# Dense Gas History of the Universe: from ASPECS to the ngVLA

C.L. Carilli  F. Walter  R. Decarli  M. Aravena  Dominik A. Riechers  J.
Gonzalez-Lopez  Yali Shao  L. Boogaard  R. Bouwens  M. Neeleman

###### Abstract

We review the evolution of the cosmic average molecular gas density to large
look-back times, using observations of rotational transitions of CO. Molecular
gas is the fuel for star formation in galaxies. Deep searches for CO emission
from distant galaxies have delineated the density of molecular gas back to
$z\sim 5$, or within 1 Gyr of the Big Bang. The results show a rise and fall
in the gas density that parallels, and likely drives, the rise and fall of the
cosmic star formation rate density. We present the potential for the next
generation Very Large Array to image the distribution and dynamics of the
molecular gas in early galaxies, and to make a precise measurement of the
dense gas history of the Universe.

##  1 Introduction

Radio astronomy plays key roles in studies of galaxy formation in the distant
Universe ($z>1$, or t${}_{univ}<6$ Gyr), in many ways111This paper is
synthesized from an invited review on radio observations of galaxy formation
presented at the URSI General Assembly in Rome, 2021.. The primary
contributions in the last decade include, as a function of observing
frequency:

  * •

At frequencies $<1$ GHz: observations of HI 21 emission and absorption, using
primarily the Giant Meterwave Radio Telescope222 https://www.gmrt.org/, and
more recently, ASKAP333 https://www.atnf.csiro.au/projects/askap/index.html,
and MeerKAT444 https://www.sarao.ac.za/gallery/meerkat/, and in the future,
the Square Kilometer Array (SKA)555https://www.skatelescope.org/, determine
the extent, temperature, and cosmic mass density of the neutral atomic gas
across cosmic time [1].

  * •

At 1 GHz to 10 GHz: observations of thermal Free-Free and non-thermal
Synchrotron continuum emission with $\mu$Jy sensitivity surveys, using the
Jansky Very Large Array (VLA)666https://science.nrao.edu/facilities/vla,
ASKAP, MeerKat, and in the future, the next generation VLA
(ngVLA)777https://ngvla.nrao.edu/, and SKA, determine dust obscuration-free
star formation rates, and radio AGN demographics [2, 3, 4].

  * •

At 30 GHz to 300 GHz: observations of cool molecular gas via the rotational
transitions of CO, using the VLA, the Northern Extended Millimeter Array
(NOEMA)888https://www.iram-institute.org/EN/noema-project.php, the Australia
Telescope Compact Arrray999https://www.narrabri.atnf.csiro.au/, and the
Atacama Large Millimeter Array
(ALMA)101010https://www.almaobservatory.org/en/home/ [5, 6], determine the
mass of the dense gas that acts as the immediate fuel for star formation in
galaxies.

  * •

At $>100$ GHz: observations of thermal emission from warm dust, with ALMA and
NOEMA, and single dish telescopes, reveal the dust-obscured star formation in
early galaxies [7, 9].

  * •

At $>250$ GHz: observations of [CII] 158 $\mu$m, and other fine structure
lines with ALMA, at $z=3$ to 10, reveal the atomic gas, and galaxy dynamics
back to the first galaxies in the Universe [11, 10].

In this short contribution, we focus on the latest results for the evolution
of the molecular gas density, as measured in the 30 GHz to 300 GHz range,
particularly by ALMA with the ASPECS survey. We then discuss the potential for
the ngVLA to advance these studies, in the age of high precision cosmology.

##  2 ASPECS Survey

Molecular gas is the immediate fuel for star formation in galaxies, and hence
constitutes a key constituent in models of galaxy formation [5, 6]. Only
recently, measurements of the cosmic volume average density of molecular gas
have been made, through the advent of large, wideband interferometers, such as
the Jansky Very Large Array, the Atacama Large Millimeter Array and NOEMA.
While the molecular gas mass is dominated by $H_{2}$, for practical reasons, a
primary (although not exclusive), method for tracing molecular gas mass
entails observation of the rotational transitions of CO, and adoption of
calibrated conversion factors [12].

The ALMA Spectroscopic Survey (ASPECS), is the definitive deep blind search
for molecular gas via CO emission in the distant Universe [13]. The survey
covered 5 arcmin2 in the Hubble Ultra-Deep field, scanning the full ALMA 90
GHz and 230 GHz bands at $\sim 1"$ resolution, covering multiple CO
transitions down to gas mass limits of a few 109 M⊙, and covering a comoving
volume of 42,000 Mpc3. Figure 1 shows a three-dimensional view of the ASPECS
field (two sky coordinates, and frequency = redshift = distance), with an
example of a CO detected galaxy [7, 13, 8]. A total of 32 CO emission lines
were detected from galaxies at $z=0.5$ to 3.6.

Properties of this unique galaxy sample selected by molecular gas mass
include:

  * •

60% are detected in dust continuum in the associated deep ALMA image, with
star formation rates $>10$ M⊙ year-1 [7].

  * •

100% are detected in the deep optical and near-IR images of the UDF [15, 7].

  * •

70% are ‘main sequence’ disk galaxies, and the rest are compact or irregular
[15].

  * •

Metallicities are typically Solar or greater [15].

  * •

Chandra X-ray observations imply 20% of the galaxies host AGN [15].

  * •

Multi-line CO observations imply modest excitation [18].

A key result from ASPECS is the confirmation that the typical gas mass to
stellar mass ratio in massive disk galaxies increases from $\sim 0.1$ in the
nearby Universe, to $\geq 1$ at $z\sim 3$ [16]. Also, the deep 230 GHz
continuum image from the survey implies a clear flattening of the source
counts below 0.1 mJy, which has important implications for future submm deep
fields [19].

The ASPECS survey, along with other deep blind surveys, such as the VLA COLDZ
large program [20], and targeted observations of known galaxies [6], have
determined the evolution of the cosmic volume average density of molecular gas
to within $\sim 1$ Gyr of the Big Bang, or a look-back time of $\sim 13$ Gyr.
Figure 2 shows a compendium of molecular gas measurements, along with the
evolution of the cosmic star formation rate density [21]. Both quantities show
a rise and fall with cosmic time, peaking in the range $z\sim 1.5$ to 2.5
(tuniv = 3.3 Gyr). This peak in the cosmic star formation rate density, during
which about half of the stars in the current Universe form, has long been
known [23, 22], but not fully explained. The parallel rise and fall in the
molecular gas density provides strong circumstantial evidence that the
evolution of the cosmic star formation rate density is a consequence of the
evolution of the molecular gas content of galaxies.

Figure 1: A 3D view of the ASPECS survey (RA, Dec, Frequency = distance), in
the ALMA 240 GHZ bands, and an example image and spectra of one galaxy [17, 8,
15]. The bright knots are emission lines, and the broad streak indicates a
thermal dust continuum signal.

The molecular gas measurements were the last piece in the puzzle of the
evolution of the baryonic matter associated with galaxies. A consistent
picture is emerging in which galaxies accrete ionized material from the
intergalactic medium, which passes through a neutral atomic phase, then builds
up as the molecular phase, which then collapses to from stars [25]. The need
for gas replenishment to fuel continued star formation in early galaxies was
pointed out in the early paper by [26]. The latest measurements confirm this
requirement.

Unfortunately, the molecular gas measurements remain limited, with large
errors at each redshift, and the ability to perform high resolution imaging of
the gas in distant galaxies remains prohibitively expensive, even with the
biggest existing interferometers.

Figure 2: Left: Evolution of the cosmic volume average density of star
formation rate [21]. The upper points, highlighted by the orange curve,
include a correction for dust obscuration, while the lower blue-highlighted
curve does not. Right: Evolution of the cosmic volume average molecular gas
density [24, 20, 25, 27]. The black and grey points are current measurements,
and the blue points and errors are predictions for the ngVLA deep fields.

