# Photoproduction of lepton pair in ultra-relativistic heavy-ion collisions

Kewei Yu  College of Science, China Three Gorges University, Yichang 443002,
China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
Jiazhen Peng  College of Science, China Three Gorges University, Yichang
443002, China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
Shuang Li<EMAIL_ADDRESS>College of Science, China Three Gorges University,
Yichang 443002, China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
Kejun Wu<EMAIL_ADDRESS>College of Science, China Three Gorges University,
Yichang 443002, China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
Wei Xie  College of Science, China Three Gorges University, Yichang 443002,
China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China
Fei Sun  College of Science, China Three Gorges University, Yichang 443002,
China
Center for Astronomy and Space Sciences, China Three Gorges University,
Yichang 443002, China

###### Abstract

Dilepton production provides a unique probe of the strong electromagnetic
field produced in heavy-ion collisions. To map out the behavior of its
transverse momentum broadening, we present a theoretical model based on the
equivalent photon approximation, and then we update it to make direct
comparisons with the recent experimental measurements. We find that the model
calculations can describe well, not only the average transverse momentum
squared of $e^{+}e^{-}$ pairs in Au–Au collisions at $\sqrt{s_{\rm NN}}=200$
GeV, but also the acoplanarity of $\mu^{+}\mu^{-}$ pairs in Pb–Pb collisions
at$\sqrt{s_{\rm NN}}=5.02$ TeV. Furthermore, the model predictions are also
able to reproduce the measured dependencies of the pair mass and the
transverse momentum squared.

##  I INTRODUCTION

Relativistic heavy-ion collisions provide a unique opportunity to investigate
the properties of strongly-interacting nuclear matter in extreme conditions of
temperature and energy density where the Quark-Gluon Plasma (QGP) is formed
Shuryak (1980); Gyulassy and McLerran (2005); Shuryak (2005). Among the
various probes of the QGP, dileptons are of particular interest since they are
produced during the whole evolution of QGP while not being subject to strong
interactions with it G. Agakichiev et al. (1995); D. Adamová et al. (2003); R.
Arnaldi et al. (2006); L. Adamczyk et al. (2014, 2015); A. Adare et al.
(2016).

In 1934 Breit and Wheeler first proposed to study the creation of electron-
positron pair via the fusion of two real photons,
$\gamma+\gamma\rightarrow{e^{+}+e^{-}}$, which was difficult to be observed in
laboratory experiments at that period Breit and Wheeler (1934). They
calculated the relevant total cross section, including photon polarization,
and these results were further extended and generalized in Ref. J. S. Toll
(1952). The above process is then realized as the Breid-Wheeler process for
the photoproduction of dilepton pairs. Indeed in 1924 Fermi described the
electromagnetic particle production in terms of an equivalent photon method
Fermi (1924), which was later improved by Williams Williams (1934) and
Weizs$\rm\ddot{a}$cker C. F. v. Weizs$\rm\ddot{a}$cker (1934). Within this
method, the electromagnetic fields of a moving nucleus can be described as a
swarm of quasi-real photons, and the electromagnetic production cross section
in nuclear collisions $\sigma^{WW}_{A_{1}A_{2}\rightarrow
A_{1}A_{2}l^{+}l^{-}}$, can be approximated by the elementary two-photon
fusion cross section from the Breid-Wheeler process
$\sigma_{\gamma\gamma\rightarrow l^{+}l^{-}}$, folded with the equivalent
photon distributions $n(\omega)$,

$\displaystyle\sigma^{WW}_{A_{1}A_{2}\rightarrow A_{1}A_{2}l^{+}l^{-}}=\int
d\omega_{1}d\omega_{2}~{}n_{1}n_{2}\sigma_{\gamma\gamma\rightarrow
l^{+}l^{-}}.$ (1)

This is known as the equivalent photon approximation (EPA Budnev  _et al._
(1975a); Bertulani and Baur (1988); Vidovic  _et al._ (1993);
G$\rm\ddot{u}$cl$\rm\ddot{u}$  _et al._ (1999); Baur  _et al._ (2002);
Bertulani  _et al._ (2005); Baur  _et al._ (2007); Klusek-Gawenda  _et al._
(2019); C. Li, J. Zhou and Y. J. Zhou (2019); Zha  _et al._ (2020); L. A.
Harland-Lang, V. A. Khoze and M. G. Ryskin (2019); Klusek-Gawenda  _et al._
(2021); Brandenburg  _et al._ (2023); Lin  _et al._ (2023); Wang  _et al._
(2023)), and it has been widely employed to interpret the physical phenomena
in ultra-relativistic heavy-ion collisions J. Adam et al. (2004, 2018, 2021);
A. M. Sirunyan et al. (2021); Acharya  _et al._ (2023); M. Aabound et al.
(2018); G. And et al. (2023).

As discussed in Ref. Brandenburg  _et al._ (2023), there are now various
theoretical approaches for EPA, such as the classical field approximation R.
J. Wang, S. Pu and Q. Wang (2021), $2^{nd}$ order QED Zha  _et al._ (2020) and
the higher correction Zha and Tang (2021); Li  _et al._ (2023). See Ref.
Brandenburg  _et al._ (2023) for details (see references therein). Among the
various approaches, the traditional one (see Eq. LABEL:eq:TradEPA) is widely
used due to its simplicity Vidovic  _et al._ (1993); Klein  _et al._ (2017);
Klein (2018). It allows to provide a fairly direct strategy for the
calculations of the Breid-Wheeler process in nuclear collisions. However, as
pointed out in Ref. Brandenburg  _et al._ (2023), the traditional approach has
the difficulty in computing the kinematic distributions of single lepton,
which are challenged by the recent measurements performed at RHIC and LHC
energies. In this work, we try to cover this issue by utilizing a Monte-Carlo-
based strategy. The latest experimental data, such as the transverse momentum
broadening and the invariant mass spectrum, will be used to examine the
relevant theoretical calculations.

The paper is organized as follows: Sec. II is dedicated to the description of
the theoretical framework, including the calculation of the electromagnetic
field, the equivalent photon spectra and the Monte-Carlo-based setup. Sec. III
shows the results about transverse momentum broadening and the invariant mass
spectrum. A summary section can be finally found. Notation: we use natural
units $c=1$ and Greek indices run from 0 to 3 while Latin indices run from 1
to 3.

##  II Theoretical method and configuration

###  II.1 The equivalent photon approximation

To calculate the energy flux of the classical equivalent photons, we first
determine the electromagnetic field of a charge distribution moving at high
velocity ($v\sim c$), and then calculate the single photon distributions. Here
we just show the final results, and the detailed aspects are relegated to
Appendix-A.

In the observer’s frame, we assume a nucleus moves with constant velocity $v$
on straight lines along $z$-axis (i.e. longitudinal direction), and being
located by the displacement $\vec{b}$ in $xy$-plane (i.e. transverse plane).
The relevant potentials $A(x)=(t,\vec{x}_{\perp}-\vec{b},z)$ of the
electromagnetic waves are determined by d’Alembert’s equation

$\displaystyle\partial_{\mu}\partial^{\mu}A^{\nu}(x)=J^{\nu}(x)$ (2)

in the Lorentz gauge

$\displaystyle\partial_{\nu}A^{\nu}(x)=0.$ (3)

We perform the Fourier transformation of Eq. LABEL:eq:dAlbertEq, yielding (Eq.
A4)

$\displaystyle A^{\nu}(k)=-\frac{1}{k^{2}}J^{\nu}(k)=2\pi
Ze~{}\delta(ku)\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}u^{\nu}e^{i\vec{k}\cdot\vec{b}},$
(4)

with $u=\gamma(1,\vec{v})=\gamma(1,\vec{0}_{\perp},v)$ and
$k=(\omega,\vec{k})=(\omega,\vec{k}_{\perp},\omega/v)$. $Z$ is the charge
number of the nucleus, and $\mathcal{F}(\sqrt{{-k^{2}}})$ is the corresponding
form factor (Eq. LABEL:eq:A7 and A5)

$\mathcal{F}(\sqrt{{-k^{2}}})=\rho_{m}(\sqrt{{-k^{2}}}),$ (5)

where, $-k^{2}\doteq\vec{k}^{\;2}_{\perp}+(\omega/\gamma)^{2}$ in the ultra-
relativistic limit $v\sim c=1$. The nuclear matter density function reads

$\rho_{m}(r)=\rho_{m0}\frac{1+\Omega\cdot(r/R)^{2}}{1+\exp[(r-R)/a]},$ (6)

where, $\rho_{m0}$ is set to satisfy the normalization $\int
d^{3}\vec{r}\rho_{m}(|\vec{r}|)=1$, $r=|\vec{r}|$ is the distance concerning
the nucleus center, $R$ is the nucleus radius, $a$ is the skin depth, and
$\Omega$ corresponds to deviations from a spherical shape for a given nucleus.
For a Au nucleus the Woods-Saxon parameters in Eq. (6) are $R=6.380$ fm,
$a=0.535$ fm and $\Omega=0$, while for a Pb nucleus, $R=6.620$ fm, $a=0.546$
fm and $\Omega=0$ Loizides  _et al._ (2018). The form factor in Eq. 5 can be
rewritten as Magnus and Oberhettinger (1949)

$\displaystyle\mathcal{F}(q^{2})=$
$\displaystyle\frac{4\pi^{2}\rho_{m0}a^{3}}{(qa)^{2}~{}sinh^{2}(\pi qa)}$ (7)
$\displaystyle{\bigr{[}}\pi qa~{}cosh(\pi
qa)~{}sin(qR)-qR~{}cos(qR)~{}sinh(\pi qa){\bigr{]}}$
$\displaystyle+8\pi\rho_{m0}a^{3}\Sigma^{\infty}_{n=1}(-1)^{n-1}\frac{ne^{-nR/a}}{[n^{2}+(qa)^{2}]^{2}},$

in which the term on the third line is expected to be much smaller than the
others. We have checked its validity and therefore neglect it in this
analysis. A similar approximation is widely employed in the literatures Şengül
_et al._ (2016).

Here we treat the electromagnetic field of the two nuclei classically by
assuming both nuclei move with constant velocity on straight lines being
separated by the impart parameter $\vec{b}$. It is argued that the
differential production for $\gamma\gamma\rightarrow l^{+}l^{-}$, may be cast
in a form being apparently closer to the equivalent photon approximation by
folding the elementary cross section $\sigma_{s}(\omega_{1},\omega_{2})$ and
$\sigma_{ps}(\omega_{1},\omega_{2})$ with the two-photon distribution
$n_{s}(\omega_{1},\omega_{2},\vec{b})$ and
$n_{ps}(\omega_{1},\omega_{2},\vec{b})$

$\displaystyle\frac{d^{4}\sigma}{d\omega_{1}d\omega_{2}d^{2}\vec{b}}=$
$\displaystyle~{}n_{s}(\omega_{1},\omega_{2},\vec{b})\sigma_{s}(\omega_{1},\omega_{2})$
(8)
$\displaystyle+~{}n_{ps}(\omega_{1},\omega_{2},\vec{b})\sigma_{ps}(\omega_{1},\omega_{2}).$

It is realized Vidovic  _et al._ (1993) that the differential production cross
section consists of a scalar part $n_{s}$, where the electromagnetic fields
have to be parallel $\vec{E}_{1}\parallel\vec{E}_{2}$, and a pesudo-scalar
part $n_{ps}$, where the electromagnetic fields have to be perpendicular
$\vec{E}_{1}\perp\vec{E}_{2}$.

$\displaystyle n_{s}(\omega_{1},\omega_{2},\vec{b})$ (9)
$\displaystyle=\frac{1}{\pi^{2}\omega_{1}\omega_{2}}\int
d^{2}\vec{x}_{\perp}{\biggr{[}}\vec{E}_{1\perp}(\omega_{1},\vec{x}_{\perp}-\vec{b},z)\cdot\vec{E}_{2\perp}(\omega_{2},\vec{x}_{\perp},z){\biggr{]}}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:C3})}}{{=}}\int
d^{2}\vec{x}_{\perp}n(\omega_{1},\vec{x}_{\perp}-\vec{b})~{}n(\omega_{2},\vec{x}_{\perp}){\biggr{[}}\frac{(\vec{x}_{\perp}-\vec{b})\cdot\vec{x}_{\perp}}{|\vec{x}_{\perp}-\vec{b}|\cdot|\vec{x}_{\perp}|}{\biggr{]}}^{2}$

in the ultra-relativistic limit $v\sim c=1$.

