Potential performance for Pb-Pb, p-Pb and p-p collisions in a future circular collider
aa r X i v : . [ phy s i c s . acc - ph ] M a r Potential performance for Pb-Pb, p-Pb and p-p collisions in a future circular collider
Michaela Schaumann ∗ CERN, Geneva, Switzerland and RWTH Aachen University, Aachen, Germany
The hadron collider studied in the Future Circular Collider (FCC) project could operate with pro-tons and lead ions in similar operation modes as the LHC. In this paper the potential performancesin lead-lead, proton-lead and proton-proton collisions are investigated. Based on average latticeparameters, the strengths of intra-beam scattering and radiation damping are evaluated and theireffect on the beam and luminosity evolution is presented. Estimates for the integrated luminosityper fill and per run are given, depending on the turnaround time. Moreover, the beam-beam tuneshift and bound free pair production losses in heavy-ion operation are addressed.
MOTIVATION
The Future Circular Collider (FCC) is a recently pro-posed collider study in a new 80–100 km tunnel at CERNin the Geneva area [1]. The design study includes threecollider options: FCC-ee (formerly known as TLEP),a e + e − collider with a center-of-mass energy of 90–400 GeV, seen as a potential intermediate step; FCC-hh,a hadron collider with a centre-of-mass energy of the or-der of 100 TeV in proton-proton collisions as a long-termgoal; and FCC-he, combining both as a hadron-electroncollider.The beam energy of the hadron machine is expectedto be E b = 50 Z TeV, where Z is the charge number ofthe circulating nuclei. Its main purpose will be to searchfor new physics in energy regimes which have never beenreached before. The FCC-hh will therefore spend most ofits physics time providing proton-proton collisions to itsexperiments. Nevertheless, operating this machine withheavy ions is being considered. It would provide, forexample, Pb-Pb and p-Pb collisions at √ s NN = 39 and63 TeV, respectively. From the heavy-ion physics pointof view, using the FCC-hh as a heavy-ion collider wouldopen a whole new regime of research opportunities [2].This paper discusses potential FCC-hh beam parame-ters for heavy-ion operation. The dominating beam dy-namic effects and estimates for the time evolution of lu-minosity, intensity, emittances and bunch length by an-alytic equations and Collider Time Evolution (CTE) [3]simulations are presented. An approximated smooth lat-tice model is assumed. Lead-lead (Pb-Pb) and proton-lead (p-Pb) operation are considered. We close with ashort discussion of proton-proton (p-p) operation, basedon the same techniques. GENERAL ASSUMPTIONS
It is foreseen to operate the FCC-hh with differenttypes of particles, e.g., protons (p) and lead-ions (Pb),but potentially also other ion species. The choice ofcertain parameters and hardware components has to en-sure the compatibility with all potential beams. As men- tioned, the production of p-p collisions will be the maintask, restricting the heavy-ion run time to a few weeksper year, similar to the current Large Hadron Collider(LHC) schedule. In order to optimise time and cost, theoperation with different species should share mostly thesame equipment and machine settings should be kept assimilar as possible. For this reason, the parameters tobe chosen for the heavy-ion operation are in line withthose for p-p operation documented in [4], where pos-sible. This work focuses on the baseline option of aring with C ring = 100 km circumference requiring 16 TNb Sn dipoles to provide a maximum beam energy of E b = 50 Z TeV.
Pre-Accelerator Chain
The study of this new hadron collider began only re-cently and the requirements for the pre-accelerator chainare still undefined. Assuming the same ratio of injectionto full energy as for the LHC, the injection energy of theFCC-hh would be E b, inj = 3 . Z TeV.Taking the existing CERN infrastructure into account,reference [4] tentatively suggests three options for thelast accelerator injecting into the FCC: a machine builteither in the SPS, the LHC or the FCC tunnel. The mag-net strength required for an injection energy of 3 . Z TeVwould be 1 T, for an injector with normal conductingmagnets in the 100 km FCC tunnel. 3 . . Sn magnets replacing theSPS. A choice has not been made, but using the exist-ing superconducting LHC magnets seems to be the mostfavoured and cost effective option today. Equipping theLHC magnets with new power converters and rampingto only about half their maximum field could reduce theramp time to an acceptable value of a few minutes.Based on this, it will be assumed here that the exist-ing pre-accelerator complex, including the LHC, is usedto accelerate the particles up to 3 . Z TeV before injec-tion into the new ring. Both LHC rings are filled and thebeams are injected in opposite direction into the FCC.This is a reliable but conservative assumption. Majorupgrades are essential in the injector chain to satisfythe requirements of the FCC experiments and to ob-tain a realistic filling time. The heavy-ion programmewill benefit from the efforts made. It can be expectedthat the performance and turnaround time will be signif-icantly improved compared to the current situation, butthe amount of improvement would be speculative today.
Smooth Lattice Approximation
At the time of this study, the lattice design is stillpreliminary [5]. However, for the calculation of manyparameters and effects, the knowledge of certain latticeproperties is required. In the design of a new machine,one has to respect some constraints, from which at leasta first approximation of the range of these quantities canbe derived.As a baseline it is assumed that the lattice would bea similar FODO design as in the LHC. The maximum(and minimum) β -function in a FODO cell is directlyproportional to the cell length, L c [6]: β ± = L c (1 ± sin µ )sin µ ∝ L c , (1)where µ is the phase advance per cell. To keep the beamsize in the arcs at a reasonable value, L c should not ex-ceed twice the LHC value of L c, LHC = 106 . L c, LHC , seems to be favoured as a compromisebetween magnet aperture and strength.The horizontal dispersion is produced in the bendingmagnets and is therefore proportional to the bending an-gle per cell, θ c , times L c . The average dispersion in aFODO cell, h D x i , is given by [6]: h D x i = L c θ c µ − ! ∝ L c θ c . (2)The total bending angle of the ring, the sum over θ c,i ofall cells, is 2 π : 2 π = Σ θ c,i = N c θ c ⇒ θ c = 2 πN c , where N c is the total number of FODO cells in the ring.The length of the circumference, filled by the arcs, is: L arcs = N c L c = 2 πθ c L c . Of this length, the dipoles themselves only occupy thefraction F arc , giving: L dipole = 2 πρ = F arc L arcs = F arc πθ c L c , TABLE I: Assumed beam parameters for heavy-ionoperation in Pb-Pb and p-Pb collisions.
Parameter Symbol Unit Lead ProtonNo. of particles per bunch N b [10 ] 1.4 115Normalised transv. emittance ǫ n [ µ m] 1.5 3.75RMS bunch length σ s [m] 0.08 0.08No. of bunches per beam k b - 432 432 β -function at IP β ∗ [m] 1.1 1.1 with ρ as the dipole bending radius. It follows that theaverage horizontal dispersion is related to the cell lengthas: θ c L c = L c F arc ρ ∝ L c (3) ⇔ h D x i ∝ L c . (4)The vertical dispersion is in general very small andcorrected for. Therefore, it is assumed to be zero: h D y i = 0 . Assuming a phase advance of µ = π/ F arc = 0 .
