J. Gao, M. Xiao, F. Su, S. Jin, D. Wang, S. Bai, T.J. Bian Analytical estimation of maximum beam-beam tune shifts for electron–positron and hadron circular colliders J. Gao, M. Xiao, F. Su, S. Jin, D. Wang, S. Bai, T.J. Bian IHEP The 55th ICFA Advanced Beam Dynamics Workshop on High Luminosity Circular e+e- Colliders – Higgs Factory IHEP, Oct. 9-12, 2014

Luminosity from colliding beams For equally intense Gaussian beams Expressing luminosity in terms of our usual beam parameters Particles in a bunch Collision frequency Geometrical factor: - crossing angle - hourglass effect Transverse beam size (RMS) where

Importance to understand beam-beam effects The is found experimentally that for a given machine could not surpass a definite value so-called due to beam-beam effects The existing of is the main purpose of the beam-beam effect study, and is a measure of the beam-beam effects. There are three ways to find out , i.e. 1) experiments 2) numerical simulations 3) theoretical analyses

Importance to understand In addition experiments, numerical simulation to understand (Ohmi’s talk later), it is very important to understand theoretically. There are different views face to in our community: 1) is same for all different machines; 2) depends on machine parameter. Apparently, 1) is wrong; but questions for 2) are depends on which machine parameters? And how it depends on these machine parameters?

Three expressions of for electron-positron circular colliders (1) The first analytical expression for was Gao’s formula in 1998 (J. Gao, “Analytical expression for the maximum beam-beam tune shift in electron storage rings”, Nuclear Instruments and Methods A 413 (1998) 431-434), which was improved in 2004 (J. Gao, “Emittance growth and beam lifetime limitations due to beam–beam effects in e+e- storage ring colliders”, Nuclear Instruments and Methods in Physics Research A 533 (2004) 270–274).

The first attempt was in 1998 The actual formula was in 2004 (general expression) (isomagnetic case)

1) Gao’s formulae: For lepton collider: For hadron collider: re is electron radius γ is normalized energy R is the dipole bending radius NIP is number of interaction points J. Gao, Nuclear Instruments and Methods in Physics Research A 533 (2004) 270–274 J. Gao, Nuclear Instruments and Methods in Physics Research A 463 (2001) 50–61 For hadron collider: rp is proton radius where Formulae from private note of J. Gao

Three expressions of for electron-positron circular colliders (2) The second analytical expression for was Assmann and Cornelis’s formula in 2000 (R. Assmann and K. Cornelis, “The beam-beam interaction with presence of strong radiation damping”, EPAC2000, P. 1187

EPAC2000 P. 1187

2) R. Assmann and K. Cornelis’s formula       ib : bunch current e : charge of electron frev : revolution frequency re electron radius βx*βy* : β function at IP εx0εy0 : zero current emittance B: is zero if there is no beam-beam blow-up R.Assmann and K.Cornelis, “The Beam-beam interaction in presence of strong radiation damping”, EPAC2000,P1187

Three expressions of for electron-positron circular colliders (3) The third analytical expression for was Talman’s formula in 2002 (R. Talman, “Specific luminosity limit of e+e- colliding rings”, PRST-AB, Vol. 5, 081001 (2002).

A paper of 19 pages.

3) R. Talman’s formula where Richard Talman, “Specific luminosity limit of e+e- colliding rings”, Physical review special topics-accelerators and beams,Vloume5,08001,2002

Comparisons with experimental results

1) Gao’s theory: For lepton collider: Machine E（GeV） R(m) B(T) γ NIP ξ B-B tune shift (experimental value) B-B tune Shift (parameter list) DAFNE 0.51 1.42 1.2 998 1 0.0292 0.02 0.044 BEPC 1.89 9.23 0.903 3698 0.0423 0.04 0.035 BEPCII 9.31 0.677 0.0422 0.04  0.0327 PEP-II(L) 3.12 13.87 0.75 6106 0.0570 0.06 0.0510/0.0727 PEP-II(H) 8.99 166.48 0.18 17593 0.0474 0.048 0.0703/0.0498 KEKB(L) 3.5 16.20 0.72 6849 0.0592 0.069 0.127/0.129 KEKB(H) 8.0 106.667 0.25 15656 0.0527 0.052 0.122/0.09 SuperKEKB(L) 4.0 70.18 0.19 7828 0.0325   0.0028/0.0881 SuperKEKB(H) 7.0 106.06 0.22 13699 0.0463 0.0012/0.0807 SuperB(L) 4.2 56 8219 0.0382 0.002/0.095 SuperB(H) 6.7 42.95 0.52 13111 0.0696 LEP-I 45.6 3096.175 0.0491 88062 4 0.0275 0.033 LEP-II 104.5 0.1112 191781 0.0639 0.079 0.025/0.065 LEP3 120 2620  0.153 234834 0.0798 0.09/0.08 CEPC 6094 0.066  2 0.0739 0.104/0.074

