#  Neutrino Anarchy from Flavor Deconstruction

Admir Greljo<EMAIL_ADDRESS>Department of Physics, University of
Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Gino Isidori
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerland

###### Abstract

Flavor deconstruction refers to non-universal gauge extensions where the
original gauge symmetry is deconstructed into separate copies, one for each
family. A hierarchical chain of symmetry breaking provides an attractive low-
scale solution to the flavor puzzle, consistent with flavor-changing neutral
currents and finite naturalness. Although successful in explaining the origin
of flavor hierarchies in the quark and charged lepton sectors, existing models
have struggled with the large and seemingly anarchic mixing observed in
neutrino oscillations. This letter identifies conditions under which neutrino
anarchy may arise from flavor deconstruction in generic models with right-
handed neutrinos. When deconstruction is applied to carefully chosen subgroups
of the extended gauge symmetry, hierarchies in the seesaw formula cancel out.

##  1 Introduction

Flavor deconstruction refers to a framework wherein a gauge group $G$ is
extended in the ultraviolet to $G^{3}$, incorporating one $G$ factor for each
family. Such non-universal gauge extensions of the Standard Model (SM) have
gained significant attention recently [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18]. The chain of spontaneous symmetry breaking initially
proceeds through $G_{1}\times G_{2}\to G_{12}$, facilitated by a scalar link
field $\phi_{21}$ acquiring a non-zero vacuum expectation value (VEV)
$\langle\phi_{21}\rangle$. This is followed by $G_{12}\times G_{3}\to G$,
mediated by another scalar link field $\phi_{32}$, which connects $G_{2}\times
G_{3}$. Fermions $f_{i}$ are individually charged under the respective $G_{i}$
factor associated with each flavor $i=1,2,3$.111Since the anomaly cancelation
takes place within a family, such gauge extensions are automatically anomaly-
free.

Spontaneous symmetry breaking to a diagonal subgroup for simple groups is
robustly predicted for a nontrivial scalar representation under both factors.
However, it is economical to choose link fields such that $\phi_{21}f_{1}$ and
$f_{2}$ belong to the same gauge representation, likewise for $\phi_{32}f_{2}$
and $f_{3}$. In addition, the Higgs field $H$ is only charged under $G_{3}$.
This sets the basis for explaining the flavor hierarchies: the inter-family
mass rations are controlled by the suppression factors
$\epsilon_{1}=\langle\phi_{21}\rangle/\Lambda_{21}$ and
$\epsilon_{2}=\langle\phi_{32}\rangle/\Lambda_{32}$, of $O(10^{-2})$, where
$\Lambda_{ij}$ generically denote the scales of the effective operators
yielding at low energies the SM Yukawa couplings. In addition, separating the
light family dynamics from the third family by
$\langle\phi_{21}\rangle\gg\langle\phi_{32}\rangle\gg\langle H\rangle$ allows
for a low-scale flavor model consistent with the flavor-changing neutral
currents and finite naturalness [10].222Flavor deconstruction could originate
from a warped extra dimension, see e.g. [19].

While hierarchical singular values are observed in the quark and charged-
lepton mass matrices, the neutrino mass matrix appears to be anarchic. In
contrast to the small and hierarchical mixing between quark generations,
neutrino oscillation experiments reveal large $\mathcal{O}(1)$ mixing angles,
see e.g. [20, 21]. Even the ratio of mass differences $|\Delta m^{\nu}_{\rm
atm}|/|\Delta m^{\nu}_{\rm sol}|\approx 5$ is not particularly large compared
to the mass hierarchies in the charged fermion sector between consecutive
generations, which are on average $\mathcal{O}(100)$. The question addressed
in this letter is whether and under which conditions flavor deconstruction can
lead to flavor anarchy in the neutrino sector together with the observed
hierarchical Yukawa couplings for quarks and charged leptons.

This question cannot be addressed without making hypotheses about the origin
of neutrino masses. We analyze the problem under the general assumption that
the set of chiral fermions is extended by at least three right-handed
neutrinos. Under this assumption, we first analyze in detail the
representative case of the type-I seesaw mechanism (Section 2). We then show
how the conclusions derived in that case can easily be generalized in the
presence of an inverse seesaw mechanism (Section 3).

##  2 Deconstructing the type-I seesaw

In this section, we will limit ourselves to the type-I seesaw mechanism [22,
23, 24, 25, 26], where neutrino masses are described by the following
Lagrangian

$-\mathcal{L}\supset
Y^{ij}_{\nu}\bar{\ell}_{i}\tilde{H}\nu_{j}+\frac{1}{2}M_{\rm
M}^{ij}\nu_{i}\nu_{j}+{\rm h.c.}$ (1)

Here, $\ell_{i}$ and $\nu_{i}$ are the left-handed lepton doublets and right-
handed neutrinos, while $\tilde{H}=i\sigma_{2}H^{*}$. After integrating out
heavy $\nu_{i}$, the mass matrix for active neutrinos becomes

$m_{\nu}\approx-\langle H\rangle^{2}~{}Y_{\nu}\,M_{\rm
M}^{-1}(Y_{\nu})^{T}~{}.$ (2)

To understand how flavor deconstruction acts on $m_{\nu}$, we need to specify
which gauge group is deconstructed. It is easy to realize that deconstructing
any of the three simple subgroups of the SM gauge symmetry prevents anarchic
neutrino mixing and hierarchical quark and charged-lepton masses
simultaneously. More precisely, producing the desired $Y_{u,d,e}$ singular
value hierarchies requires deconstructing either $\mathrm{SU}(2)_{\rm L}$ or
$\mathrm{U}(1)_{\rm Y}$ or both.333The deconstruction of QCD, which is a
vector-like group, does not predict mass hierarchies. This inevitably imprints
hierarchies in $Y_{\nu}$ while keeping $M_{\rm M}$ intact since $\nu_{i}$ are
SM gauge singlets. In particular,

$Y_{\nu}\sim\begin{bmatrix}\epsilon^{2}&\epsilon^{2}&\epsilon^{2}\\\
\epsilon&\epsilon&\epsilon\\\ 1&1&1\end{bmatrix}~{},$ (3)

for both $\mathrm{SU}(2)_{\rm L}$ and $\mathrm{U}(1)_{\rm Y}$ deconstruction,
where we assumed $\epsilon_{1}=\epsilon_{2}=\epsilon$ for simplicity. In this
case, diagonalising the complex matrix $m_{\nu}$ in Eq. (2) via
$U_{\nu}^{T}m_{\nu}U_{\nu}=\hat{m}_{\nu}$, leads to $U_{\nu}\approx{\bf 1}$.
The left-handed charged lepton mixing has a similar form, implying a neutrino
mixing matrix [27] close to the identity: $V_{\rm
PMNS}=U_{\nu}^{\dagger}U_{e_{L}}\approx{\bf 1}$.444This is not specific to the
type-I seesaw mechanism and can be formulated at the level of the Weinberg
operator $\mathcal{L}\supset\frac{c_{ij}}{\Lambda}\ell_{i}\ell_{j}HH$. Since
$\ell_{i}$ are charged under both $\mathrm{SU}(2)_{\rm L}$ and
$\mathrm{U}(1)_{\rm Y}$, $c_{ij}$ is necessarily hierarchical under their
deconstruction. Obviously, the same holds if neutrinos are Dirac, see Eq. (3),
though this case is less motivated as it does not address the overall
smallness of neutrino masses.

A necessary ingredient to achieve our goal is an extended gauge group where
the $\nu_{i}$ are not singlets, allowing $M_{\rm M}$ to acquire a non-trivial
flavor pattern that can compensate for the (unavoidably) hierarchical
structure of $Y_{\nu}$. Without loss of generality, given the fermion content
under consideration, we enlarge the gauge group to

$\mathrm{SU}(3)_{\rm C}\times\mathrm{SU}(2)_{\rm L}\times\mathrm{U}(1)_{\rm
R}\times\mathrm{U}(1)_{\rm B-L}\,,$ (4)

where hypercharge is a subgroup of the two $\mathrm{U}(1)$ factors:555Both
$\mathrm{U}(1)$ groups can be embedded in motivated non-Abelian groups;
however, this makes no difference for the present discussion. $\rm
Y=R+(B-L)/2$. In particular, the right-handed neutrinos carry ${\rm R}=+1/2$
and ${\rm L}=1$ such that $\rm Y=0$. The full deconstruction of either
$\mathrm{U}(1)_{\rm R}$ or $\mathrm{U}(1)_{\rm B-L}$, with the other remaining
universal, leads to the following texture for the Majorana mass matrix:

$M_{\rm M}\sim\begin{bmatrix}\epsilon^{4}&\epsilon^{3}&\epsilon^{2}\\\
\epsilon^{3}&\epsilon^{2}&\epsilon\\\
\epsilon^{2}&\epsilon&1\end{bmatrix}~{}.$ (5)

In order to get an anarchic $m_{\nu}$, the following form of the Dirac mass
matrix is required (Appendix A):

$Y_{\nu}\sim\begin{bmatrix}\epsilon^{2}&\lesssim\epsilon&\lesssim 1\\\
\lesssim\epsilon^{2}&\epsilon&\lesssim 1\\\
\lesssim\epsilon^{2}&\lesssim\epsilon&1\end{bmatrix}~{}.$ (6)

No entry in a given column should be parametrically larger than the diagonal
one. Additionally, the diagonal entries must strictly adhere to the
$(\epsilon^{2},\epsilon,1)$ power counting.

