# Four-loop cusp anomalous dimension from obstructions

## Abstract

We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals that depend on the kinematical invariants. We show that the anomalous dimension only receives contributions from the obstructions introduced in [F. Cachazo, M. Spradlin, and A. Volovich, J. High Energy Phys. 07 (2006) 007]. We illustrate this method by extracting the two- and three-loop anomalous dimensions analytically and the four-loop one numerically. The four-loop result was recently guessed to be f{sup (4)}=-(4{zeta}{sub 2}{sup 3}+24{zeta}{sub 2}{zeta}{sub 4}+50{zeta}{sub 6}-4(1+r){zeta}{sub 3}{sup 2}) with r=-2 using integrability and string theory arguments in [N. Beisert, B. Eden, and M. Staudacher, J. Stat. Mech. (2007) P021]. Simultaneously, f{sup (4)} was computed numerically in [Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, Phys. Rev. D 75, 085010 (2007)] from the four-loop amplitude obtaining, with best precision at the symmetric point s=t, r=-2.028(36). Our computation is manifestly s/t independent and improves the precision to r=-2.000 02(3), providing strong evidence in favor of the conjecture. The improvement is possible due to a large reduction in the number of contributing terms, as well as a reduction in the number of integrationmore »

- Authors:

- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9 (Canada)
- Brown University, Providence, Rhode Island 02912 (United States)

- Publication Date:

- OSTI Identifier:
- 20935270

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.75.105011; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANOMALOUS DIMENSION; CALCULATION METHODS; CUSPED GEOMETRIES; GLUONS; INTEGRALS; STRING MODELS; STRING THEORY; YANG-MILLS THEORY

### Citation Formats

```
Cachazo, Freddy, Spradlin, Marcus, and Volovich, Anastasia.
```*Four-loop cusp anomalous dimension from obstructions*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.105011.

```
Cachazo, Freddy, Spradlin, Marcus, & Volovich, Anastasia.
```*Four-loop cusp anomalous dimension from obstructions*. United States. doi:10.1103/PHYSREVD.75.105011.

```
Cachazo, Freddy, Spradlin, Marcus, and Volovich, Anastasia. Tue .
"Four-loop cusp anomalous dimension from obstructions". United States.
doi:10.1103/PHYSREVD.75.105011.
```

```
@article{osti_20935270,
```

title = {Four-loop cusp anomalous dimension from obstructions},

author = {Cachazo, Freddy and Spradlin, Marcus and Volovich, Anastasia},

abstractNote = {We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals that depend on the kinematical invariants. We show that the anomalous dimension only receives contributions from the obstructions introduced in [F. Cachazo, M. Spradlin, and A. Volovich, J. High Energy Phys. 07 (2006) 007]. We illustrate this method by extracting the two- and three-loop anomalous dimensions analytically and the four-loop one numerically. The four-loop result was recently guessed to be f{sup (4)}=-(4{zeta}{sub 2}{sup 3}+24{zeta}{sub 2}{zeta}{sub 4}+50{zeta}{sub 6}-4(1+r){zeta}{sub 3}{sup 2}) with r=-2 using integrability and string theory arguments in [N. Beisert, B. Eden, and M. Staudacher, J. Stat. Mech. (2007) P021]. Simultaneously, f{sup (4)} was computed numerically in [Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, Phys. Rev. D 75, 085010 (2007)] from the four-loop amplitude obtaining, with best precision at the symmetric point s=t, r=-2.028(36). Our computation is manifestly s/t independent and improves the precision to r=-2.000 02(3), providing strong evidence in favor of the conjecture. The improvement is possible due to a large reduction in the number of contributing terms, as well as a reduction in the number of integration variables in each term.},

doi = {10.1103/PHYSREVD.75.105011},

journal = {Physical Review. D, Particles Fields},

number = 10,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}