On a possibility to determine the sign
of the polarized gluon distribution (*)
W.-D. NOWAK(1), A. V. SIDOROV(2) and M. V. TOKAREV(3) (1) DESY-IfH Zeuthen - 15735 Zeuthen, Germany
(2) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research
141980 Dubna, Moscow Region, Russia
(3) Laboratory of High Energies, Joint Institute for Nuclear Research
141980 Dubna, Moscow Region, Russia
(ricevuto il 6 Agosto 1997; approvato il 7 Ottobre 1997)
Summary. — We investigate the possibility to draw conclusions on the sign of the spin-dependent gluon distribution, DG(x , Q2), from existing polarized DIS data. The
spin-dependent parton distributions Duv, Ddv, D u, D d, Ds, and DG are constructed
in the framework of a phenomenological procedure utilizing some assumptions on signs of valence and sea parton distributions and taking into account the axial gluon anomaly. Results on integral quark contributions to the proton spin, D uA, DdA, and DsA, are used as input. Our predictions for x- and Q2-dependencies of the polarized proton
and neutron structure functions, g1pand g1n, are compared to experimental data. It is
shown that the neutron structure function, gn
1, is especially sensitive to the sign of
DG(x , Q2). The result of our analysis strongly supports the conclusion that this sign
should be positive.
PACS 12.38 – Quantum chromodynamics. PACS 12.38.Qk – Experimental tests. PACS 13.88 – Decays of intermediate bosons.
1. – Introduction
Parton distributions in the nucleon are of universal nature, hence their parametrizations obtained from deep inelastic lepton-nucleon scattering can be utilized for simulations of processes outside lepton nucleon scattering; the polarized parton distributions are especially useful to predict the behaviour of pp interactions with polarized proton beams to facilitate future research programs at the RHIC, HERA and LHC colliders.
Recent results deep inelastic lepton-nucleon scattering experiments at SLAC [1, 2] and CERN [3] on spin-dependent structure functions for proton and deuteron targets,
(*) The authors of this paper have agreed to not receive the proofs for correction.
g1p and g1d, stimulate the interest in determining the spin-dependent gluon and quark distributions in a polarized nucleon. Since a complete solution of this problem is beyond the scope of perturbative QCD and there are still no sufficiently precise non-perturbative calculations available, the usual procedure is to fit numerous parametrizations of both spin-independent and spin-dependent parton distributions to the data. Up to now there is no unique solution; the results depend in one way or another on the method used.
Polarized parton distributions can be extracted in an indirect manner from doubly polarized deep inelastic lepton-proton and lepton-deuteron scattering; the measurable observables are the asymmetries Apand Ad. The structure function gp
1 can be extracted from Apaccording to g1p(x , Q2) 4Ap(x , Q2) Q F2p(x , Q2) 2 x[ 1 1R(x, Q2) ] , (1)
where additional information on the unpolarized structure function F2p[4] and on the ratio of longitudinal to transverse photon cross-section R(x , Q2) 4sL/sT[5] is required. The deuteron structure function is determined in a similar way taking into account appropriate nuclear corrections.
Since there is no practical way at present to directly extract polarized parton distributions from experimental data it is important to develop flexible procedures to construct these distributions incorporating relevant features of the data as well as reasonable constraints derived from our present theoretical understanding of the nucleon.
At present there is no strong argument favouring a positive or negative sign of the spin-dependent gluon distribution, DG(x , Q2). Several sets of spin-dependent parton distributions were constructed utilizing rather different approaches [6-9] mostly assuming a positive sign of DG(x , Q2). Different parameter choices leading to a different behaviour of DG(x , Q2) at x K1 (GHAGI, GHcGI, GHbGI) were studied in [10]. Both positive and negative values of the sign of DG(x , Q2) over a wide kinematical range 1023
E x E 1 were considered in [9]. A detailed NLO QCD analysis of the proton and deuteron data on g1was performed in [8] concluding that the size of the gluon distribution drives the perturbative evolution and, due to the fact that the SMC and E143 data were taken at different values of Q2, the polarized gluon distribution turned out to be large and positive.
