High intensity beam dynamics assessment and challenges for HL-LHC

The High Luminosity (HL-LHC) project aims to increase the integrated luminosity of CERN's Large Hadron Collider (LHC) by an order of magnitude compared to its initial design. This requires a large increase in bunch intensity and beam brightness compared to the first three LHC runs, and hence poses serious collective-effects challenges, related in particular to electron cloud, instabilities from beam-coupling impedance, and beam-beam effects. Here, we present the associated constraints and the mitigation measures proposed to achieve the baseline performance of the upgraded LHC machine. We also discuss the interplay of these mitigation measures with other aspects of the accelerator, such as optics, physical and dynamic apertures, the collimation system, and crab cavities. Additional potential sources of intensity limitations are also briefly discussed.


Introduction
HL-LHC operation is based on a levelled luminosity that reaches 5 × 10 34 cm −2 s −1 [1,2] to be able to integrate 250 fb −1 per year of proton-proton luminosity in ATLAS and CMS [2].In terms of beam properties, the luminosity L ∝  b  2 n depends mainly on the number of colliding bunches  b , the number of protons per bunch , and the normalised transverse emittance  n [1], which is assumed to be the same in the horizontal and vertical planes.
A high brightness / n of the initially injected beam, and its preservation throughout the machine cycle, are obviously critical ingredients to maximise luminosity.In view of HL-LHC, the entire LHC injector chain has undergone a series of improvements through the LHC Injectors Upgrade (LIU) project [3].LIU was designed to provide HL-LHC with  = 2.3 × 10 11 protons per bunch (p + /b) within an emittance of approximately 2.1 µm.In 2023, nominal trains with up to 2.2 × 10 11 p + /b within about 2 µm have been accelerated to 450 GeV/ in the CERN Super Proton Synchrotron (SPS), i.e. just before extraction into the LHC [4,5].The preservation of brightness in the HL-LHC era is a subject in itself and will not be specifically addressed in this article -we simply note that, based on previous experience, from injection to collision, beam losses are assumed to remain below 5% and an emittance blow-up of 20% is assumed (i.e. n =2.5 µm is foreseen at the start of collisions, as the HL-LHC target that takes into account some blow-up at injection, during the energy ramp, and finally when the separation between the two beams is collapsed), while experience in the ongoing Run 3, which started in 2022, points to an emittance growth of 30% [6].HL-LHC beam parameters are summarised in table 1, together with their 2023 counterparts.
The main focus of this paper will be on two crucial beam-dependent parameters that drive machine performance, namely, the bunch intensity and the number of bunches per beam.We will -1 -Table 1. Beam parameters at the start of collisions (i.e. at the end of the collapse process) for HL-LHC [7, table 12]), compared to those for the LHC in 2023 [6], after commissioning and intensity ramp up.HL first discuss the limitations arising from electron-cloud (or e-cloud) effects, in particular, in terms of the number of bunches, and their mitigation measures.Then we will turn to transverse impedance and stability, detailing the constraints imposed by the collimation system, the crab cavities and the dynamic aperture, as well as possible mitigation measures.Beam-based measurements will also be shown, before providing the global picture about beam stability.Subsequently, several additional considerations with impact on intensity reach will be listed, and our conclusion will follow.

