# User-Pair Selection for QoS-Aware Secrecy Rate Maximization in Untrusted
NOMA

Sapna Thapar1, Deepak Mishra2, Ravikant Saini1, and Zhiguo Ding3 1Department
of Electrical Engineering, Indian Institute of Technology Jammu, Jammu $\&$
Kashmir, India
2School of Electrical Engineering and Telecommunications, University of New
South Wales, Sydney, Australia
3School of Electrical and Electronic Engineering, University of Manchester,
Manchester, United Kingdom
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Non-orthogonal multiple access (NOMA) has been recognized as one of the key
enabling technologies for future generation wireless networks. Sharing the
same time-frequency resource among users imposes secrecy challenges in NOMA in
the presence of untrusted users. This paper characterizes the impact of user-
pair selection on the secrecy performance of an untrusted NOMA system. In this
regard, an optimization problem is formulated to maximize the secrecy rate of
the strong user while satisfying the quality of service (QoS) demands of the
user with poorer channel conditions. To solve this problem, we first obtain
optimal power allocation in a two-user NOMA system, and then investigate the
user-pair selection problem in a more generalized four user NOMA system.
Extensive performance evaluations are conducted to validate the accuracy of
the proposed results and present valuable insights on the impact of various
system parameters on the secrecy performance of the NOMA communication system.

##  I Introduction

Owning to the exponential growth in the Internet of Things, achieving
spectral-efficient communication is an inevitable challenge in wireless
networks. Further, cyber-threats have drastically expanded due to a large
number of devices getting connected to the Internet. To address these critical
issues, the coexistence of non-orthogonal multiple access (NOMA) and physical-
layer security (PLS) is being recognized as a promising solution [1], [2].
However, due to significant co-channel interference in NOMA systems, it is not
practical and realistic to apply NOMA jointly on all users in the network.
Here, user pairing/grouping, in which the users of the network are split into
many pairs/groups and NOMA is implemented within each pair/group, is an
interesting research alternative [1]. This, from the perspective of securing
untrusted users in a NOMA system [3], opens up a novel research challenging of
finding which users of the system should be grouped together?

###  I-A State-of-the-Art

To derive maximum benefits from NOMA, identifying the best user pairing
strategy has been an active research direction. In [1], user pairing for two
scenarios, i.e., NOMA with fixed power allocation and cognitive-radio inspired
NOMA was studied. In [4], user pairing in cognitive-radio based NOMA system
was investigated, where the conventional distributed matching algorithm was
applied. In [5], two user pairing strategies were proposed to maximize the
ergodic sum capacity of the system. In [6], a novel user pairing strategy was
provided to improve the performance of the weak user in the presence of two
strong users. A virtual user pairing NOMA strategy was proposed in [7] to
utilize spectrum efficiently.

Utilizing the concept of PLS, various works have investigated the secrecy
performance of NOMA systems in the presence of untrusted users. An untrusted
users’ network is a hostile but realistic situation where users do not have
mutual trust. Therefore, they focus on securing their own data from other
users, which leads to complex resource allocations [8], [9]. In [3], strong
and weak users of a two-user NOMA system were assumed to be trusted and
untrusted, respectively, and the secrecy outage probability was analyzed for
the strong user against the weak user. In [10], assuming strong and weak users
of a multiple-input single-output NOMA system as trusted and untrusted,
respectively, the sum secrecy rate for the strong user was analyzed. In [11],
two optimal relay selection schemes were proposed to attain secure
communication for a strong user in the presence of an untrusted weak user.

###  I-B Motivation and Contributions

We observed that [1], [4]-[7] worked on the user pairing problem in NOMA.
Whereas, [3], [10], [11] analyzed PLS performance of an untrusted NOMA system.
Thus, PLS and user pairing for NOMA, have been studied independently in the
literature. However, to our best knowledge, exploring the best user pairing
scheme from the perspective of improving the secrecy rate in an untrusted NOMA
system has not been considered yet. Addressing this, our key contributions
are:

  * •

Assuming a downlink communication system, we focus on finding out the best
user pairing scheme from the perspective of improving the secrecy rate of the
stronger user while fulfilling the QoS demands of the weaker user.

  * •

Firstly, we consider a two-user NOMA pair and find out a closed-form solution
of optimal power allocation that can maximize the secrecy rate for the strong
user while ensuring a minimum data rate guarantee to the weak user.

  * •

Then, we consider a four-user system and explore the best pairing scheme for
three different scenarios. The first one maximizes the secrecy rate of the
strong user, the second scenario maximizes the weak uesr’s data rate, and the
third one maximizes the secrecy rate of the strong user while satisfying weak
user’s minimum rate requirement.

  * •

Lastly, numerical results verify the correctness of the analytical derivations
and present valuable insights on the secrecy rate performance achieved through
our schemes.

##  II Secure NOMA with Untrusted Users

###  II-A System Model and NOMA Transmission Principle

We consider the downlink system with one base station and $N$ users. The
$i$-th user of the system is denoted as $U_{i}$, where
$i\in\mathcal{N}=\\{1,2,...,N\\}$. Each node in the system has a single
antenna. The channel coefficient between the base station and $U_{i}$, which
is assumed to be Rayleigh distributed, is denoted by $h_{i}$. The
corresponding channel power gain $|h_{i}|^{2}$ between base station and
$U_{i}$ experiences exponential distribution with mean
$\lambda_{i}=L_{p}d_{i}^{-e}$, where $L_{p}$, $e$, and $d_{i}$, respectively,
denote path loss constant, path loss exponent and distance between BS and
$U_{i}$. Without loss of generality, the channel power gains are assumed to be
sorted as $|h_{1}|^{2}>|h_{2}|^{2}>...>|h_{N}|^{2}$.

To achieve high spectral efficiency, users are multiplexed in the same
resource block of a NOMA system. Considering an interference-limited NOMA
system [1], we restrict to the simplest case of two users in a pair. In
general, the user with a better channel gain is paired with the user having
poorer one. However, the key question is - which two users should be selected
to be paired together? For the sake of our discussion, we consider two paired
users as $U_{m}$ and $U_{n}$, out of all $N$ users with
$|h_{m}|^{2}>|h_{n}|^{2}$, where $m,n\in\mathcal{N},n>m$. Using power domain
NOMA, for pair of users $U_{m}$ and $U_{n}$, the base station transmits a
superimposed signal
$x=\sqrt{\alpha_{m}P_{t}}x_{m}+\sqrt{\alpha_{n}P_{t}}x_{n}$, where
$x_{i}\forall i\in\\{m,n\\}$ is the signal associated with the $i$-th user,
$P_{t}$ is base station’s transmission power, $\alpha_{i}$ denotes the
coefficient of signal power for $U_{i}$, satisfying $\alpha_{m}+\alpha_{n}=1$.
We assume received noise to be additive white Gaussian with mean 0 and
variance $\sigma^{2}$ for both users.

###  II-B Achievable Secrecy Rates at Users

In NOMA, the stronger user $U_{m}$ first decodes and removes $x_{n}$ by
exploiting successive interference cancellation, and then decodes $x_{m}$. We
assume that the weaker user is an untrusted user and may try to get the
message of the stronger user. Thus, the weaker user $U_{n}$ will first decode
its own signal $x_{n}$, and then may decode $x_{m}$ [3]. Using superscript
$(m,n)$ to denote the indices of the paired users $U_{m}$ and $U_{n}$, where
$m,n\in\mathcal{N},n>m$, the received signal-to-interference-plus-noise-ratio
(SINR) at $U_{a}$, when signal of $U_{a}$ is decoded by $U_{b}$, where
$(a=m,n)$ and $(b=m,n)$, is denoted as $\Gamma_{ab}^{(m,n)}$. As a result, for
the pair $(m,n)$, the SINRs can be obtained as

$\displaystyle\Gamma^{(m,n)}_{nm}=\frac{\alpha_{n}|h_{m}|^{2}}{\alpha_{m}|h_{m}|^{2}+\frac{1}{\rho_{t}}},\quad$
$\displaystyle\Gamma^{(m,n)}_{nn}=\frac{\alpha_{n}|h_{n}|^{2}}{\alpha_{m}|h_{n}|^{2}+\frac{1}{\rho_{t}}},$
$\displaystyle\Gamma^{(m,n)}_{mm}=\alpha_{m}|h_{m}|^{2}\rho_{t},\quad$
$\displaystyle\Gamma^{(m,n)}_{mn}=\alpha_{m}|h_{n}|^{2}\rho_{t}.$ (1)

Here, $\rho_{t}=\frac{P_{t}}{\sigma^{2}}$ indicates the base station transmit
signal-to-noise ratio (SNR). Further, to analyze the secrecy performance from
the perspective of PLS, the achievable secrecy rate for $U_{a}$, where
$(a=m,n)$ can be mathematically expressed as

$\displaystyle R_{sa}^{(m,n)}=R_{aa}^{(m,n)}-R_{ab}^{(m,n)},a\neq b.$ (2)

In (2), $R_{ab}^{(m,n)}$ can be given by Shannon’s capacity formula as

$\displaystyle R_{ab}^{(m,n)}=\log_{2}(1+\Gamma^{(m,n)}_{ab}).$ (3)

###  II-C Feasibility Conditions for Positive Secrecy Rate

The key idea of achieving a positive secrecy rate for a user is to ensure that
the data rate over the desired link is higher than that of the eavesdropper’s
link. Mathematically, for $R_{sa}^{(m,n)}>0$, $R_{aa}^{(m,n)}>R_{ab}^{(m,n)}$,
simplified as $\Gamma^{(m,n)}_{aa}>\Gamma^{(m,n)}_{ab}$, must be satisfied.
Lemma 1 provides a key result on achievable secrecy performance in an
untrusted NOMA system.

###### Lemma 1

For every pair of users in an untrusted NOMA system, the message of the
stronger user is always safe from the untrusted weaker user. In comparison,
the data of the weaker user is not secured from the stronger user.

###### Proof:

For users $U_{m}$ and $U_{n}$, we assume $|h_{m}|^{2}>|h_{n}|^{2}$, where
$m,n\in\mathcal{N},n>m$. To check positive secrecy rate for stronger user
$U_{m}$ against $U_{n}$, using (II-B), we solve
$\Gamma^{(m,n)}_{mm}>\Gamma^{(m,n)}_{mn}$, and get a feasible condition
$|h_{m}|^{2}>|h_{n}|^{2}$. This indicates that always positive secrecy rate is
achieved for the strong user $U_{m}$ against the weak user $U_{n}$. Similarly,
to achieve positive secrecy rate for $U_{n}$, we solve
$\Gamma^{(m,n)}_{nn}>\Gamma^{(m,n)}_{nm}$ and obtain
$|h_{n}|^{2}>|h_{m}|^{2}$, using (II-B), which is an infeasible condition.
Thus, data of $U_{n}$ cannot be secured from $U_{m}$. ∎

##  III Proposed Optimal Power Allocation Policy

To maximize the secrecy rate of the stronger user $U_{m}$, under the
constraint of fulfilling QoS demands of the weaker user $U_{n}$, the
optimization problem for user pair $(m,n)$ can be formulated using (2) and
(3), as

$\displaystyle\mathcal{P}1:$
$\displaystyle\mathop{\mathrm{maximize}}\limits_{\alpha_{m},\alpha_{n}}\quad
R_{sm}^{(m,n)},\quad\text{subject to}$
$\displaystyle\mathcal{C}1\\!:R_{nn}^{(m,n)}\geq
R_{\mathrm{th}},\;\mathcal{C}2\\!:\alpha_{m},\alpha_{n}>0,\;\mathcal{C}3\\!:\alpha_{m}+\alpha_{n}=1,$

where $R_{\mathrm{th}}$ denotes the targeted data rate to be guaranteed for
the weaker user $U_{n}$. The problem $\mathcal{P}$1 can be reformulated via
considering the power allocation constraints $\mathcal{C}2$ and
$\mathcal{C}3$, which can be simplified in terms of $\alpha_{m}$ and
$\alpha_{n}=1-\alpha_{m}$. Thus, the reformulated optimization problem can be
stated as

$\displaystyle\mathcal{P}2:\mathop{\mathrm{maximize}}\limits_{\alpha_{m}}\;$
$\displaystyle R_{sm}^{(m,n)},\quad\;\text{subject
to}\quad\mathcal{C}1,\mathcal{C}4:0<\alpha_{m}<1.$

The optimal solution of the problem $\mathcal{P}2$ is given below.

