Phenomenology of hadronic collisions

Transcription

1 Phenomenology of hadronic collisions Grégory Soyez CERN & IPhT, CEA Saclay BND summer school Oostende, Belgium September p. 1

2 Plan You are now experts in computing Feynman diagrams You (hopefully) want to know how to compute things at hadronic colliders (the LHC in particular) p. 2

3 Disclaimer The physics of hadronic colliders is a very vast topic: p. 3

4 Disclaimer The physics of hadronic colliders is a very vast topic: ATLAS TDR (Detector and Physics Performance): 1852 pages CMS TDR (2 volumes): 1317 pages A good coverage of basic topics in collider physics: QCD and Collider Physics, R. K. Ellis, W. J. Stirling and B. R. Webber (447 pages) p. 3

5 Disclaimer The physics of hadronic colliders is a very vast topic: ATLAS TDR (Detector and Physics Performance): 1852 pages CMS TDR (2 volumes): 1317 pages A good coverage of basic topics in collider physics: QCD and Collider Physics, R. K. Ellis, W. J. Stirling and B. R. Webber (447 pages) I won t be able to cover all that in 6+2 hours! p. 3

6 Plan #2 How to describe a collision between 2 hadrons? p. 4

7 The very fundamental collision σ = f a f b ˆσ f a take a parton out of each proton f a parton distribution function (PDF) for quark and gluons a big chapter of these lectures ˆσ hard matrix element perturbative computation Forde-Feynman rules f b p. 5

8 The more realistic version Hard ME perturbative p. 6

9 The more realistic version Hard ME perturbative Parton branching initial+final state radiation p. 6

10 The more realistic version Hard ME perturbative Parton branching initial+final state radiation Hadronisation q,g hadrons p. 6

11 The more realistic version Hard ME perturbative Parton branching initial+final state radiation Hadronisation q,g hadrons Multiple interactions Underlying event (UE) p. 6

12 The more realistic version Hard ME perturbative Parton branching initial+final state radiation Hadronisation q,g hadrons Multiple interactions Underlying event (UE) Pile-up 25 pp at the LHC p. 6

13 Step by step... We shall investigate those effects one by one: e + e collisions for QCD final state (and hadronisation) ep collisions aka Deep Inelastic scattering (DIS) for the Parton Distribution Functions pp collisions: put everything together kinematics Monte-Carlo jets + various processes (W/Z, Higgs, top,...) p. 7

14 Step by step... We shall investigate those effects one by one: e + e collisions for QCD final state (and hadronisation) ep collisions aka Deep Inelastic scattering (DIS) for the Parton Distribution Functions pp collisions: put everything together kinematics Monte-Carlo jets + various processes (W/Z, Higgs, top,...) p. 7

15 Tutorial The plan is to play with Pythia 8 (the C++ version) and FastJet. You can get them (and a few sample codes) from the link at p. 8

16 e + e collisions p. 9

17 QCD final state e + e collisions give QCD final state without initial-state/beam contamination e + q s e q Useful for many QCD studies Intermediate state can be γ or Z, we only consider γ for simplicity p. 10

18 QCD final state: basic QCD p 1 p 2 e + e s q q k 1 k 2 p 1 p 2 k 1 k 2 s 2 (0,0,1,1) s 2 (0,0, 1,1) s 2 (sin(θ),0,cos(θ),1) s 2 ( sin(θ),0, cos(θ),1) p. 11

19 QCD final state: basic QCD p 1 p 2 e + e s q q k 1 k 2 p 1 p 2 k 1 k 2 s 2 (0,0,1,1) s 2 (0,0, 1,1) s 2 (sin(θ),0,cos(θ),1) s 2 ( sin(θ),0, cos(θ),1) dσ dcos(θ) = e2 qn c πα 2 e 2s [1+cos2 (θ)] σ(e + e q q) = N c ( q e2 q ) σ 0 σ 0 = 4πα2 e 3s p. 11

20 QCD final state: basic QCD p 1 e + s q k 1 p 1 e + s µ + k 1 p 2 e q k 2 p 2 e µ k 2 dσ dcos(θ) = e2 qn c πα 2 e 2s [1+cos2 (θ)] σ(e + e q q) = N c ( q e2 q ) σ 0 σ(e + e µ + µ ) = σ 0 σ 0 = 4πα2 e 3s p. 11

21 QCD final state: basic QCD R = σ(e+ e hadrons) σ(e + e µ + µ ) N c ( q e 2 q ) u,d,s: R = 3 ( ) = 2 Test of u,d,s,c: R = 3 ( ) = 10 3 u,d,s,c,b: R = 3 ( ) = 14 3 The 3 colours in QCD (N c = 3) The number of quark flavours p. 12

22 QCD final state: basic QCD Test of 10 2 R = σ(e+ e 10 hadrons) σ(e + ρ e µ + µ ) Sum of exclusive measurements u,d,s: R = 3 ( ) 1 J/ψ ψ(2s) 6 Mark-I = 2 5 u, d, s ( ρ N c q u,d,s,c: R = 3 ( R ) 4 = 10 4 Crystal Ball 3 u,d,s,c,b: R = 3 ( ) 4 = The 3 colours in QCD (N c = 3) 4 3 MD-1 The number of quark flavours 2 ω φ 3 3 loop pqcd Naive quark model 3 e 2 q Inclusive measurements ) Mark-I + LGW Mark-II PLUTO DASP BES ψ 3770 ψ 4040 ψ 4160 ψ Υ(1S) Υ(2S) Υ(3S) ARGUS CLEO CUSB DHHM Crystal Ball CLEO II DASP LENA s [GeV] Υ(4S) c b p. 12

23 QCD final state: basic QCD Test of 10 2 R = σ(e+ e 10 hadrons) σ(e + ρ e µ + µ ) Sum of exclusive measurements u,d,s: R = 3 ( ) 1 J/ψ ψ(2s) 6 Mark-I = 2 5 u, d, s ( ρ N c q u,d,s,c: R = 3 ( R ) 4 = 10 4 Crystal Ball 3 u,d,s,c,b: R = 3 ( ) 4 = The 3 colours in QCD (N c = 3) 4 3 MD-1 The number of quark flavours 2 ω φ 3 3 loop pqcd Naive quark model 3 e 2 q Inclusive measurements ) Mark-I + LGW Mark-II PLUTO DASP BES ψ 3770 ψ 4040 ψ 4160 ψ Υ(1S) Υ(2S) Υ(3S) ARGUS CLEO CUSB DHHM Crystal Ball CLEO II DASP LENA s [GeV] Υ(4S) Q: why σ(e + e µ + µ ) and not σ(e + e e + e )? p. 12 c b

24 QCD final state: QCD dynamics p 1 e + k 1,x 1 3 (4 1) 4 = 5 d.o.f. s k 3,x 3 3 Euler angles p 2 e k 2,x 2 x i = 2E i / s, x 1 +x 2 +x 3 = 2 or θ 13, θ 23 dφ 3 = = 3 d 3 k i (2π) 3 (2π) 4 δ (4) (p 1 +p 2 k 1 k 2 k 3 ) 2E i i=1 s 32(2π) 5 dαdcosβdγ dx 1 dx 2 cos(θ 13 ) = x2 1 +x2 3 x2 2 2x 1 x 3 cos(θ 23 ) = x2 2 +x2 3 x2 1 2x 2 x 3 p. 13

25 QCD final state: QCD dynamics e p + 1 e k 1,x 1 s k 3,x 3 + p 2 k 2,x 2 k 1,x 1 k 3,x 3 k 2,x 2 M 2 = 4(4π) 3 α 2 eα s C F N c (p 1.k 1 ) 2 +(p 1.k 2 ) 2 +(p 2.k 1 ) 2 +(p 2.k 2 ) 2 s(k 1.k 3 )(k 2.k 3 ) d 2 σ = e 2 α s C F dx 1 dx qn c σ 0 2 2π x 2 1 +x2 2 (1 x 1 )(1 x 2 ) p. 14