The key to a successful description of neutrino, charged-lepton, and quark
mass matrices is deconstructing some of the simple groups in Eq. (4) in order
to achieve both Eqs. (5) and (6), together with hierarchical (and properly
aligned) $Y_{u,d,e}$.

The Majorana mass matrix assumes the inverse hierarchical form under the
following three hypotheses:

  1. A)

$\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)_{\rm B-L}$,

  2. B)

$\mathrm{U}(1)_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$,

  3. C)

$\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$.

In the following, we proceed to analyze these three options in detail. As we
will show, all of them can lead to a phenomenologically viable pattern for all
mass matrices if accompanied by suitable deconstructions of either
$\mathrm{SU}(3)_{C}$ (option A) or $\mathrm{SU}(2)_{\rm L}$ (options B and C).

###  2.1 Model A: $\mathrm{SU}(3)^{2}_{\rm C}\times\mathrm{SU}(2)_{\rm
L}\times\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)_{\rm B-L}$

The scalar field content of this option is illustrated in Table 1. Note its
minimality, leaving no uneaten Goldstone bosons.

[] Scalars | $\mathrm{U}(1)^{(1)}_{\rm R}$ | $\mathrm{U}(1)^{(2)}_{\rm R}$ | $\mathrm{U}(1)^{(3)}_{\rm R}$ | $\mathrm{U}(1)_{\rm B-L}$
---|---|---|---|---
$\phi_{21}$ | $1/2$ | $-1/2$ | $0$ | $0$
$\phi_{32}$ | $0$ | $1/2$ | $-1/2$ | $0$
$\chi$ | $0$ | $0$ | $-1$ | $+2$
$H$ | $0$ | $0$ | $1/2$ | $0$
Table 1: Charges of the symmetry-breaking scalars under
$\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)_{\rm B-L}$.

The effective theory operators generating the leptonic Yukawa interactions are

$\begin{split}-\mathcal{L}_{Y}\supset&\quad\overline{\ell}_{i}He_{3}+\frac{1}{\Lambda_{32}}\overline{\ell}_{i}H\phi_{32}e_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}}\overline{\ell}_{i}H\phi_{32}\phi_{21}e_{1}\\\
&+\overline{\ell}_{i}\tilde{H}\nu_{3}+\frac{1}{\Lambda_{32}}\overline{\ell}_{i}\tilde{H}\phi_{32}^{*}\nu_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}}\overline{\ell}_{i}\tilde{H}\phi_{32}^{*}\phi_{21}^{*}\nu_{1}~{},\end{split}$
(7)

where we have omitted to show $\mathcal{O}(1)$ parameters in front of the
operators. Similarly, the Majorana mass matrix gets generated from

$\begin{split}-\mathcal{L}_{M}=&\quad\chi\nu_{3}\nu_{3}+\frac{1}{\Lambda_{32}}\chi\phi_{32}^{*}\nu_{3}\nu_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}}\chi\phi_{32}^{*}\phi_{21}^{*}\nu_{3}\nu_{1}\\\
&+\frac{1}{\Lambda_{32}^{2}}\chi\phi_{32}^{*}\phi_{32}^{*}\nu_{2}\nu_{2}+\frac{1}{\Lambda_{32}^{2}\Lambda_{21}}\chi\phi_{32}^{*}\phi_{32}^{*}\phi_{21}^{*}\nu_{2}\nu_{1}\\\
&+\frac{1}{\Lambda_{32}^{2}\Lambda_{21}^{2}}\chi\phi_{32}^{*}\phi_{32}^{*}\phi_{21}^{*}\phi_{21}^{*}\nu_{1}\nu_{1}~{}.\end{split}$
(8)

The common effective scales $\Lambda_{21}$ and $\Lambda_{32}$ lead to the
universal suppression factors
$\epsilon_{i}=\langle\phi_{ij}\rangle/\Lambda_{ij}$.666A straightforward UV
completion which ensures the effective neutrino operators in Eqs. (7) and (8)
are suppressed by the same scales $\Lambda_{21}$ and $\Lambda_{32}$ involves a
tree-level mediation of heavy vector-like fermions that transform in the same
gauge representations as $\nu_{2}$ and $\nu_{3}$, respectively. See Fig. 1.
For simplicity, we assume universal $\epsilon_{1}=\epsilon_{2}=\epsilon$ as
before.

Figure 1: A representative UV completion of selected operators in Eqs. (7) and
(8). A heavy vectorlike fermion $N=N_{L}\oplus N_{R}$ in the gauge
representation of $\nu_{3}$, and mass
$m_{N}\gg\langle\chi\rangle,\langle\phi_{i}\rangle$, when integrated out,
ensures the same suppression factor
$\epsilon_{2}=\langle\phi_{2}\rangle/m_{N}$ in $Y_{\nu}$ and $M_{\rm M}$
matrices. Similarly, adding another vectorlike fermion in the gauge
representation of $\nu_{2}$ allows the completion of the first-family
structure.

The deconstruction of $\mathrm{U}(1)_{\rm R}$ leads to Yukawa couplings of the
form

$Y_{u,d,e,\nu}^{\mathrm{U}(1)^{3}_{\rm
R}}\sim\begin{bmatrix}\epsilon^{2}&\epsilon&1\\\ \epsilon^{2}&\epsilon&1\\\
\epsilon^{2}&\epsilon&1\end{bmatrix}~{},$ (9)

while the Majorana mass matrix takes the form in Eq. (5). The singular values
of $Y_{e}$ are $(\epsilon^{2},\epsilon,1)$ producing the observed mass
hierarchies. Furthermore, given the condition in Eq. (6) is satisfied, the
resulting neutrino mass matrix $m_{\nu}$ is anarchic.

However, the left-handed mixing matrices obtained from the singular value
decomposition have $\mathcal{O}(1)$ angles. While this works well for $V_{\rm
PMNS}$, it is not a viable structure for $V_{\rm CKM}$ [28, 29] obtained from
the left-handed mixing in the quark sector. This problem can be addressed
through the flavor deconstruction of QCD without compromising the leptonic
sector. Note that since QCD is a vector-like group, its deconstruction does
not affect quark masses that remain similar to the charged-lepton ones.

Consider, for concreteness, a partial QCD deconstruction
$\mathrm{SU}(3)^{(12)}_{\rm C}\times\mathrm{SU}(3)^{(3)}_{\rm C}$. A scalar
field $\Omega\sim({\bf 3},{\bf\bar{3}})$ breaks the symmetry down to the QCD.
The effective Lagrangian for the quark sector becomes

$\begin{split}-\mathcal{L}_{Y}\supset&\quad\overline{q}_{3}Hd_{3}+\frac{1}{\Lambda^{\prime}}\overline{q}_{p}H\Omega
d_{3}\\\
&+\frac{1}{\Lambda_{32}\Lambda^{\prime}}\overline{q}_{3}H\phi_{32}\Omega^{\dagger}d_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}\Lambda^{\prime}}\overline{q}_{3}H\phi_{32}\phi_{21}\Omega^{\dagger}d_{1}\\\
&+\frac{1}{\Lambda_{32}}\overline{q}_{p}H\phi_{32}d_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}}\overline{q}_{p}H\phi_{32}\phi_{21}d_{1}\\\
&+\overline{q}_{3}\tilde{H}u_{3}+\frac{1}{\Lambda^{\prime}}\overline{q}_{p}\tilde{H}\Omega
u_{3}\\\
&+\frac{1}{\Lambda_{32}\Lambda^{\prime}}\overline{q}_{3}\tilde{H}\phi_{32}^{*}\Omega^{\dagger}u_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}\Lambda^{\prime}}\overline{q}_{3}\tilde{H}\phi_{32}^{*}\phi_{21}^{*}\Omega^{\dagger}u_{1}\\\
&+\frac{1}{\Lambda_{32}}\overline{q}_{p}\tilde{H}\phi_{32}^{*}u_{2}+\frac{1}{\Lambda_{32}\Lambda_{21}}\overline{q}_{p}\tilde{H}\phi_{32}^{*}\phi_{21}^{*}u_{1}~{}.\end{split}$
(10)

Here, $p=1,2$ and $\Lambda^{\prime}$ is the effective scale of operators
containing insertions of $\Omega$. The corresponding suppression factor
$\epsilon^{\prime}=\langle\Omega\rangle/\Lambda$ shows up only in the mixing.
The quark Yukawa matrices, therefore, take the form

$Y^{\rm I}_{u,d}\sim\begin{bmatrix}\epsilon^{2}&\epsilon&\epsilon^{\prime}\\\
\epsilon^{2}&\epsilon&\epsilon^{\prime}\\\
\epsilon^{\prime}\epsilon^{2}&\epsilon^{\prime}\epsilon&1\end{bmatrix}~{}.$
(11)

The singular values of $Y_{u,d}$ are $(\epsilon^{2},\epsilon,1)$ in agreement
with the charged leptons. The predicted Cabbibo angle is large while $V_{cb}$
and $V_{ub}$ are of order $\epsilon^{\prime}$. A full deconstruction of QCD
would predict $V_{us}\sim\epsilon^{\prime}_{1}$,
$V_{cb}\sim\epsilon^{\prime}_{2}$ and $V_{ub}\sim V_{us}V_{cb}$.

