Feynman parametrization is a technique for evaluating
loop integral
In quantum field theory and statistical mechanics, loop integrals are the integrals which appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. These integrals are used to determine counterter ...
s which arise from
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s with one or more loops. However, it is sometimes useful in integration in areas of
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
as well.
Formulas
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
observed that:
:
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:
:
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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
:
:
This formula is valid for any complex numbers ''A
1'',...,''A
n'' as long as 0 is not contained in their
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
.
Even more generally, provided that
for all
:
:
where the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
was used.
Derivation
:
By using the substitution
,
we have
, and
,
from which we get the desired result
:
In more general cases, derivations can be done very efficiently using the
Schwinger parametrization
Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.
Using the well-known observation that
:\frac=\frac\int^\infty_0 du \, u^e^,
Julian Schwinger noticed that one may ...
. 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:
:
and rewrite,
:
Then we perform the following change of integration variables,
:
:
to obtain,
:
where
denotes integration over the region
with
.
The next step is to perform the
integration.
:
where we have defined
Substituting this result, we get to the penultimate form,
:
and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,
:
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,
:
and then proceed exactly along the lines of previous case.
Alternative form
An alternative form of the parametrization that is sometimes useful is
:
This form can be derived using the change of variables
.
We can use the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
to show that
, then
:
More generally we have
:
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
This form can be useful when combining a linear denominator
with a quadratic denominator
, such as in
heavy quark effective theory
In quantum chromodynamics, heavy quark effective theory (HQET) is an effective field theory describing the physics of heavy (that is, of mass far greater than the QCD scale) quarks. It is used in studying the properties of hadrons containing a s ...
(HQET).
Symmetric form
A symmetric form of the parametrization is occasionally used, where the integral is instead performed on the interval
, leading to:
:
References
Quantum field theory
Richard Feynman
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