# Rapid decline in the lightcurves of luminous supernovae by jet-driven
bipolar explosions

Muhammad Akashi Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Kinneret College
on the Sea of Galilee, Samakh 15132, Israel  Amir Michaelis Department of
Physics, Ariel University, P.O. Box 3, Ariel 4070000, Israel;
<EMAIL_ADDRESS>Noam Soker Department of Physics, Technion, Haifa,
3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We calculate the lightcurves of jet-driven bipolar core collapse supernova
(CCSN) explosions into a bipolar circumstellar mater (CSM) and show that an
equatorial observer finds the lightcurves to possess a rapid, and even an
abrupt, drop. The scenario that might lead to such an explosion morphology is
a common envelope evolution (CEE) where shortly before the CCSN explosion the
red supergiant progenitor interacts with a more compact companion that
spirals-in and spins-up the core. The companion can be a main sequence star, a
neutron star, or a black hole. The binary interaction ejects a shell through
an intensive wind and the CEE ejects a denser gas in the equatorial plane. We
assume that the companion accretes mass and launches jets. We conduct three-
dimensional (3D) hydrodynamical simulations where we launch weak jets, the
shaping jets, into the dense shell and show that the interaction forms a
bipolar CSM. As a result of the rapid pre-collapse core rotation jets drive
the CCSN explosion. We simulate the interaction of the jets with the bipolar
CSM and use a simple scheme to calculate the lightcurves. We show that the
abrupt drop in the lightcurve of an observer not too close to the polar
directions can account for the lightcurve of the hydrogen poor luminous
supernova (LSN) SN 2018don.

stars: jets - Supernova: general - transients: supernovae - supernovae:
individual: SN 2018don

##  1 Introduction

Many studies have reached the conclusion that superluminous supernovae
(SLSNe), i.e., core collapse supernovae (CCSNe) with peak r-band magnitude of
$M_{\rm r}<-20$, and luminous supernovae (LSNe) that are CCSNe with $M_{\rm
r}=-19$ to $-20$ (see Gomez et al. 2022 for definitions), contain an extra
energy source or another ingredient in addition to the explosion itself. We
can distinguish between two classes of explanations.

In the first class of explanations the explosion itself is the delayed
neutrino mechanism. It seems that the heavily studied delayed neutrino
mechanism (Bethe & Wilson 1985; Heger et al. 2003; Janka 2012; Nordhaus et al.
2012; Bruenn et al. 2016; Janka et al. 2016; O’Connor & Couch 2018; Müller et
al. 2019; Burrows & Vartanyan 2021; Fujibayashi et al. 2021; Boccioli et al.
2022; Nakamura, Takiwaki, & Kotake 2022; for a recent brief optimistic review
of this mechanism see Mezzacappa 2022) cannot reach CCSN explosion energies of
$E_{\rm SN}\gtrsim 2\times 10^{51}{~{}\rm erg}$ (e.g., Fryer 2006; Fryer et
al. 2012; Papish, Nordhaus, & Soker 2015; Sukhbold et al. 2016; Sukhbold &
Woosley 2016; Gogilashvili et al. 2021). Many studies of SLSNe and LSNe that
take the explosion itself to be by the delayed neutrino mechanism attribute
the extra energy of LSNe and SLSNe to a magnetar, i.e., a rapidly rotating
magnetized neutron star (e.g., Greiner et al. 2015; Metzger et al. 2015; Yu et
al. 2017; Margutti et al. 2018; Nicholl et al. 2017).

According to the second class of explanations jets explode most (or even all)
CCSNe (for a review see Soker 2022d). Even if the pre-collapse core is slowly
rotating, accretion of material with stochastic angular momentum onto the
newly born neutron star leads to the launching of jittering jets (jets with
stochastic variation of directions), i.e., the jittering jets explosion
mechanism (Soker 2010; Papish & Soker 2014b; Gilkis & Soker 2015; Quataert et
al. 2019; Soker 2019, 2020; Shishkin & Soker 2021; Antoni & Quataert 2022;
Soker 2022a, b). Therefore, according to this class of models jets also
explode all LSNe and SLSNe, as well as the CCSNe that accompany gamma ray
bursts as Izzo et al. (2019) also claim from their observation of a gamma ray
burst and in line with earlier studies of jet-driven gamma ray bursts (e.g.,
Shaviv & Dar 1995; MacFadyen & Woosley 1999). Further motivation to consider
jet-driven explosions is the problems of magnetar modelling of some SLSNe,
e.g., Soker & Gilkis (2017a) who analysed the magnetar modelling by Nicholl et
al. (2017), and LSNe, e.g., Soker (2022c) who analysed the magnetar modelling
by Gomez et al. (2022). Moreover, some studies find that the formation of an
energetic magnetar most likely leads to the launching of jets during the
explosion and in some cases also after the explosion, and that these jets
might carry more energy that the magnetar (Soker, 2016a, 2017a; Soker &
Gilkis, 2017a); for some possible energy sources of SLSNe see, e.g., Wang,
Wang, & Dai 2019.)

According to the jittering jets explosion mechanism the jets that explode LSNe
and SLSNe keep more or less a fixed axis. Namely, the jittering is small. This
requires the pre-collapse core to have fast rotation. For example, a binary
companion that spirals-in inside the red supergiant progenitor to a very small
orbital separation both spins-up the core and removes the hydrogen-rich
envelope to lead eventually to striped-envelope CCSN (CCSN-I), i.e., a type Ib
or a type Ic CCSN. The fixed axis makes the efficiency of the jet feedback
mechanism (for a review see Soker 2016b) inefficient, namely, the explosion
energy will be many times the binding energy of the exploding star. Therefore,
the explosion is very energetic (Gilkis et al., 2016; Soker, 2017b).

Note that unlike many studies that require the pre-collapse core to have very
fast rotation so its specific angular momentum by itself allows the formation
of accretion disk around the newly born neutron star or black hole (e.g.,
Bisnovatyi-Kogan 1971; Duncan & Thompson 1992; Khokhlov et al. 1999; MacFadyen
& Woosley 1999; Aloy et al. 2000; MacFadyen, Woosley, & Heger 2001; Wheeler,
Meier, & Wilson 2002; Akiyama et al. 2003; Burrows et al. 2007; Wheeler,
Maund, & Couch 2008; Maeda et al. 2012; Winteler et al. 2012; López-Cámara et
al. 2013; Mösta et al. 2014; Bromberg & Tchekhovskoy 2016; Nishimura et al.
2017; Sobacchi et al. 2017; Grimmett et al. 2021; Gottlieb et al. 2022; Perley
et al. 2022), in the jittering jets explosion mechanism convective
fluctuations in the pre-collapse core can supply additional stochastic angular
momentum and therefore the core might have a slower rotation (e.g., Gilkis &
Soker 2014, 2016; Shishkin & Soker 2022).

Kaplan & Soker (2020b) proposed a toy model to explain the puzzling lightcurve
of the LSNe SN 2018don. SN 2018don is a hydrogen-poor LSN that has a sharp
drop in its lightcurve (Lunnan et al., 2020). It has a long rise time to
maximum light relative to other LSNe (Gomez et al., 2022). Kaplan & Soker
(2020b) built simple bipolar ejecta models for jet-driven CCSNe and showed
that for an observer in the equatorial plane of the bipolar morphology the
lightcurve has a rapid luminosity decline, and even an abrupt drop. In some
cases there is no abrupt drop, but rather a ‘knee’ in the lightcurve where a
rapid drop changes to a moderate drop in luminosity. Soker (2022c) proposes
that such a bipolar explosion might explain the ‘knee’ in the lightcurve of
the SLSN SN 2020wnt (for its properties see Tinyanont et al. 2021; Gutiérrez
et al. 2022; Tinyanont et al. 2022).

Kaplan & Soker (2020b) did not simulate the bipolar jet-driven explosion, but
rather simply assumed that the ejecta has an axisymmetrical structure composed
of an equatorial ejecta and faster polar ejecta, each which is a part of a
sphere (different spheres), and that it has a uniform effective temperature.
In this study we conduct hydrodynamical simulations to simulate the bipolar
explosion (section 2), and then estimate the lightcurve, assuming a uniform
effective temperature (because we do not calculate radiative transfer) but
only if the photosphere is hot (section 3). We summarize our study that
strengthens the claim that jets power (at least some) LSNe and SLSNe, and
maybe even most CCSNe (section 4).

##  2 The bipolar jet-driven explosion

In this section we describe numerical simulations of a jet-driven bipolar CCSN
explosion into several different compact bipolar circumstellar matter (CSM)
morphologies.

