The mass of the bottom quark is analyzed in the context of QCD finite energy sum rules. In contrast to the conventional approach, we use a large momentum expansion of the QCD correlator including terms to order alpha(S)(2)(m(b)(2)/q(2))(6) with the upsilon resonances from e(+)c(-) annihilation data as main input. A stable result m(b)(m(b)) = (4.19 +/- 0.05) GeV for the bottom quark mass is obtained. This result agrees with the independent calculations based on the inverse moment analysis

Bordes Villagrasa, José M.

Peñarrocha Gantes, José Antonio

Schilcher, K.

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arXiv:hep-ph/0212083v2  21 Mar 2003MZ-TH/02-31FTUV-0412/02December 2002Bottom Quark Mass and QCD Duality1J. Bordesa, J. Pen˜arrochaa and K. SchilcherbaDepartamento de F´ısica Teo´rica-IFIC, Universitat de ValenciaE-46100 Burjassot-Valencia, SpainbInstitut fu¨r Physik, Johannes-Gutenberg-Universita¨tD-55099 Mainz, GermanyAbstractThe mass of the bottom quark is analyzed in the context of QCDfinite energy sum rules. In contrast to the conventional approach,we use a large momentum expansion of the QCD correlator includ-ing terms to order α2s(m2b/q2)6 with the upsilon resonances frome+e− annihilation data as main input. A stable result mb(mb) =(4.19 ± 0.05) GeV for the bottom quark mass is obtained. This re-sult agrees with the independent calculations based on the inversemoment analysis.PACS: 12.38.Bx, 12.38.Lg.1Supported by MCYT-FEDER under contract FPA2002-00612, EC-RTN under contractHPRN-CT2000/130 and partnership Mainz-Valencia Universities.11 IntroductionWith the Standard Model being so well established the major theoretical ef-fort of today’s theoretical and experimental investigations goes into a precisedetermination of the parameters of the model. An important example is thedetermination of the CKM matrix elements and their relative phases in currentB-physics experiments. A better knowledge of these parameters will lead to abetter understanding of the structure of weak interaction including CP viola-tion, and possibly the discovery of ”new physics”. For an analysis of most ofthe experiments in the B-sector a precise value of the bottom quark mass isessential.In the last few years, there has been increased activity with the aim of de-termining the bottom quark mass [1−10] to higher accuracy. The most popularmethod was the one based global duality in QCD. In so-called inverse momentsum rules derivatives of the two point correlation function of the b quark vectorcurrent are compared with experimental information from electron-positron an-nihilation into hadrons [12]. These sum rules seem to be a suitable method sinceit is very sensitive to the heavy (b-quark) quark mass. Recent analysis usinginverse moments involving 10 to 20 inverse powers of the squared energy can befound in [2, 8] and, using a lower power inverse moments, in [9]. Non relativisticQCD, potential models (for a review see [13]) and lattice techniques [11] havealso been considered in order to determine the heavy quark masses. However,in the case of the lattice techniques and potential models, there exist problemswith the choice of the mass definition (running-, pole-, threshold mass) and thesize of non-perturbative effects. So far, the available experimental informationrelies mainly on the knowledge of the masses and widths of six upsilon reso-nances, three of them lying below the continuum threshold of the BB mesonproduction.The method we propose here is somewhat orthogonal and complementary tothe one based on the inverse moment sum rules: we employ on the theoreticalside the large momentum expansion of QCD, i.e. an expansion in m2b/s, wheres is the square of the CM energy. Such an expansion makes sense as longas s is far enough away from the continuum threshold. We will consider theperturbative expansion up to second order in the strong coupling constant andtwelve powers of the b-mass over energy ratio using the results of the reference[14]. An expansion to the third order in the strong coupling but with only fourpowers in the mass-energy ratio is also known [15], which we will not make useof for reasons of consistency. On the phenomenological side of the sum rule2we will consider the six upsilon resonances. In addition, our method should besensitive to the poorly known continuum data. We circumvent this problemby a judicious use of quark-hadron duality. Above the resonance