Particle Colliders and Concept of Luminosity. Werner Herr, CERN

Transcription

1 Particle Colliders and Concept of Luminosity (or: explaining the jargon...) Werner Herr, CERN luminosity.pdf Werner Herr, Particle Colliders, CAS 2012, Granada

2 Particle Colliders and Concept of Luminosity (or: explaining the jargon )...) ) (beta*, squeeze, femtobarn, inverse femtobarn, lumi scan, crossing angle, filling schemes, pile-up, hour glass effect, crab crossing...) Werner Herr, Particle Colliders, CAS 2012, Granada

3 Particle colliders? Used in particle physics Look for rare interactions Want highest energies Many interactions (events) Figures of merit for a collider: energy number of collisions

4 Rare interactions and high energy 2 σ(pp H+X) [pb] LHC HIGGS XS WG pp H+X at 10 pp H+X at s=14 TeV 1 s=8 TeV pp H+X at s=7 TeV M H [GeV] Often seen: cross section σ for Higgs particle in LHC Two parameters: s and pb

5 Energy: why colliding beams? Two particles: E 1, p 1,E 2, p 2,m 1 = m 2 = m E cm = s = (E 1 +E 2 ) 2 ( p 1 + p 2 ) 2 Collider versus fixed target: Fixed target: p 2 = 0 s = 2m2 +2E 1 m Collider: p 1 = p 2 s = E1 +E 2 LHC (pp): 8000 GeV versus 87 GeV LEP (e + e ): 210 GeV versus?

6 Rare interactions and cross section Cross section σ measures how often a process occurs Characteristic for a given process Measured in: barn = b = m 2 (pb = m 2 ) More common: barn = b = cm 2 (pb = cm 2 ) We have for the LHC energy: σ(pp X) 0.1 b and σ(pp X +H) b σ(pp X+H γγ) b = 50 fb (femtobarn) VERY rare (one in ), need many collisions...

7 Types of particle colliders Basic requirement: (at least) two beams Can be realized as: Double ring colliders Single ring colliders Linear colliders Some more exotic: gas jets,... Remark: ring colliders are usually storage rings

8 Double ring colliders (LHC, RHIC, ISR, HERA,...) IP5 IP4 IP6 IP3 beam1 beam2 IP7 IP2 IP8 IP1 Can accelerate and collide (in principle) any type of particles (p-p, p-pb, e-p,..) Usually requires crossing angle

9 Single ring colliders (ADA, LEP, PEP, SPS, Tevatron,...) e+, p e, p Can accelerate and collide particle and anti-particle Usually no crossing angle

10 Linear colliders (SLC, CLIC, ILC,...) X e+ e Mainly used (proposed) for leptons (reduced synchrotron radiation) With or without crossing angle

11 Collider challenges Circular colliders, beams are reused many time, efficient use for collisions Requires long life time of beams (up to 10 9 turns) Linear colliders, beams are used once, inefficient use for collisions Very small beam sizes for collisions, beam stabilization Power consumption becomes an issue

12 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: pile-up) Space structure of interactions (size of interaction region: vertex density) Quality of interactions (background, dead time etc.)

13 Figure of merit: 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

14 Fixed target luminosity dr dt = Φ ρ l {T σ p L ρ T = const. l Flux: Φ = N/s Interaction rate from flux and target density and size

15 Fixed target luminosity dr dt = Φ ρ l {T σ p L ρ T = const. l Flux: Φ = N/s In a collider: target is the other beam... (and it is moving!)

16 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 KN 1 N 2 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 = ( v 1 v 2 ) 2 ( v 1 v 2 ) 2 /c 2

17 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 f n b + 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 Mostly use Gaussian ρ for analytic calculation (in general: it is a good approximation)

18 Gaussian distribution functions Transverse: ρ z (u) = 1 σ u 2π exp ( u2 2σ 2 u ) u = x,y Longitudinal: ρ s (s±s 0 ) = 1 σ s 2π exp ( (s±s 0) 2 2σ 2 s For non-gaussian profiles not always possible to find analytic form, need a numerical integration )

19 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

20 Integration (head-on) for beams of equal size: σ 1 = σ 2 ρ 1 ρ 2 = ρ 2 : L = 2 N 1N 2 fn b ( 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 fn b 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 fn b 4πσ x σ y

21 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 = L = N 1N 2 fn b 4πσ x σ y L = N 1 N 2 fn b 2π σ1x 2 +σ2x 2 σ 2 1y +σ2y 2 Here comes β : σ x,y = ǫ β x,y β is the β-function at the collision point!

