# Mass spectra of $0^{--}$ and $0^{+-}$ hidden-heavy baryoniums

Bing-Dong<EMAIL_ADDRESS>1Department of Physics, Liaoning Normal
University, Dalian 116029, China
2 School of Physics, University of Chinese Academy of Science, Yuquan Road
19A, Beijing 100049, China
3 School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute
for Advanced Study, UCAS, Hangzhou 310024, China

###### Abstract

In this work, the spectra of the prospective exotic hidden-charm and hidden-
bottom baryonium, viz. the baryon-antibaryon states, with $J^{PC}=0^{--}$ and
$0^{+-}$ are investigated in the framework of QCD sum rules. The non-
perturbative contributions up to dimension 12 are taken into account.
Numerical results indicate that there might exist 3 possible $0^{--}$ hidden-
charm baryonium states with masses $(5.21\pm 0.15)$, $(5.53\pm 0.15)$, and
$(5.45\pm 0.12)$ GeV, and 5 possible $0^{+-}$ hidden-charm baryonium states
with masses $(4.76\pm 0.15)$, $(5.25\pm 0.15)$, $(5.16\pm 0.17)$, $(5.51\pm
0.14)$, and $(5.69\pm 0.14)$ GeV, respectively. The corresponding hidden-
bottom partners are found lying in the range of $11.68-12.28$ GeV and
$11.38-12.33$ GeV, respectively. The possible baryonium decay modes are
analyzed, which are hopefully measurable in LHC experiments.

###### pacs:

11.55.Hx, 12.38.Lg, 12.39.Mk

##  I Introduction

Since the observation of X(3872) Choi:2003ue , many novel hadronic states or
candidates, denoted as $XYZ$ and $P_{c}$ states, have been observed in
experiments. These novel hadronic states can hardly be well understood by
conventional quark model GellMann:1964nj ; Zweig , and exploring their
properties and finding more possible states may greatly enrich the hadron
family and our knowledge of the nature of quantum chromodynamics(QCD). With
the more novel hadronic states discovering and the deepening of our
understanding of strong interaction, it is reasonable to believe that the
renaissance of hadron physics will come sooner or later, which attracts more
and more interests from theorists and experimentalists.

Among the novel hadronic states, special attention ought be paid to the states
possess some exotic quantum numbers (like $0^{--}$, $0^{+-}$, $1^{-+}$,
$2^{+-}$, and so on), since they cannot mix with the conventional ones. In the
literature, many theoretical investigations on exotic hadronic states were
made, including tetraquarks Jiao:2009ra ; LEE:2020eif ; Shen:2010ky ;
Wang:2021lkg ; General:2007bk ; Huang:2016rro ; Wan:2022xkx , hybrid tates
Liu:2005rc ; Chetyrkin:2000tj ; Govaerts:1984bk ; General:2006ed ;
Ishida:1991mx , glueballs Qiao:2014vva ; Tang:2015twt ; Cotanch:2006wv ;
Bellantuono:2015fia ; Chen:2021cjr ; Zhang:2021itx . On the other hand, exotic
hadronic states have gradually been observed in experiment, like the most
recently observed $\eta_{1}(1855)$ BESIII:2022riz ; BESIII:2022iwi . It is
highly expected that more novel hadronic structures with exotic quantum
numbers will emerge soon afterwards.

While the theoretical investigations on hexaquark with exotic quantum number
are still rare. Up to now, most of studies on hexaquark are focus on
conventional quantum numbers, such as the deuteron. Deuteron created at the
beginning of Universe and its stability is responsible for the production of
other elements, is a well-established dibaryon molecular state composed of a
proton and a neutron, withe $J^{P}=1^{+}$ and binding energy $E_{B}=2.225$ MeV
Weinberg:1962hj . On the other hand, the baryon-antibaryon systems (baryonium)
is another special class of heaxquark configuration. Actually, the history of
the investigation on baryon-antibaryon system can date back to the proposal by
Fermi and Yang that $\pi$-mesons may be composite particles formed by a
nucleon-antinucleon pair in 1940s Fermi:1949voc , and their scenario was later
on replaced by the quark model. Entering the new millennium, the heavy
baryonium were proposed and employed to explain the extraordinary nature of
$Y(4260)$ Qiao:2005av ; Qiao:2007ce  and other charmonium-like states observed
in experiments. Later on, more investigations on baryonium are performed from
various aspects Chen:2011cta ; Chen:2013sba ; Wan:2019ake ; Chen:2016ymy ;
Liu:2007tj ; Wang:2021qmn ; Wan:2022uie ; Liu:2021gva ; Wan:2021vny .
Especially, our previous work Wan:2021vny  of the investigation of light
baryonium states predicted that the mass of the $\Lambda-\bar{\Lambda}$
baryonium state with $J^{PC}$ of $1^{--}$ is about $2.34$ GeV, which is
consistent with the experimental observation of the structure in
$\Lambda\bar{\Lambda}$ mass spectra with mass of $(2356\pm 24)$ MeV with
quantum number of $1^{--}$ by BESIII BESIII:2022tvj .

In this work, we investigate the hidden-heavy baryonium states with the exotic
quantum numbers of $J^{PC}=0^{--}$ and $0^{+-}$ in the framework of QCD sum
rules (QCDSR). In past 40 years, the QCDSR technique Shifman  as a model
independent approach has achieved a lot in the study of hadron spectroscopy.
To establish the QCDSR, one need to construct the proper interpolating
currents corresponding to hadrons of interest, which possesses the foremost
information of the concerned hadrons, like quantum numbers and structure
components. With the interpolating currents, the two-point correlation
function can be readily established and by equating the operator product
expansion (OPE) side and the phenomenological side of the two-point
correlation, we can then obtain the mass of the hadron.

The rest of the paper is arranged as follows. After the introduction, a brief
interpretation of QCD sum rules and some primary formulas in our calculation
are presented in Sec. II. The numerical analysis and results are given in Sec.
III. The possible decay modes of the exotic baryonium states are analyzed in
Sec. 3. The last part is left for a brief summary.

##  II Formalism

For the $0^{--}$ baryonium state, the interpolating currents in molecular
configuration can be constructed as:

$\displaystyle j_{0^{--}}^{A}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (1) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{--}}^{B}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (2) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{--}}^{C}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (3) $\displaystyle+$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{--}}^{D}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]$ (4) $\displaystyle+$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{--}}^{E}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (5) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{--}}^{F}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]$ (6) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;.$

The interpolating currents for $0^{+-}$ in molecular configuration are found
to be in forms:

$\displaystyle j_{0^{+-}}^{A}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (7) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{+-}}^{B}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (8) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{+-}}^{C}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]$ (9) $\displaystyle+$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{+-}}^{D}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (10) $\displaystyle+$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{+-}}^{E}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]$ (11) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;,$ $\displaystyle j_{0^{+-}}^{F}(x)$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{abc}\epsilon_{def}}{\sqrt{2}}{\Big{\\{}}[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)C\gamma_{\mu}\gamma_{5}q^{\prime}_{b}(x)][\bar{q}_{e}(x)C\bar{q}^{\prime
T}_{f}(x)]$ (12) $\displaystyle-$
$\displaystyle[\bar{Q}_{d}(x)\gamma_{\mu}\gamma_{5}Q_{c}(x)][q^{T}_{a}(x)Cq^{\prime}_{b}(x)][\bar{q}_{e}(x)\gamma_{\mu}\gamma_{5}C\bar{q}^{\prime
T}_{f}(x)]{\Big{\\}}}\;.$

Here, the subscripts $a\cdots f$ are color indices, $q$ and $q^{\prime}$ stand
for light quark $u$ and $d$, respectively, $Q$ represents the heavy quark $c$
or $b$, and $C$ is the charge conjugation matrix.

With the currents (1)-(12), the two-point correlation function can be readily
established, i.e.,

$\displaystyle\Pi_{J^{PC}}^{k}(q^{2})$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{iq\cdot x}\langle
0|T\\{j_{J^{PC}}^{k}(x),\;j_{J^{PC}}^{k}(0)^{\dagger}\\}|0\rangle\;,$ (13)

where $|0\rangle$ denotes the physical vacuum, and $k$ runs from $A$ to $F$.

On the OPE side, correlation function $\Pi(q^{2})$ can be expressed by the
dispersion relation as

$\displaystyle\Pi^{OPE,\;k}_{J^{PC}}(q^{2})=\int_{s_{min}}^{\infty}ds\frac{\rho^{OPE,\;k}_{J^{PC}}(s)}{s-q^{2}}\;.$
(14)

Here, $s_{min}$ is the kinematic limit, which usually corresponds to the
square of the sum of current-quark masses of the hadron Albuquerque:2013ija ,
$\rho^{OPE,\;k}_{J^{PC}}(s)=\text{Im}[\Pi^{OPE,\;k}_{J^{PC}}(s)]/\pi$, and

$\displaystyle\rho^{OPE}(s)$ $\displaystyle=$
$\displaystyle\rho^{pert}(s)+\rho^{\langle
G^{2}\rangle}(s)+\rho^{\langle\bar{q}Gq\rangle}(s)+\rho^{\langle\bar{q}q\rangle^{2}}(s)+\rho^{\langle
G^{3}\rangle}(s)$ (15) $\displaystyle+$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)+\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)+\rho^{\langle\bar{q}Gq\rangle^{2}}(s)\;.$

The Feynman diagrams corresponding to each term of Eq. (15) are schematically
shown in Fig. 1. The analytical expressions of $\rho^{OPE,\;k}_{J^{PC}}(s)$
can be calculated and given in appendix A.

Figure 1: The typical Feynman diagrams related to the correlation function,
where the thick solid line represents the heavy quark, the thin solid line
stands for the light quark, and the spiral line denotes the gluon. There is no
heavy quark consendsate due to the large heavy quark mass.

On the phenomenological side, the correlation function $\Pi(q^{2})$ can be
expressed as a dispersion integral over the physical regime after isolating
the ground state contribution from the baryonium state, i.e.,

$\displaystyle\Pi^{phen,\;k}_{J^{PC}}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{{\lambda^{k}_{J^{PC}}}^{2}}{{M^{k}_{J^{PC}}}^{2}-q^{2}}+\frac{1}{\pi}\int_{s_{0}}^{\infty}ds\frac{\rho^{k}_{J^{PC}}(s)}{s-q^{2}}\;,$
(16)

where $M$ denotes the mass of baryonium state, $\lambda$ is the coupling
constant, and $\rho(s)$ is the spectral density that contains the
contributions from higher excited states and the continuum states above the
threshold $s_{0}$.

