measurement interaction. This follows immediately
from the fact the evolution during the interaction is unitary, hence a generic
state ends up in a generic state, while the states (2) are not generic. To end
up in the form (2), the combined system (system+apparatus) must be in a tensor
state, namely uncorrelated, before the interaction. But how can we know that
it is so, if not by measurement? So, to prepare a state, we already need to
know a state. This appears to open a regress to infinity.

Furthermore, even if we get to know the effective state, how do we know that
other components of the universal wave function don’t come-in, and affect the
evolution governed by effective states? Here I try to fill these gaps.

## The key preliminary corollary

Consider a single particle. The wave is $\Psi(x)$ and the particle position is
$X$. Imagine that at some time the wave has two components disjoint in
configuration space. That is $\Psi(x)=\Psi_{1}(x)+\Psi_{2}(x)$ where the
support of $\Psi_{1}$ is a compact region $R_{1}$ and the support of
$\Psi_{2}$ is a compact region $R_{2}$ that does not overlap with $R_{1}$.
Then by the probability assumption the particle must be either in $R_{1}$ or
in $R_{2}$. Say it is in $R_{1}$. Then, as long as $\Psi_{1}$ and $\Psi_{2}$
do not interfere, we can replace $\Psi$ with $\Psi_{1}$ in the guidance
equation for $X$.

This is immediate, because the guidance equation depends only on the value of
$\Psi$ in the neighbourhood of $x=X$ and $\Psi_{2}$ vanishes there. Since the
Schrödinger equation is linear, we can evolve $\Psi_{1}$ with this equation
and disregard $\Psi_{2}$ entirely.

All this stops to hold, of course, if there is interference, namely if at some
time $R_{1}$ and $R_{2}$ get to overlap.

Next consider the same single particle, but now interacting with an
environment of many degrees of freedom $e_{n}$. The wave is $\Psi(x,e_{n})$
and the positions are $(X,E_{n})$. Imagine, as before, that at some time
$\Psi$ has two components $\Psi_{1}$ and $\Psi_{2}$ such that the supports of
$\Psi_{1}(x,E_{n})$ and $\Psi_{2}(x,E_{n})$ are disjoint in the space of the
variable $x$. Then, as before, we can disregard $\Psi_{2}$ for the dynamics of
the particle, as long as there is no interference. But let’s furthermore,
imagine that the interaction between the particle and the environment is non
negligible (in a precise sense specified below). Then, interference is
suppressed. This is because the probability distribution for the particle
position is given by

$\displaystyle\rho(x)$ $\displaystyle=$
$\displaystyle|\Psi(x,E_{n})|^{2}=|\Psi_{1}(x,E_{n})+\Psi_{2}(x,E_{n})|^{2}$
$\displaystyle=$
$\displaystyle|\Psi_{1}(x,E_{n})|^{2}+|\Psi_{2}(x,E_{n})|^{2}$
$\displaystyle+Re[\overline{\Psi_{1}(x,E_{n})}\Psi_{2}(x,E_{n})].$

Interference is given by the last term, but if the coupling with the
environment is sufficient, the two states $\Psi_{1}$ and $\Psi_{2}$ affect the
environment differently and therefore $\Psi_{1}(x,e_{n})$ and
$\Psi_{2}(x,e_{n})$ are going to have different support in the $e_{n}$
variables. But then there is no value of $E_{n}$ that can prevent this term
from vanishing. This is essentially the Bohmian version of decoherence.

The consequence is that, once more, $\Psi_{2}$ can be discarded, as far the
motion of $X$ is concerned, but now without limitations: there is no risk of
interferences bringing back $\Psi_{2}$ into the dynamics of $X$.

## The effective loss of quantum entanglement

Consider now two systems, with position variables $S$ and $A$, and state
$\psi(s,a)$, in an entangled state. That is

$\Psi(s,a)\neq\Psi(s)\Phi(a).$ (4)

For simplicity, consider an entangled state of the form

$\Psi(s,a)=\Psi_{1}(s,a)+\Psi_{2}(s,a)=\Psi_{1}(s)\Phi_{1}(a)+\Psi_{2}(s)\Phi_{2}(a).$
(5)

Immagine that the system $A$ interacts sufficiently strongly with an
environment. Say $\Phi_{1}(a)$ and $\Phi_{2}(a)$ have disjoint support, as
before, and the variable $A$ is in the support of $\Phi_{1}(a)$. Then, for the
argument above, the state $\Psi_{2}$ can be discarded, for the future
evolution of the variable $A$ and $S$.