###### Lemma 2

The optimal $\alpha_{m}$ that maximize the secrecy rate $R^{(m,n)}_{sm}$ of
$U_{m}$ against $U_{n}$, under the minimum data rate (QoS) guarantee for
$U_{n}$ and definition
$\Pi\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{\mathrm{th}}}$, can be written
as

$\displaystyle\alpha_{m}^{*}=\frac{|h_{n}|^{2}\rho_{t}-\Pi+1}{\Pi|h_{n}|^{2}\rho_{t}}.$
(4)

###### Proof:

From, $\eqref{SINRT}$, $\eqref{secrecy_rate}$ and $\eqref{info_rate}$, we see
that secrecy rate $R^{(m,n)}_{sm}$ is a function of $\alpha_{m}$. We check the
first order derivative of $R^{(m,n)}_{sm}$ with respect to $\alpha_{m}$, which
is given by

$\displaystyle\frac{\text{d}R^{(m,n)}_{sm}}{\text{d}\alpha_{m}}=\dfrac{\left(|h_{m}|^{2}-|h_{n}|^{2}\right)\rho_{t}}{\ln\left(2\right)\left(1+\alpha_{m}|h_{m}|^{2}\rho_{t}\right)\left(1+\alpha_{m}|h_{n}|^{2}\rho_{t}\right)}.$
(5)

It can be easily observed from (5), that
$\frac{\text{d}R^{(m,n)}_{sm}}{\text{d}\alpha_{m}}>0$, since
$|h_{m}|^{2}>|h_{n}|^{2}$ is assumed in Section II-A. Thus, $R^{(m,n)}_{sm}$
is an increasing function with respect to $\alpha_{m}$. Thus, to obtain
maximum secrecy rate for $U_{m}$, $\alpha_{m}$ should be maximum.

On the other hand, the range of the $\alpha_{m}$ can be obtained by solving
the constraint $\mathcal{C}1$ directly. The constraint $\mathcal{C}1$ which is
given as $R^{(m,n)}_{nn}\geq R_{\mathrm{th}}$ can be simplified using (3) as
$\Gamma^{(m,n)}_{nn}\geq(\Pi-1)$. Simplifying it further, boundary condition
on $\alpha_{m}$ is

$\displaystyle\alpha_{m}\leq\frac{|h_{n}|^{2}\rho_{t}-\Pi+1}{\Pi|h_{n}|^{2}\rho_{t}}\triangleq{\alpha^{u}}.$
(6)

Thus, the optimal solution $\alpha_{m}^{*}$ of $\mathcal{P}2$ is
$\alpha_{m}^{*}=\alpha^{u}$. ∎

Since, $\alpha_{m}^{*}$ is the power allocation coefficient, we investigate
the feasibility of optimal solution $\alpha_{m}^{*}$ through Proposition 1.

###### Proposition 1

With $\alpha_{m}^{*}\in(0,1)$ being a real number, the minimum rate guarantee
$R_{\mathrm{th}}$ of the weaker user is bounded as:
$0<R_{\mathrm{th}}<\log_{2}(|h_{n}|^{2}\rho_{t}+1)$.

###### Proof:

Solving $\alpha_{m}^{*}>0$, we obtain a feasibility condition
$\Pi<|h_{n}|^{2}\rho_{t}+1$, which can be further simplified as
$R_{\mathrm{th}}<\log_{2}(|h_{n}|^{2}\rho_{t}+1)$, since
$\Pi=2^{R_{\mathrm{th}}}$. Similarly, on solving $\alpha_{m}^{*}<1$, we obtain
$\Pi>1$, which gets simplified to $R_{\mathrm{th}}>0$. Thus, for
$\alpha_{m}^{*}$ to be in between 0 and 1, the condition
$1<\Pi<|h_{n}|^{2}\rho_{t}+1$ must be fulfilled. Simplifying the conditions in
terms of $R_{\mathrm{th}}$, we obtain
$0<R_{\mathrm{th}}<\log_{2}(|h_{n}|^{2}\rho_{t}+1)$. ∎

##  IV User-Pair Selection

To analyze the impact of different user-pairs on the secrecy performance
achieved in an untrusted NOMA system, we consider a network with four users
and choose a pair to apply NOMA while keeping other two users unpaired. We
start with discussing the possible pairs to eventually find best user pair.

###  IV-A Possible User-Pairs in a 4-User System

With the intention of choosing two users out of four for making a pair, there
are ${}^{4}C_{2}=6$ ways to form user pairs. Let us define all possible pairs
by a set $\mathbb{S}=\\{(m,n)|m,n\in\mathcal{N},n>m\\}$. Here, $(m,n)$ denotes
the indices of the paired users $U_{m}$ and $U_{n}$. Thus, the set of all
possible pairs for a system with four users can be written as
$\mathbb{S}=\\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\\}$.

TABLE I: Order of pairs in terms of secrecy achieved at strong user ($\mathcal{A}$ and $\mathcal{B}$) and data rate achieved at the weak user ($\mathcal{C})$ | $\mathcal{A}$ | $\mathcal{B}$ | $\mathcal{C}$
---|---|---|---
Pair | $|h_{i}|^{2}\\!=\\!(N\\!-\\!i+1)|h_{N}|^{2}$ | $|h_{i}|^{2}\\!=\\!\frac{|h_{1}|^{2}}{i}$ | $|h_{i}|^{2}>|h_{i+1}|^{2}$
(1,2) | VI (Worst) | III | I (Best)
(1,3) | III | II | II
(1,4) | I (Best) | I (Best) | III
(2,3) | V | V | II
(2,4) | II | IV | III
(3,4) | IV | VI (Worst) | III

###  IV-B Best User-Pair Selection

To find the best pair out of set $\mathbb{S}$, we compare each pair with the
other pair one by one. For example, pair $(m,n)\in\mathbb{S}$ is compared with
the pair $(k,l)\in\mathbb{S}$, where $(m,n)\neq(k,l)$. Considering all
possible combinations, we observe that the total possible comparisons are
${}^{6}C_{2}=15$. In the following, we investigate the best pair under three
different scenarios.

####  IV-B1 Best User-Pair to Maximize Secrecy Rate of the Strong User

In this case, we focus on maximizing the secrecy rate of the strong user only
without focusing on the data rate demands of the weak user. Therefore, we
neglect the QoS constraint of the weak user. We note from (5) that the secrecy
rate $R_{sm}^{(m,n)}$ for $U_{m}$ is an increasing function of $\alpha_{m}$.
It means that, to achieve maximum secrecy rate for $U_{m}$, $\alpha_{m}$
should be maximum. Therefore, in this case, we take $\alpha_{m}=1$ and
$\alpha_{n}=0$. Below a key result on the best pair from a secrecy perspective
is provided through Proposition 2.

###### Proposition 2

Ignoring the data rate demands of the weak user, the optimal secrecy rate
performance for the strong user can be achieved by selecting those two users
for which the channel power gains difference is maximum.

###### Proof:

Considering $\alpha_{m}=1$ for the pair $(m,n)\in\mathbb{S}$, the secrecy
rate, using (II-B), and (2), can be expressed as
$R_{sm}^{(m,n)}=\log_{2}\Big{(}\frac{1+|h_{m}|^{2}\rho_{t}}{1+|h_{n}|^{2}\rho_{t}}\Big{)}$.
It can be noted that $R_{sm}^{(m,n)}$ increases with $|h_{m}|^{2}$, and
decreases with $|h_{n}|^{2}$. Thus, the secrecy rate of the strong user is
dependent on the ratio of channel power gains of the strong user and the weak
user. Therefore, when $|h_{1}|^{2}>|h_{2}|^{2}>|h_{3}|^{2}>|h_{4}|^{2}$, the
user with the best channel condition, i.e., $U_{1}$, should be paired with the
user having the worst channel condition, i.e., $U_{4}$. ∎

Next, we focus on obtaining the ordered sequence of user pairs in terms of
decreasing the secrecy rate performance of the strong user from the highest to
the lowest. To investigate this order, we consider a specific scenario, where
channel gains are assumed to follow $|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$, $\forall
i\in\mathcal{N}$. In this regard, a key result is provided in Proposition 3.

###### Proposition 3

For an untrusted NOMA system, where channel power gains are such that
$|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$, $\forall i\in\mathcal{N}$, the highest to
the lowest order of the secrecy rate achieved for the strong user in all
possible pairs is given as

$R_{s1}^{(1,4)}>R_{s2}^{(2,4)}>R_{s1}^{(1,3)}>R_{s3}^{(3,4)}>R_{s2}^{(2,3)}>R_{s1}^{(1,2)}$
(7)

###### Proof:

The secrecy rate of a user in each pair is in the form of
$\log_{2}\big{(}\frac{1+A}{1+B}\big{)}$. On comparing two pairs, we notice
that a given secrecy rate is higher than its counterpart if either $A$ is
higher and/or $B$ is lower, with other respective parameters being the same.
We investigate each comparison one by one and find out the feasibility
conditions. Like, we compare two user pairs $(1,4)$ and $(2,4)$. The secrecy
rate for $U_{1}$ in the case of $(1,4)$ with $\alpha_{1}=1$ and
$|h_{1}|^{2}=4|h_{4}|^{2}$, can be given as

$R_{s1}^{(1,4)}\\!=\\!\log_{2}\Bigg{(}\frac{1+|h_{1}|^{2}\rho_{t}}{1+|h_{4}|^{2}\rho_{t}}\Bigg{)}=\log_{2}\Bigg{(}\frac{1+4|h_{4}|^{2}\rho_{t}}{1+|h_{4}|^{2}\rho_{t}}\Bigg{)}.$
(8)

Similarly, the secrecy rate for $U_{2}$ in the pair $(2,4)$, with
$\alpha_{2}=1$ and $|h_{2}|^{2}=3|h_{4}|^{2}$, can be given as

$R_{s2}^{(2,4)}\\!=\\!\log_{2}\Bigg{(}\frac{1+|h_{2}|^{2}\rho_{t}}{1+|h_{4}|^{2}\rho_{t}}\Bigg{)}=\log_{2}\Bigg{(}\frac{1+3|h_{4}|^{2}\rho_{t}}{1+|h_{4}|^{2}\rho_{t}}\Bigg{)}.$
(9)

Here, $A=4|h_{4}|^{2}$ for the user-pair $(1,4)$, while $A=3|h_{4}|^{2}$ in
the user-pair $(2,4)$. On comparing, we find that $A$ is higher in user-pair
$(1,4)$ than the $A$ in the user-pair $(2,4)$. Thus, it is clear that
$R_{s1}^{(1,4)}>R_{s2}^{(2,4)}$. Similarly, we compare all other possible
combinations and obtain the order of secrecy rate achieved at strong users in
the user-pairs as
$R_{s1}^{(1,4)}>R_{s2}^{(2,4)}>R_{s1}^{(1,3)}>R_{s3}^{(3,4)}>R_{s2}^{(2,3)}>R_{s1}^{(1,2)}$.
∎

###### Remark 1

Since the highest to the lowest order of the secrecy rate performance obtained
at the strong user is dependent on the relative channel power gains of the
users (refer Proposition 3), the order gets changed if the power gains get
changed. For example, if the power gains are such that
$|h_{i}|^{2}=\frac{|h_{1}|^{2}}{i}$, where $\forall i\in\mathcal{N}$, then the
obtained order of the pairs in terms of secrecy rate is provided in Table I.

