# The diffusion metrics of African Swine Fever in Wild Boar

Hartmut H. K<EMAIL_ADDRESS>Hannes Bergmann  Carola Sauter-
Louis
_Friedrich-Loeffler-Institut, Federal Research Institute for Animal Health,
Institute of Epidemiology, 17493 Greifswald - Insel Riems_

Abstract. To control African swine fever (ASF) efficiently, easily
interpretable metrics of the outbreak dynamics are needed to plan and adapt
the required measures. We found that the spread pattern of African Swine Fever
cases in wild boar follows the mechanics of a diffusion process, at least in
the early phase, for the cases that occurred in Germany. Following incursion
into a previously unaffected area, infection disseminates locally within a
naive and abundant wild boar population. Using real case data for Germany, we
derive statistics about the time differences and distances between consecutive
case reports. With the use of these statistics, we generate an ensemble of
random walkers (continuous time random walks, CTRW) that resemble the
properties of the observed outbreak pattern as one possible realization of all
possible disease dissemination patterns. The trained random walker ensemble
yields the diffusion constant, the affected area, and the outbreak velocity of
early ASF spread in wild boar. These quantities are easy to interpret, robust,
and may be generalized or adapted to different regions. Therefore, diffusion
metrics can be useful descriptors of early disease dynamics and help
facilitate efficient control of African Swine Fever.

Keywords: Epidemiological model, spatial model, spatio-temporal model,
communicable disease model.

##  1 Introduction

African swine fever virus (ASFV) causes an internationally spreading
haemorrhagic pig disease with a massive socio-economic impact [1, 2]. The
current African swine fever (ASF) pandemic originated from disease incursion
of genotype II ASFV in Georgia during 2007 [3]. From there, ASF spread
northwards into the Caucasus region, then further disseminated westwards into
Europe, eastwards into Southeast Asia [2], and even jumped across the Atlantic
to threaten the Americas with outbreaks reported in the Dominican Republic and
Haiti in 2021 [4]. Since the start of the pandemic, an estimated quarter of
the global domestic pig population has been decimated by the disease, causing
food insecurity and economic losses on an unprecedented global scale [5, 6,
7]. Particularly during the early phases following new ASF incursion, well
informed anticipation of disease spread is critical for controlling the
disease efficiently.

As a consequence of the incursion into Georgia in 2007, ASF (genotype II)
reached the territory of the European Union (EU) in 2014, when first ASF cases
were reported in wild boar in Lithuania and Poland [8, 9, 10]. Since then, and
despite ongoing control efforts as well as intensive study of disease
dynamics, ASF has been moving predominantly in western direction, affecting
many more EU countries [11]. Among them, in 2020, ASF has also reached Germany
[12], where the disease continues to spread in initially distinct spatial
clusters [13].

In eastern and central Europe, wild boar seem to represent the predominant,
disease-sustaining reservoir host in the current European ASF scenario. This
is based on the spatial extent of cases in this pig type [14], as well as
their critical role in disease transmission through persistence of virus in
the environment [15, 16, 2, 17]. Infected wild boar that succumb to the
disease, which is characterized by a case/fatality ratio of $>90$ %, may
harbor infectious virus for weeks, if not months, after the death of the
animals.

Unexpected occurrences of wild boar-ASF cases in locations that are a long
distance away from the nearest previously affected area, such as suspected
point incursions into the Czech Republic [18], Belgium [19], into the western
part of Poland [17], or into Northern Italy [4], indicate that ASF can be
relocated in association with human activities. However, typically ASF spreads
in a gradual manner through infections and dissemination of disease in wild
boar at a local scale. Whilst ASF outbreaks in domestic pigs appear to be
manageable in most countries, the gradual disease spread in wild boar is very
difficult to control and often persists [20, 21, 22]. Based on historic ASF
case reports, average disease spread velocities of approximately 1 to 1.5 km
per month have been estimated [23, 11, 24]. Control measures that efficiently
manage ASF dissemination following new incursions require risk-based
allocation of limited resources and rely on disease spread predictions that
are locally applicable to the acute outbreak situation in the field.

Most models for African swine fever in wild boar depend on a large number of
parameters and assumptions (see [25] for a comprehensive overview). Therefore,
they suffer from high complexity and it is difficult to draw practical
conclusions from these models. For controlling African swine fever, simple and
easily understandable metrics are needed, such as the following: Given a new
occurrence of ASF: (1) What is the affected area for the next time?, (2) How
far does the epidemic reach from the index case over time?, and (3) What is
the velocity of spread?

In order to answer these questions, we take a perspective different from most
predictive models: What if the underlying process is not relevant, and the
outbreak points are just generated by a random-process? If we understand this
process, we can compute all the desired metrics that are described above.

Even though ASF dynamics seem to be complex in general, disease dissemination
appears to follow a remarkably simple pattern when considered on a local
scale. In fact, the local dynamics of cases appear as growing areas of new
cases emerging, and long-distance jumps are extreme events [11]. On the one
hand, long-distance jumps are extremely hard to predict as they are presumably
caused by human activity [3]. On the other hand, short distance spread is
mainly caused by wild boar and understanding its dynamics is crucial for
efficient counter measures. Therefore, robust metrics are needed to quantify
the dynamics of local ASF outbreaks.

In order to provide such metrics, we follow the idea to describe the epidemic
as a pure diffusion process. Logically, an epidemic is actually not a pure
diffusion process – i.e. a reaction-diffusion process – but if an epidemic can
be modelled in an accurate way by a diffusion process, this allows to
interpret the results mathematically in a relatively simple fashion.

Diffusion is a macroscopic process that can be microscopically described as a
stochastic process, also known as Brownian motion [26, 27, 28]. In the context
of ASF, the microscopic process is the single local outbreak, i.e. a spatial
pattern of cases. Once the logic of this microscopic process is understood and
calibrated for the data, diffusion of disease spread can be extrapolated
spatially and over time.

For a purely diffusive process, a similar approach has been used on human
mobility data [29]. In the context of ASF, a probabilistic model considering
random walks by wild boar with infection dynamics has been proposed previously
[30]. In contrast to that model, we consider the outbreak pattern generating
process itself as a random walk.

Another model considers the diffusion around a primary case and including a
habitat-suitability component [31]. The diffusion component in [31], however,
is not time dependent and therefore the model is not suitable for temporal
predictions. A predictive model for ASF has been proposed in [32]. This
approach is based on a compartment model and is therefore suitable for a
prediction, even though it is mainly driven by assumptions instead of data
which is typical for this model type. Moreover, it does not consider a spatial
component.

Besides mathematical models, individual based models have been used in order
to estimate the transition parameters of ASF, based on real outbreak data
[33]. This model contains detailed data, and the movement and infection
dynamics are considered explicitly. In contrast to the model proposed in the
present paper, however, the model in [33] requires a rather large number of
assumptions. Finally, the local wave front velocity of ASF has been modelled
for Belgium in [34].

All of the mentioned models provide good insights on the dynamics of ASF. What
has been missing so far is a model capturing the ’physics’ behind the outbreak
pattern. As stated above, despite the fact that ASF is an infection process,
it appears as a pure diffusion process on the map. For this reason, we fit a
diffusion model to the disease data in order to measure the diffusion
parameters of the ASF epidemic directly.

##  2 Material and Methods

###  2.1 Data

We used the official ASF case data for Germany covering all cases from 10
September 2020 to 9 July 2021 from the national animal disease database
(Tierseuchennachrichtensystem) [35]. Being in the early phase of the outbreak,
we were able to separate the data into clusters [13], as shown in Figure 1. In
this paper, we will analyze one representative cluster in detail (Cluster 1),
for reasons of simplicity. All other clusters show similar microscopic
patterns (see Supplementary Information) and we compare all clusters briefly
in the results section.

Each instance in the data set represents a _case_ , i.e. time and coordinates
of a detected ASF-positive wild boar. To distinguish between real data and
data generated by our model, we use the term _event_ for the random walk model
instead of _case_. For clarity, we refer to a cluster of ASF-cases in wild
boar as an _outbreak_ (which should not be confused with occurrence of ASF in
domestic pigs).

Figure 1: African swine fever case data and its separation into clusters.

In order to get a first simple estimate of the outbreak velocity, we calculate
the distance from each case to the index case over time. This is shown in
Figure 2. Using a linear fit with vanishing intercept, we obtain a velocity of
$0.042\;\mathrm{km/day}$.

Figure 2: Distance to index case as a function of time. Every data point
represents one case. Where multiple cases were detected at one day, the y-axis
presents the mean distance. Fitted slope is $0.042\;\mathrm{km/day}$.

As shown below, this approach gives a good first estimate of the velocity,
although it does not capture all features of the dynamics.

For a deeper understanding of the outbreak pattern, we consider the causal
ordering of the cases in more detail. It is important to stress that the data
set itself does not contain any causal information between the cases. Indeed,
the measured data points represent an underlying – and unknown – infection
tree that describes in detail, which case has caused which other case(s).

Since the exact relationships in this infection tree data are unknown, we
estimate causality in the following way (a similar idea was used in [36]):

  1. 1.

Sort cases by time.

  2. 2.

Generate a directed acyclic graph (DAG) $T=(V(t),E)$ with edge set
$E=\emptyset$, where each node $v(t)\in V(t)$ is a case with time stamp $t$.

  3. 3.

Connect nodes in $T$ with directed edges from case $s$ to case $t$ as follows:
whenever the target case $t$ is after or at the same time as the source case
$s$, draw a directed edge $(s,t)$. Thus, the added edges $E\neq\emptyset$ in T
comprise all possible causal connections between the cases.

  4. 4.

Weight all edges with the reciprocal geographical distance between the
respective nodes/cases. (Vanishing distances are assigned a weight of zero.)

  5. 5.

Finally, compute a minimum spanning tree on the now weighted DAG. For this, we
used the Chu-Liu/Edmond Algorithm [37, 38] implemented in [39].

This procedure arranges the cases in a causally and geographically plausible
order. Using the minimum spanning tree, we obtain the distances between the
cases, and subsequently the jump length distribution.

The empirical distribution of waiting times was directly derived from the
outbreak data. We sort the cases by time and compute the differences between
consecutive cases yielding the waiting time distribution.

###  2.2 Brownian Motion

In the present work we consider the outbreak data as points that are seemingly
generated randomly in space. The only constraint is that new data points are
generated in geographical and temporal closeness to existing points. We hereby
establish the simple assumption that new data points are somehow related to
existing data points. If we assume in the first place that every existing data
point generates exactly one new data point at the next time step, the
generating process would be Markovian. On closer consideration, however, this
assumption is not valid, since waiting times occur between the cases, and thus
secondary outbreaks can be later in the future.

Random walk processes that consider waiting times are called _continuous time
random walks_ (CTRW). Such a process works as follows: A random walker is
initiated at time $t=0$ (time of the first case) at a location, say
$(x_{0},y_{0})=(0,0)$ (location of the first case). Then it waits for a random
time $\tau_{1}$ and makes a jump of random length $l_{1}$ in a random
direction. Thereafter, it waits for a random time $\tau_{2}$ and performs
another jump of random length $l_{2}$ and so forth. We assume that jump
lengths and waiting times are uncorrelated. Jump lengths are sampled from a
distribution $\phi(l)$ and the waiting times from a distribution $\psi(\tau)$.
In this work, the distributions of $\phi(l)$ and $\psi(\tau)$ are determined
from the outbreak data. Hence, we generate synthetic outbreak data that is
statistically equivalent to the observed data.

The CTRW is implemented as follows:

  * •

Start at the coordinates of the index case. Set these $(x_{0},y_{0})=(0,0)$.

  * •

Sample time jumps from the waiting-time distribution $\psi(\tau)$ and generate
a sequence of time points (event points) following the time jumps.

  * •

For each event point: sample a jump length from the jump-length distribution
$\phi(l)$ and perform a step in a random direction.

The latest event determines the duration $T$ of the random walk. We refer to
one realization of a CTRW as a _trajectory_ $X(t)$.

It is important to stress the fact that, in contrast to an epidemic process, a
random walker trajectory can only be at one location at a time. In order to
address this factor, we correct the available time for the random walker.