A relevant question to address is the identification of the smallest scale at
which deconstruction occurs. The EFT description in Eq. (8) is valid only when
$\Lambda_{1,2}\gtrsim\langle\chi\rangle$. The absolute neutrino mass scale
naturally points to $\langle\chi\rangle\approx\langle
H\rangle^{2}/(0.1\,\rm{eV})\sim 10^{14}$ GeV. On the other hand, the charged
fermion hierarchies are best fitted for $\epsilon\sim 4\times 10^{-3}$ (see
the Supplemental Material of [30]). Assuming the same $\epsilon$ for neutrinos
and charged leptons, this points to a high scale for the $\mathrm{U}(1)_{\rm
R}$ deconstruction, which makes it challenging for experiments. Moreover, this
scenario would require a huge tuning of the Higgs mass term in the absence of
a protection mechanism such as low-energy supersymmetry or compositeness.

One way to lower the scale would be to introduce a universal suppression
factor in the $Y_{\nu}$ matrix. A possible option is a flavor-universal
$Z_{2}$ symmetry under which only the $\nu_{i}$ are odd. Denoting $\mu_{\nu}$
the corresponding symmetry-breaking spurion,
$\langle\chi\rangle\sim(\mu_{\nu}/10^{-4})^{2}\times 1000$ TeV. A more
motivated option is to implement flavor deconstruction in a symmetry-protected
seesaw mechanism, such as the inverse seesaw. We discuss this option in
Section 3.

###  2.2 Model B: $\mathrm{SU}(3)_{\rm C}\times\mathrm{SU}(2)^{3}_{\rm
L}\times\mathrm{U}(1)_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$

A minimal option to deconstruct $\mathrm{U}(1)_{\rm B-L}$ and generate the
effective SM Yukawa couplings is via the scalar field content in Table 2. This
way, all Goldstone bosons are gauged away. However, one can consider less
minimal options also introducing the link fields
$\phi^{\ell}_{ij}\sim(\phi_{ij}^{q*})^{3}$. This is why, to keep the
discussion more general, we define
$\epsilon^{q,\ell}_{i}=\langle\phi^{q,\ell}_{ij}\rangle/\Lambda^{q,\ell}_{ij}$.

[] Scalars | $\mathrm{U}(1)^{(1)}_{\rm B-L}$ | $\mathrm{U}(1)^{(2)}_{\rm B-L}$ | $\mathrm{U}(1)^{(3)}_{\rm B-L}$ | $\mathrm{U}(1)_{\rm R}$
---|---|---|---|---
$\phi_{21}^{q}$ | $-1/3$ | $1/3$ | $0$ | $0$
$\phi_{32}^{q}$ | $0$ | $-1/3$ | $+1/3$ | $0$
$\chi$ | $0$ | $0$ | $+2$ | $-1$
$H$ | $0$ | $0$ | $0$ | $1/2$
Table 2: Charges of the minimal set of symmetry-breaking scalars under
$\mathrm{U}(1)_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$.

With these assumptions, $\mathrm{U}(1)^{3}_{\rm B-L}$ implies

$Y^{\mathrm{U}(1)^{3}_{\rm
B-L}}_{u,d}\sim\begin{bmatrix}1&\epsilon_{q}&\epsilon_{q}^{2}\\\
\epsilon_{q}&1&\epsilon_{q}\\\
\epsilon_{q}^{2}&\epsilon_{q}&1\end{bmatrix},\quad Y^{\mathrm{U}(1)^{3}_{\rm
B-L}}_{e,\nu}\sim\begin{bmatrix}1&\epsilon_{\ell}&\epsilon_{\ell}^{2}\\\
\epsilon_{\ell}&1&\epsilon_{\ell}\\\
\epsilon_{\ell}^{2}&\epsilon_{\ell}&1\end{bmatrix},$ (12)

where in the minimal model $\epsilon_{\ell}=\epsilon^{3}_{q}$. The Majorana
mass matrix takes the form in Eq. (5) with the replacement
$\epsilon\to\epsilon_{\ell}$. This texture does not generate neutrino anarchy
nor charged fermion mass hierarchy, calling for an additional deconstruction
of the $\mathrm{SU}(2)_{\rm L}$ group. To this purpose, we introduce another
pair of scalar link fields, bi-doublets under
$\mathrm{SU}(2)^{(1)}_{L}\times\mathrm{SU}(2)^{(2)}_{L}$ and
$\mathrm{SU}(2)^{(2)}_{L}\times\mathrm{SU}(2)^{(3)}_{L}$, respectively. The
Higgs is charged under $\mathrm{SU}(2)^{(3)}_{L}$. Denoting $\epsilon_{L}$ the
suppression factor associated to the $\mathrm{SU}(2)_{\rm L}$ deconstruction,
one has

$Y^{\mathrm{SU}(2)^{3}_{\rm L}}_{u,d,e,\nu}\sim\begin{bmatrix}\epsilon_{\rm
L}^{2}&\epsilon_{\rm L}^{2}&\epsilon_{\rm L}^{2}\\\ \epsilon_{\rm
L}&\epsilon_{\rm L}&\epsilon_{\rm L}\\\ 1&1&1\end{bmatrix}~{}.$ (13)

Hence the combined effect of $\mathrm{SU}(2)^{3}_{\rm L}\times U(1)_{\rm
B-L}^{3}$ is

$Y^{\rm II}_{u,d}\sim\begin{bmatrix}\epsilon_{\rm L}^{2}&\epsilon_{\rm
L}^{2}\epsilon_{q}&\epsilon_{\rm L}^{2}\epsilon_{q}^{2}\\\ \epsilon_{\rm
L}\epsilon_{q}&\epsilon_{\rm L}&\epsilon_{\rm L}\epsilon_{q}\\\
\epsilon_{q}^{2}&\epsilon_{q}&1\end{bmatrix},~{}Y^{\rm
II}_{e,\nu}\sim\begin{bmatrix}\epsilon_{\rm L}^{2}&\epsilon_{\rm
L}^{2}\epsilon_{\ell}&\epsilon_{\rm L}^{2}\epsilon_{\ell}^{2}\\\ \epsilon_{\rm
L}\epsilon_{\ell}&\epsilon_{\rm L}&\epsilon_{\rm L}\epsilon_{\ell}\\\
\epsilon_{\ell}^{2}&\epsilon_{\ell}&1\end{bmatrix}~{},$ (14)

without affecting the Majorana mass matrix. This scenario satisfies the
neutrino criteria in Eq. (6) if

$\epsilon_{\rm L}=\epsilon_{\ell}~{},$ (15)

and successfully generates hierarchical masses for quarks and charged leptons
with singular values $(\epsilon_{\rm L}^{2},\epsilon_{\rm L},1)$. In addition,
it predicts a perturbative $V_{\rm CKM}$ matrix which is close to the unit
matrix with $V_{us}\sim V_{cb}\sim\epsilon_{\rm L}\epsilon_{q}$ and
$V_{ub}\sim V_{us}V_{cb}$. The predicted off-diagonal elements are arguably
too small. However, in the minimal realization, where
$\epsilon_{q}=\epsilon_{\rm L}^{1/3}$, this could be acceptable.

###  2.3 Model C: $\mathrm{SU}(3)_{\rm C}\times\mathrm{SU}(2)^{3}_{\rm
L}\times\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$

While $\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$ yields a
successful description of quark and charged-lepton Yukawa couplings, it does
not satisfy the neutrino conditions in Eqs. (5) and (6). However, this can be
achieved via the deconstruction of the full electroweak group:
$\mathrm{SU}(2)^{3}_{\rm L}\times\mathrm{U}(1)^{3}_{\rm
R}\times\mathrm{U}(1)^{3}_{\rm B-L}$. This deconstruction implies

$Y^{\rm III}_{u,d}\sim\begin{bmatrix}\epsilon_{\rm L}^{2}\epsilon_{\rm
R}^{2}&\epsilon_{\rm L}^{2}\epsilon_{\rm R}\epsilon_{q}&\epsilon_{\rm
L}^{2}\epsilon_{q}^{2}\\\ \epsilon_{\rm L}\epsilon_{\rm
R}^{2}\epsilon_{q}&\epsilon_{\rm L}\epsilon_{\rm R}&\epsilon_{\rm
L}\epsilon_{q}\\\ \epsilon_{\rm R}^{2}\epsilon_{q}^{2}&\epsilon_{\rm
R}\epsilon_{q}&1\end{bmatrix},$ (16)

and

$Y^{\rm III}_{e,\nu}\sim\begin{bmatrix}\epsilon_{\rm L}^{2}\epsilon_{\rm
R}^{2}&\epsilon_{\rm L}^{2}\epsilon_{\rm R}\epsilon_{\ell}&\epsilon_{\rm
L}^{2}\epsilon_{\ell}^{2}\\\ \epsilon_{\rm L}\epsilon_{\rm
R}^{2}\epsilon_{\ell}&\epsilon_{\rm L}\epsilon_{\rm R}&\epsilon_{\rm
L}\epsilon_{\ell}\\\ \epsilon_{\rm R}^{2}\epsilon_{\ell}^{2}&\epsilon_{\rm
R}\epsilon_{\ell}&1\end{bmatrix}~{},$ (17)

while the Majorana mass matrix has the form in Eq. (5) with the replacement
$\epsilon\to\epsilon_{\ell}\epsilon_{R}$. When

$\epsilon_{\rm L}=\epsilon_{\rm R}=\epsilon_{\ell}~{},$ (18)

this texture produces neutrino anarchy while predicting quark and charged-
lepton mass hierarchy, with singular values $(\epsilon_{\rm
L}^{4},\epsilon_{\rm L}^{2},1)$ and the $V_{\rm CKM}$ matrix close to the unit
matrix with $V_{us}\sim V_{cb}\sim\epsilon_{\rm L}\epsilon_{q}$ and
$V_{ub}\sim V_{us}V_{cb}$. This relation works better when relating quark
masses to the $V_{\rm CKM}$ elements than in Section 2.2.

