text:
\section{Introduction}

In the rapidly evolving field of semiconductor quantum dots (QDs),
which provide many new applications in optoelectronics, nitride-based
systems are of growing interest due to their large range of possible
emission frequencies.\cite{Nakamura:00,Cho:02,Kalliakos:04} The
interaction of carriers with LO phonons is an important scattering
process which determines the redistribution of carriers between the
localized QD states and between QD and wetting-layer (WL) states.  Furthermore, these
interaction processes lead to optical dephasing which is of central
importance in coherent optical experiments and determines the
homogeneous broadening of absorption and gain spectra.  In
nitride-based QDs, the role of interaction with LO phonons is
increased due to the intermediate polar coupling strength in this
material system.  On the other hand, for QDs with wurtzite crystal
structure grown along the c-axis, strong built-in fields modify the
electronic states via the quantum-confined Stark
effect.\cite{Bernardini:98} The aim of this paper is to determine the
efficiency of carrier-phonon interaction under these conditions.

While a simplified treatment of carrier-phonon interaction in QDs,
based on Fermi's golden rule, leads to the prediction of a
phonon-bottleneck (see \cite{Benisty:95} and references therein),
recently it has been emphasized that perturbation theory is not
applicable for QD systems and that the interaction needs to be treated
in the polaron picture.\cite{Inoshita:97,Verzelen:02} A
quantum-kinetic theory has been used to describe carrier scattering
processes due to QD polarons.\cite{Seebeck:05} We extend this model by
including also the optical generation of carriers by means of a short laser pulse.
Naturally linked to this is the dephasing of the coherently driven interband polarization, which is described on the same quantum-kinetic level as the carrier scattering.
Furthermore, we study intermediate polar coupling and include a self-consistent description of the quantum-confined Stark effect for nitride-based QDs.

\section{Quantum-dot model}

We consider a system of self-assembled InGaN/GaN QDs, grown on the
c-plane in wurtzite crystal structure where the spontaneous and
piezoelectric polarization along the c-axis cause large electrostatic
fields.  The calculation of single-particle properties for these QDs
is a challenging task on its own.  Considerable recent progress has
been made in $\V{k} \cdot \V{p}$ and tight-binding models for realistic
QD geometries of various material systems to determine the
single-particle wave functions and energies and the resulting
free-particle optical transitions.\cite{Ranjan:03,Bagga:03,Fonoberov:03,Andreev:00} With the investigation of many-body interaction
effects for given single-particle states, this paper follows a
different aim.  Previous calculations of
carrier-carrier\cite{Nielsen:04,Nielsen:05} and
carrier-phonon\cite{Seebeck:05} scattering in QDs showed that
particular details of the wavefunctions are less important, while
results are more sensitive to the energy stucture (level spacing and
degeneracy).

For our purpose, we use a simplified model for the single-particle
states of the QD system based on a separation ansatz
$\Phi(\vec{r})=\varphi(\vec{\rho})~\xi(z)$ for the wave functions.
The in-plane part $\varphi(\vec{\rho})$ describes either the localized
QD states or the delocalized WL states which form a
continuum at higher energies.  In growth direction,
perpendicular to the WL plane, the carrier confinement potential is
modified by intrinsic electrostatic fields.  Due to the
quantum-confined Stark effect, charges become partly separated and a
screening field builds up which reduces the intrinsic electrostatic
field.  Therefore $\xi(z)$ has to be calculated from a self-consistent
solution of the Schr\"odinger and Poisson equation.  For details on
the QD model, the calculation of wavefunctions and material
parameters, see \cite{Nielsen:05}.

\begin{figure}
   \includegraphics[width=.9\columnwidth]{fig1.eps}
   \caption{Confinement potential $U(z)$ (solid line) and wave function
            $|\xi(z)|^2$ (dashed line) for electrons (a) and holes (b)
            in growth direction for an electrostatic field of
            2 MV/cm.\label{fig1}}
\end{figure}

We consider QDs with two confined shells which due to their
angular momentum properties are referred to as the s-shell (ground
state) and the p-shell (excited state).  The level spacing between s-
and p-states is 1.2 $\hbar\omega_\text{LO}$ for electrons and 0.4
$\hbar\omega_\text{LO}$ for holes.  For the intrinsic electrostatic
field, we consider a typical field strength of 2 MV/cm.

\section{Many-body theory}

A perturbative treatment of the carrier interaction with LO phonons
is only possible when modifications of the single-particle
states and energies due to the polar coupling remain small.
In QDs the localized states with discrete energies are subject to
hybridization effects even in materials with weak polar coupling.\cite{Inoshita:97,Verzelen:02,Seebeck:05}
The QD polaron states are broadened due to the interaction with carriers in the WL
continuum, which ensures dissipation.\cite{Seebeck:05} Quasi-particle renormalizations can
be described using the retarded Green's function, which obeys
the Dyson equation
\begin{align}
   & \left[ i\hbar\pdiff{}{\tau}{}-\epsilon^{\text{c}}_\alpha\right]
   G^{\text{c},\text{R}}_\alpha(\tau) = \delta(\tau) \nonumber\\
&+ \int_0^\tau d\tau'
   \sum_\beta G^{\text{c},\text{R}}_\beta(\tau-\tau')
 ~D^{\text{cc},
   >}_{\alpha\beta}(\tau-\tau')~G^{\text{c},\text{R}}_\alpha(\tau')
   ~,
\end{align}
for electrons in the conduction band and a similar equation with
$D^{\text{vv},<}$ for the valence band.
This equation describes the polaron in the self-consistent, diagonal,
random-phase approximation (RPA) \cite{Kral:98,Seebeck:05}.
The single-particle energies for either localized or
extended states are given by $\epsilon^{\lambda}_\alpha$ , with $\; \lambda =
\text{c,v}$ being the band index.
Here, the corresponding wave functions (which incorporate effects of the built-in
field) enter via the interaction matrix elements $M^{\lambda}_{\alpha,\beta}$.
They are included in the phonon propagator
\begin{align}
   D^{\lambda \mu,\gtrless}_{\alpha\beta}(\tau)
 &= \sum_{\vec{q}} ~ M^{\lambda}_{\alpha\beta}(\vec{q})
 M^{\mu}_{\beta\alpha}(-\vec{q})\nonumber\\
 &\times ~\left[~N_\text{LO} ~e^{\pm i\omega_\text{LO}\tau}
        + (1+N_\text{LO}) ~e^{\mp i\omega_\text{LO}\tau}\right] ~,
\end{align}
where $N_\text{LO}$ is a Bose-Einstein function for the population of
the phonon modes (assumed to act as a bath in thermal equilibrium) and
$\omega_\text{LO}$ is the LO-phonon frequency.  The Fourier transform
of the retarded GF directly provides the spectral function
$\widehat{G}_\alpha(\omega)=-2~\text{Im}~G^\text{R}_\alpha(\omega)$
which reflects the density of states (DOS) for a carrier in state $\alpha$.

For non-interacting carriers, the spectral function contains
$\delta$-functions at the free-particle energies, reflecting the
infinite lifetime of these states (if recombination on a much
longer timescale is neglected).  The spectral functions for carriers in
the QD-WL system interacting with lattice distortions are shown in
Fig.~\ref{fig2}. For a WL electron with zero momentum (at the band
edge), the $\delta$-like spectral function in the absence of
interaction (indicated by a vertical line in Fig.~\ref{fig2}a) is
broadened. The corresponding finite lifetime of the state is due to
possible scattering events which are self-consistently calculated in
terms of renormalized states and in arbitrary order in the interaction
by means of a direct numerical solution of Eqs.~(1) and (2). Furthermore,
the energy lowering of the interacting state (polaron shift) and weak
LO phonon satellites are obtained.

\begin{figure}
   \includegraphics[width=.9\columnwidth]{fig2.eps}
   \caption{Spectral function for a WL electron with zero momentum (a)
            and for a QD electron in the p-shell (b) and in the
            s-shell (c). Vertical lines indicate the positions of the
            $\delta$-functions for a non-interacting carriers.
           \label{fig2}}
\end{figure}

For the discrete QD states, Figs.~\ref{fig2}b) and c) show that the free-particle picture is strongly modified.
On this level, a direct comparison between the the weak-polar coupling material system InGaAs (Fig. 1 of Ref. \cite{Seebeck:05}) and the InGaN system with intermediate polar coupling is possible.
In the first case, one can clearly distinguish between phonon replicas of the QD states which are spaced by the LO-phonon energy $\hbar\omega_\text{LO}$ around the central peak, and the hybridization effect which leads to a splitting of these peaks that is smaller than the LO-phonon energy (when the detuning between the QD level spacing and the LO-phonon energy is small).
In the InGaN material system, the hybridization is much stronger and one can no longer distinguish between phonon replicas and their splitting.
Moreover, the obtained rich multi-peak structure provides a stronger overlap between the spectral functions which increases the scattering efficiency discussed below.
Broadening of the resonances is provided by the coupling to the WL states and the finite LO-phonon lifetime (assumed to be 5 ps).

In the markovian limit, the transition rate between two states
$\alpha$ and $\beta$ is proportional to the overlap
$\Lambda_{\alpha\beta}=\int d\omega \; \widehat{G}_\alpha(\omega) \;
\widehat{G}_\beta(\omega\pm\omega_\text{LO})$.  In the free-particle
picture, considering the $\delta$-like DOS, the results of Fermi's
golden rule are recovered.  The level-spacing has to match exactly the
LO-phonon energy $\hbar\omega_\text{LO}$, otherwise the overlap
vanishes and carrier transitions are not possible.  In the polaron
picture, strong overlap between the spectral functions of the QD
states as well as between the QD and WL states is obtained which already
indicates the possibility of efficient scattering processes. On
ultrafast timescales, the Markov approximation is not valid and
quantum-kinetic effects additionally increase the scattering
rates.

