# Reweighting the Sivers function with jet data from STAR

M. Boglione<EMAIL_ADDRESS>U. D’Alesio<EMAIL_ADDRESS>C. Flore<EMAIL_ADDRESS>J. O. Gonzalez-Hernandez
<EMAIL_ADDRESS>F. Murgia<EMAIL_ADDRESS>A.
Prokudin<EMAIL_ADDRESS>Dipartimento di Fisica Teorica, Università di
Torino, Via P. Giuria 1, Torino, I-10125, Italy  INFN, Sezione di Torino, Via
P. Giuria 1, Torino, I-10125, Italy  Dipartimento di Fisica, Università di
Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy  INFN,
Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA), Italy
Université Paris-Saclay, CNRS, IJCLab, 91405 Orsay, France  Division of
Science, Penn State University Berks, Reading, Pennsylvania 19610, USA  Theory
Center, Jefferson Lab, 12000 Jefferson Avenue, Newport News, Virginia 23606,
USA

###### Abstract

The reweighting procedure that using Bayesian statistics incorporates the
information contained in a new data set, without the need of re-fitting, is
applied to the quark Sivers function extracted from Semi-Inclusive Deep
Inelastic Scattering (SIDIS) data. We exploit the recently published single
spin asymmetry data for the inclusive jet production in polarized $pp$
collisions from the STAR Collaboration at RHIC, which cover a much wider $x$
region compared to SIDIS measurements. The reweighting method is extended to
the case of asymmetric errors and the results show a remarkable improvement of
the knowledge of the quark Sivers function.

###### keywords:

Sivers function , SIDIS , Single Spin Asymmetry , Reweigthing

††journal: Physics Letters B

##  1 Introduction

Single spin asymmetries (SSAs) in inclusive and semi-inclusive processes are
an invaluable tool to deepen our knowledge of the internal structure of
nucleons as well the hadronization mechanism. Despite the wealth and richness
of available data and an extensive theoretical effort carried out in the last
decades, the understanding of their origin still represents a formidable
challenge from the phenomenological point of view. Related to this issue,
there is nowadays a general consensus that the three-dimensional (3D) picture
of hadrons and the corresponding multi-parton correlations would lead to a
better comprehension of the hadronic structure.

The 3D hadron structure in momentum space is encoded in a new class of parton
distribution and fragmentation functions, the so-called Transverse Momentum
Dependent distributions and fragmentation functions (TMDs), which depend on
the collinear momentum fraction carried by the parton and its intrinsic
transverse momentum. Among the eight leading-twist nucleon TMD distributions,
the Sivers function [1, 2] is one of the most studied and plays a seminal
role. In fact, the Sivers distribution has several distinctive features: it is
naively time-reversal odd, its existence is related to a nonvanishing parton
orbital angular momentum and, more importantly, it is expected to be process
dependent, having opposite signs in SIDIS and Drell-Yan (DY) processes [3, 4].

For two-scale processes, such as SIDIS and DY, where $Q_{2}\gg
Q_{1}\sim\Lambda_{\rm QCD}$, TMD factorization [5, 6, 7] is proven and SSAs
are described in terms of convolutions of TMDs. These two scales are the
virtuality of the exchanged boson and the transverse momentum of the observed
hadron in SIDIS, and the invariant mass and transverse momentum of the lepton
pair in DY. Therefore, such measurements are sensitive to the non perturbative
transverse motion of bound partons in the nucleon encoded in TMDs. For
processes with one characteristic perturbative scale, or $Q_{1}\simeq
Q_{2}\gg\Lambda_{\rm QCD}$, the collinear factorization at twist-3 level [8,
9] plays the central role and SSAs are generated by the correlations of multi-
parton densities in the nucleon, the so-called collinear twist-3 functions
[10, 11, 8, 9]. It was theoretically proven and demonstrated
phenomenologically that in the intermediate region, $Q_{2}\gtrsim
Q_{1}\gg\Lambda_{\rm QCD}$, the two formalisms are related [12, 13, 14, 15,
16, 17]. TMDs can be expressed in terms of collinear and twist-3 functions via
Operator Product Expansion [18, 19, 20]. The integral relations [21, 22, 23]
based on operator definitions also show the close relation between TMD and
twist-3 distributions.

Inclusive jet production in $pp$ collisions is one example of single-scale
processes that can be described within the twist-3 approach, see for instance
Ref. [24]. Nevertheless, other approaches, such as the so-called generalized
parton model (GPM) [25, 26, 27], where a factorized formulation in terms of
TMDs is assumed as the starting point, have been successfully applied in the
analysis of SSAs for inclusive particle production in $pp$ collisions, as well
as in inclusive jet production [28]. The GPM is indeed a very powerful method
to study processes where factorization is not established and can, hopefully,
shed light on factorization breaking effects.

