Feynman parametrization

Short description: 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. It was introduced by Julian Schwinger[1] and Richard Feynman[2] in 1949 to perform calculations in quantum electrodynamics.[3][4][lower-alpha 1]

Hung Cheng and T.T. Wu proved in 1987[5] that the sum in the Dirac delta function can be reduced to a subset of Feynman parameters. This result is known as Cheng–Wu theorem.[6]

## Contents

* 1 Formulas
* 1.1 Two denominators
* 1.2 Multiple denominators
* 2 Derivation
* 3 Alternative form
* 4 Symmetric form
* 5 Notes
* 6 References
* 7 Further reading

## Formulas

### Two denominators

Richard Feynman observed that[2]

1AB=∫01du[uA+(1−u)B]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:

∫dpA(p)B(p)=∫dp∫01du[uA(p)+(1−u)B(p)]2=∫01du∫dp[uA(p)+(1−u)B(p)]2.

If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution.

### Multiple denominators

More generally, using the Dirac delta function δ:[7]

1A1⋯An=(n−1)!∫01du1⋯∫01dunδ(1−∑k=1nuk)(∑k=1nukAk)n=(n−1)!∫01du1∫0u1du2⋯∫0un−2dun−11[A1un−1+A2(un−2−un−1)+…+An(1−u1)]n.

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 Re(αj)>0 for all 1≤j≤n:

1A1α1⋯Anαn=Γ(α1+…+αn)Γ(α1)⋯Γ(αn)∫01du1⋯∫01dunδ(1−∑k=1nuk)u1α1−1⋯unαn−1(∑k=1nukAk)∑k=1nαk

where the Gamma function Γ was used.[8]

## Derivation

1AB=1A−B(1B−1A)=1A−B∫BAdzz2.

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

1AB=∫01du[uA+(1−u)B]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 1A1...An, we first reexpress all the factors in the denominator in their Schwinger parametrized form:

1Ai=∫0∞dsie−siAi for i=1,…,n

and rewrite,

1A1⋯An=∫0∞ds1⋯∫0∞dsnexp⁡(−(s1A1+⋯+snAn)).

Then we perform the following change of integration variables,

α=s1+...+sn,
αi=sis1+⋯+sn; i=1,…,n−1,

to obtain,

1A1⋯An=∫01dα1⋯dαn−1∫0∞dα αn−1exp⁡(−α{α1A1+⋯+αn−1An−1+(1−α1−⋯−αn−1)An}).

where ∫01dα1⋯dαn−1 denotes integration over the region 0≤αi≤1 with ∑i=1n−1αi≤1.

The next step is to perform the α integration.

∫0∞dα αn−1exp⁡(−αx)=∂n−1∂(−x)n−1(∫0∞dαexp⁡(−αx))=(n−1)!xn.

where we have defined x=α1A1+⋯+αn−1An−1+(1−α1−⋯−αn−1)An.

Substituting this result, we get to the penultimate form,

1A1⋯An=(n−1)!∫01dα1⋯dαn−11[α1A1+⋯+αn−1An−1+(1−α1−⋯−αn−1)An]n,

and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,

1A1⋯An=(n−1)!∫01dα1⋯∫01dαnδ(1−α1−⋯−αn)[α1A1+⋯+αnAn]n.

Similarly, in order to derive the Feynman parametrization form of the most general case,1A1α1...Anαn one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,

1A1α1=1(α1−1)!∫0∞ds1s1α1−1e−s1A1=1Γ(α1)∂α1−1∂(−A1)α1−1(∫0∞ds1e−s1A1)

and then proceed exactly along the lines of previous case.

## Alternative form

An alternative form of the parametrization that is sometimes useful is

1AB=∫0∞dλ[λA+B]2.

This form can be derived using the change of variables λ=u/(1−u). We can use the product rule to show that dλ=du/(1−u)2, then

1AB=∫01du[uA+(1−u)B]2=∫01du(1−u)21[u1−uA+B]2=∫0∞dλ[λA+B]2

More generally we have

1AmBn=Γ(m+n)Γ(m)Γ(n)∫0∞λm−1dλ[λA+B]n+m,

where Γ 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:

1AB=2∫−11du[(1+u)A+(1−u)B]2.

