# Synchro: Block-generation Protocol to Synchronously Process Cross-shard
Transactions in State Sharding

Takaki Asanuma, Takeshi Miyamae, Yuji Yamaoka T. Asanuma
<EMAIL_ADDRESS>is a researcher at Fujitsu Limited, Japan.T.
Miyamae<EMAIL_ADDRESS>is a senior researcher at Fujitsu
Limited, Japan.Y. Yamaoka<EMAIL_ADDRESS>is a senior project
director at Fujitsu Limited, Japan.

###### Abstract

Traditional blockchains cannot achieve the same transaction throughput as
Web2, so their use cases are limited. Therefore, state sharding has been
proposed to improve transaction throughput by dividing the blockchain network
and managing states and transactions in parallel. However, Nightshade in the
NEAR Protocol, a type of state sharding, provides a rollback protocol to
cancel the generation of blocks containing inconsistent transaction results
because processing cross-shard transactions (CSTXs) in a 2-phase commit may
cause state inconsistency. We present a new attack that interferes with the
generation of new blocks by repeatedly executing CSTXs that certainly causes
state inconsistency, causing continuous rollback. We also propose a block-
generation protocol called Synchro to incorporate all the state changes of
each CSTX into the same block by coordinating the block prior to approving
transactions in each shard. Synchro eliminates the occurrence of the state
inconsistency caused by the CSTXs and the necessity of the rollback protocol.
We use zero-knowledge proof to make Synchro scalable in the global validation
phase. Although the actual overhead of the zero-knowledge proof has not yet
been evaluated, we show that Synchro could achieve the same transaction
throughput as Nightshade theoretically, depending on the future innovations in
zero-knowledge proof techniques.

##  1 Introduction

###  1.1 Background

Issues have emerged with traditional Web2 platforms, such as the concentration
of personal information and profits in the hands of a few social netwworking
service (SNS) companies. Therefore, Web3 technology, based on blockchain and
characterized by its decentralized nature, is attracting increasing attention.
However, traditional blockchains cannot achieve the same transaction
throughput as Web2 SNS. For example, Twitter has historically achieved a
transaction throughput of 143,199 transactions per second (TPS), while
Ethereum, the leading smart contract platform, has only achieved about 13 TPS
(as of 07/13/2023).[1, 2]

Many scalability solutions have been proposed to increase blockchain-
transaction throughput. Layer-1 scalability solutions can be broadly
classified into two categories.

Scale-up Methods
Solana and Algorand increase transaction throughput by requiring higher
hardware requirements, such as for graphics processing units, for
validators.[3, 4] However, the improvement in transaction throughput is also
limited by the limited improvement in hardware computing power.

Scale-out Methods
These methods improve transaction throughput by approving transactions in
parallel. For example, a directed acyclic graph (DAG), used in IOTA and
Avalanche, allows multiple transactions to be approved in parallel because the
blocks are connected in a DAG-like fashion.[5, 6] State sharding, which is
used in Ethereum 2.0 and the NEAR Protocol, applies sharding technology, a
database-partitioning method, to blockchains.[7, 8, 9, 10, 11, 12] State
sharding improves transaction throughput by dividing blockchain states and
validators into subsets called shards and approving transactions in parallel
on several shards.

The transaction throughput of scale-up methods depends on the hardware
performance of the validator. Scale-out methods of DAG parallelize only the
transaction-approval process and does not partition the state management.
State sharding divides the blockchain state, allowing for parallelizing a
theoretically almost unlimited number of transaction approvals.111There are
variants of sharding blockchains that do not split and manage up to state such
as Zilliqa and Elastico.[13, 14] Therefore, this paper focuses on scale-out
methods in terms of their potential concerning transaction throughput.

In state sharding, state inconsistency can occur as a result of failed state
changes. State sharding involves transactions processed across several shards,
called cross-shard transactions (CSTXs).[12, 9, 10] State sharding, such as
Ethereum 2.0 and NEAR Protocol, uses a 2-phase commit (2PC) to process
CSTXs.[10, 8] However, 2PC changes multiple states serially and
asynchronously, which can compromise state consistency.

There are two solutions to this problem. The first is to send an additional
transaction to resolve state inconsistency due to CSTX inconsistency.[15]
However, if the processing of the additional transaction fails, further
additional transactions are required, making the protocol more complex. The
second solution is to roll back the ledger to undo the generation of blocks
containing transactions that do not meet consistency.[8] This solution avoids
the protocol complications associated with further additional transactions.
However, an attack can sabotage block generation by exploiting rollback. A
denial of service (DoS) attack that repeatedly executes CSTX, which generates
state inconsistencies, could prevent new blocks from being generated if
successive rollbacks occur.

###  1.2 Contributions

Our contributions are summarized below.

  * •

We present a new attack that interferes with the generation of new blocks by
repeatedly executing CSTXs that certainly causes state inconsistency, causing
continuous rollback.

  * •

We propose a block-generation protocol called Synchro to incorporate all the
state changes of each CSTX into the same block by coordinating the block prior
to approving transactions in each shard. We also show that this protocol could
achieve the same transaction throughput as NEAR Protocol theoretically.

###  1.3 Organization of This Paper

Chapter 2 describes prior work on state sharding. Chapter 3 introduces a new
attack that interferes with the generation of new blocks by repeatedly
executing CSTXs that certainly causes state inconsistency, causing continuous
rollback. Chapter 4 presents a block-generation protocol called Synchro to
incorporate all the state changes of each CSTX into the same block by
coordinating the block prior to approving transactions in each shard. Chapter
5 summarizes our evaluation of the security and scalability of Synchro.
Finally, Chapter 6 presents the conclusions of this work.

##  2 Previous Work

###  2.1 Cross-Shard Transaction in State Sharding

State sharding defines a CSTX as a transaction that changes states in several
shards.[12, 9, 10] Since a validator is only expected to manage the states in
the shard to which it belongs, there is no validator that manages all the
states. In other words, one validator cannot handle any CSTXs by itself.

In Ethereum 2.0, 2PC is used to handle CSTX.[10, 8] In 2PC, the state is
changed serially on the two shards. The validator on a shard approves the
former half of state changes of a CSTX sends the execution result to the
validator of the next shard. The transaction information required to approve
the latter half of state changes, including the CSTX and the former execution
result, is called a receipt.

###  2.2 Challenges in Cross-Shard Transaction

In 2PC of Ethereum 2.0, an inconsistency in the blockchain state occurs if the
former state changes were approved but the latter ones were not. In such a
case, a new independent transaction is automatically invoked to resolve the
inconsistency.[15] However, there is a possibility that the additional
transaction may also fail, exponentially increasing the number of rare anomaly
cases that must be assumed in the system design.

The NEAR Protocol’s Nightshade also uses 2PC, but the white paper proposes a
rollback protocol for resolving inconsistencies in the blockchain state.[8]
The rollback protocol eliminates the need to invoke additional transactions,
making the NEAR Protocol less complicated.

##  3 Block Generation Interference Attack

In NEAR Protocol, a DoS attack that repeatedly executes CSTXs, which causes
state inconsistency, could result in a series of rollbacks, which could
interfere with block generation. This chapter introduces an attack that
exploits the rollback implemented in the NEAR Protocol to halt block
generation semi-permanently. We first provide an overview of the NEAR Protocol
then summarize the attack and its steps.

###  3.1 Overview of Nightshade

Before describing the block-generation interference caused by DoS attacks in
Nightshade, this section describes the block-generation protocol in
Nightshade. Figure reffig:near shows an overview of the block-generation flow
of the NEAR Protocol. The user sends a transaction to a validator in the shard
to which he or she belongs. Several validators called producers belong to each
shard, and one producer validates the transaction received from the user in
turn. When processing CSTX, the producer also creates a receipt. The producer
then compiles the approved transactions and receipts into sub-blocks called
chunks and distributes the created chunks to other validators inside and
outside the shard.

