# AC False Data Injection Attacks in Power Systems: Design and Optimization

Mohammadreza Iranpour∗, Mohammad Rasoul Narimani∗ ${\ast}$: Department of
Electrical and Computer Engineering, California State University Northridge
(CSUN<EMAIL_ADDRESS>Support from NSF contract #2308498.

###### Abstract

False Data Injection (FDI) attacks are one of the challenges that the modern
power system, as a cyber-physical system, is encountering. Designing AC FDI
attacks that accurately address the physics of the power systems could
jeopardize the security of power systems as they can easily bypass the
traditional Bad Data Detection (BDD) algorithm. Knowing the essence of the AC
FDI attack and how they can be designed gives insight about detecting the
system again these attacks. Moreover, recognition of the nature of these
attacks, especially when they are designed optimally, is essential for
benchmarking various defensive approaches to increase the resilience of power
systems. This paper presents a unified approach to demonstrate the process of
designing optimal AC FDI attack. In this connection, we first define the
process of designing an AC-based FDI attack that satisfies AC power flow
equations. We then formulate an optimization problem to design an optimal AC
FDI attack that both satisfies AC power flow equations and overloads a
specific line in the system. The objective function is defined to optimize the
magnitude of the attack vector in such a way that it can evade residue-based
BDD approaches. The proposed approach for designing AC FDI attacks is applied
to the IEEE 118-bus test case system. Various comparisons are conducted to
elaborate on the impact of optimally designing AC FDI attacks on the residual
for the AC state estimation algorithm. Comparing the results of optimal and
non-optimal AC FDI attacks demonstrates the impact on the difficulty of
detecting FDI attacks and the importance of optimally designing these attacks.

##  I Introduction

Although cyber-physical networks have the potential to enhance the efficiency,
controllability, and reliability of power systems by leveraging the advantages
of Information and Communication Technologies (ICT), these systems also
introduce vulnerabilities. Specifically, the integration of communication
systems into power systems can jeopardize their security, as attackers may
hack measurement and communication devices to intrude false data into power
system measurements, raising concerns about the secure performance of these
networks [1]. False Data Injection (FDI) attacks, which involve the deliberate
insertion of incorrect data into a system’s measurements, have emerged as a
significant threat to the security of power grids. These attacks pose a major
challenge to the use of ICT applications in power systems [1].

The state estimator, a core component of the Supervisory Control and Data
Acquisition (SCADA) system, continuously monitors the operating state of a
power system to ensure safe and reliable operation. However, it has been
demonstrated that intruders can hack multiple sensors and measurement devices,
compromising the state estimator by injecting pre-designed false data vectors
into the meters which is known as the FDI attacks [2]. If this attack vector
meets certain conditions, it can bypass the commonly used residue-based Bad
Data Detectors (BDD), falsifying network topology and misleading the control
center. This undetected manipulation can lead to various disruptive
consequences, such as intentional branch outages or power grid frequency
excursions, which may result in blackouts or damage to electrical equipment
[3].

Over the past decade, numerous studies have reviewed FDI attacks, covering
background materials, construction methods, and detection and defense
strategies [4]. The construction methods have been examined from various
perspectives, including FDI attacks with limited budgets, those based on state
estimation with incomplete system knowledge[5], and those using a data-driven
approach with incomplete system knowledge[6]. The focus of FDI attacks with
limited budgets is on constructing valid attacks with minimal effort, such as
compromising at most k meters, known as the k-sparse problem[2]. In this
scenario, the attack designer needs to formulate an optimal strategy tailored
to specific situations and targets.

Building on this background, two realistic categories of FDI attacks have been
considered from a design perspective. The first is random false data injection
attacks, where the attacker aims to find any attack vector that can lead to
incorrect estimation of state variables. The second is targeted false data
injection attacks, where the attacker aims to find an attack vector that can
inject a specific error into certain state variables. Research by Liu et al.
[2] demonstrates that attackers can systematically and efficiently construct
attack vectors for both scenarios. Additionally, in [7] is shown that these
systematic approaches can be implemented by formulating an FDI attack as an
optimization problem, to find a stealthy and sparse data injection vector on
the sensor measurements to produce spurious and misleading state estimates.

Numerous studies have focused on optimization approaches that emphasize
sparsity and minimizing the number of compromised measurements, aiming to
incorporate constraints related to limited resources and stealthy operations
[8, 9]. Other studies have analyzed these optimization approaches from
different perspectives. For instance, semi-definite programming (SDP)
relaxation and sparsity penalties have been used to recover a near-optimal
solution [8]. A reduced row-echelon (RRE) form-based greedy method has been
employed to compute the minimum number of targeted measurements needed for an
attack [10]. Additionally, genetic algorithms and neural networks have been
applied to construct the least-effort attack vector [7]. The minimum number of
sensors required to stage stealth FDI attacks has also been quantified by
formulating a minimum cardinality problem, with various algorithms proposed
for efficient computation [7].

In addition to the various approaches to designing an optimal False Data
Injection (FDI) attack, another perspective involves optimizing the magnitudes
of the attack vector. In this scenario, attackers aim to adjust the magnitudes
of injected false data to ensure they fall within acceptable bounds, evading
detection by traditional state estimation algorithms. This paper addresses
this aspect by formulating an objective function to optimize the magnitudes of
injected false data in power flow measurements within the power system.
Specifically, it focuses on AC-based FDI attacks, known for their complexity
in design, and difficulty in detection compared to DC-based FDI attacks [1,
11, 12]. These attacks systematically leverage non-linear power flow equations
and Kirchhoff’s laws in a compromised power system. Apart from defining an
objective function for optimizing attack vector magnitudes, this paper also
considers the non-linear power flow equations of the compromised system as
constraints within the. This simultaneous optimization of a successful FDI
attack while optimizing the attack vector magnitudes enhances the stealthiness
of the attack, making it challenging to detect using existing methods.

The remainder of this paper is organized as follows: In Section II, we present
a systematic approach to designing a successful AC-based False Data Injection
(FDI) attack, outlining the definition of attacked and non-attacked zones
within the power system and corresponding power flow equations. Section III
formulates an appropriate objective function and necessary constraints for
designing an optimal AC-based FDI attack. The implementation and analysis of
this approach on the IEEE 118-bus test system are discussed in Section IV.
Finally, Section V offers concluding remarks on the study’s findings and
implications.

##  II Preliminary Concepts for AC False Data Injection Attack in Power
Systems

To ensure continuous operation of power systems, engineers use monitoring
systems that provide measurement data to state estimation platforms in control
centers. State estimators determine various parameters, such as voltage levels
and angles at buses, enabling appropriate decision-making and actions.
Fundamentally, accurate sensor readings usually yield state variable
estimations that closely match their true values, while abnormal measurements
can greatly alter these estimates. To detect these inconsistencies,
researchers in power systems suggest calculating the measurement residual
$r=z-H(x)$, which demonstrates the difference between the observed
measurements ($z$) and estimated measurements ($h(x)$). When
$\|z-H(x)\|>\tau$, where $\tau$ is a specified threshold, it is concluded that
there are erroneous measurements.

Although Bad Data Detectors (BDD) are engineered to detect and eliminate
measurements that appear incorrect or fall outside the normal data
distribution range, adversaries can inject meticulously crafted measurements
that disrupt the data distribution while eluding detection. Assuming the
residue of state-estimated variables under attack is:

$\displaystyle r_{attack}=\|z_{attack}-h(x_{attack})\|$
$\displaystyle=\|z_{attack}-h(x_{attack})+h(x)-h(x)\|$
$\displaystyle=\|(z+a-h(x_{attack})+h(x)-h(x)\|$
$\displaystyle=\|(r+a-h(x_{attack})+h(x)\|.$ (1)

where the attack vector $a$ is defined as:

$\displaystyle a=\|h(x_{attack})-h(x)\|.$ (2)

The residue-based bad data detection tests cannot detect the attack vector $a$
since the injected false data no longer affects the residue.

The primary obstacle in developing AC attacks involves defining the function
$h(x_{attack})$. This task requires certain foundational assumptions which,
when considered, allow for defining this function and executing a successful
attack. $h$ is the set of nonlinear power flow equations that relate
measurement and variable states. To design a successful FDI attack, the non-
linear relationship in power flow equations that connects state variables with
various quantities, including power flow in lines and power injections into
buses, should be considered. In the following sections the process of
determining function $h(x_{attack})$, is presented.

###  II-A Identifying Attack Zone

In general, an AC FDI attack focuses on a particular sub-grid region, leading
to the division of the entire grid into two distinct areas: the “attack area”
and the “normal area”. The attack area encompasses the region directly
targeted by the attack and consists of a set of buses forming a closed region
where all alterations due to the attack occur internally. Conversely, the
normal area remains unaffected by the attack.

As a fundamental aspect in designing a successful FDI attack to circumvent the
BDD algorithm, every modification resulting from such an attack must be
justified. The initial assumption crucial for designing such an attack is to
meet specific criteria between the normal and attacked areas, considering the
multiple power transfers occurring between the affected and unaffected
regions. Consequently, during the execution of an FDI attack on the targeted
region, it becomes imperative to maintain equivalent total power transfers
between this area and the unaffected regions. According to [13], fulfilling
this requirement entails enclosing the attack area with buses equipped for
power injection, i.e. buses with non-zero power injection, thus justifying
every power change solely within the attack zone. The ability to justify power
changes is attributed to buses equipped with power injection measurement
devices, because they can justify the power changes through the crafting of
injection measurements, as opposed to zero injection buses, which only extend
the attack zones’s influence. Furthermore, to confine all alternations within
the attack area during an FDI attack, it is essential to keep the state
variables on the boundary unchanged.

In general, determining the attack zone involves designating specific buses,
either individually or as groups, as primary focal points. These focal buses
are then expanded by considering neighboring non-zero injection busses and
passing through zero-injection buses to delineate the boundary of the attack
zone. For example, consider Figure 1, where bus $2$ is the targeted bus for
attack. Altering the state variables at bus $2$ affects the power flow between
busses $2$ and $1$. Since bus $1$ is a zero-injection bus, these alternations
propagate to other links connected to it, expanding the attack effects.
Similarly, the alternation of power flow between buses $2$ and $3$ occurs,
with bus $3$ being a non-zero injection bus. Bus $3$ is a no-zero injection
bus and this alternation of power flow can be justified by power injection
measurement of this bus. By fixing the state variables (voltage magnitude and
angle) at bus $3$, the power flow does not alter on the other link that is
connected to this bus, thereby confining the attack region and preventing its
expansion to other buses. This process is repeated for buses $4$ and $6$,
where the attack effects are bonded. In Figure 1, the power flow and power
injection measurements that need alternation due to the expanding attack
effects are highlighted in red. In the following sections, the second
assumption is presented, outlining a unified method for generating constraints
to provide function $h(x)$ and subsequently designing and executing an AC FDI
attack.

