# Spin-orbit entanglement in the Color Glass Condensate

Shohini Bhattacharya<EMAIL_ADDRESS>Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  Renaud Boussarie
<EMAIL_ADDRESS>CPHT, CNRS, Ecole Polytechnique, Institut
Polytechnique de Paris, 91128 Palaiseau, France  Yoshitaka Hatta
<EMAIL_ADDRESS>Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton,
NY 11973, USA

###### Abstract

We compute the spin-orbit correlations of quarks and gluons at small-$x$ and
show that the helicity and the orbital angular momentum of individual partons
are strongly anti-aligned even in unpolarized or spinless hadrons and nuclei.
Combined with the fact that gluons in the Color Glass Condensate are linearly
polarized, our finding indicates that the helicity and the orbital angular
momentum of single gluons are maximally entangled in a quantum mechanical
sense.

††preprint: LA-UR-24-22492

##  I Introduction

One of the core missions of the future Electron-Ion Collider (EIC) experiment
Abdul Khalek  _et al._ (2022) at Brookhaven National Laboratory is to
unambiguously discover and then explore the phenomenon of the gluon saturation
Gribov  _et al._ (1983); Gelis  _et al._ (2010); Morreale and Salazar (2021)
that universally occurs in all hadrons and nuclei alike when they are boosted
to sufficiently high energies. This was clearly expressed in the influential
National Academy of Science report National Academies of Sciences,
Engineering, and Medicine 2018  which concluded that the EIC can uniquely
answer the profound question “What are the emergent properties of dense
systems of gluons?” where ‘dense systems’ refer to gluon saturated matter
commonly dubbed the Color Glass Condensate (CGC). Already a number of
remarkable properties of the CGC have been exposed through decades of
extensive theoretical studies Gribov  _et al._ (1983); Gelis  _et al._ (2010);
Morreale and Salazar (2021). In this paper we uncover a novel emergent
property of the CGC that has been unnoticed so far: Perfect spin-orbit anti-
correlation for quarks and gluons at small-$x$.

Spin-orbit correlation in the present context concerns the relative
orientation of the helicity and orbital angular momentum (OAM) of individual
quarks and gluons inside a hadron or a nucleus, similar to the spin-orbit
coupling of an electron in a hydrogen atom. Such a notion was introduced in
the nucleon structure physics in Lorce and Pasquini (2011) and can be
quantified by certain Generalized Transverse Momentum Dependent (GTMD)
distributions. However, despite some progress in the literature Lorcé (2014);
Rajan  _et al._ (2018); Tan and Lu (2022); Engelhardt  _et al._ (2022); Kim
_et al._ (2024), at the moment very little is known about the properties of
the correlation, especially for gluons, let alone their experimental
verification Bhattacharya  _et al._ (2017, 2022a); Boussarie  _et al._ (2018);
Bhattacharya  _et al._ (2023). In this paper, we demonstrate that spin-orbit
correlations for both quarks and gluons are explicitly calculable at small-$x$
in the CGC framework. (See a related work Boer  _et al._ (2018).) We find that
the correlation not only survives in the eikonal approximation, but also grows
strongly with decreasing $x$ in the same manner as the BFKL pomeron Kuraev
_et al._ (1977); Balitsky and Lipatov (1978) and eventually saturates. This
may sound surprising at first given that distributions associated with
polarization are often labeled as ‘sub-eikonal’ and suppressed by a factor of
$x\ll 1$ compared to unpolarized ones (e.g. Cougoulic  _et al._ (2022) and
references therein). Moreover, the sign of the spin-orbit correlation is
consistently negative, meaning that the helicity and OAM of individual
small-$x$ partons point to opposite directions. This is similar to the known
sign difference between the helicity and OAM parton distributions at small-$x$
inside a polarized hadron Hatta  _et al._ (2017); More  _et al._ (2018); Hatta
and Yang (2018); Boussarie  _et al._ (2019); Kovchegov and Manley (2024);
Manley (2024). However, the anti-correlation exists even in unpolarized or
spinless hadrons, and is therefore an intrinsic property of the CGC.