###  2.3 Time correction in random walk

Besides the above-mentioned waiting times, another crucial assumption for
random walks does not hold for the outbreak data. On the one hand, there can
be multiple cases at every time step, that is, an epidemic can be at multiple
locations at the same time. On the other hand, a classic random walker can
only be at one location at a time. In order to resolve this issue, we use the
following idea: Let the random walk have a maximum duration of $T$. In the
easiest situation, exactly one case would occur at each day. However,
considering the situation where on average $M$ cases occur per day, the random
walker must have the ability to generate these cases/events without spending
time. We call $M$ the _multiplicity_ of the process. As an example, if we have
$M=3$ cases per day and the maximum duration is $T=100$ days, then the random
walker has $MT=300$ available days for generating these 300 events in total.
Finally, in order to return to the original time scale, we rescale the new
maximum duration (300 days) back to the initial value (100 days).

###  2.4 Diffusion Coefficient, expected radius, and velocity

The Brownian motion described above is a single realization of a microscopic
random process. Averaging over a large number of random walks yields the
macroscopic properties of the process. Since every random walker can walk in a
different direction, the expected location is
$\left<X(t)\right>=(x_{0},y_{0})=0$ for all times $t$ (the brackets
$\left<\cdot\right>$ refer to the average over all random walkers).

For large times $t$ a single random walker is expected to be located at a
great distance from the origin. Therefore, the _mean squared displacement_
(MSD) $\left<X(t)^{2}\right>$ increases with time. The detailed form of the
MSD has to be determined empirically. In case the MSD follows a linear
relation, i.e. $\left<X(t)^{2}\right>\sim t$, the corresponding macroscopic
process is called _normal diffusion_. In that case

$\left<X(t)^{2}\right>=4Dt$ (1)

and the constant $D$ is the _diffusion coefficient_. Equation (1) represents
the variance of the random walkers’ positions after time $t$.

The square root of the MSD is the expected radius, on which all random walkers
are expected to be after time $t$, i.e.

$r(t)=\sqrt{\left<X(t)^{2}\right>}=\sqrt{4Dt}.$ (2)

We identify this quantity with the radius of the affected area or the distance
between the index case and the wave front.

Finally, the velocity of the wave front $v(t)$ can be defined as the change of
the radius with respect to time, thus

$v(t)=\frac{dr(t)}{dt}=\sqrt{\frac{D}{t}}.$ (3)

Note that $r(t)$ and $v(t)$ are not linear.

Our implementation of the mentioned methods is available online [40].

##  3 Results

To bring all cases in a plausible order, we first sort the outbreak data by
detection time and compute the minimum spanning tree. This tree provides us
with the distribution of shortest jump lengths. We then generate the waiting
time distribution directly from the outbreak data. Both distributions are
shown in Figure 3.

Figure 3: Jump length distribution (A) and waiting time distribution (B)
derived from the outbreak data of Cluster 1. (The jump length distribution is
for cases ordered using the minimum spanning tree).

As mentioned above, a random walker cannot be at multiple locations at the
same time, as opposed to epidemic processes, where multiple cases can occur
simultaneously. This is exemplified by Cluster 1 in Figure 4.

Figure 4: Number of cases per day of cluster 1. The multiplicity is 3.4.

On average 3.4 cases occur per day over 300 days in total, i.e. the
multiplicity of the process is $M=3.4$. Therefore, we multiply the available
time for the random walker by $M$. This yields 1020 time steps which are
afterwards rescaled to 300 days.

Using the distributions from Figure 3 and the multiplicity $M$ we generate an
ensemble of 10,000 random walkers to guarantee statistical stability. In order
to get an impression of the microscopic properties of the random walks, we
show one realization in Figure 5.

Figure 5: Real outbreak data vs. one realization of a random walk for Cluster
1. The index case is set to coordinates (0, 0). Outbreak data from 300 days,
random walk with multiplicity 3.4 resulting in 1020 steps that represent 300
days.

This realization appears to show considerable structural similarity to the
observed outbreak data, in the sense that both sets of points appear to be
sampled from a similar distribution. Note that a random walk is an isotropic
process, i.e. all directions are equally likely. It is therefore on purpose
that the outbreak data and the synthetic data points can be in different
directions as long as they have a similar structure.

We now study the macroscopic (diffusion) properties of the random walker
ensemble. Figure 6 shows the mean squared displacement (MSD) over an ensemble
of 10,000 random walkers. The MSD follows a linear form indicating that the
measured distributions result in a normal diffusion process. Using a linear
fit, we obtain a diffusion coefficient of $D=(0.22\pm
0.01)\;\mathrm{km}^{2}/\mathrm{day}$. This value is a median over all
realizations and the error is the inter-quartile range.

Figure 6: Mean squared displacement for an ensemble of 10,000 random walkers
(red line). The resulting diffusion constant $D$ follows from a linear fit
(blue dashed line) which gives $D=(0.22\pm
0.01)\;\mathrm{km}^{2}/\mathrm{day}$.

The radius of the affected area follows the square root shaped relation shown
in Figure 7 A. After a steep increase in the early phase of the epidemic, the
radius grows over time, but the front velocity decreases. Note that the
slowing down of the wave front cannot be captured by the simple linear
approach used in Figure 2. The wave front velocity over time is shown in
Figure 7 B. The latter shows a quasi-constant behavior in the time scale of
interest, i.e. roughly 0.04 km/day measured 150 days after the first case.

Figure 7: A: Radius of the area where all random walkers are likely to be
contained after time $t$. B: Radial velocity of the area growth. Red lines are
mean values over the random walker ensemble, blue dashed lines are analytical,
using the diffusion constant.

###  3.1 Comparison between the clusters

So far, we have only studied one selected cluster. In Figure 8 we show the
diffusion coefficients for all clusters. Each value is a median over 10,000
simulations. The error bars represent the inter quartile ranges.

Figure 8: Diffusion constants for all clusters. Error bars show the inter-
quartile distance. Each data point is for 10,000 simulations.

The figure demonstrates that even if there are certain differences in the
clusters, their diffusion coefficients show a remarkable similarity. Most
diffusion constants lie in a band between 0.2 and 0.5
$\mathrm{km}^{2}/\mathrm{day}$.

We provide a detailed description of the diffusion metrics for all clusters in
the Supplementary Information.

##  4 Discussion

In the present study we have considered the outbreak propagation of African
swine fever as a diffusion process. Instead of making assumptions on wild boar
movements, we focussed on the process generating the outbreak data directly.
Although this is an abstract concept, it allows us to measure the physical
properties of the observed outbreak pattern.

Assuming that the outbreak propagation follows a random walk appears drastic,
since in contrast to a random walk, new cases can appear at multiple locations
at the same time. This could be modelled by a branching process, where the
random walker can reproduce itself. As it has turned out, however, such a
branching process becomes irrelevant, whenever all random walkers are not
restricted in their possible location. This property of the model is supported
by the fact that infected animals can freely return to already infected areas,
that is, the disease cannot be pushed out of already infected areas in the
considered early phase of the outbreak.

Although our results provide simple metrics for the propagation of ASF, the
computation of these metrics is not trivial in general. On the one hand,
estimating the wave front velocity using the simple linear distance to the
index case has turned out to give a value remarkably similar to that of our
model. On the other hand, this simple approximation does not capture the
slowing down of the wave front over time that is predicted by our model. For
the random walk model, besides the needed Monte-Carlo-simulations, finding the
distributions for waiting times and jump lengths requires manual adjustments.
These could be optimized using a hyper-parameter-tuning scheme. To obtain
interpretable results, the real outbreak clusters should be not constrained,
as it may be the case due to geographical barriers (rivers, roads, fences,
etc.). The more the real outbreak clusters are constrained, the more manual
adjustment is required. Considering the time data for the cases, the random
walk approach has proven useful, although we have used the date, when ASF
infection in dead wild boar were confirmed – and not the date, when the
animals died. In a next step, the estimated death times of the wild boar could
be used, by applying the minimal post-mortem-interval [41].

As we have demonstrated in Figure 8, the properties of some of the clusters
are remarkably similar. This seems to be reasonable, as the counter measures
implemented overall are similar in all of the clusters.

Nevertheless, Clusters 4 and 6 show higher diffusion coefficients. In the case
of Cluster 4, this could be due to the fact that the time needed for fences to
be erected was longer than in other cluster areas. Moreover, the first cases
occurred along an extended area of the border without any expansion for the
first 80 days. For Cluster 6, this could be caused by the fact that the
disease occurred in an urban area, which did not allow for implementation of
the same control measures as in the other clusters. Moreover, the different
diffusion coefficient might be caused by the fact that the cases occurred
along an extended area at the German-Polish border, thus showing a high degree
of constraint (see Figure 1, and Supplementary Information for more details)
It is important to stress the fact that this constraint is caused primarily by
the data availability and not by the underlying process. That is, we would
expect to get a more consistent picture here, if Polish data would have been
included in the analysis.

Consequently, we state that the observed patterns follow a general mechanism,
at least for this data set representing a particular area in Germany. In
conclusion, it seems possible to derive a general diffusion law for this kind
of setting, which might be helpful for disease control.

##  5 Acknowledgement

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 773701.

##  6 Author contributions

H.L. designed the study and conducted the computations. H.L., H.B., and C.S.
wrote the manuscript. C.S. contributed the data.

##  7 Competing interests

The authors declare no competing interests.

##  8 Data availability

The dataset used in this study can be made available for researchers upon
request.

##  9 Code availability

The code is available online on
https://github.com/hartmutlentz/RandomWalker2D.

