60. Quark Masses

1

60. Quark Masses
Revised August 2023 by R.M. Barnett (LBNL), L.P. Lellouch (CNRS & Aix-Marseille U.) and A.V.
Manohar (UC San Diego).

60.1 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 cannot be measured directly, but must be determined indirectly through their influenceon hadronic properties. Although one often speaks loosely of quark masses as one would of the massof the electron or muon, any quantitative statement about the value of a quark mass must makecareful reference to the particular theoretical framework that is used to define it. It is importantto 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. These
are usually called constituent quark masses and are of order 350 MeV for the u and d quarks.
Constituent quark masses model the effects of dynamical chiral symmetry breaking discussed below,
and are not directly related to the quark mass parameters mq of the QCD Lagrangian of Eq. (60.1).
The resulting masses only make sense in the limited context of a particular quark model, and cannot
be related to the quark mass parameters, mq , of the Standard Model. In order to discuss quark
masses at a fundamental level, definitions based on quantum field theory must be used, and the
purpose of this note is to discuss these definitions and the corresponding determinations of thevalues of the masses.

60.2 Mass parameters and the QCD Lagrangian
The QCD [1] Lagrangian is
L=

X

q (iD
/ − mq ) q − 12 tr Gµν Gµν ,(60.1)

q=u,d,s,...,twhere the sum runs over the quark flavors u, d, s, c, b and t. D
/ = (∂µ − igAµ ) γ µ is the gaugeicovariant derivative, Aµ is the SU(3)-valued gluon field, Gµν = g [Dµ , Dν ] is the gluon field strength,
mq is the mass parameter of quark flavor q, and q is the quark Dirac field. After renormalization, the
QCD Lagrangian Eq. (60.1) gives finite values for physical quantities, such as scattering amplitudes.
Renormalization is a procedure that invokes a subtraction scheme to render the amplitudes finite,and requires the introduction of a dimensionful scale parameter µ. The mass parameters in the QCD
Lagrangian Eq. (60.1) depend on the renormalization 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. Thissymmetry is spontaneously broken by dynamical chiral symmetry breaking, and explicitly broken
by the quark masses. The nonperturbative scale of dynamical chiral symmetry breaking, Λχ , isaround 1 GeV [2]. It is conventional to call quarks heavy if mq > Λχ , so that explicit chiral symmetry
breaking dominates (c, b, and t quarks are heavy), and light if mq < Λχ , so that spontaneous chiralsymmetry breaking dominates (the u and d are light and the s is considered to be light when using
SU(3)L ×SU(3)R chiral perturbation theory). The determination of light- and heavy-quark masses
is considered separately in Sec. 60.5 and Sec. 60.6 below.
At high energies or short distances, nonperturbative effects, such as chiral symmetry breaking,become small and one can, in principle, determine quark masses by analyzing mass-dependent
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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60. Quark Masses

2

effects using QCD perturbation theory. Such computations are conventionally performed using the
MS scheme at a scale µ  Λχ , and give the MS “running” mass m(µ). We use the MS scheme whenreporting quark masses; one can readily convert these values into other schemes using perturbation
theory.
The µ dependence of m(µ) at short distances can be calculated using the renormalization group(RG) equation,µ2

dm (µ)= −γ(αs (µ)) m (µ) ,dµ2(60.2)

where γ is the anomalous dimension which is now known to four-loop order in perturbation theory[3] [4]. αs is the coupling constant [1] in the MS scheme. Defining the expansion coefficients γr by
γ (αs ) ≡

∞
X



γr

r=1

αs
4π

r

,

(60.3)

the first four coefficients are given by
γ1 = 4,202 20NL
γ2 =
−
,
3
9

2216 160
140 2
γ3 = 1249 + −−ζ (3) NL −
N ,
27
3
81 L
4603055 135680+
ζ (3) − 8800ζ (5)
γ4 =
27
 162

91723 34192
18400
+ −
−
ζ (3) + 880ζ (4) +ζ (5) NL27
995242 800
160++ζ (3) −
ζ (4) NL2
243
9
3


332 64+ −+ ζ (3) NL3 ,243 27
where NL is the number of active light quark flavors at the scale µ, i.e. flavors with masses ≤ µ,and ζ is the Riemann zeta function (ζ(3) ' 1.2020569, ζ(4) ' 1.0823232, and ζ(5) ' 1.0369278).
Eq. (60.2) must be solved in conjunction with the RG equation for αs (µ) given in [1]. In addition,
as the renormalization scale crosses quark mass thresholds one needs to match the scale dependenceof m below and above the threshold. There are finite threshold corrections; the necessary formulae
can be found in Ref. [5].

60.3 Lattice QCD
Ab initio lattice QCD calculations of the fundamental parameters of QCD, i.e. the coupling
constant and quark masses (except for the top-quark mass), has become a precision science (seealso the sections on Lattice Quantum Chromodynamics and on Quantum Chromodynamics inthis Review). Here we only briefly recall those features which are required for the determinationof quark masses. For more details on lattice QCD, please see the section on Lattice Quantum
Chromodynamics.
The inputs into lattice calculations are bare, dimensionless versions of the fundamental parameters of QCD: the bare strong coupling and the bare quark masses in lattice units, i.e. multipliedby the lattice spacing a, the distance between neighboring points of the lattice. The physical values of the quark masses are determined by computing as many physical quantities as there are
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60. Quark Masses

3

parameters, including one for the gauge coupling. In practice, the computations are performed for
multiple lattice spacings, determined by the bare coupling via dimensional transmutation, and anumber of bare quark masses. After renormalizing these masses and converting them to physicalunits, the resulting renormalized masses are interpolated to their physical values. This is achievedby requiring that the computed physical quantities reproduce their measured values as the a → 0
limit to our continuum world is taken. Such a tuning of fundamental parameters is required in any
regularization of QCD: lattice QCD is simply a tool to perform QCD calculations in the low-energy,
nonperturbative domain.
As discussed in the previous paragraph, one must renormalize the bare quark masses of lattice
QCD simulations. The values of these bare masses refer to a particular discretization of QCD withthe lattice spacing as the ultraviolet cut-off. In order for the resulting physical masses to be useful inphenomenological applications, it is necessary to give their values in some standard renormalization
scheme such as MS. Provided that both the ultraviolet cut-off a−1 and the renormalization scaleµ are much greater than ΛQCD , the bare masses can be renormalized in perturbation theory.
However, in order to avoid uncertainties due to the unknown higher-order coefficients in lattice
perturbation theory, most results obtained recently use nonperturbative renormalization. This
procedure relates the bare masses to those defined in renormalization schemes which can be realized
directly in lattice and continuum QCD, e.g. those obtained from quark and gluon Green functionsat specified momenta in the Landau gauge [6] or those defined using finite-volume techniques andthe Schrödinger functional [7], but not MS that is only defined for dimensional regularization. Thesemethods require µ  ΛQCD so that unwanted (nonperturbative) corrections proportional to inverse
powers of µ, which appear in some approaches, remain small corrections that can be identified andremoved. This condition is also necessary so that matching to other schemes can be performed
reliably in perturbation theory. Moreover these methods require a−1  µ so that cutoff effects are
small enough to be extrapolated away when the continuum limit is taken. The conversion to the
MS scheme is then performed using continuum perturbation theory, which is more readily obtained
to higher orders and is usually better behaved than its lattice counterpart.
It is important to note that the final renormalized values for the quark masses must be the samein the continuum limit for any valid discretization of QCD and for any sensible choice of the physicalquantities used for calibration, as long as the calculation is performed with a sufficient number of
sea-quark flavors and with the relevant electromagnetic and strong-isospin breaking corrections forthe claimed precision. It is also worth noting that issues surrounding the renormalization of quark
masses disappear when considering pairwise ratios of these masses in QCD alone. Indeed, if thesame renormalization scheme and scale are implemented, also for the definition of QCD in theabsence or electromagnetism,1 QCD renormalization factors are identical for all quark flavors, andthese factors therefore cancel exactly in quark-mass ratio 2 . In particular, this means that theseratios are scheme and scale independent up to possible QED corrections. Moreover these ratios
may suffer less from the uncertainties in the determination of the lattice scale, in particular incases where the quantities used to fix the two quarks masses depend on these masses in a similarway, because these ratios are dimensionless. Thus, quark-mass ratios are often determined withsignificantly higher precision using lattice QCD than are the individual masses.
The determination of quark masses using lattice simulations is well established and the current emphasis is on the reduction and control of uncertainties. With better methods, improved
algorithms and access to more powerful computing resources, the precision of the results has improved significantly in recent years. Vacuum polarization effects are included with Nf = 2, 2 + 1
or Nf = 2 + 1 + 1 flavors of sea quarks. The number 2 here indicates that the up and down quarks
1

