# Synergy between Spin and Orbital Angular Momenta on a Möbius Strip

Lei Liu  National Laboratory of Solid State Microstructures and Department of
Materials Science and Engineering, Nanjing University, Nanjing 210093, China
Xiao-Chen Sun  National Laboratory of Solid State Microstructures and
Department of Materials Science and Engineering, Nanjing University, Nanjing
210093, China  Yuan Tian  National Laboratory of Solid State Microstructures
and Department of Materials Science and Engineering, Nanjing University,
Nanjing 210093, China  Xiujuan Zhang Corresponding author
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and
Department of Materials Science and Engineering, Nanjing University, Nanjing
210093, China  Ming-Hui Lu Corresponding author<EMAIL_ADDRESS>National
Laboratory of Solid State Microstructures and Department of Materials Science
and Engineering, Nanjing University, Nanjing 210093, China  Jiangsu Key
Laboratory of Artificial Functional Materials, Nanjing 210093, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China  Yan-Feng Chen Corresponding author
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and
Department of Materials Science and Engineering, Nanjing University, Nanjing
210093, China  Collaborative Innovation Center of Advanced Microstructures,
Nanjing University, Nanjing 210093, China

###### Abstract

Spin and orbital angular momenta are fundamental physical characteristics
described by polarization and spatial degrees of freedom, respectively.
Polarization is a feature of vector fields while spatial phase gradient
determines the orbital angular momentum ubiquitous to any scalar field. Common
wisdom treats these two degrees of freedom as distinct and independent
principles to manipulate wave propagations. Here, we demonstrate their
synergy. This is achieved by introducing two orthogonal $p$-orbitals as
eigenbases, whose spatial modal features are exploited to generate orbital
angular momenta and the associated orbital orientations provide means to
simultaneously manipulate polarizations. Through periodic modulation and
directional coupling, we realize a full cyclic evolution of the synchronized
and synergized spin-orbital angular momenta. Remarkably, this evolution
acquires a nontrivial geometric phase, leading to its representation on a
Möbius strip. Experimentally, an acoustic cavity array is designed, whose
dipole resonances precisely mimic the $p$-orbitals. The acoustic waves,
uniquely, see the pressure (scalar) field as a spatial feature and carry an
intrinsic polarization defined by the velocity (vector) field, serving as an
ideal platform to observe the synergy of spin and orbital angular momenta.
Based on such a property, we further showcase a spin-orbital-Hall effect,
highlighting the intricate locking of handedness, directionality, spin density
and spatial mode profile. Our study unveils a fundamental connection between
spin and orbital angular momenta, promising avenues for novel applications in
information coding and high-capacity communications.

Introduction.—Exploring the intricate interplay between spin (SAM) and orbital
angular momentum (OAM) has been a longstanding and compelling quest in wave
physics. Spin is an intrinsic form of angular momentum carried by elementary
particles like electrons[1]. Over a century ago, Poynting predicted a
circularly polarized light carries angular momentum[2], nowadays attributed to
the $\hbar$ spin of the photons, known as SAM. In a broader context, SAM
characterizes the orientations of the electric or magnetic fields, i.e., the
polarizations (the vector aspects of the electromagnetic waves)[3]. Apart from
SAM, light also carries OAM, which was identified by Allen et al. in 1992[4].
They recognized that unlike SAM being more like an intrinsic characteristic of
light, OAM can be generated by twisted azimuthal phase gradients which are
associated with spatial degrees of freedom and ubiquitous in all types of
waves. Benefited from such a property, OAM has been vastly explored in various
realms of waves, including structured light[5], quantum optics[6], light
manipulations based on meta-structures (metamaterials and metasurfaces)[7, 8,
9], and even longitudinal waves like acoustic waves[10, 11, 12].

While SAM and OAM are often treated as independent principles governing
distinct degrees of freedom (SAM for the vector aspects and OAM for the
spatial scalar aspects), they can interact with each other, pertaining to the
so-called spin-orbital interactions (SOIs)[13]. There are primarily two types
of SOIs. The first one is associated with the helicity-dependent position or
momentum for wave propagations, known as the spin-Hall effect[14]. One famous
example is the classical version of the topological quantum spin-Hall effect.
Therein, the propagations of the topological edge states are found to depend
on the helicity of SAM[15, 16]. The second type of SOI concerns the
combination of the vector features of SAM with different orders of OAM,
expanding scalar OAM waves to more general types of vectorial vortices[17,
18]. In these vectorial states, the vector polarization constantly changes
direction when tracing the azimuthal phase gradient, suggesting a SAM-OAM
conversion. Despite different features in SOIs, the essence is the discussion
on how one angular-momentum component affects the other.

Historically, SAM and OAM can be independently manipulated or interact with
each other. This forms the traditional understanding. However, a crucial piece
has been missing—whether SAM and OAM can be synchronized or synergized. Here,
we bridge this gap by demonstrating the synergy of SAM and OAM on a Möbius
strip, which exhibits a full cyclic evolution from linear-polarization into
circular-polarization and back to itself, accompanied by changes in OAM from
linear momentum to a perfect vortex and back. The revelation of such a
fundamental and unique connection introduces additional degrees of freedom to
manipulate waves, with high potential in information coding and high-capacity
communications.

Figure 1:  (a) Schematics of two orthogonal $p$-orbitals. (b) A 1D lattice
constructed by periodically arranging the $p$-orbitals. (c) Band structures of
the lattice in (b), incorporating $\mathcal{D}$. The left panel is for Site A
and the right for Site B. (d) One cyclic evolution of the eigenstates in (c)
by tracing $\mathcal{D}$ from $0$ to $2\pi$, mapped onto a Poincaré sphere and
exemplified for Site A in the $E_{1}$ group. As the cyclic evolution acquires
a Pancharatnam-Berry geometric phase $\theta_{g}=\pi$, it can be further
represented on a Möbius strip as shown in (e) where both the OAM states and
polarization ellipses are presented.

Synergy between SAM and OAM.—We start with two orthogonal $p$-orbitals,
labelled by $|p_{a}\rangle$ and $|p_{d}\rangle$, as schematically illustrated
in Fig. 1(a). Without loss of generality, the orientation of $|p_{a}\rangle$
($|p_{d}\rangle$) is set along the antidiagonal (diagonal) direction. In the
Jones vector representation, the corresponding vector bases are
${\frac{\sqrt{2}}{2}\left[\begin{matrix}1&1\\\ \end{matrix}\right]}^{T}$ and
${\frac{\sqrt{2}}{2}\left[\begin{matrix}1&-1\\\ \end{matrix}\right]}^{T}$. The
orbitals are arranged into a quasi-one-dimensional (1D) lattice along the
$x$-direction, as depicted in Fig. 1(b). Each unit cell comprises two sites,
labeled by A and B, each hosting a pair of orthogonal $|p_{a}\rangle$ and
$|p_{d}\rangle$. Sites A and B are further dislocated along the $y$-direction
to introduce a directional coupling (characterized by the angle $\vartheta$),
which effectively generates distinct gauge potentials for $|p_{a}\rangle$ and
$|p_{d}\rangle$, leading to a phase difference between them and therefore
giving rise to nonzero OAM. Concurrent with the same phase change, the vector
bases are superposed, leading to nonzero SAM in synergy with the OAM. Due to
the periodicity, the phase change depends on the wave vector and is periodic
as well. Correspondingly, the OAM and SAM undergo a full cycle of synergy.

The proposed model is characterized by three coupling coefficients, i.e.,
$t_{t}$, $t_{l}$ and $t_{m}$ for the transverse, longitudinal and crossed
coupling, respectively (see illustrations in Fig. 1(b)). For this model, the
Hamiltonian is written as $H(k)=\begin{bmatrix}\mathbf{0}_{2\times 2}&h(k)\\\
h^{\dagger}(k)&\mathbf{0}_{2\times 2}\end{bmatrix}$, with $\
h(k)=(t_{t}+t_{l}e^{ik})\sigma_{0}+(t_{m}+t_{m}e^{ik})\sigma_{x}$
($\sigma_{0}$ and $\sigma_{x}$ are the Pauli matrices), corresponding to
eigenfunction
$|\psi\rangle=(\phi_{A,p_{a}},\phi_{A,p_{d}},\phi_{B,p_{a}},\phi_{B,p_{d}})^{T}$.
Here, $\phi_{i,p_{j}}$ with $i=A,B$ and $j=a,d$ represents the $|p_{j}\rangle$
component on Site $i$, the lattice constant is assumed to be $1$, $k$ denotes
the wave vector, and ${\dagger}$ indicates the complex conjugate transpose.

