The meson Wave Functions in QCD Sum Rules Approach with Nonlocal Condensates

We apply the nonlocal condensate formalism to construct general ized sum rules including O s radiative corrections for and meson wave functions of twist Besides we predict the lepton decay constant f and estimate the mass of meson For all these mesons we obtain the rst moments of longitudinal wave func tions suggest the models for them and discuss their properties We consider the peculiarities of the QCD sum rules with nonlocal conden sates for transverse and b meson wave functions These results are compared with those found by Ball and Braun and by Chernyak and Zhitnitsky Introduction An important problem in the theory of strong interac tions is to calculate hadronic wave functions x x N x x x etc from the rst principles of QCD These phenomenological distributions of partons on the fraction xP of a hadron momentum P are a natural result of factorization theorems applied to hard exclusive processes They accumulate all the necessary information about non perturbative long distance dynamics of partons in hadrons E mail bakulev thsun jinr dubna su E mail mikhs thsun jinr dubna su In the standard QCD sum rule SR calculation of light meson wave functions WF s rst introduced by Chernyak and Zhitnitsky C Z and recently re estimated by Ball and Braun B B for the meson it is assumed that the correlation length of vacuum uctuations is large compared to a typical hadronic scale m Thus one can replace the original nonlocal objects like M z h q E z q z i by constant quan tities of h q q i type Based on this hypothesis the well known QCD SR approach has been applied in to calculate the rst two moments h Ni R x x dx with N for WF s of light mesons And just from these moments the whole WF s have been reconstructed which are now referred to as C Z WF s Now it is known that hadronic WF s are rather sensitive to the widths of the function M z and of other nonlocal condensates and the crucial parameter m Therefore one should use nonlocal condensates NLC s like M z whose forms re ect the complicated structure of QCD vacuum Certainly these objects can subsequently be expanded over the local condensates h q q i h q r q i etc here r ig A and one can come back to the standard SR by truncating this series Our strategy is to avoid an expansion of that sort because we thus lose an important physical property of non perturbative vacuum the possibility of vacuum quarks and gluons to ow through vacuum with non zero virtuality k Indeed the average virtuality of vacuum quarks hk i q is not small whereas the standard approach inevitably suggests that k is exact zero The value of q can be extracted from the QCD SR analysis and is connected with the condensate of the next dimension d q h qr qi h qqi h q ig G qi h qqi GeV that is of an order of the typical hadronic scale m GeV An estimate for q in the framework of the instanton liquid model yields a similar number and the lattice QCD calculations give q GeV a similar estimate q GeV has been obtained in Careful inspection of consequences of that approach to QCD SR for pion WF has revealed that the introduction of the correlation length Here E z P exp i R z dt A a t a is the Schwinger phase factor required for gauge invariance q into condensate distributions produces much smaller values for the rst moments of pion WF than C Z values As a result the pion WF is strongly di erent in shape from the C Z and approaches the asymptotic WF as x x x i e x as x Later this WF was con rmed by independent consideration of the QCD SR directly for the function x based on the non diagonal correlator and on the advanced smooth distribu tion function for the quark nonlocal condensate The key element of both the ways was to take into account the main physical reason vacuum correlation length q is of an order of m Our goal here is to show that in the case of QCD SR for the meson channel the situation is similar to the pion one All the predictions of the standard QCD SR the C Z ones for the longitudinal case and B B ones for the transverse case we call them the Local QCD SR for moments h Ni with N could not be considered as reliable as we show in our analysis even the value of h i is inde nite We apply the NLC formalism to calculate the diagonal correlators for meson currents introduced in and construct generalized SR including O s radiative corrections to obtain WF s of twist The rst ten moments of longitudinal WF s of and mesons are estimated Table and the models for them see Figs are suggested Also we predict the lepton decay constant f and estimate the mass of meson We discuss the strong dependence of results for moments of transverse WF s in channel on the model of nonlocal gluon condensate contribution The calculation technique is the same as in Refs there fore the corresponding details are omitted below but we shall through the text keep the connection with the well understood pion case Generalized sum rules for the and meson channels vs the standard version For the helicity zero charged vector mesonMj j ML ML the leading twist WF is de ned as h j d z u jML p i z f MLp Z dxe zp LML x In accordance with the B B de nition in this case  p p mML as pz where p is the polarization vector and for axial mesons Note here other independent evidences that the pion WF is close to its asymptotic form lattice calculations of the corresponding h i in QCD SR estimate of the magnitude of x at the midpoint x in