JLAB-THY-20-3186

The three-dimensional distribution of quarks in momentum space

Alessandro Bacchetta,1, 2, ∗ Filippo Delcarro,3, † Cristian Pisano,4, 5, ‡ and Marco Radici2, §

1Dipartimento di Fisica Nucleare e Teorica, Università di Pavia2INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy

3Jefferson Lab, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA4Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy

5INFN Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy

We present the distribution of unpolarized quarks in a transversely polarized proton in three-dimensional momentum space. Our results are based on consistent extractions of the unpolarizedand Sivers transverse momentum dependent parton distributions (TMDs).

The antipode of taking a picture of a black hole is tak-ing a picture of the inside of a proton, unveiling its inter-nal constituents, confined in the most common elementof the visible universe by the strong forces of QuantumChromodynamics (QCD). Using data obtained from thescattering of a hard virtual photon off a proton, we mapthe density of quarks in three dimensions, i.e., as a func-tion of their longitudinal momentum (along the photon’sdirection) and their transverse momentum (orthogonalto the photon). If the proton is unpolarized, the distri-bution is cylindrically symmetric: we determine it usingrecent results from our group [1]. If the proton is po-larized in the transverse plane, the distributions of upand down quarks turn out to be distorted in oppositedirections. This distortion, known as Sivers effect [2],is related to quark orbital angular momentum. We de-termine its details with the same formalism used for theunpolarized distribution. In this way, we obtain a consis-tent picture of the full 3-dimensional momentum distri-bution of quarks in a transversely polarized proton. Ourstudy constitutes a benchmark for future determinationsof multi-dimensional quark distributions, one of the maingoals of existing and planned experimental facilities [3–5].

We consider a frame where the proton has momentumP with space component in the +z direction, is polar-ized in the +y direction, and is probed by a spacelikevirtual photon with momentum q (with Q2 = −q2) inthe −z direction. We define the xy plane as transverseand we denote it with the subscript T . We consider thelight-cone + direction (t+ z)/

√2 and we define it as lon-

gitudinal. If Q2 is much larger than the proton’s massM2, the proton’s momentum is approximately longitudi-nal (P+ is the dominant component).

Our goal is to reconstruct the distribution of unpolar-ized quarks inside the nucleon as a function of three com-ponents of their momentum. In the frame we are consid-ering, the distribution of a quark with flavor a in a trans-versely polarized nucleon N↑ can be written in terms of

∗ alessandro.bacchetta@unipv.it† delcarro@jlab.org‡ cristian.pisano@unica.it§ marco.radici@pv.infn.it

two Transverse Momentum Distributions (TMDs) as [6]

ρaN↑(x, kx, ky;Q2) = fa1 (x, k2T ;Q2)− kxMf⊥a1T (x, k2T ;Q2) ,

(1)

where fa1 is the unpolarized TMD and f⊥a1T is the SiversTMD [2], k is the momentum of the quark, kT its trans-verse component, and x = k+/P+ is its longitudinal mo-mentum fraction. Q2 plays the role of a resolution scale.

Recent extractions of f1 have been published inRefs. [1, 7–9]. Several parametrizations of f⊥1T have beenreleased up to now [10–18]. In this work, we start froma recent determination of f1 by our group [1] and we ex-tract f⊥1T using the same formalism. Thus, for the firsttime we consistently reconstruct the full 3-dimensionalquark density of Eq. (1).

Both unpolarized and Sivers TMDs appear in the crosssection of polarized Semi-Inclusive Deep-Inelastic Scat-tering (SIDIS), i.e., the process `(l) + N(P ) → `(l′) +h(Ph) + X, where a lepton ` with momentum l scattersoff a nucleon target N with mass M and momentum P .In the final state, the scattered lepton with momentuml′ = l− q is detected, together with a hadron h with mo-mentum Ph and transverse momentum PhT . We definethe usual SIDIS variables xBj = Q2

2P ·q , z = P ·Ph

P ·q . In thisstudy, we neglect power corrections of order M2/Q2 andP 2hT /Q

2, which allow us also to identify xBj = x.The SIDIS cross section can be written in terms of

structure functions [19] that can be measured experimen-tally. Factorization theorems make it possible to writethe structure functions at small transverse momentum(P 2hT � Q2) in terms of TMDs and to derive evolution

equations that predict how TMDs change as functions oftwo scales µ2 and ζ [20]. These two scales are usuallychosen to be equal to the virtual photon mass Q2.

