Effects of transversity in deep-inelastic scattering by polarized protons

Single-spin asymmetries for pions and charged kaons are measured in semi-inclusive deep-inelastic scattering of positrons and electrons off a transversely nuclear-polarized hydrogen target. The dependence of the cross section on the azimuthal angles of the target polarization (phi_S)and the produced hadron (phi) is found to have a substantial sin(phi+phi_S) modulation for the production of pi+, pi- and K+. This Fourier component can be interpreted in terms of non-zero transversity distribution functions and non-zero favored and disfavored Collins fragmentation functions with opposite sign. For pi0 and K- production the amplitude of this Fourier component is consistent with zero.

π + , π − and K + . This Fourier component can be interpreted in terms of non-zero transversity distribution functions and non-zero favored and disfavored Collins fragmentation functions with opposite sign. Its amplitude is found to be consistent with zero for π 0 and K − production.
PACS numbers: 13.60.-r, 13.88.+e, 14.20.Dh, Most of our knowledge about the internal structure of nucleons comes from deep-inelastic scattering (DIS) experiments. The dominant process in DIS of charged leptons by nucleons is the exchange of a single space-like photon with a squared four-momentum −Q 2 much larger than the typical hadronic scale, for which the squared mass M 2 of the nucleon is chosen. The cross section for the DIS process can be decomposed in a modelindependent way in terms of structure functions. Factorization theorems (see [1,2] and references therein) based on quantum chromodynamics (QCD) allow the interpretation of these structure functions in terms of parton distribution functions (PDFs), which ultimately reveal crucial aspects of the dynamics of confined quarks and gluons.
Polarized inclusive DIS of charged leptons on nucleons, lN → l ′ X (where X denotes the undetected hadronic final state), can be described by four structure functions (see, e.g., [3]) which can be interpreted using collinear factorization theorems (see, e.g., [4] and references therein). Three of the structure functions contain contributions already at leading order in an expansion in M/Q (twist expansion). They include the leadingtwist (twist-2) parton distribution functions f q 1 (x), also denoted as q(x), and g q 1 (x), also denoted as ∆q(x) [3] (for semplicity, the dependence on the factorization scale has been dropped). The variable x represents the fraction of the nucleon momentum carried by the parton in a frame where the nucleon moves fast in the direction opposite to the probe (e.g., in the Breit frame). The presence of the hard probe is essential to defining a specific direction ( q in Fig. 1), which is usually denoted as longitudinal, and a plane perpendicular to it, which is usually denoted as transverse. The x-integrals of f q 1 (x) and g q 1 (x) are related to the vector and axial charge of the nucleon, respectively. In a parton-model picture, f q 1 (x) describes the number density of quarks of flavor q in a fast-moving nucleon without regard to their polarization. The PDF g q 1 (x) describes the difference between the number densities of quarks with helicity equal or opposite to that of the nucleon if the nucleon is longitudinally polarized.
There is a third leading-twist PDF, the function h q 1 (x) (also denoted as δq(x)), called the transversity distribution (see [5] for a review on the subject). Its x-integral is related to the tensor charge of the nucleon. It can be interpreted [6] as the difference between the densities of quarks with transverse (Pauli-Lubanski) polarization parallel or anti-parallel to the transverse polarization of the nucleon. In contrast to f q 1 (x) and g q 1 (x), due to helicity conservation h q 1 (x) does not exist for gluons in case of spin-1 2 targets and cannot mix with gluons under evolution.
The transversity distribution does not appear in any structure function in inclusive DIS because it is odd under chirality transformations. It must be combined with another chiral-odd nonperturbative partner to appear in a cross section since perturbative QED and QCD interactions preserve chirality. For this reason, in spite of decades of inclusive DIS studies, no experimental information on the transversity distribution was available until recently. The transversity distribution can be accessed experimentally only in polarized semi-inclusive DIS (SIDIS), where it can appear in combination with the chiral-odd Collins fragmentation function [7]. This letter presents a measurement of the associated signal.
