CERN-ATS HiLumi LHC. FP7 High Luminosity Large Hadron Collider Design Study PUBLICATION

CERN-ATS-2012-290 HiLumi LHC FP7 High Luminosity Large Hadron Collider Design Study PUBLICATION INTRA-BEAM SCATTERING AND LUMINOSITY EVOLUTION FOR HL-LHC PROTON BEAMS MICHAELA SCHAUMANN (RWTH AACHEN & CERN), JOHN M. JOWETT, RODERIK BRUCE (CERN) 22 OCTOBER 2012 GENEVA, SWITZERLAND CERN-ATS-2012-290 10/12/2012 The HiLumi LHC Design Study is included in the High Luminosity LHC project and is partly funded by the European Commission within the Framework Programme 7 Capacities Specific Programme, Grant Agreement 284404. This work is part of HiLumi LHC Work Package 2.4: Intensity Limitations The electronic version of this HiLumi LHC Publication is available via the HiLumi LHC web site <http://cdsweb.cern.ch/record/1498420> CERN-ATS-2012-290

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN ACCELERATORS AND TECHNOLOGY SECTOR CERN-ATS-2012-290 Intra-beam Scattering and Luminosity Evolution for HL-LHC Proton Beams Michaela Schaumann (RWTH Aachen & CERN) John M. Jowett, Roderik Bruce (CERN) Abstract Intra-beam scattering (IBS) in the LHC proton beams will be stronger in the future HL-LHC than at present because of the higher beam intensities, small emittances and new optics. The intensity decay will be due to both IBS and burn-off by the luminosity. We calculate IBS emittance growth rates with MADX and the Collider Time Evolution (CTE) program for two ATS optics versions, ATS-V6.503 and SLHCV3.1b, and different settings of the crossing angle and required corrections. The calculations are done for injection (450 GeV) and collision (7 TeV) energy for various beam conditions. Moreover, CTE simulations of the emittance, bunch length, intensity and luminosity evolution during a fill are presented. Geneva, Switzerland 22 October 2012

Contents 1 Introduction 5 2 ATS Optics 6 3 Intra-Beam Scattering Considerations 8 3.1 IBS Growth Rates at Injection....................... 8 3.2 MADX Calculations at Collision Energy.................. 9 3.2.1 Without Crossing Angles and Separation Bumps......... 9 3.2.2 With Crossing Angles....................... 14 3.3 CTE Simulations at Collision Energy.................... 21 3.4 Comparison of MADX and CTE IBS Growth Rate Determination.... 21 4 CTE Simulations of the Beam Evolution 24 5 Conclusions 26 List of Figures 1 Horizontal and vertical β-function for ATS optics............. 7 2 Horizontal β-function and Dispersion for ATS optics........... 8 3 Local IBS growth rates, ATS-V6.503 optics, β = 0.1m......... 11 4 Local IBS growth rates, ATS-V6.503 optics, β = 0.1m and 11m.... 12 5 IBS growth rates as a function of β, ATS-V6.503 optics......... 13 6 Horizontal H -function, ATS-V6.503 optics, β = 0.1m......... 14 7 IBS growth rates as a function of β, ATS-V6.503 and SLHCV3.1b optics 16 8 Vertical IBS growth rates as a function of β, ATS-V6.503 and SLHCV3.1b optics.................................... 17 9 Local IBS contribution around the ring for β = 0.1m, without correction 18 10 Local IBS contribution around the ring for β = 0.1m, with correction. 19 11 Simulated IBS growth rates evolution................... 22 12 Comparison IBS growth rates from MADX and CTE........... 23 13 Simulated absolute bunch length evolution and its growth........ 24 14 Simulated horizontal emittance evolution and its growth......... 25 15 Simulated vertical emittance evolution and its growth........... 25 16 Simulated intensity evolution and losses.................. 26 17 Simulated debunching and luminosity losses............... 26 18 Simulated evolution of the luminosity per bunch crossing......... 27 3

List of Tables 1 Beam parameters for the HL-LHC..................... 6 2 IBS growth rates at 450 GeV, ATS-V6.503 optics, β = 11m....... 9 3 IBS growth rates at 450 GeV, SLHCV3.1b optics, β = 5.5m...... 9 4 IBS growth rates calculated with MADX at 7 TeV, optics SLHCV3.1b.. 20 5 IBS growth rates at 7 TeV, optics ATS-V6.503 and SLHCV3.1b, β = 0.1m.................................... 21 6 IBS growth rates calculated at 7 TeV, CTE and MADX.......... 23 7 Summary IBS growth rates and beam parameters............. 28 4