The unpolarized TMD f1 enters the structure functionFUU,T . The Sivers TMD f⊥1T enters the structure func-tion F sin(φh−φS)

UT,T , which occurs in the polarized part of thecross section weighted by sin(φh−φS), where φh and φSindicate the azimuthal orientations of PhT and the tar-get polarization ST in the transverse plane, respectively.Both structure functions can be defined as convolutionsof TMDs upon quark transverse momenta. Their Fouriertransform can be written more conveniently as a product

arX

iv:2

004.

1427

8v1

[he

p-ph

] 2

9 A

pr 2

020

2

of the Fourier transforms of TMDs, i.e., 1

F̃UU,T (x, z, b2T , Q2) =∑

a

e2axf̃a1 (x, b2T ;Q2)D̃a→h

1 (z, b2T ;Q2) ,(2)

F̃sin(φh−φS)UT,T (x, z, b2T , Q

2) =

−M∑a

e2axf̃⊥(1)a1T (x, b2T ;Q2)D̃a→h

1 (z, b2T ;Q2) ,(3)

where we introduced the first derivative of the Siversfunction in Fourier space [21]:

f̃⊥(1)a1T (x, b2T ;Q2) = − 2

M2∂b2T f̃

⊥a1T (x, b2T ;Q2). (4)

The above equation, evaluated at bT = 0, gives the firsttransverse moment of the Sivers function

f⊥(1)a1T (x;Q2) = f̃

⊥(1)a1T (x, 0;Q2) , (5)

which is an x-dependent function and corresponds to theso-called Qiu-Sterman function [22, 23].

Apart from the unpolarized TMD f1 and the SiversTMD f⊥1T , the above equations contain also the Fouriertransform of the unpolarized transverse-momentum-dependent fragmentation function D1, another essentialingredient for the extraction of the Sivers function.

In this work, we take the unpolarized functions f1 andD1 from our own extraction of Ref. [1], which we de-note as Pavia17. We extract the Sivers function usingthe same approach, based on the work of Collins, Soper,Sterman (CSS) [20, 24]. The analysis is done at the next-to-leading-logarithmic (NLL) accuracy, as defined in de-tail in [9].2 We avoid diverging perturbative contribu-tions using the so-called b∗ prescription and introducinga universal nonperturbative term in the TMD evolution,common to the unpolarized and Sivers TMDs. At vari-ance with the standard CSS approach, we also modify thehigh-transverse-momentum behavior of TMDs throughthe so-called bmin prescription.

We write the Sivers function at the initial scale (Q0 = 1GeV) as a product of a suitably normalized kT -dependentfunction and the first transverse moment, f⊥(1)1T . Thefunction reads

f⊥a1T (x, k2T ;Q20) = f

⊥(1)a1T (x;Q2

0) f⊥1TNP(x, k2T ) . (6)

The nonperturbative term f⊥1TNP is given by

f⊥1TNP(x, k2T ) =(1 + λS k

2T ) e−k

2T /M

21

Kπ (M21 + λSM4

1 )f1NP(x, k2T ), (7)

where the f1NP is consistently taken from the Pavia17extraction. The M1, λS are free parameters, and K is

1 More details are given in the supplemental material2 At this accuracy, the hard functions and the matching coefficientscan be neglected.

the normalization factor. The first transverse moment isparametrized as

f⊥(1)a1T (x;Q2

0) =Na

Siv

Gamax

xαa(1− x)βa

× [1 +Aa T1(x) +Ba T2(x)] fa1 (x;Q20) ,

(8)

where Tn(x) are Chebyshev polynomials of order n, andf1 are collinear parton densities consistently taken fromthe same set used in the Pavia17 fit [25]. The flavor-dependent factor Gamax is introduced to guarantee thepositivity bound of the Sivers function of Eq. (6) [26].The free parameters NSiv (varying only between −1 and1), α, β,A,B are different for up, down, and sea quarks.The total number of free parameters is 17.