However, recent calculations in the context of a non–relativistic quark model and the bag model, respectively, [11] indicate that the integral over DG(x , Q2) might be negative (20.7 and 20.4, respectively). In addition, an instanton-based calculation strongly favors DG(x , Q2) being negative over the full x-range [12]. Recent detailed calculations of instanton contribution to unpolarized structure function in [13] was found to be very small. However the authors of [13] consider that the deep-inelastic processes could be used to study QCD-instantons. The aim of the present paper is to separate in a phenomenological way experimental observables being sufficiently sensitive to allow a conclusion on the sign of DG(x , Q2). As we shall show later, the neutron structure function gn
1 seems to be one of those observables.
A phenomenological approach proposed in [9] is used to construct spin-dependent parton distributions. This method uses all available data on polarized deep inelastic scattering as input, incorporates some constraints on shape and sign of parton distributions, and utilizes as additional input results on the integral quark
contributions to the nucleon spin obtained in other analyses. Finally, the effect of the axial anomaly is taken into account. We study the x and Q2dependence of g1n(x , Q2) for two different scenarios: DG(x , Q2
) D0 and DG(x, Q2
) E0. The calculated predictions are compared to experimental data; a x2criterion is chosen to judge in which of the two scenarios theoretical curves are better describing the experimental data on gn
1(x , Q2). Eventually, the choice for a positive sign of DG(x , Q2) will turn out to be the more likely one, i.e. the polarized structure function of the neutron will be shown to be sensitive to the sign of DG(x , Q2).
2. – Method
The spin-dependent proton structure function g1p is expressed through spin-dependent parton distributions in a simple way
g1p(x , Q2) 4 1 2 Q
m
4 9D uA(x, Q 2 ) 1 1 9D dA(x, Q 2 ) 1 1 9D sA(x , Q 2 )n
, (2)where Dqf4 qf12 qf2, and the qf6 are the probability distributions to find a quark
having positive (1) or negative (2) helicity relatively to positive proton helicity. The neutron structure function gn
1(x , Q2) can be written in a similar form using the replacement D uA DDdA. The valence distributions Dqv, Dqv are then obtained from
Dqv4 Dq 2 2 D q.
In this paper we shall use the spin-dependent parton distributions constructed in [9]. We shall briefly describe in the following sections the main features of the applied method.
2.1. Shape of parton distributions. – For the general form of a spin-dependent parton distribution Dqfwe use
Dqf(x , Q2) 4sign (qf) Q xafQ ( 1 2x)bfQ qf(x , Q2), qf4 uv, dv, u, d, s , G .
(3)
Here qf is the spin-independent parton distribution, af, bf are free parameters which
are to be found by comparison with experimental data. From the restriction NDqfN E qf
(4)
follows that both probability distributions q1
f , qf2 as well as their sum qf4 qf11 qf2
need to be positive; moreover bf should not be negative. To avoid the latter constraint
we introduce a renormalized parton distribution qR
f 4 ( 1 2 x)bfQ qf. This leads to the
following general form of a spin-dependent parton distribution Dqf4 sign (qf) Q xafQ qfR.
(5)
We note that all presently available procedures to construct both spin-independent and spin-dependent distributions do imply fitting procedures and have consequently no unique solution. Hence we believe that at present it is recommended to incorporate general restrictions on Dqf like the one above; this makes it easier to develop flexible
schemes to construct the helicity parton distributions q1
f and qf2, separately.
2.2. Signs of parton distributions. – Up to now there exists neither a running experiment to directly measure the polarized gluon distribution nor does the variety of indirect analyses give a unique result. Hence there is no completely convincing
argument on the sign of DG(x , Q2) yet, although a recent NLO analysis [8] clearly favors a positive sign. Our approach will be to allow for both signs of DG(x , Q2) and compare in both cases our model-dependent predictions to the experimental data.