Performing Borel transform on Eqs. (14) and (16), and matching the OPE side
with the phenomenological side of the correlation function $\Pi(q^{2})$, one
can finally obtain the mass of the tetraquark state,

$\displaystyle
M^{k}_{J^{PC}}(s_{0},M_{B}^{2})=\sqrt{-\frac{L^{k}_{J^{PC},\;1}(s_{0},M_{B}^{2})}{L^{k}_{J^{PC},\;0}(s_{0},M_{B}^{2})}}\;.$
(17)

Here $L_{0}$ and $L_{1}$ are respectively defined as

$\displaystyle
L^{k}_{J^{PC},\;0}(s_{0},M_{B}^{2})=\int_{s_{min}}^{s_{0}}ds\;\rho^{OPE,\;k}_{J^{PC}}(s)e^{-s/M_{B}^{2}}\;,$
(18)

and

$\displaystyle
L^{k}_{J^{PC},\;1}(s_{0},M_{B}^{2})=\frac{\partial}{\partial{\frac{1}{M_{B}^{2}}}}{L^{k}_{J^{PC},\;0}(s_{0},M_{B}^{2})}\;.$
(19)

##  III Numerical analysis

In performing the numerical calculation, the broadly accepted inputs are taken
Albuquerque:2013ija ; Matheus:2006xi ; Cui:2011fj ; Narison:2002pw ; P.Col ;
Tang:2019nwv :

$\displaystyle\begin{aligned} &m_{c}(m_{c})=\overline{m}_{c}=(1.275\pm
0.025)\;\text{GeV}\;,&&m_{b}(m_{b})=\overline{m}_{b}=(4.18\pm
0.03)\;\text{GeV}\;,\\\ &\langle\bar{q}q\rangle=-(0.24\pm
0.01)^{3}\;\text{GeV}^{3}\;,&&\langle g_{s}^{2}G^{2}\rangle=(0.88\pm
0.25)\;\text{GeV}^{4}\;,\\\ &\langle\bar{q}g_{s}\sigma\cdot
Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle;,&&\langle
g_{s}^{3}G^{3}\rangle=(0.045\pm 0.013)\;\text{GeV}^{6}\;,\\\
&m_{0}^{2}=(0.8\pm 0.1)\;\text{GeV}^{2}\;.\end{aligned}$ (20)

Here, $\overline{m}_{c}$ and $\overline{m}_{b}$ represent heavy-quark running
masses in $\overline{\text{MS}}$ scheme. For light quark, the chiral quark
limit mass $m_{q}=0$ is adopted.

Moreover, two additional parameters $s_{0}$ and $M_{B}^{2}$ introduced in
establishing the sum rules need to be exacted. We can fix them in light of the
so-called standard procedures by fulfilling the following two criteria Shifman
; Reinders:1984sr ; P.Col . One asks for the convergence of the OPE, which is
to compare relative contribution of higher dimension condensate to the total
contribution on the OPE side, and then a reliable region for $M_{B}^{2}$ will
be chosen to retain the convergence. The other one requires that the pole
contribution (PC) is more than $15\%$ of the total for hexaquark states
Wan:2019ake ; Wan:2022uie ; Wan:2021vny . Mathematically, the two criteria can
be formulated as:

$\displaystyle
R^{OPE}=\left|\frac{L_{J^{PC}\;,0}^{k\;,dim=12}(s_{0},M_{B}^{2})}{L_{J^{PC}\;,0}^{k}(s_{0},M_{B}^{2})}\right|\,,$
(21) $\displaystyle
R^{PC}=\frac{L_{J^{PC}\;,0}^{k}(s_{0},M_{B}^{2})}{L_{J^{PC}\;,0}^{k}(\infty,M_{B}^{2})}\;.$
(22)

To find a proper value for continuum threshold $s_{0}$, a similar analysis as
in Refs. Qiao:2013dda ; Tang:2016pcf ; Qiao:2013raa  is performed. Therein,
one needs to find the proper value, which has an optimal window for the mass
curve of the baryonium state. Within this window, the physical quantity, that
is the mass of the baryonium state, should be independent of the Borel
parameter $M_{B}^{2}$ as much as possible. In practice, we will vary
$\sqrt{s_{0}}$ by $0.2$ GeV to obtain the lower and upper bounds, and hence
the uncertainties of $\sqrt{s_{0}}$ Wan:2020oxt ; Wan:2020fsk .

With the above preparation the mass spectrum of exotic baryonium states can be
numerically evaluated. For the $0^{--}$ hidden-charm exotic baryonium state in
Eq. (1), as an example, the ratios $R^{OPE\;,A}_{0^{--}}$ and
$R^{PC\;,A}_{0^{--}}$ are presented as functions of Borel parameter
$M_{B}^{2}$ in Fig. 2(a) with different values of $\sqrt{s_{0}}$, i.e., $5.4$,
$5.6$ and $5.8$ GeV. The reliant relations of $M^{A}_{0^{--}}$ on parameter
$M_{B}^{2}$ are displayed in Fig. 2(b). The optimal Borel window is found in
range $3.1\leq M_{B}^{2}\leq 4.1\;\text{GeV}^{2}$, and the mass $M^{A}$ can
then be obtained:

$\displaystyle M^{A}_{0^{--}}$ $\displaystyle=$ $\displaystyle(5.21\pm
0.15)\;\text{GeV}.$ (23)

Figure 2:  (a) The ratios of ${R_{A}^{OPE}}$ and ${R_{A}^{PC}}$ as functions
of the Borel parameter $M_{B}^{2}$ for different values of $\sqrt{s_{0}}$,
where blue lines represent ${R_{A}^{OPE}}$ and red lines denote ${R_{A}^{PC}}$
. (b) The mass $M^{A}$ as a function of the Borel parameter $M_{B}^{2}$ for
different values of $\sqrt{s_{0}}$.

With the same analyses, and the OPE, pole contribution and the masses as
functions of Borel parameter $M_{B}^{2}$ are given in the appendix B the
masses of exotic hidden-charm baryonium states in Eqs. (2)-(12) can be
extracted and tabulated in Table 1.

Similarly, we can evaluate the exotic hidden-bottom baryonium states. By using
the obtained analytical results but with $m_{c}$ being replaced by $m_{b}$,
the corresponding masses are readily obtained and tabulated in Table 2

The errors in results mainly stem from the uncertainties in quark masses,
condensates and threshold parameter $\sqrt{s_{0}}$. For the currents
$j_{0^{--}}^{D}$, $j_{0^{--}}^{E}$, $j_{0^{--}}^{F}$, and $j_{0^{+-}}^{F}$, no
matter what values of $M_{B}^{2}$ and $\sqrt{s_{0}}$ take, no optimal window
for stable plateaus exist. That means the currents in Eqs. (4), (5), (6), and
(12) do not support the corresponding baryonium states.

Table 1: The continuum thresholds, Borel parameters, and predicted masses of hidden-charm baryonium. $J^{PC}$ | Current | $\sqrt{s_{0}}\;(\text{GeV})$ | $M_{B}^{2}\;(\text{GeV}^{2})$ | $M^{X}\;(\text{GeV})$
---|---|---|---|---
$0^{--}$ | $A$ | $5.6\pm 0.2$ | $3.1-4.1$ | $5.21\pm 0.15$
| $B$ | $5.9\pm 0.2$ | $3.2-4.3$ | $5.53\pm 0.15$
| $C$ | $5.8\pm 0.2$ | $4.1-5.1$ | $5.45\pm 0.12$
$0^{+-}$ | $A$ | $5.1\pm 0.2$ | $2.8-3.8$ | $4.76\pm 0.15$
| $B$ | $5.6\pm 0.2$ | $3.0-4.1$ | $5.25\pm 0.15$
| $C$ | $5.6\pm 0.2$ | $3.3-4.3$ | $5.16\pm 0.17$
| $D$ | $5.9\pm 0.2$ | $3.9-4.9$ | $5.51\pm 0.14$
| $E$ | $6.1\pm 0.2$ | $3.8-5.2$ | $5.69\pm 0.14$
Table 2: The continuum thresholds, Borel parameters, and predicted masses of hidden-bottom baryonium. $J^{PC}$ | Current | $\sqrt{s_{0}}\;(\text{GeV})$ | $M_{B}^{2}\;(\text{GeV}^{2})$ | $M^{X}\;(\text{GeV})$
---|---|---|---|---
$0^{--}$ | $A$ | $12.3\pm 0.2$ | $8.3-10.2$ | $11.82\pm 0.14$
| $B$ | $12.6\pm 0.2$ | $8.5-10.6$ | $12.14\pm 0.14$
| $C$ | $12.5\pm 0.2$ | $9.5-12.5$ | $11.95\pm 0.18$
$0^{+-}$ | $A$ | $12.0\pm 0.2$ | $8.0-10.0$ | $11.53\pm 0.15$
| $B$ | $12.4\pm 0.2$ | $8.5-10.9$ | $11.92\pm 0.15$
| $C$ | $12.2\pm 0.2$ | $7.9-10.3$ | $11.69\pm 0.17$
| $D$ | $12.5\pm 0.2$ | $9.4-11.4$ | $12.02\pm 0.14$
| $E$ | $12.7\pm 0.2$ | $10.0-12.1$ | $12.19\pm 0.14$

##  IV Decay analyses

To finally ascertain these exotic hidden-heavy baryonium states, the
straightforward procedure is to reconstruct them from their decay products,
though the detailed characters stills ask for more investigation. The typical
decay modes of these exotic baryonium are given in Table 3, and these
processes are expected to be measurable in the running LHC experiments.

