Transverse–Spin Quark Distributions from Asymmetry Data and Symmetry Arguments

Abstract

The transversity and the Sivers distribution functions of quarks incorporate important information about the transverse-spin and transverse-momentum structure of nucleons. We show how these distributions can be directly determined point by point from leptoproduction asymmetry data collected for various targets and produced hadrons by the COMPASS Collaboration. Only simple symmetry relations are used in the extraction.

1. Introduction

The investigation of the transverse-spin structure of nucleons has been an extremely active research area of hadronic physics in the past two decades [1,2]. Single-spin asymmetries in leptoproduction of hadrons from a transversely polarized target,ℓ^N↑→^ℓ′hX , have been discovered and measured by various experiments (for a review, see, e.g., in [3]), and represent the primary way to access the transverse-spin distributions of quarks.

One of these asymmetries—the so-called Collins asymmetry—involves the transversity distribution_h1 , a leading-twist and chirally odd distribution function which measures the transverse polarization of quarks inside a transversely polarized nucleon [4]. In the Collins asymmetry, transversity couples to a transverse-momentum-dependent chirally odd fragmentation function,_H1⊥ (the “Collins function” [5]), which describes the fragmentation of a transversely polarized quark into a spinless hadron. Collins asymmetries have been measured by the HERMES [6,7] and the COMPASS experiments [8,9,10] on a proton target, and by COMPASS on a deuteron target [11,12,13].

Another single-spin asymmetry, associated with a different angular modulation of the cross section, originates from a correlation between the transverse spin of the nucleon and the transverse momentum of quarks, described by a leading-twist transverse-momentum dependent distribution (TMD), the “Sivers function”_f1T⊥ [14,15]. A non-zero Sivers function causes the distributions of quark transverse momentum to be asymmetric with respect to the plane given by the directions of nucleon spin and momentum. This asymmetry, known as the Sivers effect, has been experimentally observed by the HERMES [6,16] and COMPASS collaborations [8,10,11,12,13,17] in the case of pion and kaon production.

Many phenomenological analyses of these asymmetries have been performed so far (see, for a review, the work in [18]). Most of them extract the quark distributions by fitting the data with a given functional form for their dependence on the Bjorken variable x. We adopt a more direct approach, taking advantage of the fact that the COMPASS experiment has provided data for different targets (proton and deuteron) and produced hadrons (positive and negative pions) with the same kinematics. By simple general arguments based on isospin symmetry and sea flavor symmetry, the asymmetries of the various leptoproduction processes can be related to each other and combined in such a way that the valence distributions_uvand_dv are separately extracted point by point in x (for details we refer the reader to the papers where this procedure was originally proposed and applied [19,20]). The approach is almost model-independent. In fact, although we adopt a Gaussian Ansatz for the transverse-momentum dependence of quark distributions (in order to factorize them from fragmentation functions), the Gaussian widths—representing the average transverse momenta of quarks—do not appear in the final results.

The transversity and Sivers quark sea distributions can be determined in the same way, but in this paper we will not consider them. As shown in [19,20], they turn out to be quite uncertain and compatible with zero within their errors.

The Sivers effect manifests itself also in the gluon sector [21]. The gluon Sivers function_f1T⊥g can be probed for instance in the inclusive leptoproduction of hadron pairs with large transverse momenta, as done by COMPASS [22]. There are three different elementary reactions contributing to this process: the leading-order scattering^γ*q→q, the QCD Compton scattering^γ*q→qg, and the photon–gluon fusion^γ*g→qq¯. As only the photon–gluon fusion involves the gluon Sivers distribution, the Sivers asymmetry data for dihadron production cannot be directly used to extract_f1T⊥g . In [22], the photon–gluon fusion component of the Sivers asymmetry has been disentangled by means of a Monte Carlo method. The result, restricted to a single x value, shows that the gluonic contribution to the Sivers effect is definitely non-vanishing (and negative). Work is now in progress to determine the magnitude of_f1T⊥g.

