Luminosity Performance of the Compact Linear Collider at 380 GeV with Static and Dynamic Imperfections
C. Gohil, P. N. Burrows, N. Blaskovic Kraljevic, A. Latina, J. Ögren, D. Schulte
LLuminosity Performance of theCompact Linear Collider at 380 GeVwith Static and Dynamic Imperfections
C. Gohil
1, 2 , P. N. Burrows , N. Blaskovic Kraljevic ∗ , A. Latina , J.¨Ogren , and D. Schulte John Adams Institute (JAI), University of Oxford, Oxford, OX1 3RH, United Kingdom The European Organization for Nuclear Research (CERN), Geneva 23, CH-1211, Switzerland(September 3, 2020)
Abstract
The Compact Linear Collider is one of the two main European options for acollider in a post Large Hadron Collider era. This is a linear e + e − collider withthree centre-of-mass energy stages: 380 GeV, 1.5 TeV and 3 TeV. The luminosityperformance of the first stage at 380 GeV is presented including the impact of staticand dynamic imperfections. These calculations are performed with fully realistictracking simulations from the exit of the damping rings to the interaction point andincluding beam-beam effects in the collisions. A luminosity of 4 . × cm − s − can be achieved with a perfect collider, which is almost three times the nominalluminosity target of 1 . × cm − s − . In simulations with static imperfections,a luminosity of 2 . × cm − s − or greater is achieved by 90% of randomlymisaligned colliders. Expressed as a percentage of the nominal luminosity target,this is a surplus of approximately 57%. Including the impact of ground motion,a luminosity surplus of 53% or greater can be expected for 90% of colliders. Theaverage expected luminosity is 2 . × cm − s − , which is almost twice the nominalluminosity target. One of the top priorities of the 2013 European Strategy for Particle Physics Update [1]was to perform R&D for a high-energy e + e − collider in Europe. The Compact LinearCollider (CLIC) is one of the two main options for an e + e − collider in Europe. Recently,several reports were submitted to the 2018 European Strategy for Particle Physics Update,describing the accelerator complex [2] and physics potential [3, 4] of this collider. Thispaper reports on the luminosity performance of this collider. ∗ Present address: ESS, Lund, Sweden. a r X i v : . [ phy s i c s . acc - ph ] S e p .1 CLIC CLIC [2, 5, 6] is a TeV-scale linear e + e − collider under development by the CLIC Collabo-ration. CLIC incorporates a staged approach with three centre-of-mass energies: 380 GeV,1.5 TeV and 3 TeV. The first stage at 380 GeV, which is the focus of this paper, has beenoptimised for studies of the Higgs boson and top-quark physics [7, 8].The integrated luminosity goal for the 380 GeV stage of CLIC is 180 fb − per year [5].Assuming 185 days of operation and 75% availability [5], this corresponds to a nominalluminosity of L = 1 . × cm − s − . (1)The 380 GeV stage of CLIC is described in detail in [5]. A schematic of the beamlineis shown in Fig. 1. The baseline is the drive-beam-based design with the Beam DeliverySystem (BDS) described in [9].CLIC utilises a novel two-beam acceleration scheme [6]. This scheme involves usingthe power from a high-current, low-energy drive beam to accelerate a low-current mainbeam to high energies. Each beam has its own accelerator complex. In this paper, westudy the main beam.The main beam is transported from the Damping Ring (DR) to the Interaction Point(IP) through three sections: the Ring to Main Linac (RTML), Main Linac (ML) andBDS. The RTML contains all the sub-systems between the DR and ML shown in Fig. 1.The geometry of the electron beamline is shown in Fig. 2. The beam is generated on thesurface and is transported 100 m underground in the RTML.Figure 1: Schematic diagram of CLIC at 380 GeV.
Imperfections in beamline elements degrade the luminosity of a collider. Simulation stud-ies are performed to determine the impact of imperfections. The ability of CLIC to reachits luminosity target in the 3 TeV stage has been studied in detail in [6]. In this paper,we study the luminosity performance of the 380 GeV stage.2 a i n A x i s [ k m ] ° ° ° ° ° ° ° P e r p e n d i c u l a r A x i s [ k m ] ° . ° . ° . . . D e p t h [ m ] ° ° ° ° M a i n A x i s [ k m ] ° ° ° ° ° ° ° P e r p e n d i c u l a r A x i s [ k m ] ° . ° . ° . . . D e p t h [ m ] Figure 2:
Geometry of the electron beamline of CLIC at 380 GeV. Main and perpendicular axisvs depth: RTML (dashed blue), ML (solid orange) and BDS (dotted red). The black arrowsshow the direction of the beam. The IP is at (0,0,-100).
Beam dynamics studies for CLIC at 380 GeV are summarised in [5]. Most effortshave focused on tuning studies of individual sections with static imperfections, namelythe RTML [10], ML [11] and BDS [12]. Previous studies of dynamic imperfections inCLIC at 380 GeV are limited to ground motion in individual sections. The impact ofATL motion in the ML and BDS is studied in [13] and [12] respectively. Intra-train IPfeedback simulations with ground motion in the BDS are presented in [14].
