LUMINOSITY LIMITATIONS AT HADRON COLLIDERS
F. Zimmermann, CERN, Geneva, Switzerland
Abstract
I discuss fundamental luminosity limitations at hadron colliders, addressing head-on and long-range beam-beam interaction, empirical scaling, synchrotron radiation, intrabeam scattering, dynamic evolution during the store, flatbeams, heat load, power consumption, and electron cloud.
Parameters of past or operating colliders — ISR, SPS,
Tevatron, RHIC — , are compared with the LHC and further extrapolated to an LHC upgrade and to the VLHC.

1 PAST AND FUTURE
So far 4 hadron colliders have been in operation (ISR,
SPS, Tevatron, and RHIC) and a 5th is under construction (LHC). The CERN ISR started operation in 1970.
A double ring collider, it reached a peak luminosity of
2.2 × 1032 cm−2 s−1 and a maximum beam energy of 31
GeV with coasting beams of 38–50 A current each. The
ISR luminosity was limited by space-charge tune shift andspread, coherent beam-beam effects, proton-electron twostream instabilities, pressure bumps, detector background,and accumulation efficiency [1]. The ISR also produced
the first pp̄ collisions, and, when operated with bunchedbeams, it reached a beam-beam tune shift of ξ = 0.0035
per interaction point (IP) with 8 crossings [2]. The second hadron collider was the CERN Spp̄S operating since
1981 at ten times higher energy than the ISR. The Spp̄S
discovered the W and Z bosons. Its luminosity was limited by beam-beam interaction, loss of longitudinal Landau damping, number of available antiprotons, hourglass
effect, and intrabeam scattering [3]. Typical beam-beamtune shift was ξ = 0.005 at each of three interaction points.
The FNAL Tevatron is the first collider based on superconducting magnets. Colliding-beam operation here started
in 1987 [4]. Tevatron luminosity is limited by antiproton
intensity, beam-beam interaction including long-range effects, luminosity lifetime, number of events per crossing,and intrabeam scattering. It reached an antiproton beambeam tune shift above ξ = 0.009. The Tevatron discovered the b and t quarks. RHIC at BNL, the first heavy-ion
collider, delivers luminosity since 2000. The main limiting factor is intrabeam scattering. Other factors again are
beam-beam interaction, luminosity lifetime, and the number of events per crossing. LHC will start operation in2006. As for the Tevatron, limits will be beam-beam interaction, luminosity lifetime, and events per crossing. Possibly, in addition, the electron cloud produced by photoemission or beam-induced multipacting [5], and local magnet
quenches induced by the collision products [6] may prove
important. Centre-of-mass energy is 14 TeV and design
luminosity 1034 cm−2 s−1 . The LHC will be the first ma-chine where radiation damping is stronger than intrabeamscattering. The scarcity of antiprotons is no longer a problem, as LHC and all future machines will collide protons
on protons. If stronger magnets become available in the future, the LHC energy could be raised, e.g., by a factor of
2. In the following, we call this energy increase, combinedwith a luminosity upgrade to 10 35 cm−2 s−1 , the ‘LHC-II’.
Finally, there exist low-field (LF) and high-field (HF) design concepts for a Very Large Hadron Collider (VLHC)[7], reaching an energy of 100 TeV centre of mass, and the
Eloisatron Project [8].
Tables 1 and 2 list parameters for all these colliders, except for the ISR and the Eloisatron. The ISR was a rather
special machine. The properties of the Eloisatron are similar to those considered for the LF or HF VLHC.
Table 1: Example parameters for heavy-ion ion colliders:gold collisions at RHIC and lead ions in LHC.
accelerator
RHIC
LHC
ion species
gold
lead
energy per charge E/Z [TeV]0.25
7energy per nucleon E/A [TeV]
0.1
2.76
39
1148
total centre of mass ECM [TeV]
dipole field B [T]3.46
8.4
circumference C [km]
3.83
26.66
57
608
no. of bunches n b
100
6.8
number of ions per bunch N b [107 ]∗rms beam size at IP σx,y
[µm]
110
15
∗
[m]
2
0.5
IP beta function β x,ytune shift per IP ξx,y
0.0023 0.00015
rms bunch length σ z [cm]
18
7.5
63.9
124.8
bunch spacing L sep [m]rms transv. emittance γ x,y [µm]1.71.5rms longit. emittance  L /Z [eVs]
0.12
0.2
0.4
9.8
IBS emittance growth τ IBS [hr]initial luminosity L
0.2
1.0[1027 cm−2 s−1 ]luminosity lifetime τ [hr]
∼10
9.3

