The mass of the charm quark is analyzed in the context of QCD finite energy sum rules using recent BESII e+e- annihilation data and a large momentum expansion of the QCD correlator which incorporates terms to order (alpha_s)^2 (m_c^2/q^2)^6. Using various versions of duality, we obtain the consistent result m_c(m_c)=(1.37 +- 0.09)GeV. Our result is quite independent of the ones based on the inverse moment analysis

Peñarrocha Gantes, José Antonio

Schilcher, K.

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arXiv:hep-ph/0105222v3  3 Jul 2001FTUV/01-0410IFIC/01-21MZ-TH/01-12April 2001QCD Duality and the Mass of the CharmQuark1J. Pen˜arrochaa and K. SchilcherbaDepartamento de F´isica Teo´rica-IFIC, Universitat de ValenciaE-46100 Burjassot-Valencia, SpainbInstitut fu¨r Physik, Johannes-Gutenberg-Universita¨tD-55099 Mainz, GermanyAbstractThe mass of the charm quark is analyzed in the context of QCDfinite energy sum rules using recent BESII e+e− annihilationdata and a large momentum expansion of the QCD correlatorwhich incorporates terms to order α2s(m2c/q2)6. Using variousversions of duality, we obtain the consistent result mc(mc) =(1.37 ± 0.09)GeV . Our result is quite independent of the onesbased on the inverse moment analysis.1Supported by CICYT under contract AEN99/0692, EC-RTN under contract HPRN-CT2000/130, Generalitat Valenciana under contract GV98-1-80 and partnership Mainz-Valencia Universities11 IntroductionRecently the BESII collaboration has presented new data on the total e+e−annihilation cross section above the charm threshold [1]. Data in this en-ergy region are particularly relevant for the extraction of the charm quarkmass, one of the fundamental parameters of QCD. The charm quark canbe determined by comparing suitable positive moments of these data withthe corresponding moments of QCD perturbation theory. This direct quark-hadron duality approach was originally applied to the charm region in Ref.[2]. A reanalysis along these lines seems indicated in view of the fact thatapart from new data, enormous progress has been made in the theoreticalcalculation of the relevant QCD correlator in the region q2 ≫ m2c . The cor-relator is now known to O(α2s) and O(m12/q12) [3], so that the question ofconvergence can be meaningfully discussed. There exist also a result to O(α3s)for the quartic mass correction [4] which we will not consider for reasons ofconsistency. We believe that in the case of the charm quark mass the directduality approach employed by us is less prone to theoretical uncertainties asthe more popular one based on inverse moments [5],[6],[7],[8],[9]. It shouldbe pointed out, that we use as phenomenological input only the new BE-SII data. This is because older data in this region are plagued by unknownsystematical errors and appear to be mutually inconsistent. An alternativewould have been to adjust the normalizations of the various data sets so thatthey agree for large q2 with QCD [10].We will manipulate our data rather on the basis of QCD duality. Usingsuitable linear combinations of moments the emphasis of the hadronic integralcan be shifted at will to experimental regions where the data errors aresmall. This technique, which was originally proposed in [11], allows in manycases more accurate prediction, and supplies in addition beautiful consistencychecks. This will be explained in the following.2 Cauchy sum ruleQCD duality means that the theoretical and phenomenological informationis being related by means of Cauchy sum rule∫ Rs01piImΠ(s)p(s)ds = −12pii∮|s|=RΠQCD(s)p(s)ds (1)where ImΠ(s) is defined in terms of the total e+e− annihilation cross-sectionbyR(e+e− → hadrons) = 12pi∑flavorsQ2f ImΠ(s) (2)2The experimental charm physical threshold in Eq.(1) is taken from the J/Ψresonance masss0 = (3.097Gev)2 (3)We have included in the sum rule a polynomial weight function p(s)p(s) =∑ansn (4)which makes the sum rule a linear combination of moments of Cauchy sumrules. The polynomial may be chosen in a suitable way to enhance or removepart of the phenomenological input in the calculation.3 QCD integralThe two-point function ΠQCD(s) is known to O(α2s) as a series expansion inpowers of m2/s up the sixth power,ΠQCD(s) =6∑i=03∑j=0Aij(m,µ)(m2s)i (ln−sµ2)j(5)where the coefficients Aij(m,µ) may contain powers of mass logarithms ln(m2/µ2).The convergence of the series is not seriously affected by the mass logarithmsprovidedlnµ2m2≪ lnµ2Λ2QCD. (6)This is always the case for the scales of µ (∼ 5GeV ) we use.At tree level, the first few terms in the expansion of ΠQCD(s) are givenbyΠQCD(s) =316pi2209−43ln−sµ2+ 8m2s+(m2s)2 (4− 8 lnm2µ2+ 8 ln−sµ2)+ ..We use the strong coupling constant α2s and running MS mass renormalizedat the scale µ. The lengthy full expression to O(α2s) and O(m12/q12) may befound in Ref.