Longitudinal double-spin asymmetry $A_1^{\rm p}$ and spin-dependent structure function $g_1^{\rm p}$ of the proton at small values of $x$ and $Q^2$

We present a precise measurement of the proton longitudinal double-spin asymmetry $A_1^{\rm p}$ and the proton spin-dependent structure function $g_1^{\rm p}$ at photon virtualities $0.006~({\rm GeV}/c)^2<Q^2<1~({\rm GeV}/c)^2$ in the Bjorken $x$ range of $4 \times 10^{-5}<x<4 \times 10^{-2}$. The results are based on data collected by the COMPASS Collaboration at CERN using muon beam energies of $160~{\rm GeV}$ and $200~{\rm GeV}$. The statistical precision is more than tenfold better than that of the previous measurement in this region. In the whole range of $x$, the measured values of $A_1^{\rm p}$ and $g_1^{\rm p}$ are found to be positive. It is for the first time that spin effects are found at such low values of $x$.


Introduction
The spin-dependent structure function of the proton, g p 1 , has been extensively studied in the last few decades. Precise measurements of g p 1 (x, Q 2 ) were realised in the deep inelastic regime of charged lepton nucleon scattering at photon virtualities Q 2 > 1 (GeV/c) 2 [1,2] over a wide range of the Bjorken scaling variable x. On the contrary, the behaviour of g p 1 at lower Q 2 is largely unknown. For fixed-target experiments, the values of Q 2 < ∼ 1 (GeV/c) 2 imply small values of x. This low-Q 2 region is governed by 'soft' processes and the transition to the region of higher Q 2 is still not understood.
Quantum Chromodynamics (QCD) allows for a description of 'hard' interactions using a perturbative expansion that is known to be applicable for Q 2 values as low as about 1 (GeV/c) 2 . For lower values of Q 2 , soft interactions become relevant and 'non-perturbative' mechanisms dominate the reaction dynamics. In order to provide a suitable description of the non-perturbative region and also of the transition region between 'soft' and 'hard' physics, it is tried in phenomenological calculations to extrapolate ideas based on the parton model to the low-Q 2 region and add mechanisms like (generalised) vector meson dominance, (G)VMD, supplemented by the Regge model (see Refs. [3][4][5][6]). New and precise data on g p 1 (x, Q 2 ) in the low-Q 2 region are hence essential to improve and validate such calculations. Measurements at low x and low Q 2 are scarce as they put very high demands on event triggering and reconstruction. In spin-dependent leptoproduction they were performed only by the Spin Muon Collaboration (SMC) using proton and deuteron targets [7] and by the COMPASS Collaboration using a deuteron target [8]. The latter, very precise results do not reveal any spin effects in g d 1 over the whole measured interval of x. In this Letter, we present new results obtained on the longitudinal double-spin asymmetry A p 1 and the spin-dependent structure function g p 1 for the proton, in the kinematic region 0.0062 (GeV/c) 2 < Q 2 < 1 (GeV/c) 2 and 4 × 10 −5 < x < 4 × 10 −2 . The data are analysed in four 2dimensional grids of kinematic variables, i.e. (x, Q 2 ), (ν, Q 2 ), (x, ν) and (Q 2 , x), where ν denotes the virtual-photon energy in the target rest frame. Note that the last grid differs from the first one in the number of bins chosen per variable. The lower limit in x coincides with that used in the COMPASS low-Q 2 deuteron analysis [8]. The low-Q 2 results presented in this Letter complement our published proton measurements covering the high-Q 2 region [2,9]. This Letter is organised as follows. We briefly describe the experimental set-up in Sec. 2, the event selection in Sec. 3 and the method of asymmetry calculation in Sec. 4. The results are presented in Sec. 5 and the summary is given in Sec. 6.

Experimental set-up
The measurements were performed using the COMPASS fixed-target set-up and positively charged muons provided by the M2 beam line of the CERN SPS. In 2007, the beam had a momentum of 160 GeV/c with 5 × 10 7 µ + /s in 4.8 s long spills every 16.8 s and in 2011 a momentum of 200 GeV/c with 10 7 µ + /s in 10 s long spills every 40 s. The beam had a momentum spread of 5%. It was naturally polarised with a polarisation P b of about −0.8, which is known with a precision of 5%. Momentum and trajectory of each incident muon were measured before the target by scintillator hodoscopes, scintillating fibre and silicon microstrip detectors.
