Diapositiva 1

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Kinematic variables
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Rapidity distribution of K0s in PbPb collisions
at SPS energy (158 A GeV/c), compared to SS
collisions at 200 A GeV.
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Rapidity distributions of pions, kaons, protons
and Lambdas in central AuAu collisions at 10 A
GeV
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Rapidity distribution (normalized to beam
rapidity) of negative hadrons in PbPb, SS and
NN collisions
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Invariant cross section for positive pions in pp
reactions at vs30.6 GeV as a function of
transverse momentum
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Invariant pion production cross section as a
function of transverse momentum, in AuAu
collisions at 1 GeV/A, measured at midrapidity
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Notemt-m0
Transverse mass distributions for positive and
negative particles in PbPb collisions at 158 A
GeV/c
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Pseudorapidity distribution of charged particles
simulated by the HIJING event generator in Pb-Pb
collisions at LHC energies (5.5 A TeV)
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Transverse momentum distribution of charged
particles simulated by the HIJING event generator
in Pb-Pb collisions at LHC energies (5.5 A TeV)
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Separation in longitudinal and transverse
directions Longitudinal velocities are mostly
relativistic and give information on the slowing
down of nucleons and on the transparency of
nuclear matter at high energies Transverse
momenta measure the internal energy of the system
and its thermal properties Both provide valuable
information on the reaction dynamics
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Rapidity variable y
E total energy Pl longitudinal momentum
y ½ ln (Epl)/(E-pl)
Dimensionless quantity It can be either positive
or negative (-8, 8) In the nonrelativistic limit
y ? ß Depends on the reference frame, but it
has additive properties under Lorentz
transformation from frame F to frame F (moving
with velocity ß with respect to F)
y y-yß Suitable variable
to describe the dynamics of relativistic particles
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To find the relation between the rapidity y in
frame F and the rapidity y in frame F, see
Exercise 2.5 in Wong
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Example Transformation from lab to cm system
ya ya ycm See Exercise 2.6 in
Wong Different regions Projectile rapidity
Target rapidity Midrapidity (central
rapidity) Separation between proj. rapidity and
target rapidity increases with incident energy
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Useful relations between the components of
momentum and rapidity variable
p0energy pzz-component of momentum
Adding the previous two equations,
p0mT cosh y
pz mT sinh y
where pT is the transverse momentum and mT is the
transverse mass, defined by
mt m2 pt21/2
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Pseudorapidity variable y
To evaluate rapidity, two quantities are
required energy and longitudinal momentum. This
implies the knowledge of momentum and mass of
detected particle Some experiments are not able
to have particle identification, but only charge
and momentum In such case a suitable variable is
the pseudorapidity ? -
ln tan(?/2) At large momenta (p E)
pseudorapidity approaches the rapidity

• 45o ? ? 1
• 90o ? ? 0
• ? 10o ? ? ? 2.4
• ? 170o ? ? ? -2.4

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Kinematical transformations
To transform from (y,pT) to (?,pT)
Adding the two
with


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From previous equations, one gets relations
between rapidity and pseudorapidity
Transformation of distributions

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In many practical situations, only the
pseudorapidity distribution dN/d? is measured, as
the integral of dN/d?dpT Relation between dN/d?
and dN/dy may be obtained from previous
equation For ygtgt0 the two distributions are
almost the same For y close to 0, the
transformation results in a small dip at y0 for
dN/d? where dN/dy has a plateau.
In the c.m. frame the peak of the dN/d?
distribution has a maximum near y ? 0
In the lab frame the peak of the dN/d?
distribution has a maximum around half of beam
rapidity ? yb/2
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Differential cross section
but is not Lorentz invariant
is used instead
is Lorenz-invariant for a boost along z
invariant cross section
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Integrating over f
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Kinematical acceptance
The NA57 experiment at the CERN SPS
Position of telescope determines rapidity and
transverse momentum acceptance windows for the
different particles
Telescope
Target
Beam
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Kinematical acceptance
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Kinematical acceptance
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Activity project
Design a program or a spreadsheet for kinematical
calculations Given projectile-target
combination in a fixed target experiment
Evaluates projectile and c.m. rapidities
as a function of incident energy
Ref. Cheuk-Yin Wong, Introduction to High Energy
Heavy-Ion Collisions, World Scientific, Chapter 2.