Table 3: typical decay modes of the exotic hidden-heavy baryonium. $J^{PC}$ |  $c$-sector |  $b$-sector
---|---|---
$0^{--}$ | $\Lambda_{c}^{+}\Lambda_{c}(2595)^{-}$ | $\Lambda_{b}\overline{\Lambda}_{b}(5912)$
| $\Lambda_{c}^{-}\Lambda_{c}(2595)^{+}$ | $\overline{\Lambda}_{b}\Lambda_{b}(5912)$
| $\Lambda_{c}(2625)^{+}\Lambda_{c}(2860)^{-}$ | $\Sigma^{\ast}_{b}\overline{\Lambda}_{b}(5920)$
| $\Lambda_{c}(2625)^{-}\Lambda_{c}(2860)^{+}$ | $\overline{\Sigma}^{\ast}_{b}\Lambda_{b}(5920)$
| $\Lambda_{c}^{+}\Lambda_{c}(2860)^{-}\gamma$ | $\Sigma^{\ast}_{b}\overline{\Lambda}_{b}\gamma$
| $\Lambda_{c}^{-}\Lambda_{c}(2860)^{+}\gamma$ | $\overline{\Sigma}_{b}\Lambda_{b}\gamma$
$0^{+-}$ | $\Lambda_{c}^{+}\Lambda_{c}(2595)^{-}$ | $\Lambda_{b}\overline{\Lambda}_{b}(5912)$
| $\Lambda_{c}^{-}\Lambda_{c}(2595)^{+}$ | $\overline{\Lambda}_{b}\Lambda_{b}(5912)$
| $\Lambda_{c}(2625)^{+}\Lambda_{c}(2860)^{-}$ | $\Sigma^{\ast}_{b}\overline{\Lambda}_{b}(5920)$
| $\Lambda_{c}(2625)^{-}\Lambda_{c}(2860)^{+}$ | $\overline{\Sigma}^{\ast}_{b}\Lambda_{b}(5920)$
| $\Lambda_{c}(2595)^{+}\Sigma_{c}(2520)^{-}\gamma$ | $\Lambda_{b}\overline{\Lambda}_{b}(5920)\gamma$
| $\Lambda_{c}(2595)^{-}\Sigma_{c}(2520)^{+}\gamma$ | $\overline{\Lambda}_{b}\Lambda_{b}(5920)\gamma$
| $\Lambda_{c}^{+}\Lambda_{c}(2625)^{-}\gamma$ | $\Sigma^{\ast}_{b}\overline{\Lambda}_{b}\gamma$
| $\Lambda_{c}^{-}\Lambda_{c}(2625)^{+}\gamma$ | $\overline{\Sigma}^{\ast}_{b}\Lambda_{b}\gamma$
| $\Lambda_{c}^{+}\Lambda_{c}(2860)^{-}\gamma$ |
| $\Lambda_{c}^{-}\Lambda_{c}(2860)^{+}\gamma$ |

##  V Summary

In summary, we have investigated the exotic hidden-heavy baryonium states with
$J^{PC}=0^{--}$ and $0^{+-}$ in the framework of QCD sum rules. Our numerical
results are tabulated in Table 1 and 2 for charm and bottom sector,
respectively. Results indicate that there might exist 3 possible $0^{--}$
hidden-charm baryonium states with masses $(5.21\pm 0.15)$, $(5.53\pm 0.15)$,
and $(5.45\pm 0.12)$ GeV, and 5 possible $0^{+-}$ hidden-charm baryonium
states with masses $(4.76\pm 0.15)$, $(5.25\pm 0.15)$, $(5.16\pm 0.17)$,
$(5.51\pm 0.14)$, and $(5.69\pm 0.14)$ GeV, respectively. The corresponding
hidden-bottom partners are found lying in the range of $11.68-12.28$ GeV and
$11.38-12.33$ GeV, respectively. Moreover, the possible exotic baryonium decay
modes are analyzed, which might serve as a guide for experimental exploration.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China (NSFC) under the Grants 12247113 and Project funded by China
Postdoctoral Science Foundation under the Grants 2022M723117.

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##  Appendix A The spectral densities for $0^{--}$ and $0^{+-}$ baryonium
states

###  A.1 The spectral densities for $0^{--}$ baryonium state in Eqs. (1) and
(2)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\bigg{\\{}\frac{{\cal
F}_{\alpha\beta}^{6}(\alpha+\beta-1)^{4}(-5{\cal
F}_{\alpha\beta}+7m_{Q}^{2}(\alpha+\beta-1)}{3\times 7\times 5^{2}\times
2^{19}\pi^{10}\alpha^{6}\beta^{6}}\bigg{\\}}\;,$ (24)
$\displaystyle\rho^{\langle G^{2}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
G^{2}\rangle}{\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\bigg{\\{}{\cal
N}_{i}\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}(5m_{Q}^{2}(\alpha+\beta-1)-3{\cal
F}_{\alpha\beta})}{3\times 5\times 2^{19}\alpha^{4}\beta^{4}}$ (25)
$\displaystyle+$ $\displaystyle\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{5\times 3^{2}\times
2^{21}\alpha^{6}\beta^{6}}\left(4m_{Q}^{2}\left(\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+(\beta-1)\beta^{3}\right)\right.$
$\displaystyle-$ $\displaystyle{\cal
F}_{\alpha\beta}\left(2\alpha^{3}-3\alpha^{2}(\beta-1)-3\alpha\beta^{2}+\beta^{2}(3+2\beta))\right)\bigg{\\}}\;,$
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{-\langle G^{3}\rangle}{5\times 3^{2}\times
2^{23}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}\frac{d\alpha}{\alpha^{6}}\int_{\beta_{min}}^{1-\alpha}\frac{d\beta}{\beta^{6}}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}$ (26) $\displaystyle\times$
$\displaystyle\bigg{\\{}5{\cal
F}_{\alpha\beta}^{2}(\alpha^{3}+\beta^{3})+8{\cal
F}_{\alpha\beta}m_{Q}^{2}(2\alpha^{4}-3\alpha^{3}(\beta-1)-3\alpha\beta^{3}$
$\displaystyle+$ $\displaystyle
3\beta^{3}+2\beta^{4})-24m_{Q}^{4}(\alpha+\beta-1)(\alpha^{4}+\beta^{4})\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{{\cal
H}_{\alpha}}{48\pi^{2}}\langle\bar{q}q\rangle^{4}\;,$ (27)

where $M_{B}$ is the Borel parameter introduced by the Borel transformation,
$Q=c$ or $b$, and the factor ${\cal N}_{i}$ has the following definition :
${\cal N}_{A}=1$ and ${\cal N}_{B}=0$. Here, we also have the following
definitions:

$\displaystyle{\cal F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(\alpha+\beta)m_{Q}^{2}-\alpha\beta s\;,{\cal
H}_{\alpha}=m_{Q}^{2}-\alpha(1-\alpha)s\;,$ (28) $\displaystyle\alpha_{min}$
$\displaystyle=$
$\displaystyle\left(1-\sqrt{1-4m_{Q}^{2}/s}\right)/2,\;,\alpha_{max}=\left(1+\sqrt{1-4m_{Q}^{2}/s}\right)/2\;,$
(29) $\displaystyle\beta_{min}$ $\displaystyle=$ $\displaystyle\alpha
m_{Q}^{2}/(s\alpha-m_{Q}^{2}).$ (30)

###  A.2 The spectral densities for $0^{--}$ baryonium state in Eqs. (3) and
(4)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{{\cal
F}^{6}_{\alpha\beta}(\alpha+\beta-1)^{4}\Big{(}7m_{Q}^{2}(\alpha+\beta-1)+10{\cal
F}_{\alpha\beta}\Big{)}}{3\times 7\times 5^{2}\times
2^{18}\pi^{10}\alpha^{6}\beta^{6}}\;,$ (31) $\displaystyle\rho^{\langle
G^{2}\rangle}(s)$ $\displaystyle=$ $\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle}{\pi^{10}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\bigg{\\{}-\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}}{3\times 5\times
2^{21}\alpha^{5}\beta^{5}}\Big{(}5m_{Q}^{2}(\alpha+\beta-1)^{2}$ (32)
$\displaystyle\times$ $\displaystyle(\alpha+\beta)+4{\cal
F}_{\alpha\beta}\big{(}\alpha^{2}+\beta(\beta-1)-\alpha(4\beta+1)\big{)}\Big{)}+\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{3^{2}\times 5\times
2^{20}\alpha^{6}\beta^{6}}$ $\displaystyle\times$
$\displaystyle\bigg{(}4m_{Q}^{2}\Big{(}\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+\beta^{3}(\beta-1)\Big{)}+{\cal
F}_{\alpha\beta}\Big{(}13\alpha^{3}+3\alpha^{2}(\beta-1)$ $\displaystyle+$
$\displaystyle
3\alpha\beta^{2}+\beta^{2}(13\beta-3)\Big{)}\bigg{)}\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}(s)$ $\displaystyle=$
$\displaystyle{\cal
N}_{i}\frac{\langle\bar{q}q\rangle^{2}}{2^{10}\pi^{6}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{2m_{Q}^{2}{\cal
F}_{\alpha\beta}^{3}(\alpha+\beta-1)^{2}+{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)}{\alpha^{3}\beta^{3}}\;,$ (33)
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle G^{3}\rangle}{5\times 3^{2}\times
2^{21}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}}{\alpha^{6}\beta^{6}}$ (34)
$\displaystyle\times$ $\displaystyle\bigg{(}5(\alpha^{3}+\beta^{3}){\cal
F}_{\alpha\beta}^{2}+4{\cal
F}_{\alpha\beta}m_{Q}^{2}\Big{(}13\alpha^{4}+3\alpha^{3}(\beta-1)+3\alpha\beta^{3}$
$\displaystyle+$
$\displaystyle\beta^{3}(13\beta-3)\Big{)}+12m_{Q}^{4}\Big{(}\alpha^{5}+\alpha^{4}(\beta-1)+\alpha\beta^{4}+\beta^{4}(\beta-1)\Big{)}\bigg{)}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)$
$\displaystyle=$ $\displaystyle\frac{-{\cal
N}_{i}\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}{2^{9}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{3}+3m_{Q}^{2}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)}{\alpha^{2}\beta^{2}}\;,$ (35)
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}$ $\displaystyle=$
$\displaystyle\frac{{\cal
N}_{i}\langle\bar{q}Gq\rangle^{2}}{2^{11}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\bigg{\\{}\frac{3{\cal
H}_{\alpha}^{2}}{2(1-\alpha)\alpha}+\int_{\beta_{min}}^{1-\alpha}d\beta\frac{3{\cal
F}_{\alpha\beta}m_{Q}^{2}}{\alpha\beta}\bigg{\\}}\;,$ (36)
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{{\cal
H}_{\alpha}-m_{Q}^{2}}{24\pi^{2}}\langle\bar{q}q\rangle^{4}\;,$ (37)

where ${\cal N}_{C}=1$ and ${\cal N}_{D}=-1$.

