The scalar radius of the pion

The pion scalar radius is given by $=(6/\pi)\int_{4M^2_\pi}^\infty{\rm d}s \delta_S(s)/s^2$, with $\delta_S$ the phase of the scalar form factor. Below $\bar{K}K$ threshold, $\delta_S=\delta_\pi$, $\delta_\pi$ being the isoscalar, S-wave $\pi\pi$ phase shift. At high energy, $s>2 {\rm GeV}^2$, $\delta_S$ is given by perturbative QCD. In between I argued, in a previous letter, that one can interpolate $\delta_S\sim\delta_\pi$, because inelasticity is small, compared with the errors. This gives $=0.75\pm0.07 {\rm fm}^2$. Recently, Ananthanarayan, Caprini, Colangelo, Gasser and Leutwyler (ACCGL) have claimed that this is incorrect and one should have instead $\delta_S\simeq\delta_\pi-\pi$; then $=0.61\pm0.04 {\rm fm}^2$. Here I show that the ACCGL phase $\delta_S$ is pathological in that it is discontinuous for small inelasticity, does not coincide with what perturbative QCD suggests at high energy, and only occurs because these authors take a value for $\delta_\pi(4m^2_K)$ different from what experiment indicates. If one uses the value for $\delta_\pi(4m^2_K)$ favoured by experiment, the ensuing phase $\delta_S$ is continuous, agrees with perturbative QCD expectations, and satisfies $\delta_S\simeq\delta_\pi$, thus confirming the correctness of my previous estimate, $=0.75\pm0.07 {\rm fm}^2$.Comment: Version to be published in Phys. Letters. A few typos corrected. Plain YeX file. 5 figure

Gluon Condensate from Superconvergent QCD Sum Rule

Sum rules for the nonperturbative piece of correlators (specifically, the vector current correlator) are discussed. The sum rule subtracting the perturbative part is of the superconvergent type. Thus it is dominated by the bound states and low energy production cross section. It leads to a determination of the gluon condensate of $= 0.048 \pm 0.039 GeV^4$Comment: plain TeX, no figure

The quadratic scalar radius of the pion and the mixed π−K\pi-K radius

We consider the quadratic scalar radius of the pion, , and the mixed $K-\pi$ scalar radius, . With respect to the second, we point out that the more recent (post-1974) experimental results in $K_{l3}$ decays imply a value, $=0.31\pm0.06 {\rm fm}^2$, which is about $2 \sigma$ above estimates based on chiral perturbation theory. On the other hand, we show that this value of suggests the existence of a low mass S$\tfrac{1}{2}$ $K\pi$ resonance. With respect to , we contest the central value and accuracy of current evaluations, that give $=0.61\pm0.04 {\rm fm}^2$. Based on experiment, we find a robust lower bound of $<r^2_{{\rm S},\pi}>\geq0.70\pm0.06 {\rm fm}^2$ and a reliable estimate, $<r^2_{{\rm S},\pi}>=0.75\pm0.07 {\rm fm}^2$, where the error bars are attainable. This implies, in particular, that the chiral result for $$ is $1.4 \sigma$ away from experiment. We also comment on implications about the chiral parameter $\bar{l}_4$, very likely substantially larger (and with larger errors) than usually assumed.Comment: PlainTeX file. Corrected asymptotic phase; numerical results unaffecte

The l=1l=1 Hyperfine Splitting in Bottomium as a Precise Probe of the QCD Vacuum.

By relating fine and hyperfine spittings for l=1 states in bottomium we can factor out the less tractable part of the perturbative and nonperturbative effects. Reliable predictions for one of the fine splittings and the hyperfine splitting can then be made calculating in terms of the remaining fine splitting, which is then taken from experiment; perturbative and nonperturbative corrections to these relations are under full control. The method (which produces reasonable results even for the $c{\bar c}$ system) predicts a value of 1.5 MeV for the $(s=1)-(s=0)$ splitting in $b{\bar b}$, opposite in sign to that in $c{\bar c}$. For this result the contribution of the gluon condensate $$ is essential, as any model (in particular potential models) which neglects this would give a negative $b{\bar b}$ hyperfine splitting.Comment: 12 pages, 2 postscript figures, typeset with ReVTe

