# xURLLC IN 6G WITH MESHED RAN

Mohammad Ali Khoshkholghi  Centre for Telecommunications Research, Department
of Engineering, King’s College London, UK  Toktam Mahmoodi  Centre for
Telecommunications Research, Department of Engineering, King’s College London,
UK  Subhankar Pal  Capgemini, India  Subhash Chopra  Capgemini, India  Mayuri
Tendulkar  Capgemini, India  Sandip Sarkar  Capgemini, USA

(NOTE: Corresponding author: Mohammad Ali Khoshkholghi,
<EMAIL_ADDRESS>
)

###### Abstract

5G Ultra-Reliable Low Latency Communications Technology (URLLC) will not be
able to provide extremely reliable low latency services to the complex
networks in 6G. Moreover, URLLC that began with 5G has to be refined and
improved in 6G to provide xURLCC (extreme URLCC) with sub-millisecond latency,
for supporting diverse mission-critical applications. This paper aims to
highlight the importance of peer-to-peer mesh connectivity for services that
require xURLLC. Deploying mesh connectivity among RAN nodes would add
significant value to the current 5G New Radio (5G NR) enabling 6G to increase
flexibility and reliability of the networks while reducing the inherent
latency introduced by the core network. To provide a mesh connectivity in RAN,
the nodes should be able to communicate with each other directly and be
independent from the mobile core network so that data can be directly
exchanged between base stations (gNBs) whereas certain aspects of signalling
procedure including data session establishment will be managed by RAN itself.
In this paper, we introduce several architectural choices for a mesh network
topology that could potentially be crucial to a number of applications. In
addition, three possible options to create mesh connectivity in RAN are
provided, and their pros and cons are discussed in detail.

Keywords – 6G, mesh connectivity, RAN, xURLCC.

## 1\. Introduction

Introduced in 3GPP release 15 to address the requirements of ITU-R M.2083,
ultra-reliable low latency communication is, arguably, the most promising
addition to the 5G capabilities. It has been further enhanced as part of 3GPP
Release-16. This new URLLC wireless connectivity guarantees latency to be
approximately 1 millisecond (ms) with 99.99% reliability and is a complete
game-changer for communications technology in the modern age. With it, we can
conduct remote surgeries, have our cars drive for us, and increase machine
productivity by large-scale factories [1].

With the advent of B5G and 6G, networks will become even more complex. There
will be a network of networks, where many subnetworks roll up to a bigger
network. Machine area networks such as a car area network or a body area
network can have hundreds of sensors over an area of less than 100 meters.
These sensors will need to communicate within 100 microseconds with extreme
high reliability for the operation of that machine system [2]. Making networks
within cars or on robots truly wireless will open a new era for the designers
of those devices as they would no longer need to install lengthy and bulky
cable systems. Also with introduction of non-terrestrial networks, satellite
and High Altitude Platform Systems (HAPS), drone networks will integrate with
terrestrial networks, to create an even more complex network of networks, to
provide extreme coverage also to the remotest parts of the planet. These
challenges are further enhanced by elements such as expectations of node-to-
node or machine-to-machine communications requirements, the uncertainty of
topology, diverse application requirements, backward compatibility, user
equipment resource limitations, and the rapidly increasing number of devices.
These elements exacerbate the technical complications of the implementation of
future B5G/6G networks.

5G URLCC technology will not be able to provide an extremely reliable low
latency service to this complex network of networks in 6G. Moreover, URLLC
that began with 5G has to be refined and improved in 6G to provide xURLCC.
xURLLC promises to offer uninterrupted connectivity for a few of the
interesting new services like remote surgery/patient diagnosis, live reporting
of critical events like natural calamities, live sports events, metaverse
applications, tactile Internet, cloud-based entertainment (VR/AR) and online
gaming [3], [4]. These services impose stringent QoS requirements in terms of
delay ($\leq 1$ ms) with reliability (in terms of error rate) within $10^{-3}$
to $10^{-9}$. Both radio access technology as well as the core needs to work
in tandem to ensure these high reliability and low latency requirements are
met for the respective applications. One of the key components which is not
completely addressed by standards pertains to communication delays and errors
introduced during signalling between RAN and the core. Therefore, this needs a
special focus to ensure those stringent delay and reliability requirements are
met end to end. This paper proposes different options that can be included in
6G to address these aspects to ensure a seamless xURLLC user experience in 6G.

To provide xURLCC service uniformly, for such extreme network and coverage, we
need an advanced network architecture that natively supports: i) mesh
connectivity to minimize the hops to send data between devices, and there is
no need to rely on centralized core network anchor. ii) procedures to avoid
delay due to complex signalling procedures in RAN and core networks.

Hence, we aim at defining network architectures and approaches which make the
mesh connectivity possible in 6G networks while taking into account the
variety of applications. As our contributions, first, we introduce a
particular use case considered in this research named as UE-to-UE use case
with low latency communication (Section 2). Then, given this use case, we
propose several architectural designs to fulfil the diverse service demands
(Section 3). Next, we discuss the general concept of mesh connectivity as well
as investigate the existing mesh connectivity provided by other wireless
technologies (Section 4). Furthermore, we design various architectures with a
focus on the Integrated Access and Backhaul (IAB) technology where mesh
connectivity is enabled among RAN nodes, although these architectures can be
simply apply to regular RAN including base stations (gNBs) each of which
consists of Central Units (CU) and Distributed Units (DU) (Section 5). We also
propose three solution approaches in order to create mesh connectivity in RAN
as i) import core functions into RAN, ii) a peer-to-peer (P2P) RRC-based
connection, and iii) a P2P XnAP-based connection, in Section 6, Section 7,
Section 8, respectively. To the best of our knowledge, this paper is the first
work to devise a mesh-connected RAN for 6G networks in order to support xURLCC
services.

## 2\. Use Case: UE-to-UE Communication With Low Latency

Although significant progress has been made in reducing network latency,
future applications may demand that we make much more progress in this
direction. Fig. 1 depicts the particular use case considered in this paper
where different end-to-end latency targets achievable for UE-to-UE
communication have been presented. The accessible latency for UE-to-UE
communication relies upon the placement of User Plane Function (UPF) anchoring
the Protocol Data Unit (PDU) sessions of UEs. Consider the scenario in Fig. 1
where UPF is located in the remote core network, then data should pass through
the core network (green line) even if the UEs are connected to two
neighbouring gNBS. This may increase the end-to-end latency for the UEs’
communications to tens of ms. In the other case, UPF could be placed in the
aggregation site closer to UEs forming the purple data path where the end-to-
end latency can be decreased up to few ms. Finally, by deploying UPF in the
RAN nodes, a direct communication between gNBs (orange data path) allows even
lower latency targets required for the new applications of 6G networks. This
scenario relies on deploying a mesh connectivity among RAN nodes which is our
main objective in this work.

It is worth mentioning that Dual Connectivity (DC) and Side Link (SL)
communications (via PC5 interface) are other options which could be considered
for UE-to-UE communication. In DC, each UE is connected to two base stations
simultaneously whereas SL deploys a direct communication between two UEs
without the participation of a base station in data transmission [5]. However,
since the main scope of this research is to study the possibility of
establishing mesh connectivity in NR RAN (among base stations), we leave DC
and SL communications for our future works.

Each of these scenarios can be more appropriate for a sort of use cases and
applications with particular requirements, so that we can design different
architectures to support the corresponding services. In addition, deployment
of a multi-connectivity scheme derived from a mesh-based network architecture
creates multiple redundant paths between UEs enabling reliable and stable
communications.

Fig. 1: The reference use case considered in this paper

## 3\. Consideration of Network Architecture

Fig. 2: Different architectural designs for the introduced use case.

In this section we present four architecture formats, which are designed for
different 6G use cases with various requirements in terms of latency,
reliability, data rate and RAN node complexity; this diversity is seen in Fig.
2. Generally, the enhanced mobile broadband (eMBB) services need high
bandwidths, that are not necessarily latency-sensitive. Hence, a high degree
of centralization for tight coordination across gNBs should be provided to
achieve high data rates. Fig. 2(a) shows the appropriate architecture for
these types of services where the core network functions (e.g. Access and
Mobility management Function (AMF), Session Management Function (SMF) and User
Plane Function (UPF)) are co-located with RAN Central Unit User Plane (CU-UP)
and Central Unit Control Plane (CU-CP) functions. We consider RAN functional
split option 7 to centralize most of the functions in the core network. In
option 7 [6], all the higher layers of both the CU-CP and CU-UP are located in
the core, while the access site including distributed unit and radio unit only
host lower functionalities of the physical layer (PHY-L) and Radio Frequency
(RF) function, respectively. DUs are kept as light components decreasing the
RAN node complexity in this architecture. Light components refer to the
functions where most or some of their functionalities are shifted to the core
or CU. Since most of the functions are centralized in the core in this
architecture, the data path as well as signalling messages for the session
management pass through the core (green line in the figure).