22 Special optics for colliders We had in the simple case: L = N 1N 2 fn b 4πσ x σ y and: σ x,y = ǫ β x,y For high luminosity need small β x,y Done with special regions Special arrangement of quadrupoles etc. So-called low β insertions

23 Example low β insertion (LHC) 4500 low beta insertion LHC CMS beta x (m) s (m) In regular lattice: 180 m At collision point (squeezed optics) : 0.5 m Must be integrated into regular optics

24 Examples: some circular colliders 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/

25 What else? What about linear colliders? See later...

26 Complications Crossing angle Hour glass effect Collision offset (wanted or unwanted) Non-Gaussian profiles Dispersion at collision point Displaced waist ( β / s = α 0) Strong coupling etc.

27 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

28 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

29 Crossing angle Assume crossing in horizontal (x, s)- plane. Transform to new coordinates: x 1 = xcos φ 2 ssin φ 2, s 1 = scos φ 2 +xsin φ 2, x 2 = xcos φ 2 +ssin φ 2, s 2 = scos φ 2 xsin φ 2 + L = 2cos 2 φ 2 N 1N 2 fn b 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 )

30 Integration (crossing angle) use as before: and: + + e at2 dt = π/a e (at2 +bt+c) dt = π/a e b2 ac a Further: 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

31 Crossing angle Crossing Angle L = N 1N 2 fn b 4πσ x σ y S S is the geometric factor For small crossing angles and σ s σ x,y S = 1 1+( σ s σ x tan φ 2 )2 1 1+( σ s φ σ x 2 )2 Example LHC (at 7 TeV): Φ = 285 µrad, σ x 17 µm, σ s = 7.5 cm, S = 0.84

32 Large crossing angle Large crossing angle: large loss of luminosity crab crossing can recover geometric factor

33 crab crossing scheme Done with transversely deflecting cavities (if you wondered what they can be used for) Feasibility needs to be demonstrated

34 Offset and crossing angle x beam 1 beam 2 x 1 x 2 φ/2 s s 2

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

36 Offset and crossing angle Transformations with offsets in crossing plane: x 1 = d 1 +xcos φ 2 ssin φ 2, s 1 = scos φ 2 +xsin φ 2, x 2 = d 2 +xcos φ 2 +ssin φ 2, s 2 = scos φ 2 xsin φ 2 Gives after integration over y and s 0 : L = L 0 2πσ s σ x 2cos 2 φ 2 2 cos 2 (φ/2)+s 2 sin 2 (φ/2) e x 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 )xcos(φ/2) 2(d 2 d 1 )ssin(φ/2) 2σ 2 x dxds.

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

38 Luminosity with correction factors L = N 1N 2 fn b 4πσ x σ y W e B 2 A S W: correction for beam offset (one per plane) S: correction for crossing angle e B2 A : correction for crossing angle and offset (if in the same plane)

39 Hour glass effect 4500 low beta insertion LHC CMS beta x (m) s (m) Remember the insertion: β-functions depends on position s

40 Hour glass effect 4500 low beta insertion LHC CMS beta x (m) s (m) Remember the insertion: β-functions depends on position s

41 Hour glass effect hourglass effect beta = 0.05 m beta = 0.50 m 3 beta function s (m) In our low β insertion we have: β(s) β (1+ ( ) 2) s β For small β the beam size grows very fast!

42 Hour glass effect beam size (micron) hourglass effect beta = 0.05 m beta = 0.50 m s (m) Beam size σ ( β (s)) depends on position s

43 Hour glass effect Beam size has shape of an Hour Glass

44 Hour glass effect beam size (micron) hourglass effect beta = 0.05 m beta = 0.50 m s (m) Beam size σ ( β (s)) depends on position s

45 Hour glass effect - short bunches hourglass effect beam size (micron) beta = 0.05 m beta = 0.50 m s (m) Small variation of beam size along bunch

46 Hour glass effect - long bunches hourglass effect beam size (micron) beta = 0.05 m beta = 0.50 m s (m) Significant effect for long bunches and small β

47 Hour glass effect β-functions depends on position s Need modification to overlap integral 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 (or smaller!)

48 Hour glass effect Using the expression: u x = β /σ s Without crossing angle and for symmetric, round Gaussian beams we get the relative luminosity reduction as: L(σ s ) + L(0) = 1 π e u2 [1+( u u x ) 2 ] du = π u x e u2 x erfc(ux ) L(σ s ) = L(0) H with : H = π u x e u2 x erfc(ux )

49 Hour glass effect 1 Hourglass effect - head on collisions L(s)/L(0) L(s)/L(0) beta*/sigma_s Hourglass reduction factor as function of ratio β /σ s. A lesson: small β does not always lead to high luminosity!

