Threshold resummation for polarized (semi-)inclusive deep inelastic scattering
TThreshold resummation for polarized (semi-)inclusive deep inelastic scattering
Daniele P. Anderle, Felix Ringer, Werner Vogelsang
Institute for Theoretical Physics, T¨ubingen University,Auf der Morgenstelle 14, 72076 T¨ubingen, Germany (Dated: November 11, 2018)We explore the effects of the resummation of large logarithmic perturbative corrections to double-longitudinal spin asymmetries for inclusive and semi-inclusive deep inelastic scattering in fixed-targetexperiments. We find that the asymmetries are overall rather robust with respect to the inclusionof the resummed higher-order terms. Significant effects are observed at fairly high values of x ,where resummation tends to decrease the spin asymmetries. This effect turns out to be morepronounced for semi-inclusive scattering. We also investigate the potential impact of resummationon the extraction of polarized valence quark distributions in dedicated high- x experiments. PACS numbers: 12.38.Bx, 13.85.Ni, 13.88.+e
I. INTRODUCTION
Longitudinal double-spin asymmetries in inclusive andsemi-inclusive deep inelastic scattering have been primesources of information on the nucleon’s spin structure forseveral decades. They may be used to extract the helicityparton distributions of the nucleon,∆ f ( x, Q ) ≡ f + ( x, Q ) − f − ( x, Q ) , (1)where f + and f − are the distributions of parton f = q, ¯ q, g with positive and negative helicity, respectively,when the parent nucleon has positive helicity. x denotesthe momentum fraction of the parton and Q the hardscale at which the distribution is probed. Inclusive po-larized deep inelastic scattering (DIS), (cid:126)(cid:96)(cid:126)p → (cid:96)X , offersaccess to the combined quark and antiquark distributionsfor a given flavor, ∆ q + ∆¯ q , whereas in semi-inclusivedeep inelastic scattering (SIDIS), (cid:126)(cid:96)(cid:126)p → (cid:96)hX , one ex-ploits the fact that a produced hadron h (like a π + )may for instance have a quark of a certain flavor as avalence quark, but not the corresponding antiquark [1].In this way, it becomes possible to separate quark andantiquark distributions in the nucleon from one another,as well as to better determine the distributions for thevarious flavors. HERMES [2] and recent COMPASS [3]measurements have marked significant progress concern-ing the accuracy and kinematic coverage of polarizedSIDIS measurements. The inclusive measurements haveimproved vastly as well [4–8]. Some modern analyses ofspin-dependent parton distributions include both inclu-sive and semi-inclusive data [9–11]. In addition, high-precision data for polarized SIDIS will become availablefrom experiments to be carried out at the Jefferson Labafter the CEBAF upgrade to a 12 GeV beam [12]. Herethe focus will be on the large- x regime.A good understanding of the theoretical framework forthe description of spin asymmetries in lepton scatteringis vital for a reliable extraction of polarized parton dis-tributions. In a recent paper [13] we have investigatedthe effects of QCD threshold resummation on hadronmultiplicities in SIDIS in the HERMES and COMPASS kinematic regimes. SIDIS is characterized by two scalingvariables, Bjorken- x and a variable z given by the energyof the produced hadron over the energy of the virtualphoton in the target rest frame. Large logarithmic cor-rections to the SIDIS cross section arise when the cor-responding partonic variables become large, correspond-ing to scattering near a