We can extend the architecture proposed for the eMBB case further to the
architecture depicted in Fig. 2(b) leveraging RAN-core convergence which is
one of the objectives defined for 6G networks [7]. In this architecture, core
functions and the majority of RAN functionalities are moved to a central cloud
and developed as cloud-native micro-services (Control Plane Micro-services
(CPMS) and User Plane Micro-services (UPMS)). In this architecture, each RAN
node consists of an RU as well as the Lower Layer Function (LLF) as a light
entity including all the air-interface-related RAN functionalities that are
not included in the RU. These micro-services could be shifted from the remote
cloud to an edge node closer to the users as well.

Fig. 2(c) shows another architecture proposed for the use cases where latency
matters but is not vital such as mMTC services. To this end, although the
control plane core functions (AMF and SMF) are kept in the core, the UPF is
co-located with RAN CU functions in the aggregation site considering an
intermediate RAN functional split option (option 2 or 3). In this case,
although the signalling messages still go through the core, data passes
through aggregation site with lower latency than the previous architectures
(purple line in the figure).

The last proposed architecture is shown in Fig. 2(d) for the Ultra-Reliable
and Low Latency Communication (URLLC) use cases. The URLLC users require a
decentralized split, which minimizes the latency experienced by these
services. Hence, most functions are decentralized and situated in the access
site. Low latency and high reliability could be ensured by the increased
degree of decentralization. The core control plane functions are located in an
aggregation site (e.g. edge node) while UPF and RAN functions are
decentralized in the access network. This scenario is based on deploying a
mesh connectivity among the base stations upon which gNBs are able to send
data to each other directly (orange path in the figure). However, signalling
messages still pass through the core functions located in the aggregation
site.

## 4\. Mesh Connectivity in other wireless communication Technologies

In this section, we explain the general concept of mesh connectivity and
explore the existing mesh connectivity provided by other communication
technologies including Bluetooth and WiFi. A mesh network is an interconnected
communication network made up of different devices and nodes (physical
redistribution points which receive and transmit wireless signals). Because of
their decentralized nature, mesh networks can continue to scale almost
endlessly, maintaining signal strength and the ability to send and receive
data with a high degree of reliability. As the number of connected devices
scales rapidly (a predicted growth of almost 5x in a 10-year span), mesh
networks will enable consumers and businesses to connect all their devices
without the need for dedicated hubs. This allows for the proliferation of
networks of connected things. Mesh networks are self-healing, meaning that if
there is any disruption to the connectivity of a certain device, the network
just connects to other devices and the network is not dropped.

Mesh networks are very popular in Wi-Fi and Bluetooth. Bluetooth mesh
networking [8] is a new topology available for Bluetooth Low Energy (LE)
devices that enables many-to-many communications. It’s optimized for creating
large-scale node networks and is ideally suited for building automation,
sensor networks, and asset tracking solutions. It is a reliable method of
sharing information in large networks. Bluetooth mesh operates under one
principle: a flood network. This is where the nodes relay messages in a
certain manner, either in uncontrolled flooding or controlled flooding.
Bluetooth mesh has a message cache that prevents conveying similar messages or
messages relayed recently. It’s more reliable and easier to install, and
enables vendor interoperability, range extension and power saving.

In the context of WiFi, a Wireless Mesh Network (WMN) [9] is a mesh network
created through the connection of Wireless Access Point (WAP) nodes installed
at each network user’s locale. It often consists of mesh clients, mesh routers
and gateways. The networking infrastructure is decentralized and simplified
because each node needs only to transmit data as far as the next node. The
network topology of a wireless mesh network may be full or partial mesh. A
full mesh network means every node communicates with every other node. In a
partial mesh topology, nodes only communicate with nearby nodes. When data is
transmitted between two nodes that do not communicate with each other, data
hops from one node to the next until it reaches the destination. The nodes are
programmed to use adaptive routing algorithms to constantly determine the
optimal route between nodes for data transmission. This networking
architecture provides collaborative, redundant backup technology, which
ensures data security in the event of disk failure. It offers increased
reliability, as each node is connected to several other nodes and, if one
drops out of the network, its neighbours simply find another route and its
scalable.

## 5\. IAB architectural design

Fig. 3: Different architectural designs for IAB.

With 5G millimeter Wave (mmWave) bringing very high speeds and capacities, it
also brings the limitation due to its limited coverage at high frequencies.
So, the carriers need many more cells to reach denser areas. To overcome this
challenge and facilitate faster 5G deployments, 3GPP has standardized a
solution for multi-hop relaying support over 5G NR in Release 16 called
Integrated Access and Backhaul (IAB) that is of particular interest for dense
deployment of street-level radio nodes. IAB architecture [10] allows for
multi-hop backhauling and includes an IAB-donor node which is wire-connected
to the core network and hosts RAN-CU functionalities, along with IAB nodes
connected to IAB-donor node and to each other wirelessly. Each IAB-node
consists of two entities as: i) DU which connects the node to both end users
and downlink IAB nodes, and ii) Mobile Terminal (MT) connecting the node to
uplink IAB-nodes or an IAB-donor node. The structure and functionalities of
entities in the IAB architecture is similar to WiFi. However, mesh
connectivity between gNBs has not yet been explored in the IAB architecture,
unlike WiFi. Therefore, considering the IAB architecture, in this section we
try to find an appropriate approach that enables mesh connectivity.

As discussed in Section 3, based on the placement of the UPF, there might be
different path options for data forwarding between base stations, each could
be efficient for a sort of services. However, to enable the shortest path
communication in order to achieve tight latency requirements of 6G use cases,
a direct connection between gNBs is necessary to create a mesh connectivity in
RAN. In the last architecture introduced in Fig. 2(d), although data can be
sent directly between gNBS, signalling messages still need certain core
network procedures to manage the session establishment. However, in order to
establish a mesh connectivity in RAN, we aim at designing architectures which
handle the direct connection between gNBs without the need for the core
functions. In this section, we propose four different IAB architectures
designed with/without core functions as shown in Fig. 3. It is worth
mentioning that we consider scenarios that an entire call can be contained
inside RAN or a group of RANs cooperating with each other. In other words, our
work considers coreless RAN communication.

Fig. 3(a) shows the first proposed architecture where RAN-CU functions are
moved from an IAB-donor to the core network. This architecture is in line with
the first and second architectural options introduced in Section 3, where most
RAN functionalities are co-located in a core/cloud while keeping RAN nodes as
light entities. In this architecture, both data and signalling messages go
through the core/cloud to serve use cases requiring a high data rate.

The second architecture is depicted in Fig. 3(b). In this option, we import
and adapt all the functionalities of a core network into a RAN-CU located in
IAB-donors (or gNBs) so that each RAN node will be able to connect to another
node independently and without a need to traverse the core network. We will
discuss the details of this approach in Section 6. Hence, in this architecture
neither data nor signalling messages go through the core in case of a direct
communication between nodes. Using this architecture, two IAB-nodes are able
to send data directly to each other while signalling messages pass through a
CU located in the IAB-donor. DUs are light entities in this case.

In the third architecture (Fig. 3(c)) we import the core functionalities
directly into the DUs of IAB-nodes so that they can create a mesh connectivity
among themselves even without passing an IAB-donor for signalling. It can
provide fast connection between IAB nodes and increase the flexibility of IAB
architecture, however, DUs will not be light entities anymore since we should
centralize RAN and core functionalities in DUs which consequently increases
the node complexity and deployment cost.

Although we are able to remove core dependency for creating mesh connectivity
using the second and third architecture, but we need a considerable amount of
modifications on the current 5G NR technology and standards since we should
develop all the required core functions in the RAN nodes. To avoid this, in
the fourth architecture we present a solution upon which we can provide mesh
connectivity among RAN nodes (gNBs or IAB-donors/nodes) without importing core
functionalities into RAN. As seen in Fig. 3(d), a CU including usual RAN-CU
functions is located in the IAB-donor, but there will be direct P2P
connections among RAN nodes (e.g. IAB-nodes) without interference of core
functions. We will discuss this solution in detail in Section 7 and Section 8.

