###### Abstract

Plasma wakefield acceleration is a promising technology to reduce the size of
particle accelerators. Use of high energy protons to drive wakefields in
plasma has been demonstrated during Run 1 of the AWAKE programme at CERN.
Protons of energy 400 GeV drove wakefields that accelerated electrons to 2 GeV
in under 10 m of plasma. The AWAKE collaboration is now embarking on Run 2
with the main aims to demonstrate stable accelerating gradients of 0.5–1 GV/m,
preserve emittance of the electron bunches during acceleration and develop
plasma sources scalable to 100s of metres and beyond. By the end of Run 2, the
AWAKE scheme should be able to provide electron beams for particle physics
experiments and several possible experiments have already been evaluated. This
article summarises the programme of AWAKE Run 2 and how it will be achieved as
well as the possible application of the AWAKE scheme to novel particle physics
experiments.

###### keywords:

AWAKE; proton-driven plasma wakefield acceleration; dark photons; strong-field
QED; electron–proton physics.

100 1 1 0 https://doi.org/ The AWAKE Run 2 programme and beyond Title Edda
Gschwendtner 1,†, Konstantin Lotov 2,†, Patric Muggli 1,3,†, Matthew Wing
4,†,∗, Riccardo Agnello 5, Claudia Christina Ahdida 1, Maria Carolina Amoedo
Goncalves 1, Yanis Andrebe 5, Oznur Apsimon 6,7, Robert Apsimon 7,8, Jordan
Matias Arnesano 1, Anna-Maria Bachmann 3, Diego Barrientos 1, Fabian Batsch 3,
Vittorio Bencini 1,9, Michele Bergamaschi 3, Patrick Blanchard 5, Philip
Nicholas Burrows 9, Birger Buttenschön 10, Allen Caldwell 3, James Chappell 4,
Eric Chevallay 1, Moses Chung 11, David Andrew Cooke 4, Heiko Damerau 1, Can
Davut 7,12, Gabor Demeter 13, Amos Christopher Dexter 7,8, Steffen Doebert 1,
Francesa Ann Elverson 1, John Farmer 1,3, Ambrogio Fasoli 5, Valentin
Fedosseev 1, Ricardo Fonseca 14,15, Ivo Furno 5, Spencer Gessner 1,16,
Aleksandr Gorn 2, Eduardo Granados 1, Marcel Granetzny 17, Tim Graubner 18,
Olaf Grulke 10,19, Eloise Daria Guran 1, Vasyl Hafych 3, Anthony Hartin 4,
James Henderson 7,20, Mathias Hüther 3, Miklos Kedves 13, Fearghus Keeble 4,
Vadim Khudiakov 21,22, Seong-Yeol Kim 11,1, Florian Kraus 18, Michel Krupa 1,
Thibaut Lefevre 1, Linbo Liang 7,12, Shengli Liu 23, Nelson Lopes 15, Miguel
Martinez Calderon 1, Stefano Mazzoni 1, David Medina Godoy 1, Joshua Moody 3,
Kookjin Moon 11, Pablo Israel Morales Guzmán 3, Mariana Moreira 15, Tatiana
Nechaeva 3, Elzbieta Nowak 1, Collette Pakuza 9, Harsha Panuganti 1, Ans
Pardons 1, Kevin Pepitone 24, Aravinda Perera 7,6, Jan Pucek 3, Alexander
Pukhov 21, Rebecca Louise Ramjiawan 1,9, Stephane Rey 1, Adam Scaachi 4,
Oliver Schmitz 17, Eugenio Senes 1, Fernando Silva 25, Luis Silva 15,
Christine Stollberg 5, Alban Sublet 1, Catherine Swain 7,6, Athanasios
Topaloudis 1, Nuno Torrado 15, Petr Tuev 2, Marlene Turner 1,26, Francesco
Velotti 1, Livio Verra 3,1,27, Victor Verzilov 23, Jorge Vieira 15, Helmut
Vincke 1, Martin Weidl 28, Carsten Welsch 7,6, Manfred Wendt 1, Peerawan
Wiwattananon 1, Joseph Wolfenden 7,6, Benjamin Woolley 1, Samuel Wyler 1,
Guoxing Xia 7,12, Vlada Yarygova 2, Michael Zepp 17, Giovanni Zevi Della Porta
1 (The AWAKE Collaboration) Firstname Lastname, Firstname Lastname and
Firstname Lastname Correspondence<EMAIL_ADDRESS>These authors contributed
equally to this work.

##  1 Introduction

When a compact particle bunch or laser pulse enters a plasma column, this
drive beam disturbs the free plasma electrons which can then set up an
oscillatory motion that leads to strong electric fields (“wakefields”) in the
direction of the bunch propagation and also transverse to it. By injecting a
witness bunch of charged particles into the correct phase of the plasma
electron oscillation, the system acts as a particle accelerator. This offers a
compelling alternative to conventional microwave radio-frequency acceleration
which is limited to accelerating gradients of about 100 MV/m, at which point
the metallic structures where the particles are accelerated start to break
down. As plasma is already ionised, it does not suffer from this limitation
and accelerating gradients many orders of magnitude higher are possible. As
such, plasma wakefield acceleration is a possible solution to developing
accelerators of significantly reduced size for high energy particle physics,
or indeed for other applications.

The concept of accelerating particles in plasma was first proposed in the
1970s prl:43:267 . The field has undergone significant development since
prl:54:693 ; pp:14:055501 ; rmp:81:1229 ; rast:9:63 ; NJP23-031101 , with
progress experimentally, theoretically and in simulation. This has been aided
by technology development in high-power lasers and high-performance computing.
Many experiments using a laser pulse as a driver have shown that wakefields of
10s of GV/m and beyond are sustainable nature:377:606 ; nature:431:535 ;
nature:431:538 ; nature:431:541 , with electrons accelerated up to 7.8 GeV in
one acceleration stage of 20 cm of plasma the highest final energy achieved so
far prl:122:084801 . Similar accelerating gradients have been achieved when an
electron bunch is used as a driver nature:445:741 ; nature:515:92 , with
energy gains of 42 GeV achieved for particles in a single bunch where the head
of the bunch drives the wakefields nature:445:741  and 9 GeV per particle
achieved for a witness bunch of electrons ppcf:58:034017 . However, the laser
pulses and electron bunches both suffer from a low stored energy meaning that
multiple acceleration stages rast:9:63 ; prab:13:101301  are being
investigated in order to achieve the high energies needed for particle physics
experiments.

The possibility to use proton bunches allows the acceleration to take place in
one stage given the high stored energy available in some proton accelerators.
The original proposal np:5:363  considered, in simulation, TeV protons in
bunches of length 100 $\mu$m which are not currently available. High energy
proton bunches at CERN are typically 10 cm long, however such bunches can
undergo a process called self-modulation (SM) in plasma bib:kumar ;
prl:107:145002 ; bib:pukhov  in which the long proton bunch is split into a
series of microbunches. These microbunches are regularly spaced and hence can
constructively interfere to drive strong wakefields. The SM process allows the
use of proton beams that currently exist in order to develop proton-driven
plasma wakefield acceleration into a technology for future particle physics
experiments.

The advanced wakefield (AWAKE) experiment at CERN ppcf:56:084013 ; NIMA-829-3
; nim:a829:76 ; bib:muggliready  was developed in order to initially verify
the concept of proton-driven plasma wakefield acceleration. Proton bunches
from the super proton synchrotron (SPS), in which each proton has an energy of
400 GeV, have a total energy of 19 kJ per bunch. The bunches are typically
6–12 cm long and undergo the SM process in a rubidium plasma. Witness bunches
of electrons can then be accelerated in the wakefields driven by the proton
microbunches. Initial experiments were performed in 2016–18 in order to
demonstrate the proof of concept (Run 1). The scheme is now being developed
with a series of experiments (Run 2) to be performed in this decade. These
will demonstrate it as a usable technology for high energy particle
acceleration which already has several potential applications in particle
physics.

The outline of the article is as follows. After this introduction, the
highlights of the AWAKE Run 1 programme are summarised in Section 2. The
physics programme of AWAKE Run 2 is discussed in Section 3 followed by a
discussion of the setup for AWAKE Run 2 in Section 4. Section 5 then
summarises the possible particle physics experiments that could be realised
after Run 2 with electrons provided by the AWAKE scheme. A brief summary is
then given in Section 6.

##  2 Summary of experimental results from AWAKE Run 1

In the first round of experiments bib:muggliready , we have demonstrated the
existence of the SM process and the possibility to accelerate electrons in SM-
driven wakefields. We have also observed a number of expected and unexpected
characteristics of SM. An overview of the experimental setup for Run 1 is
shown in Fig. 1.

Figure 1: Schematic of the AWAKE Run 1 (2016–18) layout. The laser and proton
beams are merged before entering the plasma source. A beam of 10–20 MeV
electrons is also merged with the beam line and injected into the entrance of
the plasma source. The plasma source contains rubidium vapour at about 200 ∘C
with precise temperature control over the full 10 m. The beams exit the plasma
source and a series of diagnostics are used to characterise them. There are
two imaging stations to measure the transverse profile of the proton bunch and
screens emitting optical and coherent transition radiation (OTR and CTR) to
measure the longitudinal profile of the proton bunch. Electrons are separated
from the protons using a dipole magnet which also induces an energy-dependent
spread which is measured on a scintillator screen, imaged by a camera.
Diagrams of the proton bunch self-modulation and electron capture are shown in
the bottom left. A typical image of the accelerated electron bunch as observed
on the scintillator screen is shown in the top right. From Ref. bib:nature .

