# On a conjecture on shifted primes with large prime factors

Yuchen Ding  (Yuchen Ding) School of Mathematical Science, Yangzhou
University, Yangzhou 225002, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{P}$ be the set of all primes and $\pi(x)$ be the number of
primes up to $x$. For any $n\geq 2$, let $P^{+}(n)$ be the largest prime
factor of $n$. For $0<c<1$, let

$T_{c}(x)=\\#\\{p\leq x:p\in\mathcal{P},P^{+}(p-1)\geq p^{c}\\}.$

In this note, we prove that there exists some $c<1$ such that

$\limsup_{x\rightarrow\infty}\frac{T_{c}(x)}{\pi(x)}<\frac{1}{2},$

which disproves a conjecture of Chen and Chen.

###### Key words and phrases:

Shifted prime; The Bombieri–Vinogradov theorem.

###### 2010 Mathematics Subject Classification:

Primary 11N05.

*Corresponding author

##  1\. Introduction

The investigations of the largest prime factors of shifted primes start from a
cute article of Goldfeld [6], where he proved that

$\\#\\{p\leq
x:p\in\mathcal{P},P^{+}(p-1)>p^{1/2}\\}\geq\frac{1}{2}\frac{x}{\log
x}+O\left(\frac{x\log\log x}{(\log x)^{2}}\right),$

where $\mathcal{P}$ is the set of all primes and $P^{+}(n)$ denotes the
largest prime factor of integer $n$. Conventionally, we set $P^{+}(1)=1$. For
$0<c<1$, let

$T_{c}(x)=\\#\\{p\leq x:p\in\mathcal{P},P^{+}(p-1)\geq p^{c}\\}.$

Goldfeld further remarked that one can show

$\displaystyle\liminf_{x\rightarrow\infty}\frac{T_{c}(x)}{\pi(x)}>0$ (1.1)

provided that $c<7/12$, where $\pi(x)$ is the number of primes up to $x$.
Fouvry [5] improved this by showing that there is some $c_{0}\in(2/3,1)$ such
that inequality (1.1) holds.

In the same direction, Luca et al. [9] considered the lower bound for
$T_{c}(x)$ for small values of $c$. They proved that for $1/4\leq c\leq 1/2$

$T_{c}(x)\geq(1-c)\frac{x}{\log x}+E(x),$

where

$E(x)\ll\begin{cases}x\log\log x/(\log x)^{2},&\text{for~{}}1/4<c\leq 1/2,\\\
x/(\log x)^{5/3},&\text{for~{}}c=1/4.\end{cases}$

As pointed out by Chen and Chen [3], the method of Luca et al. (essentially
due to Goldfeld) cannot be applied to $c\in(0,1/4)$. By some refinements of
the argument employed by Luca et al., Chen and Chen could extend $c$ to the
interval $(0,1/2)$ with slightly better error terms of the order of magnitude
$O(x/(\log x)^{2})$. Moreover, Chen and Chen proved the following interesting
result. For any integer $k\geq 2$, there exists at most one
$c\in[\frac{1}{k+1},\frac{1}{k})$ such that

$T_{c}(x)=(1-c)\frac{x}{\log x}+o\left(\frac{x}{\log x}\right).$

Based on this result, Chen and Chen made the following conjecture.

######  Conjecture 1.1.

For any integer $k\geq 1$ and any $c\in[\frac{1}{k+1},\frac{1}{k})$, we have

$T_{c}(x)\geq\left(1-\frac{1}{k+1}\right)\frac{x}{\log x}+o\left(\frac{x}{\log
x}\right).$

Let $\rho(u)$ be the Dickman function, defined as the unique continuous
solution of the equation differential–difference

$\displaystyle\begin{cases}\rho(u)=1,&0\leq u\leq 1,\\\
u\rho^{\prime}(u)=-\rho(u-1),&u>1.\end{cases}$

Let $\theta_{1}\thickapprox 0.3517$ be the unique solution of equation

$\theta-4\int_{1/\theta-1}^{1/\theta}\frac{\rho(t)}{t}dt=0.$

Feng and Wu [4] proved that

$T_{c}(x)\geq\left(1-4\int_{1/\theta-1}^{1/\theta}\frac{\rho(t)}{t}dt+o(1)\right)\pi(x)$

provided $0<c<\theta_{1}$, which confirmed Conjecture 1.1 for $k\geq 3$ by
numerical values involving Dickman’s function. Later, the lower bound for
$T_{c}(x)$ was further improved to

$T_{c}(x)\geq\left(1-4\rho(1/\theta)+o(1)\right)\pi(x)$

by Liu, Wu and Xi [8] provided that $0<\theta<\theta_{2}\thickapprox 0.3734$,
where $\theta_{2}$ is the unique solution to equation
$\theta-4\rho(1/\theta)=0$.

In the present note, we show that Conjecture 1.1 is actually incorrect for
$k=1$. Precisely,

######  Theorem 1.1.

There is an absolute constant $c<1$ such that

$\limsup_{x\rightarrow\infty}\frac{T_{c}(x)}{\pi(x)}<1/2.$

##  2\. Proofs

Before presenting the proof of Theorem 1.1, we first fix some notations and
then state a few lemmas to be used later. Let $\Lambda(n)$ be the von Mangoldt
function. As usual, $\varphi(n)$ denotes the Euler totient function. Let
$\pi(x;b,a)$ be the number of primes $p\equiv a\ ({\rm{mod}}\ b)$ up to $x$.
Following Goldfeld and Luca et al., we define

$L(x;u,v)=\sum_{u<m\leq v}\Lambda(m)\pi(x;m,1).$

The first two lemmas are included in the proof of Goldfeld (see also [3, Lemma
2.1]).

######  Lemma 2.1.

[6, bottom of page 23] For sufficiently large $x$, we have

$L(x;1,x)=x+O\left(x/\log x\right).$

######  Lemma 2.2.

[6, equations (2) and (3)] For sufficiently large $x$, we have

$L(x;1,x^{1/2})=x/2+O\left(\frac{x\log\log x}{\log x}\right).$

The next lemma is another conjecture of Chen and Chen confirmed by Wu later.

######  Lemma 2.3.

[14, Theorem 2] For $0<c<1$, let

$T^{\prime}_{c}(x)=\\#\\{p\leq x:p\in\mathcal{P},P^{+}(p-1)\geq x^{c}\\}.$

Then for sufficiently large $x$ we have

$T_{c}(x)=T^{\prime}_{c}(x)+O\left(\frac{x\log\log x}{(\log x)^{2}}\right).$

The proof of our theorem, among other things, is based on the following deep
result which appeared first in the paper of Banks and Shparlinski [2, Lemma
2.1]. Here, we use the version stated by Wu with a slight adjustment from
prime modulus $q$ to integer modulus $m$.

######  Lemma 2.4.

[14, Lemma 2.2] There exist two functions $K_{2}(\theta)>K_{1}(\theta)>0$,
defined on the interval $(0,17/32)$ such that for each fixed real $A>0$, and
all sufficiently large $Q=x^{\theta}$, the inequalities

$K_{1}(\theta)\frac{\pi(x)}{\varphi(m)}\leq\pi(x;m,1)\leq
K_{2}(\theta)\frac{\pi(x)}{\varphi(m)}$

hold for all integers $m\in(Q,2Q]$ with at most $O\left(Q(\log Q)^{-A}\right)$
exceptions, where the implied constant depends only on $a,A$ and $\theta$.
Moreover, for any fixed $\varepsilon>0$, these functions can be chosen to
satisfy the following properties:
$\bullet$ $K_{1}(\theta)$ is monotonic decreasing, and $K_{2}(\theta)$ is
monotonic increasing.
$\bullet$ $K_{1}(1/2)=1-\varepsilon$ and $K_{2}(1/2)=1+\varepsilon$.

Remark. The above lemma is due to the Bombieri–Vinogradov theorem for
$0<\theta<1/2$, Baker–Harman [1] for $1/2\leq\theta\leq 13/25$ and Mikawa [10]
for $13/25\leq\theta\leq 17/32$.

Now we turn to the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Let $33/64<c<1$ be a fixed constant. Throughout our proof, $p$ and $q$ will
always denote primes unless indicated otherwise. The lower bound on
$T_{c}(x)$, as observed by Goldfeld, starts from

$T_{c}(x)\geq\sum_{p\leq x}\sum_{\begin{subarray}{c}x^{c}\leq q\leq x\\\
q|p-1\end{subarray}}1\geq\frac{1}{\log x}\sum_{p\leq
x}\sum_{\begin{subarray}{c}x^{c}\leq q\leq x\\\ q|p-1\end{subarray}}\log q.$

For an upper bound of $T_{c}(x)$, we begin with the dual observation inspired
by Goldfeld with the help of Lemma 2.3

$\displaystyle T_{c}(x)$ $\displaystyle=\sum_{p\leq
x}\sum_{\begin{subarray}{c}x^{c}\leq q\leq x\\\
q|p-1\end{subarray}}1+O\left(\frac{x\log\log x}{(\log x)^{2}}\right)$
$\displaystyle\leq\frac{\log x}{c}\sum_{p\leq
x}\sum_{\begin{subarray}{c}x^{c}\leq q\leq x\\\ q|p-1\end{subarray}}\log
q+O\left(\frac{x\log\log x}{(\log x)^{2}}\right).$

Changing the order of summations above, we get

$\displaystyle T_{c}(x)\leq\frac{\log x}{c}\sum_{x^{c}\leq q\leq
x}\pi(x;q,1)\log q+O\left(\frac{x\log\log x}{(\log x)^{2}}\right).$ (2.1)

Following Goldfeld, the above estimate can be handled firstly by manipulating
the weighted sum $L(x;x^{c},x)$. For $33/64<c<1$, from Lemmas 2.1 and 2.2 we
clearly have

$\displaystyle L(x;x^{c},x)$
$\displaystyle=L(x;1,x)-L(x;1,x^{1/2})-L(x;x^{1/2},x^{c})+O(\log x)$
$\displaystyle=x/2-L(x;x^{1/2},x^{c})+O\left(\frac{x\log\log x}{\log
x}\right)$ $\displaystyle\leq
x/2-L(x;x^{1/2},x^{33/64})+O\left(\frac{x\log\log x}{\log x}\right).$ (2.2)

We now employ Lemma 2.4 to give a nontrivial lower bound of
$L(x;x^{1/2},x^{33/64})$. For integers $1\leq j\leq\left\lfloor\frac{\log
x}{65\log 2}\right\rfloor$, let $Q_{j}=2^{j}x^{1/2}$. Then we have
$Q_{j}>x^{1/2}$ and $2Q_{j}<x^{33/64}$ for sufficiently large $x$. For any
$1\leq j\leq\left\lfloor\frac{\log x}{65\log 2}\right\rfloor$, from Lemma 2.4
(with $A=2$) we have

$\displaystyle\pi(x;m,1)>\frac{K_{1}(33/64)}{2}\frac{x}{\varphi(m)\log x}$
(2.3)

for all integers $m\in(Q_{j},2Q_{j}]$ with at most $O\left(Q_{j}/(\log
x)^{2}\right)$ exceptions. For $1\leq j\leq\left\lfloor\frac{\log x}{65\log
2}\right\rfloor$, let $S_{j}$ be the set of exceptions of $m$ in the interval
$(Q_{j},2Q_{j}]$. Thus, in view of equation (2.3), we have

$\displaystyle L(x;x^{1/2},x^{33/64})$ $\displaystyle=\sum_{x^{1/2}<m\leq
x^{33/64}}\Lambda(m)\pi(x;m,1)$
$\displaystyle\geq\sum_{j=1}^{\left\lfloor\frac{\log x}{65\log
2}\right\rfloor}\sum_{\begin{subarray}{c}Q_{j}<m\leq 2Q_{j}\\\ m\not\in
S_{j}\end{subarray}}\Lambda(m)\pi(x;m,1)$
$\displaystyle\geq\frac{K_{1}(33/64)}{2}\frac{x}{\log
x}\sum_{j=1}^{\left\lfloor\frac{\log x}{65\log
2}\right\rfloor}\sum_{\begin{subarray}{c}Q_{j}<m\leq 2Q_{j}\\\ m\not\in
S_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}.$ (2.4)

For $1\leq j\leq\left\lfloor\frac{\log x}{65\log 2}\right\rfloor$, we clearly
have

$\sum_{\begin{subarray}{c}Q_{j}<m\leq
2Q_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}\geq\sum_{\begin{subarray}{c}Q_{j}<p\leq
2Q_{j}\end{subarray}}\frac{\log p}{p}\geq\log
Q_{j}\sum_{\begin{subarray}{c}Q_{j}<p\leq 2Q_{j}\end{subarray}}\frac{1}{p}.$

Chebyshev’s estimate or the prime number theorem gives us

$\sum_{\begin{subarray}{c}Q_{j}<p\leq
2Q_{j}\end{subarray}}\frac{1}{p}\geq\frac{1}{2Q_{j}}\sum_{\begin{subarray}{c}Q_{j}<p\leq
2Q_{j}\end{subarray}}1\geq\frac{1}{2Q_{j}}\frac{Q_{j}}{2\log
Q_{j}}=\frac{1}{4\log Q_{j}},$

from which we deduce that

$\sum_{\begin{subarray}{c}Q_{j}<m\leq
2Q_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}\geq\frac{1}{4}.$

Moreover, we have

$\sum_{\begin{subarray}{c}m\in
S_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}\ll\log(2Q_{j})\sum_{\begin{subarray}{c}m\in
S_{j}\end{subarray}}\frac{\log\log m}{m}\ll\log x\log\log
x\sum_{\begin{subarray}{c}m\in S_{j}\end{subarray}}\frac{1}{m}$

since $\varphi(m)\gg\frac{m}{\log\log m}$ (see for example [11, Theorem 2.9]),
where the implied constants are all absolute. Noticing that
$S_{j}\subset(Q_{j},2Q_{j}]$ and $|S_{j}|\ll Q_{j}/(\log x)^{2}$ by its
definition, we get

$\sum_{\begin{subarray}{c}m\in
S_{j}\end{subarray}}\frac{1}{m}\ll\frac{1}{Q_{j}}\frac{Q_{j}}{(\log
x)^{2}}=1/(\log x)^{2}.$

It follows that

$\sum_{\begin{subarray}{c}m\in
S_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}\ll\frac{\log\log x}{\log
x},$

where the implied constant is absolute. Thus, we have

$\sum_{\begin{subarray}{c}Q_{j}<m\leq 2Q_{j}\\\ m\not\in
S_{j}\end{subarray}}\frac{\Lambda(m)}{\varphi(m)}\geq 1/5$ (2.5)

provided that $x$ is sufficiently large. Combining equation (2), inequalities
(2) and (2.5) we conclude that there exists some $\delta>0$ such that

$\displaystyle L(x;x^{c},x)\leq(1/2-\delta)x+O\left(\frac{x\log\log x}{\log
x}\right).$ (2.6)

The transition from $L(x;x^{c},x)$ to $\sum_{x^{c}\leq q\leq x}\pi(x;q,1)\log
q$ is somewhat standard. It is clear that

$\displaystyle L(x;x^{c},x)-\sum_{x^{c}\leq q\leq x}\pi(x;q,1)\log
q=\sum_{\begin{subarray}{c}x^{c}\leq q^{k}\leq x\\\
k>1\end{subarray}}\pi(x;q^{k},1)\log q+O(\log x).$ (2.7)

