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.
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
![{\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.
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).
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
- ^ Feynman, R. P. (1949-09-15). "Space-Time Approach to Quantum Electrodynamics". Physical Review. 76 (6): 769–789. doi:10.1103/PhysRev.76.769.
- ^ Weinberg, Steven (2008). The Quantum Theory of Fields, Volume I. Cambridge: Cambridge University Press. p. 497. ISBN 978-0-521-67053-1.
- ^ Kristjan Kannike. "Notes on Feynman Parametrization and the Dirac Delta Function" (PDF). Archived from the original (PDF) on 2007-07-29. Retrieved 2011-07-24.
Richard Feynman
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