## References

  * [1] LK Dixon, H Sun, and H Roberts.  African swine fever.  Antiviral research, 165:34–41, 2019.
  * [2] L. K. Dixon, K. Stahl, F. Jori, L. Vial, and D. U. Pfeiffer.  African swine fever epidemiology and control.  Annu Rev Anim Biosci, 8:221–246, 2020.
  * [3] Carola Sauter-Louis, Franz J Conraths, Carolina Probst, Ulrike Blohm, Katja Schulz, Julia Sehl, Melina Fischer, Jan Hendrik Forth, Laura Zani, Klaus Depner, Thomas C Mettenleiter, Martin Beer, and Sandra Blome.  African swine fever in wild boar in europe-a review.  Viruses, 13(9):1717, 08 2021.
  * [4] World Organisation for Animal Health (OIE).  World animal health information system (wahis), 2021.
  * [5] Shibing You, Tingyi Liu, Miao Zhang, Xue Zhao, Yizhe Dong, Bi Wu, Yanzhen Wang, Juan Li, Xinjie Wei, and Baofeng Shi.  African swine fever outbreaks in china led to gross domestic product and economic losses.  Nature Food, 2(10):802–808, 2021.
  * [6] Thinh Nguyen-Thi, Linh Pham-Thi-Ngoc, Que Nguyen-Ngoc, Sinh Dang-Xuan, Hu Suk Lee, Hung Nguyen-Viet, Pawin Padungtod, Thuy Nguyen-Thu, Thuy Nguyen-Thi, Thang Tran-Cong, and Karl M. Rich.  An assessment of the economic impacts of the 2019 african swine fever outbreaks in vietnam.  Frontiers in Veterinary Science, 8, 2021.
  * [7] Jarkko K. Niemi.  Impacts of african swine fever on pigmeat markets in europe.  Frontiers in Veterinary Science, 7, 2020.
  * [8] Grzegorz Woźniakowski, Edyta Kozak, Andrzej Kowalczyk, Magdalena Łyjak, Małgorzata Pomorska-Mól, Krzysztof Niemczuk, and Zygmunt Pejsak.  Current status of african swine fever virus in a population of wild boar in eastern poland (2014-2015).  Archives of Virology, 161(1):189–195, 2016.
  * [9] A. Pautienius, J. Grigas, S. Pileviciene, R. Zagrabskaite, J. Buitkuviene, G. Pridotkas, R. Stankevicius, Z. Streimikyte, A. Salomskas, D. Zienius, and A. Stankevicius.  Prevalence and spatiotemporal distribution of african swine fever in lithuania, 2014-2017.  Virol J, 15(1):177, 2018.
  * [10] P. Maciulskis, M. Masiulis, G. Pridotkas, J. Buitkuviene, V. Jurgelevicius, I. Jaceviciene, R. Zagrabskaite, L. Zani, and S. Pileviciene.  The african swine fever epidemic in wild boar (sus scrofa) in lithuania (2014-2018).  Vet Sci, 7(1), 2020.
  * [11] EFSA.  Epidemiological analyses of african swine fever in the european union (november 2018 to october 2019).  Efsa Journal, 18(1), 2020.
  * [12] C. Sauter-Louis, J. H. Forth, C. Probst, C. Staubach, A. Hlinak, A. Rudovsky, D. Holland, P. Schlieben, M. Goldner, J. Schatz, S. Bock, M. Fischer, K. Schulz, T. Homeier-Bachmann, R. Plagemann, U. Klaass, R. Marquart, T. C. Mettenleiter, M. Beer, F. J. Conraths, and S. Blome.  Joining the club: First detection of african swine fever in wild boar in germany.  Transbound Emerg Dis, 2020.
  * [13] C. Sauter-Louis, K. Schulz, M. Richter, C. Staubach, T. C. Mettenleiter, and F. J. Conraths.  African swine fever: Why the situation in Germany is not comparable to that in the Czech Republic or Belgium.  Transbound Emerg Dis, Jul 2021.
  * [14] H. Bergmann, K. Schulz, F. J. Conraths, and C. Sauter-Louis.  A Review of Environmental Risk Factors for African Swine Fever in European Wild Boar.  Animals (Basel), 11(9), Sep 2021.
  * [15] J. Carlson, M. Fischer, L. Zani, M. Eschbaumer, W. Fuchs, T. Mettenleiter, M. Beer, and S. Blome.  Stability of african swine fever virus in soil and options to mitigate the potential transmission risk.  Pathogens, 9(11), 2020.
  * [16] E. Chenais, K. Stahl, V. Guberti, and K. Depner.  Identification of wild boar-habitat epidemiologic cycle in african swine fever epizootic.  Emerg Infect Dis, 24(4):810–812, 2018.
  * [17] N. Mazur-Panasiuk, J. Zmudzki, and G. Wozniakowski.  African swine fever virus - persistence in different environmental conditions and the possibility of its indirect transmission.  J Vet Res, 63(3):303–310, 2019.
  * [18] J. Cukor, R. Linda, P. Vaclavek, P. Satran, K. Mahlerova, Z. Vacek, T. Kunca, and F. Havranek.  Wild boar deathbed choice in relation to asf: Are there any differences between positive and negative carcasses?  Preventive Veterinary Medicine, 177, 2020.
  * [19] A. Linden, A. Licoppe, R. Volpe, J. Paternostre, C. Lesenfants, D. Cassart, M. Garigliany, M. Tignon, T. van den Berg, D. Desmecht, and A. B. Cay.  Summer 2018: African swine fever virus hits north-western europe.  Transboundary and Emerging Diseases, 66(1):54–55, 2019.
  * [20] E. Chenais, K. Depner, V. Guberti, K. Dietze, A. Viltrop, and K. Stahl.  Epidemiological considerations on african swine fever in europe 2014-2018.  Porcine Health Manag, 5:6, 2019.
  * [21] A. Gogin, V. Gerasimov, A. Malogolovkin, and D. Kolbasov.  African swine fever in the North Caucasus region and the Russian Federation in years 2007-2012.  Virus Res, 173(1):198–203, Apr 2013.
  * [22] K. Schulz, E. Olsevskis, C. Staubach, K. Lamberga, M. Serzants, S. Cvetkova, F. J. Conraths, and C. Sauter-Louis.  Epidemiological evaluation of latvian control measures for african swine fever in wild boar on the basis of surveillance data.  Sci Rep, 9(1):4189, 2019.
  * [23] EFSA.  Epidemiological analyses of african swine fever in the european union (november 2017 until november 2018).  EFSA Journal, (11), 2018.
  * [24] T. Podgorski and K. Smietanka.  Do wild boar movements drive the spread of african swine fever?  Transbound Emerg Dis, 65(6):1588–1596, 2018.
  * [25] Brandon H Hayes, Mathieu Andraud, Luis G Salazar, Nicolas Rose, and Timothée Vergne.  Mechanistic modelling of african swine fever: A systematic review.  Preventive Veterinary Medicine, page 105358, 2021.
  * [26] Crispin Gardiner.  Stochastic methods, volume 4.  Springer Berlin, 2009.
  * [27] Ralf Metzler and Joseph Klafter.  The random walk’s guide to anomalous diffusion: a fractional dynamics approach.  Physics reports, 339(1):1–77, 2000.
  * [28] Joseph Klafter and Igor M Sokolov.  First steps in random walks: from tools to applications.  Oxford University Press, 2011.
  * [29] Dirk Brockmann, Lars Hufnagel, and Theo Geisel.  The scaling laws of human travel.  Nature, 439(7075):462–465, 2006.
  * [30] Rachel A Taylor, Tomasz Podgórski, Robin RL Simons, Sophie Ip, Paul Gale, Louise A Kelly, and Emma L Snary.  Predicting spread and effective control measures for african swine fever—should we blame the boars?  Transboundary and Emerging Diseases, 68(2):397–416, 2021.
  * [31] FI Korennoy, VM Gulenkin, JB Malone, CN Mores, SA Dudnikov, and MA Stevenson.  Spatio-temporal modeling of the african swine fever epidemic in the russian federation, 2007–2012.  Spatial and spatio-temporal epidemiology, 11:135–141, 2014.
  * [32] Mike B Barongo, Richard P Bishop, Eric M Fèvre, Darryn L Knobel, and Amos Ssematimba.  A mathematical model that simulates control options for african swine fever virus (asfv).  PloS one, 11(7):e0158658, 2016.
  * [33] Martin Lange and Hans-Hermann Thulke.  Elucidating transmission parameters of african swine fever through wild boar carcasses by combining spatio-temporal notification data and agent-based modelling.  Stochastic environmental research and risk assessment, 31(2):379–391, 2017.
  * [34] Simon Dellicour, Daniel Desmecht, Julien Paternostre, Céline Malengreaux, Alain Licoppe, Marius Gilbert, and Annick Linden.  Unravelling the dispersal dynamics and ecological drivers of the african swine fever outbreak in belgium.  Journal of Applied Ecology, 57(8):1619–1629, 2020.
  * [35] K Kroschewski, M Kramer, A Micklich, C Staubach, R Carmanns, and FJ Conraths.  Animal disease outbreak control: the use of crisis management tools.  Revue Scientifique et Technique-Office International des Epizooties, 25(1):211, 2006.
  * [36] European Food Safety Authority (EFSA), Daniel Desmecht, Guillaume Gerbier, Christian Gortázar Schmidt, Vilija Grigaliuniene, Georgina Helyes, Maria Kantere, Daniela Korytarova, Annick Linden, Aleksandra Miteva, et al.  Epidemiological analysis of african swine fever in the european union (september 2019 to august 2020).  EFSA Journal, 19(5):e06572, 2021.
  * [37] Yoeng-Jin Chu.  On the shortest arborescence of a directed graph.  Scientia Sinica, 14:1396–1400, 1965.
  * [38] Jack Edmonds.  Optimum branchings.  Journal of Research of the National Bureau of Standards, B, 71:233–240, 1967.
  * [39] Aric Hagberg, Pieter Swart, and Daniel S Chult.  Exploring network structure, dynamics, and function using networkx.  Technical report, Los Alamos National Lab.(LANL), Los Alamos, NM (United States), 2008.
  * [40] https://github.com/hartmutlentz
/RandomWalker2D, 2021.

  * [41] Carolina Probst, Jörn Gethmann, Jens Amendt, Lena Lutz, Jens Peter Teifke, and Franz J Conraths.  Estimating the postmortem interval of wild boar carcasses.  Veterinary sciences, 7(1):6, 2020.

##  10 Supplementary Information

###  10.1 Wave front velocities

We compare the estimated velocities found by simple linear regression as done
in the main text (Figure 2). The result is shown in Figure A.1. As expected,
the velocities are related to the diffusion coefficients for the considered
clusters, i.e. they show similar values, except Clusters 4 and 6.

Figure A.1: Velocities for all clusters estimated using linear regression of
the distances to the index case.

###  10.2 Mean squared displacements of Clusters 2–6

####  10.2.1 Cluster 2

Figure A.2 shows a realization of a random walk trained on the outbreak data.
The multiplicity of the process is 4.1.

Figure A.2: Real outbreak data vs. one realization of a random walk for
Cluster 2. The index case is set to coordinates (0, 0). Outbreak data from 265
days, random walk with multiplicity 4.1 resulting in 1087 steps.

We show the mean squared displacement in Figure A.3. The diffusion coefficient
is $D=(0.30\pm 0.01)\;km^{2}/day$.

Figure A.3: Mean squared displacement for Cluster 2. Mean over 10,000 random
walkers (red line). The resulting diffusion constant is $D=(0.30\pm
0.01)\;km^{2}/day$.

####  10.2.2 Cluster 3

Figure A.4 shows a realization of a random walk trained on the outbreak data.
The multiplicity of the process is 3.6.

Figure A.4: Real outbreak data vs. one realization of a random walk for
Cluster 3. The index case is set to coordinates (0, 0). Outbreak data from 245
days, random walk with multiplicity 3.6 resulting in 882 steps.

We show the mean squared displacement in Figure A.5. The diffusion coefficient
is $D=(0.50\pm 0.01)\;km^{2}/day$.

Figure A.5: Mean squared displacement for Cluster 3. Mean over 10,000 random
walkers (red line). The resulting diffusion constant is $D=(0.50\pm
0.01)\;km^{2}/day$.

####  10.2.3 Cluster 4

Figure A.6 shows a realization of a random walk trained on the outbreak data.
The multiplicity of the process is 4.7.

Figure A.6: Real outbreak data vs. one realization of a random walk for
Cluster 4. The index case is set to coordinates (0, 0). Outbreak data from 249
days, random walk with multiplicity 4.7 resulting in 1170 steps.

We show the mean squared displacement in Figure A.7. The diffusion coefficient
is $D=(1.69\pm 0.09)\;km^{2}/day$.

Figure A.7: Mean squared displacement for Cluster 4. Mean over 10,000 random
walkers (red line). The resulting diffusion constant is $D=(1.69\pm
0.09)\;km^{2}/day$.

####  10.2.4 Cluster 5

Figure A.8 shows a realization of a random walk trained on the outbreak data.
The multiplicity of the process is 3.0.

Figure A.8: Real outbreak data vs. one realization of a random walk for
Cluster 5. The index case is set to coordinates (0, 0). Outbreak data from 121
days, random walk with multiplicity 3.0 resulting in 363 steps.

We show the mean squared displacement in Figure A.9. The diffusion coefficient
is $D=(0.16\pm 0.01)\;km^{2}/day$.

Figure A.9: Mean squared displacement for Cluster 5. Mean over 10,000 random
walkers (red line). The resulting diffusion constant is $D=(0.16\pm
0.01)\;km^{2}/day$.

####  10.2.5 Cluster 6

Figure A.8 shows a realization of a random walk trained on the outbreak data.
The multiplicity of the process is 6.2.

Figure A.10: Real outbreak data vs. one realization of a random walk for
Cluster 6. The index case is set to coordinates (0, 0). Outbreak data from 127
days, random walk with multiplicity 6.2 resulting in 787 steps.

We show the mean squared displacement in Figure A.11. The diffusion
coefficient is $D=(1.54\pm 0.07)\;km^{2}/day$.

Figure A.11: Mean squared displacement for Cluster 6. Mean over 10,000 random
walkers (red line). The resulting diffusion constant is $D=(1.54\pm
0.07)\;km^{2}/day$.

####  10.2.6 Summary and discussion of Clusters 2–6

The Clusters 2, 3, and 5 show a behavior similar to Cluster 1 in the main
text. Their multiplicities are relatively low and most realizations of random
walks appear very similar to the real outbreak data.

Clusters 4 and 6 show remarkable differences between synthetic and real
outbreak data, as shown in Figures A.6 and A.10. The figures demonstrate that
the generated data points cover a larger area when compared to the more
compact outbreak data. This is caused by the fact that both clusters – and
Cluster 6 in particular – are strongly geographically constrained by data
being restricted to within the German country borders. In particularly,
Cluster 6 is located along the river Oder. This implies that a large
proportion of cases on the polish side is missing in the cluster.