In the last decade, a color gauge invariant formulation of the GPM, named CGI-
GPM, has been proposed [29] and then extensively developed in Refs. [30, 31,
32]. Its main feature is the inclusion of initial- (ISI) and final- (FSI)
state interactions within a one-gluon exchange approximation. As a result, the
Sivers function becomes non-universal, and its calculable process dependence
can be absorbed into the partonic cross sections. Hence, in the evaluation of
physical observables, one can still use the quark Sivers function obtained
from SIDIS fits, but now convoluted with modified partonic cross sections,
such that the expected sign change from SIDIS to DY is restored. Moreover,
this modified GPM formalism has a very close connection [29] with the
collinear twist-3 approach. In all these respects, it is extremely interesting
to explore its potentiality and its implications.

In the spirit of testing the compatibility of the extraction of the Sivers
function, as obtained by best-fitting the SIDIS azimuthal asymmetries, we
analyze the recent SSA data for inclusive jet production in $pp$ collisions
from the STAR Collaboration at RHIC [33], within the GPM and the CGI-GPM
approaches. These data cover a wide region of $x$, expanding the range
explored in SIDIS measurements, and will allow us to improve and extend our
current knowledge on the quark Sivers function.

As a global fit would at present require prohibitive machine power, here we
employ an equivalent, but less numerically costly procedure, the so-called
reweighting method [34, 35, 36, 37, 38, 39] within Bayesian statistics.
Reweighting allows us to properly include the information from the new set of
data in the phenomenological analysis, estimate their impact on the extraction
of the quark Sivers distributions, and determine via Bayesian statistics the
fitted parameters and their errors. At the same time, this will also allow us
to test the relevance of the expected process dependence of the Sivers
function.

The paper is organized as follows. In Section 2 we recall the formalism
applied to compute the SIDIS azimuthal Sivers asymmetry and the corresponding
single spin asymmetries for inclusive jet production in $pp$ collisions. In
Section 3, we discuss the reweighting procedure and extend it to the case of
asymmetric errors. In Section 4 we apply reweighting to obtain the statistical
impact of the new STAR data sets to the extraction of the Sivers function.
Finally, in Section 5, we draw our conclusions.

##  2  The formalism

Here we briefly recall the main aspects of the theoretical formalism employed
in this study. All details can be found in the papers quoted below.

The expression for the azimuthal Sivers asymmetry in SIDIS, $\ell
p^{\uparrow}\to\ell^{\prime}hX$, is given by [40, 41]

$A_{UT}^{\sin(\phi_{h}-\phi_{S})}\equiv\frac{F_{UT}^{\sin(\phi_{h}-\phi_{S})}}{F_{UU,T}}\,,$
(1)

where $F_{UU,T}=\mathcal{C}[f_{1}^{q}D_{1}^{q}]$ is the TMD unpolarized
structure function, and
$F_{UT}^{\sin(\phi_{h}-\phi_{S})}=\mathcal{C}[f_{1T}^{\perp q}D_{1}^{q}]$ is
the azimuthal modulation triggered by the correlation between the nucleon spin
and the quark intrinsic transverse momentum. This effect is embodied in the
Sivers function [42]

$\displaystyle\Delta\hat{f}_{q/p^{\uparrow}}(x,\bm{k}_{\perp})$
$\displaystyle=$
$\displaystyle\Delta^{N}f_{q/p^{\uparrow}}(x,k_{\perp})\sin(\phi_{S}-\varphi)$
(2) $\displaystyle=$
$\displaystyle\\!\\!\\!\\!-\frac{2k_{\perp}}{M_{p}}f^{\perp
q}_{1T}(x,k_{\perp})\sin(\phi_{S}-\varphi)\,,$

which appears in the number density of unpolarized quarks, $q$, with intrinsic
transverse momentum $\bm{k}_{\perp}=k_{\perp}(\cos\varphi,\sin\varphi)$,
inside a transversely polarized proton $p^{\uparrow}$, with polarization
vector $\bm{S}_{T}=S_{T}(\cos\phi_{S},\sin\phi_{S})$ and mass $M_{p}$, moving
along the $z$ direction. The dependence of structure functions and TMDs on the
hard scale $Q^{2}$ is omitted for brevity.

For the inclusive jet production in $pp$ collisions, the SSA is defined as

$A_{N}\equiv\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\uparrow}+d\sigma^{\downarrow}}\equiv\,\frac{d\Delta\sigma}{2d\sigma}\,,$
(3)

where $d\sigma^{\uparrow(\downarrow)}$ denotes the single-polarized cross
section, in which one of the protons is polarized along the transverse
direction $\uparrow$($\downarrow$) with respect to the production plane. For
this process, within the GPM as well as the CGI-GPM approach, only the Sivers
effect, from quarks and gluons, can be at work.111A second possible TMD
effect, coming from the convolution of the transversity distribution and the
Boer-Mulders function, turns out to be negligible. The gluon contribution is
negligible in the region of moderate and forward rapidities, as well as at
small $x_{F}$ ($x_{F}=2P_{jL}/\sqrt{s}$, with $P_{jL}$ the longitudinal jet
momentum) as shown, for both approaches, in Refs. [43, 32]. The gluon Sivers
effect can therefore be safely ignored in this study.

