# Simultaneous reweighting of Transverse Momentum Dependent distributions

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
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 Bayesian reweighting procedure is extended to the case of multiple
independent extractions of transverse momentum dependent parton distributions
(TMDs). By exploiting the data on transverse single spin asymmetries, $A_{N}$,
for inclusive pion production in polarized proton-proton collisions measured
at RHIC, we perform a simultaneous reweighting of the quark Sivers,
transversity and Collins TMD functions extracted from semi-inclusive deep
inelastic scattering (SIDIS) and $e^{+}e^{-}$ annihilation into hadron pairs.
The impact of the implementation of the Soffer bound, as well as the
differences between older and newer $A_{N}$ data, are investigated. The
agreement with $A_{N}$ data at large-$x_{F}$ values, a kinematical region
complementary to those explored in SIDIS measurements, is enhanced, improving
the knowledge of the polarized quark TMDs in the large-$x$ region.

###### keywords:

Bayesian reweighting , TMDs , Single Spin Asymmetry , Sivers effect , Collins
effect

††journal: Physics Letters B

##  1  Introduction

The idea of incorporating intrinsic transverse motion into the parton
distribution functions dates back to the papers by Feynman, Field, and Fox who
proposed to use it for the description of the transverse momentum dependent
Drell-Yan cross-sections [1, 2]. These functions were later named Transverse
Momentum Dependent distribution and fragmentation functions (TMD-PDFs and TMD-
FFs), collectively referred to as TMDs, and the TMD formalism was developed
for (polarized) Semi-Inclusive Deep Inelastic Scattering (SIDIS), Drell-Yan,
and $e^{+}e^{-}$ annihilation into hadron pairs [3, 4, 5, 6, 7, 8, 9, 10]. QCD
factorisation theorems were developed for TMDs [11, 12, 13, 14] and they were
probed experimentally in various processes [15, 16, 17, 18, 19, 20]. TMD
physics is one of the pillars of the experimental programs of JLab 12 [21] and
the future Electron-Ion Collider (EIC) [22, 23], as well as of RHIC [24] at
BNL, COMPASS/AMBER [25, 26, 27] at CERN, BABAR [19] at SLAC, Belle II [17] at
KEK and BESIII [28] in Beijing, of the fixed-target program at the LHC [29]
and at Tevatron with the SpinQuest [30] Drell-Yan program.

Historically, TMDs played a crucial role in explaining spin asymmetries [31,
32, 33, 34, 35, 36, 37] and, in particular, the large value of the so-called
left-right ($A_{N}$) or single-spin asymmetry (SSA) observed in proton-proton
collisions [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54]. Later on, it was argued that the so-called twist-3 formalism [55, 56] is
appropriate for the description of $A_{N}$, and it was shown that TMD
factorization is, at least formally, violated in hadron production in proton-
proton collisions [57]. Nevertheless, TMD and twist-3 formalisms are
intimately connected [58], and TMD and twist-3 functions can be related either
via specific integral expressions [59, 60, 61, 62, 63, 64] or through an
operator product expansion [14, 65, 66]. Recently it was demonstrated [67, 68]
that spin asymmetries can be successfully fitted using TMD and twist-3
formalisms.

By extending our previous study [69], in this paper we will attempt, for the
first time, to simultaneously analyse the available experimental data for spin
asymmetries in SIDIS, $e^{+}e^{-}$ scattering, and proton-proton collisions,
assuming factorised expressions in terms of TMDs for all those processes. We
will exploit two models for the TMD description of $A_{N}$: the usual
Generalized Parton Model (GPM) [35, 36, 37] which assumes that all TMDs are
universal, and the Color Gauge Invariant GPM (CGI-GPM) [70, 71, 72, 73] that
takes into account the process dependence of TMDs due to the direction of
gauge links in their corresponding operator definitions. The study will be
performed by extending the Bayesian reweighting technique [74, 75, 76, 77, 78,
79] to simultaneously reweight the results of global extractions of the
transversity and Sivers distribution functions [80, 81] and of the Collins
fragmentation functions (FFs) [82], using presently available data on $A_{N}$
in proton-proton collisions.

The paper is organized as follows: in Section 2 we recall the TMD formalism,
within the GPM and CGI-GPM approaches, then in Section 3 we summarize the
reweighting procedure. A suitable method to treat Monte Carlo sets is
discussed in Section 4. Results are presented in Section 5, while our
conclusions and remarks are gathered in Section 6.

##  2  Formalism

In this Section we summarize the formalism which will be used throughout our
phenomenological analysis.

Starting with the SIDIS processes, $\ell p^{\uparrow}\to\ell^{\prime}hX$, the
two azimuthal asymmetries we are interested in are related to the Sivers and
the Collins effects, properly defined within a TMD factorization theorem. For
the first one we have [83]

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

where $F_{UU}\sim f_{1}^{q}\otimes D_{1}^{q}$ is the TMD unpolarized structure
function, and $F_{UT}^{\sin(\phi_{h}-\phi_{S})}\sim f_{1T}^{\perp q}\otimes
D_{1}^{q}$ [3, 84, 8] is the azimuthal modulation coming from the correlation
between the nucleon spin and the intrinsic transverse momentum of the
unpolarized quark. This effect is encoded in the Sivers function. For the
second one, involving also the transversity distribution, one has

$A_{UT}^{\sin(\phi_{h}+\phi_{S})}=\frac{2(1-y)}{1+(1-y)^{2}}\frac{F_{UT}^{\sin(\phi_{h}+\phi_{S})}}{F_{UU}}\;,$
(2)

where $y$ is the fractional energy loss of the incident lepton, and
$F_{UT}^{\sin(\phi_{h}+\phi_{S})}\sim h_{1}^{q}\otimes H_{1}^{\perp q}$ [3,
84, 8] is the polarized structure function of the SIDIS cross section, given
as a convolution of the TMD transversity distribution, $h_{1}^{q}$, and the
Collins FF, $H_{1}^{\perp q}$.

As is well known, in order to access this TMD fragmentation function, one has
to gather information from another complementary process, namely
$e^{+}e^{-}\to h_{1}h_{2}X$. Here one analyses the transverse momentum
inbalance of the two hadron produced in opposite hemispheres. In such a case,
still within a TMD factorization scheme, a convolution of two Collins FFs
appears via a $\cos(2\phi_{0})$ modulation [7]:

$A_{0}^{UL(C)}\sim H_{1}^{\perp{\bar{q}}}\otimes H_{1}^{\perp q}\,.$ (3)

This was measured at the energy $\sqrt{s}\simeq 10.6$ GeV by the Belle [17]
and BABAR [19] collaborations as well as by the BESIII [28] collaboration, at
a lower energy, $\sqrt{s}\simeq 3.65$ GeV.

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

$A_{N}\equiv\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\uparrow}+d\sigma^{\downarrow}}\equiv\,\frac{d\Delta\sigma}{2d\sigma}\,,$
(4)

where $d\sigma^{\uparrow(\downarrow)}\equiv
E_{h}\,d\sigma^{\uparrow(\downarrow)}/d^{3}\bm{P}_{h}$ stands for the single-
polarized cross section, in which one of the initial-state protons is
transversely polarized ($\uparrow$($\downarrow$)) with respect to the
production plane. Here, we adopt the GPM, a phenomenological model where a
factorized formulation in terms of TMDs is assumed as the starting point, and
in which one includes spin and transverse momentum correlation effects. For
completeness, we will also consider an extension of this approach, the CGI-
GPM, where initial- and final-state interactions are properly included in a
one-gluon-exchange approximation. Notice that this model allows for the well-
known process dependence of the Sivers function expected when one moves from
SIDIS to Drell-Yan processes [85].

