

 8 Nov 1995

DESY 95-164 August 1995

On the Measurability of the Structure Function g1(x; Q2) in ep collisions

at HERA

Johannes Bl"umlein

DESY - Zeuthen, Platanenallee 6, D-15735 Zeuthen, Germany

Abstract The possibility is investigated to measure the polarized structure function g1(x; Q2) in the collider mode of HERA operating with a polarized lepton and proton beam. The x dependence of g1 can be measured at a statistical precision of , 20% to 80% in the range 0:0005 ! x ! 0:5 correlated to virtualities 15 ! hQ2i ! 3500 GeV2 at beam polarizations *p , *e = 0:8 and Lint = 60pb\Gamma 1. At a low energy option, Ep = 300 GeV, the statistical accuracy improves to values of , 15% to 50 % in the range 10\Gamma 3 ! x ! 0:5 and 10 ! Q2 ! 800 GeV2.

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 x than ever probed in fixed target experiments would become accessible and the behaviour of polarized structure functions in the range down to x , 0:0001 could be investigated. This is of particular importance since the present analysis of sum-rules relies on extrapolations in this range only. Moreover little is known on the scaling violations of polarized structure functions from experiment in the whole x range so far. Due to the kinematical domain of HERA a far wider Q2 range can be accessed. Furthermore both charged and neutral current reactions can be studied from which constraints on the flavour 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 Q2 range dominated by photon exchange, i.e. Q2 !, 1000 GeV2 [2]. Here the two polarized structure functions g1(x; Q2) and g2(x; Q2) and the two unpolarized structure functions 2xF1(x; Q2) and F2(x; Q2) determine the polarization asymmetries. We analyse the possibilities to measure the polarization asymmetries Ak and A? in the HERA collider mode. An estimate is given of the statistical accuracy to which the structure function g1(x; Q2) can be measured.

2 Polarization Asymmetries The structure functions g1(x; Q2) and g2(x; Q2) can be determined from the measurement of the polarization asymmetries Ak and A? ([3, 4]):

Ak = d

2oe

!) \Gamma d2oe!(

d2oe

!) + d2oe!( (1)

A? = d

2oe

!* \Gamma d

2oe

!+

d2oe

!* + d

2oe

!+ (2)

with

d2oe

!)

dxdy \Gamma

d2oe

!(

dxdy = \Gamma *e*p

8ssff2

Q2 ("2 \Gamma y \Gamma

2M 2xy

S # g

1(x; Q

2) \Gamma 4M

2x

S g

2(x; Q

2)) (3)

d3oe

!*

dxdydOE \Gamma

d3oe

!+

dxdydOE = *e*p

4ff2

Q2 cos OE "

4M 2x(1 \Gamma y)

yS #

1=2 "

1 \Gamma M

2xy

(1 \Gamma y)S #

1=2

\Theta nyg1(x; Q2) + 2g2(x; Q2)o (4) and

d2oe

!)

dxdy +

d2oe

!(

dxdy j

d3oe

!*

dxdy +

d3oe

!+

dxdy = 4ssff

2S

Q4 (y

22xF

1(x; Q

2) + 2 1 \Gamma y \Gamma xyM

2

S ! F

2(x; Q

2)) : (5)

Here ! denotes the orientation of the electron and proton ()) polarization. x and y are the Bjorken variables, S = 4EeEp, Q2 = xyS, M is the proton mass, and *e and *p are the electron

2

and proton beam polarizations. OE denotes the angel between the planes ( k; k0) and ( k; s), where k; k0, and s are the 3-momenta of the incoming and outgoing electron, and the proton spin, respectively.

The contribution due to the structure function g2(x; Q2) in Ak is suppressed by a factor of,

4M 2x=(S(2 \Gamma y)) relative to g1(x; Q2). In the HERA collider mode this term is very small and can be disregarded as well as other terms of O(M 2=S) leading to

Ak = \Gamma *e*p Y\Gamma 2xg

1(x; Q

2)

Y+2xF1(x; Q2) \Gamma 2(1 \Gamma y)FL(x; Q2) ; (6)

where FL(x; Q2) = F2(x; Q2) \Gamma 2xF1(x; Q2), and Y\Sigma = 1 \Sigma (1 \Gamma y)2.

