Plasma-based acceleration of electrons has produced high-energy beams at high accelerating gradients with a narrow energy spread and high efficiency both in experiments and simulations. It is now being considered as a complementary approach to the use of radiofrequency cavities in next-generation lepton accelerators. However, compared with electrons, plasma-based positron acceleration is at the present time much less advanced. Although high-gradient positron acceleration in a plasma has been achieved, we are one to three orders of magnitude away from delivering the high-quality positron beams needed for a future high-energy linear collider. Here we review the status of plasma-based acceleration of electrons and positrons and discuss the prospects for substantial progress towards developing the positron arm of a plasma-based electron-positron linear collider in the next decade.

High-energy particle colliders use microwave cavities to accelerate and collide two tightly focused particle beams 1 . With a demand for colliders operating at higher and higher energies, they have become larger and thus more expensive, as evidenced by the Large Hadron Collider with its circumference of 27 km, the largest and most energetic particle collider built so far. Looking ahead, the realization of two of the proposed colliders, the International Linear collider 2 and the Future Circular Collider 3 (with its 100 km circumference), is by no means certain owing to their projected costs. The need for smaller and cheaper colliders for high-energy particle physics was already recognized in the 1980s 4 , which led to proposals for a plasma-based accelerator [5][6][7] , a two-beam accelerator, such as the Compact Linear Collider 8,9 , and a muon collider 10,11 . In this Review, we discuss plasma wakefield acceleration-based linear colliders with particular emphasis on the most pressing current challenges in developing the positron arm of an electron-positron linear collider.

There are some crucial differences between the plasma accelerating structures discussed here and the radiofrequency-powered metallic cavities used in today's accelerators. The accelerating structure in plasma-based acceleration is a space-charge density wave or wake, which is produced or driven in plasma by either an intense, short laser beam (laser wakefield acceleration) 6 or a highly relativistic and high-current charged particle beam (plasma wakefield acceleration) 7 .

Here, wakefield refers to the electric field of the charge density wave that moves at the speed of light. It has a longitudinal field component that can accelerate or decelerate charged particles co-moving with the wake and a transverse component that can focus or defocus the same charged particles. Such plasma wakes have been shown to have ultrahigh accelerating fields that can exceed 100 GeV per metre-thousands of times larger than the typically 10-50 MeV per metre achievable with radiofrequency cavities 12 used in conventional accelerators.

So far, however, the know-how to deliver ultra-relativistic electron and positron bunches with high current and high brightness needed for developing both electron and positron arms of a plasma-based linear collider exists only at conventional accelerator facilities 1 . Therefore, we focus on the status of and the decadal challenges faced in developing the positron arm of a plasma wakefield-based linear collider enabled by a charged particle beam, keeping in mind that the development of a laser wakefield-based linear collider will face similar challenges.

In a plasma-based accelerator, the energy contained in the drive beam is first transferred to the plasma wake and then extracted by a single, sufficiently high-current trailing beam-this process is called beam loading 13,14 . If the trailing bunch contains a sufficient amount of charge and is placed at the right location inside the wake, the net energy-transfer efficiency from the drive to the trailing bunch is expected to be as high as 40% in experiments and greater than 50%

https://doi.org/10.1038/s41567-025-02910-z drivers. After these achievements, electron acceleration using plasma wakes progressed rapidly. Energy gains on the order of 10 GeV with a metre-scale plasma wake, required as a first step towards realizing a multi-stage plasma-based collider, were achieved by using high-current electron bunches less than 100 fs in duration from a conventional particle accelerator [20][21][22] . This achievement brought the possibility of realizing the electron arm of a plasma-based electron-positron linear collider to the attention of the accelerator community as a complementary approach to using radiofrequency cavities 23,24 . Around the same time, the development of lasers with petawatt peak power enabled similar advances in laser wakefield acceleration 25 . There are many excellent review articles [26][27][28][29][30] that give a detailed overview of the status of electron acceleration and associated diagnostic techniques using both laser and plasma wakefield acceleration schemes.

In the first two decades of this century, experimental advances in electron acceleration by plasma wakes led to preliminary designs of a plasma-based electron-positron linear collider for high-energy particle physics 31,32 (see Fig. 1 for the layout and Table 1 for the key beam parameters).

