# Monte-Carlo Based QCD Sum Rules Analysis of $X_{0}(2900)$ and $X_{1}(2900)$

Halil Mutuk<EMAIL_ADDRESS>Department of Physics, Faculty of Arts and
Sciences, Ondokuz Mayis University, 55139, Samsun, Turkey

###### Abstract

Motivated by the very recent discovery of fully open-flavor exotic states
$X_{0}(2900)$ and $X_{1}(2900)$ by the LHCb Collaboration, we study possible
interpretations of these exotic states by QCD Sum Rules method. We have
calculated mass and decay constant by traditional QCD Sum Rules analysis and
Monte-Carlo based analysis. $X_{0}(2900)$ and $X_{1}(2900)$ are studied in
molecular and diquark-antidiquark tetraquark pictures, respectively. Obtained
mass results agree well with the masses observed in the experiment. We also
made a Monte-Carlo based analysis within QCD Sum Rules for the mass and decay
constant distributions to investigate the possible assignments for the
structures of $X_{0}(2900)$ and $X_{1}(2900)$.

###### pacs:

Later

##  I Introduction

The conventional quark model presents part of the properties of QCD in
agreement with experimental data up to 2003 in which the first exotic hadron
candidate $X(3872)$ was observed by Belle Collaboration [1]. According to
quark model, hadrons can be classified into two groups: mesons made of quark-
antiquark $(q\bar{q})$ pair and baryons made of three quarks, $(qqq)$. The
spectrum of QCD is more richer than the spectrum of conventional quark model
since any color neutral (colorless) configuration is possible in QCD.

LHCb Collaboration announced an important and exciting discovery of exotic
particles with open quark flavors in the $D^{-}K^{+}$ of the $B^{\pm}\to
D^{+}D^{-}K^{\pm}$ channel [2]. The observed structure has been parametrized
in terms of two Breit-Wigner resonances:

$\displaystyle X_{0}(2900)$ $\displaystyle:$ $\displaystyle
J^{P}=0^{+},~{}M=2866\pm 7~{}\text{MeV},~{}\Gamma_{0}=57\pm 13~{}\text{MeV},$
(1) $\displaystyle X_{1}(2900)$ $\displaystyle:$ $\displaystyle
J^{P}=1^{-},~{}M=2904\pm 5~{}\text{MeV},~{}\Gamma_{0}=110\pm 12~{}\text{MeV},$
(2)

with global significance of more than 5$\sigma$. $X_{0}(2900)$ is a narrow
state whereas $X_{1}(2900)$ is a broader one. These states are
$502~{}\text{MeV}$ and $540~{}\text{MeV}$ above than their corresponding $DK$
threshold, respectively. Both of these states decay into $D^{-}K^{+}$ and as a
result their quark content should be $[ud\bar{s}\bar{c}]$. Since they have
four different flavors, they could have an exotic nature.

Actually, this is not the first state consisting of four different flavor. D0
collaboration reported an exotic open quark flavor state $X(5568)$ decaying
into $B_{s}^{0}\pi$ in 2016 [3] but not confirmed by LHCb [4], CMS [5], CDF
[6] and ATLAS[7]. For recent studies on this exotic state see [8, 9]. Based on
this phenomena, confirmation of discovery of recent open quark flavor states
by other collaborators may help understanding of the QCD, especially low-
energy region of it.

After the observation of the LHCb Collaboration, these two new states were
studied by simple quark model [10]. In [11], the molecular nature of
$X_{0}(2900)$ state is investigated. They extracted the mass positions of its
heavy quark spin partners. Possible partners of $X_{0,1}(2900)$ were studied
by two-body Coulomb and chromomagnetic interactions in [12]. In [13], triangle
singularity of $X_{0}(2900)$ and $X_{1}(2900)$ was studied in the $B^{+}\to
D^{+}D^{-}K^{+}$ decay via the $\chi_{c1}K^{\ast+}D^{\ast-}$ and
$D_{sJ}^{+}\bar{D}_{1}^{0}K^{0}$ rescattering diagrams. QCD Sum Rules method
was applied in [14] by using four possible interpolating currents with
$J^{P}=0^{+}$. The mass spectra of open charm and bottom tetraquarks
$qq\bar{q}\bar{Q}$ within an extended relativized quark model is calculated in
[15]. $D^{(\ast)}K^{(\ast)}$ system was studied within one-boson exchange
model in molecular picture [16]. Molecular of compact tetraquark pictures of
$X_{0}(2900)$ and $X_{1}(2900)$ were studied by QCD Sum Rules in [17]. They
concluded $X_{0}(2900)$ as a molecule state and $X_{1}(2900)$ as compact
diquark-antidiquark tetraquark state. In [18], axialvector-diquark-
axialvector-diquark type and scalar-diquark-scalar-diquark type fully open
flavor tetraquark states with $J^{P}=0^{+}$ were studied in QCD Sum Rules
method. $X_{0}(2900)$ and $X_{1}(2900)$ states are studied whether two-body
strong decays into $D^{-}K^{+}$ via triangle diagrams and three-body decays
into $D^{\ast}\bar{K}\pi$ [19]. In [20], $X_{0}(2900)$ and $X_{1}(2900)$ are
studied in qBSE approach. Tetraquarks composed of $ud\bar{s}\bar{c}$ are
investigated with meson-meson and diquark-antidiquark structures in the quark
delocalization color screening mode [21]. Two states ($1^{+}$ and
$2^{+}$)stemming from the $D^{\ast}\bar{K}^{\ast}$ interaction was studied in
[22]. In [23], the LHCb vector $ud\bar{c}\bar{s}$ state X(2900) was studied
whether it can be interpreted as a triangle cusp effect arising from
$\bar{D}^{\ast}K^{\ast}$and $D_{1}K^{(}\ast)$ interactions. Mass and coupling
of $X_{0}(2900$) are determined using the QCD two-point sum rule method in
[24]. Branching ratios of $B^{-}\to D^{-}X_{0,1}(2900)$ was studied in [25].

An intriguing property of this experimental study is the widths of the
resonances. The higher resonance $X_{1}(2900)$ with $J^{P}=1^{-}$ has a
significantly larger width than the lower resonance $X_{0}(2900)$ with
$J^{P}=0^{+}$. Considering the $X_{1}(2900)\to D^{-}K^{+}$ decay is $P-$wave
and $X_{0}(2900)\to D^{-}K^{+}$ is $S-$wave, this difference of widths may
help to understand the possible interpretations of these resonances. For this
purpose, we use QCD Sum Rules, an effective nonperturbative method to handle
these resonances.

The paper is organized as follows. In Section II, a brief introduction to QCD
Sum Rules (QCDSR) and Monte-Carlo analysis are given. In Section III, Monte-
Carlo based analysis is done and numerical results are presented for
$X_{0}(2900)$. Section IV is the same as previous section but for
$X_{1}(2900)$. Section V is a brief summary of this work.

##  II QCD Sum Rules

QCD Sum Rules are introduced by Shifman, Vainshtein and Zakharov in 1979 [26]
for mesons and generalized to baryons by Ioffe [27] in 1981. Due to the its
relevance of QCD Lagrangian, it is one of the most used method in
nonperturbative methods. In this model, hadrons are represented by suitably
chosen interpolating currents made of quarks and in some cases together with
gluons. Since these quarks are related to QCD Lagrangian, they are not
constituent quarks which makes QCD Sum Rules model-independent since
constituent quark masses change in different models. This model allows one to
relate the hadron spectrum to the fundamental QCD Lagrangian. The fundamental
idea behind the sum rules is to extrapolate correlation functions from large
$q^{2}$ momentum transfer (short distances) where perturbative calculations
can be done due to the asymptotic freedom, to low $q^{2}$ momentum transfer
(large distances) where correlation functions are dominated by the resonances
as a result of nonperturbative effects. Provided that there is some region
where large $q^{2}$ and low $q^{2}$ representations overlap, one can compute
physical properties of the resonances.

In the first step, the correlation function is calculated taking into account
short distances. This is the region where negative large momentum transfer
occurs and therefore correlation function can be calculated by operator
product expansion (OPE). In the second step, the correlation function is
calculated at large distances where low momentum transfer occurs. In this
region, correlation function can be evaluated by properties of hadron
themselves. To match and overlap these two different correlation function
which are written in two different kinematical region, spectral representation
is used and hadronic properties can be extracted from this matching.

A QCD Sum Rule study starts with writing the correlation function

$\Pi(q^{2})=i\int d^{4}xe^{iqx}\langle 0|T[j(x)j^{\dagger}(0)]|0\rangle.$ (3)

Here $j(x)$ is the interpolating quark current, $q$ is the momentum of the
related state, $T$ is the time ordering operator and $|0\rangle$ is the
nonperturbative vacuum. Writing interpolating quark current is related to the
quark content of the related hadron. As mentioned above, the correlation
function in Eq. 3 can be calculated in two different ways: the first way
includes evaluation of correlation function by using OPE and the second way
includes a phenomenological usage in terms of related hadron parameters.

###  II.1 The OPE Side

Quark and gluon fields are the degrees of freedom of QCD, and we have to take
care the populating of these fields into the QCD vacuum which is the complex
structure of QCD. We can work analytically in the perturbative regime so that
we can calculate perturbative part of $\Pi(q)$ in Eq. (3). Due to the complex
structure of QCD vacuum, expectation values of the operators of quarks fields
and gluon fields are non-zero resulting what we call condensates. Condensate
(vacuum field) contributions to the correlation function can be calculated via
Wilson OPE.

Applying OPE to the Eq. (3) gives

$\Pi(q^{2})=i\int d^{4}xe^{iqx}\langle
0|T[j(x)j^{\dagger}(0)]|0\rangle=\sum_{d}C_{d}(Q^{2})\left\langle
O_{d}\right\rangle,$ (4)

where $C_{d}(Q^{2})(Q^{2}=-q^{2})$ are the Wilson’s coefficients, and
$\left\langle O_{d}\right\rangle$ is the expectation value of the local
operators. The coefficients include only short distance effects and can be
calculated perturbatively. Long-distance effects which are non-perturbative
are contained in the local operators. In the right hand side of Eq. (4), the
operators are ordered by their dimensions $d$.

In the OPE side of correlation function, one evaluate it in terms of the OPE
expansion. Since this expansion is simply an expansion of the operators
contributed from QCD condensates, it must convergence in order to obtain
physical results. Using dispersion relation, the OPE side of Eq. (3) can be
represented as

$\Pi^{\text{OPE}}(q^{2})=\int_{s_{min}}^{\infty}\frac{\rho^{\text{OPE}}(s)ds}{s-q^{2}},$
(5)

where

$\rho^{\text{OPE}}(s)=\frac{1}{\pi}\text{Im}[\Pi^{\text{OPE}}(s)],$ (6)

is the spectral density function and $s_{min}$ is a kinematical limit. This
kinematical limit can be viewed as an adequate energy to create studying
hadron.

