Chapter 2Overview
2.1

Introduction

KEKB is an electron-positron collider with asymmetric beam energies of 3.5 GeV(positron) and 8 GeV (electron). It was operated at KEK from 1998 to 2010 andachieved the world-highest peak luminosity of 2.018 ×1034 cm−2 s−1 , which is twice
higher than the design luminosity of KEKB; 1 ×1034 cm−2 s−1 . The Belle detector
collected the integrated luminosity of more than 1 ab−1 . With this large amount of
data, the Belle experiment accomplished various important physics outcomes including
the verification of the Kobayashi-Masukawa theory as is described in the previouschapter.
SuperKEKB is the upgrade project of KEKB and aims at the peak luminosityof 8 ×1035 cm−2 s−1 , which is 40 times higher than the achievement of KEKB and theintegrated luminosity of 50 ab−1 . The Belle detector is also upgraded to Belle II and its
physics goals are to search for New Physics (NP) in the flavour sector at the luminosity
frontier as is described in the previous chapter. The configuration of SuperKEKB isshown in Fig. 2.1. The choice of machine parameters for achieving this design peakluminosity is explained in the next section.

2.2

Idea of nano beam scheme and choice of fundamental parameters

The fundamental machine parameters of SuperKEKB have been chosen based on the“nano beam scheme”, which was first proposed for the Super B factory in Italy [1].
The basic idea of the nano beam scheme is to squeeze the vertical beta function atIP to extremes by minimizing the size of an overlap region of the two beams at IP,
30

Figure 2.1: Configuration of the SuperKEKB accelerator system.

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which generally limits the minimum value of the vertical beta function at IP throughthe “hourglass eﬀect”. Fig 2.2 shows a schematic view of the beam collision, which is aplane figure, in the nano beam scheme. The size of the overlap region is not determinedby the bunch length (σz ), which is the case of the head-on collision but by the muchsmaller region d, which is considered to be the eﬀective bunch length for the nano beamscheme. The length d is determined by the half crossing angle (ϕ) and the horizontal
beam size at IP (σx∗ ) with the following equation;d∼=σx∗.
ϕThe hourglass condition in the nano beam scheme is expressed asβy∗ ≳ d,instead of that for a usual head-on collision ofβy∗ ≳ σz .
In the nano beam scheme, a relatively large horizontal crossing angle, extremely
small horizontal emittances and horizontal beta functions at IP for both beams arerequired to shorten the length d. The ratio between σz and d is equal to the Piwinski
angle defined asσz
ϕP iwinski = ∗ ϕ.
σx
The nano beam scheme is characterized by a large Piwinski angle and the angle determines to what extent βy∗ can be squeezed beyond the hourglass limit. Figure 2.3
shows the schematic view of the beam collision in the boosted frame in the horizontal
direction so that the tilted two bunches (electron and positron) on the outside of thefigure collide without the crossing angle. As for the luminosity and the beam-beam parameters, a collision of the two bunches on the inside of the figure, which are projectedbunches of the outside bunches, gives the same values as the tilted bunches. Here,the eﬀective horizontal beam size σxef f ective and the eﬀective bunch length σzef f ective aredefined asσxef f ective = σz sinϕ,andσzef f ective = σx /sinϕ.
The luminosity is expressed as
L=1Np Neef f ective ∗ fcol RL .
4π σxσy32

Figure 2.2: Schematic view of beam collision in nano beam scheme.

Here, Np , Ne , σy∗ , fcol and RL are denote the number of positrons and electrons in abunch, the vertical beam size at IP, the collision frequency and a reduction factor forthe luminosity due to the crossing angle and the hourglass eﬀect. In this formula, weassume that the horizontal and vertical beam sizes are the same for the two beams. Alsoin the formula of the beam-beam parameters, we have to use the eﬀective horizontal
beam size instead of the usual horizontal beam size. The parameter d is equal to theeﬀective bunch length σzef f ective and the Piwinski angle is written as
ϕP iwinski =σz
ef f ective .
σzThe luminosity of colliders is also expressed by the following well-known formula
assuming the flat beams and equal horizontal and vertical beam sizes for two beams
at IP;()()
γ± I± ξy±
RLL=.
∗2ere
βy±
Rξy
Here, the suﬃx ± specifies the positron (+) or the electron (−). The parametersγ, e and re are the Lorenz factor, the elementary electric charge and the electronclassical radius, respectively. These are constant or a parameter which can not betaken freely. The parameter Rξy represents a reduction factor for the vertical beambeam parameter, which arises from the crossing angle and the hourglass eﬀect. The
ratio of these parameters is usually not far from unity. Therefore, the luminosity
is mainly determined by three fundamental parameters; i.e. the total beam current(I), the vertical beam-beam parameter (ξy ) and the vertical beta function at IP (βy∗ ).
Choice of these three parameters, the beam energy and the luminosity is shown in
Table 2.1 together with those of KEKB.
As for the vertical beam-beam parameter ξy , we assume almost the same values
as that achieved in KEKB. The vertical beta functions at IP βy∗ for SuperKEKB arealmost by a factor 20 smaller than those of the KEKB owing to the adoption of thenano beam scheme. Assuming these parameters, to achieve the luminosity goal of33

Figure 2.3: Schematic view of beam collision in nano beam scheme in a boosted frame.

