COHERENT BEAM-BEAM EFFECTS IN THE LHC
Y. Alexahin, JINR, Dubna
H. Grote, W. Herr and M.P. Zorzano, CERN, Geneva, Switzerland
Abstract
In the Large Hadron Collider (LHC) two proton beams ofsimilar intensities collide in several interaction points. It
is well known that the head-on collision of two beams ofequal strength can excite coherent modes whose frequencies are separated from the incoherent spectrum of oscillations of individual particles. This can lead to the lossof Landau damping and possibly to unstable motion. The
beam-beam effect in the LHC is further complicated by alarge number of bunches (2808 per beam), a finite crossing
angle and gaps in the bunch train. The coherent beam-beameffects under various conditions and operational scenarios
are studied analytically and with multiparticle simulations.
We give an overview of the studies and present proposals
to overcome these difficulties together with possible side
effects.

1

INTRODUCTION

Two colliding beams exert a force on each other which
is defocusing for beams of equal charge as in the case of
LHC. Solutions of the linearized Vlasov equation show thatfor round beams and in the case of one bunch per beam withequal parameters (intensity, beam size, betatron tune) two
coherent dipole modes of oscillations appear: the σ mode,with a frequency equal to the unperturbed betatron tune,
and the π-mode with a tune shift of Y = 1.21, where Y is
the Yokoya factor [1], times the beam-beam parameter ξ.
In the LHC the situation is complicated by some specificfeatures which affect coherent beam-beam effects:• The LHC is operated in the strong-strong regime, i.e.
equally strong beams.
• It has four interaction regions and two independentrings.
• It has parasitic (long range) interactions and many
(2808) bunches.
• The beams cross at horizontal and vertical angles atthe collision points.
It has been predicted [2, 3] that the coherent π-mode may
not be Landau damped for certain strong-strong conditions.
Therefore an accurate knowledge of the Yokoya factor andthe conditions for the excitation or suppression of coherent modes is highly desirable. In this report we present a
selection of the main coherent effects.

mutual interaction and are used at the interaction points tocalculate the force on the individual particles. To evaluate the coherent effects we employ two basic types of approaches: we study the solutions of the Vlasov equation
using analytic and numerical models such as perturbation
theory or its numerical integration [3, 4]. We further study
the effects using multiparticle tracking. A number of studies have been done for LHC using the so-called “soft Gaussian model” [5]. This model assumes the force experienced
by a particle when traversing the counter- rotating beam as
originating from a Gaussian beam distribution with variable barycenters and rms beam sizes. This allows the useof an analytical expression for the forces. This Gaussianmodel cannot take into account the non-Gaussian deformations of the distribution and as a result underestimates
the force and yields a Yokoya factor that is slightly smaller(Y = 1.1 in our case). We therefore use a different approch
for the field evaluation to get quantiatively correct results.
A Hybrid Fast Multipole Method is applied [10] and theresults agree extremely well with the analytical prediction.

3 COHERENT BEAM-BEAM MODESAs a first results we show in Fig.1 the Fourier spectrumof two bunches colliding head on in one interaction point(IP). We show the spectral density of the barycentric motion of 10 4 particles as a function of the distance to theunperturbed tune Q, normalized by the beam-beam parameter ξ: ω = ν − Q/ξ. The two modes (0-mode, right peak;
π-mode, left peak) are clearly visible with the incoherent
spectrum between, extending between 0 and −ξ. The 0mode is at the unperturbed tune (ω = 0) and the π-mode isoutside the incoherent spectrum ( ω = −1.21 ± 0.002) andtherefore cannot be Landau damped. This result accurately
confirms the analytic calculation.

Fourier spectrum of head-on coherent modes
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2 SIMULATION AND TOOLS
In a self-consistent model of the coherent interaction, thedistributions of both beams evolve as a consequence of the

Figure 1: Head on beam-beam coherent modes.

4

EFFECT OF BROKEN SYMMETRIESAn isolated, discrete π-mode is a potentially dangerous situation and various proposals were made to suppress thismode [7, 8]. All proposals rely on the breaking of somesymmetry between the beams. This makes it more difficult
to organize and maintain a collective motion and the modecannot develope. Such proposals in pure or modified formare:• Different fractional tunes of the two beams.

5 OPTICS MODIFICATIONSBreaking the symmetry between the bunches is an efficientmethod to avoid coherent motion of bunches. Changes tothe beam optics such as different tunes of the two beams,
phase advance differences between the interaction pointetc., help to suppress the coherence. Some of these possible measures we investigate in some more detail.

5.1• Different integer tunes of the two beams for multiple
interaction points.

Tune differences
Fourier spectrum of coherent modes, Q = 0.312
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0.7• Phase advance adjustments of different type for multiple interaction points.

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Although all proposals can succeed to suppress the coherent modes, they may have unwanted side effects which maylead to operational difficulties or loss of luminosity.

