Ulrich Haisch

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

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.

Effective Hamiltonian for non-leptonic decays at NNLO in QCD

Abstract We compute the effective Hamiltonian for non-leptonic | Δ F | = 1 decays in the standard model including next-to-next-to-leading order QCD corrections. In particular, we present the complete three-loop anomalous dimension matrix describing the mixing of current–current and QCD penguin operators. The calculation is performed in an operator basis which allows to consistently use fully anticommuting γ 5 in dimensional regularization at an arbitrary number of loops. The renormalization scheme dependences and their cancellation in physical quantities is discussed in detail. Furthermore, we demonstrate how our results are transformed to a different basis of effective operators which is frequently adopted in phenomenological applications. We give all necessary two-loop constant terms which allow to obtain the three-loop anomalous dimensions and the corresponding initial conditions of the two-loop Wilson coefficients in the latter scheme. Finally, we solve the renormalization group equation and give the analytic expressions for the low-energy Wilson coefficients relevant for non-leptonic B meson decays beyond next-to-leading order in both renormalization schemes.

Anomalous dimension matrix for radiative and rare semileptonic B decays up to three loops

Abstract We compute the complete O ( α s 2 ) anomalous dimension matrix relevant for the b → sγ , b → sg and b → s l + l − transitions in the standard model and some of its extensions. For radiative decays we confirm the results of Misiak and Munz, and of Chetyrkin, Misiak and Munz. The O ( α s 2 ) mixing of four-quark into semileptonic operators is instead a new result and represents one of the last missing ingredients of the next-to-next-to-leading-order analysis of rare semileptonic B meson decays.

Complete NNLO QCD Analysis of ¯ B ! X s ℓ + ℓ and Higher Order Electroweak Effects

We complete the next-to-next-to-leading order QCD calculation of the branching ratio for B -> X_s l^+ l^- including recent results for the three-loop anomalous dimension matrix and two-loop matrix elements. These new contributions modify the branching ratio in the low-q^2 region, BR_ll, by about +1% and -4%, respectively. We furthermore discuss the appropriate normalization of the electromagnetic coupling alpha and calculate the dominant higher order electroweak effects, showing that, due to accidental cancellations, they change BR_ll by only -1.5% if alpha(mu) is normalized at mu = O(m_b), while they shift it by about -8.5% if one uses a high scale normalization mu = O(M_W). The position of the zero of the forward-backward asymmetry, q_0^2, is changed by around +2%. After introducing a few additional improvements in order to reduce the theoretical error, we perform a comprehensive study of the uncertainty. We obtain BR_ll(1 GeV^2 <= q^2 <= 6 GeV^2) = (1.57 +- 0.16) x 10^-6 and q_0^2 = (3.76 +- 0.33) GeV^2 and note that the part of the uncertainty due to the b-quark mass can be easily reduced.

Three-loop mixing of dipole operators

We calculate the complete three-loop O(alpha(3)(s)) anomalous dimension matrix for the dimension-five dipole operators that arise in the standard model after integrating out the top quark and the heavy electroweak bosons. Our computation completes the three-loop anomalous dimension matrix of operators that govern low-energy |DeltaF| = 1 flavor-changing processes, and represents an important ingredient of the next-to-next-to-leading order QCD analysis of the B--> X(s)gamma decay.

Erratum: charm quark contribution to {K^{+}}\to {\pi^{+}}\nu \overline{\nu} at next-to-next-to-leading order

A bstractWe correct the treatment of anomalous triangle diagrams occuring in the effective theory in which the heavy top quark is integrated out. To this end we determine the initial conditions and anomalous dimensions of the operator describing the Z-mediated coupling of neutrinos to quarks and further rectify the bilocal renormalization group evolution. Our changes affect the charm-quark contribution Pc(X) at the next-to-leading and next-to-next-to-leading orders, but are numerically negligible as they amout to relative shifts below a permille.

Charm quark contribution to K+→π+νν̄ at next-to-next-to-leading order

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.