Beam-beam interactions in a linear collider can be strongly modified by correlations in thecolliding beams. A well known example of this is the banana effect [15]. This motivatesthe need for start-to-end simulations that integrate each section of a linear collider. Inthis paper, we present for the first time integrated simulations of the RTML, ML andBDS of CLIC at 380 GeV. We report a comprehensive study of the impact of static anddynamic imperfections on this collider. The following improvements have been made tothe studies of static imperfections referenced in the previous section: • We simulate a more complete list of static imperfections. Specifically in the ML, wenow include magnet strength errors and beam position monitor (BPM) rolls. • We use the latest lattice, which is that submitted to the 2018 European Strategyfor Particle Physics Update. Changes include a re-optimisation of the RF systemsin the RTML [5] and an updated ML lattice [5]. • Simulation of the BDS collimation section. Most tuning simulations of the BDSfocused on the final-focus system. In this work, we also simulate static imperfectionsin the collimation section. • Updated tuning procedures, in particular for the RTML and BDS. These are dis-cussed further in Sec. 4.2. 3e also study the impact of dynamic imperfections on this collider. For the first time, weperform integrated simulations including ground motion and stray magnetic fields fromnatural sources. Additional details of these simulations can be found in [16]. Tolerancesfor dynamic errors such as beam jitter, RF phase stability and magnet strength ripplesare also calculated.
Details of the simulations performed in this work are given in Sec. 2. A perfect beamlineis simulated in Sec. 3. Following this, the impact of different imperfections is studied.Sec. 4 looks at integrated simulations with static imperfections and the effectiveness oftuning procedures in recovering luminosity. Sec. 5 looks at dynamic imperfections. Here,tolerances for dynamic errors are presented along with integrated simulations of groundmotion and stray magnetic fields. Future work is discussed in Sec. 6 and the luminosityperformance of CLIC at 380 GeV is summarised in Sec. 7.
In an integrated simulation, the beam is tracked from the exit of the DR to the IP. Thesimulation codes used in this work and the beam extracted from the DR are describedbelow.
The particle tracking code PLACET [17] was used to transport each beam from the DRto the IP. The tracking simulations include the emission of synchrotron radiation andshort-range wakefields in the accelerating cavities.A single bunch-crossing luminosity was calculated with a full simulation of the collisionwith the beam-beam effects code GUINEA-PIG [18]. This was multiplied by the repetitionfrequency and number of bunches per train to calculate the total luminosity.The luminosity calculated with GUINEA-PIG is sensitive to the particle distributionof the colliding bunches. A small number of macro-particles leads to a high variance in thecalculated luminosity. It was found that using 100,000 macro-particles leads to a standarddeviation of less than 3% of the mean value. In this paper, the luminosity is calculatedwith several hundred different beam distributions at the IP. The mean luminosity will begiven. Each IP distribution was calculated with a tracking simulation in PLACET bysampling a new beam from the DR.
Parameter Symbol Value Unit
Horizontal/vertical emittance (cid:15) x /(cid:15) y σ x /σ y µ mHorizontal/vertical beam divergence σ x (cid:48) /σ y (cid:48) µ radBunch length σ z µ mEnergy E σ E Parameters of the beam extracted from the DR. .2 DR Beam In simulations, a Gaussian beam with 100,000 macro-particles is extracted from the DR.The simulated beam parameters at the exit of the DR are summarised in Table 1.
The maximum luminosity obtainable with this design of CLIC can be calculated by sim-ulating a perfect collider. The luminosity achieved with a perfect collider is L = 4 . × cm − s − . (2)This is almost three times the nominal luminosity target (Eq. (1)).The beam parameters at the end of each section are shown in Table 2. There isan emittance growth of approximately 85 nm in the horizontal direction and 0.8 nm inthe vertical direction that occurs in the RTML. This is from coherent and incoherentsynchrotron radiation in the bends [19]. A very small amount of emittance growth occursin the ML due to imperfect matching to the RTML. The emittance growth in the BDS isdue to correlations in the beam, which are described below. Section (cid:15) x [nm] (cid:15) y [nm] σ x [ µ m] σ y [ µ m] σ z [ µ m] E [GeV] σ E [%] RTML 785 5.82 18.9 0.63 70 9.00 1.0ML 791 5.85 8.05 0.29 70 190 0.35BDS 2,220 6.36 0.13 0.0013 70 190 0.35Table 2:
Simulated beam parameters at the end of each section for perfect beamline. − . − . . . . x [ µ m] N u m b e r o f M a c r o - P a rt i c l e s − . − . − . . . . . y [ µ m] N u m b e r o f M a c r o - P a rt i c l e s − −
100 0 100 200 300 z [ µ m] N u m b e r o f M a c r o - P a rt i c l e s − − −
50 0 50 100 150 x [ µ rad] N u m b e r o f M a c r o - P a rt i c l e s − − −
20 0 20 40 60 y [ µ rad] N u m b e r o f M a c r o - P a rt i c l e s . . . . . . . E [GeV] N u m b e r o f M a c r o - P a rt i c l e s Figure 3:
Histogram of the horizontal position x (top left), vertical position y (top centre),longitudinal position z (top right), horizontal angle x (cid:48) (bottom left), vertical angle y (cid:48) (bottomcentre) and energy E (bottom right) of a beam tracked through a perfect beamline. Fig. 3 shows the IP beam distribution generated by tracking a beam through a perfectbeamline. There are two correlations in the IP beam distribution: between the z - E and x (cid:48) - E coordinates. All other coordinates are uncorrelated. The z - E and x (cid:48) - E correlations are5hown in Figs. 4 and 5 respectively. The z - E correlation arises from short-range wakefieldsin the ML cavities and from off-crest acceleration, which is optimised to minimise theenergy spread at the end of the ML without compromising beam stability.In the BDS, sextupoles are placed in dispersive regions to correct chromaticity [20].This results in a correlation between the energy and horizontal angle. This can be seenin Figs. 3 and 5. The correlation leads to a horizontal emittance growth in the BDS. Asthis emittance growth is from the angular distribution, it does not significantly impactthe IP beam size and luminosity. − −
100 0 100 200 z [ µ m] . . . . . . . E [ G e V ] N u m b e r o f M a c r o - P a rt i c l e s Figure 4:
Energy E vs longitudinal position z of the IP beam distribution. This beam wastracked through a perfect beamline. − −
50 0 50 100 150 x [ µ rad] . . . . . . . E [ G e V ] N u m b e r o f M a c r o - P a rt i c l e s Figure 5:
Energy E vs horizontal angle x (cid:48) of the IP beam distribution. This beam was trackedthrough a perfect beamline. Static imperfections include errors in the alignment of accelerator elements, which areillustrated in Fig. 6, and static errors in the attributes of elements.6 oll TiltOffset Beam
Figure 6:
Illustration of different types of misalignments: roll, tilt and offset. In each case, thedashed line is the reference.
The most important static imperfection is the misalignment of BPMs. BPMs define theideal trajectory of a beam. Therefore, if they are offset with respect to a straight-linetrajectory, the beam will not follow a straight path. Furthermore, if the BPMs are rolled,a horizontal beam offset will appear partially as a vertical offset and vice versa, whichcomplicates the centring of the beam in each plane.Additionally, an important static imperfection is the noise of a BPM. Each BPMreading is corrupted by an error, which in simulations is assumed to be Gaussian. Thestandard deviation of this error is the BPM resolution. Corrections are applied to thebeam based on BPM readings. Therefore, a good resolution is desired to minimise theintroduction of noise from the BPM readings to the beam.
The misalignment of accelerating cavities is another important static imperfection. Acavity offset with respect to the beam excites wakefields, which lead to emittance growth.Novel wakefield monitors [21] are used to measure the wakefield in CLIC cavities.Additionally, tilts are an important alignment error for cavities. If a cavity is tilted, acomponent of the accelerating voltage is applied in a transverse direction with respect tothe beam. This results in the beam being kicked.
Important static imperfections for magnets are strength errors and misalignments withrespect to the ideal beam. Offset quadrupole and sextupole magnets kick the beam andlead to emittance growth. Additionally, magnet rolls lead to an xy -coupling, which resultsin emittance growth. CLIC will utilise the pre-alignment procedure described in [6]. Elements are placed ongirders, which are attached to movers equipped with sensors. A system of stretched wiresis used as a reference to align elements to a root-mean-square (RMS) offset of 10 µ m7ver distances of 200 m. Girders can be misaligned with respect to the reference line andarticulation points. All errors are assumed to have a Gaussian distribution. A summary of the errors simulatedin each section of CLIC is given in Table 3. In PLACET, misalignments are simulated withrespect to a perfect straight-line trajectory, which in reality corresponds to the system ofwires used as the reference.The imperfections listed in Table 3 are based on previous tuning studies and have beendefined in discussion with instrumentation, magnet and RF experts [5]. The RMS errorslisted for the RTML have been achieved or exceeded in existing accelerator facilities [5].The errors in the ML have been deemed achievable by experts [5]. The requirements inthe ML and BDS are the same for the 380 GeV and 3 TeV stages of CLIC to avoid systemupgrades in later stages [5, 6].
Imperfection Value
RTML Magnet and BPM offset 30 µ mMagnet and BPM roll 100 µ radBPM resolution 1 µ mCA and TA quadrupole strength errors 0.01%All other magnet strength errors 0.1%ML Magnet and BPM offset 14 µ mMagnet and BPM roll 100 µ radBPM resolution 0.1 µ mMagnet strength errors 0.01%Girder end point with respect to reference wire 12 µ mGirder end point with respect to articulation point 5 µ mAccelerating structure offset 14 µ mAccelerating structure tilt 141 µ radWakefield monitor offset 3.5 µ mBDS Magnet and BPM offset 10 µ mMagnet and BPM roll 100 µ radBPM resolution 20 nmMagnet strength errors 0.01%Table 3: RMS values for static imperfections implemented in integrated simulations. CA is thecentral arc and TA is the turn around (see Fig. 1).
