Energy-frontier TeV colliders based on plasma accelerators are attracting much attention due to the recent achievements in multi-stage laser acceleration as well as the remarkable advances in electron- and proton-driven plasma accelerators. Such colliders may suffer a fundamental energy loss due to the radiation reaction (RR) effect, as the electrons lose energy through betatron radiation emission. Although the RR may not be critical for low-energy accelerators, it will exert limitations on TeV-class plasma-based colliders that need to be considered. In this paper, we have provided an extensive study of the RR effect in all pathways toward such colliders, including multi-stage plasma acceleration driven by the state-of-the-art lasers and the relativistic electron beam as well as the single-stage plasma acceleration with the energetic proton beams available at the CERN accelerator complex. A single-particle Landau–Lifschitz approach is used to consider the RR effect on an electron accelerating in the plasma blow-out regime. The model determines the boundaries where RR plays an energy limiting role on such colliders. The energy gain, the radiation loss, and the validity of the model are numerically explored.

1.
A.
Abada
,
M.
Abbrescia
,
S. S.
AbdusSalam
,
I.
Abdyukhanov
,
J. A.
Fernandez
,
A.
Abramov
,
M.
Aburaia
,
A. O.
Acar
,
P. R.
Adzic
,
P.
Agrawal
et al, “
FCC-ee: The lepton collider
,”
Eur. Phys. J.: Spec. Top.
228
,
261
623
(
2019
).
2.
T.
Behnke
,
J. E.
Brau
,
B.
Foster
,
J.
Fuster
,
M.
Harrison
,
J. M.
Paterson
,
M.
Peskin
,
M.
Stanitzki
,
N.
Walker
, and
H.
Yamamoto
, “
The international linear collider technical design report—Volume 1: Executive summary
,” arXiv:1306.6327v1 (
2013
).
3.
T.
Charles
,
P.
Giansiracusa
,
T.
Lucas
,
R.
Rassool
,
M.
Volpi
,
C.
Balazs
,
K.
Afanaciev
,
V.
Makarenko
,
A.
Patapenka
, and
I.
Zhuk
et al, “
The compact linear collider (CLIC)—2018 summary report
,”
Report No. CERN-2018-005-M
,
2018
.
4.
F.
Zimmermann
, “
Accelerator technology and beam physics of future colliders
,”
Front. Phys.
10
,
888395
(
2022
).
5.
V.
Shiltsev
and
F.
Zimmermann
, “
Modern and future colliders
,”
Rev. Mod. Phys.
93
,
015006
(
2021
).
6.
V. D.
Shiltsev
, “
High-energy particle colliders: Past 20 years, next 20 years, and beyond
,”
Phys.-Usp.
55
,
965
976
(
2012
).
7.
E.
Esarey
,
C. B.
Schroeder
, and
W. P.
Leemans
, “
Physics of laser-driven plasma-based electron accelerators
,”
Rev. Mod. Phys.
81
,
1229
(
2009
).
8.
M. J.
Hogan
, “
Electron and positron beam-driven plasma acceleration
,”
Rev. Accel. Sci. Technol.
9
,
63
83
(
2016
).
9.
E.
Adli
and
P.
Muggli
, “
Proton-beam-driven plasma acceleration
,”
Rev. Accel. Sci. Technol.
9
,
85
104
(
2016
).
10.
S.
Gessner
,
E.
Adli
,
J. M.
Allen
,
W.
An
,
C. I.
Clarke
,
C. E.
Clayton
,
S.
Corde
,
J.
Delahaye
,
J.
Frederico
,
S. Z.
Green
et al, “
Demonstration of a positron beam-driven hollow channel plasma wakefield accelerator
,”
Nat. Commun.
7
,
11785
(
2016
).
11.
Y.
Li
,
G.
Xia
,
K. V.
Lotov
,
A. P.
Sosedkin
, and
Y.
Zhao
, “
High-quality positrons from a multi-proton bunch driven hollow plasma wakefield accelerator
,”
Plasma Phys. Controlled Fusion
61
,
025012
(
2019
).
12.
A.
Doche
,
C.
Beekman
,
S.
Corde
,
J. M.
Allen
,
C. I.
Clarke
,
J.
Frederico
,
S. J.
Gessner
,
S. Z.
Green
,
M. J.
Hogan
,
B.
O'Shea
et al, “
Acceleration of a trailing positron bunch in a plasma wakefield accelerator
,”
Sci. Rep.
7
,
14180
(
2017
).
13.
S.
Diederichs
,
C.
Benedetti
,
E.
Esarey
,
J.
Osterhoff
, and
C. B.
Schroeder
, “
High-quality positron acceleration in beam-driven plasma accelerators
,”
Phys. Rev. Accel. Beams
23
,
121301
(
2020
).
14.
W. P.
Leemans
,
B.
Nagler
,
A. J.
Gonsalves
,
C.
Tóth
,
K.
Nakamura
,
C. G. R.
Geddes
,
E.