1 Introduction For the LHC upgrade, the HL-LHC (High Luminosity Large Hadron Collider) [1], very high bunch currents combined with small emittances are necessary to reach the desired high luminosities. The achievable luminosity L is proportional to the single bunch intensities, N b1 and N b2, of Beams 1 and 2, the revolution frequency f 0 and the number of bunches in the machine k b, and inversely proportional to the β-function at the interaction point (IP) β and the bunch emittances ε: N b1 N b2 f 0 k b L = 2π F = N2 b f 0k b γ βx (ε x1 + ε x2 ) βy (ε y1 + ε y2 ) 4πβ F, (1) ε n where the second equality holds for round beams with equal transverse beam sizes and intensities. The normalised emittance is given as ε n = εγ, with γ as the relativistic Lorentz factor. The factor F describes the geometric luminosity reduction due to the crossing angle [2]: F = 1/ 1 + ( ) θc σ 2 z 2σ. (2) Table 1 quotes the possible beam properties to be considered in the upgrade plans. The official parameters for the LHC high-luminosity upgrade are set by the HiLumi Parameter and Lay-out Committee [3] and can also be found in [4]. The official choice of β is 0.15m in combination with a half-crossing angle of 295 µrad, which differs from the values given in Table 1. In this report β = 0.1m with a corresponding half-crossing angle of 360 µrad was used as a baseline for the calculations to treat the extreme case. Two different bunch spacings (25 ns and 50 ns) should be investigated. Because of the way the beams are produced in the injectors, it is not possible to achieve as good bunch properties for the 25 ns spaced beam as for the 50 ns. A larger bunch spacing, on the other hand, decreases the maximum number of bunches in the machine proportionally reducing the total luminosity. Furthermore, a special 25 ns beam with shorter bunches should be studied. For this beam an additional higher harmonic RF system operating at 800 MHz and providing a peak voltage of 24MV must be installed to modulate the main RF system (400 MHz, 16 MV); with appropriate relative phases, this can be used to provide smaller bunch lengths. Clearly the smallest possible β-function at the interaction point, β, will produce the highest possible luminosity. New optics are currently being developed to provide β values down to 0.1 m. In the following, the so-called ATS (Achromatic Telescopic Squeezing [5]) optics are used for the analysis. This new optic scheme has already been tested during several dedicated machine studies in the LHC [6]. Moreover, while the emittance has a lower limit, the intensity might be further increased to gain more luminosity. However, with increasing intensity, collective effects like beam-beam, space charge, instabilities 5

driven by the machine impedance, electron cloud and intra-beam scattering (IBS) become stronger and will limit the reachable bunch brightness ( N b /ε n ). Bunch Spacing 25 ns 25 ns (short) 50 ns Energy [TeV] E 7 7 7 β-function in IP1 and 5 [m] β 0.1 0.1 0.1 Half-Crossing Angle IP1 & 5 [µrad] α C 360 360 360 Intensity per Bunch [10 11 charges] N b 2.2 2.2 3.5 RMS Transverse Emittance [ µmrad] ε n 2.5 3.0 3.0 RMS Longitudinal Emittance [evs] ε l 2.5 1.4 2.5 RMS Bunch Length [ns] τ l = 4σ z /c 7.5 4.0 7.5 RMS Momentum Spread [10 4 ] p/p 1 1 1 RF Total Voltage [MV] V RF 16 16+24 16 RF Frequency [MHz] f RF 400 400+800 400 Table 1: Beam parameters for the HL-LHC. Three possible beam setups are under investigation. Following the customary 4-sigma convention at the LHC, the bunch length is given in time units as τ l = 4σ z /c where σ z is the RMS bunch length and c the speed of light. The special 25 ns option with short bunches requires a RF system which runs at 24 MV and 800 MHz in addition to the already existing main RF system at 400 MHz and 16 MV. This document studies the effects of IBS on the beam evolution and the limitations that arise for the high brightness bunches. IBS growth rates and the evolution of the emittances, bunch length, intensity and luminosity from CTE (Collider Time Evolution program) [7, 8] simulations and MADX [9] calculations are presented. Reference [10] describes a similar study for the proton beams used in the LHC in 2012. 2 ATS Optics The luminosity can be improved by reducing the β-function at the interaction point (IP), as follows from (1). However, as β at the IP is reduced to small values, the β-function in the inner triplets, which are the first magnets before and after the IP, rises. To approximate the β-function in the inner triplets Equation (3) can be used, which describes the increase 6

