# Spin alignment of vector mesons as a probe of spin hydrodynamics and freeze-
out

Kayman J. Gonçalves,Giorgio Torrieri  Universidade Estadual de Campinas -
Instituto de Fisica ”Gleb Wataghin”
Rua Sérgio Buarque de Holanda, 777
CEP 13083-859 - Campinas SP

###### Abstract

We argue that a detailed analysis of the spin alignment of vector mesons can
serve as a probe of two little-understood aspects of spin dynamics in the
vortical fluid: The degree of relaxation between vorticity and particle spin
polarization, and the degree of coherence of the hadron wavefunction at
freeze-out. We illustrate these with a coalescence model.

##  I Introduction

The experimental detection of event plane polarization in heavy ion collisions
has generated a great amount of experimental and theoretical interest lisa .
The data on global polarization is reasonably described by an ideal
hydrodynamic model where angular momentum is transferred to spin at freeze-out
following an extension of the Cooper-Frye equilibrium formula cf . While
experimentally this is a great success (although the local rather than global
polarization is less well described lisa ), from a theory point of view such a
model cannot be the whole story: It is now a commonly accepted idea
uscausality ; uscas2 ; relax ; jeon ; noronha ; analspin1 ; analspin1.5 ;
analspin2 that spin and vorticity are not always in equilibrium or aligned (in
non-relativistic physics this relaxation is known as the Fert-Valet equation
fart ; fart2 ). There is however little phenomenology associated with this
lack of alignment in the context of $\Lambda$ polarization. Such a
phenomenology is complicated by the fact that spin and angular momentum are
not separately conserved. Hence, the detail of how angular momentum is
partitioned at freeze-out can not be straight-forwardly be linked by a Cooper-
Frye type ansatz, that depends on entropy conservation and conservation laws
only cf . In principle more complicated dynamics (such as coalescence coal1 ;
coal2 ; coal3 ), even if close to global equilibrium, will imprint a non-
trivial signature.

In this work we would like to propose a detailed measurement of the spin space
density matrix of vector meson, projected on the global event coordinate
system (with $\theta$ defined by the reaction plane and $\phi$ on beam axis).
The experimental measurement of spin alignment of vector mesons ($K^{*}$ and
$\phi$) along the reaction plane in heavy ion collisions is a nice complement
to the measurement of hyperon polarization alice ; star . However, reconciling
them in a single framework is still an open problem, both from the theoretical
and the experimental point of viewlisa .

Experimentally, spin alignment and polarization are not really the same
observable, although they both depend on analyses of angular distributions of
decay products in the mother-particle’s rest frame. Polarization measurements
are done using weak decays of hyperons, and, because of parity violation in
such decays, the experimentalist can get a direct handle on the spin
direction. Spin alignment on the other hand is done using parity-conserving
strong $p-$wave decays. As such, as the name suggests, it is impossible to
distinguish “up pointing” from “down-pointing decays”, but just distinguish
up/down third quantum number $m_{s}\pm 1$ (aligned) vs $m_{s}=0$ spin
Eigenstates.

However, this is not to say that spin alignment potentially provides less
information than polarization. p-wave decays do not just have one reference
axis but two: In the rest frame of the decaying meson seyboth ; LI the angle
$\theta$ is defined as the angle between the polarization vector and the
momentum of the daughter particle (either of them, since spin alignment will
be the same for $\theta,\pi-\theta$). There is however an additional azimuthal
angle $\phi$ which, provided an event-by-event reference can be provided, can
be used to get information about the decaying meson and the underlying event.
in the reference frame at rest with the meson decay, given a definition for an
axial angle $\theta$ and azimuthal angle $\phi$ the distribution of decay
products in terms of $\theta$ and $\phi$ seyboth ; LI , presented as a
polynomial of harmonic coefficients, is:

$W(\theta,\phi)\propto\cos^{2}\theta\rho_{00}+\sin^{2}\theta\left(\frac{1-\rho_{00}}{2}+r_{1,-1}\cos(2\phi)+\alpha_{1,-1}\sin(2\phi)\right)+\sin(2\theta)\left(r_{10}\cos\phi+\alpha_{10}\sin\phi\right)$
(1)

where the coefficients to the trigonometric formulae are related to the
density matrix elements of the meson ensemble via a normalization coefficient
$3/(4\pi)$ and

$\begin{array}[]{ccc}\mathrm{Variable}&\mathrm{Element}&\mathrm{coefficient}\times\frac{3}{4\pi}\\\
\rho_{00}&\rho_{00}&\cos^{2}\theta\\\
\frac{1-\rho_{00}}{2}&\frac{\rho_{11}+\rho_{-1-1}}{2}&\sin^{2}\theta\\\
r_{10}&Re[\rho_{-10}-\rho_{10}]&\sin(2\theta)\cos(\phi)\\\
\alpha_{10}&Im[-\rho_{-10}+\rho_{10}]&\sin(2\theta)\sin(\phi)\\\
r_{1,-1}&Re[\rho_{1,-1}]&\sin^{2}\theta\cos(2\phi)\\\
\alpha_{1,-1}&Im[\rho_{1,-1}]&\sin^{2}\theta\sin(2\phi)\end{array}$ (2)

It is clear that for a mixture of up/down and zero states w.r.t. a single
direction and this direction is known (the reaction plane) $\phi$ is
irrelevant because the density matrix will be in a diagonal form and there
will be no $\phi$ dependence unless the “correct” spin axis is mis-identified
(xia points out this is indeed the case if both longitudinal and transverse
polarization occurs and the reaction plane is used an axis). If, however, more
than two sources of angular momentum (vorticity and spin currents!) somehow
combine to produce vector meson polarization, $\phi$ distribution could be
non-trivial in any axis because the density matrix reconstructed from Eq. 2 is
not pure.

So far, experimental collaborations alice ; star only measured the $\theta$
angle distribution w.r.t. the reaction plane (with the $\rho_{00}$ alignment
factor), since this measurement can be linked to global vorticity. $\rho_{00}$
can then be used as a related parameter to the polarization probability. One
event-independent easy to implement definition for the other angle would be in
terms of the beam axis (Fig. 2 ), although others have bee used in Charmonium
studies in p-p collisions jpsi . However, although extra analysis has been
suggested by theorists oliva ; phicoal ; xia , no systematic interpretation of
the extra parameters has so far been proposed. In this work we attempt to do
this, in terms of simple popular models of hadron production, such as
coalescence coal1 ; coal2 ; coal3 and Cooper-Frye (cf and references therein),
as well as some general properties of spin and angular momentum.

The spin state of a particle of spin $N=2s+1$, projected on spin space, will
be an $N\times N$ density matrix, described as gendens

$\rho=U\Lambda U^{-1}\phantom{AA},\phantom{AA}U\in
U(2s+1)\phantom{AA},\phantom{AA}\lambda=diag[\lambda_{i}]\phantom{AA},\phantom{AA}\sum_{1}^{2s+1}\lambda_{i}=1$
(3)

$N$ here generically corresponds to the internal state space, but of course it
reduces to rotations between spin Eigenstates when the density matrix
considered is one projected in spin space.

Assuming we have a measured distribution of spin observables we could attempt
to reconstruct such a density matrix. The amount of information that can be
extracted from it ( to what extent it is coherent w.r.t. any particular
direction, the parameter space constraints) will depend on ”Coadjoint orbits”,
coset spaces w.r.t. the subgroups of $U(N)$ describing rotations between
degenerate Eigenstates. Rotations within the same orbit keep the density
matrix diagonal.

For a spin 1/2 particle $\hat{\rho}_{2}$ the density matrix reduces to one
“qubit” of information, a point on the the Bloch sphere. One crucial issue,
specific to qubits, is that the Bloch sphere describes the parameter space of
both pure and mixed qubits. This is however a specific fact of qubits preskill
, that there always exists a unit 3-vector $\hat{n}$ such that

$\hat{\rho}_{2}=\frac{1}{2}\left(I+\hat{n}.\vec{\sigma}\right)$ (4)

where $\hat{\sigma}$ are the Pauli matrices. No matter what dynamics is used
to combine spin (call it $\hat{n}_{1}$) and angular momentum (call it
$\hat{n}_{2}$) from the quark-gluon plasma into the hadron spin, the resulting
density distribution for $\hat{\rho}_{2}$ will be a pure density state for
some axis (defined by $\hat{n}$, perhaps a mixture of $\hat{n}_{1}$ and
$\hat{n}_{2}$ as in xia ). A rotation to this axis will purify the density
matrix. Hence, how spin and angular momentum of quarks and gluons combine to
produce the $\Lambda$ spin (is it a unitary process or interaction with an
environment?) is in principle inaccessible. Of course the dependence of that
qubit on that particle’s momentum and rapidity might tell us something in this
direction but this necessitates fitting and will always be model-dependent in
a non-straight-forward way.