To compute the Sivers function at a generic scale Q2,we apply TMD evolution at NLL. This is more conve-niently written in bT space and leads to

f̃⊥(1)a1T (x, b2T ;Q2) = eS(µ

2b ,Q

2) egK(bT ) ln(Q2/Q20)

× f⊥(1)a1T (x;µ2b) f̃

⊥(1)a1TNP (x, b2T ) ,

(9)

where µb is a scale proportional to 1/bT . With our pre-scriptions, we always have Q0 ≤ µb ≤ Q. At the ini-tial scale Q0, the exponentials reduce to unity and theabove equation indeed corresponds to the derivative ofthe Fourier transform of Eq. (6). For the transverse mo-ment f⊥(1)1T , we apply the same evolution as the collinearPDF f1 using the HOPPET code [27]. This is an approx-imation of the full evolution [28], but we checked thatmodifying this part of the evolution does not lead to sig-nificant changes. Much more relevant for TMD evolutionare the Sudakov form factor S and the function gK(bT ):they are present also in the unpolarized TMD function f1and are again taken from the Pavia17 fit. Without thisinformation, it would not be possible to reliably calculatethe Sivers function at the experimental scales.

We fix the free parameters of our functional form byfitting experimental data for single transverse-spin asym-metries [6]

Asin(φh−φS)UT (x, z,P 2

hT , Q2) ≈

Fsin(φh−φS)UT,T

FUU,T. (10)

An accurate extraction requires the inclusion of asym-metry measurements taken by different experimental col-laborations, covering different ranges of kinematic vari-ables, using different type of targets and final-statehadrons. In our fit we include measurements publishedby the HERMES [29], COMPASS [30, 31] and JLab collab-orations [32]. Usually, the asymmetries are presentedas projections of the same dataset in x, z, and PhT .To avoid fully correlated measurements, we fit only thex projections, since we are mainly interested in thex-dependence of the Sivers function. We select databy applying the same criteria used for the unpolarizedTMD fit, i.e., Q2 > 1.4 GeV2, 0.20 < z < 0.74 andPhT < min[0.2Q, 0.7Qz]+0.5 GeV. With these kinematic

3

cuts, we have a total of 118 data points: 30 from HERMES,82 from COMPASS (32 from the 2009 analysis, and 50 fromthe 2017 analysis), and 6 from JLab.

Similarly to our previous Pavia17 extraction and toother studies of parton densities [33–35], we perform thefit using the bootstrap method. The method consistsin creating M different replicas of the original data byrandomly shifting them with a Gaussian noise with thesame variance as the experimental measurement. Eachreplica represents the possible outcome of an independentmeasurement. We then fit each replica separately and weobtain a vector ofM results for each free parameter. Thenumber M is fixed by accurately reproducing the meanand standard deviation of the original data points. Inour case, it turns outM = 200, which is also consistentwith our Pavia17 fit [1].

The maximal information about our results is givenby the full ensemble of 200 replicas, combined with thecorresponding unpolarized TMD replicas. To report ourresults in a concise way, we adopt the following choice:for any result (χ2 values, parameter values, resulting dis-tribution functions) we quote intervals containing 68% ofthe replicas, obtained by excluding the upper 16% andlower 16% values. These intervals correspond to the 1σconfidence level only if the observable’s values follow aGaussian distribution, which is not true in general. Whenit is not possible to draw uncertainty bands, we reportthe results obtained using replica 105, which was selectedas the representative replica in the extraction of the un-polarized TMDs.

We obtain an excellent agreement between the ex-perimental measurements and our theoretical prediction,with an overall value of χ2/d.o.f.= 1.12 ± 0.06 (totalχ2 = 113 ± 6). Our parametrization is able to describevery well the COMPASS 2009 data set (32 points withχ2 = 28.8 ± 6.3), the COMPASS 2017 data set (50 pointswith χ2 = 31.8 ± 4.2), and the JLab data set (6 pointswith χ2 = 4±0.6). The agreement with the HERMES dataset is worse (30 points with χ2 = 50.4±4.3). We checkedthat the largest contribution to the χ2 comes from thesubset of data with K− in the final state. Our predic-tions well describe also the z and PhT distributions, evenif those projections of the data were not included in thefit. (More information about the fit procedure, the best-fit parameters and the agreement with data can be foundin the Supplemental Material.)