Within the framework of the CGI-GPM, the numerator of the asymmetry is given
by [29]

$\displaystyle d\Delta$ $\displaystyle\sigma^{\rm CGI-
GPM}\,\equiv\,\frac{E_{j}\,d\sigma^{\uparrow}}{d^{3}\bm{P}_{j}}-\frac{E_{j}\,d\sigma^{\downarrow}}{d^{3}\bm{P}_{j}}$
(4)
$\displaystyle=\frac{2\alpha_{s}^{2}}{s}\sum_{a,b,c,d}\int\frac{dx_{a}\,dx_{b}}{x_{a}\,x_{b}}\;d^{2}\bm{k}_{\perp
a}\,d^{2}\bm{k}_{\perp b}$ $\displaystyle\times\left(-\frac{k_{\perp
a}}{M_{p}}\right)f^{\perp a}_{1T}(x_{a},k_{\perp a})\cos\varphi_{a}$
$\displaystyle\times f_{b/p}(x_{b},k_{\perp b})\,H^{\rm Inc}_{ab\to
cd}\>\delta(\hat{s}+\hat{t}+\hat{u})\>,$

where $\alpha_{s}$ is the strong coupling constant, $s$ is the center-of-mass
(c.m.) energy, $\bm{P}_{j}$ is the jet momentum, and $\hat{s}$, $\hat{t}$,
$\hat{u}$ are the usual Mandelstam variables for the partonic subprocess
$ab\to cd$. Furthermore, $f_{b/p}(x_{b},k_{\perp b})$ is the TMD distribution
for an unpolarized parton $b$ inside the unpolarized proton. Notice that in a
leading-order (LO) approach the jet is identified with the final parton $c$.
Moreover, $H^{\rm Inc}_{ab\to cd}$ are the perturbatively calculable hard
scattering functions. In particular, the $H^{\rm Inc}_{ab\to cd}$ functions
where $a$ is a quark or an antiquark can be found in Ref. [29]. The GPM
results can be obtained from Eq. 4 by simply replacing $H^{\rm Inc}_{ab\to
cd}$ with the standard unpolarized partonic cross sections, $H^{U}_{ab\to
cd}$.

Finally, the unpolarized cross section, $d\sigma$, at denominator in Eq. 3, is

$\displaystyle d\sigma$
$\displaystyle=\frac{\alpha_{s}^{2}}{s}\sum_{a,b,c,d}\int\frac{dx_{a}\,dx_{b}}{x_{a}\,x_{b}}\;d^{2}\bm{k}_{\perp
a}\,d^{2}\bm{k}_{\perp b}$ (5) $\displaystyle\times f_{a/p}(x_{a},k_{\perp
a})f_{b/p}(x_{b},k_{\perp b})\,H^{U}_{ab\to
cd}\>\delta(\hat{s}+\hat{t}+\hat{u})\>.$

In the computation of the jet SSA we will adopt the jet transverse momentum
$P_{jT}$ as the factorization scale.

##  3  The reweigthing method for TMDs

We now briefly illustrate the reweighting method that we will apply in our
study. This is a well established technique in the context of collinear PDF
extractions. Since the seminal work of Giele and Keller [34], the method has
been used in both the Bayesian framework [35, 44, 36, 37, 38, 39] and in the
Hessian approach [45, 46, 47]. The reweighting procedure allows to assess the
impact of new data sets on extractions of distributions describing hadron
structure, avoiding a new global fit. It also indicates whether these
additional data are consistent with the data sets used for the original
extraction.

Let us consider a model for TMDs depending on a $n$-dimensional set of
parameters $\boldsymbol{a}=\\{a_{1},\ldots,a_{n}\\}$. Traditionally, a
$\chi^{2}$ minimization procedure is employed in order to estimate the values
of parameters that describe the experimental data. Let us suppose that a set
of data $\boldsymbol{y}=y_{1},\ldots,y_{N_{\rm dat}}$ (with an associated
covariance matrix $C$) is measured. Then, the $\chi^{2}$ is defined as 222If
only uncorrelated uncertainties, $\sigma_{i}$, are given, the new $\chi^{2}$
reduces simply to
$\chi^{2}[\boldsymbol{a},\boldsymbol{y}]=\sum\limits_{i=1}^{N_{\rm
dat}}\dfrac{(y_{i}[\boldsymbol{a}]-y_{i})^{2}}{\sigma_{i}^{2}}$.