The designs for a plasma-based linear collider-as do other collider designs 33 -typically work backwards from the luminosity and the energy required at the collision point to achieve a desired reaction rate for a certain process, such as the number of Higgs bosons produced per year, while keeping the necessary wall plug power as small as possible. Future electron-positron colliders at the energy frontier of particle physics-taken here as a centre-of-mass energy exceeding 1 TeV at a in simulations-far larger than in conventional accelerators. The high accelerating gradient implies a much smaller footprint and the expected high efficiency of a plasma-based accelerator promises a reduced operating cost. These two benefits are the principal motivators behind developing this technology.

Unfortunately, such high-gradient accelerating wakes come with some major drawbacks. First, the wakes are transient and mesoscopic; they have periods of about 100 fs, lifetimes of about 10 ps and have submillimetre diameters. Second, the accelerating trailing beam is on the order of 20 fs in duration and has a diameter of less than 1 μm. These ultrashort bunches necessitate an exquisite control over shot-to-shot temporal variation between the separation, and the transverse alignment accuracy between the trailing beam and the drive beam. The temporal variations called jitter will have to be reduced from the current 10 fs level to about 1 fs to ensure that the trailing beam experiences the same electric field as the wake throughout all the stages of the accelerator. Similarly, the transverse spatial alignment accuracy will need to be reduced from the current 1 μm level to be on the order of 10 nm in each module of a multi-stage plasma-based electron-positron linear collider to minimize the growth of any transverse instabilities. A schematic of the collider 15 is illustrated in Fig. 1.

The history of plasma accelerators goes back about four decades 16 . Between the first proposals for wakefield acceleration and experimental demonstration of electron acceleration by more than one group, it took two decades. This demonstration was achieved with millimetre-to centimetre 17 -long plasma wakes with energy gains of less than 1 MeV in a plasma wakefield accelerator 18 to typically 100 MeV in a laser wakefield accelerator 19 using approximately 100-fs-long Bottom: the basic design follows the architecture of a conventional radiofrequency-powered linear collider. The electron drive beams (black arrows, marked driver) are produced by a separate conventional linear accelerator, not shown in the schematic. The trailing electron (grey) and positron (beige) beams come from conventional damping rings. In the collider concept, each acceleration stage is about 1 m long, and the stages are joined together by interstage magnetic optics (red) in a serial fashion. The collision point is shown in orange-yellow. Overall, the two arms are each roughly 2,250 m long, which includes 1,500-m-long beam delivery systems (not drawn to scale) and 500-m-long accelerator sections. For a centre-of-mass energy of 1 TeV, the accelerator sections would each typically have 25 electron and 25 positron acceleration stages that add 20 GeV energy per stage over a distance of 1 m to the trailing beam. Here, electron and positron arms are considered to be the same, although the positron arm may require more stages owing to a lower average accelerating gradient. Further details, such as the drivers of the electron and positron arms, require dedicated research. Insets: details of the nonlinear plasma wakes produced by identical electron beams for electron (left) and positron (right) acceleration. In the electron arm, both the plasma and the wake are generated by the drive electron beam itself, whereas in the positron arm, a preformed plasma column is established before the arrival of the drive electron beam, which generates a wake with an electron density filament on-axis of the ion cavity. The wake produces a net accelerating field, while the plasma ions and the electron filament together exert a net focusing force to keep the positron beam guided throughout the acceleration stage. In both cases, the trailing electron (purple) or positron (red) trailing beam efficiently removes the energy from the wake by beam loading as seen by the flattening of the longitudinal electric field of the wake.

https://doi.org/10.1038/s41567-025-02910-z peak luminosity of 2.35 × 10 34 cm -2 s -1 (ref. 31)-will probably have to be linear colliders 34 because an unacceptable amount of energy is lost to synchrotron radiation 35 when the high-energy electrons and positrons are bent by dipole magnets. The downside of a linear collider is that the average acceleration gradient must be large and that the two beams collide with each another just once, increasing the repetition rate of the machine to greater than 1 kHz to achieve the design luminosity.

