# Transverse $\Lambda$ polarization in $e^{+}e^{-}$ annihilations and in SIDIS
processes at the EIC within TMD factorization

Umberto D’Alesio<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA),
Italy  INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato
(CA), Italy  Leonard Gamberg<EMAIL_ADDRESS>Division of Science, Penn State
Berks, Reading, PA 19610, USA  Francesco Murgia<EMAIL_ADDRESS>INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA),
Italy  Marco Zaccheddu<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (CA),
Italy  INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato
(CA), Italy

###### Abstract

We present a phenomenological study on the role of charm contribution and
$SU(2)$ isospin symmetry in the extraction of the $\Lambda$ polarizing
fragmentation functions from
$e^{+}e^{-}\to\Lambda^{\uparrow}(\bar{\Lambda}^{\uparrow})\,h+X$ annihilation
processes. We adopt the well-established transverse-momentum-dependent
factorization formalism, within the Collins-Soper-Sterman evolution scheme at
next-to-leading logarithm accuracy, carefully exploiting the role of the
nonperturbative component of the polarizing fragmentation function. We then
discuss the impact of these results on the predictions for transverse
$\Lambda$, $\bar{\Lambda}$ polarization in semi-inclusive deep inelastic
scattering processes at typical energies of the future Electron-Ion Collider.

##  I Introduction

The study of the fragmentation mechanism of partons into hadrons within the
field theoretic framework of quantum chromodynamics (QCD), along with
factorization theorems, which connect perturbative parton dynamics to
universal hadron fragmentation functions, is fundamental to unfolding the
quark and gluon structure of hadrons. When one includes also spin and its
correlations with intrinsic transverse momentum the information one can
extract is much richer and the description is more complete. This can be
achieved, for instance, by studying the spontaneous transverse $\Lambda$
polarization in processes where factorization theorems, in terms of transverse
momentum dependent distributions (TMDs), hold. We refer, in particular, to
double-hadron production in $e^{+}e^{-}$ annihilation and semi-inclusive deep
inelastic scattering (SIDIS) processes [1, 2, 3]. These are characterized by
the presence of two ordered energy scales, a small one (the transverse
momentum unbalance of the two hadrons in $e^{+}e^{-}$ processes or the
transverse momentum of the final hadron in SIDIS) and a large one, the
virtuality of the exchanged photon.

We emphasize that the understanding of the transverse $\Lambda$ polarization,
originally measured in inclusive _unpolarized_ proton-proton and proton-
nucleus collisions in the late 70’s [4, 5, 6, 7, 8, 9, 10], still represents a
challenging problem in hadron physics. One of the earliest attempts to
describe this phenomenon within a phenomenological model was presented in Ref.
[11] and further extended to SIDIS processes in Ref. [12]. Recently,
experimental data collected by the Belle Collaboration [13], for the
transverse $\Lambda,\bar{\Lambda}$ polarization in almost back-to-back two-
hadron production in $e^{+}e^{-}$ processes, has triggered a renewed interest
in the subject matter. Preliminary studies within a simplified TMD model at
fixed scale were discussed in Refs. [14, 15]. Moreover, a series of
phenomenological analyses within the TMD factorization framework adopting the
Collins-Soper-Sterman (CSS) approach [16, 17, 18] has been carried out [19,
20, 21, 22, 23]. The general TMD formalism, following the Lorentz
decomposition or the helicity approach, was developed and presented in Refs.
[24, 25, 26] for $e^{+}e^{-}$ processes, and in Refs. [27, 28, 29] for SIDIS.

One of the main goals of these phenomenological studies, besides the
description of data, is the extraction of the polarizing fragmentation
function (pFF) for $\Lambda$ hyperons, that provides information on the
correlations between the intrinsic transverse momentum in the parton-to-hadron
fragmentation process and the final hadron polarization. In this respect, this
TMD function represents a window towards a deeper understanding of the
nonperturbative fragmentation mechanism when also spin-polarization effects
are taken into account.

In this paper, that represents a natural extension of Ref. [23], we reanalyze
Belle data for the transverse $\Lambda$ polarization limiting this study to
the associated production case, and paying special attention to two issues,
mentioned in our previous work that here we will study in depth: namely, the
role of the $SU(2)$ isospin symmetry (see also Refs. [22, 30]) and the charm
contribution in the fragmentation of $\Lambda$ hyperons.

We consider three different scenarios, discussing their statistical
significance in the data description and the difference in the extracted
polarizing fragmentation functions. We then employ these results to give
predictions for the same observable in SIDIS processes at the energies and
kinematics typical of the future Electron-Ion Collider (EIC). We will show how
new measurements could help in disentangling among the different scenarios.
The role of intrinsic charm in the proton [31, 32] will be also addressed.

This analysis will allow us to check, at the same time, other fundamental
issues, like the universality of the TMD fragmentation functions and their QCD
evolution with the energy scale.

The paper is organized as follows: in Sec. II we present the formalism and the
cross sections for the production of a transversely polarized spin-$1/2$
hadron in $e^{+}e^{-}$ collisions, in association with a light hadron, and in
semi-inclusive deep inelastic scattering processes. The main results are then
employed in the phenomenology part in Sec. III, where we discuss the role of
the charm quark contribution and the issue of $SU(2)$ isospin symmetry in the
re-analysis of Belle data [13]. Estimates for the transverse
$\Lambda/\bar{\Lambda}$ polarization in $e^{+}e^{-}$ collisions and in SIDIS
processes, at different center of mass energies, are presented with particular
focus on how these are influenced by the choice of the pFF parametrization and
of the nucleon PDF set. Lastly, in Sec. IV we collect our concluding remarks.

##  II Formalism

In this section, we briefly recall the formalism for the production of a
transversely polarized spin-1/2 hadron in $e^{+}e^{-}$ annihilation processes,
in association with an unpolarized light-hadron, and in semi-inclusive deep
inelastic scattering processes. The main equations will be used in the
following section to study the production of transversely polarized $\Lambda$
hyperons in both processes.

###  II.1 Double-hadron production in $e^{+}e^{-}$ processes

We start considering double-hadron production in $e^{+}e^{-}$ collisions:

$e^{+}(l_{e^{+}})\,e^{-}(l_{e^{-}})\to h_{1}(P_{1},S_{1})\,h_{2}(P_{2})+X\,,$
(1)

where $h_{1}$ is a spin-1/2 hadron, with momentum $P_{1}$, spin-polarization
vector $S_{1}$ and mass $M_{1}$, while $h_{2}$ is a light unpolarized hadron
with momentum $P_{2}$ (we will neglect its mass), and they are produced almost
back-to-back in the center-of-mass frame of the incoming leptons. For more
details we refer the reader to Refs. [23, 24].

Figure 1: Kinematics for the process $e^{+}e^{-}\to h_{1}\,h_{2}+X$ in the
hadron-frame configuration.

In Fig. 1 we show the kinematics of the process in the hadron-frame
configuration, where we fix the momentum of the second hadron, $h_{2}$, along
the $\hat{z}_{L}$ axis, while the first one, $h_{1}$, moving in the opposite
hemisphere, has a small transverse momentum $\bm{P}_{1T}$ with respect to the
second hadron direction.

From the theoretical point of view, it is however more convenient to adopt a
different frame, where the two hadrons are exactly back to back, along a new
$\hat{\bm{z}}$ axis, and the hadron transverse unbalance ($\bm{P}_{1T}$) is
now carried out by the virtual photon. In this frame, the differential cross
section can be expressed, neglecting terms not relevant in the present study,
as [24, 15, 26]

$\frac{d\sigma^{e^{+}e^{-}\to
h_{1}(S_{1})h_{2}\,X}}{2dy\,dz_{h_{1}}dz_{h_{2}}d^{2}\bm{q}_{T}}=\sigma^{e^{+}e^{-}}_{0}\bigg{[}F_{UU}-|S_{1T}|\sin(\phi_{1}-\phi_{S_{1}})F^{\sin(\phi_{1}-\phi_{S_{1}})}_{TU}+\cdots\bigg{]}\,,$
(2)

where $\phi_{S_{1}}$ is the azimuthal angle of the spin of the hadron $h_{1}$.
Here $\bm{q}_{T}$ is the transverse momentum of the virtual photon (of
momentum $q$), related to the transverse momentum of the hadron $h_{1}$ as
$\bm{P}_{1T}=-z_{1}\bm{q}_{T}$, being $z_{1}$ its light-cone momentum
fraction, defined for both hadrons as

$z_{1}=\frac{P^{-}_{1}}{p_{q}^{-}},\qquad
z_{2}=\frac{P^{+}_{2}}{p_{\bar{q}}^{+}}\,,$ (3)

where $p_{q}$ and $p_{\bar{q}}$ are the four-momenta of the quark and the
antiquark fragmenting into the hadron $h_{1}$ and $h_{2}$, carrying a
transverse momentum $\bm{k}_{\perp}$ and $\bm{p}_{\perp}$ with respect to the
parent quark momenta, respectively.

