Feynman parameters

As Feynman parameter parameters are referred to which are imported temporarily in integrals to solve this. The parameters are used in particular for the calculation of Feynman diagrams with inner loops ("loops"). Both Richard Feynman and Julian Seymour Schwinger used analogous methods.

Simple example

If you want to solve the integral , you find that the integrand can also be written as at the point . Suddenly the parameter appears which has no "physical meaning", but is only needed to solve the integral. By interchanging the integral and derivative, a simple integral over the exponential function remains, which is easy to solve. The derivation according to is feasible. After replacing , the parameter disappears and the integral is solved. ${\displaystyle \int te^{t}dt}$ ${\displaystyle \partial _{u}e^{ut}\ }$ ${\displaystyle u=1\ }$ ${\displaystyle u\ }$ ${\displaystyle u\ }$ ${\displaystyle u=1\ }$ ${\displaystyle u}$

${\displaystyle \int te^{t}dt=\int (\partial _{u}e^{ut}|_{u=1})dt=(\partial _{u}(\int e^{ut}dt))|_{u=1}=(\partial _{u}(e^{ut}/u))|_{u=1}=(e^{ut}(ut-1)/u^{2})|_{u=1}=e^{t}(t-1)}$

The electron vertex

1-loop contribution to the vertex function of the electron

When solving the 1-loop contribution to the vertex function of the electron, one encounters integrals of the form

${\displaystyle \int {\frac {d^{4}k}{(2\pi )^{4}}}{\frac {N(k)}{A(k)\ B(k)\ C(k)}}\ .}$

Although , and are simple quadratic terms of the four-momentum , these integrals cannot easily be solved. After using the corresponding equation below and linear substitution , one obtains instead of the above integral ${\displaystyle A(k)\ }$ ${\displaystyle B(k)\ }$ ${\displaystyle C(k)\ }$ ${\displaystyle k\ }$ ${\displaystyle l=k+...\ }$

${\displaystyle \int {\frac {d^{4}l}{(2\pi )^{4}}}\left({\frac {l^{2}}{|l^{2}-\Delta |^{3}}}-{\frac {l^{2}}{|l^{2}-\Delta _{\Lambda }|^{3}}}\right)={\frac {i}{(4\pi )^{2}}}\log({\frac {\Delta _{\Lambda }}{\Delta }})}$

and can then also solve the integrals via the Feynman parameters.

Example with only two factors in the denominator

The trick with the factors in the denominator is to introduce two Feynman parameters and , which, unlike in the example above, also integrate. First you use ${\displaystyle u}$ ${\displaystyle v}$

${\displaystyle {\frac {1}{AB}}=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}\;.}$

The above equation can easily be shown by substitution in the integral. With the help of the delta function this is transformed into a symmetrical form: ${\displaystyle y=uA+(1-u)B}$

${\displaystyle {\frac {1}{AB}}=\int _{0}^{1}du\int _{0}^{1}dv{\frac {\delta (1-u-v)}{\left[uA+vB\right]^{2}}}\ .}$

Appear here and now additively side by side, which significantly simplifies integration. ${\displaystyle A}$ ${\displaystyle B}$

Generalizations

For more than two factors

${\displaystyle {\frac {1}{A_{1}\cdots A_{n}}}=(n-1)!\int _{0}^{1}du_{1}\cdots \int _{0}^{1}du_{n}{\frac {\delta (u_{1}+\dots +u_{n}-1)}{\left[u_{1}A_{1}+\dots +u_{n}A_{n}\right]^{n}}}.}$

For calculations within the framework of the dimensional renormalization , a further generalization is necessary:

${\displaystyle {\frac {1}{A_{1}^{\alpha _{1}}\cdots A_{n}^{\alpha _{n}}}}={\frac {\Gamma (\alpha _{1}+\dots +\alpha _{n})}{\Gamma (\alpha _{1})\cdots \Gamma (\alpha _{n})}}\int _{0}^{1}du_{1}\int _{0}^{1-u_{1}}du_{2}\cdots \int _{0}^{1-u_{1}-\ldots -u_{n-2}}du_{n-1}{\frac {u_{1}^{\alpha _{1}-1}\cdots u_{n-1}^{\alpha _{n-1}-1}(1-u_{1}-\ldots -u_{n-1})^{\alpha _{n}-1}}{\left[u_{1}A_{1}+\dots +u_{n-1}A_{n-1}+(1-u_{1}-\ldots -u_{n-1})A_{n}\right]^{\alpha _{1}+\dots +\alpha _{n}}}},}$

where the exponents can be complex numbers (with a positive real part). With the help of the delta function this can be written as ${\displaystyle \alpha _{i}}$

${\displaystyle {\frac {1}{A_{1}^{\alpha _{1}}\cdots A_{n}^{\alpha _{n}}}}={\frac {\Gamma (\alpha _{1}+\dots +\alpha _{n})}{\Gamma (\alpha _{1})\cdots \Gamma (\alpha _{n})}}\int _{0}^{1}du_{1}\cdots \int _{0}^{1}du_{n}{\frac {\delta (1-u_{1}-\dots -u_{n})u_{1}^{\alpha _{1}-1}\cdots u_{n}^{\alpha _{n}-1}}{\left[u_{1}A_{1}+\dots +u_{n}A_{n}\right]^{\alpha _{1}+\dots +\alpha _{n}}}}.}$

application

An integral with a product in the denominator of the integrand can be transformed as follows:

${\displaystyle \int {\frac {dp}{A(p)B(p)}}=\int dp\int _{0}^{1}{\frac {du}{\left[uA(p)+(1-u)B(p)\right]^{2}}}=\int _{0}^{1}du\int {\frac {dp}{\left[uA(p)+(1-u)B(p)\right]^{2}}}.}$

Typically, after further transformations, the integrand only depends on the square of the integration variable, which makes a transition to (n-dimensional) polar coordinates possible.

literature

1. ↑ Michael E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Field Theory , Westview Press, 1995

Web links

• Kristjan Kannike: Notes on Feynman Parametrization and the Dirac Delta Function (PDF; 142 kB) Archived from the original on July 29, 2007. Retrieved on June 3, 2009.