Concept of Luminosity

Transcription

1 oncept of Luminosity in particle colliders Slide Werner Herr, ERN runo Muratori, Daresbury Laboratory Werner Herr, Luminosity, ockcroft Institute Lectures Why colliding beams Two beams: E, p, E 2, p 2, m = m 2 = m E cm = E E 2 2 p p 2 2 Slide 2 ollider versus fixed target: Fixed target: p 2 = E cm = 2m 2 2E m ollider: p = p 2 E cm = E E 2 LH pp: 4 GeV versus 5 GeV LEP e e : 2 GeV versus

2 ollider performance issues vailable energy Number of interactions per second useful collisions Total number of interactions Slide 3 Secondary issues: Time structure of interactions how often and how many at the same time Space structure of interactions size of interaction region Quality of interactions background, dead time etc Luminosity: We want: Slide 4 Proportionality factor between cross section σ p and number of interactions per second dr dt dr dt = L σ p units : cm 2 s Relativistic invariant Independent of the physical reaction Reliable procedures to compute and measure

3 Fixed target luminosity dr dt = Φ ρ l {T σ p Slide 5 L ρ T = const l Flux: Φ = Ns ollider luminosity per bunch crossing dr dt = L σ p s Slide 6 N ρx,y,s, s N 2 ρ 2 x,y,s,s N particles bunch ρ density = const L KN N 2 ρ x, y, s, s ρ 2 x, y, s, s dxdydsds s is time -variable: s = c t Kinematic factor: K = v v 2 2 v v 2 2 c 2

4 ollider luminosity per beam Slide 7 ssume uncorrelated densities in all planes factorize: ρx, y, s, s = ρ x x ρ y y ρ s s ± s For head-on collisions v = v 2 we get: L = 2 N N 2 f n b dxdydsds ρ x xρ y yρ s s s ρ 2x xρ 2y yρ 2s s s In principle: should know all distributions Mostly use Gaussian ρ for analytic calculation in general: it is a good approximation ρ iz z = Gaussian distribution functions exp z2 σ z 2π 2σz 2 i =, 2, z = x, y Slide 8 ρ is s ± s = exp s ± s 2 σ s 2π 2σs 2 For non-gaussian profiles not always possible to find analytic form, need a numerical integration

5 Luminosity for two beams and 2 Simplest case : equal beams Slide 9 σ x = σ 2x, σ y = σ 2y, σ s = σ 2s but: σ x σ y, σ 2x σ 2y is allowed Further: no dispersion at collision point Integration head-on for σ = σ 2 ρ ρ 2 = ρ 2 : L = 2 N N 2 fn b 2π 6 σ 2 s σ2 x σ2 y e x2 σ x 2 e y2 σ y 2 e s2 σ s 2 e s 2 σ s 2 dxdydsds Slide integrating over s and s, using: e at2 dt = L = 2 N N 2 fn b 8 π 4 σ 2 x σ2 y πa e x 2 σ x 2 e y 2 σ 2 y dxdy finally after integration over x and y: = L = N N 2 fn b 4πσ xσ y

6 Luminosity for two equal beams Simplest case : σ x = σ 2x, σ y = σ 2y, σ s = σ 2s Slide = L = N N 2 fn b 4πσ x σ y Typical example: round beams, eg hadrons Luminosity for two unequal beams Generalized for unequal beam sizes: σ x σ 2x σ y σ 2y, but : σ s σ 2s Slide 2 = L = N N 2 fn b 2π σx 2 σ2x 2 σ 2 y σ2y 2 Typical example: flat beams, eg leptons

7 Examples Energy L max rate σ xσ y Particles GeV cm 2 s s µmµm per bunch SPS p p 35x Slide 3 Tevatron p p x HER e p 3x LH pp 7x LEP e e 5x PEP e e 9x3 8 3 N What else What about linear colliders Slide 4 See later

8 rossing angle Hour glass effect omplications Slide 5 Displaced waist: δβ δs = α ollision offset wanted or unwanted Non-Gaussian profiles Dispersion at collision point Strong coupling etc ollisions at crossing angle Slide 6 ρ x,y,s, s ρx,y,s,s 2 Needed to avoid unwanted collisions For colliders with many bunches: LH, ESR, KEK For colliders with coasting beams Φ