In polarized semi-inclusive DIS, lN → l ′ hX, where a hadron h is detected in the final state in coincidence with the scattered lepton, the cross section depends on, among other variables, the hadron transverse momentum and its azimuthal orientation with respect to the lepton scattering plane and about the virtual-photon direction. If the polarization of the final state is not measured, the SIDIS cross section can be decomposed in terms of 18 semi-inclusive structure functions (see, e.g., [8]).
When the transverse momentum of the produced hadron is small compared to the hard scale Q, semi-inclusive DIS can be described using transversemomentum-dependent factorization [9,10]. The semiinclusive structure functions can be interpreted in terms of convolutions involving transverse-momentumdependent (TMD) parton distribution and fragmentation functions [11]. The former encode information about the distribution of partons in a three-dimensional momentum space, whereas the latter describe the hadronization process in a three-dimensional momentum space. Hence, the study of semi-inclusive DIS not only opens the way to the measurement of transversity, but also probes new dimensions of the structure of the nucleons and of the hadronization process, thus offering new perspectives to our understanding of QCD.
When performing a twist expansion, eight structure functions contain contributions at leading order, related to the eight leading-twist TMD PDFs [8]. One of these structure functions is interpreted as the convolution of the transversity distribution function h q 1 (x, p 2 T ) (not integrated over the transverse momentum) and the Collins fragmentation function H ⊥q 1 (z, k 2 T ), which acts as a polarimeter since it is sensitive to the correlation between the transverse polarization of the fragmenting quark and k T [7]. Here, z denotes the fraction of the virtual pho- ton energy carried by the produced hadron in the laboratory frame, p T denotes the transverse momentum of the quark with respect to the parent hadron direction, and k T denotes the transverse momentum of the fragmenting quark with respect to the direction of the produced hadron. This structure function manifests itself as a sin(φ + φ S ) modulation in the SIDIS cross section with a transversely polarized target. Its Fourier amplitude, henceforth named Collins amplitude, is denoted as 2 sin(φ+φ S ) UT , where φ (φ S ) represents the azimuthal angle of the hadron momentum (of the transverse component of the target spin) with respect to the lepton scattering plane and about the virtual-photon direction, in accordance with the Trento Conventions [12] (see Fig. 1). The subscript UT denotes unpolarized beam and target polarization transverse with respect to the virtual-photon direction. Other azimuthal modulations have different origins and involve other distribution and fragmentation functions. They can be disentangled through their specific dependence on the two azimuthal angles φ and φ S (see, e.g., [13]). Non-zero Collins amplitudes were previously reported for charged pions from a hydrogen target [14], based on a small subset (about 10%) of the data reported here (consisting of about 8.76 million DIS events). Similar amplitudes, albeit consistent with zero, were measured on a deuterium target by the Compass Collaboration [15][16][17]. In this letter, in addition to much improved statistical precision on the charged pion results, the Collins amplitudes for K + , K − , and π 0 are presented for the first time for a proton target. In Refs. [18,19] the first joint extraction of the transversity distribution function and the Collins fragmentation function was carried out, under simplifying assumptions, using preliminary results from a subset of the present data in combination with SIDIS data from the Compass collaboration [15][16][17] and e + e − annihilation data from the Belle collaboration [20,21].
Recently, significant amplitudes for two-hadron production in semi-inclusive DIS, which constitutes an independent process to probe transversity, were measured at the Hermes experiment [22] providing additional evidences for a non-zero transversity distribution function.
