application/xmlFinite energy sum rules for the cross section of e+e−-annihilation into hadrons in QCDK.G. ChetyrkinN.V. KrasnikovA.N. TavkhelidzePhysics Letters B 76 (1978) 83-84. doi:10.1016/0370-2693(78)90107-7journalPhysics Letters BCopyright © 1978 North-Holland Publishing Company. All Rights Reserved. Published by Elsevier B.V. All rights reserved.Elsevier B.V.0370-26937618 May 19781978-05-0883-84838410.1016/0370-2693(78)90107-7http://dx.doi.org/10.1016/0370-2693(78)90107-7doi:10.1016/0370-2693(78)90107-7http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/Volume 7613, number 1 PHYSICS LETTERS 8 May 1978 FINITE ENERGY SUM RULES FOR THE CROSS SECTION OF a+e--ANNIHILATION INTO HADRONS IN QCD K.G. CHETYRKIN, N.V. KRASNIKOV and A.N. TAVKHELIDZE The Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow 117312, USSR Received 10 January 1978 Finite energy sum rules (FESR) for the cross section of a+e--annihilation into hadrons are obtained within the framework of QCD. The FESR for the standard 4-quark model are in satisfactory agreement with experiment. The best description of the data is given by the 5-quark model, with the electric charge of the 5th quark QS = -1/3. Currently one of the most popular models of strong interactions is quantum chromodynamics (QCD), i.e. the gauge theory of colored quarks and gluons. However, due to the lack of clear understanding of the "confinement puzzle" QCD is not directly applicable to the calculations of the cross sections of physical processes on mass-shell even at high energies. For instance as applied to the process of a+e--annihilation into hadrons QCD gives information about the behaviour of the hadron vacuum polarization function H(z) only far enough from the cut [4m2, -) at which the imaginary part of H(z) is proportional to the quantity R(S) = a(e+e--hadrons)/a(e+e--p+u ) [1-41. Let Hh(z) be the value of H(z) evaluated in QCD up to Nth order in the quark-gluon coupling constant as. In ref. [4] convincing arguments were given in favour of the validity of the approximate equality 11(z) - Hth(z), lm z > G1, (1) so where A is a parameter dependent on N. In this paper we would like to call attention to the obvious fact that the analytical properties of CI(z) expressed by the Kallen-Lehman representation H(z) = z f R) 7r S ) > 4m and eq. (1) involve a host of integral sum rules con- necting R(s) and Rth (s) - 21ri (Ilth (s + ie) - IIth (s - ie)). Indeed, integrating the function H(z)zn, n = 0, +1, +2, ... , over the contour shown in fig. la and the function H(z)/zk, k = 0, ±1, t2, ... , over the contour of fig. lb and using, together with equalities (1) and (2), the Cauchy theorem, we obtain finite energy sum rules (FESR) s s R(s) s 'n d s' -::~ f Rth (s) s'n d s', (3) 4mñ s s f R(s')/s'k ds' f R(s')/sk d s'. 0 so (4) G2 Fig. 1. The integration path used in the derivation of the FESR (3) and (4). 83 Volume 76B, number I PHYSICS LETTERS 8 May 1978 Notice that the error due to invalidity of perturbation theory for Im z < 0 is equal in order of magnitude to A/(ors) for the FESR (3) and 0/ [7r(s - so)] for the FESR (4) (to within possible logarithmic factors). The sum rules (3) were discussed in refs. [5,6] on the basis of analogy with duality sum rules for strong interactions [7]. Those works, however, contained no convincing arguments in favour of a particular choice of Rth(s). In quantum chromodynamics the FESR (3) at n = 0 was used in refs. [8,9] to find the parameter defining the scale of the quark-gluon coupling constant as from experimental data for s < 1 GeV2. We compare the FESR (4) with experiment at so = 9 GeV2, s = 60 GeV2 for the values of k variable within -10 < k c 10. For Rth we used the formula (4] Rth(s) - z E Qi (3 - ut) vi(1 + 3 as.Ì(Ui))~ (5) 4m 2 ) 1/2 vß_(1- S` 7r 3 + U 7r 3 2u ( 4 ) ( 2 4~r)' s + 5m? -1 as = 121r~33 In(! 