Insights into the higher-twist distribution e(x) at CLAS
aa r X i v : . [ h e p - ph ] M a y Insights into the higher-twist distribution e ( x ) at CLAS A. Courtoy ∗ IFPA, AGO Department, Universit´e de Li`ege,Bt. B5, Sart Tilman B-4000 Li`ege, BelgiumINFN, Laboratori Nazionali di Frascati,Via E. Fermi, 40, I-00044 Frascati (Roma), Italy (Dated: May 7, 2018)
Abstract
We present the extraction of the twist-3 PDF, e ( x ), through the analysis of the preliminary datafor the sin φ -moment of the beam-spin asymmetry for di-hadron Semi-Inclusive DIS at CLAS at6 GeV. Pion-pair production off unpolarized target in the DIS regime provide an access to thehigher-twist Parton Distribution Functions e ( x ) and to Di-hadron Fragmentation Functions. Thelatter have been extracted from the semi-inclusive production of two hadron pairs in back-to-backjets in e + e − annihilation at Belle. The e ( x ) PDF offers important insights into the physics ofthe largely-unexplored quark-gluon correlations, and its x -integral is related to the marginally-known scalar-charge of the nucleon, and to the pion-nucleon σ -term, a fundamental property ofthe nucleon. PACS numbers: ∗ Electronic address: [email protected] . INTRODUCTION Hard processes are described in QCD by envisaging a perturbative stage (pQCD) wherea hard collision involving quark and gluons occurs, followed by a non-perturbative stagecharacterizing hadron structure. For example, in Deep Inelastic Scattering (DIS) the hardscattering part of the process, γ ∗ q → X , occurs at very short light cone distances, or forsmall configurations of quarks and gluons which can be presently described within pQCD.The large distance contribution is parameterized in terms of Parton Distribution Functions(PDF), which contain the structural information on the target. Formally, this factorizationcan be achieved in an Operator Product Expansion (OPE) style. In the collinear approach,the leading order of the non-perturbative contributions, called leading-twist (twist-2), iscomposed of three PDFs depending only on the fraction x of the longitudinal momentum ofthe target and on the photon virtuality. The subleading-twist is equivalently composed ofthree PDFs.The experimental determination of collinear structure of the proton is not complete. Atleading-twist, only the unpolarized PDF, f ( x ), is well known. The helicity distribution, g ( x ), is less constrained, while the transversity distribution h ( x ) is known to some extent.The subleading–twist picture consists in three collinear PDFs, e ( x ) , h L ( x ) and g T ( x ). Whilethese functions provide direct and unique insights into the dynamics inside hadrons [1],experimental information is still scarce [2]. In particular, the chiral-odd twist-3 PDF e ( x )encloses important knowledge on the largely unexplored quark-gluon correlations. In gen-eral, higher-twist PDFs describe multiparton distributions corresponding to the interferenceof higher Fock components in the hadron wave functions, and as such have no probabilisticpartonic interpretations. Yet they offer fascinating doorways to studying the structure ofthe nucleon. Higher-twist contributions are also indispensable to correctly extract twist-2components from data. Although suppressed with respect to twist-2 observables by 1 /Q ,twist-3 observables are not small in the kinematics of fixed target experiments. The CLASexperiment, installed in the Hall-B of the Jefferson Laboratory, represents the ideal envi-ronement to study O (1 /Q ) contributions, thanks to the low average photon virtuality Q explored in its experiments, and its capability to extract the observables of interest in a widekinematic range.The golden channel to access e ( x ) is Semi-Inclusive production of pion pairs in the DeepInelastic regime. In di-hadron SIDIS Single-Spin Asymmetries the PDF e ( x ) appears cou-pled to the chiral-odd Interference Fragmentation Function H ∢ [3], that, together with