###  2.1 The numerical code

We use version 4.6.2 of the adaptive-mesh refinement (AMR) hydrodynamical
FLASH code (Fryxell et al., 2000) in three dimension (3D). We turn off the
radiative cooling since the regions we simulate are optically thick. We
include radiation pressure, electrons pressure, and ions pressure as we assume
an adiabatic index of $\gamma=5/3$ and $\mu=0.5$, as we assume a fully ionised
pure hydrogen. We set outflow conditions on all the boundary surfaces of the
3D grid. We use resolution with 7 refinement levels, i.e., the ratio of the
length of the sides of the grid to the shortest cell in the grid is $2^{9}$.
The finest cell in the grid has a side of $1.05AU$. We simulate the whole
space with a total size of the Cartesian numerical grid of $(534AU)^{3}$,
i.e., $(L_{x},L_{y},L_{z})=\pm 267AU$. We simulate the entire space, rather
than only one half of the symmetry plane $z=0$, to avoid numerical artifacts
at the symmetry plane had we simulated only one half of the interaction
region. We will present the flow structure in the entire computational grid to
emphasize the bipolar structure that the jets shape and the flow in the
equatorial plane.

###  2.2 Building the bipolar explosion

We build the bipolar explosion by three numerical phases. These might result
from a common envelope evolution (CEE) or a grazing envelope evolution (GEE).
We base our assumption of a bipolar CSM on the scenario that we study here
that involves a CEE, and on the bipolar structure of many planetary nebulae
that are descendant of CEE (e.g., Corradi et al. 2014 for observations, Zou et
al. 2020 for a theoretical study, and De Marco & Izzard 2017 for a review).
However, for the present study the only important phase is the general bipolar
explosion with dense equatorial CSM gas and fast bipolar outflows.

####  2.2.1 The bipolar CSM

We build a bipolar CSM into which the jet-driven explosion occurs. The manner
by which we built the CSM is not so important. We expect a CEE or a GEE to
form such a bipolar CSM. The explosion then takes place into the bipolar CSM.
Namely, the CSM is formed and shaped, for example, by a precursor where the
CCSN progenitor expands few years before the explosion and engulfs a companion
that launches jets (e.g., Mcley & Soker 2014; Danieli & Soker 2019; Soker
2022d). Core activity of excitation of waves (e.g., Quataert & Shiode 2012;
Shiode & Quataert 2014; Fuller 2017; Fuller & Ro 2018; Wu & Fuller 2022) or
magnetic activity (e.g., Soker & Gilkis 2017b) causes the envelope expansion
years to months before explosion.

We set the initial (at $t=0$) grid to have an expanding dense shell of mass
$M_{\rm wind}$ from the center to an outer radius of $R_{\rm
sh}=10^{15}{~{}\rm cm}=67{~{}\rm AU}$. The shell was formed by an intensive
wind due to the onset of a CEE/GEE, i.e., the initial phase of a CEE/GEE that
lasts for a few dynamical times on the surface of the red supergiant. For the
wind velocity of $v_{\rm wind}=100{~{}\rm km}{~{}\rm s}^{-1}$ that we take
here, the binary interaction has been blowing this wind for a period of about
three years. This is about few times the dynamical time on the surface of a
red supergiant star. We present the values of this mass in the second column
of Table 1. In the third column we list the velocity of the wind for each case
$v_{\rm wind}$, in the forth column the corresponding mass loss rate of the
wind, and in the fifth column we list the total kinetic energy in this wind.
The duration of the spherical wind is $3.2(v_{\rm wind}/100{~{}\rm km}{~{}\rm
s}^{-1})^{-1}{~{}\rm yr}$. The density in this shell goes as $\rho_{\rm
wind}\propto r^{-2}$. In the region $r>67{~{}\rm AU}$ we have at $t=0$ a low
density expanding gas with the same velocity as the wind. The spherical wind
that composes the initial condition at $t=0$ is the first phase of our
numerical study. We assume that pre-explosion evolution processes of shocks
and radiation from the CEE heat the CSM to a typical temperature of
$10000{~{}\rm K}$, as in the warm ISM and in planetary nebulae. As well, the
optically thick shell implies a slow radiative cooling as photon diffusion
time is long relative to the flow time. We therefore set the initial
temperature of the simulation box and the jets to $10000{~{}\rm K}$. The jets
are highly supersonic such that their initial temperature does not influence
the results.

Sim. | $M_{\rm wind}$ | $v_{\rm wind}$ | $\dot{M}_{\rm wind}$ | $E_{\rm wind}$ | $M_{\rm jets}$ | $E_{\rm jets}$
---|---|---|---|---|---|---
| $M_{\odot}$ | ${~{}\rm km\;s^{-1}}$ | $M_{\odot}{~{}\rm yr}^{-1}$ | ${~{}\rm erg}$ | $M_{\odot}$ | ${~{}\rm erg}$
EXP1 | 0.25 | 100 | 0.08 | $2.5\times 10^{47}$ | $3.3\times 10^{-5}$ | $8.2\times 10^{47}$
EXP2 | 2.5 | 100 | 0.8 | $2.5\times 10^{48}$ | $3.3\times 10^{-4}$ | $8.2\times 10^{48}$
EXP3 | 0.75 | 57.7 | 0.24 | $2.5\times 10^{47}$ | $10^{-4}$ | $8.2\times 10^{47}$
EXP4 | 0.75 | 40.8 | 0.24 | $1.25\times 10^{47}$ | $10^{-4}$ | $4.1\times 10^{47}$
Table 1: Summary of the CSM properties for the four simulations that we
present in the paper. The columns list, from left to right and for each
simulation, the name of the simulation, total mass of the spherical wind, the
velocity of the wind, the mass loss rate of the wind and the total kinetic
energy of this wind. We start the simulations at $t=0$ by launching two
opposite jets into the wind (shell) for 60 days. We list the total mass of
these jets that shape the bipolar CSM just before the explosion and their
total kinetic energy in the sixth and seventh column, respectively. In all
simulations the jet-driven explosion energy that we launch later is $E_{\rm
SN}=2.5\times 10^{51}{~{}\rm erg}$, and the explosion duration, basically the
exploding-jet activity duration, is $3{~{}\rm days}$.

At $t=0$ we launch two opposite jets to form a bipolar CSM. The shaping of the
CSM by these jets is the second phase of our numerical study. These jets are
active for 60 days in all cases. To launch the jets we set a fast outflow,
velocity of $V_{\rm j}=5\times 10^{4}{~{}\rm km}{~{}\rm s}^{-1}$ inside two
opposite cones, each cone with a length of $r=7{~{}\rm AU}$ and a half-opening
angle of $\alpha=40^{\circ}$. The density of the jets inside the cones is
$\rho_{j}(r)=\dot{M}_{\rm jets}/[4\pi r^{2}v_{\rm j}(1-\cos\alpha)]$, where
$\dot{M}_{\rm jets}=M_{\rm jets}/60~{}{\rm day}$ is the mass loss rate into
the jets and $M_{\rm jets}$ is the total mass the two shaping jets carry. In
the sixth and seventh column of Table 1 we list the combined mass and energy
of these jets, respectively, for the four simulations.

During the entire simulation we inject a dense equatorial outflow due to the
binary interaction. This equatorial outflow has a half opening angle of
$30^{\circ}$ from the equatorial plane, a mass loss rate of $1M_{\odot}{~{}\rm
yr}^{-1}$, and a velocity of $100{~{}\rm km\;s^{-1}}$. During the entire
simulation (including the later explosion phase) we inject a mass of $\approx
0.4M_{\odot}$ in the equatorial outflow. The motivation to include the dense
equatorial outflow comes from the presence of tori (equatorial rings) in many
bipolar planetary nebulae (e.g., De Marco & Izzard 2017 for a review), and the
dense ring alongside the bipolar (outer) rings of SN 1987A. In any case, the
bubbles that the shaping jets inflate in the wind compress gas towards the
equatorial plane, such that a dense equatorial outflow is formed by the jet-
inflated bubbles in the wind (e.g., Akashi et al. 2015). Therefore, our
qualitative results hold even without a pre-set equatorial outflow (as we
wrote, we expect such an equatorial outflow to take place in many cases).

In Fig. 1 we present the formation of the bipolar CSM for simulation EXP1. All
panels show the density in the meridional plane, i.e., a plane along the jets’
axis. The shaping-jets are active for 60 days starting at $t=0$. In the upper,
middle, and bottom panels we present the density maps at $t=1$ day, $t=45$
day, and $t=89$ day, respectively.

Figure 1: The density maps in the meridional plane at three times (as
indicated on the panels) during the formation of the bipolar CSM into which we
later explode the star. The shaping-jets are active from $t=0$ for 60 days.
The logarithm of the density in units of $\rm g~{}cm^{-3}$ is according to the
color-bar.

####  2.2.2 Jet-driven explosion

The third numerical phase is a second, much more energetic, jet-launching
episode. This jet-launching phase is the jet-driven CCSN explosion, as we
assume that jets power this CCSN (see section 1). We assume that the newly-
born neutron star accretes fallback material and launches the jets for three
days starting 30 days after we turned off the jets that shaped the CSM
(section 2.2.1). Namely, we launch the exploding jets during the time period
$90{~{}\rm days}<t<93{~{}\rm days}$. We launch two opposite jets into two
opposite cones each cone with a length of $r=7{~{}\rm AU}$ and a half-opening
angle of $\alpha=40^{\circ}$. During these three days we linearly increase the
jets’ velocity from zero to $100,000{~{}\rm km\;s^{-1}}$, $v_{\rm
ej}(t)=100,000[t({\rm day})-90~{}{\rm day}]/3{~{}\rm km\;s^{-1}}$, and we
linearly decrease the mass loss rate into the exploding jets from
$10M_{\odot}{~{}\rm yr}^{-1}$ to zero. During the explosion we continue to
inject the dense equatorial outflow of a half opening angle of $30^{\circ}$
from the equatorial plane, a mass loss rate of $1M_{\odot}{~{}\rm yr}^{-1}$,
and a velocity of $100{~{}\rm km\;s^{-1}}$.