of the β-function in a drift space with the distance, s, from a symmetry point, e.g. the IP: β(s) = β + s2 β. (3) Already for the nominal LHC setup (optics version V6.503, squeezing down to β = 0.55m [2]) the β-function in the triplets enters the km regime. For β as small as 10 cm the triplets will reach β > 20km. This will lead to aperture problems, since the beam size σ = εβ at each location in the machine depends on the β-function in that place. To accommodate these beams, new magnets with a larger aperture are required close to the IP. Huge peak field at the coils of the magnets are necessary to provide focusing from several kilometres to a few centimetre considering the large apertures. The ATS (Achromatic Telescopic Squeezing) scheme [5] uses the matching quadrupoles from the neighbouring interaction regions (IR), additionally to those located directly around, for a telescopic squeeze from the neighbouring IP to reduce the required magnet strength in the triplets. As shown in Figure 1, this drastically increases the maximum value of β in the enclosed arcs. Note that the right plot was cut at β x,y = 1000m, but actually reaches about 24 km around the main IPs. The procedure is achromatic, since the effect on the beam of the higher order magnets will be enhanced where the β-function is large. Hence, the arising chromaticity can be corrected more effectively by the sextupoles. Within the sequence of optics that occur in the squeeze, this feature emerges only at β = 0.4m; above that value the optics are basically identical with the standard ones, as currently used in the LHC. Figure 1: Horizontal and vertical β-function for ATS optics around the ring for β = 11m (left) and β = 0.1m (right) in IP1 and 5. The increase of β in the arcs around IP1 and 5 for small β arising from the telescopic squeeze is clearly visible in the right plot. Note that the right plot is truncated at β x,y = 1000m, but actually reaches about 24 km around the IPs. At the time of writing two versions of the ATS optics are available. The first one is still based on the nominal sequence of the LHC (version ATS-V6.503 [11]), and the second 7

version (SLHCV3.1b [12]) already includes a set of new triplet magnets around the IPs, providing larger aperture and higher field strength [13]. In this report, both versions are used for the analysis and the results are compared. However, from the example of the flat machine, without crossing angles or bumps, in Figure 2, one can see that there are only minor differences between their optical functions. The figure shows the β-function (left) and the dispersion (right) around the ring for ATS-V6.503 in blue and SLHCV3.1b in red; in fact the red lines are mostly covered by the blue, so small are the differences. Therefore, we do not expect to find large discrepancies in the IBS analysis of both optics. Figure 2: Comparison between the two ATS optics versions ATS-V6.503 (blue) and SL- HCV3.1b (red). Left: horizontal β-function, right: horizontal dispersion. In both plots the blue line mostly covers the red curve due to the small differences. Note that the left plot is truncated at β x,y = 1000m, but actually reaches about 24 km around the IPs. 3 Intra-Beam Scattering Considerations 3.1 IBS Growth Rates at Injection The IBS growth rates are inversely proportional to the third power of the relativistic gamma factor, thus for the low energy at injection of 450 GeV the effect is drastically enhanced and needs special attention. Assuming the intensity and emittance could be preserved from injection into collisions at 7 TeV, the values from Table 1 are used for the calculations. However, the bunches are injected with a smaller longitudinal emittance ( 1eVs) which is blown up during the ramp. Moreover, the additional RF system, which would be necessary for the 25 ns option with shorter bunches, would also be switched off at injection where only the main RF system is used. Table 2 gives MADX calculations for the ATS-V6.503 optics version at β = 11m and Table 3 for version SLHCV3.1b at β = 5.5m. A perfectly flat machine with zero crossing angles and no orbit bumps was assumed. In this ideal case, the vertical IBS 8

growth rate has a small negative value. Comparison of the tables shows that the two optics produce nearly identical IBS growth rates even with slightly different β values. If the crossing angles and parallel separation bumps are switched on, the differences in the IBS growth rates are still small in the range of 10 4 1/h. The vertical growth rates become positive but they are still of the order of 10 4 1/h. Furthermore, as expected, the growth rates increase with increasing intensity and decreasing emittance. ATS-V6.503 Bunch Spacing 25 ns 25 ns (short) 50 ns Cases 2.2e11, 2.5 µrad, 1eVs 2.2e11, 3 µrad, 1eVs 3.5e11, 3 µrad, 1eVs α IBS,l [1/h] 0.096 0.078 0.124 α IBS,x [1/h] 0.096 0.065 0.103 α IBS,y [1/h] 6.384 10 4 4.327 10 4 6.884 10 4 Table 2: IBS growth rates α IBS,i (i = x,y,l) calculated at 450 GeV injection energy based on optics version ATS-V6.503 at β = 11m for the beams given in Table 1, except that the longitudinal emittance was assumed to be 1eVs. SLHCV3.1b Bunch Spacing 25 ns 25 ns (short) 50 ns Cases 2.2e11, 2.5 µrad, 1eVs 2.2e11, 3 µrad, 1eVs 3.5e11, 3 µrad, 1eVs α IBS,l [1/h] 0.096 0.077 0.123 α IBS,x [1/h] 0.096 0.065 0.103 α IBS,y [1/h] 6.402 10 4 4.342 10 4 6.908 10 4 Table 3: IBS growth rates α IBS,i (i = x,y,l) calculated at 450 GeV injection energy based on optics version SLHCV3.1b at β = 5.5m for the beams given in Table 1, except that the longitudinal emittance was assumed to be 1eVs. 3.2 MADX Calculations at Collision Energy 3.2.1 WITHOUT CROSSING ANGLES AND SEPARATION BUMPS In Figure 3 and 3.2.1 the local contribution of the IBS growth due to a specific element is plotted as a function of the position s around the ring. The black and red line show the 9

