Updating spin-dependent Regge intercepts

We use new high statistics data from CLAS and COMPASS on the nucleon's spin structure function at low Bjorken x and low virtuality, Q^2<0.5 GeV^2, together with earlier measurements from the SLAC E-143, HERMES and GDH experiments to estimate the effective intercept(s) for spin dependent Regge theory. We find alpha_{a_1} = 0.31 \pm 0.04 for the intercept describing the high-energy behaviour of spin dependent photoabsorption together with a new estimate for the high-energy part of the Gerasimov-Drell-Hearn sum-rule, -15 \pm 2 mu b from photon-proton centre-of-mass energy greater than 2.5 GeV. Our value of alpha_[a_1} suggests QCD physics beyond a simple straight-line a_1 trajectory.


I. INTRODUCTION
The high-energy behaviour of the spin dependent part of the photon-proton total cross section is important for determining the Gerasimov-Drell-Hearn sum-rule for polarised photoabsorption with real photons [1,2], as well as studies of the transition from polarised photoproduction to deep inelastic scattering [3].
Here we investigate this behaviour using the new high statistics measurements from CLAS at Jefferson Laboratory [4] and COMPASS at CERN [5] of the spin asymmetry for polarised photon-proton collisions at low photon virtuality Q 2 < 0.5 GeV 2 and centre-of-mass energy √ s ≥ 2.5 GeV, together with earlier measurements from the E-143 experiment at SLAC [6], HERMES at DESY [7] and the GDH Collaboration in Bonn [8].
The large s dependence of hadronic total cross-sections is usually described in terms of Regge exchanges [9,10], e.g. summing the exchanges of hadrons with given quantum numbers that occur along Regge trajectories with slope (often taken as a straight line) related to the confinement potential. Regge phenomenology has considerable success in describing unpolarised high-energy scattering processes [11].