The data reported here were recorded during the 2002-2005 running period of the Hermes experiment with a transversely nuclear-polarized hydrogen target stored in an open-ended target cell internal to the 27.6 GeV Hera polarized positron/electron storage ring at Desy. The two beam helicity states are almost perfectly balanced in the present data, and no effects arising from the residual net beam polarization were observed. The target cell was fed by an atomic-beam source [23] which uses Stern-Gerlach separation combined with radio-frequency transitions of hyperfine states. The target cell was immersed in a transversely oriented magnetic holding field. The effects of this magnetic field on the vertex positions and the scattering angles of charged particles were accounted for in the track reconstruction. The nuclear polarization of the atoms was flipped at 1-3 min time intervals, while both the polarization and the atomic fraction inside the target cell were continuously measured [24]. The average magnitude of the proton-polarization component perpendicular to the beam direction was 0.725±0.053. Scattered leptons and coincident hadrons with vertical and horizontal scattering angles in the ranges 40 < |θ y | < 140 mrad and |θ x | < 170 mrad, respectively, were detected by the Hermes spectrometer [25]. Leptons were identified with an efficiency exceeding 98% and a hadron contamination of less than 1%. Charged hadrons detected within the momentum range 2-15 GeV were identified using a dual-radiator RICH [26] by means of a hadronidentification algorithm that takes into account the event topology. The detection of the neutral pions is based on the measurements of photon pairs in the calorimeter. These were only accepted if E γ > 1 GeV and 0.10 GeV < M γγ < 0.17 GeV, where E γ and M γγ denote the photon energy and the photon-pair invariant mass, respectively. Events were selected according to the kinematic requirements W 2 > 10 GeV 2 , 0.023 < x < 0.4, 0.1 < y < 0.95, and Q 2 > 1 GeV 2 , where W 2 ≡ (P + q) 2 , Q 2 ≡ −q 2 ≡ −(k −k ′ ) 2 , y ≡ (P ·q)/(P ·k), and x ≡ Q 2 /(2P ·q) are the conventional DIS kinematic variables with P , k and k ′ representing the four-momenta of the initial state target proton, incident and outgoing lepton, respectively. In order to minimize target fragmentation effects as well as to exclude kinematic regions where contributions from exclusive channels become sizable, coincident hadrons were only included if 0.2 < z < 0.7, where z = (P · P h )/(P · q) and P h is the four-momentum of the produced hadron. The cross section for semi-inclusive production of hadrons using an unpolarized lepton beam and a transversely polarized target includes a polarization-averaged part and a polarization-dependent part. The former contains two cosine modulations and the latter contains a total of five sine modulations [8,27,28]: 2 cos(nφ) UU cos(nφ) The subscript UU denotes unpolarized beam and unpolarized target, and σ UU represents the φ-independent part of the polarization-averaged cross section.
The Collins amplitude 2 sin(φ+φ S ) UT can be interpreted in the quark-parton model as [27] 2 sin(φ+φ S ) UT (x, y, z, P h⊥ ) where P h⊥ ≡ |P h − (P h ·q)q |q| 2 | is the transverse momentum of the produced hadron, and D q 1 is the polarizationaveraged quark fragmentation function. The notation C denotes the convolution [8] where the sum runs over the quark flavors q, and e q are the quark electric charges in units of the elementary charge. Note that, as the quark flavors enter the cross section with the square of their electric charge, the uquarks are likely to provide the dominant contribution for proton targets (u-quark dominance). Similar expressions hold for the other azimuthal modulations in eq. (1) [8].
