Quark Masses on the Lattice

7/28/2019 Quark Masses on the Lattice

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arXiv:hep-lat/0012003v14

Dec2000

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Quark Masses on the Lattice: Light and Heavy

RM3-TH/00-21

V. Lubicza

aDipartimento di Fisica, Universita di Roma Tre and INFN, Sezione di Roma III,Via della Vasca Navale 84, I-00146 Rome, Italy

I review the current status of lattice calculations of light and heavy quark masses. Significant progresses, inthese studies, have been allowed by the introduction of improved actions and non-perturbative renormalizationtechniques. Current determinations of light quark masses are accurate at the level of 20%, where the main sourceof uncertainty is represented by the quenching error. The determination of the bottom quark mass is accurateat the impressive level of 2%. As final averages of lattice results, I quote mud(2 GeV) = (4.5 1.0) MeV,ms(2 GeV) = (110 25) MeV and mb(mb) = (4.26 0.09) GeV.

1. INTRODUCTION

Calculations of quark masses are one of themost intensive subject of investigation for lat-tice QCD. An accurate determination of theseparameters is in fact extremely important, forboth phenomenological and theoretical applica-tions. The charm and bottom masses, for in-stance, enter through the heavy quark expansion,the theoretical expressions of several cross sec-tions and decay rates. From a theoretical point of

view, an accurate determination of quark massesmay give insights on the physics of flavour, re-vealing relations between masses and mixing an-gles, or specific textures in the quark mass matrix,which may originate from still uncovered flavoursymmetries.

The values of quark masses cannot be directlymeasured in the experiments because quarks areconfined inside the hadrons. On the other hand,quark masses are fundamental (free) parametersof the theory and, as such, they cannot be com-puted on the basis of purely theoretical conside-rations. The values of quark masses can be only

determined by comparing the theoretical evalua-tion of a given physical quantity, which dependson quark masses, with the corresponding experi-mental value. Typically the pion, kaon and me-son masses are used on the lattice to compute thevalues of the light quark masses, whereas the b-quark mass is determined by computing the mass

of the B or the mesons. Different choices areall equivalent in principle, and the differences inthe results, obtained by using different hadronmasses as input parameters, give an estimate ofthe systematic error.

As all other parameters of the Standard Modellagrangian, quark masses can be defined as ef-fective couplings, which are both renormaliza-tion scheme and scale dependent. A schemecommonly adopted for quark masses is the MSscheme, with a renormalization scale chosen in

the short-distance region in order to make thisquantity accessible to perturbative calculations.It is a common practice to quote the values of thelight quark masses at the renormalization scale = 2 GeV, whereas the heavy quark masses areusually given at the scale of the quark mass itself,e.g. mb(mb). This convention will be followed inthe present review.

At Lattice 99, no plenary talk was dedicatedto review lattice calculations of quark masses.For this reason, I will mainly report on re-sults presented in the last two years. With re-spect to previous determinations, significant pro-gresses in controlling the systematic errors havebeen allowed by the extensive implementationof non-perturbative O(a)-improvement [1] andnon-perturbative renormalization techniques [2,3]. The effect of these tools, on lattice calcula-tions of quark masses, will be discussed in sec. 2.
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The recent determinations of light quark masseswill be presented in sec. 3, and of the b-quark

mass in sec. 4. Some comments on the deter-mination of the charm quark mass will be donein sec. 5, where I will also summarize the latticeworld averages of quark masses and present myconclusions.

2. DEFINITIONS AND THEORETICAL

PROGRESSES

2.1. Quark Masses and O(a)-improvementNon perturbative definitions of quark masses

are provided by the chiral Ward identities (WI) ofQCD [4]. These definitions allow us to express the

renormalized quark mass, in a given scheme andat a given renormalization scale, in terms of lat-tice renormalization constants and bare quanti-ties (action parameters, matrix elements or corre-lation functions). Three independent definitionshave been proposed so far which are all equiva-lent in principle, in the sense that they lead tothe same value of the renormalized mass. Let usdiscuss these definitions in some details.

1) Vector Ward Identity definition:

In terms of renormalized quantities, the chiralvector WI reads:

|V| = (m1() m2()) |S()| (1)

where | and | are arbitrary external states.Vector current conservation, within the latticeregularization, implies a lattice version of the vec-tor WI, from which the following definition of therenormalized quark mass, in terms of the (bare)Wilson parameter, can be derived:

m() = Z1S (a)m = Z1S (a)

1

2a

1

K

1

Kc

(2)

An O(a)-improved definition is obtained with thesimple replacement [1]:

m m (1 + a bmm) , (bm = bS/2) (3)

2) Axial Ward Identity definition:

In the Wilson formulation of lattice QCD, axialsymmetry is explicitly broken. The axial WI canbe then imposed only for renormalized quantities,

and, in the case of degenerate masses, takes theform:

|A| = 2 m() |P()| (4)

From this identity, an alternative definition of thequark mass is derived:

m() =ZA

ZP(a)mAWI =

ZAZP(a)

|A|

2 |P|(5)

In this case, O(a)-improvement is achieved withthe replacement [1]:

mAWI [1 + a (bA bP) m]

| (A + a cAP) |

2 |P()|

(6)

In the absence of explicit chiral symmetry brea-king induced by the Wilson term in the action,the vector and axial vector WI definitions are notindependent anymore. This is the case, for in-stance, of staggered fermions and of the proposedrealizations of Ginsparg-Wilson fermions (domainwall and overlap fermions).

