# Lorentz and permutation invariants of particles III:
constraining non-standard sources of parity violation

Christopher G. Lester  Cavendish Laboratory, University of Cambridge  Ward
Haddadin  Cavendish Laboratory, University of Cambridge  Ben Gripaios
Cavendish Laboratory, University of Cambridge

###### Abstract

Comparisons of the positive and negative halves of the distributions of
parity-odd event variables in particle-physics experimental data can provide
sensitivity to sources of non-standard parity violation. Such techniques
benefit from lacking first-order dependence on simulations or theoretical
models, but have hitherto lacked systematic means of enumerating all
discoverable signals. To address that issue this paper seeks to construct sets
of parity-odd event variables which may be proved to be able to reveal the
existence of any Lorentz-invariant source of non-standard parity violation
which could be visible in data consisting of groups of real non space-like
four-momenta exhibiting certain permutation symmetries.

###### Contents

  1. 1 Preamble concerning the scope and aims of this paper
  2. 2 Introduction
    1. 2.1 Non-standard parity violation, and why searches for it are important
    2. 2.2 The lack of frameworks for structuring such searches
    3. 2.3 Reducing a problem from infinite size to finite size
    4. 2.4 Our objectives
    5. 2.5 The continuity requirement in more detail
    6. 2.6 Our use of illustrative examples
    7. 2.7 The bigger picture
    8. 2.8 Alternative data-only constraints on non-standard sources of parity violation
    9. 2.9 The structure of the rest of the paper
  3. 3 Preliminary definitions and calculations.
  4. 4 Conditions under which certain types of event are chiral
    1. 4.1 Non-collision events $e\in\Omega_{(pq)(ab)X}$
    2. 4.2 Non-collision events $e\in\Omega_{(pq)(abc)X}$ for which at least one of $p$ and $q$ is massive
    3. 4.3 Non-collision events $e\in\Omega_{(pq)(abc)X}$ for which both $p$ and $q$ are massless
    4. 4.4 Collision events in either $\Omega_{(pq)\rightarrow(abc)X}$ or $\Omega_{(pq)\rightarrow(ab)X}$
  5. 5 Continuous Lorentz-invariant permutation-invariant parity-odd event variables for chiral events in various classes
    1. 5.1 Events in $\Omega_{(pq)(ab)X}$
    2. 5.2 Collision events in $\Omega_{(pq)\rightarrow(abc)X}$

##  1 Preamble concerning the scope and aims of this paper

  * •

This paper’s main purpose is to derive two sets of ‘event variables’
satisfying particular requirements which, the authors believe, could make then
useful to collider experiments seeking to look for non-standard sources of
parity violation.

  * •

The particular properties we demand for these sets of event variables are set
out in Section 2.4.

  * •

Although the paper’s introduction provides arguments explaining why the
demanded particular properties might expected to be useful, the primary
purpose of the paper is simply to derive a set of event variables satisfying
the stated properties. No evidence of the utility of these event variables (or
the lack thereof) is presented here.111Such evidence will be for later papers.
The purpose of this paper is only to provide falsifiable arguments in support
of our claim that the two sets of event variables we derive do indeed have the
properties which were demanded of them.

  * •

One set of event variables is designed to analyse collisions resulting in two
main final-state particles which are indistinguishable except by their four-
momenta (e.g. these could be dijet events). This set is found to comprises
three event variables ($X_{1}$, $X_{2}$ and $X_{3}$) which are defined in the
steps leading to equations (269) to (271).

  * •

The second set of event variables is designed to analyse collisions resulting
in three main final-state particles which are indistinguishable except by
their four-momenta. This set is found to comprises nineteen event variables
($V_{1}$ to $V_{19}$) contained within the four sub-sets shown in
(LABEL:eq:s19ALL).

  * •

The ‘take home’ messages for a physicist reading this paper are as follows:

    1. 1.

That if he or she has a collider with unpolarised beams, and has a detector
which is blind to polarisations or flavours but not four-momenta of detected
particles, and has access to all the events having three or fewer particles in
the final state, then every single possible source of non-standard parity-
violation, no matter how bizarre, provided that it discoverable ‘at all’
within that dataset, would generate an observable asymmetry in at least one of
the event variables derived in this paper, given enough luminosity, and given
an appropriate (parity-even) event selection.222The comment about the need for
‘an appropriate (parity-even) event selection’ refers forward to remarks made
in Corollary 5.3.

    2. 2.

More succinctly: he or she could search model independently for all potential
sources of non-standard parity violation by looking for asymmetries in only
the event variables described in this paper. These variables ‘cover all
bases’.

    3. 3.

Computer code to evaluate the event variables may be found as ancillary
support files attached to [15].

    4. 4.

Not all event variables will be relevant to all colliders. For example: some
event variables collapse to zero if the two beams of the collider have
particles of equal mass333For example: the two proton beams of the Large
Hadron Collider would satisfy this constraint while the proton-electron beams
of HERA would not., while others collapse to zero if final state particles are
treated as massless or degenerate in mass. In most analyses, therefore, the
number of variables needing to be considered would be much smaller than the
above counts of ‘three’ and ‘nineteen’ might suggest.

    5. 5.

If the initial or final state particles could be distinguished by more than
just their four-moment (e.g. if flavour or helicity information were
available) then the variables derived herein would no longer be sufficient to
catch all forms of parity violation which would then be (in principle)
observable. Nonetheless, the procedures used to generate the event variables
in this paper could be adapted to the new symmetries and new variables
generated which would have the desired coverage.

  * •

Some results are have been generalised to situations of relevance beyond
colliders444For example, (LABEL:eq:s28ALL) provides a set of twenty-eight
variables which could be used in dark matter detection experiments with five
particles in the final state., but these are not of primary interest to the
paper.

##  2 Introduction

###  2.1 Non-standard parity violation, and why searches for it are important

It is rightly beyond doubt that the laws of physics violate parity. The
elegant experiments of the 1950s [21, 7, 8] unambiguously showed that the weak
interaction of the Standard Model can tell the difference between our universe
and its mirror image. Nonetheless, only a single attempt [16] has yet been
made to look in Large Hadron Collider (LHC) data to see whether they provide
evidence of non-standard parity violating mechanisms which operate only at
high energies probed by that machine. As one of us has already noted in the
introduction of [16], the scarcity of tests of non-standard sources of parity
violation at these new energies:

> …is largely a pragmatic response to the obstacles presented by the LHC: its
> beams are not polarised, its detectors are not sensitive to polarizations,
> and it is mathematically impossible to construct a parity violating spin-
> averaged matrix element within any $CP$-conserving Locally Lorentz-Invariant
> quantum field theory (LLIQFT), effective or otherwise.555To reveal the
> existence of parity-violation a model must possess at least one matrix
> element having both a parity-even and a parity-odd part. After trace
> identities have removed spinor sums, the only parity-odd expressions which
> can remain in a Lorentz-invariant $|M|^{2}$ are contractions of the totally
> antisymmetric alternating tensor with groups of four linearly independent
> four-momenta: $\epsilon_{\mu\nu\sigma\tau}a^{\mu}b^{\nu}c^{\sigma}d^{\tau}$.
> While such terms are parity-odd, they are also time-odd. Assuming
> $CPT$-symmetry, such a matrix element therefore also violates $CP$.
> $CP$-conserving local Lorentz-invariant quantum field theories therefore
> cannot generate parity violating differential cross sections.  Given that
> the only route to probing parity-violation in the Standard Model at the LHC
> would be from within the $C$-even part of its (very small) $CP$-violating
> sector,666One could, in principle, demonstrate parity-violation
> unambiguously by using a ‘genuine $CP$-odd’ observable (such as one of those
> described in [11]) on a $C$-even final state. it is not surprising that no
> such analyses have yet been performed.777This limitation does not prevented
> the LHC from making measurements of parity-violating parameters within
> models in which a particular mechanism of parity-violation is present by
> assumption. For example, the differences between the axial and vector
> couplings of the $Z$-boson in the Standard Model violate parity and were
> measured in [18, 19]. However, neither of these papers incontrovertibly
> demonstrates that nature violates parity The reason is simple: the angles
> from which forward-backward asymmetries are calculated are even under
> parity, unlike primary observables from the experiments of the 1950s. The
> very same forward backward asymmetries could therefore also be explained, at
> least in principle, by some alternative parity conserving theory.

It would be wrong, however, to conclude that genuine tests of parity violation
are therefore of limited value. On the contrary: a purely data-driven search
for non-standard sources of parity violation was proposed and tested [16] on
CMS Open Data [CMS:QCDData] and demonstrated that genuine tests of parity-
invariance are very straightforward to make at the LHC and can (in principle)
provide very large signatures for models which are either (a) strongly
$CP$-violating LLIQFTs, or (b) not even LLIQFTs at all! While it is true that
there is an overwhelming theoretical preference for LLIQFTs (the Standard
Model itself is one, as are most popular extensions including those featuring
supersymmetry, leptoquarks, technicolor, axions, additional gauge
interactions, etc.) there is no law of nature which demands that new physics
be describable only by such theories. Moreover, given the lack of evidence for
new physics found at LHC thus far, the need for the community to search in all
possible hiding places is surely greater than ever. In particular, it is hard
to imagine any reason why every possible attempt should not be made to test
and re-test the fundamental symmetries of nature every time a door opens onto
a new energy range.

We argue, therefore, that it is important to have tools which allow us to
systematically discover and/or exclude as-yet unconstrained source of parity
violation both in non-LLIQFTs and in strongly $CP$-violating LLIQFTs. Among
those, we choose to restrict ourselves to considering only Lorentz-invariant
theories, however there is no requirement that others need make the same
choice.

###  2.2 The lack of frameworks for structuring such searches

Having established the need for such searches, the next concern is whether a
framework already exist which support their execution. This question was faced
by [16] and answered in the negative. A significant obstacle to the generality
of the results of that particular search was the observation that the number
of potential parity-odd variables with which it could have looked for signals
was unlimited in size and lacked any coherent structure. Consequently the
event-variables which it used were poorly motivated. A secondary consequence
of this arbitrariness was that, although that search saw no evidence of non-
standard parity violation, it could not make quantitative statements about the
extent of areas remaining most untested after its publication, nor give any
guide as to where best future searches should probe.

It is to find solutions to such problems that the present paper directs
itself.

###  2.3 Reducing a problem from infinite size to finite size

We seek to bring some order to the field of searches for non-standard sources
of Lorentz-invariant parity violation so that such searches may become more
tractable, and so that they may lose some of the arbitrariness that would
otherwise be present in their choices of search strategy or event variables.
We will attempt to do so by proposing a set of soon-to-be-defined requirements
on groups of parity-odd event variables (they shall be called sufficiency,
necessity/irreducibility, minimality, reality, continuity and invariance under
certain symmetries) using which we will prove that the number of event-
variables needing to be considered (given a class of events, such as ‘events
with three jets’) may be reduced from a potentially infinite size to one of of
finite size, without any loss of generality.

So far as the authors are aware there are no existing works aiming to achieve
the same objective. The closest in spirit is perhaps [11], which reminds its
readers that ‘genuine’ tests of $CP$-violation may be made at colliders like
the LHC. Nonetheless, its primary target is $CP$ not $P$, and even to the
extent to which its results apply to pure parity violation, it has other
objectives in its sights. Other works (such as [4] and [3]) share our interest
in answering questions involving the coupling of Lorentz invariance and
permutation invariances. However they choose (actively in the case of
[4]888Indeed, the top-tagger of [4] actively inverts the parity 50% of its jet
clusters (aiming to put any tertiary source of jet substructure in the first
quadrant of the its jets’ cones) in order to gain performance from the
resulting standardization of the data it inputs to its neural net. That such a
regularisation step may be performed by that work is because no non-trivial
parity structure is expected in lone quark or gluon jets., and implicitly in
the case of [3]) to make themselves completely blind to any potential sources
of parity violation. [It should be emphasised that the last remark is not
intended as a criticism of either of the aforementioned works! On the
contrary: given the lack of relevance of parity-violation to the goals that
each work targets, each work has a good reason for avoiding sensitivity to
parity as an extraneous complicating factor.]

###  2.4 Our objectives

We set out to find sets $S=\\{V_{i}\mid i=1,2,\ldots\\}$ of event variables
$V_{i}$ (functions on particle physics events) with the following properties:

  * •

[reality]: that each event variable $V_{i}$ shall map an event of a (suitably
defined) class $\Omega$ to a real number,

  * •

[parity oddness]: that each event variable $V_{i}$ shall be parity odd (that
is to say, would change sign under $\vec{x}\rightarrow-\vec{x}$),

  * •

[continuity]: that the mapping shall be continuous (that is to say: small
changes to the energy, or the mass or to any component of the momentum of a
particle in the event shall lead to only small changes in the value of the
event variable $V_{i}$),

  * •

[invariance under specific symmetries]: that all event variables $V_{i}$ shall
be invariant under a common symmetry group of choice (for example in each of
the illustrative derivations in this paper we require Lorentz-invariance
together with invariance under arbitrary permutation of the momenta of objects
belonging to a common class such as ‘all jets’ or or ‘all photons’),

  * •

[sufficiency]: that the set $S=\\{V_{i}\mid i=1,2,\ldots\\}$ of all the
variables shall have the collective property that at least one $V_{i}$ shall
evaluate to a non-zero value for every event in $\Omega$ which is chiral (an
event will be defined to be chiral if it cannot be mapped onto itself by a
parity reversal followed by the action of an arbitrary element of the chosen
symmetry group),

  * •

[necessity/irreducibility]: that every variable in the set $S$ shall be
necessary (or, equivalently, that the set itself shall be irreducible), by
which we mean that there shall be no variable $V\in S$ such that the set
$S\setminus\\{V\\}$ has the same sufficiency property possessed by $S$, and

  * •

[minimality]: that where two sets $S_{1}$ and $S_{2}$ share all the above
properties but differ in cardinality, we shall favour the one with the smaller
number of elements.

The reality requirement is present purely to ensure that we can count event
variables on a well defined basis. Any event variable could be decomposed into
a parity-even and parity-odd part. The former parts cannot ever help us
identify instances of parity-violation so the presence of the requirement of
parity oddness in the variables which we seek should therefore be self
explanatory. We hope that the motivation for the sufficiency and necessity
criteria is also readily apparent: without the former property the set of
variables would lack sensitivity to any parity-violating theory able to map
all of its chiral events to zeros in the elements $V_{i}$, while without the
latter property sets of variables could be large and wasteful. The minimality
constraint likewise seeks to push us toward simple solutions.

The benefits of requiring invariance under specific symmetries are numerous.
If the intended use of the event variables is as inputs to a machine learning
algorithm, then a clear benefit is that such an algorithm will not need to use
precious training data to learn to about symmetries which are already obvious
from context.999For example: knowing that there is no information content in
the ordering of $n$ jet momenta in some set $s$ has the effect of multiplying
the value of a training set by a factor of $n$-factorial. Knowing that there
is no information in a global rotation allows one training data sample to
stand in for an infinite number of copies of itself sitting at all possible
rotations. In short: such algorithms should generalise better. This could be
especially important if the training set is size-constrained or costly to
produce. If the event variable is destined for some other process, then it
remains the case that use of invariant variables removes the opportunity for
an analysis (whether intentionally or accidentally) to make selections or
predictions which are unphysical in a set of ways we can choose.

###  2.5 The continuity requirement in more detail

The reason for including the requirement of continuity, however, deserves in
depth discussion. In truth we impose this constraint simply because we are
prejudiced toward believing that it is ‘a good thing’ for any two events to
share near-identical descriptions when projected onto event-variables-space if
those events, for all practical purposes, are almost indistinguishable from
each other experimentally. Why? Because any subsequent analysis can only see
data through the lens provided by the event variables, and if near-
indistinguishable events were to map to separated parts of event-variable-
space, then trivial perturbations in data-space map into large perturbations
in the analysis which must then be un-learned or compensated for by the
analyser, be he or she a human or a neural net or some other machine learning
algorithm. This seems to add at best an unnecessary complication, and at worst
a dangerous one.101010The correction can fail to be done well, or may not be
done at all, or may use precious resources of either the analyser or the
neural net, etc.

The requirement of continuity is, however, a very strong one. Most of the
difficulty in the rest of the paper comes from our having demanded it.

Continuity is a requirement which is not present in the most common
descriptions of complex particle physics events, all of which record contain
data structures which store jet momenta (or the momenta of other groups of
similar particles) in order of decreasing transverse momentum. While this (or
any similar) sorting process renders any subsequent event-variables invariant
with respect to permutations of the orders of the jets (prior to sorting) –
this being a property which we also desire – it is a not-often-appreciated
fact that this same sorting process has the negative consequence that it makes
the map from events to event variables discontinuous. For example: the dot
product between the momenta of the first and second most energetic jets in an
event (or the related variable $\cos{(\Delta\theta(j_{1},j_{2}))}$) can change
discontinuously when the properties of those three jets are continuously
varied. All that is necessary for this to happen is for the the second and
third most energetic jets to approach each other’s energy while each has a
different angles to the most energetic jet. This means that machine learning
algorithms given such inputs are being asked not only to do important tasks
(separating $b$-jets from light quark jets, say) but are also being asked to
waste resources in learning to do similar things in widely separated part of
the input space which superficially appear to represent very different events
but which, in actual fact, may actually represent events which could be almost
indistinguishable from each other! Continuity, therefore, might be expected to
be beneficial in allowing machine learning algorithms to focus their finite
resources on the relevant rather than the irrelevant tasks in hand.

