Feynman Diagrams
Richard Feynman invented a method for describing amplitude calculations in Quantum Field Theory. Feynman Diagrams are graphs that represent terms, in the perturbative expansion for the amplitude, of a process. From this amplitude other observables can be calculated. As experiments become increasingly more sophisticated and accurate, we require higher accuracy from computational methods. To achieve this increased accuracy, higher orders in the perturbative expansion need to be calculated.
Computing Feynman Integrals
Below is a systematic approach to performing the integrals that occur in Feynman diagrams involving loops: • Introduce Feynman Parameters denominators of the propagators. to combine the
Polylogarithms
As mentioned earlier, Feynman Integrals lead to the appearence of special functions. We define the polylogarithm function as: Lim (z) =
∞
• Complete the Square in the new denominator by shifting to a new loop momentum variable . • Perform a Wick rotation. • Introduce Generalised Spherical Co-ordinates. • Perform the Angular Integration and using definitions of Euler’s Gamma and Beta functions perform the Radial Integration. • Perform the non-trivial Integration over Feynman Parameters. N.B. Many loop integrals are divergent and must undergo some kind of regularisation. There are many well defined methods for doing this, such as Dimensional Regularisation presented in [SW].
n =1
∑
zn nm
z C, |z| < 1, m N
When m = 2 we have the dilogarithm: Li2 (z) =
n =1
∑
∞
zn n2
z C, |z| < 1
which can be analytically continued to the cut complex plane via the integral representation:
z
Li2 (z) = −
0
log(1 − t)dt t
z C − [1, ∞)
Here is an example of an integral that is given in terms of the function above.
Electron interaction via exchange of a virtual photon
During the computation of these higher order terms (involving diagrams with increasing loop number) we come across complicated integrals which lead to special functions such as Polylogarithms.
The One-Loop Three Point Function
The diagram below is a one-loop diagram with three external lines. It has been labelled following momentum conservation and the associated integral is written below using a similar set of rules to those of ϕ4 theory. Using the systematic method mentioned above it is possible to achieve a result for this integral.
Feynman Parametrisation
Feynman found a technique for combining the denominators of the integral by introducing new parameters. This version below is most useful for basic loop integrals: 1 = ( n − 1) ! A1 · · · A n
1
Feynman Rules
Feynman Rules allow us to translate a Feynman Diagram into its associated mathematical formula. The rules are derived from the Lagrangian of the specific theory. Here is an example of the rules for ϕ4 theory:
0
dx1 · · ·dxn
δ ( x1 + · · · + x n ) ( x1 A1 + · · · + x n A n ) n
It makes a given denominator suitable to complete the square on by a shift of integration variable.
L=
1 1 λ ( ∂ µ φ )2 − m2 φ2 − φ4 2 2 4!
• For each internal scalar field line, associate a propagator i given by q2 −m2 +i . • At each vertex respect momentum conservation and place a factor of −ig. • For each closed loop in the diagram, integrate over
d4 q , (2π )4
Wick Rotation
Another technique used is a Wick Rotation which transforms the Minkowski Space vector q into a Eulclidean one Q. This allows us to perform the integral using generalised spherical co-ordinates. In four dimensions the vector q is: q2 = q2 − q2 − q2 − q2 0 2 3 1 whereas after changing variables: q0 = iQ0 , q j = Q j for j = 1, 2, 3, d4 q = id4 Q
C0 ( pi , mi ) = dn q (q2 + m2 )((q + p1 )2 + m2 )((q + p1 + p2 )2 + m2 ) 2 3 1
where q is the momentum associated with the
The Feynman Integral, after following the systematic approach, has a solution given in terms of the dilogarithm defined above, as shown in [t’HV].
loop. • For any Feynman Diagram F built from these propagators and vertices, divide by the symmetry factor of F, which is assumed to contain n vertices. The symmetry factor is given by the number of possibilities to connect n vertices to give the F divided by (4!)n .
References
[SW] Stefan Weinzierl - The Art of Computing Loop Integrals - hepph/0604068 [t’HV] t’Hooft and Veltman - Scalar One-Loop Integrals - Nucl Phys B153
we get the Eluclidean signature: Q2 = Q2 + Q2 + Q2 + Q2 0 2 3 1
DJ Schofield 4H Project
February 2008