###  II.2 The Phenomenological Side

In previous subsection, QCD side of the correlation function was evaluated by
the QCD degrees of freedom, which are quark and gluon fields. In the
phenomenological side, hadrons themselves are being used as degree of freedom.
In order to do this, intermediate states of the studying hadron are inserted
in the correlation function. The correlation can be written as follows:

$\Pi^{\text{Phen}}(q^{2})=\frac{\langle 0|j|h(q)\rangle\langle
h(q)|j^{\dagger}|0\rangle}{q^{2}-m^{2}}+\int_{s_{0}}^{\infty}ds\frac{\rho(s)}{s-q^{2}}+\mbox{subtraction
terms},$ (7)

where $\rho(s)$ is the spectral density of higher states and continuum. In
order to QCDSR be useful, one must parametrize spectral density with a small
number of parameters. The lowest resonance is often fairly narrow, where as
higher-mass resonances are broader. Therefore, one can parametrize the
spectral function as a single sharp pole decoding the lowest resonance of mass
$m$, plus a smooth continuum representing higher mass resonances as

$\rho(s)=\lambda^{2}\delta(s-m^{2})+\rho^{\text{cont}}(s),$ (8)

where $\lambda$ gives the coupling of the current to the lowest mass hadron
$m$ as $\langle 0|j|H\rangle=\lambda$. Putting this relation into the Eq. (7),
one can get the following expression for the phenomenological side:

$\Pi^{\text{Phen}}(q^{2})=\frac{\lambda^{2}}{q^{2}-m^{2}}+\int_{s_{0}}^{\infty}\frac{ds\rho^{\text{OPE}}(s)}{s-q^{2}}+\mbox{subtraction
terms}.$ (9)

Subtraction terms are suppressed when Borel transform is applied:

$\mathcal{B}_{M}^{2}[\Pi(q^{2})]=\lim_{\begin{subarray}{c}-q^{2},n\to\infty\\\
-q^{2}/n=M^{2}\end{subarray}}\frac{(-q^{2})^{n+1}}{n!}\left(\frac{d}{dq^{2}}\right)^{n}\Pi(q^{2}).$
(10)

Borel transformation kills the subtraction terms. It also kills or better to
say suppress contributions of excited resonances and continuum in the
phenomenological side. Furthermore, in the OPE side, it suppresses factorially
the contribution from higher dimension condensates which have inverse power of
$q^{2}$. After Borel transformation, one can able to extract the decay
constant $\lambda$ and mass $m$ from the sum rule. The sum rule can be written
as

$\lambda^{2}e^{-\frac{m_{H}^{2}}{M^{2}}}=\int_{s_{min}}^{s_{0}}ds\rho^{\text{OPE}}(s)e^{-\frac{s}{M^{2}}}$
(11)

By taking the derivative of Eq. (11) with respect to $1/M^{2}$ and dividing
the result by Eq. (11), one can obtain the mass

$m_{H}^{2}=\frac{\int_{s_{min}}^{s_{0}}dss\rho^{\text{OPE}}(s)e^{-s/M^{2}}}{\int_{s_{min}}^{s_{0}}ds\rho^{\text{OPE}}(s)e^{-s/M^{2}}},$
(12)

and decay constant $\lambda$ as

$\lambda^{2}m_{H}^{2}e^{-\frac{m_{H}^{2}}{M^{2}}}=\int_{s_{min}}^{s_{0}}ds\rho^{\text{OPE}}(s)e^{-\frac{s}{M^{2}}}.$
(13)

For more details of the QCDSR technique see [28, 29].

###  II.3 Monte-Carlo Analysis

In the standard QCDSR analysis, the spectral density of the higher states and
the continuum are parameterized by using quark-hadron duality which assumes
$\rho(s)=\rho^{\text{QCD}}(s)$ for $s>s_{0}$. This means that higher states
and continuum contributions to the spectral density is approximated by the
spectral density which is written in terms of QCD parameters and this
parameters are sources of uncertainties for the method.

The major source of uncertainty resulting in the QCDSR technique is the vacuum
condensates. Nonperturbative quantum fluctuations generate condensates and
these condensates emerge in the OPE side of the correlation function. Since it
is possible to study higher dimension operators in the OPE side of the
correlation function, one needs to assume factorization principle to replace
vacuum expectation values of higher dimensional operator by products of lower
dimensional operators. This assumption become problematic in the calculations.
Due to the lack of complete understanding of QCD, many features of these
condensates are not yet well understood [30]. (Although it is beyond the scope
of this work, it was claimed by Brodsky et al. in [31] that QCD condensates
are properties of hadrons themselves. This conclusion was challenged in [32].
Brodsky et al. in another work suggest that confinement is related to
condensates and conclude that if quark-hadron duality is a fact in QCD, then
condensates are contained within hadrons [33].)

The other sources of uncertainties of the QCDSR approach are unfactorized
condensate values, factorization of higher dimensional operators, $\alpha_{s}$
corrections, truncation of the OPE, and choosing of the working regions for
matching the QCD and phenomenological sides of the obtained sum rules. The
conventional QCDSR analysis include the following steps:

  * •

choosing appropriate values for the Borel parameters which give good results
generally with respect to experimental observables. A working region of Borel
parameters should be selected in order to obtain the stability and reliability
of the results.

  * •

a selection of the Borel window without giving a careful attention to OPE
convergence nor maintaining ground state dominance of the phenomenological
side of the sum rules.

  * •

fixing of input parameters (such as the continuum threshold) to preferred
values.

  * •

claiming an accuracy for QCDSR predictions without supporting calculations.
Generally it is seeked a mild dependence of the results with respect to model
parameters. This is called stability analysis. In this analysis, fit
parameters are monitored as a single condensate value is varied by a specific
amount.

This traditional QCDSR analysis can be found in any paper of QCDSR studies.
The sum rules could be analyzed in a certain range of the Borel parameter
$M^{2}$ (Borel window) and continuum threshold $s_{0}$, in which the OPE
series reaches convergences and the continuum contribution is suppressed. The
OPE convergence is ensured with two criteria: contributions of the highest
dimensional operators in the OPE side should not be too large (geenrally less
than 10 % of the total OPE contributions). This give an upper bound of Borel
parameter. On the other side, the continuum contributions should not be larger
than the pole contributions. Excited state contributions over total
contribution should be less than 50 % which gives the lower bound for Borel
parameter. One important point here is that excited state contributions depend
on continuum threshold and this value will be determined in QCDSR analysis.
Therefore, we cannot determine lower bound of Borel parameter before carrying
out QCDSR analysis. The common attitude is to make an initial guess for lower
bound of Borel parameter. If the extracted physical properties are not
sensitive to the variation in the Borel window with respect to different
continuum threshold values, reliable predictions can be obtained by demanding
the OPE convergence. This was done for exotic states in [34, 35, 36].

Apart from systematic uncertainties, QCDSR method is a powerful technique
among nonperturbative methods since it is relativistic, an analytic method and
related to the QCD Lagrangian. The main purpose of QCDSR technique is to
extract physical properties, mostly resonance mass and coupling constant. As
mentioned above, especially fixing $s_{0}$ by hand, systematic uncertainties
related to the calculation of OPE side and determination of phenomenological
quantities with uncertainties related to the sum rule calculations are
addressed and an alternative method is presented by a Monte-Carlo analysis by
Leinweber in [37].

Leinweber introduced Monte-Carlo based uncertainty analysis for QCDSR
predictions. For some applications of this method see [38, 39, 41, 40]. To do
this, we first need to estimate the standard deviation
$\sigma_{\text{OPE}}(M^{2})$ of $\Pi^{\text{OPE}}(M^{2})$ at any point in the
sum rule interval. In this work, this estimation can be done by randomly
generating 250 set of Gaussian distributed input parameters of QCD
(condensates and $\Lambda_{\text{QCD}}$) with given uncertainties. Once the
standard deviation is obtained the phenomenological output parameters
($s_{0}$, $\lambda$, $m$) can be obtained by minimizing a weighted $\chi^{2}$:

$\chi^{2}=\sum_{j=1}^{n_{B}}\frac{(\Pi^{\text{OPE}}(M^{2})-\Pi^{\text{Phys}}(M^{2},s_{0},\lambda,m))^{2}}{\sigma_{\text{OPE}}(M^{2})},$
(14)

where $M^{2}=M^{2}_{min}+(M^{2}_{max}-M^{2}_{min})(j-1)/(n_{B}-1)$ which means
dividing the sum rule window into $(n_{B}-1)$ even parts. After obtaining
$\sigma_{\text{OPE}}(M^{2})$, 3000 sets of Gaussian distributed input
parameters with the same given uncertainties will be generated. One can
minimize $\chi^{2}$ to extract a set of phenomenological output parameters for
each set and the uncertainties of continuum threshold $s_{0}$, decay constant
$\lambda$, and mass $m$. Since the distribution of input parameters occurs
randomly we can impose an important constraint: results obtained with the
$s_{0}<m^{2}$ condition will be excluded since the original sum rules of QCD
is formulated with the ansatz $s_{0}>m^{2}$.

In the next two sections, we will make a traditional analysis vs. Monte-Carlo
analysis comparison for the $X_{0}(2900)$ and $X_{1}(2900)$ resonances.

##  III Traditional and Monte-Carlo Analysis of $X_{0}(2900)$ with
$J^{P}=0^{+}$

The observed peak of $X_{0}(2900)$ by LHCb is around 2.9 GeV. This is very
close to the $D^{\ast-}K^{\ast+}$ threshold at 2902 MeV. In this section, we
assume $X_{0}(2900)$ has a molecular picture and created in the vacuum by the
following interpolating current with $J^{P}=0^{+}$:

$j(x)=\left[\bar{c}^{a}(x)\gamma_{5}d^{a}(x)\right]\left[\bar{s}^{b}(x)\gamma_{5}u^{b}(x)\right].$
(15)

Here the subscripts $a$ and $b$ are color indices and, $u$, $d$, $s$ and $c$
represent the $up$, $down$, $strange$ and $charm$ quark fields, respectively.
The components $\bar{c}_{a}\gamma_{5}d_{a}$ and $\bar{s_{b}}\gamma_{5}u_{b}$
in the interpolating current are two generic meson operators. They couple to
the $D^{\ast-}$ and $K^{\ast+}$ mesons, respectively. The current can couple
to the $S$ wave $D^{\ast-}K^{\ast+}$ molecular state with $J^{P}=0^{+}$. The
coupling of current $j(x)$ to the $X_{0}$ state can be defined as

$\left\langle 0|j|X_{0}\right\rangle=\lambda_{X_{0}}.$ (16)

The QCDSR calculation starts with obtaining correlation function in terms of
the physical degrees of freedom. This is the first step and ends up with Borel
transformed form of the function $\Pi^{\text{Phys}}(q)$:

$\mathcal{B}_{q^{2}}\Pi^{\text{Phys}}(q)=m_{X_{0}}^{2}\lambda_{X_{0}}^{2}e^{-m_{{X_{0}}}^{2}/M^{2}}+\cdots.$
(17)

The next step is to find the theoretical expression for the same function,
$\Pi^{\text{OPE}}(q)$. Contracting the quark fields yields

$\displaystyle\Pi^{\text{OPE}}(q)=i\int
d^{4}xe^{iqx}\text{Tr}[\gamma_{5}S_{c}^{aa^{\prime}}(x)S_{d}^{aa^{\prime}}(-x)]\text{Tr}[\gamma_{5}S_{u}^{aa^{\prime}}(x)S_{s}^{aa^{\prime}}(-x)],$
(18)

where $S_{q}^{ab}(x)$ ($q=u,d,s$) and $S_{c}^{ab}(x)$ are the light and heavy
quark propagators, respectively. The spectral densities are given in the
appendix. The light quark propagator $S_{q}^{ab}(x)$ reads as

$\displaystyle S^{ab}_{q}(x)$ $\displaystyle=$ $\displaystyle
i\delta_{ab}\frac{\not{x}}{2\pi^{2}x^{4}}-\delta_{ab}\frac{\langle\bar{q}q\rangle}{12}+i\delta_{ab}\frac{\not{x}m_{q}\langle\bar{q}q\rangle}{48}-\delta_{ab}\frac{x^{2}}{192}\langle\bar{q}g_{s}\sigma
Gq\rangle+i\delta_{ab}\frac{x^{2}\not{x}m_{q}}{1152}\langle\bar{q}g_{s}\sigma
Gq\rangle$ (19) $\displaystyle-$ $\displaystyle
ig_{s}\frac{G^{\alpha\beta}_{ab}}{32\pi^{2}x^{2}}[\not{x}\sigma_{\alpha\beta}+\sigma_{\alpha\beta}\not{x}]-i\delta_{ab}\frac{x^{2}\not{x}g_{s}^{2}\langle\bar{q}q\rangle^{2}}{7776}-\delta_{ab}x^{4}\frac{\langle\bar{q}q\rangle\langle
g_{s}^{2}G^{2}\rangle}{27648}+\cdots.$

The heavy quark propagator $S^{ab}_{c}(x)$ can be written as

$\displaystyle S^{ab}_{c}(x)$ $\displaystyle=$ $\displaystyle
i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}[\frac{\delta_{ab}(\not{k}+m_{c})}{k^{2}-m_{c}^{2}}-\frac{g_{s}G^{\alpha\beta}_{ab}}{4}\frac{\sigma_{\alpha\beta}(\not{k}+m_{c})+(\not{k}+m_{c})\sigma_{\alpha\beta}}{(k^{2}-m_{c}^{2})^{2}}$
(20) $\displaystyle+$
$\displaystyle\frac{g_{s}^{2}G^{2}}{12}\delta_{ab}m_{c}\frac{k^{2}+m_{c}\not{k}}{(k^{2}-m_{c}^{2})^{4}}+\cdots].$

Here $a,b=1,2,3$ and $A,B,C=1,2,\cdots,8$ are color indices. $t^{A}$ is
defined as $t^{A}=\lambda^{A}/2$, where $\lambda^{A}$ are the Gell-Mann
matrices. $G^{A}$ which is the gluon field strength tensor, defined as
$G^{A}_{\alpha\beta}=G^{A}_{\alpha\beta}(0)$ and is fixed at $x=0$.

