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HHPPP XXX hhhh (((( ��� IFT Instituto de F́ısica Teórica

Universidade Estadual Paulista

DISSERTAÇÃO DE MESTRADO IFT–D.001/19

Quantum Field Theory at high multiplicity:
The Higgsplosion mechanism.

Carlos Henrique Costa Duarte de Lima

Advisor

Eduardo Pontón

July, 2019


               Lima, Carlos Henrique Costa Duarte de 

 L732q            Quantum field theory at high multiplicity: the Higgsplosion mechanism / 
Carlos Henrique Costa Duarte de Lima. – São Paulo, 2019 

91 f.: il. (algumas color.) 
 

Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), 
Instituto de Física Teórica (IFT), São Paulo 

Orientador: Eduardo Pontón 
 

1. Teoria quântica de campos. 2. Higgs, Bosons de.  3. Teoria quântica 
relativista I. Título  

 
Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca 

do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo 
autor(a). 

 
QUANTUM FIELD THEORY AT HIGH MULTIPLICITY: 

THE HIGGSPLOSION MECHANISM 

 
Dissertação de Mestrado apresentada ao 
Instituto de Física Teórica do Câmpus de 
São Paulo, da Universidade Estadual 
Paulista “Júlio de Mesquita Filho”, como 
parte dos requisitos para obtenção do título 
Mestre em Física, Especialidade Física 
Teórica.  

 
Comissão Examinadora: 

 
Prof. Dr. RICADO D’ELIA MATHEUS (Orientador) 

Instituto de Física Teórica/UNESP  

 
Prof.  Dr. GASTÃO INÁCIO KREIN 

 Instituto de Física Teórica/UNESP 

 
Prof. Dr. ERICO BERTUZZO 

IFUSP/Instituto de Física da USP 

 
Conceito: Aprovado 

 
São Paulo, 26 de julho de 2019. 


To the memory of Eduardo Pontón


ii

Acknowledgments

Firstly, I would like to express my sincere gratitude to my advisor Prof. Eduardo Pontón, for

the continuous support of my MSc study and related research, for his patience, motivation,

and immense knowledge. His guidance helped me in all the time of research and writing of this

thesis. I could not have imagined having a better advisor and mentor for my MSc study.

I would like to say a massive thank you to Prof. Ricardo D’Elia Matheus for the fantastic

courses that I attended, and the continuous help throughout my formation in the Institute of

Theoretical Physics (IFT).

My thanks also go out to the support I received from Prof. José Helayël Neto and Prof.

Sebastião Alves Dias from Brazilian Center for Research in Physics (CBPF). Having always

the door open for me, and giving me opportunities to improve as a physicist.

I gratefully acknowledge the funding received towards my MSc from CAPES.

I am indebted to all my friends at IFT. The uncountable coffee hours discussions, and who

were always so helpful in numerous ways. Special thanks to Victor, Dean, Maria Gabriela,

Guilherme, Andrei, Renato, Vinicius, Lucas, Matheus, Felippe and André.

Some special words of gratitude go to my friends from UERJ, who have always been a major

source of support: Leonardo, João Gabriel, Lucas Braga, Lucas Toledo and Nathan.

I must express my very profound gratitude to my parents for providing me with unfailing

support and continuous encouragement throughout my years of study and through the process

of researching and writing this thesis. This accomplishment would not have been possible

without them. Thank you.

And finally to Camila Fernandes, who has been by my side throughout this MSc and all

this trajectory, living every single minute of it, and without whom, I would not have had the

courage to embark on this journey in the first place. I can only be grateful for your support for

all that time and more to come.


iii

Resumo

O presente trabalho busca entender o que acontece com uma teoria Quântica de Campos

quando estamos em um regime de alta multiplicidade. A motivação para esta busca é em

grande parte vinda de um novo (2017) proposto mecanismo que ocorreria em teorias escalares

neste regime: Higgsplosion. Será revisado o que se conhece até então dos calculos perturbativos

e alguns outros resultados vindo de aproximações semiclassicas. Por fim, será estudado qual as

consequencias desse mecanismo para uma teoria escalar e se pode haver contribuições para o

Modelo Padrão. O foco deste trabalho é entender se esse mecanismo realmente pode acontecer

em uma teoria de campos usual, essa pergunta será respondida no regime perturbativo pois

uma resposta mais geral ainda é desconhecida. Adiconalmente, uma nova posśıvel interpretação

do mecanismo de Higgsplosion é proposta e discutida.

Palavras Chaves: Teoria Quântica de Campos; Boson de Higgs; Teoria Quântica relativista

Áreas do conhecimento: Ciências Exatas e da Terra; F́ısica Teórica


iv

Abstract

The current work seeks to understand what happens to a Quantum Field Theory when we

are in the high multiplicity regime. The motivation for this study comes from a newly (2017)

proposed a mechanism that would happen in scalar theories in this limit, the Higgsplosion. We

review what it is known so far about the perturbative results in this regime and some other

results coming from different approaches. We study the consequences of this mechanism for a

normal scalar theory and if it can happen in the Standard Model. The goal is to understand

if this mechanism can really happen in usual field theory, this question will be answered in

the perturbative regime because a more general solution is still unknown. Aditionally, a new

possible interpretation for the Higgsplosion mechanism is proposed and discussed.

Key words: Quantum Field Theory; Higgs Boson; Relativistic Quantum Theory

Areas: Natural Sciences; Theoretical Physics.


v

Contents

1 Introduction 1

1.1 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Dyson Resummation and the Full Propagator . . . . . . . . . . . . . . . 7

1.1.3 Källén-Lehmann Spectral Representation . . . . . . . . . . . . . . . . . . 8

1.1.4 Functional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Perturbative Investigation of High Multiplicity Amplitudes 17

2.1 Tree Level Amplitude at Threshold . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 λϕ4

4!
in the Unbroken Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.2 λϕ4

4!
in the Broken Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 One-Loop Amplitude at Threshold . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 λϕ4

4!
in the Unbroken Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Eigenvalues and Eigenfunctions of the Poschl-Teller Potential . . . . . . . 31

2.2.3 Back to λϕ4

4!
in the Unbroken Phase . . . . . . . . . . . . . . . . . . . . . 34

2.2.4 λϕ4

4!
in the Broken Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.5 Discussion About the Factorial Growth . . . . . . . . . . . . . . . . . . . 45

2.3 Investigation of Beyond Threshold Amplitude . . . . . . . . . . . . . . . . . . . 47

2.3.1 Naive Generalization of Brown’s Methods and its Problems . . . . . . . . 47

2.3.2 Tree Level Investigation of Beyond Threshold (1→ 3) . . . . . . . . . . . 49

2.3.3 Tree Level Investigation of Beyond Threshold (1→ 5) and its Generalities 52

2.3.4 Recurrence Relations and General Results of Beyond Threshold Ampli-

tudes at High Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Higgsplosion and Higgspersion 59

3.1 The Rise of the Higgsplosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.1 The Higgsplosion Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.2 The Potential Power of the Higgsplosion . . . . . . . . . . . . . . . . . . 62

3.2 Some Questions about Higgsplosion . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.1 Criticism From “Problems with Higgsplosion” . . . . . . . . . . . . . . . 65

3.2.2 Criticism From ”Inconsistencies with Higgsplosion” . . . . . . . . . . . . 66

3.2.3 Lattice Results About Higgsplosion and Some Comments. . . . . . . . . 69


vi

3.3 Analysis of the Decay Rate Expression, What we can say about E∗ . . . . . . . 70

3.4 0-Dimensional Case of Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Exponential Decay of the Propagator, String Inspired Toy Model . . . . . . . . . 81

4 Conclusion 85


1

Chapter 1

Introduction

The discovery of the Higgs Boson [1] opened a new era for particle physics [2]. The last

fundamental piece necessary for the Standard Model (SM) to work as intended. Since its

discovery, we entered a new phase of precision measurement and confirmation of the Standard

Model. Despite its great achievements, it is known that the Standard Model cannot be the full

history. That is motivated by the lack of understanding of observational results that comes

from other sources, such as, Dark Matter [3, 4] and Dark Energy [5], which are not accounted

for by the Standard Model. Even the lack of a theoretical understanding of what Quantum

Gravity[6, 7, 8, 9] looks like can enter as evidence for the need of Beyond Standard Model

Physics(BSM). These results indicate that maybe the Standard Model is not the end.

Even if one ignores these hints, some unsolved puzzles can be identified already within the

Standard Model. One of them is the fine-tuning problem associated with the weak scale [10, 11].

Naively it is expected that the squared mass parameter of a scalar particle should be of the

order of the cutoff of the theory. In the Standard Model, the cutoff can be assumed as the

Planck scale Λp, the scale where Quantum Gravity becomes relevant:

Λp ≈ 1019GeV . (1.1)

This expectation happens because there is no symmetry to protect the theory from receiving

large contributions to the squared mass term. The story is different from a fermion particle,

where the presence of chiral symmetry in the mf → 0 limit shields the fermions from being

quadratically sensitive to the Ultra-Violet (UV). This does not occur for the scalar particle

without any additional symmetry. Thus, it is surprising that the measured Higgs mass mh ≈
125 GeV [1] and the absence of other states or signs of new physics, indicates the presence

of a scalar much lighter than a cutoff. That means the occurrence of a fine-tuning of the

contribution from BSM physics in such a way that the Higgs mass is small. The theory has

large numbers that conspire to give a small physical contribution:

m2
h ≈ m2

0 + δm2
BSM . (1.2)


2

There are a few potential solutions to the fine-tuning problem. The most famous are Super-

symmetry [12, 13, 14, 15] and Composite Higgs [16, 17, 18, 19, 20].

In this thesis, we review a new possible mechanism that can render the Higgs mass param-

eter small naturally and potentially make the Standard Model UV finite. This new mechanism

is called Higgsplosion and was proposed in 2017 by Valentin V. Khoze and Michael Span-

nowsky [21, 22]. This thesis aims to understand the proposal in detail and learn more about

Quantum Field Theory at high multiplicity as well as the applicability of ordinary perturbation

theory in such a regime. The study of Higgsplosion is intimately related to the question of what

happens to a Quantum Field Theory when we have high multiplicity processes.

Inside the Quantum Field Theory framework, people developed powerful tricks [23, 24] that

made possible to compute high multiplicity processes. In all of the computation, one finds the

unusual feature that the leading order is already growing exponentially with the number of

final states. At the time it was interpreted as implying that the perturbation theory is not

valid in this regime [25]. The leading term would not be a good approximation, and any partial

sum would not reproduce the correct answer. In this thesis, this claim is reviewed, so we can

understand precisely how the perturbation theory works in such a regime of a Quantum Field

Theory. The picture changed later when Son [26] developed a semiclassical computation to

obtain expressions for high multiplicity processes that, in principle, can be trusted in a fixed

limit of the theory. The decay rate for a high multiplicity process obtained in [26] had an

exponential form, but in the region of applicability it does not have an exponential growth.

The next breakthrough came when these semiclassical computations were generalized to the

strong ’t Hooft like coupling (λn) regime for ϕ4 in the broken phase in (1+3)D. In such a limit,

the same behavior of exponential growth of some objects at high multiplicity appears. This

fact is what motivated Valentin V. Khoze and Michael Spannowsky to propose the Higgsplosion

mechanism. It was not well understood what the limitations of their results were, and we

discuss this in detail. These result gave strong evidence that Higgsplosion may happen at least

in this model, which we discuss also. In the Higgsplosion mechanism, these results are used

together with some basic Quantum Field Theory to show that unitarity is preserved even with

this exponential growth, but with the price of the propagator vanishing exponentially fast at

high energies. This exponential suppression renders loops finite, and the theory stays at a UV

interacting fixed point. That is a strong claim, and the role of this thesis is to investigate this

and understand better if Higgsplosion can happen in a Quantum Field Theory. If this is true,

then there will be consequences to the Higgs sector of the Standard Model that could explain

the fine-tuning problem and in some sense UV complete the whole theory. Even if it turns out

that it does not apply to the Standard Model, it could be right in some limiting case of other

models, and we can learn more about Quantum Field Theory in a different regime.

The structure of this thesis is the following. In chapter 1, we review the notation and tools

that we use. In chapter 2, we calculate some high multiplicity amplitudes at the threshold (the

limit where all outgoing particles are at rest) and explore beyond threshold amplitudes. At


3

the end of chapter 2, we show some recent results that we use later to discuss the possibility

of Higgsplosion. We choose to focus more on the perturbative approach to see if Higgsplosion

happens in this regime, but these results coming from the semiclassical calculation are useful

to understand the current state of the Higgsplosion Proposal. In chapter 3, we present the

Higgsplosion mechanism itself and what it can bring to the table. After that, we discuss some

problems with the claims of Higgsplosion, as well as the known criticism of it, and present

some potential solutions. At the end of chapter 3, we present two toy models that are useful to

understand the applicability of perturbation theory and a new proposed interpretation of the

Higgsplosion mechanism. We try to point out which directions are worth exploring to settle

the open questions that have been raised about this mechanism.

1.1 Toolbox

1.1.1 Green Functions

In this thesis, we use different types of n-point correlators, so it is worth defining the notation

here. These correlators are used to construct physical amplitudes through the standard LSZ

reduction formula [27]. Knowing these n-point functions, we can construct any S-matrix ele-

ment of the theory. In this point, there is no mention of perturbation theory aside from the

assumption of asymptotic states that enters in LSZ*.