Following pre-alignment, several well known beam-based alignment methods are used totune the beamline. These are described below.8 .2.1 One-to-One (121) Steering
This is the first tuning step. The beam is electrically centred in each BPM using thenearest upstream corrector. In the RTML dipoles are used to apply the correction. Inthe ML and BDS, quadrupoles mounted on movers are displaced to apply the correction.The corrector settings θ are found by minimising the objective function [12, 22] χ = | ∆ u − R θ | + β | θ | , (3)where ∆ u = u − u , u is a vector containing the BPM readings of a beam trackedthrough an imperfect beamline, u is a vector containing the BPM readings of an idealbeam, R is the response matrix and | . | denotes the magnitude. β is a free parameterthat is included to avoid large corrector strengths. Following 121 steering, DFS is performed. Here, the correctors are used to minimise thedifference in the trajectory of two beams of differing energy. The corrector settings arefound by minimising the objective function [12, 22] χ = | ∆ u − R θ | + ω | η − D θ | + β | θ | , (4)where η = u ∆ E − u , u ∆ E is a vector containing the BPM readings using an off-energybeam, u is the same as in the previous equation and D is the dispersion response matrix. β is another free parameter to avoid large corrector settings and ω is a weight factor forthe dispersion term, which can be calculated as [12, 22] ω = σ + σ σ , (5)where σ mis is the RMS BPM offset and σ res is the BPM resolution. Usually, the optimumvalue for ω is slightly different to Eq. (5) due to non-linear effects, such as wakefields andsynchrotron radiation. A scan is performed to find the optimum value for ω . Values of ω , β and β from [11, 12, 22] were used in this work. Following 121 steering and DFS, specific tuning procedures are performed that dependon the section.
RF realignment is performed in the ML. This involves offsetting a cavity to minimisethe reading from a wakefield monitor [11].
Sextupole tuning for chromaticity correction is performed in the RTML and BDS.In the RTML, a simplex algorithm [23] is used to minimise the emittance at the end of thesection by moving the last five sextupoles in the central arc and the last five sextupolesin the turn-around loop.Previous tuning studies for the RTML simultaneously minimised the horizontal andvertical emittance at the end of the section (see [22]). This procedure would often finda solution that minimised the vertical emittance only. Here, the horizontal and verticalemittances were minimised separately. This produced beams with a lower horizontal andvertical emittance compared to the previous procedure.9he BDS collimation section uses a simplex algorithm that displaces sextupoles inorder to minimise the emittance at the start of the final-focus system.For the final-focus system, the tuning signal used is the luminosity instead of theemittance. Two beams are required to calculate a luminosity. Ideally, each beam wouldbe tracked through its own beamline. However, at the time of these studies the latesttuning procedures for the final-focus system simulated a single beam and mirrored it atthe IP to calculate a luminosity. This method of calculating the luminosity was only usedfor tuning. To estimate the luminosity of the collider two beams were tracked throughdifferent tuned beamlines, this is discussed further in Sec. 4.2.4.A combination of a random walk of sextupole offsets and sextupole knobs is used totune the final-focus system after 121 steering and DFS. These procedures are describedin [12]. When tuning the final-focus system a small number of cases get trapped in alocal optimum, which prevents them from reaching the maximum possible luminosity. Anew step is introduced for these beamlines: a random walk of quadrupole and sextupoleoffsets [12]. This puts the beamlines into a new configuration, which can make the sex-tupole tuning knobs more effective. Reapplying the tuning knobs further increases theluminosity of these beamlines.
The beam-based tuning procedures described above were applied to 100 beamlines con-taining static imperfections. We use each of these beamlines to track the electron beam.We then randomly select a beamline from the remaining 99 beamlines to track the positronbeam. Therefore, we have 100 unique beamline pairs, which we will refer to as colliders.Each beamline in a collider has been tuned independently. Therefore, the achievedluminosity is not necessarily the optimum. Furthermore, for colliders that have a high dis-ruption, correlations in the beam can influence the luminosity and result in the maximumluminosity occurring with a beam-beam offset. To optimise the luminosity, we perform avertical beam-beam offset scan, a vertical crossing angle scan and waist scan. Optimisingthe horizontal beam-beam offset and crossing angle had a small effect (less than 1% lu-minosity gain) so was not included in the tuning procedure. Because each beamline wastuned independently, the luminosity that is achieved is a conservative estimate, whichmay be improved by performing two-beam tuning.
Fig. 7 shows the luminosity of 100 colliders after the full tuning procedure. The meanluminosity and its standard deviation is L = (3 . ± . × cm − s − . (6)90% of colliders achieve a luminosity greater than 2 . × cm − s − , which expressed asa percentage of the nominal luminosity target is 157%. This means there is a significantsurplus of 57%, providing a margin for the impact of dynamic imperfections. This section reviews short-term dynamic imperfections. These are processes that impactthe beam on a train-to-train basis, which are difficult to correct because of their fast10
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Collider Number . . . . . . . . L [ c m − s − ] × Figure 7:
Luminosity L vs collider number for 100 tuned colliders with static imperfections.Colliders are ordered in ascending luminosity. temporal variation. The most important dynamic imperfections for CLIC are beam jitter,RF phase errors, magnetic field ripples, ground motion and stray magnetic fields. For beam jitter, phase errors and magnetic field ripples, tolerances to limit luminosityloss are calculated. These are presented below.
