Simultaneous extraction of transversity and Collins functions from new SIDIS and e+e- data
M. Anselmino, M. Boglione, U. D'Alesio, S. Melis, F. Murgia, A. Prokudin
JJLAB-THY-13-1704
Simultaneous extraction of transversity and Collins functionsfrom new SIDIS and e + e − data M. Anselmino,
1, 2
M. Boglione,
1, 2
U. D’Alesio,
3, 4
S. Melis,
1, 2
F. Murgia, and A. Prokudin Dipartimento di Fisica, Universit`a di Torino, Via P. Giuria 1, I-10125 Torino, Italy INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy Dipartimento di Fisica, Universit`a di Cagliari,Cittadella Universitaria, I-09042 Monserrato (CA), Italy INFN, Sezione di Cagliari, C.P. 170, I-09042 Monserrato (CA), Italy Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA (Dated: October 3, 2018)We present a global re-analysis of the most recent experimental data on azimuthal asymmetries insemi-inclusive deep inelastic scattering, from the HERMES and COMPASS Collaborations, and in e + e − → h h X processes, from the Belle Collaboration. The transversity and the Collins functionsare extracted simultaneously, in the framework of a revised analysis in which a new parameterisationof the Collins functions is also tested. PACS numbers: 13.88.+e, 13.60.-r, 13.66.Bc, 13.85.Ni
I. INTRODUCTION AND FORMALISM
The spin structure of the nucleon, in its partonic collinear configuration, is fully described, at leading-twist,by three independent Parton Distribution Functions (PDFs): the unpolarised PDF, the helicity distributionand the transversity distribution. While the unpolarised PDF and the helicity distribution, which have beenstudied for decades, are by now very well or reasonably well known, much less information is available on thelatter, which has been studied only recently. The reason is that, due to its chiral-odd nature, a transversitydistribution can only be accessed in processes where it couples to another chiral-odd quantity.The chiral-odd partner of the transversity distribution could be a fragmentation function, like the Collins func-tion [1] or the di-hadron fragmentation function [2–4] or another parton distribution, like the Boer-Mulders [5]or the transversity distribution itself. A chiral-odd partonic distribution couples to a chiral-odd fragmenta-tion function in Semi-Inclusive Deep Inelastic Scattering processes (SIDIS, (cid:96) N → (cid:96) h X ). The coupling of twochiral-odd partonic distributions could occur in Drell-Yan processes (D-Y, p N → (cid:96) + (cid:96) − X ) but, so far, no dataon polarised D-Y is available. Information on the convolution of two chiral-odd fragmentation functions (FFs)can be obtained from e + e − → h h X processes.The u and d quark transversity distributions, together with the Collins fragmentation functions, have beenextracted for the first time in Refs. [6, 7], from a combined analysis of SIDIS and e + e − data. Similar results onthe transversity distributions, coupled to the di-hadron, rather than the Collins, fragmentation function, havebeen obtained recently [8]. These independent results establish with certainty the role played by the transversitydistributions in SIDIS azimuthal asymmetries.Since the first papers [6, 7], new data have become available: from the COMPASS experiment operating on atransversely polarised proton (NH target) [9, 10], from a final analysis of the HERMES Collaboration [11] andfrom corrected results of the Belle Collaboration [12]. This fresh information motivates a new global analysisfor the simultaneous extraction of the transversity distributions and the Collins functions.This is performed using techniques similar to those implemented in Refs. [6, 7]; in addition, a second, differentparameterisation of the Collins function will be tested, in order to assess the influence of a particular functionalform on our results.Let us briefly recall the strategy followed and the formalism adopted in extracting the transversity and Collinsdistribution functions from independent SIDIS and e + e − data. A. SIDIS