phase space boundary, wherereal-gluon emission is suppressed. This is typically thecase for the presently relevant fixed-target kinematics.Threshold resummation addresses these logarithms to allorders in the strong coupling. In [13] we found fairly sig-nificant resummation effects on the spin-averaged multi-plicities. Since the spin-dependent cross section is sub-ject to similar logarithmic corrections as the unpolarizedone, it is worthwhile to explore the effects of resumma-tion on the spin asymmetries. This is the goal of thepresent paper. Our calculations will be carried out bothfor inclusive DIS and for SIDIS. We note that previouswork [14, 15] has addressed the large- x resummation forthe inclusive spin-dependent structure function g , witha focus on the moments of g and their Q -dependence.In this paper we are primarily concerned with spin asym-metries and with semi-inclusive scattering.Our work will use the framework developed in [13]. InSection II, we briefly review the basic terms and defi-nitions relevant for longitudinal spin asymmetries, andwe describe the extension of threshold resummation tothe polarized case. In Section III our phenomenologicalresults are presented. We compare our resummed inclu-sive and semi-inclusive spin asymmetries with availableHERMES, COMPASS and Jefferson Lab data. We alsodiscuss the relevance of resummation for the extractionof ∆ u/u and ∆ d/d at large values of x . II. RESUMMATION FOR LONGITUDINALSPIN ASYMMETRIES IN DIS AND SIDISA. Leading and next-to-leading order expressions
We first consider the polarized SIDIS process (cid:126)(cid:96) ( k ) (cid:126)p ( P ) → (cid:96) ( k (cid:48) ) h ( P h ) X with longitudinally polarized a r X i v : . [ h e p - ph ] A p r beam and target and with an unpolarized hadron in thefinal state. The corresponding double-spin asymmetry isgiven by a ratio of structure functions [2]: A h ( x, z, Q ) ≈ g h ( x, z, Q ) F h ( x, z, Q ) , (2)where Q = − q with q the momentum of the virtualphoton, x = Q / (2 P · q ) is the usual Bjorken variable,and z ≡ P · P h /P · q the corresponding hadronic scalingvariable associated with the fragmentation process.Using factorization, the polarized structure function g h , which appears in the numerator of Eq. (2), can bewritten as2 g h ( x, z, Q ) = (cid:88) f,f (cid:48) = q, ¯ q,g (cid:90) x d ˆ x ˆ x (cid:90) z d ˆ z ˆ z ∆ f (cid:16) x ˆ x , µ (cid:17) × D hf (cid:48) (cid:16) z ˆ z , µ (cid:17) ∆ C f (cid:48) f (cid:18) ˆ x, ˆ z, Q µ , α s ( µ ) (cid:19) , (3)where ∆ f ( ξ, µ ) denotes the polarized distribution func-tion for parton f of Eq. (1), whereas D hf (cid:48) (cid:0) ζ, µ (cid:1) is thecorresponding fragmentation function for parton f (cid:48) go-ing to the observed hadron h . The ∆ C f (cid:48) f are spin-dependent coefficient functions. We have set all factor-ization and renormalization scales equal and collectivelydenoted them by µ . In (3) ˆ x and ˆ z are the partoniccounterparts of the hadronic variables x and z . Settingfor simplicity µ = Q , we use the short-hand-notation2 g h ( x, z, Q ) ≡ (cid:88) f,f (cid:48) = q, ¯ q,g (cid:2) ∆ f ⊗ ∆ C f (cid:48) f ⊗ D hf (cid:48) (cid:3) ( x, z, Q )(4)for the convolutions in (3). A corresponding expressionfor the “transverse” unpolarized structure function 2 F h can be written by replacing the polarized parton distri-butions with the unpolarized ones, and using unpolarizedcoefficient functions which we denote here by C f (cid:48) f .The spin-dependent hard-scattering coefficient func-tions ∆ C f (cid:48) f in (3) can be computed in perturbation the-ory: ∆ C f (cid:48) f = ∆ C (0) f (cid:48) f + α s ( µ )2 π ∆ C (1) f (cid:48) f + O ( α s ) . (5)At leading order (LO), we have∆ C qq (ˆ x, ˆ z ) = ∆ C ¯ q ¯ q (ˆ x, ˆ z ) = e q δ (1 − ˆ x ) δ (1 − ˆ z ) , (6)with the quark’s fractional charge e q . All other coefficientfunctions vanish. The same result holds for the LO coef-ficient function for the spin-averaged structure function2 F h . Hence the asymmetry in Eq. (2) reduces to A h = (cid:80) q e q (cid:2) ∆ q ( x, Q ) D hq ( z, Q ) + ∆¯ q ( x, Q ) D h ¯ q ( z, Q ) (cid:3)(cid:80) q e q (cid:2) q ( x, Q ) D hq ( z, Q ) + ¯ q ( x, Q ) D h ¯ q ( z, Q ) (cid:3) . (7) At next-to-leading order (NLO), Eq. (3) becomes2 g h ( x, z, Q ) = (cid:88) q e q (cid:26) ∆ q ( x, Q ) D hq ( z, Q ) + ¯ q ( x, Q ) D h ¯ q ( z, Q )+ α s ( Q )2 π (cid:104)(cid:0) ∆ q ⊗ D hq + ∆¯ q ⊗ D h ¯ q (cid:1) ⊗ ∆ C (1) qq + (∆ q + ∆¯ q ) ⊗ ∆ C (1) gq ⊗ D hg +∆ g ⊗ ∆ C (1) qg ⊗ ( D hq + D h ¯ q ) (cid:105) ( x, z, Q ) (cid:27) , (8)where the symbol ⊗ denotes the convolution definedin Eqs. (3),(4). The explicit expressions for the spin-dependent NLO coefficients ∆ C (1) f (cid:48) f have been derivedin [16, 17]. The corresponding spin-averaged NLO co-efficient functions C (1) f (cid:48) f may be found in [13, 16–21].In the case of inclusive polarized DIS, the longitudinalspin asymmetry A is given in analogy with (2) by A ( x, Q ) ≈ g ( x, Q ) F ( x, Q ) . (9)The inclusive structure functions g and F have expres-sions analogous to their SIDIS counterparts, except forthe fact that they do not contain any fragmentation func-tions, of course. The unpolarized and polarized NLOcoefficient functions for inclusive DIS may be found atmany places; see, for example [20, 22]. B. Threshold resummation
As was discussed in [13], the higher-order terms inthe spin-averaged SIDIS coefficient function C qq intro-duce large terms near the “partonic threshold” ˆ x → z →
1. The same is true for the spin-dependent ∆ C qq .At NLO, choosing again for simplicity the scale µ = Q ,one has∆ C (1) qq (ˆ x, ˆ z ) ∼ e q C F (cid:34) + 2 δ (1 − ˆ x ) (cid:18) ln(1 − ˆ z )1 − ˆ z (cid:19) + + 2 δ (1 − ˆ z ) (cid:18) ln(1 − ˆ x )1 − ˆ x (cid:19) + + 2(1 − ˆ x ) + (1 − ˆ z ) + − δ (1 − ˆ x ) δ (1 − ˆ z ) (cid:35) , (10)where the “+”-distribution is defined as usual. Theexpression on the right-hand side is in fact identicalto the one for the unpolarized coefficient function nearthreshold [13]. At the k th order of perturbation the-ory, the coefficient function contains terms of the form α ks δ (1 − ˆ x ) (cid:16) ln k − (1 − ˆ z )1 − ˆ z (cid:17) + , α ks δ (1 − ˆ z ) (cid:16) ln k − (1 − ˆ x )1 − ˆ x (cid:17) + , or“mixed” distributions α ks (cid:16) ln m (1 − ˆ x )1 − ˆ x (cid:17) + (cid:16) ln n (1 − ˆ z )1 − ˆ z (cid:17) + with m + n = 2 k −
2, plus terms less singular by one ormore logarithms. Again, each of these terms will appearequally in the unpolarized and in the polarized coeffi-cient function. The reason for this is that the terms areassociated with emission of soft gluons [13], which doesnot care about spin. Threshold resummation addressesthe large logarithmic terms to all orders in the strongcoupling. The resummation for the case of SIDIS wascarried out in [13]. Given these results and the equal-ity of the spin-averaged and spin-dependent coefficientfunctions near threshold, it is relatively straightforwardto perform the resummation for the polarized case. Hav-ing the resummation for both g h and F h , we obtain re-summed predictions