Concept of Luminosity

Transcription

1 Concept of Luminosity (in particle colliders) Werner Herr, CERN, AB Department Bruno Muratori, Daresbury Laboratory (/afs/ictp/home/w/wfherr/public/cas/doc/luminosity.pdf) ( luminosity.pdf) Werner Herr, concept of luminosity, CAS 2005, Trieste

2 Why colliding beams? Two beams: E 1, p 1, E 2, p 2, m 1 = m 2 = m E cm = (E 1 + E 2 ) 2 ( p 1 + p 2 ) 2 Collider versus fixed target: Fixed target: p 2 = 0 E cm = 2m 2 + 2E 1 m Collider: p 1 = p 2 E cm = E 1 + E 2 LHC (pp): GeV versus 115 GeV LEP (e + e ): 210 GeV versus?

3 Collider performance issues Available energy Number of interactions per second (useful collisions) Total number of interactions 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.)

4 Luminosity: We want: Proportionality factor between cross section σ p and number of interactions per second dr dt dr dt = L σ p ( units : cm 2 s 1 ) Relativistic invariant Independent of the physical reaction Reliable procedures to compute and measure

5 Fixed target luminosity dr dt = Φ ρ l {T σ p L ρ T = const. l Flux: Φ = N/s

6 Collider luminosity (per bunch crossing) dr dt = L σ p s 0 N ρ (x,y,s, s ) N 2 ρ 2 (x,y,s,s ) 0 N particles / bunch ρ density = const. L K + ρ 1 (x, y, s, s 0 )ρ 2 (x, y, s, s 0 )dxdydsds 0 s 0 is time -variable: s 0 = c t Kinematic factor: K = 1/c ( v 1 v 2 ) 2 ( v 1 v 2 ) 2 /c 2

7 Collider luminosity (per beam) Assume uncorrelated densities in all planes factorize: ρ(x, y, s, s 0 ) = ρ x (x) ρ y (y) ρ s (s ± s 0 ) For head-on collisions ( v 1 = v 2 ) we get: L = 2 N 1 N 2 fb + dxdydsds 0 ρ 1x (x)ρ 1y (y)ρ 1s (s s 0 ) ρ 2x (x)ρ 2y (y)ρ 2s (s + s 0 ) In principle: should know all distributions Use Gaussian ρ for analytic calculation (in general: they are a good approximation for most realistic distributions)

8 ρ iz (z) = Gaussian distribution functions ( 1 exp z2 σ z 2π 2σz 2 ) i = 1, 2, z = x, y ρ is (s ± s 0 ) = ( 1 exp (s ± s 0) 2 ) σ s 2π 2σs 2 For non-gaussian profiles not always possible to find analytic form, need a numerical integration

9 Luminosity for two beams (1 and 2) Simplest case : equal beams σ 1x = σ 2x, σ 1y = σ 2y, σ 1s = σ 2s but: σ 1x σ 1y, σ 2x σ 2y is allowed Further: no dispersion at collision point

10 Integration (head-on) for σ 1 = σ 2 ρ 1 ρ 2 = ρ 2 : L = 2 N 1N 2 fb ( 2π) 6 σ 2 s σ2 x σ2 y e x 2 σ x 2 e y 2 σ 2 y e s2 σ s 2 e s 2 0 σ 2 s dxdydsds 0 integrating over s and s 0, using: L = + 2 N 1N 2 fb 8( π) 4 σ 2 x σ2 y e at2 dt = π/a e x2 σ 2 x e y2 σ 2 y dxdy finally after integration over x and y: = L = N 1N 2 fb 4πσ x σ y

11 Luminosity for two (equal) beams (1 and 2) Simplest case : σ 1x = σ 2x, σ 1y = σ 2y, σ 1s = σ 2s or: σ 1x σ 2x σ 1y σ 2y, but : σ 1s σ 2s Further: no dispersion at collision point = L = N 1N 2 fb 4πσ x σ y L = N 1 N 2 fb 2π σ1x 2 + σ2x 2 σ 2 1y + σ2y 2

12 Examples Energy L max rate σ x /σ y Particles (GeV) cm 2 s 1 s 1 µm/µm per bunch SPS (p p) 315x / Tevatron (p p) 1000x /30 30/ HERA (e + p) 30x /50 3/ LHC (pp) 7000x / LEP (e + e ) 105x / PEP (e + e ) 9x NA 150/5 2/

13 Complications Crossing angle Hour glass effect Collision offset (wanted or unwanted) Non-Gaussian profiles Dispersion at collision point δβ /δs = α 0 Strong coupling etc.