We consider, at O ( k ⊥ /Q ), the SIDIS process (cid:96) p ↑ → (cid:96) (cid:48) h X and the single spin asymmetry, a r X i v : . [ h e p - ph ] M a r A sin( φ h + φ S ) UT = 2 (cid:82) dφ h dφ S [ dσ ↑ − dσ ↓ ] sin( φ h + φ S ) (cid:82) dφ h dφ S [ dσ ↑ + dσ ↓ ] , (1)where dσ ↑ , ↓ is a shorthand notation for dσ ↑ , ↓ ≡ d σ (cid:96)p ↑ , ↓ → (cid:96)hX dx dy dz d P T dφ S and x, y, z are the usual SIDIS variables: x = x B = Q P · q ) y = ( P · q )( P · (cid:96) ) = Q x s z = z h = ( P · P h )( P · q ) · (2)We adopt here the same notations and kinematical variables as defined in Refs. [6, 13], to which we refer forfurther details, in particular for the definition of the azimuthal angles which appear above and in the followingequations.By considering the sin( φ h + φ S ) moment of A UT [14], we are able to single out the effect originating fromthe spin dependent part of the fragmentation function of a transversely polarised quark, embedded in theCollins function, ∆ N D h/q ↑ ( z, p ⊥ ) = (2 p ⊥ /z m h ) H ⊥ q ( z, p ⊥ ) [15], coupled to the TMD transversity distribution∆ T q ( x, k ⊥ ) [6]: A sin( φ h + φ S ) UT = (cid:88) q e q (cid:90) dφ h dφ S d k ⊥ ∆ T q ( x, k ⊥ ) d (∆ˆ σ ) dy ∆ N D h/q ↑ ( z, p ⊥ ) sin( φ S + ϕ + φ hq ) sin( φ h + φ S ) (cid:88) q e q (cid:90) dφ h dφ S d k ⊥ f q/p ( x, k ⊥ ) d ˆ σdy D h/q ( z, p ⊥ ) , (3)where p ⊥ = P T − z k ⊥ , and d ˆ σdy = 2 πα sxy [1 + (1 − y ) ] d (∆ˆ σ ) dy ≡ d ˆ σ (cid:96)q ↑ → (cid:96)q ↑ dy − d ˆ σ (cid:96)q ↑ → (cid:96)q ↓ dy = 4 πα sxy (1 − y ) . (4)The usual integrated transversity distribution is given, according to some common notations, by:∆ T q ( x ) ≡ h q ( x ) = (cid:90) d k ⊥ ∆ T q ( x, k ⊥ ) . (5)This analysis, performed at O ( k ⊥ /Q ), can be further simplified adopting a Gaussian and factorized parame-terization of the TMDs. In particular for the unpolarized parton distribution (TMD-PDFs) and fragmentation(TMD-FFs) functions we use: f q/p ( x, k ⊥ ) = f q/p ( x ) e − k ⊥ / (cid:104) k ⊥ (cid:105) π (cid:104) k ⊥ (cid:105) (6) D h/q ( z, p ⊥ ) = D h/q ( z ) e − p ⊥ / (cid:104) p ⊥ (cid:105) π (cid:104) p ⊥ (cid:105) , (7)with (cid:104) k ⊥ (cid:105) and (cid:104) p ⊥ (cid:105) fixed to the values found in Ref. [16] by analyzing unpolarized SIDIS azimuthal dependentdata: (cid:104) k ⊥ (cid:105) = 0 .
25 GeV (cid:104) p ⊥ (cid:105) = 0 .
20 GeV . (8)The integrated parton distribution and fragmentation functions, f q/p ( x ) and D h/q ( z ), are available in theliterature; in particular, we use the GRV98LO PDF set [17] and the DSS fragmentation function set [18].For the transversity distribution, ∆ T q ( x, k ⊥ ), and the Collins FF, ∆ N D h/q ↑ ( z, p ⊥ ), we adopt the followingparameterizations [6]: ∆ T q ( x, k ⊥ ) = 12 N T q ( x ) [ f q/p ( x ) + ∆ q ( x )] e − k ⊥ / (cid:104) k ⊥ (cid:105) T π (cid:104) k ⊥ (cid:105) T (9)∆ N D h/q ↑ ( z, p ⊥ ) = 2 N C q ( z ) D h/q ( z ) h ( p ⊥ ) e − p ⊥ / (cid:104) p ⊥ (cid:105) π (cid:104) p ⊥ (cid:105) , (10)with N T q ( x ) = N T q x α (1 − x ) β ( α + β ) ( α + β ) α α β β (11) N C q ( z ) = N C q z γ (1 − z ) δ ( γ + δ ) ( γ + δ ) γ γ δ δ (12) h ( p ⊥ ) = √ e p ⊥ M h e − p ⊥ /M h , (13)and − ≤ N T q ≤ − ≤ N C q ≤
