marcus said:
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http://www.sciencemag.org/cgi/content/abstract/sci;322/5905/1224
Ab Initio Determination of Light Hadron Masses
S. Dürr,1 Z. Fodor,1,2,3 J. Frison,4 C. Hoelbling,2,3,4 R. Hoffmann,2 S. D. Katz,2,3 S. Krieg,2 T. Kurth,2 L. Lellouch,4 T. Lippert,2,5 K. K. Szabo,2 G. Vulvert4
"More than 99% of the mass of the visible universe is made up of protons and neutrons. Both particles are much heavier than their quark and gluon constituents, and the Standard Model of particle physics should explain this difference. We present a full ab initio calculation of the masses of protons, neutrons, and other light hadrons, using lattice quantum chromodynamics. Pion masses down to 190 mega–electron volts are used to extrapolate to the physical point, with lattice sizes of approximately four times the inverse pion mass. Three lattice spacings are used for a continuum extrapolation. Our results completely agree with experimental observations and represent a quantitative confirmation of this aspect of the Standard Model with fully controlled uncertainties."
1 John von Neumann–Institut für Computing, Deutsches Elektronen-Synchrotron Zeuthen, D-15738 Zeuthen and Forschungszentrum Jülich, D-52425 Jülich, Germany.
2 Bergische Universität Wuppertal, Gaussstrasse 20, D-42119 Wuppertal, Germany.
3 Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary.
4 Centre de Physique Théorique (UMR 6207 du CNRS et des Universités d'Aix-Marseille I, d'Aix-Marseille II et du Sud Toulon-Var, affiliée à la FRUMAM), Case 907, Campus de Luminy, F-13288, Marseille Cedex 9, France.
5 Jülich Supercomputing Centre, FZ Jülich, D-52425 Jülich, Germany.
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Here are some excerpts from that article. This is severely abbridged and many symbols and subscripts are missing. It can give a taste of the article, and some main conclusions, but for most of the content one must look up the article.
===exerpts from Duerr et al, Science 20 November===
...The Standard Model of particle physics predicts a cosmological, quantum chromodynamics (QCD)–related smooth transition between a high-temperature phase dominated by quarks and gluons and a low-temperature phase dominated by hadrons. The very large energy densities at the high temperatures of the early universe have essentially disappeared through expansion and cooling. Nevertheless, a fraction of this energy is carried today by quarks and gluons, which are confined into protons and neutrons. According to the mass-energy equivalence E = mc^2, we experience this energy as mass. Because more than 99% of the mass of ordinary matter comes from protons and neutrons, and in turn about 95% of their mass comes from this confined energy, it is of fundamental interest to perform a controlled ab initio calculation based on QCD to determine the hadron masses.
QCD is a generalized version of quantum electrodynamics (QED), which describes the electromagnetic interactions. The Euclidean Lagrangian with gauge coupling g and a quark mass of m can be written as... , where Fµ = µA – Aµ + [Aµ,A]. In electrodynamics, the gauge potential Aµ is a real valued field, whereas in QCD it is a 3 x 3 matrix field. Consequently, the commutator in Fµ vanishes in QED but not in QCD. The fields also have an additional "color" index in QCD, which runs from 1 to 3. Different "flavors" of quarks are represented by independent fermionic fields, with possibly different masses. In the work presented here, a full calculation of the light hadron spectrum in QCD, only three input parameters are required: the light and strange quark masses and the coupling g.
The action S of QCD is defined as the four-volume integral of ... Green's functions are averages of products of fields over all field configurations, weighted by the Boltzmann factor exp(–S). A remarkable feature of QCD is asymptotic freedom, which means that for high energies (that is, for energies at least 10 to 100 times higher than that of a proton at rest), the interaction gets weaker and weaker (1, 2), enabling perturbative calculations based on a small coupling parameter. Much less is known about the other side, where the coupling gets large, and the physics describing the interactions becomes nonperturbative. To explore the predictions of QCD in this nonperturbative regime, the most systematic approach is to discretize (3) the above Lagrangian on a hypercubic space-time lattice with spacing a, to evaluate its Green's functions numerically and to extrapolate the resulting observables to the continuum (a0). A convenient way to carry out this discretization is to place the fermionic variables on the sites of the lattice, whereas the gauge fields are treated as 3 x 3 matrices connecting these sites. In this sense, lattice QCD is a classical four-dimensional statistical physics system.
