Extracting muon momentum scale corrections for hadron collider experiments

The quark condensate is calculated within the world-line effective-action formalism, by using for the Wilson loop an ansatz provided by the stochastic vacuum model. Starting with the relation between the quark and the gluon condensates in the heavy-quark limit, we diminish the current quark mass down to the value of the inverse vacuum correlation length, finding in this way a 64?% decrease in the absolute value of the quark condensate. In particular, we find that the conventional formula for the heavy-quark condensate cannot be applied to the c-quark, and that the corrections to this formula can reach 23?% even in the case of the b-quark. We also demonstrate that, for an exponential parametrization of the two-point correlation function of gluonic field strengths, the quark condensate does not depend on the non-confining non-perturbative interactions of the stochastic background Yang?CMills fields.     

关于夸克凝聚和混合凝聚平移不变性的讨论

讨论夸克凝聚和混合凝聚在Lorentz规范下的泰勒展开式,这两个展开式可用来计算算符乘积展开中非微扰修正项的Wilson系数.与目前在QCD求和规则方法中所应用的、在固定点规范下的相应展开式相比,新的展开式具有平移不变性的优点.     

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.     

Current correlators to all orders in the quark masses

The contributions to the coefficient functions of the quark and the mixed quark-gluon condensate to mesonic correlators are calculated for the first time to all orders in the quark masses, and to lowest order in the strong coupling constant. Existing results on the coefficient functions of the unit operator and the gluon condensate are reviewed. The proper factorization of short- and long-distance contributions in the operator product expansion is discussed in detail. It is found that to accomplish this task rigorously the operator product expansion has to be performed in terms ofnon-normal-ordered condensates. The resulting coefficient functions are improved with the help of the renormalization group. The scale invariant combination of dimension 5 operators, including mixing with the mass operator, which is needed for the renormalization group improvement, is calculated in the leading order.Supported by the German Bundesministerium für Forschung und Technologie, under the contract 06 TM 761     

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.     

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.     

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.     

Quark Gluon Condensate,Virtuality and Susceptibility of QCD Vacuum

We study vacuum of QCD in this work. The structure of non-local quark vacuum condensate, values of various local quark and gluon vacuum condensates, quark-gluon mixed vacuum condensate, quark and gluon virtuality in QCD vacuum state, quark dynamical mass and susceptibility of QCD vacuum state to external field are predicted by use of the solutions of Dyson-Schwinger equations in “rainbow” approximation with a modeling gluon propagator and three different sets of quark-quark interaction parameters. Our theoretical predictions are in good agreement with the correspondent empirical values used widely in literature, and many other theoretical calculations. The quark propagator and self-energy functions are also obtained from the numerical solutions of Dyson-Schwinger equations. This work is centrally important for studying non-perturbative QCD, and has many important applications both in particle and nuclear physics.     

Quark condensate contributions to the gluon self-energy and the ϱ meson sum rule

The operator-product expansion will be employed to obtain the lowest-order, quark condensate component of both the gluon self-energy and the ? meson correlation function to all orders in the quark mass parameter. Field-theoretic aspects of the self-energy and correlation function will be considered, and physical effects of the quark condensate upon gluon mass generation will be examined.     

Nonperturbative condensate contributions to the effective quark-gluon coupling

We study the effective quark-gluon coupling at low-energy scale, which is defined as the amplitude of a quark emitting or absorbing a gluon with some momentum at low-energy scale. This amplitude is determined from the fermionic three-point Green’s functions of QCD including the leading order contributions of nonperturbative condensates through use of the operator-product expansion. By this approach, we discuss the relationship between the constituent quark and the quark of QCD Lagrangian, and estimate the scale of chiral symmetry breaking and the size of a constituent quark in participating the strong interaction process, such as form factors and radii.     