####  IV-B2 Best User-Pair to Maximize Data Rate of the Weaker User

In this case, we intend to select the best user-pair to maximize the data rate
of the weak user without worrying about the secrecy performance of the strong
user. The first order derivative of $R_{nn}^{(m,n)}$ with respect to
$\alpha_{m}$ is given as

$\displaystyle\frac{\text{d}R^{(m,n)}_{nn}}{\text{d}\alpha_{m}}=\frac{-|h_{n}|^{2}(|h_{n}|^{2}\rho_{t}+1)\rho_{t}}{(\alpha_{m}|h_{n}|^{2}\rho_{t}+1)^{2}},$
(10)

which is negative. Thus, $R^{(m,n)}_{nn}$ is a decreasing function of
$\alpha_{m}$. Therefore, in this case, we take the lower bound on
$\alpha_{m}$, i.e., $\alpha_{m}^{(m,n)}=0$ and thus, $\alpha_{n}^{(m,n)}=1$.
The best user-pair is presented through Proposition 4 in the following.

###### Proposition 4

Regardless of the secrecy of the strong user, the maximum data rate at the
weak user is obtained when the strongest user $U_{1}$ is paired with the
second strongest user $U_{2}$.

###### Proof:

Using (II-B), (3), $\alpha_{m}=0$ and $\alpha_{n}=1$, the data rate
$R^{(m,n)}_{nn}$ of the weak user $U_{n}$ in a pair $(m,n)$, can be given as

$\displaystyle
R_{nn}^{(m,n)}=\log_{2}(1+\Gamma^{(m,n)}_{nn})=\log_{2}(1+|h_{n}|^{2}\rho_{t}).$
(11)

$R_{nn}^{(m,n)}$ is an increasing function of $|h_{n}|^{2}$. Thus, while
comparing two pairs, the pair having better channel condition of the weak user
will provide a better data rate for the weak user. Since, we assume a system
where $|h_{2}|^{2}>|h_{3}|^{2}>|h_{4}|^{2}$, it is clear that $(1,2)$ will be
the optimal user-pair. ∎

###### Remark 2

Based on Proposition 4, the highest to the lowest order of the user-pairs in
terms of the data rate performance of the weak user has been presented in
Table I. Since the ordering is based on the rate of the weak user, the
ordering does not change with a change of the strong user.

####  IV-B3 Best User-Pair for QoS-Aware Secure NOMA

Lastly, we consider the third case, where the goal is to find out the best
user pair that maximizes the secrecy rate of the strong user while ensuring
QoS demands of the weak user. To fulfil the minimum threshold rate
requirements of the paired weak user in each user-pair, we consider the
optimal power allocation as obtained in Section III. Here, we consider a
specific scenario, where channel power gains are assumed to follow
$|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$, $\forall i\in\mathcal{N}$. To investigate
the order of the pairs, we compare each pair with another, one by one. For
example, let us compare two pairs $(2,4)$ and $(1,3)$. In pair $(2,4)$ secrecy
rate for $U_{2}$ with $|h_{2}|^{2}=3|h_{4}|^{2}$, is given as

$\displaystyle
R_{s2}^{(2,4)}\\!\\!=\\!\log_{2}\left(\frac{1+\alpha_{2}^{*}|h_{2}|^{2}\rho_{t}}{1+\alpha_{2}^{*}|h_{4}|^{2}\rho_{t}}\right)\\!=\\!\log_{2}\left(\frac{1+\alpha_{2}^{*}3|h_{4}|^{2}\rho_{t}}{1+\alpha_{2}^{*}|h_{4}|^{2}\rho_{t}}\right)\\!$
(12)

where
$\alpha_{2}^{*}=\frac{|h_{4}|^{2}\rho_{t}-\Pi+1}{\Pi|h_{4}|^{2}\rho_{t}}$ is
the optimal power allocation from (4). Similarly, the secrecy rate for $U_{1}$
in the pair $(1,3)$, with $|h_{1}|^{2}=4|h_{4}|^{2}$ and
$|h_{3}|^{2}=2|h_{4}|^{2}$, can be given as
$R_{s1}^{(1,3)}=\log_{2}\Big{(}\frac{1+\alpha_{1}^{*}4|h_{4}|^{2}\rho_{t}}{1+\alpha_{1}^{*}2|h_{4}|^{2}\rho_{t}}\Big{)}$,
with
$\alpha_{1}^{*}=\frac{|h_{3}|^{2}\rho_{t}-\Pi+1}{\Pi|h_{3}|^{2}\rho_{t}}=\frac{2|h_{4}|^{2}\rho_{t}-\Pi+1}{\Pi|2h_{4}|^{2}\rho_{t}}$.
To find out, which pair performs better in terms of secrecy rate of the strong
user, we solve $R_{s2}^{(2,4)}>R_{s1}^{(1,3)}$, resulting in a condition:
$R_{\mathrm{th}}<\log_{2}\big{(}1+\frac{2|h_{4}|^{2}|h_{4}|^{2}\rho_{t}^{2}}{1+3|h_{4}|^{2}\rho_{t}}\big{)}$.

It shows that $(2,4)$ provides more secrecy rate than $(1,3)$ with a suitable
constraint on threshold rate $R_{\mathrm{th}}$. In a similar manner, bounds on
threshold rate can be obtained for other comparisons as well. However, in some
of the comparisons, we also get a feasible condition that indicates that a
pair is always better than the other pair. For example, to compare two user-
pairs $(1,4)$ and $(3,4)$, we solve $R_{s1}^{(1,4)}>R_{s3}^{(3,4)}$, and get a
feasible condition $|h_{4}|^{2}\rho_{t}+1$. It means that pair $(1,4)$ always
performs better in terms of secrecy rate for the strong user than the pair
$(3,4)$ Thus, it can be concluded that in this case, the comparison between
two pairs is dependent on the channel power gain conditions and threshold
rate. Therefore, we observe the order of the pairs with optimal power
allocation for different channel power gain ratios via simulations.

##  V Numerical Results

Here, numerical results under various system settings are presented to
validate the proposed results and present useful insights on the system
performance. We consider a downlink system with four users, where the base
station communicates with two users in a pair. Noise signal at both users is
assumed to follow Gaussian distribution with an average noise power of -120
dBm. Besides, $L_{p}=1$, $e=3$, $R_{\mathrm{th}}=0.5$, and $P_{t}=1$ mW are
considered. Simulation results have been averaged over $10^{6}$ randomly
generated channel realizations.

Figure 1: Validation of order of user-pairs with the (a) average secrecy rate
of strong user with $\alpha_{m}=1$ and $\alpha_{n}=0$ and (b) average data
rate for the weak user $\alpha_{m}=0$ and $\alpha_{n}=1$, for channels
satisfying $|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$.  Figure 2: Impact of different
channel power gain ratios on average secrecy rate performance with optimal
power allocation for various user-pairs.  Figure 3: Impact of different
threshold rate requirements of a weak user on the average secrecy rate of the
strong user for $|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$.  Figure 4: Performance
comparison of optimal user-pair with random user-pair selection for channel
power gain conditions as $|h_{i}|^{2}=(N-i+1)|h_{N}|^{2}$.

###  V-A Insights on Optimal User-Pair Selection and Order

We first observe the probability of occurrence of different user pairings for
channel power conditions as $|h_{1}|^{2}>|h_{2}|^{2}>|h_{3}|^{2}>|h_{4}|^{2}$
under three different cases. Case 1: maximizing secrecy rate for strong user
only, Case 2: maximizing data rate of weak user only, and Case 3: maximizing
secrecy rate for strong user while satisfying QoS requirements of weak user.
Results validate that $(1,4)$ is the optimal pair to maximize the secrecy rate
for the strong user as proposed in Proposition 2. Similarly, simulations also
confirm that $(1,2)$ is the optimal pair for maximizing the data rate for the
weak user as proposed in Proposition 4. Through numerical observations, here
we see that for the third case also, with optimal power allocation to users,
pair $(1,4)$ wins. Thus, we conclude that the optimal user pair solution
differs with the objective.

Through the results shown in Fig. 1(a) and Fig. 1(b), we validate the order
proposed in Proposition 3 and Section IV-B2, respectively. Here, Fig. 1(a) is
plotted for the case when the secrecy performance of the strong user is
focused, and, Fig. 1(b) is plotted for the second case focusing on the data
rate demands for the weak user.

###  V-B Impact of Key Parameters and Performance Comparison

Fig. (2) is plotted to highlight the impact of various channel power gain
conditions on the optimal secrecy rate performance of the strong user. Here,
we observe that even with optimal power allocation, the highest to the lowest
order of the secrecy rate performance for different pairs varies with the
relative channel power gain conditions. Fig. 3 is plotted to present the
impact of increasing threshold rate requirements of the weak user on the
secrecy rate performance achieved by the strong user. It is clearly observed
that on increasing the QoS demands, the secrecy rate performance of the strong
user decreases. This happens because to fulfil the QoS demands of the weak
user, more power needs to be allocated to the weak user, due to which the rate
of decoding the strong user will also increase. As a result, the secrecy rate
of the strong user decreases.

To observe the performance improvement achieved by the selection of optimal
user-pair for maximizing secrecy rate at strong user, Fig. 4 demonstrates its
performance comparison with random user-pair selection for various threshold
rates. It can be easily observed that that optimal user-pair performs best
over the random user-pair selection. Here, the average percentage improvement
of about $112\%$ is obtained.

##  VI Conclusions

This work studied the impact of user-pair selection on improving the secrecy
performance of an untrusted NOMA system. The best user pair to maximize the
secrecy rate of the strong user while satisfying the QoS demands of the weaker
user is identified. To solve this problem, first optimal power allocation in a
two-user NOMA system is obtained, and then a generalized problem with four
users is considered. Numerical results validated the correctness of the
proposed results and presented key insights on the impact of various system
parameters on the system secrecy performance.

## Acknowledgement

This work was supported in parts by the SERB, DST, under the Grant no.
CRG!2021/002464, and TCS RSP Fellowship.

## References

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# Machine-Generated Text Detection using Deep Learning

Raghav Gaggar*† Ashish Bhagchandani11footnotemark: 1*† Harsh
Oza11footnotemark: 1*†<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>University
of Southern California† These authors contributed equally to this work.

###### Abstract

Our research focuses on the crucial challenge of discerning text produced by
Large Language Models (LLMs) from human-generated text, which holds
significance for various applications. With ongoing discussions about
attaining a model with such functionality, we present supporting evidence
regarding the feasibility of such models. We evaluated our models on multiple
datasets, including Twitter Sentiment, Football Commentary, Project Gutenberg,
PubMedQA, and SQuAD, confirming the efficacy of the enhanced detection
approaches. These datasets were sampled with intricate constraints
encompassing every possibility, laying the foundation for future research. We
evaluate GPT-3.5-Turbo against various detectors such as SVM, RoBERTa-base,
and RoBERTa-large. Based on the research findings, the results predominantly
relied on the sequence length of the sentence.