###  A.3 The spectral densities for $0^{--}$ baryonium state in Eqs. (5) and
(6)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{{\cal
F}^{6}_{\alpha\beta}(\alpha+\beta-1)^{4}\Big{(}10{\cal
F}_{\alpha\beta}-7m_{Q}^{2}(\alpha+\beta-1)\Big{)}}{3\times 7\times
5^{2}\times 2^{18}\pi^{10}\alpha^{6}\beta^{6}}\;,$ (38)
$\displaystyle\rho^{\langle G^{2}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle}{\pi^{10}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\bigg{\\{}\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}}{3\times 5\times
2^{21}\alpha^{5}\beta^{5}}\Big{(}5m_{Q}^{2}(\alpha+\beta-1)^{2}$ (39)
$\displaystyle\times$ $\displaystyle(\alpha+\beta)-4{\cal
F}_{\alpha\beta}\big{(}\alpha^{2}+\beta(\beta-1)-\alpha(4\beta+1)\big{)}\Big{)}+\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{3^{2}\times 5\times
2^{20}\alpha^{6}\beta^{6}}$ $\displaystyle\times$
$\displaystyle\bigg{(}-4m_{Q}^{2}\Big{(}\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+\beta^{3}(\beta-1)\Big{)}+{\cal
F}_{\alpha\beta}\Big{(}7\alpha^{3}-3\alpha^{2}(\beta-1)$ $\displaystyle-$
$\displaystyle
3\alpha\beta^{2}+\beta^{2}(7\beta+3)\Big{)}\bigg{)}\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}(s)$ $\displaystyle=$
$\displaystyle-\frac{\langle\bar{q}q\rangle^{2}}{2^{10}\pi^{6}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{2m_{Q}^{2}{\cal
F}_{\alpha\beta}^{3}(\alpha+\beta-1)^{2}+{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)}{3\alpha^{3}\beta^{3}}\;,$ (40)
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle G^{3}\rangle}{5\times 3^{2}\times
2^{21}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}}{\alpha^{6}\beta^{6}}$ (41)
$\displaystyle\times$ $\displaystyle\bigg{(}5(\alpha^{3}+\beta^{3}){\cal
F}_{\alpha\beta}^{2}+4{\cal
F}_{\alpha\beta}m_{Q}^{2}\Big{(}13\alpha^{4}+3\alpha^{3}(\beta-1)+3\alpha\beta^{3}$
$\displaystyle+$
$\displaystyle\beta^{3}(13\beta-3)\Big{)}+12m_{Q}^{4}\Big{(}\alpha^{5}+\alpha^{4}(\beta-1)+\alpha\beta^{4}+\beta^{4}(\beta-1)\Big{)}\bigg{)}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)$
$\displaystyle=$ $\displaystyle\frac{-{\cal
N}_{i}\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}{2^{9}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{3}+3m_{Q}^{2}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)}{\alpha^{2}\beta^{2}}\;,$ (42)
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}$ $\displaystyle=$
$\displaystyle\frac{{\cal
N}_{i}\langle\bar{q}Gq\rangle^{2}}{2^{11}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\bigg{\\{}\frac{3{\cal
H}_{\alpha}^{2}}{2(1-\alpha)\alpha}+\int_{\beta_{min}}^{1-\alpha}d\beta\frac{3{\cal
F}_{\alpha\beta}m_{Q}^{2}}{\alpha\beta}\bigg{\\}}\;,$ (43)
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{{\cal
H}_{\alpha}-m_{Q}^{2}}{24\pi^{2}}\langle\bar{q}q\rangle^{4}\;.$ (44)

###  A.4 The spectral densities for $0^{+-}$ baryonium state in Eqs. (7) and
(8)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\bigg{\\{}\frac{{\cal
F}_{\alpha\beta}^{6}(\alpha+\beta-1)^{4}(5{\cal
F}_{\alpha\beta}+7m_{Q}^{2}(\alpha+\beta-1)}{3\times 7\times 5^{2}\times
2^{19}\pi^{10}\alpha^{6}\beta^{6}}\bigg{\\}}\;,$ (45)
$\displaystyle\rho^{\langle G^{2}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
G^{2}\rangle}{\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\bigg{\\{}{\cal
N}_{i}\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}(5m_{Q}^{2}(\alpha+\beta-1)+3{\cal
F}_{\alpha\beta})}{3\times 5\times 2^{19}\alpha^{4}\beta^{4}}$ (46)
$\displaystyle+$ $\displaystyle\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{5\times 3^{2}\times
2^{21}\alpha^{6}\beta^{6}}\left(4m_{Q}^{2}\left(\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+(\beta-1)\beta^{3}\right)\right.$
$\displaystyle+$ $\displaystyle{\cal
F}_{\alpha\beta}\left(2\alpha^{3}-3\alpha^{2}(\beta-1)-3\alpha\beta^{2}+\beta^{2}(3+2\beta))\right)\bigg{\\}}\;,$
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle G^{3}\rangle}{5\times 3^{2}\times
2^{23}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}\frac{d\alpha}{\alpha^{6}}\int_{\beta_{min}}^{1-\alpha}\frac{d\beta}{\beta^{6}}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}$ (47) $\displaystyle\times$
$\displaystyle\bigg{\\{}5{\cal
F}_{\alpha\beta}^{2}(\alpha^{3}+\beta^{3})+8{\cal
F}_{\alpha\beta}m_{Q}^{2}(8\alpha^{4}+3\alpha^{3}(\beta-1)+3\alpha\beta^{3}$
$\displaystyle-$ $\displaystyle 3\beta^{3}+8\beta^{4})+24\alpha
m_{Q}^{4}(\alpha+\beta-1)(\alpha^{4}+\beta^{4})\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle-\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{3{\cal
H}_{\alpha}-2m_{Q}^{2}}{144\pi^{2}}\langle\bar{q}q\rangle^{4}\;,$ (48)

where ${\cal N}_{A}=1$ and ${\cal N}_{B}=0$.

###  A.5 The spectral densities for $0^{+-}$ baryonium state in Eqs. (9) and
(10)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{{\cal
F}^{6}_{\alpha\beta}(\alpha+\beta-1)^{4}\Big{(}7m_{Q}^{2}(\alpha+\beta-1)+10{\cal
F}_{\alpha\beta}\Big{)}}{3\times 7\times 5^{2}\times
2^{18}\pi^{10}\alpha^{6}\beta^{6}}\;,$ (49) $\displaystyle\rho^{\langle
G^{2}\rangle}(s)$ $\displaystyle=$ $\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle}{\pi^{10}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\bigg{\\{}-\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}}{3\times 5\times
2^{21}\alpha^{5}\beta^{5}}\Big{(}5m_{Q}^{2}(\alpha+\beta-1)^{2}$ (50)
$\displaystyle\times$ $\displaystyle(\alpha+\beta)+4{\cal
F}_{\alpha\beta}\big{(}\alpha^{2}+\beta(\beta-1)-\alpha(4\beta+1)\big{)}\Big{)}+\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{3^{2}\times 5\times
2^{20}\alpha^{6}\beta^{6}}$ $\displaystyle\times$
$\displaystyle\bigg{(}4m_{Q}^{2}\Big{(}\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+\beta^{3}(\beta-1)\Big{)}+{\cal
F}_{\alpha\beta}\Big{(}13\alpha^{3}+3\alpha^{2}(\beta-1)$ $\displaystyle+$
$\displaystyle
3\alpha\beta^{2}+\beta^{2}(13\beta-3)\Big{)}\bigg{)}\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}(s)$ $\displaystyle=$
$\displaystyle\frac{{\cal N}_{i}\langle\bar{q}q\rangle^{2}}{3\times
2^{10}\pi^{6}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{2m_{Q}^{2}{\cal
F}_{\alpha\beta}^{3}(\alpha+\beta-1)^{2}-{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)}{\alpha^{3}\beta^{3}}\;,$ (51)
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle G^{3}\rangle}{5\times 3^{2}\times
2^{21}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}}{\alpha^{6}\beta^{6}}$ (52)
$\displaystyle\times$ $\displaystyle\bigg{(}5(\alpha^{3}+\beta^{3}){\cal
F}_{\alpha\beta}^{2}+4{\cal
F}_{\alpha\beta}m_{Q}^{2}\Big{(}7\alpha^{4}-3\alpha^{3}(\beta-1)-3\alpha\beta^{3}$
$\displaystyle+$
$\displaystyle\beta^{3}(7\beta+3)\Big{)}-12m_{Q}^{4}\Big{(}\alpha^{5}+\alpha^{4}(\beta-1)+\alpha\beta^{4}+\beta^{4}(\beta-1)\Big{)}\bigg{)}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)$
$\displaystyle=$
$\displaystyle\frac{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}{3\times
2^{9}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{3}+3m_{Q}^{2}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)}{\alpha^{2}\beta^{2}}\;,$ (53)
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}$ $\displaystyle=$
$\displaystyle\frac{\langle\bar{q}Gq\rangle^{2}}{2^{11}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\bigg{\\{}\frac{{\cal
H}_{\alpha}^{2}}{2(\alpha-1)\alpha}-\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}m_{Q}^{2}}{\alpha\beta}\bigg{\\}}\;,$ (54)
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle-\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{m_{Q}^{2}+3{\cal
H}_{\alpha}}{72\pi^{2}}\langle\bar{q}q\rangle^{4}\;,$ (55)

where ${\cal N}_{C}=1$ and ${\cal N}_{D}=-3$.