Figure 1: A toy example of defining the attack zone in a sample test case. The
red line indicates the branches within the attack zone. The red dashed line
separates the attack zone from the rest of the system.

###  II-B Constraints-based AC False Data Injection Attack

By considering this assumption that the algebraic sum of the generated and
consumed power in the attack region must remain unchanged, it is possible to
establish two types of constraints that can define function $h(x)$, for
designing the attack vector. In this regard:

  1. 1.

In zero-injection buses within the attack zone, the algebraic sum of
active/reactive power flows after the attack must be equal to zero. For
example in the Figure 1, since bus $1$ is a zero-injection bus, we can express
this as:
$\tilde{p}_{1,2}+\tilde{p}_{2,1}+\tilde{p}_{1,5}+\tilde{p}_{5,1}+\tilde{p}_{1,4}+\tilde{p}_{4,1}=0$
and
$\tilde{q}_{1,2}+\tilde{q}_{2,1}+\tilde{q}_{1,5}+\tilde{q}_{5,1}+\tilde{q}_{1,4}+\tilde{q}_{4,1}=0$.
In these equations, $\tilde{p}_{i,j}$ and $\tilde{q}_{i,j}$ represent the
active and reactive power flow from bus $i$ to bus $j$, while
$\tilde{p}_{j,i}$ and $\tilde{q}_{j,i}$ represent the active and reactive
power flow for to bus $i$ from bus $j$.

  2. 2.

In non-zero injection busses within the attack zone, the power injection after
the attack equals the primary injection power plus the sum of all changes in
power flows of lines connected to this non-zero injection bus and existing in
the attack zone. For example in Figure 1, since bus $4$ is a nonzero-injection
bus, we can express its power flow injection after the attack as
follows:$\tilde{P}_{4}={P}_{4}+(\tilde{p}_{1,4}+\tilde{p}_{4,1}-{p}_{1,4}-{p}_{4,1})$
and
$\tilde{Q}_{4}={Q}_{4}+(\tilde{q}_{1,4}+\tilde{q}_{4,1}-{q}_{1,4}-{q}_{4,1})$.
In these equations, $P_{4}$ and $Q_{4}$ represent the active and reactive
power injections at the bus $4$. It is important to note that in equations
concerning the boundary non-zero injection busses, only changes in the power
flow of lines within the attack zone and connected to these busses should be
considered. Eventually, based on these two assumptions and by solving these
equations, the attack vector $a$, as per Equation (1), can be calculated.

##  III Optimal Designing of False Data Injection Attack

To increase the chance of the FDI attack in bypassing the BDD algorithms,
especially the residue-based BDD approaches, it is important to design the
attacked vector properly. The main contribution of this paper is to optimize
the elements of the attack vectors in order to have less impact on the
residuals. By optimally designing the AC FDI attack, we not only design an AC
FDI attack but also optimize its magnitude to make it more stealthy. In this
regard, and based on the descriptions in the section II, We can design the
attack vector $a$ in Equation (2) by optimizing its elements. That is, we
inject values into the measurement to minimize its impact on the AC state
estimator. This minimization entails reducing the difference between corrupted
measurements and the real measurements, as indicated in Equation (3).

Suppose the sets of buses, and lines of a system are demonstrated by
$\mathcal{B}$, and $\mathcal{L}$, respectively. Also, $\mathcal{B_{A}}$, and
$\mathcal{L_{A}}$, are corresponding sets of buses and lines in the attack
zone, respectively. Let $S_{m}={P}_{m}+j{Q}_{m}$ represents the complex power
injection, $V_{m}$ and $\theta_{m}$ represent the voltage magnitude and angle
at bus $m\in\mathcal{B_{A}}$, each line $\left(m,l\right)\in\mathcal{L_{A}}$
is modeled as a $\Pi$ circuit with mutual admittance $g_{ml}+jb_{ml}$ and
shunt admittance $jb_{c,ml}$ and the voltage angle difference between buses
$m$ and $l$ for $(m,l)\in\mathcal{L}$ is denoted as
$\theta_{ml}=\theta_{m}-\theta_{l}$. The difference between state variables,
including both voltage magnitudes and angles and their values prior to
conducting FDI attack is shown in Equation (3) as vector $c$.

$c=[\tilde{V_{m}}-V_{m},\tilde{\theta_{m}}-\theta_{m}]$ (3)

Where $\tilde{V_{m}}$ and $\tilde{\theta_{m}}$ are variables representing
voltage magnitude and angle that need to be calculated to conduct the attack,
and $V_{m}$ and $\theta_{m}$ are the known values of voltage magnitude and
angle before the attack. By minimizing the sum of squared differences between
the variable states before and after the attack as the objective function, we
can optimize the attack vector in designing the AC FDI attack problem, as
represented in Equations (4)-(12).

$\displaystyle\min\quad\textstyle\sum_{\small{{m}\in\mathcal{B_{A}}}}\left[(\tilde{V_{m}}-V_{m})^{2},(\tilde{\theta_{m}}-\theta_{m})^{2}\right]$
(4) $\displaystyle\text{subject to}\quad\left(\forall
i\in\mathcal{N},\forall\left(l,m\right)\in\mathcal{L}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!g_{sh,i}\,\tilde{V}_{i}^{2}+\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}\\!\tilde{P}_{lm}+\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptm=i\end{subarray}}\\!\\!\tilde{P}_{ml}=P_{m,G}-P_{m,D},$ (5)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!-b_{sh,i}\,\tilde{V}_{i}^{2}+\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptl=i\end{subarray}}\\!\\!\tilde{Q}_{lm}+\\!\\!\\!\sum_{\begin{subarray}{c}(l,m)\in\mathcal{L},\\\
\text{s.t.}\hskip
3.0ptm=i\end{subarray}}\\!\\!\\!\\!\tilde{Q}_{ml}=Q_{m,G}-Q_{m,D},$ (6)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{P}_{lm}\\!=\\!g_{lm}\tilde{V}_{l}^{2}\\!-\\!g_{lm}\tilde{V}_{l}V_{m}\cos\left(\tilde{\theta}_{l}-\theta_{m}\right)\\!$
$\displaystyle\qquad\qquad-\\!b_{lm}\tilde{V}_{l}V_{m}\sin\left(\tilde{\theta}_{l}-\theta_{m}\right),$
(7)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{Q}_{lm}=-\left(b_{lm}+b_{c,lm}/2\right)\tilde{V}_{l}^{2}+b_{lm}\tilde{V}_{l}V_{m}\cos\left(\tilde{\theta}_{l}-\theta_{m}\right)$
$\displaystyle\qquad\qquad-
g_{lm}\tilde{V}_{l}V_{m}\sin\left(\tilde{\theta}_{l}-\theta_{m}\right),$ (8)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{P}_{ml}\\!=\\!g_{lm}V_{m}^{2}\\!-\\!g_{lm}\tilde{V}_{l}V_{m}\cos\left(\tilde{\theta}_{l}-\theta_{m}\right)\\!$
$\displaystyle\qquad\qquad+\\!b_{lm}\tilde{V}_{l}V_{m}\sin\left(\tilde{\theta}_{l}-\theta_{m}\right),$
(9)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{Q}_{ml}=-\left(b_{lm}+b_{c,lm}/2\right)V_{m}^{2}+b_{lm}\tilde{V}_{l}V_{m}\cos\left(\tilde{\theta}_{l}-\theta_{m}\right)$
$\displaystyle\qquad\qquad+g_{lm}\tilde{V}_{l}V_{m}\sin\left(\tilde{\theta}_{l}-\theta_{m}\right)$
(10) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{P}_{ml}=W_{lm}*{P}_{lm}^{PF},$
(11) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\tilde{Q}_{ml}=W_{lm}*{Q}_{lm}^{PF}.$
(12)

In these equations $\tilde{P}_{ml}$, $\tilde{Q}_{ml}$,
$\tilde{V}_{l}(\tilde{V}_{m})$ and $\tilde{\theta}_{l}(\tilde{\theta}_{m})$
are respectively active and reactive power flow between busses $m$ and $l$
$(m,l)\in\mathcal{L}$ and voltage values of bussess $m$ and $l$, after the
attack. Similarly, $P_{ml}$, $Q_{ml}$, $V_{l}(V_{m})$ and
$\theta_{l}(\theta_{m})$ are the same quantities above in before the attack
condition. Also, $P_{m,G}$, $Q_{m,G}$, $P_{m,D}$, and $Q_{m,D}$ are active and
reactive power generation and active and reactive power demanded in the bus
$m$ respectively. It is noticeable that in the Equations (5) and (III) when we
write these equations for zero injection busses, they satisfy assumption $1$
in the section II-B. Also, Equations 11 and 12, are additional constraints to
overloading a specified line in the attack zone with a predefined coefficient
$W$. By considering this constraint we can design an optimal AC FDI attack for
a specific aim like overloading a certain line by a predefined coefficient.
${P}_{lm}^{PF}$ and ${Q}_{lm}^{PF}$ are the active and reactive power flow
before the attack.

After solving this optimization problem, we can calculate the power injection
values of non-zero injection busses within the attack zone as follows:

$\displaystyle\tilde{P}_{m}=P_{m}+\sum_{(m,l)\in\mathcal{L}_{A}}(\tilde{P}_{m,l}-P_{m,l}),$
(13a)
$\displaystyle\tilde{Q}_{m}=Q_{m}+\sum_{(m.l)\in\mathcal{L_{A}}}(\tilde{Q}_{m,l}-Q_{m,l}).$
(13b)

Here, $\tilde{P_{m}}$, $\tilde{Q_{m}}$, $P_{m}$, and $Q_{m}$ represent the
active and reactive power injections at bus $m$ after and before the attack,
respectively. Notably, in Equations (13a), $\tilde{P}{m,l}$ and
$\tilde{Q}{m,l}\in\mathcal{L}_{A}$ should be considered only for the lines
within the attack zone that are connected to bus $m$. Furthermore, when
utilizing PMU, the current flow measurement values in the lines should be
adjusted based on the corresponding values after the attack. Since the current
flows in the lines are related to voltage variables, they can be calculated
using Equations (14a) and (14b), as follows:

$\displaystyle\tilde{I_{f}}=Y_{f}*\tilde{V}$ (14a)
$\displaystyle\tilde{I_{t}}=Y_{t}*\tilde{V}$ (14b)

Here, $\tilde{I_{f}}$ and $\tilde{I_{t}}$ represent the current flow in the
lines, from and to buses respectively, while $Y_{f}$ and $Y_{t}$ are the line
admittance matrices that account for the admittance of lines from and to the
buses, respectively. After calculating all of these values, we can define the
attack vector $a$ based on (2) as follows:

$\displaystyle
a=[\tilde{P_{ml}}-P_{ml},\tilde{Q_{ml}}-Q_{ml},\tilde{P_{m}}-P_{m},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{Q_{m}}-Q_{m},\tilde{V_{m}}\angle\tilde{\theta_{m}}-V_{m}\angle\theta_{m},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{I_{ml}}\angle\tilde{\theta_{ml}}-I_{ml}\angle\theta_{ml},\tilde{\theta_{m}}-\theta_{m}]^{T}$
(15)

Figure 2: One-line diagram depicting the IEEE 118-bus test system, with the
attack zones indicated.