2

EMPIRICAL SCALING

Figure 1 illustrates that the circumference and the dipole
field are both increasing roughly with the square root of
the beam energy, which implies that half of the energy gain
has been realized by advances in magnet technology andthe other half by expanding the real estate.
At the same time, the luminosity has roughly followed
the ideal scaling, L ∝ E 2 , as is demonstrated in the left
picture of Figure 2. The right picture shows that the betatron tune Q β grows with the square root of the circumfer-

Table 2: Example parameters for pp or pp̄ colliders: Spp̄S, Tevatron run IIa (‘TeV2a’), LHC, LHC-II, HF VLHC, and LF
VLHC. † The bunches are split in 3 trains, separated by 2.62 µs; ‡ Total LHC dipole heat load is about 0.8 W/m including
the electron cloud. ∗ Assuming a dipole packing factor 0.8 for HF-VLHC, and 0.65 for LHC and LHC-II, and ignoring
possible contributions from electron cloud. Equilibrium determined by radiation damping and intrabeam scattering.
Arrows refer to dynamic changes during the store. The suffix ‘in’ indicates initial values.
accelerator
Spp̄S
TeV2a
LHC
LHC-IIHF VLHC LF VLHC
beam energy E [TeV]
0.32
0.98
7
14
50
50
dipole field B [T]
1.4
4.34
8.39
16.8
12.5
2
total energy/beam [MJ]
0.05
1
334
1320
2800
3700
circumference C [km]
6.9
6.28
26.7
26.7
89
520
6
36
2800
5600
20000
21000
number of bunches n bbunch population N b [1011 ]
1.7 (p)2.7 (p)
1.05
1.05
0.125
0.22
0.8 (p̄)∼1.0 (p̄)
no. of IPs
3
2
2 (4)
2 (4)
2
2
∗
[µm]
80, 4032
15.9
7.4
9.0, 0.9
2.0
rms IP beam size σx,y
rms IP div. σx∗ ,y [µrad]136, 272
91
31.7
34
1.8
8.7
∗
[m]
0.6, 0.15
0.35
0.5
0.22
5.0, 0.5
0.25
IP beta βx,y
0.005
0.01
0.00340.005
0.00300.0026beam-beam tune shift / IP ξ x,ycrossing angle θ c [µrad]
0
0
300
300
18
50
30
37
7.7
4.0
2.3
20
rms bunch length σ z [cm]bunch spacing L sep [m]
1150
119 †
7.48
3.74
4.45
24.6
< 10−3
3.6
114
492
38
SR power PSR [kW]dipole heat load dP/ds [W/m] 10 −3
0.2 ‡
6.6∗
7.9∗
0.09
26
∼20
63
63
115
400
betatron tune Q β
rms transv. emittance γ x,y [µm]
3.75
∼3
3.75
3.75→1.0 1→0.3,0.03
1.0
∼ 10
2.03
1.07
0.29
0.27
eq. horiz. emittance γ eq
x [µm]
longit. emittance  L (σ) [eVs]
0.11
0.11
0.2
0.15
0.09
1.00 (?)
damp. time τx,SR [hr]
1200
52
6.5
3.3
128
10
50(?)
142
345 (in.)
224 (in.)
216
IBS growth time τx,IBS [hr]damping decrement per IP [10 −10 ]
0.025
2.5
20
190
32
events per crossing
∼6
18
90
9
50
0.0006∼ 0.02
1.00
10.
1.0
1.0
peak luminosity L [10 34 cm−2 s−1 ]lum. lifetime τ [hr]
9
9
10
3.2
13
25

Figure 1: Left: circumference vs. beam energy; right:
dipole field vs. beam energy.

ence, implying a similar scaling for the cell length and thearc beta function [11].

