#
Artificial Dynamical Effects in Quantum Field Theory

Stanley J. Brodsky  1,∗ Alexandre Deur  2 Craig D. Roberts3,4

1SLAC National Accelerator Laboratory, Stanford University, Stanford,
California 94309, USA
2 Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
3 School of Physics, Nanjing University, Nanjing, Jiangsu 210093, China
4 Institute for Nonperturbative Physics, Nanjing University, Nanjing, Jiangsu
210093, China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
∗Corresponding author

(2021 July 20)

###### Abstract

In Newtonian mechanics, inertial pseudoforces – or fictitious forces – appear
in systems studied in non-Galilean reference frames; e.g., a centrifugal force
seems to arise if the dynamics is analyzed in a rotating reference frame. The
equivalent of Galilean invariance for relativistic kinematics is Poincaré
invariance; analogous artificial effects may arise in relativistic quantum
field theory (QFT) if a system is studied in a framework violating Poincaré
invariance. We highlight how such issues complicate the traditional canonical
quantization of QFTs and can lead to a subjective description of natural
phenomena. In fact, if the system involves the strong interaction, obtaining
objective results can become an intractable problem using canonical
quantization because the pseudoforces are essentially nonperturbative. In
contrast, the treatment of the same problem using light-front (LF)
quantization is free of spurious pseudoeffects because Poincaré invariance is
manifest; thus the treatment of strong interaction problems becomes simpler.
These statements are illustrated using several examples: the Gerasimov-Drell-
Hearn (GDH) relation, a fundamental feature of QFT; the absence of any
measurable impact of Lorentz contraction in high-energy collisions; and the
fictitious character of vacuum fluctuation contributions to the cosmological
constant.

##  1 Introduction

A key scientific principle is that an observer’s choice of conventions cannot
affect physical phenomena. Effects that depend on such choices are artificial,
rather than objective features of Nature; e.g., physics cannot depend on the
choice of units or a particular scheme to renormalize a quantum field theory.
Most critically, physical effects cannot depend on the observer’s Lorentz
frame: valid phenomena should display Galilean invariance in classical
mechanics and Poincaré invariance for relativistic systems.

In Newtonian dynamics, inertial (Galilean) frames are essential, because they
provide a minimal description of Nature: only fundamental forces appear. In
non-Galilean frames, artificial inertial forces can arise, e.g., centrifugal
forces which seem to act on an object from the perspective of an observer in a
rotating frame. Such _ad-hoc_ frame-dependent pseudoforces are not innate to
the dynamics of the system. Although non-Galilean frames and pseudoforces are
sometimes intuitively useful, the fundamental description of a system in
Newtonian dynamics must be determined in a Galilean frame.

Just as breaking Galilean invariance in Newtonian dynamics introduces
artificial pseudoforces, any approach to relativistic QFT which breaks
Poincaré invariance may also lead to pseudophysics. In order to identify
situations with such potential, recall the two common methods of field
quantization: the Hamiltonian-based canonical approach; and Lagrangian path
integrals.

Canonical. This approach uses the Hamiltonian $\mathcal{H}=i{d\over{d{\mathpzc
t}}}$ as the generator of evolution in time ${\mathpzc t}$. Since it involves
time-ordering, it is also called “time-ordered perturbation theory”. However,
by singling-out the time coordinate $t$, which is a non-Lorentz-invariant
variable, one breaks Poincaré invariance; thus potentially introducing
artificial pseudodynamical effects, as discussed below.

Path integral. This Lagrangian approach leads to “covariant perturbation
theory”, centered on Feynman diagrams; time-ordering is not required. Thus,
for a reaction involving $n$ vertices, a single Feynman diagram corresponds to
the sum of $n!$ time-ordered diagrams. Since time or coordinate frame
definitions are not needed, covariant perturbation theory is free of
pseudodynamics. It is natural to make frame and metric choices to produce a
particular prediction from the covariant calculation. This is also standard in
Newtonian dynamics. For example, to connect with a practical situation, one
may choose a specific frame and spacetime metric, such as the Minkowski
metric, $(t,x,y,z)$, or the LF metric [1], $(x^{-},x,y,x^{+})$, $x^{\pm}=ct\pm
z$. (Henceforth, we set $\hbar\equiv 1\equiv c$.)

Although the canonical and path integral quantizations can be used
independently, they are often combined. To avoid the consequences of breaking
Poincaré invariance in the canonical approach, a perturbative calculation is
typically developed by employing the Hamiltonian to generate the dynamics,
summing over all time-orderings, and then deriving the Feynman rules for use
in calculations, thereby guaranteeing covariant results. Using solely the
Hamiltonian approach may seem a poor choice, since it exposes practitioners to
the issues of Poincaré invariance violation, in contrast to covariant
perturbation theory or the Hamiltonian approach supplemented by Feynman
graphs. However, physical processes may also involve nonperturbative dynamics,
which is not calculable using Feynman diagrams at any finite order.

Nonperturbative dynamics can emerge from Feynman graphs when infinitely many
diagrams are resummed, as expressed, e.g., in the Dyson-Schwinger equations
(DSEs) [2]. The perspective can be reversed, with nonperturbative DSEs serving
as generating tools for Feynman graphs. However, applying DSEs involves the
development of sound, symmetry preserving truncations, which is a complex task
[3, 4, 5, 6].

The Hamiltonian approach also offers a natural framework to formalize
nonperturbative dynamics, albeit with the caveat that pseudoforces may arise.
Herein, we concern ourselves with this method and how to avoid pseudodynamics
[1, 7], offering new perspectives by drawing parallels with well-known
Newtonian dynamics. This discussion will aid in identifying the origin of the
caveats that ride on the Hamiltonian approach and how to avoid them.

##  2 Source of Poincaré invariance violation

Canonical quantization is typically completed in Minkowski spacetime and
called “instant form quantization” (IF) [1]. IF is widely used because of its
links to nonrelativistic systems and ensuing intuitive nature. However,
applying IF quantization to systems with relativistic dynamics is problematic
because it manifestly violates Poincaré invariance. LF quantization, the other
common canonical procedure, avoids the issue since it retains boost invariance
and is independent of the observer’s Lorentz frame.

Both forms of relativistic dynamics [1] are initially constructed in
compliance with Poincaré invariance. In both cases, one identifies 10
operators which fulfill the Poincaré algebra, expressing the spacetime
symmetries, viz. invariances under spacetime translations, spatial rotations,
and boosts. However, Poincaré invariance concerns the worldlines of four-
dimensional (4D) spacetime. To study the time evolution of a system in 3D
space, spacetime must be foliated to specify the time variable. The choice of
foliation is the difference between IF and LF.

In general, a symmetry manifest in 4D need not be preserved in a subspace of
lower dimension; and in 3D, no canonical quantization form will satisfy all
the requirements of Poincaré invariance. The following analogy reveals how
Poincaré invariance becomes lost in IF, but is preserved in practice by LF. In
3D, a cylinder is rotationally invariant around one axis, conventionally
labelled $\bm{z}$. If the cylinder is projected onto a plane, then the
symmetry can either be maintained or broken. When $\bm{z}$ is perpendicular to
the plane, the projection is a circle and the symmetry is preserved.
Otherwise, the projection is an ellipse and rotational invariance is lost. In
this analogy, LF corresponds to a “perpendicular” slicing that preserves the
symmetries of primary interest; whereas IF corresponds to an “elliptical”
slicing, that breaks the desired symmetries; a crucial problem-solving asset
is thus lost. Notably, LF retains the maximum number of kinematical operators:
7 out of 10, that do not involve dynamics (genuine or pseudo).

Crucially, the three dynamical LF operators are the LF Hamiltonian, which
describes evolution in LF time $x^{+}=ct+z$, and two rotation operators in
$\bm{x}$ and $\bm{y}$ which are often unimportant for reactions in high-energy
physics. Consequently, for practical purposes, LF dynamics is effectively
Poincaré invariant. In contrast, IF has only 6 kinematical operators (the
operators for space translations and rotations). Its boost operators are
dynamical: they introduce interactions in Lorentz transformations,
complicating the IF treatment of reactions. Thus, Poincaré invariance can in
practice be violated; and a phenomenon observed in two distinct frames related
by different boosts can have contrasting dynamical descriptions. This
introduces severe problems: the dynamical description becomes subjective,
depending on the observer’s motion. Furthermore, it may not be possible in
practice to identify and compute the boost-dependent pseudoforce.

Figure 1:  Graphs used to compute the nucleon electromagnetic form factors in
the IF (left) and LF (right) cases. A virtual photon $\gamma^{*}$ (wavy line)
probes a nucleon (thick black line) with initial momentum $p$ and final
momentum $p+q$. The electromagnetic current at spacetime location $x$ is
$J^{\mu}(x)$. The red solid and blue dashed lines denote valence quarks and
gluons, respectively. Acausal, frame-dependent vacuum-induced currents are
required in IF calculations of current matrix elements – left. They are absent
in LF – right.

The mixing between dynamical and kinematical effects makes the IF boost of a
bound state wave function a complex dynamical problem which may not be
resolved within the IF framework alone [8, 9], as highlighted in Sec. 4. The
fact that IF boosts are dynamical is a major hurdle because boosts are
essential in analyzing high-energy reactions, e.g., elastic electron+proton
scattering,$e+P(p)\to e^{\prime}+P^{\prime}(p+q)$, where the electron
transfers a 4-momentum $q$ to a proton of initial 4-momentum $p$, as
illustrated in Fig. 1. To study such reactions, the proton wave function must
be boosted from $p\to p+q$. For IF, this leads to a complicated dynamical
problem, compounded by the fact that the number of particles describing the
proton can change, as will be discussed next. The boost of a bound state
combines nonperturbative dynamics with kinematics, typically making the IF
boost of a bound-state wave function in quantum chromodynamics (QCD)
incalculable, and thus the analytic determination of the boost-dependent
pseudoforce intractable.

