1 1 QUARK MASSES b,andtquarks are heavy, and the u, d and s quarks are light. Written by A. Manohar (University of California, San Diego). The computations for light quarks involve an expansion in mq/Λχ about the limit mq = 0, whereas for heavy quarks, they A. Introduction involve an expansion in Λχ/mq about mq = ∞. The corrections This note discusses some of the theoretical issues involved in are largest for the s and c quarks, which are the heaviest light the determination of quark masses. Unlike the leptons, quarks quark and the lightest heavy quark, respectively. are confined inside hadrons and are not observed as physical At high energies or short distances, nonperturbative effects particles. Quark masses cannot be measured directly, but must such as chiral symmetry breaking are unimportant, and one be determined indirectly through their influence on hadron can in principle analyze mass-dependent effects using QCD properties. As a result, the values of the quark masses depend perturbation theory to extract the quark mass values. The on precisely how they are defined; there is no one definition QCD computations are conventionally performed using the MS that is the obvious choice. Though one often speaks loosely of scheme at a scale µ Λχ,andgivetheMS “running” mass quark masses as one would of the electron or muon mass, any m(µ). The µ dependence of m(µ) at short distances can be careful statement of a quark mass value must make reference calculated using the renormalization group equations. to a particular computational scheme that is used to extract For heavy quarks, one can obtain useful information on the the mass from observations. It is important to keep this scheme quark masses by studying the spectrum and decays of hadrons dependence in mind when using the quark mass values tabulated containing heavy quarks. One method of calculation uses the in the data listings. heavy quark effective theory (HQET), which defines a HQET The simplest way to define the mass of a quark is by making quark mass mQ. Other commonly used definitions of heavy a fit of the hadron mass spectrum to a nonrelativistic quark quark masses such as the pole mass are discussed in Sec. C. model. The quark masses are defined as the values obtained QCD perturbation theory at the heavy quark scale µ = mQ can from the fit. The resulting masses only make sense in the be used to relate the various heavy quark masses to the MS limited context of a particular quark model. They depend on mass m(µ), and to each other. the phenomenological potential used, and on how relativistic For light quarks, one can obtain useful information on effects are modelled. The quark masses used in potential models the quark mass ratios by studying the properties of the light also cannot be connected with the quark mass parameters in pseudoscalar mesons using chiral perturbation theory, which the QCD Lagrangian. Fortunately, there exist other definitions utilizes the symmetries of the QCD Lagrangian Eq. (1). The of the quark mass that have a more general significance, though quark mass ratios determined using chiral perturbation theory they also depend on the method of calculation. The purpose of are those in a subtraction scheme that is independent of the this review is to explain the most important such definitions quark masses themselves, such as the MS scheme. and their interrelations. A more detailed discussion of the masses for heavy and B. Mass parameters and the QCD Lagrangian light quarks is given in the next two sections. The MS scheme The QCD Lagrangian for NF quark flavors is applies to both heavy and light quarks. It is also commonly used for predictions of quark masses in unified theories, and XNF for computing radiative corrections in the Standard Model. For L = q (iD/ − m ) q − 1 G Gµν , (1) k k k 4 µν k=1 this reason, we use the MS scheme as the standard scheme in reporting quark masses. One can easily convert the MS masses whereD=( / ∂ −igA ) γµ is the gauge covariant derivative, A µ µ µ into other schemes using the formulæ given in this review. is the gluon field, Gµν is the gluon field strength, mk is the mass th C. Heavy quarks parameter of the k quark, and qk is the quark Dirac field. The QCD Lagrangian Eq. (1) gives finite scattering amplitudes after The commonly used definitions of the quark mass for heavy renormalization, a procedure that invokes a subtraction scheme quarks are the pole mass, the MS mass, the Georgi-Politzer to render the amplitudes finite, and requires the introduction of mass, the potential model mass used in ψ and Υ spectroscopy, a dimensionful