# Mass and decay width of $T_{ccs}$ from symmetries

Mitsuru Tanaka<EMAIL_ADDRESS>Department of Physics, Nagoya
University, Nagoya 464-8602, Japan  Yasuhiro Yamaguchi
<EMAIL_ADDRESS>Department of Physics, Nagoya University,
Nagoya 464-8602, Japan  Kobayashi-Maskawa Institute for the Origin of
Particles and the Universe, Nagoya University, Nagoya, 464-8602, Japan  Meson
Science Laboratory, Cluster for Pioneering Research, RIKEN, Hirosawa, Wako,
Saitama 351-0198, Japan  Masayasu Harada<EMAIL_ADDRESS>Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya, 464-8602, Japan  Department of Physics, Nagoya
University, Nagoya 464-8602, Japan  Advanced Science Research Center, Japan
Atomic Energy Agency, Tokai 319-1195, Japan

###### Abstract

We analyze the mass and width of the doubly heavy tetraquark $T_{ccs}$
composed of a heavy diquark and a light quark cloud with strangeness with
assuming that a color anti-triplet heavy diquark is a dominant component of
the doubly charmed tetraquarks $T_{cc}$ and $T_{ccs}$. We construct an
effective Lagrangian for masses of heavy hadrons based on the superflavor
symmetry between the doubly heavy tetraquarks and the singly heavy baryons
with including the terms which simultaneously break the heavy-quark and light
flavor symmetries, and predict the mass of $T_{ccs}$ as $M(T_{ccs})=4057\pm
40$ MeV. The comparison of this prediction with future experimental
observation will give a clue to understand the color structure of the heavy
diquark. We also predict the spin-averaged mass of $\Omega_{cc}$
($J^{P}=1/2^{+},3/2^{+})$ as $M(\Omega_{cc})=3760\pm 18\,$MeV. We next
calculate the decay width of $T_{ccs}$, based on solely the light flavor
symmetry, as $\Gamma(T_{ccs})=1.2\pm 0.3$ MeV.

Keywords

superflavor symmetry, heavy quark symmetry, flavor symmetry, diquark, doubly
heavy tetraquark

##  I INTRODUCTION

The discovery of $X(3872)$ [1] marked the beginning of numerous exotic hadron
discoveries in the heavy quark sectors, yet their structure remains poorly
understood. Hadrons with exotic structures beyond ordinary baryons ($qqq$) and
mesons ($q\bar{q}$) were already indicated by Gell-Mann and Zweig in the 1960s
[2, 3, 4, 5, 6]. Possible structures of the multiquark state being a color
singlet have been discussed in many literatures (See, for a reviews, e.g.
Refs. [7, 8]). The compact multiquark has been investigated as a color singlet
state of few-body multiquark systems by the constituent quark model etc (See
e.g. Refs. [9, 10, 11, 12, 13, 14, 15, 16]). The emergence of the hadronic
molecules as a deuteron-like state, discussed as a deuson in Ref. [17], is
expected near the thresholds. In fact many candidates of a hadronic molecule
have been reported in the experimental studies as the $XYZ$ tetraquarks being
a meson-meson state and the $P_{c}$ pentaquarks being a meson-baryon one.
Investigating the exotic structures would lead to an understanding of the QCD
phenomena such as color confinement.

The doubly charmed tetraquark $T_{cc}^{+}$ was reported in the LHCb experiment
in 2021 [18, 19]. The reported state is consistent with a genuine exotic
hadron having a flavor structure $cc\bar{u}\bar{d}$. The spin and parity of
$T_{cc}^{+}$ are determined to be $J^{P}=1^{+}$, and the LHCb considers
$T_{cc}^{+}$ as an isoscalar. The mass of $T_{cc}^{+}$ is
$3874.817\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ close to the
$D^{0}D^{\ast+}$ threshold. The decay to $D^{0}D^{0}\pi^{+}$ has been
confirmed, with a decay width of $48\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
[18] or $410\,$keV [19]. Furthermore, the LHCb analysis supports that
$T_{cc}^{+}$ decays to $D^{0}D^{0}\pi^{+}$ via the intermediate state
$D^{*+}$.

Since the discovery of $T_{cc}^{+}$, research on the doubly heavy tetraquarks
(DHTs) has been actively conducted [7]. However at present, no clear answer
has been obtained regarding the structure of DHTs. For example, analyses based
on the hadronic molecular model, which assumes that $T_{cc}^{+}$ is a loosely
bound state with $D$ and $D^{*}$, have been conducted [20, 21]. This is due to
the fact that the mass of $T_{cc}^{+}$ resides in the vicinity of the
$DD^{\ast}$ threshold. On the other hand, a compact tetraquark structure of
DHT is considered [22], based on the diquark picture proposed by Jaffe [23].

Symmetries such as flavor symmetry and chiral symmetry play a crucial role in
the classification of hadronic states. For exotic hadrons including heavy
quarks, symmetries that emerge in the heavy quark limit, such as heavy quark
symmetry (HQS) (see e.g. Ref. [24]) and superflavor symmetry (SFS) [25, 26,
27], are considered potentially useful for understanding the structures of
exotics. The superflavor symmetry emerges in an exchange of an anti-heavy
quark and a heavy diquark with the same color configuration $\bar{\boldmath
3}$ in the heavy quark limit. Due to the sufficiently large mass of the heavy
diquark, the spin–dependent color magnetic force is negligible, and does not
contribute in the heavy quark limit. In this context, the heavy diquark and an
anti-heavy quark behave as the static color $\bar{3}$ source and are
equivalent in terms of color interaction. The property of hadrons remains
invariant under the interchange of the heavy diquark and anti heavy quark. We
will refer to hadrons related under this superflavor symmetry as superflavor
partners. For instance, an anti-heavy meson(HM) $\bar{Q}q$ and a doubly heavy
baryon(DHB) $QQq$ are superflavor partners. Another color representation for
diquarks, $6$, is also allowed in DHTs but not in ordinary hadrons. If
diquarks take color $6$ representation, the superflavor symmetry does not
arise. Hence, superflavor symmetry might shed light on the color configuration
of exotic hadrons. We think that understanding the color configuration of the
diquark in DHTs is important because the interaction changes according to the
color representation.

In order to analyze the mass spectrum of DHTs in terms of superflavor
symmetry, this study assumes that DHTs consist of a color heavy diquark,
treated as a spatially compact object, and a light quark cloud surrounding it.
Although it may exist in a mixed state within a DHT, we also assume that the
color anti-triplet state is the dominant state of the diquark because the
color $\bar{3}$ diquark is likely realized in the ground state [28, 29, 7].
Here, “being spatially compact” means that the heavy diquark can be
approximated as a point-like particle, with no radial excitation occurring
between two heavy quarks. The analyses of $T_{bb}$, being the bottom
counterpart of $T_{cc}$, based on the quark model [29, 30, 31] suggest that
the distance between the two bottom quarks is shorter compared to other quark
distances. This observation appears to support the notion that $T_{bb}$ is
composed of a heavy diquark and a light quark cloud. In the present analysis,
we assume that $T_{cc}$ also holds such heavy diquark structure. For
simplicity, this analysis focuses solely on diquarks where both heavy quarks
are of the same flavor. Under these conditions, a color $\bar{3}$ diquark
possesses spin one ($S_{QQ}=1$), whereas a color $6$ diquark has spin zero
($S_{QQ}=0$).

If $T_{cc}^{+}$ is the superflavor partner of $\Lambda_{c}^{+}$, we can
naturally expect the existence of $T_{ccs}$ as the superflavor partner of
$\Xi_{c}$, which belongs to the same flavor multiplet as $\Lambda_{c}^{+}$. We
consider that investigation of $T_{ccs}$ is useful to understand the nature
not only of $T_{ccs}$ itself but also of $T_{cc}^{+}$. We analyze $T_{ccs}$ by
using experimental result of $T_{cc}^{+}$ as an input. Thus if the results
obtained in this paper are eventually consistent with results in future
experiments, it can be interpreted that the color anti-triplet state is
dominant in DHTs $T_{cc}^{+}$ and $T_{ccs}$. In the following, we first derive
simple mass relations assuming that the heavy diquark is spatially compact and
color anti-triplet state together with superflavor symmetry. We note that the
relations agree with the ones derived in Ref. [32]. However, we find a
discrepancy between the prediction by the simple mass relation and the recent
experimental data of $T_{cc}^{+}$. Thus we invent the improved mass relations
with correction terms violating the symmetries and a mixing term of color
$\bar{3}$ and $6$ states of the $cc$ diquark. Then, we obtain new relations
among HMs, DHBs, SHBs and DHTs, and predict the mass of $T_{ccs}$.
Furthermore, we also predict the decay width of $T_{ccs}$ from the
$\rm{SU(3)}$ flavor symmetry.

This paper is organized as follows. In Sec, II, we derive the simple mass
relations among singly heavy and doubly heavy hadrons based on the heavy quark
and superflavor symmetries. We point out the existence of discrepancy between
the theoretical formulas and the experimental results. In Sec. III, we
construct the effective Lagrangians including corrections and obtain the
improved mass relations. The mass of $T_{ccs}$ and in addition, the one of
$\Omega_{cc}$ are predicted. In Sec. IV, the decay width of $T_{ccs}$ is
predicted by using the effective Lagrangian approach respecting the flavor
symmetry. Finally, Sec. V is devoted to the summary.