We note that direct access to DG(x , Q2) /G(x , Q2) will be possible in future experiments. Utilizing polarized protons at RHIC for the (approved) experiments STAR and PHENIX [14] and, possibly, for the suggested internal polarized target experiment HERA–NK[15] at HERA, the measurement of DG(x , Q2) /G(x , Q2) seems feasible in the range 0.1 GxgluonG 0.35. Also new doubly polarized lepton-nucleon scattering experiments approved at CERN [16] and suggested at SLAC [17, 18] will provide very valuable information on DG(x , Q2) /G(x , Q2).
For valence quark distributions the situation is much better defined; we take the sign of Duv as positive and that of Ddv as negative, respectively. This choice is
motivated by the fact that the dominant configuration in the proton wave function is
u(H) u(H) d(I), here the arrow denotes the quark spin direction. The same choice is
made in most analyses of experimental data extracting integral quark contributions to the proton spin [1-3], [19]-[22].
We assume that the signs of D u and D d are positive and negative, respectively. The choice of the signs is based on the property of spin-flip mechanism proposed by ’t Hooft [23] to change helicity of the incoming polarized quark and create at the same time the polarized quark-antiquark pair. The helicity of the outgoing quark and the produced sea quarks are opposite to that of the initial quark. In other words, the spin flip of the valence quarks of dominant configuration in the proton wave function,
u1 and d2, determines the signs of the corresponding sea quark distributions— negative for D d and positive for D u. The negative sign of Ds is in agreement with the ansatz mentioned above and is supported by the results of several analyses of g1p data [1-3, 19-22].
2.3. Inclusion of axial anomaly. – It was shown in [24] that the flavour-singlet axial current
Am04
!
f 4u, d, s
qfgmg5qf
(6)
diverges at the quark level due to the one-loop triangle anomaly ¯mAm4 as p Q NfQ tr ]FmnF Amn ( , (7)
where FAmn4 emnbgFbg, Fmn4 ¯mAn2 ¯nAm1 [AmAn], Am4 AmaQ la, as is the strong coupling constant, and Nf is the number of flavours. The anomaly induces a mixing
between the gluon and the flavour singlet axial current of quarks. For this reason, the helicity carried by each flavour undergoes a renormalization
D qAf(x , Q2) 4Dqf(x , Q2) 2 as(Q 2 ) 2 p Q DG(x , Q 2) . (8)
It was suggested in [24] that the axial anomaly might play a key role and modify the naive quark model predictions; hence parton distributions will presumably become much more sensitive to the sign of the polarized gluon distribution. Consequently, the spin-dependent structure functions g1p and g1n would become more sensitive to DG(x , Q2), as well.
2.4. Integral parton contributions to the proton spin. – A further input required to our analysis is the total contribution of each quark species to the proton spin. We utilize the results of a recent analysis [25] of the structure functions g1pand g1dfrom SMC and SLAC data incorporating 3rd-order pQCD corrections to the Bjorken sum rule. The relative quark contributions to the proton spin were determined as D uA 40.83 6 0.03, D dA4 20.43 6 0.03, D sA 4 20.10 6 0.03 at a renormalization scale Q2
04 10 (GeV/c)2. Using these values and the equation
0 1 [Dqf(x , Q02) 2 as(Q02) 2 p Q DG(x , Q 2 0) ] dx 4DfA , f 4u, d, s , (9)the free parameters af, bf in the parametrization of our spin-dependent parton
distributions DuV, DdV, D u, D d, Ds , DG can be determined both for the positive and
negative sign of DG(x , Q2).
3. – Results and discussion
The procedure described above have been used to determine the parameters af, bf
of valence and sea quark distributions for two scenarios–for positive and negative sign of DG(x , Q2). The shape of DG(x , Q2) is chosen as follows:
DG(x , Q2
) 4sign (G)QxafQ ( 1 2x)bfQ G(x , Q2) .