The phase difference between $|\left.p_{a}\right\rangle$ and
$|\left.p_{d}\right\rangle$ components is obtained as
$\mathcal{D}=\arg{\left(\frac{\phi_{i,p_{a}}}{\phi_{i,p_{d}}}\right)}$. To
track the evolution of $\mathcal{D}$, we present the energy bands in Fig. 1(c)
taking $\vartheta=60^{\circ}$, $t_{t}=0$, $t_{l}=-1$ and $t_{m}=0.4$, where
$\mathcal{D}$ is visualized through color-coding (more details can be found in
Supplementary Material[19], which includes Refs. [20, 21, 22, 23, 24, 25, 26,
27]). There are four bands grouped in two, denoted by $E_{1}$ and $E_{2}$ with
$+$ and $-$ signs representing positive and negative group velocity,
respectively. In the $E_{1}$ group, $\mathcal{D}$ undergoes an evolution from
$0$ to $2\pi$, with one half following the $E_{1,+}$ band and the other
following the $E_{1,-}$ band. In the $E_{2}$ group, the evolution is symmetric
to the $E_{1}$ group with respect to the zero-energy. For different Sites A
and B, $\mathcal{D}$ exhibits opposite evolutions. These observations
elucidate the intricate dependence of $\mathcal{D}$ on $k$, which is
intimately tied to the symmetries of the system[19].

Associated with the evolution of $\mathcal{D}$, the OAM and SAM exhibit a full
cycle of synergy. To visualize the cyclic evolution, we employ the Poincaré
sphere representation[28, 29]. Typically, a polarized state (or an OAM state)
can be described on a sphere spanned by the Stokes parameters
$\\{S_{1},S_{2},S_{3}\\}=\\{\cos{2\varphi}\cos{2\chi},\sin{2\varphi}\cos{2\chi},\sin{2\chi}\\}$,
with $2\varphi$ and $2\chi$ respectively the azimuthal and ellipticity angles.
In our scenario, the OAM and polarization states are superpositions of the
$|\left.p_{a}\right\rangle$ and $|\left.p_{d}\right\rangle$ orbital bases and
their orientation vector bases, yielding
$|\left.p_{d}\right\rangle+e^{i\mathcal{D}}|\left.p_{a}\right\rangle$ and
$\frac{\sqrt{2}}{2}\left(\begin{bmatrix}1&-1\end{bmatrix}^{T}+e^{i\mathcal{D}}\begin{bmatrix}1&1\end{bmatrix}^{T}\right)$,
respectively. Such a choice of eigenbases pins the evolution onto the
$S_{1}$-$S_{3}$ plane (corresponding to $\varphi=0$).

A full cycle of evolution can be tracked by $\mathcal{D}=2\chi$ running over
$[0,2\pi]$, as illustrated in Fig. 1(d), exemplified for Site A in the $E_{1}$
group. Initially, at $E_{1,+}(k=0)$, the OAM state is expressed as
$|\left.p_{d}\right\rangle+|\left.p_{a}\right\rangle$, corresponding to a
$p$-orbital with orientation along the $x$-direction. This state can be mapped
to the east of the equator of the Poincaré sphere, in synergy with a
horizontal linear-polarization. As $k$ increases, the linear-polarization
transforms into elliptical-polarizations, associated with distorted
$p$-orbitals. Upon reaching the poles, the evolution state becomes a perfect
vortex and is simultaneously circularly-polarized. For different sites, the
evolution occurs in opposite directions, resulting in vortices with
topological charges of $l=\pm 1$ and circular-polarizations with opposite
handedness, i.e., $|\left.p_{d}\right\rangle+i|\left.p_{a}\right\rangle$ and
$\frac{\sqrt{2}}{2}e^{i\frac{\pi}{4}}\begin{bmatrix}1&i\end{bmatrix}^{T}$ for
Site A at the north pole and
$|\left.p_{d}\right\rangle-i|\left.p_{a}\right\rangle$ and
$\frac{\sqrt{2}}{2}e^{-i\frac{\pi}{4}}\begin{bmatrix}1&-i\end{bmatrix}^{T}$
for Site B at the south pole[19]. Continuous increase in $k$ leads to the
evolution from the north pole to the west of the equator for Site A and from
the south pole to the west of the equator for Site B. Correspondingly, the OAM
state returns to the $p$-orbital and the polarization to the linear-
polarization, only with a vertical orientation.This completes half of the
evolution. By following the $E_{1,-}$ band, the other half unfolds, precisely
reversing the first half of A-evolution for Site B and the first half of
B-evolution for Site A. In this way, a full cycle of SAM-OAM synergy is
realized. For the $E_{2}$ group, a similar analysis can be performed.

It is known that a cyclic evolution on a Poincaré sphere generates a
nontrivial Pancharatnam-Berry geometric phase $\theta_{g}=\Omega/2$, where
$\Omega$ is the solid angle enclosed by the cyclic evolution[30, 31]. In our
case, the cyclic evolution encloses half of the Poincaré sphere, giving rise
to $\theta_{g}=\pi$. Such nonzero geometrical phase in the momentum space
indicates nontrivial topology of the periodic lattice[19]. It is also
suggested that the eigenstates can be represented on a Möbius strip with
$4\pi$ periodicity[32, 33, 34]. This is consistent with our observation, i.e.,
the eigenstates return to their initial states only after $4\pi$ change of $k$
(or $2\pi$ change of $\mathcal{D}$). Figure 1(e) illustrates how one cycle of
the SAM-OAM synergy is represented on the Möbius strip.

Simultaneously manipulating anisotropic orbitals and their orientations opens
up opportunities to synchronize OAM and SAM, uncovering a fundamental
connection between these angular momenta. We argue that such a principle is
universal and can be extended to a general selection of orbitals and
orientations, offering versatile control over the synergy between OAM and
SAM[19].

Figure 2:  (a) Schematics of an acoustic cavity. (b) Resonant spectra of the
cavity with two degenerate dipole resonances marked by the blue dots. (c)
Distributions of acoustic pressure (color map) and velocity (thin arrows)
fields of the two dipole modes. The orientations of the modes are indicated by
two thick arrows.  Figure 3:  (a) Schematics of the quasi-1D acoustic lattice.
The blue lines frame a unit cell. (b) Numerically calculated (green curves)
and experimentally measured (color map) band structures for the lattice in
(a). (c) Measured acoustic pressure fields $|P|$, their phase distributions
$\mathrm{Arg}(P)$, and polarization ellipses of velocity fields for cavities A
and B following $E_{1,+}$ evolution. Both representations on the Möbius strip
and indication on the $S_{1}$-$S_{3}$ plane of the Poincaré sphere are
provided. (d) The same as (c), only for $E_{2,+}$ evolution.

Experimental demonstration.—Applicable as a general principle, the revelation
of synergy between SAM and OAM is particularly noteworthy for scalar waves
such as acoustic waves, which are conventionally considered polarization-free.
Here, we show acoustic waves, with their pressure fields as spatial degrees of
freedom and velocity fields as vector degrees of freedom, inherently embody
the conditions outlined in our theoretical framework. This positions them as
an ideal platform for demonstrating the SAM-OAM synergy and exploring the
associated physical phenomena.

We consider an acoustic cylindrical cavity illustrated in Fig. 2(a), which
supports two degenerate dipole resonances, as shown by the eigenspectra and
field distributions in Figs. 2(b-c) (see Ref.[19] for geometric and material
parameters). Their pressure fields precisely mimic the spatial features of two
orthogonal $p$-orbitals and the velocity fields exactly align with the
orientations of the $p$-orbitals. Such a correspondence is inherently rooted
in the incompressible linear Euler equation where the vector velocity field is
connected with the gradient of the scalar pressure field[35].

To observe the SAM-OAM synergy, we further construct a 1D periodic array of
the acoustic cavities connected via a rectangular air channel, as illustrated
in Fig. 3(a). The band structures are presented in Fig. 3(b), both numerically
calculated and experimentally measured (see details in Ref.[19]), which
exhibit a remarkable agreement with the energy bands in Fig. 1(c). The cyclic
evolution and SAM-OAM synergy along the $E_{1,+}$ band are displayed in Fig.
3(c), where the pressure fields and their phase distributions are measured as
spatial OAM features and the polarization ellipses are presented based on the
measurements of the velocity fields, as vector polarization features. Tracking
the A-evolution, it is observed to precisely agree with the theoretical
prediction, i.e., from the east of the equator passing the north pole to the
west of the equator, an $x$-oriented $p$-orbital (accompanying the horizontal
linear-polarization) transforms into a vortex with $l=+1$ (accompanying the
right-circular-polarization), then returns to a $y$-oriented $p$-orbital
(accompanying the vertical linear-polarization). For the B-evolution, it is
also from an $x$-oriented $p$-orbital to a $y$-oriented $p$-orbital, only
passing a vortex with $l=-1$ (at the south pole).

Similar to Fig. 3(c), Fig. 3(d) presents the cyclic evolution along the
$E_{2,+}$ band. We point out that due to the technical difficulties at the
band edges (with zero group velocity), our measurements only cover the $k$
regime close to $[0,\ 2\pi]$. Nonetheless, the measurements evidently reflect
the fundamental features of the cyclic evolutions with high agreements with
the theory (see Ref.[19] for simulations covering a complete $k$ regime).