First we define the n-point function as:

G(n)(x1, . . . , xn) ≡ ⟨Ω|T (ϕ(x1) . . . ϕ(xn))|Ω⟩ , (1.3)

where T is the time ordering operator. We can use a diagrammatic representation for this

object inside perturbation theory of the form represented in Figure 1.1.

Figure 1.1: Diagrammatic representation for n-point green functions inside
perturbation theory.

*This excludes theories that we cannot separate the particles from the interaction, for instance, a confined
system.


4

Given this Green function we can define its connected part diagrammatically were all ex-

ternal points are connected (interacting):

G(n)
c (x1, . . . , xn) ≡ G(n)(x1, . . . , xn)− disconnected parts . (1.4)

We will see later that we can define generating functional for these connected Green functions

in such a way that we can work only using G
(n)
c . This definition is a generalization of the

concept of cumulants in probability theory [28]. In particular, for n = 2 (the propagator) all

diagrams are connected:

G(2)
c (x1, x2) = G(2)(x1, x2) . (1.5)

Finally, we can define a Green function that cannot be separated into subprocesses by

cutting a single line. These are called one-particle irreducible Green functions(1PI): G
(n)
1PI . We

will see how to obtain these objects using functional methods in Section 1.1.4. They are the

fundamental blocks that we can use to construct arbitrary processes. For instance, the process

represented in 1.2 is not 1PI.

Figure 1.2: Example of a process that is not 1PI.

With these objects we can define its Fourier transform:

(2π)4δ4(p1 + · · ·+ pn)G
(n)
c (p1, . . . , pn) =

∫
d4x1 . . . d

4xn e
−i(p1·x1+···+pn·xn)G(n)

c (x1, . . . , xn) .

(1.6)

Usually, we work in Fourier space and with the definition that +p are entering momenta. It

can be seen above in Eq (1.6), that we have a total momentum conservation delta function. We

can use these objects to compute off-shell amplitudes by picking any momentum in a physical

amplitude and letting it be virtual. In other words, work only with the Green function in

momentum space and ignore the overall conservation delta function. The full propagator in

momentum space is then:

G(2)(p,−p) ≡ G(2)(p) , (1.7)


5

and with this we can define the amputated Green function that plays an important role in

constructing amplitudes:

G(n)
amp(p1, . . . , pn) ≡

n∏
1

(
G(2)(pi)

)−1
G(n)

c (p1, . . . , pn) . (1.8)

Diagramatically we are removing all the external legs of an amplitude using the full prop-

agator. This can be used to generalize LSZ to off-shell amplitudes. We will not re-derive LSZ

as this is standard textbook material [27]. Nonetheless, to understand this last statement let

us consider the LSZ reduction formula for a real scalar field:

iTn→m(p1, . . . , pn; p
′
1, . . . , p

′
m) = lim

p2i→m2
(2π)4δ4

(
n∑
1

pi −
m∑
1

p′i

)(
i
√
Zϕ

)−(n+m)

× (1.9)

×(p21 −m2) . . . (p′2m −m2)G(n+m)
c (p1, . . . , pn; p

′
1, . . . , p

′
m) ,

where Zϕ is the wave function normalization, and Tn→m is the transfer matrix elements that

are related to the interacting part of the S-matrix:

S = I+ iT . (1.10)

The object to the right of the delta function in Eq. (1.9) is the invariant amplitudeM for this

process:

Tn→m(p1, . . . , pn; p
′
1, . . . , p

′
m) = (2π)4δ4

(
n∑
1

pi −
m∑
1

p′i

)
(−iM(n→ m)) . (1.11)

The generalization goes as follows, instead of removing the propagator near the mass shell,

we remove the full propagator and generate an off-shell amplitude. This amplitude became the

physical amplitude when we put all particles on-shell. We can see that we are almost removing

the inverse of the full propagator near the on-shell limit in Eq. (1.9), just changing the overall

Zϕ factor:

G(2)(p)
p2→m2

=
iZϕ

p2 −m2 + iϵ
, (1.12)

this m is the physical mass, different than the bare mass m0 that appear in the free propagator.

The inverse is defined in such away that:

G(2) ·
(
G(2)

)−1
= −i . (1.13)


6

The off-shell generalization is direct, we change these propagators near the on-shell limit to

the full propagators and use the definition of the amputated Green function:

−iM(n→ m) = (i
√
Zϕ)

(n+m)G(n+m)
amp (p1, . . . , pn; p

′
1, . . . , p

′
m) . (1.14)

It is possible to see that the amputated Green function are the off-shell amplitudes aside from

an overall normalization factor�. We can use this definition to work out a case that is used in

this thesis, the 1→ 1 “scattering”:

M(1→ 1) = −Zϕ

(
G(2)

)−1
. (1.15)

This case is interesting because we can use the Optical Theorem [27] to relate the imaginary

part of this amplitude to the total decay rate:

Im (M(1→ 1)) = mΓtotal(p) = −Zϕ Im
(
(G(2)(p))−1

)
, (1.16)

where the first equality comes from the Optical Theorem, while the second one comes from the

1 → 1 “scattering” amplitude obtained trough generalized LSZ, Eq. (1.14). Using the inverse

of the full propagator (we will comment further on this form in the Section 1.1.2):(
G(2)

)−1
= −

(
p2 −m2 − Σ(p2)

)
, (1.17)

we arrive at one of the most important relations that we will use in this thesis:

Γtotal(p
2) = −Zϕ

m
Im
(
Σ(p2)

)
. (1.18)

Thus, if the off-shell total decay rate grows exponentially, the imaginary part of Σ(p2) grows

as well. The physical total decay rate can be recovered by going to the mass shell. This feature

of working with off-shell quantities let us gain more information about the theory in general,

and it is a powerful tool in Quantum Field Theory. The exponential growth of the imaginary

part of Σ(p2) can in principle suppress the propagator, Eq. (1.17), at a scale even when the

real part of this function is well behaved (we cannot say much about its real part). Now, let

us investigate further Eq. (1.17) because this is a central point of this thesis.

�If you work with a theory where Zϕ is zero up to some loop order, then the amputed Green function is the
amplitude directly up to the same order.


7

1.1.2 Dyson Resummation and the Full Propagator

Here we discuss important properties of the full propagator presented in Eq. (1.17). Using the

interacting part of the 1PI two-point function that we define as:

Σ(p2) , (1.19)

we can recover all the information about the full propagator G(2). With Σ(p2) we can re-

construct the full propagator as a geometric sum of these graphs�:

G(2) = G
(2)
0 +G

(2)
0 (−iΣ)G(2)

0 +G
(2)
0 (−iΣ)G(2)

0 (−iΣ)G(2)
0 + . . . . (1.20)

Figure 1.3: Diagrammatic representation of the 1PI resummation.

Diagrammatically this is represented in Figure 1.3. If we do this resummation we get the

representation of the full propagator used before in Eq. (1.17). Indeed, one has:

G(2)(p2) =
G

(2)
0 (p2)

1 + iΣ(p2)G
(2)
0 (p2)

, (1.21)

and using the usual free propagator:

G
(2)
0 =

i

p2 −m2
0 + iϵ

, (1.22)

�It is interesting to note that Eq. (1.20) is not so straightfoward. It almost comes as a definition in Quantum
Field Theory. This happens because we cannot reorganize terms in a divergent series, since summation is
not infinitely associative and commutative. The 1PI organization of the perturbative series that appears in
Quantum Field Theory is not immediate from perturbation theory alone. However, it is possible to justify this
ordering using the definition of a Quantum Action as we show in Section 1.1.4. Therefore, it is a consequence
of the meaning of what a Quantum Field Theory is, and not something additional to that.


8

we get the representation for the full propagator as:

G(2) =
i

p2 −m2
0 − Σ(p2) + iϵ

. (1.23)

It is important to note that the standard perturbation theory is inside Σ(p2). We are free

to do this resummation, and any non-perturbative effect is not lost but rendered inside the

1PI function Σ(p2). This resummation can be done even when Σ(p2) is large because we can

interpret this geometric series as a divergent series representation of the full propagator. Being

a divergent series representation and having this geometric nature we can, in this case, find a

region where the series converges. For instance, in a given renormalization scheme, we can fix

Σ(m2) = 0, summing this series near the mass shell condition means that we are inside the

convergence radius. After we resum this series, the expression can be expanded to the whole

complex plane just like the regular geometric series for x = 2 or any other complex value:

1 + 2 + 4 + 8 + · · · = 1

1− 2
= −1 (1.24)

Typically in a divergent series, this is not the whole story, because of non-perturbative effects.

For the expansion in Σ(p2) given in Eq. (1.20), there is no effect of this kind. That does not

mean that we solved the theory because we do not know how to calculate Σ(p2) exactly. This

result, Eq. (1.23), is one of the few non-perturbative results in Quantum Field Theory that we

currently have. With that information, we can guarantee the relation between the imaginary

part of the 1PI function and the total decay rate as defined above, provided that the theory

is unitary. This relation is re-derived without talking about this resummation (this is called

Dyson Resummation in the literature) at the end of this section when we introduce functional

methods. With this solved, we can start to investigate what else we can say about the full

propagator, Eq. (1.23).

1.1.3 Källén-Lehmann Spectral Representation

The object of interest here is the two-point function:

⟨Ω|T (ϕ(x)ϕ(y))|Ω⟩ . (1.25)

To explore this, we pick one time configuration and then in the end recover Eq. (1.25)

constructing the time ordering. Choosing an ordering where x0 < y0, we have:

⟨Ω|ϕ(x)ϕ(y)|Ω⟩ . (1.26)

Although we cannot compute this exactly in a interacting theory, we can extract much informa-

tion from it. If we introduce a set of complete states between the operators and use translation


9

invariance we can write:

⟨Ω|ϕ(x)ϕ(y)|Ω⟩ =
∑
n

⟨Ω|ϕ(0)|n⟩ ⟨n|ϕ(0)|Ω⟩ e−ipn·(x−y) , (1.27)

where n runs over all states in the theory, discrete and continuous (the sum becomes an integral

over the continuous states). As expected this object depends only on the difference between

the points x and y. Now, we can introduce a delta function in a suggestive way to re-write this

expression:

⟨Ω|ϕ(x)ϕ(y)|Ω⟩ =
∫

d4p

(2π)4
e−ip·(x−y)

(∑
n

(2π)4δ4(p− pn) | ⟨Ω|ϕ(0)|n⟩|2
)

. (1.28)

We can now define the spectral density:

ρ̃(p) =
∑
n

(2π)4δ4(p− pn) | ⟨Ω|ϕ(0)|n⟩|2 , (1.29)

this measures the contribution to the two-point function of the states with momentum p. It

receives contributions from bound states as well as multi-particle ones. This density is a Lorentz

invariant object and vanishes when p is not in the future lightcone [27]. Using this we can write

it as:

ρ̃(p) = 2πρ(p2)θ(p0) . (1.30)

Assuming there are no negative norm states it follows that the spectral density is positive

semi-definite for all p inside the lightcone:

ρ(p2) ≥ 0 . (1.31)

We can write the non-ordered two-point function with this spectral decomposition:

⟨Ω|ϕ(x)ϕ(y)|Ω⟩ =
∫

d4p

(2π)3
e−ip·(x−y)ρ(p2)θ(p0) , (1.32)

and using the propagator in position space:

∆(x, y,m2) =

∫
d4p

(2π)3
e−ip·(x−y)θ(p0)δ(p2 −m2) (1.33)

it is possible to write this ordered two-point function as:

⟨Ω|ϕ(x)ϕ(y)|Ω⟩ =
∫ ∞

0

ds ρ(s)∆(x, y, s) . (1.34)


10

To recover the time ordering in this two-point function we can use:

⟨Ω|T (ϕ(x)ϕ(y))|Ω⟩ = θ(x0 − y0) ⟨Ω|ϕ(x)ϕ(y)|Ω⟩+ θ(y0 − x0) ⟨Ω|ϕ(y)ϕ(x)|Ω⟩ (1.35)

together with the following relation:

e−iEp(x0−y0)θ(x0 − y0) + e+iEp(x0−y0)θ(y0 − x0) = lim
ϵ→0

−Ep

πi

∫ ∞

−∞

dE

E2 − E2
p + iϵ

eiE(x0−y0) . (1.36)

To construct the ordered two-point function Eq. (1.25), i.e. the Feynman propagator:

⟨Ω|T (ϕ(x)ϕ(y))|Ω⟩ =
∫

d4p

(2π)3
eip·(x−y)i∆F (p

2) , (1.37)

were we have the full Feynman propagator in momentum space as§

∆F (p
2) =

∫ ∞

0

ds ρ(s)
1

p2 − s+ iϵ
. (1.38)

The Eq. (1.38) is known as Källén-Lehmann spectral representation of the full propagator.

We did not use any information about the interaction or any expansion. This decomposition

is a non-perturbative result. With this representation, we can derive a significant amount of

information about the interacting theory. In an interacting theory, the spectral density will

have singularities at locations of physical particles, illustrated in Figure 1.4.

Figure 1.4: Example of a spectral density. An usual theory has a single pole
at its first excitation and then some bound states before having a continuum

spectrum for the 2 particle states and above.