QCD Calculations of Heavy Quarkonium States

Recent results on the QCD analysis of bound states of heavy $\bar{q}q$ quarks are reviewed, paying attention to what can be derived from the theory with a reasonable degree of rigour. We report a calculation of $\bar{b}c$ bound states; a very precise evaluation of $b, c$ quark masses from quarkonium spectrum; the NNLO evaluation of $\Upsilon\to e^+e^-$; and a discussion of power corrections. For the $b$ quark {\sl pole} mass we get, including $O(m_c^2/m_b^2)$ and $O(\alpha_s^5\log \alpha_s)$ corrections, $m_b=5.020\pm0.058 GeV$; and for the $\bar{MS}$ mass the result, correct to $O(\alpha_s^3)$, $O(m_c^2/m_b^2)$, $\bar{m}_b(\bar{m}_b)=4.286\pm0.036 GeV$. For the decay $\Upsilon\to e^+e^-$, higher corrections are too large to permit a reliable calculation, but we can predict a toponium width of $13\pm1 keV$.Comment: PlainTex file; one figur

Basic Parameters and Some Precision Tests of the Standard Model

We present a review of the masses (except for neutrino masses) and interaction strengths in the standard model. Special emphasis is put on quantities that have been determined with significantly improved precision in the last few years. In particular, a number of determinations of $\alpha_s$ and the electromagnetic coupling on the $Z$, $\alpha_{\rm QED}(M_Z^2)$ are presented and their implications for the Higgs mass discussed; the best prediction that results for this last quantity being $$M_H=102^{+54}_{-36} GeV/c^2.$$ Besides this, we also discuss a few extra precision tests of the standard model: the electron magnetic moment and dipole moment, and the muon magnetic moment.Comment: Plain TeX file. 6figure

Theory of Small xx Deep Inelastic Scattering NLO Evaluations, and low Q2Q^2 Analysis

We calculate structure functions at small $x$ both under the assumption of a hard singularity (a power behaviour $x^{-\lambda}, \lambda$ positive, for $x\rightarrow 0$) or that of a soft-Pomeron dominated behaviour, also called double scaling limit, for the singlet component. A full next to leading order (NLO) analysis is carried for the functions $F_2, F_{\rm Glue}$ and the longitudinal one $F_L$ in $ep$ scattering, and for $x F_3$ in neutrino scattering. The results of the calculations are compared with data (HERA) in the range $x\leq 0.032, 10 gev^2\leq Q^2\leq 1 500 gev^2$. We get reasonable fits, with a chi-squared/d.o.f.$\sim 2$, for both assumptions, but none of them gives a fully satisfactory description. The results improve substantially if combining a soft and a hard component; in this case it is even possible to extend the analysis, phenomenologically, to small values of $Q^2$, $0.31 gev^2\leq Q^2\leq 8.5 gev^2$, and in the $x$ range 6\times10^{-6}\lsim x \lsim 0.04, with the same hard plus soft Pomeron hypothesis by assuming a saturating expression for the strong coupling, $\tilde{\alpha}_s(Q^2)=4\pi/\beta_0\log[(Q^2+\Lambda_{eff}^2)/\Lambda_{eff}^2]$ The description for low $Q^2$ implies self-consistent values for the parameters in the exponents of $x$. One gets, for the Regge intercepts, $\alpha_{\rho}(0)=0.48$ and $\alpha_P(0)=1.470$ [$\lambda=0.470$], in uncanny agreement with other determinations of these parameters, in particular the results of the large $Q^2$ fits. The fit to is so good that we may look (at large $Q^2$) for signals of a "triple Pomeron" vertex; some evidence is found.Comment: Tex file plus .ps figures. This paper includes the results from FTUAM 96-39 [hep-ph/9610380] and FTUAM 96-44 [hep-ph/9612469

Chiral Symmetry and Diffractive Neutral Pion Photo- and Electroproduction

We show that diffractive production of a single neutral pion in photon-induced reactions at high energy is dynamically suppressed due to the approximate chiral symmetry of QCD. These reactions have been proposed as a test of the odderon exchange mechanism. We show that the odderon contribution to the amplitude for such reactions vanishes exactly in the chiral limit. This result is obtained in a nonperturbative framework and by using PCAC relations between the amplitudes for neutral pion and axial vector current production.Comment: 22 pages, 7 figure

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 of non-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.Comment: 24 pages, LateX file, TUM-T31-21/92, 1 postscript file include

Bound states of heavy quarks in QCD

Bound states of heavy $\bar{q}q$ quarks are reviewed within the context of QCD, paying attention to what can be derived from the theory with a reasonable degree of rigour. This is compared with the results of semiclassical arguments. Among new results, we report a very precise $O(\alpha_s^4)$ evaluation of $b, c$ quark masses from quarkonium spectrum with a potential to two loops.Comment: Plain TeX, 5 figure