Hard Exclusive Electroproduction of $\pi^+\pi^-$ Pairs

Hard exclusive electroproduction of $\pi^+\pi^-$ pairs off hydrogen and deuterium targets has been studied by the HERMES experiment at DESY. Legendre moments $$ and $$ of the angular distributions of $\pi^+$ mesons in the center-of-mass frame of the pair have been measured for the first time. Their dependence on the $\pi^+\pi^-$ invariant mass can be understood as being due to the interference between relative $P$-wave (isovector) and $S,D$-wave (isoscalar) states of the two pions. % The increase in magnitude of $$ as Bjorken $x$ increases is interpreted in the framework of Generalized Parton Distributions as an enhancement of flavour {non-singlet} $q\bar{q}$ exchange for larger values of $x$, which leads to a sizable admixture of isoscalar and isovector pion pairs. % In addition, the interference between {$P$-wave} and {$D$-wave} states separately for transverse and longitudinal pion pairs has been studied. The data indicate that at {$\la Q^2 \ra = 3 $ GeV$^2$} higher twist effects can be as large as the leading-twist longitudinal component

Much of our current knowledge of the quark-gluon structure of the nucleon comes from inclusive and semi-inclusive deep inelastic scattering experiments, from which parton distribution functions can be extracted. However, our understanding of quark-gluon dynamics can be extended considerably by measurements sensitive to the generalized parton distributions (GPD) [1, 2,3], which also describe the dynamical correlations between partons with different momenta. Experimentally, GPDs can be investigated through the analysis of hard exclusive processes such as the production of mesons by longitudinal virtual photons. Under these conditions the amplitude factorizes into a hard scattering term governed by perturbative QCD and two soft parts, the GPDs for the nucleon and the distribution amplitude for meson formation [4,5]. Hard exclusive electroproduction of π + π − pairs is sensitive to the interference between isospin I = 1 and I = 0 channels, and provides a new constraint on certain combinations of GPDs.
This letter reports the first experimental data for hard exclusive π + π − pair production e + p → e + p π + π − and e + d → e + d π + π − . (1) For the proton target, the results are interpreted in the GPD framework by comparing with predictions [6,7,8], thus providing valuable information for further modelling of GPDs. So far, predictions exist only for the proton target. Exclusive pair production includes contributions from both two-gluon and quark-antiquark (qq) exchange mechanisms.
The relevant diagrams at leading twist, which may involve both resonant and non-resonant channels, are shown in Fig. 1. The Primakoff process γ ⋆ γ ⋆ → π + π − is not shown, because it is expected to contribute negligibly to the production of pions pairs with helicity zero or one [9], and the analysis reported here is insensitive to helicity two. Previous work [10] has shown that resonant π + π − production via longitudinal ρ 0 decay in the kinematical region covered by the HERMES experiment occurs primarily through two-quark exchange with the target. In the present more general case, the qq exchange mechanism gives rise to pion pairs with the values of the strong isospin I, total angular momentum J, and C-parity of either a ρ-meson (I = 1, J = 1, 3..., C = −1), or an f -meson (I = 0, J = 0, 2..., C = +1). The qq exchange with C = +1 (C = −1) is described by flavour singlet (non-singlet) parton combinations [11], and due to C-parity conservation the π + π − pairs so formed have C = −1 (C = +1). The competing two-gluon channel gives rise to pion pairs with the quantum numbers of the ρ-meson family only. Pion pairs are formed from either quarks ( Fig. 1-a,b,c) or gluons ( Fig. 1-d) produced in the perturbative hard part of the reaction. Since the cross section for isovector π + π − production is much larger than for the isoscalar case, it is difficult to obtain experimental data on the isoscalar channel. One possible solution would be to study exclusive π 0 π 0 production, but this requires a large experimental acceptance. With charged pions, the interference between the two isospin channels can also provide information on the weaker isoscalar channel at the amplitude level.
