Download Beam-Beam Effect

Beam-BeamEffect
KaiShih
(*graphcitedfrom“Beam-BeamEffectsinParticleColliders”,MauroPivi,USParticleAcceleratorSchool,2011)
• BeamexperiencesmainlyEMforce insideacollider.
• BeamgenerateselfEMfield.
• FortheforcethatproducedbyselfEMfield,callSpace-Chargeeffect
• Fortheforcethatproducedbyotherbeam’sEMfield,callbeam-beam
effect
• Beam-beameffectis2D forrelativisticbeams.
Space-Chargeeffect
beam-beameffect
• Beam-BeamForceDerivation
• Beam-BeamEffect
• Beam-BeamTuneShift
• Hour-GlassEffect
• Conclusion
Beam-BeamForceDerivation
Let’sconsideralongroundGaussianProtonbeam:
𝑁𝑒
𝑟)
𝜌 𝑟 =
𝐸𝑥𝑝[− ) ]
)
2𝜋𝜎 𝐿
2𝜎
Transversechargedistributionofbeam
Thebeamchargeisuniformlongitudinally,
withtotalcharge(Ne)
For𝜎 =1,Ne=100
AssumeLongBeam!
Efield:
𝑄
1 𝐸 2 𝑑𝑎⃑ =
𝜀9
1
𝐸: ×2𝜋𝑟𝐿 = = 𝜌(𝑟)𝑟𝑑𝜙𝑑𝑟𝑑𝑧
𝜀9
G
:
:F
1
𝑁𝑒
D G
)H 𝑑𝑟′
𝐸: ×2𝜋𝑟𝐿 = ×2𝜋 B
𝑟′
𝑒
)
𝜀9
2𝜋𝜎
9
:FG
𝑁𝑒
D G
𝐸: =
1 − 𝑒 )H
2𝜋𝜀9 𝑟𝐿
LabFrame
Er
𝑧̂
r
L
beam
AssumeLongBeam!
Bfield:
B𝜙
1 𝐵 2 𝑑𝑙⃑ = 𝜇9 𝐼
r
beam
𝐵N ×2𝜋𝑟 = 𝜇9 λ𝑣
)Q
:
:FG
D G
𝑟′ 𝑒 )H 𝑑𝑟 F 𝑑𝜙
𝜇9 𝑣
𝑁𝑒
B B
𝐵N ×2𝜋𝑟 =
𝐿 9 9 2𝜋𝜎 )
:FG
𝑁𝑒𝜇9 𝑣
D G
𝐵N =
1 − 𝑒 )H
2𝜋𝑟𝐿
LabFrame
𝑧̂
G
Self-fieldof
thebeam
:
𝑁𝑒
D G
𝐸=
1 − 𝑒 )H 𝑟̂
2𝜋𝜀9 𝑟𝐿
∵ 𝐹⃑ =−e(𝐸 + 𝑣⃑×𝐵)
G
:
𝑁𝑒𝜇9 𝑣
D G
𝐵=
1 − 𝑒 )H 𝜙R
2𝜋𝑟𝐿
For𝛽 ≈ 1,singleelectron
beam-beameffect:
Space-Chargeeffect:
𝐹⃑WX =−e(𝐸 + 𝑣𝑧̂ ×𝐵)
𝐹⃑VV =−e(𝐸 − 𝑣𝑧̂ ×𝐵)
)
G
:G
)
)
:
𝑁𝑒 )
𝑣
D G
𝑁𝑒
𝑣
D G
=−
1 − 𝑒 )H (1 + ) )
)H
=−
1
−
𝑒
(1 − ) )
2𝜋𝜀9 𝑟𝐿
𝑐
2𝜋𝜀9 𝑟𝐿
𝑐
G
)
:
)
:G
𝑁𝑒
D G
𝑁𝑒
D G
D)
≈−
1 − 𝑒 )H 𝑟̂
)H
=−
1
−
𝑒
𝛾
𝑟̂
𝜋𝜀9 𝑟𝐿
2𝜋𝜀9 𝑟𝐿
LabFrame
≈0
Beam-beamforceagainstr
20
𝐹
𝑁𝑒
=
𝑟 2𝜋𝐿𝜎 )
15
F(r )
• Itactsasafocusing/defocusingforce,
dependsonthespeciesofthebeams.
• Insmallr,thebeam-beamforce islinear
inthedirectionof+-𝑟̂ .
• Beam-beameffectforGaussianbeam
justlikeasmallquadrupoleerror!
