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In HEP the particles (e.g. protons, pions, electrons) we are concerned with are usually moving at speeds close to the speed of light. The classical relationship for the kinetic energy of the particle in terms of its mass and velocity is not valid:

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relativistic kinematics
Richard Kass

In HEP the particles (e.g. protons, pions, electrons) we are concerned with are usually moving at speeds close to the speed of light.

The classical relationship for the kinetic energy of the particle in terms of its mass and velocity is not valid:

Kinetic Energy ≠ 1/2 mv2

Thus we must use special relativity to describe the energies and momentum of the particles.

The total energy (E=rest +kinetic) of a particle with rest mass, mo, is:

Relativistic Kinematics


Appendix A of M&S

Here v is its speed, c = speed of light, and m is sometimes called the

relativistic mass. The total momentum, p, of a particle with rest mass, mo, is:

We can also relate the total energy, E, to a particle's total momentum,

p, using:

relativistic kinematics 4 vectors
Richard KassRelativistic Kinematics 4-vectors

It is sometimes convenient to describe a particle (or a collection of particles) by a 4-vector. The components of the momentum and energy 4-vector, p, are given by:

The length of the 4-vector is given by:

A particle is said to be

“on the mass shell” if

m0=rest mass

This relationship is true in ALL reference frames

(lab, center of mass,…) because it is a Lorentz invariant.

A 4-vector with length L2 is classified as follows:

Time like if L2 >0

Space like if L2<0

Light like if L2=0 (think photon!)

lorentz invariant vs conserved quantity
Richard KassLorentz Invariant Vs. Conserved quantity

With a Lorentz Invariant you get the same number in two different reference systems.

Let EL and pL be energy and momentum measured in LAB frame

Let Ecm and pcm be energy and momentum measured in center of mass frame

Then: E2cm-p2cm= E2L-p2L

Since (E, p) is a Lorentz invariant (as long as both are measured in same system)

With a conserved quantity you get the same number in the same

reference system but at a different time.

Let piL=initial momentum in lab (before collision)

Let pfL=final momentum in lab (after collision)

Let picm=initial momentum in CM (before collision)

Let pfcm=final momentum in CM (after collision)

Momentum conservation says:

piL= pfL AND picm= pfcm BUT NOTpiL= pfcm

relativistic kinematics 4 vectors1
Richard Kass

We can also manipulate 4-vectors using contravariant/covariant

(up/down) notation: m02=pupu=guvpvpu

In this notation guv is a metric and is given (e.g.) by:

Relativistic Kinematics 4-vectors

Thus the (scalar) product of two 4-vectors (a, b) is given by:

The sum of two 4-vectors is also a 4-vector.

The length of the sum of the 4-vectors of two particles (1,2) is:

m12 is called the

invariant mass

or effective mass

q=angle between particles, m1, m2 are rest masses

relativistic kinematics 4 vectors2
Richard Kass

Example: Consider a proton at rest in the lab frame and an antiproton

with 10GeV/c of momentum also in the lab frame.

What is the energy of the antiproton in the lab frame?

Since the rest energy of a particle is a Lorentz invariant we can make us of:

Relativistic Kinematics 4-vectors


For an antiproton the rest mass, m0, = 938 MeV/c2. We can re-write the above as:

Thus at high energies (E>>m0c2) E » |p|.

How fast is the anti-proton moving in the lab frame ?

We need to remember the energy/momentum relationship between the rest frame

(particle at rest) of the anti-proton and the lab frame:

Thus v = 0.996c (fast!)

more relativistic kinematics
Richard Kass

Colliding beam vs fixed target Collisions

As we will see later in the course cross sections and the energy available for new

particle production depend on the total energy in the center of mass (CM) frame. We

define the CM where the total vector momentum of the collision is zero.