A large solid-state target of ammonia (NH 3 ) inside a large-aperture superconducting solenoid provided longitudinally polarised protons. The proton polarisation was achieved by dynamic nuclear polarisation and reached an average value of |P t |≈0.85. The dilution factor f , which accounts for the presence of unpolarisable material in the target, is about 0.16 for ammonia. The target material was contained in three cylindrical cells of 4 cm diameter with 30 cm, 60 cm and 30 cm length, which were separated by 5 cm gaps and located along the beam one after the other. Neighbouring cells were polarised in opposite directions in order to use both target polarisations simultaneously during data taking. The polarisation directions were inverted on a regular basis by rotating the direction of the target magnetic field, thus compensating for acceptance differences between different cells and thereby minimising possible systematic effects. Once per year the direction of the polarisation with respect to the solenoid field was reverted by repolarisation in opposite direction keeping the solenoid field unchanged. Ten NMR coils surrounding the target material allowed for a measurement of P t with a precision of 2% in 2007 and 3.5% in 2011.
Momentum and angle of scattered muons and other produced particles were measured in a two-stage open forward spectrometer with large angle and momentum acceptance using two dipole magnets with tracking detectors upstream and downstream of the magnets. Scintillating fibre and micropattern gaseous detectors were employed in and close to the beam region, while multiwire proportional chambers, drift chambers and straw tube detectors covered the outer areas. Scattered muons were identified by drift tube planes behind iron and concrete absorbers in both first and second stage of the spectrometer. Particle identification is not used in the current analysis. Two different types of triggers were employed. "Inclusive" triggers were based on coincidences of hodoscope signals produced by scattered muons. "Semi-inclusive" triggers required an energy deposit in one of the calorimeters with an optional coincidence with an inclusive trigger. The reader is referred to Ref. [10] for the detailed description of the muon beam, the three-cell polarised NH 3 target and the COMPASS spectrometer.

Event selection
Events selected for the analysis are required to have a reconstructed incoming muon, a scattered muon and an interaction vertex. As scattering angles in the laboratory frame are very small for low-Q 2 events, at least one additional track attached to the vertex is required to improve the vertex resolution in beam direction. For the 2007 data, incoming muon momenta are required to range between 140 GeV/c and 160 GeV/c, and for the 2011 data between 185 GeV/c and 215 GeV/c. In order to equalise the beam flux through all target cells, the extrapolated track of the incoming muon is required to pass through all target cells. Interactions originating from the unpolarised material surrounding the target are rejected by imposing appropriate constraints on the position of the interaction vertex. The scattered muon is identified by requiring that it has passed more than 15 radiation lengths of material and it has to point back to the hodoscope that triggered the event. Kinematic constraints are applied on the photon virtuality, Q 2 < 1 (GeV/c) 2 , and on the Bjorken scaling variable, x > 4 × 10 −5 , as it was done in the analysis of the COMPASS deuteron data [8]. The latter constraint is used to avoid the region where x cannot be determined with sufficient accuracy. In addition, the fraction of the energy lost by the incoming muon has to fulfill the condition 0.1 < y < 0.9, where the lower limit removes badly reconstructed events and the upper limit removes events with large radiative corrections as well as low-momentum muons resulting from pion decay-in-flight. These kinematic constraints lead to a minimum value of about 5 GeV/c 2 for W , where W is the invariant mass of the γ * p system of virtual photon and proton.
For a given primary interaction vertex with incident and scattered muon, we require at least one additional (hadron candidate) track that has to carry a fraction z of the virtual photon energy with 0.1 < z < 1 and a momentum p < 140 GeV/c (2007) or p < 180 GeV/c (2011). Here, the condition on z rejects poorly reconstructed tracks and the condition on p removes beam halo muons. This "hadron method" [11] does not only improve the resolution of the primary interaction vertex but also allows the reduction of radiative background.
