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.
Formulas[edit]
Richard Feynman observed that:[1]
![{\displaystyle {\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
:[2]
![{\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}^{u_{1}}du_{2}\cdots \int _{0}^{u_{n-2}}du_{n-1}{\frac {1}{\left[A_{1}u_{n-1}+A_{2}(u_{n-2}-u_{n-1})+\dots +A_{n}(1-u_{1})\right]^{n}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e94fd2533e0bfeda618f754c11269f59a7e3ae2e)
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.[3]
Derivation[edit]
![{\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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4743252ca77e44d06fc812c80cf26fd6af6d357e)
By using the substitution
,
we have
, and
,
from which we get the desired result
![{\displaystyle {\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,
![{\displaystyle \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,
![{\displaystyle {\frac {1}{A_{1}^{\alpha _{1}}}}={\frac {1}{\left(\alpha _{1}-1\right)!}}\int _{0}^{\infty }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[edit]
An alternative form of the parametrization that is sometimes useful is
![{\displaystyle {\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
![{\displaystyle {\begin{aligned}{\frac {1}{AB}}&=\int _{0}^{1}{\frac {du}{\left[uA+(1-u)B\right]^{2}}}\\&=\int _{0}^{1}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d4182ef7a1d46a52f9f40f0c9cea3988f7c028)
More generally we have
![{\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}}},}](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[edit]
A symmetric form of the parametrization is occasionally used, where the integral is instead performed on the interval
, leading to:
![{\displaystyle {\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[edit]