##  IV ANALYSIS OF ATTACK SUCCESS RATE

In this section, the proposed approach for designing an optimal AC-FDI attack
is applied to the IEEE 118-bus test system from the PGLib-OPF v18.08 benchmark
library [14] to test its effectiveness in designing AC FDI attacks. We
demonstrate the impact of the designed AC FDI attack on the residuals of
estimations. Additionally, we compare optimal and non-optimal FDI attacks to
show how the proposed approach makes detecting the designed attack more
difficult for traditional residue-based BDD approaches.

Figure 3: Attack Vector for voltage angle. The red and gray bars indicate the
voltage angles that need to be added to the measurement for optimal and non-
optimal FDI attacks, respectively. Figure 4:  Attack Vector for voltage
magnitude. The red and gray bars indicate the voltage magnitudes that need to
be added to the measurement for optimal and non-optimal FDI attacks,
respectively.

We considered two scenarios to demonstrate the effects of the proposed attack
vector on the residuals of estimations. In the first scenario, we designed the
FDI attack without optimizing the attack vector. This was done by substituting
the objective function in Equation (4) with a fixed quantity. In this
situation, by solving the proposed optimization problem with a fixed objective
function, we designed an attack vector ($a$) that satisfied all the
constraints required for a successful AC FDI attack. In the second scenario,
by considering the objective function in Equation (4) and enforcing the same
set of constraints as in the previous scenario, we optimally designed an
attack vector ($a$).

In these two scenarios, as shown in Figure 2, we considered 22 buses in the
attack zone. Additionally, we aimed to overload line $26$, which connects
buses $15$ and $19$, by $1.5$ times its nominal value as a specific goal of
this AC-based optimal FDI attack. Based on this consideration, we have seven
boundary buses, including buses $13,14,17,23,70,37,$ and $44$. To bind the
attack zone, we fixed the voltage magnitude and angle of these boundary buses.
Therefore, there are $15$ values for voltage magnitudes and 15 values for
voltage angles corresponding to the buses within the attack zone that do not
have fixed voltages. These values are shown in Figures 3 and 4. From these
figures, it is evident that the proposed approach for designing an AC FDI
attack overloads a line with smaller changes in the magnitude of voltage and
angle compared to the non-optimal AC FDI attack approach. Additionally, there
are 20 non-zero injection buses in the attack zone. The values of the active
and reactive attack vectors for these buses are shown in Figures 5 and 6 for
active and reactive power injections, respectively. The changes in the power
flow injections of these buses are smaller for the proposed approach compared
to the non-optimal approach.

Figure 5: Attack Vector for active power injection. The red and gray bars
indicate the active power injections that need to be added to the measurement
for optimal and non-optimal FDI attacks, respectively. Figure 6: Attack Vector
for reactive power injection. The red and gray bars indicate the reactive
power injections that need to be added to the measurement for optimal and non-
optimal FDI attacks, respectively. Figure 7: Attack Vector for active power
flow in the lines from the “sending” end to the “receiving” end. The red and
gray bars indicate the active power flows that need to be added to the
measurement for optimal and non-optimal FDI attacks, respectively.

There are $25$ active power flow attack values for active power flow, $25$
reactive power flow attack values for reactive power flow, and $25$ current
magnitude attack values for current flow in $25$ lines in the attack zone of
Figure 2, which are indicated in red in this Figure. The corresponding values
for these quantities are shown in Figures 7, 8, and 9 for both optimal and
non-optimal approaches. As can be seen in these figures, the values of the
optimal attack vector are smaller than the non-optimal values. In these
figures, gray indicates the non-optimal values, and red indicates the optimal
values.

To assess the impact of the designed attacks on state estimation, we
considered PMU measurements, including voltage magnitude and angle, and
current magnitude in the system. By injecting the attack vector into the
corresponding measurements and using the MATGRID toolbox [15] to perform AC
state estimation, the residuals for these quantities were calculated and are
shown in Figures 10, 11, and 12 for voltage magnitude, voltage angle, and
current magnitude, respectively. The residuals with optimal attack vectors are
significantly smaller than those for non-optimal attack vectors. We
demonstrated that although an optimal attack vector could lead to erroneous
estimations deviating from the actual system conditions, the small residuals
make it challenging to detect using residual-based BDD detection strategies.

Figure 8: Attack Vector for reactive power flow in the lines from the
“sending” end to the “receiving” end. The red and gray bars indicate the
reactive power flows that need to be added to the measurement for optimal and
non-optimal FDI attacks, respectively. Figure 9: Attack Vector for current
magnitude for different lines. The red and gray bars indicate the current
magnitudes that need to be added to the measurement for optimal and non-
optimal FDI attacks, respectively. Figure 10: Residuals of AC state estimator
for voltage magnitudes. The red and gray bars indicate the residuals of the AC
state estimator for voltage magnitudes under optimal and non-optimal FDI
attacks, respectively. Figure 11: Residuals of AC state estimator for voltage
angles. The red and gray bars indicate the residuals of the AC state estimator
for voltage angles under optimal and non-optimal FDI attacks, respectively.
Figure 12: Residuals of AC state estimator for current magnitudes. The red and
gray bars indicate the residuals of the AC state estimator for current
magnitudes under optimal and non-optimal FDI attacks, respectively.

##  V Conclusion

This paper proposes a unified approach for designing optimal AC FDI attacks,
which can pose challenges for traditional residue-based BDD approaches. We
address the system’s physics by incorporating power flow equations into the
attack design procedure to maintain the undetectability of these proposed
attacks by the AC state estimator. Moreover, we define an appropriate
objective function to optimize the designed AC FDI attack. Additionally, we
impose a constraint on the optimization problem to enforce overloading a line
up to a predetermined value. To evaluate the proposed algorithm’s ability to
design an optimal AC FDI attack, we applied it to the 118-bus test system. The
residuals for the AC state estimator with the optimal attack vector are
significantly smaller than their counterparts with the non-optimal attack
vector. While the proposed AC FDI attack design could lead to erroneous
estimations deviating from the actual system conditions, the small residuals
in the AC state estimator make it difficult to detect the designed attack by
residue-based BDD approaches.

## References

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warn luatex=false

Saclay-t21/094

# The IR-Side of Positivity Bounds

Brando Bellazzini  Université Paris-Saclay, CNRS, CEA, Institut de Physique
Théorique, 91191, Gif-sur-Yvette, France.  Theoretical Particle Physics
Laboratory (LPTP), Institute of Physics, EPFL, Lausanne, Switzerland  CERN,
Theoretical Physics Department, Geneva, Switzerland  Marc Riembau  Theoretical
Particle Physics Laboratory (LPTP), Institute of Physics, EPFL, Lausanne,
Switzerland  Francesco Riva  Départment de Physique Théorique, Université de
Genève, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland

###### Abstract

We show how calculable IR loop effects impact positivity bounds in Effective
Field Theories with causal and unitary UV completions. We identify infrared
singularities which appear in dispersion relations at $|t|\lesssim m^{2}$. In
the massless limit, they weaken two-sided bounds based on crossing symmetry,
such as the lower bound on the amplitude for Galileon scattering. For
amplitudes that are analytic in $s$ even for large negative $t$, i.e. $|t|\gg
m^{2}$, we propose a new simple analytic approach to dispersive bounds, which
are instead insensitive to the singularities, and explicitly compute the
finite contributions from loops. Finally we show that the singularity do not
affect the bounds based on smearing in impact parameter.

##  I Introduction

The space of low energy relativistic effective field theories (EFT) is subject
to fundamental constraints known as positivity bounds. They encode, in the
form of inequalities among scattering amplitudes evaluated in the infrared
(IR), the conditions that EFTs emerge from the renormalization-group (RG)
evolution of unitary and causal microscopic dynamics. IR-consistent EFT
failing the positivity bounds belong to the “swampland”.

Positivity bounds stem from dispersion relations that provide this UV/IR
connection Pham and Truong (1985); Ananthanarayan  _et al._ (1995); Pennington
and Portoles (1995). While rooted in the old S-matrix bootstrap, the deeper
understanding and implications of positivity bounds was put forward in Adams
_et al._ (2006), where the connection with the requirement of subluminal
propagation of the low-energy degrees of freedom was also made explicit.

Generalization to arbitrary spinning particles Bellazzini (2017); de Rham  _et
al._ (2018a) and to finite scattering angle Vecchi (2007); Nicolis  _et al._
(2010); Bellazzini  _et al._ (2014); de Rham  _et al._ (2017), along with
interesting applications in particle phenomenology Distler  _et al._ (2007);
Manohar and Mateu (2008); Low  _et al._ (2010); Falkowski  _et al._ (2012);
Bellazzini  _et al._ (2017, 2014); Bellazzini and Riva (2018); Zhang and Zhou
(2019); Remmen and Rodd (2019); Bellazzini  _et al._ (2019a); Remmen and Rodd
(2019); Englert  _et al._ (2019); Trott (2020); Bonnefoy  _et al._ (2021);
Davighi  _et al._ (2021); Chala and Santiago (2021); Azatov  _et al._ (2021),
and in gravitational physics Gruzinov and Kleban (2007); Bellazzini  _et al._
(2016, 2018); Hamada  _et al._ (2019); de Rham  _et al._ (2018b); Alberte  _et
al._ (2020); Bellazzini  _et al._ (2019b); Kim  _et al._ (2019); Tokuda  _et
al._ (2020); Herrero-Valea  _et al._ (2020); Bern  _et al._ (2021), or in
connection to the positivity of time delay Camanho  _et al._ (2016); Afkhami-
Jeddi  _et al._ (2018); Bellazzini  _et al._ (2021), have recently lead to the
uncovering of the architecture behind all positivity bounds, (called EFT-
Hedron in Arkani-Hamed  _et al._ (2020)), where infinitely many Wilson
coefficient are constrained, at tree-level, by double-sided bounds Bellazzini
_et al._ (2020); Tolley  _et al._ (2020); Caron-Huot and Van Duong (2021);
Arkani-Hamed  _et al._ (2020); Sinha and Zahed (2021); Caron-Huot  _et al._
(2021); Chiang  _et al._ (2021); Henriksson  _et al._ (2021).