3 LUMINOSITY FORMULAE
Ignoring crossing angle and hour-glass effect, the luminosity L0 is given byL=

Nb2 nb frev γ
4πx,N βx∗ κ(1)Figure 2: Left: luminosity vs. beam energy; right: betatrontune vs. circumference.

where Nb denotes the bunch population, n b the number ofbunches per ring, f rev the revolution frequency, γ the beamenergy divided by the rest mass,  x,N = γx the normalized emittance, and κ = σ y /σx the aspect ratio at the collision point. The correction factor due to crossing angle θ c
and finite bunch length σ z can be found, e.g., in Ref. [9].
An unavoidable luminosity constraint arises from thebeam-beam interaction. The tune shift induced in the colli-

sion with the opposing beam is characterized by

LHC collision, IP1 and IP5 only
head−on and parasitic at +− 150 murad
0.324

(2)

where rA = Z 2 e2 /(4π0 Amp c2 ) is the classical particle
radius, A the mass of the ion in units of the proton mass,
and Z the charge in units of e. Below we assume thatβy∗ /βx∗ = y /x = κ, since this gives the maximum tune
shift in both planes: ξ x = ξy ≡ ξ. Using the beam-beam
parameter ξ, we can rewrite the luminosity, Eq. (1), as
L = (frev nb Nb )

1+κ
ξ
γ
.
βy∗
2rA

(3)

The first factor, in parentheses, is the total beam current.
Three further variables are the IP beta function β y∗ = βx∗ κ,
the emittance ratio κ, and the maximum beam-beam tune
shift ξ. For flat beams (κ  1) the luminosity is a factorof two lower than for round beams (κ = 1), unless smaller
values of βy∗ = κβx∗ can be achieved. This seems difficultin proton-proton collisions, where both beams pass throughthe same low-β triplets [10].
In Section 4, we consider limits on the IP beta functions.
Section 5 addresses the effects of synchrotron radiation andintrabeam scattering, and, in particular, the variation ofbeam parameters during the store. Interplay of radiation
damping and collective instabilities is examined in Section
6. In Section 7, we investigate bounds on the total beam
current, before we draw conclusions in Section 8.

4 MINIMUM β ∗The IP beta function should be larger than the rms bunch∗≥ σz , to avoid luminosity reduction due to thelength, βx,y∗hourglas effect. A tighter limit on β x,yarises from longrange beam-beam encounters at parasitic collision points
on both sides of the IP.
In order to ensure a sufficient dynamic aperture causedby parasitic collisions, the separation of the two beams
should be nsep ≥ 10. For the LHC, this separation results in a dynamic aperture of about 8σ [12]. On the other
hand, to avoid luminosity degradation due to the crossing,not included in Eq. (1), the total crossing angle θ c should besmaller than 2σx /σz . Combining these two requirements,
we find βx∗ ≥ nsep σz /2 ≈ 5σz . For the LHC, this limit
evaluates to about 0.38 m, not far from the design beta
function of 0.5 m.
The long-range collisions induce additional tune shifts;
see Fig. 3 (left picture). The LHC beams are crossed alternatingly in the horizontal and vertical plane at the various
IPs, so that the long-range tune shifts induced at different
IPs approximately cancel each other [13]. In this case thetighter limit on the beta function applies for both planes,
For flat beams the beta functions are unequal, and thelong-range beam-beam tune shift in the vertical plane isenhanced by a factor 1/κ [11],11 nLR= − ∆QLR
ξ,
∆QLR
y
x =
κ
κ n2sep

(4)

0.322

fractional vertical tune

∗
βx,y
rA Nb
ξx,y =∗
2πγσx,y (σx∗ + σy∗ )

0.320

0.318

0.316

0.314
0.303

0.305

0.307
0.309
fractional horizontal tune

0.311

0.313

Figure 3: Left: tune footprints due to head-on and longrange beam-beam effects in LHC IPs 1 and 5, respectively(courtesy H. Grote); right: tune shift parameter vs. damping decrement δ (LEP data courtesy of R. Assmann).
where nLR denotes the total number of parasitic collisions
on either side of the interaction point, and n sep = θc /σxthe normalized separation. In view of Eq. (4), the cancellation of tune shifts between IPs is not possible at flat-beamcolliders. However, advanced techniques to compensate effects of the long-range collisions, using either an electron
lens with modulated current [14] or pulsed electric wires[15], would be applicable to flat beams as well.
Limits on the beta functions could also arise from optical considerations, such as the available aperture, maximum quadrupole gradients, and chromaticity [16]. For all
hadron colliders in Tables 1 and 2 these optical limits aremuch looser than those imposed by the long-range collisions. They may be relaxed even further by decreasingenergy spread (adiabatic and radiation damping), and by
novel final-focus concepts [17].