##  1 Introduction

In recent years, the field of natural language processing (NLP) has witnessed
revolutionary advancements, primarily fueled by the advent of large language
models like OpenAI’s ChatGPT (Brown et al., 2020) (Radford et al., 2019).
These models have demonstrated remarkable capabilities in generating human-
like text, leading to widespread applications in various domains, including
customer service, content creation, and education. However, the increasing
sophistication of these models also poses significant challenges, particularly
in distinguishing machine-generated text from human-generated content. The
ability to make this distinction is crucial for maintaining the integrity and
trustworthiness of digital communication.

This paper presents evidence which helps to support the challenge that human-
generated sentences can be differentiated from sentences generated from
GPT-3.5-Turbo. Leveraging state-of-the-art machine learning models such as
RoBERTa-Base, RoBERTa-Large, and SVM, our approach aims to detect subtle
differences in language patterns, stylistic features, and semantic nuances.
Previous work in this area has primarily focused on identifying generic
characteristics of AI-generated text, often overlooking the specific
attributes of advanced models like ChatGPT (Chakraborty et al., 2023). Our
work builds upon these foundations but introduces a targeted analysis aligned
with the unique linguistic features of ChatGPT-generated content.

To develop and evaluate our model, we compiled a diverse dataset comprising
thousands of sentences from human sources in domains including Sports,
Medical, Twitter reviews, Text comprehension, and Literature spanning various
genres and styles. Further, using these sentences to generate similar
sentences from the GPT-3.5-Turbo API of ChatGPT. This dataset was used in the
pre-processing stage of natural language embeddings and feature extraction
before using it for any classification model.

The significance of this research lies not only in its immediate application
for content verification but also in its broader implications for the field of
digital forensics and the ethics of AI in communication. As language models
continue to evolve, distinguishing between human and AI-generated content will
become increasingly challenging, necessitating ongoing research and
development in this area (Crothers et al., 2023).

In this paper, we describe our methodology, the architecture of our detection
model, and the results of our experiments. We also discuss the broader
implications of our findings in context with sentence features and patterns.

##  2 Related Work

Numerous investigations have been undertaken to discern the features
differentiating human-generated text from machine-generated text. Let’s delve
into each of these approaches.

Recent LLMs like ChatGPT can generate text to compose essays, describe art in
detail, create AI art prompts, have philosophical conversations, and even code
for you. To detect such intricate patterns, a new methodology was proposed in
DetectGPT, where they discovered how LLMs operate where the models tend to
create text that falls into specific patterns, particularly in regions where
the model’s calculations show negative curvature. DetectGPT utilizes log
probabilities from the LLM and random perturbations that it generates to
determine whether the text is machine-generated (Mitchell et al., 2023). It is
particularly good at identifying fake news articles made by models like GPT-
NeoX and outperforms most zero-shot methods. They test LLM performance with
six diverse datasets: XSum for fake news detection, SQuAD for academic essays,
Reddit WritingPrompts for creative writing, WMT16 in English and German, and
PubMedQA for long-form expert answers. The future work that authors in
DetectGPT wanted to explore was to see how watermarking algorithms worked with
detection algorithms like DetectGPT.

(Guo et al., 2023) proposed an HC3 (Human ChatGPT Comparison Corpus) dataset,
which consists of nearly 40K questions and their corresponding human/ChatGPT
answers, and developed a ChatGPT detection model to differentiate human and
Chatgpt generated text. The detector is primarily created by fine-tuning
RoBERTa model on the dataset, and the authors propose two methods for training
it. The first method uses only the pure answered text, while the second method
utilizes the question-answer text pairs for joint model training.

In 2019, (Solaiman et al., 2019) built a detector for GPT2-generated output by
fine-tuning RoBERTa using outputs from the largest GPT2 Model, which was with
1.5 billion parameters and was capable of detecting if the text was a machine-
generated text or not. They conducted this research by using RoBERTa-base with
125 million parameters and RoBERTa-large with 356 million parameters as the
foundation for their sequence classifier. RoBERTa, distinct from GPT-2 in
terms of architecture and tokenizer, is a masked and non-generative language
model. They achieved an accuracy of approximately 95% for this research.

(Kirchenbauer et al., 2023) proposed a watermarking framework for language
models. Watermarking basically means embedding signals in the machine-
generated text so it is not detected by the human eye, but detected
algorithmically and helps to decrease potential harm from these LLM models.
Kirchenbauer tested this technique on an Open Pretrained Transformer (OPT)
model with multi-billion parameters. The authors used an approach of
categorizing watermark tokens into green and red lists for distinct patterns.
This approach helps in use cases like plagiarism check and copyright
protection. To mimic diverse language modeling situations in their dataset,
they extracted random text portions from the news-like section of the C4
dataset (Raffel et al., 2020). Each extracted string has a fixed number of
tokens removed from the end, forming a baseline completion, while the rest of
the tokens serve as the prompt.

On the other hand, on social media applications like Twitter, which are
vulnerable to generating misinformation and fake news, (Kumarage et al., 2023)
authors proposed a novel stylometric detection algorithm to detect AI-
generated tweets that are human-like. Authors use BERT and ensemble approach,
incorporating stylometric features as their baseline study. they test their
models on two datasets: one we made to mimic human-to-AI author changes on a
Twitter user’s timeline and the publicly available TweepFake dataset. They use
stylometric features to analyze text for stylistic signals, categorizing them
into phraseology, punctuation, and linguistic diversity. Using this with a
model like RoBERTa which classifies on top of the extracted features, they
found that this method performs the best with XGBoost compared to logistic
regression and random forest classifiers.

There are also tools that are developed, like (GPTZero, 2023) which is a tool
used for analyzing text, employing two primary metrics, namely perplexity and
burstiness, to differentiate between machine-generated and human text. It
offers a publicly accessible API that generates a confidence score indicating
the likelihood of a given text being machine-generated or not.

We plan to investigate the potential impact of different data sources and
types of machine-generated text on the AUCROC of our model. It may help us
understand the limitations of our approach and identify areas for future
research. By building on the existing work in this field and extending it in
these ways, we hope to contribute to developing more accurate and effective
methods for distinguishing between human-generated and machine-generated text
(ChatGPT). The model can provide a valuable addition to the literature on
identifying fake and synthetic data, as it addresses a specific type of data
generation method and can help improve the accuracy of existing methods. To do
this, we build a custom dataset by combining multiple types of text like
PubMed data, SQuAD, Twitter feed data, Football commentary, and novels.

##  3 Proposed Approach

This section presents a nuanced and multi-faceted approach for discerning
text’s origin, specifically focusing on the distinction between human-
generated and ChatGPT-generated sentences. This requires training and fine-
tuning state-of-the-art NLP models on a diverse dataset of human-generated
text and ChatGPT-generated text. The ultimate goal is to enable effective
moderation and filtering of text content across different platforms and
applications, thus ensuring the safety and integrity of online communication.
The models should be able to accurately identify text generated by ChatGPT
across multiple domains, including Sports, Medical, Twitter reviews, Text
comprehension, and Literature. The research aims to provide insights into the
decision-making process of the models and the characteristics of ChatGPT-
generated text that distinguish it from human-generated text.

###  3.1 Dataset Description

The dataset used in this study is a combination of five different sources of
human-generated text, including Twitter Sentiment, Football Commentary,
Project Gutenberg, PubMedQA, and SQuAD datasets. The purpose of using these
datasets is to improve the model’s ability to understand and analyze both
short-form and long-form text and complex medical jargon.

We split the Football Commentary dataset into 100-word and 150-word sentences,
truncated SQuAD questions to a maximum of 200 words, and extracted 200-word
sentences from the Project Gutenberg dataset to form custom literature
sentences.

The merged human-generated text resulted in 2,534,498 sentences, which were
grouped into categories of 10-200 words with 5-word increments. Normalization
was performed to ensure 16% of sentences in each category, resulting in a
final dataset of 400,015 sentences as shown in figure 1.

To generate machine text, we rephrased the selected sentences using the
GPT-3.5-turbo chatgpt API from OpenAI, with a prompt to ensure that the length
of the rephrased sentence is the same as the original sentence as shown in
Table 1. The total number of rows in our dataset is 800,030, consisting of
both human and machine generated text, with sentences ranging from 10 to 200
words in length. We truncated sentences longer than 200 words and removed
those with fewer than 10 words.

Figure 1: Dataset distribution across different range of lengths Prompt |  ChatGPT Response
---|---
Please rephrase this |  Rephrased sentence
sentence make sure the |  from Chatgpt
words length is equal to |
the given sentence. |
Table 1: Example Generating Rephrase Data.

###  3.2 Model Description

Our methodology encompasses three distinct cases, each tailored to leverage
the strengths of a specific model. In the first case, we employ Support Vector
Machines (SVM) to conduct a meticulous analysis, followed by using RoBERTa-
base in the second case and RoBERTa-large in the third case.

The models should be able to accurately identify text generated by ChatGPT
across multiple domains, including Sports, Medical, Twitter reviews, Text
comprehension, and Literature. The research aims to provide insights into the
decision-making process of the models and the characteristics of ChatGPT-
generated text that distinguish it from human-generated text.

####  3.2.1 SVM

In the case of the SVM classifier, a powerful tool in machine learning, we
employed a radial basis function (RBF) kernel to discern patterns within the
feature space. The RBF kernel is known for capturing complex relationships in
non-linear data, making it particularly well-suited for distinguishing between
human-generated and ChatGPT-generated sentences. We used the TF-IDF (Term
Frequency-Inverse Document Frequency) vectorization technique to represent the
textual data. TF-IDF accounts for the frequency of terms within each document
and considers the importance of terms in the broader corpus. This combination
of the RBF kernel and TF-IDF vectorization ensures that our SVM model is
equipped to handle the intricacies of natural language, capturing both
semantic nuances and contextual variations for accurate and robust
classification.

The choice of the SVM classifier with an RBF kernel and TF-IDF stems from its
status as a well-established and widely-used baseline approach in the domain
of text classification. This approach has been successfully employed in
previous text classification studies, as demonstrated by its use in studies
such as (Das and Chakraborty, 2018).

Figure 2: SVM Pipeline. Figure 3: RoBERTa Pipeline.

It serves as a benchmark against which the performance of more complex models
can be evaluated. In contrast to the RoBERTa models, the SVM-RBF-TF-IDF
approach is lightweight, requiring less time and computational resources for
both training and inference. This characteristic renders it more amenable to
scenarios with resource constraints. Additionally, the SVM model proves to be
robust, showcasing competitive performance compared to other machine learning
models of similar size and complexity in the realm of text classification.

However, it is essential to note that while the SVM model offers efficiency
and robustness, it does not attain state-of-the-art performance levels
achieved by the more advanced RoBERTa models, as detailed in the subsequent
sections of our analysis.

####  3.2.2 RoBERTa-base

In our second case, we employed the RoBERTa-base model, a transformer-based
architecture renowned for its exceptional natural language processing
capabilities. To tailor this pre-trained language model to our specific binary
classification task—distinguishing between human-generated and ChatGPT-
generated sentences we augmented RoBERTa-base with a fully connected (FC)
layer followed by a sigmoid activation function at the end. This modification
allowed us to transform the contextualized representations learned by RoBERTa
into a binary classification decision. The RoBERTa model, with its deep
bidirectional architecture and extensive pre-training on vast corpora,
captures intricate linguistic patterns and semantic relationships, enhancing
our ability to discern nuanced differences in text origins. Adding the
FC+sigmoid layer enables the model to map these representations to a binary
decision space, facilitating accurate classification. This hybrid approach
leverages the strengths of transformer-based language models while tailoring
them to the specific demands of our binary classification task.