We turn now to analyse the flow properties of the ejecta-CSM interaction,
postponing the calculation of the lightcurve to section 3.

###  2.3 The explosion flow structure

We describe the flow that results from the interaction of the jet-driven
ejecta (section 2.2.2) with the bipolar CSM (section 2.2.1). We recall that
the exploding jets are active for three days from $t=90~{}{\rm day}$ to
$t=93~{}{\rm day}$. The jets’ symmetry axis is along the $z$ axis through the
origin.

In Figs. 2, 3 and 4 we present the density, velocity, and temperature maps,
respectively, in the meridional plane at four times as the insets indicate in
the panels.

Figure 2: Density maps in the meridional plane at four times all after
explosion, as indicated on the panels. Time is measured from the onset of the
jets that shape the CSM. The exploding jets are active from $t=90~{}{\rm day}$
to $t=93~{}{\rm day}$. The logarithm of the density in units of $\rm
g~{}cm^{-3}$ is according to the color-bar.  Figure 3: Similar to Fig. 2 but
showing the velocity maps. The direction is according the arrows and the
logarithm of the velocity magnitude in units of $\rm cm~{}s^{-1}$ is according
to the color-bar.  Figure 4: Similar to Fig. 2 but showing the temperature
maps. The color-bar gives the logarithm of the temperature in K.

As expected, the jet-driven explosion into a bipolar CSM forms a highly non-
spherical bipolar explosion. The faster expansion along and near the polar
directions has implications to the lightcurve (e.g., Kaplan & Soker 2020b).
The first one is that the inflated polar lobes imply that at early times an
equatorial observer sees a larger photosphere than a polar observer. Later,
because of the faster expansion of the lobes the photosphere of the lobes
retreats faster. This implies that the photosphere area that an equatorial
observer sees decreases faster than that of a polar observer.

The outcome is a rapid, and even abrupt, drop in the lightcurve for an
equatorial observer, as Kaplan & Soker (2020b) noticed with their toy model.
We next demonstrate this behaviour of the lightcurve with our more
sophisticated model of bipolar explosions.

##  3 The lightcurves

###  3.1 Assumption and scheme

Our first assumption is that the ejecta and CSM radiates as a black body with
a uniform temperature. To calculate the luminosity we need therefore to find
the effective photosphere area that an observer sees. Since the morphology is,
up to instabilities, axi-symmetric, we only vary the angle of the observer
from the symmetry axis $\theta$ (i.e., $\theta=0$ for a polar observer and
$\theta=90^{\circ}$ for an equatorial observer).

For a given observation angle $\theta$ we follow many lines parallel to the
line of sight that cover the entire nebula (ejecta + CSM) and find the point
in the nebula along each line where the optical depth from infinity is

$\tau=\int^{\infty}_{R_{ph}}\kappa\rho_{\rm H}dl=2/3,$ (1)

where $dl$ is a length element along the line of sight. We take the density
$\rho_{\rm H}(x,y,z)$ from the hydrodynamical simulations (section 2), and we
assume a constant opacity $\kappa$. To these we supply two conditions. First
we take only points that are in the half space closer to the observer. Namely,
we take a plane perpendicular to the line of sight and through the explosion
site, and consider only points with $\tau=2/3$ that are between this plane and
the observer. Points in the other half of space will radiate away from the
observer. Secondly, we assume that cool areas, as the hydrodynamical results
give, will further cool and will not contribute to the luminosity in the
visible and near IR. Therefore, in calculating the photosphere area we
consider only photospheric points with a temperature of

$T_{\rm H}>T_{\rm H,c}=15,000{~{}\rm K},$ (2)

where the temperature is from the hydrodynamical simulations. The value of
$T_{\rm H,c}$ is a parameter of our lightcurve calculations that we must use
because we do not include radiative transfer. Changing the value of $T_{\rm
H,c}$ within the reasonable range of $T_{\rm H,c}\lesssim 20,000{~{}\rm K}$
changes the lightcurve quantitatively in a small manner, but not its
qualitative behavior. The later is the goal of our present study.

The above calculation gives us the cross section of the photosphere on the
plane of the sky, i.e., the effective photospheric area $A_{\rm ef,ph}$ based
on the hydrodynamical simulations and our assumptions. To estimate the total
explosion luminosity we assume that because of radiative cooling the
photosphere temperature is much lower than what the hydrodynamical simulations
give, and it is as observed in many CCSNe. We take $T_{\rm ef,ph}=8000{~{}\rm
K}$, but note it might be higher or a little lower and non-uniform. The
luminosity is then

$L=4A_{\rm ef,ph}\sigma T^{4}_{\rm ef,ph},$ (3)

where $\sigma$ is Stefan–Boltzmann constant. In this study we adopt $T_{\rm
ef,ph}=8000{~{}\rm K}$. Later we compare to observations and calculate the
absolute r-band magnitude as $m_{r}=4.64-2.5\log(L/L_{\odot})$.

We justify the values of the two temperature parameters of $T_{\rm
ef,ph}=8000{~{}\rm K}$ and $T_{\rm H,c}=15,000{~{}\rm K}$. Observations show
that the effective temperature is not constant with time. For example, in SN
2018don that we study in section 3.3, Lunnan et al. (2020) infer the effective
temperatures at three times to be $7500{~{}\rm K}$, $8000{~{}\rm K}$ and
$6500{~{}\rm K}$, when they include host galaxy extinction. In the other three
superluminous CCSNe that Lunnan et al. (2020) study the temperature decreases
with time from values of $>10,000{~{}\rm K}$ to $\simeq 6000{~{}\rm K}$ weeks
after explosion. As we do not include radiative transfer in this study we use
a constant effective temperature. The value of $T_{\rm ef,ph}=8000{~{}\rm K}$
reasonably represents the type of systems we study here, and in particular SN
2018don. However, we note that generally the effective temperature decreases
over time. The highest effective temperature that Lunnan et al. (2020) infer
is $\simeq 15,000{~{}\rm K}$ in the first point of SN 1018bym. It is therefore
reasonable to assume that the explosion itself heats the ejecta to
temperatures of $\gtrsim 15,000{~{}\rm K}$ in these superluminous CCSNe.
Ejecta parts that start with lower temperatures cool to lower temperatures and
do not contribute much to the emission. However, since we have no radiative
transfer nor radiative cooling, we simply assume that the parts with
temperatures of $T<T_{\rm H,c}=15,000{~{}\rm K}$ as we derive directly from
the hydrodynamical simulations do not contribute to the lightcurve. These two
values of the parameters are approximate. As we noted already, we could as
well use higher values of up to $T_{\rm H,c}\simeq 20,000{~{}\rm K}$.

We note that due to numerical limitations in simulating the very inner ejecta
regions our results at late times are less accurate.

###  3.2 Results

In presenting the lightcurves we recall that we set $t=0$ when we start to
shape the bipolar CSM inside which the jet-driven explosion occurs. The jet-
driven explosion itself takes place during the time period $90{~{}\rm
days}<t<93{~{}\rm days}$. Namely, the explosion time is $t=90{~{}\rm days}$.

Before we present our results we point out again the limitations of our simple
scheme to calculate the lightcurve. (1) One limitation of the hydrodynamical
simulations is that we could not follow the very inner regions of the ejecta.
As a result of that the lightcurves at late times are less accurate and can
even be qualitatively wrong, e.g., a large increase of luminosity with time at
late times. (2) Another limitation of our lightcurve calculation is that we do
not follow radiative transfer. As a result of that we do not have the correct
temperature nor the ionisation fraction in the outer regions and cannot
calculate the opacity at each point. The opacity is basically scatter on free
electrons with various degrees of ionisation. We therefore take a uniform
opacity in the grid and constant in time. (3) Also, in calculating the
relevant area of the photosphere, $A_{\rm ef,ph}$, we avoid low temperature
regions with $T<T_{\rm H,c}$ (equation 2), where $T_{\rm H,c}$ is a parameter
of our calculations. (4) We take a uniform photosphere temperature $T_{\rm
ef,ph}$ that does not evolve with time. It reality it does evolve with time.

In Fig. 5 we present the lightcurves (r-band magnitude) of case EXP1 for
observers at four angles (measured from the polar direction), and for three
opacities of $\kappa=0.06$, $0.1$, and $0.2{~{}\rm cm}^{2}{~{}\rm g}^{-1}$,
from top to bottom panel, respectively. As expected, the higher the opacity is
the longer the broad luminosity peak is. Namely, with increasing opacity the
abrupt drop in the lightcurve occurs later.

