# Suppressing defection by increasing temptation: the impact of smart
cooperators on a social dilemma situation

Hsuan-Wei Lee  Colin Cleveland  and Attila Szolnoki  Institute of Sociology,
Academia Sinica, Taiwan  Department of Physics, National Taiwan University,
Taiwan  Department of Informatics, King’s College London, London, UK
Institute of Technical Physics and Materials Science, Centre for Energy
Research, P.O. Box 49, H-1525 Budapest, Hungary

###### Abstract

In a social dilemma situation, where individual and collective interests are
in conflict, it sounds a reasonable assumption that the presence of super or
smart players, who simultaneously punish defection and reward cooperation
without allowing exploitation, could solve the basic problem. The behavior of
such a multi-strategy system, however, is more subtle than it is firstly
anticipated. When exploring the complete parameter space we find that the
emergence of cyclic dominance among strategies is rather common, which results
in several counter-intuitive phenomena. For example, the defection level can
be lowered at higher temptation, or weaker punishment provides better
conditions for smart players. Our study indicates that smart cooperators can
unexpectedly thrive under high temptation, emphasizing the complexity of
strategic interactions. This study suggests that the principles governing
these interactions can be applied to other moral behaviors, such as truth-
telling and honesty, providing valuable insights for future research in multi-
agent systems.

###### keywords:

social dilemmas , cooperation , cyclic dominance

††journal: Applied Mathematics and Computation

##  1 Introduction

To understand why we cooperate while defection offers short-term benefits is
the central problem not only in social science, but also in biology, ecology,
or economics [1, 2, 3, 4, 5]. Because of its importance, as a natural reaction
from research community, it has collected significant scientific interest from
the mentioned fields [6, 7, 8, 9, 10, 11, 12]. Evidently, there is no a single
answer and over the last decades several mechanisms or external conditions
have been identified which support the emergence of cooperation [13, 14, 15,
16, 17, 18, 19, 20]. A quite straightforward idea could be to directly support
those who cooperate or punish the rival participants hence establishing
competitive payoff situation for good players [21, 22, 23, 24, 25, 26].
However, the devil is in the details because there is no free lunch and
maintaining the abovementioned incentives requires extra care, hence extra
cost from the supporting group [27, 28, 29, 30, 31]. This often makes these
institutions vulnerable by generating a second-order free riding dilemma where
simple cooperators enjoy the benefits of these incentives without contributing
to the related expenses [32, 33, 34, 35].

Our present study targets exactly this situation where we extend the
traditional two-strategy prisoner’s dilemma model by adding a new strategy who
has some extra capacity and skill. In particular, these players, referred to
as smart cooperators, not only cooperate with partners but also are capable to
reward those partners who also cooperate. Furthermore they recognize
defection, hence they avoid being exploited and punish defectors partners
simultaneously. On the other hand, the model becomes realistic only if we
assume that these smart players should bear and extra permanent cost, which
makes it possible to “cover” the incentives and their extra information to
avoid being exploited. The competition of these strategies is studied within
the framework of prisoner’s dilemma game which is capable to grab the essence
of the basic dilemma we have mentioned above.

Despite the simplicity of this social game and the applied strategies we still
have several parameters, including reward, punishment, temptation, and extra
cost of smart players, which offer a huger liberty how system may evolve. To
explore these scenarios we have systematically scan the whole parameter space
and identify the stationary fractions of competing strategies. Our key
questions include whether the presence of smart players can always drive the
system toward the expected direction, or is it always beneficial to apply
large punishment? Or, is it always good for smart players if their extra cost
is minimal? These questions seem to be straightforward, but as we demonstrate
below, the answers are more delicate. Moreover, despite the simplicity of our
model it is capable to provide the general system behavior that is frequently
observed when multi strategies are present in the fight for space.

The rest of this paper is organized as follows. In Section 2, we define the
details of our model and specify the methodology of our computational
experiments. It is followed by consequent results and observations which are
presented in Section 3. Last, our conclusions and some general comments are
given in Section 4.

##  2 Model

Our spatial model is an extension of traditional prisoner’s dilemma game where
unconditional defector ($D$) and unconditional cooperator ($C$) players
compete for space on an $L\times L$ square lattice where periodic boundary
conditions are applied. In agreement with previous studies, in the payoff
matrix elements we fix the reward for mutual cooperation as $R=1$, and the
punishment for mutual defection $P=0$, while the remaining two elements, such
as the $T$ temptation to defect and $S$ sucker’s payoff are free parameters
[36, 37, 38, 39]. The third strategy represents a sort of “smart” cooperator
($M$) who cooperates with other cooperators, but is also capable to recognize
defectors. Consequently, when an $M$ player meets with a $D$ player then the
latter remains empty handed and the former avoids being exploited.
Furthermore, an $M$ player applies both positive and negative incentives to
support cooperation. Therefore cooperator players are awarded by a reward $W$
and defector players are punished by $U$ when they face with an $M$ partner.
Evidently, to maintain the incentives and to monitor the behavior of neighbors
require extra efforts from smart cooperators, which is considered via an
additional $G$ cost they have to bear. In this way their advantage, or their
positive impact on general cooperation is less straightforward. The relations
of different strategies are summarized in the following payoff matrix:

|  $M$ | $C$ | $D$
---|---|---|---
$M$ |  1+$W$-$G$ |  1-$G$ | -$G$
$C$ | 1+$W$ | 1 |  $S$
$D$ |  $-U$ | $T$ |  0

.

It is worth noting that a slightly similar payoff parametrization was
suggested recently by Zhao et. al in Ref. [40]. But in their model a defector
can still utilize a connection with a smart player. Furthermore, the latter
strategy is also rewarded by an unconditional cooperator which is unjustified
in the absence of additional resources. We strongly believe that our payoff
matrix, shown above, expresses the role of strategy $M$ more accurately: these
players possess the requested resources to motivate cooperation for an
additional price they have to bear.

When the strategy evolution is monitored we follow the standard protocol. More
precisely, we randomly select a player $i$ and a neighboring player $j$. If
their strategy $s_{i}$ and $s_{j}$ are different then we calculate their
$\Pi_{i}$ and $\Pi_{j}$ payoff values by considering their nearest neighbor
links to their neighbors. After player $i$ adopts the $s_{j}$ strategy of
player $j$ with the probability

$\Gamma(s_{j}\to s_{i})=\frac{1}{1+\exp[(\Pi_{i}-\Pi_{j})/K]}\,\,.$ (1)

Here, we apply the broadly used $K=0.1$ noise parameter value that generally
allows player $i$ to imitate the more successful strategy, but seemingly
irrational strategy choice is also possible with a low probability [41, 42,
43, 44, 45, 46]. This process is repeated $N=L\times L$ times which
establishes a full Monte Carlo (MC) step. Typically we use $10^{4}$ MC steps
to reach the stationary state where the linear system size is $L=1000$ or
$L=1200$. There is, however, an additional technical detail which has been
proven to be very useful to obtain reliable numerical data. In particular, we
apply a very small $\mu=10^{-6}$ mutation rate which allows players to update
their strategy randomly. It technically means that one in a million elementary
updating process is random where the imitation probability defined by Eq. 1 is
ignored. This tiny mutation rate helps us to avoid very heavy finite-size
effects without influencing the pattern formation relevantly. Evidently, the
usage of mutation prevents the system from being trapped in a homogeneous
state, hence we cannot talk “phases” anymore, but a strategy is not really
viable if its fraction is proportional to the mutation rate.

In the rest of this work we fix the sucker’s payoff $S=-0.1$ which means a
proper prisoner’s dilemma situation. According to the payoff matrix, four
other parameters remain that characterize the $T$ temptation to defect, the
$W$ extra reward to support cooperation, the $U$ punishment for defection, and
the $G$ extra cost of smart players to maintain the incentives. In the
following we explore the full parameter space and recognize the typical system
behavior thanks to the presence of smart cooperator players.

##  3 Results

But first, as a motivation plot, we present the fractions of all strategies at
representative parameter values, which illustrate nicely how unexpected the
consequence of an advanced strategy could be. This is shown in Fig. 1 where we
plot the stationary fractions of strategies in dependence of temptation level.
At $T=1$ $M$ players practically die out due to the large $G$ cost value they
have to sustain. More precisely, their fraction is so small that is
proportional to the mutation rate. As a consequence, defectors dominate the
system. The naïve expectation would be that this situation becomes even worse
if we increase the temptation value because it typifies a direct form of
support to defector strategy. Interestingly, however, the opposite happens and
the portion of defectors decays gradually by increasing $T$. In parallel, the
fraction of $M$ players increases significantly despite of the aforementioned
large cost level.

Figure 1: The stationary fractions of competing strategies in dependence of
$T$. Interestingly, the portion of defectors decays as we increase the
temptation value. Other parameters are $U=1$, $W=0.5$, $G=0.6$, and $S=-0.1$.

The above-discussed phenomenon, as we explain it later, is not necessarily
counter-intuitive if we consider the relation of competing strategies more
carefully. But before doing so we first give a general overview about the
system behavior in the vast sea of parameters. It can be done systematically
if we present the general level of a specific strategy in a $T-W$ parameter
plane at fixed $G$ and $U$ values. After we modify the values of the latter
parameters systematically in a reasonable interval. The repeat of this loop
results in $11^{2}=121$ different heat maps for each strategy. We stress again
that traditional phase diagrams cannot be presented because of the nonzero
mutation rate, hence we need to present the portions of all three strategies
separately. The results are shown in Fig. 2.

