Beam-helicity and beam-charge asymmetries associated with deeply virtual Compton scattering on the unpolarised proton

Beam-helicity and beam-charge asymmetries in the hard exclusive leptoproduction of real photons from an unpolarised hydrogen target by a 27.6 GeV lepton beam are extracted from the HERMES data set of 2006-2007 using a missing-mass event selection technique. The asymmetry amplitudes extracted from this data set are more precise than those extracted from the earlier data set of 1996-2005 previously analysed in the same manner by HERMES. The results from the two data sets are compatible with each other. Results from these combined data sets are extracted and constitute the most precise asymmetry amplitude measurements made in the HERMES kinematic region using a missing-mass event selection technique.


Introduction
Generalised Parton Distributions (GPDs) [1][2][3] encompass the familiar Parton Distribution Functions (PDFs) and nucleon Form Factors (FFs) to provide a comprehensive description of the structure of the nucleon. A thorough description of the nucleon in terms of GPDs would allow the deduction of the total angular momentum of partons in the nucleon, and the construction of a longitudinal-momentum-dissected transverse spatial map of parton densities [4]. The GPDs appear in experimental measurements in the form of complexvalued Compton Form Factors (CFFs), which are flavour-sums of convolutions of GPDs with hard scattering kernels. Constraints on these CFFs, and thus GPDs, can be obtained from measurements of exclusive leptoproduction processes. In particular, the exclusive leptoproduction of a single real photon from a nucleon or nucleus that remains intact (e N → e N γ; see figure 1) is the simplest to describe and is the most widely-used reaction channel for such work (see ).
Generalised parton distributions depend upon four kinematic variables: the Mandelstam variable t = (p−p ) 2 , which is the squared momentum transfer to the target nucleon in the exclusive scattering process with p (p ) representing the initial (final) four-momentum of the nucleon; the average fraction x of the nucleon's longitudinal momentum carried by the active quark throughout the scattering process; half the difference of the fractions of the nucleon's longitudinal momentum carried by the active quark at the start and end of the process, written as the skewness ξ; and Q 2 = −(q 2 ), i.e. the negative square of the four-momentum of the virtual photon that mediates the lepton-nucleon scattering process.  The leading Bethe-Heitler process, i.e. the emission of a real photon from the incoming or outgoing lepton. This process has the same initial and final states as DVCS.
In the Bjorken limit of Q 2 → ∞ with fixed t, the skewness ξ is related to the Bjorken variable x B = −q 2 2p·q as ξ ≈ x B 2−x B . The results are presented as a function of x B because there is no consensus on an experimentally observable representation of ξ. Exclusive leptoproduction of real photons arises from two experimentally indistinguishable processes: the Deeply Virtual Compton Scattering (DVCS) process, which is the emission of a real photon by the struck quark from the nucleon, and the Bethe-Heitler (BH) process, which is elastic lepton-nucleon scattering with the emission of a bremsstrahlung photon by the lepton. The BH process is calculable in the QED framework; this process is dominant at the kinematic conditions of the Hermes experiment. The two processes interfere and the large BH amplitude amplifies the interference term, which is proportional to the DVCS amplitude. It is through the study of this interference term that useful information for the constraint of certain GPDs can be obtained at Hermes kinematic conditions, especially since the interference term is the only part of the squared scattering amplitude that is linear in CFFs [28].
The four-fold differential cross section for the exclusive leptoproduction of real photons from an unpolarised hydrogen target can be written as [28] where e is the elementary charge, = 2x B M Q with M the target mass, and φ is the azimuthal angle between the scattering and production planes [29]. The square of the scattering amplitude |τ | 2 can be written as |τ | 2 = |τ BH | 2 + |τ DVCS | 2 + I, (1.2) with contributions from the BH process (|τ BH | 2 ), the DVCS process (|τ DVCS | 2 ) and their interference term (I). These contributions can be written as where P 1 (φ) and P 2 (φ) are the lepton propagators of the BH process, λ is the helicity of the lepton beam and e is the sign of the charge of the beam lepton. The quantities K BH = 1/(x 2 B t(1 + 2 ) 2 ), K DVCS = 1/Q 2 and K I = 1/(x B yt) are kinematic factors, where y is the fraction of the beam energy carried by the virtual photon in the target rest frame. A full explanation of the Fourier coefficients [c V unp,n , s W unp,n ], where V (W) denotes BH, DVCS or I (DVCS or I), can be found in ref. [28].