Were the requirement for continuity to be removed, many of the task of the
paper would becomes almost trivial. Construction of the desired sets of event-
variables would begin by sorting the momenta within each class of identical
particles so as to produce intermediate momenta with the requisite permutation
invariance.111111This step is not entirely trivial as structures are needed to
cope with every possible tie-breaker situation that can arise during the sort
process. Tie-breaker situations cannot be brushed under the carpet either as
they can be the norm rather than exceptions. For example, the lack of a frame-
independent way of distinguishing one beam from another in a collider
colliding identical particles creates many problems. Then all pseudo-scalar
Lorentz-contractions of those intermediate momenta would be formed and would
themselves constitute a sufficient though not usually irreducible set of
event-variables. Further work could be done to try to pare that set down to
one which was irreducible – if the resulting reduction were felt necessary.
Alternatively one could simply accept the additional computational cost of
using a sufficient but not irreducible set of event-variables.

It remains to be determined whether our prejudice motivating the requirement
of continuity is worth the additional complication and cost it imposes on the
construction of event variables. Our hunch is that the requirement is an
important one, but that it only becomes so when the number $n$ of momenta
subject to a permutation symmetry has grown sufficiently large that the cost
of coping with a number of discontinuities likely to grow as $n!$ (when
working with naïvely constructed event variables) outweighs the costs of
deriving a much smaller number of continuous but individually more powerful
event variables.121212The previously mentioned [4] perhaps takes a different
view. It comments that “While one could use advanced pre-processing beyond
some kind of ordering of the input 4-momenta, our earlier study [by which it
means [13]] suggests that this is not necessary.” This statement is made more
interesting by the fact that [13] appears to make no statements about the
benefits or dis-benefits of four-momentum ordering strategies. It appears to
make no statements about four-momentum ordering strategies of any kind.
Instead [13] processes images of jets projected onto eta-phi space. Such
objects have lost all notion of momentum ordering by an entirely different
means. If background work in support of [13] had indeed considered issues
connected with ordering of four-momenta, but the conclusions from those
studies were simply not mentioned the final paper, it is unclear whether that
work considered small sets of four-momenta (in which case we would likely
agree) or large sets of four-momenta (in which case there could be scope for
disagreement or debate between us). It is therefore possible that the most
complex permutation symmetry we have used in our illustrations ($S_{2}\times
S_{3}$) is simply too simple to see any significant benefit deriving from the
continuity requirement we have imposed.131313In the $(pq)\rightarrow(abc)+X$
scenario which we consider toward the end of the paper, a naïve transverse-
momentum-based sorting approach for the $(abc)$ momenta in the $(pq)$ rest
frame would result in only one discontinuity among the sorted $(abc)$ dot
products, and one ‘separated duplication’ caused by the inability to
distinguish in a frame-independent way between the concepts of positive and
negative rapidity when $p$ and $q$ are presumed to represent identical
particles. It would be interesting for future work to try to create variables
for $S_{2}\times S_{n}$ symmetries so as to probe performance for all values
of $n$ between 2 and $O(10)$ – or the maximum number of jets recorded in
appreciable numbers within LHC runs.

###  2.6 Our use of illustrative examples

Unfortunately we are not able to present a turn-handle algorithm for
generating the appropriate set of event variables, having all the
aforementioned properties, given a ‘general’ or ‘arbitrary’ class of events
$\Omega$. Instead, this paper illustrates the process of how one can go about
constructing a set of event variables with the right properties given a
concrete class of event, $\Omega_{1}$, which one is interested in analysing.
This process is then repeated for a number of other event classes
$\Omega_{2}$, $\Omega_{3}$, …. The event classes have (we hope) been chosen to
be sufficiently interesting that they illustrate the structure and nature of
the calculations which would need to be performed for more complicated event
classes beyond those considered.

The simplest event classes we consider are those one could use when looking
for non-standard sources of Lorentz-invariant parity violation in two-jet or
three-jet events at colliders. The most general class of events we consider is
one in which each event is assumed to contain five important momenta among
which an $S_{2}$ interchange symmetry is present between two of them, and an
$S_{3}$ interchange symmetry is present between the others. In this class of
events there are no restrictions on whether those momenta are in the initial
or final state, or are even observed at a collider.

The only high-level physics-based requirements which have been woven into to
the very fabric of our framework (and which might be absent from a purely
abstract mathematical attempt at solving the same problem) are: (i) that
momenta which comprise event-data are assumed to be time-like or null (not
space-like) and (ii) they are assumed to have entirely real (not complex)
components.

We caution the reader against becoming anxious upon seeing the long list of
special cases presented in the mid sections of the paper. Although some of
these sections (e.g. Section 4.2) have alarming titles, the final results
which they lead to are more general than they may at first appear. Although it
was necessary to break many proofs down into small components, each acting on
very narrow classes of events, these are only ingredients forming part of a
divide-and-conquer strategy aiming at investigating a simpler whole.

###  2.7 The bigger picture

As the bulk of the paper comprises theorems and proofs tailored very much to
the details of the specific event classes used as illustrations, it may be
helpful to contrast those direct attacks on the problem with a more general
description of the fundamental ideas underlying our results. This might both
help readers to see the ‘wood for the trees’ and pave the way for future
generalizations (e.g. with additional symmetries or more complicated events).

To that end: suppose we have some physical system, whose dynamics is known to
be invariant under a symmetry group $H$ and which is known to have some notion
of parity, under which the dynamics may or may not be invariant.141414$H$ will
later be the Lorentz group without parity, together with some permutation
symmetry. Rather than thinking of parity as some definite transformation of
the physical system (say ‘$\vec{x}\rightarrow-\vec{x}$’), it is helpful to
think of it as an extension of the group $H$ by $\mathbb{Z}/2$. That is to
say, we have another group $G$, of which $H$ is a normal subgroup, such that
$G/H$ is isomorphic to $\mathbb{Z}/2$, and that the non-trivial element of
this $\mathbb{Z}/2$ subgroup is what we call parity.151515$G$ will later be
the Lorentz group with parity, together with some permutation symmetry. The
phenomenologically interesting question then becomes: we know that $H$ is a
symmetry of our system, but is $G$?

To give a somewhat trivial example, suppose our system is invariant under
translations in space. Then, rather than arbitrarily choosing coordinates $x$
on our space and defining parity as the transformation
$\vec{x}\rightarrow-\vec{x}$ reflecting points in our arbitrarily chosen
origin, we should think about the reflections through all possible origins,
each of which is related to every other by a translation.

Now suppose we collect some data, which we call events, each of which is a
point in a space $X$ with the property that $G$ (and $H$) have a well-defined
action on $X$ (for example, we might choose some space of particles’ energy
and momenta, measured in a particular reference frame).161616$X$ will later be
the space of all momenta.  Because we know $H$ to be a symmetry of the system,
it makes sense to collate events which lie on a common orbit of $H$, since
there is nothing to be learnt by keeping them separate. Now, because $H$ is a
normal subgroup of $G$, there is a well-defined action of
$G/H\cong\mathbb{Z}/2$, i.e. parity, on the space $X/H$ of $H$-orbits of $X$.
Moreover, we can partition the orbits themselves into non-chiral orbits, which
are fixed under the action of $\mathbb{Z}/2$ and the chiral orbits, which are
not fixed (and which therefore contain precisely two distinct $H$-orbits of
$X$). The chiral orbits are the ones that are of interest to us, because they
allow us the possibility of testing directly whether parity is indeed a
symmetry of our dynamical system. One can do this, for example, by examining
whether, in a run of observations of events, we obtain a statistically-
significant discrepancy in the number of events observed in each member of any
such pair of orbits.

This simple observation is hindered, in general, by the fact that both the
space of possible events and the resulting space of possible $H$-orbits will
be infinitely large, requiring us to somehow combine many orbits in order to
have a chance of reaching a statistically significant discrepancy in a finite
number of observations. We would like to do so in a way which discards as
little information as possible. One way to do so is to look for observables
(i.e. nice functions on the underlying space $X$, e.g. polynomials) which are
not only $H$-invariant (so induce well-defined functions on $X/H$ which are
oblivious to useless information), but which are also parity odd (meaning that
they return values of opposite sign on each of an orbit pair), giving one the
hope of establishing whether parity is a symmetry or not. Since any function
can be decomposed into a piece which is parity odd and parity even, there is
again no loss of information here; rather, we are simply throwing away the
useless, parity even part.

Ideally, one would like to have a set of functions of this nature that is
large enough to be able to separate all of the $H$-orbits from one another, so
that again no physical information is lost in passing from events to
observables. This is, in essence, the goal of this paper. In order to achieve
it, we make full use of the freedom to restrict the possible events to those
which are physically observable at colliders and elsewhere, enabling us to
reduce the size of the set of functions as much as possible.

###  2.8 Alternative data-only constraints on non-standard sources of parity
violation

If it were desired to retain the ability to make statements about non-standard
parity-violation which are primarily based on data alone (not using
comparisons of data and Monte Carlo at first order), and if it were acceptable
to lose the requirements connected with sufficiency, an interesting approach
would be to randomly flip the parity of 50% of LHC events in software, and
then attempt to train a machine learning algorithm to distinguish the flipped
from the non-flipped events. Such methods would be many in number, limited
only by the creativity of the algorithms used. They would frequently come with
neither the qualified coverage guarantees provided by the sets of variables
proposed in the present paper, nor with assurances that algorithms is blind to
(perhaps permutation) symmetries known to be uninteresting, nor with
assurances that the algorithms respect symmetries known or presumed by choice
to be fundamental (perhaps Lorentz-invariance). Such methods would, however,
have set up costs for new event classes which are orders of magnitude below
those associated with the construction of sets of variables in the manner
suggested in the current paper. While the present paper is the third in a
series beginning with [10] and [9], the fourth171717Likely to be ‘Mastandrea
and Lester et.al.’ may well be one which compares the current proposal to such
alternative methods.

###  2.9 The structure of the rest of the paper

  * •

Section 3 prepares the ground for many subsequent calculations, most
importantly by providing axiomatic definitions of what we wish to consider as
events at colliders, what we mean by classes of events, a nomenclature for
describing the interchange symmetries on them which may be of interest, and
what it means for events to be chiral or non-chiral under parity, etc.

  * •

Section 4 then determines explicit conditions under which events in certain
classes actually are chiral. The classes of events considered include (in no
particular order) the following:

    1. 1.

$(pq)\rightarrow(ab)+X$,

    2. 2.

$(pq)\rightarrow(abc)+X$,

    3. 3.

$X\rightarrow(pq)+(ab)+Y$, and

    4. 4.

$X\rightarrow(pq)+(abc)+Y$,

  * •

For each of the classes of events just mentioned, Section 5 then obtains sets
of parity-odd event variables about which strong statements can be made
concerning their sufficiency and irreducibility properties. This section is,
arguably, the most important of the paper, since its outputs are the main
results contained herein. In particular:

  * •

For events of the form $(pq)\rightarrow(ab)+X$, $(ab)\rightarrow(pq)+X$ or
$X\rightarrow(ab)+(pq)+Y$, Section 5.1 identifies a set $S$ containing three
Lorentz-invariant event variables ($S=\\{X_{1},X_{2},X_{3}\\}$) which are
insensitive to $(pq)$\- and $(ab)$-permutations and which are proved to
satisfy the desired reality, parity oddness, continuity, invariance,
necessity, and sufficiency properties defined in the introduction. Each of the
variables in $S$ is defined in a manifestly frame independent way.

  * •

Section 5.1 also identifies a related set $\hat{S}$ containing three similar
Lorentz-invariant variables
($\hat{S}=\\{\hat{X}_{1},\hat{X}_{2},\hat{X}_{3}\\}$). These may only be used
to ascribe parities to events of the form $(pq)\rightarrow(ab)+X$. Despite
this limitation, the variables of $\hat{S}$ are arguably simpler and easier to
interpret than those of $S$, and so may be of use to some.

  * •

No claim is made that either set $S$ or $\hat{S}$ is globally minimal for the
class of events with which it is concerned. That is to say: it has not been
proved that there does not exist some other set $S_{?}$ containing only one or
two event variables but which retains all the stated properties. Nonetheless,
we conjecture that there is indeed no smaller set $S_{?}$ and that $S$ and
$\hat{S}$ are therefore examples of minimal sets.

  * •

Section 5.2 directs itself toward the generation of a set, $S_{19}$,
containing nineteen Lorentz-invariant event variables
($S_{19}=\\{V_{1},\ldots,V_{19}\\}$) which are insensitive to $(pq)$\- and
$(abc)$-permutations and which, for events of the form
$(pq)\rightarrow(abc)+X$, are proved to satisfy the reality, parity-oddness,
continuity, invariance, necessity and sufficiency properties already
described. The event variables in $S_{19}$ are defined in a manifestly frame
independent way.

  * •

No claim or conjecture is made that $S_{19}$ is globally minimal. That is to
say it has not been proved that there does not exist some other set $S_{\\#}$
containing eighteen or fewer event variables which retains all the stated
properties including sufficiency. Nonetheless, Section
LABEL:sec:discOfMinimialityOfS19 discusses the circumstantial evidence which
suggests that $S_{19}$ might not be a minimal set.

  * •

Sections LABEL:sec:bringbothnoncolltypestogeterinpqabc and
LABEL:sec:pqabcbothcollandnoncollvars work toward finding a superset of
$S_{19}$ named $S_{28}$ containing twenty-eight Lorentz-invariant event
variables ($S_{28}=\\{V_{1},\ldots,V_{28}\\}$) which are insensitive to
$(pq)$\- and $(abc)$-permutations. For events having an interchangeable pair
of particles $(pq)$ and an interchangeable triplet of particles $(abc)$
(neither of which need be in the initial state) the event variables in
$S_{28}$ are proved to satisfy the reality, parity-oddness, continuity,
invariance and sufficiency properties already described. The event variables
in $S_{28}$ are defined in a manifestly frame independent way. The set
$S_{28}$ is not demonstrated to be irreducible.

  * •

Section LABEL:sec:supportmaterials describes downloadable support materials
which may assist readers wishing to validate calculations using the event
variables $V_{1}$ to $V_{28}$.

  * •

Section LABEL:sec:discussion and beyond then discuss the potential uses of the
event variables previously defined, their relationship to related ideas, and
scope for future development.

  * •

Core notation may be found in Appendix LABEL:appendix:notation and a small
number of non-standard mathematical identities may be found in Appendix
LABEL:app:mathematicalidentitissdf.

##  3 Preliminary definitions and calculations.

This Section 3 contains important definitions and notation for key physical
concepts which are needed to support later proofs. Inevitably, the definitions
of these higher-level concepts depend on deeper notation which some (but
perhaps not all) readers may regard as ‘standard’. An example might include
the four-vector notation ‘$p=(E,\vec{p})$’ seen in Definition 3.1. Such
‘underlying’ or ‘semi-standard’ notation is important to define, but it could
be a distraction to define it here. We therefore provide it instead in a
dedicated Appendix LABEL:appendix:notation.

######  Definition 3.1.

$\mathbb{A}$ is defined to be the set containing all four-vectors having real
energies and real momentum components: $\mathbb{A}=\left\\{p\mid
p=\left(E,\vec{p}\right),\vec{p}\in\mathbb{R}^{3},E\in\mathbb{R}\right\\}.$

######  Definition 3.2.

$\mathbb{V}$ is defined to be the proper subset of $\mathbb{A}$ containing all
forward-time-directed non-spacelike four-vectors: $\mathbb{V}=\left\\{p\mid
p=\left(\sqrt{m^{2}+|\vec{p}|^{2}},\vec{p}\right),\vec{p}\in\mathbb{R}^{3},m\geq
0\right\\}\subset\mathbb{A}$.

######  Definition 3.3.

$\mathbb{M}$ is defined to be the proper subset of $\mathbb{V}$ containing all
forward-time-directed timelike non-massless four-vectors:
$\mathbb{M}=\left\\{p\mid
p=\left(\sqrt{m^{2}+|\vec{p}|^{2}},\vec{p}\right),\vec{p}\in\mathbb{R}^{3},m>0\right\\}\subset\mathbb{V}$.

######  Definition 3.4.

$\mathbb{N}$ is defined to be the proper subset of $\mathbb{V}$ containing all
forward-time-directed null (massless) four-vectors: $\mathbb{N}=\left\\{p\mid
p=\left(|\vec{p}|,\vec{p}\right),\vec{p}\in\mathbb{R}^{3}\right\\}\subset\mathbb{V}$.

######  Remark.

$\mathbb{M}\cup\mathbb{N}=\mathbb{V}$ and
$\mathbb{M}\cap\mathbb{N}=\emptyset$. The zero-vector is contained within
$\mathbb{N}$.