5

EMITTANCE

Already at the LHC, radiation damping surpasses the intrabeam scattering growth rate. For post-LHC hadron colliders, synchrotron radiation may decide the choice of machine parameters. The energy loss per turn is U 0 =Cγ E 4 /ρ where ρ is the bending radius, E the beamenergy, and C γ ≈ 4π/3 rA /(mA c2 )3 ≈ 0.778 ×
10−17 Z 2 /A4 m/GeV3 . The product of amplitude damping time and partition number is [18]
3(mA c2 )31Cτz Jz =
(5)e2 c3 rA Z 2 B 2 E 2πρ
where z labels either plane, and C/(2πρ) denotes the reciprocal of the dipole filling factor. The damping time decreases inversely with energy and the square of the dipole
field.
A maximum tune shift of 0.009 was achieved at theTevatron, where radiation damping is negligible and thetwo beams have unequal emittances. Strong radiation
damping would allow for larger beam-beam tune shifts.
Both measurements and simulations have been fitted by[19] ξmax ∝ 0.009 + 0.021 (δ/10−4 )0.5 , where δ =T0 /(nIP τx,y ) is the damping decrement, T 0 = 1/frev therevolution period and τ x,y the transverse damping time.
In order to attain a noticeable gain, the damping decrement must be at least 10−4 , and for δ ≈ 10 −3 thebeam-beam tune shift limit increases by an order of magnitude. Using Eq. (5), ρ = E/(ZeBc), and n IP =2, we rewrite the damping decrement as δ ≈ 5.7 ×
10−13 E[TeV]2 B[T]Z 3 /A4 . In all our examples, δ is far
too small to affect the beam-beam limit; see Table 2 andthe right picture in Fig. 3.
A more important consequence of the synchrotron radiation is the shrinkage of the beam, which allows forhigher luminosity. The situation differs from electron storage rings in that, even at highest energies presently envisioned, the damping times are still of the order of hours
and not milliseconds. Therefore, the synchrotron radiation
likely leads to a continuous change of the beam emittanceduring the store. For a careless choice of parameters, thismay result in unwantedly high beam-beam tune shifts, consequent blow-up, halo generation and background.
In electron storage rings, an equilibrium emittance is established as a balance of quantum excitation and damping.
The corresponding normalized emittance is [18]


3
C
55 λ̄A γ 3SR(6)x,N ≈ √2πρ
32 3 Jx Q3βwhere λ̄A = h̄/(mA c) is the Compton wavelength of theparticle (λ̄A ≈ 2.1 × 10−16 /A m). We have employed thesmooth approximations β x,y ≈ C/(2πQβ ), D̂ ≈ β 2 /ρ andH ≈ β 3 /ρ2 ≈ ρ/Q3β (C/(2πρ))3 , which we use throughout this paper. For LHC-II parameters, the horizontal equilibrium emittance is about 3 orders of magnitude smaller
than the design emittance, for the HF VLHC the differenceis still a factor 10–100.
Thus, for any reasonable bunch current, the equilibrium
emittance will not be determined by quantum fluctuations,but instead it will result from a balance of radiation damping and intrabeam scattering. For γ
1, the horizontal
emittance growth rate, 1/τ x,IBS ≡ 1/σx dσx /dt, due tointrabeam scattering (IBS) is [20]1τx,IBS≈

2
Nb ALccrA√√
216Qβ x,N κ κ + 1γσz σδ

(7)where Lc (Lc ≈ 20) denotes a Coulomb logarithm.
Asymptotically, for γ
Q β , the longitudinal growthrate 1/τδ,IBS ≡ 1/σδ dσδ /dt approaches the same value as
1/τx,IBS , and the rms relative momentum spread becomes3/2 
x /ρ. Combining this with Eqs. (5), (7), andσδ ≈ Qβσs = cαc /Ωs σδ (Ωs is the synchrotron frequency), we cansolve for the equilibrium emittance:
IBS
x,N