cumulative sum of the corresponding (blue or red, respectively) local contribution. The maximum value of the cumulative sum can be found at the end of the ring (maximum s) and is equivalent to the total IBS growth rate usually quoted for the whole machine. The calculations were done with MADX using the ATS-V6.503 optics version for a flat machine without crossing angles and separation bumps. Figure 3 compares the effect on the 25 ns (red) and 50 ns (blue) beam based on the β = 0.1m optic. As a first observation it is clear that the impact on the 25 ns beam is weaker, due to the relaxed single bunch parameters in this case. Furthermore, the cumulative sum, which accumulates the local growth rates at each element with increasing s, for the longitudinal plane (left) increases approximately linear. This indicates that the spikes close to the IPs are quite narrow and thus the contribution to the total growth rate is similar for all elements. The local horizontal growth rates (right plot) are increased for the high β regions in the arcs around IP1 and 5, while the contribution of the straight sections is almost zero, due to the small dispersion (see below). A comparison of the injection optics at β = 11m (blue) and the fully squeezed optics at β = 0.1m (red) is displayed in Figure 3.2.1 for the example of the 50 ns beam. For β = 11m all arcs are similar and show the same behaviour. For β = 0.1m the ATS scheme induces an increased amplitude in the arcs around IP1 and 5, while the remaining arcs are unchanged, similar to the effect on the β-functions shown in Figure 1. However, closer inspection of the red curve in the left plot reveals that the average of the oscillation in the high β regions is decreased with respect to β = 11m. This indicates that the total IBS growth in the longitudinal plane is slowed down by the squeeze. On the other hand, the strong contribution of the high β regions increases the total growth rate in the transverse planes. The evolution of this effect along the squeeze is shown in Figure 5 where the total IBS growth rate is plotted as a function of β for the longitudinal (dashed lines) and horizontal (solid lines) plane. The IBS growth rates α IBS,i (i = x,l,y) in the horizontal, longitudinal and vertical plane can be written [14, 15] as with and α IBS,x α IBS,l α IBS,y = A γ 2 H x ε x γ 2 σ 2 δ β y ε y 0 A = [a λ 1/2 x λ + b x ] (λ 3 + aλ 2 + bλ + c) 3/2 [a l λ + b l ] dλ (4) [a y λ + b y ] r 2 0 cn(log) 8πγ(γ 2 1) 3/2 ε x ε y σ δ σ z (5) H x = D2 x β x (1 + α 2 x ) + β x D 2 x + 2α x D x D x (6) where γ the relativistic Lorentz factor, ε x,y the horizontal and vertical emittance, σ z the RMS bunch length, σ δ the RMS energy spread, r 0 the classical particle radius, c the speed 10

Figure 3: Longitudinal (top) and horizontal (bottom) local IBS growth rates for the ATS- V6.503 optics with β = 0.1m and without crossing angles and separation bumps. Each plot compares the nominal beams given in Table 1 with 25 ns (red) and 50 ns (blue) bunch spacing. 11

Figure 4: Longitudinal (top) and horizontal (bottom) local IBS growth rates for the ATS- V6.503 optics with β = 0.1m (red) and β = 11m (blue) without crossing angles and separation bumps. The 50 ns beam with 2.5eVs longitudinal emittance was chosen as an example for both cases. The β = 11m describes the situation at the end of the ramp and not as the situation with injection optics and 450GeV. 12