As discussed in Refs. [36, 37], in $p^{\uparrow}p\to hX$ processes only two
effects survive the integration over the intrinsic transverse momenta and
their relative azimuthal phases: the Sivers and Collins effects. In the first
case, formally, also a contribution from gluons could appear. On the other
hand, in the region of moderate and forward rapidity we are interested in, as
shown in Refs. [86, 73], the gluon Sivers effect can be safely ignored.

It is important to stress that in inclusive processes these two TMD effects
cannot be separated. Therefore the numerator of $A_{N}$ will be

$d\Delta\sigma\simeq d\Delta\sigma_{\rm Siv}+d\Delta\sigma_{\rm Col}\,.$ (5)

Starting with the Sivers effect, within the CGI-GPM, the numerator of the
asymmetry can be schematically given as [70]

$\displaystyle d\Delta\sigma_{\rm Siv}^{\rm CGI-GPM}$ $\displaystyle\propto$
$\displaystyle\sum_{a,b,c,d}f^{\perp a}_{1T}(x_{a},k_{\perp
a})\cos\varphi_{a}\otimes f_{b/p}(x_{b},k_{\perp b})$ (6)
$\displaystyle\mbox{}\otimes H^{\rm Inc}_{ab\to cd}\otimes
D_{h/c}(z,{k}_{\perp h})\,,$

where $f_{b/p}(x_{b},k_{\perp b})$ is the TMD distribution for an unpolarized
parton $b$ inside the unpolarized proton. 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. [70]. The GPM result can be obtained from Eq. (6) by
simply replacing $H^{\rm Inc}_{ab\to cd}$ with the standard tree-level
unpolarized partonic cross sections, $H^{U}_{ab\to cd}$. Finally, the
unpolarized cross section, $d\sigma$, in the denominator in Eq. 4, can be
obtained by further replacing the Sivers function and its phase in the GPM
expression with the corresponding unpolarized TMD-PDF for the parton $a$.

Focusing now on the Collins effect, we have to consider only the GPM
expression, given as

$\displaystyle d\Delta\sigma_{\rm Col}^{\rm GPM}$ $\displaystyle\propto$
$\displaystyle\sum_{a,b,c,d}h_{1a}(x_{a},k_{\perp a})\otimes
f_{b/p}(x_{b},k_{\perp b})$ (7) $\displaystyle\mbox{}\otimes
d\Delta\sigma^{a^{\uparrow}b\to c^{\uparrow}d}\otimes H_{1}^{\perp
c}(z,k_{\perp h}),$

where $d\Delta\sigma^{a^{\uparrow}b\to c^{\uparrow}d}\equiv
d\sigma^{a^{\uparrow}b\to c^{\uparrow}d}-d\sigma^{a^{\uparrow}b\to
c^{\downarrow}d}$ is the transverse spin transfer at the partonic level. In
fact, in this case the functions are universal and are not modified by the
direction of the gauge links [87, 88].

##  3  Simultaneous reweighting

We now illustrate the method we have developed for a simultaneous Bayesian
reweighting of functions initially extracted from fits to independent
datasets. For simplicity, we will focus on the case of two functions, although
this approach can be easily generalized to $n$ independent extractions.

Let us consider two statistically independent functions, $f({\boldsymbol{a}})$
and $g({\boldsymbol{b}})$ depending, respectively, on $n_{a}$\- and
$n_{b}$-dimensional sets of parameters
${\boldsymbol{a}}=\left\\{a_{1},\dots,a_{n_{a}}\right\\}$ and
${\boldsymbol{b}}=\\{b_{1},\dots,b_{n_{b}}\\}$. The value of these parameters
is determined by performing two distinct fits to independent datasets
$\bm{E}^{a}$ and $\bm{E}^{b}$. For each of these fits, a $\chi^{2}$, defined
as 111If (e.g. for the fit $\boldsymbol{a}$) only uncorrelated uncertainties
$\sigma_{i}^{a}$ are given, the new $\chi^{2}$ reduces simply to
$\chi^{2}_{a}[\boldsymbol{a};\bm{E}^{a}]=\sum\limits_{i=1}^{N_{\rm
dat}^{a}}\dfrac{(T_{i}[\boldsymbol{a}]-E_{i}^{a})^{2}}{({\sigma_{i}^{a}})^{2}}$.:

$\displaystyle\chi^{2}_{a}\equiv\chi^{2}[\boldsymbol{a};{\bm{E}}^{a}]=\sum_{i,j=1}^{N_{{\rm
dat}}^{a}}(T_{i}[\boldsymbol{a}]-E_{i}^{a})\,(C_{ij}^{a})^{-1}(T_{j}[\boldsymbol{a}]-E_{j}^{a})\,,$
(8)
$\displaystyle\chi^{2}_{b}\equiv\chi^{2}[\boldsymbol{b};{\bm{E}}^{b}]=\sum_{i,j=1}^{N_{{\rm
dat}}^{b}}(T_{i}[\boldsymbol{b}]-E_{i}^{b})\,(C_{ij}^{b})^{-1}(T_{j}[\boldsymbol{b}]-E_{j}^{b})\,,$

is minimized and the best fit $\boldsymbol{a}_{0}$ and $\boldsymbol{b}_{0}$,
corresponding to the minima $\chi^{2}_{0,a}$ and $\chi^{2}_{0,b}$, are
determined. In the equations above, $T_{i}[\boldsymbol{a}]\equiv
T_{i}(f(\boldsymbol{a}))$ are the theoretical estimates corresponding to the
experimental data points $E_{i}^{a}$, and $C_{ij}^{a}$ is the covariance
matrix for the fit $\boldsymbol{a}$ (and similarly for the fit
$\boldsymbol{b}$). The fit uncertainties can then be computed via a Hessian
method or with a suitable Monte Carlo (MC) procedure. Using the latter method,
the probability density functions $\pi(\boldsymbol{a})$ and
$\pi(\boldsymbol{b})$ are reconstructed by generating $N_{\rm set}^{a}$ sets
$\boldsymbol{a}_{i}$ and $N_{\rm set}^{b}$ sets $\boldsymbol{b}_{j}$
respectively. Notice that these distributions are statistically independent
from each other. Then, expectation values and variances for any quantity
$\mathcal{O}$ depending on one of the parameter sets (e.g. $\boldsymbol{a}$)
can be computed as

$\displaystyle{\rm E}[{\mathcal{O}}]=\frac{1}{N_{\rm
set}^{a}}\sum\limits_{\mathit{i}=1}^{N_{\rm
set}^{a}}{\mathcal{O}}(\boldsymbol{a}_{\mathit{i}})\,,$ (9) $\displaystyle{\rm
V}[\mathcal{O}]=\frac{1}{N_{\rm set}^{a}}\sum\limits_{\mathit{i}=1}^{N_{\rm
set}^{a}}\left(\mathcal{O}(\boldsymbol{a}_{\mathit{i}})-{\rm{E}}[\mathcal{O}]\right)^{2}\,.$

