# Charm quark contribution to K+ ---> pi+ nu anti-nu at next-to-next-to-leading order

## Abstract

The authors calculate the complete next-to-next-to-leading order QCD corrections to the charm contribution of the rare decay K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}. They encounter several new features, which were absent in lower orders. They discuss them in detail and present the results for the two-loop matching conditions of the Wilson coefficients, the three-loop anomalous dimensions, and the two-loop matrix elements of the relevant operators that enter the next-to-next-to-leading order renormalization group analysis of the Z-penguin and the electroweak box contribution. The inclusion of the next-to-next-to-leading order QCD corrections leads to a significant reduction of the theoretical uncertainty from {+-} 9.8% down to {+-} 2.4% in the relevant parameter P{sub c}(X), implying the leftover scale uncertainties in {Beta}(K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}) and in the determination of |V{sub td}|, sin 2{beta}, and {gamma} from the K {yields} {pi}{nu}{bar {nu}} system to be {+-} 1.3%, {+-} 1.0%, {+-} 0.006, and {+-} 1.2{sup o}, respectively. For the charm quark {ovr MS} mass m{sub c}(m{sub c}) = (1.30 {+-} 0.05) GeV and |V{sub us}| = 0.2248 the next-to-leading order value P{sub c}(X) = 0.37 {+-} 0.06 is modified to P{sub c}(X) = 0.38 {+-} 0.04 at the next-to-next-to-leading order level with themore »

- Authors:

- Publication Date:

- Research Org.:
- Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 897292

- Report Number(s):
- FERMILAB-PUB-05-512-T

arXiv eprint number hep-ph/0603079; TRN: US0701382

- DOE Contract Number:
- AC02-07CH11359

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: JHEP 0611:002,2006; Journal Volume: 11

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANOMALOUS DIMENSION; DECAY; EVALUATION; ISOSPIN; MATRIX ELEMENTS; PHYSICS; QUANTUM CHROMODYNAMICS; QUARKS; RENORMALIZATION; Phenomenology-HEP

### Citation Formats

```
Buras, Andrzej J., /Munich, Tech. U., Gorbahn, Martin, /Durham U., IPPP /Karlsruhe U., TTP, Haisch, Ulrich, /Fermilab /Zurich U., Nierste, Ulrich, and /Karlsruhe U., TTP
/Fermilab.
```*Charm quark contribution to K+ ---> pi+ nu anti-nu at next-to-next-to-leading order*. United States: N. p., 2006.
Web. doi:10.1088/1126-6708/2006/11/002.

```
Buras, Andrzej J., /Munich, Tech. U., Gorbahn, Martin, /Durham U., IPPP /Karlsruhe U., TTP, Haisch, Ulrich, /Fermilab /Zurich U., Nierste, Ulrich, & /Karlsruhe U., TTP
/Fermilab.
```*Charm quark contribution to K+ ---> pi+ nu anti-nu at next-to-next-to-leading order*. United States. doi:10.1088/1126-6708/2006/11/002.

```
Buras, Andrzej J., /Munich, Tech. U., Gorbahn, Martin, /Durham U., IPPP /Karlsruhe U., TTP, Haisch, Ulrich, /Fermilab /Zurich U., Nierste, Ulrich, and /Karlsruhe U., TTP
/Fermilab. Wed .
"Charm quark contribution to K+ ---> pi+ nu anti-nu at next-to-next-to-leading order". United States.
doi:10.1088/1126-6708/2006/11/002. https://www.osti.gov/servlets/purl/897292.
```

```
@article{osti_897292,
```

title = {Charm quark contribution to K+ ---> pi+ nu anti-nu at next-to-next-to-leading order},

author = {Buras, Andrzej J. and /Munich, Tech. U. and Gorbahn, Martin and /Durham U., IPPP /Karlsruhe U., TTP and Haisch, Ulrich and /Fermilab /Zurich U. and Nierste, Ulrich and /Karlsruhe U., TTP
/Fermilab},

abstractNote = {The authors calculate the complete next-to-next-to-leading order QCD corrections to the charm contribution of the rare decay K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}. They encounter several new features, which were absent in lower orders. They discuss them in detail and present the results for the two-loop matching conditions of the Wilson coefficients, the three-loop anomalous dimensions, and the two-loop matrix elements of the relevant operators that enter the next-to-next-to-leading order renormalization group analysis of the Z-penguin and the electroweak box contribution. The inclusion of the next-to-next-to-leading order QCD corrections leads to a significant reduction of the theoretical uncertainty from {+-} 9.8% down to {+-} 2.4% in the relevant parameter P{sub c}(X), implying the leftover scale uncertainties in {Beta}(K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}) and in the determination of |V{sub td}|, sin 2{beta}, and {gamma} from the K {yields} {pi}{nu}{bar {nu}} system to be {+-} 1.3%, {+-} 1.0%, {+-} 0.006, and {+-} 1.2{sup o}, respectively. For the charm quark {ovr MS} mass m{sub c}(m{sub c}) = (1.30 {+-} 0.05) GeV and |V{sub us}| = 0.2248 the next-to-leading order value P{sub c}(X) = 0.37 {+-} 0.06 is modified to P{sub c}(X) = 0.38 {+-} 0.04 at the next-to-next-to-leading order level with the latter error fully dominated by the uncertainty in m{sub c}(m{sub c}). They present tables for P{sub c}(X) as a function of m{sub c}(m{sub c}) and {alpha}{sub s}(M{sub z}) and a very accurate analytic formula that summarizes these two dependences as well as the dominant theoretical uncertainties. Adding the recently calculated long-distance contributions they find {Beta}(K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}) = (8.0 {+-} 1.1) x 10{sup -11} with the present uncertainties in m{sub c}(m{sub c}) and the Cabibbo-Kobayashi-Maskawa elements being the dominant individual sources in the quoted error. They also emphasize that improved calculations of the long-distance contributions to K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}} and of the isospin breaking corrections in the evaluation of the weak current matrix elements from K{sup +} {yields} {pi}{sup 0}e{sup +}{nu} would be valuable in order to increase the potential of the two golden K {yields} {pi}{nu}{bar {nu}} decays in the search for new physics.},

doi = {10.1088/1126-6708/2006/11/002},

journal = {JHEP 0611:002,2006},

number = ,

volume = 11,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2006},

month = {Wed Mar 01 00:00:00 EST 2006}

}