Table 2.1: Fundamental parameters of SuperKEKB and KEKB.

Energy (GeV) (LER/HER)
ξy (LER/HER)βy∗ (mm) (LER/HER)
I (A) (LER/HER)
Luminosity (1034 cm−2 s−1 )KEKB Achieved SuperKEKB
3.5/8.0
4.000/7.007
0.129/0.090
0.0881/0.0807
5.9/5.9
0.27/0.30
1.64/1.19
3.6/2.62.1180

SuperKEKB, 8 × 1035 cm−2 s−1 , we need to double the total beam currents comparedwith those of KEKB. The choice of the beam energy is explained below.

2.2.1

Machine parameters of SuperKEKB

Machine parameters of SuperKEKB including the three fundamental parameters areshown in Table 2.2. In the following, it is shown how these parameters are determined.

2.2.2

Emittance, crossing angle, beta functions at IP

To realize the nano beam scheme, the eﬀective bunch length d(= σx∗ /ϕ) should be small.
Of the two parameters of σx∗ and ϕ, smaller σx∗ is more important than larger ϕ, since it
becomes diﬃcult to obtain the design beam-beam parameter if we decrease d only byenlarging ϕ. In the nano beam scheme, each particle in a bunch interacts with a small
portion of the other colliding bunch. To obtain the design value of ξy , extremely small
horizontal and vertical beam sizes are needed. In the optics design of SuperKEKB,
we have made eﬀorts to minimize the horizontal emittance as much as possible witha constraint that most of magnets of KEKB will be reused. The design values of thehorizontal emittance shown in Table 2.2, which are by a factor 5 smaller than thoseof KEKB, include some enlargements due to the intra-beam scattering. Values at zero34

Table 2.2: Machine Parameters of SuperKEKB.

Beam Energy
ECircumference
CHalf Crossing Angle
ϕEmittanceεx
Emittance ratio
εy /εxBeta Function at IP
βx∗ /βy∗Horizontal Beam Sizeσx∗
Vertical Beam Sizeσy∗
Betatron tune
νx /νyMomentum Compaction
αcEnergy Spreadσε
Beam CurrentI
Number of Bunches/ringnb
Energy Loss/turnU0
Total Cavity Voltage
Vc
Harmonic number
hSynchrotron TuneνsBunch Lengthσz
Beam-Beam ParameterξyLuminosity
L

LER (e+)4.000

HER (e-)7.007
3016.315
41.5

3.2(1.9)
4.6(4.4)
0.27
0.28
32 / 0.27
25 / 0.30
1011486244.53/46.57
45.53/43.57−4
3.20 × 10
4.55 × 10−4
7.92(7.53) × 10−4 6.37(6.30) × 10−4
3.6
2.6
2500
1.76
2.43
9.4
15.0
5120
-0.0245-0.0280
6.0(4.7)
5.0(4.9)
0.08810.0807358 × 10units
GeVmmradnm%mmµm
nmAMeVMVmmcm−2 s−1*) Values in parentheses denote parameters at zero beam currents. The vertical beam
sizes include the beam-beam blowup.

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beam currents are shown in the parentheses. The horizontal beta functions at IP are
also very small compared with those of KEKB the typical value of which is 1.2 m.
Even with the very small horizontal emittances and the horizontal beta functions atIP, rather small x-y coupling of 0.27 ∼ 0.28 % is needed to obtain the design valuesof ξy . The half crossing angle ϕ is 41.5 mrad and is about 4 times larger than that ofKEKB. This choice of ϕ also contributes to decreasing the eﬀective bunch length d.
However, the design value of ϕ has been determined mainly considering the IR optics
and magnet design and the detector background. With a large crossing angle, the final
focus quadrupole magnets can be independent for the two beams which brings a great
merit of much lower detector background due to the synchrotron radiation. Anothermerit of a larger crossing angle is that the final focus quadrupole magnets can be placed
closer to the IP, which contributes to widening dynamic aperture. Dynamic apertureis one of the most serious issues of SuperKEKB in the nano beam scheme. Narrow
dynamic aperture shortens the beam lifetime from the Touschek eﬀect and the lostparticles can not be compensated by the injector if the lifetime is too short. With theparameters in Table 2.2, the eﬀective bunch length d is ∼ 0.24 mm and ∼ 0.25 mm
for LER and HER, respectively. From the viewpoint of the hourglass condition, evensmaller values of βy∗ are possible than those in Table 2.2. However, there exists anotherrestriction for βy∗ , dynamic aperture. The achievable values of βy∗ in SuperKEKB are
restricted more strictly by dynamic aperture than by the hourglass eﬀect.