In general, one distinguishes between different regimes of plasmabased wakefield acceleration using either linear or nonlinear wakes excited in either preformed (Fig. 2), self-ionized (top right in Fig. 1) or hollow-channel (Fig. 2c,f) plasmas. Preformed plasmas are produced either by ionizing a gas column by an auxiliary laser pulse or by using an electric discharge before the arrival of the drive pulse. Self-ionized plasmas are generated by the transverse electric field of the drive bunch itself. Although linear plasma wakes excited by either an electron or a positron drive beam can be used to accelerate both electron and positron trailing beams (Fig. 2), the accelerating gradients of linear wakes at a few gigaelectronvolts per metre or less are smaller compared with tens of gigaelectronvolts per metre offered by nonlinear wakes. Furthermore, the physics behind the energy extraction from the wakes also differs for the electron and positron acceleration in nonlinear wakes. In the next two paragraphs, we discuss the similarities and differences in electron and positron acceleration using linear wakes produced by charged particle beams.

For driver beams that are wider than the skin depth of the plasma, which is the characteristic thickness of a plasma that is penetrated by the electric field of the beam, a linear wake is a space-charge density wave produced by the longitudinal displacement of the plasma electrons by the drive beam and their subsequent oscillation about the more massive, and therefore less mobile, ions. Because the particles in the drive beam are highly relativistic, the wake has a similarly relativistic phase velocity, almost the speed of light. Linear wakes are produced when the drive beam density is less than the plasma electron density. The wake's sinusoidal longitudinal electric field, illustrated by green curves in Fig. 2, can therefore accelerate an appropriately placed or phased trailing positron bunch in the positive field region, similar to acceleration by the longitudinal electric field of a transverse magnetic wave in a conventional accelerator. Both accelerating and focusing fields of the wake can be described by gradients of a single potential whose phase differs by a factor of π for linear wakes produced by electrons and positrons. This means that an individual accelerating stage of electron and positron arms would look identical in the case of a linear wake, except for the location of the accelerating trailing beam within the wake. An electron trailing bunch would be accelerated over distances where the wakefield is negative and focusing whereas a positron bunch would be accelerated where the wakefield is positive and focusing. The energy-extraction efficiency of a linear wake can be calculated by linear superposition of the wakes generated by the drive and the trailing beams 13 .

The main drawback of the linear wake is that the longitudinal field varies not only in the direction of propagation but also in the transverse direction. This means that operating in the linear acceleration regime requires precise shaping of the beams in both longitudinal and transverse directions to flatten or load the wake. A flat wake accelerates all particles at a constant electric field, thereby minimizing the particles' energy spread. A small-energy-spread beam is necessary for achieving a small focused beam spot size and thereby a high luminosity in the collision volume. This, however, is extremely challenging to achieve in practice. Because the accelerating gradient of a linear wake is typically one order of magnitude lower at a given plasma density than that of a nonlinear wake (Fig. 2), a trade-off between efficiency and beam quality cannot be avoided. In the linear regime, it may be necessary to use a wide drive beam and a narrow trailing beam to meet the requirements for a collider. However, wider beams are more susceptible to beam break-up instabilities, whereas a narrow-but-dense trailing beam might generate its own nonlinear wake 35 that will affect the energy spread and violate linear superposition, and make achieving high efficiency challenging.

Nonlinear wakes (Fig. 2d,e) can be produced by a narrow drive beam, whose radius is small compared with the skin depth of the plasma but whose beam density exceeds the plasma electron density 36,37 . Unlike in the case of linear wakes (Fig. 2a,b), nonlinear wakes produced by an electron or by a positron beam are very different from each other. Because the magnitude of the electron and ion density of an undisturbed plasma is the same, in the case of an electron drive beam shown in Figs. 1 and 2d, the excess negative charge contained in the highly relativistic electron drive beam repels and blows out the much lighter plasma electrons in the transverse direction with respect to the drive electron beam. These plasma electrons are then attracted back towards the beam axis by the Coulomb force of the plasma ions forming a bubble-like accelerating wake structure. This structure, often referred to as a bucket, contains plasma ions that are surrounded by a thin sheath of returning plasma electrons forming a narrow localized high-density electron spike 36 at the back of the bucket. In this 'blowout regime', the accelerating longitudinal field is constant in the transverse direction inside the bucket 37 , which means that all the particles in a given slice of the trailing electron beam gain energy at the same rate, thus minimizing the increase of the energy spread of the trailing beam. The transverse focusing field (not shown in Fig. 2), which results from the uniformly distributed plasma ions, is zero on the wake axis and increases linearly with radial distance 37 , and is constant in the longitudinal direction once the wake is fully formed. Consequently, all particles contained in a given longitudinal slice of the trailing beam oscillate about the wake axis, which is the same as the drive beam's axis, with almost the same frequency, called the betatron frequency 38 .