79, as in the LHC, Eq. (1),(2) and (3) can be used to express the dispersion and β -functions in terms of the cell length L c .The momentum compaction factor, α c , and the rela-tivistic gamma factor at transition energy, γ T , can beapproximated via the average horizontal dispersion: α c ≡ γ T = 1 C ring I D x ρ d s ≈ π h D x i C ring . (5) Beam Parameters
The potential beam parameter space is constrained bymany different limitations, including the injector perfor-mance and dynamic effects in the whole operational cy-cle. The beam parameters presented in the following arean example of what could be possible from today’s knowl-edge. Further studies should be performed to confirmtheir validity and to determine the optimum parameterset.Using the existing pre-accelerator chain, it can be ex-pected that beam parameters at least as good as in theLHC can be achieved. For the moment, the bunch-by-bunch differences observed in LHC operation [7] are ne-glected. Average bunch parameters measured in the 2013proton-lead run [8, 9] are taken as a conservative baseline.The assumed beam parameters for the lead and protonbeams for heavy-ion operation of the FCC-hh are givenin Table I.For the number of bunches per beam, k b , given in Ta-ble I, one injection per beam from the LHC is assumed.The LHC filling is assumed to be the planned ”baseline”filling scheme after LS2 [10]. One shot from the LHCfills only about one quarter of the total circumferenceof the FCC. This implies that either only one experi-ment, clusters of experiments or two experiments, placedat opposite positions in the ring, could be provided withcollisions. The reason for this choice is related to theturnaround time of the LHC as an injector, which will beexplained in the discussion of the luminosity evolution.The β ∗ -values are the same as during p-p operation.Intensity losses and emittance growth at injection, dur-ing the ramp and while preparing collisions are neglected. RF System and Longitudinal Parameters
An RF system similar to the one currently used in theLHC, which has a frequency of f RF = 400 . h = f RF f rev = 133692 (= 2 × × × C ring = 100 km.In reality, the circumference will be adjusted to give an h with more small factors, but this is not important inthe following. Injection
When the beam is injected, assuming bunch to buckettransfer, the longitudinal beam parameters, i.e., the rela-tive RMS momentum spread, σ p , the RMS bunch length, σ s , and the longitudinal emittance, ǫ s , are defined bythe previous accelerator. To conserve the beam quality,the RF bucket has to be matched to the arriving beam.Assuming an injected bunch length of σ s = 0 . σ p and ǫ s arriving from the LHC can becalculated as σ p = 2 π f s σ s c | η | = 1 . × − , (6) ǫ s = 4 πσ p σ s β rel E b / ( Zc ) = 2 . / charge , (7)where f s is the synchrotron frequency given by f s = f rev s | η | V RF hZe πβ rel E b , (8)with f rev as the revolution frequency, β rel = v/c and E b as the energy of the synchronous particle. η = γ T − γ is the slip factor with γ as the relativistic Lorentz factor, Ze is the particles’ charge. At E b = 3 . Z TeV, an RFvoltage of V RF = 12 MV was used in the LHC.From Eq. (7), it follows that ǫ s is constant, if σ s and σ p are constant. If σ s can be preserved during the transfer,
120 140 160 180 200051015 L c @ m D V R F @ M V D FIG. 1: RF voltage dependence on lattice at injectionin the FCC for matched bucket condition.Eq. (6) and (8) show that for a given lattice the RF volt-age is the only free parameter to match the momentumspread.Because of the preliminary stage of the lattice design,the effect of a varying cell length should be investigated. γ T is the only parameter in Eq. (6) depending on thelattice. From Eq. (5), (2) and (4) follows γ T ∝ L c ⇒ σ p ∝ γ T p V RF ∝ √ V RF L c (9)for γ ≫ γ T . To obtain a matched distribution with σ p equal to the injected value, V RF has to be increased pro-portionally to the square of the cell length as shown inFig. 1.We define a baseline FCC-hh lattice with a FODO celllength of L c ≈
203 m for the calculations in the follow-ing. With this, Eq. (2) and (5) estimate γ T ≈ V RF = 13 MV is required at injection in the FCC. Top Energy
To counteract the adiabatic damping of the bunchlength during the energy ramp, white RF noise is appliedto keep σ s at a constant value of 0 .
08 m. This value istaken from the p-p parameter list and is based on the res-olution limits of the experiments, imposing a minimumlength of the luminous region.Using an RF voltage of V RF = 32 MV, twice the LHCdesign value [11], at top energy of the FCC-hh, the syn-chrotron frequency, the relative RMS momentum spreadand the longitudinal emittance are f s = 3 . ,σ p = 0 . × − ,ǫ s = 10 . / charge . V RF @ MV D
120 140 160 180 2000.40.60.81.01.21.41.6 L c @ m D - Σ p FIG. 2: RMS momentum spread dependence on latticeand RF voltage at top energy, as described by Eq. (9).The bucket height, (∆ p/p ) max , and area, A bucket , eval-uate to [6] (cid:18) ∆ pp (cid:19) max = s ZeV RF πh | η | β rel E b = 1 . × − ,A bucket = 8 C ring hπc s ZeV RF E b πh | η | = 28 . / charge . At injection energy these values are (∆ p/p ) max = 4 . × − and A bucket = 4 . / charge. The calculation isbased on the baseline lattice defined in the previous para-graph.An energy spread of 0 . × − seems small and it has tobe investigated in detail, if this would cause instabilities.As Eq. (9) states and Fig. 2 visualises, increasing the RFvoltage could be advantageous, but the gain in σ p is smallfor L c on the order of twice the LHC cell length. In thedesign stage of the machine, it could as well be an optionto increase γ T by decreasing the cell length to obtain ahigher σ p . Nevertheless, the benefit has to be weighedagainst other design criteria relying on the cell length.For a chosen bunch length, the longitudinal emittancewill behave proportionally to the momentum spread.In general, it seems reasonable to aim for a similarmomentum spread as in the LHC, around σ p = 1 . × − . This however would require an unrealistically highRF voltage of about V RF ≈
100 MV.
LEAD-LEAD OPERATION
Based on the assumptions made above, approximationsof relevant beam properties and effects are calculated inthe following section. Because of the preliminary stateof the accelerator design, simplifying assumptions had tobe made in several places, therefore the study presentedhere can only give a first indication of what could beexpected from heavy-ion operation of such a machine.
Intra-Beam Scattering
Intra Beam Scattering (IBS) is a dynamic effect withina bunch of charged particles, where multiple small-angleCoulomb scattering leads to particle losses and emittancegrowth. This effect can become very strong and reducethe potential luminosity.
Formalism and Scaling
Several formalisms are available describing the physicaleffects derived by Piwinski, Bjorken and Mitingwa, Bane,Nagaitsev or Wei [12–16], based on different assumptionsand suitable for different situations. To estimate the ef-fect in the FCC-hh, the methods of Piwinski [12] and Wei[16] are used.Piwinski’s equations for the IBS emittance growthrates, α IBS , can be found in [17]. In his formalism the α IBS are proportional to A p = 2 r N b m c πγǫ n,x ǫ n,y ( Zǫ s ) , (10)where ǫ s is the invariant longitudinal emittance percharge given by Eq. (7), ǫ n,xy = β rel γǫ xy are the trans-verse normalised emittances, r the classical particle ra-dius, which relates to the classical proton radius, r p , as r = Z /A ion r p , and m is the rest mass of the particle.This factor gives an indication of the scaling and quanti-ties most important for the IBS strength. Equation (10)scales inversely with the energy, meaning the IBS growthis strongly suppressed at higher energies. On the otherhand, the rates increase with bunch intensity and de-crease with growing emittances ( α IBS ∝ N b / ( ǫ n,x ǫ n,y ǫ s )),implying that the higher the bunch brightness, desired forluminosity production, the stronger the IBS. A third rel-evant proportionality is the relation to r , which dependson the particles’ mass and charge ( α IBS ∝ Z /A ion ),hence the effect is stronger for heavy ions compared toprotons. The remaining factors in Piwinski’s equationsare complicated and depend mainly on lattice parame-ters, like the dispersion and β -functions, and the beamdivergences in all dimensions.In a simplified formalism J. Wei derived analyticalequations of the IBS emittance growth rates of hadronbeams [16], provided that the lattice of the acceleratormainly consists of regular FODO cells. For full couplingbetween the horizontal and vertical motion, the growthrates average in the transverse dimension. For roundbeams ( ǫ = ǫ x = ǫ y ) and if the motion is fully coupled,Wei’s formulae for the IBS emittance growth rates are α IBS ,x,y = C N b σ s ǫ q ǫ + C σ p (11) α IBS ,s = C ǫσ p α IBS ,x,y , (12)where C , C and C are constant during operation: C = 5 √ cZ r p A γ β D x γ − β x ( β x + β y ) β x p β x + β y C = D x β x C = 4 γ β x D x γ − β x ( β x + β y ) . Following the smooth lattice approximation, the averageof the dispersion and β -functions around the ring areused in the equations. In this form the longitudinal andtransverse growth rates are directly related. Calculation of IBS Growth Rates
