In experimental particle physics, what does "parton-level" , "particle-level" and "detector-level" exactly mean ?
PS : detailed explanations, links, etc .. would be deeply appreciated
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This is a vocabulary used in proton-proton collisions (or proton-antiproton). Each proton is modelled as a bag of non-interacting constituents, the partons. Let's look at the production of one jet $J$ of momentum $P$, and whatever else, for example. To fix the idea, a jet is a bunch of particles whose momenta are within a cone of given angle around some direction to be determined, and its momentum is the sum of the momenta of all the particles it is made of: in practice, no jet has been defined like that for a long time but this answer is already too long.
The cross section can be written as follow
$$\sigma_{pp\to J} = \sum_{ab,\,cde\cdots} \int_{x_a=0}^1 \int_{x_b=0}^1 f_a(x_a) f_b(x_b) |M_{ab\to cde\cdots}|^2\ g(p_c, p_d, p_e, \cdots; P)dxdy \tag{I}$$
This may look scary but I'll break it down.
So first $a$ is a parton from the first proton $A$ (momentum $p_A$) and $b$ is a parton from the second proton $B$ (momentum $p_B$). Then $f_a(x)$ is the density of probability to find a parton of type $a$ in $A$ which carries a fraction $x_a$ of the proton momentum, i.e. $p_a = x p_A$. Same for $f_b(x_b)$ with $p_b=x_bp_B$. The set of all $f_a$'s are called the Parton Density Functions (PDF's).
Then comes the parton-level amplitude $M_{ab\to cde\cdots}$. It is computed using Feynman rules for the Standard Model or whatever theory one wishes to explore (MSSM, etc). This computation is for a process with incoming partons $a$ and $b$ and outgoing partons $c,d,e,\cdots$. By the way, it is time I say that partons are either quarks or gluons! Then the function $g(p_c,p_d,p_e,\cdots; P)$, function of the momenta of the outgoing partons, models the jet algorithm: it is the probability that the given set of parton momenta gives a jet with the given momentum $P$.
That was for the parton-level. But because of confinement by QCD, the outgoing quarks and gluons will form particles, specifically hadrons, and the detector will see those (or their decay products), not the partons. Especially, the jet algorithms will be applied to the momenta of those outgoing hadrons. Therefore the parton-level picture above can only be a rough approximation: it relies on the fact that if a bunch of outgoing quarks or gluons fall within the cone of a jet, then the hadrons they will form will also be roughly flying in the same direction and therefore end up in the same jet. However, experimentalists usually want a more realistic model and they therefore replace $g(p_c,p_d,p_e,\cdots; P)$ with a simulation of the hadronisation of $c, d, e, \cdots$ followed by the jet algorithm. This sort of computation are the realm of Monte-Carlo generators. They use the theoretical $M_{ab\to c,d,e,\cdots}$ to get probabilities of production, which are then convoluted with a semi-phenomelogical hadronisation model. This is the particle-level.
Finally, the detector-level. I was a theorist, so I have very little to say here! The detector will measure energy deposits (calorimeter), radii of curvatures (trackers with a magnetic field), hits on a the silicon strip of a micro-vertex detector, etc. The work is then to simulate what signal a particle with a given 4-momentum will leave in the detector. So this is a simulation of the physics of the detector itself here, not of the particle physics one wishes for study.
A few remark to go deeper. The formula (I) should come as very surprising. We are indeed happily multiplying probabilities, which should be a no-go in quantum mechanics. The reasons why it works are far too involved to be explained here but know that there are good theoretical explanations.
More importantly, the PDF's have to be measured. We still do not know how to compute them from first principles. Fortunately they are process-independent. So for example, we can measure them in $e^-p$ collisions using the same kind of formula and using well-known established $M_{ab\to cde\cdots}$. And then reuse them at LHC to probe other matrix elements we are less sure about.