14 Collisions at crossing angle Φ ρ 0 0 (x,y,s, s ) 1 ρ 2 (x,y,s,s ) Needed to avoid unwanted collisions For colliders with many bunches: LHC, CESR, KEKB For colliders with coasting beams

15 Collisions angle geometry (horizontal plane) x = x cos φ/2 s sin φ/2 1 x s = s cos 2 φ/2 + x sin φ/2 s = s cos 1 φ/2 + x sin φ/2 φ/2 φ/2 s x = x cos 2 beam 2 φ/2 s sin φ/2 beam 1

16 Crossing angle Assume crossing in horizontal (x, s)- plane. Transform to new coordinates: x 1 = x cos φ 2 s sin φ 2, s 1 = 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 1N 2 fb dxdydsds 0 ρ 1x (x 1 )ρ 1y (y 1 )ρ 1s (s 1 s 0 )ρ 2x (x 2 )ρ 2y (y 2 )ρ 2s (s 2 + s 0 )

17 Integration (crossing angle) use as before: and: Further: + + e at2 dt = π/a e (at2 +bt+c) dt = π/a e b2 ac a Since σ x, x and sin(φ/2) are small: drop all terms σ k xsin l (φ/2) or x k sin l (φ/2) for all: k+l 4 Approximate: sin(φ/2) tan(φ/2) φ/2

18 Crossing angle Crossing Angle L = N 1N 2 fb 4πσ x σ y S S is the reduction factor For small crossing angles and σ s σ x,y S = ( σ s σ x tan φ 2 ) ( σ s φ σ x 2 )2 Example LHC: Φ = 285 µrad, σ s = 7.5 cm, S = 0.81

19 Offset and crossing angle x 1 x x 2 beam 1 beam 2 x 1 x 2 s 1 d 2 d 1 s 1 φ/2 s s 2 s 2

20 Offset and crossing angle Transformations with offsets in crossing plane: x 1 = d 1 + x cos φ s sin φ, s = 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 0 : L = L 0 2πσ s σ x 2 cos 2 φ 2 e x 2 cos 2 (φ/2)+s 2 sin 2 (φ/2) σx 2 e x2 sin 2 (φ/2)+s 2 cos 2 (φ/2) σs 2 e d 2 1 +d2 2 +2(d 1 +d 2 )x cos(φ/2) 2(d 2 d 1 )s sin(φ/2) 2σx 2 dxds.

21 Offset and crossing angle After integration over x: with: L = N 1N 2 fb 8π 3 2 σ s 2 cos φ 2 A = sin2 φ 2 σ 2 x + cos2 φ 2 σ 2 s + W e (As2 +2Bs) σ x σ y B = (d 2 d 1 ) sin(φ/2) 2σ 2 x ds and W = e 1 4σx 2 (d 2 d 1 ) 2 = Luminosity with correction factors

22 Luminosity with correction factors L = N 1N 2 fb 4πσ x σ y W e B 2 A S W : correction of beam offset S: correction for crossing angle e B2 A : correction for crossing angle and offset

23 Hour glass effect 4 beta at IP: 0.50 m beta at IP: 0.15 m 2 beta function Distance from bunch centre β-functions depends on position s

24 Hour glass effect β-functions depends on position s Usually: β(s) = β (1 + ( ) s 2) β i.e. σ = σ(s) const. σ(s) = σ (1 + ( ) s 2) β Important when β comparable to the r.m.s. bunch length σ s

25 Hour glass effect Take it easy: β x = β y, crossing angle, but no offset Replace σ by σ(s) in standard formulae ( ) N1 N 2 fb 2 cos φ + 2 e s2 A L = 8π πσs σxσ y[1 + ( s ) β 2 ] ds A = sin 2 φ 2 (σx) 2 [1 + ( s ) β 2 ] + cos2 φ 2 σ 2 s Numerical Integration

26 Calculations for the LHC N 1 = N 2 = particles/bunch B = 2808 bunches/beam f = khz, φ = 285 µrad β x = β y = 0.55 m σ x = σ y = 16.6 µm, σ s = 7.7 cm

27 Simplest case (Head on collision): L = cm 2 s 1 Effect of crossing angle: L = cm 2 s 1 Effect of crossing angle & Hourglass: L = cm 2 s 1

28 Exercise: If the beams are not Gaussian?? Assume flat distributions (normalized to 1) ρ 1 = ρ 2 = 1 2a, for [ a z a], z = x, y Calculate r.m.s. in x and y: and < (x, y) 2 > = L = + + Compute: L < x 2 > < y 2 > (x, y) 2 ρ(x, y)dxdy ρ 1 (x, y)ρ 2 (x, y) dxdy Repeat for various distributions and compare

29 L int (s) = Integrated luminosity T 0 +s s L(s, t)ds dt The real figure of merit: L int (s) σ p = number of events Experiments: continuous recording of L For studies: assume some life time behaviour. E.g. L(t) L 0 exp ( ) t τ Contributions to life time from: intensity decay, emittance growth etc.