The proton bunch propagates in a plasma created by a relativistic ionisation
front (RIF). The RIF is the result of the propagation of a short and intense
laser pulse in a rubidium vapour bib:oz ; bib:fabiandensity ; bib:gabor ;
pr:a104:033506 . When the RIF is placed within the proton bunch, the part of
the bunch behind the RIF travelling in plasma is transformed into a train of
microbunches. This is shown in Fig. 2 (a) where a clear periodic charge
density structure at $t>0$ ps is observed. The front ($t<0$ ps on Fig. 2(a))
is unaffected. The period of the train or the modulation frequency is
determined by the plasma electron frequency, $f_{pe}$ bib:karl , measured over
one order of magnitude in plasma electron density $n_{e0}$:
$f_{pe}\propto\sqrt{n_{e0}}$ 111The electron plasma frequency in a plasma with
electron density $n_{e0}$ is:
$\omega_{pe}=\left(\frac{n_{e0}e^{2}}{\varepsilon_{0}m_{e}}\right)^{1/2}$,
$f_{pe}=\omega_{pe}/2\pi$, where $e$ is the elementary electric charge,
$\varepsilon_{0}$ is the permittivity of free space and $m_{e}$ is the
electron mass.. The train formation is a transverse process; protons between
microbunches leave the bunch axis and form an expanding halo. The halo radius
measured 10 m downstream from the plasma exit indicates that protons gained
radial momentum from transverse wakefields with amplitudes reaching hundreds
of MV/m bib:marleneprl . This amplitude exceeds their initial amplitude driven
at the RIF at the plasma entrance ($<$10 MV/m). This indicates that wakefields
grow along the plasma, whereas the increase in radius reached by halo protons
along the bunch shows that they also grow along the bunch, with both growths
expected NIMA-829-3 . Correspondingly, large longitudinal wakefields lead to a
2 GeV energy gain of externally injected 19 MeV test electrons bib:nature ;
bib:marleneroyal . Acceleration experiments also suggest that wakefields may
break in the back of the bunch, due to the large amplitude of the wakefields
and to the finite radial extent of the plasma bib:james ; PPCF63-055002-halo .
Combined halo radius and acceleration results in experiment and simulations
show that the SM process saturates a distance between three and five metres
along the plasma bib:marleneprab .

Figure 2:  (a) Time-resolved image of the SM proton bunch with the RIF placed
125 ps (0.5$\sigma_{t}$, where $\sigma_{t}$ is the RMS duration of the proton
bunch) ahead of bunch centre (front of the bunch at $t<0$ ps), and plasma
electron density $n_{e0}=1.81\times$1014 cm-3 (other parameters in
bib:fabianprl ). The RIF is at $t=0$ ps on the image. (b) Relative RMS phase
variation $\Delta\Phi$ of the modulated bunches (in % of 2$\pi$ or of a
modulation period) for each set of images acquired every 50 ps along the bunch
and aligned in time using a reference laser pulse signal visible as a vertical
line at the bottom of image (a) ($x>2$ mm). From Ref. bib:fabianprl .

Numerical simulation results suggest that injection and acceleration of
electrons are ineffective during the SM growth. If the electrons are injected
along the proton beam axis, they are defocused in the region of longitudinally
varying plasma density near the entrance to the plasma section PoP21-123116 ;
NIMA-829-3 ; PoP25-063108 . If the electrons are injected at a small angle,
they cross the transverse boundary of the plasma column which reflects or
scatters most of the injected charge PPCF61-104004 . This is the likely reason
why the measured accelerated charge bib:nature ; bib:marleneprab  is
significantly smaller than that simulated without the effect of boundary
crossing NIMA-829-3 .

The phase velocity of the wakefields during SM is less than the speed of light
$c$ bib:pukhov ; prl:107:145002 , which prevents electrons from being
accelerated to high energies IPAC14-1537 . This also causes microbunch
destruction at late stages of SM development PoP18-024501 ; PoP22-103110 .
Fortunately, the phase velocity of the wakefields can be influenced by plasma
density gradients along the beam trajectory NIMA-829-3 . It also changes the
number of protons remaining in the microbunches and the length of the train:
more charge and a longer train with positive density gradients, as
demonstrated experimentally bib:falk  and detailed in simulations bib:pablo .

Experimental and simulation results also reveal that with a density gradient
the modulation frequency is not unique and varies radially across the time-
resolved charge distribution of the train and halo observed 3.5 m from the
exit of the plasma bib:pablo . The structure of the distribution confirms that
protons forming the halo left the wakefields over the first few metres of
plasma, their distribution carrying the plasma frequency near the plasma
entrance. The microbunch train, having travelled through the entire plasma,
carries the frequency at the end of the plasma. We observe this with negative,
linear density gradients bib:falk  with which we also observe the largest
frequency variations, with shorter trains, better able to adjust their
modulation frequency to the local plasma frequency.

The reproducibility of the accelerating structure in the plasma is essential
for the controlled acceleration of an externally-injected electron bunch. The
electron bunch must be placed at the proper phase within the wakefields. That
is, at a position within the train where wakefields have reached their
maximum, and more precisely, within a fraction of a period of the wakefields,
where fields are accelerating and focusing. The precise location is
determined, for example, by loading of the wakefields for minimisation of
energy spread and for emittance preservation bib:veronica . While we do not
measure the reproducibility of the wakefields from event to event, we
demonstrated that when the RIF provides initial wakefields with sufficient
amplitude, the phase of the bunch modulation with respect to the RIF is
reproducible, despite variation of the incoming bunch parameters bib:fabianprl
. That is, the SM process is seeded, i.e., reproducible and driven away from
its instability (SMI) regime bib:kumar . In this seeded mode, we measure RMS
variations of the modulation phase smaller than 8% of a modulation period
(Fig. 2 (b)) all along the bunch train. We observe the instability when the
RIF is placed further than $\approx 2\sigma_{t}$ ahead of the centre of the
bunch with RMS duration $\sigma_{t}$ (typically $\cong 250$ ps). When we place
the RIF $\gg 2\sigma_{t}$ ahead of the centre of the bunch, the bunch
propagates in a pre-formed plasma whose density is decaying in time because of
radial expansion and recombination. Recording the modulation frequency as a
function of RIF timing provides a measurement of plasma density as a function
of time after ionisation bib:gessner .

While the SM process is the lowest-order, symmetric mode of interaction
between the long incoming bunch and the plasma, signs of the non-axi-symmetric
mode, the hose instability bib:whittum ; thesis:huether , were also observed.
However, this mode was only observed at low plasma densities, $n_{e0}\leq
0.5\times 10^{14}\,{\rm cm}^{-3}$, much lower than those that led to
significant energy gain, $n_{e0}>1.8\times 10^{14}\,{\rm cm}^{-3}$.

The above results, in particular the occurrence and the saturation of the SM
process of the 400 GeV proton bunch over a distance less than 10 m of plasma,
with its phase reproducibility, the possibility to accelerate electrons in the
wakefields, the absence of hosing instability and the generally excellent
agreement between experimental and simulation results bib:james ; bib:gorn ;
bib:marleneprab ; bib:pablo , allow for the planning of the next experiments
bib:mugglirun2 . These will be conducted in a number of steps geared towards
experiments with two plasmas aimed at producing a multi-GeV electron bunch
with charge, emittance and relative energy spread sufficient for the
applications described in this manuscript.

##  3 The AWAKE Run 2 physics programme

The Run 2 physics programme is driven mostly by the long-term goals presented
in this paper: producing a high-energy electron bunch with quality sufficient
for high-energy or particle physics applications. Run 2 will again use proton
bunches from the SPS. The main difference with Run 1 is the use of two plasma
sources, one for SM and one for acceleration, thereby allowing for on-axis
injection of the electron bunch into the accelerator plasma and for better
control of parameters run2a-plan .

The first part of Run 2 focuses on the self-modulator, i.e., on the generation
of the self-modulated proton bunch to drive the accelerator. The second part
of Run 2 focuses on the accelerator, i.e., on external injection of the
electron bunch and on scaling of its energy gain to higher energies.

We determined in Run 1 that, as predicted by numerical simulations NIMA-829-3
; PoP21-083107 , the SM process saturates over a distance of 3–5 m
bib:marleneprab . Experiments with two plasmas will thus include a 10 m-long
self-modulator plasma, followed by a 10 m-long accelerator plasma. In the
current plan, the two plasma sources will be based on laser ionisation of a Rb
vapour bib:oz . These are the only sources known so far that provide a plasma
density step in the self-modulator bib:plyushchev  and the desired density
uniformity PoP20-013102  in the accelerator.

###  3.1 Self-modulator

The self-modulator will have two new features: the ability of seeding the SM
process using an electron bunch and the ability of imposing a plasma density
step. These two features will be first tested independently, and then
together.