§The i was removed from the definition of this propagator to facilitate the analysis in the complex plane.
Everything will be similar if we study −i∆F and kept the initial definition.


11

Because the spectral density is positive we can calculate the imaginary part using Cauchy

theorem. Given the analytic structure of the propagator (simple poles for one-particle state

and bound states, branch cuts for multi-particle states) we can write a contour integral repre-

sentation where the contour is a circle around the real line where the cut lives:

∆F (p
2) =

1

2πi

∫
γ

ds
∆F (s)

p2 − s
. (1.39)

This expression holds as long as p2 is inside the contour and the contour path does not

cross any singularity. Taking the radius of the contour R→∞ and assuming that ρ(s)→ 0 as

s→∞ we can write:

∆F (p
2) =

1

2πi

∫ ∞

s0

dsDisc (∆F (s))

p2 − s+ iϵ
, (1.40)

were s0 is the localization of the first singularity that we assume to be the single particle state,

and the discontinuity of the function along the cut is equal to the imaginary part of it:

Disc (∆F (s)) = 2i Im (∆F (s)) . (1.41)

With this information, we can use the definition of the spectral decomposition Eq. (1.38) to

write:

ρ(s) = − 1

π
Im
(
∆F (p

2)
)
. (1.42)

This feature is a consequence that the propagator is real everywhere unless when the particle

is on-shell.

Another important feature is a constraint on the power with which the propagator can

vanish at large momentum. Assuming that we can Wick rotate Eq. (1.38) (this is a non-trivial

statement in an interacting theory) we get:

p0 → ip0E , p2 → −p2E . (1.43)

This means that the modulo of the full propagator is:

∣∣∆F (−p2E)
∣∣ = ∣∣∣∣∫ ∞

0

ds ρ(s)
1

p2E + s

∣∣∣∣ , (1.44)

this implies:

∣∣∆F (−p2E)
∣∣ ⩾ ∣∣∣∣∫ s0

0

ds ρ(s)
1

p2E + s

∣∣∣∣ , (1.45)


12

for any s0 because the density is positive semi-definite. Taking the limit of Euclidean momentum

going to infinity we arrive at some point in p2E > s0 for a fixed s0. Then:

lim
p2E→∞

p2E
∣∣∆F (−p2E)

∣∣ ⩾ lim
p2E→∞

p2E

∣∣∣∣∫ s0

0

ds ρ(s)
1

p2E + s

∣∣∣∣ ⩾ lim
p2E→∞

p2E

∣∣∣∣∫ s0

0

ds ρ(s)
1

p2E

∣∣∣∣ = C (1.46)

for some finite positive number C:

C =

∫ s0

0

ds ρ(s) . (1.47)

Thus, with these assumptions, the propagator cannot fall faster than 1/p2 as p2 →∞. This

does not mean that the propagator cannot “look” as it is falling faster than p−2 in intermediary

regions (if we fix a s0 such that C = 0). This feature will be important in this thesis because

it is a strong non-perturbative constraint to the two-point function. Possible loopholes of this

are discussed in chapter 3. An important aspect of this derivation is that we can use any local

operator to get similar non-perturbative information:

∆O(x) = ⟨Ω|O(x)O(0)|Ω⟩ , (1.48)

as longs as O(x) is invariant under translations:

∆O(x) =

∫
ds

2π

∫
d3p

(2π)3
ρO(s)

2Es,p

e−ip·x (1.49)

where we define the spectral density of this operator as:

ρO(s) =
∑
n

δ(s− p0)| ⟨Ω|O(0)|n⟩ |2 . (1.50)

Now it is important to make some remarks. The first thing is to know that Eq. (1.38) and

Eq. (1.49) may contain UV divergences. This is always the case for a 4-dimensional theory.

This ultimately mean that Eq. (1.38) and Eq. (1.49) are ill-defined. The usual step to take is

to regularize and renormalize these expressions. The nature of the distribution ρ(s) dictates

how we handle these divergences. If we assume ρ(s) is a tempered distribution¶ then ρ(s) must

grow at s→∞ no faster than a polynomial. In this case we can re-arrange the expression by

adding m unknown coefficients to improve the behavior of the integral. These coefficients will

be fixed by the renormalization condition:

∆F (p
2) = Pm−1(p

2) + p2m
∫

ds ρ(s)
∆(p2, s)

sm
, (1.51)

¶Space of tempered distributions or Schwartz space is the function space of all infinitely differentiable func-
tions that are rapidly decreasing at infinity along with all partial derivatives.


13

where Pm−1 is a polynomial of degree m− 1. We have to do as many subtractions as needed to

render the integral finite. The coefficients of these polynomials are fixed by experimental input.

One way to justify Eq. (1.51) is to consider the case where we have logarithmic divergence.

Doing one subtraction for the propagator, Eq. (1.38), we get:

∆F (p
2)−∆F (p

2
0) =

∫ ∞

0

ds ρ(s)

(
1

p2 − s+ iϵ
− 1

p20 − s+ iϵ

)
, (1.52)

with the object on the right side now finite, and we can then define the once subtracted

propagator as:

∆F (p
2) = ∆F (p

2
0) + (p20 − p2)

∫ ∞

0

ds ρ(s)
1

(p2 − s+ iϵ)(p20 − s+ iϵ)
. (1.53)

Then, using one renormalization condition, we can fix the part depending on the arbitrary

parameter p0 and everything will be similar to doing renormalization in the usual way. In this

language, regularizing the integral with a finite number of subtractions means that the theory

is renormalizable. The story changes if we let ρ(s) be a different kind of distribution [29, 30],

for instance, growing exponentially at large s. Then, what follows is that we need to do an

inifinite number of subtractions. This can mean, in a worst case scenario, the necessity to fix

an infinite number of constants using boundary or renormalization conditions. Theories with

these kind of distributions normally are non-local, quasi-local or non-renormalizable. It can

happen that after an infinite number of subtractions, only a finite number of constants need to

be fixed in these three cases, but this is not a general feature. The behavior of ρ(s) at infinity

is fundamental to write the relation in Eq. (1.42). Doing this step with caution, we use the

subtracted propagator where the new distributions vanish at large s.

Lastly, we can relate Zϕ with the spectral distribution if we remove the one-particle state

from it. Using the fact that the field operators obey canonical commutation relations, this gives

a constraint that one-particle plus multi-particle states should add up to one:

1 = Zϕ +

∫ ∞

m2

dsη(s) , (1.54)

with η(s) being the spectral density after the removal of the one-particle state. This fixes Zϕ

to be a number smaller than 1:

0 < Zϕ < 1 . (1.55)

The closer Zϕ is to zero, the more the multi-particle states dominate. In the limit of

Zϕ = 0, we need to change the description of the theory because we do not have one-particle

states anymore. That would appear as a reorganization of the degrees of freedom in the theory.

With the previous understanding of the propagator, we need to introduce one more tool to


14

start doing calculations.

1.1.4 Functional Methods

Let us introduce important objects that we use throughout the thesis. The first one is the

generating functional for a n-point Green function:

Z[ρ] = ⟨Ω|T
(
ei

∫
d4xρ(x)ϕ(x)

)
|Ω⟩ = ⟨Ω|Ω⟩ρ . (1.56)

With this object, we can generate any Green function by differentiating with respect to the

external source ρ(x):

(−i)n δn

δρ(x1) . . . δρ(xn)
Z[ρ] = ⟨Ω|T

(
ϕ(x1) . . . ϕ(xn)e

i
∫
d4xρ(x)ϕ(x)

)
|Ω⟩ (1.57)

in such a way that in the limit where the source goes to zero, we recover the n-point function

Eq. (1.3). Given Eq. (1.56), we can construct the generating functional for the connected

n-point function defined in Eq. (1.4) as:

Z[ρ] = eiW [ρ] . (1.58)

This means that:

G(n)
c = i(−i)n δn

δρ(x1) . . . δρ(xn)
W [ρ]

∣∣∣∣∣
ρ=0

. (1.59)

The last important definition is the quantum action, the generating functional for the 1PI

Green function. To get this object we do a Legendre transform of W [ρ]:

ϕ(xi) ≡
δW

δρ(xi)
= ⟨Ω|ϕ(xi)|Ω⟩ρ , (1.60)

ρ(xi) ≡ −
δΓ[ϕ]

δϕ(xi)
, (1.61)

Γ[ϕ] ≡ W [ρ]−
∫

d4x ρ(x)ϕ(x) , (1.62)


15

where we trade the ρ dependence by a ϕ dependence. The n-point 1PI Green function is

obtained by taking functional derivatives with respect to the field:

G
(n)
1PI =

δn

δϕ(x1) . . . δϕ(xn)
Γ[ϕ]

∣∣∣∣∣
ϕ=0

. (1.63)

With these objects defined, we can start to analyze some relevant results. The first result

is the importance of the expectation value of the field with the presence of a source. If we find

a way to compute this in the presence of an arbitrary source, then we have all the information

needed to recover the n-point functions:

⟨Ω|ϕ(xi)|Ω⟩ρ ←→
δW [ρ]

δρ(xi)
. (1.64)

The only object that we cannot recover is the vacuum-vacuum amplitude ⟨Ω|Ω⟩, which is

irrelevant when dealing with particle physics. It turns out that it is possible to find an equation

for this object. It is precisely the expectation value of the classical equation of motion. That

why Eq. (1.60) is called the classical field. It is not, in fact, all classical because non-linearity

appears as n-point functions in this equation of 1-point function.

The next result is about the full propagator and its relation to the 1PI two-point function.

Given the connected two-point function:

Gc(xi, x2) = −i
δ2

δρ(x1)δρ(x2)
W , (1.65)

we can relate to the 1PI two point function using the identity:

δρ(xi)

δρ(xj)
= δ4(xi − xj) =

δρ(xi)

δϕ(xk)

δϕ(xk)

δρ(xj)
= (1.66)

= −
(

δ2Γ

δϕ(xi)δϕ(xk)

)(
δ2W

δρ(xk)δρ(xj)

)
. (1.67)

This is just the inversion equation for the connected propagator written in terms of the 1PI

propagator. This means that the 1PI propagator is the inverse of the connected propagator.

In momentum space:

G(2)
c (p2) = −

(
G

(2)
1PI(p

2)
)−1

, (1.68)

this shows the overall consistency of Eq. (1.17).

The last thing that is worth pointing out about these objects is that the quantum action

at tree level is the classical action, so we can use functional derivatives to derive the Feynman


16

rules of any theory:

Γ[ϕ] = S[ϕ] +O(ℏ) . (1.69)

Usually, at this point, we would introduce a path integral representation for these generating

functional and start calculating processes. However, here we go a different route because we

are interested in high multiplicity amplitudes, and usually, Feynman diagrams do not help.

Even at tree level, there are too many diagrams to count, and this method would not be

useful. The method that we use is introduced in the next chapter. Now we are ready to start

calculating some high multiplicity amplitudes and try to understand what is happening in this

regime. After the exploration of these processes, we introduce the newly proposed mechanism

of Higgsplosion and discuss the possibility of its occurrence in a scalar Quantum Field Theory.


17

Chapter 2

Perturbative Investigation of High

Multiplicity Amplitudes

The focus of this chapter is the study of high multiplicity processes. The primary motivation for

it comes from trying to understand the Higgsplosion proposal [21]. Nevertheless, this is not the

only reason to look for these processes. We typically do not explore this regime in a Quantum

Field Theory, and it is not clear what to expect. Maybe the particle interpretation of the field

excitation ceases to be valid or useful in this regime. Because this is a complicated problem,

first we do a perturbative investigation of this limit. The goal is to obtain enough information

such that we can understand the applicability of perturbation theory in this regime. Ultimately

this is answered at the end of chapter 3.

In the presence of those perturbative results, we can start to explore different approaches for

high multiplicity calculations. We do not work these additional results deeply because we chose

to focus on the perturbative calculations. After we recover most of the essential results for high

multiplicity scalar Quantum Field Theory, we start to work out the Higgsplosion framework.

2.1 Tree Level Amplitude at Threshold

We are interested in calculating the decay rate at high multiplicity of final states in a scalar

theory. This could, in principle, be calculated with Feynman diagrams. However, the high

number of final states makes this a tedious and challenging task. For example, if we are

interested in 1→ 5 processes at tree level in an unbroken ϕ4 theory, we have only ten diagrams,

showed in Figure 2.1. If we go for 1→ 7 processes, we get 280 diagrams, as it is represented in

Figure 2.2. Going beyond nine particles in the final state, we rapidly pass the 1000 diagrams

and becomes increasingly hard. This counting is only at tree level, adding quantum corrections

creates more diagrams, and the Feynman diagrammatic approach becomes almost useless.

Here we use a different approach that was proposed by Brown [23]. In this approach, we

calculate the amplitude that enters in the decay rate taking advantage of the LSZ reduction


18

Figure 2.1: Diagrams contributing for the 1→ 5 process in a ϕ4 theory in the
unbroken phase. Generated with FeynArts [31].

formula, Eq. (1.9). The decay rate that we are interested in calculating is of the form:

Γ1→n(p
2) =

1

2m

∫
dΠn |A(1→ n)|2 , (2.1)

where dΠn is the Lorentz invariant phase space factor, including the 1/n! factor since the end

particles are identical. This process is a highly virtual one because there is only one scalar field,

and it is stable. However, in the middle of a process, this could contribute as is represented in

the Figure 2.3.