II. SPIN DEPENDENT REGGE THEORY
Let σ A and σ P denote the two cross-sections for the absorption of a transversely polarised photon with spin antiparallel σ A or parallel σ P to the spin of the target nucleon. The Regge prediction for the isovector and isoscalar parts of (σ A − σ P ) for a real photon, Q 2 = 0, with s → ∞ is [12][13][14]: Here, the α i denote the Regge intercepts for isovector a 1 (1260) Regge exchange and the a 1 -pomeron cuts [12]. The α fi denote the intercepts for the isoscalar f 1 (1285) and f 1 (1420) Regge trajectories and their f 1 -pomeron cuts. The logarithm ln s/s term comes from two nonperturbative gluon exchange in the t-channel [13] with a vector short-range exchange-potential [14] and the mass parameter µ is taken as a typical hadronic scale. The coefficients N and N g are to be determined from experiment.
If one makes the usual assumption that the a 1 Regge trajectories are straight lines parallel to the (ρ, ω) trajectories then one finds α a1 −0.4 for the leading trajectory, within the range of possible α a1 values between -0.5 and zero discussed in Ref. [15]. Fitting straight line trajectories through the a 1 (1260) and a 3 (2030) states, the a 1 (1640) and a 3 (2310) states, and the f 1 (1285) and f 3 (2050) states yields near parallel trajectories with slopes 0.79 GeV −2 , 0.76 GeV −2 and 0.78 GeV −2 respectively. The two leading trajectories then have slightly lower intercepts, α a1 = −0.25 and α f1 = −0.29. With this value of α a1 the effective intercepts corresponding to the a 1 soft-pomeron cut and the a 1 hard-pomeron cut are -0.17 and +0.15 respectively if one takes the soft pomeron with intercept 1.0808 and hard pomeron proposed in Ref. [16] with intercept 1.4 as two distinct exchanges. Values of α a1 close to zero could be achieved with curved Regge trajectories; α a1 = −0.03 ± 0.07 is found in the model of Ref. [17]. For this value the intercepts of the a 1 soft-pomeron cut and the a 1 hard-pomeron cut are ∼ +0.05 and ∼ +0.37.
Before presenting our new results, we first recall the challenge of understanding the proton's internal spin structure in high Q 2 deep inelastic scattering and Q 2 dependence of the intercepts α i describing the asymptotic high-energy behaviour.
In deep inelastic kinematics the nucleon's g 1 spin structure function is related to (σ A − σ P ) by where p and q are the proton and photon four-momenta respectively and α QED is the electromagnetic coupling. The Regge prediction for the isovector g p−n with all data taken at the same Q 2 . Eq. (3) follows from x + M 2 where M is the proton mass and s Q 2 /x in the small x limit. There is possible Q 2 dependence in the α i and N (3) i . The COMPASS experiment found corresponding to an effective intercept α a1 (Q 2 ) = 0.22 ± 0.07 at Q 2 = 3 GeV 2 , with small x data down to x min ∼ 0.004 [18]. The isoscalar spin structure function g p+n 1 ∼ 0 for x < 0.03 at deep inelastic Q 2 [19], in sharp contrast to the unpolarised structure function F 2 where the isosinglet part dominates through gluonic exchanges. The proton spin puzzle, why the quark spin content of the proton is so small ∼ 0.3, concerns the collapse of the isoscalar spin sum structure function to near zero at this small x. The spin puzzle is now understood in terms of pion cloud effects with transfer of quark spin to orbital angular momentum in the pion cloud [20], a modest polarised gluon correction −3 αs 2π ∆g with ∆g less than about 0.5 at the scale of the experiments [19], and a possible topological effect at x = 0 [21].
The observed rise in g p−n 1 at deep inelastic values of Q 2 is required to reproduce the area under the fundamental Bjorken sum rule, Here g A = 1.270 ± 0.003 is the isovector axial-charge measured in neutron beta-decays and C N S (Q 2 ) is the perturbative QCD Wilson coefficient, 0.85 with QCD coupling α s = 0.3 [19]. The Bjorken sum-rule is connected to pion physics and chiral symmetry through the Goldberger-Treiman relation 2M g f π is the pion decay constant and g πN N is the pionnucleon coupling constant. The sum-rule has been confirmed in polarised deep inelastic scattering experiments at the level of 5% [18]. About 50% of the sum-rule comes from x values less than about 0.15. The g p−n 1 data is consistent with quark model and perturbative QCD predictions in the valence region x > 0.2 [22]. The size of g A forces us to accept a large contribution from small x and the observed rise in g p−n 1 is required to fulfill this non-perturbative constraint.
Perturbative QCD evolution acts to push the weight of the distribution to smaller Bjorken x with increasing Q 2 with perturbative calculations predicting rising at small x and deep inelastic Q 2 [23,24]. Regge phenomenology should describe the high-energy part of g 1 close to photoproduction and provide the input for perturbative QCD evolution at deep inelastic values of Q 2 . One then applies perturbative QCD, typically above Q 2 > 1 GeV 2 . These perturbative QCD calculations involve DGLAP evolution and double logarithm, α m s ln n 1 x , resummation at small x [25], in possible combination with vector meson dominance terms at low Q 2 [26]. For g p−n 1 with DGLAP evolution this approach has the challenging feature that the input and output (at soft and hard scales) are governed by non-perturbative constraints with perturbative QCD evolution in the middle unless the a 1 Regge input has information about g A and chiral symmetry built into it. One possibility is a separate hardexchange contribution (perhaps an a 1 hard-pomeron cut) in addition to the soft a 1 term [27].