Experimentally, the asymmetry amplitudes for opposite target-spin states ↑, ↓ were measured, using a maximum-likelihood fit alternately binned in x, z, and P h⊥ , but unbinned in φ and φ S . The asymmetry amplitudes for neutral pions were corrected for the combinatorial background evaluated in the side-bands of the photon-pair invariant mass spectrum. In addition to the five sine terms in eq. (1), the fit also included a sin(2φ + φ S ) term, arising from the small but non-vanishing target-polarization component that is longitudinal to the virtual-photon direction when the target is polarized perpendicular to the beam direction [29]. In order to avoid cross contamination arising from the limited spectrometer acceptance, the six amplitudes were extracted simultaneously. The fit did not include the cos(nφ) modulations of eq. (1). As a consequence, the Fourier amplitudes extracted from the asymmetry in eq. (4) do not coincide with those of eq. (1). However, in the following they will be considered to be equivalent because inclusion in the fit of estimates [30] for the cos(φ) and cos(2φ) amplitudes of the unpolarized cross section resulted in negligible effects on the extracted amplitudes. The extracted Collins amplitudes are shown in Fig. 2 as a function of x, z, or P h⊥ . They are positive for π + and K + , negative for π − , and consistent with zero for π 0 and K − at a confidence level of at least 95% based on a Student's t-test including the systematic uncertainties. In general, the non-vanishing amplitudes increase in magnitude with x. This is consistent with transversity mainly receiving contributions from the valence quarks. A large contribution from the sea quarks, which would dominate in the low-x region, was indeed not expected due to the fact that transversity cannot be generated in gluon splitting. The amplitudes are also found to increase with z, in qualitative agreement with the results of the Collins fragmentation function from the Belle experiment [20,21]. Note that the x, z, and P h⊥ dependencies in Fig. 2 are three projections of the same data and are thus fully correlated. A multidimensional extraction of the Collins amplitudes will be addressed in a future pa- A scale uncertainty of 7.3% on the extracted amplitudes, not shown in Fig. 2, arises from the accuracy in the measurement of the target polarization. Effects from acceptance, smearing due to detector resolution, higher order QED processes and hadron identification procedure based on the RICH are not corrected for in the data. Rather, the size of all these effects was estimated using a Pythia6 Monte Carlo simulation [31] including a full reconstruction of the Hermes spectrometer and based on fragmentation parameters tuned to Hermes hadron multiplicity data [32]. A polarization state was assigned to each generated event using a model that reflects the (transverse target) polarization dependent part of the cross section (see eq. (1)). This model was obtained through a fully differential (i.e. differential in the four relevant kinematic variables x, Q 2 , z and P h⊥ ) 2 nd order polynomial fit [33,34] of real data. The asymmetry amplitudes, extracted from the simulated data by means of the same analysis used for the real data, were then compared with the model, evaluated in each bin at the mean kinematics, to obtain an estimate of the global impact the effects listed above. The result was included in the systematic uncertainty and constitutes the largest contribution. Due to the non linearity of the model in the kinematic variables and to the finite bin size, an approximation is introduced given by the difference, in each bin, between the average model and the model evaluated at the average kinematics. This approximation is accounted for in the systematic uncertainties. The impact on the extracted amplitudes of contributions [29] from the non-vanishing longitudinal target-spin component was estimated based on previous measurements of single-spin asymmetries for longitudinally polarized protons [35,36]. The resulting relatively small effect was included in the systematic uncertainty.
A Monte Carlo simulation was used to estimate the fraction of pions and kaons originating from the decay of exclusively produced vector mesons, updating previous results reported in Ref. [37]. For charged pions, this fraction is dominated by the decay of ρ 0 mesons and, in the kinematic region covered by the present analysis, is of the order of 6-7%. The vector-meson fractions for neutral pions and charged kaons are of the order of 2-3%. The z and P h⊥ dependences of the fraction of pions and kaons stemming from the decay of exclusively produced vector mesons are shown in [13] for the two kinematic regions Q 2 < 4 GeV 2 and Q 2 > 4 GeV 2 (the x dependence was not reported due to the strong correlation between x and Q 2 in the data). They exhibit maxima at high z and low P h⊥ . These contributions are considered part of the signal and were not used to correct the pion and kaon yields analysed in the present work.