The correlation function $\Pi^{\text{OPE}}(q)$ can be written by a dispersion
integral

$\Pi^{\text{OPE}}(q)=\int_{(m_{c}+m_{s})^{2}}^{\infty}\frac{\rho^{\text{OPE}}(s)}{s-q^{2}},$
(21)

where we parametrize $s_{min}=(m_{c}+m_{s})^{2}$. The mass and decay constant
(residue) sum rules can be obtained as mentioned before by equating
$\Pi^{\text{Phys}}(q)$ and $\Pi^{\text{OPE}}(q)$. The mass can be found as

$m_{X_{0}}^{2}=\frac{\int_{(m_{c}+m_{s})^{2}}^{s_{0}}dss\rho^{\text{OPE}}(s)e^{-s/M^{2}}}{\int_{(m_{c}+m_{s})^{2}}^{s_{0}}ds\rho^{\text{OPE}}(s)e^{-s/M^{2}}},$
(22)

whereas for decay constant $\lambda_{X_{0}}$ we use the formula

$\lambda_{X_{0}}^{2}m_{X_{0}}^{2}e^{-m_{{X_{0}}}^{2}/M^{2}}=\int_{(m_{c}+m_{s})^{2}}^{s_{0}}dss\rho^{\text{OPE}}(s)e^{-s/M^{2}}.$
(23)

The physical properties can be extracted from the sum rules which depend on
vacuum condensates and quark masses. We use the following values of quark
masses and condensates [42]:

$\displaystyle m_{c}$ $\displaystyle=$
$\displaystyle(1.275^{+0.025}_{-0.035})~{}\text{GeV},$ $\displaystyle
m_{s}(2~{}\text{GeV})$ $\displaystyle=$ $\displaystyle
95^{+9}_{-3}~{}\text{MeV},$ $\displaystyle m_{0}^{2}$ $\displaystyle=$
$\displaystyle(0.8\pm 0.2)~{}\text{GeV}^{2},$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle$ $\displaystyle=$
$\displaystyle m_{0}^{2}\langle\bar{s}s\rangle,$
$\displaystyle\langle\bar{q}q\rangle$ $\displaystyle=$ $\displaystyle(-0.24\pm
00.1)^{3}~{}\text{GeV}^{3},$ $\displaystyle\langle\bar{s}s\rangle$
$\displaystyle=$ $\displaystyle(0.8\pm 0.1)\langle\bar{q}q\rangle,$
$\displaystyle\langle g^{2}G^{2}\rangle$ $\displaystyle=$
$\displaystyle(0.012\pm 0.004)~{}\text{GeV}^{4}.$

The resulting sum rules for mass and decay constant is a function of Borel
parameter $M^{2}$ and continuum threshold $s_{0}$. The correct choice of these
parameters is an important task for sum rule calculations. One needs to find
working regions for Borel and continuum threshold parameters where physical
quantities do not extremely depend on these values or have a weak dependence
on them. The QCDSR method have unavoidable theoretical uncertainties due to
the its formulation. There exist some procedures to extract $M^{2}$ and
$s_{0}$ values which are well defined in the context of the QCDSR method
itself. In order to maintain desired results, working regions for $M^{2}$ and
$s_{0}$ should satisfy some constraints on the pole contribution (PC)

$\text{PC}=\frac{\Pi(M^{2},s_{0})}{\Pi(M^{2},\infty)},$ (25)

and OPE convergence

$R(M^{2})=\frac{\Pi^{\text{DimN}}(M^{2},s_{0})}{\Pi(M^{2},s_{0})}.$ (26)

Here, $\Pi^{\text{DimN}}(M^{2},s_{0})$ is a last term in the correlation
function. We used quark, gluon, and mixing vacuum condensates up to dimension
eight.

The continuum threshold $s_{0}$ is related to the energy of the first excited
state of the studying hadron. If this first excited state exist then $s_{0}$
can be parametrized since the energy of this excited state is known. In the
case of exotic resonances or states, it is not always possible to observe
excited levels. Therefore it is difficult to determine $s_{0}$ precisely.
However, some recipes exist to determine this energy level for exotic
resonances and states.

In standard recipes, the continuum threshold $s_{0}$ is taken to be as
$s_{0}=(m_{ground}+\delta)^{2}$ where $\delta$ varies between 0.3 and 0.8 GeV.
Here we consider the LHCb Collaboration resonance as a ground state. The value
of $s_{0}$ is determined from condition that the sum rule with 10 % accuracy
reproduce the experimentally measured mass. The working window for Borel
parameter $M^{2}$ can be found by using PC and $R(M^{2})$. PC is used to fix
upper bound of $M^{2}$ and $R(M^{2})$ can be used to find lower bound of
$M^{2}$. In QCDSR formalism, the pole contribution of the ground state is
required to be greater than the contribution of the continuum states. OPE
convergence is not guaranteed for $M^{2}>3~{}\text{GeV}^{2}$ in the present
work. Our analysis shows that for $\text{PC}>0.5$ and $\text{R}<0.2$, the
working regions are

$M^{2}\in[2,3]~{}\text{GeV}^{2},~{}s_{0}\in[10,12]~{}\text{GeV}^{2}.$ (27)

The predicted value for the mass of $X_{0}$ is

$M_{X_{0}}=2792\pm 124~{}\text{MeV},$ (28)

and decay constant is

$\lambda=3.35\pm 0.15\times 10^{-3}~{}\text{GeV}^{4}.$ (29)

where the uncertainty results from the Borel mass parameter $M^{2}$, continuum
threshold $s_{0}$, and various quark and gluon parameters. The extracted mass
agree well with the mass of experimental result for $X_{0}(2900)$. In Figs. 1
and 2, we display prediction of sum rules for mass $m$ and decay constant
$\lambda$ as a function of Borel parameter $M^{2}$, respectively.

Figure 1:  The mass of the $X_{0}$ as a function of the Borel parameter
$M^{2}$ at fixed $s_{0}$ values. Figure 2:  The decay constant of the $X_{0}$
as a function of the Borel parameter $M^{2}$ at fixed $s_{0}$ values.

It can be seen from both figures that in the given working regions, there
exist a mild dependence on the Borel parameter $M^{2}$ for the mass and decay
constant results.

The next task is to study mass and decay constant of $X_{0}(2900)$ by using
QCD sum rules with Monte-Carlo analysis. We set $n_{B}=50$ and generated 3000
Gaussian distributed input parameters with given uncertainties (10%
uncertainties, which are typical uncertainties in QCDSR). The QCD input
parameters are quark condensate, gluon condensate, mixed condensate values
which are given in Eq. LABEL:qcdinput and
$\Lambda_{\text{QCD}}=0.353~{}\text{GeV}$. We have selected the physical
results since in the sum rules some constraints exist such as $s_{0}>m^{2}$.
We also plot the histogram for 3000 different $X_{0}(2900)$ matches obtained
in the least-squares fitting procedure. Fig. 3 shows the distribution of
$X_{0}$ meson masses.

Figure 3:  The histogram of the $X_{0}$ masses obtained from 3000 matches.

The distribution corresponds to

$M_{X_{0}}=2865^{+20}_{-18}~{}\text{MeV},$ (30)

which is in good agreement with the experimental value. The uncertainty is
lower than the commonly assumed 10 %. It can be seen from Fig. 3 that the
distribution of mass is very close to Gaussian curve.

Fig. 4 shows the distribution of $X_{0}$ decay constants.

Figure 4:  The histogram of the $X_{0}$ decay constant obtained from 3000
matches.

The distribution corresponds to

$\lambda_{X_{0}}=3.30^{+0.12}_{-0.14}~{}\times 10^{-3}~{}\text{GeV}^{4}.$ (31)

It can be seen from Fig. 4 that the distribution of decay constant is very
close to Gaussian curve.

##  IV Traditional and Monte-Carlo Analysis of $X_{1}(2900)$ with
$J^{P}=1^{-}$

In this section, we investigate the diquark-antidiquark structure for the
observed peak $X_{1}(2900)$ with quantum number $J^{P}=1^{-}$. We use the
following interpolating current

$j_{\mu}(x)=s_{a}^{T}(x)Cc_{b}(x)\left[\bar{u}_{a}(x)\gamma_{\mu}\gamma_{5}C\bar{d}_{b}^{T}(x)-\bar{u}_{b}(x)\gamma_{\mu}\gamma_{5}C\bar{d}_{a}^{T}(x)\right],$
(32)

where $a$ and $b$ are color indices and $C$ is the charge conjugation matrix.

The correlation function can be written as

$\displaystyle\Pi_{\mu\nu}(q)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iqx}\langle 0|T[j_{\mu}(x)j_{\nu}^{\dagger}(0)]|0\rangle$ (33)
$\displaystyle=$
$\displaystyle\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\Pi_{1}(q)+\frac{q_{\mu}q_{\nu}}{q^{2}}\Pi_{0}(q),$

where $\Pi_{1}(q)$ and $\Pi_{0}(q)$ are the invariant functions related to the
spin-1 and spin-0 intermediate states, respectively. Spin-1 intermediate
states contribute only to $\Pi_{1}(q)$, therefore we use it to perform
numerical analysis. In order to obtain sum rules expression, at first step
correlation function needs to be calculated in terms of the physical
parameters of the related hadron. Inserting a complete set of the
$X_{1}(2900)$ state into the correlation function, we find

$\Pi_{\mu\nu}^{\text{Phys}}(q)=\frac{\langle 0|j_{\mu}|X_{1}(q)\rangle\langle
X_{1}(q)|j_{\nu}^{\dagger}|0\rangle}{m_{X_{1}}^{2}-q^{2}}+\cdots,$ (34)

where the dots represents contributions from the higher resonances and
continuum states. We can write coupling $\lambda_{X_{1}}$ using the matrix
element

$\langle 0|j_{\mu}|X_{1}(q)\rangle=\lambda_{X_{1}}m_{X_{1}}\epsilon_{\mu},$
(35)

where $\epsilon_{\mu}$ is the polarization vector of the $X_{1}(2900)$ state.
With this definition correlation function can be written as

$\Pi_{\mu\nu}^{\text{Phys}}(q)=\frac{\lambda_{X_{1}}^{2}m_{X_{1}}^{2}}{m_{X_{1}}^{2}-q^{2}}\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{X_{1}}^{2}}\right)+\cdots.$
(36)

Applying Borel transformation to Eq. (36) yields

$\mathcal{B}_{q^{2}}\Pi_{\mu\nu}^{\text{Phys}}(q)=\lambda_{X_{1}}^{2}m_{X_{1}}^{2}e^{-m_{X_{1}}^{2}/M^{2}}\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{X_{1}}^{2}}\right)+\cdots$
(37)

At second step, the correlation function has to be calculated from the QCD
(OPE) side. Contracting the heavy and light quark fields yields

$\displaystyle\Pi_{\mu\nu}^{\text{OPE}}(q)$ $\displaystyle=$ $\displaystyle
i\int
d^{4}xe^{iqx}\text{Tr}[\gamma_{5}\tilde{S}^{aa^{\prime}}_{s}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)]\\{\text{Tr}[\gamma_{\mu}\tilde{S}^{a^{\prime}b}_{d}(-x)\gamma_{\nu}S_{u}^{b^{\prime}a}(-x)]-\text{Tr}[\gamma_{\mu}\tilde{S}^{b^{\prime}b}_{d}(-x)\gamma_{\nu}S_{u}^{a^{\prime}a}(-x)]$
(38) $\displaystyle+$
$\displaystyle\text{Tr}[\gamma_{\mu}\tilde{S}^{b^{\prime}a}_{d}(-x)\gamma_{\nu}S_{u}^{a^{\prime}b}(-x)]-\text{Tr}[\gamma_{\mu}\tilde{S}^{a^{\prime}a}_{d}(-x)\gamma_{\nu}S_{u}^{b^{\prime}b}(-x)]\\}.$

