aainstitutetext: Dipartimento di Fisica, Università di Roma Tor Vergata, Via
della Ricerca Scientifica 1, 00133, Roma, Italybbinstitutetext: INFN sezione
di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

# The chaotic emergence of thermalization in highly excited string decays

Maurizio Firrotta<EMAIL_ADDRESS>
###### Abstract

We analyse the most general process of a generic highly excited string that
decays into a less excited, yet generic, highly excited string emitting a
tachyon. We provide a simple and compact analytic description of the decay
process which discriminates between and within the structure of every single
microstate of the initial and final highly excited string. Taking into account
the random nature of the decay process we extract the energy spectrum of
highly excited strings, microstate by microstate, finding a behavior which
corresponds to the greybody emission spectrum. In addition, by exploiting the
analytic control of the decay process, we identify the origin of thermal
effects which are triggered by the chaotic nature of the highly excited string
interactions modeled by the microstates structure.

## Introduction

The present paper is focused on enlightening the connection between chaos and
thermal effects within the physical systems provided by highly excited string
(HES) interactions. Motivated by the intriguing interplay between chaos,
thermal effects and quantum information Gibbons:1977mu -Shenker:2013pqa ,
which are three milestones of black hole (BH) physics, we first used HES as
promising candidates of BH states Susskind:1993ws -Sundborg:2000wp  and then
we computed their energy spectra. The main goal was to detect a manifest
connection between the chaotic behavior of HES interactions and the
thermalization of their energy spectra which emerges naturally. In line with
past studies on string decays and the produced Hawking radiation Amati
-Cornalba:2006hc , we used and improved the most general process of an HES
that decays into an HES emitting a tachyon Bianchi:2019ywd Firrotta:2022cku ,
providing an analytic description of the decay process which discriminates
between and within the structure of every single microstate of the initial and
final HES. Considering the random nature of the decay process we extracted the
spectrum of the HES, microstate by microstate, finding a behavior which
corresponds to the greybody emission spectrum. In addition, exploiting the
analytic control of the decay process, we identified the origin of thermal
effects, finding that they are triggered by the chaotic nature of HES
interactions.

The setup we adopted relies on the recent improvement of the Di Vecchia, Del
Giudice and Fubini (DDF) formalism DelGiudice:1971yjh Brower:1972wj , where
its spectrum generating algebra was recasted in a manifestly covariant form
Skliros:2011si Aldi:2019osr , for both bosonic string and superstring
theories. The possibility of identifying each state of the string spectrum
with the associated physical vertex operator gave rise to a wide range of
applications: the realization of the scattering of string coherent vertex
operators111A very powerful application of the scattering amplitudes of
coherent string states is their nature of generating amplitudes of any desired
string states, obtained by a simple derivative projection over mass
eigenstates. Bianchi:2019ywd , the non perturbative string footprint in the
gravitational wave (GW) signal produced in the merging phase of BHs collision
Addazi:2020obs , the non perturbative spinning corrections to the
electromagnetic wave produced in the collision of heavy sized objects
Aldi:2021zhh , such as BHs and neutron stars NSs, the two body decay of HES
Firrotta:2022cku  and finally the indications about the chaotic behavior of
HES interactions Gross:2021gsj Rosenhaus:2021xhm .

About the chaotic behavior of HES interactions, quite recently a novel measure
of chaos for scattering amplitudes was proposed ChaosScatt , where the
behavior of HES amplitudes was compared with the chaotic distribution of the
zeros of the Riemann zeta function and the quantum mechanical scattering on a
leaky torus, finding a common pattern among them.

The scope of the present paper was to continue the study of the physical
applications connected to the possibility of exploring the interactions of the
whole tower of string excitations, or string microstates, proceeding beyond
the physics of light string states and the first Regge trajectory.

The paper is organized as follows: in section 1 we explained the connection
between the shape of classical string configurations and the structure of
quantum string configurations, in particular we studied the classical string
profiles as a function of the number of harmonics and the respective
coefficients and we compared their shape with the degenerate quantum string
partitions of generic mass levels. We followed the logic that the structure of
a quantum string state can be probed through its interaction, we selected the
simplest one $i.e$ the decay of HES into two tachyons, in order to preserve
the HES structure. After a brief review of the chaotic analysis, developed in
ChaosScatt , we compared the shape of classical string configurations with the
chaotic behavior of the scattering amplitudes relative to the quantum analog
of classical string configurations finding a common pattern.

In section 2 we studied the most general process of a generic HES that decays
into a less excited, yet generic, HES emitting a tachyon in the thermalization
regime. Exploiting the analytic control of the decay process, we identify the
origin of thermal effects with the chaotic structure of the process. Finally
we gave a description of how to compute the emission spectrum of HES.

In section 3 we presented the results of HES spectra, microstate by
microstate. In particular we found that a generic string excitation is
characterized by a greybody emission, while for the extreme case where only
excitations of the first Regge trajectory (FRtj) are considered, the thermal
nature is highly suppressed as to be negligible.

In appendix Acknowledgements we reviewed the computation developed in
Firrotta:2022cku , and we described the analytical implications of the
thermalization regime, or more precisely the regime in which the ratio between
the energy of the emitted state and the mass of the decaying state is enough
small in such a way that the energy loss of the decaying state is smooth.

##  1 Chaos in highly excited string processes

The aim of this section is to review recent results about the chaotic behavior
of HES interactions.

###  1.1 Classical string vs quantum string configurations

Classical three dimensional222In the specific case of classical string, it was
chosen $j=1,2,3$ in order to plot 3D profiles, but in general $j=1,...,D-2$.
bosonic open string profiles with Neumann boundary conditions, in the temporal
gauge ($X^{0}=t$), are given by Green:1987sp

$X^{j}_{\\{a_{n}\\}}(\sigma,\tau)=\sum_{n=1}^{n^{*}}x^{j}_{\\{n\\},\\{a_{n}\\}}(\sigma,\tau)$
(1)

$\sigma\in[0,1]$ and $\tau\in[0,\infty)$ are the worldsheet variables and
$X^{j}(\sigma,\tau)$ is the map between the worldsheet and the target space
(${\mathbb{R}}_{3}$), representing the 3-D string profile at any value of
$\tau$333We did not include the center of mass position of the string $x_{0}$,
and also the center of mass momentum $p_{0}\tau$, because are not relevant in
our investigation. . Classical string configurations can be classified by the
set of harmonics $\\{n\\}$ and the respective set of coefficients
$\\{a^{j}_{n}\\}$ which are respectively the harmonic label and the relative
weight coefficient of the classical mode $x^{j}$ :

$x^{j}_{\\{n\\},\\{a_{n}\\}}(\sigma,\tau)={a_{n}^{j}\over
n}\cos{(n\pi\sigma)}\sin{(n\pi\tau)}$ (2)

In figure 1 there are some 3-D string profiles for different choices of the
set of harmonics and coefficients, from which one can observe how the string
profile becomes more involved if the number of harmonics is increased.

Figure 1: Examples of 3D string profiles for different combinations of
harmonics $n^{*}=1,5,10,50$ with uniformly distributed random parameters
$a_{n}^{j}\in(0,1)$.

From figure 2 one can observe the dependence of string profiles from the set
of coefficients $\\{a_{n}^{j}\\}$. In the present case it was assumed that
coefficients are normalized to the identity, and they are uniformly
distributed in the interval $(0,1)$. One can observe that the behavior of
string profiles is unchanged also for non normalized integer coefficients. The
features of string profiles are unaffected by different parametrizations of
the coefficients $a_{n}^{j}$.

Figure 2: Examples of 3D string profiles for different combinations of
harmonics $n^{*}=10,50,100$. For each value of $n^{*}$ there are five choices
of string profiles for different choices of uniformly distributed random
parameters $a_{n}^{j}\in(0,1)$.

At this level a trivial observation is that, if one considers a classical
string profile

$X^{j}_{\\{a_{n}\\}}(\sigma,\tau)=x^{j}_{\\{n\\},\\{a_{n}\\}}(\sigma,\tau)$
(3)

with the same single generic harmonic $n$ in the three spatial directions
$j=1,2,3$, one obtains the smoothest profile, which is a straight line, such
as the first profile of figure 1.

This is the general picture of how the classical string profiles are modeled
by the harmonic set $\\{n\\}$ and the coefficient set $\\{a_{n}\\}$.

