e + e − Beam-beam Parameter Study for a TeV-scale PWFA Linear Collider
aa r X i v : . [ phy s i c s . acc - ph ] S e p e + e − BEAM-BEAM PARAMETER STUDY FOR A TeV-SCALE PWFALINEAR COLLIDER
J. B. B. Chen , D. Schulte, CERN, Geneva, SwitzerlandE. Adli, University of Oslo, Oslo, Norway also at University of Oslo, Oslo, Norway Abstract
We perform a beam-beam parameter study for a TeV-scale PWFA (particle-driven plasma wakefield accelera-tion) e + e − linear collider using GUINEA-PIG simulations.The study shows that the total luminosity follows the 1 /√ σ z -scaling predicted by beamstrahlung theory, where σ z is therms beam length, which is advantageous for PWFA, as shortbeam lengths are preferred. We also derive a parameter setfor a 3 TeV PWFA linear collider with main beam paramet-ers optimised for luminosity and luminosity spread intro-duced by beamstrahlung.Lastly, the study also compare the performance for scen-arios with reduced positron beam charge at 3 TeV and14 TeV with CLIC parameters. INTRODUCTION
In the blow-out regime of PWFA (particle-driven plasmawakefield acceleration), a dense ultra-relativistic drivebeam is used to excite a plasma wake, where plasma elec-trons are expelled from the region close to the propagationaxis, leaving only positively charged ions behind to form aplasma ion bubble cavity. Inside the plasma ion cavity ac-celerating gradients in the multi-GV/m level [1] can be usedto accelerate a trailing main beam.A previous parameter study on a 1.5 TeV PWFA (particle-driven plasma wakefield acceleration) accelerator [2] de-rived a parameter set that can provide reasonable stabil-ity, energy spread and efficiency for electron acceleration.This study adopted the parameter set in [2], assuming thatpositrons can be accelerated in a similar manner, and op-timised the main beam parameters at the interaction point(IP) for a e + e − collider with respect to luminosity and lumin-osity spread introduced by beam-beam effects. The prefer-ence of short beams in PWFA can be exploited to reducethe beam sizes accordingly without increasing the level ofbeamstrahlung, while achieving a higher luminosity.Furthermore, this study also examined asymmetric colli-sion scenarios with reduced numbers of positrons at 3 TeVand 14 TeV. BEAMSTRAHLUNG THEORY
Colliding beams in a linear collider are focused to smalltransverse dimensions in order to reach high luminosity.This gives rise to intense electromagnetic fields that willbend the trajectories of of particles in the opposite beam,and cause the particles to emit radiation in the form ofbeamstrahlung, and hence lose energy. A large fraction of particles will therefore collide with a less than nominal en-ergy, and form a luminosity spectrum.
Beamstrahlung Parameter
Beamstrahlung can be charaterised by the critical energydefined at half power spectrum [3] E c = ~ ω c = ~ γ cR , (1)where R is the bending radius of the particle trajectory.It is however more convenient to use the dimensionlessLorentz invariant beamstrahlung parameter defined as [4,5] Υ = e ~ m c ( p µ F µλ p ν F λν ) / , (2)where p µ is the four-momentum of the particle, and F µν is the electromagnetic field tensor of the beam field. Thebeamstrahlung parameter can also be written as Υ = ~ ω c E = γ h E + cB i B c , (3)where E is the energy of a particle before emitting radiationand B c = m c /( e ~ ) = . Υ can be interpreted as a measure for the strength ofthe electromagnetic fields in the rest frame of the electron inunits of B c . Since fields above B c are expected to cause non-linear QED effects, Υ ≪ Υ ≫ Υ is not constant during collision. For Gaussian beamswith N particles, horizontal rms beam size σ x , vertical rmsbeam size σ y and rms beam length σ z , the average and max-imum Υ can be approximated as h Υ i ≈ Nr γασ z ( σ x + σ y ) Υ max ≈ h Υ i , (4)where r e is the classical electron radius and α is the finestructure constant. Beamstrahlung and Luminosity
In the quantum regime with Υ ≫
1, the average numberof emitted photons per electron during the collision for aGaussian beam can be approximated as [5] n γ ≈ . α σ z r e γ h Υ i / = . α √ r e σ z N √ γ ( σ x + σ y ) ! / . (5)he total luminosity for a linear collider is given by L = H D N πσ x σ y n b f r = H D N πσ x σ y P b E b , (6)where n b is the number of beams per pulse, f r is the re-petition rate of pulses, P b = n b f f N E b is the beam powerper beam, E b is the beam energy and H D is a correctionfactor usually in the range 1 . − L ∝ /( σ x σ y ) and n γ ∝ /( σ x + σ y ) / , choosing a flat beam with σ x ≫ σ y can limit n γ without sacrificing luminosity. This gives thefollowing relation on σ x and n γ : σ x = . α Nn / γ r r e σ z γ . (7)Inserting this into the equation for the total luminosity, weobtain L = . H D πα r γ r e σ z n / γ σ y η P AC E b , (8)where η is the total (wall-plug to beam) conversion effi-ciency, P AC the wall-plug power for beam acceleration and E b is the beam energy.Equation (5) shows that for Υ ≫
1, a shorter beam cansuppress beamstrahlung. This implies that σ x can be re-duced accordingly for a flat beam, as described by equation(7), without increasing n γ . Consequently, the luminositycan be increased for shorter beams, as outlined by equation(8). This is particularly advantageous for PWFA, since shortbeams are preferred in PWFA due to the high plasma fre-quency. E.g. for a plasma with density n = cm − ,the plasma wavelength is λ p =
334 µm. For comparison, λ RF = .