######  Remark.

A generic element of $\mathbf{x}\in\mathbb{A}^{n}$ (or $\mathbb{V}^{n}$ or
$\mathbb{M}^{n}$ or $\mathbb{N}^{n}$) will frequently use bold-face to
emphasise that it has $n$ components. Where these components themselves need
to be indexed, the most common notation will take the form
$\mathbf{x}=(p_{1},\ldots,p_{n})$ for $p_{i}\in\mathbb{A}$ (or
$\mathbb{V}^{n}$ or $\mathbb{M}^{n}$ or $\mathbb{N}^{n}$).

######  Remark.

The set $\mathbb{A}$ can easily be extended to a vector space over the field
$\mathbb{R}$ by the addition of the usual operators allowing vector addition
and scalar multiplication. The same is not true for $\mathbb{V}$ or
$\mathbb{M}$ or $\mathbb{N}$ since these do not contain inverse elements for
the vector addition operator. Nonetheless, Lemmas 3.6 and 3.7 are related and
still hold.

######  Lemma 3.5.

If $p$ and $q$ are both in $\mathbb{V}$ then $p.q\geq 0$.

###### Proof.

$p.q=\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{q}|^{2}}-\vec{p}\cdot\vec{q}\geq|\vec{p}||\vec{q}|-|\vec{p}||\vec{q}|\cos{\theta_{pq}}\geq|\vec{p}||\vec{q}|-|\vec{p}||\vec{q}|=0.$
∎

######  Lemma 3.6.

If $p$ and $q$ are both in $\mathbb{V}$, and $\lambda$ and $\mu$ are non-
negative real numbers, then $\lambda p+\mu q\in\mathbb{V}$.

###### Proof.

It is evident from the condition on $\lambda$ and $\mu$ that the time
component of $\lambda p+\mu q$ will be non-negative (as required). The only
non-trivial check required, therefore, is that $\lambda p+\mu q$ has a non-
negative squared-mass. This may be shown by considering $(\lambda p+\mu
q)^{2}=\lambda^{2}m_{p}^{2}+\mu^{2}m_{q}^{2}+2\lambda\mu(p\cdot q)\geq
2\lambda\mu(p\cdot q)\geq 0$. (The last inequality uses Lemma 3.5.) ∎

######  Lemma 3.7.

If $p$ and $q$ are both in $\mathbb{M}$, and $\lambda$ and $\mu$ are positive
real numbers, then $\lambda p+\mu q\in\mathbb{M}$.

###### Proof.

The proof for this lemma is a trival extension of the proof of Lemma 3.6. The
only difference is that this time use may be made of the fact that
$\lambda^{2}m_{p}^{2}$ and $\mu^{2}m_{q}^{2}$ are both strictly greater than
zero. ∎

######  Lemma 3.8.

If $p\in\mathbb{V}$ and $p\neq 0$ then $E_{p}>0$ in every frame.181818We
exclude from the concept of ‘frame’ anything reached only as a limit (i.e.
with an infinite Lorentz ‘$\gamma$’).

###### Proof.

If $p$ is in $\mathbb{V}$ and $p\neq 0$ the either $\vec{p}\neq 0$ or $m\neq
0$. Since $E_{p}=\sqrt{m^{2}+|\vec{p}|^{2}}$ then $E_{p}>0$. ∎

######  Lemma 3.9.

If $p$ and $q$ are both in $\mathbb{V}$ then $p.q\geq m_{p}m_{q}$.

###### Proof.

If $m_{p}=0$ or $m_{q}=0$ then the result is already proved by Lemma 3.5. It
is only necessary to prove the result, therefore, in the case that $m_{p}\neq
0\neq m_{q}$. In this case:

$\displaystyle p.q$
$\displaystyle=\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{q}|^{2}}-\vec{p}\cdot\vec{q}$
$\displaystyle=\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{q}|^{2}}-|\vec{p}||\vec{q}|\cos{\theta_{pq}}$
$\displaystyle\geq\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{q}|^{2}}-|\vec{p}||\vec{q}|$
$\displaystyle=m_{p}m_{q}\left(\sqrt{1+x^{2}}\sqrt{1+y^{2}}-xy\right)$

where $x=|\vec{p}|/m_{p}$ and $y=|\vec{q}|/m_{q}$. The lemma shall therfore be
proved if it can be demonstrated that the function
$f(x,y)=\sqrt{1+x^{2}}\sqrt{1+y^{2}}-xy$ is greater than or equal to one for
all non-negative $x$ and $y$. Trivially $f(x,y)\geq 1$ if either $x=0$ or
$y=0$. We will only need to show that this is also true when both $x$ and $y$
are strictly positive: $x>0$ and $y>0$. At extrema of $f$:

$\displaystyle 0=\frac{\partial f}{\partial x}$
$\displaystyle=\frac{x\sqrt{1+y^{2}}}{\sqrt{1+x^{2}}}-y\qquad\text{and}$
$\displaystyle 0=\frac{\partial f}{\partial y}$
$\displaystyle=\frac{y\sqrt{1+x^{2}}}{\sqrt{1+y^{2}}}-x.$

These constraints are redundant with each other and may be jointly written as

$\displaystyle y\sqrt{1+x^{2}}=x\sqrt{1+y^{2}}$ (1)

whose solutions are contained within those of $y^{2}(1+x^{2})=x^{2}(1+y^{2})$
which is equivalent to $y^{2}=x^{2}$ or even $x=y$ given, as we have already
noted, that it suffices to consider only those solutions having $x\geq 0$ and
$y\geq 0$. We may limit ourselves therefore to checking for extrema among the
cases $(x,y)=(\lambda,\lambda)$ for $\lambda\geq 0$ – and moreover note that
all such potential solutions do indeed solve $\frac{\partial f}{\partial
x}=\frac{\partial f}{\partial y}=0$ (meaning that they are not artefects
introduced when (1) was squared). We therefore try such solutions in $f(x,y)$:

$\displaystyle f(\lambda,\lambda)$
$\displaystyle=\sqrt{1+\lambda^{2}}\sqrt{1+\lambda^{2}}-\lambda^{2}=1$

independently of $\lambda$. We therefore see that $f(x,y)$ has a degenerate
extremum along the line $(x,y)=(\lambda,\lambda)$ and has no other extrema,
except perhaps at infinity. The lemma will therefore be proved if we can
demonstrate that the degenerate extremum just found is a minimum. To do this
it suffices to find one point above the line and one point below
it191919Finding two points excludes the possibility of a degenerate
inflection. having values of $f(x,y)$ which are greater than 1. The proof is
concluded, therefore, by noting that $f(1,0)=f(0,1)=\sqrt{2}>1$. ∎

######  Definition 3.10.

A partition of $n$ things into $m$ classes having $n_{i}>0$ things in each
class $i$ will be denoted $P(n,m,\mathbf{n})$ where
$\mathbf{n}=(n_{1},\ldots,n_{m})$ and $n=\sum_{i=1}^{m}n_{i}.$

######  Remark.

When analysing an event of the form

$pp\rightarrow jjj+\mu^{+}+\mu^{-}+X$

then the partition of interest would be $P(7,4,(2,3,1,1))$. In this, the $n=7$
shows us we understand the momenta of seven objects (two incoming protons and
five outgoing identified particles), $m=4$ tells us that there are four
identifiable classes of particles in the event (namely: (i) incoming protons;
(ii) jets, (iii) muons; and (iv) anti-muons) while the $\mathbf{n}=(2,3,1,1)$
shows us how many of those particles there are in each successive category.

######  Definition 3.11.

We define the group
$S(\mathbb{V},P(n,m,\mathbf{n}))\in\operatorname{Aut}(\mathbb{V}^{n})$ to be

$S(\mathbb{V},P(n,m,\mathbf{n}))=SO^{+}(1,3)\times S_{n_{1}}\times
S_{n_{2}}\times\cdots\times S_{n_{m}}$

where $SO^{+}(1,3)$ is the part of the Lorentz Group which is connected to the
identity202020$SO^{+}(1,3)$ is sometimes called the group of proper (i.e. not
parity-altering) orthochronous (i.e. not time-reversing) Lorentz
transformations., $S_{n_{1}}$ is the symmetric group of order $n_{1}$ which
permutes the first $n_{1}$ momenta within any $\mathbf{x}\in\mathbb{V}^{n}$,
and $S_{n_{2}}$ is the symmetric group of order $n_{2}$ which permutes the
next $n_{2}$ momenta within any $\mathbf{x}\in\mathbb{V}^{n}$, and so on. The
Lorentz group acts on every element of $\mathbf{x}\in\mathbb{V}^{n}$
coherently.

######  Definition 3.12.

For every integer $n\geq 1$, an event $e$ in an event class
$\mathscr{Ec}(\mathbb{V}^{n},G)$ is an element $\mathbf{x}\in\mathbb{V}^{n}$
together with a symmetry group $G\in\operatorname{Aut}(\mathbb{V}^{n})$:

$\displaystyle\mathscr{Ec}(\mathbb{V}^{n},G)$
$\displaystyle=\mathbb{V}^{n}\times\operatorname{Aut}(\mathbb{V}^{n})$ (2)
$\displaystyle=\left\\{e\mid
e=(\mathbf{x},G),\mathbf{x}\in\mathbb{V}^{n},G\in\operatorname{Aut}(\mathbb{V}^{n})\right\\}.$
(3)

######  Remark.

We are primarily interested in event classes $\mathscr{Ec}(\mathbb{V}^{n},G)$
for which the symmetry group $G$ is of the form
$G=S(\mathbb{V},P(n,m,\mathbf{n}))$. In a small abuse of notation, we
therefore extend the notation of an event class
$\mathscr{Ec}(\mathbb{V}^{n},G)$ by making the following short-hand
definition:

######  Definition 3.13.

$\mathscr{Ec}(P(n,m,\mathbf{n}))\underset{\text{def}}{\equiv}\mathscr{Ec}(\mathbb{V}^{n},S(\mathbb{V},P(n,m,\mathbf{n}))).$

######  Definition 3.14.

We define two events $e$ and $f$ to be equivalent (written ‘$e\sim f$’) if and
only if (i) they are in the same event class, and (ii) they can be transformed
into each other by an element of the group each contains. More formally, if
$e=(\mathbf{x},G_{e})\in\mathscr{E}(\mathbb{V}^{n(e)},G_{e})$ and
$f=(\mathbf{y},G_{f})\in\mathscr{E}(\mathbb{V}^{n(f)},G_{f})$ then

$(e\sim f)\iff(\text{$n(e)=n(f)$ and $G_{e}=G_{f}$ and $\exists$ $g\in G_{e}$
s.t.~{}$\mathbf{x}=g\mathbf{y}$}).$

######  Remark.

The relation $\sim$ just given is an equivalence relation.

######  Definition 3.15.

The parity operator $\mathscr{P}$ is defined to act on $p\in\mathbb{V}$,
$\mathbf{x}\in\mathbb{V}^{n}$ and $e\in\mathscr{E}(V^{n},G)$ as follows:

$\displaystyle\mathscr{P}\cdot p$
$\displaystyle\equiv\mathscr{P}\cdot\left(E,\vec{p}\right)\underset{\text{def}}{\equiv}\left(E,-\vec{p}\right),$
(4) $\displaystyle\mathscr{P}\cdot\mathbf{x}$
$\displaystyle\equiv\mathscr{P}\cdot\left(p_{1},\ldots,p_{n}\right)\underset{\text{def}}{\equiv}\left(\mathscr{P}\cdot
p_{1},\ldots,\mathscr{P}\cdot p_{n}\right),\qquad\text{and}$ (5)
$\displaystyle\mathscr{P}\cdot e$
$\displaystyle\equiv\mathscr{P}\cdot\left(\mathbf{x},G\right)\underset{\text{def}}{\equiv}\left(\mathscr{P}\cdot\mathbf{x},G\right).$
(6)

######  Definition 3.16 (event chirality).

An event $e\in\mathscr{E}(\mathbb{V}^{n},G)$ is defined to be non-chiral if
and only if $e\sim\mathscr{P}\cdot e,$ where $\sim$ is the equivalence
relation given in Definition 3.14. Similarly, an event for which this is not
true may be termed chiral.

######  Definition 3.17.

If $\mathscr{E}$ is a set of events, we define $\aleph(\mathscr{E})$ to be the
set of chiral events in $\mathscr{E}$, and $\heartsuit(\mathscr{E})$ to be the
set of non-chiral events in $\mathscr{E}$:

$\displaystyle\aleph(\mathscr{E})$ $\displaystyle=\left\\{e\mid
e\in\mathscr{E},e\not\sim\mathscr{P}\cdot e\right\\},\text{\qquad and}$ (7)
$\displaystyle\heartsuit(\mathscr{E})$ $\displaystyle=\left\\{e\mid
e\in\mathscr{E},e\sim\mathscr{P}\cdot e\right\\}.$ (8)

######  Remark.

$\heartsuit(\mathscr{E})\cup\aleph(\mathscr{E})=\mathscr{E}$ and
$\heartsuit(\mathscr{E})\cap\aleph(\mathscr{E})=\emptyset$.

######  Definition 3.18.

Since every event $e=(\mathbf{x},G)$ contains a group
$G\in\operatorname{Aut}(\mathbb{V}^{n})$, we can define an action of $G$ on
the events in $\mathscr{E}(\mathbb{V}^{n},G)$ in a natural way. Concretely,
for any $G\in\operatorname{Aut}(\mathbb{V}^{n})$, we define the action of
$g\in G$ on $e\in\mathscr{E}(\mathbb{V}^{n},G)$ as follows:

$g\cdot e\equiv
g\cdot(\mathbf{x},G)\underset{\text{def}}{\equiv}(g\cdot\mathbf{x},G).$

######  Corollary 3.19.

It follows from Definitions 3.16 and 3.18 that an event
$e\in\mathscr{E}(\mathbb{V}^{n},G)$ will be non-chiral if and only if there
exists a $g\in G$ such that $g\cdot e=\mathscr{P}\cdot e.$

######  Lemma 3.20.

If $p\in\mathbb{V}$ and $q\in\mathbb{V}$ then $p^{2}q^{2}\leq(p.q)^{2}$ (or
equivalently $\Delta_{2}(p,q)\leq 0$).

###### Proof.

The result may be obtained by squaring Lemma 3.9. ∎

######  Lemma 3.21.

If $p\in\mathbb{V}$ and $q\in\mathbb{V}$ then $p^{2}q^{2}<(p.q)^{2}$ (or
equivalently $\Delta_{2}(p,q)<0$) is a necessary and sufficient condition for
$p$ and $q$: (a) to have a centre-of-mass frame; and (b) to have equal and
opposite non-zero three-momenta in that frame.

###### Proof.

To prove necessity, we first assume that $p$ and $q$ have a centre of mass
frame in which $\vec{p}$ and $\vec{q}$ are not only equal and opposite (which
would be required by definition of such a frame) but that they also have
$|\vec{p}|>0$ in that frame. Concretely, we assume
$p^{\mu}=\left(\sqrt{m_{p}^{2}+|\vec{p}|^{2}},\vec{p}\right)$ and
$q^{\mu}=\left(\sqrt{m_{q}^{2}+|\vec{p}|^{2}},-\vec{p}\right)$ for some
$\vec{p}\neq\vec{0}$. Hence, in such a case:

$\displaystyle(p.q)^{2}-p^{2}q^{2}$
$\displaystyle=(\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{p}|^{2}}+|\vec{p}|^{2})^{2}-m_{p}^{2}m_{q}^{2}$
(9)
$\displaystyle=(m_{p}^{2}+|\vec{p}|^{2})(m_{q}^{2}+|\vec{p}|^{2})+2|\vec{p}|^{2}\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{p}|^{2}}+|\vec{p}|^{4}-m_{p}^{2}m_{q}^{2}$
(10)
$\displaystyle=(m_{p}^{2}+m_{q}^{2})|\vec{p}|^{2}+2|\vec{p}|^{4}+2|\vec{p}|^{2}\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{p}|^{2}}$
(11)
$\displaystyle=|\vec{p}|^{2}\left((m_{p}^{2}+m_{q}^{2})+2|\vec{p}|^{2}+2\sqrt{m_{p}^{2}+|\vec{p}|^{2}}\sqrt{m_{q}^{2}+|\vec{p}|^{2}}\right)$
(12) $\displaystyle\geq 4|\vec{p}|^{4},$ (13)

which is greater than zero (by assumption), so concluding the proof of
necessity.

To prove sufficiency we will need to show that $(p^{2}q^{2}<(p.q)^{2})$
implies (($p$ and $q$ have a centre of mass frame) and ($p$ and $q$ are not
stationary in that frame)). This may be done by proving the contrapositive
statement: “If (($p$ and $q$ do not have a centre of mass frame) or ($p$ and
$q$ have a centre of mass frame but are stationary in it)) then
$(p^{2}q^{2}\geq(p.q)^{2})$”. If $p$ and $q$ have a centre of mass frame but
are stationary in it, then equation (12) shows that $p^{2}q^{2}=(p.q)^{2}$
agreeing with half of the contrapositive statement. It therefore only remains
to show what happens when $p$ and $q$ fail to have a rest frame. We need only
consider the case in which $p+q$ is non-zero, since if $p+q=0$ then
$p^{2}q^{2}\geq(p.q)^{2}$ is trivially satisfied. We therefore assume in what
follows that $p+q\neq 0$.