####  3.1.1 Electron-bunch seeding

A major result of Run 1 was the demonstration of the seeding of the SM process
using a RIF bib:karl . However, this seeding method leaves the front of the
bunch, ahead of the RIF, un-modulated. Since AWAKE long-term plans call for an
accelerator plasma tens to hundreds of metres long, this plasma will have to
be pre-formed. The laser ionisation process of Run 1 does not scale to such
long plasma because of energy depletion of the laser pulse and because of the
focusing geometry. In this preformed plasma, the un-modulated front of the
bunch could experience SMI in the accelerator plasma. The wakefields driven by
this front SMI could interfere with the self-modulated back of the bunch and
with the acceleration process.

The SM process can also in principle be seeded by a preceding driver of
wakefields, such as an electron bunch or a laser pulse. In this case, the
entire proton bunch becomes self-modulated and the possible issue with the un-
modulated front would be avoided.

The programme thus consists of demonstrating that the SM process can indeed be
seeded by the electron bunch available in the Run 1 experimental setup. The
method to be used is similar as that of Run 1 bib:fabianprl , i.e.,
determining the timing of microbunches appearing along the bunch with respect
to the time of the electron bunch, after 10 m of plasma.

####  3.1.2 Plasma density step

Numerical simulation results suggest that in a plasma with constant density
along the beam path, the continuous evolution of the bunch train and
wakefields leads to a decay of the amplitude of wakefields after their
saturation PoP18-024501 ; PoP22-103110 . These results also suggest that when
applying a density step, some distance into the plasma, within the growth of
the SM process, wakefields maintain a near-saturation amplitude for a long
distance along the plasma. Figure 3 illustrates how the density step changes
the wakefield amplitude in the SM and acceleration plasma sections. These
simulations are performed in the axi-symmetric geometry with the quasi-static
code LCODE NIMA-829-350 ; PRST-AB6-061301 . The parameters of the density step
were optimised for the strongest wakefield at $z=20$ m with no gap between the
sections PPCF62-115025 . The density step is seen to strongly increase the
wakefield at the acceleration stage even in the presence of a 1 m gap.

The AWAKE plasma source is based on a rubidium vapour, along which a uniform
temperature is imposed to obtain a correspondingly uniform vapour density.
Laser-pulse ionisation then turns this uniform vapour density into an equally
uniform plasma density bib:karl . One can therefore simply impose a
temperature step along the column to obtain the corresponding plasma density
step bib:plyushchev . Measurements of the effect of the plasma density step on
the amplitude of the wakefields will include effects of the size and shape of
the bunch halo formed by defocused protons, measurements of plasma light
signals, and measurements of electron acceleration.

###  3.2 Accelerator

The length of the accelerator plasma is 10 m for the first experiments. This
is much longer than the distance it takes for the amplitude of wakefields to
settle to steady values after the injection point (see Fig. 3). This distance
is $\sim$2 m and results from the transverse evolution of the proton bunch in
the vacuum gap between the two plasmas. The plasma is thus long enough for the
expected energy gain to be in the multi-GeV range along the ensuing length of
plasma where the driving of the wakefields is stationary.

Figure 3: The amplitude of the excited wakefield $E_{\text{z,\,max}}(z)$ in
the uniform plasma and in plasma with the optimised density step with and
without a 1 m gap between SM and acceleration plasma sections. The SM process
is seeded by an electron bunch. The density step is the linear growth of the
plasma density from $7\times 10^{14}\,\text{cm}^{-3}$ to $7.21\times
10^{14}\,\text{cm}^{-3}$ at the interval between $z=0.8$ m and $z=2.8$ m.

####  3.2.1 External injection

In Run 1, acceleration was obtained with an off-axis injection geometry
NIMA-829-3 . This geometry was chosen to avoid defocusing of the injected
electrons in the density ramp located at the entrance of the plasma
PoP21-123116 ; PoP25-063108 . In this region, the yet un-modulated proton
bunch drives transverse fields which are focusing for its own positive-charge
sign, but defocusing for injected electrons. A scheme that avoids these issues
is to inject electrons on axis after the SM process has saturated. The
parameters of the electron bunch must be such that it reaches high energies,
low final relative energy spread and preserves its incoming emittance. The SM
process does not lead to blow-out of plasma electrons from the accelerating
structure because the resonant wave drive stops when the plasma wave becomes
nonlinear and its period elongates PoP20-083119 . The initial electron bunch
density $n_{b0}$ must therefore exceed the plasma electron density: $n_{b0}\gg
n_{e0}$ to reach blow-out. Blow-out of plasma electrons is necessary for the
focusing force of the plasma to become that of the pure ion column, with
strength increasing linearly with radius PRA44-6189 . The normalised emittance
$\epsilon_{N}$ and focus RMS size of the electron bunch at the plasma entrance
$\sigma_{r0}$ must be adjusted to satisfy matching to the ion column focusing
force:

$\frac{\gamma_{0}n_{e0}}{\epsilon_{N}^{2}}\sigma_{r0}^{4}=\frac{2\varepsilon_{0}m_{e}c^{2}}{e^{2}}.$

For typical parameters ($\epsilon_{N}=2$ mm mrad, $n_{e0}=7\times
10^{14}\,\text{cm}^{-3}$), electrons must have a large enough energy or
relativistic factor $\gamma_{0}$ to be focused to the small transverse size
($\sigma_{r0}=5.65\,\mu$m, $\gamma_{0}\cong 300$). In addition, the bunch
length and timing in the wakefields must be optimised to load wakefields to
minimise the final relative energy spread $\Delta E/E$.

An example set of parameters was developed using a toy-model for the proton
bunch bib:veronica . This example shows that with parameters satisfying the
above conditions, about 70% of the initial 100 pC bunch charge preserve their
emittance and reach 1.67 GeV/c over 4 m of plasma with $\Delta E/E\cong 1\%$
(core).

####  3.2.2 Scalable plasma sources

Assuming the success of experiments on electron injection into the wakefields
of a 10 m-long plasma, energy gain suitable for applications can in principle
be achieved by extending the length of the accelerator plasma. However, the
distance over which laser ionisation can occur is limited by depletion of the
energy of the laser pulse and by the focusing geometry. We are therefore
developing other plasma sources that do not suffer from length limitations:
direct-current electrical discharge in noble gases Torrado2022  and helicon
argon plasma bib:helicon .

The direct-current discharge plasma source (DPS) uses a short pulse ($\sim
10\,\mu$s) high-current pulse ($\sim$ kA) through the length of a glass tube,
filled with a high atomic number noble gas at low pressure ($\sim$ 10 Pa)
Torrado2022 . This follows a fast-rising high-voltage pulse ($50-100$ kV) able
to ignite long tubes ($L>5$ m) into uniform plasma densities. We are currently
developing a 10 m long plasma source consisting of a double discharge from a
mid length common cathode to two anodes in the extremities. Operation at the
plasma densities relevant for AWAKE has been demonstrated. We are currently
engaged in demonstrating the ability of the plasma source to reach the
required plasma density uniformity. With this source, the length scalability
can potentially be reached by stacking together multiple plasma sections (with
lengths of a few metres to a few tens of metres) using common cathodes and
anodes.

Helicon plasma belong to the class of magnetised wave heated plasmas helicon1
. They consist of a dielectric vacuum vessel and an external wave excitation
antenna, which is powered by radio frequencies. The excited helicon waves heat
the plasma and have been demonstrated to generate discharges with high plasma
densities helicon2 . Its length can thus in principle be extended over long
lengths by stacking cells. Measurements show that the plasma density typical
of AWAKE ($n_{e0}=7\times 10^{14}\,{\rm cm}^{-3}$) can be reached bib:helicon
. The challenge is to demonstrate that the required plasma density uniformity
can also be reached. This demonstration requires highly accurate and localised
plasma density measurements. Specific diagnostics such as Thomson scattering
epfl-paper  and optical emission spectroscopy are being developed.

##  4 Overview of AWAKE Run 2 setup

The AWAKE Run 2 scheme including the two plasma sources, i.e., a self-
modulator and an accelerator, and a new electron beam system is shown in Fig.
4.

Figure 4: Layout of AWAKE Run 2.

The AWAKE Run 2 programme is subdivided into four phases Run 2a, 2b, 2c and
2d, following the physics programme described above.

The current AWAKE experiment is installed upstream of the previous CERN
neutrinos to Gran Sasso (CNGS) facility bib:cngs . The $\sim$ 100 m long CNGS
target cavern, which currently houses the CNGS target and secondary beam line
while their activation levels decay, is separated from the AWAKE experiment by
a shielding wall.

Phases Run 2a and Run 2b will be carried out in the existing AWAKE facility,
having started in 2021 and are foreseen to continue until 2024. For Run 2c and
Run 2d, however, the CNGS target area needs to be dismantled; around three
years are required for the removal of the CNGS target area and the
installation and commissioning of the additional equipment for AWAKE Run 2c,
where AWAKE takes advantage of the general 1–2 years shutdown period of the
CERN injector complex in 2026/27. It is planned to start AWAKE Run 2c with
first protons after LS3 in 2028.

###  4.1 AWAKE Run 2a

The Run 2a experiments, focusing on electron bunch seeding of the SM process,
use the same infrastructure as that of AWAKE Run 1; bunches of protons are
extracted from the CERN SPS and each consists of $3\times 10^{11}$ protons,
each with energy 400 GeV. The bunch length can be adjusted between 6 and 12 cm
and the bunch is focused at the plasma entrance to a transverse RMS size of
$\sigma_{r}\approx 0.2\,$mm. The plasma is 10 m long, has a radius of
approximately 1 mm and a density uniformity better than 0.1% bib:fabiandensity
. To create the plasma, rubidium is evaporated in a heat exchanger and the
outermost electron of each rubidium atom is ionised with a laser pulse with a
pulse length of 120 fs and a pulse energy of $\leq$ 450 mJ. The vapour (and
thus also the plasma) density is controlled by the temperature of the source
and is adjustable between 0.5 and 10 $\times 10^{14}$ atoms/cm3.