##  3 Next Generation Very Large Array

The next generation Very Large Array (ngVLA) constitutes an order of magnitude
increase in collecting area over the current VLA and ALMA, covering the
frequency range from $\sim 1$ GHz to 116 GHz, with baselines from tens of
meters to thousands of kilometers [29, 28]. A precise measurement of the dense
gas history, and high resolution imaging of molecular gas in distant galaxies,
are primary science drivers for the ngVLA [31, 32]. In just a few hours
integration, the ngVLA can detect a gas mass of $\sim 10^{9}$ M⊙ at $z\sim 2$,
at a resolution of $0.15^{\prime\prime}$. The wide frequency range of the
ngVLA also allows for multi-transition studies of molecular gas excitation,
including the low order transitions which provide perhaps the best measure of
total molecular gas mass [30].

In terms of the cosmic density of molecular gas, Figure 2 shows the capability
of the ngVLA vs. the best current measurements. The increased bandwidth and
sensitivity of the ngVLA will increase the survey speed by almost two orders
of magnitude, revealing thousands of galaxies in blind surveys (as opposed to
the current tens). The results will provide a precise measurement of the
evolution of the molecular mass density [31], to complement future
measurements of the cosmic star formation rate density by JWST and the ELTs.

For CO imaging, Figure 3 shows a simulation of ngVLA observations (30 hours)
of CO 2-1 emission from a massive disk galaxy at $z=4.2$ at a resolution of
$0.2"$ (total gas mass $\sim$ few$\times 10^{10}$ M⊙), including velocity
integrated CO emission (column density), and the mean velocity. The gas can be
easily traced over the 10 kpc disk, and the velocities are well determined.
Also shown is a rotating disk model fit to the data, from which a rotational
velocity, and even radial profile, can be derived, potentially determining the
dark matter content of the first galaxies. Such observations are well beyond
the capability of existing facilities [32].

Figure 3: Simulation of an ngVLA observation at 44 GHz of a $z=4.2$ massive
disk galaxy. Left is the total molecular gas column density, center is the
intensity weighted mean velocity, and right is a best-fit model of a rotating
disk galaxy [32].

Figure 4 shows an ngVLA simulated observation of CO 1-0 emission of a forming
massive cluster of galaxies at $z=2$. A recent exciting discovery has been the
detection of CO emission on tens to 100 kpc-scales in forming clusters of
galaxies [33]. The ngVLA has the ability to resolve the molecular gas
distribution down to 1 kpc resolution, and to delineate the extended CO
emission out to $\sim 100$ kpc [34].

Figure 4: Simulation of an ngVLA observation at 38 GHz of CO 1-0 emission from
a $z=2$ forming giant galaxy and proto-cluster. Left is the model. Center is
the total molecular gas column density at $0.13"$ = 1 kpc resolution for the
simulated observations. Right is the column density at $1"$ resolution
(adapted from [35]).

##  4 Discussion: Future Context

We have reviewed the current status of measurements of the evolution of the
molecular gas content of galaxies back to within $\sim 1$Gyr of the Big Bang,
and it’s relationship to the star formation history of the Universe. We then
present the capabilities of the next generation VLA to advance these studies
into the next decade, to obtain a precise measurement of the dense gas history
of the Universe. We briefly discuss the broader radio astronomy landscape in
the immediate future, and beyond.

A number of telescopes arrays are either starting operation, or expected to
attain first light in the coming 5 years. These include the current operations
of the $\mu$GMRT, MeerKAT, and ASKAP, and the future operation of phase I of
the SKA. Regardless of sensitivity, all of these facilities operate at
frequencies below 15 GHz. Hence, while they are, or will be, powerful devices
to study the HI 21cm emission and radio continuum emission from distant
galaxies, they will be unable to study the primary rotational molecular
transitions, except for CO 1-0 at $z>7$. At such high redshift, this
transition is highly suppressed due to depopulation of the lower excitation
states by the CMB, and effectively undetectable. The SKA phase II may
eventually go up to 30 GHz, still being limited to only CO 1-0 at high
redshift ($z>3$).

Study of the rotational molecular transitions remains the territory of
millimeter telescopes, such as the current NOEMA and ALMA, and, toward the end
of the decade, the ngVLA. We have shown that, in terms of measuring the total
gas masses, and imaging of the molecular gas on kpc-scales, the ngVLA, with
its order of magnitude improvement in collecting area, will open unique
parameter space critical for advancing our understanding of galaxy formation.

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Chris Carilli, NRAO, Socorro, NM, USA, 87801<EMAIL_ADDRESS>
Fabian Walter and Marcel Neeleman, Max-Planck Institute for Astronomy,
Heidelberg, DE<EMAIL_ADDRESS><EMAIL_ADDRESS>
R. Decarli, INAF - Osservatorio di Astrofisica, Bologna, Italy,
<EMAIL_ADDRESS>
M. Aravena and J. Gonzalez-Lopez, Facultad di Ingenieria y Ciencias, Univ.
Diego-Portales, Santiago, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS>
D. Riechers, Physics Institute, Univ. of Cologne, Cologne, Germany,
<EMAIL_ADDRESS>
Yali Shao, Max-Planck Institute for Radio Astronomy, Bonn, Germany,
<EMAIL_ADDRESS>
R. Bouwens, L. Boogaard (Leiden Observatory, Leiden University, Leiden, The
Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>


#  Entanglement in joint $\Lambda\bar{\Lambda}$ decay; cont.

Göran Fäldt<EMAIL_ADDRESS>Department of physics and astronomy,
Uppsala University, Box 516, S-751 20 Uppsala,Sweden

###### Abstract

We have previously investigated joint $\Lambda\bar{\Lambda}$ decay in the
reaction $e^{+}e^{-}\rightarrow\gamma\Lambda(\rightarrow
p\pi^{-})\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})$. The cross-section-
distribution functions encountered were relativistically covariant and
expressed in terms of scalar products of the four-momentum vectors of the
particles involved. In the present, sequel investigation, we show that by
working with three-momentum scalars instead results could possibly become more
transparent.

##  I Introduction

The BABAR Collaboration BaBar  has measured initial-state-radiation in the
annihilation reaction $e^{+}e^{-}\rightarrow\gamma\Lambda(\rightarrow
p\pi^{-})\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})$. Such measurements are
interesting since they offer opportunities to determine electromagnetic form
factors of the Lambda hyperons in the time-like region.

A theoretical analysis of this annihilation reaction is presented in refs.Novo
and Czyz . A complete determination of the cross-section-distribution
functions, including those describing joint Lambda anti-Lambda decays, is also
given in ref.Ent1 . The arguments of those functions, and there are several,
are the scalar products of the four-momentum vectors of the particles
involved. To determine all those functions is a formidable task, and we have
therefore decided on an alternative approach, to work with three-momentum
vectors instead.

Replacing four-dimensional arguments by three-dimensional ones also requires
considerable work, but this work is worth-while, as we shall see. The result
is transparent.

##  II Cross-section distribution

Our notation follows Pilkuhn Pil . The cross-section distribution for the
reaction $e^{+}e^{-}\rightarrow\gamma\Lambda(\rightarrow
p\pi^{-})\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})$ is written as

$\textrm{d}\sigma=\frac{1}{2\sqrt{\lambda(s,m_{e}^{2},m_{e}^{2})}}\,\overline{|{\cal{M}}|^{2}}\,\textrm{dLips}(k_{1}+k_{2};q,l_{1},l_{2},q_{1},q_{2}),$
(1)

where the average over the squared matrix element indicates summation over
final proton and anti-proton spins and average over initial electron and
positron spins. The definitions of the particle momenta are explained in
fig.1.