3) Definition from the Quark Propagator:

This relatively new method has been proposedby APE [5]. It is based on the study of the quarkpropagator, computed in a fixed gauge, at large

momentum. In this limit, an OPE can be derivedfor the trace of the quark propagator which, upto known logarithmic corrections, reads [6] :

Tr S(p) m

p2+

4s3p4

+O(1/p6) (7)

The leading term of this expansion is proportionalto the renormalized quark mass. The Wilson co-efficient of this term is fixed by an axial WI which,in RI-MOM scheme, implies [5]:

m() =1

12Tr

S1(p, )

p2=2

(8)

Note that the relevant renormalization constant,in this case, is the quark field renormalizationconstant Zq. Cancellation of leading O(a) ef-fects is obtained by improving the quark field.Since this field is a non gauge-invariant operator,and the quark propagator is an off-shell correla-tion function, improvement involves mixing with


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both non gauge-invariant operators and opera-tors vanishing by the equation of motion. The

O(a)-improvement of the quark field is obtainedthrough the replacement [7]:

q (1 + abqm) [1 + acq( /D + m0) + acNGI/]q (9)

0.5 1.0 1.5 2.0

(ap)2

0.0

0.5

1.0

1.5Tr[S(p)] ~ ac

q+m/p

2

O(a)Chiral Limit

0.0 0.5 1.0 1.5 2.0

(a)2

0.00

0.05

0.10

0.15

0.20

K=0.1333

K=0.1344

K=0.1349

K=0.1352

(1/12) Tr[S(p)1

]

Figure 1. Large momentum behaviour of the lat-tice quark propagator. The upper figure showsthe large O(a) term which has to be subtractedbefore extracting the subleading contribution pro-portional to the quark mass. The behaviour ofthe inverse quark propagator, after the subtrac-tion, is shown below. The results have been ob-tained by APE [5] using the non-perturbativelyO(a)-improved Wilson action at = 6.2.

In the determination of quark masses with thepropagator method, the improvement procedureis a crucial requirement. The reason is that theOPE of the quark propagator on the lattice isdominated by a large lattice artifact. From eq. (9)

one finds, in the large momentum limit and up tosmall logarithmic corrections [5]:

112

Tr S(p) 2a cq + 2a cNGIZq a cq (10)

The lattice quark propagator does not vanish atlarge p2, as dictated in the continuum limit byeq. (7), but it is dominated by terms of O(a).This is illustrated in fig. 1. The numerical studyof the propagator at large p2, in the chiral limit,provides the value of the coefficient cq, which canthus be subtracted before computing the sublead-ing contribution to the OPE proportional to thequark mass.

The improved calculations of quark masses,with the vector and axial-vector WI methods, re-quire the knowledge of several coefficients, cA,bA, bS and bP, besides the coefficient cSW ofthe clover term in the action. Non-perturbativedeterminations of these coefficients have beenperformed in refs. [1,8,9]. One finds that theeffect of improvement, in the determination ofquark masses, is quite significant, even for lightquarks. In the case of the strange quark mass,for instance, an estimate of the O(a) effects,with unimproved Wilson fermions, has been per-formed by Bhattacharya and Gupta [10] at Lat-tice 97, based on the analysis of the continuum

extrapolation. They find that discretization er-rors, at two typical values of the lattice cou-pling, = 6.0 and = 6.2, are of the orderof 25% and 18% respectively. These estimatescan be compared with those obtained with non-perturbatively O(a)-improved Wilson fermions,from a recent study by ALPHA-UKQCD [11].In this case, residual discretization effects are re-duced at the level of 7% and 3% respectively. Im-plementation ofO(a)-improvement is therefore acrucial ingredient to significantly increase the ac-curacy of lattice calculations of quark masses.

2.2. Non-perturbative Renormalization

Another potential source of systematic errors,in lattice calculations of quark masses, is intro-duced by the truncation of the perturbative ex-pansion in the evaluation of the quark mass renor-malization constant. Depending on the defini-tion considered for the bare mass, this constantis given by Z1S , with the vector WI method,


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Table 1Values of the strange quark mass, ms(2 GeV), in MeV, as obtained by computing the quark massrenormalization constant either in one loop (boosted) perturbation theory (PT) or using the RI-MOMnon-perturbative method (NP). The relative difference between the two determinations is also shown.

Action ms [PT] ms [NP]

NP-CLOVER 6.295(10)

81(9)109(11)

111(13)

15%

37%APE 98 [13]

KOGUT-SUSSKIND

6.0

6.2

6.4

80(4)

81(6)

83(4)

114(5)

110(8)

108(5)

43%

36%

30%

JLQCD 99 [14]

DOMAIN WALL 6.0 106(8)(14) 130(11)(18) 23% RBC 99 [15] From vector WI; from axial WI. The error is my estimate from the RBC data.