Collider Number . . . . . . . . σ y , j [ n m ] Figure 8:
Vertical position jitter at the IP σ y,j that corresponds to a luminosity loss of 3 × cm − s − for 100 tuned colliders with static imperfections. Colliders are ordered in ascendingposition jitter. The luminosity with a vertical beam-beam offset is influenced by beam-beam interactions,which themselves depend on many factors, such as correlations in the beam and the IPemittance. Each collider has its own sensitivity to luminosity loss due to a beam-beam11ffset. A vertical position jitter σ y,j was simulated at the IP for each beam. Fig. 8 showsthe vertical position jitter that corresponds to a luminosity loss of 3 × cm − s − (2% ofthe nominal luminosity target) for 100 tuned colliders. The mean vertical position jittertolerance and its standard deviation is σ y,j = (0 . ± .
01) nm . (7)90% of colliders have a tolerance greater than 0.20 nm. The IP beam jitter is largelydetermined by the stability of the final doublet, which measurements have shown can bestabilised to an RMS jitter of less than 0.2 nm [6]. It is also possible to relax this toleranceby including an intra-train IP feedback, such as the FONT system [24, 25]. This section examines the impact of coherent RF phase errors. For incoherent RF phaseerrors, there is an averaging effect that generally leads to much larger tolerances, whichmeans they are less important [26].Coherent RF phase errors in the ML cavities are equivalent to a global error in theaccelerating gradient. This leads to an off-energy beam at the end of the ML. Due tochromaticity, an off-energy beam in the BDS has a larger beam size and yields a lowerluminosity. Fig. 9 shows the energy error at the end of the ML that corresponds to aluminosity loss of 1 . × cm − s − (1% of the nominal luminosity target) for 100 tunedcolliders. The energy of the beam was varied by changing the effective gradient of theML cavities. The RF phase errors were only applied to one of the beamlines. The meantolerance and its standard deviation is | ∆ E ML | = (0 . ± .
01) GeV . (8)This corresponds to an RF phase error of ± . ◦ in the ML cavities, which have an RFfrequency of 12 GHz. 90% of colliders have a tolerance greater than ± .
17 GeV, which isequivalent to an RF phase error of ± . ◦ in the ML cavities. In CLIC, the RF phasestability is determined by the arrival time of the drive beam, which has a demonstratedstability of (0 . ± . ◦ [27]. Magnetic field ripples arise from power supply ripples. In simulations, a relative RMSerror σ B /B was applied to the strength of every quadrupole. A tolerance was chosenfor each section as the relative RMS error that results in a luminosity loss of less than1 . × cm − s − (1% of the nominal luminosity target). These tolerances are presentedin Table 4. The BDS, particularly the final doublet, has the tightest requirements. Thetightest tolerances for CLIC are similar to those found in the Large Hadron Collider [28]. This section describes models used to simulate ground motion and the mitigation systemsused in CLIC to limit luminosity loss. 12
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Collider Number . . . . . . | ∆ E M L | [ G e V ] Figure 9:
ML beam energy error | ∆ E ML | that corresponds to a luminosity loss of 1 . × cm − s − vs collider number for 100 tuned colliders with static imperfections. Collidersare ordered in ascending energy error. Section σ B /B RTML 10 − ML 10 − BDS (Excluding FD) 10 − FD 10 − Table 4: Quadrupole ripple tolerances σ B /B for specific sections. FD is the final doublet. Ground motion is modelled as a set of travelling waves with differing wavelength andfrequency. The amplitude of these waves is determined by a 2D power spectral density(PSD) [29]. There are several models which specify a 2D PSD [6, 29]. Ground motionmodel D is studied in this work. Model D represents a higher level of ground motion thanCLIC is expected to experience. It is based on measurements at SLAC [30], Fermilab [31]and in the CMS detector cavern [32].The 1D PSD of model D is shown in Fig. 10. There are two broadband peaks inthis PSD. One at 0.14 Hz, which arises from ocean waves, and another at approximately20 Hz, which arises from technical equipment in the accelerator tunnel. The correlationof ground motion at different locations is shown in Fig. 11. Low frequencies have a highcorrelation across large distances, whereas high frequencies are only correlated over shortdistances. There are two systems that are essential to mitigate the impact of ground motion: abeam-based feedback system and a quadrupole stabilisation system. These are describedbelow.
