Beam-beam interaction in SuperKEKB: simulations
and experiments
Demin Zhou

Acknowledgments
K. Ohmi, Y. Zhang, Y. Ohnishi, Y. Funakoshi, K. Matsuoka, M. Koratzinos, S. Uehara,
SuperKEKB commissioning team,SuperKEKB ITF team (K. Oide, D. Shatilov, M. Zobov, T. Nakamura, T. Browder, Y. Cai, C.
Lin, A.V. Bogomyagkov, P. Raimondi, P. Kicsiny, X. Bu at, et al.)ff

The 26th KEKB Accelerator Review Committee meeting, Dec. 14, 2022, KEKOutline• Scaling laws for luminosity and beam-beam tune shifts• Comparison of simulations and experiments• Beam-beam parameters• Status of beam-beam simulations• Limitations on the current performance of SuperKEKB• Beam-beam perspective on achieving target luminosity in SuperKEKB• Summary2

Scaling laws for luminosity and beam-beam tune shifts• “Nano-beam” + crab waist in SuperKEKB- Simple scaling laws are good enough to discuss luminosity andbeam-beam parameters/tune shifts for the case of β*y = 1 mm [1].

L≈

NbN+N− f*2 + σ *22π σy+
y−

Lsp ≈

θc

2 + σ 2 tan
σz+
z−
2

e

Δ2
− *2 *22(σy+ + σy−)

σz

θc
tan
≫1
Piwinski angle: ΦP =
σ*
2
x
β*
y

θc
tan
≳1
Hourglass condition:σ*2x
Schematic view of collision schemes

1*2 + σ *2
2πe 2f σy+y−

2 + σ 2 tan
σz+
z−
2

1 γ±I± LL=ξy±2ere β*y±Beam-beamtune shift [3]

r
e
i
ξy+ ≈
2πγ+

θc

SuperKEKB (2021c)
Beam-beam parameter [2]

N−β*
y+
σ*
y−

θc

2 tan2
*2
σz−
+
σ
x−
2

SuperKEKB (Final design)[1] See backup slides for validations (p.30-42). [2] K. Ohmi et al., PRST-AB 7, 104401 (2004). [3] BB parameter and tune shift refer to different terms in this talk.

3

Achieved luminosity record• The luminosity record 4.71 × 1034

cm−2s−1 was achieved with Belle II HV OFF and injection

stopped (Jun. 22, 2022).

Total currents: I+ /I−=1.4/1.14 A
Total (Yellow) and speci c luminosity (Green) by ECL
Instantaneous total luminosity by ZDLM (Yellow) andECL (Green)e- emittance by XRM: ~50 pm
e- beam BB parameter: ~0.028

e+ emittance by XRM: ~60 pm
e+ beam BB parameter: ~0.04

fi

4Comparison of simulations and experiments

Speci c luminosity

2021• HBCC (High Bunch Current Collision) machine studies
with β*y = 1 mm in 2021 and 2022:•

HBCC machine studies were done to extract the luminosity
performance.

•

Lsp (speci c luminosity) slope vs. product of beam currents
improved in 2022 but still drops quickly due to vertical blowup.
2021.12.21

Ibunch (mA)

HER

LER

HER

LER

Ie

1.25*Ie

Ie

1.25*Ie393# bunch393

Comments

Assumed value

εx (nm)

4.6

4.0

4.6

4.0

w/ IBSεy (pm)

35

20

30

35

Estimated from XRM dataβx (mm)

60

80

60

80

Calculated from latticeβy (mm)

1

1

1

1

Calculated from lattice

σz0 (mm)

5.05

4.60

5.05

4.60

Natural bunch length (w/o MWI)

νx

45.53

44.52445.53244.524

Measured tune of pilot bunchνy43.57246.58943.57246.589Measured tune of pilot bunchνs0.02720.02330.02720.0233Calculated from lattice

Crab waist

40%

80%

40%

80%

Lattice designfi

fi

2022.04.05

2022

5

Comparison of simulations and experiments

KEKB, 2010 [1]• The “Lsp puzzle” (large discrepancy between beambeam simulations and experiments)•••

The “Lsp puzzle” appeared in KEKB.
The “Lsp puzzle” re-appeared in SuperKEKB.
Was there such a “Lsp puzzle” in PEP-II?

PEP-II [3]PEP-II [2]SuperKEKB, 2022

[1] Y. Funakoshi, KEKB MAC 2010; [2] Y. Cai, KEKB MAC 2006; [3] U. Wienands, PAC07.

6

Comparison of simulations and experiments• HBCC machine studies with β*y = 1 mm in 2021 and 2022:• After ne-tuning of BxB FB system in 2022, the observed vertical beam sizes blowup became much more “normal” (abreakthrough in 2022) and closer to simulations. The origin of vertical blowup remains to be explained.
2021
In 2021, there was a“ ip- op” correlation
between σ*y−y+ and σ*Electron σ*ySpeci c luminosity

Positron σ*y2022
In 2022, this “ ip- op”
problem was cured
after ne-tuning of FB
system

fl

fl

fi

fi

fi

fl

fl

7

Comparison of simulations and experiments• HBCC machine studies with β*y = 1 mm in 2021 and 2022:• Weak blowup of horizontal beam size: qualitative agreements between simulations and experiments• Horizontal blowup is sensitive to horizontal tune (see p.48 for simulation results)
2021
Measured data
might havesystematic offsets

Electron σ*xSpeci c luminosity

Positron σ*
x

2022

fi

8

Comparison of simulations and experiments• Filling the gap between simulated andmeasured Lsp• BBSS+PIC simulation showed 5% less Lsp at
Ib+Ib− = 0.8 mA2 [see p.12].
• Impedance e ects:- Simulations showed less bunch lengthening thanExperience ofphysics runs
2018 - 2022

No experience ofphysics runs with high currents

measurements. If measured bunch lengthening isapplied, it gives ~10% extra loss of Lsp at Ib+Ib−=0.8 mA2.