The electron-cloud challenge
Since the first LHC runs, the electron-cloud effect was found to significantly affect machine performance [9][10][11][12], essentially through its impact to the heat load on the magnet beam screens, which have to be maintained at a temperature of 20 K through an active cooling system whose capacity is limited.
In addition, the electron cloud may also be responsible for emittance blow-up and instabilities.The e-cloud is generally a self-healing phenomenon, i.e. it is gradually mitigated through a progressive conditioning of the inner surface of the vacuum pipe (or that of the beam screens, in the case of the superconducting magnets).Surface conditioning occurs during operation, as well as in dedicated runs in the presence of an e-cloud.This beam-induced scrubbing process effectively reduces the secondary electron emission yield (SEY), rapidly at its beginning, but then slows down in an asymptotic way.It stops at a level that may depend on the surface properties and/or on the beam parameters.Unfortunately, LHC conditioning became less and less effective for several sectors after being vented during successive long shutdowns.The reason has been identified in the degradation of the copper-plated surface of the beam screens, related to the unwanted formation of CuO on conditioned surfaces exposed to air [13,14].This oxide increases the effective SEY value obtained after reconditioning in comparison to that of pure copper.As a consequence, the high heat load in the sector most affected, 78 [12,15], has been limiting the number of bunches in LHC during Run 3. The degradation of the heat load from Run 2 (2015-2018) to Run 3 (from 2022 onwards), for sector 78, is illustrated in figure 1, and the slowdown of conditioning during the first year of Run 3 (2022) in figure 2, where the evolution of scrubbing was reconstructed by comparing the fills separated by the same heat load dose, choosing moments with similar intensity and length of the bunches.Therefore, significant further scrubbing is not expected during Run 3.
-2 -  For HL-LHC, several options are being studied to limit the formation of excessive electron cloud, involving an amorphous carbon coating (performed in situ), possibly after CuO reduction by surface treatment [18].Although these could already be partially implemented during long shutdown 3 (LS3, 2026-2029), it is likely that the first run of HL-LHC (Run 4) will still be limited by electron-cloud-induced effects, in particular the heat load, as in Run 3, and possibly also by instabilities.Consequently, several options are considered [16,17] -they are summarised in table 2 with their consequences in terms of the achievable filling scheme.These range from the absence of any limitation (the case of an e-cloud sufficiently mitigated through beam-screen coating) to a strongly degraded situation where the SEY has intolerably increased in so many cells that the e-cloud must be limited through additional means.This can be achieved thanks to the use of an "8b4e" -3 -filling scheme [19,20], made of short trains of eight bunches with 25 ns bunch spacing separated by four empty slots (see figure 3), whose structure is able to strongly decrease electron multipacting and hence e-cloud effects, at the cost of a reduction in the number of bunches by almost 30%.In between these two extremes, hybrid schemes (containing 8b4e units and standard 25 ns trains) are envisaged for intermediate situations, i.e. moderate degradation during LS3 with a partial coating of the beam screens, or status quo with respect to Run 3.These schemes are tailored to make the heat load comply with the hard limit given by the cooling capacity of the cryogenic system.On top of that, vertical instabilities during collisions also need to be considered [21] -these are related to a degradation of the stability situation, triggered on one side by the decrease of the Landau damping from the beam-beam head-on tune spread, consequence of the burn-off, and on the other side by an instability maximum occurring for bunch intensities just below 10 11 p + /b.In 2022, these actually proved to be worse than in previous runs [22] and were not fully suppressed by conditioning, as was the case in Run 2. This means that if the e-cloud is not strongly mitigated (by using the full 8b4e scheme), chromaticity will have to be maintained high during collisions ( ′ > 15) to avoid such instabilities, although this has consequences on the dynamic aperture (see below).Table 2. Several filling scheme options considered to mitigate the heat load from e-cloud effects depending on the beam screen surface treatment and/or degradation during LS3.We indicate the corresponding limitation in number of bunches per beam [17], the requirement in terms of chromaticity during collisions ( ′ ) and the overall impact on integrated luminosity.In all cases, a bunch intensity of 2.3 × 10 11 p + /b is assumed.A possibility to go beyond this number in the 8b4e case, is currently being explored.In the LHC injectors, the pure 8b4e scheme is currently being tested, and a bunch intensity of 2.15 × 10 11 p + /b has already been reached in the SPS at top energy [4].
-4 -In the above considerations, only the limitations from heat load and beam instabilities in collisions are considered.In principle, additional issues could arise at injection, such as instabilities and emittance growth, which are under scrutiny; for the latter, a mitigation with new optics was implemented during the 2023 operation [23,24].