Now it is helpful to study the comparison between classical and quantum string
configuration. In particular, promoting the coefficients $a_{n}^{j}$ to be
creation operators ${\cal A}_{-n}^{j}$ one has the quantum analog of the
string mode, and in addition to the set of harmonics $\\{n\\}$, one has to
include the number of excitation $g_{n}$ of each harmonic so much so that for
a given quantized level $N$ of the string spectrum, one has a set of states
spanned by the set of solutions $\\{g_{n}\\}$, representing all the partitions
of the integer $N$ with occupation number $J$:

$N=\sum_{n=1}^{N}ng_{n}\,,\quad J=\sum_{n=1}^{N}g_{n}$ (4)

The promotion from classical to quantum string is summarized as

$x_{\\{n\\},\\{a_{n}\\}}(\sigma,\tau)\,\Rightarrow\,{\cal N}_{n,g_{n}}{\cal
A}_{-n}^{g_{n}}|0\rangle\,\,;\quad
X_{\\{a_{n}\\}}(\sigma,\tau)\Rightarrow\Pi^{(N)}_{\\{g_{n}\\}}=\prod_{n=1}^{N}{\cal
N}_{n,g_{n}}{\cal A}_{-n}^{g_{n}}|0\rangle$ (5)

where ${\cal N}^{-1}_{n,g_{n}}=\sqrt{n^{g_{n}}\,g_{n}!}$ is the normalization
constant of each mode444For simplicity we have suppressed the space time index
$j$. In particular due to its quantum nature, the quantum string configuration
for a fixed level $N$ has a large degeneracy555The degeneracy of states at
large $N$ grows like $\sim e^{\sqrt{N}}$ which produces the microstate
structure, so in order to identify the final quantum string state one can take
the simplest linear combination of microstates introducing the average over
microstates in the second expression of (5)

$X(\sigma,\tau)\Rightarrow\sum_{\\{g_{n}\\}}\Pi^{(N)}_{\\{g_{n}\\}}=\sum_{\\{g_{n}\\}}\prod_{n=1}^{N}{\cal
N}_{n,g_{n}}{\cal A}_{-n}^{g_{n}}|0\rangle$ (6)

Looking at the complicated classical string profiles, parametrized by
$\\{n\\}$ and $\\{a_{n}\\}$, a natural question is how the quantum string
configurations reflect their classical characteristic shape as a function of
$\\{n\\}$ and $\\{g_{n}\\}$.

A possible way of testing the features of quantum string profiles as a
function of the microstate structure, is to probe the implications of their
shapes through their interactions. In particular one can choose the simplest
string decay amplitude and study the microstate dependence.

In the next subsection there will be a review of the analysis of string
configurations leading to a chaotic behavior of their interactions ChaosScatt
.

###  1.2 Probing chaotic behavior of quantum strings through their
interactions

Inspired by the systematic of classical string profiles, the logic proposed
for the analysis of quantum string configurations is the following: the main
observable is the decay amplitude provided by a level $N$ string microstate
$\Pi_{\\{g_{n}\\}}^{(N)}$ decaying into two tachyons. Along the line of
indications and improvements introduced in Skliros:2011si  and developed in
Bianchi:2019ywd , the decay amplitude ${\cal A}_{\Pi^{(N)}_{\\{g_{n}\\}}}$ for
the most general HES was computed, with a remarkably simple procedure based on
coherent state techniques. The informations about the structure of
$\Pi_{\\{g_{n}\\}}^{(N)}$ are translated into the decay amplitude, and they
are manifested through the profile of the decay amplitude. The choice of
looking at tachyons is connected to their simple vertex operators, in such a
way that all the information inside the decay amplitude is governed by the
structure of $\Pi_{\\{g_{n}\\}}^{(N)}$. In figure 3 there is a representative
picture of the decay amplitude from which one can see that the decay amplitude
profile is only a function of the angle $\alpha$. Now comparing decay profiles
of different microstates one can extract a general behavior associated to
choices of $\\{n\\}$ and $\\{g_{n}\\}$, as it was pointed out in Gross:2021gsj
. Most of the information of the decay profile can be codified in terms of its
extrema, so it is useful to introduce the logarithmic derivative
$F_{\\{g_{n}\\}}(\alpha)$ and study its distribution of zeros. A suitable
parameterization of the distribution is intimately connected with the chaotic
behavior of the decay ChaosScatt . Before describing the chaotic analysis in
detail, a fast presentation of the resulting decay amplitude is discussed.

Figure 3: Picture of the decay amplitude where a representative microstate
$\Pi_{\\{g_{n}\\}}^{(N)}$ decays into two tachyons $T$. The only kinematical
freedom of the decay amplitude is the emission angle $\alpha$ which starts
from the reference dashed line.

The HES state of the level $N$ with polarizations $\\{\zeta_{n}\\}$ and
momentum $p$ is described by

$\Pi_{\\{g_{n}\\}}^{(N)}(\\{\zeta_{n}\\},p)=\prod_{n=1}^{N}{\left(\zeta_{n}{\cdot}{\cal
A}_{-n}\right)^{g_{n}}\over\sqrt{g_{n}!\,n^{g_{n}}}}|\widetilde{p}\rangle$ (7)

where $\widetilde{p}$ is the tachyonic DDF reference momentum, that combined
with the action of the creation operators reproduce the momentum of the final
state $p=\widetilde{p}-q\sum_{n}ng_{n}$. Following the DDF formalism one can
write the corresponding BRST vertex operator and compute the decay amplitude
of figure 3 using circular polarizations666The general amplitude is made of
contributions both linear and bilinear in $\zeta_{n}$ Bianchi:2019ywd . The
bilinear contribution is proportional to the square of the linear
contribution, so up to an irrelevant polynomial the functional dependence on
the microstate structure is preserved by the choice
$\zeta^{(a)}_{n}{\cdot}\zeta^{(b)}_{m}=0$.:

${\cal
A}_{\Pi^{(N)}_{\\{g_{n}\\}}}(\alpha)\simeq\Big{(}\zeta{\cdot}(p_{1}{-}p_{2})\Big{)}^{J}\prod_{n=1}^{N}\left({(1{-}n\sin^{2}\alpha)_{n{-}1}\over\Gamma(n)}\right)^{g_{n}}$
(8)

all the information about the microstate $\Pi_{\\{g_{n}\\}}^{(N)}$ is encoded
in the dressing factor of the coupling $\zeta{\cdot}(p_{1}{-}p_{2})$

$\Pi_{\\{g_{n}\\}}^{(N)}{-}\text{structure}\Rightarrow\prod_{n=1}^{N}\left({(1{-}n\sin^{2}\alpha)_{n{-}1}\over\Gamma(n)}\right)^{g_{n}}$
(9)

Using the properties of the Pochhammer factor and the explicit parametrization
of the polarizations the decay amplitude can be written as

${\cal
A}_{\Pi^{(N)}_{\\{g_{n}\\}}}(\alpha)\simeq\prod_{n=1}^{N}\left({\sin{\alpha}\over\Gamma(n)}\,\sin{\left(n\pi\cos^{2}\alpha/2\right)}\,\Gamma\left(n\cos^{2}\alpha/2\right)\,\Gamma\left(n\sin^{2}\alpha/2\right)\right)^{g_{n}}$
(10)

Finally the logarithmic derivative of the decay amplitude

$F_{\\{g_{n}\\}}(\alpha)={d\over d\alpha}\log{{\cal
A}_{\Pi^{(N)}_{\\{g_{n}\\}}}(\alpha)}$ (11)

has the following form

$\begin{split}F_{\\{g_{n}\\}}(\alpha)=&J\cot\alpha-\pi{\sin\alpha\over
2}\sum_{n=1}^{N}ng_{n}\cot{\left(n\pi\cos^{2}{\alpha\over 2}\right)}\\\
&{-}{\sin\alpha\over
2}\sum_{n=1}^{N}ng_{n}\Big{(}\psi\left(n\cos^{2}{\alpha\over
2}\right){-}\psi\left(n\sin^{2}{\alpha\over 2}\right)\Big{)}.\end{split}$ (12)

This is the final observable which will be subjected to the analysis described
below.

#### Chaotic analysis: setup

Random Matrix Theory (RMT) provides a very powerful tool to make the bridge
between quantum chaos and universal statistical properties RMTbook ChaosNucl .
In particular a quantitative connection between chaos and probability
distributions was conjectured in ChaosStat  and subsequently the link between
chaos and statistical properties was widely studied in many contexts such as
quantum chromodynamics (QCD)Verbaarschot:2000dy , nuclear physics
Weidenmuller:2008vb , black holes Cotler:2016fpe  and condensed matter
Guhr:1997ve . In what follows we laid out the identification strategy of the
target distribution used as a discriminant of the chaotic behavior.

  * •

Starting from the Hermite $\beta$-ensemble of $N\times N$ random matrices,
given the set $\\{\alpha\\}=(\alpha_{1},....,\alpha_{N})$ of matrix
eigenvalues, the associated joint probability distribution is given by

$P_{\beta}(\\{\alpha\\})={e^{-\sum_{j=1}^{N}{\alpha_{j}^{2}\over 2}}\over
Z_{N,\beta}}\prod_{\ell<v}|\alpha_{\ell}-\alpha_{v}|^{\beta}$ (13)

  * •

A very useful approximation was given in WigSurm , where it was considered the
joint probability distribution of nearest-neighbor spacing
$\alpha_{j+1}-\alpha_{j}=\delta_{j}$ : the Wigner surmise distribution

$P_{\beta}(\\{\delta\\})=C_{\beta}\,\delta^{\beta}\,e^{-d_{\beta}\delta^{2}}$
(14)

with constants

$C_{\beta}=2\,{\Gamma(\beta/2+1)^{\beta+1}\over\Gamma(\beta/2+1/2)^{\beta+2}}\,,\quad
d_{\beta}={\Gamma(\beta/2+1)^{2}\over\Gamma(\beta/2+1/2)^{2}}$ (15)

A very nice application of the Wigner surmise is the prediction of the zeros
distribution of the Riemann zeta function Rzeros .