theunpolarized Di-hadron Fragmentation Function (DiFF) D , constitutes a crucial ingredi-ent to obtain information on PDFs. Such a process can be analyzed in the framework ofthe collinear factorization, making the di-hadron SIDIS a unique tool to study the higher-twist effects appearing as sin φ modulations in beam-spin dependent azimuthal moments ofthe SIDIS cross section. The Interference Fragmentation Function H ∢ has been recentlyextracted [4] from Belle data [5], providing an important ingredient toward the PDF extrac-tion. As to D , in the absence of data for di-hadron multiplicities related to the unpolarizedDiFF, it has been fitted to the output of PYTHIA [6] tuned for Belle kinematics. TheDiFF framework has proven its efficiency in the transverse target case, leading to the firstextraction of the collinear transversity PDF, h ( x ), for HERMES and COMPASS data [7, 8].In this paper we present an extraction of the higher-twist PDF e ( x ). An extended studyand review on the chiral-odd parton distribution has been published 10 years ago [9]. Afirst attempt to access the e ( x ) PDF was proposed in Ref. [10] through the analysis of the2ingle-hadron SIDIS Beam-Spin Asymmetry measured by the CLAS Collaboration [11],which involves TMD factorization and four terms in the structure function. Recent data forthe Beam Spin Asymmetry for single-pion semi-inclusive electro-production [12, 13] shouldbring more light on the TMD e ( x, k T ).The paper is organized as follows. In Section II, we describe the higher-twist physics,especially e ( x ). Section III is devoted to the framework for di-hadron Beam Spin Asymmetry(BSA). In Section IV, we present the analysis and extraction of e ( x ). The results arediscussed. We then conclude. II. PARTONIC QUANTITIESA. The chiral-odd twist-3 e ( x ) The chiral-odd e ( x ) twist-3 distribution is defined as e q ( x ) = 12 M Z dλ π e iλx h P | ¯ ψ q (0) ψ q ( λn ) | P i , (1)for quarks, and for antiquarks, e ¯ q ( x ) = e q ( − x ). n is a light-like vector ; Eq. (1) is expressedin the light-cone gauge, i.e. where the gauge link becomes unity.Twist-3 PDFs are suppressed in the OPE expansion by a factor M/P + w.r.t. the twist-2 PDFs. The origin of that suppression can be either kinematical, dynamical or due toquark mass terms. The separation between these three contributions comes from a QCDoperator identity for the non-local quark-quark operator, ¯ ψψ , [14–18]. Kinematical twist-3can be reduced to an expression containing only twist-2 PDFs via QCD equations of motion,it is the so-called Wandzura-Wilczek (WW) approximation [19]. The PDF e ( x ) vanishes inthis approximation. QCD equations of motion allow to decompose the chiral-odd twist-3distributions into 3 terms, e q ( x ) = e q loc ( x ) + e q gen ( x ) + e q mass ( x ) . (2)The first term comes from the local operator: e q loc ( x ) = 12 M Z dλ π e iλx h P | ¯ ψ q (0) ψ q (0) | P i = δ ( x )2 M h P | ¯ ψ q (0) ψ q (0) | P i ; (3)the second term is a dynamical or genuine twist-3 contribution, e.g. it is interaction depen-dent and contains explicit gluon fields; the last term is proportional to the quark mass andits Mellin moments are expressed as Z − dx x n − e q mass ( x ) = m q M Z − dx x n − f q ( x ) , (4)for n > n = 0. 3he QCD evolution of e ( x ) has been studied up to NLO [16–18]. Due to the chiral-oddnature of the current, there is no mixing with gluons. Evolution of twist-3 operators iscomplex but can be reduced to a DGLAP-like scheme in the large- N c limit.The PDF e ( x ) has been calculated in various models. We cite the chiral quark solitonmodel, e.g. [20, 21], the MIT bag model [15, 22], the spectator model [23, 24], the instantonQCD vacuum calculus and the perturbative light-cone Hamiltonian approach to O ( α s )with a quark target [25, 26]. In Ref. [21] the non-relativistic limit of e q ( x ) was studied. Acalculation in the light-front quark model is ongoing [27].The chiral-odd twist-3 PDF e ( x ) carries important hadronic information. It offers aunique road to the determination of the scalar charge, i.e. the first Mellin moment of e ( x ): Z − dx e q ( x, Q ) = Z − dx e q loc ( x, Q ) = 12 M h P | ¯ ψ q (0) ψ q (0) | P i ( Q ) = σ q ( Q ) . (5)The isoscalar combination of the scalar charge is related to the pion-nucleon σ -term σ u ( Q ) + σ d ( Q ) ≡ σ πN ( m u ( Q ) + m d ( Q )) / . (6)The pion-nucleon σ -term is normalization point invariant. It is related to the strangenesscontent of the proton. The σ -term represents the contribution from the finite quark massesto the mass of the nucleon [28]. The value σ πN = 79 ± σ -term [30]. General model-independent bounds on direct dark matter detection include allpossible effective operators, beyond the V − A electroweak structure [31]. A classificationof these operators and their implications include scalar form factors, that are related tothe scalar charges in the forward limit. Also, the isovector scalar charge is related to “newcurrents” in beta decays, in the sense that the leptonic current allows the weak V − A currentstructure in the Standard Model. New structures, such as scalar and tensor, would give hintof physics Beyond the Standard Model [32] if detected.The sum rule in Eq. (5) is not strickly speaking related to a charge, as that charge isnot scale invariant. Moreover the contribution to the charges comes only from the singular– local – part of the twist-3 PDF. While little can be told experimentally on the singularcontribution, it has been studied in various models. In chiral models, the presence of thissingular term in the distribution is inseparably connected with the nonzero value of quarkcondensate in the spontaneously-breaking QCD vacuum [9, 21, 33].The second moment of e q is proportional to the number of valence quarks of flavor q , Z − dx xe q ( x ) = Z dx x ( e q − e ¯ q )( x ) = m q ( Q ) M N q , (7)and it vanishes in the chiral limit.The third moment of the chirally odd twist-3 parton distribution involves the genuine part and can be related to the transverse force experienced by a transversely polarized quarkejected from a transversely polarized nucleon [34].4 . Dihadron Fragmentation Functions The twist-2 π + π − -DiFFs are the unpolarized D and the chiral-odd H ∢ . The latter is T -odd. The D q is the unpolarized DiFF describing the hadronization of a parton with flavor q into an unpolarized hadron pair plus anything else, averaging over the parton polarization.The H ∢ q is a chiral-odd DiFF describing the correlation between the transverse polariza-tion of the fragmenting parton with flavor q and the azimuthal orientation of the planecontaining the momenta of the detected hadron pair. In a Partial Wave Analysis (PWA),the physical interpretation of the dominant contribution to H ∢ is related to the interferencebetween relative p and s wave of the pion-pairs, while, for D , the pion-pairs are in relative s waves [35].DiFFs depend on the fraction of longitudinal momentum, z = z + z , of fragmentingquark carried by the pion-pair, on the ratio ζ = ( z − z ) /z —that can be expressed in termsof the polar angle θ , formed bewteen the direction of the back-to-back emission of the twohadrons in the center of mass frame and the direction of average momentum of the hadronpair in the target rest frame— and on the invariant mass of the pair, m ππ [35].DiFFs have been studied in models [36–38] and have been analyzed for π + π − productionfrom Belle data [4]. In particular, H ∢ was extracted from the Artru-Collins asymmetrymeasured at Belle, using D fitted from the output of the MonteCarlo event generatortuned for Belle [4]. A functional form at the hadronic scale Q = 1 GeV was found, fittingthe 100 GeV data. The range of validity of the DiFF fits reflects the kinematic range ofthe Belle data. In particular, the integrated range in invariant mass considered for thefit is limited to 2 m π ≤ m ππ ≤ .
29 GeV, the upper cut excluding scarcely populated orfrequently empty bins for the Artru-Collins asymmetry. This limit varies bin by bin andthe upper limit in m ππ can be as low as 0 . z = 0 . At twist-3, the number of DiFFs increase. In particular there are four genuine twist-3DiFFs, e D ∢ , e G ∢ , e E and e H [3]. The functions e D ∢ , e G ∢ are also Interference FragmentationFunctions, like H ∢ . The genuine twist-3 DiFFs describe the fragmentation of a quark, thepropagator of which is corrected by gluon fields up to order O (1 /Q ). They vanish in theWandzura-Wilzcek approximation. Up to date, there is no clear experimental informationabout higher-twist DiFFs. III. BEAM-SPIN ASYMMETRY IN SIDIS OFF PROTON TARGET
We consider the process ℓ ( l ) + N ( P ) → ℓ ( l ′ ) + h ( P ) + h ( P ) + X, (8)where ℓ denotes the beam lepton, N the nucleon target, h and h the produced hadrons,and where four-momenta are given in parentheses. We work in the one-photon exchangeapproximation and neglect the lepton mass. The momentum transferred to the nucleontarget is q = l − l ′ . The masses of the of final hadrons are m , m and their momentaare, respectively, P , P . The total momentum of the pair is P h = P + P ; the relativemomentum R = ( P − P ) / P h is R T ≡ R − ( R · ˆ P h ) ˆ P h . See Fig. 6 of Ref. [4] and Ref. [5]. P h = m hh . The SIDIS process is definedby the kinematic variables: x = Q P · q ≡ x B , y = P · qP · l , z = P · P h P · q = z + z . (9)The kinematics and the definition of the angles can be be found in, e.g. , Refs. [3, 8]. Wemention the azimuthal angle φ R formed between the leptonic plane and the hadronic planeidentified by the vector R T and the virtual photon direction. The cross section for twoparticle SIDIS can be written in terms of modulations in the azimuthal angle φ R [39].In the limit m hh ≪ Q the structure functions of interest can be written in terms of PDFsand DiFFs, to leading-order, in the following way [3]: F UU,T = X q e q xf q ( x ) D q (cid:0) z, cos θ, m hh (cid:1) , (10) F cos φ R UU = − X q e q x | R | sin θQ z f q ( x ) e D ∢ q (cid:0) z, cos θ, m hh (cid:1) , (11) F sin φ R LU = − X q e q x | R | sin θQ (cid:20) Mm hh x e q ( x ) H ∢ q (cid:0) z, cos θ, m hh (cid:1) + 1 z f q ( x ) e G ∢ q (cid:0) z, cos θ, m hh (cid:1)(cid:21) , (12) F LL = X q e q xg q ( x ) D q (cid:0) z, cos θ, m hh (cid:1) , (13) F cos φ R LL = − X q e q x | R | sin θQ z g q ( x ) e D ∢ q (cid:0) z, cos θ, m hh (cid:1) , (14)with the first subindex of the structure function corresponding to the beam polarization,the second to the target. We now consider the structure function F sin φLU in Eq. (12) for π + π − pair production. The relevant spin asymmetry can be built as ratios of structure functions.For the longitudinal polarization of the beam, i.e. the LU combinations, one can define thefollowing BSA: A sin φ R LU ( z, m ππ , x ; Q, y ) = π p ε (1 − ε ) R d cos θ F sin φ R LU R d cos θ ( F UU,T + ǫF UU,L ) , (15)where ε is the ratio of longitudinal and transverse photon flux and can be expressed in termsof y . Combining Eqs. (10,12), to leading-order in α s and leading term in the PWA, the BSAbecomes A sin φ R LU ( x, z, m ππ ; Q, y )= − W ( y ) A ( y ) MQ | R | m ππ P q e q h xe q ( x, Q ) H ∢ ,q ,sp ( z, m ππ , Q ) + m ππ zM f q ( x, Q ) ˜ G ∢ ,qsp ( z, m ππ , Q ) iP q e q f q ( x, Q ) D q ,ss + pp ( z, m ππ , Q ) , (16)6he dependence in ( z, m ππ ) is factorized in the DiFFs and kinematical factors, leaving thedependence in x for the PDFs. The twist-2 functions are f ( x ) , H ∢ ( z, m ππ ) and D ( z, m ππ ),while the twist-3 functions are e ( x ) and ˜ G ∢ ( z, m ππ ). IV. RESULT: EXTRACTION OF THE TWIST-3 PDF e ( x ) The longitudinal Beam-Spin Asymmetry A sin φ R LU ( z, m ππ , x ; Q, y ) in Eq. (15) has beenrecently extracted by the CLAS Collaboration on data collected by impinging the CEBAF5 . H hydrogen target [40].In Fig. 2 the measured asymmetry is shown in two sets of 1D bins [40], representingrespectively the z and m ππ dependence of the BSA. In Fig. 1 the x -dependence of the BSAshows the data points used in the present extraction. The two plots of Fig. 2 are used tocheck the validity of the framework and its assumptions.The twist-3 chiral-odd PDF e ( x ) is accessed through the x -dependent 1D projection ofthe BSA. The variables ( z, m ππ ), proper to the DiFFs, do not enter in convolutions, so that,following the notation of Ref. [8], we can define the following quantities: n q, i ( Q i ) = Z z max , i z min , i dz Z ( m ππ, max ) i ( m ππ, min ) i dm ππ D q ( z, m ππ ; Q i ) , (17) n ↑ q, i ( Q i ) = Z z max , i z min , i dz Z ( m ππ, max ) i ( m ππ, min ) i dm ππ | R | m ππ H ∢ q ( z, m ππ ; Q i ) , (18) n ˜ G ∢ q ( Q i ) = Z z max , i z min , i dz Z ( m ππ, max ) i ( m ππ, min ) i dm ππ | R | M ˜ G ∢ q ( z, m ππ ; Q i ) , (19)where the index i refers to the bin number and its respective integration limits. Theseintegrated DiFFs need to be evaluated in the kinematical range of the experiment, whichvalues are given in Tab. I. The 1D projection of the BSA in Eq. (16) can be rewritten as: A sin φ R LU ( x i , m ππ i, , z i ; Q i , y i )= − W ( y i ) A ( y i ) MQ i P q e q h x i e q ( x i , Q i ) n ↑ q, i ( Q i ) + z i f q ( x i , Q i ) n ˜ G ∢ q, i ( Q i ) iP q e q f q ( x i , Q i ) n q, i ( Q i ) . (20)The variables with subindex i refer to average values for the bin i . The twist-2 DiFFs are evaluated using the Pavia fit [4] and the unpolarized PDFs, theMSTW08LO set [41]. The Q evolution of the twist-2 DiFFs has been properly includedby using standard evolution equations in a collinear framework [42] and by implementingleading-order (LO) chiral-odd splitting functions in the HOPPET code [43]. The evolutionis performed from Q = 1 GeV to the average scale of the bin Q i . QCD evolution of the From now on, we will drop the indices refering to the partial waves. The integrated average differs from the bin average value only for bumpy distributions, e.g. the m ππ behavior. We have checked that this difference was negligible for the unpolarized PDFs. In this paper, evolution effects of the twist-3 DiFFsare assumed to be at most of the order of magnitude of the running of n ↑ u ( Q ).We use some approximations to further develop the sum over flavors —we allow ourselvesto 1 −
2% relative error on the DiFF contributions, which is negligible w.r.t. the experimentalerror bars. The approximations are: • The charm contribution to f q = c ( x ) is negligible w.r.t q = u, d, s at JLab scales. • Invoking charge conjugation yields to D u → π + π − = D ¯ u → π + π − ,D d → π + π − = D ¯ d → π + π − ; (21)together with isospin symmetry: H ∢ u → π + π − = − H ∢ d → π + π − = − H ∢ ¯ u → π + π − = H ∢ ¯ d → π + π − . (22) • The Interference FF for strange and charm is zero as there is no interference from seaquarks [36]. For ˜ G ∢ we expect the same relations as for H ∢ .The BSA in Eq. (20) becomes A sin φ R LU ( x i , m ππ i, , z i ; Q i , y i ) = − W ( y i ) A ( y i ) MQ i × x i (cid:2) e u V ( x i , Q i ) − e d V ( x i , Q i ) (cid:3) n ↑ u,i ( Q i ) + h f u V ( x i , Q i ) − f d V ( x i , Q i ) i /z i n ˜ G ∢ u,i ( Q i ) P q = u,d,s e q f q ( x i , Q i ) n q, i ( Q i ) . = − W ( y i ) A ( y i ) MQ i x i (cid:2) e V ( x i , Q i ) (cid:3) n ↑ u, i ( Q i ) + (cid:2) f V ( x i , Q i ) (cid:3) /z i n ˜ G ∢ u, i ( Q i ) P q = u,d,s e q f q ( x i , Q i ) n q, i ( Q i ) , (23)where f q V ≡ f q − f ¯ q . In Eq. (23), we have defined the combinations e V and f V . Theremaining unknown are then the twist-3 functions, e ( x ) and ˜ G ∢ ( z, m ππ ). While the twist-2DiFFs are known, there is so far no study of the twist-3 DiFFs. A further assumption needsto be taken. In order to gain some insights onto the behavior of the genuine twist-3 DiFFs,we will define two extreme scenarios. A. The Wandzura-Wilzcek scenario
In the Wandzura-Wilzcek approximation, the genuine twist-3 DiFFs vanish. This ap-proximation is inspired by the preliminary data on Double Spin Asymmetry (DSA) fromCLAS, incorporating the structure functions Eqs. (11, 14). These structure functions areexpressed in terms of twist-2 PDFs and the twist-3 DiFF e D ∢ , i.e. a genuine twist-3 DiFF Studies of the evolution of twist-3 fragmentation functions in the multicolor limit show that there is noreciprocity in the anomalous dimensions for e ( x ) and the twist-3 FF G tw-3 ⊥ [44]. à à A LU H x L à à à - H x L FIG. 1: On the left panel, the x -dependent projection of the preliminary BSA used to extract e ( x ). On the right panel, the extraction of the combination e V ≡ e u V ( x i , Q i ) / − e d V ( x i , Q i ) / that we expect to be bigger than e G ∢ . These results indicate that the cos φ modulation ofthe Double Spin Asymmetry (DSA) is very small w.r.t. the constant term Eq. (13) [45].In this scenario, the BSA Eqs. (23) is straightfowardly inverted to get x i e V WW ( x i , Q i ) = − A ( y i ) W ( y i ) Q i M A sin φ R LU (cid:0) x i , m ππ, i , z i ; Q i , y i (cid:1) ×
19 4 x i f u +¯ u ( x i , Q i ) n u,i ( Q i ) + x i f d + ¯ d ( x i , Q i ) n d,i ( Q i ) + x i f s ( x i , Q i ) n s,i ( Q i ) n ↑ u,i ( Q i ) . (24)The results are given in Tab. I and shown in Fig. 1. Notice that the range of integration in m ππ goes beyond the range of known validity of the DiFF data set, i.e. the Belle data with2 m π < m ππ < .
29 GeV. The error ∆ (cid:0) e V ( x ) (cid:1) reflects the propagation of the experimen-tal –statistical and systematical– error from Ref. [40] and the error on H ∢ taken from Ref. [4].To check the presence of a possible twist-3 DiFF contribution, we have tried to reproducethe ( z, m ππ )-dependences with the DiFF fits of Ref. [4]. In the approximation of D u = D d and neglecting the strange quark contributions, we can write each projection as: A sin φ R LU, fit ( x i , m ππ,i , z i ; Q i , y i ) = − W ( y i ) A ( y i ) MQ i n x R z max , i z min , i dz R ( m ππ, max ) i ( m ππ, min ) i | R | m ππ H ∢ ,u ( z, m ππ , Q i ) R z max , i z min , i dz R ( m ππ, max ) i ( m ππ, min ) i D u ( z, m ππ , Q i ) , (25)where the respective values of bins for the z projections and the m ππ ’s are given in Table. II.Within that approximation, the x -dependence is then only a scaling factor, n x = R x max x min dx e V ( x, Q ) P q e q R x max x min dxf q +¯ q ( x, Q ) , (26)that in principle depends on Q i and on the interval [ x min , i , x max , i ]. This number is not known,but is related to the scale of the 1D projections. We show the result on Fig. 2 for n x = 0 . w.r.t. the data.9 à à A LU H z L à à à m þþ @ GeV D A LU H m þþ L FIG. 2: 1D projections of the BSA, repectively, in z and m ππ . The red squares are the preliminarydata of Ref. [40]. For the z and the m ππ projections, the blue circles represent the estimate forBSA Eq. (25) from the DiFF fit [4] for an integrated x -dependence, n x = 0 . x h Q i [GeV ] z m ππ [GeV] n ↑ u /n u ( Q ) e V WW ( x, Q ) e V lead. ( x, Q ) ∆ (cid:0) e V ( x ) (cid:1) . , .
95] [0 . , .
66] 0.199 2.611 -0.263 0.5782 0.244 1.60 [0 . , .
95] [0 . , .
50] 0.201 0.687 -0.850 0.2383 0.356 2.27 [0 . , .
92] [0 . , .
38] 0.203 0.271 -0.243 0.091TABLE I: The ratio of the integrated DiFFs and the corresponding value for the flavor combination e V ( x ) = (4 e u V ( x ) − e d V ( x )) /
9. Note that the effect of evolution is of 1-2% at most when consideringthe ratio n ↑ u /n u as the values of Q i are low. The predictions from the DiFFs fits are compatible, within error bars, with the prelimi-nary data. The descrepancy observed in the low m ππ behavior could be due to the limitedrange of validity of the DiFF fits. However, since the behavior of e G ∢ is not known, no realconclusion can be driven. B. Beyond the WW scenario
For completeness, we also consider the case in which the twist-3 DiFF e G ∢ is non-zero. Weconsider that it is of the order of magnitude of e D ∢ . As mentionned above, the preliminaryresults of CLAS [45] indicate that the cos φ modulation of the DSA is small w.r.t leading-twist contributions.The crucial observation is that the order of magnitude of e D ∢ necessary to reproducean integrated DSA is of a few percent of the integrated H ∢ , Eq. (18). Within the presentassumption, it translates in: n e G ∢ u ( Q i ) assump. ≡ n e D ∢ u ( Q i ) ∼ = κ n ↑ u ( Q i ) , (27) n e D ∢ u ( Q i ), integrated as Eqs. (17,18). -bin z m ππ [GeV] h Q i [GeV ] m ππ -bin z m ππ [GeV] h Q i [GeV ]1 [0 . , .
45] [0 . , . ] 1.74 1 [0 . , .
94] [0 . , .
46] 1.812 [0 . , .
65] [0 . , .
36] 1.78 2 [0 . , .
94] [0 . , .
66] 1.773 [0 . , .
95] [0 . , .
66] 1.74 3 [0 . , .
95] [0 . , .
66] 1.69TABLE II: Binning, respectively, for the z
1D projection and for the m ππ
1D projection. with κ ∼ . A sin φ R LU, leading ( x i , m ππ, i , z i ; Q i , y i ) = − W ( y i ) A ( y i ) MQ i (cid:2) x i e V ( x i , Q i ) + κf V ( x i , Q i ) /z i (cid:3) n ↑ u, i ( Q i ) P q = u,d,s e q f q ( x i , Q i ) n q, i ( Q i ) . (28)Since the ( z, m ππ )-dependence is integrated, a non-zero twist-3 PDF becomes manifest indeviations from the trend in x given by the unpolarized PDF contribution: A sin φ R LU ( x i ; Q i ) ∝ (cid:16) f u V − f d V (cid:17) ( x i , Q i ) (cid:16) f u +¯ u + f d + ¯ d (cid:17) ( x i , Q i ) , (29)the trend and size of which can be estimated, e.g. with the MSTW08LO set. Therefore,going beyond the WW approximation, the BSA is straightfowardly inverted to get x i e V lead. ( x i , Q i ) = − A ( y i ) W ( y i ) Q i M A sin φ R LU (cid:0) x i , m ππ, i , z i ; Q i , y i (cid:1)
19 4 x i f u +¯ u ( x i , Q i ) n u,i ( Q i ) + x i f d + ¯ d ( x i , Q i ) n d,i ( Q i ) + x i f s ( x i , Q i ) n s,i ( Q i ) n ↑ u,i ( Q i ) − κ x i z i f V ( x i , Q i ) , (30)with κ = 0 .