Figure 5: Our calculated lightcurves for the case EXP1 for observers at four
different angles relative to the symmetry (polar) axis. The upper, middle, and
lower panels are for opacities of $\kappa=0.06$, $0.1$, and $0.2{~{}\rm
cm}^{2}{~{}\rm g}^{-1}$, respectively, that we use in calculating the
photosphere (equation 1). We note the rapid decline in the lightcurve for
observers that are not close to the polar axis.

Another way to extend the lightcurve is to increase the mass of the CSM and/or
reduce its velocity. In Fig. 6 we present the lightcurves of cases EXP2, EXP3,
and EXP4 which have larger masses and EXP3 and EXP4 have also lower velocities
(Table 1). These lightcurves show that a more massive CSM can prolong the wide
peak up to $\simeq 80{~{}\rm days}$ for the parameters we use.

Figure 6:  The lightcurves for the cases EXP2, EXP3, and EXP4, all for an
opacity of $\kappa=0.1{~{}\rm cm}^{2}{~{}\rm g}^{-1}$.

Our main result is that the lightcurve for an observer that is not too close
to the polar direction might have an abrupt drop. This is similar to the
results of Kaplan & Soker (2020b) who built a bipolar explosion toy model. We
went beyond the toy model by conducting hydrodynamical simulations of jet-
driven supernova explosion into a bipolar CSM (section 2) and used the bipolar
ejecta-CSM regions to calculate the luminosity as we explained in section 3.1.
Despite the limitation of our scheme, the qualitative results are robust.
Exact fittings of specific cases will require more accurate calculations, in
particular the inclusion of radiative transfer and a more systematic search of
the (large) parameter space. Nonetheless, as we show next we can qualitatively
explain the lightcurve of SN 2018don.

###  3.3 The case of SN 2018don

Below we compare our results with the lightcurve of SN 2018don (Lunnan et al.,
2020) as Kaplan & Soker (2020b) did. The most similar lightcurve shapes to the
shape of the lightcurve of SN 2018don are the lightcurves at
$\theta=30^{\circ}$, $\theta=50^{\circ}$ and $\theta=80^{\circ}$ in EXP1 with
$\kappa=0.1{~{}\rm cm}^{2}{~{}\rm g}^{-1}$ that we present in the middle panel
of Fig. 5. However, the wide peak duration in these lightcurves is much
shorter than that of SN 2018don. We can prolong the wide peak duration by
increasing the CSM mass, e.g., four times longer in the lower panel of Fig. 6.
However, the shapes of the lightcurves of the cases with larger masses (the
three cases that we present in Fig. 6) are not as similar to the lightcurve of
SN 2018don. To obtain the same shape as in the middle panel of Fig. 5 (case
EXP1 with $\kappa=0.1{~{}\rm cm}^{2}{~{}\rm g}^{-1}$) and with a longer wide
peak we will have to conduct a very extensive search of the parameter space.
For example, we will have to modify also the explosion properties, which are
the same in all cases here. Namely, we will have to change in a systematic way
the properties of the exploding jets, like their opening angle, energy, and
duration. We also will have to consider a CSM that might be lighter in mass
and that does not contain a hydrogen-rich gas. These additions are beyond our
specific goal here. In future studies that will include also radiative
transfer we will scan the parameter space.

Our goal here is to present the possibility of bipolar explosions to account
for lightcurves with an abrupt drop. We therefore take the lightcurves from
the middle panel of Fig. 5 and simply multiply the duration by 4 (we stretch
and shift the time). We then shift the curves in time to match the zero time
in the observations of SN2018don. We present our comparison in Fig. 7. We also
note that Lunnan et al. (2020) estimate the maximum luminosity of SN 2018don
to be $-20.1$ mag according to the extinction.

Figure 7: The green and red points are the lightcurves of SN 2018don in the
g-band and r-band, respectively, as reported by (Lunnan et al., 2020). The
four lines are based on our calculated lightcurves for the case EXP1 at four
observer angles, measured from the symmetry(polar) axis as indicated in the
inset, but that we stretched in time by a factor of 4. Namely, these are the
four lines of the middle panel of Fig. 5 stretched in time by a factor of 4.
This figure demonstrates that we can reproduce the lightcurve shape for
observers at $\theta\gtrsim 30^{\circ}$, i.e., observers that are not close to
the polar axis. To reproduce the correct timescale we have to take a more
massive nebula as we demonstrate in Fig. 6.

Our conclusion from Fig. 7 is that when we observe bipolar supernova
explosions from directions that are not too close to the polar directions, the
exact angle depends on the morphologies of the bipolar CSM and of the jet-
driven bipolar explosion, we might have an abrupt drop in the lightcurve.
Specific fitting of each observed lightcurve will require an extensive search
of the parameter space of both the bipolar CSM, like its composition (adding
hydrogen-poor gas) and other morphologies, and that of the jet-driven bipolar
explosion, like other explosion energies and other jets’ opening angles.

##  4 Summary

We conducted hydrodynamical simulations of bipolar CCSN explosions and used a
simple scheme to calculate the expected lightcurves. The explosion is
applicable mainly (but not only) to stripped-envelope CCSNe according to the
following scenario (section 2).

A companion, main sequence, neutron star or a black hole, interacts with a red
supergiant star and enters a CEE with the red supergiant. The initial
interaction leads to a several-years long enhanced mass loss rate by the red
supergiant to all directions. In addition, the CEE that lasts weeks to months
leads to an even higher mass loss rate in the equatorial plane, and at the
onset of the CEE, and/or during the CEE, and/or at the exit from the CEE, the
companion launches relatively weak jets that shape the CSM to possess a
bipolar structure. The companion spirals-in to close distances to the core, or
even merges with the core, and therefore it spins-up the core. Shortly after
that the core of the red supergiant explodes. Because of the rapid pre-
collapse rotation the explosion is driven by jets that have more or less a
constant axis.

We conducted four different cases of bipolar CSM hydrodynamical simulations by
varying the properties of the spherical wind and/or those of the shaping jets
(Table 1; section 2.2.1). We present the formation of the bipolar CSM in the
case EXP1 in Fig. 1. The time $t=0$ corresponds to the time when we start
launching the shaping jets into the dense spherical wind (shell) to form the
bipolar CSM.

We then set the explosion by launching two energetic jets for three days
starting at $t=90~{}{\rm day}$ (section 2.2.2). We launch both the shaping
jets and the exploding jets (in each phase two opposite jets) along the
$z$-axis. We present the evolution of the ejecta-CSM interaction for case EXP1
at four times in Figs. 2-4.

From the bipolar structure we calculate the projected area of the photosphere
on the plane of the sky, $A_{\rm ef,ph}$, for an observer at an angle $\theta$
to the polar direction by applying conditions (1) and (2), as we explain in
section 3.1. We then calculate the luminosity and r-band magnitude as function
of time by equation (3). We present the different lightcurves in Figs. 5 and
6.

Our new results confirm the results of Kaplan & Soker (2020b) that the
lightcurve of an equatorial observer of a bipolar CCSN might have a rapid, and
even an abrupt, drop. Kaplan & Soker (2020b) used a bipolar toy model. We here
calculated the lightcurves from bipolar explosions that we built with 3D
hydrodynamical simulations that we base on a CEE scenario. In that respect our
calculation are more realistic. However, since we did not include radiative
transfer we had to assume a uniform and constant opacity in equation (1), a
minimum temperature $T_{\rm H,c}$ (equation 2) from the hydrodynamical
simulations for a photosphere area element to be included in the effective
projected area $A_{\rm ef,ph}$, and a photospheric temperature $T_{\rm ef,ph}$
in the calculation of the luminosity (equation 3). All the values that we
assume are within a very reasonable range of possible parameters of CCSNe. Due
to numerical limitations we could not follow in details the inner regions of
the ejecta, and for that our lightcurves at late times are not accurate.

Like Kaplan & Soker (2020b) we try to fit the lightcurve of the hydrogen-poor
SN 2018don. The lightcurves shapes that best fit the lightcurves are those of
case EXP1 (Fig. 5). However, these lightcurves evolve too rapidly. We showed
that by increasing the mass in the CSM and/or reducing its velocity we can
prolong the lightcurve (Fig. 6). Because of the numerical difficulties we
could not obtain lightcurves with similar shapes to that of SN2018do. For
that, we simply took the lightcurve from the middle panel of Fig. 5 and
stretched the time by a factor of 4. In order to fit the lightcurve of SN2018
we will have to make an extensive search of the parameter space, including
changing the properties of the exploding jets and compositions of the CSM and
ejecta, something we did not do here.