Figure 2: Heat map of the stationary fraction of $D$ (top-left), $C$ (top-
right), and $M$ (below-central) strategy on the full $T-W$ parameter plane.
Each micro panel was obtained at a fix value of $G$ and $U$, as indicated on
the axis. The value of $S=-0.1$ was fixed for all cases.

By checking these panels some general observations can be made. If the $G$
additional cost of the $M$ strategy is too high, then smart players are not
really functioning and the traditional relation between $C$ and $D$ players
results in a complete dominance of the latter strategy almost everywhere. More
precisely, the only exception is when $U$ punishment is high enough , thereby
availing a chance for $M$ strategy at the expense of defectors. This
phenomenon is more pronounced at higher temptation values; hence, the
phenomenon we presented in Fig. 1 is a general effect in a broad range of
parameters. When $G$ is small, then defection can be kept at a minimal level
and the real victors of this process are unconditional cooperators.
Interestingly, smart player can dominate the population only at very small $G$
cost values, but only if the $W$ reward remains below a threshold level,
otherwise it is better for a player to choose the unconditional cooperation
strategy. The latter effect is almost independent of the $U$ punishment level,
but it is straightforward because $D$ players are present just minimally at
these parameter values. However, the threshold value of $W$, where $C$ players
can beat $M$ players, depends sensitively on the $T$ temptation value, as the
low row illustrates in Fig. 2c. The only reason why this fact is worth noting
is because the payoff elements characterizing the relation of $C$ and $M$
strategies do not contain parameter $T$. Nevertheless, this observation gives
a clue to understand the above described phenomena more deeply.

These seemingly paradoxical effects can be explained if we recognize the
potential intransitive relations of competing strategies. Evidently, defectors
beat unconditional cooperators because of the negative $S$ sucker’s payoff
value. However, $C$, outperforms $M$ because the former enjoys being rewarded
by $W$ without paying its cost $G$. Last, $M$ can exceed $D$ because the
former punishes the latter and avoids being exploited. This closes the loop
and establishes a rock-scissors-paper-like cyclic dominance among the three
strategies. To support this conjecture, we present a front propagation process
between homogeneous domains in Fig. 3.

Figure 3: Propagation of invasion fronts starting from a prepared initial
state, shown in panel (a). Color coded identically to Fig. 1. Blue cooperators
invade the green bulk of smart cooperators, while red defectors invade blue
cooperators. To close the loop, the green domain gently intrudes into the red
field. Time in MC steps are shown below. Parameter values are $T=1.6$,
$W=0.3$, $G=0.5$, $U=0.2$, and $S=-0.1$.

As Figure 3a shows, we launch the evolution from a prepared initial state
where homogeneous domains of competing strategies are arranged. They are red
defectors, blue cooperators, and green smart players. The subsequent panels
show the stages of front propagation after different MC steps, as indicated in
the label. Already, panel (b) illustrates the key effects that confirm our
basic conjecture. Namely, the border separating the red and blue areas starts
moving to the right, indicating that $D$ players start invading $C$ players.
Simultaneously, there is a certain chance that the seed of defectors emerge in
the bulk of unconditional cooperators due to mutation and these “small red
domains” start growing. Hence, we can conclude that $D$ beats $C$, as
expected. Of course, green $M$ players can also emerge due to mutation in the
blue sea, but they cannot spread because they are inferior to $C$ players.
Actually, the opposite is true, as the shift of border between green and blue
domains indicates. Furthermore, a small seed of blue payers can easily grow in
the green ocean. Both movements underline the superiority of $C$ against $M$.
Finally, the green domain can invade the red domain of defectors. This process
is weaker than the previously discussed cases, because the $M\to D$ invasion
is less effective, but it still exists, as it can be seen clearly on the top
part of the (b)-(d) panels. In sum, the movement of domain walls confirm the
$D\to C\to M\to D$ intransitive relation undoubtedly.

Based on our knowledge about cyclically dominant systems [47, 48, 49, 50], the
above-described counter-intuitive behaviors can be understood in the following
way. It is a frequently observed and broadly confirmed effect that when we
support of species in a loop directly, then its predator will benefit from
this intervention mostly [51, 52, 53, 54]. Exactly the same system behavior
can be seen in our motivation figure, as shown in Fig. 1: when we increase the
$T$ temptation level and seemingly support defectors, then their “predators’,’
namely, smart players will benefit from this action. According to this
argument, we can expect other apparently unexpected system behaviors. For
example, it is expected that the increase of punishment $U$ will weaken
defectors. Indeed, it is true in the high or significant $G$ cost region where
the role of smart players is negligible. When the latter strategy is viral,
however, we can see that by weakening $D$, we indirectly penalize its
“predator” that is the mentioned strategy $M$. This effect can be seen in Fig.
4, where we plot the fractions of strategies in dependence of parameter $U$.
The figure suggests that the growth of $U$ just slightly reduces the portion
of $D$ and its predator $M$ strategy suffers more. However, the “prey” of
strategy $D$, which is unconditional $C$, benefits from harsh punishment. As
$M$ is the predator, while $C$ is the prey of strategy $D$, it explains why
the former two cooperating strategies behave differently by increasing
punishment exclusively.

Figure 4: The stationary fractions of competing strategies in dependence of
$U$. Interestingly, smart cooperators benefit the most from low punishment
level. The increase of punishment, however, just slightly modify the defection
level. Instead, cooperators enjoy it at the expense of smart cooperators.
Other parameters are $W=0.2$, $G=0.2$, $T=2$, and $S=-0.1$.

It also seems a straightforward expectation that lowering the $G$ cost of the
$M$ strategy should directly support smart players. Especially because the
mentioned parameter is present only in the top role of payoff matrix that
determine the incomes of strategy $M$. Indeed, it is generally true, but the
real beneficiary of the mentioned intervention should be $M$’s predator, which
is an unconditional cooperator. Figure 5 confirms our “advanced expectation”
because the portion of strategy $C$ increases more drastically when we lower
the cost value $G$ of smart players.

Figure 5: The stationary fractions of competing strategies in the dependence
of $G$. Interestingly, simple cooperators benefit the most from the low cost
of smart cooperation and the increase of this parameter value smites strategy
$C$ more drastically. Other parameters are $W=0.6$, $U=0.6$, $T=1.5$, and
$S=-0.1$.

##  4 Conclusion

One of the typical features of complex systems is an unexpected scenario in
response to external interventions [55]. Evidently, when there are more than
two options for participants to become successful in a multi-agent population,
then the mentioned complexity is almost an inevitable ingredient of the
collective behavior [56, 57, 58, 59, 60]. By introducing a seemingly trivial
strategy into the intensively studied enigma where unconditional strategies
compete [61, 62, 63, 64], our principal motivation is to check whether the
increment of the number of strategies may result in a system response that
cannot be predicted in advance. What can we expect from a “super player” who
is armed with a specific skill to recognize the strategy of partners? Hence,
it is protected against being exploited? On top of that, it can directly
support positive acts and retaliate against negative ones and thus can modify
competitors’ incomes. Evidently, these players are expected to be the
champions of an evolutionary process. To make the model more reliable,
however, we assume that they have to pay the price of this superpower.

Our first key observation is that this super strategy is generally unable to
prevail and $M$ players are rarely dominant. More precisely, they can only win
when their additional cost is extremely small and the reward they provide for
cooperation is also moderate. Notably, the level of punishment and how they
penalize defection have no particular role in their final dominance. In all
other cases, the real benefit of their presence is unconditional cooperators
who can be considered as second-order free riders. At this point, it is worth
noting that the very low mutation rate, which does not affect the pattern
formation relevantly, but helps to avoid serious finite-size problems, has an
additional positive consequence: second-order free riders cannot repair the
original dilemma situation that is frequently reported in other models where
cooperation supporting incentives are introduced [65].

The explanation as to why smart players become just one of the options is the
fact that their presence introduces a cyclical dominance among competing
strategies. In a vast variety of full parameter space, the $M$ strategy beats
$D$ who beats $C$ who beats $M$, hence closing the loop. Importantly, a
similar emergence of intransitive relations among competing strategies has
been reported quite often in social dilemma situations [66, 67, 68, 69, 70,
71, 72, 73, 74, 75].

Having realized the above-described relations, the seemingly counter-intuitive
system behavior becomes understandable immediately. We already knew for more
than two decades that a direct aid for a loop member will mostly support its
predator partner, and similarly, the main loser of a weakening process will be
the target’s superior fellow. Accordingly, increasing the temptation to defect
can result in the blooming of smart players in our present model. Or, the
latter strategy would suffer more by increasing the severity of punishment.

We also recognize that the principles explored in this study could extend
beyond cooperation to other aspects of moral behavior, such as truth-telling
or honesty. Recent reviews, such as [76], provide a foundational basis for
this line of inquiry. Future research could explore additional strategies,
different types of social dilemmas, and the application of our findings to
other aspects of moral behavior, such as truth-telling and honesty. These
directions offer valuable opportunities to further understand strategic
interactions in multi-agent systems. The main message of our present study is
that when we introduce more strategies to interpret a real-life situation more
faithfully, it always gives a chance for a loop to emerge among competitors.
Therefore, the consequence of a new rival on global behavior will not
necessarily be our first naïve expectation; hence, the external conditions
that can determine the payoff elements of participants should be designed with
special care.

This research was supported by the National Research, Development and
Innovation Office (NKFIH) under Grant No. K142948 and by the National Science
and Technology Council of the Republic of China (Taiwan) under Grant No.
NSTC112-2410-H-001-026-MY2.