The beam-based feedback system aims to correct the beam offset. As described Also known as model B10. − f [Hz] − − − − − − − − − P ( f ) [ m H z ] Figure 10:
PSD P ( f ) vs frequency f of ground motion model D. − f [Hz] − . − . − . − . . . . . . C ( f ) L = 1 m L = 10 m L = 100 m L = 1000 m Figure 11:
Correlation C ( f ) vs frequency f of ground motion model D for different separations L . in [29], the average impact of a beam-based feedback system can be estimated by applyinga transfer function T ( f ) to the ground motion PSD to give an effective PSD, P eff ( f ) = | T ( f ) | P ( f ) , (9)which is then used in simulations to generate the ground displacement. This approachassumes that a perfect correction is applied by the feedback system and that the transferfunction depends only on the frequency, i.e. the feedback system has the same effectacross the entire accelerator. This is a simplification, however if the feedback control isdesigned well, this is a good approximation [33, 34, 35].The transfer function of the beam-based feedback system used in CLIC is shown inFig. 12. The feedback system is effective for mitigating low frequencies, below 1 Hz. Itamplifies frequencies in the range 4-25 Hz. The repetition frequency of the CLIC beam is50 Hz. Therefore, dynamic imperfections at harmonics of 50 Hz appear static to the beamand the transfer function for the beam-based feedback system is zero. The quadrupole stabilisation system is described in [36]. This is an active sys-tem which reduces the quadrupole motion. The impact of the quadrupole stabilisation14 − f [Hz] . . . . . . . . T ( f ) Figure 12:
Transfer function T ( f ) vs frequency f for the beam-based feedback system used inCLIC. system is included with the transfer function shown in Fig. 13. The quadrupole stabilisa-tion system is effective for suppressing high-frequency ground motion, above 10 Hz. Lowfrequency, long wavelength motion is not harmful to the beam. Therefore, the quadrupolestabilisation system was designed to have a transfer function of unity for low frequencies. − f [Hz] − T ( f ) Figure 13:
Transfer function T ( f ) vs frequency f for the quadrupole stabilisation system [36]. Ground motion model D was simulated with 100 tuned colliders with static imperfections.The impact of the beam-based feedback system and quadrupole stabilisation system wasincluded in these simulations. Figure 14 shows the luminosity of the 100 colliders. Themean luminosity and its standard deviation is L = (2 . ± . × cm − s − . (10)This is a luminosity loss of approximately 0 . × cm − s − compared to the meanluminosity achieved with static imperfections (Eq. (6)).15
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Collider Number . . . . . . . . L [ c m − s − ] × Without Ground MotionWith Ground Motion
Figure 14:
Luminosity L vs collider number for 100 tuned colliders with static imperfections:with ground motion (orange square) and without ground motion (blue circle). Collider areordered in ascending luminosity using the colliders without ground motion. Figure 14 shows colliders with a higher luminosity suffer from larger luminosity lossesdue to ground motion. This reflects the fact that colliders with higher luminosities havesmaller IP beam sizes and a larger disruption, which makes them more sensitive to lumi-nosity loss due to a beam-beam offset. A luminosity above 2 . × cm − s − is achievedby 90% of colliders with ground motion. Stray magnetic fields are external dynamic magnetic fields experienced by the beam.Tolerances for stray fields in CLIC at 380 GeV are presented in [37, 38]. Stray fields canbe divided into three classifications: • Natural: stray fields from non-man-made sources. E.g. the Earth’s magnetic field. • Environmental: stray fields from man-made objects that are not elements of CLIC.E.g. the electrical grid (power lines, sub-power stations) or transport infrastructure(trains, trams, cars, etc.). • Technical: stray fields from elements of CLIC.Measurements of each type are described in [38, 39, 40].Unfortunately, no realistic model exists for stray fields from technical sources. This isbecause the spatial distribution of technical sources is a priori unknown. However, strayfields from natural sources exhibit a coherent variation across the length scale of CLIC.Such stray fields can be modelled with a fixed spatial profile across the entire beamline.
Stray fields from natural sources are discussed in [41]. One of the worst case naturalstray field is from a geomagnetic storm, which arises from an interaction of the Earth’smagnetic field and solar wind [41]. 16 representative geomagnetic storm was measured on the 8th June, 2014 in Tihany,Hungary [42]. This location has a similar magnetic environment to CERN. The orientationof the sensor and geometry of CLIC were used to calculate the component of the strayfield in the horizontal and vertical direction with respect to the beam. The PSD of thestray field in each direction is shown in Fig. 15. There is one broadband peak at a lowfrequency. . . . . . . . . . f [Hz] − − − − P ( f ) [ n T H z ] HorizontalVertical
Figure 15:
PSD P ( f ) vs frequency f of the stray field during a geomagnetic storm in thehorizontal (solid blue) and vertical (dot-dashed orange) direction with respect to the beam. A stray field can be impeded by a beam pipe. A reasonable model for a CLIC beampipe is a steel cylinder with a 1 cm inner radius, 1 mm thickness and a 10-100 µ m innercopper coating. High-frequency stray fields can induce eddy currents in the beam pipe,which will generate magnetic fields that oppose the external field, thus shielding the beam.However, a 10-100 µ m copper coating will only be effective at shielding frequencies in thekHz range. As stray fields from geomagnetic storms are at much lower frequencies, thebeam pipe will not prevent the stray field from reaching the beam. Collider Number . . . . . . . . L [ c m − s − ] × Without Geomagnetic StormWith Geomagnetic Storm
Figure 16:
Luminosity L vs collider number for 100 tuned colliders with static imperfections:with a geomagnetic storm (orange square) and without a geomagnetic storm (blue circle). Col-lider are ordered in ascending luminosity using the colliders without a geomagnetic storm.