1. We assume (cid:104) k ⊥ (cid:105) T = (cid:104) k ⊥ (cid:105) . The combination [ f q/p ( x ) + ∆ q ( x )], where ∆ q ( x )is the helicity distribution, is evolved in Q according to Ref. [19]. Notice that with these choices both thetransversity and the Collins function automatically obey their proper positivity bounds. A different functionalform of N C q ( z ) will be explored in Section II B.Using these parameterizations we obtain the following expression for A sin( φ h + φ S ) UT : A sin( φ h + φ S ) UT = P T M h − ysxy √ e (cid:104) p ⊥ (cid:105) C (cid:104) p ⊥ (cid:105) e − P T / (cid:104) P T (cid:105) C (cid:104) P T (cid:105) C (cid:88) q e q N T q ( x ) (cid:2) f q/p ( x ) + ∆ q ( x ) (cid:3) N C q ( z ) D h/q ( z ) e − P T / (cid:104) P T (cid:105) (cid:104) P T (cid:105) [1 + (1 − y ) ] sxy (cid:88) q e q f q/p ( x ) D h/q ( z ) , (14)with (cid:104) p ⊥ (cid:105) C = M h (cid:104) p ⊥ (cid:105) M h + (cid:104) p ⊥ (cid:105) (cid:104) P T (cid:105) ( C ) = (cid:104) p ⊥ (cid:105) ( C ) + z (cid:104) k ⊥ (cid:105) . (15)When data or phenomenological information at different Q values are considered, we take into account, atleading order (LO), the QCD evolution of the integrated transversity distribution. For the Collins FF, ∆ N D h/q ↑ ,as its scale dependence is unknown, we tentatively assume the same Q evolution as for the unpolarized FF, D h/q ( z ). B. e + e − → h h X processes Remarkably, independent information on the Collins functions can be obtained in unpolarized e + e − processes,by looking at the azimuthal correlations of hadrons produced in opposite jets [20]. This has been performed bythe Belle Collaboration, which have measured azimuthal hadron-hadron correlations for inclusive charged pionproduction, e + e − → π π X [12, 21, 22]. This correlation can be interpreted as a direct measure of the Collinseffect, involving the convolution of two Collins functions.Two methods have been adopted in the experimental analysis of the Belle data. These can be schematicallydescribed as (for further details and definitions see Refs. [6, 20, 22]): i ) the “cos( ϕ + ϕ ) method” in the Collins-Soper frame where the jet thrust axis is used as the ˆ z direction andthe e + e − → q ¯ q scattering defines the (cid:99) xz plane; ϕ and ϕ are the azimuthal angles of the two hadrons aroundthe thrust axis; ii ) the “cos(2 ϕ ) method”, using the Gottfried-Jackson frame where one of the produced hadrons ( h ) identifiesthe ˆ z direction and the (cid:99) xz plane is determined by the lepton and the h directions. There will then be anotherrelevant plane, determined by ˆ z and the direction of the other observed hadron h , at an angle ϕ with respectto the (cid:99) xz plane.In both cases one integrates over the magnitude of the intrinsic transverse momenta of the hadrons with respectto the fragmenting quarks. For the cos( ϕ + ϕ ) method the cross section for the process e + e − → h h X reads: dσ e + e − → h h X dz dz d cos θ d ( ϕ + ϕ )= 3 α s (cid:88) q e q (cid:110) (1 + cos θ ) D h /q ( z ) D h / ¯ q ( z )+ sin θ ϕ + ϕ ) ∆ N D h /q ↑ ( z ) ∆ N D h / ¯ q ↑ ( z ) (cid:111) , (16)where θ is the angle between the lepton direction and the thrust axis and∆ N D h/q ↑ ( z ) ≡ (cid:90) d p ⊥ ∆ N D h/q ↑ ( z, p ⊥ ) . (17)Integrating over the covered values of θ and normalizing to the corresponding azimuthal