Extracting the transversity distributions from linear combinations of the Collins asymmetries requires some knowledge of the Collins fragmentation function_H1⊥, which must be obtained independently from another class of processes, namely, inclusive dihadron production in^{e+ e−} annihilation, studied by various experiments [23,24,25,26]. An alternative way to determine the transversity from the Collins asymmetries alone is via the so-called “difference asymmetries”, which allow extracting combinations of the u and d valence quark transversity without knowing the Collins fragmentation function. This method was proposed long time ago [27,28,29] to access the helicity distribution functions. Recently it has been revisited in the context of Sivers, Boer–Mulders and transversity distributions [30]. Here, we report the results of a recent paper of ours [31], where, using again the COMPASS measurements with proton and deuteron targets, we determined the transversity ratio_h1uv /_h1dv .

2. SIDIS with a Transversely Polarized Target

The process we are interested in is semi-inclusive DIS (SIDIS) with a transversely polarized target,ℓ^N↑→^ℓ′hX. The produced hadrons h (of mass_Mhand momentum_Ph) we will consider are positive and negative pions. Conventionally, all azimuthal angles are referred to the lepton scattering plane in a reference system in which the z axis is the virtual photon direction, while the x axis is directed along the transverse momentum of the outgoing lepton:_ϕhis the azimuthal angle of_Ph,_ϕSis the azimuthal angle of the nucleon spin vector. The transverse momenta are defined as follows._kTis the transverse momentum of the quark inside the nucleon,_pTis the transverse momentum of the hadron with respect to the direction of the fragmenting quark, and_Ph⊥is the measurable transverse momentum of the produced hadron with respect to the z axis.

The SIDIS cross section for a transversely polarized target can be synthetically written as

_σt±=_σ0,t±+_STσS,t±sin(_ϕh−_ϕS)+_σC,t±sin(_ϕh+_ϕS)+…,

where_STis the transverse polarization of the target. The signs ± refer to the pion charge andt=p,d is the target type. In Equation (1) we retained only two terms: the Sivers term, associated to thesin(_ϕh−_ϕS)modulation, and the Collins term, associated to thesin(_ϕh+_ϕS)modulation. The corresponding asymmetries are the Collins asymmetry

_AC,t±=_{σC,t± σ0,t±},

and the Sivers asymmetry

_AS,t±=_{σS,t± σ0,t±}.

At leading twist and leading order in QCD, the Sivers component of the cross section couples the Sivers distribution_f1T⊥to the transverse-momentum-dependent unpolarized fragmentation function_D1 , yielding the asymmetry [32,33,34]

_AS(x,z,^Q2)=_{∑q,q¯ eq2}x∫^d2 _Ph⊥C_Ph⊥·_kTM_{Ph⊥f1T⊥D1∑q,q¯ eq2}x∫^d2 _Ph⊥C_{f1 D1},

where the convolutionCis defined as (w is a function of transverse momenta)

C[wfD]=∫^d2 _kT∫^d2 _pT^δ2(z_kT+_pT−_Ph⊥)×w(_kT,_pT)f(x,_kT2,^Q2)D(z,_pT2,^Q2).

The Collins term in the SIDIS cross section couples the transversity distribution_h1to the Collins fragmentation function_H1⊥ , and the resulting asymmetry is [32,34]

_AC(x,z,^Q2)=_{∑q,q¯ eq2}x∫^d2 _Ph⊥C_Ph⊥·_pTz_{Mh Ph⊥h1H1⊥∑q,q¯ eq2}x∫^d2 _Ph⊥C_{f1 D1}.

In order to perform the convolutions, one should know the transverse-momentum-dependence of distribution and fragmentation functions. As there is no information on that, one must make some assumption. It is usual, and convenient for computational purpose, to use a Gaussian form, with its width as a free parameter. However, as we will see, Gaussian widths will not play any rôle in our analysis, so we do not need to know their values. 3. Sivers Distributions We start from the extraction of the Sivers distributions, which is somehow easier, as it does not involve any unknown fragmentation function.

