Scaling Behavior of Circular Colliders Dominated by Synchrotron Radiation
SSCALING BEHAVIOR OF CIRCULAR COLLIDERS DOMINATED BYSYNCHROTRON RADIATION
Richard TalmanLaboratory for Elementary-Particle Physics Cornell UniversityWhite Paper at the 2015 IAS Program onthe Future of High Energy Physics
Abstract
The scaling formulas in this paper—many of which in-volve approximation—apply primarily to electron colliderslike CEPC or FCC-ee. The more abstract “radiation dom-inated” phrase in the title is intended to encourage use ofthe formulas—though admittedly less precisely—to protoncolliders like SPPC, for which synchrotron radiation beginsto dominate the design in spite of the large proton mass.Optimizing a facility having an electron-positron Higgsfactory, followed decades later by a p,p collider in the sametunnel, is a formidable task. The CepC design study con-stitutes an initial “constrained parameter” collider design.Here the constrained parameters include tunnel circumfer-ence, cell lengths, phase advance per cell, etc. This approachis valuable, if the constrained parameters are self-consistentand close to optimal. Jumping directly to detailed designmakes it possible to develop reliable, objective cost estimateson a rapid time scale.A scaling law formulation is intended to contribute to“ground-up” stage in the design of future circular colliders.In this more abstract approach, scaling formulas can be usedto investigate ways in which the design can be better opti-mized. Equally important, by solving the lattice matchingequations in closed form, as contrasted with running com-puter programs such as MAD, one can obtain better intuitionconcerning the fundamental parametric dependencies. Theground-up approach is made especially appropriate by theseemingly impossible task of simultaneous optimization oftunnel circumference for both electrons and protons. Thefact that both colliders will be radiation dominated actuallysimplifies the simultaneous optimization task.All GeV scale electron accelerators are “synchrotron radi-ation dominated”, meaning that all beam distributions evolvewithin a fraction of a second to an equilibrium state in which“heating” due to radiation fluctuations is canceled by the“cooling” in RF cavities that restore the lost energy. To thecontrary, until now, the large proton to electron mass ratiohas caused synchrotron radiation to be negligible in pro-ton accelerators. The LHC beam energy has still been lowenough that synchrotron radiation has little effect on beamdynamics; but the thermodynamic penalty in cooling thesuperconducting magnets has still made it essential for theradiated power not to be dissipated at liquid helium temper-atures. Achieving this has been a significant challenge. Forthe next generation p,p collider this will be even more true.Furthermore, the radiation will effect beam distributions on time scales measured in minutes, for example causing thebeams to be flattened, wider than they are high [1] [2] [3].In this regime scaling relations previously valid only forelectrons will be applicable also to protons.This paper concentrates primarily on establishing scalinglaws that are fully accurate for a Higgs factory such as CepC.Dominating everything is the synchrotron radiation formula ∆ E ∝ E R , (1)relating energy loss per turn ∆ E , particle energy E and bendradius R . This is the main formula governing tunnelcircumference for CepC because increasing R decreases ∆ E .The same formula will possibly dominate future protoncolliders as well. But the strong dependence of cost on super-conducting magnetic field causes the optimization of SPPCto be more complicated. Nevertheless scaling laws previ-ously applicable only to electron rings, will apply also toSPPC. In particular, like CepC, as the SPPC helium coolingcost becomes fractionally more important, its proportionalityto 1 / R favors increased ring circumference.With just one exception (deconstructing Yunhai Cai’sintersection region (IR) design) this paper makes no use ofany accelerator design code such as MAD. Apart from thefact that none is necessary, this is to promote the attitudethat scaling from existing facilities (mainly LEP in this case)is more reliable than “accurate” numerical investigation.This paper is intended to promote “ground up optimiza-tion” (as contrasted with “constrained parameter design”) offuture circular colliders, especially the Higgs factory. Butnot to perform an optimization. To the extent the investiga-tion has been started, the main suggestion is that far longercells, than have been used in existing studies, are favored,and should be investigated. From scaling relations, the op-timal cell length is approximately 200 m, which is severaltimes greater than some current designs. Scaling relationsalso suggest that an IP vertical beta function β ∗ y ≈ Scaling formulas in this paper are indicated by a broad bar in the leftmargin. In some cases a constant of proportionality is included. ONTENTS
Radius × Power Scale Invariant. . . . . . 9
Parameter Scaling with Radius. . . . . . 92.3
Staged Optimization Cost Model. . . . . 92.4
Scaling of Higgs Factory Magnet Fabri-cation . . . . . . . . . . . . . . . . . . . . 102.5
Cost Optimization . . . . . . . . . . . . . 112.6 Luminosity Limiting Phenomena . . . . . . 11Saturated Tune Shift. . . . . . . . . . . . . 11Beamstrahlung. . . . . . . . . . . . . . . . 12Reconciling the Luminosity Limits. . . . . 122.7 One Ring or Two Rings? . . . . . . . . . . 132.8 Predicted Luminosities . . . . . . . . . . . 132.9 Reconciling the Luminosity Formulas . . . 132.10 Qualitative Comments . . . . . . . . . . . 162.11 Advantages of Vertical Injection andBumper-Free, Kicker-Free, Top-Off Injection 17
Constant Dispersion Scaling with R . . . 22Linear Lattice Optics. . . . . . . . . . . . . 22Scaling with R of Arc Sextupole Strengthsand Dynamic Aperture. . . . . . . . 234.3 Revising Injector and/or Collider Parame-ters for Improved Injection . . . . . . . . . 23Implications of Changing Lattices for Im-proved Injection. . . . . . . . . . . 23 L × L ∗ Luminosity × Free Space Invariant 25 6 Transverse Sensitivity Length 27
Estimating β max y (and from it L ) . . . . . 276.2 Why β y must not be too large . . . . . . . . 276.3 Maximum β y Phenomenology Based onTransverse Orbit Sensitivity . . . . . . . . . 28
11 E. Deconstructing Yunhai Cai’s IR Optics 38
INTRODUCTION
For a cicular e+e- colliding beam storage ring to serve as a“Higgs Factory” its single beam energy has to be significantlyhigher than the E max ≈
100 GeV LEP energy. At such highenergies, as Telnov [4] has pointed out, beamstrahlung limitsthe performance with a severity that increases rapidly withincreasing E max .The basis for most of this paper, (the first formula inAppendix A, “Synchrotron Radiation Preliminaries”) gives U , the energy loss per turn, per electron, as a function ofring radius R , and electron beam energy E ; U [GeV] = C γ E R , (2)where, for electons, C γ = . × − m / GeV . For pro-tons C γ = . × − m / GeV , For proton colliderspreceeding LHC synchrotron radiation (SR) was always neg-ligibly small owing to the large proton mass. For the LHC,SR influenced the design only through the efforts neededto avoid dissipating the radiated energy at liquid Heliumtemperature. The post-LHC future circular collider will bethe first for which beam dynamics and ring optimization willbe dominated by SR. This has always been true for electroncolliders.In this paper a simulation program is used to calculatethe achievable luminosity for bend radii ranging up to fourtimes the LEP/LHC ring radius, for single-beam energiesfrom 100 GeV to 300 GeV. There are three phenomena givingluminosity limits: L RF , RF-power limitation ; L bs , beam-strahlung limitation ; and L bb , beam-beam interaction limi-tation , all of which have complicated dependencies on ringparameters. Since the achievable luminosity is equal tothe smallest of these limits, the optimal choice of param-eters requires them all to be equal.
To be specially ex-ploited is a scaling law to be obtained according to whichthe optimized luminosity is a function only of the product RP RF , tunnel-radius multiplied by RF power.Because a next-generation p,p collider is to follow theHiggs factory in the same tunnel, it is necessary to optimizethe tunnel for both purposes. The proton energy for sucha collider will be so large that, like all electron rings (ina first for protons) the performance will be dominated bysynchrotron radiation and (for this reason) luminosity willdepend primarily on a radius, power product RP wall , muchlike the similar electron ring radius, power product. Thisdiffers from the electron case in that the power goes intorefrigeration rather than into RF generation, but the impacton luminosity optimization will be similar.Increasing either R or P is expensive but many acceler-ator costs are more sensitive to P than to R . This can leadto the (possibly counter-intuitive) result that the Higgs lu-minosity can be increased, at modest increase in cost, byincreasing R and decreasing P . This will be even more true for the p,p collider. But it already suggests that theexcess cost incurred in tunnel circumference needed foreventual p,p operation at energy approaching 100 TeV(over and above what could be minimally adequate forthe Higgs factory) may not be exhorbitant. Most of the paper consists of formulations largely specificto CepC, with emphasis on their scaling behavior. The re-mainder consists of more technical appendices. Some of thematerial has been copied with little change from reports pro-duced over the last two years. As such the parameters in thevarious appendices are in some cases outdated and may bemutually inconsistent. This should be no more disconcertingthan the fact that many important parameters, in fact, remainseriously uncertain. In any case it is the dependencies, notthe numerical values, that are important.The luminosity depends importantly on an inconvenientlylarge number of collider parameters, and it is the detailedabsolute numerical values of luminosity that are important.For that reason I have attempted to make all luminositypredictions self-consistent, even when the tables appear indifferent sections of the paper.On the other hand, the ring parameters assumed in dif-ferent sections may not be consistent. The most extremeinstance of this is the “Single Beam Multibunch Operationand Beam Separation” section, in which the beam separa-tion scheme assumes the quite short cell length adopted bythe CEPC lattice design. To ease the electric separationchallenge a multiple separator scheme was introduced, con-sistent with the CEPC parameters of early 2014 (and up tothe present). But the “Lattice Optimization for Top-Off Injec-tion” section advocates much longer cells, for which fewer,longer, electric separators may be adequate. So these twosections are mutually inconsistent. The "Scaling Law Depen-dence of Luminosity on Free Space L ∗ also assumes an un-accountably small CEPC cell length in its semi-quantitativeluminosity estimation.Another open question concerns chromatic compensa-tion of the interaction region optics. Scaling laws in the“Achromatic Higgs Factory Intersection Region Optics” and“Scaling Law Dependence of Luminosity on Free Space L ∗ ”appendices assume local chromatic compensation internalto the intersection region optics. Though the L ∗ scaling lawis not in question, the example optics used to “derive” it areby no means settled. Problems associated with introducingstrong bends within the intersection region optics may favor“old fashioned” chromatic compensation in the arcs, as inLEP.The subject matter of the appendices can be discernedfrom the Table of Contents. They contain little originalmaterial, but are justified by the fact that the original sourcesare scattered and hard to merge consistently. The quite low Higgs particle mass makes a circular elec-tron collider an effective Higgs factory. Furthermore, just asLHC followed LEP in the same tunnel, building first an elec-tron collider, and later a proton collider in the same tunnel,3epresents a natural future for elementary particle physics.Though this paper is almost entirely devoted to the circularelectron Higgs factory, it is appropriate to consider the ex-tent to which the parameter choices for the Higgs factorycan be biased to improve the ultimate proton collider. Apossible modest initial cost increase can be far more thancompensated by the improvement in ultimate proton colliderperformance.
The parameter most implicated in this discussion is,of course, the ring circumference.
Once fixed this choicewill constrain the facility for its entire, at least half century,life. Furthermore this choice needs to be made before anyof the many remaining design decisions have to be made.To focus the discussion one can attempt to define a rangeof circumferences running from C min , a circumference suchas 70 km, large enough to define a guaranteed frontier rolein high energy physics through the life of the facility, to C max , such as 100 km, a circumference small enough tohold the project cost low enoungh to make project approvallikely. Different high energy communities (all representedat the IAS Program) are differently qualified to determinethese limits. C max depends largely on cost; it is the projectdirectorate (with the assistance of accelerator scientists) whoare most qualified to assess the cost of the facility. But thecost is not the only factor influencing the approval probability.It is the particle theorists and experimentalists who are mostqualified to establish the high level of enthusiasm withoutwhich project approval would be unlikely. Certainly there isa circumference value C min below which their enthusiasmfor the project would decline significantly.Discussing and determining this regrettably subjectiverange of possible tunnel circumferences can be one theme ofthe IAS program, certainly with the hope that C min < C max .Important ingredients of this formulation are bottom-updesigns of the relation between tunnel circumference andbeam energy for both electrons and protons. The naturalorder of construction is electron ring first, proton ring sec-ond. This is because the electron ring itself is much lesschallenging, much less expensive, and much less dependenton continuing improvements in superconducting magnettechnology than is the proton ring.Minimizing the initial cost (and thereby improving theapproval likelihood) makes optimizing the electron ringdesign more urgent than optimizing the proton ring de-sign. In fact, since the ideal circumference for protons issurely greater than for electrons, what is needed is to max-imize the electron ring circumference while minimizingits cost—a seemingly impossible task.The thesis of this paper is that this optimization is notas hard as it seems. More concretely, it will be shown thatmaking the electron ring circumference “unnecessarily large”(from the point of view of minimally adequate Higgs particleproduction) can increase its cost less than proportionally,if at all, provided the RF power is reduced proportionally.This argument relies on a scaling law according to whichthe optimized luminosity is a function only of the productof circumference times RF power. The “unnecessarily large” qualifying phrase in the pre-vious paragraph may, itself, be unnecessarily conservative.Depending on initial discoveries with the Higgs factory, itmay well be found appropriate in a second Higgs phase, to in-crease the RF power in order to increase the e+e- luminosityproportionally. This would be valuable not only to producemore Higgs particles, but also, for example, to increase thebeam energy well above the Higgs production threshold tostudy other Higgs particle production channels.
Three important parameters of a circular colliding beamstorage ring are N ∗ , the number of collision points, N b ,the number of circulating bunches, and N p the number ofelectrons or positrons in each bunch, assumed equal for thetwo beams. The optimal values for these parameters dependon the beam energy E and on cost considerations. The case N ∗ = N b = N b = N b = N b = N ∗ = N b to decrease as beam energy E increases. Thebunches are separated horizontally at the four quadrant arcmidpoints. In the one ring design the beams share the samevacuum chamber, lattice elements and RF cavities.Parameters for large ranges of single beam parameters aregiven in Table 1. Note: the R = U ≈ eV maxrf =
65 GV is assumed, and “excess voltage”is defined by eV excess = eV maxrf − U . The RF voltage is setto such a high value in order to be sufficient to compensatefor U , even at quite high beam energies E . As a result thetaulated RF over-voltages eV excess are highly extravagant forlow beam energies. For actual running at E =
300 GeV ahigher value would be required. Subsequent tables in thepaper simply extend the rows of this table for specific β ∗ y values, for example β ∗ y = .
004 m, β ∗ y = .
006 m, and β ∗ y = .
008 m, in order to establish trends.
According tothe simulation model, the optimum is near β ∗ y ≈ mmat the Higgs energy. Numerical examples in the text areusually taken from the shaded rows.Like Table 1, Table 2 contains single beam parameters,but specialized to 100 km circumference, with rows limitedto physically significant energies; namely “Z” for the Z resonance energy, “W” for the W-pair threshhold, “LEP”as the “nominal” LEP” beam energy, “H” for the Higgsproduction threshhold, and “tt” as the top-pair threshhold,representing high energy Higgs production channels.Even in quite favorable cases the energy loss per turn U is as much as several percent of the total energy. To keepthe energy within 1% will then require a dozen or moreRF accelerating sections. Because of its high energy loss,the Higgs factory will actually resemble a slowly curvinglinac. Nevertheless, it represents an economy, relative to alinear collider, to retain electrons along with most of theirenergy and restore their radiated energy every turn, ratherthan discarding and replacing them, as is required in a linearcollider. This paper pays special attention to the beamstrahlunglimitation pointed out by Telnov [4], and proceeds to quan-tify the limitation by a “beamstrahlung penalty” P bs . Thispenalty turns out to be so severe, and its onset (with increas-ing beam energy E ) so sudden (see Figure 14) that a sensiblestrategy is to fix parameters so that P bs remains just barelyconsistent with the capability to replenish the lost particles.In previous, lower energy e+e- colliders, it has been cus-tomary to keep the r.m.s. bunch length σ z comparable to thevertical beta function β y , in order to minimize the hourglasseffect. Because of the beamstrahlung effect this strategy mayno longer be optimal for a Higgs factory. Rather it may bemore optimal to accept a higher hourglass penalty in orderto reduce the beamstrahlung penalty. Lengthening the bunch“softens” the x-ray spectrum proportionally, which stronglyreduces the likelihood of emission of a single photon of en-ergy high enough for the radiating electron to be lost. Aswell as softening the beamstrahlung spectrum, increasingthe bunch length also has the beneficial effect of reducingwall impedance effects. It must be kept in mind, however,that the bunch length is largely determined by the lattice design, and is not easily changed, for example as the beamenergy is changed.According to the simulation result given in Eq. (80), thesaturated tune shift value ξ sat . is proportional to √ r yz where r yz = β y /σ z . Figure 21 shows the hourglass correctionfactor H ( r yz ) to be quite accurately equal to √ r yz . Theproduct of ξ sat . H ( r yz ) appearing, for example, in Eq. (93) istherefore proportional to 1 /σ z (for fixed β y ). This tends tofrustrate efforts to increase luminosity by increasing bunchlength for the purpose of decreasing beamstrahlung.In all cases the luminosity is limited by available RF powerper beam. Following recent designs that have adopted P rf =
50 MW as a kind of nominal choice, some tables in this paperuse this value.
Other tables reflect my recommendationto reduce power to P rf = MW while doubling the ringcircumference.
Fixing P rf fixes the maximum total number N tot of particles stored in each beam. At pre-LEP beamenergies all other parameters would then have been adjustedto “saturate the beam-beam tune shift [5]”. At Higgs factoryenergies the RF power limitation, in conjunction with thebeamstrahlung constraint, could make this impossible whichwill limit the luminosity accordingly.Total power is not the only significant RF parameter. Forsome tables this paper, I also choose the maximum voltagedrop to be V rf =
65 GV, which is almost certainly muchhigher than will actually be provided for initial operation;it is about 20 times higher than the maximum voltage interminal LEP operation. A given value of V rf sets an absolutemaximum beam energy. To have non-zero luminosity at E max =
300 Gev, which is the highest energy appearing inTable 1, V rf has to be at least 60 GV, for the maximum bendradius considered in the table.At sub-LEP energies there will be ample RF power tosaturate the vertical tune shift and and the luminosity can befurther increased with multiple bunches.It will be shown that the total Higgs particle production,summed over all detectors, increases with the number ofdetectors N ∗ . However, following the tentative CEPC design,this paper usually assumes N ∗ = (cid:80) N ∗ i = ξ i , where ξ i = ξ is the vertical tune shift incollision point i . Nonlinear dynamics will limit the value ξ i to the same maximum value at every intersection point.But, once stable circulating beams have been established, Iassume that N ∗ ξ can be arbitrarily high, even greater than 1,for example.Weiren Chou and Tenaji Sen have pointed out that thetopping off repetition rate has to be quite high. For example,if the beam lifetime (without topping off) is 30 minutes, thebeam has to be topped off on the order of once per minuteto keep the beam current constant to better that one percent.The exact sequence of operations by which the stable,high current steady state is obtained will not be easy, nor5 C R f U eV excess n U / ( D / ) δ = α u c (cid:15) x σ arc x GeV km km KHz GeV GeV elec./MW MV/m GeV nm mm100 28 3.0 10.60 3.0 62 2.00e+11 0.626 0.0074 0.00074 6.354 0.523150 28 3.0 10.60 14.9 50 3.94e+10 3.169 0.0249 0.00249 14.297 0.784200 28 3.0 10.60 47.2 18 1.25e+10 10.016 0.0590 0.00591 25.417 1.05250 28 3.0 10.60 115.2 -50 5.11e+09 24.453 0.1152 0.01155 39.715 1.31300 28 3.0 10.60 239.0 -1.7e+02 2.46e+09 50.707 0.1991 0.01995 57.189 1.57100 57 6.0 5.30 1.5 64 7.98e+11 0.157 0.0037 0.00037 3.177 0.37150 57 6.0 5.30 7.5 58 1.58e+11 0.792 0.0124 0.00125 7.149 0.554200 57 6.0 5.30 23.6 41 4.99e+10 2.504 0.0295 0.00296 12.709 0.739250 57 6.0 5.30 57.6 7.4 2.04e+10 6.113 0.0576 0.00577 19.857 0.924300 57 6.0 5.30 119.5 -54 9.85e+09 12.677 0.0996 0.00998 28.595 1.11100 75 8.0 3.98 1.1 64 1.42e+12 0.088 0.0028 0.00028 2.383 0.32150 75 8.0 3.98 5.6 59 2.80e+11 0.446 0.0093 0.00094 5.361 0.48200 75 8.0 3.98 17.7 47 8.87e+10 1.409 0.0221 0.00222 9.532 0.64250 75 8.0 3.98 43.2 22 3.63e+10 3.439 0.0432 0.00433 14.893 0.8300 75 8.0 3.98 89.6 -25 1.75e+10 7.131 0.0747 0.00748 21.446 0.96100 94 10.0 3.18 0.9 64 2.22e+12 0.056 0.0022 0.00022 1.906 0.286150 94 10.0 3.18 4.5 61 4.38e+11 0.285 0.0075 0.00075 4.289 0.429200 94 10.0 3.18 14.2 51 1.39e+11 0.901 0.0177 0.00177 7.625 0.573250 94 10.0 3.18 34.6 30 5.68e+10 2.201 0.0346 0.00346 11.914 0.716300 94 10.0 3.18 71.7 -6.7 2.74e+10 4.564 0.0597 0.00599 17.157 0.859100 113 12.0 2.65 0.7 64 3.19e+12 0.039 0.0018 0.00018 1.589 0.261150 113 12.0 2.65 3.7 61 6.31e+11 0.198 0.0062 0.00062 3.574 0.392200 113 12.0 2.65 11.8 53 2.00e+11 0.626 0.0148 0.00148 6.354 0.523250 113 12.0 2.65 28.8 36 8.17e+10 1.528 0.0288 0.00289 9.929 0.653300 113 12.0 2.65 59.7 5.3 3.94e+10 3.169 0.0498 0.00499 14.297 0.784
Table 1: Ring parameters for rings of various bending radii, assuming 2/3 fill factor, with half of total straight section length D taken up by RF. The U / ( D / ) column therefore indicates the minimum required energy gain per meter to be suppliedby the RF. u c is the critical energy of the synchrotron radiation energy spectrum. α is the appropriate damping decrementfor N ∗ = name E C R f U eV excess n δ = α u c (cid:15) x † σ arc x GeV km km KHz GeV GeV elec./MW GeV nm mmZ 46 100 10.6 3.00 0.04 20 5.81e+13 0.00020 0.00002 0.573 2W 80 100 10.6 3.00 0.34 20 6.08e+12 0.00107 0.00011 1.771 1.19LEP 100 100 10.6 3.00 0.83 19 2.49e+12 0.00209 0.00021 2.767 0.972H 120 100 10.6 3.00 1.73 18 1.20e+12 0.00361 0.00036 3.984 0.824tt 175 100 10.6 3.00 7.83 12 2.66e+11 0.01119 0.00112 8.473 0.585
Table 2: Single beam parameters, assuming 100 km circumference. The second last column (†) lists the value of (cid:15) x appropriate only for β ∗ y = (cid:15) x has to be adjusted, according to the value of β ∗ y , tooptimize the beam shape at the IP. Other cases can be calculated from entries in other tables. U is the energy loss per turnper particle. u c is the critical energy for bending element synchrotron radiation. δ is the synchrotron radiation dampingdecrement.will it be discussed here. Once in this state, there will be afar more easily-met requirement that the variation of N ∗ ξ over the steady-state filling sequence remain less than thedistance to the nearest destructive resonance.It is certainly optimistic to permit arbitrarily large coherentbeam-beam tune shift. But, from the point of view of linearlattice dynamics, the crossing points are indistinguishablefrom idealy thin lenses, focusing in both planes. They are not,however, “high quality” lenses, in that they have significantoctupole, and higher multipole components, which may ormay not limit the dynamic aperture. The “optimistic” aspectof our optimistic conjecture being made is that the dynamicaperture limitation does not worsen with multiple collisionpoints. This is tantamount to assuming that a phasor sumneed not have magnitude greater than any one of its terms. The simulation model [5] used to establish saturated tuneshift conditions, though ten years old, has been fired upagain for the Higgs factory study. Operationally, saturationis specified, only semi-quantitatively, by the tune shift valueat which the quadratic dependence of luminosity on beamcurrent transitions from quadratic to linear. In the simu-lation model this transition point is quantitatively precise.Described in Appendix C, the model predicts the depen-dence of luminosity on damping decrement δ , vertical betafunction β y , bunch length σ z , the three tune values Q x , Q y and Q s , and the three bunch dimensions, σ x , σ y , and σ z . Some of these dependencies are exhibited in figures inAppendix C.Requiring beamstrahlung to be “barely acceptable” in thesense described so far influences the task of fixing the bendradius R , the total length of straight sections D , and the6otal circumference C . From a cost perspective these are themost important parameters. Before these parameters can bedetermined, the maximum energy E max has to be specified.Certainly the optimal ring size and cost increase more thanproportionally with increasing E max .Apart from its reduced cost compared to a linear collider(which is due to the surprisingly low mass the Higgs particlehas been found to have) the greatest advantage of a circularcollider is its well-understood behavior and correspondinglysmall risk. The only significant uncertainty concerns theparameter η which gives the fractional energy at which abeamstrahlung radiation causes the radiating electron to belost. For numerical estimates in this report, following Telnov,I have adopted the value η = . The main Higgs factory cost-driving parameter choicesinclude: tunnel circumference C , whether there is to beone ring or two, what is the installed power, and what is the“Physics” for which the luminosity deserves to be maximized.This section discusses some of the trade-offs among thesechoices, and attempts to show that the optimization goals forthe Higgs factory and the later p,p collider are consistent. A good way to fix the circumference C is to simply ex-trapolate from earlier colliding beam rings as is done inFigure 2. Choosing E =
300 GeV to be the nominal beamenergy yields circumference
C ≈
100 km. Nothing in thispaper is incompatible with this choice.