Since QCD without QED is not the real world, it makes a difference, for example, which hadron mass is used totune a quark mass and the QED correction will differ for different choices.
2
The same remains true in the presence of QED if the two quarks in the ratio have the same charge.
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460. Quark Masses

are degenerate. The first +1 corresponds to the inclusion of strange sea-quark effects and thesecond +1 to those of the charm. Simulations with 2 + 1 and 2 + 1 + 1 flavors represent controlled
approximations to physical QCD at the low energies considered for quark mass determinations, up
to corrections of O((ΛQCD /mc )2 /Nc ) and O((ΛQCD /mb )2 /Nc ), respectively. This is not the case forsimulations with Nf = 2 or in which vacuum polarization effects are treated as a mean field (thisis the so-called quenched approximation) and results obtained in those frameworks will not enter
the discussion here.
Particularly pleasing is the observation that different formulations of lattice QCD and different
approaches, with different systematic uncertainties, yield results which are largely consistent witheach other. This gives us broad confidence in the estimates of the systematic errors. As theprecision of the results approaches (or even exceeds in some cases) 1%, isospin breaking effects,
including electromagnetic corrections need to be considered and this is beginning to be done as will
be discussed below. In particular, a reliable estimate of these effects is required for determining
the individual u and d quark masses.
Members of the lattice QCD community have organized a Flavour Lattice Averaging Group(FLAG) which, against stated quality criteria, critically reviews quantities computed in lattice
QCD that are relevant to flavor physics and presents its view of the current status of the results.
This includes the determination of quark masses. The latest edition reviews lattice results thatappeared in print before April 30th 2021 [8]. Since that deadline and the writing of the previous
edition of the present review, the only new results on quark masses that have been published arethose of Ref. [9].

60.4 Averaging procedure for lattice QCD results
In this review we present summary numbers for each quark mass which are solely based on
averages of lattice results. Those summary numbers differ from those presented in the particle
listings, for three reasons. The first is that they do not include non-lattice estimates of the quark
masses. The second is that they may differ slightly in the lattice results retained, as made explicit
here and in the listings. The third difference comes from the averaging procedures, which aredescribed in the listings and below. In the listings, summary numbers are obtained from averagesof the individual quark-mass estimates, weighted by theirp total errors. Moreover, the resulting
one-standard-deviation errors are rescaled by the usual χ2 /dof factor, which is further multipliedby 1.645 to obtain the uncertainty corresponding to a 90% confidence interval. Nevertheless, theaverages presented here and in the listings are fully consistent, with comparable uncertainties,
signalling the robustness of the two combination procedure.
The results on which the lattice summary numbers presented here are those used in the FLAG
reviews, supplemented by any new calculations that satisfy their selection criteria. The methodology that we use is also close to that of FLAG, but here we make an attempt to separate uncertainties
into their statistical and systematic components, to provide some idea of what the relative weightof each is for different quark masses. In some cases, also, slightly different choices are also made.
Thus, agreement between the results obtained is a further sign of the robustness of the combination
procedures used here, as well as in FLAG.
We now lay out our averaging procedure. For each of mud , ms or ms /mud , and for each of
Nf = 2 + 1 or Nf = 2 + 1 + 1, we perform averages of the relevant, individual lattice resultswith weights that include the sum in quadrature of the statistical and systematic uncertainties.
The weights also account for correlations by assuming that they are 100% on any uncertainty
components that may be correlated. This procedure yields a central value and a total perror foreach of the quantities of interest. The total uncertainty is rescaled by the usual factor of χ2 /dof
if that factor is larger than 1, where χ2 is the one corresponding to the weighted average procedure.
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60. Quark Masses

5

To isolate the statistical component of the uncertainty on any given average, like for the total
uncertainty we use the standard error formula for correlated data, this time only taking into accountthe statistical uncertainties on the different collaborations’ results. If the total error has been
rescaled, this statistical uncertainty is multiplied by the same factor. The candidate systematic
error on an average is then obtained by subtracting in quadrature the statistical uncertainty fromthe total one. For each quantity, the final systematic error on the average is the largest of two
uncertainties: the candidate systematic error and the minimum of the systematic uncertainties
on the individual lattice determinations. This guarantees that the final systematic error is nosmaller than the one of any given calculation. We then repeat this whole procedure to average
the Nf = 2 + 1 and Nf = 2 + 1 + 1 averages, for reasons discussed below. In cases where theerror separation is not explicit in a particular collaboration’s result, we assume that the quoted
error is statistical and take, as the minimum of systematic error, the smallest non-vanishing one
among those of the other individual results. While this procedure may change the balance between
statistical and systematic errors, it does change the average (which is weighted by the total errorsof the individual entries and the correlations between them) and does not decrease the total error.
When quantities involving mc or mb are averaged, we consider only the averages of Nf = 2 + 1 + 1
calculations, for reasons also explained below.
The values of the individual results used in the averages obtained below can be found in thequark properties listings of this Review.

60.5 Light quarks
In this section we review the determination of the masses of the light quarks u, d and s fromlattice simulations and then discuss the consequences of the approximate chiral symmetry.
60.5.1 Lattice QCD results
The most reliable determinations of the strange quark mass, ms , and of the up and downquark masses, mu and md , are obtained from lattice simulations. This is reflected in part by theuncertainties of individual determinations of these masses given in this Review’s listings of quark
properties: lattice results typically have smaller error bars than phenomenological extractions. As
explained in the previous section, the simulations are generally performed in the isospin-symmetriclimit of QCD, i.e. with degenerate up and down quarks (mu = md ) and the electromagnetic coupling
α set to zero. Thus, it is initially mud = (mu + md )/2 which is obtained from the computations.
The numerical values of these masses depend on the choice of renormalization scale and scheme. 3
Below we discuss the derivation of mu and md separately, but we start by briefly presenting ourestimates for mud , ms and their ratio, using the latest lattice results. FLAG [8] bases its summary
numbers on references that the authors consider to have the most reliable estimates of systematic
uncertainties, i.e. [12–18] for Nf = 2 + 1 and [19–24] for Nf = 2 + 1 + 1 flavors of sea quarks. Notethat the Nf = 2 + 1 + 1 results of Ref. [9] did not make the FLAG deadline. In the estimates given
below, we use the fact that Ref. [9] supersedes Ref. [22].
Applying the averaging procedure described in the previous section and the results of Refs.
[12–18], for Nf = 2 + 1 we obtain mud = (3.382 ± 0.026 ± 0.040[±0.050]) MeV, ms = (92.24 ±0.24 ± 1.00[±1.03]) MeV and (ms /mud ) = 27.42 ± 0.05 ± 0.12[±0.12]. These numbers are mud =3