Table 1 | Key parameters of a conceptual plasma wakefield acceleration-based electron-positron linear collider Centre-of-mass energy 1,000 GeV Number of electrons/positrons per bunch 1.0 × 10 10 Energy gain per stage 20 GeV Number of electron/positron stages 25 Repetition rate 15 KHz Peak accelerating gradient (2-20) GeV m -1 Average accelerating gradient 1,000 MV m -1 Overall length of each arm 500 m Beam delivery system (total; both sides) 3,000 m Overall facility length 4,500 m Electron/positron bunch size (in two transverse directions) (33.5 nm,15.0 nm) Beam length 20 μm Overall drive to trailing bunch efficiency 40% Peak luminosity 2.35 × 10 34 cm -2 s -1

To keep the overall power requirement around 150 MW and 200 MW, a wall plug efficiency of more than 15% is needed. The 40% energy-transfer efficiency from the drive beam to the trailing beam assumes that both the electron and positron arms have the same average accelerating gradient. In the designs that have been put forward, a high degree of symmetry is assumed between the two arms. The physical processes involved in the propagation of high-current dense electron and positron bunches through plasmas, however, are different, meaning that this is a simplistic assumption and more refined future designs will have to take this into account. Data from ref. 31.

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In an optimally loaded wake, the betatron frequency is almost the same for all particles contained in the beam, because their transverse positions and momenta are highly correlated. This helps to preserve the beam's transverse emittance, which refers to the area occupied by the beam in position-and-momentum phase space, even if the electron beam has a small energy spread. The smaller the transverse emittance, the smaller will be the focused beam size and therefore the higher the luminosity at the collision point of the two counterpropagating beams. This, however, requires that the accelerating beam is injected such that the focusing force of the plasma ions is balanced by the inherent defocusing caused by the transverse emittance of the beam. This process is known as beam matching [39][40][41] . An additional advantage of the blowout regime is that the focusing field arises from the plasma ions and thus is not affected by beam loading, assuming that ions do not move during the passage of the accelerated beam. There is little phase slippage between the drive and acceleration beams because both are highly relativistic and thus moving almost at the speed of light. This means that the first bucket of the wake is extremely robust and does not change its shape 20 as the wake propagates until the drive bunch is nearly fully depleted of its energy, thereby transferring much of its energy to the wake.

Although nonlinear wakes have been shown to be extremely successful for acceleration of a trailing electron bunch, its usefulness for accelerating positrons was thought to be rather limited. This is because a positron trailing beam must be accelerated in a region of the wake that is both positive and focusing 42 to ensure that there is no loss of charge caused by defocusing induced by plasma ions. As shown in the top-left inset in Fig. 1, the longitudinal component of the wakefield can be both positive and negative, but the transverse component of the wakefield is strongly defocusing for the positron beam because the first few buckets mostly contain positively charged plasma ions; the region suitable for positron acceleration consists of a very narrow region between the two buckets (Fig. 1, top-left inset).

Instead, one can consider a nonlinear wake created by a positron beam 43 , which is qualitatively different from the one created by an electron beam 44 . The reason is that plasma electrons blown out by an electron drive beam are mostly attracted by the plasma ions in the form of a thin sheath, whereas they are pulled inwards by a positron beam from a range of radial locations in the plasma. These electrons thus arrive on the plasma axis at different times and in different locations. This in turn produces a weaker accelerating field and a non-uniform net electron plasma density, which leads to a highly nonlinear focusing field. The latter is a potential showstopper for a collider application because it causes rapid emittance growth of the accelerated positron beam.

To get around this issue, one approach is to use nonlinear wakes produced by a much shorter positron beam 4 that attracts plasma

https://doi.org/10.1038/s41567-025-02910-z electrons while reducing the spread in their arrival position on the axis, thus forming a reasonably coherent wake, as shown in Fig. 2e.

