Theory of unitarity bounds and low-energy form factors

Abstract

Abstract. We present a general formalism for deriving bounds on the shape parameters of the weak and electromagnetic form factors using as input correlators calculated from perturbative QCD, and exploiting analyticity and unitarity. The values resulting from the symmetries of QCD at low energies or from lattice calculations at special points inside the analyticity domain can be included in an exact way. We write down the general solution of the corresponding Meiman problem for an arbitrary number of interior constraints and the integral equations that allow one to include the phase of the form factor along a part of the unitarity cut. A formalism that includes the phase and some information on the modulus along a part of the cut is also given. For illustration we present constraints on the slope and curvature of the K l3 scalar form factor and discuss our findings in some detail. The techniques are useful for checking the consistency of various inputs and for controlling the parameterizations of the form factors entering precision predictions in flavor physics.

Constraints on the omega-pi form factor from analyticity and unitarity

“…Using standard techniques [12,13,14,15,16,17,18,19,20,21,22,23], we start with the QCD vacuum polarization tensor…”

“…a function analytic and without zeros in |z| < 1, its modulus on |z| = 1 being equal to w ωπ (t(z), −Q 2 )|dt(z)/dz|, wheret(z) is the inverse of (16). The construction of the outer functions is explained in [46] (see also the review [22]). Using the expression (11) of w ωπ (t, −Q 2 ), we obtain for C(z) the exact analytic expression…”

“…We end this section with a remark that might be useful for improving the bounds. From general arguments [22] and the expressions given above, it follows that the bounds depend monotonically on the value of I in the L 2 -norm constraint (12): smaller values of I lead to narrower allowed intervals for |f ωπ (t)| at t < t + . Therefore, the bounds can be made tighter in principle by taking into account more intermediate states, besides the ππ pairs, in the unitarity relation (9) for the QCD correlator.…”

“…We use a method proposed originally by Okubo [12,13] (before the advent of QCD), which leads to bounds on form factors by exploiting the positivity of the spectral function of a suitable currentcurrent correlator. This technique, known as method of unitarity bounds, has been resuscitated in the QCD era and was applied to a variety of form factors of heavy and light mesons [14,15,16,17,18,19,20,21] (for a review and more references see [22,23]). In the present study, we use a dispersion relation for the polarization function of two isovector vector currents, calculated by operator prod-uct expansion (OPE) in the Euclidean region, and exploit unitarity for the spectral function.…”

Analysis Tools for Next-Generation Hadron Spectroscopy Experiments

“…A rigorous dispersive implementation of this theorem can be achieved via the MuskhelishviliOmnès (MO) method [113,114], where the amplitude is expressed in terms of an Omnès factor uniquely determined by the phase of the scattering process of the final state. This method is particularly well-suited for the study of meson form factors, not only of pions, kaons, but charmed D mesons as well, see for instance [115][116][117][118][119][120][121][122] and references therein. In addition to the right-hand cut accounted for by the MO method, the description of production amplitudes involves a left-hand cut.…”