Lattice QCD and CLEO-c tackle the CP challenge

Problems in theories such as quantum chromodynamics (QCD) that involve strong coupling are among the most intractable in physics. The difficulties of accurate calculations are particularly vexing when it comes to studying charge-parity (CP) violation – a necessary ingredient for explaining the absence of the antimatter produced in the Big Bang, and a vital topic in particle physics. Progress in making accurate QCD calculations in this sector of the Standard Model could have far-reaching consequences, because the larger theory in which the Standard Model is embedded, even if not strongly coupled ab initio, will almost certainly have strongly coupled sectors.

The amount of CP violation that is consistent with our current understanding of the Standard Model is not enough to account for the disappearance of the antimatter produced along with the matter. The decay of B mesons is the most promising arena in which to search for other sources of CP violation, and the B-factory experiments, BaBar and Belle, have been spectacularly successful in observing and studying CP violation in those decays.

However, the search for CP violation beyond the Standard Model involves comparison of the angles of the “unitarity triangle” measured in CP violation experiments, with the lengths of the sides determined from more conventional measurements. These lengths are determined from elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes the relative strengths of the weak decays of quarks. The problem is that quarks do not appear alone, but in strongly interacting combinations such as mesons and baryons. So, there is always a strong-interaction parameter that relates decay measurements to the underlying CKM matrix element. So far, the uncertainties in these parameters severely limit the precision of measurements of the CKM matrix elements.

One example of such a strong-interaction parameter, called f_B, is required to extract the CKM matrix element called V_td from measurements of B⁰B⁰bar mixing. This parameter is a measure of the separation of the b quark and the anti-d quark in a B meson, and could, in principle, be measured in B⁺ → μ⁺ν decay. However, this decay is so slow that an accurate measurement is impossible, even with the enormous data samples that the BaBar and Belle collaborations have accumulated. This is one of the parameters that lattice QCD (LQCD) theorists can calculate, but accuracy has been limited. Recent progress in LQCD holds the promise of precise calculations of the parameters, including f_B, that are required to determine CKM matrix elements, but until now there has been no effective direct experimental test of the precision of these calculations.

Five years ago, some LQCD theorists and members of the CLEO collaboration at the Cornell Laboratory for Elementary-Particle Physics realized that comparing measurements of D meson decays with LQCD calculations could test the LQCD calculations needed for extracting CKM matrix elements. In particular, in the decay D⁺ → μ⁺ν there is a parameter called f_D⁺ that is analogous to f_B. The rate for D⁺ → μ⁺ν decay is larger than the corresponding B meson decay rate, so it can be measured accurately. If good agreement is found between the experimental and LQCD values for f_D⁺, that would inspire confidence in LQCD calculations of f_B. It is also generally believed that the ratio f_B:f_D⁺ can be calculated more accurately than either one, so f_B could be determined from an experimental measurement of f_D⁺ and a LQCD calculation of the ratio. This caused the CLEO collaboration and a group of LQCD theorists to embark on D meson decay experiments and the corresponding LQCD calculations, with goals of accuracies of a few per cent. However, they had technical difficulties to overcome.

Although there have been LQCD calculations since the 1970s, their accuracy has been limited to 20% because of a simplification – the “quenched approximation”. Predictions from “unquenched” LQCD calculations that match experimental results to a few per cent are needed to demonstrate that the calculations can be done at the desired level of accuracy. The LQCD theorists were the first to reach the goal of a few per cent in their calculations of important parameters (not f_D⁺ or f_B) in the b and c quark systems. However, their results were “postdictions” not predictions, because the corresponding experimental results had already been published. Still, this success motivated the group of LQCD theorists to embark on their calculations of the strong-interaction parameters needed to determine CKM matrix elements.

Measurement of f_D⁺ is one of the main goals of the CLEO-c physics programme at the Cornell Electron Storage Ring (CESR). However, obtaining high luminosity was a major challenge for the CESR accelerator group. Although CESR had made progress in luminosity since its first operation in 1979, much of those gains would be lost in reducing the energy from the b quark threshold region, near 5 GeV per beam, to the c quark threshold region near 2 GeV per beam, where the measurement could be made most readily. As the energy of the beams is decreased, the synchrotron radiation “damping” required for high luminosity is substantially reduced. The solution was the installation of 12 “wiggler” magnets to increase the damping at the lower energies. These magnets were installed in 2003 and 2004 in CESR-c, and the CLEO-c programme then began in earnest, funded by a five-year grant from the National Science Foundation.

The first engineering run of CESR-c with six wigglers yielded an integrated luminosity of 60 pb^-1 in e⁺e^– collisions at a total energy of 3.77 GeV, the peak of the ψ (3770) resonance. This is substantially more than the luminosities available to either the MARK III or the BES II collaborations, at SLAC and the Institute of High Energy Physics in Beijing respectively, which previously took data at the same energy. Subsequent runs with 12 wigglers brought the total integrated luminosity to 281 pb^-1. This is the most desirable energy for measuring D meson decays because the ψ (3770) decays only to D⁺D^– or D⁰D⁰bar pairs, making very clean “tagged” measurements possible. In tagged measurements pioneered by the MARK III collaboration, if one D meson, D^– for example, is reconstructed in an event, then the rest of the particles in that event must be from the decay of a D⁺ meson. Coupled with the excellent resolution and large acceptance of the CLEO-c detector, tagging provides a very clean sample of D⁺ meson decays, which is an ideal arena for searching for rare decays such as D⁺ → μ⁺ν.

The CLEO collaboration found eight D⁺ → μ⁺ν events (with an estimated background of one event) in the first 60 pb^-1 of CLEO-c data. This provided a rough measurement of f_D⁺ to an accuracy of 20%, which has now been published (Bonvicini 2004). The larger data sample and improvements in selection criteria produced a yield of 50 candidates for D⁺ → μ⁺ν decay with an estimated background of 2.9 ± 0.5, enough candidates to yield an error in f_D⁺ below 10%.

While the CLEO collaboration, with the help of their CESR colleagues, was accumulating and analysing these data, a group of LQCD theorists was also hard at work calculating f_D⁺. It became clear that both groups could have substantial results just in time for the Lepton-Photon Symposium in Uppsala at the end of June. Since both communities felt that it was very important for the LQCD result to be a real prediction, they agreed to embargo both of their results until the conference. On the second day of the symposium, Marina Artuso of Syracuse University reported the preliminary CLEO-c result f_D⁺ = 223 ± 16_-9⁺⁷ MeV (CLEO-c 2005), and Iain Stewart of MIT reported the LQCD result from the Fermilab, MIMD Lattice Calculation (MILC) and High Precision QCD (HPQCD) collaborations, f_D⁺ = 201 ± 3 ± 17 MeV (Aubin et al. 2005). For both results the errors are statistical and systematic, respectively. The two results agree well within the errors of about 8% for each.

The agreement between the results motivates both communities to continue comparing LQCD calculations with experiments. On the LQCD side, important next steps include improvements in algorithms that can reduce systematic errors and precision calculations of the “form factors” involved in semileptonic decays of D and B mesons. The CLEO collaboration plans to utilize its data sample to measure form factors in semileptonic D decay and take more data to reduce errors. The LQCD theorists and the CLEO collaboration both aim to reduce errors to below 5%. The CLEO collaboration is also planning to explore the threshold region for D_sD_sbar production to search for an energy at which the tagging techniques can be applied to make the first accurate measurements of D_s meson decay, including f_Ds and D_s semileptonic decay form factors. The Fermilab, MILC and HPQCD group has already predicted the value of f_Ds.