Feynman parametrization

Richard Feynman observed in 1949, that:{{Cite journal |last=Feynman |first=R. P. |date=1949-09-15 |title=Space-Time Approach to Quantum Electrodynamics |url=https://link.aps.org/doi/10.1103/PhysRev.76.769 |journal=Physical Review |volume=76 |issue=6 |pages=769–789 |doi=10.1103/PhysRev.76.769|bibcode=1949PhRv...76..769F }}

:$\backslash frac\{1\}\{AB\}=\backslash int^1\_0\; \backslash frac\{du\}\{\backslash left[uA\; +(1-u)B\backslash right]^2\}$

which is valid for any complex numbers A and B as long as 0 is not contained in the line segment connecting A and B. The formula helps to evaluate integrals like:

:$\backslash begin\{align\}$

\int \frac{dp}{A(p)B(p)}

&= \int dp \int^1_0 \frac{du}{\left[uA(p)+(1-u)B(p)\right]^2} \\

&= \int^1_0 du \int \frac{dp}{\left[uA(p)+(1-u)B(p)\right]^2}.

\end{align}

If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution.

More generally, using the Dirac delta function $\backslash delta$:{{cite book|last=Weinberg|first=Steven|title=The Quantum Theory of Fields, Volume I|date=2008|publisher=Cambridge University Press|location=Cambridge|pages=497|isbn=978-0-521-67053-1}}

:$\backslash begin\{align\}$

\frac{1}{A_1\cdots A_n}&= (n-1)! \int^1_0 du_1 \cdots \int^1_0 du_n \frac{\delta(1-\sum_{k=1}^{n}u_{k})\;}{\left(\sum_{k=1}^{n}u_{k}A_{k}\right)^n} \\

&=(n-1)! \int^1_0 du_1 \int^{u_1}_0 du_2 \cdots \int^{u_{n-2}}_0 du_{n-1} \frac{1}{\left[A_1u_{n-1}+A_2(u_{n-2} - u_{n-1})+\dots+A_n(1 - u_1)\right]^n}.

\end{align}

This formula is valid for any complex numbers A₁,...,A_n as long as 0 is not contained in their convex hull.

Even more generally, provided that $\backslash text\{Re\}\; (\; \backslash alpha\_\{j\}\; )\; >\; 0$ for all $1\; \backslash leq\; j\; \backslash leq\; n$:

:$\backslash frac\{1\}\{A\_\{1\}^\{\backslash alpha\_\{1\}\}\backslash cdots\; A\_\{n\}^\{\backslash alpha\_\{n\}\}\}\; =\; \backslash frac\{\backslash Gamma(\backslash alpha\_\{1\}+\backslash dots+\backslash alpha\_\{n\})\}\{\backslash Gamma(\backslash alpha\_\{1\})\backslash cdots\backslash Gamma(\backslash alpha\_\{n\})\}\backslash int\_\{0\}^\{1\}du\_\{1\}\backslash cdots\backslash int\_\{0\}^\{1\}du\_\{n\}\backslash frac\{\backslash delta(1-\backslash sum\_\{k=1\}^\{n\}u\_\{k\})\backslash ;u\_\{1\}^\{\backslash alpha\_\{1\}-1\}\backslash cdots\; u\_\{n\}^\{\backslash alpha\_\{n\}-1\}\}\{\backslash left(\backslash sum\_\{k=1\}^\{n\}u\_\{k\}A\_\{k\}\backslash right)^\{\backslash sum\_\{k=1\}^\{n\}\backslash alpha\_\{k\}\}\}$

where the Gamma function $\backslash Gamma$ was used.{{cite web|url=http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf|title=Notes on Feynman Parametrization and the Dirac Delta Function|author=Kristjan Kannike|archive-url=https://web.archive.org/web/20070729015208/http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf|archive-date=2007-07-29|access-date=2011-07-24}}

:$\backslash frac\{1\}\{AB\}\; =\; \backslash frac\{1\}\{A-B\}\backslash left(\backslash frac\{1\}\{B\}-\backslash frac\{1\}\{A\}\backslash right)=\backslash frac\{1\}\{A-B\}\backslash int\_B^A\; \backslash frac\{dz\}\{z^2\}.$

By using the substitution $u=(z-B)/(A-B)$, we have $du\; =\; dz/(A-B)$, and $z\; =\; uA\; +\; (1-u)B$, from which we get the desired result

:$\backslash frac\{1\}\{AB\}\; =\; \backslash int\_0^1\; \backslash frac\{du\}\{\backslash left[uA\; +\; (1-u)B\backslash right]^2\}.$

In more general cases, derivations can be done very efficiently using the Schwinger parametrization. For example, in order to derive the Feynman parametrized form of $\backslash frac\{1\}\{A\_1...A\_n\}$, we first reexpress all the factors in the denominator in their Schwinger parametrized form:

:$\backslash frac\{1\}\{A\_i\}=\; \backslash int^\backslash infty\_0\; ds\_i\; \backslash ,\; e^\{-s\_i\; A\_i\}\; \backslash \; \backslash \; \backslash text\{for\; \}\; i\; =1,\backslash ldots,n$

and rewrite,

:$$

\frac{1}{A_1\cdots A_n}=\int_0^\infty ds_1\cdots \int_0^\infty ds_n \exp\left(-\left(s_1A_1+\cdots+s_nA_n\right)\right).

