# Direct Extraction of One-loop Integral Coefficients

## Abstract

We present a general procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted by considering two-particle and triple unitarity cuts of the corresponding bubble and triangle integral functions. After choosing a specific parameterization of the cut loop momentum we can uniquely identify the coefficients of the desired integral functions simply by examining the behavior of the cut integrand as the unconstrained parameters of the cut loop momentum approach infinity. In this way we can produce compact forms for scalar integral coefficients. Applications of this method are presented for both QCD and electroweak processes, including an alternative form for the recently computed three-mass triangle coefficient in the six-photon amplitude A{sub 6}(1{sup -}, 2{sup +}, 3{sup -}, 4{sup +}, 5{sup -}, 6{sup +}). The direct nature of this extraction procedure allows for a very straightforward automation of the procedure.

- Authors:

- Publication Date:

- Research Org.:
- Stanford Linear Accelerator Center (SLAC)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 902482

- Report Number(s):
- SLAC-PUB-12455

arXiv:0704.1835; TRN: US200717%%354

- DOE Contract Number:
- AC02-76SF00515

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review D

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AMPLITUDES; AUTOMATION; BUBBLES; QUANTUM CHROMODYNAMICS; SCALARS; UNITARITY; Phenomenology-HEP,HEPPH, HEPTH

### Citation Formats

```
Forde, Darren.
```*Direct Extraction of One-loop Integral Coefficients*. United States: N. p., 2007.
Web. doi:10.1103/PhysRevD.75.125019.

```
Forde, Darren.
```*Direct Extraction of One-loop Integral Coefficients*. United States. doi:10.1103/PhysRevD.75.125019.

```
Forde, Darren. Mon .
"Direct Extraction of One-loop Integral Coefficients". United States.
doi:10.1103/PhysRevD.75.125019. https://www.osti.gov/servlets/purl/902482.
```

```
@article{osti_902482,
```

title = {Direct Extraction of One-loop Integral Coefficients},

author = {Forde, Darren},

abstractNote = {We present a general procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted by considering two-particle and triple unitarity cuts of the corresponding bubble and triangle integral functions. After choosing a specific parameterization of the cut loop momentum we can uniquely identify the coefficients of the desired integral functions simply by examining the behavior of the cut integrand as the unconstrained parameters of the cut loop momentum approach infinity. In this way we can produce compact forms for scalar integral coefficients. Applications of this method are presented for both QCD and electroweak processes, including an alternative form for the recently computed three-mass triangle coefficient in the six-photon amplitude A{sub 6}(1{sup -}, 2{sup +}, 3{sup -}, 4{sup +}, 5{sup -}, 6{sup +}). The direct nature of this extraction procedure allows for a very straightforward automation of the procedure.},

doi = {10.1103/PhysRevD.75.125019},

journal = {Physical Review D},

number = ,

volume = ,

place = {United States},

year = {Mon Apr 16 00:00:00 EDT 2007},

month = {Mon Apr 16 00:00:00 EDT 2007}

}