Two sets of asymmetries measured at Hermes with an unpolarised hydrogen target and a polarised electron or positron beam are considered here: beam-helicity asymmetries and beam-charge asymmetries. This paper, like ref. [9], presents results related to the following asymmetries: c DVCS unp,n cos(nφ) , c DVCS unp,n cos(nφ) , where dσ(φ) + (dσ(φ) − ) refers to the differential cross section with positive (negative) beam charge and dσ(φ) → (dσ(φ) ← ) refers to the differential cross section taken with beam spin parallel (anti-parallel) to the beam momentum. The s W unp,n and c W unp,n Fourier coefficients depend on "C-functions" [28], each of which is a combination of CFFs. Contributions to the cross section are suppressed by factors that may be kinematic in nature or due to the twist-level of the GPDs appearing in that contribution. Leading twist is twist-2. Typically, the contribution of a twist-n GPD, and hence the corresponding CFF, is suppressed by O(1/Q n−2 ).
The Fourier coefficients that receive leading-twist contributions are c I unp,0 , c I unp,1 and s I unp,1 . All of these Fourier coefficients have a dominant contribution from the C I unp -function: The definition of the kinematic factor K is [28]: The factor of 1 − t min t implies that amplitudes proportional to these Fourier coefficients vanish as −t approaches its minimum value. The C I unp -function can be written [28] where F 1 and F 2 are respectively the Dirac and Pauli form factors of the nucleon and H, H and E are CFFs that relate respectively to the GPDs H, H and E. In Hermes kinematic conditions (where x B and −t 4M 2 are of order 0.1), the contributions of CFFs H and E can be neglected in eq. 1.13 with respect to H (in first approximation) since they are kinematically suppressed by an order of magnitude or more. Hence, the behaviour of C I unp is determined by CFF H and therefore GPD H can be constrained through measurements of the sin φ and cos φ terms of the A I LU (φ) and A C (φ) asymmetries respectively. Compared to the analysis in ref. [9], the analysis presented here additionally includes a larger, independent data set taken during the years 2006 and 2007 and makes use of the same missing-mass technique for event selection as was used in ref. [9]. The work covered in this publication further combines the data taken in 1996-2005 with this newer data set to produce the statistically most precise DVCS measurements that will be presented by Hermes.

Experiment and data selection
The new data presented in this work were collected in 2006 and 2007. As in ref. [9], the data were collected with the Hermes spectrometer [30] using the longitudinally polarised 27.6 GeV electron and positron beams incident upon an unpolarised hydrogen gas target internal to the Hera lepton storage ring at Desy. The integrated luminosities of the electron and positron data samples are approximately 246 pb −1 and 1460 pb −1 , with average beam polarisations of 0.303 and 0.392 respectively. The procedure used to select events is similar to that used in ref. [9]. A brief summary of this procedure is outlined in the following; more details are given in refs. [31,32].
Events in the 2006-2007 data set were selected if having exactly one lepton track detected within the acceptance of the spectrometer and exactly one photon depositing > 5 GeV in the electromagnetic calorimeter. This photon is taken to be the photon arising from the process under investigation. The latter selection criterion differs from the photon selection criterion used for the 1996-2005 data set as an intermittent hardware fault in 2006-2007 can cause spurious noise signals in the calorimeter that are misinterpreted as very low energy photons. The event selection is subject to the kinematic constraints 1 GeV 2 < Q 2 < 10 GeV 2 , 0.03 < x B < 0.35, −t < 0.7 GeV 2 , W 2 > 9 GeV 2 and ν < 22 GeV, where W is the invariant mass of the γ * p system and ν is the energy of the virtual photon in the target rest frame. The polar angle between the directions of the virtual and real photons was required to be within the limits 5 mrad < θ γ * γ < 45 mrad.
An event sample was selected requiring that the squared missing-mass M 2 X = (q + M p − q ) 2 of the e p → e γ X measurement corresponded to the square of the proton mass, M p , within the limits of the energy resolution of the Hermes spectrometer (mainly the calorimeter). Recall that q is the four-momentum of the virtual photon, p is the initial four-momentum of the target proton and q is the four-momentum of the produced photon. The "exclusive region" was defined as −(1.5 GeV) 2 < M 2 X < (1.7 GeV) 2 , as in ref. [9]. This exclusive region was shifted by up to 0.17 GeV 2 for certain subsets of the data in order to reflect observed differences in the distributions of the electron and positron data samples [32]. The data sample in the exclusive region contains events not only involving the production of real photons in which the proton remains intact, but also events involving the excitation of the target proton to a ∆ + resonant state ("associated production"). The recoiling proton is not considered and the calorimeter resolution does not allow separation of all of the latter events from the rest of the data sample. No systematic uncertainty is assigned for the contributions from these events; they are treated as part of the signal. A Monte Carlo calculation based on the parameterisation from ref. [33] is used to estimate the fractional contribution to the event sample from resonant production in each kinematic bin; the uncertainty on this estimate cannot be adequately quantified because no measurements have been made in the Hermes kinematic region. The results of the estimate, called the associated fractions and labelled "Assoc. fraction", are shown in the last row of figures 4-7 in the results section. The method used to perform this estimation is described in detail in ref. [8].

