Review of lattice results concerning low energy particle physics

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The complete review is accessible here.


The introduction can be downloaded here.

We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f + (0), arising in semileptonic K → π transition at zero momentum transfer, as well as the decay constant ratio fK / fπ of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)L×SU(2)R and SU(3)L×SU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, for this review, we focus on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant αs.

Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and in constraining possible extensions of theories â "Beyond the Standard Model". As the LHC explores a new energy frontier, the importance of flavour physics will grow still further, in searches for signatures of new physics through precision measurements and/or in helping to unravel the theoretical framework behind direct discoveries of new particles. The major theoretical limitation arises from the precision with which strong interaction effects can be quantified and large-scale numerical simulations of lattice QCD provide the opportunity of computing these effects from first principles. In this paper we review the current status of lattice results for a variety of physical quantities in low-energy physics; our aim is to provide the answer to the frequently posed question "What is currently the best lattice value for a particular quantity?", in a way which is readily accessible to non-lattice-experts. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the u and d quarks. Not only are the systematic errors therefore different, but also the methodology used to estimate these uncertainties vary between collaborations. Below we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result which provides the "best" value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results.

The first edition of the FLAG review was published in 2011 [1]. It was limited to lattice results related to pion and kaon physics: light-quark masses (u-, d- and s-flavours), the form factor f + (0) arising in semileptonic K → π transitions at zero momentum transfer and the decay constant ratio fK / fπ, as well as their implications for the CKM matrix elements Vus and Vud. Furthermore, results were reported for some of the low-energy constants of SU(2)L×SU(2)R and SU(3)L×SU(3)R Chiral Perturbation Theory and the BK parameter of neutral kaon mixing. Results for all of these quantities have been updated in the present paper. Moreover, the scope of the present review has been extended by including lattice results related to D- and B-meson physics. We focus on B- and D-meson decay constants, form factors, and mixing parameters, which are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Last but not least, the present status of lattice results on the QCD coupling αs is also reviewed. Bottom- and charm-quark masses, though important parametric inputs to Standard-Model calculations, have not been covered in the present edition. They will be included in a future FLAG report.

Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a biannual basis. This effort is supplemented by our more frequently updated website here, where figures as well as pdf-files for the individual sections can be downloaded. The papers reviewed in the present edition have appeared before the closing date 30 November 2013.

Finally, we draw attention to a particularly important point. As stated above, our aim is to make lattice QCD results easily accessible to non-lattice-experts and we are well aware that it is likely that some readers will only consult the present paper and not the original lattice literature. We consider it very important that this paper is not the only one which gets cited when the lattice results which are discussed and analyzed here are quoted. Readers who find the review and compilations offered in this paper useful are therefore kindly requested to also cite the original sources. The bibliography at the end of this paper should make this task easier. Indeed we hope that the bibliography will be one of the most widely used elements of the whole paper.

FLAG enlargement

Upon completion of the first review, it was decided to extend the project by adding new phys- ical quantities and co-authors. FLAG became more representative of the lattice community, both in terms of the geographical location of its members and the lattice collaborations to which they belong. At the time a parallel effort had been carried out [2] [3]; the two efforts have now merged in order to provide a single source of information on lattice results to the particle-physics community.

The experience gained in managing the activities of a medium-sized group of co-authors taught us that it was necessary to have a more formal structure and a set of rules by which all concerned had to abide, in order to make the inner workings of FLAG function smoothly. The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and seven Working Groups (WG). The role of the Advisory Board is that of general supervision and consultation. Its members may interfere at any point in the process of drafting the paper, expressing their opinion and offering advice. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing up the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for criticisms and suggestions.

The most important internal FLAG rules are the following:

  • members of the AB have a 4-year mandate (to avoid a simultaneous change of all members at once, some of the current members of the AB will have a shorter mandate);
  • the composition of the AB will respect a geographical balance among the main areas in which lattice collaborations are active: one member will come from America, one from Asia/Oceania and one from Europe;
  • the mandate of regular members is not limited in time, but we expect that a certain rotation will happen naturally;
  • whenever a replacement becomes necessary this has to at least keep, and possibly improve, the balance in FLAG;
  • in all working groups the three members must belong to three different lattice collaborations;[4]
  • a paper is in general not reviewed (nor colour-coded) by an author;
  • lattice collaborations not represented inside FLAG will be asked to check whether the colour coding of their calculation is correct.

Currently, the complete list of FLAG members and their Working Group assignment is:

  • Advisory Board (AB)
S. Aoki, C. Bernard, C. Sachrajda
  • Editorial Board (EB)
G. Colangelo, H. Leutwyler, T. Vladikas, U. Wenger
  • Working Groups (WG)
    • u, d and s quark masses
      L. Lellouch, T. Blum, V. Lubicz
    • Vus,Vud
      A. Jüttner, T. Kaneko, S. Simula
    • LEC
      S. Dürr, H. Fukaya, S. Necco
    • BK
      H. Wittig, J. Laiho, S. Sharpe
    • αs
      R. Sommer, R. Horsley, T. Onogi
    • f_{B_{(s)}}, f_{D_{(s)}}, BB
      A. El Khadra, Y. Aoki, M. Della Morte
    • B(s), D semileptonic and radiative decays
      R. Van de Water, E. Lunghi, C. Pena, J. Shigemitsu[5]


  1. G. Colangelo et al., Review of lattice results concerning low energy particle physics, Eur. Phys. J. C71 (2011) 1695, arXiv:1011.4408.
  2. J. Laiho, E. Lunghi and R. S. Van de Water, Lattice QCD inputs to the CKM unitarity triangle analysis, Phys. Rev. D81 (2010) 034503, arXiv:0910.2928.
  3. J. Laiho, E. Lunghi and R. Van de Water, 2+1 Flavor Lattice QCD Averages,
  4. The WG on semileptonic B decays has four members, but only three of them belong to lattice collaborations;
  5. J. Shigemitsu has withdrawn from FLAG, immediately after completion of the first version of the present paper (arXiv:1310.8555), of which she is a co-author. She is listed here in recognition of her full involvement in the review of B(s) and D semileptonic and radiative decays, as well as for her valuable contribution of the whole FLAG effort.

Quality criteria

The section on the quality criteria can be downloaded here.

The masses of the light quarks

The section on the light-quark masses can be downloaded here.

V(ud) and V(us)

The section on V(ud) and V(us) can be downloaded here.

Low-energy constants

The section on the Low Energy Constants can be downloaded here.

Kaon B-parameter

The section on the Kaon B-parameter can be downloaded here.

D-meson decay constants and form factors

The section on the D-meson decay constants and form factors can be downloaded here.

B-meson decay constants, mixing parameters, and form factors

The section on the B-meson decay constants, mixing parameters, and form factors can be downloaded here.

The strong coupling

The section on the strong coupling αs can be downloaded here.


The glossary can be downloaded here.


Notes to the various sections can be downloaded here.

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