Adopting a Gaussian Ansatz for the transverse-momentum dependence of functions, i.e.,

_f1T⊥(x,_kT2,^Q2)=_f1T⊥(x,^Q2)^{e−_kT2/⟨_kT2⟩}π⟨_kT2⟩,

_D1(z,_pT2,^Q2)=_D1(z,^Q2)^{e−_pT2/⟨_pT2⟩}π⟨_pT2⟩,

the Sivers asymmetry (4) takes the form [35,36]

_AS(x,z,^Q2)=G_{∑q,q¯ eq2}x_f1T⊥(1)q(x,^Q2)z_D1q(z,^Q2)_{∑q,q¯ eq2}x_f1q(x,^Q2)_D1q(z,^Q2).

Here,_f1T⊥(1)is the first_kT2moment of the Sivers function, defined as

_f1T⊥(1)(x,^Q2)≡∫^d2 _kTkT22^M2_f1T⊥(x,_kT2,^Q2),

and_D1(z,^Q2) is the fragmentation function integrated over the transverse momentum. The factor G, resulting from the Gaussian integrations, is given by [35,36]

G=πM⟨_pT2⟩+^z2⟨_kT2⟩,

where⟨_kT2⟩and⟨_pT2⟩are the widths of the Sivers distribution and of the unpolarized fragmentation function, respectively. In the Gaussian model, G can be approximately related to the average transverse momentum of the produced hadrons,⟨_Ph⊥⟩, by

G≃πM2⟨_Ph⊥⟩.

As⟨_Ph⊥⟩is an experimentally determined quantity, the values of the average transverse momenta of quarks are irrelevant.

Our purpose is to extract from the data the_kT2moment of the Sivers distribution, thus we integrate over z,

_D˜1(^Q2)=∫dz_D1(z,^Q2),_D˜1(1)(^Q2)=∫dzz_D1(z,^Q2),

and consider the integrated asymmetry

_AS(x,^Q2)=G_{∑q,q¯ eq2}x_f1T⊥(1)q(x,^Q2)_D˜1q(1)(^Q2)_{∑q,q¯ eq2}x_f1q(x,^Q2)_D˜1q(^Q2).

Imposing isospin symmetry and SU(2) flavor symmetry of the pion sea, we can distinguish between favored and unfavored fragmentation functions as follows (superscripts ± refer to the pion charge),

_D1,fav≡_D1u+=_D1d−=_D1u¯−=_D1d¯+

_D1,unf≡_D1u−=_D1d+=_D1u¯+=_D1d¯−.

For the strange sector, following the work in [37] we set

_D1s±=_D1s¯±=_NsD1,unf,

where_Nsis a constant coefficient.

The denominators of the asymmetries_{∑q,q¯ eq2}x_{f1q D˜1q}for a proton and a deuteron target (p,d) and for charged pions, multiplied by 9, are given by (we use again isospin symmetry and ignore the charm components of the distribution functions, which are negligible in the kinematic region we will be considering)

p,^π+:x[4(_f1u+β_f1u¯)+(β_f1d+_f1d¯)+_Nsβ(_f1s+_f1s¯)]_D˜1,fav≡x_fp+D˜1,fav,

d,^π+:x[(4+β)(_f1u+_f1d)+(1+4β)(_f1u¯+_f1d¯)+2_Nsβ(_f1s+_f1s¯)]_D˜1,fav≡x_{fd^π+ D˜1,fav},

p,^π−:x[4(β_f1u+_f1u¯)+(_f1d+β_f1d¯)+_Nsβ(_f1s+_f1s¯)]_D˜1,fav≡x_{fp−D˜1,fav},

d,^π−:x[(1+4β)(_f1u+_f1d)+(4+β)(_f1u¯+_f1d¯)+2_Nsβ(_f1s+_f1s¯)]_D˜1,fav≡x_{fd−D˜1,fav},

with

β(^Q2)≡_D˜1,unf(^Q2)_D˜1,fav(^Q2).

Similar expressions can be written for the numerator of Equation (14),_{∑q,q¯ eq2}x_{f1T⊥(1)q D˜1q(1)}, with the replacements_D˜1→_D˜1(1),_f1→_f1T⊥(1), andβ→^β′, where

^β′(^Q2)=_D˜1,unf(1)(^Q2)_D˜1,fav(1)(^Q2).