In AWAKE Run 1, the relativistic ionisation front of the laser was also used
to seed the self-modulation process by placing it close to the centre of the
proton bunch. In AWAKE Run 2a the laser pulse is placed in front of the proton
bunch so that the entire proton bunch interacts with plasma.

In AWAKE Run 1, electron acceleration in the proton-driven plasma wakefield
was demonstrated with externally injected 10–20 MeV electrons bib:nature .
These 100–600 pC electron bunches have a duration of $\sigma_{t}\geq 4$ ps and
are produced in a RF photo injector based on an S-band structure. In AWAKE Run
2a, these electrons are used to seed the proton bunch SM.

###  4.2 AWAKE Run 2b

In the Run 2b experiments, the effect of the plasma density step on the SM
process will be measured. This requires a new vapour source and corresponding
new diagnostics. The new vapour source is in its design phase and will be
exchanged with the current source for Run 2b. It includes additional
observation ports in order to diagnose the electron plasma density that
sustains wakefields. The experimental programme will focus on direct
measurements of the plasma wakefields. To this end, different diagnostics are
currently being evaluated (e.g. THz shadowgraphy diagnostics and plasma light
diagnostics).

###  4.3 AWAKE Run 2c

The CNGS cavern needs to be emptied in order to house the baseline AWAKE Run
2c and Run 2d experiments, which includes the second electron source, beam
line, klystron system and the second vapour source.

In order to be able to integrate the entire Run 2c experiment in the AWAKE
facility, the first plasma cell will be shifted by around 40 m downstream of
its current location and consequently the new equipment will also be
accordingly moved downstream. This change also needs some downstream shifting
of the proton beam line final dipole magnets, however, despite challenging
aperture constraints, no extra magnets are required. The second electron
source needs to deliver electrons with 150 MeV energy, bunch charge of a few
100 pC, beam emittance of 2 mm mrad and a short bunch length of 200 fs
duration. The baseline proposal is a novel RF gun and two X-band structures
for velocity bunching and acceleration. A prototype system is currently being
developed in order to demonstrate the required beam parameters for AWAKE and
to study the mechanical and integration aspects in the AWAKE facility.

The beam line design for the new 150 MeV electron beam from the electron
source to the plasma source is very challenging, given the tight beam
specifications Ramjiawan:2021gpy : the beam must be matched to the plasma at
the plasma merging point, with a RMS beam size satisfying
$\sigma_{x,y}=\sqrt{0.00487\epsilon_{x,y}}$, zero dispersion and
$\alpha_{x,y}=0$, for emittance, $\epsilon_{x,y}$, and Twiss parameter,
$\alpha_{x,y}$, where 0.00487 is the required $\beta$ function in metres. In
addition, the gap between the two vapour sources must be as short as possible
IPAC16-2557 , i.e., $\leq 1\,$m. Also considering integration limits from the
width of the tunnel, the baseline proposal of the electron transfer line is a
dogleg design. Studies on the injection tolerances of the proton and electron
beam are currently ongoing and are key to controlled plasma wakefield
acceleration.

The accelerator plasma source will have a length of about 10 m and will be
based on the laser ionised Rb vapour source (as used in AWAKE Run 1 and in the
first vapour source). The laser beam for the ionisation in the second vapour
source will be injected from its downstream end, counter-propagating to the
proton beam. Although the same laser as for the first vapour source can be
used in Run 2c by splitting its output beam on two branches, additional laser
transport lines and a compressor chamber need to be integrated.

###  4.4 AWAKE Run 2d

Once Run 2c has demonstrated electron acceleration to high energies while
controlling beam quality, the second plasma source can be exchanged with a
different plasma technology. These sources will be scalable to long distances
in order to accelerate electrons to energies of several 10s of GeV and beyond,
allowing for the first particle physics applications.

As discussed in the previous section, the plasma technologies currently under
study at CERN are helicon plasma sources and discharge plasma sources.

With the CNGS target cavern fully dismantled, there is enough space available
to install a plasma source of 10s of metres in length. Therefore
infrastructure changes for Run 2d concern mainly the different services such
as powering, cooling, etc. needed for the plasma source.

First studies also show that enough space is available for a prototype fixed-
target experimental setup, allowing the first particle physics experiments to
be conducted with electrons accelerated from AWAKE.

##  5 Particle physics applications of AWAKE

As outlined in the previous sections, the AWAKE scheme aims to develop an
acceleration technology that can be used to provide beams for particle physics
experiments. The ultimate goal for novel acceleration schemes is to provide
the technology for a high energy, high luminosity, linear electron–positron
collider with centre-of-mass energies $O({\rm TeV})$ and luminosities
$10^{34}$ cm-2 s-1. In principle, though, any experiment that requires a
source of bunched high energy electrons could utilise the AWAKE scheme.
Initial application focuses on experiments with less challenging beam
parameters than a linear $e^{+}e^{-}$ collider, although still having a strong
and novel particle physics case.

By the conclusion of Run 2, the AWAKE collaboration should have demonstrated
acceleration of electrons with stable GV/m gradients. Scalable plasma sources
should have been developed that can be extendable up to even kilometres in
length. The acceleration process should preserve the beam quality resulting in
bunches with transverse emittance of below 10 mm mrad. With these
developments, using proton bunches from the SPS, acceleration of electrons to
10s of GeV, and even up to $\sim$ 200 GeV, should be possible Lotov:2021nob .
Use of the LHC protons with energy 7 TeV would enable acceleration of
electrons up to about 6 TeV bib:caldwell . A limitation of the current proton
drivers is their repetition rate and hence the luminosity of any application
of the AWAKE scheme. Given this, high energy applications are considered and
also those where the luminosity is less critical.

First ideas for particle physics experiments based on electron bunches from
the AWAKE scheme were proposed in previous papers Wing:rsta ; Caldwell:2018atq
. The applications are discussed below, along with extensions and new ideas
that have been developed since. The first application would be to use a high
energy electron beam impinging on a target in order to search for new
phenomena related to dark matter, see Section 5.1. Another potential first
application is collision of an electron bunch with a high-power laser pulse to
investigate strong-field QED, see Section 5.2. Both of these experiments would
require electrons of 10s of GeV, although higher energies could be considered.
An electron–proton collider would be the potential first use of the AWAKE
scheme for a high-energy collider. There is a physics case for such a collider
with electrons of 10s of GeV, and up to the TeV scale, when in collision with
protons from the LHC. Each has a strong particle physics case but with less
strict demands on beam quality than an $e^{+}e^{-}$ collider, see Section 5.3.
Finally, a new development is consideration of the proton beam at Brookhaven
National Laboratory (BNL) as the wakefield driver to develop a compact
electron injector which is discussed in Section 5.4.

###  5.1 A beam-dump experiment for dark photon searches

Dark photons are postulated particles jetp:56:502 ; pl:b136:279 ; pl:b166:196
which could provide the link to a dark or hidden sector of particles. This
hidden sector could explain a number of issues in particle physics, not least
of which is that they are candidates for dark matter which is expected to make
up about 80% of known matter in the Universe. Dark photons are expected to
have low masses (sub-GeV) pl:b513:119 ; pr:d79:015014  and couple only weakly
to Standard Model particles and so would have not been seen in previous
experiments. The dark photon, labelled $A^{\prime}$, is a light vector boson
which results from a spontaneously broken new gauge symmetry and kinetically
mixes with the photon and couples to the electromagnetic current with strength
$\epsilon\ll 1$.

A common approach to search for dark photons is through the interaction of an
electron with a target in which the dark photon is produced and subsequently
decays. Many experiments, current and proposed arXiv:1707.04591 , are
searching for dark photons and other feebly interacting particles using
electrons impinging on a target. The initial electron beam energy varies,
although only the NA64 experiment at CERN pr:d89:075008 ; arxiv:1312.3309 ;
prl:118:011802  has access to high energy electrons ($O(100\,{\rm GeV}$)) with
other experiments limited to below ($O(10\,{\rm GeV}$)). A limitation of
experiments is the rate of electrons on target which in the case of NA64 is
about $10^{6}$ electrons on target per second as they are produced in
secondary interactions of the SPS proton beam. Given the limitations of the
number of electrons on target for a high energy beam, the AWAKE acceleration
scheme could enable an experiment to extend the search for dark photons as the
number of electrons is expected to be several orders of magnitude higher.
Assuming awake++  a bunch of $5\times 10^{9}$ electrons, each of 50 GeV in
energy, and a running period of 3 months gives $10^{16}$ electrons on a target
of centimetre transverse size. As the AWAKE scheme produces bunches of
electrons, an experiment based on this will run as a beam-dump experiment in
which electrons are absorbed in a target and a search for dark photons
decaying to an $e^{+}e^{-}$ pair is performed. This is in contrast to other
fixed-target experiments in which single electrons impact on a target and
other decay channels can be searched for such as dark photons decaying to a
pair of dark matter particles which do not leave deposits in the detectors and
so have a signal of missing energy.