$\displaystyle n_{ps}(\omega_{1},\omega_{2},\vec{b})$ (10)
$\displaystyle=\frac{1}{\pi^{2}\omega_{1}\omega_{2}}\int
d^{2}\vec{x}_{\perp}{\biggr{[}}\vec{E}_{1\perp}(\omega_{1},\vec{x}_{\perp}-\vec{b},z)\times\vec{E}_{2\perp}(\omega_{2},\vec{x}_{\perp},z){\biggr{]}}^{2}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:C3})}}{{=}}\int
d^{2}\vec{x}_{\perp}n(\omega_{1},\vec{x}_{\perp}-\vec{b})~{}n(\omega_{2},\vec{x}_{\perp}){\biggr{[}}\frac{(\vec{x}_{\perp}-\vec{b})\times\vec{x}_{\perp}}{|\vec{x}_{\perp}-\vec{b}|\cdot|\vec{x}_{\perp}|}{\biggr{]}}^{2}$

$\displaystyle
n(\omega,\vec{x}_{\perp})=\frac{Z^{2}\alpha_{EM}}{\pi^{2}\omega}{\biggr{|}}\int^{\infty}_{0}dk_{\perp}k_{\perp}^{2}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}J_{1}(x_{\perp}k_{\perp}){\biggr{|}}^{2},$
(11)

where, $\alpha_{EM}=e^{2}/(4\pi)\approx 1/137$ is the coupling constant;
$J_{1}$ is the Bessel function of the first kind (Eq. LABEL:eq:C5).

The elementary two photon fusion cross section in Eq. 8 are expressed as
Budnev  _et al._ (1975b)

$\displaystyle\sigma^{\gamma\gamma\rightarrow l^{+}l^{-}}_{s}$
$\displaystyle=\frac{4\pi\alpha^{2}_{EM}}{s}{\biggr{[}}(1+\frac{4m^{2}_{l}}{s}-\frac{12m^{4}_{l}}{s^{2}})2ln(\frac{\sqrt{s}}{2m_{l}}+\sqrt{\frac{s}{4m^{2}_{l}}-1})-(1+\frac{6m^{2}_{l}}{s})\sqrt{1-\frac{4m^{2}_{l}}{s}}){\biggr{]}}\Theta(s-4m^{2}_{l})$
(12)

$\displaystyle\sigma^{\gamma\gamma\rightarrow l^{+}l^{-}}_{ps}$
$\displaystyle=\frac{4\pi\alpha^{2}_{EM}}{s}{\biggr{[}}(1+\frac{4m^{2}_{l}}{s}-\frac{4m^{4}_{l}}{s^{2}})2ln(\frac{\sqrt{s}}{2m_{l}}+\sqrt{\frac{s}{4m^{2}_{l}}-1})-(1+\frac{2m^{2}_{l}}{s})\sqrt{1-\frac{4m^{2}_{l}}{s}}){\biggr{]}}\Theta(s-4m^{2}_{l})$
(13)

at the leading order in $\alpha_{EM}$, where, $s=4\omega_{1}\omega_{2}$
indicates the center-of-mass squared-energy and $m_{l}$ denotes the mass of
single lepton $l$. The step function $\Theta(s-4m^{2}_{l})$ guarantees the
center-of-mass energy of the two photons is no smaller than twice the lepton
mass.

The angular profile of the produced lepton pairs reads

$\displaystyle
G(\theta)=2+4\biggr{(}1-\frac{4m^{2}}{W^{2}}\biggr{)}\frac{(1-\frac{4m^{2}}{W^{2}})sin^{2}\theta\;cos^{2}\theta+\frac{4m^{2}}{W^{2}}}{\bigr{[}1-(1-\frac{4m^{2}}{W^{2}})cos^{2}\theta\bigr{]}^{2}}$
(14)

where, $\theta$ is the angle between the beam direction and one of the
leoptons in the local rest frame of the lepton pair. The result is adopted
from Ref. Brodsky  _et al._ (1971); Klein  _et al._ (2017) by neglecting the
effect of the photon momentum on the angular distribution.

It is convenient to show the results in terms of the invariant mass
$M^{\gamma\gamma}$ and rapidity $Y^{\gamma\gamma}$ of the $\gamma\gamma$
system, so that we implement further calculations by performing a simple
change of variables. The center-of-mass energy reads

$\displaystyle W\equiv M^{\gamma\gamma}$
$\displaystyle=\sqrt{{(E^{\gamma\gamma})}^{2}-(\vec{p}^{\;\gamma\gamma})^{2}}\doteq\sqrt{4\omega_{1}\omega_{2}}$
(15)

and the rapidity is

$\displaystyle Y\equiv
Y^{\gamma\gamma}=\frac{1}{2}ln\frac{E^{\gamma\gamma}+p^{\gamma\gamma}_{z}}{E^{\gamma\gamma}-p^{\gamma\gamma}_{z}}=\frac{1}{2}ln\frac{\omega_{1}}{\omega_{2}}.$
(16)

Therefore, the single photon energy can be determined by

$\displaystyle\omega_{1}=\frac{W}{2}~{}e^{Y}\qquad\omega_{2}=\frac{W}{2}~{}e^{-Y}$
(17)

and the left hand side of Eq. 8 can be rewritten as

$\displaystyle\frac{d^{4}\sigma}{d\omega_{1}d\omega_{2}d^{2}\vec{b}}=\frac{\partial(W,Y)}{\partial(\omega_{1},\omega_{2})}\frac{d^{4}\sigma}{dWdYd^{2}\vec{b}}=\frac{2}{W}\frac{d^{4}\sigma}{dWdYd^{2}\vec{b}}.$
(18)

###  II.2 Monte-Carlo-based setup

It is realized that, in Eqs. 8 and 18, the relevant dependence on the single
lepton momentum and (pseudo-)rapidity is washed out throughout the
integrations, which are restricted in the available measurements. To better
perform the comparison with data, we propose a Monte-Carlo-based strategy to
overcome this issue.

In this sub-section, we describe the numerical framework utilized for the
photoproduction of lepton pairs, $\gamma\gamma\rightarrow l^{+}l^{-}$, in
ultra-relativistic heavy-ion collisions. Generally, in the Center-of-Mass
(CoM) frame of the $\gamma\gamma$ system, the energy of each photon can be
obtained at a given rapidity, while its transverse momentum is sampled
according to the single photon energy spectrum. Then, the fusion process for
$\gamma\gamma\rightarrow l^{+}l^{-}$ is performed by employing a Monte-Carlo-
based setup. Finally, we can boost the above results from CoM to the
labroratory (LAB). The steps of this numerical procedure are

  1. (1)

Calculate the differential production cross section
$\frac{{d^{4}\sigma}}{{dWdYd^{2}\vec{b}}}$ (Eq. 18) at a given range of
$(W,Y,b)$; note that the varying ranges for $W$ and $Y$ are determined by the
corresponding measurements, while $b$ for a desired centrality class is given
in Ref. Loizides  _et al._ (2018);

  2. (2)

Sample a set of $\gamma\gamma$ pairs according to the profile of the above
spectra via Monte-Carlo, and then initialize their four-momentum, in the CoM
frame of each pair, at a given point $(W,Y,b)$

     * $\bullet$

the total energy of $\gamma\gamma$ system: $E^{Tot}=\omega_{1}+\omega_{2}$,
with the energy of single photon $i$, $\omega_{i}$, given by Eq.
LABEL:eq:W1W2;

     * $\bullet$

the total transverse momentum:
$\vec{P}^{Tot}_{\perp}=\vec{k}_{1,\perp}+\vec{k}_{2,\perp}$, with transverse
momentum of photon $i$, $\vec{k}_{i\perp}$, sampled according to the single
photon energy spectrum (Eq. LABEL:eq:C9) at a given energy $\omega_{i}$; the
azimuthal angle of $\vec{P}^{Tot}_{\perp}$ is assumed to be uniformly
distributed in the transverse plane, and it will be constrained by the
kinematics cuts on the decaying lepton pairs;

     * $\bullet$

the total longitudinal momentum:
$P^{Tot}_{z}=\sqrt{(W)^{2}+(\vec{P}^{Tot}_{\perp})^{2}}\cdot sinh(Y)$;

  3. (3)

Perform the fusion process $\gamma\gamma\rightarrow l^{+}l^{-}$

     * $\bullet$

calculate the four-momentum for the $i$-th lepton $(E_{i}$,
$\vec{p}_{i,\perp}$, $p_{i,z})$

       * –

initialize its three-momentum according to the two-body decay kinematics
$|\vec{p}_{i}|=\sqrt{W^{2}-2m^{2}_{l}}/2$ in the CoM of $\gamma\gamma$ system,
as well as the energy $E_{i}=\sqrt{(\vec{p}_{i})^{2}+m^{2}_{l}}$;

       * –

sample $\vec{p}_{i,\perp}$ and $p_{i,z}$ with Eq. 14 and a uniform
distribution, respectively, in transverse and longitudinal planes; they will
be further constrained by the additional kinematics cuts, which are imposed in
the corresponding final observable;

     * $\bullet$

boost the above results from the CoM of $\gamma\gamma$ system to the LAB
according to $\vec{\beta}=(\beta_{x},\beta_{y},\beta_{z})$, where
$\beta_{i}=-P^{Tot}_{i}/E^{Tot}$ ($i=x,y,z$); note that $\vec{P}^{Tot}$ and
$E^{Tot}$ are already determined in the previous step;

     * $\bullet$

boost further the obtained results from the LAB to CoM for the asymmetry
colliding system $A_{1}A_{2}$, according to $\vec{\beta}=(0,0,\beta_{z})$,
where $\beta_{z}=tanh(Y_{shift})$ with the rapidity shift
$Y_{shift}=\frac{1}{2}ln{\bigr{(}}\frac{Z_{1}}{A_{1}}\frac{A_{2}}{Z_{2}}{\bigr{)}}$
and $Z_{i}$ and $A_{i}$ are the charge number and mass number of nucleus $i$,
respectively;

  4. (4)

Collect the four-momentum of lepton pairs $l^{+}l^{-}$ and single leptons
within the desired acceptance, which can be used to calculate the observables
such as $\frac{dN^{l^{+}l^{-}}}{dW}$, $\frac{dN^{l^{+}l^{-}}}{dp^{2}_{\perp}}$
and $\sqrt{<(p^{l^{+}l^{-}}_{\perp})^{2}>}$, and then compared to the relevant
measurements within the same acceptance.

##  III Numerical results

Figure 1 depicts the impact parameter dependent cross section for the
electromagnetic production of an electron-positron pair in Au–Au collisions at
$\sqrt{s_{\rm NN}}=200$ GeV. The scalar and pseudoscalar contributions (Eq. 8)
are displayed as dashed black and dotted blue curves, respectively, while the
combined result is presented as a solid red curve. We observe that, for the
scalar component, a dip structure lays at $b\sim 9$ fm, while for the
pseudoscalar component, a maximum found at $b\sim 12$ fm.

Figure 1: (Color online) Impact parameter dependent differential cross section
(solid red curve) for the electromagnetic production of an electron-positron
pair with scalar (dashed black curve) and pseudoscalar contribution (dotted
blue curve), in Au–Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV. See legend
for details. Figure 2: Illustration of the electric field produced by two
colliding heavy ions.

These behaviors may be induced by the fact that: (1) the electron-positron
pairs are more produced in the vicinity of the nuclear surfaces, where the
strongest electric field densities of the two colliding heavy ions exist Deng
and Huang (2012); (2) for the two colliding heavy ions $AB$ (see the
illustrating Fig. 2), the overlap nuclear surface is shown as the blue circle,
with an arbitrary point $P$ on top of it to display the point of interest
($b\leq R_{A}+R_{B}$); the constellation of the electric fields are mostly
perpendicular, $\vec{E}_{\rm A}\perp\vec{E}_{\rm B}$, when $b^{2}=R_{\rm
A}^{2}+R_{\rm B}^{2}$, resulting in $b=\sqrt{2}R_{\rm Au}\sim 9$ fm for Au–Au
collisions; similarly, they are mostly parallel, $\vec{E}_{\rm
A}\parallel\vec{E}_{\rm B}$, when $b^{2}=(R_{\rm A}+R_{\rm B})^{2}$, resulting
in $b=2R_{\rm Au}\sim 13$ fm; (3) as mentioned in Eq.
LABEL:eq:TwoPhoton_scaler, the electromagnetic fields of the two colliding
heavy ions have to be parallel and perpendicular to produce the scalar and
pseudoscalar two-photon fusion cross section, respectively, thus, one can
expect that a dip structure exists around $b=\sqrt{2}R_{\rm Au}\sim 9$ fm for
the scalar contribution, while a maximum nearby $b=2R_{\rm Au}\sim 13$ fm for
the pseudoscalar component in Au–Au collisions. Similar results can be found
in Refs. Vidovic  _et al._ (1993); Krauss  _et al._ (1997).

