Jets, Kinematics, and Other Variables A Tutorial for Physics With p-p (LHC/Cern) and p-p (Tevatron/FNAL) Experiments Drew Baden University of Maryland World Scientific Int.J.Mod.Phys.A13:1817-1845,1998 10-Dec-2008 D. Baden, U. Geneve 1
Nucleon-nucleon Scattering Elastic scattering Forward-forward scattering, no disassociation (protons stay protons) b >> 2 r p 10-Dec-2008 D. Baden, U. Geneve 2
“Single-diffractive” scattering One of the 2 nucleons disassociates into a spray of particles – Mostly π ± and π 0 particles – Mostly in the forward direction following the parent nucleon ʼ s momenum b ~ 2 r p 10-Dec-2008 D. Baden, U. Geneve 3
“Double-diffractive” scattering Both nucleons break up Active detector – Resultant spray of particles is in the forward direction b < r p Active detector 10-Dec-2008 D. Baden, U. Geneve 4
Proton-(anti)Proton Collisions • At “high” energies we are probing the nucleon structure – “High” means Compton wavelength λ beam ≡ hc/E beam << r proton ~ hc/”1GeV” ~ 1fm • E beam =1TeV@FNAL 5-7 TeV@LHC – We are really doing parton – parton scattering ( parton = quark, gluon) • Look for scatterings with large momentum transfer, ends up in detector “central region” (large angles wrt beam direction) – Each parton has a momentum distribution – • CM of hard scattering is not fixed as in e + e - will be move along z-axis with a boost • This motivates studying boosts along z – What ʼ s “left over” from the other partons is called the “underlying event” • If no hard scattering happens, can still have disassociation – An “underlying event” with no hard scattering is called “minimum bias” 10-Dec-2008 D. Baden, U. Geneve 5
“Total Cross-section” • By far most of the processes in nucleon-nucleon scattering are described by: “inelastic” “elastic” – σ (Total) ~ σ (scattering) + σ (single diffractive) + σ (double diffractive) • This can be naively estimated…. – hard sphere scattering, partial wave analysis: ‒ σ ~ 4xArea proton =4 π r p 2 = 4 π × (1fm) 2 ~ 125mb • But! total cross-section stuff is NOT the reason we do these experiments! • Examples of “interesting” physics @ Tevatron – W production and decay via lepton • σ⋅ Br(W → e ν ) ~ 2nb, 1 in 50x10 6 collisions – Z production and decay to lepton pairs • About 1/10 that of W to leptons – Top quark production • σ (total) ~ 5pb, 1 in 20x10 9 collisions • Rates for similar things at LHC will be ~10x higher 10-Dec-2008 D. Baden, U. Geneve 6
Needles in Haystacks • What determines number of detected events N(X) for process “X”? – Or the rate: R(X)=N(X)/sec? • N(X) per unit cross-section should be a function of the brightness of the beams – And should be constant for any process: N(X)/ σ (X) = constant==L (luminosity) R(X)/ σ (X) = L (instantaneous luminosity) • Units of luminosity: – “Number of events per barn” – Note: 1nb = 10 -9 barns = 10 -9 x10 -24 cm 2 = 10 -33 cm 2 – LHC instantaneous design luminosity 10 34 cm -2 s -1 = 10 nb -1 /s, or 10 events per nb cross-section per second, or “10 inverse nanobarns per second” • e.g. 10 t-tbar events per second 10-Dec-2008 D. Baden, U. Geneve 7
Coordinates x Detector φ φ θ θ r z Proton beam direction Proton or anti-proton beam direction y 10-Dec-2008 D. Baden, U. Geneve 8
Detect the “hard scattering” Protons Anti- Protons Transverse E ≡ E T E 10-Dec-2008 D. Baden, U. Geneve 9
Phase Space d τ = d 3 p E = dp x dp y dp z • Relativistic invariant phase-space element: E – Define pp or pp collision axis along z-axis : – Coordinates p µ = (E,p x ,p y ,p z ) – Invariance with respect to boosts along z? • 2 longitudinal components: E & p z (and dp z /E) NOT invariant • 2 transverse components: p x p y , (and dp x , dp y ) ARE invariant • Boosts along z-axis – For convenience: define p µ where only 1 component is not Lorentz invariant – Choose p T , m, φ as the “transverse” (invariant) coordinates • p T ≡ p sin ( θ ) and φ is the azimuthal angle y ≡ 1 2 ln E + p z – For 4 th coordinate define “rapidity” (y) p z = E tanh y or E − p z • …How does it transform? 10-Dec-2008 D. Baden, U. Geneve 10