Introducing the ratio of the first to the zeroth moment of the fragmentation functions,

ρ(^Q2)=_D˜1,fav(1)(^Q2)_D˜1,fav(^Q2),

we find for the pion asymmetries with a proton target (for simplicity we drop the S of Sivers)

_Ap+=Gρ4(_f1T⊥(1)u+^β′ _f1T⊥(1)u¯)+(^β′ _f1T⊥(1)d+_f1T⊥(1)d¯)+_Ns ^β′(_f1T⊥(1)s+_f1T⊥(1)s¯)_fp+,

_Ap−=Gρ4(^β′ _f1T⊥(1)u+_f1T⊥(1)u¯)+(_f1T⊥(1)d+^β′ _f1T⊥(1)d¯)+_Ns ^β′(_f1T⊥(1)s+_f1T⊥(1)s¯)_fp−,

and for the deuteron target

_Ad+=Gρ(4+^β′)(_f1T⊥(1)u+_f1T⊥(1)d)+(1+4^β′)(_f1T⊥(1)u¯+_f1T⊥(1)d¯)+2_Ns ^β′(_f1T⊥(1)s+_f1T⊥(1)s¯)_fd+,

_Ad−=Gρ(1+4^β′)(_f1T⊥(1)u+_f1T⊥(1)d)+(4+^β′)(_f1T⊥(1)u¯+_f1T⊥(1)d¯)+2_Ns ^β′(_f1T⊥(1)s+_f1T⊥(1)s¯)_fd−.

The combinations

_{fp+ Ap+}−_{fp− Ap−}=Gρ(1−^β′)(4_f1T⊥(1)uv−_f1T⊥(1)dv),

_{fd+ Ad+}−_{fd− Ad−}=3Gρ(1−^β′)(_f1T⊥(1)uv+_f1T⊥(1)dv),

select the valence Sivers distributions. From Equations (29) and (30), we get the valence distributions for u and d quarks, separately:

x_f1T⊥(1)uv=15Gρ(1−^β′)(x_{fp+ Ap+}−x_{fp− Ap−})+13(x_{fd+ Ad+}−x_{fd− Ad−}),

x_f1T⊥(1)dv=15Gρ(1−^β′)43(x_{fd+ Ad+}−x_{fd− Ad−})−(x_{fp+ Ap+}−x_{fp− Ap−}).

The asymmetry data we use to extract the Sivers distributions come from COMPASS measurements on proton [10] and deuteron targets [13] (we treat deuteron as the incoherent sum of a proton and a neutron). The unpolarized distribution functions_f1q are taken from the CTEQ5D global fit [38]. The unpolarized fragmentation functions are taken from the DSS parametrization [37]. Notice that in the DSS fit_D1u+is not assumed to be equal to_D1d¯+, but their difference is rather small. Thus, we identify_D1,favwith(_D1u++_D1d¯+)/2as given by DSS. In the DSS parametrization the factor_Nsis found to be 0.83.

The normalization of the Sivers distributions is determined by the quantityG=πM/2⟨_Ph⊥⟩. The values of⟨_Ph⊥⟩, measured by COMPASS, slightly depend on x, so that G ranges from 2.8 to 3.1.

We can now use Equations (31) and (32) to extract point-by-point the valence Sivers distributions from asymmetry data. The results are displayed in Figure 1. The error bars are the statistical uncertainties of the measured asymmetries. The x points correspond to different^Q2values, ranging from 1.2 GeV²to 20 GeV², with an average value⟨^Q2⟩≈4GeV².

The_uvSivers distribution is determined more precisely than the_dvdistribution, as the asymmetry measurements on the proton are considerably more accurate than the corresponding ones on the deuteron, in particular in the valence region (the COMPASS Collaboration has taken much less data on deuterons than on protons). Although affected by larger uncertainties, the_dv distribution appears to be negative. The COMPASS experiment has also provided data on kaon leptoproduction. These measurements have been analyzed in [20], where the resulting Sivers distributions were shown to be well compatible with those extracted from pion data and presented here.

