Spin effects in hard exclusive electroproduction of mesons
aa r X i v : . [ h e p - ph ] O c t WU B 09-11October, 15 2009
SPIN EFFECTS IN HARD EXCLUSIVEELECTROPRODUCTION OF MESONS
P. KROLL
Fachbereich Physik, Universit¨at Wuppertal, D-42097 Wuppertal, GermanyandInstitut f¨ur Theoretische Physik, Universit¨at Regensburg,D-93040 Regensburg, GermanyEmail: [email protected]
Abstract
In this talk various spin effects in hard exclusive electroproduction ofmesons are briefly reviewed. The data are discussed in the light of recenttheoretical calculations within the frame work of the handbag approach.This talk has been presented at the Conference in Honor of Prof. AnatolyEfremov’s 75th Birthday held at Trento, July, 2009.
PACS Nos. 12.38.Bx, 13.60.Le, 13.88.+e
Electroproduction of mesons allows for the measurement of many spin effects.For instance, one may measure the dependence of the cross sections on the po-larization of the virtual photon by separation. Through the decay of vectormesons, e.g. ρ → π + π − , one can measure the spin density matrix elements(SDME) of the decaying meson which also provide a wealth of information onspin effects. Last not least one may work with longitudinally or transverselypolarized targets and/or longitudinally polarized beams and measure variousspin asymmetries. The investigation of spin-dependent observables allows for adeep insight in the underlying dynamics. Provided a sufficient number of themhas been measured the strength of the various contributing amplitudes and eventheir relative phases can be determined from the experimental data. Here, inthis article, it will be reported upon some spin effects and their dynamical in-terpretation in the frame work of the so-called handbag approach which offersa partonic description of meson electroproduction provided the virtuality of theexchanged photon, Q , is sufficiently large. The theoretical basis of the handbagapproach is the factorization of the process amplitudes into a hard partonic sub-process and in soft hadronic matrix elements, the so-called generalized partonistributions (GPDs) as well as wave functions for the produced mesons, seeFig. 1. In collinear approximation factorization has been shown to hold rigor-ously for hard exclusive meson electroproduction.[1, 2] It has been also shownby these authors that the transitions from a longitudinally polarized photon to alikewise polarized vector meson or a pseudoscalar one, γ ∗ L → V L ( P ), dominatesfor large Q . Other photon-meson transitions are suppressed by inverse powersof the hard scale.As mentioned spin effects in hard exclusive meson electroproduction will bebriefly reviewed and their implications on the handbag approach and above allfor the determination of the GPDs, discussed. In Sect. 2 evidences for con-tributions from transversely polarized photons in vector-meson production areintroduced. Next, in Sect. 3, the role of target spin asymmetries in meson elec-troproduction is examined and results for vector mesons shown. In Sect. 4 anestimated of the GPD E , needed for a calculation of the target spin asymme-tries for vector mesons, is presented. Sect. 5 is devoted to a discussion of thethe target spin asymmetries in pion electroproduction. Finally, in Sect. 6, asummary is given. In a number of experiments, e.g. Refs. [3, 4, 5], the ratio of the longitudinal andtransversal cross sections has been determined from the SDME r : R = σ L σ T = 1 ε r − r , (1)where ε is the ratio of the longitudinal and transversal photon fluxes. Withregard to the factorization properties of meson electroproduction one expects R ∝ Q . In Fig. 2 the HERA data for R are displayed. One observes that R is not at all large. At Q ≃ it is about 2, i.e. σ L ≃ σ T only. Forlarger Q the ratio seems to increase slowly. Evidently, there are substantialcontributions from γ ∗ T → V T transitions to vector-meson electroproduction.In a series of papers [6, 7, 8] a handbag approach has been advocated forin which the subprocess amplitudes are calculated within the modified per-turbative approach [9], and the GPDs are constructed from reggeized doubledistributions [10, 11]. In this approach the quark transverse momenta are re-tained in the subprocess and Sudakov suppressions are taken into account. Thepartons are still emitted and re-absorbed by the proton collinearly. For themeson wave functions Gaussians in the variable k ⊥ / ( τ (1 − τ )) are assumed withtransverse size parameters fitted to experiment [12]. The variable τ denotes thefraction of the meson’s momentum the quark entering the meson, carries. Itis to be emphasized that the γ ∗ T → V T transitions which are infrared divergentin collinear approximation, are regularized by the quark transverse momenta inthe modified perturbative approach. 2 (+) n ( − )+ − γ ∗ (+) ++ π + R ( r ) Q [GeV ] Figure 1: A typical lowest order Feynman graph for meson electroproduction.The signs indicate helicity labels for the contribution from transversity GPDsto the amplitude M − , ++ , see text.Figure 2: The ratio R for ρ production versus Q at W = 90 GeV. Datataken from H1 [5] (filled symbols) and ZEUS [4] (open symbols). The solid linerepresents the handbag results with the shaded bands indicating the theoreticaluncertainties [8].With this model the available data on cross sections and SDME for ρ and φ production have been fitted in the kinematical range Q > ∼ , W > ∼ ξ ≃ x Bj / < ∼ .
1) and for the squared invariantmomentum transfer − t ′ = − t + t < ∼ . where t is the value of t forforward scattering. Good agreement with experiment is found. As an examplethe results obtained in Ref. [6] for R ( ρ ) are displayed in Fig. 2 and compared toexperiment. The data are well described within that approach and the increaseof R according to Eq. (1) is clearly visible. Results of similar quality have beenobtained for φ production. The analysis carried through in [6, 7, 8] fix the GPD H for quarks and gluons. The other GPDs do practically not contribute to thecross sections and SDME at small skewness.In experiment, e.g. [3, 5], there have also been observed small but clearlynon-zero contributions from γ ∗ T → V L transitions for instance in the SDME r . Such transitions, which violate s -channel helicity conservation, are not yetunderstood in the handbag approach. They are suppressed by ∝ √− t/Q ascompared to the leading amplitude M , . For the even stronger suppressed γ ∗ L → V T and γ ∗ T → V T transitions there is no indication in experiment. The electroproduction cross sections measured with a transversely or longitu-dinally polarized target consist of many terms, each can be projected out bya sin ϕ or cos ϕ moment where ϕ is a linear combination of φ , the azimuthal3able 1: Features of the asymmetries for transversally and longitudinally polar-ized targets. The angle θ γ describes the rotation in the lepton plane from thedirection of the incoming lepton to the virtual photon one; it is very small.observable dominant amplitudes low t ′ interf. term behavior A sin( φ − φ s ) UT LL Im (cid:2) M ∗ − , M , (cid:3) ∝ √− t ′ A sin( φ s ) UT LT Im (cid:2) M ∗ − , ++ M , (cid:3) const. A sin(2 φ − φ s ) UT LT Im (cid:2) M ∗ ∓ , − + M ± , (cid:3) ∝ t ′ A sin( φ + φ s ) UT TT Im (cid:2) M ∗ − , ++ M , ++ (cid:3) ∝ √− t ′ A sin(2 φ + φ s ) UT TT ∝ sin θ γ ∝ t ′ A sin(3 φ − φ s ) UT TT Im (cid:2) M ∗ − , − + M , − + (cid:3) ∝ ( − t ′ ) (3 / A sin( φ ) UL LT Im (cid:2) M ∗ − , ++ M − , (cid:3) ∝ √− t ′ angle between the lepton and the hadron plane and φ s , the orientation of thetarget spin vector [13]. In Tab. 3 the features of some of these moments aredisplayed. As the dominant interference terms reveal the target asymmetriesprovide detailed information on the γ ∗ p → V B amplitudes and therefore on theunderlying dynamics that generates them.A number of these moments have been measured recently. A particularlystriking result is the sin φ s moment which has been measured by the HERMEScollaboration for π + electroproduction [14]. The data on this moment, shownin Fig. 3, exhibit a mild t -dependence and do not show any indication for aturnover towards zero for t ′ →
0. Inspection of Tab. 3 reveals that this be-havior of A sin φ s UT at small − t ′ requires a contribution from the interference termIm (cid:2) M ∗ − , ++ M , (cid:3) . Both the contributing amplitudes are helicity non-flipones and are therefore not forced to vanish in the forward direction by angularmomentum conservation. Thus, we see that also for pion electroproduction thereare strong contributions from γ ∗ T → π transitions. The underlying dynamicalmechanism for such transitions will be discussed in Sect. 5.For ρ production the sin ( φ − φ s ) moment has been measured by HER-MES [15] and COMPASS [16]; the latter data being still preliminary. TheHERMES data are shown in Fig. 4. In the handbag approach A sin ( φ − φ s ) UT canalso be expressed by an interference term between the convolutions of the GPDs H and E with hard scattering kernels A sin φ − φ s UT ∼ Im h E i ∗ h H i (2)instead of the helicity amplitudes. Given that H is known from the analysis ofthe ρ and φ cross sections and SDMEs, A UT provides information on E [8].Let us recapitulate what we know about the GPD E .4 .0 0.2 0.4 0.60.00.20.40.60.81.0 A U T s i n f S ( p + ) -t'[GeV2] A U T ( r ) -t' [GeV ] Figure 3: The sin φ s moment for a transversely polarized target at Q ≃ .
45 GeV and W = 3 .
99 GeV for π + production. The predictions from thehandbag approach of Ref. [18] are shown as a solid line. The dashed line isobtained disregarding the twist-3 contribution. Data are taken from [14].Figure 4: The asymmetry A sin ( φ − φ s ) UT for ρ production at W = 5 GeV and Q = 2 GeV . Data taken from Ref. [15]. The lines represent the resultspresented in Ref. [19]. For further notations see text and Ref. [19]. E In Ref. [17] the electromagnetic form factors of the proton and neutron havebeen utilized in order to determine the zero-skewness GPDs for valence quarksthrough the sum rules which for the case of the Pauli form factor, reads F p ( n )2 = Z dx h e u ( d ) E uv ( x, ξ = 0 , t ) + e d ( u ) E dv ( x, ξ = 0 , t ) i . (3)In order to determine the GPDs from the integral a parameterization of theGPD is required for which the ansatz E av ( x, , t ) = e av ( x ) exp h t ( α ′ v ln(1 /x ) + b ae ) i (4)is made in a small − t approximation [17]. The forward limit of E is parameter-ized analogously to that of the usual parton distributions: e av = N a x α v (0) (1 − x ) β av , (5)where α v (0) ( ≃ .
48) is the intercept of a standard Regge trajectory and α ′ v inEq. (4) its slope. The normalization N a is fixed from the moment κ a = Z dxE av ( x, ξ, t = 0) , (6)where κ a is the contribution of flavor- a quarks to the anomalous magnetic mo-ments of the proton and neutron ( κ u = 1 . κ d = − . β uv = 4 and β dv = 5 . β uv = 10 and β dv = 5. The analysis performed in [17] should be repeated sincenew form factor data are available from Jefferson Lab, e.g. G nE and G nM are nowmeasured up to Q = 3 . . , respectively [20, 21]. These new dataseem to favor β uv < β dv . The zero-skewness GPDs E v are used as input to adouble distribution from which the valence quark GPDs for non-zero skewnessare constructed [19].In Ref. [19], following Diehl and Kugler [22], E for gluons and sea quarks hasbeen estimated from positivity bounds and a sum rule for the second momentsof E which follows from a combination of Ji’s sum rule [23] and the momentumsum rule of deep inelastic lepton-nucleon scattering. It has turned out thatthe valence quark contribution to that sum rule is very small, in particular if β uv < β dv , with the consequence of an almost exact cancellation of the gluon andsea quark moments. The GPDs E g and E sea are parameterized analogously to E v , see Eqs. (4), (5). The normalization of E sea is fixed by assuming that anappropriate positivity bound [24, 25] is saturated while that of E g is determinedfrom the sum rule. Several variants of E have been exploited in Ref. [19] in acalculation of A sin ( φ − φ s ) UT within the handbag approach. The results for a fewvariants are compared to the HERMES data on ρ production [15] in Fig. 4.Agreement between theory and experiment is to be noted. Similar agreement isobtained for the preliminary COMPASS data [16]. Combining both the experi-ments a negative value of A sin ( φ − φ s ) UT for ρ production is favored in agreementwith the theoretical results obtained in [19], only the extreme variant β uv = 10and β dv = 5 (dashed-dotted line in Fig. 4) seems to be ruled out. In Ref. [19]predictions for ω , ρ + , K ∗ and φ productions are also given. Their comparisonwith forthcoming data from HERMES and COMPASS may provide valuablerestrictions on the GPD E .With E at hand one may exploit Ji’s sum rule for the parton angular mo-menta. At zero skewness the sum rule reads h J a i = 12 (cid:2) q a + e a (cid:3) , h J g i = 12 (cid:2) q g + e g (cid:3) . (7)From a variant with β uv = 4, β dv = 5 . E g and E sea (solid line inFig. 4) for instance one obtains h J u i = 0 . , h J d i = 0 . , h J s i = 0 . , h J g i = 0 . , (8)at the scale of 4 GeV . The angular momenta sum up to ≃ /
2, the spin ofthe proton. A very characteristic stable pattern is obtained in [19]: For allvariants investigated, J u and J g are large while the other two angular momentaare very small. The angular momenta of the valence quarks are h J uv i = 0 . h J dv i = − . Target spin asymmetries in π + production In Ref. [18] electroproduction of positively charged pions has been investigatedin the same handbag approach as applied to vector meson production [6, 7, 8]To the asymptotically leading amplitudes for longitudinally polarized photonsthe GPDs e H and e E contribute in the isovector combination e F (3) = e F uv − e F dv . (9)instead of H and E for vector mesons. In deviation to work performed incollinear approximation the full electromagnetic form factor of the pion as mea-sured by the F π − (see alsothe recent work by Bechler and Mueller, Ref. [28]). The GPDs e H and e E areagain constructed with the help of double distributions with the forward limitof e H being the polarized parton distributions while that of e E is parameterizedanalogously to Eq. (5) ˜ e u = − ˜ e d = e N e x − . (1 − x ) . (10)The normalization e N e is fitted to experiment.As is mentioned in Sect. 2 experiment requires a strong contribution fromthe helicity-non-flip amplitude M − , ++ which does not vanish in the forwarddirection. How can this amplitude be modeled in the frame work of the hand-bag approach? From the usual helicity non-flip GPDs H, E, . . . one obtains acontribution to M − , ++ that vanishes ∝ t ′ if it is non-zero at all. However,there is a second set of GPDs, the helicity-flip or transversity ones H T , E T , . . . [29, 30]. As inspection of Fig. 1 where the helicity configuration of the process isspecified, reveals the proton-parton vertex is of non-flip nature in this case and,hence, is not forced to vanish in the forward direction by angular momentumconservation. One also sees from Fig. 1, that the helicity configuration of thesubprocess is the same as for the full amplitude. Therefore, also the subprocessamplitude has not to vanish in the forward direction and so the full amplitude.The prize to pay is that quark and antiquark forming the pion have the same he-licity. Therefore, the twist-3 pion wave function is needed instead of the familiartwist-2 one. The dynamical mechanism building up the amplitude M − , ++ isso of twist-3 order consisting of leading-twist helicity-flip GPDs and the twist-3pion wave function. This mechanism has been first proposed in [31] for photo-and electroproduction of mesons where − t is considered as the large scale [32].In Ref. [18] the twist-3 pion wave function is taken from [33] with thethree-particle Fock component neglected. This wave function, still contain-ing a pseudoscalar and a tensor component, is proportional to the parameter µ π = m π / ( m u + m d ) ≃ m u and m d are current quark masses).It is further assumed that the dominant transversity GPD is H T while the otherthree can be neglected. The forward limit of H aT is the transversity distribution As compared to other work e E contains only the non-pole contribution. a ( x ) which has been determined in [34] in an analysis of data on the asym-metries in semi-inclusive electroproduction of charged pions measured with atransversely polarized target. Using these results for δ a ( x ) the GPDs H aT havebeen modeled in a manner analogous to that of the other GPDs ( see Eq. (4)) .It is shown in [18] that with the described model GPDs, the π + cross sec-tions as measured by HERMES [35] are nicely fitted as well as the transversetarget asymmetries [14]. This can be seen for A sin φ s UT from Fig. 2. Also thesin( φ − φ s ) moment which is dominantly fed by an interference term of the thetwo amplitudes for longitudinally polarized photons (see Tab. 3), is fairly welldescribed, as is obvious from Fig. 5. Very interesting is also the asymmetry for alongitudinally polarized target which is dominated by the interference term be-tween M − , ++ which comprises the twist-3 effect, and the nucleon helicity-flipamplitude for γ ∗ L → π transition, M − , . Results for A sin φUL are displayed inFig. 6 and compared to the data [36]. Also in this case good agreement betweentheory and experiment is to be seen. In both the cases, A sin φ s UT and A sin φUL theprominent role of the twist-3 mechanism is clearly visible. Switching it off oneobtains the dashed lines which are significantly at variance with experiment. Inthis case the transverse amplitudes are only fed by the pion-pole contribution.The other transverse target asymmetries quoted in Tab. 3 are predicted to besmall in absolute value which is in agreement with experiment [14]. Thus, insummary, there is strong evidence for transversity in hard exclusive pion electro-production. It should be considered as a non-trivial result that the transversitydistributions determined from data on inclusive processes lead to a transver-sity GPD which is nicely in agreement with target asymmetries measured inexclusive pion electroproduction.It is to be stressed that information on the amplitude M − , ++ can also ob-tained from the asymmetries