justified

# Impact of bunch intensity asymmetry in colliders featuring strong
beamstrahlung

Khoi Le Nguyen Nguyen<EMAIL_ADDRESS>European Organisation for Nuclear
Research (CERN), CH 1211 Geneva 23, Switzerland  École Polytechnique Fédérale
de Lausanne (EPFL), Route Cantonale, 1015 Lausanne, Switzerland  Cavendish
Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
Xavier Buffat<EMAIL_ADDRESS>European Organisation for Nuclear
Research (CERN), CH 1211 Geneva 23, Switzerland  Peter Kicsiny
<EMAIL_ADDRESS>European Organisation for Nuclear Research (CERN), CH
1211 Geneva 23, Switzerland  École Polytechnique Fédérale de Lausanne (EPFL),
Route Cantonale, 1015 Lausanne, Switzerland  Tatiana Pieloni
<EMAIL_ADDRESS>École Polytechnique Fédérale de Lausanne (EPFL), Route
Cantonale, 1015 Lausanne, Switzerland

###### Abstract

An analytical investigation of beamstrahlung-induced blow-up in Gaussian beams
with arbitrary dimensions is presented, using various approximations for the
strength of the hourglass effect and crab waist scheme. The results, applied
to the FCC-ee resonances, are compared with simulations and previous
calculations, and relative luminosity values are also calculated. The
stability of resultant conformations are analysed to rule out the existence of
a flip-flop phenomenon in the longitudinal plane analogous to the well-known
transverse counterpart. Implications for the top-up injection procedures are
discussed, and a phenomenological model is proposed to study the transverse-
longitudinal coupling in the blowup dynamics.

††preprint: APS/123-QED

##  I Introduction

In high-energy electron-positron circular colliders, particles emit
synchrotron radiation when their trajectories are bent by external magnetic
fields [1] and by the electromagnetic field generated by the opposing beam at
the Interaction Point (IP) [2]. In the next generation of machines such as the
proposed Future Circular Collider (FCC), the strength of the latter, so-called
beamstrahlung, leads to a variety of new phenomena deteriorating the beams
quality and thus the luminosity [3]. One consequence of beamstrahlung is the
increase of the bunch length many times above the equilibrium given by the
synchrotron radiation in the magnetic lattice only, thereby increasing the
energy spread as well as lowering the luminosity. This effect has been studied
analytically for the case of identical machine and beam parameters for the two
rings [4]. While configurations featuring asymmetric machine or beam
parameters can be addressed with numerical simulations featuring beamstrahlung
[3], analytical methods to rapidly approximate equilibrium lengths in the
general case are not available.

In the presence of an asymmetry in the intensity of two colliding bunches, an
instability mechanism analogous to the flip-flop effect [5], termed 3D flip-
flop, was observed in simulations [3]: As the electromagnetic fields generated
by the lowest intensity bunch are reduced, the beamstrahlung of the highest
intensity bunch is reduced. Consequently, the bunch length of the high
intensity bunch is reduced, thus increasing the strength of the beam-beam
force exerted on the low intensity bunch. This results in an increase of the
beamstrahlung of the low intensity bunch, thus increasing its length and
further weakening the beam-beam force on the strong bunch. Beyond a certain
threshold the beam-beam force experienced by the low intensity bunch becomes
too strong, such that non-linear resonances drive a transverse emittance
growth that again weakens the low intensity bunch, thus enhancing the
phenomenon. We obtain a runaway situation in which the low intensity bunch
blows up in all directions while the high intensity bunch shrinks to the
equilibrium bunch properties defined by the magnetic lattice, as the
beamstrahlung due to the blown up low intensity bunch becomes negligible.

This paper presents an analytical treatment for the effect of beamstrahlung on
the equilibrium bunch lengths of two colliding Gaussian electron-positron
bunches with arbitrary dimensions and intensities. The dispersive effects at
the IP are ignored. Beam losses due to the limited energy acceptance of the
ring are also ignored. Corrections due to the hourglass effect and the crab-
waist collision scheme will be considered up to first order. At first only
variations of the longitudinal bunch dimension are considered to obtain
luminosity values and equilibrium bunch behaviours at different mean
intensities and intensity asymmetry. The results obtained are compared to past
studies for symmetric configurations. The stability of the solutions is then
discussed. The effect of beamstrahlung on the vertical size of the low
intensity bunch is then accounted for based on a phenomenological approach,
leading to a model of the transverse-longitudinal flip-flop effect.

The presented model has a predictive power limited to small, but realistic
parameter asymmetries. Therefore the main focus of the paper is an attempt to
use analytical derivations to understand the flip-flop effect, which has so
far been demonstrated only in complicated, numerically heavy tracking
simulations. It can be useful to build confidence in simulation results in
scenarios, such as the FCC, where experimental data cannot be obtained before
the machine is built.

##  II Equilibrium bunch lengths

In the following, we describe the motion of a particle in the so-called weak
bunch through the force of the so-called strong bunch. At this stage of the
derivation the approach is thus ‘weak-strong’, _i.e._ the perturbation of the
beam-beam force during the interaction is neglected. At a later stage, the
weak-strong equations derived will be combined into a system of equations
where the low and high intensity bunch take the role of the weak and strong
beam alternatively. The system of equations thus obtained will be effectively
‘strong-strong’, i.e the perturbations of the two beams by the beam-beam force
are taken into account self-consistently.

Consider two ultra-relativistic Gaussian bunches of electrons and positrons
moving towards each other with relativistic Lorentz factors $\gamma_{w/s}$.
(Subscripts $w$ and $s$ throughout this paper denote quantities pertinent to
the weak and strong bunch respectively.) In the lab frame, the position of a
charge in either bunch can be parameterised by three fixed Cartesian
coordinates $x$, $y$ and $s$ with the origin at the centre of the weak bunch,
and a comoving coordinate $z=s-ct$, such that $s$ points along the axis of the
weak bunch towards the strong bunch, $x$ and $y$ are the transversal
directions in and perpendicular to the plane of collision respectively, and
$z$ measures the deviation of a particle of the weak bunch from its centre. A
charged particle in one beam moving through the electromagnetic field of the
other beam will accelerate and emit radiation, known as beamstrahlung; this
exists in addition to the synchrotron radiation emitted due to the bending of
the trajectories in the magnetic lattice. There exists a Lorentz
transformation [6] to an inertial frame of reference in which the beams
collide head-on; the energies of the beams as well as of the emitted photons
can be shown to only be slightly different from that measured in the lab frame
for a small crossing angle [4], and this small deviation will be ignored. In
this transformed frame, the electromagnetic field is now in the collision
plane and perpendicular to the direction of motion.