At this point, it is tempting to ask what happens if, in addition, we also
deconstruct $\mathrm{SU}(3)_{\rm C}$ such that the full underlying gauge group
is

${\rm PS}^{(1)}\times{\rm PS}^{(2)}\times{\rm PS}^{(3)}~{},$ (19)

as initially conjectured in [1]. Staying at the level of
$\mathrm{SU}(3)^{3}_{\rm C}\times\mathrm{SU}(2)^{3}_{\rm
L}\times\mathrm{U}(1)^{3}_{\rm R}\times\mathrm{U}(1)^{3}_{\rm B-L}$, we simply
assume $\phi_{21}^{q}$ in Table 2 to be a $({\bf 3},{\bf\bar{3}})$ under
$\mathrm{SU}(3)^{(1)}_{\rm C}\times\mathrm{SU}(3)^{(2)}_{\rm C}$ and the same
for $\phi_{32}^{q}$ under $\mathrm{SU}(3)^{(2)}_{\rm
C}\times\mathrm{SU}(3)^{(3)}_{\rm C}$. Here, however, we need to have
$\phi^{\ell}_{ij}$ fields as well. In fact, $\phi^{q}_{ij}$ and
$\phi^{\ell}_{ij}$ unify into a pair of bifundamentals $\Sigma_{ij}$ of
$\mathrm{SU}(4)^{3}\supset\mathrm{SU}(3)^{3}_{\rm
C}\times\mathrm{U}(1)^{3}_{\rm B-L}$. Remarkably, this model predicts exactly
the flavor structure of Model C!

##  3 Deconstructing the inverse seesaw

The inverse seesaw [31, 32] belongs to a class of symmetry-protected seesaw
mechanisms that predict lighter sterile neutrinos with larger couplings. This
provides an interesting target for probing the origin of neutrino masses in
ongoing and upcoming experiments. In addition to left-chiral $\ell_{i}$ and
right-chiral $\nu_{i}$, the model is extended by (left-chiral) gauge singlets
$S_{i}$, where for concreteness, we consider three flavors ($i=1,2,3$). Let us
define $n_{i}=(\ell^{1}_{i},\nu^{c}_{i},S_{i})$. Here, $\ell^{1}$ stands for
the neutral component of the lepton doublet and $c$ is for the charge
conjugation. The neutral lepton mass matrix ($9\times 9$) is

$-\mathcal{L}\supset\frac{1}{2}\bar{n}_{i}M^{ij}_{\nu}n_{j}^{c},\quad
M_{\nu}=\begin{bmatrix}0&M_{D}&0\\\ M_{D}^{T}&0&M_{R}^{T}\\\
0&M_{R}&\mu_{S}\end{bmatrix}~{},$ (20)

The global lepton number is an approximate symmetry softly broken by
$\mu^{ij}_{S}$. For $\mu_{S}\ll M_{D}<M_{R}$,777The PMNS non-unitarity sets
limit on $M_{D}/M_{R}$, see e.g. [33]. the active neutrino mass matrix takes
the form

$m_{\nu}\approx M_{D}M_{R}^{-1}\mu_{S}(M_{R}^{-1})^{T}M^{T}_{D}~{}.$ (21)

For the same arguments presented in Section 2, a deconstruction of the SM
gauge symmetry cannot reconcile hierarchical charged leptons with anarchic
neutrinos. Therefore, we are again led to consider the $\mathrm{SU}(3)_{\rm
C}\times\mathrm{SU}(2)_{\rm L}\times\mathrm{U}(1)_{\rm
R}\times\mathrm{U}(1)_{\rm B-L}$ gauge symmetry. Introducing the scalar field
$\chi\sim({\bf 1},{\bf 1},-1/2,+1)$ that breaks the two $\mathrm{U}(1)$’s down
to the SM hypercharge, allow us to write

$-\mathcal{L}\supset Y^{ij}_{R}\bar{S}_{i}\chi\nu_{j}~{},\quad
M^{ij}_{R}=Y^{ij}_{R}\langle\chi\rangle~{}.$ (22)

Note that the Majorana mass term for $\nu_{i}$ is absent at the renormalizable
level. As before, $M^{ij}_{D}=Y^{ij}_{\nu}\langle H\rangle$ stems from
$\mathcal{L}\supset-Y^{ij}_{\nu}\bar{\ell}_{i}\tilde{H}\nu_{j}$. The remaining
renormalizable operator
$\mathcal{L}\supset-\frac{1}{2}\mu_{S}^{ij}\bar{S}^{c}_{i}S_{j}$ finally
justifies Eq. (20).

Since the $S_{i}$ are gauge singlets, the natural expectation for
$\mu_{S}^{ij}$ is flavor anarchy. On the other hand, flavor deconstruction
imprints hierarchical patterns in $Y_{\nu}$ and $Y_{R}$. When put together,
Eq. (21) requires $Y_{\nu}Y^{-1}_{R}$ to be an anarchic matrix for $m_{\nu}$
also to be anarchic. Under $\mathrm{U}(1)_{\rm R}$ or $\mathrm{U}(1)_{\rm
B-L}$ deconstruction, $Y_{R}$ takes the form of Eq. (9). As a result,
$Y_{\nu}$ should satisfy the same condition as in Eq. (6), see Appendix B.

Since the condition in Eq. (6) is unchanged, the same gauge symmetries that
worked for the high-scale seesaw discussed in Sections 2.1 and 2.2 are also
applicable in the inverse seesaw case. This is not surprising given the
additional fermions we have introduced (the $S_{i}$), being gauge singlets do
not participate in the flavor deconstruction. The observed neutrino mass
splittings suggest $\mu\sim 0.1\,{\rm eV}\langle\chi\rangle^{2}/\langle
H\rangle^{2}$. Therefore, the major advantage of the inverse seesaw is that
the deconstruction can easily take place at the TeV scale, which is consistent
with finite naturalness (see also [34]). Detailed phenomenological
investigations are left for future studies.

##  4 Conclusions

Seemingly anarchic neutrino flavor structure, when contrasted with the
hierarchies observed in the charged fermion sector, poses a significant
challenge for many flavor models, especially those involving flavor
deconstruction. The SM gauge group’s deconstruction fails to reconcile both
phenomena simultaneously. However, as demonstrated in this letter, minimal
extensions where right-handed neutrinos are not gauge singlets allow for
reconciliation, enabling the correct generation of charged fermion mass and
mixing hierarchies while predicting anarchy for the light-active neutrinos.
Focusing on the type-I and inverse seesaw mechanisms, we have conducted a
comprehensive survey of possible options and identified promising models for
further study. While these models strictly lead to neutrino anarchy, other
deconstruction models with moderate tuning should not be disregarded.

## Acknowledgments

We thank Joe Davighi for his useful comments on the manuscript. This work has
received funding from the Swiss National Science Foundation (SNF) through the
Eccellenza Professorial Fellowship “Flavor Physics at the High Energy
Frontier,” project number 186866. In addition, this project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program under grant agreement 833280
(FLAY) and by the SNF under contract 200020_204428.

##  Appendix A Type-I seesaw: Hierarchy cancellation

Consider the matrices

$X_{\rm
M}=\begin{pmatrix}x_{11}\epsilon^{2}_{1}\epsilon_{2}^{2}&x_{12}\epsilon_{1}\epsilon_{2}^{2}&x_{13}\epsilon_{1}\epsilon_{2}\\\
x_{12}\epsilon_{1}\epsilon_{2}^{2}&x_{22}\epsilon_{2}^{2}&x_{23}\epsilon_{2}\\\
x_{13}\epsilon_{1}\epsilon_{2}&x_{23}\epsilon_{2}&x_{33}\end{pmatrix}\,,\quad
Y_{\rm D}=\begin{pmatrix}\epsilon_{1}\epsilon_{2}&0&0\\\ 0&\epsilon_{2}&0\\\
0&0&1\end{pmatrix}\,.$ (23)

The following relation holds (with no approximations):

$Y_{\rm D}X^{-1}_{\rm M}Y_{\rm D}^{T}=\begin{pmatrix}x_{11}&x_{12}&x_{13}\\\
x_{12}&x_{22}&x_{23}\\\ x_{13}&x_{23}&x_{33}\end{pmatrix}^{-1}~{}.$ (24)

For $x_{ij}\sim\mathcal{O}(1)$ and $\epsilon_{i}\ll 1$, this provides a
mechanism to generate an anarchic mass matrix for active neutrinos starting
from hierarchical Yukawa couplings in the type-I seesaw mechanism.

##  Appendix B Inverse seesaw: Hierarchy cancellation

Consider the matrices

$Y_{\rm
D}=\begin{pmatrix}y_{11}\epsilon_{1}\epsilon_{2}&y_{12}\epsilon_{2}&y_{13}\\\
y_{21}\epsilon_{1}\epsilon_{2}&y_{22}\epsilon_{2}&y_{23}\\\
y_{31}\epsilon_{1}\epsilon_{2}&y_{32}\epsilon_{2}&y_{33}\end{pmatrix}\,,\quad
Y_{R}=\begin{pmatrix}\epsilon_{1}\epsilon_{2}&0&0\\\ 0&\epsilon_{2}&0\\\
0&0&1\end{pmatrix}\,.$ (25)

The following relation holds (with no approximations):

$Y_{\rm D}Y^{-1}_{\rm R}=\begin{pmatrix}y_{11}&y_{12}&y_{13}\\\
y_{21}&y_{22}&y_{23}\\\ y_{31}&y_{32}&y_{33}\end{pmatrix}~{}.$ (26)

For $y_{ij}\sim\mathcal{O}(1)$ and $\epsilon_{i}\ll 1$, this provides a
mechanism to generate an anarchic mass matrix for active neutrinos in the
inverse seesaw mechanism.