The large parameter space, originating from the un-certainties of the lattice design, defines a range of IBSgrowth rates. Equation (1) and (2) are used to estimatethe average dispersion, h D x i , and β -functions, h β x,y i , re-quired to approximate the IBS growth rates.Figure 3 shows 1/ α IBS as a function of L c at (a)3 . Z TeV (injection energy) and at (b, c) 50 Z TeV (col-lision energy) for the initial bunch parameters given inTable I. Only the longitudinal and horizontal plane areshown. IBS in the vertical plane is negligible withoutcoupling, as assumed in the calculations. For the plotsat top energy the dependence of σ p on L c is taken intoaccount, while σ p is constant at injection energy. Theresults are calculated for a set of RF voltages. σ p canbecome very small for long cells (Fig. 2) and larger RFvoltage can mitigate this effect.Note that the horizontal IBS strength increases (=decreasing 1 /α IBS ) and the longitudinal decreases withincreasing L c at injection, but at top energy both, α IBS ,s and α IBS ,x , become stronger for longer cells. Thefactors in the IBS calculation depending on the cell lengthare h D x i ∝ L c , h β x i ∝ L c and σ p ∝ /L c . If σ p is in-dependent of L c (as in the case of the injected beam), α IBS ,s only has a weak dependence on the lattice, while α IBS ,x features a second term ∝ D x ∝ L c [17]. At topenergy, σ p is influenced by the lattice conditions and be-comes a function of L c . Thus the strong dependence of α IBS ,s on σ p takes over and the longitudinal IBS growthis enhanced for long cells.In general, IBS could lead to longitudinal emittancegrowth at injection energy, while the transverse growthrates are moderate. At collision energy IBS should stillbe modest, with growth times above 20 h. The situationeven improves, if the energy spread could be kept at theLHC design value, see Table II.However, as it will be shown in the next section, thisis only true for the initial beam parameters right afterarriving at top energy. Because of the strong radiation damping, the beam emittances will shrink and the IBSwill become strong enough to balance the damping.Table II summarises the IBS growth times for a bunchwith initial parameters, assuming (a) σ p = 0 . × − (ob-tained with γ T of baseline lattice) and (b) σ p = 1 . × − (LHC design) at collision energy, σ p = 1 . × − at injec-tion energy. The dispersion and β -functions are taken ascalculated from the baseline lattice with L c = 203 m. Thecomparison of the formalisms by Piwinski and Wei showsthat Wei estimates a systematically slightly stronger IBSrate. The overall agreement is better than 10% at highenergy and 20% at injection energy for the given param-eters. Radiation Damping
A charged particle travelling in a storage ring will radi-ate energy, when it is bent on its circular orbit. Becauseof the average energy loss into this synchrotron radiation,the betatron and synchrotron oscillation amplitudes aredamped like A i = A ,i e − α i t , where i = x, y, s , with theradiation damping rates α i given in Chapter 3.1.4 of [6].For a flat, isomagnetic ring with separated functionmagnets and zero vertical dispersion, the radiation emit-tance damping rates can be approximated by α rad ,s = 2 α rad ,x,y ≈ E b C α πρ C ring , (13)where C α = r c/ (3( m ion c ) ). Those quantities do notdepend on the beam parameters. The strongest depen-dence is on the third power of the energy, the machinesize and the particle type. Note that the longitudinaldamping is twice as fast as the transverse.To get an impression how strong the radiation dampingwill be in the FCC, the damping rates are compared tothe LHC design values: α rad,FCC α rad,LHC ≈ E /C E /C ≈ ≈ . This scaling is valid for all planes, because of rela-tion (13). The circumference of the accelerator was cho-sen such that the required dipole field does not exceed theexpected technical limits. Therefore, the bending radiuscan be approximated to be proportional to the circum-ference, ρ ∝ C ring . The new machine will be about afactor 4 longer than the LHC. Moreover, the energy willbe increased by about a factor 7 (= 50 Z TeV / Z TeV).Table III quotes the radiation damping times at injectionand collision energy. Note that the horizontal equilibriumemittance from quantum excitation for lead beams at topenergy is of the order of 10 − µ m, the effect is thus stillnegligible. LongitudinalHorizontal
120 140 160 180 200010203040 L c @ m D (cid:144) Α I B S @ h D (a) IBS at 3 . Z TeV and V RF = 13 MV. V RF @ MV D
120 140 160 180 200020406080100120140 L c @ m D (cid:144) Α I B S @ h D (b) Longitudinal IBS at 50 Z TeV. V RF @ MV D
120 140 160 180 200020406080100120140 L c @ m D (cid:144) Α I B S @ h D (c) Horizontal IBS at 50 Z TeV.
FIG. 3: Range of initial IBS growth times as a function of the cell length, L c , at injection (a) and collision (b, c)energy, evaluated with Piwinski’s equations [17]. (b) longitudinal, (c) horizontal growth times. For given values ofthe total RF voltage, V RF . No transverse coupling assumed.TABLE II: Initial IBS growth times for Pb-ions calculated with the Piwinski [17] and Wei [16] formulae, assumingbaseline lattice ( L c = 203 m) and no transverse coupling. Assumption for momentum spread: injection σ p = 1 . × − at V RF = 13 MV, collision (a) σ p = 0 . × − (obtained with γ T of baseline lattice), (b) σ p = 1 . × − (LHC design) at V RF = 32 MV. Growth Times Unit Injection CollisionPiwinski Wei (a) (b)Piwinski Wei Piwinski Wei1 /α IBS ,s [h] 6.3 5.1 29.1 27.3 141.4 132.01 /α IBS ,x [h] 10.0 8.2 30.0 28.0 43.9 41.01 /α IBS ,y [h] − − − − − − Luminosity
The quantity that measures the ability of a particleaccelerator to produce the required number of interac-tions is the luminosity . It represents the proportionalityfactor between the number of produced events per unitof time, d R/ d t , and the production cross-section of the considered reaction, σ c :d R d t = σ c L . (14)In the specific case of a circular collider and when theparticle density distribution can be approximated to aGaussian, the luminosity of two beams, colliding exactlyTABLE III: Emittance radiation damping times forPb-ions. Damping Times Unit Injection Collision1 /α rad ,s [h] 852 0.241 /α rad ,x [h] 1704 0.491 /α rad ,y [h] 1704 0.49 head-on, is given by: L = N b N b f rev k b π p σ x + σ x q σ y + σ y = N b f rev k b γ πǫ n β ∗ , (15)where N b and N b are the two colliding bunch intensi-ties, k b the number of colliding bunches per beam, σ xi and σ yi are the transverse beam-sizes in the horizontaland vertical direction, respectively. The second equalityfollows in the approximation of equal and round distri-butions and optics of both beams: N b = N b = N b , σ xy = σ xi = σ yi = p ǫ n β ∗ /γ . With ǫ n as the normalisedemittance and β ∗ as the β -function at the interactionpoint (IP).Using Table I and Eq. (15) the initial luminosity at thebeginning of collisions computes to L initial = 2 . × cm − s − . Which is, due to the higher intensity and energy, already2.6 times higher than the design luminosity for Pb-Pb ofthe LHC.The total event cross-section, σ c, tot , is given by thesum over the cross-sections of all possible interactionsremoving particles from the beam in collision (burn-off).Apart from the inelastic hadronic interactions, the effectsof Bound Free Pair Production (BFPP) and Electromag-netic Dissociation (EMD) are very important for Pb-Pbcollisions: σ c, tot = σ c, BFPP + σ c, EMD + σ c, hadron (16) ≈
354 b + 235 b + 8 b = 597 b . The numerical values in Eq. (16) are estimated for E b =50 Z TeV with the aid of References [18, 19].