Transverse impedance
The transverse impedance is one of the sources of bunch intensity limitations in the current LHC machine at the top energy [22,27], its dominant contribution being the collimation system.It is also the case in HL-LHC, as can be seen in figures 4 and 5, but crab cavities are new and potentially high contributors that need to be carefully evaluated [28].On top of these, dynamic aperture (DA) is a subject of concern, as it can significantly limit the octupole current used to stabilise the beam at flat top before collisions are established.In this section, we shall hence describe the parameter space available for each of these factors, before discussing the level of understanding of the current LHC impedance, and finally assessing the transverse stability of HL-LHC.

Collimation system
In the LHC, the collimation system protects the magnets and other sensitive elements of the machine from halo particles, or in general from any particle that starts deviating strongly from the nominal orbit.Primary (TCP) and secondary (TCS) collimators were originally made of poorly conducting carbon-reinforced carbon (CFC), which represents a trade-off between robustness and impedance considerations.To reduce the large impedance of these collimators, an upgrade programme has been launched for HL-LHC.In particular, four TCS per beam installed in insertion region 7 (IR7) were replaced during long shutdown 2 (LS2, 2019-2021) by low-impedance counterparts (TCSPM) with molybdenum graphite (MoGr) jaws coated with 5 µm of molybdenum metal [29][30][31][32].Moreover, two TCP per beam were replaced by higher conductivity MoGr ones (TCPPM) [31].Furthermore, during LS3, five other secondary collimators in CFC will be exchanged with even lower impedance ones featuring copper-coated isostatic graphite jaws and taperings [33].The decrease of the total dipolar impedance (horizontal and vertical) after the two successive upgrades is illustrated in figure 6.
For the IR7 collimators, two sets of settings are currently being considered at the top energy: tight and relaxed [34].Although tight settings are similar (slightly larger) than those successfully deployed in LHC Run 2 (2015-2018) and 3 (from 2022 onwards), relaxed settings were introduced to decrease transverse impedance [7,35,36] at the cost of reducing the margins for the  * reach and cleaning efficiency.Both configurations are summarised in table 3, for two possible end-of-levelling  * .The aperture of the tertiary collimators (TCT) at interaction points (IP) 1 and 5 is also given, as it drives the cold-protected aperture (1 larger than the TCT gaps).To respect the collimator hierarchy, the TCT gaps must be opened at least by 1 more than the secondary collimators, and must be larger than those of the single-jaw absorbers (TCDQ) in the dump region (IR6) that protects downstream machine elements from asynchronous beam dump failures [37,38].Aperture bottlenecks in the triplet region of HL-LHC are also given as an interval ranging from the ideal configuration to the worst-case scenario, assuming mechanical, alignment, and optical imperfections [39].
Even in the worst-case scenario and for relaxed settings, the triplets are well within the protected aperture.Note that for flat optics, i.e. smaller  * in the separation plane than in the crossing plane, the situation requires more studies (in particular, for  * = 7.5 cm in the separation plane [38]).
-5 -  3) as a function of frequency, for the horizontal dipolar impedance, real part (top) and imaginary part (bottom).The frequencies corresponding to the HL-LHC bunch spectrum lie in the GHz range.Reproduced with permission from [25].
In addition to aperture protection, collimator settings are driven by considerations of cleaning efficiency (including the cleaning of halo-induced backgrounds to experiments), and tight settings are clearly beneficial in that respect.However, a highly populated halo could damage collimators in case of fast failures and affect the machine availability through loss spikes in the betatron collimation system.For these aspects, the relaxed settings could be favorable [40,41], especially without the hollow electron lens (HEL) that has been removed from the HL-LHC project.
Finally, studies are ongoing to further reduce the impedance of the IR7 and IR3 collimators through optics changes and the review of collimator gap settings [42,43] together with global phase-advance optimisations [44].