ZA/ZP, with the axial WI method, or the quarkfield renormalization constant Zq with the prop-agator method. In perturbation theory, all thesequantities are known only at one loop. In thelast two years, however, in most of lattice quarkmasses calculations, the systematic error associ-ated with the evaluation of the renormalizationconstant has been reduced to a negligible amountby the use of non-perturbative renormalizationtechniques [2,3].

Non-perturbative renormalization is reviewedby Sint at this conference [12]. For this reason, Iwill only discuss here those particular aspects of

this topic which are more closely related to thedetermination of quark masses.To show the impact of non-perturbative renor-

malization in lattice calculations of quark masses,I present in table 1 the values of the strangequark mass, as obtained by computing the quarkmass renormalization constant either in one-loop(boosted) perturbation theory or with the RI-MOM non-perturbative method of ref. [2]. Forillustrative purposes, the results of three calcula-tions are considered, which employed three diffe-rent fermionic actions: non-perturbative Clover,Kogut-Susskind and domain wall fermions. As

can be seen from table 1, the difference betweenthe results obtained with perturbative and non-perturbative renormalization is rather large. De-pending on the action, the value of the couplingconstant and the specific renormalization con-stant (ZS or ZP for Wilson fermions), the re-lative error varies in the range between 15%

and 40%. Moreover, at least in the case ofstaggered fermions, this error decreases as oneget closer to the continuum limit, but at a veryslow rate. Therefore, non-perturbative renorma-lization is required to improve the accuracy of lat-tice calculations of quark masses at a significantlevel (below 15%).

I believe that a more detailed discussion de-serves the non-perturbative calculation of thepseudoscalar renormalization constant, ZP, in theRI-MOM scheme. Although rather technical, thisdiscussion is of relevance for the determination ofquark masses with the axial WI method. Within

the non-perturbative approach of ref. [2], ZP isevaluated from the amputated correlation func-tion of the pseudoscalar density, computed be-tween external off-shell quark states of momen-tum p. This function, P(p), once convenientlyprojected onto the matrix 5, is related to thetrace of the inverse quark propagator by the ax-ial WI: mP(p) = Tr S1(p). The OPE of thequark propagator, given in eq. (7), thus implies:

P(p) C1(p2) + C2(p2) m p2

+ O(1/p4) (11)

The renormalization constant ZP is determinedfrom the coefficient function C1. Eq. (11) showsthat the first power correction, which vanishes atlarge momentum, is however divergent in the chi-ral limit. Physically, this divergence is introducedby the coupling of the pseudoscalar density to thepion field, which is the Goldstone boson of thespontaneously broken chiral symmetry. It has


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been emphasized, particularly in ref. [16], that,because of the singular behaviour of this term

in the chiral limit, its contribution to the cor-relation function P(p) may not be suppressedenough at large p2, at least for those values of p2

in actual calculations at which discretization ef-fects are still under control. This may affect thedetermination of ZP and, according to ref. [16],this systematic error may in fact be the reason ofthe large discrepancy between the perturbativeand the non-perturbative determination of ZP.Since Zm Z

1P , the observation of ref. [16] is of

relevance for lattice calculations of quark masses.I believe that the warning of ref. [16] deserves

some consideration and a careful investigation.

I would like to mention, therefore, three differ-ent methods which can be used to verify whetherthe contribution of the Goldstone pole, in the ac-tual calculation of ZP, is suppressed at the re-quired level. To study this problem, I will usethe same APE data which have been used for thenon-perturbative calculation of the renormaliza-tion constant of bilinear quark operators (ZV, ZA,ZS, ZP and ZT) with the non-perturbative Cloveraction, at = 6.2 [17]. The first method is basedon the study of the scale dependence of the ratioZP/ZS, computed non-perturbatively in the RI-MOM scheme. In the large momentum region,

where power corrections are expected to be negli-gible, this ratio should exhibit a plateau, since thetwo renormalization constants have equal anoma-lous dimensions. This ratio is shown in fig. 2,and in the large-p2 region a plateau is clearly vi-sible. This suggests that all power corrections,including the contribution of the Goldstone pole,are suppressed enough. The second method isbased on the study of the renormalization groupinvariant combination ZP(a)/C(a), where therunning factor C(a), computed in perturbationtheory at the NLO, expresses the predicted lo-garithmic scale dependence of the renormaliza-tion constant. Again, in the region where non-perturbative power corrections are suppressedenough, the ratio ZP(a)/C(a) should behaveas a constant. Indeed, this constant behaviour isobserved in fig. 2. Finally, the most efficient wayto get rid of the Goldstone boson contribution inthe calculation ofZP (but the method can be ea-

0 0.5 1 1.5 2

(a)2

0

0.5

1

1.5

ZP/ZSZP (a)/C(a)

ZP~ (a)/C(a)

Figure 2. The ratios ZP/ZS, ZP(a)/C(a) and

ZP(a)/C(a) (see text for details) as a functionof the scale (a)2.