###  A.6 The spectral densities for $0^{+-}$ baryonium state in Eqs. (11) and
(12)

$\displaystyle\rho^{pert}(s)$ $\displaystyle=$
$\displaystyle\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{{\cal
F}^{6}_{\alpha\beta}(\alpha+\beta-1)^{4}\Big{(}10{\cal
F}_{\alpha\beta}-7m_{Q}^{2}(\alpha+\beta-1)\Big{)}}{3\times 7\times
5^{2}\times 2^{18}\pi^{10}\alpha^{6}\beta^{6}}\;,$ (56)
$\displaystyle\rho^{\langle G^{2}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle
g_{s}^{2}G^{2}\rangle}{\pi^{10}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\bigg{\\{}\frac{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)^{2}}{3\times 5\times
2^{21}\alpha^{5}\beta^{5}}\Big{(}5m_{Q}^{2}(\alpha+\beta-1)^{2}$ (57)
$\displaystyle\times$ $\displaystyle(\alpha+\beta)-4{\cal
F}_{\alpha\beta}\big{(}\alpha^{2}+\beta(\beta-1)-\alpha(4\beta+1)\big{)}\Big{)}+\frac{{\cal
F}_{\alpha\beta}^{3}m_{Q}^{2}(\alpha+\beta-1)^{4}}{3^{2}\times 5\times
2^{20}\alpha^{6}\beta^{6}}$ $\displaystyle\times$
$\displaystyle\bigg{(}-4m_{Q}^{2}\Big{(}\alpha^{4}+\alpha^{3}(\beta-1)+\alpha\beta^{3}+\beta^{3}(\beta-1)\Big{)}+{\cal
F}_{\alpha\beta}\Big{(}7\alpha^{3}-3\alpha^{2}(\beta-1)$ $\displaystyle-$
$\displaystyle
3\alpha\beta^{2}+\beta^{2}(7\beta+3)\Big{)}\bigg{)}\bigg{\\}}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle^{2}}(s)$ $\displaystyle=$
$\displaystyle\frac{{\cal
N}_{i}\langle\bar{q}q\rangle^{2}}{2^{10}\pi^{6}}\int^{\alpha_{max}}_{\alpha_{min}}d\alpha\int^{1-\alpha}_{\beta_{min}}d\beta\frac{{{\cal
F}_{\alpha\beta}^{4}(\alpha+\beta-1)-2m_{Q}^{2}\cal
F}_{\alpha\beta}^{3}(\alpha+\beta-1)^{2}}{\alpha^{3}\beta^{3}}\;,$ (58)
$\displaystyle\rho^{\langle G^{3}\rangle}(s)$ $\displaystyle=$
$\displaystyle\frac{\langle G^{3}\rangle}{5\times 3^{2}\times
2^{21}\pi^{10}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)^{4}}{\alpha^{6}\beta^{6}}$ (59)
$\displaystyle\times$ $\displaystyle\bigg{(}5(\alpha^{3}+\beta^{3}){\cal
F}_{\alpha\beta}^{2}+4{\cal
F}_{\alpha\beta}m_{Q}^{2}\Big{(}7\alpha^{4}-3\alpha^{3}(\beta-1)-3\alpha\beta^{3}$
$\displaystyle+$
$\displaystyle\beta^{3}(7\beta+3)\Big{)}-12m_{Q}^{4}\Big{(}\alpha^{5}+\alpha^{4}(\beta-1)+\alpha\beta^{4}+\beta^{4}(\beta-1)\Big{)}\bigg{)}\;,$
$\displaystyle\rho^{\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}(s)$
$\displaystyle=$ $\displaystyle\frac{{\cal
N}_{i}\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle}{2^{9}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\int_{\beta_{min}}^{1-\alpha}d\beta\frac{-{\cal
F}_{\alpha\beta}^{3}+3m_{Q}^{2}{\cal
F}_{\alpha\beta}^{2}(\alpha+\beta-1)}{\alpha^{2}\beta^{2}}\;,$ (60)
$\displaystyle\rho^{\langle\bar{q}Gq\rangle^{2}}$ $\displaystyle=$
$\displaystyle\frac{-3{\cal
N}_{i}\langle\bar{q}Gq\rangle^{2}}{2^{11}\pi^{6}}\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\bigg{\\{}\frac{{\cal
H}_{\alpha}^{2}}{2(\alpha-1)\alpha}+\int_{\beta_{min}}^{1-\alpha}d\beta\frac{{\cal
F}_{\alpha\beta}m_{Q}^{2}}{\alpha\beta}\bigg{\\}}\;,$ (61)
$\displaystyle\rho^{\langle\bar{q}q\rangle^{4}}$ $\displaystyle=$
$\displaystyle\int_{\alpha_{min}}^{\alpha_{max}}d\alpha\frac{m_{Q}^{2}+3{\cal
H}_{\alpha}}{72\pi^{2}}\langle\bar{q}q\rangle^{4}\;,$ (62)

where ${\cal N}_{C}=1$ and ${\cal N}_{D}=-1$.

##  Appendix B pictures

For the hidden-charm exotic baryonium states in Eqs. (2)-(3) and Eqs.
(7)-(11), the OPE, pole contribution and the masses as functions of Borel
parameter $M_{B}^{2}$ are given in Figs 3-9. For the hidden-bottom exotic
baryonium states in Eqs. (1)-(3) and Eqs. (7)-(11), the OPE, pole contribution
and the masses as functions of Borel parameter $M_{B}^{2}$ are given in Figs
10-17.

Figure 3: The same caption as in Fig 2, but for the current in Eq. (2).

Figure 4: The same caption as in Fig 2, but for the current in Eq. (3).

Figure 5: The same caption as in Fig 2, but for the current in Eq. (7).

Figure 6: The same caption as in Fig 2, but for the current in Eq. (8).

Figure 7: The same caption as in Fig 2, but for the current in Eq. (9).

Figure 8: The same caption as in Fig 2, but for the current in Eq. (10).

Figure 9: The same caption as in Fig 2, but for the current in Eq. (11).

Figure 10: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (1).

Figure 11: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (2).

Figure 12: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (3).

Figure 13: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (7).

Figure 14: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (8).

Figure 15: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (9).

Figure 16: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (10).

Figure 17: The same caption as in Fig 2, but for the hidden-bottom baryonium
state with current in Eq. (11).


# On the origin of galactic cosmic rays

Ya. N. Istomin<EMAIL_ADDRESS>P. N. Lebedev Physical Institute, Leninsky
Prospect 53, Moscow, 119991 Russia

###### Abstract

It is shown that the relativistic jet, emitted from the center of the Galaxy
during its activity, possessed power and energy spectrum of accelerated
protons sufficient to explain the current cosmic rays distribution in the
Galaxy. Proton acceleration takes place on the light cylinder surface formed
by the rotation of a massive black hole carring into rotation the radial
magnetic field and the magnetosphere. Observed in gamma, x-ray and radio bands
bubbles above and below the galactic plane can be remnants of this bipolar
get. The size of the bubble defines the time of the jet’s start, $\simeq
2.4\cdot 10^{7}$ years ago. The jet worked more than $10^{7}$ years, but less
than $2.4\cdot 10^{7}$ years.

###### keywords:

cosmic rays , galactic center , relativistic jets

###### PACS:

98.70.Sa , 98.62.Nx

††journal: Astroparticle Physics

,

##  1 Introduction

The traditional point of view on the origin of cosmic rays in the Galaxy is
the concept of acceleration of charged particles at fronts of shocks from
supernova explosions. Arguments in favour of this mechanism are sufficient
mechanical energy that is released when the supernova explodes, as well as
universal index of power law spectrum of particles, accelerated by strong
shocks. Total power of cosmic ray sources in order to maintain their observed
density of energy is $5\cdot 10^{40}erg/s$, which equals approximately 15% of
the kinetic energy of supernova explosions. When the gas compression in a
shock is equal to 4, the index of the power law energy spectrum of accelerated
particles is equal to -2, $N(E)\propto E^{-2}$. It is in a good agreement with
observed cosmic ray spectrum at energies $E<3\cdot 10^{15}eV$. Beginning from
the first works by Krymskii, 1977; Bell, 1978; Blandford & Ostriker, 1978, who
proposed the mechanism of acceleration of charged particles on fronts of
shocks propagating in the turbulent environment, much progress has been made
to explain the observed characteristics of galactic cosmic rays in the belief
that they are accelerated at the front of shocks.

On the other hand, there is no objections to generate galactic cosmic rays in
one source in the Galaxy (Ptuskin & Khazan, 1981). This potential source can
be the center of the Galaxy, which is the massive black hole of $M\simeq
4\cdot 10^{6}M_{\odot}$ mass. And while the luminosity of Sgr A* is small now,
it is only $10^{36}$ erg/s, in the past the center could be much brighter
because its Eddington luminosity equals $L_{Edd}=5.2\cdot 10^{44}$ erg/s. On
the past activity of the center of the Galaxy shows newly discovered by Fermi
Gamma-ray Space Telescope above and below the galactic plane big bubbles
emitting gamma radiation in the range of $0.1-1000$ GeV (Su et al., 2010).
Such formations was previously observed in the x-ray range $(1.5-2)$ KeV by
ROSAT All-Sky Survey (Snowden et al., 1997) and in the microwave range
$(20-40)$ GHz by WMAP (Finkbeiner, 2004). Estimated energy stored in bubbles
is of $10^{54}-10^{55}$ erg (Sofue, 2000). As we will see below, bubbles of a
relativistic gas could be formed by the jet, emitted from surroundings of the
massive black hole. Here we provide an alternative mechanism of origin
galactic cosmic rays, in which the nucleus of the Galaxy in the active phase
injected the relativistic jet, which was the source of cosmic rays.

In the following sections we will calculate the power of the jet and the
energy spectrum of protons in the relativistic jet, as well as describe the
remnants of the relativistic jet injected from the center of the Galaxy,
having the form of bubbles above and below the galactic plane. In the final
section we will discuss correspondence of the scenario of the galactic cosmic
rays origin provided with cosmic rays characteristics observed.

