Beam-beam effect with an external noise in LHC K. Ohmi ∗ KEK, 1-1 Oho, Tsukuba, 305-0801, Japan Abstract
Table 1: Basic parameters of LHC
Proton beam do not have any damping mechanism for an incoherent betatron motion. A noise, which kicks beam particles in the transverse plane, gives a coherent betatron amplitude. If the system is linear, the coherent motion remains in an amplitude range. Nonlinear force, beam-beam and beam-electron cloud interactions, causes a decoherence for the betatron motion with keeping an amplitude of each beam particle, with the result that an emittance growth arises. We focus only a fast noise, the correlation time is 1-100 turns. Slower noise is less serious, because it is regarded as an adiabatic change like closed orbit change. As sources of the noise, we consider the bunch by bunch feedback system and phase jitter of cavities which turns to transverse noise via Crab cavity.
INTRODUCTION An external noise, which kicks beam in transverse, induces an offset on the beam-beam collision. When the centroids between two colliding beams deviate an amplitude δx at the collision point, the luminosity degrades geometrically as δx2 L(δx) = L0 exp − 2 (1) 2σx where L0 and σx are the luminosity without deviation and the beam size. When δx fluctuates with a rms value, δx 2 , an averaged luminosity is given by δx2 L = L0 1 − 2σx2
symbol L E Nb θ ∗ β x,y εr ξ σz νs ν x(y) f0
nominal upgrade 26,658 m 7 TeV 1.15 × 1011 1.7 × 1011 0.14 mrad 0.22 mrad 0.55 m 0.25 m 5.07 × 10−10 m 0.0033 7 cm 3.78 cm 0.0019 63.31/59.32 109 /day
ANALYTICAL THEORY OF EMITTANCE GROWTH DUE TO AN OFFSET NOISE Weak-strong model In the weak-strong model, the beam-beam interaction is regarded as a static potential. A beam offset at a collision gives a change of action variable of a beam particle. Random fluctuation of the offset transferred to fluctuation of the action variable, with the result that an emittance growth is arisen [1]. The Hamiltonian for the weak-strong model is expressed by H(Jx , Jy ) = µx Jx + µy Jy + δP (s)U (Jx , φx , Jy , φy ) (3)
(2)
The degradation is negligible for δx/σ 1. As everybody know, the beam-beam interaction is strongly nonlinear. The collision offset caused by the noise gives an diffusion of particle motion, and induces a coherent oscillation between two beams. The coherent motion is transferred to emittance growth due to a decoherence. We treat an emittance growth and luminosity degradation induced by an external noise in the beam-beam interaction. The emittance growth is analyzed by the weak-strong and strong-strong models, in which beam particles moves in a potential given by colliding beam as a fixed charged distribution, and two beams moves with interacting each other, respectively. Parameter of LHC is shown in Table 1. ∗
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variable circumference beam energy bunch population half crossing angle beta function at IP emittance beam-beam tune shift bunch length synchrotron tune betatron tune revolution frequency
where U is the potential of the beam-beam interaction. The target beam is assumed to be a fixed Gaussian distribution with an axial symmetry, N rp ∞ du x2 + y 2 U (x, y) = exp − (4) γ 2σ 2 + u 2σ 2 + u 0 where rp and γ are the classical radius and the relativistic factor of the beam particle, and σ is rms size of the target beam. We consider a small offset of the target beam. The potential is expanded for δx as U (x + δx) = U (x) + U (x)δx.
(5)
The change of action variable induced by the offset is given by ∂U δx. (6) δJ = ∂φx
The fluctuation of J is given as follows 2 δJ 2 1 ∂U = δx2 T0 2 ∂φx
(7)
This formula includes J in the RHS:i.e., it gives a fluctuation of J as a function of J. The emittance growth is evaluated by the diffusion rate at J = ε, δε/T 0 = δJ 2 (ε0 )/2ε. When the noise has a correlation for time, δx(t1 )δx(t2 ) = Dexp(−|t1 − t2 |/tcor )
(8)
where ∆Ψ(s∗ , sc ) and δψRF are the betatron phase difference between the collision point and the crab cavity and the deviation of the RF phase of the crab cavity, respectively. In the both cases, the jitter of transverse offset is given by δx ≈ c tan φδψ/ωRF . Transverse bunch by bunch feedback system damps a transverse dipole motion of the beam. Read error of the position monitor and kicker noise give a transverse kick on the beam. Betatron amplitude is transferred as follows, Xi+1
2
The formula on δJ (ε0 ) for this noise is seen in Ref.[1].