A spin-orbital-Hall effect.—As a fundamental property enabled by SOIs, SAM can
be locked with the linear momentum[13]. In our case, remarkably, SAM is
synchronized with the OAM, promising a novel spin-orbital-Hall effect
highlighting more interesting and intricate locking of handedness,
directionality, spin density and spatial mode profile. For demonstration, an
acoustic sample is fabricated as shown in Fig. 4(a). To probe the locking
properties, we employ a chiral source generated by coupling four acoustic
loudspeakers with a phase gradient[19]. This kind of source carries nonzero
angular momentum that can couple with the SAM and correspondingly with the
synchronized OAM, resulting in an SAM-OAM-dependent sound propagation.

Such dependence can be quantified by a directional contrast
$\Gamma=\frac{\Gamma_{f}-\Gamma_{b}}{\Gamma_{f}+\Gamma_{b}}$, with
$\Gamma_{f}$ and $\Gamma_{b}$ describing the likelihood of the wave
propagation in the forward and backward directions, respectively. According to
Fermi’s golden rule[36, 37, 38],
$\Gamma_{f}\propto|\mathbf{s}^{*}\cdot\mathbf{v}_{f}|^{2}$ and
$\Gamma_{b}\propto|\mathbf{s}^{*}\cdot\mathbf{v}_{b}|^{2}$[19]. Here,
$\mathbf{s}=\frac{\sqrt{2}}{2}\begin{bmatrix}1&\pm i\end{bmatrix}^{T}$ denote
the chiral sources carrying SAM with opposite handedness. $\mathbf{v}_{f}$ and
$\mathbf{v}_{b}$ indicate the velocity fields for eigenstates with positive
and negative group velocity, respectively. Physically, this determines how
much of the energy in the source $\mathbf{s}$ is transferred into the
polarization states defined by $\mathbf{v}_{f}$ and $\mathbf{v}_{b}$.

Figures 4(b)-(c) present $\Gamma$ calculated based on the band structures in
Fig. 3(b), for Sites A and B, respectively. It is seen that a right-handed
chiral source with
$\mathbf{s}=\frac{\sqrt{2}}{2}\begin{bmatrix}1&i\end{bmatrix}^{T}$ (denoted by
red lines) transfers energy into Site A (B) as a forward (backward)
propagating state in the high-frequency regime, but as a backward (forward)
propagating state in the low-frequency regime. When the source chirality is
reversed, with
$\mathbf{s}=\frac{\sqrt{2}}{2}\begin{bmatrix}1&-i\end{bmatrix}^{T}$ (denoted
by blue lines), the sign of $\Gamma$ flips. This suggests that the right-
(left-) handedness couples to the polarization states with a preference in the
north (south) hemisphere, which is reasonable given the polarization features
analyzed in the cyclic evolutions.

Enabled by the cyclic evolutions, more interestingly, the polarization-
dependent wave propagation is tunable with respect to different frequencies.
Tracking the frequency axis in both Figs. 4(b)-(c), we see that the
$\Gamma$-curves display two maxima around 4.4 kHz and 5.6 kHz, precisely
corresponding to the poles accommodating circular-polarizations. At these
points, the source energy is maximally transferred into a single mode, leading
to unidirectional sound propagation. Away from the maxima, the source energy
is split, with a large portion transferred into the preferable direction and a
small portion into the opposite direction with reversed handedness. The
transition ratio is quantified by $\Gamma$. In fact, such tunable
polarization-splitting is related to the change of spin density during the
cyclic evolutions. The spin density is defined as the angular momentum carried
by the velocity fields, yielding
$\mathbf{S}=\frac{\rho}{2\omega}\mathrm{Im}(\mathbf{v}^{\ast}\times\mathbf{v})$[39,
40]. It characterizes the acoustic polarizations of local velocity field
rotations, with normalized $\mathbf{S}$ being zero representing linear-
polarizations, unity representing circular-polarizations (positive for the
right-circular-polarization and negative for the left-circular-polarization)
and in between representing elliptical-polarizations. Using the experimental
data measured in Fig. 3, we calculate $S_{z}$ ($S_{x}=S_{y}=0$ due to zero
$v_{z}$). As presented in Figs. 4(c)-(d), $S_{z}$-curves are consistent with
the $\Gamma$-curves (with oscillations from the finite size effect).

Figure 4:  (a) The experiment setup. (b) Numerically calculated directional
contrast $\Gamma$ (left panel), experimentally measured spin density $S_{z}$
(middle panel), and directional contrast $\Gamma_{\text{exp}}$ (right panel)
as functions of frequencies, for Site A. (c) The same as (b), only for Site B.

Remember that in our system, the SAM and OAM are synchronized and synergized,
suggesting a tunable OAM-splitting, accompanying the tunable polarization-
splitting. For experimental demonstration, we place two detectors at the right
and left ends of the sample in Fig. 4(a) to measure the output pressure
signals in the forward ($P_{f}$) and backward ($P_{b}$) directions,
respectively. An experimental directional contrast is obtained as
$\Gamma_{exp}=\frac{|P_{f}|-|P_{b}|}{\left|P_{f}\right|+|P_{b}|}$ and plotted
in Figs. 4(b)-(c). Again, $\Gamma_{exp}$-curves exhibit high agreements with
the calculated $\Gamma$ and measured $S_{z}$, indicating the spatial OAM modes
indeed exhibit a tunability in synergy with the polarization states (see more
details in Ref.[19]).

Conclusions.—We have demonstrated a full cycle of synergy between SAM and OAM
on a Möbius strip by exploiting anisotropic $p$-orbitals as eigenbases, whose
spatial mode profiles and inherent orientations serve as ingredients to
simultaneously manipulate OAM and SAM. The uncovering of such a unique and
fundamental connection deepens our understanding of these angular momenta,
which traditionally are considered independent and separate. It is especially
prominent for scalar waves like acoustic waves, which are revealed to
naturally carry synchronized SAM and OAM due to the intrinsic connection
between scalar pressure and vector velocity fields. The current model with
simple acoustic cavities already glimpses into the rich dynamical physics of
the SAM-OAM synergy and tunable SAM-OAM-locked wave propagations. Facilitated
with artificial designs, more comprehensive models taking into account various
physical principles like topology, gauging, and pumping would promise
versatile wave controls. While investigating $p$-orbitals in one dimension,
our principle can be applied to a more general selection of orbitals,
orientations, and spatial dimensions, with more degrees of freedom for rich
SAM-OAM interplay. In addition, synchronized SAM and OAM feature non-separable
states imprinting simultaneously spatial scalar and vector signatures with
high potential in information coding and high-capacity communications.
Targeting on the on-chip information technologies, our principle may offer an
integrated solution for coupled SAM-OAM control with reduced fabrication
complexity.

###### Acknowledgements.

This work is supported by the Key R&D Program of Jiangsu Province (Grant No.
BK20232015), the National Natural Science Foundation of China (Grant No.
12222407), and the National Key R&D Program of China (Grants No.
2023YFA1407700 and No. 2023YFA1406904). X.Z. thanks Z. Zhang for helpful
discussions.

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the trends in $R_{\rm Pb}^{\pi^{0}}$ at the LHC kinematics. However, here the
ratio $p_{T}/\sqrt{s}$ does not become large enough to reach the EMC region,
and thus the ratio remains larger than one for most of the $p_{T}$ range as a
result of anti-shadowing effects. Lastly, the EPPS16 predictions agree with
the nNNPDF2.0 result well within uncertainties, reflecting the underlying
consistency at the nPDF level.

Overall, the results of Fig. 5.4 confirm that inclusive hadron production in
proton-nucleus collisions can provide a handle on the nuclear PDF
modifications at medium and large-$x$, although an optimal interpretation of
the experimental data can only be achieved by the simultaneous determination
of the nPDFs together with the hadron fragmentation functions.

##  6 Summary and outlook

In this work we have presented a model-independent global determination of
nuclear parton distributions by incorporating the constraints from nuclear DIS
structure functions and gauge boson production in proton-lead collisions. We
have demonstrated that a satisfactory description of all the fitted data sets
can be achieved, highlighting the reliability of the QCD factorization
paradigm in the heavy nuclear sector. Our results demonstrate significant
nuclear effects among the quark flavors in nuclei, in particular a shadowing
of the up and down quark distributions in heavy nuclei such as lead. Nuclear
modifications are found also in the strangeness of heavier nuclei, displaying
a suppression with respect to the free proton across a large region of $x$. In
addition, we have shown that upon releasing the momentum and valence sum rule
constraints, the data prefer integral values that agree with QCD expectations
for all values of $A$.