##  II  Simple mass relations from superflavor and light-flavor symmetries

In this section, we first introduce simple mass relations among superflavor
partners, primarily derived from superflavor, heavy quark, and light-flavor
symmetries. Then, we demonstrate that these mass relations are somewhat broken
among real hadrons, indicating the need for improvements to the mass
relations. Under the superflavor symmetry, a DHT is related to an anti-SHB,
and a DHB to an anti-HM. Considering that $T_{cc}^{+}$ has isospin $I=0$, the
superflavor partners of $T_{cc}^{+}$, $T_{ccs}^{+}$, and $T_{ccs}^{++}$ are
respectively the anti-baryons of $\Lambda_{c}^{+}$, $\Xi_{c}^{+}$, and
$\Xi_{c}^{0}$. Similarly, the superflavor partners of DHBs like $\Xi_{cc}$ and
$\Omega_{cc}$ are the heavy mesons (HMs), $\bar{D}$ and $\bar{D}_{s}$,
respectively.

We first study the relation between the masses of the DHBs and HMs in terms of
the superflavor symmetry, and then extend the analysis to the DHTs and SHBs.
Based on the HQS, we divide a heavy hadron into a heavy object and light-quark
cloud which includes the interaction between them. Therefore, the dynamics of
these hadrons are determined by the properties of the light quark cloud. As a
result, the masses of heavy hadrons treated in the present analysis are
expressed as a sum of the mass of heavy objects and the energy of light quark
cloud in the heavy quark limit. Let us first estimate the masses of HMs. Here,
we consider the spin average of the doublet under the HQS. We note that due to
the spin average, the first-order term in $1/m_{Q}$ expansion that breaks only
the heavy quark spin symmetry does not appear in the mass formulas. As a
result, the spin-averaged mass can be expressed as

$\displaystyle M_{\rm ave}\left(\bar{Q}q\right)$
$\displaystyle=M\left(\bar{Q}\right)+E\left(q\right),$ (1)

where $M(\bar{Q})$ ($\bar{Q}=\bar{c},\bar{b}$) is the mass of anti-heavy
quark, and $E(q)$ ($q=u,d,s$) denotes the contribution from the light-quark
cloud. From Eq. (1), we obtain the meson mass difference between flavor
partners as

$M_{\rm ave}(\bar{Q}s)-M_{\rm ave}(\bar{Q}n)=E(s)-E(n)\ ,$ (2)

where $n=u,d$.

In the DHBs, the heavy diquark $QQ$ takes the color $\bar{3}$ representation,
so that the anti-HMs and the DHBs include common light-quark cloud in the
heavy quark limit. Thus, the spin-averaged mass of the doublet members of DHBs
is expressed in a similar formula as for the HMs:

$\displaystyle M_{\rm ave}\left(QQq\right)$
$\displaystyle=M\left(QQ\right)+E\left(q\right)\ .$ (3)

We stress that the term $E\left(q\right)$ is common in Eqs. (1) and (3). The
mass difference between flavor partners is $E(\bar{q}_{1})-E(\bar{q}_{2})$,
where $q_{1},q_{2}=u,d,s$. This leads to the following mass relation [33]:

$\displaystyle M_{\rm ave}\left(QQq_{1}\right)-M_{\rm
ave}\left(QQq_{2}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(\bar{Q}q_{1}\right)-M_{\rm ave}\left(\bar{Q}q_{2}\right)\ .$ (4)

This implies that the mass differences between flavor partners are the same in
the superflavor partners.

Next, we consider the masses of anti-SHBs and DHTs. By a similar argument as
above, the mass of an anti-SHB ($\bar{Q}\bar{q}\bar{q}$) is expressed as

$\displaystyle M_{\rm ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)$
$\displaystyle=M\left(\bar{Q}\right)+E\left(\bar{q}_{1}\bar{q}_{2}\right)\ .$
(5)

We note that we generally employ the notation $M_{\rm ave}$ even though the
anti-SHB considered here belongs to a HQS singlet.

As we stated above, we assume that two charm quarks inside $T_{cc}$ form a
compact diquark and that the diquark belonging to the color $\bar{3}$
representation is dominated. Therefore, in the heavy quark limit, the DHT
shares a common light-quark cloud with the anti-SHB according to the
superflavor symmetry. Consequently, the mass of DHT is expressed as

$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)$
$\displaystyle=M\left(QQ\right)+E\left(\bar{q}_{1}\bar{q}_{2}\right)\ .$ (6)

From the mass formulas in Eqs. (5) and (6), we obtain the following mass
relation corresponding to the mass reletion (4):

$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(QQ\bar{q}_{3}\bar{q}_{4}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(\bar{Q}\bar{q}_{3}\bar{q}_{4}\right)\ .$ (7)

where $\bar{q}_{i}=\bar{u},\bar{d},\bar{s}$ ($i=1,2,3,4$).

We further combine Eqs. (1), (3), (5) and (6) to derive the following simple
mass relation:

$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(QQq\right)-M_{\rm ave}\left(\bar{Q}q\right)\ ,$ (8)

Now, we compare the obtained simple mass relations with existing experimental
data. We first note that the relation (2) implies that the mass difference
between $D$ and $D_{s}$ is equal to that between $B$ and $B_{s}$, namely

$\displaystyle M_{\rm ave}(D_{s})-M_{\rm ave}(D)=M_{\rm ave}(B_{s})-M_{\rm
ave}(B)\,,$ (9)

because the energy difference between light clouds, $E(s)-E(n)$, is
independent of the heavy flavor.

Hadrons | Mass (MeV) | Input Value (MeV)
---|---|---
$T_{cc}^{+}$ | 3874.74 | 3875
$\Xi_{cc}^{+}$ | 3623.0 | 3622
$\Xi_{cc}^{++}$ | 3621.55 | 3622
$D^{0}$ | 1864.84 | 1867
$D^{\pm}$ | 1869.66 | 1867
$D^{*0}$ | 2006.85 | 2009
$D^{*\pm}$ | 2010.26 | 2009
$D_{s}^{\pm}$ | 1968.35 | 1968
$D_{s}^{*\pm}$ | 2112.2 | 2112
$\Lambda_{c}^{+}$ | 2286.46 | 2286
$\Xi_{c}^{+}$ | 2467.71 | 2469
$\Xi_{c}^{0}$ | 2470.44 | 2469
$D^{*0}D_{s}^{\pm}$ | 3975.20 | 3979
$D^{*\pm}D_{s}^{\pm}$ | 3978.61 | 3979
$D^{0}D_{s}^{*\pm}$ | 3977.0 | 3980
$D^{\pm}D_{s}^{*\pm}$ | 3981.9 | 3980
$B^{\pm}$ | 5279.34 | 5280
$B^{0}$ | 5279.66 | 5280
$B^{*}$ | 5324.71 | 5325
$B_{s}^{0}$ | 5366.92 | 5367
$B_{s}^{*}$ | 5415.4 | 5415
Table 1:  Experimental values [18, 19, 34] and input values in this study.

However, from the experimental values of masses shown in Table 1, the mass
differences are obtained as

$\displaystyle M_{\rm ave}\left(D_{s}\right)-M_{\rm
ave}\left(D\right)=$103\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$,$ (10)
$\displaystyle M_{\rm ave}\left(B_{s}\right)-M_{\rm
ave}\left(B\right)=$78\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$.$ (11)

This discrepancy implies the necessity of considering correction terms that
break both the heavy quark flavor symmetry and the light flavor symmetry.

Similarly, applying the mass relation (8) to $T_{cc}$, we obtain the following
mass relation:

$\displaystyle M_{\rm ave}\left(T_{cc}^{+}\right)-M_{\rm
ave}\left(\Lambda_{c}^{+}\right)=M_{\rm ave}\left(\Xi_{cc}\right)-M_{\rm
ave}\left(D\right)\ .$ (12)

Since $\Xi_{cc}^{*}$ has not experimentally confirmed yet, we estimate its
mass using the following mass relation obtained from the superflavor symmetry
[35, 27]:

$\displaystyle
M_{\Xi_{cc}^{*}}-M_{\Xi_{cc}}=\frac{3}{4}\left(M_{D^{*}}-M_{D}\right)\ ,$ (13)

leading to $M_{\Xi_{cc}^{*}}=$3728\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
and thus $M_{\rm ave}(\Xi_{cc})=3693$ MeV. Substituting this value into Eq.
(12), we get $M\left(T_{cc}^{+}\right)=3970\>\rm{MeV}$. This value is clearly
different from experimental value, $3875\>\rm{MeV}$, which motivates us to
consider the mixing between a state with a heavy diquark in the color $3$
representation and one in the color $6$ representation.

##  III IMPROVED MASS RELATIONS

In this section, we construct effective Lagrangian terms for solving the
problems raised in the previous section for heavy mesons and the $T_{cc}$. As
we stated in the previous section, we need to include the terms which
simultaneously break the heavy quark flavor symmetry and SU(3)-flavor symmetry
for light quarks to cure the problem of masses of heavy mesons. For the
problem of the mass of $T_{cc}$, we include a term leading to the mixing
between the states constructed from the heavy-diquark in the color $\bar{3}$
representation and the one in the color $6$ representation.