Experimental extraction of asymmetry amplitudes
The expectation value of the experimental yield N is parameterised as N (e , P , φ) = L(e , P )η(e , φ)σ UU (φ)[1 + P A DVCS LU (φ) + e P A I LU (φ) + e A C (φ)], (3.1) where P is the beam polarisation, L is the integrated luminosity, η is the detection efficiency and dσ UU denotes the cross section for an unpolarised target summed over both beam charges and beam helicities. The asymmetries A DVCS LU (φ), A I LU (φ) and A C (φ) are expanded in φ as where the approximation is due to the truncation of the infinite Fourier series that would describe exactly the fitted distribution. Only the sin(nφ) terms of the A LU asymmetries and the cos(nφ) terms of the A C asymmetry are motivated by the physical processes under investigation. The other terms are included both as a consistency check for any off-phase extraneous harmonics in the data and as a test of the normalisation of the fit. These terms are expected to be consistent with zero and are found to be so. A maximum-likelihood fitting technique [34] was used to extract the asymmetry amplitudes in each kinematic bin of −t, x B and Q 2 . This method, described in ref. [8], fits the expected azimuthal distribution function to the data without introducing binning effects in φ. Event weights are introduced in the fitting procedure to account for luminosity imbalances with respect to the beam charge and polarisation.
The asymmetry amplitudes A sin(nφ) LU,I/DVCS and A cos(nφ) C relate respectively to the Fourier coefficients s W unp,n and c I unp,n from the interference and DVCS terms in eqs. 1.6-1.8. The asymmetry amplitudes may also be affected by the lepton propagators and the other φdependent terms in the denominators in eqs. 1.6-1.8.
The DVCS asymmetry amplitude A sin φ LU,DVCS receives a contribution from the C DVCS unpfunction, which is bilinear in CFFs. However, this twist-3 amplitude is inherently small in Hermes kinematic conditions due to the size of the s DVCS unp,1 Fourier coefficient compared to the contributions from the c BH unp,n coefficients in the denominator of eq. 1.7. As a result of the more complicated dependence on the CFFs and this suppression, it is more difficult to constrain GPDs via the measurement of A sin φ LU,DVCS than from the kinematically-unsuppressed leading twist amplitudes.
The leading-twist asymmetry amplitudes are A cos(0φ) C , A cos φ C and A sin φ LU,I , which are proportional to the Fourier coefficients c I unp,0 , c I unp,1 and s I unp,1 defined in eqs. 1.9-1.11. Whilst all of these amplitudes receive contributions from C I unp , c I unp,0 is kinematically suppressed in comparison to c I unp,1 , so A cos φ C and A sin φ LU,I are expected to have the largest magnitude in Hermes kinematic conditions. Although strictly dependent on higher-twist quantities, the asymmetry amplitudes A sin(2φ) LU,I and A cos(2φ) C can also be expressed as having a dependence on C I unp using the Wandzura-Wilzcek approximation [35], i.e. neglecting antiquark-gluon-quark contributions; these amplitudes that are dependent on higher-twist objects can therefore be con- Table 1. Asymmetry amplitudes that can be extracted from the available data set, the related Fourier coefficients, dominant C-functions and twist-levels.

Asymmetry Amplitude Fourier Coefficient Dominant CFF Dependence Twist-Level
sidered as being functionally similar, but kinematically suppressed, when compared to the amplitudes that are dependent only on leading-twist objects.
The A cos(3φ) C amplitude depends on the c I unp,3 Fourier coefficient and hence the C I T,unpfunction. Although the CFFs in this function are of leading twist, they relate to gluon helicity-flip GPDs and are thus suppressed by α S /π, where α S is the strong coupling constant. Table 1 presents the asymmetry amplitudes extracted in this analysis and, for each of them, the related dominant Fourier coefficient and C-function, and the twist-level at which the contributing GPDs enter.