The parameters af, bf were found and presented in table I for positive and negative
sign of DG(x , Q2), respectively. In the present analysis the parametrizations of unpolarized parton distributions qf(x , Q2) obtained [26] in the leading-order QCD have
been used. (Note that the proposed method is phenomenological and therefore it allows to utilize different parametrizations of parton distributions.) Thus the formula (3) provides the phenomenological description of Q2- and x-dependence of the
spin-dependent parton distribution Dqf(x , Q2). The parametrizations could be used to
describe the Q2-dependence of spin-dependent nucleon structure function in the same way as the leading and next-to-leading order QCD predicts the Q2-evolution.
TABLE I. – The parameters af, bf and signs of spin-dependent parton distributions Dqf4
sign (qf) Q xafQ ( 1 2x)bfQ qffor positive (a) and negative (b) DG. The quark contributions to proton
spin is denoted by D qA. a) b) f Sign af bf D qA f Sign af bf D qA DG + 0.62790 0.000 — DG 2 0.50000 0.000 — DuV + 0.43763 20.450 0.83 DuV + 0.57443 20.900 0.83 D u + 0.54637 0.000 — Du + 1.0088 0.000 — DdV 2 0.71597 20.450 20.43 DdV 2 0.54871 2 0.900 2 0.43 Dd 2 0.57590 0.000 — Dd 2 0.45256 0.000 — Ds 2 0.96418 0.000 20.1 Ds 2 0.34023 0.000 20.1
Fig. 1. – Dependence of the x DG(x , Q2) spin-dependent gluon distributions on x and Q2 for
negative and positive sign of DG(x , Q2). Theoretical curves (- - - 5 ( GeV/c )2, —— 10 ( GeV/c )2)
are obtained with parameters af, bftaken from table I, respectively.
Figure 1 shows the dependence of the x DG(x , Q2) on x at Q2
4 5 , 10 ( GeV/c )2 for positive and negative sign of the polarized gluon distribution. The magnitudes of
x DG(x , Q2) for two scenarios are different. The absolute value of x DG(x , Q2) for
positive sign of DG(x , Q2) is less than for negative one in the range x E0.1 and both of them increase with decreasing x.
Figure 2 shows the x-dependence of the proton structure function xg1p(x , Q2) for two sets of polarized parton distributions, constructed for DG(x , Q2
) D0 (fig. 2a) and DG(x , Q2
) E0 (fig. 2b), respectively, for different values of four-momentum transfer,
Q24 1 , 10 , 100 (GeV/c)2. The parameters af, bf are taken from table I. From fig. 2 it
can be concluded that all theoretical curves for the proton structure function g1p are in reasonable agreement with experimental data [1, 19, 22], i.e. there seems to be no apparent sensitivity to the sign of DG(x , Q2). The Q2-behaviour of xgp
1 appears qualitatively different for DG(x , Q2
) D0 and DG(x, Q2
) E0, respectively. In the first case the maximum of the curve is moved to lower x with increasing Q2, in the second one the position of the maximum is almost not affected. If DG(x , Q2
) D0, the prediction increases with Q2 for x E0.01; if DG(x, Q2) E0, the prediction decreases with Q2over the full x-range.
Figure 3 displays the x-dependence of the neutron structure function xgn
1(x , Q2) at
Q24 3 , 5 , 10 (GeV/c)2; at each Q2 two curves are given for both possible signs of DG(x , Q2), calculated from the same 2 sets of polarized parton distribution functions as above. In each figure the experimental data presently available at the corresponding (average) Q2-value is shown in addition. Please note that to cope with the largely different experimental errors we have chosen a different ordinate scale in fig. 3c). The behaviour of xgn
1 vs. x turns out to be qualitatively and quantitatively different for the
two scenarios DG(x , Q2
) D0 and DG(x, Q2
) E0, especially at lower Q2-values. Apparently, in the range x E0.1 the neutron structure function is especially sensitive to the sign of DG(x , Q2) and therefore sufficiently accurate experimental data on gn
1 at lower x and Q2might be able to discriminate between a positive and a negative sign of the polarized gluon distribution.
Fig. 2. – Deep-inelastic proton structure function xg1p(x , Q2). Experimental data: x [1],m [19], j [22]. Theoretical curves: (a) DG D0, (b) DGE0 and – – – 1 ( GeV/c )2, —— 10 ( GeV/c )2, — — 100 ( GeV/c )2. Parametrizations of parton distributions are taken from table I,
respect-ively.
