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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