hadron collider: Machine E(TeV) R(m) γ NIP f(x) ξ SppbarS 0.315 741 B-B tune shift (experimental value) B-B tune Shift (parameter list) SppbarS 0.315 741 335.75 3 0.00956328 0.00060448 Tevatron 0.98 682 1048 2 0.0141217 0.00149268 0.012 HERA(p) 0.92 588 984 0.0140073 0.00147705 0.0009 LHC 7 2801 7458 0.0252262 0.00320658 0.0034 0.005 SSC 22 9824 23400 0.0313979 0.00431618 0.0021 HL-LHC 0.0075 HE-LHC 16.5 2750 17581 0.0363833 0.00528936 FCC-hh 50 10416 53277 0.0437486 0.0068494 SppC 37.4 6236 39872 0.0431489 0.0067169 0.006

2) R. Assmann’s theory For lepton collider: Machine E(GeV) γ C (km) Ib Bunch number frev β*x β*y ε0x (10^-9π rad-m) ε0y ξy (calculate) B-B tune shift (experimental ) (parameter list) DAFNE 0.510 998 0.098 1000 120 3.06*10^6 0.26 0.009 260 2.6 0.0549 0.02 0.044 BEPC 1.89 3698 0.2404 40 1 1.25*10^6 1.2 0.05 660 28 0.0364 0.04 0.035 BEPCII 0.2375 725 88 1.26*10^6 1.0 0.015 144 2.2 0.0341 0.04  0.0327 PEP-II(L) 3.12 6106 3026 1732 136364 0.5 0.012 24 1.8 0.1386 0.06 0.0510/0.0727 PEP-II(H) 8.99 17593 1960 48 0.02204 0.048 0.0703/0.0498 KEKB(L) 3.5 6849 3.016 1637 1585 99469 0.0059 18 0.56 0.09385 0.069 0.127/0.129 KEKB(H) 8.0 15656 1188 0.61 0.02472 0.052 0.122/0.09 SuperKEKB(L) 4.0 7828 3600 2500 0.032 0.00027 3.2 0.0086 2.8704?   0.0028/0.0881 SuperKEKB(H) 7.0 13699 2600 0.025 0.0003 4.6 0.013 0.9584 0.0012/0.0807 SuperB（L） 4.2 8219 1.258 2400 978 238474 0.026 0.00025 2.0 0.005 3.4413? 0.002/0.095 SuperB（H） 6.7 13111 1900 0.00021 2.5 0.006 1.1520 LEP-I 45.6 88062 26.66 1.28 4 11253 55.6 0.25 0.0383 0.033 LEP-II 104.5 204501 1.5 0.0642 0.079 0.025/0.065 LEP3 234834 7.2 0.2 0.001 25 0.10 0.0854 0.09/0.08 CEPC 53.6 16.6 50 5597 0.8 0.0012 6.79 0.0204 0.07368 0.104/0.074

3) R. Talman’s theory For lepton collider:

The comparision of the three theories Machine Gao’s Theory Assmann’s theory Talman’s theory B-B tune shift (experimental value) B-B tune Shift (parameter list) DAFNE 0.0292 0.0549   0.02 0.044 BEPC 0.0423 0.0364 0.068 0.04 0.035 BEPCII 0.0422 0.0341 0.0327 PEP-II(L) 0.0570 0.1386 0.06 0.0510/0.0727 PEP-II(H) 0.0474 0.02204 0.056 0.048 0.0703/0.0498 KEKB(L) 0.0592 0.09385 0.042 0.069 0.127/0.129 KEKB(H) 0.0527 0.02472 0.060 0.052 0.122/0.09 SuperKEKB(L) 0.0325 2.8704? 0.0028/0.0881 SuperKEKB(H) 0.0463 0.9584 0.0012/0.0807 SuperB（L） 0.0382 3.4413? 0.002/0.095 SuperB（H） 0.0696 1.1520 LEP-I 0.0275 0.0383 0.128 0.033 LEP-II 0.0639 0.0642 0.12 0.079 0.025/0.065 LEP3 0.0798 0.0854 0.09/0.08 CEPC 0.0739 0.07368 0.104/0.074

Comparisons of three formulae with experiments from three theories Experimental Value