- “-1 mode instability” due to the interplay of FB••

and vertical impedance [see K. Ohmi’s talk].
Found a large systematic in ECL luminosity at highinjection background. This could explain a ~10%
di erence between simulation and measured data at
2
Ib+Ib−=0.3 mA . There remains a di erence of ~10%[see p.20-24]. No physics data was taken at highbunch currents, and this systematic’s impact is
unknown.
The machine conditions for HBCC experiments were
not optimal due to the limited beam time for machinestudies.

ff

ff

ff

9

Beam-beam parameters• Overview of beam-beam parameters with crab waist [1]• The achieved beam-beam parameters during the physics run of SuperKEKB (i.e., the high voltage of Belle II
was on.) in 2022 were 0.0407/0.0279 in LER/HER (γ+Ib+ ≠ γ−Ib−, β*y =1 mm).

￼• In 2022, 0.0565/0.0434 were achieved in LER/HER during HBCC machine studies (β*y =1 mm).
• There was no clear evidence showing SuperKEKB had already reached the beam-beam limit.

1 γ±I± LL=ξy±2ere β*y±[1] Y. Funakoshi, “The SuperKEKB Has Broken the World Record of the Luminosity”, IPAC’22.

∂Fy+
1
ih
ξy+ =
dsβy+(s)4πp0c ∫−∞
∂y′
∞

10

Status of beam-beam simulations• Beam-beam (BB) simulations-

Available tools: BBWS (weak-strong BB model + simple one-turn map + perturbation maps); BBSS and IBB (strong-strong BB model + simple
one-turn map + perturbation maps); SAD (BBWS’s BB model + complete lattice + perturbation maps).

-

SuperKEKB is challenging the predictability of BB simulations: It requires reliable models of multiple physics (BB, impedances, latticenonlinearity, crab waist, realistic machine errors, space charge, etc.), not only BB.

L≈

Nb N+N− f*2 + σ *22π σy+y−θc

2 + σ 2 tan
σz+
z−
2

e

Δ2
− *2 *22(σy+ + σy−)This schematic plot shows my private viewpoint
11

Status of beam-beam simulations• Beam-beam (BB) simulations-

Tools under development: SCTR-CUDA (K. Ohmi, BBSS + complete
lattice + GPU acceleration); APES (Z. Li, Y. Zhang (IHEP), IBB +complete lattice + GPU acceleration); XSUITE (P. Kicsiny, X. Bu at(EPFL/CERN)).

-

The ultimate goal: PIC SS BB model + complete lattice + GPU
acceleration (CPU-based parallel computing is not optimal for BB
simulations).

-

Progress has been achieved in developing GPU-based BB codes.
Preliminary tests showed a speed-up factor of ~25 for PIC BB
simulations based on the CUDA compiler (K. Ohmi, in collaboration with
Z. Li and Y. Zhang (IHEP), T. Yasui (J-PARC)).

“Vertical blowup”“Longitudinal blowup”

Simulation example with machineparameters of Dec. 21, 2021

• International collaboration-

SuperKEKB, CEPC, and FCC-ee teams are working in the same
direction: SS BB + multiple physics (US: BNL+SLAC are interested injoining).
We invite full international collaboration on beam-beam simulation andrelated physics.

4 months for 6000 turns of tracking
using 8 cores of 3-GHz workstation

12
ff

-

Limitations on current performance of SuperKEKB• From the beam-beam perspective, we list some important issues:- Issue 1: Limits on bunch currents (see talks by Y. Ohnishi, H. Nakayama, H. Ikeda, and T. Ishibashi)- Issue 2: Multi-bunch e ects
- Issue 3: Optics distortion at high beam currents (see talks by Y. Ohnishi and H. Sugimoto)- Issue 4: Impedance e ects (see K. Ohmi’s talk)- Issue 5: Lsp injection correlation (see K. Matsuoka’s talk)#1,2,3,4,5
#5

L≈

#2,5

NbN+N− f*2 + σ *22π σy+
y−#1,2,3,4,5

θc

2 + σ 2 tan
σz+
z−
2
#4

e

Δ2
− *2 *22(σy+ + σy−)BB, CW, …

ff

ff

13

Issue-1: Limit on bunch currents by Sudden Beam Losses (SBLs)• Severe machine failures occurred at high beam currents when Ib+ > 0.7 mA/bunch• Bunch current Ib+ ≲ 0.7 mA (keeping Ib− /Ib+ = 0.8) was respected in 2022ab run [1]L≈

NbN+N− f2π*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)

For SBLs, see H. Ikeda’s talk.

Courtesy of K. Matsuoka[1] K. Matsuoka, “Belle II Report”, SuperKEKB 2022ab summary meeting, https://kds.kek.jp/event/42954/.

14

L≈

Issue-2: Multi-bunch effects• No clear evidence of Lsp degradation due to multi-bunch e ects

NbN+N− f2π*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)

- Coupled-bunch instabilities were suppressed by the BxB FB system (M. Tobiyama).
- Flat BxB luminosity was observed (S. Uehara).
- Electron-cloud instability for e+ beam was not observed (Y. Suetsugu et al., see K. Shibata’s talk).
Electron σ*yPhysics run with β*y =1 mm
IP knobs were routinely
optimized to achieve thebest luminosity performancearound Ib+Ib− ≈ 0.3 mA2.

Positron σ*
y

ff

15

Issue-3: Optics distortion at high beam currents

L≈• Current-dependent optics distortion- Beta-beat and global coupling become worse at high currents.
- An unexpected β*y squeeze explains the Lsp gain (see Y. Ohnishi’s talk).