If QED is not included, then quark masses also depend on the conditions used to match isospin-symmetric
QCD to the low energy effective theory of the Standard Model, which includes QCD and QED with mu 6= md . Thosematching conditions, which define a matching scheme, are quite similar in the calculations reported on below. Moreover, they have been designed to minimize the impact of isospin-breaking effects on isospin-symmetric combinationsof quark masses, such mud and ms here, and mc and mb below. Thus, as does FLAG [8], we generally assume thatany remaining differences between schemes are within the quoted errors. Such an assumption is certainly true formc (mc ) [10], where QED effects have been studied explicitly. On the other hand, in the case of mb /mc , where theeffects of QED have also been studied explicitly [11] and where uncertainties are less than 0.3%, these effects are nolonger negligible.
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60. Quark Masses

6(3.421 ± 0.035 ± 0.049[±0.060]) MeV, ms = (92.93 ± 0.51 ± 0.37[±0.063]) MeV and (ms /mud ) =27.25 ± 0.03 ± 0.07[±0.08] using the latest Nf = 2 + 1 + 1 calculations [9, 19–21, 23, 24]. The masses
are given in the MS scheme at a renormalization scale of 2 GeV. In these averages, the first error
is statistical, the second systematic and the third one in brackets is the sum in quadrature of thefirst two. Also, QED corrections have been accounted for only to the extent that they have been
removed from the hadron mass inputs phenomenologically.
A few comments are in order concerning the compatibility of the results which enter these
averages. We begin with the case of Nf = 2 + 1. For mud the results show a good degree ofcompatibility with a χ2 /dof = 5.7/4 and the final systematic uncertainty is given by the one ofRef. [17]. The level of compatibility is similar in the case of ms and the final systematic uncertainty
is also given by the one of Ref. [17]. For ms /mud the results are even more compatible and here thefinal systematic error is derived from the averaging procedure. In the case of Nf = 2+1+1, there are
only two results for mud [9,20] and their level of compatibility is poor, with χ2 /dof = 6.4/1. This isdue to the value of Ref. [9], mud = (3.636 ± 0.066 ± 0.059) MeV, which is significantly larger than allother determinations, including those obtained in Nf = 2 + 1 determinations. The scenario repeats
for ms , but is diluted by the fact that there are four results [9, 19–21] which enter the average.
However, compatibility is recovered for ms /mud , suggesting a problem in the determination of
the renormalization constants or the lattice spacing in at least one of the calculations. Since thedifferent collaborations use different formulations of lattice QCD, the (relatively) small variations of
the results between the groups provides important information about the reliability of the estimates.
Furthermore, the conservative nature of our method for estimating uncertainties is confirmed by
the excellent agreement of the averages obtained for Nf = 2 + 1 and Nf = 2 + 1 + 1 calculations.
Despite being reported in the MS scheme at a renormalization scale of 2 GeV, the schemes insimulations with 2 + 1 and 2 + 1 + 1 sea-quark flavors actually differ. This is because the former are
renormalized with NL = 3 and the latter with NL = 4. Thus for a comparison one should convert
the results to the same scheme. That is not the case for (ms /mud ), where renormalization factors
cancel in the absence of electromagnetic corrections. The conversion of the Nf = 2 + 1 results to
the NL = 4 scheme can be performed, for instance, by running them down to the charm thresholdin the NL = 3 theory, matching the results to the NL = 4 theory at that scale, and running themback up to 2 GeV with four active flavors. Such a conversion leads to an upward shift in the valuesof the quark masses of only around 0.3%, well within the quoted uncertainties on the quark masses
themselves. Quark-mass ratios are not concerned by this upward shift, because it is identical forthe numerator and denominator. There are also nonperturbative contributions to this procedure, ofO((ΛQCD /mc )2 /Nc ) ∼ 2%. However lattice QCD studies of these effects have shown that they are
typically an order of magnitude smaller on hadronic quantities or on ΛQCD [25, 26]. Given that the
total errors on the Nf = 2+1 results for mud and ms are larger than 1%, we consider these matchingeffects negligible. In addition, for mud in particular there are six Nf = 2 + 1 calculations in which
all sources of uncertainty have been accounted for, while there are only two for Nf = 2 + 1 + 1.
Thus, we average the averages from the two frameworks in the same way as above (see Sec. 60.4),
yielding as a final lattice QCD estimate in the MS scheme at µ = 2 GeV in the NL = 4 theory:
mud = (3.397 ± 0.021 ± 0.040[±0.045]) MeV,

(60.4)

ms = (92.74 ± 0.22 ± 0.49[±0.54]) MeV,

(60.5)

ms= 27.30 ± 0.03 ± 0.07[±0.08].
mud(60.6)

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7

60. Quark Masses

In these averages, the first error is statistical, the second systematic and the third one in brackets
is the sum in quadrature of the first two. QED corrections have been accounted for only to theextent that they have been removed from the hadron mass inputs phenomenologically. It is worth
noting that the result for rs in Eq. (60.6) is identical to the ratio of ms and mud from Eqs. (60.4)
and (60.5), within the four digits for rs given in Eq. (60.6), albeit with a significantly smaller total
uncertainty.
To obtain the individual values of mu and md requires the introduction of isospin breaking
effects, including electromagnetism. This is now being done completely using lattice field theory,
albeit neglecting electromagnetic effects in the sea in most cases (for an exception, see Ref. [27]
2 = M2 − M2
which includes a calculation of the valence and sea QED contributions to ∆MK
K0
K+that is critical for determining md − mu ). The effect of neglecting QED contributions from thesea on the u and d quark masses has been estimated in Ref. [28] to induce a contribution to
the uncertainty that ranges from about 3% in mu /md to less than 1% in md . FLAG has also
reviewed results for these masses [8]. Again, they separate results obtained from Nf = 2 + 1 and
Nf = 2 + 1 + 1 simulations. For the former, they use the results of Ref. [28] and for the latter,
those of Refs. [20, 29, 30].
To obtain our estimates, we proceed as for the light quark masses above. For Nf = 2 + 1, ourestimates are the results of the only calculation in which all sources of uncertainty are accounted
for [28]. These are mu = (2.27 ± 0.06 ± 0.06[±0.09]) MeV, md = (4.67 ± 0.06 ± 0.06[±0.09]) MeV
and mu /md = (0.485 ± 0.011 ± 0.016[±0.020]). For Nf = 2 + 1 + 1, they rely on Refs. [20, 29, 30].
Our estimates are mu = (2.14 ± 0.04 ± 0.07[±0.08]) MeV, md = (4.70 ± 0.03 ± 0.04[±0.05]) MeV andmu /md = (0.460 ± 0.008 ± 0.018[±0.020]). These masses are given the the MS scheme at 2 GeV in
the NL = 3 and 4 theories, respectively. Given the size of the uncertainties on these averages, tiny
corrections due to QED effects in the renormalization constants have been neglected.
In obtaining the Nf = 2 + 1 + 1 averages, there is a significant χ2 /1 = 4.7 tension between thetwo results for mu [20, 29] and a χ2 /1 = 3.4 one between those for mu /md [29, 30]. These tensions
are accounted for by the usual rescaling of uncertainties. However, agreement is found between thetwo results for md [20, 29]. Moreover, at the level of our conservative uncertainties, agreement isfound with the Nf = 2 + 1 results.
Again, given the small number of results and the few percent uncertainties, we average theNf = 2 + 1 and Nf = 2 + 1 + 1 results, as explained in Sec. 60.4, to obtain our final estimates:
mu = (2.20 ± 0.04 ± 0.06[±0.07]) MeV,

