Feynman parametrization

Short description: Parametrization used for loop integrals

Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.

## Contents

* 1 Formulas
* 2 Derivation
* 3 Alternative form
* 4 Symmetric form
* 5 References

## Formulas

Richard Feynman observed that:[1]

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

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:

[math]\displaystyle{ \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} }[/math]

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 [math]\displaystyle{ \delta }[/math]:[2]

[math]\displaystyle{ \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} }[/math]

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

Even more generally, provided that [math]\displaystyle{ \text{Re} ( \alpha_{j} ) \gt 0 }[/math] for all [math]\displaystyle{ 1 \leq j \leq n }[/math]:

[math]\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-\sum_{k=1}^{n}u_{k})\;u_{1}^{\alpha_{1}-1}\cdots u_{n}^{\alpha_{n}-1}}{\left(\sum_{k=1}^{n}u_{k}A_{k}\right)^{\sum_{k=1}^{n}\alpha_{k}}} }[/math]

where the Gamma function [math]\displaystyle{ \Gamma }[/math] was used.[3]

## Derivation

[math]\displaystyle{ \frac{1}{AB} = \frac{1}{A-B}\left(\frac{1}{B}-\frac{1}{A}\right)=\frac{1}{A-B}\int_B^A \frac{dz}{z^2}. }[/math]

By using the substitution [math]\displaystyle{ u=(z-B)/(A-B) }[/math],

we have [math]\displaystyle{ du = dz/(A-B) }[/math], and [math]\displaystyle{ z = uA + (1-u)B }[/math],

from which we get the desired result

[math]\displaystyle{ \frac{1}{AB} = \int_0^1 \frac{du}{\left[uA + (1-u)B\right]^2}. }[/math]

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 [math]\displaystyle{ \frac{1}{A_1...A_n} }[/math], we first reexpress all the factors in the denominator in their Schwinger parametrized form:

[math]\displaystyle{ \frac{1}{A_i}= \int^\infty_0 ds_i \, e^{-s_i A_i} \ \ \text{for } i =1,\ldots,n }[/math]

and rewrite,

[math]\displaystyle{ \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). }[/math]

Then we perform the following change of integration variables,

[math]\displaystyle{ \alpha = s_1+...+s_n, }[/math]
[math]\displaystyle{ \alpha_{i} = \frac{s_{i}}{s_1+\cdots+s_n}; \ i=1,\ldots,n-1, }[/math]

to obtain,

[math]\displaystyle{ \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). }[/math]

where [math]\displaystyle{ \int_{0}^{1}d\alpha_1\cdots d\alpha_{n-1} }[/math] denotes integration over the region [math]\displaystyle{ 0 \leq \alpha_i \leq 1 }[/math] with [math]\displaystyle{ \sum_{i=1}^{n-1} \alpha_i \leq 1 }[/math].

The next step is to perform the [math]\displaystyle{ \alpha }[/math] integration.

[math]\displaystyle{ \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}}. }[/math]

where we have defined [math]\displaystyle{ x= \alpha_1A_1+\cdots+\alpha_{n-1}A_{n-1}+ \left(1-\alpha_{1}-\cdots-\alpha_{n-1}\right)A_{n}. }[/math]

Substituting this result, we get to the penultimate form,

[math]\displaystyle{ \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} , }[/math]

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

[math]\displaystyle{ \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} . }[/math]

Similarly, in order to derive the Feynman parametrization form of the most general case,[math]\displaystyle{ \frac{1}{A_1^{\alpha_1}...A_n^{\alpha_n}} }[/math] one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,

[math]\displaystyle{ \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) }[/math]

and then proceed exactly along the lines of previous case.

## Alternative form

An alternative form of the parametrization that is sometimes useful is

[math]\displaystyle{ \frac{1}{AB} = \int_{0}^{\infty} \frac{d\lambda}{\left[\lambda A + B\right]^2}. }[/math]

This form can be derived using the change of variables [math]\displaystyle{ \lambda = u / ( 1 - u ) }[/math]. We can use the product rule to show that [math]\displaystyle{ d\lambda = du/(1-u)^{2} }[/math], then

[math]\displaystyle{ \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} }[/math]

More generally we have

[math]\displaystyle{ \frac{1}{A^{m}B^{n}} = \frac{\Gamma( m+n)}{\Gamma(m)\Gamma(n)}\int_{0}^{\infty} \frac{\lambda^{m-1}d\lambda}{\left[\lambda A + B\right]^{n+m}}, }[/math]

where [math]\displaystyle{ \Gamma }[/math] is the gamma function.

This form can be useful when combining a linear denominator [math]\displaystyle{ A }[/math] with a quadratic denominator [math]\displaystyle{ B }[/math], such as in heavy quark effective theory (HQET).

## Symmetric form

A symmetric form of the parametrization is occasionally used, where the integral is instead performed on the interval [math]\displaystyle{ [-1,1] }[/math], leading to:

[math]\displaystyle{ \frac{1}{AB} = 2\int_{-1}^1 \frac{du}{\left[(1+u)A + (1-u)B\right]^2}. }[/math]

## References

1. ↑ Feynman, R. P. (1949-09-15). "Space-Time Approach to Quantum Electrodynamics". Physical Review 76 (6): 769–789. doi:10.1103/PhysRev.76.769. https://link.aps.org/doi/10.1103/PhysRev.76.769.
2. ↑ Weinberg, Steven (2008). The Quantum Theory of Fields, Volume I. Cambridge: Cambridge University Press. pp. 497. ISBN 978-0-521-67053-1.
3. ↑ Kristjan Kannike. "Notes on Feynman Parametrization and the Dirac Delta Function". http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf.

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Richard Feynman

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* Feynman diagram
* Feynman–Kac formula
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* "There's Plenty of Room at the Bottom" (1959)
* The Feynman Lectures on Physics (1964)
* The Character of Physical Law (1965)
* QED: The Strange Theory of Light and Matter (1985)
* Surely You're Joking, Mr. Feynman! (1985)
* What Do You Care What Other People Think? (1988)
* The Motion of Planets Around the Sun (1997)
* The Meaning of It All (1999)
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* Perfectly Reasonable Deviations from the Beaten Track (2005)

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* Namesakes
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* Feynman Prize in Nanotechnology
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