Let us place our results within a broader scope. Some recent studies argue
that modelling lightcurves with the neutrino-driven explosion mechanism and
then adding a magnetar and/or ejecta-CSM interaction cannot explain a large
fraction of LSNe and SLSNe (for a review of this see Soker 2022d). Soker &
Gilkis (2017a) find that in about half of the SLSNe lightcurves that Nicholl
et al. (2017) fit with a magnetar the CCSN explosion energy is $E_{\rm
SN}>2\times 10^{51}{~{}\rm erg}$. This is more than what the delayed neutrino
mechanism can account for, and therefore Soker & Gilkis (2017a) conclude that
jets exploded these SLSNe. The magnetohydrodynamic simulations by Reichert et
al. (2022) support this claim for jet-powering. Similarly, Soker (2022c)
examines the the lightcurves fitting with magnetars of 40 LSNe by Gomez et al.
(2022), and finds inconsistency in 18 LSNe out of 40. Further to these, Soker
(2022d) examined the fitting to the lightcurves of 70 hydrogen-poor SLSNe that
Chen et al. (2022) perform with magnetars and/or ejecta-CSM. Soker (2022d)
argues that even the slower magnetars in their fitting most likely launched
jets at explosion. In 7 out of 14 SLSNe for which Chen et al. (2022) apply
ejecta-CSM interaction to power the lightcurve, Soker (2022d) finds the
kinetic energy of the ejecta to be larger than what the delayed neutrino
explosion mechanism can supply. With these claims for difficulties of the
magnetar and ejecta-CSM fittings to account for about half of SLSNe, and in
particular for the neutrino delayed mechanism to account for the required
large explosion energy in many cases, we accept the possibility that jets
explode a large fraction of SLSNe. Our study strengthens the claim that jet-
driven explosions can account for some properties of LSNe and SLSNe.

## Acknowledgments

We thank an anonymous referee for the good suggestions and comments. This
research was supported by the Amnon Pazy Research Foundation. AM acknowledges
support from the R&D Authority in Ariel University. Numerical simulations
presented in this work were per-formed on the Hive computer cluster at the
University of Haifa.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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# Variational Bayes and non-Bayesian Updating

Tomasz Strzalecki Department of Economics, Harvard University. Date: . I thank
Ryota Iijima for alterting me to a very close result of Dominiak, Kovach, and
Tserenjigmid (2023), which I was unaware of when writing this note.

###### Abstract

I show how variational Bayes can be used as a microfoundation for a popular
model of non-Bayesian updating.

##  1 Bayesian Updating

Let $S$ be a finite state space (parameter space) and $p\in\Delta(S)$ be the
prior of the agent. The agent gets information: let $X$ (finite) be the set of
possible signal realizations (messages). Let $f:S\to\Delta(X)$ be the
_likelihood_ function (a theorist may call $f$ a Blackwell experiment). Let
$q_{x}\in\Delta(S)$ be the agent’s belief upon observing data $x$. As we all
know, the Bayes rule says that

$q_{x}(s)=\frac{f(x|s)p(s)}{\sum_{s^{\prime}\in
S}f(x|s^{\prime})p(s^{\prime})}$ (1)

The denominator, the marginal on $x$, is often hard to compute in practice
when $X$ is continuously distributed and the sum is an integral. In the next
note I will talk about one particular computational approach to this problem,
called _variational Bayes_. For now, to get rid of the denominator, define
$\propto$ as “equal up to a multiplicative factor” and write (1) as

$q(s)\propto p(s)f(x|s).$ (2)

##  2 Non-Bayesian Updating

Researchers in psychology have found that experimental subjects deviate from
the Bayes rule in systematic ways.111Edwards and Phillips (1964); Phillips and
Edwards (1966); Tversky and Kahneman (1971); Kahneman and Tversky (1972). Such
deviations are classified as normatively incorrect (biases, errors, etc.).
Benjamin (2019) reviews this evidence and uses the following two-parameter
generalization of (2) as a descriptive model to categorize various errors in
probability updating. Augenblick and Rabin (2021) also use this model to
obtain a similar classification.

For lack of better terminology I will call this the _exponential updating
rule_

$q_{x}(s)\propto\big{[}p(s)\big{]}^{\alpha}\big{[}f(x|s)\big{]}^{\beta},$ (3)

where $\alpha,\beta>0$. The exponent $\alpha$ measures how much the agent
over/under weighs the prior and the exponent $\beta$ measures how much they
overweigh the evidence. In particular, $\alpha<1$ is classified as
underinference (or conservatism), $\alpha>1$ as overinference, $\beta<1$ as
base-rate neglect, and $\beta>1$ as confirmation bias. As noticed by Grether
(1980) if the researcher runs a log linear regression of posteriors over
priors and likelihoods (without an intercept), then they are de facto
estimating a exponential updating rule.

Interestingly, the case of $\alpha=1$, $\beta<1$ has been proposed as a
normative model in Bayesian statistics222 Grünwald (2012); Bissiri, Holmes,
and Walker (2016); Bhattacharya, Pati, and Yang (2019). and machine
learning.333Huang, Tan, Lacoste, and Courville (2018); Higgins, Matthey, Pal,
Burgess, Glorot, Botvinick, Mohamed, and Lerchner (2017); Medina, Olea, Rush,
and Velez (2022). Thus, what would be characterized by a behavioral economist
as an error in probability updating, to a statistician may be an appealing way
to guard against model misspecification. This way of modeling concern for
misspecification is distinct from the economics literature, where the focus is
not directly on misspecification of the likelihood function $f$, but on
misspecification of the of predictive posterior distribution.444Hansen and
Sargent (2001); Cerreia-Vioglio, Hansen, Maccheroni, and Marinacci (2022);
Lanzani (2023).

##  3 Variational Bayes

Consider now a different approach, where instead of an explicit formula for
updating beliefs, the agent is choosing her beliefs to minimize some loss
function. Fix the prior $p\in\Delta(S)$. Recall the _relative entropy_
(Kullback-Leibler divergence) $D:\Delta(S)\times\Delta(S)\to[0,+\infty]$,
given by

$D(q\parallel p)=\sum_{s\in S}q(s)\log\frac{q(s)}{p(s)}$

when $q\ll p$ and $+\infty$ otherwise.555We use the convention that
$0\log\frac{0}{c}=0$ for any $c\geq 0$. For any $q\in\Delta(S)$ we write $q\ll
p$ whenever $p(s)=0$ implies $q(s)=0$ (that is $q$ does not invent new states
that $p$ does not believe in). The Bayesian posterior has this property and
any alternative updated belief considered by the agent will also have it.
Relative entropy is a notion of “distance” between $q$ and $p$. If we solve

$\min_{q\in\Delta(S)}D(q\parallel p),$

we will get $q=p$.666This is known as the “information inequality,” Theorem
2.6.3. of Cover and Thomas (1991).

So far we have gotten nowhere in terms of updating. We started with a prior
and our procedure ended up with a prior, so we have not updated at all. This
is because we have not used the data yet. Consider now a different
optimization problem:

$\max_{q\in\Delta(S)}\sum_{s\in S}q(s)\log f(x|s).$

Here we are choosing $q$ to best fit the data. The solution will concentrate
all its mass on the states that lead to maximum likelihood $f(x|s)$. Maximum
likelihood estimation is the classical approach to statistics.

Now the key point is that we get Bayesian updating if we combine the two
problems:

$\min_{q\in\Delta(S)}D(q\parallel p)-\sum_{s\in S}q(s)\log f(x|s).$ (4)

######  Proposition 1.

The solution to (4) is unique and equal to the Bayesian posterior (1).

Thus, Bayesian updating can be understood as striking a balance between being
close to the prior and being close to the evidence. This way of looking at
things is known as _variational Bayes_ (Peterson 1987; Hinton and Van Camp
1993; Blei, Kucukelbir, and McAuliffe 2017).777The adjective “variational” is
here because we are solving an optimization problem (by calculus of variations
when $S$ is infinite). To be more precise: It is as hard to solve (4) as it is
to compute (1) (since we are getting the same answer). The computational
advantage in variational Bayes comes from _regularizing_ the problem, which
means adding a set of constraints (e.g., $q$ is a product measure on
$S=\mathbb{R}^{d}$) or adding additional terms to the objective function. I
will abstract from this and focus on the exact equivalence as in Proposition
1.

##  4 Model misspecification

So far we understood that Bayesians are striking a balance between priors and
data. But why equal weights? Consider this problem:

$\min_{q\in\Delta(S)}D(q\parallel p)-\lambda\sum_{s\in S}q(s)\log f(x|s),$ (5)

where $\lambda>0$ is a parameter.

Here $\lambda$ measures the attitude of our agent toward model
misspecification. If the agent has full confidence in the model, she sets
$\lambda=1$ and gets Bayes. But with imperfect confidence she might want to
set $\lambda<1$. As mentioned before, this has been proposed as a normative
model in Bayesian statistics. On the other extreme is $\lambda=\infty$, which
as we saw above, corresponds to maximum likelihood estimation, a procedure in
classical statistics.

######  Proposition 2.

The solution to (5) satisfies

$q_{m}(s)\propto p(s)\big{[}f(x|s)\big{]}^{\lambda}.$

This is exponential updating with $\alpha=1$ and $\beta=\lambda$. Thus, taking
Bayes as a point of reference, $\lambda<1$ leads to underweighting the
evidence, whereas $\lambda>1$ leads to overweighting the evidence.