We have also explored some phenomenological implications of the nNNPDF2.0
determination. We first compared nNNPDF2.0 theoretical predictions with ATLAS
measurements of isolated photon production at the LHC, an important hard probe
in proton-lead collisions. We then studied the impact on the small-$x$ nuclear
gluon PDF from the forward isolated photons production at the FoCal upgrade of
the ALICE detector. Lastly, we analyzed our theory predictions for neutral
pion production in proton-lead collisions at RHIC and LHC center-of-mass
energies. Apart from these applications, the nNNPDF2.0 PDF set can be used as
input to theoretical predictions for a range of other hard processes in pPb
and PbPb collisions, in particular for heavy ion collisions involving lighter
nuclei, in comparisons with non-perturbative nuclear models, and with QCD
calculations at small-$x$ involving dense nuclear matter. We also expect the
nNNPDF2.0 release to be used in future proton global PDF fits to estimate the
theory uncertainties associated with neutrino scattering data Ball:2018twp ,
and also in high-energy astroparticle physics processes that involve hard
scattering on nuclei Garcia:2020jwr ; Bertone:2018dse .

While the input data set used in this work allowed for a state-of-the-art
determination of the nuclear quarks and anti-quarks, it only provided loose
constraints on the nuclear gluon PDF, especially for heavier nuclei where
uncertainties are relatively large. To bypass this limitation, the next step
in the nNNPDF family of nuclear PDF fits will be to include additional
datasets that provide direct information on the nuclear gluon modifications.
In addition to the isolated photon production measurements discussed in Sect.
5.1, perhaps the most attractive candidate in this respect is dijet production
in pPb collisions. Measurements of dijet production from Run I in pp
collisions have been recently analyzed in the framework of NNLO QCD theory in
Ref. AbdulKhalek:2020jut , demonstrating a good compatibility with the global
dataset and a marked constraining power on the large-$x$ gluon. In the
corresponding pPb case, an EPPS16-based profiling analysis Eskola:2019dui  of
CMS dijet data at $\sqrt{s}=5.02$ TeV Sirunyan:2018qel  revealed a significant
pull of this measurement on the nuclear gluon modifications.

Another process that is known to provide important information on the nuclear
gluon PDF is charmed meson production, in particular from the LHCb
measurements in the forward region Aaij:2017gcy . This process offers unique
sensitivity to the small-$x$ (n)PDFs down to $x\simeq 10^{-6}$, as was
demonstrated by proton Zenaiev:2015rfa ; Bertone:2018dse ; Gauld:2016kpd  and
nuclear studies Kusina:2017gkz ; Eskola:2019bgf . Fully exploiting the
constraints provided by these measurements requires, as for the rest of hard
probes in nuclear collisions, a consistent theoretical and methodological
treatment of charm production in both proton and nuclear global QCD analyses.

On a longer timescale, one might aim to achieve a determination of the proton
and nuclear PDFs simultaneously from a universal analysis, thus bypassing the
need to include proton information by means of the proton boundary condition
penalty. In the same spirit of the QCD analyses of proton PDFs and
fragmentation functions presented in Refs. Sato:2019yez ; Ethier:2017zbq ,
such an integrated fit of proton and nuclear PDFs would ensure the ultimate
theoretical and methodological consistency of the determination of the nuclear
modifications of the free-nucleon quark and gluon structure.

The nNNPDF2.0 determination is available in the LHAPDF6 library
Buckley:2014ana  for all relevant nuclei from $A=1$ to $A=208$. The nNNPDF2.0
sets are available both for the nPDFs of bound protons, $f^{(p/A)}(x,Q^{2})$,
and those of bound nucleons, $f^{(N/A)}(x,Q^{2})$, following the conventions
in Sect. 3.1. Each of these sets is composed by $N_{\rm rep}=250$ correlated
replicas, see Sect. 5 of AbdulKhalek:2019mzd  for their usage prescriptions.
The naming convention used for the sets is the following:

$\qquad\qquad f^{(N/A)}(x,Q^{2})$ | $\qquad\qquad f^{(p/A)}(x,Q^{2})$
---|---
nNNPDF20_nlo_as_0118_N1 | nNNPDF20_nlo_as_0118_N1
nNNPDF20_nlo_as_0118_D2 | nNNPDF20_nlo_as_0118_p_A2_Z1
nNNPDF20_nlo_as_0118_He4 | nNNPDF20_nlo_as_0118_p_A4_Z2
nNNPDF20_nlo_as_0118_Li6 | nNNPDF20_nlo_as_0118_p_A6_Z3
nNNPDF20_nlo_as_0118_Be9 | nNNPDF20_nlo_as_0118_p_A9_Z4
nNNPDF20_nlo_as_0118_C12 | nNNPDF20_nlo_as_0118_p_A12_Z6
nNNPDF20_nlo_as_0118_N14 | nNNPDF20_nlo_as_0118_p_A14_Z7
nNNPDF20_nlo_as_0118_Al27 | nNNPDF20_nlo_as_0118_p_A27_Z13
nNNPDF20_nlo_as_0118_Ca40 | nNNPDF20_nlo_as_0118_p_A40_Z20
nNNPDF20_nlo_as_0118_Fe56 | nNNPDF20_nlo_as_0118_p_A56_Z26
nNNPDF20_nlo_as_0118_Cu64 | nNNPDF20_nlo_as_0118_p_A64_Z29
nNNPDF20_nlo_as_0118_Ag108 | nNNPDF20_nlo_as_0118_p_A108_Z47
nNNPDF20_nlo_as_0118_Sn119 | nNNPDF20_nlo_as_0118_p_A119_Z50
nNNPDF20_nlo_as_0118_Xe131 | nNNPDF20_nlo_as_0118_p_A131_Z54
nNNPDF20_nlo_as_0118_Au197 | nNNPDF20_nlo_as_0118_p_A197_Z79
nNNPDF20_nlo_as_0118_Pb208 | nNNPDF20_nlo_as_0118_p_A208_Z82

Additional variants of the nNNPDF2.0 NLO fit present in this work, such as the
fits without the momentum and valence sum rules and a $N_{\rm rep}=1000$
replica set for lead, are available on the NNPDF collaboration website:

http://nnpdf.mi.infn.it/for-users/nuclear-pdf-sets/

### Acknowledgments

We are grateful to our colleagues of the NNPDF collaboration for support and
stimulating discussions, and in particular to Emanuele R. Nocera for his
continuous assistance during this work. We thank Kari Eskola, Petja Paakkinen,
and Hannu Paukkunen for discussions about the EPPS16 analysis. We thank Marco
van Leeuwen for providing the predictions for FoCal and Ilkka Helenius for
those of pion production in heavy ion collisions. R. A. K., J. E., and J. R.
are supported by the Netherlands Organization for Scientific Research (NWO).
We acknowledge grants of computer capacity from the Finnish Grid and Cloud
Infrastructure (persistent identifier urn:nbn:fi:research-infras-2016072533).

##  Appendix A PDF sensitivity of input cross-sections

In this appendix we indicate the PDF sensitivity of all processes used as
input in the nNNPDF2.0 determination by providing their explicit cross-section
expressions at leading order (LO) in perturbative QCD. We provide these
expressions for a general nuclear target with $Z$ protons and $(A-Z)$ neutrons
in terms of bound proton distributions of the physical and evolution bases,
and in terms of the average bound nucleon PDFs. In what follows, we further
simplify the conventions adopted in Sect. 3.1 to

$\displaystyle f$ $\displaystyle\equiv f^{(p/A)}\,,$ (A.1) $\displaystyle
f_{A}$ $\displaystyle\equiv f^{(N/A)}\,.$ (A.2)

#### 1\. Lepton-nucleus scattering.

The double differential cross-section for the DIS of a charged lepton off a
nucleus with mass number $A$ is given by

${{d^{2}\sigma^{i,l^{\pm}}}\over{dxdQ^{2}}}(x,Q^{2},A)={{2\pi\alpha^{2}}\over{xQ^{4}}}\eta^{i}\left[Y_{+}F_{2}^{i}(x,Q^{2},A)\mp
Y_{-}xF_{3}^{i}(x,Q^{2},A)-y^{2}F_{L}^{i}(x,Q^{2},A)\right]\,,$ (A.3)

with $i=\rm NC,\,CC$, $\eta^{\rm NC}=1$, $\eta^{\rm
CC}=(1\pm\lambda)^{2}\eta_{W}$, where $\eta_{W}$ denotes the squared ratio of
the W-boson couplings and propagator with respect to those of the photon, and
$\lambda$ is the helicity of the incoming lepton. In the case of NC DIS we
restrict ourselves to photon-mediated processes, for which $F^{\rm NC}_{3}=0$.
The usual DIS kinematic variables are defined as $Y_{\pm}=1\pm(1-y)^{2}$ and

$x={{Q^{2}}\over{2P\cdot q}},\quad Q^{2}=-q^{2},\quad y={{q\cdot
P}\over{k\cdot P}}\,.$ (A.4)