The amplitude as written in Eq. (2.1) is the one without amputating the incoming leg. The

reduction formula for this case is:

A(1→ n)[x, p1, . . . , pn] = (iZϕ)
−(n+1)/2 lim

p2i→m2

∫
d4x1 . . . d

4xn e
−ip1·x1 . . . (2.2)

e−ipn·xn(p21 −m2) . . . (p2n −m2) ⟨Ω|T (ϕ(x1) . . . ϕ(xn)ϕ(x))|Ω⟩ .

Here we have to be careful because of the use of a nonstandard notation. The initial off-shell

particle is in the position representation, and the rest are in the momentum representation. This

amplitude has mixed momentum and position dependence. All of this dependence vanishes at

threshold:

p⃗i = 0 , for all i , (2.3)

where pi are the outgoing momenta in the amplitude. We calculate first these amplitudes in


19

Figure 2.2: Diagrams contributing for the 1→ 7 process in a ϕ4 theory in the
unbroken phase. Generated with FeynArts [31].

Figure 2.3: Process where the off-shell amplitude can contribute.

the threshold limit, and in the end, try to recover the momentum dependence. Another point

to notice is that, if we want to relate this amplitude to the Feynman diagram computation, we


20

need to amputate the virtual particle. The Feynman amplitude would beM, and its relation

to A is:

M(p, pi) =
(
p2 −m2

)
A(p, pi) . (2.4)

From now on, we can can ignore the Zϕ factor, because all the computations are done up

to one-loop. The effects of the Zϕ enters only at higher loops for the class of theories that we

work in this thesis. We can drop out also the overal phase factor for this process because we

square this amplitude in the end. The interesting observation used by Brown is that we can

re-write the (n+ 1)-point correlator in terms of functional derivatives using Eq. (1.60):

⟨Ω|T (ϕ(x1) . . . ϕ(xn)ϕ(x))|Ω⟩ =
δ

δρ(x1)
. . .

δ

δρ(xn)
⟨Ω|ϕ(x)|Ω⟩ρ

∣∣∣∣∣
ρ=0

, (2.5)

where the expectation value is taken in presence of an arbitrary source. Using Eq. (2.5), we

write the amplitude as:

A(1→ n) =
n∏

i=1

[
lim

p2i→m2

∫
d4xi e

−ipixi(p2i −m2)
δ

δρ(xi)

]
⟨Ω|ϕ(x)|Ω⟩ρ

∣∣∣∣∣
ρ=0

. (2.6)

It is possible to find a differential equation for ⟨Ω|ϕ(x)|Ω⟩ρ

∣∣∣∣
ρ=0

that is just like the classical

equation of motion, and then find an analytic solution for A(1→ n). This equation simplifies

when we want to calculate only the tree-level contribution. Let us specialize for this case now,

for the ϕ4 theory (any other interaction or kind of matter would follow a similar path).

2.1.1 λϕ4

4! in the Unbroken Phase

It is known that the tree level approximation for the expectation value is just the classical

solution in presence of an arbitrary source [23]:

⟨Ω|ϕ(x)|Ω⟩ρ → ϕcl(x)[ρ] . (2.7)

This statement will be made a little more rigorous when we compute loop corrections at Sec-

tion 2.2. Choosing the ϕ4 theory and using the usual particle physics normalization we get the

equation of motion:

(□+m2)ϕcl(x) +
λ

3!
ϕ3
cl(x) = ρ(x) . (2.8)

We want to find a solution of this equation to get the ρ dependence of the field and then

do n-functional derivatives to obtain the amplitude, Eq. (2.6). Because this is a differential


21

equation, we need boundary conditions. They are set by the Feynman prescription of the

propagator that tell us how to project to the right vacuum. We have transformed the problem

of finding the tree level amplitude into solving a non-linear second order differential equation

with an arbitrary source, which is still a difficult problem. The approach developed by Brown

is to focus on the threshold limit, where the source can be taken to have a simple enough form

that the equation can be solved, as we will show. Note that we want the dependence of the

field with respect to the source, not the actual form of the field solution by itself. For this, we

need to be careful because the source needs to be able to excite all modes of the field. Since

we want the amplitude at threshold, this means that there is no spatial momentum in the final

states. In that limit, the source and the field are homogeneous in space and depend only on

time. Hence, the threshold limit simplifies the equation to only one dimension.

The next step to find a solution for Eq. (2.8) is to choose a simple exponential source:

ρ(t) = ρ0e
iωt . (2.9)

The equation of motion then becames:

(∂2
t +m2)ϕtree +

λ

3!
ϕ3
tree = ρ(t) . (2.10)

This is still a non-linear problem but now is solvable. We look for a solution in perturbation

theory, using λ as our deformation parameter that turns on and off the non-linearity. We first

consider the free equation:

(∂2
t +m2)ϕ0

tree = ρ0e
iωt , (2.11)

whose solution is:

ϕ0
tree = −

ρ0
ω2 −m2 + iϵ

eiωt = z(t) . (2.12)

Turning the coupling on generates a series of the form:

ϕtree = z(t) + λϕ
(1)
tree + λ2ϕ

(2)
tree + . . . . (2.13)

Plugging this in the equation of motion:

(∂2
t +m2)(z(t) + λϕ

(1)
tree + λ2ϕ

(2)
tree + . . .) +

λ

3!
(z(t) + λϕ

(1)
tree + λ2ϕ

(2)
tree + . . .)3 = ρ0e

iωt , (2.14)


22

shows that the ϕtree(t) can be written only in terms of z(t), and the source dependence comes

only from it, for instance the first term in the expansion is:

ϕ
(1)
tree = λ

z0(ρ0, ω)

3!(9ω2 −m2)
e3iωt = −λ z(t)3

3!(9ω2 −m2)z0(ρ0,m)2
. (2.15)

The notation that we used is:

z0(ρ0, ω) =
ρ0

ω2 −m2 + iϵ
, (2.16)

in such a say that the double limit of ρ0 → 0 and ω → m this function becomes a constant z0.

We can trade the functional derivative with respect to the source for ordinary z(t) derivatives

in the amplitude, Eq (2.6):

A(1→ n) =
n∏

i=1

[
lim

p2i→m2

∫
d4xi e

−ipixi(p2i −m2)
δ

δρ(xi)

]
⟨Ω|ϕ(x)|Ω⟩ρ

∣∣∣∣∣
ρ=0

= (2.17)

=
n∏

i=1

[
lim
ω→m

∫
d4xi e

−iωti(ω2 −m2)
δ

δρ(ti)

]
⟨Ω|ϕ(x)|Ω⟩ρ

∣∣∣∣∣
ρ=0

.

Now we can use the dependence of the source from Eq. (2.12) and Eq. (2.13):

δϕ[z]

δρ(ti)
=

δz[ρ]

δρ(ti)

∂ϕ(z)

∂z
= − 1

ω2 −m2
δ(t− ti)δ

3(xi)
∂ϕ(z)

∂z
. (2.18)

Using Eq. (2.18) in Eq. (2.17) we can see the simplification of the problem, after doing the

delta integrations and ignoring the overall phase:

A(1→ n) =

(
∂

∂z

)n

ϕtree[z(t)]

∣∣∣∣∣
z=0

, (2.19)

where the on-shell condition corresponds to the limit ω → m and ρ → 0, in such a way that

z0 remains finite. The solution for ϕtree can be obtained order by order in λ like the first term

obtained in Eq (2.15). One finds that the perturbative series Eq. (2.13) can be ressumed:

ϕtree(t) =
z(t)

1− z2 λ
48m2

. (2.20)


23

It is easy to check that this is indeed a solution in a well-defined limit. Defining α = λ
48m2

we compute:

ϕ̇tree = ż
(1 + αz2)

(1− αz2)2
, (2.21)

ϕ̈tree =
z̈(1− αz4) + ż2(6αz + 2α2z3)

(1− αz2)3
, (2.22)

and using the definition of z(t), Eq. (2.12) we have that z ∝ eiωt so:

ϕ̇tree = iωz
(1 + αz2)

(1− αz2)2
, (2.23)

ϕ̈tree = −ω2 (α
2z5 + 6αz3 + z)

(1− αz2)3
. (2.24)

Putting this in the equation of motion, Eq. (2.10), and simplifying the denominator we get:

α2z5(m2 − ω2) + z3(
λ

3!
− 6ω2α− 2m2α) + z(m2 − ω2) = −(ω2 −m2)z(1− 2αz2 + α2z4) .

(2.25)

We can see that this is a solution when ω → m since the middle term is just the definition of

α. Here we can take ρ→ 0 in such a way that z(t) remains finite. Even though we arrived at

a solution using perturbation theory, we have obtained a representation, Eq (2.20) that can be

trivially continued to the full complex z(t) plane. An interesting thing comming from the form

of the amplitude Eq. (2.19) is that we can work without the source if we change the boundary

conditions. In the solution this can be seen trivially in the right side of the Eq. (2.25), since the

source contribution vanishes on-shell. This happens because we only want the solution on-shell,

and before doing this limit the ρ0 → 0 is similar to z → 0. Then, we can solve without any

source to find these amplitudes. Doing this changes the boundary conditions of the solution

until we set ρ = 0, because we are solving with a source at all times. This dictates that the

solution, Eq. (2.20), should vanish in positive Euclidean times:

Im(t)→∞ . (2.26)

This boundary condition remains for the broken case and loop corrections. Another interesting

point is that we started with a real field but got a complex solution. This complexification is

happening because of the source, Eq. (2.9). If we had chosen a real source, the solution would

be real as well. However, we only use the source as a trick to get the scattering amplitude. It


24

is arbitrary and can be chosen to be of this particular form.

With that in mind we can now find the decay amplitude, Eq. (2.19), by taking n-derivatives

with respect to z(t) in Eq. (2.20). To facilitate this, we can use a series representation of this

expression and pick up the nth-term in it:

z

1− αz2
=

∞∑
0

z2n+1αn . (2.27)

The amplitude is then:

A(1→ n) =
n!

2α
[(α)(1+n)/2) + (−α)(1+n)/2] . (2.28)

It is possible to draw some conclusions about this amplitude at tree level and threshold.

First it is necessary to remember that this is a partially off-shell amplitude and because of

that it is not a physical object by itself. Nevertheless we can use this amplitude to construct

physical observables. This amplitude has the interesting features:

� It vanishes for even final states:

A(1→ 2k) = 0 . (2.29)

� For odd final states it has the factorial growth:

A(1→ 2k + 1) = (2k + 1)!

(
λ

48m2

)k

. (2.30)

This factorial growth persists to the decay rate even after dividing by the n! coming from the

identical nature of the final states. That could be an indication that in the high multiplicity

limit, the decay rate grows or that perturbation theory cannot be trusted for a high multiplicity

computation. The total final state energy of the system at rest is:

E = (2k + 1)m. (2.31)

and the phase space in this case is zero (it is just a point). If we assume an infinitesimal sphere

around E and that the momentum dependence is constant in this region we get only an overall

small term trying to combat the factorial growth, that we will call Vn(E):

Γ2k+1(E) = (2k + 1)!

(
λ

48m2

)2k

V2k+1(E) . (2.32)

This is just a naive approximation if we want to know the phase space contribution we

need to be able to go beyond the threshold in such a way that we can get results around a


25

specific configuration. This is already potentially problematic for the perturbative unitarity of

the theory. The square amplitude divided by the symmetry factor grows factorially and can,

in principle, pass any unitary bounds of these processes:

|M|2

n!
∝ n! (2.33)

2.1.2 λϕ4

4! in the Broken Phase

The ϕ4 has another regime where we can explore this processes. In this phase the mass term is

negative so it is convenient to use the definition:

m2 = −µ2 . (2.34)

The reflection symmetry is broken and the configuration that minimizes the potential from

Eq. (2.8) is no longer zero:

ϕ2
min =

3!µ2

λ
. (2.35)

If we want to find the tree level amplitude at threshold it is easier to work in the shifted field,

where we do not have an expectation value [23, 32]:

σtree = ϕtree − ϕmin . (2.36)

Using this definition the equation of motion becames:

(□+ 2µ2)σtree(x) +
λϕmin

2!
σ2
tree(x) +

λ

3!
σ3
tree(x) = ρ(x) . (2.37)

The steps from Eq. (2.37) to Eq. (2.40) are essentially the same as detailed in the unbroken

phase above. We need to solve this equation to find σ as a functional of the source ρ. Choosing

again the same exponential source Eq. (2.9), transforms this problem into finding the solution

for the sourceless case with the boundary condition that:

σ → 0 as Im(t)→∞ . (2.38)

Now, we look for a perturbative solution for the spatially homogeneous case of Eq. (2.37).

Doing that we find the perturbative series in terms of the unperturbed solution:

z(t) = z0e
i
√
2µt , (2.39)


26

where the physical mass is now
√
2µ in the on-shell limit. It is easy to check that a solution

for Eq. (2.37), taking the ρ→ 0 limit but keeping z0 finite is:

σ(t) =
z

1− z
2ϕmin

. (2.40)

Keep in mind that ϕmin has all the coupling dependence. We find such a solution performing a

perturbative expansion, ressuming, and analytically continuing to the full complex z(t) plane.

