Definitions: luminosity, cross secion, number of events
Number of events/time $=\sigma L$
- $L [cm^{-2}s^{-1}]$: Luminosity. In LHC: $L = 10^{34} cm^{-2}s^{-1}$ (high) or $L = 10^{33} cm^{-2}s^{-1}$ (low)
- $\sigma [cm^2]$: Cross section
- 1 $TeV^{-1} $\approx$ 10^{-17} cm$, hence typically $\sigma \approx \alpha^2/TeV^2 \approx 10^{-36} cm^2$, so Luminosity at the LHC chosen to have reasonable event rates (some events per minute). Just like in the Rutherford experiment with the atom, in the vast majority of proton collisions the quarks missed, new bunch of particles 'produced' at the same axis of collision; only 1 in about 1 million or 1 billon collisions is an interesting event with large scattering angles.
- 1 $cm^2=10^{24}$ barn; 1 milibarn $\approx$ size of proton; 1pb = $10^{-36} cm^2 \approx$ 1 event/minute, at high $L$ of LHC
They are huge SM backgrounds. They are reduced compared to new SUSY in the region of large pt.
Examples at LHC high $L$:
- LHC total cross section: around proton size squared, i.e. $10^9$ collisions per second! <- Grid
- Z/W production $\approx 10^5$ pb $\approx$ 1000 ev/s
- Top quark pair production $\approx$ 1000 pb $\approx$ 10 ev/s
- WW production $\approx$ 100 pb $\approx$ 1 ev/s
Detecting particles
Electrons, muons, charged pions and kaons… are detectable, they are stable on the scale of the detectors. We don't see pi0 that decays rapidly into gamma gamma pair; similarly:
W: 30% decays leptonically (to lepton + neutrino).
Z: 10% leptonic, 20% missing, 70% hadronic.
t -> b W
Onion structure
- Tracking Chamber (charged particles): e+/e-, pi+/pi-, p…
- EM Calorimeter: photon, e+/e-…
- Hadronic Calorimeter: pi+/pi-, p, n…
- Muon Chamber
Misidentification for particles above is less than a percent. b, tau and charm have an intermediate lifetime, they are silicon tracker, closer to the beam. Not so good for the c; efficiency for b and tau is about 50%.
Lightcone coordinates
$P^{\pm}=P^0 \pm P^z$ and the usual transverse momentum $\vec{P}_T$
$P=(P^+,P^-,\vec{P})$
$A\cdot B = \frac{1}{2} (A^+ B^- + A^- B^+) - \vec{A}_T\vec{B}_T$
Under z boost $P^{\pm} \rightarrow e^{\pm \mu} P^{\pm}$
It is natural to write $P^{\pm} = \rho e^{\pm y}$, with rapidity [[y -> y + \mu]] under z boost. This is the analogue of spherical coordinates; when we have a rotationally invariant system it is convenient to use spherical coordinates, when we have Lorentz invariance we use these coordinates, and the rapidity of the analogue of the angle.
$P^2=\rho^2 - P_T^2= m^2$, hence $\rho = \sqr{m^2 + P_T^2} \equiv E_T$ and is called the transverse energy. Therefore
$P^0 = E_T \cosh y$
$P^z = E_T \sinh y$
the rapidity y is the measure of the angle of the particle wrt the beam.
Reference
PiTP-LHC: a Crash Course. Institute for Advanced Study (Princeton, New Jersey)