The two scaling variables in Eq. (2), $z_{h_{1}}$, $z_{h_{2}}$, are the usual
invariants (energy fractions), related to the light-cone momentum fractions as

$z_{h}=\frac{2P_{h}\cdot q}{Q^{2}}=\frac{2E_{h}}{Q}\simeq
z\Bigg{(}1+\frac{M^{2}_{h}}{z^{2}{Q}^{2}}\Bigg{)}\,,$ (4)

where $Q$ is the center-of-mass energy of the process, $Q^{2}=q^{2}$, and
where in the last relation we have neglected terms of the order
$\mathcal{O}(\bm{k}_{\perp}^{2}/(zQ)^{2})$. Another scaling variable, usually
adopted in phenomenological analyses, is the hadron momentum fraction

$\begin{split}z_{p}=\frac{2|\bm{P}_{h}|}{Q}\simeq
z\Bigg{(}1-\frac{M^{2}_{h}}{z^{2}{Q}^{2}}\Bigg{)}\,.\end{split}$ (5)

Notice that since for the light hadron $h_{2}$ we neglect its mass, in the
following we will use $z_{2}=~{}z_{h_{2}}=~{}z_{p_{2}}$, within this
approximation.

The remaining variable is the fraction $y=P_{2}\cdot l_{e^{+}}/P_{2}\cdot q$,
related to the polar angle $\theta$ in the hadron frame (see Fig. 1). Lastly
we have

$\sigma^{e^{+}e^{-}}_{0}=\frac{3\pi\alpha^{2}}{Q^{2}}\big{[}y^{2}+(1-y)^{2}\big{]}\,.$
(6)

In Eq. (2), the $F$ terms are convolutions of two fragmentation functions,
where the subscripts denote the polarization states of, respectively, the
first and the second hadron ($U$ = unpolarized, $T$ = transversely polarized).
These have the following expressions [23, 24, 15]:

$\displaystyle F_{UU}$ $\displaystyle=$ $\displaystyle
z_{p_{1}}^{2}z_{p_{2}}^{2}\mathcal{H}^{(e^{+}e^{-})}(Q)\,\mathcal{F}[D_{1}\bar{D}_{1}]\,,$
(7) $\displaystyle F^{\sin(\phi_{1}-\phi_{S_{1}})}_{TU}$ $\displaystyle=$
$\displaystyle
z_{p_{1}}^{2}z_{p_{2}}^{2}\mathcal{H}^{(e^{+}e^{-})}(Q)\,\mathcal{F}\bigg{[}\frac{\hat{\bm{h}}\cdot\bm{k}_{T}}{M_{1}}D^{\perp}_{1T}\bar{D}_{1}\bigg{]}\,,$
(8)

where $\mathcal{H}^{(e^{+}e^{-})}(Q)$ is the hard scattering part for the
massless on-shell process $e^{+}e^{-}\to q\bar{q}$ (normalized to one at
leading order), at the center-of-mass energy $Q$, $D_{1}(z,{k}_{\bot})$ is the
unpolarized TMD fragmentation function (FF) and $D^{\perp}_{1T}(z,{k}_{\bot})$
is the polarizing FF, with $\hat{\bm{h}}=\bm{P}_{1T}/|\bm{P}_{1T}|$ and
$\bm{k}_{T}=-\bm{k}_{\bot}/z_{p_{1}}$ (and similarly
$\bm{p}_{T}=-\bm{p}_{\bot}/z_{p_{2}}$), where $\bm{k}_{T}$ ($\bm{p}_{T}$) is
the transverse momentum of the quark (antiquark) with respect to the hadron
$h_{1}$ ($h_{2}$) direction of motion. The $\mathcal{F}$ are proper
convolutions of TMD-FFs, defined as follows:

$\mathcal{F}[\omega D\bar{D}]=\sum_{q}e^{2}_{q}\int
d^{2}\bm{k}_{T}d^{2}\bm{p}_{T}\,\delta^{(2)}(\bm{k}_{T}+\bm{p}_{T}-\bm{q}_{T})\,\omega(\bm{k}_{T},\bm{p}_{T})D(z_{1},\bm{k}_{\bot})\bar{D}(z_{2},\bm{p}_{\bot})\,,$
(9)

where $\omega$ is a suitable weight factor depending on the two transverse
momenta and $D$ and $\bar{D}$ are the TMD-FFs.

In order to employ the Collins-Soper-Sterman (CSS) evolution equations, it is
useful to write the convolutions in the conjugate $\bm{b}_{T}$-space:

$F_{UU}=z_{p_{1}}^{2}z_{p_{2}}^{2}\mathcal{B}_{0}\Big{[}\widetilde{D}_{1}\widetilde{\bar{D}}_{1}\Big{]}=z_{p_{1}}^{2}z_{p_{2}}^{2}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{2\pi}\,b_{T}J_{0}(b_{T}\,q_{T})\widetilde{D}_{1}(z_{1},b_{T})\widetilde{\bar{D}}_{1}(z_{2},b_{T})\,,$
(10) $\displaystyle F^{\sin(\phi_{1}-\phi_{S_{1}})}_{TU}$ $\displaystyle=$
$\displaystyle
M_{1}z_{p_{1}}^{2}z_{p_{2}}^{2}\mathcal{B}_{1}\Big{[}\widetilde{D}^{\perp(1)}_{1T}\widetilde{\bar{D}}_{1}\Big{]}$
(11) $\displaystyle=$ $\displaystyle
M_{1}z_{p_{1}}^{2}z_{p_{2}}^{2}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{2\pi}\,b^{2}_{T}J_{1}(b_{T}\,q_{T})\widetilde{D}^{\perp(1)}_{1T}(z_{1},b_{T})\widetilde{\bar{D}}_{1}(z_{2},b_{T})\,,$

where $\widetilde{D}_{1}(z_{1},b_{T})$ is the Fourier transform of the
unpolarized FF, $\widetilde{D}^{\perp(1)}_{1T}(z_{1},b_{T})$ is the first
moment of the polarizing fragmentation function in $\bm{b}_{T}$-space, and
$J_{i}$ is the Bessel function of the first kind of $i$-th order. Notice that
we have already used $\mathcal{H}^{(e^{+}e^{-})}(Q)=1$ and all light-cone
momentum fractions have to be properly understood in terms of the
corresponding energy fractions, $z_{h}$.

After solving the CSS evolution equations, as discussed in Refs. [1, 23], the
convolutions can be written again as:

$\displaystyle\mathcal{B}_{0}\Big{[}\widetilde{D}_{1}\widetilde{\bar{D}}_{1}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{z^{2}_{1}z^{2}_{2}}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{2\pi}\,b_{T}J_{0}(b_{T}\,q_{T})\,d_{h_{1}/q}(z_{1};\bar{\mu}_{b})\,d_{h_{2}/\bar{q}}(z_{2};\bar{\mu}_{b})$
(12) $\displaystyle\times$ $\displaystyle
M_{D_{1}}(b_{c}(b_{T}),z_{1})\,M_{D_{2}}(b_{c}(b_{T}),z_{2})\>e^{\\!\\!-g_{K}(b_{c}(b_{T});b_{\text{max}})\ln{\big{(}\frac{Q^{2}z_{1}z_{2}}{M_{1}M_{2}}\big{)}}-S_{\rm
pert}(b_{*};\bar{\mu}_{b})}\,,$
$\displaystyle\mathcal{B}_{1}\Big{[}\widetilde{D}^{\perp(1)}_{1T}\widetilde{\bar{D}}_{1}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{z^{2}_{1}z^{2}_{2}}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{2\pi}\,b^{2}_{T}J_{1}(b_{T}\,q_{T}){D}^{\perp(1)}_{1T}(z_{1};\bar{\mu}_{b})\,d_{h_{2}/\bar{q}}(z_{2};\bar{\mu}_{b})$
(13) $\displaystyle\times$ $\displaystyle
M^{\perp}_{D_{1}}(b_{c}(b_{T}),z_{1})\,M_{D_{2}}(b_{c}(b_{T}),z_{2})\>e^{\\!\\!-g_{K}(b_{c}(b_{T});b_{\text{max}})\ln{\big{(}\frac{Q^{2}z_{1}z_{2}}{M_{1}M_{2}}\big{)}}-S_{\rm
pert}(b_{*};\bar{\mu}_{b})}\,,$

where the $d_{h/j}$’s are the $p_{\perp}$-integrated unpolarized fragmentation
functions. $M_{D_{i}}$ and $M^{\perp}_{D_{1}}$ are, respectively, the
nonperturbative functions of the unpolarized and of the polarizing FFs, and
$g_{K}$ is the nonperturbative function of the Collins-Soper Kernel. All other
quantities appearing in the above equations, necessary to properly separate
the perturbative from the nonperturbative region, are defined and discussed in
detail in Ref. [23]. See also below.