## 6\. Development of core functionalities inside RAN nodes

To deploy mesh connectivity in RAN where nodes are able to communicate with
each other directly without the need for core functions, one approach is to
centralize all the connectivity features within RAN nodes. The core
functionalities can be added to and/or converged with RAN functions co-located
with RAN CU or DU entities. Given the standard user plane and control plane
protocol stacks of RAN nodes [10], one protocol layer named the mesh layer
should be adopted and developed on top of the user plane and control plane so
that it covers all the required connectivity features previously supplied by
the core network in addition to mesh-related features. This layer comprises
several components as follows:

  * •

Access and handover management: this component is implemented on top of the
RRC layer in the control plane protocol stack and is in charge of access
authentication and authorization of UEs, policy enforcement, security context
management and handover management of UEs.

  * •

Resource scheduling and QoS management: this component schedules and manages
the radio resources of RAN nodes while assuring the QoS required by different
services. This layer is developed on top of the XnAP layer in the control
plane protocol stack and the session establishment between RAN nodes will be
handled and managed by this component.

  * •

Dynamic network management: deployed on top of XnAP, this component organizes
and manages the mesh connectivity-related features in a self-organized RAN.

  * •

Data forwarding: this component supports packet routing and forwarding towards
RAN nodes across the network, and is developed on top of the SDAP layer in the
user plane protocol stack.

This approach is one of the most efficient in terms of reduction in delays and
also error rates by collocating key core functions alongside CU/DU and IAB
donors. In addition, this approach can be suitable for the use cases which
require an adaptive RAN where connecting to a remote core or backhaul node is
difficult or costly like rural or remote areas (e.g a maritime use case). On
the contrary this approach introduces complexity in terms of introducing new
components co-located with CU/DU which are currently not part of 3GPP SA work
items.

## 7\. P2P RRC-based Connection

Fig. 4: P2P RRC-based connection: a) direct connection between base stations
or IAB donors; b) direct connection procedure and protocol stacks Fig. 5: P2P
RRC-based connection: signalling

As discussed earlier, the solution introduced in Section 6 is able to create a
coreless mesh-connected RAN. However, in order to deploy this solution, the
current 5G NR standards need to be modified significantly. Therefore, another
solution which provides the same benefits with less modifications is
introduced by which we can take advantage of mesh connectivity in less time
and deployment cost. In this section, we introduce this approach which extends
a vehicular communication approach [11] used to transmit localised V2X
traffic, into the context of IAB architecture. This approach is able to create
direct communication between RAN nodes (gNBs or IAB-donors) without core
network interference using an RRC-based connection as shown in Fig. 4. In this
approach, we install a P2P data path between a pair of nodes to send data
through air interface ($U_{u}$) (Fig. 4(a)), whereas signalling procedure
happens through the RRC layer on top of the control plane protocol stack in
order to establish and manage PDU sessions between pairs.

The procedure of this approach is summarized in Fig. 4(b). Consider a UE (left
one in the figure) aims to communicate with another UE via a direct connection
between gNBs. In this case, a session request from the RRC of the UE is sent
to the RRC of the connected gNB. Then, the RRC of the gNB sends a notification
message to the corresponding gNB and gets the response back from it. The RRC,
then, is able to update and configure a mapping table based on the information
received from other gNBs by which the gNB will direct and transmit the UE’s
data to the destination. Since this approach works independently from the core
functions, a Data Transfer Function (DTF) is developed to perform the
functionalities similar to the UPF in order to transmit data through a P2P
path, such as application detection, packet routing and forwarding, per-flow
QoS handling, and traffic usage reporting. As can be seen, two new components
(DTF and mapping table) are adopted as a sublayer on top of the user plane
protocol stack which helps to transmit data directly between gNBs without the
need to use the core network. Fig. 5 shows the signalling procedure of the P2P
RRC-based connection between UEs and gNBs. Once a UE aims to connect to
another UE, it sends an RRC session request to the source gNB including user
ID, service type, destination ID and channel quality. The service type
information provides the QoS requirements to the gNB such as latency and
reliability requirements. The channel quality information provides the
required data plane resources to be allocated for the session being
established. The RRC of the source gNB checks the available user plane
resources and sends back a response including the session ID which is the
identifier of radio bearers allocated to the given service request.

In the next step, the gNB sends a notification for the session request to the
target gNB where its RRC checks the possibility of creating a connection
according to the available resources as well as QoS requirements for the given
service request and sends back the response. Next, the source and target gNBs
perform several tasks including radio bearer establishment, P2P path creation
between UEs through gNBs using the information derived from the mapping table,
allocating the user plane resources for the session, as well as configuring
and updating the mapping table. In the next step, source and target gNBs send
a path configuration message to the corresponding UEs. The path configuration
message includes session ID, radio bearer configuration, path forwarding
information and allocated user plane resources. Radio bearer configuration
provides required information for establishment of radio bearers and path
forwarding information includes mapping table information to transmit the data
over the P2P path. In the last step, UEs send back a completion message
confirming the session establishment. Finally, the data flow will be
transferred over the P2P path.

Since this approach uses air interface ($u_{u}$) to provide mesh connectivity
among RAN nodes, it is more suitable for IAB technology and V2X communication
applications where nodes are connected through ($u_{u}$).

## 8\. P2P XnAP-based Connection

Fig. 6:  Direct connection between base stations or IAB donors via XnAP  Fig.
7:  Signalling procedure of the Xn-based connection

In the previous section, we explained the RRC-based connection in order to
create direct communication between RAN nodes without the need for core
functions. Although this approach needs less modifications on the current 5G
NR technology compared with the first approach introduced in Section 6, still
we need to develop a sublayer on top of the user plane protocol stack as well
as to modify the current $U_{u}$ interface according to the signaling
procedure between gNBs discussed earlier. However, as mentioned before, in
order to accelerate the deployment process and reduce the deployment cost upon
the existing 5G technology, in this section we propose another solution which
tries to create mesh connectivity with the least modifications.

In this approach, as shown in Fig. 6, unlike the RRC-based connection which
establishes a P2P path through air interface ($U_{u}$) between RAN nodes, the
direct connection will happen through the regular and standard interface
connecting gNBs ($X_{n}$ interface) in 5G NR[12]. This approach uses Xn
application layer ($X_{n}AP$) [13] of the $X_{n}$ interface within gNBs to
connect them to each other so that data and signalling messages forward
through the $X_{n}$ interface without traversing the core network. However, to
be able to send data directly between gNBs, we consider a UPF instance is
located in each gNB enabling data forwarding over the Xn. At the current 5G
network architectures, UPFs are located on a central cloud or possibly on edge
nodes closer to the RAN. However, in this approach UPF instances are shifted
towards the RAN in order to provide a direct, fast and reliable connection
between gNBs. Therefore, we do not need to add any sublayer to the existing
user plane protocol stack anymore.

In the current 5G NR technology, the Xn interface has been used for two
purposes: a) user handover between gNBs, and b) in the Non-Standalone (NSA)
architecture to connect gNBs and 4G eNBs. However, in this approach, we adapt
the Xn interface in a way that it can be used for direct communication between
RAN nodes and consequently to enable mesh connectivity in the RAN. For this,
we consider the user handover procedure currently deployed in 5G NR [14], when
a UE moves from one gNB to another; and extend and modify it so that it can be
applied for establishing direct gNB connections. The signalling procedure of
the proposed $X_{n}AP$-based connection is presented in Fig. 7 modifying the
existing handover procedure. In the initial step, XnAP modules in the source
and target gNBs create an active connection using XnSetupRequest and
XnSetupResponse messages, respectively. In addition, the UE is in the RRC-
connected state to send and receive uplink and downlink data to/from the
source gNB. In the next step, the UE aiming to connect another UE through a
direct connection between gNBs, sends an RRC session request to XnAP of the
source gNB over the Xn interface. This session request includes user ID,
service type, destination ID, and the list of requested PDU sessions in the
form of a transparent RRC container.

The XnAP module in the source gNB checks whether there are available resources
to admit the session request and if so, it determines which target gNB should
be connected with for the given request, and then sends a connection request
to the XnAP module of the target gNB. In the next step, the target gNB
performs the admission control and if it decides to admit the request,
allocates the resources to the UE request and sends an RRC acknowledge message
to the source gNB. The acknowledge message is transferred as a transparent RRC
container including the list of admitted and not admitted PDU sessions. Next,
the source and target gNBs send the RRC session configuration messages to the
corresponding users containing the required information for the PDU session
establishment including session ID, bearer information, path forwarding
information and allocated user plane resources. Finally, UEs send a completion
message confirming the session establishment to the gNBs. After successful
establishment of the session and connection of the UEs, data is forwarded over
the Xn interface using UPF instances located in RAN nodes.

This approach uses the Xn interface (the existing interface between gNBs) to
provide mesh connectivity within the RAN. Hence, it could be more suitable for
general, and a wide range of 6G applications, where gNBs connecting to each
other through the Xn will create a mesh-enabled RAN. Table 1 summarizes and
compares three proposed solutions and outlines the benefits and drawbacks or
considerations of each solution.