To study the carrier relaxation from p-shell into s-shell, we
consider the optical excitation of carriers with a short (100 fs)
laser pulse having spectral overlap only with the p-shell.
The
coupled carrier and polarization dynamics are described by the kinetic
equation for the single-particle density matrix
${\rho}_\alpha(t)$, which contains optically induced
interband transition amplitudes $\psi_\alpha$ and occupation
probabilities $f_\alpha^{\text{c,v}}$ according to \cite{Haug_Jauho}
\begin{align}\label{eq:gk}
   \pdiff{}{t}\rho_\alpha(t) &=\pdiff{}{t}\begin{pmatrix} f_{\alpha}^\text{c}
  & \psi_{\alpha} \\ \psi_{\alpha}^* & f_{\alpha}^\text{v}
  \end{pmatrix}=\nonumber \\
  &  = \left[ \begin{pmatrix} \tilde\epsilon^\text{c}_\alpha
  & -\Omega_\alpha \\ -\Omega_\alpha^*
  & \tilde\epsilon^\text{v}_\alpha \end{pmatrix},\rho_\alpha \right]
  + \pdiff{}{t} \rho_\alpha(t)\Big|_\text{coll.} ~.
\end{align}
In this matrix notation of the semiconductor Bloch equations,
Hartree-Fock renormalizations due to the carrier Coulomb interaction
are included in the single-particle
energies $\tilde\epsilon^\lambda_\alpha=\epsilon^\lambda_\alpha+\sum_\beta v^{\lambda\lambda}_{\alpha\beta} (f^\lambda_\alpha-\delta_{\lambda\text{v}})$ and in the Rabi energy
$\Omega_\alpha=\vec{d}\vec{E}(t)+\sum_\beta v^\text{cv}_{\alpha\beta}\psi_\alpha$.
Here $\vec{d}$ is the dipole coupling, $\vec{E}(t)$ the laser pulse and $v^{\lambda\lambda'}_{\alpha\beta}$ are the matrix elements of the Coulomb interaction \cite{Nielsen:05}.
The quantum kinetic description of the carrier-phonon interaction, based on the generalized Kadanoff-Baym ansatz
\cite{Haug_Jauho},
\begin{align}\label{eq:gk_coll}
   \pdiff{}{t}&\rho^{\lambda\lambda' }_\alpha(t)\Big|_\text{coll.}
  = \int_{-\infty}^{t} dt' ~\sum_{\beta\mu} \nonumber\\
  &\times \Big\{
  G^{\lambda,\text{R}}_\beta(t,t')~\left[G^{\lambda',\text{R}}_\alpha(t,t')
  \right]^* \nonumber\\
  & ~~~~~    \times\Big[
  -~
  [\delta_{\lambda\mu}-\rho^{\lambda\mu}_\beta(t')]
  ~\rho^{\mu\lambda'}_\alpha(t')~
  D^{\lambda\mu,>}_{\alpha\beta}(t,t')       \nonumber\\
  & ~~~~~       ~~~~~    +~
  \rho^{\lambda\mu}_\beta(t')~
  [\delta_{\mu\lambda'}-\rho^{\mu\lambda'}_\alpha(t')]~
  D^{\lambda\mu,<}_{\alpha\beta}(t,t') \Big] \nonumber\\
  & +
  G^{\lambda,\text{R}}_\alpha(t,t')~\left[G^{\lambda',\text{R}}_\beta(t,t')
  \right]^* \nonumber\\
  & ~~~    \times\Big[ +~ [\delta_{\lambda\mu}-\rho^{\lambda\mu}_\alpha(t')]~
  \rho^{\mu\lambda'}_\beta(t')
  ~D^{\mu\lambda',>}_{\alpha\beta}(t,t') \nonumber\\
  & ~~~       ~~~~~    -~ \rho^{\lambda\mu}_\alpha(t')~
  [\delta_{\mu\lambda'}-\rho^{\mu\lambda'}_\beta(t')]
  ~D^{\mu\lambda',<}_{\alpha\beta}(t,t') \Big] \Big\} ~,
\end{align}
represents a generalization of the formulation in
Ref.~\cite{Seebeck:05} to the coupled carrier and polarization
dynamics and describes scattering, dephasing, and energy
renormalization contributions.
The RPA self-energy was used, and Hartree terms were neglected, since we consider only low-density populations.
In the markovian limit, the slowly varying part of $\rho(t')$ is taken out of the integral and memory kernels of the type
$\Lambda^{\lambda\mu}_{\alpha\beta}=\int~dt'
G^{\lambda,\text{R}}_\alpha(t') \big[G^{\mu,\text{R}}_\beta(t')\big]^*
e^{\pm i\omega_\text{LO}t'}$
determine the scattering efficiency between two states $\alpha$ and
$\beta$ according to the overlap of spectral functions in the
frequency domain, as stated above.  In the non-markovian case, the
memory kernel contains also the single particle density matrix.  This gives
rise to memory effects since the actual time evolution depends
explicitly on the history of the system.

\begin{figure}
   \includegraphics[width=.9\columnwidth]{fig3.eps}
   \caption{(a) Time evolution of the electron population in the
   p-shell (dashed line) and s-shell (solid line) after optical
   excitation with a 100 fs pulse resonant with the
   p-shell.(b) Total carrier density (solid line), and carrier
   density in the WL (dotted line) and QD (dashed line) versus time.
   \label{fig3}}
\end{figure}

The considered level-spacing of 1.2 $\hbar\omega_\text{LO}$ for electrons and 0.4 $\hbar\omega_\text{LO}$ for holes provides no carrier transitions in a free-particle picture.
Results of the quantum-kinetic calculation are shown in Fig.~\ref{fig3}.
Due to pulse excitation, first the p-shell is populated.
The carrier relaxation is so fast that already during the optical pulse a strong redistribution from the p-shell to the s-shell takes place.
The ultrafast scattering processes are connected with strong dephasing of the coherent interband polarization, which is included on the same quantum-kinetic level in Eq. (4).
The strong dephasing leads to a time evolution of the interband polarization, which practically follows the optical pulse (not shown).
Correspondingly, the coherent regime is limited to the duration of the 100 fs pump pulse.
A transient strong increase of the total carrier density is obtained during the coherent regime, as can be seen in Fig. 3 b).

The time-domain oscillations in Fig. 3a) indicate the strong coupling between carriers and phonons.
The oscillation frequency is connected to the hybridization in the DOS.
The phenomenon is analogous to Rabi-oscillations, though not due to coupling of carriers to the optical field, as recently discussed for QDs in Refs. \cite{Foerstner:03,Wang:05}, but rather due to coupling of carriers to the bosonic field of the phonons.
Note that the oscillations appear after the optical pump pulse in the incoherent regime.
Hence they are not connected to optical coherence.
At larger times, the possibility of carrier transitions between QD and WL states leads to a redistribution of carriers from the QD states into the WL.
The corresponding reduction of the QD population in connection with the increasing WL carrier density can be seen in Figs. 3 a) and b), respectively.
Of course, the total carrier density remains constant after the pulse.

The ultrafast scattering and dephasing processes are the direct result of the polaronic renormalizations.
Only within a non-markovian calculation the above mentioned oscillations in connection with the strong-coupling regime for the carrier-phonon interaction can be obtained.\cite{Seebeck:05}
In this respect, there is no qualitative difference between the InGaAs and InGaN material systems.
In the latter case, the larger polar coupling increases the scattering efficiency while the electron and hole charge separation induced by the built-in fields reduces only inter-band but not intra-band processes.

In the presented example, a weak excitation pulse has been used to
justify the restriction of the interaction to LO phonons. At elevated
excitation densities, carrier-carrier scattering additionally
contributes to fast scattering processes.\cite{Nielsen:05}

\section{Conclusion}

It has been shown that in nitride-based QD systems with intermediate
polar coupling strong modifications of the free-particle states are
obtained due to the interaction of carriers with LO phonons.
Therefore, Fermi's golden rule is not applicable and a quantum-kinetic
description is necessary, which provides ultrafast scattering channels.

This work was supported by the Deutsche Forschungsgemeinschaft.  We
acknowledge a grant for CPU time at the NIC, Forschungszentrum J\"ulich.

redpajama_set_name: RedPajamaArXiv


text:
\section*{Introduction}
Quantum information science is fundamentally changing the way sensitive information is distributed, shared, utilized, and perceived. While quantum key distribution has been a global focus of research and development in this field, there are many other areas where quantum physics principles can be applied to solve previously prohibitive problems and create significant impacts. Among them is the generation and distribution of shared resources amongst distrusted parties over decentralized, unsecured networks.