In Nightshade, smart contracts, like states, belong to one of the shards.[16]
When one account calls a function of a contract in another shard, the call is
made via a receipt, as in 2PC.[17]

Nightshade takes into account each chunk as one block, but the validator holds
only the chunks of the shard to which it belongs and does not centrally manage
the entire block. A CSTX is processed using 2PCs, so receipts will be stored
in the block after the block from which the CSTX was taken.

A rollback is also triggered by a challenge by a validator that discovers an
incorrect transaction, such as one that causes inconsistency in the state, and
all validators cancel the target transaction from their ledgers.222NEAR
Protocol rollback is described in the white paper.[8] But as of June 2023, the
details of the rollback are unknown as it has not yet been implemented.

Figure 1: Overview of NEAR Protocol’s block-generation flow

###  3.2 Attack Summary

The attacker uses a contract with a denial function that rejects calls by
receipts under certain conditions. The attacker deploys a homemade attack
contract and sends a CSTX that causes the invocation to be rejected by the
contract, thus rejecting the receipt and causing inconsistency in the state.
It is also possible to cause a series of rollbacks by repeatedly sending CSTX
that cause state inconsistency. All users can carry out this DoS attack, and
by using this attack, it is possible to interfere with block-generation at any
given time.

###  3.3 Attack Procedure

The attack procedure is shown below.

Step 1: Develop smart contracts to be used in the attack
The attacker develops a smart contract for the attack that implements the
following functions. An example implementation of the contract is shown in
Appendix1.

  * •

Functionality that enables tokens to be sent to this contract

  * •

If this contract holds more than 1NEAR333NEAR represents the monetary unit of
the native tokens issued on the NEAR Protocol network., a function is executed
to reject the call using the revert function and cancel the transaction
process in the middle.

Step 2: Deploy the contract
The attacker deploys the developed contract.

Step 3: Send 1NEAR to the contract
The attacker executes the deposit function using an account belonging to a
different shard from the attacking contract and sends 1NEAR.

Step 3 causes the attacker’s contract to hold more than 1NEAR, so the revert
function rejects the receipt call and cancels the state change resulting from
the contract’s deposit. Therefore, only the CSTX will be approved, resulting
in an inconsistency in the blockchain state, and a rollback will occur as a
response to the abnormal system. In this attack, the 1NEAR sent by the
rollback will be refunded, so it is possible to cause a rollback with only the
cost of gas. Repeating this attack makes it possible to interfere with new-
block generation by ordinary users. 444As of June 2023, the validity of the
attack has not been confirmed through demonstration testing, as it has not yet
been implemented.

##  4 Proposed Consistent Block-generation Protocol

###  4.1 Construction of Synchro

We propose a block-generation protocol that incorporates all the state changes
of each CSTX into the same block by coordinating the block prior to approving
transactions in each shard. We explain the differences from the NEAR Protocol.

Introduction of global validator
To prevent inconsistency in the blockchain state, a role is needed to ensure
that CSTXs and receipt approvals are completed in a same block. Therefore,
Synchro adds a new role called global validator. The global validator does not
belong to any shard and does not maintain a state. The global validator
receives chunks from the producers of each shard and validates all chunks
using zero-knowledge proofs to ensure that CSTX and receipt approvals are
complete in a same block. Thus, the global validator can prevent inconsistency
in the blockchain state by verifying that the CSTX and receipt authorization
is complete within a same block.

Creation of zero-knowledge proofs by producers
The global validator validates a chunk of all shards to ensure the producer
validates correctly. Therefore, the global validator must maintain the state
of all shards, which requires storage costs and computational load associated
with maintaining and updating the state. Therefore, to achieve state sharding,
Synchro produces a zero-knowledge proof for the producer to prove the validity
of the chunks.[18] Zendoo, proposed in a previous work, is an extension
framework for the blockchain developed by Horizon.[19] It enable
interoperability between the main chain and side chains. When a side chain
sends a transaction to the main chain, it uses zero-knowledge proofs to prove
the validity of the transaction, allowing the main chain to validate the
transaction without determining the state of the side chain. In Synchro, to
allow the global validator to validate chunks statelessly, the producer
creates a zero-knowledge proof, similar to Zendoo, to prove the validity of
the chunk’s state changes. Thus, the global validator’s storage cost and
computational load are reduced.