Luminosity Evolution
While the beams are in collision, the instantaneousvalue of the luminosity will change, through intensitylosses and emittance variations, L ( t ) = A N b ( t ) p ǫ x ( t ) ǫ y ( t ) , (17)where all time independent factors are merged in A = f rev k b / (4 πβ ∗ ). For simplification, equal beam popula-tions and sizes of both beams are assumed. To obtain the beam evolution with time, a system of four differ-ential equations for the intensity, emittances and bunchlength evolution has to be solved. The solutions can beinserted into Eq. (17) to obtain the luminosity evolution.d N b d t = − σ c, tot A N b √ ǫ x ǫ y (18)d ǫ x d t = ǫ x ( α IBS ,x − α rad ,x ) (19)d ǫ y d t = ǫ y ( α IBS ,y − α rad ,y ) (20)d σ s d t = 12 σ s ( α IBS ,s − α rad ,s ) (21)The factor 1/2 in Eq. (21) was introduced because theemittance growth rates are twice the amplitude growthrates. The change in particle number with time, d N b / d t ,is linked to the luminosity production rate described inEq. (14). Now, σ c = σ c, tot is the sum of cross-sectionsfor all processes that remove particles. A minus sign isintroduced, since for each collision event generated oneparticle is lost: d R/ d t = − d N b / d t . The time evolutionof the emittances and bunch length is influenced by dy-namic IBS growth and constant radiation damping. Thetotal emittance growth rate α ǫ = α IBS − α rad , thus variesdynamically in time and it is impossible to find an ana-lytic solution of this system.In the given form, Eq. (18) assumes that all beamlosses are from luminosity burn-off. In LHC p-p oper-ation, a large fraction of particles is lost on the collima-tors. The amount, however, strongly depends on the col-limator settings, which in past runs (2012) were tight inorder to clean the beam halo. Nevertheless, experiencefrom RHIC shows that, owing to the applied stochas-tic cooling, it is possible to achieve very low loss ratesfrom non-luminous processes [20]. The strong radiationdamping at FCC energies will have a similar effect as thestochastic cooling in RHIC, supporting the assumptionof negligible non-luminous losses made in Eq. (18).In the following, approximations will be made forwhich an analytic description is possible. To simplifythe situation, round beams and fully coupled transversemotion is assumed, such ǫ ( t ) = ǫ x ( t ) = ǫ y ( t ) at all times,reducing the ordinary differential equation (ODE) systemto three equations.(I) In the first case, ǫ ( t ) = ǫ = const. should be con-sidered, which is achieved when α IBS − α rad = 0 and thusd ǫ/ d t = 0. For zero crossing angle, the bunch length evo-lution is (in first order) decoupled from the luminosity.Eq. (18) simplifies tod N d t = − σ c, tot A N b ǫ This can easily be solved for the intensity evolution and,in combination with Eq. (17), for the luminosity evolu-tion with time: N b ( t ) = N b AN b σ c, tot t/ǫ + 1 ⇒ L ( t ) = L (cid:18) AN b σ c, tot t/ǫ + 1 (cid:19) . By investigating those equations, it becomes clear thatthe only non-constant factor is the time t , which appearsonly in the denominator, i.e. the intensity and with itthe luminosity can only decay.(II) In the second case, the total emittance dampingrate should be constant, α ǫ = const., with α IBS ≪ α rad .It is implicitly approximated that IBS is independent ofthe beam parameters, decoupling the bunch length andemittance evolutions. Simultaneously solving the two re-maining differential equations (18) and (19) gives ǫ n ( t ) = ǫ exp[ − α ǫ t ] (22) N b ( t ) = N b ǫ ǫ + AN b σ c, tot (exp[ α ǫ t ] − /α ǫ (23) L ( t ) = L ǫ exp[ α ǫ t ]( ǫ + AN b σ c, tot (exp[ α ǫ t ] − /α ǫ ) . (24)Again t is the only non-constant parameter. As expected, ǫ n ( t ) (Eq. (22)) and N b ( t ) (Eq. (23)) can only decay.However, the combination of both, the luminosity evolu-tion (Eq. (24)), features the exponentially growing factorexp[ α ǫ t ] in the numerator and denominator. This means,as long as the numerator ǫ exp[ α ǫ t ] predominates the de-nominator ( ǫ + AN b σ c, tot (exp[ α ǫ t ] − /α ǫ ) a growthof the initial luminosity to a higher peak is possible. Itshould be noted that the assumption of a constant damp-ing leads to emittances asymptotically approaching zero,which is non-physical. Because of this effect, the lumi-nosity peak computed with Eq. (24) is overestimated.(III) In reality the IBS growth rate changes dynami-cally with the intensity and emittance, thus it will be-come stronger, while the emittances shrink due to radi-ation damping. Since the total emittance growth rate isgiven by α ǫ = α IBS − α rad , the emittance will approacha value where the growth from IBS balances the damp-ing. This balance is not a real equilibrium, where theemittance and bunch length would be constant. But theIBS strength keeps decreasing due to intensity burn-off,leading to a slowly shrinking emittance and bunch lengthto maintain the balance.An analytical expression for the balance value of theemittance and bunch length can be derived from Wei’sIBS formalism given by Eq. (11) and (12). Even in thissimplified form, the transverse growth rate shows a rathercomplicated dependence on ǫ , providing only a numeri-cal solution. Both factors under the square root in thedenominator of Eq. (11) depend on evolving beam prop-erties. ǫ ∝ − /γ ≈ − and C σ p ≈ D x /β x (10 − ) ∝ − are in the same order of magnitude, therefore weapproximate q ǫ + C σ p −→ q C σ p . Eq. (11) can be Σ s , B @ c m D N b @ particles D Ε n , B @ Μ m D Ε n , B µ N b (cid:144) Σ s , B µ N b (cid:144) FIG. 4: Normalised emittance (red) and bunch length(blue) for balanced IBS and radiation damping.set equal to α rad ,x to satisfy the balance condition andbe solved for the emittance ǫ B = ǫ n,B /γ : ǫ n,B ∼ = γ s C N b √ C α rad ,x D p σ s,B . (25) D p is the proportionality factor between the momentumspread and the bunch length given by Eq. (6). ǫ n,B stilldepends on the balance value of the bunch length, σ s,B ,which is determined by replacing ǫ −→ ǫ n,B /γ in Eq. (12)and applying α IBS ,s = α rad ,s : σ s,B ∼ = C p C N b α rad ,x D / p (2 C ) / α rad ,s ! / ∝ N / b . (26)Inserting Eq. (26) into Eq. (25) leads to an equivalentequation for the emittance: ǫ n,B ∼ = γ C N b D p α rad ,s C √ C α ,x ! / ∝ N / b . (27)When a balance between IBS and radiation damping isreached, the emittance and bunch length depend onlyon the bunch intensity. The higher the number of parti-cles, the larger the beam dimensions as shown in Fig. 4.The balanced normalised emittance (red) and the bunchlength (blue) are plotted as a function of the intensity.The plot shows that those quantities become small in theexpected range of bunch charge. Longitudinal, and po-tentially transverse, blow-up might become necessary tokeep the beam sizes in a reasonable range.The intensity evolution, where α IBS = α rad , can beobtained by inserting Eq. (27) into Eq. (18): N b,B ( t ) ∼ = 3 √ N ′ b / Aσ c,tot N ′ b / t " C √ C α ,x C D p α rad ,s / − / . (28)Note that N ′ b = N b is not the initial intensity at the be-ginning of the fill, but should be the number of particlesleft when the balanced regime is reached. The luminos-ity evolution can then be calculated by inserting Eq. (28)and (27) into (17).Figure 5 shows the beam and luminosity evolution forcase (II) and (III) as discussed above in comparison withtracking simulations done with the CTE program [3].The results are displayed for two colliding lead bunchesfeaturing the beam parameters given in Table I. One ex-periment is taking data. The black line shows the calcu-lations with Eq. (22)-(24) for the approximation where α ǫ = const. and α IBS ≪ α rad . The dashed green lineshows the calculations done with Eq. (26)-(28) in theregime where IBS and radiation damping balance eachother ( α IBS = α rad ). The two red lines are CTE simu-lations with (solid) and without (dashed) IBS coupling.The simulations are based on the assumption of a smoothlattice and Piwinski’s IBS formalism.It is clearly visible that the bunch length and emit-tances of the analytical calculations for α IBS ≪ α rad (black) asymptotically approach zero, which is non-physical, leading to a strong over-estimation of the lu-minosity. While the simulation with uncoupled planes(dashed red line) shows a realistic horizontal and lon-gitudinal behaviour, the vertical emittance still dampsto zero. In the coupled simulation (solid red lines)all three beam dimensions settle at a balanced valueabove zero. The transverse normalised emittance reachesaround 0 . µ m, corresponding to a beam size of σ ∗ ≈ µ m at the IP for β ∗ = 1 . σ s ≈ Mathematica . Coupled transverse motionand round beams are assumed. The solid red line in- dicates again the CTE simulation shown in Fig. 5. Theblack dashed line shows the corresponding solution of theODE system. The agreement between the numerical so-lution and the tracking result is excellent. Hence, the an-alytic calculation in the balanced regime (with coupling)is in excellent agreement with the ODEs. The small dif-ferences, are explained by the difference in IBS growthrates calculated with Piwinski’s and Wei’s algorithms forthe same beam conditions. To prevent the bunch lengthfrom shrinking to too low values and to model the evolu-tion under