Matching long and short distances in large-N_c QCD

Abstract

It is shown, with the example of the experimentally known Adler function, that there is no matching in the intermediate region between the two asymptotic regimes described by perturbative QCD (for the very short-distances) and by chiral perturbation theory (for the very long-distances). We then propose to consider an approximation of large-N c QCD which consists in restricting the hadronic spectrum in the channels with J P quantum numbers 0 − , 1 − , 0 + and 1 + to the lightest state and to treat the rest of the narrow states as a perturbative QCD continuum; the onset of this continuum being fixed by consistency constraints from the operator product expansion. We show how to construct the low-energy effective Lagrangian which describes this approximation. The number of free parameters in the resulting effective Lagrangian can be reduced, in the chiral limit where the light quark masses are set to zero, to just one mass scale and one dimensionless constant to all orders in chiral perturbation theory. A comparison of the corresponding predictions, to O(p 4 ) in the chiral expansion, with the phenomenologically known couplings is also made.

A novel integral representation for the Adler function

“…Similarly to the perturbative approach, the QCD scale parameter Λ remains the basic characterizing quantity of the theory. Besides, all the effects due to the interrelation between the spacelike and timelike observables are automatically accounted for by the representations (14) and (17). It is important to mention also that in the limit of the vanishing pion mass m π = 0 the derived relations (14) and (17) become identical to those obtained within the massless APT [22], namely…”

“…This is so because, in addition to the possible ambiguities stemming from the truncation and general treatment of the Schwinger-Dyson equations, the final numerical implementation of the procedure of [34] seems to require rather elevated constituent masses for the light quarks (m u = m d = 250 MeV); instead, the only phenomenological parameter appearing in Eqs. (14) and (17) is the measurable mass of the pion. Finally, again in the framework of massless APT, the infrared behavior of the Adler function has been brought into qualitative agreement with experiment, but at the expense of resorting to additional assumptions, such as the vector meson dominance [35].…”

“…In turn, the Adler function can be represented in terms of the spectral function ρ D (σ) by integrating the dispersion relation (7) with R(s) given by Eq. (14). Carrying out the integration by parts, one arrives at…”

Large-Nc QCD meets Regge theory: the example of spin-one two-point functions

“…In section 4 we make a comparison with the Lowest-Meson Dominance approximation of ref. [12], and we conclude in section 5.…”

“…from the absence of a dimension-two operator in the OPE, 12) from the contribution of the gluon condensate, and…”