Feynman parametrization

Formulas

Two denominators

Richard Feynman observed that

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

:\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.

Multiple denominators

More generally, using the Dirac delta function \delta:

:\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,...,An as long as 0 is not contained in their convex hull.

Even more generally, provided that \text{Re} ( \alpha_{j} ) 0 for all 1 \leq j \leq n :

:\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}}} where the Gamma function \Gamma was used.

Derivation

:\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}. 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 :\frac{1}{AB} = \int_0^1 \frac{du}{\left[uA + (1-u)B\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 \frac{1}{A_1...A_n} , we first reexpress all the factors in the denominator in their Schwinger parametrized form:

:\frac{1}{A_i}= \int^\infty_0 ds_i , e^{-s_i A_i} \ \ \text{for } i =1,\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 \int_{0}^{1}d\alpha_1\cdots d\alpha_{n-1} denotes integration over the region 0 \leq \alpha_i \leq 1 with \sum_{i=1}^{n-1} \alpha_i \leq 1 .

The next step is to perform the \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= \alpha_1A_1+\cdots+\alpha_{n-1}A_{n-1}+ \left(1-\alpha_{1}-\cdots-\alpha_{n-1}\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, \frac{1}{A_1^{\alpha_1}...A_n^{\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.

Alternative form

An alternative form of the parametrization that is sometimes useful is :\frac{1}{AB} = \int_{0}^{\infty} \frac{d\lambda}{\left[\lambda A + B\right]^2}.

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

Symmetric form

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

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

Notes

References

References

1. Schwinger, Julian. (1949-09-15). "Quantum Electrodynamics. III. The Electromagnetic Properties of the Electron---Radiative Corrections to Scattering". Physical Review.
2. Kim, U-Rae. (2023). "The art of Schwinger and Feynman parametrizations". Journal of the Korean Physical Society.
3. Peskin, Michael E.. (2018-05-04). "An Introduction To Quantum Field Theory". CRC Press.
4. Feynman, R. P.. (1949-09-15). "Space-Time Approach to Quantum Electrodynamics". Physical Review.
5. Weinberg, Steven. (2008). "The Quantum Theory of Fields, Volume I". Cambridge University Press.
6. Kristjan Kannike. "Notes on Feynman Parametrization and the Dirac Delta Function".