Figure 5: Schematic layout of an experiment to search for dark photons. In the
AWAKE scheme, a bunch of electrons enters from the left and impacts on a
target of $O(1\,{\rm m})$ in length. A produced dark photon travels through a
vacuum tube of length $O(10\,{\rm m})$ in which it decays to an $e^{+}e^{-}$
pair which are then measured in a detector system such as a tracking detector
and calorimeter.

The potential production of dark photons is sensitive to the experimental
setup. Using initial electrons of 50 GeV, an experimental setup, as shown in
Fig. 5, has been simulated using GEANT4 geant4-1 ; geant4-2 ; geant4-3 . To
characterise the performance of the experiment, the sensitivity to the
coupling strength, $\epsilon$, and mass, $m_{A^{\prime}}$, is considered and
usually represented in a plot of the two. Examples are shown in Fig. 6 in
which the sensitivity is shown for the number of electrons on target and the
thickness of the target. These results show the value of having as many
electrons on target where the sensitivity to both $\epsilon$ and
$m_{A^{\prime}}$ is increased with an increasing number of electrons. They
also show that the sensitivity is reduced with increasing target thickness,
however, having a thicker target is necessary in order to keep the background
rate under control.

Figure 6: Sensitivity to dark photon production shown for the coupling
strength, $\epsilon$, and mass, $m_{A^{\prime}}$. The varied parameters of the
proposed beam-dump experiment are (left) the number of electrons on target and
(right) the thickness of the solid metal target which the electrons hit. The
initial electron energy is assumed to be 50 GeV.

Expected sensitivities to dark photons in an experiment using an electron beam
from the AWAKE scheme are shown in Fig. 7 in comparison with previous, current
and proposed experiments. Limits from previous experiments are shown as grey-
shaded areas in the $\epsilon-m_{A^{\prime}}$ plane. Expected sensitivities
from current and future experiments are shown as coloured lines. Using
electrons of energy 50 GeV will allow dark photon searches to be extended
towards higher masses in the range of couplings, $10^{-5}<\epsilon<10^{-3}$.
As there is the possibility of producing much higher energy electron beams, at
the TeV scale, with the AWAKE scheme using the LHC protons as the wakefield
driver, the sensitivity is shown for 1 TeV electrons; such an experiment could
run as part of a future collider facility, such as the very high energy
electron–proton (VHEeP) collider epj:c76:463 , after collisions, and as the
beam is dumped. The sensitivity is extended to much higher mass values as well
as lower $\epsilon$. The mass values reached approach 1 GeV, far beyond the
capability of any other experiment, current or planned.

Depending on the future running of the SPS accelerator which feeds the AWAKE
experiment, a larger area of parameter space could be investigated if more
electrons on target were to be possible. Also, recent investigations indicate
that higher energy electrons, up to about 200 GeV, are possible Lotov:2021nob
using the SPS protons as the drive beam which would also extend the
sensitivity beyond that shown in Fig. 7 for 50 GeV electrons. Additionally,
other decay channels, such as $A^{\prime}\to\mu^{+}\mu^{-}$ or
$A^{\prime}\to\pi^{+}\pi^{-}$ could also be considered and the experiment
optimised to be sensitive to these additional channels.

###  5.2 Investigation of strong-field QED in electron–laser collisions

Progress in high-power laser technology has revived the study of strong-field
quantum electrodynamics (QED) since the pioneering experiment, E144
pr:d60:092004 , that investigated this area of physics in electron–laser
collisions in the 1990s. As electrons pass through the intense laser pulse and
so experience strong fields (the higher the intensity, the stronger the
fields), QED becomes non-linear and experiments mimic the conditions that
occur on the surface of neutron stars, at a black hole’s event horizon or in
atomic physics.

The E144 experiment at SLAC investigated electron–laser collisions with
bunches of electrons, each of energy $\sim$ 50 GeV. Experiments (E320 at SLAC
and LUXE at DESY with the European XFEL) are underway or planned with high-
quality electron bunches, with energies in the 10–20 GeV range luxe ; e320 .
Given the expectation of bunches of electrons, each of energies in the 10s of
GeV range, from the AWAKE scheme, experiments investigating strong-field QED
are an obvious initial application with the possibility to also have higher
electron energies. As the rate of electron–laser collisions is limited by the
roughly 1 Hz repetition frequency of high-power lasers, high-rate electron
bunches are not required.

Figure 7: Limits on dark photon production decaying to an $e^{+}e^{-}$ pair in
terms of the mixing strength, $\epsilon$, and dark photon mass,
$m_{A^{\prime}}$, from previous measurements (light grey shading). The
expected sensitivity for the NA64 experiment is shown for a range of electrons
on target, $10^{10}-10^{13}$. Expectations from other potential experiments
are shown as coloured lines. Expected limits are also shown for $10^{15}$
(orange line) or $10^{16}$ (green line) electrons of 50 GeV (“AWAKE50”) on
target and $10^{16}$ (blue line) electrons of 1 TeV (“AWAKE1k”) on target
provided to an experiment using the future AWAKE accelerator scheme. From Ref.
Alemany:2019vsk .

###  5.3 High energy electron–proton/ion colliders

The HERA accelerator complex in DESY, Hamburg, has so far provided the only
electron–proton collider. With electrons and protons at maximal energies of
27.5 GeV and 920 GeV, respectively, a centre-of-mass energy of about 318 GeV
was achieved. An electron–ion collider (EIC) eic  is to start operation in BNL
in about a decade with lower centre-of-mass energy than HERA, up to about 140
GeV. However, it will have significantly higher luminosity, highly polarised
beams and variation in the centre of mass energy and ion species, providing a
rich physics programme through its great flexibility. A higher energy
electron–proton collider, LHeC (large hadron–electron collider) lhec1 ; lhec2
, has been proposed using electrons of 50 GeV in collision with LHC protons so
yielding centre-of-mass energies just above the TeV scale. The possibility of
an LHeC-like collider based on the AWAKE scheme to produce 50 GeV electrons is
outlined here. A significantly more compact design should be possible,
although with a much reduced luminosity performance. Even more compelling is
the possibility of using the AWAKE scheme to provide TeV electrons and so have
electron–proton collisions at centre-of-mass energies of 9 TeV epj:c76:463
and this is briefly summarised. Electrons at 3 TeV could also be used in
fixed-target mode and provide a centre-of-mass energy of $\sim$ 80 GeV,
thereby achieving similar values to the EIC.

A high energy $ep/eA$ collider could be the first collider application of the
AWAKE scheme. In comparison to a high energy $e^{+}e^{-}$ collider, an $ep/eA$
collider poses fewer challenges to the AWAKE scheme as only one beam is
required, such low emittances are not needed (as the proton emittance
dominates) and potentially positrons are not needed, although they are
desirable. Given the possibility of providing $O(50\,{\rm GeV})$ electrons
using the SPS protons to drive wakefields, this was formulated in the PEPIC
(plasma electron–proton/ion collider) project. The electrons would collide
with protons from the LHC and using their expected parameters during the high-
luminosity phase, this would lead to an instantaneous luminosity of $1.5\times
10^{27}\,{\rm cm}^{-2}{\rm s}^{-1}$, where parameters expected from a future
AWAKE facility have been assumed for the electron bunches awake++ . So
although PEPIC has the same energy reach (and possibly even beyond) as the
LHeC, it would have a luminosity many orders of magnitude lower. Such a low
luminosity will not allow investigation of the Higgs sector, detailed
measurements of electroweak physics or other phenomena that occur at high
$Q^{2}$, where $Q^{2}$ in the virtuality of the photon emitted by the electron
in the $ep/eA$ collision. However, processes that occur at low $x$, where $x$
is the fraction of the proton’s momentum carried by the struck parton, have
very high cross sections and so even with a low luminosity, large event
samples will be produced. The focus of the physics programme would then be on
understanding the structure of matter and quantum chromodynamics (QCD), the
theory of the strong force, in a new kinematic regime.

The PEPIC collider would be an option for CERN should the LHeC project not go
ahead, where the initial study shows that it is possible, could be housed
within the current CERN site, and has a novel particle physics programme.
Schemes to increase the luminosity should be considered where the current
design is mainly limited by the filling time of the SPS. Mechanisms to
increase the SPS bunch repetition frequency would lead to a corresponding
increase in luminosity and so should be studied, along with other parameters
relevant for the luminosity such as the electron bunch population and proton
bunch size. The possibility to accelerate electrons up to 200 GeV using the
SPS protons as the drive beam would lead to a doubling of the centre-of-mass
energy (2.4 TeV) and so an increased kinematic range. This provides a larger
phase space to search for new physics and larger lever arm to investigate the
energy dependence of high-energy cross sections.

Using the LHC protons to drive wakefields could lead to electrons at the TeV
scale and so an $ep$ collider with centre-of-mass energies $O(10\,{\rm TeV})$.
This has been formulated as the VHEeP collider epj:c76:463  in which electrons
at 3 TeV are collided with the LHC protons at 7 TeV giving a centre-of-mass
energy of 9 TeV. This represents a factor of 30 increase compared to HERA and
hence an extension in $x$ and $Q^{2}$ of a factor of 1000. As with PEPIC, the
luminosity of VHEeP is relatively low, estimated vheep-dis2015  to be
$10^{28}-10^{29}\,{\rm cm}^{-2}{\rm s}^{-1}$ or around 1 pb-1 per year. This
is mainly limited by the LHC protons needed to drive wakefields which will
need to be dumped after this process and so protons will need to be refilled
for the further acceleration of electrons by the AWAKE scheme. Schemes need to
be considered such as squeezing the proton or electron bunches, having
multiple interaction points, etc. that will increase the luminosity.