$\chi^{2}[\boldsymbol{a},\boldsymbol{y}]=\sum_{i,j=1}^{N_{\rm
dat}}(y_{i}[\boldsymbol{a}]-y_{i})\,C_{ij}^{-1}(y_{j}[\boldsymbol{a}]-y_{j})\,,$
(6)

where we have indicated by $y_{i}[\boldsymbol{a}]$ the values computed using
the theoretical model. The best-fit set $\boldsymbol{a}_{0}$, determined
through the $\chi^{2}$ minimization procedure, will have a corresponding
minimum value $\chi^{2}_{0}[\boldsymbol{a}_{0}]$. The uncertainty on the
extracted TMDs can either be calculated in terms of Hessian eigensets or
generated by applying Monte Carlo (MC) procedures. In an ideal case, both
methods yield the correct results and can be used in phenomenology. However,
the former relies on the Gaussianity of the underlying distributions and does
not necessarily account for the tails of distributions or for the potential
presence of multiple solutions to the minimum of $\chi^{2}$ in the parameter
space. In order to circumvent these complications, we will use a reweighting
procedure based on the Bayesian inference. This will allow us to exploit the
well known advantages of Bayesian inference, i.e. the ability to construct
probability distributions for the parameters, and study the influence of the
new data on the prior knowledge.

Let us assume that some prior knowledge, say theoretical, exists on the
parameters and that they are described by probability density functions
$\pi(\boldsymbol{a})$, i.e. the prior distributions. If no prior knowledge
exists, $\pi(\boldsymbol{a})$ are flat distributions.

We now generate the parameter sets, $\boldsymbol{a}_{k}$ ($k=1,\ldots,N_{\rm
set}$), with corresponding
$\chi^{2}_{k}\in[\chi^{2}_{0},\chi^{2}_{0}+\Delta\chi^{2}]$, where
$\Delta\chi^{2}$ is the desired tolerance corresponding to $n$ parameters and
at a certain confidence level (CL). For this purpose, we adopt a Markov Chain
Monte Carlo procedure and produce $N_{\rm set}$ parameter sets, employing a
Metropolis-Hastings algorithm with an auto-regressive generating density [48].
Such approach allows for an efficient exploration and reconstruction of the
parameter space, starting from the information contained in the error matrix
obtained from the minimization procedure. The $\chi^{2}_{k}$ for every
parameter set $\boldsymbol{a}_{k}$ is calculated as:

$\chi^{2}_{k}[\boldsymbol{a}_{k},\boldsymbol{y}]=\sum_{i,j=1}^{N_{\rm
dat}}(y_{i}[\boldsymbol{a}_{k}]-y_{i})\,C_{ij}^{-1}(y_{j}[\boldsymbol{a}_{k}]-y_{j})\,.$
(7)

Using Bayes theorem, one can then calculate the posterior density given the
data set as

$\cal{P}(\boldsymbol{a}|\boldsymbol{y})=\frac{{\cal
L}(\boldsymbol{y}|\boldsymbol{a})\,\pi(\boldsymbol{a})}{\mathit{Z}}\;,$ (8)

where ${\cal L}(\boldsymbol{y}|\boldsymbol{a})$ is the likelihood, and
$Z={\cal P}(\boldsymbol{y})$ is the evidence, that ensures a normalized
posterior density. $\cal{P}(\boldsymbol{a}|\boldsymbol{y})$ will therefore
incorporate the impact of the data on our knowledge of TMDs.

Different choices for the likelihood have been discussed in the literature.
Following Refs. [34, 36, 37, 38, 39], here we adopt the likelihood definition
as obtained by taking ${\cal
L}(\boldsymbol{y}|\boldsymbol{a})\,d\boldsymbol{y}$ as the probability to find
the new data confined in a differential volume $d\boldsymbol{y}$ around
$\boldsymbol{y}$. This results in defining the weights
$\mathit{w}_{\mathit{k}}$ as follows:

$\mathit{w}_{\mathit{k}}(\chi^{2}_{k})=\frac{\rm{exp}\left\\{-\frac{1}{2}\,\chi^{2}_{\mathit{k}}[\boldsymbol{a}_{\mathit{k}},\boldsymbol{y}]\right\\}}{\sum\limits_{i}\mathit{w}_{\mathit{i}}}\,.$
(9)

Such weights are normalized to 1, and can be used to calculate, for any given
observable $\cal{O}$, its expectation value $\rm{E}[\cal{O}]$ and variance
$\rm{V}[\cal{O}]$ as

$\rm{E}[\mathcal{O}]\simeq\sum_{\mathit{k}}\mathit{w}_{\mathit{k}}\,\mathcal{O}(\boldsymbol{a}_{\mathit{k}})\,,$
(10)
$\rm{V}[\mathcal{O}]\simeq\sum_{\mathit{k}}\mathit{w}_{\mathit{k}}\left(\mathcal{O}(\boldsymbol{a}_{\mathit{k}})-{\rm{E}}[\mathcal{O}]\right)^{2}.$
(11)

Note that Eqs. 10 and 11 are related to a symmetric error calculation. It is
well known that when a parameter is symmetrically distributed (e.g. according
to a Gaussian distribution) the mean value will coincide with the median value
and the uncertainty at a certain confidence level, for instance at $68\%$
(1$\sigma$) or $95\%$ (2$\sigma$) will be symmetric. Still, this does not
ensure that any distribution for any observable
$\mathcal{O}(\boldsymbol{a}_{k})$, depending on Gaussian distributed
parameters $\boldsymbol{a}_{k}$, is itself Gaussian.

