Functional approach to QCD finite-energy sum rules

A new method for the determination of the QCD condensates from low-energy hadronic data is proposed. It generalizes the usual QCD finite-energy sum rules, taking into account explicitely the truncation error of the high-energy QCD expansion. The method is applied to the e⁺e^– annihilation intoI= 1 hadrons, indicating a rather large domain for the values of the gluon and four quark condensates.     

Condensates in quantum chromodynamics

The paper presents a short review of our knowledge today on vacuum condensates in quantum chromodynamics (QCD). The condensates are defined as vacuum averages of the operators which arise due to nonperturbative effects. The important role of condensates in determining physical properties of hadrons and of their low-energy interactions in QCD is underlined. The special value of the quark condensate, connected to the existence of baryon masses, is mentioned. Vacuum condensates induced by external fields are discussed. QCD at low energy is checked on the basis of the data on hadronic τ decay. In theoretical analysis, the terms of perturbation theory (PT) up to α _s ³ are accounted for; in the operator product expansion (OPE), those up to dimension 8. The total probability of the decay τ → hadrons (with zero strangeness) and of the τ-decay structure functions are best described at α _s (m _τ ² )=0.330±0.025. It is shown that the Borel sum rules for τ-decay structure functions along the rays in the q ²-complex plane are in agreement with experiment, having an accuracy of ～2% at the values of the Borel parameter |M ²|>0.8 GeV². The magnitudes of dimension 6 and 8 condensates were found, and the limitations on gluon condensates were obtained. The sum rules for the charmed-quark vector-current polarization operator were analyzed in three loops (i.e., in order α _s ² ). The value of the charmed-quark mass (in an \(\overline {MS} \) regularization scheme) was found to be \(\bar m_c (\bar m_c^2 ) = 1.275 \pm 0.015\) GeV, and the value of gluon condensate was estimated as 〈0|(α _s/π)G ²|0〉=0.009±0.007 GeV⁴. The general conclusion is that the QCD described by PT + OPE is in good agreement with experiment at Q ²?1 GeV².     

Theoretical Description and Measurement of the Pion–Photon Transition Form Factor

Detailed predictions for the scaled pion–photon transition form factor are given, derived with the method of light-cone sum rules and using pion distribution amplitudes with two and three Gegenbauer coefficients obtained from QCD sum rules with nonlocal condensates. These predictions agree well with all experimental data that are compatible with QCD scaling (and collinear factorization), but disagree with the high-Q ² data of the BaBar Collaboration that grow with the momentum. A good agreement of our predictions with results obtained from AdS/QCD models and Dyson–Schwinger computations is found.     

Isospin breaking in nuclear physics: The Nolen-Schiffer effect

Using the QCD sum rules we calculate the neutron-proton mass difference at zero density as a function of the difference in bare quark massm _d—m _u. We confirm results of Hatsuda, Høgaasen and Prakash that the largest term results from the difference in up and down quark condensates, the explicitC(m _d—m _u) entering with the opposite sign. The quark condensates are then extended to finite density to estimate the Nolen-Schiffer effect. The neutron-proton mass difference is extremely density dependent, going to zero at roughly nuclear matter density.The Ioffe formula for the nucleon mass is interpreted as a derivation, within the QCD sum rule approach, of the Nambu-Jona-Lasinio formula. This clarifies theN _c counting and furthermore provides an alternative interpretation of the Borel mass.     

Sum rule estimate of the subleading non-perturbative contributions to Bs–$\bar{ B}_{s}$ mixing

We use QCD sum rules to compute the matrix elements of the ΔB=2 operators appearing in the heavy-quark expansion of the width difference of the B _s mass eigenstates. The main focus of our analysis is on the subleading operators R ₂ and R ₃, which appear at next-to-leading order in the 1/m _b expansion. The matrix elements of these operators are already essential for precise phenomenology, but their calculation in lattice QCD is lacking and the values given here provide a first estimate of their values. We conclude that the violation of the factorization approximation for these matrix elements due to non-perturbative vacuum condensates is as low as 1–2%.     