For simplicity, we now require that $3/4<c<1$. Trivial estimates lead to

$\displaystyle\sum_{\begin{subarray}{c}x^{c}\leq q^{k}\leq x\\\
k>1\end{subarray}}\pi(x;q^{k},1)\log q$
$\displaystyle\leq\sum_{\begin{subarray}{c}x^{3/4}\leq q^{k}\leq x\\\ q\leq
x^{1/2}\\\ k>1\end{subarray}}\pi(x;q^{k},1)\log q\ll
x^{1/4}\sum_{\begin{subarray}{c}x^{3/4}\leq q^{k}\leq x\\\ q\leq x^{1/2}\\\
k>1\end{subarray}}\log q$ $\displaystyle\ll x^{1/4}\sum_{q\leq x^{1/2}}\log
q\log x\ll x^{3/4}(\log x)^{2}.$ (2.8)

From equations (2.6), (2.7) and (2), we obtain

$\displaystyle\sum_{x^{c}\leq q\leq x}\pi(x;q,1)\log
q\leq(1/2-\delta)x+O\left(\frac{x\log\log x}{\log x}\right)$ (2.9)

provided that $3/4<c<1$. Hence, equation (2.9) together with equation (2.1)
yield

$T_{c}(x)\leq\frac{1}{c}(1/2-\delta)x/\log x+O\left(\frac{x\log\log x}{(\log
x)^{2}}\right).$

Now the theorem follows from taking $c=\max\\{1-\delta,4/5\\}$. ∎

##  3\. Final remarks

As early as 1980, Pomerance [12] conjectured that for any $c\in(0,1)$ we have

$\sum_{\begin{subarray}{c}p\leq x\\\ P^{+}(p-1)\leq
x^{c}\end{subarray}}1\sim\rho(1/c)\pi(x),\quad\text{as~{}}x\rightarrow\infty.$

This is surely a rather difficult conjecture which is beyond the power of
present mathematics. Granville [7] claimed that it follows from the
Eillott–Halberstam conjecture. Recently, Wang [13] offered an alternative
proof of this claim. Therefore, assuming the Eillott–Halberstam conjecture,
one can prove

$\lim_{x\rightarrow\infty}T_{c}(x)/\pi(x)\rightarrow
0,\quad\text{as~{}}c\rightarrow 1$

by Lemma 2.3. This immediately leads to a negative answer of Conjecture 1.1
for $k=1$ under the Eillott–Halberstam conjecture. So this note can be viewed
as an unconditional proof of this matter. The exceptional result for level
greater than $1/2$ (see Lemma 2.4) is crucial for this goal.

## Acknowledgments

The author would like to thank the anonymous referee for his/her helpful
comments which improved the quality of this note greatly.

The author is supported by National Natural Science Foundation of China under
Grant No. 12201544, Natural Science Foundation of Jiangsu Province, China,
Grant No. BK20210784, China Postdoctoral Science Foundation, Grant No.
2022M710121, the foundations of the projects ”Jiangsu Provincial
Double–Innovation Doctor Program”, Grant No. JSSCBS20211023 and ”Golden
Phoenix of the Green City–Yang Zhou” to excellent PhD, Grant No.
YZLYJF2020PHD051.

## References

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  * [5] Étienne Fouvry, Théorème de Brun–Titchmarsh: application au théorème de Fermat, Invent. Math., 79 (1985) 383–407.
  * [6] M. Goldfeld, On the number of primes $p$ for which $p+a$ has a large prime factor, Mathematika, 16 (1969) 23–27.
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  * [9] F. Luca F, R. Menares, A. Pizarro–Madariaga, On shifted primes with large prime factors and their products, Bul.l Belg. Math. Soc. Simon Stevin, 22 (2015) 39–47.
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Non-analyticity of the $S$-matrix
with spontaneously broken Lorentz invariance

Paolo Creminelli${}^{\,\rm a,\rm b}$, Matteo Delladio${}^{\,\rm c}$, Oliver
Janssen${}^{\,\rm d}$,
Alessandro Longo${}^{\,\rm e,\rm f}$ and Leonardo Senatore${}^{\,\rm c}$

a ICTP, International Centre for Theoretical Physics, 34151 Trieste, Italy

b IFPU, Institute for Fundamental Physics of the Universe, 34014 Trieste,
Italy

c Institut für Theoretische Physik, ETH Zürich, 8093 Zürich, Switzerland

d Laboratory for Theoretical Fundamental Physics, EPFL, 1015 Lausanne,
Switzerland

e SISSA, International School for Advanced Studies, 34136 Trieste, Italy

f INFN, National Institute for Nuclear Physics, 34127 Trieste, Italy

Abstract We study the $S$-matrix of Goldstones in the renormalizable theory of
a $U(1)$ complex scalar at finite charge, i.e. in a state that breaks Lorentz
invariance. The theory is weakly coupled so that this $S$-matrix exists at all
energies. Unlike the Lorentz invariant case, the resulting $S$-matrix is not
analytic in the exchanged (complexified) four-momentum. The non-analyticities
stem from the LSZ reduction formula, as a consequence of the energy-dependent
mixing between the radial and Goldstone modes.

###### Contents

  1. 1 Introduction
  2. 2 The $U(1)$ model
    1. 2.1 Canonical quantization
  3. 3 LSZ reduction formula and lack of analyticity
  4. 4 The $S$-matrix
    1. 4.1 Interesting limits
      1. 4.1.1 $\mu=0$
      2. 4.1.2 High energies
      3. 4.1.3 Low energies
  5. 5 Decay rate of the Goldstone
    1. 5.1 $\omega\ll M$
    2. 5.2 $\omega\gg M$
  6. 6 Conclusions
  7. A Second derivation of LSZ
  8. B Feynman rules
    1. B.1 Scattering amplitudes
  9. C $S$-matrix in the low-energy EFT
  10. D Estimate of the decay width

##  1 Introduction

Locality and unitarity impose stringent constraints on the $S$-matrix of
consistent effective field theories (EFTs). The derived positivity bounds [1,
2, 3, 4, 5] and the general program of $S$-matrix bootstrap [6, 7] are under
intense investigation. Nearly all research has been conducted under the
assumption of (linearly realized) Lorentz invariance, which is a natural
presumption in the context of particle physics. However, Lorentz invariance is
spontaneously broken in cosmology (inflation, dark energy, …), in condensed
matter and also when studying perturbations of localized objects like black
holes (see for instance [8, 9]). It would be valuable to derive bounds on EFTs
without relying on Lorentz invariance.

The simplest approach would be to focus on the $S$-matrix and follow similar
steps as in the Lorentz invariant case [10, 11]. However, it is evident that
studying the $S$-matrix is not appropriate for a general UV completion, as
discussed in [12]. In the presence of Lorentz invariance one can scatter
asymptotic states at any energy, so that the $S$-matrix is well-defined both
in the UV and in the IR. This does not hold in general when Lorentz invariance
is absent: the low energy excitations around a vacuum that spontaneously
breaks Lorentz (phonons, inflaton perturbations, quasinormal modes of a black
hole, …) simply do not exist at high energy, so that one cannot define the
$S$-matrix at arbitrary energies. In general one has to look for other
quantities which are well-defined both in the UV and in the IR to derive
universal bounds: one possibility is to look at conserved currents as was done
in [12].

In this paper we study the $S$-matrix in a model that can be considered the
simplest example of spontaneous breaking of Lorentz invariance: the
renormalizable theory of a complex scalar with a $U(1)$ symmetry in a state
with finite charge. Boosts are spontaneously broken, while space and time
translations are not.111The unbroken time translations actually correspond to
a linear combination of the original time translation and a $U(1)$
transformation [13]. The model features two modes: a massless Goldstone and
the radial massive excitation, whose dispersion relations will be studied in
section 2. The model is weakly coupled so that, contrary to the general case
discussed above, one can study the $S$-matrix of the Goldstones at all
energies. Before venturing to make very general claims, we think it is
important to have at least one non-trivial example of an $S$-matrix in the
absence of Lorentz invariance.

Before calculating the $S$-matrix and evaluating it in various limits of
physical interest (section 4 and appendices B and C), in section 3 (and
appendix A) we will derive an LSZ reduction formula that connects the
$S$-matrix to correlation functions. Here we encounter a striking difference
compared to the Lorentz invariant case: the relation between the $S$-matrix
and correlation functions involves operators that are non-local in space. This
spatial non-locality arises from the mixing of the two modes, which depends on
the momentum. The presence of this non-locality spoils the analyticity
properties of the $S$-matrix compared to the Lorentz invariant case. We
conclude that even in cases where the $S$-matrix exists at all energies one
cannot derive positivity bounds using the conventional dispersive arguments.

In section 5 and appendix D we study the decay rate of Goldstones, while
conclusions and future directions are discussed in section 6.

Note added: We understand that Hui, Kourkoulou, Nicolis, Podo and Zhou have
been working on essentially the same questions and model. We did not exchange
results, but we did coordinate the arXiv submission.

##  2 The $U(1)$ model

We start with the renormalizable Lagrangian (we use the mostly-minus
signature)

$\mathcal{L}=\partial\Phi^{\dagger}\cdot\partial\Phi+m^{2}\,\Phi^{\dagger}\Phi-\lambda(\Phi^{\dagger}\Phi)^{2}\,.$
(1)

We set

$\Phi=\frac{\rho}{\sqrt{2}}e^{i\theta/v}\,,$ (2)

where $v$ is a constant with dimensions of energy that will be defined in (6).
This way we obtain

$\mathcal{L}=\frac{1}{2}(\partial\rho)^{2}+\frac{1}{2v^{2}}(\partial\theta)^{2}\rho^{2}+\frac{m^{2}}{2}\rho^{2}-\frac{\lambda}{4}\rho^{4}\,.$
(3)

We are going to study this model around a time-dependent solution
$\theta=\mu^{2}t/2$, i.e. with spontaneous breaking of Lorentz invariance.
Expanding as $\theta=\mu^{2}t/2+\pi$, we get

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}(\partial\rho)^{2}+\frac{1}{2v^{2}}\left(\mu^{2}\dot{\pi}+(\partial\pi)^{2}\right)\rho^{2}+\frac{1}{2}\left(m^{2}+\frac{\mu^{4}}{4v^{2}}\right)\rho^{2}-\frac{\lambda}{4}\rho^{4}\,,$
(4)

which can be rewritten as

$\mathcal{L}=\frac{1}{2}(\partial\rho)^{2}+\frac{1}{2v^{2}}\left(\mu^{2}\dot{\pi}+(\partial\pi)^{2}\right)\rho^{2}-\frac{\lambda}{4}(\rho^{2}-v^{2})^{2}+\text{constant}\,,$
(5)

where $v$ is implicitly defined through

$v^{2}=\frac{1}{\lambda}\left(m^{2}+\frac{\mu^{4}}{4v^{2}}\right)\,.$ (6)

It is useful to notice that for $\mu\ll m/\lambda^{1/4}$, $v\approx
m/\lambda^{1/2}$, i.e. the Lorentz invariant result, while for $\mu\gg
m/\lambda^{1/4}$, $v\approx\mu/(4\lambda)^{1/4}$. Expanding around the minimum
by setting $\rho=v+h$, up to the constant and a total derivative we find

$\mathcal{L}=\frac{1}{2}(\partial
h)^{2}+\frac{1}{2}(\partial\pi)^{2}+\frac{1}{2v^{2}}\left(\mu^{2}\dot{\pi}+(\partial\pi)^{2}\right)(h^{2}+2vh)-\frac{\lambda}{4}(h^{2}+2vh)^{2}\,.$
(7)

Let us focus on the quadratic part of the Lagrangian,

$\mathcal{L}_{(2)}=\frac{1}{2}(\partial
h)^{2}+\frac{1}{2}(\partial\pi)^{2}+\frac{\mu^{2}}{v}\dot{\pi}h-\lambda
v^{2}h^{2}\,,$ (8)

which in Fourier space reads

$\displaystyle\mathcal{L}_{(2)}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{c
c}\tilde{\pi}_{-k}&\tilde{h}_{-k}\end{array}\right)\left(\begin{array}[]{cc}k^{2}&i\mu^{2}\omega/v\\\
-i\mu^{2}\omega/v&k^{2}-M^{2}\end{array}\right)\left(\begin{array}[]{c}\tilde{\pi}_{k}\\\
\tilde{h}_{k}\end{array}\right)\,,$ (9)

where

$M^{2}\equiv 2\lambda v^{2}\,.$ (10)

By looking at the zeros of the determinant one can find the spectrum of the
theory, namely one can find the dispersion relations for the various
particles:

$(\omega^{2}-\boldsymbol{k}^{2})(\omega^{2}-\boldsymbol{k}^{2}-M^{2})-\frac{\mu^{4}\omega^{2}}{v^{2}}=0\,.$
(11)

There are two branches of solutions,

$\omega^{2}=E_{\pm}(\boldsymbol{k})^{2}\equiv\boldsymbol{k}^{2}+\frac{1}{2}\left(M^{2}+\frac{\mu^{4}}{v^{2}}\right)\pm\sqrt{\frac{\mu^{4}}{v^{2}}\boldsymbol{k}^{2}+\frac{1}{4}\left(M^{2}+\frac{\mu^{4}}{v^{2}}\right)^{2}}\,.$
(12)

Thus the Fock space contains two types of particles which are labelled by
their three-momentum $\boldsymbol{k}$ and by an extra label $\pm$ indicating
which dispersion relates their energy and momentum, see figure 1. The $-$
state is the Goldstone: for small enough $\boldsymbol{k}$ its dispersion
relation is linear, with a speed of propagation smaller than unity. The $+$
state is gapped and, for energies much smaller than this gap, one integrates
out this state and writes an EFT for the Goldstone only: we will come back to
this in section 4.1.3.

Now that we have established what the asymptotic states are we must choose
which fields to use to interpolate such states. One possibility is to use $h$
and $\pi$. They clearly can interpolate both $\pm$ states. The purpose of the
next section is to quantify how much they actually do.

Figure 1: Plot of the dispersion relations of the $+$ (black) and $-$ (orange)
states in units of $v$ for $\lambda=10^{-2}$ and $\mu/v=3/10$. The blue dotted
line is the null curve $\omega(\boldsymbol{k})=|\boldsymbol{k}|$, and we have
$E_{\pm}(\boldsymbol{k})=|\boldsymbol{k}|\pm\mu^{2}/2v+\mathcal{O}\left(|\boldsymbol{k}|^{-1}\right)$
as $|\boldsymbol{k}|\rightarrow\infty$. Thus the $+$ branch is always timelike
while the $-$ branch is spacelike.

###  2.1 Canonical quantization

We begin by introducing the ladder operators of the asymptotic states with
commutation relations

$[a_{l}(\boldsymbol{k}),a^{\dagger}_{m}(\boldsymbol{q})]=(2\pi)^{3}2E_{l}(\boldsymbol{k})\delta_{lm}\delta^{(3)}(\boldsymbol{k}-\boldsymbol{q})$
(13)

so that a single particle state of type $l$ with three-momentum
$\boldsymbol{k}$ is obtained by

$\ket{l,\boldsymbol{k}}=a^{\dagger}_{l}(\boldsymbol{k})\ket{0}\,,~{}~{}l\in\\{-,+\\}\,.$
(14)

The fields can be written as222$\phi^{\pi}\equiv\pi,\phi^{h}\equiv h$.