Since the random walk model does not take into account such geographical
constraints, the random walkers move in all directions ignoring the
constraints. As a consequence, they cover a much larger area (take the eastern
regions in Figure A.10 as an example) and thus the diffusion constant is
magnified. Cluster 4 shows a similar behavior, even if to a weaker extent
(Figure A.6).

An additional bias in Cluster 6 is the high multiplicity of 6.2. This is the
highest value among all clusters and it causes a strong bias in the random
walk model, since the random walker has to cover more than 6 events occurring
in the data each day. However, the random walk assumption only holds for
multiplicities close to 1. Moreover, Cluster 6 is located in an urban area and
consequently this restriction did not allow for the same control measures as
in the other clusters.


its double discontinuity. In general, due to the Froissart-Martin bound, we
have to consider double-subtracted dispersion relations

$M(s,t,u)=c_{0}+\int_{4}^{\infty}\sigma(x)K_{1}(x;s,t,u)+\iint_{\mathcal{D}}\rho(x,y)K_{2}(x,y;s,t,u),$
(B.1)

where

$K_{1}=\frac{s-s_{0}}{(x-s)(x-s_{0})}+\frac{t-t_{0}}{(x-t)(x-t_{0})}+\frac{u-u_{0}}{(x-u)(x-u_{0})},$
(B.2)

and

$K_{2}=\frac{(s-s_{0})(t-t_{0})}{(x{-}s)(x{-}s_{0})(y{-}t)(y{-}t_{0})}+\frac{(s-s_{0})(u-u_{0})}{(x{-}s)(x{-}s_{0})(y{-}u)(y{-}u_{0})}+\frac{(t-t_{0})(u-u_{0})}{(x{-}t)(x{-}t_{0})(y{-}u)(y{-}u_{0})}.$
(B.3)

That is the reason why in the above representation depends on a subtraction
constant, and also on the single discontinuity.

The imaginary part in the $s$-channel of the amplitude can be read off by
using the standard identity
$\tfrac{1}{x-s-i\epsilon}=\mathcal{P}\frac{1}{x-s}-i\pi\delta(x-s)$ yielding

$\text{Im}\,M=\pi\sigma(s)+\pi\int_{\bar{y}(s)}^{\infty}\rho(s,y)\left(\frac{t-t_{0}}{(y-t)(y-t_{0})}+\frac{u-u_{0}}{(y-u)(y-u_{0})}\right)dy,$
(B.4)

where $\bar{y}(s)$ is the starting point of the support of the double
discontinuity$\rho(s,y)$ at fixed-$s$ (for physical amplitudes is a point on
the Karplus curve). Projecting this equation into partial waves we obtain the
expression

$\text{Im}\,f_{J}(s)=\frac{\sigma(s)}{16}\delta_{J,0}+\frac{1}{32}\int_{\bar{y}(s)}^{\infty}\rho(s,y)\left(\frac{8}{s-4}Q_{J}(1+\tfrac{2y}{s-4})-2\delta_{J,0}(\tfrac{1}{y-t_{0}}+\tfrac{1}{y-u_{0}})\right)dy,$
(B.5)

where the $Q_{J}$ are the usual Legendre functions of second kind such that
$\tfrac{1}{2i}\text{disc}_{x}Q_{J}(x)=P_{J}(x)$ for $-1<x<1$. The above
relation for $J=0$ allows us to replace the single discontinuity $\sigma$ with

$\sigma(s)=16\text{Im}\,f_{0}(s)-\int_{\bar{y}(s)}^{\infty}\rho(s,y)\left(\frac{4}{s-4}Q_{0}(1+\tfrac{2y}{s-4})-(\tfrac{1}{y-t_{0}}+\tfrac{1}{y-u_{0}})\right)dy.$
(B.6)

The higher spins, on the other hand, depend only on the double discontinuity

$\text{Im}\,f_{J\geqslant
2}(s)=\frac{1}{4}\int_{\bar{y}(s)}^{\infty}\rho(s,y)\frac{Q_{J}(1+\tfrac{2y}{s-4})}{s-4}dy\equiv\int_{\bar{y}(s)}^{\infty}\rho(s,y)K_{J}^{I}(s,y)dy.$
(B.7)

For completeness we write down also the projection onto partial waves of the
real part of the amplitude

$\text{Re}\,f_{J}(s)=\frac{c_{0}}{16\pi}\delta_{J,0}+\int_{4}^{\infty}\text{Im}\,f_{0}(x)\mathcal{K}_{J}^{1}(x;s)dx+\iint_{\mathcal{D}}\rho(x,y)\mathcal{K}^{2}_{J}(x,y;s)dxdy,$
(B.8)

where

$\mathcal{K}_{J}^{1}(x;s)=\frac{\delta_{J,0}}{\pi}\left(\frac{1}{x-s}-\frac{1}{x-s_{0}}-\frac{1}{x-t_{0}}-\frac{1}{x-u_{0}}\right)+\frac{4}{\pi(s-4)}Q_{J}(1+\tfrac{2x}{s-4}),$
(B.9)

and

$\displaystyle\mathcal{K}_{J}^{2}(x,y;s)=\frac{1}{32\pi}\frac{s-s_{0}}{(x{-}s)(x{-}s_{0})}\left(\frac{8}{s-4}Q_{J}(1+\tfrac{2y}{s-4})+2\delta_{J,0}\left(\frac{1}{(x{-}t_{0})(y{-}u_{0})}-\frac{1}{y{-}t_{0}}-\frac{1}{y{-}u_{0}}\right)\right)$
$\displaystyle+\frac{1}{32\pi}\frac{4}{s-4}\left(\frac{1}{s{-}4{+}x{+}y}(Q_{J}(1+\tfrac{2x}{s-4})+Q_{J}(1+\tfrac{2y}{s-4}))-\frac{1}{x{-}t_{0}}Q_{J}(1+\tfrac{2y}{s-4})-\frac{1}{y{-}u_{0}}Q_{J}(1+\tfrac{2x}{s-4})\right)$
$\displaystyle-\frac{1}{32\pi}\left(\frac{4}{x-4}Q_{0}(1+\tfrac{2y}{x-4})-\frac{1}{y-t_{0}}-\frac{1}{y-u_{0}}\right)(2\pi\mathcal{K}_{J}^{1}(x;s)).$
(B.10)

All in all, the double-subtracted Mandelstam representation depending only on
the spin zero imaginary part and the double discontinuity takes the form

$\displaystyle
M(s,t,u)=c_{0}+16\int_{4}^{\infty}\text{Im}\,f_{0}(x)K_{1}(x;s,t,u)+$
$\displaystyle+\iint_{\mathcal{D}}\rho(x,y)\left(K_{2}(x,y;s,t,u)-\left(\frac{4}{x-4}Q_{0}(1+\tfrac{2y}{x-4})-(\tfrac{1}{y-t_{0}}+\tfrac{1}{y-u_{0}}\right)\right)dxdy.$
(B.11)

##  Appendix C $g_{0}\phi^{4}$ theory perturbatively

It is possible to compute the imaginary part of the amplitude, $\text{Im }M$,
up to $O(g_{0}^{3})$ perturbatively in $g_{0}\phi^{4}$ theory. In this section
we are going to review this computation. Finally we extract contributions into
$c_{2}$ and $c_{3}$ coefficients up to $O(g_{0}^{3})$.

###  One-loop contributions

Remember that the one-loop imaginary part of the amplitude can be obtained by
a phase space integration of tree-level amplitudes. At leading order, it is
given by

$\displaystyle M_{z}(z,t)$
$\displaystyle=\frac{1}{2}\frac{g_{0}^{2}}{16\pi}\sqrt{\frac{z-4}{z}}\,,$
(C.1)

in $m^{2}=1$ units. Plugging this into dispersion integrals (2.23) and (2.24)
enables us to compute the contributions to the coefficients $c_{2}$ and
$c_{3}$ up to $O(g_{0}^{2})$.

$\displaystyle c_{2}^{O(g_{0}^{2})}$
$\displaystyle=\frac{9g_{0}^{2}}{8192\pi^{2}}\left(14-15\sqrt{2}\,\text{arctan}\frac{1}{\sqrt{2}}\right)\quad\approx
1.05\cdot 10^{-4}\,g_{0}^{2}\,,$ (C.2) $\displaystyle c_{3}^{O(g_{0}^{2})}$
$\displaystyle=\frac{27g_{0}^{2}}{131072\pi^{2}}\left(-130+153\sqrt{2}\,\text{arctan}\frac{1}{\sqrt{2}}\right)\quad\approx
0.66\cdot 10^{-4}\,g_{0}^{2}\,.$ (C.3)

We can also calulate the ratio $c_{3}/c_{2}$ through these numbers, giving us
(3.10).

###  Two-loop contributions

Analogously, the two-loop imaginary part can be obtained by a phase space
integration involving tree-level and one-loop amplitudues. For this reason, we
need not just the imaginary part, but also the full one-loop amplitude
$M_{\text{1-loop}}(s,t)$.

Fortunately, fixed-$t$ dispersion relation (2.19) enables us to obtain the
full amplitude from the knowledge of one-loop discontinuity and a subtraction
constant $M(s_{0},t_{0})$. Plugging in (C.1) and choosing $M(4/3,4/3)=-g_{0}$
gives us

$\displaystyle M_{\text{1-loop}}(s,t)$
$\displaystyle=\frac{g_{0}^{2}}{16\pi^{2}}\Big{[}f(s)+f(t)+f(4-s-t)\Big{]}$
(C.4) $\displaystyle\text{where}\quad f(s)$
$\displaystyle=\left(\sqrt{2}\,\arctan\frac{1}{\sqrt{2}}-\sqrt{\frac{4-s}{s}}\,\arctan\sqrt{\frac{s}{4-s}}\right)\,.$
(C.5)

A separate computation with the Feynman diagrams confirms above result.

Notice that $f(s>4)$ has a non-zero imaginary part as expected and
$M_{\text{1-loop}}(4/3,4/3)=0$ such that constant piece of the total amplitude
corresponds to $-g_{0}$.

Now it is time to use the unitarity equation to get the two-loop imaginary
part. We consider the two particle unitarity cuts of two-loop Feynman
diagrams343434There exists a three particle cut diagram, but its support on
the phase space for physical external particles is zero and therefore we don’t
consider it. and we express the equation in terms of partial-wave coefficients
$f_{\ell}$

$\displaystyle 2\,\text{Im}\,M(s,t)$
$\displaystyle=\sum_{\ell}n_{\ell}^{(4)}\text{Im}\,f_{\ell}(s)P_{\ell}(1+\tfrac{2t}{s-4})=\rho^{2}(s)\,\sum_{\ell}n_{\ell}^{(4)}\,|f_{\ell}(s)|^{2}P_{\ell}(1+\tfrac{2t}{s-4})+O(g_{0}^{4})$
$\displaystyle=\rho^{2}(s)\left(\frac{g_{0}^{2}}{16\pi}-g_{0}\,2\text{Re}f^{\text{1-loop}}_{0}+O(g_{0}^{4})\right)$
(C.6)

where $\rho^{2}(s)=\sqrt{s-4}/\sqrt{s}$ is two-particle phase space factor,
and $n_{\ell}^{(4)}=16\pi(2\ell+1)$. The first term gives us the leading
contribution as mentioned in (C.1), and the second term is what we need at
$O(g_{0}^{3})$, so plug in the following:

$\displaystyle\text{Re
}f^{\text{1-loop}}_{0}=\frac{1}{32\pi}\int_{-1}^{1}dx\,\frac{g_{0}^{2}}{16\pi^{2}}\Big{[}\text{Re
}f(s)+f(t^{\prime})+f(u^{\prime})\Big{]}$ (C.7) $\displaystyle
t^{\prime}=(s-4)(-1+x)/2\quad,\quad\quad u^{\prime}=(s-4)(-1-x)/2$ (C.8)