The RoBERTa-base model, augmented with an (FC+sigmoid) layer, is chosen for
its excellence in NLP tasks, supported by its deep bidirectional architecture.
It showcases versatility and is successfully utilized in text detection tasks
by studies like (Mitchell et al., 2023) and [DistilBERT]. With 125 million
parameters, RoBERTa-base balances complexity and efficiency compared to
RoBERTa-large (355 million parameters). This choice facilitates an exploration
of model size-performance trade-offs, offering valuable insights into
transformer-based models for binary classification tasks.

The RoBERTa-base model, enhanced with an (FC+sigmoid) layer, proves effective
in our binary classification, nearing state-of-the-art performance observed in
similar studies. Its adeptness in capturing intricate linguistic nuances is
countered by its larger size (125 million parameters), demanding more training
and inference resources. While excelling in accuracy, its computational
complexity introduces challenges in efficiency when compared to the
lightweight SVM model. This highlights the nuanced trade-offs between model
sophistication, performance, and resource requirements within the context of
our classification task.gths of transformer-based language models while
tailoring them to the specific demands of our binary classification task.

####  3.2.3 RoBERTa-large

In our third case, we employed the formidable RoBERTa-large model, a
transformer-based architecture renowned for its extensive depth and superior
natural language understanding capabilities. Augmented with an additional
fully connected (FC) layer and a sigmoid activation function at the end,
RoBERTa-large was tailored for our binary classification task, distinguishing
between human-generated and ChatGPT-generated sentences. With a substantial
parameter count of 355 million, RoBERTa-large surpasses its base counterpart
in both depth and complexity.

RoBERTa-large, enhanced with an appended (FC+sigmoid) layer, represents a
powerful extension of its base counterpart and proves instrumental in our
binary classification task. Acknowledged as a highly proficient transformer
for NLP tasks, this model, with its extensive depth and attention mechanisms,
has been a preferred choice in recent studies addressing machine-generated
text detection tasks, as demonstrated by its implementation in studies such as
(Mitchell et al., 2023) and [DistilBERT]. The RoBERTa-large model, with its
355 million parameters, boasts an intricate understanding of contextual
relationships within textual data, enhancing its ability to discern subtle
distinctions between human-generated and ChatGPT-generated sentences. Notably,
it has exhibited outstanding performance in similar studies, achieving state-
of-the-art results on our dataset.

However, the heightened complexity and larger size of RoBERTa-large introduce
computational challenges, requiring substantial training and inference
resources. This trade-off between enhanced capabilities and increased
computational demands prompts a nuanced consideration of its suitability for
our specific classification task, especially in comparison to the more
lightweight SVM and RoBERTa-base models.

##  4 Experimental Setup

This section outlines the experimental configuration for both the dataset and
the model.

###  4.1 Dataset Experiments

For our experimental approach, we organized our datasets into distinct
training, testing, and validation sets, each comprising sentences ranging in
length from 10 to 200 words. To ensure a granular examination of model
performance across different sentence lengths, we further divided the data
within each set into specific ranges, such as 10-14 words, 15-19 words, and so
on.

This stratification allowed us to scrutinize the models’ proficiency in
handling varying sentence lengths, offering insights into their adaptability
across a spectrum of linguistic contexts. By systematically categorizing the
data based on sentence length ranges, our experimental design aimed to
comprehensively evaluate model robustness and effectiveness in capturing
nuances across diverse sentence lengths.

###  4.2 Model Experiments

Furthermore, to comprehensively assess the discriminatory power of each model
across different sentence length ranges, we recorded the Area Under the
Receiver Operating Characteristic curve (AUC-ROC) for each specific range and
the cumulative performance across all ranges. This recording strategy allowed
us to capture the variations in model performance at distinct sentence lengths
and derive an aggregate measure of effectiveness.

The AUC-ROC analysis is a robust metric, offering a holistic understanding of
each model’s ability to discriminate between human-generated and ChatGPT-
generated sentences within the specified length categories. Combining results
across all ranges, our evaluation framework thoroughly examines the models’
overall discriminatory performance, contributing valuable insights to
sentence-length-specific classification.

##  5 Results and Discussion

Figure 4: Performance of SVM over texts of different lengths. Figure 5:
Performance of RoBERTa-base over texts of different lengths. Figure 6:
Performance of RoBERTa-large over texts of different lengths.

In our first experimental results in table 2, we present the AUC-ROC scores
for different models across various datasets, including training (Train),
testing (Test), and validation (Val) sets. The SVM model demonstrates strong
discriminatory performance with AUC-ROC scores on the Train, Test, and Val
sets. Moving to transformer-based models, RoBERTa-base exhibits elevated
performance compared to SVM, while RoBERTa-large further enhances this
performance across the Train, Test, and Val sets. These scores provide a
quantitative overview of each model’s effectiveness in distinguishing between
human-generated and ChatGPT-generated sentences across different datasets,
highlighting the robust discriminatory capabilities of the RoBERTa models.

Model | Train | Test | Val
---|---|---|---
SVM | 91.83% | 84.81% | 84.73%
RoBERTa-base | 96.85% | 95.24% | 95.53%
RoBERTa-large | 97.11% | 95.53% | 95.76%
Table 2: AUC-ROC comparison for models. Range | SVM | RoBERTa- | RoBERTa-
---|---|---|---
|  | base | large
10-14 | 0.799119 | 0.932474 | 0.934662
15-19 | 0.826536 | 0.945634 | 0.949327
20-24 | 0.854538 | 0.954775 | 0.957018
25-29 | 0.884957 | 0.960358 | 0.964959
30-34 | 0.87853 | 0.96152 | 0.970518
35-39 | 0.839002 | 0.95069 | 0.952666
40-49 | 0.85617 | 0.953383 | 0.956988
50-59 | 0.839941 | 0.96173 | 0.969639
60-69 | 0.869463 | 0.972758 | 0.973368
70-79 | 0.82854 | 0.971184 | 0.972567
80-89 | 0.807112 | 0.977258 | 0.982704
90-99 | 0.868776 | 0.985965 | 0.991071
100-109 | 0.951641 | 0.997299 | 0.999021
109-119 | 0.813498 | 1 | 0.966667
120-129 | 0.567105 | 0.9375 | 1
130-139 | 0.842593 | 1 | 1
140-159 | 0.924376 | 0.997717 | 0.990476
160-179 | 0.99 | 1 | 1
180-199 | 0.984375 | 1 | 0.5
Table 3: AUCROC scores achieved w.r.t ranges.

For our second experiment, we recorded AUC-ROC scores for each range length of
sentences across three different models: SVM, RoBERTa-base, and RoBERTa-large.
The table 3 displays the AUC-ROC scores for each model within specific
sentence length ranges.

As sentence lengths increase, RoBERTa models consistently outperform SVM,
showcasing robustness in capturing nuanced patterns. In table 4 RoBERTa-large
maintains competitive F1 scores even in longer sentence ranges, indicating its
proficiency across diverse linguistic contexts. These results offer valuable
insights into the models’ efficacy at different scales of textual complexity,
emphasizing the performance variations observed across various sentence length
intervals.

These scores provide a detailed breakdown of each model’s performance within
distinct sentence length intervals, offering valuable insights into their
discriminatory abilities across varying linguistic contexts.

Range | SVM | RoB-base | RoB-large
---|---|---|---
10-14 | 0.774232 | 0.921676 | 0.922112
15-19 | 0.824198 | 0.94617 | 0.948374
20-24 | 0.854733 | 0.9543 | 0.954323
25-29 | 0.875496 | 0.954439 | 0.956495
30-34 | 0.901613 | 0.968602 | 0.972937
35-39 | 0.869093 | 0.95452 | 0.954179
40-49 | 0.866435 | 0.952179 | 0.957542
50-59 | 0.875376 | 0.966087 | 0.973392
60-69 | 0.921739 | 0.980781 | 0.978584
70-79 | 0.941176 | 0.99111 | 0.993544
80-89 | 0.954955 | 0.996504 | 0.997619
90-99 | 0.963504 | 0.998038 | 0.998717
100-109 | 0.884438 | 0.995527 | 0.9665
109-119 | 0.963151 | 1 | 0.998805
120-129 | 0.959459 | 0.998361 | 0.996774
130-139 | 0.9642 | 1 | 1
140-159 | 0.760638 | 0.887805 | 0.630081
160-179 | 0.989899 | 1 | 1
180-199 | 0.984127 | 1 | 0.989247
Table 4: F1 scores achieved w.r.t ranges.

##  6 Conclusion and Future Work

In our research, we effectively create a well-constructed labeled dataset
encompassing texts originated by both humans and ChatGPT (GPT-3.5 turbo) from
five distinct sources. The sentences in the dataset exhibit a varied length
range, spanning from 10 to 200 words. Additionally, we have designed and
trained multiple classifiers to differentiate between texts generated by
humans and ChatGPT.

In the future, when creating datasets, we can aim to add a wider range of text
from more sources. This approach will make the data more versatile, suitable
for different fields, and capable of handling texts of different lengths. To
improve the results of classification, progress can be achieved by developing
and training more advanced models. Considering the increasing development of
Language Model Models (LLMs), our work can be expanded to incorporate all
noteworthy LLMs, such as Llama, Orca, Falcon, and Palm, for both dataset
creation and detection purposes. For further model improvement, we can design
a system to identify phrases within document sentences, aiming to distinguish
between machine-generated and human-generated text. Finally, exploring zero-
shot and one-shot learning systems like Mitchell et al. (2023) can eventually
aid in saving resources and time needed for training complex classifiers.

## References

  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901.
  * Chakraborty et al. (2023) Souradip Chakraborty, Amrit Singh Bedi, Sicheng Zhu, Bang An, Dinesh Manocha, and Furong Huang. 2023.  On the possibilities of ai-generated text detection.  _arXiv preprint arXiv:2304.04736_.
  * Crothers et al. (2023) Evan Crothers, Nathalie Japkowicz, and Herna L Viktor. 2023.  Machine-generated text: A comprehensive survey of threat models and detection methods.  _IEEE Access_.
  * Das and Chakraborty (2018) Bijoyan Das and Sarit Chakraborty. 2018.  An improved text sentiment classification model using tf-idf and next word negation.  _arXiv preprint arXiv:1806.06407_.
  * GPTZero (2023) GPTZero. 2023.  GPTZero Website.
  * Guo et al. (2023) Biyang Guo, Xin Zhang, Ziyuan Wang, Minqi Jiang, Jinran Nie, Yuxuan Ding, Jianwei Yue, and Yupeng Wu. 2023.  How close is chatgpt to human experts? comparison corpus, evaluation, and detection.  _arXiv preprint arXiv:2301.07597_.
  * Kirchenbauer et al. (2023) John Kirchenbauer, Jonas Geiping, Yuxin Wen, Jonathan Katz, Ian Miers, and Tom Goldstein. 2023.  A watermark for large language models.  _arXiv preprint arXiv:2301.10226_.
  * Kumarage et al. (2023) Tharindu Kumarage, Joshua Garland, Amrita Bhattacharjee, Kirill Trapeznikov, Scott Ruston, and Huan Liu. 2023.  Stylometric detection of ai-generated text in twitter timelines.  _arXiv preprint arXiv:2303.03697_.
  * Mitchell et al. (2023) Eric Mitchell, Yoonho Lee, Alexander Khazatsky, Christopher D Manning, and Chelsea Finn. 2023.  Detectgpt: Zero-shot machine-generated text detection using probability curvature.  _arXiv preprint arXiv:2301.11305_.
  * Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. 2019.  Language models are unsupervised multitask learners.  _OpenAI blog_ , 1(8):9.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _The Journal of Machine Learning Research_ , 21(1):5485–5551.
  * Solaiman et al. (2019) Irene Solaiman, Miles Brundage, Jack Clark, Amanda Askell, Ariel Herbert-Voss, Jeff Wu, Alec Radford, Gretchen Krueger, Jong Wook Kim, Sarah Kreps, et al. 2019.  Release strategies and the social impacts of language models.  _arXiv preprint arXiv:1908.09203_.