9 ollisions angle geometry horizontal plane x = x cos φ2 s sin φ2 x Slide 7 s = s cos 2 φ2 x sin φ2 φ2 s = s cos s φ2 x sin φ2 φ2 x = x cos 2 beam 2 φ2 s sin φ2 beam rossing angle ssume crossing in horizontal x, s- plane Transform to new coordinates: Slide 8 x = x cos φ 2 s sin φ 2, s = s cos φ 2 x sin φ 2, x 2 = x cos φ 2 s sin φ 2, s 2 = s cos φ 2 x sin φ 2 L = 2 cos 2 φ 2 N N 2 fn b dxdydsds ρ x x ρ y y ρ s s s ρ 2x x 2 ρ 2y y 2 ρ 2s s 2 s

10 Integration crossing angle tutorial Hint: Slide 9 use as before: and: e at2 dt = e at2 btc dt = πa πa e b2 ac a rossing angle fter integration over t and transverse coordinates: Slide 2 with: L = N N 2 fn b 8π 3 2 σ s 2 cos φ 2 = sin2 φ 2 σ 2 x cos2 φ 2 σ 2 s e s2 σ x σ y ds

11 rossing angle rossing ngle L = N N 2 fn b 4πσ x σ y S S is the reduction factor For small crossing angles and σ s σ x,y S = σs σx tan φ 2 2 σs σx φ 2 2 Example LH: Φ = 285 µrad, σ s = 75 cm, S = 84 Slide 2 Large crossing angle Large crossing angle: large loss of luminosity Slide 22

12 crab crossing scheme!!!!!!!!!!! " " " " " " " " " " # # # # # # # # # # $ $ $ $ $ $ $ $ $ % % % % % % % % % & & & & & & & & & & ' ' ' ' ' ' ' ' ' ' crab crossing recovers geometric loss factor feasibility needs to be demonstrated Slide 23 Offset and crossing angle φ2 s x s s s 2 2 d 2 x 2 x 2 d s x x beam beam 2 Slide 24

13 Offset and crossing angle Slide 25 Transformations with offsets in crossing plane: x = d x cos φ s sin φ, s 2 2 = s cos φ x sin φ, 2 2 x 2 = d 2 x cos φ 2 s sin φ 2, s 2 = s cos φ 2 x sin φ 2 Gives after integration over y and s : L = L 2πσ sσ x 2 cos 2 φ 2 e x 2 cos 2 φ2s 2 sin 2 φ2 σx 2 e x2 sin 2 φ2s 2 cos 2 φ2 σs 2 e d 2 d2 2 2d d 2 x cosφ2 2d 2 d s sinφ2 2σx 2 dxds Offset and crossing angle fter integration over x: Slide 26 with: L = N N 2 fn b 8π 3 2 σ s 2 cos φ 2 = sin2 φ 2 σ 2 x cos2 φ 2 σ 2 s and W = e 4σx 2 d 2 d 2 2s W e s2 ds σ x σ y = d 2 d sinφ2 2σ 2 x = Luminosity with correction factors

14 Luminosity with correction factors Slide 27 L = N N 2 fn b 4πσ x σ y W e 2 W : correction for beam offset S S: correction for crossing angle e 2 : correction for crossing angle and offset Hour glass effect 4 35 hourglass effect beta = 5 m beta = 5 m 3 Slide 28 beta function s m β-functions depends on position s 2 βs β s β

15 Hour glass effect Slide 29 β-functions depends on position s Hour glass effect hourglass effect Slide 3 beam size micron beta = 5 m beta = 5 m s m eam size σ β s depends on position s For smaller β : very fast increase with distance to interaction point

16 Hour glass effect - short bunches hourglass effect Slide 3 beam size micron beta = 5 m beta = 5 m s m Small variation of beam size along bunch Hour glass effect - long bunches hourglass effect Slide 32 beam size micron beta = 5 m beta = 5 m s m Significant effect for long bunches and small β

17 Hour glass effect - short bunches 8 Hourglass effect Slide 33 amplitude p p s Small variation of beam size along bunch Hour glass effect - long bunches 8 Hourglass effect 6 p 4 2 Slide 34 amplitude p p p s Significant effect for long bunches and small β