It is worth recalling that Eqs. (12) and (13) are obtained by using the
leading term of the operator product expansions (OPEs), for small-$\bm{b}_{T}$
values, of the TMD distribution functions [23, 1]. Lastly, $S_{\rm pert}$ is
the perturbative Sudakov factor, defined as (see also Appendix A for more
details):

$S_{\rm
pert}(b_{*};\bar{\mu}_{b})=-\widetilde{K}(b_{*};\bar{\mu}_{b})\ln\frac{Q^{2}}{\bar{\mu}_{b}^{2}}-\int^{Q}_{\bar{\mu}_{b}}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\bigg{[}2\gamma_{D}(g(\mu^{\prime});1)-\gamma_{K}(g(\mu^{\prime}))\ln{\frac{Q^{2}}{\mu^{\prime
2}}}\bigg{]}\,.$ (14)

The expression of the transverse polarization for the hadron $h_{1}$ is
defined as:

$P^{h_{1}}_{n}=\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\uparrow}+d\sigma^{\downarrow}}=\frac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\rm
unp}}\,,$ (15)

where $d\sigma^{\uparrow(\downarrow)}$ is the differential cross section, Eq.
(2), for the production of a transversely polarized hadron along the up(down)
direction ($\hat{\bm{n}}$) with respect to the production plane,111Notice that
in such a configuration $\sin(\phi_{1}-\phi_{S_{1}})=-1$. and $d\sigma^{\rm
unp}$ is the unpolarized cross section.

Finally, we can write the $\bm{q}_{T}$-integrated transverse polarization as
the ratio of the two convolutions in $\bm{b}_{T}$-space [23]:

$P^{h_{1}}_{n}(z_{h_{1}},z_{h_{2}})=\frac{\int
d^{2}\bm{q}_{T}\,F^{\sin(\phi_{1}-\phi_{S_{1}})}_{TU}}{\int
d^{2}\bm{q}_{T}\,F_{UU}}=\frac{M_{1}\int
dq_{T}\;q_{T}\;d\phi_{1}\,\mathcal{B}_{1}\Big{[}\widetilde{D}^{\perp(1)}_{1T}\widetilde{\bar{D}}_{1}\Big{]}}{\int
dq_{T}\;q_{T}\;d\phi_{1}\,\mathcal{B}_{0}\Big{[}\widetilde{D}_{1}\widetilde{\bar{D}}_{1}\Big{]}}\,.$
(16)

The integration over the azimuthal angle, $\phi_{1}$, is trivial. Moreover,
since the only terms inside the convolutions depending on $q_{T}$ are the
Bessel functions, we can separately integrate them, obtaining

$\int^{q_{T_{\text{max}}}}_{0}dq_{T}\,q_{T}J_{0}(b_{T}\,q_{T})=\frac{q_{T_{\text{max}}}}{b_{T}}J_{1}(b_{T}\,q_{T_{\rm
max}})\,,\\\ $ (17)
$\int^{q_{T_{\text{max}}}}_{0}dq_{T}\,q_{T}J_{1}(b_{T}\,q_{T})=\frac{\pi
q_{T_{\text{max}}}}{2b_{T}}\\{J_{1}(b_{T}\,q_{T_{\text{max}}})\bm{H}_{0}(b_{T}\,q_{T_{\rm
max}})-J_{0}(b_{T}\,q_{T_{\text{max}}})\bm{H}_{1}(b_{T}\,q_{T_{\text{max}}})\\}\,,$
(18)

where ${\bm{H}}_{0,1}$ are the Struve functions of order zero and one
respectively. Notice that in the above integration we have introduced a
maximum value $q_{T_{\text{max}}}$, that has to fulfil the condition
$q_{T_{\text{max}}}\ll Q$, in order to guarantee the validity of the TMD
factorization [33].

###  II.2 Semi-inclusive Deep Inelastic Scattering

Here, we present the formal expressions for the production of a transversely
polarized massive hadron $h_{1}$ in unpolarized SIDIS processes:

$e(l)\,N(P)\to e(l^{\prime})\,h_{1}(P_{1},S_{1})+X\,,$ (19)

where $N$ is an unpolarized nucleon with momentum $P$.

Figure 2: Kinematics for the process $e\,N\to e\,h_{1}+X$ in the hadron-frame
configuration.

In Fig. 2 we show the kinematics of the process in the $\gamma^{*}N$ c.m.
frame, where the virtual photon, with momentum $q=l-l^{\prime}$ (virtuality
$q^{2}=-Q^{2}$), and the nucleon collide along the $\hat{z}_{L}$ axis, while
the hadron $h_{1}$ moves towards the negative $\hat{z}_{L}$ direction with
transverse momentum $\bm{P}_{1T}$ with respect to the $\gamma$-$N$ direction.
Notice that at variance with the configuration adopted in the “Trento
Conventions” paper [34], the photon moves along $-\hat{z}_{L}$. As for the
case of the $e^{+}e^{-}$ annihilation process, it is more convenient to adopt
a frame where the nucleon and the hadron $h_{1}$ move back to back, along a
new $\hat{z}$ axis, and the hadron transverse unbalance is again carried out
by the virtual photon. In this frame, the differential cross section, limiting
to the terms relevant in the present study, can be written as:

$\frac{d\sigma^{eN\to
eh_{1}(S_{1})X}}{dydx_{B}dz_{h}d^{2}\bm{q}_{T}}=\sigma^{\rm
DIS}_{0}\bigg{[}F_{UU}-|S_{1T}|\sin(\phi_{1}-\phi_{S_{1}})F^{\sin(\phi_{1}-\phi_{S_{1}})}_{UT}+\dots\bigg{]}\,,$
(20)

with

$x_{B}=\frac{Q^{2}}{2P\cdot q}=x\,,\quad y=\frac{P\cdot q}{P\cdot l}\,,\quad
z_{h}=\frac{P\cdot P_{1}}{P\cdot q}=z=z_{p}\,,$ (21)

where $x=p^{+}/P^{+}$ is the light-cone momentum fraction of the nucleon
momentum carried by the parton with momentum $p$, and $z$ is the light-cone
momentum fraction, defined in Eq. (3), for the final-state hadron. Notice that
the last equalities are exactly true when neglecting the nucleon and the
hadron masses, together with terms of order
$\mathcal{O}(\bm{k}_{\perp}^{2}/Q^{2})$. It can be shown that if we keep the
final hadron mass (relevant in some kinematical regions)222The nucleon mass
can be safely neglected in our study. we have

$z_{p}\simeq
z_{h}\Bigg{(}1-\frac{M^{2}_{1}}{z^{2}_{h}Q^{2}}\frac{x_{B}}{1-x_{B}}\Bigg{)}\,.$
(22)

Another set of invariants adopted in SIDIS, useful from the phenomenological
point of view, are the following:

$s=(P+l)^{2}\,,\quad
Q^{2}=-q^{2}=x_{B}ys\,,\quad(P+q)^{2}=W^{2}=\frac{1-x_{B}}{x_{B}}Q^{2}\,,$
(23)

where $s$ is the total c.m. energy squared and $W$ is the c.m. energy of the
photon-nucleon system. In the lepton-nucleon c.m. frame they can be expressed
as:

$s=4E_{N}E_{e}\,,\quad Q^{2}=4x_{B}yE_{N}E_{e}\,,$ (24)

where $E_{N,e}$ are respectively the nucleon and electron beam energy. Lastly,
for the elementary cross section, we have [28, 29]:

$\sigma^{\rm DIS}_{0}=\frac{2\pi\alpha^{2}}{Q^{2}}\frac{1+(1-y)^{2}}{y}\,.$
(25)

In Eq. (20) the $F$ terms are now convolutions of a TMD-PDF and a TMD-FF,
where again the subscripts denote the polarization states of the initial-state
nucleon and the final-state hadron. These are defined as follows [27, 35]:

$\displaystyle F_{UU}$ $\displaystyle=$ $\displaystyle
z_{p}^{2}\,\mathcal{H}^{(\rm DIS)}(Q)\mathcal{F}[f_{1}D_{1}]\,,$ (26)
$\displaystyle F^{\sin(\phi_{1}-\phi_{S_{1}})}_{UT}$ $\displaystyle=$
$\displaystyle z_{p}^{2}\,\mathcal{H}^{(\rm
DIS)}(Q)\mathcal{F}\bigg{[}\frac{\hat{\bm{h}}\cdot\bm{k}_{T}}{M_{1}}f_{1}D^{\perp}_{1T}\bigg{]}\,,$
(27)

where $f_{1}(x,p_{\perp})$ is the TMD unpolarized parton distribution function
and $\mathcal{H}^{(\rm DIS)}(Q)$ is the hard scattering part for the massless
on-shell process $eq\to eq$, at the center-of-mass energy $Q$. Once again at
LO this last quantity is normalized to one and will be dropped in the
following.

The convolutions can be written in the conjugate $\bm{b}_{T}$-space as Fourier
transforms:

$\displaystyle F_{UU}$ $\displaystyle=$ $\displaystyle
z_{p}^{2}\,\mathcal{B}_{0}\Big{[}\widetilde{f}_{1}\widetilde{{D}}_{1}\Big{]}=z_{p}^{2}\,\sum_{q}e^{2}_{q}\int\frac{db_{T}}{(2\pi)}\,b_{T}J_{0}(b_{T}\,q_{T})\widetilde{f}_{1}(x,b_{T})\widetilde{{D}}_{1}(z,b_{T})\,,$
(28) $\displaystyle F_{UT}^{\sin(\phi_{1}-\phi_{S_{1}})}$ $\displaystyle=$
$\displaystyle
M_{1}\,z_{p}^{2}\mathcal{B}_{1}\Big{[}\widetilde{f}_{1}\widetilde{D}^{\perp(1)}_{1T}\Big{]}$
(29) $\displaystyle=$ $\displaystyle
M_{1}z_{p}^{2}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{2\pi}\,b^{2}_{T}J_{1}(b_{T}\,q_{T})\widetilde{{f}}_{1}(x,b_{T})\widetilde{D}^{\perp(1)}_{1T}(z,b_{T})\,,$

and, after solving the CSS evolution equations, they can be expressed in their
full form as:

$\displaystyle\mathcal{B}_{0}\Big{[}\widetilde{f}_{1}\widetilde{{D}}_{1}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{z^{2}}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{(2\pi)}\,b_{T}J_{0}(b_{T}\,q_{T})\,f_{q/N}(x;\bar{\mu}_{b})\,d_{h/q}(z;\bar{\mu}_{b})$
(30) $\displaystyle\times$ $\displaystyle
M_{f_{1}}(b_{c}(b_{T}),x)\,M_{D_{h}}(b_{c}(b_{T}),z)\>e^{\\!\\!-g_{K}(b_{c}(b_{T});b_{\text{max}})\ln{\big{(}\frac{Q^{2}z}{xM_{P}M_{h}}\big{)}}-S_{\rm
pert}(b_{*};\bar{\mu}_{b})}\,,$
$\displaystyle\mathcal{B}_{1}\Big{[}\widetilde{f}_{1}\widetilde{D}^{\perp(1)}_{1T}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{z^{2}}\sum_{q}e^{2}_{q}\int\frac{db_{T}}{(2\pi)}\,b^{2}_{T}J_{1}(b_{T}\,q_{T})\,f_{q/N}(x;\bar{\mu}_{b})\,{D}^{\perp(1)}_{1T,q}(z;\bar{\mu}_{b})$
(31) $\displaystyle\times$
$\displaystyle\,M_{f_{1}}(b_{c}(b_{T}),x)M^{\perp}_{D_{1}}(b_{c}(b_{T}),z)\>e^{\\!\\!-g_{K}(b_{c}(b_{T});b_{\text{max}})\ln{\big{(}\frac{Q^{2}z}{xM_{P}M_{h}}\big{)}}-S_{\rm
pert}(b_{*};\bar{\mu}_{b})}\,,$

where $f_{q/N}$ is the integrated unpolarized parton distribution function,
and $M_{f_{1}}$ is the nonperturbative component of the unpolarized PDF. All
the remaining terms that appear in Eq. (30) and (31) are the same defined in
the previous section.

The operative expression of the transverse polarization can be obtained from
Eq. (15), where now $d\sigma^{\uparrow(\downarrow)}$ is the differential cross
section for a transversely polarized hadron along the up(down) $\hat{\bm{n}}$
direction, with respect to the production plane, in Eq. (20). For nucleons, we
can directly write the transverse polarization of the final state hadron and
the $\bm{q}_{T}$-integrated one as the ratio of the two convolutions in
$\bm{b}_{T}$-space:

$P^{h_{1}}_{n}(x_{B},z_{h},q_{T})=\frac{F^{\sin(\phi_{1}-\phi_{S_{1}})}_{UT}}{F_{UU}}=\frac{M_{1}\int
d\phi_{1}\,\mathcal{B}_{1}\Big{[}\widetilde{f}_{1}\widetilde{D}^{\perp(1)}_{1T}\Big{]}}{\int
d\phi_{1}\,\mathcal{B}_{0}\Big{[}\widetilde{f}_{1}\widetilde{{D}}_{1}\Big{]}}\,,$
(32) $P^{h_{1}}_{n}(x_{B},z_{h})=\frac{\int
d^{2}\bm{q}_{T}\,F^{\sin(\phi_{1}-\phi_{S_{1}})}_{UT}}{\int
d^{2}\bm{q}_{T}\,F_{UU}}=\frac{M_{1}\int
dq_{T}\;q_{T}\;d\phi_{1}\,\mathcal{B}_{1}\Big{[}\widetilde{f}_{1}\widetilde{D}^{\perp(1)}_{1T}\Big{]}}{\int
dq_{T}\;q_{T}\;d\phi_{1}\,\mathcal{B}_{0}\Big{[}\widetilde{f}_{1}\widetilde{{D}}_{1}\Big{]}}\,.$
(33)

To compute the cross section for the scattering off nuclei, we adopt a simple
approach taking the incoherent sum of the contribution of every nucleon that
composes the nucleus, neglecting nuclear effects. That is, for the scattering
off a nucleus with $A$ nucleons and $Z$ protons we use:

$d\sigma^{eA\to eh_{1}(S_{1})X}=Z\,d\sigma^{ep\to
eh_{1}(S_{1})X}+(A-Z)\,d\sigma^{en\to eh_{1}(S_{1})X}\,.$ (34)

##  III Phenomenology

In this section, after recalling the main results of the analysis of Belle
data [13] presented in Ref. [23], we will focus more extensively on the role
of the charm contribution and of the $SU(2)$ isospin symmetry. Then we will
give predictions for the transverse $\Lambda$ polarization in $e^{+}e^{-}$
collisions, for different values of the c.m. energy. Finally, we will present
estimates for the same observable in semi-inclusive deep inelastic scattering
processes, for different values of the lepton and nucleon beam energies.

###  III.1 Two-hadron production data fit: charm and $SU(2)$ isospin symmetry

We begin giving the setup for the phenomenological analysis of Belle data.
This is mainly based on our previous work [23]. Here we consider only the
Belle data set for the polarization of $\Lambda$/$\bar{\Lambda}$ hyperons
produced in association with a light hadron, $\pi^{\pm}$ or $K^{\pm}$,
measured at $\sqrt{s}=10.58\,\text{GeV}$. The 128 data points are given as a
function of $z_{\Lambda}$ and $z_{\pi/K}$, the energy fractions of
$\Lambda$/$\bar{\Lambda}$ and $\pi/K$ particles. For the current analysis, we
impose a cut on large values of the light-hadron energy fractions,
$z_{\pi/K}<0.5$, keeping only 96 data points, as discussed and motivated in
Ref. [23]. We will come back on this point below.

We will use the following expression to parametrize the $z$ dependence of the
first transverse moment of the polarizing $\Lambda$ FF,
${D}^{\perp(1)}_{1T,\,\Lambda/q}$:

${D}^{\perp(1)}_{1T,\,\Lambda/q}(z;\mu_{b})=\mathcal{N}^{\rm
p}_{q}(z)\,d_{\Lambda/q}(z;\mu_{b})\,,$ (35)

with, as adopted and motivated in Ref. [14],
$q=u,\,d,\,s,\,\bar{u},\,\bar{d},\,\bar{s}$, and where $\mathcal{N}^{\rm
p}_{q}(z)$ (the superscript here refers to the polarizing FF) is parametrized
as:

$\mathcal{N}^{\rm
p}_{q}(z)=N_{q}z^{a_{q}}(1-z)^{b_{q}}\frac{(a_{q}+b_{q})^{(a_{q}+b_{q})}}{a_{q}^{a_{q}}b_{q}^{b_{q}}}\,.$
(36)

In Eq. (35), $d_{\Lambda/q}$ is the collinear unpolarized $\Lambda$
fragmentation function for which we employ the AKK08 set [36]. This
parametrization is given for $\Lambda+\bar{\Lambda}$ and adopts the
longitudinal momentum fraction, $z_{p}$, as scaling variable. In order to
separate the two contributions we assume

$d_{\bar{\Lambda}/q}(z_{p})=d_{\Lambda/\bar{q}}(z_{p})=(1-z_{p})\,d_{\Lambda/q}(z_{p})\,.$
(37)

This is a common way to take into account the expected difference between the
quark and antiquark FF with a suppressed sea at large $z_{p}$ as compared to
the valence component. Other similar choices have a very little impact on the
fit.

Concerning the nonperturbative function $M^{\bot}_{D,\Lambda}$ we employ the
Gaussian model:

$M^{\bot}_{D,\Lambda}(b_{T},z)=\exp{\bigg{(}-\frac{\langle
p_{\perp}^{2}\rangle_{\text{p}}b^{2}_{T}}{4z^{2}_{p}}\bigg{)}}\,,$ (38)

where $\langle p_{\perp}^{2}\rangle_{\text{p}}$ is the Gaussian width, a free
parameter that we extract from the fit. Regarding the collinear FFs of the
unpolarized light hadrons, $\pi$ and $K$, we adopt the DSS07 set [37], while
for $M_{D}$ we consider the PV17 model [38]:

$M_{D}(b_{T},z)=\frac{g_{3}\,e^{-b^{2}_{T}\frac{g_{3}}{4z^{2}}}+\frac{\lambda_{F}}{z^{2}}g^{2}_{4}(1-g_{4}\frac{b_{T}^{2}}{4z^{2}})e^{-b^{2}_{T}\frac{g_{4}}{4z^{2}}}}{g_{3}+\frac{\lambda_{F}}{z^{2}}g^{2}_{4}}\,,$
(39)

where

$g_{3,4}=N_{3,4}\frac{(z^{\beta}+\delta)(1-z)^{\gamma}}{(\hat{z}^{\beta}+\delta)(1-\hat{z})^{\gamma}}\quad$
(40) $\begin{split}\hat{z}&=0.5\,;\quad N_{3}=0.21\,\text{GeV}^{2}\,;\quad
N_{4}=0.13\,\text{GeV}^{2}\,;\\\
\beta&=1.65\,;\quad\delta=2.28\,;\quad\gamma=0.14\,;\quad\lambda_{F}=5.50\,\text{GeV}^{-2}\,.\end{split}$
(41)

For the $g_{K}$ function, we use the one extracted in Ref. [38]:

$g_{K}(b_{T};b_{\text{max}})=\frac{g_{2}b^{2}_{T}}{2}\,;\quad
g_{2}=0.13\,\text{GeV}^{2}\,.$ (42)

For what concerns the $\Lambda$ unpolarized FF, for $M_{D}$ we use a Power-Law
model, see Refs. [23, 39, 40, 41]:

$M_{D}(b_{T},z,p,m)=\frac{2^{2-p}}{\Gamma(p-1)}\,(b_{T}m/z_{p})^{p-1}{K}_{p-1}(b_{T}m/z_{p})\,,$
(43)

with $p=2$ and $m=1\,\text{GeV}$. Notice that in the above equations all
conversions among the different scaling variables ($z,z_{p},z_{h}$) involved
are properly taken into account.

In Eqs. (12), (13), (30) and (31) we use the following definition for the
$\bar{\mu}_{b}$ variable:

$\bar{\mu}_{b}=\frac{C_{1}}{b_{*}(b_{T})}\,,$ (44)

where $C_{1}=2e^{-\gamma_{E}}$ (with $\gamma_{E}$ being the Euler-Mascheroni
constant), and the $b_{*}$ prescription of Ref. [38]:

$b_{*}\equiv b_{*}(b_{T};b_{\rm min},b_{\rm max})=b_{\rm
max}\bigg{(}\frac{1-e^{-b_{T}^{4}/b^{4}_{\rm max}}}{1-e^{-b_{T}^{4}/b^{4}_{\rm
min}}}\bigg{)}^{1/4}\,.$ (45)

Moreover, we adopt

$b_{c}(b_{T})=\sqrt{b^{2}_{T}+b^{2}_{\text{min}}}\,,$ (46)

with $b_{\text{min}}=2e^{-\gamma_{E}}/Q$ and
$b_{\text{max}}=0.6\,\text{GeV}^{-1}$ where, in this analysis,
$Q=10.58\,\text{GeV}$.

Since our present goal is a phenomenological analysis at NLL accuracy, for the
perturbative Sudakov factor in Eq. (14) we use $\alpha_{s}$ at LO, the
anomalous dimension $\gamma_{K}$ at the second order, and the Collins-Soper
kernel $\tilde{K}$ and $\gamma_{D}$ at the first order (see Appendix A for the
explicit expressions of the Sudakov factor and of the anomalous dimensions). A
complete N2LL extraction could be achieved only by adopting the coefficient
functions, in the OPE, at the next order.

Lastly, for the integration in Eqs. (17) and (18), we use
$q_{T_{\text{max}}}=0.27\,Q$. This specific value is chosen on the basis of
the results obtained in Ref. [23] (Fig. 7), where, as shown for this
particular choice of nonperturbative functions, the $\chi^{2}_{\rm dof}$
reaches its minimum.

Concerning the phenomenological analysis and the extraction of the polarizing
FFs from Belle data, we consider three different scenarios, exploiting the
role of the charm quark contribution and the $SU(2)$ isospin symmetry:

  1. 1.

Scenario 1. Here we do not include the charm contribution in the unpolarized
cross section and we do not impose the $SU(2)$ isospin symmetry. We then
extract different $\Lambda$ pFFs for the $u,d,s$ quarks and a single pFF for
the sea ($\bar{u}=\bar{d}=\bar{s}$) antiquarks. As discussed in our prevous
analysis [23], the optimal choice turns out to be an eight-parameter fit:
$N_{u},N_{d},N_{s},N_{\rm sea},a_{s},b_{u},b_{\rm sea}$ and $\langle
p_{\perp}^{2}\rangle_{\rm p}$.

  2. 2.

Scenario 2. We include the charm contribution in the unpolarized cross
section, but we still do not impose the $SU(2)$ isospin symmetry. We continue
to extract different $\Lambda$ pFFs for the $u,d,s$ quarks and a single pFF
for the sea ($\bar{u}=\bar{d}=\bar{s}$) antiquarks, as in the first scenario.
Here, we need to include an extra parameter resulting in a nine-parameter fit:
$N_{u},N_{d},N_{s},N_{\rm sea},a_{d},a_{s},b_{u},b_{\rm sea}$ and $\langle
p_{\perp}^{2}\rangle_{\rm p}$.

  3. 3.

Scenario 3. We include the charm contribution in the unpolarized cross section
and impose $SU(2)$ isospin symmetry for the $u$, $d$ quark pFFs , while still
adopting different pFFs for the $s$ and $\bar{s}$ quarks. Notice that the
AKK08 FF set allows for a slight violation of the $SU(2)$ symmetry; therefore,
even imposing $\mathcal{N}_{u}^{\rm p}=\mathcal{N}_{d}^{\rm p}$ and
$\mathcal{N}_{\bar{u}}^{\rm p}=\mathcal{N}_{\bar{d}}^{\rm p}$, the extracted
pFFs will be still slightly different, see below. In such a case the nine free
parameters are:
$N_{u,d},N_{\bar{u},\bar{d}},N_{s},N_{\bar{s}},a_{u,d},a_{s},b_{u,d},b_{\bar{s}}$
and $\langle p_{\perp}^{2}\rangle_{\rm p}$. Notice that the inclusion of a
further parameter for the sea pFFs, namely $b_{\bar{u},\bar{d}}$, does not
improve the quality of the fit.

As already discussed in our previous analyses, the imposition of the $SU(2)$
symmetry alone within a three-flavor scheme would lead to a very poor quality
of the fit.

The best-fit parameters extracted for the first moment of the pFFs are given
in Tab. 1, together with the $\chi^{2}_{\rm dof}$s for each scenario, while in
Fig. 3 we show the corresponding estimates of the transverse $\Lambda$,
$\bar{\Lambda}$ polarizations, produced in association with a light-hadron,
compared against Belle data [13].

Table 1: Best-fit parameter values for the first moment of the polarizing FF
and the nonperturbative function employed to fit the double-hadron data set,
adopting the three scenarios.

| Scenario
---|---
Parameters | (1) | (2)
$N_{u}$ | $0.144_{-0.076}^{+0.122}$ | $0.178_{-0.096}^{+0.171}$
$N_{d}$ | $-0.140_{-0.057}^{+0.034}$ | $-0.376_{-0.439}^{+0.262}$
$N_{s}$ | $-0.151_{-0.102}^{+0.071}$ | $-0.121_{-0.098}^{+0.053}$
$N_{\rm sea}$ | $-0.09_{-0.099}^{+0.054}$ | $-0.127_{-0.129}^{+0.079}$
$a_{d}$ |  | $0.774_{-0.773}^{+1.074}$
$a_{s}$ | $2.046_{-0.732}^{+0.967}$ | $0.848_{-0.553}^{+0.913}$
$b_{u}$ | $3.57_{-1.471}^{+2.017}$ | $2.71_{-1.511}^{+2.387}$
$b_{\rm sea}$ | $2.606_{-1.596}^{+2.629}$ | $1.59_{-1.294}^{+2.138}$
$\langle p_{\perp}^{2}\rangle_{\rm p}$ | $0.097_{-0.034}^{+0.045}$ | $0.093_{-0.045}^{+0.054}$
$\chi^{2}_{\rm dof}$ | 1.174 | 1.259

| Scenario
---|---
Parameters | (3)
$N_{u,d}$ | $0.130_{-0.031}^{+0.036}$
$N_{\bar{u},\bar{d}}$ | $-0.174_{-0.048}^{+0.052}$
$N_{s}$ | $-0.263_{-0.183}^{+0.136}$
$N_{\bar{s}}$ | $-0.150_{-0.06}^{+0.056}$
$a_{u,d}$ | $2.838_{-2.213}^{+4.565}$
$a_{s}$ | $2.164_{-0.903}^{+1.158}$
$b_{u,d}$ | $10.58_{-7.004}^{+13.56}$
$b_{\bar{s}}$ | $1.545_{-0.818}^{+0.926}$
$\langle p_{\perp}^{2}\rangle_{p}$ | $0.116_{-0.042}^{+0.042}$
$\chi^{2}_{\rm dof}$ | 1.361

Some comments are in order here: Comparing the parameter values obtained
within the first and second scenario, we see a significant difference in their
magnitudes, somehow due to the inclusion of the charm quark contribution in
the second one. However, as already shown in our previous works [14, 23], in
both cases only the up pFF is positive, while the remaining pFFs are all
negative. On the contrary, within the third scenario, we observe that both the
up and down pFF are positive, having an opposite sign with respect to the
anti-up and anti-down pFFs. The strange and anti-strange pFFs come out still
negative. The main point in this comparison is that if we allow for different
normalization factors for the up and down pFFs (Sc. 1 and Sc. 2), they come
out opposite in sign, leading to a strong violation of the $SU(2)$ symmetry.
And this happens even if we allow for independent normalization factors for
the sea contributions. In other words, only imposing $N_{d}=N_{u}$ we can
restore, at least approximately with this set of unpolarized FFs, the
symmetry.

Figure 3: Best-fit estimates of the transverse polarization for $\Lambda$,
$\bar{\Lambda}$ in $e^{+}e^{-}\to\Lambda(\bar{\Lambda})h+X$, for
$\Lambda\pi^{\pm}$ (a), $\bar{\Lambda}\pi^{\pm}$ (b), $\Lambda K^{\pm}$ (c),
$\bar{\Lambda}K^{\pm}$ (d), as a function of $z_{h}$ ($h=\pi,K$) for different
$z_{\Lambda}$ bins, adopting the three different scenarios: (1) green, (2)
orange and (3) violet bands. Data are from Belle [13]. The statistical
uncertainty bands, at $2\sigma$ level, are also shown. Data for
$z_{\pi,K}>0.5$ are not included in the fit.

Despite these differences, within all scenarios we obtain similar sizes for
the Gaussian width. The first $k_{\perp}$-moments of the polarizing FFs are
shown in Fig. 4 for scenarios 1 and 2, and in Fig. 5 for scenario 3. In
scenarios 1 and 2 the first moments are all compatible, at least within the
uncertainty bands, with the exception of the strange pFF. When we move to the
third scenario the up pFF comes out still compatible with the results in the
other scenarios, with the strange pFF somehow in between. The most interesting
finding is that since in this scenario the down pFF is positive ($SU(2)$
constrained), the negative sea contributions are larger in size.

It is important to stress here that in the extraction of the first moment of
the polarizing FFs we do not impose any positivity bound, that, in principle,
could prevent a proper sampling of the parameter space. On the other hand we
have checked, _a posteriori_ , that this is fulfilled in all scenarios
considered.

Moving to the comparison with data, we can generally say that all three
scenarios are able to describe reasonably, or even quite, well the
$\Lambda\pi^{\pm}$, $\bar{\Lambda}\pi^{\pm}$, $\Lambda K^{-}$ and
$\bar{\Lambda}K^{+}$ polarization data. However, as already pointed out in our
first works, where we did [23] or did not [26] employ the full TMD machinery,
within scenario 1 one cannot describe, at variance with the $\Lambda\pi$ case,
the $\Lambda K^{+}$ and $\bar{\Lambda}K^{-}$ data with $z_{K}>0.5$.

Figure 4: First moments of the polarizing FFs, for the up (a), down (b),
strange (c) and sea (d) quarks, as obtained from the fit within the first (red
dashed lines) and the second (blue dot-dashed lines) scenarios. The
corresponding statistical uncertainty bands, at $2\sigma$ level, are also
shown.

Figure 5: First moments of the polarizing FFs, for the up/down (a), anti-
up/down (b), strange (c) and anti-strange (d) quarks, as obtained from fit
within the third scenario. The corresponding statistical uncertainty bands, at
$2\sigma$ level, are also shown.

Quite interestingly, when we include the charm contribution, imposing or not
the $SU(2)$ isospin symmetry, we can still obtain similar good fits with a
simultaneous very good description of these data points (even if not included
in the analysis), see Figs. 3c and 3d, lower panels. This result, focusing on
$\Lambda K^{+}$ for simplicity, can be understood as follows: in scenarios 2
and 3 the inclusion of the charm contribution in the denominator, with non
negligible charm FFs both for $K^{+}$’s and $\Lambda$’s, requires larger, in
size, pFFs in the fit. Moreover, since this extra piece in the denominator
happens to be a decreasing function in $z_{K}$ the polarization eventually
increases in size with $z_{K}$.

From the present study, it is very likely that the inclusion of the charm
contribution, at least in the unpolarized cross section, must be considered
necessary for the analysis of Belle data. Several attempts to include this
contribution also in the numerator of the transverse polarization (that is
parametrizing also pFFs for charm quarks) have been carried out but no
significant improvement on the $\chi^{2}_{\rm dof}$ value or in the
description of data has been found.

Similar conclusions, even if on a more qualitative ground since they do not
provide any $\chi^{2}$ value and any uncertainty band, have been obtained in
Ref. [22]. Here, by including the charm contribution, also for the polarizing
FFs (resulting in a 20-parameter fit) they show that Belle data can be
described reasonably well even without any isospin symmetry violation.

In this respect, we agree that the issue of $SU(2)$ symmetry has to be taken
with care and that cannot be solved by analysing only the data on the
transverse polarization of $\Lambda/\bar{\Lambda}$ produced in $e^{+}e^{-}$
processes. More experimental information is therefore certainly needed.

###  III.2 Predictions for the transverse $\Lambda$ polarization in
$e^{+}e^{-}$ collisions at different energies

Here we give some predictions at different energies, focusing on $\Lambda$-$K$
production, with the aim to look for possible significant differences among
the three scenarios. In Fig. 6 we show the estimates for the transverse
polarization of $\Lambda$’s produced with $K^{\pm}$ mesons, at different
energies, namely, 8.48 GeV (left panel) and 12.58 GeV (right panel). Notice
that in the first case we cannot have the $z_{\Lambda}=0.25$ bin for
kinematical reasons. At both energies, only the first $z_{\Lambda}$ bins show
some discrepancies at large $z_{K}$ values between the predictions obtained
within scenarios 2 and 3. On the other hand, for higher values of
$z_{\Lambda}$ all predictions become very similar, within the uncertainties.
This, also true for high energy values, could prevent the distinction between
the two scenarios in future $e^{+}e^{-}$ measurements.

Figure 6: Estimates of the transverse $\Lambda$ polarization in
$e^{+}e^{-}\to\Lambda K^{\pm}+X$ at $Q=8.48\,\text{GeV}$ (left panel), and at
$Q=12.58\,\text{GeV}$ (right panel), as a function of $z_{K}$ for different
$z_{\Lambda}$ bins, for the three different scenarios: (1) green, (2) orange
and (3) violet bands. The statistical uncertainty bands, at $2\sigma$ level,
are also shown.