For the purpose of studying the interference between π + π − production in P -wave (I = 1) and S, D-wave states (I = 0), the Legendre moments P 1 (cos θ) and P 3 (cos θ) are particularly useful because they are sensitive only to such interference. The Legendre moment of order n is given by where θ is the polar angle of the π + meson with respect to the direction of the π + π − pair in the center-of-momentum frame of the virtual photon and target nucleon. The moments P 1 and P 3 have been evaluated as a function of the pion pair invariant mass m ππ , and the Bjorken variable x = Q 2 2νMP , where −Q 2 is the squared four-momentum of the initial virtual photon, M P is the proton mass and ν is the virtual photon energy in the target rest frame. Experimentally, P n is the average of P n (cos θ i ) for all events i grouped in bins of m ππ or x.
In general, in which ρ is the spin density matrix of the pion pair, whose diagonal entries ρ JJ λλ give the probability of producing it with angular momentum J and longitudinal projection λ, and whose off-diagonal terms describe the corresponding interference terms. If parity is conserved [12]. The contributions for J > 2 are expected to be negligible in the m ππ -range covered by HERMES. The Legendre moments then are 21 11 + 4ρ 21 00 + 2 √ 5ρ 10 00 , (4a) In particular, P 1 is sensitive to P -wave interference with S and D-waves, whereas P 3 is sensitive to only P -wave interference with a D-wave.
The relevant factorization theorem [4] has been proved only for longitudinal virtual photons γ * L in leading twist. Contributions from transverse photons γ * T and other higher-twist effects are suppressed by powers of 1/Q. Therefore, the longitudinal terms ρ 21 00 and ρ 10 00 in Eqs. 4a and 4b are expected to be dominant in the m ππ region far from the f 2 meson, where the higher-twist term ρ 21 11 can be neglected. On the other hand, in the region of the f 2 resonance the possible ρ 21 11 contribution can be eliminated by taking a combination of P 1 and P 3 that projects out the longitudinal terms: Assuming s-channel helicity conservation, such that the 0-helicity photon γ * L produces a π + π − pair with 0-helicity, only ρ 00 states are populated by γ * L . In this case, the combination P 1 + 7 3 P 3 would be sensitive to longitudinal photons only. In the f 2 region, far from the ρ 0 and f 0 resonances, the term ρ 10 00 is expected to vary very slowly with m ππ , making no contribution to any structure appearing in this combination.
In the m ππ region of the f 2 meson, another combination eliminates the contribution of longitudinal tensor pairs: Hence, the transverse higher-twist ρ 21 11 and longitudinal leading-twist ρ 21 00 contributions to the Legendre moments in the f 2 domain can be disentangled by comparing the combinations given above.
The data were collected with the HERMES spectrometer [13] during the running period 1996-2000. The 27.6 GeV HERA positron beam at DESY was scattered off hydrogen and deuterium targets. Events were selected with exactly one positron track and two oppositely charged hadron tracks with momentum > 1 GeV, requiring that no additional neutral clusters occur in the calorimeter. Positrons were distinguished from hadrons with an average efficiency of 98%, and a hadron contamination below 1%, over the whole kinematic range. In order to ensure a hard scattering process, the constraints Q 2 > 1 GeV 2 and W > 2 GeV were imposed, where W is the invariant mass of the virtual photon-nucleon system. When studying the m ππ -dependence of the Legendre moments, the requirement x > 0.1 was imposed to suppress the contribution from gluon-exchange relative to that from qq exchange [6]. However, when analyzing the x-dependence of the Legendre moments, the whole x-range accessible to HERMES was used.
Since the recoiling target nucleon is not detected in the present HERMES apparatus, exclusive events were selected by restricting the quantity ∆E = , in which M X is the missing mass, and M targ is the nucleon target mass. A ∆E distribution peaked at zero is a clear signature of exclusive production, while larger ∆E values indicate non-exclusive events. For scattering off nuclei, one can have either incoherent scattering from individual nucleons inside the target (M targ ≈ M N ) or coherent scattering from the entire nucleus A (M targ ≈ M A ). For scattering off deuterium, incoherent scattering is found to dominate for HERMES kinematics [14]; therefore M targ was chosen to be the proton mass throughout the entire analysis. All detected hadrons have been treated as pions.