2. In other words, the x -dependence coming from f changes the extracted resultsof Fig. 1. The results are given in Tab. I for both scenarios and illustrated on Fig. 3.Wouldthe BSA only contain a contribution from f e G ∢ , the twist-3 PDF e ( x ), as extracted in thisscenario, would be a constant. Our result shows that it is not the case. Hence, the behaviorin x of the BSA cannot be reproduced by the unpolarized PDF, Eq. (29). We interpret thisresult as the first evidence for a non-zero e ( x ) in the range x ∈ [0 . , . i.e. varying κ = 0 to κ = 0 .
2. These two extreme scenarios set theconstraints on the twist-3 PDF e ( x ).Thought the evolution of the twist-3 PDF is usually not applied in models, in Fig. 4we propose an interpretive comparison with three –standard– model predictions, e.g. theMIT bag model [15], the spectator model [23] and the chiral quark soliton model [20].The hadronic scale of models symbolize the scale at which the model mimicks QCD for a A constant which value would be related to the uncertainty on κ . à àò ò ò - e V H x L FIG. 3: Extractions on the combination e V ≡ e u V ( x i , Q i ) / − e d V ( x i , Q i ) /
9. The red squares cor-respond to the
WW scenario , the blue triangles to the leading scenario . The error bars correspondto the propagation of the experimental and DiFF errors. à à àò ò ò à à àò ò òæ æ æ à à àò ò ò - e V H x L FIG. 4: Model predictions for combination e V ≡ e u V ( x i , Q i ) / − e d V ( x i , Q i ) / Q , compared to the error bands for the two scenarios (light blue). The blue triangles correspondto the bag model [15, 22], the red squares to the spectator model [23] and the green circles thechiral quark soliton model [20]. given partonic representation, e.g. its partonic content. To purely valence quark modelscorrespond a scale Q ∼ . for Λ LO = 0 .
27 GeV [46, 47]. The author of [48] refers toa scale Q = 0 . . 12 . CONCLUSIONS We have presented the extraction of chiral-odd PDF e ( x ) using the preliminary data forthe Beam Spin Asymmetry in di-hadron SIDIS off proton target at CLAS. The asymmetryconsists in 2 terms: the first involves the twist-3 PDF of interest multiplied by a twist-2DiFF and the second involves the usual unpolarized PDF multiplied by an unknown twist-3DiFF. We have considered two extreme scenarios: a scheme where twist-3 DiFFs do notcontribute and a second scenario in which the twist-3 DiFF is non-zero. While there arestill non-negligible theoretical as well as experimental uncertainties, we show that the trendin Bjorken- x cannot be reproduced by the PDF f ( x ). It is an experimental evidence for anon-vanishing e ( x ).We have studied the BSA dependence in the DiFF variables and show that the trendin z is compatible, within error bars, with the data. The behavior in m ππ is not clear.Detailed studies of DiFF should be done in the future to enlarge the range of validity forthe fits, improve the low energy functional form through a better Q spanning of the data.Meanwhile, we cannot conclude on the presence of the twist-3 DiFF.This manuscript has been prepared in collaboration with H. Avakian, M. Mirazitaand S. Pisano. We are grateful to A. Bacchetta, D. Hasch, M. Radici, P. Schweitzerand M. Wakamatsu for useful discussions. This work was funded by the Belgian FundF.R.S.-FNRS via the contract of Charge de recherches (A.C.). [1] R. L. Jaffe, Comments Nucl. Part. Phys. , 239 (1990).[2] A. Airapetian et al. (The HERMES Collaboration), Eur.Phys.J. C72 , 1921 (2012), 1112.5584.[3] A. Bacchetta and M. Radici, Phys.Rev.
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