In fact, asymmetric distributions may arise. Thus, mean and median for
$\mathcal{O}(\boldsymbol{a}_{k})$ would, in general, be different: the more
the distribution is asymmetric, the more the uncertainty on the observable is
not properly described by a symmetrized error, as discussed for example in
Ref. [49]. To overcome this potential issue, we extend the reweighting method,
providing asymmetric uncertainties. Such errors are calculated using the
rv_discrete function of the SciPy Python library [50]. This function is able
to build a discrete weighted distribution, providing automatically the mean
value, the median and the asymmetric uncertainty interval at a specific
confidence level. The interval is estimated considering equal areas around the
median, and the endpoints of these areas are given automatically by
rv_discrete. For our analysis we will adopt the median as central value, and
the uncertainty will be provided at a $2\sigma$ CL. We will perform this
procedure with the data and parametrizations considered in Ref. [51] and
explicitly demonstrate the equivalence of the reweighting procedure and the
global fit.

The Bayesian inference is easily generalized to evaluate the impact of the new
data $\boldsymbol{y}_{\rm new}$, in our case the data from STAR Collaboration
at RHIC [33]. As a matter of fact, the new evidence, i.e. the new data, will
change the weights $\mathit{w}_{\mathit{k}}\to\mathit{w}_{\mathit{k}}^{\rm
new}\equiv\mathit{w}_{\mathit{k}}(\chi^{2}_{k}+\chi^{2}_{\rm{new},k})$ for
each parameter set $\boldsymbol{a}_{k}$, where
$\chi^{2}_{\rm{new},k}=\chi^{2}_{k}[\boldsymbol{a}_{k},\boldsymbol{y}_{\rm
new}]$, Eq. (7). Therefore, the probability distribution for every parameter,
as well as any other observable that depends on the TMDs, is expected to
change. They can be calculated either with initial priors
$\pi(\boldsymbol{a})$ and weights
$\mathit{w}_{\mathit{k}}(\chi^{2}_{k}+\chi^{2}_{\rm{new},k})$ or,
equivalently, with posteriors $\cal{P}(\boldsymbol{a}|\boldsymbol{y})$ after
the global fit as the priors of reweighting and weights
$\mathit{w}_{\mathit{k}}(\chi^{2}_{\rm{new},k})$. Notice that, as expected,
weights are multiplicative,
$\mathit{w}_{\mathit{k}}(\chi^{2}_{k}+\chi^{2}_{\rm{new},k})\propto\mathit{w}_{\mathit{k}}(\chi^{2}_{k})\mathit{w}_{\mathit{k}}(\chi^{2}_{\rm{new},k})$,
as $\chi^{2}$ is additive. The resulting posteriors will contain the impact of
the new data and the new values of the parameters, and the observables can be
evaluated with Eqs. (10), (11) or with rv_discrete function. As already
mentioned, we will adopt the latter in our analysis.

In the following, we will apply the reweighting technique for the first time
to a TMD function, and in particular to the quark Sivers functions.

##  4  Results

We will now present and discuss our results on the reweighting of the quark
Sivers function. Before showing the outcome of the reweighting procedure using
jet $A_{N}$ data from STAR, let us briefly illustrate the model
parametrization chosen to describe the Sivers function.

Here we adopt the quark Sivers function extracted from SIDIS data in Ref.
[51]. More precisely, we use the so called “reference fit”, which results in a
minimum $\chi^{2}_{\rm dof}=0.99$ for $N_{\rm dat}^{\rm SIDIS}=220$ data
points (see Table 2 of Ref. [51]). Notice that this choice is motivated by the
simplicity of this fit, which makes it particularly suitable for the purposes
of this work. The relatively small number of parameters will allow us to
highlight the effects of reweighting the Sivers function. Another important
aspect is that this fit was performed paying special attention to the amount
of information one can actually infer from data, reducing the assumptions
which could bias the extraction. In this respect, we will be able to show and
quantify the impact of a new set of data on our knowledge of the Sivers
function using Bayesian inference.

The “reference fit” consists in a factorized $x$ and $k_{\perp}$ dependence
(the latter being Gaussian-like and flavor independent) for the up- and down-
quark Sivers function. Within this parametrization, the Sivers function reads

$\Delta^{N}\\!f_{q/p^{\uparrow}}(x,k_{\perp})=\frac{4M_{p}k_{\perp}}{\langle
k_{\perp}^{2}\rangle_{S}}\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)\frac{e^{-k_{\perp}^{2}/\langle
k_{\perp}^{2}\rangle_{S}}}{\pi\langle k_{\perp}^{2}\rangle_{S}}\,,$ (12)

where $q=u,d$, and where $\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)$ is the
Sivers first $k_{\perp}$-moment:

$\displaystyle\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)$ $\displaystyle=\int
d^{2}\boldsymbol{k}_{\perp}\frac{k_{\perp}}{4M_{p}}\Delta^{N}\\!f_{q/p^{\uparrow}}(x,k_{\perp})\equiv-
f_{1T}^{\perp(1)q}(x)$ (13) $\displaystyle=N_{q}\,(1-x)^{\beta_{q}}\,.$

Thus, the model depends on a total of five parameters:
$N_{u},N_{d},\beta_{u},\beta_{d},$ and $\langle k_{\perp}^{2}\rangle_{S}$.