30 Integrated luminosity Knowledge of preparation time allows optimization of L int

31 Integrated luminosity Typical run times LEP: 8-10 hours t r For LHC long preparation time t p expected Optimum combination of t r and t p gives maximum luminosity t r is usually a free parameter, i.e. can be chosen

32 Maximising Integrated Luminosity Assume exponential decay of luminosity L(t) = L 0 e t/τ Average luminosity < L > < L >= t r 0 dtl(t) t r +t p = L 0 τ 1 e t r/τ t r +t p (Theoretical) maximum for: t r τ ln(1 + 2t p /τ + t p /τ) Example LHC (OB): t p 10h, τ 15h, t r 15h Exercise: Would you improve τ (long t r ) or t p?

33 Luminous region 2 s s +s Density distribution of interaction vertices Fraction of collisions occur ± s from the IP? Important for experiments!

34 Luminous region Depends on σ x, σ y, σ s and crossing angle φ Integrate only along a finite longitudinal length: L 0 = L(S) = + L(s )ds L(S) = ( N1 N 2 fb 8πσ xσ y +S S L(s )ds ) 2 cos φ 2 πσs π A erf ( A S )

35 Some results for LHC σ s = 7.7 cm, β = 0.55 m, φ = 285 µrad: 100% lumi s = ± 12 cm 90% lumi s = ± 7 cm 80% lumi s = ± 5.5 cm

36 Interactions per crossing Luminosity/fB N 1 N 2 In LHC: crossing every 25 ns Per crossing approximately 20 interactions May be undesirable (pile up in detector) = more bunches B, or smaller N?? Beware: maximum (peak) luminosity L max is not the whole story...!

37 Luminosity measurement One needs to get a signal proportional to interaction rate Beam diagnostics Large dynamic range: cm 2 s 1 to cm 2 s 1 Very fast, if possible for individual bunches Used for optimization For absolute luminosity need calibration

38 Luminosity calibration (e + e ) Use well known and calculable process e + e e + e elastic scattering (Bhabha scattering) Have to go to small angles (σ el Θ 3 ) Small rates at high energy (σ el 1 E 2 )

39 Luminosity calibration x e+ e e+ e monitors Measure coincidence at small angles Low counting rates, in particular for high energy! Background may be problematic

40 Luminosity calibration (hadrons, e.g. pp or p p) Must measure beam current and beam sizes Beam 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)

41 Luminosity optimization Van der Meer scan Record counting rates R(d) as function of movement d 0.8 Counting rate Since R(d) is proportional to luminosity L(d) Get ratio of luminosity amplitude L(d)/L(0)

42 Luminosity optimization From ratio of luminosity L(d)/L 0 Remember: W = e 1 4σ 2 (d 2 d 1 ) 2 Determines σ... and centres the beams! beam-beam deflection scans... =

43 Absolute value of L (pp or p p) By 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 0 dt L = (1 + ρ2 ) 16π = (1 + ρ 2 ) σ2 tot 16π = 1 L (N inel + N el ) 2 (dn el /dt) t=0 dn el dt t=0 Luminosity determined from experimental rates

44 Absolute value of L (pp or p p) By Coulomb normalization: Coulomb amplitude exactly calculable: = 1 dn el L dt t=0 = π f C + f N 2 dσ el lim t 0 dt π 2α em t + σ tot (ρ + 4π i)eb t 2 Fit gives: σ tot, ρ, B and L 2 4πα2 em t 2 t 0 Can be done measuring only elastic scattering (No N inel needed!) = Roman pots

45 dn/dt Differential elastic cross section dn/dt UA4/ Fit strong part Fit Coulomb part Measure dn/dt at small t (0.01 < GeV ) and extrapolate to t = 0.0 Needs special optics to allow measurement at very small t 1000 Measure total counting rate N el + N inel Needs good detector coverage Often use slightly modified method, precision 1 2 % t (GeV ) 10

46 Not treated : Coasting beams (e.g. ISR) Asymmetric colliders (e.g. PEP, HERA) Linear colliders (SLC, TESLA etc.)

47 How to cook high Luminosity? Get high intensity Get small beam sizes (small ɛ and β ) Get short bunches Get many bunches Get exact head-on collisions