These considerations are however specific to spin 1/2 fermions with two
Eigenstates111In a physically very different but mathematically similar
context, the explanation as to why two flavors are not enough for CP violation
in the Cabibbo-Kobayashi-Maskawa matrix is similar. The density matrix of a
spin 1 particle, in contrast is (left side of Fig. 2 ) projected to internal
spin space is a qutrit qutrit1 ; qutrit , where pure and mixed states do not
have this degeneracy and coherence can be evaluated by geometrical means. It
is well known that qutrit1 that a qutrit‘s density matrix can be parametrized
as

$\rho=\frac{1}{3}\left(1+\sqrt{3}\vec{n}_{8}.\vec{\lambda}\right)$ (5)

where $\vec{n}_{8}$ is an 8-dimensional vector (not representable in
configuration space) and $\vec{\lambda}$ represents the Gell-Mann density
matrices. This is a parametrization, rather than a representation, since the
coefficients do not in any way transform under $SU(3)$ algebra and spin
operators transform as SU(2) representations, but this parametrization can be
used to characterize the density matrix, specifically its degree of coherence
and alignment.

Aligned in the $8$ direction ($\vec{n}_{8}$ has only the 8th component), this
matrix is going to look like

$\rho_{8}(n_{3},n_{8})=\frac{1}{3}\left(\begin{array}[]{ccc}1+\sqrt{3}\;n_{3}+n_{8}&0&0\\\
0&1-\sqrt{3}\;n_{3}+n_{8}&0\\\ 0&0&1-2n_{8}\end{array}\right)$ (6)

this contains two qubits (more exactly, a “qutrit”) of information qutrit ;
geom . The additional purity constraint
$\mathrm{Tr}\left(\hat{\rho}^{2}\right)=\hat{\rho}$ gives

$\left(1+n_{8}+\sqrt{3}\;n_{3}\right)\left(1+n_{8}-\sqrt{3}\;n_{3}\right)\left(1-2\;n_{8}\right)=0$
(7)

Thereby, we will have the three points
$\left(n_{3},n_{8}\right)=(0,-1),\;\frac{1}{2}(\pm\sqrt{3},1)$ that represent
the pure states. In the figure Fig. 1 , we show that the points where the
triangle touches the outer circle. Conversely, the points on the inner circle
of Fig. 1 correspond to maximum entropy

$\hat{\rho}=\frac{\mathcal{F}}{\mathcal{F}^{2}+\mathcal{F}+1}\mathrm{Diag}\left[\mathcal{F},1,\frac{1}{\mathcal{F}}\right]\phantom{AA},\phantom{AA}\mathcal{F}\sim\exp\left[\frac{\Omega}{T}\right]$
(8)

expected from the Cooper Frye type freezeout. Note that in the frame co-moving
with the flow the density matrix examined in xia has this form.

Figure 1: (color online) A representation of the states of the density matrix
Eq. 6. The points touching the outer circle correspond to pure states, and the
points inside the blue triangle are the mixed states, for example, the dotted
circle points inner the triangle.

If one can

associate

each qutrit to an axis direction, with both $\theta$ and $\phi$ angles
according to Eq. 1

understand

how plasma spin and orbital angular momentum combine into hadronic spin in a
calculable way

one could potentially come up with a phenomenology of spin-orbit misalignment
from a fit to Eq. 1.

The association is in fact straight-forward: It is known experimentally from
$\Lambda$ polarization lisa that spin polarization has both a transverse and a
longitudinal component. Physically, their origins are very different: The
transverse component is the result of the initial impact parameter, while the
longitudinal component is the result of the same hydrodynamic evolution that
generates azimuthally anisotropic flow coefficients.

This means that if there is diffusion between vorticity and spin, the
admixture of spin vs angular momentum in the two components is generally very
different. Naively for the transverse component vorticity and spin will be in
equilibrium, the longitudinal component less so (the opposite of flow, where
longitudinal flow is formed initially, transverse flow is not). Considering
that this difference in equilibration is directly correlated to axes, it is
worth seeing if the extra information in vector mesons is obtainable from an
angular analysis including these axes.

Now, the second requirement, understanding how plasma spin and angular
momentum combine into vector meson spin, is a bit more complicated and model
dependent. The rest of the paper is about this topic. In section II, we relate
experimental coefficients to group-theoretical parameters. In section III , we
relate these parameters to the spin density and vorticity within a coalescence
type picture.

Figure 2:  Relation between the angles $\theta,\phi$ as measured in experiment
and the meson’s density matrix parametrized as in Eq. 5

##  II From qutrits to experimental angular distributions

Consider a spin 1 density matrix aligned in a certain direction. We need to
understand the relation between a physically natural reference frame, e.g. Eq.
6, to the laboratory coordinate system. The two are related by an additional
transformation parametrized by angles $\theta_{r},\phi_{r}$ (Fig. 2 )

$\rho(n_{3},n_{8},\theta_{r},\phi_{r})=U(\theta_{r},\phi_{r})\rho_{8}(n_{3},n_{8})U(\theta_{r},\phi_{r})^{-1}$
(9)

where222Note the use of Clebsch-Gordon coefficients in this and the next
section, to underline that while one can parameterize the density Matrix using
Gell-Mann matrices, it still transforms as a representation of SU(2).

$U(\theta_{r},\phi_{r})=D\left(\alpha,\beta,\gamma\right)=exp\left(\frac{-iJ_{z}\alpha}{\hbar}\right)exp\left(\frac{-iJ_{y}\beta}{\hbar}\right)exp\left(\frac{-iJ_{z}\gamma}{\hbar}\right)$
(10)

Making $\alpha=\phi_{r}$, $\beta=\theta_{r}$ and $\gamma=0$ we can decompose

$D^{j}_{m^{{}^{\prime}}m}(\phi_{r},\theta_{r})=\left<j,m^{{}^{\prime}}\right|D\left(\phi_{r},\theta_{r},0\right)\left|j,m\right>$
(11)

with

$D^{j}_{m^{{}^{\prime}}m}(\phi_{r},\theta_{r})=e^{-im^{{}^{\prime}}\phi_{r}}d^{j}_{m^{{}^{\prime}}m}(\theta_{r})\phantom{AA},\phantom{AA}\left[D^{j}_{m^{{}^{\prime}}m}(\phi_{r},\theta_{r})\right]^{-1}=e^{im\phi_{r}}\left[d^{j}_{m^{{}^{\prime}}m}(\theta_{r})\right]^{-1}$
(12)

and for $j=1$, we have explicitly

$d^{1}(\theta_{r})=\left(\begin{array}[]{ccc}\frac{1}{2}(1+\cos\theta_{r})&-\frac{1}{\sqrt{2}}\sin\theta_{r}&\frac{1}{2}(1-\cos\theta_{r})\\\
\frac{1}{\sqrt{2}}\sin\theta_{r}&\cos\theta_{r}&-\frac{1}{\sqrt{2}}\sin\theta_{r}\\\
\frac{1}{2}(1-\cos\theta_{r})&\frac{1}{\sqrt{2}}\sin\theta_{r}&\frac{1}{2}(1+\cos\theta_{r})\end{array}\right)$
(13)

Thereby, the right hand side of Eq. 9 becomes

$=\sum_{m^{\prime\prime},m}\left<j,m^{\prime}\right|D\left|j,m^{\prime\prime}\right>\left<j,m^{\prime\prime}\right|\rho_{8}\left|j,m\right>\left<j,m\right|D^{-1}\left|j,m^{\prime\prime\prime}\right>$
(14) $\
=\sum_{m^{\prime\prime},m}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}d^{j}_{m^{{}^{\prime}}m^{{}^{\prime\prime}}}(\theta_{r})\left<j,m^{\prime\prime}\right|\rho_{8}\left|j,m\right>\left[d^{j}_{mm^{{}^{\prime\prime\prime}}}(\theta_{r})\right]^{-1}=\sum_{m^{\prime\prime},m}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}T_{m^{\prime},m^{\prime\prime\prime}}(\theta_{r},n_{3},n_{8})$

Now, as discussed in the introduction given two event-invariant axes,
experimentalists measure (right panel of Fig. 2 ) all the coefficients of Eq.
2 via a Fourier analysis of Eq. 1 . Comparing Eq. 2 with Eq. 9 we get a system
of complicated-looking but actually solvable equations (although in practice
systematic effects might be difficult to disentangle practical ). We can take
advantage of the fact that $\hat{\rho}$ in Eq. 5 is parametrized by Gell-Mann
matrices, by choosing appropriate two-sections and three-sections . The
sections represent the choice of the coefficients of the density matrix that
we will consider, but your choice depends on the geometries possibles on the
Block sphere generalization of a qutrit state qutrit .