Esarey
,
C. B.
Schroeder
, and
S. M.
Hooker
, “
GeV electron beams from a centimetre-scale accelerator
,”
Nat. Phys.
2
,
696
699
(
2006
).
15.
A. J.
Gonsalves
,
K.
Nakamura
,
J.
Daniels
,
C.
Benedetti
,
C.
Pieronek
,
T. C. H.
de Raadt
,
S.
Steinke
,
J. H.
Bin
,
S. S.
Bulanov
,
J.
van Tilborg
et al, “
Petawatt laser guiding and electron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide
,”
Phys. Rev. Lett.
122
,
084801
(
2019
).
16.
S.
Steinke
,
J.
van Tilborg
,
C.
Benedetti
,
C. G. R.
Geddes
,
J.
Daniels
,
K. K.
Swanson
,
A. J.
Gonsalves
,
K.
Nakamura
,
B. H.
Shaw
,
C. B.
Schroeder
et al, “
Staging of laser-plasma accelerators
,”
Phys. Plasmas
23
,
056705
(
2016
).
17.
W.
Leemans
and
E.
Esarey
, “
Laser-driven plasma-wave electron accelerators
,”
Phys. Today
62
(
3
),
44
(
2009
).
18.
C. B.
Schroeder
,
E.
Esarey
,
C. G. R.
Geddes
,
C.
Benedetti
, and
W. P.
Leemans
, “
Physics considerations for laser-plasma linear colliders
,”
Phys. Rev. Accel. Beams
13
,
101301
(
2010
).
19.
C. B.
Schroeder
,
C.
Benedetti
,
E.
Esarey
, and
W. P.
Leemans
, “
Laser-plasma-based linear collider using hollow plasma channels
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
829
,
113
116
(
2016
).
20.
I.
Blumenfeld
,
C. E.
Clayton
,
F.-J.
Decker
,
M. J.
Hogan
,
C.
Huang
,
R.
Ischebeck
,
R.
Iverson
,
C.
Joshi
,
T.
Katsouleas
,
N.
Kirby
et al, “
Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator
,”
Nature
445
,
741
744
(
2007
).
21.
M.
Litos
,
E.
Adli
,
W.
An
,
C. I.
Clarke
,
C. E.
Clayton
,
S.
Corde
,
J. P.
Delahaye
,
R. J.
England
,
A. S.
Fisher
,
J.
Frederico
et al, “
High-efficiency acceleration of an electron beam in a plasma wakefield accelerator
,”
Nature
515
,
92
95
(
2014
).
22.
E.
Adli
, “
Plasma wakefield linear colliders-opportunities and challenges
,”
Philos. Trans. R. Soc., A
377
,
20180419
(
2019
).
23.
E.
Adli
, “
Towards a PWFA linear collider—Opportunities and challenges
,”
J. Instrum.
17
,
T05006
(
2022
).
24.
A.
Caldwell
,
K.
Lotov
,
A.
Pukhov
, and
F.
Simon
, “
Proton-driven plasma-wakefield acceleration
,”
Nat. Phys.
5
,
363
367
(
2009
).
25.
G.
Xia
,
O.
Mete
,
A.
Aimidula
,
C. P.
Welsch
,
S.
Chattopadhyay
,
S.
Mandry
, and
M.
Wing
, “
Collider design issues based on proton-driven plasma wakefield acceleration
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
740
,
173
179
(
2014
).
26.
E.
Adli
,
A.
Ahuja
,
O.
Apsimon
,
R.
Apsimon
,
A.-M.
Bachmann
,
D.
Barrientos
,
F.
Batsch
,
J.
Bauche
,
V. K. B.
Olsen
,
M.
Bernardini
et al, “
Acceleration of electrons in the plasma wakefield of a proton bunch
,”
Nature
561
,
363
367
(
2018
).
27.
E.
Gschwendtner
,
K.
Lotov
,
P.
Muggli
,
M.
Wing
,
R.
Agnello
,
C. C.
Ahdida
,
M. C. A.
Goncalves
,
Y.
Andrebe
,
O.
Apsimon
,
R.
Apsimon
et al, “
The awake run 2 programme and beyond
,”
Symmetry
14
,
1680
(
2022
).
28.
A.
Pukhov
and
J. M.
ter Vehn
, “
Laser wake field acceleration: The highly non-linear broken-wave regime
,”
Appl. Phys. B
74
,
355
361
(
2002
).
29.
S.
Corde
,
K.
Ta Phuoc
,
G.
Lambert
,
R.
Fitour
,
V.
Malka
,
A.
Rousse
,
A.
Beck
, and
E.
Lefebvre
, “
Femtosecond x rays from laser-plasma accelerators
,”
Rev. Mod. Phys.
85
(
1
),
1
48
(
2013
).
30.
J. D.
Jackson
,
Classical Electrodynamics
,
3rd ed
. (
John Wiley and Sons
,
New York
,
1999
).
31.
A.