[3].There is also a small non-perturbative contribution arising from the gluoncondensate, which is known to O(αs) [12]. The first few terms areΠnp(s) =〈αspiGG〉− 112(m2s)2−13(m2s)3−(m2s)4 (−73− lnm2µ2+ ln−sµ2)+ ..3The Cauchy-Integral in Eq.(1) needs the evaluation ofJ(k, j) =12pii∮|s|=Rsk(ln−sµ2)jds (7)for j = 0, 1, 2, 3 and k = −6,−5,−4, ...These integrals can be evaluatedanalytically with the resultJ(k 6= −1, j) =12pii∮|s|=Rsk(ln−sµ2)jds =Rn+1n+ 1×[j lnj−1Rµ2−j(j − 1)n + 1lnj−2Rµ2−(pi26−1(n+ 1)2)j(j − 1)(j − 2) lnj−3Rµ2]andJ(k = −1, j) =12pii∮|s|=R1s(ln−sµ2)jds= lnjRµ2−pi26j(j − 1) lnj−2Rµ2See also Ref.[13]4 Data integralThe BESII data for the total e+e− annihilation cross section is representedin figure 1.The errors given by the authors of Ref. [1] distinguish between systematicand statistical errors. In our analysis this distinction is absolutely essential asthe statistical errors average out almost completely in the integration processwhile the systematic errors prevail. The threshold of the charm continuumis at twice the D0 mass which corresponds to s0cont = 13.9 GeV2. Fordefiniteness we discuss here only the integration up to smax = 25GeV2, themaximum value measured by BESII. If we require for duality that the datawithin their errors agree with QCD perturbation theory then it is seen thatsmax may also be chosen somewhat lower. We defineI1 =∫ 2513.9Rcontinuum(s)p(s)ds (8)From this integral over the total cross section the contribution of the lightquarks must be subtracted to isolate the pure charm contribution. As s >4x2422201816144.543.532.5Figure 1: R(e+e− → hadrons) from BES data as a function of x = (cm −energy)2.13.9GeV 2 it is perfectly safe to use QCD perturbation theory for contributionof the light flavors. We use the four loop result for the massless correlatorI2 = 2∫ 2513.9(1 +αspi+ ...)p (x) dx (9)where terms up to O(α4s) may be found in refs. [14], [15],[16]. We use here thehighest known order in αs because I2 may be considered as an experimentalinput. The charm continuum contribution is then given byIcont = I1 − I2 (10)Finally we have to take into account the two J/Ψ and Ψ′ charmoniumresonances below the continuum thresholdIres = 9pi(137.04)2(m1Γ1p(m21)+m2Γ2p(m22))(11)where the resonance masses and widths, are given bym1 = 3.0969GeVΓ1 = (5.26± 0.37)× 10−6GeV (12)m2 = 3.6860GeVΓ2 = (2.12± 0.18)× 10−6GeV.The complete hadronic contribution to the sum rule is sum of these twointegralsIcharm = Icont + Ires (13)55 Polynomial weight functionsThere is much freedom in the choice of the polynomial in the duality relation,Eq.(1). In this note we will use the polynomial to reduce the importance ofthe continuum contribution relative to the well established sub-thresholdresonances. We impose the following conditions:The polynomial p(s) = 1 at s = m2J/Ψ, it vanishes at the end of theintegration range (s = 25Gev2), and it is small in the continuum region. Tobe specific we choose an n-th order polynomial pn(s) = a0 + a1s + .. + ansnand determine the coefficients by the constraintspn(25) = 0pn(m2J/Ψ) = 1 (14)∫ 253.9skpk(s)ds = 0, k = 0, 1, .., n− 1For a 3-degree polynomial we obtain, for examplep3(s) = 7. 933 787 5− 1. 209 157 911s+6. 015 360 076× 10−2s2 (15)−9. 792 537 729× 10−4s3This result is plotted in Fig.2.x242220181614121010.80.60.40.20Figure 2: Polynomial fit vanishing at the continuum range (13.9− 25)GeV 2.It is obvious from the figure that the continuum data will almost cancelwhen integrated with this polynomial.66 ResultsFrom the duality sum rule Eq.(1) we obtain with the help of Eqs.(8-13) andEqs.(5, 7)Icharm = −n∑k=06∑i=03∑j=0akAij(mc, µ)(m2c)iJ(k − i, j) (16)which can be solved for mc.We present here predictions for simple duality, i.e. p(s) = 1, and for the3rd-degree polynomial above. The unknown mc in Eq.(16) is the runningcharm mass at a scale µ., which we fix to be µ = 5GeV , the relevant scale ofthe problem.For the coupling constant αs we take as an input its value at the mass ofthe tau lepton [17]αs(mτ ) = 0.345± 0.020 (17)with mτ = 1.777GeV. After appropriate matching [18] from 3 to 4 flavorsthis corresponds toα4f (5GeV ) = 0.224± 0.013 (18)For simple duality, we plot in Fig.3 the contribution of the QCD integral(lhs of Eq.(16)), at tree level and first and second order in the strong coupling,as a function of the charm quark mass mc.m21.510.503530252015105Figure 3: QCD integral without polynomial weight as a function of mass.The three curves (from top to bottom at m = 2) represent tree, first andsecond order calculations in the strong coupling constant.7For the Data integral we have the continuum contribution Icont = 14.06GeV2and the resonance contribution Ires = 12.80GeV2 with their sum being thetotal charm data integralIcharm = 26.86GeV2 (19)Solving Eq.