Figure 5: Horizontal (solid lines) and longitudinal (dashed lines) IBS growth rates as a function of β. Red: nominal 25 ns spacing, blue: 50 ns spaced beam. The calculation uses the ATS-V6.503 optics version without crossing angles and separation bumps. of light, N the number of particles per bunch, β x,y the horizontal and vertical β-function, (log) ln(r max /r min ) a Coulomb logarithm, with r max denoting the smaller of σ x and the Debye length and r min the larger of the classical distance of closest approach and quantum diffraction limit from the nuclear radius, typically assuming values of (log) 15 20. The coefficients a, b, c, a x, b x, a l, b l, a y and b y inside the integral can be found in Table 1 of [15], they as well depend on the optics and beam parameters mentioned above. In this form the horizontal IBS growth rate α IBS,x is proportional to the horizontal H - function from Equation 6 which only depends on lattice parameters, namely on the horizontal dispersion D x and its derivative D x and the optical functions β x and α x = β x/2. This function is plotted in Figure 6 for β = 11m (blue) and 0.1 m (red). The H -function is increased in the four arcs on either side of the two main IPs for the squeezed optics. From this, the increase of the horizontal IBS growth rate during the squeeze can now be understood. Nevertheless, the longitudinal growth rate does not depend linearly on the lattice functions and thus the influence of the changed optics is small; it actually improves the situation. The calculations assume uncoupled transverse planes, thus the IBS growth in the vertical plane is very small. Analysis of data from present LHC operations indicates that this 13

is a good approximation for well corrected physics optics. With the further influence of radiation damping, the net growth of the vertical emittance can even be negative. Figure 6: Horizontal H -function for the ATS-V6.503 optics with β = 0.1m (red) and 11 m (blue) and no crossing angles and separation bumps. 3.2.2 WITH CROSSING ANGLES By switching on the crossing angle bumps in the IPs the dispersion is enhanced. Due to those orbit bumps, the particles pass the triplet quadrupoles on an off-centred orbit and they see a stronger dipole component of the field, which induces dispersion. In the alternating crossing scheme the beams cross in the vertical in IP1 and IP2 and in the horizontal plane in IP5 and IP8, so the dispersion is increased in both planes. The induced off-centred orbit will lead to a wrong deflection of the beam in the magnets which can have a major influence on the beam stability and must be corrected. Via orbit bumps in the arcs the beam is again set on an off-centred orbit, due to which it sees a strong quadrupole component in the sextupoles and in combination with the phase advance of π between the sextupoles this cancels the dispersion created by the crossing angle bumps in the inner triplet [16]. In the following a comparison of the calculations with and without crossing angle and dispersion correction is given. 14

Figure 7 shows the horizontal and longitudinal IBS growth rates as a function of β for the ATS-V6.503 optics version on the left and for the SLHCV3.1b on the right, comparing on or off crossing angles and dispersion correction. When comparing both plots the reader has to be careful since the SLHCV3.1b plot does not show a continuous squeeze sequence of β values but only a few intermediate steps and as well two values for flat optics (i.e. with β aspect ratio not equal to unity, but still round emittances) at the beginning. However, for both optics the curves for the flat machine (zero crossing angle and separation, no dispersion correction) in red and the one with crossing angle and dispersion correction in blue are in very good agreement, whereas the rates for a crossing angle with uncorrected dispersion (green) are raised. In the vertical plane this is even more evident, as it can be seen in Figure 8, where the IBS growth rate jumps from a slight damping to a value comparable with the other planes. The uncorrected case in green shows a slightly higher rate for the SLHCV3.1b version, whereas the other cases are equivalent. The origin can again be seen by looking at the local IBS contributions around the ring as displayed in Figure 9 for the ATS-V6.503 on the left and SLHCV3.1b on the right. The plots show a comparison of the optics without crossing angle in blue and with crossing angle on in red. The blue curves are mostly identical for both optics, except for the height of some peaks around the IPs. After the crossing angle was switched on the differences between the two optics keep small. Nevertheless, a clear increase in the amplitude of the growth rates is visible leading to higher total growth rates in the horizontal and vertical plane. This change mainly arises from the enhanced dispersion in the transverse planes, therefore the longitudinal plane is less affected. Figure 10 shows the situation after the dispersion was corrected. The correction is not perfect, thus also the IBS local contributions show remaining perturbations close to the locations where the additional dispersion was produced. The horizontal growth rates shows peaks around IP5, which is crossing in the horizontal plane, and vice versa the vertical growth rate is heightened around IP1 with a vertical crossing angle. Table 5 compares the calculated IBS growth rates from MADX for the two optics cases and different crossing angle settings at β = 0.1m. Table 4 gives an overview of the IBS growth rates based on SLHCV3.1b optics for four different values of β including round and flat beams. 15

Figure 7: Horizontal (solid lines) and longitudinal (dashed lines) IBS growth rates as a function of β for the ATS-V6.503 (top) and SLHCV3.1b (bottom) optics version on the example of the 50 ns beam. Red: without crossing angle and dispersion correction, green: with crossing angle, but without dispersion correction, blue: with crossing angle and dispersion correction. 16