By definition then, in the CM frame we have for two 4-vectors (p1, p2):

(p1+p2)= (E1+E2, p1+p2) = (E1+E2, 0)

If the masses of the two particles are equal as in the case of proton anti-proton

collisions then the above reduces to:

(p1+p2)= (E1+ E2, p1+ p2) = (E1+ E2, 0)= (2E, 0) (twice energy of either particle)

NOTE: the square of the energy in the CM is often called s. s= magnitude of (p1+p2)

More Relativistic Kinematics

How much energy is available in the CM from a 10 GeV/c anti-proton

colliding with a proton at rest?

Since (p1+p2) is a Lorentz invariant we evaluate in any frame we please!

We are given values in the lab frame: (p1+ p2) =(E1+mp, p1+0)

The magnitude of this 4-vector is: s = (E1+mp)2- p12= (10.044+0.938)2-102= 20.6GeV2

Thus the total energy in the CM is 4.54GeV

We could have gotten the same CM energy with two beams = 2.27 GeV !

In general the energy available for new particle production increases as:

(2mtargetEbeam)1/2 for fixed target experiments

2Ebeam for colliding beam experiments

colliding beams are

more efficient for

heavy particle production

even more relativistic kinematics
Richard KassEven More Relativistic Kinematics

Our final example for this section concerns the discovery of the anti-proton. In the

early 1950’s many labs were trying to find evidence of the anti-proton. At Berkeley

a new proton accelerator (BEVATRON) was being designed for this purpose.

Assuming fixed target proton-proton collisions would be used to create the

antiproton what energy proton beam (Eb) is necessary ?

The simplest reaction that conserves all the necessary quantities (energy, momentum,

electric charge, baryon number) is:

The total energy in the CM is given by: (pb+pt)2 the sum of the 4-vectors of the

beam and target proton (assumed to be at rest):

(pb+pt)2 =(Eb+mp, pb)2= mb2+mt2+2mtEb=2mp2+2mpEb

The trick now is to remember that (pb+pt)2 is Lorentz invariant and can be evaluated in any frame we choose. The most convenient frame is the one where all the final state particles are produced at rest. Here we have:

(pb+pt)2inirtial=(Eb+mp, pb)2= (pb+pt)2final = (4mp)2

2mp2+2mpEb = (4mp)2

Eb = 7mp= 6.6 GeV

The anti-proton was discovered at Berkeley in 1955 (Nobel Prize 1959)

time dilation
Richard Kass

Most of the particles we are concerned with in HEP are not stable, i.e. they

spontaneously decay into other particles after a certain amount of time. For example

consider the lepton family:

Lepton Mean Lifetime (s)

electron stable

muon (m) »2x10-6

tau (t) »3x10-13

The above table gives the average lifetime of the leptons in their rest frame. However,

often we need to know how long a particle will live (on average) in a frame where the

particle is moving close to the speed of light (c). We need to use special relativity!

Consider a particle moving with speed v as pictured below:

Time Dilation

The particle is moving in the lab frame

with speed v along the x-axis. The

relationship between time and distance

measured in the lab and CM frame is:

tlab= g(tcm+ b/c xcm) with b= v/c.

xlab= g(xcm+ bc tcm)

In the lab frame the time between life and death of the particle is:

tlab= t2lab- t1lab= g(t2cm + b/c x2cm)- g(t1cm + b/c x1cm) = g(t2cm-t1cm)=gt

Note: t is called the proper lifetime and t£tlab

more time dilation
Richard Kass

Consider a muon (m0=0.106 GeV/c2) with 1 GeV energy in the lab frame.

On average how long does this particle live in the LAB?

In the muon’s rest frame it only lives (on average) t = 2 msec.

But in the lab frame it lives (on average):

tlab=gt, and g=E/(m0c2) = 1/0.106 » 10

tlab=gt =(10)(2 msec) = 20 msec

On average how far does this particle travel in the LAB before decaying ?

Dxlab= gbct with gb= p/(m0c) »1/0.106 » 10

Dxlab= gbct = 10ct = (10)(3x108 m/s)(2x10-6 s) = 6x103 m

More Time Dilation