The quite low Higgs mass (125 GeV) makes a circulare+e- collider (FCC-ep) ideal for producing background-freeHiggs particles. There is also ample physics motivation forplanning for a next-generation proton-proton collider withcenter of mass energy approaching 100 TeV. This suggests atwo-step plan: first build a circular e+e- Higgs factory; laterreplace it with a 100 TeV pp collider (or, at least, center ofmass energy much greater than LHC). This paper is devotedalmost entirely to the circular Higgs factory step, but keep-ing in mind the importance of preserving the p,p colliderpotential. To illustrate this possibility the LHC and FCC-pp(scaled up from LHC based on radiation-dominated scaling)are also plotted on Figure 2.The main Higgs factory cost-driving parameter choicesinclude: tunnel circumference C , whether there is to be onering or two, what is the installed power, and what are thephysics priorities. From the outset I confess my prejudicetowards a single LEP-like ring, optimized for Higgs produc-tion at E =
120 Gev, with minimum initial cost, and highestpossible eventual p,p energy. This section discusses some
10 100 1000 10000 100000 1e+06 1 10 100 1000 l og ( c i r c u m f e r en c e [ m ] ) log(single beam energy) 81.45 E[GeV] m5 300ep GeV15T pp TeV12T pp TeV 12 36 499.6 29 39BEPC-I DORISVEPP-4,KEKCESR PETRA,PEP LEPCepCZ W H tt H+LHC FCC-pp25.760 80140 Figure 2: Relation between beam energy E and circumfer-ence C for numerous colliding beam rings. Because theFCC-pp collider will be synchrotron radiation dominated,its scaling up (from LHC) should follow the same trend.In this plot p,p colliders, LHC and FCC-pp are labeled onthe right hand side of the linear fit. But the p,p horizontalaxis depends on the assumed magnetic field value. The axisare labeled for 12 T and 15 T options. Like the earlier elec-tron colliders, CESR, PEP, PETRA, and LEP, the dynamicranges are about one octave in energy. The same will, pre-sumeably, be true for the Higgs factory— from Z at thelow end, through W -pairs, Higgs, t-tbar, to associated Higgsproduction channels at the high energy end.of the trade-offs among these choices, and attempts to showthat electron/positron and proton/proton optimization goalsare consistent.Both Higgs factory power considerations and eventual p,pcollider favor a tunnel of the largest possible radius R . Obvi-ously one ring is cheaper than two rings. For 120 GeV Higgsfactory operation (and higher energies) it will be shownthat one ring is both satisfactory and cheaper than two. Buthigher luminosity (by a factor of five or so) at the (45.6 GeV) Z energy, requires two rings.Unlike the Z , there is no unique “Higgs Factory energy”.Rather there is the threshold turn-on of the cross section,shown, along with other applicable cross sections in Figure 3.We arbitrarily choose 120 GeV per beam as the Higgsparticle operating point and identify the single beam energythis way in subsequent tables. Similarly identified are the Z energy (45.6 GeV), the W-pair energy of 80 GeV, the LEPenergy (arbitrarily taken to be 100 GeV) and the t ¯ t energyof 175 GeV to represent high energy performance.7igure 3: Higgs particle cross sections up to √ s = . L ≥ × / cm / s, will produce400 Higgs per day in this range. 8 .2 Scaling up from LEP to Higgs Factory Radius × Power Scale Invariant.
Most of the conclu-sions in this paper are based on scaling laws, either withrespect to bending radius R or with respect to beam en-ergy E . Scaling with bend radius R is equivalent to scalingwith circumference C . (Because of limited “fill factor”, RF,straight sections, etc., R ≈ C / This should make the extrapolation from LEP to Higgsfactory quite reliable. In such an extrapolation it is in-creased radius more than increased beam energy that ismainly required.
One can note that, for a ring three times the size ofLEP, the ratio of E / R (synchrotron energy loss per turn) is1 . / = . less than final LEP operation . Also,for a given RF power P rf , the maximum total number ofstored particles is proportional to R —doubling the ringradius cuts in half the energy loss per turn and doublesthe time interval over which the loss occurs. Expressedas a scaling law n = number of stored electrons per MW ∝ R . (3)This is boxed to emphasize its fundamental importance. Fol-lowing directly from Eq. (1), it is the main considerationfavoring large circumference for both electron and radiation-dominated proton colliders.These comments should completely debunk a long-heldperception that LEP had the highest energy practical for anelectron storage ring.There are three distinct upper limit constraints on the lumi-nosity. As explained in Appendix D, “Luninosity Formulas”,maximum luminosity results when the ring parameters havebeen optimized so the three constraints yield the same upperlimit for the luminosity. For now we concentrate on just thesimplest luminosity constraint L RFpow , the maximum luminos-ity for given RF power P rf . With n being the number ofstored particles per MW; f the revolution frequency; N b thenumber of bunches, which is proportional to R ; σ ∗ y the beamheight at the collision point; and aspect ratio σ ∗ x /σ ∗ y fixed(at a large value such as 15); L RFpow ∝ fN b (cid:18) n P rf [MW] σ ∗ y (cid:19) . (4)Consider variations for which P rf ∝ R . (5)Dropping “constant” factors, the dependencies on R are, N b ∝ R , f ∝ / R , and n ∝ R . With the P rf ∝ / R scaling of Eq. (5), L is independent of R . In other words, the luminosity depends on R and P rf only through theirproduct RP rf . Note though, that this scaling relation doesnot imply that L ∝ P at fixed R ; rather L ∝ P rf .In this paper this scaling law will be used in the form L ( R , P rf ) = f ( RP rf ) , (6)the luminosity depends on R and P rf as a function f ( RP rf ) of only their product.This radius/power scaling formula can be checked numer-ically by comparing Tables 6 and 8. The comparison is onlyapproximate since other parameters and the scalings fromLEP are not exactly the same in the two cases. Parameter Scaling with Radius.
For simplicity, evenif it is not necessarily optimal, let us assume the Higgsfactory arc optics can be scaled directly from LEP values,which are: phase advance per cell µ x = π/
2, full cell length L c =
79 m. (The subscript “c” distinguishes the Higgsfactory collider lattice cell length from injector lattice celllength L i .)Constant dispersion scaling formulas are given in Table 3.These formulas are derived in Section 4.2 “Lattice Opti-mization for Top-Off Injection”. They are then applied toextrapolate from LEP to find the lattice parameters for Higgsfactories of (approximate) circumference 50 km and 100 km,shown in Table 5. Parameter Symbol Proportionality Scalingphase advance per cell µ L c R / bend angle per cell φ = L c / R R − / quad strength ( / f ) q / L c R − / dispersion D φ L c β L c R / tunes Q x , Q y R /β R / Sands’s “curly H” H = D /β R − / partition numbers J x / J y / J (cid:15) = 1/1/2 1horizontal emittance (cid:15) x H / ( J x R ) R − / fract. momentum spread σ δ √ B R − / arc beam width-betatron σ x ,β √ β(cid:15) x R − / -synchrotron σ x , synch . D σ δ R − / sextupole strength S q / D R − / dynamic aperture x max q / S x max /σ x R / pretzel amplitude x p σ x R − / Table 3:
Constant dispersion Constant dispersion scaling isthe result of choosing cell length L ∝ R / . The entry “1”in the last column of the shaded “dispersion” row, indicatesthat the dispersion is independent of R when the cell length L c varies proportional to √ R with the phase advance percell µ held constant. Staged Optimization Cost Model.
For best likelihood of initial approval and best eventual p,pperformance, the cost of the first step has to be minimizedand the tunnel circumference maximized. Surprisingly, theserequirements are consistent. Consider optimization princi-ples for three collider stages:9
Stage I, e+e-:
Starting configuration. Minimize costat “respectable” luminosity, e.g. 10 . Constrain thenumber of rings to 1, and the number of IP’s to N ∗ = Stage II, e+e-:
Maximize luminosity/cost for produc-tion Higgs (etc.) running. Upgrade the luminosity bysome combination of: P rf → P rf or 4 P rf , one ring → two rings, increasing N ∗ from 2 to 4, or decreasing β ∗ y .• Stage III, pp:
Maximize the ultimate physics reach,i.e. center of mass energy, i.e. maximize tunnel cir-cumference.
Scaling of Higgs Factory Magnet Fabrica-tion
Unlike the rest of the paper, this section is conjectural andidiosyncratic. It contains my opinions concerning how bestto construct the Higgs factory room temperature magnets. Itdoes not pretend to understand the economics of supercon-ducting magnet technology. But it is also not ruled out thatsimilar arguments and conclusions may be applicable to theeventual p,p collider.As a disciple of Robert Wilson, one cannot avoid ap-proaching the Higgs factory design challenge by imagininghow he would have.
Certainly Bob Wilson would have en-dorsed Nima Arkani-Hamed’s attitude that we strive for100 TeV collisions “because the project is big”, ratherthan “in spite of the fact the project is big”. “How would Bob do it?” also suggests unconventionaldesign approaches. At the early design stage, based on good,but limited, understanding of the task, one of his princi-ples can be stated as “It is better for the tentative parameterchoices to be easy to remember than to be accurate”. In thecurrent context he would certainly have liked the round num-bers in a statement such as “To obtain 100 TeV collisions weneed a ring with 100 km circumference”, especially becauseof (or, possibly, in spite of) the fact that the CERN FCCgroup favors just these values.Another Wilson attitude was that, if a competent physi-cist (where he had himself in mind) could conceptualize anelegant solution to a mechanical design problem, consistentwith the laws of physics, then a competent engineer (wherehe again had himself in mind) could certainly successfullycomplete the design.In extrapolating the room temperature magnet design fromLEP to CepC one must first acquire a prejudice as to thevacuum chamber bore diameter. Many of the scaling formu-las in this paper are devoted to determining this, along withother self-consistent parameters. To make the subsequentdiscussion as simple as possible one can accept, as a firstiteration, the choice of making the magnet bore the sameas LEP, promising to later improve this choice, in a second,or third, iteration, as necessary. It is my guess that the firstiteration will be close.In round numbers, the 100 km Higgs factory ring mag-net length is four times as great as LEP’s, and the Higg’sfactory energy is greater than the maximum LEP energy in the ratio 120 / . / = .
3. The stored magnetic energy density scales asthe square of this ratio. With the magnetic bore constant, theHiggs factory stored magnetic energy is less than for LEP inthe ratio 4 × . = .
36. Ferromagnetic magnets are oftencosted in Joules per cubic meter. If this were valid the Higgsfactory magnet would be three times cheaper than the LEPmagnet.When one actually looks into magnet costs one finds thecalculation in the previous paragraph to be entirely mislead-ing. The actual costs tend to be dominated by end effects,fabrication, transportation and installation. Accepting thesecosts as dominant would, one might think, force one to acceptthe Higgs factory magnet cost being proportional to tunnelcircumference; this would be the cost of simply replicatingLEP magnets. One reason this might be too conservativeis that, with the Higgs factory cell length being longer, themagnets could be longer. But this would also be misleadingsince the LEP magnets were already as long as economi-cally practical (because of fabrication, transportation andinstallation costs).To hold down magnet costs, the inescapable conclusionto be drawn from this discussion is that the magnets have tobe built in situ , in their final positions in the Higgs factorytunnel. This is the only possible way to prevent the magnetcost from scaling proportional to the tunnel circumference,or worse. (The same is probably true for superconductingmagnets in the later p,p phase of the project.)It is not at all challenging to build the Higgs factory col-lider magnets in place. With top-off injection these magnetsdo not have to ramp up in field. As a result they have noeddy currents and therefore do not need to be laminated.Regrettably the same is not true for the injector magnet,which will be more challenging, and may be more expensive,than the collider magnet.An even more quixotic argument for building the magnetin place is to compare the arcs of the collider to high voltageelectrical power lines, which carry vast amounts of powerover vast distances. For example a 10 V line, carrying10 A, carries 10 W of power over a distance of 100 Km,with fractional energy loss of 1%. The arcs of the Higgsfactory will similarly carry 10 V at 10 − A over a distanceof 100 Km with fractional energy loss of 1%. Same power,same loss. One would not even think of building overlandpower lines in a factory before transporting them to wherethey are needed. The same should be true for acceleratormagnets.
SPPC:
For superconducting magnetic fields B in the rangefrom 4 to 7 Tesla the cost per unit volume [7] is roughlyproportional B / but increasing “more than linearly forhigher magnetic fields”, perhaps proportional to B at, say,12 T. If true, at fixed bore diameter and fixed energy themagnet cost would be more or less independent of tunnelradius R , and there would be little need to worry about thetunnel circumference being “too big” from this point of view.10s discussed previously the synchrotron radiation heatload cost is proportional to 1 / R at fixed E . In principle,none of the synchrotron radiation has to be stopped at liq-uid helium temperature but, in practice, this is very hard toachieve. As with electrons, the reduced synchrotron radia-tion power load can be exploited to increase the stored beamcharge by increasing R . This has the further beneficial effectof increasing the beam burn-off (interaction) lifetime. Prob-ably a more important interaction lifetime effect is that thestored charge can be proportional to R , causing the burn-offlifetime to be proportional to R . Cost Optimization
Treating the cost of the 2 detectors as fixed, and letting C be the cost exclusive of detectors, the cost can be expressedas a linearized fit, the sum of a term proportional to size anda term proportional to power; C = C R + C P ≡ c R R + c P P rf (7)where c R and c P are unit cost coefficients. As given byscaling formula (5), for constant luminosity, the RF power,luminosity, and ring radius, for small variations, are relatedby P rf = L k R . (8)Minimizing C at fixed L leads to R opt = (cid:114) k c P c R L . (9)Conventional thinking has it that c P is universal world widebut, at the moment, c R is thought to be somewhat cheaperin China than elsewhere. If so, the optimal radius shouldbe somewhat greater in China than elsewhere. Exploiting P rf ∝ L / R , some estimated costs (in arbi-trary cost units) and luminosities for Stage I and (HiggsFactory)Stage-II are given in Table 4.The cost ratios in this table were originally extracted fromthe LEP “Pink Book” [8]. They have now been more reli-ably up-dated using values from the CEPC Pre-CDR designreport [9]. The most significant finding is that doubling thecircumference while cutting the power in half increases thecost by a factor of 1.4.Being quite weak ferromagnets, the bending magnet costscould, in principle, be proportional to stored magnetic en-ergy. For this assumption to be at all realistic the magnetshave to be constructed in situ in the tunnel, in order to elim-inate transportation and installation costs. I am confidentthat sophisticated engineering can accomplish this.The luminosity estimates are from Table 8 and are ex-plained in later sections, especially Section 4“Lattice Opti-mization for Top-Off Injection”. Note that doubling the radius, while cutting the powerin half, increases the cost only modestly, while leaving
R P rf C tun C acc Phase-I L I L I cost (Higgs) ( Z ) km MW arb. arb. arb. Table 4: Estimated costs, one ring in the upper table, two inthe lower. C tun is the tunnel cost, C acc is the cost of the rest ofthe accelerator complex. Costs have been extrapolated fromthe CEPC pre-CDR proposal. *With one ring, changes R → R and P → P / R and P held fixed is estimated to increase the cost bya factor 1.64. The ratio in the table, 3.87/2.87=1.35, is thecost ratio of doubling the ring while cutting the RF power inhalf. From an authoritative CEPC source, this ratio is morereliably calculated to be 1.40. generous options for upgrading to maximize Higgs lu-minosity, as well as maximizing the potential p,p physicsreach. The shaded row in Table 4 seems like the best deal.Both Higgs factory and, later, p,p luminosities are maxi-mized, and the initial cost is (almost) minimized. Of coursethis optimization has been restricted to a simple choice be-tween 50 km and 100 km circumference.
Saturated Tune Shift.
My electron/positron beam-beam simulation [5] dead reckons the saturation tune shift ξ max which is closely connected to the maximum luminos-ity. For an assumed R ∝ E / tunnel circumference scaling, ξ max is plotted as a function of machine energy E in Figure 4.This plot assumes that the r.m.s. bunchlength σ z is equal to β ∗ y , the vertical beta function at the intersection point (IP).The physics of the simulation assumes there is an equilib-rium established between beam-beam heating versus radi-ation cooling of vertical betatron oscillations. Under idealsingle beam conditions the beam height would be σ y ≈ but this is unphysical . In fact beam-beam forcescause the beam height to grow into a new equilibrium withnormal radiation damping. It is parametric modulation ofthe vertical beam-beam force by horizontal betatron andlongitudinal synchrotron oscillation that modulates the ver-tical force and increases the beam height. The resonancedriving strength for this class of resonance is proportional to1 /σ y and would be infinite if σ y =0— this too is unphysical .Nature, “abhoring” both zero and infinity, plays off beam-beam emittance growth against radiation damping. Howeveramplitude-dependent detuning limits the growth, so there isonly vertical beam growth but no particle loss (at least fromthis mechanism). In equilibrium the beam height is propor-tional to the bunch charge. The simulation automaticallyaccounts for whatever resonances are nearby.11 M a x i m u m s a t u r a t ed t une s h i ft, ξ m a x " Beam Energy, E m [GeV] Figure 4: Plot of maximum tune shift ξ max as a function ofmaximum beam energy for rings such that E ∝ R / . Thenon-smoothness has to be blamed on statistical fluctuationsin the Monte Carlo program calculation. The maximumachieved tune shift parameter 0.09 at 100 GeV at LEP wasless than shown, but their torturous injection and energyramping seriously constrained their operations.To estimate Higgs factory luminosity the tune plane isscanned for various vertical beta function values and bunchlengths, as well as other, less influential, parameters. Theresulting ratio ( ξ sat / β ∗ y ) is plotted in Figure 5. The ratio ξ t y p . / β y σ z [m]’SIGMASeqBETYST’ u 1:5 Figure 5: Plot of ξ typ . / β y , the “typical” tune shift value ξ typ . inversely weighted by β y , as a function of σ z , with β y = σ z , δ = . Q s = . A just small enough for the tune shift to be saturatedwith the available number of electrons N p in each beambunch. ξ sat . / β ∗ y determines the beam area A β y just sufficient forvertical saturation according to the formula, (see Eq. (10)for this and other of the following formulas), A β y = πσ x σ y = N p r e γ ( ξ sat . / β y ) . (10) This fixes the tune-shift-saturated charge density (per unittransverse area). It is only the product σ x σ y that is fixedbut there is a broad optimum in luminosity for aspect ratio a xy = σ x /σ y ≈
15. To within this ambiguity all trans-verse betatron parameters are then fixed. β ∗ x is adjustedto make horizontal and vertical beam-beam tune shifts ap-proximately equal. The lattice optics is adjusted so that the(arc-dominated) emittance (cid:15) x gives the intended aspect ratio a xy ; (cid:15) x = σ x / β ∗ x .(Incidentally, it will not necessarily be easy to optimize (cid:15) x for each beam energy. Section “Lattice Optimizationfor Top-Off Injection” discusses tailoring cell length L c toadjust (cid:15) x . Unfortunately other considerations influence thechoice of L c and, in any case, once optimized for one energy, L c remains fixed at all energies.) Beamstrahlung. “Beamstrahlung” is the same as syn-chrotron radiation, except that it occurs when a particle inone beam is deflected by the electric and magnetic fieldsof the other beam. The emission of sychrotron radiationx-rays is inevitable and the lost energy has to be paid for.Much worse is the occasional radiation of a single photon(or, by chance, the sum of two) of sufficiently high energythat the reduction in momentum causes the particle itself tobe lost. This magnifies the energy loss by the ratio of thex-ray energy lost to the energy of the circulating electron bysome two orders of magnitude. It is this process that makesbeamstrahlung so damaging. It contributes directly to theso-called “interaction lifetime”. The damage is quantifiedby the beamstrahlung-dominated beam lifetime τ bs .The important parameter governing beamstrahlung is the“critical energy” u ∗ c which is proportional to 1/bunch-length σ z ; beamstrahlung particle loss increases exponentially with u ∗ c . To decrease beamstrahlung by increasing σ z also en-tails increasing β ∗ y which reduces luminosity. A favorablecompromise can be to increase charge per bunch along with β ∗ y . Reconciling the Luminosity Limits.