We recall that the Sivers functions can be disentangled from the transverse momentum convolution and extracted with no need for the Gaussian model by considering the asymmetries weighted with_Ph⊥ [33]. The COMPASS Collaboration has recently performed this analysis obtaining a set of Sivers distributions in agreement with those presented here [39].

4. Transversity Distributions from Collins Asymmetries Let us now move to the point-by-point determination of transversity from the Collins asymmetry data.

Using again a Gaussian Ansatz for the transversity distribution and the Collins fragmentation function,

_h1(x,_kT2,^Q2)=_h1(x,^Q2)^{e−_kT2/⟨_kT2⟩}π_⟨kT2⟩S,

_H1⊥(z,_pT2,^Q2)=_H1⊥(z,^Q2)^{e−_pT2/⟨_pT2⟩}π⟨_pT2⟩,

the Collins asymmetry (6) becomes [40]

_AC(x,z,^Q2)=G_{∑q,q¯ eq2}x_h1q(x,^Q2)_H1q⊥(1/2)(z,^Q2)_{∑q,q¯ eq2}x_f1q(x,^Q2)_D1q(z,^Q2).

The “half-moment” of_H1⊥is defined as

_H1⊥(1/2)(z,^Q2)≡∫^d2 _pTpTz_MhH1⊥(z,_pT2,^Q2),

and in the Gaussian model is proportional to_H1⊥(z,^Q2) , as defined in Equation (34):

_H1⊥(1/2)(z,^Q2)=π⟨_pT2⟩2z_MhH1⊥(z,^Q2).

The factor G in Equation (35) is

G=11+^z2⟨_kT2⟩/⟨_pT2⟩.

With the reasonable assumption^z2⟨_kT2⟩/⟨_pT2⟩≪1, we can approximately setG≃1.

Being interested in the extraction of the transversity distributions, we can integrate over z,

_H˜1⊥(1/2)(^Q2)=∫dz_H1⊥(1/2)(z,^Q2),_D˜1(^Q2)=∫dz_D1(z,^Q2),

and write the integrated asymmetry as

_AC(x,^Q2)=_{∑q,q¯ eq2}x_h1q(x,^Q2)_{H˜1q⊥(1/2)}(^Q2)_{∑q,q¯ eq2}x_f1q(x,^Q2)_D˜1q(^Q2).

The favored and unfavored fragmentation functions_D1 are the same as in Equations (15) and (16). The corresponding relations for_H1⊥, based on isospin and flavor symmetries, are

_H1,fav⊥=_H1u⊥+=_H1d⊥−=_H1u¯⊥−=_H1d¯⊥+

_H1,unf⊥=_H1u⊥−=_H1d⊥+=_H1u¯⊥+=_H1d¯⊥−.

We assume_H1s⊥=_H1s¯⊥=0, as suggested by some models, and we ignore the c components of the distribution functions, which are negligible at thex,^Q2values of interest here. The denominators of the asymmetries_{∑q,q¯ eq2}x_{f1q D˜1q}for a proton and a deuteron target (p,d) and for charged pions, multiplied by 9, are -4.6cm0cm

p,^π+:x[4(_f1u+β_f1u¯)+(β_f1d+_f1d¯)+Nβ(_f1s+_f1s¯)]_D˜1,fav≡x_fp+D˜1,fav,

d,^π+:x[(4+β)(_f1u+_f1d)+(1+4β)(_f1u¯+_f1d¯)+2Nβ(_f1s+_f1s¯)]_D˜1,fav≡x_fd+D˜1,fav,

p,^π−:x[4(β_f1u+_f1u¯)+(_f1d+β_f1d¯)+Nβ(_f1s+_f1s¯)]_D˜1,fav≡x_{fp−D˜1,fav},

d,^π−:x[(1+4β)(_f1u+_f1d)+(4+β)(_f1u¯+_f1d¯)+2Nβ(_f1s+_f1s¯)]_D˜1,fav≡x_{fd−D˜1,fav},

whereβ is defined in Equation (22) and can be taken from standard parametrizations of fragmentation functions.

