# LONG-RANGE BEAM–BEAM EXPERIMENTS IN THE RELATIVISTIC HEAVY ION COLLIDER
(Published in Phys. Rev. ST Accel. Beams 14, 091001)

R. Calaga  CERN  Geneva  Switzerland
W. Fischer  N. Milas  G. Robert-Demolaize  BNL  Upton  NY  USA

###### Abstract

Long-range beam–beam effects are a potential limit to the LHC performance with
the nominal design parameters, and certain upgrade scenarios under discussion.
To mitigate long-range effects, current carrying wires parallel to the beam
were proposed and space is reserved in the LHC for such wires. Two current
carrying wires were installed in RHIC to study the effect of strong long-range
beam–beam effects in a collider, as well as test the compensation of a single
long-range interaction. The experimental data were used to benchmark
simulations. We summarize this work.

##  1 Introduction

The reader should note that this is an identical copy of an article first
published in [1]. Beam–beam effects have limited the performance of previous
and existing hadron colliders [2, 3, 4] such as the ISR [5, 6],
Sp$\mathrm{\bar{p}}$S [7, 8, 9, 10], Tevatron [11, 12, 13] and RHIC [14, 15],
and are also expected to limit the performance of the LHC [16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Beam–beam effects can be categorized as either incoherent (dynamic aperture
and beam lifetime), PACMAN (bunch-to-bunch variations), or coherent (beam
oscillations and instabilities) [25]. These effects can be caused by both
head-on and long-range interactions. Head-on effects, leading to tune shifts
and spreads, are important in all hadron colliders. Total beam–beam induced
tune shifts as large as 0.028 were achieved in the Sp$\mathrm{\bar{p}}$S [10]
and Tevatron [13], although operational tune shift values are somewhat lower.
The LHC in its early stages of commissioning has already reached a total head-
on beam–beam tune shift of 0.02 [32].

Long-range effects, however, differ in previous and existing colliders. In the
ISR the beams collided under a large crossing angle of 15 deg [33] that
greatly reduced long-range effects. In the Sp$\mathrm{\bar{p}}$S, with both
beams in the same aperture and only three bunches per beam, there were a few
long-range interactions distributed over the ring circumference. Due to the
difference in the bunch intensities, the effect on the anti-protons was
stronger. In the Tevatron, also with both beams in the same aperture but 36
bunches per beam, there are more long-range interactions. With increased
intensity of the anti-proton bunches, protons can also be affected.

In RHIC (Fig. 1), where both beams share the same aperture only in the
interaction regions, there is only one long-range interaction per interaction
region without an experiment (a total of four in the current configuration),
with a 10 mm separation (corresponding to 30 rms beam sizes for protons at 250
GeV energy). Long-range interactions have affected the RHIC ramp transmission
in the past [14].

Figure 1: Beam–beam interactions in RHIC and locations of wires and electron
lenses.

##  2 Long-range effects and compensation in the LHC

In the LHC there are 32 long-range beam–beam interactions localized in each of
four interaction regions [25]. A major upgrade of the LHC interactions region
is foreseen by the end of the decade with the primary objective to increase
the average luminosity of the machine by about a factor of 5 to 10 above the
design performance. Among the various upgrade scenarios a crab crossing scheme
(CC), an early beam separation scheme (ES), and a large Piwinski angle (LPA)
are considered. In the CC scheme, crab cavities placed on either side of the
interaction region impart a transverse kick to effectively compensate the
crossing angle. This scheme allows for a large crossing angle that greatly
reduces long-range beam–beam effects. In the ES scheme [28, 27] the number of
long-range interactions is greatly reduced but four parasitic collisions at
4–5 $\sigma$ per IP remain. In the LPA scheme [26] the small crossing angle
will be maintained, and long bunches of intensities up to $4$–$5\times
10^{11}$ protons are used. All schemes aim at higher than nominal bunch
currents and reduced $\beta^{*}$. Therefore, long-range effects tend to become
more problematic and require more aperture for larger crossing angles or
compensation to mitigate these effects. The LPA scheme would most benefit from
long-range beam–beam compensation. The compensation of long-range effects in
the Tevatron was proposed with electron lenses [34], and in the LHC with wires
[35]. Electron lenses were also considered for the LHC [36], and the use of
wires was also studied for the Tevatron [37]. Implementation of long-range
beam–beam compensation in the Tevatron is challenging because the effect is
distributed over the whole ring. In the LHC the effect is localized in the
interaction regions. A partial long-range beam–beam compensation was
successfully implemented in the $\mathrm{e}^{+}\mathrm{e}^{-}$ collider
DA$\Phi$NE [38]. Beam–beam compensation and related issues were reviewed at a
workshop in 2007 [39].

##  3 RHIC as a test bench for long-range studies

Figures 1 and 2 show the basic layout of the beam–beam interaction and
compensation studies in RHIC. At store there are nominally two head-on
interactions in points 6 and 8 (IP6 and IP8), and long-range interactions with
a large separation in the other interaction points. Three bunches in the Blue
ring are coupled to three bunches in the Yellow ring through the head-on
beam–beam interaction. For studies, two DC wires were installed in the Blue
and Yellow rings respectively in interaction region 6 (IR6). Table 1 shows the
main beam parameters for polarized proton operation, both achieved and design.
In RHIC the beam–beam effect is strongest in proton operation.

Figure 2: Schematic of the RHIC interaction regions. Table 1: Main RHIC parameters achieved in polarized proton operation that are relevant for beam–beam effects protons (2009). Note that the polarized proton bunch intensity is also limited by intensity dependent depolarization effects in the AGS. Quantity | Unit |  |
---|---|---|---
Beam energy, $E_{b}$ | GeV | 100 | 250
Bunch intensity, $N_{b}$ | $10^{11}$ | 1.35 | 1.1
Norm emittance, $\epsilon$ |  $\mu$m | 2.5 | 3.0
rms bunch length, $\sigma_{z}$ | m | 0.85 | 0.60
Beam–beam parameter $\xi$/IP | … | 0.0056 | 0.0045
No of IPs | … | 2 | 2
$\beta^{*}$ at IP6, IP8 | m | 0.7 | 0.7

In the LHC locations in warm sections of the interaction regions are reserved
to accommodate long-range beam–beam wire compensators (Fig. 3), or electron
lenses. These locations have about equal horizontal and vertical
$\beta$-functions. With the expected strong long-range beam–beam effects in
the LHC, and the proposed wire compensation, experimental data and simulations
of long-range effects are highly desirable. Operational and experimental data
exist from the Sp$\mathrm{\bar{p}}$S and the Tevatron. In the SPS, wires were
installed to further investigate strong long-range beam–beam interactions, to
test the compensation scheme, and to benchmark simulations [40, 41, 42, 30].

Figure 3: LHC interaction region schematically showing the common focusing
channel with the 32 long-range interactions on the left and the right of the
collision point (top) and the optics functions in the region.

The wire experiments in RHIC complement these studies. The beam lifetime in
RHIC is typical for a collider and better than in the SPS wire experiments. In
addition, and unlike in the SPS, head-on effects can be included, and with
properly placed long-range interactions and wires, the compensation of a
single long-range interaction is possible.

##  4 Wires in RHIC

The RHIC wire design is based on experience gained with the SPS units. Design
considerations are: the location in ring, the integrated strength $(IL)$, the
wire temperature $T$ in operation, the positioning range and accuracy, power
supply requirements, controls, and diagnostics [43, 44]. The wire parameters
are shown in Table 2.

