Physics 2170: Introduction to Particle Physics

1
Physics 2170 Introduction to Particle Physics

• Lecture 15
• Greg Landsberg
• March 16, 2009

2
Feynman Rules Recapped

• For EACH vertex in the diagram, introduce a
  factor of ig, where g is the coupling
• In real theories coupling is always
  dimensionless e.g., e is EM coupling and e2/4p
  a
• In toy theory, coupling has dimensions of energy
• For EACH vertex, further write down four-momentum
  conservation via a delta function
  (2p)4d4(k1k2kn), where ki are four-momenta
  coming INTO vertex (i.e., for the outgoing
  momenta use ki)

3
Feynman Rules Recapped

• For EACH internal line, write a propagator
  i/(qj2 mj2)
• For EACH internal line put a factor d4qi/(2p)4
  and integrate over all the internal momenta
• Cancel the delta function the final result will
  include a delta function of the type
  (2p)4d4(p1p2pn), which is part of the Fermi
  Golden rule erase it and multiply by i to get M

4
Example Lifetime of A

• Feynman rules
• Hence, M i (-ig) g
• Decay rate
• Here is the amplitude of either outgoing
  momentum
• The lifetime is then (g
  GeV)

5
Example AA ? BB Scattering

• Feynman rules
• Lets integrate over the second delta function,
  which sends q ? p4 p2
• Therefore
• Note that M 1, as expected

6
AA ? BB Scattering (contd)

• Not a full story yet there isanother diagram
  obtained bytwisting the outgoing legs
• Note that twisting incoming legs doesnt give
  any additional diagrams as the only important
  thing is whether p3 connects to p1 or p2
• One can get the amplitude via p3 ? p4
• As usual, the overall amplitude is the sum of the
  two

7
AA ? BB Cross Section

• Suppose we would like to get differential cross
  section in the c.o.m. frame for a simpler case
  mA mB m and mC 0
• Here p is the absolute value of particle 1
  momentum, which is the same as the momenta of
  particles 2, 3, 4
• Hence

8
Higher-Order Diagrams

• So far we considered only the lowest order (tree
  level) diagrams for AA ? BB scattering
• There are many more
• There are 8 next-to-leading order diagrams

9
Next-to-Leading Order Diagrams

• We wont calculate them all, lets just consider
  one self-energy diagram
• Feynman rules give

Integrate over q1 and q4 using the first and last
deltas q1 ? p1 p3 q4 ? p4 p2
Now integrate over q2 using the first delta q2
? p1 p3 q3 cancel last delta
10
How to Calculate This Integral

• d4q q3dq dW (similar to 3D d3r r2dr dW)
• Thus the integral at large q can be approximated
  as follows
• This integral diverges at high q, thus making
  quantum corrections infinite!
• The problem held QED development for some 20
  years, until Feynman, Tomonaga, Schwinger and
  others found a solution, known as
  renormalization procedure
• The idea was to regularize the integral by using
  an arbitrary cutoff, L, thus making it finite

11
Dealing with Infinities

• In the case of our problem, the regularization
  can be achieved by inserting the following factor
  under the integral
• This of course appears completely arbitrary,
  except that if one takes L to infinity, the
  fudge-factor approaches 1
• However, with this factor, the integral can be
  rewritten as a sum of two parts the finite one
  and the diverging one lnL
• Now, a magic thing is that ALL the divergent
  pieces appear in the final expression ONLY as
  additive terms to masses and couplings m ? m
  dm g ? g dg

12
Renormalization

• The renormalization procedure basically replaces
  bare masses and couplings with their
  renormalized physical values
• There is a minor problem these masses and
  couplings approach infinity at infinite energies,
  but as we measure them only at finite energies,
  this is really an unphysical infinity
• Cf. with self-energy of electric field around the
  point-like electric charge which is infinite and
  thus corresponds to the infinite mass via E m
• Apart from these infinite terms, there are
  perfectly calculable and finite terms that are
  added to masses and couplings from loop diagrams
• They cause running of the couplings and masses

13
Still Concerned?

• Renormalization procedure can be considered as a
  way to parameterize unknown modifications to
  quantum mechanics at very short scales
• Usually when an infinity comes as a part of
  solution, this merely means a limitation of the
  theory
• For example, if space-time is quantized there
  exists a minimum length scale
• Think about renormalized masses and couplings as
  effective massive and couplings for a particle
  propagating through a complex structure of
  quantum vacuum
• These effective variables parameterize properties
  of quantum vacuum, not the particles per se