Figure 3 shows the normalized differential yield as a function of the
invariant mass of $e^{+}e^{-}$ in mid-central ($40-60\%$, upper) and
peripheral ($60-80\%$, lower) Au–Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV.
The results (curves) are restricted within the kinematics region
($p_{\perp}^{e}>0.2~{}{\rm GeV}/{\it c}$, $|\eta^{e}|<1$,
$p_{\perp}^{ee}>0.15~{}{\rm GeV}/{\it c}$ and $|y^{ee}|<1$), as applied in the
STAR measurements, for direct comparisons with the experimental data (points).
It is known that the invariant mass of the electron-positron system reads
$M^{ee}=2p_{\perp}^{e}cosh(\frac{y^{e^{+}}-y^{e^{-}}}{2})\geqslant
2p_{\perp}^{e}$, where $p_{\perp}^{e}$ and $y^{e}$ are the single electron
transverse momentum and rapidity, respectively, in the center-of-mass frame of
the electron-positron system. Thus, as expected, the spectra are zero when the
invariant mass less than twice the transverse momentum of the single electron,
$M^{ee}\lesssim 0.4~{}{\rm GeV}/c^{2}$. It peaks at $M^{ee}\sim 0.5~{}{\rm
GeV}/c^{2}$ and then decreases exponentially towards larger $M^{ee}$. It can
be seen that the model calculations provide a very good description of the
measured $M^{ee}$-dependent spectra data in both $40-60\%$ (upper) and
$60-80\%$ (lower) centrality classes. Similar results are found at LHC energy,
see Fig. 4.

Figure 3: (Color online) The differential yield with respect to dielectron
invariant mass in mid-central (upper) and peripheral (lower), indicating as
the centrality percentile $40-60\%$ and $60-80\%$, respectively, Au–Au
collisions at $\sqrt{s_{\rm NN}}=200$ GeV. The results are filtered within the
selected acceptance. The relevant data (solid black point J. Adam et al.
(2018)) are shown for comparison. See legend and text for details. Figure 4:
Same as Fig. 3 but in $50-70\%$ (upper) and $70-90\%$ (lower) Pb–Pb collisions
at $\sqrt{s_{\rm NN}}=5.02$ TeV. The relevant data (solid black point Acharya
_et al._ (2023)) are shown for comparison.

In Fig. 5, the differential electron-positron pair $p_{\perp}^{2}$ spectrum
$d^{2}N/(dp_{\perp}^{2}dy)$ are calculated within three invariant mass regions
in $60-80\%$ Au–Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV. See the legend
for details. The numerical calculations (curves) are filtered with the STAR
acceptance ($p_{\perp}^{e}>0.2~{}{\rm GeV}/{\it c}$, $|\eta^{e}|<1$ and
$|y^{ee}|<1$), allowing direct comparison with the experimental measurements
(points). Within the experimental uncertainties, the measured $p_{\perp}^{2}$
dependencies are better described by the model predictions in the range
$p_{\perp}^{2}\gtrsim 0.0005~{}({\rm GeV}/\it{c})^{\rm 2}$, while a slightly
larger discrepancy observed at extremely lower $p_{\perp}^{2}$.

Figure 5: (Color online) The dielectron pair-$p_{\perp}^{2}$ spectrum within
the mass regions $0.4<M^{ee}<0.76~{}{\rm GeV}/c^{2}$ (solid black curve),
$0.76<M^{ee}<1.2~{}{\rm GeV}/c^{2}$ (dotted red curve) and
$1.2<M^{ee}<2.6~{}{\rm GeV}/c^{2}$ (dashed blue curve) in $60-80\%$ Au–Au
collisions at $\sqrt{s_{\rm NN}}=200$ GeV. The comparisons with available STAR
measurements J. Adam et al. (2018) are plotted as well.

Figure 6 presents the average transverse momentum squared
$\sqrt{<p_{\perp}^{2}>}$ of electron-positron pairs as a function of $M^{ee}$.
It is observed that $\sqrt{<p_{\perp}^{2}>}$ is more significant at larger
$M^{ee}$, where the electron-positron pairs are generated predominantly in the
vicinity of the stronger electromagnetic field, which, in turn, creates larger
$p_{\perp}$. The available STAR measurements J. Adam et al. (2018) are
displayed as well for comparison. Both the shape and the magnitude of the
measured data can be described reasonably well by this calculation, which
indicated that the extra contributions, such as the higher-order QED, could be
trivial for the $p_{\perp}$ broadening effect. The same conclusion can be
drawn for the comparison in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=5.02~{}{\rm
TeV}$ Acharya  _et al._ (2023).

Figure 6: The dielectron pair-$\sqrt{<p_{\perp}^{2}>}$ distribution as a function of invariant mass in $60-80\%$ Au–Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV. The measurements J. Adam et al. (2018) are also shown for comparisons. Mass region | ALICE Data Acharya  _et al._ (2023) | QED Zha  _et al._ (2020); Brandenburg  _et al._ (2021) | Wigner Klusek-Gawenda  _et al._ (2021) | STARLight Klein  _et al._ (2017); ALI (2022) | This work
---|---|---|---|---|---
($\rm Gev/{\it{c}}^{2}$) |  ($\rm MeV/{\it{c}})$ |  ($\rm MeV/{\it{c}})$ |  ($\rm MeV/{\it{c}})$ |  ($\rm MeV/{\it{c}})$ |  ($\rm MeV/{\it{c}}$)
$0.4\leq M^{ee}\leq 0.7$ | $\qquad 44\pm 28(stat.)\pm 6(syst.)\qquad$ | 44 | 45 | 30 | 39
$0.7\leq M^{ee}\leq 1.1$ | $\qquad 45\pm 36(stat.)\pm 8(syst.)\qquad$ | 48 | 48 | 38 | 43
$1.1\leq M^{ee}\leq 2.7$ | $\qquad 69\pm 36(stat.)\pm 8(syst.)\qquad$ | 50 | 50 | 42 | 45
Table 1: Summary of the different models for $\sqrt{<p_{\perp}^{2}>}$ at
desired invariant mass regions in $70-90\%$ Pb–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. The relevant data are presented as well for comparison.

To investigate further the broadening of $p_{\perp}$ via
$\sqrt{<p_{\perp}^{2}>}$, we study its impact parameter $b$ dependence, as
displayed in Fig. 7. It can be seen that: (1) $\sqrt{<p_{\perp}^{2}>}$
increases with decreasing $b$ and reaches a maximum value 1.5 times larger
than that in ultra-peripheral collisions. This result supports the statement
Acharya  _et al._ (2023) that the $p_{\perp}$ broadening originates
predominantly from the initial electromagnetic field strength that varies
significantly with impact parameters; (2) moreover, as discussed in Fig. 6,
the broadening depends on the invariant mass of the electron-positron pair,
behaving an increasing trend with $M^{ee}$. Similar results can be found in
Refs. Zha  _et al._ (2020); Wang  _et al._ (2022). In this figure, the STAR
measurements J. Adam et al. (2018) are also plotted for comparison and show
good agreement within uncertainties.

Figure 7: (Color online) Same as Fig. 6 but with respect to the impact
parameter in the mass regions $0.4<M^{ee}<0.76~{}{\rm GeV}/{\it c}^{2}$ (solid
black curve), $0.76<M^{ee}<1.2~{}{\rm GeV}/{\it c}^{2}$ (dotted red curve) and
$1.2<M^{ee}<2.6~{}{\rm GeV}/{\it c}^{2}$ (dashed blue curve). For visibility,
the data points are shifted horizontally.

With higher beam energy and intensity, the ATLAS Collaboration utilizes the
dimuon pair acoplanarity, $\alpha$, to characterize the $p_{\perp}$ broadening
effects.

$\displaystyle\alpha$ $\displaystyle\equiv
1-\frac{|\phi^{\mu^{+}}-\phi^{\mu^{-}}|}{\pi},$ (19)

where, $\phi^{\mu^{\pm}}$ are the azimuthal angles of the single muons. The
model predictions are shown in Fig. 8 with each distribution normalized to
unity over the $\alpha$ range of interest, $\int
d\alpha(\frac{1}{N}\frac{dN}{d\alpha})=1$. The results are also filtered with
the fiducial cuts ($p_{\perp}^{\mu}>4~{}{\rm GeV}/{\it c}$, $|\eta^{\mu}|<2.4$
and $4<M^{\mu\mu}<45~{}{\rm GeV}/{\it c}^{2}$) to allow a direct comparison
with the ATLAS measurements M. Aabound et al. (2018). The panel-a in Fig. 8
presents the model calculations within various centrality classes. We observe
that the $\alpha$ distributions are narrower in peripheral collisions than
those from more central collisions, indicating that the transverse momentum
broadening effect is more pronounced for central collisions, where the impact
parameters are small. The same conclusion was found in Fig. 7. This
centrality-dependent broadening behavior are compared with the available data
for different centrality classes, as shown from panel-b to panel-f in Fig. 8.
Within the experimental uncertainties, our calculations can describe well the
measured $\alpha$ dependencies. Note that, even though the statistics of ATLAS
data is limited, a centrality-dependent depletion of the dimuon yield is
observed at small $\alpha$, where the discrepancy between model and data is
visible.

Figure 8: (Color online) The dimuon pair-$\alpha$ distribution for the
kinematics of the ATLAS measurements M. Aabound et al. (2018). The panel-a
(upper left) shows the model predictions within the different centrality
classes as the curves in different styles, while the other ones display the
comparisons with the relevant experimental data (solid black point).

Figure 9: (Color online) The dimuon pair-$\alpha$ (left-hand panels) and
$k_{T}$ (right-hand panels) distributions within different $\bar{p}_{\perp}$
intervals, $4<\bar{p}_{\perp}<5~{}\rm GeV$ (solid black curve),
$5<\bar{p}_{\perp}<6~{}\rm GeV$ (long dashed red curve),
$\bar{p}_{\perp}>6~{}\rm GeV$ (dashed blue curve), for three centrality
intervals: $0-20\%$ (upper), $20-40\%$ (middle) and $60-80\%$ (bottom). The
relevant experimental data G. And et al. (2023) (points) are presented as well
for comparison.

The centrality-dependent broadening of the $\alpha$ distributions is further
studied according to the higher luminosity measurement performed by ATLAS
Collaboration G. And et al. (2023). Meanwhile, it allows to investigate the
$\alpha$ broadening behavior within different average transverse momentum
$\bar{p}_{\perp}$,

$\displaystyle\bar{p}_{\perp}$
$\displaystyle\equiv\frac{p_{\perp}^{\mu^{+}}+p_{\perp}^{\mu^{-}}}{2},$ (20)

of the muons in the pair, where, $p_{\perp}^{\mu^{\pm}}$ are the transverse
momentum of single muons. As shown in the left panels of Fig. 9, the dimuon
pair-$\alpha$ distributions within three $\bar{p}_{\perp}$ ranges,
$4<\bar{p}_{\perp}<5~{}\rm GeV$ (solid black curve),
$5<\bar{p}_{\perp}<6~{}\rm GeV$ (long dashed red curve),
$\bar{p}_{\perp}>6~{}\rm GeV$ (dashed blue curve), for three centrality
intervals, $0-20\%$ (panel-a), $20-40\%$ (panel-b) and $60-80\%$ (panel-c).
The relevant data (points) are shown as well for comparisons. We can see that:
(1) the centrality-dependent broadening of the $\alpha$ distributions is
confirmed, and for a given $\bar{p}_{\perp}$ range, the broadening is more
pronounced in central collisions, as observed in Fig. 8; (2) the $\alpha$
broadening seems to have a visible $\bar{p}_{\perp}$-dependence with a
considerable decrease from lower toward higher $\bar{p}_{\perp}$; (3) the
$\alpha$ distributions shown less depletion at $\alpha=0$ at higher
$\bar{p}_{\perp}$. Consequently, the broadening of the $\alpha$ distributions
are affected by both the centrality and the average transverse momentum.

It is argued G. And et al. (2023) that the distributions of the transverse
momentum scale $k_{\perp}$,

$\displaystyle k_{\perp}$
$\displaystyle\equiv\pi\alpha\bar{p}_{\perp}=\frac{p_{\perp}^{\mu^{+}}+p_{\perp}^{\mu^{-}}}{2}\bigr{(}\pi-|\phi^{\mu^{+}}-\phi^{\mu^{-}}|\bigr{)},$
(21)

which effectively quantify the component of the dimuon $\vec{p}_{\perp}$, show
no significant dependence on $\bar{p}_{\perp}$. The relevant results are
presented in the right panels of Fig. 9, the dimuon pair-$k_{\perp}$
distributions within three $\bar{p}_{\perp}$ ranges,
$4<\bar{p}_{\perp}<5~{}\rm GeV$ (solid black curve),
$5<\bar{p}_{\perp}<6~{}\rm GeV$ (long dashed red curve),
$\bar{p}_{\perp}>6~{}\rm GeV$ (dashed blue curve), for three centrality
intervals, $0-20\%$ (panel-d), $20-40\%$ (panel-e) and $60-80\%$ (panel-f).
The relevant data (points) are shown as well for comparisons. We can see that:
(1) in a given centrality range, the $k_{\perp}$ distributions show a weak
$\bar{p}_{\perp}$-dependence from lower toward higher $\bar{p}_{\perp}$,
confirming that $k_{\perp}$ distributions have trivial
$\bar{p}_{\perp}$-dependence; (2) the $k_{\perp}$ broadening is more
pronounced in central collisions; (3) within the experimental uncertainties,
the data can be well described by the relevant model calculations. Thus,
$k_{\perp}$ variable is better suited for assessing the centrality-dependent
modifications of the dimuon alignment.