Suppose Alice, Bob, Dave, and Charles, who are four validators in a blockchain, need to choose one consensus leader among themselves to create the next block in a secure and fair manner, without relying on any centralized randomness source. Suppose a lottery service must prove to claimants that the process of declaring winners is purely random and not controlled by any central entity. The common ground of such scenarios is, in the absence of a trusted third party, how a group of mutually distrusted participants can agree on some genuine random choices for a public settlement among themselves \cite{WassermanProbibility}. This is the problem of constructing a random number generator in a decentralized environment (dRNG) where RNs are not only unbiased, unpredictable, tamper-resistant, but also \textit{publicly verifiable}.
Post-election auditing, creating public parameters for cryptographic protocols, \cite{cryptoeprint:2015:366, cryptoeprint:2015:1249}, gambling and lotteries services, and private preserved messages \cite{Vuvuzela} are some applications requiring dRNGs. In recent years, with the booming of blockchain applications such as distributed computing platforms, smart contracts, and Proof-of-Stake (PoS) based consensus algorithms \cite{Vitalik, Cardano}, dRNG has significantly become more demanding. With the rapid rise in the field of quantum random number generators (QRNGs), one might wonder why not exploit the fundamentally unpredictable randomness from quantum process \cite{Gabriel2010, Shi_2016, samsonov2020vacuumbased, https://doi.org/10.1002/que2.8} \cite{Qi:10,HongPhysRevE2010}, \cite{YouPanAPL2014, Xu:16} to build a beacon for dRNG.
In the past years, most QRNs research focused on creating either a high speed or a self-tested, device-independent generator. In the first scheme, QRNGs are built by fully characterized devices, offering up to Gbits speeds, comparatively as high as classical pseudo-random number generators (PRNGs), to accommodate numerical simulations or gaming service applications \cite{https://doi.org/10.1063/1.5078547,Kim:2021}. Meanwhile, the latter QRNGs type serves mostly cryptography purposes by delivering rather slower speeds but self-tested capability in which quantum randomness can be monitored and verified without depending on any trusted physical implementations \cite{PhysRevLett.114.150501, Avesani2018, Liu2018}.
Nevertheless, these demonstrations benefit only individual users that have physical access to collect QRNs directly from their own devices. To solve the problem of dRNGs, a , users must remotely receive numbers broadcasted from a centralized source of quantum randomness. Because there is no way to verify if those outcomes truly originated from quantum process or whether they are protected in transit, the randomness of these numbers become subjective.  Beacon users have no choice but to trust the honesty of QRNG manufacturers, or rely on verification from a third party and security of the datatransfer process.
These two considerable approach however do not solve the problem of dRNG. From the point of view of dRNG users, a QRNG remains a \textit{centralized source of randomness that they send request and receive digital RNs}, therefore, it only meets the technical trust of users but not the \textit{social trust}. dRNGs users still have to count on the honesty of some third party. For example, in 2018, NIST built a QRNG beacon where randomness is certified by the impossibility of superluminal signals. This brought RNG beacon to the next level where users no longer worry of adversarial activities on measurement settings, measurement devices, or randomness seeds. Nevertheless, the loophole Bell tests must be isolated from hackers and customers \cite{Bierhorst2018}, forcing public users to put trust in the beacon owners. Clearly, a crucial element is missing in current QRNG technology: not all or no RNs users participate in verifying the randomness publicly. Building on these observations, we recognize an urgent need to unlock more potential of quantum properties into solving these problems. In this paper, we propose the first dQRNG protocol that, on-demand, produces one or one set of random numbers used for decision-making on a matter involving N parties. Our protocol is motivated from both quantum and classical cryptography. We show that this novel decentralized quantum random number generator (dQRNG) protocol is:\newline
- \textit{Genuine}: no participant has control over the final outcomes.\newline
- \textit{Flexible in randomness distribution}: without post-processing, QRNs follow arbitrary probability distribution specified and agreed upon by all N parties.\newline
- \textit{Decentralized}: anyone is able to participate in the QRN generation process.\newline
- \textit{Publicly verifiable}: all N parties have the power to verify the genuineness of the outcome RN(s). \newline
- \textit{Highly tolerant}: outcomes are unpredictable if at least one party is honest among N parties. \newline
- \textit{Quantum secured}: immune to quantum computer attacks.\par
We provide a general framework of how to implement our protocol into current classical and quantum network infrastructure. Along the way, we demonstrate experimentally dQRNG with four participants and report the robustness of the protocol by showing randomness testing results, probability distribution verification, and fairness for users in QRNs generation process.
\section*{Results}
\subsection*{Previous works}
To our best knowledge, there is currently no quantum protocol for dRNG. Thus, we would like to briefly explain how classical techniques generate RNs over a decentralized network before presenting our quantum solution. Many classical consensus protocols have been developed and implemented the past years \cite{9152802, GINAR2019, cryptoeprint:2016:1067}. A popular approach is producing RNs based on a commit-reveal scheme \cite{RANDAO}. In the commit phase, each participant submits the hash of their own secret RN. In the reveal phase, all participants announce the RNs of their choices so that others can compare with the encrypted RN they sent earlier. The final RN is the outcome of a previously agreed upon function combining all secret RNs from the entire pool. A challenge this method faces is that as each participant publicizes their RNs, the last participant to reveal has an advantage of knowing the resulting RN ahead of time. Therefore, the protocol could fail because the last revealer either manipulates the output RN or refuses to submit their secret RN, thus terminating the whole RN generation process. To avoid this, RNs are fed into a verifiable delay function (VDF) with the assumption that the malicious participant does not possess specialized hardware capable of cracking the inherently slow hash computation of the VDF \cite{DanVDF}. Other directions for dRNG are built upon threshold digital signature methods where RNs are obtained by amalgamating majority of random signatures from participants \cite{NEAR}. These methods rely on public-private keys scheme and require distributed keys generation process, hence, are vulnerable against quantum computer attacks \cite{ThresholdSig}.
\subsection*{Model}
\begin{figure}[h]
\includegraphics[width=\linewidth]{dQRNGarchitecture.png}
\caption{Decentralized quantum random number generator (dQRNG) architecture}
\label{fig:dQRNGarchitecture}
\centering
\end{figure}
All communication layers of dQRNG are implemented in parallel corresponding to those in classical communication \cite{Cerf1983TheDI}. Figure \ref{fig:dQRNGarchitecture} depicts quantum related layers of dQRNG. Physical layers are identical to quantum network where there must be an entanglement source, private quantum channels between communication parties, and entanglement verification setup. Detailed schematic illustration of dQRNG physical layers are described in figure \ref{fig:physicalLayers}a and \ref{fig:physicalLayers}b.
\begin{figure}[h]
\includegraphics[width=\linewidth]{dQRNGphysicalLayers.pdf}
\caption{Details of necessary devices inside physical 1 (a) and 2 (b) layers of dQRNG architecture. (c) explains the timing diagram of the synchronized reference signal for all nodes. A detected photon carries an identification (ID) corresponding to the closest reference pulse arriving before it.}
\label{fig:physicalLayers}
\centering
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{dQRNGprotocoldiagramfixedcropped.pdf}
\caption{Diagram of the dQRNG protocol procedure involving four parties, Alice, Bob, Charles, and Dave.}
\end{figure*}
We apply a model that has been reported in many previous studies as a solution for large-scale quantum networks, in which entangled photons are passed through a beamsplitter of N, distributing to N nodes such that each party shares photon pairs with every other party (total of $P_{num} = \frac{N(N - 1)}{2}$ number of possible channel pairs) \cite{doi:10.1063/5.0002595}. Each communication node must have local measurement and data processing devices such as single photon detectors and classical information processors. Transport layer in classical communication is co-used in this protocol for transmission of QRNs between nodes. Unlike quantum network architecture, our protocol can be built without the assumption of secured authentication. Consensus algorithm of dQRNG is operated on classical application layer. All parties in the communication pool are synchronized to a common reference signal of period $T$ to keep track order of their detected photon state. Such a reference can be supplied through the same optical communication channels to distribute the entangled photons (see figure \ref{fig:physicalLayers}b and \ref{fig:physicalLayers}c).
\subsection*{Construction}

We first introduce the notations necessary for reading this protocol. Let $N$ be the number of participants, and $k$ be $(N-1)$. This pool would like to agree on a list of random numbers $R$ of length $l$ within a bounded range $B$. Depending on the application, parameter $\mu$, representing the probability of being generated of a random numbers, is decided publicly in advance \cite{Nguyen2019ProofofStakeCM,10.1093/rfs/hhaa075}. Parameter ${I}$ indicates reference identification number or pulse index representing the time slot number of a detected photon. $R\textsuperscript{c}$ is a list of classical true random numbers generated by participants' choices, $R\textsuperscript{q}$ is a list of QRNs generated by quantum process, and $R\textsuperscript{qc}$ is a list representing RNs produced from the dQRNG protocol. $F$ is a function, such as $XOR$ operation or summation operation, agreed ahead of time by participants such that $R = F(R\textsuperscript{qc})$.
We assume the existence of a quantum network infrastructure, as described in the previous section, connecting all communication nodes. All participants publicly agree upon a common photon measurement basis. We do not assume any necessary authentication between nodes nor existence of any encryption and decryption, given quantum or classical method, for data transferring during the protocol operation. A set of random numbers $R$ is generated in five steps:

\textbf{Reveal $\&$ Consensus 1}. Each participant generates and publicizes their own list of random values of length $m$ ($m >> l$) called $R\textsuperscript{c}_{i} = {\{r\textsuperscript{c}_{i,m}\}}$ and probability weight $\mu_i$ ($i$ indicates participant such as Alice (A), Bob (B), Charles(C), Dave(D), etc; this parameter is optional) $r\textsuperscript{c}_{i,m} \in N$, $r\textsuperscript{c}_{i,m} \leq B$. List of all $R\textsuperscript{c}_{i}$ are combined to create $R\textsuperscript{c}$ = $R\textsuperscript{c}_{A} \cup R\textsuperscript{c}_{B} \cup R\textsuperscript{c}_{C} \cup R\textsuperscript{c}_{D} ...\cup R\textsuperscript{c}_{N} = \{r\textsuperscript{c}_{m_c}\}$, where $m_c = m \times N$. A probability density function (pdf) $C$ is constructed by combining probability weight $\mu_i$.

\textbf{Quantum measurements}. Entangled photon pairs are sent to all participants. Each participant then performs photon measurement privately to obtain their collapsed photon states $R\textsuperscript{q}_{i} = \{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\}$ with corresponding indices $I_i = \{ind_{i,m\prime_{i}}\}$ ($m\prime$ is the number of detected photons. In practice, length $m\prime$ is different for each node depending on the loss of the quantum channel and measurement noise, $m\prime_{i} >> m >> l$).

\textbf{Quantum results reveal}. Measurement results of photon states with corresponding indices are publicly revealed by each participant. Quantum entanglement verification is performed by extracting detected photon results previous step from $P_{num}$ number of channel pairs, $R\textsuperscript{q}_{ij} = R\textsuperscript{q}_{i} \cap R\textsuperscript{q}_{j} = \{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\} \cap \{\ket{r\textsuperscript{q}_{j,m\prime_{j}}}\} = \{\ket{r\textsuperscript{q}_{ij,m\prime_{ij}}}\}$ with corresponding list of indices $I_{ij} = I_i \cap I_j$. In the ideal case, $m\prime_{ij} = m\prime_{i} = m\prime_{j}$, however, $m\prime_{ij} < m\prime_{i}$ and $m\prime_{j}$ due to quantum channel loss and measurement noise. For example, for pair Alice and Bob, correlated list of $R\textsuperscript{q}_{AB} = R\textsuperscript{q}_{A} \cap R\textsuperscript{q}_{B} = \{\ket{r\textsuperscript{q}_{A,m\prime_A}}\} \cap \{\ket{r\textsuperscript{q}_{B,m\prime_B}}\}$ with corresponding list of indices $I_{AB} = I_A \cap I_B = \{ind_{A,m\prime}\} \cap \{ind_{B,m\prime}\} = \{ind_{AB,m\prime_{AB}}\}$. \\
Similarly, using the same measurement results, probability distribution verification is performed to compare with pdf $C$.

\textbf{Consensus 2}. A list of QRNs is obtained by appending lists of correlated photon measurement results $R\textsuperscript{q}_{ij}$ from all $P_{num}$ channel pairs, forming $R\textsuperscript{q} = R\textsuperscript{q}_{AB} \cup R\textsuperscript{q}_{AC} \cup R\textsuperscript{q}_{AD} \cup R\textsuperscript{q}_{BC} ... =  \{\ket{r\textsuperscript{q}_{m \prime \prime}}\}$, for all $P_{num}$ and $I = I_{AB} \cup I_{AC} \cup I_{AD} \cup I_{BC} ...= \{ind_{m\prime \prime}\}$, for all $P_{num}$ where $m^{\prime\prime} = \sum_{P_{num}} m\prime_{ij}$.\\ Quantum randomness is combined with users' randomness by first publicly perform $I \cap R\textsuperscript{c}$ to assemble a new list of indices called $I_{final} = \{ind_{m\prime \prime}\} \cap \{r\textsuperscript{c}_{m_c}\} $. Then, the final list of random numbers is filtered so that only values with corresponding indices in $I_{final}$ remain, called $R_{qc} = \{r\textsuperscript{qc}_{m_c}\}$. It can be seen that $R_{qc}$ are selected as subset of QRNs harvested in the quantum process by consensus happened before quantum measurements were operated, and therefore, still are QRNs.