If the positron beam is placed in the first half of the second bucket, where the z component of the electric field is positive, it can attract and confine a substantial fraction of these plasma electrons on the axis and produce a dense electron column that can exert a focusing force on the positron beam. The net result is that the nonlinear wake can now both accelerate and focus the positron beam. Another is producing a wake with a tightly focused electron beam in either a hollow-channel plasma 45,46 (Fig. 2f) or in a preformed plasma (Fig. 2d and top-right inset in Fig. 1), and placing the positron beam in the second bucket. Using electron drive beams in both the positron and electron arms simplifies the architecture of the plasma collider. In all these cases (Fig. 2d-f), the produced nonlinear wake has a dense electron filament 44 resulting from confinement of some of the returning sheath electrons on the wake axis by a trailing positron beam placed in the first half of the second bucket. If the filament density is axially varying so as to flatten the longitudinal accelerating field, a beam with a narrow energy spread is obtained. Numerous other schemes to generate such an electron filament have been explored using particle-in-cell codes [47][48][49][50][51][52][53] . However, these approaches add further complications to the basic idea of creating an on-axis electron density filament and await proof-of-concept demonstrations.

As discussed earlier, the main shortcomings of the linear regime is that the longitudinal field of the wake is smaller and it varies in the transverse direction, which makes it necessary to use a precisely shaped trailing beam in all three dimensions to preserve the beam quality. In addition, if there is phase slippage between the accelerating particles and the drive beam as in a laser wakefield accelerator, then precise shaping of the beam must account for variation of the accelerating and focusing fields over the acceleration distance. However, this difficulty can be bypassed altogether through the use of a hollow plasma channel. This is because both loaded and unloaded wakes in perfectly circular channels have a radially uniform, but longitudinally sinusoidal accelerating field (Fig. 2c), and no transverse focusing field 54,55 . In the absence of a transverse electric focusing field in the channel, conventional magnetic lenses surrounding the hollow channel may be necessary to guide the drive and trailing beams. Although it was considered impractical to generate metre-scale hollow plasma channels for the propagation of positron beams, there has been much progress in this area [56][57][58][59] . This advantage offered by a hollow plasma channel has given impetus to theoretical and experimental studies of the use of such hollow channels for positron acceleration. In the example in Fig. 2f, the intense electron drive beam perturbs the channel wall and generates large surface charge and current densities, leading to a nonlinear wakefield inside the channel that can support an extremely high accelerating field as well.

The realization that ultrahigh gradient (tens of gigaelectronvolts per metre) nonlinear wakes with an on-axis electron density filament in the second bucket (Fig. 2f) can be generated in a hollow plasma channel may turn out to be the needed breakthrough for the development of the positron arm of a plasma-based electron-positron linear collider. There are, however, multiple challenges regarding the successful implementation of this idea. First, such high gradients require metre-long plasma channels with a radius less than 100 μm (Fig. 2f) and substantial wall plasma densities. Second, off-axis propagation, asymmetries in the beam and channel profiles, and wall imperfections can lead to the drive and trailing beams being susceptible to the growth of the head-to-tail instability, also known as the beam hosing instability 60,61 , in which any transverse tilt along the beam grows exponentially with propagation. This would cause an unacceptable increase of the beam emittance.

In summary, placing a positron beam in a nonlinear wake produced by an electron drive beam either in a preformed plasma or inside a narrow hollow plasma channel, both with a density filament on the beam axis, may be a way to achieve high-gradient positron acceleration, while obtaining a high energy-extraction efficiency

The highly nonlinear blowout regime 36,37 may simultaneously meet all the needs for a single electron stage of a plasma-based electron-positron linear collider. We now briefly describe the experimental achievements that led to this realization.

By the late 1990s, the Stanford Linear Collider was able to provide up to 50 GeV, highly compressed 100-fs-long electron and positron beams for the development of advanced acceleration concepts. A series of experiments on beam-driven plasma wakefield acceleration in the nonlinear blowout regime [20][21][22]62,63 culminated in demonstrating acceleration by highly nonlinear wakefields of 50 GeV per metre in a 1-m-long beam ionized plasma. The predicted increase in the accelerating gradient 64 for short (20-30 μm) drive bunches in high-density (3 × 10 16 -2 × 10 17 cm -3 ) plasmas was verified in these experiments. These accelerating fields ultimately doubled the energy of 42 GeV electrons in the tail of the same drive beam-obtained from the 3-km-long linear accelerator, over a distance of just 85 cm (ref. 20)-albeit with a 100% energy spread and a negligible energy-transfer efficiency.

This unacceptably high energy spread was remedied by sending two appropriately spaced shorter bunches into the plasma-the first produced the wakefield and the second extracted a significant fraction of the energy from the wake 21 , as envisioned by theoretical and computational predictions.