Table 1: Comparison of three approaches proposed in this work Approach |  Interface
|  Application |  Benefits |  Considerations
---|---|---|---|---
Developing core
functionalities inside RAN nodes
|  $X_{n}$ |  For the use cases which require an adaptive RAN where connecting to remote core or backhaul node is difficult or costly like rural or remote areas (e.g maritime use case)
|  Most efficient in terms of reduction in delays and error rates as well as providing self-organized mesh-connected RAN |  Introduces complexity in terms of introducing new components co-located with CU/DU which are currently not part of 3GPP SA work items
P2P RRC-based Connection
|  $U_{u}$ |  More suitable for IAB technology and V2X communication applications where nodes are connected through $U_{u}$
|  Providing direct wireless connection between RAN nodes using air interface |  Needs to develop a sublayer on top of user plane protocol stack as well as to modify the current $U_{u}$ interface
P2P XnAP-based connection
|  $X_{n}$ |  For general and a wide range of 6G applications where gNBs connect to each other through $X_{n}$
|  Providing mesh connectivity in RAN with the current 5G NR technologies |  Needs to modify the current $X_{n}$ interface

## 9\. Conclusions and Future Work

Recognizing its importance to facilitate faster 5G deployments, in this paper,
we aimed to enable mesh connectivity among RAN nodes. We proposed several
architectural choices where mesh connectivity can be supported in 6G networks.
In order to create a mesh connection between gNBs, they should communicate
with each other in a way that data can be transferred directly between gNBs,
whereas the signalling procedure should be handled within the RAN, without
passing through the core network. To this end, we introduce three approaches
to manage PDU session management inside the RAN, as well as providing the
capability of direct data forwarding among gNBs without core network
interference. In the first approach, the core functionalities should be
imported and adapted within the RAN so that data and signalling messages can
be sent directly between gNBs. In the second approach, an RRC-based connection
is used to make a direct communication between gNBs and then data will be sent
through a P2P local path. Finally, an XnAP-based connection approach is
proposed where it makes possible direct connection among RAN nodes with the
least amount of modifications on the current 5G NR technology. For future work
we aim to implement our work on a real-world testbed to evaluate the
efficiency of the proposed solutions compared with the current URLLC services
provided by 5G networks. In addition, we will deploy mesh connectivity
considering sidelinks connecting UEs as well, in order to support other use
cases that will emerge in future mobile networks.

## References

  * [1] J Cosmas et al.  “View on 5G Architecture: Version 3.0”  In _5GPPP Architecture Working Group_ , 2019
  * [2] Marco Giordani et al.  “Toward 6G networks: Use cases and technologies”  In _IEEE Communications Magazine_ 58.3 IEEE, 2020, pp. 55–61
  * [3] Oliver Holland et al.  “The IEEE 1918.1 “tactile internet” standards working group and its standards”  In _Proceedings of the IEEE_ 107.2 Ieee, 2019, pp. 256–279
  * [4] Kwang Soon Kim et al.  “Ultrareliable and low-latency communication techniques for tactile internet services”  In _Proceedings of the IEEE_ 107.2 IEEE, 2018, pp. 376–393
  * [5] Mehdi Harounabadi et al.  “V2X in 3GPP standardization: NR sidelink in release-16 and beyond”  In _IEEE Communications Standards Magazine_ 5.1 IEEE, 2021, pp. 12–21
  * [6] Line MP Larsen, Aleksandra Checko and Henrik L Christiansen  “A survey of the functional splits proposed for 5G mobile crosshaul networks”  In _IEEE Communications Surveys & Tutorials_ 21.1 IEEE, 2018, pp. 146–172
  * [7] Volker Ziegler et al.  “6G architecture to connect the worlds”  In _IEEE Access_ 8 IEEE, 2020, pp. 173508–173520
  * [8] Martin Woolley  “Bluetooth Mesh Networking”  URL: www.bluetooth.com/wp-content/uploads/2019/03/Mesh-Technology-Overview.pdf
  * [9] Syed Yasmeen Shahdad, Asfia Sabahath and Reshma Parveez  “Architecture, issues and challenges of wireless mesh network”  In _2016 International Conference on Communication and Signal Processing (ICCSP)_ , 2016, pp. 0557–0560  IEEE
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## Authors

Mohammad Ali Khoshkholghi received his Ph.D. degree in computer science from
University Putra Malaysia in 2017, and bachelor’s and master’s of computer
science, in 2007 and 2011, respectively. He is currently a research associate
at the Centre for Telecommunication Research (CTR), Department of Engineering,
King’s College London (KCL), UK. Before joining KCL, he worked as a
postdoctoral research fellow with the DISCO Research Group, Department of
Computer science, Karlstad University, Sweden, from 2018 to 2020. He has also
worked as a researcher and university lecturer within computer science in
industry and academia. He serves as a referee, TPC and editorial board member
for many prestigious journals and conferences. His research interests lie in
the area of edge and cloud computing, 6G networks and machine learning.

Toktam Mahmoodi is a professor of communication engineering and Director of
Centre for Telecommunications Research (CTR) at King’s College London. She
received a B.Sc. degree in electrical engineering from the Sharif University
of Technology, Iran, and a Ph.D. degree in telecommunications from King’s
College London, UK, in 2002 and 2009 respectively. She was a visiting research
scientist with F5 Networks, San Jose, CA, USA, in 2013, a postdoctoral
research associate with the ISN Research Group, Electrical and Electronic
Engineering Department, Imperial College, from 2010 to 2011, and a mobile VCE
researcher, from 2006 to 2009. She has also worked in the mobile and personal
communications industry, from 2002 to 2006. She has contributed to, and led a
number of FP7, H2020, and EPSRC funded projects, advancing mobile and wireless
communication networks. Her research interests include 6G mobile networks,
open networking, critical communication and and low latency networking.

Subhankar Pal is Senior Director and global Innovation leader for the
Intelligent Networks program in Capgemini Engineering. He has close to 22
years of experience in the telecommunication industry. His responsibilities
include technology product incubation, product strategy and roadmap
development, and consulting services offering definition for telecommunication
and related markets. His interest areas include advanced network automation,
optimization and sustainability using cloud native principles and machine
learning for 5G and beyond networks. Prior to joining Capgemini, he worked at
Nokia Networks and C-DOT. Subhankar has extensive experience in speaking at
international conferences and presenting technical papers in various forums.
He has several blogs and whitepapers published in international journals and
event proceedings.

Subhash Chopra has 23 years of professional experience in the product
engineering services and telecommunications industry with a focus on
technology and innovation. He has extensive experience in solution
architecture development, system integration and technical consulting for
complex telecommunication networks and related services. Subhash is associated
with Altran, a Capgemini Group company, since 2001. He is part of the Research
and Innovation Team with a focus on “5G ORAN RIC” and “Industry 4.0”
solutions. His responsibilities include incubating new solutions, research on
new areas related to B5G/6G, developing proof-of-concepts, developing roadmaps
for productization of solutions, and services offering definitions and
technology consulting.

Mayuri Tendulkar received a B.E. degree in electronics engineering from Mumbai
University, India and M.S. degree in electrical engineering,
telecommunications from the University of Maryland Baltimore County, USA. She
is currently part of the Capgemini Global Innovation Team Connectivity Group
and working on intelligent network solutions with 5G O-RAN RIC and NWDAF. Her
research interests include 5G and beyond, network virtualization, network
sustainability, intelligent networking and embedded platforms.

Sandip Sarkar graduated from the Indian Institute of Technology in Kanpur,
India with a bachelor’s degree in technology. He earned his Ph.D. in
electrical engineering from Princeton University, and is currently the lead of
5G strategy for Capgemini. Before that Sandip was a senior director at
Qualcomm the worldwide lead for 5G/LTE. Sarkar was initially hired by Qualcomm
as a senior engineer and worked on Globalstar modem design and Condor
encryption algorithm design. Later, he became the cdma2000 reverse link
systems lead, and subsequently served as the editor for the cdma2000 physical
layer and band class specifications. Then, he served as the MIMO lead for
3GPP, the UMB and LTE standards delegate for Qualcomm, and the editor for UMB
physical layer specifications as well. He has served as a chairman of the
physical layer text group as well as an editor for 3GPP2 specifications.
Sarkar is a senior member of IEEE, and holds more that 300+ worldwide patents.