The chief purpose of this article is to show how calculable IR effects and
loops deform the EFT-Hedron beyond the idealized tree-level limit. Discussing
realistic EFTs at finite coupling in the IR is important in order to render
positivity bounds actually sharp: throwing theories in the swampland must be
irreversible, should not be undone by weak coupling corrections possibly
enhanced by IR divergences.

Broadly speaking, there are two implications of IR loop effects on positivity
bounds for Wilson coefficients, when evaluated at the EFT cutoff where 2-sided
bounds are most relevant. First of all, $n$-subtracted dispersion relations
involve (via the integral along the IR discontinuity) all EFT couplings, in
contrast to the tree-level limit where they select the coefficient of $s^{n}$
in the amplitude. Therefore sharp dispersive bounds deliver not so sharp
constraints on Wilson coefficients, and for practical applications require
perturbative assumptions about the EFT convergence Bellazzini  _et al._
(2020). Secondly, and perhaps more importantly, in the massless limit the non
analyticities extend to $s\to 0$ and $t\to 0$. There, the most relevant
interactions and their IR loops always dominate the amplitude, screening,
potentially, information on less relevant couplings. Moreover, the amplitude
being non-analytic in $t$ as the mass goes to zero restricts the use of the
dispersion relations based on the near-forward region.

In this article we present a thorough study of 1-loop effects within the EFT
of a scalar with 4-point interactions, focusing on forward singularities and
their impact on 2-sided bounds. As a case study we discuss bounds on the ratio
$g_{3,1}s/g_{2}$ between the coefficients $g_{3,1}$ of $stu$ in the amplitude
(which would-be dominant for Galileons Nicolis  _et al._ (2009)) and the
coefficient $g_{2}$ of $s^{2}+t^{2}+u^{2}$ (ordinary Goldstone boson), and
discuss how the arguments generalize to other couplings.

Expressing _near-forward_ $t\approx 0$ dispersion relations as 2D-moments of a
positive measure in the UV, generalizing Bellazzini  _et al._ (2020) along the
lines of Chiang  _et al._ (2021), we show that the lower bound on
$g_{3,1}s/g_{2}$ that holds at tree level actually disappears, in the massless
limit, regardless of how small the coupling is taken, as shown in Fig. 1. The
same conclusion holds for the approaches of Refs. Arkani-Hamed  _et al._
(2020); Tolley  _et al._ (2020); Caron-Huot and Van Duong (2021).

This pessimistic result is however based on the near-forward positivity bounds
only, that posit amplitude’s analyticity only within the rigorously proven
domain. We contrast such conclusions with those we obtain by working with an
enlarged domain of analyticity in $s$ for negative $t\ll-m^{2}$ which, despite
not being rigorously proved, is supported by the perturbative analysis of
Landau equations within the EFT. We present a simple approach to dispersion
relations that shows that the tree-level bounds do actually survive also at
loop level, as long as the coupling remains perturbative, under the assumption
of maximal analyticity. Our approach is analytic and simple, and nicely
complements the numerical technique of Caron-Huot  _et al._ (2021).

##  II Dispersion Relations at Finite $t$

We study $2\to 2$ scattering of a single particle of spin-0. We assume that
the amplitude ${\cal M}(s,t)$ is analytic in the entire complex $s$ plane
except the physical branch-cut at $s\geq 4m^{2}$ and its crossing symmetric
counterpart at $s<-t$, for fixed values of $t$ as specified in the following
sections. We focus on the amplitude with $n+2$ ($n\geq 0$) subtractions in
$s=0$ and $n+1$ subtractions in $s=-t$, and define “arcs” the following
$\hat{s}$\- and $t$-dependent contour integral

$a_{n}(\hat{s},t)\equiv\int_{\footnotesize\begin{array}[]{c}\smallfrown\\\\[-7.0pt]
\smallsmile\end{array}}\frac{d{\hat{s}^{\prime}}}{2\pi
i{\hat{s}^{\prime}}}\frac{{\hat{\cal
M}}({\hat{s}^{\prime}},t)}{[{\hat{s}^{\prime}}({\hat{s}^{\prime}}+t)]^{n+1}}\,,\quad
n\geq 0$ (1)

where

$\hat{s}\equiv s-2m^{2}\,,\qquad{\hat{\cal M}}(\hat{s},t)\equiv{\cal M}(s,t)$
(2)

and ${\footnotesize\begin{array}[]{c}\smallfrown\\\\[-7.0pt]
\smallsmile\end{array}}$ is the circle with radius $\hat{s}+t/2$ and centered
at $-t/2$ (minus its interception with the real axis). Arcs probe the theory
at energy $\hat{s}$ and momentum transfer $q^{2}=-t$, and are suited to
capture the RG flow within EFTs.

We deform the contour $\footnotesize\begin{array}[]{c}\smallfrown\\\\[-7.0pt]
\smallsmile\end{array}$ into a contour that encompasses the discontinuities on
positive ${\hat{s}^{\prime}}\geq\hat{s}$ and negative
${\hat{s}^{\prime}}\leq-\hat{s}-t$ real axis, together with upper and lower
semicircles at infinity. We further assume that the latter vanish, because,

$\lim_{|s|\to\infty}{\cal M}(s,t)/s^{2}=0$ (3)

as implied by the Froissart-Martin bound Froissart (1961); Martin (1963); Jin
and Martin (1964). From crossing symmetry, real analyticity, and the partial-
wave expansion,

$\displaystyle{\hat{\cal M}}(\hat{s},t)={\hat{\cal
M}}(-\hat{s}-t,t)\,,\quad{\hat{\cal M}}(\hat{s},t)={\hat{\cal
M}}^{*}(\hat{s}^{*},t)\,,$ $\displaystyle{\hat{\cal
M}}(\hat{s},t)=\sum_{l=0}^{\infty}P_{\ell}(1+\frac{2t}{\hat{s}\\!\\!-\\!2m^{2}})\hat{f}_{\ell}(\hat{s})$
(4)

where $P_{\ell}(\cos\theta)$ are the Legendre Polynomials and $\ell$ is even
for identical scalars, we can express the _arcs_ in terms of their UV integral
representation,

$\displaystyle a_{n}(\hat{s},t)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\int_{\hat{s}}^{\infty}\sum_{\ell=0}^{\infty}d{\hat{s}^{\prime}}\textrm{Im}\hat{f}_{\ell}({\hat{s}^{\prime}})\frac{I^{n}_{t}(\ell,{\hat{s}^{\prime}})}{\hat{s}^{\prime
2n+3}}$ (5)

with the kernel $I^{n}_{t}(\ell,{\hat{s}^{\prime}})$ given by,

$I^{n}_{t}(\ell,{\hat{s}^{\prime}})\equiv\frac{\left(1+\frac{t}{2{\hat{s}^{\prime}}}\right)}{\left(1+\frac{t}{{\hat{s}^{\prime}}}\right)^{n+2}}P_{\ell}(1+\frac{2t}{{\hat{s}^{\prime}}-2m^{2}})\,.$
(6)

In the following we are interested in the limit

$m^{2}\ll s$ (7)

where also $\hat{s}\to s$. In practice we will set $m\rightarrow 0$ everywhere
except in the presence of IR divergences where the mass acts as an explicit
regulator.

###  II.A IR Arcs

Arcs can be computed within the EFT via Eq. (1) in terms of Wilson
coefficients. Consider and EFT which, in a weak-coupling regime, can be well
approximated by the tree-level expression,

${\cal
M}(s,t)=\sum_{n,q}g_{n,q}\left(\frac{s^{2}+t^{2}+u^{2}}{2}\right)^{\frac{n-3q}{2}}\cdot\left(stu\right)^{q}$
(8)

where $n$ sets the overall energy-squared scaling, whereas $q$ tells how fast
the amplitude vanishes for $t\rightarrow 0$ and fixed $s$. Then, at $t=0$ the
arcs read

$a_{n}(\hat{s},0)=g_{2n+2,0}$ (9)

while at finite $t$,111The subtraction choice
$\frac{1}{[s(s+t)]^{n+1}}=\frac{(-t)^{n+1}}{(stu)^{n+1}}$ in Eq. (1) implies
that the coefficient of $(stu)^{q}$ only appears in $a_{n\geq q-1}$, while
$(s^{2}+t^{2}+u^{2})^{p}$ appears in all $a_{n\leq p}$. This coincides with
the choice of Ref. Bellazzini  _et al._ (2020) only at $t=0$.

$a_{0}(\hat{s},t)=\sum_{n=1}^{\infty}[nt^{2n-2}g_{2n,0}-t^{2n-1}g_{2n+1,1}]\,.$
(10)

with similar expressions for higher arcs. At tree-level, arcs are in one-to-
one correspondence with couplings. In particular, knowledge of

$a_{n}|_{t=0},\partial_{t}a_{n}|_{t=0},\cdots,\partial_{t}^{n+1}a_{n}|_{t=0}$
(11)

is enough to reconstruct the whole series of Wilson coefficients.

From Eq. (8), IR loop effects are calculable. In this article we will focus on
the most important 1-loop effect, from two insertions of $g_{2,0}$:

$\delta{\cal
M}=\frac{\beta_{4}}{2}s^{2}(s^{2}-\frac{tu}{21})\log(-s)+(s\leftrightarrow
t,u)\,,$ (12)

where the renormalisation scale is implicit and,

$\beta_{4}=-\frac{7}{10}\frac{g_{2,0}^{2}}{16\pi^{2}}$ (13)

which must be small $|\beta_{4}\hat{s}^{2}|\ll 1$ for the perturbative
expansion to make sense. This gives an additional contribution to the first
arc,

$\displaystyle\delta a_{0}(\hat{s},t)$ $\displaystyle=$
$\displaystyle\beta_{4}\big{(}\hat{s}^{2}-\frac{41}{21}\hat{s}t+\frac{t^{2}}{2}-\frac{\hat{s}^{3}}{2(\hat{s}+t)}$
$\displaystyle+t^{2}(2\log(\hat{s}+t)+\frac{1}{42}\log\frac{-t}{\hat{s}+t})\big{)}\,,$

which is non-analytic in either $s,t,u\to 0$. In particular, the term
proportional to $t^{2}\log-t$ will play an important role in what follows,
since present bounds rely on arc $t$-derivatives in the forward limit,
$\partial_{t}^{m}a_{n}|_{t=0}$. In fact all couplings generate non-
analyticities in $t=0$. The most singular effects in the 1-loop contributions
proportional to the marginal coupling $g_{0,0}$ are,

$\displaystyle\delta{\cal M}=$
$\displaystyle\frac{g_{0,0}}{32\pi^{2}}\,\log\frac{-t}{s}\Big{(}-g_{0,0}$
$\displaystyle+t^{2}\frac{5g_{2,0}}{3}+t^{3}\frac{g_{3,1}}{3}+t^{4}\frac{7g_{4,0}}{5}+\cdots\Big{)}\,.$

These (and all other 1-loop contributions involving $g_{0,0}$) carry no powers
of $s$, and therefore do not appear in arcs and do not enter in the dispersion
relations.