The number ofelectrons per bunch N p is itself fixed by the available RFpower and the number of bunches N b . For increasing theluminosity N b needs to be reduced . To keep beamstrahlungacceptably small N b needs to be increased . The maximumachievable luminosity is determined by this compromisebetween beamstrahlung and available power.Three limiting luminosities can be defined: L RFpow is theRF power limited luminosity (introduced earlier to analyseconstant luminosity scaling); L bbsat is the beam-beam satu-rated luminosity; L bstrans is the beamstrahlung-limited lumi-nosity. Single beam dynamics gives σ y = L RFpow = ∞ ? Nonsense. Recalling the earlier discussion, theresonance driving force, being proportional to 1 /σ y wouldalso be infinite. As a result the beam-beam force expands σ y = Saturation is automatic (unless the sin-gle beam emittance is already too great for the beam-beamforce to take control—it seems this condition was just barely12atisfied in highest energy LEP operation [10]). Formulasfor the luminosity limits are: L RFpow = N ∗ N b H ( r yz ) a xy f π (cid:16) n P rf [MW] σ y (cid:17) , (11) N tot = n P rf [MW] , (12) A β y = πσ x σ y = N p r e γ ( ξ sat . / β y ) = πσ x σ y , (13) L bbsat = N ∗ N b N p H ( r yz ) f γ r e ( ξ sat . / β y ) , (14) L bstrans = N ∗ N b H ( r yz ) a xy σ z f (cid:16) √ π . × . √ /π (cid:17) × r e (cid:72) E (cid:16) η ln (cid:0) /τ bs f n ∗ γ, R Gauasunif . (cid:1) (cid:17) , (15) N b = (cid:115) L bbsat L bstrans . (16)Here H ( r yz ) is the hourglass reduction factor. If L bstrans < L bbsat we must increase N b . But L bstrans ∝ N b , and L RFpow ∝ / N b . We accept the better of the compro-mises N b , new / N b , old = L bbsat / L bstrans or N b , new / N b , old = (cid:113) L bbsat / L bstrans as good enough.Parameter tables, scaled up from LEP, are given for100 km circumference Higgs factories in Tables 6 and 8.The former of these tables assume the number of bunches N b is unlimited. The latter table derates the luminosity un-der the assumtion that N b cannot exceed 200. Discussionof the one ring vs two rings issue can therefore be based onTable 8.Some parameters not given in tables are: Optimistic=1.5(a shameless excuse for actual optimatization), η Telnov =0.01(lattice fractional energy acceptance), τ bs =600 s, R GauUnif =0.300, P r f =
25 MW, Over Voltage=20 GeV, aspect ratio a xy =15, r yz = β ∗ y /σ z =1, and β arc max =198.2 m.With the exception of the final table, which is specificto the single ring option, the following tables apply equallyto single ring or dual ring Higgs factories. The exceptionrelates to N b , the number of bunches in each beam. With N b unlimited (as would be the case with two rings) all parame-ters are the same for one or two rings (at least according tothe formulas in this paper). With one ring, the maximum number of bunches is limitedto approximately ≤ N b >
200 the luminosity L has to be de-rated accordingly; L → L actual = L × / N b . This correction is applied inTable 8. This table, whose entries are simply drawn fromTable 6, makes it easy to choose between one and two rings.Entries in this table have been copied into the earlier Table 4.When the optimal number of bunches is less than (roughly) 200, single ring operation is satisfactory, and hence favored.When the optimal number of bunches is much greater than200, for example at the Z energy, two rings are better.Note though, that the Z single ring luminosities are stillvery healthy. In fact, with β ∗ y =10 mm, which is a moreconservative estimate than most others in this paper and inother FCC reports, the Z single ring penalty is substantiallyless.Luminosities and optimal numbers of bunches in a secondgeneration scaled-up-luminosity Higgs factory running areshown in Figure 6. Lu m i no s i t y [ / c m / s ], N b / Single beam energy E [GeV]’luminosity’ u 1:2’NbBy5’ u 1:($3/60)
Figure 6: Dependence of luminosity on single beam en-ergy (after upgrade to Stage II luminosity). The numberof bunches (axis label to be read as N b /
60) is also shown,confirming that (as long as the optimal value of N b is 1or greater) the luminosity is proportional to the number ofbunches. There is useful luminosity up to E =
500 GeV CMenergy.
With one 100 km circumference ring, the maximum num-ber of bunches is limited to about 200. For N b < L has to be reduced proportionally. L →L actual = L × N b / C =50 km, the RF power50 MW per beam, and the number of bunches N b =112. Theresults are shown in Table 7 (unlimited N b ) and Table 9(with N b =112).The values of parameters not shown in the tables are η Telnov =0.01, β ∗ y = xi typ . / β ∗ y =22.8, τ bs =600 s, Opti-mistic= 1.5, R Gau − unif =0.30, eV rf =20 GeV, OV req . =20 GV, a xy =15, r yz =1, β x , arcmax =120 m. Several formulas have been given for the luminosity. Theluminosity actually predicted is the smallest of the entriesintries in the three luminosity columns, for example in13 arameter Symbol Value Unit Energy-scaled Radius- scaledbend radius R m R / E /91.5 GeV
120 120 120
Circumference C L c m 79 108 153Momentum compaction α c Q x Q y J x / J y / J (cid:15) B U τ x τ x / T σ δ x (cid:15) x y (cid:15) y β max x
125 m 125 171 242Max. arc dispersion D max β ∗ x , β ∗ y σ ∗ x , σ ∗ y µ m 178/11 N/Sc. N/Sc.Beam-beam parameters ξ x , ξ y N b L − s − V RF
380 MV 3500 N/Sc. N/Sc.Synchrotron tune Q s σ z α c R σ e Q s E N/Sc. N/Sc.
Table 5: Higgs factory parameter values for 50 km and 100 km options. The entries are mainly extrapolated from Jowett’s,45.6 Gev report [12] [13], and educated guesses. “N/Sc.” indicates (important) parameters too complicated to be estimatedby scaling. Duplicate entries in the third column, such as 45.6/91.5 are from Jowett [12]; subsequent scalings are based onthe 45.6 Gev values.Table 6. For the middle shaded row the lowest value is L = . × cm − s − .Under ideal single beam conditions, the beam height σ y is vanishingly small and Eq. (11) predicts infinite luminosity,even for arbitrarily small RF power. Of course this is non-sense; nature “abhors” both zero and infinity. In fact, whenin collision, the beam-beam force causes σ y to grow (asthe simulation model assumes). In the current context thisimplies that it is always possible to saturate the tune shiftoperationally. In this circumstance Eq. (13) is applicable,and gives the beam area A β y small enough for the tune shiftto be saturated with the available number of electrons, whichis given by Eq. (13). Tentatively we assume N b = N p = N tot . Then σ y = (cid:115) A β y π a xy , and σ x = a xy σ y . (17)With the beam aspect ratio a xy being treated as if known, thispermits the bunch height and width to be determined. Butthis determination is only preliminiary since the number ofbunches N b is not yet fixed. Then, for a tentatively adoptedvalue of bunch length σ z , with ( ξ sat . / β y ) read from Figure 5,Eq. (14) gives the predicted luminosity with all the beam inone bunch. But this has neglected the beamstrahlung limitation;Eq. (15) gives the maximum luminosity allowed by beam-strahlung. (Factors have not been collected in thisembarrassingly-cluttered formula so they can be traced fromearlier formulas.) This beamstrahlung-limited luminositywill usually be less than the beam-power limited luminosity.The only recourse in this case is to split the beam into N b bunches. Changing N b does not change L bbsat , because N b N p is fixed, but it increases L bstrans , and it decreases L RFpow by thesame factor. Unfortunately, not yet definitively knowing σ y ,we cannot yet reckon the optimal value of N b . As a com-promise we use the square-rooted ratio in Eq. (16) to fix N b .This increases L bstrans and decreases L RFpow by the same factor(assuming N b > N b = L bbsat / L bstrans . This is justifiable, since L bbsat depends only on N tot . and is unaffected by changing N b This has not beendone for the tables since it leads to unacchievably largevalues of N b at low beam energy. It would, however, giveluminosity more than twice as great in some cases.For N ∗ = N ∗ =
2, if Eq. (16) gives a value of N b less than 1 it means that N b = N ∗ =
4, ifEq. (16) gives a value of N b less than 2 it means that N b = ame E (cid:15) x β ∗ y (cid:15) y ξ sat N tot σ y σ x u ∗ c n ∗ γ, L RF L bstrans L bb N b β ∗ x P rf GeV nm mm pm µ m µ m GeV m MWZ 46 0.949 2 63.3 0.094 1500 0.356 5.34 0.000 2.01 52.5 103 52.5 65243 0.03 25W 80 0.336 2 22.4 0.101 150 0.212 3.17 0.001 2.10 9.66 17.2 9.6 10980 0.03 25LEP 100 0.223 2 14.9 0.101 62 0.172 2.59 0.002 2.13 4.95 8.46 4.94 5421 0.03 25H 120 0.159 2 10.6 0.102 30 0.146 2.19 0.003 2.17 2.86 4.74 2.86 3044 0.03 25tt 175 0.078 2 5.33 0.118 6.6 0.103 1.55 0.006 2.24 0.923 1.43 0.92 920 0.03 25Z 46 17.2 5 1140 0.094 1500 2.39 35.89 0.001 2.16 21 35.1 21. 3605 0.075 25W 80 6.11 5 408 0.101 150 1.43 21.42 0.003 2.26 3.86 5.83 3.86 602 0.075 25LEP 100 4.07 5 271 0.101 62 1.16 17.47 0.005 2.31 1.98 2.86 1.97 296 0.075 25H 120 2.92 5 195 0.102 30 0.987 14.80 0.008 2.35 1.15 1.6 1.14 166 0.075 25tt 175 1.47 5 98.1 0.118 6.6 0.7 10.51 0.017 2.43 0.369 0.479 0.37 49 0.075 25Z 46 155 10 10300 0.094 1500 10.2 152.3 0.002 2.29 10.5 15.5 10.5 400 0.15 25W 80 55.4 10 3690 0.101 150 6.08 91.17 0.007 2.41 1.93 2.55 1.93 66 0.15 25LEP 100 37.0 10 2470 0.101 62 4.97 74.48 0.011 2.46 0.989 1.25 0.99 32 0.15 25H 120 26.6 10 1770 0.102 30 4.21 63.15 0.016 2.50 0.573 0.696 0.57 18.3 0.15 25tt 175 13.5 10 898 0.118 6.6 3.0 44.94 0.036 2.60 0.185 0.207 0.19 5.5 0.15 25 Table 6: The major factors influencing luminosity, assuming 100 km circumference and 25 MW/beam RF power. Thepredicted luminosity is the smallest of the three luminosities, L RF , L bstrans , and L bb . All entries in this table apply to eitherone ring or two rings, except where the number of bunches N b is too great for a single ring. name E (cid:15) x β ∗ y (cid:15) y ξ sat N tot σ y σ x u ∗ c n ∗ γ, L RF L bstrans L bb N b β ∗ x P rf GeV nm mm pm µ m µ m GeV m MWZ 46 0.916 2 61.1 0.094 7.3e+14 0.35 5.24 0.000 1.97 52.5 96.8 52.513 33795 0.03 50W 80 0.323 2 21.6 0.101 7.6e+13 0.208 3.12 0.001 2.06 9.66 16.2 9.661 5696 0.03 50LEP 100 0.215 2 14.3 0.101 3.1e+13 0.169 2.54 0.002 2.10 4.95 8 4.947 2814 0.03 50H 120 0.153 2 10.2 0.102 1.5e+13 0.143 2.15 0.003 2.13 2.86 4.48 2.863 1581 0.03 50tt 175 0.077 2 5.12 0.118 3.3e+12 0.101 1.52 0.006 2.19 0.923 1.35 0.923 478 0.03 50Z 46 16.5 5 1100 0.094 7.3e+14 2.35 35.21 0.001 2.12 21 33.2 21.005 1872 0.075 50W 80 5.88 5 392 0.101 7.6e+13 1.4 20.99 0.003 2.22 3.86 5.52 3.864 313 0.075 50LEP 100 3.91 5 261 0.101 3.1e+13 1.14 17.12 0.005 2.26 1.98 2.71 1.979 154 0.075 50H 120 2.80 5 187 0.102 1.5e+13 0.966 14.50 0.007 2.30 1.15 1.52 1.145 86 0.075 50tt 175 1.41 5 94 0.118 3.3e+12 0.686 10.28 0.016 2.38 0.369 0.455 0.369 26 0.075 50Z 46 149 10 9900 0.094 7.3e+14 9.95 149.28 0.002 2.24 10.5 14.7 10.503 208 0.15 50W 80 53.1 10 3540 0.101 7.6e+13 5.95 89.26 0.007 2.36 1.93 2.42 1.932 34 0.15 50LEP 100 35.4 10 2360 0.101 3.1e+13 4.86 72.88 0.011 2.41 0.989 1.19 0.989 17 0.15 50H 120 25.4 10 1700 0.102 1.5e+13 4.12 61.78 0.016 2.45 0.573 0.663 0.573 9.5 0.15 50tt 175 12.9 10 857 0.118 3.3e+12 2.93 43.92 0.035 2.54 0.185 0.198 0.185 2.9 0.15 50 Table 7: Luminosity influencing parameters and luminosities with unlimited number of bunches N b , assuming 50 kmcircumference ring and 50 ˙MW per beam RF power. E β ∗ y ξ sat L actual N b , actual P rf GeV m MW/beam46 0.002 0.094 0.161 200 2580 0.002 0.1 0.176 200 25100 0.002 0.1 0.182 200 25120 0.002 0.1 0.188 200 25175 0.002 0.12 0.200 200 2546 0.005 0.094 1.165 200 2580 0.005 0.1 1.282 200 25100 0.005 0.1 1.334 200 25120 0.005 0.1 1.145 166 25175 0.005 0.12 0.369 50 2546 0.010 0.094 5.247 200 2580 0.010 0.1 1.932 66.5 25100 0.010 0.1 0.989 32.7 25120 0.010 0.1 0.573 18.3 25175 0.010 0.12 0.185 5.5 25
Table 8: Luminosites achievable with a single ring for whichthe number of bunches N b is limited to 200, assuming100 km circumference and 25 MW/beam RF power. En-tries in this table have been distilled down to include onlythe most important entries in Table 6, as corrected for therestricted number of bunches. The luminosity entries inTable 4 have been obtained from this table.is optimal, since N b = E β ∗ y ξ sat L actual N b , actual P rf GeV m MW46 0.002 0.094 0.174 112 5080 0.002 0.1 0.190 112 50100 0.002 0.1 0.197 112 50120 0.002 0.1 0.203 112 50175 0.002 0.12 0.216 112 5046 0.005 0.094 1.256 112 5080 0.005 0.1 1.380 112 50100 0.005 0.1 1.434 112 50120 0.005 0.1 1.145 86.6 50175 0.005 0.12 0.369 26.1 5046 0.010 0.094 5.644 112.0 5080 0.010 0.1 1.932 34.7 50100 0.010 0.1 0.989 17.1 50120 0.010 0.1 0.573 9.5 50175 0.010 0.12 0.185 2.9 50
Table 9: Luminosity influencing parameters and luminositieswith the number of bunches limited to N b = R and beam energy E ) is β y or, equivalently, σ z , since r yz is being held fixed. These dependencies areexhibited in the tables. The nature of the required compromises can be inferredfrom the luminosity formulas. Except for beam-beam effectsthe beam height can, in principle, be arbitrarily small. (Thisassumes perfect decoupling and zero vertical dispersion.Both of these conditions are adequately achieved in lowenergy rings, but neither was persuasively achieved at LEP.)Because the Higgs factory will be operationally simpler thanLEP, we assume that σ y will actually be driven by the beam-beam effect. In this case Eq. (11) implies that the tune shiftcan be saturated irrespective of the beam power. At toohigh energy this will surely cease to be applicable, and theformulas will have to be modified accordingly.For now we assume that saturated operation is alwayspossible. Then, except for possible beamstrahlung limitation,the luminosity given by Eq. (14) is, in principle, achievable,and the beam area A β y is given by Eq. (13). (Note, though,that a graph such as Figure (18) assumes σ z = .
01 m whichis not the result of any optimization. Also, repeating whathas already been implied, the luminosity will be very smallif the area A β y has to be made excessively small to achievesaturation.)Starting from some given parameter set, to increase lum-nosity by reducing beamstrahlung favors increasing a xy and σ z . But increasing a xy reduces L RFpow and inreasing σ z re-duces L bbsat —it can be seen by multiplying by ( β y /σ z ) / r yz ≡
1. Based on a cursory preliminary investigation there seem tobe broad optima roughly centered on a xy =
15 and r yz = . r yz by a factoras great as ten (which may very well be demanded by highermode considerations) decreases the luminosity by less thana factor of three.To compare predicted luminosities let us take, for example,the case with R =
10 km, E =
250 GeV, which is a higherenergy than exhibited in tables of the present paper. I havefound luminosity 0 .
11 compared, for comparable parameters,to Telnov’s storage ring value of 0 .
087 (in the final columnof Telnov’s Table II [4], which is his most nearly comparablecase).A much-debated question (prior to the surprisingly lowHiggs mass discovery) concerned the relative effectivenessof circular and linear colliders at very high beam ener-gies. My optimized luminosity value of 1 . × cm − s − (summed over 4 IP’s) is comparable with the advertised ILCluminosity value of about 1.8 at 250 GeV [14]. This suggests that, as well as being based on routine technology, circularstorage rings can remain competitive with linear collidersup to almost three times the LEP energy. Telnov’s number of bunches per beam is N b = N b =
2. My β y = β y = .
26 mm, for the same beam energy; (see the finalcolumn of Telnov’s Table I). Oide assumes 2 bunches perbeam and obtains luminosity 0 . × m − s − , much lessthan my value.For simplicity my RF acceleration has been patternedafter the final-configuration LEP design, which had totalvoltage drop V rf = .
63 GV. My value of 65 GV is 18 timesgreater than this; I chose this value just big enough to avoidhaving all tabulated luminosity values at E =
250 GeV fromvanishing. To actually run at E =
300 GeV would require themaximum RF voltage to be increased. Conversely, duringa first phase of operation, a far smaller value would be ade-quate. During final LEP operation there were 44 klystrons,with each having power dissipation of roughly 1 MW. MyRF power dissipation is only two or three times this great.This reflects the fact that I assume that, as in all previouscircular electron rings, there is a substantial “overvoltage”compared to the average energy loss per turn.An exhaustive investigation of the parameter space hasnot been attempted. However, within my model, it seemsthe parameters in the tables in this report are fairly close tooptimal.Fortunately most of the entries in the tables for the larger R values show performance ranging from respectable toexcellent. This is true in spite of the fact that (except for eV rf and P rf ) none of the parameters seem to be especiallychallenging. Even a ring only twice the circumference ofLEP (except for possibly unachievably low β y ) could serveas a respectable Higgs factory. For various reasons a largercircumference would be more conservative short term, andhave far greater long term potential.Parameters in the tables have been “tuned” for the shadedrow, E =
250 GeV, twice the energy for maximum Higgsproduction. The large range from minimum to maximumluminosities at E =
125 GeV imply that, after retuning, aluminosity significanly higher than 10 cm − s − will bepossible, and the 200 per day Higgs particle productionincreased proportionally. This too improves with increasedcircumference.Especially satisfying is the finding that bunch lengths σ z ≈ N b . A heuristically plausible mnemonic: circular colliders are superior aslong as their energy loss per turn is an order of magnitude less than theirfull energy. At this point the number of bunches in the collider cannot befurther reduced while still serving all of the intersection regions. (cid:15) y ≈ . (cid:15) y to an acceptably small value. I find this not tobe the case.The intersection region optics seems to be manageable aswell. In a presentation at the FNAL Higgs Factory Work-shop in early 2013, Yunhai Cai [17] showed, for example, afinal focus with β x =
50 mm, β y = I proposed kicker-free, septum free, vertical injection atBeijing in April 2014, and described it in paper SAT4A3,“Lattice Optimization for Top-Off Injection” at the 55th ICFAAdvanced Beam Dynamics Workshop on High LuminosityCircular e+e- Colliders, in the WG 6 “Injection” workinggroup for HF2014 October 11.Handling the synchrotron radiation at a Higgs Factoryis difficult and replenishing the power loss is expensive.Otherwise the RF power loss is purely beneficial, espe-cially for injection. Betatron damping decrements δ (frac-tional amplitude loss per turn) are approximately half theenergy loss per turn divided by the beam energy, (e.g. δ ≈ . × . / = .
25 %.) Also the energy depen-dence is large enough for injection efficiency to improvesignificantly with increasing energy.According to Liouville’s theorem, increasing the beamparticle density by injection is impossible for a Hamiltoniansystem. The damping decrement δ measures the degree towhich the system is not Hamiltonian. Usually bumpers andkickers are needed to keep the already stored beam capturedwhile the injected beam has time to damp. If δ is largeenough one can, at least in principle, inject with no bumpersor kickers.The most fundamental parameter limiting injection effi-ciency is the emittance of the injected beam. The verticalemittance in the booster accelerator can be very small, per-haps (cid:15) y < − m. This may require a brief flat top atfull energy in the booster. For injection purposes the beamheight can then be taken to be effectively zero. The nextmost important injector parameter is the septum thickness.For horizontal injection this septum normally also has tocarry the current to produce a horizontal deflection. Typi-cally this requires the septum thickness to be at least 1 mm.For vertical injection, with angular deflection not necessarilyrequired, the septum can be very thin, even zero. The remain-ing (and most important) injection uncertainty is whetherthe ring dynamic aperture extends out to the septum. If not, it may be possible to improve the situation by moving theclosed orbit closer to the wall using DC bumpers. (Howeverthis may be disadvantageous for vertical injection as verticalbends contribute unwanted vertical emittance to the storedbeams.)A virtue of top-off injection is that, with beam currentsalways essentially constant, the linear part of the beam-beamtune shift can be designed into the linear lattice optics. Onebeam “looks”, to a particle in the other beam, like a lens (fo-cusing in both planes). Large octupole moments makes thislens far from ideal. Nevertheless, if the beam currents areconstant the pure linear part can be subsumed into the linearlattice model. And the octupole component, though non-linear, does not necessarily limit the achievable luminosityvery severely.With injection continuing during data collection therewould be no need for cycling between injection mode anddata collection mode. This could avoid the need for thealways difficult “beta squeeze” in transitioning from injec-tion mode to collision mode. Runs could then last for days,always at maximum luminosity. This would improve bothaverage luminosity and data quality.Kicker-free vertical injection is illustrated schematically inFigure 7. Let n inj . be a small integer indicating the number ofturns following injection before the injected beam threatensto wipe out on the injection septum. The fractional shrinkageof the Courant-Snyder invariant after n inj . turns is n inj . δ . Byjudicious choice of vertical, horizontal, and synchrotrontunes this shrinkage may be great enough that less than, say,10 % of the beam is lost on the septum. β = 30 m2 31 y’ vertical phase space.98.96.94.921.00 beam tubewallinjected beam"perfect" ribbon y yz Figure 7: A cartoon of kicker-free, vertical injection. Thedashed line shows the Courant-Snyder amplitude of the in-jected particle with the fractional shrinking per turn drawnmore or less to scale.17
SINGLE RING MULTIBUNCHOPERATION AND BEAMSEPARATION
As illustrated in Figure 8, the counter-circulating electronsand positrons in a circular Higgs Factory have to be separatedeverywhere except at the N ∗ intersection points (IP). Theseparation has to be electric and, to avoid unwanted increaseof vertical emittance (cid:15) y , the separation has to be horizontal.This section considers only head-on collisions at N ∗ = Operating Energies.