$\phi^{a}(t,\boldsymbol{x})\equiv\sum_{l=\pm}\int\frac{\mathrm{d}^{3}\boldsymbol{k}}{(2\pi)^{3}2E_{l}(\boldsymbol{k})}\left(Z^{a}_{l}(\boldsymbol{k})a_{l}(\boldsymbol{k})e^{-i(E_{l}(\boldsymbol{k})t-\boldsymbol{k}\cdot\boldsymbol{x})}+\text{h.c.}\right)\,,~{}~{}a\in\\{\pi,h\\}\,.$
(15)

We will now obtain an expression for the $Z$-factors using the canonical
commutation relations and the free equations of motion. The momenta conjugate
to $\pi$ and $h$ are

$\Pi^{\pi}=\dot{\pi}+\frac{\mu^{2}}{2v}h\,,~{}~{}\Pi^{h}=\dot{h}-\frac{\mu^{2}}{2v}\pi\,,$
(16)

so that333$\epsilon^{\pi h}\equiv 1\equiv-\epsilon^{h\pi}$ and zero otherwise.

$\Pi^{b}(t,\boldsymbol{x})=\sum_{l=\pm}\int\frac{\mathrm{d}^{3}\boldsymbol{k}}{(2\pi)^{3}2E_{l}(\boldsymbol{k})}\left[\left(-iE_{l}(\boldsymbol{k})Z^{b}_{l}(\boldsymbol{k})+\epsilon^{bc}\frac{\mu^{2}}{2v}Z^{c}_{l}(\boldsymbol{k})\right)a_{l}(\boldsymbol{k})e^{-i(E_{l}(\boldsymbol{k})t-\boldsymbol{k}\cdot\boldsymbol{x})}+\text{h.c.}\right]\,.$
(17)

The equal-time commutation relations are

$\displaystyle[\phi^{a}(t,\boldsymbol{x}),\phi^{b}(t,\boldsymbol{y})]=0\,,$
(18)
$\displaystyle[\phi^{a}(t,\boldsymbol{x}),\Pi^{b}(t,\boldsymbol{y})]=i\delta^{ab}\delta^{(3)}(\boldsymbol{x}-\boldsymbol{y})\,,$
$\displaystyle[\Pi^{a}(t,\boldsymbol{x}),\Pi^{b}(t,\boldsymbol{y})]=0\,.$

The first and second equations read444Here we use invariance under spatial
rotations – that $Z^{a}_{l}(\boldsymbol{k})$ is only a function of
$|\boldsymbol{k}|$.

$\displaystyle\sum_{l=\pm}\frac{Z^{a}_{l}\bar{Z}^{b}_{l}-\bar{Z}^{a}_{l}Z^{b}_{l}}{E_{l}}=0\,,$
(19)
$\displaystyle\sum_{l=\pm}Z^{a}_{l}\bar{Z}^{b}_{l}+\bar{Z}^{a}_{l}Z^{b}_{l}=2\delta^{ab}\,,$

where we have suppressed the $\boldsymbol{k}$-dependence and indicated with
$\bar{Z}$ the complex conjugate of $Z$. The third equation boils down to

$\sum_{l=\pm}\frac{E_{l}}{2}\left(Z^{a}_{l}\bar{Z}^{b}_{l}-\bar{Z}^{a}_{l}Z^{b}_{l}\right)=-i\frac{\mu^{2}}{v}\epsilon^{ab}\,.$
(20)

The free equations of motion imply that for fixed $l$ the vector $Z^{a}_{l}$
is an eigenvector of the matrix in Eq. (9). This gives

$Z^{h}_{l}=\frac{iE_{l}\mu^{2}}{vk_{l}^{2}}Z^{\pi}_{l}\,,$ (21)

where $k^{2}_{l}=E_{l}(\boldsymbol{k})^{2}-\boldsymbol{k}^{2}$ and this
equation holds for $l\in\\{-,+\\}$ separately. Using Eq. (21) in (19) gives

$\sum_{l=\pm}\frac{|Z^{\pi}_{l}|^{2}}{k_{l}^{2}-M^{2}}=0\,,~{}~{}\sum_{l=\pm}|Z^{\pi}_{l}|^{2}=1\,,$
(22)

whose solutions are

$|Z^{\pi}_{l}|^{2}=l\frac{k_{l}^{2}-M^{2}}{k_{+}^{2}-k_{-}^{2}}\,,~{}~{}l\in\\{-,+\\}\,.$
(23)

Explicitly in terms of the spatial momentum,

$\displaystyle|Z^{\pi}_{\pm}(\boldsymbol{k})|^{2}$
$\displaystyle=\frac{1}{2}\pm\frac{\mu^{4}/v^{2}-M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}\,,$
(24) $\displaystyle|Z^{h}_{\pm}(\boldsymbol{k})|^{2}$
$\displaystyle=\frac{1}{2}\pm\frac{\mu^{4}/v^{2}+M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}\,.$
(25)

With the relations (21), there are two remaining unfixed phases in the
solution. These correspond to our freedom in (15) to rotate
$a_{\pm}(\boldsymbol{k})$. Choosing $Z^{\pi}_{-}$ and $Z^{h}_{+}$ to be
positive real, the explicit expressions for the interpolating factors are

$\displaystyle Z^{\pi}_{-}(\boldsymbol{k})$
$\displaystyle=\sqrt{\frac{1}{2}-\frac{\mu^{4}/v^{2}-M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}}\,,$
(26) $\displaystyle Z^{h}_{-}(\boldsymbol{k})$
$\displaystyle=-i\sqrt{\frac{1}{2}-\frac{\mu^{4}/v^{2}+M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}}\,,$
(27) $\displaystyle Z^{\pi}_{+}(\boldsymbol{k})$
$\displaystyle=-i\sqrt{\frac{1}{2}+\frac{\mu^{4}/v^{2}-M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}}\,,$
(28) $\displaystyle Z^{h}_{+}(\boldsymbol{k})$
$\displaystyle=\sqrt{\frac{1}{2}+\frac{\mu^{4}/v^{2}+M^{2}}{2\sqrt{4\mu^{4}\boldsymbol{k}^{2}/v^{2}+\left(\mu^{4}/v^{2}+M^{2}\right)^{2}}}}\,.$
(29)

From these expressions we see

$Z^{\pi}_{-}(\boldsymbol{k})\sim
Z^{h}_{+}(\boldsymbol{k})\sim\frac{1}{\sqrt{2}}\,,~{}~{}Z^{\pi}_{+}(\boldsymbol{k})\sim
Z^{h}_{-}(\boldsymbol{k})\sim-\frac{i}{\sqrt{2}}~{}~{}\text{as
}|\boldsymbol{k}|\rightarrow\infty\,.$ (30)

This means that at high energies the two fields $\pi,h$ interpolate the two
asymptotic states $-,+$ equally well. This can also be understood by
considering the quadratic Lagrangian (8) and noticing that, in the high energy
limit, the splitting is dominated by the one-derivative operator
$(\mu^{2}/v)h\dot{\pi}$ while the mass can be neglected. So at high energies
the quadratic Lagrangian reads

$\mathcal{L}_{(2)}\sim\frac{1}{2}\left(\begin{array}[]{c
c}\pi&h\end{array}\right)\left(\begin{array}[]{cc}-\partial^{2}&-(\mu^{2}/v)\partial_{t}\\\
(\mu^{2}/v)\partial_{t}&-\partial^{2}\end{array}\right)\left(\begin{array}[]{c}\pi\\\
h\end{array}\right)$ (31)

which can be diagonalized yielding

$\mathcal{L}_{(2)}\sim\frac{1}{2}\times\frac{1}{\sqrt{2}}\left(\begin{array}[]{c
c}\pi-
ih&h-i\pi\end{array}\right)\left(\begin{array}[]{cc}-\partial^{2}+i(\mu^{2}/v)\partial_{t}&0\\\
0&-\partial^{2}-i(\mu^{2}/v)\partial_{t}\end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\pi+ih\\\
h+i\pi\end{array}\right).$ (32)

In the two eigenvalues we recognize the high energy limits of the dispersion
relations (12) and in the eigenvectors we recognize the high energy limits of
the $Z$-factors (30). Since the leading splitting operator
$(\mu^{2}/v)h\dot{\pi}$ is Lorentz-breaking, going to very high energies one
does not recover the same eigenvectors as in the Lorentz invariant limit
$\mu\rightarrow 0$. At low energies, conversely, the splitting is dominated by
the mass. The crossover between the two regimes happens at

$\frac{\mu^{2}\omega}{v}\sim
M^{2}\quad\Rightarrow\quad\omega_{c}\sim\frac{M^{2}v}{\mu^{2}}\sim\lambda\frac{v^{3}}{\mu^{2}}\,.$
(33)

For $\mu\lesssim m/\lambda^{1/4}$, this crossover scale is $\omega_{c}\sim
m^{3}/(\mu^{2}\lambda^{1/2})$, otherwise, for $\mu\gtrsim m/\lambda^{1/4}$,
$\omega_{c}\sim\lambda^{1/4}\mu$.

##  3 LSZ reduction formula and lack of analyticity

In this section we will derive the LSZ reduction formula [14], following
Weinberg’s “polology” arguments [15]. This is well-known material in the
Lorentz invariant case, and we will highlight the differences in the absence
of Lorentz invariance.

We want to prove the following relation between time-ordered Green’s functions
and $S$-matrix elements:

$\begin{split}\prod_{i=1}^{n}\int\mathrm{d}^{4}y_{i}\,e^{ip_{i}\cdot
y_{i}}\prod_{j=1}^{m}\int\mathrm{d}^{4}x_{j}\,e^{-ik_{j}\cdot x_{j}}\langle
0|T(\pi(y_{1})\ldots\pi(y_{n})\pi(x_{1})\ldots\pi(x_{m}))|0\rangle\cong\\\
\prod_{i=1}^{n}\frac{iZ_{-}^{\pi}(\boldsymbol{p}_{i})}{(p_{i}^{0})^{2}-E_{-}(\boldsymbol{p}_{i})^{2}+i\varepsilon}\prod_{j=1}^{m}\frac{i\bar{Z}_{-}^{\pi}(\boldsymbol{k}_{j})}{(k_{j}^{0})^{2}-E_{-}(\boldsymbol{k}_{j})^{2}+i\varepsilon}\langle\boldsymbol{p}_{1}\ldots\boldsymbol{p}_{n}|S|\boldsymbol{k}_{1}\ldots\boldsymbol{k}_{m}\rangle\,.\end{split}$
(34)

The $\cong$ symbol indicates the equality holds in the limit all particles go
on-shell: $p_{i}^{0}\to E_{-}(\boldsymbol{p}_{i})$ and $k_{j}^{0}\to
E_{-}(\boldsymbol{k}_{j})$. Notice we are taking the convention that
$p_{i}^{0},k_{j}^{0}>0$. The matrix elements were defined in (15) as

$Z_{\pm}^{a}(\boldsymbol{k})\equiv\braket{0}{\phi^{a}(0)}{\pm,k}\,,~{}~{}a\in\\{h,\pi\\}\,.$
(35)

(In the standard Lorentz invariant treatment, these would reduce to a factor
$\sqrt{Z}$, which describes the wavefunction normalization.) Here we are
considering the scattering of $-$ particles; if one were interested in $+$
particles, one should use $Z_{+}$’s and also use the dispersion relations
$E_{+}(\,\cdot\,)^{2}$. In this way of deriving the LSZ reduction formula, one
is free to use any field to interpolate the particles of interest provided
there is a nonzero overlap. For instance, one could write a similar formula by
replacing some (or all) the fields $\pi$ with $h$. Since the overlap between a
$-$ particle and the $h$-field goes to zero at low energies (see (29)), it is
more natural to use $\pi$.

To prove (34) one focuses on the regions of integration $y_{i}^{0}\to+\infty$
and $x_{i}^{0}\to-\infty$ since this is the region that gives rise to the
poles [15]. In this regime all the $y$’s are after all the $x$’s and the time-
ordered product factorizes as

$\langle
0|T(\pi(y_{1})\ldots\pi(y_{n}))T(\pi(x_{1})\ldots\pi(x_{m}))|0\rangle\,.$ (36)

We can thus insert a complete set of states (twice) in the middle,

$\sum_{\alpha,\beta}\langle 0|T(\pi(y_{1})\ldots\pi(y_{n}))|\alpha_{\rm
out}\rangle\langle\alpha_{\rm out}|\beta_{\rm in}\rangle\langle\beta_{\rm
in}|T(\pi(x_{1})\ldots\pi(x_{m}))|0\rangle\,.$ (37)

The object $\langle\alpha_{\rm out}|\beta_{\rm in}\rangle$ is the $S$-matrix.
In the late-time limit, after the wave packets of the various particles are
separated, the matrix element on the left is dominated by a state with $n$
particles:

$\begin{split}\lim_{y_{i}^{0}\to+\infty}\langle
0|T(\pi(y_{1})\ldots\pi(y_{n}))|\alpha_{\rm out}\rangle&\cong\langle
0|T(\pi(y_{1})\ldots\pi(y_{n}))|\boldsymbol{q}_{1}\ldots\boldsymbol{q}_{n}\rangle_{\rm
out}\cong\\\ \cong\prod_{i=1}^{n}\langle
0|\pi(y_{i})|\boldsymbol{q}_{i}\rangle_{\rm
out}\cong&\prod_{i=1}^{n}Z_{-}^{\pi}(\boldsymbol{q}_{i})e^{-iq_{i}\cdot
y_{i}}\,.\end{split}$ (38)

(The resolution of identity contains also $+$ particles, but those would not
contribute to the pole when one goes on-shell for $-$ states.) Notice that if
the theory were invariant under boosts, $Z$ would not depend on
$\boldsymbol{q}$: as we will see, this is a crucial difference. (We have been
sloppy here with permutations, which will eventually cancel with the
normalization of states with identical particles.) Each term in the product
gives

$\int\mathrm{d}^{4}y_{i}\;e^{ip_{i}\cdot
y_{i}}Z_{-}^{\pi}(\boldsymbol{q}_{i})e^{-iq_{i}\cdot
y_{i}}=\int\mathrm{d}y_{i}^{0}(2\pi)^{3}\delta^{(3)}(\boldsymbol{p}_{i}-\boldsymbol{q}_{i})Z_{-}^{\pi}(\boldsymbol{q}_{i})e^{i(p_{i}^{0}-q_{i}^{0})y_{i}^{0}}\,,$
(39)

where we did the integral over the spatial variables. Then doing the integral
over the phase space, $\int{\rm
d}^{3}\boldsymbol{q}_{i}/((2\pi)^{3}2E_{-}(\boldsymbol{q}_{i}))$, one gets

$\int\mathrm{d}y_{i}^{0}\frac{1}{2E_{-}(\boldsymbol{p}_{i})}Z_{-}^{\pi}(\boldsymbol{p}_{i})e^{i(p_{i}^{0}-E_{-}(\boldsymbol{p}_{i}))y_{i}^{0}}\cong\frac{iZ_{-}^{\pi}(\boldsymbol{p}_{i})}{(p_{i}^{0})^{2}-E_{-}(\boldsymbol{p}_{i})^{2}+i\varepsilon}\,,$
(40)

where the last equality is valid up to terms that vanish as one goes close to
the pole. To make the time integral well-defined one has to slightly deform
the contour of integration, hence the $i\varepsilon$. The same arguments work
for the incoming particles, proving the LSZ formula (34).