Following derivations from Refs. [4, 7], the time-dependent curvature, to the
power of some positive integer $n$ and averaged over the entire weak bunch, of
the path of a charge influenced by the electromagnetic field of a Gaussian
strong bunch in the transformed frame is given by the integral,

$\displaystyle\mathcal{I}_{n,w}\equiv\int_{-\infty}^{+\infty}ds\Bigg{\langle}\dfrac{1}{\rho_{w}^{n}}(s,t)\Bigg{\rangle}$
$\displaystyle=\dfrac{1}{(2\pi)^{3/2}}\iiiint\limits_{x,y,z,s\in\mathbb{R}^{4}}\dfrac{dxdydzds}{\sigma_{x,w}^{*}\sigma_{y,w}^{*}G(x,s)_{w}\sigma_{z,w}}\rho(x,y,z,s;t)_{w}^{-n}$
$\displaystyle\hskip
10.0pt\times\exp\left[-\dfrac{\left(x+{z\theta_{c}}/{2}\right)^{2}}{2\sigma_{x,w}^{*2}}-\dfrac{y^{2}}{2\sigma_{y,w}^{*2}G(x,s)_{w}^{2}}-\dfrac{z^{2}}{2\sigma_{z,w}^{2}}\right]$
$\displaystyle:=\frac{1}{(2\pi)^{3/2}}\iiiint\limits_{x,y,z,s\in\mathbb{R}^{4}}dxdydzds\frac{\mathcal{G}}{\rho_{w}^{n}},$
(1)

where the local curvature is

$\displaystyle\dfrac{1}{\rho(x,y,z,s;t)_{w}}$
$\displaystyle=\left|\mathcal{F}\left[x-\left(s-\dfrac{z}{2}\right)\theta_{c},y,s\right]_{s}\right|$
$\displaystyle\times\sqrt{\dfrac{2}{\pi}}\dfrac{2r_{e}N_{s}}{\gamma_{w}\sigma_{z,s}}\exp\left[-\dfrac{(2s-z)^{2}}{2\sigma_{z,s}^{2}}\right],$
(2)

and $\mathcal{G}$ denotes the Gaussian density of the weak bunch. The bunch
parameters in Eq. II are taken in the Lorentz transformed, head-on reference
frame. In particular, $x$ and $\sigma_{x,w}^{*}$ are the horizontal
coordinates and r.m.s. size of a bunch tilted by the half crossing angle
$\theta_{c}/2$. The classical electron radius is $r_{e}$, $N_{s}$ is the
number of particles in the strong bunch. The factor
$G(s,x)_{w/s}=\sqrt{1+[({s\mp x/\theta_{c}})/{\beta_{y}^{*}}]^{2}}$ takes into
account the dominant vertical hourglass effect (via the $s/\beta_{y}^{*}$
term) and crab waist collision scheme (via the $\pm
x/(\theta_{c}\beta_{y}^{*})$ term) experienced by the two beams, with optical
$\beta$ function in the y-direction $\beta_{y}^{*}$ at the IP, and the + ($-$)
sign for the strong (weak) beam respectively. The exact form of the function
$\mathcal{F}(x,y,z)_{s}$, can be found in Refs. [1, 4], and is not reproduced
here, as the focus of this paper is on machines featuring a much smaller
vertical beam size at the IP with respect to the horizontal size. This feature
allows for approximations that enable a fully analytical approach and will be
discussed in the next section. In Eq. (II), the integral is taken over the
entire interacting length, and the Gaussian kernel accounts for the particle
distribution of the weak bunch. Starred quantities denote values at the IP.

This integral plays an important role in controlling a variety of quantities
associated with beamstrahlung [1, 4, 8]. Here, for convenience and
completeness, we reiterate these quantities. With $n=1$, Eq. (II) leads to the
average number of photons emitted per collision

$\mathcal{N}_{\text{ph},w}\approx\dfrac{5}{2\sqrt{3}}\alpha\gamma_{w}\mathcal{I}_{1,w},$
(3)

where $\alpha$ is the fine structure constant. With $n=2$, Eq. (II) gives the
average relative energy loss per collision due to beamstrahlung,

$\delta_{\text{BS},w}\approx\dfrac{2}{3}r_{e}\gamma_{w}^{3}\mathcal{I}_{2,w}.$
(4)

With $n=3$, Eq. (II) relates to the energy spread of the photons emitted and
the resulting quantum excitation,

$\\{\mathcal{N}_{\text{ph},w}\langle
u^{2}\rangle\\}_{z,w,\text{BS}}\approx\dfrac{55}{24\sqrt{3}}\dfrac{r_{e}^{2}\gamma_{w}^{5}}{\alpha}\mathcal{I}_{3,w}.$
(5)

In writing down these expressions, we have implicitly assumed that both beams
have the same $\beta_{y}^{*}$ parameter.

###  II.1 Evolution of the bunch lengths towards equilibrium

The rate of change of the bunch length with time is affected by the emission
of synchrotron radiation in the magnetic lattice and beamstrahlung radiation
at the IP. Their contributions to the damping time and the quantum excitation
are labelled $SR$ and $BS$ respectively, we have:

$\displaystyle\odv{\sigma_{z,w}^{2}}{t}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\\{\dot{\mathcal{N}}_{\text{ph}}\langle
u^{2}\rangle\\}_{z,w,\text{SR}}+\dfrac{1}{2}\\{\dot{\mathcal{N}}_{\text{ph}}\langle
u^{2}\rangle\\}_{z,w,\text{BS}}$ (6) $\displaystyle-$
$\displaystyle\left(\dfrac{2}{\tau_{z,w,\text{SR}}}+\dfrac{2}{\tau_{z,w,\text{BS}}}\right)\sigma_{z,w}^{2}.$

We define the equilibrium bunch length due to synchrotron radiation in the
arcs alone, _i.e._

$\dfrac{1}{2}\\{\dot{\mathcal{N}}_{\text{ph}}\langle
u^{2}\rangle\\}_{z,w,\text{SR}}=\dfrac{2}{\tau_{z,w,\text{SR}}}\sigma_{z,w,\text{SR}}^{2}.$
(7)