Table 2: Parameters for RHIC wires. The wire material is Cu at 20∘C. The nominal wire strength is for a single long-range interaction with a proton bunch intensity of $2\times 10^{11}$. Quantity | Unit | Value
---|---|---
Strength $(IL)$, nominal |  A m | 9.6
Max. strength $(IL)_{\mathrm{max}}$ |  A m | 125
Length of wire $L$ | m | 2.5
Radius of wire $r$ | mm | 3.5
Number of heat sinks $n$ | … | 3
Electrical resistivity $\rho_{e}$ |  $\Omega\,$m | $1.72\\!\\!\times\\!\\!10^{-8}$
Ceat conductivity $\lambda$ |  W m-1K-1 | 384
Thermal expansion coeff. |  K-1 | $1.68\\!\\!\times\\!\\!10^{-5}$
Radius of existing pipe $r_{p}$ | mm | 60
Current $I$, nominal | A | 3.8
Max. current $I_{\mathrm{max}}$ | A | 50
Current ripple $\Delta I/I$ (at 50 A) | $10^{-4}$ | $<1.7$
Electric resistance $R$ |  m$\Omega$ | 1.12
Max. voltage $U_{\mathrm{max}}$ | mV | 55.9
Max. power $P_{\mathrm{max}}$ | W | 2.8
Max. temp. change $\Delta T_{\mathrm{max}}$ | K | 15
Max. length change $\Delta L_{\mathrm{max}}$ | mm | 0.4
Vertical position range |  mm/$\sigma_{y}$ | 65/10.6

###  4.1 Location in the Ring

For a successful compensation the phase advance between the long-range
interaction and the compensator should be no larger than about 10 degrees
[45]. Lattices with $\beta^{*}\leq 1.0$ m have such small phase advances
between the entrance to the DX and the exit of Q3. Thus it is possible to
place a wire in the warm region after Q3 to compensate for a long-range
beam–beam interaction near the DX magnet (Fig. 4). Since the beam paths must
cross horizontally, it is easier to control the distance between the beams in
an experiment through vertical separation. To compensate for a vertical long-
range interaction near the DX magnet, one wire can be installed in each ring
(see Fig. 5). In the Blue ring the wire is installed below the beam axis, in
the Yellow ring above the beam axis.

Figure 4: Location of wires in RHIC and location of long-range beam–beam
interaction for compensation. Figure 5: The two long-range beam–beam wires in
the RHIC tunnel during installation.

###  4.2 Integrated Strength

To compensate a single long-range interaction, the compensator’s integrated
strength $(IL)$ must be the same as the opposing bunch’s current integrated
over its length $(IL)=N_{b}ec$, where $I$ is the current in the wire, $L$ its
length, $N_{b}$ the bunch intensity, $e$ the elementary charge, and $c$ the
speed of light (see Table. 2).

In the LHC, an integrated strength of 80 A m is required to correct for the 16
long-range interactions on either side of an IR [35]. Such a strength is also
expected to lead to enhanced diffusion at amplitudes larger than 6 rms
transverse beam sizes [45]. To study the enhanced diffusion in RHIC, the wire
is designed for $(IL)_{\mathrm{max}}=125$ A m.

###  4.3 Wire Temperature

The wire temperature should not exceed 100∘C to avoid increased outgassing of
the vacuum components. We use a number of air cooled heat sinks to limit the
wire temperature.

Assume first a wire in vacuum of radius $r$ and length $l$, with electrical
resistivity $\rho_{e}$ and heat conductivity $\lambda$. A current $I$ flows
through the wire, and at both ends there are heat sinks that maintain the
temperature $T_{0}$. Further we assume that the temperature rise $\Delta T$ in
the wire is small enough so that the material coefficients $\rho_{e}$ and
$\lambda$ are constants. In each length element $\mathrm{d}x$ heat
$\mathrm{d}Q$ is produced through the wire’s resistivity at the rate

$\frac{\mathrm{d}Q}{\mathrm{d}t}=\rho_{e}\frac{\mathrm{d}x}{\pi r^{2}}I^{2},$
(1)

and the heat flow is connected to the temperature gradient
$\mathrm{d}T(x)/\mathrm{d}x$ via the heat equation

$\frac{\mathrm{d}Q}{\mathrm{d}t}=-\lambda\pi
r^{2}\frac{\mathrm{d}T}{\mathrm{d}x}.$ (2)

Combining Eqs. (1) and (2) yields the differential equation for the
temperature

$\frac{\mathrm{d}T^{2}(x)}{\mathrm{d}x^{2}}=-\frac{\rho_{e}}{\lambda}\frac{I^{2}}{\pi^{2}r^{4}}$
(3)

with the solution

$T(x)=-\frac{1}{2\pi^{2}}\frac{\rho_{e}}{\lambda}\frac{I^{2}}{r^{4}}x^{2}+ax+b.$
(4)

The coefficients $a$ and $b$ can be determined from the boundary conditions
$T(0)=T(l)=T_{0}$ yielding

$T(x)=T_{0}+\frac{1}{2\pi^{2}}\frac{\rho_{e}}{\lambda}\frac{I^{2}}{r^{4}}(xl-x^{2}).$
(5)

The maximum temperature increase $\Delta T_{\mathrm{max}}$ is in the centre of
the wire, $x=l/2$, and is

$\Delta
T_{\mathrm{max}}=\frac{1}{8\pi^{2}}\frac{\rho_{e}}{\lambda}\frac{(Il)^{2}}{r^{4}}.$
(6)

If we now assume a wire of length $L$ with $n$ heat sinks, we can replace $l$
by $L/(n-1)$ in Eq. (6) and arrive at

$\Delta
T_{\mathrm{max}}=\frac{1}{8\pi^{2}}\frac{\rho_{e}}{\lambda}\frac{(IL)^{2}}{(n-1)^{2}r^{4}}.$
(7)

We use $n=3$ heat sinks cooled with forced air. To move the wire compensator
close to the beam, its radius should not be much larger than an rms transverse
beam size. The calculated temperature change is shown in Table 2. Figure 6
shows a drawing of the end of a wire. Visible are the wire support, the
electrical feed-through which is also a heat sink, and a connecting loop
allowing for thermal expansion of the wire.

Figure 6: Drawing of the end of a long-range beam–beam wire in RHIC.

###  4.4 Power Supply Requirements

To limit emittance growth, a current ripple of $\Delta I/I<10^{-4}$ is
required [45]. A measurement shows a current ripple of $\Delta I/I<1.7\times
10^{-4}$ where the upper limit is given by the noise floor of the current
measurement.

##  5 Long-range experiments at RHIC

More than 30 dedicated Long-Range (LR) beam–beam experiments were performed at
different energies, with different species and various machine configurations.
They span a variety of long-range conditions which help benchmark simulation
tools. The main parameters that were varied were the strength of the long-
range interactions (wire current), the distance between the beam and the wire
(or other beam), the tune and chromaticity. All experimental sessions to study
long-range beam–beam interactions in RHIC can be broadly classified into three
categories approximately in chronological order:

  * •

measurement of a single long-range interaction between the two proton bunches
at 23 and 100 GeV in IP6;

  * •

effect of the DC wires on a single beam either by varying the current at a
fixed distance or varying the distance to the beam with fixed current on both
protons at 100 GeV and gold at 100 GeV/nucleon;

  * •

effect of long-range interaction either with a wire in the presence of head-on
collisions or long-range interactions between the two beams in IP6 with
simultaneous compensation using a wire at 100 GeV.

A summary of all long-range experiments performed in the RHIC accelerator
between 2005 to 2009 is listed with corresponding beam conditions in Table 3.