longitudinal blow-up, the ODEs are solved forconstant bunch length (d σ s / d t = 0, green dashed line).This enhances the intensity burn-off and the luminositypeak, since the IBS is weakened, reducing further the bal-ance value of the emittance. Introducing an additionalconstant term in Eq. (19) can constrain the emittanceabove a certain value, ǫ min , similar to the equilibriumbetween radiation damping and quantum excitation inlepton machines [6]:d ǫ d t = α IBS ,x ǫ − α rad ( ǫ − ǫ min ) . Solving the equations for both, constant bunch lengthand a minimum emittance of e.g. ǫ min = 0 . µ m, resultsin the blue dashed-dotted curve. As intended, the emit-tance stops decaying at about 0 . µ m, naturally comingalong with a luminosity reduction.Looking back at Fig. 5, the intensity decay is very fast,because of the high burn-off rates going along with thesmall emittances. In the analytical case (black) the totalbeam intensity is converted into luminosity in about 4 h.In the simulation the finite emittances reduce the peakluminosity and spread out the luminosity events over alonger period, however, the event production is still veryefficient: only about 20% of the initial particles are leftafter 6 h collision time.For comparison, in a normal LHC fill, the natural cool-ing from radiation damping is much weaker and not suf-ficient to increase the luminosity above its initial value.After about 6 h, the luminosity has decayed so much thatit is necessary to refill. At that time, the beam popula-tion has only decreased to 40 or 50% of its initial value.Those particles have to be thrown away to be replacedwith fresh beam. To maximise the integrated luminosity,the time in collisions has to be optimised.In a very high energy hadron collider, the event pro-duction efficiency will be close to its optimum, whereall particles are converted into luminosity. Under equaloperational conditions, this will lead to a constant filllength. In this regime the integrated luminosity per fill0 Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad
Analytical Eq. for Α IBS =Α rad @ h D L b @ c m - s - D Instantaneous Bunch Luminosity
Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad @ h D L b , i n t @ m b - D Integrated Bunch Luminosity
Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad
Analytical Eq. for Α IBS =Α rad @ h D Ε n , x @ Μ m D Horizontal Normalised Emittance
Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad
Analytical Eq. for Α IBS =Α rad @ h D Ε n , y @ Μ m D Vertical Normalised Emittance
Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad
Analytical Eq. for Α IBS =Α rad @ h D N b @ i on s D Bunch Intensity
Simulation H with coupling L Simulation H no coupling L Analytical Eq. for Α IBS <<Α rad
Analytical Eq. for Α IBS =Α rad @ h D Σ s @ m D Bunch Length
FIG. 5: Pb-Pb beam and luminosity evolution. Top: instantaneous (left) and integrated luminosity (right), middle:horizontal (left) and vertical (right) normalised emittance, bottom: intensity (left) and bunch length (right). Oneexperiment is in collisions. The black lines show the calculations done with Eq. (22)-(24) for α IBS ≪ α rad , thedashed green lines show the calculations done with Eq. (26)-(28) in the regime where IBS and radiation dampingbalance each other ( α IBS = α rad ), the two red lines are CTE simulations with (solid) and without (dashed) IBScoupling. Note that the dashed red line in the middle right plot is hidden behind the black line.is given by L int = N b k b σ c, tot . (29)The simulations show that the luminosity evolution is not symmetric to the maximum, but it drops ratherslowly once the balanced regime is reached. Dependingon the turnaround time, t ta , it is advantageous to dumpthe beams before all particles are burned-off and refill.1 Simulation, IBS PiwinskiODE, IBS WeiODE, Σ s = Σ s = Ε n ³ Μ m @ h D L b @ c m - s - D Instantaneous Bunch Luminosity
Simulation, IBS PiwinskiODE, IBS WeiODE, Σ s = Σ s = Ε n ³ Μ m @ h D N b @ i on s D Bunch Intensity
Simulation, IBS PiwinskiODE, IBS WeiODE, Σ s = ODE, Σ s = Ε n ³ Μ m @ h D Ε n , x @ Μ m D Horizontal Normalised Emittance
Simulation IBS PiwinskiApprox. Equation IBS WeiNum. Solution of ODE IBS Wei
Num. Solution of ODE H Σ s = L @ h D Σ s @ m D Bunch Length
FIG. 6: Pb-Pb beam evolution derived from ODE in comparison with simulation result. Top: Luminosity (left) andintensity (right), bottom: normalised emittance (left) and bunch length (right).The turnaround time is the time required to go back intocollision after a beam abort. The average integrated lu-minosity defined as h L int i = 1 t coll + t ta Z t coll L int ( t )d t (30)can be used to estimate the luminosity outcome per hour,depending on the expected turnaround time and timein collision, t coll . In fact, for a given t ta this equationcan be used to find the duration t coll for which h L int i is maximized. The Fig. 7a shows h L int i /h as a func-tion of t coll . For t ta = 2 h the maximum is reached afteraround t coll = 3 h, which is about the time when the lu-minosity has decreased back to its initial value. Underoptimal running conditions, without failures and earlybeam aborts, from this point on it is more efficient todump and refill, rather than collecting at low rates. AsFig. 7c displays, around 8 nb − (red solid line) could becollected during such an idealised 30 days Pb-Pb run. Itis assumed that only one injection with two beams of 432bunches each is taken from the LHC.In general, a maximum of four injections would fit intothe FCC. The total turnaround time consists of two com- ponents, t ta = t ta,FCC + ( n inj − t ta,LHC firstly t ta,FCC , including everything done in the FCC(cycling to go back to injection energy, ramp, prepar-ing collisions etc.), and secondly t ta,LHC , being the timebetween injections. n inj is the number of LHC injec-tions. The current LHC turnaround time is on averageabout t ta,LHC = 3 h. Consequently, the already injectedbunches would have to wait many hours at the FCC injec-tion plateau. At this energy, the Pb bunches lose about R loss = 5% of their intensity per hour from IBS. For moreintense bunches, the loss rate is enhanced. Approximat-ing the intensity loss at the injection plateau as linearand neglecting losses during t ta,FCC , the total collidingbeam intensity can be estimated with N beam = k b N b n inj X i =1 (1 − R loss t ta ( i − . Dividing this by the injected beam intensity, n inj k b N b ,gives the fractional part of the intensity surviving untilcollision. Taking into account that L ∝ N , one can2 t coll @ h D X L i n t \ (cid:144) h @ Μ b - D Turn Around Time t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = n inj = (a) Average integrated luminosity per h. n inj t ta,LHC @ h D t c o ll @ h D t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = (b) Optimal time in collision. n inj t ta,LHC @ h D X L i n t \ (cid:144) r un @ nb - D t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = (c) Optimised integrated luminosity. FIG. 7: (a) Average integrated luminosity per hour,(b) optimal time in collision, assuming different numberof LHC injections, n inj , (c) optimised integratedluminosity in a 30 days Pb-Pb run.approximate that the potential luminosity is reduced bya factor ( N beam /n inj k b N b ) due to IBS at the injectionplateau.Multiplying h L int i by this factor leads to the estimatesof h L int i /run shown in Fig. 7c for up to n inj = 4. Thecorresponding optimal time in collision is displayed in TABLE IV: Pb-Pb luminosity. The maximumintegrated luminosity per bunch calculated withEq. (29) is L int,fill = 0 . µ b − . Unit per Bunch k b Bunches L initial [Hz/mb] 0.006 2.6 L peak [Hz/mb] 0.017 7.3 L int,fill [ µ b − ] 0.13 57.8 L int,run [nb − ] 0.02 8.3 Fig. 7b. The total luminosity per run is shown as a func-tion of the LHC turnaround time. This in an essentialquantity to be improved for FCC, as it significantly in-fluences the operation strategy. t ta,FCC = 2 h is assumed.The plot makes clear, that the longer t ta,LHC the less at-tractive it becomes to inject more than once. It has to beconsidered that the larger n inj , the higher the risk of los-ing an LHC fill during the injection process. This wouldlengthen the injection plateau by several hours and hencereduce the achievable luminosity. Moreover, for shorter t ta,FCC , the crossing point of the curves shifts to the left,meaning that even for faster LHC cycles the potentialluminosity outcome might be higher for fewer injectionsper fill. The unknown turnaround time imposes a largeuncertainty on the estimates of h L int i per hour and run.Any operational problems leading to delays will reducethe overall efficiency and reduce the estimated perfor-mance.Table IV collects the numerical values for the initial,peak and integrated luminosity per fill in Pb-Pb opera-tion. The values quoted are taken from the simulation in-cluding coupling, to treat the most realistic case. The op-timisation is taken into account and the values are givenfor n inj = 1, t coll = 3 h, t ta = 2 h and t run = 30 days.The initial luminosity value is already 2.6 times over thenominal LHC, the peak could go up to around 7 timesnominal LHC, which would be of the order of the re-quested LHC Pb-Pb luminosity for Run 3. Luminosity Lifetime