for the experimentally relevant spinasymmetry A h .In [13, 23, 24] threshold resummation for SIDIS wasderived using an eikonal approach, for which exponen-tiation of the threshold logarithms is achieved in Mellinspace. One takes Mellin moments of g h separately in thetwo independent variables x and z [18, 25]:˜ g h ( N, M, Q ) ≡ (cid:90) dxx N − (cid:90) dzz M − g h ( x, z, Q ) . (11)With this definition, Eq. (4) takes the form (again atscale µ = Q )2˜ g h ( N, M, Q ) = (cid:88) f,f (cid:48) = q, ¯ q,g ∆ ˜ f N ( Q ) × ∆ ˜ C f (cid:48) f ( N, M, α s ( Q )) ˜ D h,Mf (cid:48) ( Q ) , (12)where the moments of the polarized parton distributionsand the fragmentation functions are defined as∆ ˜ f N ( Q ) ≡ (cid:90) dxx N − ∆ f ( x, Q ) , ˜ D h,Mf (cid:48) ( Q ) ≡ (cid:90) dzz M − D hf (cid:48) ( z, Q ) , (13)and the double Mellin moments of the polarized coeffi-cient functions are∆ ˜ C f (cid:48) f (cid:0) N, M, α s ( Q ) (cid:1) ≡ (cid:90) d ˆ x ˆ x N − (cid:90) d ˆ z ˆ z M − × ∆ C f (cid:48) f (cid:0) ˆ x, ˆ z, , α s ( Q ) (cid:1) . (14)Large ˆ x and ˆ z in ∆ C f (cid:48) f correspond to large N and M in∆ ˜ C f (cid:48) f , respectively.The resummed spin-dependent coefficient function isidentical to the spin-averaged one of [13] and reads tonext-to-leading logarithmic (NLL) accuracy in the MS-scheme:∆ ˜ C res qq ( N, M, α s ( Q )) = e q H qq (cid:0) α s ( Q ) (cid:1) × exp (cid:34) (cid:90) Q Q N ¯ M dk ⊥ k ⊥ A q (cid:0) α s ( k ⊥ ) (cid:1) ln (cid:18) k ⊥ Q (cid:112) ¯ N ¯ M (cid:19)(cid:35) , (15)where ¯ N ≡ N e γ E , ¯ M ≡ M e γ E , with γ E the Euler con-stant, and A q ( α s ) = α s π A (1) q + (cid:16) α s π (cid:17) A (2) q + . . . (16)is a perturbative function. The coefficients required toNLL read A (1) q = C F , A (2) q = 12 C F (cid:20) C A (cid:18) − π (cid:19) − N f (cid:21) , (17)where C F = 4 / C A = 3 and N f is the number of activeflavors. Furthermore, H qq ( α s ) = 1 + α s π C F (cid:18) − π (cid:19) + O ( α s ) . (18)The explicit NLL expansion of the exponent in (15) isgiven by [13] (cid:90) Q Q N ¯ M dk ⊥ k ⊥ A q (cid:0) α s ( k ⊥ ) (cid:1) ln (cid:18) k ⊥ Q (cid:112) ¯ N ¯ M (cid:19) ≈ h (1) q (cid:18) λ NM (cid:19) λ NM b α s ( µ ) + h (2) q (cid:18) λ NM , Q µ , Q µ F (cid:19) , (19)where λ NM ≡ b α s ( µ ) (cid:0) log ¯ N + log ¯ M (cid:1) ,h (1) q ( λ ) = A (1) q πb λ [2 λ + (1 − λ ) ln(1 − λ )] ,h (2) q (cid:18) λ, Q µ , Q µ F (cid:19) = − A (2) q π b [2 λ + ln(1 − λ )]+ A (1) q b πb (cid:20) λ + ln(1 − λ ) + 12 ln (1 − λ ) (cid:21) + A (1) q πb [2 λ + ln(1 − λ )] ln Q µ − A (1) q πb λ ln Q µ F , (20)with b = 11 C A − T R N f π ,b = 17 C A − C A T R N f − C F T R N f π . (21)The functions h (1) q , h (2) q collect all leading-logarithmicand NLL terms in the exponent, which are of the form α ks ln n ¯ N ln m ¯ M with n + m = k + 1 and n + m = k ,respectively. Note that we have restored the full depen-dence on the factorization and renormalization scales inthe above expressions.The polarized moment-space structure function ˜ g h, res1 resummed to NLL is obtained by inserting the resummedcoefficient function into in Eq. (12). To get the physi-cal hadronic structure function g h, res1 one needs to takethe Mellin inverse of the moment-space expression. Asin [13], we choose the required integration contours incomplex N, M -space according to the minimal prescrip-tion of [26], in order to properly deal with the singular-ities arising from the Landau pole due to the divergenceof the perturbative running strong coupling constant α s at scale Λ QCD . Moreover, we