##  2 Relativistic jet

Sources of energy of active galactic nuclei are the accretion on a massive
black hole, in which the gravitational energy of a falling gas transforms into
radiation and heat, as well as the rotation of a black hole. Mechanism of
extraction of energy and angular momentum from the black hole is called as the
mechanism of Blandford-Znajek (1977). The energy of a rotating black hole is a
large value, $E_{rot}=Mr_{H}^{2}\Omega_{H}^{2}/2=a^{2}Mc^{2}/8=2.25\cdot
10^{53}a^{2}(M/M_{\odot})$ erg. For the Galaxy $E_{rot}\simeq 9\cdot
10^{59}a^{2}$ erg. Here we have introduced the dimensionless parameter of $a$,
describing rotation of the black hole, $a=Jc/M^{2}G,\,a<1$. $J$ is the black
hole angular momentum, $G$ is the gravitational constant. Angular velocity of
rotation of a black hole is proportional to the value of $a$,
$\Omega_{H}=ac/2r_{H}$, $r_{H}$ is the gravitational radius of not rotating
black hole $(a=0)$, $r_{H}=2MG/c^{2}$. Energy extraction is possible when
there is a poloidal magnetic field $B$ near the black hole horizon. In this
case, rotating black hole acts as a Dynamo machine, creating a voltage
$U=f_{H}\Omega_{H}/2\pi c$ (Landau & Lifshits, 1984). The value of $f_{H}$ is
the flux of the poloidal magnetic field reaching the horizon of a black hole,
$f_{H}\simeq\pi Br_{H}^{2}$. Voltage $U$ generates the electric current
$I=U/(R+R_{H})$, which on the one hand is closed on the horizon of a black
hole that has the resistance $R_{H}=4\pi/c\simeq 377$ ohm. Resistance of the
outer part of the current loop is $R$. Thus, the power, extracting from a
rotating black hole, is
$L=RI^{2}=U^{2}R/(R+R_{H})^{2}=a^{2}B^{2}r_{H}^{2}R/16(R+R_{H})^{2}$, and
reaches the maximum $L_{m}$ at $R=R_{H}$, $L_{m}=a^{2}B^{2}r_{H}^{2}c/256\pi$.
The value of $L$ is proportional to the energy of the poloidal magnetic field
near a black hole and can reach the Eddington luminosity at sufficiently large
magnetic fields $B\simeq 10^{6}a^{-1}$ Gauss in the center of the Galaxy. This
field is accumulated near the horizon of a black hole in the process of
accretion of a disk matter in which the magnetic field is frozen. Thus, for
the effective work of the mechanism of Blandford-Znajek an accretion disk
around a massive black hole is required, not as a source of the energy, but as
the agent bearing the magnetic field to a black hole. In addition, the
electric current $I$ flows in the disk, this is the part of the current loop:
in the disk, in the black hole horizon, then in the jet, closing at large
distances in the interstellar matter (see Figure 1).

Figure 1: Configuration of the magnetic field and electric currents in the jet
and in the disk.

In the disk, in addition to the radial electric current $I_{\rho}=\int
j_{\rho}ds=-I$, stronger toroidal current $j_{\phi}$, $j_{\phi}\simeq
10^{2}j_{\rho}$, flows also (Istomin & Sol, 2011), it generates the radial
magnetic field $B$.

The rotating black hole brings into rotation the radial magnetic field in the
magnetosphere of a black hole above and below the disk. Angular velocity of
rotation of the magnetic field lines, the same as rotation of the
magnetospheric plasma, $\Omega_{F}$, is proportional to the angular velocity
of rotation of the black hole $\Omega_{H}$, $\Omega_{F}=\Omega_{H}R/(R+R_{H})$
(Thorne et al., 1986). Plasma rotation is the drift motion in crossed radial
magnetic field and electric field of plasma polarization. Thus, there appears
so-called the light cylinder surface in the black hole magnetosphere, where
the magnitude of the electric field is compared with that of the magnetic
field and the rotation velocity approaches the speed of light $c$. The radius
of the light surface is $r_{L}=c/\Omega_{F}=2a^{-1}r_{H}(R+R_{H})/R>r_{H}$. On
the light surface charged particles get considerable energy and angular
momentum of rotation. Energy density of particles on the light surface is
compared with the energy density of the electromagnetic field
$(E_{L}^{2}+B_{L}^{2})/8\pi=B_{L}^{2}/4\pi$ (Istomin, 2010). All energy passes
to protons, $\gamma={\cal E}_{p}/m_{p}c^{2}>>1$. Electrons are practically not
accelerated due to large synchrotron losses in a strong magnetic field
(Istomin & Sol, 2009). Energetic protons, accelerated near the light surface,
and whose energy is mainly in the azimuthal motion, create the jet. Jet’s
power is $L_{J}=B^{2}r_{H}^{2}c(\omega_{cH}r_{H}/c)^{-1/4}/2$ (Istomin & Sol,
2011). Here $\omega_{cH}$ is the non relativistic cyclotron frequency of
protons in the magnetic field near the black hole, $\omega_{cH}=eB/m_{p}c$.
Jet arises when the power extracted from the rotating black hole $L$ becomes
greater than the jet power $L_{J}$, $L>L_{J}$. This imposes a limitation on
the value of the magnetic field

$\frac{\omega_{cH}r_{H}}{c}\geq(128\pi)^{4}a^{-8}\left[\frac{(R+R_{H})^{2}}{4RR_{H}}\right]^{4}.$
(1)

For $R=R_{H}$ it gives

$B\geq 2.7\cdot 10^{11}a^{-8}\frac{M_{\odot}}{M}{\rm Gauss}.$ (2)

For the center of the Galaxy, the magnetic field must satisfy the condition
$B\geq 6.75\cdot 10^{4}a^{-8}$ Gauss. We see that to generate a jet, less
massive black holes should have a stronger poloidal magnetic field near the
horizon, $B\propto M^{-1}$. In addition, rotation must be fast, close to the
critical value of $a\simeq 1$, because of strong dependence of the expression
(2) on $a$. It should also be noted that the resistance of the external
current loop $R$, on which the jet power $L_{J}$ depends, is not the ohmic one
$R_{c}$, which turns out to be small, $R_{c}<<R_{H}$ (Istomin & Sol, 2011),
but is the effective resistance $R_{J}$, which can be attributed to the jet,
receiving energy from the rotating black hole. If $L=L_{J}$ the resistance of
the jet is $R_{J}=R_{H}$. Under the equality in the expression (1) when a
rotating black hole can begin to generate a jet, the expression for the jet’s
power becomes universal

$L_{J}=2^{48}\pi^{7}m_{p}c^{2}\left(\frac{m_{p}c^{3}}{e^{2}}\right)a^{-14}=2.5\cdot
10^{41}a^{-14}\,erg/s.$ (3)

All jet energy are in the energy of protons, and, as we can see, is sufficient
for the production of galactic cosmic rays.

##  3 Energy spectrum of fast particles in jet

Istomin and Sol (2009) had shown that on the light surface, produced by the
rotating radial magnetic field, which is carried into rotation by a black
hole, protons gain considerable energy. The Lorentz factor $\gamma$ becomes
equal to $\gamma=(\gamma_{0}\gamma_{i})^{1/2}$. The value of $\gamma_{i}$ is
the Lorentz factor of particles in the magnetosphere of a black hole before
crossing the light surface. And the value of $\gamma_{0}$ is the maximum of
the Lorentz factor, which could be achieved by a particle in this acceleration
mechanism, $\gamma_{0}=\omega_{cL}/\Omega_{F}$. Here $\omega_{cL}$ is the
cyclotron frequency of rotation of protons in the poloidal magnetic field near
the light surface. When $\gamma=\gamma_{0}$ the cyclotron radius of a proton
is compared with the radius of the light surface. For non relativistic
particles of the black hole magnetosphere, $\gamma_{i}\simeq 1$, the Lorentz
factor of accelerated particles is equal to
$\gamma=\gamma_{0}^{1/2}=(\omega_{cL}/\Omega_{F})^{1/2}$. Crossing the light
surface at different distances $z$ from the accretion disk plane, particles
gain different energies, since the magnetic field decreases with distance from
the black hole. For a radial magnetic field $B\propto(z^{2}+r_{L}^{2})^{-1}$.
Thus, $\gamma=\gamma_{m}(1+z^{2}/r_{L}^{2})^{-1/2}$, where $\gamma_{m}$ is the
maximal Lorentz factor of accelerated particles near the accretion disk.
Accelerated protons of the jet are collected from various parts of the light
cylinder surface of $r_{L}$ radius, but located at different distances $z$.
Therefore, the number of particles is $dN\propto ndz$, where $n$ is the
density of protons in the magnetosphere near the light surface. Connecting
values $z$ and $\gamma$, we get

$\frac{dz}{r_{L}}=-\frac{\gamma_{m}d\gamma}{\gamma^{2}(1-\gamma^{2}/\gamma_{m}^{2})^{1/2}}.$

Considering that the vertical size of the magnetosphere is larger than the
light surface radius, the density $n$ can be taken as constant. As a result we
get the distribution function of relativistic protons in the jet,
$F(\gamma)=dN/d\gamma$,

$F(\gamma)=const\cdot\gamma^{-2}(1-\gamma^{2}/\gamma_{m}^{2})^{-1/2},\,\gamma<\gamma_{m}.$
(4)

We see that in the range $\gamma<<\gamma_{m}$ the spectrum of relativistic
protons is the power law spectrum with the index -2. This spectrum is observed
in gamma radiation from bubbles above and below the Galactic plane by Fermi
Gamma-ray Space Telescope (Su et al., 2010). Considering that the gamma
radiation occurs due to collisions of relativistic protons with interstellar
gas through meson production (Crocker & Aharonian, 2011), and the distribution
of photons is similar to the distribution of protons, one can conclude that
the jet from the center of the Galaxy actually existed, and bubbles are filled
with relativistic protons of the jet. The value of $\gamma_{m}$ is (Istomin &
Sol, 2011)

$\gamma_{m}=\left(\frac{\omega_{cH}r_{H}}{c}\right)^{1/2},$ (5)

and for $R=R_{H}$ equals (see the expression (1))

$\gamma_{m}=(128\pi)^{2}a^{-4}=1.6\cdot 10^{5}a^{-4}.$

The spectrum of protons (4) breaks at $\gamma=\gamma_{m}$ and has there the
root singularity (integrable) that is smoothed considering the thermal
dispersion of particles in the magnetosphere of the black hole,
$\Delta\gamma_{i}={\cal E}_{p}/m_{p}c^{2}$. The distribution (4) is shown on
Figure 2. The value of $\gamma_{m}$ is chosen to be equal to the Lorentz
factor of the break in the observed spectrum of cosmic rays at the energy
$E=3\cdot 10^{15}$ GeV, $\gamma_{m}=3.2\cdot 10^{6}$. This corresponds to the
rotation parameter $a=0.47$. The power of the jet (3) is $L_{J}\simeq 8.9\cdot
10^{45}$ erg/s. That is in agreement with estimated from observations powers
of jets ejected from active galactic nuclei–$10^{45}-10^{46}$ erg/s (Mao-Li et
al., 2008).

Figure 2: Distribution function of relativistic protons in the jet,
$\gamma<\gamma_{m}$. The value of $\gamma_{m}$ corresponds to the break in the
spectrum of the cosmic ray. The slope is equal to -2.