Failure to have a rest frame means that the velocity of $(p+q)$ in the any
frame must be equal to the velocity of light, or else it would be possible to
catch up with $(p+q)$ from some frame. Let us therefore evaluate the velocity
of $(p+q)$ in an arbitrary frame defined (up to rotation) by unit-mass four-
momentum $\Lambda$ which is taken to be at rest in that frame. In that frame,
we know from Lemma 3.8 that $E_{p+q}>0$ since Lemma 3.6 places $p+q$ in
$\mathbb{V}$, and $p+q\neq 0$ by assumption. Therefore we may safely say that:

$\displaystyle\left.\beta_{p+q}\right|_{\Lambda}$
$\displaystyle=\left.\frac{|{\vec{p}}|_{p+q}}{E_{p+q}}\right|_{\Lambda}=\left.\frac{\sqrt{E_{p+q}^{2}-m_{p+q}^{2}}}{E_{p+q}}\right|_{\Lambda}=\frac{\sqrt{(\Lambda.(p+q))^{2}-(p+q)^{2}}}{\Lambda.(p+q)}$
(14)

and so a failure of $(p+q)$ to have a rest frame must imply that

$\displaystyle(\left.\beta_{p+q}^{2}\right|_{\Lambda}=1)$
$\displaystyle\implies\frac{{(\Lambda.(p+q))^{2}-(p+q)^{2}}}{(\Lambda.(p+q))^{2}}=1$
(15)
$\displaystyle\implies{{(\Lambda.(p+q))^{2}-(p+q)^{2}}}={(\Lambda.(p+q))^{2}}$
(16) $\displaystyle\implies(p+q)^{2}=0$ (17) $\displaystyle\implies
p^{2}+q^{2}+2(p.q)=0$ (18) $\displaystyle\implies p^{2}=q^{2}=(p.q)=0$ (19)

using Lemma 3.5 three times in the last step. Accordingly, we see that it is
indeed the case that $p^{2}q^{2}\geq(p.q)^{2}$ since both left and right hand
sides are identically zero. ∎

######  Lemma 3.22.

If $p\in\mathbb{V}$ and $q\in\mathbb{V}$ then $p^{2}q^{2}=(p.q)^{2}$ (or
equivalently $\Delta_{2}(p,q)=0$) is a necessary and sufficient condition for
there to exist non-negative real numbers $\lambda\geq 0$ and $\mu\geq 0$, not
both zero, such that $\lambda p=\mu q$.

###### Proof.

To prove necessity, assume first that $\lambda\geq$, $\mu\geq 0$, $\lambda
p=\mu q$ and that $\lambda$ and $\mu$ are not both zero. Without loss of
generality, assume that $\mu$ is non-zero. This assumption permits $q$ to be
written as $q=\frac{\lambda}{\mu}p$ and so both $p^{2}q^{2}$ and $(p.q)^{2}$
are equal to $(p^{2})^{2}\frac{\lambda^{2}}{\mu^{2}}$.

To prove sufficiency, we begin with the assumption that $(p.q)^{2}=p^{2}q^{2}$
and the knowledge that $p\in\mathbb{V}$ and $q\in\mathbb{V}$. The latter two
permit us to write $p$ and $q$ as by
$p^{\mu}=\left(\sqrt{m_{p}^{2}+|\vec{p}|^{2}},\vec{p}\right)$ and
$q^{\mu}=\left(\sqrt{m_{q}^{2}+|\vec{q}|^{2}},\vec{q}\right)$. To complete the
proof we will consider two cases separately: (i) both $p$ and $q$ are
massless, and (ii) at least one of $p$ and $q$ (without loss of generality
$p$) has mass $m>0$.

In case (i) our parameterisation changes to $p^{\mu}=(|\vec{p}|,\vec{p})$ and
$q^{\mu}=(|\vec{q}|,\vec{q})$ and so:

$\displaystyle 0$ $\displaystyle=(p.q)^{2}-p^{2}q^{2}$ (20)
$\displaystyle=(|\vec{p}||\vec{q}|-\vec{p}.\vec{q})^{2}-0^{2}0^{2}$ (21)
$\displaystyle=(|\vec{p}||\vec{q}|(1-\cos\theta))^{2}$ (22)

which tells us that $\vec{p}=0$, $\vec{q}=0$, or
$((\cos\theta=0)\land(\vec{p}\neq 0\neq\vec{q}))$. In the sub-case that
$\vec{p}=0$ then $(\lambda,\mu)=(1,0)$ satisfies the lemma. In the sub-case
that $\vec{q}=0$ then $(\lambda,\mu)=(0,1)$ satisfies the lemma. In the
remaining sub-case, $(\lambda,\mu)=(|\vec{q}|,|\vec{p}|)$ satisfies the lemma.

In case (ii) at least one of $p$ and $q$ has a non-zero mass. Without loss of
generality, assume that it is $p$ which has a non-zero mass $m_{p}\neq 0$. If
the relationship $\lambda p=\mu q$ can be proved in one frame it will be true
in all frames, so we choose to prove it in a convenient frame, namely the
rest-frame of $p$. In that frame our parameterization becomes:
$p^{\mu}=(m_{p},\vec{0})$ and
$q^{\mu}=\left(\sqrt{m_{q}^{2}+|\vec{q}|^{2}},\vec{q}\right)$, and so

$\displaystyle 0$ $\displaystyle=(p.q)^{2}-p^{2}q^{2}$ (23)
$\displaystyle=\left(m_{p}\sqrt{m_{q}^{2}+|\vec{q}|^{2}}-0\right)^{2}-m_{p}^{2}m_{q}^{2}$
(24) $\displaystyle=m_{p}^{2}(m_{q}^{2}+|\vec{q}|^{2})-m_{p}^{2}m_{q}^{2}$
(25) $\displaystyle=m_{p}^{2}|\vec{q}|^{2}$ (26)

and so either $m_{p}=0$ or $\vec{q}=0$. In the sub-case in which $m_{p}$=0,
then $p=(0,\vec{0})$ and so $(\lambda,\mu)=(1,0)$ satisfies the lemma. In the
sub-case in which $\vec{q}=0$ we have $q=(m_{q},\vec{0})$ and so
$(\lambda,\nu)=(m_{q},m_{p})$ satisfies the lemma (recall that $m_{p}$ is non-
zero). ∎

######  Definition 3.23 (collision events).

Motivated by Lemma 3.21 we define the set of all collision events with a
symmetry group $G$ to be the set of all events of the form
$e=((p,q,\ldots),G)\in\mathscr{E}(\mathbb{V}^{n},G)$ for which
$p^{2}q^{2}<(p.q)^{2}$ (equivalently $\Delta_{2}\left({p},{q}\right)<0$):

$\mathscr{E}^{c}(\mathbb{V}^{n},G)=\left\\{e\mid
e=((p,q,\ldots),G)\in\mathscr{E}(\mathbb{V}^{n},G),p^{2}q^{2}<(p.q)^{2}\right\\}\subset\mathscr{E}(\mathbb{V}^{n},G).$

######  Remark.

Note to have enough space to hold $p$ and $q$ every collision event will need
to have $n\geq 2$.

######  Lemma 3.24.

If $p\in\mathbb{V}$ and $q\in\mathbb{V}$ and there exist $\lambda\geq 0$ and
$\mu\geq 0$, not both zero, such that $\lambda p=\mu q$ then one of the
following seven non-overlapping conditions pertains:

  1. 1.

$p=q=0$

  2. 2.

$p=0$ while $q\in\mathbb{M}$ (and so $q\neq 0$),

  3. 3.

$q=0$ while $p\in\mathbb{M}$ (and so $p\neq 0$),

  4. 4.

$p=0$ while $q\neq 0$ is an otherwise unconstrained element of $\mathbb{N}$,

  5. 5.

$q=0$ while $p\neq 0$ is an otherwise unconstrained element of $\mathbb{N}$,

  6. 6.

$p=\mu q\neq 0$ and $q=\lambda p\neq 0$ for some $\lambda>0$, $\mu>0$, and
both $p$ and $q$ are in $\mathbb{M}$,

  7. 7.

$p=\mu q\neq 0$ and $q=\lambda p\neq 0$ for some $\lambda>0$, $\mu>0$, and
both $p$ and $q$ are in $\mathbb{N}$.

###### Proof.

Left as an exercise for the reader. ∎

######  Lemma 3.25.

If $p$ and $q$ are both in $\mathbb{V}$ then the following five statements
(one of which makes used of a symbol $\Delta_{2}$ for a symmetric Gram
Determinant, defined in Appendix LABEL:app:gramnotation) are equivalent:

  * •

$(p.q)=m_{p}m_{q}$,

  * •

$(p.q)^{2}=p^{2}q^{2}$,

  * •

$\Delta_{2}(p,q)=0$,

  * •

there exists a $\lambda\geq 0$ and a $\mu\geq 0$, not both zero, such that
$\lambda p=\mu q$,

  * •

the event falls into any of the categories listed in Lemma 3.24.

###### Proof.

The equivalence of the first three statements is a trivial consequence of the
definitions of invariant mass and Gram Determinants. Equivalence of the fourth
to the second follows from Lemma 3.22. Equivalence of the fifth to the fourth
follows from Lemma 3.24. ∎

######  Definition 3.26 (non-collision events).

Lemmas 3.9, 3.20 and 3.22 show us that the only events with symmetry group $G$
which the definition of collision event excludes are: (i) events with symmetry
group $G$ having fewer than two particles (i.e. events in
$\mathscr{E}(\mathbb{V}^{0},G)\cup\mathscr{E}(\mathbb{V}^{1},G)$), and (ii)
events $e=((p,q,\ldots),G)\in\mathscr{E}(\mathbb{V}^{n},G)$ for which any of
the five (equivalent) statements in Lemma 3.25 applies. We therefore define an
event in either category (i) or category (ii) above to be a non-collision
event.

######  Lemma 3.27 (every collision event has an axis in a frame in which
$(p+q)$ is stationary).

If follows from Lemma 3.21 and Definition 3.23 that the directions of mutual
approach or departure of $p$ and $q$ in a collision event can always be used
to define a unique axis in the rest frame of ($p+q$).

######  Remark (The relevance of of non-collision events).

Collider physicists short on time might wish to skip the parts of this paper
dealing with non-collision events. These events are included more for
mathematical completeness than for their usefulness in actual experiments. The
main reason that non-collision events are irrelevant to most real physics
detectors is that non-collision events have $p$ and $q$ in a configuration in
which each ‘overlaps’ with the other (either as massive particles sharing the
same rest frame, or as light-like momenta going in the same direction as each
other, albeit with possibly different energies). Not only do these properties
prevent $p$ and $q$ from representing colliding particles, they often also
prevent $p$ and $q$ from being observed as momenta of final-state particles.
An exception could be if the detector is one which can distinguish collinear
photons of different energies via the use of (say) scintillators with narrow-
band sensitivities. Another exception could be the case in which overlapping
massive-particle momenta can be inferred from subsidiary decay products (e.g.
two $Z$-bosons sharing a rest frame could be simultaneously observed if one
$Z$-boson decayed to a dimuon pair while the other decaying to a dielectron
pair).212121Examples of concrete non-collision events are given in Remarks at
the end of Corollaries 4.4 and 4.6.

######  Corollary 3.28.

If $e=((p,q,\cdots),G)$ is a collision event then $p+q\in\mathbb{M}.$

###### Proof.

That $e=((p,q,\cdots),G)$ is a collision event tells us that
$p,q\in\mathbb{V}$ and $p^{2}q^{2}<(p.q)^{2}$. From Lemma 3.6 it is already
clear that $p+q\in\mathbb{V}$, so it only remains to show that $(p+q)^{2}>0.$
Since $p,q\in\mathbb{V}$, then $p^{2}\geq 0$, $q^{2}\geq 0$ and Lemme 3.5
tells us that $(p.q)\geq 0.$ Therefore $p^{2}q^{2}<(p.q)^{2}$ may be re-
written $\sqrt{p^{2}}\sqrt{q^{2}}<(p.q)$ and so
$(p+q)^{2}=p^{2}+q^{2}+2(p.q)>p^{2}+q^{2}+2\sqrt{p^{2}}\sqrt{q^{2}}=\sqrt{p^{2}}+\sqrt{q^{2}})^{2}\geq
0$, and thus $(p+q)^{2}>0.$ ∎

######  Definition 3.29 (transverse plane).

If $e$ is a collision event, we define its transverse plane,
$\operatorname{TP}(e)$, to be the set containing all momenta in $\mathbb{V}$
whose spatial parts are perpendicular to $\vec{p}-\vec{q}$ in the $(p+q)$ rest
frame:

$\operatorname{TP}(e)=\left\\{\ r\ \middle|\ r\in\mathbb{V},\
{G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\ \\!}c}{r}&{p+q}\\\
{p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}=0\right\\}$

in which $G(\cdots)$ is a ‘Gram Determinant’ as defined in Appendix
LABEL:app:gramnotation, not a symmetry group $G$ such as that used in
Corollary 3.28.

######  Remark.

This definition makes sense as Lemma 3.32 tells us that a Gram determinant of
the form shown measures the dot product between $\vec{r}$ and
$\vec{p}-\vec{q}$ in the $(p+q)$-rest frame.

######  Lemma 3.30.

The momenta $a$ and $b$ in a collision event $e=((p,q,a,b,\cdots),G)$ are
reflections of each other in $\operatorname{TP}(e)$ if and only if

$\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\ \\!}c}{a+b}&{p+q}\\\
{p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}=0\right)\land(\Delta_{3}\left({a-b},{p},{q}\right)=0).$

###### Proof.

The first criterion says that in the $(p+q)$-rest frame $\vec{a}+\vec{b}$ is
perpendicular to the beam axis (see Lemma 3.32). The second criterion says
that $(\vec{a}-\vec{b})$ has no transverse component in the $(p+q)$-rest frame
(this may be seen by coupling Lemma 3.34 with invariance properties of Gram
determinants). If the former is true, it asserts that $a_{z}=-b_{z}$ in the
$(p+q)$-rest frame. If the latter is true it asserts that
$\vec{a}_{T}=\vec{b}_{T}$ in the same frame. If both are true it asserts that
if $\vec{a}=(x,y,z)$ then $\vec{b}=(x,y,-z)$ which indeed shows that $\vec{a}$
and $\vec{b}$ are reflections of each other in the transverse plane.
Conversely, assuming that $\vec{a}=(x,y,z)$ then $\vec{b}=(x,y,-z)$ the given
condition may be shown to be true by reversing the argument just given. ∎

######  Lemma 3.31.

If $a\in\mathbb{A}$ and $\Lambda\in\mathbb{M}$ then the energy of $a$ in the
rest frame of $\Lambda$ is given by:

$\left.E_{a}\vphantom{\int}\right|_{\Lambda}=\frac{a.\Lambda}{\sqrt{\Lambda^{2}}}.$

######  Lemma 3.32.

If $a,b\in\mathbb{A}$ and $\Lambda\in\mathbb{M}$ then use of Lemma 3.31 shows
that $\vec{a}.\vec{b}$ in the rest frame of $\Lambda$ is given by:

$\left.(\vec{a}.\vec{b})\
\vphantom{\int}\right|_{\Lambda}=\left.E_{a}E_{b}\vphantom{\int}\right|_{\Lambda}-\left.(E_{a}E_{b}-\vec{a}.\vec{b})\vphantom{\int}\right|_{\Lambda}=\frac{a.\Lambda}{\sqrt{\Lambda^{2}}}\frac{b.\Lambda}{\sqrt{\Lambda^{2}}}-a.b=\frac{(a.\Lambda)(b.\Lambda)-(a.b)\Lambda^{2}}{\Lambda^{2}}=-\frac{1}{\Lambda^{2}}G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Lambda}\\\ {b}&{\Lambda}\end{array}\\!\\!\\!\\!\right)$

in which we have used a Gram determinant and notation from Definition
LABEL:def:gram.

######  Lemma 3.33.

If $a\in\mathbb{A}$ and $\Lambda\in\mathbb{M}$, then using Lemma 3.32 the
square of the magnitude of the three momentum of $a$ in the rest frame of
$\Lambda$ is:

$\left.|\vec{a}|^{2}\vphantom{\int}\right|_{\Lambda}=\left.(\vec{a}.\vec{a})\vphantom{\int}\right|_{\Lambda}=-\frac{1}{\Lambda^{2}}G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Lambda}\\\
{a}&{\Lambda}\end{array}\\!\\!\\!\\!\right)=\frac{-\Delta_{2}\left({a},{\Lambda}\right)}{\Lambda^{2}}$

in which we have used a symmetric Gram determinant and notation from
Definition LABEL:def:symgram.