# Can $1/N_{c}$ corrections destroy the saturation of dipole densities?

Eugene Levin<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Particle Physics, Tel Aviv University, Tel Aviv 69978, Israel  Departemento de
Física, Universidad Técnica Federico Santa María, and Centro Científico-
Tecnológico de Valparaíso, Avda. Espana 1680, Casilla 110-V, Valparaíso, Chile

###### Abstract

In this paper we discuss a well known QCD result: the steep increase of
Green’s function for exchange of $n$ BFKL Pomerons
$G_{nI\\!\\!P}\left(Y\right)\propto\exp\left(\frac{n^{2}}{N^{2}_{c}}\Delta_{\mbox{\tiny
BFKL}}Y\right)$ where $N_{c}$ is the number of colours ,Y is the rapidity and
$\Delta_{\mbox{\tiny BFKL}}$ is the intercept of the BFKL Pomeron. We consider
this problem in the framework of the simple Pomeon models in zero transverse
dimensions, which have two advantages :(i) they allow to take into account all
shadowing corrections, including the summation of the Pomeron loops and (ii)
they have the same as in QCD striking increase of
$G_{nI\\!\\!P}\left(Y\right)$. We found that the strength of shadowing
corrections is not enough to stop the increase of the scattering amplitude
with energy in contradiction to the unitarity constraints. Hence, our answer
to the question in the title is positive. .

###### pacs:

13.60.Hb, 12.38.Cy

###### Contents

  1. I Introduction
  2. II Setting the problem
  3. III  The structure of the parton cascade
    1. III.1 BFKL cascade
    2. III.2 Asymptotic solution
    3. III.3 Exact solution
    4. III.4 Energy growth from the general structure of equation.
  4. IV Conclusions

##  I Introduction

The only candidate for the effective theory of high energy QCD is Colour Glass
Condensate(CGC) approach (see Ref.KOLEB  for a review). Two main ideas of CGC:
the saturation of the dipole density and the new dimensional scale ($Q_{s}$),
which increases with energy, have become widely accepted language for
discussing the high energy scattering in QCD. However the CGC approach suffers
several problems. The most known of them is the power-like behaviour of the
scattering amplitude at large impact parametersKW1 ; KW2 ; KW3 ; FIIM  that
violated the Froissart theoremFROI . We have to introduce non-perturbative
corrections at large impact parameters and the embryonic stage of our
understanding of confinement of quarks and gluons does not allow us to come up
with a reasonable theoretical approach to the problem. The second well known
problem is summing of the BFKL Pomeron loops111 The abbreviation BFKL Pomeron
stands for Balitsky, Fadin, Kuraev and Lipatov Pomeron.. This problem is a
technical one, but in spite of intensive work BFKL ; LIP ; LIREV ; LIFT ; GLR
; GLR1 ; MUQI ; MUDI ; Salam ; NAPE ; BART ; BKP ; MV ; MUSA ; KOLE ; BRN ;
BRAUN ; BK ; KOLU ; JIMWLK1 ; JIMWLK2 ; JIMWLK3 ; JIMWLK4 ; JIMWLK5 ; JIMWLK6
; JIMWLK7 ; JIMWLK8 ; AKLL ; KOLU1 ; KOLUD ; BA05 ; SMITH ; KLW ; KLLL1 ;
KLLL2 ; LEN  it is still far away from being solved.This situation makes the
problem one of the principle problems, without solving which we cannot
consider the dilute-dilute and dense-dense parton densities collisions. As has
been recently shownKLLL1 ; KLLL2 , even the Balitsky-Kovchegov (BK) equation,
that governs the dilute-dense parton density scattering (deep inelastic
scattering (DIS) of electron with proton), has to be modified due to
contributions of Pomeron loops.

In this paper we wish to draw the attention of the reader to a different
problem of the CGC approach. In the CGC approach the scattering amplitude does
not exceed the unitarity limit due to shadowing corrections. The Balitsky-
Kovchegov BK  non-linear equation, which sums the ‘fan’ diagrams of
interacting BFKL Pomerons (see Fig. 1-a), generates the amplitude, which tends
to unity (the unitary limit) at high energies. However, it has been shown in
Refs.LRS ; BART0 ; BARTU ; LLS ; LLR  that Green’s function of the exchange of
$n$ BFKL Pomerons does not increase as $\exp\left(n\,\Delta_{\mbox{\tiny
BFKL}}Y\right)$ where $\Delta_{\mbox{\tiny BFKL}}$ is the intercept of the
BFKL Pomeron and $Y$ is the rapidity of two colliding dipoles. It turns out
that Green’s function grows as
$\exp\left(\frac{n^{2}}{N^{2}_{c}}\Delta_{\mbox{\tiny BFKL}}Y\right)$ ($N_{c}$
is the number of colours) and this increase cannot be suppressed by the
shadowing corrections 222 Actually, in Refs.LRS ; BART0 ; BARTU ; LLS ; LLR
the exchange of $n$-BFKL Pomerons were considered in the double log
approximation, which leads to the anomalous dimension
$\frac{\bar{\alpha}_{S}}{(N^{2}-1)^{2}\omega}n^{4}$ instead of
$\frac{\bar{\alpha}_{S}}{\omega}n^{2}$ for the exchange of $n$ BFKL Pomerons
.. The $N_{c}$ suppression gives rise for the hope that such an increase can
manifest itself only at very high energies, but recently A. Kovner and M. Li
showed that for nucleus-nucleus scattering in the CGC approach there exist
large $1/N^{2}_{c}$ corrections that are larger than the shadowing
suppression333 We thank A, Kovner and M. Li for sharing with us their
finding.. In spite of these results for the CGC approach, the referencesLRS ;
BART0 ; BARTU ; LLS ; LLR  show that these large corrections can be treated in
the BFKL Pomeron calculus if we introduce the vertices of interaction of four
Pomerons ($I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$).

Figure 1: Fig. 1-a: the ‘fan’ diagrams of the Balitsky-Kovchegov non-linear
equation. The wavy lines denote the BFKL Pomerons. The blob shows the triple
Pomeron vertex. Fig. 1b: is the Green’s function of the exchange of $n$ BFKL
Pomerons.

Bearing this in mind we wish to return to discussion of these corrections in
simple, but exactly solvable, two dimensional modelsMUSA ; ACJ ; AAJ ; JEN ;
ABMC ; CLR ; CIAF ; BIT ; RS ; KLremark2 ; SHXI ; KOLEV ; nestor ; LEPRI ; utm
; utmm . In these models we can sum Pomeron loops and we have the same kind of
corrections, which go under slang name of many particle Regge poles DISASTER .
In the next section we will discuss these corrections in details. Here we wish
to tell that the main goal of this paper is to find the scattering amplitude
taking into account the $I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$ vertices in
the simple two dimensional models to get experience what we can expect in QCD
in the CGC approach for these disastrous contributions.

##  II Setting the problem

Green’s function for one Pomeron in the two dimensional model can be viewed as
a sum of the contributions:

$G_{I\\!\\!P}\left(Y\right)\,\,=\,\,\sum_{k=0}^{\infty}\,\frac{\left(\Delta\,Y\right)^{n}}{n!}\,\,=\,\,e^{\Delta\,Y}$
(1)

Eq. (1) sums the ladder diagrams in leading log(1/x) approximation in which
all produced dipoles have a strong ordering in the fractions of total momentum
$x_{i}$:

$1\,\gg\,x_{1}\,\gg\,x_{2}\,\gg\,\dots\,x_{i}\,\gg\,x_{i+1}\,\gg\,\dots\,\gg\,\,x_{n-1}\,\gg\,x_{n}$
(2)

this leads to $1/n!$ in Eq. (1). The two Pomeron exchange is shown in Fig. 2-a
and corresponds to the Feynman diagram of Fig. 2-b in which all dipoles
emitted by the ladder 1 are absorbed by ladder 1’, and all dipoles produced by
ladder 2 are absorbed by ladder 2’. These diagrams lead to the Green’s
function of the exchange of two Pomerons:
$G_{2I\\!\\!P}=G^{2}_{I\\!\\!P}\left(Y\right)$. However, the diagrams in which
the dipole, emitted from ladder 1, will be absorbed by ladder 2’ are not small
in the two dimensional models. We are going to call these diagrams “switch
diagrams" , using the terminology suggested in Refs.LRS ; BART0 ; BARTU ; LLS
; LLR . Indeed, after first exchange ladder 1 and 2’ will give the Pomeron
exchange, leading to the diagram of Fig. 2-d. Since at given rapidity we have
two ‘switch diagrams: dipole emitted between ladders 1 and 2’ and the dipole
emitted between ladders 2 and 1’ we can obtain the vertex
$I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$ , which is equal to $2\Delta$.

The diagrams of Fig. 2-d can be summed in $\omega$-representation:

$G_{2}\left(\omega\right)\,\,=\,\,\sum_{k=0}^{\infty}\frac{1}{\omega\,-\,2\,\Delta}\left(\frac{2\,\Delta}{\omega\,-\,2\,\Delta}\right)^{k}\,=\,\frac{1}{\omega\,-\,4\,\Delta}$
(3)

Where $1/\left(\omega-2\Delta\right)$ is the contribution of the exchange of
two Pomerons (see Fig. 2-b). Coming back to $Y$ representation one can see
that $G_{2}\left(Y\right)\,\,=\,\,\exp\left(4\,\Delta\,Y\right)$ instead of
$G_{2}\left(Y\right)\,\,=\,\,\exp\left(2\,\Delta\,Y\right)$ which is expected
for two Pomeron exchange.

|
---|---
Figure 2:  Summing the switch diagrams for the exchange of two BFKL Pomerons.
Fig. 2-a: the exchange of two BFKL Pomerons. Fig. 2-b: the exchange of two
BFKL Pomeron in the two dimensional model. Fig. 2-c: the switch diagram for
two ladder exchange in which the produced dipoles from the Pomeron 1 are
absorbed by the Pomeron 2’ and vise versa. Fig. 2-d: the sum of all diagrams
in the two dimensional model that contribute in the Green’s function of the
exchange of two Pomerons in t-channel. Fig. 2-e: the Green’s function of the
exchange of n$n$ Pomerons.The wavy lines denote the exchange of one Pomeron
with the Green’s function $\exp\left(\Delta Y\right)$. The
$I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$ vertex is denoted by blob and equals
to $2\Delta$.

Summing the diagrams of Fig. 2-e we obtain the Green’s function for the
exchange of $n$ Pomerons: $G_{n}\left(Y\right)\,=\,\,\exp\left(n^{2}\Delta
Y\right)$. This Green’s function was derived in Ref.DISASTER  for the parton
approach to high energy scattering. In QCD the structure of the corrections
remain to be the same and the only difference is that the ‘switch’ diagrams as
well the intercept of $n$ Pomeron state have the smallness of the order of
$1/N^{2}_{c}$. Therefore, the $I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$ vertex
in QCD is equal to $\frac{2}{N^{2}_{c}}\,\Delta_{\mbox{\tiny BFKL}}$. leading
to
$G_{n}\left(Y\right)\,=\,\,\exp\left(n\Delta\,Y\,+\,\frac{n(n-1)}{N^{2}_{c}}\Delta
Y\right)$.