2) - 2 ri In ( 2 t2), , n = n +smi where mt, Qi are the mass and charge of the ith flavour quark. (We note that in eq. (S) the quark masses correspond to the position of the poles of quark propagators; for the discussion of other possible ways of introducing the parameters corresponding to quark masses in perturbation theory, see ref. [9].) The masses of light u, d and s quarks were taken to be zero and for the quantity A we used the value A = 500 meV which is preferable as regards the description of deep inelastic lepton-hadron reactions [10] and the mean value of R(s) at s < 1 GeV2 in terms of quantum chromodynamics. To calculate the integrals with R(s), we used an analytical fit for R, obtained by combining the formulas of refs. [11,12] at -\/s < 5 GeV, and for ,\/s > 5 GeV we put R = 5 [13]. In this case for Rth we used the expression (5) with the addition corresponding to production of a pair of heavy leptons T+T- [14] . For the standard four-quark model with me = 1.25 GeV/c2 [15] the FESR (4) holds to within 20%(20%)*t . For the five-quark model with QS = -1/3 the best agreement between theory and experiment is achieved *t Given here and below in brackets are the relative values of errors in cases where the interaction is disregarded (as = 0). 84 with masses of the fourth and fifth quarks m4 = ms = 1.6 (GeV/c)2, the FESR (4) holding to within 10%(12%). However, it should be noted that the values m4 and ms are very sensitive to the particular choice of A. So, if we take A = 700 meV, we should get m4 = 1.6 GeV/c2, ms = 2.3 GeV/c2. If we assume that the contribution of charmed and usual hadrons to R are additive and confine ourselves to the zero-order approximation of perturbation theory, then it follows from eq. (4) that in the standard model (k > 2) Rc(s) ds' 4mS ~k i ß ß 4m where Re is the contribution to R from production of hadrons containing c-quarks. The FESR (6) were first obtained in ref. [15] by calculating (dnlIc/dsn) (0). A comparison of eq. (6) with experiment made in that work showed that the FESR (6) are in reasonable agreement with experiment only when k < 7. We thus come to the conclusion that rejection of the hypothesis of additivity and inclusion of the interaction substantially improve the agreement between the FESR and experiment. The authors are grateful to A.A. Logunov and D.V. Shirkov for interest in this work and to V.A. Kuz'min for helpful discussions. [ 1 ] T. Appelquist and H. Georgi, Phys. Rev. D8 (1973) 4000. [2] A. Zee, Phys. Rev. D8 (1973) 4038. [3] S.L. Adler, Phys. Rev. D10 (1974) 132. [4] E. Poggio, H. Quinn and S. Weinberg, Phys. Rev. D13 (1976) 1 [SJ S.B. Gerasimov, Proc. Intern Seminar on Vector mesons and em interactions (Dubna,1969) p. 367. [6] LI. Sakurai, Phys. Lett. B46 (1973) 207. [7J A.A. Logunov, L.D. Soloviev and AX Tavkhelidze, Phys. Phys. Lett. B24 (1967) 181. [8] R. Shankar, Phys. Rev. D15 (1977) 755. [91 R.G. Moorhose, M.R. Pennington and G.G. Ross, Nucl. Phys. B124 (1977) 285. [ 10] A. De Rújula, H. Georgi and H.D. Politzer, Ann. Phys. (NY) 103 (1977) 315. [11] V. Barger, W.F. Long and M.G. Olsson, Phys. Lett. B57 (1975) 452. [121 N. Martinescu and Stech, Phys. Rev. D14 (1976) 3237. [ 13 ] V. Luth, Talk at the Intern. School on High-energy Physics (Baku, 1976). [14] M. Perl, Talk at "Neutrino 76" (Aachen, 1976). [15] V.A. Novikov et al., Plays. Rev. Lett. 38 (1977) 626. Re h'ksl ) ds', s' (6) JournalsS350.3PLB901070370-2693(78)90107-710.1016/0370-2693(78)90107-7North-Holland Publishing Company. All Rights Reserved.Finite energy sum rules for the cross section of e+e−-annihilation into hadrons in QCDK.G.ChetyrkinN.V.KrasnikovA.N.TavkhelidzeThe Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow 117312, USSRAbstractFinite energy sum rules (FESR) for the cross section of e+e−-annihilation into hadrons are obtained within the framework of QCD. The FESR for the standard 4-quark model are in satisfactory agreement with experiment. The best description of the data is given by the 5-quark model, with the electric charge of the 5th quark Q5 = −1/3.References[1]AppelquistT.GeorgiH.Phys. Rev.D819734000[2]ZeeA.Phys. Rev.D819734038[3]AdlerS.L.Phys. Rev.D101974132[4]PoggioE.QuinnH.WeinbergS.Phys. Rev.D1319761958[5]GerasimovS.B.Proc. Intern Seminar on Vector mesons and em interactionsDubna1969367[6]SakuraiI.I.Phys. Lett.B461973207[7]LogunovA.A.SolovievL.D.TavkhelidzeA.N.Phys. Phys. Lett.B241967181[8]ShankarR.Phys. Rev.D151977755[9]MoorhoseR.G.PenningtonM.R.RossG.G.Nucl. Phys.B1241977285[10]De RújulaA.GeorgiH.PolitzerH.D.Ann. Phys. (NY)1031977315[11]BargerV.LongW.F.OlssonM.G.Phys. Lett.B571975452[12]MartinescuN.StechPhys. Rev.D1419763237[13]LuthV.Talk at the Intern. School on High-energy PhysicsBaku1976[14]PerlM.Talk at “Neutrino 76”Aachen1976[15]NovikovV.A.Phys. Rev. Lett.381977626