High-gradient, high-efficiency plasma wakefield acceleration experiments have since demonstrated excellent beam loading that increased the energy of the initial 20 GeV trailing beam by up to 9 GeV in about 1 m while demonstrating that energy spreads of around 4%, instead of the aforementioned 100%, were possible 21,22 (Fig. 3a). These experiments were carried out in a preformed plasma. Furthermore, more than 30% of the energy contained in the wake was transferred back to the trailing beam in this experiment 22 , confirming the wake flattening suggested by the small energy spread. Subsequent experiments at the FLASHForward facility at DESY (Deutsches Elektronen-Synchrotron) in Germany demonstrated the possibility of achieving a high repetition rate 65 and optimal beam loading of a nonlinear plasma wakefield, resulting in simultaneous preservation of energy spread and high energy-transfer efficiency without loss of beam charge 66 .

Before the year 2000, there were only a handful of theoretical works on how high-current (>10 kA) positron bunches interacted with plasmas, but no experiments. This changed when intense positron bunches from the Stanford linear accelerator became available for plasma acceleration experiments. Following the first experiment that showed plasma focusing of a 28 GeV positron beam by a short section of plasma 67 , acceleration of positrons at the tail of a positron drive beam in a quasilinear wake (bordering the linear and the nonlinear regimes) was demonstrated in a preformed plasma. A modest-but larger than in radiofrequency cavities-accelerating gradient with a peak amplitude of 80 MeV per metre was measured along a positron bunch 68 .

By increasing the product of the plasma density and length by two orders of magnitude and by using a high-current positron beam to generate the wake, another regime for positron acceleration was discovered 43 . When a dense positron bunch enters the plasma, it self-bifurcates into two distinct populations of positrons, which quickly settle into near equilibrium as the bunch propagates through the remaining part of the plasma. Roughly speaking, the front half of the beam attracts plasma electrons towards the axis, creating a wake, and thus loses energy to the wake. However, the back half of the beam together with the plasma ions attracts and confines a significant

https://doi.org/10.1038/s41567-025-02910-z fraction of the plasma electrons that are drawn in by the front half of the beam close to the axis. These electrons act like a dense electron filament that flattens the wakefield and simultaneously confines and accelerates this second group of positrons in the back half of the original bunch. This unexpected result was followed by an explosion of ideas for positron acceleration in nonlinear wakes. This experiment also was the first example of self-loading of the wake by a positron beam. This second group of positrons are now accelerated by the flattened, uniform wakefield at a high accelerating gradient (exceeding 3 GeV per metre). The accelerated positron beam had about 5% energy spread and a small divergence angle of around 100 μrad (ref. 43; Fig. 3b).

This experiment showed that creating a high-density electron region within the nonlinear wake was the key to successfully employing the high-gradient nonlinear wake regime for positron acceleration. However, the amount of the accelerated charge was one order of magnitude and the repetition rate and emittance of the accelerated beam were three orders of magnitude away from what is needed for a future high-energy linear collider. Nevertheless, this experiment 43 was key to the realization that a dense cylindrical electron filament was necessary for keeping the positrons confined on the nonlinear wake axis where they could also be accelerated using either a positron or an electron drive beam 44,[47][48][49][50][51][52][53]68 .

Hollow-channel plasmas have recently been created by ionizing a lithium vapour column with a high-order Bessel laser beam that has an intensity minimum on-axis. Longitudinal and transverse fields within the channel have been characterized by measuring the energy loss of a positron drive and energy gain as well as transverse deflections of a slightly off-axis high-energy positron trailing beam 61 . In this case, the plasma channel was roughly 20 cm long-short enough not to require auxiliary external focusing to keep the positron beam focused during its passage through the channel. For a more complete description of these and other experiments, as well as a complete list of references, the reader is referred to ref. 69.

Any credible proposal for a future electron-positron linear collider based on novel technology will have to demonstrate successful serial linking of at least two stages 31 without loss of charge, high-gradient acceleration at high energy-transfer efficiency between the drive and accelerated beams, as well as preservation of the beam energy spread and emittance 32 of both the electron and positron stages. The rationale behind this is that if we can demonstrate linking of two stages, we have a chance to serially link many more stages that are needed to reach the final design energy. One of the principle objectives of the BELLA facility at Lawrence Berkeley National Laboratory 27 is to show serial linking of two laser-driven electron acceleration stages. In this context, our understanding of how to design the electron arm is more advanced than the positron arm based on ongoing experiments and computer simulations. In the following, we surmise what progress we can expect if adequate positron facilities are made available in the next decade and how simulations can help this progress. We also describe alternative approaches to construct a future collider.