##  IV Summary

In this work we have studied the photoproduction of lepton pair in high energy
nuclear collisions by employing a theoretical model with the equivalent photon
approximation, which can be derived from a full QED calculation at leading
order in $\alpha_{\rm EM}$ Vidovic  _et al._ (1993). In this literature, the
form factor is the one for a homogeneously charged shpere. Moreover, the
relevant calculations, such as the production cross section, are obtained by
integrating over full phase space of the single leptons.

We update it by taking into account the widely used Woods-Saxon distribution
to quantify well the nucleon profile. Furthermore, to filter with the fiducial
cut on the decaying leptons as applied in the current experimental
measurements and allow direct comparisons, we propose a Monte-Carlo-based
strategy to achieve this goal. The calculated results can describe well the
centrality-dependent broadening of both the acoplanarity ($\alpha$) and the
transverse momentum scale ($k_{\perp}$) distributions, which are measured at
the highest RHIC and LHC energies. It is interesting to note that the $\alpha$
distributions vary with the average transverse momentum $\bar{p}_{\perp}$ of
muons in the pair, while the $k_{\perp}$ distributions do not. Therefore, the
$k_{\perp}$ variable is preferred for the study of the centrality-dependent
broadening effect. We also examine the updated model by calculating the
differential spectra as functions of pair mass $M^{ee}$ and the transverse
momentum squared $p_{\perp}^{2}$, and it is clearly found that the relevant
experimental data can be well described, which indicated that the employed
model is powerful to characterize the photoproduction of lepton pairs in
ultra-relativistic heavy-ion collisions.

###### Acknowledgements.

The authors are grateful to Prof. Jinfeng Liao and Prof. Wangmei Zha for
helpful discussions and communications. This work is supported by the National
Science Foundation of China (NSFC) under Grant Nos.12005114 and 12375137.

##  Appendix A Derivation of the energy flux of the photons in equivalent
photon approximation (EPA)

For the d’Alembert’s equation (Eq. LABEL:eq:dAlbertEq), one can perform the
Fourier transformation from momentum to position space, resulting in Krauss
_et al._ (1997)

$A^{\nu}(k)=-\frac{1}{k^{2}}J^{\nu}(k),$ (A1)

where, the current density reads

$\displaystyle
J^{\nu}(k)=2\pi~{}\delta(ku)~{}\rho_{c}(\sqrt{-k^{2}})~{}u^{\nu}~{}e^{i\vec{k}\cdot\vec{b}},$
(A2)

with $u^{\nu}=\gamma(1,\vec{0}_{\perp},v)$ and
$k=(\omega,\vec{k}_{\perp},k_{z})$. $\rho_{c}$ is the nuclear charge density
function

$\displaystyle\rho_{c}\equiv Ze\rho_{m}$ (A3)

and $\rho_{m}$ is the nuclear matter density function. Here we have chosen the
$z$-axis as the direction of the Lorentz-boost $\gamma=1/\sqrt{1-v^{2}}$
between the two frames. With the $\delta$-function in Eq. LABEL:eq:A6, we have
$-k^{2}=\vec{k}_{\perp}^{2}+(\frac{\omega}{\gamma})^{2}$.

The electromagnetic potential can be obtained by inserting Eq. LABEL:eq:A6
into Eq. A1, yielding

$\displaystyle A^{\nu}(k)$ $\displaystyle=2\pi
Ze~{}\delta(ku)~{}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}~{}u^{\nu}~{}e^{i\vec{k}\cdot\vec{b}},$
(A4)

where, $Z$ is the nuclear charge number and $\mathcal{F}(\sqrt{{-k^{2}}})$ is
the relevant normalized form factor defined as the Fourier transform of the
charge distribution.

$\mathcal{F}(\sqrt{{-k^{2}}})\equiv\frac{1}{Ze}\int^{\infty}_{-\infty}d^{3}\vec{r}e^{-i\vec{k}\cdot\vec{r}}\rho_{c}(\vec{r})=\frac{1}{Ze}\rho_{c}(\sqrt{{-k^{2}}}).$
(A5)

The electromagnetic field strength tensor in momentum space reads Krauss  _et
al._ (1997)

$\displaystyle F^{\mu\nu}(k)$
$\displaystyle=-i{\biggr{[}}k^{\mu}A^{\nu}(k)-ik^{\nu}A^{\mu}(k){\biggr{]}}$
(A6) $\displaystyle=\left(\begin{array}[]{cccc}0&-E_{x}&-E_{y}&-E_{z}\\\
E_{x}&~{}~{}0&-B_{z}&~{}~{}B_{y}\\\ E_{y}&~{}~{}B_{z}&~{}~{}0&-B_{x}\\\
E_{z}&-B_{y}&~{}~{}B_{x}&~{}~{}0\end{array}\right).$

Note that

  * (1)

the transverse components of the electric field is given by

$\displaystyle\vec{E}_{\perp}=(F^{10},F^{20})=-i{\bigr{[}}A^{0}\vec{k}_{\perp}-k^{0}\vec{A}_{\perp}{\bigr{]}}=-iA^{0}\vec{k}_{\perp}$
(A7)

since $\vec{u}_{\perp}=\vec{A}_{\perp}=0$ (Eq. A4); the transverse components
of the magnetic field read

$\displaystyle\vec{B}_{\perp}=\vec{v}\times E_{\perp}$
$\displaystyle=-ivA^{0}~{}(-k_{y},k_{x},0)^{T}.$ (A8)

  * (2)

the longitudinal component of the electric field is

$\displaystyle E_{z}=F^{30}=-i{\bigr{[}}k^{3}A^{0}-k^{0}A^{3}{\bigr{]}}\doteq
0,$ (A9)

since $k^{3}\doteq k^{0}=\omega$ and $A^{0}\doteq A^{3}$ in the ultra-
relativistic limit ($v\sim c=1$); consequently, the longitudinal components of
the electromagnetic fields vanish in this limit, resulting in the relations as

$\displaystyle\vec{E}\doteq\vec{E}_{\perp},~{}~{}\vec{B}\doteq\vec{B}_{\perp},$
(A10) $\displaystyle|\vec{E}_{\perp}|\doteq|\vec{B}_{\perp}|,~{}~{}E_{z}\doteq
B_{z}\doteq 0,~{}~{}|\vec{E}|\doteq|\vec{B}|$
$\displaystyle\vec{E}\perp\vec{B},~{}~{}\vec{E}\perp\vec{v},~{}~{}\vec{B}\perp\vec{v};$

  * (3)

the energy flux of the electromagnetic field through a plane perpendicular to
the direction of the moving nucleus, is provided by the Poynting vector

$\displaystyle\vec{S}(\vec{r},t)\equiv\vec{E}(\vec{r},t)\times\vec{B}(\vec{r},t)\doteq{\bigr{|}}\vec{E}_{\perp}(\vec{r},t){\bigr{|}}^{2}$
(A11)

in the limit $v\sim c=1$.

As discussed above, the electromagnetic field of a charged nucleus moving at
high velocities becomes more and more transverse with reference to the
direction of propagation, see Eq. LABEL:eq:B6. As a consequence, an observer
in the laboratory cannot distinguish between the electromagnetic of a
relativistic nucleus and a bunch of equivalent photons. To extract the
equivalent photon spectrum $n(\omega,\vec{x}_{\perp})$ which depends on the
photon frequency $\omega$ and the displacement $\vec{x}_{\perp}$ in the
transverse plane, we can require that the energy flux of the electromagnetic
fields through a transverse plane is identical to the energy flux of the
equivalent photons Krauss  _et al._ (1997), i.e. equivalent photon
approximation (EPA).

$\displaystyle\int^{\infty}_{-\infty}dt\int^{\infty}_{-\infty}d^{2}\vec{x}_{\perp}\vec{S}(\vec{r},t)\equiv\int^{\infty}_{0}d\omega~{}\omega\int^{\infty}_{-\infty}d^{2}\vec{x}_{\perp}n(\omega,\vec{x}_{\perp}).$
(A12)

With Eq. LABEL:eq:B7, the left hand side (LHS) of Eq. LABEL:eq:C1 can be
expressed as

$\displaystyle LHS$
$\displaystyle=\frac{1}{\pi}\int^{\infty}_{0}d\omega\int^{\infty}_{-\infty}d^{2}\vec{x}_{\perp}{\bigr{|}}\vec{E}_{\perp}(\vec{r},\omega){\bigr{|}}^{2}$
(A13)

Compared with the right hand side (RHS) of Eq. LABEL:eq:C1, we have

$\displaystyle
n(\omega,\vec{x}_{\perp})=\frac{1}{\pi\omega}{\bigr{|}}\vec{E}_{\perp}(\vec{r},\omega){\bigr{|}}^{2}$
(A14)

in the ultra-relativistic limit $v\sim c=1$. The electric field in Eq.
LABEL:eq:C3 is given by

$\displaystyle{\bigr{|}}\vec{E}_{\perp}(\vec{r},\omega){\bigr{|}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:A8},\ref{eq:B3})}}{{=}}\frac{Ze}{2\pi}{\biggr{|}}\int^{\infty}_{0}dk_{\perp}~{}k_{\perp}^{2}~{}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}~{}J_{1}(x_{\perp}k_{\perp}){\biggr{|}}$
(A15)

with the Bessel function

$\displaystyle J_{n}(z)\equiv\frac{1}{2\pi i^{n}}\int^{2\pi}_{0}d\theta
e^{izcos\theta}e^{in\theta}$ (A16)

of the first kind ($n=1$). The energy flux of the equivalent photons can be
obtained by inserting Eq. A15 into Eq. LABEL:eq:C3, yielding

$\displaystyle
n(\omega,\vec{x}_{\perp})=\frac{Z^{2}\alpha_{EM}}{\pi^{2}\omega}{\biggr{|}}\int^{\infty}_{0}dk_{\perp}k_{\perp}^{2}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}J_{1}(x_{\perp}k_{\perp}){\biggr{|}}^{2}$
(A17)

with the electromagnetic coupling factor $\alpha_{EM}=e^{2}/(4\pi)$. Similar
results can be found in Ref. Vidovic  _et al._ (1993).

Note that

  * (1)

the convergence of the integration part in Eq. LABEL:eq:C6, is guaranteed if
the form factor vanishes fast enough to zero at $k_{\perp}\rightarrow\infty$
Vidovic  _et al._ (1993); we take
$k^{2}_{\perp}\leqslant\frac{1}{R^{2}}-{\bigr{(}}\frac{\omega}{\gamma}{\bigr{)}}^{2}$
in the numerical calculations Niu  _et al._ (2022), where $R$ is the nuclear
radius of the heavy-ion, see Eq. 6;

  * (2)

the transverse plane integrated energy flux in Eq. LABEL:eq:C3, can be
obtained via

$\displaystyle n(\omega)$
$\displaystyle=\int^{\infty}_{-\infty}d^{2}\vec{x}_{\perp}n(\omega,\vec{x}_{\perp})$
(A18)
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:B3},\ref{eq:A8})}}{{=}}\frac{2Z^{2}\alpha_{EM}}{\pi\omega}\int^{\infty}_{0}dk_{\perp}|\vec{k}_{\perp}|^{3}{\biggr{[}}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}{\biggr{]}}^{2},$

resulting in

$\displaystyle\frac{dn(\omega,k_{\perp})}{dk_{\perp}}=\frac{2Z^{2}\alpha_{EM}}{\pi\omega}|\vec{k}_{\perp}|^{3}{\biggr{[}}\frac{\mathcal{F}(\sqrt{{-k^{2}}})}{-k^{2}}{\biggr{]}}^{2}.$
(A19)

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# Large Angular Momentum

Kenichi Konishi(1,2), Roberto Menta(3,4)
(1)Department of Physics “E. Fermi”, University of Pisa
Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy
(2)INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy
(3) Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56127 Pisa, Italy
(4)Laboratorio NEST, Piazza San Silvestro, 12, 56127 Pisa, Italy

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum states of a spin $\tfrac{1}{2}$ (a qubit) are parametrized by the
space ${\mathbf{CP}}^{1}\sim S^{2}$, the Bloch sphere. A spin $j$ (a
$2j+1$-state system) for generic $j$ is represented instead by a point of a
larger space, ${\mathbf{CP}}^{2j}$. Here we study the angular momentum/spin in
the limit, $j\to\infty$. The state,
$({\hat{\mathbf{J}}}\cdot{\mathbf{n}})\,|j,{\mathbf{n}}\rangle=j\,|j,{\mathbf{n}}\rangle$,
where ${\hat{\mathbf{J}}}$ is the angular momentum operator and ${\mathbf{n}}$
stands for a generic unit vector in ${\bf R}^{3}$, is found to behave as a
classical angular momentum, $j\,{\mathbf{n}}$. We discuss this phenomenon, by
analysing the Stern-Gerlach experiments, the angular-momentum composition
rule, and the rotation matrix. This problem arose from the consideration of a
macroscopic body under an inhomogeneous magnetic field. Our observations help
to explain how classical mechanics (with unique particle trajectories) emerges
naturally from quantum mechanics in this context, and at the same time, make
the widespread notion that large spins somehow become classical, a more
precise one.