Introduction of coordinator
If several producers in each shard were free to create chunks, the global
validator would need to combine chunks so that CSTXs and receipts could be
processed within a same block. However, a considerable number of chunk
combinations would have to be validated. Therefore, Synchro adds a new role,
called coordinator, to limit the number of chunk combinations. The coordinator
does not belong to any shard and does not maintain a state. The coordinator
receives transactions from users and generates blocks. If a CSTX is included
in the chunk that makes up the block, the producer creates the corresponding
receipt conventionally. However, in Synchro, the coordinator creates the
corresponding receipt and stores it in a chunk of the same block. Thus, the
coordinator generates the entire block, and the global validator only needs to
verify the combination of chunks created by the same coordinator.

###  4.2 Block-generation Flow of Synchro

Figure 2 shows an overview of the Synchro’s block-generation flow. This
section describes the procedure of Synchro in chronological order.

1\. Transaction submission by user
The user sends the created transaction to all coordinators.

2\. Block-generation by coordinator
Coordinators receive transactions from users and generate blocks
independently. However, as multiple coordinators generate blocks
independently, the global validator receives many chunks and requires help in
identifying the correct combination. Therefore, after generating a block, the
coordinator calculates a block hash and stores it in the chunk. In this way,
the global validator can rely on the block hash to combine chunks and
reconstruct the block.

3\. Chunk verification and zero-knowledge proof production by producer
The producer selects one coordinator and requests chunks and block hashes. The
received chunks are verified using a state to confirm the validity of the
chunks. If the chunk is approved, the producer creates a zero-knowledge proof
of the chunk’s validity to allow the global validator to validate the chunk
statelessly. The producer then receives a request from the global validator
and sends the chunk and zero-knowledge proof.

4\. Chunk validation and block-generation by the global validator
The global validator designates a producer for each shard to request chunks
and zero-knowledge proofs. It combines the chunks from the received chunks by
relying on the block hash and reconstructing the block. By reconstructing the
block generated by the coordinator, it is possible to easily search for a
combination of chunks that allows CSTX and receipt approval to be completed
within a same block. The global validator also verifies the validity of state
changes using zero-knowledge proofs for all chunks that comprise the
reconstructed block. If all chunks are approved, the global validator combines
the chunks to generate a block.

Generating blocks by the global validator may result in multiple types of
blocks. Therefore, the blockchain may fork, but Synchro will determine the
legitimate chain in accordance with the fork-choice rule.

Figure 2: Overview of Synchro’s block-generation flow

###  4.3 Incentives for each role

Synchro prevents fraud by multiplexing each role. However, such fraud cannot
be prevented unless a sufficient number of honest roles are gathered.
Therefore, we introduce incentives to induce each role to participate
actively.

Specifically, each role will be given a digital signature for the data it
generates so that it can later be verified by an unspecified number of
participants who generated the data. An incentive rule is also added in which
all participants involved in the generation of the approved block will be
granted a token. Thus, each role is incentivized to follow the protocol for
incentive acquisition. Since incentive acquisition is a joint effort of all
roles, producers will come to trust the coordinator that has proposed several
blocks that have been approved in the past and will receive chunks from that
coordinator. The global validator will also come to trust the producer that
has proposed several chunks for previously approved blocks and will
preferentially validate chunk combinations from that producer. Thus, malicious
coordinators and producers that attempt to propose incorrect blocks or chunks
will eventually be weeded out, as honest producers and global validators will
ignore them.

###  4.4 Effectiveness of Synchro

In conventional state sharding, CSTX is handled using 2PC. However, 2PCs
process CSTXs and receipts serially, resulting in the inconsistency of
blockchain states. Therefore, Synchro introduces a new coordinator and global
validator. Synchro can achieve 1PC without sacrificing efficiency and
decentralization by having the coordinator propose candidate blocks so that
CSTX and receipt approvals can be completed in a same block and having the
global validator verify the consistency of CSTXs and receipts. Therefore,
Synchro does not require rollback for inconsistencies in the blockchain state,
making DoS attacks that cause continuous rollback impossible.