The asymmetry A? contains a factor fA? = 4M=pS leading to a strong suppression in the case of the collider mode at HERA in comparison with the kinematics in fixed target experiments at SLAC or HERMES.

A? = *e*p cos OE2ss fA?qxy(1 \Gamma y) yxg

1(x; Q

2) + 2xg

2(x; Q

2)

Y+2xF1(x; Q2) \Gamma 2(1 \Gamma y)FL(x; Q2) (7)

Since fA? , 0:013 to 0:022 for Ep = 820 GeV to 300 GeV a measurement of A? at the same precision as for Ak would require luminosities much larger than the HERA design value. Due to this a measurement of g2(x; Q2) in the HERA collider mode is not possible based on inclusive polarization asymmetries.

3 Parton Parametrizations The dominant contribution to the structure function g1(x; Q2) is of twist 2. It may be described in the parton model by

g1(x; Q2) = 12 (e2u X

i h\Delta

ui(x; Q2) + \Delta ui(x; Q2)i + e2d X

i h\Delta

di(x; Q2) + \Delta di(x; Q2)i) (8)

with e2u = 4=9; e2d = 1=9 and \Delta f (x; Q2) = f "(x; Q2) \Gamma f #(x; Q2), where f "# are the parton densities at a given nucleon polarization. Here we assumed contributions from the light flavours only.

So far most of the constraints on the different flavour contributions to (8) are due to measurements of gp1(x; Q2) and gd1(x; Q2) at low values of Q2 in the range of 0:01 !, x. The accuracy of the present data still leaves considerable freedom on the parametrization of gp1(x; Q2) and gn1 (x; Q2) both in the small x range and at larger values of Q2 which are both accessible in possible later HERA experiments.

In figure 1 and 2 we compare four recent leading order parametrizations [5-8]1 of gp1(x; Q2) and gn1 (x; Q2) in the range 10\Gamma 4 ! x ! 1 and 10 ! Q2 ! 104 GeV2. Although the x shape of gp;n1 in the range 0:01 ! x is quite similar for all parametrizations both the extrapolation to the small x range and to higher Q2 turn out to be rather different. This is due to the particular

1If the LO Q2 dependence was not provided by the authors of the respective parametrization it was derived from their Ansatz at Q20 by LO QCD evolution using the CTEQ program [9]. I am indebted to Glenn Ladinsky for providing me with according parametrizations prior to publication [10]. In the case of parametrization [8] the set with ffG = 1 was used choosing \Delta us = \Delta u = \Delta ds = \Delta d = 4\Delta s = 4\Delta s. 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 / \Delta G was added to eq. (8). In all other cases eq. (8) was used.

3

parametrization of the individual flavour contributions x\Delta fi, and partly due to the starting point of the evolution choosen.

Whereas gp1(x; Q2) takes negative values for x ! 10\Gamma 3 and shows a falling behaviour in the case of parametrization [5] for 10 ! Q2 ! 104 GeV2, it remains positive and rising for parametrization [7] in the same Q2 range. For parametrization [6] the falling behaviour is only observed for Q2 ? 103 GeV2. Also parametrization [8] yields negative values in the range x , O(10\Gamma 4 ::: 10\Gamma 3) at larger Q2 which are, however, larger than those obtained from parametrization [5]. Moreover at still smaller x values gp1 takes positive values again in the case of parametrization [8], while a falling behavior is obtained by [5, 6].

A similar uncertainty extrapolating from the kinematical range of the present measurements to the domain accessible at HERA is found comparing different parametrizations of gn1 (cf. figure 2.). However, the range in which gn1 takes positive values at large x is about similar. The parametrization [8] predicts a rising behaviour of gn1 in the small x range again.

A measurement of the x shape of gp1(x) at HERA should allow to distinguish between these parametrizations. Also the study of polarized ed scattering would be interesting to constrain the behaviour of gn1 at smaller values of x and larger values of Q2.