Notice that $f(t^{\prime})$ and $f(u^{\prime})$ are purely real functions in
the region $s>4$, which is the same interval as in the dispersion integrals
for $c_{2}$ and $c_{3}$. For the real part of the first term, we used the
following nice formula: $\arctan y=1/2i\log(1+iy)/(1-iy)$.
After analytically obtaining $\text{Im }M$ given above, we can plug it in the
dispersion integrals (2.23) and (2.24), then evaluate them numerically. This
will give us

$c_{2}^{O(g_{0}^{3})}\approx-1.53\cdot 10^{-7}\,g_{0}^{3}\quad,\quad\quad
c_{3}^{O(g_{0}^{3})}\approx-3.09\cdot 10^{-7}\,g_{0}^{3}\ .$ (C.9)

##  Appendix D Subtracted dispersion relations for the $O(n)$ theory

In this appendix we provide further details for the derivation of the sum rule
(5.6). The starting point the the fixed-$t$ doubly-subtracted dispersion
relation of Roy [73] adapted to global $O(n)$ symmetry. After blowing up the
contour $\frac{s^{2}}{2\pi i}\oint
dz\frac{1}{z^{2}}\frac{1}{z-s}\overrightarrow{M}(z,t)$, we are lead to

$\overrightarrow{M}(s,t)=C_{st}.\left[\vec{c}(t)+(s-u)\vec{d}(t)\right]+\frac{1}{\pi}\int_{4}^{\infty}dzK(z;s,u).\overrightarrow{M}_{z}(z,t)$
(D.1)

where the kernel is a matrix given by

$K(z;s,u)=\frac{1}{z^{2}}\left(\frac{s^{2}}{z-s}\mathds{1}+\frac{u^{2}}{z-u}C_{su}\right)\,.$
(D.2)

and recall that we are working in $m^{2}=1$ units, and that
$M_{z}(z,t)\equiv\text{disc}_{z}M(z,t)/(2i)$. The vector
$\overrightarrow{M}=(M^{\text{sing}},M^{\text{sym}},M^{\text{anti}})$ is given
by the irreps

$\displaystyle M^{\text{sing}}$
$\displaystyle=n\,M(\bar{s}|\bar{t},\bar{u})+M(\bar{t}|\bar{u},\bar{s})+M(\bar{u}|\bar{s},\bar{t})\,,$
(D.3) $\displaystyle M^{\text{sym}}$
$\displaystyle=M(\bar{t}|\bar{u},\bar{s})+M(\bar{u}|\bar{s},\bar{t})\,,$ (D.4)
$\displaystyle M^{\text{anti }}$
$\displaystyle=M(\bar{t}|\bar{u},\bar{s})-M(\bar{u}|\bar{s},\bar{t})\,.$ (D.5)

The crossing matrices are given by

$C_{st}=\left(\begin{array}[]{ccc}\frac{1}{n}&\frac{n^{2}+n-2}{2n}&\frac{n-1}{2}\\\
\frac{1}{n}&\frac{1}{2}-\frac{1}{n}&-\frac{1}{2}\\\
\frac{1}{n}&-\frac{n+2}{2n}&\frac{1}{2}\\\ \end{array}\right)\quad,\quad\quad
C_{su}=\left(\begin{array}[]{ccc}\frac{1}{n}&\frac{n^{2}+n-2}{2n}&\frac{1-n}{2}\\\
\frac{1}{n}&\frac{1}{2}-\frac{1}{n}&\frac{1}{2}\\\
-\frac{1}{n}&\frac{1}{n}+\frac{1}{2}&\frac{1}{2}\\\ \end{array}\right)\,,$
(D.6)

and satisfy $C_{st}^{2}=C_{su}^{2}=1$. The dispersion relation in (D.1) is
composed of the integral over the discontinuity $\overrightarrow{M}_{z}(z,t)$
and the subtraction functions

$\vec{c}(t)=\left(\begin{matrix}c^{\text{sing}}(t)\\\ c^{\text{sym}}(t)\\\
0\end{matrix}\right)\quad,\quad\quad\vec{d}(t)=\left(\begin{matrix}0\\\ 0\\\
d^{\text{anti}}(t)\end{matrix}\right)\,.$ (D.7)

This form of the vectors are dictated by the explicit $s\leftrightarrow u$
symmetry property of (D.1).

Next we would like to eliminate these subtraction functions in (D.1) in favour
of the amplitude $\overrightarrow{M}(s_{0},t_{0})$ and the integrals over the
discontinuities $\overrightarrow{M}_{z}(z,t)$. In order to do so we follow
Roy’s strategy [73]. Using first crossing-symmetry
$\overrightarrow{M}(t_{0},t)=C_{st}.\overrightarrow{M}(t,t_{0})$ we have

$C_{st}.\left[\vec{c}(t)+(2t_{0}-4+t)\,\vec{d}(t)\right]=\left[\vec{c}(t_{0})+(2t-4+t_{0})\,\vec{d}(t_{0})\right]+\int\text{\emph{absorptive
pieces}}$ (D.8)

the second term schematically denotes an integral of the absortive part of the
amplitude $M_{z}(z,t)$ or $M_{z}(z,t_{0})$ against various kernels. The second
piece $\vec{c}(t_{0})+(-4+2t+t_{0})\vec{d}(t_{0})$ can be expressed in terms
of absorptive pieces and $\vec{M}(s_{0},t_{0})$ using (D.1)

$\vec{c}(t_{0})+(t_{0}-4+2s_{0})\vec{d}(t_{0})=\overrightarrow{M}(t_{0},s_{0})+\int\text{\emph{absorptive
pieces}}$ (D.9)

Because the components of the vectors (D.7) do not mix we can easily solve for
them. Namely, we plug the last equation into (D.8) leading to
$c^{\text{sing}},c^{\text{sym}},d^{\text{anti}}\sim\overrightarrow{M}(t_{0},s_{0})+\int\text{\emph{absorptive
pieces}}$. Plugging this solution in (D.1) we arrive to a dispersion relation
expressed in terms of an arbitrary subtraction point $M(s_{0},t_{0})$, i.e.
$\overrightarrow{M}(s,t)\sim
C^{\prime}.\overrightarrow{M}(s_{0},t_{0})+\int\text{\emph{absorptive
pieces}}$, where $C^{\prime}(s,t;s_{0},t_{0})$ is a matrix easy to determine.

Having outlined the logic, it is now a matter of algebra to derive (5.6) from
$\vec{M}(s,t)\sim
C^{\prime}.\overrightarrow{M}(s_{0},t_{0})+\int\text{\emph{absorptive
pieces}}$ and
$c_{H}=\frac{1}{(n-1)}\frac{\partial}{\partial\bar{s}}M^{\text{sing}}(\bar{s},\bar{t})\big{|}_{\bar{s}=\bar{t}=0}$.
We find that the vectors and integration kernels in (5.6) are given by

$\displaystyle C(s,t)$
$\displaystyle=\frac{1}{2n}\frac{(3s-4)}{(3t-4)}\frac{1}{(s-u)}\left(-1\,,\,\frac{n+2}{2}\,,\,\frac{n(9s+6t-20)}{2(3s-4)}\right)\,,$
(D.10) $\displaystyle D_{1}(z,s,t)$
$\displaystyle=\frac{1}{2n}\frac{(3s-4)}{(3t-4)}\frac{(3s+3t-8)}{(3t+3z-8)}\frac{1}{(z-s)(z-u)}\,\left(1,\frac{-n-2}{2},-\frac{n(6t+9z-20)}{2(3z-4)}\right),$
(D.11) $\displaystyle D_{2}(z,t)$
$\displaystyle=\frac{3}{n}\frac{(3t-4)^{2}}{(3z-4)^{2}}\frac{1}{(t-z)(3t+3z-8)}\left(1\,,\,\frac{-n-2}{2}\,,\,\frac{n}{2}\right)\,.$
(D.12)

In terms of crossing-symmetry matrices we find that the $c_{H}$ sum rule is
given by following vector-like dispersion relation

$c_{H}\left(\begin{matrix}n-1\\\ -1\\\
1\end{matrix}\right)=F_{2}(t).F_{1}(s,t).\overrightarrow{M}(s,t)+\int_{4}^{\infty}\frac{dz}{\pi}\left[K_{1}(z,t).\overrightarrow{M}_{z}(z,t)+K_{2}(z,t).\overrightarrow{M}_{z}(z,4/3)\right]\
,$ (D.13)

where

$\displaystyle K_{1}(z,s,t)$
$\displaystyle=F_{2}(t).\left[K(z;4/3,8/3-t)-F_{1}(s,t).K(z;4-s-t)\right]\,,$
$\displaystyle K_{2}(z,t)$
$\displaystyle=K^{\prime}(z)-F_{2}(t).C_{st}.K(z;t,8/3-t)\,,$

and $K^{\prime}(z)=\partial_{s}K(z;s,4-t-s)\big{|}_{{t,s=4/3}}$ and

$F_{1}(s,t)=\mathds{1}+\frac{4/3-s}{s-u}(\mathds{1}-C_{su})\quad,\quad\quad
F_{2}(t)=\frac{1}{2}\frac{1}{t-4/3}C_{st}.(\mathds{1}-C_{su})\,.$ (D.14)

##  Appendix E A toy model

In this section we develop the ideas presented in the main text in a
simplified setting where we scatter massive particles in a line, i.e. in
$d=1+1$ spacetime dimensions. The theory is invariant under a spacetime parity
transformation and there is no scattering angle, therefore there is a single
kinematical invariant given by the Mandelstam variable $s=(p_{1}+p_{2})^{2}$.
We assume a $Z_{2}$ symmetry under which the particle we scatter is odd, thus
forbidding the triple vertex interaction.

The analytic structure of $M$ is sketched in Fig. 20. Crossing symmetry
implies $M(s-2m^{2})=M(2m^{2}-s)$, and there is a unitary branch-cut starting
at $s=(2m)^{2}$ (and its crossed starting at $s=0)$ and extending all the way
to $s=\infty$ (and $s=-\infty)$. The scattering matrix element $M$ is assumed
to be analytic everywhere in the complex plane away from the real axis and in
the segment $s\in(0,4m^{2})$. In particular, it is analytic in the crossing-
symmetric point $s-2m^{2}=0$, and thus we can represent the amplitude in terms
of a series around this point. Due to crossing-symmetry, this series involves
even powers of $s-2m^{2}$ only. It is convenient to define

$2m^{2}\bar{s}\equiv
s-2m^{2}\quad\text{and}\quad\epsilon\equiv\frac{2m^{2}}{\Lambda^{2}}\,,$ (E.1)

with $\Lambda$ an additional scale. With these two definitions, we can
represent the scattering amplitude by

$M(\bar{s})=m^{2}\left(-c_{0}+c_{2}(\epsilon\bar{s})^{2}+c_{4}(\epsilon\bar{s})^{4}+c_{6}(\epsilon\bar{s})^{6}+\dots\right)\,.$
(E.2)

where we have factored out $m^{2}$ to account for the dimensions – recall that
the two-to-two scattering amplitude has energy dimension $4-d$. The
coefficients $c_{i}$ are in general functions $c_{i}(\epsilon)$; and we have
factored out a suitable powers of $\epsilon$ to match the power of $\bar{s}$,
$(\epsilon\bar{s})^{n}$. As we show below, this power counting is useful in
order to interpret the amplitude (E.2) in terms of an Effective Field Theory.

Figure 20:  Sketch of the analytical structure of the two-dimensional
amplitude.