As far as experiments are concerned, the SLAC National Accelerator Laboratory had been the only facility worldwide with both gigaelectronvolt-class electron and positron beams where research on plasma-based acceleration of electrons and positrons could be carried out. With the repurposing of the SLAC linear accelerator from powering the Stanford Linear Collider to the laboratory's two X-ray free-electron lasers, the positron damping ring of the Stanford Linear Collider, which collects and cools the positrons, was decommissioned around 2015. The central one-third of its accelerator has been augmented with a low-emittance photo-injector electron source for continued studies of plasma-based acceleration at the FACET-II National User Facility with a 10 GeV electron beam. Plans to restore the positron capability have been developed 70 (Fig. 4), including new features such as the ability to study positron acceleration in the wake of an electron beam. For a timely progress of the positron arm of both a beam-and a laser-driven plasma-based linear collider, more such facilities are needed in the next decade, for the production of collider-quality positron bunches.

The availability of collider-quality positron beams is also an issue of some urgency for the laser-wakefield-acceleration community. The EuPRAXIA facility, which is currently under construction in Europe, plans to deliver gigaelectronvolt-class positron beams to users as a test-bed facility for experimental studies of acceleration of positrons using plasma wakefields 71 . There are now around a dozen petawatt-class laser facilities all having laser wakefield acceleration of electrons as one of the centrepieces of their research programme, but none of them have access to a gigaelectronvolt-class positron facility to meaningfully participate in the development of the positron arm of a plasma-based linear collider. Therefore, it is important that one or more such laser facilities are equipped with a suitable positron source. In the meantime, experimental and computational research on the generation of a positron test beam derived from a 100-MeV-class electron beam from a laser-plasma wakefield accelerator is beginning. Although these positron beams are unlikely to contain a sufficient charge to load the wake to study critical collider issues, such as beam quality, they may prove to be useful for understanding issues, such as positron injection, acceleration and extraction from laser-produced wakes 72,73 . In this context, we point out that a recent proof-of-concept experiment has shown energy gain of a witness positron beam using laser-produced wakes 74 for both positron generation and acceleration.

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Currently, few facilities can provide multi-gigaelectronvolt positron beams together with either a 10-20 GeV electron beam or a petawatt-class laser to conduct plasma-based positron acceleration experiments with a future linear collider in mind. Therefore, researchers must rely on high-fidelity particle-in-cell simulations (refs. 15,75-79 and references therein) to develop new acceleration or collider concepts. The successful validation of many plasma wakefield acceleration experiments have made such simulations an indispensable tool for simulating a single electron and positron stage of a plasma-based linear collider without any approximations. Such a design can then be continuously improved based on the latest experimental data. Furthermore, novel concepts for matching positron beams into and out of one and possibly two stages will need to be tested. Issues, such as emittance preservation, are likely to be qualitatively different between the electron and positron arms and will need to be addressed with simulations. Fortunately, computer resources and software capability have kept up with the needs of the field and the trend is likely to continue for the foreseeable future.

Polarized positrons. In proposed future colliders, high-charge, spinpolarized positron bunches are derived from multi-gigaelectronvolt electron bunches 80,81 . In one concept, the multi-gigaelectronvolt electron bunch produces circularly polarized photons in a magnetic helical undulator. The photons with energies of tens of megaelectronvolts produce spin-polarized positrons in a target with a high atomic number 82 . The positrons are then collected, cooled in a damping ring and then accelerated to produce a spin-polarized, high-energy positron beam. Methods for producing polarized electrons in situ for the electron arm of a plasmabased linear collider have been proposed 83,84 , but creating a polarized positron beam within the plasma wake is an unsolved problem 85,86 . Current plasma collider designs assume (Fig. 1) that the existing acceleratorbased technology can be adapted for injecting spin-polarized positron bunches into a plasma accelerator and that the subsequent highgradient acceleration will not significantly depolarize the positrons. However, this assumption will need to be verified in experiments.