26 QCD final state: QCD dynamics e p + 1 e k 1,x 1 s k 3,x 3 + p 2 k 2,x 2 k 1,x 1 k 3,x 3 k 2,x 2 M 2 = 4(4π) 3 α 2 eα s C F N c QCD coupling e + e q q (p 1.k 1 ) 2 QCD +(p 1.k colour 2 ) 2 +(p 2.k 1 ) 2 +(p 2.k 2 ) 2 s(k 1.k 3 )(k 2.k 3 ) QCD dynamics d 2 σ = e 2 α s C F dx 1 dx qn c σ 0 2 2π x 2 1 +x2 2 (1 x 1 )(1 x 2 ) p. 14

27 QCD final state: QCD dynamics p 1 e + k 1,x 1 M 2 (p 1.k 1 ) 2 +(p 1.k 2 ) 2 +(p 2.k 1 ) 2 +(p 2.k 2 ) 2 s(k 1.k 3 )(k 2.k 3 ) s k 3,x 3 d 2 σ dx 1 dx 2 = e 2 α qn c σ s C F 0 2π x 2 1 +x2 2 (1 x 1 )(1 x 2 ) p 2 e k 2,x 2 c θ13= x2 1 +x2 3 x2 2 x i = 2E i s x 1 +x 2 +x 3 =2 2x 1 x 3 p. 15

28 QCD final state: QCD dynamics p 1 e + k 1,x 1 M 2 (p 1.k 1 ) 2 +(p 1.k 2 ) 2 +(p 2.k 1 ) 2 +(p 2.k 2 ) 2 s(k 1.k 3 )(k 2.k 3 ) s k 3,x 3 d 2 σ dx 1 dx 2 = e 2 α qn c σ s C F 0 2π x 2 1 +x2 2 (1 x 1 )(1 x 2 ) p 2 e k 2,x 2 c θ13= x2 1 +x2 3 x2 2 x i = 2E i s x 1 +x 2 +x 3 =2 2x 1 x 3 Divergent when k 1.k 3 0 or k 2.k 3 0 k 1,x 1 k 1.k 3 0 (k 1 +k 3 ) 2 0 i.e. θ 13 parent quark propag = 1 (k 1 +k 3 ) 2 k 3,x 3 Physical origin of the divergence! They are infrared divergences ((k 1 +k 3 ) 2 0, not ) (one power cancelled by phase-space log divergence) p. 15

29 QCD final state: QCD dynamics p 1 e + k 1,x 1 M 2 (p 1.k 1 ) 2 +(p 1.k 2 ) 2 +(p 2.k 1 ) 2 +(p 2.k 2 ) 2 s(k 1.k 3 )(k 2.k 3 ) s k 3,x 3 d 2 σ dx 1 dx 2 = e 2 α qn c σ s C F 0 2π x 2 1 +x2 2 (1 x 1 )(1 x 2 ) p 2 e k 2,x 2 c θ13= x2 1 +x2 3 x2 2 x i = 2E i s x 1 +x 2 +x 3 =2 2x 1 x 3 Divergent when x 1 (or x 2 ) 1 k 1,x 1 1 x 2 = 1 2 x 1x 3 [1 cos(θ 13 )] θ 13 0 (or θ 23 ): collinear divergence divergence x 3 0 (i.e. E g 0): soft divergence θ 13 k 3,x 3 p. 15

30 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) we will meet them often through these lectures p. 16

31 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) also present for g gg ( QED; C F C A ) p. 16

32 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) also present for g gg ( QED; C F C A ) cancelled by virtual corrections p 1 e + k 1,x 1 p 1 e + k 1,x 1 s k 3,x 3 s p 2 e k 2,x 2 p 2 e k 2,x 2 Real Virtual p. 16

33 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) also present for g gg ( QED; C F C A ) cancelled by virtual corrections Dimensional regularisation d = 4 2ε: σ (q qg) real σ (q qg) virt = e 2 qn c σ 0 α s C F 2π T(ǫ) [ 2 ε ε O(ε) ] = e 2 qn c σ 0 α s C F 2π T(ǫ) [ 2 ε 2 3 ε 8+O(ε) ] σ (q qg) O(α s ) = e2 qn c σ 0 3α s C F 4π = e 2 qn c σ 0 α s π p. 16

34 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) also present for g gg ( QED; C F C A ) cancelled by virtual corrections cancellation order-by-order in perturbation theory Block-Nordsieck, Kinoshita-Lee-Nauenberg theorems p. 16

35 QCD final state: coll and soft divergences Collinear and soft divergences fundamental/omnipresent in QCD! (also in QED) also present for g gg ( QED; C F C A ) cancelled by virtual corrections cancellation order-by-order in perturbation theory Block-Nordsieck, Kinoshita-Lee-Nauenberg theorems Terminology issue: soft divergence sometimes called infrared divergence (though both soft and coll are infrared) p. 16

36 QCD final state: IRC safety Cancellation of divergence not true for any observable Example: number of partons in the final state, dp/dn LO (O(α 0 s)): dp/dn = δ(n 2) NLO (O(α 1 s)): (i) real emission: n = 3 (ii) virtual correction: n = 2 dp/dn = [1 α s ]δ(n 2)+ α s δ(n 3) p. 17

37 QCD final state: IRC safety Cancellation of divergence not true for any observable Example: number of partons in the final state, dp/dn LO (O(α 0 s)): dp/dn = δ(n 2) NLO (O(α 1 s)): (i) real emission: n = 3 (ii) virtual correction: n = 2 dp/dn = [1 α s ]δ(n 2)+ α s δ(n 3) Observables for which cancellation happens are called INFRARED-AND-COLLINEAR SAFE Necessary for perturbative QCD computation to make sense!! p. 17

38 QCD final state: IRC safety Observable O: O = dψ n (k 1,...,k n ) }{{} n=0 phasespace dσ dψ n (k 1,...,k n ) }{{} matrix element O n (k 1,...,k n ) }{{} observable IR safety: adding a soft particle does not change O O n+1 (k 1,...,k n,k n+1 ) k n+1 0 = O n (k 1,...,k n ) Collinear safety: a collinear splitting does not change O O n+1 (k 1,...,λk n,(1 λ)k n ) = O n (k 1,...,k n ) for 0 < λ < 1 p. 18

39 QCD final state: IRC safety Example #1: event-shapes in e + e thrust, sphericity, thrust major, thrust minor,... Thrust: T n = max u =1 n i=0 k i. u n i=0 k i pencil-like: T 1 spherical: T 1/2 p. 19

40 QCD final state: IRC safety Example #1: event-shapes in e + e thrust, sphericity, thrust major, thrust minor,... Thrust: T n = max u =1 n i=0 k i. u n i=0 k i the thrust is infrared safe: for k n+1 0 T n+1 = max u =1 n+1 i=0 k i. u n+1 i=0 k i = max u =1 n i=0 k i. u n i=0 k i = T n the thrust is collinear safe 0 < λ < 1 { u.(λ k +(1 λ) k) = u. k λ k +(1 λ) k = k p. 19

41 QCD final state: IRC safety Example #1: event-shapes in e + e thrust, sphericity, thrust major, thrust minor,... Thrust: T n = max u =1 n i=0 k i. u n i=0 k i Computation in perturbative QCD (from the matrix element given earlier) 1 σ dσ dt = α sc F 2π [ 2(2 3T +3T 2 ) log T(1 T) ( 2T 1 1 T Allows for test of QCD (e.g. at LEP) ) 3(2 T)(3T 2) 1 T log is a reminiscence from the soft and collinear divergence ] p. 19

42 Thrust (1-T) 1/σ had dσ/d T NNLO NLO LO Q = M Z α s (M Z ) = T comparison with LEP data: peaked at T = 1 p. 20

43 e + e : QCD divergences Typical behaviour of divergences: 1 z θ Collinear limit: z 1 dσ α s 1+(1 z) 2 σ 0 2π z } {{ } splitting proba dθ 2 θ 2 }{{} coll.div For different situations (different parton types), the branching probability changes but the dθ/θ is generic! p. 21