NbN+N− f*2 + σ *22π σy+
y−

r
e
i
ξy+ ≈
2πγ+

θc

2 + σ 2 tan
σz+
z−
2

e

Δ2
− *2 *22(σy+ + σy−)

N−β*
y+
σ*
y−

θc

2 tan2
*2
σz−
+
σ
x−
2

Courtesy of Y. Ohnishi
16Issue-3: Optics distortion at high beam currents

r
e
i
ξy+ ≈
2πγ+

• Current-dependent orbit o sets at SLY* magnets (see H. Sugimoto’s talk)

N−β*
y+
σ*
y−

θc

2 tan2
*2
σz−
+
σ
x−
2

Observed offsets with beam
The following horizontal offsets of SLY*
magnets are used to estimate the beta beats:

Resulting beta-beat(SAD simulation)β*y squeezedff

Courtesy of H. Koiso

17

Issue-4: Impedance effects (LER)

L≈• Current-dependent single-beam blowup in LER

NbN+N− f2π*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)- This problem was solved by ne-tuning the FB system in Mar. 2022. After new damage to collimators-(D06V1 and D02V1), the LER beam blowup problem re-appeared.
On Jun. 21, 2022, tunings were done to improve the blowup threshold (from 0.5 mA/bunch to ~0.87
34mA/bunch). This contributed to achieving the luminosity record 4.71 × 10 cm−2s−1 on Jun. 22, 2022.
KCG shift report on LER vertical blowup study
By S. Terui, T. Ishibashi, K. Yoshihara, M. Nishiwaki
Jun. 21, 2022

LER vertical tune:
νy .582 → .595;

I+ (mA)

Threshold <0.5mA/bunchϵy+ (pm)ϵy+ (pm)

Machine conditions:Single-beam, 393 bunches

D06V1 gap:
TOP 4.2 mm,
BTM -5.1 → -8.0 mm.

I+ (mA)

After tunings ~0.87mA/bunchfi18Issue-4: Impedance effects (HER)

L≈• Current-dependent single-beam vertical emittance in HER

NbN+N− f2π*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)- No clear evidence of single-beam blowup (up to 0.64 mA/bunch) in HER

Machine conditions:Single-beam, 393 bunches

KCG shift report on high bunch-current collision study
By D. Zhou, R. Ueki, M. Nishiwaki
Jun. 21, 2022

19

Issue-5: Lsp-Injection correlation• Luminosity record of 4.65 × 10

34

L≈

NbN+N− f*2 + σ *22π σy+y−cm−2s−1 was achieved with BelleII HV ON when the injection was intentionally stopped (Jun. 8, 2022).

θc

2 + σ 2 tan
σz+
z−
2

e

Δ2
− *2 *22(σy+ + σy−)Yellow: Total luminosity ECL (20-second average)
Green: Speci c luminosity by ECL (Lsp)
Yellow: Total luminosity ZDLM
Green: Total luminosity ECL (updated per 2.5 second)* Peak luminosity always appears
after injection stopped* Lsp always jump up after injection
stoppedfi

20

Issue-5: Lsp-Injection correlation

Lsp ≈• The phenomenon: 2022-06-02 21:05 PM1
2πe 2f*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)- All luminosity PVs gave a similar jump response to injection stop/start.
- Lsp ⋅*2*2
σy+ + σy− still shows jump-response. It means there is a geometric loss of luminosity.

Blue: B2_nsm:get:ECL_LUM_MON:lum_acc_corrected
Red: B2_nsm:get:ECL_LUM_MON:lum_acc_20
Green: B2_nsm:get:MONZDLMINT:ZDLM_INTVAL:value
Black: CG_OPR:Speci cLuminosity

Lsp degradation by ~10%, independent to vertical emittances

CG_OPR:Speci cLuminosity * Cap-Sigmay
LER injection
OFF
ON

fi

fi

21

Issue-5: Lsp-Injection correlation

Lsp ≈

1
2πe 2f*2 + σ *2
σy+y−• Injection background a ected ECL luminosity [1]• Data of Jun. 2022: Injection background contributed to ~5% luminosity “loss”

θc2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)

Lsp degradation by ~10%

Courtesy of K. Matsuoka

ff[1] K. Matsuoka, Oct. 14, 2022, https://kds.kek.jp/event/44070/.

22

Lsp ≈

Issue-5: Lsp-Injection correlation1
2πe 2f*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)• Injection background on ECL was identi ed [1]• Data of Jun. 2022: LER injection kicker (leakage elds) contributed to ~3% of luminosity loss
About 20% of the stored bunches are excited by
the leakage elds of injection kickers in LER

One turn
Two trains of bunches

Courtesy of K. Matsuoka
23
fi

fi

fi

[1] K. Matsuoka, Oct. 14, 2022, https://kds.kek.jp/event/44070/.

Lsp ≈

Issue-5: Lsp-Injection correlation• Lsp with β*y = 1 mm in 2021 and 2022:1
2πe 2f*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2eΔ2− *2 *22(σy+ + σy−)• The fast drop of measured Lsp vs. Ib+Ib−: ECL data of the physics run shows a slightly faster drop than ZDLM data.
• It is consistent with K. Matsuoka’s analysis: Injection background a ects ECL luminosity more than ZDLM.
2021

2022

ff

ZDLM gives relative luminosity, while ECL gives absolute luminosity.

24

Beam-beam perspective on achieving target luminosity
35• Achieving 10 cm−2s−1: SBLs, “-1 mode instability”, etc. → Non-Linear Collimator (NLC)35• Achieving 6 × 10 cm−2s−1: DA (Dynamic aperture), lifetime, perfect CW, etc. → IR model(better understanding of the current IR) and upgrade (“Clean IR”)

Luminosity measurement:
1) Fake luminosity loss in ECLThis is not a problem at all

1 γ±I± LL=ξy±2ere β*y±IR optics:
We achieved β*y = 1 mmIf we can achieve β*
y = 0.3 mm, we will gain by 3.3
Obstacles:
1) DA and lifetime resulted from IR nonlinearity (+BB+CW)
2) Optics tuning at high currents

Total beam currents:
We achieved 1.4 A in LER (Jun. 2022)
If we can achieve 3.6 A, we will gain by 2.5
Obstacles:1) Sudden beam losses (SBLs)
2) Short lifetime (challenging injection power)

Beam-beam limit:
We achieved 0.04 in Jun. 2022
We expect the upper limit is ~0.1 (including the hourglass
effect), then we will gain by 2.5
Obstacles:
1) Vertical blowup by “-1 mode instability” (NLC is thehoped solution)2) Vertical blowup by BB (+Lattice nonlinearity+Impedance)3) Imperfect crab waist (to be veri ed)

fi25

Beam-beam perspective on achieving target luminosity

Crab waist

Courtesy of K. Oide

[1] K. Oide, https://kds.kek.jp/event/44644/.