measured with a longitudinally polarized beam orwith a longitudinally polarized beam and target. The first asymmetry, A sin φLU ,is dominated by the same interference term as A sin φUL but diluted by the factor p (1 − ε ) / (1 + ε ). Also the second asymmetry, A cos φLL , is dominated by the in-terference term M ∗ − , ++ M − , . However, in this case its real part occurs. ForHERMES kinematics it is predicted to be rather large and positive at small − t ′ and changes sign at − t ′ ≃ . [18]. A measurement of these asymmetrieswould constitute a serious check of the twist-3 effect.Although the main purpose of the work presented in [18] is focused on theanalysis of the HERMES data one may be also interested in comparing thisapproach with the Jefferson Lab data on the cross sections [27]. With the GPDs e H, e E and H T in their present form the agreement with these data is reasonablefor the transverse cross section while the longitudinal one is somewhat too small.It is however to be stressed that the approach advocated for in [6, 18, 19] isdesigned for small skewness. At larger values of it the parameterizations ofthe GPDs are perhaps to simple and may require improvements. It is also While the relative signs of δ u and δ d is fixed in the analysis performed by Ref. [34] theabsolute sign is not. Here, in π + electroproduction a positive δ u is required by the signs ofthe target asymmetries. .0 0.2 0.4 0.6-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 A U T s i n ( f - f S ) ( p + ) -t'[GeV2] A U L ( p + ) -t'[GeV2] Figure 5: Left: Predictions for the sin ( φ − φ s ) moment at Q = 2 .
45 GeV and W = 3 .
99 GeV shown as solid lines [18]. The dashed line represents thelongitudinal contribution to the sin ( φ − φ s ) moment. Data are taken from [14].Figure 6: Right: The asymmetry for a longitudinally polarized target at Q ≃ . and W ≃ . x less than about 0.6. One may therefore change tosome extent the GPDs for large x without changing the results for cross sectionsand asymmetries in the kinematical region of small skewness. For Jefferson Labkinematics, on the other hand, such changes of the GPDs may matter. Recent measurements of spin effects in hard meson electroproduction has beenreviewed. The spin effects include separated electroproduction cross sections,SDME and target as well as beam asymmetries. The data clearly show that aleading-twist calculation of meson electroproduction within the handbag ap-proach is insufficient. They demand higher-twist and/or power correctionswhich manifest themselves through substantial contributions from γ ∗ T to me-son transitions.A most striking effect is the target asymmetry A sin φ s UT in π + electroproduc-tion. The interpretation of this effect requires a large contribution from thehelicity non-flip amplitude M − , ++ . Within the handbag approach such a con-tribution is generated by the helicity-flip or transversity GPDs in combinationwith a twist-3 pion wave function [18]. This explanation establishes an in-teresting connection to transversity parton distributions measured in inclusiveprocesses. Further studies of transversity in exclusive reactions are certainlydemanded. For instance, data on the asymmetries obtained with a longitudi-nally polarized beam and with likewise polarized beam and target would be veryhelpful in settling this dynamical issue.9ood data on π electroproduction would also be highly welcome. Theywould not only allow for an additional test of the twist-3 mechanism but alsogive the opportunity to verify the model GPDs e H and e E as used in [18].One may wonder whether the twist-3 mechanism does not apply to vector-meson electroproduction as well and offers an explanation of the experimentallyobserved γ ∗ T → V L transitions mentioned in Sect. 2. It however turned outthat this effect is too small in comparison to the data, for instance, r . Thereason is that instead of the parameter µ π the mass of the vector meson setsthe scale of the twist-3 effect. This amounts to a reduction by about a factorof three. Further suppression comes from the unfavorable flavor combination of H T occurring for uncharged vector mesons, e.g. e u H uT − e d H dT for ρ productioninstead of H uT − H dT for π + production. Perhaps the gluonic GPD H gT may leadto a larger effect. Acknowledgements
This work is supported in part by the Heisenberg-Landau program and by the BMBF, contract number 06RY258.
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