\textbf{Computing output}. To obtain the list of $R$ random numbers of length $l$ as desired, an aggregation function $F$ (e.g modular addition) is used to calculate publicly and can be confirmed by any participants if needed, $R = F(\{r\textsuperscript{qc}_{m_c}\})$.

\subsection*{Analysis}
\begin{figure*}[t]
\includegraphics[width=\textwidth]{DQRNGexperimentalsetup2.pdf}
\caption{Experimental setup to demonstrate four parties performing random number generation consensus}
\label{Fig:experimentalSetup}
\centering
\end{figure*}
\textbf{Correctness}: For as long as at most $k$ participants are malicious, meaning at least 1 participants follow the protocol honestly, the results will still be unbiased and unpredictable, the protocol will be executed and all participants will agree upon the final result when the protocol is completed. \\
\textit{Proof}. Assume the case of four participants Alice, Bob, Charlie, and Dave where they need to agree on one truly random number ($l = 1$) in the range from 0 to 6, given each of number is equally likely to be selected. Alice is the only one being honest ($k = 3$). We proceed the dQRNG protocol to show that the final result is still random and incalculable. \\
\textit{Reveal $\&$ Consensus 1}. Each participant Alice (A), Bob (B), Charles(C), Dave(D) generates and publicizes their own list of random values of length $m$, let's make $m = 10$:
\begin{align*}
    R\textsuperscript{c}_{A} = {\{r\textsuperscript{c}_{A,1\rightarrow{10}}\}},
    R\textsuperscript{c}_{B} = {\{r\textsuperscript{c}_{B,1\rightarrow{10}}\}},\\
    R\textsuperscript{c}_{C} = {\{r\textsuperscript{c}_{C,1\rightarrow{10}}\}},
    R\textsuperscript{c}_{D} = {\{r\textsuperscript{c}_{D,1\rightarrow{10}}\}},\\
    R\textsuperscript{c} = R\textsuperscript{c}_{A} \cup R\textsuperscript{c}_{B} \cup R\textsuperscript{c}_{C} \cup R\textsuperscript{c}_{D}
   = \{r\textsuperscript{c}_{1\rightarrow{40}}\}, m_c = 40\\
    C = 1
\end{align*}
\textit{Quantum measurements}.
\begin{align*}
    R\textsuperscript{q}_{A} = {\{r\textsuperscript{q}_{A,m\prime_{A}}}\},
    R\textsuperscript{q}_{B} = {\{r\textsuperscript{q}_{B,m\prime_{B}}}\},\\
    R\textsuperscript{q}_{C} = {\{r\textsuperscript{q}_{C,m\prime_{C}}}\},
    R\textsuperscript{q}_{D} = {\{r\textsuperscript{q}_{D,m\prime_{D}}}\},\\
     (m\prime_{i} >> 10 >> 1).
\end{align*}
\textit{Quantum results reveal}.
Alice reveals her results honestly while Bob, Charles, and Dave do not. The three of them might cheat without knowing there are other cheaters in the pool. They could alter their results partially or entirely with artificial data. This method, however, does not favor the cheaters in anyway. Quantum entanglement verification easily displays them because their announced results must be compared with each others and with Alice.\\
In another possible scenario, all three cheaters might consult each other and assemble set of data such that they can pass the quantum entanglement verification with each other in public. At the same time, they still embedded some honest results to verify with Alice. In such situation,
\begin{align*}
   R\textsuperscript{q}_{AB} = R\textsuperscript{q}_{A} \cap R\textsuperscript{q}_{B} (\text{non-deterministic}),\\\nonumber
   R\textsuperscript{q}_{AC} = R\textsuperscript{q}_{A} \cap R\textsuperscript{q}_{C} (\text{non-deterministic}),\\\nonumber
   R\textsuperscript{q}_{AD} = R\textsuperscript{q}_{A} \cap R\textsuperscript{q}_{D} (\text{non-deterministic}),\\\nonumber
   R\textsuperscript{q}_{BC} = R\textsuperscript{q}_{B} \cap R\textsuperscript{q}_{C} (\text{deterministic}),\\\nonumber
   R\textsuperscript{q}_{BD} = R\textsuperscript{q}_{B} \cap R\textsuperscript{q}_{D} (\text{deterministic}),\\\nonumber
   R\textsuperscript{q}_{CD} = R\textsuperscript{q}_{C} \cap R\textsuperscript{q}_{D} (\text{deterministic}),\\\nonumber
\end{align*}
with corresponding indices behave in the same manner $I_{AB}, I_{AC}, I_{AD}$ are non-deterministic; $I_{BC}, I_{BD}, I_{CD}$ are deterministic. \\
\textit{Consensus 2}. $R\textsuperscript{q} = R\textsuperscript{q}_{AB} \cup R\textsuperscript{q}_{AC} \cup R\textsuperscript{q}_{AD} \cup R\textsuperscript{q}_{BC} \cup R\textsuperscript{q}_{BD} \cup R\textsuperscript{q}_{CD}$ and therefore, it's a combination of random numbers extracted from quantum process and falsified ones made up by Bob, Charles, and Dave. Likewise, $I = I_{AB} \cup I_{AC} \cup I_{AD} \cup I_{BC} \cup I_{BD} \cup I_{CD}$ is a combination of random indices and fabricate ones. Assume in the first step, $R\textsuperscript{c}_{B},R\textsuperscript{c}_{C}, R\textsuperscript{c}_{D}$ are agreed and arranged by Bob, Charles, and Dave before they proclaim, $R\textsuperscript{c}$ still carries private randomness comes from Alice. Thus, the outcome indices of $I_{final} = I \cap R\textsuperscript{c}$ remains unpredictable. By post-selecting only numbers with indices $I_{final}$ in $R\textsuperscript{q}$ list,  $R_{qc} = \{r\textsuperscript{qc}\}$ is composed from mixture of quantum random numbers and arbitrary subset of predetermined numbers.\\
\textit{Computing output}. As a consequence, computing $R = F(\{r\textsuperscript{qc}\})$ returns an unpredictable number.

Now let's examine some common attacks on quantum networks and how dQRNG protocols are immune against them.

\textbf{Digital emulation attack}. The quantum results reveal step occurs over the classical network, it is vulnerable to digital emulation attack. Without the access to the quantum channel, an adversary can pretend to be one of the participant because the consensus happens in the application layer of the classical channel. Observing others revealing their results first, he then fabricates $R\textsuperscript{q}_{i} = \{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\}$ with corresponding indices $I_i = \{ind_{i,m\prime_{i}}\}$ to emulate that he has matching measurement results with every other participants. This malicious action is rather meaningless because it neither helps him know the final random numbers result before hand nor manipulate them.

\textbf{Man-in-the-middle attack (MITM)}. Since we do not assume any secure quantum authentication requirement, an outside adversary can perform MITM attack and take the identity of that node. In this case, the adversary gains full access to quantum measurement results and has the choice to publicize any type of results he wants. His dishonest reveal, if he chooses to do so, does not affect the unpredictability of the final results as proved previously. Participants do not hide their quantum measurement results and corresponding indices, therefore, knowing these information ahead of time does not give the adversary any advantages either.

\subsection*{Results}
\begin{table}[h]
    \centering
    \begin{tabular}{|c | c | c |}
    \hline
     Case  &  1 & 2\\
     \hline \hline
     N    & 4 & 4\\
     \hline
     R    &  8-bit (binary) value & Number represents A, B, C or D\\
     \hline
     B     &  [1,256] & [0,3]\\
     \hline
     l     &    1    &  1 \\
     \hline
     $\mu_i$ & N/A & $\mu_A = \mu_B = \mu_C = \mu_D = 0.25$\\
     \hline
     $R\textsuperscript{c}_i$ & generated by PRNG & generated by PRNG\\
     \hline
     m     &   8388607     & 8388607\\
     \hline
     $R\textsuperscript{q}_{i}$ & collected from & collected from\\
     & quantum measurements & quantum measurements\\
    \hline
    \end{tabular}
    \caption{For this typical example, users A, B, C, and D received $m\prime_A=1029054$, $m\prime_B=964265$, $m\prime_C=952180$, and $m\prime_D=1019553$ photons respectively. This parameter will vary each time the protocol is run. Other parameters that will vary are user-input random number list $R_c$, matching pairs $R^q_{ij}$ with indices $I_{ij}$ where $i,j=A,B,C,D$}
    \label{tab:dQRNG_experimtal_parameters}
\end{table}

\begin{figure}[h]
\includegraphics[width=\linewidth]{uniform100cyclesprobabilityplot.png}
\caption{Case 1: dQRNG protocol is used when four parties must together repeatedly generate an 8-bit random number after a certain time interval for cryptography purposes. The protocol is run 100 times with various amount of 8-bit random numbers produced each time. Typical results of the probability that an 8-bit random number (total 256 possible values) is generated by dQRNG protocol with four participants is recorded. Comparing to the theoretical probability of  $\frac{1}{256} = 0.0039$, experimental results fluctuate between 0.0032 and 0.0047 (this gives errors less than $20.5\%$).}
\label{Fig:dQRNGSimulationUniformCase1}
\centering
\end{figure}

\begin{figure}[h]
\includegraphics[width=\linewidth]{dQRNGsimulationuniformcase2.png}
\caption{Case 2: dQRNG protocol is used when four parties must together select a winner among themselves for some voting protocol purposes. The protocol is run 100 times with one party is chosen each time. Comparing to the theoretical probability of 0.25 experimental results only fluctuate less than $5.28 \%$ showing the protocol is unbiased.}
\label{Fig:dQRNGsimulationuniformcase2}
\centering
\end{figure}
Here we realize the dQRNG protocol in setups purposing for different applications. Table \ref{tab:dQRNG_experimtal_parameters} records parameters used and measured in each experiment. The first type of scenario occurs when a pool of parties repeatedly shares a new random string for cryptography protocols such as private messages or multi-party computation\cite{Private}. In particular, we create a case where four parties must generate an 8-bit string periodically. We use pseudo-random number generators (PRNG) to simulate choices of $R\textsuperscript{c}_i$ list from each party. The rest of the dQRNG steps are executed following the raw data collected from the experiment whose details are explained in section Methods. To examine the uniformity of the dQRNG, we run protocol 100 times with multiple random numbers string are produced each time. Figure \ref{Fig:dQRNG_simulation_uniform_case1} indicates the average  probability of generating a certain 8-bit value calculated from 100 experiment runs comparing to the theoretically probability.
The second type of dQRNG use case happens in scenario requiring a random fair voting mechanism. Specific examples could be a winner must be decided in a lottery or a validator must be selected to create the next block in a blockchain protocol. In our experimental demonstration, a pool of four parties must decide who is chosen with the condition that each party equally likely has chance of being chosen ($\mu_i = 0.25$). Similar to case 1, PRNGs are used to generate list $R\textsuperscript{c}_i$ in step 1 of dQRNG while quantum measurement outcomes are used for rest of the steps. We run our protocol 100 times and recorded the average probability results in Figure \ref{Fig:dQRNG_simulation_uniform_case2}.