We remove some trivial factors from the squared matrix element, collected in a
factor denoted ${\cal{K}}$,

$\overline{|{\cal{M}}|^{2}}={\cal{K}}\overline{|{\cal{M}}_{red}|^{2}}.$ (2)

##  III Previous analysis

We start where our previous analysis ended, ref.Ent1 , but before we can do so
it is necessary to repeat some of the important definitions and results.

The form factors of the hyperon-electromagnetic couplings are denoted $G_{1}$
and $G_{2}$, a standard choice. The designations of particle four-momenta can
be seen in the Feynman diagrams of fig. 1.

Figure 1: Graphs included in our calculation of the reaction
$e^{+}e^{-}\rightarrow\gamma\Lambda(\rightarrow
p\pi^{-})\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})$.

The cross-section-distribution function, or rather the covariant square of the
annihilation matrix element $\overline{|{\cal{M}}_{red}|^{2}}$, is the
contraction of hadronic $H_{\mu\nu}$ and leptonic $L^{\mu\nu}$ tensors,

$\overline{|{\cal{M}}_{red}|^{2}}=L^{\mu\nu}H_{\mu\nu}.$ (3)

Its right-hand side is naturally decomposed as

$\overline{|{\cal{M}}_{red}|^{2}}=\bar{R}_{\Lambda}R_{\Lambda}M^{RR}+\bar{R}_{\Lambda}S_{\Lambda}M^{RS}+\bar{S}_{\Lambda}R_{\Lambda}M^{SR}+\bar{S}_{\Lambda}S_{\Lambda}M^{SS},$
(4)

with suffixes $RS$ and $\bar{R}\bar{S}$ referring to $\Lambda$ and
$\bar{\Lambda}$ decay constants, with $R$ the spin-independent and $S$ the
spin-dependent one.

From the structure of the lepton tensor, eq.(24) of ref.Ent1 , one concludes
that each of the $M^{XY}$ functions has two parts,

$M^{XY}=-a_{y}A^{XY}(G_{1},G_{2})-b_{y}B^{XY}(G_{1},G_{2}),$ (5)

where the $A^{XY}$ factor is obtained by contracting the hadron tensor with
the symmetric tensor $k_{1\mu}k_{1\nu}+k_{2\mu}k_{2\nu}$, and the $B^{XY}$
factor by contraction with the tensor $g_{\mu\nu}.$ For details see ref.Ent1 .
The weight factors $a_{y}$ and $b_{y}$ are defined in appendix A.

The functions $A^{XY}$ and $B^{XY}$ are bilinear forms of $G_{1}$ and $G_{2}$,
and we expand them accordingly, for $A^{XY}$,

$\displaystyle A^{XY}(G_{1},G_{2})$ $\displaystyle=$
$\displaystyle|G_{1}|^{2}{\cal{K}}^{AXY}_{1}+|G_{2}|^{2}{\cal{K}}^{AXY}_{2}$
(6) $\displaystyle+$ $\displaystyle
2\Re(G_{1}G_{2}^{\star}){\cal{K}}^{AXY}_{3}+2\Im(G_{1}G_{2}^{\star}){\cal{K}}^{AXY}_{4},$

and similarly for $B^{XY}$. We refer to the functions $\\{{\cal{K}}\\}$ as co-
factors. They are Lorentz covariant functions of the particle four-momenta and
the functions of our attention.

##  IV Previous results

The leading term of eq.(4) is $M^{RR}$ as it is independent of variables that
relate to spin dependences in the hyperon decay distributions. We have

$\displaystyle A^{RR}$ $\displaystyle=$ $\displaystyle
2|G_{1}|^{2}\bigg{[}(k_{1}\cdot P)^{2}+(k_{2}\cdot P)^{2}-(k_{1}\cdot
Q)^{2}-(k_{2}\cdot Q)^{2}\bigg{]}$ (7)
$\displaystyle+4\Re(G_{1}G_{2}^{\star})\bigg{[}(k_{1}\cdot Q)^{2}+(k_{2}\cdot
Q)^{2}\bigg{]}-|G_{2}|^{2}\frac{Q^{2}}{2M^{2}}\bigg{[}(k_{1}\cdot
Q)^{2}+(k_{2}\cdot Q)^{2}\bigg{]},$

with $Q=p_{1}-p_{2}.$ Furthermore,

$\displaystyle B^{RR}$ $\displaystyle=$
$\displaystyle-4|G_{1}|^{2}(P^{2}+2M^{2})+4\Re(G_{1}G_{2}^{\star})Q^{2}-|G_{2}|^{2}\frac{(Q^{2})^{2}}{2M^{2}}.$
(8)

Thus, the distribution function $M^{RR}$ does not depend on the decay momenta
$l$ and $q$ of the Lambda hyperons.

Next in order are terms linear in the spin variables,

$\displaystyle A^{RS}$ $\displaystyle=$
$\displaystyle-4\Im(G_{1}G_{2}^{\star})\bigg{[}k_{1}\cdot
Q\,\mbox{det}(p_{2}p_{1}l_{1}k_{1})+k_{2}\cdot
Q\,\mbox{det}(p_{2}p_{1}l_{1}k_{2})\bigg{]},$ (9) $\displaystyle A^{SR}$
$\displaystyle=$ $\displaystyle-4\Im(G_{1}G_{2}^{\star})\bigg{[}k_{1}\cdot
Q\,\mbox{det}(p_{2}p_{1}l_{2}k_{1})+k_{2}\cdot
Q\,\mbox{det}(p_{2}p_{1}l_{2}k_{2})\bigg{]},$ (10)

with
$\mbox{det}(abcd)=\epsilon_{\alpha\beta\gamma\delta}a^{\alpha}b^{\beta}c^{\gamma}d^{\delta}$
and

$\displaystyle B^{RS}$ $\displaystyle=$ $\displaystyle 0,$ (11) $\displaystyle
B^{SR}$ $\displaystyle=$ $\displaystyle 0.$ (12)

The expressions for the spin-spin contributions are more complicated. We have
for the $A$ contribution

