Quantum Chromodynamics and Hadrons: an Elementary Introduction

Notes of five lectures given at the 2003 European School of High-Energy Physics, Tsakhkadzor, Armenia, September 2003Comment: 52 pages, 25 figures, latex, cernrep.cls; lectures at the 2003 European School of High-Energy Physics, Tsakhkadzor, Armenia, September 200

B→ππB \to \pi\pi Decay in QCD

A new method is suggested to calculate the $B\to \pi\pi$ hadronic matrix elements from QCD light-cone sum rules. To leading order in $\alpha_s$ and $1/m_b$, the sum rule reproduces a factorizable matrix element, in accordance with the prediction of the QCD factorization approach. Whereas the QCD factorization can only take into account the nonfactorizable corrections induced by hard-gluon exchanges, the method suggested here also allows a systematic inclusion of soft-gluon effects. In this paper, I concentrate on the latter aspect and present a calculation of the nonfactorizable soft-gluon exchange contributions to $B\to \pi\pi$. The result, including twist 3 and 4 terms, is suppressed by one power of $1/m_b$ with respect to the factorizable amplitude. Despite its numerical smallness, the soft effect is at the same level as the $O(\alpha_s)$ correction to the QCD factorization formula for $B\to \pi\pi$. The method suggested here can be applied to matrix elements of different topologies and operators and to various other B-decay channels. I also comment on the earlier applications of QCD sum rules to nonleptonic decays of heavy mesons.Comment: 20 pages, 2 figures, Latex, epsfig.sty, a few sentences reworded, two comments added, version to appear in Nucl. Phys.

Applications of QCD Sum Rules to Heavy Quark Physics

In these lectures, I present several important applications of QCD sum rules to the decay processes involving heavy-flavour hadrons. The first lecture is introductory. As a study case, the sum rules for decay constants of the heavy-light mesons are considered. They are relevant for the leptonic decays of $B$-mesons. In the second lecture I describe the method of QCD light-cone sum rules used to calculate the heavy-to-light form factors at large hadronic recoil, such as the $B\to \pi \ell \nu_\ell$ form factors. In the third lecture, the nonlocal hadronic amplitudes in the flavour-changing neutral current decays $B\to K^{(*)}\ell\ell$ are discussed. Light-cone sum rules provide important nonfactorizable contributions to these amplitudes.Comment: Lectures at the Helmholtz International Summer School "Physics of Heavy Quarks and Hadrons'', July 2013, Dubna, Russia. SI-HEP-2013-1

Form factors in B→ππℓνˉℓB\to \pi\pi\ell\bar\nu_\ell from QCD light-cone sum rules

The form factors of the semileptonic $B\to \pi\pi\ell\bar\nu$ decay are calculated from QCD light-cone sum rules with the distribution amplitudes of dipion states. This method is valid in the kinematical region, where the hadronic dipion state has a small invariant mass and simultaneously a large recoil. The derivation of the sum rules is complicated by the presence of an additional variable related to the angle between the two pions. In particular, we realize that not all invariant amplitudes in the underlying correlation function can be used, some of them generating kinematical singularities in the dispersion relation. The two sum rules that are free from these ambiguities are obtained in the leading twist-2 approximation, predicting the $\bar{B}^0\to \pi^+\pi^0$ form factors $F_{\perp}$ and $F_{\parallel}$ of the vector and axial $b\to u$ current, respectively. We calculate these form factors at the momentum transfers $0<q^2\lesssim 12$ GeV$^2$ and at the dipion mass close to the threshold $4m_\pi^2$. The sum rule results indicate that the contributions of the higher partial waves to the form factors are suppressed with respect to the lowest $P$-wave contribution and that the latter is not completely saturated by the $\rho$-meson term.Comment: 17 pages, 4 figures, a few comments and references added, version to be published in Nucl.Phys.

Counting contact terms in B→VγB\to V\gamma decays

We clarify the origin and cancellation of contact terms in the weak annihilation amplitudes contributing to $B\to V \gamma$. It is demonstrated that the photon emission from the final-state quarks vanishes in the chiral limit of massless quarks. The contact terms in the QCD light-cone sum rule evaluation of the weak annihilation amplitudes are also discussed.Comment: 16 pages, 2 figures, Latex, ws-p8-50x6-00.cls, to be published in Sergei Matinian Festschrift ``From Integrable Models to Gauge Theories.'', Eds. V. Gurzadyan, A. Sedrakyan, World Scientific, 200

QCD Sum Rules, a Modern Perspective

An introduction to the method of QCD sum rules is given for those who want to learn how to use this method. Furthermore, we discuss various applications of sum rules, from the determination of quark masses to the calculation of hadronic form factors and structure functions. Finally, we explain the idea of the light-cone sum rules and outline the recent development of this approach.Comment: 84 pages, 14 figures, Latex,epsfig,sprocl.sty. To be published in the Boris Ioffe Festschrift ''At the Frontier of Particle Physics / Handbook of QCD'', edited by M. Shifman (World Scientific, Singapore, 2001)

Exploring Light-Cone Sum Rules for Pion and Kaon Form Factors

We analyze the higher-twist effects and the SU(3)-flavour symmetry breaking in the correlation functions used to calculate form factors of pseudoscalar mesons in the QCD light-cone sum rule approach. It is shown that the Ward identities for these correlation functions yield relations between twist-4 two- and three-particle distribution amplitudes. In addition to the relations already obtained from the QCD equations of motions, we have found a new one. With the help of these relations, the twist-4 contribution to the light-cone sum rule for the pion electromagnetic form factor is reduced to a very simple form. Simultaneously, we correct a sign error in the earlier calculation. The updated light-cone sum rule prediction for the pion form factor at intermediate momentum transfers is compared with the recent Jefferson Lab data. Furthermore, from the correlation functions with strange-quark currents the kaon electromagnetic form factor and the $K\to \pi$ weak transition form factors are predicted with $O(m_s)\sim O(m_K^2)$ accuracy.Comment: 26 pages, Latex, 6 figure

Bs→KℓνℓB_{s}\to K \ell \nu_\ell and B(s)→π(K)ℓ+ℓ−B_{(s)} \to \pi (K) \ell^+\ell^- decays at large recoil and CKM matrix elements

We provide hadronic input for the $B$-meson semileptonic transitions to a light pseudoscalar meson at large recoil. The $B_s\to K$ form factor calculated from QCD light-cone sum rule is updated, to be used for a $|V_{ub}|$ determination from the $B_s\to K \ell \nu$ width. Furthermore, we calculate the hadronic input for the binned observables of $B \to \pi \ell^+ \ell^-$ and $B \to K \ell^+ \ell^-$. In addition to the form factors, the nonlocal hadronic matrix elements are obtained, combining QCD factorization and light-cone sum rules with hadronic dispersion relations. We emphasize that, due to nonlocal effects, the ratio of branching fractions of these decays is not sufficient for an accurate extraction of the $|V_{td}/V_{ts}|$ ratio. Instead, we suggest to determine the Wolfenstein parameters $A,\rho,\eta$ of the CKM matrix, combining the branching fractions of $B \to K \ell^+ \ell^-$ and $B \to \pi \ell^+ \ell^-$ with the direct $CP$-asymmetry in the latter decay. We also obtain the hadronic matrix elements for a yet unexplored channel $B_s \to K \ell^+ \ell^-$.Comment: 21 pages, 4 figures, 5 tables, a few comments and references addded, version accepted for publication in JHE