Crab cavities
The crab cavities (CC) are an essential improvement for HL-LHC [2], their goal being to increase the geometric reduction factor of the luminosity, induced by the crossing angle [1, chap. 4].However, they are detrimental to transverse impedance and stability, as a result of their numerous narrow-band resonant modes and the high  functions in the crabbing plane at the location of the CC.Although high-order modes (HOM) are well under control [45,46], the impedance from the fundamental mode was only recently considered very significant [28,[47][48][49].In the most critical part of the LHC cycle in terms of impedance-related stability, i.e. right before collisions are established at top energy, the CC fundamental mode would lead to a dramatic increase of the octupole current required to stabilise the beam through Landau damping [50], even with the standard RF CC feedback.Several -7 - -8 -Table 3. Collimator settings [34] (in  units, with the convention  n =2.5 µm), in IR6, IR7 and for the TCT close to ATLAS and CMS, for two different  * , together with protected aperture in the triplets and aperture bottleneck, given as a range (all in number of ) [38].The apertures margins require experimental validations (at the start of the first HL-LHC run), specially for the cases with relaxed settings.1. Switch off the cavities and detune the resonant frequency between unstable betatron lines (and switch them back on only when collisions are established).This option needs further evaluation, especially after the observation made during dedicated machine development studies in the SPS of transient effects when cavities and RF feedbacks are switched on with the circulating high-intensity beam [51].

Relaxed
2. Use a standard RF feedback [52] that will broaden and reduce the height of the peak from the fundamental mode, together with optics modifications to decrease the  function in the crabbing plane at the CC location, as the gain from this feedback is limited [53, slide 4].In particular, moving to flat optics, with a higher  * in the crossing plane than in the separation plane (e.g.2.8 m vs. 0.7 m at flat top [44]), thus lowering  at the CC location in the crabbing plane would help [28].
3. Use a RF feedback together with a betatron comb filter to specifically reduce impedance at betatron frequencies, similar to what is done for RF cavities to mitigate their longitudinal fundamental mode [54,55].Nevertheless, a large uncertainty remains in the case of the betatron comb filter: as the betatron lines are offset by the tune, one needs to ensure that the tune remains within a given bandwidth.Hence, tune jitter or bunch-by-bunch tune variations (from collective effects) are a matter of concern and are being studied.
Currently, the third option (betatron comb filter) appears to be the most promising mitigation of the fundamental mode impedance, as it decreases the additional octupole current needed by 80 %, vs. 60 % for the second option, with flat optics [28].

Dynamic aperture considerations
LHC octupoles are used to damp instabilities through the Landau damping mechanism, and can be set to a current as large as 570 A [27].In HL-LHC, such a high octupole current may become a limiting factor in reaching the DA target, particularly in the critical phase when beams are brought into collisions -few beam dumps related to losses have already been observed in this phase during Run 3 [57, slide 39].Indeed, at that point of the cycle, the octupoles are strongly powered because they are needed in the preceding phase (flat top), where they are the main source of Landau damping, and their current cannot be decreased very rapidly.At the same time, the beam-beam head-on tune spread will increase rapidly, while the separation is reduced towards zero.In HL-LHC, DA issues during collisions are enhanced with respect to the LHC, as a consequence of beam-beam effects due to a higher brightness.In addition, the interplay between magnet imperfections and a large crossing angle should be carefully considered.Finally, an additional constraint is the need for a minimum tune separation of 5 × 10 −3 (above the diagonal), to avoid loss of transverse Landau damping related to the  −  coupling [58] (assumed to be such that | − | ≤ 10 −3 ).
In the aforementioned end-of-collapse phase, with as much as 460 A in the octupoles,  * =1 m,  = 2.3 × 10 11 p + /b and  n =2 µm (anticipating a very good emittance preservation), a DA marginally above 6 can be found for only a few working points [7, figure 4], with the baseline filling scheme (see table 2).The DA situation is better with negative octupole current, with the 8b4e scheme [59], or with a smaller chromaticity (e.g. ′ = 5 instead of 15) [60].As an illustration, the impact of the filling scheme and octupole current on DA, in the flat-top phase, is shown in figure 7.
There are various optics strategies to mitigate the impact from Landau octupoles on DA, or in general to increase DA.Optimisation of phase advance, between IP 1 and 5 or even between arcs, as done at injection, can improve the situation [24,38,61].Within this strategy, the octupole strengths could also be optimised arc-by-arc.Another possibility would be the use, with optimised phase advances, of additional octupolar correctors, such as those from the interaction regions [35, Appendix A], or the octupole spool pieces from the arcs, although their impact is limited by the relatively high  * at the end of collapse for the first and by their intrinsically small strengths for the second.