50 Luminosity with (more) correction factors L = N 1N 2 fn b 4πσ x σ y W e B 2 A S H W: correction for beam offset S: correction for crossing angle e B2 A : correction for crossing angle and offset H: correction for hour glass effect

51 Calculations for the LHC N 1 = N 2 = particles/bunch n 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

52 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 Most important: effect of crossing angle

53 1]1e3 Luminosity in LHC as function of time Luminosity [( bs) CMS ATLAS LHCb Time [h] (Courtesy X. Buffat) Luminosity evolution in LHC during 2 typical days Run time up to 15 hour Preparation time 3-4 hours

54 What really counts: Integrated luminosity L int = T 0 L int σ p = L(t)dt T 0 events observed of process p L(t)dt σ p = total number of Unit is: cm 2, i.e. inverse cross-section Often expressed in inverse barn 1 fb 1 (inverse femto-barn) is cm 2 for 1 fb 1 : requires 10 6 s at L = cm 2 s 1

55 What really counts: Integrated luminosity What does it mean? Assume: You have accumulated 20 fb 1 (inverse femto-barn) You are interested in σ(pp X +H γγ) 50 fb (femtobarn) You have 20 fb 1 50 fb = 1000 You have 1000 events of interest in your data!!

56 Integrated luminosity Luminosity decays as function of time 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. Aim: optimize integrated luminosity, taken into account preparation time

57 Integrated luminosity Knowledge of preparation time and luminosity decay allows optimization of L int

58 Integrated luminosity Typical run times LHC: t r 8-15 hours For optimization: Need to know preparation time t p very important: good model of luminosity evolution as function of time L(t) depends on many parameters!!

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

60 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

61 Luminosity calibration Remember the basic definition: dr dt = L σ p For a well known and calculable process we know σ p The experiments measure the counting rate dr dt for this process Get the absolute, calibrated luminosity

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

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

64 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, ISR LHC 2012)

65 Luminosity optimization 1.2 Van der Meer scan Record counting rates R(d) 1 as function of movement d 0.8 Counting rate Since R(d) is proportional to luminosity L(d) amplitude Get ratio of luminosity L(d)/L(0)

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

67 Absolute value of L (pp or p p) By Coulomb normalization: Coulomb amplitude exactly calculable: dσ el lim = 1 dn el t 0 dt L dt t=0 = π f C +f N 2 π 2α em + σ tot t 4π (ρ+i)eb t 2 2 4πα2 em t 2 t 0 Fit gives: σ tot,ρ,b and L Can be done measuring elastic scattering at small angles

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

69 Luminosity in linear colliders Mainly (only) e+e colliders Past collider: SLC (SLAC) Under consideration: CLIC, ILC Special issues: Particles collide only once (dynamics)! Particles collide only once (beam power)! Must be taken into account

70 Luminosity in linear colliders Basic formula: From : L = N2 f n b 4πσ x σ y to : L = N2 f rep n b 4πσ x σ y Replace frequency f by repetition rate f rep. And introduce effective beam sizes σ x,σ y : L = N2 f rep n b 4πσ x σ y

71 Final focusing in linear colliders β 1/2 [m 1/2 ] Diagnostics Energy collimation Transverse collimation β x 1/2 β y 1/2 D x Longitudinal location [km] Final Focus system D [m] (Courtesy R. Tomas) Final Focus and Beam delivery System At the end of the beam line only! Smaller β in linear colliders (10 mm 0.2 mm!!)

72 Luminosity in linear colliders Using the enhancement factor H D : L = N2 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 )

73 Pinch effect - disruption beam-beam collision s Additional focusing by opposing beams

74 Pinch effect - disruption beam-beam collision s Additional focusing by opposing beams

75 Luminosity in linear colliders For weak disruption D 1 and round beams: H D = π D + O(D2 ) For strong disruption and flat beams: computer simulation necessary, maybe can get some scaling

76 Beamstrahlung Disruption at interaction point is basically a strong bending Results in strong synchrotron radiation: beamstrahlung This causes (unwanted): Spread of centre-of-mass energy Pair creation and detector background Again: luminosity is not the only important parameter

77 Beamstrahlung Parameter Y Measure of the mean field strength in the rest frame normalized to critical field B c : with: Y = < E +B > B c 5 6 r 2 eγn ασ z (σ x +σ y ) B c = m2 c 3 e h G

78 Energy loss and power consumption Average fractional energy loss δ E : δ E = 1.24 ασ zm e λ C E Y (1+(1.5Y) 2/3 ) 1/2 where E is beam energy at interaction point and λ C the Compton wavelength.