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# Searches for $p_{\rm T}$ dependent fluctuations of flow angle and magnitude
in Pb–Pb and p–Pb collisions

You Zhou (for the ALICE111A list of members of the ALICE Collaboration and
acknowledgements can be found at the end of this issue.Collaboration)  Nikhef,
Science Park 105, 1098 XG Amsterdam, The Netherlands
Utrecht University, P.O.Box 80000, 3508 TA Utrecht, The Netherlands
<EMAIL_ADDRESS>
###### Abstract

Anisotropic azimuthal correlations are used to probe the properties and the
evolution of the system created in heavy-ion collisions. Two-particle
azimuthal correlations were used in the searches of $p_{\rm T}$ dependent
fluctuations of flow angle and magnitude, measured with the ALICE detector.
The comparison of hydrodynamic calculations with measurements will also be
presented in this proceedings.

###### keywords:

flow angle , flow fluctuations , factorization

##  1 Introduction

The primary goal of ultra relativistic heavy-ion collisions is to understand
the properties of the quark-gluon plasma (QGP), a new state of matter whose
existence under extreme conditions is predicted by quantum chromodynamics. An
important experimental observable toward this goal is the study of anisotropic
flow using a Fourier expansion of the azimuthal anisotropy [1],

$E\frac{d^{3}N}{d^{3}{\bf p}}=\frac{1}{2\pi}\frac{d^{2}N}{p_{\rm T}dp_{\rm
T}d\eta}(1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\varphi-\Psi_{n})])$ (1)

where $E$ is the energy, ${\bf p}$ the momentum, $p_{\rm T}$ the transverse
momentum, $\varphi$ the azimuthal angle, $\eta$ the pseudorapidity of the
particle. The $v_{n}$ coefficients and $\Psi_{n}$ represent the magnitude and
angle (symmetry plane) of the $n^{th}$-order harmonic, respectively. The
elliptic flow $v_{2}$, triangular flow $v_{3}$, quadrangular flow $v_{4}$, and
pentagonal flow $v_{5}$ have been measured at the CERN Large Hadron Collider
(LHC). It provides compelling evidence that strongly interacting matter
appears to behave like an almost perfect fluid [2]. Recent hydrodynamic
simulations predict the $p_{\rm T}$ dependent fluctuations of flow angle and
magnitude. Two new observables, $v_{n}\\{2\\}/v_{n}[2]$ and $r_{n}$ were
proposed in hydrodynamic calculations [3, 4]. They are used to check whether
the factorization of two-particle correlations into the product of single
particle anisotropy harmonics is valid, which probes the $p_{\rm T}$ dependent
flow angle and magnitude fluctuations.

In this proceeding, we present the measurements of $v_{2}\\{2\\}/v_{2}[2]$ and
factorization ratio $r_{n}$ in Pb–Pb collisions at $\sqrt{s_{\rm NN}}=$ 2.76
TeV and p–Pb collisions at $\sqrt{s_{\rm NN}}=$ 5.02 TeV in ALICE.

##  2 Analysis Details

The data sample collected by ALICE in the first Pb–Pb run and p–Pb run at the
Large Hadron Collider were used in this analysis. For more details of ALICE
detector, refer to [5]. About 16 million Pb–Pb events and 97 million p–Pb
events were recorded with a minimum-bias trigger, based on signals from two
VZERO detectors (-3.7$\textless\eta\textless$-1.7 for VZERO-C and
2.8$\textless\eta\textless$5.1 for VZERO-A) and on the Silicon Pixel Detector.
The VZERO detectors were also used for the determination of the collision
centrality in Pb–Pb collisions while VZERO-A was taken for the determination
of the multiplicity classes in p–Pb collisions. Charged particles are
reconstructed using the Inner Tracking System and the Time Projection Chamber
with full azimuthal coverage for pseudo-rapidity range $|\eta|\textless$0.8.
The two-particle correlations are measured using the Q-Cumulant method [6]
with a pseudorapidity gap $|\Delta\eta|>$ 0.8, which is expected to suppress
the non-flow effects (azimuthal correlations not related to the symmetry
plane) as much as possible but still allow good statistical precision within
the TPC acceptance.

##  3 Results

Figure 1:  The ratio $v_{2}\\{2,|\Delta\eta|>0.8\\}/v_{2}[2,|\Delta\eta|>0.8]$
for various centralities of Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV.
The error bars correspond to statistical uncertainties, while the shaded color
bands correspond to the systematic uncertainties. Hydrodynamic calculations
with MC-Glauber initial condition and $\eta/s=$ 0.08 and with MC-KLN initial
condition and $\eta/s=$ 0.20 are shown in green and orange lines.  Figure 2:
The ratio $v_{3}\\{2,|\Delta\eta|>0.8\\}/v_{3}[2,|\Delta\eta|>0.8]$ for
various centralities of Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV.

Figure 1 presents the ratio
$v_{2}\\{2,|\Delta\eta|>0.8\\}/v_{2}[2,|\Delta\eta|>0.8]$ as a function of
$p_{\rm T}$ in different centrality classes. It is observed that the ratio is
consistent with unity up to $p_{\rm T}\approx$ 2 GeV/$c$ in most central
collisions and up to $p_{\rm T}\approx$ 3 GeV/$c$ in the non-central
collisions. The deviations from unity become weaker but occur at higher
$p_{\rm T}$ range toward peripheral collisions. The result indicates that if
$p_{\rm T}$ dependent flow angle ($\Psi_{2}$) and magnitude ($v_{2}$)
fluctuations exist, such effect is within 10$\%$ in non-central collisions in
the presented $p_{\rm T}$ range. The comparison to hydrodynamic calculations
shows that both calculations overestimate the deviation of
$v_{2}\\{2,|\Delta\eta|>0.8\\}/v_{2}[2,|\Delta\eta|>0.8]$ in most central
collisions, and the data is better described by the one with MC-KLN initial
condition and $\eta/s=$ 0.20.

The ratios of $v_{3}\\{2,|\Delta\eta|>0.8\\}/v_{3}[2,|\Delta\eta|>0.8]$ and
$v_{4}\\{2,|\Delta\eta|>0.8\\}/v_{4}[2,|\Delta\eta|>0.8]$ together with
various hydrodynamic calculations are shown in Fig. 2. It is found that both
ratios agree with unity over a wider $p_{\rm T}$ range than
$v_{2}\\{2,|\Delta\eta|>0.8\\}/v_{2}[2,|\Delta\eta|>0.8]$. The $p_{\rm T}$
dependent fluctuations of flow angle $\Psi_{3}$ (and $\Psi_{4}$) and magnitude
$v_{3}$ (and $v_{4}$) are not significant in the presented $p_{\rm T}$ range.
Meanwhile it’s still difficult to conclude which hydrodynamic calculations
describe the measurements better, due to limited statistics.

Figure 3:  The factorization ratio $r_{2}$, as a function of $p_{\rm T}^{a}$
in bins of $p_{\rm T}^{t}$ for 0-5 $\%$, 20-30 $\%$ and 40-50 $\%$ in Pb–Pb
collisions at $\sqrt{s_{{}_{\rm NN}}}=2.76$ TeV, is presented by solid
circles. CMS measurements [7] are presented by open squares, hydrodynamic
calculations with MC-Glauber initial condition and $\eta/s=$ 0.08 and with MC-
KLN initial condition and $\eta/s=$ 0.20 are shown in green and orange curves.

Figure 3 shows the factorization ratio $r_{2}$ for three $p_{\rm T}$ of
triggle particles (denoted as $p_{\rm T}^{t}$), as a function of $p_{\rm T}$
of associate particles (denoted as $p_{\rm T}^{a}$) for centrality classes 0-5
$\%$, 20-30 $\%$ and 40-50 $\%$ in Pb–Pb collisions. It significantly deviates
from unity as the collisions become more central, this effect becomes stronger
with an increase of $|p_{\rm T}^{t}-p_{\rm T}^{a}|$. The deviation is within
10 $\%$ for the lowest $p_{\rm T}^{a}$ in the 0-5 $\%$ for
2.5$\textless~{}p_{\rm T}^{t}~{}\textless$ 3.0 GeV/$c$. This can be due to
$p_{\rm T}$ dependent fluctuations of flow angle ($\Psi_{2}$) and magnitude
($v_{2}$) generated by initial event-by-event geometry fluctuations, and
possible remaining non-flow effects. Various hydrodynamic calculations are
compared to data for the presented centrality classes. Both hydrodynamic
calculations with MC-Glauber initial condition and $\eta/s=$ 0.08 and MC-KLN
initial condition and $\eta/s=$ 0.20 qualitatively predict the trend of
$r_{2}$, while the latter agrees better with data. In addition, recent CMS
measurements [7], which are based on $|\Delta\eta|>$ 2.0 (asymmetry cuts in
pseudorapidity), are consistent with our measurements. Notice that the ALICE
analysis focus on the mid-pseudorapidity range and the selected two correlated
particles are always from symmetric sub-events in pseudorapidity, while this
is not the case in CMS measurements. The agreement suggest that the possible
additional pseudorapidity dependent fluctuations of flow angle ($\Psi_{2}$)
and magnitude ($v_{2}$) is not accessible in the presented pseudorapidity
range. The factorization of $r_{3}$ (not shown here) in Pb–Pb collisions is
valid over a wider range of $p_{\rm T}^{a}$, $p_{\rm T}^{t}$ and centrality
ranges, compared to $r_{2}$. The factorization is broken within 10 $\%$ for
$p_{\rm T}^{a}$, $p_{\rm T}^{t}$ below 4 GeV/$c$. CMS measurements
quantitatively agree with our $r_{3}$ measurements, additional pseudorapidity
dependent fluctuations of flow angle ($\Psi_{3}$) and magnitude ($v_{3}$)
might be negligible in the presented pseudorapidity range.