######  Lemma 3.34.

If $a,p\in\mathbb{A}$, $\Lambda\in\mathbb{M}$, and
$-\Delta_{2}\left({p},{\Lambda}\right)>0$, then the square of the transverse
momentum of $\vec{a}$ with respect to the axis which $\vec{p}$ defines is:

$\left.|{\vec{a}}_{T}|^{2}\vphantom{\int}\right|_{\Lambda}=\frac{\Delta_{3}\left({a},{p},{\Lambda}\right)}{-\Delta_{2}\left({p},{\Lambda}\right)}.$

###### Proof.

An ugly argument follows:

$\displaystyle\left.|{\vec{a}}_{T}|^{2}\vphantom{\int}\right|_{\Lambda}$
$\displaystyle=\left.|{\vec{a}}|^{2}\vphantom{\int}\right|_{\Lambda}-\left.\frac{(\vec{a}.\vec{p})^{2}}{|\vec{p}|^{2}}\vphantom{\int}\right|_{\Lambda}=\frac{-\Delta_{2}\left({a},{\Lambda}\right)}{\Lambda^{2}}-\frac{\frac{1}{\Lambda^{4}}G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Lambda}\\\
{p}&{\Lambda}\end{array}\\!\\!\\!\\!\right)^{2}}{\frac{-\Delta_{2}\left({p},{\Lambda}\right)}{\Lambda^{2}}}$
$\displaystyle=\frac{-\Delta_{2}\left({a},{\Lambda}\right)}{\Lambda^{2}}-\frac{G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Lambda}\\\
{p}&{\Lambda}\end{array}\\!\\!\\!\\!\right)^{2}}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}=\frac{\Delta_{2}\left({a},{\Lambda}\right)\Delta_{2}\left({p},{\Lambda}\right)-G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Lambda}\\\
{p}&{\Lambda}\end{array}\\!\\!\\!\\!\right)^{2}}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}$
$\displaystyle=\frac{\left|\begin{array}[]{cc}a^{2}&a.\Lambda\\\
a.\Lambda&\Lambda^{2}\end{array}\right|\left|\begin{array}[]{cc}p^{2}&p.\Lambda\\\
p.\Lambda&\Lambda^{2}\end{array}\right|-\left|\begin{array}[]{cc}a.p&a.\Lambda\\\
p.\Lambda&\Lambda^{2}\end{array}\right|^{2}}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}$
$\displaystyle=\frac{(a^{2}\Lambda^{2}-(a.\Lambda)^{2})(p^{2}\Lambda^{2}-(p.\Lambda)^{2})-((a.p)\Lambda^{2}-(a.\Lambda)(p.\Lambda))^{2}}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}$
$\displaystyle=\frac{(a^{2}\Lambda^{2}p^{2}\Lambda^{2}-a^{2}\Lambda^{2}(p.\Lambda)^{2}-(a.\Lambda)^{2}p^{2}\Lambda^{2}+(a.\Lambda)^{2}(p.\Lambda)^{2})-((a.p)^{2}\Lambda^{4}+(a.\Lambda)^{2}(p.\Lambda)^{2}-2(a.p)\Lambda^{2}(a.\Lambda)(p.\Lambda))}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}$
$\displaystyle=\frac{a^{2}\Lambda^{2}p^{2}\Lambda^{2}-a^{2}\Lambda^{2}(p.\Lambda)^{2}-(a.\Lambda)^{2}p^{2}\Lambda^{2}+(a.\Lambda)^{2}(p.\Lambda)^{2}-(a.p)^{2}\Lambda^{4}-(a.\Lambda)^{2}(p.\Lambda)^{2}+2(a.p)\Lambda^{2}(a.\Lambda)(p.\Lambda)}{-\Delta_{2}\left({p},{\Lambda}\right){\Lambda^{2}}}$
$\displaystyle=\frac{a^{2}\Lambda^{2}p^{2}-a^{2}(p.\Lambda)^{2}-(a.\Lambda)^{2}p^{2}-(a.p)^{2}\Lambda^{2}+2(a.p)(a.\Lambda)(p.\Lambda)}{-\Delta_{2}\left({p},{\Lambda}\right)}=\frac{\Delta_{3}\left({a},{p},{\Lambda}\right)}{-\Delta_{2}\left({p},{\Lambda}\right)}.$

A more elegant argument would start by proving that

$\left.(\vec{a}\times\vec{b}).(\vec{c}\times\vec{d})\vphantom{\int}\right|_{\Lambda}=\frac{G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
}c@{,\ }c}{a}&{b}&{\Lambda}\\\
{c}&{d}&{\Lambda}\end{array}\\!\\!\\!\\!\right)}{\Lambda^{2}}$

and would then use that result to evaluate the RHS of:

$\left.|{\vec{a}}_{T}|^{2}\vphantom{\int}\right|_{\Lambda}=\left.\frac{\left|\vec{a}\times\vec{p}\right|^{2}}{|\vec{p}|^{2}}\vphantom{\int}\right|_{\Lambda}.$

∎

######  Lemma 3.35.

In the rest frame of $\Lambda$ the vector triple product
$\vec{a}\times\vec{b}\cdot\vec{c}$ of the spatial momenta within $a^{\mu}$,
$b^{\mu}$ and $c^{\mu}$ is given by:

$\displaystyle\left.\left(\vec{a}\times\vec{b}\cdot\vec{c}\right)\right|_{\Lambda}$
$\displaystyle=\epsilon_{0123}\frac{\epsilon_{\mu\nu\sigma\tau}\Lambda^{\mu}a^{\nu}b^{\sigma}c^{\tau}}{\sqrt{\Lambda^{2}}}\qquad\text{(see
notes on `epsilon notation' in Section~{}\ref{sec:epsilonNotation})}$ (27)
$\displaystyle=\epsilon_{0123}\frac{[\Lambda,a,b,c]}{\sqrt{\Lambda^{2}}}\qquad\qquad\qquad\qquad\text{(see
same section for notation).}$ (28)

###### Proof.

The expression of (27) is manifestly Lorentz invariant. It is therefore true
if it is true in some frame. In the rest frame of $\Lambda$ it takes the form
$\Lambda^{\mu}=\left(\sqrt{\Lambda^{2}},(0,0,0)\right)$. In this form the
required result follows naturally, with the $\epsilon_{0123}$ needed to ensure
the sign of the answer is independent of the choice of sign convention within
$\epsilon$.∎

######  Lemma 3.36.

In the $(p+q)$ rest frame of a collision event $e=((p,q,a,\cdots),G)$ the
square of the transverse momentum of $a$ with respect to the beam axis is
given by:

$\displaystyle\left.|{\vec{a}}_{T}|^{2}\vphantom{\int}\right|_{\Lambda}$
$\displaystyle=\frac{\Delta_{3}\left({a},{p},{q}\right)}{-\Delta_{2}\left({p},{q}\right)}.$
(29)

###### Proof.

Lemma 3.34 gives us the first step:

$\displaystyle\left.|{\vec{a}}_{T}|^{2}\vphantom{\int}\right|_{\Lambda}$
$\displaystyle=\frac{\Delta_{3}\left({a},{p},{p+q}\right)}{-\Delta_{2}\left({p},{p+q}\right)}$

This simplifies to the result given using the property that matrix
determinants (in this case Gram determinants) are insensitive to the addition
of any multiple of some row (or column) to another row (or column). Note also
that collision events have $\Delta_{2}\left({p},{q}\right)<0$ by Definition
3.23, so there is no risk of division by zero in Equation (29). ∎

######  Lemma 3.37.

For any two vectors $\vec{a},\vec{b}\in\mathbb{R}^{n}$:
$(|\vec{a}|=|\vec{b}|)\iff((\vec{a}-\vec{b}).(\vec{a}+\vec{b})=0).$

###### Proof.

$(|\vec{a}|=|\vec{b}|)\iff(|\vec{a}|^{2}=|\vec{b}|^{2})\iff(|\vec{a}|^{2}-|\vec{b}|^{2}=0)\iff(\vec{a}.\vec{a}-\vec{b}.\vec{b}=0)\iff(\vec{a}.\vec{a}+\vec{a}.\vec{b}-\vec{b}.\vec{a}-\vec{b}.\vec{b}=0)\iff((\vec{a}-\vec{b}).(\vec{a}+\vec{b})=0).$
∎

######  Definition 3.38.

$\qquad\Omega_{(pq)\rightarrow(abc)X}\underset{\text{def}}{\equiv}\mathscr{E}^{c}(P(5,2,(2,3)))\qquad\text{with}\qquad\mathbf{x}=(p,q,a,b,c).$

######  Remark.

This definition is useful for referring quickly and concisely to collision
events of the form $(pq)\rightarrow(abc)+X$ which live in
$\mathscr{E}^{c}(P(5,2,(2,3)))$ with $\mathbf{x}=(p,q,a,b,c)$ and have the
symmetry group $G=SO^{+}(1,3)\times S_{2}(pq)\times S_{3}(abc).$

######  Definition 3.39.

$\qquad\Omega_{(pq)(abc)X}\underset{\text{def}}{\equiv}\mathscr{E}(P(5,2,(2,3)))\qquad\text{with}\qquad\mathbf{x}=(p,q,a,b,c).$

######  Remark.

This definition is useful for referring quickly and concisely to events having
entities $p$, $q$, $a$, $b$, $c$ and possibly also other things $X$, which
live in $\mathscr{E}(P(5,2,(2,3)))$ with $\mathbf{x}=(p,q,a,b,c)$ and have the
symmetry group $G=SO^{+}(1,3)\times S_{2}(pq)\times S_{3}(abc).$

######  Definition 3.40.

$\qquad\Omega_{(pq)\rightarrow(ab)X}\underset{\text{def}}{\equiv}\mathscr{E}^{c}(P(4,2,(2,2)))\qquad\text{with}\qquad\mathbf{x}=(p,q,a,b).$

######  Remark.

This definition is useful for referring quickly and concisely to collision
events of the form $(pq)\rightarrow(ab)+X$ which live in
$\mathscr{E}^{c}(P(4,2,(2,2)))$ with $\mathbf{x}=(p,q,a,b)$ and have the
symmetry group $G=SO^{+}(1,3)\times S_{2}(pq)\times S_{2}(ab).$

######  Definition 3.41.

$\qquad\Omega_{(pq)(ab)X}\underset{\text{def}}{\equiv}\mathscr{E}(P(4,2,(2,2)))\qquad\text{with}\qquad\mathbf{x}=(p,q,a,b).$

######  Remark.

This definition is useful for referring quickly and concisely to events having
entities $p$, $q$, $a$, $b$ and possibly also other things $X$, which live in
$\mathscr{E}(P(4,2,(2,2)))$ with $\mathbf{x}=(p,q,a,b)$ and have the symmetry
group $G=SO^{+}(1,3)\times S_{2}(pq)\times S_{2}(ab).$

##  4 Conditions under which certain types of event are chiral

###  4.1 Non-collision events $e\in\Omega_{(pq)(ab)X}$

######  Lemma 4.1.

Every non-collision event in $\Omega_{(pq)(ab)X}$ is non-chiral.

###### Proof.

Definition 3.26 reminds us that for every non-collision event in
$\Omega_{(pq)(ab)X}$ there will exist a $\lambda\geq 0$ and a $\mu\geq 0$ (not
both zero) such that $\lambda p=\mu q$. Equivalently, $p$ and $q$ must be
linearly dependent. This means that of the four vectors in the set
$\\{p^{\mu},q^{\mu},a^{\mu},b^{\mu}\\}$ at most three are linearly
independent. This is too few independent momenta to create a non-zero
pseudoscalar though dotting with an $\epsilon_{\mu\nu\sigma\tau}$. ∎

###  4.2 Non-collision events $e\in\Omega_{(pq)(abc)X}$ for which at least one
of $p$ and $q$ is massive

In this section we will find a condition which specifies whether or not a non-
collision event $e\in\Omega_{(pq)(abc)X}$ is non-chiral given that which at
least one of $p$ and $q$ is massive. The condition is given in Corollary 4.4.

######  Lemma 4.2.

For each non-collision event $e\in\Omega_{(pq)(abc)X}$ for which at least one
of $p$ and $q$ is massive, there is a frame in which both $p$ and $q$ have no
spatial momentum. It is the rest frame of $(p+q)$.

###### Proof.

Since at least one of $p$ and $q$ has a mass assume, without loss of
generality, that $m_{p}>0$, i.e. $p\in\mathbb{M}$. The particle $p$ therefore
has a rest frame. We will show that in this frame the particle $q$ also has no
spatial momentum. Definition 3.26 tells us that our non-collision event must
be in one of the seven configurations listed in Lemma 3.24. Our additional
requirement that (without loss of generality) $p\in\mathbb{M}$ further
constrains our event $e$ to be in configuration 3 or configuration 6 of Lemma
3.24. In configuration 3 we see that $q$ has no spatial momentum since it is
the zero four-vector: $q=0$. In configuration $6$ we are told that there
exists a $\lambda>0$ such that $q=\lambda p$ which shows us that in a frame in
which $p$ has no spatial momentum then $q$ shall also have none. ∎

######  Corollary 4.3.

In the $(p+q)$ rest frame every non-collision event $e\in\Omega_{(pq)(abc)X}$
for which at least one of $p$ and $q$ is massive may be parameterized by the
five non-negative masses $m_{p}$, $m_{q}$, $m_{a}$, $m_{b}$ and $m_{c}$
together with the three spatial momenta $\vec{a}$, $\vec{b}$ and $\vec{c}$
which in that frame $\vec{p}=\vec{q}=\vec{0}$. At least one of $m_{p}$ and
$m_{q}$ is non-zero. We may represent such an event notationally as follows:

$\displaystyle e=\left\\{\begin{array}[]{cc|ccc}p&q&a&b&c\\\ \hline\cr
m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
\vec{0}&\vec{0}&\vec{a}&\vec{b}&\vec{c}\end{array}\right\\}.$ (33)

######  Remark.

The task in-hand is to identify which events of the form (33) are chiral and
which are not. Recall that Definition 3.16 says that an event is non-chiral if
and only if its parity inverted form,

$\displaystyle\mathscr{P}\cdot e=\left\\{\begin{array}[]{cc|ccc}p&q&a&b&c\\\
\hline\cr m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
\vec{0}&\vec{0}&-\vec{a}&-\vec{b}&-\vec{c}\end{array}\right\\},$ (37)

can be mapped onto $e$ by the action of an element of the symmetry group,
which in our case consists of Lorentz boosts, rotations, permutations of
$(pq$) and permutations of $(abc)$. To get a match with $p$ and $q$ it is
clear that by working in the $(p+q)$ frame we can now exclude Lorentz boosts
from further consideration. Likewise, $p$ and $q$ do not need permuting, and
even if they did, such permutations would only affect $m_{p}$ and $m_{q}$. We
therefore only need consider permutations of $(abc)$ and global rotations $R$.
In principle there are six permutations of $(abc)$ to consider: $1$, $(ab)$,
$(bc)$, $(ca)$, $(abc)$ and $(cba)$. We shall only consider $h_{1}=R$,
$h_{2}=(ab)\cdot R$ and $h_{3}=(abc)\cdot R$ since the effects of the
remaining ones may be inferred from these three by symmetry.

$\displaystyle h_{1}\cdot\mathscr{P}\cdot e$ $\displaystyle=$ $\displaystyle
R\cdot\mathscr{P}\cdot e$
$\displaystyle=\left\\{\begin{array}[]{cc|ccc}p&q&a&b&c\\\ \hline\cr
m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
\vec{0}&\vec{0}&-R\vec{a}&-R\vec{b}&-R\vec{c}\end{array}\right\\},$ (41)
$\displaystyle h_{2}\cdot\mathscr{P}\cdot e$ $\displaystyle=$
$\displaystyle(ab)\cdot R\cdot\mathscr{P}\cdot e$
$\displaystyle=\left\\{\begin{array}[]{cc|ccc}p&q&a&b&c\\\ \hline\cr
m_{p}&m_{q}&m_{b}&m_{a}&m_{c}\\\
\vec{0}&\vec{0}&-R\vec{b}&-R\vec{a}&-R\vec{c}\end{array}\right\\},$ (45)
$\displaystyle h_{3}\cdot\mathscr{P}\cdot e$ $\displaystyle=$ $\displaystyle\
(abc)\cdot R\cdot\mathscr{P}\cdot e$
$\displaystyle=\left\\{\begin{array}[]{cc|ccc}p&q&a&b&c\\\ \hline\cr
m_{p}&m_{q}&m_{b}&m_{c}&m_{a}\\\
\vec{0}&\vec{0}&-R\vec{b}&-R\vec{c}&-R\vec{a}\end{array}\right\\}.$ (49)

####  4.2.1 Case $h_{1}$: ‘$R\cdot\mathscr{P}\cdot e=e$’

For events to be non-chiral under this case we shall need $\vec{a}=-R\vec{a}$,
$\vec{b}=-R\vec{b}$ and $\vec{c}=-R\vec{c}$. If one or more of $\vec{a}$,
$\vec{b}$ or $\vec{c}$ is zero, then the two remaining vectors will always be
in a common plane, and so $R$ could be taken to be a 180-degree rotation about
an axis normal to that common plane. The only non-trivial constraint therefore
comes when all of $\vec{a}$, $\vec{b}$ and $\vec{c}$ are non-zero. In such a
case, those three vectors must be required to be in a common plane, and this
may be enforced by requiring that $[a,b,c,p+q]=0$ using the square-bracket
contraction notation defined in (LABEL:eq:epscontractionnotation). Since this
statement is also trivially true in the case that any of $\vec{a}$, $\vec{b}$
or $\vec{c}$ is zero, we can say that Case $h_{1}$ is satisfied if and only if

$[a,b,c,p+q]=0.$

####  4.2.2 Case $h_{2}$: ‘$(ab)\cdot R\cdot\mathscr{P}\cdot e=e$’

For events to be non-chiral under this case we shall need $m_{a}=m_{b}$,
$\vec{a}=-R\vec{b}$, $\vec{b}=-R\vec{a}$ and $\vec{c}=-R\vec{c}$. Note that
these relations require that: $(\vec{a}=R^{2}\vec{a})$;
$(\vec{b}=R^{2}\vec{b})$ and;
$((\vec{a}\neq\vec{0})\iff(\vec{b}\neq\vec{0})\iff(R^{2}=1))$. Case $h_{2}$
will be satisfied, therefore, if $m_{a}=m_{b}$ and least one of the following
is true:

  1. 1.