Following Ref. [4], we obtain the average energy loss due to beamstrahlung:

$\displaystyle\dfrac{1}{2}\\{\dot{\mathcal{N}}_{\text{ph}}\langle
u^{2}\rangle\\}_{z,w,\text{BS}}\approx\dfrac{2}{\tau_{z,w,\text{SR}}}\times\dfrac{n_{\text{IP}}\tau_{z,w,\text{SR}}}{4T_{\text{rev}}}\\{\mathcal{N}_{\text{ph}}\langle
u^{2}\rangle\\}_{z,w,\text{BS}}$
$\displaystyle\approx\dfrac{2}{\tau_{z,w,\text{SR}}}\times\dfrac{n_{\text{IP}}\tau_{z,w,\text{SR}}}{4T_{\text{rev}}}\left(\dfrac{\alpha_{p}C}{2\pi
Q_{\text{s}}}\right)^{2}\dfrac{55}{24\sqrt{3}}\dfrac{r_{e}^{2}\gamma_{w}^{5}}{\alpha}\mathcal{I}_{3,w},$
$\displaystyle\equiv\dfrac{2}{\tau_{z,w,\text{SR}}}\mathcal{A}_{w}\mathcal{I}_{3,w},$
(8)

where $n_{\text{IP}}$ is the number of interaction points within one
revolution of period $T_{\text{rev}}$ in a ring of circumference $C$, with
momentum compaction factor $\alpha_{p}$ and synchrotron tune $Q_{\text{s}}$.
For convenience, we have introduced

$\mathcal{A}_{w}\equiv\frac{n_{\text{IP}}\tau_{z,w,\text{SR}}}{4T_{\text{rev}}}\left(\frac{\alpha_{p}C}{2\pi
Q_{\text{s}}}\right)^{2}\frac{55}{24\sqrt{3}}\frac{r_{e}^{2}\gamma_{w}^{5}}{\alpha}.$
(9)

The beamstrahlung damping time, in units of turns, can be deduced from Eq. (4)
to be

$\dfrac{1}{\tau_{z,w,\text{BS}}}=n_{\text{IP}}\delta_{w,\text{BS}}\approx\dfrac{2}{3}n_{\text{IP}}r_{e}\gamma_{w}^{3}\mathcal{I}_{2,w}.$
(10)

The rate of change of the bunch length can be re-written as

$\displaystyle\odv{\sigma_{z,w}^{2}}{t}$
$\displaystyle=\dfrac{2}{\tau_{z,\text{SR}}}(\sigma_{z,w,\text{SR}}^{2}+\mathcal{A}_{w}\mathcal{I}_{3,w})$
$\displaystyle\hskip
10.0pt-\left(\dfrac{2}{\tau_{z,\text{SR}}}+\mathcal{B}_{w}\mathcal{I}_{2,w}\right)\sigma_{z,w}^{2},$
(11)

where $\mathcal{B}_{w}\equiv n_{\text{IP}}\frac{4}{3}r_{e}\gamma_{w}^{3}$. The
equilibrium is attained when $\odv{\sigma_{z,l/h}^{2}}{t}=0$ leading to a set
of two implicit, non-linear equations that can be solved for the equilibrium
bunch length of the low and high intensity bunches, with subscripts $l$ and
$h$ respectively.

###  II.2 Analytical approximations for $\mathcal{I}_{n,w}$

In the flat-beam approximation, $\sigma_{x,s}^{*}\gg\sigma_{y,s}^{*}$, the
function $\mathcal{F}(x,y,z)_{s}$ can be approximated as [4]

$|\mathcal{F}(x,y,s)_{s}|\approx\left|\dfrac{y}{\sigma_{y,s}^{*}G(s,x)_{s}}+\dfrac{ix}{\sigma_{x,s}^{*}}\right|\dfrac{\exp\left[-{x^{2}}/({2\sigma_{x,s}^{*2}})\right]}{\sigma_{x,s}^{*}}.$
(12)

Considering $|x|<\sigma^{*}_{x,s}$ and $|y/G|<\sigma^{*}_{y,s}$, one may
further approximate the local curvature [4],

$\displaystyle\frac{1}{\rho(x,y,z,s;t)_{w}}=\sqrt{\frac{2}{\pi}}\frac{2r_{e}N_{s}}{\gamma_{w}\sigma_{z,s}}$
$\displaystyle\times\frac{\exp[-(x-(s-z/2)\theta_{c})^{2}/(2\sigma_{x,s}^{*2})]}{\sigma_{x,s}^{*}}\exp\left[-\frac{(2s-z)^{2}}{2\sigma_{z,s}^{2}}\right]$
$\displaystyle\times\left[\left(\frac{y}{\sigma_{y,s}^{*}G(s,x-(s-z/2)\theta_{c})_{s}}\right)^{2}+\left(\frac{x-(s-z/2)\theta_{c}}{\sigma_{x,s}^{*}}\right)^{2}\right]^{1/2}.$
(13)

At this point, it is worth assessing the overall impact of the various
approximations on the integrand in Eq. (II). The local curvature is a function
of four space variables $x$, $y$, $z$ and $s$, as well as of time $t$ (which
is not an integration variable), and so a meaningful comparison between the
approximation of Eq. (13) and, say, that which is obtained from numerical
simulations must involve comparing two three-dimensional objects embedded in
$\mathbb{R}^{4}$. Instead of attempting to do this, we have compared the
approximation of the integrand of Eq. (II), $\mathcal{G}/\rho_{w}^{n}$ (by
substituting Eq. (13)) with numerical values obtained from Xsuite [9] for
$n\in\\{1,2,3\\}$ along the $x$-, $y$\- and $z$-axes for $G(x,s)=1$. The plots
in the $x$-direction are reproduced here as an illustration. The upper part of
Fig. 1 shows that for $n=1$, our analytical expressions well approximate the
simulated behaviour; the $1\sigma$ cut-off of the approximated integrand is
also fairly justifiable by the exponential suppression. Results for cuts along
the $y$\- and $z$-directions show similar good approximations for $n=1$;
however, for higher values of $n$, our analytical formulae do not agree as
well with the simulated values (see the middle and bottom part of Fig. 1 for
the $x$-direction cuts). The behaviour of the integrand for cuts in the $y$\-
and $z$-directions show similar trends; notably, the analytical approximation
overestimates the numerical values for the $y$-direction cuts (while
underestimating it for cuts in the other two directions).