###### Contents

  1. 1 Introduction
  2. 2 Stern-Gerlach experiments
    1. 2.1 Spin $1/2$ atom
    2. 2.2 Large spin
      1. 2.2.1 Density matrix
      2. 2.2.2 The state preparation and successive SG set-ups
      3. 2.2.3 States almost directed towards ${\vec{\mathbf{n}}}$
  3. 3 Quantum and classical angular momentum states
    1. 3.1 Orbital angular momentum
  4. 4 Addition of angular momenta
  5. 5 Rotation matrix
  6. 6 Conclusion
  7. A Newton’s equation for a macroscopic body

##  1 Introduction

It is a widely shared view that in the limit of large spin (angular momentum)
the quantum-mechanical spin somehow becomes classical. After all, the large
spin limit ($S/\hbar\to\infty$) is equivalent to the $\hbar\to 0$ limit, and,
in accordance with the general idea of the semi-classical limit, physics of
large angular momentum is expected to become that of classical angular
momentum. But to the best of the authors’ knowledge, it has not been clearly
exhibited in the literature how exactly this works 111Understandably, there is
a vast literature about large angular momentum, from the early days of quantum
mechanics to today, in a wide range of research fields, in various contexts
and problems, and also based on different motivations. In order to avoid being
biased and necessarily incomplete, we shall refrain in this work from
attempting to cite or comment on any of them.. It is the aim of this note to
fill in this gap.

The question came under a sharp focus in a general discussion on emergence of
classical mechanics for a macroscopic body from quantum mechanics [1, 2].
Knowing about the typical Stern-Gerlach (SG) process for a spin $1/2$ atom, we
ask what happens if a macroscopic body consisting of many (e.g.
$N\sim\mathcal{O}(10^{23}))$ spins is set under an inhomogeneous magnetic
field of a SG set-up. In general, with many spins oriented in different
directions, the problem does not arise, as the bound-state Hamiltonian does
not allow each component particle to split à la SG into spatially separated
sub-wavepackets. Logically, however, one cannot exclude the special situation
in which a large number of spins are all oriented in the same direction. Such
a large spin system might continue to behave quantum-mechanically, e.g. with a
typical SG spreading of the wavepackets, even in the $S/\hbar\to\infty$ limit.

It will be shown that the state $|S,{\mathbf{n}}\rangle$, the state in which
the large spin is oriented in some direction ${\mathbf{n}}$, behaves instead
as a classical angular momentum, yielding the unique trajectory in the SG
processes (Sec. 2). This result leads us to consider more carefully in Sec. 3
the relation between the space of quantum-mechanical angular-momentum states
and that of classical angular momenta, and to identify the special class of
quantum angular momentum states to be put in correspondence with the classical
angular momentum (see Eqs. (3.11), (3.12), (3.13) below). The content of the
present work is to check the consistency of this picture, through the analysis
of the SG experiments (Sec. 2), the angular-momentum composition rule (Sec.
4), and the rotation matrix (Sec. 5). We conclude with a brief reflection in
Sec. 6.

##  2 Stern-Gerlach experiments

###  2.1 Spin $1/2$ atom

For definiteness let us take an incident atom with spin $S=\tfrac{1}{2}$
directed along a definite, but generic, direction ${\bf
n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, i.e.

$\Psi=\psi({\bf r},t)\,|{\bf n}\rangle\;,$ (2.1) $|{\bf
n}\rangle=c_{1}|\\!\uparrow\rangle+c_{2}|\\!\downarrow\rangle\;,\qquad
c_{1}=e^{-i\phi/2}\cos\tfrac{\theta}{2}\;,\qquad
c_{2}=e^{i\phi/2}\sin\tfrac{\theta}{2}\;.$ (2.2)

When the atom passes through a region with an inhomogeneous magnetic field,

${\mathbf{B}}=(0,0,B(z)),\qquad\frac{\partial B(z)}{\partial z}\neq 0\;,$
(2.3)

the wave packet splits into two subpackets, with respective weights,
$|c_{1}|^{2}$ and $|c_{2}|^{2}$: they manifest as the relative intensities of
the image bands on the screen at the end.

The original Stern-Gerlach experiment [3] has had a fundamental impact to the
development of quantum theory, in proving the quantization of the angular
momentum, the existence of half-integer spin, and more generally,
demonstrating the quantum-mechanical nature of the center of mass (CM) of the
silver atom [2].

###  2.2 Large spin

A classical particle with a magnetic moment directed towards

${\boldsymbol{\mu}}={\rm const}.\,{\bf n}\;,\qquad{\bf
n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\;,$ (2.4)

moves in an inhomogeneous magnetic field (2.3), according to Newton’s
equations,

$m\,{\dot{\bf r}}={\bf
p}\;,\qquad\frac{dp_{x}}{dt}=\frac{dp_{y}}{dt}=0\;,\quad\frac{dp_{z}}{dt}=F_{z}=\frac{\partial}{\partial
z}{\boldsymbol{\mu}}\cdot{\bf B}\;.$ (2.5)

The way the unique trajectory for a classical particle emerges from quantum
mechanics has been discussed in Ref. [1], where the magnetic moment is the
expectation value

$\sum_{i}\langle\Psi|({\hat{\boldsymbol{\mu}}}_{i}+\frac{e_{i}{\hat{\boldsymbol{\ell}}}_{i}}{2m_{i}c})|\Psi\rangle={\boldsymbol{\mu}}\;,$
(2.6)

taken in the internal bound-state wave function $|\Psi\rangle$ where $\mu_{i}$
and $\frac{e_{i}{\bf\ell}_{i}}{2m_{i}c}$ denote the intrinsic magnetic moment
and the one due to the orbital motion of the $i$-th constituent atom or
molecule ($i=1,2,\ldots,N$), respectively. Clearly, in general, the
considerations made in Sec. 2.1 for a spin $1/2$ atom, with a doubly split
wavepacket, cannot be simply generalized to (or compared with) a classical
body (2.6) with $N\sim\mathcal{O}(10^{23})$ constituents.

Typically, the spins are oriented along different directions, but the
particles carrying them are bound in the macroscopic body made of atomic,
molecular and larger structures. The wave function of particles in bound
states does not split in sub-wavepackets, as that of an isolated spin
$\tfrac{1}{2}$ atom does, under an inhomogeneous magnetic field. The bound-
state Hamiltonian does not allow it 222This discussion is indeed closely
parallel to the question of quantum diffusion. Unlike free particles,
particles in bound states (the electrons in atoms; atoms in molecules, etc.)
do not diffuse, as they move in binding potentials. This is one of the
elements for the emergence of the classical mechanics, with unique
trajectories for macroscopic bodies. As for the center-of-mass (CM) wave
function of an isolated macroscopic body, its free quantum diffusion is
suppressed by mass.. What determines the motion of the CM of the body is the
average magnetic moment, (2.6). Logically, nevertheless, one cannot exclude
special cases (e.g. a magnetized piece of metal) of all spins of the body
oriented in the same direction. One might wonder how the wave function of such
a particle with spin $S\to\infty$ behaves, under an inhomogeneous magnetic
field.

The question is whether the conditions enlisted in Ref. [1] (see Appendix A),
for the emergence of classical mechanics for a macroscopic body with a unique
trajectory, are sufficient, or whether some extra condition (or a new
mechanism) is required, to suppress possible wide spreading of the wave
function into many sub-packets (see Fig. 1 for a small spin, $S=13/2$) under
an inhomogeneous magnetic field.

Figure 1: The distribution $|c_{k}|^{2}$ in $k$, i.e. in possible values of
$S_{z}$, $-S\leq S_{z}\leq S$, for a spin $S=13/2$ particle in the state
(2.7), with $\theta=\pi/2$ (center, orange dots) and $\theta=\pi/4$ (right,
blue dots).

The answer turns out to be simple but somewhat unexpected. The following
discussion, up to Eq. (2.15), is taken almost verbatim from Sec. 3.1.3 of Ref.
[2]; for the present work it represents the starting point, and serves as the
main tool of analysis below.

Consider the state of a spin $S$ directed towards a direction ${\mathbf{n}}$,

$\left({\hat{\mathbf{S}}}\cdot{\mathbf{n}}\right)\,|S,{\mathbf{n}}\rangle=S\,|S,{\mathbf{n}}\rangle\;,\qquad{\bf
n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\;.$ (2.7)

Choosing the magnetic field (and its gradients) in the ${z}$-direction, we
need to express $|S,{\mathbf{n}}\rangle$ as a superposition of the eigenstates
of ${\hat{S_{z}}}$ ($S=N/2$),

$|S,{\mathbf{n}}\rangle=\sum_{k=0}^{N}c_{k}\,|S,S_{z}=M\rangle\;,\qquad
M=-\frac{N}{2}+k\;,\quad(k=0,1,\ldots,N)\;.$ (2.8)

The expansion coefficients $c_{k}$ are known:

$c_{k}={\binom{N}{k}}^{1/2}\,e^{-iM\phi/2}\left(\cos\tfrac{\theta}{2}\right)^{k}\left(\sin\tfrac{\theta}{2}\right)^{N-k}\;,\qquad\sum_{k=0}^{N}|c_{k}|^{2}=1\;,$
(2.9)

where $\binom{N}{k}=N!/k!(N-k)!$ are the binomial coefficients.

To get (2.9), let us consider (2.7) as a direct product of $N$ spin
$\tfrac{1}{2}$ particles, (2.2),

$|S,{\mathbf{n}}\rangle=|{\bf n}\rangle_{1}\otimes|{\bf
n}\rangle_{2}\otimes\cdots|{\bf n}\rangle_{N}\;,$ (2.10)

all oriented in the same direction. Expanding and collecting terms with a
fixed $k$ (the number of spin-up particles) one arrives at Eq. (2.9).

Using Stirling’s formula, one can derive, for large values of both $N$ and $k$
with $x=k/N$ fixed, the distribution across different values of $S_{z}=M$,

$|c_{k}|^{2}\simeq e^{Nf(x)}\;,\qquad x=k/N\;,$ (2.11)

with

$f(x)=-x\log
x-(1-x)\log(1-x)+2x\log\cos\tfrac{\theta}{2}+2(1-x)\log\sin\tfrac{\theta}{2}\;.$
(2.12)

The saddle-point approximation, valid at $N\to\infty$, yields

$f(x)\simeq-\frac{(x-x_{0})^{2}}{x_{0}(1-x_{0})}\;,\qquad
x_{0}=\cos^{2}\tfrac{\theta}{2}\;,$ (2.13)

and thus

$\sum_{k}|c_{k}|^{2}(\cdots)\longrightarrow\int_{0}^{1}dx\,\delta(x-x_{0})\,(\cdots)$
(2.14)

in the $N\to\infty$ ($x=k/N$ fixed) limit. The narrow peak position $x=x_{0}$
corresponds to (see Eq. (2.8))

$S_{z}=M=N(x-\tfrac{1}{2})=S\,(2\cos^{2}\tfrac{\theta}{2}-1)=S\,\cos\theta\;.$
(2.15)

This means that a large spin ($S\gg\hbar$) quantum particle with spin directed
towards ${\mathbf{n}}$, in a Stern-Gerlach setting with an inhomogeneous
magnetic field (2.3), moves along a single trajectory of a classical particle
with $S_{z}=S\,\cos\theta$, instead of spreading over a wide range of split
sub-packet trajectories covering $-S\leq S_{z}\leq S$. See Fig. 2, cfr. Fig.
1.

Figure 2: The distribution $|c_{k}|^{2}$ in $k$, i.e. in possible values of
$S_{z}$, for a spin $S=1\cdot 10^{3}$ particle in the state (2.7), with
$\theta=\pi/2$ (center, orange peak) and $\theta=\pi/4$ (right, blue peak).
The particle starts looking like classical.

This (perhaps) somewhat surprising result shows that quantum mechanics (QM)
takes care of itself, so to speak, in ensuring that a large spin particle
($S/\hbar\to\infty$) traces a classical trajectory. This is consistent with
the known general behavior of the wave function in the semi-classical limit
($\hbar\to 0$) 333Of course this does not mean that the classical limit
necessarily requires or implies $S/\hbar\to\infty$, though.. Note that if the
value $S\gg\hbar$ is understood as due to the large number of spin $1/2$
particles composing it (see the comment after (2.9)), the spike (2.13), (2.14)
can be understood as due to the accumulation of a huge number of microstates
giving $S_{z}\simeq M$.