##  III  The structure of the parton cascade

The useful tool for discussing the structure of the parton cascade is the
generating function, which has the following formMUDI ; LELU :

$Z\left(Y,u\right)\,\,=\,\,\sum_{n}P_{n}(Y)\,u^{n}$ (4)

where $P_{n}(Y)$ is the probability to find $n$ dipoles with rapidity $Y$. For
the scattering of one dipole with the target we have the following initial and
boundary conditions:

$\mbox{initial
condition:}\,\,Z\left(Y=0,u\right)=u;~{}~{}~{}~{}~{}\mbox{boundary
condition:}\,\,Z\left(Y,u=1\right)\,=\,1;$ (5)

The boundary condition follows from $P_{n}(Y)$ being probabilities.

We need to write the evolution equation for $Z\left(Y,u\right)$ taking into
account two Pometon vertices: $\Gamma_{3I\\!\\!P}=\Delta$ for $I\\!\\!P\to
I\\!\\!P+I\\!\\!P$ and $\Gamma^{2I\\!\\!P}_{2I\\!\\!P}\,=\,2\,\Delta$ for
$I\\!\\!P+I\\!\\!P\to I\\!\\!P+I\\!\\!P$. Fortunately the equation for $Z$ for
such a cascade has been written in Ref.LELU  and it takes the form:

$\frac{\partial\,Z\left(Y,u\right)}{\partial Y}\,\,=\,\,-\Delta
u(1-u)\frac{\partial\,Z\left(Y,u\right)}{\partial
u}\,\,+\,\,\Delta\,u(u-1)\frac{\partial^{2}\,Z\left(Y,u\right)}{\partial
u^{2}}$ (6)

As has been mentioned that Eq. (6) has been suggested in Ref.LELU  , but Eq.
(6) has a remarkable difference in comparison with equations in this
reference: the sign in front of $u^{2}\partial^{2}/\partial\,u^{2}$ is plus,
not minus as in our previous attempts.

Below we discuss a bit different equation:

$\frac{\partial\,Z\left(\tilde{Y},u\right)}{\partial\tilde{Y}}\,\,=\,\,-\
u(1-u)\frac{\partial\,Z\left(\tilde{Y},u\right)}{\partial
u}\,\,+\,\,\kappa\,u(u-1)\frac{\partial^{2}\,Z\left(\tilde{Y},u\right)}{\partial
u^{2}}$ (7)

where $\tilde{Y}=\Delta\,Y$ and factor $\kappa=1/N^{2}_{c}$ takes into account
a suppression for $\Gamma^{2I\\!\\!P}_{2I\\!\\!P}=1/N^{2}_{c}$ in QCD. For two
dimensional models $\kappa=1$.

###  III.1 BFKL cascade

Neglecting $\Gamma^{2I\\!\\!P}_{2I\\!\\!P}$ we obtain the well known equation
for the BFKL cascade.

$\frac{\partial\,Z\left(\tilde{Y},u\right)}{\partial\tilde{Y}}\,\,=\,\,-\
u(1-u)\frac{\partial\,Z\left(\tilde{Y},u\right)}{\partial u}$ (8)

It is instructive to observe that Eq. (8) leads to a non-linear equation for
$Z\left(Y,u\right)$MUDI ; LELU . Indeed, the general solution to Eq. (8) is of
the form $Z(Y,u)=Z(u(Y))$; if we substitute this function into Eq. (8) , the
derivatives $\partial Z/\partial Y$ on the l.h.s. and r.h.s. of Eq. (8)
cancel, and we obtain a differential equation for the function $u(Y)$. Using
the initial condition of Eq. (15) we can re-write Eq. (8) in the form:

$\frac{\partial\,Z\left(\tilde{Y},u\right)}{\partial\tilde{Y}}\,\,=\,\,-\,Z\left(Y,u\right)\,\,+\,\,Z^{2}\left(Y,u\right)$
(9)

Note, that the scattering amplitude in our models is equal to
$N\left(Y\right)\,\,=\,\,1\,\,-\,\,Z\left(Y,1-\gamma\right)$ where $\gamma$ is
the amplitude of the interaction of the dipole with the target at low energy.
For $N\left(Y\right)$ we have the nonlinear equation:

$\frac{\partial\,N\left(\tilde{Y}\right)}{\partial\tilde{Y}}\,\,=\,\,-\,N\left(Y\right)\,\,+\,\,N^{2}\left(Y\right)$
(10)

which is the Balitsky-Kovchegov equationBK  for our simple models.

The solution to Eq. (9) , which satisfies the initial and boundary conditions
of Eq. (15) has the following form:

$Z\left(Y,u\right)\,\,=\,\,\frac{u\,e^{-\Delta\,Y}}{1\,\,+\,\,u\left(e^{-\Delta\,Y}\,\,-\,\,1\right)}$
(11)

One can see that at large values of $Y$ $Z\,\to\,0$ ($N\,\to\,1$). In other
words, the nonlinear corrections, which stem from triple Pomeron interactions
, suppress the increase of the scattering amplitude
($N\propto\,e^{\Delta\,Y}$) and lead to the scattering amplitude which reaches
the unitarity bound. We call this phenomenon the saturation of parton
densities.

###  III.2 Asymptotic solution

The general solution to Eq. (8) has been found in Ref.KOLEV . However, before
applying the developed technique to this particular equation we wish to see a
qualitative changes in the behaviour of $Z$ at large values of $Y$ that stems
from a different sign of $\Gamma^{2I\\!\\!P}_{2I\\!\\!P}$ than in previous
attempts to develop a similar cascade.

For this purpose we are going to find the asymptotic solution at large $Y$
from the following equation:

$\frac{\partial\,Z^{\rm asymp}\left(\tilde{Y},u\right)}{\partial
u}\,\,+\,\,\kappa\frac{\partial^{2}\,Z^{\rm
asymp}\left(\tilde{Y},u\right)}{\partial u^{2}}\,\,=\,\,0$ (12)

It has an obvious solution

$Z^{\rm
asymp}\left(u\right)=\,\,\frac{1\,\,-\,\,e^{-\frac{u}{\kappa}}}{1\,\,-\,\,e^{-\frac{1}{\kappa}}},$
(13)

which satisfies the boundary condition : $Z^{\rm
asymp}\left(u=1\right)\,\,=\,\,1$ and $Z^{\rm asymp}\left(u\right)\propto u$
at $u\ll 1$. To find, how the solution approaches the asymptotic one, we are
looking for the solution in the form:
$Z\left(Y,u\right)\,\,=\,\,\left(1\,-\,\,e^{-\frac{u}{\kappa}-\phi(Y,u)}\right)/\left(1\,\,-\,\,e^{-\frac{1}{\kappa}}\right)$,
assuming that $\phi^{\prime\prime}_{uu}$ and $\phi^{\prime 2}_{u}$ are small.
The equation for $\phi$ takes the form:

$\phi^{\prime}_{\tilde{Y}}\left(\tilde{Y},u\right)\,\,=\,\,u\,(1-u)\,\phi^{\prime}_{u}\left(\tilde{Y},u\right)$
(14)

with the initial condition at Y=0:

$\phi\left(\tilde{Y}=0,u\right)\,\,=\,\,\left(\frac{1}{C}\,-\,\frac{1}{\kappa}\right)\,u(1-u)~{}~{}~{}\mbox{with}~{}~{}C=\frac{1}{1-\exp\left(1/\kappa\right)}$
(15)

which corresponds to $Z\left(Y=0,u\right)=u$ and $Z\left(Y=0,u=1\right)=1$. A
general solution to Eq. (14) has the form:

$\phi\left(Y,u\right)\,\,=\,\,\Phi\left(\tilde{Y}+\ln\frac{u}{1-u}\right)$
(16)

where the arbitrary function $\Phi$ has to be found from the initial
conditions of Eq. (15).

Finally, the solution takes the form:

$Z\left(Y,u\right)\,\,=\,\,\frac{1}{1-e^{-1/\kappa}}\Bigg{(}1-\exp\left(-\frac{u}{\kappa}-\left((1-\exp\left(-1/\kappa\right))-\frac{1}{\kappa}\right)\frac{u(1-u)\,e^{\tilde{Y}}}{(1-u+ue^{\tilde{Y}})^{2}}\right)\Bigg{)}$
(17)

Fig. 3 shows that this solution approaches the unitarity limit, giving a hope
that the shadowing corrections could suppress the increase of the Green’s
function of the $n$-Pomerons in $t$-channel. However, this figure shows that
$Z\left(\tilde{Y},u\right)\,<\,0$ in the limited range of $u$, where this
solution violates the unitarity constraints. Hence, we need to find exact
solution for the final conclusions.

|
---|---
Fig. 3-a | Fig. 3-b
Figure 3: Graphic form of the solution of Eq. (17).Fig. 3-a: $\kappa=1$. Fig.
3-b: $\kappa=1/8$. $\Delta=0.2$.

###  III.3 Exact solution

Fortunately, the general solution to Eq. (8) has been found in Ref.KOLEV . We
discuss this solution here, repeating all steps of Ref.KOLEV  and paying
special attention to the sign in the r.h.s. of Eq. (8). First, we consider
$Z\left(Y,u\right)$ in $\omega$ representation:

$Z\left(Y,u\right)\,\,=\,\,\int\limits^{\epsilon+i\infty}_{\epsilon-i\infty}\frac{d\omega}{2\,\pi\,i}e^{\omega\,\tilde{Y}}z\left(\omega,u\right)$
(18)

For $z\left(\omega,u\right)$ Eq. (8) takes the form:

$\omega\,z\left(\omega,u\right)\,\,=\,\,-u(1-u)\Big{(}\frac{\partial\,z\left(\omega,u\right)}{\partial\,u}\,\,+\,\,\kappa\frac{\partial^{2}\,z\left(\omega,u\right)}{\partial\,u^{2}}\Big{)}$
(19)

Plugging
$z\left(\omega,u\right)\,\,=\,\,\exp\left(-\frac{u}{2\kappa}\right){\tilde{z}}\left(\omega,u\right)$
in Eq. (19) we obtain:

$4\,\kappa\,\omega\,{\tilde{z}}\left(\omega,u\right)\,\,=\,\,-u(1-u)\Big{(}-{\tilde{z}}\left(\omega,u\right)\,\,+\,\,4\kappa^{2}\,{\tilde{z}}^{\prime\prime}_{uu}\left(\omega,u\right)\Big{)}$
(20)

Introducing $\tilde{z}\left(\omega,u\right)=u(1-u){\cal
G}\left(\omega,u\right)=\frac{1-v^{2}}{4}{\cal G}\left(\omega,v\right)$ with
$1-2u=v$, we can rewrite Eq. (20) in the following form:

$\displaystyle 4\,\kappa\,\omega{\cal G}\left(\omega,u\right)\,\,$
$\displaystyle=$ $\displaystyle\,\,u(1-u){\cal
G}\left(\omega,u\right)\,\,-\,\,4\kappa^{2}\left(u(1-u)\,{\cal
G}\left(\omega,u\right)\right)^{\prime\prime}_{uu}$ (21) $\displaystyle=$
$\displaystyle\left(u(1-u)+8\kappa^{2}\right){\cal
G}\left(\omega,u\right)\,+\,8\,\kappa^{2}(2u-1){\cal
G}^{\prime}_{u}\left(\omega,u\right)\,\,-\,\,4\kappa^{2}u(1-u){\cal
G}^{\prime\prime}_{uu}\left(\omega,u\right)$

Using $v$ we have

$(1-v^{2})\,{\cal G}^{\prime\prime}_{vv}\left(\omega,v\right)\,-\,4\,v\,{\cal
G}^{\prime}_{v}\left(\omega,v\right)\,+\,\Bigg{\\{}-2+\frac{1}{\kappa}\omega-\frac{1-v^{2}}{4\,\kappa^{2}}\Bigg{\\}}\,{\cal
G}\left(\omega,v\right)\,\,=\,\,0$ (22)

In Ref.KOLEV  it is noted, that function ${\cal G}\left(\omega,u\right)$ are
intimately related to prolate spheroidal wave functions
$S_{n,m}\left(c,v\right)=(1-v^{2})^{\frac{m}{2}}\,{\cal
G}\left(c,v\right)$ABST ; SPHF , which satisfy the following equation:

$\frac{d}{dv}\Bigg{(}(1-v^{2})\frac{d\,S_{n,m}\left(c,v\right)}{d\,v}\Bigg{)}\,\,+\,\,\left(\lambda^{m}_{n}\,-\,c^{2}\,v^{2}-\frac{m}{1-v^{2}}\right)S_{n,m}\left(c,v\right)\,\,=\,\,0$
(23)

with $n$ and $m$ being integer numbers.