Figure 8: Vertical IBS growth rates as a function of β for the ATS-V6.503 (top) and SLHCV3.1b (bottom) optics version for the example of the 50 ns beam. Red: without crossing angle and dispersion correction, green: with crossing angle, but without dispersion correction, blue: with crossing angle and dispersion correction. 17

Figure 9: Longitudinal (top), horizontal (middle) and vertical (bottom) local IBS contribution around the ring for β = 0.1m. Blue: without crossing angle, separation and dispersion correction, red: with crossing angle, but without separation and correction. On the left for ATS-V6.503 and on the right for SLHCV3.1b. 18

Figure 10: Longitudinal (top), horizontal (middle) and vertical (bottom) local IBS contribution around the ring for β = 0.1m. Blue: without crossing angle, separation and dispersion correction, red: with crossing angle and dispersion correction. On the left for ATS-V6.503 and on the right for SLHCV3.1b. 19

(0,0,0) (1,0,0) (1,0,1) (0,0,0) (1,0,0) (1,0,1) SLHCV3.1b 25 ns βx /βsep [cm] α IBS,l [1/h] α IBS,x [1/h] α IBS,y [1/h] 10/10 0.046 0.060-1.6e-6 15/15 0.049 0.054-1.6e-6 5/20 0.045 0.065-1.4e-6 7.5/30 0.047 0.058-1.5e-6 10/10 0.039 0.079 0.029 15/15 0.045 0.063 0.015 5/20 0.042 0.079 0.016 7.5/30 0.046 0.065 0.008 10/10 0.046 0.060 0.001 15/15 0.049 0.054 0.001 5/20 0.045 0.065 0.001 7.5/30 0.048 0.059 4.4e-4 50 ns 10/10 0.059 0.064-1.7e-6 15/15 0.062 0.058-1.7e-6 5/20 0.057 0.070-1.5e-6 7.5/30 0.061 0.063-1.6e-6 10/10 0.050 0.086 0.031 15/15 0.058 0.068 0.016 5/20 0.053 0.085 0.017 7.5/30 0.058 0.070 0.008 10/10 0.058 0.064 0.002 15/15 0.062 0.058 0.001 5/20 0.057 0.070 0.001 7.5/30 0.060 0.063 4.6e-4 Table 4: IBS growth rates calculated with MADX at 7 TeV collision energy bases on optics version SLHCV3.1b for a set of β values for the beams given in Table 1. β sep and β x are the values of the β-function in the separation and crossing plane, respectively. Three different settings of the crossing angle, separation and dispersion correction are given: (xangle, sep, disp) = (0,0,0) or (1,0,0) or (1,0,1), where 0 and 1 mean off and on. 20

Optics ATS-V6.503 SLHCV3.1b Bunch Spacing 25 ns 50 ns 25 ns 50 ns (0,0,0) (1,0,0) (1,0,1) α IBS,l [1/h] 0.046 0.059 0.046 0.059 α IBS,x [1/h] 0.060 0.064 0.060 0.064 α IBS,y [1/h] -1.6e-6-1.7e-6-1.6e-6-1.7e-6 α IBS,l [1/h] 0.041 0.053 0.039 0.050 α IBS,x [1/h] 0.074 0.080 0.079 0.086 α IBS,y [1/h] 0.022 0.024 0.029 0.031 α IBS,l [1/h] 0.046 0.059 0.046 0.058 α IBS,x [1/h] 0.060 0.065 0.060 0.064 α IBS,y [1/h] 0.001 0.001 0.002 0.002 Table 5: IBS growth rates calculated at 7 TeV collision energy based on optics version ATS-V6.503 and SLHCV3.1b at β = 0.1m for the beams given in Table 1. Three different settings of the crossing angle, separation and dispersion correction are given: (xangle, sep, disp) = (0,0,0) or (1,0,0) or (1,0,1), where 0 and 1 mean off and on. 3.3 CTE Simulations at Collision Energy From the simulations with the CTE program the evolution with time of the IBS growth rates shown in Figure 11 are obtained. The CTE program provides the option to include two RF systems with different voltages and harmonic numbers, so it was possible to simulate the 25 ns case with short bunches for which the modulation of the main RF system with an additional one at 24MV and 800MHz is necessary, shown as the green dashed curve. The nominal 25 ns are displayed in red and the 50 ns case in blue. The initial values can be found in Table 6 together with the calculations of MADX. It is interesting that the evolution of the green curve is more similar to the blue (50 ns) than to the red (25 ns) one. This can be understood by looking at the plots 13 to 17 of the evolution of the beam parameters in Section 4. Those show that the relative evolution of the parameters is more similar for the green and blue curve than for the green and the red curve. 3.4 Comparison of MADX and CTE IBS Growth Rate Determination The agreement of the IBS calculations between the CTE program and MADX is very good, as Figure 12 shows. Here calculations for the 25 ns (red) and 50 ns (blue) beams are done as a function of time. The CTE simulation (solid line) was started with the input parameters given in Table 1 and once per hour the data was extracted, to be used as input for the MADX (dashed lines) calculation. In this way it is ensured that the calculations for a given time are based on the same beam parameters and thus they can be compared 21