match the resummed g h, res1 to its NLO value, i.e. we subtract the O ( α s ) expansionfrom the resummed expression and add the full NLO re-sult: g h, match1 ≡ g h, res1 − g h, res1 (cid:12)(cid:12)(cid:12) O ( α s ) + g h, NLO1 . (22)The final resummed and matched expression for the spinasymmetry A h is then given by A h, res1 ( x, z, Q ) ≡ g h, match1 ( x, z, Q ) F h, match1 ( x, z, Q ) . (23)Similar considerations can be made for inclusive DIS,where again the resummation for g proceeds identicallyto that of F in moment space. Only single Mellin mo-ments of the structure function have to be taken:˜ g ( N, Q ) ≡ (cid:90) dxx N − g ( x, Q ) . (24)The threshold resummed coefficient function is the sameas in the spin-averaged case and is discussed for examplein [13]. We note that the outgoing quark in the pro-cess γ ∗ q → q remains “unobserved” in inclusive DIS. Athigher orders this is known to generate Sudakov suppres-sion effects [27] that counteract the Sudakov enhance-ment associated with soft-gluon radiation from the initialquark. This is in contrast to SIDIS, where the outgoingquark fragments and hence is “observed”, so that boththe initial and the final quark contribute to Sudakov en-hancement. As a result, resummation effects are gener-ally larger in SIDIS than in DIS, for given kinematics. III. PHENOMENOLOGICAL RESULTS
We now analyze numerically the impact of thresh-old resummation on the semi-inclusive and inclusive DISasymmetries A h and A . Given that the resummedexponents are identical for the spin-averaged and spin-dependent structure functions, we expect the resumma-tion effects to be generally very modest. On the other FIG. 1:
Spin asymmetry for semi-inclusive π + production off aproton target. The data points are from [2] and show statisticalerrors only. The (cid:104) x (cid:105) and (cid:104) Q (cid:105) values were taken accordingly tothe HERMES measurements. hand, it is also clear that the effects will not cancel iden-tically in the spin asymmetries: Even though the re-summed exponents for g and F are identical in Mellin-moment space, they are convoluted with different partondistributions and hence no longer give identical resultsafter Mellin inversion. Moreover, the matching proce-dure also introduces differences since the NLO coefficientfunctions are somewhat different for g and F . It istherefore still relevant to investigate the impact of re-summation on the spin asymmetries. We will compareour results to data sets from HERMES [2] and COM-PASS [3, 5]. In addition, we present some results relevantfor measurements at the Jefferson Laboratory [6, 7], inparticular those to be carried out in the near future afterthe CEBAF upgrade to 12 GeV [12].For our calculations we use the NLO polarized par-ton distribution functions of [9] and the unpolarized onesof [28]. Our choice of the latter is motivated by the factthat this set was also adopted as the baseline unpolarizedset in [9], so that the two sets are consistent in the sensethat the same strong coupling constant is used. Addi-tionally, in the case of SIDIS we choose the “de Florian-Sassot-Stratmann” [29] NLO set of fragmentation func-tions. In this work, we choose to focus only on pions inthe final state. Resummation effects for other hadronswill be very similar. The factorization and renormaliza-tion scales are set to Q .Figures 1 and 2 present comparisons of our resummedcalculations with HERMES data [2] for semi-inclusive( π + ) and inclusive DIS, respectively, both off a protontarget at √ s ≈ .