Calculations have been performed using the quenched approximation, which assumes that the fermion determinant (obtained after integrating over the fields) is independent of the gauge field. Although this approach omits the most computationally demanding part of a full QCD calculation,
a thorough determination of the quenched spectrum took almost 20 years. It was shown (4) that the quenched theory agreed with the experimental spectrum to approximately 10% for typical hadron masses and demonstrated that systematic differences were observed between quenched and two-flavor QCD beyond that level of precision (4, 5).
Including the effects of the light sea quarks has dramatically improved the agreement between experiment and lattice QCD results. Five years ago, a collaboration of collaborations (6) produced results for many physical quantities that agreed well with experimental results. Thanks to continuous progress since then, lattice QCD calculations can now be performed with light sea quarks whose masses are very close to their physical values (7) (though in quite small volumes). Other calculations, which include these sea-quark effects in the light hadron spectrum, have also appeared in the literature (8–16). However, all of these studies have neglected one or more of the ingredients required for a full and controlled calculation. The five most important of those are, in the order that they will be addressed below:
The inclusion of the up (u), down (d), and strange (s) quarks in the fermion determinant with an exact algorithm and with an action whose universality class is QCD. For the light hadron spectrum, the effects of the heavier charm, bottom, and top quarks are included in the coupling constant and light quark masses.
A complete determination of the masses of the light ground-state, flavor nonsinglet mesons and octet and decuplet baryons.
Three of these are used to fix the masses of the isospin-averaged light (m_ud) and strange (m_s) quark masses and
the overall scale in physical units.
...
...
Controlled interpolations and extrapolations of the results to physical mud and ms (or eventually directly simulating at these mass values). Although interpolations to physical m_s, corresponding to M_K 495 MeV, are straightforward, the extrapolations to the physical value of mud, corresponding to M? 135 MeV, are difficult. They need computationally intensive calculations, with M? reaching down to 200 MeV or less.
Controlled extrapolations to the continuum limit, requiring that the calculations be performed at no less than three values of the lattice spacing, in order to guarantee that the scaling region is reached.
Our analysis includes all five ingredients listed above, thus providing a calculation of the light hadron spectrum with fully controlled systematics as follows.Owing to the key statement from renormalization group theory that higher-dimension, local operators in the action are irrelevant in the continuum limit, there is, in principle, an unlimited freedom in choosing a lattice action. There is no consensus regarding which action would offer the most cost-effective approach to the continuum limit and to physical mud. We use an action that improves both the gauge and fermionic sectors and heavily suppresses nonphysical, ultraviolet modes (19). We perform a series of 2 + 1 flavor calculations;
that is, we include degenerate u and d sea quarks and an additional s sea quark. We fix m_s to its approximate physical value. To interpolate to the physical value, four of our simulations were repeated with a slightly different m_s. We vary m_ud in a range that extends down to M 190 MeV.
QCD does not predict hadron masses in physical units:
Only dimensionless combinations (such as mass ratios) can be calculated. To set the overall physical scale, any dimensionful observable can be used. However, practical issues influence this choice. First of all, it should be a quantity that can be calculated precisely and whose experimental value is well known. Second, it should have a weak dependence on m_ud, so that its chiral behavior does not interfere with that of other observables. Because we are considering spectral quantities here, these two conditions should guide our choice of the particle whose mass will set the scale. Furthermore, the particle should not decay under the strong interaction. On the one hand, the larger the strange content of the particle, the more precise the mass determination and the weaker the dependence on m_ud. These facts support the use of the ...?baryon, the particle with the highest strange content. On the other hand, ..
... Typical effective masses are shown in Fig. 1.
...
Fig. 1. Effective masses aM = log[C(t/a)/C(t/a + 1)], where C(t/a) is the correlator at time t, for ?, ?K, ?N,...
Fig. 2. Pion mass dependence of the nucleon (N) and for all three values of the lattice spacing. ...
...
Table 1. Spectrum results in giga–electron volts.nd bands are the experimental values with their decay widths. Our results are shown by solid circles. Vertical error bars represent our combined statistical (SEM) and systematic error estimates.
?, ?K, and ? have no error bars, because they are used to set the light quark mass, the strange quark mass and the overall scale, respectively. ...
...
Thus, our study strongly suggests that QCD is the theory of the strong interaction, at low energies as well, and furthermore that lattice studies have reached the stage where all systematic errors can be fully controlled. This will prove important in the forthcoming era in which lattice calculations will play a vital role in unraveling possible new physics from processes that are interlaced with QCD effects.
References and Notes
1. D. J. Gross, F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).
2. H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).
3. K. G. Wilson, Phys. Rev. D Part. Fields 10, 2445 (1974).
...
...
30. Computations were performed on the Blue Gene supercomputers at FZ Jülich and at IDRIS and on clusters at Wuppertal and CPT. ...
==endquote==