At first, we define the effective DHB($QQq$) fields with quantum numbers
$J^{P}=\frac{1}{2}^{+}$ and $J^{P}=\frac{3}{2}^{+}$ in the heavy quark limit
by combining the heavy diquark with $J^{P}_{\rm heavy}=1^{+}$ to light quark
cloud with $J^{P}_{\rm light}=\frac{1}{2}^{+}$. This doubly-heavy baryon field
${\mathcal{B}}$ is expressed as

$\displaystyle\left[{\mathcal{B}}\right]_{hh^{\prime}l}=\left[P_{+}\gamma_{\mu}CP_{+}^{T}\right]_{hh^{\prime}}\psi^{\mu}_{l}\
,$ (14)

where $h$ and $h^{\prime}$ are spinor indices for heavy quarks and $l$ is the
spinor index for the light quark cloud, and $\psi^{\mu}$ is the field for the
heavy-quark spin doublet with $J^{P}=(1/2^{+},3/2^{+})$. We should note that
the field ${\mathcal{B}}$ carries the index to specify the heavy quark flavor
$Q$. But here and henceforce, we omit the index to avoid too many indices for
one field. The projection operator for heavy quark $P_{+}$ is defined as
$P_{+}=(1+v^{\mu}\gamma_{\mu})/2$, where $v^{\mu}$ is the velocity of the
heavy-diquark. The $\psi^{\mu}$ field is further decomposed into
$J^{P}=1/2^{+}$ field $\psi_{1/2}$ and $J^{P}=3/2^{+}$ field
$\psi_{3/2}^{\mu}$ field as

$\psi^{\mu}=\frac{1}{\sqrt{3}}\sigma^{\mu\nu}v_{\nu}\,\psi_{1/2}+\psi^{\mu}_{3/2}\
,$ (15)

where

$\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu}\,,\,\gamma^{\nu}\right]\ ,$
(16)

and the spin-$3/2$ Rarita-Schwinger field $\psi^{\mu}_{3/2}$ satisfies

$\displaystyle v^{\mu}\gamma_{\mu}\,\psi^{\mu}_{3/2}=\psi^{\mu}_{3/2}\ ,$
$\displaystyle v_{\mu}\psi^{\mu}_{3/2}=\gamma_{\mu}\psi^{\mu}_{3/2}=0\ .$ (17)

For later use, we define the conjugate field $\bar{\mathcal{B}}$ for DHB as

$\left[\bar{\mathcal{B}}\right]_{hh^{\prime}l}=\left[\gamma_{0}\right]_{hh_{1}}\left[{\mathcal{B}}^{\dagger}\right]_{h_{1}h_{2}l_{1}}\left[\gamma_{0}\right]_{h_{2}h^{\prime}}\left[\gamma_{0}\right]_{l_{1}l}\
.$ (18)

For realizing the superflavor symmetry, we take the effective DHB field
$\mathcal{B}$ and the effective anti-HM field $\bar{H}(\sim\bar{Q}q)$ into a
unified field $\bar{\Psi}$ as

$\displaystyle\bar{\Psi}=\begin{pmatrix}\bar{H}\\\ B\end{pmatrix}\ .$ (19)

The parity transformations are given by

$\displaystyle\bar{H}\rightarrow\gamma_{0}\bar{H}\gamma_{0},$ (20)
$\displaystyle[B]_{hh^{\prime}l}\rightarrow[\gamma_{0}]_{ll_{1}}[\gamma_{0}]_{hh_{1}}[B]_{h_{1}h_{2}l_{1}}[\gamma_{0}]_{h_{2}h^{\prime}},$
(21)

where $h,h^{\prime},h_{1},h_{2}$ are heavy quark spinor index and $l,l_{1}$
are spinor indices for light cloud. Since the field $\bar{\Psi}$ belongs to
$3$ representation of $\rm{SU(3)_{f}}$ light flavor, the $\rm{SU(3)_{f}}$
transformation is given by

$\displaystyle\bar{\Psi^{i}}\rightarrow U^{i}_{\>j}\bar{\Psi}^{j}\ ,$ (22)

where $i,j$ are the light flavor indices and $U\in\rm{SU(3)_{f}}$.

Next, we define the effective DHT($\sim\bar{Q}\bar{Q}qq$) fields. As we stated
in the previous section, we include two DHT fields: One is constructed from
the heavy diquark with color $\bar{3}$ representation carrying $J_{\rm
heavy}^{P}=1^{+}$, which is combined with the light-quark cloud with
$J^{P}_{\rm light}=0^{+}$ to make the DHT with $J^{P}=1^{+}$. Another one is
constructed from the heavy diquark carrying color $6$ representation and
$J^{P}_{\rm heavy}=0^{+}$ combined with $J^{P}_{\rm light}=1^{+}$ to make the
one with $J^{P}=1^{+}$. The former one is denoted as $T_{\mu}^{(\bar{3})}$ and
the latter as $T_{\mu}^{(6)}$. These fields are defined as

$\displaystyle\left[T_{\mu}^{(\bar{3})}\right]_{hh^{\prime}}=\left[P_{+}\gamma_{\mu}CP_{+}^{T}\right]_{hh^{\prime}}\,\phi\
,$ (23)
$\displaystyle\left[T_{\mu}^{(6)}\right]_{hh^{\prime}}=\left[P_{+}\gamma_{5}CP_{+}^{T}\right]_{hh^{\prime}}\,\varphi_{\mu}\
,$ (24)

where $h$ and $h^{\prime}$ are spinor-indices for heavy quarks, and the upper
indices of $T$, $(\bar{3})$ and $(6)$, represent the color representation of
heavy diquark in DHTs. We note that $\phi$ and $\varphi_{\mu}$ stand for the
annihilation operators and the same applies to $T$. We also note that these
fields have the index of light quark flavor. As we said in the introduction,
we consider heavy diquarks made from two heavy quarks with the same flavor.
The light quark cloud is made from two anti light quarks in the flavor anti-
symmetric representation. Then, the fields $T$ belong to $3$ representation of
$\rm{SU(3)_{f}}$ light flavor symmetry.

Based on the superflavor symmetry, the field $T_{\mu}^{(3)}$ and the effective
anti-SHB field $\bar{S}(\sim\bar{Q}\bar{q}\bar{q})$ are arranged into a
unified field $\Phi$ as

$\displaystyle\bar{\Phi}=\begin{pmatrix}\bar{S}\\\
T_{\mu}^{(\bar{3})}\end{pmatrix}\ .$ (25)

The parity transformations are given by

$\displaystyle\bar{S}$ $\displaystyle\rightarrow\bar{S}\gamma_{0}\ ,$ (26)
$\displaystyle[T]_{hh^{\prime}}$
$\displaystyle\rightarrow[\gamma_{0}]_{hh_{1}}[T]_{h_{1}h_{2}}[\gamma_{0}]_{h_{2}h^{\prime}}\
.$ (27)

The conjugate fields are defined as

$\displaystyle S=\gamma_{0}\bar{S}^{\dagger},$ (28)
$\displaystyle[\bar{T}]_{hh^{\prime}}=[\gamma_{0}]_{hh_{1}}[T^{\dagger}]_{h_{1}h_{2}}[\gamma_{0}]_{h_{2}h^{\prime}}\
.$ (29)

The $\rm{SU(3)_{f}}$ transformation is given by

$\displaystyle\bar{\Phi}_{ij}\rightarrow\left(U^{\ast}\right)_{i}^{\>k}\bar{\Phi}_{kl}\left(U^{\dagger}\right)_{\>j}^{l}$
(30)

Let us construct the effective Lagrangian for the DHT, SHB, DHB and HM. As we
said in the previous section, we need to include the terms which break
simultaneously the heavy-quark flavor symmetry and the light flavor symmetry.
We first note that the terms which break the heavy quark flavor symmetry are
inversely proportional to the heavy quark mass, i.e. $\propto 1/m_{Q}$. We
also include the term for generating the mixing between $\bar{T}^{(3)}_{\mu}$
and $\bar{T}^{(\bar{6})}_{\mu}$, which must include $1/m_{Q}$ since the mixing
vanishes at the heavy quark limit. For the light-flavor symmetry breaking, we
use the spurion field corresponding to the light quark mass matrix
${\mathcal{M}}_{q}$ as

${\mathcal{M}}_{q}=\begin{pmatrix}m_{u}&&\\\ &m_{d}&\\\ &&m_{s}\\\
\end{pmatrix}\ ,$ (31)

where $m_{u}$, $m_{d}$ and $m_{s}$ are the current quark masses of $u$, $d$
and $s$ quarks, respectively. This ${\mathcal{M}}_{q}$ transforms as

$\left({\mathcal{M}}_{q}\right)^{i}{}_{j}\to\left(U\right)^{i}{}_{k}\,\left({\mathcal{M}}_{q}\right)^{k}{}_{\ell}\,\left(U^{\dagger}\right)^{\ell}{}_{j}\
.$ (32)

Now, the kinetic and mass terms invariant under the heavy quark symmetry and
SU(3) light flavor symmetry are given by

$\displaystyle{\mathcal{L}}_{0}=$
$\displaystyle-\mbox{Tr}\left[\bar{\Psi}iv\cdot\partial\,\Psi+\bar{\Phi}iv\cdot\partial\,\Phi\right]$
$\displaystyle{}-\Lambda_{\Psi}\,\mbox{Tr}\left[\bar{\Psi}\Psi\right]-\Lambda_{\Phi}\,\mbox{Tr}\left[\bar{\Phi}\Phi\right]\
,$ (33)

where $\Lambda_{\Psi}$ and $\Lambda_{\Phi}$ are constants with mass dimension
one, and Tr implies that the traces in the light-flavor space in addition to
the spinor space and heavy-spin space are taken. In the following, we
explicitly write the indices for light flavor while we omit the indices for
heavy quark spin and flavor.

A possible term which breaks the SU(3) light flavor symmetry for $\Psi$ field
is written as

$\displaystyle{\mathcal{L}}_{\Psi{\rm-br}}=$ $\displaystyle-
c_{\Psi}\,\mbox{tr}\,\left[\bar{\Psi}^{i}\Psi_{j}\right]\left(\mathcal{M}_{q}\right)^{j}{}_{i}\
,$ (34)

where $c_{\Psi}$ is a constant with mass dimension zero and tr indicates that
the traces in spinor space and heavy-spin space are taken.