To check that this is a solution is direct:

σ̈ = −2µ2

z2

2ϕmin
− z

(1− z
2ϕmin

)3
. (2.41)

Plugging this in the equation of motion and simplifying the denominator we get:

z3(− λ

12
+

µ2

2ϕ2
min

) + z2(−3 µ2

ϕmin

+
λϕmin

2
) + z(−2µ2 + 2µ2) = 0 , (2.42)

where we already used that ω2 → 2µ2 and set ρ0 → 0. The l.h.s vanishes using Eq. (2.35).

Now that we have this solution, we can compute the tree level threshold amplitude for the

broken phase, following the same logic as already explained in the unbroken case:

AB(1→ n) =

(
∂

∂z

)n

σ[z]

∣∣∣∣∣
z=0

. (2.43)

As before we find an series expansion and pick up the nth-term to get:

AB(1→ n) = n!

(
λ

24µ2

)(n−1)/2

. (2.44)

We can see that the factorial growth is still present in this phase. The difference is that now

we can have even final states. We also get a factorially growing decay rate:

ΓB(1→ n) = n!

(
λ

24µ2

)(n−1)

Vn(E) . (2.45)

As highlighted before this can be a potential danger for the unitarity of the theory. If these

processes start to dominate with factorial power, then the cross section for a process of few

particles going to n start to grows as well. This growth is inconsistent with the perturbative

unitarity of the theory. There are a few possible explanations for this feature and ways to save

this growth. We will continue working within the perturbative approach, at threshold, and see

if the factorial growth persists after quantum corrections, at the one-loop level. Later on, we

explore going beyond threshold such that the assumption that Vn(E) is constant can be checked


27

and obtain a better decay rate expression.

2.2 One-Loop Amplitude at Threshold

Until now, we saw that the tree level amplitude at the threshold for the ϕ4 theory displays a

factorial growth already in the first term of the series. We expect that these series that appear

in Quantum Field Theory to be divergent. However, it is not the usual case when the first

term of the series is already large. This, in principle, could mean that we cannot even trust the

first term of the series as a good approximation. If we want to understand this better, we need

to compute quantum corrections for this theory to see if somehow these factorial growths get

tamed. It is possible to implement loop corrections in this formalism if we expand the operators

in a ℏ expansion, where the first term corresponds to the tree level. In this expansion, we have

to be careful because ℏ has dimension, and so far have been using units such that:

ℏ = 1 . (2.46)

To get around this, we can introduce a deformation parameter ϵ where ℏ would appear, such

that the limit ϵ→ 0 is the classical limit that ℏ→ 0.

The expansion of the field operator is:

ϕ̂(x) = ϕ0Î +
√
ϵϕ̂1/2 + ϵϕ̂1 +O(ϵ3/2) . (2.47)

The expansion is in
√
ϵ because we are working at the level of the equation of motion.

We will see from the calculation that the one-loop correction appears in the ϵ term using this

convention. With this definition, we can find the expectation value of the field order by order in

ϵ, and then the amplitude can be computed up to that same order by an appropriate functional

derivative. Now, let us specialize in the cases that we have considered above to see how quantum

corrections modify them.

2.2.1 λϕ4

4! in the Unbroken Phase

The first case of interest is the ϕ4 theory in the unbroken phase [33], the equation of motion in

terms of the field operator, before expanding in ϵ is:

⟨Ω|T
[
(□+m2)ϕ̂+

λ

3!
ϕ̂3

]
|Ω⟩ρ = 0 . (2.48)

We will use the notation:

⟨Ω|ϕ̂i|Ω⟩ρ = ϕi[ρ] . (2.49)


28

Expanding the field operator up to O(ϵ3/2) the equation of motion is*[
(□+m2)ϕ0(x) +

λ

3!
ϕ3
0(x)− ρ(x)

]
+ (2.50)

+
√
ϵ

[(
□+m2 +

λ

2
ϕ2
0(x)

)
ϕ1/2(x)

]
+

+ϵ

[(
□+m2 +

λ

2
ϕ2
0(x)

)
ϕ1(x) +

λϕ0

2
⟨Ω|T

(
ϕ̂1/2(x)ϕ̂1/2(x)

)
|Ω⟩ρ

]
+O(ϵ3/2) = 0 .

Now we do the threshold limit. This means that ϕ0 is as calculated before, Eq. (2.20):

ϕ0 =
z(t)

1− λ
48m2 z(t)2

. (2.51)

The equation of order
√
ϵ defines the two-point function that appears at order ϵ in Eq. (2.50).

It is the zero mode equation for the differential operator:

♢ = □+m2 +
λ

2
ϕ2
0(t) , (2.52)

where the tree level and one-loop fields are at threshold, and the ϕ1/2 is allowed to have spatial

dependence that appears in ♢ and its Green function. We will see that the boundary conditions

kill all zero modes of ♢, so this equation give us that ϕ1/2 is zero as expected. This is a common

feature since we are treating things at the level of the equation of motion. The order ϵ is what

we are interested in and gives us the one-loop contribution for the amplitude. In Eq. (2.50),

there is a contribution of the two-point Green function of ϕ̂1/2. The operator that we need to

invert to find this Green function is Eq. (2.52), and in the end, set x′ = x. This Green function

will be taken at the same point so we can expect divergences to appear. These divergences are

familiar from more standard Quantum Field Theory arguments.

Using ϕ0 we can write the operator, Eq. (2.52), as:

♢ = □+m2 +
λ

2

(
z(t)

1− λ
48m2 z2

)2

. (2.53)

From the start we have a problem, this operator is not Hermitian because z(t) is complex. This

makes our job a little harder. To deal with this fact, we proceed as proposed in [33]. Going to

Euclidian time, we can make this problem simpler. However, in Euclidean time we have a pole

*The source that appear in the equation is the classical external one. It appears after we bring the d’Alembert
operator out of the expectation value, passing trough the time ordering.


29

on the countour of integration. To adress this, we can do a shift in the Euclidian time variable

to avoid the pole and then analytically continue the solution to the whole complex Euclidean

plane. It is convenient to work in terms of [33]:

u(τ) = emτ , (2.54)

where u(τ) is defined as:

− λ

48m2
z(t)2 = u(τ)2 , (2.55)

and the new Euclidian time coordinate is:

τ = it+ i
iπ

2m
+

ln
(

λz20
48m2

)
2m

. (2.56)

The tree level solution takes the form:

ϕ0(t) = i

√
48m2

λ

u(τ)

1 + u(τ)2
= i

√
48m2

λ

sech(mτ)

2
. (2.57)

Then the operator to be inverted, Eq. (2.53), reads:

♢ =
(
−∂2

τ −∇2 +m2 − 6m2 sech2(mτ)
)
. (2.58)

Doing a partial Fourier transform only in the spatial section we can write the Green function

of this operator as:

G(x⃗, x⃗′; τ, τ ′) =

∫
d3k

(2π)3
Gk(τ, τ

′)eik⃗.(x⃗−x⃗′) , (2.59)

the Green function satisfying:

♢(x⃗, τ)G(x⃗, x⃗′; τ, τ ′) = δ3(x⃗− x⃗′)δ(τ − τ ′) . (2.60)

Using Eq. (2.59) we focus on the following Green function:(
−∂2

τ + θ2 − 6m2 sech2(mτ)
)
Gθ(τ, τ

′) = δ(τ − τ ′) , (2.61)

where we defined θ2 = k⃗2 +m2. From Gθ we can then find the full Green functions by doing

momenta integrals using Eq. (2.59). We are doing this computation in (3 + 1)D, but the

generalization to other dimensions is straightforward only changing the numbers of k integrals.

In fact, the Green function that appears in Eq. (2.50) is evaluated at coincident space-time


30

points, therfore, we have:

⟨Ω|T
(
ϕ̂1/2(x)ϕ̂1/2(x)

)
|Ω⟩ρ =

∫
d3k

(2π)3
Gθ(τ, τ) . (2.62)

Surprisingly this Green function is very similar to a known quantum mechanical potential

(Poschl-Teller Potential) [34]. We will transform this problem into that quantum mechanical

problem and then show how to solve this potential exactly. After that, we continue with the

solution to find the Green function and in the end, the loop correction. To look for this Green

function, we search for two regular solutions to the homogeneous equation, one regular at

τ →∞ the other at τ → −∞. With both solutions f−(τ), f+(τ) and with the Wronskian W :

W = f+(τ)f−(τ)
′ − f+(τ)

′f−(τ) , (2.63)

we then construct the Green function:

Gθ(τ, τ
′) =

f+(τ)f−(τ
′)

W
for τ > τ ′ , (2.64)

Gθ(τ, τ
′) =

f+(τ
′)f−(τ)

W
for τ ′ > τ . (2.65)

Both solutions can be written in a Schrodinger like form:(
−∂2

τ − 6m2 sech2(mτ)
)
f(τ) = −θ2f(τ) , (2.66)

where we identify Ẽθ = −θ2 and the Hamiltonian being the operator on the left. We can solve

this only using algebra. To facilitate we can work using coordinates without dimension by doing

a change of variables, x = mτ :

Ĥ =
(
−∂2

x − 6 sech2(x)
)
. (2.67)

The dimensionless energy is defined as:

Eθ = −
θ2

m2
, (2.68)

such that we need to solve the eigenvalue problem:

Ĥf(x) = Eθf(x) . (2.69)


31

It turns out that to find the solution for this Hamiltonian we need to generalize it to:

Ĥl = p2 − l(l + 1) sech2(x) . (2.70)

Our case is l = 2. Now we will do a little sidetrack to solve this eigenvalue problem because

even though this is an exactly solvable quantum mechanical system, it is not so trivial to find

the solution.

2.2.2 Eigenvalues and Eigenfunctions of the Poschl-Teller Potential

This section can be skipped without affecting the core of the thesis. Here we want to solve the

quantum mechanical problem, Eq. (2.70):

Ĥl |p, l⟩ = El |p, l⟩ . (2.71)

We want to find the eigenfunctions for the special case of l = 2. For each l we have a quantum

system with eigenvalues El. The spectrum that we are interested in is the continuum band of

the Poschl-Teller potential:

V (x) = −l(l + 1) sech2(x) . (2.72)

The esiest one is l = 0, the system is the free particle, and the energy does not depend on l

trivially:

Ĥ0 |p, 0⟩ = p2 |p, 0⟩ . (2.73)

The special propriety of this system is that it belongs to a class of factorizable potentials.

This feature is reminiscent from the supersymmetric version of this potential [35]:

Vs(x) = −l(l + 1) sech2(x) + l2 . (2.74)

Because we are interested in the continuum spectrum of Eq. (2.72), we will not introduce

Supersymmetric Quantum Mechanics [12, 13]. Nevertheless, Supersymmetry is the basis of

why this process work and why this potential is solvable. The factorization of the Hamiltonian

Eq. (2.70) can be archived using the following operators:

al = p− il tanh(x) , (2.75)

a†l = p+ il tanh(x) . (2.76)


32

They are choosen in this form to satisfy:[
a†l , al

]
= 2l sech2(x) , (2.77)

With the operators Eq. (2.75) and Eq. (2.76) we can construct the initial Hamiltonian

Eq. (2.70):

H+
l ≡ a†lal = Hl + l2 . (2.78)

The object in the l.h.s of Eq. (2.78) is the Hamiltonian for the supersymmetric description of

this potential. We can define its parter Hamiltonian exchanging the order of the operators:

H−
l ≡ ala

†
l = Hl−1 − l2 . (2.79)

In this case both supersymmetric partners are related only by a change in constants inside the

Hamiltonian. This class of systems is called Shape Invariant Potentials [36], and this plays a

pivot role in making this potential solvable. With the definitions of Eq. (2.78) and Eq. (2.79)

we can see that, if we have an eigenstate of H+ with contiuum spectrum:

H+
l

∣∣E+, l
〉
= E+

l

∣∣E+, l
〉
, (2.80)

there exists an eigenstate of H−
l with the same eigenvalue, except for the ground state:

alH
+
l

∣∣E+, l
〉
= H−

l

(
al
∣∣E+, l

〉)
= E+

l

(
al
∣∣E+, l

〉)
. (2.81)

This occurs the other way around as well. This is a consequence of the supersymmetry on the

system, both spectrums are related by the operators al and a†l as it is represented in Figure 2.4.

