rpp2010-rev-quark-masses - 1 QUARK MASSES Updated Jan 2010...

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– 1– QUARK MASSES Updated Jan 2010 by A.V. Manohar (University of California, San Diego) and C.T. Sachrajda (University of Southampton) A. Introduction This note discusses some of the theoretical issues relevant for the determination of quark masses, which are fundamental parameters of the Standard Model of particle physics. Unlike the leptons, quarks are confined inside hadrons and are not observed as physical particles. Quark masses therefore can- not be measured directly, but must be determined indirectly through their influence on hadronic properties. Although one often speaks loosely of quark masses as one would of the mass of the electron or muon, any quantitative statement about the value of a quark mass must make careful reference to the particular theoretical framework that is used to define it. It is important to keep this scheme dependence in mind when using the quark mass values tabulated in the data listings. Historically, the first determinations of quark masses were performed using quark models. The resulting masses only make sense in the limited context of a particular quark model, and cannot be related to the quark mass parameters of the Standard Model. In order to discuss quark masses at a fundamental level, definitions based on quantum field theory be used, and the purpose of this note is to discuss these definitions and the corresponding determinations of the values of the masses. B. Mass parameters and the QCD Lagrangian The QCD [1] Lagrangian for N F quark flavors is L = N F k =1 q k ( i /D m k ) q k 1 4 G μν G μν , (1) where /D = ( μ igA μ ) γ μ is the gauge covariant derivative, A μ is the gluon field, G μν is the gluon field strength, m k is the mass parameter of the k th quark, and q k is the quark Dirac field. After renormalization, the QCD Lagrangian Eq. (1) gives finite values for physical quantities, such as scattering amplitudes. Renormalization is a procedure that invokes a sub- traction scheme to render the amplitudes finite, and requires CITATION: K. Nakamura et al. (Particle Data Group), JPG 37 , 075021 (2010) (URL: http://pdg.lbl.gov) July 30, 2010 14:34
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– 2– the introduction of a dimensionful scale parameter μ . The mass parameters in the QCD Lagrangian Eq. (1) depend on the renor- malization scheme used to define the theory, and also on the scale parameter μ . The most commonly used renormalization scheme for QCD perturbation theory is the MS scheme. The QCD Lagrangian has a chiral symmetry in the limit that the quark masses vanish. This symmetry is spontaneously broken by dynamical chiral symmetry breaking, and explicitly broken by the quark masses. The nonperturbative scale of dynamical chiral symmetry breaking, Λ χ , is around 1 GeV [2]. It is conventional to call quarks heavy if m > Λ χ , so that explicit chiral symmetry breaking dominates ( c , b , and t quarks are heavy), and light if m < Λ χ , so that spontaneous chiral symmetry breaking dominates ( u , d and s quarks are light). The determination of light- and heavy-quark masses is considered separately in sections D and E below.
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