Figure 4:  The $v_{2}\\{2,|\Delta\eta|>0.8\\}$ and $v_{2}[2,|\Delta\eta|>0.8]$
for various multiplicity classes of p–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. Hydrodynamic calculations (MUSIC) of $v_{2}\\{2\\}$ and
$v_{2}[2]$ are shown by solid and dash lines.

The $v_{2}\\{2,|\Delta\eta|>0.8\\}$ and $v_{2}[2,|\Delta\eta|>0.8]$ are also
measured in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV in ALICE. It is
found that $v_{2}\\{2,|\Delta\eta|>0.8\\}$ and $v_{2}[2,|\Delta\eta|>0.8]$
deviate at $p_{\rm T}\sim$ 3 GeV/$c$ for the presented multiplicity classes.
Hydrodynamic calculation [8] show a similar trend as a function of $p_{\rm T}$
but shows better agreement in central than in peripheral p–Pb collisions. Due
to limited statistics, the $v_{3}\\{2,|\Delta\eta|>0.8\\}$ and
$v_{3}[2,|\Delta\eta|>0.8]$ are investigated up to $p_{\rm T}\sim$ 3 GeV/$c$.
The difference of this two measurements are not observed for the the presented
multiplicity and $p_{\rm T}$ range.

##  4 Summary

In summary, searches of $p_{\rm T}$ dependent flow angle and magnitude
fluctuations are performed by measuring $v_{n}\\{2\\}/v_{n}[2]$ and $r_{n}$.
It is found that $v_{2}\\{2,|\Delta\eta|>0.8\\}/v_{2}[2,|\Delta\eta|>0.8]$ is
consistent with unity up to $p_{\rm T}\approx$ 2 GeV/$c$ in most central Pb–Pb
collisions, the deviation becomes weaker but occur at higher transverse
momenta towards peripheral collisions. A significant deviation of
$v_{3}\\{2,|\Delta\eta|>0.8\\}/v_{3}[2,|\Delta\eta|>0.8]$ and
$v_{4}\\{2,|\Delta\eta|>0.8\\}/v_{4}[2,|\Delta\eta|>0.8]$ from unity is not
observed. The factorization ratio $r_{2}$ significantly deviates from unity as
the collisions become more central, such effect becomes stronger as $|p_{\rm
T}^{t}-p_{\rm T}^{a}|$ is increasing. The comparison to hydrodynamic
calculations shows that the one with MC-KLN initial condition and $\eta/s=$
0.20 has better agreement to the data than with MC-Glauber initial condition
and $\eta/s=$ 0.08, but none of them can quantitatively describe the data. The
$v_{n}\\{2\\}/v_{n}[2]$ has been measured also in p–Pb collisions, it deviates
from unity for $p_{\rm T}>$ 3 GeV/$c$ and is under predicted in hydrodynamic
calculations.

## References

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  * [5] K. Aamodt et al. [ALICE Collaboration], JINST 3, S08002 (2008).
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# Temperature effects for $\displaystyle
e^{-}+e^{+}\rightarrow\mu^{-}+\mu^{+}$ scattering in very special relativity

A. F. Santos<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  Faqir C. Khanna 111Professor Emeritus
- Physics Department, Theoretical Physics Institute, University of Alberta
Edmonton, Alberta, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Victoria,BC V8P 5C2, Canada

###### Abstract

The electron-positron scattering process is investigated in the context of
very special relativity (VSR). This theory assumes that the true symmetry of
nature is not the full Lorentz group, but some of its subgroups, such as the
subgroups $\displaystyle SIM(2)$ and $\displaystyle HOM(2)$. In this context,
the cross-section for electron-positron scattering at finite temperature is
calculated. The effects of temperature are introduced using the Thermo Field
Dynamics (TFD) formalism. Our result shows that the cross-section is changed
due to both effects, the VSR contributions and temperature effects. An
estimated value for the VSR parameter using experimental data available in the
literature is discussed.

##  I Introduction

The structure of the standard model, as well as all fundamental physics, is
based on Lorentz symmetry. However, the search for a new physics beyond the
standard model leads to the possibility that Lorentz symmetry is not an
absolute symmetry of nature, but only approximate, valid up to a certain limit
of energy. Thus, in recent years, Lorentz invariance and its breakdown have
been intensively explored. One motivation, among many others, for such an
investigation is the quantization of gravity, the deepest puzzle of
contemporary physics. Numerous attempts to construct a consistent theory of
quantum gravity lead to the Lorentz violation, such as string theory, loop
quantum gravity, non-commutative geometry, among others Kost ; Gambini ;
Carroll ; Jacobson . Therefore, the search for a fundamental theory that
unifies all interactions in nature indicates that Lorentz symmetry may fail at
high energy scales. As a consequence, a new physics with different degrees of
freedom can be explored. In order to investigate these new possibilities,
theories that can accommodate deviations from these symmetries, in the context
of standard model particle physics or gravitational phenomena, have been
constructed. An example is the standard model extension (SME) SME1 ; SME2 .
The SME is a theoretical framework that includes standard model, general
relativity and all possible operators that break Lorentz symmetry. Another
interesting alternative to investigate the Lorentz violation is the very
special relativity (VSR) Cohen1 ; Cohen2 .

The basic idea of VSR is that Lorentz symmetry is not a fundamental symmetry
of nature, this role is reserved for one of its subgroups preserving the basic
elements of special relativity. The $\displaystyle SIM(2)$ and $\displaystyle
HOM(2)$ are examples of these subgroups. The subgroup $\displaystyle HOM(2)$
consist of three parameters and is generated by $\displaystyle
T_{1}=K_{x}+J_{y}$, $\displaystyle T_{2}=K_{y}-J_{x}$ and $\displaystyle
K_{z}$, where $\displaystyle\vec{J}$ and $\displaystyle\vec{K}$ are the
generators of rotations and boosts, respectively. While the subgroup
$\displaystyle SIM(2)$ is composed of the $\displaystyle HOM(2)$ group plus
the $\displaystyle J_{z}$ generator. An interesting feature of these subgroups
is that they both preserve the direction of a light-like four-vector
$\displaystyle n_{\mu}$, that transform as $\displaystyle n\rightarrow
e^{\varphi}n$ under boost in the z direction, and have no invariant tensor
fields. This implies that invariant theories under these subgroups have a
preferred direction in Minkowski space-time. Local operators preserving
$\displaystyle HOM(2)$ or $\displaystyle SIM(2)$ also preserve Lorentz
symmetry. On the other hand, non-local terms can be constructed as ratios of
contractions of $\displaystyle n_{\mu}$ with other kinematic vectors. Although
these non-local terms violate Lorentz symmetry, they are invariant under
$\displaystyle HOM(2)$ or $\displaystyle SIM(2)$. There are many theoretical
and phenomenological studies that investigate interesting aspects of the
effects of VSR Cohen3 ; Vo ; MM ; Das ; Riv ; Bufalo ; Bufalo2 ; Soto ; Alf ;
Bufalo3 ; Bufalo4 ; Our ; Alfaro ; Dimakis . In this paper, the well-known QED
(quantum electrodynamics) process, $\displaystyle
e^{-}+e^{+}\rightarrow\mu^{-}+\mu^{+}$ scattering, in VSR at finite
temperature is studied. The Thermo Field Dynamics (TFD) formalism is used to
introduce temperature effects.

Two different approaches can be used to introduce the effects of temperature
into a quantum field theory. (i) The imaginary time formalism Matsubara  and
(ii) the real time formalism. The latter consist of two approaches: (a) the
closed time path formalism Schwinger  and (b) the TFD formalism Umezawa1 ;
Umezawa2 ; Umezawa22 ; Khanna1 ; Khanna2 ; Kbook . Here the TFD formalism is
considered. The main characteristic of this formalism is the duplication of
the original Fock space and the use of Bogoliubov transformation. The doubling
space is defined as $\displaystyle{\cal S}_{T}={\cal S}\otimes\tilde{\cal S}$,
where $\displaystyle{\cal S}$ and $\displaystyle\tilde{\cal S}$ are the
original Fock space and tilde space, respectively. This doubling is defined by
the tilde or dual conjugation rules. The Bogoliubov transformation introduces
a rotation in the tilde and non-tilde variables, and then temperature effects
are implemented in the doubled Fock space. Another important feature of this
approach is that the temporal evolution of the system and the effects of
temperature can be considered simultaneously.

This paper is organized as follows. In section II, the quantum electrodynamics
in VSR is presented. A new interaction Lagrangian due to the VSR modifications
is discussed. In section III, TFD formalism is briefly introduced. In section
IV, the transition amplitude for the electron-positron scattering at finite
temperature is calculated. Then the cross-section in VSR at finite temperature
for this process is obtained. A numerical estimate for the VSR parameter is
analyzed. In section V, some concluding remarks are presented.

##  II QED in Very Special Relativity

In this section, the main characteristics of QED in VSR are presented. The
$\displaystyle SIM(2)$ VSR-invariant Lagrangian that describes the interaction
between fermions and photons is given as

$\displaystyle\displaystyle{\cal
L}=-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\bar{\psi}\left(i\gamma^{\mu}\nabla_{\mu}-m_{e}\right)\psi-\frac{1}{2\alpha}\left(\tilde{\partial}_{\mu}A^{\mu}\right)^{2},$
(1)

where

$\displaystyle\displaystyle\tilde{F}_{\mu\nu}=\tilde{\partial}_{\mu}A_{\nu}-\tilde{\partial}_{\nu}A_{\mu}$
(2)

is the field strength defined in terms of the wiggled derivative which is
given as

$\displaystyle\displaystyle\tilde{\partial}_{\mu}=\partial_{\mu}+\frac{m^{2}}{2}\frac{n_{\mu}}{n\cdot\partial}.$
(3)

Here, $\displaystyle n_{\mu}=(1,0,0,1)$ is a light-like four-vector that
represents the preferred null direction and the $\displaystyle m$ parameter
sets the scale for the VSR effects. It is to be noted that, using the wiggle
derivative in the field strength the VSR-Maxwell equation leads to

$\displaystyle\displaystyle\left(\partial^{2}+m^{2}\right)A^{\nu}=0.$ (4)

This implies that $\displaystyle A^{\nu}$ is a massive field. Furthermore, it
has been shown that the gauge symmetry associated with the VSR provides
massive modes as in the standard case, i.e., without changing the number of
physical polarization states of the photon Cheon ; Rivelles . It is important
to note that, in this context, the photon mass coming from a term that is
gauge invariant, unlike the case where a term of the type $\displaystyle
m^{2}A^{\mu}A_{\mu}$ is not gauge invariant.

In the VSR framework, the minimal coupling between fermions and photons is
determined by a new gauge invariant covariant derivative that is given as

$\displaystyle\displaystyle\nabla_{\mu}\psi=D_{\mu}\psi+\frac{1}{2}\frac{m^{2}}{n\cdot
D}n_{\mu}\psi,$ (5)

where $\displaystyle D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the usual covariant
derivative. The term $\displaystyle 1/(n\cdot D)$ in the covariant derivative
has a non-local character that leads to an infinite number of interactions in
the coupling $\displaystyle e$. Details about these interactions and Feynman
rules have been constructed, see Dunn . In order to calculate the cross-
section for electron-positron scattering, the most important part of the
Lagrangian is the interaction part that is given as

$\displaystyle\displaystyle{\cal L}_{int}=\bar{\psi}{\cal
V}^{\mu}(p_{1},p_{2})\psi A_{\mu},$ (6)

where $\displaystyle{\cal V}^{\mu}(p_{1},p_{2})$ is the vertex which is
defined as

$\displaystyle\displaystyle{\cal
V}^{\mu}(p_{1},p_{2})=-ie\left[\gamma^{\mu}+\frac{m^{2}}{2}\frac{\not{n}\,n^{\mu}}{(n\cdot
p_{1})(n\cdot p_{2})}\right].$ (7)

Observe that in this vertex the contribution due to the VSR has a non-local
form. Another important quantity for calculations that will be developed is
the photon propagator. Then the gauge propagator with massive pole is given as

$\displaystyle\displaystyle
i\Delta^{\mu\nu}(k)=\frac{\eta^{\mu\nu}}{\tilde{k}^{2}}$ (8)

with $\displaystyle\tilde{k}^{2}=k^{2}-m^{2}$. It is important to emphasize
that the photon propagates massive modes in a gauge invariant framework
without changing its number of physical degree of freedom.

In the next section, the TFD formalism is introduced and then the cross-
section at finite temperature for $\displaystyle
e^{-}+e^{+}\rightarrow\mu^{-}+\mu^{+}$ scattering in very special relativity
is calculated.

##  III TFD formalism

Here a brief introduction to TFD formalism is presented. The main motivation
for constructing the TFD formalism, which is a real time formalism of quantum
field theory at finite temperature, is to obtain that $\displaystyle\langle
A\rangle=\langle 0(\beta)|A|0(\beta)\rangle$, i.e., the thermal average of any
operator A is equal to its temperature dependent vacuum expectation value,
with $\displaystyle|0(\beta)\rangle$ being a thermal vacuum. Such a
development requires two essential elements: (i) the doubling the degrees of
freedom in a Hilbert space and (ii) the Bogoliubov transformation.

The doubling is defined by the tilde (∼) conjugation rules, that associate
each operator in $\displaystyle S$ to two operators in $\displaystyle S_{T}$,
where the expanded space is $\displaystyle S_{T}=S\otimes\tilde{S}$, with
$\displaystyle S$ being the standard Hilbert space and
$\displaystyle\tilde{S}$ the tilde (dual) space. The map between the tilde
$\displaystyle\tilde{A_{i}}$ and non-tilde $\displaystyle A_{i}$ operators is
defined by the following tilde (or dual) conjugation rules:

$\displaystyle\displaystyle(A_{i}A_{j})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}\tilde{A_{j}},\quad(\tilde{A_{i}})^{\thicksim}=-\xi
A_{i},$ (9) $\displaystyle\displaystyle(A_{i}^{\dagger})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}^{\dagger},\quad(cA_{i}+A_{j})^{\thicksim}=c^{*}\tilde{A_{i}}+\tilde{A_{j}},$

with $\displaystyle\xi=-1$ for bosons and $\displaystyle\xi=+1$ for fermions.

The other fundamental ingredient of the TFD formalism is the Bogoliubov
transformation that introduces a rotation in the tilde and non-tilde
variables, thus the effects of temperature are introduced. The Bogoliubov
transformations for fermions, with $\displaystyle c_{p}^{\dagger}$ and
$\displaystyle c_{p}$ being creation and annihilation operators respectively,
are

$\displaystyle\displaystyle c_{p}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}(\beta)c_{p}(\beta)+\mathsf{v}(\beta)\tilde{c}_{p}^{\dagger}(\beta),$
(10) $\displaystyle\displaystyle c_{p}^{\dagger}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}(\beta)c_{p}^{\dagger}(\beta)+\mathsf{v}(\beta)\tilde{c}_{p}(\beta),$
(11) $\displaystyle\displaystyle\tilde{c}_{p}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}(\beta)\tilde{c}_{p}(\beta)-\mathsf{v}(\beta)c_{p}^{\dagger}(\beta),$
(12) $\displaystyle\displaystyle\tilde{c}_{p}^{\dagger}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}(\beta)\tilde{c}_{p}^{\dagger}(\beta)-\mathsf{v}(\beta)c_{p}(\beta),$
(13)

with $\displaystyle\mathsf{u}(\beta)=\cos\theta(\beta)$ and
$\displaystyle\mathsf{v}(\beta)=\sin\theta(\beta)$. The anti-commutation
relations for creation and annihilation operators are similar to those at zero
temperature

$\displaystyle\displaystyle\left\\{c(k,\beta),c^{\dagger}(p,\beta)\right\\}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\delta^{3}(k-p),$
$\displaystyle\displaystyle\left\\{\tilde{c}(k,\beta),\tilde{c}^{\dagger}(p,\beta)\right\\}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\delta^{3}(k-p),$
(14)

and other anti-commutation relations are null.

The Bogoliubov transformations for bosons are given as

$\displaystyle\displaystyle a_{p}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}^{\prime}(\beta)a_{p}(\beta)+\mathsf{v}^{\prime}(\beta)\tilde{a}_{p}^{\dagger}(\beta),$
(15) $\displaystyle\displaystyle a_{p}^{\dagger}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}^{\prime}(\beta)a_{p}^{\dagger}(\beta)+\mathsf{v}^{\prime}(\beta)\tilde{a}_{p}(\beta),$
(16) $\displaystyle\displaystyle\tilde{a}_{p}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}^{\prime}(\beta)\tilde{a}_{p}(\beta)+\mathsf{v}^{\prime}(\beta)a_{p}^{\dagger}(\beta),$
(17) $\displaystyle\displaystyle\tilde{a}_{p}^{\dagger}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathsf{u}^{\prime}(\beta)\tilde{a}_{p}^{\dagger}(\beta)+\mathsf{v}^{\prime}(\beta)a_{p}(\beta),$
(18)

where $\displaystyle a_{p}^{\dagger}$ and $\displaystyle a_{p}$ are creation
and annihilation operators, respectively. Here,
$\displaystyle\mathsf{u}^{\prime}(\beta)=\cosh\theta(\beta)$,
$\displaystyle\mathsf{v}^{\prime}(\beta)=\sinh\theta(\beta)$ and algebraic
rules satisfy the relations

$\displaystyle\displaystyle\left[a(k,\beta),a^{\dagger}(p,\beta)\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\delta^{3}(k-p),$
$\displaystyle\displaystyle\left[\tilde{a}(k,\beta),\tilde{a}^{\dagger}(p,\beta)\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\delta^{3}(k-p),$
(19)

and other commutation relations are null.

In the next section, the cross-section for the electron-positron scattering at
finite temperature in VSR is calculated.

##  IV Cross section for $\displaystyle e^{-}+e^{+}\rightarrow\mu^{-}+\mu^{+}$
scattering at finite temperature

Here, the scattering process $\displaystyle
e^{-}+e^{+}\rightarrow\mu^{-}+\mu^{+}$ is considered. The main objective is to
analyze how the new vertex changes this process. The Feynman diagrams, that
describe this scattering process in VSR, are given as:

Figure 1: Feynman diagrams to electron-positron scattering. The usual QED
vertex is denoted by $\displaystyle\bullet$ and the new vertex due to VSR
contributions is $\displaystyle\otimes$. Diagram (a) describes the standard
QED interaction, while diagrams (b), (c), and (d) describe the new
interactions.