25 GeV. The error bars show the sta-
FIG. 2:
Spin asymmetry for inclusive polarized DIS off a protontarget. The data points are from [4] and show statistical errorsonly. The (cid:104) x (cid:105) and (cid:104) Q (cid:105) values were taken accordingly to theHERMES measurements. tistical uncertainties only. For the SIDIS asymmetry, weintegrate the numerator and the denominator of Eq. (2)separately over a region of 0 . < z < .
8. We plot thetheoretical results at the average values of x and Q ofeach data point and connect the points by a line. Thefigures show the NLO (dashed lines) and the resummed-matched (solid lines) results. As one can see, the higher-order effects generated by resummation are indeed fairlysmall, although not negligible. They are overall moresignificant for SIDIS, which is expected due to the ad-ditional threshold logarithms in SIDIS (see discussion atthe end of Sec. II B). We expect the resummed resultsto be most reliable at rather high values of x (cid:38) . √ s ≈ neutron spin asymmetry is particularlyinteresting from the point of view of resummation, sinceit is known [6] to exhibit a sign change at fairly largevalues of x . Near a zero of the polarized cross sectionresummation effects are expected to be particularly rel- evant. Figure 5 shows the asymmetry at NLO and forthe NLL resummed case. For illustration we show thepresently most precise data available, which are from theHall-A Collaboration [6] at the Jefferson Laboratory. Inorder to mimic the correlation of x and Q for the presentJefferson Lab kinematics, we choose Q = x × inthe theoretical calculation. As one can see, the effects ofresummation are indeed more pronounced than for theinclusive proton structure functions considered in Figs. 2and 4. Evidently the zero of the asymmetry shifts slightlydue to resummation. On the other hand, the asymmetryis overall still quite stable with respect to the resummedhigher order corrections.The latter observation is quite relevant for the ex-traction of polarized large- x parton distributions fromdata for proton and neutron spin asymmetries in lep-ton scattering. For instance, to good approximation [6]one may use the inclusive structure functions to directlydetermine the combinations (∆ u + ∆¯ u ) / ( u + ¯ u ) and(∆ d + ∆ ¯ d ) / ( d + ¯ d ). At lowest order, and neglecting thecontributions from strange and heavier quarks and anti-quarks, one has R u ≡ ∆ u + ∆¯ uu + ¯ u ( x, Q ) = 4 g , p − g , n F , p − F , n ( x, Q ) ,R d ≡ ∆ d + ∆ ¯ dd + ¯ d ( x, Q ) = 4 g , n − g , p F , n − F , p ( x, Q ) , (25)where the subscripts p,n denote a proton or neutrontarget, respectively. One may therefore determine(∆ u + ∆¯ u ) / ( u + ¯ u ) and (∆ d + ∆ ¯ d ) / ( d + ¯ d ) directlyfrom experiment by using measured structure functions FIG. 3:
Same as Fig. 1 but comparing to the COMPASS mea-surements [3].