(60.7)

md = (4.69 ± 0.03 ± 0.04[±0.05]) MeV,mur≡
= (0.473 ± 0.007 ± 0.016[±0.017]),md(60.8)(60.9)

where the three uncertainties have the same meaning as described after Eq. 60.6. r in Eq. (60.9) is
the combination of direct lattice computations of that ratio in which some systematics cancel. The
averages are given in the MS scheme at 2 GeV in the NL = 4 theory and, as above, tiny corrections
due to QED effects in the renormalization constants have been neglected. Within one standard
deviation it agrees with the ratio of mu over md from Eqs. (60.7) and (60.8).
It is also worth noting that mu differs from zero by more than 30 times the quoted error, making
a scenario in which mu = 0 very unlikely. This is important because there would be no strong CP
problem if mu were to vanish.
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860. Quark Masses

60.5.2 Chiral Perturbation Theory
For light quarks, one can use the techniques of chiral perturbation theory [31–33] to extract
quark mass ratios. The mass term for light quarks in the QCD Lagrangian isΨ M Ψ = Ψ L M Ψ R + Ψ R M † ΨL ,(60.10)where M is the light quark mass matrix,




mu 0
0


M =  0 md 0  ,0
0 ms(60.11)

Ψ = (u, d, s), and L and R are the left- and right-chiral components of Ψ given by ΨL,R = PL,R Ψ ,PL = (1−γ5 )/2, PR = (1+γ5 )/2. The mass term is the only term in the QCD Lagrangian that mixes
left- and right-handed quarks. In the limit M → 0, there is an independent SU(3)×U(1) flavor
symmetry for the left- and right-handed quarks. The vector U(1) symmetry is baryon number; theaxial U(1) symmetry of the classical theory is broken in the quantum theory due to the anomaly.
The remaining Gχ = SU(3)L × SU(3)R chiral symmetry of the QCD Lagrangian is spontaneously
broken to SU(3)V , which, in the limit M → 0, leads to eight massless Goldstone bosons, the π’s,
K’s, and η.
The symmetry Gχ is only an approximate symmetry, since it is explicitly broken by the quark
mass matrix M . The Goldstone bosons acquire masses which can be computed in a systematic
expansion in M , in terms of low-energy constants, which are unknown nonperturbative parametersof the effective theory, and are not fixed by the symmetries. One treats the quark mass matrix
M as a uniform, external field that transforms under Gχ as M → LM R† , where ΨL → LΨLand ΨR → RΨR are the SU(3)L and SU(3)R transformations, and writes down the most general
Lagrangian invariant under Gχ . Then one sets M to its given constant value Eq. (60.11), which
implements the symmetry breaking. To first order in M one finds that [34]m2π0 = B (mu + md ) ,m2π± = B (mu + md ) + ∆em ,m2K 0 = m2 0 = B (md + ms ) ,K

m2K ± = B (mu + ms ) + ∆em ,1m2η = B (mu + md + 4ms ) ,3(60.12)

with two unknown constants B, related to the light quark condensate, and ∆em , the electromagnetic
mass difference. From Eq. (60.12), one can determine the quark mass ratios [34]mu 2m2π0 − m2π+ + m2K + − m2K 0== 0.56,md
m2K 0 − m2K + + m2π+
ms m2K 0 + m2K + − m2π+=
= 20.2,
md m2K 0 + m2π+ − m2K +(60.13)

to lowest order in chiral perturbation theory, with an error which will be estimated below. Since themass ratios extracted using chiral perturbation theory use the symmetry transformation propertyof M under the chiral symmetry Gχ , it is important to use a renormalization scheme for QCD that
does not change this transformation law. Any mass-independent subtraction scheme such as MS is
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60. Quark Masses

9suitable. The ratios of quark masses are scale independent in such a scheme (up to electromagnetic
corrections), and Eq. (60.13) can be taken to be the ratio of MS masses. Chiral perturbation theory
cannot determine the overall scale of the quark masses, since it uses only the symmetry propertiesof M , and any multiple of M has the same Gχ transformation law as M .
Chiral perturbation theory is a systematic expansion in powers of the light quark masses. The
typical expansion parameter is m2K /Λ2χ ∼ 0.25 if one uses SU(3) chiral symmetry, and m2π /Λ2χ ∼ 0.02
if instead one uses SU(2) chiral symmetry. Electromagnetic effects at the few percent level also
break SU(2) and SU(3) symmetry. The mass formulæ Eq. (60.12) were derived using SU(3) chiralsymmetry, and are expected to have approximately a 25% uncertainty due to second order corrections. This estimate of the uncertainty yields results consistent with the lattice results summarizedin Eq. (60.5)–(60.6).
There is a subtlety which arises when one tries to determine quark mass ratios at second orderin chiral perturbation theory. The second order quark mass term [35]


M†

−1

det M †

(60.14)(which can be generated by instantons) transforms in the same way under Gχ as M . Chiral perturbation theory cannot distinguish between M and (M † )−1 det M † ; one can make the replacement
M → M (λ) = M + λM (M † M )−1 det M † in the chiral Lagrangian,M (λ) = diag (mu (λ), md (λ), ms (λ))(60.15)= diag (mu + λmd ms , md + λmu ms , ms + λmu md ) ,and leave all observables unchanged.
The combination

where
Q2 =

mu
md

2

+

m2s − m2ud,
m2d − m2u

1
Q2



ms
md

2

mud =

=1

(60.16)1(mu + md ) ,2

is insensitive to the transformation in Eq. (60.15). Eq. (60.16) gives an ellipse in the mu /md -ms /mdplane. The ellipse is well-determined by chiral perturbation theory, but the location on the ellipse,
and the absolute normalization of the quark masses, have larger uncertainties.
A leading-order result for Q in SU(3) chiral perturbation theory can be derived using Eq. (60.12)
and the values for the relevant meson masses. This result actually holds to next-to-leading order,
NLO
yielding Q = 24.3. Phenomenologically, the preferred way to determine Q is from η → 3π decay,
giving the smaller value Q = 22.1(7) [36]. Lattice QCD collaborations have also reported determinations of Q. Using Nf = 2 + 1 simulations, Ref. [28] obtains Q = 23.4 ± 0.6 and Ref. [29]determines Q = 23.8 ± 1.1 with Nf = 2 + 1 + 1 simulations, results which are fully compatible.
These results are also compatible with Q = 23.4 ± 0.5, obtained from Q2 = (rs2 − 1)(1 + r)/[4(1 − r)]and from our lattice averages for rs and r from Eqs. (60.6) and (60.9), respectively. Given the sizeof the uncertainties on the lattice results for Q, it is safe to neglect effects due to the definition of
isosymmetric QCD (see footnote 3). On the whole, the lattice results are larger than the one fromphenomenology [36]. This difference could be due to surprisingly large corrections to the NLO
substitution of meson masses by Q in the phenomenological determination, as suggested by theauthors of [36].
The absolute normalization of the quark masses cannot be determined using chiral perturbation
theory. Other methods, such as lattice simulations discussed above, or spectral function sum
rules [37, 38] for hadronic correlation functions reviewed next, are necessary.
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60. Quark Masses

10

60.5.3 Sum rules
Sum rule methods have been used extensively to determine quark masses and for illustration
we briefly discuss here their application to hadronic τ decays [39]. Other applications involve very
similar techniques.
■✁ s