Choosing the two-sections we obtain, after some Algebra333 The operations
leading to Eq. 15 ,Eq. 16 and Eq. 17 are not $SU(2)$ transformations

$\displaystyle\frac{1}{12}\left(3\left(n_{8}-\sqrt{3}\;n_{3}\right)\cos\left(2\theta_{r}\right)-\sqrt{3}\;n_{3}+n_{8}+4\right)=\rho_{00}$
(15)
$\displaystyle\frac{\left(n_{8}-\sqrt{3}\;n_{3}\right)\sin\left(\theta_{r}\right)\cos\left(\theta_{r}\right)\cos\left(\phi_{r}\right)}{\sqrt{2}}=r_{10}$
(16)
$\displaystyle-\frac{\left(\sqrt{3}\;n_{3}+3n_{8}\right)\sin\left(\theta_{r}\right)\sin\left(\phi_{r}\right)}{3\sqrt{2}}=\alpha_{10}$
(17)
$\phi_{r}=-\frac{1}{2}\tan^{-1}\left(\frac{\alpha_{1,-1}}{r_{1,-1}}\right)$
(18)

Which allows us obtain the elements of the array in Eq. 2

Eq. 18 is a trivial combination of observables. The rest, Solving the systems
equations 15 to 17 we obtain the expressions for $n_{3,8},\theta_{r}$ as
rational functions given in Appendix A .

It is a good point to pause and understand the physical meaning of these
formulae. The dynamics of transfer of angular momentum from collective
vorticity to spin is complicated and still not understood ( section III will
give a simple example). The relativistic regime complicates this considerably,
since angles are not Lorentz invariant and hence spin directions potentially
change when one moves from the co-moving frame with the resonance (where spin
is imprinted) to the lab frame (where it is measured). In this work, we
“evade” such difficulties by only considering the density matrix in spin
space, with momentum projected out. The vorticity dynamics, whatever it is,
will imprint itself into the density matrix elements, measurable
experimentally through Eq. 1, from which the expectation value (over all
momentum space in the detector acceptance) of $\theta_{r}$ and $\phi_{r}$ will
be obtained via Eq. 26 and Eq. 18 . While of course for non-linear processes
the average of a function is not a function of the average, we can reasonably
hope that reconstructing information about both hydrodynamics and coherence is
possible, i.e., $\theta_{r}$ and $\phi_{r}$ might not be wholly washed by
event-by-event fluctuations. The amount of such washing-out also depends on
the coherence of the freeze-out process. For instance, for a totally
incoherent freeze-out, Since the only axis that enters the Cooper-Frye formula
is the vorticity axis, one expects $\phi_{r}$ to be washed out if freeze-out
happens in this approach (this can be seen straight-forwardly from Eq. 18 ,
since the ratio of the coefficients depends on $\phi_{r}$ only) and
$\theta_{r}$ will have a peak consistent with zero (up to a shift by $\pi$ of
course). However in more complicated freeze-out dynamics this does not have to
be the case. A similar analysis is impossible for a Fermion density matrix
projected on this system of coordinates as per the discussion around Eq. 4 .
This is a good illustration of the potential in looking at the full vector
meson decay angular distribution in the global coordinate system.

Physically, $\theta_{r},\phi_{r}$ are related to the transverse and
longitudinal axis. The first represents “initial state” vorticity, the second
is created concurrently with longitudinal vorticity, during the hydrodynamic
evolution stage lisa . A relative non-equilibrium therefore will manifest
itself by comparing the $\rho_{00}$ and $\rho_{0i},\rho_{ij}$. We therefore
expect $\theta_{r}$ to be indeed peaked around the reaction plane
$\theta_{r}\simeq 0$. Assuming non-equilibrium between spin and angular
momentum at freeze-out, $\phi_{r}$ will also exhibit a peak at some value
different from zero (again, a distribution symmetric by $\pi$), set in detail
by the non-equilibrium spin-vorticity diffusion.

While a quantitative comparison is outside the scope of this work, in the next
section III we will demonstrate how “early vorticity” (expected to be mainly
transverse) combines with later vorticity (which is also longitudinal) in a
coalescence model. .

##  III From plasma spin and vorticity to qutrits via coalescence

Figure 3:  Coalescence of a vector meson from two quarks, including the effect
of an out-of-equilibrium spin density $\Omega$ with vorticity $\omega$. Quark
spin is determined early on and vorticity (angular momentum number) at
freezeout

We shall implement a coalescence picture in spin space, related to that in
phicoal but allowing vorticity and quark spin to combine in different
directions to reflect the fact that spin and angular momentum are generally
not in equilibrium, and are related by a diffusion equation with a
characteristic timescale.

Putting in coalescence, this means $\Omega$ represents the angular momentum at
the early stage, when quark angular momentum is fixed, while $L$ represents
the angular momentum at hadronization, carried by the vorticity $\omega$.
Without loss of generality, therefore, $L$ can be assumed to point in the
$x-z$ direction while $\vec{\hat{\Omega}}$ is a vector in the direction $z$.
(See Fig. 3 )

$\vec{\omega}=\omega_{x}\;\hat{x}+\omega_{z}\;\hat{z}\phantom{AA},\phantom{AA}\vec{\hat{\Omega}}=\Omega\hat{z}$
(19)

The estimate we make here is on the vector resonance spin density matrix only,
with any momentum space dependence traced over in the co-moving frame of the
resonance. Hence, our estimate is at best a qualitative one, since it neglects
the correlation between vorticity and the local velocity w.r.t. the lab frame,
that would show up as “weights” in the transformation between the two. We
expect this issue could become important w.r.t. longitudinal flow and
transverse vorticity uslisa , and a detailed analysis of the pseudorapidity
dependence of the various coefficients of Eq. 1 is necessary to disentangle
it.

Nevertheless, a non-trivial dependence of the coefficients of Eq. 1 on the
lack of equilibrium between spin and vorticity angular momentum is sufficient
to seriously consider these coefficients as a promising experimental
observable.

The interplay between these two angular momenta can be implemented at the
wavefunction level using a coalescence formalism coal1 ; coal2 ; coal3 . In
spin space, in the frame of reference aligned with the final vorticity
$\Omega$, we can parametrize the coalesced vector meson density matrix from
the spinor quark density matrices as well as coefficients that factorize the
usual way (Clebsch-Gordan, 6j symbols), isolating all model dependence in a
single coefficient. The expression for the vector meson density matrix is

$\begin{array}[]{c}\left(\hat{\rho}^{M}\right)_{mn}=\sum_{ijkl}\left(P_{12}^{L}\right)_{ijklmn}U_{S}(\phi_{r},\theta_{r})\left(U_{\omega}({\mu_{1},\nu_{1}})\rho^{1}(\Omega)U^{-1}_{\omega}(\mu_{1},\nu_{1})\right)_{ij}\\\
\times\left(U_{\omega}(\mu_{2},\nu_{2})\rho^{2}(\Omega)U^{-1}_{\omega}(\mu_{2},\nu_{2})\right)_{kl}U^{-1}_{S}(\phi_{r},\theta_{r})\end{array}$
(20)

Where $M$ refers to the meson, $S_{1,2}$ are the spins of the constituent
quarks and $L$ is the spatial angular momentum given by the vorticity. The
matrices $U_{\omega}$ are SU(2) rotation matrices that rotate the spins in the
direction of the vorticity, depending on the relative angles $\nu_{1,2}$ of
each quark’s spin w.r.t. the vorticity frame. Physically the matrices $U$
reflect that $\Omega$ (connected to the quark spin $S_{1,2}$ quantum numbers)
and $\omega$ (connected to the spacetime angular momentum quantum number $L$)
are both angular momenta generated by the vortical evolution of the fluid, but
they are not necessarily the same, nor they point in the same direction.

Finally, the qutrit rotation matrix $U_{S}$ transforms the system from the
reference frame where $\omega$ is in the plane $x-z$ to the lab frame where
$\theta$ and $\phi$ are defined, examined in section II (the weights discussed
after Eq. 19 would be attached to $U_{s}$).