Comparison of the impedance model with LHC beam-based measurements
Studies have been performed since the first LHC run to compare the impedance model with its effects in the machine, in terms of tune shifts, growth rates, and octupole thresholds (i.e. the minimum octupole current necessary to stabilise the beam through Landau damping), gradually improving the measurement procedure over the years [27,[68][69][70][71][72][73].
The impedance model was also refined over time, in particular that of individual collimators, which currently contains both the large resistive-wall contribution of the jaws [69], and the impedance of geometric features such as small angle taperings [74], including their resistive-wall contribution [33].Finally, for low-impedance primaries and secondaries, the geometric impedance now provides a relatively important contribution compared to the resistive-wall part which has been considerably reduced; hence, recently significant effort was made to further refine their model, ultimately using electromagnetic simulations and measurements [75][76][77].
The latest studies show reasonable agreement between simulations and measurements, in terms of intensity-dependent tune shifts [73] and tune shifts from individual collimators [72,77].Additional measurements of growth rates and octupole thresholds were performed in 2022 and are compared with -10 -     Minimum DA ( ) simulations using the impedance model in figures 8 and 9, showing reasonable agreement as well.In the latter case, the effect of noise (detrimental to beam stability [67]) is removed by quickly decreasing the octupole current until it reaches the threshold, hence the lower threshold values compared to the -11 - operational case where latency effects have to be taken into account (see next section).Also, the behaviour at high chromaticy  ′ is still under study -recently a good agreement with measurements was obtained, as visible in figure 9, by including the non-linearity of the RF bucket in the simulations.
In general, we observe a better stability of the real machine with respect to simulations, which could be the symptom of non-linearities that are not yet considered in the model (from e.g.space charge, quadrupolar impedance, or lattice non-linearities).

Transverse stability limits
The situation of transverse stability, in terms of the minimum octupole current needed to stabilise the beam at the flat-top energy (before collisions), is summarised in figure 10, where we explore the most relevant options described above, in terms of collimator settings and mitigation of the fundamental mode of crab cavities.Positive octupole polarity is assumed and a wide chromaticity range is explored due to the uncertainty on  ′ .Negative octupoles are more effective, in principle, but the tune spread is partly compensated for by the long-range beam-beam effects; hence, further studies are needed.Note that, contrary to previous estimates [7], transverse Gaussian tails are taken into account, after HEL descoping2 and studies in the injectors during Run 3 indicating that beams sent to LHC have significant tails [78].This reduces the threshold by up to 20 % [79].Also, note that latency effects [67,80] are taken into account.The octupole threshold with the RF comb filter at the most unstable chromaticity in the range 10 <  ′ < 20 is 364 A and 443 A for relaxed and tight collimator settings, respectively.With a standard RF feedback and flat optics, we get 419 A for relaxed settings and 495 A for tight ones3.Therefore, all cases are compatible with DA (see above), although tight settings do not provide much margin.
Note that the transverse mode coupling instability (TMCI) threshold is not an issue for HL-LHC: the latest estimates put it around 6 × 10 11 p + /b (in single bunch) [31]; moreover TMCI is strongly attenuated by transverse feedback [82] and thus does not have to be considered as a hard limit.
The octupole thresholds presented in figure 10 are computed for an energy of 7 TeV.The option of going to 7.5 TeV would a priori increase the threshold by ∼7 % as long as the collimator settings remain unchanged in mm, as assumed in ref. [83].

