hep-ph/ Aug 1995

Transcription

1 DESY 95{164 August 1995 On the Measurability of the Structure Function g 1 (; Q in ep collisions at HERA hep-ph/ Aug 1995 Johannes Blumlein DESY { Zeuthen, Platanenallee 6, D{15735 Zeuthen, Germany Abstract The possibility is investigated to measure the polarized structure function g 1 (; Q in the collider mode of HERA operating with a polarized lepton and proton beam. The dependence of g 1 can be measured at a statistical precision of 15% to 7% in the range :5 <<:5 correlated to virtualities 15 < hq i < 35 GeV at beam polarizations p e =:8 and L int = 6pb 1.

2 1 Introduction The possibility to study deep inelastic electron{proton scattering with both polarized electron and proton beams at the HERA collider would open up a new era in the investigation of the nucleon spin structure [1]. Far smaller values of Bjorken than ever probed in ed target eperiments would become accessible and the behaviour of polarized structure functions in the range down to :1 could be investigated. This is of particular importance since the present analysis of sum{rules relies on etrapolations in this range only. Moreover little is known on the scaling violations of polarized structure functions from eperiment in the whole range so far. Due to the kinematical domain of HERA a far wider Q range can be accessed. Furthermore both charged and neutral current reactions can be studied from which constraints on the avour structure of polarized structure functions may be obtained. In the present paper we concentrate on the case of neutral current deep inelastic scattering in the Q range dominated by photon echange, i.e. Q < 1 GeV []. Here the two polarized structure functions g 1 (; Q and g (; Q and the two unpolarized structure functions F 1 (; Q and F (; Q determine the polarization asymmetries. We analyse the possibilities to measure the polarization asymmetries A k and A? in the HERA collider mode. An estimate is given of the statistical accuracy to which the structure function g 1 (; Q can be measured. Polarization Asymmetries The structure functions g 1 (; Q and g (; Q can be determined from the measurement of the polarization asymmetries A k and A? ([3, 4]): with and d! ) ddy d 3! * ddyd d! ( ddy = e p 8 A k = d!) d (! d!) + d (! (1)!! A? = d * d +! () d! * + d + Q (" d 3! + ddyd = e p 4! ( d! ddy + d ddy y M y S # g 1 (; Q " 4M # 1= " Q cos (1 y) ys 1 ) 4M S g (; Q # M 1= y (1 y)s n yg 1 (; Q +g (; Q o (4)! d3 *! + ddy + d3 ddy ( = 4 S y F Q 4 1 (; Q + 1 y ym S! (3) ) F (; Q : (5) Here! denotes the orientation of the electron and proton ()) polarization. and y are the Bjorken variables, S =4E e E p,q =ys, M is the proton mass, and e and p are the electron

3 and proton beam polarizations. denotes the angel between the planes ( k; k ) and ( k; s), where k; k, and s are the 3{momenta of the incoming and outgoing electron, and the proton spin, respectively. The contribution due to the structure function g (; Q ina k is suppressed by a factor of 4M =(S( y)) relative tog 1 (; Q. In the HERA collider mode this term is very small and can be disregarded as well as other terms of O(M =S) leading to A k = e p Y g 1 (; Q Y + F 1 (; Q (1 y)f L (; Q ; (6) where F L (; Q =F (; Q F 1 (; Q, and Y = 1 (1 y). The asymmetry A? contains a factor f A? =4M= p S leading to a strong suppression in the case of the collider mode at HERA in comparison with the kinematics in ed target eperiments at SLAC or HERMES. A? = e p cos f A? q yg 1 (; Q +g (; Q y(1 y) Y + F 1 (; Q (1 y)f L (; Q Since f A? :13 to : for E p = 8 GeV to 3 GeV a measurement ofa? at the same precision as for A k would require luminosities much larger than the HERA design value. Due to this a measurement ofg (; Q in the HERA collider mode is not possible based on inclusive polarization asymmetries. (7) 3 Parton Parametrizations The dominant contribution to the structure function g 1 (; Q isoftwist. It may be described in the parton model by g 1 (; Q = 1 ( e u X i h ui (; Q +u i (; Q i + e d X h i) di (; Q +d i (; Q (8) with e u =4=9;e d =1=9and f(; Q = f " (; Q f # (; Q, where f "# are the parton densities at a given nucleon polarization. Here we assumed contributions from the light avours only. So far most of the constraints on the dierent avour contributions to (8) are due to measurements of g1(; p Q and g1(; d Q atlowvalues of Q in the range of :1 <. The accuracy of the present data still leaves considerable freedom on the parametrization of g1(; p Q and g n 1 (; Q both in the small range and at larger values of Q which are both accessible in possible later HERA eperiments. In gure 1 and we compare four recent leading order parametrizations [5{8] 1 of g1(; p Q and g n 1 (; Q in the range 1 4 <<1 and 1 <Q <1 4 GeV. Although the shape of g p;n 1 in the range :1 <is quite similar for all parametrizations both the etrapolation to the small range and to higher Q turn out to be rather dierent. This is due to the particular 1 If the LO Q dependence was not provided by the authors of the respective parametrization it was derived from their Ansatz at Q by LO QCD evolution using the CTEQ program [9]. I am indebted to Glenn Ladinsky for providing me with according parametrizations prior to publication [1]. In the case of parametrization [8] the set with G = 1 was used choosing u s =u=d s =d=4s=4s. The numerical illustrations below correspond to the 'standard scenario' in the case of parametrization [5] and the set 'gluon A' in ref. [6]. Note that in the case of ref. [6] a contribution / G was added to eq. (8). In all other cases eq. (8) was used. i 3