  * •

In considering the joint probability distribution of consecutive spacings,
there is a technical issue related to the unfolding procedure of the data,
that in general cases can be difficult to implement, as explained in rIndex .
In order to avoid the unfolding procedure one can introduce a more robust
index rIndex : the ratio of consecutive level spacing

$r_{j}={\delta_{j+1}\over\delta_{j}}$ (16)

where the joint probability distribution was intensively studied in rIndexAna3
rIndexAna4 , and for $3\times 3$ block diagonal matrices takes the following
form

$P_{\beta}(r)={3^{3(1+\beta)/2}\Gamma(1+\beta/2)^{2}\over
2\pi\Gamma(1+\beta)}\,{(r+r^{2})^{\beta}\over(1+r+r^{2})^{1+3\beta/2}}$ (17)

which is valid for any value of $\beta>0$ rIndexAnaExt  and also for large
asymptotic values of $\beta$ rIndexAnaAsym . For $\beta=1,2,4$ one has the
standard GOE, GUE and GSE respectively, which are the gaussian
orthogonal/unitary/symplectic ensembles.

#### Chaotic analysis: results

Following the discussion related to figure 1 one can expect that the
microstate with the maximal number of harmonics will produce a less smooth
decay amplitude. In particular one can measure the chaotic behavior of the
decay profile computing the distribution of the index (16) for the zeros of
(12), which is the study of how the unbiased $r$-index indicator(16) for the
extrema of the amplitude is distributed in agreement with the target chaotic
class of $\beta$-distributions (17). From the results in figure 4, relative to
microstates of $N=100$, one can observe the profile of the logarithmic
derivative $F_{\\{g_{n}\\}}(\alpha)$ and the respective joint probability
distribution of the microstate with the maximal number of harmonics (which are
the first two plots respectively). The five small plots represent how the
measured joint probability distributions deviate from the target distribution
(17), respect the variation of the number of harmonics. Quite similar to the
case of classical string profiles where the number of harmonics triggers the
complexity of shape of the profiles, the chaotic behavior of the decay profile
is triggered by the number of harmonics. An additional check of this kind of
harmonic hierarchy is provided in figure 5, where we plot the logarithmic
derivative of the decay amplitude and the joint probability distribution of
the single harmonic microstate of $N=100$. In particular one can observe that
the distribution totally deviates from the chaotic jGUE, which is a hint about
the connection between classical single harmonic string profiles (straight
lines) and the absence of chaos in the decay amplitude of single harmonic
microstates.

Figure 4: Microstates of $N=100$. From left to right the profile of the
logarithmic derivative of the decay amplitude and the joint probability
distribution both relative to the microstate with maximal number of harmonics.
The dashed line is the joint probability distribution with $\beta=2$, which
represents the joint GUE distribution (jGUE). Below there are five examples of
joint probability distributions for microstates with less number of harmonics.
Figure 5: The logarithmic derivative of the decay amplitude and the joint
probability distribution both relative to the microstate of $N=100$ with only
one harmonic. The dashed line is the jGUE. Figure 6: Comparison between
classical string profiles and decay amplitudes

##  2 Thermalization emergence in highly excited string decays

In the previous section we presented a systematic study of how the microstate
structure emerges through the profile of its decay process. In particular we
described how the chaotic behavior of the decay process is triggered by the
microstate structure. The aim of this section is to study how the chaotic
behavior is related to the thermalization process of the most general HES
decay (fig.7).

Figure 7: Picture of the decay amplitude where a representative microstate
$\Pi_{\\{g_{n}\\}}^{(N)}$ decays into
$\Pi_{\\{g^{\prime}_{n^{\prime}}\\}}^{(N^{\prime})}$ emitting a tachyon $T$.

Following Firrotta:2022cku  and appendix (A) the decay rate of the present
process, in a d-dimensional phase space, is given by the following expression

${\Gamma}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\Omega_{s}\,E_{k}^{d-3}\over
16(N{-}1)(2\pi)^{d-2}}{\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}}$
(18)

This is the decay rate of the single microstate at level $N$ decaying into a
single microstate at level $N^{\prime}$ through the emission of a tachyon of
energy $E_{k}$, where the solid angle
$\Omega_{s}=2\pi^{(d{-}1)2}/\Gamma((d{-}1)/2)$ is introduced.

The non trivial dependence of the decay rate is due to the square of the
absolute value of the amplitude, which is the main quantity that will be
analyzed. In particular in the region where the ratio between the energy of
the emitted state and the mass of the decaying state is enough small, in such
a way that the energy loss of the decaying state is smooth, the absolute value
square of the amplitude assumes the following form

${\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}}=\widetilde{{\cal
N}}_{\\{g_{n}\\}}^{2}\widetilde{{\cal
N}^{\prime}}_{\\{g^{\prime}_{n^{\prime}}\\}}^{2}\,e^{-{\cal
C}_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\}){E_{k}\over
T_{H}}-2\mu_{N}\left(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\};E_{k}/T_{H}\right)}$
(19)

where there is a weight factor ${\cal C}_{N}$ that depends on the level $N$ of
the decaying state, and also depends on the microstates geometry through the
relation

${\cal
C}_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})={2\over\sqrt{N}}\left(\sum_{n=1}^{N}g_{n}n\log{n}-\sum_{n^{\prime}=1}^{N^{\prime}}g^{\prime}_{n^{\prime}}n^{\prime}\log{n^{\prime}}\right)$
(20)

the other function in the exponent is given by

$\mu_{N}\left(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\};{E_{k}\over
T_{H}}\right)=\sum_{n=1}^{N}g_{n}\log\Gamma\left(1{-}{n\over\sqrt{N}}{E_{k}\over
T_{H}}\right)+\sum_{n^{\prime}=1}^{N^{\prime}}g^{\prime}_{n^{\prime}}\log\Gamma\left(1{+}{n^{\prime}\over\sqrt{N}}{E_{k}\over
T_{H}}\right).$ (21)

and finally the dimensional temperature $T_{H}=1/\ell_{s}$ is the Hagedorn
temperature which is the inverse of the string length
$\ell_{s}=\sqrt{\alpha^{\prime}}$.

The thermal nature of the characteristic expression (19) is intimately related
to the chaotic behavior of the decay, in fact the non trivial dependence on
the microstates is directly related to chaos (section 1) and it will play also
a crucial role in the thermalization of the decay, as will be explained in the
next part 2.1 of the present section.

When is considered the decay rate of a state of the level $N$ made of many
microstates (see fig.8)

$|BH\rangle_{N}=\sum_{\\{g_{n}\\}}\Pi^{(N)}_{\\{g_{n}\\}}$ (22)

which can be interpreted as a black hole state, the final decay rate results
to be the sum over all the possible microstate configurations of the decay
rates

${\Gamma}\Big{(}{|BH\rangle_{N}\Rightarrow|BH\rangle_{N^{\prime}}+E_{k}}\Big{)}={\Omega_{s}\,E_{k}^{d-3}\over
16(N{-}1)(2\pi)^{d-2}}\sum_{\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\}}{\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}}.$
(23)

This is a very rich observable that is characterized by the highly non trivial
functional dependence of the microstates. In the last part 2.2 of the present
section, we will describe the behavior of such observable and the derivation
of the non trivial energy spectrum leading to the greybody radiation of highly
excited string states.

Figure 8: Qualitative picture of the decay rate
$|BH\rangle_{N}\Rightarrow|BH\rangle_{N^{\prime}}+E_{k}$. The microstates of
the decaying state are represented by colorful small strings, each color is
associated to the possible microstate decay. The red region represents the
random superposition of all the possible decays which is the mechanism through
which the thermalization process emerge.

###  2.1 Chaos driven thermalization

In section 1 we analyzed the chaotic behavior of a string microstate decaying
into two tachyons, where we described the connection between the microstate
structure and the chaotic behavior of the decay. The sensitivity of the decay
to the microstate structure was essentially encoded in the dressing factor
(9). Now considering the direct extension of the decay in (fig.3), which is
the one of (fig.7), one has the generalization of (9) to the case of two
microstate structures: one for the decaying microstate $\Pi^{N}_{\\{g_{n}\\}}$
and the other for the final microstate
$\Pi^{N^{\prime}}_{\\{g^{\prime}_{n^{\prime}}\\}}$. Using the general setup
described in appendix A one finds a remarkably compact expression of the decay
rate777In the expression we used $\alpha^{\prime}=1/2$ that means
$T_{H}=\sqrt{2}$.

${\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}}\simeq\prod_{n=1}^{N}\left({\left(1{-}n(1{-}E_{k}/\sqrt{2N})\right)_{n{-}1}\over\Gamma(n)}\right)^{2g_{n}}\prod_{n^{\prime}=1}^{N^{\prime}}\left({\left(1{-}n^{\prime}(1{+}E_{k}/\sqrt{2N})\right)_{n^{\prime}{-}1}\over\Gamma(n^{\prime})}\right)^{2g^{\prime}_{n^{\prime}}}$
(24)

comparing this expression with (9) one can see the systematic generalization
of the decay rate due to the presence of an additional microstate. Following
the expansion (83) and the rest of the appendix, one recovers (19). This is
the systematics of how the chaotic factors make the decay rate thermal, giving
a precise identification between chaos and the emergence of thermalization.