Maximum Beam-Beam tune shifts Choices for CEPC and SppC

Main parameters for CEPC Unit Value Beam energy [E] GeV 120 Circumference [C] km 53.6 Number of IP[NIP] 2 SR loss/turn [U0] 3 Bunch number/beam[nB] 50 Bunch population [Ne] 3.71E+11 SR power/beam [P] MW Beam current [I] mA 16.6 Bending radius [r] m 6094 momentum compaction factor [ap] 4.15E-05 Revolution period [T0] s 1.83E-04 Revolution frequency [f0] Hz 5991.66 emittance (x/y) nm 6.12/0.018 bIP(x/y) mm 800/1.2 Transverse size (x/y) 69.97/0.15 xx,y/IP 0.1/0.074 Beam length SR [ss.SR] 2.14 Beam length total [ss.tot] 2.66 Lifetime due to Beamstrahlung min 80 lifetime due to radiative Bhabha scattering [tL] 56 RF voltage [Vrf] GV 6.87 RF frequency [frf] MHz 650 Harmonic number [h] 116244 Synchrotron oscillation tune [ns] 0.199 Energy acceptance RF [h] % 5.56 Damping partition number [Je] Energy spread SR [sd.SR] 0.13 Energy spread BS [sd.BS] 0.07 Energy spread total [sd.tot] 0.15 ng 0.22 Transverse damping time [nx] turns 81 Longitudinal damping time [ne] 40 Hourglass factor Fh 0.679 Luminosity /IP[L] cm-2s-1 1.8E+34

CEPC maximum Beam-beam tune shift analytical estimation For lepton collider: J. Gao, Nuclear Instruments and Methods in Physics Research A 533 (2004) 270–274 re is electron radius γ is normalized energy R is the dipole bending radius NIP is number of interaction points J. Gao, Nuclear Instruments and Methods in Physics Research A 463 (2001) 50–61

CEPC Beam-beam simulation result Luminosity behavior depends on tune operating points. The current main parameters has been checked with beam-beam simulation, by Ohmi, Zhang Yuan, Demitry Shatilov, and it is recommended that

SppC main parameters Parameter Value Unit Circumference 56 km Bunch separation 25 ns Beam energy 37.4 TeV Number of bunches 5973 Lorentz gamma 39891 Bunch population 2.0E+11 Dipole field 20 T Accumulated particles per beam 1.2E+15 Dipole curvature radius 6236 m Normalized rms transverse emittance 4.1 mm Arc filling factor 0.79 Beam life time due to burn-off 9.3 hour Total dipole magnet length 39184 Total / inelastic cross section 140 mbarn Arc length 49600 Reduction factor in luminosity 0.96 Total straight section length 6400 Full crossing angle 71 mrad Energy gain factor in collider rings 17.8 rms bunch length 75.5 Injection energy 2.1 rms IP spot size 9.0 Number of IPs 2 Beta at the 1st parasitic encounter 19.5 Revolution frequency 5.36 kHz rms spot size at the 1st parasitic encounter 46.1 Peak luminosity per IP 1.3E+35 cm-2s-1 Stored energy per beam 6.3 GJ Beta function at collision 0.75 SR power per beam MW Circulating beam current 1.0 A SR heat load at arc dipoles 63.9 W/m Max beam-beam tune shift per IP 0.006 Energy loss per turn 2.45 MeV

SppC beam-beam tune shift limit analytical estimations For hadron collider: where rp is proton radius SppC (actual parameter list) Formulae from private note of J. Gao

SppC beam-beam effects simulation result According to the SppC parameters , we use BBSIM code to do the beam-beam effects simulation study. A beam-beam tune footprint (provided by T. Sen (Fermi) and Ming Xiao) with 2 head-on Interaction Points in SPPC (using the LHC tunes) is shown in below: And the achievable beam-beam tune shift from beam-beam simulation is 0.0061 with the LHC tunes. So it is reasonable to choose the SppC nominal beam-beam tune shift be 0.006, which is a little lower than the maximum beam-beam tune shift limit value from Gao’s analytical formulae result, 0.0064.

Discussion

Difference between an e+e- Linear Collider and an e+e- storage ring collider P0 is single beam radiation power Pb is single beam power where for storage ring luminosity expression, has been used

Conclusions (1) For electron-positron circular colliders, three formulae have be reviewed and compared with experiments and numerical results It is shown that or agrees well with experiments and numerical simulation results is a function de pending on , , and For a given constructed machine, is increasing linearly with (general expression) (isomagnetic case)

Conclusions (2)   (general expression) (isomagnetic case)  

Thank you for your attention

Back up slides

E. Keil and R. Talman (1983) S. Peggs (1999) LEP contribution R. Assman and K. Cornelis (2000)