Additional considerations 4.1 Local heating in sensitive devices
Heat load is systematically checked for any new device added to the machine, but non-conformities may be present.In particular, after an incident on an RF vacuum module (A4L1) in 2023, several RF vacuum modules have been found to be non-conforming and will be replaced [84]; intensity has been limited to 1.6 × 10 11 p + /b, and studies are ongoing to determine the role of impedance in the problem.
3In the flat optics case, octupoles have been rescaled to a telescopic index (see [81] for a definition) of one for comparison purposes, as these optics would otherwise induce a telescopic index that strongly lowers the octupole threshold.

Limitations on the RF power
New high-efficiency klystrons are needed to achieve the HL-LHC baseline and hybrid filling schemes, to cope with the strong injection transients and the high average power required, which is beyond the capabilities of the present system [85].The exact limitations of the current system and its upgrade are still being studied.

Beam-beam wire compensation
The wire could be used to gain a margin on the DA during collisions, in particular if the TCT can be moved to maintain a constant gap in  during luminosity levelling [86], as done in 2023.On the other hand, before collisions,  * is much higher, which reduces the potential gain in DA.

Conclusions
The limitations in number of bunches and bunch intensity have been reviewed, in particular those stemming from electron-cloud and impedance effects.Depending on the state of the beam screen surface after LS3, several options in terms of number of bunches and filling schemes are envisaged, from baseline to 8b4e, via an intermediate hybrid scheme.In terms of the bunch intensity limit, the baseline is achievable, but the octupole currents, and hence the dynamic aperture and lifetime during the separation collapse, will ultimately depend on the collimator settings chosen and the mitigation strategy used to reduce the impedance contribution of the crab cavities.

Figure 1 .
Figure 1.Evolution of the heat load along all half-cells of sector 78, between the end of Run 2 (2018) and the beginning of Run 3 (2022), with similar beam parameters.Reproduced with permission from [16].

Figure 2 .
Figure 2. Evolution of the heat load for all sectors during 2022.Fills along the year were chosen with similar bunch length and intensity (parameters are provided below the plot).Reproduced from[17].CC BY 4.0.

Figure 4 .
Figure 4. Breakdown of the relative contributions to the HL-LHC impedance (relaxed settings case, see table3) as a function of frequency, for the horizontal dipolar impedance, real part (top) and imaginary part (bottom).The frequencies corresponding to the HL-LHC bunch spectrum lie in the GHz range.Reproduced with permission from[25].

Figure 5 .
Figure 5. Breakdown of the relative contributions to the HL-LHC impedance (relaxed settings case, see table3) as a function of frequency, for the vertical dipolar impedance, real part (top) and imaginary part (bottom).The frequencies corresponding to the HL-LHC bunch spectrum lie in the GHz range.Reproduced with permission from[26].

Figure 6 .
Figure 6.LHC and HL-LHC horizontal (top) and vertical (bottom) dipolar impedances, at flat top, for various stages of the collimation system upgrade.For HL-LHC, relaxed settings (see table3) are assumed, and the crab-cavities (see section 3.2) are not included.

Figure 8 .
Figure 8. Growth rates measured at injection during a dedicated experiment in the LHC in 2022, compared with simulations performed with the DELPHI Vlasov solver [62, 63] using the LHC impedance model.

Figure 9 .
Figure 9. Octupole thresholds measured at flat top during a dedicated experiment in the LHC in 2022, compared with macroparticle simulations, performed with PyHEADTAIL[65,66] using the LHC impedance model.In this experiment the octupole current was decreased in a fast way, which effectively removes latency effects related to noise[67], hence the lower threshold compared to standard operation.Reproduced with permission from[64].