There is another complication when using IF: it cannot describe a system’s
dynamics while respecting causality. This has far-reaching consequences. For
instance, the form factor (Fig. 1) of a bound state involves current matrix
elements, like $\bra{p+q}J^{\mu}(0)\ket{p}$, which cannot be computed solely
from the bound state IF wave function because additional currents arise from
pair creation from the vacuum. For example for the muonium atom
($\mu^{+}e^{-}$), one must also couple the external current $J^{\mu}$ to the
leptons that are created or annihilated from the vacuum by the
$e^{+}e^{-}\gamma$ interaction. These “vacuum-induced” effects appear when one
relates Feynman diagrams to the sum of canonical IF perturbation theory
diagrams, as illustrated in Fig. 1. Such contributions are acausal because the
pair-creation time is uncorrelated with the time of the scattering event.
Thus, Poincaré-invariant physical results cannot be obtained using the IF
without including acausal dynamical contributions. In contrast, pair
contributions are absent in the LF framework, and form factors can be
correctly computed [10, 11, 12].

##  3 The Light Front

The fundamental problems arising from IF vanish with LF quantization [1, 7,
13], which is effectively frame-independent and causal. The Galilean time $t$
of IF is replaced by LF time $\tau=x^{+}:=ct+z$, where $z$ can be chosen as
any one of the spatial coordinates. Measurements at fixed LF time correspond
to observations at the front of a light wave moving in the $-z$ direction, as
in a flash photograph. The LF foliation of spacetime in terms of
$(\tau,x^{-})$, where $x^{-}:=ct-z$, is distinct from $(t,z)$; however, the
other spatial coordinates $(x,y):=\bf{x_{\bot}}$ are the same in LF and IF.
Most important, the relation between $(\tau,x^{-})$ and $(t,z)$ implies that
physical descriptions in IF and LF are _not_ related to each other via a
Lorentz transformation.

The $7$ kinematical LF generators correspond to invariances under spatial
translations ($p^{+}:=p^{0}+p^{z}$, $\bf{p_{\perp}}$), rotation about $\bf{z}$
($J^{z}$), and Lorentz boosts in the $\bf x$, $\bf y$ or $\bf x^{+}$
directions ($K_{x}$, $K_{y}$, $K_{x^{+}}$). One of the $3$ dynamical
generators is the LF Hamiltonian: the LF time evolution operator is
$p^{-}\equiv p^{0}-p^{z}=i{d\over d\tau}$ and the eigenstates are
eigenfunctions of the LF Hamiltonian,
$\mathcal{H}_{LF}|\Psi\rangle=M^{2}|\Psi\rangle$,
$\mathcal{H}_{LF}=p^{-}p^{+}-\bf{p}_{\perp}^{2}$. The other two dynamical
generators are the rotations about $\bf x$ or $\bf y$ ($J^{x}$, $J^{y}$). LF
not only maximizes the number of kinematic generators of the Poincaré algebra,
it also utilizes those which are typically most useful. These features allow a
complete, and in practice, Poincaré-invariant description of the system and
its evolution. In contrast, dynamical operators (beside the Hamiltonian) are
necessary in IF; and since they describe spatial symmetries of the system,
problematic pseudoforces must be included for a complete analysis.

A hadron, e.g. a proton, is an eigenstate of the QCD LF Hamiltonian:
$\mathcal{H}_{LF}|\Psi_{p}\rangle=M_{p}^{2}|\Psi_{p}\rangle$. The LF wave
functions (LFWFs) for each Fock state of the proton,
$\psi_{p}^{n_{0}}(x_{i},\bf p_{\perp i},\lambda_{i})$, are projections
$\langle n_{0}|\Psi_{p}\rangle$ of the eigenstate, where $\lambda_{i}$ are the
LF helicity eigenvalues, which are kinematical quantities, and the
$|n_{0}\rangle$ are the quark and gluon Fock state eigenstates of the free LF
Hamiltonian $\mathcal{H}^{0}_{LF}$, which have the same quantum numbers as the
proton. One can identify the free (kinetic) and interaction-dependent
contributions to the LF Hamiltonian

${\cal H}_{LF}|\Psi_{p}\rangle=\mathcal{H}^{0}_{LF}|\Psi_{p}\rangle+{\cal
H}^{\rm int}_{LF}|\Psi_{p}\rangle=M_{p}^{2}|\Psi_{p}\rangle.$ (1)

The interactions, expressed in ${\cal H}^{\rm int}_{LF}$, couple the Fock
states to each other:

$\langle
n_{0}|\mathcal{H}^{0}_{LF}|\Psi_{p}\rangle+\sum_{n_{0}^{\prime}}\langle
n_{0}|\mathcal{H}^{\rm int}_{LF}|n_{0}^{\prime}\rangle\langle
n_{0}^{\prime}|\Psi_{p}\rangle=M_{p}^{2}\langle n_{0}|\Psi_{p}\rangle\,.$ (2)

Consequently, each $\psi_{P}^{n_{0}}(x_{i},\bf p_{\perp i},\lambda_{i})$ is a
nontrivial Fock state LFWF, which is dynamically coupled to all the others.
Crucially, the wave functions of the eigenstates of $\mathcal{H}_{LF}$ are
independent of the hadron’s LF momenta $(p^{+},\bf{p_{\perp}})$, _viz_. they
are independent of the frame defined by these quantities. This property is
manifest, e.g., in LF scattering formulae, such as the Drell-Yan–West
expression for a form factor [7]. The LFWFs of any composite system are
similarly frame-independent, in contrast to the analogous IF wave functions.
Thus the LF provides a fundamental, objective description of both elementary
and bound-state systems without makeshift, frame-dependent dynamical
corrections.

For instance, using LF, hadron form factors can rigorously be computed from
the overlap integral of the product of the initial- and final-state LFWFs [10,
11, 12], see Fig. 1. The simplifying frame $q^{+}=0$ for the photon momentum
$q$ eliminates time-ordered contributions where the photon creates or
annihilates quark-antiquark pairs. Thus, unlike IF, vacuum-induced currents
are not generated. Note, however, that if one evaluates the electromagnetic
current in a frame with $q^{+}\neq 0$; one must allow for LF Fock states with
extra $q\bar{q}$ pairs.

Factorization theorems, like the Drell-Yan formula [14] for lepton production
$p\bar{p}\to\ell\bar{\ell}X$, or the DGLAP [15, 16, 17, 18] and ERBL [19, 20]
evolution equations are all derived using the LF, which also enables a
straightforward proof [21] that the anomalous gravitomagnetic moment of
composite systems must vanish [22].

##  4 The GDH relation

The GDH relation [23, 24, 25], Fig. 2, connects the anomalous magnetic moment,
$\kappa$, of a particle, such as a proton, to its helicity-dependent photo-
production cross-sections, $\sigma_{P}$, $\sigma_{A}$:

$\int_{0}^{\infty}{{\sigma_{P}(\nu)-\sigma_{A}(\nu)}\over{\nu}}d\nu={{4\pi^{2}S\alpha\kappa^{2}}\over{M^{2}}}.\vspace{-0.2cm}$
(3)

Here, $\nu=\nicefrac{{(k+p)^{2}}}{{2M}}$ is the invariant energy of the
absorbed photon of momentum $k^{\mu}$, with $p^{\mu}$ the momentum of the
studied particle, $M$ is its rest mass, $S$ is its spin, and $\alpha$ is the
QED fine-structure constant. In $\sigma_{P}$ ($\sigma_{A}$), the photon spin
is parallel (antiparallel) to the particle’s spin. This relation, which is a
fundamental feature of QFT, can be used to illustrate how pseudoforces arise
in IF from the dynamical character of IF boosts. If pseudoforces are
neglected, an incomplete description follows [26]. When they are incorporated
[8, 9], one arrives at the correct description in IF.

Figure 2:  The GDH relation. Polarized photons (wavy lines) impinging on a
target particle (grey sphere) can produce secondary particles (thin lines).
The flux of particles produced depends on the polarization direction of the
photon (circular arrow) with respect to the target spin direction (straight
arrow). The GDH relation connects the square of anomalous magnetic moment of
the target particle, symbolized here by magnetic field lines, to the
difference in the flux of produced particles for the two opposite photon
polarizations. The relation holds after integrating the production difference
over the photon energy $\nu$ (mathematical symbols in magenta).

###  4.1 Pseudoforces in the IF derivation.

The GDH relation has been derived in several ways: (A) using a dispersion
relation for the forward Compton amplitude [23, 24] together with the low
energy theorem [27, 28], the latter resting on Lorentz and gauge invariances;
(B) using LF current algebra [29]; and (C) using IF current algebra [25].
Lorentz invariance is respected in (A) and (B), but not in (C) where a
Lorentz-violating term appears on the right-hand side of Eq. (3) [30, 31]:
$4\pi^{2}S\alpha\big{(}{{\kappa^{2}}\over{M^{2}}}-{{(Z+\kappa)^{2}}\over{(p^{0})^{2}}}\big{)},$
where $p^{0}$ is the initial state particle energy and $Z$ its charge. The
$p^{0}$ dependence violates Lorentz invariance, revealing $(Z+\kappa)^{2}$ as
a fictitious elastic contribution to Compton scattering.