18 Hour glass effect β-functions depends on position s Usually: βs = β s 2 β ie σ = σs const σs = σ s 2 β Important when β comparable to the rms bunch length σ s or smaller! Hour glass effect Without crossing angle and for symmetric, round Gaussian beams we get the relative luminosity reduction as: Lσ s L = e u2 π [ u 2 ] du = π u x e u2x erfcu x ux Using the expression: u x = β σ s Slide 35 Slide 36

19 Hour glass effect betasigma_s LsL Hourglass effect - head on collisions LsL Hourglass reduction factor as function of ratio β σs Luminosity loss with longitudinal displacement Loss of luminosity if beams collide with longitudinal displacement: ie β-waist not at collision point δβ δs = α : Lσ s L = e u uw2 π [ u 2 ] du ux Using the expression: u w = s w σ s Slide 37 Slide 38

20 Hour glass effect Hourglass effect - longitudinal displacement LsL Slide 39 LsL displacement ssigma_s Hourglass reduction factor with longitudinal displacement s w as function of ratio s w σ s Hour glass effect Hourglass effect - longitudinal displacement LsL Slide 4 LsL displacement ssigma_s Where could that become important

21 alculations for the LH N = N 2 = 5 particlesbunch Slide 4 n b = 288 bunchesbeam f = 2455 khz, φ = 285 µrad β x = β y = 55 m σ x = σ y = 66 µm, σ s = 77 cm Simplest case Head on collision: L = 2 34 cm 2 s Effect of crossing angle: Slide 42 L = cm 2 s Effect of crossing angle & Hourglass: L = cm 2 s What about large crossing angle and long bunches

22 Large crossing angle - long bunches Slide 43 IP ssume crossing angle in horizontal plane Large crossing angle: large loss of luminosity Large crossing angle P x Slide 44 IP For large amplitude particles: collision point longitudinally displaced

23 Large crossing angle P x Slide 45 IP For large amplitude particles: collision point longitudinally displaced an introduce coupling transverse and synchro betatron, bad for flat beams Large crossing angle β y Slide 46 x x IP x particle s collision point amplitude dependent Different vertical β functions at collision points

24 Large crossing angle Slide 47 x x IP x particle s collision point amplitude dependent Different β functions at collision points hour glass! crab waist scheme Slide 48 x x IP x Make vertical waist βy min amplitude x dependent ll particles in both beams collide in minimum β y region

25 crab waist scheme optical setup SEXTUPOLE β x β y β x β y IP x SEXTUPOLE β x β y Slide 49 µ y = π 2 µ y = π 2 µ x = π µ x = π for the sextupole strength: k = 2θ β y βy β x β x and: β y = β s xθ2 β y crab waist scheme Make vertical waist minimum of β amplitude x dependent an be done with two sextupoles Slide 5 First tried at DPHNE Frascati in 28 Geometrical gain small Smaller vertical tune shift as function of horizontal coordinate Less betatron and synchrotron coupling Good remedy for flat ie lepton beams with large crossing angle

26 If the beams are not Gaussian Exercise: ssume flat distributions normalized to ρ = ρ2 = 2a, for [ a z a], z = x, y alculate rms in x and y: < x, y 2 = x, y 2 ρx, ydxdy and L = ρx, yρ2x, y dxdy ompute: L < x 2 < y 2 Repeat for various distributions and compare Slide 5 What about unequal bunch lengths,,,,,,,,,,,,,,,,, Overlap not optimum Typical case: lepton hadron colliders HER, LHe Slide 52

27 What about unequal bunch lengths Relevant with a crossing angle For lepton-hadron colliders usually: Slide 53 σ p s σ e s For example HER and LHe: σ p s 8 cm, σ e s cm Same procedures applied, but now with σ p s σ e s Luminosity correction Result from numerical integration for σ p s σ e s Vary protonelectron bunch length ratio Slide 54 9e32 8e32 7e32 6e32 Luminosity versus bunch length ratio mrad 5 mrad 3 mrad L 5e32 4e32 3e32 2e32 e sigma-p sigma-e

28 Integrated luminosity Slide 55 L int = T Ltdt The figure of merit: L int σ p = number of events Experiments: continuous recording of L For studies: assume some life time behaviour Eg Lt L exp t τ ontributions to life time from: intensity decay, emittance growth etc Integrated luminosity Knowledge of preparation time allows optimization of L int Slide 56

