The running charm-quark mass in the $\bar{MS}$ scheme is determined from
weighted finite energy QCD sum rules (FESR) involving the vector current
correlator. Only the short distance expansion of this correlator is used,
together with integration kernels (weights) involving positive powers of $s$,
the squared energy. The optimal kernels are found to be a simple {\it pinched}
kernel, and polynomials of the Legendre type. The former kernel reduces
potential duality violations near the real axis in the complex s-plane, and the
latter allows to extend the analysis to energy regions beyond the end point of
the data. These kernels, together with the high energy expansion of the
correlator, weigh the experimental and theoretical information differently from
e.g. inverse moments FESR. Current, state of the art results for the vector
correlator up to four-loop order in perturbative QCD are used in the FESR,
together with the latest experimental data. The integration in the complex
s-plane is performed using three different methods, fixed order perturbation
theory (FOPT), contour improved perturbation theory (CIPT), and a fixed
renormalization scale $\mu$ (FMUPT). The final result is $\bar{m}_c (3\, {GeV})
= 1008\,\pm\, 26\, {MeV}$, in a wide region of stability against changes in the
integration radius $s_0$ in the complex s-plane.Comment: A short discussion on convergence issues has been added at the end of
  the pape

A. Pich

C. A. Dominguez

C. Pahl

J. Bordes

J. Peñarrocha

K. Schilcher

P. Colangelo

S. Bodenstein

English

arXiv

Crossref

Physical Review D

Charm-quark mass from weighted finite energy QCD sum rules

The running charm-quark mass in the scheme is determined from weighted finite energy QCD sum rules involving the vector current correlator. Only the short distance expansion of this correlator is used, together with integration kernels (weights) involving positive powers of s, the squared energy. The optimal kernels are found to be a simple pinched kernel and polynomials of the Legendre type. The former kernel reduces potential duality violations near the real axis in the complex s plane, and the latter allows us to extend the analysis to energy regions beyond the end point of the data. These kernels, together with the high energy expansion of the correlator, weigh the experimental and theoretical information differently from e. g. inverse moments finite energy sum rules. Current, state of the art results for the vector correlator up to four-loop order in perturbative QCD are used in the finite energy sum rules, together with the latest experimental data. The integration in the complex s plane is performed using three different methods: fixed order perturbation theory, contour improved perturbation theory, and a fixed renormalization scale mu. The final result is (m) over bar (c)(3 GeV) = 1008 +/- 26 MeV, in a wide region of stability against changes in the integration radius s(0) in the complex s plane.This work was supported in part by the European FEDER and Spanish MICINN under Grant No. MICINN/FPA 2008-02878, by the Generalitat Valenciana under Grant No. GVPROMETEO 2010-056, by NRF (South Africa), and by DFG (Germany). One of us (S. B) thanks P. Maier, M. Steinhauser, and C. Sturm for helpful correspondence.Peer reviewe

Bodenstein, S.

Bordes, José

Domínguez, C. A.

Peñarrocha, J. A.

Schilcher, K.

Digital.CSIC

arXiv:1009.4325v2  [hep-ph]  1 Dec 2010UCT-TP-281/10MZ-TH/10-34Charm-quark mass from weighted finite energy QCD sum rulesS. Bodenstein,1 J. Bordes,2 C. A. Dominguez,1, 3 J. Pen˜arrocha,2 and K. Schilcher41Centre for Theoretical & Mathematical Physics,University of Cape Town, Rondebosch 7700, South Africa2Departamento de F´ısica Teo´rica, Universitat de Valencia,and Instituto de F´ısica Corpuscular, Centro Mixto Universitat de Valencia-CSIC3Department of Physics, Stellenbosch University, Stellenbosch 7600, South Africa4Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Staudingerweg 7, D-55099 Mainz, Germany(Dated: December 2, 2010)The running charm-quark mass in the MS scheme is determined from weighted finite energyQCD sum rules (FESR) involving the vector current correlator. Only the short distance expansionof this correlator is used, together with integration kernels (weights) involving positive powers of s,the squared energy. The optimal kernels are found to be a simple pinched kernel, and polynomialsof the Legendre type. The former kernel reduces potential duality violations near the real axisin the complex s-plane, and the latter allows to extend the analysis to energy regions beyond theend point of the data. These kernels, together with the high energy expansion of the correlator,weigh the experimental and theoretical information differently from e.g. inverse moments FESR.Current, state of the art results for the vector correlator up to four-loop order in perturbative QCDare used in the FESR, together with the latest experimental data. The integration in the complexs-plane is performed using three different methods, fixed order perturbation theory (FOPT),contour improved perturbation theory (CIPT), and a fixed renormalization scale µ (FMUPT). Thefinal result is m¯c(3GeV) = 1008 ± 26MeV, in a wide region of stability against changes in theintegration radius s0 in the complex s-plane.PACS numbers: 12.38.Lg, 11.55.Hx, 12.38.Bx, 14.65.DwConsiderable progress has been made over the years toextract accurate values of the quark masses by con-fronting QCD with experimental data in the frameworkof QCD sum rules [1]. We consider in this paper thecase of the charm-quark mass. Following pioneering ap-proaches [2], it has been customary to write the OperatorProduct Expansion (OPE) of the vector correlator withthe scale invariant massmc(mc) as the expansion param-eter. This is in contrast to the case of light quarks wherethe square of the four-momentum Q2 ≡ −q2 is the nat-ural large expansion parameter. If one were to considerthe charm quark mass in e.g. theMS scheme at a typicalscale of 3 GeV, then the fact that m¯c(3 GeV) ≃ 1 GeVshould be a matter of some concern regarding this ex-pansion [3]. In any case, the latest determinations basedon inverse moment QCD sum rules claim an accuracy atthe 1% level [4]. These inverse moments require QCDknowledge of the vector correlator in the low energy re-gion, around the open charm threshold, as well as in thehigh energy region. An alternative approach involvingpositive moment sum rules, requiring QCD informationonly at high energy, and naturally suited to determinem¯c(µ) in theMS scheme, was proposed some time ago in[5]. We follow this approach here, and make use of stateof the art QCD information up to four-loop level [6]-[17],together with the latest experimental data [18]-[21]. Weconsider Finite Energy Sum Rules (FESR) weighted bytwo types of integration kernels. The first type is theso called pinched kernel, which has been shown to min-imize potential duality violations close to the real axisin the complex s-plane [22]. The second type is a poly-nomial (Legendre) kernel tuned to minimize the impactof the energy region where the data is either poor ornon-existent. Effectively, these Legendre-type integra-tion kernels, involving several positive powers of the en-ergy, allow for an extension of the analysis beyond theend point of the data. Both types of kernels have beenused successfully in the light quark sector to study thesaturation of chiral sum rules [23], to quantify dualityviolations [24], to extract the values of the vacuum con-densates in the OPE [25], and to determine the quarkcondensates [26] and light quark masses [27].We begin by considering the vector current correlatorΠµν(q2) = i∫d4x eiqx〈0|T (Vµ(x) Vν(0))|0〉= (qµ qν − q2gµν) Π(q2) , (1)2where Vµ(x) = c¯(x)γµc(x). Invoking Cauchy’s theoremin the complex s-plane (−q2 ≡ Q2 ≡ s) one has∫ s00p(s)1piIm Π(s) ds = − 12pii∮C(|s0|)p(s) Π(s) ds ,(2)where p(s) is an arbitrary but analytical integration ker-nel, andIm Π(s) =112piRc(s) , (3)with Rc(s) the standard R-ratio for charm production.The perturbative QCD (PQCD) expression of Π(s) athigh energies can be written asΠ(s)|PQCD =∑n=0(αs(µ2)pi)nΠ(n)(s) , (4)whereΠ(n)(s) =∑i=0(m¯2cs)iΠ(n)i , (5)and m¯c stands for the running charm-quark mass in theMS-scheme. The complete analytical PQCD result forΠ(s)|PQCD up to order O [α2s(m¯2c/s)6] is given in [6], andexact results for Π(3)0 and Π(3)1 are from [7]. The func-tion Π(3)2 is known exactly up to a constant [8] whichhas been estimated using Pade´ approximants in [9]. Atfive-loop order O(α4s) the full logarithmic terms for Π(4)0are given in [10], and for Π(4)1 in [11]. Since there is in-complete knowledge at this loop-order, we shall use theavailable information as a measure of the truncation er-ror in PQCD.The fundamental parameters entering the QCD corre-14 15 16 17 18 19 200.80.91.01.11.2s0 HGeV2LmcH3GeVLHGeVLFIG. 1: The running mass m¯c(3 GeV) in theMS scheme for afixed µ = 3.0 GeV (FMUPT), as a function of s0 and using thepinched kernel Eq. (6). Results from FOPT and CIPT leadto similar stability regions. Errors are due to uncertaintiesin the data, to the values of αs and the gluon condensate, toPQCD truncation, and to changes in µ of ± 50%.lator are the running strong coupling αs(µ2), and thegluon condensate. Regarding the strong coupling, weuse the latest comprehensive update analysis at theτ -scale [28] which gives αs(M2τ ) = 0.342 ± 0.012, orαs(M2Z) = 0.1213 ± 0.0014. This value agrees withinerrors with a recent determination [29] from e+e− anni-hilation data: αs(M2Z) = 0.1172 ± 0.0051. However, aworld average with much smaller errors [30] gives the re-sult αs(M2Z) = 0.1184±0.0007, in agreement with latticeQCD results [31]. We shall include both sets of values ofαs in our determination of m¯c. In the non-perturbativesector the leading power correction in the OPE involvesthe gluon condensate, i.e.〈(αs/pi)G2〉. The latest valueof the gluon condensate [25], extracted from the ALEPHdata on τ -decays, is〈(αs/pi)G2〉= (0.046±0.012) GeV4,for ΛQCD = 300 MeV, and〈(αs/pi)G2〉= (0.021 ±0.006) GeV4 for ΛQCD = 350 MeV, where ΛQCD standsfor the QCD scale in the MS-scheme. While the gluoncondensate is renormalization group invariant, there is anunavoidable dependence on αs when its value is extractedfrom data using QCD sum rules. In fact, the sum rules in-volve the difference between hadronic data integrals andPQCD integrals, with the latter being strongly depen-dent on the value of αs. In other words, the uncertaintyin αs induces an uncertainty in the condensate. Ex-trapolating the above results to include a value ΛQCD =380 MeV, in line with the latest determination of αs(M2τ )[28], leads to〈(αs/pi)G2〉= (0.01 ± 0.01) GeV4. Thislarge uncertainty in the value of the gluon condensatewill have a very small impact on our results for m¯c. Thisis because the high energy expansion of the vector corre-lator, together with integration kernels involving positivepowers of s, tend to reduce considerably the importanceof this term. This is in contrast to the inverse momentsmethod which enhances this contribution with increasinginverse powers of s. Something similar happens with theimpact of the uncertainty in αs. Results for m¯c from in-verse moments are far more sensitive to αs than resultsfrom sum rules involving the high energy behaviour ofthe correlator and positive powers of s, as with pinchedor Legendre integration kernels.Turning to the experimental data, our analysis followsclosely that of [4], [32]. For the first two resonanceswe use the latest data from the Particle Data Group[33], MJ/ψ = 3.096916(11) GeV, ΓJ/ψ = 5.55(14) keV,Mψ(2s) = 3.68609(4) GeV, Γψ(2s) = 2.35(4) keV. Thefirst two resonances are followed by the open charm re-gion where the contribution from the light quark sectorRuds must be subtracted as background from the to-tal R-ratio Rtot. This we do as in [32]. In the region3.97 GeV ≤ √s ≤ 4.26 GeV we only use CLEO data[21] as they have the least error. Regarding the threedata sets from BES [18]-[20], we assume conservativelythat the systematic uncertainties are not entirely inde-pendent and add them linearly, rather than in quadra-ture, but we treat them as independent from the CLEOdata set [21], and thus add these in quadrature. The re-gion s = 25− 49 GeV2 has no data, beyond which thereis CLEO data up to s ≃ 110 GeV2. The latter is fullycompatible with PQCD.3We discuss next the integration kernels p(s) in Eq.(2),100 120 140 160 180 2000.80.91.01.11.2s0 HGeV2LmcH3GeVLHGeVLFIG. 2: The running mass m¯c(3 GeV) in the MS scheme asa function of s0, using the Legendre-type polynomial P5 withs1 = 24 GeV2. Polynomials of different order lead to equallywide stability regions. The errors are due to uncertaintiesin the data, the values of αs and the gluon condensate, andchanges in µ of ± 50%.and first introduce the pinched kernelp(s) = 1− ss0. (6)As shown in the past [22] - [25], this kernel suppressespotential duality violations close to the real axis in thecomplex s-plane. Clearly, there is a large variety of al-ternative functional forms vanishing at s = s0. Havingconsidered many of these alternatives, the simple choiceabove turns out to be the optimal. In any case, higherpowers of s will be part of the Legendre-type kernels wediscuss next. The purpose of these kernels is to extendthe analysis beyond the end point of the data, as achievede.g. in the analysis of duality violations using the ALEPHτ -decay data [24]. In the present case, there is no data inthe interval 25 GeV2 . s . 50 GeV2, while the data fors & 50 GeV2 agrees with PQCD. Hence, we introduce akernel p(s) such thatp(s) = Pn[x(s)] , (7)wherex(s) =2s− (s0 + s1)s0 − s1 , (8)with s0 > s1, and Pn(x) are the standard Legendre poly-nomials, i.e. P1(x) = x, P2(x) = (5x3 − 3x)/2, etc.,which satisfy the constraint∫ s0s1sk Pn[x(s)] ds = 0 , (9)where s1 ≃ 24 GeV2, and s0 varies in the region wherethere is no data. In the hadronic sector, the Legendre-type kernels provide extra weight to the well known reso-nance region on account of their rapid growth for s < s1.Uncertainties (MeV)Method m¯c(3GeV) Exp. ∆αs Trunc. NP Var. TotalFOPT 996 9 5 9 1 11 25FMUPT 988 9 3 6 1 9 25CIPT 983 9 1 2 1 16 25TABLE I: Results for m¯c(3 GeV) (in MeV) in the MS scheme usingthe pinched kernel Eq.(6) in FOPT, CIPT, and in FMUPT using a fixedscale µ = 3 GeV. Listed results are at s0 = 17 GeV2, a representativevalue inside the wide stability region (see Fig. 1). The uncertainties aredue to the data (Exp.), to αs (∆αs), to truncation of PQCD (Trunc.),to the gluon condensate (NP), and to variation inside the stability re-gion in s0 (Var.). All uncertainties are added in quadrature except forthat in s0 which is conservatively added linearly. The FMUPT totalerror includes an uncertainty in µ of ± 50%.Uncertainties (MeV)Kernel s0 m¯c(3GeV) Exp. ∆αs ∆µ NP TotalP3 100 1008 24 1 8 1 25P4 160 1008 24 1 8 1 26P5 200 1007 22 1 8 2 23P6 300 1008 23 1 9 2 25TABLE II: Results for m¯c(3 GeV) (in MeV) in the MS scheme ob-tained using various Legendre polynomial-type kernels, Eqs. (8)-(10),with s1 = 24 GeV2, µ = 3 GeV, and representative values of s0 in theremarkably wide stability region (see Fig. 2). The uncertainties aredue to the data (Exp.), to αs (∆αs), to changes in µ by ± 50% (∆µ),and to the gluon condensate (NP), which are added in quadrature. Thevariation in the stability region is negligible, and so is the truncationuncertainty.In Fig. 1 we show the result for m¯c(3 GeV) as a functionof s0 using a fixed scale µ = 3 GeV (FMUPT). The errorsshown are the result of adding in quadrature the uncer-tainties due to experiment, to the values of αs and thegluon condensate, and to the truncation of PQCD whichis taken as the difference between the known four-loopresult and the partially known five-loop expression (tozeroth order in mc). This FMUPT total error includesan uncertainty in µ of ± 50% (not present in FOPT orCIPT). This combined error is then conservatively addedlinearly to the uncertainty due to variation of s0 in thestability region. Results from FOPT and CIPT are simi-lar, and fully compatible within errors. Numerical valuesare given in Table I. In Fig. 2 we show m¯c(3GeV) asa function of s0 for the kernel P5 with s1 = 24 GeV2.Legendre-type polynomials of other order lead to basi-cally the same results, which are very stable in a remark-ably wide region. Numerical values are listed in TableII where the uncertainties are due to the data (Exp.),to αs (∆αs), to changes in µ by ± 50% (∆µ), and tothe gluon condensate (NP). Variation within the stabil-ity region gives a negligible uncertainty, and so does thePQCD truncation error. In comparison with values inTable I, one notices the larger impact of experimentaluncertainties. This is because s0 is much larger, thus in-cluding more data in the hadronic integral. Since quarkmass terms in PQCD are suppressed by inverse powersof s, the Legendre-type kernels are expected to improveconvergence for increasing energies. We choose as ourfinal result the one from the Legendre-type kernels due4to its stability against changes in s0, and use the pinchedkernel values in Table I as confirmation. In this casem¯c(3GeV) = 1008 ± 26 MeV , (10)in good agreement within errors with the result from in-verse moments QCD sum rules [4], other recent deter-minations [32], [34], as well as lattice QCD [31]. Trans-lated into a scale invariant mass, the above result givesm¯c(m¯c) = 1319 ± 26 MeV for the value used here forthe strong coupling, αs(M2Z) = 0.1213 ± 0.0014. Usinginstead αs(M2Z) = 0.1189± 0.0020, as in [4], changes thismass to m¯c(m¯c) = 1299 ± 26 MeV.In closing we briefly discuss two convergence issues, (a)the convergence pattern in αs of the perturbative QCDintegral as a function ofmc, and (b) the convergence pat-tern of the values of mc resulting from the FESR trun-cated at different orders in αs. In the first case the per-turbative contour integralI(s0) = − 12pii∮C(|s0|)p(s)Π(s) ds (11)is a function of both the mass m¯ and the cou-pling αs. One can investigate the convergence ofthis integral as a function of αs for different val-ues of m¯c. Using a typical integration kernel, e.g.p(s) = P4 and the representative value m¯ = 1GeV,we find I(0) = 1.4176GeV2, I(1) = 1.4563GeV2,I(2) = 1.4541GeV2, I(3) = 1.4512GeV2 andI(4) = 1.4524GeV2, where the upper index in I(j)indicates the power of αs. If one were to use preciselythe same input parameters, but with a higher quarkmass, e.g. m¯ = 5GeV, the convergence would expect-edly be lost. To restore it one only needs to increasecorrespondingly the scale µ. Turning to the secondissue, and using again the typical kernel P4, one cancalculate m¯c(3GeV) truncated at different orders in αsas a further check on the convergence. The results arem¯(0)c = 986MeV, m¯(1)c = 1011MeV, m¯(2)c = 1010MeV,m¯(3)c = 1008MeV, and m¯(4)c = 1009MeV, where theupper index in m¯(j)) indicates the power of αs. Consid-ering the difference between m¯(3)c and m¯(4)c would givean uncertainty of roughly 1MeV, which is very muchsmaller than that from the 50% variation in the scale µconsidered in our analysis.This work was supported in part by the Euro-pean FEDER and Spanish MICINN under grantMICINN/FPA 2008-02878, by the Generalitat Valen-ciana under grant GVPROMETEO 2010-056, by NRF(South Africa) and by DFG (Germany). One of us (SB)wishes to thank P. Maier, M. Steinhauser, and C. Sturmfor helpful correspondence.[1] For a recent review see e.g. P. Colangelo, A. Khod-jamirian, in: ”At the Frontier of Particle Physics/ Hand-book of QCD”’, M. Shifman, ed. (World Scientific, Sin-gapore 2001), Vol. 3, 1495-1576.[2] V. A. Novikov, et al. Phys.Rep. 41, 1 (1978).[3] C. A. Dominguez and N. Paver, Phys. Lett. B 293, 197(1992); ibid. B 333, 184 (1994).[4] J. H. Ku¨hn, M. Steinhauser, and C. Sturm, Nucl. Phys.B 778, 192 (2007); K. G. Chetyrkin et al., Phys. Rev. D80, 074010 (2009).[5] J. Pen˜arrocha, and K. Schilcher, Phys. Lett. B 515, 291(2001); J. Bordes, J. Pen˜arrocha, and K. Schilcher, ibid.B 562, 81 (2003).[6] K. G. Chetyrkin, R. Harlander, J. H. Ku¨hn, and M.Steinhauser, Nucl. Phys. B 503, 339 (1997).[7] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Nucl.Phys. B (Proc. Suppl.) 189, 49 (2009).[8] K. G. Chetyrkin, R. Harlander, J. H. Ku¨hn, Nucl. Phys.B 586, 56 (2000).[9] Y. Kiyo, A. Maier, P. Maierho¨fer, and P. Marquard, Nucl.Phys. B 823, 269 (2009).[10] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Phys.Rev. Lett. 101, 012002 (2008).[11] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Nucl.Phys. B (Proc. Suppl.) 135, 243 (2004).[12] K. G. Chetyrkin, J. H. Ku¨hn, and M. Steinhauser, Phys.Lett. B 371, 93 (1996); Nucl. Phys. B 482, 213 (1996);ibid. B 505, 40 (1997).[13] R. Boughezal, M. Czakon, and T. Schutzmeier, Phys.Rev. D 74, 074006 (2006); Nucl. Phys. B (Proc. Suppl.)160, 164 (2006).[14] A. Maier, P. Maiero¨fer, and P. Marquard, Nucl. Phys. B797, 218 (2008); Phys. Lett. B 669, 88 (2008).[15] K. G. Chetyrkin, J. H. Ku¨hn, and C. Sturm, Eur. Phys.J. C 48, 107 (2006).[16] A. Maier, P. Maierho¨fer, P. Marquard, and A. V.Smirnov, Nucl. Phys. B 824, 1 (2010).[17] A. H. Hoang, V. Mateu, and S. Mohammad Zebarjad,Nucl. Phys. B 813, 349 (2009).[18] J. Z. Bai et al. (BES 2000), Phys. Rev. Lett. 84, 594(2000).[19] J. Z. Bai et al. (BES 2002), Phys. Rev. Lett. 88, 101802(2002).[20] J. Z. Bai et al. (BES 2006), Phys. Rev. Lett. 97, 262001(2006).[21] D. Cronin-Hennessy et al. (CLEO 2009), Phys. Rev. D80, 072001 (2009).[22] K. Maltman, Phys. Lett. B 440, 367 (1998) ; C. A.Dominguez, and K. Schilcher, Phys. Lett. B 448, 93(1999).[23] C. A. Dominguez, and K. Schilcher, Phys. Lett. B 581,193 (1999); J. Bordes, C. A. Dominguez, J. Pen˜arrocha,and K. Schilcher, J. High Energy Phys. 0602, 037 (2006).[24] C. A. Dominguez, N. F. Nasrallah, and K. Schilcher,Phys. Rev. D 80, 054014 (2009).[25] C. A. Dominguez, and K. Schilcher, J. High Energy Phys.0701, 093 (2007).[26] C. A. Dominguez, N. F. Nasrallah, and K. Schilcher, J.5High Energy Phys. 0802, 072 (2008); J. Bordes et al., J.High Energy Phys. 1005, 064 (2010).[27] C. A. Dominguez, N. F. Nasrallah, R. Ro¨ntsch, and K.Schilcher, J. High Energy Phys. 0805 (2008) 020; Phys.Rev. D 79, 014009 (2009).[28] A. Pich, arXiv:hep-ph/1001.0389.[29] C. Pahl et al., Proc. of Sci. EPS-HEP 2009, 292 (2009).[30] S. Bethke, Eur. Phys. J. C 64, 689 (2009).[31] C. T. H. Davies et al., HPQCD Collab., Phys. Rev. D 78,114507 (2008); C. McNeile, PQCD Collab., Phys. Rev.D 82, 034512 (2010).[32] A. Hoang, and M. Jamin, Nucl. Phys. B 594, 127 (2004).[33] K. Nakamura et al., Particle Data Group, J. Phys. G 37,075021 (2010).[34] A. Signer, Phys. Lett. B 672, 333 (2009).

The running charm-quark mass in the MS¯ scheme is determined from weighted finite energy QCD sum rules (FESR) involving the vector current correlator. Only the short distance expansion of this correlator is used, together with integration kernels (weights) involving positive powers of s, the squared energy. The optimal kernels are found to be a simple {\it pinched} kernel, and polynomials of the Legendre type. The former kernel reduces potential duality violations near the real axis in the complex s-plane, and the latter allows to extend the analysis to energy regions beyond the end point of the data. These kernels, together with the high energy expansion of the correlator, weigh the experimental and theoretical information differently from e.g. inverse moments FESR. Current, state of the art results for the vector correlator up to four-loop order in perturbative QCD are used in the FESR, together with the latest experimental data. The integration in the complex s-plane is performed using three different methods, fixed order perturbation theory (FOPT), contour improved perturbation theory (CIPT), and a fixed renormalization scale μ (FMUPT). The final result is m¯c(3GeV)=1008±26MeV, in a wide region of stability against changes in the integration radius s0 in the complex s-plane

Bodenstein, Sebastian

Bordes Villagrasa, José M.

Domínguez, Cesáreo A.

Peñarrocha Gantes, José Antonio

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

arXiv:1009.4325v2  [hep-ph]  1 Dec 2010UCT-TP-281/10MZ-TH/10-34Charm-quark mass from weighted finite energy QCD sum rulesS. Bodenstein,1 J. Bordes,2 C. A. Dominguez,1, 3 J. Pen˜arrocha,2 and K. Schilcher41Centre for Theoretical & Mathematical Physics,University of Cape Town, Rondebosch 7700, South Africa2Departamento de F´ısica Teo´rica, Universitat de Valencia,and Instituto de F´ısica Corpuscular, Centro Mixto Universitat de Valencia-CSIC3Department of Physics, Stellenbosch University, Stellenbosch 7600, South Africa4Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Staudingerweg 7, D-55099 Mainz, Germany(Dated: November 18, 2013)The running charm-quark mass in the MS scheme is determined from weighted finite energyQCD sum rules (FESR) involving the vector current correlator. Only the short distance expansionof this correlator is used, together with integration kernels (weights) involving positive powers of s,the squared energy. The optimal kernels are found to be a simple pinched kernel, and polynomialsof the Legendre type. The former kernel reduces potential duality violations near the real axisin the complex s-plane, and the latter allows to extend the analysis to energy regions beyond theend point of the data. These kernels, together with the high energy expansion of the correlator,weigh the experimental and theoretical information differently from e.g. inverse moments FESR.Current, state of the art results for the vector correlator up to four-loop order in perturbative QCDare used in the FESR, together with the latest experimental data. The integration in the complexs-plane is performed using three different methods, fixed order perturbation theory (FOPT),contour improved perturbation theory (CIPT), and a fixed renormalization scale µ (FMUPT). Thefinal result is m¯c(3GeV) = 1008 ± 26MeV, in a wide region of stability against changes in theintegration radius s0 in the complex s-plane.PACS numbers: 12.38.Lg, 11.55.Hx, 12.38.Bx, 14.65.DwConsiderable progress has been made over the years toextract accurate values of the quark masses by con-fronting QCD with experimental data in the frameworkof QCD sum rules [1]. We consider in this paper thecase of the charm-quark mass. Following pioneering ap-proaches [2], it has been customary to write the OperatorProduct Expansion (OPE) of the vector correlator withthe scale invariant massmc(mc) as the expansion param-eter. This is in contrast to the case of light quarks wherethe square of the four-momentum Q2 ≡ −q2 is the nat-ural large expansion parameter. If one were to considerthe charm quark mass in e.g. theMS scheme at a typicalscale of 3 GeV, then the fact that m¯c(3 GeV) ≃ 1 GeVshould be a matter of some concern regarding this ex-pansion [3]. In any case, the latest determinations basedon inverse moment QCD sum rules claim an accuracy atthe 1% level [4]. These inverse moments require QCDknowledge of the vector correlator in the low energy re-gion, around the open charm threshold, as well as in thehigh energy region. An alternative approach involvingpositive moment sum rules, requiring QCD informationonly at high energy, and naturally suited to determinem¯c(µ) in theMS scheme, was proposed some time ago in[5]. We follow this approach here, and make use of stateof the art QCD information up to four-loop level [6]-[17],together with the latest experimental data [18]-[21]. Weconsider Finite Energy Sum Rules (FESR) weighted bytwo types of integration kernels. The first type is theso called pinched kernel, which has been shown to min-imize potential duality violations close to the real axisin the complex s-plane [22]. The second type is a poly-nomial (Legendre) kernel tuned to minimize the impactof the energy region where the data is either poor ornon-existent. Effectively, these Legendre-type integra-tion kernels, involving several positive powers of the en-ergy, allow for an extension of the analysis beyond theend point of the data. Both types of kernels have beenused successfully in the light quark sector to study thesaturation of chiral sum rules [23], to quantify dualityviolations [24], to extract the values of the vacuum con-densates in the OPE [25], and to determine the quarkcondensates [26] and light quark masses [27].We begin by considering the vector current correlatorΠµν(q2) = i∫d4x eiqx〈0|T (Vµ(x) Vν(0))|0〉= (qµ qν − q2gµν) Π(q2) , (1)2where Vµ(x) = c¯(x)γµc(x). Invoking Cauchy’s theoremin the complex s-plane (−q2 ≡ Q2 ≡ s) one has∫ s00p(s)1piIm Π(s) ds = − 12pii∮C(|s0|)p(s) Π(s) ds ,(2)where p(s) is an arbitrary but analytical integration ker-nel, andIm Π(s) =112piRc(s) , (3)with Rc(s) the standard R-ratio for charm production.The perturbative QCD (PQCD) expression of Π(s) athigh energies can be written asΠ(s)|PQCD =∑n=0(αs(µ2)pi)nΠ(n)(s) , (4)whereΠ(n)(s) =∑i=0(m¯2cs)iΠ(n)i , (5)and m¯c stands for the running charm-quark mass in theMS-scheme. The complete analytical PQCD result forΠ(s)|PQCD up to order O [α2s(m¯2c/s)6] is given in [6], andexact results for Π(3)0 and Π(3)1 are from [7]. The func-tion Π(3)2 is known exactly up to a constant [8] whichhas been estimated using Pade´ approximants in [9]. Atfive-loop order O(α4s) the full logarithmic terms for Π(4)0are given in [10], and for Π(4)1 in [11]. Since there is in-complete knowledge at this loop-order, we shall use theavailable information as a measure of the truncation er-ror in PQCD.The fundamental parameters entering the QCD corre-14 15 16 17 18 19 200.80.91.01.11.2s0 HGeV2LmcH3GeVLHGeVLFIG. 1: The running mass m¯c(3 GeV) in theMS scheme for afixed µ = 3.0 GeV (FMUPT), as a function of s0 and using thepinched kernel Eq. (6). Results from FOPT and CIPT leadto similar stability regions. Errors are due to uncertaintiesin the data, to the values of αs and the gluon condensate, toPQCD truncation, and to changes in µ of ± 50%.lator are the running strong coupling αs(µ2), and thegluon condensate. Regarding the strong coupling, weuse the latest comprehensive update analysis at theτ -scale [28] which gives αs(M2τ ) = 0.342 ± 0.012, orαs(M2Z) = 0.1213 ± 0.0014. This value agrees withinerrors with a recent determination [29] from e+e− anni-hilation data: αs(M2Z) = 0.1172 ± 0.0051. However, aworld average with much smaller errors [30] gives the re-sult αs(M2Z) = 0.1184±0.0007, in agreement with latticeQCD results [31]. We shall include both sets of values ofαs in our determination of m¯c. In the non-perturbativesector the leading power correction in the OPE involvesthe gluon condensate, i.e.〈(αs/pi)G2〉. The latest valueof the gluon condensate [25], extracted from the ALEPHdata on τ -decays, is〈(αs/pi)G2〉= (0.046±0.012) GeV4,for ΛQCD = 300 MeV, and〈(αs/pi)G2〉= (0.021 ±0.006) GeV4 for ΛQCD = 350 MeV, where ΛQCD standsfor the QCD scale in the MS-scheme. While the gluoncondensate is renormalization group invariant, there is anunavoidable dependence on αs when its value is extractedfrom data using QCD sum rules. In fact, the sum rules in-volve the difference between hadronic data integrals andPQCD integrals, with the latter being strongly depen-dent on the value of αs. In other words, the uncertaintyin αs induces an uncertainty in the condensate. Ex-trapolating the above results to include a value ΛQCD =380 MeV, in line with the latest determination of αs(M2τ )[28], leads to〈(αs/pi)G2〉= (0.01 ± 0.01) GeV4. Thislarge uncertainty in the value of the gluon condensatewill have a very small impact on our results for m¯c. Thisis because the high energy expansion of the vector corre-lator, together with integration kernels involving positivepowers of s, tend to reduce considerably the importanceof this term. This is in contrast to the inverse momentsmethod which enhances this contribution with increasinginverse powers of s. Something similar happens with theimpact of the uncertainty in αs. Results for m¯c from in-verse moments are far more sensitive to αs than resultsfrom sum rules involving the high energy behaviour ofthe correlator and positive powers of s, as with pinchedor Legendre integration kernels.Turning to the experimental data, our analysis followsclosely that of [4], [32]. For the first two resonanceswe use the latest data from the Particle Data Group[33], MJ/ψ = 3.096916(11) GeV, ΓJ/ψ = 5.55(14) keV,Mψ(2s) = 3.68609(4) GeV, Γψ(2s) = 2.35(4) keV. Thefirst two resonances are followed by the open charm re-gion where the contribution from the light quark sectorRuds must be subtracted as background from the to-tal R-ratio Rtot. This we do as in [32]. In the region3.97 GeV ≤ √s ≤ 4.26 GeV we only use CLEO data[21] as they have the least error. Regarding the threedata sets from BES [18]-[20], we assume conservativelythat the systematic uncertainties are not entirely inde-pendent and add them linearly, rather than in quadra-ture, but we treat them as independent from the CLEOdata set [21], and thus add these in quadrature. The re-gion s = 25− 49 GeV2 has no data, beyond which thereis CLEO data up to s ≃ 110 GeV2. The latter is fullycompatible with PQCD.3We discuss next the integration kernels p(s) in Eq.