FIG. 4:
Same as Fig. 2 but comparing to the COMPASS mea-surements [5]. g , p , g , n , F , p , F , n in (25). Up to certain refinementsrequired by the fact that measurements of the ratios g , p /F , p and g , n /F , n are more readily available thanthose of the individual structure functions, this is es-sentially the approach used by the Hall-A Collabora-tion (alternatively, one may also use the correspondingspin asymmetry for the deuteron instead of the neutronone [7]). In the following we explore the typical size ofthe corrections to the ratios due to higher orders. Fig-ure 6 shows first of all the structure function ratios onthe right-hand side of (25), computed at NLO using asbefore the polarized and unpolarized parton distributionfunctions of [9] and [28], respectively (solid lines). Wehave again chosen Q = x × . Using (25), theseratios would correspond to the “direct experimental de-terminations” of R u and R d . The dashed lines in thefigure show the actual ratios (∆ u + ∆¯ u ) / ( u + ¯ u ) and(∆ d + ∆ ¯ d ) / ( d + ¯ d ) as given by the sets of parton distri-bution functions that we use. Any difference betweenthe solid and dashed lines is, therefore, a measure ofthe significance of effects related to strange quarks andantiquarks, and to NLO corrections. As one can see,these have relatively modest size. Finally, we estimatethe potential effect of resummation on R u , R d : Follow-ing [30, 31], we define ‘resummed’ quark (and antiquark)distributions by demanding that their contributions tothe structure functions g , F match those of the corre-sponding NLO distributions, which is ensured by setting˜ q N, res ( Q ) ≡ ˜ C NLO q ( N, α s ( Q ))˜ C res q ( N, α s ( Q )) ˜ q N, NLO ( Q ) (26)in Mellin-moment space. Here, ˜ C NLO q and ˜ C res q are the FIG. 5:
Spin asymmetry for inclusive polarized DIS off a neutrontarget. The data points are from [6] and show statistical errorsonly. The Q values in the theoretical calculation were chosenas Q = x × GeV . NLO and resummed quark coefficient functions for theinclusive structure function F , respectively. We matchthe resummed coefficient function to the NLO one bysubtracting out its NLO contribution and adding the fullNLO one, in analogy with (22). Equation (26) can bestraightforwardly extended to the spin-dependent case.The ratios R u , R d for these ‘resummed’ parton distribu-tions are shown by the dotted lines in Fig. 6. As onecan see, they are quite close to the other results, indicat-ing that resummation is not likely to induce very largechanges in the parton polarizations extracted from fu-ture high-precision data. For illustration, we also showthe Hall-A [6] and CLAS [7] data in the figure, whichhave been obtained using parton-model relations for theinclusive structure functions, similar to (25). One can seethat the error bars of the data are presently still largerthan the differences between our various theoretical re-sults. This situation is expected to be improved withthe advent of the Jefferson Lab 12-GeV upgrade [12] oran Electron Ion Collider [32]. As is well-known, SIDISmeasurements provide additional information on R u , R d ,albeit so far primarily at lower x [2]. IV. CONCLUSIONS
We have investigated the size of threshold resumma-tion effects on double-longitudinal spin asymmetries forinclusive and semi-inclusive deep inelastic scattering infixed-target experiments. Overall, the asymmetries are
FIG. 6:
High- x up and down polarizations (∆ u + ∆¯ u ) / ( u + ¯ u ) and (∆ d + ∆ ¯ d ) / ( d + ¯ d ) . The solid lines show the ratios ofstructure functions on the right-hand sides of Eq. (25), whilethe dashed lines show the actual parton distribution ratios asrepresented by the NLO sets of [9] and [28]. The dotted linesshow the expected shift of the distributions when resummationeffects are included in their extraction, using Eq. (26). The Q values in the theoretical calculation were chosen as Q = x × GeV . We also show the present Hall-A [6] and CLAS [7]data obtained from inclusive DIS measurements. Their error barsare statistical only. rather stable with respect to resummation, in particularfor the inclusive case. Towards large values of x , resum-mation tends to cause a decrease of the spin asymmetries,which is more pronounced in the semi-inclusive case andfor asymmetries measured off neutron targets.The relative robustness of the spin asymmetries bodeswell for the extraction of high- x parton polarizations(∆ u +∆¯ u ) / ( u + ¯ u ) and (∆ d +∆ ¯ d ) / ( d + ¯ d ), which are con-sequently also rather robust. Nevertheless, knowledge ofthe predicted higher-order corrections should be quiterelevant when future high-statistics large- x data becomeavailable. On the theoretical side, it will be interestingto study the interplay of our perturbative correctionswith power corrections that are ultimately also expectedto become important at high- x [14, 15, 33–35], althoughit appears likely that present data are in a windowwhere the perturbative corrections clearly dominate.Finally, we note that related large- x logarithmic effectshave also been investigated for the nucleon’s light conewave function [36], where they turn out to enhancecomponents of the wave function with non-zero orbitalangular momentum, impacting the large- x behaviorof parton distributions. It will be very worthwhile toexplore the possible connections between the logarithmiccorrections discussed here and in [36]. V. ACKNOWLEDGMENTS
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