29 Integrated luminosity Typical run times LEP: 8 - hours t r Slide 57 For LH long preparation time t p expected Optimum combination of t r and t p gives maximum luminosity t r is usually a free parameter, ie can be chosen Maximising Integrated Luminosity ssume exponential decay of luminosity Lt = L e tτ Slide 58 verage luminosity < L < L = tr dtlt t rt p = L τ e trτ t rt p Theoretical maximum for: t r τ ln 2t p τ t p τ Example LH: t p h, τ 5h, t r 5h Exercise: Would you improve τ long t r or t p

30 verage luminosity for different run times 5 45 verage luminosity tau = 5 hrs tau = 5 hrs tau = 25 hrs Slide 59 <Ltr runtime tr verage luminosity, τ p = hrs, for different beam lifetimes verage luminosity for different run times 9 8 verage luminosity tau = 5 hrs tau = 5 hrs tau = 25 hrs 7 6 Slide 6 <Ltr runtime tr verage luminosity, τ p = 5 hr, shorter run times preferred

31 Luminous region 2 s Slide 6 s s Density distribution of interaction vertices Fraction of collisions occur ± s from the IP Important for experiments! Luminous region Depends on σ x, σ y, σ s and crossing angle φ Integrate only along a finite longitudinal length: Slide 62 L = Ls ds LS = S S Ls ds N N 2 fn b 2 cos φ 2 π LS = 8πσxσ y πσs erf S Real figure of merit: L int S = T S S Ls, tds dt

32 Some results for LH σ s = 77 cm, β = 55 m, φ = 285 µrad: Slide 63 % lumi S = ± 2 cm 9% lumi S = ± 7 cm 8% lumi S = ± 55 cm Interactions per crossing Luminosityfn b N N 2 In LH: crossing every 25 ns Slide 64 Per crossing approximately 2 interactions May be undesirable pile up in detector = more bunches n b, or smaller N eware: maximum peak luminosity L max is not the whole story!

33 Luminosity measurement Slide 65 One needs to get a signal proportional to interaction rate eam diagnostics Large dynamic range: 27 cm 2 s to 34 cm 2 s Very fast, if possible for individual bunches Used for optimization For absolute luminosity need calibration Luminosity calibration e e Slide 66 Use well known and calculable process e e e e elastic scattering habha scattering Have to go to small angles σ el Θ 3 Small rates at high energy σ el E 2

34 Luminosity calibration : : : : : : : : : : : : : : : : : : ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; < < < < < < < < < < < < < < < < < < = = = = = = = = = = = = = = = = = = x e e e e monitors Measure coincidence at small angles Low counting rates, in particular for high energy! ackground may be problematic Slide 67 Luminosity calibration hadrons, eg pp or p p Must measure beam current and beam sizes eam size measurement: Wire scanner or synchrotron light monitors Measurement with beam remember luminosity with offset Move the two beams against each other in transverse planes van der Meer scan Slide 68

35 D E D E H I H I F G F G D E D E Luminosity optimization 2 Van der Meer scan Record counting rates Rd as function of movement d Slide 69 ounting rate Since Rd is proportional to luminosity Ld amplitude Get ratio of luminosity LdL Luminosity optimization From ratio of luminosity LdL Remember: W = e 4σ 2 d 2 d 2 Slide 7 Determines σ and centres the beams! Others: eam-beam deflection scans LEP eam-beam excitation

36 L Scan in LEP with bunch trains N N N Nš N N N Families Luminosity 3 cm -2 s Luminosity 3 cm -2 s IP2 Scan a N b c Slide 7 Luminosity 3 cm -2 s Vernier setting µm IP N Nª Luminosity 3 cm -2 s Vernier setting µm IP4 Nº¹ N µ Offset a 96-3 Offset b 3-5 Offset c 6-7 IP4 Scan Offset a Offset b Offset c IP6 Scan Offset a 9-25 Offset b Offset c IP8 Scan Vernier setting µm IP6 2 Vernier setting µm IP8 Offset a Offset b Offset c bsolute value of L pp or p p Slide 72 y total rate and optical theorem also: luminosity independent determination of σ tot : σ tot L = N inel N el Total counting rate dσ el lim t dt = ρ 2 σ2 tot 6π = dn el L dt t= L = ρ2 N inel N el 2 6π dn el dt t= Luminosity determined from experimental rates