$\vec{a}=\vec{b}=\vec{c}=\vec{0}$; or

  2. 2.

$\vec{a}=\vec{b}=0$ and $\vec{c}\neq\vec{0}$ (in which case $R$ must be a
rotation of 180 degrees about some axis perpendicular to $\vec{c}$); or

  3. 3.

$\vec{a}\neq\vec{0}\neq\vec{b}$ and $R=1$ (in which case it must be the case
that $\vec{a}=-\vec{b}$ and $\vec{c}=\vec{0}$); or

  4. 4.

$\vec{a}\neq\vec{0}\neq\vec{b}$ and $R$ is a rotation of 180 degrees about
some axis (in which case it must be the case that $\vec{c}$ is in the plane
orthogonal to the axis of rotation).

Sub-cases 1, 2 and 3 all have $[a,b,c,p+q]=0$ and so are also sub-cases of
$h_{1}$ (though $R$ may have a different interpretation). The only ‘new’ way
of being non-chiral provided by Case $h_{2}$ is therefore sub-case 4. It says
that $m_{a}=m_{b}$ and that $\vec{a}$ and $\vec{b}$ are reflections of each
other in a plane which also holds $\vec{c}$. We may exclude from consideration
both the case in which $\vec{a}=\vec{b}$ and the case $\vec{c}=\vec{0}$, since
both of these place $\vec{a}$, $\vec{b}$ and $\vec{c}$ into a common plane,
thus making them sub-cases of Case $h_{1}$. The geometry of the new non-chiral
case is shown in Figure 1.

Figure 1:  Diagram of a non-chiral configuration of momenta described in the
text.

The condition shown can be summarised by the requirement:

$\displaystyle\left(m_{a}=m_{b}\right)\land\left(|\vec{a}|=|\vec{b}|\right)\land\left(\vec{a}\cdot\vec{c}=\vec{b}\cdot\vec{c}\right)$
(50)

or in Lorentz-invariant form as

$\displaystyle\left(a^{2}=b^{2}\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{p+q}\\\
{a}&{p+q}\end{array}\\!\\!\\!\\!\right)=G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\
{b}&{p+q}\end{array}\\!\\!\\!\\!\right)\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{p+q}\\\
{c}&{p+q}\end{array}\\!\\!\\!\\!\right)=G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\ {c}&{p+q}\end{array}\\!\\!\\!\\!\right)\right).$ (59)

Using Lemma LABEL:lem:tetanotherlemmaaboutgramdetsfughfjdn (having set $s$
therein to $p+q$) we note that (59) may be written more symmetrically as

$\displaystyle\left(a^{2}-b^{2}=0\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{p+q}\\\
{a}&{p+q}\end{array}\\!\\!\\!\\!\right)-G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\
{b}&{p+q}\end{array}\\!\\!\\!\\!\right)=0\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a-b}&{p+q}\\\ {a+b+c}&{p+q}\end{array}\\!\\!\\!\\!\right)=0\right)$
(66)

which, in the compressed notation later to be introduced in (138), may be
written as

$\displaystyle\left(a^{2}-b^{2}=0\right)\land\left(g^{a}_{a}-g^{b}_{b}=0\right)\land\left(g^{a-b}_{a+b+c}=0\right)$
(67)

or, if using also (LABEL:eq:gaagbbdifferencelemma), may be written as:

$\displaystyle\left(a^{2}-b^{2}=0\right)\land\left(g^{a-b}_{a+b}=0\right)\land\left(g^{a-b}_{a+b+c}=0\right)$
(68)

according to taste. Note the similarity of the above constraints to those in
the relevant row(s) of (150).

####  4.2.3 Case $h_{3}$: ‘$(abc)\cdot R\cdot\mathscr{P}\cdot e=e$’

For events to be non-chiral under this case we shall need $m_{a}=m_{b}=m_{c}$,
$\vec{a}=-R\vec{b}$, $\vec{b}=-R\vec{c}$ and $\vec{c}=-R\vec{a}$. Note that
these relations require that:
$(\vec{a},\vec{b},\vec{c})=-R^{3}\cdot(\vec{a},\vec{b},\vec{c})$. By Lemma
LABEL:lem:whenonplaneperttorot, therefore, $\vec{a}$, $\vec{b}$ and $\vec{c}$
must all lie on a common plane which is perpendicular to the axis of rotation
of $R$. This means that any configuration satisfying Case $h_{3}$ is already
to be found as a special case of Case $h_{1}$. Case $h_{3}$ therefore adds no
new ways of being non-chiral.

####  4.2.4 Summary of all cases

######  Corollary 4.4.

A non-collision event $e\in\Omega_{(pq)(abc)X}$ for which at least one of $p$
and $q$ is massive is non-chiral if and only if

$\displaystyle[a,b,c,p+q]=0$
$\displaystyle\raisebox{2.27621pt}{\rule{0.0pt}{0.56905pt}}\lor$
$\displaystyle\left\\{\begin{array}[]{c}\left(a^{2}-b^{2}=0\right)\land\left(g^{a}_{a}-g^{b}_{b}=0\right)\land\left(g^{a-b}_{a+b+c}=0\right)\\\
\raisebox{2.27621pt}{\rule{0.0pt}{0.56905pt}}\lor\raisebox{2.27621pt}{\rule{0.0pt}{0.56905pt}}\\\
\left(b^{2}-c^{2}=0\right)\land\left(g^{b}_{b}-g^{c}_{c}=0\right)\land\left(g^{b-c}_{a+b+c}=0\right)\\\
\raisebox{2.27621pt}{\rule{0.0pt}{0.56905pt}}\lor\raisebox{2.27621pt}{\rule{0.0pt}{0.56905pt}}\\\
\left(c^{2}-a^{2}=0\right)\land\left(g^{c}_{c}-g^{a}_{a}=0\right)\land\left(g^{c-a}_{a+b+c}=0\right)\end{array}\right\\}.$

Note that the conditions inside the curly brakets have been written in the
form of (67) however they could just have easily been written using the
alternative variations (66) or (68). Note also use of the square-bracket
contraction notation defined in (LABEL:eq:epscontractionnotation).

###### Proof.

This result is just an ‘or’ of the conditions obtained after consideration of
cases $h_{1}$ to $h_{3}$ above, together with the other cases that would be
obtained from them by symmetry. ∎

######  Remark.

The following event

$\displaystyle
e=\left[\begin{array}[]{l}p^{\mu}=\left(10,\left(0,0,0\right)\right),\\\
q^{\mu}=\left(10,\left(0,0,0\right)\right),\\\
a^{\mu}=\left(3,\left(1,0,0\right)\right),\\\
b^{\mu}=\left(2,\left(0,1,0\right)\right),\\\
c^{\mu}=\left(1,\left(0,0,1\right)\right]\end{array}\right]$ (74)

(which is written using the Lorentz vector the display conventions given in
Appendix LABEL:sec:lorentzvectornotation) is an example of a non-collision
event $e\in\Omega_{(pq)(abc)X}$ for which at least one of $p$ and $q$ is
massive and which is chiral. The non-collision status may be verified by
checking that $p$ and $q$ meet the requirements of Definition 3.26. The chiral
status may be checked by noting that since $a^{2}=3^{2}-1^{2}=8$,
$b^{2}=2^{2}-1^{2}=3$ and $c^{2}=1^{2}-1^{2}=0$, every line of the condition
given in Corollary 4.4 is violated because:

$\displaystyle[a,b,c,p+q]\propto\left|\ \begin{matrix}3&2&1&20\\\ 1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\end{matrix}\ \right|=-20$ $\displaystyle\neq 0,$
$\displaystyle a^{2}-b^{2}=8-3=+5$ $\displaystyle\neq 0,$ $\displaystyle
b^{2}-c^{2}=3-0=+3$ $\displaystyle\neq 0,$ $\displaystyle c^{2}-a^{2}=0-8=-8$
$\displaystyle\neq 0.$

###  4.3 Non-collision events $e\in\Omega_{(pq)(abc)X}$ for which both $p$ and
$q$ are massless

In this section we will find a condition which specifies whether or not a non-
collision event $e\in\Omega_{(pq)(abc)X}$ is non-chiral given that both of $p$
and $q$ are massless. The condition is given in Corollary 4.6.

If $(a+b+c)^{2}=0$ then all of $a$, $b$ and $c$ must be massless and must all
point in the same direction (i.e. there will exist $\alpha\geq 0$ and
$\beta\geq 0$, not both zero, such that $\alpha a=\beta b$, etc). In such a
case there are at most three linearly independent Lorentz-vectors in the
event, namely $p$ and $q$ together with at most one of $a$, $b$ and $c$. These
are too few to construct a non-zero pseudoscalar from contraction with an
epsilon alternating tensor. Such an event therefore non-chiral.

Similarly, if $p=0=q$ then again there are too few linearly independent
Lorentz-vectors to construct a non-zero pseudoscalar. Such events are
therefore also non-chiral.

If $(a+b+c)^{2}>0$ and at least one of $p$ and $q$ is not the zero-vector,
then more work needs to be done to determine whether or not the event is
chiral. However, in this case the non-zero mass of $(a+b+c)$ means there is a
natural frame in which we can work, namely that in which $(a+b+c)$ is at rest.
In the remainder of Section 4.3 (and its subsections) we will work in that
frame and will assume (without loss of generality) that $p$ is not the zero-
vector.

Although we have already chosen a rest-frame in which to work, we have not yet
used the freedom to orient that frame. Definition 3.26 tells us that, since we
are considering only non-collision events, there must exist a $\lambda\geq 0$
and a $\mu\geq 0$, not both zero, such that $\lambda p=\mu q$. This means that
in the frame in which we have chosen to work, at least one of $\vec{p}$ and
$\vec{q}$ is non-zero, and furthermore if both are non-zero then both point in
the same direction. We therefore choose to orient our frame such that
$\vec{p}$ and $\vec{q}$ are parallel to the $z$-axis and so that neither of
them has a negative $z$-component.222222At least one of them will have a
positive $z$-component.

######  Corollary 4.5.

Every non-collision event $e\in\Omega_{(pq)(abc)X}$ having $(a+b+c)^{2}>0$ and
having at least one of $p$ and $q$ non-zero may, in the $(a+b+c)$-rest frame,
be parameterized by the three non-negative masses $m_{a}$, $m_{b}$ and
$m_{c}$, two non-negative scalars $p_{z}$ and $q_{z}$ (at least one of which
is non-zero), three transverse momenta $\vec{a}_{T}$, $\vec{b}_{T}$ and
$\vec{c}_{T}$ (each having two components) jointly satisfying
$a_{T}+\vec{b}_{T}+\vec{c}_{T}=\vec{0}_{T}$ and three other scalars $a_{z}$,
$b_{z}$ and $c_{z}$ jointly satisfying $a_{z}+b_{z}+c_{z}=0$. We may represent
such an event notationally as follows:

$\displaystyle\hat{e}=\left[\begin{array}[]{cc|ccc}0&0&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}&\mathbf{b}&\mathbf{c}\\\
p_{z}&q_{z}&a_{z}&b_{z}&c_{z}\end{array}\right]\qquad\text{with}\qquad\left(\begin{array}[]{l}p_{z},q_{z},m_{a},m_{b},m_{c}\geq
0,\\\ p_{z}+q_{z}>0,\\\ \mathbf{a},\mathbf{b},\mathbf{c}\in\mathbb{C},\\\
\mathbf{a}+\mathbf{b}+\mathbf{c}=0,\\\ a_{z},b_{z},c_{z}\in\mathbb{R}\
\text{and}\ a_{z}+b_{z}+c_{z}=0\end{array}\right).$ (83)

######  Remark.

The task in-hand is to identify which events of the form (83) are chiral and
which are not. Recall that Definition 3.16 says that an event is non-chiral if
and only if its parity inverted form,

$\displaystyle\mathscr{P}\cdot
e=\left[\begin{array}[]{cc|ccc}0&0&m_{a}&m_{b}&m_{c}\\\
0&0&-\mathbf{a}&-\mathbf{b}&-\mathbf{c}\\\
-p_{z}&-q_{z}&-a_{z}&-b_{z}&-c_{z}\end{array}\right],$ (87)

can be mapped onto $e$ by the action of an element of the symmetry group,
which in our case consists of Lorentz boosts, rotations, permutations of
$(pq$) and permutations of $(abc)$. Since every element $g$ in the symmetry
group has an inverse, we may instead choose to apply the above definition not
to $\mathscr{P}\cdot e$ but to $g\cdot\mathscr{P}\cdot e$. We will do so,
choosing $g$ to be a rotation of 180 degrees about the $y$-axis, this being
something that will ensure that the momenta of $p$ and $q$ are thereby already
mapped into agreement:

$\displaystyle R_{y}(\pi)\cdot\mathscr{P}\cdot
e=\left[\begin{array}[]{cc|ccc}0&0&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}^{*}&\mathbf{b}^{*}&\mathbf{c}^{*}\\\
p_{z}&q_{z}&a_{z}&b_{z}&c_{z}\end{array}\right].$ (91)

What elements of the symmetry group can be used to map
$R_{y}(\pi)\cdot\mathscr{P}\cdot e$ back onto $e$? Since we already have a
match for $p$ and $q$ we must not break this match. This rules out the use of
rotations except for those about the $z$-axis. Boosts are also ruled out for
those same reason. Interchanges of $p$ and $q$ could, in principle, be applied
if $p_{z}=q_{z}$. However such swaps achieve nothing. All we need consider,
therefore, are rotations $R_{z}(\theta)$ about the $z$-axis by an arbitrary
angle $\theta$, together with any permutation of $a$, $b$ and $c$. Though
there are six such permutations, it will only be necessary to examine three of
them (1, (ab) and (abc)) since the behaviour of the others may be determined
by symmetry from these. We therefore consider three cases as follows:

$\displaystyle R_{z}(\theta)\cdot R_{y}(\pi)\cdot\mathscr{P}\cdot e$
$\displaystyle=\left[\begin{array}[]{cc|ccc}0&0&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}^{*}e^{i\theta}&\mathbf{b}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}\\\
p_{z}&q_{z}&a_{z}&b_{z}&c_{z}\end{array}\right],$ (95) $\displaystyle(ab)\cdot
R_{z}(\theta)\cdot R_{y}(\pi)\cdot\mathscr{P}\cdot e$
$\displaystyle=\left[\begin{array}[]{cc|ccc}0&0&m_{b}&m_{a}&m_{c}\\\
0&0&\mathbf{b}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}\\\
p_{z}&q_{z}&b_{z}&a_{z}&c_{z}\end{array}\right],$ (99)
$\displaystyle(abc)\cdot R_{z}(\theta)\cdot R_{y}(\pi)\cdot\mathscr{P}\cdot e$
$\displaystyle=\left[\begin{array}[]{cc|ccc}0&0&m_{b}&m_{c}&m_{a}\\\
0&0&\mathbf{b}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}\\\
p_{z}&q_{z}&b_{z}&c_{z}&a_{z}\end{array}\right].$ (103)

The above cases will be considered with the complex number $\mathbf{a}$,
$\mathbf{b}$ and $\mathbf{c}$ in polar form, vis
$\mathbf{a}=|\mathbf{a}|e^{i\alpha}$, $\mathbf{b}=|\mathbf{b}|e^{i\beta}$ and
$\mathbf{c}=|\mathbf{c}|e^{i\gamma}$.