Furthermore, we shall argue that ‘correlation’ actually implies ‘entanglement’
between the spin and the OAM of individual gluons in a quantum mechanical
sense. This provides a novel perspective on the CGC as a correlated quantum
state beyond the usual characterization as a classical Yang-Mills system with
random color charges. We contrast our finding with the analogous entangled
photon states that have been recently observed in the field of quantum optics
Stav  _et al._ (2018).

##  II Gluon spin-orbit correlation

For a high energy hadron or nucleus single particle state $|p\rangle$, we
define the quark and gluon spin-orbit correlation $C_{q,g}$ as

$\displaystyle\int\frac{d^{3}z}{2(2\pi)^{3}}e^{ixP^{+}z^{-}-ik_{\perp}\cdot
z_{\perp}}\langle
p^{\prime}|\bar{\psi}(-z/2)\gamma^{+}\gamma_{5}W_{\pm}\psi(z/2)|p\rangle$
$\displaystyle=$
$\displaystyle-i\frac{\epsilon_{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}C_{q}(x,\xi,k_{\perp},\Delta_{\perp}),$
(1)
$\displaystyle\frac{i}{x}\int\frac{d^{3}z}{(2\pi)^{3}P^{+}}e^{ixP^{+}z^{-}-ik_{\perp}\cdot
z_{\perp}}\langle p^{\prime}|2{\rm
Tr}[W_{+}\tilde{F}^{+\mu}(-z/2)W_{\pm}F^{+}_{\ \mu}(z/2)]|p\rangle$
$\displaystyle=$
$\displaystyle-i\frac{\epsilon_{ij}k_{\perp}^{i}\Delta_{\perp}^{j}}{M^{2}}C^{[+\pm]}_{g}(x,\xi,k_{\perp},\Delta_{\perp}),$
(2)

where $p^{\mu}=P^{\mu}-\Delta^{\mu}/2$, $p^{\prime\mu}=P^{\mu}+\Delta^{\mu}/2$
with $P^{\mu}\approx\delta^{\mu}_{+}P^{+}$. $\xi=\frac{-\Delta^{+}}{2P^{+}}$
is the skewness variable which will be set to zero in the following. For
definiteness, we have assumed a spinless hadron or nucleus in the above
parametrizations. In the case of nonzero spin, additional terms and
appropriate spinor/tensor structures should be supplemented. However, such a
generalization is not necessary for the present purpose. $W_{\pm}$ is a
staple-shaped Wilson line connecting the two points $\pm z/2$ via the light
cone infinity $z^{-}=\pm\infty$. In the quark case, the difference $W_{+}$ or
$W_{-}$ does not matter for the present purpose. In the gluon case, the two
different configurations of $W_{\pm}$ lead to physically different
distributions Bomhof  _et al._ (2006) as is well known in the analysis of the
unpolarized gluon distributions Dominguez  _et al._ (2011). Following the
usual nomenclature, we shall refer to $C_{g}^{[++]}$ and $C_{g}^{[+-]}$ as the
Weiszäcker-Williams (WW) distribution and the dipole distribution,
respectively, and use superscripts ‘WW’ and ‘dip’ instead of $[+\pm]$.

The existence of the term proportional to
$\epsilon_{ij}k_{\perp}^{i}\Delta_{\perp}^{j}$ was pointed out in Meissner
_et al._ (2008, 2009) which was later interpreted as the spin-orbit
correlation Lorce and Pasquini (2011). A quick way to see this is to notice
that $\bar{\psi}\gamma^{+}\gamma_{5}\psi$ and $i\tilde{F}^{+\mu}F^{+}_{\ \mu}$
are the quark and gluon helicity operators, and the product
$-i\epsilon_{ij}k^{i}_{\perp}\Delta_{\perp}^{j}\sim\vec{b}_{\perp}\times\vec{k}_{\perp}$
with $b_{\perp}$ being the impact parameter Fourier conjugate to
$\Delta_{\perp}$ represents the longitudinal component of OAM. They may be
loosely written as $C\sim\langle s^{z}l^{z}\rangle$ where $s^{z}$ and $l^{z}$
denote the longitudinal component of the helicity and OAM of individual
partons. Previous studies of the quark spin-orbit correlation can be found in
Lorcé (2014); Rajan  _et al._ (2018); Tan and Lu (2022) assuming a straight
(not staple-shaped) Wilson line, and in Engelhardt  _et al._ (2022) using
lattice QCD. It has also been demonstrated that the quark and gluon spin-orbit
correlations enter certain exclusive observables Bhattacharya  _et al._ (2017,
2022a); Boussarie  _et al._ (2018); Bhattacharya  _et al._ (2023) (denoted by
$G_{1,1}$ or ${\cal G}_{4}$ in these references).