averaged unpolarizedcross section one has: R ( z , z , ϕ + ϕ ) ≡ (cid:104) dσ (cid:105) dσ e + e − → h h X dz dz d ( ϕ + ϕ )= 1 + 14 (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) cos( ϕ + ϕ ) (cid:80) q e q ∆ N D h /q ↑ ( z ) ∆ N D h / ¯ q ↑ ( z ) (cid:80) q e q D h /q ( z ) D h / ¯ q ( z ) (18) ≡ (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) cos( ϕ + ϕ ) P ( z , z ) · For the cos(2 ϕ ) method, with the Gaussian ansatz (10), the analogue of Eq. (18) reads R ( z , z , ϕ ) ≡ (cid:104) dσ (cid:105) dσ e + e − → h h X dz dz dϕ = 1 + 1 π z z z + z (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) cos(2 ϕ ) (cid:80) q e q ∆ N D h /q ↑ ( z ) ∆ N D h / ¯ q ↑ ( z ) (cid:80) q e q D h /q ( z ) D h / ¯ q ( z ) (19) ≡ π z z z + z (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) cos(2 ϕ ) P ( z , z ) , where θ is now the angle between the lepton and the h hadron directions.In both cases, Eqs. (18) and (19), the value of (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) ≡ C ( θ ) (20)can be found in the experimental data (see Tables IV and V of Ref. [22]).To eliminate false asymmetries, the Belle Collaboration consider the ratio of unlike-sign ( π + π − + π − π + )to like-sign ( π + π + + π − π − ) or charged ( π + π + + π + π − + π − π + + π − π − ) pion pair production, denotedrespectively with indices U , L and C . For example, in the case of unlike- to like-pair production, one has R U R L = 1 + 14 C ( θ ) cos( ϕ + ϕ ) P U C ( θ ) cos( ϕ + ϕ ) P L (cid:39) C ( θ ) cos( ϕ + ϕ ) ( P U − P L ) (21) ≡ ϕ + ϕ ) A UL (22)and R U R L = 1 + 1 π z z z + z C ( θ ) cos(2 ϕ ) P U π z z z + z C ( θ ) cos(2 ϕ ) P L (cid:39) π z z z + z C ( θ ) cos(2 ϕ ) ( P U − P L ) (23) ≡ ϕ ) A UL (24)and similarly for R U /R C and R U /R C . Explicitely, one has: P U = (cid:80) q e q [∆ N D π + /q ↑ ( z ) ∆ N D π − / ¯ q ↑ ( z ) + ∆ N D π − /q ↑ ( z ) ∆ N D π + / ¯ q ↑ ( z )] (cid:80) q e q [ D π + /q ( z ) D π − / ¯ q ( z ) + D π − /q ( z ) D π + / ¯ q ( z )] ≡ ( P U ) N ( P U ) D (25) P L = (cid:80) q e q [∆ N D π + /q ↑ ( z ) ∆ N D π + / ¯ q ↑ ( z ) + ∆ N D π − /q ↑ ( z ) ∆ N D π − / ¯ q ↑ ( z )] (cid:80) q e q [ D π + /q ( z ) D π + / ¯ q ( z ) + D π − /q ( z ) D π − / ¯ q ( z )] ≡ ( P L ) N ( P L ) D (26) P C = ( P U ) N + ( P L ) N ( P U ) D + ( P L ) D (27) A UL,C ( z , z ) = 14 (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) ( P U − P L,C ) (28) A UL,C ( z , z ) = 1 π z z z + z (cid:104) sin θ (cid:105)(cid:104) θ (cid:105) ( P U − P L,C ) . (29)For fitting purposes, it is convenient to introduce favoured and disfavoured fragmentation functions, assumingin Eq. (10):∆ N D π + /u ↑ , ¯ d ↑ ( z, p ⊥ ) D π + /u, ¯ d ( z ) = ∆ N D π − /d ↑ , ¯ u ↑ ( z, p ⊥ ) D π − /d, ¯ u ( z ) = 2 N C fav ( z ) h ( p ⊥ ) e − p ⊥ / (cid:104) p ⊥ (cid:105) π (cid:104) p ⊥ (cid:105) (30)∆ N D π + /d ↑ , ¯ u ↑ ( z, p ⊥ ) D π + /d, ¯ u ( z ) = ∆ N D π − /u ↑ , ¯ d ↑ ( z, p ⊥ ) D π − /u, ¯ d ( z ) = ∆ N D π ± /s ↑ , ¯ s ↑ ( z, p ⊥ ) D π ± /s, ¯ s ( z ) = 2 N C dis ( z ) h ( p ⊥ ) e − p ⊥ / (cid:104) p ⊥ (cid:105) π (cid:104) p ⊥ (cid:105) , (31)with the corresponding relations for the integrated Collins functions, Eq. (17), and with N C fav , dis ( z ) as given inEq. (12) with N C q = N C fav , dis .We can now perform a best fit of the data from HERMES and COMPASS on A sin( φ h + φ S ) UT and of the data,from the Belle Collaboration, on A UL,C and A UL,C . Their expressions, Eqs. (14) and (25)–(31), contain thetransversity and the Collins functions, parameterised