In-medium operator product expansion for heavy-light-quark pseudoscalar mesons

The operator product expansion (OPE) for heavy-light-quark pseudoscalar mesons (D -mesons and B -mesons) in medium is determined, both for a moving meson with respect to the surrounding medium as well as for a meson at rest. First of all, the OPE is given in terms of normal-ordered operators up to mass dimension 5, and the mass of the heavy quark and the mass of the light quark are kept finite. The Wilson coefficients of such an expansion are infrared (IR) divergent in the limit of a vanishing light-quark mass. A consistent separation of scales necessitates an OPE in terms of non-normal-ordered operators, which implies operator mixing, where the IR-divergences are absorbed into the operators. It is shown that the Wilson coefficients of such an expansion are IR-stable, and the limit of a vanishing light-quark mass is perfomed. Details of the major steps for the calculation of the Wilson coefficients are presented. By a comparison with previous results obtained by other theoretical groups we have found serious disagreements.     

The Nielsen identities for the two-point functions of QED and QCD

We consider the Nielsen identities for the twopoint functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon selfenergy and ghost propagator in terms of the gauge dependence of other, independent Green functions.     

Effective potential evaluations in a modified PQCD

A procedure discussed in a previous work for properly defining the Feynman diagrams at any number of loops in a modified version of PQCD, is employed here to evaluate some zero- and one-loop corrections to the effective potential, as functions of the gluon and quark condensate parameters. The calculated terms indicate an instability of massless QCD under the development of quark condensates even in the absence of the gluon one. Therefore, a mechanism is suggested for the dynamical generation of quark masses and condensates. The absence of indications coming from lattice calculations to this possibility could be determined by the current limitations in treating fermion determinants. PACS 12.38.Aw, 12.38.Bx, 12.38.Cy, 14.65.Ha     

Calculations in external fields in quantum chromodynamics. Technical review

We review the technique of calculation of operator expansion coefficients. The main emphasis is put on gluon operators which appear in expansion of n-point functions induced by colourless quark currents. Two convenient schemes are discussed in detail: the abstract operator method and the method based on the Fock-Schwinger gauge for the vacuum gluon field. We consider a large number of instructive examples important from the point of view of physical applications.     

Chiral-symmetry breaking and gluon condensation

With the help of the Ward-Takahashi identities associated with a specific non linear chiral transformation of the quark fields we show, in the framework of massless QCD, that in addition to scale invariance breaking gluon condensation implies dynamical chiral symmetry breaking, that is to say the appearance of a non vanishing quark condensate. Moreover the dynamically generated quark mass is directly connected to the quark and gluon condensates.     

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.     

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.     

Light 0- Meson Solution of Bethe-Salpeter Equation Involving the Vacuum Condensates in QCD

We dkcuss the Bethe-Salpeter equation involving the vacuum condensates in QCD in the backgrouud fields. The bound state equation for the light 0^- meson (π and K) is solved in the ladder approximation of single gluon exchange, where the gluon propagator is calculated in tree level of a gluon condensate and a quark condensate. The kernel is divided into two parts: perturbative and non-pert,urbative. The non-perturbative part is determined by the vacuum condensate. Pseudo-scalar meson spectrum and decay constants are in good agreement with data.     

Determination of the renormalized heavy-quark mass in lattice QCD

We study on the lattice the correlator of heavy-quark currents in the vicinity of vanishing momentum. The renormalized charmed quark mass, the renormalized strong coupling constant and gluon condensate can be defined in terms of the derivatives of that correlator at zero momentum. We analyze quenched Monte Carlo data on a small 8³ × 16 lattice for β = 6. We generalize dispersion relations to the lattice theory in a simple way and use them successfully to fit both the short-distance part and long-distance part of the correlator in such a small volume.     

Mixed quark–gluon condensate at finite temperature and density in the global color symmetry model

We study the properties of mixed quark–gluon condensate at finite temperature and chemical potential in the framework of global color symmetry model. In comparing with the quark condensate, we confirm that both of these condensates give the same information about chiral phase transition. We also find that the ratio of these two condensates is insensitive to the temperature T and the chemical potential μ, which supports the conclusion obtained recently by the authors using quenched lattice QCD.