We can now proceed to study the analyticity of the $2\to 2$ $S$-matrix. In
this case the LSZ formula, in which only one in-going and one out-going leg is
reduced, gives

$S=-\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,e^{i(q_{2}\cdot y-q_{1}\cdot
x)}\frac{-\partial_{y^{0}}^{2}-E_{-}(-i\partial_{y_{i}})^{2}}{Z_{-}^{\pi}(-i\partial_{y_{i}})}\frac{-\partial_{x^{0}}^{2}-E_{-}(-i\partial_{x_{i}})^{2}}{\bar{Z}_{-}^{\pi}(-i\partial_{x_{i}})}\langle\boldsymbol{p}_{2}|T(\pi(y)\pi(x))|\boldsymbol{p}_{1}\rangle\,.$
(41)

We slightly changed the notation calling $\boldsymbol{q}_{1}$ and
$\boldsymbol{p}_{1}$ the incoming momenta ($\boldsymbol{q}_{2}$ and
$\boldsymbol{p}_{2}$ are outgoing). This simplifies the comparison with
Itzykson & Zuber’s book [16], whose logic we will follow. As in the Lorentz
invariant case one can replace the time-ordered correlation function with the
retarded two-point function: the difference only contributes to a disconnected
piece of the $S$-matrix. Therefore one has

$S=-\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,e^{i(q_{2}\cdot y-q_{1}\cdot
x)}\frac{-\partial_{y^{0}}^{2}-E_{-}(-i\partial_{y_{i}})^{2}}{Z_{-}^{\pi}(-i\partial_{y_{i}})}\frac{-\partial_{x^{0}}^{2}-E_{-}(-i\partial_{x_{i}})^{2}}{\bar{Z}_{-}^{\pi}(-i\partial_{x_{i}})}\langle\boldsymbol{p}_{2}|\theta(y^{0}-x^{0})[\pi(y),\pi(x)]|\boldsymbol{p}_{1}\rangle\,.$
(42)

Using translation invariance one can then factor out the delta function of
energy and momentum conservation,

$S=(2\pi)^{4}\delta^{(4)}(p_{2}+q_{2}-p_{1}-q_{1})i\mathcal{T}\,,$ (43)

so that, in terms of $q\equiv\frac{1}{2}(q_{1}+q_{2})$, one has

$\mathcal{T}=i\int\mathrm{d}^{4}z\,e^{iq\cdot
z}\langle\boldsymbol{p}_{2}|\theta(z^{0})\left[\frac{\partial_{z^{0}/2}^{2}+E_{-}(-i\partial_{z_{i}/2})^{2}}{Z_{-}^{\pi}(-i\partial_{z_{i}/2})}\pi\left(\frac{z}{2}\right),\frac{\partial_{z^{0}/2}^{2}+E_{-}(-i\partial_{z_{i}/2})^{2}}{\bar{Z}_{-}^{\pi}(-i\partial_{z_{i}/2})}\pi\left(-\frac{z}{2}\right)\right]|\boldsymbol{p}_{1}\rangle\,.$
(44)

In the Lorentz invariant case the $Z$’s are just constants and the
differential operators are the Klein-Gordon ones, $\Box+m^{2}$. Since these
operators are local, the integrand is nonzero only for $z^{\mu}$ in the
forward lightcone. Therefore $\mathcal{T}(q^{\mu})$ is analytic if
$\mathrm{Im}\;q^{\mu}$ lies in the forward lightcone. (In this region all the
derivatives of $\mathcal{T}$ are convergent integrals, assuming polynomial
boundedness of the matrix elements, and it thus defines an analytic function.)
This primitive domain of analyticity is the stepping stone for the use of
dispersive arguments in the $S$-matrix [17, 16]. Here, however, the
differential operators acting on the fields are non-local. In particular, from
the explicit expressions of $Z_{-}^{\pi}$ and $E_{-}^{2}$ one sees the
presence of branch points for complex spatial momenta. These singularities are
present for any value of the time components of the momenta. This means there
is no cone of analyticity for $\mathrm{Im}\;q^{\mu}$, contrary to the Lorentz
invariant case. The only guaranteed region of analyticity, which is universal
and independent of the model at hand, is for $\mathrm{Im}\;q^{0}>0$: this is
just a consequence of the fact the integrand above is retarded, i.e. different
from zero only for $z^{0}\geq 0$.

Another way to see the absence of analyticity is to consider fields built only
out of creation/annihilation operators of $-$ (or respectively $+$) states:

$\varphi_{l}(t,\boldsymbol{x})\equiv\int\frac{\mathrm{d}^{3}\boldsymbol{k}}{(2\pi)^{3}2E_{l}(\boldsymbol{k})}\left(a_{l}(\boldsymbol{k})e^{-i(E_{l}(\boldsymbol{k})t-\boldsymbol{k}\cdot\boldsymbol{x})}+\text{h.c.}\right)\,,~{}l\in\\{-,+\\}\,.$
(45)

These fields interpolate only one of the two eigenstates, so that $\langle
0|\varphi_{l}(0)|l,q\rangle_{\rm out}=1$. The $\varphi$’s however are not
microcausal, i.e. they do not commute outside the lightcone. The simplest way
to see this is to look at the Fourier representation of the retarded Green’s
function,

$G_{R,l}(\omega,\boldsymbol{k})=\frac{i}{(\omega+i\varepsilon)^{2}-E_{l}(\boldsymbol{k})^{2}}\,.$
(46)

A necessary and sufficient condition for the vanishing of the retarded Green’s
function outside the forward lightcone is that its Fourier representation is
analytic when the imaginary part of the 4-vector $(\omega,\boldsymbol{k})$ is
in the forward lightcone. This property does not hold here since
$E_{l}(\boldsymbol{k})$ has branch points, as we saw above. On the other hand,
the fields $\pi(x)$ and $h(x)$ commute outside the lightcone.555Notice that
$\pi(x)$ is not a Lorentz scalar, since Lorentz invariance is spontaneously
broken. However the relation between the original Lorentz invariant field
$\Phi(x)$ and $\pi(x),h(x)$ is local, so that microcausality is preserved. In
order to create a single energy eigenstate, one has to act non-locally in the
fields and this is the ultimate reason for the absence of analyticity.

The relation between the $S$-matrix and correlation functions is uniquely
defined only in the limit when the external momenta are real and on-shell.
Therefore the definition of Eq. (44) is ambiguous for generic complex
$q^{\mu}$.666For fixed $\boldsymbol{p}_{1}$ and $\boldsymbol{p}_{2}$ the
$S$-matrix depends on 3 real variables, for instance the vector
$\boldsymbol{q}_{1}$ (in the presence of Lorentz invariance it would depend on
a single real variable). Therefore there is no unique analytic extension to a
function of $q^{\mu}\in\mathbb{C}^{4}$. In appendix A we derive an LSZ
reduction formula starting from the explicit expression of the creation and
annihilation operators of the $+$ and $-$ states. The final reduction formula
gives the same $S$-matrix, but it is different from the one obtained above.
The conclusion about the lack of analyticity remains the same, however. Given
this ambiguity, one may wonder whether a “clever” LSZ procedure exists that
gives an analytic off-shell $S$-matrix. In particular analyticity is preserved
if one acts on each external leg with a local differential operator, i.e. a
polynomial in Fourier space. This is indeed what happens in the Lorentz
invariant case. We will now prove that this does not work: there is no way to
extract the correct $S$-matrix by acting with local differential operators.

Let us assume that such an operator exists. This corresponds to a polynomial
$P(\omega,|\boldsymbol{k}|^{2})$ that multiplies each external leg. (For
rotational invariance $P$ must depend only on $|\boldsymbol{k}|^{2}$.) In
order to give the correct $S$-matrix this polynomial must satisfy

$P(\omega,|\boldsymbol{k}|^{2})=\frac{2E_{-}(|\boldsymbol{k}|)}{Z_{-}^{\pi}(|\boldsymbol{k}|)}\left(\omega-
E_{-}(|\boldsymbol{k}|)\right)+\mathcal{O}\left(\left(\omega-
E_{-}(|\boldsymbol{k}|)\right)^{2}\right)~{}~{}\text{as }\omega\rightarrow
E_{-}(|\boldsymbol{k}|)\,.$ (47)

At fixed $|\boldsymbol{k}|$, the polynomial in $\omega$ must have
$E_{-}(|\boldsymbol{k}|)$ as a simple root. This is surely possible: indeed
the dispersion relations in the model at hand are solutions of the polynomial
equation (11). To get also the correct normalization, $P$ must satisfy

$\left.\frac{\partial}{\partial\omega}P(\omega,|\boldsymbol{k}|^{2})\right|_{\omega=E_{-}(|\boldsymbol{k}|)}=\frac{2E_{-}(|\boldsymbol{k}|)}{Z_{-}^{\pi}(|\boldsymbol{k}|)}\,.$
(48)

Taking the partial derivative with respect to $\omega$ of the polynomial $P$
gives another polynomial $Q(\omega,|\boldsymbol{k}|^{2})$, which should
satisfy

$Q(\omega=E_{-}(|\boldsymbol{k}|),|\boldsymbol{k}|^{2})=\frac{2E_{-}(|\boldsymbol{k}|)}{Z_{-}^{\pi}(|\boldsymbol{k}|)}\,.$
(49)

Let us look at the two sides of this equation as a function of the complex
variable $|\boldsymbol{k}|^{2}$. The RHS is analytic around the positive real
axis $|\boldsymbol{k}|^{2}>0$: if the two sides of the equation above coincide
for physical values $|\boldsymbol{k}|^{2}>0$, they must have the same analytic
extension in the complex $|\boldsymbol{k}|^{2}$ plane (properly extended to
take into account branch points). However it is impossible that the two
functions coincide in the complex plane. The RHS has poles at the zeros of
$Z_{-}^{\pi}(|\boldsymbol{k}|)$, while the LHS has no poles being a polynomial
in $|\boldsymbol{k}|^{2}$ and $E_{-}(|\boldsymbol{k}|)$ (see its explicit
expression Eq. (12)). Therefore $P$ cannot exist.

In conclusion, one cannot extract the $S$-matrix from correlation functions
acting with local differential operators, preserving in this way the
analyticity of the retarded Green’s function. One cannot exclude some more
exotic way of defining the $S$-matrix for complex momenta that preserves
analyticity.

##  4 The $S$-matrix

The object of interest in this work is the scattering amplitude for the
process $\ket{-,\boldsymbol{p}_{1}}\ket{-,\boldsymbol{q}_{1}}$ into
$\ket{-,\boldsymbol{p}_{2}}\ket{-,\boldsymbol{q}_{2}}$. In what follows we
will denote this scattering amplitude by $\mathcal{M}$ (details about the
calculation are deferred to appendix B.)

This scattering amplitude has four contributions, from contact interactions
and from exchange diagrams in the three channels:

$\mathcal{M}=\mathcal{M}_{c}+\mathcal{M}_{s}+\mathcal{M}_{t}+\mathcal{M}_{u}\,.$
(50)

These terms read

$\displaystyle\mathcal{M}_{c}$
$\displaystyle=-\frac{\mu^{4}}{v^{4}}\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\Big{[}6\lambda\mu^{4}p_{1}^{0}q_{1}^{0}p_{2}^{0}q_{2}^{0}+$
(51) $\displaystyle-2p_{1}\cdot
q_{1}(p_{1}^{2}-M^{2})(q_{1}^{2}-M^{2})p_{2}^{0}q_{2}^{0}-2p_{1}\cdot
q_{2}(p_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})q_{1}^{0}p_{2}^{0}+$
$\displaystyle-2p_{1}\cdot
p_{2}(p_{1}^{2}-M^{2})(p_{2}^{2}-M^{2})q_{1}^{0}q_{2}^{0}-2q_{1}\cdot
q_{2}(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})p_{1}^{0}p_{2}^{0}+$
$\displaystyle-2p_{2}\cdot
q_{1}(p_{2}^{2}-M^{2})(q_{1}^{2}-M^{2})p_{1}^{0}q_{2}^{0}-2q_{2}\cdot
p_{2}(p_{2}^{2}-M^{2})(q_{2}^{2}-M^{2})p_{1}^{0}q_{1}^{0}\Big{]}\,,$
$\displaystyle\mathcal{M}_{s}=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\frac{v^{-9}}{(p_{1}+q_{1})^{2}((p_{1}+q_{1})^{2}-M^{2})-\frac{\mu^{4}}{v^{2}}(p_{1}^{0}+q_{1}^{0})^{2}}$
(52) $\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(p_{1},q_{1})B(p_{2},q_{2})+[B(p_{1},q_{1})A(p_{2},q_{2})+B(p_{2},q_{2})A(p_{1},q_{1})]\mu^{4}(p_{1}^{0}+q_{1}^{0})-$
$\displaystyle
v[A(p_{1},q_{1})A(p_{2},q_{2})+\mu^{4}B(p_{1},q_{1})B(p_{2},q_{2})](p_{1}+q_{1})^{2}\Big{]}\,,$
$\displaystyle\mathcal{M}_{t}=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\frac{v^{-9}}{(q_{1}-q_{2})^{2}((q_{1}-q_{2})^{2}-M^{2})-\frac{\mu^{4}}{v^{2}}(q_{1}^{0}-q_{2}^{0})^{2}}$
(53) $\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(q_{1},-q_{2})B(-p_{1},p_{2})+[B(q_{1},-q_{2})A(-p_{1},p_{2})+B(-p_{1},p_{2})A(q_{1},-q_{2})]\mu^{4}(q_{1}^{0}-q_{2}^{0})-$
$\displaystyle
v[A(q_{1},-q_{2})A(-p_{1},p_{2})+\mu^{4}B(q_{1},-q_{2})B(-p_{1},p_{2})](q_{1}-q_{2})^{2}\Big{]}\,,$

and

$\displaystyle\mathcal{M}_{u}=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\frac{v^{-9}}{(q_{1}-p_{2})^{2}((q_{1}-p_{2})^{2}-M^{2})-\frac{\mu^{4}}{v^{2}}(q_{1}^{0}-p_{2}^{0})^{2}}$
(54) $\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(q_{1},-p_{2})B(q_{2},-p_{1})+[B(q_{1},-p_{2})A(q_{2},-p_{1})+B(q_{2},-p_{1})A(q_{1},-p_{2})]\mu^{4}(q_{1}^{0}-p_{2}^{0})-$
$\displaystyle
v[A(q_{1},-p_{2})A(q_{2},-p_{1})+\mu^{4}B(q_{1},-p_{2})B(q_{2},-p_{1})](q_{1}-p_{2})^{2}\Big{]}\,.$

Here, as specified in section 2.1, we have chosen $Z^{\pi}_{-}$ to be real, so
we do not make distinction between $Z^{\pi}_{-}$ and $\bar{Z}^{\pi}_{-}$. In
order to write the blocks in a slightly more compact form we have defined the
functions

$A(p_{1},q_{1})\equiv 2v^{3}(p_{1}^{2}-M^{2})(q_{1}^{2}-M^{2})p_{1}\cdot
q_{1}-v\mu^{4}p_{1}^{0}q_{1}^{0}(p_{1}^{2}+q_{1}^{2}+M^{2})\,,$ (55)

and

$B(p_{1},q_{1})\equiv\mu^{4}(p_{1}^{0})^{2}q_{1}^{0}-2v^{2}q_{1}^{0}(p_{1}^{2}-M^{2})(p_{1}^{2}+p_{1}\cdot
q_{1})+(p_{1}\leftrightarrow q_{1})\,.$ (56)

It is straightforward to check that the amplitude would be analytic in
$q\equiv\frac{1}{2}(q_{1}+q_{2})$ for $\mathrm{Im}\;q^{\mu}$ in the forward
lightcone, if the LSZ procedure did not introduce the non-analyticities
discussed the previous section. Notice that under the exchange
$p_{1}\leftrightarrow-q_{2}$ we have
$\mathcal{M}_{s}\leftrightarrow\mathcal{M}_{t}$ and under
$p_{1}\leftrightarrow-p_{2}$ we have
$\mathcal{M}_{s}\leftrightarrow\mathcal{M}_{t}$, whereas $\mathcal{M}_{c}$ is
always unchanged. This shows that $\mathcal{M}$ is invariant under the
exchange of the external momenta, i.e. it is crossing symmetric.