37 bsolute value of L pp or p p Slide 73 y oulomb normalization: oulomb amplitude exactly calculable: dn el dσ el lim = t dt L dt t= = π f f N 2 π 2αem σtot ρ t 4π ieb t 2 2 4πα2 em t 2 t Fit gives: σ tot, ρ, b and L an be done measuring only elastic scattering No N inel needed! Optics for luminosity measurement Slide 74 Want to measure small momentum transfer reactions: t Small beam divergence σ needed! Since σ = ɛβ Large β at collision point needed

38 High β optics for luminosity measurement 25 High beta optics for luminosity measurement beam beam 2 2 Slide 75 beta s For LH: β 5-26 m foreseen dndt Differential elastic cross section dndt U42 Fit strong part Fit oulomb part Measure dndt at small t < GeVc2 and extrapolate to t = Needs special optics to allow measurement at very small t Slide 76 Measure total counting rate N el N inel Needs good detector coverage Often use slightly modified method, precision 2 % t GeVc

39 Luminosity in linear colliders Mainly only e e colliders Past collider: SL SL Slide 77 Under consideration: LI, IL Special issues: Particles collide only once dynamics! Particles collide only once beam power! Must be taken into account Luminosity in linear colliders asic formula: Slide 78 From : L = N 2 f n b 4πσ x σ y to : L = N 2 f rep n b 4πσ x σ y Replace frequency f by repetition rate f rep nd introduce effective beam sizes σ x, σ y : L = N 2 f rep n b 4πσ x σ y

40 Luminosity in linear colliders Using the enhancement factor H D : Slide 79 L = N 2 f rep n b 4πσ x σ y L = H D N 2 f rep n b 4πσ x σ y Enhancement factor H D takes into account reduction of nominal beam size by the disruptive field pinch effect Related to disruption parameter D: D x,y = 2r e Nσ z γσ x,y σ x σ y Pinch effect - disruption beam-beam collision 4 2 Slide s dditional focusing by opposing beams

41 Pinch effect - disruption beam-beam collision 4 2 Slide s dditional focusing by opposing beams Luminosity in linear colliders For weak disruption D and round beams: Slide 82 H D = 2 3 π D OD2 For strong disruption and flat beams: computer simulation necessary, maybe can get some scaling

42 eamstrahlung Slide 83 Disruption at interaction point is basically a strong bending can be very strong! Results in strong synchrotron radiation: beamstrahlung 4 2 beam-beam collision with beamstrahlung -2 γ γ s eamstrahlung Disruption at interaction point is basically a strong bending can be very strong! Results in strong synchrotron radiation: beamstrahlung Slide 84 This causes unwanted: Spread of centre-of-mass energy Pair creation and detector background gain: luminosity is not the only important parameter

43 eamstrahlung Parameter Y Measure of the mean field strength in the rest frame normalized to critical field c : Slide 85 with: Y = < E 5 re 2γN c 6 ασ z σ x σ y c = m2 c 3 e h 44 3 G Energy loss and power consumption verage fractional energy loss δ E : Slide 86 δ E = 24 ασ zm e λ E Y 5Y 23 2 where E is beam energy at interaction point and λ the ompton wavelength

44 Luminosity in linear colliders Using the beam power P b and beam energy E in the luminosity: Slide 87 L = H D N 2 f rep n b 4πσ x σ y L = H D N P b ee 4πσ x σ y eam power P b related to power consumption P via efficiency η b P b = η b P Figure of merit in linear colliders Luminosity at given energy normalized to power consumption and momentum spread due to beamstrahlung: Slide 88 M = LE δb P With previous definition and reasonably small beamstrahlung this becomes: M = LE η b δb P ɛ y These are optimized in the linear collider design

45 Figure of merit in linear colliders Luminosity at given energy normalized to power consumption and momentum spread due to beamstrahlung: Slide 89 M = LE δb P With previous definition and reasonably small beamstrahlung this becomes: M = LE η b δb P ɛ y These are optimized in the linear collider design Not treated : oasting beams eg ISR Slide 9 symmetric colliders: symmetric energies eg PEP symmetric charges eg p-pb Ring - LIN colliders

46 How to cook high Luminosity Get high intensity Get small beam sizes small ɛ and β Slide 9 Get many bunches Get small crossing angle if any Get exact head-on collisions Get short bunches