In the computation of any asymmetry, special care should be taken in the
calculation of the corresponding unpolarized cross section. Here we follow the
same approach adopted in Ref. [51], and compute the unpolarized SIDIS cross
section, which appears at denominator in Eq. (1), by applying the
$k_{\perp}$-widths extracted in the multiplicity analysis of Ref. [52]. For
the jet single spin asymmetry, we adopt the corresponding $k_{\perp}$-width
resulting from the fit of HERMES SIDIS data.

For the collinear parton densities, we use the CTEQ6L1 set of PDFs [53] and
the DSS set of fragmentation functions [54]. Finally, for this fit, we
generate $N_{\rm set}=2\cdot 10^{5}$ parameter sets, adopting the
corresponding $\Delta\chi^{2}$ at $2\sigma$ CL for $N=5$ parameters, i.e.
$\Delta\chi^{2}=11.31$, as tolerance.

###  4.1  Impact of jet data on the quark Sivers extraction

We now proceed by illustrating our final results. Very recently, the STAR
Collaboration at RHIC has published new measurements of the single-spin
asymmetries in inclusive jet production from polarized proton-proton
collisions, at two different c.m. energies: $\sqrt{s}=200\,\text{GeV}$ and
$\sqrt{s}=500\,\text{GeV}$ [33]. This new data set, amounting to $N_{\rm
dat}^{\rm{jet}}=18$ points, covers a wide range in $x_{F}$ ($0.1\lesssim
x_{F}\lesssim 0.6$), and is useful to further constrain the quark Sivers
function in the large-$x$ region, where information from SIDIS data is either
scarce or even absent.

Figure 1: Results for the reweighting procedure from SIDIS and $A_{N}$ jet
data in the GPM formalism, compared with STAR measurements [33] at
$\sqrt{s}=200\,\text{GeV}$ (upper panel) and $\sqrt{s}=500\,\text{GeV}$ (lower
panel). Uncertainty bands are at $2\sigma$ CL. The results before (hatched
grey bands) and after (solid red bands) reweighting are shown.

STAR measurements refer to electromagnetic jet production from polarized $pp$
scattering. Two sets of data are collected at each energy: one fully inclusive
and one imposing a cut on the photon multiplicity ($n_{\gamma}>2$). The latter
is the most suitable for our analysis since it is not contaminated by single-
photon and isolated $\pi^{0}$ production. Nevertheless, for completeness, we
have also considered this data set and we will comment on the corresponding
results.

In the following, apart from showing the impact of the $A_{N}$ jet data on the
extraction of the Sivers function, we will also address whether any signal
pointing towards a process dependence of the Sivers function itself can be
observed.

As far as SIDIS is concerned, for the extraction of the Sivers function we
will refer to Ref. [51], applying a LO approximation. Indeed, extractions of
the Sivers function at higher perturbative orders exist, as in Refs. [55, 56,
57], but all the extracted Sivers functions are in good agreement, confirming
the findings of Ref. [51] on the weak dependence of the asymmetries on the
scale. We plan to perform similar TMD phenomenological analyses and
reweighting of the Sivers functions to higher perturbative orders in the
future.

In what follows, the predictions based on the Sivers functions as extracted by
fitting only the SIDIS data are dubbed as “SIDIS”, while the asymmetries
computed after the reweighting procedure, by using the jet data as the new
evidence as described in Section 3, are indicated by a “SIDIS+jet” label. The
central value and the asymmetric uncertainty bands at $2\sigma$ CL are
calculated using the procedure explained in Section 3, and adopting the
corresponding weights for the “SIDIS” and the “SIDIS+jet” (GPM and CGI-GPM)
cases.

Figs. 1 and 2 show the results of the reweighting procedure for the Sivers
function in the GPM and the CGI-GPM, respectively. On the upper (lower)
panels, the comparison with data at $\sqrt{s}=200\,(500)\,\text{GeV}$ is
shown.