4 Kinematical Range Constraints on the kinematical range for neutral current deep inelastic scattering at HERA have been discussed in [11]. The different boundaries are implied by angle and energy cuts, bounds due to resolution effects [12], and large QED radiative corrections [13]. In figure 3 the accessible x; Q2 range is shown for the HERA beam energies Ee = 27:6 GeV and Ep = 820 GeV demanding EJet ? 5 GeV, `Jet ? 5o, `e ! 175o, 0:01 ! y ! 0:9, and x ! 0:7. The values of hQ2i are correlated with x. 2 They are indicated by stars in figure 3 for the case of neutral current deep inelastic scattering. In the following we will investigate the sensitivity to measure the structure function g1(x; Q2) in this range.

5 Accuracy of a measurement of gp1 (x; Q2) In figure 4 the statistical accuracy of a measurement of Ak is illustrated as a function of x assuming Lint = 30pb\Gamma 1/beam polarization and *e = *p = 0:8. We used parametrization [6] as a reference value for the polarized parton densities, and [9] for the unpolarized densities3.

The statistical error of g1(x; Q2) measured from Ak is given by

ffixg1(x; Q2) = 12*

e*p

Y+ Y\Gamma "2xF

1(x; Q

2) \Gamma 2(1 \Gamma y)

Y+ FL(x; Q

2)#

\Theta h2Lintd2oe0=dxdQ2\Delta x\Delta Q2i

\Gamma 1=2 q1 \Gamma A2

k(x; Q

2) (9)

where d2oe0=dxdQ2 denotes the unpolarized differential scattering cross section, \Delta x\Delta Q2 is the bin size, and Lint the integrated luminosity per beam polarization. ffigp1(x; Q2) does only weakly depend on the value of g1(x; Q2) itself as long as A2k o/ 1.

2For Q2 ?, 1000 GeV2 the neutral current deep inelastic scattering cross section contains also conributions due to flZ interference and Z excange terms which are related to new structure functions. As in the case of the measurement of F2(x; Q2) (cf. [2]) these terms may be delt with as corrections in the measurement of gp1(x; Q2).

3In the x and Q2 range considered the different parametrizations for unpolarized parton densities agree to a

wide extent. The contribution due to FL was disregarded in the numerical calculation of Ak.

4

It turns out that at the value of Lint considered the product of the beam polarizations should take values of

*e*p ?, 0:5 (10)

to obtain a sufficient resolution for gp1.

In figure 5 the accuracy of a x shape measurement of gp1(x; Q2) in the kinematical range of HERA (cf. figure 3) is shown. The values of g1(x) represent averages over Q2. With rising values of x hQ2i rises. Assuming the parametrization ref. [6] as a reference value relative errors for gp1(x) between 20% and 80% in the x range from 0:5 to 5 \Delta 10\Gamma 4 at 3500 ? Q2 ? 15 GeV2 are obtained under the above conditions. Predictions for gp1(x; hQ2i) by other parametrizations are shown for comparison. We compare possible measurements at proton beam energies Ep = 820 GeV and Ep = 300 GeV. The statistical precision of the measurement improves with rising values of x and is larger in the case of the low energy option at which the range down to x , 10\Gamma 3 can be probed still. The measurement of gp1 in this kinematical range will allow to further constrain existing parametrizations of polarized parton densities. Particularly it will be interesting to see whether gp1(x; Q2) takes negative values in the small x range.

In figure 6 the statistical accuracy of gp1(x; hQ2i) is illustrated in the range x ? 0:01. Differences between the parametrizations [5-8] are still observed. The size of the scaling violations in LO in this x range is shown comparing the values of gp1(x) for Q2 = hQ2i and Q2 = 4 GeV2, the range of the data from fixed target experiments. For x ! 0:1 the scaling violations of gp1(x; Q2) are of the size of ffigp1(x) for a measurement under the conditions mentioned above. Towards larger x values the scaling violations shrink due to the fix point at x , 0:15. Thus, to see scaling violations of gp1(x; Q2) in a clear way requires a higher luminosity.