as in Eqs. (9)–(13). They depend on the free parameters α, β, γ, δ, N T q , N C q and M h . Following Ref. [6] we assume the exponents α, β and the mass scale M h to be flavourindependent and consider the transversity distributions only for u and d quarks (with the two free parameters N T u and N T d ). The favoured and disfavoured Collins functions are fixed, in addition to the flavour independentexponents γ and δ , by N C fav and N C dis . This makes a total of 9 parameters, to be fixed with a best fit procedure.Notice that while in the present analysis we can safely neglect any flavour dependence of the parameter β (whichis anyway hardly constrained by the SIDIS data), this issue could play a significant role in other studies, likethose discussed in Ref. [23]. II. BEST FITS, RESULTS AND PARAMETERISATIONSA. Standard parameterisation
We start by repeating the same fitting procedure as in Refs. [6, 7], using the same “standard” parameterisation,Eqs. (6)–(13), with the difference that now we include all the most recent SIDIS data from COMPASS [10] andHERMES [11] Collaborations, and the corrected Belle data [12] on A UL and A UC . Notice, in particular, thatthe A UC data are included in our fits for the first time here. In fact, a previous inconsistency between A UL and A UC data, present in the first Belle results [21], has been removed in Ref. [12].The results we obtain are remarkably good, with a total χ . o . f of 0 .
80, as reported in the first line of Table I,and the values of the resulting parameters, given in Table II, are consistent with those found in our previousextractions. Our best fits are shown in Fig. 1 (upper plots), for the Belle A data, in Fig. 2 for the SIDISCOMPASS data and in Fig. 3 for the HERMES results.We have not inserted the A Belle data in our global analysis as they are strongly correlated with the A results, being a different analysis of the same experimental events. However, using the extracted parameterswe can compute the A UL and A UC azimuthal asymmetries, in good qualitative agreement with the Bellemeasurements, although the corresponding χ values are rather large, as shown in Table I. These results arepresented in Fig. 1 (lower plots).The shaded uncertainty bands are computed according to the procedure explained in the Appendix of Ref. [24].We have allowed the set of best fit parameters to vary in such a way that the corresponding new curves have atotal χ which differs from the best fit χ by less than a certain amount ∆ χ . All these (1500) new curves lieinside the shaded area. The chosen value of ∆ χ = 17 .
21 is such that the probability to find the “true” resultinside the shaded band is 95.45%.
TABLE I: Summary of the χ values obtained in our fits. The columns, from left to right give the χ per degree offreedom, the total χ , and the separate contributions to the total χ of the data from SIDIS, A UL , A UC , A UL and A UC .“NO FIT” means that the χ for that set of data does not refer to a best fit, but to the computation of the correspondingquantity using the best fit parameters fixed by the other data. The four lines show the results for the two choices ofparameterisation of the z dependence of the Collins functions (standard and polynomial) and for the two independentsets of data fitted (SIDIS, A UL , A UC and SIDIS, A UL , A UC ).FIT DATA SIDIS A UL A UC A UL A UC
178 points 146 points 16 points 16 points 16 points 16 pointsStandardParameterization χ = 135 χ = 123 χ = 7 χ = 5 χ = 44 χ = 39 χ . o . f = 0 .