###  4.2 Pseudoforces in the GDH relation for composite systems.

Another example of pseudodynamics illustrated with the GDH relation appears
when one explicitly accounts for particle compositeness.

The GDH relation was originally thought to be invalid for composite particles
[26]. However, the purported invalidity originated from neglect of IF
pseudoforces. The proof of Eq. (3) for the deuteron was given in Refs. [8, 9].
Verifying Eq. (3) for the deuteron involves the analysis of a bound-state of
two nucleons in an external electromagnetic field. A boost is required because
the deuteron momentum is affected by the probing photon, integrated in Eq. (3)
over all values of $\nu$. The covariant Bethe-Salpeter formalism [32] provides
the correct IF boost of the deuteron wave function, which includes
pseudodynamics. In contrast, assuming a product of boosts of two independent
nucleons [26] overlooks pseudodynamics and produces an erroneous IF result.

The boosted IF deuteron wave function includes Dirac spinors of the form [8,
9]:

$\biggl{[}\begin{matrix}1+{{\bm{\sigma_{i}\cdot
P}}\over{\mathcal{M}+E}}{{\bm{\sigma_{i}\cdot p}}\over{2m_{i}+k_{i}}}\\\
\bm{\sigma_{i}\cdot}\big{(}{{\bm{P}\over{\mathcal{M}+E}}+{{\bm{p}}\over{2m_{i}+k_{i}}}\big{)}}\end{matrix}\biggr{]}$
(4)

where: $\bm{\sigma_{i}}$, $i=a,b$, is the spin of nucleon $i$;
$(E,\bm{P})=p_{a}+p_{b}$ is the deuteron 4-momentum, with $p_{i}$ the center-
of-mass nucleon 4-momenta;
$\bm{p}=(m_{a}\bm{p_{a}}-m_{b}\bm{p_{b}})/\mathcal{M}$ is the relative
momentum; $\mathcal{M}$ is the deuteron rest mass; $m_{1,2}$ the nucleon
masses; and $k_{i}\approx\bm{p}^{2}/(2m_{i})$;

The boost generates the $\bm{\sigma}\cdot(\bm{p},\bm{P})$ terms. They would be
absent if one erroneously assumes that the net boost is mrely the product of
boosts for two independent nucleons. They arise from the boost-induced mixing
of the triplet state $\ket{1}$ of the deuteron $\ket{1/2}\times\ket{1/2}$
system with its singlet state $\ket{0}$ [33]. Thus a pseudointeraction appears
in the resulting boosted Pauli equation for a composite system in addition to
the sum of the individual one-body interactions with the external field. The
extra “spin-orbit” terms express the spin coupling to the induced magnetic
field $\bm{B}_{\rm ind}=\bm{v}\times\bm{E}$ owing to motion in the external
electric field.

The pseudoterms $\bm{\sigma}\cdot(\bm{p},\bm{P})$ indicate that energy from
the external field is changed into deuteron internal energy. Specifically, the
force acting on the two-body system is modified during a boost owing to
triplet-singlet state mixing, since only the triplet state $\ket{1}$ couples
to the external magnetic field. Thus, a kinematic action – the IF boost –
results in a dynamical modification: the change of force, from the reduced
coupling to the external magnetic field owing to the transfer of spin-1 $\to$
spin-0. Without this effect, the GDH relation, as dynamically determined in
IF, cannot be recovered for composite objects. The kinematical and dynamical
mixing is analogous to classical mechanics, where pseudoforces mix kinetic and
potential energies, e.g., when the kinetic energy is converted into the energy
of a centrifugal force if a rotating non-Galilean frame is chosen for the
analysis.

We have discussed Eq. (3) within the IF framework. It can be derived for
composite systems using the frame-independent LF formalism, and pseudoforces
do not appear. This offers a concrete example of the advantage of LF. Although
IF derivation of the GDH relation is possible – with considerably more
challenges than in LF, as illustrated above – this is not so for many other
observables.

##  5 Lorentz Contraction

Lorentz contraction is clearly a subjective effect because it depends
explicitly on the observer’s motion relative to the contracted object; but is
it fictitious? Lorentz contraction is often invoked when describing
relativistic collisions of composite objects. However, mistakenly ascribing an
objective meaning to Lorentz contraction can lead to erroneous descriptions.
In fact, since Lorentz contraction is not a true observable, it cannot
influence any observable properties of a relativistic collision. Working with
the LF approach enables one to avoid such pitfalls.

In discussions of collisions involving composite objects moving with
relativistic velocities, one commonly depicts the colliding objects as thin
discs (pancakes), in an attempt to highlight Lorentz contraction [34].
However, if such contractions were fundamental to the dynamical description of
the collision, a paradox would ensue because the internal dynamics of hadrons
is frame independent. For instance, the structure of the proton measured in
deep inelastic lepton scattering in the laboratory frame, where the proton is
at rest, is the same [35] as that when measured at a lepton-proton collider,
where the proton is, supposedly, Lorentz-contracted.

This seeming conflict between the Lorentz contraction of a moving object
versus the frame-independence of hadron observables is resolved by carefully
distinguishing the character of the measurement [36, 37, 38]. The length of an
object is defined by two observations performed at identical Galilean time,
$t$. Thus, in addition to violating causality, the Lorentz contraction would
only be applicable for an idealized experiment in which a prodigious number of
detectors is placed close to the object and arranged to image it at fixed $t$.
However, in actual measurements, such as lepton-proton scattering, the
exchanged photon interacts with the quarks of a proton at fixed LF time,
$\tau$, i.e., along the front of a light-wave, in analogy with a flash
photograph, not at fixed $t$. One also measures at fixed $\tau$ when observing
radiation from a system at a detector far from the moving source. That no
Lorentz contraction is observed in actual measurements is immediately clear
when using LF, but it requires non-trivial reasoning when using IF. The often-
used illustration of relativistic nuclei as colliding “pancakes” is therefore
not only subjective, but also erroneous.

##  6 Vacuum energy

Estimating the “vacuum energy”, a conjectured source for the cosmological
constant, $\Lambda$, has been a notorious problem: conventional arguments
yield values up to $10^{120}$-times greater than observation [39, 40, 41, 42,
43, 44]. However, such analyses rely on the concept of IF vacuum fluctuations,
which does not provide objective, frame-independent, properties of spacetime.
In contrast, LF dynamics predicts $\Lambda=0$ [45, 46, 47, 48, 49] because the
LF vacuum does not fluctuate: the vacuum has $p^{+}=0$, by definition;
momentum conservation demands that some virtual particles participating in
fluctuations would have $p^{+}<0$, in conflict with the LF property that
$p^{+}>0$ for any particles.

Just as pseudodynamics emerge to preserve fundamental principles, in classical
mechanics and in IF dynamics, IF vacuum fluctuations arise to preserve
causality, a keystone of QFT. To see this explicitly, one may trace how IF
vacuum fluctuations arise; this will also provide an intuitive understanding
of why the LF vacuum is trivial, complementing the formal reason that
$p^{+}>0$ for all particles.

###  6.1 The origin of the IF vacuum fluctuations.

|  |
---|---|---
Figure 3:  Reordering of event times by an IF boost. Left panel. IF spacelike
propagation from events $E_{1}$ to $E_{2}$, in frame $(t,z)$ (with
$t_{2}-t_{1}>0$). Middle panel. Same, but in a frame $(t^{\prime},z^{\prime})$
(with $t^{\prime}_{2}-t^{\prime}_{1}<0)$, boosted with respect to $(t,z)$:
since an IF boost in the $z$-direction rotates clockwise the $t$-axis and
counterclockwise the $z$-axis by the same angle, $E_{2}$ may occur before
$E_{1}$ in the boosted $(t^{\prime},z^{\prime})$ frame:
$t^{\prime}_{2}<t^{\prime}_{1}$. Right panel. A) Single particle propagation
in frame $(t,z)$. B) Same process in frame $(t^{\prime},z^{\prime})$ (pair
creation).

IF vacuum fluctuations arise when one allows propagators to link spacelike
events. This is explained in Ref. [50], which considers a scalar particle,
$\phi$, interacting with an external scalar potential, $U$. The amplitude
describing the state survival probability after scattering events
$E_{1}(x_{1})$ and $E_{2}(x_{2})$ is $\mathcal{A}=1-{\mathcal{I}}$,

${\mathcal{I}}=\int
dx_{1}dx_{2}\int{{dP}\over{(2\pi)^{3}4E^{2}}}U^{*}(x_{2})\phi^{*}(x_{2})e^{-iP_{\mu}(x_{2}^{\mu}-x_{1}^{\mu})}U(x_{1})\phi(x_{1}),$
(5)

with $P$ the four-momentum of particle $\phi$, of positive energy.
${\mathcal{I}}$ can be nonzero for spacelike $x_{2}^{\mu}-x_{1}^{\mu}$ because
any function, $f(t)$, that can be Fourier decomposed using only positive
energies:

$f(t)=\int_{0}^{\infty}dEe^{-iEt}F(E),$ (6)

cannot be zero for a finite $t$ range (unless it vanishes everywhere); so,
spacelike processes are allowed.

For such processes, one may have $t_{2}>t_{1}$ in some frames, such as the
frame $(t,z)$ of Fig. 3 – left. Here, ${\mathcal{I}}$ describes the
propagation of a single particle, see image A in Fig. 3 – right. In another
frame $t_{2}<t_{1}$, as for frame $(t^{\prime},z^{\prime})$ in Fig. 3 –
center, where $\mathcal{I}$ describes an initially propagating particle, which
then annihilates at $t_{1}$ with an antiparticle from a pair created at
$t_{2}$; see image B in Fig. 3 – right. However, additional amplitudes from
disconnected vacuum diagrams must then be added to preserve unitarity: the
vacuum diagrams compensate the negative amplitudes of the Z-graphs of image B.
Disconnected vacuum processes are thus directly coupled to the causality-
violating processes imposed by Eq. (6). In contrast, in LF dynamics, a boost
simply rescales the $\tau$ and $x^{-}$ axes and thus preserves the sign of the
LF time interval $\tau_{2}-\tau_{1}$. Therefore causality is preserved, no
Z-graphs appear; hence, disconnected vacuum processes are not required to
enforce unitarity.