Table 3: Summary of long-range beam–beam experiments in RHIC. The wires in the Blue and Yellow ring are named B-BBLR and Y-BBLR respectively. Fields are left blank when the experimental value could not be determined. fill | ring | scan | species | rel. | bunches | $Q_{x}$ | $Q_{y}$ | LR | LR | LR | fitted |  $d$ for | comment
---|---|---|---|---|---|---|---|---|---|---|---|---|---
no |  |  |  | $\gamma$ | per ring |  |  | location | strength | separation | exponent |  $\tau<20$ h |
|  |  |  |  |  |  |  |  | $(IL)$ | $d$ | $p$ |  |
… | … | … | … | … | … | … | … | … |  A m | $\sigma$ | … | $\sigma$ |
2005 |  |  |  |  |  |  |  |  |
6981 | B | 1 | p | 25.963 | 1 | 0.7331 | 0.7223 | IP4 | 5.3 | B moved |  |  | weak signal
6981 | Y | 1 | p | 25.963 | 1 | 0.7267 | 0.7234 | IP4 | 5.3 | B moved |  |  | weak signal
6981 | B | 2 | p | 25.963 | 1 | 0.7351 | 0.7223 | IP4 | 5.8 | B moved |  |  | weak signal
6981 | Y | 2 | p | 25.963 | 1 | 0.7282 | 0.7233 | IP4 | 5.8 | B moved |  |  | weak signal
6981 | B | 3 | p | 25.963 | 1 | 0.7383 | 0.7247 | IR4 DX | 8.6 | Y moved |  |  | weak signal
6981 | Y | 3 | p | 25.963 | 1 | 0.7271 | 0.7218 | IR4 DX | 8.6 | Y moved |  |  | weak signal
6981 | B | 4 | p | 25.963 | 1 | 0.7394 | 0.7271 | IR4 DX | 8.9 | Y moved | 4.9 | 6.5 |
6981 | Y | 4 | p | 25.963 | 1 | 0.7264 | 0.7388 | IR4 DX | 8.9 | Y moved | 2.8 |  |
2006 |  |  |  |  |  |  |  |  |
7707 | B | 1 | p | 106.597 | 10 |  |  | IR6 DX | 6.7 | B moved |  |  | weak signal
7707 | Y | 1 | p | 106.597 | 10 |  |  | IR6 DX | 6.7 | B moved |  |  | weak signal
7707 | B | 2 | p | 106.597 | 10 |  |  | IR6 DX | 6.7 | Y moved |  |  | weak signal
7707 | Y | 2 | p | 106.597 | 10 |  |  | IR6 DX | 6.7 | Y moved |  |  | weak signal
7747 | B | 1 | p | 106.597 | 8 |  |  | IR6 DX | 7.9 | B moved |  |  | weak signal
7747 | Y | 1 | p | 106.597 | 10 |  |  | IR6 DX | 7.9 | B moved |  |  | weak signal
7747 | B | 2 | p | 106.597 | 8 |  |  | IR6 DX | 7.0 | Y moved |  |  | weak signal
7747 | Y | 2 | p | 106.597 | 10 |  |  | IR6 DX | 7.0 | Y moved |  |  | weak signal
7807 | B | 1 | p | 106.597 | 12 | 0.6912 | 0.6966 | IR6 DX | 8.2 | Y moved | 2.5 | 3.5 | additional octupoles
7807 | Y | 1 | p | 106.597 | 12 | 0.7092 | 0.6966 | IR6 DX | 8.2 | Y moved | 1.5 | 3.5 | additional octupoles
2007 |  |  |  |  |  |  |  |  |
8231 | B | 1 | Au | 10.520 | 6 | 0.2327 | 0.2141 | B-BBLR | 12.5 | B-BBLR moved | 7.2 | 6.5 |
8231 | B | 1 | Au | 10.520 | 6 | 0.2322 | 0.2140 | B-BBLR | 125 | B-BBLR moved | 7.8 | 9.0 |
8405 | B | 1 | Au | 107.369 | 56 | 0.2260 | 0.2270 | B-BBLR | 125 | B-BBLR moved | 1.7 | 15.0 | background test
8609 | B | 1 | Au | 107.369 | 23 | 0.2340 | 0.2260 | B-BBLR | 12.5 | B-BBLR moved | 7.4 | 6.0 |
8609 | B | 2 | Au | 107.369 | 23 | 0.2340 | 0.2260 | B-BBLR | 125 | B-BBLR moved | 16.0 | 5.5 |
8609 | Y | 1 | Au | 107.369 | 23 | 0.2280 | 0.2350 | Y-BBLR | 12.5 | Y-BBLR moved | 4.8 | 9.5 |
8609 | Y | 2 | Au | 107.369 | 23 | 0.2280 | 0.2350 | Y-BBLR | 125 | Y-BBLR moved | 4.1 | 7.5 |
8727 | B | 1 | Au | 107.369 | 23 | 0.2200 | 0.2320 | B-BBLR | 12.5 | B-BBLR moved | 5.2 | 9.5 |
8727 | B | 2 | Au | 107.369 | 23 | 0.2200 | 0.2320 | B-BBLR | 125 | B-BBLR moved | 8.1 | 10.0 |
8727 | B | 1 | Au | 107.369 | 23 | 0.2320 | 0.2280 | Y-BBLR | 12.5 | Y-BBLR moved | 6.3 | 4.5 |
8727 | B | 2 | Au | 107.369 | 23 | 0.2320 | 0.2280 | Y-BBLR | 125 | Y-BBLR moved | 10.8 | 5.0 |
8727 | B | 3 | Au | 107.369 | 23 | 0.2320 | 0.2280 | Y-BBLR | 125-0 | -6.5 |  |  |
8727 | B | 4 | Au | 107.369 | 23 | 0.2320 | 0.2280 | Y-BBLR | 125 | -6.5 |  |  | ver. chromaticity 2-8
8727 | B | 5 | Au | 107.369 | 23 | 0.2320 | 0.2280 | Y-BBLR | 125-0 | -6.5 |  |  | ver. chromaticity 8
2008 |  |  |  |  |  |  |  |  |
9664 | B | 1 | d | 107.369 | 12 | 0.2288 | 0.2248 | B-BBLR | 125 | B-BBLR moved | 3.8 | 17.0 | end of physics store
9664 | B | 2 | d | 107.369 | 12 | 0.2288 | 0.2248 | B-BBLR | 75-125 | 5.8 |  |  | end of physics store
2009 |  |  |  |  |  |  |  |  |
10793 | B | - | p | 106.597 | 36 | 0.691 | 0.688 | B-BBLR | 125 | B-BBLR moved |  |  | with head-on collisions
10793 | Y | - | p | 106.597 | 36 | 0.695 | 0.692 | Y-BBLR | 125 | Y-BBLR moved |  |  | with head-on collisions
10793 | B | - | p | 106.597 | 36 | 0.691 | 0.688 | IR6 DX | 12.5 | B-BBLR moved |  |  | LR compensation
10793 | Y | - | p | 106.597 | 36 | 0.695 | 0.692 | IR6 DX | 12.5 | Y-BBLR moved |  |  | LR compensation

The main observables in long-range beam–beam experiments are orbits, tunes,
Beam Transfer Functions (BTFs), and the beam lifetime. Several simulations
were performed for a subset of measurements which show successful
reconstruction of all measurable quantities and the onset of losses [46].
Specific examples for each of the three categories with detailed results are
presented in the next sections to summarize all the long-range experiments
performed at RHIC.

###  5.1 Single Long-range Measurements

The first set of long-range beam–beam experiments were performed with proton
beams in 2006. The motivation of these experiments was to characterize the
effect of one parasitic interaction on beam losses for a future compensation
demonstration. The Blue and Yellow beams were vertically separated in the IR6
region close to the DX magnet (Fig. 2). The RHIC beams are very stable at the
nominal working point and the effect of a single long-range (weak effect) is
not visible in the beam lifetime. An effect of a compensation effect will not
be possible to detect with the available instrumentation.

Therefore, a finite strength in lattice octupoles and a working point close to
the 10${}^{\text{th}}$ order resonance was used as shown in Fig. 7. At this
modified working point, the beams are marginally stable as the introduction of
the single parasitic interaction increases the tune spread of the large
amplitude particles on to the 10${}^{\text{th}}$ order resonance, thus
enhancing the effect. This setup of marginally stable beams is only used for
experiments with single long-range interaction between the two beams. Some
relevant lattice and beam parameters are listed in Table 4.

Figure 7: Tunes in the resonance diagram for both beams and both planes during a scan. Table 4: RHIC parameters for experiments with long-range interactions with proton beams. Quantity | Enit | Blue | Yellow
---|---|---|---
Beam energy $E$ | GeV/n | 100
Rigidity $(B\rho)$ |  T m | 831.8
Number of bunches | … | 12
LR interaction from IP6 | m | 10.6
Norm. Emittances ($\epsilon_{x,y}$) |  $\mu$m | 15-20
$\beta_{x}$ at wire location | m | 105
Tunes (Qx,y) | … | 0.69/0.7 | 0.71/0.69
$\beta_{x}$ at wire location | m | 1060 | 342
$\beta_{y}$ at wire location | m | 357 | 1000
Octupole Strength (kl) |  m-2 |  6.3 $\times 10^{-3}$

The marginally stable beams were essential as the effect of the single long-
range interaction on the rather stable RHIC beams is subtle. In one such
experiment, the effect on the beam losses on both beams as a function of the
separation is shown in Fig. 8. To increase the signal-to-noise ratio the
losses are averaged over the 12 bunches.

Figure 8: Beam losses due to a single parasitic interaction of the Blue and
Yellow beam. The Yellow beam moved closer to the Blue beam from an initially
large separation.

Note that the Yellow beam was moved while the Blue beam was kept stationary.
Therefore, the effect on the Blue beam is of relevance as the losses in the
Yellow beam may also be affected by orbit and tunes shifts. A small effect is
visible when the beams are approximately 5 $\sigma$ or closer.

Compensation of such small effects is difficult as the losses are smaller than
the natural reproducibility of the machine for a given beam setup. Therefore,
it was important to significantly enhance the loss due to the long-range
interactions to clearly demonstrate compensation with a DC wire. Increased
chromaticity and introduction of head-on collisions were utilized to enhance
the effect of the LR interaction with the DC wires [46].

###  5.2 Wire Scans on Single Beam

After the installation of the DC wires in 2007, the majority of the
experiments were carried out using the individual wires of the Blue and the
Yellow ring to characterize the onset of the losses under certain beam
conditions [44, 46]. Most of the wire experiments were done with gold beams.
Table 5 shows the main beam parameters for the wire experiments at store with
gold beams.