In this paper, we study the small-$x$-behavior of the $k_{\perp}$-moment of
the spin-orbit correlations defined through staple-shaped Wilson lines

$\displaystyle C_{q,g}(x)\equiv\int
d^{2}k_{\perp}\frac{k_{\perp}^{2}}{M^{2}}C_{q,g}(x,k_{\perp},\Delta_{\perp}=0).$
(3)

For antiquarks, $C_{\bar{q}}(x)=-C_{q}(-x)$. It is expected that
$C_{q,g}(x)>0$ if the helicity and OAM of individual quarks and gluons with a
given momentum fraction $x$ tend to be aligned with each other, and
$C_{q,g}(x)<0$ if they are anti-aligned. Note that this interpretation makes
no reference to the spin of the parent particle. The correlation exists even
in unpolarized or spinless hadrons and nuclei.

###  II.1 Dipole gluon distribution

We demonstrate that $C_{q,g}(x)$ at small-$x$ is calculable in the CGC
framework. We begin with the dipole gluon distribution $C_{g}^{{\rm dip}}$. In
the limit $x\ll 1$ one can employ the eikonal approximation
$e^{ixP^{+}z^{-}}\approx 1$ and write (2) as

$\displaystyle\frac{\epsilon_{ij}\int
d^{2}w_{\perp}d^{2}z_{\perp}}{(2\pi)^{3}P^{+}g^{2}\int
d^{3}x}e^{-i\left(k_{\perp}-\frac{\Delta_{\perp}}{2}\right)\cdot
z_{\perp}+i\left(k_{\perp}+\frac{\Delta_{\perp}}{2}\right)\cdot
w_{\perp}}\langle p^{\prime}|2{\rm
Tr}\partial^{i}U(w_{\perp})\partial^{j}U^{\dagger}(z_{\perp})|p\rangle$
$\displaystyle=-\frac{N_{c}}{\pi\alpha_{s}}\epsilon_{ij}k_{\perp}^{i}\Delta_{\perp}^{j}\frac{\int
d^{2}w_{\perp}d^{2}z_{\perp}}{(2\pi)^{3}}e^{-i\left(k_{\perp}-\frac{\Delta_{\perp}}{2}\right)\cdot
z_{\perp}+i\left(k_{\perp}+\frac{\Delta_{\perp}}{2}\right)\cdot
w_{\perp}}\frac{\langle p^{\prime}|\frac{1}{N_{c}}{\rm
Tr}U(w_{\perp})U^{\dagger}(z_{\perp})-1|p\rangle}{\langle p|p\rangle},$ (4)

where $\langle p|p\rangle=2P^{+}\int d^{3}x$ and we have integrated by parts
in $z_{\perp}$ and $w_{\perp}$. $U(w_{\perp})={\rm
P}\exp\left(-ig\int_{-\infty}^{\infty}dw^{-}A(w^{-},w_{\perp})\right)$ is an
infinite Wilson line along the light-cone and their product
$\frac{1}{N_{c}}{\rm Tr}U(w_{\perp})U^{\dagger}(z_{\perp})$ is called the
color dipole operator frequently encountered in the CGC literature. We have
used the trick $\langle p^{\prime}|\frac{1}{N_{c}}{\rm
Tr}UU^{\dagger}|p\rangle=\langle p^{\prime}|p\rangle+\langle
p^{\prime}|\left(\frac{1}{N_{c}}{\rm Tr}UU^{\dagger}-1\right)|p\rangle$ and
dropped the first term since $\Delta^{j}_{\perp}\langle
p^{\prime}|p\rangle\propto\Delta^{j}_{\perp}\delta^{(2)}(\Delta_{\perp})=0$.
The forward limit $p^{\prime}\to p$ can be safely taken in the second term. We
can thus identify

