Charm quark contribution to K+ > pi+ nu antinu at nexttonexttoleading order
Abstract
The authors calculate the complete nexttonexttoleading 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 twoloop matching conditions of the Wilson coefficients, the threeloop anomalous dimensions, and the twoloop matrix elements of the relevant operators that enter the nexttonexttoleading order renormalization group analysis of the Zpenguin and the electroweak box contribution. The inclusion of the nexttonexttoleading 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 nexttoleading order value P{sub c}(X) = 0.37 {+} 0.06 is modified to P{sub c}(X) = 0.38 {+} 0.04 at the nexttonexttoleading 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):
 FERMILABPUB05512T
arXiv eprint number hepph/0603079; TRN: US0701382
 DOE Contract Number:
 AC0207CH11359
 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; PhenomenologyHEP
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 antinu at nexttonexttoleading order. United States: N. p., 2006.
Web. doi:10.1088/11266708/2006/11/002.
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 antinu at nexttonexttoleading order. United States. doi:10.1088/11266708/2006/11/002.
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 antinu at nexttonexttoleading order". United States.
doi:10.1088/11266708/2006/11/002. https://www.osti.gov/servlets/purl/897292.
@article{osti_897292,
title = {Charm quark contribution to K+ > pi+ nu antinu at nexttonexttoleading 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 nexttonexttoleading 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 twoloop matching conditions of the Wilson coefficients, the threeloop anomalous dimensions, and the twoloop matrix elements of the relevant operators that enter the nexttonexttoleading order renormalization group analysis of the Zpenguin and the electroweak box contribution. The inclusion of the nexttonexttoleading 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 nexttoleading order value P{sub c}(X) = 0.37 {+} 0.06 is modified to P{sub c}(X) = 0.38 {+} 0.04 at the nexttonexttoleading 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 longdistance 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 CabibboKobayashiMaskawa elements being the dominant individual sources in the quoted error. They also emphasize that improved calculations of the longdistance 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 = {10.1088/11266708/2006/11/002},
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}
}

The authors calculate the complete nexttonexttoleading order QCD correction of the charm quark contribution to the branching ratio for the rare decay K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}} in the standard model. The inclusion of these {Omicron}({alpha}{sub s}) contributions leads to a significant reduction of the theoretical uncertainty from {+} 10.1% down to {+} 2.4% in the relevant parameter P{sub c}, implying the left over 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%, {+} 0.006more »

CharmQuark Contribution to K{sub L}{yields}{mu}{sup +}{mu}{sup } at NexttoNexttoLeading Order
We calculate the charmquark contribution to the decay K{sub L}{yields}{mu}{sup +}{mu}{sup } in nexttonexttoleading order of QCD. This new contribution reduces the theoretical uncertainty in the relevant parameter P{sub c} from {+}22% down to {+}7%, corresponding to scale uncertainties of {+}3% and {+}6% in the shortdistance part of the branching ratio and the determination of the Wolfenstein parameter {rho} from K{sub L}{yields}{mu}{sup +}{mu}{sup }. The error in P{sub c}=0.115{+}0.018 is now in equal shares due to the combined scale uncertainties and the current uncertainty in the charmquark mass. We find B(K{sub L}{yields}{mu}{sup +}{mu}{sup }){sub SD}=(0.79{+}0.12)x10{sup 9}, with the present uncertaintymore » 
{epsilon}{sub K} at nexttonexttoleading order: The charmtopquark contribution
We perform a nexttonexttoleading order QCD analysis of the charmtopquark contribution {eta}{sub ct} to the effective {Delta}S=2 Hamiltonian in the standard model. {eta}{sub ct} represents an important part of the short distance contribution to the parameter {epsilon}{sub K}. We calculate the threeloop anomalous dimension of the leading operator Qtilde{sub S2}, the threeloop mixing of the currentcurrent and penguin operators into Qtilde{sub S2}, and the corresponding twoloop matching conditions at the electroweak, the bottomquark, and the charmquark scale. As our final numerical result we obtain {eta}{sub ct}=0.496{+}0.047, which is roughly 7% larger than the nexttoleadingorder (NLO) value {eta}{sub ct}{sup NLO}=0.457{+}0.073. Thismore » 
Rare Decay K{sup +}{yields}{pi}{sup +}{nu}{nu} at the NexttoNexttoLeading Order in QCD
We calculate the charm quark contribution to the rare decay K{sup +}{yields}{pi}{sup +}{nu}{nu} in the nexttonexttoleading order of QCD. This new contribution reduces the theoretical uncertainty in the relevant parameter P{sub c} from {+}10.1% down to {+}2.4%, corresponding to scale uncertainties of {+}1.3%, {+}1.0%, {+}0.006, and {+}1.2 deg. in B(K{sup +}{yields}{pi}{sup +}{nu}{nu}) and in V{sub td}, sin2{beta}, and {gamma} extracted from the K{yields}{pi}{nu}{nu} system. The error in P{sub c}=0.37{+}0.04 is now fully dominated by the current uncertainty of {+}3.8% in the charm quark mass m{sub c}. We find B(K{sup +}{yields}{pi}{sup +}{nu}{nu})=(8.0{+}1.1)x10{sup 11}, where the quoted error stems almost entirely frommore » 
Electroweak corrections to the charm quark contribution to K{sup +}{yields}{pi}{sup +}{nu}{nu}
We compute the leadinglog QED, the nexttoleadinglog QEDQCD, and the electroweak corrections to the charm quark contribution relevant for the rare decay K{sup +}{yields}{pi}{sup +}{nu}{nu}. The corresponding parameter P{sub c}(X) is increased by up to 2% with respect to the pure QCD estimate to P{sub c}(X)=0.372{+}0.015 for m{sub c}(m{sub c})=(1.286{+}0.013)GeV, {alpha}{sub s}(M{sub Z})=0.1176{+}0.0020, and V{sub us}=0.2255. For the branching ratio we find B(K{sup +}{yields}{pi}{sup +}{nu}{nu})=(8.5{+}0.7)x10{sup 11}, where the quoted uncertainty is dominated by the CabibboKobayashiMaskawa elements.