Typical energies for “Higgs Fac-tory” operation are established by the cross sections shownin Figure 3. We arbitrarily choose 120 GeV per beam as theHiggs particle operating point and identify the single beamenergy this way in subsequent tables. Similarly identifiedare the Z energy (45.6 GeV), the W-pair energy of 80 GeV,the LEP energy (arbitrarily taken to be 100 GeV) and the t ¯ t energy of 175 GeV to represent high energy performance. Bunch Separation at LEP.
Much of the material in thissection has been drawn from John Jowett’s article “BeamDynamics at LEP” [12]. When LEP was first commissionedfor four bunches ( N b =4) and four IPs ( N ∗ =4) operation,bunch collisions at the 45 degree points were avoided byvertical electric separation bumps. It was later realized thatvertical bumps are inadvisable because of their undesirableeffect on vertical emittance (cid:15) y , which needs to be minimized.We therefore consider only horizontal separation schemes.Various horizontal pretzel separation schemes were triedat LEP. They were constrained by the need to be superim-posed on an existing lattice. LEP investigations in the early1990’s mainly concentrated on what now would be calledquite low energies, especially the Z energy, E = . everywhere . Even inthis case there are periodic “nodes” at which the distortedorbits cross. To achieve the desired beam separation, one all RF ccavities arecentered at bump nodes C redred blueredblue redblue RFRFRFRF IPIP blue12 element closed electric bump6 closing elements of RFRF R Figure 8: Cartoon illustrating beam separation in one arc ofa Higgs factory. There are N b =4 bunches in each beamand N ∗ =2 interaction points (IP). The bend radius R issignificantly less than the average radius C / ( π ) ; roughly C = π R . For scaling purposes R and C are taken tobe strictly proportional. Far more separation loops andcrossovers are actually needed than are shown.has only to arrange for the desired crossing points to be atnodes and the parasitic crossing points to be at “loops” ofthe respective closed orbits. Raphael Littauer introduced thepicturesque term “pretzel” to distill the entire discussion ofthis paragraph into a single word.The original pretzel conception was not good enough,however, since the crossing angle at the IP was damagingto the luminosity. It was Littauer who fleshed out the ideaand led its successful implementation. It proved necessaryto introduce four electric separators in matched pairs aboutthe North and South IP’s. This invalidated the original name“pretzel” since the scheme amounted to closed electric bumpseparation separately in the East and West arcs. Nevertheless,by that time the “pretzel” name had caught on and the schemecontinued to be called pretzel separation in CESR and in allsubsequent rings.A disadvantage of the metaphorical terminology is that itconveys a picture of the whole ring being a single “pretzel”,18bscuring the fact that the separation bumps are closed ineach arc—two closed pretzels, if one prefers. To emphasizethis point, for this paper only, I discuss closing electric multi-bumps, arc-by-arc. But what is being described is a pretzelseparation scheme.Separating the beam in a pre-existing ring is significantlymore difficult than designing beam separation during theplanning stage, as was done, for example, for the 45 degreeseparation points in the initial LEP ring. Obviously theseparators have to be electric and therefore probably quitelong. At CESR there was no free space long enough, soexisting magnets had to be made shorter and stronger tofree up space for electric separators. Even so, the requiredelectric field was uncomfortably large.With N b equally-spaced bunches in each of the counter-rotating beams the beams need to be separated at the 2 N b − N ∗ “parasitic” crossing points. Standard closed bumps aretypically π -bumps or 2 π bumps. But, with 4 deflectors, twoat each end of a sector, bumps can easily be designed to be n π bumps, where n is an arbitrary integer matched to thedesired number of separation points.This discussion is illustrated pictorially in Figure 9 us-ing a space-time plot introduced (in this context) by JohnJowett. The beams are plotted as “world trajectories”, whosecrossings in space do not, in general, coincide with theircrossings in time. Separated world events with the samelabel correspond to the same point at different times.In the figure, associating point 4 with point 1 would cor-respond to the original McDaniel pretzel scheme in whichthe counter-circulating orbits are different everywhere in thering. With the “closed pretzels” there is no such association.The separated beams are smoothly merged onto commonorbits at both ends.(With care) one can associate the transverse bump dis-placement pattern with the space-time diagram, interpretingthe vertical axis as bump amplitude. A head-on collisionoccurs when two populated bunches pass through the samespace-time event. To avoid parasitic crossings the minimumbunch separation distance is therefore twice the closed bumpperiod.Another separation scheme tried at LEP was local electricbumps close to the 4 IP’s and angle crossing to permit “trains”with more than one bunch per train. This permitted as manyas 4 bunches per train though, in practice, more than 3 werenever used. For lack of time this option is not considered inthis paper.The primary horizontal separation scheme at LEP is illus-trated in Jowett’s clear, but complicated, Figure 3 [12]. Thescheme used 8 primary separators and 2 trim separators withthe separation bumps continuing through the 4 IP’s, but withvanishing crossing angles at all 4 IP’s. Starting from scratchin a circular collider that is still on the drawing board, onehopes for a simpler separation scheme. Separated Beams and RF Cavities.
By introducingslightly shortened, slightly strengthened, special purposebending magnets to make space for electric separators, multi- e+ e+ e+ e+ e+e e e ee+ e1 21 2LONGITUDINAL POSITION 3 43 4TIME
Figure 9: A minimal and modified “Jowett Toroidal Space-Time Beam Separation Plot” illustrating the separation ofcounter-circulating beams. Points with the same label at thetop and the bottom of the plot are the same points (at differenttimes). Though drawn to suggest a toroid the plot is purelytwo dimensional. The original McDaniel pretzel encom-passed the whole ring—that is, in this figure, points 1 and 4would also be identified. But this identification is not essen-tial. It is important to interpret “toroid space-time separation”in topological terms, not as a synonym for “barber-pole sep-aration” in which the orbits are actually toroidal in space.Though thinkable, such a separation scheme brings with itserious problems that almost inevitably lead to increasedvertical emittance.element electric bumps can be located arbitrarily withoutseriously perturbing any existing lattice design. But there isan issue with separated orbits and RF cavities. Probably bothbeams should pass through the centers of the RF cavities.But it seems safe to assume that the closed orbit anglesthrough the RF can be (symmetrically) different from zero.Otherwise, far more electric separators will be required,and far fewer bunches would be possible. RF cavities aretherefore to be placed at beam separation nodes.“Topping-off” injection is essential; especially to permitlarge beam-beam tune shifts. As long as the beam currentis constant the beam-beam deflection is equivalent to a thinlens, focusing in both planes, though with strong octupolesuperimposed. The linear focusing part can be incorpo-rated into the (linear) lattice optics. And the superimposedoctupole is not necessarily very damaging. Strong damp-ing makes bump-free, kicker-free, bunch-by-bunch, high-efficiency, vertical injection possible. Then steady-state,continuous operation without fill cycling may be possible.Somewhat reduced beam separation at bump ends maybe acceptable. With crossing angle the number of bunchesmay later be increased.
Bunches must not collide in arcs. They should be sepa-rated by at least 10 beam width sigmas when they pass. Withboth beams passing through the same RF, the path lengthsbetween RF cavities probably have to be equal. A single19ing is as good as dual rings if the total number of bunchescan be limited to, say, less than 200. Here it is proposed tosupport only head-on collisions at each of the two IP’s. Theminimum bunch spacing will then be slightly greater thanthe total length of the intersection region (IR).The half ring shown in Figure 8 shows a closed electricbump in the west arc. Orbits are common only in the twoIR’s. On the exit from each IR an electric bump is started andthe bump is closed just before the next IR. These “bumps”are very long, almost half the circumference. As alreadyexplained, this is not “pretzel” beam separation, as that termwas initially understood. Other than being horizontal ratherthan vertical and having multiple avoided parasitic collisions,these are much like the four separation bumps in the originalLEP design.Closed bumps require at least 3, or for symmetry, 4 con-trollable deflectors. Here a 12-bump scheme is described,with 6 electrostatic separators at the bump start and 6 at thebump stop. This scheme could be needed if the lattice celllengths are too short to contain sufficiently strong electricseparators. In Section ”Lattice Optimization for Top-OffInjection”, I show that the optimal collider cell length L i issignificantly longer than was assumed when this separationscheme was designed. With longer cells a simpler 4 or 6kicker bump may be adequate.The design orbit spirals in significantly; this requires theRF acceleration to be distributed quite uniformly. Basicallythe ring is a “curved linac”. The only betatron tune tunabilityis in the arcs. As the arc phase advances are changed (by apercent or so) the bumps have to be closed (very accurately)by tuning phase advance per cell and trim electric separators.As with beam separation in LEP, trim separators may berequired.Sketches and design formulas for a multi-element electricbump are shown in Figure 10. Figure 11 exhibits the sepa-ration of up to 112 bunches in a 50 km ring. Notice that, toavoid head-on parasitic collisions, the bunch separations areequal to two wavelengths of the electric bump pattern. (Mangling Jowett’s careful formulation [12] for brevity)the longitudinal partition number J (cid:15) depends on focusingfunction K , dispersion D , and on fractional momentumoffset, δ = ¯ δ + δ s . o . (where “s.o.” stands for synchrotronoscillation) and separator displacement x p ( s ) ; J (cid:15) ( δ, x p ) = + (cid:72) K D ds (cid:72) ( / R ) ds (cid:0) ¯ δ + δ s . o . (cid:1) ++ (cid:72) K (cid:0) D ( s ) − D ( s ) (cid:1) x p ( s ) ds (cid:72) ( / R ) ds ; (18)here D / D are the separator-on/separator-off horizontal dis-persion functions. The middle term here can be used to shift J (cid:15) away from 2, as proved useful at LEP, but it does not de-pend on x p ; it is shown only as protection against confusingit with the final term. Setting ¯ δ + δ s . o . =0, and averaging, the separator-displacedpartition number is J (cid:15) ( | x p | ) = + < K (cid:0) D − D (cid:1) x p >< / R > . (19)In spite of x p averaging to zero, there is a non-vanishingshift of J (cid:15) ( | x p | ) because K , D , and x p are correlated. AtLEP this shift was observed to be significanly damaging andto be dominated by sextupoles. The factors in Eq. (19) scaleas x p ∝ σ x ∝ R / , K = ql q ∝ / R / R / ∝ R , D − D ∝ S ∝ R / , ∆ J (cid:15) ( | x p | ) ∝ R . (20)These scaling formulas (derived in Section ”Lattice Op-timization For Top-Off Injection”) indicate that the seri-ousness of this partition number shift actually decreaseswith increasing R . Even if this were not true, should thepartition number shift be unacceptably large, it can be re-duced by increasing the quadrupole length l q to decrease K .The partition number shift is due to excess radiation in thequadrupoles. Since this radiation intensity is proportionalto the square of the magnetic field, doubling the quadrupolelength halves the radiation and the partition number shift. Section “Lattice Optimization for Top-Off Injection”) de-scribes the scaling of lattice parameters obtained after re-designing both injector and collider for efficient injection.The resulting collider cell length is L c =
213 m. Becausethe cells are so long, there may be no need for multiple elec-trostatic separators. Instead one may use, for example, twoor three electric kickers to launch each electric bump, withtwo or three matched kickers to terminate it. A large in-crease in cell length will surely also require a correspondingincrease in longitudinal separation of circulating bunches.The single beam luminosity will be correspondingly reducedif the luminosity is already limited by the maximum numberof bunches, as in the case of Z production. The luminosityreduction should be little affected at the Higgs energy andabove.Irrespective of bunch separation schemes, the minimumbunch separation will still be at least equal to the lengthof the intersection region. For single ring operation thiswill probably be less restrictive than the bunch separationrequired for the separation scheme. This section discusses Higgs factory injection. Full en-ergy, top-off injection is assumed. Vertical injection seems20 lectricseparator G E L E H E V E L Edipole magnet sextupoles, etc are not shown E quadrupole l There is a (conservatively weak) electric separator in each of 6 cells at each end of each arc. − + θ
1N N2 N1 N2 φ j φ k − θ β ^ x θ φ j φ k − θ β ^ x N2 θ N2 +++ ++ 600 120 180 240 300 360 420 480 540positive kicknegative kick0 1 2 3 4 5 6 7 8 9angle = φ(φ) = number of accumulating bump stagessin( )=displacement at "j" due to kick angle at "k" = sin( ) ( for 60 degree lattice ) x positive ramp start negative (or positive) ramp start = number of effective kicks per half bump = 4 (for 60 degree lattice)= 4 x 69.3m x positive kick effective at 360negative kick effective at 360RESONANT BUMP PHASE ADVANCESNOTE: deflections by arc quadrupoles are typically greater than electric separator deflections Figure 10: Sketches and design formulas for a multi-element electric bump.preferable to horizontal (as will be shown). Kicker-free,bunch-by-bunch injection concurrent with physics runningmay be feasible. Achieving high efficiency injection justifiesoptimizing injector and/or collider lattices for maximuminjection efficiency. Stronger focusing in the injector andweaker focusing in the collider improves the injection effi-ciency. Scaling formulas (for the dependence on ring radius R ) show injection efficiency increasing with increasing ringcircumference. Scaling up from LEP, more nearly optimalparameters for both injector and collider are obtained. Max-imum luminosity favors adjusting the collider cell length L c for maximum luminosity and choosing a shorter injectorcell length, L i < L c , for maximizing injection efficiency. Introduction.
I take it as given that full energy top-off injection will be required for the FCC electron-positronHiggs factory. Without reviewing the advantages of top-offinjection, one has to be aware of one disadvantage. The costin energy of losing a full energy particle due to injectioninefficiency is the same as the cost of losing a circulatingparticle to Bhabha scattering or to beamstrahlung or to anyother loss mechanism. Injection efficiency of 50% is equiv-alent to doubling the irreducible circulating beam loss rate.To make this degradation unimportant one should thereforetry to achieve 90% injection efficiency.Achieving high efficiency injection is therefore sufficientlyimportant to justify optimizing one or both of injector andcollider lattices to improve injection. The aspect of this op-21igure 11: Short partial sections of the multibump beam separation are shown: one at the beginning, one at an RF locationin the interior, and one at the far end of a long arc in Figure 8. The bunch separations are 480 m in a 50 km ring with celllength L c =
60 m. IP’s are indicated by vertical black bars, RF cavities by blue rectangles, electron bunches are greenrectangles moving left to right, positron bunches are open rectangles moving from right to left. Counter-circulating bunchesare separated at closed bump loop locations, and they must not pass through the nodes at the same time.timization to be emphasized here is shrinking the injectorbeam emittances and expanding the collider beam accep-tances by using stronger focusing in the injector than in thecollider. What are the dynamic aperture implications? Theywill be shown to be progressively more favorable as the ringradius R is increased relative to the LEP value. The dynamic-aperture/beam-width ratio increases as R / and is the samefor injector and collider. Constant Dispersion Scaling with R Linear Lattice Optics.
Most of the following scalingformulas come from Jowett [12] or Keil [20] or from ref-erence [21]. The emphasis on parameter scaling is in verymuch the spirit emphasized by Alex Chao [22]. For simplic-ity, even if it is not necessarily optimal, assume the Higgsfactory arc optics can be scaled directly from LEP values,which are: phase advance per cell µ x = π/
2, full cell length L c =
79 m. (The subscript “c” distinguishes the colliderlattice cell length from the injector lattice cell length L i .)At constant phase advance, the beta function β x scales as L c and dispersion D scales as bend angle per cell φ = L c / R multiplied by cell length L c ; D ∝ L c R . (21) (For 90 degree cells, the constant of proportionality in thisformula is approximately 0.5, for the average dispersion < D > .) Holding L c constant as R is increased woulddecrease the dispersion, impairing our ability to controlchromaticity. Let us therefore tentatively adopt the scaling L c ∝ R / , corresponding to φ ∝ R − / . (22)This is tantamount to holding dispersion D constant, and isconsistent with electron storage ring design trends over thedecades.These quantities and “Sands curly H” H then scale as β x ∝ R / , D ∝ , H ∝ D β x ∝ R / . (23)Copied from Jowett [12], the fractional energy spread isgiven by σ (cid:15) = √ (cid:126) m e c γ e F (cid:15) , where F (cid:15) = < / R > J x < / R > ∝ R , (24)22nd the horizontal emittance is given by (cid:15) x = √ (cid:126) m e c γ e F x , where F x = < H / R > J x < / R > ∝ R / . (25)The betatron contribution to beam width scales as σ x , betatron ∝ (cid:112) β x (cid:15) x ∝ / R / . (26)Similarly, at fixed beam energy, the fractional beam energy(or momentum) spread σ δ scales as σ δ ∝ √ B ∝ / R / . (27) Scaling with R of Arc Sextupole Strengths and Dy-namic Aperture. At this stage in the Higgs Factory design,it remains uncertain whether the IP-induced chromaticitycan be cancelled locally, which promises more than a fac-tor of two increase in luminosity, but would require strongbends close to the IP. For the time being I assume the IPchromaticity is cancelled in the arcs. Individual sextupolestrengths can be apportioned as S = S arc chr . + S IP chr . (28)The IP-compensating sextupole portion S IP chr . depends onthe IP-induced chromaticity. A convenient rule of thumbhas the IP chromaticity equal to the arc chromaticity. Bythis rule doubling the arc-compensating sextupole strengthscancels both the arc and the IP chromaticity.With dispersion D ∝
1, quad strength q ∝ / R / ,and S arc chr . ∝ q / D , one obtains the scaling of sextupolestrengths and dynamic aperture; S ∝ R / , and x dyn . ap . ∝ qS arc chr . ∝ . (29)The most appropriate measure of dynamic aperture is itsratio to beam width, x dyn . ap . σ x ∝ / R / ∝ R / . (30)The increase of this ratio with increasing R would allow theIP optics to be more aggressive for the Higgs factory thanfor LEP. Unfortunately it is the chromatic mismatch betweenIP and arc that is thought to be more important in limitingthe dynamic aperture than is the simple compensation oftotal chromaticity. The constant dispersion scaling formulasderived so far are given in Table 3. What has been discussed so far has been “constant disper-sion scalling”. But, as already stated, we wish to differentiatethe injector and collider optics such that the injector emit-tances are smaller and the collider acceptances are larger.This can be accomplished by shortening injector length L i and lengthening collider cell length L c . The resulting R -dependencies and scaling formulas are shown in Table 10.They yield the lattice parameters in Table 11 for both the50 km and 100 km circumference options. Implications of Changing Lattices for Improved Injec-tion.
According to these calculations there is substantialadvantage and little disadvantage to strengthening the in-jector focusing and weakening the collider focusing. Thisis achieved by shortening the injector cell length L i andincreasing the collider cell length L c . Weakening the col-lider focusing has the effect of increasing the equilibriumtransverse beam dimensions.As indicated in the caption to Table 11, the improvementin the injector emittance/collider acceptance ratio is probablyunnecessaily large, at least for a 100 km ring, where theimprovement in the injector/collider emittance ratio is afactor of seven.Furthermore there is at least one more constraint thatneeds to be met. Maximum luminosity results only whenthe beam aspect ratio at the crossing point is optimal. Amongother things this imposes a condition of the horizontal emit-tance (cid:15) x . At the moment the preferred method for controlling (cid:15) x is by the appropriate choice of cell length L c . With latticemanipulations other than changing the cell length it may bepossible to increase, but probably not decrease (cid:15) x .According to Table 2 of Section “Ring Circumferenceand Two Rings vs One Ring”, with β ∗ y = (cid:15) x is 3.98 nm. According to Table 11 the actualvalue will be (cid:15) x = .
82 nm. These values can be considered“close enough for now”, or they can be considered differentenough to argue that further design refinement is required(which is obvious in any case). But the suggestion is thatthe L c =
213 m collider cell length choice in Table 11 maybe somewhat too long.Unfortunately the optimal value of (cid:15) x depends strongly onthe optimal value of β ∗ y , which is presently unkown. Theseconsiderations show that the arc and intersection regiondesigns cannot be separately optimized. Rather a full ringoptimization is required.23 arameter Symbol Proportionality L ∝ R / L ∝ R / L ∝ R / injector colliderphase advance per cell µ x ◦ ◦ ◦ cell length L R / R / R /
110 m 153 m 213 mbend angle per cell φ = L / R R − / R − / R − / momentum compaction φ R − / R − R − / quad strength (1 / f ) q / L R − / R − / R − / dispersion D φ L R − / R / beta β L R / R / R / tune Q x R / β R / R / R / Q y R / β R / R / R / H = D / β R − / R − / R / partition numbers J x / J y / J (cid:15) (cid:15) x H / ( J x R ) R − / R − / R − / fract. momentum spread σ δ √ B R − / R − / R − / arc beam width-betatron σ x ,β = √ β(cid:15) x R − R − / σ x , synch . = D σ δ R − R − / S q / D R / R − / R − / dynamic aperture x da q / S R − / R / relative dyn. aperture x da /σ x R / R / R / separation amplitude x p σ x N/A R − / Table 10: To improve injection efficiency (compared to constant dispersion scaling) the injector cell length can increaseless, for example L i ∝ R / , and the collider cell length can increase more, for example L i ∝ R / . The shaded entriesassume circumference C =100 km, R / R LEP =3.75.