Similar expressions are obtained for the numerator of Equation (40),_{∑q,q¯ eq2}x_{h1q H˜1q⊥(1/2)}, with the replacements_D˜1→_H˜1⊥,_f1→_h1, andβ→α, where

α(^Q2)≡_{H˜1,unf⊥(1/2)}(^Q2)_{H˜1,fav⊥(1/2)}(^Q2).

Introducing the analyzing power

_aP(^Q2)=_{H˜1,fav⊥(1/2)}(^Q2)_D˜1,fav(^Q2),

we find for the proton target (for simplicity we drop the C of Collins)

_Ap+=_aP4(_h1u+α_h1u¯)+(α_h1d+_h1d¯)_fp+,

_Ap−=_aP4(α_h1u+_h1u¯)+(_h1d+α_h1d¯)_fp−,

and for the deuteron target

_Ad+=_aP(4+α)(_h1u+_h1d)+(1+4α)(_h1u¯+_h1d¯)_fd+,

_Ad−=_aP(1+4α)(_h1u+_h1d)+(4+α)(_h1u¯+_h1d¯)_fd−.

The combinations

_{fp+ Ap+}−_{fp− Ap−}=_aP(1−α)(4_h1uv −_h1dv )

_{fd+ Ad+}−_{fd− Ad−}=_aP3(1−α)(_h1uv +_h1dv )

select the valence transversity distributions. From Equations (53) and (54), we get the valence distributions for u and d quarks, separately:

x_h1uv =151_aP(1−α)(x_{fp+ Ap+}−x_{fp− Ap−})+13(x_{fd+ Ad+}−x_{fd− Ad−}),

x_h1dv =151_aP(1−α)43(x_{fd+ Ad+}−x_{fd− Ad−})−(x_{fp+ Ap+}−x_{fp− Ap−}).

The analyzing power_a˜Phis obtained from inclusive two-hadron production in electron–positron annihilation,^e+e−→_h1h2X, with the two hadrons in different hemispheres. In this process, the Collins effect is observed in the combination of the fragmenting processes of a quark and an antiquark, resulting in the product of two Collins functions.

The ratio of the unfavored to favored Collins function, Equation (47), is not constrained by the data, so we have to make some hypothesis. We assume the unfavored Collins function to be equal and opposite to the favored one,

_{H1,fav⊥(1/2)}(z,^Q2)=−_{H1,unf⊥(1/2)}(z,^Q2),

that is, we setα(^Q2)=−1. This assumption is suggested by the fact that the asymmetries for positive and negative pions are found to have approximately the same size but an opposite sign.

Using Equation (57), we find that the favored Collins function extracted from the Belle data [23] can be fitted as

_{H1,fav⊥(1/2)}(z,_QB2)=Nz^(1−z)γ_D1,fav(z,_QB2),_QB2=110^GeV2/^c2,

withC=0.46±0.03andγ=0.49±0.07. The fragmentation functions from the Belle value of the momentum transfer_QB2=110GeV²/^c2to the^Q2values of COMPASS data. The evolution of_H1⊥(1/2)(z,^Q2)involves unknown twist-3 fragmentation functions and cannot be implemented. Therefore, we simply assume that the analyzing power is constant in^Q2. The value we obtain is_aP=0.122.

Using the CTEQ5D unpolarized distribution functions [38] and the DSS unpolarized fragmentation functions [37], and the asymmetries measured by COMPASS into Equations (55) and (56), we find the valence transversity distributions plotted in Figure 2.

The valence u quark transversity distribution is positive and well determined, while the d quark has about the same size but an opposite sign and considerably larger uncertainties. We have checked the robustness of our results against different assumptions about the relation between the favored and the unfavored Collins function, and different hypotheses on the evolution of the fragmentation functions. The effects of all these changes are very small and negligible within the present uncertainties. 5. Transversity Distributions from Difference Asymmetries

Some information on the transversity distributions, with no need for an independent measurement of the Collins function, can be obtained from SIDIS by considering the so-called difference asymmetries, namely,

_AC,t≡_σC,t+−_{σC,t−σ0,t+}+_σ0,t−.