Even though the luminosity of the VHEeP collider is modest, the very high
energy provides a compelling particle physics cases. At low values of $Q^{2}$,
10s of millions of events are expected and so high precision measurements and
searches for new physics will be possible. This will allow investigation of
hadronic cross sections and the structure of matter at very high energies. The
collisions are also equivalent to a photon of energy of 20 PeV on a fixed
target and so has synergy with cosmic-ray physics. Searches for physics beyond
the Standard Model, such as quark substructure or leptoquarks, will also be
possible. Even with modest luminosities, the very high energy ensures that the
sensitivity to leptoquarks exceeds that possible in proton–proton collisions
at the LHC. The particle physics case is discussed in more detail elsewhere
epj:c76:463 ; Wing:rsta ; Caldwell:2018atq . As well as $ep$ collisions which
have been the focus so far, $eA$ collisions should be considered.

###  5.4 Use of BNL proton beams for a compact electron injector for a future
electron–ion collider

The AWAKE acceleration scheme has been considered as a possible technology for
the injector for the high energy electron beam for the future EIC. The EIC is
expected to collide electrons of up to 20 GeV with protons of up to 275 GeV –
a possible site at BNL in the US already has a circular, 4 km long proton
accelerator, but will require a new electron accelerator of similar size. It
has been proposed Chappell:2019ovd  to use the AWAKE technology at BNL by
using their high-intensity proton bunches to generate large wakefields and
hence accelerate electrons to high energies over short distances. Along with
CERN, where AWAKE is currently based, BNL is one of the few places in the
world with high energy proton bunches that can be used for the AWAKE
acceleration scheme.

It has been assumed that the proton bunch parameters expected for the future
EIC, namely a proton energy of 275 GeV, $2\times 10^{11}$ protons/bunch, a
bunch length of 5 cm and a bunch radius of $40$ or $100\,\mu$m. These are
similar values to currently used at AWAKE using CERN’s SPS accelerator except
that the smaller bunch size allows for a higher plasma density to be used
which should yield larger accelerating fields. Using these parameters, the
process has been simulated and the accelerating electric field determined for
the two values of the bunch radius. The results are shown in Fig. 8, where the
larger beam radius is given by the red line and smaller beam radius by the
blue line. In the more promising scenario, with a bunch radius of $40\,\mu$m,
the peak field is almost 7 GV/m, which although falls rapidly, levels off
above 1 GV/m, an accelerating gradient well above current conventional
accelerator technology.

Figure 8:  Evolution of the peak longitudinal fields driven by the BNL proton
drive beams over 10 m using bunch parameters which differ only in their
transverse size, $\sigma_{r}=40\,\mu$m or $100\,\mu$m. From Ref.
Chappell:2019ovd .

This confirms that the high, $>1$ GV/m gradients, can be harnessed and that
the required maximum energy for the EIC of 20 GeV could be achieved in under
20 m. There are indications that the accelerating gradient can be frozen
PoP22-103110  soon after the peak field, which will be studied at AWAKE (see
Sections 3.1.2 and 4.2), and values more like $5-6$ GV/m would be observed in
Fig. 8 in which case electrons could be accelerated to the required 20 GeV in
under 4 m. Alternatively, significantly higher electron energies than
currently planned, of 50 GeV and beyond, could be achieved and so extend the
kinematic reach in the investigation of parton dynamics in the proton at the
EIC.

##  6 Summary

This article summarised the plans for AWAKE Run 2, which will take the scheme
of proton-driven plasma wakefield acceleration from proof of concept to a
technology which can provide beams for particle physics experiments.

The proof of concept, AWAKE Run 1, showed that a long proton bunch can be
modulated into a series of microbunches, which are regularly and reproducibly
spaced. These microbunches constructively interfere to generate strong
electric fields in their wake. The wakefields were sampled by an externally-
injected bunch of electrons which were accelerated from about 20 MeV up to
about 2 GeV within 10 m of plasma, representing an average field of $>200$
MV/m, with peak fields of up to $\sim 1$ GV/m expected.

In AWAKE Run 2, the proof of concept will be significantly extended to address
the requirements needed to develop beams for use in particle physics
experiments. The primary aims of AWAKE Run 2 are to sustain the expected peak
fields of $0.5-1$ GV/m over long distances, thereby increasing the accelerated
electron energy; to demonstrate that the emittance of the electron bunch is
preserved during acceleration in plasma; and to develop plasma sources that
are scalable to 100s of metres and beyond. This will be achieved in a staged
approach during the 2020s which will require significant extension to the
current AWAKE facility, in particular the development of short witness
electron bunches for injection, new plasma sources and a suite of diagnostics
to measure the physics of the acceleration process.

After completion of AWAKE Run 2, at the end of the 2020s, the scheme should
have been sufficiently demonstrated such that it can be used to provide beams
for particle physics experiments. Given the challenge in producing high energy
electron bunches by conventional means, electrons in the 20–200 GeV range, as
driven by protons from the SPS, or even at the TeV scale if using LHC protons
as the wakefield driver, can be used in a variety of particle physics
experiments. Such electron bunches can be used in experiments to search for
dark photons, to measure QED in strong fields or as the injector or main
accelerator for the electron arm of an electron–proton or electron–ion
collider. These first applications place less stringent requirements on the
parameters of the electron bunch than for a high energy, high luminosity
linear electron–positron collider, although they will provide a useful
stepping stone, along with continued R&D, to such ultimate applications whilst
also providing beams for novel particle physics experiments.

E.G, K.L., P.M and M.W. conceived, wrote and edited this article. Members of
the AWAKE collaboration contributed to the review of the manuscript and to the
work here summarised.

This work was supported in parts by a Leverhulme Trust Research Project Grant
RPG-2017-143 and by STFC (AWAKE-UK, Cockcroft Institute core, John Adams
Institute core, and UCL consolidated grants), United Kingdom; the Russian
Science Foundation, project 20-12-00062, for Novosibirsk’s contribution; the
National Research Foundation of Korea (Nos. NRF-2016R1A5A1013277 and
NRF-2020R1A2C1010835); the Wolfgang Gentner Programme of the German Federal
Ministry of Education and Research (grant no. 05E15CHA); M. Wing acknowledges
the support of DESY, Hamburg. Support of the National Office for Research,
Development and Innovation (NKFIH) under contract numbers
2019-2.1.6-NEMZ_KI-2019-00004 and MEC_R-140947, and the support of the Wigner
Datacenter Cloud facility through the Awakelaser project is acknowledged. The
work of V. Hafych has been supported by the European Union’s Framework
Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie
Sklodowska-Curie Grant Agreement No. 765710. TRIUMF contribution is supported
by NSERC of Canada.

No new data is provided in this article which presents a review of previous
work.

###### Acknowledgements.

The AWAKE collaboration acknowledge the SPS team for their excellent proton
delivery. The authors declare no conflict of interest. The following
abbreviations are used in this manuscript:
AWAKE | Advanced wakefield experiment
---|---
BNL | Brookhaven National Laboratory
CERN | European Organisation for Nuclear Research
| (Conseil Européen pour la Recherche Nucléaire)
CNGS | CERN neutrinos to Gran Sasso
DESY | Deutsches Elektronen-Synchrotron
EIC | Electron–ion collider
GEANT | Geometry and tracking
HERA | Hadron–electron ring accelerator
LHC | Large hadron collider
LHeC | Large hadron–electron collider
LUXE | Laser und XFEL experiment
PEPIC | Plasma electron–proton/ion collider
QCD | Quantum chromodynamics
QED | Quantum electrodynamics
RIF | Relativistic ionisation front
RMS | Root mean square
SLAC | Stanford Linear Accelerator
SM | Self-modulation
SMI | Self-modulation instability
SPS | Super proton synchrotron
VHEeP | Very high energy electron–proton
XFEL | X-ray free electron laser
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11institutetext: Universidad Pablo de Olavide, Km. 1 Utrera road, 43013
Seville, Spain 22institutetext: Tessella, Altran World-Class Center for
Analytics, c/ Campezo 1, 28022 Madrid, Spain 22email<EMAIL_ADDRESS>
22email<EMAIL_ADDRESS>
# A privacy-preserving, distributed and cooperative FCM-based learning
approach for Cancer Research

Jose L. Salmeron 1122 Irina Arévalo 11

###### Abstract

Distributed Artificial Intelligence is attracting interest day by day. In this
paper, the authors introduce an innovative methodology for distributed
learning of Particle Swarm Optimization-based Fuzzy Cognitive Maps in a
privacy-preserving way. The authors design a training scheme for collaborative
FCM learning that offers data privacy compliant with the current regulation.
This method is applied to a cancer detection problem, proving that the
performance of the model is improved by the Federated Learning process, and
obtaining similar results to the ones that can be found in the literature.