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# Photon centroids and their subluminal propagation

Konstantin Y. Bliokh  Theoretical Quantum Physics Laboratory, Cluster for
Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Centre of
Excellence ENSEMBLE3 Sp. z o.o., 01-919 Warsaw, Poland  Donostia International
Physics Center (DIPC), Donostia-San Sebastián 20018, Spain

###### Abstract

We examine properties and propagation of the energy-density and photon-
probability centroids of electromagnetic wavepackets in free space. In the
second-order paraxial approximation, both of these centroids propagate with
the same subluminal velocity because of the transverse confinement of the
wavepacket and its diffraction. The tiny difference between the energy and
probability centroid velocities appears only in the fourth order. We consider
three types of wavepackets: Gaussian, Bessel, and non-diffracting Bessel. In
all these cases, the subluminal propagation is clearly visible in the
intensity distributions and can be measured experimentally in both classical-
light and single-photon regimes. For Gaussian wavepackets, the half-wavelength
delay is accumulated after propagation over about 12 Rayleigh lengths.

the of and to

##  1 Introduction

Subluminal and superluminal velocities in the propagation of light has
intrigued scientists for several decades [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].
There are different velocities characterizing propagation of light (phase,
group, signal, energy-transfer, etc.) and different mechanisms for the
subluminal and superluminal propagations: interactions with matter [2, 3, 4,
5, 9], local phase gradients in structured free-space light [6, 7, 8], tilt of
the plane-wave Fourier components in a transversely-confined light beams [10],
and specially space-time structured wavepackets [11].

In this work, we examine propagation of the energy and photon-probability
centroids of regular optical wavepackets and analyse their subluminal
velocities. This phenomenon of subluminal propagation originates from the
transverse confinement of the wavepackets, and hence is related to the
mechanism considered in Ref. [10]. However, while Ref. [10] considered single-
photon wavepackets within monochromatic (i.e., longitudinally delocalized)
optical-beam setup, we consider polychromatic transversely and longitudinally
confined wavepackets with well defined centroids. Thus, our problem is equally
well defined in the single-photon and classical-light frameworks.

A fundamental difficulty with the definition of a photon centroid is that the
photon wavefunction, and, hence, the probability density, are ill-defined in
real space [12, 13]. (This is a consequence of the Weinberg-Witten theorem for
massless spin-1 particles [14].) Only the energy density is well defined, and
the corresponding energy centroid is calculated straightforwardly. It moves
rectilinearly and uniformly in free space, as a consequence of the
conservation of the boost momentum, i.e., spatiotemporal components of the
relativistic angular momentum tensor [15, 16]. Nonetheless, despite the ill-
defined probability density, the probability centroid can be determined as the
expectation value of the position operator in the momentum representation [17,
18].

In this work, we explicitly calculate the motion of both the energy and
probability centroids of a ‘semiclassical’ (i.e., large as compared with the
central wavelength) electromagnetic wavepacket in free space. We show that
both centroids move rectilinearly and uniformly with velocities slightly less
than the speed of light $c$. The difference between these velocities and $c$
is of the second order in the small paraxiality parameter (the characteristic
polar angle of the wavevectors in the wavepacket), while the difference
between the energy and probability centroid velocities is of the fourth order.
We consider different examples of wavepackets: Gaussian, spatiotemporal
Bessel, and non-diffracting spatiotemporal Bessel.

##  2 Energy and photon-probability centroids

The first-quantization wave formalism for photons is the classical Maxwell
electromagnetism [19, 20, 13]. The energy centroid of an electromagnetic
wavepacket is

${\bf R}_{E}=\frac{\int W{\bf r}\,d^{3}{\bf r}}{\int Wd^{3}{\bf r}}\,,$ (1)

where $W=\left({\bf E}^{2}+{\bf H}^{2}\right)/2$ is the energy density,
whereas ${\bf E}({\bf r},t)$ and ${\bf H}({\bf r},t)$ are the real-valued
electric and magnetic fields, and we use the Gaussian units omitting
inessential numerical constants. The velocity of the energy centroid
propagation is obtained by differentiating (1) with respect to time:

${\bf V}_{E}=\frac{\partial}{\partial t}\frac{\int W{\bf r}\,d^{3}{\bf
r}}{\int Wd^{3}{\bf r}}=-\frac{\int({\bm{\nabla}}\cdot{\bm{\Pi}})\,{\bf
r}\,d^{3}{\bf r}}{\int Wd^{3}{\bf r}}=\frac{\int{\bf\Pi}\,d^{3}{\bf r}}{\int
Wd^{3}{\bf r}}={\bf const}\,.$ (2)

Here ${\bf\Pi}=c({\bf E}\times{\bf H})=c^{2}{\bf P}$ is the energy flux
(Poynting vector), ${\bf P}$ is the kinetic momentum density, and we used the
energy and momentum conservation laws [15, 21]:

$\frac{\partial W}{\partial
t}+{\bm{\nabla}}\cdot{\bm{\Pi}}=0\,,\quad\frac{\partial}{\partial t}\int
Wd^{3}{\bf r}=0\,,\quad\frac{\partial}{\partial t}\int{\bm{\Pi}}\,d^{3}{\bf
r}={\bf 0}\,.$ (3)

Equation (2) shows that the energy centroid moves uniformly and rectilinearly
in free space. The energy-centroid velocity ${\bf V}_{E}$ is given by the
ratio of the integral Poynting vector (i.e., momentum multiplied by $c^{2}$)
to the integral energy. This form is equivalent to the velocity of a free
relativistic classical particle: ${\bf v}=c^{2}{\bf p}/{\cal E}$ [15]. In some
electromagnetic problems (e.g., for quasi-1D transversely-localized
monochromatic guided modes), the ratio of the integral Poynting vector to the
integral energy is associated with the group velocity ${\bf
v}_{g}=\partial\omega({\bf k})/\partial{\bf k}$ [22, 23, 24, 25]. However, as
we will see, the situation becomes more sophisticated for transversely and
longitudinally localized free-space wavepackets. Note also that Eq. (2)
follows from the conservation of the integral electromagnetic boost momentum
[15, 16, 17]:

$\int({\bf r}W-t{\bm{\Pi}})\,d^{3}{\bf r}={\bf const}\,,$ (4)

Let us now define the photon probability centroid. Since the photon
wavefunction is well defined only in the momentum representation, we use the
complex Fourier (plane-wave) amplitudes of the electric and magnetic fields,
$\tilde{\bf E}({\bf k})e^{-i\omega({\bf k})t}$ and $\tilde{\bf H}({\bf
k})e^{-i\omega({\bf k})t}$, satisfying $\tilde{\bf H}=c\,{\bf
k}\times\tilde{\bf E}/\omega$ and $\omega({\bf k})=ck$. In all calculations
below, the magnetic- and electric-field amplitudes make equal contributions,
so that we can use only the electric-field amplitudes. Then, the photon
wavefunction in the momentum representation is given by ${\bm{\psi}}({\bf
k},t)\propto\tilde{\bf E}({\bf k})e^{-i\omega({\bf k})t}/\sqrt{\omega({\bf
k})}$ [12, 13, 17, 18], and the probability centroid is given by the
normalized expectation value of the canonical position operator $\hat{\bf
r}=i{\bm{\nabla}}_{\bf k}$:

${\bf R}=\frac{\langle{\bm{\psi}}|\hat{\bf
r}|{\bm{\psi}}\rangle}{\langle{\bm{\psi}}|{\bm{\psi}}\rangle}=\frac{\int\omega^{-1}\tilde{\bf
E}^{*}e^{i\omega t}\cdot(i{\nabla}_{\bf k})\tilde{\bf E}e^{-i\omega
t}\,d^{3}{\bf k}}{\int\omega^{-1}|\tilde{\bf E}|^{2}\,d^{3}{\bf k}}\,.$ (5)

This centroid has been previously employed for the analysis of the intrinsic
and extrinsic contributions to the orbital angular momentum of vortex wave
states [26, 27, 17, 18]. Calculating the time derivative of Eq. (5), we find
the photon probability-centroid velocity:

${\bf V}=\frac{\partial}{\partial t}\frac{\int\omega^{-1}\tilde{\bf
E}^{*}e^{i\omega t}\cdot(i{\nabla}_{\bf k})\tilde{\bf E}e^{-i\omega
t}\,d^{3}{\bf k}}{\int\omega^{-1}|\tilde{\bf E}|^{2}\,d^{3}{\bf
k}}=\frac{\int\omega^{-1}|\tilde{\bf E}|^{2}{\bf v}_{g}\,d^{3}{\bf
k}}{\int\omega^{-1}|\tilde{\bf E}|^{2}\,d^{3}{\bf k}}\equiv\langle{\bf
v}_{g}\rangle={\bf const}\,.$ (6)

Here ${\bf v}_{g}=\partial\omega/\partial{\bf k}=c{\bf k}/k$ is the ${\bf
k}$-dependent group velocity. Equation (6) is manifestly time-independent,
which indicates the uniform rectilinear propagation of the probability
centroid with the mean group velocity $\langle{\bf v}_{g}\rangle$.