Following the same logic as in (fig.6), where a comparison between classical
strings and decay amplitudes was argued, one can complete the scenario
describing the mechanism that originates the thermalization of the precess.

In fact one can analyze two extreme cases: the first one is to consider
microstates of the first Regge trajectory. Classically their profile is just a
straight line (fig.6), because of the fact that they are single harmonic
states, so they do not produce chaos. The associated thermalization is given
by the expression (24) with $n=1,\,g_{1}=N$ and
$n^{\prime}=1,\,g^{\prime}_{1}=N^{\prime}$, which trivializes to unity, so the
decay is not thermal.

The second extreme case, which is less trivial, is to consider still single
harmonic states but with the maximal harmonic excited, $n=N,\,g_{N}=1$ and
$n^{\prime}=N^{\prime},\,g^{\prime}_{N^{\prime}}=1$. Classically they are a
straight lines and do not produce chaos (fig.5). From (24) and (19) one can
finds

${\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{N}=1\\}},\Pi^{N^{\prime}}_{\\{g_{N^{\prime}}=1\\}}}\Big{|}^{2}}\simeq
e^{-2\log\left({N\over
N^{\prime}}\right)\sqrt{N}E_{k}/T_{H}}\left({\sin\left({\pi\sqrt{N}E_{k}/T_{H}}\right)\over\pi\sqrt{N}E_{k}/T_{H}}\right)^{2}$
(25)

this behavior, at large $N$, is very suppressed even if $N^{\prime}\sim N$,
and it reflects how the thermal behavior of the decay is subdominant in the
case of states with only the maximal harmonic excited.

Beyond the two extremal cases discussed, one can have a more qualitative
picture of the microstates functional dependence of the decay looking at the
explicit distributions in (fig.9) and (fig.10).

The fluctuations of the decay reflect the connection between the geometry of
microstates, which is clear in the classical picture of string profiles, and
the associated chaos which is the trigger of the thermal behavior.

Figure 9: Microstates population of the decay rate in logarithmic scale for
100 random partitions $\\{g_{n}\\}$ of $N=100$ and 100 random partitions
$\\{g^{\prime}_{n^{\prime}}\\}$ of $N^{\prime}=99$. The red points are the
values of the decay rate. Figure 10: Microstates population of the decay rate
in logarithmic scale for 500 random partitions $\\{g_{n}\\}$ of $N=100$ and
500 random partitions $\\{g^{\prime}_{n^{\prime}}\\}$ of $N^{\prime}=99$. The
red points are the values of the decay rate.

###  2.2 Thermal spectrum: the greybody emission of highly excited strings

In the previous part we described the connection between chaos and thermal
behavior of the decay rate, where the intrinsic microscopical structure of
microstates was fundamental for the origin of such connection. Here the goal
will be the analysis of the energy spectrum radiated from HES states. The
first thing to note is that the decay rate (19) is not a function of the
emitted energy $E_{k}$, the kinematics of the two body decay fixes the energy
$E_{k}$ as a function of the masses of the present states

$E_{k}={M_{N}^{2}-M_{N^{\prime}}^{2}+M_{T}^{2}\over
2M_{N}}={N-N^{\prime}-1\over\sqrt{2N-2}}$ (26)

In order to extract the energy spectrum of an HES state, one has to explore
the energy range of the decay rate point by point, producing a discrete energy
trajectory of the decay process. Since the target observable is the radiation
of the HES state at generic level $N$, one can reproduce the energy trajectory
varying the mass $M_{N^{\prime}}=2N^{\prime}-2$ of the final state. The energy
region of interest is identified by the range in which the ratio between the
energy of the emitted state $E_{k}$ and the mass of the decaying state
$M_{N}=2N-2$ is enough small in such a way that the energy loss of the
decaying state is smooth, which is the situation where a thermal spectrum is
expected. Given $N$ and $N^{\prime}$, the decay rate of a state made of
microstates is given in (23), and even if the energy $E_{k}$ is fixed there is
a non trivial microstates distribution that mediates the decay rate (fig.10).
Therefore in order to reproduce the energy spectrum of a given HES state with
${\small\big{\\{}\Pi^{(N)}_{\\{g_{n}\\}}\big{\\}}}$ microstates, one can
generate an energy trajectory of random decays for each microstate
${\small\Pi^{(N)}_{\\{g_{n}\\}}}$ of the level N that can decay into a random
microstate ${\small\Pi^{(N^{\prime})}_{\\{g^{\prime}_{n^{\prime}}\\}}}$ of the
whole set
${\small\big{\\{}\Pi^{(N^{\prime})}_{\\{g^{\prime}_{n^{\prime}}\\}}\big{\\}}}$
of the level $N^{\prime}$. Finally the spectrum is obtained averaging over the
energy trajectories (fig.11). In general each microstate of the level $N$ can
decay into each microstate of the level $N^{\prime}$, and this is the reason
why one has to mimic the randomness of the decay process in order to describe
the intrinsic nature of a degenerate sized object such as the HES.

Figure 11: Picture of the random microstates energy trajectories ${\cal
P}(\\{E_{k}\\})$.

The general behavior of the energy spectrum turns out to be described by the
greybody radiation

$\begin{split}\Sigma^{\small{\text{grey}}}_{N}(E_{k}/T_{eff})&=\langle{\Gamma_{N\Rightarrow
N^{\prime}}}\rangle_{{\cal
P}(\\{E_{k}\\})}={\sigma^{\text{grey}}_{N}(E_{k}/T_{eff})\over e^{{E_{k}\over
T_{eff}}}-1}\end{split}$ (27)

with an effective temperature which depends on the decaying state through

$T_{eff}=T_{H}/\sqrt{N}$ (28)

and with the greybody factor which is sensitive to the nature of the
microstates of $N$ and $N^{\prime}$ and also depends on the randomness of the
decay process

$\sigma^{\text{grey}}_{N}(E_{k}/T_{eff})={C_{N}(\\{g_{n}\\}.\\{g^{\prime}_{n^{\prime}}\\})\Omega_{s}E_{k}^{d-3}\over
16(N{-}1)(2\pi)^{d-2}}{\Big{(}e^{{E_{k}\over
T_{eff}}}-1\Big{)}\left({E_{k}\over
T_{eff}}\right)^{r_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})}\over
e^{\nu_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\}){E_{k}\over T_{eff}}}-1}$
(29)

The randomness of the decay process is incorporated in the coefficients

$C_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})\,,\quad
r_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})\,,\quad\nu_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})$
(30)

that also reflect the intrinsic dependence on the chosen microstates that
determine the decaying state and the final state.

In the next section we will perform the spectrum analysis of different states,
in particular a state with definite mass and occupation number $(N,J)$
decaying into a generic state of the level $N^{\prime}$, a state $(N,J)$
decaying into a state $(N^{\prime},J^{\prime})$ and finally a generic state
$N$ decaying into a state $N^{\prime}$. For simplicity it will be considered
the greybody spectrum (27) without the phase space factor, which is equal for
each case, therefore without loss of generality one can redefine the final
observable as

$\gamma^{\text{grey}}_{N}(E_{k}/T_{eff})=C_{N}(\\{g_{n}\\}.\\{g^{\prime}_{n^{\prime}}\\}){\left({E_{k}\over
T_{eff}}\right)^{r_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\})}\over
e^{\nu_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\}){E_{k}\over T_{eff}}}-1}$
(31)

##  3 Results of random generated spectra

Let’s start by considering a simple example in which only a single microstate
can decay, in particular let’s consider the specific microstate
${\small\Pi^{(100)}_{\pi_{1}(g_{n})}}$ of the level $N=100$ given by

$\pi_{1}(g_{n})=\\{g_{1}=1,g_{2}=1,g_{5}=2,g_{7}=1,g_{8}=1,g_{9}=1,g_{10}=1,g_{25}=1,g_{28}=1\\}$
(32)

The first energy point of the spectrum is given by the decay of
$\Pi^{(100)}_{\pi_{1}(g_{n})}$ into a generic microstate of the level
$N^{\prime}=N-1$ and corresponds to $E_{k}=0$. The fact that $E_{k}=0$ is
constrained by the kinematics of the emitted tachyonic particle which make the
first point of the spectrum to be trivial.

The first non trivial energy point of the spectrum is given by the decay of
$\Pi^{(100)}_{\pi_{1}(g_{n})}$ into a generic microstate of the level
$N^{\prime}{=}N{-}2$, and the consecutive points are obtained from
$N^{\prime}{=}N{-}3$, $N^{\prime}{=}N{-}4$ and so on. Since the HES of the
level $N^{\prime}$ is generically composed by many degenerate microstates, one
can introduce the intrinsic random nature of the process taking the average
over different sets of random microstates for each energy point of the
spectrum.

Figure 12: Thermal spectrum of the representative microstate
$\Pi^{(100)}_{\pi_{1}(g_{n})}$ computed for two different random sample of
microstates of $N^{\prime}$.

In order to make clear the relation between chaos and thermalization one can
adopt the same logic of (fig.4), in particular one can compare the spectrum of
the microstate with the maximal number of harmonics and the extreme case of
the first Regge trajectory (fig.13).

Figure 13: Spectra of microstates of $N=100$. Comparison between the spectrum
of the first Regge trajectory (blue points) and the spectrum of the state with
the maximal number of harmonics (red points). The black dashed line represents
the fitted behavior of the red spectrum.