Another, equivalent way to see the shrinkage of the distribution in $S_{z}$,
(2.14), is to study its dispersion. Given the relative weights $|c_{k}|^{2}$,
(2.9), where $S_{z}=-S+k$ and $S=\tfrac{N}{2}$, with $k=0,1,\ldots,N$, it is a
simple exercise to find that

$\langle S_{z}\rangle=S\,\cos\theta\;,\qquad\langle
S_{z}^{2}\rangle=S^{2}\cos^{2}\theta+\frac{S}{2}\sin^{2}\theta\;,$ (2.16)

so that the dispersion (the fluctuation width) is given by

$(\Delta S_{z})^{2}=\langle(S_{z}-\langle S_{z}\rangle)^{2}\rangle=\langle
S_{z}^{2}\rangle-\langle S_{z}\rangle^{2}=\frac{S}{2}\sin^{2}\theta\;.$ (2.17)

It follows that

$\lim_{S/\hbar\to\infty}\frac{|\Delta S_{z}|}{S}=0\;,$ (2.18)

i.e. it is an infinitely narrow Gaussian. Note that the SG set-up is indeed a
measurement of $S_{z}$ in the state (2.7). The fact that the result is always
$S_{z}\sim\langle S_{z}\rangle=S\,\cos\theta$ with no dispersion, means that
it is a classical angular momentum $\sim S{\mathbf{n}}$ (see Eqs. (3.11),
(3.12) below).

####  2.2.1 Density matrix

The same phenomenon can be seen also by using the density matrix. Being the
state (2.7) a pure state, its $(2S+1)\times(2S+1)$ density matrix is given by

$\rho_{k\ell}=c_{k}c_{\ell}^{*}\,,$ (2.19)

where $c_{k}(c_{\ell})$ are given in Eq. (2.9). In the limit as $S\to\infty$,
it can be observed that it essentially reduces to a single diagonal element,
provided that the fluctuations of the order of $\sqrt{S}$ are neglected in
comparison to $S$. The density matrix ${\boldsymbol{\rho}}$ for $S=500$, with
$(\theta,\phi)=(\frac{\pi}{2},0)$ and $(\theta,\phi)=(\frac{\pi}{4},0)$, is
depicted in Figs. 3.

Figure 3: The density matrix ${\boldsymbol{\rho}}$ for the state (2.7), of
spin $S=5\cdot 10^{2}$, with ($a$): $(\theta,\phi)=(\tfrac{\pi}{2},0)$ and
($b$): $(\theta,\phi)=(\tfrac{\pi}{4},0)$. The height of the plots indicates
the matrix elements $\rho_{k\ell}$. Note that ${\boldsymbol{\rho}}$ is real
for $\phi=0$.

####  2.2.2 The state preparation and successive SG set-ups

For a small $S$, as it is well known, the SG set-up can be used to prepare any
desired state $|S,S_{z}=M\rangle$, by an intentionally biased SG apparatus,
e.g. in which all the split sub-wavepackets are blocked, except for the one
corresponding to $|S,S_{z}=M\rangle$. Equivalently, one may use the so-called
null-measurements (see Ref. [4] for a recent review). In any case, for a small
finite $S$, where there are $2S+1$ well-separated sub-packets (see Fig. 1),
there is no difficulty in extracting any desired quantum state for subsequent
experiments (state preparation).

In the large-spin limit ($S\to\infty$) we are considering, since there is
essentially only one wavepacket present (see (2.15)), there is no possibility
of extracting or selecting the desired sub-wavepacket. The experimental
resolution would not allow it. We have in mind a macroscopic scenario where,
in comparison to the finite region (in principle) occupied by the waves
$-S\leq M\leq S$, the fluctuations of the values of $M$ around (2.15) are of
the order of $\sqrt{S}\ll S$, i.e. of zero width.

We might consider introducing a second set of Stern-Gerlach (SG) magnetic
fields directed in a different direction, denoted by ${\hat{z}}^{\prime}$, as
a new quantization axis. This setup allows us to study the state emerging from
the first set of SG magnetic fields (without the presence of an imaging
screen),

$|S,{\mathbf{n}}\rangle\sim c_{k}\,|S,S_{z}=M\rangle\;,\qquad k\sim
2S\cos^{2}\tfrac{\theta}{2}\;,\qquad M\sim S\,\cos\theta\;.$ (2.20)

Since the sum of states around $k\sim 2S\cos^{2}\left(\frac{\theta}{2}\right)$
is essentially equal to the complete sum, this state can be rewritten in terms
of the eigenvalues $M^{\prime}=-S+k^{\prime}$, where $k^{\prime}=0,1,\ldots
2S$ correspond to the eigenvalues of $S_{z^{\prime}}$ as:

$\displaystyle|S,{\mathbf{n}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k,k^{\prime}}c_{k}\,|S,M^{\prime}\rangle\langle
S,M^{\prime}|S,M\rangle$ (2.21) $\displaystyle=$
$\displaystyle\sum_{k,k^{\prime}}|S,M^{\prime}\rangle\langle
S,M^{\prime}|S,M\rangle\langle S,M|S,{\mathbf{n}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime}}|S,M^{\prime}\rangle\langle
S,M^{\prime}|S,{\mathbf{n}}\rangle=\sum_{k^{\prime}}c_{k^{\prime}}|S,M^{\prime}\rangle\;.$

This represents an expansion of the original state $|S,{\mathbf{n}}\rangle$ as
a linear combination of the eigenstates of $S_{z^{\prime}}$. In the limit
$S\to\infty$, the results from Section 2.2 tell us that the sum is dominated
by the small region around $S_{z^{\prime}}\sim
S,\cos({\mathbf{n}}\cdot{\hat{z}}^{\prime})$, with negligible fluctuations.
The trajectory remains unique, regardless of the choice of quantization axis
(i.e. the direction of the magnetic field) ${\hat{z}}^{\prime}$.

####  2.2.3 States almost directed towards ${\vec{\mathbf{n}}}$

Obviously, in the large spin limit, other states close to
$|S,{\mathbf{n}}\rangle$,

$\left({\hat{\mathbf{S}}}\cdot{\mathbf{n}}\right)\,|S,{\mathbf{n}},q\rangle=(S-q)\,|S,{\mathbf{n}},q\rangle\;,\qquad
q=1,2,\ldots\,\,\,{\rm fixed}$ (2.22)

(i.e. the spin roughly oriented towards ${\mathbf{n}}$) will behave similarly.
For instance, the $q=1$ state is (cfr. Eq. (2.10))

$\displaystyle|S,{\mathbf{n}},1\rangle$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{N-1}\frac{1}{\sqrt{N}}\,|{\bf
n}\rangle_{1}\otimes\cdots|{\bf n}\rangle_{\ell}\otimes|-{\bf
n}\rangle\otimes|{\bf n}\rangle_{\ell+2}\otimes\cdots|{\bf n}\rangle_{N}\;,$
(2.23) $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{N-1}\frac{1}{\sqrt{N}}|S_{1},{\mathbf{n}}\rangle\otimes|-{\bf
n}\rangle\otimes|S_{2},{\mathbf{n}}\rangle\;,$

where $|S_{1},{\mathbf{n}}\rangle$ and $|S_{2},{\mathbf{n}}\rangle$
($S_{1}=\tfrac{\ell}{2}\;,S_{2}=\tfrac{N-\ell-1}{2}$), are two spins oriented
towards ${\mathbf{n}}$, (2.8), as already studied in Sec. 2.2. In the
$S\to\infty$ limit, the sum is dominated by terms with
$\ell\sim\mathcal{O}(N)\to\infty$. The SG projection of this state onto the
$S_{z}$ eigenstates is therefore approximated by

$S_{z}\sim S_{1\,z}+S_{2\,z}=S_{1}\cos\theta+S_{2}\cos\theta\simeq
S\,\cos\theta\;,$ (2.24)

where the results of Sec. 2.2 have been used. A similar result holds for
$|S,{\mathbf{n}},q\rangle$ for any finite $q$, showing that all these states
behave as a classical angular momentum, $S\,{\mathbf{n}}$.

##  3 Quantum and classical angular momentum states

Before proceeding, we must discuss the correspondence between the quantum
mechanical angular momentum states and their classical counterparts, more
carefully. The spin $1/2$ (two-state system) wave function is given by

$|\psi\rangle=c_{1}|\\!\uparrow\rangle+c_{2}|\\!\downarrow\rangle\;,$ (3.1)

where $(c_{1},c_{2})\sim\lambda(c_{1},c_{2})$ with
$\lambda\in{\mathbf{C}}/\\{0\\}$, describes the points of $\mathbf{CP}^{1}\sim
S^{2}$. We note that for a spin $1/2$, any pure state (3.1) can be interpreted
as the state in which the spin is oriented in some space direction
${\mathbf{n}}$, as in Eq. (2.2), without loss of generality. This is because
the manifold of pure spin $1/2$ quantum states and that of the unit space
vector are both $S^{2}\sim\mathbf{CP}^{1}$. Indeed, given the state (2.2),
there is always a rotation matrix such that

$D^{1/2}(0,\theta,\phi)\cdot\left(\begin{array}[]{c}e^{-i\phi/2}\cos\tfrac{\theta}{2}\\\
e^{i\phi/2}\sin\tfrac{\theta}{2}\end{array}\right)=\left(\begin{array}[]{c}1\\\
0\end{array}\right)\;.$ (3.2)

We recall that the general rotation matrix for a spin $1/2$ is given, using
Euler angles 444Note that the third Euler angle (the rotation angle $\gamma$
about the final $z$-axis) is redundant here, as it gives only the phase
$e^{i\gamma/2}$, and has been set to $0$ in (3.2). ($\gamma,\theta,\phi$), as
follows

$D^{1/2}(\gamma,\theta,\phi)=U_{z}(\gamma)U_{y}(\theta)U_{z}(\phi)=\left(\begin{array}[]{cc}e^{i(\gamma+\phi)/2}\cos\tfrac{\theta}{2}&e^{(\gamma-\phi)/2}\sin\tfrac{\theta}{2}\\\
-e^{i(-\gamma+\phi)/2}\sin\tfrac{\theta}{2}&e^{-i(\gamma+\phi)/2}\cos\tfrac{\theta}{2}\end{array}\right)\;.$
(3.3)

The situation is different for a generic spin $j=N/2$ 555In this section we
denote the general angular momentum (including orbital angular momentum) by
$j$, rather than $S$, a symbol adopted commonly to indicate the spin (the
intrinsic angular momentum of a particle). In this work we will use both $j$
and $S$. Likewise, we shall use the terms “spin” and “angular momentum”,
interchangeably.. The wave function now has the form,

$|\psi\rangle=\sum_{k=0}^{N}b_{k}|j,j_{z}=-j+k\rangle$ (3.4)

where

$(b_{0},b_{1},\ldots,b_{N})\sim\lambda(b_{0},b_{1},\ldots,b_{N})\;,\qquad\lambda\in{\mathbf{C}}/{(0)}\;,$
(3.5)

are the coordinates of the space $\mathbf{CP}^{N}$. On the other hand, the
variety of the directions in the three-dimensional space, ${\mathbf{n}}$, is
always $S^{2}\sim\mathbf{CP}^{1}$. It means that the state of the spin
oriented towards a definite direction (2.8), is not a general pure spin $j$
state, but belongs to a small subset,
$\mathbf{CP}^{1}\subset\mathbf{CP}^{2j}$. In other words, not all states of
spin $j$, as described in (3.4), can be transformed by a rotation matrix
(selecting an appropriate new ${\hat{z}}$ axis) into the form:

$(1,0,\ldots,0)^{T}\;.$ (3.6)

These considerations highlight a potential subtlety in our argument that
quantum mechanical angular momentum states somehow become classical in the
large $j$ limit.

Let us recall that because of the commutation relations

$[{\hat{J}}_{k},{\hat{J}}_{\ell}]=i\hbar\,\epsilon^{k\ell m}{\hat{J}}_{m}\;,$
(3.7)

only one of the components, e.g. ${\hat{J}}_{z}$, can be diagonalized together
with the total (Casimir) angular momentum,
${\hat{J}}^{2}={\hat{J}}_{x}^{2}+{\hat{J}}_{y}^{2}+{\hat{J}}_{z}^{2}$,

${\hat{J}}^{2}|j,m\rangle=j(j+1)|j,m\rangle\;,\qquad{\hat{J}}_{z}|j,m\rangle=m|j,m\rangle\;.$
(3.8)

In the state, $|j,m\rangle$, where ${\hat{J}}_{z}$ has a definite value,
$m\,\hbar$, ${\hat{J}}_{x}$ and ${\hat{J}}_{y}$ are fluctuating. Their
magnitudes $\Delta{J}_{x,y}=\langle j,m|{\hat{J}}_{x,y}^{2}|j,m\rangle^{1/2}$
are found to be

$\Delta{J}_{x}=\Delta{J}_{y}=\left(\langle
j,m|\big{(}{\hat{J}}^{2}-{\hat{J}}_{z}^{2}\big{)}|j,m\rangle/2\right)^{1/2}=\left(j(j+1)-m^{2})/2\right)^{1/2}\hbar\sim\mathcal{O}(j)\hbar\;.$
(3.9)

Therefore, in a generic state $|j,m\rangle$, there are strong quantum
fluctuations of $J_{x}$ and $J_{y}$ at $j\to\infty$. There is no way such a
state can be associated with a classical angular momentum, which requires all
three components to be well-defined simultaneously.