In the photon-wavefunction ${\bf k}$-space formalism, the integral
electromagnetic energy and momentum take the forms $\int Wd^{3}{\bf
r}\propto\langle{\bm{\psi}}|\omega|{\bm{\psi}}\rangle$ and $\int{\bf
P}d^{3}{\bf r}\propto\langle{\bm{\psi}}|{\bf k}|{\bm{\psi}}\rangle$, so the
energy centroid (1) and its velocity (2) can be written as

${\bf R}_{E}=\frac{\langle{\bm{\psi}}|\omega\hat{\bf
r}|{\bm{\psi}}\rangle}{\langle{\bm{\psi}}|\omega|{\bm{\psi}}\rangle}=\frac{\int\tilde{\bf
E}^{*}e^{i\omega t}\cdot(i{\nabla}_{\bf k})\tilde{\bf E}e^{-i\omega
t}\,d^{3}{\bf k}}{\int|\tilde{\bf E}|^{2}\,d^{3}{\bf k}}\,,$ (7) ${\bf
V}_{E}=\frac{\int|\tilde{\bf E}|^{2}{\bf v}_{g}\,d^{3}{\bf k}}{\int|\tilde{\bf
E}|^{2}\,d^{3}{\bf k}}\equiv\langle{\bf v}_{g}\rangle_{E}={\bf const}\,.$ (8)

Here, the averaged group velocity $\equiv\langle{\bf v}_{g}\rangle_{E}$ is
weigthed by the electric-field rather than wavefunction intensity. The
velocities of the energy and probability centroids, in general, differ from
each other due to the extra $\omega({\bf k})$ factors in the integrals
(5)–(8).

##  3 Propagation of the centroids of paraxial wavepackets

###  3.1 Gaussian wavepacket in the second-order approximation

We now examine propagation of paraxial electromagnetic wavepackets in free
space. First, consider a Gaussian wavepacket, which propagates along the
$z$-axis and is localized in the $(x,z)$-plane but, for the sake of
simplicity, infinite and uniform in the $y$-direction. This enables us to omit
inessential $y$-integrals and assume uniform $y$-polarization with a single
electric-field component $E_{y}$ satisfying the transversality condition
${\bm{\nabla}}\cdot{\bf E}=\partial E_{y}/\partial y=0$. We write the Gaussian
Fourier spectrum, Fig. 1(a), as

$\tilde{E}_{y}({\bf
k})\propto\exp\\!\left[-\frac{(k_{z}-k_{0})^{2}l^{2}}{2}-\frac{k_{x}^{2}w^{2}}{2}\right],$
(9)

where $k_{0}$ determines the central wavevector ${\bf k}_{0}=k_{0}\bar{\bf z}$
(the overbar denotes the unit vector of the corresponding axis), whereas $l\gg
k_{0}^{-1}$ and $w\gg k_{0}^{-1}$ denote the $z$-length and $x$-width of the
‘semiclassical’ wavepacket.

Figure 1: Plane-wave Fourier spectrum (9) and real-space field (12) of the
Gaussian wavepacket with $k_{0}w=5$ and $k_{0}l=7.5$. Here and hereafter, we
use representation of complex fields with brightness proportional to its
intensity and colours indicating the phase [28]. The panels (b) and (c) show
the real-space field at $t=0$ and after propagation over 10 Rayleigh lengths:
$ct=10k_{0}w^{2}$. The white dot indicates the position of the energy and
probability centroids $Z\simeq Z_{E}<ct$ propagating with subluminal velocity
(11).

In the second-order approximation in small parameters
$\kappa_{z}=(k_{z}-k_{0})/k_{0}$ and $\kappa_{x}=k_{x}/k_{0}$, using relations
$\omega=ck$ and ${\bf v}_{g}=c{\bf k}/k$, we find:

$\displaystyle\omega^{-1}\simeq\omega_{0}^{-1}\\!\left(1-\kappa_{z}+\kappa_{z}^{2}-\frac{\kappa_{x}^{2}}{2}\right),\quad{\bf
v}_{g}\simeq c\\!\left(1-\frac{\kappa_{x}^{2}}{2}\right)\\!\bar{\bf
z}+c\left(\kappa_{x}-\kappa_{x}\kappa_{z}\right)\bar{\bf x}\,,$
$\displaystyle\omega^{-1}{\bf
v}_{g}\simeq\omega_{0}^{-1}c\left(1-\kappa_{z}+\kappa_{z}^{2}-{\kappa_{x}^{2}}\right)\bar{\bf
z}+\omega_{0}^{-1}c\left(\kappa_{x}-2\kappa_{x}\kappa_{z}\right)\bar{\bf
x}\,.$ (10)

Substituting expressions (9) and (3.1) into Eqs. (6) and (8) and calculating
standard integrals over $(\kappa_{x},\kappa_{z})$, we derive:

${\bf V}={\bf V}_{E}\simeq
c\left(1-\frac{1}{4k_{0}^{2}w^{2}}\right)\\!\bar{\bf z}\,.$ (11)

Thus, in the second-order approximation, the energy and probability centroids
propagate with the same velocity which is slightly less than speed of light.
This effect originates from the tilt of the wavevectors in the transversely-
localized wavepacket. That is why it is determined by the transversal width
$w$ and can be equally measured on a single-photon level in $z$-delocalized
wave beams [10]. To estimate the magnitude of this effect, note that
propagation over 12 Rayleigh lengths $z_{R}=k_{0}w^{2}$, $t=12k_{0}w^{2}/c$,
yields the centroid delay of about half the central wavelength: $\Delta
Z=(V-c)t\simeq-3/k_{0}\simeq-\lambda_{0}/2$. This is quite feasible for
experimental measurements.

Figure 1 shows the Fourier spectrum $\tilde{E}_{y}({\bf k})$ of the Gaussian
beam, Eq. (9), as well as its real-space intensity-phase distribution via
complex field [29]

${\mathbb{E}}_{y}({\bf
r},t)\propto\exp\\!\left[-\frac{(z-ct)^{2}}{2l^{2}}-\frac{x^{2}}{2(w^{2}+ict/k_{0})}+ik_{0}(z-ct)\right],$
(12)

This field is the complex Fourier transform of $\tilde{E}_{y}({\bf k})$, while
the real-valued field is $E_{y}={\rm Re}\,{\mathbb{E}}_{y}$. At $z=t=0$, the
field presents the usual Gauassian wavepacket, Fig. 1(b), but after
propagation over several Rayleigh lengths diffraction transforms it into an
arc-like far-field shape, Fig. 1(c). The edges of this shape propagate slower
than its centre at $x=0$, so that the resulting centroid becomes shifted to
$Z-ct<0$. This corresponds to the subluminal propagation with the velocity
(11).

###  3.2 Bessel wavepacket in the fourth-order approximation

Let us examine another type of a wavepacket and perform calculation in a
higher-order approximation. Bessel-type wavepackets provide the simplest
example for ${\bf k}$-space calculations. Their spectrum can be presented by
an ellipse centered at ${\bf k}_{0}=k_{0}\bar{\bf z}$ in the $(k_{x},k_{z})$
plane [18, 30], Fig. 2(a):

$k_{z}=k_{0}+l^{-1}\cos\phi\,,~{}~{}k_{x}=w^{-1}\sin\phi\,,~{}~{}E_{y}\propto\exp(im\phi)\,,$
(13)

where $\phi$ is a parameter such that
$\varphi=\arctan\\!\left(\dfrac{l}{w}\tan\phi\right)$ is the azimuthal angle
with respect to the ellipse center. Here we consider Bessel wavepackets with
the spatiotemporal vortex in the $(x,z)$ [or $(x,t)$] plane characterized by
the integer quantum number $m=0,\pm 1,\pm 2,...$. The zeroth-order wavepacket
with $m=0$ does not have a vortex. Because of the one-dimensional character of
the Fourier spectrum, the ${\bf k}$-space integration should be performed over
the single azimuthal coordinate $\phi$.

Figure 2: Same as in Fig. 1 but for the Bessel-type wavepacket (13) and (18)
with $m=0$ and the same values of parameters $w$, $l$, and $t$.

Similar to the Gaussian wavepacket case, we first approximate the main
quantities with respect to small parameters
$(\kappa_{x},\kappa_{z})=(k_{0}^{-1}w^{-1}\sin\phi,k_{0}^{-1}l^{-1}\cos\phi)$,
but now in the fourth-order approximation. This results in the higher-order
corrections to Eqs. (3.1):

$\displaystyle\omega^{-1}\to\omega^{-1}+\omega_{0}^{-1}\\!\left(-\kappa_{z}^{3}+\frac{3}{2}\kappa_{z}\kappa_{x}^{2}+\kappa_{z}^{4}-3\kappa_{z}^{2}\kappa_{x}^{2}+\frac{3}{8}\kappa_{x}^{4}\right),$
$\displaystyle{\bf v}_{g}\to{\bf
v}_{g}+c\\!\left(\kappa_{x}^{2}\kappa_{z}-\frac{3}{2}\kappa_{x}^{2}\kappa_{z}^{2}+\frac{3}{8}\kappa_{x}^{4}\right)\\!\bar{\bf
z}+c\\!\left(\kappa_{z}^{2}\kappa_{x}-\frac{\kappa_{x}^{3}}{2}-\kappa_{x}\kappa_{z}^{3}+\frac{3}{2}k_{x}^{3}k_{z}\right)\\!\bar{\bf
x}\,,$ $\displaystyle\omega^{-1}{\bf v}_{g}\to\omega^{-1}{\bf
v}_{g}+\omega_{0}^{-1}c\left(-\kappa_{z}^{3}+3\kappa_{x}^{2}\kappa_{z}+\kappa_{z}^{4}-6\kappa_{x}^{2}\kappa_{z}^{2}+\kappa_{x}^{4}\right)\bar{\bf
z}$
$\displaystyle\qquad+\omega_{0}^{-1}c\left(3\kappa_{x}\kappa_{z}^{2}-\kappa_{x}^{3}-4\kappa_{x}\kappa_{z}^{3}+4\kappa_{x}^{3}\kappa_{z}\right)\bar{\bf
x}\,.$ (14)

