A brief review of measurements and expectations concerning the spin structure
function g_1 of the nucleon at low values of the scaling variable x is given.Comment: 4 pages, 3 figures. Invited talk presented at the ``International
  Workshop on the Spin Structure of the Proton and Polarized Collider
  Physics'', ECT*, Trento, Italy, July 23-28, 200

Adeva

B. Badełek

Badełek

Bartels

Bass

Blümlein

Burkert

Close

De Roeck

Heimann

Ioffe

Kuti

Kwieciński

Stratmann

Windmolders

English

arXiv

A brief review of measurements and expectations concerning the spin structure function g_1 of the nucleon at low values of the scaling variable x is given

Badelek, B

CERN Document Server

1Spin structure function g1 at low x: status and plansB. Bade lekaMCSD]Physics Department, Uppsala University, Box 530, S - 751 21 Uppsala, Sweden, andInstitute of Experimental Physics, Warsaw University, Hoza 69, PL - 00 681 Warsaw, Polanda[A brief review of measurements and expectations concerning the spin structure function g1 of the nucleon atlow values of the scaling variable x is given.1. INTRODUCTIONThe spin-dependent structure function g1 isspecially interesting at low x, i.e. at high par-ton densities, where new dynamical mechanismsmay be revealed. Its knowledge there is neededfor evaluating the spin sum rules necessary tounderstand the origin of nucleon spin. The be-haviour of g1 at x < 0.001 and in the scalingregion, Q2 > 1 GeV2, is unknown. Spin indepen-dent structure function F2 rises there, in agree-ment with QCD and contrary to the Regge modelpredictions. A great opportunity to explore thespin dependent phenomena in the region of lowx would be through polarising the proton beamat HERA. In the xed target experiments, lowvalues of x are correlated with low values of Q2.Theoretical analysis of these results thus requiresa suitable extrapolation of g1 to the low Q2 regionwhere the dynamical mechanisms, like the Vec-tor Meson Dominance (VMD), can be important.For large Q2 the VMD contribution to g1 vanishesas 1=Q4 and can usually be neglected. Moreover,the partonic contribution to g1 which controls thestructure functions in the deep inelastic domainand which scales there modulo logarithmic cor-rections, has to be suitably extended to the lowQ2 region. In the Q2=0 limit g1 should be a -nite function of W 2, free from any kinematicalsingularities or zeros.∗Supported in part by the KBN grants 2 P03B 05119 andSPUB nr 621/E-78/SPUB-M/CERN/P-03/DZ298/20002. PRESENT INFORMATION ON THEg1 OF THE NUCLEON2.1. Results of measurementsAs a result of a large experimental eort overthe years, proton and deuteron g1 was measuredfor 0.000 06 < x < 0.8, cf. Fig. 1, [1] (and refe-rences therein). Direct measurements on the neu-tron are limited to x > 0.02 (see e.g. [2]). Recentmeasurements by HERMES [2] are not shown inFig. 1; those results cover the region x > 0.005and do not change the overall picture. The re-gion of lowest x values was explored only by theSMC due to the high energy of the muon beam,the demand of a nal state hadron in the analy-sis [3] and the implementation of a dedicated lowx trigger with a calorimeter signal [4], the twolatter requirements eciently removing the dom-inant e scattering background. No signicantspin eects were observed there.As seen in Fig.1 the scaling violation ing1(x; Q2) is weak: the average Q2 is about 10GeV2 for the SMC and almost an order of mag-nitude less for the SLAC and HERMES experi-ments. For the SMC data [4], hxi = 0.0001 corres-ponds to hQ2i = 0.02 GeV2; Q2 becomes largerthan 1 GeV2 for x > 0.003. At low x resultson g1 have large errors but it seems that gp1 ispositive and gd1 and gn1 are negative there. Sta-tistical errors dominate in that kinematic inter-val. It should be noted that a direct result ofall measurements is the longitudinal cross sectionasymmetry, Ak which permits to extract the vir-tual photon { proton asymmetry, A1 and nally,using F2 and R, to get g1.2x gp. 1xEMCSMCE143HERMES-0.0200.020.040.060.0810 -4 10 -3 10 -2 10 -1 1Q2 < 1 GeV2-0.00200.0020.0040.00610-410-3Figure 1. Summary of the xgp1 as a function ofx at the measured Q2. The insets show the SMCdata for Q2 < 1 GeV2. Errors are statistical. Thecurves, calculated at (x; Q2) values of the SMCdata, result of the model described in Section 2.5.Figure taken from [1].2.2. Regge pole model expectationsThe low x behaviour of g1 for xed Q2 reflectsthe high energy behaviour of the virtual Comp-ton scattering cross section with centre-of-massenergy squared, W 2. This is the Regge limit ofthe (deep) inelastic scattering where the Reggepole exchange model should be applicable. Ac-cording to this model, g1(x; Q2)  x− for x ! 