KUCP-76 January 1995 Target Mass Corrections to QCD Bjorken Sum Rule for 5 9 9 Nucleon Spin Structure Functions 1 ∗ n a J 5 2 1 Hiroyuki KAWAMURA and Tsuneo UEMATSU v 8 Department of Fundamental Sciences 6 3 FIHS, Kyoto University 1 0 Kyoto 606-01, JAPAN 5 9 / h p - p e h : v i X r a Abstract We discuss the possible target mass corrections in the QCD analysis of nucleon’s spin-dependent structure functions measured in the polarized deep-inelastic leptoproduction. The target mass correction for the QCD Bjorken sum rule is obtained from the Nachtmann moment and its magni- tude is estimated employing positivity bound as well as the experimental data for the asymmetry parameters. We also study the uncertainty due to target mass effects in determining the QCD effective coupling constant α (Q2) from the Bjorken sum rule. The target mass effect for the Ellis-Jaffe s sum rule is also briefly discussed. ∗TalkpresentedbyT.UematsuattheYITPWorkshop“FromHadronicMattertoQuarkMatter”, Kyoto, Japan, Oct. 30 - Nov. 1, 1994. To appear in the proceedings. 1. Introduction I would like to talk about possible target mass corrections in the QCD analysis of spin-dependent structure functions which can be measured by the deep inelastic scat- tering of polarized leptons on polarized nucleon targets [1, 2]. We especially investigate those for the QCD Bjorken sum rule. As discussed in the previous talks by Dr. Kodaira and Dr. Sloan, the Bjorken sum rule with QCD corrections is given by 1 1G α (Q2) dx[gp(x,Q2) gn(x,Q2)] = A[1 s + (α2)], (1) 1 − 1 6G − π O s Z0 V where gp(x,Q2) and gn(x,Q2) are the spin structure function g of proton and neutron, 1 1 1 respectively, with x and Q2 being the Bjorken variable and the virtual photon mass squared. On the right-hand side, G /G g is the ratio of the axial-vector to vector A V A ≡ coupling constants. The QCD correction of the order of α (Q2) was first obtained some s years agobasedonoperatorproduct expansion (OPE) andrenormalizationgroup(RG) method in ref. [3, 4]. The higher order corrections were calculated in refs. [5, 6, 7, 8]. Now recent experiments on the spin structure function g (x,Q2) for the deuteron, 1 3He and the proton target at CERN and SLAC [9, 10, 11, 12] together with EMC data have provided us with thedata for testing the Bjorken sum rule[13]aswell asEllis-Jaffe sum rule [14]. In order to confront the QCD prediction with the experimental data at low Q2 where the QCD corrections are significant, we have to take into account the corrections due to the mass of the target nucleon, which we denote by M. In this Q2 region, we cannot neglect the order M2/Q2 term, which consists of higher-twist effects as well as target mass effects. Here we shall not discuss the higher-twist effect which was mentioned by Dr. Kodaira and will be discussed by Dr. Mueller this afternoon. Here we confine ourselves to the target mass effects. Here we observe that 1) The target mass effects are calculable without any ambiguity ; 2) The infinite power series in M2/Q2 can actually be summed up into a closed analytic form. In the framework of OPE, the target mass effects of structure functions can be evaluatedbytakingaccountoftracetermsofcompositeoperatorstohaveadefinitespin projection. Thisamountstoreplacetheordinarymoments ofthestructurefunctionsby the Nachtmann moments [15]. The Nachtmann moments for spin structure functions 1 were obtained in refs.