The luminosity lifetime is defined as the time atwhich the luminosity has decreased to 1 /e of its ini-tial value. This time can easily be extracted from thesimulated data, by searching for the first time where L ( t = τ L ) ≤ L ( t = 0) /e : τ L = 6 . , with one experiment in collisions including burn-off, ra-diation damping and IBS. In case of two exactly oppositeexperiments, taking data under the same conditions, theluminosity lifetime will decrease accordingly, since theparticle losses per turn are doubled.3 Beam-Beam Tune Shift
The two beams travel in separated beam pipes. Onlyin the interaction regions they do pass through a com-mon pipe to bring them into collisions in the local exper-iment. In those regions of interaction the beams exertelectromagnetic forces on each other, the so-called beam-beam force. Especially during the passage of one bunchthrough the other in the IP, during a so-called head-on interaction, those forces can be very strong. In a simpli-fied picture, each single particle of one bunch receives akick from the opposite beam and is deflected by a cer-tain angle. In a linear approximation this kick acts as aquadrupole lens and thus introduces a tune shift, whichcan be approximated with the linear beam-beam param-eter ξ [6]: ξ i,u = N b,j r p Z i Z j β ∗ πA ion ,i γ i σ j,u ( σ j,u + σ j,v ) , (31)where the beam receiving the kick is labelled with i andthe beam exerting the force is labelled j . u and v describethe two transverse planes. r p is the classical proton ra-dius, Z and A ion the charge and atomic mass number ofthe corresponding beams, σ the beam size in the corre-sponding plane.For equal and round beams Eq. (31) simplifies to ξ = N b r β ∗ πγσ = N b r πǫ n = 3 . × − , (32)with r as the classical radius of the considered particleand ǫ n the normalised emittance. As it is easy to see,this equation only depends on the beams themselves andis independent of energy and lattice parameters. Equa-tions (31) and (32) describe the tune shift introduceddue to one head-on collision per turn, if the beams col-lide in more than one place, ξ has to be multiplied by thenumber of experiments in which the investigated bunchis colliding. The numeric value in Eq. (32) was obtainedwith the initial parameters given in Table I.The beam-beam tune shift can be a limiting factor forthe luminosity, since, if it becomes too large, the particlescould cross resonances and get lost. If this is the case, theintensities have to be reduced or the emittances blown-up to force the tune shift below its limit, consequentlythe luminosity will be reduced simultaneously. N b and ǫ n change during the fill and thus the beam-beam tuneshift. This is especially important in the case discussedhere, since with the damped emittance, the tune shift in-creases. From the simulation results displayed in Fig. 5the intensity and emittance evolutions are combined todetermine the variation of the beam-beam tune shift dur-ing a fill with one experiment in collisions, see Fig. 8.The peak value reaches ξ ≈ . × − . If more thanone experiment is taking data, this tune shift should bemultiplied by the number of experiments. However, thisis not exactly true for the curve in Fig. 8, since it was @ h D - Ξ FIG. 8: Evolution of the beam-beam tune shift for oneexperiment in collision.obtained from the simulated beam evolution consideringone active IP. The curve would change slightly (to lowervalues), due to the faster intensity burn-off and thus thebeam evolution for two or more experiments would bedifferent.Only during operation does it become certain wherethe beam-beam limit of a collider exactly is. For the p-poperation in the LHC, for instance, a beam-beam limitof 0.015 was expected, based on the Sp¯pS experience.Nevertheless, the tune shifts achieved in p-p in dedicatedexperiments exceeded the nominal value by almost a fac-tor of 5 and the value reached in normal operation byalready a factor of 2 [21].Comparing to those factors, and taking into accountthat the usual tune stability in the LHC is in the orderof 10 − , the beam-beam tune shift in Pb-Pb operationfor FCC is not negligible, but probably also not at thelimit. Bound-Free Pair Production Power
Ultraperipheral electromagnetic interactions dominatethe total cross-section during heavy-ion collisions, seeEq. (16), and cause the initial intensity to decay rapidly[22]. The most important interactions in Pb collisionsare Bound-Free Pair-Production (BFPP) Pb + Pb −→ Pb + Pb + e + and Electromagnetic Dissociation (EMD) Pb + Pb −→ Pb + Pb + n . Those interactions change the charge state or mass of oneof the colliding ions, creating a secondary beam emerg-ing from the collision point. The resulting momentumdeviation of the secondary beam lies outside the momen-tum acceptance of the ring, resulting in an impact onthe beam screen in a localised position (depending on thelattice) most probably around a superconducting magnet4 @ h D P o w e r o f B FPP B ea m @ W D FIG. 9: BFPP1 beam ( Pb ions) power evolutionin Pb-Pb operation.downstream the IP. This occurs on each side of every IPwhere ions collide.Following Eq. (14), the production rate of those pro-cesses is proportional to the instantaneous luminosityand will thus change during the fill. Nevertheless, themagnets would suffer from a continuous high exposure.Already under LHC conditions, the risk of quenching asuperconducting magnet due to these losses is high [23].In the FCC the peak luminosity could be an order ofmagnitude higher, increasing the risk even further. Thepower, P , in those secondary beams can be calculated asthe production rate times the particle energy: P = σ c L γm ion c . Figure 9 shows the power evolution of the BFPP1beam ( Pb ions, capture of one e − ), which has thehighest cross-section and accordingly the highest inten-sity and damage potential. For the calculation the totalBFPP cross-section, σ BFPP = 354 b at 50 Z TeV, esti-mated with [18], was used. The probability of higherorder interaction, i.e., capturing two or more electronsand leading to a charge state of ≤ + is much smallerand ignored for the purpose of estimating the upper limitof the stored power.For the computation of the beam power, the simu-lated luminosity, shown in Fig. 5, was used. The maxi-mum power goes up to P ≈ . Pb ions, emission of one neutron) might as well hit the beamscreen, depositing additional energy. For comparison,the BFPP1 beam power in the nominal LHC is about26 W, which is already expected to cause operationalproblems and, possibly, long-term damage. Countermea-sures would definitely be required to absorb those parti-cles before they can impact on the superconducting mag-nets. It has to be studied, if a highly resistant collimator in the dispersion suppressor region, as discussed for HL-LHC heavy-ion operation [24, 25], would be sufficient tostop the beams produced in the collisions at the highestenergy of the FCC. PROTON-LEAD OPERATIONBeam and Luminosity Evolution
IBS approximately scales with r ∝ ( Z /A ion ) and istherefore weaker for protons than for lead ions. In fact,IBS is negligible for the (initial) proton beam parame-ters used in p-Pb operation at top energy. The radiationdamping rates in Eq. (13) show a dependence on the par-ticle type as ( E b Z ) r /m ∝ Z /A . Calculating thisratio shows that the radiation damping for lead is abouttwice as fast as for protons at the same equivalent en-ergy. Thus, the emittances of both beams evolve withdifferent time constants. Consequently, eight differentialequations, four per beam, have to be solved simultane-ously to describe the beam and luminosity evolution forp-Pb collisions. Those could be reduced to six equationsby assuming fully coupled transverse motion and roundbeams, in this case ǫ ( t ) = ǫ x ( t ) = ǫ y ( t ) holds at all times.Rewriting Eq. (15) under this approximation leads to theinstantaneous luminosity for p-Pb L = A N b (p) N b (Pb) ǫ (p) + ǫ (Pb) , (33)with A = f rev k b / (2 πβ ∗ ). With this, the differential equa-tion system followsd N b ( i )d t = − σ c, tot A N b ( j ) N b ( i ) ǫ ( j ) + ǫ ( i ) (34)d ǫ ( i )d t = ǫ ( i )( α IBS ,x,y ( i ) − α rad ,x,y ( i )) (35)d σ s ( i )d t = 12 σ s ( i )( α IBS ,s ( i ) − α rad ,s ( i )) , (36)where only the three equations of beam i (either Pb orp) are noted. The equations for beam j have an equiv-alent form with different initial conditions and growthrates. The dependences of the IBS growth rates on N b , ǫ and σ s couple the three equations for each beam. Thedependence of the luminosity on both beams’ emittancesand intensities couple the Pb and p beam evolution. Anexact analytic solution of this coupled differential equa-tion system does not exist. Unfortunately, the CTE pro-gram does not feature simulations with different particlespecies, so only approximated analytical and numericalsolutions of the ODE system are available to perform es-timates.At the beginning of the fill, α IBS ≪ α rad and it canbe approximated that α ǫ = α ǫ ( P b ) = 2 α ǫ ( p ) = const. in5all three planes. This constant emittance decay rate, ne-glects the dynamically changing IBS with damped emit-tance. As seen in the Pb-Pb analysis, the peak and in-tegrated luminosity estimates