$\displaystyle\frac{xC^{{\rm
dip}}_{g}(x,k_{\perp})}{M^{2}}=-\frac{2N_{c}}{\alpha_{s}}\int\frac{d^{2}w_{\perp}d^{2}z_{\perp}}{(2\pi)^{4}}e^{-ik_{\perp}\cdot(z_{\perp}-w_{\perp})}\frac{\langle
p|\frac{1}{N_{c}}{\rm
Tr}U(w_{\perp})U^{\dagger}(z_{\perp})-1|p\rangle}{\langle p|p\rangle}.$ (5)

Notice that the $x$-dependence has seemingly disappeared on the right hand
side. In this representation, it has been encoded in the regularization of $U$
(see, e.g., Balitsky and Chirilli (2008)). A new formalism which explicitly
keeps $x$ and thereby facilitates the matching to moderate-$x$ approaches has
been recently developed Boussarie and Mehtar-Tani (2022, 2023). On the other
hand, the unpolarized gluon TMD is obtained by simply replacing
$\epsilon_{ij}\to\delta_{ij}$ and we find

$\displaystyle\frac{C^{{\rm dip}}_{g}(x,k_{\perp})}{M^{2}}=-\frac{f^{{\rm
dip}}_{g}(x,k_{\perp})}{k_{\perp}^{2}},$ (6)

so that

$\displaystyle C^{{\rm dip}}_{g}(x)=-G(x),$ (7)

where $G(x)$ is the unpolarized gluon PDF. The result (6) was previously
obtained in Boer  _et al._ (2018), but its physical meaning was not explored.
We see that the spin-orbit correlation does not vanish in the eikonal
approximation despite its association with the polarized gluon operator
$\tilde{F}^{+\mu}F^{+}_{\ \mu}$. Moreover, it grows rapidly at small-$x$ and
eventually reaches gluon saturation in exactly the same way as the unpolarized
gluon distribution, but with an opposite sign. (7) can be interpreted that the
gluon helicity and OAM $s^{z},l^{z}=\pm 1$ are ‘maximally’ anti-aligned at
small-$x$. Namely, for every single gluon at small-$x$, its helicity and OAM
are ‘locked-up’ in opposite directions $s^{z}l^{z}=-1$. An intuitive way to
understand this phenomenon is to notice that already in the leading order
production processes $q\to qg,g\to gg$, the net angular momentum of the
emitted soft gluon vanishes $(s^{z},l^{z})=(\pm 1,\mp 1)$. The total spin-
orbit correlation (7) is therefore $-1$ times the number of gluons.

###  II.2 WW gluon distribution

Next we turn to the WW gluon distribution $C_{g}^{\rm WW}$. Again in the
eikonal approximation, the matrix element in (2) can be written as Dominguez
_et al._ (2011), up to a prefactor,

$\displaystyle\epsilon_{ij}\frac{\langle p^{\prime}|{\rm
Tr}[U^{\dagger}(w_{\perp})\partial_{i}U(w_{\perp})U^{\dagger}(z_{\perp})\partial_{j}U(z_{\perp})]|p\rangle}{\langle
p|p\rangle}.$ (8)

In contrast to the dipole case, one cannot integrate by parts in
$\partial_{i,j}$. As a result, a model-independent relation between
$C_{g}^{{\rm WW}}$ and $f_{g}^{{\rm WW}}$ is unavailable in the present
approach. We thus work in the McLerran-Venugopalan model McLerran and
Venugopalan (1994a, b) where the correlator of four Wilson lines (8) can be
evaluated as Jalilian-Marian  _et al._ (1997); McLerran and Venugopalan
(1999); Dominguez  _et al._ (2011)