sily adapted to the calculation of other renormal-ization constants) has been recently suggested inref. [18]. The idea is to consider a new definition

ofZP, which I will denote with ZP, involving twoflavours of non-degenerate quarks:

ZP()Zq()

=m1 m2

m1P(, m1) m2P(, m2)(12)

On the r.h.s. of eq. (12), the limit m1, m2 0is understood. For m2 = 0 (and at large valuesof the scale), the above definition reduces to the

standard RI-MOM definition of ZP. However,one immediately realizes that, at the leading or-der in the quark mass, the Goldstone boson con-tribution of eq. (11) cancels in the definition ofZP, which is thus well behaved in the chiral limit.To the calculation of ZP, the warning of ref. [16]does not apply anymore. The ratio ZP()/C(a)is plotted in fig. 2 as a function of the scale. Onecan see, in this case, that the desired constant be-haviour is reached much faster than with the ratioZP()/C(a). Moreover, the two ratios convergeto the same value at large scales, within less than5%, an uncertainty which is of the same order ofthe other systematic errors involved in the cal-culation. This analysis supports the assumptionof an adequate suppression of the Goldstone polecontribution in previous calculations of ZP.

A significant accuracy in the non-perturbativedetermination of the quark mass renormalizationconstant has been achieved by ALPHA [3], within


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the Schrodinger functional approach. Besides de-termining the value of mass renormalization con-

stant at a low hadronic scale, ALPHA has alsocomputed, non-perturbatively, the running factorC() over a large region extending from the lowhadronic scale up to a very high scale. This cal-culation, which is based on a recursive finite-sizescaling technique, allows a fully non-perturbativedetermination of the renormalization group in-variant quark mass. The non-perturbative calcu-lation ofC() turns out to be in very good agree-ment with the predictions of NLO perturbationtheory. The advantage of this approach is thatthe matching factor, relating the renormalizationgroup invariant mass to the MS mass, is known

at N4LO in perturbation theory [19,20]. The re-maining perturbative uncertainty in this determi-nation is thus reduced at a completely negligiblelevel. The ALPHA calculation of the mass renor-malization constant has been discussed by Sintat this conference, and I refer to his review forfurther details [12].

3. LIGHT QUARK MASSES

3.1. Quenched Calculations

The determination of light quark masses is, atpresent, one of the main fields of activity of lat-

tice QCD simulations. This is illustrated in fig. 3,which presents a compilation of all lattice resultsfor the strange quark mass, in the quenched ap-proximation, starting from the first lattice predic-tion for ms obtained with a NLO accuracy [21].These results have been produced by using dif-ferent actions, different orders of improvement,different renormalization techniques and differentchoices of the input parameters, used to fix boththe scale and the strange quark mass itself (ei-ther mK or m). In spite of these differences, wefind good consistency among the several determi-nations. In addition, all the results, with the pos-sible exception of the rather large value obtainedby SESAM [22], suggest an interval of allowed va-lues for the strange quark mass which is smallerthan the one quoted by the particle data group(PDG) [23], indicated by the band in the figure.One can also observe that the lattice determina-tions of the strange quark mass cluster close to

50

100

150

200

50

100

150

200

m

s(2 GeV) [Quenched]

O(a2)

NPR

O(a2) + NPR

1994 1996 1997 1998 1999 2000

Figure 3. Results of all lattice calculations of thestrange quark mass obtained, in the quenched ap-proximation, since 1994. The band shows the es-timate of ms quoted by the particle data group.The dashed line indicates the unitarity lowerbound on ms.

the unitarity lower bound, ms > 100 MeV [24],obtained by using unitarity and the positivity ofthe pseudoscalar spectral function.1 Finally, the

results which are plotted on the same vertical linein the figure have been obtained by using eitherthe kaon or the meson mass as input parameter.It has been shown that a systematic difference ofapproximately 20% is found between the two de-terminations within the quenched approximation.

In order to derive an average value for thestrange quark mass, I will concentrate in the fol-lowing on the more recent and accurate results,which are given in table 2 and shown in fig.4 withfull circles. In this case, only the results obtainedby using mK as experimental input have beenconsidered.

The comparison among the results presentedin table 2 becomes more significant if one triesto correct them by taking into account for thedifferent systematic errors involved in the calcu-

1The N2LO bound of ref. [24] slightly decreases to ms >95 MeV, when the N3LO perturbative corrections to thepseudoscalar spectral function are taken into account [ 25].


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Table 2Quenched lattice results for the strange quark mass obtained by using in input the value of the kaonmass. Details concerning the action, the value of the lattice spacing, the renormalization procedure andthe perturbative accuracy are also given.

Action a1 Zm PT ms(2 GeV)/ MeVAPE 98 [13] NP-Clover 2.7 GeV (m) NP-RI N

2LO 111 12JLQCD 99 [14] KS a 0 (m) NP-RI N

2LO 106 7CP-PACS 99 [26] Wilson a 0 (m) PT NLO 116 3ALPHA-UKQCD 99 [11] NP-Clover a 0 (r0, fK) NP-SF N

4LO 97 4QCDSF 99 [27] NP-Clover a 0 (r0) NP-SF N4LO 105 4APE 99 [5] NP-Clover 2.7 GeV (m) NP-RI N3LO 111 9CP-PACS 00 [28] MF-Clover a 0 (m) PT NLO 110

+34

RBC 00 [29] DWF 1.9 GeV (m) NP-RI N3LO110 2 22

105 6 21

From vector WI; from axial WI.