\begin{figure*}[t]
  \centering
  \subcaptionbox{}[.3\linewidth][c]{%
    \includegraphics[width=\linewidth]{uniformDist.png}}
 \subcaptionbox{}[.3\linewidth][c]{%
    \includegraphics[width=\linewidth]{gaussianDist.png}}
  \subcaptionbox{}[.3\linewidth][c]{%
    \includegraphics[width=\linewidth]{rayleighDist.png}}
\caption{Consider the independent timing jitters of the SPD (33 ps) and the TDC (20 ps) at each communication node, photon arrival times collected are rounded to 250 ps per bin-width to compensate the overall timing jitter of all nodes in the network. Joint photon arrival time plots which present correlated QRNs of a typical channel pair are showed in (a), (b), and (c) following uniform, gaussian, and rayleigh probability distributions.}
\label{Fig:JTI}

\end{figure*}

\begin{figure*}
\centering
\begin{minipage}[b]{.4\textwidth}
\includegraphics[width=\linewidth]{cc250ps.png}
\label{fig:Coincidence}
\end{minipage}\qquad
\begin{minipage}[b]{.4\textwidth}
\includegraphics[width=\linewidth]{car250ps.png}
\label{fig:CAR}
\end{minipage}
\caption{Left figure: Temporal cross-correlation between photon arrival times of each channel pair in different probability distributions. From left to right are uniform, Gaussian, and Rayleigh distributions. Total coincidence photon pairs are collected over 60s interval delaying within 500 ps around the maximum peak. Right figure: Accidental coincidence counts with photon arrival times rounded to 250 ps binwidth }
\label{fig:CAR-coin}
\end{figure*}

\section*{Discussion}
The invention of distributed applications and blockchain have promised to bring security, fairness, and transparency to digital technologies, business solutions, as well as socioeconomic and political structure by enabling consensus decision in a global scale. Nonetheless, this relies on dRNG and heavily depends on keeping it conspiracy-resistant using current classical communication techniques. It will be deleterious when a fast enough quantum computer breaks the current dRNG and from there, vandalizes this decentralized-everything idea completely. One might argue, a natural approach to quantum proof dRNG could be applying quantum cryptography protocols such as quantum keys distribution, quantum secret sharing, or quantum digital signatures to protect data transfer or signatures encryption \cite{Hillery_1999, Collins_2017}. However, in trying to make a scalable and applicable dRNG protocol, we create and present a quantum version that is simple, lightweight. Our dQRNG avoids the cumbersome procedures in quantum cryptography including keys distillation, quantum error correction, or slow key rates, but still holds the quantum advance in being provably random and information-theoretic secure. At the same time, comparing to classical domain, our protocol operates without colluding given only one honest party while requiring no encryption algorithm, and thus, reduces the communication complexity between parties. Furthermore, with the arbitrary probability distribution QRNs generation feature, our protocol outperforms others by avoiding an extra step of complex mathematics and not wasting random numbers when transforming QRNs from uniform distribution to others.

\section*{Methods}
We realize our dQRNG protocol by constructing a simple experimental model comprised of four participants depicted in Figure \ref{Fig:experimentalSetup}. We first apply the same technique of generating QRNs with arbitrary probability distribution as we introduced in our previous work to create programmable QRN\cite{NguyenQRNG}. The difference is that the dQRNG protocol requires broadcasting QRNs to all participants, thus an entangled photons generation and distribution process must follow afterwards. The procedure is described as follow: first, intensity of an 1559.67 nm continuous wave (CW) laser is modulated using an electro-optical modulator (EOM) before coupled into a PPLN waveguide for second-harmonic generation (SHG) at 779.8 nm ; next, the SHG light is coupled into  another identical PPLN waveguide for generating entangled photon pairs via spontaneous-parametric-down-conversion (SPDC) process; and lastly, a system of three $50/50$ fiber beamsplitters are used to randomly transmit photons into four different nodes. Note that quantum entanglement verification is an essential procedure for dQRNG protocol. SPDC is ubiquitous for many entanglement-based quantum networks demonstrated thus far. At each node, photons are passed through fiber polarization controller (FPC) before detected by superconducting nanowire single photon detector (SNSPD). In this experiment, using a time-to-digital converter (TDC), we record the photon arrival times compared to a network synchronized 10 MHz reference signal thus harvest the high dimensional temporal mode of each single photon detection. Moreover, the dQRNG protocol is scalable to multiple degrees of freedom in photonics states (ie. polarization, spectral, and spatial modes ) for photon efficient generation of QRNs. Data are then transferred to a local machine which is connected to others in the classical network to perform consensus tasks in the application layer.

In the experimental demonstration of our dQRNG protocol, three different pump shapes including CW, Gaussian, and Rayleigh pulses are sent into the SPDC source, showing the versatility in probability distribution of our QRNs. Correlated QRNs of each channel pair $R\textsuperscript{q}_{ij}$ must satisfy the designated distribution to pass the \textit{quantum results reveal} step. Figure 5 displays joint QRNs between a typical channel pair when photon arrival times are rounded to 250 ps binwidth. Figure 6 (left) describes the time resolved coincidence peaks of all available channel pairs AB, AC, AD, BC, BD, and CD with corresponding CW, Rayleigh, and Gaussian SPDC pump pulses, respectively. Discrepancies in coincidence counts between channel pairs are due to the imperfection of beamsplitters as well as contributions by different detection efficiency and timing jitters of each SPD channel. Using one SPDC source to distribute photon pairs to the entire network reduces the coincidence count rate or correlated QRNs rate. The dQRNG produces one or one small set of RNs at a time, thus, even though the average $R\textsuperscript{q}_{ij}$ rate we achieve is rather low, it does not affect the performance of the protocol. With the same time bin's width, coincidence to accidental coincidence ratios (CARs) of all possible channel pairs is described in Figure \ref{fig:CAR-coin}b. Security threshold of dQRNG protocol depends on CAR to decide when the SPDC process should end. For example, assuming CAR threshold is 50 for gaussian pump shape, photon pairs generation continue until all channel pairs achieve this value. From our experiment data, this means Alice, Bob, Charles, and Dave each has length $m\prime_A = , m\prime_B = , m\prime_C = , m\prime_D = $, respectively. It can be seen that $m\prime_i >> l = 1$ in this case. Lastly, following \textit{consensus 2} step, we append all $R\textsuperscript{q}_{ij}$ into one list $R\textsuperscript{q}$ and examine this sequence using National Institute of Standards and Technology (NIST) SP 800-22 Statistical Test Suite \cite{8986}. Results of a typical QRNs sequence in uniform distribution is presented in figure \ref{NIST}, showing our QRNs pass these test suites with high confidence level.

\begin{figure}[H]
\includegraphics[width=\linewidth]{NISTPlotedit1.pdf}
\centering
\vspace{-2cm}
\caption{NIST results of typical 63182 correlated QRNs of uniform probability distribution. QRNs are harvested from 250 ps binwidth photon arrival times and then QRNs are converted into 8-bit representation of 505456 bits and inputted into the NIST test suite. P-values greater than significance level 0.01 certifies this sequence passes the randomness threshold rigorously for standard cryptography purposes.}
\label{NIST}
\end{figure}

\nocite{*}


text:
\section{Introduction}
\label{intro}

Search for supersymmetry at accelerators of the previous
decade did not result in discovery of any new physics but
rather pushed forward the boundary of unknown territory up to
a few hundred GeV. Still the low energy supersymmetry and
first of all the MSSM~\cite{MSSM1,MSSM2,MSSM3,MSSM4,MSSM5}
proved to be a consistent
model compatible with all experimental data and promissing new
discoveries round the corner. Numerous attempts to fit the
theoretical and experimental requirements with the MSSM have
led to a consistent picture of restricted parameter space
where prediction of the particle spectra and of the
cross-sections of supersymmetry production is
possible~\cite{CMSSM1,CMSSM2,CMSSM3,CMSSM4,CMSSM5,CMSSM6,CMSSM7,CMSSM8,CMSSM9,CMSSM10,CMSSM11,CMSSM12,CMSSM13}.
Though the details depend on the choice
of constraints and the mechanism of supersymmetry breaking,
the allowed region of parameter space still indicates the
presence of light superpartners within the reach of modern
accelerators.

The main hopes of the last decade were connected with the LEP
II $e^+e^-$ collider, where the light charginos, the
superpartners of the weak gauge bosons and charged Higgses,
might be produced together with light sleptons, the
superpartners of the three generations of leptons. As for the
strongly interacting particles, squarks and gluino, they are
typically much heavier, at least within mSUGRA models, and the
corresponding production cross-sections are suppressed.

The situation has changed after LEP shutdown since now we have
the Tevatron and soon coming LHC hadron colliders. There one
expects that first of all the strongly interacting particles
will be produced because the cross-section is enhanced by the
strong coupling. However, the severe background coming from
the SM particles is essential and one has a problem in
extracting the signal from the background.

Recently, a new ingredient to this scheme came from
astroparticle physics where considerable experimental
activities, mostly in space, has led to remarkably precise
data. In particular, the WMAP collaboration measuring the
thermal fluctuations of the Cosmic Microwave Background
determined the matter content of the Universe resulting in $23
\pm 4$\% attributed to the Dark Matter~\cite{WMAP1,WMAP2}. It
perfectly fits supersymmetry, since SUSY provides an excellent
candidate for the Dark Matter particle, namely, the lightest
neutralino, a mixture of the superpartners of the photon,
\mbox{$Z$-boson} and neutral Higgses. Due to a high precision the
WMAP data provide a very restrictive constraint on SUSY
models, and allow one to further restrict the parameter space.

The other contribution comes from the cosmic ray data. It
refers to the measurement of the flux and the spectrum of the
gamma rays and antiparticles coming to the Earth. In
particular, the diffuse gamma rays were measured in a
satellite experiment by the EGRET collaboration and result in
an excess above the background for the energies above 1
GeV~\cite{EGRET1,EGRET2,EGRET3,EGRET4,EGRET5,EGRET6}.
Being interpreted as additional contribution
coming from the SUSY Dark Matter annihilation it leads to
rather restrictive constraint on the value of the neutralino
mass and, respectively, to the restriction on the parameter
space of the MSSM~\cite{new1,new2,new3,new4,new5,new6}.