#### A1. Neutral-current DIS.

For NC DIS, the underlying process is lepton-nucleus scattering mediated by a
virtual photon exchange, $l^{\pm}+A\xrightarrow[]{\gamma}l^{\pm}+X$. Since the
available data is at $Q^{2}<<M_{Z}$, the contributions from $Z$ boson exchange
can be neglected. The double-differential cross-section for the scattering of
a charged lepton off a nucleus with atomic and mass numbers $Z$ and $A$ is
proportional at leading order to the $F_{2}$ structure function which can be
expressed as

$\displaystyle\begin{split}F_{2}^{\rm
NC}&={{x}\over{9A}}\bigg{[}(4A-3Z)d^{+}+As^{+}+(A+3Z)u^{+}\bigg{]}={{x}\over{9}}\bigg{[}4u^{+}_{A}+d^{+}_{A}+s^{+}_{A}\bigg{]}\\\
&={{x}\over{18A}}\bigg{[}4A\Sigma+AT_{8}+3(2Z-A)T_{3}\bigg{]}\,,\end{split}$
(A.5)

Here we define the structure function per nucleon, rather than the structure
function of the nucleus as a whole. We can see from the above expression that
for an isoscalar nucleus $A=2Z$ the contribution proportional to $T_{3}$
vanishes and the nuclear structure functions depend only on $\Sigma$ and
$T_{8}$ via the $\Sigma+T_{8}$ combination. At LO only Eq. (A.5) is relevant
for the description of neutral-current DIS since the longitudinal structure
function vanishes due to the Callan-Gross relation, $F_{L}^{\rm NC}=F_{2}^{\rm
NC}-2xF_{1}^{\rm NC}=0$.

#### B1. Charged-current DIS.

For CC DIS, the underlying process is neutrino scattering off of a nucleus via
the exchange of a W boson, $\nu+A\xrightarrow[]{W^{-}}l^{-}+X$ and
$\bar{\nu}+A\xrightarrow[]{W^{+}}+l^{+}+X$. Due to the fact that W+ and W-
bosons couple to different quark flavors, the difference between neutrino and
anti-neutrino structure functions provides a handle on quark flavor
separation. Here we provide the expressions for $\nu(\bar{nu})A$ scattering,
the expressions for the conjugate process involving the CC scattering of
charged leptons is the same. The expressions for the inclusive CC structure
functions via $W^{-}$-boson exchange,
$\bar{\nu}+A\xrightarrow[]{W^{-}}+l^{+}X$, are the following:

$\displaystyle\begin{split}F_{2}^{\bar{\nu}A\xrightarrow[]{W^{-}}l^{+}X}={{2x}\over{A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}Z(u+\bar{d})+(A-Z)(d+\bar{u})\bigg{)}+|V_{us}|^{2}\bigg{(}Zu+(A-Z)d+A\bar{s}\bigg{)}\bigg{]}\\\
=2x\bigg{[}&|V_{ud}|^{2}\bigg{(}u_{A}+\bar{d}_{A}\bigg{)}+|V_{us}|^{2}\bigg{(}u_{A}+\bar{s_{A}}\bigg{)}\bigg{]}\\\
={{x}\over{6A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}2A(2\Sigma+T_{8})+6(2Z-A)V_{3}\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}3(2Z-A)T_{3}+3(2Z-A)V_{3}+A(4\Sigma-
T_{8}+3V_{8})\bigg{)}\bigg{]}\end{split}$ (A.6)

$\displaystyle\begin{split}F_{3}^{\bar{\nu}A\xrightarrow[]{W^{-}}l^{+}X}={{2}\over{A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}Z(u-\bar{d})+(A-Z)(d-\bar{u})\bigg{)}+|V_{us}|^{2}\bigg{(}Zu+(A-Z)d-A\bar{s}\bigg{)}\bigg{]}\\\
=2\bigg{[}&|V_{ud}|^{2}\bigg{(}u_{A}-\bar{d}_{A}\bigg{)}+|V_{us}|^{2}\bigg{(}u_{A}-\bar{s_{A}}\bigg{)}\bigg{]}\\\
={{-1}\over{6A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}6(A-2Z)T_{3}-2A(2V+V_{8})\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}+3(A-2Z)T_{3}+3(A-2Z)V_{3}+A(-3T_{8}-4V+V_{8})\bigg{)}\bigg{]}\end{split}$
(A.7)

Here we consider only the contribution from up-, down-, and strange-initiated
processes for simplicity. The generalization to heavy-quark initiated
processes is straightforward. An interesting observation from these
expressions is that $F_{2}$ and $F_{3}$ provide complementary information on
quark flavor separation.

The corresponding expressions for the inclusive charged-current structure
functions in the case of neutrino scattering via $W^{+}$-boson exchange,
$\nu+A\xrightarrow[]{W^{+}}+l^{-}X$, are given by

$\displaystyle\begin{split}F_{2}^{\nu
A\xrightarrow[]{W^{+}}l^{-}X}={{2x}\over{A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}Z(\bar{u}+d)+(A-Z)(u+\bar{d})\bigg{)}+|V_{us}|^{2}\bigg{(}Z\bar{u}+(A-Z)\bar{d}+As\bigg{)}\bigg{]}\\\
=2x\bigg{[}&|V_{ud}|^{2}\bigg{(}\bar{u}_{A}+d_{A}\bigg{)}+|V_{us}|^{2}\bigg{(}\bar{u}_{A}+s_{A}\bigg{)}\bigg{]}\\\
={{x}\over{6A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}2A(2\Sigma+T_{8})+6(A-2Z)V_{3}\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}A(4\Sigma-
T_{8}-3V_{8})+3(2Z-A)T_{3}+3(A-2Z)V_{3}\bigg{)}\bigg{]}\end{split}$ (A.8)
$\displaystyle\begin{split}F_{3}^{\nu
A\xrightarrow[]{W^{+}}l^{-}X}={{2}\over{A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}Z(d-\bar{u})+(A-Z)(u-\bar{d})\bigg{)}+|V_{us}|^{2}\bigg{(}-Z\bar{u}-(A-Z)\bar{d}+As\bigg{)}\bigg{]}\\\
=2\bigg{[}&|V_{ud}|^{2}\bigg{(}-\bar{u}_{A}+d_{A}\bigg{)}+|V_{us}|^{2}\bigg{(}-\bar{u}_{A}+s_{A}\bigg{)}\bigg{]}\\\
={{1}\over{6A}}\bigg{[}&|V_{ud}|^{2}\bigg{(}2A(2V+V_{8})+6(A-2Z)T_{3}\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}A(-3T_{8}+4V-V_{8})+3(A-2Z)T_{3}+3(2Z-A)V_{3}\bigg{)}\bigg{]}\end{split}$
(A.9)

We now turn to the exclusive CC charm production structure functions, required
for the description of the NuTeV cross-sections. Following the FONLL treatment
of heavy quark structure functions, the charm contribution to the inclusive
structure function is defined by all terms where the charm quark couples to
the W boson. Note that this definition is somewhat different from the
experimental definition, where charm structure functions are identified by
requesting the presence of charm in the final state, while the theory
definition includes charm-initiated contributions as well. The charm
production structure functions in the case of antineutrino- and neutrino-
initiated scattering reads for $W^{-}$ as

$\displaystyle\begin{split}F_{2}^{\bar{\nu}A\xrightarrow[]{W^{-}}l^{+}cX}={{2x}\over{A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}Z\bar{d}+(A-Z)\bar{u}\bigg{)}+|V_{cs}|^{2}A\bar{s}\bigg{]}\\\
=2x\bigg{[}&|V_{cd}|^{2}\bar{d}_{A}+|V_{cs}|^{2}\bar{s}_{A}\bigg{]}\\\
={{x}\over{6A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}A(2\Sigma+T_{8}-2V-V_{8})+3(A-2Z)T_{3}+3(2Z-A)V_{3}\bigg{)}\\\
+&|V_{cs}|^{2}2A\bigg{(}\Sigma-T_{8}-V+V_{8}\bigg{)}\bigg{]}\end{split}$
(A.10)
$\displaystyle\begin{split}F_{3}^{\bar{\nu}A\xrightarrow[]{W^{-}}l^{+}cX}={{2}\over{A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}-Z\bar{d}-(A-Z)\bar{u}\bigg{)}-|V_{cs}|^{2}A\bar{s}\bigg{]}\\\
=-2\bigg{[}&|V_{cd}|^{2}\bar{d}_{A}+|V_{cs}|^{2}\bar{s}_{A}\bigg{]}\\\
={{-1}\over{6A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}3(A-2Z)T_{3}+3(2Z-A)V_{3}+A(2\Sigma+AT_{8}-2AV-
AV_{8})\bigg{)}\\\ +&|V_{cs}|^{2}2A\bigg{(}+\Sigma-
T_{8}-V+V_{8}\bigg{)}\bigg{]}\end{split}$ (A.11)

and for $W^{+}$:

$\displaystyle\begin{split}F_{2}^{\nu
A\xrightarrow[]{W^{+}}l^{-}cX}={{2x}\over{A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}Zd+(A-Z)u\bigg{)}+|V_{cs}|^{2}As\bigg{]}\\\
=2x\bigg{[}&|V_{cd}|^{2}d_{A}+|V_{cs}|^{2}s_{A}\bigg{]}\\\
={{x}\over{6A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}A(2\Sigma+T_{8}+2V+V_{8})+3(A-2Z)T_{3}+3(A-2Z)V_{3}\bigg{)}\\\
+&|V_{cs}|^{2}2A\bigg{(}+\Sigma-T_{8}+V-V_{8}\bigg{)}\bigg{]}\end{split}$
(A.12) $\displaystyle\begin{split}F_{3}^{\nu
A\xrightarrow[]{W^{+}}l^{-}cX}={{-2}\over{A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}+Zd+(A-Z)u\bigg{)}+|V_{cs}|^{2}As\bigg{]}\\\
=-2\bigg{[}&|V_{cd}|^{2}d_{A}+|V_{cs}|^{2}s_{A}\bigg{]}\\\
={{-1}\over{6A}}\bigg{[}&|V_{cd}|^{2}\bigg{(}A(2\Sigma+T_{8}+2V+V_{8})+3(A-2Z)T_{3}+3(A-2Z)V_{3}\bigg{)}\\\
+&|V_{cs}|^{2}2A\bigg{(}+\Sigma-T_{8}+V-V_{8}\bigg{)}\bigg{]}\end{split}$
(A.13)

#### 2\. Weak boson production.

Here we provide the corresponding leading order expressions for weak gauge
boson production in proton-nucleus collisions. Experimental measurements for
NC observables are provided in terms of the invariant mass $M$ and rapidity of
the dilepton final state, while CC measurements are binned in terms of the
charged lepton rapidity. Here we show the LO expressions without accounting
for the gauge boson decay, so in the case of W± production the connection with
the experimental data is somewhat less direct than for $Z$ boson production.

#### A2. Neutral current DY

We start by discussing NC Drell-Yan (DY), namely $Z$-boson production in
proton-nucleus collisions, $p+A\xrightarrow[]{Z}l^{+}l^{-}$. The leading order
expressions are given by

$\displaystyle\begin{split}{{d\sigma_{Z}^{pA}}\over{dM^{2}dy}}&\propto
A\bigg{[}a_{u}\bigg{(}u_{1}\bar{u}_{2}+\bar{u}_{1}u_{2}\bigg{)}+a_{d}\bigg{(}d_{1}\bar{d}_{2}+\bar{d}_{1}d_{2}+s_{1}\bar{s}_{2}+\bar{s}_{1}s_{2}\bigg{)}\bigg{]}\end{split}$
(A.14)

where $q_{1}=q(x_{1},M^{2})$ and $q_{2}=q(x_{2},M^{2})$ and the Drell-Yan
kinematics at LO are:

$x_{1,2}={{M}\over{\sqrt{s}}}e^{\pm y}\,,\quad\quad M^{2}=x_{1}x_{2}s\,,$
(A.15)

with $\sqrt{s}$ being the center-of-mass energy of the collision, $M$ being
the invariant mass of the final state, and $y$ the rapidity of the final state
system. The effective weak couplings are given by

$a_{q}=(\bar{g}^{q}_{V}+\bar{g}^{q}_{A}),\quad\quad\bar{g}^{f}_{V}=(t^{(f)}_{3}-2Q_{f}\sin^{2}\theta_{W}),\quad\quad\bar{g}^{f}_{A}=t^{(f)}_{3}$

where $t^{(f)}_{3}$ is the weak isospin of fermion $f$, $Q_{f}$ is the
electric charge and $\theta_{W}$ is Weinberg’s angle.

#### B2. Charged current DY

The corresponding LO expressions in the case of W- and W+ boson production in
pA collisions are given by

$\displaystyle\begin{split}{{d\sigma_{W^{-}}^{pA,}}\over{dM^{2}dy}}\propto\,\,&|V_{ud}|^{2}\bigg{(}Z(d_{1}\bar{u}_{2}+\bar{u}_{1}d_{2})+(A-Z)(\bar{d}_{1}u_{2}+u_{1}\bar{d}_{2})\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}Z(d_{1}\bar{s}_{2}+\bar{s}_{1}d_{2})+(A-Z)(u_{1}\bar{s}_{2}+\bar{s}_{1}u_{2})\bigg{)}\end{split}$
(A.16)

and:

$\displaystyle\begin{split}{{d\sigma_{W^{+}}^{pA,}}\over{dM^{2}dy}}\propto\,\,&|V_{ud}|^{2}\bigg{(}Z(u_{1}\bar{d}_{2}+\bar{d}_{1}u_{2})+(A-Z)(d_{1}\bar{u}_{2}+\bar{u}_{1}d_{2})\bigg{)}\\\
+&|V_{us}|^{2}\bigg{(}Z(u_{1}\bar{s}_{2}+\bar{s}_{1}u_{2})+(A-Z)(d_{1}\bar{s}_{2}+\bar{s}_{1}d_{2})\bigg{)}\end{split}$
(A.17)

where in this case $y$ stands for the $W$ boson rapidity, which in general is
different from the pseudo-rapidity of the charged lepton that is measured by
experiments. However, in either case the PDF combinations that enter at LO
remain the same. From the comparison between the LO expressions of the DIS
structure functions and the $W$ and $Z$ production cross-sections in pA
collisions, it is clear that each group of process is sensitive to a different
combination of quark and antiquark PDFs. Therefore, their combination (due to
the PDF universality in QCD) into a global QCD analysis provides a unique
handle for a robust separation between quark and antiquark flavors in nuclei.

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# Deep Metric Learning-Based Semi-Supervised Regression
With Alternate Learning

###### Abstract

This paper introduces a novel deep metric learning-based semi-supervised
regression (DML-S2R) method for parameter estimation problems. The proposed
DML-S2R method aims to mitigate the problems of insufficient amount of labeled
samples without collecting any additional sample with a target value. To this
end, it is made up of two main steps: i) pairwise similarity modeling with
scarce labeled data; and ii) triplet-based metric learning with abundant
unlabeled data. The first step aims to model pairwise sample similarities by
using a small number of labeled samples. This is achieved by estimating the
target value differences of labeled samples with a Siamese neural network
(SNN). The second step aims to learn a triplet-based metric space (in which
similar samples are close to each other and dissimilar samples are far apart
from each other) when the number of labeled samples is insufficient. This is
achieved by employing the SNN of the first step for triplet-based deep metric
learning that exploits not only labeled samples but also unlabeled samples.
For the end-to-end training of DML-S2R, we investigate an alternate learning
strategy for the two steps. Due to this strategy, the encoded information in
each step becomes a guidance for learning phase of the other step. The
experimental results confirm the success of DML-S2R compared to the state-of-
the-art semi-supervised regression methods. The code of the proposed method is
publicly available at https://git.tu-berlin.de/rsim/DML-S2R.

Index Terms—  Semi-supervised regression, parameter estimation, metric
learning, deep learning.

##  1 Introduction

The accurate estimation of parameters from specific data (e.g., estimation of
carbon monoxide concentration in the environment, estimation of forest
parameters, estimation of glucose concentration in diabetic patients, etc.) is
an important research field in machine learning [1]. The use of regression
methods, which aim at learning functional relations between a set of variables
and corresponding target values, is an effective way for parameter estimation
problems. Accordingly, several regression methods (e.g., random forest
regression [2], support vector regression [1], deep learning-based regression
[3, 4], etc.) have been introduced in the literature. The success of these
methods depends on the quantity and quality of training samples for which the
respective target values are available. The quantity of training samples
depends on the number of available labeled samples, while the quality of
training samples depends on their capability to represent the real sample
distribution. A small amount of labeled samples with insufficient quality can
lead to inadequate modeling of the regression task. However, collecting
labeled samples is often expensive and complex as producing reference measures
may require significant time [1]. To address this issue, semi-supervised
regression (SSR) methods, which aim at jointly using the information of both
labeled and unlabeled samples in the learning phase of the regression
algorithm, have been introduced in [5, 6, 7].

In recent years, deep metric learning (DML) has been utilized in the framework
of SSR. DML operates on sample tuples (e.g., triplets) to learn a metric
space, in which similar samples are mapped close to each other and dissimilar
samples are mapped apart from each other. In [8], a metric-learning based SSR
method that utilizes a Siamese neural network (SNN) with a contrastive loss
function is introduced to learn a pairwise metric space. In this method,
dissimilar and similar pairs, which are constructed from both labeled and
unlabeled samples, are utilized for DML. To define sample similarity, the
absolute differences of target values are exploited with a thresholding
strategy for labeled samples. The similarity of unlabeled samples to labeled
samples is estimated based on Euclidean distances between data points. Once
the SNN is trained with dissimilar and similar sample pairs, the target value
estimation of samples is performed on the learned metric space by the
k-nearest neighbors (k-NN) algorithm. This method only considers the pairwise
similarities, which may not be sufficient in accurately learning a metric
space. In [9], an SSR method that exploits a long short-term memory (LSTM)
based SNN is proposed for continuous emotion recognition. This method applies
a two-stage training. In the first stage, the SNN is trained with the log-
ratio loss function (which operates on sample triplets) by using only labeled
samples. Then, the SNN is used to generate pseudo-labels for unlabeled
samples. In the second stage, the SNN is trained with the mean squared error
loss function by using pseudo-labels. This method utilizes only labeled
samples for learning metric space in the first stage. This may lead to the
inaccurate characterization of sample similarities in the case of the
availability of small-sized labeled training sets. To address the limitations
of above-mentioned methods, we introduce a novel Deep Metric Learning-based
Semi-Supervised Regression (DML-S2R) method.