Figure 2: Results for the reweighting procedure from SIDIS and $A_{N}$ jet data in the CGI-GPM formalism, compared with STAR measurements [33] at $\sqrt{s}=200\,\text{GeV}$ (upper panel) and $\sqrt{s}=500\,\text{GeV}$ (lower panel). Uncertainty bands are at $2\sigma$ CL. The results before (hatched grey bands) and after (solid green bands) reweighting are shown. | $N_{u}$ | $N_{d}$ | $\beta_{u}$ | $\beta_{d}$ | $\langle k_{\perp}^{2}\rangle_{S}$ | $\chi^{2}_{\rm dof}$
---|---|---|---|---|---|---
SIDIS | $0.40^{+0.05}_{-0.04}$ | $-0.65^{+0.13}_{-0.15}$ | $5.52^{+0.93}_{-0.83}$ | $6.77^{+2.29}_{-1.85}$ | $0.30^{+0.08}_{-0.08}$ | $1.01^{+0.03}_{-0.02}$
SIDIS+jet, GPM | $0.36^{+0.04}_{-0.03}$ | $-0.55^{+0.07}_{-0.10}$ | $4.98^{+0.34}_{-0.30}$ | $6.45^{+0.63}_{-0.52}$ | $0.28^{+0.07}_{-0.07}$ | $1.05^{+0.03}_{-0.01}$
SIDIS+jet, CGI-GPM | $0.35^{+0.02}_{-0.01}$ | $-0.43^{+0.01}_{-0.02}$ | $4.79^{+0.28}_{-0.19}$ | $4.48^{+0.17}_{-0.13}$ | $0.26^{+0.03}_{-0.02}$ | $1.25^{+0.04}_{-0.01}$
Table 1: Summary of the results of the reweighting procedure for the fitted
parameters. The quoted asymmetric uncertainties are at $2\sigma$ CL.

The results in Figs. 1 and 2 clearly show the effect of the reweighting
procedure. As we will see later, the jet SSA data measured by the STAR
Collaboration allow to drastically reduce the uncertainty on the quark Sivers
functions (especially for $d$ quarks) and, in turn, on the corresponding
estimates for the single-spin asymmetries in $p^{\uparrow}p\to{\rm jet}\,X$.
It is important to emphasize the role of the STAR data, that extend up to
$x_{F}\simeq 0.6$ with remarkably small uncertainties, offering valuable
information on the large $x$-region which in SIDIS remains largely uncovered.
The reweighting procedure clearly indicates that jet data offer a crucial
complementary information to SIDIS data. This analysis also points in favor of
a good compatibility between the two data sets.

Figure 3: Parameters and $\chi^{2}_{\rm dof}$ probability densities. Hatched
histograms refer to the priors coming from SIDIS data only; the red (green)
ones are the posterior densities, reweighted using jet data, in the GPM (CGI-
GPM) formalism.

To better interpret the results shown in Figs. 1 and 2, in Fig. 3 we present
the probability densities before and after reweighting for the five parameters
of the “reference fit” and the $\chi^{2}_{\rm dof}$. The corresponding values
and uncertainties, computed according to the procedure described at the end of
Section 3, are gathered in Table 1. Notice that the results for the “SIDIS”
case are fully compatible with the findings of Ref. [51]. Some comments are in
order:

  1. 1.

The flavor independent Sivers Gaussian width (lower left panel in Fig. 3) does
not vary significantly. This signals a mild role of TMD-evolution effects in
the available jet SSA data. It is indeed worth to notice that at large $x_{F}$
we probe large $P_{jT}$ values (up to $4\div 5\,\text{GeV}$ depending on the
data set). Since $P_{jT}$ represents the factorization scale adopted for this
observable, we reach $Q^{2}$ scales significantly larger than those probed in
SIDIS. However, asymmetries are ratios of cross sections where evolution and
higher order effects tend to cancel out [58]. Although our parametrization
does not have the complete features of TMD evolution, results of Refs. [51]
are compatible with full TMD evolution at higher logarithmic accuracy [55, 56,
57].

  2. 2.

The $\beta_{q}\,(q=u,d)$ parameters (mid panels in Fig. 3) that govern the
large-$x$ behavior of the Sivers function change, but in a different way when
applying the GPM or the CGI-GPM formalisms: this is due to the fact that in
the CGI-GPM approach color factors change the role of the $u$\- and $d$-quark
Sivers contributions. In particular, for the dominant channels in the forward
rapidity region, like $qg\to qg$, $H^{\rm inc}$ in the CGI-GPM approach
presents, roughly, a change of sign w.r.t. $H^{U}$ in the GPM (which are all
positive). This implies that while the slightly positive $A_{N}$ at large
$x_{F}$ in the GPM is driven by the positive sign of the up-quark Sivers
function, in the CGI-GPM is given by the negative down-quark Sivers function.

  3. 3.

Concerning the normalization parameters (upper panels in Fig. 3) we see that
while $N_{u}$ changes slightly, $N_{d}$ turns out to be smaller in size in the
CGI-GPM approach. On the other hand, as mentioned above, the corresponding
Sivers function for $d$-quarks is less suppressed in the large-$x$ region.

  4. 4.

The $\chi^{2}_{\rm dof}$ after the reweighting, calculated for $N_{\rm
dat}^{\rm{SIDIS+jet}}=238$ points, is different for the GPM and the CGI-GPM
cases, slightly favoring the former approach.

(a)

(b)

Figure 4: Comparison between the Sivers first $k_{\perp}$-moments (a) and
their values normalized to the corresponding central value (b) from SIDIS data
and their reweighted SIDIS+jet counterparts in the GPM (left panels) and CGI-
GPM (right panels) framework. In both plots, results for $u$\- (upper panels)
and $d$\- (lower panels) quarks are shown. Bands correspond to a $2\sigma$ CL.
Figure 5: Ratio of relative errors for the “SIDIS+jet” and the “SIDIS” cases
in the GPM (left panels) and the CGI-GPM (right panels) approaches, for
$u$-(upper panels) and $d$-(lower panels) quarks.