Boosts Along beam-axis • Form a boost of velocity β along z axis – p z ⇒ γ (p z + β E) ( ) + γ p z + β E ( ) 2 ln γ E + β p z 2 ln E + p z y = 1 ⇒ 1 – E ⇒ γ (E+ β p z ) ( ) − γ p z + β E ( ) E − p z γ E + β p z – Transform rapidity: ( ) 1 + β ( ) 2 ln E + p z = 1 ( ) = y + ln γ 1 + β ( ) 1 − β ( ) E − p z y ⇒ y + y b • Boosts along the beam axis with v= β c will change y by a constant y b – (p T ,y, φ ,m) ⇒ (p T ,y+y b , φ ,m) with y ⇒ y+ y b , y b ≡ ln γ (1+ β ) simple additive to rapidity – Relationship between y, β , and θ can be seen using p z = p cos ( θ ) and p = β E y = 1 2 ln1 + β cos θ tanh y = β cos θ or where β is the CM boost 1 − β cos θ 10-Dec-2008 D. Baden, U. Geneve 11
d τ ≡ d 3 p E = dp x dp y dp z Phase Space (cont) E • Transform phase space element d τ from (E,p x ,p y ,p z ) to (p t , y, φ , m) dp x dp y = 1 ∂ y + ∂ y ∂ E 2 d φ & 2 dp T dy = dp z using ∂ p z ∂ E ∂ p z y ≡ 1 2 ln E + p z E − p z p z p z E = dp z 2 − E 2 − p z E 2 − p z 2 E = dp z d τ = 1 2 d φ dy 2 dp T • Gives: E • Basic quantum mechanics: d σ = | M | 2 d τ – If | M | 2 varies slowly with respect to rapidity, d σ / dy will be ~constant in y – Origin of the “rapidity plateau” for the min bias and underlying event structure – Apply to jet fragmentation - particles should be uniform in rapidity wrt jet axis: • We expect jet fragmentation to be function of momentum perpendicular to jet axis • This is tested in detectors that have a magnetic field used to measure tracks 10-Dec-2008 D. Baden, U. Geneve 12
Transverse Energy and Momentum Definitions • Transverse Momentum: momentum perpendicular to beam direction: 2 = p x 2 + p y 2 p T = p sin θ p T or • Transverse Energy defined as the energy if p z was identically 0: E T ≡ E(p z =0) 2 = p x 2 + p y 2 + m 2 = p T 2 + m 2 = E 2 − p z 2 E T • How does E and p z change with the boost along beam direction? p z = E tanh y tanh y = β cos θ p z = p cos θ – Using and gives 2 = E 2 − E 2 tanh 2 y = E 2 sech 2 y 2 = E 2 − p z E T then or which also means p z = E T sinh y E = E T cosh y – (remember boosts cause y → y + y b ) E T = E sin θ – Note that the sometimes used formula is not (strictly) correct! – But it ʼ s close – more later…. 10-Dec-2008 D. Baden, U. Geneve 13
Invariant Mass M 1,2 of 2 particles p 1 , p 2 2 = m 1 2 = p 1 + p 2 2 + m 2 2 + 2 E 1 E 2 − p 1 ⋅ p 2 ( ) ( ) • Well defined: M 1,2 • Switch to p µ =(p T ,y, φ ,m) (and do some algebra…) ( ) p 1 ⋅ p 1 = p x 1 p x 2 + p y 1 p y 2 + p z 1 p z 2 = E T 1 E T 2 β T 1 β T 2 cos Δ φ + sinh y 1 sinh y 2 with and β T ≡ p T E T 2 = m 1 2 + m 2 2 + 2 E T 1 E T 2 cosh Δ y − β T 1 β T 2 cos Δ φ ( ) M 1,2 • This gives – With β T ≡ p T /E T – Note: • For Δ y → 0 and Δφ → 0, high momentum limit: M → 0: angles “generate” mass 2 = 2 E T 1 E T 2 cosh Δ y − cos Δ φ • For β → 1 (m/p → 0) ( ) M 1,2 This is a useful formula when analyzing data… 10-Dec-2008 D. Baden, U. Geneve 14
Invariant Mass, multi particles • Extend to more than 2 particles: 2 = p 1 + p 2 2 + 2 p 1 + p 2 2 2 ( ) ( ) ( ) p 3 + m 3 M 1,2,3 = p 1 + p 2 + p 3 2 + 2 p 1 p 3 2 [ ] + 2 p 2 p 3 [ ] + m 3 = M 1,2 2 + p 1 2 + 2 p 1 p 3 + p 3 2 − m 3 2 + p 2 2 + 2 p 2 p 3 + p 3 2 − m 3 2 + m 3 [ ] − m 1 [ ] − m 2 2 2 2 = M 1,2 2 + M 1,3 2 + M 2,3 2 − m 1 2 − m 2 2 − m 3 2 = M 1,2 • In the high energy limit as m/p → 0 for each particle: 2 + M 2,3 2 + M 1,3 2 2 M 1,2,3 = M 1,2 ⇒ Multi-particle invariant masses where each mass is negligible – no need to id ⇒ Example: t → Wb and W → jet+jet – Find M(jet,jet,b) by just adding the 3 2-body invariant masses in quadriture – Doesn ʼ t matter which one you call the b-jet and which the “other” jets as long as you are in the high energy limit 10-Dec-2008 D. Baden, U. Geneve 15
Pseudo-rapidity 10-Dec-2008 D. Baden, U. Geneve 16
“Pseudo” rapidity and “Real” rapidity Definition of y: tanh (y) = β cos( θ ) • – Can almost (but not quite) associate position in the detector ( θ ) with rapidity ( y ) • But…at Tevatron and LHC, most particles in the detector (>90%) are π ʼ s with β ≈ 1 Define “pseudo-rapidity” defined as η ≡ y( θ , β =1 ), or tanh ( η ) = cos( θ ) or • η = 1 2 ln1 + cos θ 1 − cos θ = ln cos θ 2 ( ) sin θ 2 = − ln tan θ 2 ( η =5, θ =0.77°) CMS HCAL CMS ECAL 10-Dec-2008 D. Baden, U. Geneve 17
Rapidity (y) vs “Pseudo-rapidity” ( η ) • From tanh ( η ) = cos( θ ) = tanh (y)/ β – We see that | η | ≥ | y | – Processes “flat” in rapidity y will not be “flat” in pseudo-rapidity η • (y distributions will be “pushed out” in pseudo-rapidity) 1.4 GeV π π 10-Dec-2008 D. Baden, U. Geneve 18
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