=

1/6
1/3 
ρ Nb
Zfrf eVrf
cEA κ(κ + 1)Qβ γ 7/6
5/6



C
2πρ

1/6 

3rA Lc A
16

(8)

1/3

,

the vertical emittance  y continues to decrease, until it approaches a value κ x , where κ is determined by linear coupling and spurious vertical dispersion. Following Keil [18],we assume that LHC-II is operated on or near the coupling
resonances, so that the horizontal and vertical emittances
are approximately equal.
Assuming a total rf voltage of 45 MV at 400 MHz, κ = 1
for LHC-II and LF-VLHC, κ = 0.1 for the HF-VLHC, andJx,y = 1, we obtain the equilibrium emittances and bunch
lengths listed in Table 2. Since the equilibrium emittancedepends on the beam current, which, in collision, decays ona time scale comparable to the damping time, no real steadystate is established, and the luminosity lifetime is longer
than it would be for constant emittances. Figure 4 illustrates the simulated variation of emittances, beam current,
beam-beam tune shifts and luminosity during a 10-hr store
in LHC-II. The simulation includes radiation damping, intrabeam scattering, and particle consumption in collisions
at two IPs. The maximum tune shift approaches 0.005.
This is only half the peak value reached at the Tevatron.
Emittance evolution and tune shift excursions could be further optimized. A constant beam-beam tune shift may be
maintained, e.g., by varying the damping partition numbers
and the IP beta functions during the store.

(9)

where frf is the rf frequency and V rf the total rf voltage.
In a flat-beam configuration, the horizontal and longitudinal emittances may already halt their decline, while

Figure 4: Dynamic changes during a store in LHC-II for theparameters of Table 2 and damping partition numbers J x =Jy = 1, Js = 2; emittances (top left), beam current (topright), beam-beam tune shifts (bottom left), and luminosity(bottom right) vs. time.

6

COLLECTIVE EFFECTS

If the coherent synchrotron tune shift exceeds the tune
spread due to the rf curvature, Landau damping is lost forhigher-order longitudinal modes. Introducing the effective
impedance (Z L /n)eﬀ and harmonic number h rf the condition for stability is [21, 22]
σs ≥



 1/5ZL
C π 3 Nb frev eIm.
2π 6 h3rf Vrf
n eﬀ

(10)

If synchrotron radiation damping reduces the rms bunch
length, the beam could become unstable during the store.
An example for the LHC-II parameters is shown in theleft picture of Fig. 5. Assuming an effective impedance ofIm(ZL /n)eﬀ ≈ 0.1 Ω, similar to the present LHC value,
Landau damping is lost after about 3 hours. One approachof maintaining a bunch length above the threshold is longitudinal excitation using ‘pink noise’ [23]. The bunchlength evolution for such scenario is shown on the right.

9 ACKNOWLEDGEMENTS
I thank J. Gareyte, R. Assmann, P. Bagley, M. Battaglia,
O. Brüning, H. Burkhardt, W. Fischer, O. Gröbner,
H. Grote, G. Guignard, W. Herr, J.B. Jeanneret, J. Jowett,
E. Keil, J.-P. Koutchouk, S. Peggs, F. Pilat, F. Ruggiero,
G. Rumolo, F. Schmidt, R. Schmidt, E. Shaposhnikova,
R. Thomas, A. Verdier, R. v. Weelderen, and J. Wei for
discussions and informations.

10

REFERENCES[1] E. Keil, “Intersecting Storage Rings,” CERN 72-14, 1972.
[2] A. Hofmann, et al., CERN ISR-OP-TH/80-19, 1980.
[3] L. Evans, 1987 IEEE PAC, Washington, 1987; M. Harrison,
R. Schmidt, EPAC 88 Nice, 1988; V. Hatton, 1991 IEEE PAC,San Francisco, 1991.
[4] G. Dugan, 14th HEACC, Tsukuba, 1989.

Figure 5: Evolution of the rms bunch length during a store
in LHC-II compared with the threshold values for loss of
Landau damping, Eq. (10) and for longitudinal microwave
instability [22], for the same parameters as in Fig. 4 (left)
and when after 3 hours noise is added to maintain a constant value L ≥ 0.104 eVs (right).
For rings with large circumference coupled-bunch instabilities driven by the resistive wall are also a concern[18, 21]. Another problem is the electron cloud [5].