region, whereexperimental data are poorly known, we incorporate a real polynomial in thesum rule [16] or, what is the same, a suitable combination of positive moments, insuch a way that the contribution of the data in the region above the resonancescan be practically eliminated. The method has been successfully checked inthe charm sector [17], where the charm quark mass was predicted by using theexperimental data and the polynomial insertion in the intermediate region. Theresult for the charm mass was found in good agreement with the ones obtainedusing other methods. Of course, in the b-sector one is in a truly heavy quarkregime and it is not clear that the same approach may lead to a result for theb-mass of competitive accuracy. We will show that this is actually the case.The plan of this note is the following: in the second section we briefly reviewthe theoretical method used here which is based on weighted finite energy sumrules, in the third section we discuss the theoretical and experimental inputsused in the calculation and present our results for the bottom quark mass witha discussion of the errors. We finish the paper giving the conclusions.2 The calculationIn order to write down the sum rule relevant to our case, we apply Cauchy’stheorem to the two point correlation function Π(s), with the appropriate flavorcontent in the quark currents, and weight the integration with a polynomialP (s). The integration path extends along a circle of radius |s| = sR, and alongboth sides of the physical cut [s0, sR]. The polynomial P (s) does not change theanalytical properties of Π(s) in the integration region, so we obtain the followingsum rule1pi∫ sRs0P (s) ImΠ(s) ds = −12pii∮|s|=sRP (s) Π(s) ds, (1)where s0 is the physical threshold of the bb channel, starting at the first bbresonance below the continuum threshold sC > s0. The integration radius sRis chosen in such a way that on the circle the asymptotic expansion of QCD,ΠQCD(s), constitutes a good approximation to the two point correlator. Theleft hand side of equation (1), is related to the experimental e+e− annihilationcross section via the unitarity relation,R(s; e+e− → hadrons) = 12pi∑flavorsQ2f ImΠ(s). (2)In order to perform analytically the contour integration in (1) we take, as amatter of convenience, the two point correlation function ΠQCD(q2) which hasbeen calculated in [14] as an expansion up to second order (three loops) in the3strong coupling constant αs and as a power series of m2/q2 up to the sixthorder,ΠQCD(q2) = Π(0)(q2) +(αs(µ)pi)Π(1)(q2) +(αs(µ)pi)2Π(2)(q2), (3)where the different terms of the expansion in αs have the form:Π(i)(q2) =6∑j=03∑k=0A(i)j,k(m,µ)(ln−q2µ2)k (m2q2)j. (4)In equations (3,4) µ is the renormalization scale of the perturbative calculation,m is the running quark mass and A(i)j,k(m,µ) are the coefficients of the QCDperturbative expansion which depend on m and µ through powers of lnm2/µ2[14]. In the asymptotic expansion of equation (4) one should include the non-perturbative terms due to the quark and gluon condensates, which are knownup to order αs [19]. Their contribution, however, is completely negligible in therange of energies of interest to us. For this reason we will discard them fromnow on.As commented in the introduction, the experimental side of the sum ruleis dominated by the six well established bb upsilon resonances. With this ex-perimental information in the narrow width approximation we have the hadroncross section of equation (2) given byR(s) =9piα2em∑resMres Γres δ(s−M2res),where Mres and Γres are respectively the masses and electronic widths of the sixresonances, and αem = 1/131.8 is the electromagnetic coupling constant takenat the typical scale of 10 GeV, where the resonances are produced.Furthermore, in the theoretical side of the sum rule, equation (1), P (s) istaken to be a third degree polynomialP (s) = a0 + a1s+ a2s2 + a3s3,whose coefficients are fixed by imposing a normalization condition P (s0) = 1,and requiring that it should minimize the contribution of the continuum [sC , sR]in a least square sense, i.e.,∫ sRsCsnP (s) ds = 0 for n = 0, 1, 2,where sC is the value of the continuum physical threshold for the BB mesonproduction. The choice of this polynomial guarantees that the contribution ofthe experimental data in the continuum region (which is experimentally poorly4known) will be negligible in the sum rule as compared to the contribution of