By noting that two light quarks in the $\Phi$ field are anti-symmetric in
flavor, the light flavor breaking term is expressed as

$\displaystyle{\mathcal{L}}_{\Phi{\rm-br}}=$ $\displaystyle-
c_{\Phi}\,\mbox{tr}\,\left[\bar{\Phi}_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}\Phi^{ki}\right]\
,$ (35)

where $c_{\Phi}$ is a constant with mass dimension zero.

We next construct possible terms which break both SU(3) light flavor symmetry
and the heavy quark flavor symmetry. We note that the terms also break the
superflavor symmetry, so that the terms should be written separately for
superflavor partners. The terms for $\bar{H}$ (anti-HM) and $B$ (DHB) included
in $\Psi$ are expressed as

$\displaystyle{\mathcal{L}}_{\rm HB-br}=$
$\displaystyle-\frac{\Lambda_{q}}{m_{Q}}\mbox{tr}\left[\bar{H}^{i}\,H_{j}\left(\mathcal{M}_{q}\right)^{j}{}_{i}\right]$
$\displaystyle{}-\frac{\Lambda_{q}^{\prime}}{2m_{Q}}\mbox{tr}\left[B^{i}\,\bar{B}_{j}\left(\mathcal{M}_{q}\right)^{j}{}_{i}\right]\
,$ (36)

where $\Lambda_{q}$ and $\Lambda_{q}^{\prime}$ are constants with mass
dimension one. We put an extra factor $1/2$ in the second line since the DHB
includes two heavy quarks. Similarly, possible breaking terms for $\bar{S}$
(anti-SHB) and $T$ (DHT) in $\Phi$ field are expressed as

$\displaystyle{\mathcal{L}}_{\rm ST-
br}=-\frac{\Lambda_{qq}}{m_{Q}}\,\mbox{tr}\left[\bar{S}_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}S^{ki}\right]$
$\displaystyle\quad{}-\frac{\Lambda_{qq}^{\prime}}{2m_{Q}}\,\mbox{tr}\left[\left(T_{\mu}^{(\bar{3})}\right)_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}\left(\bar{T}^{(\bar{3})\mu}\right)^{ki}\right]\
,$ (37)

where $\Lambda_{qq}$ and $\Lambda_{qq}^{\prime}$ are constants with mass
dimension one.

Finally, we consider a DHT field $T_{\mu}^{(6)}$ constructed from the heavy
diquark with color ${\boldmath 6}$ representation. The kinetic and mass terms
are written as

${\mathcal{L}}_{T^{(6)}}=-\mbox{Tr}\,\left[\bar{T}_{\mu}^{(6)}\left(iv\cdot\partial+\Lambda_{6}\right)T^{(6)\mu}\right]\
,$ (38)

where $\Lambda_{6}$ is a constant with mass dimension one. The term for the
mixing between two DHT fields $T_{\mu}^{(3)}$ and $T_{\mu}^{(6)}$ is expressed
as

$\displaystyle{\mathcal{L}}_{\rm mix}=$
$\displaystyle-\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\mbox{Tr}\Bigg{[}\left(\bar{T}^{(6)\mu}\right)\left(T_{\mu}^{(\bar{3})}\right)+\left(\bar{T}^{(\bar{3})\mu}\right)\left(T_{\mu}^{(6)}\right)\Bigg{]}\
.$ (39)

From the above Lagrangian terms, the masses of HM, DHB and SHB are modified
from Eqs. (1), (3) and (5) as

$\displaystyle M_{\rm ave}\left(\bar{Q}q\right)=$ $\displaystyle
M\left(\bar{Q}\right)+E\left(q\right)+\frac{m_{q}}{m_{Q}}\Lambda_{q},$ (40)
$\displaystyle M_{\rm ave}\left(QQq\right)=$ $\displaystyle
M\left(QQ\right)+E\left(q\right)+\frac{m_{q}}{2m_{Q}}\Lambda_{q}^{\prime},$
(41) $\displaystyle M_{\rm ave}\left(\bar{Q}\bar{q_{1}}\bar{q_{2}}\right)=$
$\displaystyle
M\left(\bar{Q}\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)+\frac{m_{q_{1}}+m_{q_{2}}}{m_{Q}}\Lambda_{qq},$
(42)

where

$\displaystyle E(q)$ $\displaystyle=\Lambda_{\Psi}+c_{\Psi}m_{q}\ ,$
$\displaystyle E(\bar{q}_{1}\bar{q}_{2})$
$\displaystyle=\Lambda_{\Phi}+c_{\Phi}\left(m_{q_{1}}+m_{q_{2}}\right)\ .$
(43)

The square mass matrix for two DHT fields is expressed as

$\begin{pmatrix}M_{3}^{2}&\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\sqrt{M_{6}M_{3}}\\\
\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\sqrt{M_{6}M_{3}}&M_{6}^{2}\\\
\end{pmatrix}\ ,$ (44)

where

$M_{3}=M\left(QQ\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)+\frac{m_{q_{1}}+m_{q_{2}}}{2m_{Q}}\Lambda_{qq}^{\prime}\
,$ (45)

and $M_{6}$ is the mass of $T_{\mu}^{(6)}$ field before mixing. By
diagonalizing this matrix up to $1/m_{Q}$ order, the mass of lightest DHT is
obtained as

$\displaystyle M_{\rm
ave}\left(QQ\bar{q_{1}}\bar{q_{2}}\right)=M\left(QQ\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)$
$\displaystyle\quad{}+\frac{m_{q_{1}}+m_{q_{2}}}{2m_{Q}}\Lambda_{qq}^{\prime}-\frac{M_{6}}{2(M_{6}^{2}-M_{3}^{2})}\left(\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\right)^{2}\
.$ (46)

Combining the above mass formulas, we obtain the following modified mass
relations:

$\displaystyle M_{\rm ave}\left(QQs\right)-M_{\rm
ave}\left(QQn\right)-\frac{m_{s}-m_{n}}{2m_{Q}}\Lambda_{q}^{\prime}$
$\displaystyle=M_{\rm ave}\left(\bar{Q}s\right)-M_{\rm
ave}\left(\bar{Q}n\right)-\frac{m_{s}-m_{n}}{m_{Q}}\Lambda_{q},$ (47)
$\displaystyle M_{\rm ave}\left(QQ\bar{s}\bar{n}\right)-M_{\rm
ave}\left(QQ\bar{u}\bar{d}\right)-\frac{m_{s}-m_{n}}{2m_{Q}}\Lambda_{qq}^{\prime}$
$\displaystyle=M_{\rm ave}\left(\bar{Q}\bar{s}\bar{n}\right)-M_{\rm
ave}\left(\bar{Q}\bar{u}\bar{d}\right)-\frac{m_{s}-m_{n}}{m_{Q}}\Lambda_{qq},$
(48) $\displaystyle M_{\rm ave}\left(QQ\bar{u}\bar{d}\right)-M_{\rm
ave}\left(\bar{Q}\bar{u}\bar{d}\right)$
$\displaystyle+\frac{M_{6}}{2\left(M_{6}^{\>2}-M_{3}^{\>2}\right)}\left(\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\right)^{2}-\frac{m_{n}}{2m_{Q}}\left(\Lambda_{qq}^{\prime}-2\Lambda_{qq}\right)$
$\displaystyle=M_{\rm ave}\left(QQn\right)-M_{\rm
ave}\left(\bar{Q}n\right)-\frac{m_{n}}{2m_{Q}}\left(\Lambda_{q}^{\prime}-2\Lambda_{q}\right)\
,$ (49)

where $n=u,d$ and $m_{n}=(m_{u}+m_{d})/2$ is the isospin averaged mass of up
and down quarks. Furthermore, from Eq. (40), we obtain the following mass
relation:

$\displaystyle\left[M_{\rm ave}(D_{s})-M_{\rm ave}(D)\right]-\left[M_{\rm
ave}(B_{s})-M_{\rm ave}(B)\right]$
$\displaystyle\quad=\left(m_{s}-m_{n}\right)\left(\frac{1}{m_{c}}-\frac{1}{m_{b}}\right)\Lambda_{q}\
,$ (50)

where $n=u,d$.

Quarks | Mass (MeV)
---|---
$u,d$ | 3.415
$s$ | 93.40
$c$ | 1270
$b$ | 4180
Table 2:  Mass of quarks [34] used as inputs.

Using the mass differences in Eqs. (10) and (11) together with the current
quark masses shown in Table 2, we determine the value of $\Lambda_{q}$ as

$\Lambda_{q}=506\,\mbox{MeV}\ .$ (51)

The other parameters $\Lambda^{\prime}_{q}$, $\Lambda_{qq}$ and
$\Lambda^{\prime}_{qq}$ cannot be determined due to lack of input hadron
masses. We therefore assume that they are of the same order as $\Lambda_{q}$.
Specifically, we set their median values as $0$ and assign an error margin of
$\pm 506\,\rm{MeV}$, resulting in
$\Lambda^{\prime}_{q}=\Lambda_{qq}=\Lambda^{\prime}_{qq}=0\pm 506\,\rm{MeV}$.