44 e + e : QCD divergences Typical behaviour of divergences: 1 z θ Collinear limit: z Soft limit: 1 dσ α s 1+(1 z) 2 σ 0 2π z } {{ } splitting proba dθ 2 θ 2 }{{} coll.div α s C F (k 1.k 2 ) dσ q qg = dσ q q π 2 (k 1.k 3 )(k 2.k 3 ) d4 k 3 δ(k 2 ) de 3 dz E 3 z Antenna formula soft-gluon emission p. 21

45 e + e : QCD divergences Frequent appearance in computations: Both soft and collinear divergences are logarithmic the emission of a gluon comes with a factor α s log Example: soft emissions for the thrust : α s log(1 T) At some point, α s log 1 i.e. NLO LO in the perturbative series At order n, we will have α n s log n all of the same order ALL have to be considered: resummation p. 22

46 Other interests in e + e collisions Fragmentation functions parton hadron transition, D p/π (z,p t ) Hadronisation e.g. Lund strings Jets Collinear divergence a parton develops into a bunch of collimated particles We will postpone (part of) this to the hadronic collisions chapter p. 23

47 e + e : Summary e + e collisions: good framework to test QCD (final state) emission of a gluon has 2 divergences: soft and collinear cancel between real and virtual daigrams... provided the observable is IRC safe give rise to logarithms in perturbative computations... resummed to all orders when α s log 1... done analytically or by parton cascade MC collinear divergence+parton branching jets p. 24

48 Time for questions! p. 25

49 <interlude hadronic collisions> kinematics jets p. 26

50 The very fundamental collision σ = f a f b ˆσ f a take a parton out of each proton f a parton distribution function (PDF) for quark and gluons hard matrix element perturbative computation Forde-Feynman rules ˆσ f b p. 27

51 Kinematics Incoming partons: p 1 x 1 s 2 (0,0, 1,1) p 2 x 2 s 2 (0,0, 1,1) carry a fraction of the beam s (longitudinal) momentum Energy 2 in the hard collision: (p 1 +p 2 ) 2 = x 1 x 2 s s the partonic centre-of-mass is shifted/boosted compared to the lab/pp centre-of-mass need variables (longitudinally) boost-invariant p. 28

52 Kinematics Final-state particles: commonly-used variables k (k x,k y,k z,e) E(sin(θ)cos(φ),sin(θ)sin(φ),cos(θ),1) E and θ are not suited! p. 29

53 Kinematics Final-state particles: commonly-used variables Transverse plane azimuthal angle φ transverse momentum p t = p 2 x +p 2 y p. 29

54 Kinematics Final-state particles: commonly-used variables Transverse plane azimuthal angle φ transverse momentum p t = p 2 x +p 2 y Longitudinal variable ( ) Rapidity: y = 2 1 log E+pz E p z Boost: y 1 ( ) γ(e 2 log βpz )+γ(p z βe) γ(e βp z ) γ(p z βe) = 1 2 log ( γ(1 β)(e +pz ) γ(1+β)(e p z ) ) = y log ( (1 β) (1+β) ) not boost-invariant itself but y = y 2 y 1 is ( θ is not) p. 29

55 Kinematics Final-state particles: commonly-used variables Transverse plane azimuthal angle φ transverse momentum p t = p 2 x +p 2 y Longitudinal variable ( ) Rapidity: y = 2 1 log E+pz E p z k (k t cos(φ),k t sin(φ),m t sinh(y),m t cosh(y)) Transverse mass: m 2 t = k2 t +m2 Pseudo-rapidity: η = 1 2 log(tan(θ/2)) η boost-invariant if massless For massless particles: y = η p. 29

56 Jets We have seen in the e + e studies (thrust) that the final state is pencil-like Consequence of the collinear divergence QCD branchings are most likely collinear (dp/dθ α s /θ) p. 30

57 Jets We have seen in the e + e studies (thrust) that the final state is pencil-like jet 1 jet 1 jet 2 jet 3 jet 2 Consequence of the collinear divergence QCD branchings are most likely collinear (dp/dθ α s /θ) Jets bunch of collimated particles = hard partons p. 30

58 Jets We have seen in the e + e studies (thrust) that the final state is pencil-like q q g q q Consequence of the collinear divergence QCD branchings are most likely collinear (dp/dθ α s /θ) Jets bunch of collimated particles = hard partons p. 30

59 Jets Jets bunch of collimated particles = hard partons obviously 2 jets q q p. 31

60 Jets 3 jets Jets bunch of collimated particles = hard partons p. 31

61 Jets Jets bunch of collimated particles = hard partons 3 jets... or 4? collinear is arbitrary p. 31

62 Jets Jets bunch of collimated particles = hard partons 3 jets... or 4? collinear is arbitrary parton concept strictly valid only at LO p. 31

63 Jets Partons/Particles/Calorimeter towers/tracks Jet definition Jet algorithm Parameters Recomb. scheme Jets p. 32

64 Jets A jet definiton is supposed to be (as) consistent (as possible) across different view of an event p π π φ K LO partons NLO partons parton shower hadron level Jet Def n Jet Def n Jet Def n Jet Def n jet 1 jet 2 jet 1 jet 2 jet 1 jet 2 jet 1 jet 2 p. 33

65 Jet definitions: constraints SNOWMASS accords (FermiLab, 1990) p. 34

66 Jet definitions: constraints SNOWMASS accords (FermiLab, 1990) 30 years later, these are only recently satisfied!!! p. 34

67 Jet definitions: cone Cone algorithm Concept of stable cone as a direction of energy flow cone : circle of fixed radius R in the (y,φ) plane stable : sum of the particles (4-mom.) inside the cone points in the direction of its centre p. 35

68 Jet definitions: cone Cone algorithm Concept of stable cone as a direction of energy flow cone : circle of fixed radius R in the (y,φ) plane stable : sum of the particles (4-mom.) inside the cone points in the direction of its centre Iterative stable-cone search (aka seeded cone): start from an initial direction (seed) for the cone centre the sum of particles in the cone gives a new direction iterate until stable p. 35

69 Jet definitions: cone Cone algorithm Concept of stable cone as a direction of energy flow cone : circle of fixed radius R in the (y,φ) plane stable : sum of the particles (4-mom.) inside the cone points in the direction of its centre Iterative stable-cone search (aka seeded cone): start from an initial direction (seed) for the cone centre the sum of particles in the cone gives a new direction iterate until stable Stable cones jets... up to overlaps! p. 35

70 Jet definitions: cone with SM Cone algorithm: (1) cone with split merge Step 1: find the stable cones with the seeds 1. input particles (over a seed threshold) 2. midpoints of the stable cones found above Step 2: split merge (with threshold f) p t,common > f p t,hard p t,common < f p t,hard p. 36

71 Jet definitions: cone with SM Cone algorithm: (1) cone with split merge Step 1: find the stable cones with the seeds 1. input particles (over a seed threshold) 2. midpoints of the stable cones found above Step 2: split merge (with threshold f) Examples: main algorithm at the Tevatron CDF JetClu (1) CDF MidPoint (1+2) D0 Run II Cone (1+2) ATLAS Cone (1) p. 36

72 Jet definitions: cone with SM Cone algorithm: (1) cone with split merge Step 1: find the stable cones with the seeds 1. input particles (over a seed threshold) 2. midpoints of the stable cones found above Step 2: split merge (with threshold f) Examples: main algorithm at the Tevatron CDF JetClu (1) IR unsafe (2 hard+1 soft) CDF MidPoint (1+2) IR unsafe (3 hard+1 soft) D0 Run II Cone (1+2) IR unsafe (3 hard+1 soft) ATLAS Cone (1) IR unsafe (2 hard+1 soft) p. 36

