1. Lecture 2 - Feynman Diagrams & Experimental Measurements
From drawing diagrams of interactions to calculating scattering cross-sections
In this lecture only for electromagnetic interactions between (hypothetical) spinless charged elementary particles
15/1/10
Particle Physics Lecture 2 Steve Playfer

2. Particle Physics Lecture 2 Steve Playfer
Two-Body Collisions
p3 (scattered)
LAB Frame
p2= 0
“Fixed target”
experiments q y
p1 (beam)
p4 (recoil)
p3=pf*
p1=pi* q*
CM Frame
“Collider”
experiments
p2= -pi*
p4= -pf*
Elastic scattering has pi*= pf*
15/1/10
Particle Physics Lecture 2 Steve Playfer

3. Drawing Feynman Diagrams
“virtual” particles
are exchanged
In the middle
Initial state particles
are on the left
Final state particles
are on the right
Each interaction vertex
has a coupling constant
Warning – do not interpret diagrams literally as time (x) and space (y) coordinates!
fermions
antifermions
photons
gluons
W, Z, H
bosons
15/1/10
Particle Physics Lecture 2 Steve Playfer

4. Particle Physics Lecture 2 Steve Playfer
The Feynman Rules
Initial and final state particles have wavefunctions:
Spin 0 bosons are plane waves
Spin 1/2 fermions have Dirac spinors
Spin 1 bosons have polarization vectors em
Vertices have dimensionless coupling constants:
Electromagnetism has √(4pa) = e
Strong interaction has √as = gs
Weak interactions have gL and gR (or cA and cV)
Virtual particles have propagators:
Virtual photon propagator is 1/q2
Virtual W boson propagator is 1/(q2 –MW2)
Virtual fermion propagator is (gmqm + m)/(q2 – m2)
(see Lecture 3 for definition of gamma matrices gm)
15/1/10
Particle Physics Lecture 2 Steve Playfer

5. Relativistic Kinematics
Four momenta of initial and final state particles:
pm = [ E, p ] with m2 = E2 – |p|2
Four momenta of virtual particles are “off the mass shell”
qm = [ q0, q ] with q2 = q02 – |q|2 ≠ m2
Each component of four momentum is conserved
Four momentum is conserved at each vertex of a Feynman diagram
Lorentz transformations between LAB and CM frames
p’x = g(px – bE) E’ = g(E – bpx) with b = p*/E* and g2= 1/(1–b2)
Apply same Lorentz transformation to all initial and final state particles
Amplitudes must be Lorentz invariant.
All products of two four vectors are Lorentz invariants.
Examples on this page are q2 and m2
15/1/10
Particle Physics Lecture 2 Steve Playfer

6. Particle Physics Lecture 2 Steve Playfer
Mandelstam Variables
The variables s,t and u are Lorentz invariants p1
s-channel p3 p1
t-channel p3
t = (p1- p3)2
s = (p1+ p2)2 p2 p4 p2 p4
For highly relativistic elastic scattering
where p~E, m << E:
s = 4 p*2
t = -2p*2 (1-cosq*) u = -2p*2 (1+cosq*)
where p* is the CM momentum of the
particles, and q* is the CM scattering angle p1
u-channel p4
u = (p1- p4)2 p2 p3
15/1/10
Particle Physics Lecture 2 Steve Playfer

7. Particle Wavefunctions
A free particle is a plane wave:
y = N e –ip.x where p.x = pmxm = h(k.x – wt)
(this wavefunction contains no spin, charge or colour information)
Probability density: r = i [ y* dy - y dy* ]
dt dt
Normalisation of density of states in a box of Volume V:
∫V r d3x = 2E (usually take V=1, N=1)
Probability current: jm = i [ y* dmy - y dmy* ]
For a particle transition p1 ® p3: jm = (p1 + p3 ) e –i(p3-p1).x
15/1/10
Particle Physics Lecture 2 Steve Playfer

8. Scattering Cross-sections
A cross-section describes the rate of scattering during a collision
between two particles. Units are barns (1 b = m2).
Cross-section: ds = Wfi = 2p |Mfi|2 rf
ri ri
where Wfi is the transition rate and Mfi is the matrix element
ri is an initial state flux factor (number of states and relative velocity)
rf = dNf/dE is the final state phase space factor (density of final states)
Total cross-section s=∫ds is a Lorentz invariant physical observable
15/1/10
Particle Physics Lecture 2 Steve Playfer