10
5
0
0
1
2
3
4
5
r
For𝜎 =1,Ne=100
Beam-BeamEffect
• ConsiderbeamA,B(P+ ,e- )isshortround Gaussianbeam
• ItbeamAactasalens tobeamB,viceversa.
beamB
beamA
𝑟
𝑝:
=
𝑓
𝑝9
r
f
BeamAFrame
𝑟
𝑝:
=
𝑓
𝑝9
AssumeSmallr,forceislinearinr.
a
∫9 𝐹𝑑𝑡
+0
1 1
= 2
𝑓 𝑟
𝛾𝑚𝑐 )
1
𝐹𝐿
= 2
𝑟 2𝛾𝑚𝑐 c
1 2𝑁𝑟d
= 2
1−
𝑟 𝛾𝑟
𝑒)
𝑟d =
4𝜋𝜀9 𝑚𝑐 )
BeamAFrame
𝐿
T≈
2𝑐
:G
D G
𝑒 )H
Ingeneral, 𝜎f ≠ 𝜎h
1 2𝑁𝑟d
𝑟)
∴ ≈
1−1+ )
)
𝑓
𝛾𝑟
2𝜎
𝑁 𝑟d
= )2
𝜎 𝛾
!!!beamAcontribution(lens)
!!!beamBcontribution
1
2𝑁𝑟d
=
𝑓f,h 𝛾𝜎f,h (𝜎f + 𝜎h )
𝜎m
𝑁𝑟d 𝜎m
Wecandefineadisruptionparameter 𝐷 =
=
𝑓
𝛾𝜎 )
ForD<1
ForD>>1
𝜎m
𝜎m
D=1
D=1
• Collisionoftwolongbeam
• Bothbeamexperiencesafocusingforce
• Beamsenvelopeareoscillating whentheymet
(*graphcitedfrom“Beam-BeamEffectsinParticleColliders”,MauroPivi,USParticleAcceleratorSchool,2011)
Beam-BeamTuneShift
cos 2𝜋𝑄h
𝑀 = −1
sin 2𝜋𝑄h
∗
𝛽
𝛽 ∗ sin 2𝜋𝑄h
cos 2𝜋𝑄h
ForD<1
Beam–beameffectlikeaangularkick,
thereforetheoneturnmapping𝑀a :
1 0
𝐾 = −1 1
𝑓
cos 2𝜋𝑄h
𝑀a = −1
sin 2𝜋𝑄h
∗
𝛽
𝛽 ∗ sin 2𝜋𝑄h
cos 2𝜋𝑄h
1 0
2 −1 1
𝑓
cos 2𝜋𝑄h
𝑀a = −1
sin 2𝜋𝑄h
∗
𝛽
𝛽 ∗ sin 2𝜋𝑄h
cos 2𝜋𝑄h
1 0
2 −1 1
𝑓
𝛽∗
cos 2𝜋𝑄h − sin 2𝜋𝑄h
𝛽 ∗ sin 2𝜋𝑄h
𝑓
=
−1
1
sin 2𝜋𝑄h − cos 2𝜋𝑄h
cos 2𝜋𝑄h
∗
𝛽
𝑓
However,ingeneral,beam-beameffectcanalso
beingtreatedasaperturbativetuneshiftinoneturn:
cos 2𝜋(𝑄h + 𝜉h )
𝑀v = −1
sin 2𝜋 cos 2𝜋(𝑄h + 𝜉h )
∗
𝛽
𝛽 ∗ sin 2𝜋 cos 2𝜋(𝑄h + 𝜉h )
cos 2𝜋(𝑄h + 𝜉h )
∵ 𝑇𝑟 𝑀a = 𝑇𝑟(𝑀v )
𝛽∗
2cos 2𝜋𝑄h − sin 2𝜋𝑄h = 2cos 2𝜋(𝑄h + 𝜉h )
𝑓
𝛽∗
2cos 2𝜋𝑄h − sin 2𝜋𝑄h = 2cos 2𝜋𝑄h cos 2𝜋𝜉h
𝑓
− 2sin 2𝜋𝑄h sin 2𝜋𝜉h
𝛽∗
2cos 2𝜋𝑄h − sin 2𝜋𝑄h ≈ 2cos 2𝜋𝑄h − 4𝜋𝜉h sin 2𝜋𝑄h 𝑓
𝛽∗
∴ 𝜉h =
4𝜋𝑓
Inorderforthebeamtobestable :
𝑇𝑟 𝑀a
≤2
−2 ≤ 2cos 2𝜋𝑄h − 4𝜋𝜉h sin 2𝜋𝑄h ≤ 2
0 ≤ 2cos) 𝜋𝑄h − 4𝜋𝜉h sin𝜋𝑄h 𝑐𝑜𝑠 𝜋𝑄h ≤ 2
−1
1
tan𝜋𝑄h ≤ 𝜉h ≤
cot𝜋𝑄h
2𝜋
2𝜋
Ingeneral,thetwobeamcentroids canhaveaoffset
Theangularkickmatrix forthebeamcentroids:
beamB
beamA
∴ 𝑥V€ = 𝑥V9
𝑥V9 − 𝑥•9
F
F
F
∴ 𝑥V€
=−
+ 𝑥V9
− 𝑥•9
𝑓•
𝑥V9
𝑥•9
f
𝑥•€
1
0
F
𝑥•€
−1/𝑓V 1
=
𝑥V€
0
0
F
𝑥V€
1/𝑓• −1
0
0
1/𝑓V −1