On the other hand, 1-loop effects involving less relevant couplings have
singularities:

$\displaystyle\delta\mathcal{M}=-\frac{s^{2}t^{2}}{16\pi^{2}}\log\frac{-t}{s}\Big{(}-t\frac{g_{2,0}g_{3,1}}{30}+t^{2}\frac{2g_{2,0}g_{4,0}}{35}$
$\displaystyle+t^{2}\frac{g_{3,1}g_{3,1}}{60}-t^{3}\frac{2g_{3,1}g_{4,0}}{35}+t^{2}s^{2}\frac{g_{4,0}^{2}}{1260}+\cdots\Big{)}\,.$

Of these, the first four will appear in the first arc $a_{0}(s,t)$ and will be
subdominant to the term $\propto g_{2,0}^{2}$ from Eq. (II.A). Only the term
$\propto g_{4,0}^{2}$ has enough powers of $s$ to appear in the second arc
$a_{1}$, but it is proportional to $t^{4}\log(-t)$. Therefore, the second arc
and its first three derivatives are regular at $t\to 0$. This trend propagates
to higher arcs, with leading divergences,

$\delta a_{n}(\hat{s},t)\propto t^{2n+2}\log\frac{-t}{\hat{s}}\,.$ (16)

We deduce that arc $n$ and its first $2n+1$ derivatives are regular in $t\to
0$. This is important in light of Eq. (11): it is possible to reconstruct all
coefficients in the forward limit, without encountering $t=0$
singularities.222Note that the pattern of IR divergences Eq. (16) translates
into a pattern of UV divergences, as
$\partial_{t}^{(k)}a_{n}(\hat{s},t)|_{t=0}$ is mapped to a UV integral
proportional to $\sum_{\ell}\ell^{2k}/s^{n}$. At fixed $n$, for high enough
$k$, the IR part diverges, implying a measure with support on arbitrarily
large $\ell$. At fixed $k$, for high enough $n$, the IR converges, implying
that contributions from large $\ell$ also have large $s$. With the ansatz
$m^{2}\sim\ell^{\alpha}$ for the UV spectrum, Eq. (16) fixes $\alpha=2$
exactly ($\alpha=1$ would correspond to $a_{n}(\hat{s},t)\propto
t^{n+1}\log\frac{-t}{\hat{s}}$). We illustrate this in Table 1.

Table 1: In green, a schematic representation of the arcs and their
derivatives (both at $t=0$) needed to reconstruct all Wilson coefficients, Eq.
(11). Red cells correspond instead to arc derivatives that are singular at
$t=0$, according to Eq. (16).

##  III 2D Moments and Dispersion relations around $t=0$

We now discuss bounds on arcs, and how they translate to bounds on Wilson
coefficients, given the results of the previous section. In this first section
we take a conservative approach and rely only on analyticity of the amplitude
for $t$ within the domain established (assuming unitarity) by Martin Martin
(1965, 1969); Sommer (1970), whose size is set by the scattered particle mass
$m$ . In the limit of small mass $m\to 0$, which we consider here, analyticity
requires $t\to 0$.

One efficient way of deriving near-forward bounds, involves mapping UV arcs to
moments of a positive distribution in $\hat{s}$ and $\ell$ Bellazzini  _et
al._ (2020); Chiang  _et al._ (2021). Bounds on moments then provide bounds
from the UV, see Chiang  _et al._ (2021); Lasserre (2009). The interpretation
in terms of moments is possible because the coefficients in the $t$-Taylor
expansion of the arc integrand Eq. (6), are themselves polynomials in
$1/{\hat{s}^{\prime}}$ and $J^{2}\equiv\ell(\ell+1)$,

$I^{n}_{t}(\ell,{\hat{s}^{\prime}})=\sum_{m=0}^{\infty}\left(\frac{t}{{\hat{s}^{\prime}}}\right)^{m}\sum_{k=0}^{m}d_{n,m-k}\sum_{i=0}^{k}\alpha^{k}_{i}J^{2i}$
(17)

with

$\displaystyle d_{n,i}$ $\displaystyle\equiv$
$\displaystyle(-1)^{i}\frac{(i+n)!}{i!n!}\frac{2n+i+2}{2n+2}\,,$ (18)
$\displaystyle\sum_{i=0}^{k}\alpha^{k}_{i}J^{2i}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(k!)^{2}}\prod_{j=0}^{k-1}(J^{2}-j(j+1))$ (19)

from expanding respectively the ratio and Legendre polynomials in Eq. (6).
Changing variables,

$x=(\hat{s}/{\hat{s}^{\prime}})\,,$ (20)

we can thus write the arc $t$-derivatives as linear combinations,

$\frac{1}{m!}\partial_{t}^{m}a_{n}(t,s)\big{|}_{t=0}=\sum_{k=0}^{m}d_{n,m-k}\sum_{j=0}^{k}\alpha^{k}_{j}\,\,\mu^{j}_{2n+m}$
(21)

of two-dimensional moments,

$\mu_{n}^{q}=\frac{1}{\hat{s}^{n+2}}\int d\mu(x,J^{2})\,\,\,x^{n}J^{2q}$ (22)

with respect to a positive measure $d\mu(x,J^{2})$ (proportional to
$x\textrm{Im}\hat{f}_{\ell(J^{2})}(\hat{s}/x)dx>0$ by unitarity) with support
on $x\in[0,1]$, times the non-compact discrete set of positive integer numbers
(for even $\ell$, $J^{2}=0,6,20,\cdots$). More explicitly, the arcs
$a_{n}(\hat{s},t)$ at finite $t$ are written via their UV representation as
linear combinations of 2D moments,

$\displaystyle
a_{n}(\hat{s},t)=\mu_{2n}^{0}+t\left(\mu_{2n+1}^{1}-\frac{3+n}{2}\mu_{2n+1}^{0}\right)$
(23)
$\displaystyle+t^{2}\left(\frac{\mu_{2n+2}^{2}}{4}-(2+n)\mu_{2n+2}^{1}+\frac{(2+n)^{2}}{2}\mu_{2n+2}^{0}\right)+\cdots$

where dots denote higher powers of $t$.

###  III.A Bounds on moments

Once arcs are expressed in terms of moments, their positivity constraints stem
from the bounds on the 2D moments, which, in turn, are in one-to-one
correspondence with the space of all polynomials in $x$ and $J^{2}$ that are
positive on the integration domain. Indeed, every positive polynomial
$\sum\bar{\beta}_{i,j}x^{i}J^{2j}\equiv p(x,J^{2})>0$ implies a positivity
condition among moments, $\sum\beta_{i,j}\mu_{i}^{j}=\int
d\mu(x,J^{2})p(x,J^{2})>0$.

We will first take the limit in which $J^{2}$ is continuous, which allows us
to find _conservative_ bounds in terms of a _finite_ number of conditions. The
continuum bounds (which are quantitatively very similar to exact bounds) will
be more conservative because the space of positive polynomials in the
continuum contains those positive in the discrete. Moreover, this approach
will enable us to obtain analytic bounds without having to rely on numerical
extrapolations to large $\ell$. In appendix A, we will show how to include the
countably infinite conditions that define the bounds for $\ell$ discrete.

The set $(x,J^{2})\in[0,1]\times\mathbb{R}^{+}$ can be described by the
conditions,

$x\geq 0\,,\quad 1-x\geq 0\,,\quad xJ^{2}\geq 0\,.$ (24)

From this, a theorem due to Schmüdgen Schmuedgen (1991); Lasserre (2009)
classifies all positive polynomials $p(x,J^{2})$ in a domain in terms of
squares of polynomials as,333Eqs. (24,25) parametrize polynomials of
$(x,xJ^{2})$ rather than $(x,J^{2})$. This is an efficient set of polynomials
to characterize moments $\mu_{n}^{q}$ with $n\geq q$, as they appear in arcs
Eq. (21). It will provide simpler expression when considering truncations to
polynomials of finite order (a finite number of moments), but does not make
any difference once polynomials of arbitrary order are taken into account.

$p(x,xJ^{2})=\sum_{k}q_{k}(x,J^{2})\left(\sum_{i,j}\beta^{k}_{i,j}x^{i+j}J^{2j}\right)^{2}$
(25)

where $q_{k}$ stem from products of the monomials in Eq. (24) defining the
domain as $q_{k}\geq 0$, and in our case belongs to the set,

$\\{1,x,xJ^{2},J^{2}x^{2},1-x,x(1-x),J^{2}x(1-x),J^{2}x^{2}(1-x)\\}\,.$

These capture terms that cannot be written as squares (recall $J^{2}$ rather
than $J$ is the variable entering in the polynomials). The conditions for each
individual $q_{k}$ can be written in compact form. For instance, for
$q_{k}=1$, integrating Eq. (25) against the positive measure leads to the
condition $\sum_{i,j,m,n}\beta^{1}_{i,j}\mu^{j+n}_{i+j+m+n}\beta^{1}_{m,n}>0$,
which implies positive definiteness of the infinitely sized matrix,

$H_{(0,0)}=\left(\begin{array}[]{c|cc|ccc|c}\mu_{0}^{0}&\mu_{1}^{0}&\mu_{1}^{1}&\mu_{2}^{0}&\mu_{2}^{1}&\mu_{2}^{2}&\cdots\\\
\hline\cr\mu_{1}^{0}&\mu_{2}^{0}&\mu_{2}^{1}&\mu_{3}^{0}&\mu_{3}^{1}&\mu_{3}^{2}&\cdots\\\
\mu_{1}^{1}&\mu_{2}^{1}&\mu_{2}^{2}&\mu_{3}^{1}&\mu_{3}^{2}&\mu_{3}^{3}&\cdots\\\
\hline\cr\mu_{2}^{0}&\mu_{3}^{0}&\mu_{3}^{1}&\mu_{4}^{0}&\mu_{4}^{1}&\mu_{4}^{2}&\cdots\\\
\mu_{2}^{1}&\mu_{3}^{1}&\mu_{3}^{2}&\mu_{4}^{1}&\mu_{4}^{2}&\mu_{4}^{3}&\cdots\\\
\mu_{2}^{2}&\mu_{3}^{2}&\mu_{3}^{3}&\mu_{4}^{2}&\mu_{4}^{3}&\mu_{4}^{4}&\cdots\\\
\hline\cr\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\end{array}\right)\succ
0$ (26)

where the indices of $H_{(0,0)}$ denote the indices of the first entry, and
the blocks correspond to monomials of given order $k$ in $x$:
$x^{k-i}(xJ^{2})^{i},i=0,\dots,k$.