$\hat{\rho}_{M}$ is a vector meson $3\times 3$ density matrix, to be compared
with Eq. 9 and $\hat{\rho}^{1,2}$ are spinor $2\times 2$ density matrices with
distributions analogous to zanna ; tinti ,specifically for a temperature $T$
and vorticity angular momentum $\hat{\Omega}$ at the early stage when quarks
are polarized (not the final vorticity at freezeout). Since quarks are close
to local equilibrium when polarized

$\hat{\rho}^{1,2}_{kl}=\frac{1}{Z}\exp\left[\frac{\vec{\hat{\Omega}}\cdot\vec{\hat{\sigma}}}{2T}\right]$
(21)

with the usual normalization $\hat{\rho}_{kk}=1$ and $\sigma$ the usual Pauli
spin matrices.

$P^{L}_{12}$ are the coefficients of the coalescence Wigner function projected
into spin space. They, and even their normalization, are only partially
calculable since they incorporate all configuration space and momentum space
dynamics. However, the fact that this dynamics is rotationally invariant
allows us to use the Wigner-Eckart theorem to parametrize these unknowns into
only a few normalization coefficients.

Assuming rotational invariance in the meson rest frame, coalescence will
project an eigenstate which in spin space is equivalent to the addition of 3
angular momenta $s_{1},s_{2},L$ for which one can use Wigner 6j symbols. We
shall assume, a reasonable guess in line with coalescence, that first the
hadron wavefunction is formed ($s_{1}$ and $s_{2}$ are added) and then it
aligns with $L$. Hence pratt , by the Wigner-Eckart theorem the array
$P_{12}^{L}$ in Eq. 20 is identified with

$P_{12}^{L}\equiv\left(C_{S_{1}S_{2}}^{L}\right)_{S}=\sum_{m_{1},m_{2},m_{L},m_{12}}P_{L}(\omega)C^{S_{1}+S_{2},L}_{m_{12},m_{L},S,M_{S}}C^{S_{1},S_{2}}_{m_{1},m_{2},S_{1}+S_{2},m_{12}}$
(22)

where $C^{...}_{...}$ are the Clebsh-Gordon coefficients and index-counting
matches Eq. 20 and $P_{L}(\omega)$ is the probability of a thermalized fluid
with vorticity $W$ to transfer momentum quantum number $L$ to the coalescence
Wigner function. This we do not know anything about, beyond the reasonable
expectation that larger $\omega$ will favor higher $L$ and the fact that
angular momentum conservation will only allow $L=0,1,2$ to be transferred to a
meson of spin 1.

After some algebra, described in Appendix B and based on the same rotation
operations described in the last section (Eq. 11 and Eq. 12 ), we arrive at
the meson density matrix, which is given by:

$\left(\hat{\rho}^{M}\right)_{m^{\prime},m^{\prime\prime\prime}}=\int_{0}^{2\pi}P_{W}(\nu)d\nu\sum_{m^{\prime\prime},m,\tilde{m}^{\prime\prime},\tilde{m}}P^{2}_{L}(\omega)\tilde{C}^{L}_{m^{{}^{\prime\prime}},m}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}d^{j}_{m^{\prime}\tilde{m}^{\prime\prime}}(\theta_{r})d^{j}_{\tilde{m}^{\prime\prime}m^{{}^{\prime\prime}}}(\nu)\times$
(23)
$\times\left<j,m^{\prime\prime}\right|\rho^{1}(\Omega)\rho^{2}(\Omega)\left|j,m\right>\left[d^{j}_{m\tilde{m}}(\nu)\right]^{-1}\left[d^{j}_{\tilde{m}m^{\prime\prime\prime}}(\theta_{r})\right]^{-1}$

with each coefficient given explicitly in Appendix B as well as Eq. 11 and Eq.
12 and $P_{W}(\nu)$ is the angular Wigner function of two quarks aligned
w.r.t. angle $\nu$ to coalesce. Note that the $L=0$ with $\theta_{r}=0$,
$\phi_{r}=0$ and $P_{W}(\nu)\rightarrow\delta(\nu)$ case reduces to the one
examined in xia , as we explicitly show in appendix C.

We can now calculate the distribution of coefficients of Eq. 2 in the case
that spin and angular momentum are not necessarily aligned, as expected in
uscausality , via Eq. 9 and Eq. 22 and 20. Without loss of generality, we
define the plane defined as $\nu$ as $\phi_{r}=0$. The full expressions are
rather long, and we have put them in Appendix D , assuming a Gaussian Wigner
function $P_{W}(\nu)$. The main message is the sensitivity of the observables
of the array in Eq. 2 to both quark polarization and vorticity within a
coalescence scenario.

To illustrate this, we choose a set of parameters that fit $\rho_{00}$ close
to $1/3$ in the “trivial” $L=0$ limit, $\sigma_{\nu}=0$, $\Omega_{3}/T=1/5$
and $\phi_{r}=0.5$, with negligible $\sigma_{\nu}$. We obtain Fig. 4 . We can
see from figures that both $L=1$ and $L=2$ contributions for coefficients
$\rho_{00}$ and $r_{10}$ oscillate, with opposite signs. $\alpha_{1,-1}$’s
width changes, and the baseline amplitudes of $\alpha_{10}$ and $r_{1,-1}$ are
very different for the different $L$ contributions. All these coefficients are
therefore potentially sensitive probes of spin-angular momentum non-
equilibrium.

Figure 4:  Coefficients of Eq. 1 calculated in the coalescence model for a
representative choice of parameters. The $L=0,1,2$ represents the angular
momentum number the meson obtains from the vorticity, something that we can
not calculate within the current formalism

The plots are not meant to be quantitative, this will depend sensitively on
the details of the hydrodynamic evolution, vorticity formation, coalescence
Wigner function etc. Rather, they are indicative of which component is most
sensitive to non-equilibrium between spin and vorticity.

These equations would need to be integrated over probabilities to get
$\theta_{r},\phi_{r}$ as a function of $\Omega$ and $w$ and the momentum
region, which need to be read off a hydrodynamic model. Initial vorticity
going into quark spin, $\Omega_{1,2,3}$ and $T$, as well as final vorticity
$\omega$, would also need to be obtained from such a model.

Summarizing this section, we are now in a position to get some information on
the coefficients of Eq. 1 based on the dynamics of the system. The requirement
of rotational invariance, implemented by Wigner function techniques, gives us
quite a lot of, although not quite the full information linking spin 1 spin
alignment to the evolution of angular momentum and spin currents. What is the
angular momentum corresponding to this vorticity, parametrized by
$P_{L}(\omega)$ of course is not known, but all coefficients are proportional
to it in the same way. Since one expects higher vorticity to transmit more
angular momentum to the hadron it is expected to be monotonic in
$\omega,L,\sim w^{\alpha_{1}}L^{\alpha}_{2},\alpha_{1,2}\geq 0$.

The figures in this section indicate which Fourier coefficients of Eq. 1 are
more sensitive to non-equilibrium between spin and vorticity. By eye,
$r_{1,0},r_{1,-1}$ and $\alpha_{1,-1}$ are approximately equally constraining.
But what is sure is that a simultaneous fit of the coefficients of Eq. 2 and
their dependence on rapidity and reaction plane angle have the potential of
thoroughly constrain the degree of non-equilibrium between vorticity and
polarization.

##  IV Discussion

Eq. 1 with Eq. 9, make it clear that, given a good distribution of
$\theta,\phi$ and given an non-chaotic variation of $\theta_{r},\phi_{r}$
between events, one should be able to recover a distribution of $n_{3,8}$. It
is a matter of Fourier analysis of Eq. 1 and algebra.

In section II we showed that this is indeed the expectation, and computed a
complicated but closed form function relating experimental observables to the
spin density matrix, Eq. 24 and Eq. 25 and, crucially, the rotation angles
$\theta_{r},\phi_{r}$ (given respectively by Eq. 18 and Eq. 26 ) between
between the experimentally accessible frame and one optimized w.r.t. the
qutrit parametrization of Eq. 6 These, if centered around a particular region,
can give information about the spin-hydrodynamic evolution, and crucially the
degree of coherence of the spin wavefunction of the meson.

What can we learn from all this? Here we have to get a bit more qualitative,
with two limiting cases.

(i) Cooper-Frye freezeout

If spin and vorticity are in equilibrium, one expects that Cooper-Frye is a
good estimate. This means that the spin of this meson is not a coherent state,
but rather a superposition of Eigenstates of the type of Eq. 8 . As Eq. 18
makes clear, in this case $\phi_{r}$ should have a uniform event-by-event
distribution.

(ii) Wigner function coalescence

If spin and vorticity are not in equilibrium, of course tings become a bit
more model-dependent. But a coalescence formalism (see conclusion of
uscausality ) means the matrix is a pure state for some choice of
$\theta_{r},\phi_{r}$. It is a good guess that One of these angles is related
to vorticity and the other to spin. In this case we could make a direct link
to the non-equilibrium between vorticity and polarization from the analysis of
the coefficients in Eq. 2 as illustrated in Fig. 4 .

