Course Outline for Physics 880.05
II. A. Kinematic Variables (Ch. 2)
 Generalities about collisions at high energy
 We have choices about what units we use
and what variables we use as the
independent variables when presenting measurements or
calculations of observables > make useful or convenient
choices!
 "Natural" units: \hbar = c = 1
 What are the most "natural" units to use? Usually we
start with the 3 independent dimensions of mass, length, and time
and pick units that give us reasonable numbers.
 [M], [L], [T] are basic ingredients of everyday world
 Combine these uniquely to make other quantities such as
energy, velocity, momentum, angular momentum
 So MKS, CGS, or English system of units.
 In other applications we can use years instead of
seconds, and so on.
 However, we could choose different combinations as the
basis, and different, more useful, units based on the physics
at hand.
 Consider relativistic heavyion physics
 relativistic > speed of light
c is fundamental. Basic conversion
factor between time and distance or energy and momentum.
A fundamental constant, and therefore a natural unit.
 energy is an accessible quantity. If a charged
particle is accelerated through so many volts, we know
directly its energy in electron volts. For us, MeV or
GeV are the convenient units (numbers are not too big or
small).
 quantum mechanics  fundamental constant is \hbar,
which has units of action or angular momentum. It is the
quantization unit (for spin, for example) and the
scale for uncertainty principle relations (between energy
and time or momentum and position)
and therefore is a natural unit to choose.
 So instead of the 3 basic dimensions of mass, length, and
time, we decompose dimensional quantities into energy, action
(or angular momentum), and velocity.
 Other choices are more useful in other circumstances.
E.g., for
atomic physics, c is not fundamental since
nonrelativistic. But the charge and mass of the electron
are, so choose them ("atomic units"). Makes the Bohr
radius 1, for example.
 We choose units for each. MeV or GeV for
energy is a convenient scale (reasonable sized number).
The actual constants c and \hbar are reasonable scales for
the velocity and action because we are dealing with
relativistic phenomena (so velocities close to c are the rule)
and quantum phenomena (so the uncertainty relations, which have
\hbar setting the scale, are a key feature).
 Now we make the conversion between [L],[M],[T] and
[E],[c],[\hbar]
 Take any dimensional quantity Q
Q \sim [M]^\alpha [L]^\beta [T]^\delta
e.g., momentum p \sim [M]^1 [L]^1 [T]^{1}
 But we can also write
Q \sim [E]^\alpha' [\hbar]^\beta' [c]^\delta' ,
where [\hbar] is action (multiples of \hbar) and [c] is
velocity (fraction of c).
 There is a onetoone conversion from unprimed to
primed since it is linear (and linearly independent).
To find it, substitute into the second expression the
decompositions of energy, action, and velocity in terms
of [M], [L], and [T] and equate powers of each.
 Result:
\alpha' = \alpha  \beta  \delta
\beta' = \beta + \delta
\delta' = \beta  2\alpha
So Q \sim [M]^\alpha [L]^\beta [T]^\delta
\sim
[E]^{\alpha\beta\delta} [\hbar]^{\beta+\delta}
[c]^{\beta2\alpha}
 Try it out for velocity, length, time, mass, momentum.
They are all in MeV to some power
with some multiple of \hbar and c,
which we usually suppress.
e.g., momentum p \sim [E]^1 [\hbar]^0 [c]^{1} => MeV/c
 Suppressing the \hbar's and c's, mass, energy, momentum
are all in MeV and length and time are in 1/MeV.
 Naturalness
 If we build quantities from natural units for a
physics problem, we will typically find the magnitudes are
O(1), giving us an estimate for the quantity.
 To do so here, we will need to pick an appropriate
characteristic energy, such as a mass or an energy scale
of the physics (like \Lambda_QCD \sim 200 MeV).
 What is a typical time scale for strong interaction
processes? Turns out to be around 10^{23} seconds, but
where does this come from? Easier to say that a typical
distance scale is 1 fm (size of hadron) or a typical
energy scale is \Lambda_QCD \sim 200 MeV.
 We will often want to have length rather than energy as
our third dimension. No problem converting, since \hbar c has
dimensions of energy times length.
 Operationally:
 Putting back c's and \hbar's. Just use formulas above.
 \hbar = 6.58 x 10^{22} MeVsec to put things
in sec.

Conversion between MeV or GeV and fm.