$\displaystyle A^{SS}$ $\displaystyle=$
$\displaystyle-2|G_{1}|^{2}\bigg{[}\bigg{(}(k_{1}\cdot P)^{2}+(k_{2}\cdot
P)^{2}-(k_{1}\cdot Q)^{2}-(k_{2}\cdot Q)^{2}\bigg{)}\bigg{(}p_{1}\cdot
l_{1}p_{2}\cdot l_{2}+M^{2}l_{1}\cdot l_{2}\bigg{)}$ (13)
$\displaystyle\qquad\quad+2M^{2}\bigg{(}P^{2}(k_{1}\cdot l_{1}k_{1}\cdot
l_{2}+k_{2}\cdot l_{1}k_{2}\cdot l_{2})-2P\cdot l_{2}(k_{1}\cdot
l_{1}k_{1}\cdot p_{2}+k_{2}\cdot l_{1}k_{2}\cdot p_{2})$
$\displaystyle\qquad\qquad\qquad\quad-2P\cdot l_{1}(k_{1}\cdot l_{2}k_{1}\cdot
p_{1}+k_{2}\cdot l_{2}k_{2}\cdot p_{1})\bigg{)}\bigg{]}$
$\displaystyle-4\Re(G_{1}G_{2}^{\star})\bigg{[}M^{2}l_{1}\cdot
l_{2}\bigg{(}(k_{1}\cdot Q)^{2}+(k_{2}\cdot Q)^{2}\bigg{)}$
$\displaystyle\qquad\quad-M^{2}\bigg{(}P\cdot l_{2}(k_{1}\cdot Qk_{1}\cdot
l_{1}+k_{2}\cdot Qk_{2}\cdot l_{1})-P\cdot l_{1}(k_{1}\cdot Qk_{1}\cdot
l_{2}+k_{2}\cdot Qk_{2}\cdot l_{2})\bigg{)}$ $\displaystyle\qquad\quad-
k_{1}\cdot Q\bigg{(}k_{1}\cdot p_{1}p_{2}\cdot l_{1}p_{2}\cdot
l_{2}-k_{1}\cdot p_{2}p_{1}\cdot l_{1}p_{1}\cdot
l_{2}-\mbox{${\textstyle\frac{1}{2}}$}P^{2}k_{1}\cdot l_{1}p_{2}\cdot
l_{2}+\mbox{${\textstyle\frac{1}{2}}$}P^{2}k_{1}\cdot l_{2}p_{1}\cdot
l_{1}\bigg{)}$ $\displaystyle\qquad\quad-k_{2}\cdot Q\bigg{(}k_{2}\cdot
p_{1}p_{2}\cdot l_{1}p_{2}\cdot l_{2}-k_{2}\cdot p_{2}p_{1}\cdot
l_{1}p_{1}\cdot l_{2}-\mbox{${\textstyle\frac{1}{2}}$}P^{2}k_{2}\cdot
l_{1}p_{2}\cdot l_{2}+\mbox{${\textstyle\frac{1}{2}}$}P^{2}k_{2}\cdot
l_{2}p_{1}\cdot l_{1}\bigg{)}\bigg{]}$
$\displaystyle-|G_{2}|^{2}\frac{1}{2M^{2}}\bigg{(}(k_{1}\cdot
Q)^{2}+(k_{2}\cdot Q)^{2}\bigg{)}\bigg{[}Q^{2}\bigg{(}p_{1}\cdot
l_{1}p_{2}\cdot l_{2}-M^{2}l_{1}\cdot l_{2}\bigg{)}+2M^{2}Q\cdot l_{1}Q\cdot
l_{2}\bigg{]},$

and for the $B$ contribution

$\displaystyle B^{SS}$ $\displaystyle=$
$\displaystyle+4|G_{1}|^{2}\bigg{[}(P^{2}+2M^{2})(p_{1}\cdot l_{1}p_{2}\cdot
l_{2}+M^{2}l_{1}\cdot l_{2})$ (14)
$\displaystyle\qquad\qquad-M^{2}\bigg{(}P^{2}l_{1}\cdot l_{2}+2P\cdot
l_{2}l_{1}\cdot p_{1}+2P\cdot l_{1}l_{2}\cdot p_{2}\bigg{)}\bigg{]}$
$\displaystyle-4\Re(G_{1}G_{2}^{\star})\bigg{[}Q^{2}M^{2}l_{1}\cdot
l_{2}-M^{2}\bigg{(}Q\cdot l_{1}P\cdot l_{2}-Q\cdot l_{2}P\cdot l_{1}\bigg{)}$
$\displaystyle\qquad\quad-\bigg{(}p_{1}\cdot Qp_{2}\cdot l_{1}p_{2}\cdot
l_{2}-p_{2}\cdot Qp_{1}\cdot l_{1}p_{1}\cdot
l_{2}-\mbox{${\textstyle\frac{1}{2}}$}P^{2}Q\cdot l_{1}p_{2}\cdot
l_{2}+\mbox{${\textstyle\frac{1}{2}}$}P^{2}Q\cdot l_{2}p_{1}\cdot
l_{1}\bigg{)}\bigg{]}$
$\displaystyle-|G_{2}|^{2}\frac{Q^{2}}{2M^{2}}\bigg{[}Q^{2}\bigg{(}p_{1}\cdot
l_{1}p_{2}\cdot l_{2}-M^{2}l_{1}\cdot l_{2}\bigg{)}+2M^{2}Q\cdot l_{1}Q\cdot
l_{2}\bigg{]}.$

The functions $A^{SS}$ and $B^{SS}$ describe the joint-decay distributions of
the Lambda and anti-Lambda hyperons. The distributions are correlated, i.e.,
they cannot be written as a product of Lambda and anti-Lambda distribution
functions. Our distribution functions are explicitly covariant, as they are
expressed in terms of the four-momentum vectors of the participating
particles. It is not necessary to work in several coordinate systems, as in
refs. BaBar  and Czyz . Another important point is that our calculation
correctly counts the number of intermediate hyperon states.

##  V Reference frames

The cross-section distribution function of sect. IV involves expressions that
are functions of scalar products of particle four-momenta. To determine the
scalar product of two four-vectors requires knowledge of those vectors in one
and the same reference frame. Our task in this section is to demonstrate how
this is achieved.

Designations of the particle four-momenta follow from the energy-momentum-
balance condition in the reaction
$e^{+}e^{-}\rightarrow\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})\Lambda(\rightarrow
p\pi\\-)\gamma$,

$k_{1}+k_{2}=p_{1}+p_{2}+q.$ (15)

Additional information is contained in fig. 1.

The gamma three-momentum $\mathbf{q}$, and electron three-momentum
$\mathbf{k}$, are momenta defined in the $e^{+}e^{-}$ centre-of-momentum
(c.m.) reference frame, in which
$\hat{\mathbf{q}}\cdot\hat{\mathbf{k}}=\cos\theta$. We refer to this frame as
$S_{0}$. In $S_{0}$ electron and positron four-momenta are
$k_{1}=\epsilon(1,\hat{\mathbf{k}})$ and
$k_{2}=\epsilon(1,-\hat{\mathbf{k}})$, with $\epsilon$ the common lepton
energy. With $\omega$ the gamma energy, the gamma four-momentum is denoted
$q=\omega(1,\hat{\mathbf{q}})$. Furthermore, the four-momenta of Lambda and
anti-Lambda are $p_{1}=(E_{1},\mathbf{p}_{1})$ and
$p_{2}=(E_{2},\mathbf{p}_{2})$.

Now, we shall not perform our calculations in $S_{0}$ but in $S_{1}$ which is
the c.m. frame of the $\Lambda\bar{\Lambda}$ pair. We indicate variables in
this frame by a prime, so that

$p_{1,2}^{\prime}=(E_{\Lambda},\pm p_{\Lambda}\mathbf{f})$ (16)

with $E_{\Lambda}=\sqrt{p_{\Lambda}^{2}+M_{\Lambda}^{2}}$ and $\mathbf{f}$ a
unit vector. The $\Lambda\bar{\Lambda}$ c.m. energy $W=2E_{\Lambda}$ may be
obtained from the identity

$W^{2}=4\epsilon(\epsilon-\omega).$ (17)

The next question concerns the relation between frames $S_{1}$ and $S_{0}$.
Since $\mathbf{p}_{1}+\mathbf{p}_{2}=-\mathbf{q}$ in $S_{0}$, we argue that
$S_{1}$ can be reached from $S_{0}$ through a boost along the direction of
motion of the gamma, and of magnitude,

$v=\frac{-(\mathbf{p}_{1}+\mathbf{p}_{2})\cdot\mathbf{q}}{E_{1}+E_{2}}=\frac{\omega}{\sqrt{\omega^{2}+W^{2}}},$
(18)

and with Lorentz-transformation (LT) coefficient

$\gamma(v)=\frac{1}{\sqrt{1-v^{2}}}=\frac{\sqrt{\omega^{2}+W^{2}}}{W}.$ (19)

Also, note that $v$ is the relative velocity between two reference frames, it
is not a particle velocity.