In the following calculations, the center of mass frame (CM) is considered.
Then

$\displaystyle\displaystyle p_{1}=(E,p^{i}),\quad\quad\quad p_{3}=(E,p^{\prime
i}),$ $\displaystyle\displaystyle p_{2}=(E,-p^{i}),\quad\quad
p_{4}=(E,-p^{\prime i}),$
$\displaystyle\displaystyle\kappa=(p_{1}+p_{2})=(\sqrt{s},0),$ (20)

with $\displaystyle p_{1},\,p_{2},\,p_{3}$ and $\displaystyle p_{4}$ being the
4-momentum of the electron, positron, muon and anti-muon, respectively and
$\displaystyle\sqrt{s}$ is the energy in the center of mass.

In order to calculate the cross-section for the scattering described in FIG.1,
the important quantity that must be determined is transition amplitude at
finite temperature. It is defined as

$\displaystyle\displaystyle{\cal S}(\beta)=\langle
f,\beta|\hat{S}^{(2)}|i,\beta\rangle,$ (21)

where

$\displaystyle\displaystyle|i,\beta\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
c_{p_{1}}^{\dagger}(\beta)d_{p_{2}}^{\dagger}(\beta)|0(\beta)\rangle,$
$\displaystyle\displaystyle|f,\beta\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
c_{p_{3}}^{\dagger}(\beta)d_{p_{4}}^{\dagger}(\beta)|0(\beta)\rangle,$ (22)

are the thermal states, with $\displaystyle c_{p_{j}}^{\dagger}(\beta)$ and
$\displaystyle d_{p_{j}}^{\dagger}(\beta)$ being creation operators. In
addition, $\displaystyle\hat{S}^{(2)}$ is the second order term of the
$\displaystyle\hat{S}$-matrix that is given as

$\displaystyle\displaystyle\hat{S}^{(2)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)^{2}}{2!}\int
d^{4}x\,d^{4}y\,\mathbb{T}\left[\hat{\cal L}_{int}(x)\hat{\cal
L}_{int}(y)\right],$ (23)

where $\displaystyle\mathbb{T}$ is the time ordering operator and
$\displaystyle\hat{\cal L}_{int}(x)={\cal L}_{int}(x)-\tilde{\cal L}_{int}(x)$
describes the interaction. Then the transition amplitude at finite temperature
becomes

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)^{2}}{2!}\int d^{4}x\,d^{4}y\,\langle
f,\beta|\mathbb{T}({\cal L}_{int}(x){\cal L}_{int}(y))|i,\beta\rangle.$ (24)

It is to be noted that, there is similar equation for the transition amplitude
that includes tilde part.

Using the interaction Lagrangian given by eq. (6), we get

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\int d^{4}x\,d^{4}y\,\langle
f,\beta|\mathbb{T}\left[\bar{\psi}(x){\cal V}^{\mu}\psi(x)\bar{\psi}(y){\cal
V}^{\nu}\psi(y)A_{\mu}(x)A_{\nu}(y)\right]|i,\beta\rangle.$ (25)

Considering that the wave function of the fermions field is

$\Psi(\mathrm{x})=\int\\!\\!d\mathrm{p}\Big{[}c_{\mathrm{p}}(s)u(\mathrm{p},s)e^{-\imath\mathrm{px}}+d_{\mathrm{p}}^{\dagger}(s)v(\mathrm{p},s)e^{\imath\mathrm{px}}\Big{]},$
(26)

with $\displaystyle c_{p}$ and $\displaystyle d_{p}$ being annihilation
operators for electrons and positrons, respectively with $\displaystyle
u(p,s)$ and $\displaystyle v(p,s)$ being Dirac spinors. It is important to
note that getting this solution into VSR is not trivial. Non-local terms are
introduced and then the canonical quantization must be analyzed. Auxiliary
fields are used to performed the quantization. This leads to a non-canonical
anti-commutation relation for the fermion field. More details about this study
are presented in Can . Using the Bogoliubov transformations eq.(10) and eq.
(11), the transition amplitude is given as

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\int\frac{d^{4}p}{(2\pi)^{4}}\int
d^{4}x\,d^{4}y\,(\mathsf{u}^{2}-\mathsf{v}^{2})^{2}\,e^{-ix(p_{1}-p_{3})}e^{-iy(p_{2}-p_{4})}$
(27) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{3}){\cal
V}^{\nu}(p_{3},p_{4})v(p_{4})\langle
0(\beta)|\mathbb{T}[A_{\mu}(x)A_{\nu}(y)]|0(\beta)\rangle.$

In the TFD approach, the photon propagator at finite temperature is given as

$\displaystyle\displaystyle\langle
0(\beta)|\mathbb{T}[A_{\mu}(x)A_{\nu}(y)]|0(\beta)\rangle=i\int\frac{d^{4}\tilde{q}}{(2\pi)^{4}}e^{-i\tilde{q}(x-y)}\Delta(\tilde{q},\beta)\,\eta_{\mu\nu},$
(28)

with

$\displaystyle\displaystyle\Delta(\tilde{q},\beta)=\Delta^{(0)}(\tilde{q})+\Delta^{(\beta)}(\tilde{q})$
(29)

where $\displaystyle\Delta^{(0)}(\tilde{q})$ and
$\displaystyle\Delta^{(\beta)}(\tilde{q})$ are zero and finite temperature
parts respectively, that are defined as

$\displaystyle\displaystyle\Delta^{(0)}(\tilde{q})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\tilde{q}^{2}}\tau,$
$\displaystyle\displaystyle\Delta^{(\beta)}(\tilde{q})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-\frac{2\pi
i\delta(\tilde{q}^{2})}{e^{\beta\tilde{q}_{0}}-1}\left(\begin{array}[]{cc}1&e^{\beta\tilde{q}_{0}/2}\\\
e^{\beta\tilde{q}_{0}/2}&1\end{array}\right).$ (32)

Here, $\displaystyle\tau=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$. More details about photon propagator at finite
temperature is given in first . By taking
$\displaystyle\mathsf{u}(\beta)=\cos\theta(\beta)$ and
$\displaystyle\mathsf{v}(\beta)=\sin\theta(\beta)$ leads to
$\displaystyle(\mathsf{u}^{2}-\mathsf{v}^{2})^{2}=\tanh^{2}\left(\frac{\beta
E_{CM}}{2}\right)$. Then the transition amplitude becomes

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-i\int\frac{d^{4}p}{(2\pi)^{4}}\int
d^{4}x\,d^{4}y\,\int\frac{d^{4}\tilde{q}}{(2\pi)^{4}}\tanh^{2}\left(\frac{\beta
E_{CM}}{2}\right)\,e^{-ix(p_{1}-p_{3}+\tilde{q})}e^{-iy(p_{2}-p_{4}-\tilde{q})}$
(33) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{3}){\cal
V}_{\mu}(p_{3},p_{4})v(p_{4})\Delta(\tilde{q},\beta).$

Considering the definition of the four-dimensional delta function,

$\displaystyle\displaystyle\int
d^{4}x\,d^{4}y\,e^{-ix(p_{1}-p_{3}+\tilde{q})}e^{-iy(p_{2}-p_{4}-\tilde{q})}=\delta^{4}(p_{1}-p_{3}+\tilde{q})\delta^{4}(p_{2}-p_{4}-\tilde{q}),$
(34)

and carrying out the $\displaystyle\tilde{q}$ integral leads to

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-i\int\frac{d^{4}p}{(2\pi)^{4}}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})\tanh^{2}\left(\frac{\beta
E_{CM}}{2}\right)$ (35) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{3}){\cal
V}_{\mu}(p_{3},p_{4})v(p_{4})\Delta(\tilde{q},\beta).$

It is interesting note that, the remaining delta function and the
$\displaystyle p$ integral express overall four-momentum conservation. By
convention, this integral is ignored and the transition amplitude is

$\displaystyle\displaystyle{\cal S}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-i\tanh^{2}\left(\frac{\beta
E_{CM}}{2}\right)\,\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{3}){\cal
V}_{\mu}(p_{3},p_{4})v(p_{4})\Delta(\tilde{q},\beta).$ (36)

Then the square of the transition amplitude is given as

$\displaystyle\displaystyle\frac{1}{4}\sum_{spin}|i{\cal S}(\beta)|^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{4}\sum_{spin}\tanh^{4}\left(\frac{\beta
E_{CM}}{2}\right)\Delta^{2}(\tilde{q},\beta)\left[\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{1}){\cal
V}^{\nu}(p_{1},p_{2})v(p_{2})\right]$ (37) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\left[\bar{u}(p_{3}){\cal
V}^{\mu}(p_{3},p_{4})v(p_{4})\bar{v}(p_{4}){\cal
V}^{\nu}(p_{3},p_{4})u(p_{3})\right],$

where averaging over the spin of incoming and outgoing particles has been
considered and $\displaystyle\Delta^{2}(\tilde{q},\beta)$ is defined as

$\displaystyle\displaystyle\Delta^{2}(\tilde{q},\beta)=\frac{1}{\tilde{q}^{4}}\left[1+\frac{(2\pi\tilde{q}^{2})^{2}\delta^{2}(\tilde{q}^{2})}{(e^{\beta
E_{CM}}-1)^{2}}\right]$ (38)

with $\displaystyle\tilde{q}^{4}=(\tilde{p}_{1}+\tilde{p}_{2})^{4}$. Using the
relation