50 70 90 110 130 150 170 19050 70 90 110 130 150 170 190

m

s(2 GeV)

QUENCHED

APE 98

JLQCD 99

CPPACS 99

ALPHA +

UKQCD 99

QCDSF 99

APE 99

RBC 00

CPPACS 00

AVERAGE

PDG

Figure 4. Values of the strange quark massobtained from recent lattice calculations in thequenched approximation (full circles). Empty cir-cles denote the values and errors obtained by theauthor attempting to correct for the different sys-tematics.

lations. In this way, I obtain the estimates shownin fig.4 as empty circles.

All results in table 2 have been obtained adopt-ing a non-perturbative renormalization techni-que, with the only exceptions of the two CP-PACS determinations [26,28], in which the quarkmass renormalization constant has been evalu-

ated using one-loop perturbation theory. An ad-ditional uncertainty, due to the use of perturba-tion theory, should then be added to these results,to account for the corresponding systematic er-ror. This uncertainty, however, is difficult to bequantified, because two different renormalizationconstants (Z1S and ZA/ZP) have been used inthe calculation and, moreover, non-perturbativedeterminations of renormalization constants withthe Iwasaki action used in ref. [28] have not beenperformed yet. In fig. 4, a systematic error of 10%has been added to the CP-PACS results. This er-

ror, however, may be underestimated, accordingto the analysis of the previous section (see ta-ble 1).

The values of the strange quark mass by AL-PHA-UKQCD [11] and QCDSF [27] have beenobtained by fixing the lattice spacing from thevalue of the phenomenological parameter r0 [32],rather than from m as in all other determina-tions. QCDSF finds that the choice of r0 isroughly equivalent to using m, while ALPHA-UKQCD finds that this is equivalent to settingthe scale from fK. ALPHA-UKQCD also esti-mates that a 10% higher value of m

swould have

been obtained by fixing the scale from the valueof the nucleon mass. Since the scale determinedfrom m turns out to be typically in between thevalues obtained from fK and mN, in order tocompare with the other determinations I have in-creased by 5% in fig.4 the values of ms obtained


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by ALPHA-UKQCD and QCDSF.The other corrections to the results presented

in table 2 are smaller. The APE determinationsof refs. [5,13], obtained with the non-perturbativeClover action at = 6.2, have not been extrapo-lated to the continuum limit. In this case, themore extensive analysis by ALPHA-UKQCD [11]shows that the value of the strange quark mass isunderestimated by approximately 3%. Finally, inthe calculations by APE [13] and JLQCD [14] theconversion from the non-perturbative RI-MOMrenormalization scheme to the MS scheme hasbeen done at the N2LO in perturbation the-ory [30], since at the time when these studieshave been performed the N3LO perturbative cal-

culation of [31] was not available yet. In fig.4, inorder to account for the difference between N2LOand N3LO, the results of refs. [13] and [14] havebeen decreased by 3%.

I would like to emphasize that the previous dis-cussion, besides exploiting the different sourcesand sizes of systematic errors, also shows that arather good level of statistical and systematic ac-curacy has been achieved in the determinations oflight quark masses within the quenched approxi-mation. Indeed, the values of ms shown in fig. 4are in remarkable agreement within each other,and even more when the differences in the sys-

tematics have been taken into account. From thespread of the results, and accounting for the typi-cal uncertainty on the scale in the quenched case( 10%), I obtain as a final average of the latticeresults, within the quenched approximation,

ms(2 GeV)QUEN = (110 15) MeV (13)

This value is shown in fig.4, together with the se-veral lattice results and the average value of msquoted by the PDG [23]. Note that the uncer-tainty in eq. (13) is approximately three timessmaller than the one quoted by the PDG, al-though, in the former, the effect of the quenchingapproximation has not been taken into accountyet.

In the light quark sector, the other quantity,besides ms, which is accessible to lattice calcu-lations is the average value of the up and downquark masses, mud (mu + md) /2. Rather thenthe value of mud itself, however, I find more con-

Table 3Quenched lattice results for the ratio of thestrange to the average up-down quark masses.