Because this potential is shape invariant, using Eq. (2.78) and Eq. (2.79) we can see that

the spectrum does not depend on l. The shape invariance tells us that a change of l relates H+

and H−. This implies that the eigenvalues will be the same as the free case, l = 0:

El = p2 . (2.82)

Now, to construct the different eigenestates we can use Eq. (2.81) to build up all the states of

Hl. For l = 0 we have:

⟨x|p, 0⟩ = eipx , (2.83)


33

Figure 2.4: Relation between the two parterns Hamiltonians..

and for l=1:

|p, 1⟩ = a†1 |p, 0⟩ . (2.84)

Just to see that indeed this is a eigenstate of H1 we can check directly:

H1 |p, 1⟩ = (η1 − 1)a†1 |p, 0⟩ = a†1ξ1 |p, 0⟩ − a†1 |p, 0⟩ = (2.85)

= (p2 + 1)a†1 |p, 0⟩ − a†1 |p, 0⟩ = p2(a†1 |p, 0⟩ = p2 |p, 1⟩ . (2.86)

The wave function for the l = 1 case is:

⟨x|p, 1⟩ = ⟨x|a†1|p, 0⟩ = ⟨x|p̂+ i tanh(x̂)|p, 0⟩ = (p+ i tanh(x)) eipx . (2.87)

Finally, the case of interest is the l = 2 case:

⟨x|p, 2⟩ = ⟨x|a†2|p, 1⟩ =
(
1 + p2 + 3ip tanh(x)− 3 tanh2(x)

)
eipx . (2.88)

Using this wave function, we can extend this solution to all complex p plane to get both

functions to construct the Green function Eq. (2.64):

p = ± θ

m
. (2.89)


34

Because we need regular solutions at ±∞, both p’s will be used in this contruction:

f−(x) =

(
1− θ2

m2
+ 3

θ

m
tanh(x)− 3 tanh2(x)

)
e

θx
m , (2.90)

f+(x) =

(
1− θ2

m2
− 3

θ

m
tanh(x)− 3 tanh2(x)

)
e−

θx
m . (2.91)

2.2.3 Back to λϕ4

4! in the Unbroken Phase

Now that we have both solutions Eq. (2.90) and Eq. (2.91) to construct the Green function of

Eq. (2.64) we just need to write them in terms of the variables that we are using:

u(x) = ex , (2.92)

f−(x) =
u4(− θ2

m2 + 3 θ
m
− 2) + u2(−2 θ2

m2 + 8)− θ2

m2 − 3 θ
m
− 2

(1 + u2)2
u

θ
m , (2.93)

f+(x) =
u4(− θ2

m2 − 3 θ
m
− 2) + u2(−2 θ2

m2 + 8)− θ2

m2 + 3 θ
m
− 2

(1 + u2)2
u− θ

m . (2.94)

Having these two solutions we can calculate the Wronskian, Eq. (2.63):

W = 2
θ

m4
(θ2 −m2)(θ2 − 4m2) . (2.95)

With this, we can construct the equal time Green function:

Gθ(τ, τ) =
f+(mτ)f−(mτ)

W
= (2.96)

=
1

2θ(1 + u2)4

(
(1 + u8) + (u6 + u2)

4(θ2 + 2m2)

(θ2 −m2)
+ u46(θ

4 −m2θ2 + 12m4)

(θ2 −m2)(θ2 − 4m2)

)
.

The Green function in this form is not so useful. We can use partial fraction expansion in the

u variable to separate the different kind of contributions to the Green function. Doing so it is

straightforward to see that we have a finite part and a divergent part when doing the Fourier

integral in Eq. (2.59). We can separate both parts in the following way that will facilitate the

renormalization latter:

Gθ(τ, τ) = Gθ(τ, τ)
div +Gθ(τ, τ)

fin , (2.97)


35

Gθ(τ, τ)
div =

1

2θ
+

6m2u2

(1 + u2)2
1

θ3
, (2.98)

Gθ(τ, τ)
fin =

6m4

θ3(θ2 −m2)

(
u2 + u6

(1 + u2)4

)
+

6m4u4

(1 + u2)4

(
14θ2 − 8m2

θ3(θ2 −m2)(θ2 − 4m2)

)
. (2.99)

Now we are ready to integrate this to get the full Green function that enters into the one-loop

equation:

G(x⃗, x⃗; τ, τ) =

∫
d3k

(2π)3
Gθ(τ, τ) , (2.100)

with θ2 = k⃗2 +m2. The divergent part of the two-point function is:

Gdiv(x⃗, x⃗; τ, τ) =
1

2
I1 +

6m2u2

(1 + u2)2
I2 , (2.101)

where I1 is quadratically divergent and I2 has an logarithmic divergence:

I1 =
∫

d3k

(2π)3
1

(k⃗2 +m2)1/2
, (2.102)

I2 =
∫

d3k

(2π)3
1

(k⃗2 +m2)3/2
. (2.103)

To get the finite part we need to do two integrals:

If1 =
∫

d3k

(2π)3
6m4

θ3(θ2 −m2)
, (2.104)

If2 =
∫

d3k

(2π)3
6m4(14θ2 − 8m2)

θ3(θ2 −m2)(θ2 − 4m2)
. (2.105)

The angular part of these integrals is trivial to solve, and the only difficult part is the

radial integration. One important thing to remember is that the fields that generate such

propagators have fixed boundary conditions to give the right vacuum projection. The boundary

conditions fix the prescription to pass through the poles, and this means that the denominator

in Eq. (2.104) and Eq. (2.105) has a −m2+ iϵ term. This information is important only for the

poles inside the domain of integration. The next step to solve these integrals is to change to a


36

dimensionless variable:

θ = ωm . (2.106)

Doing this transformation it is straightforward to find the solution for these integrals:

If1 =
3m2

π2

∫ ∞

0

dω
1

ω2(ω2 − 1)1/2
=

3m2

π2
, (2.107)

If2 = lim
ϵ→0

3m2

π2

∫ ∞

0

dω
(14ω2 − 8)

ω2(ω2 − 1)1/2(ω2 − 4 + iϵ)
= (2.108)

= lim
ϵ→0

6m2

π2

[
4i
√
−3 + iϵ

√
4i+ ϵ+ 4

√
−1(24− 7iϵ) sinh−1( 4

√
−1
√
4i+ ϵ)√

iϵ− 3(ϵ+ 4i)3/2

]
.

Doing the ϵ limit we get:

If2 =
6m2

π2
+

3im2
√
3

π
− 3m2

√
3

π2
ln

(
2 +
√
3

2−
√
3

)
. (2.109)

Thus, the full two-point function, Eq. (2.62), is:

G(x⃗, x⃗; τ, τ) =
1

2
I1 +

6m2u2

(1 + u2)2

(
I2 +

1

2π2

)
− 6m2u4

(1 + u2)4
F , (2.110)

where we use the notation from [33]:

F =

√
3

2π2

(
ln

(
2 +
√
3

2−
√
3

)
− iπ

)
. (2.111)

An important point to make is that the ϵ limit is evaluated from the right such that these

expressions make sense.

This result is in accordance with [33]. Now, we need to renormalize our theory, we could do

dimensional regularization to extract only the divergent part from the integrals using minimal

subtraction. However, it is simpler to just absorb the whole divergent part, using the same

scheme as in [33]. To visualize better the renormalization we can re-write the equation of

motion using:

λ→ λR + δλ , (2.112)


37

m2 → m2
R + δm2 , (2.113)

where the corrections start at order ϵ, being a power series in this variable. To see that these

two constants absorb the divergences, we insert back the two-point function in the equation of

motion Eq. (2.50) to get: [
(−∂2

τ +m2)ϕ0(t) +
λ

3!
ϕ3
0(t)− ρ(t)

]
+ (2.114)

+
√
ϵ

[(
□+m2 +

λ

2
ϕ2
0(x)

)
ϕ1/2(x)

]
+

+ϵ

[(
−∂2

τ +m2 +
λ

2
ϕ2
0(x)

)
ϕ1(x) +

λϕ0

2

(
1

2
I1 +

6m2u2

(1 + u2)2
(I2 +

1

2π2
)− 6m2u4

(1 + u2)4
F

)]
+O(ϵ3/2) = 0 .

(2.115)

Using the definition of the tree level solution, Eq. (2.20), the divergent part has the distinctive

forms:

ϕ0

(
λmI1
4

)
, (2.116)

ϕ3
0

(
−3λ2

48
(I2 +

1

2π2
)

)
. (2.117)

So we can see that the redefinition of mass and coupling could absorb these divergences.

We write these constants like:

m2 = m2
R + ϵδm2

1 +O(ϵ2) , (2.118)

λ = λR + ϵδλ1 +O(ϵ2) . (2.119)

Using this definition in the equation of motion, Eq. (2.114), we get:[
(−∂2

τ +m2
R)ϕ0(t) +

λR

3!
ϕ3
0(t)− ρ(t)

]
+ (2.120)

+
√
ϵ

[(
□+m2

R +
λR

2
ϕ2
0(t)

)
ϕ1/2(x)

]
+


38

+ϵ

[(
−∂2

τ +m2
R +

λR

2
ϕ2
0(τ)

)
ϕ1(τ) + ϕ0

(
δm2

1 +
λRI1
4

)
+ ϕ3

0

(
δλ1

3!
− λ2

R

16
(I2 +

1

2π2
)

)
− 6m2

Ru
4

(1 + u2)4
F

]
+

+O(ϵ3/2) = 0 .

The obvious choice for the counter terms are:

δm2
1 = −

λRI1
4

, (2.121)

δλ1

3!
=

λ2
R

16
(I2 +

1

2π2
) . (2.122)

Choosing this renormalization we can proceed to solve the one-loop generator of the ampli-

tudes. The tree level solution stays the same as before, except for the replacement of bare for

renormalized quantities. The one-loop equation simplifies to:(
−∂2

τ +m2
R +

λR

2
ϕ2
0(τ)

)
ϕ1(τ) =

6m2
Ru

4

(1 + u2)4
F
λRϕ0

2
. (2.123)

Writing the equation in terms of u(τ) = emRτ we get:(
−∂2

τ +m2
R −

24m2
Ru

2

(1 + u2)2

)
ϕ1(τ) = 12im3

RF
√
3λR

u5

(1 + u2)5
. (2.124)

From the form of the right side of Eq. (2.124) we can try to look for solutions of the form:

ϕ1(τ) = α
u5

(1 + u2)3
, (2.125)

and confirm by direct computation that this is a solution if:

α = −iFλR

8

√
48m2

R

λR

. (2.126)

Having this solution we can analytically continue back to real time using the relation

Eq. (2.55), using the renormalized mass inside z(t). This gives us:

ϕ1(t) = −
FλR

8

(
λR

48m2
R

)2
z5

(1− λR

48m2
R
z2)3

. (2.127)


39

Then the full contribution for the generator of the amplitudes at one-loop is:

ϕ0(t) + ϕ1(t) = ϕ0+1(t) =
z

(1− λR

48m2
R
z2)

[
1− ϵ

FλR

8

( λR

48m2
R
)2z4

(1− λR

48m2
R
z2)2

]
. (2.128)

To calculate the amplitude from this solution we just need to differentiate n times it with

respect to z(t), following Eq. (2.19):

A1loop(1→ 2k + 1) = (2k + 1)!

(
λR

48m2
R

)k [
1− FλRk(k − 1)

16

]
, (2.129)

where we set ϵ to one. This expression was obtained first in [33].

Here we can see that the factorial growth persists. The next correction only makes things

worse, and this can be seen as an indication that we are using the wrong approximation for this

regime. We expect that in a large k approximation of the amplitude, the corrections to be of

order 1/k. In this case, the true object that needs to be small such that this approximation is

useful is λRk
2. In the regime where this is small, this expression is well defined. It is not known

how much we can trust this expression outside this regime even though we arrived at it only

using that λR is small. This is because, for a large n, the loop correction will be larger than

the tree level one, signaling that the approximation is not good. We discuss this further in the

context of a simpler toy model at the end of chapter 3. For now, let us see if this behavior is

the same in the broken phase of this theory.

2.2.4 λϕ4

4! in the Broken Phase

In the case of broken reflection symmetry we need to be more careful in the renormalization

because the shift done at tree level in Eq. (2.36) is not the appropriate one. We use the shift

in the variables done before Eq. (2.36), and when we get to renormalization, we will do the

appropriate adjustments as in [32]. Aside from this, everything else is very similar to the

unbroken case. We expand the σ̂ operator:

σ̂(x) = σ0Î +
√
ϵσ̂1/2(x) + ϵσ̂1(x) +O(ϵ3/2) . (2.130)

The equation of motion, Eq. (2.37), using this expansion is:[
(□+M2)σ0(x) +

λϕmin

2
σ2
0(x) +

λ

3!
σ3
0(x)− ρ(x)

]
+ (2.131)

+
√
ϵ

[(
□+M2 + λϕminσ0(x) +

λ

2
σ2
0

)
σ1/2(x)

]
+


40

+ϵ

[(
□+M2 + λϕminσ0(x) +

λ

2
σ2
0

)
σ1(x) +

λ

2
(ϕmin + σ0) ⟨Ω|T

(
σ̂1/2(x)σ̂1/2(x)

)
|Ω⟩ρ

]
+O(ϵ3/2) = 0 ,

(2.132)

where ϕ2
min = 3M2

λ
and M =

√
2µ is the mass of the excitation. We are interested in solving

the one-loop contribution at threshold. Just like before the tree level is already solved when

we ask for homogeneous solution and the next order only give the information about the two-

point function that enters in the one-loop equation. The tree level solution is the one that we

calculated before:

σ0(t) =
z(t)

1− z(t)
2ϕmin

. (2.133)

The operator that we need to invert to find the two-point function now is:

♢b = □+M2 + λϕminσ0 +
λσ2

0

2
. (2.134)

Using the solution σ0(t) we get to the same problem of non-hermiticity of the operator. Doing

the same step as before, we perform a Wick rotation taking care of the pole in the Euclidean

line:

− z(t)

2ϕmin

= u(τ) = eMτ , (2.135)

τ = it+ i
π

M
+

1

M
ln

(
z0

2ϕmin

)
. (2.136)

Doing this change of variables, the operator that we want to invert becomes:

♢b = −∂2
τ −∇2 +M2 − 2ϕ2

minλ
u

(1 + u)2
. (2.137)

We can re-write the last term in a familiar form:

♢b = −∂2
τ −∇2 +M2 − 3M2

2
sech2(

Mτ

2
) . (2.138)

Now the steps are very similar, we do a partial Fourier transform in the spatial part. The

remaining operator that we need to invert is almost what we had before:(
−∂2

τ + θ2 − 3M2

2
sech2(

Mτ

2
)

)
Gθ(τ, τ

′) = δ(τ − τ ′) , (2.139)


41

where θ2 = k⃗2 + M2. Doing a change of variables to an dimensionless one we can cast the

functions in a known form:

Mτ

2
= ξ , (2.140)

(
−M2

4
∂2
ξ + θ2 − 3M2

2
sech2(ξ)

)
f(ξ) = 0 . (2.141)

This is almost the equation that we had before, Eq. (2.69), just re-scaling the θ:

θ → 2θ . (2.142)

In terms of u(τ) the solution will change the power because of the definition of ξ:

u2 → u . (2.143)

We already solved these equations in the last section, so the same point Green function

before integrating is:

Gθ(τ, τ) = Gθ(τ, τ)
div +Gθ(τ, τ)

fin , (2.144)

Gθ(τ, τ)
div =

1

4θ
+

3M2

θ3
u

(1 + u)2
, (2.145)

Gθ(τ, τ)
fin =

3M4

4θ3(4θ2 −M2)

u

(1 + u)4
+

3M4(56θ2 − 8M2)

4θ3(4θ2 −M2)(4θ2 − 4M2)

u2

(1 + u)4
+ (2.146)

+
3M4

4θ3(4θ2 −M2)

u3

(1 + u)4
.