The results of Fig. 2 show that the π − amplitude is of opposite sign to that of π + and larger in magnitude. A possible explanation is dominance of u flavor among struck quarks, in conjunction with a substantial magnitude with opposite sign of the disfavored Collins fragmentation function describing, e.g., the fragmentation of u quarks into π − mesons, as already suggested in Ref. [14]. Opposite signs for the favored and disfavored Collins fragmentation functions are not in contradiction to the Belle results [20,21] and are supported by the combined fits reported in [18]. They can be understood in the light of the string model of fragmentation [38] (and also of the Schäfer-Teryaev sum rule [39]). If a favored pion is created at the string end by the first break, a disfavored pion from the next break is likely to inherit transverse momentum in the opposite direction. The string fragmentation model, which predicts such a P h⊥ strong negative correlation between favored and disfavored pions, is implemented in the successful and widespread Jetset simulation [40]. Under the assumption of isospin invariance, the fragmentation functions for neutral pions are assumed equal to the average of those for charged pions. Using factorization of the semi-inclusive cross section results in the following isospin relation for the Collins amplitudes for pions: where C denotes the polarization-averaged cross section ratio for semi-inclusive negative and positive pion production (C = σ π − UU /σ π + UU ). The extracted pion amplitudes are found to fulfill eq. (5) within the experimental uncertainties.
Despite the expectation of similar magnitudes for the π + and K + amplitudes, based on the common u-quark dominance, the amplitudes for K + are found to be larger than those for π + at a confidence level of at least 90% (99%) based on a Student's t-test including (not including) the systematic uncertainties. On the other hand, the amplitudes for π − and K − exhibit a very different behavior, the former being significantly negative, while the latter is consistent with zero in the whole kinematic range. Here however, one should keep in mind that, in contrast to π − , a K − has no valence quarks in common with the target proton.
In interpreting the various features of the extracted amplitudes, and in particular the differences between those of pions and kaons, the largely unknown role of several concurring factors should be considered. Among these are, e.g., (i) the role of sea quarks in conjunction with possibly large fragmentation functions; (ii) the different contributions from decay of semi-inclusively produced vector-mesons, mainly ρ and ω mesons producing pions (up to 37% and 10%, respectively), and K * and φ mesons producing kaons (up to 41% and 3.5%, respctively); (iii) the k T dependences of the fragmentation functions, which can be different for different The above discussion is based on eq. (2) and is thus valid up to twist-3. It is therefore interesting to investigate the presence of possible twist-4 contributions. To this end, the Q 2 dependence of the extracted amplitudes was studied in some detail. To minimize effects arising from the strong correlation between x and Q 2 in the data, the events in each x bin were divided into two sub-bins, with Q 2 below and above the mean value Q 2 (x i ) for the original bin (see Fig. 3). However, due to the limited statistics it was not possible to exclude nor support the presence of twist-4 contributions by fitting the data in Fig. 3 with different Q 2 dependencies. The Q 2 range for each i-bin in x was divided into the two regions above and below the average Q 2 of that bin ( Q 2 (xi) ). The bottom panels show the x-dependence of the average Q 2 .
In summary, non-zero Collins amplitudes in semiinclusive DIS are measured for charged pions and positive kaons. These amplitudes arise from the transverse polarization of quarks in the target, revealed by its influence on the fragmentation of the struck quark, and thus support the existence of a non-zero transversity distribution function in the proton. They also support the existence of a non-zero Collins fragmentation functions. In particular, by comparing the Collins amplitudes of π + and π − , it appears that fragmentation that is disfavored in terms of quark flavor has a surprising importance, and enters with a sign opposite to that of the favored one. In contrast to the expectation that the π + and the K + Collins amplitudes should have similar magnitudes, based on the common u-quark dominance, the amplitude for K + is found to be significantly larger than that for π + . This could be an indication, e.g., of an important role of the sea quarks in conjunction with possibly large fragmentation functions. Collins amplitudes consistent with zero are measured for π 0 and K − . These data should considerably improve the precision of transversity extractions from future global fits.
We gratefully acknowledge the Desy management for its support, the staff at Desy and the collaborating institutions for their significant effort, and our national funding agencies and the EU RII3-CT-2004-506078 program for financial support.