For functions ${\cal G}\left(c,v\right)$ Eq. (23) takes the form:

$(1-v^{2})\,{\cal
G}^{\prime\prime}_{vv}\left(c,v\right)\,\,-\,\,2(m+1)v\,{\cal
G}^{\prime}_{v}\left(c,v\right)\,\,+\,\,\Big{\\{}\lambda^{m}_{n}\,-\,c^{2}\,v^{2}-m(m+1)\Big{\\}}{\cal
G}\left(c,v\right)\,\,=\,\,0$ (24)

Comparing Eq. (22) and Eq. (24) we obtain that

$m=1;~{}~{}c^{2}=-\frac{1}{4\,\kappa^{2}};~{}~{}\omega_{n}\,\,=\,\,\kappa\,\lambda^{1}_{n}+\frac{1}{4\,\kappa};$
(25)

Hence, the set of the eigenfunctions for the generating function
$Z\left(\omega,u\right)$ has the form:

$Z_{n}\left(v\right)\,\,=\,\,\,\sqrt{1-v^{2}}\,e^{-\frac{1-v}{4\,\kappa}}\,S_{n,1}\left(\frac{i}{2\,\kappa},v\right)$
(26)

Going back to rapidity representation, one can see that

$Z_{n}\left(Y,v\right)\,\,=\,\,\,e^{\omega_{n}\,\tilde{Y}}\,\sqrt{1-v^{2}}\,e^{-\frac{1-v}{4\,\kappa}}\,S_{n,1}\left(\frac{i}{2\,\kappa},v\right)\,=\,\,e^{\kappa\lambda^{1}_{n}\,\tilde{Y}}\sqrt{1-v^{2}}\,e^{-\frac{1-v}{4\,\kappa}}\,S_{n,1}\left(\frac{i}{2\,\kappa},v\right)$
(27)

Therefore, one can see that each eigenfunction increases as function of
$\tilde{Y}$ (see Table I for the values of $\omega_{n}$).

$\omega$ | $\kappa$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$\omega_{n}$ | 8 ($N_{c}=3$) | 1.636 | 1.766 | 2.616 | 3.579 | 4.8
$\omega_{n}$ | 1 ($N_{c}=1$) | 2.2 | 6.14 | 12.133 | 20.113 | 30.128
Table 1: First five eigenvalues $\omega_{n}$ of
$\omega_{n}=\kappa\,\lambda^{1}_{n}$.

Generally speaking the generating function is equal to

$Z\left(Y,v\right)=Z^{\rm asymp}\left(u\right)\,+\,\sum_{n=1}^{\infty}{\rm
C_{n}}e^{\omega_{n}\,\tilde{Y}}\,\sqrt{1-v^{2}}\,e^{-\frac{1-v}{4\,\kappa}}\,S_{n,1}\left(\frac{i}{2\,\kappa},v\right)$
(28)

where

${\rm C_{n}}\,\,=\,\,\int^{1}_{-1}dv\Bigg{(}Z\left(Y=0,u\right)-Z^{\rm
asymp}\left(u\right)\Bigg{)}\,\frac{e^{\frac{1-v}{4\,\kappa}}}{\sqrt{1-v^{2}}}\,\,\frac{S_{n,1}\left(\frac{i}{2\,\kappa},v\right)}{||S_{n,1}\left(\frac{i}{2\,\kappa},v\right)||}$
(29)

with

$||S_{n,1}\left(\frac{i}{2\,\kappa},v\right)||\,\,=\,\,\int^{1}_{-1}\\!\\!\\!dv\,|S_{n,1}\left(\frac{i}{2\,\kappa},v\right)|^{2}\,\,=\,\,\frac{2\,n\,(n+1)}{2\,n\,+\,1}$
(30)

$Z^{\rm asymp}\left(u\right)$ is the solution to Eq. (12) which correponds to
the minimal value of $\omega=0$. It is convenient to choose it in the
following form:

$Z^{\rm
asymp}\left(u\right)\,\,=\,\,\frac{1-e^{-\frac{u}{\kappa}}}{1-e^{-\frac{1}{\kappa}}}$
(31)

This solution gives $Z^{\rm asymp}\left(u=1\right)=1$ and $Z^{\rm
asymp}\left(u=0\right)=0$. Hence the difference $Z\left(Y=0,u\right)-Z^{\rm
asymp}\left(u\right)$ satisfies the following boundary condition:

$Z\left(Y=0,u=0\right)-Z^{\rm
asymp}\left(u=0\right)=0;~{}~{}~{}Z\left(Y=0,u=1\right)-Z^{\rm
asymp}\left(u=1\right)=0;$ (32)

This difference can be expanded in the series of Eq. (28) since functions
$S_{n,1}\left(\frac{i}{2\,\kappa},v\right)$ satisfy the boundary conditions of
Eq. (32) being equal to zero at $v=\pm 1$444We wish to refer to Refs.KOLEV ;
POLY  for more detailed analysis of solutions to since Eq. (8) belong to this
class.

From Eq. (28) we conclude that $Z\left(Y,u\right)$ increases with energy
(rapidity Y). In other words even in the simple case of the initial condition
of Eq. (15) , which corresponds to the deep inelastic scattering the shadowing
corrections failed to stop the increase of the scattering amplitude.

###  III.4 Energy growth from the general structure of equation.

In this section we demonstrate that the energy increase actually stems from
the general structure of Eq. (8). Indeed, Eq. (19) can be rewritten in the
form of the Sturm-Liouville equation:

$s(u)\,\omega\,Z\left(\omega,u\right)+\frac{d}{d\,u}\Bigg{(}p(u)Z^{\prime}_{u}\left(\omega,u\right)\Bigg{)}\,\,=\,\,0~{}~{}~{}\mbox{with}~{}~{}s(u)=\frac{\kappa}{u(1-u)}e^{\frac{u}{\kappa}}~{}~{}\mbox{and}~{}~{}p(u)=e^{\frac{u}{\kappa}}$
(33)

The Sturm- Liouville equation has the following general featuresKOLEV ; POLY :

  1. 1.

Eq. (33) has infinite set of eigenvalues $\omega_{m}=\lambda_{n}$.
$\lambda_{n}$ monotonically increases with $n$ with $\lambda_{n}\to\infty$ at
large $n$. In our case of Eq. (33) all $\lambda_{n}>0$. The least value of
$\lambda_{n}$ is $\lambda_{0}=0$ which corresponds the asymptotic solution of
Eq. (13).

  2. 2.

The multiplicity of each eigenvalue is equal to 1.

  3. 3.

The eigenfunctions $Z_{n}(u)$ are orthogonal

$\int^{1}_{0}\\!\\!\\!du\,s(u)\,Z_{n}(u)\,Z_{m}(u)\,\,=\,0~{}~{}~{}\mbox{for}~{}~{}n\neq
n$ (34)

  4. 4.

For large $n$

$\lambda_{n}\,\,\frac{\pi^{2}\,n^{2}}{\delta^{2}}~{}~{}\mbox{with}~{}~{}\delta=\int^{1}_{0}\\!\\!\\!du\sqrt{\frac{s(u)}{p(u)}}$
(35)

  5. 5.

For our equation

$\delta=\sqrt{\kappa}\int^{1}_{0}\frac{du}{\sqrt{u(1-u)}}\,=\,\,\sqrt{\kappa}\,\pi$
(36)

giving

$\lambda_{n}=\,\,\kappa\,n^{2}$ (37)

Having Eq. (37) we can conclude, that the generating function
$Z_{n}\left(\tilde{Y},u\right)$ increases with $Y$. This feature is based on
the general features of the Sturm-Liouville equation and can be stated without
finding the exact solution. Since the class of Sturm-Lioville equations is
much wider than our particular equation (see Eq. (8)) , we believe that more
complicate equations in the case of QCD will still have these property,

##  IV Conclusions

The main question, that we have answered in this paper, whether the shadowing
correction can stop the steep increase of Green’s function for the exchange of
the $n$ BFKL Pomerons:
$G_{nI\\!\\!P}\left(Y\right)\propto\exp\left(\frac{n^{2}}{N^{2}_{c}}\Delta_{\mbox{\tiny
BFKL}}Y\right)$. In this paper we considering the simple Pomeron calculus in
zero transverse dimension. This approach has two great advantages: (i) it
takes into account all shadowing corrections including the summation of the
Pomeron loops and (ii) it has the same as in QCD striking increase of
$G_{nI\\!\\!P}\left(Y\right)$. Solving exactly the evolution equation we
demonstrate that the shadowing corrections cannot stop the increase of
scattering amplitude which violates the unitarity constraints. Hence, our
answer to the question in the title is positive.

However, another phenomenon could considerably increase the shadowing
corrections: the saturation effects inside the parton cascade. In the two
dimensional Pomeron calculus such corrections have been included in UTM model
MUSA ; BIT ; utm ; utmm . On one hand, we have demonstrated in Ref.utmm  the
scattering amplitude for this model in spite of saturation in the parton
cascade coincides with the BFKL cascades at high energies. On the other hand,
we need to include $\Gamma^{2I\\!\\!P}_{2I\\!\\!P}$ in this model. It has not
been done and we consider this as a next problem to be solved. If we think
about the theoretical realization of the parton model (see for example
Ref.WEZU ), we do not expect that the unitarity would be violated. Hence,
perhaps all our problems stem from the use of the Pomeron calculus and we need
to find the theoretical description beyond this approach.

The violation of unitarity stems from the vertex
$\Gamma^{2I\\!\\!P}_{2I\\!\\!P}>0$ , which also appears in QCD (see Refs.LRS ;
BART0 ; BARTU ; LLS ; LLR  ) with the same sign. We do not believe that the
technical complications, coming with the QCD analysis, could provide the
stronger shadowing than in the simple two dimensional models. However, we are
aware that we have no idea how to sum Pomeron loops in QCD and how they
influence the strength of the shadowing corrections. On the other hand, we
need to consider a possibility to go out of the Pomeron calculus to treat the
high energy amplitude in CGC (see for example Ref.LIEFF ).

Concluding, we believe that CGC approach correctly describes the high energy
interaction at $N_{c}\to\infty$ and we do not have a clue how the shadowing
correction could suppress the growth of the scattering amplitude in $1/N_{c}$
order.

Acknowledgements

We thank our colleagues at Tel Aviv university and UTFSM for encouraging
discussions. Special thanks go A. Kovner, M. Li and M. Lublinsky for
stimulating and encouraging discussions on the subject of this paper. This
research was supported by ANID PIA/APOYO AFB180002 (Chile), and the Tel Aviv
university encouragement grant #5731.

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