1
0
−1/𝑓• 1
𝑥•9
F
𝑥•9
𝑥V9
F
𝑥V9
However,thereisnocoupling inthe
Oneturnmapping
𝑥•€
F
𝑥•€
𝑥V€ =
F
𝑥V€
𝑀a•‚•
cos 2𝜋𝑄h
−1
sin 2𝜋𝑄h
𝛽∗
0
0
𝛽∗ sin 2𝜋𝑄h
cos 2𝜋𝑄h
0
0
0
0
cos 2𝜋𝑄h
−1
sin 2𝜋𝑄h
𝛽∗
𝛽∗
cos 2𝜋𝑄h − sin 2𝜋𝑄h
𝛽∗ sin 2𝜋𝑄h
𝑓V
−1
1
sin
2𝜋𝑄
−
cos 2𝜋𝑄h
cos 2𝜋𝑄h
h
𝛽∗
𝑓V
=
𝛽∗
sin 2𝜋𝑄h −𝛽∗ sin 2𝜋𝑄h
𝑓•
1
cos 2𝜋𝑄h
−cos 2𝜋𝑄h
𝑓•
0
0
𝛽∗ sin 2𝜋𝑄h
cos 2𝜋𝑄h
𝑥•9
F
𝑥•9
𝑥V9
F
𝑥V9
𝛽∗
sin 2𝜋𝑄h −𝛽∗ sin 2𝜋𝑄h
𝑓V
1
cos 2𝜋𝑄h
− cos 2𝜋𝑄h
𝑓V
𝛽∗
cos 2𝜋𝑄h − sin 2𝜋𝑄h
𝛽∗ sin 2𝜋𝑄h
𝑓•
−1
1
sin
2𝜋𝑄
−
cos 2𝜋𝑄h
cos 2𝜋𝑄h
h
𝛽∗
𝑓•
Hour-GlassEffect
𝜎(𝑠) =
𝜀𝛽(𝑠) =
𝜀(𝛽 ∗ +
𝑠)
𝛽∗
β(s)
• Let’sconsidera“long”beam
• Duetothelongitudinalbeamsize,
IPwillhappeninaseriesofdifferent𝛽 ∗ (s)
• Withoutbeam-beameffect
)
Betafunctionoftwobeams
4
2
0
-2
-4
Collisionhappenedindifferentbeamsize
-1.0
-0.5
0.0
0.5
1.0
s
(Red->electrons,Blue->Protons)
Considerashortelectronbeamcollide
withalongProtonbeam
𝛽ƒ (𝑠)
𝜉ƒ (𝑠) =
4𝜋𝑓d
𝑁d 𝑟v 𝛽v (𝑠)
=
4𝜋𝜎d (𝑠)) 𝛾ƒ
Fromabove:
1 𝑁𝑟d
=
𝑓 𝛾𝜎 )
𝛽d 𝑠
1
𝜉d = B
𝑑
4𝜋
𝑓d
𝑁v 𝜆 𝑠 𝑟d 𝛽d (𝑠)
=B
𝑑𝑠
)
4𝜋𝜎v (𝑠) 𝛾d
Where𝜆 𝑠 isthenormalizedprotondensitydistribution
thattheelectronsmet.𝑑𝑁v (𝑠) = 𝑁v 𝜆 𝑠 𝑑𝑠
Asluminosity formuladefinedas:
𝑁d 𝑁ƒ 𝜈ℎ
𝐿=
4𝜋𝜎 )
Thereforeluminosityformulawillchangetothisform
𝑁d 𝑁ƒ 𝜆 𝑠 𝜈ℎ
𝐿=B
𝑑𝑠
)
)
2𝜋[𝜎ƒ (𝑠) + 𝜎d (𝑠) ]
Only thecenterpartofprotonbeamcollidein
minimum𝛽.Luminositywasreduced!
However,Hour-Glasseffectstarttobeimportantwhen 𝜎m ≥ 𝛽∗
Conclusion
• Beam-Beameffectisa2Dforce
• ForaGaussianbeam,itislikeaquadrupoleerror.
• Forashortbeam,itactasfocusing/defocusing thinlens
• Beam-beameffectcreatesaextratuneshift
• Beamradius willbealsoaffected,whichcancauseluminositychange.
• The Longitudinal beamsize causesHour-glasseffect ,
whichcanalsocauseluminositychange.
References
1.
“Beam-BeamInteractionStudyinERLBasedERHIC”,YueHao,Ph.D.Thesis,DepartmentofPhysics,IndianaUniversity
September,2008
2.
“Beam-BeamEffectsinParticleColliders”,MauroPivi,USParticleAcceleratorSchool,2011