4 parametrization of the individual avour contributions f i, and partly due to the starting point of the evolution choosen. Whereas g1(; p Q takes negativevalues for <1 3 and shows a falling behaviour in the case of parametrization [5] for 1 <Q <1 4 GeV, it remains positive and rising for parametrization [7] in the same Q range. For parametrization [6] the falling behaviour is only observed for Q > 1 3 GeV. Also parametrization [8] yields negativevalues in the range O(1 4 ::: 1 3 ) at larger Q which are, however, larger than those obtained from parametrization [5]. Moreover at still smaller values g p 1 takes positive values again in the case of parametrization [8], while a falling behavior is obtained by [5, 6]. A similar uncertainty etrapolating from the kinematical range of the present measurements to the domain accessible at HERA is found comparing dierent parametrizations of g n 1 (cf. gure.). However, the range in which g n 1 takes positive values at large is about similar. The parametrization [8] predicts a rising behaviour of g n 1 in the small range again. A measurement of the shape of g1() p at HERA should allow to distinguish between these parametrizations. Also the study of polarized ed scattering would be interesting to constrain the behaviour of g n 1 at smaller values of and larger values of Q. 4 Kinematical Range Constraints on the kinematical range for neutral current deep inelastic scattering at HERA have been discussed in [11]. The dierent boundaries are implied by angle and energy cuts, bounds due to resolution eects [1], and large QED radiative corrections [13]. In gure 3 the accessible ; Q range is shown for the HERA beam energies E e =7:6GeV and E p = 8 GeV demanding E Jet > 5 GeV, Jet > 5 o, e < 175 o,:1 <y<:9, and <:7. The values of hq i are correlated with. They are indicated by stars in gure 3 for the case of neutral current deep inelastic scattering. In the following we will investigate the sensitivity to measure the structure function g 1 (; Q in this range. 5 Accuracy of a measurement ofg p 1 (; Q In gure 4 the statistical accuracy of a measurement ofa k is illustrated as a function of assuming L int = 3pb 1 /beam polarization and e = p =:8. We used parametrization [6] as a reference value for the polarized parton densities, and [9] for the unpolarized densities 3. The statistical error of g 1 (; Q measured from A k is given by g 1 (; Q = 1 Y + e p Y " F 1 (; Q (1 y) Y + F L (; Q h L int d =ddq Q i 1= q 1 A k (; Q (9) where d =ddq denotes the unpolarized dierential scattering cross section, Q is the bin size, and L int the integrated luminosity per beam polarization. g1(; p Q does only weakly depend on the value of g 1 (; Q itself as long as A k 1. For Q > 1 GeV the neutral current deep inelastic scattering cross section contains also conributions due to Z interference and Z ecange terms which are related to new structure functions. As in the case of the measurement off (; Q (cf. []) these terms may be delt with as corrections in the measurement ofg p 1 (; Q. 3 In the and Q range considered the dierent parametrizations for unpolarized parton densities agree to a wide etent. The contribution due to F L was disregarded in the numerical calculation of A k. # 4