Obviously, the $x$-components in these equations are antisymmetric functions
of either $\kappa_{x}$ or $\kappa_{z}$ and all the integrals of these vanish.
Substituting Eqs. (13) and (3.2) into Eqs. (6) and (8) and calculating the
integrals of their $z$-components over $\phi$, we derive:

$\displaystyle{\bf V}_{E}\simeq
c\left(1-\frac{1}{4k_{0}^{2}w^{2}}-\frac{3}{16k_{0}^{4}w^{2}l^{2}}+\frac{9}{64k_{0}^{4}w^{4}}\right)\\!\bar{\bf
z}\,,$ (15) $\displaystyle{\bf V}\simeq
c\left(1-\frac{1}{4k_{0}^{2}w^{2}}-\frac{1}{4k_{0}^{4}w^{2}l^{2}}+\frac{11}{64k_{0}^{4}w^{4}}\right)\\!\bar{\bf
z}\,,$ (16)

These equations show that the velocities of the energy and probability
centroids differ in the fourth order in the semiclassical parameters:

$\displaystyle\delta{\bf V}\equiv{\bf V}_{E}-{\bf V}\simeq
c\left(\frac{1}{16k_{0}^{4}w^{2}l^{2}}-\frac{1}{32k_{0}^{4}w^{4}}\right)\\!\bar{\bf
z}\,.$ (17)

The transverse confinement of the wavepacket, $w$, is again crucial for this
difference. As shown in Ref. [18], the presence of a spatiotemporal vortex
induces transverse $m$-dependent shifts of the energy and probability
centroids, $X_{E}$ and $X$, but the above equations show that it does not
affect the rectilinear $z$-propagation of these centroids. Therefore, in Figs.
2 and 3 we show the simplest zeroth-order wavepckets with $m=0$.

Figure 2 shows the real-space evolution of the Bessel wavepacket numerically
calculated via complex Fourier transform of the spectrum (13):

${\mathbb{E}}_{y}({\bf
r},t)\propto\int_{0}^{2\pi}\tilde{E}_{y}(\phi)\exp\\!\left[i{\bf
k}(\phi)\cdot{\bf r}-i\omega(\phi)t\right]d\phi\,.$ (18)

At $z=t=0$ it has the elliptically deformed radial Bessel-function intensity
distribution, Fig. 2(b), but with propagation it undergoes strong deformations
(due to the ‘temporal diffraction’ [30]) into a ring-like intensity
distribution mimicking the elliptical Fourier spectrum, Fig. 2(c). Akin to the
Gaussian case, one can notice the acr-like deformations within this ring-like
structure, which are responsible for the shift of the centroids to the
subluminal area $z-ct<0$. In the second-order approximation the energy and
probability centroids propagate with the same subluminal velocity (11); the
fourth-order difference (17) is still unnoticeable in Fig. 2.

In the above calculations we assumed a uniform distribution of the density of
states over the azimuthal parameter $\phi$. For the uniform distribution over
the azimuthal angle $\varphi$, all the integrals involve extra
$d\varphi/d\phi$ factors. This modifies some numerical factors in Eqs.
(15)–(17), but does not affect the qualitative picture.

Figure 3: Same as in Figs. 1 and 2 but for the ultrashort (subwavelength) non-
diffracting Bessel wavepacket (13), (18), and (19). The parameters are: $m=0$,
$k_{0}w=3$, $k_{0}l\simeq 0.95$, and $ct=10k_{0}w^{2}$ in the panel (c). The
white and black dots indicate positions of the probability and energy
centroids $Z<ct$ and $Z_{E}<ct$, respectively, propagating with subluminal
velocities (15)–(17).

###  3.3 Non-diffracting Bessel wavepacket

A remarkable example is provided by the Bessel wavepacket described by the
spectrum (13) with

$l=\frac{w}{\sqrt{1+k_{0}^{2}w^{2}}}\,,$ (19)

shown in Fig. 3(a). Such spectrum can be transformed to a circle centred at
${\bf k}={\bf 0}$ by a suitable Lorentz boost along the $z$-direction [27,
30]. This means that such Bessel wavepacket represents a Lorentz boost of a
standing Bessel circular wave, and hence its intensity distribution does not
diffract during the propagation [27, 30], Fig. 3(b,c), providing a linear 2D
analogue of ‘light bullets’ [31, 32]. The price for this is that the
elliptical Fourier spectrum of such non-diffracting Bessel wavepacket embraces
the ${\bf k}={\bf 0}$ origin ($l^{-1}>k_{0}$), so that such solution involves
all directions of the wavevectors in the $(k_{x},k_{z})$ plane and cannot be
considered as paraxial (even when $w\gg k_{0}^{-1}$). Notably, the wavepacket
(19) is ultrashort (subwavelength) because $k_{0}l<1$, and the $z$-shifts of
the centroids look more pronounced in this solution.

Figure 3 shows the real-space evolution of a non-diffracting Bessel wavepacket
numerically calculated via Eq. (18). One can see that the wavepacket indeed
propagates with an unvarying elliptical Bessel-like intensity distribution
with subluminal velocity. This subluminal velocity allows one to perform a
Lorentz $z$-boost to its rest frame, where it becomes a standing circular
Bessel wave. Figure 3(c) shows the noticeable yet tiny fourth-order difference
between propagations of the energy and probability centroids.

##  4 Conclusions

We have examined properties of the energy and photon-probability centroids of
electromagnetic wavepackets in free space. In the second-order paraxial
approximation, these centroids propagate with the same subluminal velocity.
This effect is caused by the transverse confinement of the wavepacket and the
corresponding tilt of the wavevectors in its Fourier spectrum. A tiny
difference between the propagation of the energy and probability centroids
appears in the fourth order. Even if the experimental accuracy would allow to
measure this difference, it is not clear if there are detectors or light-
matter interaction processes specifically sensitive to the photon-probability
centroid. We have considered three different types of 2D wavepackets:
Gaussian, Bessel, and non-diffracting Bessel. In all cases, the subluminal
propagation of their energy centroids is clearly visible in the intensity
distributions and can be measured experimentally for propagation distances
over several Rayleigh lengths.

Our results shed light on the propagation of electrpmagnetic wavepackets both
in the classical-light and single-photon regimes. In the single-photon regime,
similar subluminal propagation has been measured in Ref. [10], but the
continuous-wave regime in this work corresponds to $z$-delocalized
monochromatic wave beams where the centroids’ positions and propagation are
ill-defined.

It should be noticed that the subluminal propagation of the wavepacket
centroids does not contradict propagation of signals or information transfer
with the speed of light. The wavepackets under consideration are solutions
infinitely extended in both space and time. When one considers a wavefield
which exactly vanishes in the $t<0$ and $z>0$ half-spaces, its non-zero
intensity front (precursor) always propagates with the speed of light $c$ [1,
33]. This is the case even for optical fields in media or waveguides [1, 33,
34, 35] and for massive quantum waves in free space [36, 37].

## Acknowledgements

I am grateful to Prof. Aleksandr Bekshaev for helpful discussions.

## References

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# A non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic
equivalence

Arnaud Beauville  Université Côte d’Azur
CNRS – Laboratoire J.-A. Dieudonné
Parc Valrose
F-06108 Nice cedex 2, France<EMAIL_ADDRESS>and  Chad Schoen
Department of Mathematics, Duke University Box 90320, Durham NC 27708-0320,
USA<EMAIL_ADDRESS>
###### Abstract.

We exhibit a non-hyperelliptic curve $C$ of genus $3$ such that the class of
the Ceresa cycle $[C]-[C^{-}]$ in $JC$ modulo algebraic equivalence is
torsion.

Thanks to E. Colombo and B. van Geemen for their crucial input. The second
named author thanks S. Katz and M. Reid for helpful discussions.

##  1\. Introduction

Let $C$ be a complex curve of genus $g\geq 2$, and $p$ a point of $C$. We
embed $C$ into its Jacobian $J$ by the Abel-Jacobi map $x\mapsto[x]-[p]$; we
denote by $C^{-}$ the image of $C$ under the involution $(-1)_{J}:a\mapsto-a$
of $J$. The _Ceresa class_ is the class $\mathfrak{z}(C):=[C]-[C^{-}]$ in the
group $A_{1}(J)$ of $1$-cycles on $J$ modulo algebraic equivalence (it is
independent of the choice of $p$). Since $(-1)_{J}$ acts trivially on
$H^{p}(J,\mathbb{Z})$ for $p$ even, $\mathfrak{z}(C)$ belongs to the
_Griffiths group_ $G(J)$, the kernel of the cycle class map
$A_{1}(J)\rightarrow H^{2g-2}(J,\mathbb{Z})$.

Ceresa classes have played a prominent role in the study of Griffiths groups,
especially in the development of techniques for showing that a given element
is non-zero [C, C-P, H]. In addition they played an important role in showing
that $G(J)$ can have infinite rank [N]. As the conjectures of Bloch and
Beilinson were developed and are studied $\mathfrak{z}(C)$ appears repeatedly
[Bl, BST], [Z, §1.5], always as an element of infinite order.