[16, 17]. In ref. [16] the Nachtmann moments were given as an infinite power series, while in ref. [17] they were obtained as a closed analytic form. 2. OPE and Target Mass Effects The anti-symmetric part of the virtual Compton amplitude can be written in the OPE [18]: 2 n T (p,q,s)[A] iε qλ q q En(Q2,g) p,s Rσµ1···µn−1 p,s µν ≃ − µνλσ Q2 µ1 ··· µn−1 1 h | 1 | i n=X1,3,···(cid:16) (cid:17) n 1 2 n i(ε q qρ ε q qρ q2ε ) − q q En(Q2,g) − µρλσ ν − νρλσ µ − µνλσ n Q2 µ1 ··· µn−2 2 n=X3,5,··· (cid:16) (cid:17) p,s Rλσµ1···µn−2 p,s , (2) ×h | 2 | i where the nucleon matrix element of the twist-2 operators Rn is given by 1 p,s Rσµ1···µn−1 p,s = a [ sσpµ1 pµn−1 trace terms] h | 1 | i − n { ··· }− a sσpµ1 pµn−1 , (3) n n ≡ − { ··· } which is totally symmetric in Lorentz indices σ,µ1, ,µn−1 and traceless. ( denotes ··· { } symmetrization.) While for the twist-3 operator Rn, the matrix element is given by 2 1 p,s Rλσµ1···µn−2 p,s = d [ (sλpσ sσpλ)pµ1 pµn−2 trace terms] h | 2 | i − n 2 − ··· − d sλpσpµ1 pµn−2 , (4) ≡ − n{ ··· }Mn which is symmetric in σµ1 µn−2 and anti-symmetric in λσ and also traceless in its ··· Lorentz indices. Taking account of trace terms we can project out the contribution from a defi- nite spin as follows. By taking a contraction of the tensor appearing in eq.(3) with q q we get µ1 ··· µn−1 q q [ sσpµ1 pµn−1 (trace terms)] µ1 ··· µn−1 { ··· }− 1 q s = [sσan−1C(2) (η)+qσ · an−1 4C(3) (η)+pσq san−2 2C(3) (η)], (5) n2 n−1 Q2 × n−3 · × n−2 where C(m)(η) is a Gegenbauer polynomial with n η = iν/Q, ν = p q/M, a = 1iMQ. (6) · −2 2 Using the orthogonality property of Gegenbauer polynomials, one can project out the contribution from operators with a definite spin. The closed analytic forms for the Nachtmann moments are given as [17]: Mn(Q2) 1a En(Q2,g) 1 ≡ 2 n 1 1 dx x n2 Mx Mξ 4n M2x2 = ξn+1 g (x,Q2) g (x,Q2) , x2 {ξ − (n+2)2 Q Q } 1 − n+2 Q2 2 Z0 h i (n = 1,3, ) (7) ··· Mn(Q2) 1d En(Q2,g) 2 ≡ 2 n 2 1 dx x n x2 n M2x2 = ξn+1 g (x,Q2)+ g (x,Q2) , x2 ξ 1 {n 1ξ2 − n+1 Q2 } 2 Z0 h − i (n = 3,5, ) (8) ··· where ξ is a variable given by [19]: 2x ξ . (9) ≡ 1+ 1+4M2x2/Q2 q Taking the difference between the first moment for the proton target and that for the neutron in eq.(7), we can arrive at the QCD Bjorken sum rule with target mass correction: 1 1 ξ2 4M2x2 dx 5+4 1+ gp(x,Q2) gn(x,Q2) 9 Z0 x2h s Q2 ih 1 − 1 i 4 1 ξ2 M2x2 1G α (Q2) dx gp(x,Q2) gn(x,Q2) = A 1 s +O(α2) . (10) −3 x2 Q2 2 − 2 6G − π s Z0 h i V h i Notethatinthepresenceoftargetmasscorrection, theotherspinstructurefunction gp,n(x,Q2) also comes into play in the Bjorken sum rule. Here we emphasize that 2 the target mass correction treated through the above procedure is not mere a power correction but given as a closed analytic form. It should also be noted that target mass corrections considered as the expansion in powers of M2/Q2 is not valid when M2/Q2 is of order unity [20, 21]. Our result can be compared with the target mass correction as a power correction discussed in the literatures. Expanding our Nachtmann moment in powers of M2/Q2 we get 1 1G α dxgp−n(x,Q2) = A(1 s + ) 1 6G − π ··· Z0 V 10M2 1 12M2 1 + dxx2gp−n(x,Q2)+ dxx2gp−n(x,Q2),(11) 9 Q2 1 9 Q2 2 Z0 Z0 3 which coincides with the result given by Balitsky-Braun-Kolesnichenko in ref.[21] up to the contribution from the twist-4 operator to the order of 1/Q2. Now, the difference between the left-hand side of (10) and that of (1) leads to the target mass correction ∆Γ: 1 5ξ2 4ξ2 4M2x2 ∆Γ = dx + 1+ 1 gp(x,Q2) gn(x,Q2) Z0 {9x2 9x2s Q2 − }×h 1 − 1 i 4 1 ξ2 M2x2 dx gp(x,Q2) gn(x,Q2) . (12) −3 x2 Q2 2 − 2 Z0 h i 3. Estimation of the Target Mass Effects Let us now study the size of the target mass correction ∆Γ to the Bjorken sum rule. First we note that the spin structure functions g and g are written in terms 1 2 of virtual photon asymmetry parameters A and A , which are measured at the ex- 1 2 periments, together with