Then we perform the following change of integration variables,

:$$

\alpha = s_1+...+s_n,

:$$

\alpha_{i} = \frac{s_{i}}{s_1+\cdots+s_n}; \ i=1,\ldots,n-1,

to obtain,

:$$

\frac{1}{A_1\cdots A_n} = \int_{0}^{1}d\alpha_1\cdots d\alpha_{n-1} \int_{0}^{\infty}d\alpha\ \alpha^{n-1}\exp\left(-\alpha\left\{ \alpha_1A_1+\cdots+\alpha_{n-1}A_{n-1}+ \left(1-\alpha_{1}-\cdots-\alpha_{n-1}\right)A_{n}\right\} \right).

where $\backslash int\_\{0\}^\{1\}d\backslash alpha\_1\backslash cdots\; d\backslash alpha\_\{n-1\}$ denotes integration over the region $0\; \backslash leq\; \backslash alpha\_i\; \backslash leq\; 1$ with $\backslash sum\_\{i=1\}^\{n-1\}\; \backslash alpha\_i\; \backslash leq\; 1$.

The next step is to perform the $\backslash alpha$ integration.

:$$

\int_{0}^{\infty}d\alpha\ \alpha^{n-1}\exp(-\alpha x)= \frac{\partial^{n-1}}{\partial(-x)^{n-1}}\left(\int_{0}^{\infty}d\alpha\exp(-\alpha x)\right)=\frac{\left(n-1\right)!}{x^{n}}.

where we have defined $x=\; \backslash alpha\_1A\_1+\backslash cdots+\backslash alpha\_\{n-1\}A\_\{n-1\}+\; \backslash left(1-\backslash alpha\_\{1\}-\backslash cdots-\backslash alpha\_\{n-1\}\backslash right)A\_\{n\}.$

Substituting this result, we get to the penultimate form,

:$$

\frac{1}{A_1\cdots A_n}=\left(n-1\right)!\int_{0}^{1}d\alpha_1\cdots d\alpha_{n-1}\frac{1}{[\alpha_1A_1+\cdots+\alpha_{n-1}A_{n-1}+ \left(1-\alpha_{1}-\cdots-\alpha_{n-1}\right)A_{n}]^n} ,

and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,

:$$

\frac{1}{A_1\cdots A_n}=\left(n-1\right)!\int_{0}^{1}d\alpha_1\cdots\int_{0}^{1}d\alpha_{n}\frac{\delta\left(1-\alpha_1-\cdots-\alpha_n\right)}{[\alpha_1A_1+\cdots+\alpha_{n}A_{n}]^n} .

Similarly, in order to derive the Feynman parametrization form of the most general case,$\backslash frac\{1\}\{A\_1^\{\backslash alpha\_1\}...A\_n^\{\backslash alpha\_n\}\}$ one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,

:$$

\frac{1}{A_1^{\alpha_1}} = \frac{1}{\left(\alpha_1-1\right)!}\int^\infty_0 ds_1 \,s_1^{\alpha_1-1} e^{-s_1 A_1} = \frac{1}{\Gamma(\alpha_1)}\frac{\partial^{\alpha_1-1}}{\partial(-A_1)^{\alpha_1-1}}\left(\int_{0}^{\infty}ds_1 e^{-s_1 A_1}\right)

and then proceed exactly along the lines of previous case.

An alternative form of the parametrization that is sometimes useful is

:$\backslash frac\{1\}\{AB\}\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{d\backslash lambda\}\{\backslash left[\backslash lambda\; A\; +\; B\backslash right]^2\}.$

This form can be derived using the change of variables $\backslash lambda\; =\; u\; /\; (\; 1\; -\; u\; )$.

We can use the product rule to show that $d\backslash lambda\; =\; du/(1-u)^\{2\}$, then

:$\backslash begin\{align\}$

\frac{1}{AB} & = \int^1_0 \frac{du}{\left[uA +(1-u)B\right]^2} \\

& = \int^1_0 \frac{du}{(1-u)^{2}} \frac{1}{\left[\frac{u}{1-u} A + B \right]^2} \\

& = \int_{0}^{\infty} \frac{d\lambda}{\left[\lambda A + B\right]^2} \\

\end{align}

More generally we have

:$\backslash frac\{1\}\{A^\{m\}B^\{n\}\}\; =\; \backslash frac\{\backslash Gamma(\; m+n)\}\{\backslash Gamma(m)\backslash Gamma(n)\}\backslash int\_\{0\}^\{\backslash infty\}\; \backslash frac\{\backslash lambda^\{m-1\}d\backslash lambda\}\{\backslash left[\backslash lambda\; A\; +\; B\backslash right]^\{n+m\}\},$

where $\backslash Gamma$ is the gamma function.

This form can be useful when combining a linear denominator $A$ with a quadratic denominator $B$, such as in heavy quark effective theory (HQET).

A symmetric form of the parametrization is occasionally used, where the integral is instead performed on the interval $[-1,1]$, leading to:

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

{{reflist}}

• Michael E. Peskin and Daniel V. Schroeder , An Introduction To Quantum Field Theory, Addison-Wesley, Reading, 1995.
• Silvan S. Schweber, Feynman and the visualization of space-time processes, Rev. Mod. Phys, 58, p.449 ,1986 doi:10.1103/RevModPhys.58.449
• Vladimir A. Smirnov: Evaluating Feynman Integrals, Springer, ISBN 978-3-54023933-8 (Dec.,2004).
• Vladimir A. Smirnov: Feynman Integral Calculus, Springer, ISBN 978-3-54030610-8 (Aug.,2006).
• Vladimir A. Smirnov: Analytic Tools for Feynman Integrals, Springer, ISBN 978-3-64234885-3 (Jan.,2013).
• Johannes Blümlein and Carsten Schneider (Eds.): Anti-Differentiation and the Calculation of Feynman Amplitudes, Springer, ISBN 978-3-030-80218-9 (2021).
• Stefan Weinzierl: Feynman Integrals: A Comprehensive Treatment for Students and Researchers, Springer, ISBN 978-3-030-99560-7 (Jun., 2023).

{{Richard Feynman}}

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