###  III.3 Predictions for the transverse $\Lambda$ polarization in SIDIS

In this section, by using Eqs. (30), (31), (32) and (33), we present the
predictions for the transverse $\Lambda$ polarization in unpolarized SIDIS
electron-proton and electron-deuterium collisions, for different values of
their beam energies, and considering the three previously presented scenarios.

A similar analysis, showing also estimates obtained from our previous
extraction [23], has been presented in Ref. [30]. In this paper they consider
in detail the transverse $\Lambda$ polarization only, including the charm
contribution and imposing $SU(2)$ symmetry (see Ref. [22] and our comments
above). In this respect, our predictions are indeed in qualitative agreement
with theirs. On the other hand, they do not show any result for the
$\bar{\Lambda}$ case, that, as we will discuss below, represents a much
powerful tool to discriminate different scenarios. It is also worth noticing
that a comprehensive phenomenological impact study on the transverse $\Lambda$
polarization at the future EIC, even if limited to scenario 1 and at LO
accuracy, has been carried out in Ref. [20]. In this work they include EIC
pseudodata to reweight the parametrization of the polarizing FFs as extracted
from Belle $e^{+}e^{-}$ data, leading to a significant reduction in the
theoretical uncertainties.

Concerning the present analysis, for $\Lambda$ hyperon, we employ the
unpolarized nonperturbative function in Eq. (43), and the polarizing FF first
moment and nonperturbative function in Eqs. (35), (36) and (38), adopting the
parameters given in Tab. 1. As for the proton PDFs, we use for the unpolarized
ones the CT14 NNLO set [42], and for the nonperturbative function the one
extracted in Ref. [38], which has the following form:

$M_{f_{1}}(b_{T},x)=\frac{1}{2\pi}e^{-g_{1}\frac{b_{T}^{2}}{4}}\bigg{(}1-\frac{\lambda
g^{2}_{1}}{1+\lambda g_{1}}\frac{b_{T}^{2}}{4}\bigg{)}\,,$ (47)

where

$g_{1}=N_{1}\frac{(1-x)^{\alpha}x^{\sigma}}{(1-\hat{x})^{\alpha}\hat{x}^{\sigma}}\quad$
(48) $\begin{split}\hat{x}&=0.1\quad N_{1}=0.28\,\text{GeV}^{2}\\\
\alpha&=2.95\quad\sigma=0.173\quad\lambda=0.86\,\text{GeV}^{-2}\,.\end{split}$
(49)

Regarding the neutron PDF, we use the same proton nonperturbative function and
unpolarized PDF set but with the following substitution for the up and down
quarks:

$u_{n}=d_{p}\,,\quad
d_{n}=u_{p}\,,\quad\bar{u}_{n}=\bar{d}_{p}\,,\quad\bar{d}_{n}=\bar{u}_{p}\,.$
(50)

In the following we show estimates for the transverse $\Lambda$ polarization
integrated over its transverse momentum (or, more precisely, over
$\bm{q}_{T}$), using Eq. (33).

Table 2: Nucleon and electron beam energies and the corresponding c.m. energy $E_{N}$ (GeV) |  $E_{e}$ (GeV) |  $\sqrt{s_{eN}}$ (GeV)
---|---|---
41 | 5 | 28.6
100 | 10 | 63.2

We will consider two different values of the c.m. energy $\sqrt{s_{eN}}$, Eq.
(24), reported in Tab. 2, corresponding to various combinations of nucleon
(electron) beam energies $E_{N}$ ($E_{e}$). We will keep fixed $y$ to $0.4$,
and explore different values of $x_{B}$ and $z_{\Lambda}$.

In Fig. 7 we show the estimates for the $\bm{q}_{T}$-integrated transverse
$\Lambda/\bar{\Lambda}$ polarization in electron-proton (deuterium)
collisions, see Eqs. (33) and (34), for different values of $\sqrt{s_{eN}}$,
$x_{B}$ and $z_{\Lambda}$, in the three scenarios. Since we adopt a fixed
ratio $q_{T_{\rm max}}/Q=0.27$, by exploring large values of $Q$, up to 30 GeV
in our case, for certain $x_{B}$ values we enter the region of large $q_{T}$
(up to 8 GeV).

Firstly, we notice that scenarios 1 and 2 lead to $\Lambda/\bar{\Lambda}$
polarization with similar size and behavior, for the two values of
$\sqrt{s_{eN}}$ both for proton and deuteron targets.

Figure 7: Estimates of the transverse $\Lambda/\bar{\Lambda}$ polarization in
electron-proton (a, b) and electron-deuterium (c, d) scattering at
$\sqrt{s_{eN}}=28.6$ GeV (left panels) and 63.2 GeV (right panels), at
$y=0.4$, as a function of $z_{\Lambda}$ for different $x_{B}$ bins, and the
three scenarios: (1) green, (2) orange and (3) violet. The statistical
uncertainty bands, at $2\sigma$ level, are also shown.

We can see how the $\Lambda$ polarization tends to decrease, becoming
negative, as $z_{\Lambda}$ increases, while the $\bar{\Lambda}$ polarization
is always negative. In particular, for $\sqrt{s_{eN}}=28.6$ GeV (Fig. 7.a and
7.c), the polarization has the same pattern and size in each $x_{B}$ bin,
while for greater $\sqrt{s_{eN}}$ values (Fig. 7.b and 7.d), we have a general
reduction in size of the polarization as $x_{B}$ grows.

For what concerns the third scenario, we can see that the polarization follows
a pattern similar to that illustrated for the first and second scenarios, but
with some differences. The $\Lambda$ polarization has a similar or slightly
greater size than in the other two scenarios; the most significant difference
can be found for the $\bar{\Lambda}$ polarization, which is much greater in
size, reaching values of about $40\%$ for $x_{B}=0.6$ and
$\sqrt{s_{eN}}=28.6\,\text{GeV}$.

Finally we provide a comment on the strong similarities between the
$\bar{\Lambda}$ polarizations in $ep$ and $eD$ collisions: the reasons can be
traced back to the dominant contribution driven by the up and down
distribution functions in both targets. For $\bar{\Lambda}$ production this
enters directly convoluted with the polarizing FF for sea quarks in the
numerator and the unpolarized sea FF in the denominator. In $\Lambda$
production this does not happen since the up and down parton distributions
couple in a different way to the up and down pFFs when one considers a proton
or a deuterium target.

At variance with the case of the double-hadron production in $e^{+}e^{-}$
collisions, the estimates for the transverse polarization within the second
and third scenarios are clearly separated. Thus, future measurements of
transversely polarized $\Lambda/\bar{\Lambda}$ in SIDIS will potentially allow
us to gain further insights and to distinguish between the two scenarios.

It is worth noticing that the corresponding estimates for the transverse
polarization as a function of the $\Lambda/\bar{\Lambda}$ transverse momentum,
$P_{1T}$, are not able to discriminate among the different scenarios.

###  III.4 Role of intrinsic charm contribution

From the previous discussion it is clear that the charm contribution in the
fragmentation process can be relevant for the study of the transverse
$\Lambda$ polarization. Here, we explore how the employment of collinear PDFs,
that take into account the presence of an intrinsic charm (IC) component in
the proton, can play a role in this context. For this study, we consider again
the CT14NNLO set and two recent PDF sets: the CT14NNLO IC set [43], by using
the Brodsky-Hoyer-Peterson-Sakai (BHPS) model for the intrinsic charm
component, and the NNPDF4.0 NNLO set [44].

In Fig. 8, we present a comparison of the estimates of the transverse
$\Lambda/\bar{\Lambda}$ polarization in electron-proton (deuterium) scattering
at $\sqrt{s_{eN}}=28.6\,\text{GeV}$, obtained using the second scenario
parameters for the polarizing FFs and the three PDF sets. We observe that the
estimated polarization obtained using the BHPS model for IC (violet bands) and
the NNPDF set (green bands) do not differ significantly from the predictions
shown in the previous section (Fig. 7) without the IC component (orange
bands). This behavior is also present for smaller and greater values of the
c.m. energy. For completeness, we have also explored the role of the
perturbative charm component (NNPDF set) without any significant differences.

Figure 8: Estimates of the transverse $\Lambda/\bar{\Lambda}$ polarization in
electron-proton (a) and electron-deuterium (b) scattering at
$\sqrt{s_{eN}}=28.6\,\text{GeV}$, as a function of $z_{\Lambda}$ for different
$x_{B}$ bins, obtained using the second scenario parameters for the first
moment of the pFFs and different PDF sets: CT14NNLO no IC (orange bands),
CT14NNLO IC with BHPS model for IC (violet bands) and NNPDF (green bands). The
statistical uncertainty bands, at $2\sigma$ level, are also shown.