Figure 1: Plots of the integrand of Eq. II, $\mathcal{G}/\rho_{w}^{n}$, as a
function of $x$, using the tt̄2 (182.5 GeV) parameter set. Our analytical
predictions (red) are compared to values obtained by simulation (blue) of a
single beam-beam pass in Xsuite, using 101 longitudinal slices, and calculated
as the ratio of the transverse kick over the longitudinal distance between two
consecutive slices.

General cases featuring first order approximations of the impact of the
hourglass and of the crab waist are treated in App. B. For simplicity we focus
here on the configuration neglecting both, i.e $G(s,x)_{w/s}\approx 1$, so
that we have,

$\displaystyle\mathcal{I}_{n,w}^{\text{\cancel{HG}, \cancel{CW}}}$
$\displaystyle\approx\dfrac{1}{(2\pi)^{3/2}}\iiiint\displaylimits_{x,y,z,s\in\mathbb{R}^{4}}\dfrac{dxdydzds}{\sigma_{x,w}^{*}\sigma_{y,w}^{*}\sigma_{z,w}\sigma_{x,s}^{*n}}\left(\dfrac{2r_{e}N_{s}}{\gamma_{w}\sigma_{z,s}}\sqrt{\dfrac{2}{\pi}}\right)^{n}\left[\left(\dfrac{x-\left(s-{z}/{2}\right)\theta_{c}}{\sigma_{x,s}^{*}}\right)^{2}+\left(\dfrac{y}{\sigma_{y,s}^{*}}\right)^{2}\right]^{n/2}$
$\displaystyle\times\exp\left[-\dfrac{n\left[x-\left(s-{z}/{2}\right)\theta_{c}\right]^{2}}{2\sigma_{x,s}^{*2}}-\dfrac{n(2s-z)^{2}}{2\sigma_{z,s}^{2}}-\dfrac{\left(x+{z\theta_{c}}/{2}\right)^{2}}{2\sigma_{x,w}^{*2}}-\dfrac{y^{2}}{2\sigma_{y,w}^{*2}}-\dfrac{z^{2}}{2\sigma_{z,w}^{2}}\right].$
(14)

The integral of interest was already evaluated analytically for symmetric
bunch parameters with $n=1,3$ in Ref. [4]. A more general derivation, which
closely follows that in Ref. [4] but is applicable for any $n$ as well as
asymmetric bunch parameters, is given here.

Setting $x-\left(s-{z}/{2}\right)\theta_{c}=u$ as in Ref. [4], the negative of
the exponent in the integrand can be rewritten in a form that facilitates
Gaussian-like integrals,

$\displaystyle
n\dfrac{u^{2}}{2\sigma_{x,s}^{*2}}+n\dfrac{(2s-z)^{2}}{2\sigma_{z,s}^{*2}}+\dfrac{(u+s\theta_{c})^{2}}{2\sigma_{x,w}^{*2}}+\dfrac{y^{2}}{2\sigma_{y,w}^{*2}}+\dfrac{z^{2}}{2\sigma_{z,w}^{2}}.$
(15)

It is convenient to introduce the following terms

$\displaystyle A$
$\displaystyle=\dfrac{n}{2\sigma_{x,s}^{*2}}+\dfrac{1}{2\sigma_{x,w}^{*2}},$
$\displaystyle B$ $\displaystyle=\dfrac{\theta_{c}}{\sigma_{x,w}^{*2}},$
$\displaystyle C$
$\displaystyle=\dfrac{2n}{\sigma_{z,s}^{2}}+\dfrac{\theta_{c}^{2}}{2\sigma_{x,w}^{*2}},$
$\displaystyle D$
$\displaystyle=\dfrac{n}{2\sigma_{z,s}^{2}}+\dfrac{1}{2\sigma_{z,w}^{2}},$
$\displaystyle E$ $\displaystyle=-\dfrac{2n}{\sigma_{z,s}^{2}},$
$\displaystyle G$ $\displaystyle=C-\dfrac{E^{2}}{4D},$ (16)

such that the negative of the exponent can be re-written as

$\displaystyle
G\left(s+\dfrac{Bu}{2G}\right)^{2}+\left(A-\dfrac{B^{2}}{4G}\right)u^{2}+D\left(z+\dfrac{Es}{2D}\right)^{2}+\dfrac{y^{2}}{2\sigma_{y,w}^{*2}}.$
(17)

Performing the two Gaussian integrals over z and $s$ gives

$\displaystyle\mathcal{I}_{n,w}^{\text{\cancel{HG}, \cancel{CW}}}$
$\displaystyle=H_{n}\dfrac{\pi}{\sqrt{DG}}\iint\displaylimits_{u,y\in\mathbb{R}^{2}}dudy\left[\left(\dfrac{u}{\sigma_{x,s}^{*}}\right)^{2}+\left(\dfrac{y}{\sigma_{y,s}^{*}}\right)^{2}\right]^{n/2}$
$\displaystyle\times\exp\left[-\left(A-\dfrac{B^{2}}{4G}\right)u^{2}-\dfrac{y^{2}}{2\sigma_{y,w}^{*2}}\right],$
(18)

where

$H_{n}=\dfrac{1}{(2\pi)^{3/2}}\dfrac{1}{\sigma_{x,w}^{*}\sigma_{y,w}^{*}\sigma_{z,w}\sigma_{x,s}^{*n}}\left(\dfrac{2r_{e}N_{s}}{\gamma_{s}\sigma_{z,s}}\sqrt{\dfrac{2}{\pi}}\right)^{n}.$
(19)

Changing variables to $u/\sigma_{x,s}^{*}=r\cos\phi$ and
$y/\sigma_{y,s}^{*}=r\sin\phi$ for $0<r<\infty$ and $0\leq\phi<\pi/2$, and
performing the resultant Gaussian-like integral over $r$ using Ref. [10, Eq.
(3.1.1)] gives

$\displaystyle\mathcal{I}_{n,w}^{\text{\cancel{HG}, \cancel{CW}}}$
$\displaystyle=\dfrac{2H_{n}\Gamma\left(\frac{n}{2}+1\right)}{M_{n,w}^{\frac{n}{2}+1}}\dfrac{\pi}{\sqrt{DG}}\sigma_{x,s}^{*}\sigma_{y,s}^{*}$
$\displaystyle\times\int\displaylimits_{0}^{\pi/2}d\phi\dfrac{1}{\left[1-P_{n,w}\sin^{2}\phi\right]^{\frac{n}{2}+1}},$
(20)

in which we note that due to the symmetry in the range of $\phi$, a factor of
4 has been extracted to limit the range of integration over $\phi$ to
$0\leq\phi<\pi/2$. We have also defined the new parameters