We show below that taking into account all these constraints
results in a very narrow region of the parameter space.
Choosing parameters in this region one can calculate the
spectrum of superpartners and the cross-section of their
production. These cross-sections happen to be relatively large
to be of interest for future experiments at hadron colliders.
Moreover, as it will be shown below, the cross-section for
the chargino production given by the weak processes is unexpectedly high
and approaches that for the strong processes of
squark and gluino production. This might serve as a signature
for supersymmetry in the future LHC experiments.

\section{Features of the EGRET preferred region}
\label{sec:egret}

The framework of our analysis is the Minimal Supersymmetric
Standard Model with supergravity inspired supersymmetry
breaking terms. The parameters of the model are those of the
Standard Model (three gauge couplings $\alpha_i$, and three $3
\times 3$ matrices of the Yukawa couplings $y^i_{ab}$, where
$i = L, U, D$.), Higgs mixing parameter  $\mu$, and a set of
SUSY breaking parameters (mass terms for squarks and sleptons,
mass terms for gauginos and bilinear and trilinear terms).

In the general case, the MSSM contains more than a hundred
unknown parameters. Most of them come from the SUSY breaking
sector and are the main source of uncertainties. To reduce the
number of unknown parameters, one usually imposes some
constraints, both simple and obvious, and model-dependent
ones.

One of the strictest constraints is the gauge couplings
unification, it fixes the threshold of supersymmetry breaking
$M_{SUSY} \sim 1$~TeV, thus fixing the scale of superparticle
masses~\cite{Unif1,Unif2,Unif3}. The second very hard constraint follows
from the requirement of radiative electroweak symmetry
breaking. While running from the GUT scale towards the lower
energies, one (or both) of the Higgs mass-squared parameters
$m_{H_i}^2$ becomes negative; thus the condition for
spontaneous breaking of electroweak symmetry is fulfilled.
Requiring the breaking to take place at the EW scale $M_{EW}
\sim 100$~GeV restricts the initial conditions for the
corresponding RGE at the GUT scale for the Higgs masses to be
equal to $\sqrt{m_0^2+\mu^2}$, thus expressing the value of
the $\mu$ parameter in terms of $m_0$ and
$m_{1/2}$. The sign of $\mu$ remains
undetermined. One can fix it from other constraints, for
example, from the anomalous magnetic moment of the muon which
has a small deviation from the Standard Model predictions of
the order of 2$\sigma$. This deficiency may be easily filled
with the SUSY contribution, which is proportional to $\mu$.
This requires a positive sign of $\mu$ that kills a half of
the parameter space of the
MSSM~\cite{Anom1,Anom2,Anom3,Anom4,Anom5,Anom6,Anom7,Anom8}.

Further constraints are due to flavour changing processes like
$b \to s \gamma$ responsible for the rare $B$-meson decays
which can occur at the one-loop level due to a virtual $W$-top
pair and are strongly suppressed in the Standard Model. In the
supersymmetric model contribution to the branching ratio $BR(b
\to s \gamma)$ due to superpartners exchange may be rather
big, exceeding the experimental value by a few standard
deviations. However, the next-to-leading order corrections are
essential and improve the situation. This requirement imposes
severe restrictions on the parameter space, especially in the
case of high
$\tan\beta$~\cite{bsg1,bsg2,bsg3,bsg4,bsg5,bsg6,bsg7,bsg8,bsg9,bsg10,bsg11,bsg12,bsg13,bsg14}.

Experimental lower limits on the Higgs boson mass~\cite{Higgs}
impose further constraints when taking into account two-loop
radiative corrections. This limit ($m_h~\le~113.4$~GeV)
forbids left lower corner of the $m_0 - m_{1/2}$ plane,
together with the $b\to s \gamma$
constraint~\cite{Anom7,higgscon1,higgscon2}.
Conservation of $R$-parity results in the existence of the
lightest supersymmetric particle (LSP) which is usually the
lightest neutralino $\chi^0_1$. The neutralino is a perfect
candidate for the non-baryonic cold Dark Matter particle. The
requirement that LSP is the lightest neutralino excludes the
whole area in the parameter space, where LSP is the charged
stau~\cite{LSP1,LSP2}.

Recent very precise data from WMAP collaboration, which
measured thermal fluctuations of Cosmic Microwave Background
radiation, restricted the amount of the Dark matter in the
Universe up to $23 \pm 4\%$.  This imposes further
restrictions on the model. In the early Universe all particles
were produced abundantly and were in thermal equilibrium
through annihilation and production processes. The time
evolution of the number density of the particles is given by
Boltzmann equation and can be evaluated knowing the thermally
averaged total annihilation cross section. The amount of
neutralinos should not be too large to overclose the Universe
and, at the same time, it should be large enough to produce
the right amount of the Dark matter.  This serves as a very
severe bound on SUSY parameters and leaves a very narrow band
in the $m_0 - m_{1/2}$ plane
(WMAP-band)~\cite{Wband1,Wband2,Wband3,Wband4,Wband5,Wband6}.

Having in mind the above mentioned constraints one can find
the most preferable region of the parameter space by
minimizing the $\chi^2$ function. It is remarkable
that all these constraints can be fulfilled
simultaneously.

In what follows, we consider the $m_0 - m_{1/2}$ plane and
find the allowed region in this plane. Each point at this
plane corresponds to a fixed set of parameters and allows one
to calculate the spectrum, cross-sections, etc.

\begin{figure}[b]
\centerline{\psfig{file=unexplhc_fig1a.eps,width=0.47\textwidth}
\hspace{0.03\textwidth}
\psfig{file=unexplhc_fig1b.eps,width=0.47\textwidth}}
\vspace*{8pt} \caption{1.MSSM parameter space. The light
shaded (blue) area is the region preferred by EGRET data for
$\tan\beta=50$, $\mu>0$ and $A_0=0$. The excluded regions
where the stau would be the LSP, or EWSB fails, or the Higgs
boson is too light are indicated by the dots. 2. Running
masses of squarks and gauginos with values of $m_1/2$ and
$m_0$ compatible with EGRET data. \label{r}}
\end{figure}

New constraints also comes from astrophysics. According to
recent data from EGRET collaboration on diffuse gamma ray
flux, there is a clear isotropic excess for energies above 1
GeV in comparison with the expectations from conventional
galactic models. If one conjectures that it is
due to the Dark Matter (neutralino) annihilation, then from
the position of the excess  one can fit the preferable
neutralino mass which happens to be around 80 GeV which in
turn restricts the value of $m_{1/2}$. It appears
that the most preferable values are about $m_{1/2}=180$~GeV
and $m_0 \simeq 1400$~GeV (Fig.\ref{r}).

This region has not been intensively studied yet, though it is
within the reach of Tevatron and forthcoming LHC. It appears
to be very interesting phenomenologically, because of the mass
splitting between light gauginos and heavy squarks and
sleptons (see Fig.\ref{r} right). The cross-sections for
chargino and neutralino production in this case are relatively
large not being suppressed by masses and being comparable with
squark and gluino production. The latter being enhanced by
strong interactions remains suppressed by heaviness of
squarks. This means that in the EGRET region leptonic channels
are not suppressed and so might give  clear leptonic signature
for supersymmetry in the upcoming LHC experiments. Below we
study these cross-sections and the main decay modes in detail.

\section{Neutralino and chargino production and decay modes}

Owing to small $m_{1/2}$ and large $m_0$ in the chosen region
of parameter space we deal with  relatively light gauginos and
heavy squarks (Fig.\ref{r}). Charginos and neutralinos are,
respectively, two and four eigenstates of the corresponding
mass matrices. The neutralino mass matrix is

\begin{equation}
M^{(0)}\!=\!\left(
\begin{array}{cccc}
M_1 & 0 & -M_{\!Z} c_\beta s_W & M_{\!Z} s_\beta s_W \\
0 &
M_2 & M_{\!Z} c_\beta c_W   & -M_{\!Z} s_\beta c_W  \\
-M_{\!Z} c_\beta s_W & M_{\!Z} c_\beta c_W  & 0 & -\mu \\
M_{\!Z} s_\beta s_W & -M_{\!Z} s_\beta c_W  & -\mu & 0 \\
\end{array} \right)\!,
\label{fff}
\end{equation}
where $c_\beta=\cos\beta$, $s_\beta=\sin\beta$. $s_W$ and
$c_W$ are the sine and cosine of the Weinberg angle
respectively. Four eigenstates of this matrix are the four
types of physical particles. The lightest one
$\tilde\chi_1^0$,  is the LSP. In the chosen region of small
$m_{1/2}$ the lightest neutralino mass is around $70$~GeV
which perfectly fits the EGRET data. For
$m_{1/2}=180$~GeV the values of gaugino masses are
$M_1=120$~GeV and $M_2~=~250$~GeV.

The lightest neutralino in our case is mostly bino, the second
neutralino is mostly wino, while the third and the fourth
are mostly higgsinos (see Table \ref{a}).

\begin{table}[b]
\tbl{The bino, wino, first and second higgsino fractions
of neutralinos.}
{\begin{tabular}{@{}cccccc@{}} \toprule
 & \hphantom{aaa}Bino\hphantom{aaa} & \hphantom{aaa}Wino\hphantom{aaa} & 1$^{\rm st}$ higgsino & 2$^{\rm nd}$ higgsino & mass (GeV)\\
\colrule
$\chi^0_1$ & $90\%$            & $\hphantom{0}1\%$ & $\hphantom{0}8\%$ & $\hphantom{0}1\%$ & \hphantom{0}70 \\
$\chi^0_2$ & $\hphantom{0}5\%$ & $71\%$            & $18\%$            & $\hphantom{0}6\%$ & 125 \\
$\chi^0_3$ & $\hphantom{0}1\%$ & $\hphantom{0}2\%$ & $46\%$            & $51\%$            & 220 \\
$\chi^0_4$ & $\hphantom{0}3\%$ & $25\%$            & $30\%$            & $42\%$            & 250 \\
\botrule
\end{tabular} \label{a}}
\end{table}

\begin{figure}[t]
\centerline{\psfig{file=unexplhc_fig2.eps,width=0.8\textwidth}}
\vspace*{8pt}
\caption{Leading chargino and neutralino production processes. \label{prod}}
\end{figure}

Two egienstates of the chargino mass matrix
\begin{equation}
M^{(c)}\!=\!\left(
\begin{array}{cc}
M_2 & \sqrt{2}M_W s_\beta \\ \sqrt{2}M_W c_\beta & \mu
\end{array} \right).
\label{ccm}
\end{equation}
are the two charginos $\tilde\chi^\pm_{1,2}$. Their masses are
given by the following formula:
\begin{eqnarray}
&& M^2_{1,2}\!=\!\frac{1}{2}\Big[M^2_2\!+\!\mu^2\!+\!2M^2_{\!W} \mp\\
&& \left.\! \sqrt{(M^2_2\!\!-\!\!\mu^2)^2\!\!+\!4M^4_{\!W} \cos^2{\! 2\beta}
\!\!+\!4M^2_{\!W}(M^2_2\!\!+\!\mu^2\!\!+\!2M_2\mu \sin{\! 2\beta})}\right]
\nonumber
\end{eqnarray}
The first chargino is relatively light due to the gaugino
component.