17 geomagnetic storm was simulated in 100 tuned colliders with static imperfections.The direction of the beam (as described by Fig. 2) was taken into account when calculatingthe kick from the stray field. The impact of the beam-based feedback system was included.Figure 16 shows the luminosity of the 100 colliders. The mean luminosity and 90%threshold is virtually unaltered by the geomagnetic storm. This is because the impact ofthe geomagnetic storm can be effectively corrected with the beam-based feedback system.
Performing integrated simulations has been a major step forward to realistically study thebeam dynamics in CLIC. Symmetric beamlines for the electron and positron beams weresimulated in this work. This is a simplification because the electron and positron beamshave different RTMLs. A future step is to include the correct RTML in simulations totrack the positron beam.The simulations in Sec. 4 show that there is an effective tuning procedure that canmitigate the impact of static imperfections. The reliability and accuracy of the lumi-nosity prediction with static imperfections can be improved by simulating more realisticmodels with less approximations. Furthermore, tuning studies should be performed inthe presence of dynamic errors and with realistic signals.The luminosity estimates with static imperfections are conservative because eachbeamline was tuned independently. The luminosity can be improved by performing two-beam simulations, which tune the final-focus system to maximise the luminosity for aparticular beamline pair.In this paper, we performed single-bunch simulations including short-range wakefieldsin the accelerating cavities. The impact of resistive wall wakefields should also be includedin the simulation. The impact of long-range wakefields leads to bunches in the trainhaving different offsets. The mean offset of a train is optimised during tuning, whichmeans a relative offset between colliding bunches is possible. Long-range wakefields mustbe simulated to examine their impact on luminosity.In this paper, we have focused on the impact of dynamic imperfections on the short-term luminosity stability. However, there are also processes that impact the long-termluminosity stability. The most important long-term dynamic imperfection is the drift ofaccelerator elements due to slow ground motion, which is usually modelled using the ATLlaw [43]. Usually, luminosity loss from ATL motion can be fully recovered by performingbeam-based alignment. The impact of ATL motion in the ML and the final-focus systemhas been studied in [13] and [12] respectively. The long-term stability of the collidershould be a future study.
The nominal luminosity target for CLIC is 1 . × cm − s − . In the case withoutimperfections, integrated simulations show that a luminosity of 4 . × cm − s − canbe achieved. This is almost three times the nominal luminosity target.Implementing static imperfections and simulating beam-based tuning, integrated sim-ulations of 100 colliders show that an average luminosity of 3 × cm − s − can beachieved, which is twice the nominal luminosity target. 90% of colliders achieve a lumi-nosity above 2 . × cm − s − . Expressed as a percentage of the nominal luminosity18arget this is 157%. Therefore, there is a margin of up to 57% or greater for dynamic im-perfections. This surplus also gives a margin for unforeseen processes which may impactthe luminosity.At the 90% threshold the luminosity with static imperfections and ground motion is2 . × cm − s − , which expressed as a percentage of the nominal luminosity target is153%. The luminosity loss from stray magnetic fields is negligible. If other dynamic effectssuch as beam jitter, RF phase errors, etc. are kept within their tolerance, their impactwill be on the percent level. Therefore, a significant luminosity surplus of approximately50% or greater can be expected for CLIC. A luminosity of 2 . × cm − s − is achievedby the average collider including static imperfections and ground motion. Expressed asa percentage of the nominal luminosity target, this is 187%, which is almost twice thetarget. References [1] T. Nakada, CERN Report No. CERN-ESU-003, 2013.[2] A. Robson, P. N. Burrows, N. Catalan Lasheras, L. Linssen, M. Petric, D. Schulte,E. Sicking, S. Stapnes, and W. Wuensch, arXiv:1812.07987.[3] P. Roloff, R. Franceschini, U. Schnoor and A. Wulzer, arXiv:1812.07986.