##  5 Evaluation

###  5.1 Security

While Synchro adds new roles, we ensure that cheating by each role does not
occur.

Coordinator
It is possible that a malicious coordinator could intentionally ignore
transactions involving a particular user. If this disregard were carried out,
the transactions of a particular user would be interfered with. However, since
Synchro multiplexes coordinators, other coordinators store that transaction in
a block even if an attacker ignores a particular transaction. Thus, it would
be impossible for an attacker to interfere with transactions involving a
particular user.

A malicious coordinator could generate a block containing only CSTX or
receipts. If that block were approved, only one of the states would be
changed, creating the possibility of lost tokens and double payments. However,
in Synchro, the global validator must verify that CSTX and receipt approval
are completed in a same block. Therefore, an attacker cannot change only one
of the states in the CSTX in a same block, and no token loss or double payment
can occur due to inconsistency in the blockchain states.

Producer
A malicious producer could create a malformed chunk and send it to the global
validator instead of the chunk created by the coordinator. A DoS attack by
sending malformed chunks would enable an attacker to interfere with the
generation of new blocks by ordinary users. However, since Synchro enables
each producer to select and propose chunks to the global validator
independently, the global validator can successfully generate blocks if at
least one producer belonging to the same shard as the attacker operates in
accordance with the protocol. Thus, as long as there is an honest producer in
each shard, it will be impossible to interfere with block generation.

Global Validator
A malicious global validator could generate blocks of incorrect chunk
combinations. For example, token loss or double payments could occur if an
attacker generates a block that does not contain CSTX and receipts
simultaneously. However, the Synchro is supposed to determine the authorized
chain in accordance with the fork-choice rule. Thus, if most global validators
are honest, fraudulent blockchains will be eliminated.

###  5.2 Scalability

The new roles and zero-knowledge proofs in Synchro may reduce transaction
throughput compared with Nightshade. Therefore, we theoretically compared
Nightshade and Synchro in terms of transaction throughput.

Changes associated with the introduction of coordinators
In Synchro, a new coordinator is introduced, which causes a delay in
transaction processing due to the overhead associated with it. However, the
processing of the coordinator can be pipelined with the processing of other
roles. Therefore, if the coordinator can generate a block within the block
time, the overhead associated with the introduction of the coordinator will
not affect transaction throughput. Since the coordinator’s processing is only
central-processing-unit computation and network communication, it should be
possible to complete the processing during the block time.

Changes associated with the introduction of zero-knowledge proofs
Since Synchro introduces a new zero-knowledge proof, there is an associated
overhead. However, when the average block time of Nightshade is $t_{block}$,
chunk-verification time by the producer is $t_{chunk}$ and zero-knowledge-
proof production time is $t_{zk_{p}}$, Formula (1) is satisfied. The overhead
does not affect scalability.

$t_{block}\geq t_{chunk}+t_{zk_{p}}$ (1)

As of July 7, 2023, $T_{block}$ is about 1s.[20] Prior research has found that
the time to create a zero-knowledge proof using the simplest generation
circuit is $t_{zk_{p}}=0.41$s[18]. Therefore, if Formula (1) is satisfied, the
block time of Synchro will not be reduced compared with Nightshade.

Assuming that 100,000 TPS is achieved with 1000 shards, 100 transactions per
chunk would be required (the upper limit in the specification is 1000
transactions per chunk). Assuming that a zero-knowledge proof is generated for
each transaction, a simple calculation shows that it would take
0.41s$\times$100 = 41s, a performance degradation compared with Nightshade.
Therefore, it is necessary to keep the proof-creation time within 1s by
adopting a policy of creating one zero-knowledge proof per chunk, designing
the proposition circuit as simple as possible, and adopting the most efficient
zero-knowledge-proof protocol.