Here we use the notation $\tilde{S}_{q}(x)=CS^{T}(x)C$, where $C$ is the
charge conjugation operator. The spectral densities are given in the appendix.
The $\Pi_{\mu\nu}^{\text{OPE}}(q)$ can be written by a dispersion integral

$\Pi_{\mu\nu}^{\text{OPE}}(q)=\int_{(m_{c}+m_{s})^{2}}^{\infty}ds\frac{\rho^{\text{OPE}}(s)}{s-q^{2}}.$
(39)

After normal procedure as defined previous section, the mass can be found as

$m_{X_{1}}^{2}=\frac{\int_{(m_{c}+m_{s})^{2}}^{s_{0}}dss\rho^{\text{OPE}}(s)e^{-s/M^{2}}}{\int_{(m_{c}+m_{s})^{2}}^{s_{0}}ds\rho^{\text{OPE}}(s)e^{-s/M^{2}}},$
(40)

whereas for the decay constant $\lambda_{X_{1}}$ we use the formula

$\lambda_{X_{1}}^{2}m_{X_{1}}^{2}e^{-m_{{X_{1}}}^{2}/M^{2}}=\int_{(m_{c}+m_{s})^{2}}^{s_{0}}dss\rho^{\text{OPE}}(s)e^{-s/M^{2}}.$
(41)

The working regions for continuum threshold $s_{0}$ and Borel parameter
$M^{2}$ are the same as given in previous section. The predicted value for the
mass of $X_{1}$ is

$M_{X_{1}}=2963\pm 64~{}\text{MeV}$ (42)

which is in good agreement with the mass of experimental result for
$X_{1}(2900)$. The predicted decay constant of $X_{1}(2900)$ is

$\lambda=3.39\pm 0.20\times 10^{-3}~{}\text{GeV}^{4}.$ (43)

In Figs. 5 and 6 we display prediction of sum rules for mass $m$ and decay
constant $\lambda$ as a function of Borel parameter $M^{2}$, respectively.

Figure 5:  The mass of the $X_{1}$ as a function of the Borel parameter
$M^{2}$ at fixed $s_{0}$ values. Figure 6:  The decay constant of the $X_{1}$
as a function of the Borel parameter $M^{2}$ at fixed $s_{0}$ values.

It can be seen from Figs. 5 and 6 that the predictions of sum rule for mass
$m$ and decay constant $\lambda$ as a function of Borel parameter $M^{2}$ have
a smooth dependence on the Borel parameter.

In order to obtain mass and decay constant of $X_{1}(2900)$ we set $n_{B}=50$
and generated 3000 Gaussian distributed input parameters with given
uncertainties (10% uncertainties, which are typical uncertainties in QCDSR).
The QCD input parameters are quark condensate, gluon condensate, mixed
condensate values which are given in Eq. LABEL:qcdinput and
$\Lambda_{\text{QCD}}=0.353~{}\text{GeV}$. We have selected the physical
results since in the sum rules some constraints exist such as $s_{0}>m^{2}$.
We also plot the histogram for 3000 different $X_{0}(2900)$ matches obtained
in the least-squares fitting procedure. The histogram for masses of $X_{1}$
can be seen in Fig. 7.

Figure 7:  The histogram of the $X_{1}$ masses obtained from 3000 matches.

The distribution corresponds to

$M_{X_{1}}=2900\pm 19~{}\text{MeV},$ (44)

which is in good agreement with the experimental value. The uncertainty is
lower than the commonly assumed 10 %. The distribution of mass is very close
to Gaussian curve. Fig. 8 shows the distribution of decay constants $\lambda$
for the $X_{1}(2900)$.

Figure 8:  The histogram of the $X_{1}$ decay constants obtained from 3000
matches.

The distribution corresponds to

$\lambda_{X_{1}}=3.40^{+0.15}_{-0.13}~{}\times 10^{-3}~{}\text{GeV}^{4}.$ (45)

It can be seen from Fig. 8 that the distribution of decay constant is very
close to Gaussian curve.

##  V Discussion, Summary and Concluding Remarks

In this present work, we have studied two new resonances $X_{0}$ and $X_{1}$
observed by the LHCb collaboration. This is the first time observing fully
open-flavor exotic states. We studied their possible interpretations using
traditional and Monte-Carlo based QCD sum rules method.

We used a molecular current (meson-meson tetraquark) with $J^{P}=0^{+}$ to
study mass and decay constant of $X_{0}(2900)$. In the traditional analysis,
the extracted mass value is

$M_{X_{0}}=2792\pm 124~{}\text{MeV},$ (46)

and in good agreement with the experimental mass of $X_{0}(2900)$ suggesting a
possible molecular picture as $J^{P}=0^{+}$ $D^{-}K^{+}$. This molecular
picture was supported in [43], in which the authors made a prediction for the
$0^{+}$ state with mass $2848~{}\text{MeV}$ and $\Gamma=59~{}\text{MeV}$
before the experiment. The decay constant of $X_{0}(2900)$ is obtained as

$\lambda=3.35\pm 0.15\times 10^{-3}~{}\text{GeV}^{4}.$ (47)

The Monte-Carlo analysis for the mass and decay constant of $X_{0}(2900)$
yields mass as

$M_{X_{0}}=2865^{+20}_{-18}~{}\text{MeV},$ (48)

and decay constant as

$\lambda_{X_{0}}=3.30^{+0.12}_{-0.14}~{}\times 10^{-3}~{}\text{GeV}^{4}.$ (49)

Our analysis give results which are in good agreement with the Monte-Carlo
analysis.

A diquark-antidiquark tetraquark interpolating current with $J^{P}=1^{-}$ was
used to study mass and decay constant of $X_{1}(2900)$. The extracted mass
value is

$M_{X_{1}}=2963\pm 64~{}\text{MeV}$ (50)

which is in good agreement with the experimental mass of $X_{1}(2900)$. The
obtained decay constant is

$\lambda=3.39\pm 0.20\times 10^{-3}~{}\text{GeV}^{4}.$ (51)

The Monte-Carlo analysis for the mass and decay constant of $X_{1}(2900)$
yields mass as

$M_{X_{0}}=2900\pm 19~{}\text{MeV},$ (52)

and decay constant as

$\lambda_{X_{1}}=3.40^{+0.15}_{-0.13}~{}\times 10^{-3}~{}\text{GeV}^{4}.$ (53)

Our analysis give results which are in good agreement with the Monte-Carlo
analysis. Comparison of mass results are tabulated in 1 whereas decay constant
results are tabulated in 2

Table 1:  Mass values of traditional sum rule analysis vs. Monte-Carlo analysis. All results are in MeV. | Traditional | Monte-Carlo | Experiment
---|---|---|---
$X_{0}$ | $2792\pm 124$ | $2865^{+20}_{-18}$ | $2866\pm 7$
$X_{1}$ | $2963\pm 64$ | $2900\pm 19$ | $2904\pm 5$
Table 2:  Decay constants of traditional sum rule analysis vs. Monte-Carlo analysis. | Traditional | Monte-Carlo
---|---|---
$X_{0}$ | $3.35\pm 0.15\times 10^{-3}~{}\text{GeV}^{4}$ | $3.30^{+0.12}_{-0.14}~{}\times 10^{-3}~{}\text{GeV}^{4}$
$X_{1}$ | $3.39\pm 0.20\times 10^{-3}~{}\text{GeV}^{4}$ | $3.40^{+0.15}_{-0.13}~{}\times 10^{-3}~{}\text{GeV}^{4}$

As can be seen from both tables, results for the both of the traditional
analysis and Monte-Carlo analysis are in good agreement with experimental
values.

An interesting feature of this experiment is that $X_{1}(2900)$ has larger
width than the $X_{0}(2900)$. For this purpose, we made a Monte-Carlo based
analysis for the masses of these states. It can bee seen form Figs. 3 and 7,
$X_{0}(2900)$ has a narrower shape than the $X_{1}(2900)$. In other words,
$X_{0}(2900)$ has a small width than the $X_{1}(2900)$ has. This analysis
could support the molecular picture for $X_{0}(2900)$ and diquark-antidiquark
tetraquark structure for $X_{1}(2900)$. This outcome could be tested by
calculating decay and width properties. For example, the width of the
$X_{0}(2900)$ state in molecular picture was calculated in [24]. The result is
$\Gamma=(49.6\pm 9.3)~{}\text{MeV}$ and could be considered as to favor
molecular picture. In [15], it is found that a compact tetraquark assignment
for $X_{0}(2900)$ is not likely. A QCD sum rule study suggested the tetraquark
configuration for $X_{0}(2900)$ [14]. It is pointed out that $X_{1}(2900)$ has
a width around $\Gamma\sim 100~{}\text{MeV}$ and could be interpreted as
P-wave compact tetraquark [17]. It was showed in [12] that $X_{0}(2900)$ and
$X_{1}(2900)$ states ban be explained as radial excited tetraquark and
orbitally excited tetraquark of $[ud\bar{s}\bar{c}]$ configurations,
respectively.

In QCD sum rule method, one cannot conclude if a state have a tetraquark or
molecular configuration. However, both a traditional and Monte-Carlo analysis
with mass and decay constant could support the possible assignments. The two
interpretations made above are just possible assignments for these states.
More theoretical and experimental studies are needed to investigate these
exotic states.

##  Appendix A Spectral Density of $X_{0}(2900)$ Resonance

With the definitions

$\displaystyle p$ $\displaystyle=$ $\displaystyle[m_{c}^{2}+s(x-1)],$
$\displaystyle q$ $\displaystyle=$ $\displaystyle[m_{c}^{2}+2s(x-1)],$
$\displaystyle r$ $\displaystyle=$ $\displaystyle[m_{c}^{2}+3s(x-1)],$
$\displaystyle s$ $\displaystyle=$ $\displaystyle[m_{c}^{2}+5s(x-1)],$
$\displaystyle t$ $\displaystyle=$ $\displaystyle m_{c}m_{s}\langle
s\bar{s}\rangle(1-x),$

we give the spectral density expressions of correlation functions for the
$X_{0}(2900)$ resonance.

$\displaystyle\rho^{\text{Pert}}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{13}\pi^{6}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{4}}{(1-x)^{3}}p^{3}r,$
$\displaystyle\rho^{3}$ $\displaystyle=$
$\displaystyle\frac{3}{2^{8}\pi^{4}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{2}}{(1-x)^{2}}p\left[t(\frac{m_{c}^{2}+1}{m_{c}})-m_{c}^{2}\langle
q\bar{q}\rangle-2m_{s}\langle q\bar{q}\rangle(1-x)(r-s(x-1))-2m_{s}s\langle
s\bar{s}\rangle(x-1)^{2}\right],$ $\displaystyle\rho^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{32^{6}\pi^{4}}\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{2}}{(1-x)^{3}}\left[2m_{c}^{4}(x(13x-30)+18)+3m_{c}^{2}s(6-5x)^{2}(x-1)+24s^{2}(x-1)^{3}(2x-3)\right],$
$\displaystyle\rho^{5}$ $\displaystyle=$
$\displaystyle\frac{m_{0}^{2}}{2^{8}\pi^{4}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx}{(1-x)}\left[-3m_{c}^{3}\langle
q\bar{q}\rangle+2m_{c}^{2}m_{s}\langle s\bar{s}\rangle(1-x)+3m_{c}s\langle
q\bar{q}\rangle(1-x)-3m_{s}(1-x)(\langle q\bar{q}\rangle-s\langle
s\bar{s}\rangle(1-x))\right],$ $\displaystyle\rho^{6}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{6}\pi^{4}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}dxx\left[\langle
g^{3}G^{3}\rangle\frac{x^{4}}{2^{5}5\pi^{2}(1-x)^{3}}(m_{c}^{2}(2x+3)+s(5x^{2}-7x+2))-g_{s}^{2}/3^{3}(m_{c}^{2}+r)(2\langle
q\bar{q}\rangle^{2}+\langle s\bar{s}\rangle)\right],$ $\displaystyle\rho^{7}$
$\displaystyle=$
$\displaystyle\frac{1}{32^{4}\pi^{2}}\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dx}{(1-x)^{2}}\left[m_{c}^{2}\langle
q\bar{q}\rangle(2x^{3}+7x^{2}-14x+7)+2m_{s}\langle
q\bar{q}\rangle(5x^{3}-9x^{2}+3x+1)-6m_{s}\langle
s\bar{s}\rangle(x^{3}-2x^{2}+x)\right].$

##  Appendix B Spectral Density of $X_{1}(2900)$ Resonance

In this section, we give the spectral density expressions of correlation
functions for the $X_{1}(2900)$ resonance.