73 IR unsafety of the Midpoint alg 400 p t particle event MidPoint clustering φ p. 37

74 IR unsafety of the Midpoint alg 400 p t φ 1st seed p. 37

75 IR unsafety of the Midpoint alg 400 p t φ iterate p. 37

76 IR unsafety of the Midpoint alg 400 p t φ stable; 2nd seed p. 37

77 IR unsafety of the Midpoint alg 400 p t φ iterate p. 37

78 IR unsafety of the Midpoint alg 400 p t φ stable; 3rd seed p. 37

79 IR unsafety of the Midpoint alg 400 p t φ stable; midpoint seed p. 37

80 IR unsafety of the Midpoint alg 400 p t φ iterate p. 37

81 IR unsafety of the Midpoint alg 400 p t φ iterate p. 37

82 IR unsafety of the Midpoint alg 400 p t 400 p t add an infinitely soft particle φ φ p. 37

83 IR unsafety of the Midpoint alg 400 p t 400 p t φ hard seeds + midpoint seed 2 stable cones φ p. 37

84 IR unsafety of the Midpoint alg 400 p t 400 p t new seed! φ φ p. 37

85 IR unsafety of the Midpoint alg 400 p t 400 p t iterate φ φ p. 37

86 IR unsafety of the Midpoint alg 400 p t 400 p t φ φ Stable cones: Midpoint: {1,2} & {3} {1,2} & {3} & {2,3} Seedless: {1,2} & {3} & {2,3} {1,2} & {3} & {2,3} Jets: (f = 0.5) Midpoint: {1,2} & {3} {1,2,3} Seedless: {1,2,3} {1,2,3} p. 37

87 IR unsafety of the Midpoint alg 400 p t 400 p t φ φ Stable cones: Midpoint: {1,2} & {3} {1,2} & {3} & {2,3} Seedless: {1,2} & {3} & {2,3} {1,2} & {3} & {2,3} Jets: (f = 0.5) Midpoint: {1,2} & {3} {1,2,3} Seedless: {1,2,3} {1,2,3} p. 37

88 IR unsafety of the Midpoint alg 400 p t 400 p t φ φ Stable cones: Midpoint: {1,2} & {3} {1,2} & {3} & {2,3} Seedless: {1,2} & {3} & {2,3} {1,2} & {3} & {2,3} Jets: (f = 0.5) Midpoint: {1,2} & {3} {1,2,3} Seedless: {1,2,3} {1,2,3} p. 37

89 IR unsafety of the Midpoint alg 400 p t 400 p t φ φ Stable cones: Midpoint: {1,2} & {3} {1,2} & {3} & {2,3} Seedless: {1,2} & {3} & {2,3} {1,2} & {3} & {2,3} Jets: (f = 0.5) Midpoint: {1,2} & {3} {1,2,3} Seedless: {1,2,3} {1,2,3} Stable cone missed MidPoint is IR unsafe p. 37

90 Jet definitions Cone algorithm: (1) cone with split merge Step 1: find ALL stable cones in a reasonable time MidPoint: time N 3 All-Naive: time 2 N SISCone: time N 2 log(n) Step 2: split merge (with threshold f) Example: SISCone Seedless Infrared-Safe Cone 2007!!! p. 38

91 Jet definition: cone with PR Cone algorithm: (2) cone with progressive removal Recipe: start with the hardest particle as a seed iterate to find a stable cone stable cone 1 st jet remove its constituents continue with the next hardest particle left p. 39

92 Jet definition: cone with PR Cone algorithm: (2) cone with progressive removal Recipe: start with the hardest particle as a seed iterate to find a stable cone stable cone 1 st jet remove its constituents continue with the next hardest particle left Benchmark: circular/soft-resilient hard jets p. 39

93 Jet definition: cone with PR Cone algorithm: (2) cone with progressive removal Recipe: start with the hardest particle as a seed iterate to find a stable cone stable cone 1 st jet remove its constituents continue with the next hardest particle left Benchmark: circular/soft-resilient hard jets Example: CMS Iterative Cone p. 39

94 Jet definition: cone with PR Cone algorithm: (2) cone with progressive removal Recipe: start with the hardest particle as a seed iterate to find a stable cone stable cone 1 st jet remove its constituents continue with the next hardest particle left Benchmark: circular/soft-resilient hard jets Example: CMS Iterative Cone BUT Collinear unsafe (3 hard+1 coll.splitting)!! p. 39

95 Jet definition: successive recombinations Idea: Undo the QCD cascade Define an inter-particle distance d ij and a beam distance d ib Successively Find the minimum of all d ij, d ib If d ij, recombine i+j k (remove i, j; add k) If d ib, call i a jet (remove i) Until all particles have been clustered p. 40

96 Jet definition: successive recombinations Typical choice of distances: d 2 ij = min(k 2p t,i,k2p t,j )( y2 ij + φ2 ij ) d 2 ib = k2p t,i R2 p = 1: k t algorithm (1993) pqcd p = 0: Cambridge-Aachen algorithm (1997) pqcd p = 1: anti-k t algorithm (2008) pqcd parameter R (jet separation) trivially IRC-safe p. 41

97 Jet definition: successive recombinations Typical choice of distances: d 2 ij = min(k 2p t,i,k2p t,j )( y2 ij + φ2 ij ) d 2 ib = k2p t,i R2 p = 1: k t algorithm (1993) (as close as possible to pqcd) p = 0: Cambridge-Aachen algorithm (1997) (close to pqcd; useful for substructure) p = 1: anti-k t algorithm (2008) (circular/soft-resilient jets; replaces it. cone) Variants for e + e collisions (+JADE) p. 41

98 Jet definitions: IRC safety matters As said in e + e : IRC safety matters if you want to compare to QCD computations Last OK order today s Process IR 2+1 IR/Coll 3+1 safe pqcd Incl. jet x-sect LO NLO any NLO W/Z/H+1 jet LO NLO any NLO 3-jet x-sect none LO any NLO W/Z/H+2 jet none LO any NLO jet mass in 3-jet none none any LO p. 42

99 Jet definitions: IRC safety matters As said in e + e : IRC safety matters if you want to compare to QCD computations Last OK order today s Process IR 2+1 IR/Coll 3+1 safe pqcd Incl. jet x-sect LO NLO any NLO W/Z/H+1 jet LO NLO any NLO 3-jet x-sect none LO any NLO W/Z/H+2 jet none LO any NLO jet mass in 3-jet none none any LO Use an IRC-safe algorithm like k t, C/A, anti-k t or SISCone p. 42

100 Jet definitions: comparison Quick comparison of the algorithms k t C/A anti-k t SISCone pqcd soft (UE) OK speed substruct calibr. p. 43

101 Jet clustering: usage/access FastJet [M.Cacciari, G.Salam, GS] Fast implementation of recomb. algs (N log(n)) Plugins for all common algs (SISCone; CDF, D0, ATLAS, CMS algs; e + e algs) Other tools (like jet areas) More in the tutorial part! p. 44

102 Jets: experimentally Tevatron Use of IR-unsafe JetClu or MidPoint and sometimes k t dy (pb/gev) T σ/dp 2 d 7 10 y <0.4 (x32) DØ Run II s = 1.96 TeV -1 L = 0.70 fb R cone = 0.7 NLO pqcd +non-perturbative corrections CTEQ6.5M µ = µ = p R F T 0.4< y <0.8 (x16) 0.8< y <1.2 (x8) 1.2< y <1.6 (x4) 1.6< y <2.0 (x2) 2.0< y < p T 600 (GeV) p. 45

103 Jets: experimentally Tevatron Use of IR-unsafe JetClu or MidPoint and sometimes k t LHC: anti-k t by default [pb/gev] T d σ/d p ATLAS Preliminary -1 R=0.6, L dt=17 nb Data ( s=7 TeV)+syst. Alpgen MC+scale uncert. Pythia MC 0.62 N jets 2 [pb/gev] T d σ/d p ATLAS Preliminary -1 R=0.6, L dt=17 nb Data ( s=7 TeV)+syst. Alpgen MC+scale uncert. Pythia MC 0.60 N jets 4 Data/MC p (leading jet) [GeV] T Data/MC th p (4 leading jet) [GeV] T p. 45