Overweighting the evidence seems similar to underweighting the prior. This is
true at the level of “preferences over beliefs” represented by (5): increasing
$\lambda$ puts more weight on the evidence and less relative weight on the
prior. However, at the level of the updating rule, this is not exactly true:
in the exponential updating formula there is a separate parameter, $\alpha$,
which measures how much we over/underweight the prior. What force does this
correspond to at the level of “preferences over beliefs?” I explore this in
the next section.

##  5 Attitudes toward uncertainty

Recall that the entropy of $q$ is defined by

$H(q):=-\sum_{s\in S}q(s)\log q(s),$

with the convention $0\log 0=0$. The function $H$ measures how chaotic or
uncertain the distribution $q$ is.

Suppose now that our decision maker has a taste for entropy and solves:

$\min_{q\in\Delta(S)}D(q\parallel p)-\lambda\sum_{s\in S}q(s)\log f(x|s)+\mu
H(q).$ (6)

for some parameters $\lambda>0,\mu\in\mathbb{R}$.

The case $\mu<0$ (uncertainty loving) was studied by Prat-Carrabin, Meyniel,
Tsodyks, and Azeredo da Silveira (2021) under the assumption that $\lambda=1$
(no model misspecification).888Formally, their agent minimizes $D(q\parallel
q^{Bayes}_{x})+\mu H(q)$ for some $\mu<0$. This agent dislikes having beliefs
that are close to certainty. A related notion of extreme-belief aversion was
studied by Benjamin, Rabin, and Raymond (2016). On the other hand, if the
agent prefers to have simple theories of the world and holding complex beliefs
is more costly (Enke and Graeber, 2023), then we have $\mu>0$ (uncertainty
aversion).

The next proposition shows that such an agent either follows the exponential
updating rule (if they are not too averse toward uncertainty) or puts all
posterior probability on one state (if they dislike uncertainty too much).

######  Proposition 3.

Suppose that $\lambda>0$. If $\mu<1$ the solution to (6) is the exponential
updating rule with $\alpha:=\frac{1}{1-\mu}$ and
$\beta:=\frac{\lambda}{1-\mu}$. If $\mu\geq 1$, then the solution puts all
probability on states that maximize $p(s)\left(f(x|s)\right)^{\lambda}$.

Behavior “flips” around $\mu=1$. The objective function is convex as long as
$\mu<1$. For $\mu\geq 1$ the objective function is concave and the minimum
value is attained at the extreme points of the simplex. The case of $\mu\geq
1,\lambda=1$ corresponds to Maximum a Posteriori (MAP) estimation in Bayesian
statistics, i.e., the mode of the posterior distribution (Robert, 2007).

######  Corollary 1.

Any exponential updating rule $(\alpha,\beta)$ is a solution to (6) with

$\lambda=\frac{\beta}{\alpha}\text{ and }\mu=1-\frac{1}{\alpha}.$ (7)

##  6 Related Literature

Dominiak, Kovach, and Tserenjigmid (2023) study a model of updating where the
agent has a state space $\Omega$ and receives information in the form of
$E\subseteq\Omega$. Their agent chooses their posterior to minimize a general
notion of distance between their posterior and their prior, subject to the
constraint that the posterior puts probability one on event $E$. They show
that in the case when $\Omega=S\times X$ and $E=S\times\\{x\\}$, a specific
cost function leads to the exponential updating rule considered here. My
Proposition 3 accomplishes the same, but with the added value of decomposing
the preferences into attitudes toward model misspecification and toward
uncertainty.

Zhao (2022) studies a related model of updating upon _qualitative_
information, where the update is a solution to minimizing relative entropy.
His result is an analogue of Proposition 1.

de Clippel and Zhang (2022) study non-Bayesian persuasion where the receiver
uses a updating rule such that the updated belief is a function only of the
prior belief and the correct Bayesian posterior. They show that exponential
discounting satisfies this property.

Cripps (2018) studies a general class of updating rules where the order of
signal matters. The $\alpha=1$ case of exponential updating satisfies this
property, but for $\alpha\neq 1$ the order of signals matters.

Epstein, Noor, and Sandroni 2008 study a model where the updated belief is an
affine combination of the prior and the Bayesian posterior. In this note the
mixing operates in the log-probability space, not the probability space.

In the literature on motivated beliefs agents optimally choose their beliefs
subject to some constraints or costs (Brunnermeier and Parker 2005; Bénabou
and Tirole 2016). From this point of view, Proposition 1 identifies the
objective function that leads to the benchmark case of Bayes updating.

##  Appendix A Proofs

###  A.1 A Variational Lemma

The following variational formula is well-known to economists (Stahl 1990,
Anderson, de Palma, and Thisse 1992, Fudenberg and Levine 1998, Hansen and
Sargent 2001). It also holds for infinite $S$, but I will limit my discussion
to finite spaces to avoid measurability issues.

######  Lemma 1.

For any function $g:S\to\mathbb{R}$ the minimum in

$\min_{q\in\Delta(S)}\mathbb{E}_{q}g-H(q)$

is attained uniquely by

$q(s)\propto\exp(-g(s)).$

###  A.2 Proof of Proposition 1

Fix the prior $p\in\Delta(S)$. Given that $D(p\parallel q)$ is infinite for
$q\ll p$, the minimum will be attained at $q\in\Delta(S^{*})$ where
$S^{*}:=\mathrm{supp\,}(p)$. Set $g(s):=-\left[\log p(s)+\log f(x|s)\right]$.

###  A.3 Proof of Proposition 2

Set $g(s):=-\left[\log p(s)+\lambda\log f(x|s)\right]$.

###  A.4 Proof of Proposition 3

When $\mu\leq 1$, set $g(s):=-\frac{1}{1-\mu}\left[\log p(s)+\lambda\log
f(x|s)\right]$. When $\mu\geq 1$, we are minimizing a concave function, so the
minimum value will be obtained at some extreme point. Which one? The Dirac on
state $s\in S$ that maximizes $p(s)\left(f(x|s)\right)^{\lambda}$. If there
are multiple such states, then any mixture between them is also a solution.

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# Geometry and causal flux in multi-loop Feynman diagrams

G. F. R. Sborlini1,2111 Speaker 1 Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany.  2 Instituto de Física Corpuscular,
Universitat de València - Consejo Superior de Investigaciones Científicas,
Parc Científic, E-46980 Paterna, Valencia, Spain<EMAIL_ADDRESS>
###### Abstract

In this review, we discuss recent developments concerning efficient
calculations of multi-loop multi-leg scattering amplitudes. Inspired by the
remarkable properties of the Loop-Tree Duality (LTD), we explain how to
reconstruct an integrand level representation of scattering amplitudes which
only contains physical singularities. These so-called _causal representations_
can be derived from connected binary partitions of Feynman diagrams, properly
entangled according to specific rules. We will focus on the detection of flux
orientations which are compatible with causality, describing the
implementation of a quantum algorithm to identify such configurations.

##  1 Motivation and introduction

The Standard Model (SM) is considered one of the most successful theories,
since many predictions were found in complete agreement with the experiments.
Recent technological advances are leading to a mayor improvement in the
quality of the data collected from particle accelerators, unveiling the
tiniest details of the structure of matter. The forthcoming upgrades of LHC
and the planed future colliders [1, 2] will challenge the available
theoretical results, forcing to achieve more precise predictions.

In the context of high-energy physics, perturbation theory is widely applied
to extract predictions from the models. Thus, higher orders are required in
order to increase the accuracy of the theoretical results. This involves
dealing with complicated multi-leg multi-loop Feynman diagrams and multi-
particle phase-space integrals. To increase even more the difficulty,
singularities and ill-defined expressions pop up in intermediate steps of the
calculations.

To tackle these problems, regularization techniques must be used. A frequent
choice is Dimensional Regularization (DREG) [3, 4, 5], which turns out to
preserve most of the original symmetries but at the same time introduces
issues related to the $D$-dimensional extension of the theory. Several
regularization methods are available and novel ideas are being proposed to
regularize the expressions keeping the original number of space-time
dimensions [6, 7].

Regularizing the singularities only make them explicit, but several
ingredients must be computed and added together in order to arrive to a finite
physical result. By virtue of general theorems [8, 9], virtual and real
contribution must be simultaneously considered to ensure the cancellation of
infrared (IR) divergences. The cancellation of additional IR singularities due
to initial-state radiation (ISR) requires the introduction of proper counter-
terms, which can be calculated within perturbation theory and involve the
collinear splitting functions [10, 11, 12, 13]. Also, the unrestricted loop
integration in the virtual contributions leads to ultraviolet (UV)
divergences, that can be removed through the renormalization procedure [14].

The purpose of this review is to explain novel technologies pointing towards a
more efficient calculation of the loop contributions. In particular, these
developments are done in the context of the Loop-Tree Duality (LTD) formalism
[15, 16, 17, 18, 19]. The main advantage of this formalism is that Feynman
loop integrals in Minkowski space are transformed into sums of phase-space
integrals of tree-level like objects, i.e. the so-called _dual terms_ ,
naturally defined in Euclidean space. In this way, it is more transparent to
establish a connection between the IR singularities in the real and the
virtual contributions [20], since they involve the same kind of integration
variables. The LTD is the foundation of a disruptive method known as Four-
Dimensional Unsubtraction (FDU) [21, 22, 23, 24], that achieves a purely four-
dimensional representation of physical observables by introducing proper
kinematical mappings to combine the virtual and real contributions. In this
way, the cancellation of IR singularities takes place directly at integrand
level, before integration and thus by-passing any need of using DREG (or other
regularization prescription).