done under those assump-tions will be overestimated, due to the emittances asymp-totically approaching zero.In general, the proton beams are more intense com-pared to lead. In the LHC proton-proton operation,bunches with 10 particles are regularly used. Leadbunches have in the order of 10 particles. In proton-leadoperation, it is possible to increase the initial luminosityby increasing the proton intensity (the lead intensity isassumed to be at the limit). Nevertheless, the higher theproton intensity, the stronger the beam-beam effects inthose strong-weak interactions. Therefore, it was chosenfor the LHC proton-lead run in 2013 [8] to use protonintensities around 10% of the nominal value used in p-p operation. This should also be the baseline for p-Pbcollision mode in FCC-hh.In each collision of a proton with a lead ion, thosetwo particles are removed from their beams. Therefore,the maximum integrated luminosity is reached when eachlead ion has found a collision partner in the more intenseproton beam. The number of lead ions is only about 1%of the number of protons. In the limit of burning-off allthe lead, the proton intensity is hardly changed and couldbe considered as roughly constant through the whole fill.To find an approximated analytical equation for theproton-lead luminosity evolution, the following assump- tions are made: N b (Pb) ≪ N b (p) = N b (p) = const. (37) α ǫ = α ǫ (Pb) = 2 α ǫ (p) ≈ − α rad (Pb) (38) α rad ,s = 2 α rad ,x,y (39)where Eq. (38) is assumed for all three planes andEq. (39) follows from Eq. (13). Applying those constrainsto the differential equations (34) - (36) leads to an expo-nential behaviour of the emittance and bunch length ofboth beams with related time constants ǫ (Pb , t ) = ǫ (Pb) exp[ α ǫ t ] (40) σ s (Pb , t ) = σ s (Pb) exp[ α ǫ t ] (41) ǫ (p , t ) = ǫ (p) exp[ α ǫ t/
2] (42) σ s (p , t ) = σ s (p) exp[ α ǫ t/ , (43)where the emittance growth rate of the Pb beam is takenas the reference, α ǫ ≈ − α rad ,x,y (Pb). This value is neg-ative, hence those are exponential decays. The protonintensity was assumed to be time independent, thus N b (p , t ) = N b (p) . (44)To solve the last equation for the Pb intensity evolution,Eq. (40), (42) and (44) are inserted into Eq. (34), followedby applying the method of separation of variables. Thesolution of the arising integral isln ( N b (Pb , t)) = Z d xx ( ax + b ) = − bx + ab ln (cid:18) ax + bx (cid:19) with x = exp[ α ǫ t/ N b (Pb , t ) = N Pb e − σc, tot AN p αǫǫ ( ǫ p (exp[ − α ǫ t/ − ǫ Pb ln[ ǫ p + ǫ Pb ] − ǫ Pb ln[ ǫ p exp[ − α ǫ t/ ǫ Pb ]) . The equations for the evolution of the emittance andintensity are inserted into Eq. (33) to obtain the p-Pb lu-minosity evolution. Figure 10 presents the results. Theabove derived analytical approximation is shown as thesolid lines, while the dashed lines correspond to the nu-merical solution of the ODE system. The evolution ofthe intensity (middle left), beam size at the IP (middleright) and bunch length (bottom) are displayed in blackfor the proton and in red for the lead beam.The peak luminosity is shifted to later times comparedto Pb-Pb operation, due to the slower radiation dampingfor protons, leading to longer fills. The Pb intensity burn-off is very fast, while the proton intensity hardly changes.This arises form the fact that in one collision one Pbnucleus is lost per proton. A free knob to adjust theluminosity peak and evolution is the proton intensity.Increasing N b (p) would lead to higher initial and peak rates followed by an even faster Pb burn-off and shorterfills. Decreasing N b (p) would distribute the achievableluminosity over a longer period with reduced peak rates.The 1 /e -luminosity lifetime, extracted from the nu-merical solution of the ODE system shown in Fig. 10,determines to τ L = 14 . . Optimising the Integrated Luminosity
Because of the weaker IBS for protons, their intensitylosses at injection are smaller and the proton beam canwait in the machine without deteriorating significantly.Therefore, the proton beam is injected first, followed bythe lead. Depending on the number of injections, eitherboth LHC rings are filled with the same species, or the6
Ana. Eq.: Α IBS <<Α rad
Numeric ODE @ h D L b @ c m - s - D p - Pb Bunch Luminosity
Ana. Eq.: Α IBS <<Α rad
Numeric ODE @ h D L b , i n t @ Μ b - D p - Pb Integrated Bunch Luminosity Evolution
Pb: Α Ε = const.Pb: Num. ODEp: Α Ε , N b = const.p: Num. ODE @ h D N b @ c h a r g e s D Bunch Intensity
Pb: Α Ε = const.Pb: Num. ODEp: Α Ε , N b = const. p: Num. ODE @ h D Σ * @ Μ m D Transverse Beam Size at the IP
Pb: Α Ε = const.Pb: Num. ODEp: Α Ε , N b = const.p: Num. ODE @ h D Σ s @ m D Bunch Length
FIG. 10: p-Pb beam and luminosity evolution for one experiment in collisions. Top: instantaneous (left) andintegrated (right) bunch luminosity, middle: intensity (left) and beam size at the IP (right), bottom: bunch lengthfor the proton (black) and lead (red) beam. Approximated analytic calculations (solid lines), neglect thedynamically changing IBS, leading to unrealistically small beam sizes. The numerical ODE solution is shown asdashed lines, giving more realistic estimates.filling is shared between the species and each LHC beamis injected in opposite directions in the FCC. In this waythe number of particles surviving until top energy is max-imised.From the numerical solution of the ODE system, which provides the best estimate of the beam and luminos-ity evolution available today, the average luminosity perhour is determined. Similar to Pb-Pb, an expression forthe total available p and Pb beam intensity in collision isderived, taking into account the different waiting times7TABLE V: p-Pb luminosity. σ c,tot = 2 b was used. Themaximum integrated luminosity per fill calculated withEq. (29) is L int = 30 nb − . Unit per Bunch k b Bunches L initial [Hz/mb] 0.5 213 L peak [Hz/mb] 2.8 1192 L int,fill [ µ b − ] 48.7 21068 L int,run [nb − ] 4.1 1784 and loss rates at the injection plateau. The average lumi-nosity per hour is then calculated as in Eq. (30) reducedby the factor( N beam (Pb) /n inj k b N b (Pb)) × ( N beam (p) /n inj k b N b (p))(45)for losses during injection.Figure 11 shows the results for the average integratedluminosity (a) and the corresponding time in collisions(b) to achieve the optimised integrated luminosity per30 days run (c). For n inj = 1 the maximal luminosity of1 . − /run is reached for a fill length of 6 . Beam Current Lifetime
As mentioned, the ion beam is naturally weak, whileproton beams can be produced with much higher intensi-ties. In the collision the lead beam loses Z = 82 chargesper lost proton. Thus, the ion beam will in general havethe smaller beam current lifetime, i.e., faster intensitydecay. Consequently, the ion beam lifetime determinesthe length of the fill in p-Pb operation.The beam current lifetime is given by1 τ N = − N d N d t = − N σ c, tot L , with N = k b N b and N b = N b (Pb). Inserting Eq. (33)for the luminosity, the Pb beam current lifetime in p-Pbcollisions is − τ N (Pb , t ) = 2 πβ ∗ ( ǫ (p , t ) + ǫ (Pb , t )) σ c, tot n exp f rev N b (p) . (46) t coll @ h D X L i n t \ (cid:144) h @ nb - D Turn Around Time t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = n inj = (a) Average integrated luminosity per hour. n inj t ta,LHC @ h D t c o ll @ h D t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = (b) Optimal time in collision. n inj t ta,LHC @ h D X L i n t \ (cid:144) r un @ pb - D t ta = t ta,FCC + H n inj - L t ta,LHC t ta,FCC = (c) Optimised integrated luminosity. FIG. 11: (a) Average integrated luminosity per hour,(b) optimal time in collision, (c) optimised integratedluminosity for a 30 days p-Pb run.The first factor is constant for N b (p) ≫ N b (Pb). Hence,the lifetime only varies in time proportionally to theconvoluted emittance of the two beams. As expected, τ N (Pb) decreases with increasing proton intensity, be-cause of the higher interaction probability. The initial8value evaluates to τ N (Pb , t = 0) = 39 . n exp = 1. Owing to the damping of the emittances,these values will decrease exponentially during the filland lead to a much shorter fill durations. It is interestingto note that Eq. (46) is independent of the lead beamcurrent. Beam-Beam Effects
Unequal Beam Sizes
The initial beam sizes of the proton and lead beamin p-Pb operation is assumed to be equal. Because ofthe stronger radiation damping for Pb, the Pb beam sizefalls below the proton beam size in the first period ofthe fill, see Fig. 10. After about one hour in collisionsthe Pb emittance reaches the balanced regime and doesnow change only slowly due to the intensity losses andthe thus decreasing IBS rate. Since the IBS is weaker forprotons, the emittance is damped to a lower value. Afterabout 1 . . ≤ σ Pb /σ p ≤ . . σ Pb /σ p , indicates that a potential reductionof the Pb beam lifetime due to unequal beam sizes wouldprobably affect the luminosity only in the second half ofthe fill, when the collision rates have already past themaximum.For comparison, the Tevatron ran with mismatchedbeam sizes between the proton and antiproton beam ofaround σ p /σ ¯p ≈ σ Pb /σ p ≈ p ® Ξ H Pb L Pb ® Ξ H p L @ h D - Ξ FIG. 12: p-Pb beam-beam tune shift for 1 IP incollision.
