On the R and R, Ratios at the Five-Loop Level of Perturbative &CD* ABSTRACT

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1 On the R R, Ratios at the Five-Loop Level of Perturbative &CD* SLAC-PUB-6370 October 1993 T/E MARK A. SAMUEL Department of Physics Oklahoma State University Stillwater, OK Stanford Linear Accelerator Center Stanford University, Stanford, California G. LI Department of Physics Oklahoma State University Stillwater, Oh? 4078 ABSTRACT We study perturbative QCD at the five-loop level. In particular we consider R = ctot( ese- + hadrons)/a( e+e- + p+p-) R, = I (7 + v + hadrons)/i (r + evv). We use our method to estimate the five-loop coefficients. As a result, we obtain cys(mz) = (11) a,(34 GeV) = (16), which are accurate at the 1% level. We also find R = (18) which is consistent with R, is accurate to 0.05%. 1 Submitted to Physical Review Letters. * Work supported by the Department of Energy, contract DE-AC03-76SF00515.

2 - Perturbative QCD has been used to describe the strong interaction very suc- cessfully, when the energy scale is large enough. This includes the R ratio R = a(e+e- --f hadrons/a(e+e- -+ p+p-) (1) also the R, ratio R, = r( r + u + hadrons) r(7 --f euf) (2) even though the mass scale Mr is not very large. Recently Braaten [l] presented a discussion of R, as well, a new quantity, the spin asymmetry parameter A, where A, = RF-RB RF+&' (3) RF RB are the forward backward components of R, R,=RF+RB. (4) The lowest order estimates are R, = 3 A, = l/3. (5) These estimates are changed by perturbative non-perturbative corrections as follows:

3 (6) where SEW = (7) is the electroweak correction the ft & are proportional to l/m: with coefficients that depend logarithmically on Mr. The purely perturbative QCD effects from the interactions of massless quarks gluons for Nf = 3 are f! = : (T) (7) + (df) ) (T) (8) fp = n- * ) (:) (9) where crs = os(mr) is the running coupling constant of QCD in the MS scheme (3) evaluated at the scale Mr. The coefficient d, is the fifth coefficient in the series do = 1, dl = 1, d2 = 1.64 d3 = 6.37 has not yet been calculated (perturbative expansion of -2r2s(d/ds)r(1)(s)). We will use our estimation method which makes use of Padd approximants to estimate the value of d4. From Eq. (9) the Pad& Approximant Prediction (PAP) is df) = 41. From the equation below it it is dr) = 116. Applying it directly to the d; series we &t&in df) = 31. The average is d y = 55. Finally the PAP for the (cx~/~)~ term for R, is 133. Thus df) = = 55, in agreement with the average above. 3

4 For further details, see our earlier papers [2]. Thus we take as our value, with conservative error estimates dc3) = (10) This is our result for Nf = 3. Our results for f; = fj? + # + jj! = (11) agree with. those of Braaten. The relative contribution to R, is. -_ (2f; + fi) = -1.58%. 3 (12) There are various experimental values for R,. We use the WOI d-average [3] for B, B,. From B, = 17.76(15)% we obtain R, = 3.658(31) (13) from B, = 17.53(19)% we obtain R, = 3.629(24). The weighted -_.--- average of Eqs. (13) (14) is R, = 3.640(19). (15) 4

5 From the measured T lifetime [4], rr = (32) x 10-12S one obtains R, = 3.549(39). (16) We take as our value the weighted average of Eqs. (15) (16) R, = 3.623(17) (17) (see also Ref. [5]). Now from Eqs. (4), (6), (7), (8), (9), (lo), (12) (17) we obtain our result for as(k), a&w,) = (89) (18) using Eq. (3) we obtain. _ A, = 0.413(22). (19) Equation (19) a g rees with Braaten s result. Our result for cr,(m,) is somewhat different from Braaten s cy@!f,) = 0.319(17). (20) Note, however, that the error in Eq. (18) is much smaller than the error in Eq. (20). Actually, d ue to an interpolating error Braaten s result should be as(mr) = 0.324(17). (21) We can now obtain AC31 from the running of a,, -u = 27r par, 1 1J31en2L Po2L 4L2po4 [&ln22l - &fml + P2Po - P?] 1 (22) 5

6 where PO,,&,,02 are the coefficients of the QCD p function P2=,- 18Nf +gn;. L = +!.np/a Nf is the number of fermions (quarks). For Nf = 3 we obtain. _ ffoti Eqs. (18), (22) (23) At31 = 352( 17) MeV. (24) We use the MS scheme throughout this paper In or.der to ensure continuity of cxs as Nf changes, we have derived the following relationships: 6

7 I ~(5) = ~(6) (fi.l)2 23 [In(f$..-)] (!$)174 52g AC41 = *t5) ( E!J2 25 En (f.l$)] g ( g) *(4) = *c31 ( q2 [In ( ;;2)] (??) AC3) = *t4) (5)2 27 [p,(f.$)] (g)32 81 These relations differ from those of Marciano [6] by approximately 3%. From Eqs. (25) we obtain AC41 = 304(16) MeV (26) Ac5) = 216(13) MeV (27) Ac6) = 92(6) MeV. (28) 7

8 I From Eqs. (22) (27) we obtain crs(mz) = (11) (29) whose error is less than l%! This is consistent with the experimental value from LEP [7] the latest value from SLD [8] %(MZ) = 0.120(7) LEP f O.O02(stat) f O.OOS(sys) f O.OlO(th) SLD. (30) It is clear that a more accurate experimental value is needed. For ~~~(34 GeV) our result is a,(34 GeV) = (16). (31) From the experimental value for R = 3CQ2f, T- = 1.049(7) (32) one obtains [9] ~~~(34 GeV) = 0.149(21) (33) in agreement with Eq. (31). Th ese results along with os evaluated at 5 GeV, 10 GeV, 17.3 GeV, 80.6 GeV 180 GeV are shown in Table I. It can be seen that all these results are consistent with experiment. For Nf = 5 we have ff = (?) (F) (T) + (df) ) (F?) (34)...f$ = (T) (T) (T) + (dy) ) (T). (35) We have neglected the term proportional to (CQi)2 as it should be very small. 8