Table 5: RHIC parameters for experiments with DC wires on individual gold beams. Quantity | Unit | Blue | Yellow
---|---|---|---
Beam energy $E$ | GeV/nucleon | 100
Rigidity $(B\rho)$ | Tm | 831.8
Number of bunches | … | 6–56
Norm. Emittance $\epsilon_{x,y}$ |  $\mu$rad | 17 | 17
Distance IP6 to wire | m | 40.92
centre |  |
Parameter $K$ (at 50 A) | nm | $-30.1$
Hor. tune Qx | … | 28.234 | 28.228
Ver. tune Qy | … | 28.226 | 29.235
$\beta_{x}$ at wire location | m | 1091 | 350
$\beta_{y}$ at wire location | m | 378 | 1067

The $\beta$-functions in Table 5 are the best estimate of the real
$\beta$-functions in the machine. The design lattice has $\beta^{*}=0.8$ m at
IP6. To calculate the $\beta$-functions at the wire location we use
$\beta^{*}=0.9$ m, and assume a 10% error. Figure 9 shows the MAD lattice near
the interaction region 6 where the wires are located.

Figure 9: The beta functions from MAD lattice for both rings as a function of
longitudinal position.

The measurements consisted mainly of distance and current scans and
simultaneous measurements of the beam loss rate. An overview of the beam
losses and wire position for the Blue and the Yellow ring during the course of
a scan (Fill 8727) is illustrated in Fig. 10. The beam loss rates are clearly
different for the Blue and Yellow beams. This indicates towards different
diffusion rates and re-population of tails for the two beams. The exact reason
for this difference is not identified. It should be noted that the wire
installations are identical.

Figure 10: Beam losses and wire position as a function of time during the
course of position and current scan (May 2007, Fill 8727).

Orbit, tune and chromaticity changes can be calculated as a function of the
long-range strength and distance [47]. These quantities and beam transfer
functions are usually recorded to benchmark with theory and simulations. The
vertical dipole kick $\Delta y^{\prime}$ and vertical tune change $\Delta
Q_{y}$ due to the wire for a separation $d$ in the vertical plane between the
beam and the wire are given by (assume no horizontal separation)

$\Delta y^{\prime}=\frac{K}{d}\;\;\;\;\;\mathrm{and}\;\;\;\;\;\Delta
Q_{x,y}=\pm\frac{K\beta_{x,y}}{4\pi}\frac{1}{d^{2}}$ (8)

with

$K=\frac{\mu_{0}(IL)}{2\pi(B\rho)},$ (9)

where $d$ is the distance between the wire and the beam, $\mu_{0}$ the
permeability of the vacuum, $(IL)$ the integrated wire strength, and $(B\rho)$
the beam rigidity.

Note that we take a positive sign for $d$ for a wire above the beam, and a
negative sign below the beam. We also assume that the reference vertical orbit
position at the location of the wire is zero ($y_{ref}=0$) for the wire
current off. The sign of $K$ depends on the direction of the wire current
relative to the beam direction, and the charge of the beam particles. In our
case the wire current has the opposite direction to the beam, the Blue wire is
above and the Yellow wire below the beam, and the beam particles have positive
charges. In this case the sign of $K$ is negative in Blue, and positive in
Yellow. The orbit change $\Delta y$ at the location of the wire due to the
dipole kick $\Delta y^{\prime}$, for $\Delta y\ll d$, is then

$\Delta y=\frac{K\beta_{y}}{2d}\frac{\cos{(\pi Q_{y})}}{|\sin{(\pi Q_{y})|}}.$
(10)

If the wire comes close to the beam Eq. (10) becomes inaccurate and needs to
be replaced by

$\Delta y=\frac{d}{2}-\sqrt{\frac{d^{2}}{4}-\frac{1}{2}K\beta_{y}\cot(\pi
Q_{y})}$ (11)

where $d$ is now the distance between the wire and the beam position at zero
wire current.

Orbit and tune changes agree with expectations under well controlled
experimental circumstances [46, 48]. Figure 11 shows a comparison of the
measured beam trajectories to the analytical prediction as a function of the
separation between the wire and the Blue beam.

Figure 11: Vertical orbit change (average of 3 BPMs near wire) as a function
of vertical distance, in Blue and Yellow ring at 5 A and 50 A. Solid lines in
all plots represent the analytical prediction.

Figure 12 shows a comparison of the measured tunes to the analytical
prediction as a function of the separation between the wire and the beam.

Figure 12: Horizontal and vertical tune change for 50 A and 5 A wire current,
for the Blue ring. Solid lines in all plots represent the analytical
prediction.

The beam lifetime, however, is determined through the nonlinear beam–beam
effect and can only be assessed in detailed simulations. Figure 13 (top) shows
the beam loss rate as a function of the vertical wire distance to the beam.
The onset of losses due to a long-range type interaction between the wire and
the beam is visible. Similarly the effect on beam losses due to a current scan
at a fixed distance is shown in Fig. 13 (bottom). The approximate separation
in the Blue ring is 9 $\sigma$ and in the Yellow ring is 5 $\sigma$. The
Yellow ring shows very weak or no effect with a current scan which is probably
due to a previous distance scan resulting in a cleaning of the large amplitude
particles.

Figure 13: Top: Beam loss as a function of the DC wire separation to the Blue
and the Yellow beams at 5 A and 50 A. Bottom: Beam loss due to a current scan
in the DC wire fixed at a given distance from the beam. Solid lines in all
plots show a power law fit to the losses.

It was speculated that the beam lifetime $\tau$ can be expressed as
$\tau=Ad^{p}$ where $A$ is an amplitude, $d$ the distance between wire and
beam, and $p$ an exponent that would typically be in a narrow range. For the
SPS $p$ had been found to be about 5, and for the Tevatron to be about 3 [49].
In Table 3 the fitted exponents are listed for all cases for which a fit was
possible. The fitted exponents range from 1.7 to 16, i.e. $p$ is not
constrained within a narrow range. Ten of the thirteen $p$ values are between
4 and 10. Figure 14 shows the fitted exponents $p$ as a function of the ion
tunes in the upper part, and the proton tunes in the lower part. Ion tunes
near the diagonal and away from either horizontal or vertical resonances show
smaller exponents $p$. The experiments also showed that the beam lifetime is
reduced with increased chromaticity [46].

Figure 14: Fitted exponents $p$ for long-range beam–beam experiments as a
function of the ion tunes (top) and the proton tunes (bottom). The fitted
exponents range from 1.7 to 16.

Another simple measure of assessing the long-range beam–beam effect in
experiments is the distance between the beam and wire (or other beam) at which
the beam lifetime becomes smaller than a certain value. We have chosen this
value to be 20 h, which would imply a luminosity lifetime of 10 h or less.
Table 3 shows an amplitude range between 3.5 and 17 $\sigma$. With the
available amount of data no clear correlation can be established between this
distance and the fitted coefficient $p$. In two cases the distance was found
to be as large as or larger than 10 $\sigma$, and most cases fall between 4
and 10 $\sigma$. Operation with less than 5 $\sigma$ separation appears to be
difficult [50]. Note that the beam is sometimes used for multiple scans and
that a large lifetime drop at large distances is more typical for previously
unused beams (Table 3).

Figure 15: Comparison of measured and simulated beam loss rate as a function
of distance between wire and beam. Experiment with gold beam at store, wire
strength of 125 Am [51, 53].

One important goal of the experiments is to benchmark simulations. In several
simulations the onset of large losses as a function of the distance between
wire and beam was reproduced within about 1 $\sigma$ [48, 52, 51, 53, 54, 55,
30]. One such comparison is shown in Fig. 15.

###  5.3 Long-range Effects with Head-on Collisions

End of physics fills were initially used to test the effect of the wires on
colliding gold and deuteron beams (see Table 3). It should be noted that the
beam–beam parameter of proton beams in RHIC is approximately three times
larger than the beam–beam parameter of heavy ion beams. The first dedicated
experiment with protons to compare the effect of the wire on colliding beams
and compensation of a single LR beam–beam interaction was conducted in 2009 at
100 GeV. Due to aperture considerations for decreasing $\beta^{*}$, the Blue
wire was removed during the shutdown after the 2009 run and the Yellow wire
was removed subsequently. Therefore, the experiments in 2009 serve as the
final set of measurements for LR beam–beam with RHIC as a test bed. The
relevant RHIC beam and lattice parameters are listed in Table 6 for the
experiments in 2009.