Tune Shift
With Eq. (31) the beam-beam tune shift ξ can be cal-culated for weak-strong beam-beam interactions as in thecase of p-Pb collisions. The initial beam parameters inTable I are such that the number of charges and the beamsizes of both beams are approximately equal, resulting inthe same tune shift, ξ (p) ≈ ξ (Pb) = 3 . × − , at thebeginning of the fill. However, the proton and lead beamproperties evolve differently with time, changing the forceexerted from one to the other during the fill. Figure 12shows the calculation of ξ based on the numerical solu-tion of the ODE system. The effect on the proton (black)beam is small ( ξ (p) < × − ). The increase of ξ (p)due to the shrinking lead beam emittances is negated bythe rapid Pb intensity losses. Owing to the almost con-stant proton intensity but damping emittances, the tuneshift to the Pb beam becomes significant and approachesa value of ξ (Pb) = 8 . × − in the regime where IBSand radiation damping start to balance each other. Thisvalue is close to the assumed beam-beam limit of ξ = 0 . PROTON-PROTON OPERATION
In the following the tools used in the above analysis areapplied to p-p operation in the FCC. In p-p operation twoscenarios are under investigation, namely bunches spacedby 25 or 5 ns with different beam properties. The protonbeam parameters are listed in Table VI.Radiation damping is negligible for protons at injec-tion energy. At 50 TeV the transverse and longitudinalemittance radiation damping times are 1 /α rad ,x,y = 1 . /α rad ,x,y = 0 . − µ m, which is still an order of mag-nitude smaller than the emittance ranges of the scenarios9TABLE VI: Assumed beam parameters forproton-proton operation [4]. Parameter Symbol Unit 25 ns 5 nsNo. of particles per bunch N b [10 ] 1.0 0.2Normalised transv. emittance ǫ n [ µ m] 2.2 0.44RMS bunch length σ s [m] 0.08 0.08No. of bunches per beam k b - 10600 53000 β -function at IP β ∗ [m] 1.1 1.1Total cross section σ c, tot [mb] 153 153No. of main IPs - - 2 2 TABLE VII: Initial IBS growth times for protonscalculated with Piwinski’s formalism, assuming thebaseline lattice ( L c = 203 m). Assumption formomentum spread: injection σ p = 1 . × − , collision(a) σ p = 0 . × − (obtained with γ T of baselinelattice), (b) σ p = 1 . × − (LHC design). Growth Times Unit Injection Collision25 ns 5 ns (a) (b)25 ns 5 ns 25 ns 5 ns1 /α IBS ,s [h] 25.9 21.3 283.1 264.7 1467 15341 /α IBS ,x [h] 37.7 6.2 169.5 31.7 265.4 55.51 /α IBS ,y [h] − − − − − − considered.As already explained, depending on the lattice choice,the IBS growth rates can be rather different. Figure 13shows the IBS growth times as a function of the FODOcell length, L c , at (a) injection and (b) top energy. Thesame behaviour as for Pb is observed, whereas the ratesare lower. For the chosen baseline lattice with L c ≈
203 m and γ T ≈ N b d t = − σ c, tot A N b ǫ d ǫ d t = α IBS ,x ǫ − α rad ,x ( ǫ − N b N b ǫ )d σ s d t = 0 ,
25 ns Long.25ns Hor.5ns Long.5ns Hor.
120 140 160 180 2000102030405060 L c @ m D (cid:144) Α I B S @ h D (a) Initial IBS at injection.
25 ns Long.25ns Hor.5ns Long.5ns Hor.
120 140 160 180 2000100200300400500600700 L c @ m D (cid:144) Α I B S @ h D (b) Initial IBS at top energy. FIG. 13: Initial IBS growth times and theirdependence on the FODO cell length, L c , at injection(a) and top energy (b) for p-p operation.here a constant bunch length and a transverse emittanceblow-up designed to keep the beam-beam parameter ξ at its initial value is implemented. For the FCC studyit is assumed that the peak luminosity is limited by amaximum beam-beam tune shift of ξ = 0 .
01, from whichthe initial beam parameters were derived. Leaving thebeams to evolve freely leads to an increase of up to ∼ . µ m.The peak luminosity reaches 16 × cm − s − for 25 nsand to 11 × cm − s − for the 5 ns scenario. Sincethe beam-beam parameter is proportional to N b /ǫ , theluminosity will decay exponentially, if ξ = const. Thisluminosity decay could be mitigated by β ∗ -levelling. Theminimum β ∗ is constrained by the aperture in the triplet,thus β ∗ could be lowered proportionally to the shrinkingemittance, resulting in an about constant luminosity aslong as the damping is strong enough.0 Ξ= const5ns ODE, Ξ= const @ h D L @ c m - s - D Instantaneous Luminosity, 2 IPs Ξ= const5ns ODE, Ξ= const @ h D L i n t @ f b - D Integrated Luminosity, 2 IPs Ξ= const Ξ= const @ h D N b @ c h a r g e s D Bunch Intensity Ξ= const Ξ= const @ h D Ε x @ Μ m D Horizontal Normalised Emittance Ξ= const5ns ODE, Ξ= const @ h D Σ s @ m D Bunch Length Ξ= const5ns ODE, Ξ= const @ h D Ξ t o t Beam - Beam Tune Shift
FIG. 14: p-p beam and luminosity evolution for two experiment in collisions. Top: instantaneous (left) andintegrated (right) luminosity, middle: intensity (left) and normalised emittance (right), bottom: bunch length (left)and total beam-beam tune shift (right). Solid lines show free beam evolution without artificial blow-up, dashed linesshow situation with constant bunch length and transverse emittance blow-up such ξ = const. Beams for 25 (red) and5 ns (black) bunch spacing are investigated. The instantaneous and integrated luminosity, the bunch length and tuneshift evolution are very similar (overlapping lines) for both bunch spacings if ξ = const.Figure 15 shows the average integrated luminosity asa function of the time in collisions, assuming a totalturnaround time of t ta = 5 h (as in [4]), evaluated withEq. (30) and the results shown in the upper right plot of Fig. 14. The four cases discussed in the previousparagraph are displayed. The particle losses of protonbunches on the LHC injection plateau are small and thusneglected. The optimum time in collisions calculates to1 Ξ= const5ns ODE, Ξ= const t coll @ h D X L i n t \ (cid:144) h @ f b - D t ta = FIG. 15: Average integrated luminosity per hour in p-poperation.5 . . . − (25 ns) and 3 . − (5 ns) could be (onaverage) collected per day. Considering ξ = const., thetwo options are very similar. The integrated luminosityis maximised for 11 . . − / day. If the beam-beam limit is higher thanexpected and the beams could be left to evolve freely, theluminosity outcome could potentially be doubled. SUMMARY TABLE
In Table VIII calculated and assumed parameters forPb-Pb, p-Pb and p-p operation at E b = 50 Z TeV in theFCC-hh are summarised. In case of p-Pb operation thePb beam is assumed to be the same as for Pb-Pb, there-fore the corresponding column only quotes the protonbeam parameters. The Pb beam parameters at injectionare listed as well as the LHC Pb-Pb and p-p design pa-rameters [11]. The p-p luminosity parameters given arebased on the case where the beam-beam tune shift is keptconstant to its initial value. The ” / ” separates the resultsfor two beam options. CONCLUSIONS
The FCC will enter a new regime of hadron collideroperation. Strong radiation damping will lead to smallemittances and very efficient intensity burn-off. Theemittances and bunch length become so small that ar-tificial blow-up might be necessary to avoid instabilities.An artificial blow-up might also be used as a way of lumi-nosity levelling. Because of the small beam dimensions,the peak Pb-Pb luminosity can expected to be about 7times the nominal LHC design value. The absolute in-tegrated luminosity maximum per fill, when all particlesare converted into luminosity, comes into reach, again because of the natural cooling from radiation damping.It is estimated that an integrated luminosity of about8 nb − could be expected per run of 30 days.If the LHC is used as the last pre-accelerator, its cycletime has to be drastically improved. Otherwise, the timebetween two injections into the FCC will be in the sameorder as the expected time in collisions per fill. To opti-mise the run time, the LHC could be re-filled in parallelto physics operation, maximising the time in physics andthe integrated luminosity, while filling only one fourth ofthe FCC.In p-Pb operation, the fill length is determined by theburn-off of the lead beam. The longer radiation dampingtime and weaker IBS of the proton beam, lead to longerfills in p-Pb operation. However, by adjusting the protonbeam intensity the luminosity peak and time distributioncould be levelled.The formalisms developed for the heavy-ion operationhave also been applied to p-p operation. First predic-tion of the p-p beam and luminosity evolution, underthe assumption of constant bunch length and an emit-tance blow-up, designed to keep the beam-beam tuneshift ξ = const., have been presented. Furthermore, IBScalculations show that transverse emittance growth forlong injection plateaus could become an issue for highintensity, low emittance protons. ACKNOWLEDGMENTS
It is a pleasure to thank John M. Jowett for the reveal-ing discussions and the support during the preparationof this work. I would also like to acknowledge AndreaDainese, Silvia Masciocchi and Urs Wiedemann for themotivation of this study and the opportunity to presentthe results in their collaboration meetings, providing adiscussion background on the physics potential of nuclearbeams at the FCC. I am also grateful to Daniel Schultefor discussions. This work is supported by the Wolfgang-Gentner-Programme of the Bundesministerium f¨ur Bil-dung und Forschung (BMBF), Germany. ∗ [email protected][1] O. Dominguez and F. Zimmermann, in
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Circumference [km] 26.659 100 100Field of main bends [T] 8.33 1.0 16Bending radius [m] 2803.95 10424 10424Cell length [m] 106.9 203 203Gamma transition γ T General Beam Parameters
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