At the very low values of x studied in this analysis, there exists a contamination by events that originate from elastic scattering of muons off atomic electrons of the target material. These events show up in the x distribution as a prominent peak around the value x µe = m electron /M = 5.45×10 −4 , where M is the proton mass. We remove this contamination by imposing a constraint on the product qθ , where q = +1 (−1) is used if a particle of positive (negative) charge is associated to the track and θ is the angle between the hadron candidate track and the virtual-photon direction. In the range −3.6 < log 10 (x) < −3.0, events with one hadron candidate are rejected if −0.005 rad < qθ < 0.002 rad and events with two hadron candidates if −0.001 rad < qθ < 0 rad. For the former case and either hadron charge, the distribution of the product qθ is presented in Fig. 1 (left). In Fig. 1 (right), the x distributions of accepted events are shown without and with the constraint on qθ .
After having applied all selection criteria, 447 million events taken with a beam energy of 160 GeV and 229 million taken with 200 GeV remain for analysis.

Asymmetry extraction
The longitudinal double-spin lepton-proton cross-section asymmetry is given by where the arrows refer to the longitudinal spin orientations of incoming muon (→) and target proton (⇒). It can be decomposed into a longitudinal photon-nucleon asymmetry A p 1 and a transverse photon-nucleon asymmetry A p 2 , where the longitudinal asymmetry is defined in terms of the γ * p cross sections as Here, the subscript refers to the total angular momentum of the γ * p system. The factor D in Eq. (1) is the so-called depolarisation factor and η is a kinematic factor. Full expressions for D and η are given in Ref. [8]; the behaviour of D in the kinematic region of this analysis is shown in Fig. 2 (left). In the COMPASS kinematic range, the factor η is negligible, hence the term containing the transverse asymmetry A p 2 is of negligible size and its possible contribution is included in the systematic uncertainty of A p 1 . The number of events originating from a given target cell with a given direction of the target polarisation can be expressed as Here, a i is the acceptance, φ i the beam flux, n i the number of target nuclei,σ the spin-independent cross section and f the dilution factor. The four equations of Eq.       The dilution factor f includes a correction factor ρ = σ p 1γ /σ p tot [12] that accounts for radiative events originating from unpolarised target protons. This effective dilution factor depends only weakly on the incident energy, decreases with x and reaches a value of about 0.17 at x ∼ 10 −4 and about 0.15 at x ∼ 10 −2 . Its relative uncertainty amounts to 5%. The x dependence of the average dilution factor is shown in Fig. 2 (right). The beam polarisation is a function of the beam momentum and is taken from a parametrisation based on a Monte Carlo simulation of the beam line, which was validated by SMC [13].
The final value of A p 1 is obtained as weighted average of the values calculated for the two target-spin orientations. It is corrected for spin-dependent radiative effects [14] and for the polarisation of the 14 N nuclei present in the target. It was verified that the use of semi-inclusive triggers and the requirement of a reconstructed hadron do not bias the determination of A p 1 [11,15]. More details on the analysis can be found in Ref. [16].
The additive and multiplicative systematic uncertainties of A p 1 are shown in Table 1. The largest multiplicative contribution originates from the depolarisation factor D through its dependence on the poorly known function R = σ L /σ T , which is the ratio of the absorption cross sections of longitudinally and transversely polarised virtual photons. The parameterisation of the function R described in detail in Ref. [8] takes into account all existing measurements together with an extension to very low values of Q 2 . As systematic uncertainty of R a constant value of 0.2 is taken for Q 2 < 0.2 (GeV/c) 2 . The largest additive contribution originates from possible false asymmetries, which are estimated from time-dependent instabilities in the spectrometer as described in Ref. [16]. In certain kinematic regions, it can be larger than the statistical uncertainty. The spin-dependent structure function of the proton, g p 1 , is determined from the virtual-photon asymmetry A p 1 neglecting A p 2 : g Here, F p 2 is the spin-independent structure function of the proton. For F p 2 we used the SMC parameterisation [11] within its validity limits, i.e. x > 0.0009 and Q 2 > 0.2 (GeV/c) 2 . Outside these limits, the values were calculated using the phenomenological model of Refs. [17,18], which is based on the GVMD concept. Equation (4) can be written as so that the systematic uncertainty of g p 1 can be obtained from the following three components: i) the systematic uncertainty of A p LL ≡ A p 1 /D, ii) the systematic uncertainty of F p 2 , and iii) the systematic uncertainty of the product D(1 + R). The systematic uncertainties of A LL and R were already discussed above. The systematic uncertainty of F p 2 is estimated from the difference between the SMC parameterisation and the models of Refs. [17,19,20]. It is taken as half of the maximum of the absolute differences between the used parameterisation or model and the remaining models. For Q 2 > 0.2 (GeV/c) 2 , this is always the absolute value of the difference between the SMC parameterization and the model of Refs. [17,19]. When calculating g p 1 using Eq. (5) instead of Eq. (4), we benefit from the fact that D and R are correlated (see also Ref. [8]), which results in a reduced systematic uncertainty compared to the one of A p 1 .

