The ratios among condensates from e+e− (I=1) data

We calculate the ratios among condensates from the ρ channel (I=1) e⁺e⁻ data in a systematic way. We use quotients of SVZ sum rules for different moments of the correlation functions. The results turn out to be very accurate. A factor of 1.6 for the ratio of the four quark condensate over the gluon condensate compared with the standard value is predicted.     

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.     

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.     

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.     

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.     

Gluon condensates at finite baryon densities and temperature

We derive here the equation of state for quark matter with a nontrivial vacuum structure in QCD at finite temperature and baryon density. Using thermofield dynamics, the parameters of thermal vacuum and the gluon condensate function are determined through minimisation of the thermodynamic potential, along with a self-consistent determination of the effective gluon and quark masses. The scale parameter for the gluon condensates is related to the SVZ parameter in the context of QCD sum rules at zero temperature. With inclusion of quarks in the thermal vacuum the critical temperature at which the gluon condensate vanishes decreases as compared to that containing only gluons. At zero temperature, we similarly obtain the critical baryon density for the same to be about 0.36 fm^?3.     

Plane-wave,coordinate-space,and moment techniques in the operator-product expansion: Equivalence,improved methods,and the heavy quark expansion

We compare plane-wave, coordinate-space and moment methods for evaluating operator-product expansion (OPE) coefficients of the light-quark and gluon condensates. Equivalence of these methods for quark condensate contributions is proven to all orders in the quark mass parameterm. The three methods are also shown to yield equivalent gluon condensate contributions to two-current correlation functions, regardless of the gauge chosen for external gluon fields in the coordinate space approach. An improved method for evaluating quarkcondensate OPE coefficients is presented for several (two-current) correlation functions. Gauge-dependent Green functions are also discussed. It is shown that contradictory expressions for the gluon-condensate contribution to the quark propagator occurring from the plane-wave and coordinate-space approaches yield identical relations between the heavy-quark and gluon condensates, as anticipated from the gauge invariance of the heavy-quark expansion.     

Renormalized Wick expansion for a modified PQCD

The renormalization scheme for the Wick expansion of a modified version of the perturbative QCD introduced in previous works is discussed. Massless QCD is considered by implementing the usual multiplicative scaling of the gluon and quark wave functions and vertices. However, also massive quark and gluon counterterms are allowed in this massless theory since the condensates are expected to generate masses. A natural set of expansion parameters of the physical quantities is introduced: the coupling itself and the two masses m_q and m_g associated to quarks and gluons, respectively. This procedure allows one to implement a dimensional transmutation effect through these new mass scales. A general expression for the new generating functional in terms of the mass parameters m_q and m_g is obtained in terms of integrals over arbitrary but constant gluon or quark fields in each case. Further, the one loop potential is evaluated in more detail in the case when only the quark condensate is retained. This lowest order result again indicates the dynamical generation of quark condensates in the vacuum.     

Non-perturbative QCD vacuum frome +e−→I=1 hadron data

We estimate the gluon vacuum condensate α _s 〈F ²〉 from thee ⁺e^?→I=1 hadron cross-section known below 2 GeV using moment sum rules ratios. We obtain α _s 〈F ²〉= (3.9±1.0)10^?2GeV⁴. We also re-evaluate the contribution of the dimension-six vacuum condensates to the above sum rule and test the factorization hypothesis of the four-quark operator. Useful rules for the evaluation of the dimension-six vacuum condensates contributions are given.     

Two-loop two-point functions with masses: asymptotic expansions and Taylor series,in any dimension

In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and smallq ², ind dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Padé approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon self-energy, for alld, and achieve highly accelerated convergence of its expansions in powers ofq ²/m ² orm ²/q ², ford=4.     