In the ∆E spectrum, the resolution due to instrumen-tal effects ranges between 0.260 and 0.380 GeV, depending on the data production year. Thus, even at low ∆E the sample is contaminated by non-exclusive processes. This background yield was assumed to be semi-inclusive deep inelastic scattering (SIDIS) events and was evaluated by first calculating the ∆E distribution of SIDIS events with a Lepto Monte Carlo simulation [15,16], and then normalizing it to the data in the range ∆E > 2 GeV. The effect of varying this normalization region was treated as a systematic uncertainty contribution. Fig. 2 shows the normalized Monte Carlo distribution in ∆E compared to the data, and their difference. The simulated background shape is in agreement with the data at large ∆E, while at small ∆E the data show a surplus due to the presence of the exclusive process not included in that Monte Carlo simulation. Comparison of the exclusive peak in the data with the result of a Monte Carlo simulation using the diffractive ρ 0 DIPSI generator [17] reveals an excess at ∆E ≈ 1.5 GeV. This excess can be explained by the combined contributions of ρ 0 production via single and double-dissociation of the proton as described in Ref. [18], and of radiative corrections [19], which all three are not simulated by the DIPSI Monte Carlo.
In order to evaluate the background contribution to the exclusive signal, the experimental and the normalized Monte Carlo yields were separately integrated up to a limiting ∆E value ∆E cut , resulting in N tot and N MC respectively. The value of ∆E cut was optimized by requiring the ratio of the exclusive signal N Sg = N tot − N MC over the background (N Sg /N Bg ) to be large, and the relative statistical uncertainty ∆N Sg /N Sg to be small. The optimized ∆E cut value for both targets is 0.625 GeV. Be- . For both spectra, the requirement x > 0.1 has been applied. For both targets, the mππ-spectrum for ∆E < 0.125 GeV is normalized and superimposed (shaded area) to show the suppression of the ω → π + π − π 0 contamination described in the text.
low the chosen ∆E cut value, the SIDIS contamination is found to range between 2% and 65% of the total events, depending on m ππ and x. In particular, this contamination is small at m ππ values around m ρ 0 , and increases at smaller and larger invariant mass values. The SIDIS model does not account for contamination from other processes. In order to suppress the ω → π + π − π 0 decay at low m ππ , as explained below, a more severe ∆E cut was applied than the value optimized for the SIDIS background. The final ∆E cut values used in this analysis for both targets are 0.125 GeV for m ππ ≤ 0.60 GeV, and 0.625 GeV for 0.60 < m ππ ≤ 1.40 GeV.
The limited ∆E resolution does not allow for the complete suppression of single and double-dissociation processes. An example is the process in which the nucleon is left in a ∆ resonance state that decays with an unobserved pion. The contamination from single and double-dissociation was estimated by shifting the value of ∆E cut by 0.5 GeV, from a low value of 0.125 GeV where this contamination is negligible, to a relatively large value, 0.625 GeV, where this background is possibly substantial. This effect was included in the systematic uncertainty.
The contamination from target excitations such as e + p → e + π∆ → e + pπ + π − , which have been found to contaminate the process e + p → e + pπ + π − at lower energy and W values [20], in the HERMES kinematics were found to be negligible in a Dalitz-plot analysis [21].
The contamination of exclusive K + K − pairs from φ(1020) meson decay, which appears in the event yield at m ππ ≈ 0.35 GeV, is entirely eliminated by applying the additional cut m KK > 1.06 GeV. Here m KK is the invariant mass of the two hadrons when they are treated as kaons. Similarly, the contamination of φ → K S K L , with K S detected through its decay in π + π − , by using a Monte Carlo DIPSI simulation was found to be entirely absent within the chosen ∆E cut values. The channel ω → π + π − and exclusive non-resonant K + K − (1) production were estimated to contaminate the signal by less than 0.3% and 1.5%, respectively, and were neglected. The decays φ → π + π − π 0 (2) , with the π 0 outside the acceptance, gives a contamination of less than 1%. A contamination of about 18% from the decay ω → π + π − π 0 , with only the charged tracks detected, yields a reconstructed m ππ distribution centered at 0.45 GeV with a Gaussian width of approximately 0.075 GeV [22]. This contribution to the yield was suppressed by imposing ∆E < 0.125 GeV in the region m ππ ≤ 0.6 GeV. The effect of the remaining contamination was taken into account in the systematic uncertainty of the relevant bins. All the above estimations of these additional background components are small compared to the background predicted by the SIDIS model.