The LTD approach turns out to be extremely useful to explore the causal
structure of multi-loop multi-leg Feynman amplitudes. After presenting the
basis of the LTD formulation, we will explain the deep connection between the
geometrical interpretation of LTD and the causal structure of scattering
amplitudes in Sec. 2. In particular, we will briefly introduce some geometric
rules to select configurations leading to causal entangled thresholds, in Sec.
2.2. Due to the fact that this involves testing the causality conditions on an
increasingly large number of potential configurations, we review the basis of
Grover’s search algorithm in Sec. 3. Then, in Sec. 4, we show a concrete
proof-of-concept of a quantum algorithm to select those configurations
compatible with causal momentum flow. Finally, in Sec. 5, we present the
conclusions and give an outlook of interesting future research directions
inspired by the reported results.

##  2 A brief introduction to Loop-Tree Duality

The LTD formulation is based on the application of Cauchy’s residue theorem to
multi-loop multi-leg Feynman integrals. It was first introduced at one-loop in
Ref. [15], where the connection to the classical Feynman Tree Theorem (FTT)
was achieved through a suitable modification of the customary $\imath 0$
prescription. This led to the definition of the dual propagators which allows
to re-write $N$-point one-loop integrals as sums of $N$ dual terms originated
from single cuts [16]. This approach was extended to deal with amplitudes at
two-loops and beyond in Ref. [17], and also to re-write amplitudes with
higher-powers of the denominators [18]. Besides that, LTD turns out to be
particularly useful for exploring simplifications in asymptotic expansions at
integrand level [25, 26, 27], connections with the color-kinematics duality
[28] and efficient numerical integration strategies [29, 30].

In the last years, a renewed interest on the topic attracted the attention of
several groups and motivated new discoveries [31, 32, 33, 34, 35, 36, 37, 38].
A novel formulation based on the calculation of nested residues [33, 39]
allows to smoothly extend the LTD to multi-loop multi-leg amplitudes. By
applying the Cauchy’s residue theorem on the energy component, we can remove
one degree of freedom per integral, thus reducing the number of integration
variables and transforming the integration domain into an Euclidean space.
Explicitly, given an $N$-point $L$-loop scattering amplitude in the Feynman
representation, i.e.

$\displaystyle{\cal A}_{F}^{(L)}$ $\displaystyle=$
$\displaystyle\sum_{l}\,\int_{\ell_{1}\ldots\ell_{L}}\,{\cal
N}_{l}(\\{q_{i}\\},\\{p_{j}\\})G_{F}(1,2,\ldots,n)\equiv\int_{\ell_{1}\ldots\ell_{L}}\,d{\cal
A}_{N}^{\left(L\right)}\,,$ (1)

we obtain the following integrand-level representation

$\displaystyle d{\cal A}_{D}^{(L)}(1,\ldots,r;r+1,\ldots,n)$ $\displaystyle=$
$\displaystyle\sum_{i\in r}{\rm Res}\left(d{\cal
A}_{D}^{\left(L\right)}{(1,\ldots,r-1;r,\ldots,n)},{\rm
Im}(q_{i,0})<0\right)\,,$ (2)

for the $r$-th step. The dual representation is obtained after $L$ iterations
of Eq. (2), and the corresponding integration over the three-momenta of the
$L$ loop variables. More details about the notation used here, as well as the
associated technical details can be found in Refs. [33, 40, 41].

The reformulation of LTD based on nested residues presents some remarkable
properties. In first place, after each iteration, there are some contributions
that vanish. These are the so-called _displaced poles_ , and they are
associated to unphysical contributions originated in poles of the integrand
with opposite residues. The formal proof of their cancellation was given in
Ref. [39].

Figure 1: Compatible causal entangled threshold for a one-loop four-point
function (i.e. a box). Since there are 4 vertices, $k=3$ thresholds can be
simultaneously entangled. The configuration shown also exemplifies the
geometrical compatibility rules described in Sec. 2.2.

In second place, after the $L$-th iteration of Eq. (2) and adding all the
terms, further cancellations take place. Only those contributions involving
same-sign combinations of on-shell energies remain: these are the so-called
_causal contributions_. This behaviour is ultimately related to the deep
connection between the LTD formalism and causality in quantum field theories
[42, 34, 31, 33]. In fact, LTD naturally leads to a manifestly causal
integrand-level representation of any multi-loop multi-leg scattering
amplitude [43, 33, 44, 45, 46]. After a careful exploration of several multi-
loop multi-leg topologies, we obtain

$\displaystyle{\cal A}_{N}^{\left(L\right)}$ $\displaystyle=$
$\displaystyle\int_{\vec{\ell}_{1},\cdots,\vec{\ell}_{L}}\,\sum_{\sigma\in\Sigma}\frac{{\cal
N}_{\sigma}(\\{q_{r,0}^{(+)}\\},\\{p_{j,0}\\})}{x_{n}}\,\prod_{i=1}^{k}\frac{-1}{\lambda_{\sigma(i)}}\
+\ (\sigma\leftrightarrow\bar{\sigma})\,,$ (3)

which provides a causal representation in terms _causal entangled thresholds_.
The building blocks of this representation are the causal propagators,
$\lambda_{j}^{\pm}\equiv\sum q_{i,0}^{(+)}\pm k_{j,0}$, which involve positive
on-shell energies and same-sign combinations of the energies of the external
particles. Also, each causal propagator is in one-to-one correspondence with
each possible physical threshold of the original amplitude. The number of
thresholds (or, equivalently, causal propagators) that are simultaneously
entangled determines the order of the diagram, i.e. $k$. The order is related
to the number of off-shell lines in the traditional LTD representation, as
well as the number of vertices $k=V-1$ [45]. A pictorial example of Eq. (3)
for the one-loop four-point function is given in Fig. 1. Concrete application
examples and all-order generalizations of the LTD causal representation are
given in Refs. [40, 47, 44, 46].

###  2.1 Geometry, causality and causal propagators

In the previous section, we presented two equivalent representations of the
scattering amplitudes: the Feynman representation in Eq. (1) and the causal
representation in Eq. (3). Whilst the first one is associated to Feynman
graphs made of propagators, vertices and loops, the second one can be built
from _multi-edges_ , vertices and _eloops_ [45, 48, 41]. It turns out that the
causal structure is codified in the so-called reduced Feynman graphs, which
are built from the original diagram by merging all the propagators connecting
the same vertices. These merged propagators are the multi-edges and their
associated on-shell energy is given by $q_{G,0}^{(+)}=\sum_{i\in
G}\,q_{i,0}^{(+)}$, i.e. the same-sign sum of all the on-shell energies of the
lines connecting those vertices.

From the reduced Feynman graph, we can built the _vertex matrix_ , ${\cal V}$,
which is equivalent to the adjacency matrix and contains information regarding
how the different vertices are connected. It is possible to implement
operations on ${\cal V}$ in order to identify the set of connected binary
partitions of vertices, ${\cal P}_{V}^{C}$. Graphically, $p\in{\cal
P}_{V}^{C}$ corresponds to cutting the graph into two connected diagrams.
Moreover, we can introduce the concept of _conjugated causal propagator_ ,
$\bar{\lambda}_{p}$, by summing all the momenta flowing through the partition
$p$ (or, equivalently, $p^{c}$). Since momentum conservation is enforced in
each vertex, we have $\bar{\lambda}_{p}+\bar{\lambda}_{p^{c}}=0$ if the
external particles fulfil momenta conservation.

The motivation to provide such definitions is that the binary partitions are
associated to thresholds of the diagram. According to Cutkosky rules [49],
physical thresholds are originated by splitting or cutting the diagram in two
pieces. Since causal propagators are in one-to-one correspondence to the
physical thresholds, they must be also related to the binary connected
partitions. In fact, we can build the set of causal propagators starting from
$\bar{\lambda}_{p}$ and replacing the multi-edge energies by the corresponding
on-shell energies. The presence of external momenta leads to a two-fold
mapping to generate the casual propagators, i.e. there exists a transformation
such that $\bar{\lambda}_{p}\to\lambda^{\pm}_{p}$ which depends on the
orientation of the multi-edges. More details can be found in Ref. [45].

To conclude this section, let us mention two extreme cases. Given a diagram
with $V$ vertices, the possible number of multi-edges fulfills $V\leq M\leq
V(V-1)/2$. A graph with $M=V$ is minimally connected, whilst those with
$M=V(V-1)/2$ are maximally connected. In Refs. [44, 46], maximally connected
graphs (MCG) are the basic building blocks of an all-order causal
representation. The advantage MCG is that all the binary partitions are
connected (${\cal P}_{V}^{C}={\cal P}_{V}$), which implies that the number of
causal propagators is maximal. On the opposite side, minimally connected
graphs (mCG) are always one eloop diagrams with the minimal number of causal
propagators. We will do some further comments on these cases in the next
sections.

