Feynman Diagrams and Loops


Feynman Diagrams and Loops
--------------------------

Feynman Rules
-------------

QED:

For each vertex assign a coupling constant as: -iγ^μg
    
Aside:  This comes from:
    _
L = Ψ(iγ^μD_μ - m)Ψ  where D_μ = ∂_μ + igA_μ

Which leads to:
    _
L = Ψ(iγ^μ∂_μ + igγ^μA_μ - m)Ψ  

The electron - photon interaction interaction term in the 
Lagrangian is:
      _
L = -gΨγ^μA_μΨ  

For each virtual particle: -ig_μν/(p² + iε) (photon)   
                           i(γ^μp_μ + m)/(p² - m² + iε) (fermion)

Impose momentum conservation at each vertex: (2π)⁴δ⁴(p₁ + p₂ - q₁ - q₂)
The sum of the incoming 4 momenta =  sum of the outgoing 4 momenta.
4 momenta flow through the diagrams like electric currents.

Integrate over all unconstrained momenta ∫d⁴p/(2π)⁴

Incoming electron:  u(p)
                    _          _
Outgoing electron:  u(p) where u = γ⁰u^†
                    _
Incoming positron:  v(p)
                              
Outgoing positron:  v(p)

Incoming photon: ξ  (polarization vector)
                 _
Outgoing photon: ξ

Loops and Trees
---------------

In addition to the Feynman diagram shown in (a) we can also draw 
diagrams that contain loops as shown in (b) and (c).  Diagrams 
without loops are called TREE DIAGRAMS.  Diagrams with loops are 
called LOOP DIAGRAMS.  Consider the following diagrams corresponding
to electron-electron scattering.



In these diagrams the solid red lines are internal lines and represent 
the Fermionic propagator.  The wavy line is also internal and represents 
the photon propagator.  

When a diagram contains a closed loop the energy and momentum of the 
virtual particles participating in the loop can be off-shell, meaning 
that E² - p² ≠ m².  Therefore the particles in the loop are not uniquely 
determined by the energy and momenta of the incoming and outgoing 
particles.  This means that we need to integrate over all possible 
combinations of energy and momentum that could travel around the loop.  
However, these integrals are often divergent when the energy and 
momenta of the particles are high (equivalentl to short distances and 
time^*).  In the case of (c), for example, the overall probability 
amplitude for the process is equal to:

∫d⁴p^''/(2π)⁴(-igγ^α)(γ^μq-m)/(q²-m²+iε)(-igγ^β)(γ^νr-m)/(r²-m²+iε)(-igγ^δ)g^μν/p^''2+iε)

Where q = p' - p'' and r = p - p''

The divergences that occur are referred to as ULTRAVIOLET DIVERGENCES. 

      * Using dimensional analysis we can show that the greater 
        the momentum, energy and mass, the smaller the distance.  
        When h = c = 1 we can write:

        [E] = [M] = [L]^-1 (from E = hc/λ)

        [p] = [L]^-1 (from p = hk = h/λ)

        For example, assume a particle is a charged spherical shell 
        of radius, r.  The mass-energy relationship is:

        E = mc² = ε(1/2)E²

               = (ε1/2)∫(q/4πr²)²4πr²dr

               = q²/4πεr

        Clearly the energy becomes infinite as r -> 0.   

Regularization
--------------

Removing these loop divergences is referred to as REGULARIZATION.  
Regularization involves imposing a CUT-OFF at high energies and 
momentum so that the momentum integrals converge.  To see how 
such a cut-off works, we can consider the following integral where 
Λ is the cut-off momentum:

     _Λ
I = ∫d⁴p i/(p² - m² + iε)
   ⁰

This integral is logarithmically divergent as Λ -> ∞

In theory we are supposed to draw and calculate all of the possible 
loops representing interactions between the initial and final states.   
However, if the coupling constant, g, is less than 1 then we can 
ignore higher order terms.  For example in the case of (a), the 
probability of the process is proportional to g² whereas in (b) the
probability is proportional to g⁴.  Ignoring higher order terms in 
this way is the basis of Perturbation theory.

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