As expected the spectrum of a state of the FRtj is not thermal.

A complementary picture of the thermal spectrum of string microstates is given
in fig.14, where many spectra are presented, classified by different
microstates and values of the occupation number $J$. The specific parameters
of (31) are reported in Tab.1.

Figure 14: Some examples of thermal spectra as a function of the occupation number $J$. $J$ | Microstate | $C_{N}$ | $r_{N}$ | $\nu_{N}$
---|---|---|---|---
| $\\{g_{2}{=}3,\,g_{12}{=}2,\,g_{3}{=}g_{10}{=}g_{13}{=}g_{20}{=}g_{24}{=}1\\}$ | $1.49\times 10^{-9}$ | 4.38 | 1.64
10 | $\\{g_{2}{=}g_{3}{=}g_{4}{=}g_{5}{=}g_{6}{=}g_{13}{=}g_{15}{=}g_{16}{=}g_{17}{=}g_{19}{=}1\\}$ | $4.90\times 10^{-9}$ | 4.70 | 1.60
| $\\{g_{13}{=}2,\,g_{1}{=}g_{2}{=}g_{4}{=}g_{6}{=}g_{7}{=}g_{12}{=}g_{19}{=}g_{23}{=}1\\}$ | $6.23\times 10^{-9}$ | 3.53 | 1.33
| $\\{g_{2}{=}6,\,g_{3}{=}3,\,g_{5}{=}g_{9}{=}2,\,g_{1}{=}g_{4}{=}g_{6}{=}g_{7}{=}g_{8}{=}g_{12}{=}g_{13}{=}1\\}$ | $1.27\times 10^{-14}$ | 9.84 | 1.74
20 | $\\{g_{2}{=}8,\,g_{7}{=}2,\,g_{1}{=}4,\,g_{3}{=}g_{4}{=}g_{11}{=}g_{12}{=}g_{16}{=}g_{20}{=}1\\}$ | $2.08\times 10^{-14}$ | 11.30 | 2.64
| $\\{g_{1}{=}g_{2}{=}3,\,g_{3}{=}4,\,g_{4}{=}3,\,g_{8}{=}2,\,g_{5}{=}g_{6}{=}g_{7}{=}g_{12}{=}g_{21}\\}$ | $28.9\times 10^{-14}$ | 9.52 | 2.01
| $\\{g_{3}{=}g_{4}{=}g_{6}{=}3,\,g_{1}{=}8,\,g_{2}{=}6,\,g_{5}{=}5,g_{7}{=}g_{9}{=}1\\}$ | $1.00\times 10^{-23}$ | 17.02 | 1.97
30 | $\\{g_{1}{=}8,\,g_{2}{=}11,\,g_{3}{=}g_{6}{=}3,\,g_{8}{=}2,\,g_{4}{=}g_{5}{=}g_{18}{=}1\\}$ | $7.51\times 10^{-24}$ | 17.18 | 2.56
| $\\{g_{1}{=}11,\,g_{2}{=}7,\,g_{3}{=}3,\,g_{4}{=}g_{7}{=}2,\,g_{5}{=}g_{6}{=}g_{8}{=}g_{11}{=}g_{14}{=}1\\}$ | $9.74\times 10^{-22}$ | 15.63 | 2.18
| $\\{g_{1}{=}15,\,g_{2}{=}9,\,g_{3}{=}7,\,g_{4}{=}4,g_{5}{=}3,\,g_{6}{=}g_{9}{=}1\\}$ | $7.44\times 10^{-33}$ | 18.35 | 1.74
40 | $\\{g_{1}{=}7,\,g_{2}{=}10,\,g_{3}{=}g_{4}{=}4,\,g_{6}{=}3,\,g_{5}{=}g_{12}{=}1\\}$ | $6.22\times 10^{-32}$ | 16.75 | 1.78
| $\\{g_{1}{=}17,\,g_{2}{=}9,\,g_{3}{=}7,\,g_{6}{=}3,g_{7}{=}2,\,g_{4}{=}g_{8}{=}1\\}$ | $6.27\times 10^{-32}$ | 16.75 | 1.78
Table 1: Table of specific microstates and parameters of (31) relative to the
spectra of fig.14.

##  4 Conclusion and future directions

In section 1 we have studied the interplay between classical string
configurations and quantum string configurations, where the latter are
essentially the microstates of the mass degeneracy of the HES. We have
observed how the chaotic nature of HES interactions has a common pattern with
the shape of classical string profiles, which is provided by the number of
harmonics that characterizes the microstate. We have presented explicit
results for a representative HES at mass level $N=100$, but the same
systematics holds for different mass levels. From the point of view of the
analysis of the chaotic information present in HES interactions, one can
quantitatively improve the measure of chaos with additional parametrization of
the information content based on modern techniques of quantum information
theory Witten:2018zva -Erbin:2022rgx

In section 2 we have analyzed how the thermal nature of the decay amplitude is
intimately related to its chaotic nature, in particular we have observed that
the chaotic analytical structure, which appear as a non trivial dressing
factor, originates a Boltzmann factor which encodes the thermal information of
the decay process. Exploiting the exact analytical result of the decay process
of a generic HES that decays into a less excited, yet generic HES, through the
emission of scalar particle, such as the tachyon, we have computed the energy
spectrum of HES. Starting from a definite microstate of the level $N$, we
computed the average over random microstates of the level $N^{\prime}$, for
many different values of $N^{\prime}$. In particular in section 3 we have
presented numerical results of the extracted spectra for many different
microstates of the representative level $N=100$, but the same holds for higher
levels. Lower levels, $N<100$, have less accessible energy points of interest.
We observed a non trivial dependence of the spectra on the occupation number
$J$ that we want to quantitatively address in future works, together with a
fully quantitative computation of the spectrum of degenerate HES, which means
the computation of the linear combination of the same spectra we computed, but
for all the possible microstate of the level $N$.

The chaotic nature of the decay process together with the non trivial
dependence of final microstates gave rise to a greybody emission spectrum with
an effective temperature $T_{eff}=T_{H}/\sqrt{N}$, which is different from the
temperature of the string/BH transition Horowitz:1996nw . In fact we recovered
the characteristic behavior of the Hawking temperature which is expected to be
proportional to the inverse of the mass of the decaying state. A possible
interpretation of such result can be connected to an enhancement of the
effective string Schwarzschild radius, due to the random walk nature of HES
interactions modeled by the explicit introduction of the microstates
dependence. The chaotic nature of HES interactions Gross:2021gsj -ChaosScatt ,
and also the associated thermal nature suggest a non trivial spatial
distribution of HES, which can be probed in a scattering experiment similar to
the analysis in Mitchell:1990cu . It is well known that an HES can be
described as random walk of interactions Kruczenski:2005pj -Damour:1999aw ,
and than one can measure the precise microstates structure of the spatial
distribution of HES studying the effective horizon, for example, probed in HES
Compton-like scattering processes. Implementing random surface techniques
Tan:1980zh -Charles:2018oob  to the HES form factors one can obtain a
complementary picture of the HES nature where chaotic and thermal effects can
be matched with the effective HES horizon governed by the superposition of
microstates. We leave this investigation for future works.

Alternatively quite recently it was proposed a technique to resolve the
spatial distribution of strings Hashimoto:2022ugt  and also a connected
chaotic analysis of the HES Compton scattering Hashimoto:2022bll , based on
the principle of transient chaos.

The understanding of the intrinsic structure of HES along with a complete
picture of HES interactions can be very useful in studying deep microscopical
connections between thermalization and chaos. The statistical non trivial
nature of HES provides a very rich physical system which still deserves to be
further explored.

## Acknowledgements

I would like to thank M. Bianchi, G. Rossi, J. Sonnenschein, D. Weissman, V.
Rosenhaus, D. Gross, B. Sundborg, A. Tseytlin, G. Di Russo, A. Guerrieri and
V. Niarchos for valuable discussions and comments. I would like also to thank
The Graduate Center, CUNY for the hospitality during the completion of the
manuscript.

##  Appendix A Highly excited string decay: $H_{N}\Rightarrow
H_{N^{\prime}}+T$

The present appendix concerns a detailed review and new insights about the
decay process computation based on Firrotta:2022cku . In particular we have
presented the analytical setup with the main steps of the computation and also
it will be discussed the behavior of the decay rate in the thermalization
region, which is reached when the ratio between the energy of the emitted
state and the mass of the decaying state is enough small in such a way that
the energy loss of the decaying state is smooth.

Figure 15: Picture of the decay amplitude where a representative microstate
$\Pi_{\\{g_{n}\\}}^{(N)}$ decays into
$\Pi_{\\{g^{\prime}_{n^{\prime}}\\}}^{(N^{\prime})}$ emitting a tachyon $T$.
The DDF structure of the microstates is also depicted below the decay process.