$\displaystyle
C_{F}\epsilon_{ij}\frac{\partial_{i}^{w}\partial_{j}^{z}D(w_{\perp},z_{\perp})}{D(w_{\perp},z_{\perp})}\left(1-e^{\frac{N_{c}}{C_{F}}D(w_{\perp},z_{\perp})}\right),$
(9)

where $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$ and $D$ is defined through the two-
point function

$\displaystyle\frac{\langle p^{\prime}|\frac{1}{N_{c}}{\rm
Tr}U(w_{\perp})U^{\dagger}(z_{\perp})|p\rangle}{\langle
p|p\rangle}=e^{D(w_{\perp},z_{\perp})}=e^{-i\frac{z_{\perp}+w_{\perp}}{2}\cdot\Delta_{\perp}}e^{D(w_{\perp}-z_{\perp},\Delta_{\perp})}.$
(10)

Changing variables as $r_{\perp}=z_{\perp}-w_{\perp}$ and
$b_{\perp}=\frac{z_{\perp}+w_{\perp}}{2}$, and linearizing with respect to
$\Delta_{\perp}$, we obtain

$\displaystyle k_{\perp}^{2}\frac{C^{{\rm
WW}}_{g}(x,k_{\perp})}{M^{2}}=-f_{g}^{{\rm
WW}}(x,k_{\perp})-\frac{C_{F}}{\pi\alpha_{s}x}\int\frac{d^{2}b_{\perp}d^{2}r_{\perp}}{(2\pi)^{3}}e^{-ik_{\perp}\cdot
r_{\perp}}\partial_{i}^{r}D(r_{\perp})\partial_{i}^{r}\left(\frac{1-e^{\frac{N_{c}}{C_{F}}D(r_{\perp})}}{D(r_{\perp})}\right).$
(11)

where the the WW gluon TMD in the present model is well known Jalilian-Marian
_et al._ (1997),

$\displaystyle f_{g}^{{\rm
WW}}(x,k_{\perp})=\frac{C_{F}}{\pi\alpha_{s}x}\int\frac{d^{2}b_{\perp}d^{2}r_{\perp}}{(2\pi)^{3}}e^{-ik_{\perp}\cdot
r_{\perp}}\frac{\nabla^{2}D(r_{\perp})}{D(r_{\perp})}\left(1-e^{\frac{N_{c}}{C_{F}}D(r_{\perp})}\right).$
(12)

Compare with the dipole case (6). The second term in (11) is absent in the
‘dilute’ (non-saturated) regime $D\ll 1$. We have checked numerically in a
Gaussian model $D(r_{\perp})\propto r_{\perp}^{2}$ that this term tends to
have an opposite sign from the first term but does not flip the sign of the
latter. Moreover, the second term vanishes when integrated over $k_{\perp}$
due to color transparency $D(r_{\perp}\to 0)=0$. We thus find that

$\displaystyle C^{{\rm WW}}_{g}(x)=-G(x),$ (13)

and therefore, $C_{g}^{{\rm WW}}(x)=C_{g}^{{\rm dip}}(x)$ at small-$x$. In
fact, one can actually prove that $C_{g}^{{\rm WW}}(x)=C_{g}^{{\rm dip}}(x)$
for all values of $x$ by slightly adapting a related proof in Hatta  _et al._
(2017).

##  III Quark spin-orbit coupling

The analysis of the quark spin-orbit coupling is slightly more involved but
basically follows the same strategy as for the quark TMD at small-$x$ McLerran
and Venugopalan (1999); Marquet  _et al._ (2009). Let us write (1) as