Since the integrals in Eqs. (5), (6) are suppressed for large values of
$P_{\mu}(x_{2}^{\mu}-x_{1}^{\mu})$, this discussion connects to Heisenberg’s
principle, often used to depict the origin of vacuum fluctuations: Fig. 4
illustrates how the $\Delta P_{z}\Delta z$ uncertainty for $E_{2}$, in-
principle timelike, may actually allow it to be spacelike, enabling
disconnected vacuum processes. In contrast, in LF, $\Delta x^{-}$ varies along
the light-front direction $x^{-}$, preserving the timelike nature of $E_{2}$.

Figure 4:  Effect of quantum fluctuations on the causal nature of an event.
Events in the spacelike region with respect to event $E_{1}$ generate vacuum
fluctuations at the origin of the vacuum energy. Left panel: In IF dynamics,
the Heisenberg uncertainty $\Delta z_{2}$ (horizontal segment around event
$E_{2}$) applies to fixed time $t_{2}$. Thus, $\Delta z_{2}$ may allow $E_{2}$
(which would otherwise be timelike) to be spacelike. Right panel: In LF
dynamics, the uncertainty $\Delta x^{-}$ (red segment around $E_{2}$) is at
fixed LF time $\tau$, i.e. along the border between the timelike and spacelike
regions. Therefore, $E_{2}$ remains timelike with respect to $E_{1}$, and
vacuum fluctuations are not possible.

The violation of causality in Eq. (6) can be avoided by quantizing the field
at fixed $\tau$ (LF dynamics), so that two causally linked events cannot
become spacelike owing to quantum fluctuations, see Fig. 4. Thus the initial
mechanism which leads to vacuum fluctuations, via unitarity, is eliminated:
the LF vacuum is trivial, and LF dynamics preserves causality.

Clearly, the complex IF vacuum utilizes a fictitious mechanism for restoring
causality, which in any case is broken by choosing a dynamics that violates
Poincaré invariance.

###  6.2 Confinement contains condensates.

Insofar as contributions to $\Lambda$ from vacuum fluctuations are concerned,
the answer might be straightforward [48]. The connection between vacuum
structure and $\Lambda$ is typically linked to a belief that infinitely many
distinct, spacetime-independent condensates fill the Universe. These are the
supposedly measurable consequences of IF vacuum fluctuations. However, the
problem vanishes if one jettisons the notion that condensates have a physical
existence, independent of hadronic matter [45]. This seems obligatory in a
confining theory [47, 46, 48, 49] such as QCD where all predictions can be
expressed using a hadronic basis. This is quark-hadron duality. The paradigm
drawn in Refs. [45, 47, 46, 48, 49] can therefore be summarized as follows:
“If quark-hadron duality is a reality, then condensates, those quantities
commonly viewed as constant mass-scales that fill all spacetime, are instead
wholly contained within hadrons; to wit, they are properties of hadrons
themselves, expressed, e.g., in their LFWFs.”

The chiral condensate, $\langle\bar{q}q\rangle$, is the vacuum condensate most
often discussed. It is an order parameter for dynamical chiral symmetry
breaking and, using LF quantization, it was long ago argued [51] to be a
property of hadron wave functions, not the vacuum. This quality is not
surprising: in a rigorous treatment of phase transitions in statistical
mechanics, a transition only occurs in the infinite-volume limit. However, in
empirical reality, e.g., Cooper pair condensation, the order parameters are
only observed in finite-volume samples; viz. the condensates are found within
the material that supports them, not throughout an infinite volume. Similarly,
especially because of confinement, QCD condensates should be contained within
the same domain which permits the propagation of the gluons and quarks that
produce them; namely, inside hadrons. Recognizing that QCD condensates are
“in-hadron” properties, reduces the mismatch between observation and theory by
a factor of $10^{46}$.

##  7 Conclusion

In classical dynamics, one may choose any frame to analyze a system – provided
that pseudoforces are included if the frame is non-inertial. Likewise, one may
choose the IF approach, if its advantages balance the complications of
including pseudoeffects. An example of IF dynamics, supplemented by covariant
methods to compute pseudocontributions, is the GDH relation derivation for the
deuteron [8, 9]. Yet, this is not a fundamental derivation because it not only
demands additional input beyond those intrinsic to QED and the nuclear force
framework, but also introduces subjective, observer-dependent forces. Of
greater practical concern in the case of systems described by QCD, the
treatment of the pseudoeffects in the IF is generally intractable. The
inability to provide a frame-independent derivation of the GDH relation, even
for non-composite systems; and the erroneous suggestion that Lorentz
contraction plays a dynamical role in relativistic heavy-ion collisions are
other examples of the drawbacks of IF quantization. The contrast between the
complex, causality-violating IF vacuum and the simple LF vacuum demonstrates
the usefulness of LF quantization and emphasizes that the cosmological
constant cannot receive a physical contribution from vacuum structure.

Today, no approach to QFT is perfect, but LF quantization possesses many
merits. It often provides the simplest known description of Nature. As always
in physics, one can formulate a problem using any framework one desires; but
if the wrong approach is chosen, the costs are high [52].

It is worth closing with a lesson from classical mechanics. Although one may
elect to describe the solar system using a geocentric standpoint, perhaps
because it appears intuitive, one will eventually learn that this is not the
minimal description. Ultimately, the heliocentric vision is more economical
and objective; and arguably, closer to Nature. Likewise, despite its initial
conflict with human-based intuition, the absence of subjective effects in LF
quantization often makes it the most economical choice for a description of
relativistic dynamics.

This work is supported by: U.S. Department of Energy contract DE-
AC02-76SF00515 (SJB); U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, contract DE-AC05-06OR23177 (AD); and National Natural Science
Foundation of China, grant no. 12135007 (CDR).

All authors contributed equally to the preparation of this contribution.

The authors declare that they have no competing financial interests.

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$i\in\\{0,1,\ldots,k-1\\}$, $c$ is an (initial or looping) $i$-configuration,
and $q$ is a state of $\mathcal{P}$, together with a sink state $s_{\bot}$.
The initial position is $(0,c_{\bot},q_{I})$ where $c_{\bot}$ is the
configuration with empty domain (which is the unique initial
$0$-configuration).

We define the action set for Player $i$ (for odd $i$) as
$\mathcal{B}^{\Delta_{i}}\cup\mathcal{B}$, where actions in
$\mathcal{B}^{\Delta_{i}}$ are intended for round $0$ and those in
$\mathcal{B}$ are intended for all other rounds. If the wrong action is used,
then the sink state will be reached. Next, we define the function $o$
determining which player picks an action to continue a play: we have
$o(i,c,q)=i$ for odd $i$, $o(i,c,q)=1$ for even $i$ (we will soon explain how
Player $U$ moves are simulated even though Player $1$ owns the corresponding
positions), and $o(s_{\bot})=1$.

The set $E$ of edges is defined as follows (recall that we label edges by
actions of a single player, as we define a turn-based game):

  * •

We begin by modelling the moves of Player $U$. Recall that in a turn-based
distributed game, Nature resolves the nondeterminism left after the player who
is in charge at that positions has picked an action. Thus, we simulate a move
of Player $U$ by giving the position to (say) Player $1$. Then, we define the
edges such that Player $1$’s move is irrelevant, but the nondeterminism models
the choice of Player $U$.

Formally, for $i\in\\{0,2,\ldots,k-2\\}$, an initial $i$-configuration $c$,
and a state $q$ of $\mathcal{P}$, we have the edge
$((i,c,q),a,(i+1,\mathrm{ext}(c,\overline{b}),q))$ for every action $a$ of
Player $1$ and every $\overline{b}\in\mathcal{B}^{\Delta_{i}}$: No matter
which action $a$ Player $1$ picks, Nature can for each possible $\overline{b}$
pick a successor that extends $c$ by $\overline{b}$. This indeed simulates the
move of Player $U$ in subround $(0,i)$.

  * •

For $i\in\\{1,3,\ldots,k-3\\}$, an initial $i$-configuration $c$, and a state
$q$ of $\mathcal{P}$, we have the edge
$((i,c,q),\overline{b},(i+1,\mathrm{ext}(c,\overline{b}),q))$ for each
$\overline{b}\in\mathcal{B}^{\Delta_{i}}$ (this simulates the move of Player
$i$ in subround $(0,i)$) as well as the edge $((i,c,q),b,s_{\bot})$ for each
$b\in\mathcal{B}$ (Player $i$ may not pick a single block in subround $(0,i)$
if $i<k-1$). Note that there is no nondeterminism to resolve for Nature, as
there is a unique successor position for each action.

  * •

For a $(k-1)$-configuration $c$ and state $q$ of $\mathcal{P}$, we have the
edge $((k-1,c,q),b,(0,\mathrm{shft}(c^{\prime}),q^{\prime}))$ for each
$b\in\mathcal{B}$ (recall that we have $\Delta_{k-1}=1$ here), where
$c^{\prime}=\mathrm{ext}(c,b)$ and $q^{\prime}$ is the state reached by
$\mathcal{P}$ when processing
$\mathrm{mrg}(c^{\prime}(0,0),c^{\prime}(1,0),\ldots,c^{\prime}(k-1,0))$ from
$q$.