80 NO FIT NO FITStandardParameterization χ = 190 χ = 125 χ = 20 χ = 12 χ = 35 χ = 30 χ . o . f = 1 .
12 NO FIT NO FITPolynomialParameterization χ . tot = 136 χ = 123 χ = 8 χ = 5 χ = 45 χ = 39 χ . o . f = 0 .
81 NO FIT NO FITPolynomialParameterization χ = 171 χ = 141 χ = 44 χ = 27 χ = 15 χ = 15 χ . o . f = 1 .
01 NO FIT NO FIT
We have also performed a global fit based on the SIDIS and A Belle data, and then computed the A values. We do not show the best fit plots, which are not very informative, but the quality of the results can bejudged from the second line of Table I, which shows that although this time A UL and A UC are actually fitted,their corresponding χ values remain large. This has induced us to explore a different functional shape for theparameterisation of N C q ( z ), Eq. (12), which will be discussed in the next Subsection.The difference between A and A is a delicate issue, that deserves some further comments. On the experi-mental side, the hadronic-plane method used for the extraction of A implies a simple analysis of the raw data,as it requires the sole reconstruction of the tracks of the two detected hadrons; therefore it leads to very cleandata points, with remarkably small error bars. On the contrary, the thrust-axis method is much more involved asit requires the reconstruction of the original direction of the q and ¯ q which fragment into the observed hadrons;this makes the measurement of the A asymmetry experimentally more challenging, and leads to data pointswhith larger uncertainties.On the theoretical side, the situation is just the opposite: as the thrust-axis method assumes a perfect TABLE II: Best values of the 9 free parameters fixing the u and d quark transversity distribution functions and thefavoured and disfavoured Collins fragmentation functions, as obtained by fitting simultaneously SIDIS data on the Collinsasymmetry and Belle data on A UL and A UC . The transversity distributions are parameterised according to Eqs. (9),(11) and the Collins fragmentation functions according to the standard parameterisation, Eqs. (10), (12) and (13). Weobtain a total χ / d . o . f . = 0 .
80. The statistical errors quoted for each parameter correspond to the shaded uncertaintyareas in Figs. 1–3, as explained in the text and in the Appendix of Ref. [24]. N T u = 0 . +0 . − . N T d = − . +1 . − . α = 1 . +0 . − . β = 3 . +5 . − . N C fav = 0 . +0 . − . N C dis = − . +0 . − . γ = 1 . +0 . − . δ = 0 . +0 . − . M h = 1 . +2 . − . GeV A U L z [0.2 In an attempt to fit equally well A and A (keeping in mind, however, the comments at the end of theprevious Subsection) we have explored a possible new parameterisation of the z dependence of the Collins -0.1-0.05 0 0.05 0.01 0.1 A U T s i n ( φ h + φ S ) x B COMPASS PROTON π - -0.05 0 0.05 0.1 A U T s i n ( φ h + φ S ) π + h T [GeV] -0.12-0.08-0.04 0 0.04 0.01 0.1 A U T s i n ( φ h + φ S ) x B COMPASS DEUTERON π - -0.04 0 0.04 0.08 A U T s i n ( φ h + φ S ) π + h T [GeV] FIG. 2: The experimental data on the SIDIS azimuthal moment A sin( φ h + φ S ) UT as measured by the COMPASS Collabora-tion [10] on proton (upper plots) and deuteron (lower plots) targets, are compared to the curves obtained from our globalfit. The solid lines correspond to the parameters given in Table II, obtained by fitting the SIDIS and the A asymme-tries with standard parameterisation; the shaded areas correspond to the statistical uncertainty on the parameters, asexplained in the text and in Ref. [24]. function. We notice that data on A ( z ) seem to favour an increase at large z values, rather then a decrease,which is implicitly forced by a behaviour of the kind given in Eqs. (10) and (12) (at least with positive δ values).In addition, an increasing trend of A ( z ) and A ( z ) seems to be confirmed by very interesting preliminaryresults of the BABAR Collaboration, which have performed an independent new analysis of e + e − → h h X data [25], analogous to that of Belle.This suggests that a different parameterisation of the z dependence of favoured and disfavoured Collinsfunctions could turn out to be more convenient. Then, we try an alternative polynomial parameterisationwhich allows more flexibility on the behaviour of N C q ( z ) at large z : N C q ( z ) = N C q z [(1 − a − b ) + a z + b z ] , (32) -0.1-0.05 0 0.05 0 0.1 0.2 0.3 A U T s i n ( φ h + φ S ) x B HERMES PROTON π - -0.05 0 0.05 0.1 A U T s i n ( φ h + φ S ) π + h T [GeV] FIG. 3: The experimental data on the SIDIS azimuthal moment A sin( φ h + φ S ) UT as measured by the HERMES Collabora-tion [11], are compared to the curves obtained from our global fit. The solid lines correspond to the parameters givenin Table II, obtained by fitting the SIDIS and the A asymmetries with standard