Following the picture in (fig.15) the kinematics of the process is
parametrized as follows

$p=\sqrt{2N-2}(1,\vec{0})\,,\quad
p^{\prime}=-(E^{\prime},\omega\sin\theta,\omega\cos\theta,\vec{0})\,,\quad
k=(-E_{k},\omega\sin\theta,\omega\cos\theta,\vec{0})$ (33)
$q=-{(1,0,1,\vec{0})\over\sqrt{2N-2}}\,,\quad\lambda={(0,1,0,\vec{\Lambda})\over\sqrt{1+|\vec{\Lambda}|^{2}}}\,;\quad
q^{\prime}=-{(1,0,1,\vec{0})\over\omega\cos\theta-E^{\prime}}\,,\quad\lambda^{\prime}={(0,1,0,\vec{\Lambda}^{\prime})\over\sqrt{1+|\vec{\Lambda}^{\prime}|^{2}}}$
(34)

where the momenta $\tilde{p}$ and $\tilde{p}^{\prime}$ of (fig.15) are
tachyonic DDF reference momenta, while $p=\tilde{p}-\sum_{n}ng_{n}\,q$ and
$p^{\prime}=\tilde{p}^{\prime}-\sum_{n^{\prime}}n^{\prime}g^{\prime}_{n^{\prime}}\,q^{\prime}$
are the total momenta of the microstates $\Pi_{\\{g_{n}\\}}^{(N)}$ and
$\Pi_{\\{g^{\prime}_{n^{\prime}}\\}}^{(N^{\prime})}$. The nice feature of the
DDF formalism relies in the direct identification of the final BRST vertex
operator corresponding to the microstate, where its structure is modeled by
the number of DDF photon insertions with polarizations $\lambda_{n}$,
$\lambda^{\prime}_{n}$ and momenta $q$ and $q^{\prime}$. The photon insertions
are in exact correspondence with each harmonic and every excitation number,
exactly reproducing the action of creation operators.

###  A.1 Decay amplitude

Let’s start by introducing the generating amplitude of all the possible decay
processes of $H_{N}\Rightarrow H_{N^{\prime}}+T$ discriminated by all the
possible microstate $\Pi^{(N)}_{\\{g_{n}\\}}$ and
$\Pi^{(N^{\prime})}_{\\{g_{n^{\prime}}\\}}$ originated from the partition of
integers $N=\sum_{n}ng_{n}$ and
$N^{\prime}=\sum_{n^{\prime}}n^{\prime}g_{n^{\prime}}$

$\begin{split}{\cal
A}_{gen}=\exp\Bigg{(}&\sum_{n;\,a_{n}=1}^{g_{n}}J_{n}^{(a_{n})}{\cdot}V_{n}+{J}_{n}^{{}^{\prime}(a_{n})}{\cdot}V^{\prime}_{n}+\\\
&\sum_{n,m;\,a_{n},b_{m}}^{g_{n},g_{m}}J^{(a_{n})}_{n}{\cdot}J^{(b_{m})}_{m}W_{m,n}+J^{\prime(a_{n})}_{n}{\cdot}J^{\prime(b_{m})}_{m}W^{\prime}_{m,n}+J^{(a_{n})}_{n}{\cdot}J^{\prime(b_{m})}_{m}M_{m,n}\Bigg{)}\end{split}$
(35)

where all the interaction terms are classified as follows

${\cal
V}_{n}^{\mu}={p^{\prime}}^{\mu}V_{n}={p^{\prime}}^{\mu}{(-)^{n+1}\over\Gamma(n)}{(1+nq{\cdot}p^{\prime})_{n{-}1}}\,,\quad{{\cal
V}^{\prime}}^{\mu}=p^{\mu}V^{\prime}_{n}={p^{\mu}\over\Gamma(n)}(1+nq^{\prime}{\cdot}p)_{n{-}1}$
(36) $W_{n,m}={n\,m\over
n+m}{(1+q{\cdot}p^{\prime})\,q{\cdot}p^{\prime}}\,V_{n}\,V_{m}\,,\quad
W^{\prime}_{n,m}={n\,m\over
n+m}{(1+q^{\prime}{\cdot}p)\,q^{\prime}{\cdot}p}\,V^{\prime}_{n}\,V^{\prime}_{m}$
(37) $M_{n,m}=-{nm(1+q{\cdot}p^{\prime})\over
m+nq{\cdot}p^{\prime}}V_{n}V^{\prime}_{m}$ (38)

Any particular decay amplitude can be obtained by operating with derivative
combinations representing the projection on the single amplitude with the
desired states identified by the partition set $\\{g_{n}\\}$ and
$\\{g_{n^{\prime}}\\}$ as

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}=\prod_{n}\prod_{a_{n}=1}^{g_{n}}\zeta_{n}^{(a_{n})}{\cdot}{d\over
dJ_{n}^{(a_{n})}}\,\prod_{n^{\prime}}\prod_{a_{n^{\prime}}=1}^{g_{n^{\prime}}}{\zeta^{\prime}}_{n^{\prime}}^{(a_{n^{\prime}})}{\cdot}{d\over
d{J^{\prime}}_{n^{\prime}}^{(a_{n^{\prime}})}}\,{\cal
A}_{gen}\Bigg{|}_{J=J^{\prime}=0}$ (39)

where all the polarizations
$\zeta_{n\,,\mu}^{(a_{n})}=\lambda^{(a_{n})}_{n\,,\mu}-\lambda^{(a_{n})}_{n}{\cdot}pq_{\mu}$
and
${\zeta^{\prime}}_{n^{\prime}\,,\mu}^{(a_{n^{\prime}})}={\lambda^{\prime}}_{n^{\prime}\,,\mu}^{(a_{n^{\prime}})}-{\lambda^{\prime}}_{n^{\prime}}^{(a_{n^{\prime}})}{\cdot}p^{\prime}q^{\prime}_{\mu}$
are independent.

By considering a kinematical setup where the decaying string is at rest, one
can compute the relevant scalar product $q{\cdot}p^{\prime}$ that encodes the
partition dependence of the interacting states. In particular

$E^{\prime}=M_{N}-E_{k}\,,\quad\omega=\sqrt{E_{k}^{2}-M_{T}^{2}}\,,\quad
E_{k}={N-N^{\prime}-1\over\sqrt{2N-2}}$ (40)

and using the kinematics one finds

$q{\cdot}p^{\prime}={1\over
q^{\prime}{\cdot}p}=-{E^{\prime}-\omega\cos\theta\over M_{N}}=-1+{E_{k}\over
M_{N}}+{\sqrt{2}\over M_{N}}\sqrt{1+{E_{k}^{2}\over 2}}\cos\theta$ (41)

Without loss of generality one can chose $\theta=\pi/2$, in fact the final
observable will be the decay rate, so when the modulus square of the amplitude
is considered, the exact spherical symmetry is restored. It means that one can
freely fix the value of $\theta$ without loss of generality. With this choice
one has

$q{\cdot}p^{\prime}=-1+{E_{k}\over M_{N}}$ (42)

In this framework one can easily analyze the non trivial contributions in
(36), (37) and (38):

$V_{n}=(-)^{n+1}{(1+nq{\cdot}p^{\prime})_{n{-}1}\over\Gamma(n)}={(-)^{n+1}\Gamma\left(n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1-n(1-{E_{k}\over M_{N}})\right)}$ (43)

this is the oscillating function that generates chaos, in fact it can be
written as

$V_{n}={1\over\pi\,\Gamma(n)}{\Gamma\left(n{E_{k}\over
M_{N}}\right)\Gamma\left(n-n{E_{k}\over
M_{N}}\right)}\sin\left(n\pi{E_{k}\over M_{N}}\right)$ (44)

The other term to analyze is

$W_{n,m}={n\,m\over
n+m}{(1+q{\cdot}p^{\prime})\,q{\cdot}p^{\prime}}\,V_{n}\,V_{m}={n\,m\over
n+m}{E_{k}\over M_{N}}\left({E_{k}\over M_{N}}-1\right)\,V_{n}\,V_{m}$ (45)

and also there is

$V^{\prime}_{n}={(1+nq^{\prime}{\cdot}p)_{n{-}1}\over\Gamma(n)}={\Gamma\left(-n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1-n(1+{E_{k}\over M_{N}})\right)}$ (46)

where both numerator and denominator are oscillating

$V^{\prime}_{n}={\Gamma\left(n+n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1+n{E_{k}\over
M_{N}}\right)}{\sin\left(n\pi+n\pi{E_{k}\over
M_{N}}\right)\over\sin\left(n\pi{E_{k}\over
M_{N}}\right)}=(-)^{n}{\Gamma\left(n+n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1+n{E_{k}\over M_{N}}\right)}$ (47)

then there is

$W^{\prime}_{n,m}={n\,m\over
n+m}{(1+q^{\prime}{\cdot}p)\,q^{\prime}{\cdot}p}\,V^{\prime}_{n}\,V^{\prime}_{m}={n\,m\over
n+m}{E_{k}\over M_{N}}\left({E_{k}\over
M_{N}}+1\right)\,V^{\prime}_{n}\,V^{\prime}_{m}$ (48)

finally the mixed term

$M_{n,m}=-{nm(1+q{\cdot}p^{\prime})\over
m+nq{\cdot}p^{\prime}}V_{n}V^{\prime}_{m}=-{nm{E_{k}\over M_{N}}\over
m-n+n{E_{k}\over M_{N}}}V_{n}V^{\prime}_{m}$ (49)