  * •

For $i\in\\{0,2,\ldots,k-2\\}$, a looping $i$-configuration $c$, and a state
$q$ of $\mathcal{P}$, we have the edge $((i,c,q),a,(i+1,\mathrm{ext}(c,b),q))$
for every action $a$ of Player $1$ and every $b\in\mathcal{B}$: No matter
which action $a$ Player $1$ picks, Nature can for each possible $b$ pick a
successor that extends $c$ by $b$. This simulates the move of Player $U$ in a
subround $(r,i+1)$ for $r>0$ using the same mechanism as described above for
moves of Player $U$ in initial configurations.

  * •

For $i\in\\{1,3,\ldots,k-3\\}$, a looping $i$-configuration $c$, and a state
$q$ of $\mathcal{P}$, we have the edge $((i,c,q),b,(i+1,\mathrm{ext}(c,b),q))$
for each $b\in\mathcal{B}$ (this simulates the move of Player $i$ in subround
$(r,i)$ for $r>0$) as well as the edge $((i,c,q),\overline{b},s_{\bot})$ for
each $\overline{b}\in\mathcal{B}^{\Delta_{i}}$ (Player $i$ may not pick a
sequence of blocks in a subround $(r,i)$). Again, there is no nondeterminism
to resolve for Nature, as there is a unique successor position for each
action.

  * •

For completeness, we have the edge $(s_{\bot},a,s_{\bot})$ for every action
$a$ of Player $1$.

It remains to define the observation functions $\beta^{i}$ as
$\beta^{i}(i,c,q)=c\\!\upharpoonright\\!i$ where $c\\!\upharpoonright\\!i$ is
the configuration defined as

$(c\\!\upharpoonright\\!i)(j,x)=\begin{cases}c(j,x)&\text{ if
}j\in\\{0,2,\ldots,i-1\\},\\\ \text{undefined}&\text{ otherwise,}\end{cases}$

i.e., Player $i$ can only observe the blocks picked for the variables
$\pi_{j}$ with even $j<i$. For completeness, we also define the observation of
the sink state $s_{\bot}$ to be $\bot$, where $\bot\notin C$. Then, the order
$1\succeq 3\succeq\cdots\succeq k-1$ witnesses that the game graph yields
hierarchical information.

This completes the definition of the game graph. To complete the definition of
the game we define the winning condition as follows: Let
$(i_{0},c_{0},q_{0})(i_{1},c_{1},q_{1})(i_{2},c_{2},q_{2})\cdots$ be a play
and let $(i_{0},c_{0},q_{0})(i_{k},c_{k},q_{k})(i_{2k},c_{2k},q_{2k})\cdots$
be the subsequence of all (initial and looping) $0$-configurations. Recall
that the state $q_{rk}$ in such a position (for $r>0$) is obtained by
processing some $\mathrm{mrg}(b_{0},\ldots,b_{k-1})$ from $q_{(r-1)k}$ , where
the $b_{i}$ are blocks picked by the players. We say
$(i_{0},c_{0},q_{0})(i_{1},c_{1},q_{1})(i_{2},c_{2},q_{2})\cdots$ is in the
winning condition, if $\Omega(q_{0})\Omega(q_{k})\Omega(q_{2k})\cdots$
satisfies the parity condition, which is an $\omega$-regular winning
condition. In particular, no winning play may visit the sink $s_{\bot}$.

######  Remark 9.

The (concrete) distributed game constructed here is a formalization of the
abstract game $\mathcal{G}(\mathfrak{T},\varphi)$ described in Subsection 9.2.
In particular, a winning collection of (finite-state) strategies for the
coalition of players in the abstract game corresponds to a winning (finite-
state) strategy profile in the concrete game and vice versa.

Hence, whenever convenient below, we do not distinguish between the concrete
and the abstract game.

Now, our main theorem (Theorem 3) is a direct consequence of Proposition 4 and
Lemma 2. Recall that we need to show that the following problem is decidable:
“Given a transition system $\mathfrak{T}$ and a HyperLTL sentence $\varphi$
with $\mathfrak{T}\models\varphi$, is $\mathfrak{T}\models\varphi$ witnessed
by computable Skolem functions?”, i.e., does $\mathfrak{T}\models\varphi$ have
an explanation in the cs-paradigm? If the answer is yes, our algorithm
computes bounded-delay transducers implementing such Skolem functions.

###### Proof.

Due to Lemma 2 and Proposition 4.2, the following statements are equivalent:

  * •

$\mathfrak{T}\models\varphi$ has computable Skolem functions.

  * •

$\mathcal{G}(\mathfrak{T})$ has a winning strategy profile.

  * •

$\mathcal{G}(\mathfrak{T})$ has a winning profile of finite-state strategies.

The last statement can be decided effectively due to Proposition 4.1. Thus,
the existence of computable Skolem functions is decidable.

Now, assume $\mathfrak{T}\models\varphi$ has computable Skolem functions. Due
to the equivalence above, (the concrete) $\mathcal{G}(\mathfrak{T},\varphi)$
has a winning profile $(s^{1},s^{3},\ldots,s^{k-1})$ of finite-state
strategies. We show by induction over $i$ how the finite-state strategies
$s^{i}$ can be turned into bounded-delay transducers $\mathcal{T}_{i}$
computing Skolem functions witnessing $\mathfrak{T}\models\varphi$. The proof
follows closely the analogous results shown in Lemma 2.1 for Turing machine
computable Skolem functions.

Let us fix some $i\in\\{1,3,\ldots,k-1\\}$ and let $\mathcal{S}_{i}$ be a
Moore machine for Player $i$ implementing $s^{i}$. Recall that
$\mathcal{S}_{i}$ reads observations of Player $i$ (configurations of the form
$c\\!\upharpoonright\\!i$ for initial and looping configurations) and returns
actions of Player $i$.

On the other hand, $\mathcal{T}_{i}$ reads an input
$\mathrm{mrg}(t_{0},t_{2},\ldots,t_{i-1})\in((2^{\mathrm{AP}})^{\frac{i}{2}})^{\omega}$
where we split each $t_{j}$ into blocks
$t_{j}=t_{j}^{0}t_{j}^{1}t_{j}^{2}\cdots$.

We construct $\mathcal{T}_{i}$ so that it works in two phases, an
initialization phase and a looping phase, that is repeated ad infinitum. We
begin by describing the initialization. It begins by $\mathcal{T}_{i}$ reading
$\Delta_{0}$ blocks from each $t_{j}$. These blocks can now be assembled into
a sequence of observations of Player $i$ corresponding to the play prefix of
$\mathcal{G}(\mathfrak{T},\varphi)$ in which Player $U$ picks these blocks.
Note that this requires to process each such observation twice, as Player
$i$’s observation does not get updated, when Player $j$ for odd $j<i$ makes a
move. All unused blocks are stored in the state space of $\mathcal{T}$. Now,
$\mathcal{T}_{i}$ simulates the run of the Moore machine $\mathcal{S}_{i}$
implementing $s^{i}$ on this sequence of observations, yielding an action
$a^{i}$. As $s^{i}$ is winning, this action is a sequence
$\overline{b}=b_{0},b_{1},\ldots,b_{\Delta_{i}-1}$ of blocks. Then,
$\mathcal{T}_{i}$ outputs $b_{0}b_{1}\cdots b_{\Delta_{i}-1}$. Now, we process
the last observation another $k-i$ times with $\mathcal{S}_{i}$, simulating
the moves for the remaining variables (which are hidden from Player $i$ which
implies the observation is unchanged). This concludes the initialization
phase.

The looping phase begins with $\mathcal{T}_{i}$ reading another block from
each $t_{j}$. Again, these blocks and the ones stored in the state space can
be assembled into observations of Player $i$ corresponding to a continuation
of the simulated play prefix in which Player $U$ pick these blocks. Then, the
run of the Moore machine $\mathcal{S}_{i}$ implementing $s^{i}$ can be
continued, yielding an action $a^{i}$. This is now a block $b$, which is
output by $\mathcal{T}_{i}$. Again, we process the last observation just
assembled another $k-i$ times to simulate the moves for the remaining
variables, which concludes one looping phase. Note that the delay of
$\mathcal{T}_{i}$ is bounded by $k\cdot\ell$, where $\ell$ is the block
length.

By storing blocks from the input that have been read but not yet used in
observations, by discarding blocks no longer needed, and by keeping track of
the state the simulated run of $\mathcal{S}_{i}$ ends in, this behaviour can
indeed be implemented using a finite state-space. We leave the tedious, but
straightforward, formal definition of $\mathcal{T}_{i}$ to the reader. ∎


# Cathodoluminescence excitation spectroscopy: nanoscale imaging of excitation
pathways

Nadezda Varkentina,1∗ Yves Auad,1∗ Steffi Y. Woo,1 Alberto Zobelli,1
Laura Bocher,1 Jean-Denis Blazit,1 Xiaoyan Li,1 Marcel Tencé,1
Kenji Watanabe,2 Takashi Taniguchi,3 Odile Stéphan,1 Mathieu Kociak,1†
Luiz H. G. Tizei,1†
1Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France
2Research Center for Functional Materials, National Institute for Materials
Science,
1-1 Namiki, Tsukuba 305-0044, Japan
3International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
∗These authors contributed equally
†To whom correspondence should be addressed;
E-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
> By recording the lifespan of optical excitations from their creation to
> their decay in the far-field, photoluminescence excitation spectroscopy
> (PLE) is crucial in the understanding the optical properties in materials,
> and therefore indispensable in devices design and optimization. The
> investigation of materials in general would benefit from an analogous
> technique, but with nanoscale spatial resolution. By coincidentally
> measuring the energy loss of a relativistic electron in optical materials
> (plasmonic nanoparticles, luminescent 2D materials) and their subsequent
> light emission, we demonstrate the possibility of making cathodoluminescence
> excitation spectroscopy, a natural counterpart of PLE with a resolution
> better than 40 nm. The information retrieved should prove invaluable for
> understanding material optics at the nanometer scale.