2.2.3

Beam energy

In SuperKEKB, the beam energies have been changed from the KEKB values of 3.5
and 8.0 GeV to 4.0 and 7.0 GeV. This change was decided from motivations of theaccelerator design. In the nano beam scheme, the emittance growth from the intrabeam scattering and the short beam lifetime from the Touschek eﬀect are very serious
problems particularly in LER. The increase in the beam energy of LER from 3.5 to 4.0
GeV is very helpful to mitigate these problems. An impact of this change of the beamenergy asymmetry on the physics sensitivity is discussed elsewhere in this report.

2.2.4

Beam-beam parameterAs a design value of ξy , we assumed the value of 0.09 which was actually achieved inKEKB. However, this value of KEKB was achieved by using the crab cavities whichenable eﬀectively a head-on collision in the crossing angle collision system. SupeKEKB
will adopt a large crossing angle. Therefore, we need a careful study of the eﬀect of
the crossing angle on the achievable value of ξy . In the case of the crossing angle,
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there exists another kind of hourglass eﬀect. A particle with a finite horizontal oﬀset
at IP collides with (the center of) the other beam at the place where the vertical beta
function βy is larger than its minimum value βy∗ . The diﬀerence of βy from its minimum value depends on the amount of the horizontal oﬀset. This shift of the collision
point from the vertical waist position depending on the horizontal oﬀset brings anotherkind of hourglass eﬀects. This hourglass eﬀect could possibly degrade the beam-beamperformance and lower the achievable value of ξy . It is known that this eﬀect can be
avoided by using so-called the “crab waist” scheme [1]. The crab waist scheme shifts
the vertical waist position using sextupole magnets so that the vertical waist positionsof one beam are aligned along the trajectory of the other beam around IP. Beam-beamsimulations have been done based on the strong-weak mode to investigate the beambeam performance in the nano beam scheme including eﬀectiveness of the crab waist
scheme. The simulations showed that eﬀectiveness of the crab waist scheme depends
on machine parameters and with the parameters of SuperKEKB shown in Table 2.2 theluminosity improvement with the crab waist scheme is only about 10 %. The simulation also showed that the design values for ξy is achievable with the design parametersin Table 2.2 without the crab waist scheme. In the present design of SuperKEKB,
we does not employ the crab waist scheme. However, we still consider the crab waist
scheme as a backup option. A tune survey was also done in the beam-beam simulations
to find the best working point. The fractional parts of the betatron tunes shown inTable 2.2 were determined by the simulation to maximize ξy . The optimum horizontal
tune is not so near to the half integer as the case of KEKB.

2.2.5

Beam current and beam current related parameters

To achieve the target luminosity of 8 × 1035 cm−2 s−1 with the design values of βy∗ andξy in Table 2.2, the beam currents of 3.6 and 2.6 A are needed for LER and HER,respectively. These currents are about twice as high as those of KEKB. We have been
conducting R&D’s for hardware components such as the vacuum system and the RFsystem assuming even higher beam currents of 9.4 and 4.1A. These currents had been
assumed before we adopted the nano beam scheme. Feasibility of the present design
beam currents are ensured by these long-term R&D’s. The number of bunches perring (nb ) is 2500, which implies that every other RF buckets are filled with the beams.
If we decrease nb with keeping the total beam currents, which means higher bunch
currents, then we can obtain the same luminosity with a larger x-y coupling. That
will somewhat relax diﬃculty of the optics design or the x-y coupling correction in
the actual beam operation. However, the higher bunch currents bring other problems
such as diﬃculty of handling higher HOM power, single bunch instabilities like the37micro-wave instability or the emittance growth due to the intra-beam scattering. With
the design value of nb in Table 2.2, it has been confirmed that these problems aretolerable. As for the bunch length σz , a shorter bunch is preferable, since with a longer
bunch length we need to decrease the x-y coupling or increase the bunch currents tocompensate lower particle densities at the beam overlap region of the two beams at
IP. However, several bunch lengthening eﬀects, the potential-well distortion, the microwave instability and the intra-beam scattering, prevent us from achieving a short bunch
length. The calculations of σz considering these eﬀects with the wakefield including theeﬀect of CSR (Coherent Synchrotron Radiation) showed that the design values of σz in
the Table 2.2 are attainable. The RF voltages in Table 2.2 were adjusted so that thedesign value for σz is obtained at the design bunch currents with the eﬀects mentioned
above. As for the energy spread σε , we also made calculations on the eﬀect of themicro-wave instability at the design bunch currents. Although some enlargements ofσε are expected, they are still tolerable.

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Bibliography[1] P. Raimondi, 2nd SuperB Workshop, Frascati, (2006).

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