[4] P. N. Burrows, N. Catalan Lasheras, L. Linssen, M. Petriˇc, A. Robson, D. Schulte,E. Sicking, and S. Stapnes, CERN Report No. CERN-2018-005-M, 2018.[5] M. Aicheler, P. N. Burrows, N. Catalan Lasheras, R. Corsini, M. Draper, J. Osborne,D. Schulte, S. Stapnes, and M. J. Stuart, CERN Report No. CERN-2018-010-M,2018.[6] M. Aicheler, P. N. Burrows, M. Draper, T. Garvey, P. Lebrun, K. Peach, N. Phinney,H. Schmickler, D. Schulte, and N. Toge, CERN Report No. CERN-2012-007, 2012.[7] H. Abramowicz, et al., Euro. Phys. J. C , 475 (2017).[8] H. Abramowicz, et al., J. High Energy Phys. , 3 (2019).[9] F. Plassard, A. Latina, E. Marin, R. Tom´as, and P. Bambade, Phys. Rev. ST Accel.Beams Proceedings of the 8th InternationalParticle Accelerator Conference, Copenhagen, Denmark, 2017 , (JACoW, Geneva,2017), p. TUPIK099.[11] N. Blaskovic Kraljevic, and D. Schulte, CERN Report No. CERN-ACC-2018-0053,2018.[12] J. ¨Ogren, A. Latina, R. Tom´as, and D. Schulte, Phys. Rev. ST Accel. Beams Proceedings of the 10th International Par-ticle Accelerator Conference, Melbourne, Australia, 2019 , (JACoW, Geneva, 2019),p. MOPMP017. 1914] R. M. Bodenstein, P. N. Burrows, F. Plassard and J. Snuverink, in
Proceedings of the7th International Particle Accelerator Conference, Busan, Korea, 2016 , (JACoW,Geneva, 2016), p. WEPOR009.[15] D. Schulte, CERN Report No. CERN-OPEN-2003-014, 2003.[16] C. Gohil, DPhil thesis, University of Oxford, 2020.[17] The tracking code PLACET, https://clicsw.web.cern.ch/clicsw .[18] D. Schulte, Ph.D. thesis, University of Hamburg, 1997.[19] H. H. Braun, R. Corsini, L. Groening, F. Zhou, A. Kabel, T. O. Raubenheimer, R.Li, and T. Limberg, Phys. Rev. ST Accel. Beams , 124402 (2000).[20] P. Raimondi and A. Seryi, Phys. Rev. Lett. , 3779 (2001).[21] N. Galindo Munoz, N. Catalan Lasheras, S. Zorzetti, M. Wendt, A. Faus Golfe,and V. Boria Esbert, in Proceeding of the 4th International Beam InstrumentationConference, Melbourne, Australia, 2015 , (JACoW, Geneva, 2015), p. TUPB054.[22] Y. Han, A. Latina, L. Ma and D. Schulte, J. Instrum. , P06010 (2017).[23] W. H. Press, W. T. Vetterling, and S. Teukolsky, Numerical recipes 3rd edition: Theart of scientific computing (Cambridge University Press, 2007).[24] R. J. Apsimon, D. R. Bett, N. Blaskovic Kraljevic, R. M. Bodenstein, T. Bromwich,P. N. Burrows, G. B. Christian, B. D. Constance, M. R. Davis, C. Perry, and R. Ramji-awan, Phys. Rev. Accel. Beams , 122802 (2018).[25] J. Resta-L´opez, P. N. Burrows and G. Christian, J. Instrum. , P09007 (2010).[26] D. Schulte and R. Tomas, in Proceedings of the 9th International Particle Accel-erator Conference, Vancouver, BC, Canada, 2018 , (JACoW, Geneva, 2018), p.TH6PFP046.[27] J. Roberts, P. Skowronski, P. N. Burrows, G. B. Christian, R. Corsini, A. Ghigo, F.Marcellini, and C. Perry, Phys. Rev. Accel. Beams , 011001 (2018).[28] O. S. Br¨uning, P. Collier, P. Lebrun, S. Myers, R. Ostojic, J. Poole, and P. Proudlock,CERN Report No. CERN-2004-003-V-1, 2004.[29] A. Seryi and O. Napoly, Phys. Rev. , 5323 (1996).[30] C. Adolphsen, et al., SLAC Report No. SLAC-R-474, 1996.[31] B. Baklakov, T. Bolshakov, A. Chupyra, A. Erokhin, P. Lebedev, V. Parkhomchuk,Sh. Singatulin, J. Lach, and V. Shiltsev, Phys. Rev. ST Accel. Beams , 031001(1998).[32] A. Kuzmin, Tech. CERN Report No. EDMS:1027459, 2009.[33] J. Pfingstner, Ph.D. thesis, Vienna University of Technology, 2013.2034] G. Balik, B. Caron, D. Schulte, J. Snuverink, and J. Pfingstner, Nucl. Instrum.Methods A , 163-170 (2013).[35] J. Pfingstner, J. Snuverink, and D. Schulte, Nucl. Instrum. Methods A , 71-78 (2010).[37] C. Gohil, D. Schulte, and P. N. Burrows, CERN Report No. CERN-ACC-2018-0052,2018.[38] C. Gohil, M. C. L. Buzio, E. Marin, D. Schulte, and P. N. Burrows, in Proceedingsof the 9th International Particle Accelerator Conference, Vancouver, BC, Canada,2018 , (JACoW, Geneva, 2018), p. THPAF047.[39] C. Gohil, N. Blaskovic Kraljevic, P. N. Burrows, B. Heilig, and D. Schulte, in
Proceed-ings of the 10th International Particle Accelerator Conference, Melbourne, Australia,2019 , (JACoW, Geneva, 2019), p. MOPGW081.[40] E. Marin, D. Schulte, and B. Heilig, in
Proceedings of the 8th International ParticleAccelerator Conference, Copenhagen, Denmark, 2017 , (JACoW, Geneva, 2017), p.MOPIK077.[41] B. Heilig, C. Beggan, and J. Lichtenberger, CERN Report No. CERN-ACC-2018-0033, 2018.[42] B. Heilig (private communication).[43] V. D. Shiltsev, in