A Lorentz boost from $S_{0}$ to $S_{1}$ leads to new four-momentum vectors for
the initial state leptons, namely

$k_{1,2}^{\prime}=\epsilon\gamma\bigg{[}(1\pm
v\mathbf{n}\cdot\hat{\mathbf{k}});\ v(\mathbf{n}\pm\mathbf{N})\bigg{]},$ (20)

and with $\mathbf{n}$ and $\mathbf{N}$ by definition

$\displaystyle\mathbf{n}$ $\displaystyle=$ $\displaystyle\hat{\mathbf{q}},$
(21) $\displaystyle\mathbf{N}$ $\displaystyle=$
$\displaystyle\frac{1}{v\gamma}\bigg{[}\hat{\mathbf{k}}+(\gamma-1)(\mathbf{n}\cdot\hat{\mathbf{k}})\mathbf{n}\bigg{]}.$
(22)

Relations (21) and (22) are identical to those introduced by the BaBar
collaboration BaBar .

The photon radiated in our annihilation process carries energy $\omega$ and
three-momentum $\mathbf{q}=\omega\mathbf{n}$ , when observed in $S_{0}$. A
boost from $S_{0}$ to $S_{1}$, sends this vector into
$\mathbf{q}^{\prime}=\omega^{\prime}\mathbf{n}$, with

$\omega^{\prime}=\omega\sqrt{\frac{1+v}{1-v}}.$ (23)

However, we should not forget the decay products of the hyperons, the
antiproton and the proton. In the rest system $S_{2}$ of the Lambda the proton
is represented by the four-vector

$l_{1}=(E_{g},p_{g}\,\mathbf{g}),$ (24)

with $\mathbf{g}$ a unit vector, and with decay parameters $p_{g}$ and
$E_{g}$.

Similarly, in the rest system $S_{3}$ of the anti-Lambda the anti-proton is
represented by the four-vector

$l_{2}=(E_{h},p_{h}\mathbf{h}),$ (25)

with $\mathbf{h}$ a unit vector, and decay parameters $p_{h}=p_{g}$ and
$E_{h}=E_{g}$. A passage from $S_{3}$ to $S_{1}$ is achieved by a Lorentz
boost with velocity $v_{h}$ and direction $\mathbf{f}$, whereas a passage from
$S_{2}$ to $S_{1}$ is achieved by a Lorentz boost with velocity $v_{g}$ and
direction $-\mathbf{f}$.

The boost equations for the massive hyperons are well known. Vectors
orthogonal to the boost velocity, $\mathbf{v}=v\mathbf{n}$, are unchanged,
those parallel are changed according to the Lorentz-transformation
prescription

$\displaystyle\mathbf{p}_{1,2}^{\prime}$ $\displaystyle=$
$\displaystyle\big{[}\mathbf{p}_{1,2}-\mathbf{n}(\mathbf{n}\cdot\mathbf{p}_{1,2})\big{]}+\gamma(v)\mathbf{n}\big{[}\mathbf{n}\cdot\mathbf{p}_{1,2}+vE_{1,2}\big{]},$
(26) $\displaystyle E_{1,2}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma(v)\big{[}E_{1,2}-\mathbf{v}\cdot\mathbf{p}_{1,2}\big{]}.$
(27)

The inverse-transformation equations, going from $S_{1}$ to $S_{0}$, are
obtained by changing the sign of the velocity, from $\mathbf{n}$ to
$-\mathbf{n}$.

After this elementary discussion we are ready for the proton and antiproton
four-momentum vectors in $S_{1}$; for the proton

$\displaystyle\mathbf{p}_{g}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma_{\Lambda}E_{g}\mathbf{f}\big{[}\mathbf{f}\cdot(\mathbf{v}_{g}+\mathbf{v}_{\Lambda})\big{]}+\mathbf{p}_{g\bot},$
(28) $\displaystyle E_{g}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma_{\Lambda}E_{g}\big{[}1+\mathbf{v}_{\Lambda}\cdot\mathbf{v}_{g}\big{]},$
(29)

the transverse-vector component $\mathbf{p}_{g\bot}$ being

$\mathbf{p}_{g\bot}=\mathbf{p}_{g}-(\mathbf{p}_{g}\cdot\mathbf{f})\,\mathbf{f},$
(30)

with $\mathbf{p}_{g\bot}\cdot\mathbf{f}=0;$ and for the antiproton

$\displaystyle\mathbf{p}_{h}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma_{\Lambda}E_{h}\mathbf{f}\big{[}\mathbf{f}\cdot(\mathbf{v}_{h}-\mathbf{v}_{\Lambda})\big{]}+\mathbf{p}_{h\bot},$
(31) $\displaystyle E_{h}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma_{\Lambda}E_{h}\big{[}1-\mathbf{v}_{\Lambda}\cdot\mathbf{v}_{h}\big{]}.$
(32)

Our calculations make use of the shorthand notations,

$\displaystyle G_{g}$
$\displaystyle=(\mathbf{v}_{g}+\mathbf{v}_{\Lambda})\cdot\mathbf{f},$
$\displaystyle H_{g}$
$\displaystyle=1+\mathbf{v}_{\Lambda}\cdot\mathbf{v}_{g},$ (33) $\displaystyle
G_{h}$ $\displaystyle=(\mathbf{v}_{h}-\mathbf{v}_{\Lambda})\cdot\mathbf{f},$
$\displaystyle H_{h}$
$\displaystyle=1-\mathbf{v}_{\Lambda}\cdot\mathbf{v}_{h}.$ (34)

##  VI Calculating co-factors

Co-factors can be identified in the $A^{XY}$ and the $B^{XY}$ functional
distributions of sect. IV. The results are co-factors expressed in terms of
scalar products of four-vector momenta. However, our goal was to find simpler
expressons, and this by evaluating all scalars in one and the same reference
frame, the c.m. reference frame of the $\Lambda\bar{\Lambda}$ pair.

We have evaluated cross-section distributions for two sets of form-factor
parameters. The two sets have attached form-factor sets, that we indicate by
different letters, such that $(G_{1},G_{2})\Rightarrow\\{{\cal{K}}\\}$ and
$(G_{M},G_{E})\Rightarrow\\{{\cal{L}}\\}$ . We start with the ${\cal{K}}$ set
and return to the ${\cal{L}}$ set in sect.VII.

The much needed $\Omega$ functions are defined in appendix B. The spin-
correlation functions are equally important. Their definitions are,

$\displaystyle X_{a}(\mathbf{g},\mathbf{h})$ $\displaystyle=$ $\displaystyle
2\mathbf{g}\cdot\mathbf{f}\,\mathbf{h}\cdot\mathbf{f}-\mathbf{g}\cdot\mathbf{h},$
(35) $\displaystyle X_{b}(\mathbf{g},\mathbf{h})$ $\displaystyle=$
$\displaystyle\mathbf{g}\cdot\mathbf{f}\,\mathbf{h}\cdot\mathbf{f}.$ (36)

A parameter that appears in practically every formula is the $Z$ parameter,

$Z=\frac{4M_{\Lambda}^{2}}{Q^{2}}=\frac{-1}{\gamma_{\Lambda}^{2}v_{\Lambda}^{2}}=1-\frac{1}{v_{\Lambda}^{2}}.$
(37)

Other important parameters are $v_{\Lambda}=p_{\Lambda}/E_{\Lambda}$ and
$\gamma_{\Lambda}=E_{\Lambda}/M_{\Lambda}$.

###  VI.1 Spin-independent co-factors

We start with the ${\cal{K}}$ base and the ${\cal{K}}^{ARR}$ co-factors, which
can be extracted from the $A^{RR}$ contribution to the $M^{RR}$ functional
distribution of eq.(7). Since he calculation is straightforward we are
satisfied with the result

$\displaystyle{\cal{K}}_{1}^{ARR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\bigg{[}-Z+\Omega_{\bot}/(\Omega_{f}v_{\Lambda}^{2})\bigg{]},$
(38) $\displaystyle{\cal{K}}_{2}^{ARR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\bigg{[}\frac{-1}{Z}\bigg{]},$
(39) $\displaystyle{\cal{K}}_{3}^{ARR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\bigg{[}1\bigg{]}.$
(40)

The $\Omega$ functions are described in appendix B.