The S-matrix 353535The S-matrix operator is given by
$\hat{S}(s)=\mathds{1}+(2\pi)^{d}\delta^{(d)}(p_{1}+p_{2}-p_{3}-p_{4})M(s)$,
where as usual $\mathds{1}=\langle
p_{1},p_{2}|p_{3},p_{4}\rangle=\prod_{i=1,2}((2\pi)^{(d-1)}2E_{i})(\delta^{(d-1)}(\vec{k}_{1}-\vec{k}_{3})\delta^{(d-1)}(\vec{k}_{2}-\vec{k}_{4})+[\vec{k}_{1}\leftrightarrow\vec{k}_{2}])$.
In $d=2$, using the relation
$\delta(x)\delta(y)=\delta([x+y]/2)\delta([x-y]/2)$, we can relate the total
momentum conservation delta function $\delta^{(d)}(p_{1}+p_{2}-p_{3}-p_{4})$
to the identity operator $\mathds{1}$ and obtain
$\hat{S}(s)=\mathds{1}S(s)=\mathds{1}(1+\frac{1}{2\sqrt{s}\sqrt{s-4m^{2}}}iM(s))$.
is given by
$S(\bar{s})=1+\frac{1}{4\sqrt{1-\bar{s}^{2}}}\left[\,-c_{0}+\sum^{\infty}_{i=1}c_{2i}(\epsilon\bar{s})^{2i}\,\right]$.
In order to encode the analytic structure of $M(s)$ it is convenient to map
the complex $s$-plane into the unit disk using the following map

$\rho(\bar{s})=\frac{i-\sqrt{\bar{s}+1}\sqrt{\bar{s}-1}}{i+\sqrt{\bar{s}+1}\sqrt{\bar{s}-1}}\,.$
(E.3)

The map is crossing-symmetric ($\bar{s}\rightarrow-\bar{s}$), and thus maps
only the upper half plane into the unit disk. This transformation opens the
$s$-plane branch cuts and map them into the boundary of the unit disk, it has
the fixed point $\rho(\bar{s}=0)=0$, and the thresholds of physical scattering
energy map to $\rho(\bar{s}=\pm 1)=1$. Now, upon expanding the S-matrix in the
unit disk around $\rho=0$ we have

$\displaystyle S(\rho)$
$\displaystyle=(1-c_{0}/4)+\left(c_{2}\epsilon^{2}-c_{0}/2\right)\rho+\left(4c_{4}\epsilon^{4}-c_{0}/2\right)\rho^{2}$
(E.4)
$\displaystyle+\left(c_{2}\epsilon^{2}-8c_{4}\epsilon^{4}+16c_{6}\epsilon^{6}-c_{0}/2\right)\rho^{3}+\mathcal{O}(\rho^{4})\,.$

Note that the expansion in (E.4) has even and odd powers in $\rho$, because we
have solved for crossing-symmetry through the map (E.3). Unitarity of the
S-matrix implies

$|S|^{2}\leqslant 1$ (E.5)

for physical energies $s\in[4m^{2},\infty)$; or equivalently in the
$\rho$-plane for $\arg\rho\in[0,\pi)$ and $|\rho|=1$.

The amplitude $M(s)$ looks like a higher dimensional amplitude in the forward
limit ($t=0$). This is true regarding the analytic and crossing-symmetry
properties of $M$. However, unitarity is much simpler in the $d=1+1$ setting,
see (E.5). The large simplification of this toy model can be summarised by
noting that the system of equations we are studying is as if we had a single
partial wave (E.5). In higher dimensions the presence of infinitely many
partial waves 363636Often defined by projecting with Legendre polynomials
$P_{\ell}$, $S_{\ell}\equiv
1+i\sqrt{\frac{s-4}{s}}\int_{-1}^{1}dxP_{\ell}(x)M(s,t)|_{t\rightarrow(s/2-2)(x-1)}$.
makes crossing symmetry and the corresponding unitarity equations –
$|S_{\ell}(s)|^{2}\leqslant 1$ with $\ell=0,1,2,\dots$ – much harder to
analyse.

###  Bounds on LECs

In this appendix, we shall call the parameters $c_{i}$ in (E.4) _low energy
constants_ (LECs). Clearly, in perturbation theory the LECs are identified
with couplings and Wilson coefficients of a putative Effective Field Theory
Lagrangian, we comment more on this identification later on in section E.2.

The low energy couplings $c_{i}$ in (E.4) can be optimally bounded as in ref.
[39]. 373737We emphasise that ref. [39] considered the EFT of massless
goldstones (associated to transverse fluctuation of a relativistic flux tube)
while here we are interested in the EFT of massive particles. The idea is to
exploit the nice rigidly-smooth shape that holomorphic functions have, which
we will now briefly review.

The function $S(\rho)$ is holomorphic in the unit disk $\rho\in\mathds{D}$,
therefore it is equal to the average of neighbouring points
$S(z)=\oint_{|w-z|=\epsilon}\frac{dw}{2\pi
i}\frac{S(w)}{z-w}=\int_{0}^{2\pi}\frac{d\theta}{2\pi}S(z+\epsilon
e^{i\theta})$, for $z\in\mathds{D}$. Thus, the modulus $|S(z)|$ is bounded in
the region of analyticity by the value of the function at the boundary
$\partial\mathds{D}$. This is the content of the maximum modulus principle. In
our case, the modulus of $S(z)$ is bounded by one at the boundary of the unit
disk (E.5) and therefore $S(z)$ is bounded everywhere inside the disk by one.
In particular it implies

$|S(\rho=0)|=|1-c_{0}/4|\leqslant 1\,.$ (E.6)

for our S-matrix (E.4). The bound in (E.6) is often interpreted as a bound on
the maximal coupling, defined precisely as the value of $M$ at the crossing-
symmetric point. A similar analysis can be done in the presence of bound
states, in order to constraint the maximal residue at the pole [82, 36].

It turns out that the derivatives $|\partial_{\rho}^{n}S(\rho)|_{\rho=0}$ are
also also bounded as consequence of the Schwarz-Pick multi-point lemmas. The
basic logic goes as follows. Consider the function of $z$

$S^{[1]}(z|w)=\frac{S(z)-S(w)}{1-S^{*}(w)S(z)}\left(\frac{z-w}{1-w^{*}z}\right)^{-1}$
(E.7)

with $w\in\mathds{D}$. Clearly, $S^{[1]}(z|w)$ is an holomorphic function of
$z\in\mathds{D}$. As a function of $z$ (and for $w\in\mathds{D}$), it is
bounded $|S^{[1]}(e^{i\phi}|w)|^{2}\leqslant 1$ in the boundary
$\partial\mathds{D}$. Therefore we can apply to maximal modulus principle, and
conclude $|S^{[1]}(z|w)|^{2}\leqslant 1$ for $z\in\mathds{D}$. Upon taking the
limit $w\rightarrow z\rightarrow 0$ on the last inequality we get

$\Big{|}\,\frac{c_{2}\epsilon^{2}-c_{0}/2}{1-(1-c_{0}/4)^{2}}\,\Big{|}\leqslant
1\,,$ (E.8)

for the S-matrix in (E.4). Clearly one can now recurse over this construction
and define the function $S^{[n]}(z|w_{1},\dots,w_{n})$ out of
$S^{[n-1]}(z|w_{1},\dots,w_{n-1})$, which is an holomorphic function of
$z\in\mathds{D}$ (with $w_{i}\in\mathds{D}$) and bounded in the boundary
$|S^{[n-1]}(e^{i\phi}|w_{1},\dots,w_{n-1})|\leqslant 1$. The content of the
multi-point Schwarz-Pick lemma is the bound on $|S^{[n]}|$, which implies a
bound on the $n$-th derivative of $S(z)$. See section E.3 for a summary of the
Schwarz-Pick formulas we use in this section. Further details on the theory of
Schwarz-Pick applied to S-matrices can be found in ref. [39].

In figure 21, left plot, we show the allowed region of the first two LECs
$(c_{0},c_{2}\epsilon^{2})$. The boundary of the blue region is described by
two parabolas and saturates (E.8). At the cusps (0,0) and (8,4) both of
inequalities (E.6) and (E.8) are saturated. The boundary in figure 21 is
saturated by the LECs of the following functions

$S_{\text{up}}(\rho,\rho_{0})=\frac{\rho-\rho_{0}}{1-\rho\rho_{0}}\quad\text{and}\quad
S_{\text{low}}(\rho,\rho_{0})=-\frac{\rho+\rho_{0}}{1+\rho\rho_{0}},$ (E.9)

with $\rho_{0}\in[-1,1]$, parametrising the upper and lower branch
respectively. 383838 These functions are called CDD-factors (for Castillejo-
Dalitz-Dyson) in the two-dimensional S-matrix literature or Blaschke products
in the context of the Schwarz-Pick theorems.  Indeed, after matching the
lowest LEC of our S-matrix (E.4) with $\rho_{0}=(c_{0}/4-1)$, we have

$S_{\text{up}}(\rho,c_{0}/4-1)=\left(1-c_{0}/4\right)+\big{(}\underbrace{1-\left(c_{0}/4-1\right)^{2}+c_{0}/2}_{c_{2}\epsilon^{2}}-c_{0}/2\big{)}\rho+O(\rho^{2})\,,$
(E.10)

and similarly for the lower branch. The upper branch has a resonance at
$S_{\text{up}}(\rho_{0},\rho_{0})=0$, while the lower branch at
$S_{\text{low}}(-\rho_{0},\rho_{0})=0$. 393939See for instance ref. [48] for a
discussion of resonances.  Varying $\rho_{0}$ parametrises the edges
connecting the two cusps along the upper and lower branch. The cusp at the
origin $S_{\text{up/low}}(\rho,-1)=1$ is a free bosonic theory, and the other
one $S_{\text{up/low}}(\rho,1)=-1$ is a free fermionic theory.

Figure 21: Left plot: In blue the allowed region in the
$c_{0}-c_{2}\epsilon^{2}$ plane. In orange and green we plot the values of the
$T\bar{T}$ S-matrix in (E.12) and the _Roaming Trajectories_ S-matrix,
respectively, as they approach the EFTs of a massive free boson and massive
free fermion on the lower and upper cusp. Right plot: Regions of allowed
values for $\\{c_{4}\epsilon^{4},\,c_{6}\epsilon^{6}\\}$.

Approaching the lower cusp at $(0,0)$ through the upper branch can be
described by an Effective Field Theory of a single massive scalar because, for
a small positive $\delta$,

$S_{\text{up}}(0,\delta-1)=1-\delta+O(\delta)\,,$ (E.11)

the resonance of $S_{\text{up}}$ is at $\rho=\delta-1$, which corresponds to a
resonance at $s=\infty$ as $\delta\rightarrow 0$. If instead we approach the
cusp at the origin from the lower branch
$S_{\text{low}}(0,\delta-1)=1+O(\delta)$, the resonance is at $\rho=1-\delta$,
thus at threshold $\rho(\hat{s}=1)=1$ as $\delta\rightarrow 1$. A particular
Effective Field Theory example touching the cusp at the origin is provided by
the $T\bar{T}$ deformation of a free massive boson [83, 84, 85]. In terms of
the $\rho$ variable, the S-matrix is given by

$S(\rho)=e^{-\frac{\epsilon}{4}\sqrt{\frac{(1-\rho)^{2}}{(1+\rho)^{2}}}}\,.$
(E.12)

The coefficients $c_{0}(\epsilon)$ and $c_{2}(\epsilon)\epsilon^{2}$ can be
read from the last equation after performing the Taylor series around $\rho=0$
and matching with equation (E.4). In figure 21, left, we plot with a dashed
orange curve the values of $\\{c_{0}(\epsilon),c_{2}(\epsilon)\epsilon^{2}\\}$
as we vary the parameter $\epsilon\in[0,\infty)$. When the cutoff is increased
(i.e. $\epsilon$ decreased) the values of
$\\{c_{0}(\epsilon),c_{2}(\epsilon)\epsilon^{2}\\}$ get close to the origin
monotonically.

A similar analysis holds for the upper cusp (8,4): if approached through the
lower branch it is described by an EFT of a massive fermion, while if
approached from the upper branch the scattering of the almost free fermions
features a resonance at threshold $\hat{s}=(s-2m^{2})/(2m^{2})=1$. We can
identify a theory approaching the $(8,4)$ cusp from the lower branch: the
_Roaming Trajectories_ S-matrix [86], which is in fact given by
$S_{\text{up}}(\rho,\rho_{0})$ with $\rho_{0}$ restricted to the range
$\rho_{0}\in(-1,0]$. For these values of the parameter $\rho_{0}$, the
S-matrix features a resonance at purely imaginary values, and thus $\rho_{0}$
increases monotonically as $\epsilon$ decreases. In figure 21, left, we plot
with a dashed green curve the values of
$\\{c_{0}(\epsilon),c_{2}(\epsilon)\epsilon^{2}\\}$ for this S-matrix as
$\epsilon$ is varied. The values $\rho_{0}\in[0,1)$ corresponds instead to a
resonance in the range $[0,4m^{2})$ in the Mandelstam $s$-plane, and thus it
is identified with the Sinh-Gordon S-matrix.