scale parameter µ. The mass parameters in the and the HQET mass. QCD Lagrangian Eq. (1) depend on the renormalization scheme The strong interaction coupling constant at the heavy quark used to define the theory, and also on the scale parameter µ. scale is small, and one can compute the heavy quark propagator The most commonly used renormalization scheme for QCD using QCD perturbation theory. For an observable particle such perturbation theory is the MS scheme. as the electron, the position of the pole in the propagator is the The QCD Lagrangian has a chiral symmetry in the limit definition of the particle mass. In QCD this definition of the that the quark masses vanish. This symmetry is spontaneously quark mass is known as the pole mass mP , and is independent of broken by dynamical chiral symmetry breaking, and explicitly the renormalization scheme used. It is known that the on-shell broken by the quark masses. The nonperturbative scale of quark propagator has no infrared divergences in perturbation theory [1], so this provides a perturbative definition of the dynamical chiral symmetry breaking, Λχ, is around 1 GeV. It quark mass. The pole mass cannot be used to arbitrarily high is conventional to call quarks heavy if m>Λχ, so that explicit accuracy because of nonperturbative infrared effects in QCD. chiral symmetry breaking dominates, and light if m<Λχ,so that spontaneous chiral symmetry breaking dominates. The c, The full quark propagator has no pole because the quarks are 2 2 ! 2 confined, so that the pole mass cannot be defined outside of Λ M(D)=mc+Λ+O . (2) perturbation theory. mc The MS running mass m(µ) is defined by regulating the − QCD theory using dimensional regularization, and subtracting This allows one to determine the mass difference mb mc = − 2 − the divergences using the modified minimal subtraction scheme. M(B) M(D)=3.4 GeV up to corrections of order Λ /mb 2 The MS scheme is particularly convenient for Feynman diagram Λ /mc. The extraction of the individual quark masses mb and computations, and is the most commonly used subtraction mc requires some knowledge of Λ. An estimate of Λusing scheme. QCD sum rules gives Λ=0.57 0.07 GeV [11]. The HQET The Georgi-Politzer mass mb is defined using the momentum masseswiththisvalueofΛaremb=4.74 0.14 GeV and space subtraction scheme at the spacelike point −p2 = mb 2 [2]. mc =1.4 0.2 GeV, where the spin averaged meson masses ∗ ∗ A generalization of the Georgi-Politzer mass that is often used (3M(B )+M(B))/4and(3M(D )+M(D))/4 have been used O 2 in computations involving QCD sum rules [3] is mb (ξ), defined to eliminate the spin-dependent (Λ /mQ) correction terms. 2 − 2 The errors reflect the uncertainty in Λ and the unknown spin- at the subtraction point p = (ξ +1)mP.QCDsumrules O 2 are discussed in more detail in the next section on light quark averaged (Λ /mQ) correction. The errors do not include any masses. theoretical uncertainty in the QCD sum rules, which could be Lattice gauge theory calculations can be used to obtain large. A quark model estimate suggests that Λ is the constituent heavy quark masses from ψ and Υ spectroscopy. The quark quark mass (≈ 350 MeV), which differs significantly from the masses are obtained by comparing a nonperturbative computa- sum rule estimate. In HQET, the 1/mQ corrections to heavy tion of the meson spectrum with the experimental data. The meson decay form-factors are also given in terms of Λ. Thus lattice quark mass values can then be converted into quark mass an accurate enough measurement of these form-factors could values in the continuum QCD Lagrangian Eq. (1) using lattice be used to extract Λ directly from experiment, which then perturbation theory at a scale given by the inverse lattice spac- determines the quark masses up to corrections of order 1/mQ. ing. A recent computation determines the b-quark pole mass to The quark mass mQ of HQET can be related to other quark be 5.0 0.2GeV,andtheMS mass to be 4.0 0.1 GeV [4]. mass parameters using QCD perturbation theory at the scale b Potential model calculations of the hadron spectrum also mQ. The relation between mQ and mQ(ξ) at one loop is [12] involve the heavy quark mass. There is no way to relate the αb (ξ)ξ+2 m =mb (ξ) 1+ s log (ξ +2) , (3) quark mass as defined in a potential model to the quark Q Q π ξ+1 mass parameter of the QCD Lagrangian, or to the pole mass. b Even in the heavy quark limit, the two masses can differ by where αs(ξ) is the strong interaction coupling constant in the momentum space subtraction scheme. The relation between m nonperturbative effects of order ΛQCD.