To better visualize the effect of the reweighting procedure on the Sivers
function, in Fig. 4 we show the comparison between the prior Sivers first
moments and the corresponding moments after reweighting in the GPM (left
panels) and in the CGI-GPM (right panels) frameworks. More specifically, in
Fig. 4(a) we compare the first $k_{\perp}$-moments before and after the
reweighting, while in Fig. 4(b) we show the ratio of each Sivers first moment
to its corresponding central value. As previously mentioned, the reweighting
allows to constrain the Sivers function in the large-$x$ region, not covered
by the current SIDIS data. No significant difference is observed in the
low-$x$ region, as the model parametrization does not have any parameter
controlling the low-$x$ behavior. The reduction of the uncertainty is
dramatic, especially for the $d$-quark Sivers function in the CGI-GPM
approach, that SIDIS leaves largely unconstrained. This is confirmed by the
much narrower reweighted probability density for the $\beta_{d}$ parameter
(see mid-right panel of Fig. 3).

In order to quantify the uncertainty reduction, in Fig. 5 we show the ratio
between the relative errors on the Sivers function before and after the
reweighting procedure, both for the GPM (left panels) and CGI-GPM (right
panels) formalisms. In the GPM approach, we see an uncertainty reduction of
about $60\%\,(80\%)$ for the reweighted $u$($d$)-quark Sivers function at
$x>0.2$, while in the CGI-GPM case, and in the same kinematical region, we
have $\sim 80\%$ and $\sim 90\%$ for the $u$ and $d$ Sivers function,
respectively.

Before concluding this section, let us comment on the use of the jet data set,
where no cut on the photon multiplicity is imposed. The corresponding results
of the reweighting are indeed very similar to those already shown in all
respects, apart from the fact that, in this case, the resulting $\chi^{2}_{\rm
dof}$ slightly favors the CGI-GPM approach.

##  5 Conclusions and outlook

In the present analysis we applied, for the first time, a reweighting
procedure to a TMD parton density, the quark Sivers distribution function
extracted from SIDIS data. By exploiting the recently published single spin
asymmetry data for inclusive jet production in polarized $pp$ collisions from
STAR [33], we showed the feasibility of such a procedure, which represents a
valuable alternative to a full global fit.

This allowed us, for the first time, to combine SIDIS azimuthal asymmetries
data and the single spin asymmetries measured in $p^{\uparrow}p\to{\rm
jet}\,X$ processes in a global analysis. Moreover, by using two different
approaches, the GPM and the CGI-GPM, we could also attempt to assess the
degree of process dependence of the Sivers function, beside its sign change.
Although the reweighted $\chi^{2}_{\rm dof}$ slightly favors the GPM approach,
further investigations are needed to have a clear discrimination between the
two formalisms.

By including the jet SSA data from STAR we were able to significantly improve
and extend our present knowledge of the quark Sivers function towards larger
$x$ values, not probed in current SIDIS data. In particular, their high
precision allows to remarkably reduce the uncertainties of the Sivers function
in such a kinematical region. We found that for the $u$-quark Sivers
distribution, such reduction is about $60\%$ in the GPM and $80\%$ in the CGI-
GPM frameworks, while for the $d$-quark case we observed a reduction of about
$80\%$ and $90\%$ for GPM and CGI-GPM respectively.

This work has to be considered as an exploratory study to show, on one side,
the potentiality of the reweighting procedure in the TMD sector and, on the
other side, to refine the behavior of the Sivers function in a region not
explored in SIDIS processes. The natural extension of this study will be a
global analysis including also SSAs for inclusive pion production. This will
allow us to simultaneously apply the reweighting procedure to the Collins
fragmentation function, to transversity and to the Sivers function, as
extracted by best-fitting the azimuthal asymmetries in SIDIS and $e^{+}e^{-}$
processes. We also expect that forthcoming SIDIS measurements from COMPASS
[59], JLab [60] and the future Electron Ion Collider [61, 62] will play a
crucial role in unravelling the nucleon structure in its full complexity.

This rather ambitious program will provide important information on the impact
of inclusive SSA data in the determination of these TMDs as well as a test of
the compatibility of their extractions.

## Acknowledgments

We would like to thank the STAR Collaboration for providing us with the
experimental data [33]. We are grateful to Mauro Anselmino for his involvement
in the early stages of this work. This project has received funding from the
European Union’s Horizon 2020 research and innovation programme under Grant
No. 824093 (STRONG-2020) (M.B, U.D., C.F., J.O.G.H., F.M.), by the National
Science Foundation under the Contract No. PHY-2012002 (A.P.), and by the US
Department of Energy under contract No. DE-AC05-06OR23177 (A.P.) under which
JSA, LLC operates Jefferson Lab, and within the framework of the TMD Topical
Collaboration (A.P.). C.F. is thankful to the Physics Department of Cagliari
University for the hospitality and support for his visit during which part of
the project was done.

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