51 cm in CLIC.
BEAM-BEAM PARAMETER SCAN
We performed beam-beam simulations using GUINEA-PIG [3], where we optimised collisions of e + e − beams withrespect to luminosity spread by performing parameter scansover β x , β y and σ z . Equal e + e − Beam Charges
In this study, we assumed that the number of particlesin both the e + and the e − beams are the same, and that β y can be made arbitrarily small regardless of technical con-straints. Furthermore, we define the peak luminosity L . as the part of the luminosity corresponding to centre of massenergy √ s > . √ s , where √ s is the nominal centre ofmass collision energy. The acceptable level of luminosityspread is chosen to be L . /L ≈ /
3, where L is the totalluminosity.Both beams have N = · particles and were col-lided at √ s = β y and σ z , we keptonly the results given by an optimal β x that corresponds to L . /L ≈ /
3. The corresponding results for L and L . are shown in fig. 1 and 2, respectively. The unit bx -1 de-notes “per beam crossing”. Figure 1: Contour plot of total luminosity L vs. beamlength σ z and vertical beta function β y , where the hori-zontal β x for each pair of σ z and β y has been chosen suchthat L . /L ≈ / Figure 2: Contour plot of peak luminosity L . vs. beamlength σ z and vertical beta function β y , where the hori-zontal beta function β x for each pair of σ z and β y has beenchosen such that L . /L ≈ / σ x can be made sufficiently small despite tech-nical constraints to keep n γ constant as σ z is reduced, andthat σ y is kept constant, eq. (8) gives the scaling L ∝ /√ σ z . The luminosity is plotted against σ z for a selectionof β y along with the corresponding L ∝ /√ σ z fits in fig.3. The 1 /√ σ z -scaling agree very well with simulation res-ults, especially for larger values of β y . The disagreementat small β y may be due to the hourglass effect, which im-poses β y ≥ σ z . When β y < σ z , a small beam size is onlymaintained over a small length, which reduces luminosity.Thus, using our range of σ z -values, the luminosity appearsto decrease faster than the 1 /√ σ z -scaling. Figure 3: Total luminosity L vs. rms beam length σ z forseveral vertical beta functions β y along with correspondingtheoretical 1 /√ σ z fits.In a previous parameter study [2] on a 1 . N = · electrons and a rms beam length of σ z = N = · σ z = Λ / Λ [2, 6] is ameasure for stability used to quantify the amplification ofthe transverse jitter of the main beam. It is listed along withthe relative rms energy spread σ E /h E i and drive beam tomain beam efficiency η in table 1.In deriving this parameter set, we did not consider tech-nological constraints on the vertical beta function. The ver-tical beta function β y = .
068 mm from the 3 TeV CLICparameter set [7] represents what is currently achievable,which is about one order of magnitude larger than our pro-posed value.
