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.
Formulas[]
Richard Feynman observed that:
![{\frac {1}{AB}}=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6124fb1e6f1989accc58fa8f8fdefeb8f767bf)
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:
![{\displaystyle {\begin{aligned}\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}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29ccc41783d63b9e3ea5f9e74b9bb3a76df0e872)
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
:[1]
![{\displaystyle {\begin{aligned}{\frac {1}{A_{1}\cdots A_{n}}}&=(n-1)!\int _{0}^{1}du_{1}\cdots \int _{0}^{1}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 _{0}^{1}du_{1}\int _{0}^{1-u_{1}}du_{2}\cdots \int _{0}^{1-u_{1}-\dots -u_{n-2}}du_{n-1}{\frac {1}{\left[A_{1}+u_{1}(A_{2}-A_{1})+\dots +u_{n-1}(A_{n}-A_{1})\right]^{n}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da54b9ac189e022bf67f78c9f6d6c25fc401b6ba)
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
for all
:
![{\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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f73bbf734bd1234c270cf0e5486f568e8543c1e)
where the Gamma function
was used.[2]
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}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4743252ca77e44d06fc812c80cf26fd6af6d357e)
Now just linearly transform the integral using the substitution,
which leads to
so ![z=uA+(1-u)B](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb2c60830cf00b2de1f286b5086889325081c73)
and we get the desired result:
![{\frac {1}{AB}}=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c889abc83102e38acad92e417ca8271f2b7dbf2)
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
, we first reexpress all the factors in the denominator in their Schwinger parametrized form:
![{\displaystyle {\frac {1}{A_{i}}}=\int _{0}^{\infty }ds_{i}\,e^{-s_{i}A_{i}}\ \ {\text{for }}i=1,\ldots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1edfa49cc66e4dffd2239ac37e2e0146637a723)
and rewrite,
![{\displaystyle {\frac {1}{A_{1}\cdots A_{n}}}=\int _{0}^{\infty }ds_{1}\cdots \int _{0}^{\infty }ds_{n}\exp \left(-\left(s_{1}A_{1}+\cdots +s_{n}A_{n}\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/783107414dd7997127c95d749be0d782b7d13155)
Then we perform the following change of integration variables,
![\alpha = s_1+...+s_n,](https://wikimedia.org/api/rest_v1/media/math/render/svg/524d8dd63ee1f9d69791e56ebdfabc757ef4e9a6)
![{\displaystyle \alpha _{i}={\frac {s_{i}}{s_{1}+\cdots +s_{n}}};\ i=1,\ldots ,n-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b77d09b4a08126830ea3dbae7566882c316227a3)
to obtain,
![{\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 _{1}A_{1}+\cdots +\alpha _{n-1}A_{n-1}+\left(1-\alpha _{1}-\cdots -\alpha _{n-1}\right)A_{n}\right\}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4e468d97de7bcb2cb2b10e1d286b4085761cf8)
where
denotes integration over the region
with
.
The next step is to perform the
integration.
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415c7c775445e9e28dd6949f4dac1a83caa307a2)
where we have defined
Substituting this result, we get to the penultimate form,
![{\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 _{1}A_{1}+\cdots +\alpha _{n-1}A_{n-1}+\left(1-\alpha _{1}-\cdots -\alpha _{n-1}\right)A_{n}]^{n}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f292fd8ef3883120a5bf7f93df4cabf998a83cc)
and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,
![{\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 _{1}A_{1}+\cdots +\alpha _{n}A_{n}]^{n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b11d20312b5ef0eb58f51c9cb4b61c2cc8b95f71)
Similarly, in order to derive the Feynman parametrization form of the most general case,
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)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa325ae2b9d6f17b2c7a24fde357527e32d480bc)
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}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b103991b478aa4c7f21ffc0d51b9c8e7711846)
This form can be derived using the change of variables
.
We can use the product rule to show that
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d4182ef7a1d46a52f9f40f0c9cea3988f7c028)
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}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b986497e8262710b61788c4b56e6b6753d55a3e)
where
is the gamma function.
This form can be useful when combining a linear denominator
with a quadratic denominator
, 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
, leading to:
![{\frac {1}{AB}}=2\int _{{-1}}^{1}{\frac {du}{\left[(1+u)A+(1-u)B\right]^{2}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/869668a62d68d9029027a8be5272a4ec432944ef)
References[]