In the right plot of figure 21, we show allowed values of
$\\{c_{4}\epsilon^{4},c_{6}\epsilon^{6}\\}$ for fixed values of $c_{0}$ and
$c_{2}\epsilon^{2}$. Each region A, B and C of the right plot corresponds to
the points A, B and C shown on the left plot. As we approach the boundary of
the $c_{0}-c_{2}\epsilon^{2}$ allowed region, i.e. as we move along the line
$C\rightarrow B\rightarrow A$, the region shrinks, and reduces to a point when
hitting the boundary. At the boundary of the allowed region in the
$c_{0}-c_{2}\epsilon^{2}$ plane, the values of
$\\{c_{4}\epsilon^{4},c_{6}\epsilon^{6}\\}$ are fixed by the S-matrices in
(E.9). These values are shown by the gray and black curve, which correspond to
$S_{\text{low}}$ and $S_{\text{up}}$, respectively.

###  Positivity constraints and the space of Effective Field Theories

Next we compare the optimal bounds on the LECs just obtained with positivity.
We use the so-called arc variables [20]

$a_{n}(x)\equiv\frac{1}{m^{2}}\frac{2}{\pi}\int_{x}^{\infty}du\frac{\text{Im}M(u)}{(2m^{2}u)^{2n+3}}\quad,\quad
n\geqslant 0\,,$ (E.13)

and define $a_{n}\equiv a_{n}(1)$. The following optimal constraints are
satisfied

$\left(\begin{array}[]{cc}a_{0}&a_{1}\\\
a_{1}&a_{2}\end{array}\right)\geqslant 0\,,\ \ \ a_{1}\geqslant 0\,,\ \ \
a_{0}\geqslant\hat{s}^{2}a_{1}\,,\ \ \ a_{1}\geqslant\hat{s}^{2}a_{2}\,,$
(E.14)

as a consequence of _positivity_

$\text{Im}M\geqslant 0\,.$ (E.15)

The last equation follows from the $2\rightarrow 2$ S-matrix unitarity
equation (E.5). Next we modify the contour of integration and relate the arc
variables to the LECs

$a_{n}=c_{2n+2}(\epsilon/(2m^{2}))^{2n+2}$ (E.16)

where recall that $\epsilon/(2m^{2})=1/\Lambda^{2}$. In $d=3+1$ the
convergence of (E.13) at high energies is justified thanks to the Froissart-
Martin bound, proved for theories with a mass gap. In the set up we are
discussing, $d=1+1$, the convergence of (E.13) is a simple consequence of the
unitary equation (E.5), and is valid in the massless limit. We summarise the
positivity bounds (E.14) in figure 22: all theories consistent with the
positivity constraint must take values inside the solid black region (with
$a_{0},\,a_{1}>0$).

Figure 22: We show in black the bounds from the positivity constraint (E.15),
while with dashed lines we show the bounds that follow from the exact
$2\rightarrow 2$ unitarity constraints (E.5), a.k.a. S-matrix bootsrap.

An interesting question is to compare these constraints with the optimal
bounds that follow from the exact $2\rightarrow 2$ unitary equation that we
showed in figure 21. Clearly the bootstrap bounds in figure 21 are in general
more constraining than the positivity bounds (E.14), and thus the allowed
values in figure 21 lie inside the black solid line of figure 22.

We also note that deep inside the region of validity of the EFT the positivity
bounds are not saturated, and the theory obeys stronger bounds, namely the
bootstrap bounds. In order to show this, it is convenient to define

$C_{4}\equiv\frac{c_{4}\epsilon^{4}}{c_{2}\epsilon^{2}}=\frac{a_{1}}{a_{0}}(2m^{2})^{2}\quad\text{and}\quad
C_{6}\equiv\frac{c_{6}\epsilon^{6}}{c_{2}\epsilon^{2}}=\frac{a_{2}}{a_{0}}(2m^{2})^{4}\,.$
(E.17)

These variables are singular as we take the weak coupling limit
$c_{2}\epsilon^{2}\xrightarrow{}0$. In this limit, we approach the origin in
the $c_{4}\epsilon^{4}-c_{6}\epsilon^{6}$ plane. The regions inside the
positivity plot of figure 22 depend on the angle at which we approach the
origin of the $c_{0}-c_{2}\epsilon^{2}$ plane, i.e. the free theory. We define
the angle by the $\beta\equiv c_{2}\epsilon^{2}/c_{0}$. In figure 22 we plot
with dashed lines the allowed regions for different values of $\beta$.
Interesting cases are the extremal values: $\beta=0$ filling completely the
banana region and $\beta=1$ shrinking to the origin. Both cases can be
understood by analysing the Schwarz-Pick inequalities in (E.23) and (E.24).
For $\beta\xrightarrow{}1$ these inequalities are given by

$1\geqslant|1+2C_{4}/(\beta-1)+O(c_{0},\beta-1)|\quad\text{and}\quad
1\geqslant|-1+O(c_{0},\beta-1)|\ ,$ (E.18)

which implies $C_{4}=0$ and thus $C_{6}=0$; instead, the limit
$\beta\xrightarrow{}0$ leads to

$1\geqslant|1-2C_{4}+O(c_{0},\beta)|\quad\text{and}\quad
1\geqslant|(C_{4}+C_{4}^{2}-2C_{6})/(C_{4}^{2}-C_{4})+O(c_{0},\beta)|\,,$
(E.19)

which are equivalent to the arc inequalities $C_{4}>C_{6}$ and
$C_{4}^{2}<C_{6}$, and therefore to the region plotted in black in figure 22.

Our observations provide a new interpretation of (E.14) in terms of $d=1+1$
S-matrices. These equations were analysed in ref. [20] in the context of
$d=3+1$. On one hand, there it was observed that theories in the far IR flow
to the origin of the $c_{0}-c_{2}\epsilon^{2}$ plane along trajectories
compatible with the positivity bounds (i.e. by flowing in the reverse
direction pointed by the arrows of figure 2, left, in ref. [20]). This
corresponds to our limit in (E.18). Interestingly as we approach the free
theory along various directions in the $c_{0}-c_{2}\epsilon^{2}$ plane we can
place tighter bounds than the positivity bounds. On the other hand it was
observed that tree-level theories with resonances at threshold of the lower
limit of the arc integrations do saturate the bounds. This corresponds to our
second limit in (E.19). Here however we are not working at tree-level, but
with exact S-matrices. As we approach the free theory though the lower branch
of the left plot in figure 21, the S-matrix $S_{\text{low}}$ in (E.9) features
a bound state that goes to threshold, giving
$\text{Im}M_{\text{low}}\propto\,\delta(s-4m^{2}+0^{+})$.

###  Schwarz-Pick inequalities

Let $f(z)$ be a complex function defined inside the unit disk $|z|\leqslant 1$
with real Taylor coefficients $\alpha_{i}\in\mathbb{R}$.

$f(z)=\alpha_{0}+\alpha_{1}z+\alpha_{2}z^{2}+\dots+\alpha_{i}z^{i}$ (E.20)

If it is bounded along the boundary of the disk i.e. $|f(z)|\leqslant 1$ for
$|z|=1$, its Taylor coefficients obey the following inequalities

$\displaystyle 1$ $\displaystyle\geqslant|\alpha_{0}|$ (E.21) $\displaystyle
1$
$\displaystyle\geqslant\Big{|}\,\frac{\alpha_{1}}{1-\alpha_{0}^{2}}\,\Big{|}$
(E.22) $\displaystyle 1$
$\displaystyle\geqslant\Big{|}\,\frac{\alpha_{0}\alpha_{1}^{2}+\alpha_{2}-\alpha_{0}^{2}\alpha_{2}}{1-2\alpha_{0}^{2}+\alpha_{0}^{4}-\alpha_{1}^{2}}\,\Big{|}$
(E.23) $\displaystyle 1$
$\displaystyle\geqslant\Big{|}\frac{2\alpha_{0}\alpha_{1}\alpha_{2}-2\alpha_{0}^{3}\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{2}^{2}+\alpha_{0}^{2}(\alpha_{1}^{3}-2\alpha_{3})+\alpha_{3}+\alpha_{0}^{4}\alpha_{3}-\alpha_{1}^{2}\alpha_{3}}{-1-3\alpha_{0}^{4}+\alpha_{0}^{6}+2\alpha_{1}^{2}-\alpha_{1}^{4}+2\alpha_{0}\alpha_{1}^{2}\alpha_{2}+\alpha_{2}^{2}-\alpha_{0}^{2}(-3+2\alpha_{1}^{2}+\alpha_{2}^{2})}\Big{|}$
(E.24)

These are obtained from the Maximum Modulus Principle, and the 1st, 2nd, 3rd
Schwarz-Pick inequalities, respectively.