Reduced Positron Beam Charge
In the blow-out regime of PWFA, an electron beam willbe focused by the positive ion background, while a positronbeam will be defocused. Different approaches such as hol-low channel plasma [8] and the quasi-linear [9] regime havebeen studied, but positron acceleration remain one of themain challenges in PWFA, as there are currently no self-consistent scheme for positron acceleration in plasma thatcan simultaneously provide high efficiency, low preservedemittance and mitigation of transverse instabilities.Here we examine the effects of asymmetric e + e − colli-sions on luminosity at two energy levels, where the num-ber of particles N e + in the e + beam is only a fraction of thenumber of particles N e − in the e − beam. Other beam para-meters such as σ z , β x , y and ε N x , y are identical for the e + e − beams, and can be found in table 1. The results for differ- Table 1: Main Parameters for a 3 TeV PWFA Linear e + e − Collider
Parameter Symbol [unit] Value
Plasma density n [10 cm −3 ] 2.0Particle number N [10 ] 5rms beam length σ z [ µm ] β x [ mm ] β y [ µm ] γε x [ mm mrad ] γε y [ mm mrad ] σ E /h E i [ % ] Λ / Λ η [ % ] P b /( f r n b ) [kWs] 1.2Beamstrahlung photons/e − n γ L m −2 bx − ]Peak 1% luminosity/beam L . m −2 bx − ]ent scenarios are summarised in table 2 together with CLICparameters [7].As a result of the reduced N e + , the total luminosity is re-duced by approximately the same factor compared to caseswhere N e + = N e − . However, by reducing N e + , the beam-strahlung from the electron beam is also reduced, which res-ults in a narrower luminosity spectrum. Alternatively, thisalso allows the horizontal beam size to be further reducedwithout increasing n γ .Even in the N e + = . N e − scenario, a PWFA linear col-lider using the parameter set in table 1 can still provide acomparable luminosity level per beam crossing comparedto CLIC . Furthermore, the N e + = . N e − scenario shownin table 2 has a luminosity spread that is significantly betterthan our defined tolerance of L/L . ≈ /
3, which indic-ates that the horizontal beam size can likely be reduced evenfurther to increase the total luminosity.The muon collider submission to the European ParticlePhysics Strategy [10] showed that a 14 TeV muon collidercan provide a similar effective discovery potential as the100 TeV FCC. For comparison, the same parameters for14 TeV collision energy are shown in table 3.At 14 TeV, a common luminosity goal for linear collidersis 40 · cm −2 s −1 , which can be achieved in the N e + = N e − and N e + = . N e − scenarios with total beam powersof 20 MW and 281 MW, respectively. For comparison, the n b = f r =
50 Hz for CLIC. able 2: Parameter Comparison at 3 TeV Collision Energy
Parameter Unit N e + = N e − N e + = . N e − N e + = . N e − CLIC N P b /( f r n b ) kWs 1.2 1.2 1.2 0.89 E b TeV 1.5 1.5 1.5 1.5 L m −2 bx − L . m −2 bx − Parameter Unit N e + = N e − N e + = . N e − N e + = . N e − CLIC N P b /( f r n b ) kWs 5.61 5.61 5.61 4.17 E b TeV 7.0 7.0 7.0 7.0 L m −2 bx − L . m −2 bx − requires a total beam power of 90 MWto achieve this luminosity. However, note that none of theparameter sets in table 3 have been optimised for 14 TeV,which can be seen in the large luminosity spread. Further-more, we also made the optimistic assumption that the sameemittance levels and beta functions can be maintained forthe two energy levels. CONCLUSION
The suppression of beamstrahlung with decreasing beamlength (given that the horizontal beam size can be scaled ap-propriately to limit beamstrahlung) is beneficial for PWFA,as short beams are preferred. The derived parameter set fora 3 TeV e + e − PWFA linear collider shows a promising levelof luminosity, while maintaining a reasonable luminosityspread. Even for a scenario where the positron beam onlycontains 10% of the particles in the electron beam, the de-rived parameter set can still provide luminosity per beamcrossing comparable to that of CLIC at 3 TeV.At 14 TeV, the derived parameter set is able to achievethe luminosity goal of 40 · cm −2 s −1 with a significantlylower beam power than the CLIC parameter set. These para-meters are however not optimised for this energy level, andthus give rise to a large luminosity spread.The proposed parameter set furthermore has a verticalbeta function that is an order of magnitude smaller thanwhat is achievable today. Thus, study in how to achievesuch a small vertical beta function is required.Due to the challenges of accelerating positrons in aplasma, and the possibility of achieving a higher luminosity Note that here the calculations in GUINEA-PIG were done with β x , y that do not take non-linear effects into account and thus differ from thevalues given in [7]. Here we chose β x = . β y = .
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