By applying the improved mass relation corresponding to Eq. (7) to
$T_{cc}^{+}$, we derive

$\displaystyle M_{\rm ave}\left(T_{ccs}\right)-M_{\rm
ave}\left(T_{cc}^{+}\right)+\frac{m_{s}-m_{n}}{2m_{c}}\Lambda_{qq}^{\prime}$
$\displaystyle\quad=M_{\rm ave}\left(\Xi_{c}\right)-M_{\rm
ave}\left(\Lambda_{c}^{+}\right)+\frac{m_{s}-m_{n}}{m_{c}}\Lambda_{qq}\ ,$
(52)

and we obtain the mass of $T_{ccs}$ as

$\displaystyle M_{\rm ave}\left(T_{ccs}\right)=4057\pm 40\>\rm{MeV},$ (53)

where $M_{\rm ave}(T_{ccs})$ denotes the isospin averaged mass of
$T_{ccs}^{+}$ and $T_{ccs}^{++}$.

Hadrons | Mass (MeV) |
---|---|---
$T_{ccs}$ | $4057\pm 40$ | Our result
$4106$ | NQM [36]
$T_{ccs}^{+}$ | $3969\pm 8$ | Lattice QCD [37]
Table 3:  Theoretical predictions for the masses of $T_{ccs}$. NQM imply non-
relativistic quark model.

If this result agrees with future experimental data, it means that the color
anti-triplet state of the heavy diquark is dominant in $T_{cc}^{+}$ and
$T_{ccs}$. We compare our result with other results in Table 3.

From the mass relation in Eq. (47), we further obtain

$\displaystyle M_{\rm ave}\left(\Omega_{cc}\right)$ $\displaystyle-M_{\rm
ave}\left(\Xi_{cc}\right)+\frac{m_{s}-m_{n}}{2m_{c}}\Lambda_{q}^{\prime}$
$\displaystyle=M_{\rm ave}\left(D_{s}\right)-M_{\rm
ave}\left(D\right)+\frac{m_{s}-m_{n}}{m_{c}}\Lambda_{q}.$ (54)

From this relation, the mass of $\Omega_{cc}$ which has not been
experimentally reported so far is also predicted as

$\displaystyle M_{\rm ave}\left(\Omega_{cc}\right)=3760\pm 18\>\rm{MeV}.$ (55)

This result serves as another indicator for assessing whether our mass
relations are correct. If this result does not match future experimental
results, one possible reason might be that $\Lambda_{q}^{\prime}$ is larger
than $\Lambda_{q}$. Additionally, the second-order effects of $1/m_{Q}$ might
also contribute. We show comparison with other results in Table 4.

Hadrons | Mass (MeV) |
---|---|---
$\Omega_{cc}$ | $3760\pm 18$ | Our result
3811 | NQM [38]
3753 | RQM [39]
3841 | RQM [40]
3657 | NQM [41]
3856 | NQM [42]
3773 | Lattice QCD [43]
3754 | Lattice QCD [44]
Table 4: Theoretical predictions of the mass of $\Omega_{cc}$. NQM and RQM
imply non-relativistic and relativistic quark models, respctively.

##  IV WIDTH OF $T_{ccs}^{+}$

In this section, we construct the Lagrangian solely from the $\rm{SU(3)}$
light flavor symmetry and predict the decay width of $T_{ccs}$ using the decay
width of $T_{cc}^{+}$ as an input. Our approach hinges on the fact that both
$T_{ccs}$ and $T_{cc}^{+}$ belong to the same light flavor representation.
Notably, this method is independent of both superflavor symmetry and color
representation.

Since $T_{cc}^{+}$ locates just below the $DD^{*}$ threshold, we consider the
decay process of $T_{cc}^{+}\rightarrow D^{0}D^{0}\pi^{+}$ and
$T_{cc}^{+}\rightarrow D^{0}D^{+}\pi^{0}$ with $D^{*}$ as an intermediate
state, as depicted in Fig. 1. While the decay $T_{cc}^{+}\rightarrow
D^{0}D^{+}\pi^{0}$ has not been observed in experiments [18, 19], it is not
prohibited kinematically. Hence, in this study, we also incorporate Fig.
1(b),(c). For $T_{ccs}$, we consider the decay processes $T_{ccs}\rightarrow
DD_{s}^{*}$ and $T_{ccs}\rightarrow D^{\ast}D_{s}$ as shown in Fig. 2, because
the mass of $T_{ccs}$ predicted in the previous section is above the
$DD_{s}^{*}$ and $D^{*}D_{s}$ thresholds.

Let us construct an effective Lagrangian for the interaction among the
tetraquarks and charmed mesons based on just light flavor symmetry. In the
following we use relativistic field $T^{\mu}$ for the mass eigenstates of
flavor triplet tetraquarks:

$T^{\mu}=\begin{pmatrix}0&T_{cc}^{+}&T_{ccs}^{++}\\\
-T_{cc}^{+}&0&T_{ccs}^{+}\\\ -T_{ccs}^{++}&-T_{ccs}^{+}&0\\\ \end{pmatrix}\ ,$
(56)

While the fields for charmed mesons with $J^{P}=\left(0^{-},1^{-}\right)$ are
represented by relativistic fields $\left(D,D^{*\mu}\right)$:

$D=\begin{pmatrix}D^{0}\\\ D^{+}\\\ D_{s}\\\ \end{pmatrix}\ ,\quad
D^{\ast}=\begin{pmatrix}D^{\ast 0}\\\ D^{\ast+}\\\ D_{s}^{\ast}\\\
\end{pmatrix}\ .$ (57)

From Eqs. (22) and (30), the $\rm{SU(3)}$ light flavor transformations of
these fields are given as

$\displaystyle
T_{ij}^{\mu}\rightarrow\left(U^{\ast}\right)_{i}^{\>k}T_{kl}^{\mu}\left(U^{\dagger}\right)_{\>j}^{l},$
$\displaystyle D_{i}\rightarrow
D_{j}\left(U^{{\dagger}}\right)^{j}_{\;i},\>D_{i}^{*\mu}\rightarrow
D_{j}^{*\mu}\left(U^{{\dagger}}\right)^{j}_{\;i}\ .$ (58)

The Lagrangian for $T_{cc(s)}$ and heavy mesons invariant under this
transformation is given by

$\displaystyle\mathcal{L}=g_{TDD}\bar{D}_{\mu}^{\ast
i}T_{ij}^{\mu}\bar{D}^{j}\ .$ (59)

|  |
---|---|---
(a) | (b) | (c)
Figure 1: Feynmann diagrams of the $T_{cc}^{+}\to DD\pi$ decays.

As we said above, we determine the value of $g_{TDD}$ from the decay width of
$T_{cc}$ assuming that decay is dominated by the process shown in Fig. 1. By
using the value of $D^{\ast}D\pi$ coupling constant determined from the decay
of $D^{\ast}$ meson (see e.g. Ref [24]), the value of $g_{TDD}$ is calculated
as

$\displaystyle g_{TDD}=0.68\times 10^{3}\>\rm{MeV}\ .$ (60)

We note that this value is natural when nondimensionalized by the mass of the
charm quark.

|
---|---
(a) | (b)
Figure 2: Feynmann diagrams of the $T_{ccs}\to DD_{s}^{*}$ and $D^{*}D_{s}$
decays.

Now, based on the processes shown in Fig. 2. the formula of the decay width of
$T_{ccs}$ is calculated as

$\displaystyle\Gamma\left(T_{ccs}\right)=g_{TDD}^{2}\frac{\left|P_{1}\right|+\left|P_{2}\right|}{6\pi
M\left(T_{ccs}\right)^{2}}\ ,$ (61)

where $P_{1}$ and $P_{2}$ and the momenta for $D_{s}^{\pm}$ and $D^{0}$,
respectively. By using the mass of $T_{ccs}$ predicted in Eq. (53) together
with the value of $g_{TDD}$ in Eq. (60), the decay width is predicted as

$\displaystyle\Gamma\left(T_{ccs}\right)=1.2\pm 0.3\>\rm{MeV}\ .$ (62)

We consider this decay width to be sufficiently small as to be an
experimentally observable.

##  V SUMMARY

In this study, we investigated the mass and decay width of the doubly heavy
tetraquark $T_{ccs}$ from the superflavor and $\rm{SU(3)}$ light flavor
symmetries. We assumed that doubly charmed tetraquarks are constructed from a
color anti-triplet $cc$ diquark and thus they are the superflavor partners of
the singly heavy baryons. First, we derived the simple mass relations under
heavy quark and superflavor symmetries. However, we found discrepancy between
predictions of the obtained mass relations and the experimental data. Then, we
constructed an effective Lagrangian based on these symmetries with including
correction terms violating simultaneously the heavy quark symmetry and the
light flavor symmetry. From the Lagrangian, we obtained the improved mass
relations among heavy mesons (HMs), doubly heavy baryons (DHBs), singly heavy
baryons (SHBs) and doubly heavy tetraquarks (DHTs). Based on the relations, we
predicted the mass of unobserved tetraquark $T_{ccs}$ as
$M\left(T_{ccs}\right)=4057\pm 40\>\rm{MeV}$. We also predicted the mass of
unobserved $\Omega_{cc}$ as $M(\Omega_{cc})=3760\pm 18\,\rm{MeV}$.

We then constructed an effective Lagrangian term for the decay of $T_{ccs}$
based on SU(3) light flavor symmetry. The unknown coupling constant was
determined by using the $T_{cc}$ decay data. Incorporating the predicted mass,
we derived the decay width of $T_{ccs}$ as $\Gamma\left(T_{ccs}\right)=1.2\pm
0.3\>\rm{MeV}$.

The obtained masses and widths will be useful to understand the color
configuration of DHTs. If these results agree with future experimental data,
it means that the color anti-triplet state in $T_{cc}^{+}$ and $T_{ccs}$ is
dominant.