104 Jets: hadronic colliders At hadronic colliders, many contaminations to a jet: radiation from partons in the initial state Underlying event/multiple interactions shift: UE uniform soft background i.e. contamination jet area R 2 smearing: due to UE fluctuations typical scale: a few GeV Pile-up: many pp interactions in 1 bunch-crossing: n L t bunch σ pp Again: shift + smearing Typical scale: GeV Need for subtraction techniques p. 46

105 </interlude> p. 47

106 The very fundamental collision σ = f a f b ˆσ f a take a parton out of each proton f a parton distribution function (PDF) for quark and gluons hard matrix element perturbative computation Forde-Feynman rules ˆσ f b p. 48

107 Deep Inelastic Scattering Introduce/Discuss/Study the PDFs p. 49

108 Process + kinematics e(k) e (k ) q p } X s = (e+p) 2 W 2 = (q +p) 2 Q 2 = q 2 > 0 ν = p.q = W 2 +Q 2 x = Q 2 /(2ν) y = p.q/p.k = (W 2 +Q 2 )/s ep ex with γ exchange Z and W also possible as well as ν instead of e also more exclusive meas.: ep ep, exy, eyp, e.g. jets, charm, vector-mesons, photons p. 50

109 Process + kinematics e(k) e (k ) q p } X s = (e+p) 2 W 2 = (q +p) 2 Q 2 = q 2 > 0 ν = p.q = W 2 +Q 2 x = Q 2 /(2ν) y = p.q/p.k = (W 2 +Q 2 )/s Experimentally: only the outgoing e is needed to reconstruct the kinematics Q 2 = 4EE cos 2 (θ e /2) x = EE cos 2 (θ e /2) P[E E sin 2 (θ e /2)] p. 50

110 Process + kinematics e(k) e (k ) q p } X s = (e+p) 2 W 2 = (q +p) 2 Q 2 = q 2 > 0 ν = p.q = W 2 +Q 2 x = Q 2 /(2ν) y = p.q/p.k = (W 2 +Q 2 )/s Idea: use the photon to probe the proton structure Q 2 large small distance 1/Q p. 50

111 Process + kinematics e(k) e (k ) q p } X s = (e+p) 2 W 2 = (q +p) 2 Q 2 = q 2 > 0 ν = p.q = W 2 +Q 2 x = Q 2 /(2ν) y = p.q/p.k = (W 2 +Q 2 )/s Experiments: most important results recently from HERA at DESY (H1 and ZEUS experiments) p. 50

112 A crystal-clear example Electroweak unification neutral cur γ,z e ± p e ± total x-sect differential in Q 2 Neutral currents ep ex via γ,z W ± charged cur Charged currents ep νx via W ± p. 51

113 Process + kinematics e l µν q W µν p Factorisation in a leptonic and hadronic part: M 2 = l µν W µν l µν = 4e 2 (k µ k ν +k ν k µ g µν k.k ) study the hadronic tensor W µν (W 2,Q 2 ) (or W µν (x,q 2 )) p. 52

114 Hadronic tensor Most generic structure for W µν (x,q 2 ) W µν = Ag µν +Bp µ p ν +Cq µ q ν +Dp µ q ν +Eq µ p ν. Constraints: W µν = W νµ and q µ W µν = 0 (gauge inv.) Implying W µν = ( g µν + qµ q ν Q 2 ) F 1 + 2x Q 2 ( p µ + qµ 2x )( p ν + qν 2x ) F 2 F 1,F 2 (x,q 2 ): proton structure functions p. 53

115 Structure functions (inclusive) proton interaction fully parametrised by the 2 structure functions F 1 and F 2 (x,q 2 ) dimensionless F L = F 2 2xF 1 (longitudinally-polarized γ ) For charged currents: additional F 3 (x,q 2 ) p. 54

116 Parton model Useful to consider a frame where the proton is highly boosted (P 1, p looks like a pancake) p µ (0,0,P,P) n µ (0,0, 1 2P, 1 2P ) (n2 = 0, n.p = 1) q µ q µ +νnµ (n.q = 0, q 2 = Q2 ) We obtain F 2 = νn µ n ν W µν F L = 4x2 ν pµ p ν W µν p. 55

117 Parton model Bag model The photon resolves a quark inside the proton q k k +q k q k µ = ξp µ + k2 +k 2 2ξ n µ +k µ p B(k, p) p W µν = e 2 q d 4 k (2π) 4tr(γµ (k/ +q/)γ ν B(k,p))δ ( (k +q) 2) p. 56

118 Parton model Bag model The photon resolves a quark inside the proton q k k +q k q k µ = ξp µ + k2 +k 2 2ξ n µ +k µ p B(k, p) p F 2 = νe 2 q d 4 k (2π) 4tr(n/(k/ +q/)n/b(k,p))δ( (k +q) 2) p. 56

119 Parton model Bag model The photon resolves a quark inside the proton q k k +q k q k µ = ξp µ + k2 +k 2 2ξ n µ +k µ p B(k, p) p F 2 = νe 2 q d 4 k (2π) 4tr(n/(k/ +q/)n/b(k,p))δ( (k +q) 2) tr(n/(k/ +q/)n/b(k,p)) = 2ξtr(n/B(k,p)) p. 56

120 Parton model Bag model The photon resolves a quark inside the proton q k k +q k q k µ = ξp µ + k2 +k 2 2ξ n µ +k µ p B(k, p) p F 2 = νe 2 q d 4 k (2π) 4tr(n/(k/ +q/)n/b(k,p))δ( (k +q) 2) δ ( (k +q) 2) = δ ( ) k 2 Q 2 +2ξν 2 k 2. q 2 Q 2 δ(2νξ Q 2 ) 1 2ν δ(ξ x) p. 56

121 Parton model Putting everything together: F 2 = xe 2 q d 4 k (2π) 4tr(n/B(k,p))δ(x ξ) i.e. F 2 = xe 2 qq(x) with q(x) = d 4 k (2π) 4tr(n/B(k,p))δ(x ξ) with a sum over flavours F 2 = q xe 2 q[q(x)+ q(x)] q(x): parton distribution function (PDF) p. 57

122 Parton model F 2 = xe 2 q[q(x)+ q(x)] q(x) PDF q interpreted as the probability density to find a quark carrying a fraction x of the proton s momentum (universal!!) F 2 (x,q 2 ) = F 2 (x): Q 2 -independent. Bjorken scaling F L suppressed by 1/Q 2 compared to F 2 F 2 = 2xF 1. Calan-Gross relation: spin 1/2 for q charged currents: different quark combinations p. 58

123 Bjorken scaling F 2 from BCDMS, SLAC, NMC, H1 and ZEUS ( 1990) p. 59

124 Bjorken scaling violations HERA measurements ( ) BCDMS EMC E665 H1 ZEUS F 2 p < Q 2 < GeV 2 0 1e Scaling violations!!! x log scale! p. 60

125 Bjorken scaling violations A closer look for 3 bins in x F 2 p decrease at large x 0 x= Q 2 (GeV 2 ) BCDMS EMC H1 ZEUS x=0.13 x=0.65 (strong) rise at small x p. 61

126 Bjorken scaling violations Can we describe the scaling violations in QCD? p. 62

127 Bjorken scaling violations Can we describe the scaling violations in QCD? Idea: quarks can Idea: radiate gluons p q k k +q p k p. 62

128 One-gluon emission q q k + k + + p p 4 graphs to compute Work in an axial gauge n.a = 0 (recall n 2 = 0, n.p = 1, n.q = 0): gluon of mom k µ has propagator d µν (k) = ( g µν + nµ k ν +k µ n ν n.k ) 1 k 2 p. 63

