Kinematics

Kinematics

Here we define some of the key kinematic variables used above:

• pseudorapidity --- the quantity "eta" is not quite the same as the rapidity; it is a stand-in for "theta", the latitude in spherical coordinates relative to the beam axis. In particular,  eta is defined to be -ln[tan(theta/2)].  For massless particles, eta is the same as the rapidity y, defined in terms of the energy E and z-component of momentum pz as y=1/2[ln([E+pz]/[E-pz])].
  
• azimuthal angle --- the angle "phi" is the angle around the z axis in cylindrical coordinates, or the longitude in spherical coordinates, with the beam axis oriented along the z axis.
  
• R --- a distance measure in (eta,phi) space --- if two particles have momenta pointing in the directions eta1,phi1 and eta2,phi2, respectively, then their distance in R is Sqrt[(eta2-eta1)^2+(phi2-phi1)^2]
  
• transverse momentum --- the px and py components of the momentum of an object, with the beam axis being the z direction. 
  
• invariant mass --- the square of the sum of the four-momenta of two (or more) objects.  If a particle of mass M decays to n objects, the invariant mass of the n objects will be M.  In the case of a jet, the cells of the calorimeter that lie within the jet-defining cone are viewed as having detecting the energy and direction (and therefore the momentum) of little massless mini-objects.  The invariant mass of the jet is the square of the sum of all the mini-object four-momenta.  Note this is generally much larger than the invariant mass of the charged particles (detected as tracks) in the jet, and much larger than the mass of the quark which generated it.
• Missing Transverse Energy
  
  What is meant by this term? And how precisely is it defined in the context of the PGS detector? 
  
  "Missing transverse energy" is not missing energy; indeed, what is "tranverse energy"?!  Precisely stated, it is the magnitude of the missing transverse momentum in an event. 
  
  Energy conservation cannot be used in a hadronic collider, because so much energy is carried off in unmeasurable particles --- remnants of the shattered initial protons --- inside or very near the beampipe.  For the same reason, momentum conservation along the beampipe cannot be used.  However, momentum conservation transverse to the beampipe should work.  A failure of momentum conservation in the transverse plane suggests 
  

1. the presence of a neutrino, or neutrinos, or new undetectable objects that have carried off momentum invisibly

1. a mismeasurement of the energy of a jet, which is a common occurrence
   

1. a particle sneaking through a crack in the detector structure, or otherwise evading detection for a technical reason

However, experimentally there is more than one way to define the missing momentum, because there are multiple measurements of momentum and they don't always agree.  Here is what we do, using our version of PGS:
   
Missing-Et is defined by summing (as a vector) the directed transverse energy deposited in all of the calorimeter cells (treating each cell as a massless particle) --- this combines, ideally, the momenta of all photons, electrons, hadronically-decaying taus, and jets --- and adding to this the transverse momenta of any muons, whose energy is measured using the muon detection system.   The magnitude of the resultant vector is the "missing transverse energy".

A caution: muon detection works only out to |pseudorapidity|=2, whereas the calorimeter extends to |pseudorapidity|=4, so muons at large |pseudorapidity| (very near the beampipe) can cause additional missed transverse momentum!