In this study, we have primarily focused on the charm sector. This is because
hadrons containing two $b$ quarks have not yet been experimentally discovered,
and we cannot determine the value of $M(bb)-M(b)$. However, our results are
immediately applicable to the bottom sector once hadrons such as $\Xi_{bb}$,
$\Omega_{bb}$, $T_{bb}$, and $T_{bbs}$ are found.

This work was supported in part by JSPS KAKENHI Grant Nos. JP20K03927,
JP23H05439 (MH) and JP20K14478 (YY).

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# An information-theoretic upper bound on prime gaps

Aidan Rocke
<EMAIL_ADDRESS>

###### Abstract

Within the setting of rare event modelling, the method of level sets allows us
to define an equivalence relation over rare events with distinct rates of
entropy production. This method allows us to clarify the relation between the
empirical density of primes and their source distribution which then allows us
to address Cramér’s conjecture, an open problem in probabilistic number
theory. As a natural consequence, this analysis places strong epistemic limits
on the application of machine learning to analyse the distribution of primes.

##  1 Classifying rare events using level sets

Sequences of rare events that are not finite-state compressible are recurrent,
of variable frequency, and unpredictable relative to a finite-state machine
such as a machine learning model. For these reasons, they are of great
scientific interest. But, how might we classify rare events that satisfy these
criteria?

If we identify rare events using the function
$1_{X}:\mathbb{N}\rightarrow\\{0,1\\}$, we may analyse the rate of entropy
production of the computable binary sequence
$X_{n}=\\{x_{i}\\}_{i=1}^{n}\in\\{0,1\\}^{n}$. In order to qualify as a rare
event, a randomly sampled element $x_{z}$ of the sequence $X_{n}$ must
generally pass $\pi_{x}(n)$ tests. These tests $A_{i},A_{j\neq
i}\in\\{A_{i}\\}_{i=1}^{\pi_{x}(n)}$ are de-correlated for rare events as
these are assumed to occur independently of each other:

$\forall z\sim U([1,n]),P(z\in A_{i}\land z\in A_{j\neq i})=P(z\in A_{i})\cdot
P(z\in A_{j\neq i})$ (1)

where the $P(z\in A_{i})$ have a natural frequentist interpretation. We
generally assume that the tests are consistent so
$\\{A_{i}\\}_{i=1}^{\pi_{x}(n)}\subset\\{A_{i}\\}_{i=1}^{\pi_{x}(n+1)}$.

Furthermore, $\pi_{x}(n)$ counts the number of rare events in the discrete
time interval $[1,n]$ so it is monotonically increasing. This may be deduced
from the fact that each rare event that occurs in an interval defines one
information-theoretic constraint $A_{i}$ where $A_{i}\perp A_{j\neq i}$ due to
pairwise independence. Therefore, if $m$ rare events occur in an interval, a
uniformly sampled element $x_{z}$ must simultaneously satisfy $m$ independent
constraints in order to qualify as a rare event.

Given (1), we may define the average probability density using the inclusion-
exclusion principle:

$\forall z\sim
U([1,n]),P(x_{z}=1)=P(z\in\bigcap_{k\leq\pi_{x}(n)}A_{k})=\prod_{k=1}^{\pi_{x}(n)}P(z\in
A_{k})$ (2)

as well as the entropy of the uniformly distributed random variable:

$\forall z\sim U([1,n]),H(X_{n})=-\ln\prod_{k=1}^{\pi_{x}(n)}P(z\in A_{k})$
(3)

which is a measure of expected surprise, or the expected information gained
from observing a rare event.

Having defined the entropy (3), we may compare the entropies of distinct
sequences using the typical probabilities $q_{n}\in(0,1)$, a parameter which
allows us to define an equivalence relation over entropies:

$\mathcal{L}_{q_{n}}=\\{X_{n}\in\\{0,1\\}^{\infty}:H(X_{n})\sim\ln\big{(}\frac{1}{q_{n}}\big{)}\\}$
(4)

This parameter is chosen because it corresponds to the multiplicative inverse
of the exponential of entropy known hereafter as the entropy rate
$\frac{1}{q_{n}}\approx e^{H(X_{n})}$ which is a robust measure of the
expected waiting time before rare events. As these parameters are both
parameterisation invariant i.e. don’t depend upon the choice of base for
logarithms, they are an ideal choice for classifying rare events in terms of
their rate of entropy production.

Given the definition (4), we have:

$\forall X_{n}\in\mathcal{L}_{q_{n}},\lim_{n\to\infty}-\frac{\ln
q_{n}}{H(X_{n})}=1$ (5)

so the following holds asymptotically almost surely:

$\frac{\partial}{\partial q_{n}}H(X_{n})=\frac{\partial}{\partial q_{n}}-\ln
q_{n}\implies\forall z\sim U([1,N]),P(x_{z}=1)\sim q_{N}$ (6)

and therefore for large $N\in\mathbb{N}$:

$\pi_{x}(N)\sim N\cdot q_{N}$ (7)

This leads us naturally to consider statistical generalisations of the Prime
Number Theorem.

###  1.1 Statistical generalisations of the Prime Number Theorem

From statistical considerations, Gauss and Legendre inferred that the density
of primes at $n\in\mathbb{N}$ is on the order of $\sim\frac{1}{\ln n}$. This
led them to conjecture that the number of primes less than $N$ is on the order
of:

$\pi(N)\sim\int_{2}^{N}\frac{1}{\ln x}dx\sim N\cdot\frac{1}{\ln N}$ (8)

which is equivalent to the Prime Number Theorem.

However, given our assumptions we are more generally interested in the family
of density functions that satisfy:

$0<c\ll N,\int_{c}^{N}\frac{1}{f(x)}dx\sim N\cdot\frac{1}{f(N)}$ (9)

where $f$ is analytic and $\lim\limits_{N\to\infty}\frac{1}{f(N)}=0$. Using
integration by parts, this is equivalent to the criterion:

$\int_{c}^{N}g(x)dx=N\cdot g(N)-c\cdot g(c)-\int_{c}^{N}x\cdot
g^{\prime}(x)dx\sim N\cdot g(N)$ (10)

where $g(x)=\frac{1}{f(x)}$, and $g(x)$ dominates $x\cdot g^{\prime}(x)$. So
this family naturally includes density functions of the form:

$\exists a>0\forall x\in\mathbb{R}_{+},g(x)=(\ln x)^{-a}$ (11)

and it is worth adding that (11) admits an interpretation as an average
probability density which we may now make precise.

###  1.2 The typical probability as the average probability

Given that the density $g(x)=(\ln x)^{-a}$ is Riemann-Integrable on $[2,N]$
for $N<\infty$, by the Intermediate Value Theorem there exists a sequence
$x_{n}\in[n,n+1]$ such that:

$\int_{2}^{N}(\ln x)^{-a}dx=\sum_{n=2}^{N}\frac{1}{(\ln x_{n})^{a}}$ (12)

and since $\forall n\in\mathbb{N},1\leq\big{(}\frac{\ln x_{n}}{\ln
n}\big{)}^{a}\leq\big{(}\frac{\ln(n+1)}{\ln n}\big{)}^{a}$ where $\forall
a>0,\lim\limits_{n\to\infty}\big{(}\frac{\ln(n+1)}{\ln n}\big{)}^{a}=1$ so we
have:

$\int_{2}^{N}(\ln x)^{-a}dx\sim\sum_{n=2}^{N}\frac{1}{(\ln n)^{a}}\sim
N\cdot\frac{1}{(\ln N)^{a}}$ (13)

and therefore the typical probability is asymptotically equivalent to the
average probability:

$q_{N}\sim\frac{1}{N}\sum_{n=2}^{N}\frac{1}{(\ln n)^{a}}$ (14)

In fact, for large $n$ the average probability density in (2) may be expressed
as:

$\forall z\sim U([1,n]),P(x_{z}=1)\sim\frac{1}{n}\sum_{k=1}^{n}P(x_{k}=1)$
(15)

###  1.3 A remark on applications of the level set method

In practice, the empirical data will consist of a binary sequence of rare
event observations and the actual tests are generally unknown. These may be
modelled as latent variables that are a function of discrete time(i.e. the
integers) as well as hidden variables that provide us with initial conditions.
Thus, we shall generally assume that empirical binary sequences are the output
of a discrete dynamical system.

Assuming that deep neural networks may be used as universal data compressors,
we may determine whether the data is finite-state incompressible using a
machine-learning driven overfitting test provided in the appendix [12]. As a
concrete demonstration of the analytical power of the level set method, we
shall consider its application to an open problem in probabilistic number
theory.

However, we must first carefully go over the information-theoretic derivation
of the Prime Number Theorem [9].

##  2 Information-theoretic derivation of the Prime Number Theorem

If we know nothing about the distribution of primes, in the worst case we may
assume that each prime less than or equal to $N$ is drawn uniformly from
$[1,N]$. So our source of primes is:

$X\sim U([1,N])$ (16)

where $H(X)=\ln N$ is the Shannon entropy of the uniform distribution.

Now, we may define the prime encoding of $[1,N]$ as the binary sequence
$X_{N}=\\{x_{n}\\}_{n=1}^{N}$ where $x_{n}=1$ if $n$ is prime and $x_{n}=0$
otherwise. With no prior knowledge, given that each integer is either prime or
not prime, we have $2^{N}$ possible prime encodings in
$[1,N]\subset\mathbb{N}$.