To calculate the cross-sections for sparticle production at
hadron colliders we use the CalcHEP~2.3.7.
package~\cite{calchep} which takes into account parton
distributions inside protons~\cite{MRST}. First we selected
three leading types of chargino and neutralino production
processes (Fig.~\ref{prod}).

\begin{figure}[b]
\centerline{\psfig{file=unexplhc_fig3a.eps,width=0.47\textwidth}
\hspace{0.03\textwidth}
\psfig{file=unexplhc_fig3b.eps,width=0.47\textwidth}}
\vspace*{8pt}
\caption{The cross-section for chargino and neutralino production at LHC for
$\sqrt{s}=14$ TeV as functions of $m_{1/2}$ and  $m_{0}$ for $\tan\beta=51$,
$A_0=0$ and sign($\mu$)=1. \label{b}}
\end{figure}

The calculations are made for proton-proton collisions at the
center of mass energy of $\sqrt{s}=14$~TeV (LHC).  As one can
see from Fig.~\ref{b}, the cross sections for neutralino and
chargino production processes slightly depend on $m_0$ and
strongly depend on $m_{1/2}$; thus, for the future reference
we fix the value of $m_0=1400$~GeV. The cross section
dependence on $m_{1/2}$ for chargino and neutralino production
processes is shown in Fig.\ref{c}.

\begin{figure}[tb]
\centerline{\psfig{file=unexplhc_fig4.eps,width=0.85\textwidth}}
\vspace*{8pt}
\caption{The cross-section dependence on $m_{1/2}$ for chargino and neutralino
production. The upper yellow line is for $pp(q\bar{q})\rightarrow
\tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{+}+X$, the red line in the bottom is for
$pp(q\bar{q})\rightarrow \tilde{\chi}_{2}^{0}\tilde{\chi}_{1}^{-}+X$, and the green line in the
middle is for $pp(q\bar{q})\rightarrow\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}+X$
in case of $m_{0}=1400$~GeV, $\tan\beta=51$, $A_0=0$ and sign($\mu$)=1. \label{c}}
\end{figure}

\begin{table}[b]
\tbl{The 1$^{\rm st}$ chargino and the 2$^{\rm nd}$ neutralino decay modes and partial widths for
$m_{0}=1400$~GeV, $m_{1/2}=180$~GeV, $\tan\beta=51$, $A_0=0$ and sign($\mu$)=1.}
{\begin{tabular}{@{}cccc@{}} \toprule
Initial particle & Decay mode & Branching ratio & Partial width (GeV)\\
\colrule
$\chi^\pm_1$ &
$\begin{array}{l}
\chi^0_1 \ \bar q_i \ q_k \\ \chi^{1}_0 \ \ell \ \nu
\end{array}$ &
$\begin{array}{l}
67\% \\ 33\%
\end{array}$ &
$\begin{array}{l}
0.35\times10^{-4} \\ 0.17\times10^{-4}
\end{array}$ \\
\colrule
$\chi^0_2$ &
$\begin{array}{l}
\chi^0_1 \ q \ \bar q  \\ \chi^0_1 \ \bar\ell \ \ell  \\ \chi^0_1
\ \bar\nu \ \nu
\end{array}$ &
$\begin{array}{l}
70\% \\ 10\% \\ 20\%
\end{array}$ &
$\begin{array}{l}
0.66\times10^{-5} \\ 0.10\times10^{-5} \\ 0.20\times10^{-5}
\end{array}$ \\
\botrule
\end{tabular} \label{h}}
\end{table}

\begin{table}[th]
\tbl{Production of the 1$^{\rm st}$ chargino and the 2$^{\rm nd}$
neutralino with subsequent cascade decays.}
{\begin{tabular}{@{}cc@{}}
\toprule
Process & Final states \\
\colrule
\psfig{file=unexplhc_smallfig2a.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
2\ell \\ 2\nu \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.25~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig2b.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
\ell \\ \nu \\ 2j \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.50~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig2c.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
3\ell \\ \nu \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.14~{\rm pb}
\end{array}$} \\
\botrule
\end{tabular}
\hspace{2mm}
\begin{tabular}{@{}cc@{}}
\toprule
Process & Final states \\
\colrule
\psfig{file=unexplhc_smallfig2d.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
\ell \\ 3\nu \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.28~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig2e.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
\ell \\ \nu \\ 2j \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 1.0~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig2f.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
2\ell \\ 2j \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.29~{\rm pb}
\end{array}$} \\
\botrule
\end{tabular}
\label{j}}
\end{table}

In this region of parameter space the cross section for
production of the second neutralino alongside with the first
chargino is larger, owing to the mixing parameters, than that
when the lightest neutralino is produced. After being produced
the light chargino and the second neutralino rapidly decay
into the Standard Model particles and the LSP. The
decay modes and partial widths obtained from ISAJET~\cite{isajet}
are shown in Table~\ref{h}.

\begin{figure}[tb]
\centerline{\psfig{file=unexplhc_fig5.eps,width=0.85\textwidth}}
\vspace*{8pt}
\caption{Cross section dependence on $m_{1/2}$
for chargino pair production $p\bar{p} \rightarrow\chi_{1}^{+}\chi_{1}^{-}$ at TeVatron
for $m_0$ = 1400 GeV,  $\tan\beta=51$, $A_0=0$ and sign($\mu$)=1. \label{Dd}}
\end{figure}

Chargino mostly decays into LSP and via virtual $W$ boson into either
quarks of different flavor or  a neutrino--lepton pair. The second neutralino decays into
LSP and via virtual $Z$-boson into either quark--antiquark pair or lepton--antilepton
pair. Different types of chargino and neutralino cascade decays are shown in
Table~\ref{j}.

Thus, in the chosen region of parameter space due to the
relatively large neutralino and chargino production cross
sections and branching ratios to leptons and LSP, one can have
an unexpectedly large number of outgoing leptons. Moreover,
some of the pure leptonic events, like one or three charged
leptons clearly differ from the Standard Model background
(pair $W$ and $Z$ production) by \textit{large} missing energy,
thus being a convincing indication to the new physics.

The same analysis was conducted for the pro\-ton-anti\-pro\-ton
collisions at the Tevatron
center of mass energy $\sqrt{s}=2$~TeV. The Tevatron energy is
not sufficient to produce heavy
squarks or sleptons but is sufficient to produce light
char\-gi\-nos and neutralinos. The
cross-section dependence for chargino pair production
on $m_{1/2}$ is shown in
Fig.~\ref{Dd}. As one can see the cross-section is
around 0.1--1~pb. The chargino has the
same decay modes as discussed above but ten times smaller
cross section compared
to LHC. However, the present Tevatron luminosity is not
enough to extract the signal from
the background, and one can get only lower limits on chargino masses
(117 GeV at the 95\% C.L.)~\cite{D0}.

\section{Squark and gluino production and decay modes}

Squarks of the 3rd generation are two eigenstates of the mass
matrices
\begin{eqnarray}
&&\left(
\begin{array}{cc}
\tilde m_{tL}^2& m_t(A_t-\mu\cot \beta )\\
m_t(A_t-\mu\cot \beta ) & \tilde m_{tR}^2
\end{array}
\right), \\
&&\left(
\begin{array}{cc} \tilde  m_{bL}^2& m_b(A_b-\mu\tan \beta )\\
m_b(A_b-\mu\tan \beta ) & \tilde  m_{bR}^2
\end{array}
\right),
\end{eqnarray}
where
\begin{eqnarray}
  \tilde m_{tL}^2&=&\tilde{m}_Q^2+m_t^2+\frac{1}{6}(4M_W^2-M_Z^2)\cos
  2\beta , \nonumber \\
  \tilde m_{tR}^2&=&\tilde{m}_U^2+m_t^2-\frac{2}{3}(M_W^2-M_Z^2)\cos
  2\beta ,\\
  \tilde m_{bL}^2&=&\tilde{m}_Q^2+m_b^2-\frac{1}{6}(2M_W^2+M_Z^2)\cos
  2\beta , \nonumber \\
  \tilde m_{bR}^2&=&\tilde{m}_D^2+m_b^2+\frac{1}{3}(M_W^2-M_Z^2)\cos
  2\beta . \nonumber
\label{sq}
\end{eqnarray}
The first terms here are the soft SUSY breaking parameters,
the second terms are the ordinary quark masses and the third
ones are the so called $D$-terms. Similar formulae can
be written for the first and second generations.

The nonvanishing Yukawa couplings result in the large mixing
and splitting of mass eigenstates for the third generation of
squarks. As a result one of the eigenstates becomes lighter
than the others. This leads to a remarkable consequence: the
decay branching ratios for the third generation of squarks are
several times bigger than those for the first two generations
as we will discuss below.

The cross-sections for squark production via proton collisions
at the center of mass energy of $\sqrt{s}=14$~TeV are strongly
suppressed due to the large squark masses. At the same time,
the cross-sections for squark-gluino and pair gluino
production are not that strongly suppressed due to the
relatively light gluino. The leading procceses for production
of strongly interacting spacticles are shown in
Fig.~\ref{prodq}.

\begin{figure}[ptb]
\centerline{\psfig{file=unexplhc_fig6.eps,width=0.8\textwidth}}
\vspace*{8pt}
\caption{Leading types of strongly interacting sparticles
production processes. \label{prodq}}
\end{figure}

\begin{figure}[ptb]
\centerline{\psfig{file=unexplhc_fig7.eps,width=0.9\textwidth}}
\vspace*{8pt}
\caption{Cross-section for squark production
as a function of $m_{1/2}$. The upper blue line is for
$pp\ (q\bar{q})\rightarrow
\tilde{g}\tilde{u}+X $, the purple line in the bottom is for
$pp\ (q\bar{q})\rightarrow
\tilde{g}\tilde{d}+X$ in case of $m_0$=1400~GeV $\tan\beta=51$,
$A_0=0$ and sign($\mu$)=1. \label{e}}
\end{figure}

\begin{figure}[ptb]
\centerline{\psfig{file=unexplhc_fig8.eps,width=0.9\textwidth}}
\vspace*{8pt}
\caption{Cross-section  for gluino production $pp\ (gg)\rightarrow
\tilde{g}\tilde{g}+X $ as a function of $m_{1/2}$
in case of $m_0$=1400~GeV $\tan\beta=51$, $A_0=0$ and sign($\mu$)=1. \label{f}}
\end{figure}

The cross-sections for squark and gluino production depend on
$m_{1/2}$ much stronger than on $m_0$. As it was mentioned
above we fixed the value of $m_0~=~1400$~GeV. The dependence of
squark and gluino production cross-section on $m_{1/2}$ is
shown in Fig.~\ref{e}.