Let us now suppose that a new set of data ${\bm{E}}$ (with an associated
covariance matrix $C$) is measured, and that these data can be described by a
linear combination of $f(\boldsymbol{a})$ and $g(\boldsymbol{b})$. Then, we
can compute the $\chi^{2}$ corresponding to these new data as

$\chi^{2}_{{\rm
new}}[\boldsymbol{a},\boldsymbol{b};{\bm{E}}]=\sum_{i,j=1}^{N_{\rm
dat}}(T_{i}[\boldsymbol{a},\boldsymbol{b}]-E_{i})\,C_{ij}^{-1}(T_{j}[\boldsymbol{a},\boldsymbol{b}]-E_{j})\,.$
(10)

Since $f$ and $g$ come from statistically independent fits, the uncertainty
bands for the theoretical predictions $T_{i}[\boldsymbol{a},\boldsymbol{b}]$
have to be built by taking all possible ($N_{\rm set}^{a}\times N_{\rm
set}^{b}$) combinations of the MC parameter sets $\boldsymbol{a}_{i}$ and
$\boldsymbol{b}_{j}$. Thus, the $\chi^{2}$ on the new data will depend on the
$i$-th and $j$-th MC sets:

$\chi^{2}_{{\rm new}}\equiv\chi^{2}_{ij,{\rm new}}=\chi^{2}_{{\rm
new}}[\boldsymbol{a}_{i},\boldsymbol{b}_{j};{\bm{E}}]$ (11)

leading to ($N_{\rm set}^{a}\times N_{\rm set}^{b}$) values of $\chi^{2}$.

By using Bayes theorem, we can then evaluate the impact of these new data on
our prior distributions $\pi(\boldsymbol{a})$ and $\pi(\boldsymbol{b})$. Since
these distributions are a priori independent, we can build a factorized prior
$\pi(\boldsymbol{a},\boldsymbol{b})=\pi(\boldsymbol{a})\pi(\boldsymbol{b})$
and apply Bayes theorem to compute the posterior densities:

${\cal P}(\boldsymbol{a},\boldsymbol{b}|{\boldsymbol{E}})=\frac{{\cal
L}({\boldsymbol{E}}|\boldsymbol{a},\boldsymbol{b})\,\pi(\boldsymbol{a},\boldsymbol{b})}{\mathit{Z}}\;.$
(12)

where ${\cal L}({\boldsymbol{E}}|\boldsymbol{a},\boldsymbol{b})$ is the
likelihood, and $Z={\cal P}({{\boldsymbol{E}}})$ is the evidence, that ensures
a normalized posterior density.

Various choices for the likelihood have been discussed in the literature [74,
75]. The resulting weights depend on such a choice. Following Ref. [89], we
define the weights $\mathit{w}_{\mathit{i}\mathit{j}}$ as

$\mathit{w}_{\mathit{i}\mathit{j}}(\chi^{2}_{\rm new})=\frac{{\rm
exp}\left\\{-\frac{1}{2}\frac{\chi^{2}_{ij,{\rm
new}}}{\Delta\chi^{2}}\right\\}}{\sum\limits_{k,l}{\rm
exp}\left\\{-\frac{1}{2}\frac{\chi^{2}_{kl,{\rm
new}}}{\Delta\chi^{2}}\right\\}}\,.$ (13)

where $\Delta\chi^{2}$ is the tolerance at a certain confidence level (CL) for
$n_{a}+n_{b}$ parameters. Notice that these weights are just the weights
defined in the original work by Giele and Keller [74], but by rescaling the
exponent: $\chi^{2}_{ij}\to\chi^{2}_{ij}/\Delta\chi^{2}$.

For a generic extraction with $N_{\rm set}$ sets and weights
$\mathit{w}_{\mathit{k}}$, this rescaled version is equivalent to the Hessian
reweighting [89], and allows to retain a larger effective number of sets
$N_{\rm eff}$, defined as

$N_{\rm eff}={\rm exp}\left\\{\sum\limits_{k=1}^{N_{\rm
set}}\mathit{w}_{\mathit{k}}\ln\left(\frac{1}{\mathit{w}_{\mathit{k}}}\right)\right\\}\,.$
(14)

$N_{\rm eff}$ can be considered as a measure for how many sets have
effectively a non-negligible weight. If $N_{\rm eff}\ll N_{\rm set}$, the
method is considered no longer reliable, signalling that either the new data
require a full refitting, or that they are inconsistent with the old ones
[75].

Coming back to the case of study, one can obtain a reweighted quantity as

$\displaystyle\rm{E}[\mathcal{O}]$ $\displaystyle=\sum\limits_{i=1}^{N_{\rm
set}^{a}}\sum\limits_{j=1}^{N_{\rm
set}^{b}}\mathit{w}_{\mathit{i}\mathit{j}}\,\mathcal{O}(\boldsymbol{a}_{\mathit{i}},\boldsymbol{b}_{\mathit{j}})\,,$
(15) $\displaystyle\rm{V}[\mathcal{O}]$
$\displaystyle=\sum\limits_{i=1}^{N_{\rm set}^{a}}\sum\limits_{j=1}^{N_{\rm
set}^{b}}\mathit{w}_{\mathit{i}\mathit{j}}\left(\mathcal{O}(\boldsymbol{a}_{\mathit{i}},\boldsymbol{b}_{\mathit{j}})-{\rm{E}}[\mathcal{O}]\right)^{2}\,.$

Then, to compute a reweighted quantity that depends only on
$f(\boldsymbol{a})$ (or $g(\boldsymbol{b})$), one has to evaluate the
corresponding weights $\mathit{w}_{\mathit{i}}$ (or $\mathit{w}_{\mathit{j}}$)

$\mathit{w}_{\mathit{i}}=\sum\limits_{j=1}^{N_{\rm
set}^{b}}\mathit{w}_{\mathit{i}\mathit{j}}\,,\qquad\mathit{w}_{\mathit{j}}=\sum\limits_{i=1}^{N_{\rm
set}^{a}}\mathit{w}_{\mathit{i}\mathit{j}}\,,$ (16)

and use again the weighted sums in Eq. (15) with the new, updated weights
$\mathit{w}_{\mathit{i}}$ (or $\mathit{w}_{\mathit{j}}$) for the corresponding
$\mathcal{O}(\boldsymbol{a}_{\mathit{i}})$ (or
$\mathcal{O}(\boldsymbol{b}_{\mathit{j}})$). By doing so, one is able to
evaluate the impact of the new data on the two independent prior distributions
$\pi(\boldsymbol{a})$ and $\pi(\boldsymbol{b})$ and on any quantity depending
on the parameters $\boldsymbol{a}$ and/or $\boldsymbol{b}$. Notice that, as
the weights defined in Eq. (13) are by construction normalized to one,
$\mathit{w}_{\mathit{i}}$ and $\mathit{w}_{\mathit{j}}$ are automatically
normalized to one too.