$\displaystyle-\frac{\int d^{3}zd^{3}w}{2(2\pi)^{3}\int
d^{3}x}e^{ixP^{+}(z^{-}-w^{-})-i\left(k_{\perp}-\frac{\Delta_{\perp}}{2}\right)\cdot
z_{\perp}+i\left(k_{\perp}+\frac{\Delta_{\perp}}{2}\right)\cdot
w_{\perp}}\langle p^{\prime}|{\rm
Tr}[W_{w,-\infty}W_{-\infty,z}\gamma^{+}\gamma_{5}S(z,w)]|p\rangle,$ (14)

where $S(z,w)$ is the quark propagator in the background gluon field
(‘shockwave’) created by the high energy hadron/nucleus. The trace is over
both the color and Dirac indices. We have assumed that the staple-shaped
Wilson line is past-pointing $W_{-}(w,z)=W_{w,-\infty}W_{-\infty,z}$. The case
with a future-pointing Wilson line $W_{+}$ is entirely analogous. To leading
order in $1/x$, one may assume that the background gauge field is localized at
$z^{-}=0$ and use the propagator Balitsky (1996)

$\displaystyle S(z,w)=\begin{cases}\int
d^{4}y\delta(y^{-})S_{0}(z,y)\gamma^{-}S_{0}(y,w)U(y_{\perp}),\qquad(z^{-}>0,w^{-}<0)\\\
-\int
d^{4}y\delta(y^{-})S_{0}(z,y)\gamma^{-}S_{0}(y,w)U^{\dagger}(y_{\perp}),\qquad(z^{-}<0,w^{-}>0)\end{cases}$
(15)

where $S_{0}(z,y)$ is the free fermion propagator. Note that the regions
$z^{-}w^{-}>0$ where $S(z,w)=S_{0}(z,w)$ do not contribute due to the presence
of a $\gamma_{5}$. After some manipulations, (14) is evaluated as

$\displaystyle-\frac{iN_{c}}{8\pi^{4}x}\epsilon^{ij}\int
d^{2}k_{g\perp}(k_{\perp}-k_{g\perp})_{i}\left[2\frac{k_{\perp}\cdot\Delta_{\perp}}{k_{\perp}^{2}}k_{\perp
j}+\Delta_{\perp
j}\frac{\ln\frac{k_{\perp}^{2}}{(k_{\perp}-k_{g\perp})^{2}}}{k_{\perp}^{2}-(k_{\perp}-k_{g\perp})^{2}}\right]\frac{\langle
p^{\prime}|\left(\frac{1}{N_{c}}{\rm
Tr}UU^{\dagger}-1\right)(k_{g\perp})|p\rangle}{\langle p|p\rangle},$ (16)

where ${\rm
Tr}UU^{\dagger}(k_{g\perp})\equiv\int\frac{dw_{\perp}dz_{\perp}}{(2\pi)^{2}}e^{-ik_{g\perp}\cdot(z_{\perp}-w_{\perp})}{\rm
Tr}U(w_{\perp})U^{\dagger}(z_{\perp})$ and we have already linearized in
$\Delta_{\perp}$. We can thus identify

$\displaystyle\frac{C_{q}(x,k_{\perp})}{M^{2}}=\frac{N_{c}S_{\perp}}{8\pi^{4}xk_{\perp}^{2}}\int
d^{2}k_{g\perp}(k_{\perp}-k_{g\perp})\cdot
k_{\perp}\frac{\ln\frac{k_{\perp}^{2}}{(k_{\perp}-k_{g\perp})^{2}}}{k_{\perp}^{2}-(k_{\perp}-k_{g\perp})^{2}}\frac{\langle
p|\left(\frac{1}{N_{c}}{\rm
Tr}UU^{\dagger}-1\right)(k_{g\perp})|p\rangle}{\langle p|p\rangle}.$ (17)

Comparing this with the known result for the quark TMD $f_{q}(x,k_{\perp})$
McLerran and Venugopalan (1999); Marquet  _et al._ (2009), we find the
relation