Taking into account the other $q_{k}$, we find the conditions on the shifted
matrices,

$\displaystyle H_{(0,0)}\succ 0\,\quad H_{(1,0)}\succ 0\,\quad H_{(1,1)}\succ
0\quad H_{(2,1)}\succ 0$ $\displaystyle H_{(0,0)}-\hat{s}^{2}H_{(1,0)}\succ
0\,,\quad H_{(1,0)}-\hat{s}^{2}H_{(2,0)}\succ 0\,,\quad$ $\displaystyle
H_{(1,1)}-\hat{s}^{2}H_{(2,1)}\succ 0\,.$ (27)

Contrary to the 1D moment problem Bellazzini  _et al._ (2021), in 2D and
higher there is no optimal solution involving only a finite number of moments
(the truncated 2D moment problem is solved only asymptotically). This means
that in order to find the exact bounds satisfied by, e.g., moments up to order
two in $x$ and $xJ^{2}$,
$\\{\mu_{0}^{0},\mu_{1}^{0},\mu_{1}^{1},\mu_{2}^{0},\mu_{2}^{1},\mu_{2}^{2}\\}$,
we still need to compute infinitely many bounds involving infinitely many
moments, and then project into the finite subset. Because of Sylvester’s
criterion, positive definiteness in Eq. (27) implies also positive
definiteness of all finite-size principal minors, corresponding to matrices
$H_{(i,j)}$ of finite size. This necessary but not sufficient condition leads
to conservative bounds. For instance, the bounds in Eq. (26) involving only
moments up to $\mu_{2}^{2}$ are

$\displaystyle\left(\begin{array}[]{c|cc}\mu_{0}^{0}&\mu_{1}^{0}&\mu_{1}^{1}\\\
\hline\cr\mu_{1}^{0}&\mu_{2}^{0}&\mu_{2}^{1}\\\
\mu_{1}^{1}&\mu_{2}^{1}&\mu_{2}^{2}\end{array}\right)\succ
0\,\quad\mu_{1}^{0}>0\,\quad\mu_{1}^{1}>0\,\quad\mu_{2}^{1}>0$ (31)
$\displaystyle\mu_{0}^{0}-\hat{s}^{2}\mu_{1}^{0}>0\,\quad\mu_{1}^{0}-\hat{s}^{2}\mu_{2}^{0}>0\,\quad\mu_{1}^{1}-\hat{s}^{2}\mu_{2}^{1}>0$
(32)

and represent a simple (albeit not optimal) subset of all bounds.

###  III.B Bounds on Wilson Coefficients

Now, near-forward bounds on Wilson coefficients stem from comparing, order by
order in $t$, IR and UV definitions of arcs in terms of moments. Using Eq.
(10) and expanding the loop contribution Eq. (II.A), we find from
$a_{0}(\hat{s},t)$,

$\displaystyle g_{2,0}+\frac{\beta_{4}\hat{s}^{2}}{2}=\mu_{0}^{0}\,,\quad-
g_{3,1}-\frac{61}{42}\beta_{4}\hat{s}=\mu_{1}^{1}-\frac{3}{2}\mu_{1}^{0}\,,$
$\displaystyle
2g_{4,0}+\beta_{4}\left(2\log\hat{s}+\frac{\log(m^{2}/\hat{s})}{42}\right)=\frac{\mu_{2}^{2}}{4}-2\mu_{2}^{1}+2\mu_{2}^{0}\,.$
(33)

The term proportional to $\log m^{2}$ represents the leading effect at finite
mass, which acts here as a regulator for the otherwise divergent expression as
$t\rightarrow 0$.

From Eq. (32) (in particular $\mu_{1}^{1}>0$ and
$\mu_{0}^{0}-\hat{s}^{2}\mu_{1}^{0}>0$) we read the upper bound

$g_{3,1}\hat{s}<\frac{3}{2}g_{2,0}-\frac{10}{7}\beta_{4}\hat{s}^{2}$ (34)

which for $\beta_{4}\to 0$ reduces to the tree-level values discussed in de
Rham  _et al._ (2017); Bellazzini  _et al._ (2020).

On the other hand the moments $\mu_{1}^{1},\mu^{2}_{2}$ do not have upper
bounds in Eq. (32), and consequently there appears to be no lower bound on
$g_{3,1}$.444This is a consequence of the $\ell$ domain being non-compact:
$\mu_{n}^{q}$ can be larger and larger as $q$ increases. In contrast, $0\leq
x\leq 1$, and moments $\mu_{n}^{q}$ are monotonically decreasing in $n$.  A
lower bound stems from realizing that, because of full $s-t-u$ crossing
symmetry Tolley  _et al._ (2020); Caron-Huot  _et al._ (2021), $g_{4,0}$
appears also in the second arc at $t=0$,

$a_{1}=g_{4,0}+\beta_{4}\log\hat{s}=\mu_{2}^{0}\,.$ (35)

Comparing this to the second line in Eq. (33) leads to a null constraint
which, taking into account loop effects, reads,

$\mu_{2}^{2}=8\mu_{2}^{1}+\frac{2\beta_{4}}{21}\log\frac{m^{2}}{\hat{s}}\,.$
(36)

Null constraints relate higher and lower moments in $J^{2}$ and, together with
bounds on moments, lead also to a lower bound on $g_{3,1}$ Caron-Huot and Van
Duong (2021); Tolley  _et al._ (2020); Sinha and Zahed (2021); Chiang  _et
al._ (2021). The simplest way to see this is to combine the null constraint
with the condition $\mu_{1}^{1}-\hat{s}^{2}\mu_{2}^{1}>0$ and
$\mu_{2}^{2}\mu_{0}^{0}>(\mu_{1}^{1})^{2}$ (the latter follows from positivity
of the minors in the positive definite matrix of Eq. (32)), to obtain,

$g_{3,0}\hat{s}>-4g_{2,0}\left(1+\sqrt{1-\frac{g_{2,0}\hat{s}^{2}\log\frac{m^{2}}{\hat{s}}}{240\times
16\pi^{2}}}\right)-\frac{10}{7}\beta_{4}\hat{s}^{2}\,.$ (37)

In absence of loop effects, this reduces to $g_{3,0}\hat{s}>-8g_{2,0}$;
instead using all relations from Eq. (32) we find
$g_{3,0}\hat{s}>-6.5g_{2,0}$; finally, using moments up to $\mu_{6}^{6}$, we
find $g_{3,0}\hat{s}\gtrsim-5.3g_{2,0}$ in agreement with Caron-Huot and Van
Duong (2021); Tolley  _et al._ (2020); Sinha and Zahed (2021); Chiang  _et
al._ (2021) – this is an example of how the 2D moment problem converges as
more and more moments are taken into account.

We show these bounds, as function of $\log m^{2}\hat{s}$, in Fig. 1.555At
fixed $m$, with more moments, the lower bound improves, because the solution
to the 2D moment problem is more precise, but it still diverges as $m\to 0$.
In contrast to the tree-level limit, here higher null constraints (from
expressing any $g_{p,q}$ in terms of different combinations of moments,
similarly to Eq. (36)) do not improve the bound, because higher
$t$-derivatives of Eq. (II.A) are more and more singular
$\partial^{k}_{t}a_{0}\sim m^{-2(k-2)}$ as $m\to 0$. The interesting feature
is that in the exact $m=0$ limit, although the upper bound is untouched, the
lower bound disappears completely.

Nevertheless, for practical purposes, in the case of the Goldstone, as soon as
the mass is finite the impact of loop effects is limited. Indeed, even
considering the most favorable phenomenological conditions for the EFT, in
which the cutoff is at the Planck scale $\hat{s}=M_{pl}^{2}$, while the mass
of the particle is at the lowest testable scale (Hubble) $m=H_{0}$, the
logarithm is of order $\log\frac{m^{2}}{\hat{s}}\approx-280$ and the
departures from the tree-level limit are of the size of a loop factor $\sim
g_{2,0}^{2}s^{2}/16\pi^{2}$.

Figure 1: Bounds on ratio $g_{3,1}\hat{s}/g_{2,0}$ as a function of the
particle mass $m$, for fixed $\beta_{4}\hat{s}^{2}=0.1$. Lower lying curves
correspond to bounds involving more moments in the UV: in orange only Cauchy-
Schwartz Eq. (37), in red moments ut to $\mu_{2}^{2}$ as in Eq. (32), and in
blue moments up to $\mu_{4}^{4}$ (numerical). All curves diverge as
$m\rightarrow 0$.

In the approach of this section, positive functions are approximated by
polynomials. This is a strength, because they provide a simple and systematic
path to positivity. At the same time, polynomials are the weak link of this
approach, as they do not converge uniformly to continuum functions on non-
compact domains. For instance, consider the combination of moments associated
with integrating $\cos J^{2}$ over the measure $d\mu$ in Eq. (22),
$\mu^{cos}\equiv\sum_{n=0}^{\infty}(-1)^{n}\mu_{0}^{2n}/(2n!)$. Clearly
$\mu^{cos}<\mu_{0}^{0}$, since $|\cos{J^{2}}|<1$; yet this feature is
invisible at any finite order of moments. In the next section we provide an
alternative method to extract bounds, that attacks directly the integrand
boundedness.

##  IV Dispersion Relations at large $-t$

In this section, we assume the amplitude to be analytic in the cut complex $s$
plane for fixed large negative $t$ (within the EFT validity) – as often
(though not always Guerrieri and Sever (2021)) assumed in the modern S-matrix
bootstrap approach Paulos  _et al._ (2017a, b, c). This provides a way of
regulating the singularities using finite $t$ rather than mass. Analyticity in
$s$ for finite $t<0$ has been rigorously proven Bros  _et al._ (1965), except
for a region of size $\sim(-t)^{3}$ around the origin. For $t$ within the
range of EFT validity, the amplitude is explicitly calculable and known, and
it displays in fact no non-analyticity.