129 One-gluon emission q 2 k µ = ξp µ + k2 k2 n ν +k µ 2ξ p (0,0,P,P) p k k +q p k n µ n ν M 2 = 1 e 2 2N qg 2 tr(t a t a ) 1 c k 4tr( n/(k/ +q/)n/k/γ α p/γ β k/ ) [ ] g αβ + nα (p k) β +(p k) α n β n.(p k) = 32πe 2 qα s ξp(ξ) k 2 P(ξ) = C F 1+ξ 2 1 ξ p. 64

130 One-gluon emission k µ = ξp µ + k2 k2 n ν +k µ 2ξ P(ξ) = C F 1+ξ 2 1 ξ ˆF 2 = e 2 q α s 4π 2 dξξp(ξ) p q k k +q p k d k 2 k 2 dk2 dθδ( (p k) 2) δ ( (k+q) 2) 2 p. 64

131 One-gluon emission k µ = ξp µ + k2 k2 n ν +k µ 2ξ P(ξ) = C F 1+ξ 2 1 ξ ˆF 2 = e 2 q = e 2 q α s 4π 2 2ν α s 2π 2 0 dξξp(ξ) d k 2 k 2 with ξ ± = x±o( k 2 /Q 2 ) d k 2 ξ+ ξ dξ p q k k +q p k k 2 dk2 dθδ( (p k) 2) δ ( (k+q) 2) ξp(ξ) (ξ+ ξ)(ξ ξ ) 2 p. 64

132 One-gluon emission ˆF 2 = e 2 q 2ν α s 2π 2 xp(x) 0 d k 2 k 2 = e 2 q Q 2 α s 2π 2 xp(x) 0 d k 2 k 2 other diagrams suppressed by powers of Q only kept the leading terms in Q k 2 integration DIVERGENT!! p. 65

133 One-gluon emission ˆF 2 = e 2 q 2ν α s 2π 2 xp(x) 0 d k 2 k 2 = e 2 q Q 2 α s 2π 2 xp(x) 0 d k 2 k 2 other diagrams suppressed by powers of Q k 2 integration DIVERGENT!! From δ((p k) 2 ) we get k 2 = (1 ξ) k2 Thus, k 2 0 k 0 p This is thus a collinear divergence! The same as we already encountered in e + e collisions. k p k p. 65

134 One-gluon emission ˆF 2 = e 2 q 2ν α s 2π 2 xp(x) 0 d k 2 k 2 = e 2 q Q 2 α s 2π 2 xp(x) 0 d k 2 k 2 other diagrams suppressed by powers of Q k 2 integration DIVERGENT!! From δ((p k) 2 ) we get k 2 = (1 ξ) k2 Thus, k 2 0 k 0 p k p k This is thus a collinear divergence! The same as we already encountered in e + e collisions. Not cancelled by virtual corrections Here: technique similar to renormalisation p. 65

135 Recall: renormalisation Vertex correction in QED q q + k α bare α bare q α(q) q 2 α+β 0 α 2 0 dk 2 µ 2 k 2 = α+β 0 α 2 q 2 α(µ 2 )+β 0 α 2 α(µ 2 )+β 0 α 2 (µ 2 ) α(q 2 ) 0 dk 2 q 2 k 2 +betaα2 dk2 µ k 2 2 dk2 µ k 2 2 q 2 dk2 µ k 2 2 p. 66

136 Recall: renormalisation Vertex correction in QED q q + k α bare α bare q α(q) We have defined a scale-dependent coupling µ 2 α(µ 2 ) = α+β 0.α 2 0 dk 2 k 2 p. 66

137 Recall: renormalisation Vertex correction in QED q q + k α bare α bare q α(q) We have defined a scale-dependent coupling µ 2 α(µ 2 ) = α+β 0.α 2 0 dk 2 k 2 µ 2 is arbitrary i.e. physics should not depend on it µ 2 µ 2α(µ 2 ) = β 0 α 2 (µ 2 ) renormalisation group equation p. 66

138 Reabsorption of the collinear divergence Q 2 Q 2 x + ξ x [0 : Q 2 ] q bare (x) q bare (ξ) p. 67

139 Reabsorption of the collinear divergence Q 2 Q 2 x + ξ x [0 : Q 2 ] q bare (x) q bare (ξ) Q 2 Q 2 Q 2 = x + ξ x [0 : µ 2 ] + ξ x [µ : Q 2 ] q bare (x) q bare (ξ) q bare (ξ) p. 67

140 Reabsorption of the collinear divergence Q 2 Q 2 x + ξ x [0 : Q 2 ] q bare (x) q bare (ξ) Q 2 Q 2 Q 2 = x + ξ x [0 : µ 2 ] + ξ x [µ : Q 2 ] q bare (x) q bare (ξ) q bare (ξ) q(x,µ 2 ) p. 67

141 Reabsorption of the collinear divergence Q 2 Q 2 x + ξ x [0 : Q 2 ] q bare (x) q bare (ξ) Q 2 Q 2 = x + ξ x q(x,µ 2 ) q(ξ,µ 2 ) p. 67

142 Reabsorption of the collinear divergence F 2 (x,q 2 ) = xe 2 q = xe 2 q +xe 2 q = xe 2 q 1 x 1 dξ ξ dξ ξ x 1 1 x x dξ ξ = xe 2 qq(ξ,q 2 ) [ [ δ δ dξ ξ P [ δ ( ( 1 x ξ 1 x ξ ( x ξ ( 1 x ξ ) ) +P +P ) Q 2 ) ( ) Q 2 x ξ 0 ( ) µ 2 x ξ 0 d k 2 k 2 d k 2 k 2 d k2 µ k 2 q bare(ξ) 2 ( ) Q 2 x +P d k2 ξ µ k 2 2 ] ] ] q bare (ξ) q bare (ξ) q(ξ,µ 2 ) P(x) = α s 2π C F 1+x2 1 x p. 68

143 Reabsorption of the collinear divergence We have defined q(x,µ 2 ) = q bare (x)+ 1 x dξ ξ P ( ) µ 2 x ξ 0 d k 2 k 2 q bare(ξ) p. 69

144 Reabsorption of the collinear divergence We have defined q(x,µ 2 ) = q bare (x)+ 1 x dξ ξ P ( ) µ 2 x ξ 0 d k 2 k 2 q bare(ξ) Physics independent of the choice for µ 2 µ 2 µ 2q(x,µ 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,µ 2 ) DGLAP equation p. 69

145 The DGLAP equation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi p. 70

146 The DGLAP equation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi the PDFs get some dependence on Q 2 Bjorken scaling violations p. 70

147 The DGLAP equation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi the PDFs get some dependence on Q 2 Bjorken scaling violations µ called the factorisation scale p. 70

148 The DGLAP equation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi the PDFs get some dependence on Q 2 Bjorken scaling violations µ called the factorisation scale Leading order computation in α s log(q 2 /µ 2 ) p. 70

149 The DGLAP equation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi the PDFs get some dependence on Q 2 Bjorken scaling violations µ called the factorisation scale Leading order computation in α s log(q 2 /µ 2 ) Actually resums all terms αs n log n (Q 2 /µ 2 ) (recall: α s log(q 2 /µ 2 ) 1 compute at all orders) p. 70

150 The DGLAP equation: resummation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) Q 2 x q(x,q 2 ) p. 71

151 The DGLAP equation: resummation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) Q 2 +δq 2 x q(x,q 2 +δq 2 ) p. 71

152 The DGLAP equation: resummation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) Q 2 +δq 2 Q 2 +δq 2 Q 2 +δq 2 x = x + ξ x q(x,q 2 +δq 2 ) q(x,q 2 ) q(ξ,q 2 ) p. 71