###  4.1 Interesting limits

Given the complexity of the scattering amplitude it is useful to study it in
some interesting limits.

####  4.1.1 $\mu=0$

As a sanity check we want to be sure to recover the usual Lorentz invariant
form of the $S$-matrix when we switch off the breaking $\mu$. It is
straightforward to see that for $\mu=0$,

$\displaystyle Z_{-}^{\pi}=1\,,\quad
E_{-}(\boldsymbol{k})^{2}=\boldsymbol{k}^{2}\,,\quad
E_{+}(\boldsymbol{k})^{2}=\boldsymbol{k}^{2}+M^{2}\,,$ (57) $\displaystyle
A(p_{1},q_{1})=2v^{3}M^{4}p_{1}\cdot q_{1}\,,\quad
B(p_{1},q_{1})=2v^{2}M^{2}(p_{1}^{0}+q_{1}^{0})p_{1}\cdot q_{1}\,.$

As a consequence, the various contributions to the tree-level amplitude reduce
to

$\mathcal{M}_{c}=0\,,~{}\mathcal{M}_{s}=-\frac{s^{2}}{v^{2}(s-M^{2})}\,,~{}\mathcal{M}_{t}=-\frac{t^{2}}{v^{2}(t-M^{2})}\,,~{}\mathcal{M}_{u}=-\frac{u^{2}}{v^{2}(u-M^{2})}\,,$
(58)

using the usual Maldelstam variables. The full amplitude is thus

$\mathcal{M}=-\frac{1}{v^{2}}\left[\frac{s^{2}}{s-M^{2}}+\frac{t^{2}}{t-M^{2}}+\frac{u^{2}}{u-M^{2}}\right].$
(59)

The amplitude reproduces its Lorentz invariant counterpart as soon as the
breaking is switched off.

####  4.1.2 High energies

Another interesting limit to explore is the “high-energy limit”, when the
energies of the scattered states are much larger than all the other scales of
the problem. For simplicity we specialize to a specific kinematical
configuration and set

$q_{1}^{0}=q_{2}^{0}=p_{1}^{0}=p_{2}^{0}\equiv\omega\,.$ (60)

The high-energy limit is therefore realized by

$\lim_{\omega/\Omega\to\infty}\mathcal{M}$ (61)

where

$\Omega\gg v,M,\mu$ (62)

is a reference scale whose value is much larger than all the other scales of
the problem. In this limit, the behavior of the different contributions to the
amplitude is

$\displaystyle\mathcal{M}_{c}=$
$\displaystyle-\frac{\omega^{2}(\hat{p_{1}}\cdot\hat{q_{1}}+\hat{p_{1}}\cdot\hat{q_{2}}+\hat{q_{1}}\cdot\hat{q_{2}}-3)}{v^{2}}-\frac{3\mu^{2}\omega}{v^{3}}+$
(63)
$\displaystyle+\frac{9\mu^{8}-3M^{4}v^{4}+(\hat{p_{1}}\cdot\hat{q_{1}}+\hat{p_{1}}\cdot\hat{q_{2}}+\hat{q_{1}}\cdot\hat{q_{2}})\left(\mu^{4}+M^{2}v^{2}\right)^{2}-6\lambda\mu^{4}v^{4}}{4\mu^{4}v^{4}}+\mathcal{O}\left(\frac{\Omega}{\omega}\right)\,,$
$\displaystyle\mathcal{M}_{s}=$
$\displaystyle\frac{5\omega^{2}(\hat{p_{1}}\cdot\hat{q_{1}}-1)}{2v^{2}}+\frac{\omega\left(6\mu^{4}+M^{2}v^{2}\hat{p_{1}}\cdot\hat{q_{1}}-M^{2}v^{2}\right)}{2\mu^{2}v^{3}}+$
(64)
$\displaystyle+\frac{1}{8}\left(16\lambda+\frac{3M^{4}}{\mu^{4}}+\frac{8M^{2}}{v^{2}}+\hat{p_{1}}\cdot\hat{q_{1}}\left(-\frac{3M^{4}}{\mu^{4}}-\frac{10M^{2}}{v^{2}}-\frac{5\mu^{4}}{v^{4}}\right)-\frac{19\mu^{4}}{v^{4}}\right)+\mathcal{O}\left(\frac{\Omega}{\omega}\right)\,,$
$\displaystyle\mathcal{M}_{t}=$
$\displaystyle\frac{\omega^{2}(1-\hat{q_{1}}\cdot\hat{q_{2}})}{2v^{2}}+\frac{M^{2}\omega(1-\hat{q_{1}}\cdot\hat{q_{2}})}{2\mu^{2}v}+$
(65)
$\displaystyle+\frac{-\mu^{8}+M^{4}v^{4}-16\mu^{4}M^{2}v^{2}+\hat{q_{1}}\cdot\hat{q_{2}}\left(\mu^{8}-M^{4}v^{4}+2\mu^{4}M^{2}v^{2}\right)}{8\mu^{4}v^{4}}+\mathcal{O}\left(\frac{\Omega}{\omega}\right)\,,$
$\displaystyle\mathcal{M}_{u}=$
$\displaystyle\frac{\omega^{2}(1-\hat{p_{1}}\cdot\hat{q_{2}})}{2v^{2}}+\frac{M^{2}\omega(1-\hat{p_{1}}\cdot\hat{q_{2}})}{2\mu^{2}v}+$
(66)
$\displaystyle+\frac{-\mu^{8}+M^{4}v^{4}-16\mu^{4}M^{2}v^{2}+\hat{p_{1}}\cdot\hat{q_{2}}\left(\mu^{8}-M^{4}v^{4}+2\mu^{4}M^{2}v^{2}\right)}{8\mu^{4}v^{4}}+\mathcal{O}\left(\frac{\Omega}{\omega}\right)\text{
as }\frac{\omega}{\Omega}\rightarrow\infty\,.$

where
$\hat{p_{i}}\cdot\hat{q_{j}}=\boldsymbol{p}_{i}\cdot\boldsymbol{q}_{j}/|\boldsymbol{p}_{i}||\boldsymbol{q}_{j}|$.
Momentum conservation enforces the condition

$1+\hat{p_{1}}\cdot\hat{q_{1}}-\hat{q_{2}}\cdot\hat{q_{1}}-\hat{p_{1}}\cdot\hat{q_{2}}=0\,,$

which ensures the cancellation of the divergent terms in the above blocks,
yielding a constant asymptotic value for the full amplitude:

$\mathcal{M}=-2\frac{M^{2}}{v^{2}}+\mathcal{O}\left(\frac{\Omega}{\omega}\right)=-4\lambda+\mathcal{O}\left(\frac{\Omega}{\omega}\right)~{}~{}\text{as
}\frac{\omega}{\Omega}\rightarrow\infty\,.$ (67)

This constant value is equal to the $s\to\infty$ limit of the Lorentz
invariant amplitude in Eq. (59): as expected, the high-energy limit of the
amplitude reproduces its Lorentz invariant counterpart.

####  4.1.3 Low energies

As a final check we explore the low-energy limit of the amplitude to show that
it agrees with the amplitude computed within the EFT of $\pi$. This EFT is
obtained by integrating out the radial mode $h$ at tree level and at leading
order in the derivative expansion. Starting from the equation of motion of
$h$,

$\Box
h-\frac{\mu^{2}\dot{\pi}+(\partial\pi)^{2}}{v^{2}}(h+v)+\lambda(h^{2}+2hv)(h+v)=0\,,$
(68)

at leading order in the derivative expansion we have

$h^{2}+2hv=\frac{\mu^{2}\dot{\pi}+(\partial\pi)^{2}}{\lambda v^{2}}\,.$ (69)

Plugging this expression into Eq. (7) yields the low-energy Lagrangian of
$\pi$ at leading order in the derivative expansion (neglecting a total
derivative):

$\mathcal{L}_{\text{EFT}}=\frac{(\partial\pi)^{2}}{2}+\frac{(\mu^{2}\dot{\pi}+(\partial\pi)^{2})^{2}}{4\lambda
v^{4}}\,.$ (70)

Canonically normalizing by setting $\pi=\pi_{c}c_{s}$ yields

$\mathcal{L}_{\text{EFT}}=\frac{1}{2}\left(\dot{\pi}_{c}^{2}-c_{s}^{2}(\boldsymbol{\nabla}\pi_{c})^{2}\right)+\frac{\mu^{2}c_{s}^{3}}{2\lambda
v^{4}}\dot{\pi}_{c}(\partial\pi_{c})^{2}+\frac{c_{s}^{4}}{4\lambda
v^{4}}(\partial\pi_{c})^{4}\,,$ (71)

where

$c_{s}^{2}=\frac{1}{1+\mu^{4}/2\lambda v^{4}}\equiv\frac{1}{1+\alpha}\,.$ (72)

The $S$-matrix for $\pi\pi\to\pi\pi$ can be computed easily and its explicit
expression is reported in appendix C. We will now check that the low-energy
limit of the full $S$-matrix (50) agrees with the expression obtained within
the EFT. As before we specialize to a kinematical configuration with all
energies equal. In this configuration, the low energy limit of the $S$-matrix
is realized by considering energies much smaller than the mass of the radial
mode. Keeping only the leading order terms in the derivative expansion and
making use of

$Z^{\pi}_{-}(0)=c_{s}\,,~{}E_{-}(0)^{2}-E_{+}(0)^{2}=-\frac{M^{2}}{c_{s}^{2}}\,,$
(73)

the amplitude contributions reduce to ($s\equiv(p_{1}+q_{1})^{2}$,
$t\equiv(q_{2}-q_{1})^{2}$ and $u\equiv(p_{2}-q_{1})^{2}$)

$\displaystyle\mathcal{M}_{c}\to-\alpha^{2}\frac{c_{s}^{4}}{\mu^{4}}\omega^{2}\left(7\alpha\omega^{2}-2s+2t+2u\right)\,,$
(74)
$\displaystyle\mathcal{M}_{s}\to-\frac{16c_{s}^{4}\alpha^{2}\omega^{2}}{\mu^{4}}\frac{(s+\alpha\omega^{2})^{2}}{s+4\alpha\omega^{2}}+c_{s}^{4}\frac{\alpha\left(\alpha\omega^{2}+s\right)^{2}}{\mu^{4}}\,,$
(75) $\displaystyle\mathcal{M}_{u}\to
c_{s}^{4}\frac{\alpha\left(3\alpha\omega^{2}+u\right)^{2}}{\mu^{4}}\,,$ (76)
$\displaystyle\mathcal{M}_{t}\to
c_{s}^{4}\frac{\alpha\left(3\alpha\omega^{2}+t\right)^{2}}{\mu^{4}}\,.$ (77)

As a consequence, they sum up to give

$\displaystyle\mathcal{M}_{\mathrm{low-
energy}}=\frac{c_{s}^{4}\alpha\left(s^{2}+t^{2}+u^{2}-4\alpha\omega^{2}\right)}{\mu^{4}}-\frac{16c_{s}^{4}\alpha^{2}\omega^{2}}{\mu^{4}}\frac{(s+\alpha\omega^{2})^{2}}{s+4\alpha\omega^{2}}\,,$
(78)

in agreement with the EFT computation (see Eq. (126) in appendix C).

##  5 Decay rate of the Goldstone

The states $-$ and $+$ are unstable, so they are not exact asymptotic states.
Therefore, the $S$-matrix we calculated makes sense only if the typical time
of interaction, which is of order $\omega^{-1}$, is much shorter than the
lifetime of the states. In terms of the decay rate one needs

$\frac{\Gamma(\omega)}{\omega}\ll 1\,.$ (79)

Before delving into the estimates of the decay rate, it is important to notice
that the model at hand is weakly coupled at all energies. As such, we expect
all interactions to be suppressed by powers of the coupling constant
$\lambda$: also $\Gamma$ is thus $\lambda$-suppressed. Therefore we expect
that, for $\lambda\ll 1$, the condition (79) is satisfied. We will now check
this expectation computing the decay width in the two regimes $\omega\ll M$
and $\omega\gg M$, where $M$ is the mass of the radial mode $h$.

###  5.1 $\omega\ll M$

The gap of the $+$ branch, $E_{+}(0)$, is always of the same order as $M$. For
$\omega\ll M$ the only allowed decay process with a 2-particle final state is
$-\to--$ and in this low-energy regime we can rely on the EFT Lagrangian (71):
the decay is induced by the operator

$\frac{\mu^{2}c_{s}^{3}}{2\lambda v^{4}}\dot{\pi}_{c}(\partial\pi_{c})^{2}\,.$
(80)

The ratio (79) can be estimated as

$\frac{\Gamma}{\omega}(\omega\ll
M)\sim(1-c_{s}^{-2})^{2}\left(\frac{\omega}{M}\right)^{4}\frac{\mu^{4}\lambda^{2}}{M^{4}}\sim\lambda\left(\frac{\omega}{M}\right)^{4}\left(\frac{\lambda\mu^{4}}{M^{4}}\right)^{3}\ll\lambda\,.$
(81)

We have used $1-c_{s}^{-2}\approx-\mu^{4}/\lambda v^{4}$,
$M^{2}\geq\sqrt{\lambda}\mu^{2}$ and the low-energy condition $\omega\ll M$.
The kinematical suppression $(1-c_{s}^{-2})^{2}$ requires some explanation.
For small frequency the dispersion relation is linear, so that the kinematics
of the decay is collinear. The scalar product $p_{1}^{\mu}p_{2\;\mu}$, which
arises from the operator (80), is thus proportional to $1-c_{s}^{-2}$.

###  5.2 $\omega\gg M$

To tackle the high-energy side of the problem we begin by performing a
convenient change of basis. In the UV it is useful to use a field basis where
renormalizability is manifest. In this basis we will not need to worry about
cancellations among various terms and it is thus simpler to make estimates.
Such a basis is easy to find: the Lagrangian (1) we started with is manifestly
renormalizable and we simply need to expand around a symmetry breaking
solution777Another natural choice of fields would have been
$\Phi\equiv\phi_{b}(t)+e^{i\mu^{2}t/2v}\frac{\sigma_{1}+i\sigma_{2}}{\sqrt{2}}\,.$
(82) The Lagrangian written in this basis reads
$\mathcal{L}=\frac{(\partial\sigma_{1})^{2}}{2}+\frac{(\partial\sigma_{2})^{2}}{2}-\frac{\mu^{2}}{2v}\big{(}\dot{\sigma}_{1}\sigma_{2}-\dot{\sigma}_{2}\sigma_{1}\big{)}-\frac{\lambda}{4}\big{(}2v\sigma_{1}+\sigma_{1}^{2}+\sigma_{2}^{2}\big{)}^{2}\,.$
(83)

$\Phi=\phi_{b}(t)+\frac{\phi_{1}+i\phi_{2}}{\sqrt{2}}\,,\qquad\phi_{b}(t)\equiv\frac{v}{\sqrt{2}}e^{i\mu^{2}t/2v}\;.$
(84)

Notice that one is allowed to treat $\phi_{b}$ as approximately constant in
the regime $\omega\gg M$: indeed the frequency of $\phi_{b}$ is of order
$\mu^{2}/v$, which is at most of order $M$. For our estimates we need to know
how much these new fields interpolate the two asymptotic states. As already
discussed at the end of section 2.1 it is easy to see that, above the
crossover scale $\omega_{c}$, both $\phi_{1}$ and $\phi_{2}$ interpolate to
order one for $+$ and $-$, i.e.

