# Course Outline for Physics 880.05

## II. A. Kinematic Variables (Ch. 2)

1. 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!
2. "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 heavy-ion 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 one-to-one 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]^{\beta-2\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:
1. Putting back c's and \hbar's. Just use formulas above.
• \hbar = 6.58 x 10^{-22} MeV-sec to put things in sec.
2. Conversion between MeV or GeV and fm. Use \hbar c = 200 MeV-fm = 0.2 GeV-fm (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 MeV-fm = .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.
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.)
4. 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
5. Summary of "new" variables
• light-cone fractions x+ and x-
• rapidity y
• pseudo-rapidity \eta
• Feynman scaling variable x_F
6. Four-vectors -- use Bj+D conventions
• We'll review these as we go, but not separately.
• four-momentum 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)
7. Longitudinal and transverse quantities
• beam axis breaks 3-d rotational symmetry -- distinguishes longitudinal and transverse
• "longitudinal" means along beam direction (z-axis): c_z
• "transverse" means in the x-y 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 four-momenta; relevant boost direction is along beam (z) axis. Really a two-d 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 "on-shell" or "off-shell" in the two cases.
• Transverse mass m_T defined as m^2 - c_T^2.
8. 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 light-cone momenta"
• consider how c+ and c- change under Lorentz boost in z-direction
• 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 light-cone 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 on-shell, 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.
• 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 two-d 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 four-momentum
• The definitions above for p_0 and p_z were for an on-shell particle.
• We note that p+ = m_T e^y and p- = m_T e^-y, so y is a logarithmic measure of the light-cone 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_0-p_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_b-y_a), y_a* = -1/2(y_b-y_a).
10. 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 on-shell particles, so the plot is basically of
x+ = m_cT/m_B e^(y-y_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.
11. 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)/(1-p_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.