Next, we extract the ${\cal{K}}^{BRR}$ co-factors from the $B^{RR}$
contribution (8) to the $M^{RR}$ functional distribution of eq.(5);

$\displaystyle{\cal{K}}_{1}^{BRR}$ $\displaystyle=$
$\displaystyle-2Q^{2}\bigg{[}Z-2(1-Z)\bigg{]},$ (41)
$\displaystyle{\cal{K}}_{2}^{BRR}$ $\displaystyle=$
$\displaystyle-2Q^{2}\bigg{[}\frac{1}{Z}\bigg{]},$ (42)
$\displaystyle{\cal{K}}_{3}^{BRR}$ $\displaystyle=$
$\displaystyle-2Q^{2}\bigg{[}-1\bigg{]}.$ (43)

Knowledge of these co-factors leads immediately to the $A^{RR}$ and $B^{RR}$
functions of eq.(6),

$\displaystyle A^{RR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\frac{1}{Z(1-Z)}\bigg{[}-|ZG_{1}-G_{2}|^{2}\Omega_{f}+Z(1-Z)|G_{1}|^{2}\Omega_{\bot}\bigg{]},$
(44) $\displaystyle B^{RR}$ $\displaystyle=$
$\displaystyle\frac{2Q^{2}}{Z}\bigg{[}|ZG_{1}-G_{2}|^{2}+2Z(1-Z)|G_{1}|^{2}\bigg{]},$
(45)

expressions which are well-known and also displayed in refs.Novo  and Czyz .

###  VI.2 Linearly spin-dependent co-factors

Next in order are terms linear in the spin variables, represented by the
functions of eqs.(9) and (10), and their only non-vanishing co-factors ${\cal
K}^{ARS}_{4}$ and ${\cal K}^{ASR}_{4}$. The expessions for the prefactors of
the above-mentioned equations are quite easily obtained, and equals

$Q\cdot
k_{1,2}=-\frac{2p_{\Lambda}\epsilon\omega}{W}\mathbf{f}\cdot(\mathbf{n}\pm\mathbf{N}).$
(46)

In the $S_{1}$ frame the determinant boils down to

$\mbox{det}(p_{2}^{\prime}p_{1}^{\prime}l_{1}^{\prime}k_{2}^{\prime})=2E_{\Lambda}\mathbf{p}_{1}^{\prime}\cdot(\mathbf{p}_{g}^{\prime}\times\mathbf{k}_{2}^{\prime})=2E_{\Lambda}p_{\Lambda}\mathbf{f}\cdot(\mathbf{p}_{g}^{\prime}\times\mathbf{k}_{2}^{\prime}).$
(47)

Hence, components of the vectors $\mathbf{p}_{g}^{\prime}$ or
$\mathbf{k}_{2}^{\prime}$ along $\mathbf{f}$ will not contribute to the value
of the determinant, so that by eq.(28) we may replace
$\mathbf{p}_{g}^{\prime}$ by $\mathbf{p}_{g}$. For $\mathbf{k}_{2}^{\prime}$
we make recourse to eq.(20). All this yields,

$\displaystyle{\cal K}^{ARS}_{4}$ $\displaystyle=$ $\displaystyle
2p_{\Lambda}p_{g}(2\epsilon\omega)^{2}v_{\Lambda}\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{N})\bigg{]},$
(48) $\displaystyle{\cal K}^{ASR}_{4}$ $\displaystyle=$ $\displaystyle
2p_{\Lambda}p_{g}(2\epsilon\omega)^{2}v_{\Lambda}\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{N})\bigg{]}.$
(49)

The co-factors that relate to $B^{XY}$ of eq.(5) vanish,

$\displaystyle{\cal K}^{BRS}_{4}$ $\displaystyle=$ $\displaystyle 0,$ (50)
$\displaystyle{\cal K}^{BSR}_{4}$ $\displaystyle=$ $\displaystyle 0.$ (51)

With the co-factors of eqs.(48) and (49) in hand we can determine $A^{RS}$ and
$A^{SR}$ from eq.(6). The related functions $B^{RS}$ and $B^{SR}$ vanish
identically. In agreement with refs.Novo ,

$\displaystyle A^{RS}$ $\displaystyle=$ $\displaystyle
2p_{\Lambda}v_{\Lambda}p_{g}(2\epsilon\omega)^{2}2\Im(G_{1}G_{2}^{\star})\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{N})\bigg{]},$
(52) $\displaystyle A^{SR}$ $\displaystyle=$ $\displaystyle
2p_{\Lambda}v_{\Lambda}p_{g}(2\epsilon\omega)^{2}2\Im(G_{1}G_{2}^{\star})\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{N})\bigg{]}.$
(53)

###  VI.3 Doubly spin-dependent co-factors

Now, the co-factors are suffixed $ASS$ and $BSS$. Those suffixed $ASS$ are
obtained by analysing $A^{SS}$ of eq.(13);

$\displaystyle{\cal{K}}_{1}^{ASS}$ $\displaystyle=$
$\displaystyle(2p_{\Lambda}p_{g}\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\,\bigg{[}-Z^{2}X_{a}+\frac{Z\Omega_{\bot}}{v_{\Lambda}^{2}\Omega_{f}}\big{(}X_{a}-2X_{b})+\frac{Z}{v_{\Lambda}^{2}\Omega_{f}}B_{1}\bigg{]},$
(54) $\displaystyle{\cal{K}}_{2}^{ASS}$ $\displaystyle=$
$\displaystyle(2p_{\Lambda}p_{g}\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\bigg{[}-X_{a}\bigg{]},$
(55) $\displaystyle{\cal{K}}_{3}^{ASS}$ $\displaystyle=$
$\displaystyle(2p_{\Lambda}p_{g}\epsilon\omega)^{2}\Omega_{f}v_{\Lambda}^{2}\bigg{[}ZX_{a}+\frac{Z}{v_{\Lambda}^{2}\Omega_{f}}B_{3}\bigg{]},$
(56)

with functions $B_{1}$ and $B_{3}$ as defined in appendix B. The co-factors
suffixed $BSS$ are dug out from $B^{SS}$ of eq.(14);

$\displaystyle{\cal{K}}_{1}^{BSS}$ $\displaystyle=$ $\displaystyle
2(2p_{\Lambda}^{2}p_{g})^{2}\,\bigg{[}Z^{2}X_{a}+2Z(1-Z)X_{b}\bigg{]},$ (57)
$\displaystyle{\cal{K}}_{2}^{BSS}$ $\displaystyle=$ $\displaystyle
2(2p_{\Lambda}^{2}p_{g})^{2}\,\bigg{[}X_{a}\bigg{]},$ (58)
$\displaystyle{\cal{K}}_{3}^{BSS}$ $\displaystyle=$ $\displaystyle
2(2p_{\Lambda}^{2}p_{g})^{2}\,\bigg{[}-ZX_{a}\bigg{]},$ (59)

where $Z$ is defined in eq.(37).