“Clean IR”“Clean IR”: A transparent IR with minimal amplitude-dependent and chromatic nonlinearities

26

Beam-beam perspective on achieving target luminosity• How to achieve a “clean IR”- IR remodeling (the mainstream upgrade plan (see M. Masuzawa’s talk) under investigation)- Using CCT (Canted Cosine Theta) magnets: M. Koratzinos did the rst exercise (considering constraints from thetechnology and infrastructure of SuperKEKB) and showed encouraging results. Using the CCT magnets, a compact
and cleaner IR is conceivable (Idea: “The current distribution of any canted layer generates a pure harmonic eld aswell as a solenoid that can be canceled with a similar but oppositely canted layer.” [2]).

-Courtesy of M. Koratzinos
From the beam-beam perspective, we invite full international collaboration on IR upgrades to achieve the target
luminosity of SuperKEKB.

fi

fi[1] M. Koratzinos, https://kds.kek.jp/event/44644/. [2] S. Caspi et al., “Canted-Cosine-Theta magnet (CCT)-A concept for high field accelerator magnets” , IEEE Trans. Appl. Supercond. 24, 1. (2014).

27

Summary• With progress in machine tuning, the measured luminosity of SuperKEKB is approaching

predictions of BB simulations (SS BB + Simple lattice model + Impedance models).
• Prediction of luminosity via beam-beam simulations requires reliable models of multiple dynamics,
such as the beam-beam interaction, machine imperfections, impedance models, etc.
• Several sources of luminosity degradation in the current SuperKEKB have been well identi ed.
• Many subjects will be investigated via experiments (after LS1) and simulations.

• From the beam-beam perspective, with β*
y =0.3 mm, a signi cant IR upgrade is required toachieve the target luminosity in SuperKEKB (after LS2).
• We invite full international collaboration on beam-beam simulations and an IR upgrade R&D
program.

fi

fi

28

Backup29

Luminosity
Sketch of nano-beam collision (SuperKEKB)With hourglass, without crab waist• Luminosity [1]:L = N+N− fcK d xds⃗ 0 ρ+(x,⃗ − s0)ρ−(x,⃗ s0)∫
3• 3D Gaussian distribution:(s − s0)2y2
x2− 2 − 2−2σx (s) 2σy (s, x)2σ2zeρ(x, y, s, s0) =(2π)3/2σx(s)σy(s, x)σz

Overlap region is essential for luminosityand beam-beam parameters

Tailing of charge density• Hourglass e ect:*2
βx,y(s) = β*x,y (1 + s /βx,y)22*2
1 + s /βx,yσx,y(s) = σ*x,ySketch of nano-beam collision (SuperKEKB)
Without hourglass• Crab waist:*2σy(s, x) = σ*
1
+
s
+
R
x/tan
θ
/β
(
y
CW
c)
y
2

ff

[1] W. Herr and B. Muratori, Concept of luminosity, CAS lecture note (2006).

30

Luminosity• RC is de ned as the reduction factor from the• Luminosity:
L=
L0 =NbIb+Ib−RHC22πe f0Σ*x Σ*
ycrossing angle• The reduction factor from the hourglass e ect
can be de ned as= L0RHC

RH = RHC /RC

NbN+N− fc*2 + σ *22π σy+y−

L = L0RH RC = L′0RH*2 + σ *2
σx+
x−

*2
*2
σu+ + σu−Σ*u =• RHC is the geometric reduction factor including
e ects of crossing angle and hourglass• If no hourglass e ect, there is1

RHC = RC =

θc

2
tan
*2
2

*2 + σ
σx+
x−

31
ff

ff

fi

fi

ff

￼

1+

2 + σ2σz+z−

Sketch of nano-beam collision (SuperKEKB)
Without hourglass

Luminosity• Assume no crab waist and at beams(σ*
y ≪ σ*x ); a very good approximation [1]:
RHC ≈
a=

Σ*
=
β

Σ*
y

2 b
ae K0(b)πΣ2zθcb = a 1 + *2 tan
Σx
2)
(
2

ΣzΣ*
β

*2 *2
*2 *2
σy+ /βy+ + σy− /βy−

Σz =

2

• With the crossing angle factor RC, we cancalculate the hourglass factor RH = RHC /RC• Numerical tests showed great agreements
between analytic formulae and BBSS (using
SuperKEKB baseline design parameters)• Luminosity gain from CW is less than 5%22
σz+ + σz−

• With a full crab waist, at beams, and a large

Luminosity gain from CW

Piwinski angle, I derived the analytic formula:
θc

CW
RHC
≈

f(d) =

Σ*x Σz tan 2θc22Σz tan 2 + σ*
x+σ*
x−

d=

d2

πd ⋅ e Erfc(d)

f(d)

Σ*
y Σ*
x

2Σ*
x+σ*
x−
β σ*

sin θcfl

fl[1] K. Hirata, Analysis of beam-beam interactions with a large crossing angle, Physical review letters 74, 2228 (1995).