thelower resonances below the continuum threshold. In fact, a smooth continuumcontribution which can be described by a second order polynomial vanishesidentically. In this way, we avoid the inaccurate experimental information inthe bb continuum region. This approximation procedure was checked in the ccchannel, where there is good experimental information on the continuum dueto recent results from the BES II collaboration [18]. Accurate and consistentresults for the charm mass have been obtained, by either incorporating thecontinuum or eliminating it by the procedure above [17].Notice also that the dependence on the method, which relies on the choiceof the polynomial (2), is very weak. To check that fact we have performedthe same calculation using polynomials of different degrees as well as changingthe boundary conditions of the third degree polynomial (2), although keepingthe same philosophy as commented in the paragraph above. For instance for atwo degree polynomial we find a difference of .01 Gev in the bottom mass butthe experimental error coming from the uncertainties in the resonance data isdouble than the one we find with a degree three polynomial. On the other handa polynomial of three degree changing slightly the boundary conditions givesjust the same result.Higher order polynomials, as well as the inclusion of the incomplete order α3sin equation (3) can be considered as well. However, before doing so, a detailedstudy of the different moments involved in equation (1) should be performed.We defer this study to a further publication.With our study we conclude that our selection for the polynomial do notpreclude the results found nor the errors presented in this work.The integrals that we have to calculate on the right hand side of the sumrule, equation (1), areJ(p, k) =12pii∮|s|=sRsp(ln−sµ2)kds,for k = 0, 1, 2, 3 and p = −6,−5, .., 3. Their analytic evaluation can be found inreference [20]. After integration, equation (1) can be cast into the form9pi(131.8)2∑resMres Γres P (M2res)= −4pi233∑n=02∑i=06∑j=03∑k=0an(αspi)iA(i)j,k(m,µ)(m2)jJ(n− j, k), (5)giving a non-linear equation with the quark mass (m) as the only unknown. Toproceed further and solve this equation we have to choose a suitable value ofsR for which perturbative QCD gives a good approximation for the correlationfunction. Furthermore, the scale µ can be chosen arbitrarily to give a predictionof the quark running mass at this scale. For calculational convenience, we take5µ2 = sR since most of the logarithmic terms vanish after integration. Then,after solving equation (5), we run the quark mass from that scale to the massscale itself, m(µ = m), by means of the corresponding renormalization groupequations at the appropriate loop order in the strong coupling constant [21, 22].As far as the theoretical input matches the experimental data within theerror bars, the results should be independent of the choice for sR. Nevertheless,the lack of experimental data in the continuum region of the bb channel makessensible in our method to study the sR dependence of the quark mass results.Although this dependence is going to be very small, due to the choice of ourpolynomial in the sum rule, we will take the quark mass prediction at the moststable value with sR.3 ResultsThe experimental inputs in our calculation are as follows:Firstly, the physical threshold s0 is the squared mass of the first low lyingresonances in the bb channel, Υ(9460),s0 = m2Υ = 9.462 GeV2,whereas the continuum threshold sC is taken at the squared energy of the BBmeson productionsC = (2mB)2 = (10.58)2 GeV2.Secondly, the absorptive part of the two point correlation function is ob-tained from the six known bb vector resonances Υ(1S),..,Υ(6S) with the follow-ing masses and electromagnetic widths [23, 24].M(1S) = (9460.30± 0.26) MeV,M(2S) = (10023.26± 0.31) MeV,M(3S) = (10355.2± 0.5) MeV,M(4S) = (10580.0± 3.5) MeV,M(5S) = (10865± 8) MeV,M(6S) = (11019± 8) MeV,Γ(1S) = (1.32± 0.05) keV,Γ(2S) = (0.520± 0.032) keV,Γ(3S) = (0.48± 0.04) keV,Γ(4S) = (0.248± 0.031) keV,Γ(5S) = (0.31± 0.07) keV,Γ(6S) = (0.130± 0.030) keV.