$\displaystyle\displaystyle\bar{v}(p_{2}){\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{1}){\cal
V}^{\nu}(p_{1},p_{2})v(p_{2})=\mathrm{tr}\left[{\cal
V}^{\mu}(p_{1},p_{2})u(p_{1})\bar{u}(p_{1}){\cal
V}^{\nu}(p_{1},p_{2})v(p_{2})\bar{v}(p_{2})\right]$ (39)

and the completeness relations

$\displaystyle\displaystyle\sum_{spin}u(p_{1})\bar{u}(p_{1})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{\not{p}}_{1}+m_{p},$
$\displaystyle\displaystyle\sum_{spin}v(p_{1})\bar{v}(p_{1})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{\not{p}}_{1}-m_{p},$ (40)

with $\displaystyle\tilde{p}_{\mu}=p_{\mu}-\frac{m^{2}n_{\mu}}{2(n\cdot p)}$
being the wiggle momentum and $\displaystyle m_{p}$ the particle mass.
Hereafter it is assumed that the scattering process is calculated at the high
energy limit, which corresponds to $\displaystyle m_{p}=0$. Then eq. (37)
becomes

$\displaystyle\displaystyle\frac{1}{4}\sum_{spin}|i{\cal S}(\beta)|^{2}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{{\cal
B}(\beta)}{4(\tilde{p}_{1}+\tilde{p}_{2})^{4}}\mathrm{tr}\left[{\cal
V}^{\mu}(p_{1},p_{2})\tilde{\not{p}_{1}}{\cal
V}^{\nu}(p_{1},p_{2})\tilde{\not{p}_{2}}\right]\mathrm{tr}\left[{\cal
V}_{\mu}(p_{3},p_{4})\tilde{\not{p}_{4}}{\cal
V}_{\nu}(p_{3},p_{4})\tilde{\not{p}_{3}}\right],$ (41)

where

$\displaystyle\displaystyle{\cal B}(\beta)=\tanh^{4}\left(\frac{\beta
E_{CM}}{2}\right)\left[1+\frac{(2\pi\tilde{q}^{2})^{2}\delta^{2}(\tilde{q}^{2})}{(e^{\beta
E_{CM}}-1)^{2}}\right].$ (42)

Thus, the differential cross-section for this scattering process is given as

$\displaystyle\displaystyle\frac{d\sigma}{d\Omega}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{64\pi^{2}s}\frac{1}{4}\sum_{\mathrm{spin}}|i{\cal
S(\beta)}|^{2},$ (43) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{e^{4}}{64\pi^{2}s}\frac{{\cal
B}(\beta)\,{\cal J}}{4(\tilde{p}_{1}+\tilde{p}_{2})^{4}},$

with

$\displaystyle\displaystyle{\cal J}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
32(\tilde{p}_{1}\cdot\tilde{p}_{4})(\tilde{p}_{2}\cdot\tilde{p}_{3})+32(\tilde{p}_{1}\cdot\tilde{p}_{3})(\tilde{p}_{2}\cdot\tilde{p}_{4})$
(44) $\displaystyle\displaystyle+$ $\displaystyle\displaystyle 16m^{2}{\cal
G}\Bigl{[}(\tilde{p}_{1}\cdot\tilde{p}_{3})(n\cdot\tilde{p}_{2})(n\cdot\tilde{p}_{4})+(\tilde{p}_{1}\cdot\tilde{p}_{4})(n\cdot\tilde{p}_{2})(n\cdot\tilde{p}_{3})+(\tilde{p}_{2}\cdot\tilde{p}_{3})(n\cdot\tilde{p}_{1})(n\cdot\tilde{p}_{4})$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle(\tilde{p}_{2}\cdot\tilde{p}_{4})(n\cdot\tilde{p}_{1})(n\cdot\tilde{p}_{3})-2(\tilde{p}_{3}\cdot\tilde{p}_{4})(n\cdot\tilde{p}_{1})(n\cdot\tilde{p}_{2})-2(\tilde{p}_{1}\cdot\tilde{p}_{2})(n\cdot\tilde{p}_{3})(n\cdot\tilde{p}_{4})\Bigl{]}$

and

$\displaystyle\displaystyle{\cal G}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(n\cdot\tilde{p}_{1})(n\cdot\tilde{p}_{2})}+\frac{1}{(n\cdot\tilde{p}_{3})(n\cdot\tilde{p}_{4})}.$
(45)

From these results and the CM coordinates, the differential cross-section at
finite temperature is

$\displaystyle\displaystyle\frac{d\sigma}{d\Omega}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left[\frac{e^{4}}{128\pi^{2}\,s}(3+\cos
2\theta)-\frac{e^{4}\,\Theta^{2}}{256\pi^{2}\,s}(19+\cos 2\theta)\right]{\cal
B}(\beta),$ (46)

where $\displaystyle\Theta\equiv\frac{m}{E}$ is the parameter that controls
the VSR effects. The total cross section is obtained by integration

$\displaystyle\displaystyle\sigma=2\pi\,{\cal
B}(\beta)\int_{0}^{\pi}\left[\frac{e^{4}}{128\pi^{2}\,s}(3+\cos
2\theta)-\frac{e^{4}\,\Theta^{2}}{256\pi^{2}\,s}(19+\cos
2\theta)\right]\sin\theta d\theta.$ (47)

Performing the integration, the cross-section for electron-positron scattering
in VSR at finite temperature is

$\displaystyle\displaystyle\sigma=\sigma_{QED}\left(1-\frac{7}{2}\Theta^{2}\right){\cal
B}(\beta),$ (48)

where $\displaystyle\sigma_{QED}=\frac{4\pi\alpha^{2}}{3s}$ is the usual to
QED contribution to the electron-positron scattering. This result shows
contributions due to VSR and the finite temperature. It is important to note
that the VSR correction term has a similar structure to the QED. There are
some experimental results for the cross-section of the electron-positron
scattering at zero temperature. Then, from these experimental data, a natural
question arises: is it possible to estimate a numerical value for the VSR
parameter at the zero temperature limit? In order to answer this question,
experimental data experimental  is used. The VSR correction for this
scattering can be assessed using the formula

$\displaystyle\displaystyle\frac{\sigma-\sigma_{QED}}{\sigma_{QED}}=\pm\frac{2s}{\Lambda^{2}},$
(49)

where the center-of-mass energy is $\displaystyle\sqrt{s}=29\,\mathrm{GeV}$
and $\displaystyle\Lambda=170\,\mathrm{GeV}$. Using this data, the estimated
value for the VSR parameter is $\displaystyle m\leq 1.9\,\mathrm{GeV}$. It is
important to note that, if this parameter is associated with the photon mass,
the bound found here is not significant since $\displaystyle
m_{photon}<10^{-18}\,\mathrm{eV}$ mass . On the other hand, although the
effects of VSR alter the usual QED result for electron-positron scattering,
experimental data at zero temperature do not establish strong limits on the
non-local parameter of VSR. A similar limit for the VSR parameter has been
found in the description of the Bhabha scattering Bufalo3 . Our results show
that the temperature effect on the electron-positron scattering may contribute
to a new class of constraints on the VSR parameter. Therefore, if the
electron-positron scattering is carried out at a finite temperature, a better
constraint on the parameters of the model will be obtained.

##  V Conclusion

Studies with sufficiently high energies lead to the possibility that a small
violation of the Lorentz symmetry may arise. Then, the search for theoretical
and phenomenological aspects of the Lorentz violation has increased in recent
years. Here, quantum electrodynamics in the VSR Lorentz violating framework is
considered. The most interesting feature of the VSR is that the laws of
physics are invariant under the (kinematical) subgroups of the Poincaré group,
preserving the basic elements of special relativity. In this context, the
cross-section for the electron-positron scattering at finite temperature is
calculated. The effects of temperature are introduced using the TFD formalism.
TFD is a thermal quantum field theory that is composed by two fundamental
ingredients: the doubling the degrees of freedom in a Hilbert space and the
Bogoliubov transformation. Our result shows that the VSR effects change the
usual QED cross-section for this scattering process. However, it is important
to note that the VSR effects have the same energy profile as the QED. In
addition, the cross-section is affected by the finite temperature. Using
experimental data at zero temperature, an estimate for the VSR parameter is
obtained. But this binding limit is not significant, since this parameter is
compared with the mass of the photon. On the other hand, this limit can be
improved if the electron-positron scattering is carried out at high
temperatures. Although the deviation predicted in this paper due to the VSR is
of theoretical interest, any significant constraints on the parameters of the
model will be improved from the experiments performed at non-zero temperature.

## Acknowledgments

This work by A. F. S. is supported by National Council for Scientific and
Technological Development - CNPq projects 430194/2018-8 and 313400/2020-2.

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In this brief note, we formulate Principal Component Analysis (PCA)
over datasets consisting not of points but of distributions, characterized by
their location and covariance. Just like the usual PCA on points can be
equivalently derived via a variance-maximization principle and via a
minimization of reconstruction error, we derive a closed-form solution for
distributional PCA from both of these perspectives.