$\displaystyle M_{n,w}$
$\displaystyle=\left(A-\dfrac{B^{2}}{4G}\right)\sigma_{x,s}^{*2},$
$\displaystyle\Theta_{w}$
$\displaystyle=\dfrac{\sigma_{y,s}^{*}}{\sigma_{y,w}^{*}},$ $\displaystyle
P_{n,w}$ $\displaystyle=1-\dfrac{\Theta_{w}^{2}}{2M_{n,w}}.$ (21)

To perform the remaining integral over $\phi$, we change variables to
$t=\sin^{2}\phi$ and define the constant

$R_{n,w}=\pi\Gamma\left(\frac{n}{2}+1\right)H_{n}\frac{\pi}{\sqrt{DG}}\sigma_{x,s}^{*}\sigma_{y,s}^{*}$
(22)

to obtain

$\mathcal{I}_{n,w}^{\text{\cancel{HG},
\cancel{CW}}}=\dfrac{R_{n,w}}{M_{n,w}^{\frac{n}{2}+1}}F\left(\dfrac{n}{2}+1,\dfrac{1}{2},1;P_{n,w}\right),$
(23)

where the hypergeometric function $F(\alpha,\beta,\gamma;z)$ has the integral
representation in Ref. [10, Eq.(4.5.6)]. Expressed in terms of the beam
quantities, the terms in Eq. (23) read

$\displaystyle M_{n,w}$
$\displaystyle=\dfrac{n}{2}+\dfrac{2n\sigma_{x,s}^{*2}}{\theta_{c}^{2}(n\sigma_{z,w}^{2}+\sigma_{z,s}^{2})+4n\sigma_{x,w}^{*2}},$
$\displaystyle P_{n,w}$
$\displaystyle=1-\dfrac{({\sigma_{y,s}^{*2}})/({2\sigma_{y,w}^{*2}})}{\dfrac{n}{2}+\dfrac{2n\sigma_{x,s}^{*2}}{\theta_{c}^{2}(n\sigma_{z,w}^{2}+\sigma_{z,s}^{2})+4n\sigma_{x,w}^{*2}}},$
$\displaystyle R_{n,w}$
$\displaystyle=\left(\dfrac{\pi}{2}\right)^{1/2}\Gamma\left(\dfrac{n}{2}+1\right)\dfrac{\sigma_{z,s}\sigma_{y,s}^{*}}{\sigma_{y,w}^{*}\sigma_{x,s}^{*n-1}}\left(\dfrac{2r_{e}N_{s}}{\gamma_{w}\sigma_{z,s}}\sqrt{\dfrac{2}{\pi}}\right)^{n}$
$\displaystyle\hskip
20.0pt\times\dfrac{1}{\sqrt{\theta_{c}^{2}(n\sigma_{z,w}^{2}+\sigma_{z,s}^{2})+4n\sigma_{x,w}^{*2}}}.$
(24)

A proof that this more general result reduces to those obtained for the
special case of symmetric collisions as presented in Ref. [4] is provided in
App. A.

###  II.3 Evaluating equilibrium bunch lengths using different approximations

Table 1: Relevant FCC parameters, reproduced from Ref. [11]. |  | Z | WW | ZH | tt̄1 | tt̄2
---|---|---|---|---|---|---
$C$ | [km] | 97.756
$\theta_{c}$ | [mrad] | 30
Beam energy | [GeV] | 45.6 | 80 | 120 | 175 | 182.5
$N_{0}$ | [$10^{11}$] | 1.7 | 1.5 | 1.8 | 2.2 | 2.3
$\alpha_{p}$ | [$10^{-6}$] | 14.8 | 14.8 | 7.3 | 7.3 | 7.3
$\beta_{x}^{*}$ | [m] | 0.15 | 0.2 | 0.3 | 1.0
$\beta_{y}^{*}$ | [mm] | 0.8 | 1.0 | 1.0 | 1.6
$\sigma_{x}^{*}$ | [$\mu$m] | 6.4 | 13.0 | 13.7 | 36.7 | 38.2
$\sigma_{y}^{*}$ | [nm] | 28 | 41 | 36 | 66 | 68
$\sigma_{z,\text{SR}}$ | [mm] | 3.5 | 3.0 | 3.15 | 2.01 | 1.97
$\Phi$ |  | 28.5 | 7.0 | 5.8 | 1.1 | 1.0
$L_{i}$ | [mm] | 0.42 | 0.85 | 0.90 | 1.8
$R_{\text{HG}}$ |  | 0.95 | 0.89 | 0.88 | 0.84
$Q_{s}$ |  | 0.0250 | 0.0506 | 0.0358 | 0.0818 | 0.0872
$\tau_{z,\text{SR}}$ | [turns] | 1273 | 236 | 70.3 | 23.1 | 20.4
$\tau_{z,\text{BS}}$ | [turns] | 186018 | 69011 | 26586 | 11239 | 10068
$U_{\text{SR}}$ | [GeV] | 0.036 | 0.34 | 1.72 | 7.8 | 9.2
$U_{\text{BS}}$ | [MeV] | 0.49 | 2.32 | 9.03 | 31.14 | 36.25
$\xi_{x}/\xi_{y}$ |  | $\frac{0.004}{0.133}$ | $\frac{0.010}{0.113}$ | $\frac{0.016}{0.118}$ | $\frac{0.097}{0.128}$ | $\frac{0.099}{0.126}$
Figure 2: Comparing the equilibrium bunch lengths, using FCC-ee parameters at
the Z, WW, ZH and tt̄ resonances (obtained from the various approximations,
with hg being “hourglass” and CW being “crab waist”) with those obtained from
simulations in Ref. [3] and by integrating numerically Eq. (B1) as done in
Ref. [4] (for the latter, only data for Z, WW and ZH resonances are
available).