###### Keywords:

Fuzzy Cognitive Maps Federated Learning Distributed Artificial Intelligence
Cancer Diagnosis

##  1 Introduction

111Preprint - Published in Rough Sets: International Joint Conference, IJCRS
2020

Distributed Artificial Intelligence is a subfield of Artificial Intelligence
that studies the coordination among several semi-autonomous agents called
participants. Such systems are able to solve more complex problems involving a
large amount of data, but there are privacy concerns about sharing sensitive
information.

Federated Learning is a novel approach to Distributed Artificial Intelligence
that enables privacy-preserving communications by sharing the model (or
gradients) instead of the data. A central server sends a model to be trained
by the participants with their local data, who send the parameters of the
model back to the server to be aggregated. After iterating this process, the
output is a model that has been trained with the private information of all
participants.

This method is especially useful when dealing with sensitive data, from
domains such as finance or healthcare. In this paper, the authors propose a
Federated Fuzzy Cognitive Map approach to help diagnose malignant breast tumor
cells.

The contributions of this paper can be summarized as follows:

  * •

Distributed learning. The authors propose a PSO-based FCM learning in a
distributed way.

  * •

Privacy-preserving machine learning. The authors design a training scheme for
collaborative FCM learning that offers data privacy. This proposal enables
multiple participants to learn a FCM model on their own inputs, preserving the
privacy of their own data and complying with data privacy regulations.

  * •

Implementation. The authors evaluate the performance of the proposal with a
well-known dataset of cancer diagnosis. The experimental results show that the
proposal achieve a similar performance to other non-distributed methods and
improves the performance of the non-collaborative approach.

The rest of this paper is organized as follows. We discuss existing
fundamentals of FCM and the learning approach in Section 2. Distributed
Artificial Intelligence is described in Section 3. Then, we present the
methodological proposal in Section 4. Section 5 describes the details of the
experimental approach and the results. Finally, we draw a conclusion in
Section 6.

##  2 Fuzzy Cognitive Maps

###  2.1 Fundamentals

Fuzzy Cognitive Maps (FCMs) were initially proposed by Kosko [3]. FCMs
represent concepts, variables or features as nodes, the relationships between
them as arcs, and the strengths of those relations as weights. It means that a
weight assesses how much node $X$ causes node $Y$. The fuzzy weights for arcs
are normalised on the range $\\{[0,+1]|[-1,+1]\\}$, depending if it includes
negative values or not. The maximum negative influence is $-1$ and the maximum
positive influence is $+1$. The value zero shows that there is no relationship
between the concepts. For computational purposes, FCMs can be described via a
weight matrix (connection or adjacency matrix) which contains all weight
values of edges between the concepts.

The relationships between the nodes are expressed by their weights. That is,
if there is a positive causality between two nodes, then $\varpi_{ij}>0$. If
there is a negative causality,then $\varpi_{ij}<0$ and if there is no
relationship between the two nodes, then $\varpi_{ij}=0$. The state of the
nodes together is shown in the state vector $c=[c_{1},c_{2},\ldots,c_{N}]$
that gives a snapshot of nodes at any point of the instant in the scenario.

From a formal point of view, it is possible to represent a FCM as a 4-tuple
$\Phi=\langle c,\mathcal{W},f,r\rangle$, where $c=\\{c_{i}\\}_{i=1}^{n}$ is
the state of the nodes with $n$ as the number of nodes,
$\mathcal{W}=[\varpi_{ij}]_{n\times n}$ is the adjacency matrix representing
the weights between the nodes, $f$ is the activation function, and $r$ is the
nodes’ range.

FCMs are dynamical systems involving feedback, where the effect of change in
the state of a node may affect the state of other nodes, which in turn can
affect the former node [7].

The dynamic starts with an initial vector state
$c(0)=\big{(}c_{1}(0),\ldots,c_{n}(0)\big{)}$, which represents the initial
state (value) of each node. The new state of the nodes is computed as an
iterative process. It includes an activation function [1] for mapping
monotonically the node state into a normalized range $\\{[0,+1]|[-1,+1]\\}$.
If the range is $[0,+1]$, the unipolar sigmoid is the most used activation
function, but hyperbolic tangent is the most used when the range is $[-1,+1]$.

The component $i$ of the vector state at time $t,$ $c_{i}(t),$ can be computed
as

$c_{i}(t)=f\Bigg{(}\sum_{j=1}^{n}\varpi_{ji}\cdot c_{j}(t-1)\Bigg{)}.$ (1)

Some systems include nodes whose states should be steady because their states
are not related with the dynamics of the system but their state has some
influence on the state of the other nodes (i.e. sun radiation, wind speed and
so on). In such cases, the state of the node is the same along the dynamics
$c_{i}(t)=c(t-1)\;|\;c_{i}\in\mathcal{O},$ where $\mathcal{O}$ is the set of
output concepts.

If the activation function $f$ is unipolar sigmoid, then the component $i$ of
the vector state $c_{i}(t)$ at the instant $t$ is computed as follows

$c_{i}(t)=\Big{(}1+e^{-\lambda\cdot\sum_{j=1}^{n}\varpi_{ji}\cdot
c_{j}(t-1)}\Big{)}^{-1}$ (2)

If the activation function $f$ is hyperbolic tangent, then the component $i$
of the vector state $c_{i}(t)$ at the instant $t$ is computed as follows

$c_{i}(t)=\frac{e^{\lambda\cdot\sum_{j=1}^{n}\varpi_{ji}\cdot
c_{j}(t-1)}-e^{-\lambda\cdot\sum_{j=1}^{n}\varpi_{ji}\cdot
c_{j}(t-1)}}{e^{\lambda\cdot\sum_{j=1}^{n}\varpi_{ji}\cdot
c_{j}(t-1)}+e^{-\lambda\cdot\sum_{j=1}^{n}\varpi_{ji}\cdot c_{j}(t-1)}}$ (3)

After the dynamics, the FCM reaches one of the three following states after a
number of iterations: it settles down to either a fixed pattern of node values
(the so-called hidden pattern), a limited cycle, or a fixed-point attractor.

###  2.2 Augmented FCMs

According to the FCM literature [4], an augmented adjacency matrix is built by
aggregating the adjacency matrix of each FCM. The elements’ aggregation
depends on whether there are common nodes. If the adjacency matrices had no
common nodes, the elements $\varpi_{ij}$ in the augmented matrix
($\displaystyle\otimes_{i=1}^{N}$) are computed by adding the adjacency matrix
of each FCM model ($\mathcal{W}_{i}$).

The addition method when the adjacency matrices have not common nodes is known
as direct sum of matrices, and the augmented matrix is denoted as
$\otimes_{i=1}^{N}\varpi_{i}$. Given a couple of FCMs with no common nodes and
even different number of nodes with adjacency matrices $[\varpi_{ij}]_{n\times
n}$ and $[\varpi_{kl}]_{m\times m}$, the resulting augmented adjacency matrix
is as follows

$\begin{array}[]{rl}\displaystyle\otimes_{i=1}^{N}\varpi_{i}&=\textrm{{diag}}(\varpi_{jk},\varpi_{lo})\\\\[7.11317pt]
&=\left(\begin{array}[]{cc}0&[\varpi_{jk}]_{r\times r}\\\
{[\varpi_{lo}]}_{m\times m}&0\end{array}\right)\end{array}$ (4)

where $N$ is the number or adjacency matrices to add, zeroes are actually zero
matrices and the dimension of $\displaystyle\otimes_{i=1}^{N}\varpi_{i}$ is
$[\cdot]_{m+r\times m+r}$. In the case of common nodes, they would be computed
as the average or weighted average of the states of the nodes in each
adjacency matrix.

###  2.3 FCM for classification

FCMs classification capabilities have been analysed by [8]. In general terms,
the main goal of a conventional classifier is the mapping of an input to a
specific output according to a pattern. In this proposal, the input concepts
represent the features of the dataset, while the output concepts are the
classes’ labels where the patterns belong.

Figure 1 shows the typical topology of a FCM classifier where the state of the
concepts $c_{1}$ and $c_{2}$ defines the class where the input vector state
belongs. In that sense, if $c_{1}>c_{2}$ the input vector state belongs to
class 1 but if $c_{1}<c_{2}$ the input vector state belongs to class 2. Note
that $c_{i}\in\\{[-1,+1],[0,+1]\\}$, therefore if $c_{1}=0.03$ and
$c_{2}=0.1$, then the input vector state belong to class 2.

Figure 1: Fuzzy Cognitive Maps classifier

###  2.4 PSO-based FCM learning

FCM learning endeavours are commonly focused on building the adjacency matrix
based either on the available historical raw data or on expert knowledge. FCM
learning approaches could be divided into three categories [10]: Hebbian,
population-based, and hybrid, mixing the main aspects of Hebbian-based and
population-based learning algorithm.

The goal of the Hebbian-based FCM learning approaches is to modify adjacency
matrices leading the FCM model to either achieve a steady state or converge
into an acceptable region for the target system.

Population-based approaches do not need the human intervention. They compute
adjacency matrices from historical raw data that best fit the sequence of
input state vectors (the instances of the dataset). The learning goal of FCM
evolutionary learning is to generate optimal adjacency matrix for modeling
systems behaviour.

In this sense, Salmeron et al. [11] proposed an advanced decision support tool
based on consultations with a group of experienced medical professionals using
FCMs trained with Particle Swarm Optimization (PSO). Also, Salmeron and
Froelich [9] apply PSO for time series forecasting.