The extended domain of analyticity enables us to study dispersion relations at
fixed large negative $t$, without the need to expand them around $t\approx 0$.
Unfortunately, contrary to analogous quantities defined at $t=0$, Legendre
polynomials are not positive, and therefore the integrand Eq. (5) is _not
positive_. For $-\hat{s}<t\leq 0$, however, the integrand
$I_{t}(\ell,{\hat{s}^{\prime}})$ is _bounded from above and below_ , because
the Legendre polynomials are themselves bounded for all even $\ell$ and
${\hat{s}^{\prime}}\in[\hat{s},\infty[$

$-\frac{1}{2}\leq\min_{\ell,{\hat{s}^{\prime}}}P_{\ell}(1+\frac{2t}{{\hat{s}^{\prime}}\\!\\!-\\!2m^{2}})\leq
P_{\ell}(\cos\theta)\leq 1\,.$ (38)

Since the integrand is bounded, we can pull it out of the integral and bound
arcs $a_{n}(\hat{s},t^{\prime})$ in terms of arcs at $t=0$ (where
$P_{\ell}(\cos\theta)=P_{\ell}(1)=1$),

$a_{n}(\hat{s},0)=\frac{2}{\pi}\int_{\hat{s}}^{\infty}\sum_{\ell=0}^{\infty}\frac{d{\hat{s}^{\prime}}\textrm{Im}\hat{f}_{\ell}({\hat{s}^{\prime}})}{\hat{s}^{\prime\,2n+3}}=\mu_{2n}^{0}\,,$
(39)

which, as shown above, are strictly positive moments of a 1D distribution
Bellazzini  _et al._ (2020).

Combining Eqs. (5-39) we find the constraint,

$\min_{\ell,{\hat{s}^{\prime}}}I^{n}_{t}(\ell,{\hat{s}^{\prime}})\leq\frac{a_{n}(\hat{s},t)}{a_{n}(\hat{s},0)}\leq\frac{\hat{s}+\frac{t}{2}}{[\hat{s}(\hat{s}+t)]^{n+2}}\,.$
(40)

Here, for all $t$, the upper bound is exactly saturated in $\ell=0$ or
${\hat{s}^{\prime}}\to\infty$ (corresponding to $\cos\theta=1$),
$P_{0}(\cos\theta)=P_{\ell}(1)=1$. Instead, for generic value of $t/\hat{s}$
the lower bound is determined by different points in the
$\ell,{\hat{s}^{\prime}}$ domain: for $t/\hat{s}\sim-1/2$,
$\min_{\ell,{\hat{s}^{\prime}}}I_{t}(\ell,{\hat{s}^{\prime}})$ is saturated by
the $\ell=2$ polynomial, and as $t/\hat{s}\to 0$ or $1$ it is saturated by
larger and larger values of $\ell$.

The region excluded is illustrated for $n=0$ by the grey area in Fig. 2 (the
black lines correspond to the simple $t$-independent bound $-\frac{1}{2}\leq
P_{\ell}(\cos\theta)\leq 1$).

Figure 2: Bounds on arc $a_{0}(\hat{s},t)$ as a function of $t$. In gray the
region excluded by UV positivity bounds (in black the region excluded by the
simple approximation on the extremes of Eq. (38)). Orange dashed lines
illustrate IR arcs calculated with the approximated EFT ($a_{0}(\hat{s},t)$ in
Eq. (10) truncated at $O(t)$), for extremal values of $g_{3,1}$.

We can now compare the arcs computed within the EFT Eq. (10) with the arcs
bounded by UV unitarity/causality in Eq. (40). Calling $s_{max}$ the theory’s
cutoff, we will first consider the kinematics,

$|t|\ll\hat{s}\ll\hat{s}_{max}\,.$ (41)

The second inequality labels the regime where loop effects are under control,
even in strongly coupled theories.

We first discuss the tree-level limit in which we neglect these effects
altogether such that the IR amplitude is well described by Eq. (8). We then
invoke the first inequality in Eq. (41) so that the first arc is well
approximated by,

$a_{0}(\hat{s},t)\approx g_{2,0}-tg_{3,1}\,.$ (42)

and higher-order terms can be neglected.666 While neglecting higher orders is
a customary assumption in the context of EFTs, it is plausible that $g_{3,1}$
is not suppressed w.r.t. $g_{2,0}$ in units of $\hat{s}$, while higher order
terms are: a situation that corresponds to a system with approximate Galileon
symmetry). This is possibility that we are exploring under assumption Eq.
(42).

The approximate IR tree-level EFT arc is illustrated in Fig. 2, by lines with
steepness $-\hat{s}g_{3,1}/g_{2,0}$: its extrema (orange dashed) are found by
requiring that $a_{0}(\hat{s},t)$ lies within the UV allowed region Eq. (40),
for all values of $-1\leq t/\hat{s}\leq 0$. Equating IR and UV arcs, and
dividing by $a_{0}(\hat{s},0)$, we find,

$\frac{3}{2}>\hat{s}\frac{g_{3,1}}{g_{2,0}}>-\min_{t}\left(\frac{\min_{\ell,{\hat{s}^{\prime}}}I^{0}_{t}-1}{-t}\right)\,.$
(43)

The upper bound is saturated by the $\ell=0$, ${\hat{s}^{\prime}}=\hat{s}$ UV
contribution, corresponding to a weakly coupled scalar with mass
$M^{2}=\hat{s}$. This bound comes from the near forward limit, where our
assumption of neglecting higher order terms is exact. It can be found
analytically by comparing $\partial|_{t=0}a_{0}(\hat{s},t)$ in the UV (the rhs
of Eq. (40)) and the IR, Eq. (42), and is in fact equivalent to the moment
problem approach.

The lower bound instead requires $\min_{\ell,{\hat{s}^{\prime}}}I^{0}_{t}$,
which implies finding the minima of (Legendre) polynomials, which is saturated
by larger $\ell$ as $t/\hat{s}\to 0$. Nevertheless, the bound Eq. (43) is
dominated by finite $|t|/\hat{s}$ where
$\min_{\ell,{\hat{s}^{\prime}}}I^{0}_{t}$ is at the interception of the
$\ell=2$ and $\ell=4$ contributions, which can be found by solving a 4th order
equation. It leads to,

$\frac{\hat{s}g_{3,1}}{g_{2,0}}\gtrsim-4.9\,,$ (44)

and is dominated by the region $t\approx-0.3\hat{s}$, which explains why
$s$-analyticity at large negative $-t\gg m^{2}$ is necessary. Notice that this
is a preliminary result that relies on neglecting higher order terms. We will
come back in the next section to a way (proposed in Caron-Huot  _et al._
(2021)) to partially circumvent the question of higher order terms, at least
at tree-level.

Figure 3: Bounds on the ratio $g_{3,1}\hat{s}/g_{2,0}$ as a function of the
coupling $g_{2,0}\hat{s}^{2}$. Grey area/dashed line: bounds neglecting terms
$O(t^{2})$ in $a_{0}(\hat{s},t)$ (but using the full loop contribution Eq.
(II.A)). Solid lines: bounds from improved arcs, adding Eq. (48) and Eq. (51).

We now turn to loop effects, as captured by Eq. (II.A). Despite its non-
analyticity, both the amplitude and its first derivative remain finite at all
$s,t,u$ within the EFT validity. This is important, because the bounds
presented in the previous paragraph depend on the first derivative of
$a_{0}(\hat{s},t)$ at $t=0$ and on the amplitude at finite
$t\approx-0.3\hat{s}$, where loop effects have a finite impact. Ignoring terms
at order $(t/\hat{s})^{2}\sim(0.3)^{2}$, loop effects are captured by the
substitution,

$\frac{\hat{s}g_{3,1}}{g_{2,0}}\to\frac{\hat{s}g_{3,1}+\frac{61}{42}\beta_{4}\hat{s}^{2}}{g_{2,0}+\frac{\beta_{4}\hat{s}^{2}}{2}}$
(45)

in Eq. (44). We illustrate this in Fig. 3 (the area between the upper solid
line and the lower dashed line). Contrary to bounds on moments, here loop
effects have a finite impact on the lower bound, which survives also in the
massless limit. In practice, the singularity is regulated, rather than by the
mass, by the finite value of $t$ which happens to determine the lower bound.

###  IV.A Resummed Higher Order Terms

Figure 4: As in Fig. 2, but using improved arcs $a^{imp}_{0}(\hat{s},t)$, and
comparison with the now exact expression Eq. (48) (orange) with extremal
values given in Eqs. (50).

The previous paragraph relies on the approximation that at small $|t|\ll s$
the higher order terms can be neglected, see Eq. (42). This assumption can be
made rigorous using the near forward bounds from Bellazzini  _et al._ (2020),
which imply that $0<g_{2n,0}\hat{s}^{2n-2}<g_{2,0}$ and
$(2n+1)g_{2,0}\hat{s}^{2}/2>g_{2n+1,1}\hat{s}^{2n+1}>-g_{3,1}\hat{s}$.777Notice
that, even without crossing symmetry, terms that vanish in the forward
amplitude still contribute at loop level to forward arcs, and are bounded in
absolute value to be smaller than a loop factor times $g_{2,0}$ Nicolis  _et
al._ (2010); Bellazzini  _et al._ (2018). Alternatively, _at tree-level_ ,
higher Wilson coefficients can be eliminated altogether exploiting crossing
symmetry, via the “improved” arcs defined in Ref. Caron-Huot  _et al._ (2021),
leading to expressions that are valid for large angle scattering,

$-t\leq\hat{s}\ll\hat{s}_{max}\,,$ (46)

to be contrasted with Eq. (41).

Improved arcs rely on the full $s-t-u$ crossing symmetry of the amplitude to
write terms of order $O(t^{2})$ or higher as _forward_ higher arcs $a_{n\geq
1}(\hat{s},0)$ or their first $t$-derivatives $\partial_{t}a_{n\geq
1}(\hat{s},t)|_{t=0}$. These can then be subtracted from both the IR and the
UV representation of arcs. The resulting improved arc Caron-Huot  _et al._
(2021),

$a^{imp}_{0}=a_{0}(s,t)-\sum_{n\geq 1}(na_{n}-\partial_{t}a_{n})|_{t=0}\,,$
(47)

is defined in the IR and UV by,

$a^{imp}_{0}(\hat{s},t)=g_{2,0}-tg_{3,1}=\frac{2}{\pi}\int_{\hat{s}}^{\infty}\\!\\!\sum_{\ell=0}^{\infty}\frac{d{\hat{s}^{\prime}}\textrm{Im}\hat{f}_{\ell}({\hat{s}^{\prime}})}{\hat{s}^{\prime
3}}I^{imp}_{t}\,,$ (48)

with

$I^{imp}_{t}\equiv\frac{(2{\hat{s}^{\prime}}+t)P_{\ell}(\cos\theta)}{2({\hat{s}^{\prime}}+t)^{2}}-t^{2}\left(\frac{3t+4{\hat{s}^{\prime}}}{2({\hat{s}^{\prime}}+t)^{2}}+\frac{\ell(\ell+1)t}{\hat{s}^{\prime
2}-t^{2}}\right)$ (49)

where $\ell(\ell+1)$ stems from the derivative of $P_{\ell}$ in the forward
limit.