The exception occurs for states $|j,\pm j\rangle$, for which

$\Delta{\hat{J}}_{x}=\Delta{\hat{J}}_{y}=(j/2)^{1/2}\hbar\ll j\,\hbar\;.$
(3.10)

For these state, it makes sense to interpret them “classically” as angular
momentum pointed towards $\pm{\hat{z}}$, as its fluctuations
$(\sim\mathcal{O}(\sqrt{j})$) become negligible with respect to its magnitude
$j$, in the limit $j\to\infty$. The same holds true in all states in which
spin is oriented towards a generic direction ${\mathbf{n}}$, as has been
already noted in the previous section, Eqs. (2.16)-(2.18). Accordingly, we
propose the correspondence (by setting $\hbar=1$)

$\boxed{\quad|j,{\mathbf{n}}\rangle\quad\longleftrightarrow\quad{\bf j}^{(\rm
class)}=j\,{\bf n}\;\quad}$ (3.11)

where

${\hat{J}}^{2}|j,{\mathbf{n}}\rangle=j(j+1)|j,{\mathbf{n}}\rangle\;,\qquad\left({\hat{\mathbf{J}}}\cdot{\mathbf{n}}\right)\,|j,{\mathbf{n}}\rangle=j\,|j,{\mathbf{n}}\rangle\;.$
(3.12)

Actually, all states “almost oriented towards ${\mathbf{n}}$”,
$|j,{\mathbf{n}},q\rangle,$

$\left({\hat{\mathbf{J}}}\cdot{\mathbf{n}}\right)\,|j,{\mathbf{n}},q\rangle=(j-q)\,|j,{\mathbf{n}},q\rangle\;,\qquad
q=1,2,\ldots\,,\,\,\,{\rm finite},$ (3.13)

discussed in Sec. 2.2.3 (see (2.22)), can also be interpreted as ${\bf
j}^{(\rm class)}=j\,{\bf n}$. The aim of our present investigation is to
examine the consistency of such a picture, with respect to the SG processes
(this has already been done in Sec. 2.2), the addition rule of the angular
momenta (Sec. 4 below) and the rotation matrix (Sec. 5 below).

As for the states with strong quantum fluctuations $|j,m\rangle$, (3.9) (with
$m$ generic and not close to $\pm j$), they are certainly quantum mechanical,
independently of $j$. The same can be said of a generic spin $j$ wave
function, (3.4). If such a particle is sent into a SG set-up, it could split
in an arbitrary distribution of sub wavepackets, in contrast to the cases
discussed in Sec. 2.2. This results in a macroscopic quantum system. It is
possible that such a strongly fluctuating, macroscopic quantum angular-
momentum state can be realized experimentally, if the whole system is brought
to extremely low temperatures near $T=0$ 666We recall that any system is in
its quantum-mechanical ground state at $T=0$. Accordingly many macroscopic
systems can be brought to a quantum-mechanical state, if sufficiently low
temperatures can be attained. See the references cited in [1] for the recent
efforts to fabricate such systems experimentally..

To see what happens at finite temperatures (either the body-temperature of the
macroscopic body, or a warm environment) instead, it is useful to recall the
so-called Born-Einstein dispute and its resolution (see [1]). Einstein
strongly rebuked the idea by Born, that the absence of quantum diffusion is
sufficient to explain the classical nature (the unique trajectoriy) of a
macroscopic body, arguing that a doubly or multiply split wave packet, with
their centers separated by macroscopic distances, are allowed by Schrödinger
equations even for a macroscopic particle. The latter is certainly non-
classical. The missing piece for solving this puzzle turns out to be the
temperature [1]. Even though at exceptionally low temperatures such a state is
possible, it is not so at finite temperatures. Emission of photons and the
ensuing self-decoherence (in the case of an isolated body) [1] or an
environment-induced decoherence (for an open system) [5, 6, 7] makes a split
system a mixture (a mixed state) essentially instantaneously. Also, they
cannot be prepared experimentally, e.g., by a passage through a double slit
[1]. A (macroscopic) particle passes either through one or the other slit, due
to absence of diffusion.

A similar consideration applies to the general angular momentum states with
strong fluctuations, (3.9) or (3.4). In the large $j$ limit, such a state
necessarily involves large space extension: a macroscopic quantum state. A
macroscopic pure quantum state can be realized only at exceedingly low
temperatures 777To tie up possible loose ends of the argument, let us note
that a single, spin $1/2$ isolated atom in its ground state (a microscopic
system) travelling in a good vacuum, has effectively $T=0$. Thus the fact that
its split wave packets can attain a macroscopic extension - i.e. its being in
a macroscopic quantum state - is perfectly consistent. To realize a
macroscopic system (made of many atoms and molecules) in a pure quantum state
is another story: it is much more difficult to prepare the necessary low
temperatures close to $T=0$. . See [1] for more discussions. At finite
temperatures, macroscopically split wave packet of such a particle under the
SG set-up is necessarily an incoherent mixture.

What is rather remarkable, perhaps, is the fact that the angular momentum
states of minimum fluctuations, (3.12), (3.13), though pure, effectively
become classical at $j\to\infty$, without need of any (thermal or environment-
induced) decoherence effects, hence independently of the temperatures. This is
in agreement with the known general semi-classical behavior of a (pure-state)
wave function in QM in the limit $\hbar\to 0$.

###  3.1 Orbital angular momentum

All our formal arguments (3.7)–(3.13) apply equally well, both to spin (the
intrinsic angular momentum of a particle or the total angular momentum of the
center of mass of a body in its rest frame) and to orbital angular momentum.
However, our primary tool of analysis is the Stern-Gerlach (SG) set-up (Sec.
2.2), which is more suitable for spin than for an angular momentum associated
with the orbital motion of a particle. One might wonder how the large angular
momentum limit works for the latter. Here is a brief acccount.

The (angular part of the) wave function of a particle in the state
$|\ell,m\rangle$ corresponds to the spherical harmonics,

$\langle\theta,\phi|\ell,m\rangle=Y_{\ell,m}(\theta,\phi)=(-)^{(m+|m|)}\sqrt{\tfrac{(2\ell+1)(\ell-|m|)!}{4\pi(\ell+|m|)!}}P_{\ell}^{|m|}(x)\,e^{im\phi}\;,\qquad
x\equiv\cos\theta\;,$ (3.14)

where $P_{\ell}^{m}(x)$ are the associated Legendre polynomials. Let us
consider the state of minimum fluctuation, $|\ell,\ell\rangle$ (see Eq.
(3.10)). It is easy to see that the angular distribution is strongly peaked at
$\theta=\tfrac{\pi}{2}$ at large $\ell$,

$|Y_{\ell,\ell}(\theta,\phi)|^{2}=\frac{2\ell+1}{4\pi}\frac{(2\ell)!}{2^{2\ell}(l!)^{2}}(1-x^{2})^{\ell}\stackrel{{\scriptstyle\ell\to\infty}}{{\longrightarrow}}\frac{1}{2\pi}\delta(x)\;.$
(3.15)

For instance, see Fig. 5 for $\ell=m=10^{3}$. In states $|\ell,m\rangle$ with
stronger fluctuations ($m$ not close to $\ell$), the distribution instead
covers a large portion of the $3$D solid angle, for any $\ell$ (see Fig. 5 for
$\ell=100,m=50$).

Now a classical angular momentum has its all three components well defined. By
choosing an appropriate coordinate system, it can be written as ${\bf L}^{(\rm
class)}=(0,0,L)$. It describes a particle moving in the $(x,y)$ plane, with
$z=p_{z}=0$. The large $\ell$ limit of the wave function
$|Y_{\ell,\ell}(\theta,\phi)|^{2}$, (3.15), describes precisely such a motion.

Figure 4: The distribution in $\theta$, for
the state $|\ell,\ell\rangle$ with $\ell=10^{3}$.

Figure 5: The distribution in $\theta$, for
the state $|\ell,m\rangle$ with $\ell=100$, $m=50$.

##  4 Addition of angular momenta

Let us now consider two spins, ${\mathbf{j}}_{1}$ and ${\mathbf{j}}_{2}$. The
composition-decomposition rules in QM is well known, i.e.

${\underline{j_{1}}}\otimes{\underline{j_{2}}}={\underline{j_{1}+j_{2}}}\oplus{\underline{j_{1}+j_{2}-1}}\oplus\ldots{\underline{{|j_{1}-j_{2}|}}}\;,$
(4.1)

where $\underline{j_{1}}$, $\underline{j_{2}}$, indicates the multiplet (the
irreducible representation) of SU(2) 888We recall that SU(2) is the simply-
connected double cover of the orthogonal group SO(3)., with multiplicity
$2j+1$ and with $\max\\{j_{z}\\}=j$. We set $\hbar=1$ here.

We wish to find out how the addition rule looks like in the limit
$j_{1},j_{2}\to\infty$. For the reasons discussed in Sec. 3, we are
particularly interested in the composition rule for angular momentum states
with minimum fluctuations, of the form, $|j,{\mathbf{n}}\rangle$. Therefore,
let us consider two particles (spins) in the states,

${\hat{\mathbf{j}}}_{1}\cdot{\mathbf{n}}_{1}\,|j_{1},{\mathbf{n}}_{1}\rangle=j_{1}\,|j_{1},{\mathbf{n}}_{1}\rangle\;,\qquad{\hat{\mathbf{j}}}_{2}\cdot{\mathbf{n}}_{2}\,|j_{2},{\mathbf{n}}_{2}\rangle=j_{2}\,|j_{2},{\mathbf{n}}_{2}\rangle\;,$
(4.2)

namely, they are spins oriented towards the directions, ${\mathbf{n}}_{1}$ and
${\mathbf{n}}_{2}$, respectively. Our aim is to find out the properties of the
direct product state,

$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle\;,$ (4.3)

in the large $j_{1},j_{2}$ limit. Since the choice of the axes is arbitrary,
we may take

${\mathbf{n}}_{2}=(0,0,1)\;.$ (4.4)

Then the product state is just

$|j_{1},{\tilde{\mathbf{n}}}_{1}\rangle\otimes|j_{2},m_{2}=j_{2}\rangle\;,$
(4.5)

where $|j_{2},m_{2}=j_{2}\rangle$ is the highest $\hat{j}_{2\,z}$ state. We
added the tilde sign on ${\mathbf{n}}_{1}$ to indicate that it is the vector
${\mathbf{n}}_{1}$ in the reference system of (4.4). Now
$|j_{1},{\tilde{\mathbf{n}}}_{1}\rangle$ in the $j_{1}\to\infty$ limit has
been studied in Sec. 2. It is

$|j_{1},{\tilde{\mathbf{n}}}_{1}\rangle=\sum_{m=-j_{1}}^{j_{1}}c_{m}\,|j_{1\,z}=m\rangle\;,\qquad$
(4.6)

where the sum is dominated by the values of $m$ around

$m=j_{1\,z}\simeq
j_{1}\cos\theta\;,\qquad\cos\theta={\tilde{\mathbf{n}}}_{1\,z}={\mathbf{n}}_{1}\cdot{\mathbf{n}}_{2}\;.$
(4.7)

Since the eigenvalues of $\hat{j_{z}}$ simply add up in the product state, the
expansion of
$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle$ in the
eigenstates of $\hat{j}_{{\rm tot}\,z}$ is dominated by terms

$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle\simeq|j_{\rm
tot},j_{\rm tot}^{{\tilde{n}}_{2}}\simeq j_{1}\cos\theta+j_{2}\rangle\;.$
(4.8)

Exchanging ${\mathbf{n}}_{1}$ and ${\mathbf{n}}_{2}$ and repeating the
arguments, one finds also that

$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle\simeq|j_{\rm
tot},j_{\rm tot}^{\tilde{n}_{1}}\simeq j_{1}+j_{2}\cos\theta\rangle\;.$ (4.9)

But which quantum angular-momentum state does the product state (4.3)
represent? The answer is that

$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle\simeq|j_{\rm
tot},{\mathbf{n}}\rangle\;,$ (4.10)

where $j_{\rm tot}$ and the unit vector ${\mathbf{n}}$ are such that (i.e.
defined by)

$j_{\rm tot}\,{\mathbf{n}}=j_{1}{\mathbf{n}}_{1}+j_{2}{\mathbf{n}}_{2}\;.$
(4.11)

The proof of this is as follows. By selecting the directions of the SG magnets
directions (the quantization axis) to ${\mathbf{n}}_{2}$ or to
${\mathbf{n}}_{1}$ on the state $|j_{\rm tot},{\mathbf{n}}\rangle$, (4.11),
one can apply the results of Sec. 2.1 to get exactly the results (4.8) or
(4.9), respectively. Now, according to the quantum-classical correspondence
(Eq. (3.11) and Eq. (3.12)), the state $|j_{\rm tot},{\mathbf{n}}\rangle$ (see
Eq. (4.11)) translates to

${\mathbf{j}}_{\rm tot}^{(\rm
class)}=j_{1}\,{\mathbf{n}}_{1}+j_{2}\,{\mathbf{n}}_{2}\;.$ (4.12)

But, in view of (3.11), this is precisely the addition rule of the two
classical angular momenta (i.e. a vector sum).