❈✷

♠✷✧

❘ s

❈✶
♠✷✦

✹♠✷✦

Figure 60.1: The analytic structure of Π(s) in the complex s-plane. The contours C1 and C2
are the integration contours discussed in the text, and the integral over the closed contour C1 + C2
vanishes. m2τ has not been drawn to scale; m2τ ∼ 40(4m2π ).
The experimentally measured quantity is Rτ ,dRτ dΓ/ds (τ − → hadrons + ντ (γ))=
ds
Γ (τ − → e− ν e ντ (γ))(60.17)the hadronic invariant mass spectrum in semihadronic τ decay, normalized to the leptonic τ decayrate. It is useful to define q as the total momentum of the hadronic final state, so s = q 2 is the
hadronic invariant mass. The total hadronic τ decay rate Rτ is then given by integrating dRτ /ds
over the kinematically allowed range 0 ≤ s ≤ Mτ2 .
Rτ can be written as
Z M2
τRτ = 12π
0

ds
s
1− 2
Mτ2
Mτ


2 

s
1 + 2 2 Im Π T (s) + Im Π L (s)Mτ(60.18)where the hadronic spectral functions Π L,T are defined from the time-ordered correlation functionof two weak currents (j µ (x) and j ν (0)) byZ

d4 x eiq·x h0| T j µ (x)j ν (0)† |0i ,(60.19)Π µν (q) = (−g µν + q µ q ν ) Π T (s) + q µ q ν Π L (s),(60.20)Π µν (q) =i

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60. Quark Masses

11

and the decomposition Eq. (60.19) is the most general possible structure consistent with Lorentz
invariance.
By the optical theorem, the imaginary part of Π µν is proportional to the total cross-section forthe current to produce all possible states. A detailed analysis including the phase space factors
leads to Eq. (60.18). The spectral functions Π L,T (s) are analytic in the complex s plane, withsingularities along the real axis. There is an isolated pole at s = m2π , and single- and multi-particle
singularities for s ≥ 4m2π , the two-particle threshold. The discontinuity along the real axis isΠ L,T (s + i0+ ) − Π L,T (s − i0+ ) = 2iIm Π L,T (s). As a result, Eq. (60.18) can be rewritten with thereplacement Im Π L,T (s) → −iΠ L,T (s)/2, and the integration being over the contour C1 . Finally,
the contour C1 can be deformed to −C2 without crossing any singularities, and so leaving the
integral unchanged, i.e. the integral over the closed contour C1 + C2 vanishes. One can derive aseries of sum rules analogous to Eq. (60.18) by weighting the differential τ hadronic decay rate bydifferent powers of the hadronic invariant mass [40],Rτkl =

Z M2
τ



ds

0

s
1− 2
Mτ

k 

s
Mτ2

l

dRτ
.
ds

(60.21)This leads to the final form of the sum rule(s),Rτkl = −6πi

ds
s
1− 2
2
M
M
C2
τ
τ

Z



2+k 

s
Mτ2

l 

s
Π T (s) + Π L (s) .
Mτ2


1+2



(60.22)The manipulations so far are completely rigorous and exact, relying only on the general analytic
structure of quantum field theory. The left-hand side of the sum rule Eq. (60.22) is obtained fromexperiment. The right hand-side can be computed for s far away from any physical cuts using
the operator product expansion (OPE) for the time-ordered product of currents in Eq. (60.18),
and QCD perturbation theory. The OPE is an expansion in a series of local operators, and is an
expansion about the q → ∞ limit. It gives Π L,T (s) as an expansion in powers of αs (s) and Λ2QCD /s,and is valid when s is far (in units of Λ2QCD ) from any singularities in the complex s-plane.
The OPE gives Π L,T (s) as a series in αs , quark masses, and various nonperturbative vacuum
matrix elements. By computing Π L,T (s) theoretically, and comparing with the experimental valuesof Rτkl , one determines various parameters such as αs and the quark masses. The theoreticaluncertainties in using Eq. (60.22) arise from neglected higher order corrections (both perturbativeand nonperturbative), and because the OPE is no longer valid near the real axis, where Π L,T
have singularities. The contribution of neglected higher order corrections can be estimated as forany other perturbative computation. The error due to the failure of the OPE is more difficult toestimate. In Eq. (60.22), the OPE fails on the endpoints of C2 that touch the real axis at s = Mτ2 .
The weight factor (1−s/Mτ2 ) in Eq. (60.22) vanishes at this point, so the importance of the endpoint
can be reduced by choosing larger values of k.
Light quark masses are often determined using QCD sum rules [38], which are similar to theτ sum rules. One takes the correlator of two light-quark-bilinear operators (e.g. an axial vector
current), as in Eq. (60.18), and computes their Laplace transforms or moments
Ln (τ ) =Z ∞n −τ sds s eIm Π(s),2

Mn (Q ) =

0

Z ∞
0

ds
Im Π(s)(s + Q2 )nto get Laplace or moment sum rules, respectively. The quark masses are extracted by comparing
the theoretical and experimental values of Ln (τ ) and Mn (Q2 ). Considerable theoretical effort hasgone into optimizing n and Q2 to improve the precision of the resulting light quark masses.
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60. Quark Masses

12

60.6 Heavy quarks
60.6.1 Continuum approaches and results
For heavy quark physics one can exploit the fact that mQ  ΛQCD to construct effective theories(mQ is the mass of the heavy quark Q). The masses and decay rates of hadrons containing a single
heavy quark, such as the B and D mesons can be determined using the heavy quark effective theory(HQET) [41]. The theoretical calculations involve radiative corrections computed in perturbation
theory with an expansion in αs (mQ ) and nonperturbative corrections with an expansion in powersof ΛQCD /mQ . Due to the asymptotic nature of the QCD perturbation series, the two kinds ofcorrections are intimately related; an example of this are renormalon effects in the perturbative
expansion which are associated with nonperturbative corrections.

6.0
5.8
5.6
5.4
5.2
5.0
4.8
4.6
4.4
4.2
4.0

1.5

2.0

2.5

3.0

3.5

Figure 60.2: The allowed region (shown in white) for up quark and down quark masses renormalized in the MS scheme at 2 GeV, determined from the allowed ranges of mass parameters (seeFig. 60.3). The parameter (mu + md )/2 yields the two downward-sloping white lines, while mu /mdyields the two rising white lines originating at (0, 0). The white region is the remaining allowed
parameter space after applying the limits on mu , md , (mu + md )/2 and mu /md , where those limits
are 90% confidence levels including the scale factors shown in the ideograms in the Data Listings.
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60. Quark Masses

Systems containing two heavy quarks such as the Υ or J/Ψ are treated using non-relativistic
QCD (NRQCD) [42]. The typical momentum and energy transfers in these systems are αs mQ , and
αs2 mQ , respectively, so these bound states are sensitive to scales much smaller than mQ . However,smeared observables, such as the cross-section for e+ e− → bb averaged over some range of s that
includes several bound state energy levels, are better behaved and only sensitive to scales nearmQ . For this reason, most determinations of the c, b quark masses using perturbative calculations
compare smeared observables with experiment [43–45]. The method is similar to that outlined for τ
decays. The current correlator in Eq. (60.18) is the electromagnetic current, and the experimental
data is the value of R(s) in the threshold region for e+ e− → QQ. The theoretical values for the
moments are computed using renormalization group improved calculations in non-relativistic QCD.
There are many continuum extractions of the c and b quark masses, some with quoted errorsof 10 MeV or smaller. There are systematic effects of comparable size, which are typically notincluded in these error estimates. Reference [46], for example, shows that even though the error
estimate of mc using the rapid convergence of the αs perturbation series is only a few MeV, thecentral value of mc can differ by a much larger amount depending on which algorithm (all of which
are formally equally good) is used to determine mc from the data. This leads to a systematic
error from perturbation theory of around 20 MeV for the c quark and 25 MeV for the b quark.
Electromagnetic effects, which also are important at this precision, are often not included. For thisreason, we inflate the errors on the continuum extractions of mc and mb . The average values of mc
and mb from continuum determinations are (see Sec. 60.7 for the 1S scheme)
mc (mc ) = (1.280 ± 0.025)GeV,
mb (mb ) = (4.18 ± 0.03) GeV,

m1S
b = (4.65 ± 0.03) GeV.

