Feynman parametrization explained

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:^[1]

(1)/(AB) = \int ^{1} _{0} (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 (dp)/(A(p)B(p)) & = \int dp \int ^{1} _{0} (du)/(\left[uA(p) + ( 1- u)B(p)\right] ^{2}) \\ & = \int ^{1} _{0} du \int (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.

\delta :^[2] \begin{align} (1)/(A _{1 …} A _{n}) & = (n -1 )! \int ^{1} _{0} du _{1} … \int ^{1} _{0} du _{n} (\delta( 1- \sum ^{n} _{k =1} u _{k} ))/(\left(\sum ^{n} _{k =1} u _{k} A _{k} \right) ^{n}) \\ & = (n -1 )! \int ^{1} _{0} du _{1} \int ^{u _{1}} _{0} du _{2} … \int ^{u _{n-2}} _{0} du _{n-1} (1)/(\left[A _{1 u} _{n-1} + A _{2(u} _{n-2} - u _{n-1} ) +...+ A _{n(1} - u ^{n} _{1 )\right]}) . \end{align}

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

Even more generally, provided that

Re ( \alpha _{j} ) > 0 for all 1 \leq j \leq n :(1)/(A ^{\alpha _{1}} _{1} … A ^{\alpha _{n}} _{n}) = (\Gamma(\alpha _{1} +...+ \alpha _{n} ))/(\Gamma(\alpha _{1} ) … \Gamma(\alpha _{n} )) \int ^{1} _{0} du _{1} … \int ^{1} _{0} du _{n} (\delta( 1- \sum ^{n} _{k =1} u _{k} ) u ^{\alpha _{1} -1} _{1} … u ^{\alpha _{n} -1} _{n})/(\left(\sum ^{n} _{k =1} u _{k} A _{k} \right) ^{\sum ^{n} _{k =1} \alpha _{k}}) where the Gamma function \Gamma was used.^[3]

Derivation

(1)/(AB) = (1)/(\left( A - B) (1)/(- B) (1)/(\right) = A) (1)/(A - B) \int ^{A} _{B} (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

(1)/(AB) = \int ^{1} _{0} (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

(1)/(A _{1...A} _{n}) , we first reexpress all the factors in the denominator in their Schwinger parametrized form:(1)/(A _{i}) = \int ^{inf ty} _{0} ds _{i} e ^{- s _{i} A _{i}} for i =1 ,\ldots,n and rewrite,(1)/(A _{1 …} A _{n}) = \int ^{inf ty} _{0} ds _{1 …} \int ^{inf ty} _{0} ds _{n} \exp\left( - \left(s _{1A} _{1+ … +s} _{nA} _{n\right)\right).}

Then we perform the following change of integration variables,

\alpha = s _{1+...+s} _{n,} \alpha _{i} = (s _{i})/(s _{1+ … +s} _{n}) ; i =1 ,\ldots,n -1 , to obtain,(1)/(A _{1 …} A _{n}) = \int ^{1} _{0} d\alpha _{1 …} d\alpha _{n-1} \int ^{inf ty} _{0} d\alpha \alpha ^{n-1} \exp\left( - \alpha\left\{ \alpha _{1A} _{1+ … +\alpha} _{n-1} A _{n-1} + \left( 1- \alpha _{1} - … - \alpha _{n-1} \right)A _{n} \right\} \right). where $\backslash int\_^d\backslash alpha\_1\backslash cdots\; d\backslash alpha\_$ denotes integration over the region 0 \leq \alpha _{i} \leq 1 with $\backslash sum\_^\; \backslash alpha\_i\; \backslash leq\; 1$.

The next step is to perform the

\alpha integration.\int ^{inf ty} _{0} d\alpha \alpha ^{n-1} \exp( - \alpha x) = (\partial ^{n-1})/(\partial( - x) ^{n-1}) \left(\int ^{inf ty} _{0} d\alpha\exp( - \alpha x)\right) = (\left(n -1 \right)!)/(x ^{n}) . where we have defined x = \alpha _{1A} _{1+ … +\alpha} _{n-1} A _{n-1} + \left( 1- \alpha _{1} - … - \alpha _{n-1} \right)A _{n} .

Substituting this result, we get to the penultimate form,

(1)/(A _{1 …} A _{n}) = \left(n -1 \right)!\int ^{1} _{0} d\alpha _{1 …} d\alpha _{n-1} (1)/([\alpha _{1A} _{1+ … +\alpha} _{n-1} A _{n-1} + \left( 1- \alpha _{1} - … - \alpha _{n-1} \right)A _{n} ] ^{n}) , and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,(1)/(A _{1 …} A _{n}) = \left(n -1 \right)!\int ^{1} _{0} d\alpha _{1 … \int} ^{1} ^{0} d\alpha _{n} (\delta\left( 1- \alpha _{1- … -\alpha} _{n\right)})/([\alpha _{1A} _{1+ … +\alpha} _{n} A _{n} ] ^{n}) .

Similarly, in order to derive the Feynman parametrization form of the most general case,$\backslash frac$ one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,

(1)/(A ^{\alpha _{1}} _{1}) = (1)/(\left(\alpha _{1-1\right)!}) \int ^{inf ty} _{0} ds _{1} s ^{\alpha _{1-1}} _{1} e ^{- s _{1} A _{1}} = (1)/(\Gamma(\alpha _{1)}) (\partial ^{\alpha _{1-1}})/(\partial( - A ^{\alpha _{1-1}} _{1 )}) \left(\int ^{inf ty} _{0} 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

(1)/(AB) = \int ^{inf ty} _{0} (d λ)/(\left[ λ A + B\right] ^{2}) .

This form can be derived using the change of variables

λ = u / ( 1 - u ) .We can use the product rule to show that d λ = du/( 1- u) ^{2} , then\begin{align} (1)/(AB) & = \int ^{1} _{0} (du)/(\left[uA + ( 1- u)B\right] ^{2}) \\ & = \int ^{1} _{0} (du)/(( 1- u) ^{2}) (1)/(\left[ (u)/(A + B \right] ^{2} 1- u)) \\ & = \int ^{inf ty} _{0} (d λ)/(\left[ λ A + B\right] ^{2}) \\ \end{align}

More generally we have

(1)/(A ^{m} B ^{n}) = (\Gamma( m + n))/(\Gamma(m)\Gamma(n)) \int ^{inf ty} _{0} (λ ^{m-1} d λ)/(\left[ λ 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:(1)/(AB) = 2 \int ^{1} _{-1} (du)/(\left[( 1+ u)A + ( 1- u)B\right] ^{2}) .

Notes and References

1. Feynman . R. P. . 1949-09-15 . Space-Time Approach to Quantum Electrodynamics . Physical Review . 76 . 6 . 769–789 . 10.1103/PhysRev.76.769.
2. Book: Weinberg, Steven. The Quantum Theory of Fields, Volume I. 2008. Cambridge University Press. Cambridge. 497. 978-0-521-67053-1.
3. Web site: Notes on Feynman Parametrization and the Dirac Delta Function. Kristjan Kannike. https://web.archive.org/web/20070729015208/http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf. 2007-07-29. 2011-07-24.