III. FITTING THE HIGH ENERGY SPIN ASYMMETRY
We next estimate the spin dependent Regge intercepts. Good statistics measurements of the spin asymmetry for photon-proton collisions A p 1 = (σ A − σ P )/(σ A + σ P ) at large √ s and low Q 2 have recently become available from the CLAS and COMPASS experiments, complementing earlier measurements from SLAC, HERMES and the GDH Collaboration. We make a Regge motivated fit to this data on ∆σ = σ A −σ P = A p 1 (σ A +σ P ) with the constraints √ s ≥ 2.5 GeV where Regge theory is expected to apply [11] and Q 2 < 0.5 GeV 2 . Keeping Q 2 < 0.5 GeV 2 is a compromise between keeping Q 2 as low as possible and including the maximum amount of data. This input data involves 18 points from COMPASS with √ s between 11 and 15 GeV [5], 2 data points from HERMES with √ s at 6.6 and 6.8 GeV [7], 7 points from SLAC E-143 with √ s between 2.5 and 3.1 GeV [6], and 102 points from CLAS between 2.5 and 2.9 GeV [4]. This data is consistent with A p 1 being Q 2 independent in each experiment within the chosen kinematics. We also consider the highest energy single data point from the GDH photoproduction experiment with √ s = 2.5 GeV and Q 2 = 0 [8]. Data at higher Q 2 values between 0.5 and 1 GeV 2 are in principle sensitive to the extra effects of turning on DGLAP evolution and decay of higher-twist terms with increasing Q 2 .
Here, we first assume A p 1 to be Q 2 independent in our chosen kinematics with Q 2 < 0.5GeV 2 . That is, we conjecture Second, we assume that the isoscalar deuteron asymmetry A d 1 can be taken as zero in first approximation. The deuteron data on A d 1 are consistent with zero in each experiment in our chosen kinematics [6,7,34,35] (as well as in g d 1 measurements at deep inelastic Q 2 and low x < 0.03 [19]). This means that we set the normalisation factors N (0) i = N g = 0 in Eq.(1). Third, we take σ tot from a fit to unpolarised data. We assume that the errors on σ tot can be neglected compared to the errors on A p 1 . For the total photoproduction crosssection we take (σ A + σ P ) = 67.7 s +0.0808 + 129 s −0.4545 (9) (in units of µb), which provides a good Regge fit for √ s between 2.5 GeV and 250 GeV [11]. The s +0.0808 contribution is associated with gluonic pomeron exchange and the s −0.4545 contribution is associated with the isoscalar ω and isovector ρ trajectories.
Our best fit of form (σ A − σ P ) ∼ N s α including all data is for √ s ≥ 2.5 GeV corresponding to an effective Regge intercept α a1 = +0.31 ± 0.04 (11) -see Fig. 1. The χ 2 /ndf for the fit is 0.98. Statistical and systematic errors for each data point have been added in quadrature.
and take into account that experimental measurements in different x bins are typically taken at different Q 2 values. For example, the COMPASS measurements using a 160 GeV muon beam at x = 0.000052 were taken at Q 2 = 0.0062 GeV 2 whereas their measurements at x = 0.0020 were taken at Q 2 = 0.33 GeV 2 [5], a factor of 53 greater in Q 2 . Within each x bin taken separately, Q 2 was varied over a more limited factor of about 5 and the experimental uncertainties too large to make definite conclusions about possible Q 2 dependence within individual x bins. All of our COMPASS points with Q 2 < 0.5 GeV 2 are in the range √ s between 11 and 15 GeV. One expects A p 1 to vanish in the small x limit, which follows in this data when all points are shifted to the same Q 2 by dividing out the factor (Q 2 ) αa 1 −1.0808 ∼ (Q 2 ) −0.77 from Eqs. (9)(10)(11).

IV. DISCUSSION
It is very interesting that the intercept in Eq.(11) is close to the value found in deep inelastic scattering, viz. α a1 (Q 2 ) = 0.22 ± 0.07 at Q 2 = 3GeV 2 in Eq.(4). Our new low Q 2 value signifies either the presence of a hard exchange, perhaps involving an a 1 hard pomeron cut, or a curved Regge trajectory instead of just a simple straightline a 1 Regge trajectory.
More valuable experimental input could come from the proposed future electron-ion-collider which could extend the experimental data up to √ s values between 40 GeV and 140 GeV [36,37] -that is, up to an order of magnitude higher in √ s than the present highest centre-ofmass energy COMPASS data. Estimates for the expected asymmetries are given in [3]. The fit values in Eqs. (10,11) suggest low Q 2 asymmetries Taking the fit values in Eq. (10), we estimate the highenergy contribution to the Gerasimov-Drell-Hearn sumrule from √ s ≥ 2.5 GeV to be ∞ s0 ds s − M 2 (σ P − σ A ) = −15 ± 2 µb. (12) This determination compares with previous estimates: −15 ± 10µb for √ s ≥ 2.5 GeV based on an extrapolation of lower energy photoproduction data which also gave α a1 = 0.42 ± 0.23 [38], −25 ± 10µb from an early estimate using lower statistics low Q 2 data (pre CLAS and COMPASS) for √ s ≥ 2.5GeV [39], and −26 ± 7µb for √ s ≥ 2 GeV [40] from early Regge fits to low Q 2 data. The new result in Eq. (12) is a factor of 3.5 times more accurate than the previous most accurate determination. The corresponding integral from threshold up to √ s = 2.5 GeV has been extracted from proton fixed tar-get experiments with photon energy up to 2.9 GeV. One finds 226 ± 5 ± 12µb [8,38]. Combining this number and the result in Eq. (12) gives ∞ M 2 ds s − M 2 (σ P − σ A ) = 211 ± 13 µb (13) for the Gerasimov-Drell-Hearn sum-rule. This value compares with the sum-rule prediction 2π 2 α QED κ 2 /M 2 = 205 µb with κ = 1.79 the proton's anomalous magnetic moment [1,2].