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Units, Constants and Useful Formulas
Feynman Diagrams and Loops
--------------------------
Feynman Rules
-------------
QED:
For each vertex assign a coupling constant as: -iγ^{μ}g
Aside: This comes from:
_
L = Ψ(iγ^{μ}D_{μ} - m)Ψ where D_{μ} = ∂_{μ} + igA_{μ}
Which leads to:
_
L = Ψ(iγ^{μ}∂_{μ} + igγ^{μ}A_{μ} - m)Ψ
The electron - photon interaction interaction term in the
Lagrangian is:
_
L = -gΨγ^{μ}A_{μ}Ψ
For each virtual particle: -ig_{μν}/(p^{2} + iε) (photon)
i(γ^{μ}p_{μ} + m)/(p^{2} - m^{2} + iε) (fermion)
Impose momentum conservation at each vertex: (2π)^{4}δ^{4}(p_{1} + p_{2} - q_{1} - q_{2})
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 ∫d^{4}p/(2π)^{4}
Incoming electron: u(p)
_ _
Outgoing electron: u(p) where u = γ^{0}u^{†}
_
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 E^{2} - p^{2} ≠ m^{2}. 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:
∫d^{4}p^{''}/(2π)^{4}(-igγ^{α})(γ^{μ}q-m)/(q^{2}-m^{2}+iε)(-igγ^{β})(γ^{ν}r-m)/(r^{2}-m^{2}+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 = mc^{2} = ε(1/2)E^{2}
^{ } = (ε1/2)∫(q/4πr^{2})^{2}4πr^{2}dr
^{ } = q^{2}/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 = ∫d^{4}p i/(p^{2} - m^{2} + 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 g^{2} whereas in (b) the
probability is proportional to g^{4}. Ignoring higher order terms in
this way is the basis of Perturbation theory.