$\begin{gathered}|\braket{0}{\phi_{1,2}}{\pm}|=\mathcal{O}(1)\,.\end{gathered}$
(85)

(Using this in the regime $M\ll\omega\ll\omega_{c}$ will give an upper bound
for the decay rate, as we will discuss.) By plugging (84) into (1), setting
$t=0$ and focusing on the cubic part of the Lagrangian we obtain

$\mathcal{L}_{(3)}\sim-\lambda v\phi_{1}^{2}\phi_{2}-\lambda v\phi_{1}^{3}\,.$
(86)

As expected this operator is relevant and we do not need to worry about
cancellations. We can simply write

$\frac{\Gamma}{\omega}(\omega\gg
M)\sim\lambda^{2}\frac{v^{2}}{\omega^{2}}\sim\lambda\frac{M^{2}}{\omega^{2}}\,.$
(87)

As expected, the ratio between the interaction time and the decay lifetime of
$-$ is always suppressed by $\lambda$. Notice that the two estimates (81) and
(87) do not agree when evaluated at $\omega=M$. Indeed, approaching $\omega=M$
from the low and the high-energy regime gives respectively

$\frac{\Gamma}{\omega}(\omega\ll
M)\to\lambda\left(\frac{\lambda\mu^{4}}{M^{4}}\right)^{3}\;,\quad\frac{\Gamma}{\omega}(\omega\gg
M)\to\lambda\>.$ (88)

This difference is a consequence of the fact that (87) actually yields an
overestimate of the $\Gamma/\omega$ ratio within the window of energies
$M\ll\omega\ll\omega_{c}$. Even if these estimates are enough to conclude that
the $S$-matrix is well defined since the decay is always perturbative, i.e.
suppressed by $\lambda$, it is instructive to work out a finer estimate for
$\Gamma/\omega$ in the intermediate regime $M\ll\omega\ll\omega_{c}$. We will
do this in Appendix D.

The careful reader might wonder why we only paid attention to the $-\to--$
process among the possible decay channels. What about the decay into $-+$ and
$++$? The decay into $++$ is kinematically forbidden: by energy and momentum
conservation we should have

$p^{2}=(p_{1}+p_{2})^{2}\qquad\text{where }p^{2}<0\,,\quad p_{i}^{2}>0\,.$
(89)

Given that the sum of two timelike future-directed four-momenta, see eq. (12),
is always timelike future-directed, we see that this channel is kinematically
prohibited. We can try to repeat the same argument for $-$ decaying into $-$
and $+$. In this case we have

$(p-p_{1})^{2}=p_{2}^{2}\,,$ (90)

where we have taken the first outgoing particle to be gapless. Generically now
we cannot conclude anything because the sum of two spacelike vectors is not
necessarily spacelike. If we restrict our attention to
$\omega_{1},\omega_{2}<\omega<\omega_{c}$, we can approximate the dispersion
relation as linear and then

$\displaystyle(p-p_{1})^{2}$ $\displaystyle=p^{2}+p_{1}^{2}-2p_{1}\cdot
p_{2}\simeq(c_{s}^{2}-1)(\boldsymbol{p}^{2}+\boldsymbol{p}_{1}^{2})-2(c_{s}^{2}|\boldsymbol{p}_{1}||\boldsymbol{p}_{2}|-\boldsymbol{p}_{1}\cdot\boldsymbol{p}_{2})$
(91)
$\displaystyle<(c_{s}^{2}-1)(\boldsymbol{p}^{2}+\boldsymbol{p}_{1}^{2})-2(c_{s}^{2}|\boldsymbol{p}_{1}||\boldsymbol{p}_{2}|-|\boldsymbol{p}_{1}||\boldsymbol{p}_{2}|)=(c_{s}^{2}-1)(|\boldsymbol{p}_{1}|-|\boldsymbol{p}_{2}|)^{2}<0\,.$

Since $p_{2}^{2}>0$ we conclude that for $\omega<\omega_{c}$ this decay does
not happen and the only kinematically allowed process in this regime is
$-\rightarrow--$. The decay $-\to-+$ only happens for
$\omega\gtrsim\omega_{c}$. In this range of energies, the cubic part of the
Lagrangian is again given by (86), leading to the same estimate for the
$\Gamma/\omega$ ratio of the $-\to-+$ process:

$\frac{\Gamma}{\omega}(\omega\gtrsim\omega_{c})\sim\lambda\frac{M^{2}}{\omega^{2}}\ll\lambda\,.$
(92)

##  6 Conclusions

We calculated the $S$-matrix for Goldstone scattering in a complex
$\lambda|\Phi|^{4}$ model in a state at finite charge, i.e. with spontaneous
breaking of Lorentz invariance. This is arguably the simplest model that gives
a Lorentz-violating $S$-matrix that is well-defined at all energies, and it
may be a good testing ground to discuss general properties of the $S$-matrix
when Lorentz invariance is broken. A general lesson that emerged from this
calculation is that the $S$-matrix does not enjoy the same analyticity
properties as in the Lorentz invariant case: the relation between the
$S$-matrix and correlation functions introduces non-analyticities which are a
consequence of the breaking of boosts.

It seems thus quite difficult to obtain general positivity bounds using the
$S$-matrix, at least in the most naive way. It would be worth exploring
whether one gains something looking at the limit of small breaking $\mu\to 0$:
this would correspond to studying the $S$-matrix on a weak Lorentz-breaking
background, similarly to the analysis of the propagation of signals on a
background, pioneered in [1]. Alternatively, one could abandon the $S$-matrix
and use other objects which are well-defined both in the UV and in the IR: one
example are conserved currents as discussed in [12], where some bounds were
already derived.

Perhaps the more general lesson is that we have a very limited understanding
of the basic properties of QFT when Lorentz is spontaneously broken.

## Acknowledgements

It is a pleasure to thank B. Salehian for useful discussions.

##  Appendix A Second derivation of LSZ

In this appendix we derive the LSZ reduction formula starting from the
expression of creation/annihilation operators. The final formula will be
different from Eq. (34), before going on-shell, although it gives of course
the same $S$-matrix. The conclusion about the lack of analyticity will remain
the same.

We need to write the creation/annihilation operators in terms of the fields
and their time derivative. These are given at $t=0$ by

$\displaystyle\int\mathrm{d}^{3}\boldsymbol{x}\,e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}\phi^{a}(0,\boldsymbol{x})=\sum_{l=\pm}\frac{Z^{a}_{l}a_{l}+\bar{Z}^{a}_{l}a^{\dagger}_{l}}{2E_{l}}\,,$
(93)
$\displaystyle\int\mathrm{d}^{3}\boldsymbol{x}\,e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}\dot{\phi}^{a}(0,\boldsymbol{x})=\sum_{l=\pm}-i\frac{Z^{a}_{l}a_{l}-\bar{Z}^{a}_{l}a^{\dagger}_{l}}{2}\,,$

where $a\in\\{h,\pi\\}$. We define

$Y^{a}_{l}\equiv\frac{Z^{a}_{l}}{2E_{l}}\>,\quad
X^{a}_{l}\equiv-\frac{i}{2}Z^{a}_{l}\,.$ (94)

Multiplying the first and second expression of (93) respectively by
$\bar{Y}^{-1}$ and $\bar{X}^{-1}$ we get

$\displaystyle\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}(\bar{Y}^{-1})^{m}_{a}\phi^{a}(0,\boldsymbol{x})=a^{\dagger}_{m}+\sum_{l=\pm}(\bar{Y}^{-1})^{m}_{a}Y^{a}_{l}a_{l}\,,$
(95)
$\displaystyle\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}(\bar{X}^{-1})^{m}_{a}\dot{\phi}^{a}(0,\boldsymbol{x})=a^{\dagger}_{m}+\sum_{l=\pm}(\bar{X}^{-1})^{m}_{a}X^{a}_{l}a_{l}\,.$

Subtracting the two expressions we arrive at

$\displaystyle\sum_{l=\pm}D^{m}_{l}a_{l}(\boldsymbol{k})=\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}(\bar{Y}^{-1})^{m}_{a}\phi^{a}(0,\boldsymbol{x})-\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-i\boldsymbol{k}\cdot\boldsymbol{x}}(\bar{X}^{-1})^{m}_{a}\dot{\phi}^{a}(0,\boldsymbol{x})\,,$
(96)
$\displaystyle\sum_{l=\pm}\bar{D}^{m}_{l}a^{\dagger}_{l}(\boldsymbol{k})=\int\mathrm{d}^{3}\boldsymbol{x}\;e^{i\boldsymbol{k}\cdot\boldsymbol{x}}(Y^{-1})^{m}_{a}\phi^{a}(0,\boldsymbol{x})-\int\mathrm{d}^{3}\boldsymbol{x}\;e^{i\boldsymbol{k}\cdot\boldsymbol{x}}(X^{-1})^{m}_{a}\dot{\phi}^{a}(0,\boldsymbol{x})\,,$

where

$D^{m}_{l}=(\bar{Y}^{-1})^{m}_{a}Y^{a}_{l}-(\bar{X}^{-1})^{m}_{a}X^{a}_{l}\,.$
(97)

Finally we have the expression of creation/annihilation operators we were
after (reintroducing the time dependence):

$\displaystyle
a_{l}=F^{l}_{a}\int\mathrm{d}^{3}\boldsymbol{x}\;e^{iE_{l}(\boldsymbol{k})t-i\boldsymbol{k}\cdot\boldsymbol{x}}\phi^{a}-G^{l}_{a}\int\mathrm{d}^{3}\boldsymbol{x}\;e^{iE_{l}(\boldsymbol{k})t-i\boldsymbol{k}\cdot\boldsymbol{x}}\dot{\phi}^{a}\,,$
(98) $\displaystyle
a^{\dagger}_{l}=\bar{F}^{l}_{a}\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-iE_{l}(\boldsymbol{k})t+i\boldsymbol{k}\cdot\boldsymbol{x}}\phi^{a}-\bar{G}^{l}_{a}\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-iE_{l}(\boldsymbol{k})t+i\boldsymbol{k}\cdot\boldsymbol{x}}\dot{\phi}^{a}\,,$

with

$F^{l}_{a}=(D^{-1})^{l}_{m}(\bar{Y}^{-1})^{m}_{a}\>,\quad
G^{l}_{a}=(D^{-1})^{l}_{m}(\bar{X}^{-1})^{m}_{a}\,.$ (99)

Notice that creation/annihilation operators are written as a linear
combination of the two fields (and their derivative), while in the discussion
in the main text we were using a single field to interpolate the asymptotic
states.

We can now perform the LSZ reduction:

$\displaystyle\langle q_{2},p_{2}|p_{1},q_{1}\rangle=\langle
q_{2},p_{2}|a^{\dagger}_{-,\text{in}}(\boldsymbol{q}_{1})|p_{1}\rangle=\langle
q_{2},p_{2}|(a^{\dagger}_{-,\text{in}}(\boldsymbol{q}_{1})-a^{\dagger}_{-,\text{out}}(\boldsymbol{q}_{1}))|p_{1}\rangle$
(100)
$\displaystyle=-\left(\lim_{x^{0}\to+\infty}-\lim_{x^{0}\to-\infty}\right)\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-iE_{-}(\boldsymbol{q}_{1})t+i\boldsymbol{q}_{1}\cdot\boldsymbol{x}}\langle\bar{F}^{-}_{a}\phi^{a}-\bar{G}^{-}_{a}\dot{\phi}^{a}\rangle$
$\displaystyle=-\int\mathrm{d}x^{0}\;\partial_{0}\int\mathrm{d}^{3}\boldsymbol{x}\;e^{-iE_{-}(\boldsymbol{q}_{1})t+i\boldsymbol{q}_{1}\cdot\boldsymbol{x}}\langle\bar{F}^{-}_{a}\phi^{a}-\bar{G}^{-}_{a}\dot{\phi}^{a}\rangle$
$\displaystyle=-\int\mathrm{d}^{4}x\;e^{-iE_{-}(\boldsymbol{q}_{1})t+i\boldsymbol{q}_{1}\cdot\boldsymbol{x}}(-iE_{-}(\boldsymbol{q}_{1})\bar{F}^{-}_{a}+iE_{-}(\boldsymbol{q}_{1})\bar{G}^{-}_{a}\partial_{0}+\bar{F}^{-}_{a}\partial_{0}-\bar{G}^{-}_{a}\partial_{0}^{2})\langle\phi^{a}\rangle$
$\displaystyle=\int\mathrm{d}^{4}x\;e^{-iq_{1,-}\cdot
x}\mathcal{D}^{-}_{a}\langle q_{2},p_{2}|\phi^{a}|p_{1}\rangle\,.$

The differential operator $\mathcal{D}^{-}_{a}$ achieves the LSZ reduction on
each leg. Let us look at its expression in Fourier space:

$\mathcal{D}^{-}_{a}=\frac{(\omega-
E_{-})(E_{-}^{2}-\boldsymbol{k}^{2}-M^{2})}{(\boldsymbol{k}^{2}+M^{2})(E_{+}^{2}-E_{-}^{2})\bar{Z}^{\pi}_{-}}\begin{bmatrix}i(\boldsymbol{k}^{2}\omega+M^{2}\omega+E_{-}E_{+}^{2})\;,&(v/\mu^{2})(E_{+}^{2}-\boldsymbol{k}^{2}-M^{2})(\boldsymbol{k}^{2}+M^{2}+\omega
E_{-})\end{bmatrix}_{a}\,.$ (101)

First of all let us see that this expression reduces to the usual one in the
Lorentz invariant limit, $\mu=0$. In this case,

$Y^{a}_{l}\to\frac{\delta^{a}_{l}}{2E_{l}}\,,~{}X^{a}_{l}\to\frac{-i}{2}\delta^{a}_{l}\,,~{}D^{l}_{m}\to
2\delta^{l}_{m}\,,~{}F^{m}_{a}\to
E_{m}\delta^{m}_{a}\,,~{}G^{m}_{a}\to-i\delta^{m}_{a}$ (102)

so that

$\mathcal{D}^{-}_{a}\to i\delta^{-}_{a}(E^{2}_{-}+\partial_{0}^{2})\,.$ (103)

This can be easily read from Eq. (101). Indeed, switching off the breaking
implies the following limits on the dispersion relations:

$E_{+}^{2}\to\boldsymbol{k}^{2}+M^{2}\>,\quad
E_{-}^{2}\to\boldsymbol{k}^{2}\,,$ (104)

so that

$\mathcal{D}^{-}_{a}\to(-\omega+E_{-})\begin{bmatrix}i(\omega+E_{-})\,,&0\end{bmatrix}=\begin{bmatrix}i(E_{-}^{2}-\omega^{2})\,,&0\end{bmatrix}_{a}$
(105)

which is the same as Eq. (103).

We now wish to show that this LSZ procedure is equivalent to what was obtained
in the main text, Eq. (34). In terms of the field propagator $\Delta_{ab}$
this amounts to proving that

$\lim_{\omega\to E_{-}}\mathcal{D}^{-}_{a}\Delta_{ba}=\lim_{\omega\to
E_{-}}\frac{(\omega^{2}-E_{-}^{2})}{i\bar{Z}^{\pi}_{-}}\Delta_{b\pi}\,.$ (106)

Using the explicit form of the propagator,

$\Delta_{ab}=\frac{i}{2E_{l}}\left(\frac{Z^{a}_{l}\bar{Z}^{b}_{l}}{\omega-
E_{l}}-\frac{\bar{Z}^{a}_{l}Z^{b}_{l}}{\omega+E_{l}}\right)\,,$ (107)

one sees that the RHS of Eq. (106) yields $Z^{b}_{-}$. It is straightforward
to verify that

$\lim_{\omega\to E_{-}}\mathcal{D}^{-}_{a}\Delta_{\pi
a}=Z^{\pi}_{-}\,,~{}\lim_{\omega\to
E_{-}}\mathcal{D}^{-}_{a}\Delta_{ha}=Z^{h}_{-}\,.$ (108)

Notice that the above equality holds only in the on-shell limit $\omega\to
E_{-}$. The two LSZ procedures give different results off-shell and thus
define different functions of the complex momenta.