The statistical accuracy improves as hQ2i becomes smaller since the differential scattering cross sections behave , 1=Q4. In figure 7 we illustrate the statistical precision of a possible measurement at Ep = 30 GeV and Ee = 3 GeV [14] whith *e = *p = 0:8, Lint = 100 pb\Gamma 1=polarization and applying the kinematical cuts described in section 4. The range of Q2 at these beam energies is rather small and extends from 4 to 12 GeV2 only. A very precise shape measurement of gp1(x; hQ2i) is possible in the range x ? 10\Gamma 2 at this option.

6 Conclusions The measurement of the structure function gp1(x; Q2) 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 x and larger values of Q2 compared to measurements possible in fixed target experiments. gp1(x) can be measured at a statistical precision of 80% to 20% in the range of 5 \Delta 10\Gamma 4 ! x ! 0:5 correlated with values of 15 ! hQ2i ! 3500 GeV2 for an integrated luminosity of Lint = 30pb\Gamma 1/beam polarization and polarization values of *e = *p = 0:8. The measurement of gp1 using a lower proton energy, Ep = 300 GeV, leads to an improvement of the statistical accuracy without a significant restriction in the x range. Further constraints on the x behaviour of gp1 can be obtained by these measurements. A detailed investigation of scaling violations of g1(x; Q2) requires a larger integrated luminosity.

5

References

[1] For reviews see :

E. Reya , in: Proceedings of the Workshop QCD - 20 Years Later, Aachen, 1992, Vol. 1, p. 272, eds. P.M. Zerwas and H.A. Kastrup, (World Scientific, Singapore, 1992); G. Altarelli and G. Ridolfi, in: Proceedings of the Workshop QCD '94, Montpellier, 1994, ed. S. Narison, Nucl. Phys. Proc. Suppl. 39B,C (1995) 106.

[2] J. Bl"umlein, M. Klein, T. Naumann, and T. Riemann, in: Proceedings of the HERA

Workshop, Vol. I, p. 67, ed. R. Peccei, (DESY, Hamburg, 1988); J. Bl"umlein, G. Ingelman, M. Klein, and R. R"uckl, Z. Phys. C 45 (1990) 501.

[3] E. Zijlstra and W. van Neerven, Nucl. Phys. B417 (1994) 62; Erratum B426 (1994) 245. [4] M. Anselmino, P. Gambino, and J. Kalinowski, Z. Phys. C64 (1994) 267. [5] M. Gl"uck, E. Reya, and W. Vogelsang, DO-TH 95/11 and RAL-TR-95-028. [6] T. Gehrmann and W.J. Stirling, Z. Phys. C65 (1995) 461. [7] P.M. Nadolsky, Z. Phys. C63 (1994) 601. [8] S.J. Brodsky, M. Burkhardt, and I. Schmidt, Nucl. Phys. B441 (1995) 197. [9] H. Lai, J. Botts, J. Huston, J.G. Morfin, J.F. Owens, J.W. Qui, W.K. Tung, and H. Weerts,

Phys. Rev. D51 (1995) 4763.

[10] G. Ladinsky, in: Proceedings of the Workshop Prospects of Spin Physics at HERA,

Zeuthen, 1995, eds. J. Bl"umlein 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. Buchm"uller and G. Ingelman, (DESY, Hamburg, 1992).

[12] J. Bl"umlein and M. Klein, Nucl. Instr. Meth. A329 (1993) 112. [13] D.Y. Bardin, C. Burdik, P.C. Christova, and T. Riemann, Z. Physik C42 (1989) 679;

J. Bl"umlein, Z. Phys. C47 (1990) 89; H. Spiesberger, DESY 89-175.

[14] Polarized ep scattering at these energies and luminosities could be studied at possible

future facilities which are currently being discussed, A. Sch"afer, priv. communication.