Figure 11: Simulated IBS growth rates in the longitudinal (top, left), horizontal (top, right) and vertical (bottom) plane. The ATS-V6.503 optics version with β = 0.1m was used. directly. The agreement in the horizontal plane is better than 5% and in the longitudinal plane around 2% for the initial points and it seems to become even better with time, when the beam parameters become more relaxed. In Table 6 the IBS growth rates of both methods calculated with the input parameters taken from Table 1 are summarized again. In this case it can be seen that the IBS growth rates of the MADX calculation differ more from the CTE ones as indicated in Figure 12. This is due to the fact that CTE is using an iterative procedure to exactly match the longitudinal phase space before starting the tracking and does not use the small angel approximation. For this it takes the bunch length and the RF properties as input and varies the energy spread until the desired bunch length and the iterated energy spread follow a Hamiltonian trajectory in the longitudinal phase space. Using this method the energy spread is not constrained and can be slightly different from the nominal one given in Table 1. Hence, the longitudinal emittance which is calculated from the RMS energy spread, RMS bunch length and the total energy might also not be exactly 2.5eVs, which is the value used for the MADX calculation in Table 6. 22

Bunch Spacing 25 ns 25 ns (short) 50 ns Program CTE MADX CTE CTE MADX α IBS,l [1/h] 0.060 0.046 0.075 0.075 0.059 α IBS,x [1/h] 0.065 0.060 0.079 0.069 0.065 α IBS,y [1/h] -1.543e-6-1.572e-6-1.868e-6-1.619e-6-1.671e-6 Table 6: IBS growth rates calculated at 7 TeV for the beams given in Table 1. Figure 12: Comparison of IBS growth rate calculation of MADX (dashed lines) and CTE Nagaitsev Method [18, 17] (solid lines). Red: 25 ns, blue: 50 ns spacing from Table 1. 23

4 CTE Simulations of the Beam Evolution In this section we present a collection of the beam parameter evolutions simulated with CTE for the three initial beams from Table 1. The CTE program provides a choice of several methods for calculating the IBS growth rates. Most of them take full account of the details of the optics and some allow the effects of vertical dispersion to be included, just as MADX does. However typical simulations require the growth rate integrals to be evaluated a very large number of times so we have preferred to use the fast method according to [17]. However this does not include the effect of vertical dispersion on IBS, so the case where the crossing angle is switched on but the dispersion is not corrected could not be simulated correctly. However, as can be seen form the previous analysis with MADX, the calculations for the flat machine and when the crossing angle is applied while the dispersion is corrected are in very good agreement. Therefore, it is convenient and reasonably accurate to use the optics of the flat machine, without crossing angles and separation bumps, as a basis for the CTE simulations to estimate the beam evolution. Moreover, the SLHCV3.1b optics version shows only small differences with respect to the ATS version based on the nominal sequence, thus it is enough to run the simulations with either one of them as a good approximation to the other. ATS-V6.503 was chosen as the underlying lattice for the simulations. Each figure shows the evolution on the left side on an absolute scale and on the right relative to the initial value. The red line demonstrates the nominal 25 ns beam, the blue curve the 50 ns beam and the dashed green curve the special case of the 25 ns spaced beam with shorter bunches and an additional RF system at 24MV and 800MHz. Figure 13: Simulated absolute bunch length (left) and bunch length growth (right). The bunch length (Figure 13) evolves in a similar way for all three beams. It is decreasing because the rather strong radiation damping at this energy overcomes the contrary effect of IBS in the longitudinal plane. Radiation damping only depends on the energy and the bending magnets and has the same strength for all beams and optics considered here (small contributions from the orbit bumps in quadrupoles have been neglected). 24

Figure 14: Simulated normalised horizontal emittance (left) and horizontal emittance growth (right). Figure 15: Simulated normalised vertical emittance (left) and vertical emittance growth (right). As can be seen from Figure 11 the IBS growth rates are also very similar. Only the nominal 25 ns beam (red) is slightly less affected by IBS, resulting in a slightly faster bunch length decrease. The horizontal and vertical normalised emittances are given as functions of time in Figure 14 and 15. The horizontal emittance is growing, since the radiation damping rate of the transverse plane is only half of the longitudinal damping rate and is insufficeient to compensate IBS. Moreover, the short bunch 25 ns beam has the fastest emittance growth arising from the higher IBS growth rate, due to the small bunch length. Moreover, because of the short bunch length, the debunching losses shown in the right plot of Figure 17, which are in general very small for these beams, are almost zero for the green curve. Consequently, the main source of particle losses (Figure 16 and 17 left) is the luminosity production, as desired. The relative particle losses are equivalent for the 25