* [GM95] J.P.C. Greenlees, J.P. May, _Generalized Tate Cohomology_ , Memoirs of the American Mathematical Society 543 (1995).
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  * [Lew88] L. G. Lewis, _The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear $\operatorname{\mathbb{Z}}/p$ actions_, Algebraic topology and transformation groups, Volume 1361, pg 53–122, Springer, 1988.
  * [May87] J.P. May, _Characteristic Classes in Borel Cohomology_ , Journal of Pure and Applied Algebra 44 (1987) 287-289, North-Holland
  * [Shu14] M. Schulman, _Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces _, arXiv:1405.1770
  * [Wil19] D. Wilson, _$C_{2}$ -equivariant Homology Operations: Results and Formulas_, arXiv:1905.00058
  * [Zeng17] M. Zeng, _Equivariant Eilenberg-MacLane spectra in cyclic $p$-groups _, arXiv:1710.01769v2

1

Department of Mathematics, University of Chicago
E-mail<EMAIL_ADDRESS>
Website: math.uchicago.edu/$\sim$nickg


# Hadronic $J/\psi$ Regeneration in Pb+Pb Collisions

Joseph Dominicus Lap1 and Berndt Müller1,2 1Department of Physics, Yale
University, New Haven, CT 06511, USA  2Department of Physics, Duke University,
Durham, NC 27708, USA

###### Abstract

We make use of published yields for $D$-mesons and $J/\psi$ in Pb+Pb
collisions at ALICE and a schematic description of the expansion of the hadron
gas to study $D$-meson collisions during the hadronic break-up phase as a
production mechanism for charmonium in relativistic heavy ion collisions at
the Large Hadron Collider. Our calculation is based on chemical reaction rates
with thermal cross sections for an effective meson interaction among
pseudoscalar and vector mesons. We find that due to regeneration, the newly
measured $J/\psi$ yields are consistent with anywhere from roughly $25\%$ to
$110\%$ of the total yield present at hadronization time. This allows us to
bound the fractional abundance of $J/\psi$ immediately after hadronization:
$0.28\leq\frac{dN_{0}^{J/\psi}/dy}{dN_{\rm eq}^{J/\psi}/dy}\leq 1.13$. Our
results are robust under the relaxation of the particulars of our schematic
description and imply that it will be difficult to distinguish regeneration
during hadronization from regeneration by final-state hadronic interactions.
Therefore, regeneration must be taken into account when modelling.

##  1 Introduction

The surprisingly large yield of $J/\psi$ measured in Pb+Pb collisions at the
Large Hadron Collider (LHC) [1, 2] has been interpreted as evidence for the
regeneration of $J/\psi$ mesons during hadronization of the quark-gluon
plasma, which is oversaturated with charm [3, 4, 5]. In the energy regime of
the LHC ($\sqrt{s_{\rm NN}}=2.76,5.02$ TeV) an abundant number of charm quark
pairs is created during the initial collision of two 208Pb nuclei. Assuming
sufficiently rapid thermalization, these $c$ and $\bar{c}$ quarks become an
integral part of the quark-gluon plasma (QGP) that is created in the collision
and eventually participate in the bulk hadronization of the QGP.

Under the thermal conditions at hadronization ($T_{H}\sim 155$ MeV), the
initially produced number of $c$ and $\bar{c}$ quarks is larger than the
expected thermal yield and can be characterized by a fugacity factor
$\lambda_{c}\approx 30$ [6]. In the statistical hadronization model for charm
(SHMc) this overpopulation of charm quarks results in a strong enhancement in
the number of emitted charm quark containing hadrons, with $D$-mesons being
enhanced by a factor $\lambda_{c}$ and charmonium states by a factor
$\lambda_{c}^{2}$.

While the $D$-meson and $J/\psi$ yields predicted by the SHMc agree within
uncertainties with the measured yields [6, 7, 8], $J/\psi$ production by
coalescence of $c$ and $\bar{c}$ quarks at the QGP phase boundary may not the
sole mechanism contributing to secondary $J/\psi$ production. For example,
$J/\psi$ may be formed by recombination of $c$ and $\bar{c}$ quarks inside the
QGP [4] once the temperature falls before the $J/\psi$ melting point
(approximately $1.5T_{c}$ [9]). Charmonium can also be created by reactions
among open charm hadrons in the late hadronic gas phase [10, 11, 12, 13].

This means that the measured $J/\psi$ yield in the final state will be the
result of at least four different production mechanisms: Primordial formation
by gluon-gluon fusion, recombination of $c\bar{c}$ quarks within the QGP,
recombination at the QGP-hadron phase boundary, and regeneration from
$D/\bar{D}$-mesons in the hadron phase. In our study here, we explore to what
extent the last of these sources can cloak the combined contributions from the
first three. Is it possible from the data alone to conclude what the
identifiable contribution from mechanisms other than regeneration in the
hadronic phase to the measured yield is? If regeneration from $D/\bar{D}$
alone can establish chemical equilibrium, it is principally impossible to
deduce the presence of earlier sources from the data. Since all previous
studies of hadronic regeneration of $J/\psi$ [10, 11, 12, 13] predate the
availability of LHC data for $D$-meson production [14, 15] and $J/\Psi$
production[8]; we here aim for an updated calculation that takes these data
into account.

It is sometimes argued that after hadronization the chemical reaction rates
among hadrons with open charm are too small to affect the number of produced
$J/\psi$. However, this is not necessarily true as the overpopulation of charm
quarks in the QGP also means that the produced number of mesons with open
charm is much larger than what is expected in thermal equilibrium. This
enhances the rates of $J/\psi$-producing reactions of the type $D+\bar{D}\to
J/\psi+h$, where $h$ is a light hadron, quadratically. Furthermore, for many
hadron states $h$, the regeneration reactions are exothermic and thus not
suppressed as the hadron gas cools.

The total level of hadronic $J/\psi$ regeneration depends on the number of
$D$-mesons present in the hadron gas – which can be measured – and on the
magnitude of the cross sections for the hadronic charm exchange reactions,
which are not known from experiment. This lack of knowledge naturally
introduces a significant uncertainty of the calculated regeneration
probability, which we will explore in Appendix A.

We will first briefly review the broken SU(4) symmetric model for
$D$-meson–$J/\psi$ interactions and the reaction rates derived by Abreu et al.
[16]. We next discuss the rate equation governing $J/\psi$ regeneration and
its initial conditions and describe several schematic expansion models. We
then present our results for the $J/\psi$ yield and discuss their
implications.

##  2 Chemical Reaction Model

###  2.1 $D$-meson–$J/\psi$ Interactions

There exist several models for interactions among hadrons containing $c$ and
$\bar{c}$ quarks.111An alternative example is the valence quark exchange model
[17, 18], which we do not consider here. Here we consider the class of highly-
constrained models [16, 19, 20, 21, 22, 23] that are based on an effective
interaction among pseudoscalar ($P$) and vector ($V_{\mu}$) mesons with broken
SU(4) flavor symmetry. The mesons are organized into multiplets in the adjoint
representation of SU(4) for pseudoscalars:

$P=\begin{pmatrix}P_{+}&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&P_{-}&K^{0}&D^{-}\\\ K^{-}&\bar{K}^{0}&P_{3}&D_{s}^{-}\\\
D^{0}&D^{+}&D_{s}^{+}&-\frac{3}{\sqrt{12}}\eta_{c}\end{pmatrix},$ (1)

with

$\displaystyle P_{\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}},$
$\displaystyle P_{3}$ $\displaystyle=$
$\displaystyle-\frac{2\,\eta}{\sqrt{6}}+\frac{\eta_{c}}{\sqrt{12}},$ (2)

and vector mesons:

$V_{\mu}=\begin{pmatrix}V_{+}&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\
\rho^{-}&V_{-}&K^{0*}&D^{*-}\\\ K^{*-}&\bar{K}^{*0}&V_{3}&D_{s}^{*-}\\\
D^{*0}&D^{*+}&D_{s}^{*+}&-\frac{3}{\sqrt{12}}J/\psi\end{pmatrix}$ (3)

with

$\displaystyle V_{\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{6}}+\frac{J/\psi}{\sqrt{12}},$
$\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle-\frac{2\,\omega}{\sqrt{6}}+\frac{J/\psi}{\sqrt{12}}.$ (4)

Lorentz invariance and SU(4) symmetry then dictate the form of allowed
interactions and determine vertices for the interactions of type PPV, PVV,
VVV, PPPV, PPVV, PVVV, and VVVV. The relevant interactions are:

$\displaystyle\mathcal{L}_{\rm PPV}$ $\displaystyle=$ $\displaystyle-ig_{\rm
PPV}\langle V^{\mu}[P,\partial_{\mu}P]\rangle$ $\displaystyle\mathcal{L}_{\rm
PVV}$ $\displaystyle=$ $\displaystyle g_{\rm
PVV}\epsilon^{\mu\nu\alpha\beta}\langle\partial_{\mu}V_{\nu}\partial_{\alpha}V_{\beta}P\rangle$
$\displaystyle\mathcal{L}_{\rm VVV}$ $\displaystyle=$ $\displaystyle ig_{\rm
VVV}\langle\partial_{\mu}V_{\nu}[V^{\mu},V^{\nu}]\rangle$
$\displaystyle\mathcal{L}_{PPPV}$ $\displaystyle=$ $\displaystyle-ig_{\rm
PPPV}\epsilon^{\mu\nu\alpha\beta}\langle
V_{\mu}(\partial_{\nu}P)(\partial_{\alpha}P)(\partial_{\beta}P)\rangle$
$\displaystyle\mathcal{L}_{\rm PPVV}$ $\displaystyle=$ $\displaystyle g_{\rm
PPVV}\langle PV^{\mu}[V_{\mu},P]\rangle$ $\displaystyle\mathcal{L}_{\rm PVVV}$
$\displaystyle=$ $\displaystyle ig_{\rm
PVVV}\epsilon^{\mu\nu\alpha\beta}\langle
V_{\mu}V_{\nu}V_{\alpha}\partial_{\beta}P\rangle$
$\displaystyle\mathcal{L}_{\rm VVVV}$ $\displaystyle=$ $\displaystyle g_{\rm
VVVV}\langle V^{\mu}V_{\nu}[V^{\mu},V^{\nu}]\rangle$ (5)

The coupling constants for these interactions can be deduced from experimental
data, and where such data are lacking, from SU(4) symmetry [19, 20, 23].

###  2.2 Reaction Rates

Using these interactions one can calculate the isospin-averaged cross-section
for $J/\psi$ absorption processes:

$\sigma_{r}(s)=\frac{1}{64\pi^{2}s}\frac{p_{f}}{p_{i}}\int
d\Omega\langle|\mathcal{M}_{r}(s,\theta)|^{2}\rangle_{S,I},$ (6)

where $\mathcal{M}_{r}$ denotes the invariant amplitude for reaction $r$,
$\sqrt{s}$ is the center of mass energy, $p_{i}$ ($p_{f}$) are the relative
momenta in the initial (final) state of the reaction, and the average is taken
over isospin and spin degrees of freedom weighted by their respective
degeneracies. The matrix elements are usually calculated at lowest order in
the meson coupling. The cross sections for inverse reactions can be obtained
using the detailed balance relation

$\sigma_{(3+4\to 1+2)}=\sigma_{(1+2\to
3+4)}\frac{(2S_{1}+1)(2S_{2}+1)}{(2S_{3}+1)(2S_{4}+1)}\frac{p_{i}^{2}}{p_{f}^{2}}.$
(7)

Abreu et al.[16] obtained estimates for the averaged cross-sections of various
$J/\psi$ production processes. They also calculated the thermal average of the
production cross-sections:

$\langle\sigma_{ab\to cd}v_{ab}\rangle=\frac{\int
d^{3}(p_{a})d^{3}(p_{b})f_{a}(p_{a})f_{b}(p_{b})\sigma_{ab\to cd}v_{ab}}{\int
d^{3}(p_{a})d^{3}(p_{b})f_{a}(p_{a})f_{b}(p_{b})}$ (8)

where $v_{ab}$ is the relative velocity of the incoming particles, and
$f_{i}(p_{i})$ is the temperature dependent Bose-Einstein distribution for
particles of species $i$.

As the reactions are exothermic, the resultant thermally averaged $J/\psi$
production cross sections vary little over the relevant temperature range (90
MeV $\leq T\leq$ 155 MeV). The following values are taken from Figs. 4,5 of
[16] using error bars to constrain any deviations from a constant value:

$\displaystyle\langle\sigma v\rangle(D\bar{D}\to J\Psi+\pi)$ $\displaystyle=$
$\displaystyle.03\pm.01~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D^{*}\bar{D}^{*}\to J\Psi+\pi)$ $\displaystyle=$ $\displaystyle
1~{}{\rm mb}$ $\displaystyle\langle\sigma v\rangle(D^{*}\bar{D}\to J\Psi+\pi)$
$\displaystyle=$ $\displaystyle 3.75\pm.25~{}{\rm mb}$
$\displaystyle\langle\sigma v\rangle(D\bar{D}\to J\Psi+\rho)$ $\displaystyle=$
$\displaystyle.065\pm.01~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D^{*}\bar{D}\to J\Psi+\rho)$ $\displaystyle=$
$\displaystyle.15\pm.05~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D^{*}\bar{D}^{*}\to J\Psi+\rho)$ $\displaystyle=$
$\displaystyle.95\pm.05~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D_{s}\bar{D}\to J/\psi+K)$ $\displaystyle=$
$\displaystyle.25\pm.05~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D_{s}^{*}\bar{D}\to J/\psi+K)$ $\displaystyle=$ $\displaystyle
1.7~{}{\rm mb}$ $\displaystyle\langle\sigma v\rangle(D_{s}\bar{D}^{*}\to
J/\psi+K)$ $\displaystyle=$ $\displaystyle 1.5~{}{\rm mb}$
$\displaystyle\langle\sigma v\rangle(D_{s}^{*}\bar{D}^{*}\to J/\psi+K)$
$\displaystyle=$ $\displaystyle.4~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D_{s}\bar{D}\to J\Psi+K^{*})$ $\displaystyle=$ $\displaystyle 0.3\pm
0.1~{}{\rm mb}$ $\displaystyle\langle\sigma v\rangle(D_{s}^{*}\bar{D}\to
J\Psi+K^{*})$ $\displaystyle=$ $\displaystyle.8~{}{\rm mb}$
$\displaystyle\langle\sigma v\rangle(D_{s}\bar{D}^{*}\to J\Psi+K^{*})$
$\displaystyle=$ $\displaystyle 1.2~{}{\rm mb}$ $\displaystyle\langle\sigma
v\rangle(D_{s}^{*}\bar{D}^{*}\to J\Psi+K^{*})$ $\displaystyle=$ $\displaystyle
4~{}{\rm mb}$

We note that the thermal cross sections estimated by Bratkovskaya et al. [12]
in their phenomenological model agree quite well with those obtained by Abreu
et al. [16] in the effective SU(4) chiral model. Before moving on, it is worth
mentioning that Abreu et al. also considered reactions involving the
$s$-channel $Z_{c}(3900)$ and $Z_{c}(4025)$ resonances and found them to be
small enough to safely ignore.
For the absorption cross-sections we use the full temperature dependent
expressions. However, as we note in Appendix A the final yield is fairly
insensitive to the temperature dependence of the cross-sections.

##  3 Rate Equation

The time evolution of the number of $J/\psi$ particles, $N_{J/\psi}$, is
described by the following rate equation [16]:

$\displaystyle\frac{dN_{J/\psi}(\tau)}{d\tau}$ $\displaystyle=$
$\displaystyle\sum_{p_{1},p_{2},h}\langle\sigma_{p_{1}+p_{2}\to
J/\psi+h}\rangle\frac{N_{p_{1}}(\tau)N_{p_{2}}(\tau)}{V(\tau)}$ (9)
$\displaystyle-\sum_{p_{1},p_{2},h}\langle\sigma_{J/\psi+h\to
p_{1}+p_{2}}\rangle\frac{N_{J/\psi}(\tau)N_{h}(\tau)}{V(\tau)},$

where $N_{p_{1}},N_{p_{2}}$ denote the abundances of the various species of
$D$-mesons and $N_{h}$ denotes the number of light hadrons in the initial
state of the reverse reaction. Here we assumed that the hadrons are evenly
distributed over the density profile of the fireball. This assumption allows
us to write the rate equations for volume integrated numbers of mesons $N_{i}$
instead of local particle densities $n_{i}$. The first term in (9) accounts
for formation reactions; the second term accounts for absorption processes.
Assuming boost invariance – an assumption that is well justified at
midrapidity in heavy ion collisions in the LHC energy range – we will use
particle numbers and volume per unit rapidity, $dN_{i}/dy$ and $dV/dy$, but
denote these simply as $N_{i}$ and $V$.

