1. 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

2. 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

3. 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

4. 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?

5. 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) ), 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).

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

7. 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

8. 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

9. EPAC2000
P. 1187

10. 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

11. 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, (2002).

12. A paper of 19 pages.

13. 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

14. Comparisons with experimental results

15. 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
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
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

16. 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
Tevatron
0.98
682
1048 2
0.012
HERA(p)
0.92
588
984
0.0009
LHC 7
2801
7458
0.0034
0.005
SSC 22
9824
23400
0.0021
HL-LHC
0.0075
HE-LHC
16.5
2750
17581
FCC-hh 50
10416
53277
SppC
37.4
6236
39872
0.006

17. 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
0.048
0.0703/0.0498
KEKB(L)
3.5
6849
3.016
1637
1585
99469
0.0059 18
0.56
0.069
0.127/0.129
KEKB(H)
8.0
15656
1188
0.61
0.052
0.122/0.09
SuperKEKB(L)
4.0
7828
3600
2500
0.032
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
2.0
0.005
3.4413?
0.002/0.095
SuperB（H）
6.7
13111
1900
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
0.104/0.074

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

19. 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
0.056
0.048
0.0703/0.0498
KEKB(L)
0.0592
0.042
0.069
0.127/0.129
KEKB(H)
0.0527
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
0.104/0.074

20. Comparisons of three formulae with experiments from three theories
Experimental
Value

21. Maximum Beam-Beam tune shifts Choices for CEPC and SppC

22. 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
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

23. 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

24. 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

25. 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

26. 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

27. 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 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,

28. Discussion

29. 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

30. 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)

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

32. Thank you for your attention

33. Back up slides

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