Parameter Symbol LEP(sc) Unit Injector Colliderbend radius R m beam Energy 120 GeV 120 120 120 120circumference C L
79 m 92.4 110 127 213momentum compaction α c Q x Q y J x / J y / J (cid:15) B U τ x τ x / T
37 turns 69 139 69 139fractional energy spread σ δ x (cid:15) x nm y (cid:15) y β max x
125 m 146 174 200 337max. arc dispersion D max q ≈ ± . / L p σ x = (cid:112) β max x (cid:15) x √ √ √ √ √ S = q / D x da ∼ q / S ∼ ∼ ∼ ∼ ∼ x da /σ x ∼ ∼ ∼ ∼ ∼ ± σ x ± . √ ± √ ± √ Table 11: Lattice parameters for improved injection efficiency. This table is to be compared with Table 5 to assess theeffect of lattice changes on injection efficiency. The shaded row indicates how successfully the injector emittance has beenreduced relative to the collider emittance. The factor of seven improvement, 7.82/1.08, in this ratio for a 100 km ring,seems unnecessarily large, indicating that less radical scaling should be satisfactory. As it happens the 213 m collider celllength agrees almost perfectly with the cell length adopted for the FCC-pp collider, as reported by Schulte [16] at the 2015FCC-week in Washington D.C. 24
L × L ∗ LUMINOSITY × FREE SPACEINVARIANT
Yunhai Cai’s intersection region design [17] is analysedin detail in Appendix E, “Deconstructing Yunhai Cai’s IROptics”. For maximum operational convenience in changingIP beta functions, Yunhai’s design was designed to be scal-able. This makes the IR design ideal for using dimensionalanalysis to derive scaling law dependence on the free spacelength L ∗ , which is the length of the space left free for theparticle collision reconstruction apparatus. This scaling lawcan be employed to investigate how the choice of free IPlength L ∗ affects the achievable luminosity. Yunhai’s designis probably close to optimal. But, even if it is not, the sameresults, based purely on scaling behavior, will still be valid.This prescription does not establish the absolute luminos-ity but it does determine the relative luminosity under theplausible hypothesis that the luminosity maximum will begoverned by the maximum β functions (anywhere in thering).For convenience in extracting scaling laws, Yunhai’sMAD lattice file was modified to include a scaling factor“FAC” such that FAC=1 results in no change. All lengthsare multiplied by FAC and all quadrupole focal lengths aredivided by FAC. (This means the quadrupole coefficients inthe MAD file have to be divided by FAC to account for thealtered quadrupole lengths). Similarly sextupole coefficientshave to be divided by FAC . After these changes, MADruns produced the beta function plots shown in Figure 12 for the four parameter sets given in Table 12. Other thannoting their identical shapes (confirming the scaling) onlythe maximum β max y values are extracted from the plots. FAC L ∗ β ∗ x β ∗ y β max y L ∗ β ∗ y m mm mm m depend. depend.1.0 2 200 2 4900 (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) Table 12: Parameter sets for dimensional analysis of YunhaiCai Higgs factory intersection region design.Results of the MAD runs are plotted in Figure 13. Thesmooth fitting function in the left plot of Figure 13 gives thescaling law β max y = ] β ∗ y (cid:16) L ∗ L ∗ nom (cid:17) p , (31)where L ∗ nom = L ∗ , with exponent p to be determinedlater. The right plot of Figure 13 gives the scaling law β ∗ y = β ∗ y nom (cid:16) L ∗ L ∗ nom (cid:17) . (32)For Eqs. (31) and (32) to be compatible requires p =
2. ThenEq. (31) becomes β ∗ y = . L ∗ β max y e . g . = . = . (cid:88) (33)Using Eq. (33) the luminosity is given by L static e . g . = . × cm − s − m × β max y L ∗ . (34) The constant of proportionality in this equation has not beendetermined by the scaling formula. It has been chosen tomatch preliminary CEPC estimated luminosities.
For localchromaticity I.P. design (Yunhai Cai), lengths are scalable,quad strengths scale as 1/length, beta functions scale aslength. Assuming some upper limit on β max y has been es-tablished, an upper limit for the luminosity times detectorlength has also been detemined; L ∗ × L product (upper limit) is fixed. (35)25igure 12: The upper plots differ by the choice F AC = F AC = . L ∗ is 50% greater in the lower plot. The lower plots differ by the choice β y = β y =
10 mm in the lower right plot. The important parameters are copied from the data line into the table. s q r t ( β y m a x [ m ] ) β y* [mm]dependence of β maxy on β y* sqrt(10000/x) β * y [ mm ] Ld1 [m]dependence of β y* on Ld12*(x/2)**2 Figure 13: Parameter dependencies implied by the Yunhai Cai intersection region design. The left plots β Y max (or ratherits square root) versus β ∗ y with the longitudinal scale held constant. The right plots β Y ∗ versus L ∗ with β Y max = TRANSVERSE SENSITIVITY LENGTH
Estimating β max y (and from it L ) According to Eq. (34) the achievable luminosity L is pro-portional to the maximum achievable beta function value β max y . This quantity is hard to determine, however, since itdepends on unknown parameter uncertainty, such as mis-alignments and fringe fields. Here we attempt to determine β max y by scaling from operational experience with existingrings.All of accelerator physics is based on a radial multipoleexpansion of the magnetic fields, with coefficients that arebend (constant), then quadrupole (linear), then sextupole(quadratic), and so on. The Courant-Snyder formalism isbased on the first two (linear) terms and is considered to beperfectly reliable for small amplitudes. For various reasons,such as head-tail instability, the ring must be achromatic.This requires sextupoles; i.e. nonlinearity . The perturba-tive resonance-driving effect of any individual sextupole isproportional to β / (multiplied by the sextupole strength)and each higher multipole order brings in another power of β / . Uncertain nonlinear multipole moments are magnifiedby large beta functions (raised to power β / , β , β / . . . . ).For protons (but probably not electrons) because of theircoil-dominated magnets, the very convergence of the mul-tipole series is an issue. In the design stage there are toomany unknown higher order multipoles to compensate reli- ably. Historically the dominant lattice defects can only beidentified and cured in machine studies during the first yearof operation and beyond.The maximum tolerable beta function depends on un-known errors. For purposes of estimation one can guess thatthe most sensitive lattice element is the quadrupole situatedat the location where β y is maximal, and that it producesan uncorrected angular deflection error ∆ x (cid:48) = q ∆ x , pro-portional to the quadrupole strength q and to an unknown“effective” displacement factor ∆ x . This error could be dueto the quadrupole itself being displaced or due to other ele-ments in the ring being displaced from their design locations.Dropping a factor of order 1 (or greater in case of resonance)that depends on betatron phase, the maximum displacementcaused by this error would be β max y q ∆ x tol . , where ∆ x tol . is tobe a phenomenologically determined “transverse displace-ment tolerance length”. If the resulting error exceeds thedynamic aperture the particle will be lost. Using scalingequation (29) and q ∝ / L c , these quantities are given by ∆ x ≈ β max y q ∆ x tol . , x dyn . ap . ∝ qS ∝ D , (36)Using q ∝ / L c , setting these equal, and solving for ∆ x tol . ∆ x tol . ∝ DL c β max y . (37)Dimensionally this quantity is a length. Though expressedunambiguously, for any actual ring it has to be multipliedby a factor depending on unknown transverse positioningimperfections, the absolute magnitude of which can only beinferred phenomenologically. To the extent the errors aredue to positioning and field quality construction errors theymay be expected to be quite similar in modern acceleratorsconstructed with best practically achievable precision. β y must not be too large To get higher luminosity requires reducing β ∗ y . Reducing β ∗ y increases β max y , which invariably makes the collider moreerratic, often unacceptably so. Sensitivity to beam-beameffects and other effects is greatly magnified by large β any-where in the ring. There are inevitable unknown transverseelement displacement errors ∆ y transverse . From the transversesensitivity just discussed, the limitation imposed by a large β max y at one or a few points in the ring is expressed by atransverse sensitivity length = DL C β max . (38)The optical deviation caused by ∆ y will be negligible onlyin the limit ∆ y << transverse sensitivity length . (39)27he scale factor is phenomenological but, for empirical com-parison purposes, it is to be taken to be independent ofparticle energy and type, electron or proton. β y Phenomenology Based onTransverse Orbit Sensitivity
The inverse of the sensitivity length is a “figure of de-merit,“FOD” = β max Y DL c that can be used to compare differentrings, either proton or electron, independent of their beamenergies. When β max y is large, it is always because β ∗ y is β ∗ y Ring
D L c β max y β max y DL c m m m m 1/m0.015 CESR exp. 1.1 17 95 5.10.08 PETRA exp. 0.32 14.4 225 49HERA exp. 1.5 48 2025 280.05 LEP exp. 0.8 79 441 7.00.007 KEKB exp. 0.5 20 290 29LHC exp. 1.6 79 4500 360.01 CepC des. 0.31 47 1225 840.01 CepC des. 1.03 153 1225 8.80.001 CEPC des. 0.31 47 6000 4100.001 FCC-ee des. 0.10 50 9025 1805 Table 13: ”Figures of Demerit”, inverses of “transversesensitivity lengths”, are plotted for various low and highenergy colliders, both proton and electron.small. But the value of β ∗ y is irrelevant in assessing the dy-namic aperture limitation caused by the large value of β max y .Nevertheless β ∗ y values are given in the table. Note that β ∗ y tends to be “big” for the ancient rings toward the top of thetable, and “small” toward the bottom. The two CepC rowsassume identical IP optics with β ∗ y =
10 mm, but differentarc parameters. For the CepC row the ring parameters arecopied from the CEPC, CDR design. For the CepC rowthe ring parameters are copied from Table 5. For the CEPCrow β ∗ y = L ∗ values (1.5 m for CEPC, 2.0 mfor FCC-ee).Compared in this way the transverse tolerances of KEKBand LHC are close in value, even though, as storage rings,they could scarcely be more disimilar; KEKB is a “small”electron collider, LHC is a large proton collider. The pes-simistic behavior of LEP can be blamed on the absence oftop-off injection, which led to the tortuous ramping and betasqueeze operations. This limited the β ∗ y to be not less than5 cm. Entries in Table 13 suggest an empirically determinedupper limit rule on FOD trans . sens . , FOD trans . sens . < . (40)CEPC exceeds this limit by a factor of 10, FCC-ee by a factorof 50. This is partly due to their way too short cell lengths.This transverse sensitivity discussion has been only semi-quantitative but, at least, it is dimensionally consistent, and it provides a prescription for comparing performance of verydifferent colliders. For the “transverse sensitivity length” tobe a valid comparison gauge implicitly assumes that thislength (dependent of survey and positioning precision) canbe expected to be the same for accelerators of all sizes, andfor both electrons and protons. The approach has been some-what ad hoc however, and it depends on the validity of thescaling laws emphasized in this paper. Some length otherthan DL c / β max y might provide a more valid comparison,though it would probably disrupt the good agreement be-tween two modern rings, KEKB and LHC, in the last columnof Table 13. APPENDICES
The dependence of ring radius R on maximum electronbeam energy E max is dominated by the relation giving U ,the energy loss per turn, per electron; U [GeV] = C γ E R , (41)where, for electons, C γ = . × − m / GeV . C γ can legitimately be referred to as the “Sands [23] con-stant” . In these units the formula yields an energy in GeVunits. Except as noted in a footnote all energies in thisreport are expressed in GeV. Corresponding to energy loss U , with each beam having N b bunches of N p particles, theradiated power from each beam, in GW units, is given by P rad = U N tot f , (42)where N tot = N b N p and f = c / C is the revolution fre-quency.The spectrum of radiated photons is characterized by a“critical energy” given by [24] u c [GeV] = − π hc γ R = . × − GeV . m γ R (43) For protons C γ = . × − m / GeV . Post-LHC p,p storage ringswill, for the first time, be dominated by synchrotron radiation, in much theway that electron rings always have been. This is due to a combination ofthe energy ratio (to the fourth power) and the fact that some fraction ofthe synchrotron radiation energy is invevitably dissipated at liquid heliumtemperature, “amplifying” its cost in wall power. The convention in this report is for all quantities except energies to beexpressed in SI units. Following Sands, energies are expressed in GeV.By this time an energy unit of 100 GeV would be more convenient. Inthis report, to avoid redefining the Sands constant C γ , a dimensionlessenergy (cid:72) E = E /
100 GeV is introduced. When a factor occuring on the right hand side of an equation is givennumerically (rather than symbolically) and it represents a quantity withunits, the units are appended explicitly to the numerical value, as inthe equation in the previous footnote. The units of quantities expressedsymbolically are determined by convention but, for emphasis, the unitsmay occasionally be be given in square brackets. The intention has beento make it possible to mentally check the dimensional consistency ofevery equation. − in the first expression is included sothat, when the remaining factor is evaluated using SIunits, u c will come out in GeV. The factor hc is equal to12390 eV.Å=1 . × − eV.m.For numerical convenience, γ can be written as γ = . × (cid:18) E [GeV]100 GeV (cid:19) ≡ . × (cid:72) E , (44)where (cid:72) E is the (now dimensionless) beam energy divided bya “nominal” reference energy taken to be 100 GeV. For LEPoperating at 100 GeV, which we will be using as a referenceenergy, (cid:72) E =
1. The beam energy itself, in GeV, is E =
100 GeV (cid:72) E . (45)The (cid:72) E -values to be considered will range from 1 to (at ourmost optimistic) 3. For radiated photons of energy u , theaverage energy value is related to the critical energy by < u > = . u c ≈ u c / .
2. The number of radiated photons perelectron per turn is then n γ, = . × GeV u c C γ (cid:72) E R . (46)This number is invariably much greater than 1. Later, whencomparing beamstrahlung to bremstrahlung, it will be con-venient to use this relation in the form U = . n γ, u c , (47)even if this may seem, at present, to involve circular reason-ing.The horizontal ring emittance (cid:15) x is established by theequilibrium between quantum fluctuation heating and syn-chrotron radiation damping. The important lattice parameterfor this is the “Sands curly-H parameter” H . I adopt, as anumerical value,(independent of E max for simplicity) thevalue from Cai et al. [17]; (cid:104)H (cid:105) ≈ (cid:28) η x + ( β x η (cid:48) x − β (cid:48) x η x / ) β x (cid:29) ≈ . × − m . (48)The horizontal emittance is then given by [25] (cid:15) x = . J x u c E (cid:104)H (cid:105) , (49)where I will take J x = (cid:15) x the r.m.s. beam width is givenby σ x = (cid:112) β x (cid:15) x . (50) As first explained by Telnov [4], with increasing beamenergy, beamstrahlung has an even greater impact on theenergy dependence of circular e+e- storage rings than does the energy loss per turn given by Eq. (41). Instead of beinglimited by power loss that can be restored by RF cavities,beamstrahlung causes reduced beam lifetime. This sets aluminosity limit based on the maximum beam current theinjection system can provide. This section re-derives beam-strahlung formulas.To quantify the lattice energy acceptance Telnov intro-duces a parameter η ≈ . q to a particle in the opposing bunch, causes deflection ∆ y (cid:48) = − q y . (51)The effective quadrupole strength q is related to the “beam-beam tune shift parameter” ξ ; ξ = β y π q , (52)where β y is the vertical beta function at the interaction point. ξ is given in terms of other beam parameters by [4] ξ = π γ N p r e β y σ x σ y , (53)where r e = . × − m is the classical electron radius.Note that there is no “hourglass” correction to the tune shift.For flat beams, by Gauss’s law, the vertical electric fieldis proportional to the fractional excess charge below thetest particle, and this fraction is independent of longitudinalcoordinate z .The first of two important aspects of beamstrahlung isthe energy scale of the radiated x-rays (eventually, γ -rays).The vertical displacement of a typical electron is σ y , ther.m.s. beam height, which is the approximate location wherea particle is subject to the maximum deflection force fromthe other beam. Treating the other beam as a quadrupole ofinverse focal length q , the slope change of the particle is ∆ y (cid:48) = − q σ y also ≈ − (cid:96) z ρ ∗ , (54)where (cid:96) z is the “effective” bunch length of each of the beams.To treat the bunch length as uniform, with density per unitlength equal to the actual charge density at the center of thebunch, while retaining the correct total charge, we take (cid:96) z = (cid:114) π σ z , (55)where σ z is the r.m.s. bunch length. The formula willbe used only to calculate the effective bending radius ofthe orbit for purposes of calculating the effective criticalbeamstrahlung energy u ∗ c . This treats individual bunchesas uniform, with distance from the back of the bunch tothe front of the bunch as 2 (cid:96) z . The time duration of thedeflection pulse is reduced by a factor of two by the relativespeed being 2 c ; which hardens the radiation. The hardening29ffect is being accounted for by reducing the effective bunchlength by a factor of two in Eq. (54) .The fact that the distribution is actually Gaussian, notuniform, softens the radiation. This softening effect is beingneglected. This gives a (substantial) overestimate of u ∗ c , a cor-responding overestimate of the importance of beamstrahlung,and a corresponding underestimate of the luminosity.Solving for ρ ∗ yields ρ ∗ = (cid:96) z σ y q . (56)Substituting from Eq. (52) into Eq. (56) produces ρ ∗ = (cid:96) z σ y β y πξ . (57)The qualitative behavior of radiation from a short radiatordepends on whether the radiator is technically “short” mean-ing the deflection angle is short compared to the radiationcone angle 1 /γ or “long”, in the opposite case. (Drawnfrom undulator radiation terminology) this distinction canbe quantified by a strength parameter K defined by K = ∆ x (cid:48) γ. (58)The nominal boundary between short and long bend elementlength is at K =
1. In our colliding beam beamstrahlungapplication the trend is from K < K >> K >>
1, formulation. Fortunately thisis not a calculational hardship since the formulas requiredare the same as needed to describe the effects of ordinarysynchrotron radiation. But, because the beamstrahlung radi-ation is much “harder” than the synchrotron radiation, it isthe ultrahard photons at the upper end of the beamstrahlungspectrum that need to be described.The spectrum is most compactly charactized by the “criti-cal photon wavelength” λ ∗ c [24];1 λ ∗ c = π γ ρ ∗ , (59)where ρ ∗ is the radius of curvature of a radiating electron,assuming its orbit while passing through the other beam is aperfect circular arc of radius ρ ∗ . Corresponding to λ ∗ c is the“critical energy” u ∗ c of the radiation which, as in Eq. (43), isgiven by u ∗ c [GeV] = . × − GeV . m γ ρ ∗ (60) There is a similar cancelation in the tuneshift calculation. Because therelative speed of the test particle and the opposing beam bunch is 2 c ,the pulse duration of the transverse force is reduced by a factor of twocompared to the time interval between arrival time of head and tail atthe origin. There is a compensating factor of two because the electricand magnetic forces are almost exactly equal. The beam-beam tune shiftparameter ξ accounts for both of these factors. Expressed in terms of (cid:72) E , u ∗ c = . . m (cid:112) /π σ y σ z ξβ y (cid:72) E . (61) u ∗ c is the reduction in beam particle energy caused by radia-tion of one photon having energy equal to the critical energy.The fractional energy loss caused by a single emission at thecritical energy is u ∗ c E = .
280 m (cid:112) /π σ y σ z ξβ y (cid:72) E . (62)When applied to LEP operation at 100 Gev, with ξ = . u ∗ c / E = . × − . This is in acceptableagreement, given the ambiguities, with the value of 0 . × − given in Telnov’s Table I. Appendix B discusses theeffect a Gaussian (as contrasted to uniform) longitudinalbunch distribution has on the beamstrahlung loss calculation.The second beamstrahlung parameter of importance is thetotal energy U ∗ radiated by a single electron in its N ∗ pas-sages through the counter-circulating beam as it completesone full turn. Treating the bunches as long and uniform,Eq. (41), multiplied by N ∗ (cid:96) z / ( π ρ ∗ ) to account for the frac-tion of the circumference, U ∗ [GeV] ? = F ∗ C γ E ρ ∗ N ∗ (cid:96) z π = F ∗ C γ E π ξ σ y β y N ∗ (cid:96) z . (63)Here the question mark serves as a warning that, in aquadrupole, only a fairly small fraction of the electrons areactually subject to the maximum deflecting force. FollowingTelnov [4] we estimate that about 10% of the electrons con-tribute importantly to the radiation. The factor F ∗ , whichwe tentatively set equal to 0.1, accounts for this fraction. Thefact that this estimate is so crude will ultimately have littleeffect on our results since the ring parameters will alwaysbe arranged to make beamstrahlung almost negligible.As in Eq. (46), the number of beamstrahlung photonsemitted per turn is n ∗ γ, = . × GeV u ∗ c F ∗ C γ (cid:72) E π ξ σ y β y N ∗ (cid:96) z . (64)Note (from Eq. (46)), that n ∗ γ, has the same explicit (cid:72) E factor as n γ, .What makes the synchrotron radiation damaging is theenergy loss increase proportional to γ , making it expensiveto restore the power loss. What makes beamstrahlung po-tentially even more damaging than synchrotron radiation atultra-high energy is that photons near the upper end of thebeamstrahlung spectrum can be energetic enough that, whenan electron emits one such photon, the electron’s energy isreduced to below the ring acceptance, causing the particleto be lost.At a minimum, when an electron is lost due to beam-strahlung, its energy has to be made up in accelerating itsreplacement. If we neglect other complications, such as30ositron production and other acceleration and injectionissues, the energy loss can be made commensurate withsynchrotron radiation loss by introducing a “beamstrahlungpenalty” defined by the energy loss ratio, P bs = U ∗ U = F ∗ . n ∗ γ, E . n γ, u c ; (65)it is the replacement of u ∗ c by E in the numerator that makesthis penalty large. The factor F ∗ is the fraction of beam-strahlung photons energetic enough to cause the radiatingelectron to be lost. An accurate calculation of F ∗ will not benecessary since the proposed colliding beam design strategywill be to reduce beamstrahlung to unimportance.Neglecting the energy loss from beamstrahlung photonsof too low energy to cause particle loss, the numerator ofEq. (65) gives the energy loss due to beamstrahlung. Thedenominator gives the usual energy loss due to synchrotronradiation. Substituting from Eqs. (46) and (64) P bs = π F ∗ ξ σ y β y N ∗ R (cid:96) z (cid:20) Eu ∗ c F ∗ (cid:16) u ∗ c E , η (cid:17) (cid:21) . (66)This time the arguments of function F ∗ are given. Recallthat F ∗ ( u c ∗ E , η ) is equal to the probability that radiationof a beamstrahlung photon will cause the radiatin electronto be lost. Alternatively, substituting from Eq. (61), thebeamstrahlung penalty is P bs = π F ∗ F ∗ (cid:16) u ∗ c E , η (cid:17) ξσ y β y N ∗ R .