$\displaystyle\rho^{\text{Pert}}$ $\displaystyle=$
$\displaystyle\frac{-1}{2^{13}\pi^{6}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{4}}{(1-x)^{3}}p^{3}s,$
$\displaystyle\rho^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{7}\pi^{4}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{2}}{(1-x)^{2}}p\left[2pm_{c}\langle
q\bar{q}\rangle-rm_{s}(1-x)(\langle s\bar{s}\rangle-2\langle
q\bar{q}\rangle)\right],$ $\displaystyle\rho^{4}$ $\displaystyle=$
$\displaystyle\frac{-1}{3^{2}2^{1}0\pi^{4}}\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx^{2}}{(1-x)^{3}}\left[m_{c}^{4}(x(13x+21)-9)+2m_{c}^{2}s(x(23x-40+18)(x-1))-s^{2}(1-x)^{3}(32x-27)\right],$
$\displaystyle\rho^{5}$ $\displaystyle=$
$\displaystyle\frac{m_{0}^{2}}{32^{6}\pi^{4}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dxx}{(1-x)}\left[3m_{c}p\langle
q\bar{q}\rangle-m_{s}(1-x)q(\langle s\bar{s}\rangle-3\langle
q\bar{q}\rangle)\right],$ $\displaystyle\rho^{6}$ $\displaystyle=$
$\displaystyle\frac{1}{352^{1}2\pi^{6}}\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}dxx\left[m_{c}^{2}\langle
g^{3}G^{3}\rangle xq+g_{s}^{2}x^{2}q+36m_{c}m_{s}\langle
q\bar{q}\rangle(\langle s\bar{s}\rangle-2\langle q\bar{q}\rangle)\right],$
$\displaystyle\rho^{7}$ $\displaystyle=$
$\displaystyle\frac{1}{3^{2}2^{7}\pi^{2}}\langle\frac{\alpha_{s}G^{2}}{\pi}\rangle\int_{(m_{c}^{2}+m_{s}^{2})}^{s_{0}}\frac{dx}{(1-x)}\left[8m_{c}^{\langle}q\bar{q}\rangle+m_{s}(3x\langle
s\bar{s}\rangle)+2\langle q\bar{q}\rangle(1-4x)\right].$

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Published by Kazuki Otsuka. All rights reserved.

# Computational Language $\beta$ based on Orthomodular Lattices with the Non-
distributivity of Quantum Logic

Kazuki Otsuka 111 This paper was written and published as a master’s thesis
(Jan 2023).
The Graduate School of Interdisciplinary Information Studies, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033, Tokyo, Japan

(2023; 2023)

###### Abstract

Abstract

It is argued that transformation processes (generation rules) showing evidence
of a long evolutionary history in universal computing systems can be
generalized. The explicit function class $\Omega$ is defined as follows:
‘Operators whose eigenvectors (or eigenvalues) have an irrational number in
their components constitute a class of functions with quasi-periodic
structure, $\Omega$, and the class $\Omega$ shows evidence of a long
evolutionary history.’ In order to empirically proove this theorem by
examining physical systems carrying out life activities or intellectual
outputs of developed intelligence, the basic framework of the universal
machine model $\pazocal{C}$ and the computational language $\beta$ is
presented as a model for general computational methods, which allow
transformation processes (generation rules) with deep algorithmic complexity
to be derived from generation results. $\pazocal{C}$ and $\beta$ perform
massively parallel computations on event-state systems consisting of
exponential combinations of propositional elements expressed in terms of
correlations between subsystems. The logical structure of the computational
language relies on a non-distributivity in Hilbert spaces or orthogonal
modular lattices, allowing for the manipulation and deduction of simultaneous
propositions. In this logical local structure, the propositions implying
certain consequences are not uniquely determined.

##  1 Why Beautiful Things are Beautiful

When a sequence of symbols is generated with a high degree of complexity,
there is no general method for deriving the transformation process from the
generated results. Supposing there is a sequence of binary symbols as follows:

$0101010101010101010101010101010101$ (1.1) $111110000111001110011000111100000$
(1.2) $10101101010101011010101101010101$ (1.3)

The first one is an alternating sequence of 0 and 1 and appears to be the
simplest of the three sequences. The second is symmetric if one looks
carefully. The degree of complexity of the first and the second are both
small, but the second seems to be slightly more complex. The third is a
combination of the alternation rules of the first and the symmetry rules of
the second and can be said to contain the complexity of the generative rules
of the first and the second.

Computational complexity theory has defined the complexity of generation
processes as the computing time of programs that efficiently give rise to a
given $\mathbb{Z}\rightarrow\mathbb{Z}$ on a Turing-equivalent system. Most
systems with computational universality generate computable functions for
$\mathbb{Z}\rightarrow\mathbb{Z}$ by applying basic finite procedures
recursively. This complexity corresponds to the amount of evolution required
to evolve from a precursor in the evolution of life, and similarly to the
amount of evolution required to create new knowledge from existing
intellectual output in terms of the growth of human knowledge. Therefore, it
is reasonable to assume that the "evidence" imprinting the long "causal
history" that arose in the natural world under a "certain order" is preserved
in some form or encoded information. From the principle of natural selection,
it is also reasonable to think that life acquires the ability to intuitively
discern this degree of complexity through the generated results, and that this
ability enables it to distinguish between more evolved and less evolved forms.

On the other hand, it is not generally possible to distinguish from the
results generated by a universal computing system whether those results are
the output of a complex transformation process or the result of a random
behavior.

$110101100011010101100010101010010110101$ (1.4)
$010010100100101001010010100100101001010$ (1.5)

The first of these two binary sequences is a random output, while the second
is the result of applying the Fibonacci word generation rules nine times. In
principle, one cannot rule out the possibility that a randomly generated
result may coincide with a result generated by a complex transformation
process. The second one can be generated with a probability of
$\frac{1}{2^{34}}$ . It is easy to infer that the application of consistent
rules to systems of a certain size is required in order to show evidence that
the morphology of the generated results is due to a complex evolutionary
process.

It is known that a pair of natural numbers $(p,q)$ generated by the Fibonacci
rule maximizes the computational complexity of the Euclidean algorithm for
finding the greatest common divisor (gcd). That is, if the quotient of $u$
divided by $v$ is $s$ and the remainder is $r$, then for

$\begin{array}[]{c}u_{(1)}=v_{(1)}s_{(1)}+r_{(1)}\\\
u_{(2)}=v_{(2)}s_{(2)}+r_{(2)}\\\ \ \ \ \ \ \ \ \ \vdots\\\
u_{(t)}=v_{(t)}s_{(t)}+r_{(t)}\\\ \\\ \end{array}$ (1.6)

where $u_{(1)}=q,\ v_{(1)}=p,\ \ (p,\ q,\ r,\ t\in\mathbb{N})$, the following
is obtained

$\forall t,\ s_{(t)}=1,\ u_{(t+1)}=v_{(t)},\ v_{(t+1)}=r_{(t)}$

and the Fibonacci process that generated $q$ (and $p$) is exactly reversed
until $p=1$ and $q=2$. This fact suggests that transformations by the
Fibonacci rule provide a way to show evidence of a long evolutionary history
of causality in nature. That is, the results of multiple applications of a
transformation rule are recorded in the form they are as proof of the
generation costs.

Fibonacci numbers are generated by using the operator

$\pazocal{F}=\begin{bmatrix}1&1\\\ 1&\end{bmatrix}$ (1.7)

with a recursive process

$\begin{array}[]{l}\mathbfit{u}_{t+1}=\pazocal{F}\mathbfit{u}_{t},\ \
\mathbfit{u}_{0}=\begin{bmatrix}1\\\ 0\end{bmatrix}\end{array}$ (1.8)

The matrix $\pazocal{F}$ is a self-adjunct Hermitian operator with real
eigenvalues including $\tau=\frac{1+\sqrt{5}}{2}$. It is also invertible and
the transformation $T:G\rightarrow G$ is bijective and hence logically
reversible. Its spectral decomposition is a linear combination of the
projection operators on the eigenvectors containing $\tau$.

Let the eigenvalues of $\pazocal{F}$ be
$\lambda_{1}=\frac{1+\sqrt{5}}{2}(=\tau)\approx 1.618$,
$\lambda_{2}=\frac{1-\sqrt{5}}{2}\approx-.618$ and the eigenvectors

$\mathbfit{x}_{1}=\begin{bmatrix}\lambda_{1}\\\ 1\end{bmatrix},\ \ \
\mathbfit{x}_{2}=\begin{bmatrix}\lambda_{2}\\\ 1\end{bmatrix}$ (1.9)

then the diagonalization of $\pazocal{F}$ gives

$\pazocal{F}^{n}u(0)=\frac{1}{\lambda_{1}-\lambda_{2}}\begin{bmatrix}&\\\
\mathbfit{x}_{1}&\mathbfit{x}_{2}\\\ &\\\
\end{bmatrix}\begin{bmatrix}\lambda_{1}^{n}&\\\
&\lambda_{2}^{n}\end{bmatrix}\begin{bmatrix}1\\\
-1\end{bmatrix}=\frac{1}{\lambda_{1}-\lambda_{2}}\begin{bmatrix}\lambda_{1}^{n+1}-\lambda_{2}^{n+1}\\\
\\\ \lambda_{1}^{n}-\lambda_{2}^{n}\end{bmatrix}$ (1.10)

This equation shows that, the number generated by applying the Fibonacci rule
$n$ times is

$\phi(n)=\frac{\lambda_{1}^{n+1}-\lambda_{2}^{n+1}}{\lambda_{1}-\lambda_{2}}$
(1.11)

since $\lambda_{2}<0$, the ratio converges

$\lim_{n\rightarrow\infty}\phi(n+1)/\phi(n)\rightarrow\lambda_{1}(=\tau)$
(1.12)

it means multiple applications of the Fibonacci process bring the ratio closer
to $\tau$. This $\tau$ is generally called the golden ratio. Now, note that
$\tau$ contains an irrational number $\sqrt{5}$. From the polar decomposition

$\pazocal{F}=U\Sigma V^{T}=(UV^{T})(V\Sigma V^{T})=(Q)(S)$ (1.13)

of $\pazocal{F}$, the components of the rotation matrix $Q$ are

$\mathbfit{u}_{1}=\frac{\mathbfit{x}_{1}}{|\mathbfit{x}_{1}|},\
\mathbfit{u}_{2}=\frac{\mathbfit{x}_{2}}{|\mathbfit{x}_{2}|},\
\mathbfit{v}_{1}=\frac{A^{T}\mathbfit{x}_{1}}{\sqrt{\lambda_{1}}|\mathbfit{x}_{1}|},\
\mathbfit{v}_{2}=\frac{A^{T}\mathbfit{x}_{2}}{\sqrt{\lambda_{2}}|\mathbfit{x}_{2}|},\
$ (1.14)

that each consist of the left singular vector matrix $U$ and the right
singular vector matrix $V^{T}$. Expanding this $Q$, the upper left component
is
$\cos\theta=\frac{4+\sqrt{5}}{\sqrt{2}\sqrt{35-15\sqrt{5}}}+\frac{-1+-\sqrt{5}}{\sqrt{2}\sqrt{30+15\sqrt{5}}}$
, which still contains an irrational number. In general,

$\cos\theta=\frac{a_{11}x_{11}^{2}+a_{12}x_{11}x_{12}}{\sqrt{\lambda_{1}}|\mathbfit{x}_{1}|^{2}}+\frac{a_{11}x_{21}^{2}+a_{12}x_{21}x_{22}}{\sqrt{\lambda_{2}}|\mathbfit{x}_{2}|^{2}}$
(1.15)

where $a_{ij}$ is the $(i,j)$ component of the original matrix. According to
the definition by Bohr (1946), if there exists an "_almost parallel shift_ "
$t$ satisfying

$\forall x\in\mathbb{R},\ \ \epsilon>0,\ \ \ |f(x+t)-f(x)|<\epsilon\ \ $
(1.16)

$f(x)$ is said to be an "_almost periodic_ " function. And
$f(x)=\sin(x)+sin(\tau x)$ has this quasi-periodic pattern, i.e., an ordered
but never repeated structure.