the unpolarized structure function, F (x,Q2), and the ratio 2 of the longitudinal to transverse virtual photon cross sections, R = σ /σ [1, 2]. We L T shall estimate the upper bound for the target mass correction of ∆Γ, which we denote by ∆Γ (i.e. ∆Γ ∆Γ ) in a variety of methods. u.b. u.b. | | ≤ For the first analysis (Analysis I), we apply the positivity bound for the asymmetry parameters [23]: A 1, A √R. (13) 1 2 | | ≤ | | ≤ We use the parametrization for R taken from the global fit of the SLAC data [24] and the NMC parametrization for F (x,Q2) [25]. In Fig.1 we have plotted the upper 2 bound, ∆Γ , as a function of Q2 for Analysis I by a solid line. Here the error of the u.b. upper bounds of ∆Γ due to the parametrizations R and F is typically around 10 %. 2 In our second analysis (Analysis II), we employ the experimetal data on spin asym- metry A and positivity bound for A to improve the upper bound. We take the data 1 2 on Ap from SMC data [11] together with EMC data [2] and those for Ad from SMC 1 1 group [9] to extract An, for which we can also use the E142 data [10]. For this case, 1 the upper bound is shown in Fig.1 by the short-dashed line, which is located slightly lower than ∆Γ for Analysis I. When we decompose the ∆Γ into two parts, ∆Γ u.b. u.b. 1 and ∆Γ , which are the contributions from A and A , respectively, it turns out that 2 1 2 ∆Γ is much larger than ∆Γ . The value of ∆Γ turns out be less than 10 % of ∆Γ . 2 1 1 2 4 The third analysis (Analysis III) uses the recently measured Ap by the SMC group 2 [27] in addition to the same data for Ap,n together with the positivity bound for An 1 2 as in Analysis II. We have also plotted the upper bound for Analysis III in Fig.1 by the long-dashed line. Here we took the data on Ap obtained by SMC group at the first 2 measurement of transverse asymmetries [27], where the number of data points are still four and the relative error bars are not so small. The Ap measured is much smaller 2 than the positivity bound. If the A for the neutron is also small as mentioned in 2 ref.[10], the ∆Γ becomes very small. u.b. Finally we briefly comment on uncertainty due to target mass effects indetermining theQCDcouplingconstant fromBjorkensumrulewhichhasrecently beendiscussed by Ellis and Karliner [28]. From the QCD corrections up to (α4) [6, 7, 8] they obtained O s the value α (Q2 = 2.5GeV2) = 0.375+0.062[28], by using the known g = G /G ratio s −0.081 A A V and taking the value Γ(Q2 = 2.5GeV2) = 0.161 0.007 0.015, in their analysis of E142 ± ± and E143 data [28]. The Q2 = 2.5GeV2 is the averaged value of the mean Q2 of the E142 data (< Q2 > 2GeV2) and the E143 data (< Q2 > 3GeV2). Here we shall ≃ ≃ not take into account the higher-twist effects which are considered to be rather small as claimed in refs. [28]. The uncertainty in Γ due to target mass effects gives rise to that for the QCD coupling constant α (Q2 = 2.5GeV2). Namely, ∆Γ (Q2 = 2.5GeV2)=0.029, 0.027 s u.b. and 0.011 for Analyses I, II and III, respectively, we get the ambiguities for α s 0.213 α (Q2 = 2.5GeV2) 0.474 (Analysis I), s ≤ ≤ 0.228 α (Q2 = 2.5GeV2) 0.469 (Analysis II), s ≤ ≤ 0.315 α (Q2 = 2.5GeV2) 0.424 (Analysis III). (14) s ≤ ≤ 4. Conclusion In this talk we have examined the possible target mass corrections to the Bjorken sum rule using positivity bound and experimental data on asymmetry paprmeters. We have found that at relatively small Q2 where the QCD effect is significant, the target mass effects are also non-negligible. We found that to test the target mass correction precisely, we need accurate data for A (x,Q2). In determining the QCD coupling 2 5 constant α from the Bjorken sum rule, there appears uncertainty due to target mass s effects. This uncertainty can also be removed by the experimental data on A (x,Q2). 