Gonoskov
,
T. G.
Blackburn
,
M.
Marklund
, and
S. S.
Bulanov
, “
Charged particle motion and radiation in strong electromagnetic fields
,”
Rev. Mod. Phys.
94
,
045001
(
2022
).
32.
F.
Zimmermann
, “
Possible limits of plasma linear colliders
,”
J. Phys.: Conf. Ser.
874
,
012030
(
2017
).
33.
P.
Michel
,
C. B.
Schroeder
,
B. A.
Shadwick
,
E.
Esarey
, and
W. P.
Leemans
, “
Radiative damping and electron beam dynamics in plasma-based accelerators
,”
Phys. Rev. E
74
,
026501
(
2006
).
34.
I. Y.
Kostyukov
,
E. N.
Nerush
, and
A. G.
Litvak
, “
Radiative damping in plasma-based accelerators
,”
Phys. Rev. Accel. Beams
15
,
111001
(
2012
).
35.
M.
Zeng
and
K.
Seto
, “
Radiation reaction of betatron oscillation in plasma wakefield accelerators
,”
New J. Phys.
23
,
075008
(
2021
).
36.
Y.
Liu
and
M.
Zeng
, “
Modeling of three-dimensional betatron oscillation and radiation reaction in plasma accelerators
,”
Phys. Rev. Accel. Beams
26
,
031301
(
2023
).
37.
D. A.
Burton
and
A.
Noble
, “
Aspects of electromagnetic radiation reaction in strong fields
,”
Contemp. Phys.
55
,
110
121
(
2014
).
38.
A. D.
Yaghjian
, “
Lorentz-Abraham-Dirac and Landau-Lifshitz equations of motion and the solution to a relativistic electron in a counterpropagating laser beam
,”
Phys. Rev. Accel. Beams
24
,
114002
(
2021
).
39.
L. D.
Landau
and
E. M.
Lifshitz
,
The Classical Theory of Fields
, Course of Theoretical Physics Vol.
2
(
Pergamon
,
Amsterdam
,
1975
).
40.
I.
Kostyukov
,
A.
Pukhov
, and
S.
Kiselev
, “
Phenomenological theory of laser-plasma interaction in bubble regime
,”
Phys. Plasmas
11
,
5256
(
2004
).
41.
W.
Lu
,
C.
Huang
,
M.
Zhou
,
M.
Tzoufras
,
F. S.
Tsung
,
W. B.
Mori
, and
T.
Katsouleas
, “
A nonlinear theory for multidimensional relativistic plasma wave wakefields
,”
Phys. Plasmas
13
,
056709
(
2006
).
42.
W.
Lu
,
C.
Huang
,
M.
Zhou
,
W. B.
Mori
, and
T.
Katsouleas
, “
Nonlinear theory for relativistic plasma wakefields in the blowout regime
,”
Phys. Rev. Lett.
96
,
165002
(
2006
).
43.
A.
Caldwell
and
K. V.
Lotov
, “
Plasma wakefield acceleration with a modulated proton bunch
,”
Phys. Plasmas
18
,
103101
(
2011
).
44.
G.
Xia
,
R.
Assmann
,
R. A.
Fonseca
,
C.
Huang
,
W.
Mori
,
L. O.
Silva
,
J.
Vieira
,
F.
Zimmermann
, and
P.
Muggli
, “
A proposed demonstration of an experiment of proton-driven plasma wakefield acceleration based on CERN SPS
,”
J. Plasmas Phys.
78
,
347
353
(
2012
).
45.
P.
Muggli
, “
White paper: Awake, plasma wakefield acceleration of electron bunches for near and long term particle physics applications
,” arXiv:2203.09198v2 (
2022
).
46.
A.
Pukhov
,
N.
Kumar
,
T.
Tückmantel
,
A.
Upadhyay
,
K.
Lotov
,
P.
Muggli
,
V.
Khudik
,
C.
Siemon
, and
G.
Shvets
, “
Phase velocity and particle injection in a self-modulated proton-driven plasma wakefield accelerator
,”
Phys. Rev. Lett.
107
,
145003
(
2011
).
47.
K. V.
Lotov
, “
Simulation of proton driven plasma wakefield acceleration
,”
Phys. Rev. Spec. Top. Accel. Beams
13
,
041301
(
2010
).
48.
W.
Bartmann
,
A.
Caldwell
,
M.
Calviani
,
J.
Chappell
,
P.
Crivelli
,
H.
Damerau
,
E.
Depero
,
S.
Doebert
,
J.
Gall
,
S.
Gninenko
et al, “
AWAKE++: The AWAKE acceleration scheme for new particle physics experiments at CERN
,”
Report No. CERN-PBC-REPORT-2018-005
,
2018
.
49.
K. V.
Lotov
and
P. V.
Tuev
, “
Plasma wakefield acceleration beyond the dephasing limit with 400 GeV proton driver
,”
Plasma Phys. Controlled Fusion
63
,
125027
(
2021
).
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