To be complete, one must check that the projection on any generic direction
${\mathbf{n}}_{3}$ works correctly. By using the result of Sec. 2 for the SG
projection on a new ${\mathbf{n}}_{3}$ axis, one finds

$|j_{1},{\mathbf{n}}_{1}\rangle\otimes|j_{2},{\mathbf{n}}_{2}\rangle\simeq|j_{\rm
tot}^{\prime},j_{\rm tot}^{\prime\,n_{3}}\simeq
j_{1}{\mathbf{n}}_{1}\cdot{\mathbf{n}}_{3}+j_{2}{\mathbf{n}}_{2}\cdot{\mathbf{n}}_{3}\rangle\;,$
(4.13)

where we made use of the fact that ${\hat{\mathbf{j}}}_{1}$ and
${\hat{\mathbf{j}}}_{2}$ commute and the eigenvalues of their
${\mathbf{n}}_{3}$ components just sum up in the direct product state. Also,
to be conservative, we have left $j_{\rm tot}^{\prime}$ unknown. However,

$j_{1}{\mathbf{n}}_{1}\cdot{\mathbf{n}}_{3}+j_{2}{\mathbf{n}}_{2}\cdot{\mathbf{n}}_{3}=(j_{1}{\mathbf{n}}_{1}+j_{2}{\mathbf{n}}_{2})\cdot{\mathbf{n}}_{3}\;,$
(4.14)

thus

${\mathbf{j}}_{\rm tot}^{\prime}={\mathbf{j}}_{\rm
tot}=j_{1}\,{\mathbf{n}}_{1}+j_{2}\,{\mathbf{n}}_{2}\;,$ (4.15)

by again using the results of Sec. 2.1 in Eq. (4.13). In conclusion, the
direct product state, (4.2), (4.3), has a simple classical interpretation. It
corresponds to the classical sum (the vector addition) of two angular momenta.

From the perspective of the general composition rule of two angular momentum
states (4.1), what we have seen are the results concerning a specific pair of
states (i.e. those of minimum fluctuations) in ${\underline{j_{1}}}$ and
${\underline{j_{2}}}$, characterized by the “orientations” ${\bf n}_{1}$ and
${\bf n}_{2}$. The total angular momentum magnitude in the product state
(4.11) satisfies

$j_{1}+j_{2}\geq j_{\rm
tot}=|j_{1}{\mathbf{n}}_{1}+j_{2}{\mathbf{n}}_{2}|\geq|j_{1}-j_{2}|\;,$ (4.16)

depending on the relative orientation of ${\mathbf{n}}_{1}$ and
${\mathbf{n}}_{2}$. This is consistent with the quantum-mechanical addition
law, (4.1).

##  5 Rotation matrix

The rotation matrix for a general spin $j$ can be constructed by taking the
direct products of the rotation matrix for spin $1/2$, i.e. (3.3). This is
based on the fact that any angular momentum state, $|j,m\rangle$, can be
constructed as the system made of $N$ spin $\tfrac{1}{2}$ particles,
completely symmetric under exchange of the particles, a fact already
conveniently used in Sec. 2.1. The theory of angular momentum can be fully
reconstructed this way [8] by using the operators,

${\hat{J}}^{a}=\sum_{i,j=1,2}a_{i}^{\dagger}\Big{(}\frac{\tau^{a}}{2}\Big{)}_{ij}a_{j}\;,\qquad[{\hat{J}}^{a},{\hat{J}}^{b}]=i\,\epsilon_{abc}{\hat{J}}^{c}\;,$
(5.1)

where $\tau^{a}$ are the Pauli matrices, $a_{i},a^{\dagger}_{i}$ are the
harmonic-oscillator annihilation and creation operators, of spin-up ($i=1$)
and spin-down ($i=2$) particles,

$[a_{i},a^{\dagger}_{j}]=\delta_{ij}\;.$ (5.2)

By construction, all states $\propto
a_{1}^{\dagger}a_{2}^{\dagger}a_{1}^{\dagger}a_{1}^{\dagger}\ldots|0\rangle$,
with $k$ $a_{1}^{\dagger}$’s and $(N-k)$ $a_{2}^{\dagger}$’s ($k=0,1,2,\ldots
N$) are symmetric under exchanges of the spins, therefore belong to the
irreducible representation ${\underline{j}}$, with $j=N/2$ 999In the SU(2)
group, any irreducible representation can be constructed from the direct
products of $N=2j$ spin $1/2$’s which are totally symmetric, so no generality
is lost. An anaologous construction for the SU($N$) group, $N\geq 3$, works
only for totally symmetric representations, $\yng(3)\cdots\yng(1)$..

The rotation matrix for a generic spin $j$ is simply (writing the Euler angles
symbolically as $\Omega\equiv\\{\gamma,\theta,\phi\\}$) the direct productof
the rotation matrices for spin $1/2$, (3.3),

$(D^{j}(\Omega))_{M,M^{\prime}}={\cal
S}\left[(D^{1/2}(\Omega))_{m_{1},m_{1}^{\prime}}(D^{1/2}(\Omega))_{m_{2},m_{2}^{\prime}}\cdots(D^{1/2}(\Omega))_{m_{N},m_{N}^{\prime}}\right]$
(5.3)

symmetrized (${\cal S}$) with respect to $(m_{1},m_{2},\ldots,m_{N})$, where

$N=2j\;,\qquad M=\sum m_{i}\;,\qquad M^{\prime}=\sum m_{i}^{\prime}\;.$ (5.4)

It acts on the multiplet,

$\displaystyle|j,j\rangle$ $\displaystyle=$
$\displaystyle|\\!\uparrow\rangle|\\!\uparrow\rangle|\\!\uparrow\rangle\cdots|\\!\uparrow\rangle\;,$
$\displaystyle|j,j-1\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\\{|\\!\downarrow\rangle|\\!\uparrow\rangle|\\!\uparrow\rangle\cdots|\\!\uparrow\rangle+|\\!\uparrow\rangle|\\!\downarrow\rangle|\\!\uparrow\rangle\cdots|\\!\uparrow\rangle+\ldots\\}\;,$
$\displaystyle\ldots$ $\displaystyle|j,M\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{{}_{N}C_{k}}}\\{{\rm
sum\,\,of\,\,terms\,\,with}\,\,k\,\,|\\!\uparrow\rangle^{\prime}s\\}\qquad(M=-j+k)$
$\displaystyle\ldots$ $\displaystyle|j,-j\rangle$ $\displaystyle=$
$\displaystyle|\\!\downarrow\rangle|\\!\downarrow\rangle|\\!\downarrow\rangle\cdots|\\!\downarrow\rangle\;.$
(5.5)

The action of $(D^{j}(\Omega))_{M,M^{\prime}}$ on particular states,

$|j,{\mathbf{n}}\rangle=|\tfrac{1}{2},{\mathbf{n}}\rangle\otimes|\tfrac{1}{2},{\mathbf{n}}\rangle\otimes\ldots|\tfrac{1}{2},{\mathbf{n}}\rangle\;,$
(5.6)

where $|\tfrac{1}{2},{\mathbf{n}}\rangle$ is given in Eq. (2.2), i.e. on the
states with all spin $\tfrac{1}{2}$’s oriented in the same direction, is very
simple. It is just the operation of rotating each and all spins towards the
${\hat{z}}$ direction (setting $\gamma=0$),

$(D^{j}(\Omega))|j,{\mathbf{n}}\rangle=|\\!\uparrow\rangle|\\!\uparrow\rangle|\\!\uparrow\rangle\cdots|\\!\uparrow\rangle=(1,0,\ldots,0)^{T}=|j,{\hat{z}}\rangle\;.$
(5.7)

This corresponds precisely to the rotation of a vector,

$R(\Omega)\,{\bf j}^{(\rm class)}=R(\Omega)\,j\,{\mathbf{n}}=(0,0,j)^{T}\;,$
(5.8)

where $R(\Omega)$ is the standard $3\times 3$ rotation matrix of classical
mechanics.

##  6 Conclusion

The observations made in this note render the concept that a spin ($S$) (or
angular momentum, $j$) becomes classical in the limit, $S(j)\to\infty$, a more
precise one. In particular, we found the identification between the states
with minimum fluctuations ((3.11), (3.12), and (3.13)) with classical angular
momentum vectors in such a limit, and verified its consistency through the
analyses of SG experiments, of the angular-momentum addition rule, and of the
rotation matrix.

At the same time, our analysis has revealed a subtlety in the quantum-
classical correspondence in this context. It reflects the difference between
the spaces of the quantum-mechanical and classical angular momentum states of
definite magnitude ($\mathbf{CP}^{2j}$ and $\mathbf{CP}^{1}\sim S^{2}$,
respectively). It is the angular momentum states of minimum fluctuations,
(3.11), (3.12), (3.13), that naturally smoothly transit into classical angular
momentum in the $S(j)\to\infty$ limit, independently of temperatures. On the
other hand, more general, strongly fluctuating angular momentum states
belonging to $\mathbf{CP}^{2j}$ are possible as pure quantum states only at
exceedingly low temperatures. The reason is that in the $j\to\infty$ limit
such a state necessarily involves a large space extension, in a way or other.
A macroscopic system is necessarily in a mixed state at finite temperatures
[5, 6, 7],[1]. For instance, no macroscopically split pure wave packets in a
SG setting survive decoherence effects.

The idea that a large spin is made of many spin $\tfrac{1}{2}$ particles seems
to be a fruitful one both mathematically and physically. From the mathematics
point of view, the entire theory of angular momentum can be reconstructed this
way [8], as we recalled already, and it helps to recover certain formulas
easily. It looks also useful from physics point of view, as such a large
angular momentum system may be regarded as an idealized model of a macroscopic
body, made of many atoms and molecules (spins). The sharp spike of the SG
projection of the state $|S,{\bf n}\rangle$ in the narrow region around
$S_{z}=S\cos\theta$ can then be understood as a results of a large number of
microscopic states of atoms (spins) accumulating to give such a value of
$S_{z}$.

The various form of suppression of the fluctuations of angular momentum
components we observed in this note in the $j\to\infty$ limit, is in fact
analogous to the “$\sqrt{n}$ law” invoked by Schrödinger [9], to explain the
exactness (classicality) of laws of macroscopic world based on statistical
mechanics, although, here, the relevant fluctuations are quantum-mechanical
ones.

## Acknowledgments

The work by KK is supported by the INFN special-initiative project grant GAST
(Gauge and String Theories). We are grateful to Hans-Thomas Elze, Jarah
Evslin, Riccardo Guida, Pietro Menotti and Arkady Vainshtein for useful
comments and precious advices.

## References

  * [1] K. Konishi, “Newton’s equations from quantum mechanics for a macroscopic body in the vacuum”, Int. Journ. Mod. Phys. A 38 2350080 (2023).
  * [2] K. Konishi and H. T. Elze, “The Quantum Ratio”, Symmetry, 16 (4) 427 (2024).
  * [3] W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld”, Zeitschrift für Physik. 9 349-352 (1922).
  * [4] K. Konishi, “On the negative-result experiments in quantum mechanics”, arXiv:2310.01955 [quant-ph]] (2023).
  * [5] E. Joos and H. D. Zeh, “The emergence of classical properties through interaction with the environment”, Z. Phys. B 59, 223-243 (1985).
  * [6] W. H. Zurek, “Decoherence and the Transition from Quantum to Classical”, Physics Today 44 (10), 36-44 (1991).
  * [7] M. Tegmark, “Apparent wave function collapse caused by scattering”, Found. Phys. Lett. 6, 571-590 (1993).
  * [8] J. Schwinger, “On Angular Momentum”, World Scientific Series in 20th Century Physics, A Quantum Legacy, 173-223 (2000). A reproduction of a book chapter from L. C. Biedenharn, H. van Dam, “Angular Momentum in Quantum Physics”, New York, Academic Press (1965).
  * [9] E. Schrödinger, “What is life?”, Cambridge University Press, (1944).

##  Appendix A Newton’s equation for a macroscopic body

The conditions needed for the CM of an isolated macroscopic body at finite
body temperatures to obey Newton’s equations have been discussed in great care
by one of the present authors [1]. They are

  * (i)

For macroscopic motions (for which $\hbar\simeq 0$) the Heisenberg relation
does not limit the simultaneous determination – the initial condition – of the
position and momentum;

  * (ii)

The absence of quantum diffusion, due to a large mass (a large number of atoms
and molecules composing the body);

  * (iii)

A finite body temperature, implying the thermal decoherence and mixed-state
nature of the body.

Under these conditions, the CM of an isolated macroscopic body has a unique
trajectory. Newton’s equations for it follow from the Ehrenfest theorem. See
Ref. [1] for various subtleties and for the explicit derivation of Newton’s
equation under external gravitational forces, under weak, static, smoothly
varying external electromagnetic fields, and under a harmonic-oscillator
potential. Somewhat unexpectedly, the environment-induced decoherence [5, 6,
7] which is extremely effective in rendering macroscopic states in a finite-
temperature environment a mixture, has been found not to be the most essential
elements for the derivation of classical mechanics from quantum mechanics [1,
2].