Assuming that transformations by the Fibonacci rule are evidence of a long
evolutionary history of causality, the results produced by such
transformations have the structure of an asymmetric transformation that has a
discernible basic pattern, but, at the same time, do not match no matter how
many times the transformation is repeated. On the other hand, the symmetry
group $S_{3}:G\rightarrow G$ can be described by six operations $\\{1,\ x,\
x^{2},\ y,\ xy,\ x^{2}y\\}$, and the result always coincides with one of the
rotations represented by $e^{\frac{n}{6}i2\pi}\ \ (n\in\\{1,\dots,6\\})$ . In
contrast to quasi-periodic transformations, symmetric transformations do not
increase in complexity in the sense of the example at the beginning, or only
in combination with quasi-periodic transformations. Therefore, they do not by
themselves provide direct evidence of a complex evolutionary process.

Thus, these arguments can be generalized and I propose

Theorem 1 ‘Operators whose eigenvectors (or eigenvalues) have an irrational
number in their  components constitute a class of functions with quasi-
periodic structure, $\Omega$ ’ (1.17) Therem 2 ‘The class of functions
$\Omega$ shows evidence of a long evolutionary history.’  (1.18)

Then I add a new interpretation of the _golden ratio_.

Theorem 3 ‘The golden ratio is a ratio at which the long history of evolution
by a function in the function class $\Omega$ converges.’ (1.19)

There are a total of $16$ $2\times 2$ matrices made by binary $1$ and $0$, of
which only $\pazocal{F}$ satisfies the condition (1.17). We can consider a
class of primitive recursive functions with these binary linear operators as
basic functions, and this issue is then discussed below.

The paradigm of non-reductionistic systems in which the parts and the whole
are interdependent was presented by Anderson (1972), where a new order arises
that did not exist in the subsystems alone. Cellular automata with
computational universality represent the behavior of finite interdependent
systems and provide a model for studying primitive recursive functions with
simple binary rules. For example, Cook (2004) proved that a 1-dimensional
cellular automata, where a cell becomes 1 if a right neighbor is 1 (rule 1)
and becomes 0 if both neighbors are 1 (rule 2), can emulate the Post normal
canonical system and then is Turing-complete.

$\begin{matrix}0&0&\underline{1}\end{matrix}\ \ \rightarrow\ \
\begin{matrix}0&\mathbb{\underline{1}}&1\end{matrix}\ \ \rightarrow\ \
\begin{matrix}\mathbb{\underline{1}}&1&\underline{1}\end{matrix}\ \
\rightarrow\ \ \begin{matrix}1&\mathbb{0}&1\end{matrix}$

It can be proved elementary that there is no finite binary linear operator
that expresses this production rule. For example, a linear transformation
representing (rule 1)

$\begin{bmatrix}1&1&\\\ &1&1\\\ &&1\end{bmatrix}$ (1.20)

works correctly on inputs $\begin{matrix}0&0&1\end{matrix}$,
$\begin{matrix}0&1&0\end{matrix}$, $\begin{matrix}1&0&0\end{matrix}$, and
$\begin{matrix}1&0&1\end{matrix}$, but $2$ is generated from inputs
$\begin{matrix}1&1&0\end{matrix}$, and $\begin{matrix}0&1&1\end{matrix}$ . A
complete representation does not exist under the classical algebraic group
laws.

On the other hand, if it is assumed that this output "value" simply gives a
probability distribution over a finite number of binary rules of the system,
it may be possible to claim that the operator correctly describes the
behavior. But this requires a change in logic or some different framework. One
possibility would have to assume that there is not just one proposition
$\pazocal{P}$ (cause) that implies a proposition $\pazocal{S}$ (generated
result).

Just as life was once thought to be a creation of God, but not many people
believe that now, so the idea that all mathematical logic is not the "language
of God" has become more common, especially since the establishment of non-
Euclidean geometry. With the birth of quantum theory at the beginning of the
20th century, it became clear that we live in a world with a non-classical
logic, while classical logic is organized on a very specific lattice, the
Boolean lattice.

According to Putnam (1968), unlike classical logic where

> "there was (idealizing) one proposition $\pazocal{P}$ which was true of
> $\pazocal{S}$ and such that every physical proposition about $\pazocal{S}$
> was implied by $\pazocal{P}$"

in the structure of non-distributive quantum logic on orthogonal modular
lattices, $(S_{1}\land S_{2}\land\cdots\land S_{R})$ can be true even if
$S_{1},\ S_{2},\ldots,\ S_{R}$ are all false. Therefore, if T1 is true, it is
possible for

$T_{1}\cdot(S_{1}\lor S_{2}\lor\cdots\lor S_{R})$ (1.21)

to be true and simultaneously

$T_{1}\cdot S_{1}\lor T_{1}\cdot S_{2}\lor\cdots\lor T_{1}\cdot S_{R}$ (1.22)

to be false (failure of the distribution law).

These two propositions (1.21) and (1.22), which were equivalent in classical
logic, are mapped into two different subspaces on the space
$\mathbb{L}(\pazocal{H})$ representing all propositions of the system. The
theory was formulated by Birkhoff and Neumann (1936) as a system where
arbitrary observable $A$ is described by self-adjoint operators. In this
projection logic on Hilbert spaces, the mapping of propositions represented by
state vectors to closed subspaces expresses (probabilistic) truth or
falsehood.

After a few decades, the universal machine model of quantum computers was then
formulated by Deutsch (1985) for the first time, but since then it has been
studied mainly for the complete simulation of the time evolution of physical
systems, and there are not many theoretical models of computational languages
that explicitly use the non-classical logic structure of quantum logic.
Including QTM, Many Turing-equivalent systems exist. Shannon (1956) posed the
problem of a minimal Turing machine, which was later proved by Minsky (1961)
to require two tapes and three variables. On the other hand, the logic or
language system that can be constructed on a universal computer remains a
challenge for future research.

I call a transformation that expresses rules that provide evidence of deep
evolution, "_universal algorithms_ ". These rules include a class of quasi-
periodic transformations $\Omega$. It seems reasonable to rely on computations
that assume qubits with the "genuine" microscopic logical structure of nature
to verify the existence of universal algorithms and to decode them. This is
because, in order to verify the "genuine" transformation rule, it is
reasonable to choose the natural empirical logic for obtaining the class of
basic functions that make up a general recursive function.

"_Universal_ ", in the sense of computability presented by Deutsch (1985),
i.e., a computation in which the physical system is representable, refers to a
transformation algorithm that exists universally (nonlocally) in nature. It
also means that its generation rules are independent of the decoder.
Hofstadter (1979) discussed this universality of "meaning" mainly by the Post
normal canonical system. The theoretical models of computers $\pazocal{C}$ and
their language $\beta$, that performs deductive reasoning to derive
transformation algorithms from the generated results, based on quantum logic
are presented in the following chapter. $\pazocal{C}$ and $\beta$ process
event-state systems consisting of exponential combinations of propositional
elements in a nonclassical logic structure, through massively parallel
computation. In this structure, there is no single proposition implying a
certain generated outcome.

On the issue of logical depth, Benett (1994) formulated algorithmic complexity
as an absolute measure that includes correlations between different parts. One
background is that a few percent of the code that exists between distant parts
of the genome of higher organisms does not strictly determine the structure of
any part, but at the same time contains mutual information to convey the
relationships between the parts. The detection of nonlocal correlations, or
"_coherence_ ," is essential to distinguish truly profound transformations
from coincidentally generated results. In short, if a "transformation" occurs
similarly in one place as in another, then the transformation is justified in
being non-accidental. When describing the interpretation of physical reality,
Mermin (1999) argued that correlations are the only objective and reducible
units of composition. In other words, we can say the following. There is no
absolute causality in a non-element-reductive complex system in which the
parts and the whole are interdependent, and a complete description of the
system can only be achieved through correlations between the subsystems.

The computation language $\beta$ and the computer $\pazocal{C}$ are based on
the composite system spanned by the propositional dimension of
$\pazocal{O}(\prod\ M_{(n)})$, which is exponentially larger than the
dimension of the Hilbert space created by each of the subsystems, i.e., the
sum $\pazocal{O}(\sum M_{(n)})$ of the _propositional elements_ described in
the next chapter. By this difference, the local (or nonlocal) correlation
among multiple subsystems (propositional elements) is expressed.

In the subsequent Chapter II, Section 2.1 first formulates an event-state
system and a recursive model of the universal computer $\pazocal{C}$. The
section also defines the variable types placed in the address space for the
language $\beta$, including correlated state types or combined state types
generated by multiple propositional elements. Section 2.2 discusses and
prooves Turing completeness in $\pazocal{C}$. Section 2.3 presents a
mathematical model of a linear logical operator that realizes the conditional
expressions in the non-classical logic structure presented at the end of
Section 2.1.

For a detailed description of quantum logic in Hilbert spaces (or orthomodular
lattices) and quantum information theory, see Cohen (1989) and Peres (1995).

##  2 Computational Language $\beta$

###  2.1 Composite Event-state Systems and the Model of Computers
$\pazocal{C}$

Computers $\pazocal{C}$ have an infinite memory space
$\ket{\xi}(i\in\mathbb{N}_{0})$ and a finite bit instruction register
$\ket{\gamma}$ that takes the general form of

$g(\mathbfit{i},\mathbfit{x},\mathbfit{y},\phi(t),k)=\ket{\underbrace{\mathbfit{i}}_{\text{instructions}};\
\underbrace{x_{1},\ldots,x_{\bar{M}}}_{\text{source states}};\
\underbrace{y_{1},\ldots,y_{\bar{M}}}_{\text{target states}};\
\underbrace{\phi_{1}(t),\ldots,\phi_{\bar{M}}(t)}_{\text{joint spectra at
t}};\ \underbrace{k}_{\text{indices}}}=\ket{\gamma}$ (2.1)

where $\mathbb{N}_{0}\in\\{0,1,\ldots\\}$, $\mathbb{Z}_{2}\in\\{0,1\\}$,
$x_{m},\ y_{m}\in\mathbb{Z}_{2}$, $\phi(t)\in\mathbb{N}_{0}$, $i,\
k\in\mathbb{N}$, and $\bar{M}$ is the number of computational bases to be
described below.
Their computational basis state is a composite system of $N$ subsystems, each
with $M^{(n)}(n\in\mathbb{N}_{N})$ $2$-state qubits

$\ket{\psi}=\ket{sub\ 1}\otimes\dots\otimes\ket{sub\
N}\in\pazocal{H}^{\bar{M}}\ \ (N\in\mathbb{N})$ (2.2)

defined on $\bar{M}$-dimensional Hilbert space $\pazocal{H}$. Each subsystem
is represented as

$\ket{sub\,n}=\sum_{i=0}^{M^{(n)}-1}\alpha_{i}^{(n)}\ket{i}$ (2.3)

Each $M^{(n)}$ is assumed to satisfy the condition
$M^{(n)}=2l^{(n)}(l\in\mathbb{N})$. Thus the number of standard bases for the
composite system $\ket{\psi}$ is

$\bar{M}=M^{(1)}M^{(2)}\dots M^{(N)}=\prod_{n=1}^{N}M^{(n)}$ (2.4)

This number is obviously larger than the sum of the number of states each
subsystem originally had, $\sum_{n=1}^{N}M_{(n)}$. As mentioned in Chapter 1,
this is because "_correlation information_ " is contained in this difference,
and the _correlated states_ as a variable type defined by this are described
below. This correlational information is essential and is the only universal
logical foundation that encompasses causality as its substructure.