On the contrary, when adopting the third scenario parameters, the estimates
can vary significantly as $x_{B}$ increases. As we can see in Fig. 9, the
transverse $\bar{\Lambda}$ polarization estimates obtained with the BHPS model
for IC (violet bands) and the NNPDF set (green bands) gradually move away from
the predictions without the IC component, both in electron-proton and in
electron-deuterium collisions, leading to a smaller polarization size. Notice
that this conclusion is valid also when adopting the perturbative NNPDF charm
component. Concerning the $\Lambda$ polarization, only the estimates with the
NNPDF set differ from the other two predictions. In fact, both the bands
without the IC component and the ones with the BHPS model decrease to zero, as
$z_{\Lambda}$ increases, while the NNPDF predictions become negative and grow
up to about $10\%$ in size. It is worth noting that this is mainly due to the
different PDF set adopted and not to the inclusion of the IC component.

In Fig. 10, we can see how, including the intrinsic charm contribution, the
estimates for $\bar{\Lambda}$ in $eD$ collisions in the second and third
scenarios are sufficiently well separated for both PDF sets.

Figure 9: Estimates of the transverse $\Lambda/\bar{\Lambda}$ polarization in
electron-proton (a) and electron-deuterium (b) scattering at
$\sqrt{s_{eN}}=28.6\,\text{GeV}$, as a function of $z_{\Lambda}$ for different
$x_{B}$ bins, obtained using the third scenario parameters for the first
moment of the pFFs and different PDF sets: CT14NNLO no IC (orange bands),
CT14NNLO IC with BHPS model for IC (violet bands) and NNPDF (green bands). The
statistical uncertainty bands, at $2\sigma$ level, are also shown. Figure 10:
Estimates of the transverse $\Lambda/\bar{\Lambda}$ polarization and electron-
deuterium scattering at $\sqrt{s_{eN}}=28.6\,\text{GeV}$, as a function of
$z_{\Lambda}$ for different $x_{B}$ bins, obtained using the second and third
scenario parameters for the first moment of the pFFs and different PDF sets:
CT14NNLO IC with BHPS IC (black (Sc. 2) and violet (Sc. 3) bands) and NNPDF
(orange (Sc. 2) and green (Sc. 3) bands). The statistical uncertainties, at
$2\sigma$ level, are also shown.

##  IV Conclusions

In this paper we have carried out a comprehensive reanalysis, within a TMD
framework at NLL accuracy, of the transverse $\Lambda/\bar{\Lambda}$
polarization data from Belle Collaboration in the associated two-hadron
production in $e^{+}e^{-}$ processes. In particular, we have focused on the
role of isospin symmetry and of the charm contribution in the extraction of
the polarizing fragmentation functions. While requiring $SU(2)$ simmetry alone
(within a three-flavor scheme) leads to a very unsatisfactory fit, we have
shown that all other scenarios considered allow for very similar and quite
good descriptions of the available data. We can then conclude that Belle
$e^{+}e^{-}$ data, or more generally $e^{+}e^{-}$ processes, alone are not
able to discriminate among the different scenarios and, in particular, to shed
light on the $SU(2)$ symmetry issue.

We have therefore explored this fundamental aspect by considering the same
observable in SIDIS processes. By assuming the expected universality of the
polarizing FFs we have given several predictions for the kinematical set-up
reachable at the EIC, exploiting three different scenarios. In such a case, by
including the charm contribution in the unpolarized cross section, one can
indeed distinguish between a scenario where isospin symmetry is respected or
not. We have also considered a pFF for charm quarks in the numerator of the
polarization without any improvement in the fit. For completeness, we have
discussed the role of the intrinsic charm in the proton for SIDIS processes
and shown that the above conclusion does not change.

The spontaneous transverse $\Lambda$ polarization remains a challenging
subject, but at the same time offers a unique opportunity to study the
fragmentation mechanism and, more specifically, spin and transverse momentum
correlations. The present study, focused on processes where TMD factorization
has been proven to hold, provides a further step to shed light on this very
interesting phenomenon.

As we have shown, future EIC meaurements can play a significant role in this
context: certainly in testing the phenomenological results obtained in
$e^{+}e^{-}$ annihilation processes and more generally, in testing fundamental
issues like the universality of the polarizing FFs, their scale dependence,
their flavor decomposition a well as the role of $SU(2)$ symmetry.

## Acknowledgments

We thank Carlo Flore for his suggestions on the role of the intrinsic charm.
This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement N. 824093
(STRONG-2020). U.D. and M.Z. also acknowledge financial support by Fondazione
di Sardegna under the projects “Proton tomography at the LHC”, project number
F72F20000220007 and “Matter-antimatter asymmetry and polarisation in strange
hadrons at LHCb‘”, project number F73C22001150007 (University of Cagliari).
L.G. acknowledges support from the US Department of Energy under contract No.
DE-FG02-07ER41460.

##  Appendix A Perturbative Sudakov factor

Here we give the analytic expression of the perturbative Sudakov factor
presented in Eq. (14):

$S_{\rm
pert}(b_{*};\bar{\mu}_{b})=-\widetilde{K}(b_{*};\bar{\mu}_{b})\ln\frac{Q^{2}}{\bar{\mu}_{b}^{2}}-\int^{Q}_{\bar{\mu}_{b}}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\bigg{[}2\gamma_{D}(g(\mu^{\prime});1)-\gamma_{K}(g(\mu^{\prime}))\ln{\frac{Q^{2}}{\mu^{\prime
2}}}\bigg{]}\,.$ (51)

As discussed in Section III, since our goal is a phenomenological analysis at
NLL accuracy, we take $\alpha_{s}$ at LO order:

$\alpha_{s}(\mu^{2})=\frac{1}{\beta_{0}\ln(\mu^{2}/\Lambda^{2}_{\rm QCD})}\,,$
(52)

and we expand the anomalous dimensions as follows:

$\gamma_{K}=\sum_{n}\gamma^{[n]}_{K}\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{n}\,\quad\gamma_{D}=\sum_{n}\gamma^{[n]}_{D}\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{n}\,,$
(53)

up to, respectively, the second and first order. Given that the first order
term of $\widetilde{K}(b_{*};\bar{\mu}_{b})$ is zero [1, 45], the perturbative
Sudakov factor can be written again as:

$\displaystyle S_{\rm pert}(b_{*};\bar{\mu}_{b})$ $\displaystyle=$
$\displaystyle\frac{\gamma^{[1]}_{D}}{4\pi\beta_{0}}\ln\bigg{(}\frac{\ln(Q/\Lambda_{\rm
QCD})}{\ln(\bar{\mu}_{b}/\Lambda_{\rm
QCD})}\bigg{)}+\frac{\gamma^{[1]}_{K}}{4\pi\beta_{0}}\bigg{[}\ln(Q/\bar{\mu}_{b})-\ln(Q/\Lambda_{\rm
QCD})\ln\bigg{(}\frac{\ln(Q/\Lambda_{\rm QCD})}{\ln(\bar{\mu}_{b}/\Lambda_{\rm
QCD})}\bigg{)}\bigg{]}$ (54) $\displaystyle+$
$\displaystyle\frac{\gamma^{[2]}_{K}}{2(4\pi\beta_{0})^{2}}\bigg{[}-\frac{\ln(Q/\bar{\mu}_{b})}{\ln(\bar{\mu}_{b}/\Lambda_{\rm
QCD})}+\ln\bigg{(}\frac{\ln(Q/\Lambda_{\rm
QCD})}{\ln(\bar{\mu}_{b}/\Lambda_{\rm QCD})}\bigg{)}\bigg{]}\,,$

where [46]:

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{11C_{A}-4T_{F}n_{f}}{12\pi}\,,\qquad\gamma^{[1]}_{D}=6C_{F}\,,$
$\displaystyle\gamma^{[1]}_{K}$ $\displaystyle=$ $\displaystyle
8C_{F}\,,\qquad\gamma^{[2]}_{K}=C_{A}C_{F}\bigg{(}\frac{536}{9}-\frac{8\pi^{2}}{3}\bigg{)}-\frac{80}{9}C_{F}n_{f}\,,$
(55)

with $C_{F}=4/3$, $C_{A}=3$, $T_{F}=1/2$, and $\Lambda_{\rm
QCD}=0.2123\,\text{GeV}$ for $n_{f}=3$ or $\Lambda_{\rm
QCD}=0.1737\,\text{GeV}$ for $n_{f}=4$.

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