60.6.2 Lattice approaches and results
Lattice QCD simulations of a heavy quark Q, described in a relativistic fermion formulation, leadto potentially large discretization errors which are powers of amQ (modulated by logarithms); theleading power depends on the formulation of lattice QCD being used and in most cases is quadratic.
Clearly those errors can be reduced by performing simulations at smaller lattice spacings a, butalso by using improved discretizations of the theory, in which the leading discretization errors are
systematically eliminated. Recently, with more powerful computing resources, better algorithms
and improved discretizations of QCD, it has become possible to perform simulations with quarks
more massive than the charm and even up to the b [11], decreasing or eliminating the extrapolation
which has to be performed to reach the mb . It is worth noting that computations using a relativistic
fermion formulation for the valence charm with Nf = 2 + 1 sea-quark flavors, or with Nf = 2 + 1 + 1
flavors and a relativistic b, suffer from partial-quenching effects. This is because the heavy valencequarks are absent from the sea. While these effects are not expected to be large in Nf = 2 + 1 + 1
computations, they should be investigated when quoting results below the percent level in the caseof Nf = 2 + 1 sea-quark flavors. For that reason we will omit heavy-quark results obtained with
Nf = 2 + 1 sea-quark flavors from our averages and, more generally focus on Nf = 2 + 1 + 1 results.
Note that these partial-quenching effects are accounted for systematically in lattice calculations inwhich the heavy valence quarks are described by HQET or NRQCD.
Traditionally the charm quark mass is obtained by tuning its bare, simulation input value toreproduce the physical mass of charmonium mesons or of the D, Ds mesons (requiring a more
precise tuning of the light quark masses and the computation of larger QED effects in some cases).
This mass can then be renormalized and matched to the MS scheme using the methods discussed
for the light quarks.
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14

60. Quark Masses

25

27

29

Figure 60.3: The allowed regions (shown in white) for quark masses renormalized in the MS
scheme at 2 GeV. The points (taken from the Data Listings) are in chronological order with themore recent measurements shown at the top. The white regions indicate values allowed at the 90%
confidence level including the scale factors shown in the ideograms in the Data Listings.

An alternative approach for obtaining the MS mass was proposed in [47]. Euclidean-time moments of pseudoscalar, two-point functions of cc quark-bilinear operators can readily be computedon the lattice and extrapolated to the continuum limit where they can be compared to perturbative
calculations of the same quantities at 4-loop order. In this way, both the strong coupling constantand the charm quark mass can be determined with remarkably small errors. As this approach
uses the same perturbative expressions for two-point correlators as the continuum determinations
discussed in Sec. 60.6.1, it suffers from similar perturbation-theory, systematic errors. FLAG [8]has reviewed lattice determinations of the charm-quark mass obtained using both approaches. For
their Nf = 2 + 1 average they retain the results [13, 48–50], with [51] being published after their
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60. Quark Masses

deadline. As stated above, we focus here on Nf = 2 + 1 + 1 results. For these, FLAG uses theresults of Refs. [10, 19, 20, 22, 52].
Since the FLAG review, Refs. [22, 52] have been superseded by Ref. [9]. In that reference, only
mc (3 GeV) is given and we multiply it by the running factor 0.7739 ± 0.0060 from µ = 3 GeV to
mc [8]. With this substitution and using the combination method described in Sec. 60.4, we obtain:
mc (mc ) = (1.275 ± 0.004 ± 0.008[±0.009]) GeV(60.23)where the three uncertainties have the same meaning as described after Eq. (60.6). This result isin good agreement with the continuum result given above, but with a smaller error.
It is worth noting that three [10, 19, 20] of the four results entering this average agree, while
the other [9] is over 2 standard deviations larger than the average. This is taken into account bystretching errors in the usual way, but the total uncertainty on the lattice average of mc will notsignificantly be reduced until this discrepancy is resolved. It should also be noted that [10] includes
QED directly in the lattice calculation, albeit only for valence quarks, while some of the othercomputations account for these effects using phenomenology. For mc (mc ) considered above, theseeffects are below the permil [10] and are therefore neglected here.
Historically, the main approach to controlling the discretization errors in lattice studies of bquark physics was to perform simulations of effective theories such as HQET and NRQCD. Thisremains an important technique, both in its own right and in providing additional information for
extrapolations from lower masses to the bottom region. Using effective theories, mb is obtained
from what is essentially a computation of the difference of MHb − mb , where MHb is the mass ofa hadron Hb containing a b quark. The relative error on mb is therefore much smaller than thatfor MHb − mb . The principal systematic errors are the matching of the effective theories to QCD
and the presence of power divergences in a−1 in the 1/mb corrections which have to be subtracted
numerically. A procedure for performing these subtractions fully nonperturbatively was proposed
and implemented for the first time in [53].
The most recent Nf = 2+1+1 lattice QCD determinations of the b quark mass rely on a variety
of approaches, including Euclidean-time moments of correlation functions with [54] or without
NRQCD [19] and interpolations [55, 56] (using results from HQET simulations) or extrapolations[11, 20] from above the charm mass to that of the b. One calculation [11] even includes simulations
performed with a heavy quark whose mass is very near that of the b quark. The overall agreementof the results obtained using these very different approaches, which have different systematic errors,
is a confirmation that the various groups control these uncertainties. As the range of heavy-quark
masses which can be used in numerical simulations increases, the extrapolation or interpolation of
results to the physical b mass, possibly including some obtained directly at that mass, are becoming
ever more reliable (see e.g. Refs. [11, 20]).
Using the results from Refs. [11, 19, 20, 55, 56], converting those given in the NL = 4 theory
to the NL = 5 one and applying the averaging method described in Sec. 60.4, we obtained theestimate:mb (mb ) = (4.196 ± 0.009 ± 0.009[±0.012]) GeV ,(60.24)
where the three uncertainties have the same meaning as in the estimates given above for light-quark
masses.
Among the results entering this average, only one [11] accounts for electromagnetic effects.
Using Refs. [10,11], it is straightforward to show that QED effects on the value of mb at µ = 3 GeV,
and thus at µ = mb , are smaller than one permil in the separation scheme of those two references.
Assuming that the same is true for the other results, we neglect these corrections here.
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60. Quark Masses