Changes associated with the introduction of global validators
In Nightshade, only the producer generates the block. However, Synchro adds
verification of the block by a global validator, which causes a delay in
transaction processing due to the associated overhead. However, the processing
of the producer and global validator can be pipelined. If $t_{chunk}$ denotes
the chunk-validation time by the producer, $t_{zk_{p}}$ the zero-knowledge-
proof production time, $t_{zk_{v}}$ the chunk validation by the global
validator, and $s$ the number of shards, then Formula (2) is satisfied The
overhead of adding the global validator does not affect transaction
throughput.

$t_{chunk}+t_{zk_{p}}\geq s\times t_{zk_{v}}$ (2)

In a previous work, the verification time with zero-knowledge proofs using the
simplest generation circuit was $t_{zk_{v}}=0.0043$s.[18] For $s=100$,
$s\times t_{zk_{v}}=0.43$s. Therefore, given that $t_{zk_{p}}=0.41$s, it is
possible to satisfy Formula (2) depending on $t_{chunk}$.

##  6 Conclusion

We presented a new attack that interferes with the generation of new blocks by
repeatedly executing CSTXs that certainly causes state inconsistency, causing
continuous rollback. We proposed a block-generation protocol called Synchro to
incorporate all the state changes of each CSTX into the same block by
coordinating the block prior to approving transactions in each shard. In this
protocol, we showed that avoiding DoS attacks that repeatedly execute CSTXs is
possible, which causes state inconsistency. We also showed that this protocol
could theoretically achieve the same transaction throughput as NEAR Protocol,
depending on technical and future innovations in zero-knowledge proofs.

## Author Contributions

All authors contributed equally to this work.

## Conflict of Interest

All authors have affirmed they have no conflicts of interest as described in
Ledger’s Conflict of Interest Policy.

## References

  * [1] https://blog.twitter.com/engineering/en_us/a/2013/new-tweets-per-second-record-and-how “Tweet/Tweet per Second.”.
  * [2] https://etherscan.io/chart/tx “Etherscan/Transactions per Day.”.
  * [3] https://solana.com/solana-whitepaper.pdf “Solana/whitepaper.”.
  * [4] https://www.tbstat.com/wp/uploads/2021/08/20210811_Layer1Platforms_TheBlockResearch.pdf “Layer-1 Platforms: A Framework for Comparison.”.
  * [5] Popov, S. “The tangle.” _White paper_ 1.3 30 (2018).
  * [6] https://www.avalabs.org/whitepapers “Avalanche/whitepaper.”.
  * [7] https://github.com/ethereum/wiki “Ethereum2.0/wiki.”.
  * [8] https://near.org/papers/nightshade/ “NEAR Protocol/Nightshade.”.
  * [9] Wang, G., Shi, Z. J., Nixon, M., Han, S. “Sok: Sharding on blockchain.” In _Proceedings of the 1st ACM Conference on Advances in Financial Technologies_ 41–61 (2019) .
  * [10] Han, R., Yu, J., Lin, H., Chen, S., Esteves-Veríssimo, P. “On the security and performance of blockchain sharding.” _Cryptology ePrint Archive_ .
  * [11] Mizrahi, A., Rottenstreich, O. “Blockchain state sharding with space-aware representations.” _IEEE Transactions on Network and Service Management_ 18.2 1571–1583 (2020).
  * [12] Hashim, F., Shuaib, K., Zaki, N. “Sharding for scalable blockchain networks.” _SN Computer Science_ 4.1 2 (2022).
  * [13] https://docs.zilliqa.com/whitepaper.pdf “Zilliqa/Whitepaper.”.
  * [14] Luu, L., Narayanan, V., Zheng, C., Baweja, K., Gilbert, S., Saxena, P. “A Secure Sharding Protocol For Open Blockchains.” In _Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security_ New York, NY, USA: Association for Computing Machinery 17–30 (2016) doi:10.1145/2976749.2978389 URL https://doi.org/10.1145/2976749.2978389.
  * [15] https://vitalik.ca/general/2017/12/31/sharding_faq.html#how-can-we-facilitate-cross-shard-communication “Ethereum2.0/additional transaction.”.
  * [16] https://docs.near.org/concepts/basics/accounts/state “NEAR Protocol/contracts.”.
  * [17] https://nomicon.io/RuntimeSpec/receipts “NEAR Protocol/cross-call.”.
  * [18] MIYAMAE, T., KOZAKURA, F., NAKAMURA, M., MORINAGA, M. “ZGridBC: Zero-Knowledge Proof Based Scalable and Privacy-Enhanced Blockchain Platform for Electricity Tracking.” _IEICE TRANSACTIONS on Information and Systems_ 106.7 1219–1229 (2023).
  * [19] Garoffolo, A., Kaidalov, D., Oliynykov, R. “Zendoo: A zk-SNARK verifiable cross-chain transfer protocol enabling decoupled and decentralized sidechains.” In _2020 IEEE 40th International Conference on Distributed Computing Systems (ICDCS)_ IEEE 1257–1262 (2020) .
  * [20] https://explorer.near.org/ “NEAR Protocol/blocktime.”.