We can then proceed as before: for fixed values of $-\hat{s}\leq t\leq 0$ we
can find the extrema of $I^{imp}_{t}$ in ${\hat{s}^{\prime}}$ and $\ell$. The
maximum of $I^{imp}_{t}$ is discontinuous in $t$, since for $t<0$ it is
dominated by the (positive) term proportional to $\ell(\ell+1)$, which
diverges as $\ell\to\infty$; at $t=0$ instead, it is finite and has slope
$-3/2$ (saturated by $\ell=0$ and ${\hat{s}^{\prime}}=\hat{s}$). The minimum
of $I^{imp}_{t}$, similarly to the previous paragraph, is dominated in the
interesting region by the interplay of the $\ell=2$ and $\ell=4$ polynomials.
The result is illustrated in Fig. 4: improved UV arcs must lie within the
allowed (non-gray) region, for all values of $t$.

At tree-level, a comparison with the (now) exact IR arc Eq. (48) leads to

$\frac{3}{2}>\hat{s}\,\frac{g_{3,1}}{g_{2,0}}\gtrsim-5.18\,\,,$ (50)

compatibly with Caron-Huot and Van Duong (2021); Tolley  _et al._ (2020);
Caron-Huot  _et al._ (2021); Chiang  _et al._ (2021).888In this approach we
have not included higher null constraints. Notice that this bound appears
weaker than the non-improved one Eq. (44), which ignored $O(t^{2})$ terms.

The tree-level improved bound appears to be somewhat sharper, since higher-
order tree-level Wilson coefficients have been eliminated exactly from both IR
and UV parts of the dispersion relation. Once IR effects are taken into
account, however, these higher-order Wilson coefficients reappear into arcs.
For instance, while the term $\propto g_{4,0}$ is canceled from
$a_{0}(\hat{s},t)-2t^{2}a_{1}(\hat{s},0)-3t^{4}a_{2}(\hat{s},0)-\cdots$, there
is a loop effect $\propto g_{2,0}g_{4,0}/16\pi^{2}$ which does not cancel. It
is not possible to eliminate $g_{4,0}$ altogether from $a_{0}(\hat{s},t)$.

So, contrary to tree-level arcs that could be improved into compact
expressions involving a finite number of Wilson coefficients, loop level
effects propagate all order coefficients into arcs. We will have to assume
that these terms are small for all values of $s$ at which we evaluate bounds.
We express this assumption as $\hat{s}\ll\hat{s}_{max}$ in Eq. (46).

Under this assumption, we can focus on the most relevant term, discussed
already in Eq. (12), and compute the improved arc using the algorithm Eq.
(47). In addition to the first expression in Eq. (48), we find,

$\delta
a_{0}^{imp}=\frac{g_{2,0}^{2}}{16\pi^{2}}\left(\frac{-21s^{2}+61ts+t^{2}\log\left(1-\frac{s}{t}\right)}{60}-\frac{2t^{3}}{3s}\right)\,.$
(51)

instead of the non-improved version Eq. (II.A). Including this term in the
EFT, we can rederive the bound Eq. (50), which we show as the solid line in
Fig. 4. The improved result is very similar to the approximated one.

Importantly, the improvement algorithm Eq. (47) involves higher arcs and their
first derivatives at $t\to 0$. As argued in Eq. (II.A), these are finite,
despite being evaluated in the forward kinematics. This result is not a priori
obvious. Moreover, thank to Eq. (16), it extends also to improvements of
higher arcs. Indeed, consider for example $g_{6,2}$. The necessary improvement
(the analog of Eq. (48)) to eliminate all terms $g_{n,2}$, $n>6$ from
$a_{1}(\hat{s},t)$, involves now second derivatives of higher arcs at $t=0$,
$\partial^{2}_{t}a_{n}(\hat{s},t)|_{t=0,n\geq 2}$. Eq. (16) implies that these
are all regular. More generally, terms $\propto(stu)^{q}$ in Eq. (8), appear
first in arc $a_{q-1}(\hat{s},t)$ (see footnote 1). Its improved version
requires $\partial^{q}|_{t=0}a_{n}(\hat{s},t),\,n\geq q$, which is regular
according to Eq. (16).

These results give strength to the methods pioneered in Caron-Huot  _et al._
(2021): although designed for tree-level amplitudes, they hold also at loop-
level for the simple scalar theory with no gravity we have considered in this
work.

##  V Conclusions and Outlook

We have studied the impact of calculable IR loop effects on EFT positivity
bounds, focusing on the effect of IR singularities in the massless limit. This
largely extends the initial investigations in Bellazzini  _et al._ (2020);
Arkani-Hamed  _et al._ (2020) which focused on IR-finite contributions as
running effects. The role of IR-sensitive loop correction studied in the
present work is actually more relevant for bounds relying on full crossing
symmetry of the amplitude Tolley  _et al._ (2020); Caron-Huot and Van Duong
(2021), such as the lower bound on the coefficient ratio
$g_{3,1}\hat{s}/g_{2,0}$. The singular behavior stems from a tension between
the need of probing the EFT at large enough energies $\hat{s}$, where bounds
on Wilson coefficients are stronger, and small enough $t$ where the amplitude
is analytic.

If $s$-analyticity of the amplitude is granted only for a limited range of
values taken by $t$, $|t|\lesssim m^{2}$, we have shown how to extract the
bounds, extending the approach of Bellazzini  _et al._ (2020) to the 2D moment
problem, as in Chiang  _et al._ (2021). One-loop effects involving marginal
couplings $\lambda\phi^{4}$ have no qualitative impact on the bounds. On the
other hand, loops involving the coupling $g_{2,0}$ induce a $\log
m^{2}/\hat{s}$ divergence in the derivative of the arcs, which invalidates the
bound for $m\to 0$. Since $g_{2,0}$ and $g_{3,1}$ do not run, the divergence
is physical.

If instead the amplitude is analytic also for large negative $t$, then all the
bounds are robust. We have shown this by introducing a novel and simple
analytic understanding of dispersion relations at finite $t$ (which complement
the numerical approach of Caron-Huot  _et al._ (2021)), represented by the
_trumpet_ in Fig. 2, which can be applied to any amplitude regular in the
forward limit. In this approach, the upper bound on $g_{3,1}$ is shown to be
dominated by the dispersion relation at $t/\hat{s}\approx-0.3$, where the
above singularities are absent. Moreover, this approach makes little use of
crossing symmetry, whose practical purpose is to guarantee that the higher
order coefficients $g_{n,1}(n>4)$ are bounded by the lower coefficients. It
would be interesting to bring this analytic approach into a more systematic
tool to approach all bounds.

Moreover, the improvement procedure that removes higher order terms, still
relies on evaluating infinitely many dispersion relations in the forward
limit. We have studied the structure of all 1-loop effects in the scalar
theory and found that, interestingly, all the necessary forward limits are
regular. It would be interesting to repeat this analysis in more complex
theories, including $\phi^{3}$, flavor or gravity, to establish the robustness
of this mixed forward/non-forward procedure. Moreover, the study of higher
loops of more relevant couplings as well theories with exactly massless
particles, such as Yang-Mills theory, might reveal more singular behaviors.
Overall, it would be useful to develop an improvement procedure without the
forward limit.

## Acknowledgments

We are very grateful to J. Elias-Miro for collaboration in the earl stages of
this work, and we thank R. Rattazzi and A. Zhiboedov for discussions. F.R. is
particularly grateful to J.B. Lasserre, for numerous exchanges on the higher
dimensional moment problem. The work of FR is supported by the Swiss National
Science Foundation under grant no. PP00P2-170578.

##  Appendix A Discrete Moments

Treating the distribution in $J^{2}$ as continuous gives a convenient and
manageable way to setting conservative constraints on the arcs without having
to truncate the expansion in Legendre polynomials $P_{\ell}$ up to some
$\ell_{max}$. However, we remark that it excludes polynomials that might be
negative between integer values of $\ell$, so the bounds are not optimal. In
Chiang  _et al._ (2021) the condition is imposed via a set of positive
determinants. In this Appendix propose a simple way to systematically improve
the bounds.

The two-dimensional moments

$\mu_{n}^{j}=\frac{2}{\pi}\int_{s}^{\infty}\frac{ds}{s}\text{Im}f_{\ell}(s)\sum_{\ell}\frac{(\ell(\ell+1))^{j}}{s^{n+2}}$
(52)

can be split as a sum of two terms

$\displaystyle\mu_{n}^{j}$ $\displaystyle=$
$\displaystyle\sum_{\ell}^{L-1}(\ell(\ell+1))^{j}\frac{2}{\pi}\int_{s}^{\infty}\frac{ds}{s}\text{Im}f_{\ell}(s)\frac{1}{s^{n+2}}$
(53) $\displaystyle+$
$\displaystyle\frac{2}{\pi}\int_{s}^{\infty}\frac{ds}{s}\int_{L(L+1)}^{\infty}f(J^{2},s)\frac{(\ell(\ell+1))^{j}}{s^{n+2}}$
(54)

In the first term, we are considering the two dimensional moment problem in a
grid of $L$ sites, each site being a one-dimensional moment problems with a
different measure $\text{Im}f_{\ell}(s)$. This automatically imposes the
constraints for all moments above $j>L$.

The grid in $J^{2}$ is specified by the zeroes of

$g(J^{2})=\prod_{i=0}^{L-1}(J^{2}-i(i+1))$ (55)

and gives a relation between $L$ different moments. Explicitly, for $L=2$,
$g(J^{2})=J^{2}(J^{2}-2)$ and $\mu_{n}^{j+2}-2\mu_{n}^{j+1}=0$, so there are
only two ($L$, in general) independent moments in $j$, $\mu_{n}^{0}$ and
$\mu_{n}^{1}$. Writing the two-dimensional problem as continuous and then
imposing the grid constraints in Eq. (55) is equivalent of considering $L$
different one-dimensional problems. We find more convenient using the latter
since, besides being more physical, one can impose the upper bound on the
partial waves $\text{Im}f_{\ell}\leq 16\pi(2\ell+1)$,

$\frac{2}{\pi}\int_{s}^{\infty}\frac{ds}{s}\frac{\text{Im}f_{\ell}(s)}{s^{n+2}}\,<\,\frac{1}{s^{n+2}}\frac{32(2\ell+1)}{n+2}.$
(56)

The second term in Eq. (53) is the two-dimensional moment problem already
described in the main text. Notice that now the domain is given by

$x\geq 0\,,\quad 1-x\geq 0\,,\quad J^{2}-L(L+1)\geq 0\,.$ (57)

instead of Eq. (24). This maintains information for asymptotic partial waves,
and the approximation of not considering negative polynomials between $\ell$
above $L$ is extremely good already for relatively low $L\sim 4$ in the
examples considered if numerical precision is desired, but setting $L=0$ as
done in the main text allows for a simple and accurate analytical
understanding.

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