parameterisation; the shaded areascorrespond to the statistical uncertainty on the parameters, as explained in the text and in Ref. [24]. with the subfix q = fav , dis, and − ≤ N C q ≤ a and b are flavour independent so that the total number ofparameters for the Collins functions (in addition to M h ) remains 4. Such a choice fixes the term N C q ( z ) tobe equal to 0 at z = 0 and not larger than 1 at z = 1. Notice that we do not automatically impose, as inEq. (12), the condition |N C q ( z ) | ≤ 1; however, we have explicitly checked that the best fit results and all thesets of parameters corresponding to curves inside the shaded uncertainty bands satisfy that condition.We have repeated the same fitting procedure as performed with the standard parameterisation. When fittingthe combined SIDIS, A UL and A UC Belle data, the resulting best fits (not shown) hardly exhibit any differencewith respect to those obtained with the standard parameterisation (Fig. 1). This can be seen also from the χ ’sin Table I, where the third line is very similar to the first one. As a further confirmation, the correspondingbest fit plots for N C fav , dis ( z ), in case of the standard and polynomial parameterisations, plotted in Fig. 4 (leftpanel) practically coincide up to values of z very close to 1.The situation is different when best fitting the SIDIS data together with A UL and A UC ; in such a casethe polynomial parameterisation allows a much better best fit, as shown in Fig. 5, upper plots. A reasonableagreement can also be achieved between the data and the computed values of A UL and A UC , as shown by the χ values in Table I and by the lower plots in Fig. 5. In this case the polynomial form of N C fav , dis ( z ) differs fromthe standard one, as shown in the right plots in Fig. 4.Notice, again, that the large χ values of the computed A UL is almost completely due to the last z bins,which correspond to the quasi exclusive region. Also, the larger χ values corresponding to SIDIS data aremainly due to a slightly worse description of HERMES π − azimuthal moments. The values of the parametersobtained using the polynomial shape of N C fav , dis ( z ), Eq. (32), are given in Table III. C. The extracted transversity and Collins functions; predictions and final comments Our newly extracted transversity and Collins functions are shown in Figs. 6 and 7; to be precise, in the leftpanels we show x ∆ T q ( x ) = x h q ( x ), for u and d quarks, while in the right panels we plot: z ∆ N D h/q ↑ ( z ) = z (cid:90) d p ⊥ ∆ N D h/q ↑ ( z, p ⊥ ) = z (cid:90) d p ⊥ p ⊥ z m h H ⊥ q ( z, p ⊥ ) = 4 z H ⊥ (1 / q ( z ) (33)0 -1-0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 N C ( z ) z fav. polyn. dis. polyn. fav. std dis. std -1-0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 N C ( z ) z fav. polyn. dis. polyn. fav. std dis. std FIG. 4: Plots of the functions N C fav ( z ) and N C dis ( z ) for the favoured and disfavoured Collins functions as obtained by usingthe standard, Eq. (12), and polynomial, Eq. (32), parameterisations. On the left panel we show the results obtained byfitting the SIDIS data together with the A Belle asymmetries (both with standard and polynomial parameterisation),while on the right panel we show the corresponding results obtained by fitting the SIDIS data together with the A Belleasymmetries.TABLE III: Best values of the 9 free parameters fixing the u and d quark transversity distribution functions and thefavoured and disfavoured Collins fragmentation functions, as obtained by fitting simultaneously SIDIS data on the Collinsasymmetry and Belle data on A UL and A UC . The transversity distributions are parameterised according to Eqs. (9),(11) and the Collins fragmentation functions according to the polynomial parameterisation, Eqs. (10), (32) and (13). Weobtain a total χ / d . o . f . = 1 . 01. The statistical errors quoted for each parameter correspond to the shaded uncertaintyareas in Fig. 5, as explained in the text and in the Appendix of Ref. [24]. N T u = 0 . +0 . − . N T d = − . +0 . . α = 1 . +0 . − . β = 3 . +5 . − . N C fav = 1 . +0 . − . N C dis = − . +0 . − . a = − . +1 . − . b = 2 . +0 . − . M h = 0 . +1 . − . GeV for h = π ± and q = u . The Collins results for d quark are not shown explicitly, but could be obtained fromTables II and III.Fig. 6 shows the results which best fit the COMPASS and HERMES SIDIS data on A sin( φ h + φ S ) UT , togetherwith the Belle results on A UL and A UC , using the standard parameterisation. The red solid lines correspond tothe parameters given in Table II. The shaded bands show the uncertainty region, which is the region spannedby the 1500 different sets of parameters fixed according to the procedure explained above and in the Appendixof Ref. [24]. The