As an immediate test, it is worth to check if $\phi_{r}$ in Eq. 18 has a
peaked distribution event-by-event, between the production plane and a beam
plane. As argued at the end of section II , this would be evidence that
freeze-out is happening dynamically, rather than by Cooper-Frye. Similar
points were made in a recent work xia .

A further more involved analysis would be a purity test on $\rho$, i.e. to
verify experimentally if $\rho=\rho^{2}$. For Eq. 6 this is tantamount to
verifying that for some choice of $\theta_{r},\phi_{r}$ we have
$\left(n_{3},n_{8}\right)=(0,-1),\;\frac{1}{2}(\pm\sqrt{3},1)$qutrit ; geom .
One also needs to check the relation with $\Lambda$ polarization, in other
words that the $\theta_{r},\phi_{r}$ obtained from the analysis in section II
are also close to the mean polarization axis of the $\Lambda$ baryon (see eq.
2-3 of oliva ). Eq. 26 and Eq. 18 make it clear that this will happen if both
$\theta_{r}$ and $\phi_{r}$ are well-defined and centered around zero. This
indicates spin and angular momentum are in relative equilibrium, but a non-
Cooper-Frye freezeout (For Cooper-Frye the distribution of $\phi_{r}$ would
need to be relatively homogeneous)

A maximally impure state Eq. 8 is a signature of a Cooper-Frye freeze-out,
where indeed spin and orbit are fully in equilibrium. Conversely, purity will
indicate that freeze-out dynamics is more complicated than that, as indeed
causality uscausality and the fact that hydrodynamics with spin cannot be
derived purely from conservation laws tinti seem to indicate.

The simplest hadronization model besides Cooper-Frye is coalescence. In
section III we used this model, together with considerations coming from
rotational symmetry, to understand how the coefficients derived in this
section relate to the spin and vorticity of the quark-gluon plasma. A generic
expectation is that $\theta_{r}$ and $\phi_{r}$ are indeed determined by the
interplay of primordial spin current and developed vorticity current, not
necessarily in equilibrium with each other.

The techniques developed in the second part of this work allow us to relate
the decay matrix elements of Eq. 1 to theoretical calculations where spin and
angular momentum are not in equilibrium, such as uscausality . Currently no
numerical implementation exists, although analytical calculations look quite
promising, as qualitatively they agree with global $\Lambda$ polarization
analspin2 . Weather lack of equilibrium between spin and vorticity can indeed
account for the apparent inconsistency between spin alignment and Fermion
polarization will be left to a forthcoming work, since it would require
understanding coalescence and decay dynamics in the co-moving frame as well as
a quantitative understanding of the flow profile throughout the fireball (to
transform all angles to the lab frame accurately).

Another obvious direction is to extend the current formalism to Baryons. Doing
so will determine if the current formalism can explain the experimental puzzle
that at LHC energies $\Lambda$ polarization is consistent with zero, while
spin alignment is not alice . A look at Equation 20 shows such an explanation
is indeed possible: The equivalent of Eq. 20 for baryons would be a yet more
complicated coalescence term on the right hand side (3 fermions and a
vorticity) and a yet more restrictive wavefunction on the left hand side (a
qubit instead of a qutrit). Hence, there are more ways for vorticity and quark
spin effects to cancel in a baryon rather than a spin 1 meson.

In conclusion, vector spin alignment is an interesting observable which has
not to date been fully exploited. A full harmonic analysis of the vector
meson’s decay coefficients Eq. 2 would have the potential to elucidate the
mechanism determining hadron spin at freeze-out and also to provide
experimental evidence of lack of equilibrium between spin and angular
momentum. We eagerly await experimental data in this direction.

GT thanks CNPQ bolsa de produtividade 306152/2020-7, bolsa FAPESP 2021/01700-2
and participation in tematic FAPESP, 2017/05685-2. K.J.G. is supported by
CAPES doctoral fellowship 88887.464061/2019-00

##  Appendix A Relation between Gell-Mann matrix elements and coefficients of
Eq. 1

The elements of the density matrix are parametrized according to Eq. 5 as well
as the angle $\theta_{r}$ in terms of Eq. 1 are

$n_{3}\left(\rho_{00},r_{10},\phi_{r}\right)=\frac{1}{8r_{10}^{2}}\sqrt{\frac{3}{2}}\alpha_{10}\csc\left(\phi_{r}\right)\;\lambda_{2}\left(\rho_{00},r_{10},\phi_{r}\right)\left((1-3\rho_{00})^{2}+2\;\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)\right.+$
(24)
$\left.+(1-3\rho_{00})^{2}\cos\left(2\phi_{r}\right)+12r_{10}^{2}\right)+\frac{\sqrt{3}}{24\rho_{00}-8}\;\left((1-3\rho_{00})^{2}-3\sec^{2}\left(\phi_{r}\right)\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)\right)$
$n_{8}\left(\rho_{00},r_{10},\phi_{r}\right)=\frac{3\alpha_{10}\csc\left(\phi_{r}\right)}{16r_{10}^{2}}\left((1-3\rho_{00})^{2}+2\;\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)+(1-3\rho_{00})^{2}\cos\left(2\phi_{r}\right)+\right.$
(25)
$\left.+12r_{10}^{2}\right)\;\lambda_{3}\left(\rho_{00},r_{10},\phi_{r}\right)+\frac{1}{8}\left(-3\rho_{00}+\frac{3\sec^{2}\left(\phi_{r}\right)\;\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)}{3\rho_{00}-1}+1\right)$
$\theta_{r}\left(\rho_{00},r_{10},\phi_{r}\right)=\tan^{-1}\left(\frac{\lambda_{4}\left(\rho_{00},r_{10},\phi_{r}\right)}{\lambda_{5}\left(\rho_{00},r_{10},\phi_{r}\right)}\right)$
(26)

where the functions $\lambda_{i}$ are

$\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)=\sqrt{(1-3\rho_{00})^{4}\cos^{4}\left(\phi_{r}\right)+4(1-3\rho_{00})^{2}r_{10}^{2}\cos^{2}\left(\phi_{r}\right)}$
(27)
$\lambda_{2}\left(\rho_{00},r_{10},\phi_{r}\right)=\sqrt{\frac{r_{10}^{2}}{(1-3\rho_{00})^{2}\cos^{2}\left(\phi_{r}\right)+\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)+6r_{10}^{2}}}$
(28)
$\lambda_{3}\left(\rho_{00},r_{10},\phi_{r}\right)=\sqrt{\frac{(1-3\rho_{00})^{2}-\sec^{2}\left(\phi_{r}\right)\left(\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)-6r_{10}^{2}\right)}{4(1-3\rho_{00})^{2}+18r_{10}^{2}\sec^{2}\left(\phi_{r}\right)}}$
(29)
$\lambda_{4}\left(\rho_{00},r_{10},\phi_{r}\right)=-\frac{\sec\left(\phi_{r}\right)}{(3\rho_{00}-1)r_{10}}\left((1-3\rho_{00})^{2}+2\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)+\right.$
(30)
$\left.+(1-3\rho_{00})^{2}\cos\left(2\phi_{r}\right)\right)\lambda_{3}\left(\rho_{00},r_{10},\phi_{r}\right)$
$\lambda_{5}\left(\rho_{00},r_{10},\phi_{r}\right)=-4\sqrt{\frac{(1-3\rho_{00})^{2}\cos^{2}\left(\phi_{r}\right)-\lambda_{1}\left(\rho_{00},r_{10},\phi_{r}\right)+6r_{10}^{2}}{2(1-3\rho_{00})^{2}\cos^{2}\left(\phi_{r}\right)+9r_{10}^{2}}}$
(31)

##  Appendix B Details of the density matrix calculations

From the density matrix for mesons Eq. 20 , the rotation matrix composition
explicitly form is

$U_{S}(\phi_{r},\theta_{r})U_{\omega}({\mu_{1},\nu_{1}})\rho^{1}(\Omega)\times
U^{-1}_{\omega}(\mu_{1},\nu_{1})U_{\omega}(\mu_{2},\nu_{2})\rho^{2}(\Omega)U^{-1}_{\omega}(\mu_{2},\nu_{2})U^{-1}_{S}(\phi_{r},\theta_{r})$
(32)

. Rotational symmetry allows us to approximate $\mu_{1}\simeq\mu_{2}\simeq\mu$
and $\nu_{1}\simeq\nu_{2}\simeq\nu$ and put the residual uncertainty in the
smearing width (it appears as $\sigma_{\nu}$ in Eq. 43 ). Thus Eq. 32 reduce
to