2ms/(mu + md)APE 98 [13] 24.6 2.2JLQCD 99 [14] 25.1 1.7CP-PACS 99 [26] 25.3 1.0QCDSF 99 [27] 23.9 1.1APE 99 [5] 23.4 2.4CP-PACS 00 [28] 25.2 1.0

The errors are my estimates based on the originaldata.

venient to consider the ratio of the strange tothe average up-down quark masses, because in

this ratio many of the systematic uncertainties,coming for instance from the renormalization con-stants or the choice of the scale, are expected topartially cancel. The values of ms/mud from themost recent lattice calculations are collected intable 3 and shown in fig. 5. The quoted errors

15 20 25 3013

15

17

19

21

15 20 25 30

2ms/(m

u+m

d)

QUENCHED

APE 98

JLQCD 99

CPPACS 99

QCDSF 99

APE 99

CPPACS 00

ChPT

Figure 5. Ratio of the strange to the average up-down quark masses as obtained from recent latticecalculations in the quenched approximation (cir-cles). The errors are my estimates based on theoriginal data. The lowest point (diamond) repre-sents the prediction of chiral perturbation theory.

are my estimates based on the original data, be-


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cause the value of this ratio has not been quoteddirectly in the original papers.

The relevant result, shown in fig. 5, is that thelattice predictions for ms/mud are in very goodagreement with the value 24.4 1.5 predicted bychiral perturbation theory [33]. By using this in-formation and eq. (13), I obtain the quenched ave-rage:

mud(2 GeV)QUEN = (4.5 0.6) MeV . (14)

Finally, I would like to mention that the first(quenched) lattice calculation of mud with over-lap fermions has been presented at this confer-ence [34]. The preliminary result is mud(2 GeV)=5.5(5) MeV, in good agreement with the avera-

ge (14).

3.2. Unquenched Results

Several unquenched calculations of light quarkmasses, with two flavours of dynamical quarks,have been presented at this conference. A compi-lation of the results, together with details of thesimulations, is given in table 4. With the excep-tion of the SESAM [35] and CP-PACS [28] deter-minations, all other results presented in the tableare still preliminary.

A striking result shown in table 4 is the valueof the ratio of the strange to the average up-

down quark masses obtained by SESAM [35]:ms/mud 55. This value is larger, by roughly afactor 2, than all other quenched and unquenchedlattice determinations (including the quenchedresult by SESAM [22]), and in disagreement withthe prediction of chiral perturbation theory. Thereason for such a large value of ms/mud obtainedby SESAM is the way in which the unquenchedanalysis has been performed. Indeed, as discussedbelow, by following a different procedure they ob-tain ms/mud 30 [22], which is much closer tothe expected value. I believe this point is worthto be discussed in details, because it is also ofrelevance for other unquenched studies.

In numerical simulations with dynamical fer-mions all physical quantities, as well as the latticespacing and the critical value of the hopping pa-rameter, are functions of the two parameters en-tering in the action, namely the coupling constantand the sea quark mass. Theoretical considera-

tions suggest that the dependence of the latticespacing on the value of the sea quark mass should

be exponential [40]. Indeed, the lattice scale de-pends exponentially on the effective coupling con-stant which, in turn, is affected by the values ofquark masses because quarks run in the loops. Alarge dependence of the lattice spacing on the seaquark mass is in fact observed in actual calcula-tions. For instance, JLQCD [38] finds that, inthe range of Ksea used in the simulation (corre-sponding to mPS/mV 0.6 0.8), the inverselattice spacing varies approximately by 30%, be-tween 1.5 and 2.0 GeV. The important pointis that such a dependence, if not properly takeninto account, may lead to uncontrolled systematic

effects in the various extrapolations, in both thesea and valence quark masses.

In order to control these effects, an appropriateprocedure for the unquenched analysis has beenadvocated [40,41]. It consists in performing allthe analysis, including the calibration of the lat-tice spacing and the calculation of the relevantphysical quantities, at each fixed value of Ksea.The simulations performed at fixed Ksea can bethought of as pseudo-quenched simulations, whichcome closer to the description of the real world asthe sea quark mass approaches its physical value.Only when all the quantities, computed at fixed

Ksea, are expressed in physical units, the resultscan be extrapolated in the sea quark mass.

This procedure has been followed by APE [37]in the calculation of the strange quark mass.At two fixed masses of the sea quark, both thestrange and the average up-down quark masseshave been computed. The results are shownin fig. 6, in terms of the ratios mud/mK andms/mK . Presenting these ratios is equivalentto show the quark masses in physical units. Thevalues of mud are then extrapolated in the seaquark mass, also computed in physical units, upto the point corresponding to msea = mud. Oncethe physical value ofmud has been determined inthis way, a similar extrapolation is performed forthe strange mass. From fig. 6 one can see thatthe final extrapolations in the sea quark mass areindeed smooth.2

2Since, however, only two values ofKsea have been simu-


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Table 4Simulation details and physical results of unquenched lattice calculations of light quark masses.

Action a1

#(,Ksea) Zm ms(2 GeV) ms/mud m

QUEN

s /mUNQ

s

SESAM 98 [35] Wilson 2.3 GeV 4 PT 151(30) (mK,) 55(12) 1.10(24)

MILC 99 [36] Fatlink 1.9 GeV 1 PT113(11)

125(9)

(mK)

(m)22(4) 1.08(13)

APE 00 [37] Wilson 2.6 GeV 2 NP-RI112(15)

108(26)

(mK)

(m)26(2) 1.09(20)

CP-PACS 00 [28] MF-Clover a 0 12 PT88+4690+511

(mK)

(m)26(2) 1.25(7)

JLQCD 00 [38] NP-Clover 2.0 GeV 5 PT

94(2)

88(3)

109(4)

102(6)

(mK)

(m)

QCDSF +

UKQCD 00 [39]NP-Clover 2.0 GeV 6 PT 90(5) (mK) 26(2)

From vector WI; from axial WI. The errors on the ratios ms/mud and mQUENs /m

UNQs are my estimates

based on the original data.