##  Appendix B Feynman rules

In this appendix we explicitly compute the scattering amplitude (50). The
first step is to find the Feynman rules. Let’s start with the propagators. As
we have seen the quadratic Lagrangian reads

$\displaystyle\tilde{\mathcal{L}}_{(2)}=\frac{1}{2}\Phi_{i}(-k)\Pi_{ij}^{(2)}(k)\Phi_{j}(k)\,,$
(109)

where we have defined

$\Phi_{h}(k)\equiv\tilde{h}_{k}\,,~{}~{}\Phi_{\pi}(k)\equiv\tilde{\pi}_{k}~{}\text{
and
}~{}\Pi_{ij}^{(2)}(k)\equiv\left(\begin{array}[]{cc}k^{2}&i\mu^{2}\omega/v\\\
-i\mu^{2}\omega/v&k^{2}-M^{2}\end{array}\right)_{ij}\,.$ (110)

The propagators then read

$\displaystyle=\frac{i}{(\omega^{2}-E_{+}(\boldsymbol{k})^{2})(\omega^{2}-E_{-}(\boldsymbol{k})^{2})}\left(\begin{array}[]{cc}k^{2}-M^{2}&-i\mu^{2}\omega/v\\\
i\mu^{2}\omega/v&k^{2}\end{array}\right)_{ij}$ (111)
$\displaystyle\equiv\Delta_{ij}(k)\,.$

We can now deal with the interacting Lagrangian. The cubic Lagrangian reads

$\displaystyle\tilde{\mathcal{L}}_{(3)}$
$\displaystyle=-i\frac{\mu^{2}}{2v^{2}}\omega_{1}\tilde{\pi}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}-\frac{k_{1}\cdot
k_{2}}{v}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{3}}-\lambda
v\,\tilde{h}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}$ (112)
$\displaystyle=-i\frac{\mu^{2}}{6v^{2}}\left(\omega_{1}\tilde{\pi}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}+\omega_{2}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{1}}\tilde{h}_{k_{3}}+\omega_{3}\tilde{\pi}_{k_{3}}\tilde{h}_{k_{2}}\tilde{h}_{k_{1}}\right)$
$\displaystyle-\frac{1}{3v}\left(k_{1}\cdot
k_{2}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{3}}+k_{1}\cdot
k_{3}\tilde{\pi}_{k_{3}}\tilde{\pi}_{k_{1}}\tilde{h}_{k_{2}}+k_{3}\cdot
k_{2}\tilde{\pi}_{k_{3}}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{1}}\right)$
$\displaystyle-\lambda
v\,\tilde{h}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}\equiv\frac{1}{3!}\Pi^{(3)}_{i_{1}i_{2}i_{3}}(k_{1},k_{2},k_{3})\Phi_{i_{1}}(k_{1})\Phi_{i_{2}}(k_{2})\Phi_{i_{3}}(k_{3})\,.$

Written explicitly the tensor $\Pi^{(3)}_{i_{1}i_{2}i_{3}}(k_{1},k_{2},k_{3})$
reads

$\displaystyle\frac{1}{3!}\Pi^{(3)}_{i_{1}i_{2}i_{3}}(k_{1},k_{2},k_{3})=-\frac{i\mu^{2}}{v^{2}}\frac{\omega_{1}\delta_{i_{1}\pi}\delta_{i_{2}h}\delta_{i_{3}h}+\omega_{2}\delta_{i_{1}h}\delta_{i_{2}\pi}\delta_{i_{3}h}+\omega_{3}\delta_{i_{1}h}\delta_{i_{2}h}\delta_{i_{3}\pi}}{6}$
(113) $\displaystyle-\frac{k_{1}\cdot
k_{2}\delta_{i_{1}\pi}\delta_{i_{2}\pi}\delta_{i_{3}h}+k_{1}\cdot
k_{3}\delta_{i_{1}\pi}\delta_{i_{2}h}\delta_{i_{3}\pi}+k_{2}\cdot
k_{3}\delta_{i_{1}h}\delta_{i_{2}\pi}\delta_{i_{3}\pi}}{3v}-\lambda
v\,\delta_{i_{1}h}\delta_{i_{2}h}\delta_{i_{3}h}$
$\displaystyle=-\delta_{i_{3}h}\left(\begin{array}[]{cc}\frac{k_{1}\cdot
k_{2}}{3v}&\frac{i\omega_{2}\mu^{2}}{6v^{2}}\\\
\frac{i\omega_{1}\mu^{2}}{6v^{2}}&\lambda
v\end{array}\right)_{i_{1}i_{2}}-\delta_{i_{3}\pi}\left(\begin{array}[]{cc}0&\frac{k_{2}\cdot
k_{3}}{3v}\\\ \frac{k_{1}\cdot
k_{3}}{3v}&\frac{i\omega_{3}\mu^{2}}{6v^{2}}\end{array}\right)_{i_{1}i_{2}}.$

The Feynman rule can be schematically written as

Finally we can turn to four point functions. The four-fields Lagrangian is

$\displaystyle\tilde{\mathcal{L}}_{(4)}$ $\displaystyle=-\frac{k_{1}\cdot
k_{2}}{2v^{2}}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{3}}\tilde{h}_{k_{4}}-\frac{\lambda}{4}\tilde{h}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}\tilde{h}_{k_{4}}$
(114) $\displaystyle=-\frac{1}{12v^{2}}\Big{(}k_{1}\cdot
k_{2}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{2}}\tilde{h}_{k_{3}}\tilde{h}_{k_{4}}+k_{1}\cdot
k_{3}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{3}}\tilde{h}_{k_{2}}\tilde{h}_{k_{4}}+k_{1}\cdot
k_{4}\tilde{\pi}_{k_{1}}\tilde{\pi}_{k_{4}}\tilde{h}_{k_{3}}\tilde{h}_{k_{2}}$
$\displaystyle+k_{2}\cdot
k_{3}\tilde{\pi}_{k_{2}}\tilde{\pi}_{k_{3}}\tilde{h}_{k_{1}}\tilde{h}_{k_{4}}+k_{2}\cdot
k_{4}\tilde{\pi}_{k_{2}}\tilde{\pi}_{k_{4}}\tilde{h}_{k_{1}}\tilde{h}_{k_{3}}+k_{3}\cdot
k_{4}\tilde{\pi}_{k_{3}}\tilde{\pi}_{k_{4}}\tilde{h}_{k_{1}}\tilde{h}_{k_{2}}\Big{)}$
$\displaystyle-\frac{\lambda}{4}\tilde{h}_{k_{1}}\tilde{h}_{k_{2}}\tilde{h}_{k_{3}}\tilde{h}_{k_{4}}\equiv\frac{1}{4!}\Pi^{(4)}_{i_{1}i_{2}i_{3}i_{4}}(k_{1},k_{2},k_{3},k_{4})\Phi_{i_{1}}\Phi_{i_{2}}\Phi_{i_{3}}\Phi_{i_{4}}\,.$

The Feynman rule reads

###  B.1 Scattering amplitudes

#### Four-point

$\displaystyle=$
$\displaystyle\sum_{j}\Delta_{j_{1}\pi}(p_{1})\Delta_{j_{2}\pi}(q_{1})\Delta_{\pi
j_{3}}(q_{2})\Delta_{\pi
j_{4}}(p_{2})i\Pi^{(4)}_{j_{1},j_{2},j_{3},j_{4}}(p_{1},q_{1},-q_{2},-p_{2})$
(115)
$\displaystyle=\prod_{i=1}^{2}\left(\frac{iZ^{\pi}_{-}(\boldsymbol{p}_{l})}{(p^{0}_{i})^{2}-E_{-}(\boldsymbol{p}_{l})^{2}}\frac{iZ^{\pi}_{-}(\boldsymbol{q}_{l})}{(q^{0}_{i})^{2}-E_{-}(\boldsymbol{q}_{l})^{2}}\right)i\mathcal{M}_{\text{c}}\,,$
$\displaystyle\mathcal{M}_{c}=-\frac{\mu^{4}}{v^{4}}\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\Big{[}6\lambda\mu^{4}p_{1}^{0}q_{1}^{0}q_{2}^{0}p_{2}^{0}$
(116) $\displaystyle-2p_{1}\cdot
q_{1}(p_{1}^{2}-M^{2})(q_{1}^{2}-M^{2})q_{2}^{0}p_{2}^{0}-2p_{1}\cdot
p_{2}(p_{1}^{2}-M^{2})(p_{2}^{2}-M^{2})q_{1}^{0}q_{2}^{0}$
$\displaystyle-2p_{1}\cdot
q_{2}(p_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})q_{1}^{0}p_{2}^{0}-2q_{1}\cdot
p_{2}(q_{1}^{2}-M^{2})(p_{2}^{2}-M^{2})p_{1}^{0}q_{2}^{0}$
$\displaystyle-2q_{2}\cdot
q_{1}(q_{2}^{2}-M^{2})(q_{1}^{2}-M^{2})p_{1}^{0}p_{2}^{0}-2p_{2}\cdot
q_{2}(q_{2}^{2}-M^{2})(p_{2}^{2}-M^{2})p_{1}^{0}q_{1}^{0}\Big{]}\,.$

####  $s$-channel

$\displaystyle=\sum_{j}\sum_{l}\Delta_{j_{1}\pi}(p_{1})\Delta_{j_{2}\pi}(q_{1})i\Pi^{(3)}_{j_{1}j_{2}l_{1}}(p_{1},q_{1},-p_{1}-q_{1})\Delta_{l_{2}l_{1}}(p_{1}+q_{1})$
(117)
$\displaystyle\times\Pi^{(3)}_{l_{2}j_{3}j_{4}}(p_{1}+q_{1},-q_{2},-p_{2})\Delta_{\pi
j_{3}}(q_{2})\Delta_{\pi j_{4}}(p_{2})$
$\displaystyle=\sum_{l}\Big{[}\Delta(-p_{1})i\Pi^{(3)}_{l_{1}}(p_{1},q_{1},-p_{1}-q_{1})\Delta(q_{1})\Big{]}_{\pi\pi}\Delta_{l_{2}l_{1}}(p_{1}+q_{1})$
$\displaystyle\times\Big{[}\Delta(p_{1})i\Pi^{(3)}_{l_{2}}(-p_{1},-p_{2},q_{2}+p_{2})\Delta(-p_{2})\Big{]}_{\pi\pi}\,.$

So all together

$\displaystyle\mathcal{G}_{s}$
$\displaystyle=\Bigg{[}vA(p_{1},q_{1})\left\\{v(p_{1}+q_{1})^{2}A(p_{1},q_{1})-\mu^{4}(p_{1}^{0}+q_{1}^{0})B(q_{2},p_{2})\right\\}$
(118)
$\displaystyle-\left\\{v\mu^{4}B(p_{1},q_{1})\right\\}\times\left\\{(p_{1}^{0}+q_{1}^{0})A(q_{2},p_{2})+v(M^{2}-(p_{1}+q_{1})^{2})B(q_{2},p_{2})\right\\}\Bigg{]}$
$\displaystyle\prod_{i=1}^{2}\left(\frac{1}{p_{j}^{2}(p_{j}^{2}-M^{2})-(\mu^{4}/v^{2})(p_{l}^{0})^{2}}\frac{1}{q_{j}^{2}(q_{j}^{2}-M^{2})-(\mu^{4}/v^{2})(q_{l}^{0})^{2}}\right)$
$\displaystyle\times\frac{iv^{-10}}{(p_{1}+q_{1})^{2}((p_{1}+q_{1})^{2}-M^{2})-(\mu^{4}/v^{2})(p_{1}^{0}+q_{1}^{0})^{2}}\,,$

where we have defined

$A(p_{1},q_{1})\equiv 2v^{3}(M^{2}-p_{1}^{2})(M^{2}-q_{1}^{2})p_{1}\cdot
q_{1}-v\mu^{4}p_{1}^{0}q_{1}^{0}(p_{1}^{2}+q_{1}^{2}+M^{2})\,,$ (119)

and

$B(p_{1},q_{1})\equiv\mu^{4}(p_{1}^{0})^{2}q_{1}^{0}+2v^{2}q_{1}^{0}(M^{2}-p_{1}^{2})(p_{1}^{2}+p_{1}\cdot
q_{1})+(p_{1}\leftrightarrow q_{1})\,.$ (120)

Finally we obtain

$\displaystyle\mathcal{M}_{s}$
$\displaystyle=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)$
(121)
$\displaystyle\times\frac{v^{-9}}{(p_{1}+q_{1})^{2}((p_{1}+q_{1})^{2}-M^{2})-(\mu^{4}/v^{2})(p_{1}^{0}+q_{1}^{0})^{2}}$
$\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(p_{1},q_{1})B(q_{2},p_{2})+[B(p_{1},q_{1})A(q_{2},p_{2})$
$\displaystyle+B(q_{2},p_{2})A(p_{1},q_{1})]\mu^{4}(p_{1}^{0}+q_{1}^{0})-v[A(p_{1},q_{1})A(q_{2},p_{2})$
$\displaystyle+\mu^{4}B(p_{1},q_{1})B(q_{2},p_{2})](p_{1}+q_{1})^{2}\Big{]}\,.$

####  $t$-channel

$\displaystyle=\sum_{j}\sum_{l}$
$\displaystyle\Delta_{j_{1}\pi}(q_{1})\Delta_{j_{2}\pi}(-q_{2})i\Pi^{(3)}_{j_{1}j_{2}l_{1}}(q_{1},-q_{2},-q_{1}+q_{2})\Delta_{l_{2}l_{1}}(q_{1}-q_{2})$
(122)
$\displaystyle\times\Pi^{(3)}_{l_{2}j_{3}j_{4}}(q_{1}-q_{2},p_{1},-p_{2})\Delta_{\pi
j_{3}}(-p_{1})\Delta_{\pi j_{4}}(p_{2})\,.$ $\displaystyle\mathcal{M}_{t}$
$\displaystyle=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\times$
(123)
$\displaystyle\frac{v^{-9}}{(q_{1}-q_{2})^{2}((q_{1}-q_{2})^{2}-M^{2})-(\mu^{4}/v^{2})(q_{1}^{0}-q_{2}^{0})^{2}}$
$\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(q_{1},-q_{2})B(-p_{1},p_{2})+[B(q_{1},-q_{2})A(-p_{1},p_{2})$
$\displaystyle+B(-p_{1},p_{2})A(q_{1},-q_{2})]\mu^{4}(q_{1}^{0}-q_{2}^{0})-v[A(q_{1},-q_{2})A(-p_{1},p_{2})$
$\displaystyle+\mu^{4}B(q_{1},-q_{2})B(-p_{1},p_{2})](q_{1}-q_{2})^{2}\Big{]}\,.$