In general, the new data could introduce a correlation between the a priori
statistical independent fits. The correlation is encoded in the ($N_{\rm
set}^{a}\times N_{\rm set}^{b}$) combinations computed, and is taken into
account in the evaluation of a reweighted quantity. Notice also that the
distribution can be a posteriori independent. This would imply that, if a
covariance matrix using the reweighted parameters is built, this matrix will
be factorized into two submatrices and the off-diagonal terms, describing the
correlation between the $i$-th parameter set $\boldsymbol{a}_{i}$ and the
$j$-th parameter set $\boldsymbol{b}_{j}$, will be small.

##  4  Compressing the MC sets

Here, we discuss how to faithfully represent the parameter space with a
reduced number of sets.

In our case, after the initial fitting stage, we generate (e.g. for the fit
$\boldsymbol{a}$) $N^{a}_{\rm set}$ MC sets $\boldsymbol{a}_{i}$, each one
with a corresponding
$\chi^{2}_{a,i}\in[\chi^{2}_{0,a},\chi^{2}_{0,a}+\Delta\chi^{2}_{a}]$. Again,
$\chi^{2}_{0,a}$ is the minimum found by the fit and $\Delta\chi^{2}_{a}$ is a
tolerance that depends on the number of parameters and is given at a certain
CL. These sets allow us to reliably reconstruct the parameter distribution
$\pi(\boldsymbol{a})$, provided a sufficiently large number of sets is
generated. For instance, in Refs. [90, 79, 69], the number of sets needed was
up to $O(10^{6})$. In the case we consider here, involving two independent
functions, this implies up to $O(10^{12})$ combinations in Eq. (11). In order
to reduce the computational cost, we will use a compression procedure, which
we describe in what follows.

Starting from the full sample of parameters $\boldsymbol{a}_{i}$, we select a
random sample
$\boldsymbol{a}_{k}^{\prime}=\\{\boldsymbol{a}_{1}^{\prime},\dots,\boldsymbol{a}^{\prime}_{N^{a^{\prime}}_{\rm
set}}\\}$ with $N^{a^{\prime}}_{\rm set}\ll N^{a}_{\rm set}$. If
$\pi({\boldsymbol{a}_{k}^{\prime}})\simeq\pi(\boldsymbol{a}_{i})$, then one
also expects that
$\pi({\mathcal{O}}(\boldsymbol{a}_{k}^{\prime}))\simeq\pi({\mathcal{O}}(\boldsymbol{a}_{i}))$.
In other words, if the sample $\boldsymbol{a}_{k}^{\prime}$ renders a
statistically equivalent distribution to that of $\boldsymbol{a}_{i}$, the
corresponding distributions of any quantity will not differ significantly.
Thus, this procedure would help in decreasing the number of combinations
needed to be computed for the simultaneous reweighting.

The question now is how to check that the sampled distribution is
statistically equivalent to the full one. As discussed in Section C.2 of Ref.
[91], we adopt as measure the Welch’s $t$-statistic, defined as

$t=\frac{\mu_{\boldsymbol{a}}-\mu_{\boldsymbol{a}^{\prime}}}{\sqrt{\frac{\sigma^{2}_{a}}{N^{a}_{\rm
set}}+\frac{\sigma^{2}_{a^{\prime}}}{N^{a^{\prime}}_{\rm set}}}}\,,$ (17)

which quantifies the difference of the arithmetical means of two samples (in
our case, the full sample $\boldsymbol{a}_{i}$ and the random sample
$\boldsymbol{a}_{k}^{\prime}$) with unequal variances and sizes. One has to
verify that $|t|$ is such that the corresponding $p$-values are $\gtrsim 0.1$.
Provided this condition holds, one can conclude that the sampled distribution
and the original distribution are statistically equivalent. Notice that, at
variance with Ref. [91], we do not sample in the observable space (i.e.
$A_{N}$ in our case), but rather in the parameter space. Moreover, since the
Welch’s $t$-statistic implicitly assumes underlying Gaussian distributions we
also verify the compatibility between the medians and uncertainty intervals of
the samples $\boldsymbol{a}_{k}^{\prime}$ and $\boldsymbol{a}_{i}$. This
allows us to correctly sample asymmetric distributions. To check how this
compression algorithm works, we detail below an explicit example.

###  4.1  An explicit example

Let us consider the reweighting performed for the quark Sivers function using
STAR $A_{N}$ jet data [69]. Here, we will re-perform the reweighting using the
rescaled weights as defined in Eq. (13).

In the original work, $N^{a}_{\rm set}=2\cdot 10^{5}$ MC sets were generated
and used to represent the uncertainty on the up- and down-quark Sivers
functions. Here, we will select a random sample $\boldsymbol{a}_{k}^{\prime}$
of the parameter sets $\boldsymbol{a}_{i}$, with $N^{a^{\prime}}_{\rm set}\ll
N^{a}_{\rm set}$, checking that their corresponding distribution
$\pi(\boldsymbol{a}_{k}^{\prime})$ is statistically equivalent to
$\pi(\boldsymbol{a}_{i})$ and re-perform the reweighting using only the
reduced sample of sets. By applying the compression algorithm presented above,
we select only 1% of the initial sample, i.e. $N^{a^{\prime}}_{\rm set}=2\cdot
10^{3}$ sets.

Figure 1: Comparison between reweighted curves for the full (grey bands) and
reduced (hatched bands) samples for the reweighting analysis of Ref. [69].

Fig. 1 shows the comparison between reweighted curves for the GPM (upper
panels) and the CGI-GPM (lower panels) approaches, together with STAR data. As
in Ref. [69], the central values are the median values, and the bands are at
$2\sigma$ CL. The plot clearly shows that median values and uncertainties of
the full sample (grey bands) are correctly and satisfactorily reproduced by
the reduced sample of MC sets (hatched bands). We verified that the same
happens for the unweighted curves, not shown here. For completeness, and to
make this comparison more explicit, in Fig. 2 we show the reweighted
uncertainties (normalized to their central value) for the Sivers first moment
for up- and down-quark in the GPM (left panels) and CGI-GPM (right panels).
Reweighted curves from the full sample are shown in grey with black dashed
borders, while the ones for the reduced sample are shown in hatched colors.
These results allow us to validate the compression procedure, that we will use
in what follows for simultaneously reweighting the Sivers, transversity and
Collins functions.

Figure 2: Comparison between reweighted first moments of up- (upper panels)
and down-quark (lower panels) Sivers functions, normalized to their central
value, using full (grey bands) and reduced (hatched bands) samples for the
reweighting analysis of Ref. [69].

##  5  Data selection and results

We now present and discuss the data selection and the results for the
simultaneous reweighting of the quark Sivers function, the TMD transversity
and the Collins fragmentation function. First, let us briefly illustrate the
adopted parametrizations and the updated extractions of the aforementioned
TMDs. In what follows, we employ as collinear input the MSHT20nlo proton PDFs
[92] and the DEHSS fragmentation functions for pions and kaons [93, 94].