<Emphasis Type="Italic">V</Emphasis>-<Emphasis Type="Italic">A</Emphasis> sum rules with <Emphasis Type="Italic">D</Emphasis> = 10 operators

The difference of vector and axial-vector charged current correlators is analyzed by means of QCD sum rules. The contribution of 10-dimensional 4-quark condensates is calculated and its value is estimated within the framework of the factorization hypothesis. It is compared to the result obtained from an operator fit of Borel sum rules in the complex q ²-plane, calculated from experimental data on hadronic -decays. This fit gives accurate values of the light quark condensate and the quark-gluon mixed condensate. The size of the high-order operators and the convergence of the operator series are discussed.Received: 10 May 2004, Revised: 7 September 2004, Published online: 18 November 2004     

A class of modified sum rules in QCD

We consider a class of modifications to the exponential QCD sum rules, replacing the energy variableq ² by (q ²) ^k withk positive integer. As a result of this modification, (i) the relative role of the condensates on the theoretical side is changed, (ii) the integral on the phenomenological side acquires a rapidly decreasing weight function and (iii) the way of approaching infinity in theq ² complex plane splits intok rays. We explicitly discuss the modified sum rules withk=1,k=2 andk=3 for the ρ meson and conclude that, unlike to thek=1 case, thek=2 andk=3 sum rules do not yield a satisfactory scheme for the prediction of the resonance parameters unless many higher condensates are known. We show that, for practical applications with the present knowledge of condensates, the standard exponential QCD sum rules are the most suitable ones out of the class considered.     

Analysis of the vector form factors f<Superscript>+</Superscript><Subscript>Kπ</Subscript>(Q<Superscript>2</Superscript>) and f<Superscript>-</Superscript><Subscript>Kπ</Subscript>(Q<Superscript>2</Superscript>) with light-cone QCD sum rules

In this article, we calculate the vector form factors f⁺ _Kπ(Q²) and f^- _Kπ(Q²) within the framework of the light-cone QCD sum rule approach. The numerical values of f⁺ _Kπ(Q²) are compatible with existing theoretical calculations, and the central value of f⁺ _Kπ(0) (f⁺ _Kπ(0)=0.97) is in excellent agreement with the values from chiral perturbation theory and lattice QCD. The values of |f^- _Kπ(0)| are very large compared to the theoretical calculations and experimental data, and they cannot give any reliable prediction. At large momentum transfer with Q² > 5 GeV², the form factors f⁺ _Kπ(Q²) and |f^- _Kπ(Q²)| can either show the asymptotic behavior of or decrease more quickly than ; more experimental data are needed to select the ideal sum rules. PACS 12.38.Lg; 12.38.Bx; 12.15.Hh     

Pion form factor in the NLC QCD SR approach

We present results of a calculation of the electromagnetic pion form factor within the framework of QCD sum rules with nonlocal condensates and using a perturbative spectral density which includes O(α _s ) contributions.     

The strange quark mass from QCD sum rules

The strange quark mass is calculated from QCD sum rules for the divergence of the vector as well as axialvector current in the next-next-to-leading logarithmic approximation. The determination for the divergence of the axial-vector current is found to be unreliable due to large uncertainties in the hadronic parametrisation of the two-point function.From the sum rule for the divergence of the vector current, we obtain a value of (1 GeV)=189±32 MeV, where the error is dominated by the unknown perturbativeO( _s ³ ) correction. Assuming a continued geometric growth of the perturbation series, we findm _s =178±18 MeV. Using both determinations ofm _s , together with quark-mass ratios from chiral perturbation theory, we also give estimates of the light quark massesm _u andm _d.     

Analysis of the decay B0→χclπ0 with light-cone QCD sum rules

In this article,we calculate the contribution from the nonfactorizable soft hadronic matrix element to the decay B~0→χc1~π~0 with the light-cone quantum chromo-dynamic(QCD) sum rules.The numerical results show that its contribution is rather large and should not be neglected.The total amplitudes lead to a branching fraction which is in agreement with the experimental data marginally.     

Determination of quark and gluon vacuum condensates from τ-lepton decay data

The values of the gluon and four-quark vacuum condensates are estimated using recent experimental data on the semileptonic τ-lepton decays τ→v_τ+nπ, which determine the vector and axialvector hadronic spectral functions. An optimal estimate is achieved through a systematic combined use Finite Energy, Laplace and Gaussian transform QCD sum rules. As a byproduct, the values of the dimensiond=8 vacuum condensates in the vector and axial-vector channels are also estimated.     