Baseline

2022ab32

Luminosity• Simple luminosity formula for “nano-beamscheme”:• With large Piwinski angle, the formula for
speci c luminosity is simple:

L
Lsp =NbN+N−(ef )2

L ≈ L′0 = L0RC
L0 =

NbN+N− fc*2 + σ *22π σy+
y−

RC =

Lsp ≈

*2 + σ *2
σx+x−1
2πe 2f*2 + σ *2
σy+
y−

θc
2
2
σz+ + σz− tan 2

1
1+

2 + σ2
σz+
z−

θc

2
tan
*2
2

*2 + σσx+x−• The hourglass factor L/L′0 = RH is the order of

10% for the SuperKEKB baseline design [1] andabout 2% for β*y = 1 mm (2022ab run)

• Conclusion: The simple formula L′0 is fairly good￼￼[1] SuperKEKB TDR, https://kds.kek.jp/event/15914/.
fi

￼for discussions on the scaling laws of luminosity
in SuperKEKB
33

Beam-beam parameters• The beam-beam parameters (=incoherent BB tune shifts) can be calculated from theelectromagnetic elds of 3D Gaussian beam:∞eN−γ−xEx−(x, y, z, t) =dw
2ϵ0π 3/2 ∫0e

2 (s) + w
2σ
( x−
)

∞

eN−γ−yEy−(x, y, z, t) =dw
2ϵ0π 3/2 ∫01
Bx− = − Ey−
c
∞

￼

￼

∂F
1
x+
ih
ξx+ =
dsβx+(s)4πp0c ∫−∞∂x′γ 2−(z − s)2
y2
x2
− 2
− 2
− 2 2
2σx−(s) + w 2σy−(s) + w 2γ−σz− + w

e

3/2

2 (s) + w
2σ
( y−
)

1/2

2 σ2 + w
2γ( − z−)

1/2

γ 2−(z − s)2
y2
x2
− 2
− 2
− 2 2
2σx−(s) + w 2σy−(s) + w 2γ−σz− + w

2 (s) + w
2σ
( x−
)

1/2

2 (s) + w
2σ
( y−
)

3/2

2 σ2 + w
2γ( − z−)

1/2

1
By− = Ex−
c
∂Fy+
1
ih
ξy+ =
dsβy+(s)4πp0c ∫−∞
∂y′
∞

fi

34

Beam-beam parameters• If no hourglass e ect, exact analytic formulae• Explicit formulae exist [1]:
ih
ξx+
=

ih
ξy+
=

Λ+

∞

π ∫0

Λ+

∞

π ∫0

dw

dw

∞

∫−∞
∞

∫−∞

ds

ds

γ− (1 + cos θc) βx+(s)gx+(s)e
2 (s) + w
2σ
( x−
)

3/2

2 (s) + w
2σ
( y−
)

γ− (1 + cos θc) βy+(s)e
2 (s) + w
(2σx−
)

1/2

2
2
γ2
s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

1/2

2 σ2 + w
2γ( − z−)

can be derived:
i
ξx+
=

1/2

2
2
γ2
s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

2 (s) + w
2σ
( y−
)

3/2

2 +w
(2γ−2 σz−)

gx+(s) = cos θc + 2s2sin θc2[2 +w
2γ−2 σz−*2 1 + ϕ 2 α
σx−
−
+

i
ξy+ =

1/2

reN−
Λ+ =
2πγ+
γ−2 (1 + cos θc)

Λ+β*
x+

σ*
x−σ*
y−

σz−

θc
ϕ− =
tan
σ*
2
x−
cos θc− 22σx−(s) + w ]

κ− =

σ*
y−

θ
1
2 c
α+ = 1 + 2 tan
γ+
2

σx− =

Λ+β*
y+
1 + ϕ−2 + κ− α+)Piwinski angle

θc2

Lorentz factors
Projected horizontal beam sizes[1] A. Valishev, Practical Beam-Beam Tune Shift Formulae for Simulation Cross-Check, Tech. Rep. (Fermi National Accelerator Lab.(FNAL), Batavia, IL (United States), 2013).
ff)

Flat beams: κ± ≪ 1

σ*
x−

*2
2
σx−
+ σz−
tan2

(

(

1 + ϕ−2 + κ− α+35Beam-beam parameters• With hourglass e ect, it is di cult to nd theihih
analytic solutions of ξx+ and ξy+:
ih
ξx+
=

ih
ξy+
=

Λ+

∞

π ∫0

Λ+

∞

π ∫0

dw

dw

∞

∫−∞
∞

∫−∞

ds

ds

γ− (1 + cos θc) βx+(s)gx+(s)e
2 (s) + w
(2σx−
)

3/2

2 (s) + w
(2σx−
)

1/2

1/2

2 +w
(2γ−2 σz−
)

1/2

2
2
γ2
s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

2 (s) + w
2σ
y−
(
)

3/2

2 +w
(2γ−2 σz−)

1/2

• For at beams and β*x ≫ σz (this is the case ofSuperKEKB), approximate formulae can befound:
re N−β*x+2πγ+ σ2x−• Rξu± can be calculated via numericalintegrations. But we must be careful:convergence issues appear, especially for theSuperKEKB case. Inconsistency might appear
when we use Mathematica, SAD, BBWS
(Numerical evaluation of complex error
function), etc.