2 Although in this paper we have confined ourselves to the target mass effects in the Bjorken sum rule, the similar analysis can be carried out for the Ellis-Jaffe sum rule. For the proton target, ∆Γp (Q2 = 2.5GeV2)=0.017, 0.016 and 0.0046 with typical u.b. errors of 10% for Analyses I, II and III, respectively. Those for the neutron turn out to be 0.012 and 0.011 for Analyses I and II. Finally, we note that there exists the Burkhardt-Cottingham sum rule for g (x,Q2) 2 1 dxg (x,Q2) = 0, (15) 2 Z0 which is not only protected from QCD radiative corrections [4, 29, 30] but also free from target mass effects [17]. We hope that future experiments at CERN, SLAC and DESY will provide us with data on A possessing higher statistics as well as the data on A with high accuracy 1 2 which will enable us to study g structure functions and also target mass effects more 2 in detail. References [1] M. J. Alguard et al., Phys. Rev. Lett. 37 (1976) 1261; 41 (1978) 70. G. Baum et al.,Phys. Rev. Lett. 51 (1983) 1135; 45 (1980) 2000. [2] J. Ashman et al.,Phys. Lett. B206 (1988) 364; Nucl. Phys. B328 (1989) 1. [3] J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu, Phys. Rev. D20 (1979) 627. [4] J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu, Nucl. Phys. B159 (1979) 99. [5] S. G. Gorishny and S. A. Larin, Phys. Lett. B172 (1986) 109. [6] S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B259 (1991) 345. [7] S. A. Larin, F. V. Tkachev and J. A. M. Vermaseren, Phys. Rev. Lett. 66 (1991) 862. [8] A. L. Kataev and V. Starshenko, preprint CERN-TH-7198-94. 6 [9] B. Adeva et al., Phys. Lett. B302 (1993) 533. [10] P. L. Anthony et al., Phys. Rev. Lett. 71 (1993) 959. [11] D. Adams et al., Phys. Lett. B329 (1994) 399. [12] K. Abe et al., Preprint, SLAC-PUB-6508 (August 1994). [13] J. D. Bjorken, Phys. Rev. 148 (1966) 1467;D1 (1970) 1376. [14] J. Ellis and R. L. Jaffe, Phys. Rev. D9 (1974) 1444;D10 (1974) 1669. [15] O. Nachtmann, Nucl. Phys. B63 (1973) 237; B78 (1974) 455. [16] S. Wandzura, Nucl. Phys. B122 (1977) 412. [17] S. Matsuda and T. Uematsu, Nucl. Phys. B168 (1980) 181. [18] A. J. G. Hey and J. E. Mandula, Phys. Rev. D5 (1972) 2610. [19] H. Georgi and H. D. Politzer, Phys. Rev. D14 (1976) 1829. A. De Ru´jula, H. Georgi and H. D. Politzer, Ann. of Phys. 103 (1977) 315. [20] E.V.Shuryak andA.I.Vainshtein, Phys.Lett.B105(1981)65; Nucl.Phys.B201 (1982) 141. [21] I. I. Balitsky, V. M. Braun and A. V. Kolesnichenko, Phys. Lett. B242 (1990) 245. [22] H. Kawamura and T. Uematsu, Phys. Lett. B343 (1995) 346 . [23] M. G. Doncel and E. de Rafael, Nuovo Cimento 4A (1971) 363. [24] L. W. Whitlow, S. Rock, A. Bodek, S. Dasu and E. M. Riordan, Phys. Lett. B250 (1990) 193. L. W. Whitlow, Ph.D thesis, Stanford University, SLAC-report-357 (1990). [25] P. Amaudruz et al., Phys. Lett. B295 (1992) 159. [26] J. Ellis and M. Karliner, Phys. Lett. B313 (1993) 131. [27] D. Adams et al., Phys. Lett. B336 (1994) 125. [28] J. Ellis and M. Karliner, Phys. Lett. B341 (1995) 397. [29] G. Altarelli, B. Lampe, P. Nason and G. Ridolfi, Phys. Lett. B334 (1994) 187. [30] J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu, preprint KUCP-70; HUPD- 9410;YNU-HEPTh-94-105 (1994), Phys. Lett. B345 to appear. 7 0.1 Analysis I ∆Γ u. .b. AAnnaallyyssiiss III Analysis III 0.05 0 5 10 2 2 Q (GeV ) Fig.1 Fig.1 Theupper boundforthetargetmass correction∆Γ, ∆Γ , asafunctionofQ2. u.b. The solid, short-dashed and long-dashed lines show the upper bounds for the analyses I, II and III, respectively. 8