Single phrase summary: Cathodoluminescence excitation spectroscopy (CLE)
images excitation energy transfer pathways at the nanoscale

Light emission spectroscopies reveal materials’ optical excitations. The
knowledge of the mechanisms leading from light absorption to emission, i.e,
the absorption and decay pathways, is paramount for understanding these
excitations’ physics and their applications. Photoluminescence excitation
spectroscopy (PLE) is especially suitable for this purpose. In this
spectroscopy, the emission intensity is measured as a function of excitation
energy. Quantitatively, it directly maps a system’s relative quantum
efficiency as a function of excitation energy. Qualitatively, it permits to
access the competition between different relaxation pathways from the selected
absorption states towards emission in the selected energy window.

PLE has proven to be invaluable as it provides unparalleled information on the
optical properties of materials. Examples include identification of excited
exciton states and quantitative measurement of their binding energy in 2D
materials (?, ?), determination of the energy transfer quantum efficiency in
carbon nanotube/porphyrin compounds (?), exhaustive characterisation of the
photophysics of single photon emitters in nanodiamonds (?), and deep insight
into the relaxation pathways in GaAs quantum dots (?). Also, the correlative
nature of PLE makes it extremely sensitive compared to other absorption
techniques. Despite all these advantages and their impact in all fields
relying on optical material characterization, from quantum optics to
photovoltaics, light diffraction imposes a limit onto spatial resolution for
PLE. This severely hinders its application, as the efficiency of the
excitation and decay pathways vary drastically at scales much smaller than the
wavelength of light.

Cathodoluminescence (CL) is an emission spectroscopy (?, ?, ?) that measures
the light emission spectrum under free-electron excitation. It carries the
added benefits of sub-wavelength spatial resolution, due to the small
wavelength of electrons (3.6 pm for 100 keV electrons), and of broadband
excitation, extending from the infrared to the hard X-ray range (?). The last
20 years have witnessed an impressive success of this technique for
nanosciences (?, ?), because it can be seen as a nanoscale equivalent of off-
resonance PL for semiconductors (?), and of scattering spectroscopy for
plasmonic and optical excitations (?, ?, ?). Nevertheless, as the electron
excitation is not monochromatic, a cathodoluminescence excitation spectroscopy
(CLE) could not be developed solely by mimicking the principles behind PLE.
This is a major drawback which limits the understanding at the nanometer scale
of the exact excitation, relaxation, and decay pathways and, therefore, of the
optical excitation physics.

Attempts to circumvent this problem includes the introduction of a novel light
intensity correlation methods in CL, which indicates that at least for some
excitation pathways the bulk plasmon creation and decay into multiple
electron-hole pairs have to play a role (?), as previously proposed (?). This
technique allows the nanoscale mapping of the energy-integrated relative
quantum efficiency of semiconductor nanowires (?, ?), without resolving the
absorption energy at the origin of the luminescence, therefore failing to
resolve the exact physical origin of the absorption and decay pathways.

The absorption information can in principle be retrieved with a companion
relativistic electron spectroscopy, the electron energy loss spectroscopy
(EELS). It can be described as a nanometer scale counterpart of absorption (or
more precisely extinction) spectroscopy (?). It has been used in combination
with CL (?, ?, ?) to gain insights into the physics behind light emission upon
electron scattering. Unfortunately, the intrinsic time-averaged nature of the
current technologies prevents the disentanglement of all the possible physical
processes usually available by PLE. CLE was therefore still to be invented.

Here, we demonstrate CLE with nanometer scale spatial resolution over a broad
energy range (from the visible to the soft X-ray, 2 to 620 eV) in a scanning
transmission electron microscope (STEM). Our approach relies on a new
coincidence scheme between inelastic electron scattering and photon emission
events. If the temporal information of both these events is known, correlation
can be performed to unveil the probability of each energy transfer pathways.
CLE spectra are constructed with EELS events that are time-correlated with a
photon emission, while energy relative quantum efficiency spectra are given by
the ratio of CLE and total EELS spectra. As a proof-of-principle of CLE, we
focused on representatives of the two main families of optically relevant
materials, plasmonic nanoparticles for the photonic materials and defects in
semiconducting materials for the luminescent ones. Studying CLE on Au
nanospheres embedded in SiO2, two light emission pathways are identified:
surface plasmons for the Au (SP) and transition radiation (TR) for the SiO2
and Au. The direct energy and time correlation between absorption and emission
for these excitations, known to be phase-locked with the exciting electron, is
a confirmation of the relevance of CLE. With CLE on h-BN flakes, TR was
detected. This signal is usually in the background of EELS spectra, evidencing
the extremely high sensitivity of the technique, with a typical 2 orders of
magnitude improvement. CLE was also used to explore the decay pathways leading
to the excitation and emission of the 4.1 eV defect in h-BN. All excitations,
from the near-band edge (NBE) to the core-losses, including the bulk plasmon,
are demonstrated to participate to photon emission. The bulk plasmon is
experimentally confirmed as the main absorption pathway. Nevertheless, the
relative quantum efficiency first peaks at the NBE energy and is followed by a
linear increase up to the maximum energy in the soft X-ray energy range (620
eV), which had not yet been unveiled. The NBE pathway is unexpectedly the most
efficient excitation channel up to 15 eV. Finally, spatially resolved CLE in
h–BN reveals the spatial variation of the excitation and decay pathways with a
40 nm spatial resolution. STEM-CLE, on that account, has proven to be a
nanometer-scale counterpart of PLE.

In the following, we concentrate on Au nanosphere embedded in SiO2, mainly
showing SP resonances at around 2.2 eV both in absorption and emission, and
h-BN flakes, with a dominating NBE and plasmonic absorption and strong 4.1 eV
defect emission (?, ?), as clearly seen on the absorption (EELS) and emission
(CL) spectra in Fig. 1A and B. Nevertheless, due to their time-averaged
nature, these spectra alone cannot reveal the excitation to emission pathways
shown schematically in Fig. 1C.

Figure 1:  Photon emission pathways upon electron scattering: (A-B) Time-
averaged CL (orange) and EELS (purple) spectra of an Au/SiO2 nanosphere and a
thin h–BN flake show different absorption and emission features, described in
the text. From these correlative time-averaged spectra, one cannot identify
which absorption transitions lead to light emission. The small intensity
emission at $\approx$2 eV in the h-BN CL spectrum is a replica of the 4.1 eV
defect emission due to the diffraction grating. The insets show images of the
nanosphere and the h-BN edge. CL and EELS spectra have been normalized and
shifted vertically for clarity. (C) A relativistic inelastic electron
scattering event in a solid can generate different excitations (vertical
purple arrows): direct optical transition, NBE transition, bulk plasmon
excitation, and core-electron excitation. Excitations not involving single
particles (excitons, bulk and surface plasmons, etc.) are represented between
a fundamental (F) and excited (E) state. These can relax through different
pathways, leading to the excitation of a final optically bright energy level
and to photon emission (vertical orange arrows).

Indeed, electron scattering in matter leads to light emission through
different processes, extending in a wide energy (from meV to keV) and time
(from fs to $\mu$s) ranges. In Fig. 1C, optical transitions are represented by
vertical arrows and relaxation pathways by black arrows, with qualitative
temporal axis from left to right.

In order to understand CLE, it is necessary to know how EELS and CL spectra
relate to the processes described in Fig. 1C. Every inelastically scattered
electron must undergo an initial excitation (purple arrow in Fig. 1C) that can
be measured with EELS. This encompasses TR, NBE excitations, core-electron
transitions (?), bulk plasmon (?) and SP (?) excitations, phonon excitations
(?, ?) and exciton excitations (?). TR occurs when a relativistic electron
crosses a dielectric discontinuity and is often missed in the presence of
other excitations in the same energy range, due to its small oscillator
strength (Fig. 1A-B). The NBE of semiconductors is easily detected in EELS
(Fig. 1A-B for SiO2 and h-BN), especially with modern electron monochromator
technologies (?). Core-electron spectroscopy is widely used for chemical
mapping and allotrope identification (?) down to the atomic scale (?).

As aforementioned, these distinct excitations over a wide range of energies
can lead to photon emission, detected with CL in the IR-UV range, through
different relaxation pathways, some of which are still not understood. TR and
SP are typical of photonic materials, characterized by a phase-locked relation
with the exciting electrons (?) (as it happens with photons). As a consequence
extinction (EELS) and emission (CL) spectra are similar, with only slight
shifts expected (?). In nanostructures, they emit photons in the sub-
picosecond time scale. TR is always observed to emit photons when crossing a
dielectric boundary (Fig. 1A), although it can be experimentally difficult to
disentangle in a light-emission spectrum (CL) when more intense signals occur
in the same range (?, ?) (Fig. 1B). SPs typically have larger oscillator
strengths compared to TR. In luminescent materials, absorption and decay
pathways are expected to be more complex upon electron excitation. As depicted
in Fig. 1C, NBE, bulk plasmons, core-hole excitations, or direct excitations
can lead to the emission of light. The weight of different pathways could not
be measured in the past. For this reason the most intense absorption path, the
bulk plasmon, has been attributed as the main source of multiple electron-hole
pairs generation (?, ?), leading to light emission after relaxation to point
defects (?, ?), single atoms (?) or excitons in quantum wells (?). Finally,
atomic core-level excitations can also lead to light emission, as observed in
the past (?). A microscopic description of the weight of each energy transfer
processes is still not available.