$\displaystyle
f_{q}(x,k_{\perp})=-2\frac{k_{\perp}^{2}}{M^{2}}C_{q}(x,k_{\perp}),$ (18)

or after the $k_{\perp}$ integration,

$\displaystyle C_{q}(x)=-\frac{1}{2}q(x),$ (19)

where $q(x)$ is the quark PDF. The same result holds for antiquarks
$C_{\bar{q}}(x)=-\frac{1}{2}\bar{q}(x)$. Again, (19) can be interpreted that,
for every single quark at small-$x$, the helicity and OAM are anti-aligned
$s^{z}l^{z}=-\frac{1}{2}$. Physically, this is because small-$x$ (anti-)quarks
are created by $g\to q\bar{q}$ splittings with $(s^{z},l^{z})=(\pm 1,\mp
1)\to(\pm\frac{1}{2},\mp 1)\oplus(\mp\frac{1}{2},\pm 1)$. The total spin-orbit
correlation is thus $-\frac{1}{2}$ times the number of quarks (and similarly
for antiquarks).

##  IV Quantum entanglement

We now argue that the spin-orbit anti-correlation that we have found actually
signifies a deeper underlying phenomenon; quantum entanglement between
helicity and OAM. To see this, recall that the basic idea of the CGC is to
describe small-$x$ gluons as the Weiszäcker-Williams (boosted color Coulomb)
field sourced by fast-moving valence quarks at large-$x$ McLerran and
Venugopalan (1994a). As is well known in the context of photons in QED, such
gluons are linearly polarized with their polarization vectors randomly
oriented in the transverse plane McLerran (2002); Metz and Zhou (2011). In the
helicity basis $s^{z}=\pm 1$, their spin states are therefore described by the
vectors

$\displaystyle\frac{1}{\sqrt{2}}\bigl{(}|+\rangle_{s}+|-\rangle_{s}\bigr{)},\qquad\frac{1}{\sqrt{2}i}\bigl{(}|+\rangle_{s}-|-\rangle_{s}\bigr{)},$
(20)

with vanishing expectation value $\langle s^{z}\rangle=0$. However, we have
seen above that the spin and spatial parts of the wavefunction are not
independent. Given the perfect anti-correlation between helicity and OAM
$l^{z}=-s^{z}$, it is more appropriate to represent the spin states as ‘Bell
states’

$\displaystyle|\Psi^{+}\rangle=\frac{1}{\sqrt{2}}\bigl{(}|+\rangle_{s}|-\rangle_{l}+|-\rangle_{s}|+\rangle_{l}\bigr{)},\qquad|\Psi^{-}\rangle=\frac{1}{\sqrt{2}i}\bigl{(}|+\rangle_{s}|-\rangle_{l}-|-\rangle_{s}|+\rangle_{l}\bigr{)},$
(21)

such that $\langle\Psi^{\pm}|s^{z}l^{z}|\Psi^{\pm}\rangle=-1$ even though
$\langle\Psi^{\pm}|s^{z}|\Psi^{\pm}\rangle=0$ and
$\langle\Psi^{\pm}|l^{z}|\Psi^{\pm}\rangle=0$. These states are also linearly
polarized (since $\langle s^{z}\rangle=0$) with the orientation of the
polarization vector varying spatially. For example, the so-called vortex mode
of photons carries OAM characterized by the azimuthal angular wavefunction
$|\pm\rangle_{l}\sim e^{\pm i\phi}$ Bliokh and Nori (2015). This leads to
angular-dependent polarization vectors
$|\Psi^{+}\rangle\sim(1,i)e^{-i\phi}+(1,-i)e^{i\phi}\sim(\cos\phi,\sin\phi)$
and
$|\Psi^{-}\rangle\sim-i\left((1,i)e^{-i\phi}-(1,-i)e^{i\phi}\right)\sim(-\sin\phi,\cos\phi)$.
In QCD, the OAM of gluons is associated with their literal orbital motion
inside a confined hadron. The corresponding wavefunction will be more
complicated.