The preliminary high statistics gn
1 data presented recently by the E154 Collaboration [27] is included in this analysis. Hence we expect now more definite conclusions on the sign of DG(x , Q2) than those drawn from older data in [28].
We calculate x2-values to quantitatively distinguish between the two scenarios; the structure functions calculated from our 2 sets of constructed parton distributions are separately compared to the available experimental data sets. The results are summarized in table II where the references to experimental data on gn
1, the average
Q2 values, and the number of experimental points are given in column 1, 2, and 3, respectively. If “all” is displayed in col. 2 instead of the overall average Q2-value of the experiment, the comparison takes into account that each individual data point is in fact measured at a different average Q2, i.e. here the x2-value is calculated using in the
Fig. 3. – Deep-inelastic neutron structure function xgn
1(x , Q2). Experimental data: x [2],! [3],
m[21] j [29], {[27]. Theoretical curves: – – – DG D0, —— DGE0 at Q24 3 , 5 , 10 ( GeV/c )2. Parametrizations of parton distributions are taken from table I, respectively.
TABLE II. – x2 comparison between theoretical predictions, calculated for the two scenarios
DG(x , Q2) D0 and DG(x, Q2) E0, and experimental data on gn 1(x , Q2). Experiment aQ2b (GeV/c)2 data points x2/n.d.f. DG D0 x2/n.d.f. DG E0 E142 [21] 2 8 1.20 2.05 E143 [2] 3 9 0.89 1.41 SMC [3] 10 12 1.28 1.63 HERMES [29] 3 8 0.86 1.20 E142 [21] all 8 1.45 2.30 SMC [3] all 12 1.35 2.41 E154 [27] 5 11 1.23 3.90
theoretical calculation the correct average Q2-value at each x-point. The corresponding kinematically accessible ranges are 1.1–5.2 (GeV/c)2 for E142, 1.3–48.7 (GeV/c)2 for SMC, and 1.2–15.0 (GeV/c)2 for E154. This method seems to us the closest possible description of the data by a theoretical calculation, hence we expect the x2 values obtained for the “all” comparison to yield the best possible separation.
From table II one can see that for every data set the x2 per degree of freedom is significantly better in the case DG(x , Q2
) D0 compared to the case DG(x, Q2
) E0. As it was expected, the discrimination power delivered by the new (preliminary) E154 data is superior even if all data points are taken at the same Q2value. These results can be considered as clear evidence that the case of a positive polarized gluon distribution function is the more likely scenario compared to the case of a negative one. We expect a much more significant result once the E154 data points can be utilized with their correct Q2-value.
4. – Conclusions
The possibility to draw conclusions on a positive or negative sign of the polarized gluon distribution DG(x , Q2) was studied using a phenomenological procedure to construct spin-dependent parton distributions. The method includes some constraints on the signs of valence and sea quark distributions, takes into account the axial gluon anomaly and utilizes results on integral contributions to the nucleon spin, D uA, DdA, DsA. Investigating the x- and Q2-dependencies of the structure functions gp
1 and g1n constructed by this method we introduce a x2criterion to discriminate between the two scenarios obtained for DG(x , Q2
) D0 and DG(x, Q2
) E0, respectively. The neutron structure function gn
1 was found to be an observable being sufficiently sensitive to the sign of DG(x , Q2). The results of our analysis based on experimental data of the E142, E143, E154, SMC and HERMES Collaborations strongly support a positive sign of DG(x , Q2). Especially the recent (preliminary) high statistics E154 data on g1n are much better described when the sign of the polarized gluon distribution is positive.
* * *
This work was partially supported by Grants of the Heisenberg-Landau Program for 1996 and of the Russian Foundation of Fundamental Research under No. 95-02-05061 and No. 96-02-18897. One of authors (MT) is very grateful to YU.
KOLOMENSKY, G. CATES and L. STUART for providing us with E154 data for neutron structure function.
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