Starting from the ${\cal{K}}^{ASS}$ and ${\cal{K}}^{BSS}$ co-factors we easily
derive functions $A^{SS}$ and $B^{SS}$,

$\displaystyle A^{SS}$ $\displaystyle=$ $\displaystyle
4(p_{\Lambda}p_{g}\epsilon\omega)^{2}\bigg{[}-|ZG_{1}-G_{2}|^{2}\frac{\Omega_{f}}{1-Z}X_{a}+Z\Omega_{\bot}|G_{1}|^{2}(X_{a}-2X_{b})$
(60) $\displaystyle+2Z\Re(G_{1}G_{2}^{\star})B_{3}+Z|G_{1}|^{2}B_{1}\bigg{]},$
$\displaystyle B^{SS}$ $\displaystyle=$ $\displaystyle
8(p_{\Lambda}^{2}p_{g})^{2}\bigg{[}|ZG_{1}-G_{2}|^{2}X_{a}+2Z(1-Z)|G_{1}|^{2}X_{b}\bigg{]},$
(61)

two functions which have not been investigated before. Now, we have all the
ingrediants needed to calculate the cross-section-distribution function frm
eqs.(4-6).

##  VII The $G_{E}/G_{M}$ set

Up to now, we have only considered an expansion of the cross-section-
distribution functions $A^{XY}$ and $B^{XY}$ in terms of the form factors
$G_{1}$ and $G_{2}$ and their co-factors. Other choices of form factors are
possible and we shall in particular consider the pair $G_{E}$ and $G_{M}$. The
two sets are related by

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle G_{M},$ (62)
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\frac{4M_{\Lambda}^{2}}{Q^{2}}(G_{M}-G_{E}).$ (63)

The arguments of the form factors are all equal to $P^{2}$. In particular,
when $P^{2}=4M_{\Lambda}^{2}$ then $G_{M}=G_{E}$.

The functions $A^{XY}$ and $B^{XY}$ are bilinear forms of $G_{1}$ and $G_{2}$,
and expanded according to eq.(6), but they can also be expanded in terms of
$G_{M}$ and $G_{E}$ in which case

$\displaystyle A^{XY}(G_{M},G_{E})$ $\displaystyle=$
$\displaystyle|G_{M}|^{2}{\cal{L}}^{AXY}_{1}+|G_{E}|^{2}{\cal{L}}^{AXY}_{2}$
(64) $\displaystyle+$ $\displaystyle
2\Re(G_{M}G_{E}^{\star}){\cal{L}}^{AXY}_{3}+2\Im(G_{M}G_{E}^{\star}){\cal{L}}^{AXY}_{4},$
(65)

and similarly for $B^{XY}$. The relation between the two sets of functions,
{${\cal{K}}_{i}^{XYZ}$} and {${\cal{L}}_{i}^{XYZ}$} becomes

$\displaystyle{\cal{L}}_{1}$ $\displaystyle=$
$\displaystyle{\cal{K}}_{1}+Z^{2}{\cal{K}}_{2}+2Z{\cal{K}}_{3},$ (66)
$\displaystyle{\cal{L}}_{2}$ $\displaystyle=$ $\displaystyle
Z^{2}{\cal{K}}_{2},$ (67) $\displaystyle{\cal{L}}_{3}$ $\displaystyle=$
$\displaystyle-Z^{2}{\cal{K}}_{2}-Z{\cal{K}}_{3}$ (68)
$\displaystyle{\cal{L}}_{4}$ $\displaystyle=$ $\displaystyle-Z{\cal{K}}_{4},$
(69)

and with the parameter $Z$ defined in eq.(37). The most notible fact of the
new set is that several co-factors vanish;

${\cal{L}}_{3}^{ARR}={\cal{L}}_{3}^{BRR}={\cal{L}}_{3}^{BSS}=0,$ (70)

whereas ${\cal{L}}_{3}^{ASS}\neq 0.$

Co-factors suffixed ${ARR}$;

$\displaystyle{\cal{L}}_{1}^{ARR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\,\bigg{[}(\mathbf{n}\times\mathbf{f})^{2}+(\mathbf{N}\times\mathbf{f})^{2}\bigg{]},$
(71) $\displaystyle{\cal{L}}_{2}^{ARR}$ $\displaystyle=$
$\displaystyle(2\epsilon\omega)^{2}\,\frac{1}{\gamma_{\Lambda}^{2}}\,\bigg{[}(\mathbf{n}\cdot\mathbf{f})^{2}+(\mathbf{N}\cdot\mathbf{f})^{2}\bigg{]},$
(72) $\displaystyle{\cal{L}}_{3}^{ARR}$ $\displaystyle=$ $\displaystyle 0.$
(73)

Co-factors suffixed ${BRR}$;

$\displaystyle{\cal{L}}_{1}^{BRR}$ $\displaystyle=$ $\displaystyle-4P^{2},$
(74) $\displaystyle{\cal{L}}_{2}^{BRR}$ $\displaystyle=$ $\displaystyle
8M^{2},$ (75) $\displaystyle{\cal{L}}_{3}^{BRR}$ $\displaystyle=$
$\displaystyle 0.$ (76)

Co-factors suffixed ${ARS}$ and ${ASR}$;

$\displaystyle{\cal L}^{ARS}_{4}$ $\displaystyle=$
$\displaystyle-2Zp_{\Lambda}p_{g}(2\epsilon\omega)^{2}v_{\Lambda}\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{g}\times\mathbf{N})\bigg{]},$
(77) $\displaystyle{\cal L}^{ASR}_{4}$ $\displaystyle=$
$\displaystyle-2Zp_{\Lambda}p_{g}(2\epsilon\omega)^{2}v_{\Lambda}\bigg{[}\mathbf{f}\cdot\mathbf{n}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{n})+\mathbf{f}\cdot\mathbf{N}\mathbf{f}\cdot(\mathbf{h}\times\mathbf{N})\bigg{]}.$
(78)

The co-factors that relate to $B^{XY}$ of eq.(5) vanish,

$\displaystyle{\cal K}^{BRS}_{4}$ $\displaystyle=$ $\displaystyle 0,$ (79)
$\displaystyle{\cal K}^{BSRb}_{4}$ $\displaystyle=$ $\displaystyle 0.$ (80)

With these co-factors in hand we can determine $A^{RS}$ and $A^{SR}$ from
Eq.(6) whereas the related functions $B^{RS}$ and $B^{SR}$ vanish.

##  VIII Discussion

In a previous report, ref.Ent1 , we derived a set of cross-section-
distribution functions for the reaction
$e^{+}e^{-}\rightarrow\gamma\Lambda(\rightarrow
p\pi^{-})\bar{\Lambda}(\rightarrow\bar{p}\pi^{+})$. It involved functions
whose arguments were Lorentzian scalar products of four-momentum vectors of
the particles participating in the reaction. Since some of the functons are
quite intricate we have here tried another approach, replacing the Lorentz
scalars by Euclidean scalars of three-momentum vectors. In doing so it should
be remembered that one can form a scalar product of two vectors only if they
are defined in the same reference system. The functions which multiply the
coupling constants are called co-factors, or weight factors, and can be
retrieved from eqs.(7-14). For a complete determination of the cross-section-
distribution function fourteen co-factors are needed.

The cross-section-distribution function describing our $e^{+}e^{-}$
annihilation reaction is detailed in sect. VI. However, we are also interested
in knowing the final-state-distribution function after integrating over one or
both hyperon decays. Let us start with the integral over the $\Omega_{g}$
hyperon decay, while keeping the anti-hyperon decay angle $\Omega_{h}$ fixed.
Then, terms which are linear in the three-vector $\mathbf{g}$ vanish. Thus,

$\int\frac{\textrm{d}\Omega_{g}}{4\pi}A^{SS},B^{SS},A^{RS}=0.$ (81)

Also, since $B^{RS}$ is non-excisting we can add $B^{RS}=0$ leaving the three-
vector $\mathbf{h}$ described by the co-vector of eq.(49).