The same definition of the divergent integrals I1 and I2 are used to write the divergent part of

the two-point function as:

G(x⃗, x⃗; τ, τ)div =
1

4
I1 +

u

(1 + u)2
3M2I2

4
. (2.147)

Now for the finite part, we need to solve the following two integrals:

If3 =
∫

d3k

(2π)3
1

θ3(4θ2 −M2)
, (2.148)


42

If4 =
∫

d3k

(2π)3
56θ2 − 8M2

θ3(4θ2 −M2)(4θ2 − 4M2)
. (2.149)

This time there is no pole inside the domain of integration, so we don’t need to worry about

the Feynman prescription. The finite part in terms of these integrals is:

G(x⃗, x⃗; τ, τ)fin =
3M4

4

(
If3

u+ u3

(1 + u)4
+ If2

u2

(1 + u)4

)
. (2.150)

The integrals can be solved directly, in both cases we use the dimensionless coordinates θ = ωM :

If3 =
1

M2

(
6−
√
3π

12π2

)
, (2.151)

If4 =
1

M2

(
6 +
√
3π

6π2

)
. (2.152)

Using this information we can construct the two-point function. We just write in an appropriate

form to interpret after during renormalization:

G(x⃗, x⃗; τ, τ) =
1

4
I1 +

u

(1 + u)2

(
3M2

4
I2 +

3M2

8π2
− M2

√
3

16π

)
+

M2
√
3

4π

u2

(1 + u)4
. (2.153)

We need to deal with the divergent part now. We can absorb them in the renormalization

of the constants:

M2 = M2
R + ϵδM2

1 +O(ϵ2) , (2.154)

λ = λR + ϵδλ1 +O(ϵ2) . (2.155)

To do the renormalization properly let us focus only in the contribution for the one-loop equa-

tion, that is in the form:

λ

2
(ϕmin + σ0)G(x⃗, x⃗; τ, τ) . (2.156)

As we said before it is better to work in the unshifted field because this shift receives quantum

corrections. Using the definition ϕ0

ϕmin
= γ we can write Eq. (2.156) in general as:

λϕmin

2
γ

[
A− B

4
(γ2 − 1) +

C

16
(γ2 − 1)2

]
, (2.157)


43

where the constants are:

A =
I1
4
, (2.158)

B =
3M2

4
I2 +

3M2

8π2
− M2

√
3

16π
, (2.159)

C =
M2
√
3

4π
. (2.160)

Only A and B have divergences, the contribution for the one-loop equation is written as:

γ
λϕmin

2

(
A− B

4

)
− λϕmin

2
γ3B +

λϕmin

2
C

(
γ5

16
− γ3

8
+

γ

16

)
. (2.161)

The renormalization scheme of [32] will absorb the two initial terms. Because we are using

the unshifted field it is easy to read the renormalized mass and coupling. We just need to be

careful because in the unshifted field the mass has the “wrong” sign:

m2 = m2
R + ϵδm2

1 +O(ϵ2) , (2.162)

λ = λR + ϵδλ1 +O(ϵ2) . (2.163)

Using this in the equation of motion, we see that:

δm2
1 =

λR

8

(
I1 −M2

R(
3I2
4

+
3

8π2
−
√
3

16π
)

)
, (2.164)

δλ1

3!
=

λ2
R

48

(
3I2
2

+
3

4π2
−
√
3

16π

)
. (2.165)

In this renormalization scheme we have to solve now the equation for σ1, where we shift by

the renormalized couplings. The equation for the tree level stays the same and the one-loop

became:(
−∂2

τ +M2
R + λRϕ

R
minσ0(τ) +

λR

2
σ2
0(τ)

)
σ1(τ) = −

λRϕ
R
minM

2
R

√
3

8π

(
γ5

16
− γ3

8
+

γ

16

)
. (2.166)


44

We can write everything in terms of u(ξ) where ξ = MRτ is the dimensionless time coordinate.

The equation using these variables is:(
∂2
ξ − 1 +

6u

(1 + u)
− 6u2

(1 + u)2

)
σ1(ξ) =

3MR

√
λRu

2(1− u)

8π(1 + u)5
. (2.167)

To solve this equation we look for an ansatz of the form:

σ1 = α
u2

(1 + u)3
. (2.168)

Plugging this solution in Eq. (2.167), it is immediate to see that α = MR
√
λR

8π
. This means that

the one-loop solution is:

σ1 =
MR

√
λR

8π

u2

(1 + u)3
. (2.169)

The complete solution for the generator of the tree level and threshold amplitudes at one-loop

is then:

σ0 + σ1 = σ0+1(τ) =
−2ϕR

minu

(1 + u)

(
1 + ϵ

λ
3/2
R

96πMR

(
−2ϕR

min

u

(1 + u)2

))
. (2.170)

With this solution we can analytically continue to the whole complex τ plane to get the answer

in real time, in terms of z(t):

σ0+1(t) =
z

1− z
2ϕR

min

(
1 + ϵ

λ
3/2
R

96πMR

z

(1− z
2ϕR

min
)2

)
. (2.171)

Having this solution we can find the one-loop amplitude at threshold doing n derivatives in

z(t) using Eq. (2.43) (and setting ϵ to one):

AB
1loop(1→ n) = n!

(
1

2ϕR
0

)n−1
(
1 +

λR

√
3

96π
n(n− 1)

)
. (2.172)

We can see that this contribution is real, different from the unbroken case Eq. (2.129). It

has a different sign, and it still has factorial growth even in the first order. It is clear that the

important coupling is, in this case, λn2. This is the object that needs to be small such that the

approximation makes sense. This is similar to the case before, so in a high multiplicity limit,

it is not so straightforward to recover information from these initial terms. This discussion

will be continued at the end of chapter 3, were we discuss the range of validity of perturbation

theory for high multiplicity calculation. If we want to understand this better, we need to try

to recover momentum dependence in these amplitudes. Without this, we cannot say anything


45

about the decay rate that is meaningful. It is worth to point out that all the results so far are

exact in 0 spatial dimensions, giving just quantum mechanics. There is no phase space in this

case, and we still have the factorial growth of these amplitudes. This could be problematic to

the unitarity of the theory if this equation can be trusted in the region of interest. Next, we

show a possible way to understand why this factorial growth is happening in the perturbative

regime. After that, we start to investigate how we can recover the momentum dependence

doing some trivial cases and trying to generalize for high multiplicity. In the end, we review

some general results in the literature about this regime so we can start the exploration of the

Higgsplosion proposal.

2.2.5 Discussion About the Factorial Growth

Usually, the series that we deal with in Quantum Mechanics and Quantum Field Theory are

divergent. This divergence is not a problem because we know how to deal with this kind of

series with different summation machines like Borel or Padé [37, 38]. To understand this other

kind of divergence in the amplitude, we need to understand why the series is divergent in the

first place. The analysis is done for λx4 in Quantum Mechanics, but we can generalize to the

Field Theory case. We can see that there is a connection between the graphs with N vertices

with the coefficients of a given series:∑
graphs

all graphs with N verticies = aN . (2.173)

Now, if we have N vertices in x4 theory the structure would be something as is represented in

Figure (2.5);

Figure 2.5: Combinatorics for the ϕ4 interaction.

We have to connect all these vertices. In a usual process, we have to connect some external

lines, but they are only a finite amount so we can ignore in the large-N limit. If we pick one

of the lines, we have (4N − 1) possible connections, the next line we have (4N − 3). Going all

the way we have:

aN ∼ (4N − 1)!! as N →∞ . (2.174)


46

In this case, we are over counting. There are different arrangements for the vertices that give

the same result, N !. We still are overcounting because the lines on a vertex are identical, giving

a factor of 4!. In the end, we can write the coefficient of the series as:

aN ∼
(4N − 1)!!

N !(24)N
as N →∞ . (2.175)

It could be that some of these graphs are disconnected and do not contribute to physical

processes. In the counting of the graphs, the chance that this happens is small so we do not

consider this. In the large-N limit, it does not make a difference. Now, in this limit the

coefficient of the series behaves like:

aN ∼ N !CN as N →∞ . (2.176)

The factorial growth of the series indicates the divergent nature of it. This is just a heuristic

argument, but it captures the spirit of what is happening. We could get some integrals that

give a small contribution and change this picture. The divergence of the series occurs because,

at small N , the small coefficients dominate. However, when going to large-N , the number of

diagrams is so large that no matter how small the coefficient is, it will blow up. The behavior

of a typical divergent series can be seen in the Figure 2.6.

Figure 2.6: Typical Behavior of a divergent series. The optimal truncation of
a series like this would be when it hits the lowest point because of the definition

of an asymptotic series.


47

Now, we can see why these amplitudes are growing factorially. We have too many diagrams

for large multiplicity amplitudes already at tree level as it is shown in Figure 2.1 and Figure 2.2.

There is no small contribution that wins in the large-n limit. The picture is significantly different

in this case because there is never a region where the small coefficient dominates. This means

that we still can extract information from the series, but any partial sum will give a bad

approximation for the function. If we use summation machines to the loop corrections, we get

the real behavior of these amplitudes. In this framework, we cannot trust the perturbative

expression even at the tree level, but they have information about the actual function. In

high multiplicity processes not only λ matters but n as well. This argument does not hold for

different kinds of approximations like semiclassical calculation, but in those cases, it should be

possible to do a similar analysis so we can understand the limitations of the results.

2.3 Investigation of Beyond Threshold Amplitude

So far, we only calculated threshold amplitudes. With this information only, we cannot recon-

struct the decay rate because at the threshold the phase space is just a point. Nevertheless,

this result is exact in 0 spatial dimensions, being the usual x4 theory. If we want to see if the

decay rate has an exponential growth at high multiplicity, we need to analyze if the phase space

contribution has enough strength to combat the factorial growth of Eq. (2.129) and Eq. (2.172).

Going beyond the threshold at high multiplicity is an incredibly difficult task. There are too

many momenta in the final state. To surpass this problem, we can try to construct these ampli-

tudes in the near threshold limit where all final particles are non-relativistic. In this Section, we

try to generalize Brown’s method for beyond threshold amplitudes and discuss the difficulties

of doing so. After that, we use Feynman rules for small multiplicity to try to understand what

we should expect in the near threshold limit. Finally, in the end, we comment about some

results in the literature of beyond threshold amplitude and what we can take from them.

2.3.1 Naive Generalization of Brown’s Methods and its Problems

We saw that the Brown method [23] of using the expectation value of the field with respect

to a source works well for the threshold amplitude. If we want to generalize this, a first naive

attempt is trying a source of the type:

ρ(xµ) = ρ0e
ikµxµ

. (2.177)

In the limit where k⃗ → 0 we recover the solution of before. If we do this in the tree level

solution, we need to solve the equation:

(□+m2)ϕ0 +
λ

3!
ϕ3
0 = ρ0e

ikµxµ

, (2.178)


48

it turns out that we can solve this equation in the same way as the threshold case. The only

difference is that the mass shell condition will be for the four-momentum. The solution is

similar, because:

□enikµx
µ

= −n2k2enikµx
µ

. (2.179)

Then, in the on-shell limit we get the same result as before:

ϕ0(x) =
z(xµ)(

1− λ
48m2 z(xµ)2

) . (2.180)

were z(xµ) is a similar expression compared to Eq. (2.12):

z(xµ) = z0e
ikµxµ

, (2.181)

z0 = −
ρ0

k2 −m2
. (2.182)

This is strange, because in the LSZ reduction formula we get something like:

δϕ0[ρ]

δρ(xi)
=

1

p2i −m2
δ4(x− xi)

∂ϕ0

∂z
. (2.183)

The problem is that, in the end, we set ρ to zero and all the momentum dependence vanishes,

obtaining a threshold result again. Even though this is a solution of the equation of motion

with a space-time dependent source, this is not enough to find beyond threshold amplitudes.