In each of the analyzed bins, P n data was evaluated within the chosen exclusive ∆E region, with no back-

FIG. 4:
The mππ-dependence of the Legendre moments P1 (upper panels) and P3 (lower panels) for hydrogen (left panels) and deuterium (right panels), for x > 0.1. The region 0.8 < mππ < 1.1 GeV is presented with finer bins to better investigate possible contributions from the narrow f0(980) resonance, as shown in the insert. In the upper panels, leading twist predictions for the hydrogen target including the two-gluon exchange mechanism contribution, LSPG [6,7] (solid curve) at x = 0.16 are shown. A calculation without the gluon exchange contribution is shown for limited mππ values, LPPSG [8] (open squares at x = 0.1, open triangles at x = 0.2). In these calculations, the contribution from f0 meson decay was not considered. Instead, the inset panel for the hydrogen target shows the prediction from [25], which includes the f0 meson contribution. All experimental data have x = 0.16, Q 2 = 3.2 (3.3) GeV 2 , and −t = 0.43 (0.29) GeV 2 for hydrogen (deuterium). The systematic uncertainty is represented by the error band.
ground subtraction. The values of P n SIDIS for the background events were extracted from the data for ∆E > 2 GeV, where SIDIS events dominate. These values were found to be consistent when evaluated in three different ∆E bins: 2 < ∆E < 4 GeV, 4 < ∆E < 6 GeV, and ∆E > 6 GeV. The moments were corrected for SIDIS background using in which r is the ratio of integrated exclusive data to background Monte Carlo events for ∆E < ∆E cut in the analyzed bin.
A Monte Carlo generator based on the GPD framework for the hard π + π − exclusive process does not exist. Therefore the DIPSI generator was used to evaluate the effects of geometric acceptance and instrumental smearing on the Legendre moments, which were both found to be negligible [21]. This Monte Carlo simulation is in good agreement with the kinematic distributions of exclusive ρ 0 mesons observed at HERMES.
The analyzed moments might be sensitive to radiative corrections that affect the cos θ angular distribution. For ρ 0 decay, which dominates in the cross section for exclusive π + π − production, the angular distribution depends linearly only on the vector spin density matrix element r 04 00 . In previous work [23] the relative correction of r 04 00 for radiative corrections has been evaluated, and found to be less than 0.3% at Q 2 ≈ 3 GeV 2 in the kinematics of the H1 and ZEUS experiments. At larger x, where the HERMES analysis is performed, they are even smaller. As a result of these considerations, radiative corrections effects have been neglected in this analysis. The m ππ -dependence of P 1 and P 3 for exclusive π + π − production off hydrogen and deuterium is presented in Fig. 4, for x > 0.1. The average values of Q 2 , −t, and x for both targets in this domain are reported in Tab. I. For m ππ < 1 GeV, the moments are similar for the two targets. In each panel for P 1 , the region 0.8 < m ππ < 1.1 GeV is shown as an insert with finer binning to better investigate possible contributions from the narrow f 0 (980) resonance.
The values for P 1 differ significantly from zero, and depend strongly on m ππ . At small invariant mass, i.e. close to the threshold 2m π , this non-zero moment is interpreted as originating from the interference between the lower tail of the isovector ρ 0 (770) (P -wave) with the S-wave non-resonant π + π − amplitude. At m ππ values around m ρ 0 , the absolute value of this quantity shows a minimum, which is explained in terms of the overwhelming dominance of ρ 0 vector meson production in the denominator of the moment. The increase of the size of P 1 at larger invariant mass is due to the interference of the upper tail of the ρ 0 with the non-resonant π + π − S-wave production. At m ππ ≈ 1 GeV, the observed oscillation in hydrogen P 1 suggests an interference between the ρ 0 tail and the S-wave π + π − production from the narrow f 0 (980) resonance. Moreover, in the f 2 (1270) meson region, the data suggest a sign change caused by the interference between the ρ 0 upper tail and the f 2 (D-wave).