The unpolarized TMD PDFs and FFs are parametrized using a factorized Gaussian
ansatz as:

$\displaystyle
f_{a/p}(x,k_{\perp}^{2})=f_{a/p}(x)\frac{e^{-k_{\perp}^{2}/\langle
k_{\perp}^{2}\rangle}}{\pi\langle k_{\perp}^{2}\rangle}$ (18) $\displaystyle
D_{h/q}(z,p_{\perp}^{2})=D_{h/q}(z)\,\frac{e^{-p^{2}_{\perp}/\langle
p^{2}_{\perp}\rangle}}{\pi\langle p^{2}_{\perp}\rangle}$

with $\langle k_{\perp}^{2}\rangle=0.57$ GeV2 and $\langle
p_{\perp}^{2}\rangle=0.12$ GeV2 as extracted from a fit to HERMES
multiplicities [95].

For the up- and down-quark Sivers functions, we adopt the parametrization of
Ref. [90], that consists in factorized $x$ and $k_{\perp}$ dependences (the
latter being Gaussian-like and flavor independent):

$\Delta^{N}\\!f_{q/p^{\uparrow}}(x,k_{\perp})=\frac{4M_{p}k_{\perp}}{\langle
k_{\perp}^{2}\rangle_{S}}\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)\frac{e^{-k_{\perp}^{2}/\langle
k_{\perp}^{2}\rangle_{S}}}{\pi\langle k_{\perp}^{2}\rangle_{S}}\,,$ (19)

where $q=u,d$, $M_{p}$ is the proton mass, and where
$\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)$ is the Sivers first
$k_{\perp}$-moment [83]:

$\displaystyle\Delta^{N}\\!f_{q/p^{\uparrow}}^{(1)}(x)$ $\displaystyle=\int
d^{2}\boldsymbol{k}_{\perp}\frac{k_{\perp}}{4M_{p}}\Delta^{N}\\!f_{q/p^{\uparrow}}(x,k_{\perp})\equiv-
f_{1T}^{\perp(1)q}(x)$ (20) $\displaystyle=N_{q}\,(1-x)^{\beta_{q}}\,.$

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

Following Section 3.1 of Ref. [90], in the computation of $F_{UU}$ (see Eq.
(1)), we coherently adopt the corresponding Gaussian widths extracted from
HERMES and COMPASS multiplicities. The latter have been obtained in the
framework of a Gaussian approximation, and no difference was found for the
fitted widths used in the previous extraction.

The adopted dataset is slightly different with respect to what used in Ref.
[90]. In fact, as the HERMES Collaboration published new data for a
transversely polarized proton target [96], we replaced the old, superseded
proton data with the latest ones. Notice also that we do not include any DY
data, as this would need a more extended analysis that is beyond the scope of
this study.

For the transversity and Collins functions we adopt the same parametrization
as in Refs. [97, 79]. The transversity function is parametrized as

$h_{1}^{q}(x,k^{2}_{\perp})=h_{1}^{q}(x)\frac{e^{-k^{2}_{\perp}/\langle
k^{2}_{\perp}\rangle}}{\pi\langle k^{2}_{\perp}\rangle}.$ (21)

where the Gaussian width is assumed to be the same as for the unpolarized TMD-
PDFs. As in Refs. [98, 99, 97, 79], the $x$-dependent part of the TMD
transversity is parametrized at the initial scale $Q_{0}^{2}$ in terms of the
Soffer bound [100]:

$\displaystyle h_{1}^{q}(x,Q_{0}^{2})$
$\displaystyle=\mathcal{N}^{T}_{q}(x)\frac{1}{2}\left[f_{q/p}(x,Q_{0}^{2})+g_{1L}^{q}(x,Q_{0}^{2})\right]$
(22) $\displaystyle\equiv\mathcal{N}^{T}_{q}(x)\,{\rm SB}(x,Q_{0}^{2}).$

where

${\cal
N}^{T}_{q}(x)=N^{T}_{q}x^{\alpha}(1-x)^{\beta}\,\frac{(\alpha+\beta)^{\alpha+\beta}}{\alpha^{\alpha}\beta^{\beta}},\quad(q=u_{v},\,d_{v})$
(23)

with the same $\alpha$ and $\beta$ parameters for the valence $u_{v}$ and
$d_{v}$ transversity functions, for a total of four parameters for
$h_{1}^{q}$. We stress that we do not enforce the automatic fulfillment of the
Soffer bound ($\lvert N_{q}^{T}\rvert\leq 1$), but we apply such a constraint
a posteriori on the generated MC sets. As shown in Ref. [79], this choice
allows to avoid a bias in the fitting procedure and to properly estimate the
uncertainty on the transversity functions. We also expect that such a choice
will render larger asymmetries for the Collins contribution in Eq. (7).

The Collins function is parametrized as in Refs. [98, 99, 97, 79]:

$H_{1}^{\perp q}(z,p_{\perp}^{2})={\cal
N}^{C}_{q}(z)\frac{zm_{h}}{M_{C}}\,\sqrt{2e}\,e^{-p_{\perp}^{2}/M_{C}^{2}}\,D_{h/q}(z,p_{\perp}^{2})\,,$
(24)

with $q=\rm{fav},\rm{unf}$ (favored/unfavored), $m_{h}$ is the produced hadron
mass, and where $M_{C}$ is a free parameter with mass dimension.
$D_{h/q}(z,p_{\perp}^{2})$ is again the unpolarized TMD fragmentation
function. The ${\cal N}^{C}_{q}(z)$ factors are given by

${\cal
N}^{C}_{\rm{fav}}(z)=N^{C}_{\rm{fav}}\,z^{\gamma}(1-z)^{\delta}\frac{(\gamma+\delta)^{\gamma+\delta}}{\gamma^{\gamma}\delta^{\delta}}\,,\quad{\cal
N}^{C}_{\rm{unf}}(z)=N^{C}_{\rm{unf}}\,,$ (25)

for a total of five parameters for the Collins functions.

For its importance and later use we also give the expression for the first
moment of the Collins function [101]

$\displaystyle H_{1}^{\perp(1)\,q}(z)$ $\displaystyle=z^{2}\int
d^{2}\bm{p}_{\perp}\frac{p_{\perp}^{2}}{2m_{h}^{2}}\,H_{1}^{\perp
q}(z,z^{2}p_{\perp}^{2})$ (26)
$\displaystyle=\sqrt{\frac{e}{2}}\frac{1}{zm_{h}}\frac{M_{C}^{3}\langle
p^{2}_{\perp}\rangle}{\left(\langle
p^{2}_{\perp}\rangle+M_{C}^{2}\right)^{2}}\,{\cal N}^{C}_{q}(z)D_{h/q}(z)\,,$

where the last line is obtained adopting the parametrization in Eq. (24).

The preliminary fits showed that the $\delta$ parameter in Eq. (25) is always
compatible with zero. For this reason we fix it to zero in the following.
Thus, the total number of parameters for the $h_{1}^{q}$ and $H_{1}^{\perp q}$
fit decreases from nine to eight, and the total number of free parameters
entering our analysis is $N_{\rm par}=13$.

For building the Soffer bound, we adopt the DSSV set [102] for the collinear
helicity distributions, $g_{1L}(x)$. By using an appropriately modified
version [103, 104] of the HOPPET code [105], a transversity DGLAP kernel is
employed to evolve $h_{1}(x)$ up to higher values of $Q^{2}$. We adopt
$Q_{0}^{2}=0.81\,\text{GeV}^{2}$ as the input scale in Eq. (21), with
$\alpha_{S}(M_{Z})\simeq 0.118$. Finally, for the collinear part of the
Collins function, we adopt a DGLAP evolution. As shown for instance in Ref.
[106], such a choice renders very similar results to those obtained by using
the full TMD evolution. In this respect, our scheme is well sufficient to
address the main issues of our study.