When $C$ is hyperelliptic, $\mathfrak{z}(C)=0$; in fact $C-C^{-}$ is zero as a
cycle when $p$ is a Weierstrass point. In this note we will exhibit what we
believe to be the first example of a non-hyperelliptic curve $C$ with
$\mathfrak{z}(C)=0$ in $A_{1}(J)\otimes\mathbb{Q}$. The curve $C$ has genus
$3$, and admits an automorphism $\sigma$ of order 9, such that the quotient
variety $J/\langle{\sigma}\rangle$ is uniruled. This implies that the
Griffiths group of a resolution of $J/\langle{\sigma}\rangle$ is torsion;
going back to $J$ gives the result.

##  2\. Main result

######  Theorem.

Let $C\subset\mathbb{P}^{2}$ be the genus $3$ curve defined by
$X^{4}+XZ^{3}+Y^{3}Z=0$. Then $\mathfrak{z}(C)=0$ in
$A_{1}(J)\otimes\mathbb{Q}$.

Proof : Let $\zeta$ be a primitive 9-th root of unity. We consider the
automorphism $\sigma$ of $C$ defined by
$\sigma(X,Y,Z)=(X,\zeta^{2}Y,\zeta^{3}Z)$. We use the fixed point $p=(0,0,1)$
to embed $C$ in its Jacobian $J$, so that the action of $\sigma$ on $J$
preserves $C$ and $C^{-}$. We denote by $V$ the quotient variety
$J/\langle{\sigma}\rangle$, and by $\pi:J\rightarrow V$ the quotient map. Let
$F\subset J$ be the subset of elements with nontrivial stabilizer; the
singular locus $\operatorname{Sing}V$ of $V$ is $\pi(F)$. We put
$J^{\mathrm{o}}:=J\smallsetminus F$ and
$V^{\mathrm{o}}:=V\smallsetminus\operatorname{Sing}V$.

######  Lemma 1.

$\operatorname{Sing}V$ is finite; the points $\pi(x)$ for
$x\in\operatorname{Ker}(1_{J}-\sigma)$ are non-canonical singularities.

Proof : The space $T_{0}(J)$ is canonically identified with
$H^{0}(C,K_{C})^{*}$. The elements of $H^{0}(C,K_{C})$ are of the form
$L\,\dfrac{XdZ-ZdX}{Y^{2}Z}$, with $L\in
H^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}}(1))$ [B-K, §9.3, Corollary of
Theorem 1]. It follows that the eigenvalues of $\sigma$ on $H^{0}(C,K_{C})$
are $\zeta^{5},\zeta^{7},\zeta^{8}$, and those on
$T_{0}(J)=H^{0}(C,K_{C})^{*}$ are $\zeta,\zeta^{2},\zeta^{4}$. Therefore
$\operatorname{Ker}(1_{J}-\sigma^{d})$ is finite for $0<d<9$, so $F$ is
finite. Since $1+2+4<9$, Reid’s criterion [R, Theorem 3.1] implies that the
singular points $\pi(x)$ for $x\in\operatorname{Ker}(1_{J}-\sigma)$ are not
canonical.∎

######  Lemma 2.

The variety $V$ is uniruled.

Proof : Let $\rho:\tilde{V}\rightarrow V$ be a resolution of singularities; it
suffices to prove that $\tilde{V}$ has Kodaira dimension $-\infty$ [M].
Suppose this is not the case: there exist an integer $r\geq 1$ and a nonzero
section $\tilde{\omega}$ of $K_{\tilde{V}}^{r}$. By restriction to
$\rho^{-1}(V^{\mathrm{o}})\cong V^{\mathrm{o}}$, we get a section $\omega$ of
$K^{r}_{V^{\mathrm{o}}}$, whose pull back under $\pi$ is a nonzero section of
$K^{r}_{J^{\mathrm{o}}}$; therefore $\omega$ is a generator of
$K^{r}_{V^{\mathrm{o}}}$, hence extends to a generator of the reflexive sheaf
$K_{V}^{[r]}$ (with the notation of [R]). By construction this generator
remains regular on $\tilde{V}$, which means that the singularities of $V$ are
canonical [R, Proposition 1.2], contradicting Lemma 1.∎

######  Lemma 3.

Let $X$ be a uniruled smooth projective threefold. The Griffiths group $G(X)$
is torsion.

Proof : There exists a smooth projective surface $S$ and a dominant rational
map $S\times\mathbb{P}^{1}\dasharrow X$. After blowing up some points and some
smooth curves in $S\times\mathbb{P}^{1}$, we get a smooth projective threefold
$W$ birational to $S\times\mathbb{P}^{1}$, and a generically finite morphism
$f:W\rightarrow X$. Since the Griffiths group is a stably birational invariant
(see [V, Proposition 2.30]), we have $G(W)=G(S)=0$. For $z\in G(X)$, we have
$(\deg f)z=f_{*}f^{*}(z)=0$, hence $G(X)$ is annihilated by $\deg f$.∎

_Remark_ $.-$ One can actually deduce from [Bl-S, Theorem 1 (ii)] that
$G(X)=0$ – but we will not need this fact.

_Proof of the Theorem_ : We can choose the resolution
$\rho:\tilde{V}\rightarrow V$ so that $E:=\rho^{-1}(\operatorname{SingV})$ is
a normal crossing divisor, whose irreducible components are smooth and
_rational_ [Fuj, Corollary of Theorem 1].

Let $\bar{C}$ and $\bar{C}^{-}$ be the images in $V$ of $C$ and $C^{-}$, and
let $\tilde{C}$ and $\tilde{C}^{-}$ be their proper transforms in $\tilde{V}$.
We have $[\bar{C}]-[\bar{C}^{-}]=\frac{1}{9}\pi_{*}([C]-[C^{-}])=0$ in
$H^{4}(V^{\mathrm{o}},\mathbb{Q})$. Now we have an exact sequence [D,
Corollaire 8.2.8]

$H^{2}(\tilde{E},\mathbb{Q})\xrightarrow{\ i_{*}\
}H^{4}(\tilde{V},\mathbb{Q})\rightarrow H^{4}(V^{\mathrm{o}},\mathbb{Q})\,,$

where $\tilde{E}$ is the normalization of $E$, and $i$ the composition
$\tilde{E}\rightarrow E\hookrightarrow\tilde{V}$. Therefore we have
$[\tilde{C}]-[\tilde{C}^{-}]=i_{*}z$ in $H^{4}(\tilde{V},\mathbb{Q})$ for some
class $z\in H^{2}(\tilde{E},\mathbb{Q})$. Since the components of $\tilde{E}$
are rational, $z$ is the class of an element $\mathfrak{z}$ of
$A_{1}(\tilde{E})\otimes\mathbb{Q}$. Then
$[\tilde{C}]-[\tilde{C}^{-}]-i_{*}\mathfrak{z}\in
A_{1}(\tilde{V})\otimes\mathbb{Q}$ is homologous to zero, hence equal to zero
by Lemma 3. Restricting to $\tilde{V}\smallsetminus E\cong V^{\mathrm{o}}$, we
get $[\bar{C}]-[\bar{C}^{-}]=0$ in $A_{1}(V^{\mathrm{o}}\otimes\mathbb{Q})$,
hence $[C]-[C^{-}]=\pi^{*}([\bar{C}]-[\bar{C}^{-}])=0$ in
$A_{1}(J^{\mathrm{o}})\otimes\mathbb{Q}$. But the restriction map
$A_{1}(J)\rightarrow A_{1}(J^{\mathrm{o}})$ is an isomorphism [Ful, Example
10.3.4], hence the Theorem.∎

##  3\. Complements

######  Corollary 1.

Let $\Theta$ be a Theta divisor on $J$. We have $[C]=\dfrac{[\Theta]^{2}}{2}$
in $A_{1}(J)\otimes\mathbb{Q}$ (Poincaré formula).

Proof : Indeed for _any_ genus 3 curve $C$ we have $[\Theta]^{2}=[C]+[C^{-}]$
in $A_{1}(J)\otimes\mathbb{Q}$ (if $p,q$ are two distinct points of $C$, the
intersection of $\Theta$ with its translate by $[p]-[q]$ is the union of a
translate of $C$ and a translate of $C^{-}$ — see for instance [Mu, Lecture
IV]). Thus the corollary is equivalent to the theorem.∎

Recall that the _modified diagonal cycle_ $\Gamma(C,p)$, first considered in
[G-S], is the element $\Gamma(C,p)$ of $A_{1}(C^{3})$ defined as follows. We
denote by $[x,x,x],[x,x,p],[x,p,p]$ etc. the classes in $A_{1}(C\times C\times
C)$ of the image of $C$ by the maps $x\mapsto(x,x,x)$, $x\mapsto(x,x,p)$,
$x\mapsto(x,p,p)$ etc. Then:

$\Gamma(C,p):=[x,x,x]-[x,x,p]-[x,p,x]-[p,x,x]+[x,p,p]+[p,x,p]+[p,p,x]\,.$

By [FLV, Remark 3.4], we have

######  Corollary 2.

$\Gamma(C,p)=0$ in $A_{1}(C^{3})\otimes\mathbb{Q}$.

Finally let us mention the result of [B3]: the class of $[C]-[C^{-}]$ in the
intermediate Jacobian $\mathscr{J}_{1}(J)$ is torsion. It can be also deduced
from our theorem, though the proof in [B3] is more direct.

In [BLLS] the authors construct a genus 7 curve with the same property, and
suggest that the corresponding Ceresa cycle should be torsion modulo algebraic
equivalence (Remark 1.2).

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