Relativistic protons with the spectrum (4) are formed from thermal particles
of the black hole magnetosphere, $\gamma_{i}\simeq 1$. But in addition to
thermal particles in the magnetosphere there can exist accelerated protons.
The turbulent motion of the accreting disk matter in the presence of the
frozen magnetic field leads to acceleration of particles, which have the power
law energy spectrum,
$f(\gamma)=const\cdot\gamma^{-\beta},\,\gamma<\gamma_{1},\,\beta\simeq 1$
(Istomin & Sol, 2009). The disk must be turbulent to provide for the abnormal
gas transport. Getting onto the light surface, accelerated protons are
converted to more energetic, $\gamma\rightarrow(\gamma_{0}\gamma)^{1/2}$.
Their distribution function becomes equal (Istomin & Sol 2009)
$f^{\prime}(\gamma)=2const\cdot\gamma_{0}^{\beta-1}\gamma^{-2\beta+1},\,\gamma_{0}^{1/2}<\gamma<(\gamma_{0}\gamma_{1})^{1/2}$.
Thus, there is another component of jet relativistic protons, their number is
equal to

$N\propto\int_{0}^{\infty}dz\int_{\gamma_{0}^{1/2}}^{(\gamma_{0}\gamma_{1})^{1/2}}\gamma_{0}^{\beta-1}\gamma^{-2\beta+1}d\gamma,\,\gamma_{0}=\gamma_{m}^{2}(1+z^{2}/r_{L}^{2})^{-1}.$
(6)

Transforming the integration area in (6), we get

$\displaystyle
N\propto\int_{1}^{\gamma_{m}}\gamma^{-2\beta+1}d\gamma\int_{(\gamma_{m}^{2}/\gamma^{2}-1)^{1/2}}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}+$
$\displaystyle\int_{\gamma_{m}}^{\gamma_{m}\gamma_{1}^{1/2}}\gamma^{-2\beta+1}d\gamma\int_{0}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}.$
(7)

The first term in Eq. (7) corresponds to relativistic protons with energies
$\gamma<\gamma_{m}$ similar to protons (4), accelerated from the thermal gas.
Their distribution function equals

$F(\gamma)=const\cdot\gamma^{-2\beta+1}\int_{(\gamma_{m}^{2}/\gamma^{2}-1)^{1/2}}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}.$

In the energy range $\gamma<<\gamma_{m}$ this distribution has the same power
law spectrum (4), $F(\gamma)\propto\gamma^{-2}$. But since the number of
accelerated particles in the magnetosphere of the black hole is much less than
that of thermal particles, the contribution of these particles into the total
distribution at $\gamma<\gamma_{m}$ can be neglected. The second term in Eq.
(7) describes the distribution of relativistic protons at
$\gamma<\gamma_{m}<\gamma_{m}\gamma_{1}^{1/2}$

$\displaystyle
F(\gamma)=const\cdot\gamma^{-2\beta+1}\int_{0}^{(\gamma_{m}^{2}\gamma_{1}/\gamma^{2}-1)^{1/2}}\left(1+\frac{z^{2}}{r_{L}^{2}}\right)^{1-\beta}\frac{dz}{r_{L}}=$
$\displaystyle\frac{1}{2}const\cdot\gamma^{-2\beta+1}\left[B(1,\beta-3/2,1/2)-B(\gamma^{2}/(\gamma_{m}^{2}\gamma_{1}),\beta-3/2,1/2)\right].$
(8)

Here $B(x,a,b)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt$ is the incomplete Beta
function, $B(1,\beta-3/2,1/2)=\pi^{1/2}\Gamma(\beta-3/2)/\Gamma(\beta-1)$,
$\Gamma(x)$ is the Gamma function. The distribution (8) with
$\beta=1.7,\,\gamma_{m}=3.2\cdot 10^{6}$ and $\gamma_{1}=10^{5}$ is shown on
Figure 3. For energies $\gamma<\gamma_{m}\gamma_{1}^{1/2}$ the distribution of
relativistic protons is the power law with index $-(2\beta-1)$. At
$\gamma\simeq\gamma_{m}\gamma_{1}^{1/2}$ the distribution falls down, for
$\gamma_{m}=3.2\cdot 10^{6}$ and $\gamma_{1}=10^{5}$ the maximum energy is
$\simeq 10^{18}$ eV.

Figure 3: The distribution function of relativistic protons in the jet,
$\gamma>\gamma_{m}$. The value of $\gamma_{m}$ corresponds to the break in the
cosmic ray spectrum. The spectral index equals -2.4. The maximum energy is
$10^{18}$ eV.

We have chosen the value of $\beta=1.7$ from the fact that indices of spectrum
of cosmic rays before and after the break at energy $3\cdot 10^{15}$ eV differ
on the value of 0.4 – the spectrum becomes more soft with the index $-3.1$.
The same difference in indices must be in the source of cosmic rays, which in
our case is the relativistic jet. At $\beta=1.7$ the index of the spectrum (8)
equals -2.4, while the index of the spectrum (4) is equal to -2. It should be
noted that the distribution function of relativistic protons (4) at energies
$\gamma<\gamma_{m}$, and (8) at energies $\gamma>\gamma_{m}$, is continuous,
i.e. $F(\gamma=\gamma_{m}-0)=F(\gamma=\gamma_{m}+0)$. This is because the
acceleration on the light surface undergo protons of the black hole
magnetosphere with a unique spectrum – thermal at low energies, turning into
the tail of fast particles up to the energy $\gamma=\gamma_{1}$. If their
energy distribution function is $F_{i}({\cal E}_{i})$, then the distribution
of particles, accelerated on the light surface, have also the continuous
distribution $F({\cal E})=F_{i}[{\cal E}_{i}({\cal E})]d{\cal E}_{i}/d{\cal
E,}\,{\cal E}_{i}={\cal E}^{2}/\gamma_{0}m_{p}c^{2}$.

Here and hereafter, we talk about protons, bearing in mind that they are
’heavy’ particles, unlike electrons. Nuclei of $m$ mass and $Ze$ charge will
be accelerated effectively also on the light surface. As we saw above, the
efficiency of acceleration depends on the value of
$\gamma_{0}^{1/2}=(\omega_{c}/\Omega_{F})^{1/2}$. Thus, nuclei will receive
energy per nucleon $(Zm_{p}/m)^{1/2}\simeq 2^{-1/2}$ times less than protons.

##  4 Jet’s remnants

The center of the Galaxy, being active, created a relativistic jet, particles
of which spread in the Galaxy, forming isotropic background of cosmic rays. If
one consider that the angular momentum of the massive black hole in the center
of the Galaxy coincides with that of the Galaxy, than the direction of the jet
propagation is perpendicular to the plane of the Galaxy. The jet length can be
quite large, so jet in M87 extends $\simeq 2$ kpc. Therefore, above and below
the Galactic plane (if we have two almost symmetrical jets) one can see traces
of the jet.

Let us consider a simple diffusive model of propagation of relativistic
particles, whose source is located in the center of the Galaxy $({\bf r}=0)$,
in an interstellar medium

$\frac{\partial N}{\partial t}+{\bf u}\nabla N-\nabla\hat{D}\nabla
N=Q(t)\delta({\bf r}).$ (9)

Here ${\bf u}$ is the velocity of the interstellar matter. We consider the
region outside the stellar galactic disk, then the velocity ${\bf u}$ is the
speed of the galactic wind, which is along the coordinates $z$ which is
perpendicular to the plane of the Galaxy. The value of $\hat{D}$ is the
diffusion coefficient, which can be anisotropic. Diffusion depends on the
intensity of the magnetic field $B$, falling exponentially with the distance
$z$ from the galactic plane, $B=B_{0}\exp(-z/z_{1}),\,z_{1}\simeq 2$ kpc.
Diffusion of charged particles is determined by their motion in the magnetic
field, and the diffusion coefficient is inversely proportional to the magnetic
field strength, $D\propto B^{-\alpha}$. So, for the most strong Bohm diffusion
$D=cr_{c}/3$, $r_{c}$ is the proton cyclotron radius, $\alpha=1$. Thus, the
particle diffusion increases exponentially with the coordinate $z$,
$D=D_{0}\exp(z/z_{0}),\,z_{0}=z_{1}/\alpha$. Such a strong dependence of the
diffusion on the coordinate $z$ leads to effective non diffusion expansion of
particles along this coordinate and decreasing its density. To take into
account this effect we insert the function $\varphi=N\exp(z/z_{0})$. Eq. (9)
becomes

$\displaystyle\exp\left(-\frac{z}{z_{0}}\right)\frac{\partial\varphi}{\partial
t}+u\exp\left(-\frac{z}{z_{0}}\right)\left(\frac{\partial\varphi}{\partial
z}-\frac{\varphi}{z_{0}}\right)+\frac{D_{0}}{z_{0}}\frac{\partial\varphi}{\partial
z}-$ $\displaystyle
D_{0}\left[\frac{\kappa}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\varphi}{\partial\rho}\right)+\frac{\partial^{2}\varphi}{\partial
z^{2}}\right]=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$ (10)

We consider the distribution of particles as azimuthal symmetric depending on
the distance $z$ and the cylindrical radius $\rho$. The value of $\kappa$ is
the ratio of the transverse diffusion coefficient $D_{\perp}$, perpendicular
to $z$, to the longitudinal one $D_{\parallel}$, along $z$,
$\kappa=D_{\perp}/D_{\parallel}$. We see that in the equation of particle
motion there appears the effective velocity along $z$, $u_{0}=D_{0}/z_{0}$. It
arises as a result of the exponential growth of the particle diffusion over
$z$. For characteristic values of $D_{0}=5\cdot 10^{28}$ $cm^{2}$/s and
$z_{0}=2$ kpc the velocity $u_{0}$ is of $u_{0}\simeq 10^{2}$ km/s. This speed
is much larger than the speed of the galactic wind $u$, which at distances of
several kpc from the galactic plane is less than $30$ km/s (Ptuskin, 2007).
Although the wind velocity increases with distance $z$, the exponential factor
$\exp(-z/z_{0})$ in Eq. (10), allows us to ignore the velocity of the galactic
wind in comparison with the velocity $u_{0}$. Values of $D_{0}$ and $z_{0}$
specify scales of length and time, so it is convenient to go to the
dimensionless variables in Eq. (10)
$z^{\prime}=z/z_{0},\,\rho^{\prime}=\rho/z_{0},\,t^{\prime}=t(D_{0}/z_{0}^{2}),\,\varphi^{\prime}=\varphi
z_{0}^{3}$. Eq. (10) becomes (primes are omitted)