Also in this case, we update the dataset employed in Ref. [79] by using the
newest HERMES proton data for the Collins effect in SIDIS [96] and adding the
JLab data from Ref. [20].

For the two independent extractions of TMDs from the Sivers and Collins
asymmetries we generate $\mathit{O}(10^{5})$ sets using a Markov chain MC, and
apply the compression algorithm discussed in Section 4 to select $2\cdot
10^{3}$ MC sets for each extraction. This amounts to $4\cdot 10^{6}$
combinations to be computed for each of the $A_{N}$ bins we will use for the
simultaneous reweighting. In what follows, we adopt the median as central
value, and the uncertainties for unweigthed and reweighted observables are at
2$\sigma$ CL, with corresponding $\Delta\chi^{2}=22.69$ for 13 parameters.

###  5.1 Simultaneous reweighting on $A_{N}$ data for inclusive pion
production

We start by illustrating the comparison between our predictions for $A_{N}$
for the inclusive production of charged and neutral pions in the GPM and CGI-
GPM formalisms with the experimental data used for the simultaneous
reweighting. We consider as new evidence the preliminary data for $A_{N}$
measured by BRAHMS for $\pi^{\pm}$ production at $\sqrt{s}=200$ GeV [45], the
data from STAR for $\pi^{0}$ production at $\sqrt{s}=200$ GeV [43, 46, 49] and
the latest STAR data for non-isolated $\pi^{0}$ production from Ref. [54] at
$\sqrt{s}=200$ GeV and $\sqrt{s}=500$ GeV.

In the computation of $A_{N}$, the transverse momentum of the final state
pion, $P_{T}$, is adopted as the hard scale. For the applicability of the
(CGI-)GPM formalism, we then select only the data points with $P_{T}>1$ GeV.

Hereafter, we present the unweighted curves, coming from predictions based on
the information from SIDIS and $e^{+}e^{-}$ asymmetries only, in grey. The
reweighted curves in the GPM and the CGI-GPM approaches are depicted
respectively with red and green bands. Moreover, we depict with grey points
the data having $P_{T}<1.5$ GeV. This is done to make manifest some
kinematical regions in which the perturbative approach may be questioned,
especially if one were to look at scale uncertainties (see e.g. Ref. [107] for
a recent discussion on the computation of scale uncertainties). Finally, we
anticipate that the dominant contribution to $A_{N}$ comes from the Collins
effect (see Eq. (7)).

Let us start from the results for BRAHMS for charged pion production. Before
entering into our main discussion, few comments are in order. Indeed, on the
one hand, we know that $\pi^{0}$ data are mostly sensitive to the relative
contribution of up- and down-quark TMD distributions (Sivers or transversity,
depending on the effect considered), weighted with the same suitable
fragmentation function. On the other hand, $\pi^{\pm}$ data allow for a more
direct flavor separation as compared to $\pi^{0}$ ones, giving a larger
discriminating power to any phenomenological study. In view of their relevance
we have therefore included these datasets in our analysis, even if still
unpublished and covering a limited kinematical range. Charged pion $A_{N}$
measurements at future facilities like the EIC [22, 23], the JLab 22 program
[108], AMBER [27] and the proposed fixed-target program at the LHC [29] will
surely help in improving future TMD analyses.

Figure 3: Results for the simultaneous reweighting of the Sivers, transversity
and Collins functions: unweighted and reweighted predictions for BRAHMS
$A_{N}^{\pi^{\pm}}$ data [45] in the GPM (left panels) and the CGI-GPM (right
panels) approaches are presented. Data points in grey correspond to
$P_{T}<1.5$ GeV.

In Fig. 3 we show the unweighted and reweighted bands in the GPM and the CGI-
GPM, compared against $A_{N}$ data from BRAHMS for $\pi^{+}$ (full bullet
points) and $\pi^{-}$ (empty bullet points). As expected, reweighted curves
present reduced uncertainties. The GPM describes these data better than the
CGI-GPM, and the quality of the description increases if one does not consider
the aforementioned data points with $P_{T}<1.5$ GeV. A somehow larger
discrepancy between our computations and the data is seen for $\pi^{-}$ in the
CGI-GPM formalism.

Figure 4: Results for the simultaneous reweighting of the Sivers, transversity
and Collins functions: reweighted curves for STAR data [43, 46, 49] in the GPM
(upper panels) and the CGI-GPM (lower panels) approaches are presented. Data
points in grey correspond to $P_{T}<1.5$ GeV.

The comparison with the older STAR data [43, 46, 49], collected without
separating isolated and non-isolated pion samples, is shown in Fig. 4 for the
GPM (upper panels) and the CGI-GPM (lower panels) approaches, in four
different ranges of pseudorapidity. Notice that, in the two kinematical
configurations with largest $\langle\eta\rangle$ (right plots in the two
panels), the first two data points at lower $x_{F}$ values correspond to
$P_{T}<1.5$ GeV. Both GPM and CGI-GPM estimates are in qualitative agreement
with the data. The reweighted bands are able to describe the data at moderate
$x_{F}$, and more interestingly, they present a shape that better represents
the steady increase of the asymmetry at large-$x_{F}$ values, where the
agreement is enhanced with respect to older analyses [109, 110].

We finally move to the latest STAR data [54] for non-isolated $\pi^{0}s$. The
kinematics of this dataset is more compatible with that in our initial fits in
SIDIS and $e^{+}e^{-}$, since only pions with moderate momentum fractions $z$
contribute, excluding those with $z\sim 1$ [54]. Notice also that, especially
in the large-$x_{F}$ region, the non-isolated $\pi^{0}$ data are different
with respect to the full data sample and also to the past measurements in
similar kinematical regions [43, 46, 49] (see also Figs. 6, 7 and 8 of Ref.
[54] for a more complete comparison). For these reasons, as we will discuss
below (without showing any figure for brevity), we also consider the
corresponding outcome from the reweighting limited to this dataset.

In Fig. 5 we show our estimates and compare them against STAR results for non-
isolated pions. As one can notice, at the two different energies, both GPM and
CGI-GPM describe the data quite well within uncertainties. As the reweighting
is performed including the information coming from all the aforementioned
datasets, also here the steady increase at large-$x_{F}$ values is observed.
Performing the reweighting only on the new STAR data, the shape of the
reweighted bands results flatter, as the data for non-isolated pions suggest.
Below, we will comment on the uncertainties on the TMDs and the corresponding
$N_{\rm eff}$ we obtain in this case after the reweighting procedure.

Figure 5: Results for the simultaneous reweighting of the Sivers, transversity
and Collins functions: unweighted and reweighted predictions for STAR $A_{N}$
data for non isolated neutral pions [54]. Comparison with data at
$\sqrt{s}=200$ GeV (upper panels) and $\sqrt{s}=500$ GeV (lower panels), in
the GPM (left panels) and the CGI-GPM (right panels) approaches is presented.
All the data points correspond to $P_{T}>1.5$ GeV.