Strong-strong model In the strong-strong model, an offset of the collision triggers a coherent beam-beam mode. Smear of the coherent motion due to the nonlinear beam-beam force gives emittance growth [2]. A coherent oscillation induced by an offset (δx) gives a change of invariant amplitude (J) as follows, δx2 δJ =K 2 J σx
(9)
where K is a form factor for an excitation of the beambeam mode, which is estimated to be 0.089 by Alexahin. When the feedback gain is zero, the coherent motion smear to the beam size in 2π|ξ| turns. When bunch by bunch feedback system damps the coherent motion faster than the smear, the contribution to the beam size is reduced with the factor (G/2π|ξ|), with the result that the emittance growth is expressed by 2
δx 1 δε ≈K 2 T0 σx
1 1+
G 2π|ξ|
2 .
(10)
NOISE SOURCES The crab cavity can be source of diffusion: i.e., since the crab cavity is operated by a transverse mode, the deviation and jitter of RF phase give a dipole kick to the beam, with the result that transverse offset at the collision point is generated. Both of phases of main RF and crab cavity can be source of the transverse offset. Jitters of RF phase of main cavity causes a deviation of timing of beam arrival at the crab cavity. The transverse offset, which arise from the jitter of main RF system, is expressed by c tan φ δψRF . (11) δx = ωRF where δψRF is the phase error of the main RF system. The crab cavity gives a transverse kick due to its jitters of RF phase, with the result that the offset given by the kick is expressed as follows, δx =
c tan φ cos(πνx − ∆Ψ(s∗ , sc ) δψcrab , ωRF 2 sin πνx
(12)
= Xi − G(Xi + δXmon ) + δXkick = (1 − G)Xi − G + δXmon + δXkick (13)
where G, δXmon and δXmon are the feedback damping rate, kicker noise and monitor noise, respectively. The fluctuation of the betatron amplitude is given by X 2 =
1 2 2 2 G Xmon + Xkick . 2G
(14)
These fluctuations induce a diffusion in betatron amplitude and an excitation of a coherent beam-beam mode. Two type of implementation for noise are installed in two beam-beam simulation codes. One code is based on the weak-strong model, and the other is based on strong-strong model. First type of noise is expressed by a transformation as follows, T1 (−δ) exp(−Ucol )T1 (δ)M0
where T1 (δ)
x y
=
x + δx y + δy
(15)
.
(16)
δ’s are random variables, which are common values for every macro-particles. RF feedback system of accelerating and crab cavities is closed and is little influenced by the beam-beam interaction. We use this transformation for studying a crab cavity type of noise. Bunch by bunch feedback system kicks the beam to reduce its coherent betatron amplitude. Beam-beam interaction can influence the feedback system. The transformation of the bunch by bunch feedback system is characterized by two variables, damping rate and fluctuation, as follows, exp(−Ucol )T2 (δ, τ )M0 where T2 (δ, G)
x y
=
x(1 − G) + δx y(1 − G) + δy
(17) .
(18)
SIMULATION OF BEAM-BEAM INTERACTIONS WITH THE NOISES We execute a weak-strong and strong-strong simulations to study the noise effects. In the weak-strong simulation, strong beam is represented by a static potential, while weak beam is represented by macro-particles. The macroparticles in the weak beam is tracked in the static potential of the strong beam.
1e-06
2
1e-07 ∆σ2/σ0
In the strong-strong simulation, both beams are represented by macro-particles. The strong-strong simulation contains a numerical noise due to the statistics of√macroparticles. Indeed the dipole moment fluctuates σ/ N turn by turn. The simulation gives an artificial emittance growth due to the numerical noise; the external noise less than the numerical noise is not visible. The number of particles should be increased according to the noise level and emittance growth rate to be studied.
1e-08 1e-09 1e-10 1e-11 1e-04
Crab cavity type of noise
0.01
0.1
0.01
0.1
δx 1e-06
1e-07 -∆L/L0
We first discuss the beam-beam effect with the noise given by Eq.(16). Figure 1 shows the evolution of emittance and luminosity for various noise amplitude given by the weak-strong simulation. The emittance growth rate and luminosity decrement, which are estimated in Figure 1, are summarize in Figure 2. In the weak-strong simulation, macro-particles moves in a static potential. Nonlinear beam-beam force (potential) can cause emittance growth even if no noise. Since the beam-beam tune shift is rather small (ξ = 0.0033), the motion is near solvable, therefore emittance growth is very weak, < 10 −10 . For increasing noise amplitude, the emittance growth and luminosity decrement become visible in the simulation. Since the revolution frequency is 10 9 par day, the luminosity decrement 10−9 corresponds to 1 day life time. The noise level, δx =0.1-0.2%, is limit for 1 day luminosity life time from Figure 2.
0.001
1e-08
1e-09
1e-10 1e-04
0.001 δx
Figure 2: Emittance growth rate and luminosity decrement as a function of noise amplitude given by the weak-strong simulation.
16.69
16.685
δx,y=0.006 σ
0.012
σx (µm)
16.68
16.675
0.003
16.67 0.0012 16.665
0.0006 0.