7TOTAL BEAM CURRENTCγ E 4 Nb nb c
= U0 frev nb Nb ,Cρwhich, using Eq. (5), can be rewritten as [18]



3/2
8πrA
κ E 3/2 Lβx∗C√
PSR = √.
2πρ3cEA 1 + κ2 ξ Jz τz[6] S. Klein, LBL-PUB-45566, 2000; J.B. Jeanneret, SL/AP
Beam Physics Note 41; D. Brandt, LHC Project Report 450,
2000.
[7] See web site http://vlhc.org[8] A.G. Ruggiero, Hadron Colliders at Highest Energies and Luminosities, World Scientific, Singapore, 1998.
[9] K. Yokoya, P. Chen, 1990 US-CERN School, Springer.

The first term on the right hand side of Eq. (3) is associatedwith the total beam current. This may be limited by thesustainable synchrotron radiation power,
PSR =[5] K. Ohmi, Phys. Rev. Lett. 75, 1526, 1995; F. Zimmermann,
CERN LHC PR 95, 1997; O. Gröbner, IEEE PAC97 Vancouver; K. Ohmi, F. Zimmermann, Phys. Rev. Lett. 85, 3821,
2000; G. Rumolo et al., PRST-AB 012801, 2001; CERN SL
workshops Chamonix X & XI, 2000 and 2001.

(11)[10] E. Keil, private communication (2001).
[11] A. Chao, S. Peggs, R. Talman, presentations at MiniWorkshop: “The effect of synchrotron radiation in the
VLHC,” BNL, Sept. 18–20, 2000.
[12] J. Irwin, SSC-233, 1989; Y. Papaphilippou et al., PRST-AB
2, 104001, 1999.
[13] D. Neuffer, S. Peggs, SSC-63, 1986; W. Herr, CERN-SL/9006 AP, 1990.

(12)

For a constant dipole field B, we obtain J z τz ∝ 1/E andPSR ∝ E 2 L, whereas for B ∝ E 1/2 , the damping timeshortens as Jz τz ∝ 1/E 2 , and the s.r. power increases as
PSR ∝ E 5/2 L. These scaling laws are more pessimistic
than those derived in VLHC studies [11]. If the wall-plug
power required for cryogenics is about 30 times the radiation power PSR [11], it stays below 20 MW for all projects
considered here. In Table 2 the maximum values for the
power per unit length are a factor of 4–6 above the LHC
cooling capacity, which can be taken care of, e.g., by increasing the beam-screen temperature.

8 CONCLUSIONS
Hadron colliders have performed exceedingly well in thepast. Profiting from enhanced synchrotron radiation, theyhold the promise of further substantial advancements in energy and luminosity at sustainable power levels and costs.

[14] V.D. Shiltsev et al., PRST-AB, 2:071001,1999[15] J.-P. Koutchouk, LHC PN 223; submitted to PAC2001.
[16] E.D Courant and E. Keil, Proc. Batavia 1978; E. Keil, Joint
US-CERN School, Benalmadena 1992.
[17] P. Raimondi and A. Seryi, LINAC2000, Monterey, 2000.
[18] E. Keil, “Synchotron Radiation Dominated Hadron Colliders,” Proc. PAC 97, p. 104, 1996, and Proc. 34th EloisatronWorkshop, Erice 1996.
[19] E. Keil and R. Talman, Particle Accelerators vol. 14, 1993;
S. Peggs, LHC99 beam-beam workshop, Geneva, 1999;
R. Assmann and K. Cornelis, EPAC 2000 Vienna, 2000.
[20] J. Wei, “Intrabeam Scattering Scaling for Very Large
Hadron Colliders,” unpublished draft (2001); J. Bjorken and
S. Mtingwa, Part. Acc. 13, 115 (1983).
[21] F. Ruggiero, CERN SL/95-09, 1995.
[22] J.T. Rogers, CBN 96-14, 1996.
[23] T. Toyama, et al., Proc. EPAC 2000, Vienna; similar methods were used at the Spp̄S by T. Linnecar.