It seems that this source, Eq. (2.177) can only excite one frequency mode, not having enough

information about the field to construct beyond threshold amplitudes:∫
d4xϕ0(x)ρ(x) ∝ ϕ0(k) . (2.184)

The alternative to this is to find a more comprehensive source that we can solve that has

the right threshold limit and has at least some non-relativistic generalization. It turns out that

this is a hard task because of the non-linearity and for now we cannot advance further. Before

proceeding to the next part, it is worth pointing out one thing. The limit of z = 0 in the

LSZ reduction formula may appear strange and in this case, be responsible for vanishing the

momentum dependence. If we do this computation with caution, we see that in the double

limit of k2 → m2 and ρ → 0 we are left with a constant z0 term, so it is possible to try out

instead z = z0e
ikµxµ

. However, if we do all the work before going to mass shell the expression

for z(x) is finite, and we can take the limit for the source going to zero there, taking z0 to zero

first. The order of these limits is essential, and in the LSZ reduction formula, the ρ going to


49

zero is the first one, so this should not alter the results.

2.3.2 Tree Level Investigation of Beyond Threshold (1→ 3)

If we want to recover the momenta dependence of a high multiplicity amplitude, it is worth to

calculate simpler cases. The region of interest is with all external particles in a non-relativistic

limit. Here we work the first non-trivial case of 1 particle going to 3 with non-relativistic

momenta. This case is useful to check the threshold computation done in the previous Section,

using Feynman diagrams, and see how is the shape of the first momentum correction. The

process that we are interested in is represented in Figure 2.7.

Figure 2.7: Amputated amplitude that we are interested in is the 1→ 3 process.
Generated with FeynArts [31].

Here we have to remember that we do not remove the incoming leg, like we usually do.

Then the amplitude is:

M(1→ 3) = (p2 −m2)A(1→ 3) . (2.185)

Feynman diagrams construct M(1 → 3) and pµ is the momentum of the incoming particle.

Using the usual Feynman rule for the ϕ4 theory as is represented in Figure 2.8

We get the amplitude as:

A(1→ 3) =
λ

p2 −m2
. (2.186)

To investigate the non-relativistic limit of this expression we need to expand p2 in the

appropriate manner. Calling the outgoing momenta as qi, in the non-relativistic limit we have

|q⃗i| ≪ m. The definition of the incoming momentum is:

p = q1 + q2 + q3 , (2.187)


50

Figure 2.8: Feynman rule for the ϕ4 case in the normalization that we are
using. Generated with FeynArts [31].

where we write each external momentum as:

qi = (γim, q⃗i) . (2.188)

In the non-relativistic limit we expand the first component as:

qi = (m+
q⃗ 2
i

2m
− q⃗ 4

i

8m3
+ . . . , q⃗i) , (2.189)

this means that the denominator of Eq. (2.186) can be written as:

(q1 + q2 + q3)
2 −m2 = 2

(
m2 + q1 · q2 + q1 · q3 + q2 · q3

)
= (2.190)

2

(
4m2 + q⃗ 2

1 + q⃗ 2
2 + q⃗ 2

3 − q⃗1q⃗2 − q⃗1q⃗3 − q⃗2q⃗3+

+
1

4m2
(q⃗ 2

1 q⃗
2
2 + q⃗ 2

1 q⃗
2
3 + q⃗ 2

2 q⃗
2
3 − q⃗ 4

1 − q⃗ 4
2 − q⃗ 4

3 ) + . . .

)
.

The interesting feature of this expression that remains in the high multiplicity cases is that at

leading order we can write it in terms of the non-relativistic energy of the outgoing particles in

a general frame:

E =
1

2m

∑
q⃗ 2
i −

1

2mn

(∑
q⃗i

)2
=

n− 1

2mn

∑
q⃗ 2
i −

1

nm

∑
i ̸=j

q⃗iq⃗j . (2.191)


51

Using Eq. (2.191) the denominator of Eq. (2.186) can be written as:

(q1 + q2 + q3)
2 −m2 = 8m2 + 6mE +

1

2m2
(q⃗ 2

1 q⃗
2
2 + q⃗ 2

1 q⃗
2
3 + q⃗ 2

2 q⃗
2
3 − q⃗ 4

1 − q⃗ 4
2 − q⃗ 4

3 ) + . . . .

(2.192)

The first important thing to notice is that in the next order we cannot write the denominator

in terms of only E. It is easy to see this because E2 would contain odd powers of momenta

and this does not appear in Eq. (2.192). This feature remains when we increase the number

of external particles. We can see that in the large n limit this problem vanishes, leaving only

even powers of momentum in Eq. (2.191). If we write the amplitude in a expansion of small

spatial momenta (small E) we get:

A(1→ 3) =
λ

8m2

(
1− 3

4

E

m
+

9

8

E2

m2
− (2.193)

− 1

8m4
(−q⃗ 3

1 q⃗2 − q⃗ 3
1 q⃗3 − q⃗ 3

3 q⃗1 − q⃗ 3
2 q⃗1 − q⃗ 3

2 q⃗3 − q⃗ 3
3 q⃗2 + 2q⃗ 2

1 q⃗
2
2 + 2q⃗ 2

1 q⃗
2
3 + 2q⃗ 2

2 q⃗
2
3 ) + . . .

)
.

From this result, we can see that for small momenta the first term depends only on the non-

relativistic energy. If we go to high orders in the momentum expansion, it is expected that we

cannot describe the system only with E and other functions will have an important play. From

Eq. (2.191) we can see that these other objects are subdominant, and only the non-relativistic

energy dominates. However, even with this expression, we cannot say much about the decay

rate. If we are working in a limit where the kinetic energy is small, we are in a shell of the

phase space as represented in Figure 2.9

Figure 2.9: Difference between the decay rate that we can get using a non-
relativistic approximation to what it would be useful to have.

This means that the decay rate can only be calculated for this shell that is smaller than

2m. For a good approximation of the decay rate, we expect to be able to calculate in a broader


52

region. The limit of validity of this expression does not help us much in understanding the

behavior of the decay rate, but it is a step in this direction. Next, we do the same calculation

for the 1→ 5 case where we start to see a trend in behavior. One thing that we can take from

this is that the threshold computation works, being the first term of Eq. (2.19).

2.3.3 Tree Level Investigation of Beyond Threshold (1→ 5) and its

Generalities

Let us go one step beyond and analyze the process of 1→ 5 particles in the same non-relativistic

limit. Now the problem starts to appear because we have ten different combinations of external

leg positions for the Feynman diagram, as is shown in Figure 2.10. These are all the possible

combinations that the internal propagator can have, considering the particles are identical.

Figure 2.10: Amputated amplitude that we are interested in for the 1 → 5
process. Generated with FeynArts [31].

The amplitude can be written as:

A(1→ 5) =
1

p2 −m2

∑
ijk

1

(qi + qj + qk)2 −m2
λ2 , (2.194)


53

where i, j, k are the sum of the following combinations written in the Table 2.1.

i j k
1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

Table 2.1: Possible combinations for the sum in i, j and k.

The problem now is to do the non-relativistic limit, because we have two propagators. One

of them with all the momenta and other only with three at a time. Doing the non-relativistic

limit just like before we can write the amplitude as:

A(1→ 5) =

(
λ2

2

)2
1

∆1 +∆2t2 +∆3t4

∑
ijk

1

∆4 +∆ijk
5 t2 − 1

4m2∆
ijk
6 t4

. (2.195)

In this expression t is just a fictitious parameter to help expand this in small spatial momentum,

it keeps track of the power of |q⃗i|. In the end we expand Eq. (2.195) in t and set t = 1 to get

the non-relativistic approximation. These deltas appears from doing the expansion on the

denominators of Eq. (2.194), up to quartic order:

∆1 = 12m2 , (2.196)

∆2 = 2(q⃗ 2
1 + q⃗ 2

2 + q⃗ 2
3 + q⃗ 2

4 + q⃗ 2
5 )− q⃗1q⃗2 − q⃗1q⃗3 − q⃗1q⃗4− (2.197)

−q⃗1q⃗5 − q⃗2q⃗3 − q⃗2q⃗4 − q⃗2q⃗5 − q⃗3q⃗4 − q⃗3q⃗5 − q⃗4q⃗5 ,

∆3 = −
1

2m2
(q⃗ 4

1 + q⃗ 4
2 + q⃗ 4

3 + q⃗ 4
4 + q⃗ 4

5 ) +
1

4m2
(q⃗ 2

1 q⃗
2
2 + q⃗ 2

1 q⃗
2
3 + q⃗ 2

1 q⃗
2
4 + q⃗ 2

1 q⃗
2
5 + (2.198)

+q⃗ 2
2 q⃗

2
3 + q⃗ 2

2 q⃗
2
4 + q⃗ 2

2 q⃗
2
5 + q⃗ 2

3 q⃗
2
4 + q⃗ 2

3 q⃗
2
5 + q⃗ 2

4 q⃗
2
5 ) ,


54

∆4 = 4m2 , (2.199)

∆ijk
5 = q⃗ 2

i + q⃗ 2
j + q⃗ 2

k − q⃗iq⃗j − q⃗iq⃗k − q⃗j q⃗k , (2.200)

∆ijk
6 = q⃗ 4

i + q⃗ 4
j + q⃗ 4

k − q⃗ 2
i q⃗

2
j − q⃗ 2

i q⃗
2
k − q⃗ 2

j q⃗
2
k . (2.201)

With this information we can do the non-relativistic expansion of the amplitude:

A(1→ 5) = λ2
∑
ijk

(
1

192m4
− 1

2304m6
(∆2 + 3∆ijk

5 ) + . . .

)
. (2.202)

The first term gives the right threshold amplitude, the second term we will re-write in terms of

the non-relativistic energy. Using the definition of the non-relativistic energy, Eq. (2.191) it is

a direct but boring computation to show:∑
ijk

∆2 + 3∆ijk
5 = 95mE . (2.203)

The amplitude up to first order is:

A(1→ 5) = 5!

(
λ

48m2

)2(
1− 19

24

E

m
+ . . .

)
. (2.204)

This result is in agreement with [39, 40]. This shows that indeed, the first correction beyond

threshold comes only from the non-relativistic energy. Again, if we try to go to the next order,

this ceases to be the case because of odd powers of momentum in the expression for the energy

square that does not appear in the calculation. The next step in this investigation is to show

and discuss some results coming from recursion relation and semiclassical computation. Mostly

we will comment and try to interpret inside this framework. From these cases, we can take

that the information beyond the threshold is tough to get.

2.3.4 Recurrence Relations and General Results of Beyond Thresh-

old Amplitudes at High Multiplicity

The focus so far was to study these high multiplicity processes in the perturbative regime. We

saw already why these amplitudes are growing factorially. The last piece of information on the

perturbative regime comes from recursion relations. Here we highlight the history and main

results of it. After that, we comment on some recent results coming from the semiclassical

calculation. These results were essential to motivate the studies of high multiplicity processes

after the initial wave of results that we covered so far.


55

The first improvement came in 1992 when E.N Argyres and Costa G. Papadopoulos used

recurrence relations to find amplitudes with one momentum in the final state [39]. The general

form of the recursion relation is represented in Figure 2.11. They find the solution for general

monomial interaction of the form:

V (ϕ) = λm
ϕm

m!
(2.205)

The specialization for our ϕ4 case gave the result:

Figure 2.11: Recursion relation for ϕ4 theory in the unbroken phase, n =
n1 + n2 + n3.

A(1→ 2k + 1) = (2k)!(2k)(2k + 1 + ω)

(
λ4

48

)k (
k + 1 +

2(3− ω)

3 + ω
k +

(3− ω)(1− ω)

(3 + ω)(5 + ω)
(k − 1)

)
,

(2.206)

where the mass is set to one and ω = 2E − 1 with E being the energy of the final particle with

nonzero spatial momentum. The threshold result is recovered when ω = 1. We can find with

these results 2→ n processes, if we use a negative ω, as it was shown to recover the threshold

results:

A(2→ n) = 0 n > 4 . (2.207)

In the same paper the authors calculate the case of broken phase with one particle in the final

state with arbitrary momentum:

AB(1→ n) = (n− 1)!(n− 1)(n+ ω)

(
λ4

12

)(n−1)/2(
n+ 2(n− 1)

(1− ω)

(2 + ω)
− ω(1− ω)

(2 + ω)(3 + ω)

)
,

(2.208)


56

and again the limit of ω = 1 checks out. These results seem to show that in the perturbative

region, even momentum dependence cannot save the behavior of these amplitudes.

After these results, people realized that if we restrict ourselves to the quantum mechanical

limit of this, we get the complete result. This motivated the study of transition amplitudes in

the x4 oscillator by C.A Diamantis, B.C Georgalas, A. B. Lahanas and E. Papantonopolus [41].

They looked for bounds in the transition amplitudes generated by an external source. This

would be a simplification of our case in the (1+0)D case. They introduced the concept of holy

grail function F (λn) for