$|\Psi^{+}\rangle$ and $|\Psi^{-}\rangle$ are maximally entangled states
realized on a single gluon at small-$x$ each carrying entanglement entropy
$\ln 2$. Remarkably, a quite analogous phenomenon, quantum entanglement of the
spin and the OAM of a single photon, has been experimentally realized in the
field of quantum optics Stav  _et al._ (2018). In this experiment, the
transition from (20) to (21) was artificially made by passing linearly
polarized photons through a specially engineered surface which creates or
annihilates one unit of OAM depending on the photon helicity. In contrast, in
QCD the spin-orbit entanglement is a default property of soft gluons from
birth. It is a novel form of entanglement distinct from those previously
discussed in the CGC literature Kutak (2011); Kovner and Lublinsky (2015);
Kharzeev and Levin (2017); Hagiwara  _et al._ (2018); Neill and Waalewijn
(2019); Dvali and Venugopalan (2022); Dumitru  _et al._ (2023).

##  V Conclusions

In this paper, we have uncovered a novel emergent property of the Color Glass
Condensate, the universal form of hadrons and nuclei at high energy. We have
shown that the helicity and OAM of single quarks and gluons at small-$x$ are
strongly anti-aligned even if the parent hadron/nucleus is unpolarized or
spinless. Combining this knowledge with the fact that soft gluons in the CGC
are linearly polarized, we conclude that the helicity and OAM of soft gluons
are maximally entangled in a quantum mechanical sense. In the parlance of
quantum information science, neither the measurement of helicity $\langle
s^{z}\rangle=0$ nor OAM $\langle l^{z}\rangle=0$ alone can reveal the true
quantum nature of the state which is encoded in the correlation $\langle
s^{z}l^{z}\rangle\neq 0$.

Our finding is consistent with and even explains the known anti-correlation
between the quark $\Delta q(x)$ and gluon $\Delta G(x)$ helicity distributions
and the OAM $L_{q,g}(x)$ distributions in a polarized hadron Hatta  _et al._
(2017); More  _et al._ (2018); Hatta and Yang (2018); Boussarie  _et al._
(2019); Kovchegov and Manley (2024); Manley (2024). When a hadron is
polarized, the helicities of quarks and gluons tend to align in certain
directions. Since the helicity and OAM of each parton are locked up even
before polarization, naturally the resulting distributions have opposite signs
at small-$x$ Hatta and Yang (2018); Boussarie  _et al._ (2019)

$\displaystyle\Delta q(x)\approx-\frac{1}{1+c}L_{q}(x),\qquad\Delta
G(x)\approx-\frac{2}{1+c}L_{g}(x),$ (22)

where $c$ is the Regge intercept $\Delta q(x)\sim\Delta G(x)\sim 1/x^{c}$.
Note however that perfect anti-correlation would imply $\Delta q(x)\approx-
L_{q}(x)$ and $\Delta G(x)\approx-L_{g}(x)$, but this does not agree with (22)
for any value of $c$ (see also Kovchegov and Manley (2024); Manley (2024)).
This indicates that the polarization of the parent hadron/nucleus, despite
being a sub-eikonal effect, non-trivially affects the spin-orbit entanglement
of partons.

Finally, it is of course very interesting to experimentally probe the spin-
orbit correlation. Previously the quark and gluon $G_{1,1}$ GTMDs have
appeared in certain exclusive reactions Bhattacharya  _et al._ (2017, 2022a);
Boussarie  _et al._ (2018), but a quantitative evaluation of cross sections in
a concrete experimental setup has been lacking. We have recently found that
the gluon $G_{1,1}$ at small-$x$ affects the observable proposed in
Bhattacharya  _et al._ (2022b). This will be elaborated in a forthcoming
publication S. Bhattacharya, R. Boussarie, and Y. Hatta

## Acknowledgements

The work of S. B. has been supported by the Laboratory Directed Research and
Development program of Los Alamos National Laboratory under project number
20240738PRD1. S. B. has also received support from the U. S. Department of
Energy through the Los Alamos National Laboratory. Los Alamos National
Laboratory is operated by Triad National Security, LLC, for the National
Nuclear Security Administration of U. S. Department of Energy (Contract No.
89233218CNA000001). Y. H. was supported by the U.S. Department of Energy under
Contract No. DE-SC0012704, and also by Laboratory Directed Research and
Development (LDRD) funds from Brookhaven Science Associates. R. B. and Y. H.
were also supported by the framework of the Saturated Glue (SURGE) Topical
Theory Collaboration.

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