50 ns and the short bunch 25 ns beams, but the higher initial intensity of the 50 ns bunch produces more luminosity, see Figure 18, even though the luminosity reduction due to the (half) crossing angle is weaker for shorter bunches. Figure 16: Simulated intensity (left) and intensity losses (right). Figure 17: Simulated debunching losses (left) and luminosity losses (right.) 5 Conclusions We have provided a detailed comparison of the various optics and beam parameter options for HL-LHC from the point of view of intra-beam scattering growth rates. Simulations have included the evolution of beam parameters and luminosity with self-consistent IBS calculations, including non-gaussian longitudinal distributions and losses from the RF bucket, radiation damping and the loss of intensity due to luminosity production ( burnoff ). 26

Figure 18: Simulated evolution of the luminosity per bunch crossing (without crabcavity). The growth rates increase with increasing intensity and decreasing emittance, leading to stronger IBS effects for the beam parameters of the beam spaced by 50 ns as compared to 25 ns. Furthermore, higher IBS growth rates at injection are due to the smaller energy and the smaller longitudinal emittance. Throughout the squeeze to β values low as 10cm, using the ATS optics scheme, the longitudinal IBS growth rate improves by about 20%, whereas the horizontal growth rate increases by about 20%. The reason for this was understood in detail by investigating the local IBS contributions around the ring, where the growth in the horizontal plane strongly depends on the lattice. The horizontal H -function increases in the high β regions around IP1 and 5, resulting in an increase of the accumulated IBS growth rate. The dependence on the lattice parameters of the longitudinal growth is less dominant and hence the average longitudinal IBS contribution is even reduced. The calculations for the flat machine compared to a setup with crossing angle and dispersion correction are in very good agreement. For a machine with uncorrected dispersion after introducing the crossing angle the transverse growth rates rise while the longitudinal one is improved. It has to be noted that the vertical growth rate can be neglected in a corrected machine, but may become comparable with the horizontal and longitudinal plane in the presences of vertical dispersion (even without significant betatron coupling). The correction of the spurious dispersion, in particular for the vertical plane, is a new feature of the ATS optics which is not available for the nominal optics. The two ATS optics versions ATS-V6.503 and SLHCV3.1b, which include different element sequences, show very similar behaviour as far as IBS is concerned. The calcu- 27

Inj. Coll. Inj. Coll. Spacing 25 ns 50 ns E [GeV] 450 7000 450 7000 β [m] 11 0.15 11 0.15 N b [10 11 charges] 2.2 3.5 ε n [ µmrad] 2.5 3.0 ε l [evs] 1.0 2.5 1.0 2.5 α IBS,l [1/h] 0.096 0.049 0.124 0.062 α IBS,x [1/h] 0.096 0.054 0.103 0.058 α IBS,y [1/h] 6.4 10 4 1.5 10 6 6.9 10 4 1.6 10 6 Table 7: Summary of the HL-LHC beam parameters and the corresponding IBS growth rates calculated with MADX for the ATS-V6.503 optics version at injection and collision energy. lated growth rates only differ in the order of 10 4 1/h. and are summarised in Table 5 and 4. Besides, it has been shown that the MADX and CTE (using the Nagaitsev method [18]) IBS calculations agree within 5% in the parameter range considered. Table 7 gives an overview of the HL-LHC beam parameters together with the corresponding IBS growth rates. CTE simulations of the emittance, intensity, bunch length and luminosity evolution were presented. The effect of radiation damping turns out to have a positive effect on the emittance growth, and even leads to a decreasing bunch length. The intensity losses are dominated by luminosity burn off and the intensity loss from debunching is negligible in all cases. The relative intensity and emittance evolution is very similar for the 50 ns spaced beam and the special 25 ns spaced beam with shorter bunches. However, the 50 ns beam produces more luminosity thanks to the higher initial bunch intensity, even though the luminosity reduction due to the crossing angle is weaker for shorter bunches. Acknowledgements We thank several colleagues, in particular R. De Maria, S. Fartoukh and E. Métral for useful discussions and comments on this report. One of us (MS) is supported by the Gentner programme of the BMBF (Federal Ministry of Education and Research, Germany). This work was carried out in the frame of Task 2.4 Collective Effects of the HiLumi 28

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