## Notes

1. ↑ Schwinger is credited as having published first. Feynman credits the method "suggested by some work of Schwinger’s involving Gaussian integrals".

## References

1. ↑ Schwinger, Julian (1949-09-15). "Quantum Electrodynamics. III. The Electromagnetic Properties of the Electron
Radiative Corrections to Scattering". Physical Review 76 (6): 790–817. doi:10.1103/PhysRev.76.790. https://link.aps.org/doi/10.1103/PhysRev.76.790.
2. ↑ 2.0 2.1 Feynman, R. P. (1949-09-15). "Space-Time Approach to Quantum Electrodynamics". Physical Review 76 (6): 769–789. doi:10.1103/PhysRev.76.769. Bibcode: 1949PhRv...76..769F. https://link.aps.org/doi/10.1103/PhysRev.76.769.
3. ↑ Kim, U-Rae; Cho, Sungwoong; Lee, Jungil (2023). "The art of Schwinger and Feynman parametrizations" (in en). Journal of the Korean Physical Society 82 (11): 1023–1039. doi:10.1007/s40042-023-00764-3. ISSN 0374-4884. https://link.springer.com/10.1007/s40042-023-00764-3.
4. ↑ Peskin, Michael E. (2018-05-04) (in en). An Introduction To Quantum Field Theory. CRC Press. ISBN 978-0-429-97210-2. https://www.google.fr/books/edition/An_Introduction_To_Quantum_Field_Theory/HUpaDwAAQBAJ?hl=en&gbpv=1&dq=peskin+schroeder&printsec=frontcover.
5. ↑ Cheng, H.; Wu, T.T. (1987). Expanding Protons: Scattering at High Energies. Cambridge, Massachusetts: MIT Press.
6. ↑ Abreu, Samuel; Britto, Ruth; Duhr, Claude (2022-11-04). "The SAGEX review on scattering amplitudes Chapter 3: Mathematical structures in Feynman integrals". Journal of Physics A: Mathematical and Theoretical 55 (44): 443004. doi:10.1088/1751-8121/ac87de. ISSN 1751-8113. https://iopscience.iop.org/article/10.1088/1751-8121/ac87de.
7. ↑ Weinberg, Steven (2008). The Quantum Theory of Fields, Volume I. Cambridge: Cambridge University Press. pp. 497. ISBN 978-0-521-67053-1.
8. ↑ Kristjan Kannike. "Notes on Feynman Parametrization and the Dirac Delta Function". http://www.physic.ut.ee/~kkannike/english/science/physics/notes/feynman_param.pdf.

## Further reading

* Michael E. Peskin and Daniel V. Schroeder, An Introduction To Quantum Field Theory, Addison-Wesley, Reading, 1995.
* Silvan S. Schweber, Feynman and the visualization of space-time processes, Rev. Mod. Phys, 58, p. 449 ,1986 doi:10.1103/RevModPhys.58.449
* Vladimir A. Smirnov: Evaluating Feynman Integrals, Springer, ISBN 978-3-54023933-8 (Dec., 2004).
* Vladimir A. Smirnov: Feynman Integral Calculus, Springer, ISBN 978-3-54030610-8 (Aug., 2006).
* Vladimir A. Smirnov: Analytic Tools for Feynman Integrals, Springer, ISBN 978-3-64234885-3 (Jan.,2013).
* Johannes Blümlein and Carsten Schneider (Eds.): Anti-Differentiation and the Calculation of Feynman Amplitudes, Springer, ISBN 978-3-030-80218-9 (2021).
* Stefan Weinzierl: Feynman Integrals: A Comprehensive Treatment for Students and Researchers, Springer, ISBN 978-3-030-99560-7 (Jun., 2023).

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* "There's Plenty of Room at the Bottom" (1959)
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* Surely You're Joking, Mr. Feynman! (1985)
* What Do You Care What Other People Think? (1988)
* The Motion of Planets Around the Sun (1997)
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