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Intensity differences

Fourier spectrum of coherent modes, Q = 0.314
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The coherent modes only exist in the real strong-strong
regime where the two beams have equal intensities andbeam sizes. For beams of unequal intensities the π-mode
moves closer and closer to the incoherent continuum andmerges into it [2] when the ratio becomes less than 0.6.
As a result the mode can be Landau damped and disap-

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Spectrum of coherent modes, Intensity ratio 0.65
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Figure 3: Coherent modes with different fractional tunes.

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In Fig.3 we show the effect of different fractional tunes.
While the horizontal tune of beam 1 was kept at 0.31, we
have changed the tune of beam 2 to 0.312 (upper figure) and
0.314 (lower figure). The strong coherent modes disappear
for a tune difference above 0.75ξ, proving the principle inthis simply case.

Spectrum of coherent modes, Intensity ratio 0.55
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Separate working points

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The LHC has two separate beams colliding in the interaction points, and therefore it is possible to run the two beams
with different working points. Three working points (WP)
have been proposed, based on studies of the incoherent single particle dynamics::

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WP 1 (Qx,1 , Qy,1 ) = (0.232, 0.242)
Figure 2: Head on beam-beam coherent modes for two different beam intensity ratios.
pears. This is illustrated in Fig.2 where we plot the coherent modes again, but for intensity ratios of 0.65 (top) and
0.55 (bottom). While for a ratio of 0.65 the mode is at theedge of the incoherent spectrum, it has merged for 0.55.
The analytical prediction is completely confirmed.

WP 2 (Qx,2 , Qy,2 ) = (0.310, 0.320)
WP 3 (Qx,3 , Qy,3 ) = (0.385, 0.395)
However the study [8] has shown that this strategy excites
a new type of resonances where the two beams couple toresonances of the type: nQ x,i + mQx,j = r, e.g.:
1Qx,1 + 2Qx,3 = 1.002 or 2Qx,2 + 1Qx,3 = 1.005. It

can be shown that the coherent mode is moved back into theincoherent spectrum and is Landau damped, but at the expense of a small emittance growth. Details can be found in[8]. This is demonstrated in Fig.4 where we plot the dipole
0.1<x(n)>1-<x(n)>2<x(n)>1+<x(n)>2

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5.3

Multiple interaction regions

In case of multiple interaction points additional measures
can be employed to break the symmetry. Such possibilities
are phase advance differences and integer tune changes.
Details of the many possible options are given in [11]. We

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oscillation amplitude

resonance and the growth stops.

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Figure 4: Dipole oscillations and emittance growth forworking points 1 and 3.
oscillations and the emittances of the beams as a function of
the number of turns. The working points 1 and 3 are used inthis example. The dipole oscillation does not grow in time,
but the emittance is steadily increased. In Fig.5 we show
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∆x=0.7∆x=0.82.004

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Figure 6: Coherent modes for 4 collisions and with different integer tunes. Horizontal axis as before but normalized
to 4ξ.

εx(1)+εx(2)

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Figure 5: Emittance growth for working points 2 and 3.
The emittance growth until the beams size is larger and thebeam-beam effect lower and outside the resonance stopband. ∆x is the distance to the resonance stopband as defined in [8].
another example with the combination of working points
2 and 3. We show the emittance as a function of the turnnumber for two distances to the resonance stopband. In thefirst case where the distance if large enough (∆ x = 0.8),
no oscillation or emittance growths is seen. In the secondcase (∆x = 0.7) no oscillation is observed but the emittance of one beam grows until the beam-beam parameters
becomes lower and the beam is outside the stopband of thegive one instructive example in Fig.6. In the upper figure we have used the standard tunes (Q x = 64.31, Qy =59.32) and collide 2 bunches per beam in 4 interaction
points. The modes are again very prominent. The horizontal axis is ω = ν − Q/4ξ and therefore the π-mode appears
at ω = -1.21. The symmetry hides some of the additionalmodes. In the lower figure the integer tunes are swapped
between the two planes (Q x = 59.31, Qy = 64.32) and themodes have disappeared. For the used collision symmetry
it is enough to have unequal parity (odd and even) for thetwo beams to obtain practically complete suppression.

5.4

Small tune split

Finally, we present the effect of a small tune split of 0.02
between the two beams. This time both beams have thesame working point to avoid the above problems, but are
separated by ± 0.01. No more growth is observed, howevera small beam size asymmetry has developed in both planes,
usually called a flip-flop situation (Fig.7).

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Fourier spectrum of coherent modes, from HFMM
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transverse size

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Figure 7: Coherent modes with small tune split, showing a
flip-flop effect.