In Fig. 1, we show the first transverse moment xf⊥(1)1T(Eq. (5), multiplied by x) as a function of x at Q0 = 2GeV2 for the up (upper panel) and down quark (lowerpanel). We compare our results (solid band) with other

parametrizations available in the literature [14, 16, 17](hatched bands, as indicated in the figure). In agreementwith previous studies, the distribution for the up quark isnegative, while for the down quark is positive and bothhave a similar magnitude. The Sivers function for seaquarks is very small but not compatible with zero.

In general, the result of a fit is biased whenever a spe-cific fitting functional form is chosen at the initial scale.

�0.07

�0.06

�0.05

�0.04

�0.03

�0.02

�0.01

0.00

0.01

u

xf?(1)1T

PV11 [Q2 = 1.0 GeV2]TC [Q2 = 1.2 GeV2]EIKV [Q2 = 2.4 GeV2]This work [Q2 = 4.0 GeV2]

10�2 10�1 100

x

�0.01

0.00

0.02

0.04

0.06

d

PV11 [Q2 = 1.0 GeV2]TC [Q2 = 1.2 GeV2]EIKV [Q2 = 2.4 GeV2]This work [Q2 = 4.0 GeV2]

FIG. 1. The first transverse moment xf⊥(1)1T of the Sivers

TMD as a function of x for the up (upper panel) and downquark (lower panel). Solid band: the 68% confidence inter-val obtained in this work at Q2 = 4 GeV2. Hatched bandsfrom PV11 [14], EIKV [16], TC18 [17] and at different Q2 asindicated in the figure.

In our case, we tried to reduce this bias by adopting aflexible functional form, as it is evident particularly inEq. (8). Nevertheless, we stress that our extraction isstill affected by this bias and extrapolations outside therange where data exist (0.01 . x . 0.3) should be takenwith due care. At variance with other studies, in thedenominator of the asymmetry in Eq. (10) we are us-ing unpolarized TMDs that were extracted from data inour previous Pavia17 fit, with their own uncertainties.Therefore, our uncertainty bands in Fig. 1 represent themost realistic estimate that we can currently make on thestatistical error of the Sivers function.

In Fig. 2, we show the density distribution ρap↑ of unpo-larized quarks in a transversely polarized proton definedin Eq. (1), at x = 0.1 (upper panels) and x = 0.01 (lower

panels) and at the scale Q2 = 4 GeV2. The proton ismoving towards the reader and is polarized along the +ydirection. Since the up Sivers function is negative, the

4

x = 0.1

uup

1.0 0.5 0.0 0.5 1.0kx (GeV)

2

4ky = 02 4

1.0

0.5

0.0

0.5

1.0

k y(G

eV)

kx = 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

x = 0.1

ddp

1.0 0.5 0.0 0.5 1.0kx (GeV)

123ky = 0

1 2 31.0

0.5

0.0

0.5

1.0

k y(G

eV)

kx = 0

0.5

1.0

1.5

2.0

2.5

3.0

x = 0.01

uup

1.0 0.5 0.0 0.5 1.0kx (GeV)

102030

ky = 010 20 30

1.0

0.5

0.0

0.5

1.0

k y(G

eV)

kx = 0

5

10

15

20

25

x = 0.01

ddp

1.0 0.5 0.0 0.5 1.0kx (GeV)

10

20ky = 010 20

1.0

0.5

0.0

0.5

1.0

k y(G

eV)

kx = 0

5

10

15

20

FIG. 2. The density distribution ρap↑ of an unpolarized quark with flavor a in a proton polarized along the +y direction andmoving towards the reader, as a function of (kx, ky) at Q2 = 4 GeV2. Left panels for the up quark, right panels for the downquark. Upper panels for results at x = 0.1, lower panels at x = 0.01. For each panel, lower ancillary plots represent the 68%uncertainty band of the distribution at ky = 0 (where the effect of the distortion due to the Sivers function is maximal) whileleft ancillary plots at kx = 0 (where the distribution is the same as for an unpolarized proton). Results in the contour plotsand the solid lines in the projections correspond to replica 105.

induced distortion is positive along the +x direction forthe up quark (left panels), and opposite for the downquark (right panels).