The lattice result of Eq. (60.24) is compatible with the average value of continuum results, butwith a significantly smaller uncertainty. Note that FLAG [8] also provides an average of Nf = 2 + 1
results, taken from [13, 50]. It is about one combined, standard deviations lower that the result
quoted above.
As explained in Sec. 60.3, ratios of quark masses can have significantly smaller errors than theindividual masses if they are determined within the same lattice QCD framework. This led HPQCD
to leverage their precise determination of mc [47] to determine ms and mud [57], through a precise
computation of mc /ms [57] and of ms /mud [58]. The Nf = 2 + 1 calculation [57] was updated
using Nf = 2 + 1 + 1 simulations in [19]. The ratio mc /ms was also computed in Refs [48, 59]with Nf = 2 + 1 simulations and in Refs. [20, 22] with Nf = 2 + 1 + 1 ones. On the basis of thosereferences, FLAG [8] gives average results from Nf = 2 + 1 and Nf = 2 + 1 + 1 simulations. Here
we focus on Nf = 2 + 1 + 1 results and we replace the results of Ref. [22] by the collaboration’s
update [9]. Applying to the modern Nf = 2 + 1 + 1 computations of Refs. [9,19,20] the combination
procedure described in Sec. 60.4, we obtain the following lattice average:
mc /ms = 11.769 ± 0.021 ± 0.028[±0.035],
where the three uncertainties have the some meaning as described after Eq. 60.6. The average isdominated by Ref. [20] that has a substantially smaller error than the other results. The resultof Ref. [9] is a little more that 1 standard deviation below that average and the one of Ref. [19]close to 2σ above, for a total χ2 /dof = 5.0/2 and a rescaling factor of ∼ 1.6 in the statistical andsystematic errors.
The above ratio is renormalization scheme and scale independent, in the absence of electromagnetic corrections. None of the calculations above estimates the size of those corrections. Nevertheless, they are expected to still be smaller than the total uncertainty quoted above in a standard
scheme for defining QCD in the isospin limit.
The ratio mb /mc has also been computed on the lattice. The two Nf = 2+1 results which satisfy
FLAG’s criteria are [13,50]. As explained above, we focus on Nf = 2 + 1 + 1 calculations [11,20,55].
Ref. [11] is the only calculation which accounts for QED directly in the lattice simulations, albeit
only for valence quarks, and thus must specify the renormalization scheme and scale of its result,which it chooses to be MS at 3 GeV. Here we correct the results of Refs. [20, 55] by the QED
factor 1.0017 obtained in Ref. [11], and ascribe to it a 100% systematic uncertainty with 100%
correlation. This means that we assume that the differences due to the separation schemes used ineach calculation lies within that error. Averaging the resulting numbers, as prescribed in Sec. 60.4,
yieldsmb /mc = 4.584 ± 0.006 ± 0.010[±0.012],
where exceptionally the bar indicates that thep masses are renormalized in the MS at 3 GeV in
the four-flavor theory. Here a scale factor of χ2 /dof = 1.30 has been applied to the error bar.
Indeed, [55] contributes 3.3 to the total χ2 .
60.6.3 Warnings concerning the use of the pole mass
For an observable particle such as the electron, the position of the pole in the propagator is thedefinition of its mass. In QCD this definition of the quark mass is known as the pole mass. It is
known that the on-shell quark propagator has no infrared divergences in perturbation theory [60,61],
so this provides a perturbative definition of the quark mass. However, the pole mass cannot be
used to arbitrarily high accuracy because of nonperturbative infrared effects in QCD. In fact thefull quark propagator has no pole because the quarks are confined, so that the pole mass cannot be
defined outside of perturbation theory. The relation between the pole mass mQ and the MS mass
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60. Quark Masses

17

mQ , used throughout this review, is known to three loops [62–65](

mQ = mQ (mQ ) 1 +"+ −1.0414

Xq

4αs (mQ )
3π
4 mq
1−
3 mQ

(60.25)!

#+ 13.4434

αs (mQ ) 2πi  α (m ) 3sQh+ 0.6527NL2 − 26.655NL + 190.595

π

)

,

where αs (µ) is the strong interaction coupling constants in the MS scheme, and the sum over q
extends over the NL flavors lighter than Q. The complete mass dependence of the αs2 term can befound in [62]; the mass dependence of the αs3 term is not known. For the b-quark, Eq. (60.25) reads
mb = mb (mb ) [1 + 0.10 + 0.05 + 0.03] ,(60.26)

where the contributions from the different orders in αs are shown explicitly. The two and three loop
corrections are comparable in size and have the same sign as the one loop term. This is a signalof the asymptotic nature of the perturbation series (there is a renormalon in the pole mass [66]).
Such a badly behaved perturbation expansion can be avoided by directly extracting, from data, themass defined in the MS (used in this review) or other short-distance schemes (see below), without
invoking the pole mass as an intermediate step.

60.7 Numerical values and caveats
The quark masses in the particle data listings have been obtained by using a wide variety ofmethods. Each method involves its own set of approximations and uncertainties. In most cases,
the errors are an estimate of the size of neglected higher-order corrections or other uncertainties.
The expansion parameters for some of the approximations are not very small (for example, they are
m2K /Λ2χ ∼ 0.25 for the SU(3) chiral expansion and ΛQCD /mb ∼ 0.1 for the heavy-quark expansion),so an unexpectedly large coefficient in a neglected higher-order term could significantly alter the
results. Thus, before using a particular result, it is important to understand the possible limitationsof the approach used to obtain it. It is also important to note that the quark mass values can besignificantly different in the different schemes.
We have specified all masses in the MS scheme. For light quarks, the renormalization scale
has been chosen to be µ = 2 GeV. Quoting these masses at smaller values of µ, where perturbative
corrections become significantly larger, would introduce unnecessary uncertainties in the results. In
fact, as lattice calculations, performed on finer and finer lattices, allow to determine quark masses,
fully nonperturbatively, at larger and larger values of µ, it may become advantageous to quote
quark mass results at renormalization scales above 2 GeV, where perturbative uncertainties are
smaller.
The heavy quark masses obtained using HQET, QCD sum rules, or lattice gauge theory are
consistent with each other if they are all converted into the same scheme and scale. For thesequarks it is conventional to choose the renormalization scale equal to the quark mass, so we havequoted mQ (µ) at µ = mQ for the c and b quarks. Given the small size of the charm quark mass,in the future it may become advantageous to quote its value at larger values of µ so as not to
introduce unnecessary perturbative uncertainties (see discussion above). Analyses of inclusive B
meson decays have shown that other mass definitions lead to a better behaved perturbation series
than for the MS mass, and hence to more accurate mass values [67–70]. Thus, we have chosen toalso give values for one of these, the b quark mass in the 1S scheme [67, 68]. Other schemes that
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60. Quark Masses