28 m 1 (cid:72) E . (67) The (normalized to 1) probability distribution of the (rel-ative) energy ζ = u ∗ / u ∗ c of a synchrotron-radiated photon,when the characteristic photon energy is u ∗ c is [28] s ( ζ ) = . ζ . e − . ζ . (68) ζ can be thought of as the photon energy in units of thecharacteristic energy. This formula is valid for all radiatedenergies in the range 0 < ζ < ∞ , except in the (unimpor-tant, both as regards normalization and physical effect) longwavelength, low energy limit. Defining a cumulative dis-tribution function S ( ζ ) = (cid:82) ζ s ( ζ (cid:48) ) d ζ (cid:48) , its complement is Sc ( ζ ) = − S ( ζ ) Sc ( ζ ) = (cid:90) ∞ ζ s ( ζ ) d ζ = probability that ζ > ζ . (69)The function Sc ( x ) is plotted in Figure 14; it shows a rapidlyfalling loss probability for values of x increasing beyond x =
10. The value x = .
05 causes the loss probability tobe 10 − .To achieve a beamstrahlung lifetime τ bs requires1 τ bs = n ∗ γ, f R Gaussunif . Sc (cid:16) η Eu ∗ c (cid:17) ≈ n ∗ γ, f R Gaussunif . exp (cid:16) − . η Eu ∗ c (cid:17) , (70) where the factor R Gaussunif . is explained in an appendix. Solvingthis for u ∗ c yields u ∗ c = − . η E ln (cid:16) /τ bs n ∗ γ, f R Gaussunif . (cid:17) . (71)This determines the value of u ∗ c that would correspond to abeamstrahlung lifetime of τ bs .Figure 14: The loss probability per beamstrahlung-radiatedphoton plotted with linear and logarithmic scales. On thelog plot both Sc ( x ) and exp ( − . ∗ x ) are plotted, and canbe seen to be approximately equal over the central range.Below some value such as x ≈
15, the beamstrahlung lossprobability becomes unsustainable. But the loss probabilityfalls rapidly as x is increased. In this appendix, as in the body of the paper, beam-strahlung is treated with the same formulas as synchrotronradiation in the arcs, and all corresponding parameters aredistinguished by asterisks. The charge density of the oppos-ing bunch is now allowed to depend on longitudinal position z . As a result the beamstrahlung radius of curvature ρ ∗ ( z ) and the critical energy u ∗ c ( z ) also depend on z . As in Eq. (63),the energy radiated in longitudinal interval dz is dU ∗ = F ∗ C γ E ρ ∗ ( z ) dz π ρ ∗ ( z ) , (72)31nd, as in Eq. (64), the number of photons radiated in thesame range is dn ∗ γ = . u ∗ c ( z ) ρ ∗ ( z ) F ∗ C γ E ρ ∗ ( z ) dz π . (73)The denominator factor u ∗ c ( z ) ρ ∗ ( z ) can be replaced usingEq. (60); dn ∗ γ = π . . × − GeV . m F ∗ C γ E /γ ρ ∗ ( z ) dz ≡ K ρ ∗ exp (cid:16) − z σ z (cid:17) dz . (74)In the last step constant factors have been abbreviated byfactor K , the opposing beam has been taken to be Gaussiandistributed longitudinally, and it subjects the beam beinganalysed to curvature ρ ∗ at the origin.It is only the occasional emission of a very hard beam-strahlung photon that can cause an electron to be lost. Thedistribution of photon energies u can be expressed in termsof a universal (normalized to 1) probability distribution s ( ζ ) where ζ = u / u ∗ c is the photon energy u in units of the criticalenergy u ∗ c (which now depends on z ). The cumulative proba-bility distribution corresponding to s ( ζ ) is S ( ζ ) which is theprobability of emissions for which ζ < ζ . The “complemen-tary” cumulative distribution function Sc ( ζ ) = − S ( ζ ) gives the probability that ζ > ζ . These functions are dis-cussed in greater detail in conjunction with Eqs. (68) and(69).The fraction of the dn ∗ γ photons in Eq. (74) having energyin excess of η E (the energy loss great enough for the electronto be lost) is given by (cid:90) ∞ η E / u ∗ c ( z ) s ( ζ ) d ζ = Sc (cid:16) η Eu ∗ c ( z ) (cid:17) = Sc (cid:18) η Eu ∗ c , exp (cid:16) z σ z (cid:17)(cid:19) , (75)where u ∗ c , is the critical energy at the origin and u ∗ c ( z ) = u ∗ c , exp ( − z / ( σ z )) . Finally, integrating over z , the prob-ability for an electron to be lost is equal to the number ofbeamstrahlung photons exceeding the loss threshold; P Gaussian e = K ρ ∗ (cid:90) ∞−∞ dz exp (cid:16) − z σ z (cid:17) Sc (cid:18) η Eu ∗ c , exp (cid:16) z σ z (cid:17)(cid:19) . (76)For a uniform longitudinal distribution the formula is similar,the integration is trivial, and the loss probability is P uniform e = K ρ ∗ σ z Sc (cid:18) η Eu ∗ c , (cid:17) . (77)Expressed as a ratio to the loss rate from a uniform bunch,the correction factor to account for the Gaussian profile is R Gaussunif . ≡ P Gaussian e P uniform e = Sc (cid:18) η Eu ∗ c , (cid:17) (cid:90) ∞−∞ dz σ z exp (cid:16) − z σ z (cid:17) × Sc (cid:18) η Eu ∗ c , exp (cid:16) z σ z (cid:17)(cid:19) . (78) In practice the argument η E / u ∗ c , will be a big number, like10 for example, and the value of the Sc -function in thedenominator will be a very small number, like 0.0001 forexample. As z moves away from the origin, the value ofthe argument of the Sc function in the numerator increasesquite strongly. Furthermore the function Sc ( ζ ) decreasesexponentially with increasing ζ . This tends to “cut off”the integral, which magnifies the importance of the non-uniformity of the longitudinal charge distribution. One an-ticipates, therefore, a substantially smaller beamstrahlungloss rate than is obtained assuming uniform longitudinalbeam profile. In a 2002 article published in PRST-AB [5] I describeda simulation program designed to an absolute, adjustable-parameter free, calculation of the maximum specific lumi-nosity of an e+e- ring. Though a computer simulation, thiscode provides an “absolute” calculation in the sense thatthere are no empirically-adjustable parameters. For the vari-ety of their operating tunes and energies, shown in Table 14(excerpted from my original paper), the ratio of theoreticalto observed was 1 . ± .
45, suggesting that the model canbe trusted within a factor of two.I have now applied this identical simulation to the designof a Higgs factory. To buttress the claim that there are no freeparameters I have not changed the code at all; only the valuesof the ring-specific parameters exhibited in Eq. (79). µ isthe vertical betatron phase advance between collision points,and µ x and µ s are the corresponding horizontal betatronand longitudinal (synchrotron) phase advances. a x and a s are the amplitudes in units of the r.m.s. width σ x and bunchlength σ z . δ is the “damping decrement” of vertical betatronmotion. Eq. (79) is a difference equation calculating thevertical displacement on turn t + t and t − y t + = + δ (cid:32) µ y t − y t − ( − δ ) (79) − πξ sin µ exp (cid:16) − a x cos µ x ( a x )( t + t x ) (cid:17) × (cid:115) + (cid:16) σ z β ∗ y (cid:17) a s cos (cid:0) µ s ( t + t s ) t (cid:1) (cid:114) π y t √ (cid:33) The simulation consists of nothing more than checking (re-peatedly and ad nauseum) whether the motion described bythe difference equation is “stable” or “unstable”, and notingthe ξ -value at the transition. Technical definitions of theseterms are in the original paper.The “physics” of the simulation is that the beam height σ y which would otherwise be negligibly small, is swollenby beam-beam forces. The mechanism is modulation of ver-tical focusing acting on each particle by its own (inexorable)32orizontal betatron and longitudinal synchrotron motion.This modulation provides parametric pumping force withstrength inversely proportional to σ y (which is guaranteedto countermand the “negligibly-small” natural single beam,beam height). Amplitude dependence of the vertical tunelimits the growth, however, so the resonance causes no parti-cle loss. The growth factor also depends on the distance (intune space) from the nearest resonance, and diffusive effectsare constantly moving particles closer to or away from reso-nances (which are everywhere). All this is explained in theoriginal paper. The paper analyses the growth mechanismanalytically assuming “nearest-resonance” damping, and noresonance overlap. (Fortunately) the simulation automati-cally accounts for whatever resonances are nearby.For any particular operating point of any particular ringmost of the parameters are taken from appropriate lab reports.Scans over appropriate ranges of a x and a s are performed.According to a saturation principle, the beam height adjustsitself to that value for which the least stable particle is barelystable at small amplitude. This determines the tune shiftparameter ξ max min (and its corresponding beam height) beyondwhich further increase of beam current causes the beamheight to increase proportionally.Some of the predictions from the 2002 paper are shownin Table 14. For existing rings these were “post-predictions”since the tunes Q x and Q y were given.For Higgs factory predictions, since the tunes of the Higgsfactory are free, what is needed are scans over the tune plane,for various values of damping decrement δ and vertical betafunction β y . The results for a six order of magnitude rangeof δ are shown in Figure 16. Values of ξ ( Q x , Q y ) can beobtained using the grayscale to scale down the ξ -value at ( Q x , Q y ) from the maximum value (for “all” ( Q x , Q y ) pairs)(which is white, with value ximinmax shown at the top ofthe graph). Here “all” means all of the tune plane shown inthe plots. A band of very low and a band of high Q y valuesis excluded. High Q y don’t need to be checked since theyalways give small ξ values. Very low Q y values usually givelarge ξ -values, but are almost surely excluded operationally.As the figure shows, only one quarter of the tune plane isscanned. The simulation has symmetries such that all fourquadrants of the fractional tune plane are equivalent (as areall integer tunes). As it happens the optimal tunes are almostinvariably in near the lower right corner of the fractional tunequadrants. There are always white (maximal ξ ) regions there.As it happens no storage rings I am aware of has attempted tooperate in this region. Or, more likely, there has been someunrelated problem making operation their difficult. For allsimulations in this paper I have assumed that the saturationtune shift is less than the maximum. Rather I have found theaverage value xiav and then, to determine a “typical” goodvalue, have picked a value half way between the averageand the maximum, i.e. xityp=(ximinmax+xiav)/2 . Allsimulation results are based on xi typ values determined inthis way. Dependence of ξ typ on E m , for a particular valueof β y and for a plausible scaling of ring radius R with E m ,is shown in Figure 15. T y p i c a l s a t u r a t ed t une s h i ft, ξ t y p " Beam Energy, E m [GeV] Figure 15: Plot of “typical” saturated tune shift ξ typ as afunction of maximum beam energy E m for ring radius R scaling as E . m . β y = σ z = δ ( DEL in the code ), and β ∗ y ( BETYST in the code , simply β y in this paper) the verti-cal beta function at the intersection point. The main outputis ξ maxmin ( XIMINMAX or ximinmax in the code) which is thequantity plotted in the page of density plots making up Fig-ure 16. For these plots the damping decrement (shown as delta ) ranges over six orders of magnitude. The saturationtune shifts can be read off (roughly) using the grayscale.Figure 17 plots the quantity printed at the top of theplots in Figure 16, for a series of values β ∗ y = . , . . , . , .
02 m. For very small values of damping decre-ment δ the saturation tune shift is independent of δ . But forvalues of δ exceeding 0.01, as will be the case for a Higgsfactory, there is an appreciable increase in saturated tuneshift with increasing damping decrement. Figure 19 plotsthe same data, but in the form ξ maxmin (cid:112) σ z / β ∗ y ; this providesthe simulation code input to luminosity formula (93).Figure 20 plots the ξ maxmin as a function of number of inter-section points N ∗ . This is somewhat redundant of the pre-vious plots since the only quantity changing is δ N ∗ = δ/ N ∗ where δ N ∗ is the damping decrement from one collisionpoint to the next. For Higgs factory simulations we distillsimulation results shown in the following graphs into theform ξ sat . ( δ i , β y , σ z ) = ξ sat . ( δ i , . , . ) (cid:115) β y σ z ≡ √ r yz ξ sat . ( δ i , . , . ) , (80)(with all lengths in meters). Values of ξ sat . ( δ i , . , . ) (with σ z = β y = i (with the appropriate damping decrement δ i ) and substitutedinto Eq. (80) to extrapolate to the actual values of σ z and β y . Fits of this form are shown in the appendix. Theygive ±
50% accuracy in the range from β y ≈ β y > β y < σ z . For greater accuracy33able 14: Parameters of some circular, flat beam, e+e- colliding rings, and the saturation tune shift values predicted (withno free parameters) by the simulation. Ring IP’s Q x / IP Q y / IP Q s / IP σ z β ∗ y δ y ξ th . ∆ Q y , exp . th/expVEPP4 1 8.55 9.57 0.024 0.06 0.12 1.68 0.028 0.046 0.61PEP-1IP 1 21.296 18.205 0.024 0.021 0.05 6.86 0.076 0.049 1.55PEP-2IP 2 5.303 9.1065 0.0175 0.020 0.14 4.08 0.050 0.054 0.93CESR-4.7 2 4.697 4.682 0.049 0.020 0.03 0.38 0.037 0.018 2.06CESR-5.0 2 4.697 4.682 0.049 0.021 0.03 0.46 0.034 0.022 1.55CESR-5.3 2 4.697 4.682 0.049 0.023 0.03 0.55 0.029 0.025 1.16CESR-5.5 2 4.697 4.682 0.049 0.024 0.03 0.61 0.027 0.027 1.00CESR-2000 1 10.52 9.57 0.055 0.019 0.02 1.113 0.028 0.043 0.65KEK-1IP 1 10.13 10.27 0.037 0.014 0.03 2.84 0.046 0.047 0.98KEK-2IP 2 4.565 4.60 0.021 0.015 0.03 1.42 0.048 0.027 1.78PEP-LER 1 38.65 36.58 0.027 0.0123 0.0125 1.17 0.044 0.044 1.00KEK-LER 1 45.518 44.096 0.021 0.0057 0.007 2.34 0.042 0.032 1.31BEPC 1 5.80 6.70 0.020 0.05 0.05 0.16 0.068 0.039 1.74 (within the limitations of the model) the full simulationwould have to be run at each data point.
10 D. LUMINOSITY FORMULAS
To (artificially) reduce the number of free parameters,while retaining the more important dependencies we definetwo ratios: a xy = σ x σ y = “transverse aspect ratio” >> , (81) r yz = β y σ z = ratio of “Rayleigh length” β y to σ z << . (82)We will tend to keep a xy and r yz more or less constant whileallowing σ y to vary over significantly large ranges. Builtinto all luminosity formulas is the flat beam assumption, σ y << σ x . A nominal value for the width-by-height ratiowill be a xy =
15. This is not necessarily optimal, but the“flat beam condition” a xy >> r yz << r yz = . σ z is allowed to be long comparedto β y one has to reduce the luminosity by an “hourglassfactor” [26] H ( r yz ) = r yz π exp (cid:16) r yz (cid:17) K (cid:16) r yz (cid:17) , (83)where K is a Bessel function. This is plotted in in Figure 21. As modified from reference [4] to account for multi-ple bunches and multiple interaction points, the luminositysummed over N ∗ interaction points is L summed = N ∗ N N b f π a xy σ y , (84) where the revolution frequency is f = c / C . The product N b N p = N tot is the total number of particles stored in eachbeam.All numerical values required are in Table 1 and its contin-uations. The luminosity will always be RF power limited—ifnot, the number of bunches N b can be increased, with pro-portional increase in luminosity. The column labelled n inTable 1 gives the number of particles N tot per MW of RFpower. This is obtained, using P rf [MW] = I [A] V [MV] = f N tot eU [MeV] , or (85) n = N p N b P rf [MW] = − f eU [GeV] . (86)For given P rf , N tot = n P rf can be treated as known. Thebest chance for saturating the beam-beam tune shift is withjust one bunch, N b =
1, in which case N tot = N p . But for N ∗ = N b = H ( r yz ) , L RF − powsummed = N ∗ N b H ( r yz ) n a xy f π (cid:16) P rf [MW] σ y (cid:17) . (87)This is the first of four luminosity formulas. The other lu-minosity formulas, giving only upper limits, are needed tocheck that beamstrahlung levels are tolerable and that thebeam-beam tune shift being assumed does not exceed thesaturation level. For any given choice of parameters it willbe the lowest of the four luminosity formulas that has to beaccepted. But under optimal conditions all four luminosityformulas should give approximately the same values andthere is likely to be a nearby parameter set giving betterluminosity than the lowest one. From Eq. (53) we can calculate the beam-beam tune shift, ξ = π γ N p r e β y σ x σ y . (88)34igure 16: Density plots of ξ maxmin over (most of) one quarter of the ( Q x , Q y ) tune plane for a five order of magnitude rangeof damping decrements. The symmetry of the simulation is such that the density plots for the other three quadrants of thefractional tune plane are identical to this quadrant. Starting from the upper left, in sequence left to right, the dampingdecrements are δ = − , − , − , − , − . The δ = − tune plot (which is indistinguishable from the δ = − plot)is replaced by the grayscale reference. The highest ξ maxmin value is plotted above each plot. Within each plot the ξ maxmin ( Q x , Q y ) can be inferred from this maximum value multiplied by the (lower right) grayscale value.The appearances of these density plots are absolutely typical. The hundreds of plots made all look very muchlike this, including the washing out of resonance lines when the damping decrement is large, and the optimal performancenear the lower right corner. Only one quadrant is displayed since the model is unaffected by half integer shifts in either tune.With unlimited beam power, the maximum value of ξ wouldbe the “saturated value” ξ sat . . Rearranging this equation inthis case gives, for the beam “area”, A ( y ) β y = πσ x σ y = N p r e γ ( ξ sat . / β y ) . (89)But we also have to limit ξ x , for which the formula is ξ x = π γ N p r e β x σ x = π γ N p r e β x a xy σ x σ y . (90)which gives an ξ x -limiting area A ( x ) β y = N p r e γ β x / β y a xy ( ξ sat . / β y ) . (91)To make these areas equal we adjust β x to the value β x = a xy β y . (92) Based on the simulation model, the area denominator factor ( ξ sat . / β y ) can be obtained from Figure 18. Note, though,that this figure arbitrarily assumes σ z = .
01 m, which isby no means optimal. More useful is to use Figure 5, whichgives ( ξ sat . / β y ) constrained by r yz =
1; i.e. σ z = β y .But, especially at high energies, the area A may be un-achievably small. In this case the actual ξ -value may beless because the beam currents are too small to saturate ξ .Conversely, in actual operations, especially at low energy, itmay be possible to push the luminosity to higher values bypushing the beam currents to higher values than are requiredto saturate the tune shift.Substituting into Eq. (4) yields L bb − sat . summed = N ∗ N tot f γ r e H ( r yz )( ξ sat . / β y ) . (93)35 ξ m a x m i n " Damping decrement δ ’bY01’ u 1:2’bY02’ u 1:2’bY05’ u 1:2’bY1’ u 1:2’bY2’ u 1:2 Figure 17: Plot of max/min tune shift value XIMINMAX, ξ maxmin versus damping decrement DEL ( δ ). For a range ofvalues of β y ; β y = . , .
2, 0 . , , σ z = .
01 m. ξ m a x m i n σ z / β y Damping decrement δ ’bY01’ u 1:($2*0.01/0.001)’bY02’ u 1:($2*0.01/0.002)’bY05’ u 1:($2*0.01/0.005)’bY1’ u 1:($2*0.01/0.01)’bY2’ u 1:($2*0.01/0.02) Figure 18: Plot of weighted saturated tune shift value ξ maxmin σ z / β y versus damping decrement DEL ≡ δ . Valuesfrom this plot are especially simple (e.g. ξ/ β y ≈ . /σ z =
19 /m for δ = .
01) for evaluating the luminosity usingEq. (93), which gives luminosity proportional to ξ maxmin / β y when the luminosity is limited by RF power (which is tosay, almost always, in a Higgs factory). In all cases here, σ z = .
01 m.This agrees with Telnov’s Eq. (15), except for our extrafactor of N ∗ , our inclusion of hourglass correction, and ourintention to obtain ( ξ sat . / β y ) from simulation. L bb − sat . summed isthe beam-beam saturated luminosity. With ξ sat . / β y ) givenby simulation, and N tot known, Eq. (93) fixes the summedluminosity predicted by the simulation, but only if N p islarge enough to “saturate” the tune shift.As commented earlier (according to the simulation model)both factors in the product H ( r yz ) ξ sat . appearing in Eq. (93)are proportional to the ratio √ r yz = (cid:112) β y /σ z . See Figure 21.The resulting proportionality to 1 /σ z tends to frustrate thebenefit of decreasing beamstrahlung by increasing bunchlength σ z .If the beam currents are unnecessarily large for saturating ξ , then L bb − sat . summed can be the actual summed luminosity, butonly if the beams are split into the appropriate number of ξ m a x m i n . s q r t ( σ z / β y ) β y ’DEL_1e-1’ u 1:(sqrt(0.01/$1)*$2)’DEL_1e-2’ u 1:(sqrt(0.01/$1)*$2)’DEL_1e-3’ u 1:(sqrt(0.01/$1)*$2)’DEL_1e-4’ u 1:(sqrt(0.01/$1)*$2)’DEL_1e-5’ u 1:(sqrt(0.01/$1)*$2)’DEL_1e-6’ u 1:(sqrt(0.01/$1)*$2) Figure 19: This is the same data as in Figure 17, but thesaturated tune shift value is weighted by (cid:112) σ z / β y and plottedas a function of β y with DEL held constant, instead of theother way round. In all cases here, σ z = .
01 m. ξ m a x m i n " Number of Collision Points, N * ’betYeq0.002’ u 1:3’betYeq0.005’ u 1:3’betYeq0.01’ u 1:3’betYeq0.02’ u 1:3 Figure 20: Plot of maximum/minimum tune shift valueXIMINMAX ξ maxmin versus number of collision points N ∗ .The damping decrement and synchrotron tune vary as δ = δ / N ∗ ∗ and Q s = . / N ∗ ∗ . The saturated tune shift de-pends only weakly on the number of intersection points.bunches. For determining optimal parameters the importantdifference between L bb − sat . summed and L RF − powsummed is their differentdependencies on the bunch number N b . By appropriatechoice of N b these luminosities could be made exactly equalexcept for the fact that N b has to be an integer. In the tables,to obtain smoother variation, this integer requirement for B b is not imposed. The importance of this defect can beestimated by the degree to which N b deviates from being aninteger or, for N ∗ =
4, and even integer.A significant parameter to be determined is the verticalbeta function β y . According to Figure 16, ξ/ β y is maximalnear β y = ξ/ β y can be treated as inde-pendent of N b and N p , the N ∗ N tot factor in Eq. (93) exhibitsthe expected proportionality to the number of intersectionpoints and to the total circulating charge.36igure 21: Plot of the “flat beam hourglass correction factor” H ( r yz ) , with r yz ≡ β y /σ z . (Fortuitously) the function √ r yz shown plotted red (and passing through r yz =
1) providesan excellent fit to the hourglass function.
