Feynman parametrization

As Feynman parameters parameters are referred to that are temporarily imported into integrals to solve this. The parameters are used in particular in the calculation of Feynman diagrams with internal loops ( " loop "). Both Richard Feynman and Julian Schwinger Seymour used analog methods.

Simple Example

If you want to solve the integral, it can be seen that the integrand can be written as in the place. Here, suddenly, the parameter appears, which has no " physical meaning ", but is only needed to solve the integral. By exchanging integral and derivative remains a simple integral over the exponential function, which is easy to solve. The derivative can be carried out according to. By replacing the parameter disappears and the integral is achieved.

The electron vertex

In the solution of the one -loop contribution to the vertex function of the electron one encounters integrals of the form

Although, simple and quadratic terms of the four-momentum are, then these integrals can not be easily solved. After using the appropriate equation below and linear substitution, we obtain in place of the above integral

And, the integrals over the Feynman parameters then solve.

Example with only two factors in the denominator

The trick with the factors in the denominator is two Feynman parameters and introduce is also integrated different than the example above about the. First, one uses

The above equation can easily be shown by substitution in the integral. With the help of the delta function is molded this into a symmetric form:

Here and now appear side by side on additive, which significantly simplifies the integration.

Generalizations

For more than two factors

For calculations in the dimensional renormalization, a further generalization is necessary:

Where the exponents are complex numbers ( with positive real part ) can be. With the help of the delta function one can write this as

Application

An integral of a product of the denominator of the integrands can be transformed as follows:

Typically, then the integrand depends for further transformations only quadratically on the variable of integration from what makes a transition to ( n-dimensional ) polar coordinates possible.