PSO is a bio-inspired, population-based and stochastic optimization algorithm.
The PSO algorithm generates a swarm of particles moving in an $n$-dimensional
search space which must include all potential candidate solutions.

In order to train the FCM adjacency matrices we take into account the $k^{th}$
particle’s position (a candidate solution or adjacency matrix), denoted as
$\varpi_{k}=(\varpi_{k_{1}},\ldots,\varpi_{k_{j}})$ and its velocity,
$v_{k}=(v_{k_{1}},\ldots,v_{k_{j}})$. Note that each particle is a potential
solution or FCM candidate and its position $\varpi_{k}$ represents its
adjacency matrix.

Each particle’s velocity and position are updated at each time step. The
position and the velocity of each particle is computed as follows

$\displaystyle\varpi_{k}(t+1)$ $\displaystyle=\varpi_{k}(t)+v_{k}(t)$ (5a)
$\displaystyle v_{k}(t+1)$
$\displaystyle=v_{k}(t)+U(0,\phi_{1})\otimes(\dot{\varpi}_{k}-\varpi_{k}(t))+U(0,\phi_{2})\otimes(\ddot{\varpi}_{k}-\varpi_{k}(t))$
(5b)

where $U(0,\phi_{i})$ is a vector of random numbers generated from a uniform
distribution within $[0,\phi_{i}],$ generated at each iteration and for each
particle. Also, $\dot{\varpi}_{k}$ is the best position of particle $k$ in all
former iterations and $\ddot{\varpi}_{k}$ the best position of the whole
population in all previous iterations and $\otimes$ is the component-wise
multiplication.

The PSO algorithm’s goal is to locate all the particles in the global optima
to a multidimensional hyper-volume. The fitness function used in this research
is the complement of the Jaccard similarity coefficient
($\overline{J}=(Y\times\hat{Y})\setminus J$). The Jaccard score computes the
average of Jaccard similarity coefficients between pairs of the sets of
labels. The Jaccard similarity coefficient of the $i$-th samples, with a
ground truth label set and a predicted label set. The complement operation is
needed in terms of minimization of the fitness function. The Jaccard
similarity coefficient’s complement is computed as follows

$\overline{J}(y_{i},\hat{y}_{i})=1-\frac{|y_{i}\cap\hat{y}_{i}|}{|y_{i}\cup\hat{y}_{i}|}$
(6)

The fitness function is sampled after each particle position update and is the
objective function used to compute how close a given particle is in order to
be able to achieve the set aims.

##  3 Methodological proposal

###  3.1 Fundamentals

Distributed Artificial Intelligence is a subset of Artificial Intelligence
that allows the sharing of information among several agents or participants
that interact by cooperation, by coexistence or by competition. Such system
manages the distribution of tasks, being therefore more apt to solve complex
problems, especially if they involve a large amount of data.

One of the methods available to construct a distributed artificial
intelligence system is Federated Learning, proposed by McMahan et al. [5] and
further developed in Konecny et al. [2] and McMahan and Ramage [6]. In such
system, a central server constructs a model, usually a neural network, and
sends it to the participants, who train the model in their private data. Their
data never leaves their local devices, therefore ensuring privacy and
security. The parameters of the participant’s model are then averaged to
obtain a global model. This process may be iterated till convergence.

Described in a formal way, a Federated Learning project is composed by a
central server and the participants. The central server is responsible for
managing the federated model and the communications with the participants. The
participants own the datasets and train the partial models. The whole process
is described in Figure 2 and it is as follows:

  1. 1.

The central server sends a federated model to each participant. If it is the
initial iteration the federated model is proposed by the central server.

  2. 2.

Each participant trains the received model with their own private dataset.

  3. 3.

After the partial model is trained, each participant sends the parameters of
the model or its gradients to the central server, encrypted to ensure privacy.

  4. 4.

The central server aggregates the partial model and builds the federated
model.

  5. 5.

The central server checks the termination condition and if it is accomplished
the federated model is finished, otherwise the process goes back to step 1.

Figure 2: Federated Learning process

When the researchers at Google first defined Federated Learning, their initial
idea was to allow Android mobile phones to collaborative construct a
prediction model without migrating the training data from the phone (see
McMahan et al. [6] from the Google AI Blog). A first application they had was
to use FL in Gboard on Android, the Google Keyboard, which predicts the most
probable next phrase or word based on the user-generated preceding text.
Recently, Federated Learning has improved this process, allowing the use of
more accurate models with lower latency, ensuring privacy and less power
consumption.

One of the main advantages of Federated Learning is the promise of secure and
private distributed machine learning, but there are risks associated with
sharing data among several agents, such as the reconstruction of training
examples from the neural network parameters, the uploading of private data
from the agents to the central server, and the protection of the models as
intellectual property of the companies. There is a large research interest in
privacy-preserving methods applied to Federated Learning, such as the
application of Differential Privacy, Secure Multi-Party Computation or
Homomorphic encryption.

###  3.2 FCM distributed learning

The proposed methodology combines Federated Learning with learning FCMs using
Particle Swarm Optimization. The process is shown in Figure 3 and it is
explained as follows.

  1. 1.

Triggering the Federated Learning process. The central server triggers the
process in the participants machines.

  2. 2.

Training FCM in the local dataset. Each participant trains a local FCM with
their own dataset. The authors apply PSO but this methodology is agnostic to
the learning approach. The FCM dynamics is considered steady when the
difference between two consecutive vector states is under $tol=0.00001$

  3. 3.

Sending the trained adjacency matrices and local accuracy for this stage to
the central server. The local FCM is stored in the participant devices.

  4. 4.

Weighting local FCMs using accuracy. The central server aggregates the local
FCMs weighting by the accuracy. The aggregation method have been detailed as
Section 2.2.

  5. 5.

Aggregating Federated and Local FCMs. The participants aggregate the Federated
FCM from the central server and their own local FCM.

  6. 6.

Sending adjacency matrices and accuracy. Participants send again the local
adjacency matrices and the new local accuracy.

  7. 7.

Checking termination condition. The central server checks if the Federated
process has been run 20 iterations as termination condition. If it is not
accomplished then it goes back to the step 4.

  8. 8.

If the termination condition is accomplished then a Federated FCM is achieved.

Figure 3: Proposed methodology

The main contribution of this paper is the application of Federation Learning
paradigm for privacy-preserving FCM distributed and coorperative learning.

##  4 Experimental approach

###  4.1 Dataset

Breast cancer is one of the most common cancers among women, accounting for
25% of all cancer cases that affect women worldwide. According to the American
Cancer Society, when breast cancer is detected early, and is in the localized
stage, the 5-year relative survival rate is 99%, which makes the early
diagnosis of breast cancer a main key in the prognosis and chance of survival
of such types of cancer.

In recent years the use of Machine Learning algorithms in medicine has
increased exponentially, with applications such as EEG analysis and Cancer
detection. For example, automatized algorithms have been use to examine
biological data such as DNA methylation and RNA sequencing to infer which
genes can cause cancer and which genes can instead be able to suppress its
expression.

In this paper the authors will use the Breast Cancer Wisconsin Dataset,
created by Dr. William H. Wolberg, physician at the University Of Wisconsin
Hospital at Madison, and made publicly available at the UC Irvine Machine
Learning Repository. The dataset comprises data from digitized images of the
fine-needle aspirate of a breast mass that describes features of the nucleus
of the current image of 569 patients, of which 212 are malignant and 357 are
benign cases.

The first two features correspond to the identifier number and the diagnosis
status (our target). The remaining attributes are thirty real attributes that
measure the mean, the standard error, and the worst radius, texture,
perimeter, area, smoothness, compactness, concave points, concavity, symmetry,
and fractal dimension of the nucleus of the solid breast mass These data were
obtained using a graphical computer program called Xcyt, which is capable of
perform the analysis of cytological features based on a digital scan. More
details can be found in [13], [12].

###  4.2 Results

After 20 iterations of the Federated Learning process, the Fuzzy Cognitive
Map-based classifier is able to predict whether the tumor is malignant with an
average accuracy of 0.9383 across all participants, improving the accuracy of
a single Fuzzy Cognitive Map trained in the whole data, and the accuracy in
each participant before the federation.

The goal of this paper is not the accuracy of the proposal but a distributed
and privacy-preserving approach. Nevertheless, our results are similar to the
ones found in literature [14].

Table 1: Results of the experiments | Accuracy | Accuracy
---|---|---
Participant | pre-Federated | post-Federated
| Learning | Learning
1 | 0.7727 | 0.9091
2 | 0.9130 | 0.9130
3 | 0.8696 | 0.8696
4 | 0.9565 | 1.0000
5 | 1.0000 | 1.0000

##  5 Conclusions

This paper proposes an innovative methodology for learning Fuzzy Cognitive
Maps with Federated Learning. It is a step forward for Distributed Artificial
Intelligence and accomplishes the privacy-preserving requirements of the
society.

In addition, the authors have developed a method for distributed Fuzzy
Cognitive Maps that improves the accuracy of both the algorithm trained in the
whole dataset in a local node and the participant’s algorithms before the
Federated Learning process.

This method was applied to a cancer detection problem, obtaining an accuracy
of 0.9383. The participants in this process do not share their private data,
therefore forming a privacy-preserving distributed system.

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