####  4.3.1 Case $h_{1}$: ‘$R_{z}(\theta)\cdot R_{y}(\pi)\cdot\mathscr{P}\cdot
e=e$’

This case requires

$((\mathbf{a}=0)\lor(\theta-\alpha+2n_{a}\pi=\alpha))\land((\mathbf{b}=0)\lor(\theta-\beta+2n_{b}\pi=\beta))\land((\mathbf{c}=0)\lor(\theta-\gamma+2n_{c}\pi=\gamma))$

which is equivalent to

$((\mathbf{a}=0)\lor(\alpha=\frac{1}{2}\theta+n_{a}\pi))\land((\mathbf{b}=0)\lor(\beta=\frac{1}{2}\theta+n_{b}\pi))\land((\mathbf{c}=0)\lor(\gamma=\frac{1}{2}\theta+n_{c}\pi))$

which is the same as requiring that all non-zero elements of the set
$\\{\mathbf{a},\mathbf{b},\mathbf{c}\\}$ must have the same complex argument
to within a $\pi$, This is itself the same as saying that $\vec{a}$,
$\vec{b}$, $\vec{c}$, $\vec{p}$ and $\vec{q}$ are all in a common plane. Note:
it is no surprise that $\vec{a}$, $\vec{b}$, $\vec{c}$ are in a common plane.
This is inevitable given that we are working in the $(a+b+c)$ rest frame. The
‘extra’ requirement here is only that $p$ and $q$ can be added while remaining
in one plane. Since $\vec{p}$ and $\vec{q}$ are necessarily collinear (and not
anti-parallel) it is sufficient to test that $(\vec{p}+\vec{q})$ lie in the
same plane as each of the other two vectors. I.e. in the $(a+b+c)$ rest frame
we wish to enforce the requirement that
$\vec{a}\times\vec{b}\cdot(\vec{p}+\vec{q})=\vec{b}\times\vec{c}\cdot(\vec{p}+\vec{q})=\vec{c}\times\vec{a}\cdot(\vec{p}+\vec{q})=0$.
Using the square-bracket contraction notation defined in
(LABEL:eq:epscontractionnotation) this constraint may be written in Lorentz
invariant form as $[a,b,p+q,a+b+c]=[b,c,p+q,a+b+c]=[c,a,p+q,a+b+c]=0$.
However, given standard properties of determinants these are all equivalent to
the single constraint

$[a,b,c,p+q]=0.$

####  4.3.2 Case $h_{2}$: ‘$(ab)\cdot R_{z}(\theta)\cdot
R_{y}(\pi)\cdot\mathscr{P}\cdot e=e$’

This case requires $m_{a}=m_{b}$, $a_{z}=b_{z}$ and
$|\mathbf{a}|=|\mathbf{b}|$ together with

$((\mathbf{a}=0)\lor((\theta-\beta+2n_{a}\pi=\alpha)\land(\theta-\alpha+2n_{b}\pi=\beta)))\land((\mathbf{c}=0)\lor(\theta-\gamma+2n_{c}\pi=\gamma))$

which is equivalent to $m_{a}=m_{b}$, $a_{z}=b_{z}$ and
$|\mathbf{a}|=|\mathbf{b}|$ together with

$\displaystyle((\mathbf{a}=0)\lor(\frac{1}{2}(\alpha+\beta)=\frac{1}{2}\theta+n\pi))\land((\mathbf{c}=0)\lor(\gamma=\frac{1}{2}\theta+n_{c}\pi)).$
(104)

Given that we have a constraint $|\mathbf{a}|=|\mathbf{b}|$ we may break this
down into four sub-cases:

  1. 1.

$|\mathbf{a}|=|\mathbf{b}|=|\mathbf{c}|=0$,

  2. 2.

$|\mathbf{a}|=|\mathbf{b}|=0$ and $|\mathbf{c}|\neq 0$, and

  3. 3.

$|\mathbf{a}|=|\mathbf{b}|\neq 0$.

Sub-cases 1 and 2 above are manifestly already contained within Case $h_{1}$
and so do not represent ‘new’ sources of non-chirality for these events and so
may be ignored. Only sub-case 3 tells us something new. It tells us that a
non-collision event of the type we are considering will be non-chiral if, in
the $(a+b+c)$ rest frame, $\vec{a}$ and $\vec{b}$ have transverse232323Here
‘transverse’ means with respect to the $(\vec{p}+\vec{q})$-axis in our frame.)
momenta with equal magnitudes and have the same component in the
$(\vec{p}+\vec{q})$-direction provided that, whenever $\vec{c}$ has a non-zero
transverse momentum, $\vec{c}$ lies in the plane in which $\vec{a}$ mirrors
$\vec{b}$ and which also includes $(\vec{p}+\vec{q})$.

Note that the mirror plane just mentioned is poorly defined if $\vec{a}$ and
$\vec{b}$ are axial (that is to say they are along $\vec{p}+\vec{q}$
direction) however this case may be ignored since it is a sub-case of Case
$h_{1}$. The case where $\vec{a}=\vec{b}$ and off axis may also be ignored as
another a sub-case of Case $h_{1}$, even though the mirror plane is well
defined in this case.

The forms of non-chiral event contained in the above and not already contained
within the $[a,b,c,p+q]=0$ condition of Case $h_{1}$ may be written

$\displaystyle(m_{a}=m_{b})\land(a_{z}=b_{z})\land(|\vec{a}|=|\vec{b}|)\land((\vec{a}-\vec{b})\cdot\vec{c}=0)$
(105)

which, having defined $\Sigma=a+b+c$, can re-written in Lorentz-invariant form
as

$\displaystyle(a^{2}=b^{2})\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a-b}&{\Sigma}\\\
{p+q}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=0\right)\land\left(\Delta_{2}\left({a},{\Sigma}\right)=\Delta_{2}\left({b},{\Sigma}\right)\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Sigma}\\\
{c}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{\Sigma}\\\ {c}&{\Sigma}\end{array}\\!\\!\\!\\!\right)\right).$
(112)

Using Lemma LABEL:lem:tetanotherlemmaaboutgramdetsfughfjdn (having set $s$
therein to $\Sigma$) we note that (112) may be written more symmetrically as

$\displaystyle(a^{2}=b^{2})\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a-b}&{\Sigma}\\\
{p+q}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=0\right)\land\left(\Delta_{2}\left({a},{\Sigma}\right)=\Delta_{2}\left({b},{\Sigma}\right)\right)\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{\Sigma}\\\
{\Sigma}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{\Sigma}\\\ {\Sigma}&{\Sigma}\end{array}\\!\\!\\!\\!\right)\right)$
(119)

however the properties of Gram determinants ensure that
$\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\ \\!}c}{a}&{\Sigma}\\\
{\Sigma}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{\Sigma}\\\ {\Sigma}&{\Sigma}\end{array}\\!\\!\\!\\!\right)\right)$
is always true (since the left and right hand sides of this equation are both
identically zero) so we may write our condition in the simpler and final form:

$\displaystyle(a^{2}=b^{2})\land\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a-b}&{\Sigma}\\\
{p+q}&{\Sigma}\end{array}\\!\\!\\!\\!\right)=0\right)\land\left(\Delta_{2}\left({a},{\Sigma}\right)=\Delta_{2}\left({b},{\Sigma}\right)\right).$
(122)

####  4.3.3 Case $h_{3}$: ‘$(abc)\cdot R_{z}(\theta)\cdot
R_{y}(\pi)\cdot\mathscr{P}\cdot e=e$’

In this case, although many constraints are required to be satisfied we will
only need to consider three. They are those which say $a_{z}=b_{z}=c_{z}$.
These, together with the already made requirement that $a_{z}+b_{z}+c_{z}=0$
reduce to $a_{z}=b_{z}=c_{z}=0$. These place all of $\vec{a}$, $\vec{b}$ and
$\vec{c}$ on the plane transverse to $\vec{p}+\vec{q}$ meaning that, in the
$(a+b+c)$ rest frame $\vec{a}\times\vec{b}\cdot(p+q)$,
$\vec{b}\times\vec{c}\cdot(p+q)$ and $\vec{c}\times\vec{a}\cdot(p+q)$ must all
be zero. This case is therefore a sub-case of Case $h_{1}$ and adds nothing
new.

####  4.3.4 Summary of all cases

######  Corollary 4.6.

A non-collision event $e\in\Omega_{(pq)(abc)X}$ for which both $p$ and $q$ are
massless is non-chiral if and only if

wherein $\Sigma$ stands for $a+b+c$ and where the square-bracket contraction
$[a,b,c,p+q]$ uses notation defined in (LABEL:eq:epscontractionnotation).

###### Proof.

This result is just an ‘or’ of the conditions obtained after consideration of
cases $h_{1}$ to $h_{3}$ above, together with the other cases that would be
obtained from them by symmetry, and together with the two earlier results just
preceding them concerning (i) the case $(a+b+c)^{2}=0$ and (ii) the case
$p=q=0$. ∎

######  Remark.

The following event

$\displaystyle
e=\left[\begin{array}[]{l}p^{\mu}=\left(2,\left(0,0,2\right)\right),\\\
q^{\mu}=\left(1,\left(0,0,1\right)\right),\\\
a^{\mu}=\left(3,\left(1,0,0\right)\right),\\\
b^{\mu}=\left(2,\left(0,1,0\right)\right),\\\
c^{\mu}=\left(2,\left(-1,-1,0\right)\right]\end{array}\right]$ (128)

(which is written using the Lorentz vector the display conventions given in
Appendix LABEL:sec:lorentzvectornotation) is an example of a non-collision
event $e\in\Omega_{(pq)(abc)X}$ for which both $p$ and $q$ are massless and
which is chiral. The non-collision status may be verified by checking that $p$
and $q$ meet the requirements of Definition 3.26. The chiral status may be
checked by noting that since $a^{2}=3^{2}-1^{2}=8$, $b^{2}=2^{2}-1^{2}=3$ and
$c^{2}=2^{2}-1^{2}-1^{2}=2$, every line of the condition given in Corollary
4.6 is violated because:

$\displaystyle(a+b+c)^{2}\geq a^{2}+b^{2}+c^{2}=8+3+2=13$ $\displaystyle>0,$
$\displaystyle p$ $\displaystyle\neq 0,$
$\displaystyle[a,b,c,p+q]\propto\left|\ \begin{matrix}3&2&2&3\\\ 1&0&-1&0\\\
0&1&-1&0\\\ 0&0&0&3\end{matrix}\ \right|=21$ $\displaystyle\neq 0,$
$\displaystyle a^{2}-b^{2}=8-3=+5$ $\displaystyle\neq 0,$ $\displaystyle
b^{2}-c^{2}=3-2=+1$ $\displaystyle\neq 0,$ $\displaystyle c^{2}-a^{2}=2-8=-6$
$\displaystyle\neq 0.$

###  4.4 Collision events in either $\Omega_{(pq)\rightarrow(abc)X}$ or
$\Omega_{(pq)\rightarrow(ab)X}$

######  Theorem 4.7.

A collision event $e\in\Omega_{(pq)\rightarrow(abc)X}$ is chiral if and only
if $C$ is true, where

$\displaystyle C\equiv$ (129) $\displaystyle(([a,b,p,q]\neq
0)\lor([b,c,p,q]\neq 0)\lor([c,a,p,q]\neq 0))$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle({a}^{2}\neq{b}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}-{b}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}-{b}}&{p+q}\\\ {{a}+{b}}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}-{b}}&{p+q}\\\ {a+b+c}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({a}={b}\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}$
$\displaystyle({b}^{2}\neq{c}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{b}-{c}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{b}-{c}}&{p+q}\\\ {{b}+{c}}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{b}-{c}}&{p+q}\\\ {a+b+c}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({b}={c}\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}$
$\displaystyle({c}^{2}\neq{a}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{c}-{a}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{c}-{a}}&{p+q}\\\ {{c}+{a}}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{c}-{a}}&{p+q}\\\ {a+b+c}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({c}={a}\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{c}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq 0\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor({a}^{2}\neq{b}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a+b+c}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}+{b}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(\Delta_{3}\left({{a}-{b}},{p},{q}\right)\neq 0\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor({b}^{2}\neq{c}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a+b+c}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{b}+{c}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(\Delta_{3}\left({{b}-{c}},{p},{q}\right)\neq 0\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor({c}^{2}\neq{a}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a+b+c}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{c}+{a}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(\Delta_{3}\left({{c}-{a}},{p},{q}\right)\neq 0\right)$

which makes use of square-bracket contraction notation defined in
(LABEL:eq:epscontractionnotation) and which uses notation for Gram
determinants $G$ and $\Delta$ defined in (LABEL:eq:gram) and
(LABEL:eq:symmgram).

###### Proof.

The proof of Theorem 4.7 is long and not particularly illuminating. Exposition
of it is therefore delayed until Sections 4.4.1 to 4.4.15 so as not to disrupt
the arguments about to be made. ∎

######  Corollary 4.8.

It follows from Theorem 4.7 almost by inspection242424The result may be
obtained either by substituting the zero four-vector in place of $c$ in the
statement of Theorem 4.7 and then simplifying the resulting expression, or by
considering the steps of the proof of Theorem 4.7 which would have been
required had there been no $c$ to begin with. that a collision event
$e\in\Omega_{(pq)\rightarrow(ab)X}$ is chiral if and only if $B$ is true,
where

$\displaystyle B\equiv$ (130) $\displaystyle(([a,b,p,q]\neq 0))$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle({a}^{2}\neq{b}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}-{b}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}-{b}}&{p+q}\\\ {{a}+{b}}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq
0\right)\lor\left({a}={b}\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{a}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq 0\right)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(p^{2}\neq
q^{2})\lor({a}^{2}\neq{b}^{2})\lor\left({G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{{a}+{b}}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)}\neq
0\right)\lor\left(\Delta_{3}\left({{a}-{b}},{p},{q}\right)\neq 0\right)$

which makes use of square-bracket contraction notation defined in
(LABEL:eq:epscontractionnotation) and which uses notation for Gram
determinants $G$ and $\Delta$ defined in (LABEL:eq:gram) and
(LABEL:eq:symmgram).

######  Definition 4.9.

Because the constraints $C$ and $B$ just written are rather big and bulky, we
will benefit from writing them in a compressed notation. It will be
particularly important to be able to see, at a glance, which quantities within
them change sign under interchange of $p$ and $q$ (these will be denoted with
a subscript ${\cdot}_{pq}$), which quantities are completely antisymmetric in
$a$, $b$ and/or $c$ (these will be denoted with superscripts ${\cdot}^{abc}$
or ${\cdot}^{ab}$ etc.), and which change sign under parity (these will be
denoted with a $\mathscr{P}$). Having made those symmetries clear, it will
then be easier to plot a path toward combining the ingredients into functions
which are invariant under the required interchange symmetries. We therefore
make the following definitions:

$\displaystyle\mathscr{P}^{ab}_{pq}(0)$
$\displaystyle\equiv\epsilon^{ab}_{pq}\equiv[a,b,p,q],\qquad\text{(square-
bracket contraction defined in \eqref{eq:epscontractionnotation})}$ (131)
$\displaystyle\mathscr{P}^{abc}_{pq}(1)$
$\displaystyle\equiv\epsilon^{ab}_{pq}+\epsilon^{bc}_{pq}+\epsilon^{ca}_{pq},$
(132) $\displaystyle\mathscr{P}^{abc}_{pq}(2)$
$\displaystyle\equiv\epsilon^{ab}_{pq}\epsilon^{bc}_{pq}\epsilon^{ca}_{pq},$
(133) $\displaystyle\mathscr{P}_{pq}(3)$
$\displaystyle\equiv(\epsilon^{ab}_{pq}-\epsilon^{bc}_{pq})(\epsilon^{bc}_{pq}-\epsilon^{ca}_{pq})(\epsilon^{ca}_{pq}-\epsilon^{ab}_{pq}),$
(134) $\displaystyle\mathscr{P}^{{abc}}_{pq}(\mathbb{C})$
$\displaystyle\equiv\mathscr{P}^{abc}_{pq}(1)+i\mathscr{P}^{abc}_{pq}(2)$
(135) $\displaystyle g^{x}_{y}$ $\displaystyle\equiv
G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\ \\!}c}{x}&{p+q}\\\
{y}&{p+q}\end{array}\\!\\!\\!\\!\right),\qquad\text{(using a Gram determinant
defined in \eqref{eq:gram})}$ (138) $\displaystyle
G^{\phantom{ab}}_{pq}(0)\equiv F^{\phantom{abc}}_{pq}(0)$ $\displaystyle\equiv
p^{2}-q^{2},$ (139) $\displaystyle F^{\phantom{abc}}_{pq}(1)$
$\displaystyle\equiv g^{a+b+c}_{p-q},$ (140) $\displaystyle
F^{\phantom{abc}}_{pq}(2)$ $\displaystyle\equiv
g^{a}_{p-q}g^{b}_{p-q}g^{c}_{p-q},$ (141) $\displaystyle F^{{abc}}_{pq}(3)$
$\displaystyle\equiv g^{a-b}_{p-q}g^{b-c}_{p-q}g^{c-a}_{p-q},$ (142)
$\displaystyle F^{\cdot\cdot\cdot}_{pq}(\\!\\!\\!\\!\\!\text{ \tiny{
$\begin{array}[]{c}0\\\ 1\\\ 2\end{array}$ } }\\!\\!\\!\\!)$
$\displaystyle\equiv\left(F_{pq}(0),F_{pq}(1),F_{pq}(2)\right),$ (146)
$\displaystyle G^{\phantom{ab}}_{pq}(1)$ $\displaystyle\equiv g^{a+b}_{p-q},$
(147) $\displaystyle G^{ab}_{pq}(2)$ $\displaystyle\equiv
g^{a-b}_{p-q},\qquad\qquad\text{and}$ (148) $\displaystyle f^{ab}$
$\displaystyle\equiv a^{2}-b^{2}.$ (149)

######  Lemma 4.10.

A collision event $e\in\Omega_{(pq)\rightarrow(abc)X}$ is chiral if and only
if $C$ is true where:

$\displaystyle C=$ (150) $\displaystyle(\mathscr{P}^{abc}_{pq}(1)\neq
0)\lor(\mathscr{P}^{abc}_{pq}(2)\neq 0)\lor(\mathscr{P}_{pq}(3)\neq 0)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle\left\\{\begin{array}[]{c}(f^{ab}\neq 0)\lor(g^{a-b}_{p-q}\neq
0)\lor(g^{a-b}_{a+b}\neq 0)\lor(g^{a-b}_{a+b+c}\neq 0)\lor(a=b)\\\
\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\\\
(f^{bc}\neq 0)\lor(g^{b-c}_{p-q}\neq 0)\lor(g^{b-c}_{b+c}\neq
0)\lor(g^{b-c}_{a+b+c}\neq 0)\lor(b=c)\\\
\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{151.76964pt}{0.28453pt}}\\\
(f^{ca}\neq 0)\lor(g^{c-a}_{p-q}\neq 0)\lor(g^{c-a}_{c+a}\neq
0)\lor(g^{c-a}_{a+b+c}\neq 0)\lor(c=a)\end{array}\right\\}$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(F_{pq}(0)\neq 0)\lor(F_{pq}(1)\neq 0)\lor(F_{pq}(2)\neq
0)\lor(F^{abc}_{pq}(3)\neq 0)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(F_{pq}(0)\neq 0)\lor(F_{pq}(1)\neq
0)\lor\left\\{\begin{array}[]{c}(f^{ab}\neq 0)\lor(g^{a+b}_{p-q}\neq
0)\lor(\Delta_{3}(a-b,p,q)\neq 0)\\\
\raisebox{2.27621pt}{\rule{65.04034pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{65.04034pt}{0.28453pt}}\\\
(f^{bc}\neq 0)\lor(g^{b+c}_{p-q}\neq 0)\lor(\Delta_{3}(b-c,p,q)\neq 0)\\\
\raisebox{2.27621pt}{\rule{65.04034pt}{0.28453pt}}\land\raisebox{2.27621pt}{\rule{65.04034pt}{0.28453pt}}\\\
(f^{ca}\neq 0)\lor(g^{c+a}_{p-q}\neq 0)\lor(\Delta_{3}(c-a,p,q)\neq
0)\end{array}\right\\}$

###### Proof.

This is an almost trivial re-writing of the condition $C$ (of (129)) using the
notation of Definition 4.9 together with a few applications of Lemma
LABEL:lem:tripledecomp. ∎

######  Lemma 4.11.

A collision event $e\in\Omega_{(pq)\rightarrow(ab)X}$ is chiral if and only if
$B$ is true where:

$\displaystyle B=$ (151) $\displaystyle\mathscr{P}^{ab}_{pq}(0)\neq 0$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(f^{ab}\neq 0)\lor(G^{ab}_{pq}(2)\neq 0)\lor(g^{a-b}_{a+b}\neq
0)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(G^{\phantom{ab}}_{pq}(0)\neq
0)\lor(G^{\phantom{ab}}_{pq}(1)\neq 0)\lor(G^{ab}_{pq}(2)\neq 0)$
$\displaystyle\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}\land\raisebox{2.27621pt}{\rule{195.12767pt}{0.56905pt}}$
$\displaystyle(G^{\phantom{ab}}_{pq}(0)\neq
0)\lor(G^{\phantom{ab}}_{pq}(1)\neq 0)\lor\left\\{(f^{ab}\neq
0)\lor(\Delta_{3}(a-b,p,q)\neq 0)\right\\}.$

###### Proof.

This is largely just a matter of substituting the new notation into the
expression already given in (130)). Only two things are worth remarking:

  * •

the ‘$\ldots\lor(a=b)$’ part of (130) has been omitted as it is incompatible
with the very first ‘$\epsilon^{ab}_{pq}\neq 0$’ requirement, and

  * •

naïve notational substitution for
$\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\ \\!}c}{a}&{p+q}\\\
{p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq
0\right)\lor\left(G\\!\left(\\!\\!\\!\\!\begin{array}[]{c@{,\
\\!}c}{b}&{p+q}\\\ {p-q}&{p+q}\end{array}\\!\\!\\!\\!\right)\neq 0\right)$
would result in $\left(g^{a}_{p-q}\neq 0\right)\lor\left(g^{b}_{p-q}\neq
0\right)$ however it has been rendered instead as
$(G^{\phantom{ab}}_{pq}(1)\neq 0)\lor(G^{ab}_{pq}(2)\neq 0)$ meaning
$\left(g^{a+b}_{p-q}\neq 0\right)\lor\left(g^{a-b}_{p-q}\neq 0\right)$ since
we crave statements which are either $(ab)$-even or $(ab)$-odd.

∎

####  4.4.1 Start of proof of Theorem 4.7

######  Remark.

Given Lemma 3.27 we can work with any event in
$\Omega_{(pq)\rightarrow(abc)X}$ in the $(p+q)$ rest frame, having aligned
$\vec{p}$ with the positive $z$-axis and $\vec{q}$ with the negative $z$-axis.
In such a frame, the following quantities in $\hat{e}$ uniquely describe any
event in $\Omega_{(pq)\rightarrow(abc)X}$:

$\displaystyle\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}&\mathbf{b}&\mathbf{c}\\\
p&-p&a_{z}&b_{z}&c_{z}\end{array}\right]\qquad\text{with}\qquad\left(\begin{array}[]{l}m_{p},m_{q},m_{a},m_{b},m_{c}\geq
0,\\\ p>0,\\\ \mathbf{a},\mathbf{b},\mathbf{c}\in\mathbb{C},\ \text{and}\\\
a_{z},b_{z},c_{z}\in\mathbb{R}\end{array}\right),$ (159)

provided that we take, respectively, the real and imaginary parts of
$\mathbf{a}$ to represent the $x$ and $y$ components of $a$ (and similarly for
$\mathbf{b}$ and $\mathbf{c}$).

######  Definition 4.12.

We name a matrix $\hat{e}$ which has the properties shown above to be the
event representation of an event $e\in\Omega_{(pq)\rightarrow(abc)X}$.

######  Remark.

One may naturally extend the action of the parity operator, $\mathscr{P}$, and
of elements of the symmetry group $G$, to an action on event representations
$\hat{e}$. We will make such an extensions implicitly rather than explicitly,
as happens in the next lemma:

######  Lemma 4.13.

$\displaystyle\mathscr{P}\cdot\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&-\mathbf{a}&-\mathbf{b}&-\mathbf{c}\\\
-p&p&-a_{z}&-b_{z}&-c_{z}\end{array}\right].$ (163)

###### Proof.

The result follows directly from Definition 3.15. ∎

Recall that we are attempting to find the conditions under which collision
events $e\in\Omega_{(pq)\rightarrow(abc)X}$ are non-chiral. Corollary 3.19
reminds us that in order to identify such events it suffices to find all those
$\hat{e}$ for which there exists a $g\in G$ such that
$g\cdot\hat{e}=\mathscr{P}\cdot\hat{e}$. However, because

$\displaystyle\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&-\mathbf{a}^{*}&-\mathbf{b}^{*}&-\mathbf{c}^{*}\\\
p&-p&a_{z}&b_{z}&c_{z}\end{array}\right]$ (167)

has fewer minus signs than $\mathscr{P}\cdot\hat{e}$ and is identical to
$\hat{e}$ in all but three ‘cells’, it is more convenient for us to find the
non-chiral collision events by searching for all $\hat{e}$ for which there
exists a $g\in G$ such that
$g\cdot\hat{e}=\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}$.
The inclusion of the
$\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)$ has no effect
on the set of events so obtained since $G$ is a group. Furthermore, every
element of $g\in G$ can be written as a product of an element of $SO^{+}(1,3)$
with one of the twelve elements of the following group $H$:

$H=\left\\{1,\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)\right\\}\otimes\left\\{1,(ab),(ac),(bc),(abc),(cba)\right\\}$

in which, once more, the
$\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)$ operator has
been added only for convenience, to reduce the number of minus signs appearing
in later work. We can therefore reduce the task of finding non-chiral collider
events to having to find only those events $\hat{e}$ for which there exists an
element $s\in SO^{+}(1,3)$ such that $s\cdot
h\cdot\hat{e}=\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}$
for some $h$ in $H$.

It is easy to check that every for every element $h\in H$:

$\displaystyle s\cdot h\cdot\hat{e}$ $\displaystyle=s\cdot
h\cdot\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}&\mathbf{b}&\mathbf{c}\\\
p&-p&a_{z}&b_{z}&c_{z}\end{array}\right]$ (171)
$\displaystyle=s\cdot\left[\begin{array}[]{cc|ccc}?&?&?&?&?\\\
0&0&\mathbf{?}&\mathbf{?}&\mathbf{?}\\\
p&-p&?&?&?\end{array}\right]\qquad\text{with $p>0$.}$ (175)

Since we require that the RHS of Equation (175) must match the
$\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}$
shown in Equation (167) we can see that the only elements $s\in SO^{+}(1,3)$
which need be considered are those which leave the spatial components of
$\vec{p}$ and $\vec{q}$ invariant.252525That we can make such a statement
derives ultimately from Lemma 3.27. Any boost would necessarily modify some
component of $\vec{p}$ or $\vec{q}$ we know that the only permissible options
for $s$ are rotations which, since they must leave the $z$-axis invariant, are
rotations about $z$. We have therefore reduced the task of proving Theorem 4.7
to finding the events $\hat{e}$ for which there exists an angle $\theta$ and
an element $h\in H$ such that

$\displaystyle\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)\cdot
h\cdot\hat{e}=\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}$
(176)

whose RHS we already know.

######  Definition 4.14.

Let us define $\heartsuit(h,\theta,\Omega_{(pq)\rightarrow(abc)X})$ to be the
set of non-chiral collision events
$e\in\heartsuit(\Omega_{(pq)\rightarrow(abc)X})\subset\Omega_{(pq)\rightarrow(abc)X}$
which satisfy Equation (176):

$\heartsuit(h,\theta,\Omega_{(pq)\rightarrow(abc)X})\underset{\text{def}}{\equiv}\left\\{\
\ e\ \ \middle|\ \
e\in\Omega_{(pq)\rightarrow(abc)X},\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)\cdot
h\cdot\hat{e}=\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}\right\\}.$

Similarly, let us define $\heartsuit(h,\Omega_{(pq)\rightarrow(abc)X})$ to be
the set of non-chiral collision events
$e\in\heartsuit(\Omega_{(pq)\rightarrow(abc)X})\subset\Omega_{(pq)\rightarrow(abc)X}$
which satisfy Equation (176) for some $\theta$:

$\displaystyle\heartsuit(h,\Omega_{(pq)\rightarrow(abc)X})$
$\displaystyle\underset{\text{def}}{\equiv}\left\\{\ \ e\ \ \middle|\ \
e\in\Omega_{(pq)\rightarrow(abc)X},\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)\cdot
h\cdot\hat{e}=\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e},\theta\in\mathbb{R}\right\\}$
$\displaystyle=\bigcup_{\theta\in\mathbb{R}}\heartsuit(h,\theta,\Omega_{(pq)\rightarrow(abc)X}).$

######  Corollary 4.15.

In terms of the quantities just defined, our goal,
$\heartsuit(\Omega_{(pq)\rightarrow(abc)X})$, may be written as:

$\heartsuit(\Omega_{(pq)\rightarrow(abc)X})=\bigcup_{h\in
H}\heartsuit(h,\Omega_{(pq)\rightarrow(abc)X}).$

The action of
$\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)$ on any
event representation $\hat{e}$ is nice, as it only multiplies each element of
the complex row of the representation by $e^{i\theta}$. Since there are only
twelve elements of $H$ (we will define them to be $\\{$ $h_{1}{=}1$,
$h_{2}{=}(ab)$, $h_{3}{=}(ac)$, $h_{4}{=}(bc)$, $h_{5}{=}(abc)$,
$h_{6}{=}(cba)$,
$h_{7}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)$,
$h_{8}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)(ab)$,
$h_{9}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)(ac)$,
$h_{10}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)(bc)$,
$h_{11}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)(abc)$,
$h_{12}{=}\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)(pq)(cba)$
$\\}$) we can now enumerate all twelve LHSs of Equation (176):

$\displaystyle\begin{array}[]{rr}{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{1}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}e^{i\theta}&\mathbf{b}e^{i\theta}&\mathbf{c}e^{i\theta}\\\
p&-p&{a}_{z}&{b}_{z}&{c}_{z}\end{array}\right]},&{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{7}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{a}&m_{b}&m_{c}\\\
0&0&\mathbf{a}^{*}e^{i\theta}&\mathbf{b}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}\\\
p&-p&-{a}_{z}&-{b}_{z}&-{c}_{z}\end{array}\right]},\\\
{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{2}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{b}&m_{a}&m_{c}\\\
0&0&\mathbf{b}e^{i\theta}&\mathbf{a}e^{i\theta}&\mathbf{c}e^{i\theta}\\\
p&-p&{b}_{z}&{a}_{z}&{c}_{z}\end{array}\right]},&{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{8}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{b}&m_{a}&m_{c}\\\
0&0&\mathbf{b}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}\\\
p&-p&-{b}_{z}&-{a}_{z}&-{c}_{z}\end{array}\right]},\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{3}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{c}&m_{b}&m_{a}\\\
0&0&\mathbf{c}e^{i\theta}&\mathbf{b}e^{i\theta}&\mathbf{a}e^{i\theta}\\\
p&-p&{c}_{z}&{b}_{z}&{a}_{z}\end{array}\right]},&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{9}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{c}&m_{b}&m_{a}\\\
0&0&\mathbf{c}^{*}e^{i\theta}&\mathbf{b}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}\\\
p&-p&-{c}_{z}&-{b}_{z}&-{a}_{z}\end{array}\right]},\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{4}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{c}&m_{b}\\\
0&0&\mathbf{a}e^{i\theta}&\mathbf{c}e^{i\theta}&\mathbf{b}e^{i\theta}\\\
p&-p&{a}_{z}&{c}_{z}&{b}_{z}\end{array}\right]},&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{10}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{a}&m_{c}&m_{b}\\\
0&0&\mathbf{a}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}&\mathbf{b}^{*}e^{i\theta}\\\
p&-p&-{a}_{z}&-{c}_{z}&-{b}_{z}\end{array}\right]},\\\
{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{5}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{b}&m_{c}&m_{a}\\\
0&0&\mathbf{b}e^{i\theta}&\mathbf{c}e^{i\theta}&\mathbf{a}e^{i\theta}\\\
p&-p&{b}_{z}&{c}_{z}&{a}_{z}\end{array}\right]},&{\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{11}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{b}&m_{c}&m_{a}\\\
0&0&\mathbf{b}^{*}e^{i\theta}&\mathbf{c}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}\\\
p&-p&-{b}_{z}&-{c}_{z}&-{a}_{z}\end{array}\right]},\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{6}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{c}&m_{a}&m_{b}\\\
0&0&\mathbf{c}e^{i\theta}&\mathbf{a}e^{i\theta}&\mathbf{b}e^{i\theta}\\\
p&-p&{c}_{z}&{a}_{z}&{b}_{z}\end{array}\right]},&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{12}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{q}&m_{p}&m_{c}&m_{a}&m_{b}\\\
0&0&\mathbf{c}^{*}e^{i\theta}&\mathbf{a}^{*}e^{i\theta}&\mathbf{b}^{*}e^{i\theta}\\\
p&-p&-{c}_{z}&-{a}_{z}&-{b}_{z}\end{array}\right]},\end{array}$ (219)

at least one of which will equal

$\left({\overset{\pi}{\underset{yz}{\curvearrowleft}}}\right)\cdot\mathscr{P}\cdot\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&-\mathbf{a}^{*}&-\mathbf{b}^{*}&-\mathbf{c}^{*}\\\
p&-p&a_{z}&b_{z}&c_{z}\end{array}\right].$

for some $\theta$ if $\hat{e}$ is non-chiral. In (219) six of of the
constraints have been coloured blue. This is because these constraints are
strongly related to another black one in the list. (We will see how later.)
The ‘hard work’ only needs to be invested in the black cases. Once those are
done, results relating to the remaining blue ones will follow trivially.

We now take the black constraints from (219) and re-write their complex
numbers in polar form by making the following replacements:

$\displaystyle\mathbf{a}$ $\displaystyle\rightarrow|\mathbf{a}|e^{i\alpha},$
(223) $\displaystyle\mathbf{b}$
$\displaystyle\rightarrow|\mathbf{b}|e^{i\beta},\qquad\text{and}$ (224)
$\displaystyle\mathbf{c}$ $\displaystyle\rightarrow|\mathbf{c}|e^{i\gamma}.$
(225)

This results in:

$\displaystyle\begin{array}[]{rr}\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{1}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{a}&m_{b}&m_{c}\\\
0&0&{|\mathbf{a}|}e^{i(\theta+{\alpha})}&{|\mathbf{b}|}e^{i(\theta+{\beta})}&{|\mathbf{c}|}e^{i(\theta+{\gamma})}\\\
p&-p&{a}_{z}&{b}_{z}&{c}_{z}\end{array}\right],\\\
\left({\overset{{\theta}}{\underset{xy}{\curvearrowleft}}}\right)h_{2}\hat{e}=\left[\begin{array}[]{cc|ccc}m_{p}&m_{q}&m_{b}&m_{a}&m_{c}\\\