###  2.2 Selection rules

Once the causal propagators are identified, we need to introduce a set of
geometrical rules to build all the possible compatible causal entangled
thresholds, $\bar{\Sigma}$. Given a topology with $V$ vertices and $M$ multi-
edges, we will need to entangle $k=V-1$ causal thresholds. Then, we impose the
following criteria:

  1. 1.

_No uncut multi-edges_ : If we overlap all the causal thresholds (as the ones
shown in Fig. 1), all the internal multi-edges must be crossed by the causal
cuts.

  2. 2.

_Disjoint separation_ : If we represent each causal propagator with a line
cutting the multi-edges, then these lines can not intersect each other.

  3. 3.

_Consistent causal flow_ : The multi-edges associated to a partition must be
aligned, and flow to different partitions.

It is worth appreciating that, for some topologies, there will be degenerated
contributions to Eq. (3). In order to obtain a non-degenerated subset,
$\Sigma$, additional steps are required, as carefully explained Ref. [45]. The
last criteria involves studying all the possible orientations of multi-edges,
i.e. $2^{M}$ configurations. The fact that the aligned momenta must go from
one partition to a different one implies that there can not be loops or
cycles. Thus, the situation reduces to the identification of _directed acyclic
graphs_. Once all the causal orientations are properly selected, we can dress
the graphs with the entangled thresholds selected by criteria 1-2, and
generate the set $\bar{\Sigma}$.

Figure 2: (Left) One eloop three-vertex topology with the corresponding
labelling of the multi-edges. (Right) Graphical representation of the quantum
circuit implemented in Qiskit. The different logical gates are drawn, as well
as the input ($\\{q_{i}\\}$), the auxiliary ($\\{c_{j}\\}$ and $\\{a_{j}\\}$)
and output ($|{out_{0}}\rangle$) registries.

##  3 Quantum computing and Grover’s algorithm

Testing the causality conditions described in the previous section is a very
time-consuming task. When the complexity of the diagram increases, i.e. by
adding more vertices $V$ or multi-edges $M$, the number of combinations grows
very fast. However, the criteria can be executed in parallel and enhance the
performance of the computation.

As mentioned in Sec. 2.2, we can start by looking for all the possible
directed acyclic graphs. Even if there are several well-performing algorithms
[50] for classical computers, the complexity of the problem scales very fast
with the number of multi-edges. Thus, in the search of more efficient
strategies, we tried a novel approach based on quantum algorithms in Ref.
[48]. In particular, we relied on Grover’s search algorithm [51, 52, 53] to
test the causal flow conditions simultaneously for all the configurations.

To explain the basis of the algorithm, let us consider the space $\Omega$
composed by $N$ states. A sub-space $\omega$ with dimension $r$ identifies the
_winning_ states, i.e. the ones that we want to select. Then, we can define
the two states

$|{q}\rangle=\frac{1}{\sqrt{N}}\sum_{\alpha\in\Omega}|{\alpha}\rangle\,,\quad|{w}\rangle=\frac{1}{\sqrt{r}}\sum_{\alpha\in\omega}|{\alpha}\rangle\,,$
(4)

where $|{q}\rangle$ ($|{w}\rangle$) corresponds to the uniform superposition
of all the (winning) states. We will consider $|{q}\rangle$ as the initial
state for running the algorithm, and introduce

$U_{w}={\rm Id}-2|{w}\rangle\langle{w}|\,,\quad
U_{q}=2|{q}\rangle\langle{q}|-{\rm Id}\,,$ (5)

which are the so-called oracle and diffusion operators, respectively. Notice
that $U_{w}|{x}\rangle=-|{x}\rangle$ if $|{x}\rangle\in\omega$ and leaves the
state unchanged if it belongs to the orthogonal space $\omega^{c}$ (i.e. the
space of non-winning states). Grover’s querying algorithm relies on the
repeated application of these operators, which leads to

$|{q^{\prime}}\rangle\equiv(U_{w}\,U_{q})^{n}|{q}\rangle=\sin\left((2n+1)\beta\right)|{w}\rangle+\cos\left((2n+1)\beta\right)|{w^{c}}\rangle\,,$
(6)

after $n$ iterations and $\beta=\arcsin(r/N)$ is the original projection of
$|{q}\rangle$ over the winning space. In this way, if $r/N$ is small enough,
it is possible to amplify the probability of the winning states w.r.t. those
belonging to the orthogonal space after a few iterations of the algorithm.

##  4 Quantum algorithm for causal flux identification

Finally, we explain how to profit from the quantum parallelization to
efficiently identify the casual flux configurations. In first place, we
associate a qubit to each multi-edge: $|{q_{i}}\rangle=|{1}\rangle$ if the
flow agrees with the initial choice and $|{q_{i}}\rangle=|{0}\rangle$ if it is
reversed. In this way, our quantum circuit will have the qubits
$\\{q_{0},\ldots,q_{M-1}\\}$ to describe the flow configuration. The uniform
superposition state $|{q}\rangle$ is achieved through the application of
Hadamard operators.

The explicit implementation of the oracle and diffusion operators requires to
use logical gates and introduce auxiliary qubits. This is because qubits are
quantum states: they can not be erased and re-used in the same way that we do
with classical bits. Thus, we compare the fluxes of the different adjacent
multi-edges and store the output in the auxiliary registers $\\{c_{i}\\}$.
Using this information, we introduce another logical clause to probe whether
the multi-edges (codified in the qubits $\\{q_{i}\\}$) are part of a cycle or
not: the result is stored in the registry $\\{a_{j}\\}$. With all these
elements, we build the marker function $f(a,q)$ and define Grover’s oracle.

Figure 3: Probability distribution of the different configurations for an one
eloop topology with three vertices. We appreciate that the algorithm
successfully enhanced the probability of the three states satisfying the
casual flow condition.

As a practical example, we consider the identification of all the causal-
compatible flux configurations for a three-vertex one-eloop topology (i.e. a
_triangle_). The circuit, as implemented in Qiskit, is shown in Fig. 2
(right). Since the number of solutions is roughly one half of the total
states, we include an additional qubit to achieve the amplification of the
winning states [48]. With this modification, we successfully identify the 3
causal-compatible configurations. The corresponding probability distributions
are shown in Fig. 3, where we can clearly appreciate that
$\\{|{001}\rangle,|{011}\rangle,|{101}\rangle\\}$ describe directed acyclic
graphs following the convention of Fig. 2 (left).

In general, the number of causal-compatible solutions depends on the
topological complexity of the diagram (i.e. the number of vertices and
eloops). For mCG diagrams, the presence of only one eloop leads to a weak
constraining of the orientations: in this case, the ratio of winning over
total states ($r/N$) tends to 1 as the number of vertices increases. On the
other hand, MCG are extremely constrained systems and $r/N$ goes rapidly to 0.

##  5 Conclusions and outlook

The causal structure of multi-loop multi-leg scattering amplitudes contains
all the relevant information to characterize their physical singularities. In
combination with the Loop-Tree Duality formalism [33], causal representations
lead to a more efficient numerical integration because they only contain
physical singularities: instabilities due to spurious divergences are avoided.
Moreover, the integration domain is transformed from a Minkowski to an
Euclidean space, thus paving the way for a natural integrand-level combination
with the real-radiation contributions.

Recently, there have been several developments to reconstruct dual causal
representations by-passing the direct calculation of nested residues [46, 45].
One of these strategies relies on geometrical algorithms to detect all the
possible causal propagators and identify the set of compatible causal
entangled thresholds. In order to do that, we start from the reduced Feynman
graph and associate a causal propagator to each possible connected binary
partition of vertices. If the graph has $V$ vertices, we entangle $k=V-1$
causal thresholds in such a way that they fulfill specific selection criteria.

One of these selection criteria is related with the causal-compatible flux,
which turns out to be equivalent to the identification of acyclic graphs [48].
In this review, we elaborate on the implementation of a quantum algorithm,
based on Grover’s methods, to efficiently detect acyclic directed graphs. It
constitutes the first proof-of-concept of such an algorithm for bootstrapping
the causal representation of Feynman scattering amplitudes, and we illustrate
its application with a one eloop three-vertex topology.

It is interesting to notice that the performance of the algorithm increases as
the ratio of casual versus total configurations decreases, which occurs as the
topologies become more complicated (i.e. more vertices and more multi-edges).
Moreover, since the total number of compatible causal entangled thresholds is
small compared to the total possible combinations, we expect that a variation
of this algorithm could efficiently reconstruct the whole causal
representation. This supports the claim that the irruption of quantum
technologies could allow to overcome many of the current bottlenecks in
theoretical high-energy physics.

##  6 Acknowledgments

I would like to thank G. Rodrigo, S. Ramírez-Uribe, A. Rentería-Olivo (IFIC),
R. Hernández-Pinto (UAS) and P. Marquard (DESY) for fruitful discussions and
comments on the manuscript. This article is based upon work from COST Action
PARTICLEFACE CA16201, supported by COST (European Cooperation in Science and
Technology).

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