A comprehensive revisit of the ρ meson with improved Monte-Carlo based QCD sum rules

We improve the Monte-Carlo based QCD sum rules by introducing the rigorous H¨older-inequalitydetermined sum rule window and a Breit-Wigner type parametrization for the phenomenological spectral function.In this improved sum rule analysis methodology, the sum rule analysis window can be determined without any assumptions on OPE convergence or the QCD continuum. Therefore, an unbiased prediction can be obtained for the phenomenological parameters(the hadronic mass and width etc.). We test the new approach in the ρ meson channel with re-examination and inclusion of α_s corrections to dimension-4 condensates in the OPE. We obtain results highly consistent with experimental values. We also discuss the possible extension of this method to some other channels.     

The spectrum of hot hadronic matter and finite-temperature QCD sum rules

The sum rules method, which is widely used for the investigation of the resonance physics within the QCD framework, is generalized to the case of finite temperatures and densities. Conditions are formulated under which this method is quite efficient for the determination of the spectrum of hadronic matter at T ≠ 0. The finite-temperature QCD sum rules are analysed in the vector channel J^PC = 1⁻⁻. Sharp and qualitative changes in the spectrum are found in the temperature interval T = 130–150 MeV. It is naturally explained as a consequence of the disappearance of confinement at the temperature T_c = 140±10 MeV. The finite-temperature QCD sum rules also show that the restoration of chiral symmetry at some temperature T_F cannot precede deconfinement. In the case T_F⪢T_c the sum rules indicate that the intermediate phase at T_c<T<T_F is dominated by quasi-free quarks with nonzero dynamical mass m_q^T ≈ 300 MeV.     

Gluon condensates,quark matter equation of state and quark stars

We consider here quark matter equation of state including strange quarks and taking into account a nontrivial vacuum structure for QCD with gluon condensates. The parameters of condendsate function are determined from minimisation of the thermodynamic potential. The scale parameter of the gluon condensates is fixed from the SVZ parameter in the context of QCD sum rules at zero temperature and zero baryon density. The equation of state for strange matter at zero temperature as derived is used to study quark star structure using Tolman Oppenheimer Volkoff equations. Stable solutions for quark stars are obtained with a large Chandrasekhar limit as 3.2M and radii around 17 kms.     

Contribution of the DK continuum in the QCD sum rule for D<Subscript>sJ</Subscript>(2317)

Using the soft-pion theorem and the assumption on the final-state interactions, we include the contribution of the DK continuum into the QCD sum rules for the D_sJ(2317) meson. We find that this contribution can significantly lower the mass and the decay constant of the D_s(0⁺) state. For the value of the current quark mass m_c(m_c) = 1.286 GeV, we obtain the mass of D_s(0⁺) M=2.33 ± 0.02 GeV in the interval s₀ = 7.5–8.0 GeV², being in agreement with the experimental data, and the vector current decay constant of D_s(0⁺) f₀=0.128 ± 0.013 GeV, much lower than those obtained in the previous literature. PACS  12.39.Hg; 13.25.Hw; 13.25.Ft; 12.38.Lg     

Contribution of higher gluon condensates to light-quark vacuum polarization

A method of classifying quark operators in QCD sum rules is suggested. The expansion coefficients of all thed≦8 bilinear quark condensates in gluon condensates are calculated. The coefficient functions at the gluon operators withd≦8 in the polarization operator ∏(q ²) of the light-quark vector current are obtained. A comparison is performed with the calculations in the covariantly constant fields and self-dual fields. The results obtained can be used in the sum rules for the ρ, ω and ? families.     

Analysis of the Nonet Scalar Mesons as Tetraquark States with New QCD Sum Rules

In this article, we take the scalar diquarks as point particles and describe them as basic quantum fields, then introduce the SU(3) color gauge interaction and new vacuum condensates to study the nonet scalar mesons as tetraquark states with the QCD sum rules. Comparing with the conventional quark currents, the diquark currents have the outstanding advantage to satisfy the two criteria of the QCD sum rules more easily.