As electron scattering energy is statistical, time-average spectra (Fig.1A-B)
cannot give information about the probability efficiency of each process
involved (Fig. 1C), which would give insights into possible microscopic
descriptions. This information is stored in the probability of each electron
scattering event and photon emission, which can be retrieved by generating
coincidence histograms of electron energy loss and photon emission events
(described in what follows and in the SI), such as in Fig. 2A and B.
Coincidence electron spectroscopy and microscopy have been performed in the
past (coincidence of EELS with secondary electron or X-ray emission (?, ?, ?))
including EELS-CL for discrete selected EELS energy ranges (?, ?), although
the relative quantum efficiency as a function of energy and its spatial
dependence has not been measured.

To achieve CLE, a temporal resolution below the time interval between events,
given by the electron current (typically 1 electron every 25 ns for 10 pA), is
required for all energy loss events of interest. With this in mind, we
implemented an EELS-CL set-up in a STEM, displayed in Fig. 2A. For EELS, a
Timepix3 detector was used (?). The detector provides sub 10 ns time
resolution over arbitrary energy ranges determined by the resolution power of
the electron spectrometer and the Timepix3 pixel size. Additionally, the
particular detector used (CheeTah, from Amsterdam Scientific Instruments) has
two time-to-digital converters (TDCs), allowing to append timestamps from
external signals into the original electron data flow. Photon emission events
were detected with a photomultiplier tube (PMT) working in the 2.0 to 5.0 eV
energy range. The PMT output is directly connected to one of the Timepix3 TDC
lines. For the h–BN experiments, a 3.65-4.13 eV bandpass filter was used.
Electrons and photons arrival times were stored in a list, along with the
electron energy loss. The response time of the detection scheme is $\approx$5
ns. A search algorithm (Methods in the SI) is used to find electrons which are
within $\pm 25$ ns of a detected photon, from which a 2D histogram of time
delay versus electron energy loss is reconstructed, Fig. 2C. This 2D histogram
shows the temporal evolution of the loss spectrum as function of delay to a
detected photon.

From these, a 2D histogram of electron energy loss events as a function of
time delay to a photon emission is reconstructed (Fig. 2C). From short time
delays ($\pm 5$ ns, unless a wider time range is specified) a CLE spectrum is
extracted. It resembles an EELS spectrum, but weighted by the photon emission
probability (Fig. 2D-E). The ratio of the CLE and the non-coincidence EELS
therefore provides the relative quantum efficiency (QE) of different
absorption processes. It highlights differences between competitive radiative
and non-radiative pathways (Fig. 2F).

For the nanospheres (Fig. 2D and Fig. SI1-SI2) with the electron beam incident
on the SiO2 shell photon emission is due to the Au nanosphere SP decay
(2.0-2.4 eV) and SiO2 TR (2.6-4.0 eV), while higher energy losses do not
contribute to light emission in the emission detection range (some spurious
coincidences are detected, but this cannot be avoided). This is a reassuring
observation, as photonic modes such as plasmons or TR are created in phase
with the electron field and can only be created by loss events with energies
in the same range as the emission ones (?). A similar observation is reported
by Feist et al. on micrometric photonic structures using an equivalent EELS-CL
coincidence experiment (?). The observation of these coincidence events and
decay channels essentially validate our methodology.

Figure 2: CLE in a STEM (A) Sketch of the experimental setup: A 60-100 keV
electron beam is focused in a nanometer spot that can be scanned along the
surface of a sample. Time-resolved CL events (orange) are collected through a
parabolic mirror and detected, after passing through a filter, with a PMT, and
time-resolved EELS events are measured by a Timpepix3 detector after an EELS
spectrometer. These are stored in an ordered list, used to produce coincidence
spectra (blue). (B) A search algorithm is used to find electrons which are
within $\pm 25$ ns of a detected photon, from which a 2D histogram of time
delay versus electron energy loss is reconstructed. (C) This 2D histogram
shows the temporal evolution of the loss spectrum as function of delay to a
detected photon. Detector junctions appear at around 15, 33, and 51 eV. (D-F)
With this information, a total EELS (all detected electrons), CLE ($\pm 5$ ns
from $\Delta t=0$ ns) and relative QE spectra (CLE/(total EELS)) for a Au/SiO2
nanosphere and a thin h–BN flake, respectively. For the nanosphere the SP and
TR decay channels are efficient photon emission pathways. TR is less visible
in the average EELS spectrum, it is exacerbated in the CLE and in the relative
quantum efficiency. For the h–BN thin flake TR, NBE, and all energies above it
contribute to photon emission. NBE losses are more efficient pathways for
light emission than energies below the bulk plasmon. Above 15 eV, the relative
QE increases linearly. The nanosphere relative QE was divided by 2 for
clarity, which is shown in detail in Fig. SI1. Detector junction values are
interpolated in spectra (D-F).

In a thin h-BN flake ($<$50 nm), the CLE and relative QE spectra show
contributions from TR, NBE, bulk plasmons and higher energies towards light
emission (4.1 eV defect and TR, included in our detection range). As
discussed, the contribution from TR is usually missed in EELS spectra because
of its small cross section. As a matter of fact, they are invisible in the
EELS and CL spectra of Fig. 1B. In the CLE spectra (Fig. 2E) their
contribution (at 1e-5 event counts compared to the regular EELS counts, a
typical 2 orders of magnitude better sensitivity than previously demonstrated)
is revealed, as a signature of the high sensitivity of CLE, much along the
lines of PLE.

The contribution from the defect emission can be seen in the NBE, bulk plasmon
(Fig. 2E) and core-losses up to the maximum detected energy (Fig. SI3). From
the CLE, we prove experimentally the common assumption that the bulk plasmon
(24.4 eV) is indeed the main source of electron-hole pairs which, after
relaxation, leads to emission of the 4.1 eV defect. Nevertheless, NBE
absorption is demonstrated to be a non-negligible source of emission, with
core-losses also a possible excitation path. In fact, energy-resolved relative
quantum efficiency (Fig. 2F) shows strikingly that NBE is a more efficient
excitation channel than other excitations up to 15 eV. Above this energy, the
efficiency for photon emission increases linearly, up to the maximum energy we
have measured (620 eV), i.e. over an energy range larger than achievable with
PLE. The extrapolated linear trend at low energy crosses zero at the band gap
energy. This can be tentatively explained as follows: each excitation at
energy loss $E$ can lead to the generation of at most $N$ electron-hole pairs,
where $N=E/E_{g}$, $E_{g}$, and, hence, $N$ photons. Below the bandgap the
number of electron-hole pairs generated is zero. Assuming that the last step
to 4.1 eV defect emission are NBE electron-hole pairs, the linear trend is
deduced. With this, the peak in the relative QE at the NBE energy is
reminiscent of an unforeseen resonant effect that will require further
theoretical investigation. Direct resonant excitation of defect states by fast
electrons has yet to be observed.

Now that the principle of the CLE is established, we turn to the possibility
of mapping the different pathways directly in real space. Indeed, the proposed
spectroscopy scheme allows for coincidence mapping. The 4.1 eV emission in
h–BN (Fig. 1C) is known to arise from single point defects (?). For each
single defect the CL excitation area forms an intensity spot $\approx$ 80x80
nm2 wide. We performed CLE mapping by rastering a nanometer-sized beam on the
sample, and collecting a full CLE spectrum corresponding to emission in the
3.65-4.13 eV range at each pixel of the scan. From this, CLE maps can be
created by filtering over different absorption (EELS) ranges.

These time-resolved maps permit disentangling the different decay pathways in
space and energy, with a 40 nm spatial resolution. Indeed, the CLE map
filtered above 6.5 eV energy loss shows two sharply localized intensity spots
consistent with the observation of 4.1 eV localized defects (Fig. 3A). On the
contrary, the CLE map filtered between 2 and 5 eV (Fig. 3B), on the peak
linked to TR, shows that both the h-BN flake and the thin amorphous carbon
support (of the TEM grid, see Methods in the SI) exhibit coincidence events
distributed in a relatively uniform manner. We note that we could not identify
specific absorption signature of the defects at their absorption energy.
Coincidence measurements with better EELS spectral resolution might reveal it
in the future.

Figure 3: Spatially resolved CLE maps in h–BN: (A) CLE energy filtered map
above 6.5 eV, the NBE energy, showing multiple localized absorption maxima,
that lead to emission of the 4.1 eV defect. (B) CLE energy filtered map
between 2 and 5 eV, showing where TR occurs. Both the h–BN thin flake and the
amorphous carbon support (lower right, which is the support for h–BN in the
TEM sample) show absorption leading to photon emission. (C-D) CLE spectra of
regions marked 1 and 2 in panels A-B, with marked integrated ranges for maps A
and B, respectively.