The set of standard bases for $\pazocal{C}$ is denoted by

$\pazocal{N}=\\{\ket{m}|m\in\mathbb{N}_{\bar{M}}\\}$ (2.5)

Each element $\ket{m}$ of $\pazocal{N}$ corresponds to a state $s_{m}$ on
$\bar{M}$-dimensional Hilbert space logic $\mathbb{L}(\pazocal{H})$, i.e., a
$1$-dimensional closed subspace of $\mathbb{L}$. $\mathbb{L}(\pazocal{H})$ is
the semi-ordered set of all subspaces of $\pazocal{H}\in\mathbb{C}^{\bar{M}}$.
Each state $s_{m}$ can be regarded as representing a unique logical
conjunction of the set of all subsystem states

$\Theta=\\{\overbrace{s_{0}^{(1)},\dots,\ s_{\bar{M}^{(1)}-1}^{(1)}}^{states\
of\ subsystem\ 1},\ \overbrace{s_{0}^{(2)},\dots,\
s_{\bar{M}^{(2)}-1}^{(2)}}^{states\ of\ subsystem\ 2},\ \dotsm\ ,\
\overbrace{s_{0}^{(N)},\dots,\ s_{\bar{M}^{(N)}-1}^{(N)}}^{states\ of\
subsystem\ N}\\}$ (2.6)

Elements of $\Theta$ correspond to $\ket{i}$ in (2.3). Let us refer to these
_propositional elements_ as the most primitive units of event-state systems.
Examples of propositional elements are a side of a coin, whether a right
neighbor cell in a cellular automaton is active or not, and whether a note is
written as $E$ or not.

Here a minimal composite system $\ket{\gamma_{AB}}$ with two distinct 1-qubit
subsystems $\ket{\gamma_{A}}$ and $\ket{\gamma_{B}}$ is considered. Let
$s_{A}$ and $s_{B}$ be the corresponding event states (or propositional
elements) to $\ket{\gamma_{A}}$ and $\ket{\gamma_{B}}$, respectively.
$\ket{\gamma_{A}}$ and $\ket{\gamma_{B}}$ each represent a subspace spanned by
two orthogonal $1$-dimensional closed subspaces, which can be viewed as units
representing the states $s_{A}\lor s_{A}^{\perp}$ and $s_{B}\lor
s_{B}^{\perp}$ on $\mathbb{L}(\pazocal{H})$. Thus, the composite system
$\ket{\gamma_{A}}\otimes\ket{\gamma_{B}}$ constitutes a $\sigma$-complete
Hilbert space logic whose states are all $\sigma$-additive sets of $\\{s_{a},\
s_{b}\\}$, and is represented as follows

$\ket{\gamma_{A}}\otimes\ket{\gamma_{B}}=\left(\alpha_{A}\begin{bmatrix}1\\\
0\end{bmatrix}+\beta_{A}\begin{bmatrix}0\\\
1\end{bmatrix}\right)\otimes\left(\alpha_{B}\begin{bmatrix}1\\\
0\end{bmatrix}+\beta_{B}\begin{bmatrix}0\\\
1\end{bmatrix}\right)=\begin{bmatrix}\alpha_{A}\alpha_{B}\\\
\alpha_{A}\beta_{B}\\\ \beta_{A}\alpha_{B}\\\ \beta_{A}\beta_{B}\\\
\end{bmatrix}=\begin{bmatrix}p(s_{A}\land s_{B})^{\frac{1}{2}}\\\ p(s_{A}\land
s_{B}^{\perp})^{\frac{1}{2}}\\\ p(s_{A}\land s_{B})^{\frac{1}{2}}\\\
p(s_{A}^{\perp}\land s_{B}^{\perp})^{\frac{1}{2}}\\\
\end{bmatrix}\begin{matrix}(a)\\\ (b)\\\ (c)\\\ (d)\end{matrix}$ (2.7)

where $p(s)$ is the probability of state $s$ occuring. Note that $s_{A}$ and
$s_{B}$ are distinct propositions, and if both pairs (a) and (d) (or (b) and
(c)) occur with high probability, subsystems $\gamma_{A}$ and $\gamma_{B}$ are
in "_correlated states_ ".

When some distinct propositional elements are correlated, there exist states
in the system that can be called _correlated states_. Redefining them as a
single state (or we might be able to say a "concept") helps to decipher
complex systems by making the relationship between time-varying states
clearer. Analogically, when each state refers to each finger of the right hand
being stretched, the event-state system is decoded by redefining the coupled
state as "the right hand is open”.
Let the bases corresponding to $s_{p}$ and $s_{q}$ be $\ket{p}$, $\ket{q}$.
The bases $\ket{p}$, $\ket{q}$ are orthogonal to each other on these spanned
minimal Hilbert spaces $\pazocal{H}^{4}$, but the basis $\ket{p\land q}$ by
linear combination of $\ket{p}$, $\ket{q}$ is no longer orthogonal to the
original $\ket{p}$, $\ket{q}$ (Figure 1). In non-distributive logic on
orthomodular lattices, two propositions $p$, $q$ are compatible if and only if

$p=(p\land q)\lor(p\land q^{\prime})\ and\ q=(q\land p)\lor(q\land
p^{\prime})$ (2.8)

Conversely, if $p$ and $q$ are incompatible, it means that $(p^{\prime}\land
q)$ and $(p^{\prime}\land q^{\prime})$ are true even if $p$ is true.

Figure 1:  $p$ and $q$ are orthogonal on $\pazocal{H}^{4}$ , while $p$(or $q$)
and $p\land q$ are not orthogonal (i.e., $p$ and $q$ are compatible, while
$p$(or $q$) and $p\land q$ are incompatible). $\pazocal{H}^{4}$ is the
simplest non-trivial composite system that can represent correlation
information. Note that $\ket{p}$, $\ket{q}$,$\ket{p^{\prime}}$,
$\ket{q^{\prime}}$ are orthogonal to each other in both left and right
figures.

As mentioned in Chapter 1, in the structure of non-distributive quantum logic
on orthogonal modular lattices, $(S_{1}\land S_{2}\land\cdots\land S_{R})$ can
be true even if $S_{1},\ S_{2},\ldots,\ S_{R}$ are all false. The
computational language $\beta$ provides a formalization for manipulating this
logical structure, as shown below.

Here, I take a subset $\pazocal{N}^{\prime}\subset\pazocal{N}$ and represent
its linear combination by

$\ket{\Phi}=\sum_{m\in\mathbb{Z}_{\lor}}\alpha_{m}\ket{m}\ \
(\mathbb{Z}_{\lor}=\\{m\ |\ m\ \text{is\ an\ index\ of\ basis\ states\ in\
}\pazocal{{N}^{\prime}}\\})$ (2.9)

where $\mathbb{Z}\lor$ is the set of indices of the states in
$\pazocal{N}^{\prime}$. In quantum theory, $\ket{\Phi}$ is a pure state when
it is contained in the eigenspace
$E(A,\lambda)=\\{\mathbfit{v}\in\pazocal{H}^{\bar{M}}\ and\
A\mathbfit{v}=\lambda\mathbfit{v}\\}$ of the observable (Hermitian operator),
and constitutes a closed $1$-dimensional subspace on the subspace spanned by
$\pazocal{N}^{\prime}$, which corresponds to $\ket{p\land q}$ as mentioned
above. To clearly distinguish this newly generated _combined state_ from
stochastic mixed states, in which it is simply uncertain which of
$\\{s_{m}|m\in z_{\lor}\\}$ will occur, Let us denote it by $\bigwedge_{m\in
z_{\lor}}s_{m}$. The combined states are assigned to a different variable
memory space on $\ket{\xi}$ from the original states. Thus the semantics of
combined and mixed states are

1) if $\ket{\Phi}$ is a _combined state_ , then $\bigwedge_{m\in
z_{\lor}}s_{m}$ is true (false) irrespective of $s_{m}(m\in z_{\lor})$

2) if $\ket{\Phi}$ is a _mixed state_ , then $\exists m\in z_{\lor}$, $V_{m\in
z_{v}}s_{m}$ is true.

In particular, in 1), two propositions that are equivalent in classical logic
are mapped to different closed subspaces on $\mathbb{L}(\pazocal{H})$. Thus,
the combined states connected by a conjunction are evaluated independently of
the truth or falsehood of the individual propositions that form the subset. An
example of program codes that represent this logic is

1 if any($s_{m}$) % any event in $\\{s_{m}\\}$
2 print(‘at least one event of $s_{p},s_{q},s_{r},\ldots$ is true’)
3 elif $s_{m}^{\ast}$ % combined events $\bigwedge_{m\in z_{\lor}}s_{m}$
4 print(‘$s_{p}\land s_{q}\land s_{r},\ldots$ is true’)

Note that the fourth line is never executed in a conventional framework.
Let us assume, for example, that we define the basis state $\mathbfit{c}_{1}$
of the minimal propositional element on some $\pazocal{H}^{2}$ along the
eigenvectors of any operator belonging to the class $\Omega$ defined in
(1.17). Then, another combined state $\mathbfit{c}_{2}$ is generated by a
unitary transformation with $\mathbfit{c}_{1}$ as the ground basis. This
recursive operation can generate an infinite number of basis states through
repeated rotations involving irrational numbers. As a result, an infinite
number of different $1$-dimensional subspace propositions can be generated in
the plane, which never coincide.

Let Hermitian operators that output the eigenvalue $m$ for the _standard
state_ $s_{m}\ (m\in\mathbb{Z}_{\bar{M}})$ be

$A=\sum_{m\in\mathbb{Z}_{\lor}}m\ket{m}\bra{m}$ (2.10)

then, by the Spectrum Theorem, observables that measure the combined state
$\ket{\Phi}$ can be written as

$A^{\ast}=\sum_{m\in\mathbb{Z}_{\lor}}m^{\ast}\ket{m^{\ast}}\bra{m^{\ast}}+\sum_{m\in\mathbb{Z}_{\lor}^{C}}m\ket{m}\bra{m}$
(2.11)

Let $\pazocal{N}^{\prime}\subset\pazocal{N}$ be the basis states selected to
generate the combined state $\ket{\Phi}$, and $\mathbb{Z}_{\lor}$ be its set
of indices. $\mathbb{Z}_{\lor}^{C}$ then represents the complement of
$\mathbb{Z}_{\lor}$ within $\mathbb{Z}_{\bar{M}}$. Also let

$m^{\ast}=M+\prod_{m\in\mathbb{Z}_{\lor}}e_{m}\ \ (e_{m}\ \text{is\ the\ m\
th\ prime\ number})$ (2.12)

be the index of the newly generated $|\mathbb{Z}_{\lor}|$ orthonormal bases.
New bases including $\ket{\Phi}$ can be obtained by applying the Gram-Shmidt
process

$g(\Phi,\ \Phi^{\prime},\
\cdots)=\ket{\Phi}-\left(\cdots+\frac{\braket{\Phi^{\prime\prime}}{\Phi}}{\braket{\Phi^{\prime\prime}}{\Phi^{\prime\prime}}}\ket{\Phi^{\prime\prime}}+\frac{\braket{\Phi^{\prime}}{\Phi}}{\braket{\Phi^{\prime}}{\Phi^{\prime}}}\ket{\Phi^{\prime}}\right)\
\ (\text{\'{\ }denotes\ the\ number\ of\ recursions})$ (2.13)

recursively $|\mathbb{Z_{\lor}}|-1$ times. Let the linear combination of these
newly generated bases be

$\ket{\Phi}=\sum_{m^{\ast}\in\mathbb{Z}_{\lor}}\alpha_{m^{\ast}}\ket{m^{\ast}}$
(2.14)

A unique index can then be output through the norm $m^{*}$ of
$A^{\ast}\ket{\Phi}$.

When a combined state $\Phi$ is generated, an orthogonal basis to $\Phi$ is
also automatically defined in the space $\mathbb{L}(\pazocal{H})$. In the
variable memory space corresponding to this basis, a complementary proposition
that is always false when $\Phi$ is true is assigned along with the variable
definition. These orthogonal propositions can be called "complementary pairs"
and are assigned a special type in the formalization of $\beta$.

$\pazocal{C}$ is a mathematical model for massively parallel computing that
defines combined or correlated states as logical propositions of a composite
probability distribution, and enables programmatic manipulation of these
states, which are not intuitively accessible to us. $\pazocal{C}$ is designed
to determine a class $<E,\,S>$ of event-state systems through large-scale
linear properties, while finding nonlocal correlations contained in the
complex composite system. $\pazocal{C}$ and its computational language,
$\beta$, deduce a generation or transformation process from a finite number of
possible combinations of states at a given timestep $t$ in parallel. For this
purpose, the computational basis state $\ket{\psi}(t)$ at each time step is
manipulated by _programs_
$\pazocal{P}:\ket{\psi}(t)\rightarrow\ket{\psi}(t+1)$, which are called
_hypotheses_. And spectra

$\ket{\phi}(t)=\\{\phi_{i}\\}\ \ (i\in\mathbb{N}_{0}^{2\bar{M}-1})\ \ \ \ \ \
\\\ \begin{cases}\phi_{2m-2}:\ \text{a\ frequency\ of\ states\ }s_{m}\\\
\phi_{2m-1}:\ \text{a\ frequency\ of\ states\ }s_{m}^{\perp}\end{cases}$
(2.15)

of joint probability distributions are output through multiple projection
measurements. These spectra are then separated into mutually exclusive states
$\ket{\phi}(t)\rightarrow\\{\ket{\phi}(t,k)\\}$ and stored in memory
$\ket{\xi}$ to serve as input values to the instruction register at the next
time step. The Von-Neumann entropy $\pazocal{S}$ of density operators are
computed from the normalized joint probability distribution
$\ket{\phi}^{\prime}(t)$ and the set of bases $\pazocal{N}$. When the entropy
becomes $\pazocal{S}<\epsilon$, the class $<E,\,S>$ of the event-state system
is "_decided_ " and the system is said to have "_converged_ ". The number of
steps $T$ required for the hypothesis to make the system converge can be
regarded as the _logical depth_ of the system.