## References

  * [1] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP 10 (2006) 014, arXiv:hep-th/0602178.
  * [2] T. N. Pham and T. N. Truong, “Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation,” Phys. Rev. D 31 (1985) 3027.
  * [3] M. R. Pennington and J. Portoles, “The Chiral Lagrangian parameters, l1, l2, are determined by the rho resonance,” Phys. Lett. B 344 (1995) 399–406, arXiv:hep-ph/9409426.
  * [4] B. Ananthanarayan, D. Toublan, and G. Wanders, “Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints,” Phys. Rev. D 51 (1995) 1093–1100, arXiv:hep-ph/9410302.
  * [5] A. V. Manohar and V. Mateu, “Dispersion Relation Bounds for pi pi Scattering,” Phys. Rev. D 77 (2008) 094019, arXiv:0801.3222 [hep-ph].
  * [6] I. Low, R. Rattazzi, and A. Vichi, “Theoretical Constraints on the Higgs Effective Couplings,” JHEP 04 (2010) 126, arXiv:0907.5413 [hep-ph].
  * [7] Z. Komargodski and A. Schwimmer, “On Renormalization Group Flows in Four Dimensions,” JHEP 12 (2011) 099, arXiv:1107.3987 [hep-th].
  * [8] M. A. Luty, J. Polchinski, and R. Rattazzi, “The $a$-theorem and the Asymptotics of 4D Quantum Field Theory,” JHEP 01 (2013) 152, arXiv:1204.5221 [hep-th].
  * [9] B. Bellazzini, “Softness and amplitudes’ positivity for spinning particles,” JHEP 02 (2017) 034, arXiv:1605.06111 [hep-th].
  * [10] C. Cheung and G. N. Remmen, “Positive Signs in Massive Gravity,” JHEP 04 (2016) 002, arXiv:1601.04068 [hep-th].
  * [11] J. Distler, B. Grinstein, R. A. Porto, and I. Z. Rothstein, “Falsifying Models of New Physics via WW Scattering,” Phys. Rev. Lett. 98 (2007) 041601, arXiv:hep-ph/0604255.
  * [12] C. Englert, G. F. Giudice, A. Greljo, and M. Mccullough, “The $\hat{H}$-Parameter: An Oblique Higgs View,” JHEP 09 (2019) 041, arXiv:1903.07725 [hep-ph].
  * [13] B. Bellazzini, F. Riva, J. Serra, and F. Sgarlata, “Beyond Positivity Bounds and the Fate of Massive Gravity,” Phys. Rev. Lett. 120 no. 16, (2018) 161101, arXiv:1710.02539 [hep-th].
  * [14] L. Alberte, C. de Rham, S. Jaitly, and A. J. Tolley, “QED positivity bounds,” arXiv:2012.05798 [hep-th].
  * [15] B. Bellazzini, F. Riva, J. Serra, and F. Sgarlata, “Massive Higher Spins: Effective Theory and Consistency,” JHEP 10 (2019) 189, arXiv:1903.08664 [hep-th].
  * [16] J. Gu, L.-T. Wang, and C. Zhang, “An unambiguous test of positivity at lepton colliders,” arXiv:2011.03055 [hep-ph].
  * [17] C. de Rham, S. Melville, A. J. Tolley, and S.-Y. Zhou, “Positivity Bounds for Massive Spin-1 and Spin-2 Fields,” JHEP 03 (2019) 182, arXiv:1804.10624 [hep-th].
  * [18] N. Arkani-Hamed, T.-C. Huang, and Y.-T. Huang, “The EFT-Hedron,” arXiv:2012.15849 [hep-th].
  * [19] M. B. Green and C. Wen, “Superstring amplitudes, unitarily, and Hankel determinants of multiple zeta values,” JHEP 11 (2019) 079, arXiv:1908.08426 [hep-th].
  * [20] B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau, and F. Riva, “Positive Moments for Scattering Amplitudes,” arXiv:2011.00037 [hep-th].
  * [21] B. Bellazzini, M. Riembau, and F. Riva, “The IR-Side of Positivity Bounds,” arXiv:2112.12561 [hep-th].
  * [22] B. Bellazzini, G. Isabella, M. Lewandowski, and F. Sgarlata, “Gravitational causality and the self-stress of photons,” JHEP 05 (2022) 154, arXiv:2108.05896 [hep-th].
  * [23] A. J. Tolley, Z.-Y. Wang, and S.-Y. Zhou, “New positivity bounds from full crossing symmetry,” arXiv:2011.02400 [hep-th].
  * [24] S. Caron-Huot and V. Van Duong, “Extremal Effective Field Theories,” arXiv:2011.02957 [hep-th].
  * [25] S. Komatsu, M. F. Paulos, B. C. Van Rees, and X. Zhao, “Landau diagrams in AdS and S-matrices from conformal correlators,” JHEP 11 (2020) 046, arXiv:2007.13745 [hep-th].
  * [26] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin, “Sharp Boundaries for the Swampland,” arXiv:2102.08951 [hep-th].
  * [27] Z. Bern, D. Kosmopoulos, and A. Zhiboedov, “Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude,” arXiv:2103.12728 [hep-th].
  * [28] L. Alberte, C. de Rham, S. Jaitly, and A. J. Tolley, “Reverse Bootstrapping: IR Lessons for UV Physics,” Phys. Rev. Lett. 128 no. 5, (2022) 051602, arXiv:2111.09226 [hep-th].
  * [29] L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina, and H.-C. Weng, “Into the EFThedron and UV constraints from IR consistency,” arXiv:2105.02862 [hep-th].
  * [30] J. Henriksson, B. McPeak, F. Russo, and A. Vichi, “Rigorous bounds on light-by-light scattering,” JHEP 06 (2022) 158, arXiv:2107.13009 [hep-th].
  * [31] A. Sinha and A. Zahed, “Crossing Symmetric Dispersion Relations in Quantum Field Theories,” Phys. Rev. Lett. 126 no. 18, (2021) 181601, arXiv:2012.04877 [hep-th].
  * [32] W. Knop and D. Mazac, “Dispersive Sum Rules in AdS2,” arXiv:2203.11170 [hep-th].
  * [33] P. Creminelli, O. Janssen, and L. Senatore, “Positivity bounds on effective field theories with spontaneously broken Lorentz invariance,” arXiv:2207.14224 [hep-th].
  * [34] K. Häring and A. Zhiboedov, “Gravitational Regge bounds,” arXiv:2202.08280 [hep-th].
  * [35] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap. Part I: QFT in AdS,” JHEP 11 (2017) 133, arXiv:1607.06109 [hep-th].
  * [36] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap II: two dimensional amplitudes,” JHEP 11 (2017) 143, arXiv:1607.06110 [hep-th].
  * [37] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap. Part III: higher dimensional amplitudes,” JHEP 12 (2019) 040, arXiv:1708.06765 [hep-th].
  * [38] A. Homrich, J. a. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix Bootstrap IV: Multiple Amplitudes,” JHEP 11 (2019) 076, arXiv:1905.06905 [hep-th].
  * [39] J. Elias Miró, A. L. Guerrieri, A. Hebbar, J. a. Penedones, and P. Vieira, “Flux Tube S-matrix Bootstrap,” Phys. Rev. Lett. 123 no. 22, (2019) 221602, arXiv:1906.08098 [hep-th].
  * [40] J. Elias Miró and A. Guerrieri, “Dual EFT bootstrap: QCD flux tubes,” JHEP 10 (2021) 126, arXiv:2106.07957 [hep-th].
  * [41] A. L. Guerrieri, J. Penedones, and P. Vieira, “Bootstrapping QCD Using Pion Scattering Amplitudes,” Phys. Rev. Lett. 122 no. 24, (2019) 241604, arXiv:1810.12849 [hep-th].
  * [42] A. L. Guerrieri, J. Penedones, and P. Vieira, “S-matrix bootstrap for effective field theories: massless pions,” JHEP 06 (2021) 088, arXiv:2011.02802 [hep-th].
  * [43] A. Hebbar, D. Karateev, and J. Penedones, “Spinning S-matrix bootstrap in 4d,” JHEP 01 (2022) 060, arXiv:2011.11708 [hep-th].
  * [44] A. Guerrieri, J. Penedones, and P. Vieira, “Where is String Theory?,” arXiv:2102.02847 [hep-th].
  * [45] A. Guerrieri, H. Murali, J. Penedones, and P. Vieira. To appear soon.
  * [46] M. Correia, A. Sever, and A. Zhiboedov, “An Analytical Toolkit for the S-matrix Bootstrap,” arXiv:2006.08221 [hep-th].
  * [47] M. Correia, A. Sever, and A. Zhiboedov, “Probing multi-particle unitarity with the Landau equations,” arXiv:2111.12100 [hep-th].
  * [48] N. Doroud and J. Elias Miró, “S-matrix bootstrap for resonances,” JHEP 09 (2018) 052, arXiv:1804.04376 [hep-th].
  * [49] M. F. Paulos and Z. Zheng, “Bounding scattering of charged particles in $1+1$ dimensions,” JHEP 05 (2020) 145, arXiv:1805.11429 [hep-th].
  * [50] Y. He, A. Irrgang, and M. Kruczenski, “A note on the S-matrix bootstrap for the 2d O(N) bosonic model,” JHEP 11 (2018) 093, arXiv:1805.02812 [hep-th].
  * [51] L. Córdova and P. Vieira, “Adding flavour to the S-matrix bootstrap,” JHEP 12 (2018) 063, arXiv:1805.11143 [hep-th].
  * [52] D. Karateev, S. Kuhn, and J. a. Penedones, “Bootstrapping Massive Quantum Field Theories,” JHEP 07 (2020) 035, arXiv:1912.08940 [hep-th].
  * [53] H. Chen, A. L. Fitzpatrick, and D. Karateev, “Bootstrapping 2d $\phi$4 theory with Hamiltonian truncation data,” JHEP 02 (2022) 146, arXiv:2107.10286 [hep-th].
  * [54] A. Bose, P. Haldar, A. Sinha, P. Sinha, and S. S. Tiwari, “Relative entropy in scattering and the S-matrix bootstrap,” SciPost Phys. 9 (2020) 081, arXiv:2006.12213 [hep-th].
  * [55] A. Bose, A. Sinha, and S. S. Tiwari, “Selection rules for the S-Matrix bootstrap,” arXiv:2011.07944 [hep-th].
  * [56] D. Karateev, J. Marucha, J. a. Penedones, and B. Sahoo, “Bootstrapping the $a$-anomaly in $4d$ QFTs,” arXiv:2204.01786 [hep-th].
  * [57] H. Chen, A. L. Fitzpatrick, and D. Karateev, “Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d\geqslant 2$,” arXiv:2207.12448 [hep-th].
  * [58] B. Gabai and X. Yin, “On The S-Matrix of Ising Field Theory in Two Dimensions,” arXiv:1905.00710 [hep-th].
  * [59] P. Tourkine and A. Zhiboedov, “Scattering from production in 2d,” JHEP 07 (2021) 228, arXiv:2101.05211 [hep-th].
  * [60] L. Córdova, Y. He, M. Kruczenski, and P. Vieira, “The O(N) S-matrix Monolith,” JHEP 04 (2020) 142, arXiv:1909.06495 [hep-th].
  * [61] A. L. Guerrieri, A. Homrich, and P. Vieira, “Dual S-matrix bootstrap. Part I. 2D theory,” JHEP 11 (2020) 084, arXiv:2008.02770 [hep-th].
  * [62] M. Kruczenski and H. Murali, “The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary,” JHEP 04 (2021) 097, arXiv:2012.15576 [hep-th].
  * [63] Y. He and M. Kruczenski, “S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem,” arXiv:2103.11484 [hep-th].
  * [64] A. Guerrieri and A. Sever, “Rigorous Bounds on the Analytic S Matrix,” Phys. Rev. Lett. 127 no. 25, (2021) 251601, arXiv:2106.10257 [hep-th].
  * [65] A. Martin, “Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. 1.,” Nuovo Cim. A 42 (1965) 930–953.
  * [66] C. Lopez, “A Lower Bound to the pi0 pi0 S-Wave Scattering Length,” Nucl. Phys. B 88 (1975) 358–364.
  * [67] C. Lopez, “Rigorous Lower Bounds for the pi pi p-Wave Scattering Length,” Lett. Nuovo Cim. 13 (1975) 69.
  * [68] C. Lopez and G. Mennessier, “A New Absolute Bound on the pi0 pi0 S-Wave Scattering Length,” Phys. Lett. B 58 (1975) 437–441.
  * [69] B. Bonnier, C. Lopez, and G. Mennessier, “Improved Absolute Bounds on the pi0 pi0 Amplitude,” Phys. Lett. B 60 (1975) 63–66.
  * [70] C. Lopez and G. Mennessier, “Bounds on the pi0 pi0 Amplitude,” Nucl. Phys. B 118 (1977) 426–444.
  * [71] See the Wikipedia page.
  * [72] M. Froissart, “Asymptotic behavior and subtractions in the Mandelstam representation,” Phys. Rev. 123 (1961) 1053–1057.
  * [73] S. M. Roy, “Exact integral equation for pion pion scattering involving only physical region partial waves,” Phys. Lett. B 36 (1971) 353–356.
  * [74] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in 4D CFT,” JHEP 12 (2008) 031, arXiv:0807.0004 [hep-th].
  * [75] D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,” Rev. Mod. Phys. 91 (2019) 015002, arXiv:1805.04405 [hep-th].
  * [76] P. Haldar, A. Sinha, and A. Zahed, “Quantum field theory and the Bieberbach conjecture,” SciPost Phys. 11 (2021) 002, arXiv:2103.12108 [hep-th].
  * [77] A. Zahed, “Positivity and geometric function theory constraints on pion scattering,” JHEP 12 (2021) 036, arXiv:2108.10355 [hep-th].
  * [78] J. Elias-Miro, A. Guerrieri, and M. Gumus, “to appear soon,”.
  * [79] J. Elias-Miro, J. R. Espinosa, E. Masso, and A. Pomarol, “Higgs windows to new physics through d=6 operators: constraints and one-loop anomalous dimensions,” JHEP 11 (2013) 066, arXiv:1308.1879 [hep-ph].
  * [80] J. De Blas, G. Durieux, C. Grojean, J. Gu, and A. Paul, “On the future of Higgs, electroweak and diboson measurements at lepton colliders,” JHEP 12 (2019) 117, arXiv:1907.04311 [hep-ph].
  * [81] SMEFiT Collaboration, J. J. Ethier, G. Magni, F. Maltoni, L. Mantani, E. R. Nocera, J. Rojo, E. Slade, E. Vryonidou, and C. Zhang, “Combined SMEFT interpretation of Higgs, diboson, and top quark data from the LHC,” JHEP 11 (2021) 089, arXiv:2105.00006 [hep-ph].
  * [82] M. Creutz, “Rigorous bounds on coupling constants in two-dimensional field theories,” Phys. Rev. D 6 (1972) 2763–2765.
  * [83] S. Dubovsky, R. Flauger, and V. Gorbenko, “Solving the Simplest Theory of Quantum Gravity,” JHEP 09 (2012) 133, arXiv:1205.6805 [hep-th].
  * [84] S. Dubovsky, V. Gorbenko, and M. Mirbabayi, “Natural Tuning: Towards A Proof of Concept,” JHEP 09 (2013) 045, arXiv:1305.6939 [hep-th].
  * [85] A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016) 112, arXiv:1608.05534 [hep-th].
  * [86] A. B. Zamolodchikov, “Resonance factorized scattering and roaming trajectories,” J. Phys. A 39 (2006) 12847–12862.