Interim role for plasmas in a future collider. In comparison with electrons, plasma-based positron acceleration is at the present time much less advanced. Although high-gradient acceleration of positrons has been achieved by positron-beam-produced plasma wakes, we are one to three orders of magnitude away from delivering the high-quality positron beams that are needed for a future teraelectronvolt-class linear collider. Despite the many proposals for the high-gradient acceleration of positrons in plasma wakes, it remains uncertain whether any of these ideas will be realized in time to ascertain whether a multi-stage positron arm of a plasma-based linear collider (as shown in Fig. 1) with the requirements regarding beam energy, charge, emittance, energy spread, efficiency and repetition rate (Table 1) will be possible. Nevertheless, plasma acceleration can play multiple roles in future colliders.

An alternative approach to a multi-stage plasma-based linear collider is the so-called plasma afterburner 87 , where a single plasma-based acceleration stage is added at the end of the electron and positron arms of an existing, conventional, linear collider to roughly double the energy of both beams. In this concept, each arm produces two closely separated bunches. The first beam has a high charge and is used to produce a tens-of-metres-long wake. The wake then accelerates a trailing spin-polarized beam with a lower charge but extremely high brightness with an average gradient of tens of gigaelectronvolts per metre.

In this scenario, the acceleration of a trailing positron beam could be achieved by a wake generated by either an electron or a positron drive beam propagating in a preformed, uniform plasma or in a hollow channel, as discussed earlier. The idea is attractive because it eliminates the complexity of a multi-stage plasma-based linear collider. In addition, it offers a potential path to extend the energy reach of a conventional linear collider by roughly a factor of two. It is important to demonstrate a prototype of this concept on a future <1 TeV conventional linear collider by boosting its beams' centre-of-mass energy by a factor of 2 or so because it sets the stage for increasing the energy of a 5 TeV electron-positron linear collider to 10 TeV as called for in the recent 2023 Particle Physics Project Prioritization Panel report (https://www.usparticlephysics.org/2023-p5-report/).

Another example is a plasma-radiofrequency cavity hybrid scheme for an asymmetric Higgs factory 24 . Here, a beam-driven plasma The electron beam produced by a radiofrequency gun is subsequently accelerated first by a linear accelerator (injector) and then by three sections of the SLAC linear accelerator structures (purple 'disk-loaded' cavities). Between the accelerator structures, the beam is compressed by bunch compressors (green). In addition, electron bunches are compressed in the W-chicane before being delivered to the experimental area.

The electron beam energy is around 10 GeV at the end of the third accelerating section. The positrons are produced by sending the 10 GeV electron beam through a thick target with high atomic number. The positrons are collected, pre-accelerated and sent through a positron return line (purple) to the positron damping ring, where they are cooled and form a high-quality, high-charge bunch. After the damping ring, the positrons enter a conventional accelerating cavity, where they acquire a correlated energy spread. Positrons are compressed once using a magnetic bunch compressor before injection into the second accelerating section of the electron beam line with similar energy, bunch length and charge as the electron bunches. Afterwards, they pass through the 'sailboat' chicane, which delivers them to the interaction point on the same trajectory as the electrons, but with a variable time delay.

https://doi.org/10.1038/s41567-025-02910-z wakefield accelerator generates a higher-energy electron beam that is collided against a lower-energy positron beam produced by a conventional radiofrequency-powered accelerator.

Other proposals focus on capitalizing on the already demonstrated success of plasma-based electron acceleration. These include an electron-electron collider 88 , a photon-photon collider 88,89 , or doing fixed-target experiments with an electron beam or colliding an electron beam with a high-energy proton beam 90 . Electron beams from a plasma-based accelerator could also find application in creating coherent or incoherent X-ray beams. Plasma-based free-electron lasers have recently attracted attention 91,92 in light of their successful demonstration, albeit at much lower photon energies 93,94 . Another proposal uses a short plasma section as a very thin lens for strongly focusing a teraelectronvolt-class beam right before the collision point 95 or to linearize the longitudinal phase of ultrashort, ultrahigh-energy beams 96,97 .

There is no lack of creative ideas to solve the problem of efficient high-gradient positron acceleration. With concerted experimental effort, we are perhaps one decade away from reaching the same level of preparedness with solving the issues of acceleration of a high-quality positron beam as we already have for the electron arm of a plasma-based electron-positron linear collider.