5 It turns out that at the value of L int considered the product of the beam polarizations should take values of > e p :5 (1) to obtain a sucient resolution for g1. p In gure 5 the accuracy of an shape measurement ofg1(; p Q in the kinematical range of HERA (cf. gure 3) is shown. The values of g 1 () representaverages over Q. With rising values of hq i rises. Assuming the parametrization ref. [6] as a reference value relative errors for g1() p between 15% and 7% in the range from :5 to51 4 at 35 >Q >15 GeV are obtained under the above conditions. Predictions for g1(; p hq i)by other parametrizations are shown for comparison. A measurement ofg1in p this kinematical range will allow to further constrain eisting parametrizations of polarized parton densities. Particularly it will be interesting to see whether g1(; p Q takes negative values in the small range. In gure 6 the statistical accuracy of g1(; p hq i) is illustrated in the range >:1. Dierences between the parametrizations [5{8] are still observed. The size of the scaling violations in LO in this range is shown comparing the values of g1() p for Q = hq i and Q = 4 GeV, the range of the data from ed target eperiments. For <:1 the scaling violations of g1(; p Q are of the size of g1() p for a measurement under the conditions mentioned above. Towards larger values the scaling violations shrink due to the point at:15. Thus, to see scaling violations of g1(; p Q in a clear way requires a higher luminosity. 6 Conclusions The measurement of the structure function g1(; p Q in the HERA collider mode with both longitudinally polarized electron and proton beams would allow to probe the behaviour of this structure function at smaller values of and larger values of Q compared to measurements possible in ed target eperiments. g1() p can be measured at a statistical precision of 7% to 15% in the range of <<:5correlated with values of 15 < hq i < 35 GeV for an integrated luminosity ofl int = 3pb 1 /beam polarization and polarization values of e = p =:8. Further constraints on the behaviour of g p 1 can be obtained by this measurement. A detailed investigation of scaling violations of g 1 (; Q requires a larger integrated luminosity. References [1] For reviews see : E. Reya, in: Proceedings of the Workshop QCD{Years Later, Aachen, 199, Vol. 1, p. 7, eds. P.M. Zerwas and H.A. Kastrup, (World Scientic, Singapore, 199); G. Altarelli and G. Ridol, in: Proceedings of the Workshop QCD '94, Montpellier, 1994, ed. S. Narison, Nucl. Phys. Proc. Suppl. 39B,C (1995) 16. [] J. Blumlein, M. Klein, T. Naumann, and T. Riemann, in: Proceedings of the HERA Workshop, Vol. I, p. 67, ed. R. Peccei, (DESY, Hamburg, 1988); J. Blumlein, G. Ingelman, M. Klein, and R. Ruckl, Z. Phys. C45(199) 51. [3] E. Zijlstra and W. van Neerven, Nucl. Phys. B417 (1994) 6; Erratum B46 (1994) 45. [4] M. Anselmino, P. Gambino, and J. Kalinowski, Z. Phys. C64 (1994) 67. 5

6 [5] M. Gluck, E. Reya, and W. Vogelsang, DO{TH 95/11 and RAL{TR{95{8. [6] T. Gehrmann and W.J. Stirling, Z. Phys. C65 (1995) 461. [7] P.M. Nadolsky, Z. Phys. C63 (1994) 61. [8] S.J. Brodsky, M. Burkhardt, and I. Schmidt, Nucl. Phys. B441 (1995) 197. [9] H. Lai, J. Botts, J. Huston, J.G. Morn, J.F. Owens, J.W. Qui, W.K. Tung, and H. Weerts, Phys. Rev. D51 (1995) [1] G. Ladinsky, in: Proceedings of the Workshop Prospects of Spin Physics at HERA, Zeuthen, 1995, eds. J. Blumlein and W.D. Nowak, (DESY, Hamburg, 1995), to appear. [11] e.g. M. Klein, in: Proceedings of the Workshop Physics at HERA, Vol. 1, p. 73, eds. W. Buchmuller and G. Ingelman, (DESY, Hamburg, 199). [1] J. Blumlein and M. Klein, Nucl. Instr. Meth. A39 (1993) 11. [13] D.Y. Bardin, C. Burdik, P.C. Christova, and T. Riemann, Z. Physik C4 (1989) 679; J. Blumlein, Z. Phys. C47 (199) 89; H. Spiesberger, DESY 89{175. 6

7 g 1 p (,Q.5 g 1 p (,Q g 1 p (,Q g 1 p (,Q Figure 1: The structure function g p 1 (; Q in the range >1 4.Full line: Q = 1 GeV, dashed line: Q =1 GeV, dotted line: Q =1 3 GeV, dash{dotted line: Q =1 4 GeV. The parametrizations are: (a) ref. [5], (b) ref. [6], (c) ref. [7], (d) ref. [8]. 7

8 g 1 n (,Q g 1 n (,Q g 1 n (,Q - g 1 n (,Q Figure : The structure function g n 1 (; Q in the range >1 4.Full line: Q = 1 GeV, dashed line: Q =1 GeV, dotted line: Q =1 3 GeV, dash{dotted line: Q =1 4 GeV. The parametrizations are: (a) ref. [5], (b) ref. [6], (c) ref. [7], (d) ref. [8]. 8

9 Q /GeV Figure 3: The accessible kinematical range for neutral current deep inelastic scattering at HERA; E p = 8 GeV, E e =7:6 GeV. The stars indicate the values of hq i at a given value of for neutral current deep inelastic scattering. 9

10 -A parallel (,Q Figure 4: Statistical precision of a measurement of A k (; hq i) in the kinematical domain of HERA. The data points represent averages over the accessible Q range and were calculated using the parametrizations [6, 9]. 1

11 g 1 p (,Q Figure 5: Statistical precision of a measurementofg p 1(; Q in the kinematical domain of HERA. The data points represent averages over the accessible Q range and were calculated using the parametrization [6]. The dashed, dotted line, and dash{dotted line correspond to the values of g p 1(; hq i) for the parametrizations [8], [7], and [5], respectively. 11

12 g 1 p (,Q Figure 6: Statistical precision of a measurement ofg p 1(; Q in the kinematical domain of HERA at larger values of. The data points represent averages over the accessible Q range and were calculated using the parametrization [6]. The dashed, dotted, and upper dash{dotted line correspond to the values of g p 1(; hq i) for the parametrizations [8], [7], and [5], respectively. The lower dash{dotted line shows g p 1(; Q ) for Q = 4 GeV for parametrization [5]. 1