6

-20 -17.5

-15 -12.5

-10 -7.5

-5 -2.5

0 2.5

10 -4 10 -3 10 -2 10 -1 1x g1p (x,Q 2 )

10 #

-30 -25 -20 -15 -10

-5

0 5 10

10 -4 10 -3 10 -2 10 -1 1x g1p (x,Q 2 )

10 #

0 2 4 6 8 10 12 14

10 -4 10 -3 10 -2 10 -1 1x g1p (x,Q 2 )

10 #-4 -2

0 2 4 6 8 10

10 -4 10 -3 10 -2 10 -1 1x g1p (x,Q 2 )

10 # Figure 1: The structure function gp1(x; Q2) in the range x ? 10\Gamma 4. Full line: Q2 = 10 GeV2, dashed line: Q2 = 102 GeV2, dotted line: Q2 = 103 GeV2, dash-dotted line: Q2 = 104 GeV2. The parametrizations are: (a) ref. [5], (b) ref. [6], (c) ref. [7], (d) ref. [8].

7

-30 -25 -20 -15 -10

-5

0

10 -4 10 -3 10 -2 10 -1 1x g1n (x,Q 2 ) 10 #

-40 -35 -30 -25 -20 -15 -10

-5

0 5

10 -4 10 -3 10 -2 10 -1 1x g1n (x,Q 2 )

10 #

-14 -12 -10

-8 -6 -4 -2

0

10 -4 10 -3 10 -2 10 -1 1x g1n (x,Q 2 ) 10 #

-12 -10

-8 -6 -4 -2

0 2

10 -4 10 -3 10 -2 10 -1 1x g1n (x,Q 2 )

10 #

Figure 2: The structure function gn1 (x; Q2) in the range x ? 10\Gamma 4. Full line: Q2 = 10 GeV2, dashed line: Q2 = 102 GeV2, dotted line: Q2 = 103 GeV2, dash-dotted line: Q2 = 104 GeV2. The parametrizations are: (a) ref. [5], (b) ref. [6], (c) ref. [7], (d) ref. [8].

8

1 10 10 2 10 3 10 4 10 5

10 -4 10 -3 10 -2 10 -1 x Q2 /GeV 2

10 # Figure 3: The accessible kinematical range for neutral current deep inelastic scattering at HERA; Ep = 820 GeV, Ee = 27:6 GeV. The stars indicate the values of hQ2i at a given value of x for neutral current deep inelastic scattering.

9

10 -3 10 -2 10 -1

10 -4 10 -3 10 -2 10 -1 1x -A

parallel

(x,Q 2 )

10 #

Figure 4: Statistical precision of a measurement of \Gamma Ak(x; hQ2i) in the kinematical domain of HERA for Ep = 300 GeV. The data points represent averages over the accessible Q2 range and were calculated using the parametrizations [6, 9].

10

-1 -0.5

0 0.5

1 1.5

2 2.5

3

10 -4 10 -3 10 -2 10 -1 1x g1p (x,Q 2 )

10 # Figure 5: Statistical precision of a measurement of gp1(x; Q2) in the kinematical domain of HERA. The data points represent averages over the accessible Q2 range and were calculated using the parametrization [6]. The dashed, dotted line, and dash-dotted line correspond to the values of gp1(x; hQ2i) for the parametrizations [8], [7], and [5], respectively.

11

0 0.2 0.4 0.6 0.8

1 1.2 1.4

10 -2 10 -1 x g1p (x,Q 2 )

10 # Figure 6: Statistical precision of a measurement of gp1(x; Q2) in the kinematical domain of HERA at larger values of x. The data points represent averages over the accessible Q2 range and were calculated using the parametrization [6]. The dashed, dotted, and upper dash-dotted line correspond to the values of gp1(x; hQ2i) for the parametrizations [8], [7], and [5], respectively. The lower dash-dotted line shows gp1(x; Q20) for Q20 = 4 GeV2 for parametrization [5].

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

10 -2 10 -1 1x g1p (x,Q 2 )

10 # Figure 7: Statistical precision of a measurement of gp1(x; hQ2i) for ep scattering at Ee = 3 GeV and Ep = 30 GeV assuming the same kinematical cuts as at HERA.

13