In the analysis by SESAM and CP-PACS thepseudoscalar and vector meson masses have beenfitted simultaneously in both the sea and the va-

lence quark masses, with all quantities expressedin lattice units. The large dependence of the lat-tice spacing on the sea quark mass may not betaken properly into account in this way. Indeed,SESAM finds that the value of ms/mud changesby a factor 2 when the analysis, instead, is per-formed at fixed values of Ksea, as advocated be-fore. Moreover, the result obtained with the anal-ysis performed at fixed Ksea is much closer to theexpected value of this ratio. The systematic ef-fects, introduced by the neglecting of the latticespacing dependence on the sea quark mass, aredifficult to evaluate a priori, and depend on the

specific values of the parameters used in the cal-culation. JLQCD, for instance, attempted thechiral extrapolations by using either dimension-less or dimensionful quantities [38] (although con-sidering different orders in the expansions), and

lated by APE, the results should be taken as preliminary.

obtained for ms compatible results.Since unquenched studies of light quark masses

have not yet reached the same degree of accu-

racy achieved in quenched calculations, in orderto obtain an estimate of the quenching effect it isconvenient to consider directly the ratio betweenquenched and unquenched results. Some of thesystematic errors are expected to cancel in thisratio, when both determinations are performedby using the same action, the same renormaliza-tion procedure and the same choice of physicalinputs. The results for the ratio mQUENs /m

UNQs

are presented in table 4. In the very exten-sive calculation performed by CP-PACS the valuemQUENs /m

UNQs = 1.25(8) is obtained, suggesting

a sizable decrease of the quark mass in the un-

quenched case. This result, however, is not con-firmed by the other (less extensive) studies bySESAM, APE and MILC, which find a decrease,in the unquenched case, rather of the order of10%. Given these results, and bearing in mind theuncertainties in the analysis methods, I believethat, in order to quote a final estimate of light


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0.00 0.05 0.10 0.15 0.20

mseaRI(3 GeV)/mK*

0.100

0.125

0.150

ms

RI(3GeV)/mK*

0.000

0.005

0.010

mu,d

RI(3GeV)/m

K*

ms

ea

=m

u,d

(a)

(b)

Figure 6. Extrapolation of the average up-down(a) and the strange (b) quark masses, computed at

fixedKsea, to the physical value of the sea quarkmass, msea = mud.

quark masses, it is more appropriate to includethe quenching error as an additional systematicuncertainty in the quenched averages (eqs. (13)and (14)), rather than varying the central values.Assuming this error to be of the order of 20 MeVin the case of the strange mass, I then obtain:

ms(2 GeV) = (110 15 20) MeV (15)

mud(2 GeV) = (4.5 0.6 0.8) MeV (16)

Note also that in the unquenched case (see ta-ble 4) the differences between the determinationsof the strange quark mass from mK and m, ifany, are rather small.

4. THE b-QUARK MASS

The b-quark cannot be directly simulated onthe lattice since its mass is larger than typical va-lues of the ultraviolet cutoff in present lattice cal-culations (a1 3 GeV). The b mass, however, isalso larger than the typical energy scale of stronginteractions. Thus, the heavy degrees of freedomof the b-quark can be integrated out. Beautyhadrons may be therefore simulated on the latticein the framework of a low-energy effective theory.Indeed, in the past years, many lattice simula-

tions of Heavy Quark Effective Theory (HQET)and Non-Relativistic QCD (NRQCD) have been

performed.Within the effective theory, the mass of a B-meson, MB, is related to the pole mass of the b-quark. Up to higher-order 1/mb corrections, thisrelation has the form:

MB = mpoleb + m (17)

where is the so-called binding energy, whichcan be computed non-perturbatively by a nume-rical simulation, and m is the residual mass, ge-nerated by radiative corrections, which can beevaluated in perturbation theory. The perturba-tive calculation of the residual mass and the non-

perturbative (lattice) calculation of the bindingenergy thus allow a determination of the b polemass. The result can be then translated into theMS mass, mb, by using perturbation theory.