When taking the ratios of the asymmetries on deuteron and proton, the Collins fragmentation functions cancel out:

_{AC,d AC,p}=3(4_f1u+4_f1u¯+_f1d+_f1d¯)(_D1,fav+_D1,unf)+2(_f1s+_f1s¯)_D1,s5(_f1u+_f1d+_f1u¯+_f1d¯)(_D1,fav+_D1,unf)+4(_f1s+_f1s¯)_D1,sh1uv +_h1dv 4_h1uv −_h1dv ,

and the only unknowns are the transversity distributions. Thus, by measuring_ACon p and d, one obtains the ratio_h1dv /_h1uv in terms of known quantities.

The procedure for calculating the difference asymmetries from COMPASS data is described in [31]. The quantities(_h1uv +_h1dv )/(4_h1uv −_h1dv ) have been determined by using Equation (60) and standard parametrizations for the unpolarized parton distributions [38] and fragmentation functions [37]. Finally, from the quantities(_h1uv +_h1dv )/(4_h1uv −_h1dv ), the valence ratio_h1dv /_h1uv is obtained. This ratio is plotted in Figure 3 (solid circles) for the higher x bins (centered at 0.062, 0.100, 0.161, 0.280). The points at smaller x have much too large uncertainties, as the proton asymmetries in that region are compatible with zero, and have not been plotted. As expected, the uncertainties are large, but the results agree with those obtained by the Collins asymmetry analysis presented in the previous Section, as discussed in [31]. Averaging over the four points, one finds the ratio_h1dv /_h1uv to be−0.82±0.43in the x range spanned by the measurement. The large uncertainty is mainly due to the large uncertainty of the deuteron data.

It is interesting however to apply this method using the Collins asymmetry data of the 2010 proton run [9] and the projections for the new measurements which COMPASS plans to perform in 2021 and 2022 on a transversely polarized deuteron target [41]. The new run will balance the world data on proton and deuteron, making isospin separation much easier and more precise. The projections for the ratio_h1dv /_h1uv are also plotted in Figure 3 (open circles). As one can see, the gain of accuracy is impressive: the uncertainty of the weighted mean of the four points goes from±0.43to±0.11.

6. Conclusions We determined in a simple and direct way the Sivers distributions and transversity distributions of valence quarks from the COMPASS measurements of charged pion leptoproduction on proton and deuteron targets. Taking advantage of the variety of processes investigated by the COMPASS experiment with the same kinematics, we extracted the quark distributions point by point by combining only observable quantities on the basis of isospin symmetry. In order to factorize the distribution functions from the fragmentation functions we used a Gaussian model for the transverse momentum dependence, but the final results do not depend on the Gaussian widths. Thus, our approach does not involve any free parameter.

Both the transversity and the Sivers_uvand_dvdistributions obtained in our analysis are in good agreement with the results of previous phenomenological analyses, which fitted the data with a given functional form for the distributions in x.

In general, while the_uvdistributions are determined quite accurately, the_dvdistributions are more uncertain. A better knowledge of the_dvsector would require more data with a deuteron target. This is one of the goals of the next COMPASS run.

Figure 1. The firstkT2moments of the Sivers valence distributions,xf1T⊥(1)uv(red solid circles) andxf1T⊥(1)dv(black open circles).

Figure 2. Valence transversity distributions. Black circles representxh1uvand red squares representxh1dv.

Figure 3. Ratioh1dv/h1uvfrom the difference asymmetries. Solid circles: determination from existing measurements. Open circles: projection for future COMPASS run.

Author Contributions

Data curation, F.B. and A.M.; Formal analysis, V.B.; Writing, V.B., F.B. and A.M. All authors have read and agreed to the published version of the manuscript.

Funding

V.B. has been partially supported by "Fondi di Ricerca Locale" of the University of Piemonte Orientale. F.B. and A.M. have been partially supported by the Projects FRA2015 and FRA2018 of the University of Trieste.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We thank our colleagues of the Trieste COMPASS group for their collaboration on the analysis of difference asymmetries.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this paper:

DSS	de Florian, Sassot, Stratmann
QCD	Quantum chromodynamics
SIDIS	Semi-inclusive deep inelastic scattering

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