2 r2
1
2
−
r−e K0(r−) +U,0,2r−2(2)*2 r
π2 2βy+−2 θc
β*y−σx− cos 2

Important parameter for evaluation of hourglass effect

σz−σ*
x−

36
fi

ff

fl

r− =

ih
i
Rξu± = ξu± /ξu±

*2
βy−

fi

β*
r
N
y+
e
−
ih
ξy+ ≈
2πγ+ σx−σ*
y−

ffi

ih
i
ξx+
≈ ξx+
=

parameters can be de ned as:22γ2s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

2 (s) + w
2σ
y−
(
)

γ− (1 + cos θc) βy+(s)e• The hourglass factor for beam-beamBeam-beam parameters• For the hourglass factor of luminosity, there
always is RH < 1• For the hourglass factors of BB parameters,ih
ih
there can be ξy+ < 1, or ξy+ > 1:
ih
ξy+
=

Λ+

∞

π ∫0

dw

∞

∫−∞

ds

γ− (1 + cos θc) βy+(s)e
2 (s) + w
(2σx−
)

1/2

2
2
γ2
s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

2 (s) + w
2σ
y−
(
)

3/2

2 +w
(2γ−2 σz−
)

•

ih i
The hourglass Rξy± = ξy± /ξy± is the order of 8% for the•Conclusion: The simple formula ξyi ± is fairly good for
discussions on the scaling laws of beam-beam parameters inSuperKEKBSuperKEKB baseline design and about 1% for β*y = 1 mm(2022ab run)

1/2

Baseline

2022ab37

Relation of luminosity and beam-beam parameters• The luminosity L and beam-beam parametersih
ξy± can be correlated:• If we ignore both asymmetries of the colliding

1 γ±I± ih 2σ*y±σx± RHL=ξy±
2ere β*ΣyΣx Rξy±y±

1 γ±I± LL=ξy±2ere β*y±Beam-beam parameter• If we use “measured” beam sizes at the IP to

Hourglass factor

Asymmetry factor• There are three “methods” for calculating thebeam-beam parameters from luminosity.
• If the beta functions and beam sizes at the IP
are well known, this is the standard method:
ih
ξy+
=

Λ+

∞

π ∫0

dw

∞

∫−∞

ds

γ− (1 + cos θc) βy+(s)e
2 (s) + w
(2σx−
)

ff

fl

[1] K. Ohmi et al., PRST-AB 7, 104401 (2004).

1/2

beams and hourglass e ects, this is Ohmimethod [1]:estimate the hourglass factor and use it as acalibration factor, this is the KEKB method [AsOide-san proposed](Assumed: equal beam
parameters and at beams) [2]:22γ2s2 sin2 θc−s (1 + cos θc)− 2−2σx−(s) + w
2γ2−σ 2
z− + w

2 (s) + w
2σ
y−
(
)

3/2

2 +w
(2γ−2 σz−)

1/2[2] KEKB B-Factory Design Report, KEK Report 95-7 (1995).

38

Relation of luminosity and beam-beam parameters• Note that my notation has certain di erences with the formulations in KEKB design report:
Crossing angle factors cancel
each other with symmetric beams

RL = RHC = RH RC = RH

Rξy(KEKB) = Rξy±(DZ)1
1+

2 + σ2
σz+
z−

θc

2
tan
*2
2

*2 + σ
σx+
x−

1
1+

σz2±

θc

2
tan
*2
2

σx±

ff

39

Relation of luminosity and beam-beam parameters• For the SuperKEKB baseline design, thehourglass factor RL /Rξy is 0.8.
• For the KEKB case, RL /Rξy was about 0.7 [1].
• For the 2022ab run (β*y = 1 mm) of
SuperKEKB, RL /Rξy is very close to 1. It should
be safe to use simple formulae (w/o hourglass)for luminosity and beam-beam parameters.

[1] Y. Funakoshi, Achievements of KEKB, Prog. Theor. Exp. Phys. (2013) 03A001.

40

Applications to SuperKEKB• Cap sigma Σ*y (SAD panels for SuperKEKB
operation)L≈

NbN+N− f2π*2*2σy+ + σy−θc22
σz+ + σz− tan 2(5)When machine conditions are good,the cap sigma Σ*y estimated fromluminosity and XRM data agree witheach other.

Data from XRMs and SRMs

41

Applications to SuperKEKB• Beam-beam parameters

Two ways to calculate beam-beam parameters:
Using XRM data and using luminosityreξyi+ ≈
2πγ+

N−β*
y+
σ*
y−

(

2 tan2*2 + σ*
σz−
+
σ
x−
y−
2
)
θc

1 γ±I±
L=ξy±
2ere β*y±When machine conditions are good,the beam-beam parameters estimatedfrom luminosity and XRM data agree witheach other.
42

Luminosity and beam-beam tune shifts• “Nano-beam scheme” for SuperKEKB- Beam-beam-driven footprint in tune space is

LER
Red: 2022.04.05, w/ CW
Blue: 2019.07.01, w/o CW
Notes:* Hourglass effect ignored in
calculation of BB footprint* Resonances mνx ± nνy = N
not plotted* Collective effects dynamicallyshift the resonances
HER
Red: 2022.04.05, w/ CW
Blue: 2019.07.01, w/o CW

43
ff

-

useful for understanding beam-beam e ects.
The choice of working point dynamically depends
on machine conditions.

Status of beam-beam simulations• Weak-strong model + simple one-turn map: BBWS code [1]- The weak beam is represented by N macro-particles (statistical errors ~ 1/ N ). The strong beam
has a rigid charge distribution with its EM fields expressed by the Bassetti-Erskine formula.

- The simple one-turn map contains lattice transformation (Tunes, alpha functions, beta functions, X-Y
couplings, dispersions, etc.), chromatic perturbation, synchrotron radiation damping, quantum
excitation, crab waist, etc.

• Weak-strong model + full lattice: SAD code
- The BBWS code was implemented into SAD as a type of BEAMBEAM element, where the beambeam map is called during particle tracking.

- Tracking using SAD: 1) Symplectic maps for elements of BEND, QUAD, MULT, CAVI, etc. 2) Element-;
BEAMBEAM

by-element SR damping/excitation; 3) Distributed weak-strong space-charge; 4) MAP element forarbitrary perturbation maps (such as crab waist, wakefields, artificial SR damping/excitation, etc.); …

• Strong-strong model + simple one-turn map: BBSS code [1]BMBMP =(NP=3.63776D10
BETAX=0.06 BETAY=0.001
EX=0.D0 EY=0.D0
EMIX=4.6D-9 EMIY=40.D-12
SIGZ=6.D-3 DP=6.30427D-4
ALPHAX=0.D0 ALPHAY=0.D0
DX=0.E-6 DZ=0.0
SLICE=200.D0 XANGLE=41.5D-3
STURN=1000);- Both beams are represented by N macro-particles- The one-turn map is the same as weak-strong code. The Beamstrahlung model is also available.
Choices of numerical techniques: PIC, Gaussian fitting for each slice, …

- For SuperKEKB, it is hard to include lattice.
• GPU-powered strong-strong model + full lattice: SCTR code
- Under development (K. Ohmi)- KEK/IHEP/J-PARC collaboration[1] K. Ohmi, Talk presented at the 2019 SAD workshop, https://conference-indico.kek.jp/event/75/.