For symmetric bunch parameters, the two equations of (11) for the stationary
case coincide. This is the case for the machine parameters proposed in the
FCC-ee design study [11]; relevant parameters are reproduced in Tab. 1. Fig. 2
shows the equilibrium bunch lengths obtained using these parameters and
compares them to results previously published in Ref. [3] using simulations
and in Ref. [4] using direct numerical integration. For most configurations,
the first order approximation of the hourglass and the crab waist (App. B2)
quoted “weak hg, weak CW” yields the closest equilibrium bunch length to the
simulations. This approximation is however rather poor for the Z configuration
described in Tab. 1, as it features the strongest hourglass effect
($\sigma_{z}/\beta_{y}^{*}\sim 10$). Due to the first order approximation, the
correction terms in $\mathcal{O}[(\sigma_{z}/\beta_{y}^{*})^{2}]$ (App. B)
introduce strong artefacts in the integral. Taking aside this exception, the
different approximations have a minor impact on the equilibrium bunch length
and yield estimates comparable to simulations. The impact of the
approximations was found to be minor also in the analysis of asymmetric
configurations. Thus, in the following, we only show results using Eq. (23)
which neglects both the hourglass effect and the crab waist.

###  II.4 Luminosity per interaction point at equilibrium

We may now calculate the equilibrium length for the configurations featuring
an asymmetry in the population of the two colliding bunches $\Delta N$ such
that $N_{h/l}=N_{0}\times(1\pm\Delta N)$, with $N_{h}$, $N_{l}$ and $N_{0}$
the high, low and mean bunch intensity respectively. Though it is the most
difficult configuration, due to the strong hourglass effect, we shall focus on
the Z configuration of the FCC as this is where the tolerance on the asymmetry
will turn out to be the tightest. The equilibirum bunch lengths of the two
bunches are shown in Fig. 3.

Figure 3: Equilibrium bunch lengths at the Z resonance determined with the
outlined approximations using the first set of published machine parameters
Tab. 1 but with varying intensities of the two bunches. The solid line
indicates the weak bunch length, while the dotted line gives that of the
strong bunch. The FCC design study required both beams to have an intensity of
$1.7\times 10^{11}$ particles at this resonance.

At a given mean intensity, with increasing asymmetry, the weak bunch becomes
longer without bound, while the strong bunch shortens to the length it would
have if beamstrahlung were absent; both bunches are equally long when they
have the same intensities, as would be expected from the symmetry of the
system. The equilibrium bunch lengths for a given intensity asymmetry
generally increase with the mean bunch intensity, as the effect of
beamstrahlung becomes stronger, which is also in accordance with expectation.
It should also be noted that as the mean intensity and asymmetry increase, at
some point the bunch lengths obtained by these calculations ceases to be
physical, as it exceeds the length of the longitudinal bucket. The
corresponding beam losses are not taken into account in this simplistic model.

For mean bunch intensities higher than $5\times 10^{10}$ (red curve in Fig. 3)
we observe a strong divergence of the bunch length of the low and high
intensity bunch caused by the difference in the strength of beamstrahlung.
This divergence results in a fast drop of the luminosity as a function of the
bunch intensity asymmetry, as shown in Fig. 4.

Figure 4: Luminosity, for collisions at the Z resonance, normalised to the
symmetric value at every mean intensity, determined with the outlined
approximations using the corresponding published machine parameters in Tab. 1
and the equilibrium lengths found via solving Eq. (11) for the stationary
case.

The luminosity estimates assume again that corrections linked to the hourglass
effect and the crab waist are negligible [12],

$\displaystyle\mathcal{L}$
$\displaystyle=\dfrac{N_{l}N_{h}}{T_{\text{rev}}}\dfrac{\cos(\theta_{c}/2)}{2\pi}\dfrac{1}{\sqrt{\sigma_{y,l}^{*2}+\sigma_{y,h}^{*2}}}$
$\displaystyle\times\dfrac{1}{\sqrt{(\sigma_{x,l}^{*2}+\sigma_{x,h}^{*2})\cos^{2}(\theta_{c}/2)+(\sigma_{z,l}^{2}+\sigma_{z,h}^{2})\sin^{2}(\theta_{c}/2)}}.$
(25)

We have thus obtained a precursor of the 3D flip-flop mechanism by considering
only the longitudinal effects linked to beamshtralung. In Sec. III, we will
address how the luminosity degradation is further enhanced by the transverse
blow-up of the low intensity bunch due to the strong non-linearity of the
beam-beam force generated by the short, high intensity bunch.

###  II.5 Stability of the stationary solutions

We now study the stability of the solution obtained with Eq. (11) in the
stationary case. We define the right hand side of the equation as
$f_{l/h}(\sigma_{z,l}^{2},\sigma_{z,h}^{2})$ and for simplicity we only
consider the model without hourglass and crab waist. Using Ref. [10,
Eq.(4.2.10)] in the differentiation, we find,

$\displaystyle\partialderivative{f_{l}}{\sigma_{z,l}^{2}}=\dfrac{2}{\tau_{z,\text{SR}}}\left\\{\mathcal{A}_{l}\left[\left(\dfrac{M_{3,l}^{5/2}\partialderivative{R_{3,l}}{\sigma_{z,l}^{2}}-\dfrac{5}{2}R_{3,l}M_{3,l}^{3/2}\partialderivative{M_{3,l}}{\sigma_{z,l}^{2}}}{M_{3,l}^{5}}\right)F\left(\dfrac{5}{2},\dfrac{1}{2};1;P_{3,l}\right)+\dfrac{5}{4}\dfrac{R_{3,l}}{M_{3,l}^{5/2}}\partialderivative{P_{3,l}}{\sigma_{z,l}^{2}}F\left(\dfrac{7}{2},\dfrac{3}{2};2;P_{3,l}\right)\right]-1\right\\}$
$\displaystyle-\mathcal{B}_{l}\left\\{\mathcal{I}_{2,l}+\sigma_{z,l}^{2}\left[\left(\dfrac{M_{2,l}^{2}\partialderivative{R_{2,l}}{\sigma_{z,l}^{2}}-2R_{2,l}M_{2,l}\partialderivative{M_{2,l}}{\sigma_{z,l}^{2}}}{M_{2,l}^{4}}\right)F\left(2,\dfrac{1}{2};1;P_{2,l}\right)+\dfrac{R_{2,l}}{M_{2,l}^{2}}\partialderivative{P_{2,l}}{\sigma_{z,l}^{2}}F\left(3,\dfrac{3}{2};2;P_{2,l}\right)\right]\right\\}.$
(26)
$\displaystyle\partialderivative{f_{l}}{\sigma_{z,h}^{2}}=\dfrac{2}{\tau_{z,\text{SR}}}\left\\{\mathcal{A}_{l}\left[\left(\dfrac{M_{3,l}^{5/2}\partialderivative{R_{3,l}}{\sigma_{z,h}^{2}}-\dfrac{5}{2}R_{3,l}M_{3,l}^{3/2}\partialderivative{M_{3,l}}{\sigma_{z,h}^{2}}}{M_{3,l}^{5}}\right)F\left(\dfrac{5}{2},\dfrac{1}{2};1;P_{3,l}\right)+\dfrac{5}{4}\dfrac{R_{3,l}}{M_{3,l}^{5/2}}\partialderivative{P_{3,l}}{\sigma_{z,h}^{2}}F\left(\dfrac{7}{2},\dfrac{3}{2};2;P_{3,l}\right)\right]\right\\}$
$\displaystyle-\mathcal{B}_{l}\left\\{\mathcal{I}_{2,l}+\sigma_{z,l}^{2}\left[\left(\dfrac{M_{2,l}^{2}\partialderivative{R_{2,l}}{\sigma_{z,s}^{2}}-2R_{2,l}M_{2,l}\partialderivative{M_{2,l}}{\sigma_{z,s}^{2}}}{M_{2,l}^{4}}\right)F\left(2,\dfrac{1}{2};1;P_{2,l}\right)+\dfrac{R_{2,l}}{M_{2,l}^{2}}\partialderivative{P_{2,l}}{\sigma_{z,h}^{2}}F\left(3,\dfrac{3}{2};2;P_{2,l}\right)\right]\right\\},$
(27)

with the other two derivatives for $f_{h}$ given by exchanging indices in the
above equations. We may evaluate the Jacobian at the equilibrium

$J(\sigma_{z,l,\text{eqm}}^{2},\sigma_{z,h,\text{eqm}}^{2})=\left(\begin{matrix}\partial{f_{l}}/\partial{\sigma_{z,l}^{2}}&\partial{f_{l}}/\partial{\sigma_{z,h}^{2}}\\\
\partial{f_{h}}/\partial{\sigma_{z,l}^{2}}&\partial{f_{h}}/\partial{\sigma_{z,h}^{2}}\end{matrix}\right)\Bigg{|}_{\text{eqm}}.$
(28)

It was found that, over a physical range of mean bunch intensities, at all
intensity asymmetries, all resonances had two perturbation eigenvectors,
$\bm{e}_{++}$ corresponding to the case where both bunches grow or shrink, and
$\bm{e}_{+-}$ to the case where one bunch grows and the other one shrinks. The
corresponding eigenvalues, $\lambda_{++}$ and $\lambda_{+-}$, were found to
always be negative. The sign of the eigenvalues implies that all equilibrium
configurations of the bunch lengths are stable to perturbation. One reason for
this is due to the damping effect of beamstrahlung: while the beam-beam
interaction is necessary for the blow-up to occur, as the beam intensity
increases, at higher bunch intensities and asymmetries, where beamstrahlung
damping dominates the damping effect of synchrotron radiation emitted from
bending in the arcs, the weak bunch is more strongly shortened by the emitted
radiation than it is lengthened by the resulting energy spread, and therefore
any lengthening effect due to the field of the strong bunch is quickly
dissipated away.

Nevertheless, the study of the eigenvalue coresponding to an asymmetric
perturbation of the bunch lengths $\lambda_{+-}$ in Fig. 5b shows regions
where the perturbations are expected to decay on time scales longer than the
one expected from the radiation damping in the arc only. On the other hand,
Fig. 5a shows that the decay time corresponding to a symmetric perturbation
can be significantly faster.

Consequently, one may expect that the transient perturbation due to the top-up
injection of additional particles to the bunches could decay on longer time
scale. For the configuration studied here, namely the FCC-ee at the Z pole,
the time scale remains short enough not to affect the luminosity
significantly.

(a) Z, $\lambda_{++}$

(b) Z, $\lambda_{+-}$

Figure 5: Eigenvalues obtained in performing linear stability analysis on the
equilibrium bunch lengths at different mean bunch intensities and intensity
asymmetry of collisions at the Z resonance in the “no hourglass, no crab
waist” approximation. They are normalised to the decay rate of the emittance
due to synchrotron radiation $2/\tau_{\text{{SR}}}$, indicated by the broken
line on Fig. 5b. It can be seen that as the intensity asymmetry approaches
100%, beamsstrahlung is no longer active, and the one of the eigenvalues
approaches this synchrotron-induced decay rate.

##  III Transverse blow-up

Figure 6: Equilibrium bunch lengths at the Z resonance using the 3D
phenomenological model with different $\chi/\sigma_{y,0}^{*}$. The solid line
indicates the weak bunch’s length, while the dotted line gives that of the
strong bunch. The red line, showing no $zy$-coupling, is identical to that
which would have been obtained by applying the 1D “no hourglass, no crab
waist” model to the published machine parameters in Tab. 1 for a symmetric
collision. Here, $\xi_{0}=0.20.$ Figure 7:  Equilibrium vertical width of the
weak bunch at the Z resonance using different approximations in the 3D
phenomenological model with different $\chi/\sigma_{y,0}^{*}$. The range of
intensity asymmetries has been truncated to show more clearly the behaviour of
the weak bunch’s width in the two regimes. Here, $\xi_{0}=0.20.$

The 3D flip-flop phenomenon as discussed in Ref. [3] involves an increase of
the transverse dimensions of the low intensity bunch caused by non-linear
diffusion driven by the beam-beam force of the higher intensity (and shorter)
bunch, and coupling between the longitudinal and transverse beam-beam forces
due to the collision with a large crossing angle. In quantifying this effect,
the beam-beam parameters $\xi$ is often used, which, in the case of flat
bunches ($\sigma_{y}\ll\sigma_{x}$) with a non-zero crossing angle
$\theta_{c}$ and large Piwinski angle $\Phi\gg 1$, take the form [3],

$\displaystyle\xi_{x,w}(\sigma_{z,s})$
$\displaystyle\approx\dfrac{N_{s}r_{e}}{\pi\gamma_{s}}\times\dfrac{2\beta_{x}^{*}}{(\sigma_{z,s}\theta_{c})^{2}},$
(29) $\displaystyle\xi_{y,w}(\sigma_{z,s},\sigma_{y,s}^{*})$
$\displaystyle\approx\dfrac{N_{s}r_{e}}{\pi\gamma_{s}}\times\dfrac{\beta_{y}^{*}}{\sigma_{z,s}\sigma_{y,s}^{*}\theta_{c}}.$
(30)

The flat beam geometry usually implies that $\xi_{y}>\xi_{x}$. Moreover, since
$N_{l}<N_{h}$ by assumption and $\sigma_{z,l}<\sigma_{z,h}$ as previously
found, the vertical beam-beam parameter of the low intensity bunch $\xi_{y,l}$
is the largest. Therefore, we shall coarsely model a transverse-longitudinal
flip-flop, with the blowup limited to the y-z plane, with a phenomenological
model of the vertical size of the low intensity bunch parameterised by
$\xi_{y,l}$. We assume that the vertical blow up can be neglected for a beam-
beam strength below a given threshold $\xi_{0}$. Above this threshold the
vertical beam size increases proportionally to the excess of beam-beam
parameters:

$\sigma_{y,l,\text{new}}^{*}=\left\\{\begin{array}[]{ll}\sigma_{y,l,0}^{*}&\text{if}~{}\xi<\xi_{0}\\\
\sigma_{y,l,0}^{*}+\chi\Delta\xi_{y}&\text{otherwise},\end{array}\right.$ (31)

with $\chi$ the phenomenological transversal blow-up factor and
$\Delta\xi_{y}$ the excess of vertical beam-beam tune shift:

$\Delta\xi_{y}=\xi_{y,l,\text{new}}(\sigma_{z,h,\text{eqm,
new}},\sigma_{y,h}^{*}=\sigma_{y,0}^{*})-\xi_{0}.$ (32)

The nonlinear dependence of the beam sizes on $\xi_{x,y}$ is key to the onset
of the transverse-longitudinal flip-flop effect. Note that Eq. 31 is nonlinear
in the sense that it is a threshold function, representing regular motion for
weak enough non-linear forces (_i.e._ $\xi<\xi_{0}$) with an onset of
transverse diffusion at a given $\xi_{0}$. Clearly, the behavior much beyond
the threshold is more nonlinear than the linear dependence assumed by the
model, yet its aim is to describe the dynamics close to the threshold
$\xi_{0}$.The phenomenological model has indeed no predictive power for strong
beam-beam strength ($\xi\gg\xi_{0}$).

Considering the $\xi_{y}$ values in Tab. 1, we chose $\xi_{0}=0.2$ as a
reasonable threshold beam interaction value which is not much higher than that
for symmetric collisions at the machine parameters. The threshold beam-beam
parameter defines the boundary between two different regimes: the
$zy$-decoupled (1D) and the $zy$-coupled (2D) regimes. The transition between
these regimes is visible in Figs. 6 and 7. In the 2D regime, the introduction
of coupling between the transverse and longitudinal dimensions of the weak
bunch to the original two equilibrium equations causes the blow up in
equilibrium bunch length and bunch width to occur more rapidly with increasing
asymmetry in the bunch intensities and increasing $\chi$. This corresponds to
a simplified model of the 3D flip-flop described in Ref. [3], without coupling
to the horizontal plane. We note however that this phenomenon is different in
nature to the transverse flip-flop observed in lower energies electron-
positron colliders featuring negligible beamstrahlung [5]. Indeed, here the
dynamical system does not feature multiple equilibrium solutions.
Nevertheless, the runaway of the low intensity bunch parameters towards
lowering the beam-beam force and of the high intensity bunch towards
increasing the beam-beam force in the transverse-longitudinal flip-flop
results in a sensitivity to the asymmetry between the two beams which is
comparable to the purely transverse flip-flop phenomenon.

Fig. 8 shows the tolerance for intensity asymmetry such that the luminosity
remains above 80% of that of a completely symmetric collision. The most
sensitive configuration of the FCC-ee corresponds to the Z pole: where an
asymmetry of only 8% at $\chi=1.5\sigma_{y,0}^{*}$ can be tolerated. This
tolerance is comparable to the 5% quoted for the FCC-ee design [11], yet we
note that the phenomenological parameters $\chi$ and $\xi_{0}$ used here were
coarsely estimated. Numerical simulations without approximating the beam-beam
force and the approximated phenomenological model for the transverse size
increase due to non-linear diffusion are still needed to determine suitable
values for the phenomenological parameters. Given that they are driven by the
non-linear dynamics of the particles, those parameters are expected to
strongly depend on the working point chosen.

Figure 8: Critical intensity asymmetry for which the relative intensity does
not drop below 80% using the 3D “no hourglass, no crab waist” model. The
shaded region for every resonance represents values of $\chi/\sigma_{y,0}^{*}$
lying between 0 (the upper bound, for which there is no coupling between the
z- and y-dimensions and so reducing to the 1D model) and 1.5 (the lower
bound). Here, $\xi_{0}=0.20.$

##  IV Conclusion

Integrals that give the average energy loss and quantum excitation due to
beamstrahlung effects in two crossing Gaussian bunches of arbitrary bunch
dimensions were generalised to configurations featuring asymmetric machine and
beam parameters. The hourglass effect and the crab waist were taken into
account to first order, but their impact on the equilibrium beam parameters
remained marginal.

Thanks to these integrals, the impact of the intensity asymmetry on the
equilibrium bunch lengths of the two beams could be studied. The low intensity
bunch tends to increase in length due to the increase in beamstrahlung caused
by the high intensity bunch. On the other hand, the high intensity bunch tends
to shrink towards the value determined by the magnetic lattice. The
equilibrium solutions were found to be stable. However in some configurations
an initial asymmetric perturbation, _e.g._ during top-up injection, could damp
with a rate significantly lower than the one given by synchrotron radiation
damping.

A plausible phenomenological model, which includes transverse blow up of the
weak bunch’s vertical width due to non-linear diffusion, was proposed as a way
to model the vertical blow-up observed in simulations. Though arguably coarse,
the phenomenological model yields a tolerance on the intensity asymmetry
comparable to those obtained in past studies with simulations.

###### Acknowledgements.

We thank T. v. Witzleben for a careful reading of the manuscript in its early
stages. This work was carried out in the framework of an undergraduate student
summer project as part of the Excellence Research Internship Programme hosted
by the École Polytechnique Fédérale de Lausanne (EPFL), under an exchange
agreement with the University of Cambridge, at the European Organisation for
Nuclear Research (CERN) and under the auspices of and with support from the
Swiss Accelerator Research and Technology (CHART) programme (www.chart.ch).

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