Background corrections and systematic uncertainties for the 2006-2007 data
The extracted asymmetry amplitudes are subject to systematic uncertainties that result from a combination of background processes, shifts in the missing-mass distributions, and various detector and binning effects determined in the same manner as used in refs. [8,9] in order to maintain consistency with the results published in ref. [9] and therefore facilitate the combination of the two data sets. No systematic uncertainty is assigned from the intermittent fault in the calorimeter mentioned in section 2; the number of events in which the fault occurred is very small for the 2006-2007 data sample and completely negligible in the context of the combined data sets. The contribution to the uncertainties on the amplitude measurements arising from background in the data from neutral meson production is predominantly due to the failure to identify one of the two photons from the decay of these neutral mesons. It is possible that both photons from the decay of a neutral meson could be boosted into a single calorimeter cluster and thus be reconstructed as a single photon produced in the BH or DVCS processes. It is similarly possible that the trajectory of one of the produced photons goes outwith the spectrometer acceptance and that the remaining photon is mistaken for one produced by the BH or DVCS processes. The A DVCS LU amplitudes are corrected for the fraction of the data sample and the magnitude of the asymmetry due to semi-inclusive pion production. A further uncertainty is assigned to these amplitudes for the influence of photons produced in decays of exclusively produced pions. Corrections for dilution of the amplitudes for the A I LU and A C asymmetries are applied. No asymmetry value is assigned to the influence of the dilution because influences from meson production are expected to vanish when considering the difference between beam charges. The procedure for estimating the uncertainty and correction factor for each measured amplitude value is described in detail in refs. [8,9]. Each measurement in the −t, x B and Q 2 projections has this uncertainty estimated at the centre of the relevant kinematic bin and included as part of the total systematic uncertainty.
A contribution to the systematic uncertainties of the measured amplitudes arises from shifts in the missing-mass distributions. Such shifts appear in a comparison of electron and positron data [31,32]. One quarter of the difference between the asymmetries extracted using the standard and shifted missing-mass windows is taken as the corresponding systematic uncertainty.
The predominant contribution to the systematic uncertainty arises from detector effects. These include the acceptance of the spectrometer, smearing effects due to detector resolution (e.g. the minimum opening angle between the scattered lepton and produced photon trajectories that can be resolved in the calorimeter), external radiation in detector material, potential misalignment and the finite bin width of the −t, x B and Q 2 projections. In order to quantify these effects, events were generated using a Monte Carlo simulation of the spectrometer that included them. An event generator based on the GPD model described in ref. [36] was used for the simulation because its output describes the data well and it was employed in ref. [9]. Asymmetry amplitudes were extracted from these simulated events using the same analysis procedure used to extract amplitudes from experimental data. In each kinematic bin, the systematic uncertainty was determined as the difference between the asymmetry amplitude reconstructed from the simulated data and that calculated from the GPD model at the average −t, x B and Q 2 value for that bin. The MC simulation shows that, in terms of kinematic smearing, the data sample is 99.9% pure in each of the large "overall" bins. In the kinematic projections, the best purity is found in the sixth Q 2 bin, which is 98% pure. The least pure bin is the third bin in −t, where approximately one-third of the events reconstructed in this bin are generated from outwith it. The average kinematic values in each of the bins are shown to be shifted by no more than 5% as a result of kinematic smearing and the typical effect is a shift in the average kinematic values of a bin on the order of 1%.
The total systematic uncertainty for the 2006-2007 data sample was determined for each kinematic bin by adding in quadrature the uncertainties arising from the background correction, the missing-mass shifts, and the detector effects. The 1996-2005 sample also has a systematic uncertainty from misalignment of the spectrometer [9], which has been eliminated for the 2006-2007 data sample due to improved surveying measurements. However, because the systematic uncertainty calculation for the 2006-2007 data uses the same Monte Carlo generator and reconstruction technique as was used for the 1996-2005 data, the systematic uncertainty for 2006-2007 is overestimated; this overestimate is very slight, because the systematic uncertainty contribution from potential misalignment affecting the asymmetries extracted from the 1996-2005 data set is very small [32].

