Standard Model

Mathias Jamin

The Standard Model of particle interactions is one of the major achievements of fundamental science. Within this framework a wide range of phenomena can be described to an impressive degree of accuracy. As a matter of fact, few are the branches of Physics where the predictive power of a theory has been tested to such a level of precision.

Introduction

The Standard Model (SM) subgroup of the IFAE theory division investigates the phenomenology of particle physics within the realms of the Standard Model. Even if physics going beyond the SM is expected, suggested for example by the presence of dark matter or neutrino masses, precise values of the fundamental SM parameters like couplings and masses are essential inputs for predictions within the SM, and beyond-SM physics should show up as clashes between those predictions and the experimental measurements.

STRONG COUPLING FROM electronic hadron production BELOW CHARM

Ever since the beginning of Quantum Chromodynamics (QCD), experimental data on the process $e^+e^-\to{hadrons}$ has been very instrumental in our understanding of the dynamics of quarks and gluons. With the help of a new compilation of this data, members of the Theory Division (S. Peris) together with D. Boito (Brazil), M. Golterman (USA), A. Keshavarzi (UK), K. Maltman (Canada), D. Nomura (Japan) and T. Teubner (UK) conducted a new analysis to determine the strong coupling, $\alpha_{s}$, which is the fundamental parameter governing the strength of the QCD interaction.

In this analysis use was made of all the $e^+e^-\to{hadrons}$ data from threshold to a center-of-mass energy of 2~GeV by employing finite-energy sum rules. The advantage of these sum rules is that, based on fundamental properties of Field Theory such as analyticity and unitarity, they allow us to consider the experimental data from threshold up to an arbitrary higher energy, which in our case was $4~\mathrm{GeV}^2$. Since the threshold is located at very low energy (in fact, the pion mass), the non-perturbative character of QCD would make the analysis impossible without the help of these sum rules. Data above $4~\mathrm{GeV}^2$ is also available but has less accuracy and does not provide much additional constraint. However, it is fully consistent with the values for $\alpha_s$ we obtain. We choose to quote our results at the $\tau$ mass to facilitate comparison with the results from analogous analyses of hadronic $\tau$-decay data, and obtained $\alpha(m_\tau^2)=0.298\pm 0.016\pm 0.006$ in fixed-order perturbation theory, and $\alpha_{s}(m_\tau^2)=0.304\pm 0.018\pm 0.006$ in contour-improved perturbation theory, where the first error is statistical, and the second error reflects our estimate of various systematic effects. These values are in good agreement with a recent determination from the OPAL and ALEPH data for hadronic $\t$ decays also carried out by our group.

Figure 1: Example of a comparison of the data obtained for one of the finite-energy sum rules (red points) with the theory description (black curve) as a function of the energy (squared). The two vertical lines show the window in which the analysis was made and the plot shows a little extrapolation on both ends.

Figure 2: The FOPT strong coupling $\a_s(m_\t^2)$ as a function of the energy (squared) $s_0^{\rm min}$. Black data points show an example of the values obtained for this coupling when our parametrization of Duality Violations is used. Non-black data points show the results from a variety of finite-energy sum rules when this parametrization is omitted, showing a clear divergence as the energy is lowered. The purple horizontal band is the final result of our analysis.

Figures 1 and 2 show an example of the results obtained. Figure 1 shows the agreement between the result for one of the finite-energy sum rules and our value of $\alpha_s$ for the theory description. Figure 2 focuses on the effect of including Duality Violations (DV) in the analysis. DVs are effects not captured by the Operator Product Expansion, and are supposed to be larger the lower the energy considered. As one can see, omitting these DVs pulls the values of $\alpha_s$ downwards at low energies making the results mutually incompatible. However, including DVs makes the value of $\alpha_s$ level off as the energy is lowered, substantially enlarging the region of compatibility with the results obtained at higher energies. This is further evidence for the correctness of the DV parametrization developed by our group.