At x = 0.1 the distortion due to the Sivers effect isevident, since we are close to the maximum value of thefunction shown in Fig. 1. The distortion is opposite forup and down quarks, reflecting the opposite sign of theSivers function. It is more pronounced for down quarks,because the Sivers function is larger and at the same timethe unpolarized TMD is smaller. At lower values of x, thedistortion disappears. These plots suggest that a virtualphoton hitting a transversely polarized proton effectively“sees” more up quarks to its right and more down quarksto its left in momentum space. The peak positions are ap-proximately (kx)max ≈ 0.1 GeV for up quarks and −0.15GeV for down quarks. To have a feeling of the order ofmagnitude of this distortion, we can estimate the expres-sion eq/(kx)max ≈ 2 × 10−34C ×m ≈ 0.6 × 10−4 debye,

which is about 3 × 10−5 times the electric dipole of awater molecule.

The existence of this distortion requires two ingredi-ents. First of all, the wavefunction describing quarksinside the proton must have a component with nonvan-ishing angular momentum. Secondly, effects due to finalstate interactions should be present [36], which in Feyn-man gauge can be described as the exchange of Coulombgluons between the quark and the rest of the proton [37].In simplified models [38], it is possible to separate thesetwo ingredients and obtain an estimate of the angularmomentum carried by each quark [39]. It turns out thatup quarks give almost 50% contribution to the proton’sspin, while all other quarks and antiquarks give less than10% [14]. We will leave this model-dependent study toa future publication. A model-independent estimate ofquark angular momentum requires the determination ofparton distributions that depend simultaneously on mo-

5

mentum and position [40, 41]. Nevertheless, the study ofTMDs, and of the Sivers function in particular, can pro-vide important constraints on models of the nucleon [42]and test lattice QCD computations [43].

In the near future, more data are expected from exper-iments at Jefferson Laboratory and CERN. Pioneeringmeasurements in Drell-Yan processes have been alreadyreported, but they are not included in the present analy-sis because of their relatively large uncertainties. In thelonger term, the recently approved Electron Ion Colliderproject [3] will provide a large amount of data in differ-ent kinematic regions compared to present experiments.With this abundance of data, we will be able to reducethe error bands, extend the range of validity of the ex-

tractions to lower and higher values of x, and obtain amuch more detailed knowledge of the 3-dimensional dis-tribution of partons in momentum space.

ACKNOWLEDGMENTS

This work is supported by the European ResearchCouncil (ERC) under the European Union’s Horizon2020 research and innovation program (grant agreementNo. 647981, 3DSPIN). This material is based upon worksupported by the U.S. Department of Energy, Office ofScience, Office of Nuclear Physics under contract DE-AC05-06OR23177.

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6

Appendix A: Supplemental material

1. Fourier transforms of structure functions

The structure functions of Eqs. (2) and (3) can be written more explicitly as [21]

FUU,T (x, z,P 2hT , Q

2) =1

2π

∫ ∞0

dbT bTJ0(bT |PhT |/z)F̃UU,T (x, z, b2T , Q2)

=1

2π

∑a

e2ax

∫ ∞0

dbT bTJ0(bT |PhT |/z)f̃a1 (x, b2T ;Q2)D̃a→h1 (z, b2T ;Q2) ,

(A1)

Fsin(φh−φS)UT,T (x, z,P 2

hT , Q2) =

1

2π

∫ ∞0

dbT b2TJ1(bT |PhT |/z)F̃ sin(φh−φS)

UT,T (x, z, b2T , Q2)

= −M2π

∑a

e2ax

∫ ∞0

dbT b2TJ1(bT |PhT |/z)f̃⊥(1)a1T (x, bT ;Q2)D̃a→h

1 (z, b2T ;Q2) .(A2)

The Fourier transforms of the TMDs are defined as

f̃a1 (x, b2T ;Q2) =

∫d2kT e

ibT ·kT fa1 (x, k2T ;Q2) = π

∫ ∞0

d|kT |2J0(bT |kT |)fa1 (x, k2T ;Q2) (A3)

f̃⊥(1)a1T (x, b2T ;Q2) =

∫d2kT e

ibT ·kT|kT |22M2

fa1 (x, k2T ;Q2) =π

M2

∫ ∞0

d|kT |2|kT |bT

J1(bT |kT |)f⊥a1T (x, k2T ;Q2) . (A4)

Note that there is a factor 2π difference compared to the definition in the Pavia17 extraction [1], which has beentaken into account in the rest of the article.

