1. Scaling of High-Energy e+e- Ring Colliders K. Yokoya 2012.3.15 Accelerator Seminar, KEK 2012/3/15 Accelerator Seminar Yokoya 1

2. Proposed Ring Colliders Recently several authors suggested possibilities of e+e- ring colliders for Ecm>200GeV. A) T.Sen, J.Norem, Phys.Rev.ST-AB 5(2002)031001 C=233km tunnel for VLHC B)A.Blondel and F.Zimmermann, CERN-OPEN-2011-047, Jan.2012 (Version 2.9). arXiv:1112.2518 LEP3, DLEP C)K.Oide, "SuperTRISTAN: A possibility of ring collider for Higgs factory", KEK meeting on 13 Feb 2012. SuperTRISTAN D)G.Lyons, arXiv:1112.1105 [physics.acc-ph], Feb.2012. PhD thesis. Nanobeam version of A) E)D.Summers, et.al. “Rapid Recycling Magnets - Tests & Simulations”, Muon Accelerator Program 2012 Winter Meeting, 4-8 Mar.2012. SLAC. Small ring version of D) 2012/3/15 Accelerator Seminar Yokoya 2

3. Reference Parameters 3 2012/3/15 Accelerator Seminar Yokoya

4. Common Features For reducing synchrotron radiation – Large circumference – small number of bunches compared with B Factories Bunch collision frequency ranges 5kHz to ~150kHz compared with 13kHz in ILC Luminosity similar to ILC  Luminosity by one bunch collision comparable to ILC  Beamstrahlung similar to ILC 2012/3/15 Accelerator Seminar Yokoya 4

5. Beamstrahlung for Proposed Parameter Sets 2012/3/15 Accelerator Seminar Yokoya 5

6. Beamstrahlung The average energy loss and the number of photons per electron for the head-on collision with beam energy E=  mc 2, bunch charge eN, rms bunch length  z, beam size  x,  y, are given by 2012/3/15 Accelerator Seminar Yokoya 6 For flat beams,  x +  y ~  x BB Field integration length

7. Interaction Length In the case of head-on collision, the orbit length in the on-coming beam is effecttively min(  z,  y ) In the nanobeam scheme, choose  y <<  z. The interaction length is ~min(  y,  x /  ) (  half crossing angle) Combining these, define As crude approximation, Leff can be used instead of  z in the formulas of Luminosity, tune-shift, energy loss, number of photons. Note: better to eliminate  y for beamstrahlung because the beamstrahlung is insensitive to the vertical beam size. But OK because we consider here only the case of  y close to either one the of others. 2012/3/15 Accelerator Seminar Yokoya 7 zz x/x/ 

8. Relevant Formulas 2012/3/15 Accelerator Seminar Yokoya 8

9. Beamstrahlung Limit If you raise the beam energy under tuneshift limit with fixed beam structure, the power limit of synchrotron radiation is soon reached. If the upper limit of power is set high, beamstrahlung limit is reached soon or later. Beamstrahlung limit is very much different between Ring colliders and Linear colliders – In the case of Linear colliders, the limit comes basically from physics requirements. (Very high beamstrahlung, e.g., >20%, may also be a problem of accelerator: to safely lead the beam to the dump.) – In the case of Ring colliders, the beam after beamstrahlung must circulate safely over the ring. The energy loss by one collision  BS (energy spread is comparable or larger – discuss later) will accumulate over the radiation damping time. The equilibrium energy spread will be about Sqrt(number of turns in damping time) x  BS which is order of percent even if  BS =0.1%.  Very large momentum aperture is needed. 2012/3/15 Accelerator Seminar Yokoya 9

10. Beamstrahlung Limit での Luminosity Once the beamstrahlung limit is reached, the luminosity above this energy goes down as 1/E 4 （ Or 1/E 4.5 if geometric emittance is fixed) If the bunch charge is reduced to 1/n,  BS reduces by 1/n 2 but the luminosity is also reduced by 1/n 2. To restore the luminosity the number of bunches must be increased by n 2 times, hence the required power increases by n 2 x 1/n = n. 2012/3/15 Accelerator Seminar Yokoya 10

11. Increase of dynamic aperture by a significant factor is unrealistic For given luminosity and power consumption the only cures are – Huge ring (like 233km of VLCC) – Extremely small vertical emittance (like <1pm of CW250 and Summers) 2012/3/15 Accelerator Seminar Yokoya 11

12. Energy Spread What really matters in ring colliders is not the average energy loss  BS but the energy spread  BS – The former is anyway compensated by the RF system The energy spread due to the beamstrahlung is discussed in K. Yokoya, NIM A251 (1986) 1-16 for round Gaussian beam and elliptic cylinder beam. – There are two mechanisms of energy spread Orbit in the on-coming bunch is different from particle to particle Stochastic spread even along the same orbit – It was shown the latter is dominant unless n  (number of photons) is very large, e.g., for round Gaussian beams  almost no correlation between successive collisions If the typical photon energy is , then the average energy loss is –  BS  n   and the spread due to the stochastic process is –  BS  (n     Hence, –  BS   BS / (n    2012/3/15 Accelerator Seminar Yokoya 12

13. Energy Spread (continued) According to the simulations for the above parameter sets (and others),  BS  2.4  BS / (n    For not totally unrealistic parameter sets, n  is about 1 or less. Hence  BS is significantly larger than  BS The equilibrium energy spread is the square sum of synchrotron radiation and beamstrahlung. The latter is approximately 2012/3/15 Accelerator Seminar Yokoya 13

14. Luminosity Scaling with Given   2012/3/15 Accelerator Seminar Yokoya 14

15. Conclusions The luminosity scaling of ring colliders at beamstahlung limit is established. The ring colliders (in particular for Ecm=400 and 500GeV) are scientifically impossible because of the energy spread due to the beamstrahlung, under the constraints that the luminosity and power consumption are comparable to those of ILC. The only way to solve is – Huge ring – Extremely small vertical emittance The machine for Ecm=240GeV is at the border of feasibility. It is not a trivial machine. It requires serious studies of lattice design with very large momentum aperture or very small vertical emittance. 2012/3/15 Accelerator Seminar Yokoya 15