The next step is integration over the hyperon decay angles $\Omega_{h}$,

$\int\frac{\textrm{d}\Omega_{h}}{4\pi}A^{SR}=0.$ (82)

Since by definition, $B^{SR}=0$, we end up with the cross-section-distribution
function for the reaction $e^{+}e^{-}\rightarrow\gamma\Lambda\bar{\Lambda}$,
as expected. This distribution function is proportional to the function
$M^{RR}$ of eq.(4).

The angular integrations just described can also be performed in the four-
dimensional formulation of the co-factors, as in sect. IV, by exploiting
eq.(7.48) of ref.Ent1  for the integration.

There is an alternative approach to the angular integration Eul , which
employs the Euler angles. In this case the angular measure is written as

$\textrm{d}\Omega_{h}\textrm{d}\Omega_{g}=\textrm{d}(\cos\theta_{gh}\big{)}\textrm{d}\alpha\textrm{d}(\cos\beta)\textrm{d}\gamma.$
(83)

Since we know that terms linear in the vectors $\mathbf{g}$ or $\mathbf{h}$
vanish upon angular integration, we need only concern ourselves with the co-
factors of $A^{SS}$ and $B^{SS}$ of sect. VI.3. We notice that
$\cos\theta_{gh}$ only appears in the function $X_{a}(\mathbf{g},\mathbf{h})$
of eq.(35),

$X_{a}(\mathbf{g},\mathbf{h})=2\mathbf{g}\cdot\mathbf{f}\,\mathbf{h}\cdot\mathbf{f}-\mathbf{g}\cdot\mathbf{h}.$

The first term in this expression vanishes on the $\alpha,\gamma$
integrations, and $\mathbf{g}\cdot\mathbf{h}=\cos\theta_{gh}$.

With a little help from the co-factors of sect. Eul  we get closed expressions
for $B^{SS}$ and $A^{SS}$,

$\displaystyle B^{SS}$
$\displaystyle=8\cos\theta_{gh}(p_{\Lambda}^{2}p_{g})^{2}\Big{(}-|ZG_{1}-G_{2}|^{2}\Big{)},$
(84) $\displaystyle A^{SS}$
$\displaystyle=4\cos\theta_{gh}(p_{\Lambda}p_{g}\epsilon\omega)^{2}\frac{1}{1-Z}\Big{(}|ZG_{1}-G_{2}|^{2}\Omega_{f}+Z(1-Z)|G_{1}|^{2}\Omega_{\bot}\Big{)}.$
(85)

Both functions vanish on integration over the $\cos\theta_{gh}$ variable.
Moreover, $ZG_{1}-G_{2}=ZG_{E}$ and $G_{1}=G_{M}$.

##  Appendix A Kinematics explained

The parameters describing the decay of Lambda into proton and pion are $p_{g}$
and $E_{g}$, with

$\displaystyle p_{g}$
$\displaystyle=\frac{1}{2M_{\Lambda}}\bigg{[}\big{(}(M_{\Lambda}+m_{p})^{2}-\mu^{2}\big{)}\big{(}(M_{\Lambda}-m_{p})^{2}-\mu^{2}\big{)}\bigg{]}^{1/2},$
(86) $\displaystyle E_{g}$
$\displaystyle=\frac{1}{2M_{\Lambda}}\big{(}M_{\Lambda}^{2}+m_{p}^{2}-\mu^{2}\big{)},$
(87)

representing the proton in the Lambda rest system.

The kinematic variables $P^{2},$ $y_{1},$ and $y_{2}$ of eq.(5) are defined
by,

$\displaystyle P^{2}$ $\displaystyle=$ $\displaystyle(p_{1}+p_{2})^{2},$ (88)
$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle 2k_{1}\cdot
q=2\epsilon\omega(1-\cos\theta),$ (89) $\displaystyle y_{2}$ $\displaystyle=$
$\displaystyle 2k_{2}\cdot q=2\epsilon\omega(1+\cos\theta),$ (90)

and the normalisation factors $a_{y}$ and $b_{y}$ of the same equation by

$\displaystyle a_{y}$ $\displaystyle=$ $\displaystyle 4P^{2}/(y_{1}y_{2}),$
(91) $\displaystyle b_{y}$ $\displaystyle=$
$\displaystyle(2sP^{2}+y_{1}^{2}+y_{2}^{2})/(y_{1}y_{2}).$ (92)

##  Appendix B Notations explained

The $\Omega$ fuctions are by definition, and $\mathbf{n}^{2}=1$,

$\displaystyle\Omega(\mathbf{n},\mathbf{N})$ $\displaystyle=$
$\displaystyle\mathbf{n}^{2}+\mathbf{N}^{2}=\frac{1}{v^{2}}+(\mathbf{n}\cdot\hat{\mathbf{k}})^{2},$
(93) $\displaystyle=$
$\displaystyle\Omega_{f}(\mathbf{n},\mathbf{N})+\Omega_{\bot}(\mathbf{n},\mathbf{N}),$
(94) $\displaystyle\Omega_{f}(\mathbf{n},\mathbf{N})$ $\displaystyle=$
$\displaystyle(\mathbf{n}\cdot\mathbf{f})^{2}+(\mathbf{N}\cdot\mathbf{f})^{2},$
(95) $\displaystyle\Omega_{\bot}(\mathbf{n},\mathbf{N})$ $\displaystyle=$
$\displaystyle(\mathbf{n}\times\mathbf{f})^{2}+(\mathbf{N}\times\mathbf{f})^{2}.$
(96)

The $\mathbf{N}$ vector is defined in eq.(22), and fulfils the relations,

$\displaystyle\mathbf{N}^{2}$ $\displaystyle=$
$\displaystyle\frac{W^{2}}{\omega^{2}}+(\mathbf{n}\cdot\hat{\mathbf{k}})^{2},$
(97) $\displaystyle\mathbf{n}\cdot\mathbf{N}$ $\displaystyle=$
$\displaystyle\frac{2\epsilon-\omega}{\omega}\mathbf{n}\cdot\hat{\mathbf{k}}.$
(98)

Also, introduced are co-factor functions $B_{1}$ and $B_{3}$

$\displaystyle L_{0}$ $\displaystyle=$
$\displaystyle\mathbf{n}\cdot\mathbf{g}_{\bot}\,\mathbf{n}\cdot\mathbf{h}_{\bot}+\mathbf{N}\cdot\mathbf{g}_{\bot}\,\mathbf{N}\cdot\mathbf{h}_{\bot},$
(99) $\displaystyle L_{M}$ $\displaystyle=$
$\displaystyle(\mathbf{f}\cdot\mathbf{g}\mathbf{h}_{\bot}+\mathbf{f}\cdot\mathbf{h}\mathbf{g}_{\bot})\cdot(\mathbf{n}\mathbf{f}\cdot\mathbf{n}+\mathbf{N}\mathbf{f}\cdot\mathbf{N}),$
(100) $\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle
2\bigg{[}L_{0}+\frac{1}{\gamma_{\Lambda}}L_{M}\bigg{]},$ (101) $\displaystyle
B_{3}$ $\displaystyle=$ $\displaystyle\gamma_{\Lambda}L_{M},$ (102)

with $\gamma_{\Lambda}=E_{\Lambda}/M_{\Lambda}$.

## References

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  * (2) L. V. Kardapoltzev, Bachelor’s thesis, Novosibirsk State University, 2007 (unpublished).
  * (3) H. Czyż, A. Grzelińska, and J. H. Kühn, Phys. Rev. D 75, 0740+26 (2007).
  * (4) Göran Fäldt, Eur. Phys. J. 52, 141 (2016).
  * (5) H. Pilkuhn, Relativistic Particle Physics (Springer-Verlag, Berlin, 1979).
  * (6) Göran Fäldt, Phys. Rev. D 97, 053002 (2018).