In conclusion, we demonstrated spatially resolved CLE, which encompasses the
main advantages of PLE (high sensitivity measurement of the relative quantum
efficiency and consequent insight between multiple light emission decay
pathways) with that of electron spectroscopies (wide energy range, nanometer
scale spatial resolution). Numerous applications of CLE are expected for
nanomaterials, ranging from the optimization of single photon sources (?), the
unveiling of the role of nanometer to atomic scale features on the optical
properties of transition metal dichalcogenide monolayer by mapping the
excitons biding energy (?, ?) to the characterization of new optical materials
such as hybrid perovskites (?), and others yet to be discovered and
understood. The spectromicroscopy scheme described requires only time-resolved
electron and photon detectors, being implementable in any electron microscope.
Therefore, it is applicable to any object compatible with STEM observation,
should they be photonic (plasmonics systems, photonic band gap materials,
waveguides …) or luminescent (quantum wells, quantum dots, single photon
emitters …) (?, ?). The current applications of the setup in the time domain
is limited by the electron detector temporal resolution. Improvements in the
near future are expected with the new Timepix4 detector (?), with fast
deflectors or with the use of pulsed electron guns (?, ?, ?). Photon and
electron energy resolved experiments in the core-level range with better
temporal resolution should give further hints on the microscopic physics
behind the relaxation pathways. Additionally, as the number of emitted photons
per electron per energy is lost using single pixel detectors, the use of
multiple PMTs or 2D arrays of detectors solves this, giving access to
excitation-energy-resolved Hanbury Brown and Twiss interferometry (?) for
energy-resolved retrieval of quantum statistics, energy efficiency for total
photon yield and excited-energy-resolved bunching experiments (?). As for PLE,
such technique resolved in emission and absorption energy will allow one to
assign specific energy bands to each observed transition, but now with
nanoscale spatial resolution. Finally, polarization dependent EELS (?, ?) and
CL will give us an almost ideal nano-optics to probe excitations symmetries.

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## Acknowledgement

This project has been funded in part by the National Agency for Research under
the program of future investment TEMPOS-CHROMATEM (reference no.
ANR-10-EQPX-50), the JCJC grant SpinE (reference no. ANR-20-CE42-0020), and
the BONASPES project (ANR-19-CE30-0007), and by the European Union’s Horizon
2020 research and innovation programme under grant agreement No. 823717
(ESTEEM3) and 101017720 (EBEAM). K.W. and T.T. acknowledge support from the
Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number
JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and
21H05233). The authors thank ASI for extensive discussion during the Timepix3
CheeTah implementaiton for EELS and coincidence experiments.

## Authors contributions

LHGT, OS, LB, AZ, and MK designed the experiment. YA, MT, and JDB developed
the Timepix3 implementation for EELS. NV, YA, SYW, and LHGT performed
experiments with support from XL. NV, YA, and LHGT analyzed the data. KW and
TT the provided materials. All authors contributed to writing the manuscript.

## Competing interests

Mathieu Kociak patented and licensed technologies at the basis of the
Attolight Mönch used in this study and is a part time consultant at Attolight.
All other authors declare no competing financial interests.

## Supplementary materials

Materials and Methods
Supplementary Text
Figs. S1 to S3

## Supplementary materials to Cathodoluminescence excitation spectroscopy:
nanoscale imaging of excitation pathways

Nadezda Varkentina,1∗ Yves Auad,1∗ Steffi Y. Woo,1 Alberto Zobelli,1
Laura Bocher,1 Jean-Denis Blazit,1 Xiaoyan Li,1 Marcel Tencé,1
Kenji Watanabe,2 Takashi Taniguchi,3 Odile Stéphan,1 Mathieu Kociak,1†
Luiz H. G. Tizei,1†

1Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France
2Research Center for Functional Materials, National Institute for Materials
Science,
1-1 Namiki, Tsukuba 305-0044, Japan
3International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
∗These authors contributed equally
†To whom correspondence should be addressed;
E-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
##  S1 Methods

Coincidence EELS-CL experiments were performed on a modified Vacuum Generator
(VG) HB501 scanning transmission electron microscope (STEM) equipped with a
cold field emission source, an Attolight Mönch light collection system, a
liquid nitrogen cooled sample stage, and a ASI Cheetah Timepix3 event-base
direct electron detector, CheeTah from Amsterdam Scientific Instruments. More
details about the experimental setup and event based detection can be found in
Auad et al. (?). Beam current in coincidence measurements was typically from 1
to 10 pA and convergence half-angle of 7.5mrad was used.

High energy resolution EELS and CL measurements and high angle annular dark-
field (HAADF) imaging (Fig. 1A-B) were performed on monochromated and Cs-
corrected ChromaTEM modified Hermes200 STEM from NION. Spatially resolved data
are acquired by scanning a subnanometer electron beam on the sample. Beam
current in the order of 200 pA and convergence half-angle of 25 mrad were used
for the experiments. EELS dispersion was set to either 25 meV per channel for
low losses (Fig. 1A-B) or 270 meV per channel for core-losses (Fig. 1B). CL
used a Mönch system from Attolight, fitted with a diffraction grating giving
an wavelength resolution of 0.34 nm (about 2 meV at 500 nm in wavelength).

Experiments were performed with 60 and 100 keV electron kinetic energy. h-BN
flakes suffered damage at 100 keV on experiments on the VG microscope, where
the sample chamber vacuum conditions are degraded (higher pressure and water
partial pressure) in comparison to the ChromaTEM microscope.

Data acquisition was handled with Nion Swift 1.5 (ChromaTEM) and 1.6 (VG)
python-based microscopy-control application.

##  S2 Samples description

The first sample consists of gold silica core-shell nanospheres from
nanoComposix. Their total diameter measured by TEM is $140\pm 10$ nm as stated
by manufacturer. The nanospheres solution were further diluted in spectral
quality ethanol in proportion 1:2. One drop of the final solution was then
drop cast on a conductive lacey carbon film (tens of nanometers thick)
supported on copper TEM grid (agar Scientific) and the extra volume was
absorbed by filter paper (Whatman).

The second sample is a Carbon-13 doped hexagonal Boron Nitride (h-BN) (?).
Thin h-BN flakes were prepared by liquid phase exfoliation from h-BN
monocrystals dispersed in 1 mL of spectral quality isopropanol (Carlos Erba),
and then sonicated for 15 minutes. Three drops of the solution were then
successively drop casted onto the TEM grids (suspended by tweezer) containing
the Au/SiO2 nanospheres. The grid was left to dry until total evaporation of
the solvent.

Thin h-BN flakes (below 50 nm) areas were chosen, to ensure that fast
electrons suffer statistically at most one scattering event on the sample (?)
(sample relative thickness below one mean free path for inelastic scattering).

##  S3 Description of coincidence seeking algorithm

Before applying the coincidence-seeking algorithm, electron events in Timepix3
were cluster-corrected following a procedure that can be found in ref. (?).
The time of arrival of each electron event is compared with the time of
arrival list from the photon events (after both being sorted by time). A
coincident electron is found when its time lies within a given time interval
($\pm 25$ ns or $\pm 50$ in this work) centered at the first-matching photon
time. Instead of pairing every electron with all the elements of the photon
list, a sliding window algorithm is performed, which allows to increase
performance by as much as a factor of 10 due to the effective photon list’s
reduced size. The algorithm standard output consists of a list of energy-loss
indices along their associated time delay. For the hyperspectral dataset
reconstruction (Fig 3), supplementary events from the microscope scanning unit
is used. This procedure can be found in detail in ref. (?). Time delay
($\Delta t$) was set to zero at the maximum of coincidence counts. Its
absolute value is meaningless as it includes electronic and propagation
delays.

##  S4 Data processing and availability

All data was processed using the following Python libraries: Numpy 1.21.5,
Matplotlib 3.4.3, Scipy 1.7.3, Hyperspy 1.6.5 (?).

All published data will be available at the Zenodo repository after final
publication.

##  S5 EELS, CLE, relative QE of nanosphere in Fig. 2

A new graph with the same spectra as in Fig. 2D and F is presented in Fig.
SI1, to show in detail the double peaks due to SP and TR in the relative
quantum efficiency of a Au/SiO2 nanosphere.

Figure SI1: EELS, CLE and relative QE of the nanosphere in Fig. 2D: The three
spectra were ploted in a narrower energy band and in the same graph to show
the double peak structure in the relative quantum efficiency (green). The
first peak appears due to the SP resonance, seen in the EELS spectrum, while
the second occurs due to TR. Its presence is already indicated in the CLE
spectrum (blue), by a shoulder on the SP resonance.

##  S6 CLE nanosphere in impact and aloof geometry

EELS, CLE and relative quantum efficiency for a nanosphere in the aloof
(electron beam outside the SiO2 surface) and impact (electron beam on the SiO2
layer) geometries.

Figure SI2:  EELS, CLE and relative QE of a nanosphere in aloof and impact
geometries: (A-C) EELS, CLE and relative QE for an Au/SiO2 nanosphere in aloof
geometry. (D-F) EELS, CLE and relative QE in impact geometry. (G-I) Comparison
of EELS, CLE and relative QE between aloof and impact geometries.

##  S7 CLE of h-BN up to core losses

The relative QE of h-BN for the 4.1 eV emission has a linear dependence up to
620 eV shown in Fig. SI3, within the currently achievable signal-to-noise
ratio. B and N K-edge fine structure is visible in the CLE spectra. No
spectral fine structure appears in the relative QE, but the noise level is
still high. The minimum of relative QE is visible at around 15 eV, as well as
the decrease in relative QE between 6.5 and 15 eV (pointed by green vertical
arrows in SI3E-F).

Figure SI3: (A-B) EELS and CLE of two h-BN flakes measured up to 620 eV. (C-D)
Same spectra as in A-B but with logarithmic scales. B-K and N-K mark the boron
and nitrogen EELS K edges. (E-F) Relative QE for the two measured flakes. The
minimum of relative QE is visible at around 15 eV, as well as the decrease in
relative QE between 6.5 and 15 eV (pointed by green vertical arrows). CLE
integrated time range $\pm 10$ ns.