As long as the beams remain flat and with constant widththrough the intersection region, the vertical beam-beam de-flection is approximately independent of z , the longitudinaldeviation from the intersection point. Unlike the differentialluminosity, which falls off rapidly as a function of z , thereis no significant hourglass correction to either the beam-beam tune shift or the beamstrahlung, (but the beamstrahlungpenalty is strongly reduced by increasing σ z ).The critical energy u ∗ c for beamstrahlung radiation is givenby Eq. (61); u ∗ c = . . m (cid:112) /π r yz β y (cid:72) E ( ξσ y ) , (94)and from Eq. (88), ξσ y = N b π . × (cid:72) E n r e β y a xy P rf [MW] σ y . (95)Substituting this into Eq. (94) produces u ∗ c = N b . . m √ /π π × . × r yz a xy β y (cid:72) E n r e P rf [MW] σ y . (96)Finally, this equation can be used to express the luminosityin terms of the critical beamstrahlung energy by substitutioninto Eq. (87), L bs − limitedsummed = (97) N ∗ N b H ( r yz ) r yz a xy β y f (cid:16) √ π . × . . m √ /π (cid:17) r e (cid:72) E u ∗ c . Clearly it is advantageous to make u ∗ c as large as possible,consistent with keeping the beamstrahlung loss rate accept-ably low. Increasing the number of bunches is also helpfulbut, at highest energies this avenue is closed.There are two ways the radiation of one or a few hard γ -rays can cause the radiating electron to be lost—transverselyor longitudinally.In Telnov’s calculation of beamstrahlung-limited lumi-nosity, u ∗ c is compared with the tolerable fractional energyacceptance η E . The beam decay rate ascribable to beam-strahlung is the time rate of beamstrahlung emissions per electron, multiplied by the probability per beamstrahlungphoton that its energy exceeds η E . Substituting for u ∗ c fromEq. (71), the transverse-limited luminosity corresponding tobeamstrahlung lifetime τ bs is L bstrans = N ∗ N b H ( r yz ) r yz a xy β y f (cid:16) √ π . × . √ /π (cid:17) × r e (cid:72) E (cid:16) η ln (cid:0) /τ bs f n ∗ γ, R Gauasunif . (cid:1) (cid:17) . (98)This form of calculation is valid for a low energy stor-age ring, with one RF cavity, with a synchrotron dampingdecrement per turn much less than 1. With insufficient damp-ing to pull the particle energy back toward equilibrium, aparticle’s energy deviation remains outside the lattice en-ergy acceptance for many turns. An electron emitting ananomalously-hard beamstrahlung photon can easily be lost,even if there is ample RF overvoltage to keep it in a stableRF bucket.In a Higgs factory the situation will be qualitatively dif-ferent. There will still be ample RF overvoltage, except atthe very highest beam energy, where the luminosity willnecessarily drop precipitously. But there will be multipleRF cavities distributed at equal intervals around the ring.Furthermore these cavities will be routinely restoring a sig-nificant fraction (more than one tenth for example) of thebeam energy each turn.In this circumstance it seems appropriate to compare u ∗ c to eV excess , the excess energy the RF cavities are capable ofrestoring each turn, rather than to η E . Values of eV excess aregiven in Table 1. When eV excess is negative, the luminosityobviously vanishes. But, especially for beam energies lessthan 250 GeV, eV excess is many GeV. Even a 10 or 20 GeVbeamstrahlung emission will typically not knock an electronout of its stable bucket. This luminosity further assumes(and it must be checked) that the distributed RF will save theparticle as long as it remains within the stable RF bucket.It has been seen earlier that the probability for the en-ergy of a single beamstrahlung photon to exceed 10 u ∗ c isabout 10 − . (Multiplied by the emission rate, this wouldgive an unacceptably large loss rate.) The probability for thesame electron to emit two photons with energy summing to20 u ∗ c will be about 10 − . (Multiplied by a total number ofemissions in one damping time, this gives the probability ofenergy loss in excess of 20 u ∗ c , which would cause the particleto be lost.) This probability is small enough to make beam-strahlung loss small compared to other loss mechanisms. Bythis estimate, the requirement on the critical energy will be u ∗ c < eV excess . (99)37ccepting this as an equality, and substituting into Eq. (97) L bslongit = N ∗ N b H ( r yz ) r yz a xy β y × f (cid:16) √ π . × . . m √ /π (cid:17) r e (cid:72) E (cid:16) eV excess (cid:17) . (100)This formula is deceptively simple because it ascribesall luminosity restriction to beamstrahlung. Though it issensitive to the bunch aspect ratios, it has the curious prop-erty of being independent of the overall bunch scale. Onlyafter σ y has been fixed are the absolute bunch dimensionsdetermined. Furthermore the formula suggests, unrealisti-cally, that the luminosity can be made arbitrarily large bychoosing a xy large (wide beam) and r yz small (long beam).Counter-intuitively, the formula also favors large β y . Butthese factors are constrained by other design considerations.Only realistically achievable parameters can be used, andother constraints, such as whether the assumed aspect ratiosare achievable, have to be checked.For all these reasons L bs − limited . summed will always be treated asan upper limit on the luminosity that can be achieved withbeamstrahlung still being negligible. The r.m.s. horizontal beam width at the crossing pointsis given by a xy σ ∗ y and the arc-determined horizontal emit-tance is given by Eq. (49). This value of emittance is known(from the lattice design) to give an acceptably small hori-zontal beam size in the arcs of the ring, σ arc x = (cid:112) β arc x (cid:15) x .The discussion of beam-beam saturated operation showedthat β ∗ x has to be determined by the condition that x and y beam-beam tune shifts are approximately equal. The twodeterminations of β x will not, in general, be equal. If this isthe case we have to suppose that the lattice will be modifiedto make the arc-determined horizontal emittance give the re-quired beam aspect ratio at the IP. The horizontal emittanceis then given by (cid:15) x = σ x β ∗ x . (101)One thing that can go wrong with β x , and very hard to esti-mate, is whether the intersection region optics can actuallybe designed. A possible failure mechanism is for σ arc x , max to be insufficiently small compared to the separation of thecounter-circulating beams. These concerns have been largelyallayed by preliminary lattice design by the LBNL/SLACgroup [17].
11 E. DECONSTRUCTING YUNHAICAI’S IR OPTICS
This section contains an elementary discussion of chro-maticity correction of a circular e+e- Higgs factory, withemphasis on CepC. To achieve high luminosity at the in-tersection point requires the beam to be focussed down to a “point” or a “ribbon” or (given Liouville’s phase spacerequirements) a flattened ellipse, whose dimensions are char-acterized by the (Twiss) beta-functions β x and β y .For various reasons the entire ring, including the intersec-tion region, has to be approximately“achromatic”, meaningindependent of fractional momentum offset δ . Regrettablymagnetic lenses are inherently chromatic since their focallengths are proportional to particle momentum. The onlyway known to reduce the chromaticity is to place nonlin-ear elements, namely sextupoles (loosely speaking they arelenses with focal length proportional to radial position) atlocations in the lattice at which there is non-vanishing dis-persion (radial position proportional to momentum). This isroutine, since higher momentum particles tend to circulatemore toward the outside of the vacuum chamber.In early storage rings, up to and including LEP, the IPchromaticity was cancelled “globally” using sextupoles dis-tributed more or less uniformly in the arcs of the ring. Start-ing with the B-factories, and influenced by ILC “final focusoptics” studies, schemes for distributing the chromatic com-pensation “locally” in intersection region (IR) have beendeveloped. Yunhai Cai’s intersection region optics design islocal in this sense.Local chromatic compensation has been critical to obtain-ing the astonishingly large B-factory luminosities. UnlikeB-factories, at the Higgs production threshhold energy andhigher energies, the Higgs factory will be RF-dominated (asexplained in the rest of the paper). This is a sufficiently largedifference to call into question the superiority of local overglobal chromaticity correction. But this appendix does not address this question. Rather it assumes that the local routewill be taken.Large values of either of the lattice beta-functions, β x or β y implies operational sensitivity to lattice imperfections.It will become obvious that achieving small β ∗ y at the IP (asneeded for high luminosity) implies a very large value of β y peak near the IP. And, for the same reason, a smallish valueof β ∗ x implies a largish value of β x .Primary emphasis is therefore on preserving the capabil-ity to adjust the vertical beta function β ∗ y at the crossingpoints, without changing in quadrupole locations. As Caihas explained, this precise capability is automatically builtinto his lattice design. This is important to retain the optionof running with large numbers of bunches N b and small β ∗ y on the one hand, or small N b and larger β ∗ y on the other.In section “Scaling Law Dependence of Luminosity onFree Space L ∗ ” this capability is used to investigate thedependence of luminosity on L ∗ . (This is of special im-portance for particle physics experimentalists who have tocompromise between their desire for long free space for theirdetection apparatus and their desire for high luminosity.) Us-ing dimensional analysis a scaling relation between L ∗ andthe maximum value of β max y (and hence dynamic aperture)is obtained.Copied from Yunhai Cai’s β ∗ y = π -phase advance sections,38eeded to cancel their nonlinear kicks. Tunability of thesesections, for example to reduce maximum beta functionvalues compared to the β y = In terms of Twiss parameters, the transfer matrix fromarbitrary lattice point 1 (e.g. the IP) to arbitrary lattice point2 (e.g. the start of the FODO arc) is M ( ← ) = (cid:42)(cid:46)(cid:46)(cid:44) (cid:113) β β ( C + α S ) √ β β S − S ( + α α ) + C ( α − α ) √ β β (cid:113) β β ( C − α S ) (cid:43)(cid:47)(cid:47)(cid:45) , (102)where S ≡ sin φ , C ≡ cos φ , and where φ is the betatronphase advance through the section from 1 to 2.By symmetry at the IP, α = δ . By startingand ending the regular arc with a half-quad one can, withoutloss of generality, assume that α = δ (cid:44) M from IP to arc start then has the form (cid:42)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:44) (cid:113) β x β x C x √ β x β x S x − √ β x β x S x (cid:113) β x β x C x (cid:113) β y β y C y (cid:112) β y β y S y − √ β y β y S y (cid:113) β y β y C y (cid:43)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:45) . (103)This same form can be used for matching between any sectorsthat have the property that there are simultaneous waists( α x = α y =
0) in both planes at both ends. usually thereis a maximum or minimum of one or both beta functionsat any given quadrupole location. Both α -functions changesign in these cases, and for these quads there is a “waist” inboth planes in the quad interior. Especially for thin quadsit is usually valid to approximate these waist locations ascoinciding. In terms of the beta functions at both ends of asimultaneous waist, the following equations can be derivedby combining elements of the partitioned sub-matrices M ; M β x − M β x = , M β y − M β y = , M + β x β x M = , M + β y β y M = . (104) Other equations, such as β x M M + M M / β x = , M M / β x + β x M M = , (105)plus the corresponding y equations can be derived but, beingquadratic in the matrix elements, and hence giving higherorder polynomials in the unknowns, they are less amenableto solution by polynomial solvers such as MAPLE, which iswhat I use. But they can be used for checking purposes.Here we discuss transfer matrices between lattice pointswhich can, potentially, have waists in both planes. Just be-cause there can be waists at both ends does not, however,guarantee that there will be. But the transfer matrix througha sector can be designed so that a waist at one end will guar-antee a waist also at the other end; this is not unlike requiringa differential equation to satisfy periodic boundary condi-tions. For lattice design this can be a useful approach sincesome waists have to exist by symmetry and some others canbe required to exist by design constraints which simply mustnot be compromised.We cannot afford to assume there is zero dispersionthrough the IR sector because this would make it impos-sible to compensate chromaticity locally. Neglecting thetiny radial focusing that occurs in bend elements, the trans-verse optics is determined by the drifts and quadrupoles.Then the elements M i j are polynomial functions of the quadstrengths q i and drift lengths L i . In principle, if the fourparameters β x , β x , β y , and β y are given, and four of the q i and L i are free variables, then the variables can be deter-mined to match the betas. In practice it is not so simple, asthere may be multiple solutions, or worse, none at all. Someor all of the matches may have complex strengths or complexor negative lengths. Nevertheless, with a certain amount oftrial and error, Eqs. (104) can be used to derive closed form,analytically-exact matched solutions. To obtain unique solu-tions it may be necessary to introduce intermediate pointsand to complete the matching sector by sector.Supposing that the q i and L i of the IP-to-regular-arc-region have been determined, the matrix M is known, andsatisfies Eqs. (104) and (105). It would be nice if theseequations could be satisfied identically in δ . But this isclearly impossible. (See, for example, in Steffen [27] forproof of the impossibility of designing fully momentum-independent bend-free sections.) One sometimes concen-trates on compensating just the “global chromaticities” χ x ≡ Q (cid:48) x ≡ dQ x / d δ and χ y ≡ Q (cid:48) y ≡ dQ y / d δ . Typically thechromaticities due to the IR section are comparable to or,especially after “ β squeeze” greater than the chromaticitiesdue to the rest of the ring.Various considerations have always made it important forthe dispersion to vanish at the intersection point (IP). For ane+e- Higgs factory there us another new important reason;beamstrahlung radiation. The simplest way to cancel the ringchromaticities is to uniformly increase the strengths of thearc sextupoles. Even with no intersection regions the pres-ence of arc sextupoles limits the dynamic aperture of the ring.39ut this limitation can be quite harmless and, for a Higgs fac-tory there are so many arc sextupoles that the arc-dominateddynamic aperture would probably remain acceptably largeby simply increasing the arc sextupole strengths uniformlyto cancel the IR-induced chromaticity. Though this easilyadjusts the chromaticities to zero, this approach proves to beunsatisfactory.Unfortunately it is the chromatic mismatch between arcsand IR sections that dominates the off-momentum dynamicaperture. For a Higgs factory it is critically important tomaximize the off-momentum dynamic aperture. It is theseconsiderations that makes achromatic IR design attractive.One (unfortunate) consequence of local chromaticity controlis that bend elements need to be built into the IR design inorder to produce and control the momentum dispersion.It is usually assumed therefore that the dominant phe-nomenon limiting off-momentum acceptance is “chro-matic beta-mismatch”. Because low-beta IR’s are “highlytuned”, the q i / ( + δ ) momentum dependence of the IRregion quadrupoles causes a mismatch which launches a δ -dependent “beta-wave” which seriously reduces the off-momentum aperture. The entries in Table 15 and its attached figure describe(an approximation to) the Cai IR design, with waist locationsidentified by letters A,B,C, . . . , I. With no x , y coupling thewaist-to-waist transfer matrix of Eq. (103) is block diagonalwith blocks of the form, M ( µ ) = (cid:42)(cid:46)(cid:46)(cid:44) (cid:113) β β cos µ √ β β sin µ − √ β β sin µ (cid:113) β β cos µ (cid:43)(cid:47)(cid:47)(cid:45) . (106)The following special cases of the form (106) are usefulfor designing chromaticity correcting sectors, such as thematched sectors BC and CD shown Figure 23: M ( ) = M ( π ) = (cid:42)(cid:46)(cid:46)(cid:44) (cid:113) β β (cid:113) β β (cid:43)(cid:47)(cid:47)(cid:45) = “0 (cid:48)(cid:48) , M ( π/ ) = (cid:42)(cid:44) √ β β − √ β β (cid:43)(cid:45) = “ π/ (cid:48)(cid:48) , M ( π ) = (cid:42)(cid:46)(cid:46)(cid:44) − (cid:113) β β − (cid:113) β β (cid:43)(cid:47)(cid:47)(cid:45) = “ π (cid:48)(cid:48) , M ( π/ ) = (cid:42)(cid:44) −√ β β √ β β (cid:43)(cid:45) = “3 π/ (cid:48)(cid:48) . (107)As an example, consider sections represented by the matrixproduct M BC = Q ( q ) ∗ D ( ) ∗ Q ( q ) ∗ D ( ) ∗ Q ( q ) , M CD = Q ( q ) ∗ D ( ) ∗ Q ( q ) ∗ D ( ) ∗ Q ( q ) , (108)with matrix multiplication indicated by asterisks. Here Q ( q ) is the 2x2 transfer matrix for a thin quad of strength q and D ( ) is the 2x2 transfer matrix for a drift (with length takento be 1 to simplify the formulas). These sections are to becombined to form the full chromatic adjustment sectors. Thequadrupoles strengths can be adjusted so that M BC = (cid:42)(cid:44) (cid:112) β B β C − √ β B β C (cid:43)(cid:45) , M CD = (cid:42)(cid:44) (cid:112) β C β D − √ β C β D (cid:43)(cid:45) , (109)in one or the other of the horizontal and vertical planes. Tun-ing curves for the quadrupole strengths accomplishing thisas a function of the product (cid:112) β C β D are shown in Figure 22.Because the drift lengths in the line have been taken to be1, the beta functions are in units of the drift length and thequad strengths are in units of 1/drift_length. One sees thatthe beta function β C at the sextupole locations can be variedover a large range by controlling the quadrupole strengthsand the matched beta functions β B and β D at the start andfinish. The concatenated matrix product is M CD M BC = (cid:32) − (cid:112) β D / β B − (cid:112) β B / β D (cid:33) . (110)To obtain the desired relation M CD M BC = − I thequadrupole strengths have to be symmetric about the centerand β B = β D .Figure 22: Tuning curves for adjusting β -functions at sex-tupole locations. Beta functions are in units of drift lengthand quad strengths are in units of 1/drift_length.Ideally one would have both M CDx M BCx = − I and M CDy M BCy = − I . Unfortunately no such solution exists.The best one can do is to satisfy the more important of thesetwo conditions—the y -relation for the χ y compensationmodule and the x -relation for the χ x compensation module.The impossibility of simultaneously satisfying the − I con-dition in both planes has only been demonstrated for thinquadrupoles. But it would be very curious for such a condi-tion to be possible for thick element but not for thin elements.Erratic behavior frequently reported for MAD Higgs factorysimulations can be due to striving for an unachievable result.40hat can be done, and has been done in the design, is toarrange for the sextupole locations to approximately coincidewith minimum beta locations of the “insensitive coordinate”;for example there are minima of β x near locations C and E,where the y -chromaticity is being compensated. Since thesepoints are not “ π -separated” in x , their nonlinear horizontalkicks do not cancel but, because β x is so small at theselocations, their nonlinear horizontal effect is expected to besmall.All the waist-to-waist sections are broken out, with pa-rameters defined, in the following series of figures. Themain “sacred” parameters, that will not be allowed to change(after having been negotiated between detector and accel-erator groups) are L ∗ = l I R =
20 m, the distance between same-sign quadrupolesthroughout the IR region. Of course even these dimensionscan change in the early design days, but they need to befrozen fairly early in the design process.
The preceeding discussion of the CepC intersection re-gion optics suggests that there is some tunability of the IRoptics, even with all the labeled waist locations held in fixedlocations, and some ability to alter the heights of the highbeta peaks needed for chromaticity compensation. Minorchanges to increase the flexibility will be suggested below.It is unnecessary to discuss the bend elements initially since,to a good approximation, they do not affect the linear optics.However they are critical for the chromaticity compensation.Also they represent serious sources of synchrotron radiusfrom which the detectors will need to be shielded.It is often (even usually) convenient in a lattice model torepresent a ring quadrupole by two thin quadrupoles, so thatthe waist location occurs midway between them, even whenthere is just one physical quadrupole.To assure symmetry of the y -chromatic adjustment cen-tered at D, β y has to have the same values at waist locationsB, D, and F. Then, for the same reason, the β x values at F,H, and J must also be the same, though not necessarily withthe same values as β y at B, D, and F. One eventually has tocheck that the bend elements are satisfactorily positioned tobe able to flexibly provide the required dependence of thedispersion D ( s ) .These assumptions heavily constrain the IR section op-tics, but there is still considerable freedom. The heightof the first peak at C of the y -chromaticity module can beadjusted using quad strengths qBC1=q1, qBC2=q2, andqBC3=q1, as governed by the tuning curves of Figure 22including qBC3=qBC1. By adjusting the beta functions atB the heights of the β peaks (especially β Cy which is large)can be adjusted. The relations qCD1=q1, qCD2=q2, andqCD3=q1 are then imposed by symmetry. This establishesthe χ y compensation centered at D. Quad strengths for thepeak at E can be copied: qDE1=q1, qDE2=q2, qDE3=q1,qEF1=q1, qEF2=q2, qEF3=q1.The x -chromatic adjustment peaks follow the same pat-tern. The same values of quad strength magnitudes, but with reversed signs, qFG1=qFG3=-q1 and qFG2=-q2, establish π -separated β x peaks in the χ x -compensating module. Evenso, the equal β x peak heights at D and I are not necessarilyequal to the earlier β y peaks.There is still freedom in adjusting section AB, which canusefully be thought of a forming a complex triplet with themirror-symmetric section on the other side of the IP and withthe beam-beam focusing force forming the central element.Since the beam-beam strength is variable, and can be strong,one can try to keep this section matched independent of beamcurrent. Then all the other quads can track, to maximize thedynamic aperture at every beam current.Part of the lore of beam-beam interaction effects is thatthe beam-beam tune shift parameters ξ x and (especially) ξ y are more usefully thought of as measures of the nonlinearityof the beam-beam interaction than as their technical accu-rate definitions as quantitative shifts of the betatron tunes.Nevertheless, it still makes sense to incorporate the linearpart of this focusing into the linear lattice model. Thoughthe beam-beam focusing is far from being linear, the leadingnonlinearity is octupole. This leads to amplitude-dependentdetuning which is not automatically undesirable, especiallyin conjunction with the extremely strong betatron dampingat a Higgs factory.The strong damping at a Higgs factory also motivateslooking into the possibility of avoiding the so-called “beta-squeeze” or beam separation cycling between injection anddata collection operational states. One should strive to deter-mine optics that can evolve adiabiatically as the beam cur-rents are gradually increased during initial injection. Withtopping-off injection the beam-beam tune shifts need notchange by more than a percent or so between injection cycles.Except for this change, the beam-beam force is just a linearelement, focusing in both planes.One can further strive to design turn-by-turn, bumper-free,kicker-free, injection. If this injection process can be de-signed to be continuous and consistent with simultaneousdata collection, then there will be no need for frequent cy-cling between injection and data collection. This holds outthe possible goal of day-long, or longer, steady-state datacollection runs. Unique Parameter Determinations.
The y -Chromaticity Correction Module Because the IPbeta functions satisfy β ∗ y < β ∗ x the vertical chromaticitydue to the IP is greater than the horizontal. This makes itappropriate to concentrate first on compensating the verticalchromaticity and to place its compensating module as closeto the IP as possible. The transfer matrix through an arbitrarysector has the partitioned form M BC = (cid:32) M BCx M BCy (cid:33) . (111)To simplify notation the initial and final lattice locations arelabeled B and C, but the same constraints will be applied to41
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
BA C D E F G H I J
Figure 23: Yunhai Cai IR design, with significant waist locations identified as A, B, C, . . . .all subsequent sections. It will be shown that the followingrelations unambiguously determine the entire matched y -chromaticity correcting module:for y , M y = , M y = , M y = (cid:113) β By β Cy , and for x , M x = . (112)Notice that these equations are consistent with y havingwaists at both B and C. But they do not allow x to have waistsat both positions. These constraints leave considerable flexi-bility to choose necessary adjustment parameters. From the y constraints (along with the unit determinant requirement)one sees that the precise “ π/
2” pattern of Eq. (107) will havebeen achieved for M y . From the x constraint one sees thatpart of the “ π ” pattern will have been achieved, M x = M x = M BC = (cid:42)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:46)(cid:44) − α Cx − α Bx β Bx − (cid:113) β By β Cy − √ β By β Cy (cid:43)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:47)(cid:45) . (113)In this form β Bx is free, and β Cx has the same value. Also β By is free, but the product β By β Cy is fixed. To make β Cy largeone will make β By small. The y phase advance is exactly π/ x propagation is not quite matched—i.e. there isno x -waist at C even if there is an x -waist at B. Exploiting Quadrupole Polarity Reversal.