###  5.2 Impact of $A_{N}$ data on Sivers, transversity and Collins functions

We now move to the role played by $A_{N}$ data in the extractions of the
Sivers, transversity and Collins functions. As a general feature, we can
anticipate that these data impact mostly on the TMD-PDFs, namely the Sivers
and the transversity functions.

First, we examine the Sivers case. In Fig. 6 we compare the unweighted and
reweighted first moment of the quark Sivers functions, Eq.(20), in the GPM
(left panels) and CGI-GPM (right panels). As a general trend, the reweighted
curves present reduced uncertainties. This reduction is larger for the
$d$-quark than the $u$-quark Sivers function. The relative reduction of
uncertainty of the reweighted Sivers first moments is about $20-30$% for
$f_{1T}^{\perp u}$ and $40-90$% for $f_{1T}^{\perp d}$. The effective number
of sets (see Eq. (14)) surviving after the reweighting is $N_{\rm
eff}=476\,(464)$ in the GPM (CGI-GPM) case. By looking at Fig. 7, we can see
that in both approaches the parameters for the $u$-quark Sivers function and
the Gaussian Sivers width do not change very much, while it turns out that the
GPM favors a smaller overall absolute value of the normalization for the
$d$-quark Sivers function, with a slower decrease at large $x$ (smaller
$\beta_{d}$ parameter), and that the CGI prefers a larger $N_{d}$ (in size),
but with a faster decrease at large $x$.

Figure 6: Comparison of unweighted and reweighted first moments of up- (upper
panels) and down-quark (lower panels) Sivers functions in the GPM (left
panels) and in the CGI-GPM (right panels). The relative reduction of
uncertainty is also shown in the smaller panels. Figure 7: Comparison of
unweighted (grey) and reweighted distributions of parameters for the quark
Sivers functions in the GPM (red) and in the CGI-GPM (green).

Moving now to the transversity and Collins case, we stress that, though the
Collins contribution to $A_{N}$ is formally the same in the GPM and CGI-GPM,
the results for the reweighted curves for $h_{1}^{q}$ and $H_{1}^{\perp q}$ we
show below are slightly different. This reflects the different role of the
Sivers contribution to $A_{N}$ in the two approaches.

In Fig. 8 we present the comparison bewteen unweighted and reweighted $u_{v}$
and $d_{v}$ transversity functions, together with their corresponding Soffer
bound, in the GPM (left panels) and CGI-GPM (right panels) at $Q^{2}=4$ GeV2.
One can notice that, compared to the unweighted results, $A_{N}$ data favors
on average a slightly smaller $h_{1}^{u_{v}}$ in the region $x\lesssim 0.3$
and a slightly larger $h_{1}^{u_{v}}$ in the large-$x$ region. The inclusion
of $A_{N}$ data greatly reduces its uncertainty band in the region of
$x\gtrsim 0.3$. As for the $h_{1}^{d_{v}}$, we observe that larger absolute
value is preferred by the data on $A_{N}$. This is related to the information
coming from $A_{N}^{\pi^{0}}$ data at large $x_{F}$ (which are related to
large $x$ values of the functions probed upon integration), that tends to
favor those sets rendering a large asymmetry. The uncertainty reduction is
about $20-30\%$ at smaller values of $x$, reaching up to $80-90\%$ at larger
$x$ values for $h_{1}^{u_{v}}$, both in the GPM and CGI-GPM cases, while for
$h_{1}^{d_{v}}$ the reduction is $30-40\%$ (60%) in the GPM (CGI-GPM) at small
$x$ and up to $80-90\%$ at large $x$ in both approaches. In this case, the
effective number of sets after the reweighting is $N_{\rm eff}=198$ (GPM) and
$N_{\rm eff}=62$ (CGI-GPM). This might be due to the poor description of
$\pi^{-}$ data (see Fig. 3).

Figure 8: Comparison of unweighted and reweighted $u_{v}$ and $d_{v}$
transversity functions in the GPM (left panels) and in the CGI-GPM (right
panels). The corresponding Soffer bounds and the relative reduction of
uncertainty (same color codes in the bottom panels) are also shown. Figure 9:
Comparison of unweighted and reweighted favored (upper panels) and unfavored
(lower panels) Collins functions in the GPM (left panels) and in the CGI-GPM
(right panels) at $Q^{2}=4$ GeV2. The relative reduction of uncertainties is
also shown in the bottom plots. Figure 10: Comparison of unweighted (grey) and
reweighted distributions of parameters for the transversity and Collins
extraction in the GPM (red) and in the CGI-GPM (green).

In Fig. 9 we show the unweighted and reweighted Collins first moments (see Eq.
(26)) in the GPM and CGI-GPM approaches, at $Q^{2}=4$ GeV2, a typical SIDIS
scale. For these functions, the impact of $A_{N}$ data is less strong, but
allows for a reduction of the uncertainties (in both approaches) of about 10%
for the favored Collins function and about 20% for the unfavored one.

The aforementioned slower decrease at increasing $x$ values for the
transversity is made manifest also when looking at Fig. 10, where we show the
comparison between unweighted (hatched grey histograms) and reweighted
distributions of the fit parameters in the GPM (red histograms) and CGI-GPM
(green histograms). As previously mentioned, $A_{N}$ data mainly influence the
transversity function. This is clear by looking at the four top plots of Fig.
10: reweighted values of $N^{T}_{u_{v}}$ tend to be smaller while the negative
$N^{T}_{d_{v}}$ values are larger in size, approaching the limiting value of
the Soffer bound ($\lvert N_{q}^{T}\rvert\leq 1$). At large $x$, $h_{1}^{q}$
tends to decrease more similarly to the Soffer bound itself (see the
corresponding histogram of the $\beta$ parameter distribution, where the
reweighted average values move close to zero). On the other hand, even if the
Collins parameter distributions vary much less, there is still a reduction of
the uncertainties on the reweighted Collins functions. These considerations
point again towards our initial observation: the dominant effect generating
$A_{N}$ is the Collins one. This is consistent with what was also recently
obtained within the twist-3 approach [67, 68], where it was found that the
main contribution to $A_{N}$ comes from the fragmentation process.

We finally comment on the results we obtained for the reweighting procedure
using only the new STAR data for non-isolated $\pi^{0}$s. The reduction of the
uncertainties for the reweighted TMDs turns out to be smaller for the Sivers
and the Collins functions, and similar for $h_{1}^{q}$ at large $x$. The
effective number of sets retained is larger, being $N_{\rm eff}=1632\,(1873)$
for the Sivers fit in the GPM (CGI-GPM), and $N_{\rm eff}=1845$ and $N_{\rm
eff}=1474$ for the transversity and Collins extraction in the GPM and CGI-GPM,
respectively. These considerations point again towards a larger compatibility
of these new STAR data with SIDIS and $e^{+}e^{-}$ measurements.