Table 6: Relevant RHIC beam and lattice parameters for experiments with proton beams. Quantity | Unit | Blue | Yellow
---|---|---|---
Beam energy $E$ | GeV | 100
Rigidity $(B\rho)$ | Tm | 333.5
Number of bunches | - | 36
# of colliding bunches | - | 30
Bunch intensity | $10^{11}$ | 1.7 | 1.7
Norm. Emittance $\epsilon_{x,y}$ |  $\mu$rad | 25,24 | 49,19
Horizontal tune $Q_{x}$ | … | 28.691 | 28.232
Vertical tune $Q_{y}$ | … | 29.688 | 29.692
Chromaticities ($\xi_{x},\xi_{y}$) | … | $(+2,+2)$
$\beta_{x}$ at wire location | m | 1566 | 556
$\beta_{y}$ at wire location | m | 576 | 1607

Prior to a long-range compensation attempt, a position scan of the wire on
each beam was performed with a wire current of 50 A. A 36$\times$36 bunch
pattern with six non-colliding bunches was chosen to enable a comparison of
the lifetime in the presence of the wire between single beam and colliding
beams simultaneously. The corresponding beam loss rates as a function of beam
to wire separation on both colliding and non-colliding bunches were measured.
The initial beam loss rates with colliding beams were stabilized to the
nominal 10% per hour. The maximum total beam losses for the wire movements
towards the beam at fixed current were constrained to 100–150% per hour for a
very short period to avoid disrupting the beam quality significantly for
subsequent measurements.

Figure 17 shows the evolution of the intensity between bunches with and
without head-on collisions. It is evident that the bunches with the head-on
collisions have a more severe effect from the LR forces of the wire. Several
hypotheses can be formulated to explain the increased losses for bunches with
head-on collision.

  * •

The dynamic aperture for the bunches with head-on is significantly smaller
than that of the single beam which could lead to the observed beam losses.

  * •

It was also suggested by an anonymous referee that the addition of the head-on
collisions enhances the diffusion leading to enhanced losses in the presence
of long-range interactions. Figure 17 clearly shows a larger initial slope for
bunch intensities with collisions. However, it is difficult to untangle the
contribution from the reduced dynamic aperture as opposed to enhanced
diffusion.

  * •

The additional tune shift due to the wire along with large head-on tune shift
could lead to beam losses due to very limited tune space available. No tune
optimization was performed during the experiment.

  * •

The effect of the wire on the orbit can introduce a static offset between the
two beams at the IP which is approximately proportional to the wire distance.
A large offset due to the kick from the wire can lead to emittance blow-up and
beam losses [31]. The relative offset at the collision point during the wire
scan with 50 A (see Fig. 16) is well below the 1 $\sigma$ level which is very
small.

Figure 16: Orbit offset at IP6 as a function of the wire position for Blue
(top) and Yellow (bottom) rings with a current of 50 A.

However, simulations to support each of the above hypotheses to explain its
contribution towards observed losses is beyond the scope of this paper.

Figure 17: Single bunch intensities as a function of wire position for Blue
(top) and Yellow (bottom) rings with a current of 50 A. Comparison between
bunches with head-on and no head-on collision is shown.

###  5.4 Single Long-range and Wire Compensation

The bunch spacing and the interaction region geometry in RHIC does not
inherently have LR beam–beam interactions. It is therefore necessary to shift
the collision point towards the DX magnet closest to the DC wires as noted
before. This location enables an artificially induced LR interaction between
the two beams and simultaneously allows for a minimum phase advance between
the LR interaction and DC wires (6 deg). Additionally, this location has
sufficient aperture for an orbit scan with the range of interest (3–10
$\sigma$). Figure 18 shows the trajectories of the Blue and Yellow rings with
the LR interaction set at approximately 3.1 $\sigma$.

Figure 18: Orbits right of IP6 for the Blue and the Yellow ring with the LR
interaction setup near the DX magnet at approximately 3.1 $\sigma$.

The individual bunch intensities and beam losses were recorded during the
position scan with the LR compensation [56]. Figure 19 shows the beam losses
as a function of the wire position. In the Blue ring, the losses are always
increasing as the wire approaches the beam. Therefore, no evidence of
compensation of the LR interaction from the Blue beam is visible. However, in
the Yellow ring, the beam lifetime improved as the beam to wire distance
approaches 3 $\sigma$ (Fig. 19). Consecutive retractions and restoration of
the beam to wire distance to 3 $\sigma$ show similar improvement of the beam
lifetime. This indicates a compensation of the effect of LR interaction by the
DC wire.

Figure 19: Beam loss rate for a Blue (top) and Yellow (bottom) bunch with one
long-range interaction, and an additional wire interaction. The wire position
varies, and the wire current is constant at 5 A.

In addition to beam losses, the individual bunch intensities with and without
LR interactions and simultaneous compensation is shown in Fig. 20. Note that
all 36 bunches experience the effect of the DC wire, but only 30 bunches
experience LR interactions. Therefore, only bunches with an LR interaction can
experience a compensation. In the Blue ring, the bunch intensity evolution is
similar for bunches with and without LR compensation. Hence, only the effect
from the wire is visible. The bunches with LR interaction and simultaneous
compensation have reduced beam losses as compared to the bunches that only see
the wire. This is consistent with the beam loss measurements (Fig. 19).

Figure 20: Beam intensity comparison between bunches with a single long-range
and no long-range interaction as a function of the wire position for Blue
(top) and Yellow (bottom) rings with a wire current of 5 A.

##  6 Summary

Long-range beam–beam experiments were conducted in RHIC from 2005 to 2009. The
motivation for these were two-fold. First, the experimental data can benchmark
simulation codes for situations of strong localized long-range beam–beam
interactions as they will exist in the LHC. Second, the compensation of a
single long-range beam–beam interaction can be tested in a scheme that is also
usable in the LHC.

These experiments complement the experience with long-range beam–beam
interactions in the Sp$\mathrm{\bar{p}}$S and Tevatron, wire experiments in
the SPS, and the partial long-range compensation in DA$\Phi$NE. The RHIC wires
created strong localized long-range beam–beam effects, comparable in strength
to the effect expected in the LHC, with a beam that has a lifetime typical of
hadron colliders, and including head-on beam–beam collisions. The observed
orbit and tune changes due to the wire were as expected. The effect of the
long-range beam–beam interactions on the beam loss rate is sensitive to a
number of beam parameters such as the tunes and chromaticities. Fitting the
beam lifetime $\tau$ to an exponential function $\tau\propto d^{p}$ as a
function of the distance $d$ between the beam and the wire, exponents $p$ in
the range between 1.7 and 16 were found. Distances smaller than 5 $\sigma$
created losses too large for collider operation. The experimentally observed
distance from the wire to the beam at which large beam losses set in could be
reproduced in simulations within 1 $\sigma$. The beam lifetime with long-range
interactions created by the wire was degraded further through head-on
collisions. A single attempt to compensate long-range beam–beam interaction
via a DC wire showed evidence of compensation.

##  7 Acknowledgements

For discussions and help we are most thankful to collaborators within the US
LHC Accelerator Research Program as well as several people at various
laboratories. Among these are O. Brüning, R. DeMaria, U. Dorda, W. Herr, A.
Kabel, H.-J. Kim, J.-P. Koutchouk, C. Milardi, K. Ohmi, T. Pieloni, J. Qiang,
F. Schmidt, T. Sen, G. Sterbini, and F. Zimmermann.

Work supported by Brookhaven Science Associates, LLC under Contract No. DE-
AC02-98CH10886 with the US Department of Energy, and in part by the US LHC
Accelerator Research Program.