Gluon condensate in charmonium sum rules with three-loop corrections

Charmonium sum rules are analyzed with the primary goal to obtain the restrictions on the value of the dimension 4 gluon condensate. The moments M _n (Q ² ) of the polarization operator of the vector charm currents are calculated and compared with the experimental data. The three-loop () perturbative corrections, the contribution of the gluon condensate with corrections and the contribution of the dimension 6 operator G³ are accounted. It is shown that the sum rules for the moments do not work at Q ² = 0, where the perturbation series diverges and the G³ contribution is large. The domain in the (n, Q ² ) plane where the sum rules are legitimate is found. A strong correlation of the values of gluon condensate and charm quark mass is determined. The absolute limits are found to be for the gluon condensate and for the charm quark mass in the scheme. Received: 16 July 2002 / Revised version: 6 November 2002 / Published online: 24 January 2003 RID="a" ID="a" e-mail: ioffe@vitep1.itep.ru RID="b" ID="b" e-mail: zyablyuk@heron.itep.ru     

<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     

Effects of explicit SUf(3) breaking on the quark condensates

Effects of the explicit breaking of flavour symmetry on the quark condensates in the large-N_c limit are examined with the use of a QCD-motivated effective lagrangian. It is shown that, as the current mass increases, the non-perturbative quark condensate decreases in the absolute values, which agrees well with that obtained from QCD sum rules, not only qualitatively but also quantitatively. The condensatesat finite temperature are also investigated in relation to the chiral transition.     

Hadron resonance gas and nonperturbative QCD vacuum at finite temperature

Nonperturbative QCD vacuum with two light quarks at finite temperature was studied in a hadron resonance-gas model. Temperature dependences of the quark and gluon condensates in the confined phase were obtained. It is shown that the quark condensate and one-half (chromoelectric component) of the gluon condensate are evaporated at the same temperature corresponding to the quark-hadron phase transition. With allowance for the temperature shift of hadron masses, the critical temperature was found to be T _c ?190 MeV.     

Vacuum condensates in the bag model

The consequences of having quark and gluon condensates inside a MIT bag are investigated. We show that one naturally expects a state dependent bag constant and a colour-magnetic interaction term ～R². This gives the possibility of having a small strong coupling constant α_s inside the bag.     

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.     

Calculation of electromagnetic ρπ formfactor from QCD sum rules

Electromagnetic ρπ formfactor at intermediate momentum transfer, 0.7 GeV²≦Q ²≦3 GeV², is calculated using QCD sum rules for the vertex function of two vector and one axial vector currents. In this region the results obtained are consistent within 25% accuracy with the vector meson dominance model predictions and can be regarded as its theoretical justification.     

Anomalous dimensions of tensor operators in 4−ϵ dimensions

Anomalous dimensions of tensor operators that determine the Callan-Gross and Corwall-Norton sum rules for the deep inelastic structure functions are calculated to first order in ? in the pseudoscalar Yukawa coupling model. Their values are found to be intermediate between those for the φ⁴ model and the gluon model.     

Use and misuse of QCD sum rules,factorization and related topics

The purpose of this paper is twofold. First, we are prompted by some recent publications to reply to the criticism of the QCD sum-rules approach contained therein. Hopefully, some of the discussion is of wider interest. In particular, we point out that the multi-gluon operators unlike the multi-quark ones, relevant to the sum rules, do not factorize at large N_c. This implies that the masterfield, even if it is found, will be of no immediate help in evaluating the quarkonium spectrum. Second, we derive new sum rules for light quarks which are sensitive to the mean intensity of the gluon field in the vacuum (the so-called gluon condensate, or 〈vac|G²|vac〉). New sum rules confirm the standard value of 〈vac|G²|vac〉. Some casual remarks on the π⁰ transitions into two virtual photons, π⁰ → γ^*γ^*, are also presented. Finally, we enumerate (in sect. 7) basic points of the sum-rule approach and discuss, im brief, the unsolved problems.     

Determination of QCD condensates frome + e − annihilation: anL ∞ norm approach

Results of a new determination of QCD condensates frome ⁺ e ^?→Isospin 1 hadrons data are given. Using a new method to analyse these data, we show that the range of values for the gluon condensate <α/πGG> and for the four quarks condensates, compatible with these data, is even larger than what could be expected from the already large dispersion of previous determinations. The ‘standard’ value 0.01 GeV⁴ are not completely excluded for the gluon condensate. Its value is however strongly correlated with the value of the four quarks condensates. We also find evidences that the dimension 8 condensates should be rather large.