have been proposed are the PS-scheme [69], the kinetic scheme [71] and, most recently, the minimal
renormalon-subtracted mass (MRS) [70] used in the lattice calculation [20].
If necessary, we have converted values in the original papers to our chosen scheme using
two-loop formulæ. It is important to realize that our conversions introduce significant additional
errors. In converting to the MS b-quark mass, for example, the three-loop conversions from the1S and pole masses give values about 35 MeV and 135 MeV lower than the two-loop conversions.
The uncertainty in αs (MZ ) = 0.1179 ± 0.0010 [1] gives an uncertainty of ±9 MeV and ±21 MeV
respectively in the same conversions. We have not added these additional errors when we do ourconversions. The αs value in the conversion is correlated with the αs value used in determining thequark mass, so the conversion error is not a simple additional error on the quark mass.
References[1] See the review of QCD in this volume.
[2] A. Manohar and H. Georgi, Nucl. Phys. B 234, 189 (1984).
[3] K. Chetyrkin, Phys. Lett. B 404, 161 (1997), [hep-ph/9703278].
[4] J. Vermaseren, S. Larin and T. van Ritbergen, Phys. Lett. B 405, 327 (1997), [hepph/9703284].
[5] K. Chetyrkin, B. A. Kniehl and M. Steinhauser, Nucl. Phys. B 510, 61 (1998), [hepph/9708255].
[6] G. Martinelli et al., Nucl. Phys. B 445, 81 (1995), [hep-lat/9411010].
[7] K. Jansen et al., Phys. Lett. B 372, 275 (1996), [hep-lat/9512009].
[8] Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), Eur. Phys. J. C 82, 10, 869 (2022),[arXiv:2111.09849].
[9] C. Alexandrou et al. (Extended Twisted Mass), Phys. Rev. D 104, 7, 074515 (2021),[arXiv:2104.13408].
[10] D. Hatton et al. (HPQCD), Phys. Rev. D 102, 5, 054511 (2020), [arXiv:2005.01845].
[11] D. Hatton et al., Phys. Rev. D 103, 11, 114508 (2021), [arXiv:2102.09609].
[12] A. Bazavov et al. (MILC), PoS CD09, 007 (2009), [arXiv:0910.2966].
[13] C. McNeile et al., Phys. Rev. D 82, 034512 (2010), [arXiv:1004.4285].
[14] A. Bazavov et al., PoS LATTICE2010, 083 (2010), [arXiv:1011.1792].
[15] S. Durr et al., Phys. Lett. B 701, 265 (2011), [arXiv:1011.2403].
[16] S. Durr et al., JHEP 08, 148 (2011), [arXiv:1011.2711].
[17] T. Blum et al. (RBC, UKQCD), Phys. Rev. D 93, 7, 074505 (2016), [arXiv:1411.7017].
[18] M. Bruno et al. (ALPHA), Eur. Phys. J. C 80, 2, 169 (2020), [arXiv:1911.08025].
[19] B. Chakraborty et al., Phys. Rev. D 91, 5, 054508 (2015), [arXiv:1408.4169].
[20] A. Bazavov et al. (Fermilab Lattice, MILC, TUMQCD), Phys. Rev. D 98, 5, 054517 (2018),[arXiv:1802.04248].
[21] A. Lytle et al. (HPQCD), Phys. Rev. D 98, 1, 014513 (2018), [arXiv:1805.06225].
[22] N. Carrasco et al. (European Twisted Mass), Nucl. Phys. B 887, 19 (2014), [arXiv:1403.4504].
[23] A. Bazavov et al. (Fermilab Lattice, MILC), Phys. Rev. D 90, 7, 074509 (2014),[arXiv:1407.3772].
[24] A. Bazavov et al., Phys. Rev. D 98, 7, 074512 (2018), [arXiv:1712.09262].
[25] M. Bruno et al. (ALPHA), Phys. Rev. Lett. 114, 10, 102001 (2015), [arXiv:1410.8374].
31st May, 2024

19

60. Quark Masses[26] A. Athenodorou et al. (ALPHA), Nucl. Phys. B 943, 114612 (2019), [arXiv:1809.03383].
[27] S. Borsanyi et al., Science 347, 1452 (2015), [arXiv:1406.4088].
[28] Z. Fodor et al., Phys. Rev. Lett. 117, 8, 082001 (2016), [arXiv:1604.07112].
[29] D. Giusti et al., Phys. Rev. D 95, 11, 114504 (2017), [arXiv:1704.06561].
[30] S. Basak et al. (MILC), Phys. Rev. D 99, 3, 034503 (2019), [arXiv:1807.05556].
[31] S. Weinberg, Physica A 96, 1-2, 327 (1979).
[32] J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984).
[33] A. Pich, Rept. Prog. Phys. 58, 563 (1995), [hep-ph/9502366].
[34] S. Weinberg, Trans. New York Acad. Sci. 38, 185 (1977).
[35] D. B. Kaplan and A. V. Manohar, Phys. Rev. Lett. 56, 2004 (1986).
[36] G. Colangelo et al., Eur. Phys. J. C 78, 11, 947 (2018), [arXiv:1807.11937].
[37] S. Weinberg, Phys. Rev. Lett. 18, 507 (1967).
[38] M. A. Shifman, A. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979).
[39] E. Braaten, S. Narison and A. Pich, Nucl. Phys. B 373, 581 (1992).
[40] F. Le Diberder and A. Pich, Phys. Lett. B 286, 147 (1992).
[41] N. Isgur and M. B. Wise, Phys. Lett. B 237, 527 (1990).
[42] G. T. Bodwin, E. Braaten and G. Lepage, Phys. Rev. D 51, 1125 (1995), [Erratum: Phys.Rev.D
55, 5853 (1997)], [hep-ph/9407339].
[43] A. Hoang, Phys. Rev. D 61, 034005 (2000), [hep-ph/9905550].
[44] K. Melnikov and A. Yelkhovsky, Phys. Rev. D 59, 114009 (1999), [hep-ph/9805270].
[45] M. Beneke and A. Signer, Phys. Lett. B 471, 233 (1999), [hep-ph/9906475].
[46] B. Dehnadi et al., JHEP 09, 103 (2013), [arXiv:1102.2264].
[47] I. Allison et al. (HPQCD), Phys. Rev. D 78, 054513 (2008), [arXiv:0805.2999].
[48] Y.-B. Yang et al., Phys. Rev. D 92, 3, 034517 (2015), [arXiv:1410.3343].
[49] K. Nakayama, B. Fahy and S. Hashimoto, Phys. Rev. D 94, 5, 054507 (2016),[arXiv:1606.01002].
[50] P. Petreczky and J. Weber, Phys. Rev. D 100, 3, 034519 (2019), [arXiv:1901.06424].
[51] J. Heitger, F. Joswig and S. Kuberski (ALPHA), JHEP 05, 288 (2021), [arXiv:2101.02694].
[52] C. Alexandrou et al., Phys. Rev. D 90, 7, 074501 (2014), [arXiv:1406.4310].
[53] J. Heitger and R. Sommer (ALPHA), JHEP 02, 022 (2004), [hep-lat/0310035].
[54] B. Colquhoun et al., Phys. Rev. D 91, 7, 074514 (2015), [arXiv:1408.5768].
[55] A. Bussone et al. (ETM), Phys. Rev. D 93, 11, 114505 (2016), [arXiv:1603.04306].
[56] P. Gambino, A. Melis and S. Simula, Phys. Rev. D 96, 1, 014511 (2017), [arXiv:1704.06105].
[57] C. Davies et al., Phys. Rev. Lett. 104, 132003 (2010), [arXiv:0910.3102].
[58] A. Bazavov et al. (MILC), Rev. Mod. Phys. 82, 1349 (2010), [arXiv:0903.3598].
[59] Y. Maezawa and P. Petreczky, Phys. Rev. D 94, 3, 034507 (2016), [arXiv:1606.08798].
[60] R. Tarrach, Nucl. Phys. B 183, 384 (1981).
[61] A. S. Kronfeld, Phys. Rev. D 58, 051501 (1998), [hep-ph/9805215].
[62] N. Gray et al., Z. Phys. C 48, 673 (1990).
[63] D. J. Broadhurst, N. Gray and K. Schilcher, Z. Phys. C 52, 111 (1991).
31st May, 2024

20

60. Quark Masses[64] K. Chetyrkin and M. Steinhauser, Phys. Rev. Lett. 83, 4001 (1999), [hep-ph/9907509].
[65] K. Melnikov and T. v. Ritbergen, Phys. Lett. B 482, 99 (2000), [hep-ph/9912391].
[66] M. Beneke and V. M. Braun, Nucl. Phys. B 426, 301 (1994), [hep-ph/9402364].
[67] A. H. Hoang, Z. Ligeti and A. V. Manohar, Phys. Rev. Lett. 82, 277 (1999), [hep-ph/9809423].
[68] A. H. Hoang, Z. Ligeti and A. V. Manohar, Phys. Rev. D 59, 074017 (1999), [hep-ph/9811239].
[69] M. Beneke, Phys. Lett. B 434, 115 (1998), [hep-ph/9804241].
[70] N. Brambilla et al. (TUMQCD), Phys. Rev. D 97, 3, 034503 (2018), [arXiv:1712.04983].
[71] P. Gambino and N. Uraltsev, Eur. Phys. J. C 34, 181 (2004), [hep-ph/0401063].

31st May, 2024