###  2.2 Turing Completeness of $\pazocal{C}$

The instruction register $g(\mathbfit{i},\mathbfit{x},\mathbfit{y},\phi(t),k)$
described in (2.1) is equivalent to a primitive recursive function

$\begin{cases}h(\ket{\phi},\ 0)=f(\ket{\phi})\\\ h(\ket{\phi},\
k+1)=g(\ket{\phi},\ k,\ h(\ket{\phi},k))\end{cases}$ (2.16)

Furthermore, given a function

$g:\pazocal{H}\in\mathbb{C}^{M}\rightarrow\mathbb{N}\ \ \ \ \\\
f:\mathbb{N}\rightarrow\mathbb{N}$ (2.17)

we can define a function

$f^{\ast}=gfg^{-1}\ \text{and}\ f^{\ast}:\mathbb{N}\rightarrow\mathbb{N}$
(2.18)

that maps any transformation on $\pazocal{H}\in\mathbb{C}^{N}$ to
$\mathbb{N}\rightarrow\mathbb{N}$. Thus, a set of primitive operators that can
compute any point on $\pazocal{H}$ creates a class of $\mu$-recursive
functions. Such a set of operators is trivially definable (e.g., invertible
operators in $N$-dimensional Hilbert space which gives arbitrary rotation and
expansion), and thus $\pazocal{C}$ is Turing-complete.

The important problem is that the operator giving the transformation of any
$\mathbb{N}\ \text{(time\ step\ t)}\rightarrow\mathbb{N}\ \text{(time\ step\
t+1)}$ is nontrivial. For example, it is not easy to deduce the existence of
an operator $\pazocal{F}$, just from the generated result $\mathbb{N}\
\text{(time\ step\ t >\\!> 0)}$, which gives the convergence of a point
sequence on $\pazocal{H}$ of the formula (1.7). Determining how the system is
reasonably transformed is the primary motivation of $\pazocal{C}$ and $\beta$.

###  2.3 Conditional Operators to Joint Probability Distributions

The hypothetical program to verify the transformation
$\ket{\psi}(t)\rightarrow\ket{\psi}(t+1)$ proceeds through a recursive
execution that consists mainly of a conditional expression that takes a joint
probability distribution $\ket{\psi}(t)$ at any time $t$ as an argument, and a
transformation that updates any state of the subsystem. Conditional
expressions evaluate the boolean values of variables defined as the combined
and correlated states as well as the standard state, and are expressed in
terms of input values to the instruction register
$g(\mathbfit{i},\mathbfit{x},\mathbfit{y},\phi(t),k)$. The operators that
realize these generalized representations

$\text{if}\ \bigvee_{m\in\pi(\mathbfit{x})}s_{m}\ \text{is\ true\ (false),\
perform}\ X\ \text{to}\ \mathbfit{y}$ (2.19)

in $\beta$ are discussed below.
Here, $\bigvee_{m\in\pi(\mathbfit{x})}s_{m}$ is any standard state pair
$s_{u}\lor s_{v}\lor s_{w}\lor\cdots$, and $\mathbfit{x}$ is the source states
input to $g(\mathbfit{i},\mathbfit{x},\mathbfit{y},\phi(t),k)$. The source
states are specified by $\mathbb{Z}_{2}$, and hence

$\pi(\mathbfit{x})=\\{m\in\bar{M}\ |\ x_{m}\in\mathbfit{x}\ and\ x_{m}=1\\}$
(2.20)

Firstly, a subclass of (2.19)

$\text{if}\ s_{m}\ \text{is\ true\ (false),\ perform}\ X\ \text{to}\
\mathbfit{y}$ (2.21)

is realized as follows. Since $\ket{\phi}=\\{\phi_{i}\\}\
(i\in\mathbb{N}_{0}^{2\bar{M}-1})$ outputs the spectrum of the joint
probability distribution of $s_{m}$ for $i=2m-1$ and $s_{m}^{\perp}$ for
$i=2m-1$, the $\bar{M}\times 2\bar{M}$ operator can be thought as follows:

$\mathbfit{V}(\ket{\phi})=\begin{bmatrix}1&0&\cdots&&&&&\\\
&0&1&0&\cdots&&&\\\ &&0&0&1&\cdots&&\\\ &&&&0&0&\cdots&\\\
&&&&&&\ddots&\end{bmatrix}$ (2.22)
$\mathbfit{V^{\prime}}(\ket{\phi})=\begin{bmatrix}0&1&\cdots&&&&&\\\
&0&0&1&\cdots&&&\\\ &&&0&0&1&\cdots&\\\ &&&&&0&0&\\\
&&&&&&\ddots&\end{bmatrix}$ (2.23)

such that the matrix element $a_{ij}$ is $\delta^{(i,\ j)}_{(\forall m,\
2m-1)}\ (m\in\bar{M})$ if "is true" and $\delta^{(i,\ j)}_{(\forall m,\
2m-2)}\ (m\in\bar{M})$ if "is false". Thereby, as in

$\mathbfit{V}(\ket{\phi})=\begin{bmatrix}s_{1}\\\ s_{2}\\\ \vdots\\\
s_{\bar{M}}\\\ \end{bmatrix},\
\mathbfit{V^{\prime}}(\ket{\phi})=\begin{bmatrix}s_{1}^{\perp}\\\
s_{2}^{\perp}\\\ \vdots\\\ s_{\bar{M}}^{\perp}\\\ \end{bmatrix}$ (2.24)

the output organized into $s_{m}$ and $s_{m}^{\perp}$ is

$\ket{\phi^{\prime}}=\\{\phi^{\prime}_{i}\\}\ \ (i\in\mathbb{\bar{M}}-1)$
(2.25)

Then the operator $\mathbfit{IF}^{x}_{\mathbf{y}}$, which outputs
$\phi_{2m-2}$ (or $\phi_{2m-1}$) $\in\mathbb{N}$ for the specified target
state $\mathbfit{y}$ only if the specified source state $x$ is true (or
false), and $0$ for all others, is an $\bar{M}\times\bar{M}$ matrix such that
the element $a_{i,\ j}$ is $\\{\delta^{\phi^{\prime}_{x}y_{i}}_{1}\\}\
(i\in\mathbfit{y})$ given by

$\mathbfit{IF}^{\ast}\mathbfit{V}^{(^{\prime})}(\ket{\phi})=\begin{bmatrix}0&\cdots&\delta^{\phi^{\prime}_{x}y_{1}}_{1}&\cdots&0\\\
0&\cdots&\delta^{\phi^{\prime}_{x}y_{2}}_{1}&\cdots&0\\\
\vdots&&\vdots&&\vdots\\\
0&\cdots&\delta^{\phi^{\prime}_{x}y_{m}}_{1}&\cdots&0\\\
\vdots&&\vdots&&\vdots\\\
0&\cdots&\delta^{\phi^{\prime}_{x}y_{\bar{M}}}_{1}&\cdots&0\\\
\end{bmatrix}\mathbfit{V}^{(^{\prime})}(\ket{\phi})=\begin{bmatrix}\delta^{\phi^{\prime}_{x}y_{1}}_{1}\phi_{1}^{\prime}\\\
\vdots\\\
\delta^{\phi^{\prime}_{x}y_{\bar{M}}}_{1}\phi_{\bar{M}}^{\prime}\end{bmatrix}$
(2.26)

where $\mathbfit{V}^{(^{\prime})}$ represents $\mathbfit{V}$ or
$\mathbfit{V}^{\prime}$. To generate the conditional expression given in
(2.19) as an extension to the super class of (2.21), we only need to take the
direct sum of
$\mathbfit{IF}^{x_{m}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})$
for all specified $x_{m}\in\mathbfit{x}$. Therefore, this can be written as

$\mathbfit{IF}^{x_{u}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\oplus\mathbfit{IF}^{x_{v}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\oplus\cdots\mathbfit{IF}^{x_{z}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})=\bigoplus_{m\in\pi({\mathbfit{x}})}\mathbfit{IF}^{x_{m}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})$
(2.27)

Regarding logical product

$\text{if}\ \bigwedge_{m\in\pi(\mathbfit{x})}s_{m}\ \text{is\ true\ (false),\
perform}\ X\ \text{to}\ \mathbfit{y}$ (2.28)

similarly the output
$\\{\delta^{\phi^{\prime}_{x}y_{m}}_{1}\phi_{\bar{m}}^{\prime}\\}\
(m\in\bar{M})$ can be obtained as in (2.26) by taking the Adamard product of
$\mathbfit{IF}^{x_{m}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\
(m\in\pi(\mathbfit{x}))$ , i.e.,

$\mathbfit{IF}^{x_{u}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\odot\mathbfit{IF}^{x_{v}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\odot\cdots\mathbfit{IF}^{x_{z}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})=\bigodot_{m\in\pi({\mathbfit{x}})}\mathbfit{IF}^{x_{m}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})$
(2.29)

Furthermore, the conditional expression by any combination of logical ORs
$\text{if}\ \bigvee_{m\in\pi(\mathbfit{x})}s_{m}$ and ANDs $\text{if}\
\bigwedge_{m\in\pi(\mathbfit{x})}s_{m}$ can be constructed from any
concatenation by $\oplus$ and $\odot$ for
$\mathbfit{IF}^{x_{m}}_{\mathbfit{y}}\mathbfit{V}^{(^{\prime})}(\ket{\phi})\
(m\in\pi(\mathbfit{x}))$ .

##  3 Conclusions and Further Research

To decodeh highly complex symbol sequences generated by systems with
computational universality, a general computational method is proposed that
derives a transformation process (generation rule) from the generation
results. This transformation process gives complexity or logical depth in
terms of computation time, which efficiently provides
$\mathbb{Z}\rightarrow\mathbb{Z}$ generation. I formulated a universal
diductive machine model $\pazocal{C}$ and a computational language $\beta$
that relies on the structure of non-distributive quantum logic
$\mathbb{L}(\pazocal{H})$ in Hilbert spaces or orthogonal modular lattices,
i.e., a non-boolean logic structure in which no proposition implying a
generated outcome can be uniquely determined. The $\pazocal{C}$ and its
language $\beta$ performs massively parallel computations on event-state
systems consisting of exponential combinations of propositional elements,
which are represented by correlations between subsystems. To construct and
operate this basic framework, a model of the theory of computation is
presented, as well as the conditional logical operations performed on the
framework among its other main operations. At the same time, as a
generalization of functions that show evidence of a long evolutionary history,
a class of operators with quasi-periodic structures whose eigenvectors (or
eigenvalues) include irrational numbers as components is defined as $\Omega$.

As for future prospects, first of all, theoretically, it will be necessary to
study pure mathematics, such as the theory of rings of operators for the
symmetry of transformations or something like surreal numbers with non-
classical algebraic rules of numbers. Second, from an empirical point of view,
detailed observations of real (terrestrial) natural systems will be needed,
especially observations of communication between systems that perform life
activities in the broadest sense of the word. Third, it will be necessary to
develop a computational language and computer itself to substantiate or
simulate the model.

This work has established a foothold, but it is only the beginning, and the
road ahead is endless. Is what lies beyond that something like what Mandelbrot
(2012) calls "the ultimate knowledge, likened to the Holy Grail or the Golden
Fleece"? No one knows. The question raised at the beginning of this paper is
one that I have been asking myself since I was a teenager. I would like to
thank all those who have supported and given me the opportunity to do this
research.

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