16.66 0
100000 200000 300000 400000 500000 600000 700000 800000 900000 turn
1e+06
0. 0.0006
1 0.0012
L/L0
0.998
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δx,y=0.006 σ
0.012
0.992
tion of a coherent motion is estimated by the strong-strong simulation. The simulations were done with 1,000,000 macro-particles. The simulation contains the intrinsic error due to the statistics of the number of macro-particles. Indeed the dipole moment fluctuates 0.1% due to this statistics. The fluctuation level 0.1% is critical for the luminosity decrement 10 −9 (1 day life tome) as is shown in the weak-strong simulation. Strong-strong simulation with less macro-particle does not have an ability of the prediction for the luminosity decrement 10 −9 . Figure 3 shows the luminosity decrement for the intrinsic noise. The luminosity decrement is 1.24 × 10 −9 . As is expected, the statistics is reliable for discussion of 1 day luminosity life. Figures 4 and 5 show evolutions of dipole amplitude, emittance and luminosity for the correlation time of t cor = 1 and 100 turn, respectively. These results are summarized in Figure 6.
0.99 0
100000
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600000
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900000
1e+06
turn
Figure 1: Evolutions of emittance and luminosity for various noise amplitude given by the weak-strong simulation. In the weak-strong simulation, the noise does not induce a coherent beam-beam mode. The effect due to excita-
Noise of bunch by bunch feedback system A simulation has been performed for the second type of noise. An excitation of the beam-beam mode is the source of the emittance growth, therefore strong-strong simulation is essential for this study. Figures 7 and 8 show evolutions of dipole moment and
1
1.004
δx, y=0.03 σ 0.5
x (µm)
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0.508
turn ε (nm)
0.506
Figure 3: Luminosity decrement for no external noise by the strong-strong simulation.
0.504 0.0012, 0.003, 0.006, 0.012 0.502
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0 δx, y=0.03 σ
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Figure 5: Evolutions of dipole amplitude, emittance and luminosity for the correlation time t cor = 100 turn given by the strong-strong simulation.
0.53 ε (nm)
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0.006 0.96 δx, y=0.03 0.012 σ 0.94
0.92
0.9 0
50000
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250000
Figure 4: Evolutions of dipole amplitude, emittance and luminosity for the correlation time t cor = 1 turn given by the strong-strong simulation.
emittance for various feedback gain with the kick noise δx = 0.05µm (0.3% of σ) and δx = 0.02µm (0.12% of σ), respectively. The same simulations were performed for δx = 0.2, 0.1 and 0.01µm (1.2, 0.6 and 0.06% of σ). The results are summarized in Figure 9.
1e-06 τcor=1
1e-08
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1e-10 0.001
G=0.02 G=0.05 G=0.1
0.5 x (µm)
δσ2/σ2
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0.01 δx/Symbol s
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1e-07
1e-08
0.5 τcor=100
1e-09 0.001
δx, y=0.003 σ 0.48
0.01 δx/Symbol s
0.1
Figure 6: Emittance growth rate and luminosity decrement as a function of fluctuation amplitude given by the strongstrong simulation.
0
50000
100000 turn
150000
200000
Figure 7: Evolutions of dipole moment (a) and emittance (b) for various feedback gain with the kick noise δx = 0.05µm (0.3% of σ)
10
∆x =0.012 σ 0.006 0.003 0.0012 0.0006
δx/σx
1 0.1 0.01 0.001 1e-04 0
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0
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0.04 0.06 0.08 Feedback Gain
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Figure 8: Evolutions of Dipole moment (a) and emittance (b) for various feedback gain with the kick noise δx = 0.02µm. (0.12% of σ)
δε/Symbol e0
1e-05
1e-06
1e-07
1e-08
1e-09
Figure 9: Residual dipole amplitude, emittance growth rate and luminosity decrement as a function of the feedback gain for various noise amplitude. Emittance growth given by analytical estimate.
CONCLUSIONS Emittance growth and luminosity decrement due to external noise in beam-beam collision system have been studied. To achieve 1 day luminosity life time, the noise (δx/σx ) should be 0.1% for turn by turn noise (t cor = 1turn). If the correlation time of the noise is 100 turn, the tolerance is√1%. The tolerance roughly scale the correlation time as tcor . For transverse bunch by bunch feedback, the noise level, δxkick /σ = 0.0006 and G=0.1, is about the limit of the luminosity decrement 10 −9 . The corresponding monitor resolution is δxmon = 0.0006/G = 0.006 = 0.6%, the resulting fluctuation in Eq.14 is 0.1%. For feedback system with lower gain, the monitor resolution is looser.
ACKNOWLEDGMENMT The author thanks to W. Hoffle and F. Zimmermann for fruitful discussions.
REFERENCES [1] T. Sen and J. A. Ellison, Phys. Rev. Lett., 77, 1051 (1996). [2] Y.I. Alexahin, Nucl. Instru. and Methods in Phys. Res.,A 391, 73 (1996).