## References

  * [1] N. Abreu, R. Calaga, W. Fischer and G. Robert-Demolaize, Phys. Rev. ST Accel. Beams 14 (2011) 091001.
  * [2] E. Keil, “Beam–Beam Effects in Electron and Proton Colliders,” PAC’89, Illinois, March 1989, pp. 1731–1735 (1989).
  * [3] Y. Alexahin, “Theory and Reality of Beam–Beam Effects at Hadron Colliders,” PAC’05, Tennessee, May 2005, pp. 544–548 (2005).
  * [4] J.-P. Koutchouk, “A Summary on CARE-HHH_APD Presentations on Beam–Beam Effects & Beam–Beam Compensation at the LHC,” Final CARE-HHH Workshop on Scenarios for the LHC Upgrade and Fair - HHH-2008 - Chavannes-de-Bogis, Switzerland; W. Scandale and F. Zimmermann (eds.), CERN-2009-004 (2009); CARE-Conf-08-032-HHH (2008).
  * [5] M. Jacob and K. Johnson (editors), “The Role of the ISR in the Development of Particle Physics,” CERN Yellow Report 84-23 (1984).
  * [6] J.Y. Hémery, A. Hofmann, J.-P. Koutchouk, S. Myers, and L. Vos, “Investigation of the Coherent Beam–Beam Effects in the ISR,” IEEE Trans. of Nucl. Sci. NS-28, No. 3 (1981).
  * [7] L.R. Evans and J. Gareyte, “Beam–Beam and Single Beam Effects in the SPS Proton-Antiproton Collider,” PAC’83, Santa Fee, March 1983, pp. 2397–2399 (1983).
  * [8] L. Evans, J. Gareyte, M. Meddahi, and R. Schmidt, “Beam–Beam Effects in the Strong-Strong Regime at the CERN-SPS,” PAC’89, Illinois, March 1989, pp. 1403–1405 (1989).
  * [9] K. Cornelis, M. Meddahi, and R. Schmidt, “The Beam–Beam Effect in the SPS Proton Antiproton Collider With Unequal Emittances,” EPAC’90, Nice, June 1990, pp. 1670–1672 (1990).
  * [10] K. Cornelis, “Beam–Beam Effects in the SPS Antiproton-Proton Collider,” LHC’99, Geneva, June 1999, CERN-SL-99-039 AP (1999).
  * [11] X. Zhang, T. Sen, V. Shiltsev, M. Xiao, Y. Alexahin, F. Schmidt, and F. Zimmermann, “Experimental Studies of Beam–Beam Effects in the Tevatron”, PAC’03, Oregon, 2003, pp. 1757–1759 (2003).
  * [12] V. Shiltsev, Y. Alexahin, V. Lebedev, P. Lebrun, R. S. Moore, T. Sen, A. Tollestrup, A. Valishev, and X. L. Zhang, Phys. Rev. ST Accel. Beams 8, (2005) 101001.
  * [13] A. Valishev, “Simulation of Beam–Beam Effects and Tevatron Experience”, EPAC’08, Genoa, 2008, pp. 2937–2941 (2008).
  * [14] W. Fischer et al., “Observation of Strong-Strong and Other Beam–Beam Effects in RHIC,” PAC’03, Oregon, 2003, pp. 135–137 (2003).
  * [15] V. Ptitsyn et al., “RHIC Performance With Polarized Protons in Run-6,” EPAC’06, Edinburgh, 2006, pp. 592–594 (2006).
  * [16] W. Herr, “Computer Simulation of the Coherent Beam–Beam Effect in the LHC,” PAC’91, San Francisco, May 1991, pp. 1068–1070 (1991).
  * [17] N. Gelfand, C. Johnstone, T. Sen, and W. Wan, “Effect of the Beam–Beam Interactions on the Dynamic Aperture of the LHC at Collision,” PAC’99, New York, March 1999, pp. 1677–1679 (1999).
  * [18] J. Gareyte, “Beam–Beam Design Criteria for LHC,” LHC’99, Geneva, June 1999, CERN-SL-99-039 AP (1999).
  * [19] M.A. Furman and W.C. Turner, “Beam–Beam Simulations for Separated Beams in the LHC,” EPAC’00, Vienna, June 2000, pp. 1196–1189 (2000).
  * [20] H. Grote, “Self-Consistent Orbits With Beam–Beam Effect in the LHC,” EPAC’00, Vienna, June 2000, pp. 1202–1204 (2000).
  * [21] L.H.A. Leunissen, H. Grote, and F. Schmidt, “LHC Dynamic Aperture Including the Beam–Beam Force,” EPAC’00, Vienna, June 2000, pp. 1208–1210 (2000).
  * [22] M.P. Zorzano and T. Sen, “Emittance Growth of the LHC Beam Due to the Effect of Head-On Beam–Beam Interaction and Ground Motion,” EPAC’00, Vienna, June 2000, pp. 1226–1228 (2000).
  * [23] M.P. Zorzano and F. Zimmermann, “Simulations of Coherent Beam–Beam Modes at the LHC,” EPAC’00, Vienna, June 2000, pp. 1229–1231 (2000).
  * [24] W. Herr, “Tune Spreads and Shifts Due to the Long-Range Beam–Beam Effect in The LHC,” EPAC’00, Vienna, June 2000, pp. 1630–1632 (2000).
  * [25] O. Brüning, P. Collier, P. Lebrun, S. Myers, R. Ostojic, J. Poole, and P. Proudlock (editors), CERN-2004-003 (2004).
  * [26] F. Zimmermann, “LHC Upgrade Scenarios,” PAC’07, Albuquerque, June 2007, pp. 714–718 (2007).
  * [27] J.-P. Koutchouk, “A Concept for the Luminosity Upgrade …,” PAC’07, Albuquerque, June 2007, pp. 3387–3389 (2007).
  * [28] G. Sterbini and G. Tommasini, “A Feasibility Study of Superconducting Dipole for the Early Separation Scheme of SLHC,” EPAC’08, Genoa, June 2008, pp. 2461–2463 (2008).
  * [29] K. Ohmi, “Study of Beam–Beam Effect at Various Collision Schemes in LHC,” EPAC’08, Genoa, June 2008, pp. 2593–2595 (2008).
  * [30] U. Dorda, “Compensation of Long-Range Beam–Beam Interaction at the CERN LHC,” PhD thesis, Vienna University of Technology, Austria (2008).
  * [31] T. Pieloni, “A Study of Beam–Beam Effects in Hadron Colliders With a Large Number of Bunches,” PhD thesis, Ecole Polytechnique Federale de Lausanne, Switzerland (2008).
  * [32] W. Herr, LHC Beam Operation Workshop, Evian (2010).
  * [33] The CERN study group on new accelerators, CERN AR/Int. SG/64-9 (1964).
  * [34] V. Shiltsev, V. Danilov, D. Finley, and A. Sery, Phys. Rev. ST Accel. Beams 2 (1999) 071001.
  * [35] J.-P. Koutchouk, “Correction of the Long-Range Beam–Beam Effect in LHC Using Electro-Magnetic Lenses,” PAC’01, Illinois, 2001, pp. 1681–1683 (2004).
  * [36] U. Dorda, F. Zimmermann, W. Fischer, and V. Shiltsev, “LHC Beam–Beam Compensation Using Wires and Electron Lenses,” PAC’07, Albuquerque, June 2007, pp. 1589–1591 (2007).
  * [37] T. Sen and B. Erdelyi, “Feasibility Study of Beam–Beam Compensation in the Tevatron With Wires,” PAC’05, Knoxville, 2005, pp. 2645–2647 (2005).
  * [38] C. Milardi, D. Alesini, M.A. Pregner, P. Raimondi, M. Zobov, and D. Shatilov, “Wire Compensation of Parasitic Crossing in DA$\Phi$NE,” EPAC’06, Edinburgh, 2006, pp. 2808–2810 (2006).
  * [39] W. Fischer, O. Brüning, J.-P. Koutchouk, F. Zimmermann, T. Sen, V. Shiltsev, K. Ohmi, M. Furman, Y. Cai, and A. Chao, informal note BNL C-A/AP/291; ICFA Beam Dynamics Newsletter No. 44, pp. 220-225; proceedings of BEAM’07, CERN-2008-005, CARE-Conf-08-004-HHH (2008).
  * [40] J.-P. Koutchouk, J. Wenniger, and F. Zimmermann, “Experiments on LHC Long-Range Beam–Beam Compensation in the SPS,” proceedings EPAC’04, Lucerne, 2004, pp. 1936–1938 (2004).
  * [41] F. Zimmermann, J.-P. Koutchouk, F. Roncarolo, J. Wenninger, T. Sen, V. Shiltsev, and Y. Papaphilippou, “Experiments on Long-Range Beam–Beam Compensation and Crossing Schemes at the CERN SPS in 2004,” PAC’05, Knoxville, 2005, pp. 686–688 (2005).
  * [42] U. Dorda, J.-P. Koutchouk, R. Tomas, J. Wenniger, F. Zimmermann, R. Calaga, and W. Fischer, “Wire Excitation Experiments in the CERN SPS,” EPAC’08, Genoa, pp. 3176–3178 (2008).
  * [43] W. Fischer, R. Alforque, H.C. Hseuh, R. Lambiase, C.J. Liaw, G. Miglionico, T. Russo, J.-P. Koutchouk, F. Zimmermann, and T. Sen, informal note BNL C-A/AP/236 (2006).
  * [44] W. Fischer, R. Calaga, U. Dorda, J.-P. Koutchouk, F. Zimmermann, V. Ranjbar, T. Sen, J. Shi, J. Qiang, and A. Kabel, “Observation of Long-Range Beam–Beam Effect in RHIC and Plans for Compensation,” EPAC’06, Edinburgh, 2006, pp. 2158–2160 (2006).
  * [45] F. Zimmermann, “Weak-Strong Simulation Studies for the LHC Long-Range Beam–Beam Compensation,” proceedings of the Beam–Beam Workshop at Fermilab, FERMILAB-Conf-01/390-T, CERN LHC Project Report 502 (2001)
  * [46] W. Fischer et al., “Experiments With a DC Wire in RHIC,” PAC’07, Albuquerque, 2007, pp. 1859–1861 (2007).
  * [47] T. Sen and B. Erdelyi, “Analytical Studies of the Long-Range Beam–Beam Tune Shifts and Chromaticities,” EPAC’02, Paris, 2002, pp. 1247–1249 (2004).
  * [48] H.J. Kim and T. Sen, “Simulations of Beam-Wire Experiments at RHIC,” PAC’07, Albuquerque, 2007, pp. 3492–3494 (2007).
  * [49] F. Zimmermann, private communication.
  * [50] N. Abreu, “Beam–Beam With a Few Long-Range Encounters at Long Distance,” proceedings BEAM’07, CERN-2008-005, CARE-Conf-08-004-HHH (2008).
  * [51] H.J. Kim, T. Sen, N.P. Abreu, and W. Fischer, “Simulations of Wire Compensator in RHIC,” HB2008, Nashville, 2008, pp. 72 (2008).
  * [52] H.J. Kim, T. Sen, N.P. Abreu, and W. Fischer, “Studies of Wire Compensation and Beam–Beam Interaction in RHIC,” EPAC’08, Genoa, 2008, pp. 3119–3121 (2008).
  * [53] H.J. Kim and T. Sen, Phys. Rev. ST Accel. Beams 12, 031001 (2009).
  * [54] H.J. Kim, T. Sen, and W. Fischer, “Long-Range Beam–Beam Compensation in RHIC,” IPAC’10, Kyoto, May 2010, pp. 2072–2074 (2010).
  * [55] A. Kabel, private communication.
  * [56] R. Calaga, W. Fischer, and G. Robert-Demolaize, “RHIC BBLR Measurements in 2009,” IPAC’10, Kyoto, May 2010, pp. 510–512 (2010).