The Legendre moment P 3 is sensitive only to the interference of P -wave and D-wave states in π + π − production. Consistent with the expectation that no resonance decay into π + π − pairs in D-wave states occurs for m ππ ≤ 1 GeV, no interference is observed in this invariant mass region. The P 3 moment for deuterium increases in magnitude in the f 2 (1270) meson region. A sign change is also prominently visible, reflecting the interference of the P -wave and D-wave resonant π + π − channels. On the other hand, no such signature is evident in the hydrogen data.
In Fig. 4 the m ππ -dependence of P 1 for hydrogen is compared with theoretical calculations based on the GPD framework, with [6,7] (solid curve) and without [8] (open points) the inclusion of the two-gluon exchange mechanism. A possible contribution from the f 0 meson was not considered in the calculations. The calculations include only the longitudinal component σ L of the π + π − cross section, while in this analysis no separation between the σ L and σ T contributions could be made. The σ T contribution to the total cross section for ρ 0 production is estimated to be approximately 60% [18]. The reasonable agreement of the leading twist predictions for the m ππ -dependence of the P 1 data may tentatively be understood as arising from the cancellation of higher twist effects in this moment [24].
To date, the f 0 contribution is taken into account only by Ref. [25], where the discussion is restricted to diffractive physics at center-of-mass energies larger than 100 GeV. Nevertheless, to demonstrate the possible effect of this resonance, the comparison with those predictions for P 1 on hydrogen is shown in the panel insert of Fig. 4.
In order to study the contribution of the f 2 resonance to the Legendre moments in more detail, the m ππ -dependence of the purely longitudinal combination P 1 + 7 3 · P 3 is presented in Fig. 5 for both hydrogen and deuterium. For comparison, this figure also shows the combination P 1 − 14 9 · P 3 which is believed to be dominated by the higher-twist transverse contribution to the excitation of the f 2 resonance. The comparison between these two distributions suggests that the higher-twist transverse contribution to the Legendre moments in the f 2 (1270) region is possibly as large as the longitudinal leading-twist production. The systematic uncertainty is given by the error band. Theoretical predictions (stars) from LPPSG [8] for hydrogen, which neglect two-gluon exchange mechanism, are compared with the data.
The x-dependence of P 1 is shown in Fig. 6 for both targets in two regions of m ππ : 0.30 < m ππ < 0.60 GeV and 0.60 < m ππ < 0.95 GeV. The statistical precision at larger values of m ππ is insufficient for such a presentation. The average values of Q 2 , −t, and x for both targets in these m ππ regions are reported in Table I. In both invariant mass regions and for both targets, P 1 is non-zero, which we interpret as originating from the interference of resonant ρ 0 P -wave with non-resonant S-wave π + π − production. The moment increases in magnitude with x, suggesting that the exchange of flavour non-singlet quark combinations (C = −1) becomes competitive with the dominant singlet exchange (C = +1). Predictions with only the quark exchange mechanism in the GPD framework [8] are compared with the data, and are found to be in fair agreement with them.
In summary, the Legendre moments P 1 (cos θ) and P 3 (cos θ) for exclusive electroproduction of π + π − pairs have been measured for the first time for hydrogen and deuterium targets. The data show signatures of the interference between the dominant isospin state I = 1 (P -wave) and I = 0 (S, D-wave) of these pion pairs. The interference of the ρ 0 amplitude with the non-resonant S-wave and resonant D-wave states appears to be larger than the interference with the resonant f 0 S-wave. In the f 2 region, the combinations P 1 + 7/3 · P 3 and P 1 − 14/9 · P 3 are sensitive to the longitudinal and the transverse states of a D-wave π + π − pair, respectively. Comparison of these combinations suggests that, at Q 2 = 3 GeV 2 , the higher-twist transverse contribution to the Legendre moments in the f 2 domain can be as large as the leading-twist longitudinal contribution.
These results constrain models for Generalized Parton Distributions, and may allow, by comparing the data with a larger statistical significance with the more accurate next-to-leading order predictions with and without the inclusion of the two-gluon mechanism, the separation of the contributions of two-gluon and qq exchange mechanisms, which are connected to the quark and gluon content of the nucleon.