If there are $\pi(N)$ primes less than or equal to $N$ then the average number
of bits per arrangement gives us the average amount of information gained from
correctly identifying each prime in $[1,N]$ as:

$S_{c}=\frac{\log_{2}(2^{N})}{\pi(N)}=\frac{N}{\pi(N)}$ (17)

and if we assume a maximum entropy distribution over the primes then we would
expect that each prime is drawn from a uniform distribution as in (16). So we
would have:

$S_{c}=\frac{N}{\pi(N)}\sim\ln N$ (18)

As for why the natural logarithm appears in (21), we may first note that the
base of the logarithm in the Shannon Entropy may be freely chosen without
changing its properties. Moreover, given the assumptions the expected
information gained from observing each prime in $[1,N]$ is on the order of:

$\sum_{k=1}^{N-1}\frac{1}{k}\cdot|(k,k+1]|=\sum_{k=1}^{N-1}\frac{1}{k}\approx\ln
N$ (19)

as there are $k$ distinct ways to sample uniformly from $[1,k]$ and a
frequency of $\frac{1}{k}$ associated with the event that $k\in\mathbb{P}$.

This implies that the average number of bits per prime number is given by
$\frac{N}{\pi(N)}\sim\ln N$. Rearranging, we find:

$\frac{\pi(N)}{N}\sim\frac{1}{\ln N}$ (20)

which is in complete agreement with the Prime Number Theorem. Moreover, this
derivation indicates that the prime numbers are empirically distributed as if
they were arranged uniformly so we may distinguish the empirical distribution
of the primes from their source distribution.

###  2.1 The Shannon source coding theorem and the algorithmic randomness of
prime encodings

By the Shannon source coding theorem, we may infer that $\pi(N)$ primes can’t
be compressed into fewer than $\pi(N)\cdot\ln N$ bits. Furthermore, as the
expected Kolmogorov Complexity equals the Shannon entropy for computable
probability distributions:

$\mathbb{E}[K(X_{N})]\sim\pi(N)\cdot\ln N\sim N$ (21)

where the identification of $\mathbb{E}[K(X_{N})]$ with $\pi(N)\cdot\ln N$ is
a direct consequence of the fact that $K(X_{N})$ measures the information
gained from observing $X_{N}$ and $\pi(N)\cdot\ln N$ measures the expected
information gained from observing $X_{N}$. From an information-theoretic
perspective, this implies that prime encodings are finite-state incompressible
as they have a maximum entropy distribution.

The only implicit assumption in this derivation is that all the information in
the Universe is conserved as it is only in such Universes that Occam’s razor
is generally applicable. The Law of Conservation of information and its
significance is discussed in more detail at the end of the Appendix.

##  3 Clarifying the relation between the empirical density of primes and
their source distribution

In the worst case, rare events $x_{k}\in X_{n}$ are arranged uniformly in the
discrete time interval $[1,n]$ so the expected information gained from
observing all the events in $X_{n}$ is on the order of:

$I_{N}=\sum_{n=1}^{N}q_{n}\cdot\ln n$ (22)

Furthermore, in order to model rare events it is crucial to consider the sum
of entropies (3):

$S_{N}=\sum_{n=1}^{N}H(X_{n})$ (23)

subject to the constraint $H(X_{n})\sim-\ln q_{n}$ so we may define the
Lagrangian function:

$\mathcal{L}(\lambda,q_{n})=\sum_{n=1}^{N}q_{n}\cdot\ln
n-\lambda\big{(}S_{N}+\sum_{n=1}^{N}\ln q_{n}\big{)}$ (24)

Thus, we find that:

$\frac{\partial\mathcal{L}}{\partial q_{n}}=\ln
n-\frac{\lambda}{q_{n}}=0\implies q_{n}=\frac{\lambda}{\ln n}$ (25)

which only makes sense if $\lambda=1$ so we have:

$q_{n}=\frac{1}{\ln n}$ (26)

and therefore among the rare-event densities, the empirical density of
primes(which we may observe) is a natural model for a uniform source
distribution which is not directly observable.

##  4 Application to Cramér’s random model, Part I

Using the method of level sets, we may demonstrate that the typical
probability that an integer in the interval $[1,n]\subset\mathbb{N}$ is prime
is given by:

$\forall z\sim U([1,n]),P(z\in\mathbb{P})\sim\frac{1}{\ln n}$ (27)

If $X_{n}=\\{x_{i}\\}_{i=1}^{n}\in\\{0,1\\}^{n}$ defines a prime encoding i.e.
a sequence where $x_{k}=1$ if $k\in\mathbb{P}$ and $x_{k}=0$ otherwise, then
we may define $\pi(\sqrt{n})$ primality tests as any composite integer in
$[1,n]$ has at most $\pi(\sqrt{n})$ distinct prime factors:

$\forall
A_{p}\in\\{A_{p_{k}}\\}_{k=1}^{\pi(\sqrt{n})},A_{p}=\\{z\in[1,n]:\text{gcd}(p,z)=1\\}\bigcup\\{p\\}$
(28)

and therefore:

$\forall z\sim U([1,n]),H(X_{n})=-\ln\prod_{k=1}^{\pi(\sqrt{n})}P(z\in
A_{p_{k}})$ (29)

In order to define $P(z\in A_{p_{k}})$ we shall implicitly use the
approximation that for
$p\leq\sqrt{n},\frac{1}{p}-\frac{1}{n}\approx\frac{1}{p}$ so we have:

$\forall z\sim U([1,n]),P(z\in
A_{p_{k}})=\big{(}1-\frac{1}{p_{k}}\big{)}+\frac{1}{n}\approx\big{(}1-\frac{1}{p_{k}}\big{)}$
(30)

which allows us to make (29) precise:

$H(X_{n})\sim-\ln\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}$ (31)

Using Mertens’ third theorem we have:

$\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\approx\frac{e^{-\gamma}}{\frac{1}{2}\cdot\ln
n}\approx\frac{0.9}{\ln n}$ (32)

so we may infer that:

$H(X_{n})\sim\ln\ln n$ (33)

and therefore $X_{n}\in\mathcal{L}_{q_{n}}$ where:

$\mathcal{L}_{q_{n}}=\\{X_{n}\in\\{0,1\\}^{\infty}:H(X_{n})\sim\ln\ln n\\}$
(34)

From (34), we may deduce (27):

$\forall z\sim U([1,n]),q_{n}=P(z\in\mathbb{P})\sim\frac{1}{\ln n}$ (35)

which means that the density of the primes at $x\in\mathbb{N}$ is on the order
of $\sim\frac{1}{\ln x}$ and therefore the expected number of primes less than
$n$ is given by:

$\pi(n)\sim\int_{2}^{n}\frac{1}{\ln x}dx\sim\frac{n}{\ln n}$ (36)

in complete agreement with the Prime Number Theorem.

##  5 Application to Cramér’s random model, Part II

Given the prime encoding $X_{n}$ with uniform source distribution, let’s
define the nth prime gap $G_{n}=p_{n+1}-p_{n}$ so we may consider the
cumulative probability:

$\sum_{k=1}^{G_{n}}P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)=1$ (37)

and its associated entropy:

$H_{n}=\sum_{k=1}^{G_{n}}-P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)\cdot\ln
P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)$ (38)

where $G_{n}<p_{n}$ due to Bertrand’s postulate and we’ll note that

$\exists k,P(p_{n+1}-p_{n}=k)=1\implies H_{n}=0$ (39)

so the entropy $H_{n}$ collapses if and only if we simultaneously measure the
values of both $p_{n}$ and $p_{n+1}$.

As the subsequence $\\{x_{p_{n+k}}\\}_{k=1}^{G_{n}}$ halts at $k\in[1,G_{n}]$
where $x_{p_{n}+k}=1$, there are at most $G_{n}$ possible ways for this
sequence to halt. Furthermore, in order to bound the halting probability
$P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)$ we may use the density formula (36) to
consider its components:

$\forall k\sim
U([1,G_{n}]),P(p_{n+1}=p_{n}+k)=P(p_{n}=p_{n+1}-k)\sim\frac{1}{\ln p_{n}}$
(40)

$\forall k\sim U([1,G_{n}]),P(x_{p_{n}+k}=1)\sim\frac{1}{\ln p_{n}}$ (41)

where

$P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)\geq P(x_{p_{n}+k}=1)\cdot
P(p_{n+1}=p_{n}+k)$ (42)

Using (40),(41) and (42) we may then derive the lower bound:

$\forall k\sim U([1,G_{n}]),P(x_{p_{n}+k}=1\land
p_{n+1}-p_{n}=k)\gtrsim\frac{1}{(\ln p_{n})^{2}}$ (43)

which implies:

$\frac{1}{(\ln
p_{n})^{2}}\lesssim\frac{1}{G_{n}}\sum_{k=1}^{G_{n}}P(x_{p_{n}+k}=1\land
p_{n+1}-p_{n}=k)=\frac{1}{G_{n}}$ (44)

and since $G_{n}=p_{n+1}-p_{n}$, we may conclude:

$p_{n+1}-p_{n}=\mathcal{O}((\ln p_{n})^{2})$ (45)

as conjectured by Harald Cramér in 1936 [1].