This means that the interaction of a system $A$ with the environment has the
effect to cancel any interference effect due to any entanglement between $A$
and another system $S$. Once again, this can be recognised as a typical effect
of decoherence. Hence if $A$ interacts sufficiently with an environment, we
can always effectively assume that its state is not entangled with another
system $S$.

Armed with this, we can tackle the preparation of a state in Bohmian
mechanics.

## Preparation

Consider a variable $S$ (System) and a variable $A$ (apparatus, or pointer
variable) interacting with an environment. Say the interaction hamiltonian
between $S$ and $A$ depends on a function $O$ on the phase space of the
coupled system, and is turned on at some time $t$. Assume that before this
time, $A$ has sufficiently interacted with the environment. Then, because of
the argument given above, we can assume that at time $t$ the state is a tensor
state. More precisely, as the variable $A$ has sufficiently interacted with
the environment, the system and the apparatus can be effectively considered
uncorrelated. The empty components of the universal wave are there, but do not
disturb. This is the missing ingredient in previous analysis of Bohmian
measurement that I found.

Let $\psi(s,a)=\Psi(s,a,E_{n})$ be the effective state of System+Apparatus.
Its evolution is effectively governed by a Schrödinger equation. The standard
Schrödinger evolution of an initial uncorrelated state
$\psi(s,a)=\psi_{A}(a)\psi_{S}(s)$ by an interaction hamiltonian that depends
on an observable $O$ drives the state to the well known form

$\psi(s,a)=\sum_{n}c_{v}\ \psi_{n}(s)\phi_{n}(a),$ (6)

as above, where $\phi_{n}$ are states of the pointer variable that are peaked
on well disjoint regions $R_{n}$ of the configuration space of $a$. In Bohm
theory, the pointer $A$ is going to be in one of these, say the $n$th one. By
the results of the previous section, because of the fact that $A$ interacts
with the environment, any interference between the terms of the last equation
is suppressed and therefore the future evolution of the variable $S$ is going
to be guided by the single eigenstate $\psi_{n}$ that correspond to the non
empty branch of the apparatus wave. This is preparation of a state in Bohmian
mechanics.

## Conclusion

Contrary to the expectation I had when I started looking into this, it works.
The process corresponding to decoherence plays a major role. The trick is that
it is true that knowledge of the $X_{n}$ gives us very little information
about the $\Psi$, but this can be circumvented by having decoherence rendering
most components of $\Psi$ irrelevant and arranging the dynamics so that the
relevant part of the effective state $\psi$ is a finite sum of terms we know,
so that the pointer variable being in one region restricts the effective wave
function of the system to a known form. And this is of course exactly what
experimenters do in a real laboratory experiment.

To summarize: first, the environment-apparatus interaction renders the
interference between states with different values of the pointer variable
irrelevant. Second, we can write a _generic_ state of the combined
(system+apparatus) as a sum of tensor terms peaked on different values of the
pointer, and because of the previous point, we can forget the empty branches.
Hence we can safely assume that before the start of the interaction system and
apparatus are uncorrelated. The measurement interaction correlates them.
Finally, the value of the pointer value tells us precisely that the only
relevant branch of the effective state of the system is the corresponding
eigenstate.

I embarked in this exploration thinking that Bohiam mechanics does not really
work, but, as far as I can see, it works. There is no need for this to make
any special (“past low entropy”) assumption (as I at some point suspected),
but decoherence is necessary to make predictions.

***

Many thanks to Wayne Myrvold, Davide Romano, and David Albert for comments,
enlightening discussions, and corrections to the first draft of this paper.

## References

  * Goldstein (2017) Sheldon Goldstein, “Bohmian Mechanics,” in  _The Stanford Encyclopedia of Philosophy_, edited by Edward N. Zalta (2017).
  * Valentini (2010) Antony Valentini, “Inflationary Cosmology as a Probe of Primordial Quantum Mechanics,” Phys. Rev. D 82, 063513 (2010).
  * Bohm and Hiley (1984) D. Bohm and B.J. Hiley, “Measurement Understood Through the Quantum Potential Approach,” Foundations of Physics 14, 255 (1984).
  * Bohm  _et al._ (1987) D. Bohm, B.J. Hiley, and P.N. Kaloyerou, “An Ontological basis for the Quantum Theory,” Physics reports 144, 321–375 (1987).