Pictures Paths Particles Processes

84 November 7, 2013
k
↔
¯h
| ⃗ k| 2 + m 2
k 1 k 2
↔ − λ 4
k 4 k 3
¯h (2π)D δ D ( ⃗ k 1 + ⃗ k 2 + ⃗ k 3 + ⃗ k 4 )
k 1 k 2 ↔ +
J( ⃗ k 2 )
¯h (2π)D δ D ( ⃗ k 1 + ⃗ k 2 )
At vertices, the wavevectors are considered either all incoming or
all outgoing.
Each internal wave vector ⃗ k is to be integrated over, with integration
element d D⃗ k/(2π) D .
Feynman rules, version 2.5 (2.85)
2.4.6 Loop integrals
As stated above, diagrams with loops contain internal wave vectors that have
to be integrated over, and many of these integrals are divergent. Therefore, we
have two face two technical challenges. In the first place, we have to devise a
way to quantify these divergences : this is called regularization. In the second
place, regularizing these divergences does not make them go away, and therefore
we shall have to arrive at a method of including these divergences into the
theory in such a way as to yield finite and unambiguous answers for physically
interesting quantities. This last procedure is called renormalization. In this
section we shall only address regularization, for the case of one-loop integrals.
The idea of regularization is to let the theory depend on an arbitrarily introduced
parameter, such that the divergences appear when that parameter takes
on a certain value. Different regularization schemes are available, with different
choices for the extra parameter, which may be particle masses, upper limits on
momenta, etcetera. It must be kept in mind, however, that theories may depend
sensitively on such parameters, and therefore it may be prudent to choose the
parameter in such a way that the behaviour of the theory does not depend on it
too sensitively. The most popular regularization scheme is that of dimensional
regularization : in this approach the number of dimensions, D, is chosen as the
freely varying parameter. Already anticipating that we shall study theories in
four spacetime dimensions, we therefore write
D = 4 − 2ɛ ,
with the implication that, at the end of all calculations, we shall take ɛ down to