§ INTRODUCTION

Most commonly in data science we are concerned with datasets that consist of
points \(\{x_1, \ldots, x_n\} \subset \bbR^d \).  In this note we focus on
datasets of random variables
\(\{\var{x}_1, \ldots, \var{x}_n\}\)
where each \(\var{x}_i\) has a probability distribution
summarized by its mean and variance

\[
\bbE[\var{x}_i] = \mu_i,\qquad
\bbV[\var{x}_i] = \Sigma_i.
\]

This scenario fully subsumes the standard point dataset in the limit of
\(\Sigma_i \to 0\), but allows us to further model situations such as:

* Uncertainty or measurement noise (, inherent variability of the
\(\var{x}_i\)s),

* Hierarchical data (, psychometric data where each \(\var{x}_i\) is a study
participant for whom several measurements are taken).

In this work we extend the usual pointwise PCA to distributional data.
We first recap the definition and derivation of PCA. Then, we show
two different derivations of its distributional counterpart.

§ BACKGROUND

PCA <cit.> is a workhorse of statistical analysis, data science,
and visualization. It
is a dimensionality reduction technique that summarizes a (point) dataset by
linearly transforming it to the most important dimensions of variability.
There are two typical ways to define most important, and it turns out
they both lead to the same result.
For this section we assume
a centered point dataset \(\{x_1, \ldots, x_n\}\), i.e., \(\sum_i x_i = 0\).
[PCA is typically defined after centering, but in some scenarios (,
high-dimensional sparse data) centering is sometimes skipped. While centering
is important for some statistical interpretation of the method, it makes no
difference for our derivation.]

Directions of maximal variance

One road toward PCA starts with the question: what is the direction
\(u\) that maximizes the variance of the projected dataset?
In other words, we seek:

\begin{equation}\label{eq:pca_variance}
\arg\max \left\{ \sum_i (u^\top x_i)^2 : u \in \bbR^d, \|u\|=1 \right\}.
\end{equation}

This is because \(z_i = u^\top x_i\) is the projection of \(x_i\) along the direction
of \(u\), and since we assume the \(x_i\)s are centered then so are the
\(z_i\)s, and so the objective of <ref> is the empirical
variance of the \(z_i\)s.

We may rewrite the objective of <ref> as \(u^\top S u \) where
\(S = \sum_i x_i x_i^\top \),
and therefore we recognize that the solution of <ref> is the
eigenvector of \(S\) corresponding to the largest eigenvalue.
This view readily extends to seeking the top-k principal components \(u_1,
\ldots, u_k\) by requiring additional orthogonal constraints, , \(U^\top U =
I\), and the solution is likewise given by the top-k eigenvectors of \(S\).

Minimizing the reconstruction error
If we view \(z_i = u^\top x_i\) as a 1-d encoded representation of \(x_i\),
we can map \(z_i\) back into \(\bbR^n\) as the vector \(z_i u\) in the span of
\(u\). This process will lose information.
We may then ask the question: which direction \(u\) minimizes the
reconstruction error of this encoding-decoding process? Or,

\begin{equation}
\arg\min \left\{ \sum_i \| x_i - uu^\top x_i \|^2 : u \in \bbR^d, \|u\|=1
\right\}.
\end{equation}

Denote \(Q=uu^\top\) and remark \(Q\) is a projection matrix, thus idempotent
and so is \(I-Q\). Then

\begin{equation}\label{eq:minerr}
\begin{aligned}
\sum_i \|x_i - Qx_i\|^2
&= \sum_i \|(I-Q)x_i \|^2\\
&= \sum_i x_i^\top (I-Q) x_i \\
&= -\sum_i x_i^\top (uu^\top) x_i + \text{const}\\
&= -u^\top S u + \text{const}, \\
\end{aligned}
\end{equation}

where the last step uses the same rearranging of the dot product as in the
paragraph above.

So, minimizing the reconstruction error, or maximizing projected variance, are
equivalent views that lead to the same principal component solution.

§ DERIVING DISTRIBUTIONAL PCA

We propose the following formulation for PCA over a dataset of random variables
\(\{\var{x}_1, \ldots, \var{x}_n\}\):

Given a dataset of random variables, denoted \(\{\var{x}_1, \ldots,
\var{x}_n\}\),
with means \(\mu_i\) and covariance matrices \(\Sigma_i\), the principal
components of this dataset are the leading eigenvectors of the matrix:

\[ \sum_i \mu_i \mu_i^\top + \Sigma_i. \]

We shall give two justifications of this definition.

Distributional PCA, as in <ref>, maximizes the expected projected

\begin{equation}
\arg\max \left\{
\bbE_{\var{x}_1, \ldots, \var{x}_n} \left[ \sum_i (u^\top \var{x}_i)^2\right]:
%\sum_i \bbE_{\var{x}_i}\left[ (u^\top \var{x}_i)^2\right]:
u \in \bbR^d, \|u\|=1 \right\}.
\end{equation}

Rearranging and using linearity, we may rewrite the objective as

\[
\begin{aligned}
\bbE\left[\sum_i (u^\top \var{x}_i)^2 \right]
&= \bbE\left[\sum_i u^\top(\var{x}_i \var{x}_i^\top) u\right] \\
&= \sum_i u^\top\left(\bbE[ \var{x}_i \var{x}_i^\top]\right) u\\
\sum_i \mu_i \mu_i^\top + \Sigma_i
\right) u.
\end{aligned}
\]

Distributional PCA, as in <ref>, minimizes the total squared
2-Wasserstein reconstruction error under the linear projection:

\begin{equation}
\arg\min \left\{
\sum_i W_2^2(\var{x}_i, uu^\top \var{x}_i)
: u \in \bbR^d, \|u\|=1
\right\}.
\end{equation}

To prove this result, we need the following lemma:

Let \(\var{x}\) be a random variable with mean \(\mu\) and variance
\(\Sigma\), and \(Q\) be a projection matrix.

\[
W_2^2(\var{x}, Q\var{x}) = \|\mu - Q\mu\|^2 + \tr((I-Q)\Sigma).
\]

(of the lemma).
This is a slight extension of the unnumbered result given by
<cit.> in their section 5.
First, we use the translation property of \(W_2\)
to reduce the problem to a
distance between zero-mean measures:

\[
W_2^2(\var{x}, Q\var{x}) = \|\mu - Q\mu\|^2 +
W_2^2(\var{\bar{x}}, Q\var{\bar{x}})
\]

where \(\var{\bar{x}}=\var{x}-\mu\).
If \(\alpha\) is the probability measure associated with \(\var{\bar{x}}\),
then \(\beta = Q_\sharp \alpha\) is the probability measure of
the pushforward \(Q\var{\bar{x}}\).
Since \(Q\) is a projection matrix, it is symmetric positive semidefinite and
therefore it is the gradient of a convex mapping \(x \to \frac{1}{2} x^\top Q x\)
By Brenier's theorem <cit.>, Q is the optimal transport plan
between \(\alpha\) and \(\beta\). This implies

\[
\begin{aligned}
W_2^2(\var{\bar{x}}, Q\var{\bar{x}})
&= \int_x
\|x - Qx\|^2  \\
&= \int_x d\alpha~x^\top (I-Q) x \\
%&= \int_x
%x^\top x
%- x^\top Q x
%\qquad (\text{since } Q^\top Q = Q)
&= \int_x
d\alpha~\tr \left( (I-Q) xx^\top \right)\\
&= \tr \left( (I-Q) \Sigma \right).
\end{aligned}
\]

(of the proposition).
Let \(Q=uu^\top\) denote the projection operator onto the span of \(u\).
Applying the lemma,

\begin{equation}
\begin{aligned}
\sum_i W_2^2(\var{x}_i, Q\var{x}_i) &=
\left( \sum_i \mu_i^\top (I-Q) \mu_i
+ \tr\left((I-Q)\Sigma_i\right) \right)\\
&= -\sum_i \tr\left(Q(\mu_i\mu_i^\top + \Sigma_i)\right) + \text{const}\\
&= -u^\top \left(\sum_i \mu_i \mu_i^\top + \Sigma_i\right) u + \text{const}.
\end{aligned}
\end{equation}

We have thus shown that distributional PCA can also be viewed equivalently from
a variance-maximization and error-minimization angle, just like usual pointwise
PCA. In addition, in the limit of all \(\Sigma_i \to 0\), we recover usual PCA.
Finally we remark that while we use a single principal component in the above
derivations, everything holds for \(k\) orthogonal principal components as well.

§ DISCUSSION

To demonstrate how distributional PCA works, we construct a dataset with four
Gaussian random variables. Their locations are
\(
\mu_1 = (-0.5, -2),
\mu_2 = (0.5, -1),
\mu_3 = (-0.5, 0),
\mu_4 = (-0.5, 1),
\)
and their covariances are all equal to \(\Sigma=\diag(1, 0.5)\).
<Ref> shows the principal component direction obtained by
performing the usual PCA on the four means, performing distributional PCA, and
performing PCA on a dataset obtained by drawing 1000 samples from each of the
four distributions. Our proposed formula indeed characterizes the limit case
of sampling from the distributional dataset.

Comparing PCA on distributions, means, and samples,
on a dataset of four Gaussian distributions.

Related work.
<cit.> recently proposed a transportation-based PCA between covariance
matrices. Their formulation applies PCA in the tangent space of a manifold of
covariance operators and therefore leads to a different algorithm, somewhat more
computationally intensive as it requires calculating a Fréchet mean.
While our formulation only depends on covariance matrices through their sum,
their formulation seems more suited for capturing differences between individual
covariances. On the other hand, transportation PCA does not take into account
means, just covariances. We shall explore
the relationship and tradeoffs between the two formulations in the future.

This work is partly supported by NWO VI.Veni.212.228
and the European Union's Horizon Europe research and innovation programme
via UTTER 101070631.