9. Initial & Final State Factors
Each initial state particle has 2Ei states.
Flux factor also requires relative velocity between particles vi
ri = 2EA 2EB vi
Lorentz invariant final state phase space has a density per final state particle of: d3p rf = Pj=1,n d3pj
(2p)3 2E (2p)(3n-4) Ej
For a relativistic two-body scattering process A+B ® C+D in the CM system the three momenta are pA = -pB = pi pC = -pD = pf
ri = 4pi√s rf = pf dW
16p2 √s
where √s is the total CM energy
15/1/10
Particle Physics Lecture 2 Steve Playfer

10. Differential Cross-section
The differential cross-section for A+B ® C+D in the CM system is:
ds = pf |Mfi |2
dW p2 s pi
N.B. Solid angle element dW = d(cos q) df depends on frame of reference
because cos q* ≠ cos qlab (see Dynamics & Relativity lectures)
Angular dependence of the differential cross-section is
contained in the matrix element Mfi
For elastic scattering note that pi = pf = p*
For inelastic scattering usually have pf < pi
15/1/10
Particle Physics Lecture 2 Steve Playfer

11. Matrix Element for Spinless Scattering
Hypothetical interaction in which
two spinless charged particles
exchange a virtual photon
(and we ignore higher order diagrams)
q = p1– p3 = p4 – p2 √a
Vertex
Couplings p2 p4
M = a (p1 + p3) (p2 + p4) d4(p1 + p2– p3 – p4) q2
Photon
Propagator
Four momentum
conservation
Plane Waves
M = a (s – u) (in terms of Mandelstam variables) t
15/1/10
Particle Physics Lecture 2 Steve Playfer

12. Spinless differential cross-section
The amplitude for a transition from an initial state to a final state
is given by the sum of all possible Feynman diagrams.
Spinless matrix element for two non-identical particles:
M = a (u - s) (t-channel) t
but for two identical particles it becomes:
M = a (u - s) + a (t – s)
t u (t-channel + u-channel)
because you can’t tell which final state particle is which!
For elastic scattering of identical spinless particles:
ds = a (u - s) + (t - s) 2
dW p2s t u [ ]
15/1/10
Particle Physics Lecture 2 Steve Playfer

13. Including Polarization
If the two-body scattering process A+B ® C+D involves fermions,
the Matrix element has to include the initial and final spin states:
Mfi ( ) Mfi ( ) Mfi ( ) Mfi ( )
These are not necessarily the same!
If fermion spins are observed we measure polarized cross-sections
If spins are not known the cross-section is unpolarized
To calculate unpolarized cross-sections:
Initial spin states are averaged over
Final spin states are summed over
15/1/10
Particle Physics Lecture 2 Steve Playfer

14. Particle Physics Lecture 2 Steve Playfer
Particle Decays m1 M m2
-p* p*
Two-Body
M > m1 + m p*2 = (M2 – m12 – m22)2 – 4m12m22
4M2 p2 m2 m1 M p1
Three-Body m3 p3
M > m1 + m2 + m p1 = p2 + p3
15/1/10
Particle Physics Lecture 2 Steve Playfer

15. Decay Rates & Lifetimes
Partial decay width of a particle mass M to a final state f:
Gf = h Wfi = 2p |Mfi |2 rf
For a decay to a two-body final state (m1, m2) :
rf = pf with E1 = √(m12 + pf2) = M2 + m12 – m22
16p2 M M
For several possible decay modes the total decay width
and branching fractions are:
G = Sf Gf B (i®f) = Gf / G
The proper lifetime in the rest frame is t = h/G
and the decay length in a moving frame is L = gbct
15/1/10
Particle Physics Lecture 2 Steve Playfer

16. Particle Physics Lecture 2 Steve Playfer
Three-body Decays
The final state phase space is given by:
rf = Pj=1,3 d3pj
(2p) Ej
The allowed kinematic region is determined by the available energy Q :
Q = M - Sj=1,3 mj = Sj=1,3 Tj
(where Tj are the Kinetic energies of the final state particles)
Using the centre of mass energy/momentum constraints:
√s = M = √(M122 + p122 ) + √(m32 + p122 ) (and similarly for p13 and p23)
where M122 = (E1 + E2)2 - p122 and p12 = (p1 + p2)
From this it can be shown that the invariant mass squared of 1 & 2 is:
M122 = s + m32 – 2√s E3
It is usual to represent the kinematic region by a “Dalitz plot”
The axes of these plots are either M122 and M232 or T3 and T1
15/1/10
Particle Physics Lecture 2 Steve Playfer

17. Particle Physics Lecture 2 Steve Playfer
Dalitz Plots T1 T1
Non-relativistic
limit T=p2/2m
Relativistic
limit T=E
Allowed regions for
three identical final
state particles T2 T3 T2 T3
B ® ppp
In general the Matrix
element depends on the
position in the Dalitz plot
(M122, M232) or (T3,T1)
These areas removed
because of background
15/1/10
Particle Physics Lecture 2 Steve Playfer