(16) for this value of charm data, the results for the massof the charm quark at different orders in the perturbative expansion are:m(0)c = 0.916GeV , m(1)c = 0.980GeV and m(2)c = 0.990GeV . We see that theconvergence of the QCD asymptotic expansion is extremely good. The mainsource of uncertainties come from the strong coupling constant and data.The result that we quote with this approach ismc(µ = 5GeV ) = (0.99± 0.01asymp ± 0.01αs ± 0.04res ± 0.10cont)GeV (20)The asymptotic uncertainty comes from the difference of two- and three-loopresults used for the QCD correlators. The result of Eq. 20 corresponds to aninvariant mass [19]mc(mc) = (1.40± 0.11)GeV (21)In this final result the errors have been added quadratically.The contribution of the gluon condensate is completely negligibleOur second approach consist in plug-in the 3rd degree polynomial inEq.(15) into Eq.(16) in order to minimize the contribution from the charmcontinuum data, and therefore minimize the error involved in these data. Asbefore we plot in Fig.4 the QCD integral as a function of mc for differentorders in the perturbative expansionThe contribution from the continuum data is now Icont = −0.176Gev2,whereas the resonance contribution, practically from J/Ψ resonance, is Ires =9.341GeV 2. The complete charm data contribution is thenIcharm = 9.165GeV2 (22)With this value we solve again Eq.(16) at different orders in the perturbativeexpansion of αs with the results: m(0)c = 1.153GeV ,m(1)c = 0.991GeV ,m(2)c =0.927GeV . We see that, although we have eliminate the uncertainty comingfrom continuous data, the price we pay is that the asymptotic expansion doesnot converge so nicely as before, but still good enough to make a sensibleprediction for the mass of the charm quark. We havemc(µ = 5GeV ) = (0.93± 0.06asymp ± 0.015αs ± 0.014res)GeV (23)This corresponds to an invariant mass [19]mc(mc) = (1.34± 0.08)GeV (24)8m21.510.5050403020100Figure 4: QCD integral with polynomial weight as a function of mass. Thethree curves (from top to bottom at m = 1) represent tree, first and secondorder calculations in the strong coupling constant.The influence of the gluon condensate is in this approach∼ 0.3% for 〈αsGG/pi〉 ∼.024GeV 4, still negligible.The result we find is perfectly compatible, within error-bars, with the onewe found above by the simple duality approach. This agreement constitutesa non trivial confirmation of the duality ansatz. Averaging the results ofboth approaches, we finally findmc(mc) = (1.37± 0.09)GeV. (25)Our value for mc(mc) appears to be slightly bigger than the ones given byalternative QCD sum rule methods. With our normalization point the latterresults readmc(mc) = (1.29±0.05)GeV [7] andmc(mc) = (1.28±0.06)GeV [8].Although these values agree, within error bars, with ours, it should be keptin mind that these authors use older and lower values of the QCD couplingconstant, so that the error-bar quoted should really be larger.7 ConclusionsIn this letter we have analyzed the mass of the charm quark in the context ofQCD finite energy sum rules. In the phenomenological side of the sum rulewe use recent BESII e+e− data, whereas in the theoretical side we employthe a large momentum expansion of QCD vector correlator function up toO(α2s) and O(m12c /q12) . Two approaches are considered. The first oneuses simple Cauchy sum rule for the correlator. The second one includes a9polynomial in the sum rule to minimize the contribution of the continuumdata. The results from both approaches are nicely compatible with eachother. Whereas with the first approach suffers from a substantial uncertaintyarising from the continuum data, the second one shifts this uncertainty toQCD asymptotic expansion. The results allow a nice consistency check ofQCD duality assumption. More precise results need either better data orfurther terms in QCD asymptotic expansion.10References[1] J. Z. Bai et al. (BES collaboration), hep-ex/0102003, Feb 2001.[2] S. Narison and E. de Rafael. Nucl. Phys. B169 (1980) 253.[3] K. G. Chetyrkin, R. Harlander, J. H. Ku¨hn and M. Steinhauser,Nucl.Phys. B503 (1997) 339.[4] K. G. Chetyrkin, R. V. Harlander and J. H. Ku¨hn, Nucl.Phys. B586(2000).[5] M. Eidemuller and M. Jamin, Phys.Lett. B498 (2001) 203.[6] S. Narison, Nucl.Phys.Proc.Suppl. 74 (1999) 304.[7] S. Narison, Phys.Lett. B341 (1994) 73.[8] C. A. Dominguez, G. R. Gluckman and N.Paver, Phys.Lett. B333 (1994)184.[9] Antonio Pineda, ”Determination of the bottom quark mass from theΥ(1S) system”, University of Karlsruhe preprint Nr. 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QCD duality and the mass of the Charm Quark

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura