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UCT-TP-288/11 MZ-TH/11-37 Bottom-quark mass from finite energy QCD sum rules S. Bodenstein,1 J. Bordes,2 C. A. Dominguez,1 J. Penãrrocha,2 and K. Schilcher1,3 1Centre for Theoretical & Mathematical Physics, and Department of Physics, University of Cape Town, Rondebosch 7700, South Africa 2Departamento de F´ısica Teórica, Universitat de Valencia, and Instituto de F´ısica Corpuscular, Centro Mixto Universitat de Valencia-CSIC 3Institut fu¨r Physik, Johannes Gutenberg-Universität Staudingerweg 7, D-55099 Mainz, Germany (Dated: January 17, 2012) Finite energy QCD sum rules involving both inverse and positive moment integration kernels are employed to determine the bottom quark mass. The result obtained in the MS scheme at a 2 referencescaleof10GeVism (10GeV)=3623(9)MeV. Thisvaluetranslatesintoascaleinvariant b 1 mass m (m ) = 4171(9)MeV. This result has the lowest total uncertainty of any method, and is b b 0 lesssensitivetoanumberofsystematicuncertaintiesthataffectotherQCDsumruledeterminations. 2 n PACSnumbers: 12.38.Lg,11.55.Hx,12.38.Bx,14.65.Fy a J 6 I. INTRODUCTION s = s has to be assumed. As a benefit, this procedure 1 0 reduces also the continuum contribution relative to the well known Υ narrow resonances. ] h With the availability of new cross section data on e+e− p annihilationintohadronsfromtheBABARcollaboration - p [1],thebottomquarkmasswasdeterminedrecentlywith e unprecedentedprecisionusinginversemomentQCDsum II. THEORETICAL BACKGROUND h rules[2]. TheresultintheMSschemeatareferencescale [ of 10GeV is 2 We consider the vector current correlator v mb(10GeV)=3610(16)MeV. (1) 2 4 However,aswassubsequentlypointedout[3],this result Πµν(q2) = iZ d4xeiqxh0|T(Vµ(x)Vν(0))|0i 7 relies on the assumption that PQCD is already valid at 5 theendpointoftheBABAR data,i.e. √s = 11.21GeV, = (qµqν −q2gµν)Π(q2), (2) . 1 wheresisthesquaredenergy. Thisassumptionmightbe where V (x)=b(x)γ b(x), andb(x) is the bottom-quark 1 questionable, as the prediction of PQCD for the R-ratio µ µ 1 doesnotagreewiththeexperimentallymeasuredvalueat field. Cauchy’s residue theorem in the complex s-plane 1 ( q2 Q2 s) implies that this point. This QCD sum rule result was also shown to − ≡ ≡ : v depend significantly on this assumption. Hence, further s0 1 1 Xi reductions in the error of the bottom-quark mass using p(s) ImΠ(s)ds = p(s)Π(s)ds QCDsumrules willdepend onthe ability to controlthis Z0 π −2πiIC(|s0|) r a systematic uncertainty. One way ofachieving this would + Res[Π(s)p(s),s=0], (3) beforanewexperimenttoextendtheBABAR measure- mentintoa regionwherePQCDisunquestionablyvalid. where p(s) is an arbitrary Laurent polynomial, and In this paper we follow another approach based entirely on theory. We use a finite energy QCD sum rule with 1 ImΠ(s)= R (s), (4) integration kernels involving both inverse and positive 12π b powers of the energy, as employed recently to determine the charm-quark mass [4]. We also exploit the freedom with Rb(s) the standard R-ratio for bottom production. offered by Cauchy’s theorem to reduce the dependence The power series expansion of Π(s) for large and space- of the quark mass on the above systematic uncertainty. like s can be calculated in PQCD, and has the form This is achieved by using integration kernels that re- α (µ2) n duce the contributions in the region √s 11.21GeV to Π(s) =e2 s Π(n)(s), (5) √s0, where there is no data and the ons≃et of PQCD at (cid:12)PQCD bnX=0(cid:16) π (cid:17) (cid:12) 2 where e =2/3 is the bottom-quark electric charge, and 0.6 b à 0.5 à àààà à m2 i à à aHnreteeswrtuehltremsersfbeuonlr≡tosirmm=upba(1Πlµit,z(o)·na·)ti·(sios,[tn)6αh=2seh(camqaXiv=lu2ee0a/(cid:16)rµbske).3sem0Tbn](cid:17)hacoseasbΠlotcira(iunndiln)aet.erhtdeeOdrM[ieαncS2se([n5smct],lh2bye/wms[(i6)t6eih]).], HLRsb 0000....1234 àààààààààààààààààààààààààààààààààààààààààààà àààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààà O s b At order [α3], Π(3) and Π(3) are known [7], and the à logarithmiOc tesrms i0n Π(3) ma1y be found in [8]. The con- 0.010.6 à 10.7 10.8 10.9 11.0 11.1 11.2 11.3 2 stant term in Π(3) is not known exactly, but has been s HGeVL 2 estimated using Padé approximants [9], and the Mellin- FIG. 1: The corrected BABAR data [1] and the PQCD pre- Barnestransform[10]. Atorder [α4]theexactlogarith- O s diction (solid black line) obtained using Rhad [20]. mic terms in Π(4) and Π(4) were determined in [11]-[12], 0 1 whilstthe constanttermsarenotyetknown. Giventhat these constant terms will contribute to sum rules with kernels containing powers s−1 and s0, respectively, for III. EXPERIMENTAL INPUT consistency we shall not include any five-loop order expressions. However,wefindthatifallknownfive-loopor- dertermsaretakenintoaccount,themassofthebottom- Inordertoevaluatetheleft-handsideofEq.(3)oneneeds quark only changes by roughly 0.03%, which is about a to use experimental input. First, there are the four nar- tenth of the accuracy of this determination. rowΥ-resonances,andwecalculatetheir contributionto The Taylor series expansion of Π(s) about s = 0 is usu- Eq.(3) using the zero-width approximation ally cast in the form 9πM Γ Rres = i i δ(s M2), (9) b α2 (s) − i 3e2 Xi EM Π(s) = b C¯ zn, (7) (cid:12)(cid:12)PQCD 16π2 nX≥0 n where i = 1,··· ,4, corresponding to Υ(1S), Υ(2S), Υ(3S), and Υ(4S). We use the masses and widths from the Particle Data Group [17]. The widths wherez ≡s/(4m¯2b). ThecoefficientsC¯n canbeexpanded are ΓΥ(1S) = 1.340(18)keV, ΓΥ(2S) = 0.612(11)keV, in powers of αs(µ) as ΓΥ(3S) = 0.443(8)keV and ΓΥ(4S) = 0.272(29)keV. Given that the widths of the Υ(1S),Υ(2S) and Υ(3S) were obtained at the same experimental facility, we α (µ) C¯n =C¯n(0)+ sπ (C¯n(10)+C¯n(11)lm) will assume their uncertainties to be correlated. The masses are M = 9.46030(26)GeV, M = α (µ) 2 Υ(1S) Υ(2S) + s (C¯(20)+C¯(21)l +C¯(22)l2 ) 10.02326(31)GeV, M = 10.3552(5)GeV, and (cid:16) π (cid:17) n n m n m M = 10.5794(12)ΥG(3eSV). Finally, we use the effec- Υ(4S) α (µ) 3 + s (C¯(30)+C¯(31)l +C¯(32)l2 tive electromagnetic couplings from [16]. The BABAR (cid:16) π (cid:17) n n m n m Collaboration [1] has performed direct measurements of +C¯(33)l3 )+... (8) R in the continuum threshold regionbetween10.62GeV n m b and 11.21GeV. There is also data on the full ratio R in thebottom-quarkregionbytheCLEOCollaboration[18] where l ln(m2/µ2). Up to (α2), the coefficients up to n=3m0≡ofC¯n abre known[13]O-[14]s. There is also a sub- dsuarteinmgenbtacink1t9o9189[1895].,aStuabsseinqugleenetlnye,ragyl,aster C10L.5E3OGmeVea2-, leadingcontributionoforder (α2 (m /m )2)[15]affect- ≃ O s c b gives a total R-ratio roughly 30% lower than the 1985 (20) ingthecoefficientC inEq.(8),aswellasQEDcorrec- data in this region. Since this discrepancyremains unre- n tions. Theformercontributesaround 1.0MeV,andthe solved we shall use here only the BABAR data. As was − latter roughly 2.0MeV to the result for m (10GeV). pointedoutin[2],theseBABAR datacannotbeuseddi- b − Finally, there is a non-perturbative contribution to Π(s) rectly in sum-rules, such as e.g. Eq.(3), for the following from the gluon condensate, but it has been found to be reasons. First, the initial-state radiation and the radia- completely negligible [16]. We fully agree, and thus con- tive tail of the Υ resonancemust be removed. Second, 4S firm this result. For the strong running coupling we use thevacuumpolarizationcontributionmustbetakeninto the Particle Data Group [17] value α (m )=0.1184(7), account. We follow this procedure, as detailed in [2], to s Z which corresponds to α (10GeV)=0.1792(16). correct the BABAR data with results shown in Fig. 1. s 3 3635 tion C:TheBABAR dataarecorrect,andPQCDstarts at √s = 11.21GeV. However, the PQCD prediction of 3630 Rhad is incorrect. The motivation for this option is that the exact analytical form of RPQCD is only known up to b LV one-loop level. At order (α2) already the full analytic Me 3625 O s H result has to be reconstructed using Padè approximants LV Ge to patch together information about Π(s) obtained at Hm10b 3620 √s = 0,√s = 2mb(µ) and √s → −∞. Both the Padè method, and the reliance on PQCD results obtained at 3615 threshold (√s = 2mb(µ)) could introduce unaccounted systematicerrors. Asameasureofthedependenceofthe method on the prediction of RPQCD(s) up to s (chosen 3610 b 0 15 20 25 to be large enough so that the high energy expansion s0 HGeVL becomes a rigorous prediction), we use RPQCD(s) calcu- b lated using the high energy expansion. The prediction FIG. 2: The values of m (10GeV), obtained for different b of RPQCD at √s = 11.21GeV using the high energy ex- values of s0 and using the 10 different kernels in the class b (i,j,k)(s ,s). All results lie within theshaded region. pansion is also closer to experiment than the prediction P3 0 obtained using Rhad. The high-energy expansion of Π(s), given in Eq.(5), IV. CHOICE OF INTEGRATION KERNELS is only formally guaranteed to converge above √s = 4m (µ) 15GeV, due to non-planar diagrams having b ≈ cuts starting there. Above this value the high energy To minimize the dependence of results for the bottom- expansion is an almost perfect approximationto the full quark mass on Option A and Option C, the contri- analytic PQCD result [20]. Therefore, we shall always bution from the region √s √s∗ 11.21GeV to √s0 choose √s0 > 4mb(µ) in Eq.(3) so that it is safe to use should be quenched. This c≡an be ≡achieved by borrow- thehighenergyexpansionofΠ(s)inthecontourintegral. ing from the method of [21], where a Legendre polyno- Between the end point of the data (√s = 11.21GeV) mial was used to minimize the contribution of the then and√s0 >4mb(µ),wewillusethebestavailablePQCD poorly known continuum threshold region. We choose prediction of R (s), obtained from the Fortran program b here a Legendre-type Laurent polynomial, i.e. we con- Rhad[20]. We considerthis asdata input,eventhoughit sider linear combinations of powers of s chosen from the stems from theory. The Rhad [20] prediction of Rb(s) is set = s−3,s−2,s−1,1,s . Inverse powers higher than shown in Fig. 1. s−3Slead{to a deterioration}of the convergenceof PQCD, The first uncertainties affecting the bottom-quark introducinglargeuncertaintiesfromchangesintherenor- mass are due to the uncertainty in the strong cou- malization scale µ and the strong coupling α (see also s plingαs (∆αs),theuncertaintyintheexperimentaldata [24]). We only use positive powers up to s1, as higher (∆EXP),andourlimitedknowledgeofPQCD(∆µ). The powers emphasize unknown (α3) terms in the high en- latterwasestimatedbyvaryingtherenormalizationscale O s ergy expansion. The optimal order of the Legendre-type µ = 10GeV by 5GeV, running the mass calculated Laurent polynomial was found to be 3 or 4. First, let us ± at this scale back to µ = 10GeV and then taking the consider the order 3 case and let maximum difference. The second set are systematic uncertainties stemming from the fact that the PQCD pre- p(s) (i,j,k)(s,s )=A(si+Bsj +Csk), (10) diction for R (s) does not agree with the experimen- ≡P3 0 b tally determined values at the end point of the data subject to the global constraint (√s=11.21GeV), ascanbe seenfromFig. 1. Twopos- sibilities forthis discrepancywereconsideredin[3]. Op- s0 (i,j,k)(s,s )s−n ds=0, (11) tion A: The BABAR data are correct, but PQCD only Zs∗ P3 0 starts at higher energies, say at √s =13GeV. Use then a linear interpolation between RbEXP(11.21GeV) = 0.32 where n 0,1 , i,j,k 3, 2, 1,0,1 , and i,j,k and RPQCD(13GeV) = 0.377, rather than the predic- are all di∈ffe{rent.} The ab∈ov{e−con−stra−int det}ermines the b tion from Rhad. Option B: The PQCD prediction from constants B and C. The constant A is an arbitrary Rhad is correct, but the BABAR data are incorrect,per- overall normalization which cancels out in the sum rule haps affected by an unreported systematic error. In this Eq.(3). The reasonforthe presenceofthe integrands−n case multiply all the data by a factor of 1.21 to make above is that the behavior of R (s) in the region to be b thedataconsistentwithPQCD.Inadditiontothesetwo quenchedresemblesamonotonicallydecreasinglogarith- options, we wish to consider a third possibility. Op- mic function. Hence, aninversepowerofs optimizes the 4 Uncertainties (MeV) Options A, B, C (MeV) p(s) mb(10GeV) √s0(GeV) ∆EXP. ∆αs ∆µ ∆TOTAL ∆A ∆B ∆C s−3 3612 9 4 1 10 20 -17 16 s−4 3622 ∞ 7 5 10 13 12 -12 8 ∞ (−3,−1,0)(s ,s) 3623 16 6 6 2 9 1 -6 0 P3(−3,−1,1)(s0,s) 3623 16 6 6 2 9 2 -7 0 P3(−3,0,1)(s ,0s) 3624 16 7 6 2 9 2 -7 0 P3(−1,0,1)(s0,s) 3625 16 8 5 4 10 4 -12 0 P3 0 (−3,−1,0,1)(s ,s) 3623 20 6 6 3 9 0 -4 0 P4 0 TABLE I: Results for m (10GeV) using kernels p(s) selected for producing the lowest uncertainty. Results from the kernels b p(s)=s−3 and p(s)=s−4 used in [2]-[3] are given here for comparison. The errors are from experiment (∆EXP.), the strong coupling (∆αs) and variation of therenormalization scale by 5GeV around µ=10GeV (∆µ). These sources were added in ± quadrature to give the total uncertainty (∆TOTAL). The option uncertainties ∆A, ∆B and ∆C are the differences between m (10GeV)obtainedwithandwithout Option A,B,orC.Asin[2]-[3]thesearenotaddedtothetotaluncertainty,andare b listed only for comparison purposes. quenching. As an example, taking s = (16GeV)2 (and say (i,j,k)(s,s ) only included inverse powers of s, then 0 P3 0 A=1) we find almost the entire right hand side of Eq.(3) would em- anate from the residue, and hence from the low energy P3(−3,−1,0)(s,s0) = s−3−(1.02×10−4 GeV−4)s−1 expansion of PQCD. If however P3(i,j,k)(s,s0) were com- + 3.70 10−7 GeV−6 , (12) posed of only positive powers of s, then only the high × energy expansion of PQCD would enter the right hand with s in units of GeV2. There are ten different kernels sideofEq.(3). Differentkernelscanthereforeleadtosig- P3(i,j,k), and the spread of values obtained for mb using nificantly different dependencies on the renormalization this set of different kernels will be used as a consistency scale µ. Our philosophy is to choose those kernels pro- check on the method. Outside the interval s [s∗,s0], ducing the lowest total uncertainty. The results from ∈ (i,j,k)(s,s ) will blow-up, which leads to a suppression these are displayed in Table I. We also plot in Fig.2 P3 0 of the continuum threshold region relative to the well the range of values for m (10GeV) obtained using all b measured Υ-resonances. This will minimize the depen- of the 10 kernels in the class (i,j,k)(s,s ), as a function denceoftheresultsonOptionB.Hence,thiskernelmin- of s0. Remarkably, betweenP132GeV <0√s0 < 28GeV, imizes all three sources of systematic uncertainty. The all of the masses obtained using all 10 kernels from fdoeufirntehd-obrdyetrheLacuonresntrtaipnotlyEnqo.(m1i1a)l, bPu4(ti,jw,ki,trh)(sn,s0)0i,s1,a2lso. tm¯he(1c0laGsseVP)3(−3,3−612,05)(Ms,esV0). Ofaullrimnetthheodragnigvees3a62c1onMsiesVten≤t There are also five different kernels P4(i,j,k,r)∈(s,{s0). I}n resbult even in≤the region √s0 <4mb(µ)≈15GeV where general,thehigherthe ordernof ,the betterthe con- thehigh-energyexpansionusedinthecontourintegralin n P trol over the systematic errors. However, the price to Eq.(3)isnotguaranteedtoconverge. Using,rather,the5 pay is a reduction in the rate of convergence of PQCD, kernels in the class (i,j,k,r)(s,s ), and varying s in the though this convergence can be improved by increasing range 18GeV < √sP4< 70GeV0, all of the mass0es thus 0 s0. In the Appendix we give explicit expressions for the obtained lie in the interval 3620MeV m¯b(10GeV) various kernels used in table I. 3626MeV. Theseresultsshowagreatin≤sensitivityofou≤r methodontheparameters ,andalsoonwhichpowersof 0 (i,j,k) (i,j,k,r) s are used to construct (s,s ) and (s,s ). P3 0 P4 0 This in turn demonstrates the consistency between the V. RESULTS AND CONCLUSIONS high and low energy expansions of PQCD. For our final result we choose the optimal kernel (−3,−1,0)(s ,s) to obtain We considered a total of 15 different kernels p(s) used P3 0 in Eq.(3), 10 from the class of kernels (i,j,k)(s,s ) and m (10GeV)=3623(9)MeV, (13) P3 0 b 5 from the class (i,j,k,r)(s,s ). All these are similarly P4 0 constructed (i.e they obey Eq.(11)), and hence have a m (m )=4171(9)MeV. (14) b b similar ability to reduce the dependence of the bottom- quark mass on Options A, B, C. They do, however, Thisresultisfullyconsistentwiththelatestlatticevalue place very different emphasis on theory. In particular, if m (10GeV) = 3617(25)MeV [23]. It is also consis- b 5 tent with a previous QCD sum rule precision determination[2]-[3]givingm (10GeV)=3610(16)MeV. Apart b from our novel QCD sum rule approach, the inputs in (−3,−1,1)(s,s ) = s−3 (6.875 10−5GeV−4)s−1 the latter are almost identical to ours, with the excep- P3 0 − × + (1.000 10−9GeV−8)s, (16) tion of their use of kernels of the form p(s) = s−n, × n 2,3,4,5 ,and the use of a value of the strong cou- ∈{ } pling with a larger uncertainty. Their final result was obtained using p(s) = s−3, which can be seen from Ta- ble I as being far more sensitive to possible systematic uncertainties arising from Options A, B, C. They also P3(−3,0,1)(s,s0) = s−3−7.767×10−7GeV−6 determinedmb usingp(s)=s−4,forwhichtheyobtained + (3.103 10−9GeV−8)s, (17) m (10GeV)=3619(18)MeV. This value is closer to our × b result, which may not be surprising given that it is less sensitivetoOptionsA,B,Cthanp(s)=s−3,although not as insensitive as using our kernels. Inconclusion,wehavediscussedhereafiniteenergyQCD (−1,0,1)(s,s ) = s−1 0.01129GeV−2 sum rule method with integration kernels involving in- P3 0 − verse and positive powers of the squared energy. The + (3.059 10−5GeV−4)s. (18) × result for the bottom-quark mass has a lower total uncertainty, and is far less sensitive than the popular in- versemomentmethodtothethreesystematicuncertain- Next, for s =(20GeV)2 0 tiesidentifiedearlier,i.e. OptionsA,B,C.Itshouldbe appreciated from Table I that the results Eqs.(13)-(14) are independent of the PQCD prediction from Rhad in the region between √s≃11.21GeV and √s=4mb(µ). P3(−3,−1,0,1)(s,s0)=s−3−(1.4668×10−4GeV−4)s−1 + 8.781 10−7GeV−6 (1.381 10−9GeV−8)s. (19) × − × VI. APPENDIX VII. ACKNOWLEDGEMENTS Up to an overall constant, the integration kernels (s,s ) can be obtained from Eq.(11). 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