Use \hbar c = 200 MeVfm = 0.2 GeVfm (197.33 to be more
precise). Note that this converts energy or momentum
to distance or time as expected from the uncertainty
principle. Examples:
 To convert 1 fm to MeV, divide 1 fm by one power
of \hbar c: 1 fm/200 MeVfm = .005 MeV^{1}
 To convert an energy density of 8x10^6 MeV^4 to
MeV/fm^3, divide by (\hbar c)^3 to cancel three powers
of MeV. The result is 1 MeV/fm^3.
 Consider the basic reaction
b + a > c + X
 b is the beam particle (e.g., proton or nucleus)
 a is the target particle at rest in lab
(e.g., proton or nucleus)
 c is a detected particle (e.g., proton or pion)
 X is everything else (could be many particles or none)
 "projectile fragmentation" reaction if b is "parent" of c;
forward direction in momentum
 "target fragmentation" reaction if a is "parent" of c;
near momentum region where at rest
 This is the lab frame as described. Very often we want
to transform to the com frame. (For colliding beams of
equal mass particles they are the same.)
 Choosing Kinematic Variables
 One issue is what happens under a change of Lorentz frame
(that is, a boost)
 At low energies, we can use particle velocity or
momentum.
 Not so useful at high energies
 Life for collisions with very high energy is different;
there is a need for a special choice of variables
 Examples:
 velocities are frequently very close to 1
 p_0 is often right on top of p_z or \vec p or minus it
 so we need to spread things out
 Summary of "new" variables
 lightcone fractions x+ and x
 rapidity y
 pseudorapidity \eta
 Feynman scaling variable x_F
 Fourvectors  use Bj+D conventions
 We'll review these as we go, but not separately.
 fourmomentum p^\mu = (p_0, \vec p) while
p_\mu = (p_0, \vec p) . Use g^{\mu\nu} or g_{\mu\nu}
to raise or lower an index.
 tricky one to keep straight: gradient
i\partial^\mu = i(\partial/\partial t,  \vec\nabla)
 Longitudinal and transverse quantities
 beam axis breaks 3d rotational symmetry 
distinguishes longitudinal and transverse
 "longitudinal" means along beam direction (zaxis): c_z
 "transverse" means in the xy plane perpendicular to the
beam: \vec c_T
 new notation for four vectors:
c^\mu \equiv c = (c_0,\vec c_T,c_z)
 Lorentz transformation for fourmomenta; relevant
boost direction is along beam (z) axis. Really a twod
mixing of zero and z components. Under such a boost,
c_T is invariant.
 mass shell for c (c^2 = c_0^2  c_z^2  c_T^2 = m^2).
Relevant for a free particle (for example, a beam particle
or a detected particle.
Off mass shell (c^2 \neq m^2) if the particle is interacting
or "virtual". (A "virtual" particle by definition doesn't
satisfy the mass shell relation.) The particle is also said
to be "onshell" or "offshell" in the two cases.
 Transverse mass m_T defined as m^2  c_T^2.
 Motivate use of p+ and p
 For beam particle b, plot locus of possible b_0,
b_z points (remember on shell)
 same for detected particle c
 identify p+ = p_0 + p_z and p = p_0  p_z
axes on graph (light cone)
 when energy is high, particles' four momenta are near
these axes => better basis for kinematic variables
 p+/p are "forward/backward lightcone momenta"
 consider how c+ and c change under Lorentz boost in
zdirection
 Transformation with \gamma = 1/\sqrt{1\beta^2}:
c_0' = \gamma(c_0  \beta c_z)
c_z' = \gamma(\beta c_0 + c_z)
\vec c_T' = \vec c_T
 Forming c+ and c just gives \gamma(1\beta) or
\gamma(1+\beta).
 Just a multiplicative factor, independent of the
details of the particle, only depending on \beta of the
transformation (relative velocity of the frames).
The light cones stay aligned in the same directions!
 We can eliminate the factor by identifying a
reference particle, such as b.
 Then the lightcone fraction x+=c+/b+ is invariant under
longitudinal boost. We always like to work with
Lorentz invariants if possible! Note that x+ \leq 1.
 If onshell, then x+ and \vec c_T can be used as
complete kinematic specification in terms of Lorentz
invariants (under longitudinal boosts)
 If interaction, then not on mass shell, and
(x+,c^2,\vec c_T) is a complete set.
 An invariant volume element can be formed from
x+, c^2, and \vec c_T.
 Rapidity instead of velocity
 What is the velocity of most of the particles involved?
(In natural units.)
Close to 1. Not very helpful; we'd like to spread it out
more. Also, velocity doesn't transform to other frames
conveniently at high
energies (unlike low energies, where it is simply additive).
 We also expect p+ and p to be range over very
large and small magnitudes; it
would be useful to define a logarithmic measure.