An important observation, concerning this pro-cedure, is that the binding energy is not a phy-sical quantity, and it is affected by power diver-gences proportional to the inverse lattice spacing,1/a. These divergences are canceled by similarsingularities in the residual mass m. Moreover,the pole mass mpoleb , and consequently m, arealso affected by renormalon singularities, whichintroduce in their definitions an uncertainty ofthe order of QCD [42,43]. These singularities arethen canceled by the perturbative series relatingthe pole mass and the MS mass [44]. As a result,the MS mass, mb, is a finite, well defined, short-distance quantity. In the actual calculation, how-ever, since the residual mass is computed up toa finite order in perturbation theory, only a par-tial cancellation of both power divergences andthe renormalon singularities occurs. For this rea-son, it is crucial to compute the residual mass mup to the highest possible order in perturbationtheory. Moreover, because of the incomplete can-

cellation of power divergences, the lattice spacingin such calculations cannot be taken too small.At present, the most accurate determination of

m has been obtained in the framework of HQET.In terms of the lattice bare coupling constant, theperturbative expansion can be written as:

a m = C1 0 + C2 20 + C3

30 + . . . (18)


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The two-loop coefficient, C2 has been derivedanalytically, by Martinelli and Sachrajda [45],

from the study of the asymptotic behaviour ofWilson loops. In the quenched case (Nf = 0),also the coefficient C3 has been evaluated, byDi Renzo and Scorzato [46], using the methodof numerical stochastic perturbation theory. Animportant check of this approach is obtainedfrom the comparison of their numerical resultsfor the first two coefficients, C1 = 2.09(4) andC2 = 10.7(7), with the analytically known values:C1 = 2.1173 and C2 = 11.152 [45]. The value ob-tained for the third coefficient, C3 = 86.2(5), isalso consistent with the determination by Lepageet al. [47], C3 = 81(2), obtained, within a com-

pletely different approach, by fitting the resultsof small coupling Monte Carlo calculations.

These combined theoretical efforts allow a de-termination of the b-quark mass which is accu-rate, in the quenched approximation, up to theN3LO. Two independent results have been ob-tained so far:

m QUENb = (4.30 0.05 0.05) GeV [48] (19)

m QUENb = (4.34 0.03 0.06) GeV [49] (20)

nicely in agreement within each other. Remar-kably, the two determinations have been derivedusing different approaches. The APE result (19)

has been obtained within HQET, while the deter-mination (20), by Collins et al., is obtained fromthe NRQCD study of the B-meson spectrum inthe static limit. The last error in eqs. (19) and(20) represents the residual uncertainty due to theneglecting of higher orders in the perturbative ex-pansion of m. For comparison, this uncertaintywas estimated to be of the order of 200 MeV atNLO [50] and 100 MeV at N2LO [45] respectively.

Results compatible with those in eqs. (19) and(20) have been obtained from the study of the Band the spectrum with NRQCD [51,52]. In thiscase, however, only the coefficient C1 is known,so that the achieved accuracy is at NLO. Theseresults are thus affected by a larger perturbativeuncertainty, which can be estimated, on the basisof a renormalon analysis [44], to be of the orderof 100 MeV.

The first unquenched calculation of the b-quarkmass has been performed by APE this year [40].

The perturbative accuracy is at N2LO, since thethree-loop coefficient, in the expansion of m, is

yet unknown in the unquenched case. Their re-sult,

mb = (4.26 0.06 0.07) GeV . (21)

represents to date the most accurate determina-tion of the b-quark mass from lattice QCD calcu-lations. Remarkably, the relative uncertainty isreduced at the level of 2% (90 MeV).

A preliminary analysis [49] performed in thestatic limit with NRQCD indicates a decrease ofmb, in the unquenched case, by approximately 70MeV. Combined with the quenched determina-tion (20), this result leads to an estimate of theb-quark mass in good agreement with (21).

5. CONCLUSIONS

Significant progresses in lattice calculations oflight and heavy quark masses have been allowedby the introduction of improved actions and non-perturbative renormalization techniques.

Current determinations of light quark masseshave reached a good level of statistical and sys-tematic accuracy, which is of the order of 10%within the quenched approximation. The mainsource of uncertainty, in this case, is due to the

quenched approximation. Several unquenched re-sults, with two flavours of dynamical quarks, havebeen presented at this conference. The effect ofquenching is found to be in the range of 10-20%,but further investigations are required to obtaina firmer estimate.

I have not reported, in this review, lattice cal-culations of the charm quark mass, since recentresults for this quantity are missing. The charmmass is of great phenomenological interest sinceit enters, through the heavy quark expansion,the theoretical predictions of several cross sec-tions and decay rates. For this reasons, new lat-tice calculations with the non-perturbatively im-proved action and non-perturbative renormaliza-tion techniques is highly recommended.

Enormous progresses have been achieved in lat-tice calculations of the bottom quark mass. Thepresent accuracy is at N3LO in the quenched ap-proximation, and at N2LO in the unquenched


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case. The relative uncertainty is reduced at theimpressive level of 2%.

Final averages of lattice calculations for quarkmasses have been given in eqs. (15), (16) and (21).By combining statistical and systematic errors,these correspond to:

mud(2 GeV) = (4.5 1.0) MeV

ms(2 GeV) = (110 25) MeV (22)

mb(mb) = (4.26 0.09) GeV .

6. ACKNOWLEDGEMENTS

I thank D. Becirevic, C. Bernard, R. Burkhal-ter, C. Davies, F. Di Renzo, V. Gimenez, L. Giu-

sti, T. Kaneko, H. Leutwyler, K.F. Liu, G. Mar-tinelli, R. Sommer, M. Wingate and H. Wittig fordiscussions and private communications.

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Quark Masses on the Lattice