# Imaginary Machines: A Serverless Model for Cloud Applications

Michael Wawrzoniak1, Rodrigo Bruno2, Ana Klimovic1, Gustavo Alonso1
1Systems Group, Dept. of Computer Science, ETH Zurich
2INESC-ID/Técnico, U. Lisboa

###### Abstract.

Serverless Function-as-a-Service (FaaS) platforms provide applications with
resources that are highly elastic, quick to instantiate, accounted at fine
granularity, and without the need for explicit run-time resource
orchestration. This combination of the core properties underpins the success
and popularity of the serverless FaaS paradigm. However, these benefits are
not available to most cloud applications because they are designed for
networked virtual machines/containers environments. Since such cloud
applications cannot take advantage of the highly elastic resources of
serverless and require run-time orchestration systems to operate, they suffer
from lower resource utilization, additional management complexity, and costs
relative to their FaaS serverless counterparts.

We propose _Imaginary Machines_ , a new serverless model for cloud
applications. This model (1.) exposes the highly elastic resources of
serverless platforms as the traditional network-of-hosts model that cloud
applications expect, and (2.) it eliminates the need for explicit run-time
orchestration by transparently managing application resources based on signals
generated during cloud application executions. With the Imaginary Machines
model, _unmodified cloud applications become serverless applications_. While
still based on the network-of-host model, they benefit from the highly elastic
resources and do not require runtime orchestration, just like their
specialized serverless FaaS counterparts, promising increased resource
utilization while reducing management costs.

††conference: The 2nd Workshop on Serverless Systems, Applications and
Methodologies; 22-25 April 2024, Athens, Greece;

##  1\. Introduction

Cloud users today can choose between two main options for developing and
managing their applications: renting VMs on-demand or using providers’
serverless compute and storage services. VMs come with a familiar
programming/execution environment of network-of-hosts, but provide a lower
level of resource elasticity, and users need to explicitly manage the
infrastructure at run-time (e.g., using cluster managers, turn machines on/off
based on the application load.) Serverless platforms automate infrastructure
management and provide highly elastic resources but use a restrictive FaaS
programming model with limitations that reduce its broader applicability
(Hellerstein et al., 2019). To overcome the limitations, researchers have been
exploring new serverless platform designs (Sreekanti et al., 2020), additional
infrastructure (Klimovic et al., 2018), and constructing new FaaS-specialized
applications (Müller et al., 2020).

This paper explores how to get the best of both worlds – applying the
serverless paradigm to cloud applications designed for networked VMs with the
familiar network-of-hosts programming model. We explore how to automate
infrastructure orchestration à la serverless for cloud applications. We
propose a serverless model for cloud applications, which we call Imaginary
Machines, that preserves the familiar, well-loved network-of-hosts programming
model of VM-based applications with the orchestration automation of serverless
platforms. We also discuss approaches to realize this model and propose an
overlay approach based on an evolution of Boxer (Wawrzoniak et al., 2021) to
realize the model on publicly available FaaS resources.

##  2\. Serverless Cloud Applications

We believe that cloud applications based on the network-of-hosts model can
also be serverless, taking advantage of highly elastic resources and automated
infrastructure orchestration. Our perspective is based on the following
observations and insights.

Serverless resources can suit cloud applications:  Although commonly viewed as
lightweight and restricted, resources made available by FaaS can match many
cloud application requirements. FaaS platforms continue to reduce limitations
and increase resource limits, but even today’s public platforms, such as AWS
Lambda (aws, 2020a), can match many cloud application requirements of memory,
compute capacity, state persistence, and reliability for some use cases.

Resources available in today’s FaaS platforms have been increasing in size to
the point where functions available today are comparable to the largest VMs
available 15 years ago (memory and cpu), the environment in which many of the
popular cloud applications have been originally designed for (zoo, 2023; spa,
2023). AWS Lambda functions can be configured up to 10GB of memory and 6vCPU
cores (aws, 2020b), matching largest AWS EC2 CPU instances introduced 12 years
earlier (aws, 2008).

Second, the wide adoption of disaggregated storage resulted in cloud
applications that are structured into multiple tiers of functionality, where
many services do not depend on the local persistent state, which makes these
layers a match for the ephemeral state provided by publicly available FaaS
functions. For example, a completely unmodified application logic tier of a
popular DeathStarBench (Gan et al., 2019) microservice benchmark runs in AWS
Lambda (Wawrzoniak et al., 2024a).

Third, FaaS functions are becoming more reliable and are no longer short-
lived. The maximum execution time of FaaS functions has been increasing, in
AWS Lambda, it is now 15 minutes (aws, 2018).

Unbundling serverless resources from programming model:  Serverless resources
do not have to be bundled with the FaaS programming model, even in mainstream
publicly available serverless platforms such as AWS Lambda. An evolution of
Boxer (Wawrzoniak et al., 2021), a platform providing the network-of-hosts
programming model on top of AWS Lambda FaaS, has been demonstrated to run
unmodified off-the-shelf distributed data processing systems such as Apache
Spark and Drill (Wawrzoniak et al., 2024b), unmodified Apache Zookeeper, and
the logic layer of the DeathStarBench microservice benchmark (Wawrzoniak et
al., 2022, 2024a). These unmodified cloud applications run on AWS Lambda
functions just as if they were running on VM-based networked hosts, each
function instance having a routable network address, a resolvable hostname,
and function-to-function network transport.

These projects demonstrated the feasibility of using the network-of-hosts
model with serverless resources. However, they also took a step back by
reintroducing the need for a serverful orchestration system (docker-compose
(doc, 2023) was used in (Wawrzoniak et al., 2024b)) to manage executions of
individual functions.

Serverless orchestration for cloud applications:  We believe that automated
infrastructure orchestration, an essential property of serverless, can also be
realized for network-of-hosts model cloud applications. We first describe a
perspective on automated resource orchestration of existing FaaS and then
propose how to achieve an analogous mechanism for the network-of-hosts model.