It is useful to rewrite (9) in the following form:

$\frac{dN_{J/\psi}(\tau)}{d\tau}=\frac{C_{J/\psi}-A_{J/\psi}(\tau)N_{J/\psi}(\tau)}{V(\tau)}$
(10)

where

$\displaystyle C_{J/\psi}$ $\displaystyle=$
$\displaystyle\sum_{p_{1},p_{2},h}\langle\sigma_{p_{1}+p_{2}\to
J/\psi+h}\rangle N_{p_{1}}N_{p_{2}},$ (11) $\displaystyle A_{J/\psi}(\tau)$
$\displaystyle=$ $\displaystyle\sum_{p_{1},p_{2},h}\langle\sigma_{J/\psi+h\to
p_{1}+p_{2}}\rangle(\tau)N_{h}.$ (12)

Because the thermal production cross sections are nearly temperature
independent and the number of $D$-mesons does not appreciably change during
the hadronic expansion stage, $C_{J/\psi}$ remains approximately constant
during the entire hadronic fireball expansion. On the other hand,
$A_{J/\psi}(\tau)$ drops rapidly with time as the hadronic gas expands and
cools, because the thermal $J/\psi$ absorption cross sections decrease rapidly
with falling temperature and the light hadron densities drop inversely with
the volume of the fireball.

The solution for yields as a function of time is then:

$N_{J/\psi}(\tau)=\frac{N_{J/\psi}(\tau_{H})+C_{J/\psi}\int_{\tau_{H}}^{\tau}\frac{\mu(\tau^{\prime})}{V(\tau^{\prime})}d\tau^{\prime}}{\mu(\tau)}$

Where
$\mu(\tau)=\exp\left(\int_{\tau_{H}}^{\tau}\frac{A_{J/\psi}(\tau^{\prime})}{V(\tau^{\prime})}d\tau^{\prime}\right)$.

###  3.1 Initial conditions

We base our investigation on the measured yields $dN_{i}/dy$ for $D$-mesons in
$0-10$% central Pb+Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV (see Table 4
in [15] and Table 3 of [14]). Because the system is almost net-baryon free,
particle and antiparticle yields are assumed to be identical. The $D^{0}$
yield listed in [15] includes $D^{0*}$, and the $D_{s}$ yield in [14] includes
the $D_{s}^{*}$. In order to estimate the separate yields for these species we
assume the yields are in the same proportion as the identified $D^{+}/D^{+*}$
ratio ($0.80+.69-.36$). We also assume that the fractional uncertainty on the
subdivided results is maintained. This gives the following separate yield
estimates:

$\displaystyle N(D^{0})$ $\displaystyle=$ $\displaystyle 3.03^{+.63}_{-.64}$
$\displaystyle N(D^{0*})$ $\displaystyle=$ $\displaystyle 3.79^{+.79}_{-.80}$
$\displaystyle N(D^{+})$ $\displaystyle=$ $\displaystyle 3.04^{+.63}_{-.90}$
$\displaystyle N(D^{*+})$ $\displaystyle=$ $\displaystyle
3.80^{+1.12}_{-1.34}$ $\displaystyle N(D_{s})$ $\displaystyle=$ $\displaystyle
0.84^{+.24}_{-.38}$ $\displaystyle N(D_{s}^{*})$ $\displaystyle=$
$\displaystyle 1.05^{+.35}_{-.47}$ (13)

These numbers include feed-down from multiple short-lived states with
$D$-meson quantum numbers. Estimates indicate that up to 75% of the measured
yields (13) originate from the decay of such states. Since most of the higher
lying $D$-meson states in the Particle Data Book [24] have decay widths
greater than 200 MeV,222the $D_{2}^{*}(2460)$ with a width of 47 MeV is an
exception corresponding to lifetimes less than 1 fm/c, we will here assume
that these feed-down processes are complete by the time when we begin
integrating the rate equation (10). If some of the higher excited states (such
as the $D_{2}^{*}(2460)$) remain populated, they would also contribute to
$J/\psi$ regeneration. While we do not know what the corresponding thermal
cross sections are, it is a reasonable assumption that they would be
comparable to the cross sections (5) for $J/\psi$ production from the low-
lying $D$-meson states. We will therefore ignore this complication and simply
use the yields listed in (13).

Using these values for the $D$-meson yields immediately after hadronization of
the QGP and the thermally averaged cross sections the constant prefactor
$C_{J/\psi}$ in (10) takes the value

$C_{J/\psi}=53^{+28}_{-33}~{}{\rm fm}^{2}.$ (14)

The uncertainty range accounts for the uncertainty in the number of $D$-mesons
formed during hadronization, which enters quadratically into $C_{J/\psi}$.

###  3.2 Expansion Models

The final missing piece in our calculation is the time evolution of the volume
$V(\tau)$ of the fireball per unit rapidity. As the thermal cross sections for
$J/\psi$ production are not known from experiment and have large theoretical
uncertainties, it makes sense to adopt a schematic model for the expansion of
the hadronic system instead of conducting a detailed transport simulation of
its expansion. As already stated, we assume that the hadrons are uniformly
distributed over a large volume at the moment of hadronization and maintain
this uniformity during the expansion with a prescribed velocity profile.

Both Abreu et al. [16] and Andronic et al. [6] model the expanding hadronic
system as a cylindrical fireball, but their schematic models differ in
details. Both models assume a global hadronization proper time $\tau_{H}$, at
which the QGP converts into hadrons. Here we compare the two models in order
to get a sense of how sensitive the produced $J/\psi$ yield is to those
details.

Abreu et al. [16] estimate the fireball volume as a function of the global
proper time $\tau$ by:

$V(\tau)=c\tau\pi R(\tau)^{2},$ (15)

where $R(\tau)$ denotes the expanding transverse radius of the fireball:

$R(\tau)=R_{H}+v_{H}(\tau-\tau_{H})+\frac{a_{H}}{2}(\tau-\tau_{H})^{2}.$ (16)

We adapt the parameter values given in [16] for Au+Au collisions at RHIC to
Pb+Pb collisions at LHC. We choose these values such that they correspond to a
hadronization volume $V(\tau_{H})=4,997\pm 455~{}{\rm fm}^{3}$ [6].

Andronic et al. [6] calculate the hadronization volume at chemical freeze-out
as a one-dimensional integral along the freeze-out contour in the $\tau-r$
plane, where $\tau$ is the local proper time and $r$ denotes the transverse
coordinate:

$V_{H}=2\pi\int_{0}^{R_{H}}dr\,r\,\tau(r)\,u^{\tau}\,[1-\beta(r)\frac{\partial\tau}{\partial
r}].$ (17)

where $u^{\tau}=(1-\beta^{2})^{-1/2}$ is the Lorentz factor associated with
transverse expansion and $\beta(r)$ is the radial velocity. This approach
enables them to study different freeze-out surfaces
$\tau(r)=R_{H}+\int_{0}^{r}dr^{\prime}\beta(r^{\prime})$ with a transverse
flow profile $\beta(r)$. We adopt the scaling model
$\beta(r)=\beta_{H}(r/R_{H})^{n}$ with values $n=0.85$ and $\beta_{H}=0.62$ as
advocated in [6] and $n=0.83$, $\beta_{H}=0.57$ as advocated in [25]. By
requiring that both models have same volume at hadronization we can calculate
a hadronization radius $R_{H}$ and time $\tau_{H}$. The uncertainty of the
volume corresponds to the range $10.15~{}{\rm fm}\leq R_{H}\leq 12.25~{}{\rm
fm}$, which includes the transverse radius $R_{H}=12~{}{\rm fm}$ used in the
Abreu et al. model. Based on this hadronization radius we find a hadronization
time of $9.63~{}{\rm fm}/c\leq\tau_{H}\leq 16.84~{}{\rm fm}/c$ for the Abreu
model (instead of using the values $v_{H}=0.6,a_{H}=.044$ estimated in Ref.
[16] we use the values $v_{H}=0.62,a_{H}=0$ from Andronic et al. [6]). We find
much tighter bounds of $10.85~{}{\rm fm}/c\leq\tau_{H}\leq 11.53~{}{\rm fm}/c$
and $10.15~{}{\rm fm}/c\leq\tau_{H}\leq 10.79~{}{\rm fm}/c$ for the Andronic
et al. [6] and Mazeliauskas et al. [25] volume models respectively.

In order to estimate the freeze-out volume $V_{F}$, we use the relation

$\frac{V_{F}}{V_{H}}\geq\frac{\gamma_{H}}{\gamma_{F}}\left(\frac{T_{H}}{T_{F}}\right)^{3}$
(18)

where $\gamma_{H}$ and $\gamma_{F}$ are the fugacities at hadronization and
freeze-out and equality holds for isentropic expansion (which we’d expect from
a gas of massless particles). As Xu and Ko [26] have shown, the entropy per
particle stays roughly constant during the hadronic expansion, and thus we do
not expect large deviations from equality. In addition the fugacities are for
light quarks and we assume they are approximately unity. Our results are
however insensitive to the assumption of isentropic expansion and unit
fugacities. If one remains agnostic as to the temperature dependence of the
cross sections (thereby remaining agnostic as to the form of the expansion and
fugacities) one obtains consistent results albeit with larger error bars333It
is worth pointing out that we will find that regeneration continues to occur
until $\tau\rightarrow\infty$ rather than ending at the freeze-out time
$\tau_{F}$. Therefore, the value of the freeze-out volume obtained by assuming
equality in (18) does not factor into our main result.. The hadronization
temperature and freeze-out temperature are estimated to be $T_{H}\approx 155$
MeV and $T_{F}\approx 90$ MeV, respectively [27].This yields $17~{}{\rm
fm}/c\lessapprox\tau_{F}\lessapprox 29~{}{\rm fm}/c$ and $V_{F}\approx
25,000~{}{\rm fm}^{3}$.