153 The DGLAP equation: resummation Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) Q 2 +δq 2 Q 2 +δq 2 x n x = n=0 q(x,q 2 +δq 2 ) ξ 2 1 q(ξ,q 2 0) Resumming (leading) contributions α n s log n (Q 2 /Q 2 0 ) p. 71

154 The DGLAP equation: splitting function Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) P(ξ) called the splitting function: transition from a quark of longitudinal momentum xp to a quark of momentum xξp with emission of a gluon p. 72

155 The DGLAP equation: splitting function Q 2 Q 2q(x,Q 2 ) = α s 2π 1 x dξ ξ P ( x ξ ) q(ξ,q 2 ) P(ξ) called the splitting function: transition from a quark of longitudinal momentum xp to a quark of momentum xξp with emission of a gluon Correction due to virtual-gluon emission: ] P(x) = C F [ 1+x 2 1 x NB: the 1/(1 x) behaviour is the soft QCD divergence + p. 72

156 The DGLAP equation: splitting function Q 2 Q 2 ( q(x,q 2 ) g(x,q 2 ) ) = α s 2π 1 x dξ ξ ( P qq P qg P gq P gg ) (x ξ ) ( q(ξ,q 2 ) g(ξ,q 2 ) ) P ab (ξ) called the splitting function: P qq P gq P qg P gg P ab (x) is the probability to obtain a parton of type a carrying a fraction x of the longitudinal momentum of a parent parton of type b p. 73

157 DGLAP and the factorisation theorem The result is more general: it holds at any order in perturbation theory ) with P(x) = Q 2 Q 2q(x,Q 2 ) = 1 x dξ ξ P ( x ξ q(ξ,q 2 ) ( αs ) ( P (0) αs ) 2P ( (x)+ (1) αs ) 3P (x)+ (2x) (x)+... 2π 2π 2π p. 74

158 DGLAP and the factorisation theorem The result is more general: it holds at any order in perturbation theory ) with P(x) = ( αs Q 2 Q 2q(x,Q 2 ) = 2π ) P (0) (x) + } {{ } LO ( αs 1 x dξ ξ P ( x ξ ) 2P (1) (x) + 2π } {{ } NLO q(ξ,q 2 ) ( αs ) 3P (2) (x) π } {{ } NNLO LO resums α n s log n (Q 2 /µ 2 ) (leading logarithms) NLO resums α n s log n (Q 2 /µ 2 ) and α n+1 s log n (Q 2 /µ 2 ) Note: order refers to P ; includes diagrams at all orders Note: known up to NNLO since 2004 (Moch, Vermaseren, Vogt) p. 74

159 DGLAP and the factorisation theorem The result is more general: it holds at any order in perturbation theory ) with P(x) = Q 2 Q 2q(x,Q 2 ) = 1 x dξ ξ P ( x ξ q(ξ,q 2 ) ( αs ) ( P (0) αs ) 2P ( (x)+ (1) αs ) 3P (x)+ (2x) (x)+... 2π 2π 2π Fundamental result in QCD know as the factorisation theorem Collinear divergences can be reabsorbed in the definition of the PDFs at all orders! p. 74

160 DGLAP vs. data Very nice description of the Q 2 -dependence observed in the data p. 75

161 DGLAP vs. data DGLAP only gives the Q 2 evolution of the PDFs One still needs an initial condition f a (x,µ 2 ) Global PDF fit: Parametrise q and g at an initial scale µ 2 e.g. q(x,µ 2 ) = x λ (1 x) β (A+B x+cx) Obtain the PDFs f a (x,q 2 ) at all Q 2 using DGLAP Compute a series of observables (e.g. F 2 ) Fit the experimental measurements (χ 2 minimisation) p. 76

162 DGLAP vs. data Many teams: MSTW/MRST, CTEQ, NNPDF, HERA, H1, ZEUS, Alekhin, GRV Each with many updates e.g. CTEQ4l, CTEQ4m, CTEQ5l, CTEQ5m, CTEQ6, CTEQ6l, CTEQ6m, CTEQ61, CTEQ65, CTEQ66 MRST98, MRST2001, MRST2002, MRST2003, MRST2004, MRST2006, MRST2007, MSTW2008 p. 77

163 DGLAP vs. data Many teams: MSTW/MRST, CTEQ, NNPDF, HERA, H1, ZEUS, Alekhin, GRV Each with many updates Points of difference (7): p. 77

164 DGLAP vs. data Many teams: MSTW/MRST, CTEQ, NNPDF, HERA, H1, ZEUS, Alekhin, GRV Each with many updates Points of difference (7): Choice of initial scale Choice of initial parametrisation Order of the fit (LO, NLO, NNLO) Data selection (e.g. cuts, old vs. new data) Heavy-flavour treatment Computation of PDFs uncertainties List of observables (9) p. 77

165 DGLAP vs. data Many teams: MSTW/MRST, CTEQ, NNPDF, HERA, H1, ZEUS, Alekhin, GRV Each with many updates Points of difference (7): Choice of initial scale Choice of initial parametrisation Order of the fit (LO, NLO, NNLO) Data selection (e.g. cuts, old vs. new data) Heavy-flavour treatment Computation of PDFs uncertainties List of observables (9) F p 2, Fd 2, F L, F ν 2, Fν 3, Fc 2, Fb 2, Drell-Yan, Tev. jets p. 77

166 Global fits Global fits are important for LHC physics as they affect every perturbative computation p. 78

167 Global fits Initial distributions Q 2 = µ 2 = 2 GeV (x, Q ) x S sea asym. x( d ū) NNPDF2.0 CTEQ6.6 MSTW NNPDF2.0 CTEQ6.6 MSTW x xv (x, Q ) valence x q (q q) x p. 79

168 Global fits Initial flavour-singlet distributions Q 2 = µ 2 = 2 GeV ) NNPDF2.0 CTEQ6.6 MSTW 2008 xg (x, Q x q (q + q) sea quarks 6 5 NNPDF2.0 CTEQ MSTW x xσ (x, Q ) gluons xg x p. 80

169 Global fits Impact of HERA measurements With HERA Without HERA p. 81

170 Global fits Z 0 W W + t t p. 82

171 DIS: summary DIS: γ p scattering with highly virtual γ (Q 2 Λ 2 QCD ) Parton model directly probes partons inside the proton Bjorken scaling p. 83

172 DIS: summary DIS: γ p scattering with highly virtual γ (Q 2 Λ 2 QCD ) Parton model directly probes partons inside the proton Bjorken scaling QCD collinear divergences Violations of Bjorken scaling Factorisation theorem/dglap equation (fundamental result/prediction of QCD) Parton Distribution Functions (PDF) Global fits for the PDF determination of the PDFs: mandatory for precision at the LHC p. 83

173 Time for questions! p. 84

174 pp collisions (at last!) p. 85

175 The very fundamental collision σ = f a f b ˆσ f a take a parton out of each proton f a parton distribution function (PDF) for quark and gluons a big chapter of these lectures ˆσ hard matrix element perturbative computation Forde-Feynman rules f b p. 86

176 The more realistic version Hard ME perturbative Parton branching initial+final state radiation Hadronisation q,g hadrons Multiple interactions Underlying event (UE) Pile-up 25 pp at the LHC p. 87

177 Plan A few generic considerations kinematics (done) Monte-Carlo Processes one-by-one Drell-Yan Jets (done) W/Z (+jets) top H SUSY (?) p. 88

178 Plan A few generic considerations kinematics (done) Monte-Carlo Processes one-by-one Drell-Yan Jets (done) W/Z (+jets) top H SUSY (?) p. 88

179 Parton luminosities Vary s same ME, only PDF vary σ = = dx 1 dx 2 f a (x 1 )f b (x 2 )ˆσ ij dŝ dl ij dŝ ˆσ(ŝ) NB: Tevatron: p p NB: LHC: pp p. 89