One can see that the cross-section for squark
and gluino production is of the same order
of magnitude as the cross section for chargino
and neutralino production. However,
further squarks and gluinos are the main source
of outgoing jets in sparticle decay while
charginos and second neutralinos are the main
source of leptons.

At the same time, in the small $m_{1/2}$ region
light gluino  production cross-section
in proton collisions via gluon fusion
(Fig.~\ref{f}) is by almost two orders of magnitude
larger than all other sparticle production cross-sections.

After being produced gluinos rapidly decay.
Different types of gluino decay modes
 obtained from ISAJET~\cite{isajet}
are shown in Table~\ref{g}.  There are two leading decay
modes. First of them is when gluino decays to bottom quark and
via virtual sbottom into the second neutralino and anti-bottom
quark, while in the second case it decays into quark and via
virtual squark to the first chargino and anti-quark of a
different flavor. Charginos and neutralinos produced in gluino
decay processes may decay not only to quarks but also to
leptons. Hence, gluino cascade decays might have a different
number of leptons and jets in the final states, as shown in
Table~\ref{n}. This slightly differs chargino and neutralino
decays which have pure leptonic final states.

\begin{table}[b]
\tbl{Gluino and top-quark decay modes and partial widths.}
{\begin{tabular}{@{}cccc@{}} \toprule
Initial particle & Decay mode & Branching ratio & Partial width (GeV)\\
\colrule
$\tilde g$ &
$\begin{array}{l}
\chi^0_2 \ \bar b \ b \\
\chi^0_2 \ \bar q \ q \\
\chi^\pm_1 \ \bar q_i \ q_k \\
\chi^\pm_1 \ \bar t \ b
\end{array}$ &
$\begin{array}{l}
16\% \\ 24\% \\ 30\% \\ 6.6\%
\end{array}$ &
$\begin{array}{l}
0.14\times10^{-4} \\ 0.20\times10^{-4} \\ 0.27\times10^{-4} \\ 0.55\times10^{-4}
\end{array}$ \\
\colrule
$t$ &
$\begin{array}{l}
b \ \bar q_i \ q_k \\ b \ \ell \ \nu
\end{array}$ &
$\begin{array}{l}
67\% \\ 33\%
\end{array}$ &
$\begin{array}{l}
0.8 \\ 0.4
\end{array}$ \\
\botrule
\end{tabular} \label{g}}
\end{table}

\begin{table}[th]
\tbl{Cascade decays for pair of gluinos.}
{\begin{tabular}{@{}cc@{}}
\toprule
Process & Final states \\
\colrule
\psfig{file=unexplhc_smallfig1a.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
2\ell \\ 2\nu \\ 6j \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.008~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig1b.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
4\ell \\ 4j \\ \Big/ \hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.003~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig1c.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
2\ell \\ 6j \\ \Big/ \hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.019~{\rm pb}
\end{array}$} \\
\botrule
\end{tabular}
\hspace{2mm}
\begin{tabular}{@{}cc@{}}
\toprule
Process & Final states \\
\colrule
\psfig{file=unexplhc_smallfig1d.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
2\ell \\ 2\nu \\ 8j \\ \Big/\hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.002~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig1e.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
8j \\ \Big/ \hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.40~{\rm pb}
\end{array}$} \\
\colrule
\psfig{file=unexplhc_smallfig1f.eps,width=38mm,height=30mm} &
\raisebox{15mm}{
$\begin{array}{c}
8j \\ \Big/ \hspace{-0.3cm E_T} \\ \\ \sigma \approx 0.29~{\rm pb}
\end{array}$} \\
\botrule
\end{tabular}
\label{n}}
\end{table}

\begin{figure}[tb]
\centerline{\psfig{file=unexplhc_fig9a.eps,width=0.45\textwidth}
\hspace{0.05\textwidth}
\psfig{file=unexplhc_fig9b.eps,width=0.45\textwidth}}
\centerline{\psfig{file=unexplhc_fig9c.eps,width=0.45\textwidth}
\hspace{0.05\textwidth}
\psfig{file=unexplhc_fig9d.eps,width=0.45\textwidth}}
\vspace*{8pt}
\caption{Cross-section dependence on $m_{1/2}$, $m_{0}$ for sparticle production
processes for  $\tan\beta=51$, $A_0=0$ and sign($\mu$)=1. \label{Mss}}
\end{figure}

As one can see from Fig.~\ref{Mss}, the cross-sections for all
sparticle production processes drop drastically with increase
of $m_{1/2}$. This is especially true for the production
pro\-ces\-ses of weakly interacting sparticles: charginos and
second neutralinos due to the large gaugino fraction.
Alongside with them drops the number of leptonic events above
the Standard Model background. With
increase of $m_{1/2}$ gluino production cross-section also
becomes suppressed by heavy gluino mass. Most of the final
state cross-sections in gluino cascade decays with a variety
of leptons and jets in the final states also drop by $2-3$
orders of magnitude with the increase of $m_{1/2}$ up to $400$~GeV.

Thus, it looks like only a small $m_{1/2}$  region
might bring a relatively large number
of leptoninc events coming above the SM background.
Moreover, from the analysis conducted
for the Tevatron (see Fig.~\ref{Dd}) in this region
of parameter space supersymmetry is
already within a reach of the modern hadron colliders
and the only reason why it is not
seen is the low luminosity of the Tevatron.

Remarkable that in jet production one mainly has $b$-jets with the branching ratio that
may reach 30 \%. This fact finds its explanation in kinematic factors due to the fact
that $b$-squarks are lighter than those of the first two generations.

Indeed, consider  the process of gluino decay into quarks and
squarks. There are two possibilities: either gluino is heavier
than squarks and can decay on the mass shell (see
Fig.\ref{pic1}a) or it is lighter than squarks and decays into
virtual squark which in turn decays into the second neutralino
and the quark (see Fig.\ref{pic1}b).

\begin{figure}[b]
\centerline{\psfig{file=unexplhc_fig10.eps,width=0.7\textwidth}}
\vspace*{8pt}
\caption{Leading modes of gluino decay. \label{pic1}}
\end{figure}

In the first case, the width is easily calculable and has the form
\begin{eqnarray}\label{width}
\Gamma(\tilde{g}\rightarrow\tilde{q }q)&\sim&
\frac{1}{m_{\tilde{g}}^3}\sqrt{(m_{\tilde{g}}^2
\!-\!(m_{q}\!-\!m_{\tilde{q}})^2)
(m_{\tilde{g}}^2\!-\!(m_{q}\!+\!m_{\tilde{q}})^2)}
\nonumber \\
&\times&(m_{\tilde{q}}^2-m_{\tilde{g}}^2-m_{q}^2
\pm2\sin{2\theta_{q}}m_{\tilde{g}}m_{q}),
\end{eqnarray}
where $\theta_{q}$ is the squark mixing angle proportional
to the quark mass according to
eq.(\ref{sq}) and $\pm$~sign refers to the first and the
second squark, respectively. The
square root in this equation essentially comes from the phase space.

Since the quark mass is much smaller than the squark
one and can be ignored (except for
the t quark), eq.(\ref{width}) can be reduced to
\begin{equation}
\label{width2}
\Gamma(\tilde{g}\rightarrow\tilde{q }q)
\sim (m_{\tilde{g}}^2-m_{\tilde{q}}^2)^2,
\end{equation}
which clearly shows that the lighter the squark, the bigger
the width. Since the third generation of squarks is lighter
than the first two (mainly due to the RG running induced by
Yukawa couplings and the mixing in squark mass matrix), the
width for the third generation is larger. Besides, the top
quark production is suppressed compared to the bottom one due
to the large top mass which decreases the phase space in
eq.(\ref{width}). So the $b$-quark production is enhanced
compared to other flavours.

The situation is somewhat different in the second case, when
gluino is lighter than squark, which is the case of the EGRET
point.  Here the expression for the width is more complicated
to be written explicitly~\cite{decay}; however, one can
examine the essential part. When gluino is almost on the mass
shell, a contribution to the amplitude is proportional to
\begin{equation}
\label{wid}
\Gamma(\tilde{g}\rightarrow\tilde{q }q\chi_{0}^{2}))\sim
\frac{3-body\ phase\ space}{(m_{\tilde{q}}^2-(m_{\tilde{g}}
+ m_{q})^2)^2}.
\end{equation}

One can see that the bracket in the denominator which comes
from the squark propagator is minimized for the lightest
squark, since contrary to the previous case gluino is light.
Thus it gives the largest contribution for the third
generation for the same reason as above. The contribution to
the top quark compared to the bottom one is suppressed due to
the phase space which contains only light particles and is
essentially reduced for the heavy top.

This way one can justify that in jet production $b$-jets are
dominant in both cases, which sounds promising for
experimental observation.

\section{Conclusion}

Thus, we conclude that in the EGRET preferred region of
parameter space of the MSSM (small $m_{1/2} \sim 180$~GeV and
large $m_0\sim 1400$~GeV) characterized by a considerable
splitting between the scalar superpartners and gauginos, the
cross-sections for sparticle production at hadronic machines
may reach a few \%~pb. Their decay modes lead to jets (mostly
$b$) and/or leptons in the final states with additional
missing energy carried by escaping neutralinos. These events
have an exceptional signature (see e.g. Table~\ref{n}) that
allows one to distinguish them from the SM background
providing enough integrated luminosity and may be promising
for SUSY searches within the advocated scenario.

\subsection*{Acknowledgements}
Financial support from RFBR grant \# 05-02-17603, grant of
the Ministry of Science
and Technology Policy of the Russian Federation \# 5362.2006.2,
DFG grant \mbox{\# 436 RUS 113/626/0-1} and
the Heisenberg-Landau Programme are kindly
acknowledged. We would like to
thank W. de Boer, V. Zhukov, and V. Bednyakov for valuable
discussions, and A. Pukhov for providing us with
the latest version of CalcHEP and for numerical consultations.