Normally one evolves the hadronic fireball until the thermal freeze-out time
when the elastic scattering among hadrons becomes ineffective. However, as the
$D$-meson reactions leading to $J/\psi$ production are exothermic and thus
$\langle\sigma v\rangle$ remains constant even after nominal freeze-out, there
is no justification for stopping the calculation at the freeze-out time. The
production of $J/\psi$ merely becomes less common by dilution as the volume
increases. As the integral $\int d\tau/V(\tau)$ is convergent, it makes sense
to simply integrate until $\tau\to\infty$ rather than imposing an artificial
upper time limit.

Figure 1: Comparison of volume evolution post hadronization for Bjorken
freeze-out surface

We find that the physics is not very sensitive to the details of the expansion
models as shown in Fig. 1. For example, we have used the different values for
$n$ and $\beta$ given in [6] and [25] but found that the behaviour is
insensitive to both parameters within the error bars. While we have taken into
account the temperature dependence of the cross sections this required us to
enforce isentropic expansion. However, if one remains agnostic concerning the
details of the expansion process considering instead just the upper and lower
bound of the cross sections, rather than their temperature dependence, one
finds the total yields are also insensitive to this choice, but the error bars
are increased as one would expect.

##  4 Results

For a first exploration of the importance of hadronic regeneration of
$J/\psi$, we make the extreme assumption that no $J/\psi$ is present
immediately after hadronization, i. e. $N_{J/\psi}(\tau_{H})=0$.

Figure 2: Time dependence of hadronic $J/\psi$ production by $D$-mesons. We
note that $D$-meson interactions occurring after the estimated kinetic freeze-
out time, $\sim 20$ fm/$c$, contribute significantly to the total $J/\psi$
yield. The horizontal black line indicates the chemical equilibrium yield.
N.B. These yields are provided without uncertainty due to $D$-meson yields for
legibility.

The time dependence of $N_{J/\psi}$ for the three different evolution models
is shown in Fig. 2. The number of $J/\psi$ converges to an asymptotic value in
both cases, but the yield still increases substantially after the nominal
freeze-out time $\tau_{F}$, and it takes until $\tau\sim 50~{}{\rm fm}/c$ for
the yield to remain constant with time. Even though $D$-meson collisions
becoming increasingly rare as the hadron gas expands, terminating the
calculation at the thermal freeze-out time would underestimate the total
yield.

Figure 2 shows that the final $J/\psi$ yield from $D$-meson reactions starting
from zero at $\tau_{H}$, is $0.047<N_{J/\psi}<0.073$. This is a significant
fraction of the measured equilibrium $J/\psi$ yield of $N_{\rm
exp}(J/\psi)=0.12\pm.017$ [8] and it is reasonable to wonder how much
hadronically produced $J/\psi$ is necessary to arrive at the equilibrium
yield. As such, we now study how the total $J/\psi$ yield changes if we vary
the number of $J/\psi$ assumed to be present immediately after hadronization.
The results are shown in Fig. 3 as a function of the equilibrium fraction of
$J/\psi$ at $\tau_{H}$. As expected, starting from a larger number of $J/\psi$
increases the final $J/\psi$ yield, but it is clear from Figure 3 that a wide
range of initial conditions are compatible with the measured yields. Such an
effect is irrespective of the schematically chosen volume models.

Figure 3: Ratio of total $J/\psi$ yield at $\tau=+\infty$ to equilibrium yield
including the effect of absorption and assuming a fraction of the equilibrium
yield of $J/\psi$ (shown on the abscissa) is produced during hadronization.

##  5 Summary and Discussion

Our calculation demonstrates that we cannot safely ignore the contribution
from hadronic charmonium production processes to the $J/\psi$ yield in heavy-
ion collisions at LHC energies. As such, any calculation of thermal production
of $J/\psi$ must take regeneration by $D$-meson collisions into account. The
central result of our study is that the measured $J/\psi$ yield permits us to
set lower and upper bounds on the fractional abundance of $J/\psi$ immediately
after hadronization:

$0.28\leq\frac{dN_{0}^{J/\psi}/dy}{dN_{\rm eq}^{J/\psi}/dy}\leq 1.13\,.$ (19)

These bounds could be further tightened by improving the expansion model or by
reducing the uncertainty of the measured $D$-meson and $J/\psi$ yields.

We expect our results to underestimate the true yield from hadronic
regeneration of $J/\psi$ as certain production channels have been neglected,
such as $D_{s}+\bar{D}_{s}\to J/\psi+\phi$. A non-uniform distribution of $c$
($\bar{c}$) quarks with greater concentration toward the center of the
collision, as is expected from the transverse distribution of binary nucleon-
nucleon collisions, also tends to increase the overall yield. Finally, it is
unknown what fraction of $D\bar{D}$ collisions result in the formation of
excited charmonium states, especially the $\chi_{1}$, which then would produce
additional $J/\psi$ via feed-down. There is no reason to expect that the
primary ratio $N(\chi_{1})/N(J/\psi)$ in $D$-meson mediated reactions is given
by the ratio obtained in the SHMc.

Since these additional sources of uncertainty tend to enhance the $J/\psi$
yield we surmise that our result is likely an underestimate. We therefore
conclude that $D$-meson reactions contribute substantially to the measured
$J/\psi$ yield in Pb+Pb collisions at the LHC and must be taken into account
in microscopic transport models of the final hadronic stage of the heavy-ion
reaction. As such, it is not possible to conclude that the observed
equilibrium yield of $J/\psi$ mesons provides unambiguous evidence for the
formation of all observed $J/\psi$ during the hadronization transition.

Acknowledgement: We thank A. Andronic, E. Bratkovskaya, P. Braun-Munzinger,
and J. Stachel for valuable comments on a draft of this manuscript. This work
was supported by grant DE-FG02-05ER41367 from the Office of Science of the
U.S. Department of Energy. B.M. acknowledges hospitality and financial support
during a sabbatical stay at Yale University.

## Appendix A: Robustness

In our study of how regeneration obfuscates the degree of the $J/\psi$ yield
we were forced to make some choices: assuming isentropic expansion, and using
an effective model to calculate the production and absorption cross-sections.
One might reasonably wonder how robust this effect is, that is whether it is
an artifact of the assumptions we’ve made. The purpose of this section is to
soothe the skeptical reader by showing that relaxing these assumptions
reproduces the same result albeit with larger error bars.

###  5.1 Isentropic expansion

Can we relax the assumption of isentropic expansion? The only place the
assumption of isentropic expansion factors into the final yield is in
$A_{J/\psi}(\tau)$ which falls off rapidly until freeze-out time (which is
calculated by assuming isentropic expansion) at which point it remains
constant.
If we remain agnostic on how the temperature falls between $\sim 155{\rm MeV}$
and $\sim 90{\rm MeV}$, we can bound $A_{J/\psi}(\tau)$

$155~{}{\rm MeV}\geq T(\tau\geq\tau_{H})\geq 90~{}{\rm MeV}$

This increases the error bars as shown in Fig. 4 and makes the effect even
more pronounced.

Figure 4: Ratio of total $J/\psi$ yield at $\tau=+\infty$ to equilibrium yield
showing how the error bars grow if we relax the isentropic expansion
assumption.

###  5.2 Effective Model

It is quite possible that the cross sections from the effective model are off
by up to a factor of 2. We plot in Fig. 5 a version of Fig. 3 in which we
incorporate this potential error:

Figure 5: Ratio of total $J/\psi$ yield at $\tau=+\infty$ to equilibrium yield
with $C_{J/\psi}^{\prime}=2C_{J/\psi}$ and
$A_{J/\psi}^{\prime}(\tau)=2A_{J/\psi}(\tau)$. Systematic uncertainty assumed
to be $10\%$ like (Figure 3)

If one assumes that the cross-sections are twice as large as the values quoted
in [16], then it is consistent that as much as $100\%$ of total $J/\psi$ yield
comes from hadronic regeneration.

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