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6 COHERENT MODES FROM LONG
RANGE INTERACTIONS

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A feature that is important for colliders with many bunches
is the existence of parasitic (long range) interactions. In the
LHC we have about 120 parasitic interactions around the 4interaction points. Apart from many other effects, they can
excite coherent modes, just like head on collisions. However, while in the head on case and for small separation, thebeam-beam force is basically linear, the long range forces
are rather non-linear. Their effects can be studied using
Vlasov perturbation theory or multi particle tracking. Details about the two methods are found in [3, 10]. As an
example we show in Figs.8 and 9 the coherent modes frommulti-particle tracking for single bunches with separationof 10 σ and 6 σ in the horizontal plane. The basic modes(0-mode and π-mode) are again visible as in the head-on
case and can be identified as such by analysing the sum or
difference of the centroid motion of the beams. A basic
difference is the sign of the π-mode tune shift. It has the
opposite sign as for the head on case in the plane of separation and the same sign in the other plane. This is due to thedifferent focussing properties of separating beams, a very
well-known result. One can now speculate, and it was also
believed in the past, that the tune shifts of opposite sign ofthe head on and long range modes can compensate, at least
partially, and return the π-mode into the incoherent spectrum, which would allow Landau damping. In the Fig.10
we now plot the coherent modes for combined head on andlong range interactions. The horizontal modes are plottedfor horizontal (left column) and vertical crossing (central
column). The right column shows the spectra for alternating crossings, i.e. one crossing with horizontal and anotherwith vertical crossing. The contribution from long range
interaction increases from top (no long range interactions)to bottom. We observe that while the head on modes remain visible, although their amplitude becomes smaller, asecond coherent mode appears on the other side of the incoherent spectrum. The whole picture is shifted due to thelong range tune shift. Instead of a compensation we have
now two apparently independent modes on both sides of
the incoherent continuum, a somewhat unexpected result.
An explanation of the above is given in Fig.11 where we

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Figure 8: Coherent beam-beam modes from long range interactions with 10 σ separation in the horizontal plane.
Fourier spectrum of coherent modes, from HFMM
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Figure 9: Coherent beam-beam modes from long range interactions with 6 σ separation in the horizontal plane.
show the eigenfunctions of the two discrete modes appearing at the bottom left of Fig.10. The corresponding eigenfunctions as a function of the amplitude are very different.
While in the first mode (head on case) mainly core particles
participate, the second (long range) mode involves mainly
tail particles with large amplitude J x . Therefore different
parts of the bunch contribute to the two modes and this is
why the two modes do not compensate. In the case of vertical separation (centre column) only the head on mode isvisible and rapidly increasing its amplitude with increasingdamping [3]. The finite crossing angle of ±150 µrad also
reduces the tuneshift from the head-on collision and therefore the Yokoya factor.

8

REFERENCES[1] K. Yokoya, H. Koiso, Tune shift of coherent beam-beam oscillations. Particle Accelerators, 1990. Vol. 27, pp. 181-186.
[2] Y.I. Alexahin, On the Landau damping and decoherence oftransverse dipole oscillations in colliding beams. Particle Accelerators, Volume 59, p. 43 (April 1999).
[3] Y.I. Alexahin; A study of the Coherent Beam-beam Effectin the Framework of the Vlasov Perturbation Theory. LHC
Project Report 467, 2001.
[4] M. P. Zorzano; Numerical integration of the Vlasoc equationof two colliding beams. LHC Project Report 440, (2000).

Figure 10: Head on and long range coherent modes in thehorizontal plane with increasing long range contributions(top to bottom) from crossings in the horizontal plane (left),vertical plane (centre) and alternating crossing (right).

[5] M. P. Zorzano, F. Zimmermann; Coherent beam-beam oscillations at the LHC. LHC Project Report 314, 1999.
[6] K. Yokoya, Limitation of the Gaussian approximation in
beam-beam simulations. Phys.Rev.STAB, Vol. 3, 124401(2000).
[7] A. Hofmann; Beam-beam modes for two beams with unequaltunes. Proceedings of the LHC-99 Beam-beam workshop at
CERN.
[8] Y.I. Alexahin and M. P. Zorzano; Excitation of Coherent
Beam-Beam resonances for beams with unequal tunes in the
LHC. LHC Project Note 226, 2000.
[9] S. Krishnagopal, Energy transparency and symmetries inthe beam-beam interaction. Phys.Rev.STAB, Vol. 3, 024401(2000).

Figure 11: Coherent beam-beam eigenmodes for head on(ψ1 ) and long range (ψ 2 ) modes.
long range contributions. Therefore if the separation is in
the same plane in all interaction regions, the stability in theother plane may be strongly deteriorated. The problem issolved with alternating crossings as proposed for the LHC(right column). Here we observe a small decrease of thehorizontal mode (and the same for the vertical, since now
both planes are equivalent.

7 EFFECT OF CROSSING ANGLE AND
SYNCHROTRON MOTION
So far we have ignored the effect of the crossing angle
and the synchrotron motion. The latter can couple to thetransverse motion through the crossing angle, leading tosynchro-betatron resonances. This coupling affects the coherence in two ways: first the additional degree of freedom
weakens the coherence and reduces the Yokoya factor. Secondly, since the synchrotron tune (Q s = 0.00212) is comparable to the beam-beam tune shift, the discrete modes
can overlap with synchritron sidebands, leading to Landau[10] W. Herr, M. P. Zorzano and F. Jones; A Hybrid Fast Multipole Method applied to beam-beam collisions in the strong
strong regime. LHC Project Report 466, 2001.
[11] W. Herr and M. P. Zorzano; Coherent dipole modes for multiple interaction regions. LHC Project Report 462, 2001.