###  5.3 Tensor charges

We conclude our analysis by reporting the corresponding values obtained for
the nucleon tensor charges, defined as:

$\delta
q=\int\limits_{0}^{1}\left[h_{1}^{q}(x)-h_{1}^{\bar{q}}(x)\right]dx\,,\quad
g_{T}=\delta u-\delta d.$ (27)

The values coming from the unweighted transversity functions at $Q^{2}=4$ GeV2
are (central values are the median values): $\delta u=0.46_{-0.09}^{+0.10}$,
$\delta d=-0.15_{-0.07}^{+0.10}$, $g_{T}=0.60_{-0.11}^{+0.13}$. The reweighted
tensor charges in the GPM (CGI-GPM) approach are: $\delta
u=0.44_{-0.05}^{+0.09}$ $(0.44_{-0.05}^{+0.08})$, $\delta
d=-0.19_{-0.05}^{+0.09}$ $(-0.19_{-0.05}^{+0.05})$,
$g_{T}=0.62_{-0.08}^{+0.11}$ $(0.63_{-0.05}^{+0.11})$. These values are
slightly larger as compared to older analyses (see Table 1 of Ref. [79],
“using SB” case), thanks to the new HERMES data on proton, which render larger
asymmetries and hence larger fitted functions, as already observed for
instance in Ref. [68]. A detailed comparison of our results and estimates of
the tensor charges from phenomenological analyses is presented in Fig. 11. We
observe that our current analysis and the majority of the previous studies of
Refs. [99, 111, 112, 106, 113, 114, 79, 115] yield consistent values for
$g_{T}$, $\delta u$, and $\delta d$. This corroborates the universality of the
transversity function.

Figure 11: Comparison of our results for the tensor charges with
phenomenological estimates from Refs. [99, 111, 112, 106, 113, 114, 79, 115]
at $Q^{2}=4$ GeV2 for the contribution to the tensor charge from $u$ and $d$
quarks (left panel) and the iso-vector combination $g_{T}$ (right panel). CGI
(green) and GPM (red) reweighted central values coincide, having also similar
uncertainties.

##  6  Conclusions and outlook

In this paper we have carefully investigated the Bayesian reweighting
procedure, extending it to the case of multiple, independent fits, and applied
it, for the first time, to simultaneously reweight two independent extractions
of quark TMD parton densities, the one for the Sivers function and the one for
the TMD transversity and Collins functions. Our study has been performed using
transverse single spin asymmetries data for inclusive pion production in
polarized $pp$ collisions at RHIC.

As the simultaneous reweighting involves two statistically independent fits,
each of them with a large MC sample ($\mathit{O}(10^{5})$ sets) representing
their corresponding uncertainty, and since these fits contribute to $A_{N}$ in
an additive way, all possible MC set combinations would have in principle to
be computed. To make the numerical computation faster and more efficient, we
have extended and developed a compression technique of MC sets samples, that
has allowed us to adopt only 1% of them in the reweighting, without losing any
statistical information on the distributions of parameters. Such a procedure
represents a step forward towards more efficient computations, and is flexible
enough to be extended to any study involving parameter distributions with
large samples.

Our phenomenological study, also for its peculiarities, has required an
educated selection of the experimental data to be used for the reweighting
procedure. The latest $A_{N}^{\pi^{0}}$ data from STAR Collaboration differ
from the previous measurements at RHIC, as they are provided also for non-
isolated pions. This dataset explores kinematical regions compatible with the
ones in SIDIS and $e^{+}e^{-}$ processes, for which TMD factorization holds
and from which we extract the aforementioned TMDs. Furthermore, having a
smaller magnitude at large $x_{F}$, these measurements do not show the usual
pronounced steady increase observed so far in this kinematical region.
Nevertheless, for the sake of completeness, we have included all the available
data for charged and neutral pions, obtaining an overall global good
description.

The adopted dataset is dominated by $\pi^{0}$ production data, while only a
few data points from BRAHMS for charged pions are available. As
$A_{N}^{\pi^{\pm}}$ data are more sensitive to flavor separation, they could
also help in disentangling the issue of the predicted Sivers sign change. The
description of these data seems to favor the GPM approach, where all TMDs are
assumed to be universal and in which, contrary to the CGI-GPM, the expected
Sivers sign change is not naturally recovered. More data from future
experiments as COMPASS/AMBER [25, 26, 27], JLab [21], the EIC [22, 23] and the
fixed-target programs at Tevatron at SpinQuest [30], and the LHC [29] are
needed to shed light on this fundamental issue.

The overall agreement of our estimates with the data is certainly improved
with respect to older analyses, also thanks to a thorough usage of the Soffer
bound in the fit of the TMD transversity distribution. In fact, applying such
a theoretical constraint a posteriori allows for larger asymmetries at large
$x_{F}$. As a consequence, within the (CGI-)GPM approach, the Collins
mechanism turns out to be the dominant one, as also observed in recent
analyses within the collinear twist-3 formalism [67, 68].

As in our previous work [69], the reweighted TMD distributions present reduced
uncertainties at large $x$, confirming once again the complementarity of
$A_{N}$ data with SIDIS measurements. The uncertainty reduction is about 40%
(90%) for the $u$-($d$-)quark Sivers function, and about 80 to 90% for the
$u_{v}$ and $d_{v}$ TMD transversity functions. The reduction of the
uncertainty for the Collins functions is smaller (about 10% for the favored
and 20% for the unfavored one), confirming that $e^{+}e^{-}$ data are the ones
giving stronger constraints on this polarized TMD fragmentation function.
Performing the reweighting only on non-isolated pion data, the reduction of
uncertainties is smaller for the Sivers and Collins functions, and similar for
the TMD transversity at large $x$. The retained $N_{\rm eff}$ sets in the GPM
(CGI-GPM) case is $\sim 80\%$ ($\sim 90\%$) for the Sivers fit and $\sim 90\%$
($\sim 70\%$) for the transversity and Collins extraction. This confirms the
expected enhanced compatibility between SIDIS and $e^{+}e^{-}$ data and the
new $A_{N}$ measurements from STAR for non-isolated pions.

This work is a natural extension of our previous study [69], and a proof of
concept for future TMD analyses. We envision studies on different TMDs
parametrizations, updated fits with new data from COMPASS/AMBER [26], JLab
[21] and the future Electron-Ion Collider [22, 23]. We are planning to check
the compatibility between the reweighted functions of the present work and
data for inclusive jet or pion-in-jet production, where respectively Sivers
and Collins effects individually contribute. This ambitious program not only
points to improve our current knowledge on TMDs, but also sets the ground for
more quantitative studies on TMD universality and on the size of factorization
breaking effects.

## Acknowledgments

We thank Mauro Anselmino for numerous discussions. We are grateful to Emanuele
Nocera and Andrea Signori for discussions on the statistical approaches and
technical aspects of the current analysis. C.F. and A.P. are grateful to the
Physics Department of the University of Cagliari and the INFN Cagliari
division for the kind hospitality extended to them during the development of
this project. A.P. acknowledges financial support from the University of
Cagliari under the Visiting Professor Programme. U.D. also acknowledges
financial support from Fondazione di Sardegna under the project “Matter-
antimatter asymmetry and polarisation in strange hadrons at LHCb”, project
number F73C22001150007 (University of Cagliari). This article is part of a
project that has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement STRONG-2020 - No.
824093. This work was partially supported by the Grant for
Internationalization SIGA_GFI_22_01 financed by Torino University, by the
National Science Foundation under Grants No. PHY-2012002, No. PHY-2310031, No.
PHY-2335114 (A.P.), and by the U.S. Department of Energy contract No. DE-
AC05-06OR23177, under which Jefferson Science Associates, LLC operates
Jefferson Lab (A.P.).

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