9 - From Eq. (34) fo r j$) we obta in df) = 35.2 fro m R, we get d f) = From Eq. (35) fo r j$ we get da) = 77.8 fro m th e series directly df) = We shall be conservati ve ta ke as our value d(5) = * (36) The R rati o [II] in th e MS scheme fo r Nf = 5 is 1 + (:) (T) ( )3 7r ( CQf)2 =Q; (3 + R4 (34] = 3CQ;r where-[1 21 R4 = df) (38) so +54 R4 = (39) Again we neglect th e te rm proportional to (CQf)2, q, since fo r th e case of inte rest 9 = l/3 3 th is te rm should be negligible. Now using our result fro m a,(3 4 GeV) = (16) (40) weobta in f = (5) (41) 9

10 R = (18). (42) These results are accurate at the 0.05% level! For 31.6 GeV we obtain ~~~(31.6 GeV) = (16) (43), hence, r(31.6 GeV) = (6). (44) This should be compared to the experimental result [14] ~(31.6 GeV) = (50). (45) In conclusion, we have shown how one can use our Pad& Approximant Predic- tion (PAP) Method to estimate R, at the five-loop level of PQCD. This estimate has then been used to obtain more accurate predictions for as(p) for various values of p. The agreement with experiment is excellent! We have also used our result for a,(34 GeV) t o obtain the R ratio accurate to 0.05%. It should be emphasized that once we have fixed cr,(m,) all the results in this paper are determined have been obtained with no adjustable parameters. Now we need to improve the accuracy of the experimental values! Acknowledgements One.of us (MAS) would like to thank the theory group at SLAC for its kind hospitality. He would also like to thank Eric Braaten, Stan Brodsky, David Atwood, 10

11 Richard Blankenbecler, Bill Marciano, Helen Quinn for helpful discussions. This work was supported by the U.S. Department of Energy under grant numbers DE-FG05-84ER40215 DE-AC03-76SF REFERENCES [l] E. Braaten, Phys. Rev. Lett. j l, 1316 (1993). We wish to thank Eric Braaten for sharing his results for j$) &)I, for general Nf, with us. [2] M. A. Samuel, G. Li E. Steinfelds, Phys. Rev. M, 869 (1993); M. A. Samuel, G. Li E. Steinfelds, On Estimating Perturbative Coefficients in Quantum Field Theory, Condensed Matter Theory Statistical _. _ Physics, Ok1 a h oma State University Research Note 278, August (1993); M. A. Samuel, G. Li E. Steinfelds, Estimating Perturbative Coeffi- cients in Quantum Field Theory Using Pade Approximants, Oklahoma State University Research Note 273, October (1992); M. A. Samuel G. Li, Estimating Perturbative Coefficients in High Energy Physics Condensed Matter Theory, Oklahoma State University Research Note 275, December (1992). [3] M. Davier, Proceedings of the Second Workshop on Tau Lepton Physics, The Ohio State University, Columbus, Ohio, September 8-11, (1992) pg. 514, edited by K. K. Gan, World Scientific (1993). [4] W. Trischuk, Proceedings of the Second Workshop on Tau Lepton Physics, The Ohio State University, Columbus, Ohio, September 8-11, (1992) pg. 59, edited by K. K. Gan, World Scientific (1993); W. J. Marciano, Proceedings of the DPF92 Meeting, Fermilab, November (1992). 11

12 [5] M. A. Samuel, The Tau Lepton Tests of the Stard Model, Fermilab-Pub 92/201-T (1992), Modern Physics Letts. A (1993). [6] W. J. Marciano, Phys. Rev. U, 580 (1984). [7] H. Wachsmuth, Determining the Strong Coupling Constant from e+e- Collisions at LEP, CERN-PPE/Sl-145, unpublished (1991). [8] K. Abe et al., (SLD C o 11 a b oration) Phys. Rev. Lett 71, 2528 (1993). [9] M. A. Samuel L. Surguladze, Modern Physics Letters fl, 781 (1992). [lo] T. Akesson et al, Zeit. fur Phys. m, 317 (1986). [ll] L. R. S ur g u 1 a d ze M. A. Samuel, Phys. Rev. Lett. 66, 560 (1991); S. G. Gorishny, A. L. Kataev S. A. Larin, Phys. Lett. B259, 144 (1991). [12] M. R. Pennington G. G. Ross, Phys. Lett. B102, 167 (1981). -:[13]. J. Alitti et al, Phys. Lett. B263, 563 (1991). [14] R. Marshall, Z. Phys. Q& 595 (1989). 12

13 TABLE I Experimental Predicted Values for as(p) P(GW Theoretical Values for as(p) Experimental Values for CY~( cl) (89) (20) ( 16) ( 16) ( 11) (11) (g) input 0.18(5) Ref (16) Ref (22) Ref (25) Ref (7) Ref f (stat) f (sys) f (th) Ref. 8

Local and Global Duality and the Determination of α(m Z )

MZ-TH/99-18, CLNS/99-1619, hep-ph/9905373, May 1999 Local and Global Duality and the Determination of α(m Z ) arxiv:hep-ph/9905373v1 17 May 1999 Stefan Groote Institut für Physik, Johannes-Gutenberg-Universität,