79 Luminosity in linear colliders Using the beam power P b and beam energy E in the luminosity: L = H D N 2 f rep n b 4πσ x σ y L = H D N P b ee 4πσ x σ y Beam power P b related to AC power consumption P AC via efficiency η AC b P b = η AC b P AC

80 Figure of merit in linear colliders Luminosity at given energy normalized to power consumption and momentum spread due to beamstrahlung: M = LE δb P AC With previous definition (and reasonably small beamstrahlung) this becomes: M = LE δb P AC ηac b ǫ y These are optimized in the linear collider design

81 Not treated : Coasting beams (e.g. ISR) Asymmetric colliders (e.g. PEP, HERA, LHeC) All concepts can be formally extended...

82 How to cook high Luminosity? Get high intensity Get small beam sizes (small ǫ and β ) Get many bunches Get small crossing angle (if any) Get exact head-on collisions Get short bunches

83 Luminosity in a nutshell L = N 1N 2 fn b 4πσ x σ y W e B 2 A S H

84 Luminosity in a nutshell L = N 1N 2 fn b 4πσ x σ y W e B 2 A S H Are there limits to what we can do?

85 Luminosity in a nutshell L = N 1N 2 fn b 4πσ x σ y W e B 2 A S H Are there limits to what we can do? Yes, there are beam-beam effects In LHC: collisions with the other beam per fill!!

86 Luminosity in a nutshell L = N 1N 2 fn b 4πσ x σ y W e B 2 A S H

87 Summary Colliders are used exclusively for particle physics experiments Colliders are the only tools to get highest centre of mass energies Type of collider is decided by the type of particles and use Design and performance must take into account the needs of the experiments

88 Energy: 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): 8000 GeV versus 87 GeV LEP (e + e ): 210 GeV versus 0.5 GeV!!! LC (e + e ): 2000 GeV versus 1.0 GeV!!!

89 Bibliography Some bibliography in the hand-out Luminosity lectures and basics: W. Herr, Concept of Luminosity, CERN Accelerator School, Zeuthen 2003, in: CERN (2006). A. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering, World Scientific, (1998).

90 - BACKUP SLIDES -

91 Rare interactions and high energy 10 σ(pp H+X) [pb] 10 2 LHC HIGGS XS WG s= 8 TeV pp H (NNLO+NNLL QCD + NLO EW) pp qqh (NNLO QCD + NLO EW) pp WH (NNLO QCD + NLO EW) pp tth (NLO QCD) pp ZH (NNLO QCD +NLO EW) M H [GeV] Often seen: cross section σ for Higgs particle Typical channels

92 Rare interactions and high energy

93 BR [pb] σ τ + τ VBF H τ + τ ± WH l νbb - s = 8TeV ± WW l νqq + - WW l νl ν LHC HIGGS XS WG ZZ l l qq + - ZZ l l νν + - ZZ l l + - l l ZH l l bb l = e, µ ν = ν e,ν µ,ν τ q = udscb γγ tth ttbb M H [GeV]

94 Often seen: cross section σ for Higgs particle Typical channels

95 Large crossing angle CP x IP For large amplitude particles: collision point longitudinally displaced

96 Large crossing angle CP x IP For large amplitude particles: collision point longitudinally displaced Can introduce coupling (transverse and synchro betatron, bad for flat beams)

97 Large crossing angle β y x x x IP A particle s collision point amplitude dependent Different (vertical) β functions at collision points

98 Large crossing angle x x x IP A particle s collision point amplitude dependent Different β functions at collision points (hour glass!)

99 crab waist scheme x x x IP Make vertical waist (βy min ) amplitude (x) dependent All particles in both beams collide in minimum β y region

100 crab waist scheme Make vertical waist (minimum of β) amplitude (x) dependent Without details: can be done with two sextupoles First tried at DAPHNE (Frascati) in 2008 Geometrical gain small Smaller vertical tune shift as function of horizontal coordinate Less betatron and synchrotron coupling Good remedy for flat (i.e. lepton) beams with large crossing angle

101 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

102 Maximising Integrated Luminosity Assume exponential decay of luminosity L(t) = L 0 e t/τ Average (integrated) 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: t p 10h, τ 15h, t r 15h Exercise: Would you improve τ (long t r ) or t p?