##  6 Details of the $\frac{1}{p}-\frac{1}{n}\approx\frac{1}{p}$ approximation
in Cramér’s model

If we define,

$\Lambda_{k}:=\text{set of }{\pi(\sqrt{n})\choose k}\text{ distinct subsets of
}\\{p_{k}\\}_{k=1}^{\pi(\sqrt{n})}$ (46)

then $\lvert\Lambda_{k}\rvert={\pi(\sqrt{n})\choose k}$ and we may derive the
absolute error:

$\Big{\lvert}\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}+\frac{1}{n})-\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\Big{\rvert}\leq\sum_{k=1}^{\pi(\sqrt{n})-1}\frac{1}{n^{k}}\sum_{\lambda\in\Lambda_{k}}\prod_{p\in\lambda}\big{(}1-\frac{1}{p}\big{)}+\frac{1}{n^{\pi(\sqrt{n})}}$
(47)

and given that:

$\pi(\sqrt{n})<\sqrt{n}\implies\frac{1}{n^{k}}\cdot{\pi(\sqrt{n})\choose
k}\leq\frac{1}{n^{k/2}}$ (48)

the absolute error satisfies the following inequality:

$\Delta_{n}=\Big{\lvert}\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}+\frac{1}{n}\big{)}-\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\Big{\rvert}\leq\frac{1}{\sqrt{n}}+\frac{1}{n}$
(49)

so the relative error converges to zero:

$\lim_{n\to\infty}\frac{\Delta_{n}}{\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}}\leq\lim_{n\to\infty}\big{(}\frac{\ln
n}{\sqrt{n}}+\frac{\ln n}{n}\big{)}=0$ (50)

which provides us with the necessary justifications in our derivation of
Cramér’s random model, specifically formulas (32),(33) and (34).

##  7 The incompressibility of prime encodings as a fundamental law of
physics?

> Information is physical.-Rolf Landauer

Let’s suppose we get the world’s best theoretical physicists to design a
machine learning system whose aim is to predict the location of the Nth prime
given the locations of the first $N-1$ primes. What is the best case scenario
we can hope for?

Considering that the prime numbers have a maximum entropy distribution the
true positive rate of any such system can’t exceed 50%. This state of affairs
should hold true at all times regardless of technological progress, given what
is known about Turing Machines without access to Oracles. So this fundamental
limit might as well be a Physical Law.

This Law might even have Cosmological implications. In fact, let’s suppose
that the Universe emerged from a Singularity which would be consistent with
Big Bang Cosmology. At this precise instant, when the Universe and all of its
mathematical structure came into existence the most reasonable machine
learning system would have to assume a maximum entropy distribution over the
location of the primes. This would be equivalent to the application of Occam’s
razor, a principle that is generally applicable in Universes where Quantum
Information is conserved.

Thus, at this Singularity the Minimum Description Length of prime encodings
would have been defined relative to a Universal Turing Machine as follows:

$\mathbb{E}[K(X_{N})]\sim\pi(N)\cdot\ln N\sim N$ (51)

which is consistent with present-day observations i.e. the information-
theoretic analyses on pages 4 and 5. How might we explain this mysterious
coincidence?

In accordance with Big Bang Cosmology and the principle that all the Quantum
Information in the Universe is conserved we are led to a natural Cosmological
hypothesis concerning the distribution of primes. The maximum entropy
distribution of prime encodings may be understood as a mathematical signature
that time travelled from the earliest moments of the Big Bang.

The author concedes that such a Hypothesis would appear heretical to
scientists who haven’t considered the possibility that the Universe may be
simulated by a Universal Quantum Turing Machine.

##  8 Conclusion

The finite-state incompressibility of prime encodings is of general scientific
importance as it implies that it is not possible for any finite state machine
such as a machine learning model to infer the definition of prime numbers(i.e.
Unique Factorisation Theorem) from a prime encoding $X_{n}$ of any length $n$.
Equivalently, theoretical physicists can’t design a machine learning system
which has favourable odds of locating the next prime number i.e. a true-
positive rate greater than 50%. Therefore, we may conclude that machine
learning does not confer an advantage to a mathematician investigating the
distribution of primes.

Having said this, it is worth clarifying that this analysis does not preclude
the application of machine learning to derive number-theoretic insights. In
fact, Yang-Hui He recently found machine learning to be particularly effective
at identifying interesting structures in the setting of arithmetic geometry
and he suspects that it is for the following reason [11]:

> At the most basic level, every computation in algebraic geometry, be it a
> spectral sequence or a Gröbner basis, reduces to finding kernels and
> cokernels of sets of matrices (over $\mathbb{Z}$ or even over $\mathbb{C}$),
> albeit of quickly forbidding dimensions. Matrix/tensor manipulation is the
> heart of any neural network. Number theory, on the other hand, ultimately
> involves patterns of prime numbers which, as is well known, remain elusive.

which is both consistent with our findings and gives us reason for measured
optimism.

##  Appendix A An invariance theorem for algorithmically random data

Let’s suppose we have a natural signal described by the process $X$:

$x_{n}\in\\{0,1\\},x_{n+1}=\varphi\circ x_{1:n}$ (52)

If we should use machine learning to approximate $\varphi$ given the datasets
$X_{N}^{\text{train}}=\\{x_{i}\\}_{i=1}^{N},X_{N}^{\text{test}}=\\{x_{i}\\}_{i=N+1}^{2N}$
such that for any $\hat{f}\in F_{\theta}$:

$\exists k\in[1,n-1],x_{n+1}=\hat{f}\circ
x_{n-k:n}\Rightarrow\delta_{\hat{f}(x_{n-k:n}),x_{n+1}}=1$ (53)

then $X_{N}$ is asymptotically incompressible if for large $N$ any solution to
the empirical risk minimisation problem:

$\hat{f}=\max_{f\in
F_{\theta}}\frac{1}{N-k}\sum_{n=k+1}^{N}\delta_{f(x_{n-k:n}),x_{n+1}}$ (54)

has an expected performance:

$\frac{1}{N-k}\sum_{n=N+k+1}^{2N-1}\delta_{\hat{f}(x_{n-k:n}),x_{n+1}}\leq\frac{1}{2}$
(55)

Furthermore, if the dataset is imbalanced i.e.
$\frac{1}{N}\sum_{i=1}^{N}x_{i}\neq\frac{1}{2}$ then we may generalise this
result by introducing the auxiliary definitions:

$y_{n}=x_{n+1}$ (56)

$\hat{y_{n}}=\hat{f}\circ x_{n-k:n}$ (57)

$\beta_{n}=\delta_{y_{n},\hat{y_{n}}}$ (58)

$N_{1}=\sum_{n=N+k+1}^{2N}\delta_{y_{n},1}$ (59)

$N_{0}=\sum_{n=N+k+1}^{2N}\delta_{y_{n},0}$ (60)

and so for large $N$, we have:

$\mathcal{L}_{N}[\hat{f}]=\min\Big{[}\frac{1}{N_{0}}\sum_{n=N+k+1}^{2N-1}\delta_{y_{n},0}\cdot\beta_{n},\frac{1}{N_{1}}\sum_{n=N+k+1}^{2N-1}\delta_{y_{n},1}\cdot\beta_{n}\Big{]}\leq\frac{1}{2}$
(61)

and therefore:

$\forall\hat{f}\in
F_{\theta},\lim_{N\to\infty}P(\mathcal{L}_{N}[\hat{f}]>\frac{1}{2})=0$ (62)

Finally, as these results are invariant to transformations that preserve the
phase-space dimension of $X$, this theorem may be used as an overfitting test
for binary sequences that are finite-state incompressible.

##  Appendix B Revisiting the unreasonable effectiveness of mathematics

62 years since Eugene Wigner’s highly influential essay on the unreasonable
effectiveness of mathematics in the natural sciences, it may be time for a re-
appraisal. On balance, with important theoretical advances in algorithmic
information theory and Quantum Computation it appears that the remarkable
effectiveness of mathematics in the natural sciences is quite reasonable.

By effectiveness, I am specifically referring to Wigner’s observation that
mathematical laws have remarkable generalisation power.

###  B.1 An information-theoretic perspective

An acute observer will note that the same mathematical laws with remarkable
generalisation power in the natural sciences are also constrained by Occam’s
razor. Given two computable theories, Einstein explicitly stated that a
physicist ought to choose the simplest theory that yields negligible
experimental error:

> It can be scarcely denied that the supreme goal of all theory is to make the
> irreducible basic elements as simple and as few as possible without having
> to surrender the adequate representation of a single datum of
> experience.-Einstein(1933)

In fact, from an information-theoretic perspective the remarkable
generalisation power of mathematical laws in the natural sciences is a direct
consequence of the effectiveness of Occam’s razor.

###  B.2 The Law of Conservation of Information

From an information-theoretic perspective, a Universe where Occam’s razor is
generally applicable is one where information is generally conserved. This law
of conservation of information which dates back to von Neumann essentially
states that the von Neumann entropy is invariant to Unitary transformations.
This is meaningful within the framework of Everettian Quantum Mechanics as a
density matrix may be assigned to the state of the Universe. This way
information is conserved as we run a simulation of the Universe forward in
time.

Moreover, given that Occam’s razor has an appropriate formulation within the
context of algorithmic information theory as the Minimum Description Length
principle, this information-theoretic perspective generally presumes that the
Universe itself may be simulated by a Universal Turing Machine.

###  B.3 The Physical Church-Turing thesis

The research of David Deutsch(and others) on the Physical Church-Turing thesis
explains how a Universal Quantum computer may simulate the laws of physics.
This is consistent with the general belief that Quantum Mechanics may be used
to simulate all of physics so the most important contributions to the Physical
Church-Turing thesis have been via theories of quantum computation.

More importantly, the Physical Church-Turing thesis provides us with a
credible explanation for the remarkable effectiveness of mathematics in the
natural sciences.

###  B.4 What is truly remarkable

If we view the scientific method as an algorithmic search procedure then there
is no reason, a priori, to suspect that a particular inductive bias should be
particularly powerful. This much was established by David Wolpert in his No
Free Lunch theorems [22].

On the other hand, the history of the natural sciences indicates that Occam’s
razor is remarkably effective. The effectiveness of this inductive bias has
recently been used to explain the generalisation power of deep neural networks
when applied to data that is finite-state compressible [23].

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[14] David Deutsch. Quantum theory, the Church–Turing principle and the
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