Results
In figures 2 and 3, results for the beam-helicity and beam-charge asymmetry amplitudes extracted from the 2006-2007 hydrogen data sets in this work are compared with results extracted from the 1996-2005 a data set published previously [9]. Each of the asymmetry amplitudes is shown extracted in one bin over all kinematic variables ("Overall") and also projected against −t, x B and Q 2 . The beam-helicity asymmetry amplitudes are subject to an additional scale uncertainty from the measurement of the beam polarisation, which is stated in the captions of the figures. A statistical test (Student's t-test) was applied in order to check for possible incompatibility between the asymmetry amplitudes extracted from the two data sets. Only the statistical uncertainties were employed in this test as the largest contributions to the systematic uncertainties, i.e. effects from detector resolution, acceptance, misalignment and smearing, are largely correlated. This test revealed no significant evidence for incompatibility between the data sets. The beam-helicity and beam-charge asymmetry amplitudes can therefore be extracted from the entire hydrogen data set recorded during the entire experimental operation of Hermes.  The major contributions to the systematic error bands associated with the asymmetry amplitudes extracted from the combined data set were determined using Monte Carlo simulations as explained in section 4, i.e. contributions from acceptance, smearing, finite bin widths and misalignment. The background in the combined data sample is estimated using the method from refs. [8,31,32]. The uncertainty contributions due to the observed shifts in the missing-mass distributions for the combined data sets were calculated using the procedure described in section 4 and the results were averaged. The total systematic uncertainties for the combined results are obtained by adding these three independent contributions in quadrature. The first and second harmonics of A I LU , which are sensitive to the interference term in the scattering amplitude, are shown in the first and third rows of figure 4. The leadingtwist amplitude A sin φ LU,I has the largest magnitude of any of the amplitudes when extracted in a single bin from the entire data set. This amplitude shows no strong dependence on −t, x B or Q 2 , implying a strong dependence at smaller values of −t as the amplitude must approach zero as −t approaches its minimum value because of the dependence of the amplitude on the factor K defined in eq. 1.12. The A sin φ LU,I amplitude is sensitive to the imaginary part of the CFF H and thereby can constrain GPD H. The A sin φ LU,DVCS asymmetry is shown in the second row of figure 4. Both the A sin φ LU,DVCS asymmetry amplitude and the A sin(2φ) LU,I asymmetry amplitude are compatible with zero, and neither asymmetry amplitude shows any dependence on −t, x B or Q 2 .
The A cos(nφ) C amplitudes are shown in figure 5. The systematic uncertainties are estimated using the same procedure as was used to estimate those for the beam-helicity asymmetries. The leading twist A cos(0φ) C and A cos φ  based on a quark-diquark model with a Regge-inspired term that is included in order to describe accurately parton distribution functions at low x values [41]. The "Regge" term is extended to include contributions that determine the t-dependence of the corresponding GPD. The model incorporates fits to global deep-inelastic and elastic scattering data (to account for the ξ-independent limits and moments of the underlying GPDs) and DVCS data from Jefferson Lab. (to describe the skewness dependence). It describes the t-projections of the A sin φ LU amplitude reported here well, but the projections in the other kinematic variables are not as well described. The model describes the trends of the A cos(0φ) C and A cos φ C asymmetry amplitudes well.
In order to provide more detailed information that can be used in future fits, in particular for the determination of the entanglement of the skewness and −t dependences of GPDs, the amplitudes already presented in figures 4 and 5 are shown as a function of −t for three different ranges of x B in figures 6 and 7. These figures represent the kinematic dependences of the amplitudes in a less-correlated manner than the one-dimensional projections: within experimental uncertainty, there is no evidence of a correlation between the −t and x B dependences for any of the amplitudes.
The results from this paper will be made available in the Durham Database. The results will also be made available in the same 4-bin format as used in previous analyses at Hermes [8,9,12].

Summary
Beam-helicity and beam-charge asymmetries in the azimuthal distribution of real photons from hard exclusive leptoproduction on an unpolarised hydrogen target have been presented. These asymmetries were extracted from an unpolarised hydrogen data set taken during the 2006 and 2007 operating periods of Hermes. Analogous asymmetry amplitudes were extracted previously from hydrogen data obtained during the 1996-2005 experimental period as described in ref. [9]. A comparison of the amplitudes extracted from these independent data sets has shown that they are compatible and the asymmetry amplitudes can therefore be extracted from the complete 1996-2007 data set. The asymmetry amplitudes extracted from the complete data set are the most statistically precise DVCS measurements presented by Hermes. There is a strong signal in the first harmonic of the interference contribution to the beam-helicity asymmetry. There are non-zero amplitudes in the zeroth and first harmonics of the beam-charge asymmetry. All asymmetry amplitudes related to higher harmonics are consistent with zero. The results from the complete data set are compared to calculations from ongoing work to fit GPD models to experimental data. All asymmetry amplitudes are also presented as projections in −t in bins of x B . No additional features are observed in any particular x B -bin.