0and xed Q2, where  is the intercept of theRegge trajectory, here corresponding to axial vec-tor mesons. It is expected that   0 for bothI = 0 and I = 1 trajectories, [5]. This behaviourof g1 should go smoothly to the W 2 dependencefor Q2 ! 0. It is also expected that the flavoursinglet part of the g1 should have a similar low xbehaviour as the nonsinglet one. Regge modelpredictions for g1 become unstable against theQCD evolution which generates more singular be-haviour of g1 than that given by x−.Other predictions based on the Regge modelare: g1  lnx, [6] and g1  2 ln(1/x){1, [7]. Aperverse behaviour, g1 1/(xln2x), recalled in [6],is not valid for g1, [8].The Regge model prediction, g1  x0, has inthe past often been used to obtain the x ! 0 ex-trapolation of g1 (see Fig. 2) required to extractits rst moments (cf. [9] and references therein).-30-25-20-15-10-50510152010-510-410-310-210-1SMCRegge ExtrapolationQCD-ExtrapolationQCD-fitHERA1x(ln(x))2 Extrapolationg1pX-0.5-0.2500.250.50.7511.251.51.75210-210-1Figure 2. Three scenarios of the possible be-haviour of gp1 at low x [17].The low x data of the SMC [4] include thekinematic region where W 2 = (p + q)2 is high,W 2 > 100 GeV2, and W 2  Q2. Thus the Reggemodel should be applicable there. However forthose data W 2 changes very little: from about100 GeV2 at x = 0.1 to about 220 GeV2 at x =0.0001, contrary to a strong change of Q2: fromabout 20 GeV2 to about 0.01 GeV2 respectively.Thus those data cannot test the Regge behaviourof g1 through the x dependence of the latter.2.3. QCD fit to the world data on gN1Next-to-leading (NLO) order QCD analyses ofthe Q2 dependence of g1 had been performed onthe g1 data [9{11] and have indicated a large con-tribution of gluon polarisation to the proton spinalbeit determined with very large errors.Extrapolations of QCD t results to the un-measured low x region give all three structurefunctions, gp1 , gd1 and gn1 negative (gp1 becomessuch below the lowest x data point used in thet), due to a large negative singlet contribution.3It should be stressed that the g1 results for xvalues below these of the data do not influencethe results of the t. Thus there is no reasonto expect that the partons at very low x be-have as those in the measured (larger x) region.Nevertheless the low x QCD extrapolations arepresently being used to get the x ! 0 extrap-olation of g1 [9], necessary to evaluate its rstmoments. They strongly disagree with the Reggeasymptotic form, cf. Fig. 2.2.4. ln2(1=x) corrections to g1The small x behaviour of g1(x; Q2) is con-trolled by the double logarithmic ln2(1=x) con-tributions, i.e. by those terms of the pertur-bative expansion which correspond to the pow-ers of ln2(1=x) at each order of the expansion[12]. The ln2(1=x) eects go beyond the standard(i.e. lnQ2) LO (and NLO) QCD evolution. It isconvenient to resum these terms using the un-integrated (spin-dependent) parton distributions.The resulting integrated parton distributions con-tain nonperturbative parts, p0j(x), correspond-ing to low transverse momenta of partons, to beparametrised semiphenomenologically.A complete formalism incorporating the LOAltarelli{Parisi evolution and the ln2(1=x) resum-mation at low x has been built for gp1 [13,14]. Ithas been shown that the nonsinglet part of g1 isdominated by ladder diagrams while a contribu-tion of the nonladder bremsstrahlung diagrams isimportant for the singlet part. Results are pre-sented in Fig. 3; small x eects suppress the verystrong x dependence of g1 resulting from pureAltarelli{Parisi evolution.Even if the ln2(1=x) eects are not important inthe x range of the xed target experiments theysignicantly aect g1 in the low x region whichmay be probed at the polarised HERA.2.5. Nonperturbative effects in g1At the low x, low Q2 region, g1 was representedby [1]:g1(x; Q2) = gV MD1 (x; Q2) + gpart1 (x; Q2) (1)The partonic contribution, gpart1 is at low x con-trolled by the ln2(1=x) terms resummed using un-integrated parton distributions, cf. section 2.4.This formalism is very suitable for extrapolating-300-250-200-150-100-5001e-05 0.0001 0.001 0.01 0.1 1g 1xFigure 3. gp1(x; Q2) at Q2=10 GeV2 after includ-ing the ln2(1=x) corrections. A thick solid linecorresponds to full calculations, a dashed one {only the ladder ln2(1=x) resummation with LOAltarelli-Parisi evolution, a dotted one - pure LOAltarelli-Parisi evolution and a thin solid line -the nonperturbative input to g1 [14].g1 to the region of low Q2 at xed W 2. Results forgpart1 (x; Q2) are shown as a curve in Fig.1. TheVMD contribution, gV MD1 (x; Q2), was taken as:gV MD1 (x; Q2) =pq4∑v=;!;m4vv(W2)γ2v(Q2 + m2v)2(2)where γ2v are determined from the leptonic widthsof the vector mesons and mv is the mass of thevector meson v. Unknown vector meson { nu-cleon cross sections v = (1=2 − 3=2)=2 (sub-scripts refer to the projections of the total spin onthe vector meson momentum) were taken propor-tional (with a proportionality coecient C) to theappropriate combinations of the nonperturbativecontributions p0j(x) to the polarised quark andantiquark distributions. The p0j(x) behave asx0 for x !0. As a result the cross sections vbehave as 1=W 2 at large W 2 that corresponds tozero intercepts of the appropriate Regge trajecto-ries. Exact x dependence of p0j (x) was included.The statistical accuracy of the SMC data is toopoor to constraint the value of the coecient C.The SLAC E143 data [15] preferred a small nega-tive value of C which is consistent with the results4of the phenomenological analysis of the sum rules[16]. Similar analysis of the neutron and deuteronspin structure functions was inconclusive.3. g1 AT FUTURE COLLIDERS3.1. gp1Presently the form of the x ! 0 extrapolationof the g1 provides the largest error on its rst mo-ment: the contribution from the unmeasured lowx region amounts from about 2 to 10% of the in-tegral in the measured range, depending whetherthe Regge model or the QCD is used for the ex-trapolation (cf. [9] and references therein). Anempirical insight into the validity of dierent lowx mechanisms would be possible through polaris-ing the proton beam at HERA. This would extendthe measured region of x down to approximately0.000 06 at Q2 > 1 GeV2.Polarised ep (eA) colliders (HERA, EIC) willalso have a potential to explore the transitionbetween the photoproduction and scaling region,0< Q2 < 1 GeV2 [18,19]. Photoproduction mea-surements would constraint the spin dependentRegge model and Regge contribution to the Drell-Hearn-Gerasimov sum rule. Measurements in thetransition region would impose new limits on g1models containing both perturbative and nonper-turbative components.3.2. gγ1Spin dependent structure function of the pho-ton, gγ1 , can in principle become accessible in thefuture linear e+e− or eγ linear colliders, [20]. Thelatter mode would be particularly suitable forprobing the photon structure at low values of x.The gγ1 (x; Q2) has been analysed [21] within theformalism being an extension of that developedfor the gp1 i, [14]. It was found that e.g. dierentscenarios for the nonperturbative spin dependentgluon content of the photon give signicantly dif-ferent values of gγ1 at x 10−5.4. OUTLOOKThe spin dependent structure function g1 atlow x is both fascinating in itself and necessaryfor understanding the origin of the proton spin.Future colliders, HERA and EIC (and eγ one forthe gγ1 ) would give us a great opportunity to ex-plore it.REFERENCES1. B. Bade lek, J. Kiryluk and J. Kwiecinski,Phys. Rev. D61 (1999) 014009.2. HERMES, A. Fantoni, these proceedings.3. SMC, B. Adeva et al., Phys. Rev. D58 (1998)112001.4. SMC, B. Adeva et al., Phys. Rev. D60 (1999)072004 and erratum: ibid D62 (2000) 079902.5. R.L. Heimann, Nucl. Phys. B64 (1973) 429.6. F.E. Close and R.G. Roberts, Phys. Lett.B336 (1994) 257.7. S.D. Bass and P.V. Landsho, Phys. Lett.B336 (1994) 537.8. J. Kuti in The Spin Structure of the Nucleon,World Scientic, Singapore, 1996; M.G.Ryskin, private communication (Durham,1998).9. SMC, B. Adeva et al., Phys. Rev. D58 (1998)112002.10. For a review of older QCD analyses of g1 seee.g. R. Windmolders, Nucl. Phys. B (Proc.Suppl.) 79 (1999) 51.11. J. Blu¨mlein and H. Bo¨ttcher, DESY 01-107(2001) and hep-ph/0107317.12. J. Bartels, B.I. Ermolaev and M.G. Ryskin,Z. Phys. C70 (1996) 273; C72 (1996) 627.13. B. Bade lek and J. Kwiecinski, Phys. Lett.,B418 (1998) 229.14. J. Kwiecinski and B. Ziaja, Phys. Rev. D60(1999) 054004.15. E143, K. Abe et al., Phys. Rev. D58 (1998)112003.16. V. Burkert and B.L. Ioe, Phys. Lett. B296(1992) 223; B.L. Ioe, Yad. Fiz. 60 (1997)1866 [Phys. At. Nucl. 60 (1997) 1707].17. A. De Roeck et al., Eur. Phys. J. C6 (1999)121.18. S.D. Bass and A. De Roeck, Eur. Phys. J. C18(2001) 531.19. A. Deshpande, these proceedings.20. M. Stratmann, Nucl. Phys. B (Proc. Suppl.)82 (2000) 400.21. J. Kwiecinski and B. Ziaja, Phys. Rev. D63(2001) 054022.

Spin structure function $g_{1}$ at low x: status and plans

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Spin structure function g1 at low x: status and plans

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Spin structure function g_1 at low x: status and plans

Spin structure function g_1 at low x: status and plans

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