##  2 Proposed DML-S2R Method

Let
$\mathcal{S}=\\{(\boldsymbol{x}_{i}^{l},\boldsymbol{y}_{i}^{l})\\}^{N}_{i=1}$
be a set of $N$ labeled samples, where $\boldsymbol{x}_{i}^{l}$ is the $i$th
labeled sample and $\boldsymbol{y}_{i}^{l}$ is its corresponding target value.
Let $\mathcal{U}=\\{\boldsymbol{x}_{j}^{u}\\}^{M}_{j=1}$ be the set of $M$
unlabeled samples, where $\boldsymbol{x}_{j}^{u}$ is the $j$th sample, for
which the corresponding target value is unknown. The training set
$\mathcal{T}$ consists of the labeled sample set $\mathcal{S}$ and the
unlabeled sample set $\mathcal{U}$ ($\mathcal{T}=\mathcal{S}\cup\mathcal{U}$).
We assume that $N$ is significantly lower than $M$ ($N\\!<\\!<\\!M$).

The proposed DML-S2R method aims to learn a metric space, in which similar
samples are located close to each other, by effectively exploiting abundant
unlabeled data together with scarce labeled data. This is achieved by two main
steps: i) pairwise similarity modeling with scarce labeled data; and ii)
triplet-based metric learning with abundant unlabeled data. For the end-to-end
training of DML-S2R, we investigate an alternate learning strategy for the two
steps. Fig. 1 shows a general overview of the proposed method, which is
explained in detail in the following subsections.

###  2.1 Pairwise Similarity Modeling with Scarce Labeled Data

The first step aims to model the pairwise sample similarity by using a small
amount of labeled samples. For this purpose, one could exploit a contrastive
loss function that requires the selection of similar and dissimilar pairs
based on the target values of training samples. However, in the framework of
regression problems, it is challenging to define the boundaries between the
target values of similar/dissimilar samples [10]. To overcome this problem, we
redefine the regression problem. Instead of learning a function that estimates
the target value of each sample, DML-S2R learns a function $f^{d}$ which
estimates the target value differences of two labeled samples as:

$f^{d}(\boldsymbol{x}_{i}^{l},\boldsymbol{x}_{j}^{l})=\boldsymbol{y}_{i}^{l}-\boldsymbol{y}_{j}^{l}.$
(1)

While learning to estimate target value differences, DML-S2R is also enforced
to model the sample similarity for the accurate prediction of the differences.
Due to this definition, instead of using $N$ samples in the first step,
DML-S2R exploits $N(N\\!-\\!1)$ pairs of samples (i.e., all the possible pairs
from the labeled sample set). This mitigates the limitations of labeled data
scarcity, and thus over-fitting. For $f^{d}$, we utilize an SNN, which
contains two identical sub-networks with shared weights. Two sub-networks of
the SNN take an input pair and provide the sample features, which are
concatenated to form the feature associated to the input pair. Then, this
feature is fed into a fully connected layer (FC) that directly estimates the
target value differences of sample pairs. Let $\boldsymbol{z}_{ij}$ be the
target value difference of $\boldsymbol{x}_{i}^{l}$ and
$\boldsymbol{x}_{j}^{l}$ while $i\neq j$. To learn the model parameters of
$f^{d}$, we propose the pairwise similarity modeling loss function
$\mathcal{L}_{\text{PSM}}$ as follows:

$\mathcal{L}_{\text{PSM}}=\frac{1}{N(N\\!-\\!1)}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathds{1}_{[i\neq
j]}(\boldsymbol{z}_{ij}-f^{d}(\boldsymbol{x}_{i}^{l},\boldsymbol{x}_{j}^{l}))^{2},$
(2)

where $\mathds{1}$ is the indicator function. Once the model parameters of the
SNN is learnt with (2), its feature space, which encodes the pairwise sample
similarity, forms the basis for the second step.

$\mathcal{L}_{\text{PSM}}$$\boldsymbol{x}_{i}^{l}$ SNN
$f^{d}(\boldsymbol{x}_{i}^{l},\boldsymbol{x}_{j}^{l})$$\boldsymbol{x}_{j}^{l}$$\mathbb{P}_{\boldsymbol{x}_{i}^{l}}$
SNN $\boldsymbol{x}_{i}^{l}$ SNN SNN SNN
$\mathbb{N}_{\boldsymbol{x}_{i}^{l}}$$\mathcal{L}_{\mathbb{P}}$$\mathcal{L}_{\mathbb{N}}$$\mathcal{L}_{\text{RRL}}$
First Step Second Step Fig. 1: Illustration of the proposed DML-S2R method.

###  2.2 Triplet-Based Metric Learning with Abundant Unlabeled Data

The second step aims to learn a metric space (where similar samples are
located close to each other) when the number of labeled samples is limited.
This is achieved by triplet-based DML that takes into account not only labeled
samples but also unlabeled samples. Triplet-based DML operates on triplets for
the characterization of a metric space. Accordingly, we convert the SNN of the
first step to the triplet-based SNN by replicating one of its identical sub-
networks three times without changing weights, and thus make it appropriate
for triplet-based DML. A standard triplet consists of a sample anchor and a
positive sample (which is similar to the anchor) and a negative sample (which
is dissimilar to the anchor). For the construction of triplets, anchor samples
are selected from the labeled set $\mathcal{S}$ and positive and negative
samples are selected from the unlabeled set $\mathcal{U}$. To effectively
exploit abundant unlabeled data, we create a set of positive and negative
samples per anchor. This leads to a faster convergence compared to having only
one positive and one negative sample per anchor [11]. For an anchor
$\boldsymbol{x}_{a}^{l}\in\mathcal{S}$, the set of $k$ positive samples
$\mathbb{P}_{\boldsymbol{x}_{a}^{l}}$ and the set of $k$ negative samples
$\mathbb{N}_{\boldsymbol{x}_{a}^{l}}$ are selected with $f^{d}$ from the first
step based on their estimated target value differences with the anchor. In
detail, all the possible pairs between the anchor $\boldsymbol{x}_{a}^{l}$ and
each unlabeled sample $\boldsymbol{x}_{i}^{u}\in\mathcal{U}$ are created.
Then, the target value difference of each pair is estimated by using $f^{d}$
from the first step. The $k$ unlabeled samples having the smallest difference
with the anchor are selected for $\mathbb{P}_{\boldsymbol{x}_{a}^{l}}$, while
the $k$ unlabeled samples having the highest difference with the anchor are
selected for $\mathbb{N}_{\boldsymbol{x}_{a}^{l}}$. The positive-negative set
selection procedure is shown in Algorithm 1. After the selection of the
positive and negative sets for all anchor samples, we employ the ranked list
loss function [11] as follows:

$\displaystyle w(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{j}^{u})$
$\displaystyle=\text{exp}(\tau(d(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{j}^{u})-(\alpha-m))),$
(3) $\displaystyle\mathcal{L}_{\mathbb{S}}(\boldsymbol{x}_{a}^{l},\mathbb{S})$
$\displaystyle=\sum_{\boldsymbol{x}_{j}^{u}\in\mathbb{S}}\frac{w(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{j}^{u})}{\sum_{\boldsymbol{x}_{j}^{u}\in\mathbb{S}}w(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{j}^{u})}\mathcal{L}_{m}(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{j}^{u}),$
$\displaystyle\mathcal{L}_{\text{RLL}}$
$\displaystyle=\frac{1}{2N}\sum_{i=1}^{N}\mathcal{L}_{\mathbb{P}}(\boldsymbol{x}_{a}^{l},\mathbb{P}_{\boldsymbol{x}_{a}^{l}})+\mathcal{L}_{\mathbb{N}}(\boldsymbol{x}_{a}^{l},\mathbb{N}_{\boldsymbol{x}_{a}^{l}}),$

where $\tau$ is the temperature parameter, $\alpha$ is the negative sample
boundary, $m$ is the margin parameter, $d$ measures the Euclidean distance
between two samples in the feature space and $\mathcal{L}_{m}$ is the margin
loss function. $\mathcal{L}_{\text{RLL}}$ pulls a set of positive samples
closer than the set of negatives by the margin $m$ on the feature space of the
SNN.