####  $u$-channel

$\displaystyle=\sum_{j}\sum_{l}$
$\displaystyle\Delta_{j_{1}\pi}(q_{1})\Delta_{j_{2}\pi}(-p_{2})i\Pi^{(3)}_{j_{1}j_{2}l_{1}}(q_{1},-p_{2},-q_{1}+p_{2})\Delta_{l_{2}l_{1}}(q_{1}-p_{2})$
(124)
$\displaystyle\times\Pi^{(3)}_{l_{2}j_{3}j_{4}}(q_{1}-p_{2},-q_{2},p_{1})\Delta_{\pi
j_{3}}(q_{2})\Delta_{\pi j_{4}}(-p_{1})\,.$

Thus

$\displaystyle\mathcal{M}_{u}$
$\displaystyle=\prod_{i=1}^{2}\left(\frac{Z^{\pi}_{-}(\boldsymbol{p}_{i})^{-1}}{(p^{0}_{i})^{2}-E_{+}(\boldsymbol{p}_{i})^{2}}\frac{Z^{\pi}_{-}(\boldsymbol{q}_{i})^{-1}}{(q^{0}_{i})^{2}-E_{+}(\boldsymbol{q}_{i})^{2}}\right)\times$
(125)
$\displaystyle\frac{v^{-9}}{(q_{1}-p_{2})^{2}((q_{1}-p_{2})^{2}-M^{2})-(\mu^{4}/v^{2})(q_{1}^{0}-p_{2}^{0})^{2}}$
$\displaystyle\Big{[}2\lambda
v^{3}\mu^{4}B(q_{1},-p_{2})B(q_{2},-p_{1})+[B(q_{1},-p_{2})A(q_{2},-p_{1})$
$\displaystyle+B(q_{2},-p_{1})A(q_{1},-p_{2})]\mu^{4}(q_{1}^{0}-p_{2}^{0})-v[A(q_{1},-p_{2})A(q_{2},-p_{1})$
$\displaystyle+\mu^{4}B(q_{1},-p_{2})B(q_{2},-p_{1})](q_{1}-p_{2})^{2}\Big{]}\,.$

##  Appendix C $S$-matrix in the low-energy EFT

In this appendix, the EFT amplitude for the $\pi\pi\to\pi\pi$ process is
reported. Making use of the Feynman rules coming from Lagrangian (71), the
different contributions to the full amplitude can be evaluated:

$\displaystyle\mathcal{M}_{c}=\frac{2c_{s}^{4}}{\lambda v^{4}}(p_{1}\cdot
q_{1}\,p_{2}\cdot q_{2}+q_{1}\cdot q_{2}\,p_{2}\cdot p_{1}+p_{2}\cdot
q_{1}\,p_{1}\cdot q_{2})\,,$
$\displaystyle\mathcal{M}_{s}=-\left(\frac{c_{s}^{3}\mu^{2}}{\lambda
v^{4}}\right)^{2}\frac{(2(q_{1}^{0}+p_{1}^{0})p_{1}\cdot
q_{1}+p_{1}^{0}q_{1}^{2}+q_{1}^{0}p_{1}^{2})(2(q_{2}^{0}+p_{2}^{0})p_{2}\cdot
q_{2}+p_{2}^{0}q_{2}^{2}+q_{2}^{0}p_{2}^{2})}{(q_{1}^{0}+p_{1}^{0})^{2}-c_{s}^{2}(\boldsymbol{q}_{1}+\boldsymbol{p}_{1})^{2}}\,,$
$\displaystyle\mathcal{M}_{t}=-\left(\frac{c_{s}^{3}\mu^{2}}{\lambda
v^{4}}\right)^{2}\frac{(-2(q_{1}^{0}-q_{2}^{0})q_{2}\cdot
q_{1}-q_{2}^{0}q_{1}^{2}+q_{1}^{0}q_{2}^{2})(-2(-p_{1}+p_{2}^{0})p_{2}\cdot
p_{1}+p_{2}^{0}q_{2}^{2}-p_{1}^{0}p_{2}^{2})}{(q_{1}^{0}-q_{2}^{0})^{2}-c_{s}^{2}(\boldsymbol{q}_{1}-\boldsymbol{q}_{2})^{2}}\,,$
$\displaystyle\mathcal{M}_{u}=-\left(\frac{c_{s}^{3}\mu^{2}}{\lambda
v^{4}}\right)^{2}\frac{(-2(q_{1}^{0}-p_{2}^{0})p_{2}\cdot
q_{1}-p_{2}^{0}q_{1}^{2}+q_{1}^{0}p_{2}^{2})(-2(q_{2}^{0}-p_{1}^{0})p_{1}\cdot
q_{2}-p_{1}^{0}q_{2}^{2}+q_{2}^{0}p_{1}^{2})}{(q_{1}^{0}-p_{2}^{0})^{2}-c_{s}^{2}(\boldsymbol{q}_{1}-\boldsymbol{p}_{2})^{2}}\,.$

Specializing to equal energies
$p_{1}^{0}=q_{1}^{0}=p_{2}^{0}=q_{2}^{0}=\omega$ yields a simpler expression
for the $S$-matrix:

$\displaystyle\mathcal{M}_{c}=\frac{2c_{s}^{4}}{\lambda
v^{4}}(3\alpha^{2}\omega^{4}+\frac{s^{2}+u^{2}+t^{2}}{4}+\alpha\omega^{2}(s+u+t))=\frac{c_{s}^{4}\alpha}{\mu^{4}}(-4\alpha^{2}\omega^{4}+s^{2}+u^{2}+t^{2})\,,$
(126)
$\displaystyle\mathcal{M}_{s}=-\frac{16c_{s}^{4}\alpha^{2}\omega^{2}}{\mu^{4}}\frac{(s+\alpha\omega^{2})^{2}}{s+4\alpha\omega^{2}}\,,~{}\mathcal{M}_{u}=0\,,~{}\mathcal{M}_{t}=0\,.$

##  Appendix D Estimate of the decay width

In this appendix we wish to refine the estimate the decay width
$\Gamma(\omega)$ for $\omega\gg M$. To this end, we work directly in the
diagonal field basis $\varphi_{\pm}$ introduced in section 3. As we already
pointed out there,

$\varphi_{\pm}(t,\boldsymbol{x})\equiv\int\frac{\mathrm{d}^{3}\boldsymbol{k}}{(2\pi)^{3}2E_{\pm}(\boldsymbol{k})}\left(a_{\pm}(\boldsymbol{k})e^{-i(E_{\pm}(\boldsymbol{k})t-\boldsymbol{k}\cdot\boldsymbol{x})}+\text{h.c.}\right)$
(127)

are the field operators that interpolate only the two states $+$ and $-$
respectively. Recalling Eq. (15) it is evident that $\pi$ and $h$ are linear
combinations of the $\varphi_{\pm}$ fields. In particular, the $(h,\pi)$-basis
is linked to the $\pm$ basis via the following transformation

$\begin{pmatrix}\tilde{\pi}(k)\\\
\tilde{h}(k)\end{pmatrix}=\begin{pmatrix}Z^{\pi}_{-}(\boldsymbol{k})&Z^{\pi}_{+}(\boldsymbol{k})\\\
Z^{h}_{-}(\boldsymbol{k})&Z^{h}_{+}(\boldsymbol{k})\end{pmatrix}\begin{pmatrix}\tilde{\varphi}_{-}(k)\\\
\tilde{\varphi}_{+}(k)\end{pmatrix}\equiv\mathcal{U}(\boldsymbol{k})\begin{pmatrix}\tilde{\varphi}_{-}(k)\\\
\tilde{\varphi}_{+}(k)\end{pmatrix}\,.$ (128)

The transformation $\mathcal{U}(\boldsymbol{k})$ that realizes the change of
basis is a unitary matrix that diagonalizes the quadratic Lagrangian (8):

$\displaystyle\mathcal{L}_{(2)}=\begin{pmatrix}\tilde{\varphi}_{-}(-k)&\tilde{\varphi}_{+}(-k)\end{pmatrix}\begin{pmatrix}k^{2}-\frac{M^{2}}{2}+\frac{M^{2}}{2}\sqrt{1+\frac{4\omega^{2}}{\omega_{c}^{2}}}&0\\\
0&k^{2}-\frac{M^{2}}{2}-\frac{M^{2}}{2}\sqrt{1+\frac{4\omega^{2}}{\omega_{c}^{2}}}\end{pmatrix}\begin{pmatrix}\tilde{\varphi}_{-}(k)\\\
\tilde{\varphi}_{+}(k)\end{pmatrix}\,.$ (129)

Making use of the dispersion relations (12) we can trade the
$\boldsymbol{k}^{2}$-dependence for $\omega$-dependence. For the interpolation
functions we obtain

$Z^{\pi}_{-}(\omega)=Z^{h}_{+}(\omega)=\frac{\frac{1}{2}+\frac{1}{2}\sqrt{1+4\frac{\omega^{2}}{\omega_{c}^{2}}}}{\sqrt{2\frac{\omega^{2}}{\omega_{c}^{2}}+\frac{1}{2}+\frac{1}{2}\sqrt{1+4\frac{\omega^{2}}{\omega_{c}^{2}}}}}\>,\>Z^{\pi}_{+}(\omega)=Z^{h}_{-}(\omega)=-i\frac{\omega/\omega_{c}}{\sqrt{2\frac{\omega^{2}}{\omega_{c}^{2}}+\frac{1}{2}+\frac{1}{2}\sqrt{1+4\frac{\omega^{2}}{\omega_{c}^{2}}}}}\,.$
(130)

Making use of this change of basis we can now rewrite the full Lagrangian in
terms of the diagonal fields $\varphi_{\pm}$. Thus, if we are interested in
the $-\to--$ decay process we can simply focus on the trilinear vertices
involving the $\varphi_{-}$ fields only. It is important to notice that, in
general, the full Lagrangian will be non-local. The energy scale at which this
non-locality will become important is $\omega_{c}\sim\lambda v^{3}/\mu^{2}$,
introduced in Eq. (33). If we are interested in a given process at energies
$\omega\ll\omega_{c}$, we can expand the interpolating functions

$Z^{\pi}_{-}(\omega)=Z^{h}_{+}(\omega)\sim
1+\mathcal{O}\left(\frac{\omega^{2}}{\omega_{c}^{2}}\right)\>,\>Z^{\pi}_{+}(\omega)=Z^{h}_{-}(\omega)\sim-i\frac{\omega}{\omega_{p}}+\mathcal{O}\left(\frac{\omega^{3}}{\omega_{c}^{3}}\right)\,,$
(131)

thus obtaining a local Lagrangian. Equivalently, to leading order in
derivatives, we can write

$\pi\sim\varphi_{-}-i\frac{\dot{\varphi}_{+}}{\omega_{c}}\>,\>h\sim\varphi_{+}-i\frac{\dot{\varphi}_{-}}{\omega_{c}}\,.$
(132)

Plugging these expressions back into the cubic part of the Lagrangian and
focusing on the trilinear vertices involving $\varphi_{-}$ only we obtain

$\frac{(\partial\varphi_{-})^{2}\dot{\varphi}_{-}}{v\omega_{c}}\,.$ (133)

By dimensional analysis we thus obtain

$\frac{\Gamma}{\omega}(M\ll\omega\ll\omega_{c})\sim(1-c_{s}^{-2})^{2}\frac{\omega^{4}}{v^{2}\omega_{c}^{2}}\sim\lambda\left(\frac{\omega}{\omega_{c}}\right)^{4}\frac{\lambda\mu^{4}}{M^{4}}\ll\lambda\,.$
(134)

Again we have the same kinematical suppression factor discussed in section 5.
This is a refinement of the previous overestimate (87) and has the same
functional form as the EFT decay width. We can now turn to the only missing
case, namely $\omega\gg\omega_{c}$. The first idea that comes to mind is to
expand the $Z$’s in $\omega_{c}/\omega$,

$Z^{\pi}_{-}(\omega)=Z^{h}_{+}(\omega)\sim\frac{1}{\sqrt{2}}+\mathcal{O}\left(\frac{\omega_{c}}{\omega}\right)\>,\>Z^{\pi}_{+}(\omega)=Z^{h}_{-}(\omega)\sim-\frac{i}{\sqrt{2}}+\mathcal{O}\left(\frac{\omega_{c}}{\omega}\right)\,,$
(135)

to obtain a simplified Lagrangian suitable to compute processes in this
regime. Although this looks technically very similar to what has been done
just above, in essence it is rather different. In fact the $\omega_{c}/\omega$
expansion brings powers of energy to the denominator and as such yields non-
local vertices, which are somewhat cumbersome to deal with. For these reasons
we choose to work with the unexpanded Lagrangian instead:

$\begin{gathered}\mathcal{L}_{(3)}\supset\Bigg{(}-\frac{i\omega_{1}\mu^{2}}{2v^{2}}Z^{\pi}_{-}(\omega_{1})Z^{h}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})-\frac{k_{1}\cdot
k_{2}}{v}Z^{\pi}_{-}(\omega_{1})Z^{\pi}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})\\\
-\lambda
vZ^{h}_{-}(\omega_{1})Z^{h}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})\Bigg{)}\tilde{\varphi}_{-}(k_{1})\tilde{\varphi}_{-}(k_{2})\tilde{\varphi}_{-}(k_{3})\,,\end{gathered}$
(136)

and perform the expansion only at the end. The associated matrix element reads

$\begin{gathered}\mathcal{M}=-\frac{i\mu^{2}}{v^{2}}\Bigg{(}\omega_{1}Z^{\pi}_{-}(\omega_{1})Z^{h}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})+\omega_{2}Z^{\pi}_{-}(\omega_{2})Z^{h}_{-}(\omega_{1})Z^{h}_{-}(\omega_{3})+\omega_{3}Z^{\pi}_{-}(\omega_{3})Z^{h}_{-}(\omega_{2})Z^{h}_{-}(\omega_{1})\Bigg{)}\\\
-\frac{2}{v}\Bigg{(}p_{1}\cdot
p_{2}Z^{\pi}_{-}(\omega_{1})Z^{\pi}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})+p_{1}\cdot
p_{3}Z^{\pi}_{-}(\omega_{1})Z^{\pi}_{-}(\omega_{3})Z^{h}_{-}(\omega_{2})\\\
+p_{3}\cdot
p_{2}Z^{\pi}_{-}(\omega_{3})Z^{\pi}_{-}(\omega_{2})Z^{h}_{-}(\omega_{1})\Bigg{)}-6\lambda
v\,Z^{h}_{-}(\omega_{1})Z^{h}_{-}(\omega_{2})Z^{h}_{-}(\omega_{3})\,.\end{gathered}$
(137)

In the high energy limit, using conservation of energy and momentum we find

$\mathcal{M}=-i\frac{\mu^{2}}{\sqrt{2}v^{2}}(\omega_{1}-\omega_{2})+i\frac{\mu^{2}}{\sqrt{2}v^{2}}(\omega_{1}-\omega_{2})-i\frac{v\lambda}{\sqrt{2}}-i\frac{3v\lambda}{\sqrt{2}}+\mathcal{O}\left(\frac{\omega_{c}}{\omega}\right)=-i2\sqrt{2}\lambda
v+\mathcal{O}\left(\frac{\omega_{c}}{\omega}\right)\,.$ (138)

By dimensional analysis we thus conclude that the ratio between the
interaction time and the decay time for the $-\to--$ process can be estimated
to be

$\frac{\Gamma}{\omega}(\omega\gg\omega_{p})\sim\frac{\lambda^{2}v^{2}}{\omega^{2}}\sim\lambda\frac{M^{2}}{\omega^{2}}\ll\lambda\,,$
(139)

in agreement with the result obtained in the $\phi_{1},\phi_{2}$ field basis,
Eq. (87). This is a further confirmation that the ratio between interaction
time and the decay time is always suppressed by $\lambda$.

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