44

Status of beam-beam simulations• Beam-beam simulations have shown that multiple factors can strongly interplay with beambeam interaction.
- Imperfections in linear optics: beta beat, linear couplings, dispersions, etc. at the IP- Geometric nonlinearities: It is crucial when β*y < 1 mm- Coupling impedances: Longitudinal and transverse- Space charge- BxB feedback• Predictability of beam-beam simulations: The case of SuperKEKB sets demands on- Accurate modeling of linear optics- Strong-strong model of beam-beam interaction- X-Z instability (i.e. Beam-beam head-tail instability)- Synchro-betatron resonances with working points near half integers- Reliable impedance modeling- Longitudinal impedance: potential-well distortion and synchrotron tune spread- Transverse impedance: Betatron tune shift and spread- Monopolar (longitudinal potential-well distortion and transverse beam tilt), dipole (TMCI), and quadrupolar (tuneshift)

45

Status of beam-beam simulations• Weak-strong model + simple one-turn map: BBWS code
- Pros: Fast simulation of luminosity and beam-beam effects. Not require muchcomputing resources. Used for tune survey, fast luminosity calculation, etc..

- Cons: Strong beam frozen. Crab waist of strong beam not implemented. Not sensitive tocoherent beam-beam head-tail (BBHT) instability (BBHTI).

• Weak-strong model + full lattice: SAD code
- Pros: Relatively fast to allow tracking with lattice. Interplay of beam-beam and latticenonlinearity. Space-charge modeling possible. Localized geometric wakes possible.

- Cons: Same as BBWS code. Tune survey possible but relatively slow.
• Strong-strong model + simple one-turn map: BBSS code
- Pros: Allow dynamic evolution of 3D distribution of two beams. Detect BBHTI.
- Cons: Tracking quite slow. Not feasible for tune survey. No effective method ofparallelization.

46

Status of beam-beam simulations• BBSS simulations: PIC vs. Gaussian tting model-“Vertical blowup”

PIC method predicts lower luminosity (~5%).

“Longitudinal blowup”

Using workstations(8 cores), one PIC simulation requires ~8 months, and a Gaussian- tting simulation takes ~1.2 days.
Signi cant progress has been achieved recently in developing GPU-based BB codes. Preliminary tests showed a speed-up factor of ~50 for PIC
simulations based on the CUDA compiler (K. Ohmi, in collaboration with Y. Zhang and Z. Li (IHEP), T. Yasui (J-PARC)).

-

This will speed up our investigations, especially of the interplay between beam-beam and machine imperfections.
2021.12.21
HER
LER

Ibunch (mA)# bunchεx (nm)0.8

Comments

1.0-

σz0 (mm)

4.6
35
60
1
5.05

4.0
20
80
1
4.60

w/ IBS
Estimated from XRM data
Calculated from latticeCalculated from lattice
Natural bunch length (w/o MWI)

νx

45.53

44.524

Measured tune of pilot bunchνy43.57246.589Measured tune of pilot bunchνs0.02720.0233Calculated from lattice

Crab waist

40%

80%

Lattice design

εy (pm)βx (mm)βy (mm)4 months for 6000 turns of tracking
using 8 cores of 3-GHz workstation

fi

fi

fi

47

Status of beam-beam simulations• Scan LER νx (with LER νy and HER νx,y xed as the values of the parameter table of 2021.12.21)
Coupling impedances included
Weak horizontal blowup when 0.5 + νs < [νx] < 0.5 + 1.5νsElectron σ*xElectron σ*yPositron σ*x

Positron σ*y

X-Z instability is sensitive to νx.

48
fi

-

Status of beam-beam simulations• BBSS simulations: Scan LER νy with bunch currents varied (with LER νx and HER νx,y xed as thevalues of the parameter table of 2021.12.21, BB+Wxy+Wz)

Electron σ*yPositron σ*x

Positron σ*
y

49
fi

* The interplay of BB+Wx,y+Wz causes instability,
consistent with Y. Zhang and K. Ohmi’s findings.
* This instability has a threshold that is νy-dependent.

Electron σ*xCrab waist applied to SuperKEKB• SuperKEKB nal design (β*y = 0.3/0.27 mm) with ideal crab waist• Tune scans using BBWS
• Crab waist creates large area in tune space for choice of working pointfi

50Crab waist applied to SuperKEKB• SuperKEKB nal design (β*y = 0.3/0.27 mm) with ideal crab waist• Beam-beam driven halo can be suppressedfi51

Crab waist applied to SuperKEKB

Lum. w/o crab waist in LER• SuperKEKB 2021b run (β*y = 1 mm) with ideal crab waist- Tune scan using BBWS showed that 80% crab waist ratio in LER ise ective in suppressing vertical blowup caused by beam-beamresonances (mainly νx ± 4νy + α = N).

2021.07.01
HER
LER
Ibunch (mA)

0.80

1.0
1174

# bunchComments

Assumed value

εx (nm)4.6

4.0

w/ IBSεy (pm)23

23

Estimated from XRM dataβx (mm)60

80

Calculated from latticeβy (mm)1

1

Calculated from lattice

σz0 (mm)

5.05

4.84

Natural bunch length (w/o MWI)νx45.53244.525

Measured tune of pilot bunch

νy43.58246.593

Measured tune of pilot bunchνs0.02720.0221

Calculated from lattice

Crab waist

40%

80%

Lattice design

Lum. w/ 80% crab waist in LER

ff52

Crab waist applied to SuperKEKB

Lum. w/o crab waist in HER• SuperKEKB 2021b run (β*y = 1 mm) with ideal crab waist- Tune scan using BBWS showed that 40% crab waist ratio (current
operation condition) in HER is not enough for suppressing vertical
blowup caused by beam-beam resonances (mainly
νx ± 4νy + α = N).