2. Details about the fit

We denote the replicated measurements as ASivr , with the r index running from 1 toM. Once replicas are generated,

a minimization procedure is applied to each replica separately to search for the parameter values, {pr0}, that minimizethe error function

E2r

({pr}

)=∑i

(ASivr −ASiv

th

({pr}

))2i(

∆ASivstat

)2i

+(∆ASiv

sys

)2i

+(∆ASiv

th

)2i

. (A5)

The terms in the denominator are the statistical and systematic experimental errors, assumed to be completelyuncorrelated, and the theoretical error due to the uncertainty in the unpolarized TMDs. For each replica separately,we identify the minimum and the corresponding values of best-fit parameters. The initial parameter values are chosenrandomly within reasonable intervals. For each replica, the goodness of the fit is evaluated using the usual χ2 test,which corresponds to the error function of Eq. (A5), but with the original experimental data instead of the replicatedones.

In Tab. I we give the value of the parameters obtained from our fit. For each one, we quote the central 68% of the200 replica values (by quoting the average ± the semi-difference of the upper and lower limits). Parameters of replica105, used for the multidimensional plots, are also given.

7

M1 λS αd αu αs

All replicas 0.97± 0.53 −0.45± 0.81 1.09± 0.96 0.22± 0.22 0.61± 0.54

Replica 105 0.87 −0.85 2.01 0.14 0.27

βd βu βs Ad Au As

All replicas 5.86± 4.13 2.00± 1.89 4.49± 4.45 −0.86± 21.70 −2.79± 4.37 2.87± 7.59

Replica 105 10.00 0.18 0.11 170.00 1.19 0.07

Bd Bu Bs NdSiv Nu

Siv NsSiv

All replicas 6.77± 12.90 2.24± 5.38 0.77± 3.50 1.99× 10−6 ± 1.00 −0.09± 0.52 0.04± 0.54

Replica 105 87.60 2.49 0.35 −1.00 1.00 0.31

TABLE I. Values of the best fit parameters for the Sivers distribution. Upper rows contain the central 68% confidence intervalsobtained by 200 replicas. Lower rows refer to the best fit parameters obtained from replica 105.

8

10−2 10−1 100

−0.06

−0.04

−0.02

0.00

u

xf⊥(1)1T

10−2 10−1 100−0.02

0.00

0.02

0.04

0.06

d

10−2 10−1 100

x

−0.5

0.0

0.5

1.0

1.5

2.0

2.5×10−3

s

FIG. 3. The first transverse moment of the Sivers function, xf⊥(1)1T , as a function of x calculated for the up, down and sea

quarks at the scale Q2 = 4 GeV2. The plots show all the 200 replicas obtained from the fit. For each value of x, the uncertaintybands contain the central 68% of the replicas.

9

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FIG. 4. HERMES Sivers asymmetries from SIDIS off a proton target (H) with production of π+, π0, π−, K+, K− in the finalstate, presented as a function of x, z, PhT . Only the x-dependent projections have been included in the fit.

10

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FIG. 5. COMPASS 2009 Sivers asymmetries from SIDIS off a deuteron target (6LiD) with production of π+, π−, K+, K− inthe final state, presented as function of x, z, PhT . Only the x-dependent projections have been included in the fit.

11

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FIG. 6. COMPASS 2017 Sivers asymmetries from SIDIS off a proton target (NH3) with production of positive hadrons h+,presented as function of x, z, PhT and divided in four different Q2 bins. Only the x-dependent projections have been includedin the fit.

12

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FIG. 7. COMPASS 2017 Sivers asymmetries from SIDIS off a proton target (NH3) with production of negative hadrons h−,presented as function of x, z, PhT and divided in four different Q2 bins. Only the x-dependent projections have been includedin the fit.

13

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FIG. 8. JLab Sivers asymmetries from SIDIS off a deuteron target (6LiD) with production of positive and negative π in thefinal state, presented as function of x. Only the x-dependent projections have been included in the fit.