18-Jan pp Collisions: Introduction & Kinematics Talking points based on Lectures by Hebbeker (Aachen) & Baden (Maryland) Original Notes + Extra material posted on ag

18-Jan Fixed Target Vs Collider

18-Jan Nucleon-nucleon Scattering in Colliding Environment b >> 2 r p Forward-forward scattering, no disassociation

18-Jan Single-diffractive scattering One of the 2 nucleons disassociates b ~ 2 r p

18-Jan Double-diffractive scattering Both nucleons disassociates b < r p

18-Jan Proton-(anti)Proton Collisions At “high” energies we are probing the nucleon structure –“High” means E beam >> hc/r proton ~ 1 GeV (E beam –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  CM of parton–parton system will be moving along z-axis with some boost This motivates studying boosts along z –What’s “left over” from the other partons is called the “underlying event” If no hard scattering happens, can still have disassociation –Underlying event with no hard scattering is called “minimum bias”

18-Jan “Total Cross-section” By far most of the processes in nucleon-nucleon scattering are described by: –  (Total) ~  (scattering) +  (single diffractive) +  (double diffractive) This can be naively estimated…. –  ~ 4  r p 2 ~ 100mb Total cross-section stuff is NOT the reason we do these experiments! Examples of “interesting” Tevatron (2 TeV) –W production and decay via lepton  Br(W  e ) ~ 2nb 1 in 5x10 7 collisions –Z production and decay to lepton pairs About 1/10 that of W to leptons –Top quark production  (total) ~ 5pb 1 in 2x10 10 collisions

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9 Cross Section in pp Collision

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18-Jan Phase Space & Rapidity Relativistic invariant phase-space element: –Define pp/pp collision axis along z-axis: –Coordinates p  = (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  where only 1 component is not Lorentz invariant –Choose p T, m,  as the “transverse” (invariant) coordinates p T  psin(  ) and  is the azimuthal angle –For 4 th coordinate define “rapidity” (y) oror

18-Jan Rapidity & Boosts Along beam-axis Form a boost of velocity  along z axis –p z   (p z +  E) –E   (E+  p z ) –Transform rapidity: Boosts along the beam axis with v=  will change y by a constant y b –(p T,y, ,m)  (p T,y+y b, ,m) with y  y+ y b, y b  ln  (1+  ) simple additive to rapidity –Relationship between y, , and  can be seen using p z = pcos(  ) and p =  E or where  is the CM boost

18-Jan Phase Space (cont) Transform phase space element d  from (E,p x,p y,p z ) to (p t, y, , m) Gives: Basic quantum mechanics: d  = | M | 2 d  –If | M | 2 varies slowly with respect to rapidity, d  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 & using

18-Jan “Pseudo”rapidity and “Real” rapidity Definition of y: tanh(y) =  cos(  ) –Can almost (but not quite) associate position in the detector (  ) with rapidity (y) But…at Tevatron and LHC, most particles in the detector (>90%) are  ’s with   1 Define “pseudo-rapidity” defined as  y( ,  ), or tanh(  ) = cos(  ) or  =5,  =0.77°) CMS ECAL CMS HCAL

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18-Jan  vs y From tanh(  ) = cos(  ) = tanh(y)/  –We see that    y  –Processes “flat” in rapidity y will not be “flat” in pseudo-rapidity  1.4 GeV 

18-Jan  –  y  and p T – Calorimater Cells At colliders, cm can be moving with respect to detector frame Lots of longitudinal momentum can escape down beam pipe –But transverse momentum p T is conserved in the detector Plot  y for constant m , p T   (  ) For all  in DØ/CDF, can use  position to give y: –Pions: |  |  |y| 0.1GeV –Protons: |  |  |y| 2.0GeV –As  →1, y→  (so much for “pseudo”) DØ calorimeter cell width  =0.1 p T =0.1GeV p T =0.2GeV p T =0.3GeV CMS HCAL cell width 0.08 CMS ECAL cell width 0.005

18-Jan Claudio’s HW, 1st week Fall Quarter

18-Jan Claudio’s HW, 1st week Fall Quarter

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18-Jan Transverse Energy and Momentum Definitions Transverse Momentum: momentum perpendicular to beam direction: Transverse Energy defined as the energy if p z was identically 0: E T  E(p z =0) How does E and p z change with the boost along beam direction? –Using and gives –(remember boosts cause y  y + y b ) –Note that the sometimes used formula is not (strictly) correct! –But it’s close – more later…. or then or which also means

18-Jan Invariant Mass M 1,2 of 2 particles p 1, p 2 Well defined: Switch to p  = (p T,y, ,m) (and do some algebra…) This gives –With  T  p T /E T –Note: For  y  0 and   0, high momentum limit: M  0: angles “generate” mass For   1 (m/p  0) This is a useful formula when analyzing data… with and

18-Jan Invariant Mass, multi particles Extend to more than 2 particles: In the high energy limit as m/p  0 for each particle:  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 –Doesn’t matter which one you call the b-jet and which the “other” jets as long as you are in the high energy limit

18-Jan Transverse Mass

18-Jan Measured momentum conservation Momentum conservation: and What we measure using the calorimeter: and For processes with high energy neutrinos in the final state: We “measure” p by “missing p T ” method: –e.g. W  e or  Longitudinal momentum of neutrino cannot be reliably estimated –“Missing” measured longitudinal momentum also due to CM energy going down beam pipe due to the other (underlying) particles in the event –This gets a lot worse at LHC where there are multiple pp interactions per crossing Most of the interactions don’t involve hard scattering so it looks like a busier underlying event

18-Jan Transverse Mass Since we don’t measure p z of neutrino, cannot construct invariant mass of W What measurements/constraints do we have? –Electron 4-vector –Neutrino 2-d momentum (p T ) and m=0 So construct “transverse mass” M T by: 1.Form “transverse” 4-momentum by ignoring p z (or set p z =0) 2.Form “transverse mass” from these 4-vectors: This is equivalent to setting  1 =  2 =0 For e/  and, set m e = m  = m = 0 to get:

18-Jan Transverse Mass Kinematics for W-> l nu Transverse mass distribution? Start with Constrain to M W =80GeV and p T (W)=0 –cos  = -1 –E Te = E T –This gives you E T e E T versus  Now construct transverse mass –Cleary M T =M W when  =0 

18-Jan Neutrino Rapidity Can you constrain M(e, ) to determine the pseudo-rapidity of the ? –Would be nice, then you could veto on  in “crack” regions Use M(e, ) = 80GeV and Since we know  e, we know that  =  e ±  –Two solutions. Neutrino can be either higher or lower in rapidity than electron –Why? Because invariant mass involves the opening angle between particles. –Clean up sample of W’s by requiring both solutions are away from gaps? to get and solve for  :

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End STOP HERE

18-Jan Rapidity “plateau” Constant p t, rapidity plateau means d  /dy ~ k –How does that translate into d  /d  ? –Calculate dy/d  keeping m, and p t constant –After much algebra… dy/d  =  –“pseudo-rapidity” plateau…only for   1 …some useful formulae…

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