To sum up one has the following structures

${\cal V}_{n}^{\mu}={p^{\prime}}^{\mu}V_{n}(E_{k})\,,\quad{{\cal
V}^{\prime}}_{n}^{\mu}=p^{\mu}V^{\prime}_{n}(E_{k})\,,\quad
M_{n,m}=\mu_{n,m}(E_{k})V_{n}(E_{k})V^{\prime}_{m}(E_{k})$ (50)
$W_{n,m}=w_{n,m}(E_{k})V_{n}(E_{k})V_{m}(E_{k})\,,\quad
W^{\prime}_{n,m}=w^{\prime}_{n,m}(E_{k})V^{\prime}_{n}(E_{k})V^{\prime}_{m}(E_{k})$
(51)

where

$\mu_{n,m}(E_{k})=-{nm{E_{k}\over M_{N}}\over m-n+n{E_{k}\over M_{N}}}\,,\quad
w_{n,m}(E_{k})={n\,m\over n+m}{E_{k}\over M_{N}}\left({E_{k}\over
M_{N}}-1\right)$ (52) $w^{\prime}_{n,m}(E_{k})={n\,m\over n+m}{E_{k}\over
M_{N}}\left({E_{k}\over M_{N}}+1\right)$ (53)

Finally the general decay amplitude can be written as

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\cal
P}_{\Pi_{N},\Pi_{N^{\prime}}}\left(\zeta_{n}^{(a)}\,,\zeta_{n^{\prime}}^{\,{}_{{}^{\prime}}(a^{\prime})}\,,p\,,p^{\prime}\,,w_{n,m}\,,w^{\prime}_{n^{\prime},m^{\prime}}\,,\mu_{n,n^{\prime}}\right)\,\prod_{n}\big{(}V_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{V_{n^{\prime}}^{\prime}}\big{)}^{g_{n^{\prime}}}$
(54)

where ${\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}$ is a polynomial that depends on
the partition of $N$ and $N^{\prime}$ and it can be obtained by the generating
function

$\begin{split}{\cal
P}_{gen}=\exp\Bigg{(}&\sum_{n;\,a_{n}=1}^{g_{n}}J_{n}^{(a_{n})}{\cdot}p^{\prime}+\sum_{n;\,a_{n}=1}^{g_{n}}{J}_{n}^{{}^{\prime}(a_{n})}{\cdot}p+\sum_{n,m;\,a_{n},b_{m}}^{g_{n},g_{m}}J^{(a_{n})}_{n}{\cdot}J^{\prime(b_{m})}_{m}\mu_{m,n}\\\
&\sum_{n,m;\,a_{n},b_{m}}^{g_{n},g_{m}}J^{(a_{n})}_{n}{\cdot}J^{(b_{m})}_{m}w_{m,n}+\sum_{n,m;\,a_{n},b_{m}}^{g_{n},g_{m}}J^{\prime(a_{n})}_{n}{\cdot}J^{\prime(b_{m})}_{m}w^{\prime}_{m,n}\Bigg{)}\end{split}$
(55)

the important thing to note is that the oscillating terms $V_{n}$ and
$V^{\prime}_{n}$ are factorized from the polynomial. They are the same factors
that produce chaos and that contain most of the information about the
microstate dependence.

###  A.2 Decay rate

The general structure of the absolute value square of the amplitude is given
by

$\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}=\Big{|}{\cal
P}_{\Pi_{N},\Pi_{N^{\prime}}}\left(\zeta_{n}\,,\zeta^{\prime}_{n^{\prime}}\,,p\,,p^{\prime}\,,w_{n,m}\,,w^{\prime}_{n^{\prime},m^{\prime}}\,,\mu_{n,n^{\prime}}\right)\Big{|}^{2}\,\prod_{n}\big{(}V^{2}_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{{V_{n^{\prime}}^{\prime}}^{2}}\big{)}^{g_{n^{\prime}}}$
(56)

where the polynomial is generated by

${\cal
P}_{\Pi_{N},\Pi_{N^{\prime}}}=\prod_{n}\prod_{a_{n}=1}^{g_{n}}\zeta_{n}^{(a_{n})}{\cdot}{d\over
dJ_{n}^{(a_{n})}}\,\prod_{n^{\prime}}\prod_{a_{n^{\prime}}=1}^{g_{n^{\prime}}}{\zeta^{\prime}}_{n^{\prime}}^{(a_{n^{\prime}})}{\cdot}{d\over
d{J^{\prime}}_{n^{\prime}}^{(a_{n^{\prime}})}}\,{\cal
P}_{gen}\Bigg{|}_{J=J^{\prime}=0}$ (57)

Taking the absolute value square and using the completeness relations

$\sum_{pol}\zeta_{n}^{(a)\mu}\zeta_{n}^{*(a)\mu}={\cal
L}^{(a)\mu\nu}\,,\quad\sum_{pol}{\zeta^{\prime}}_{n}^{(a)\mu}{\zeta^{\prime}}_{n}^{*(a)\mu}={{\cal
L}^{\prime}}^{(a)\mu\nu}$ (58)

where the superscript $(a)$ refers to different independent polarizations,
with the explicit completeness structures

${\cal
L}^{(a)\mu\nu}=\eta^{\mu\nu}-2p^{(\mu}q^{\mu)}+p^{2}q^{\mu}q^{\nu}\,,\quad{{\cal
L}^{\prime}}^{(a)\mu\nu}=\eta^{\mu\nu}-2p^{\prime(\mu}q^{\prime\mu)}+p^{\prime
2}q^{\prime\mu}q^{\prime\nu}$ (59)

one can study the relevant contributions of the polynomial, in particular the
first one yields

$p^{\prime}{\cdot}{\cal
L}{\cdot}p^{\prime}={p^{\prime}}^{2}-2p{\cdot}p^{\prime}\,q{\cdot}p^{\prime}+p^{2}(q{\cdot}p^{\prime})^{2}\,,\quad\text{with}\,\,\,q{\cdot}p^{\prime}=-1+{E_{k}\over
M_{N}}$ (60)

that can be written as

$p^{\prime}{\cdot}{\cal
L}{\cdot}p^{\prime}=(p+p^{\prime})^{2}-2p{\cdot}p^{\prime}{E_{k}\over
M_{N}}-2p^{2}{E_{k}\over M_{N}}+O(E_{k}^{2})$ (61)

using $p{\cdot}p^{\prime}=M_{N}E^{\prime}=M_{N}^{2}-M_{N}E_{k}$,
$p^{2}=-M_{N}^{2}$ and $(p+p^{\prime})^{2}=k^{2}$ one has

$p^{\prime}{\cdot}{\cal
L}{\cdot}p^{\prime}=k^{2}-2M_{N}E_{k}+2E_{k}^{2}+2M_{N}E_{k}=\omega^{2}+O(E_{k}^{2})\simeq-
M_{T}^{2}=2$ (62)

The second contribution is given by

$p{\cdot}{\cal
L}^{\prime}{\cdot}p=p^{2}-2p{\cdot}p^{\prime}q^{\prime}{\cdot}p+{p^{\prime}}^{2}(q^{\prime}{\cdot}p)^{2}\,,\quad\text{with}\,\,\,q^{\prime}{\cdot}p=-1-{E_{k}\over
M_{N}}$ (63)

which yields

$p{\cdot}{\cal
L}^{\prime}{\cdot}p=(p+p^{\prime})^{2}+2p{\cdot}p^{\prime}{E_{k}\over
M_{N}}+2{p^{\prime}}^{2}{E_{k}\over M_{N}}$ (64)

still using $p{\cdot}p^{\prime}=M_{N}E^{\prime}=M_{N}^{2}-M_{N}E_{k}$,
${p^{\prime}}^{2}=-{M}_{N^{\prime}}^{2}$ and $(p+p^{\prime})^{2}=k^{2}$ one
has

$p{\cdot}{\cal
L}^{\prime}{\cdot}p=k^{2}+2M_{N}E_{k}-2E_{k}^{2}-{M_{N^{\prime}}^{2}\over
M_{N}^{2}}2M_{N}E_{k}$ (65)

and finally noting that

${M_{N^{\prime}}^{2}\over M_{N}^{2}}=1-{E_{k}\over M_{N}}+O(1/M_{N}^{2})$ (66)

the expression becomes

$p{\cdot}{\cal L}^{\prime}{\cdot}p=k^{2}=2$ (67)

Using the same steps the last term can be written as

$p^{\prime}{\cdot}{\cal L}{\cdot}{\cal
L}^{\prime}{\cdot}p=M_{N}E^{\prime}+M_{N}^{2}q{\cdot}p^{\prime}+M_{N^{\prime}}^{2}q^{\prime}{\cdot}p=2(1+E_{k}/M_{N})+O(E_{k}^{2})$
(68)

This is a map from ${\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}$ and its absolute
value square, where the latter can be recasted in a new polynomial ${\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}$

$\Big{|}{\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}\Big{|}^{2}={\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}\left(p^{\prime}{\cdot}{\cal
L}^{(a)}{\cdot}p^{\prime},\,p{\cdot}{{\cal
L}^{\prime}}^{(a)}{\cdot}p,\,p^{\prime}{\cdot}{\cal L}^{(a)}{\cdot}{{\cal
L}^{\prime}}^{(b)}{\cdot}p,\,w_{n,m},\,w^{\prime}_{n^{\prime},m^{\prime}},\,\mu_{n,n^{\prime}}\right)$
(69)

Now let’s compare the amplitude with its mod square

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\cal
P}_{\Pi_{N},\Pi_{N^{\prime}}}\,\prod_{n}\big{(}V_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{V_{n^{\prime}}^{\prime}}\big{)}^{g_{n^{\prime}}}$
(70) $\Big{|}{\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}\Big{|}^{2}={\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}\,\prod_{n}\big{(}V^{2}_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{{V_{n^{\prime}}^{\prime}}^{2}}\big{)}^{g_{n^{\prime}}}$
(71)

it is clear that if one takes $n,n^{\prime}=1$ and
$g_{1}=N,\,g^{\prime}_{1}=N^{\prime}$ then $V_{1}=V^{\prime}_{1}=1$, and even
if the polynomial ${\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}$ is very complicated
there are no chaotic oscillations. This is a clear information of how the
polynomial ${\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}$ does not contain most of the
information about the microstate structures. When ${\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}$ is considered, there is still a complicated
structure, but there is a constant universal leading term plus many other
suppressed terms. The crucial point is that, if there is thermalization the
only way to generate the process is using the chaotic factors
$\\{V_{n}\\},\\{V^{\prime}_{n^{\prime}}\\}$. This is an evidence of how the
chaos is a trigger for the thermalization.