(2),100 120 140 160 180 2000.80.91.01.11.2s0 HGeV2LmcH3GeVLHGeVLFIG. 2: The running mass m¯c(3 GeV) in the MS scheme asa function of s0, using the Legendre-type polynomial P5 withs1 = 24 GeV2. Polynomials of different order lead to equallywide stability regions. The errors are due to uncertaintiesin the data, the values of αs and the gluon condensate, andchanges in µ of ± 50%.and first introduce the pinched kernelp(s) = 1− ss0. (6)As shown in the past [22] - [25], this kernel suppressespotential duality violations close to the real axis in thecomplex s-plane. Clearly, there is a large variety of al-ternative functional forms vanishing at s = s0. Havingconsidered many of these alternatives, the simple choiceabove turns out to be the optimal. In any case, higherpowers of s will be part of the Legendre-type kernels wediscuss next. The purpose of these kernels is to extendthe analysis beyond the end point of the data, as achievede.g. in the analysis of duality violations using the ALEPHτ -decay data [24]. In the present case, there is no data inthe interval 25 GeV2 . s . 50 GeV2, while the data fors & 50 GeV2 agrees with PQCD. Hence, we introduce akernel p(s) such thatp(s) = Pn[x(s)] , (7)wherex(s) =2s− (s0 + s1)s0 − s1 , (8)with s0 > s1, and Pn(x) are the standard Legendre poly-nomials, i.e. P1(x) = x, P2(x) = (5x3 − 3x)/2, etc.,which satisfy the constraint∫ s0s1sk Pn[x(s)] ds = 0 , (9)where s1 ≃ 24 GeV2, and s0 varies in the region wherethere is no data. In the hadronic sector, the Legendre-type kernels provide extra weight to the well known reso-nance region on account of their rapid growth for s < s1.Uncertainties (MeV)Method m¯c(3GeV) Exp. ∆αs Trunc. NP Var. TotalFOPT 996 9 5 9 1 11 25FMUPT 988 9 3 6 1 9 25CIPT 983 9 1 2 1 16 25TABLE I: Results for m¯c(3 GeV) (in MeV) in the MS scheme usingthe pinched kernel Eq.(6) in FOPT, CIPT, and in FMUPT using a fixedscale µ = 3 GeV. Listed results are at s0 = 17 GeV2, a representativevalue inside the wide stability region (see Fig. 1). The uncertainties aredue to the data (Exp.), to αs (∆αs), to truncation of PQCD (Trunc.),to the gluon condensate (NP), and to variation inside the stability re-gion in s0 (Var.). All uncertainties are added in quadrature except forthat in s0 which is conservatively added linearly. The FMUPT totalerror includes an uncertainty in µ of ± 50%.Uncertainties (MeV)Kernel s0 m¯c(3GeV) Exp. ∆αs ∆µ NP TotalP3 100 1008 24 1 8 1 25P4 160 1008 24 1 8 1 26P5 200 1007 22 1 8 2 23P6 300 1008 23 1 9 2 25TABLE II: Results for m¯c(3 GeV) (in MeV) in the MS scheme ob-tained using various Legendre polynomial-type kernels, Eqs. (8)-(10),with s1 = 24 GeV2, µ = 3 GeV, and representative values of s0 in theremarkably wide stability region (see Fig. 2). The uncertainties aredue to the data (Exp.), to αs (∆αs), to changes in µ by ± 50% (∆µ),and to the gluon condensate (NP), which are added in quadrature. Thevariation in the stability region is negligible, and so is the truncationuncertainty.In Fig. 1 we show the result for m¯c(3 GeV) as a functionof s0 using a fixed scale µ = 3 GeV (FMUPT). The errorsshown are the result of adding in quadrature the uncer-tainties due to experiment, to the values of αs and thegluon condensate, and to the truncation of PQCD whichis taken as the difference between the known four-loopresult and the partially known five-loop expression (tozeroth order in mc). This FMUPT total error includesan uncertainty in µ of ± 50% (not present in FOPT orCIPT). This combined error is then conservatively addedlinearly to the uncertainty due to variation of s0 in thestability region. Results from FOPT and CIPT are simi-lar, and fully compatible within errors. Numerical valuesare given in Table I. In Fig. 2 we show m¯c(3GeV) asa function of s0 for the kernel P5 with s1 = 24 GeV2.Legendre-type polynomials of other order lead to basi-cally the same results, which are very stable in a remark-ably wide region. Numerical values are listed in TableII where the uncertainties are due to the data (Exp.),to αs (∆αs), to changes in µ by ± 50% (∆µ), and tothe gluon condensate (NP). Variation within the stabil-ity region gives a negligible uncertainty, and so does thePQCD truncation error. In comparison with values inTable I, one notices the larger impact of experimentaluncertainties. This is because s0 is much larger, thus in-cluding more data in the hadronic integral. Since quarkmass terms in PQCD are suppressed by inverse powersof s, the Legendre-type kernels are expected to improveconvergence for increasing energies. We choose as ourfinal result the one from the Legendre-type kernels due4to its stability against changes in s0, and use the pinchedkernel values in Table I as confirmation. In this casem¯c(3GeV) = 1008 ± 26 MeV , (10)in good agreement within errors with the result from in-verse moments QCD sum rules [4], other recent deter-minations [32], [34], as well as lattice QCD [31]. Trans-lated into a scale invariant mass, the above result givesm¯c(m¯c) = 1319 ± 26 MeV for the value used here forthe strong coupling, αs(M2Z) = 0.1213 ± 0.0014. Usinginstead αs(M2Z) = 0.1189± 0.0020, as in [4], changes thismass to m¯c(m¯c) = 1299 ± 26 MeV.In closing we briefly discuss two convergence issues, (a)the convergence pattern in αs of the perturbative QCDintegral as a function ofmc, and (b) the convergence pat-tern of the values of mc resulting from the FESR trun-cated at different orders in αs. In the first case the per-turbative contour integralI(s0) = − 12pii∮C(|s0|)p(s)Π(s) ds (11)is a function of both the mass m¯ and the cou-pling αs. One can investigate the convergence ofthis integral as a function of αs for different val-ues of m¯c. Using a typical integration kernel, e.g.p(s) = P4 and the representative value m¯ = 1GeV,we find I(0) = 1.4176GeV2, I(1) = 1.4563GeV2,I(2) = 1.4541GeV2, I(3) = 1.4512GeV2 andI(4) = 1.4524GeV2, where the upper index in I(j)indicates the power of αs. If one were to use preciselythe same input parameters, but with a higher quarkmass, e.g. m¯ = 5GeV, the convergence would expect-edly be lost. To restore it one only needs to increasecorrespondingly the scale µ. Turning to the secondissue, and using again the typical kernel P4, one cancalculate m¯c(3GeV) truncated at different orders in αsas a further check on the convergence. The results arem¯(0)c = 986MeV, m¯(1)c = 1011MeV, m¯(2)c = 1010MeV,m¯(3)c = 1008MeV, and m¯(4)c = 1009MeV, where theupper index in m¯(j)) indicates the power of αs. Consid-ering the difference between m¯(3)c and m¯(4)c would givean uncertainty of roughly 1MeV, which is very muchsmaller than that from the 50% variation in the scale µconsidered in our analysis.This work was supported in part by the Euro-pean FEDER and Spanish MICINN under grantMICINN/FPA 2008-02878, by the Generalitat Valen-ciana under grant GVPROMETEO 2010-056, by NRF(South Africa) and by DFG (Germany). One of us (SB)wishes to thank P. Maier, M. Steinhauser, and C. Sturmfor helpful correspondence.[1] For a recent review see e.g. P. Colangelo, A. Khod-jamirian, in: ”At the Frontier of Particle Physics/ Hand-book of QCD”’, M. Shifman, ed. (World Scientific, Sin-gapore 2001), Vol. 3, 1495-1576.[2] V. A. Novikov, et al. Phys.Rep. 41, 1 (1978).[3] C. A. Dominguez and N. Paver, Phys. Lett. B 293, 197(1992); ibid. B 333, 184 (1994).[4] J. H. Ku¨hn, M. Steinhauser, and C. Sturm, Nucl. Phys.B 778, 192 (2007); K. G. Chetyrkin et al., Phys. Rev. D80, 074010 (2009).[5] J. Pen˜arrocha, and K. Schilcher, Phys. Lett. B 515, 291(2001); J. Bordes, J. Pen˜arrocha, and K. Schilcher, ibid.B 562, 81 (2003).[6] K. G. Chetyrkin, R. Harlander, J. H. Ku¨hn, and M.Steinhauser, Nucl. Phys. B 503, 339 (1997).[7] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Nucl.Phys. B (Proc. Suppl.) 189, 49 (2009).[8] K. G. Chetyrkin, R. Harlander, J. H. Ku¨hn, Nucl. Phys.B 586, 56 (2000).[9] Y. Kiyo, A. Maier, P. Maierho¨fer, and P. Marquard, Nucl.Phys. B 823, 269 (2009).[10] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Phys.Rev. Lett. 101, 012002 (2008).[11] P. A. Baikov, K. G. Chetyrkin, and J. H. Ku¨hn, Nucl.Phys. B (Proc. Suppl.) 135, 243 (2004).[12] K. G. Chetyrkin, J. H. Ku¨hn, and M. Steinhauser, Phys.Lett. B 371, 93 (1996); Nucl. Phys. B 482, 213 (1996);ibid. B 505, 40 (1997).[13] R. Boughezal, M. Czakon, and T. Schutzmeier, Phys.Rev. D 74, 074006 (2006); Nucl. Phys. B (Proc. Suppl.)160, 164 (2006).[14] A. Maier, P. Maiero¨fer, and P. Marquard, Nucl. Phys. B797, 218 (2008); Phys. Lett. B 669, 88 (2008).[15] K. G. Chetyrkin, J. H. Ku¨hn, and C. Sturm, Eur. Phys.J. C 48, 107 (2006).[16] A. Maier, P. Maierho¨fer, P. Marquard, and A. V.Smirnov, Nucl. Phys. B 824, 1 (2010).[17] A. H. Hoang, V. Mateu, and S. Mohammad Zebarjad,Nucl. Phys. B 813, 349 (2009).[18] J. Z. Bai et al. (BES 2000), Phys. Rev. Lett. 84, 594(2000).[19] J. Z. Bai et al. (BES 2002), Phys. Rev. Lett. 88, 101802(2002).[20] J. Z. Bai et al. (BES 2006), Phys. Rev. Lett. 97, 262001(2006).[21] D. Cronin-Hennessy et al. (CLEO 2009), Phys. Rev. D80, 072001 (2009).[22] K. Maltman, Phys. Lett. B 440, 367 (1998) ; C. A.Dominguez, and K. Schilcher, Phys. Lett. B 448, 93(1999).[23] C. A. Dominguez, and K. Schilcher, Phys. Lett. B 581,193 (1999); J. Bordes, C. A. Dominguez, J. Pen˜arrocha,and K. Schilcher, J. High Energy Phys. 0602, 037 (2006).[24] C. A. Dominguez, N. F. Nasrallah, and K. Schilcher,Phys. Rev. D 80, 054014 (2009).[25] C. A. Dominguez, and K. Schilcher, J. High Energy Phys.0701, 093 (2007).[26] C. A. Dominguez, N. F. Nasrallah, and K. Schilcher, J.5High Energy Phys. 0802, 072 (2008); J. Bordes et al., J.High Energy Phys. 1005, 064 (2010).[27] C. A. Dominguez, N. F. Nasrallah, R. Ro¨ntsch, and K.Schilcher, J. High Energy Phys. 0805 (2008) 020; Phys.Rev. D 79, 014009 (2009).[28] A. Pich, arXiv:hep-ph/1001.0389.[29] C. Pahl et al., Proc. of Sci. EPS-HEP 2009, 292 (2009).[30] S. Bethke, Eur. Phys. J. C 64, 689 (2009).[31] C. T. H. Davies et al., HPQCD Collab., Phys. Rev. D 78,114507 (2008); C. McNeile, PQCD Collab., Phys. Rev.D 82, 034512 (2010).[32] A. Hoang, and M. Jamin, Nucl. Phys. B 594, 127 (2004).[33] K. Nakamura et al., Particle Data Group, J. Phys. G 37,075021 (2010).[34] A. Signer, Phys. Lett. B 672, 333 (2009).

Charm-quark mass from weighted finite energy QCD sum rules

Abstract

Similar works

Full text

Available Versions

Crossref

Digital.CSIC

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