The FaaS programming model enables automated infrastructure orchestration by
embedding resource allocation signals in application control flow. The two
signals generated during application execution are (1.) function invocation
resulting in resource allocation of a function instance (and invocation of the
function event handler), and (2.) function handler exit, when resources are
released. As a ubiquitous and essential optimization, the resources are
normally not released immediately after function handlers exit, instead, to
provide low-latency warm-starts, function instances are first suspended for a
limited time before they are terminated. FaaS applications are factored into
collections of functions (possibly structured as dataflows (aws, [n. d.]))
that compose by invoking each other (directly or by producing events) based on
internal control flow. Each such function invocation is a resource allocation
signal used by FaaS platforms to dynamically rescale resources and create the
illusion of avoiding resource orchestration.

Cloud applications based on the network-of-hosts model execute processes on
hosts that ’compose’ by communicating with each other via a network. The
analogous resource allocation signals to those embedded in FaaS functions
(above) are also present in cloud applications. Generally, any signal that
indicates that any application process on a host function could be runnable is
potentially a resource allocation signal for that host function (analogous to
a function invocation). Similarly, any signal that indicates that all
application processes of a host are idle is a resource release signal for that
host function (analogous to a function event handler exiting). To achieve the
automated infrastructure orchestration for the network-of-hosts model, the
platform must observe such signals and perform the necessary on-demand
resource allocation actions of allocating and releasing host functions, just
like in FaaS.

Although not as explicit as in FaaS, cloud applications provide such signals;
these signals are already used by local operating systems of running hosts to
perform local application process scheduling (e.g., in response to a signal
indicating that data is available for a waiting blocked process, the scheduler
may choose to make the process runnable). However, in contrast to VM-based
infrastructure, to achieve serverless fine-grain scaling, host functions of
cloud applications must be allocated on demand only when needed and released
(suspended or terminated) when not used. This means that, instead of the
host’s local operating system, which may be suspended or not even started, the
appropriate resource allocation signals must be observed and acted on
_externally_ to the host function. Fortunately, assuming that cloud
applications run entirely on the serverless platform, the allocation signals
for all non-running host functions must be generated by (and observable at)
the running host functions or external network requests, all observable by the
serverless platform. As a concrete example, when application processes
initiate network communication to a valid destination host that is not running
(e.g., during initialization or scaling up), the platform must use this signal
to allocate the new destination host function automatically and in a
transparent way to the already running application processes. Conversely, as
application processes on a running host become idle, the platform suspends and
eventually possibly terminates the host, automatically releasing unused
application resources. More signaling scenarios must be handled, which we do
not describe here, but based on this approach, the resources used by network-
of-hosts applications are allocated and released on-demand and transparently
to the application, providing a form of automated infrastructure
orchestration.

In order to provide application transparency, the resource allocation
latencies (time to start or resume a host) must be sufficiently short (and
infrequent) to be within tolerable network latencies of the applications.
Fortunately, as evidenced by publicly available systems such as AWS Lambda,
latency to instantiate new host functions (cold $\sim$200ms) can be in the
range of wide-area network latencies, suggesting they are within the tolerable
range of connection timeouts that many cloud applications can handle.

##  3\. Imaginary Machines Model

We refer to the serverless execution model of cloud applications as the
Imaginary Machines (IM) model.

  * •

_From the application perspective_ , the Imaginary Machines model presents a
network-of-hosts programming model, where all possible (configured) hosts are
already instantiated. Cloud application processes run _imagining_ as if all
network destinations were already running and available.

Significantly, the model _does not_ include any explicit run-time
infrastructure orchestration just like in the FaaS model).

  * •

_From the platform perspective_ , the system must support the programming
model that the applications expect (above) while performing on-demand,
automatic, transparent, and fine-grained resource orchestration for the
application.

We are in the process of realizing a version of the IM model by extending
Boxer (Wawrzoniak et al., 2024a) serverless overlay system that already
provides network-of-hosts model on top of AWS Lambda FaaS by also including
the necessary function allocation mechanism.

##  4\. Conclusion

We motivated and described an alternative serverless model for cloud
applications called Imaginary Machines. Given that the dominant programming
model in the cloud is based on the network-of-hosts model, providing
applications based on that model with access to serverless resource elasticity
and reduced operational complexity has the potential for a great impact.

## References

  * (1)
  * aws ([n. d.]) [n. d.].  AWS Step Functions.  https://aws.amazon.com/step-functions/
  * aws (2008) 2008\.  Amazon EC2 now provides High-CPU instance types.  Retrieved 2023-12-03 from https://aws.amazon.com/articles/feature-guide-amazon-ec2-high-cpu-instance-types/
  * aws (2018) 2018\.  AWS Lambda enables functions that can run up to 15 minutes.  Retrieved 2024-01-27 from https://aws.amazon.com/about-aws/whats-new/2018/10/aws-lambda-supports-functions-that-can-run-up-to-15-minutes/
  * aws (2020a) 2020a.  AWS Lambda.  Retrieved 2020-08-17 from https://aws.amazon.com/lambda
  * aws (2020b) 2020b.  AWS Lambda now supports up to 10 GB of memory and 6 vCPU cores for Lambda Functions.  Retrieved 2024-01-27 from https://aws.amazon.com/about-aws/whats-new/2020/12/aws-lambda-supports-10gb-memory-6-vcpu-cores-lambda-functions/
  * spa (2023) 2023\.  Apache Spark history.  Retrieved 2023-12-03 from https://spark.apache.org/history.html
  * doc (2023) 2023\.  Docker Compose.  Retrieved 2023-07-27 from https://docs.docker.com/compose/
  * zoo (2023) 2023\.  Initial ZooKeeper code contribution from Yahoo!  Retrieved 2023-12-03 from https://issues.apache.org/jira/browse/ZOOKEEPER-1
  * Gan et al. (2019) Yu Gan, Yanqi Zhang, Dailun Cheng, Ankitha Shetty, Priyal Rathi, Nayan Katarki, Ariana Bruno, Justin Hu, Brian Ritchken, Brendon Jackson, Kelvin Hu, Meghna Pancholi, Yuan He, Brett Clancy, Chris Colen, Fukang Wen, Catherine Leung, Siyuan Wang, Leon Zaruvinsky, Mateo Espinosa, Rick Lin, Zhongling Liu, Jake Padilla, and Christina Delimitrou. 2019.  An Open-Source Benchmark Suite for Microservices and Their Hardware-Software Implications for Cloud & Edge Systems. In _ASPLOS_.
  * Hellerstein et al. (2019) Joseph M. Hellerstein, Jose M. Faleiro, Joseph Gonzalez, Johann Schleier-Smith, Vikram Sreekanti, Alexey Tumanov, and Chenggang Wu. 2019.  Serverless Computing: One Step Forward, Two Steps Back. In _CIDR_.
  * Klimovic et al. (2018) Ana Klimovic, Yawen Wang, Patrick Stuedi, Animesh Trivedi, Jonas Pfefferle, and Christos Kozyrakis. 2018.  Pocket: Elastic Ephemeral Storage for Serverless Analytics. In _OSDI_. 427–444.
  * Müller et al. (2020) Ingo Müller, Renato Marroquín, and Gustavo Alonso. 2020.  Lambada: Interactive Data Analytics on Cold Data Using Serverless Cloud Infrastructure. In _SIGMOD_.
  * Sreekanti et al. (2020) Vikram Sreekanti, Chenggang Wu, Xiayue Charles Lin, Johann Schleier-Smith, Joseph Gonzalez, Joseph M. Hellerstein, and Alexey Tumanov. 2020.  Cloudburst: Stateful Functions-as-a-Service.  _PVLDB_ (2020).
  * Wawrzoniak et al. (2024a) Michael Wawrzoniak, Rodrigo Bruno, Ana Klimovic, and Gustavo Alonso. 2024a.  Boxer: FaaSt Ephemeral Elasticity for Off-the-Shelf Cloud Applications.  arXiv:2204.5701258 [cs.DC]
  * Wawrzoniak et al. (2024b) Michael Wawrzoniak, Gianluca Moro, Rodrigo Bruno, and Gustavo Alonso. 2024b.  Off-the-shelf Data Analytics on Serverless. In _CIDR_.
  * Wawrzoniak et al. (2021) Michael Wawrzoniak, Ingo Müller, Rodrigo Bruno, and Gustavo Alonso. 2021.  Boxer: Data Analytics on Network-enabled Serverless Platforms. In _CIDR_.
  * Wawrzoniak et al. (2022) Michael Wawrzoniak, Ingo Müller, Rodrigo Bruno, Ana Klimovic, and Gustavo Alonso. 2022.  Short-lived Datacenters.  arXiv:2202.06646 [cs.DC]