$=U_{S}(\phi_{r},\theta_{r})U_{\omega}({\mu,\nu})\rho^{1}(\Omega)U^{-1}_{\omega}(\mu,\nu)U_{\omega}(\mu,\nu)\rho^{2}(\Omega)U^{-1}_{\omega}(\mu,\nu)U^{-1}_{S}(\phi_{r},\theta_{r})$
$=U_{S}(\phi_{r},\theta_{r})U_{\omega}({\mu,\nu})\rho^{1}(\Omega)\rho^{2}(\Omega)U^{-1}_{\omega}(\mu,\nu)U^{-1}_{S}(\phi_{r},\theta_{r})$

we can now apply Eq. 11 and Eq. 12 to reduce these equations to matrix form,
and use Eq. 14 to expand

$=\sum_{m^{\prime\prime},m,\tilde{m}^{\prime\prime},\tilde{m}}\left<j,m^{\prime}\right|D_{S}(\phi_{r},\theta_{r})\left|j,\tilde{m}^{\prime\prime}\right>\left<j,\tilde{m}^{\prime\prime}\right|D_{\omega}\left(\mu,\nu\right)\left|j,m^{\prime\prime}\right>\left<j,m^{\prime\prime}\right|\rho^{1}(\Omega)\rho^{2}(\Omega)\left|j,m\right>\times$
$\times\left<j,m\right|D_{\omega}\left(\mu,\nu\right)^{-1}\left|j,\tilde{m}\right>\left<j,\tilde{m}\right|D^{-1}_{S}(\phi_{r},\theta_{r})\left|j,m^{\prime\prime\prime}\right>$
$=\sum_{m^{\prime\prime},m,\tilde{m}^{\prime\prime},\tilde{m}}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}e^{i(\tilde{m}-\tilde{m}^{\prime\prime})\mu}d^{j}_{m^{\prime}\tilde{m}^{\prime\prime}}(\theta_{r})d^{j}_{\tilde{m}^{\prime\prime}m^{{}^{\prime\prime}}}(\nu)\times$
$\times\left<j,m^{\prime\prime}\right|\rho^{1}(\Omega)\rho^{2}(\Omega)\left|j,m\right>\left[d^{j}_{m\tilde{m}}(\nu)\right]^{-1}\left[d^{j}_{\tilde{m}m^{\prime\prime\prime}}(\theta_{r})\right]^{-1}$

Making

$T_{m^{\prime},m^{\prime\prime\prime}}(\theta_{r},\nu,\Omega)=d^{j}_{m^{\prime}\tilde{m}^{\prime\prime}}(\theta_{r})d^{j}_{\tilde{m}^{\prime\prime}m^{{}^{\prime\prime}}}(\nu)\left<j,m^{\prime\prime}\right|\rho^{1}(\Omega)\rho^{2}(\Omega)\left|j,m\right>\left[d^{j}_{m\tilde{m}}(\nu)\right]^{-1}\left[d^{j}_{\tilde{m}m^{\prime\prime\prime}}(\theta_{r})\right]^{-1}$
(33)
$=\sum_{m^{\prime\prime},m,\tilde{m}^{\prime\prime},\tilde{m}}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}e^{i(\tilde{m}-\tilde{m}^{\prime\prime})\mu}T_{m^{\prime},m^{\prime\prime\prime}}(\theta_{r},\nu,\Omega)$

So, from the Clebsch Gordan coefficients 22 and making
$M^{\prime}_{S}=m^{{}^{\prime\prime}}$ and $M_{S}=m$, we can write the
following equation:

$\tilde{C}^{L}_{m^{{}^{\prime\prime}},m}=\sum_{m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{L},m^{\prime}_{12}}\;\sum_{m_{1},m_{2},m_{L},m_{12}}\;C^{S_{1}+S_{2},L}_{m^{\prime}_{12},m^{\prime}_{L},S,M^{\prime}_{S}}C^{S_{1},S_{2}}_{m^{\prime}_{1},m^{\prime}_{2},S_{1}+S_{2},m^{\prime}_{12}}\;C^{S_{1}+S_{2},L}_{m_{12},m_{L},S,M_{S}}C^{S_{1},S_{2}}_{m_{1},m_{2},S_{1}+S_{2},m_{12}}$
(34)

So

$\left(\hat{\rho}^{M}\right)_{m^{\prime},m^{\prime\prime\prime}}=\sum_{m^{\prime\prime},m,\tilde{m}^{\prime\prime},\tilde{m}}\;P^{2}_{L}(\omega)\;\tilde{C}^{L}_{m^{{}^{\prime\prime}},m}e^{i(m^{\prime\prime\prime}-m^{\prime})\phi_{r}}e^{i(\tilde{m}-\tilde{m}^{\prime\prime})\mu}T_{m^{\prime},m^{\prime\prime\prime}}(\theta_{r},\nu,\Omega)$
(35)

putting all components of $\rho_{M}$ together Eq. 23 follows

##  Appendix C Comparison of our calculations with pre-existing models( xia )

From equation 44 for the case where $L=0$ and $\sigma_{\nu}=0$, we have, after
some algebra

$\langle\rho^{L=0}_{00}(\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{2}\left(1+\cosh\left(\frac{\Omega_{3}}{T}\right)-\left(\cosh\left(\frac{\Omega_{3}}{T}\right)-1\right)\cos\left(2\theta_{r}\right)\right)$
(36)

using
$\tilde{P}^{2}_{L}\left(\omega\right)=\frac{1}{1+2\cosh\left(\frac{\Omega_{3}}{T}\right)}$,
this simplifies to

$\langle\rho^{L=0}_{00}(\theta_{r},\Omega_{3},T)\rangle=\frac{1+\cosh\left(\frac{\Omega_{3}}{T}\right)}{2\left(1+2\cosh\left(\frac{\Omega_{3}}{T}\right)\right)}\left[1-\left(\frac{\cosh\left(\frac{\Omega_{3}}{T}\right)-1}{\cosh\left(\frac{\Omega_{3}}{T}\right)+1}\right)\cos\left(2\theta_{r}\right)\right]$

Then one can make a comparison with the $\rho_{0,0}$ coefficient as calculated
in xia (equation 28-29)

$\rho_{0,0}\left(\Delta\Psi\right)=\frac{1-F^{2}_{\perp}\cos\left(2\Delta\Psi\right)}{3+F^{2}_{\perp}}$
(37)

Since the variables $\Omega_{3}$, $T$ are linked with xia ’s $F_{\perp}$ via

$F^{2}_{\perp}=\tanh^{2}\left(\frac{\Omega_{3}}{2T}\right)$ (38)

and it is possible to identify $\theta_{r}$with the angle used as an azimuthal
correlation in xia , $\theta_{r}\rightarrow\Delta\Psi$ For the case $r_{1,-1}$

$-2\langle
r^{L=0}_{1,-1}(\theta_{r},\Omega_{3},T)\rangle=\left(\frac{-2\cos\left(2\phi_{r}\right)\sinh^{2}\left(\frac{\Omega_{3}}{2T}\right)}{1+2\cosh\left(\frac{\Omega_{3}}{T}\right)}\right)\sin^{2}\left(\theta_{r}\right)$
(39)
$\sqrt{2}\left(\text{Im}\rho_{1,0}-\text{Im}\rho_{0,-1}\right)=\left(\frac{-2\sin\left(\phi_{r}\right)\sinh^{2}\left(\frac{\Omega_{3}}{T}\right)}{1+2\cosh\left(\frac{\Omega_{3}}{T}\right)}\right)\sin\left(2\theta_{r}\right)$
(40)

putting everything together

$-2r_{1,-1}=\frac{2F^{2}_{\perp}}{3+F^{2}_{\perp}}\sin^{2}\left(\Delta\Psi\right)$
(41)
$\sqrt{2}\left(\text{Im}\rho_{1,0}-\text{Im}\rho_{0,-1}\right)=\frac{2F^{2}_{\perp}}{3+F^{2}_{\perp}}\sin\left(2\Delta\Psi\right)$
(42)

Then, from equations 39 and 40 and $\phi_{r}=-\frac{\pi}{2}$ we can
reproducible the value of paper xia as is shown in Fig. 5 .

Figure 5:  Comparison of our model with the one in xia in the appropriate
limit

For the case we have $\phi_{r}=-\pi/2$ the coefficient $\rho_{0,0}$ is the
same as shown in figure 4 and other coefficients $r_{10}$ and $\alpha_{1,-1}$
are equal to zero.