Results
We present here the results for the spin asymmetry A p 1 and the spin structure function g p 1 measured in the kinematic range Q 2 < 1 (GeV/c) 2 and 4×10 −5 < x < 4×10 −2 using the two beam energies 160 GeV and 200 GeV. For each beam energy, the data are analysed in four two-dimensional grids: (x, Q 2 ), (ν, Q 2 ), (x, ν) and (Q 2 , x), where the latter has a smaller number of x bins. The x dependence of A p 1 at the measured values of Q 2 is shown in Fig. 3 (left) for the two beam energies. A positive asymmetry is observed, which slightly rises with x. It amounts to about 0.01 at x < 10 −3 , indicating for the first time the existence of spin effects at such small values of x. Note that the COMPASS results for the deuteron [8] show an asymmetry A d 1 compatible with zero. In Fig. 3 (left), also the results for A p 1 from SMC [7,11] and HERMES [21] are shown. Within the large statistical uncertainties, their results are consistent with our present results, but also with zero. Compared to the results from SMC, which is the only other experiment that covers the low-x region, we improve the statistics by a factor of about 150. In Fig. 3 (right), the ν-dependence of A p 1 is shown. A rather flat distribution is measured, apart from a slight enhancement for ν < 50 GeV that corresponds to higher values of Q 2 . In Fig. 4, the results for A    [7,11,21] are also shown.
For the two beam energies, our results on g p 1 are shown versus Q 2 for the same 15 bins in x (Fig. 5) and versus x in 5 different bins in Q 2 (Fig. 6). Down to the smallest value of x, i.e. 4 × 10 −5 , g p 1 is positive within experimental uncertainties and does not show any trend to become negative or to grow with decreasing values of x.
All numerical values are available on HepData [22]. The numerical values for A p 1 and g p 1 versus x, averaged over Q 2 , are given together with their statistical and systematic uncertainites in Table A.1 of the appendix for the two energies separately. The data for the two energies were combined and false asymmetries reevaluated for the merged data. The values for the combined results are given in Table A.2.
In Fig. 7, the present results on g p 1 are compared with the predictions of the phenomenological models of Refs. [5,6]. The first model (BKZ) is based on GVMD ideas supplemented by the Regge formalism. The contribution of heavy vector mesons to g p 1 was treated as an extrapolation of the QCD improved parton model to arbitrarily low values of Q 2 . The magnitude of the light vector meson contribution was fixed in the photoproduction limit by relating the first moment of g p 1 to the static properties of the proton via the Drell-Hearn-Gerasimov sum rule [23], using the measurements in the region of baryonic  resonances [24]. Both perturbative and non-perturbative contributions to g p 1 are found to be present and substantial at all values of Q 2 . Reasonable agreement is observed between the BKZ model and our measurements in all four two-dimensional grids of kinematic variables. Figure 7 (left) shows a comparison of the x dependence of the BKZ model prediction with the results for g p 1 obtained combining the 160 GeV and 200 GeV results. In the model of Ref. [6] (ZR), the nonperturbative part of g 1 is also parameterised using the vector meson dominance mechanism together with Regge predictions (albeit done differently than in Ref. [5]), while in the perturbative part QCD evolution is employed together with parton recombination corrections. The g p 1 calculations of Ref. [6] are presented in Fig. 7 (right), where the broad bump at lowest values of x is almost entirely due to the VMD contribution.  14) 160 GeV 200 GeV    with their statistical and systematic uncertainties as a function of x and the average values of x, Q 2 and y, for the 160 GeV and 200 GeV data. The maximum value of Q 2 used in the analysis is 1 (GeV/c) 2 . Bins in x are of equal width in log 10 x.
x range x Q 2 y A with their statistical and systematic uncertainities as a function of x and the average values of x, Q 2 and y, shown for the combination of 160 GeV and 200 GeV data. The maximum value of Q 2 used in the analysis is 1 (GeV/c) 2 . Bins in x are of equal width in log 10 x.
x range x Q 2 y A