$\exp(-z)\frac{\partial\varphi}{\partial t}+\frac{\partial\varphi}{\partial
z}-\frac{\partial^{2}\varphi}{\partial
z^{2}}-\frac{\kappa}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial\varphi}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$
(11)

At $z>1$ the propagation (first derivative over $z$) predominates over the
diffusion (second derivative), and Eq. (11) makes easy

$\varphi=\frac{Q[t-(1-e^{-z})]}{4\pi\kappa
z}\exp\left(-\frac{\rho^{2}}{4\kappa z}\right).$

From this solution one can see that to the point $z$ come particles, which
were emitted by the source at the retarded time $t^{\prime}=t-(1-e^{z})$. If
the time of the jet’s start is $t=0$, particles will lift to the maximum
height of $z_{m}=-\ln(1-t),\,t<1$. Formally at $t=1$ particles come to
infinity during the finite time, that is impossible. The velocity limitation
implies the condition $z_{m}<(cz_{0}/D_{0})t$, $c$ is the speed of the light,
which is not difficult to hold because $cz_{0}/D_{0}\simeq 3.6\cdot 10^{3}$.
Knowing the value of $z_{m}$ from observation one can estimate the time
$t_{1}$ when the power source of cosmic rays in the center of the Galaxy
begins to work, i.e. when the jet starts, $t_{1}=1-\exp(z_{m})$. In
dimensional units $t_{1}=t_{0}[1-\exp(z_{m}/z_{0})],\,t_{0}=z_{0}^{2}/D_{0}$.
When $z_{0}=2$ kpc and $D_{0}=5\cdot 10^{28}\,cm^{2}/s$ the time $t_{0}$ is
$t_{0}=7.6\cdot 10^{14}\,s=2.4\cdot 10^{7}$ yr. The gamma radiation observed
above and below the galactic plane extends to the height of about 8 kpc (Su et
al., 2010), i.e. $z_{m}\simeq 4$. This means that in fact $t_{1}=t_{0}$. In
addition, one can find the time $t_{2}$ when the source turns off. If the jet
worked a short time, all particles would have lifted in height at $z\geq 1$,
and we would see their absence near the galactic plane. Since this is not
observed in bubbles, then $t_{2}<t_{0}(1-1/e)\simeq 0.6t_{0}$.

Knowing the solution for $\varphi$ at $z>1$ we can find the density of
relativistic particles $N(t,z,\rho)$ in the same region

$N=\frac{Q[t-(1-e^{-z})]}{4\pi\kappa
z_{0}^{3}}\exp\left[-\left(\frac{\rho^{2}}{4\kappa z}+z+\ln(z)\right)\right].$
(12)

The distribution $N(z,\rho)$ (12) for the permanent source, $Q=const(t)$, is
shown on Figure 4.

Figure 4: The density distribution of relativistic particles (12) above the
galactic plane in dimensionless coordinates: $z$, the distance from the plane
of the Galaxy, and $\rho$, the cylindrical radius.

We also draw levels of the constant density $N,\,\rho^{2}/4\kappa
z+z+ln(z)=const$, Figure 5.

Figure 5: Levels of the constant density of relativistic particles specified
by the distribution (12).

The value of $\kappa$, the anisotropy of diffusion, is chosen to be equal to
$\kappa=0.14$ from those considerations that the observed ratio of bubble’s
scales $z/\rho\simeq 8kpc/3kpc=8/3$ would be consistent with the forms of the
constant density profiles painted on Figure 5. It seems that such value of
$\kappa$ indicates that the magnetic field in the halo of the Galaxy near the
center is mostly vertical. And this is natural because for the cylindrical
symmetry radial and azimuthal magnetic field components should approach zero
on the axis $\rho=0$.

##  5 Discussion

We have shown that whereas in the past the center of the Galaxy was active and
radiated the jet, its energy and composition are sufficient to explain the
origin of cosmic rays in the Galaxy. Bubbles of a relativistic gas observed in
gamma, x-ray and radio bands above and below the galactic plane, apparently,
are remnants of the bipolar jet emitted from the vicinity of the massive black
hole in the center of the Galaxy. The vertical size of the bubble $z\simeq 8$
kpc permits us to estimate the time of switching on of the jet, $t_{1}\simeq
t_{0}=2.4\cdot 10^{7}$ years ago. We can also estimate the lower limit of the
jet’s work, $\Delta t=t_{1}-t_{2}>0.4t_{0}\simeq 10^{7}$ yr. During the time
$\Delta t$ the jet got the energy $L_{J}\Delta t\simeq 8.9\cdot 10^{45}$ erg/s
$\times$ $3\cdot 10^{14}$ s $\simeq 2.7\cdot 10^{60}$ erg, that is slightly
larger than the energy stored in the black hole rotation $\simeq 10^{60}$ erg.
However, if we estimate the mass of the accreted matter, absorbed by the black
hole at the same time, it can reach a large part of the mass of the black
hole, $\Delta M\simeq{\dot{M}}_{Edd}\Delta t=9.2\cdot
10^{-2}M_{\odot}\,yr^{-1}\times 10^{7}\,yr\simeq 10^{6}M_{\odot}$, $\Delta
M\simeq M/4$. Transmitted to the black hole by the accreted matter, the
angular momentum $\Delta J$ can even exceed its initial value (Istomin, 2004).
The jet’s energy $\simeq 10^{60}$ erg is enough to fill by cosmic rays as the
disk ($10^{55}$ erg), as the halo ($10^{57}-10^{58}$ erg) of the Galaxy.
Filling of the disk of the Galaxy by relativistic particles is described by
the same Eq. (11), but there should be $|z|<1$. Therefore, we can ignore the
dependence of the diffusion coefficient on the distances $z$, and solve the
pure diffusion equation

$\frac{\partial N}{\partial t}-D_{\parallel}\frac{\partial^{2}N}{\partial
z^{2}}-D_{\perp}\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial
N}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)\delta(z)}{2\pi\rho}.$

Here we are interested in distribution of particles in the disk on the
transverse distances $\rho$, so we average this equation over $z$ and get

$\frac{\partial{\bar{N}}}{\partial
t}-\frac{D_{g}}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial{\bar{N}}}{\partial\rho}\right)=Q(t)\frac{\delta(\rho)}{2\pi\rho},$
(13)

where the value of $D_{\perp}=D_{g}$ is the diffusion coefficient of cosmic
rays in the galactic disk. The solution is

${\bar{N}}=\frac{1}{4\pi
D_{g}}\int_{0}^{t_{1}}\frac{1}{\tau}\exp\left(-\frac{\rho^{2}}{4D_{g}\tau}\right)Q(t_{1}-\tau)d\tau.$

The time $t_{1}$ is the start time of the jet. If the jet had worked with
constant power $Q$ and switched off at the time $t_{2}$, then the density
distribution of cosmic rays in the disk is

${\bar{N}}=\frac{Q}{4\pi
D_{g}}\int_{\rho^{2}/4D_{g}t_{1}}^{\rho^{2}/4D_{g}(t_{1}-t_{2})}x^{-1}e^{-x}dx.$
(14)

The solution (14) shows that if $\rho^{2}/4D_{g}t_{1}<1$ then the distribution
of the density over the radius $\rho$ is almost uniform. For
$\rho^{2}/4D_{g}(t_{1}-t_{2})>1$ it is logarithmic,
${\bar{N}}\propto-\ln(\rho^{2}/4D_{g}t_{1})$, and for
$\rho^{2}/4D_{g}(t_{1}-t_{2})<1$ it is constant,
$N\propto\ln(t_{1}/(t_{1}-t_{2}))$. The diffusion coefficient of cosmic rays
in the disk equals $D_{g}=2.2\cdot 10^{28}\gamma^{0.6}\,cm^{2}/s$ (Ptuskin,
2007). The condition $R^{2}/4D_{g}<t_{0},\,R\simeq 15$ kpc is the disk radius,
imposes a lower limit on the energy of protons, homogeneously filling the
galactic disc, $\gamma>400$. Apart the diffusion particles can move freely
along the regular magnetic field of spiral arms of the galactic disk. The
necessary for filling velocity $R/t_{0}\simeq 6\cdot 10^{7}$ cm/s $=2\cdot
10^{-3}$ c does not contradict the observed anisotropy of cosmic rays
$\delta$, $\delta\simeq 10^{-3}$. Since the dependence of the distribution
(14) over the energy is determined not only by the energy spectrum of the
source $Q(\gamma)$ but also the dependence of the diffusion coefficient
$D_{g}\propto\gamma^{0.6}$ over the energy, the spectrum of particles in the
disk will be softer than that in the source. Thus, the discussed mechanism of
origin of galactic cosmic rays by the jet, emitted from the center of the
Galaxy, satisfactorily explains the observed spectrum, the index -2.7 (-2.6
for the jet) before the break, and the index -3.1 (-3.0 for the jet) after the
break.

Cosmic rays, filling simultaneously the galactic disk and the halo, flow out
from the Galaxy. Their lifetime $\tau$ is determined by as the energy loss
time $\tau_{E}$, as the time of diffusion leakage from the Galaxy after the
source of relativistic particles stopped, $\tau_{D}$. The time $\tau_{E}$ is
estimated as $\tau_{E}\simeq 3\cdot 10^{7}$ yr (Strong & Moskalenko, 1998). It
is larger than the time of the jet’s start $t_{1}$, i.e. beginning of the
filling of the Galaxy by cosmic rays, $\tau_{E}>t_{1}$. The diffusion time
$\tau_{D}=r^{2}/4D$ is determined by the distance $r$ that particles travel
during the period from the switching on of the source $r=(4Dt_{1})^{1/2}$.
Thus, $\tau_{D}\simeq t_{1}$ does not depend on the energy of particles. This
time is also larger than the time of jet’s switching off $t_{2}$,
$t_{2}<0.6t_{1}$. We see that to now the distribution of relativistic
particles, generated by the jet, does not change noticeable.

## Aknowlegements

This work was done under support of the Russian Foundation for Fundamental
Research (grant number 11-02-01021).

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