##  Appendix D Explicit form of the distribution coefficients in a coalescence
model

Here we give form for the coefficients in the coalescence scenario. As
explained in the main text, we can not determine the angular momentum quantum
number $L$ transferred from the macroscopic vortex to the vector meson. In the
main text it is paramterized by $P_{L}$.

The point is to write in closed form Eq. 23 , and integrate over the
coalescence angle $\nu$ weighted by a Wigner function $P_{W}(\nu)$. $\nu$ is a
parameter of the coalescence model, representing the angle of the quarks
w.r.t. the meson. Assuming the Wigner function in azimuthal space is a
Gaussian of width (depending on the internal dynamics) $\sigma_{\nu}\ll 2\pi$

${\color[rgb]{0,0,0}P_{W}(\nu)=\frac{1}{\sqrt{2\pi\sigma^{2}_{\nu}}}\exp\left[-\frac{\nu^{2}}{2\sigma_{\nu}^{2}}\right]}\phantom{AA},\phantom{AA}\int_{0}^{2\pi}P_{W}(\nu)f(\nu)d\nu\simeq\int_{-\infty}^{\infty}P_{W}(\nu)f(\nu)d\nu$
(43)

Averaging over a Gaussian Wigner function of coalesce angles, Eq. 43 gives
these coefficients:

The coefficient

$\rho_{00}$

$\langle\rho^{L=0}_{00}(\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{4}e^{-2\sigma_{\nu}^{2}-\frac{\Omega_{3}}{T}}\left(e^{2\sigma_{\nu}^{2}}\left(e^{\frac{\Omega_{3}}{T}}+1\right){}^{2}-\left(e^{\frac{\Omega_{3}}{T}}-1\right){}^{2}\cos\left(2\left(\theta_{r}\right)\right)\right)$
(44)
$\langle\rho^{L=1}_{00}(\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{8}e^{-2\sigma_{\nu}^{2}-\frac{\Omega_{3}}{T}}\left(2\sqrt{2}\left(e^{\frac{2\Omega_{3}}{T}}-1\right)\sin\left(2\left(\theta_{r}\right)\right)+\right.$
(45)
$\left.\left(-4e^{\frac{\Omega_{3}}{T}}+e^{\frac{2\Omega_{3}}{T}}+1\right)\cos\left(2\left(\theta_{r}\right)\right)+e^{2\sigma_{\nu}^{2}}\left(4e^{\frac{\Omega_{3}}{T}}+3e^{\frac{2\Omega_{3}}{T}}+3\right)\right)$
$\langle\rho^{L=2}_{00}(\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{40}e^{-2\sigma_{\nu}^{2}-\frac{\Omega_{3}}{T}}\left(2\left(\sqrt{6}-6\right)\left(e^{\frac{2\Omega_{3}}{T}}-1\right)\sin\left(2\left(\theta_{r}\right)\right)+\right.$
(46)
$\left.+2e^{\frac{\Omega_{3}}{T}}\left(\left(2\sqrt{6}-1\right)\cosh\left(\frac{\Omega_{3}}{T}\right)+4\right)\cos\left(2\left(\theta_{r}\right)\right)-\left(2\sqrt{6}-13\right)e^{2\sigma_{\nu}^{2}}\left(e^{\frac{2\Omega_{3}}{T}}+1\right)+8e^{2\sigma_{\nu}^{2}+\frac{\Omega_{3}}{T}}\right)$

The coefficient

$r_{10}$

$\langle
r^{L=0}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\tilde{P}^{2}_{L}\left(\omega\right)\sqrt{2}e^{-2\sigma_{\nu}^{2}}\cos\left(\phi_{r}\right)\sinh^{2}\left(\frac{\Omega_{3}}{2T}\right)\sin\left(2\left(\theta_{r}\right)\right)$
(47) $\langle
r^{L=1}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{4}e^{-2\sigma_{\nu}^{2}}\cos\left(\phi_{r}\right)\left(\sqrt{2}\left(\cosh\left(\frac{\Omega_{3}}{T}\right)-2\right)\sin\left(2\left(\theta_{r}\right)\right)\right.$
(48)
$\left.-4\sinh\left(\frac{\Omega_{3}}{T}\right)\cos\left(2\left(\theta_{r}\right)\right)\right)$
$\langle
r^{L=2}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{20}e^{-2\sigma_{\nu}^{2}}\cos\left(\phi_{r}\right)\left(4\sqrt{21-6\sqrt{6}}\sinh\left(\frac{\Omega_{3}}{T}\right)\cos\left(2\left(\theta_{r}\right)\right)+\right.$
(49)
$\left.+\sqrt{2}\left(\left(2\sqrt{6}-1\right)\cosh\left(\frac{\Omega_{3}}{T}\right)+4\right)\sin\left(2\left(\theta_{r}\right)\right)\right)$

The coefficient

$\alpha_{10}$

$\langle\alpha^{L=0}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\frac{\tilde{P}^{2}_{L}\left(\omega\right)\sqrt{2}\sin\left(\phi_{r}\right)\sinh\left(\frac{\Omega_{3}}{T}\right)\sin\left(\theta_{r}\right)}{\sqrt{e^{\sigma_{\nu}^{2}}}}$
(50)
$\langle\alpha^{L=1}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\frac{\tilde{P}^{2}_{L}\left(\omega\right)\sin\left(\phi_{r}\right)\left(\sqrt{2}\sinh\left(\frac{\Omega_{3}}{T}\right)\sin\left(\theta_{r}\right)+2\cosh\left(\frac{\Omega_{3}}{T}\right)\cos\left(\theta_{r}\right)\right)}{2\sqrt{e^{\sigma_{\nu}^{2}}}}$
(51)
$\langle\alpha^{L=2}_{10}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\frac{\tilde{P}^{2}_{L}\left(\omega\right)\sin\left(\phi_{r}\right)}{10\sqrt{e^{\sigma_{\nu}^{2}}}}\left(2\left(\left(3\sqrt{2}+\sqrt{3}\right)\cosh\left(\frac{\Omega_{3}}{T}\right)+2\sqrt{3}\right)\cos\left(\theta_{r}\right)\right.$
(52)
$\left.-5\sqrt{2}\sinh\left(\frac{\Omega_{3}}{T}\right)\sin\left(\theta_{r}\right)\right)$

The coefficient

$r_{1,-1}$

$\langle
r^{L=0}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{2}e^{-2\sigma_{\nu}^{2}}\cos\left(2\phi_{r}\right)\sinh^{2}\left(\frac{\Omega_{3}}{2T}\right)\left(e^{2\sigma_{\nu}^{2}}-\cos\left(2\left(\theta_{r}\right)\right)\right)$
(53) $\langle
r^{L=1}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{16}\cos\left(2\phi_{r}\right)e^{-2\sigma_{\nu}^{2}-\frac{\Omega_{3}}{T}}\left(-2\sqrt{2}\left(e^{\frac{2\Omega_{3}}{T}}-1\right)\sin\left(2\left(\theta_{r}\right)\right)\right.$
(54)
$\left.-\left(-4e^{\frac{\Omega_{3}}{T}}+e^{\frac{2\Omega_{3}}{T}}+1\right)\cos\left(2\left(\theta_{r}\right)\right)+e^{2\sigma_{\nu}^{2}}\left(4e^{\frac{\Omega_{3}}{T}}+e^{\frac{2\Omega_{3}}{T}}+1\right)\right)$
$\langle
r^{L=2}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{40}e^{-2\sigma_{\nu}^{2}}\cos\left(2\phi_{r}\right)\left(4\left(\cos\left(2\left(\theta_{r}\right)\right)+2e^{2\sigma_{\nu}^{2}}\right)+\right.$
(55)
$\left.+\cosh\left(\frac{\Omega_{3}}{T}\right)\left(\left(2\sqrt{6}-1\right)\cos\left(2\left(\theta_{r}\right)\right)+\left(1+6\sqrt{6}\right)e^{2\sigma_{\nu}^{2}}\right)+2\left(\sqrt{6}-6\right)\sinh\left(\frac{\Omega_{3}}{T}\right)\sin\left(2\left(\theta_{r}\right)\right)\right)$