180 Drell-Yan Production of a lepton pair (of mass M) Hard matrix element: f a q γ/z l + dˆσ dm 2 = e2 qn c N 2 c 4πα 2 3M 2 δ(x 1x 2 s M 2 ) q l Lowest order (PDF 1 PDF 2 ME) dσ dm 2 = dx 1 dx 2 [q(x 1,M 2 ) q(x 2,M 2 dˆσ )+(1 2)] dm 2 q f b p. 90

181 Drell-Yan Production of a lepton pair (of mass M) More differential cross-sections: f a q γ/z l + Ex. 1: lepton-pair rapidity (y) δ(x 1 x 2 s M 2 ) δ(y 1 2 log(x 1/x 2 )) q f b l d 2 σ dm 2 dy = q 4πe 2 qα 2 3N c M 2 s [ q( M s e y,m 2 ) q( M s e y,m 2 )+(y y) ] p. 90

182 Drell-Yan Production of a lepton pair (of mass M) More differential cross-sections: f a q γ/z l + Ex. 1: lepton-pair rapidity (y) δ(x 1 x 2 s M 2 ) δ(y 1 2 log(x 1/x 2 )) q f b l Ex. 2: Feynman x (x F ) x F = 2 s (p z,l + p z,l ) LO = x 1 x 2 : also 2 δ s p. 90

183 Drell-Yan f a Next order: emission of one gluon real and virtual depends on g(x,m 2 ) q γ/z l + p t,γ/z 0 q l f b p. 91

184 Drell-Yan Next order: emission of one gluon factorisation proven at ANY order dσ dm 2 = f dx 1 dx 2 dz 1 dz 2 f a (x 1,M 2 )f b (x 2,M 2 )D ab (z 1 /x 1,z 2 /x 2 ) dˆσ dm 2(z 1,z 2 ;M 2 ) p. 91

185 Drell-Yan Next order: emission of one gluon factorisation proven at ANY order dσ dm 2 = f dx 1 dx 2 dz 1 dz 2 f a (x 1,M 2 )f b (x 2,M 2 )D ab (z 1 /x 1,z 2 /x 2 ) dˆσ dm 2(z 1,z 2 ;M 2 ) ONLY case where the factorisation PDF 1 PDF 2 ME is proven, otherwise it s just a reasonable assumption p. 91

186 Monte-Carlo generators Parton cascades, hadronisation, Underlying Event, pileup: a realistic event is complicated! Use of (Monte-Carlo) event generators to simulate full events p. 92

187 Monte-Carlo generators: fixed order Perturbative computations are the base of everything But are often hard/impossible to compute analytically (especially for exclusive measurements) use a fixed-order Monte-Carlo genrator p. 93

188 Monte-Carlo generators: fixed order Perturbative computations are the base of everything But are often hard/impossible to compute analytically (especially for exclusive measurements) use a fixed-order Monte-Carlo genrator Aim: provide signals and backgrounds for LHC studies (usually needed at NLO) See the LesHouche list of completed/wanted processes, e,g, many jets W+jets H+jets top (t t and single top) SUSY p. 93

189 Monte-Carlo generators: fixed order Perturbative computations are the base of everything But are often hard/impossible to compute analytically (especially for exclusive measurements) use a fixed-order Monte-Carlo genrator Aim: provide signals and backgrounds for LHC studies (usually needed at NLO) Generate matrix elements + phase-space 2 big categories: LO (many legs) or NLO (includes virtual corrections) Tendency to automate! Plenty of them: Alpgen, MadGraph, NLOJet, MCFM, BlackHat, Golem,... p. 93

190 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) As seen in e + e, they have the form d 2 P dθdz = α sp(z) 1 θ Leading terms (α n s log n (1/θ)) have angular ordering θ 1 > θ 2 > > θ n Watch out: LO collinear branchings!!! e.g. Multi-jet processes hardly reliable (alternatives like virtuality ordered but always LO p. 94

191 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) hadronisation: non-perturbative per se! e.g. Lund string fragmentations (form strings based on colour connections and fragment them) p. 94

192 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) hadronisation: non-perturbative per se! Multiple interactions/underlying Event: hadronic beams carry colour i.e. interact strongly Modelling Then tuning to Tevatron (and LHC) data p. 94

193 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) hadronisation: non-perturbative per se! Multiple interactions/underlying Event: hadronic beams carry colour i.e. interact strongly Matching to fixed-order LO generator: better description of multi-jet final-states p. 94

194 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) hadronisation: non-perturbative per se! Multiple interactions/underlying Event: hadronic beams carry colour i.e. interact strongly Matching to fixed-order LO generator: better description of multi-jet final-states Progress towards NLO generator p. 94

195 Monte-Carlo generators: full event For full-event simulation, Monte-Carlo generators are a cornerstone parton cascade: collinear splittings (DGLAP-like) hadronisation: non-perturbative per se! Multiple interactions/underlying Event: hadronic beams carry colour i.e. interact strongly Matching to fixed-order LO generator: better description of multi-jet final-states Progress towards NLO generator Most commonly used: Pythia, Herwig, Sherpa... but others available more in the tutorials p. 94

196 W/Z production Production: q q W ± q q Z 14 TeV σ W 20 nb i.e. 200 W/s (L = cm 2 /s) Decay: W q q 2 jets (BR 2/3) W lν l (BR 1/3) Z q q 2 jets (BR 70%) Z l l (BR 10%) Z ν ν (BR 20%) leptonic channel most convenient hadronic important for statistics! p. 95

197 W/Z physics not really a discovery channel but important in many respects often W/Z+jets standard model tests/mc calibration background to many searches e.g. top ( Wb) or SUSY (E/ t ) W cross-section as a standard candle for luminosity measurements p. 96

198 W for lumi measurement W cross-section as a standard candle for luminosity measurements W + W PDF main source of uncertainty p. 97

199 top physics Production: Mostly gg t t Tevatron: σ t 4 pb: discovery! LHC: σ t 1 nb: 10/s LHC top factory Decay: Mostly t Wb t q qb ( 66%) or t lν l b ( 33%) for t t: 3 options leptonic: not-so-easy because 2 neutrinos semi-leptonic: l, 4 jets (2b) and E/ t semi-leptonic: (the most convenient) hadronic: 6 jets i.e. technical to reconstruct hadronic: but 45% of the stat! p. 98

200 top physics top very important at the LHC precision mass measurement many new physics scenario involve the top (mostly because of its large mass) need to reconstruct as many tops as possible p. 99

201 top physics top very important at the LHC precision mass measurement many new physics scenario involve the top (mostly because of its large mass) need to reconstruct as many tops as possible Issues: W +jets background b mis-tagging combinatorial background (especially for full hadr.) efforts e.g. in boosted-top reconstruction p. 99

202 Higgs: production Production at the LHC: mostly gg fusion (through top loop) gg H σ(pp H+X) [pb] s = 14 TeV M t = 174 GeV CTEQ6M qq _ HW qq Hqq 10-2 gg,qq _ Htt _ gg,qq _ Hbb _ qq _ HZ M H [GeV] m H = 120 GeV σ (L0) H 21 pb (vs 0.3 at the Tevatron) p. 100

203 Higgs: decay 1 bb _ BR(H) WW ZZ τ + τ cc _ gg tt - γγ Zγ Heavy higgs (m 2m W ): M H [GeV] mostly H WW ( ) or H ZZ the easiest situation (see e.g. Tevatron) p. 101

204 Higgs: decay 1 bb _ BR(H) WW ZZ τ + τ cc _ gg tt - Light higgs (m < 2m W ): more complicated 10-3 γγ Zγ M H [GeV] bb jets dominant but buried in the QCD bkgd γγ clean but only % of the events p. 101

205 Higgs: discovery Significance 10-1 CMS, 30 fb 30 fb 1 needed for 5σ discovery H γγ cuts H γγ opt H ZZ 4l H WW 2l2ν qqh, H WW lνjj qqh, H ττ l+jet qqh, H γγ M H,GeV/c 2 p. 102

206 Higgs: additional comments H b b may be visible/helpful for boosted H +W/Z p. 103