Input : $\boldsymbol{x}_{a}^{l},\mathcal{U},f^{d},k$

Output :
$\mathbb{P}_{\boldsymbol{x}_{a}^{l}},\mathbb{N}_{\boldsymbol{x}_{a}^{l}}$

1

2$\mathbb{P}_{\boldsymbol{x}_{a}^{l}}=\emptyset,\mathbb{N}_{\boldsymbol{x}_{a}^{l}}=\emptyset$

3while  _$|\mathbb{P}_{\boldsymbol{x}_{a}^{l}}|\leq k$_ do

4
$\operatorname*{\arg\\!\min}\limits_{\boldsymbol{x}_{i}^{u}\in\mathcal{U}\setminus\mathbb{P}_{\boldsymbol{x}_{a}^{l}}}f^{d}(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{i}^{u})\to\mathbb{P}_{\boldsymbol{x}_{a}^{l}}$
(Positive set selection)

5 end while

6

7while  _$|\mathbb{N}_{\boldsymbol{x}_{a}^{l}}|\leq k$_ do

8
$\operatorname*{\arg\\!\max}\limits_{\boldsymbol{x}_{i}^{u}\in\mathcal{U}\setminus\mathbb{N}_{\boldsymbol{x}_{a}^{l}}}f^{d}(\boldsymbol{x}_{a}^{l},\boldsymbol{x}_{i}^{u})\to\mathbb{N}_{\boldsymbol{x}_{a}^{l}}$
(Negative set selection)

9 end while

return _$\mathbb{P}_{\boldsymbol{x}_{a}^{l}}$ ,
$\mathbb{N}_{\boldsymbol{x}_{a}^{l}}$_

Algorithm 1 Positive-negative set selection for an anchor within the proposed
DML-S2R method

It is worth noting that the effectiveness of each step in DML-S2R depends on
each other. Inaccurate learning of $f^{d}$ in the first step leads to
incorrect selection of positive-negative sets in the second step. If the
metric space is not accurately learned in the second step, the weights of the
SNN can not be effectively learned in the first step due to a small number of
labeled samples. This prevents to utilize standard joint learning strategies
for DML-S2R. To accurately learn both steps, we investigate an alternate
learning strategy, in which the SNN is trained for both steps within the
consecutive training epochs. In detail, the whole learning procedure starts
with training the SNN for one epoch by the first step while minimizing
$\mathcal{L}_{\text{PSM}}$. Then, the SNN is trained for one epoch by the
second step with all the anchor samples and the associated positive-negative
sets while minimizing $\mathcal{L}_{\text{RRL}}$. Training continues while
alternating between the two steps until convergence of both loss functions.
Once the training of DML-S2R is completed, the target value estimation of a
new sample is achieved based on $f^{d}$ as follows:

$\boldsymbol{y}^{*}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{f^{d}(\boldsymbol{x}^{*},\boldsymbol{x}_{i}^{l})-f^{d}(\boldsymbol{x}_{i}^{l},\boldsymbol{x}^{*})}{2}+\boldsymbol{y}_{i}^{l}\right).$
(4)

##  3 Experimental Results

Experiments were conducted on Boston Housing [12], Superconductivity [13] and
Air Quality [14] datasets associated to the regression problems of housing
value estimation, critical temperature estimation of a superconductor and
benzene estimation for pollution monitoring, respectively. The Boston Housing
dataset includes 506 samples, each of which is associated with 13 variables.
The Superconductivity dataset includes 21263 samples, while each sample is
associated with 81 variables. The Air Quality dataset consists of 9358
samples, each of which is associated with 14 variables. For the construction
of the unlabeled set, we randomly selected 1000 samples for the
Superconductivity and Air Quality datasets, while 200 samples were randomly
chosen for the Boston Housing dataset. In the experiments, the number of
labeled samples is varied as $|S|=10,20,50$ for each dataset. The rest of the
samples were used as the test set for each dataset. All the samples from three
datasets were normalized by using the min-max normalization approach. In the
experiments, two hidden layers, each of which includes 100 neurons, were used
for the SNN of DML-S2R. The parameter $k$ was set to 5. We trained our method
for 30 epochs by using the Adam optimizer with the initial learning rate of
0.001. The results are provided in terms of mean absolute error (MAE). To
perform the ablation study of the proposed DML-S2R method, we compared it with
only using the first step of DML-S2R. To assess the effectiveness of the
proposed DML-S2R method, we compared it with the cotraining-style SSR
algorithm (COREG) [15] and metric-based SSR (denoted as MSSR) method [8].

Table 1: Mean absolute error (MAE) scores when the different steps of the proposed DML-S2R method were utilized (Superconductivity dataset). Steps of DML-S2R | Number of Labeled Samples
---|---
1st |  2nd |  10 |  20 |  50 |  100
✓ |  ✗ |  23.5 |  22.8 |  18.2 |  17.2
✓ |  ✓ |  22.3 |  18.2 |  16.2 |  15.8
Table 2: Mean absolute error (MAE) scores obtained by COREG, MSSR and the proposed DML-S2R method (Superconductivity dataset). Method | Number of Labeled Samples
---|---
|  10 |  20 |  50
COREG [15] |  23.1 |  20.5 |  17.2
MSSR [8] |  22.4 |  21.2 |  18.1
DML-S2R (Ours) |  22.3 |  18.2 |  16.2
Table 3: Mean absolute error (MAE) scores obtained by COREG, MSSR and the proposed DML-S2R method (Boston Housing dataset). Method | Number of Labeled Samples
---|---
|  10 |  20 |  50
COREG [15] |  8.2 |  7.9 |  5.1
MSSR [8] |  6.5 |  5.6 |  4.9
DML-S2R (Ours) |  6.1 |  5.4 |  4.5
Table 4: Mean absolute error (MAE) scores obtained by COREG, MSSR and the proposed DML-S2R method (Air Quality dataset). Method | Number of Labeled Samples
---|---
|  10 |  20 |  50
COREG [15] |  12.4 |  11.9 |  9.5
MSSR [8] |  17.9 |  17.3 |  12.7
DML-S2R (Ours) |  10.9 |  6.0 |  3.3

Table 1 shows the regression performances obtained on the Superconductivity
dataset when: i) only the first step of the proposed DML-S2R method is
applied; and ii) both steps of DML-S2R are applied. By analyzing the table,
one can see that utilizing both steps of DML-S2R results in significantly
higher accuracies than using only the first step of DML-S2R under different
numbers of labeled samples. As an example, DML-S2R achieves an MAE of 18.2
with 20 labeled samples, whereas using only the first step of DML-S2R results
in an MAE of 22.8 with the same number of labeled samples. This is due to the
fact that in the second step of DML-S2R, unlabeled samples are accurately
exploited to learn a deep metric space that leads to a more accurate target
value estimation. This also shows the success of the alternate learning
strategy for learning both steps. We observed similar behaviours for the other
considered datasets (not reported for space constraints). Tables 2-4 show the
regression performances of COREG, MSSR and proposed DML-S2R with the different
numbers of labeled samples for all the datasets. One can see from the tables
that our method provides the highest accuracies for all datasets. As an
example, DML-S2R achieves an MAE of 3.3 with 50 labeled samples, whereas COREG
provides an MAE of 9.5 and MSSR achieves an MAE of 12.7 with the same number
of labeled samples for the Superconductivity dataset. This is due to the fact
that our method effectively models the pairwise similarity of samples by using
scarce labeled data, while accurately learning a deep metric space by
exploiting labeled and unlabeled samples together.

##  4 Conclusion

In this paper, we have presented a novel deep metric learning-based semi-
supervised regression (DML-S2R) method for parameter estimation problems.
DML-S2R includes two consecutive steps. The first step aims at modeling the
pairwise similarities of samples when the labeled data is scarce. This is
achieved by learning to estimate the target value differences of the labeled
sample pairs with an SNN. Due to this step, the proposed DML-S2R method
overcomes the challenges of defining pairwise similar/dissimilar samples based
on the target values of labeled samples in the framework of regression
problems. The second step aims at learning a metric space that utilizes not
only labeled samples but also unlabeled samples to further enrich modeling
sample similarities with abundant unlabeled data. This is achieved by
employing the SNN of the first step for triplet-based DML where positive-
negative samples in each triplet are selected from unlabeled samples based on
the SNN. For the whole learning procedure of DML-S2R, we investigate an
alternate learning strategy, in which the SNN is trained for both steps
through consecutive training epochs. Due to this strategy, the encoded
information by the SNN in each step becomes a guidance for learning phase of
the other step. Experimental results show the effectiveness of proposed
DML-S2R compared to state-of-the-art SSR methods. It is worth noting that
DML-S2R is independent from the type of the SNN architecture, and thus
suitable to be integrated into any Siamese-based deep architecture. As a
future work, we plan to apply DML-S2R to the parameter estimation problems in
image domain (e.g., age prediction, emotion recognition, etc.).

##  5 Acknowledgments

This work is funded by the European Research Council (ERC) through the
ERC-2017-STG BigEarth Project under Grant 759764.

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