The coefficient

$\alpha_{1,-1}$

$\langle\alpha^{L=0}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{2}e^{-2\sigma_{\nu}^{2}}\sin\left(2\phi_{r}\right)\sinh^{2}\left(\frac{\Omega_{3}}{2T}\right)\left(\cos\left(2\left(\theta_{r}\right)\right)-e^{2\sigma_{\nu}^{2}}\right)$
(56)
$\langle\alpha^{L=1}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{16}\sin\left(2\phi_{r}\right)e^{-2\sigma_{\nu}^{2}-\frac{\Omega_{3}}{T}}\left(-2\sqrt{2}\left(e^{\frac{2\Omega_{3}}{T}}-1\right)\sin\left(2\left(\theta_{r}\right)\right)\right.$
(57)
$\left.-\left(-4e^{\frac{\Omega_{3}}{T}}+e^{\frac{2\Omega_{3}}{T}}+1\right)\cos\left(2\left(\theta_{r}\right)\right)+e^{2\sigma_{\nu}^{2}}\left(4e^{\frac{\Omega_{3}}{T}}+e^{\frac{2\Omega_{3}}{T}}+1\right)\right)$
$\langle\alpha^{L=2}_{1,-1}(\phi_{r},\theta_{r},\Omega_{3},T)\rangle=-\frac{\tilde{P}^{2}_{L}\left(\omega\right)}{40}e^{-2\sigma_{\nu}^{2}}\sin\left(2\phi_{r}\right)\left(4\left(\cos\left(2\left(\theta_{r}\right)\right)+2e^{2\sigma_{\nu}^{2}}\right)+\right.$
(58)
$\left.+\cosh\left(\frac{\Omega_{3}}{T}\right)\left(\left(2\sqrt{6}-1\right)\cos\left(2\left(\theta_{r}\right)\right)+\left(1+6\sqrt{6}\right)e^{2\sigma_{\nu}^{2}}\right)+2\left(\sqrt{6}-6\right)\sinh\left(\frac{\Omega_{3}}{T}\right)\sin\left(2\left(\theta_{r}\right)\right)\right)$

The figures in section III are based on these formulae. Note that, as a
consistency check, Eq. 18 is recovered from the last two equations for all
studied case of $L$.

## References

  * (1) F. Becattini and M. A. Lisa, Ann. Rev. Nucl. Part. Sci. 70 (2020), 395-423 doi:10.1146/annurev-nucl-021920-095245 [arXiv:2003.03640 [nucl-ex]].
  * (2) R. Derradi de Souza, T. Koide and T. Kodama, Prog. Part. Nucl. Phys. 86 (2016), 35-85 doi:10.1016/j.ppnp.2015.09.002 [arXiv:1506.03863 [nucl-th]].
  * (3) D. Montenegro and G. Torrieri, Phys. Rev. D 100, no.5, 056011 (2019) doi:10.1103/PhysRevD.100.056011 [arXiv:1807.02796 [hep-th]].
  * (4) D. Montenegro and G. Torrieri, [arXiv:2004.10195 [hep-th]].
  * (5) S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar and R. Ryblewski, Phys. Lett. B 814 (2021), 136096 doi:10.1016/j.physletb.2021.136096 [arXiv:2002.03937 [hep-ph]].
  * (6) S. Shi, C. Gale and S. Jeon, [arXiv:2008.08618 [nucl-th]].
  * (7) E. Speranza, F. S. Bemfica, M. M. Disconzi and J. Noronha, [arXiv:2104.02110 [hep-th]].
  * (8) W. Florkowski, A. Kumar, R. Ryblewski and R. Singh, Phys. Rev. C 99 (2019) no.4, 044910 doi:10.1103/PhysRevC.99.044910 [arXiv:1901.09655 [hep-ph]].
  * (9) R. Singh, G. Sophys and R. Ryblewski, Phys. Rev. D 103 (2021) no.7, 074024 doi:10.1103/PhysRevD.103.074024 [arXiv:2011.14907 [hep-ph]].
  * (10) A. D. Gallegos, U. Gürsoy and A. Yarom, [arXiv:2101.04759 [hep-th]], [arXiv:2203.05044 [hep-th]]
  * (11) R.Takahashi et al, Nature Physics 12, 52–56(2016)
  * (12) Valet, T. , Fert, A. Phys. Rev. B 48, 7099–7113 (1993).
  * (13) V. Greco, C. M. Ko and P. Levai, Phys. Rev. C 68 (2003), 034904 doi:10.1103/PhysRevC.68.034904 [arXiv:nucl-th/0305024 [nucl-th]].
  * (14) R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. C 68 (2003), 044902 doi:10.1103/PhysRevC.68.044902 [arXiv:nucl-th/0306027 [nucl-th]].
  * (15) R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. C 68 (2003), 044902 doi:10.1103/PhysRevC.68.044902 [arXiv:nucl-th/0306027 [nucl-th]].
  * (16) S. Acharya et al. [ALICE], [arXiv:1910.14408 [nucl-ex]].
  * (17) S. Singha [STAR], Nucl. Phys. A 1005 (2021), 121733 doi:10.1016/j.nuclphysa.2020.121733 [arXiv:2002.07427 [nucl-ex]].
  * (18) K. Schilling, P. Seyboth and G. E. Wolf, Nucl. Phys. B 15, 397-412 (1970) doi:10.1016/0550-3213(70)90070-2
  * (19) Z. T. Liang and X. N. Wang, Phys. Lett. B 629, 20-26 (2005) doi:10.1016/j.physletb.2005.09.060 [arXiv:nucl-th/0411101 [nucl-th]].
  * (20) J. Adam et al. [STAR], Phys. Rev. D 102 (2020) no.9, 092009 doi:10.1103/PhysRevD.102.092009 [arXiv:2007.04732 [hep-ex]].
  * (21) X. L. Sheng, L. Oliva and Q. Wang, [arXiv:1910.13684 [nucl-th]].
  * (22) X. L. Sheng, Q. Wang and X. N. Wang, Phys. Rev. D 102 (2020) no.5, 056013 doi:10.1103/PhysRevD.102.056013 [arXiv:2007.05106 [nucl-th]].
  * (23) X. L. Xia, H. Li, X. G. Huang and H. Z. Huang, [arXiv:2010.01474 [nucl-th]].
  * (24) I. Contreras, E. Ercolessi and M. Schiavina, J. Math. Phys. 57, no.6, 062209 (2016) doi:10.1063/1.4954328 [arXiv:1501.00054 [math-ph]].
  * (25) J.Preskill, “Lecture Notes on quantum information“ http://theory.caltech.edu/~preskill/ph229/
  * (26) A. T. Bolukbasi, T. Dereli, Journal of Physics: Conference Series, Volume 36, 28 [arXiv:quant-ph/0511111].
  * (27) GOYAL, Sandeep K; SIMON, B Neethi; SINGH, Rajeev; SIMON, Sudhavathani. qutrits. Journal Of Physics A: Mathematical and Theoretical, [s.l.], v. 49, n. 16, p. 1-31, 8 mar. 2016. IOP Publishing. http://dx.doi.org/10.1088/1751-8113/49/16/165203. [arXiv:1111.4427 [quant-ph]].
  * (28) M.Byrd, J. Math. Phys 10.1063/1.532618 [arXiv:math-ph/9807032 [path-ph]]
  * (29) A. H. Tang, B. Tu and C. S. Zhou, Phys. Rev. C 98 (2018) no.4, 044907 doi:10.1103/PhysRevC.98.044907 [arXiv:1803.05777 [nucl-ex]].
  * (30) A. Ayala, M. A. A. Torres, E. Cuautle, I. Dominguez, M. A. F. Sanchez, I. Maldonado, L. M. Montano Zetina, E. Moreno Barbosa, P. A. N. Marin, L. Rebolledo, M. Rodriguez-Cahuantzi, J. Salinas, M. E. Tejeda-Yeomans, L. Valenzuela and C. H. Zepeda Fernandez, [arXiv:2003.13757 [hep-ph]].
  * (31) F. Becattini, V. Chandra, L. Del Zanna and E. Grossi, Annals Phys. 338 (2013), 32-49 doi:10.1016/j.aop.2013.07.004 [arXiv:1303.3431 [nucl-th]].
  * (32) F. Becattini and L. Tinti, Annals Phys. 325, 1566-1594 (2010) doi:10.1016/j.aop.2010.03.007 [arXiv:0911.0864 [gr-qc]].
  * (33) S.Pratt, Quantum Mechanics lectures, 10.2
https://people.nscl.msu.edu/~pratt/phy851/lectures/lectures_full.pdf

  * (34) Quantum Mechanics Messiah, A., v. 2, Dover books on physics, https://books.google.com.br/books?id=mwssSDXzkNcC, 1999, Dover Publications
  * (35) M. A. Lisa, J. G. P. Barbon, D. D. Chinellato, W. M. Serenone, C. Shen, J. Takahashi and G. Torrieri, Phys. Rev. C 104 (2021) no.1, 011901 doi:10.1103/PhysRevC.104.L011901 [arXiv:2101.10872 [hep-ph]].