blue dashed lines show, for comparison, our previous results [7]: the difference between thesolid red and dashed blue lines is only due to the updated SIDIS and A UL data used here, with the addition of A UC , while keeping the same parameterisation. The present and previous results agree within the uncertaintyband: one could at most notice a slight decrease of the new u quark transversity distribution at large x values.Fig. 7 shows the results which best fit the COMPASS and HERMES SIDIS data on A sin( φ h + φ S ) UT , together withthe Belle results on A UL and A UC , using the polynomial parameterisation. The red solid lines correspond tothe parameters given in Table III. This is not a simple updating of our previous 2008 fit [7], as we use differentsets of data (SIDIS and A rather than SIDIS and A ) with a different polynomial parameterisation. In thiscase the comparison with the 2008 results is less significant. If comparing the results of Fig. 6 and 7, one noticesa sizeable difference in the favoured ( u/π + ) Collins function, and less evident differences in the transversitydistributions.In Fig. 8 we show, for comparison with similar results presented in Ref. [7], the tensor charge, corresponding toour best fit transversity distributions, as given in Tables II and III. Our extracted values are shown at Q = 0 . A U L z [0.2 41 GeV . The evolution to the chosen value has beenobtained by evolving at LO the collinear part of the factorized distribution and fragmentation functions. TheTMD evolution, which might affect the k ⊥ and p ⊥ dependence, is not yet known for the Collins function.Consistently, it has not been taken into account for the other distribution and fragmentation functions.As BABAR data on A and A should be available soon, we show in Figs. 9 and 10 our expectations, basedon our extracted Collins functions. Fig. 9 shows the expected values of A UL , A UC , A UL and A UC , as a functionof z for different bins of z , using the parameters of Table II, obtained by fitting the SIDIS and the A Belledata with the standard parameterisation. Fig. 10 shows the same quantities using the parameters of Table III,obtained by fitting the SIDIS and the A Belle data with the polynomial parameterisation.The Belle (and BABAR) e + e − results on the azimuthal correlations of hadrons produced in opposite jets,together with the SIDIS data on the azimuthal asymmetry A sin( φ h + φ S ) UT , measured by both the HERMES andCOMPASS Collaborations, definitely establish the importance of the Collins effect in the fragmentation of a2 -0.3-0.2-0.1 0 0.1 0.001 0.01 0.1 1 x ∆ T d ( x ) x Q =2.41 GeV x ∆ T u ( x ) -0.2-0.1 0 0 0.2 0.4 0.6 0.8 1 z ∆ N D π - / u ( z ) z Q =2.41 GeV z ∆ N D π + / u ( z ) FIG. 6: In the left panel we plot (solid red lines) the transversity distribution functions x h q ( x ) = x ∆ T q ( x ) for q = u, d ,with their uncertainty bands (shaded areas), obtained from our best fit of SIDIS data on A sin( φ h + φ S ) UT and e + e − dataon A , adopting the standard parameterisation (Table II). Similarly, in the right panel we plot the corresponding firstmoment of the favoured and disfavoured Collins functions, Eq. (33). All results are given at Q = 2 . 41 GeV . Thedashed blue lines show the same quantities as obtained in Ref. [7] using the data then available on A sin( φ h + φ S ) UT and A UL . transversely polarised quark. In addition, the SIDIS asymmetry can only be observed if coupled to a non negligi-ble quark transversity distribution. The first original extraction of the transversity distribution and the Collinsfragmentation functions [6, 7], has been confirmed here, with new data and a possible new functional shape ofthe Collins functions. The results on the transversity distribution have also been confirmed independently inRef. [8].A further improvement in the QCD analysis of the experimental data, towards a more complete understandingof the Collins and transversity distributions, and their possible role in other processes, would require taking intoaccount the TMD-evolution of ∆ T q ( x, k ⊥ ) and ∆ N D h/q ↑ ( z, p ⊥ ). Great progress has been recently achieved in thestudy of the TMD-evolution of the unpolarized and Sivers transverse momentum dependent distributions [33–37]and a similar progress is expected soon for the Collins function and the transversity TMD distribution [38]. Acknowledgments Authored by a Jefferson Science Associate, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. Weacknowledge support from the European Community under the FP7 “Capacities - Research Infrastructures”program (HadronPhysics3, Grant Agreement 283286). We also acknowledge support by MIUR under Cofi-nanziamento PRIN 2008. 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