Lum. w/ 80% crab waist in HER
Lum. w/ 40% crab waist in HER

53

Crab waist applied to SuperKEKB• SuperKEKB nal design (β*y = 0.3/0.27 mm) with practical crab waist [1]• CW scheme with CW sextupoles outside IR• CW reduces dynamic aperture and Touschek lifetime, and was not chosen as baseline for TDR

fi[1] SuperKEKB TDR.

54

Crab waist applied to SuperKEKB• SuperKEKB nal design (β*y = 0.3/0.27 mm) with practical crab waist• CW does not work well because of the nonlinear IR. The nonlinearity scales as 1/β*y [1].
• SuperKEKB design lattice includes nonlinear elds extracted from 3D model [2][1] K. Ohmi, EIC workshop, March, 2014.

fi

fi[2] N. Ohuchi, SuperKEKB ARC, 2018.

55

Crab waist applied to SuperKEKB• Optics design with crab waist for β*y = 1 mm

• In 2020, K. Oide introduced the FCC-ee CW scheme [1] to SuperKEKB [2].
• FCC-ee CW scheme utilizes the sextupoles (a-d) for local chromaticity correction and crab waist.
β*=1SuperKEKB
w/
mm
y
FCC-ee[2] Y. Ohnishi, SuperKEKB ARC 2020.
[1] K. Oide et al., PRAB 19, 111005 (2016).
56

Crab waist applied to SuperKEKB• Optics design with crab waist for β*y = 0.6 mm by K. Oide [1][1] K. Oide, SuperKEKB ARC, 2021.

57

Crab waist applied to SuperKEKB• Optics design with crab waist for β*y = 0.6 mm by K. Oide [1]• With 50% CW strength, lifetime is acceptable for beam operation[1] K. Oide, SuperKEKB ARC, 2021.

58

Crab waist applied to SuperKEKB• SuperKEKB beam operation with crab waist for β*y = 1 mm
• Operation with CW has been successful [1].

Crab waist introduced since April 2020

[1] Y. Ohnishi, The European PhysicalJournal Plus volume 136, 1023 (2021).

59

Comparison of simulations and experiments• Known sources of luminosity degradation- Bunch lengthening- Chromatic couplings- Single-beam blowup in LER (Impedance e ects and its interplay with FB, see K. Ohmi’s talk)- Optics distortion due to SR heating (see Y. Ohnishi’s talk)Identi ed in 2022
- Luminosity “loss” correlated with injection.
• Sources to be investigated via experiments- Imperfect crab waist- Beam-beam-driven synchro-betatron resonances- Interplay of BB, longitudinal and transverse impedances, and feedback system- Global couplings (side e ects of IP knobs)- The interplay of BB and nonlinear lattices- Coupled bunch instabilities

}

ff

ff

fi

60

BB simulations w/ nal design con guration• Findings [1]- K. Ohmi and K. Hirosawa developed a simple method to-

calculate the nonlinear terms. Good agreements were foundwith PTC results.
Then perturbation maps were made via MAP element in SAD tosimulate luminosity loss. Finally, the term of px2 py was found to
be important. Its sources were also well understood. Other
chromatic terms can also be important in addition to chromatic
couplings.

- Finally we arrived at a clear picture for the luminosity loss in
beam-beam simulations (weak-strong model plus design
lattice): The sources are beam-beam resonances andnonlinearity of the IR. But, the remedy is far from apparent.

fi

fi

fl

[1] K. Hirosawa et al., The in uence of higher order multipoles of IR magnets on luminosity for SuperKEKB, in Proceedings of IPAC'18, Vancouver, BC, Canada, 2018.

61

Beam-beam viewpoints on achieving higher luminosity

=
β*
=
β*
ϵ
=
ϵ
=
ϵ
,
and
the
hourglass
e
ect
is
not
• Assume balanced collision: β*
y−
y
y+
y−
y
y+
strong, we can look into the formula of beam-beam parameter and discuss the challenges• Note that we have to respect the constraints of real machines.

ξy ≤ 0.1

i
ξy± ≈

fi

Crossing angle:
1) IR layout (constraints fromoptics design)
2) Reducing θc does not createa gain of luminosity if there thebeam-beam limit exists.

To keep ξy ≤ 0.1, higher
currents requires smaller β*
y

re

Ib∓

θc σ
2πef0γ± tan 2 z∓

If we must accept ξy ≤ 0.1, thensmaller β*y is always preferred

β*
y
ϵy

Impedance effects
Longer σz0 can be bene cial

We achieved ϵy0 ≈ 20 pm
ξy ≤ 0.1 sets a lower limit on theachievable ϵy (at a given β*y ).
It is not feasible to achieve ϵy ≈ 10 pm
at β*
y = 1 mm62
ff

Beam-beam limit requires:

Beam-beam viewpoints on achieving higher luminosity• Speci c luminosity only depends on the geometric parameters (beam sizes and crossing angle).

We achieved ~9 × 10

31

−2
−2
−1
cm s mA with β*y = 1 mm
31
−2
−2 −1

The baseline design is ~21 × 10

cm

Lsp ≈

s

mA

1
*2 + σ *2
2πe 2f σy+y−

The fundamental limit lies in vertical beam sizes
Challenges: High currents, beam-beam, crab waist, latticeimperfections, …

θc

2 + σ 2 tan
σz+
z−
2

Impedance effects
modify the synchrotron motion,
indirectly playing a role in many issuesfi63