It was statedabove that the solution to Eqs. (112) is unique, and that iscorrect, but these equations have a natural symmetry thatguarantees the existence of a solution to a related set of42able 15: Twiss function values eye-balled from the Yunhai Cai Higgs factory final focus optics shown above for β ∗ y = β ∗ y = 10 mm design that woulduse the same or almost the same element locations.waist waist (or other) location Yunhai Cai IR design targetslabel identifier m β x β y η β x β y A collision point 0 50 mm 1 mm 0 1 m 10 mmfirst quad L ∗ / + lQ / < < < < < QAB1 QAB2 QAB3QA* lAB3lAB2lAB1
A B
QAB1QA* lAB3lAB2lAB1
A B
QAB2
QBC0
10 20 lAB4QAB3a,b QAB5a,blAB5 after modification
QAB6QAB7QAB4
Figure 24: The AB “demagnifying” section adjacent to the IP, shown before and after extensive redesign, lengthening theIR region by 10 meters. The beam-beam focusing element QA ∗ represents half of the beam-beam focusing, with the otherhalf provided by the mirrored section on the other side of the IP. The linear focusing part can be designed into the basiclattice design, even adapting the full ring optics as the strength of the beam-beam force varies with beam currents. Thoughshown as horizontally defocusing the beam-beam force actually focuses in both planes.equations. This symmetry is associated with reversing thepolarities of all the quadrupoles. This interchange can beassociated with interchanging the x and the y constraintequations. This means that the design of the x -chromaticitymodule can be (in fact has to be) identical to that of the y -module. In both cases the height of the beta peaks atsextupole locations are controlled by the beta functions atthe cell ends. To obtain high β y peaks the input β y valueis low. This choice also affects β x at the same locations,but the values of β x are low (and hence negligible) at thoselocations.One can now understand the function of the symme-try reversal (i.e. quadrupole polarity reversal) starting atlocation F. For full self-consistency the quadrupole at Fhas to be turned off to preserve this symmetry. That is qFG = − qEF
3. As one consequence of symmetry, the beta function pattern will be the same in the x -module as inthe y -module, with the exception that horizontal and verticalbeta functions are reversed. In particular the β y peaks canbe large in the y -module and the β x peaks can be large inthe x -module. Conservation of both beta function values at(every second waist) locations B, D, F, H, and J, makes itpossible to tune the heights differently in the y and x mod-ules by adjusting these conserved β x and β y values. Thisdegree of tunability makes it practical to use the same latticelayout over a large range of chromaticity adjustment, evenincluding large differences between χ x and χ y . Resulting Lattice Parameters.
Referring to the (com-pletely general) form in Eq. (102), one sees that the followingphase-advance-dependent determinations have been made: S BCx = , C BCx = − , S BCy = , C BCy = . (114)43lso determined have been β By = β Cy , α By = α Cy = , and β Bx = β Cx , but α Bx (cid:44) α Cx . (115)The initial beta functions β Bx and β By remain free.The numerical values of the elements in M BC depend ona master IR length scale, which is being held frozen here,and on the quadrupole strengths qBC = qBC = − / = − .
075 /m and qBC = / = . M BC = (cid:42)(cid:46)(cid:46)(cid:46)(cid:46)(cid:44) − − / − /
40 0 (cid:43)(cid:47)(cid:47)(cid:47)(cid:47)(cid:45) . (116)The reason these elements are so thoroughly numerical, isthat the basic (half-cell) length l I Rh has been specified to be10 m. The entries are not approximate, they are exact. Fordifferent values of l I Rh the entries can simply be scaled onthe basis of dimensional analysis. A numerical consequenceof the y entries is that, with β By = β Cy peak is given by β Cy = / β By e . g . = Concatenating Successive Lattice Sections.
Thetransfer matrices from B to D are given by M BDx = (cid:42)(cid:44) − ∆ α BCx β Bx − (cid:43)(cid:45) (cid:42)(cid:44) − ∆ α BCx β Bx − (cid:43)(cid:45) = (cid:42)(cid:44) − − ∆ α BCx β Bx − (cid:43)(cid:45) , M BDy = (cid:42)(cid:46)(cid:44) (cid:113) β By β Cy − √ β By β Cy (cid:43)(cid:47)(cid:45) (cid:42)(cid:46)(cid:44) (cid:113) β By β Cy − √ β By β Cy (cid:43)(cid:47)(cid:45) = (cid:32) − − (cid:33) . (117)Defining ∆ BCx = α Cx − α Bx , the transfer matrices from B toF are M BFx = (cid:42)(cid:44) ∆ α BCx β Bx (cid:43)(cid:45) , M BFy = (cid:32) (cid:33) . (118)It can be seen that the magnitude of the matrix element M x “defect” accumulates by the same amount each section.Location F marks the end of the y -chromaticity correctionmodule and the beginning of the x -chromaticity correctionmodule. To make F be a true waist one can refer back to thestarting point at B and impose, as an initial condition, α B − x = ∆ α BCx . (119) Checking the Twiss Parameter Evolution.
One canuse evolution formulas (129) (which simplify markedly) to perform various checks of the results just obtained: β Cy ? = M BCy β By = β By β Cy β By = β Cy , (cid:88) (120)0 ? = α Cy = M BCy , M BCy , α By = , (cid:88) (121) β Cx ? = M BCx , M BCx , β Bx = β Bx , (cid:88) (122) α Fx ? = − M BFx , M BFx , β Bx + M BFx , M BFx , α B − x = − ∆ α BC β Bx β Bx + ∆ α BC = . (cid:88) (123)The x -chromaticity module from F to J can just be copiedfrom the module from B to F just derived. With the modulesjoining at the common waist at F, the x -correction modulesimply inherits its input β functions from the y -correctionmodule. Suggested Lattice Alterations.
Figure 25 is almost thesame as earlier figures, but suggested modifications are in-cluded, primarily by splitting and slightly separating thepreviously-superimposed quadrupoles at all of the labeledlattice locations, B, C, D, E, F, G, H, I, and J. There aredifferent reasons for doing this in each case and some of thereasons are more important than others.The most important quadrupole splits are at sextupolelocations C, E, G, and I. By placing a sextupole at the ex-act center of its two matched quadrupoles one makes bothits chromatic compensation and its betatron phase locationrelatively insensitive to lattice errors. Without doublingsuch a quadrupole, its sextupole sits where the beta functiondepends strongly on position, making it hypersensitive to lat-tice imperfections. The effects of such errors are “amplified”by unballencing the sextupole pair cancellation.Since the quadrupole at F has to be “turned off” to complywith the symmetry reversal, this quadrupole could simplybe removed. Better, however, is to split it and separate it toform a (weak) doublet. To the extent both are turned off, thiscorresponds to no change whatsoever. But, when turned on,such a doublet can be used to alter the β x and β y values inthe separate x and y modules. This can be used to producean independent knob for tuning the χ x module relative tothe χ y module.Doubling the quadrupoles at B and J is also likely toprove useful. The main defect of the overall design is thenon-vanishing M element in Eq. (113). It will be seenin the next section how this defect is to be handled. Thisplaces extra demands on tuning the transitions into and outof the chromatic correction modules. The (newly-available)doublet at F can be involved in this same adjustment.There may be no great advantage to splitting thequadrupoles as locations D and H, at the exact centers of the χ x and χ y modules. However there is an important formaladvantage to treating these two locations the same as all theothers. It is that (until the split quadrupoles are tuned tono longer exact match) the overall transfer matrices can beobtained by concatenating identical transfer matrices. As44t happens it is only the quadrupoles with odd indices (e.g.QBC1 and QBC3) that are split. Quadrupoles with evenindices (e.g. QBC2) are not split. The individual transfermatrices then depend on only two independent quadrupolestrengths, qBC = q qBC = q
2. As a consequencethe overall transfer matrices have the same property.There is yet another advantage to splitting the quadrupolesas I am proposing. By splitting the quads at the maximumbeta locations, quadrupoles what will necessarily be imple-mented as thick quads in the physical lattice will be beingrepresented by two, slightly-separated, thin quads in the thinquad model. Being at beta function maxima, the overalloptics is more dependent on the quality of these quadrupolesthan on any of the others. Representing these quads as twothin quads is the first step in the eventual further splittingof these quads (as required to preserve symplecticity in thethick element model) into the enough zero-length quads forfaithful representation. As a result the thin element modelbeing described here will more nearly resemble the morenearly faithful thick element model.This last argument is sufficiently persuasive that I willimplement a CepC thin quadrupole model with the A,B,. . . ,Jquadrupoles split as I suggest, whether or not they are reallysplit in the eventual physical model. Such a model is certainto be symplectic, and is likely to be have properties quiteclose to other models of the expected performance (such asdynamic aperture and injection efficiency).
The effects of splitting the quads as suggested have beeninvestigated numerically by repeating the calculations ofSections 11.4 and 11.4. The quad separation distances arealways lq = . (cid:113) β By β Cy =35.9754. Corresponding to Eq. (116) the re-sulting BC transfer matrix is M BC = (cid:42)(cid:46)(cid:46)(cid:46)(cid:46)(cid:44) − − . − . − . (cid:43)(cid:47)(cid:47)(cid:47)(cid:47)(cid:45) . (124)The transfer matrices from B to D are M BDx = (cid:32) − . − (cid:33) , M BDy = (cid:32) − − (cid:33) . (125)The transfer matrices from B to F are M BFx = (cid:32) . (cid:33) , M BFy = (cid:32) (cid:33) . (126)All these results are consistent with expectations. As beforethe only (and unavoidable) blemish is the accumulating M x elements. The only significant numerical changes (comparedto the pre-modification results) is in the M y element, whichhas been reduced from 40/m to 35.99/m. As a result the max-imum β y value is, for example, 35.99 =1295 m comparedto the corresponding earlier value of 1600 m (with the β Cy value taken to be 1 m in both cases.) This reduction is pre-sumably the result of moving the end quadrupoles in slightlyand representing the quads at C and E more faithfully bytwo separated zero-length quads than by single zero-lengthquads. This change has the cosmetic benefit of eliminatingunsightly beta function kinks at sextupole locations D, E,G, and I and providing the sextupoles comfortable nests (asemphasized above). The following figures illustrate a tentative IR design basedon the principles described above.
Suppose that the arc sextupoles have been tuned to makethe ring achromatic in the Q (cid:48) x = Q (cid:48) y = δ . In particular their“period matched” end values, in both planes, are α ( δ ) and β ( δ ) . It is this limitation that forces the IR chromaticityto be compensated locally. But, to the extent this compen-sation is only approximate, the same considerations will beapplicable.Fortunately the compensation of excess chromaticity willhave only minor effect on the arc optics when spread uni-formly over the entire arcs. It is the residual chromaticdependence of the IR sections that really matters (even ifthis chromaticity has been greatly reduced in the IR design).After some iterations to optimize the overall design, assumethat the derivatives with respect to δ , β (cid:48) x , β (cid:48) y , α (cid:48) x , α (cid:48) y , (127)at the ends of the arcs are known. Perfect IR design wouldexactly compensate these dependencies to make the IP val-ues α ( δ ) and β ( δ ) independent of δ at the ends of the IRsectors. We expect this to be impossible, but it might bethought unnecessary to be quite so fussy. We could affordto let β x ( δ ) and β y ( δ ) continue to depend on δ , but withmatching good enough to preserve about the same dynamicranges as on-momentum. Even if the beta values vary sub-stantially over the momenta present in the beam, the resultingbeam distortion at the IP could be acceptable, and mightcause only a minor loss of luminosity. However, closure ofthe lattice off-momentum requires α x ( δ ) and α y ( δ ) to beindependent of δ ; that is α (cid:48) x ( δ ) = α (cid:48) y ( δ ) = . (128)If these relations are satisfied the lattice might be expectedto stay approximately matched over a substantial range of45 even if the β ’s vary with δ . This is a luxury which ismade possible for the designers of linear colliders by the factthat, after collision, the unscattered particles do not need toremain captured. Unfortunately this “freedom” will now beshown to be unavailable in a storage ring.The formulas by which Twiss functions evolve from latticeposition 1 to 2 are β = M β − M M α + M + α β ,α = − M M β + ( M M + M M ) α − M M + α β . (129)To proceed backwards through a sector such as the demag-nifying section we require the inverse matrix, M − = (cid:42)(cid:46)(cid:46)(cid:46)(cid:46)(cid:44) M − M − M M M − M − M M (cid:43)(cid:47)(cid:47)(cid:47)(cid:47)(cid:45) . (130)The elements of this matrix can then be substituted into theinverses of Eq. (129); β = M β + M M ( − α ) + M + α β ,α = M M β + ( M M + M M )( − α ) + M M + α β . (131)In this step the signs of α have been reversed since theevolution is back through the demagnifying section. A usefulspecial case of Eq. (131) applies when α is known to vanish,in which case β = M β + M β ,α = M M β + M M β . (132)On-momentum ( δ =
0) and Eqs. (131) are presumably al-ready satisfied (in both planes) because the lattice is assumedto be already matched. It is the first order momentum depen-dence we are interested in. Furthermore, as explained above,we are only insisting on the α matches. In particular, wedemand that conditions Eq. (128) be satisfied, with α s sub-stituted from Eq. (131). Since this is two fewer conditionsthan full achromaticity requires, we can remain hopeful thata bend-free IR could be designed to satisfies them.All quantities appearing on the left hand sides of Eq. (128)are known. The matrix elements M i j are all known as poly-nomials in the q i , the L i and δ . The on-momentum α ’s and β ’s are known from the matched lattice design and theirslopes are known, according to Eq. (127). The operativeword “known” is being used loosely here since, as mentionedalready, a certain amount of iteration will be required. The Twiss parameters and their first order momentum derivativeswill be known from whatever lattice fitting software is beingused.If the IR section were being matched to general arcs, ac-cording to Eqs. (128), formulas for the α -functions wouldalso be required. But since we are fitting to a pure FODOarc (neglecting the perturbing influence of the far straight)the arc α -functions vanish at the IR boundaries. ConditionsEq. (128) then reduce to dd δ (cid:16) ˜ M ˜ M ˜ β x + ˜ M ˜ M β x (cid:17) = , dd δ (cid:16) ˜ M ˜ M ˜ β y + ˜ M ˜ M β y (cid:17) = . (133)The tildes on the ˜ M i j , ˜ β x and ˜ β y indicate that allquadrupole strength parameters q i have been replaced by q i / ( + δ ) ≈ q i ( − δ ) in the formulas expressing the ma-trix elements in terms of the quadrupole strengths and driftlengths; sextupoles do not contribute in leading order. For amatch to the regular arc, valid to linear order in δ , the fac-tors ˜ β x and ˜ β y have to agree with the values in Eq. (127).(Because of the weak chromaticity of the arcs, just treating˜ β x and ˜ β y as independent of δ may be adequate.)One can note that the validity of Eqs. (133) would implythat Eqs. (105) are satisfied independent of momentum (toleading order.) Also, if points 1 and 2 are reversed in theabove argument, one obtains an equation equivalent to thefirst of Eqs. (105) (in lowest order.) So requiring Eq. (133)amounts to requiring that Eqs. (104) hold not only for δ = δ . This means that demanding amomentum-independent α match implies also a momentum-independent β match. As suggested above, such a match isprobably impossible.It is considerations like these that make it obligatory tocompensate the chromaticity locally within the IR regions tomake the compensation as local as possible. At a minimumthis has required the introduction of bending elements andsextupoles into IR lattice designs, for example to satisfyconditions Eqs. (133). Since the simplest form of achromatuses identical sextupoles separated by phase advance of π , itis appropriate to use two sections, each with phase advance π/ This section will be of little value until it contains a discus-sion of sextupole families. It is present primarily to evaluatederivatives needed in Eq. (133).
For simplicity we assume thin lenses everywhere, eventhough, ultimately, thick lens formulas have to be applied,especially to the quadrupoles adjacent to the IP. Since thesextupole strengths S and S are determined only implicitly,they have to be determined to adjust the overall chromatici-ties to zero (or whatever nearby values are called for.) Letus assume that each FODO cell starts and ends with a verti-cally focusing half quad of strength q (which is negative,46eaning horizontally defocusing), the middle quad strengthis q (which is positive), and the half-cell lengths are (cid:96) . Thehorizontal transfer matrix through the first half-cell is (cid:32) − q (cid:33) (cid:32) (cid:96) (cid:33) (cid:32) − q (cid:33) = (cid:32) − q (cid:96) (cid:96) − q − q + q q (cid:96) − q (cid:96) (cid:33) . (134)Momentum dependence can be built into this formula bythe replacements q i → q i / ( + δ ) . Sextupoles can also beincorporated if they are superimposed on the quadrupoles.Decomposing the horizontal displacement as x = x β + η x δ ,the horizontal angular deflection caused by a quadrupole ofstrength q i with a sextupole of strength S i superimposed, is S i x /
2, or ∆ x (cid:48) = (cid:16) q i + δ + S i η x δ (cid:17) x + terms to be dropped ≈ ( q i + ( S i η x − q i ) δ ) x . (135)The “terms to be dropped” require special discussion. Drop-ping the term proportional to δ is probably always valid.The term S i x , being nonlinear, is always serious. By can-celing pairs of π -separated kicks one can hope to validatedropping these terms. Let us therefore define the dimension-less parameters˜ q = (cid:0) q + ( S η x − q ) δ (cid:1) (cid:96) , ˜ q = (cid:0) q + ( S η x − q ) δ (cid:1) (cid:96) , (136)which “wrap” or “hide” the functional dependencies on δ ,the S ’s, η x and (cid:96) . The horizontal transfer matrix throughthe full cell is given by M x ( δ ) = (cid:32) − q − q + q ˜ q ( − ˜ q ) (cid:96) ( − ˜ q − ˜ q + ˜ q ˜ q )( − ˜ q ) /(cid:96) − q − q + q ˜ q (cid:33) ≡ (cid:32) cos µ x ( δ ) β x ( δ ) sin µ x ( p ) − sin µ x ( δ ) β x ( δ ) cos µ x ( δ ) (cid:33) (137)Here µ x ( δ = ) is the on-momentum horizontal phase ad-vance per cell. The β -functions, now valid to linear order in δ are obtained from − M / M , β x = l (cid:115) − ˜ q − ˜ q (cid:115) q + ˜ q − ˜ q ˜ q ,β y = l (cid:115) + ˜ q + ˜ q (cid:115) − ˜ q − ˜ q − ˜ q ˜ q ,β x = l (cid:115) − ˜ q − ˜ q (cid:115) q + ˜ q − ˜ q ˜ q ,β y = l (cid:115) + ˜ q + ˜ q (cid:115) − ˜ q − ˜ q − ˜ q ˜ q . (138)The phase advances are given bysin µ x ( δ ) = ˜ q + ˜ q − ˜ q ˜ q , sin µ y ( δ ) = − ˜ q − ˜ q − ˜ q ˜ q . (139) To use relations (136) it is necessary to have formulas forthe η x functions; η x = ( − q (cid:96)/ ) l ∆ θ ( sin µ ( x ) ) ; η x = ( − q (cid:96)/ ) l ∆ θ ( sin µ ( x ) ) . (140)For equal tunes these reduce to η x = ( + | q | (cid:96)/ ) l ∆ θ | q | (cid:96) ; η x = ( − | q | (cid:96)/ ) l ∆ θ | q | (cid:96) . (141)Since these are already the coefficients of terms first order in δ it is not necessary to allow for their momentum dependence.This means their values can simply be copied from the outputof a lattice program. In fact, since the same commentscan be made about the sextupole strengths, it will only benecessary to know the products S η x and S η x . The mostrudimentary, most local, form of chromatic correction is tochoose S = q η x , S = q η x , (142)though, of course, this compensates only the arc chromaticity.(In fact. replacement (142) does not even exactly cancel thearc chromaticities since it does not allow for differences ofthe beta functions at the sextupole locations.) Since η xi is(almost) always positive, S i will normally have the samesign as q i . The following derivatives enter Eqs. (133): d ˜ q d δ = ( S η x − q ) (cid:96), d ˜ q d δ = ( S η x − q ) (cid:96). (143)In these formulas the coefficients S η x − q and S η x − q can be regarded as the excess due to compensating also theIR chromaticity.47 B2 B3 B4 lBC2 lCD1 lEF2QBC1 QBC2 QCD2 QEF1 QEF2 QEF3lDE2 lEF1lDE1 QDE2 QDE3QCD1QBC3
10 30 40 50 60 70 80 90symmetryreversal π separation lBC3 lCD3 lDE0lCD2QCD3QDE1 C D E lDE3 lEF0 lEF3 FB lBC1 lCD0lBC0 B5 π separation B7 B8 QFG2 QFG3 QGH2lFG1 lGH2lFG2 lGH1QFG1 H QHI2lHI1 lHI2 symmetryreversal100 110 120 130 140 150 160 170 180
QGH1
F G lGH0 lGH3
QGH3 B6 QHI1 lHI0 lHI3
QHI3 QIJ2 QIJ3lIJ2lIJ1 I lIJ0 QIJ1 J lIJ3 Figure 25: Suggested modifications to improve lattice tunabiity. The changes amount to cutting the quadrupoles in halfat all labeled locations B through I and separating them symmetrically. This permits the compensating sextupoles to beoptimally situated, and other advantages are given in the text.48igure 26: Chromaticity tuning modules, vertical above,horizontal below.Figure 27: Beta function plot for matched fit to full CepCbeamline, with β ∗ y =
10 mm, from the IP to the end of thechromaticity correcting modules. Relative to the Yunhai IR,the demagnifying section is 10 m longer which shifts theremaining elements 10 m towards larger s . The variation ofthe “non-peaking” beta function through the chromatic mod-ules cannot be eliminated without disrupting the required -I compensation condition. However the -I preservation isextremely robust. Figure 28: Tentative fit to full CepC beamline from the IP tothe beginning of the chromaticity correcting sections. Thissection is probably overly complicated (to simplify the matchto the chromatic module) and it has not been optimized usingMAD or any other fitting program.49 EFERENCES [1] S. Peggs, M. Harrison, F. Pilat, M. syphers,
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