 Analogy of Lorentz transformation
to rotation and polar coordinates
 If we have a twod problem with x and y coordinates
that involves rotating motion, polar coordinates are
often more useful and intuitive than cartesian coordinates.
 So we identify the length and an angle:
x = r \cos \theta
y = r \sin \theta
so r^2 = x^2 + y^2 and we use \cos^2 + \sin^2 = 1.
 If we transform to a new coordinate frame by rotating
by angle \theta_R, then a vector transforms trivially:
(r,\theta) > (r',\theta') = (r,\theta\theta_R)
so that the angle is simply additive.
 Compare this to the tranformation to x' and y', in which
the x and y pieces mix.
 We have an analogous situation with our momenta,
since p_0 and p_z mix under Lorentz boosts while the
"length" m_T stays the same.
 So define the analog to polar coordinates, designating
the "angle" as the rapidity y:
p_0 = m_T \cosh y
p_z = m_T \sinh y
The use of cosh and sinh reflects the minus sign in
p_0^2  p_z^2 = 1 (using cosh^2  sinh^2 = 1).
 Then a Lorentz boost by velocity \beta
is just a "rotation" by
an angle y_\beta = \cosh^{1} \gamma, with
gamma = 1/\sqrt{1\beta^2}:
p_0' = \gamma(p_0\beta p_z) = \cosh y_\beta  \sinh y_\beta
p_z' = \gamma(\beta p_0 +p_z)
= \sinh y_\beta  \cosh y_\beta
 rapidity y_\beta for transformation (analog of rotation
angle)
 nonrelativistic limit gives velocity v of
transformation
cosh y_\beta = \gamma => 1 + y^2/2 + ... = 1 + \beta^2/2
+ ...
 Solving for y_\beta gives
y_\beta = 1/2 \ln(1+\beta/1\beta)
 Finding y given fourmomentum
 The definitions above for p_0 and p_z were for an
onshell particle.
 We note that p+ = m_T e^y and p = m_T e^y,
so y is a logarithmic measure of the lightcone momentum
(for fixed m_T).
 invert to find y, take as general definition
even if off mass shell:
y = 1/2 \ln (p+/p) = 1/2 \ln [(p_0+p+z)/(p_0p_z)]
 Transformation of rapidity between frames: additive
y' = y  y_\beta
 Rapidity is a relativistic measure of velocity useful
when you get close to the light cone
 Illustrate with transformation to com
 Consider equal mass beam and target particles.
 At small velocities in the center of mass,
v*_a = v*_b, and v_cm = 1/2(v_a+v_b).
 The analogous relations hold for the rapidity:
y*_a = y*_b, y_cm = 1/2(y_a+y_b), and
y_b* = 1/2(y_by_a), y_a* = 1/2(y_by_a).
 Rapidity in a collision  look at
figure 2.1 from Wong
 given incident energy, easy to determine rapidity of
projectiles and targets
 For pp collision at 100 GeV/c in lab, b_T=0 so
m_T = m = .939 GeV, and b_z = 100 GeV/c. So
y = \sinh^1(100 GeV/.938 GeV) = 5.36.
 y_a = \sinh^1(0) = 0.
 Note the target and beam rapidities on the
figure.
 x+ and x can be related to y for onshell particles,
so the plot is basically of
x+ = m_cT/m_B e^(yy_B) for two different m_cT's (for pions
and protons)
 "central rapidity" region is particle production region.
 Note that regraphing in the center of mass just
shifts the horizontal axis by y_b/2. The vertical axis
is unchanged since x+ is an invariant.
 Figure 2.1 shows which variable is more effective to use.
If x+ is near 1 it the projectile fragmentation region and
x+ spreads things out more. For particle production, the
range in x+ near central rapidity is small, so y is better.
 Measuring rapidity (motivation for pseudorapidity)
 To measure rapidity we apparently
need both p_0 (or \vec p) and p_z
 But if we rewrite the rapidity as
y = 1/2 \ln [(1+p_z/p_0)/(1p_z/p_0)],
we see that we need only
the ratio. This ratio is not
directly accessible with one measurement.
 However,
at high energy, p_0 > \vec p, so we just need the
ratio p_z/\vec p, which is just the cosine of the
scattering angle => one piece of readily available information
 We define the analog to the rapidity using \vec p
instead of p_0 to be the pseudorapidity.
 Later: Discuss the difference in having distributions in
rapidity vs. pseudorapidity using Figure 3.2.
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Copyright © 1997,1998 Richard Furnstahl and James Steele.
Email:
furnstah@mps.ohiostate.edu