To see how the ${\cal X}_{\Pi_{N},\Pi_{N^{\prime}}}$ contributes, let’s start
by considering

$\Big{|}{\cal P}_{\Pi_{N},\Pi_{N^{\prime}}}\Big{|}^{2}={\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}\left(p^{\prime}{\cdot}{\cal
L}^{(a)}{\cdot}p^{\prime},\,p{\cdot}{{\cal
L}^{\prime}}^{(a)}{\cdot}p,\,p^{\prime}{\cdot}{\cal L}^{(a)}{\cdot}{{\cal
L}^{\prime}}^{(b)}{\cdot}p,\,w_{n,m},\,w^{\prime}_{n^{\prime},m^{\prime}},\,\mu_{n,n^{\prime}}\right)$
(72)

where

$\mu_{n,m}(E_{k})=-{nm{E_{k}\over M_{N}}\over m-n+n{E_{k}\over M_{N}}}\,,\quad
w_{n,m}(E_{k})={n\,m\over n+m}{E_{k}\over M_{N}}\left({E_{k}\over
M_{N}}-1\right)$ (73) $w^{\prime}_{n,m}(E_{k})={n\,m\over n+m}{E_{k}\over
M_{N}}\left({E_{k}\over M_{N}}+1\right)$ (74)

when $E_{k}/M_{N}$ is small, which is the region where the emitted radiation
is expected to be thermal, one has

$w^{\prime}_{n,m}=-w_{n,m}={n\,m\over n+m}{E_{k}\over
M_{N}}+O\left({E_{k}^{2}\over M_{N}^{2}}\right)$ (75)

and the other terms are given by (sopra), therefore if $E_{k}/M_{N}$ is small
one has

$p^{\prime}{\cdot}{\cal L}{\cdot}p^{\prime}=p{\cdot}{\cal
L}^{\prime}{\cdot}p=p^{\prime}{\cdot}{\cal L}{\cdot}{\cal
L}^{\prime}{\cdot}p\simeq 2=-2\alpha^{\prime}M_{T}^{2}$ (76)

and the leading term of the polynomial ${\cal X}$, included the microstates
normalization is

${\cal N}_{\\{g_{n}\\}}{\cal N}^{\prime}_{\\{g^{\prime}_{n^{\prime}}\\}}{\cal
X}_{\Pi_{N},\Pi_{N^{\prime}}}\simeq{(-2\alpha^{\prime}M_{T}^{2})^{J+J^{\prime}}\over\prod_{n}n^{g_{n}}g_{n}!\,\prod_{n^{\prime}}n^{\prime
g_{n^{\prime}}}g_{n^{\prime}}!}+O\left({E_{k}\over M_{N}}\right)$ (77)

so one can absorb the polynomial contribution into a new microstates
normalization

$\widetilde{{\cal N}}_{\\{g_{n}\\}}\widetilde{{\cal
N}}^{\prime}_{\\{g^{\prime}_{n^{\prime}}\\}}={1\over\prod_{n}(n/2)^{g_{n}}g_{n}!\,\prod_{n^{\prime}}(n^{\prime}/2)^{g_{n^{\prime}}}g_{n^{\prime}}!}$
(78)

Now focusing on the chaotic factors, $V_{n}$ and $V^{\prime}_{n^{\prime}}$,
when $E_{k}/M_{N}$ is small one can write

$V_{n}={\Gamma\left(n-n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1-n{E_{k}\over M_{N}}\right)}$ (79)

Using the Legendre duplication formula

$\Gamma(nx)={\prod_{k=0}^{n-1}\Gamma\left(x+{k\over
n}\right)\over(2\pi)^{n-1\over 2}n^{1\over 2}n^{-nx}}$ (80)

one has

$\Gamma\left(n\left(1-{E_{k}\over
M_{N}}\right)\right)={\prod_{k=0}^{n-1}\Gamma\left(1-{E_{k}\over
M_{N}}+{k\over n}\right)\over(2\pi)^{n-1\over 2}n^{{1\over
2}-n}}\,e^{-n{E_{k}\over M_{N}}\log n}$ (81)

since $E_{k}/M_{N}$ is very small one can approximate the function as

$\Gamma\left(n\left(1-{E_{k}\over
M_{N}}\right)\right)=\Gamma(n)\,e^{-n{E_{k}\over M_{N}}\log n}$ (82)

and the final result is given by

$V_{n}\simeq{\Gamma\left(n-n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1-n{E_{k}\over
M_{N}}\right)}\simeq{e^{-n{E_{k}\over M_{N}}\log
n}\over\Gamma\left(1-n{E_{k}\over M_{N}}\right)}$ (83)

In a similar fashion the other term can be written as

$V^{\prime}_{n}=(-)^{n}{\Gamma\left(n+n{E_{k}\over
M_{N}}\right)\over\Gamma(n)\Gamma\left(1+n{E_{k}\over
M_{N}}\right)}\simeq(-)^{n}\,{e^{n{E_{k}\over M_{N}}\log
n}\over\Gamma\left(1+n{E_{k}\over M_{N}}\right)}$ (84)

this is how the chaotic factors play their role in the thermalization of the
decay rate.

###  A.3 Thermal nature of the decay amplitude

Let’s consider the general form of the decay amplitude

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\cal
P}_{\Pi_{N},\Pi_{N^{\prime}}}\,\prod_{n}\big{(}V_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{V_{n^{\prime}}^{\prime}}\big{)}^{g_{n^{\prime}}}$
(85)

in the limit $E_{k}/M_{N}$ small, it was shown that the decay rate, which is
the physical observable, has a simple compact dependence on the polynomial
${\cal X}_{\Pi_{N},\Pi_{N^{\prime}}}$ that can be absorbed in the microstates
normalization. Such simplification can be translated in the structure of the
decay amplitude when only circular polarizations are considered, for example
taking

$\lambda_{+}=\lambda^{\prime}_{+}=(0,1,i,\vec{0})$ (86)

one finds

$\zeta_{n}{\cdot}p^{\prime}=\zeta^{\prime}_{n^{\prime}}{\cdot}p\simeq-{\omega}\,,\quad\zeta_{n}{\cdot}\zeta_{n}=\zeta^{\prime}_{n^{\prime}}{\cdot}\zeta^{\prime}_{n^{\prime}}=\zeta_{n}{\cdot}\zeta^{\prime}_{n^{\prime}}=0$
(87)

and the decay amplitude simplifies to

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\cal
N}_{\\{g_{n}\\}}{\cal
N}^{\prime}_{\\{g^{\prime}_{n^{\prime}}\\}}\left(-{\omega}\right)^{J+J^{\prime}}\,\prod_{n}\big{(}V_{n}\big{)}^{g_{n}}\prod_{n^{\prime}}\big{(}{V_{n^{\prime}}^{\prime}}\big{)}^{g_{n^{\prime}}}$
(88)

Now using (83) and (84) one finds

${\cal
A}_{\Pi^{N}_{\\{g_{n}\\}},\Pi^{N^{\prime}}_{\\{g_{n^{\prime}}\\}}}={\cal
N}_{\\{g_{n}\\}}{\cal
N}^{\prime}_{\\{g^{\prime}_{n^{\prime}}\\}}\left(-{\omega}\right)^{J+J^{\prime}}\,e^{-{\cal
C}_{N}(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\}){E_{k}\over
2T_{H}}-\mu_{N}\left(\\{g_{n}\\},\\{g^{\prime}_{n^{\prime}}\\};E_{k}/T_{H}\right)}$
(89)

and squaring the decay amplitude one recovers the formula of the decay rate
(19). The kinematical simplifications of the decay amplitude are just the
reflection of the subleading contributions of bilinear polarization terms of
the decay rate, in the limit $E_{k}/M_{N}$ small. The possibility of computing
the most general decay amplitude, with the explicit dependence on the
microstates structure leads to the identification of the decay amplitude with
the Boltzmann factor, which is the thermal weight of the interaction between
microstates.

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