(6)Finally, for the strong coupling constant we take the input value at the massof the electroweak Z boson [23]αs(MZ) = 0.118± 0.003 (7)and run down to the scale of the contour radius sR using the four loop formulaeof reference [21].Now we proceed as follows. For µ2 = sR with sR in the range [150, 350] GeV2we determine m(i)b (µ) by solving equation (5) keeping terms in the perturbativeexpansion up to order (αs/pi)i for i = 0, 1, 2. Then we run the results from6µ2 = sR to µ = mb with the appropriate renormalization group equations [22].Finally, we choose the value of sR which is most stable in the range of energiesconsidered. In this way we find stability at sR = 240 GeV2 and the followingresults for the bottom mass:m(0)b (mb) = 4.09 GeV,m(1)b (mb) = 4.23 GeV, (8)m(2)b (mb) = 4.19 GeV.To estimate the various errors arising in the calculation of the b quark masswe consider first the uncertainties in the masses and widths of the resonances inequation (6). This gives an experimental error for the mass εEXP = 0.02 GeV2.Secondly, we consider the uncertainty in the strong coupling constant, equation(7), which leads to εα = 0.01 GeV. Finally, we include a conservative asymp-totic error originating from the higher orders in expansion of equation (8). Toestimate this error we consider the difference between the second and first orderresults which gives εASY = 0.04 GeV. Adding the errors quadratically, we findfor the b-quark massmb(mb) = (4.19± 0.05) GeV. (9)Before going on, a comment on the non perturbative contributions to the twopoint correlation function is in order. Using accepted values for the condensates,we find that their contribution to the b-quark mass for the considered range ofsR is always of the order of 0.001 GeV and, therefore, negligible in comparisonto the errors obtained before. We have also checked that the convergence of theresult (9) with respect to the power series expansion in m2/s has no influenceto the error bars in the sR stability domain.To have a flavor of the whole procedure, we include two figures which sum-marize the main steps followed in the calculation of the bottom quark mass.As a sample calculation, we have plotted in figure 1 the results of m(1,2)b (mb)as function of the radius sR. The upper (lower) curve corresponds to the first(second) order in the strong coupling constant. We find the most stable resultfor sR = 240 GeV2 . The difference between second and first order results isεASY = 0.04 GeV. The sR dependence is so tiny that all the values for the massin the whole range [150, 350] GeV2 lie within the relatively small asymptoticerror bar.In figure 2 we have plotted the right hand side of the mass equation (5) as afunction of mb(µ) with the renormalization point taken at the value of sR andthe latter in the stability point (µ2 = sR = 240 GeV2). The thin curve includesonly the calculation to one loop, whereas the medium and thick curves includecalculations to two and three loops, respectively. The straight strip correspondsto the contribution of the experimental data to the left hand side of equation (5)2Recently, it has been claimed an enhancement of the experimental data in the 4S and 5Sresonance region [25]. This could change the final value of the bottom mass by less than 0.01GeV, which is within the exprimental errors of the method.7Figure 1: Stability of the two (upper curve) and three (lower curve) loop calcu-lations of the bottom quark mass.including experimental error bars. Hence, the crossing of the straight strip withthe three curves gives the solution of equation (5), i.e. the results for the runningmass m(0,1,2)b (240 GeV2). Running this results to µ = mb we find the valuesquoted in equation (8). Notice the very good convergence of the perturbativeseries at the energies relevant to this process.4 ConclusionsThe bottom quarks mass has been determined in an unconventional fashion byemploying finite energy QCD sum rules with positive moments. By apply-ing Cauchy’s theorem to the bb vector current correlator weighted with a realpolynomial, it was possible to virtually eliminate the contribution of the yetunknown experimental data in the continuum region from just above the res-onances to the beginning of the asymptotic region where QCD is valid. Theexperimental input needed is only the resonance masses and couplings.The method we use is quite independent of the more popular one based oninverse moment sum rules. The result we obtain, mb(mb) = (4.19± 0.05) GeV,agrees with most of the recent calculations [13], especially with references [8]and [9] that study high and low inverse moment sum rules, respectively. 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Bottom quark mass and QCD duality

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