##  Appendix A Attack Contract Implementation

Listing 1: Example of Attack Contract Source Code(RUST)

⬇

1 use near_sdk::borsh::{self, BorshDeserialize, BorshSerialize};

2 use near_sdk::{env, near_bindgen, Balance};

3

4 #[near_bindgen]

5 #[derive(Default, BorshDeserialize, BorshSerialize)]

6 pub struct Contract {

7 }

8

9 #[near_bindgen]

10 impl Contract {

11 #[payable]

12 pub fn send() {

13 assert!(env::account_balance() > 1e24, "Over");

14 }

15 }


# Modifications on parameters of $Z(4430)$ in a dense medium

K. Azizi<EMAIL_ADDRESS>Department of Physics, University of Tehran,
North Karegar Ave. Tehran 14395-547, Iran  Department of Physics, Doǧuş
University, Acibadem-Kadiköy, 34722 Istanbul, Turkey  N. Er<EMAIL_ADDRESS>Department of Physics, Bolu Abant İzzet Baysal University, Gölköy Kampüsü,
14980 Bolu, Turkey

###### Abstract

The charmonium-like resonance $Z_{c}(3900)$ and its excited state $Z(4430)$
are among the particles that are serious candidates for double heavy
tetraquarks. Calculations of different parameters associated with these states
both in the vacuum and the medium with finite density are of great importance.
Such investigations help us clarify their nature, internal quark-gluon
organization and quantum numbers. In this accordance, we extend our previous
analyses on the ground state $Z_{c}(3900)$ to investigate the medium
modifications on different parameters of the excited $Z(4430)$ state. In
particular, we calculate the mass, vector self-energy and current coupling of
$Z(4430)$ in terms of density, up to a density comparable to the density of
the cores of massive neutron stars. The obtained results may help experimental
groups aiming to study the behavior of exotic states at higher densities.

##  I Introduction

Over the past two decades, the results of many experimental observations on
some resonances have shown that these states can not be put in the class of
the standard mesons and baryons since their spectrum and decay patterns differ
from the standard hadrons, considerably. These results have led us to collect
these states under the new title: the exotic states made of multiquarks,
antiquarks and valence gluons. Among these states, the tetraquarks have been
relatively more in the focus of the experimental and theoretical studies. The
charmonium-like resonances made of a $c\bar{c}$ and a light $q_{1}\bar{q_{2}}$
pair receive much attentions as they demonstrate different properties.
Investigation of these states can help us not only clarify their nature and
internal structure but also get useful knowledge on the nature of strong
interaction inside these particles.

The observation of the ground states $Z^{\pm}_{c}(3900)$ were reported
simultaneously by the BESIII Ablikim:2013mio  and Belle Liu:2013dau
Collaborations in 2013. Many aspects of these states have been previously
investigated. For details see for instance Refs. Agaev:2020zad ; Agaev:2016dev
; PhysRevD.96.034026 ; Azizi:2020itk ; Ozdem:2017jqh  and references therein.
Hence, we won’t spend time on this state in the present work and will focus on
the state $Z(4430)$. For a first time in 2008, the Belle Collaboration
reported a distinct peak in the $\pi^{\pm}\psi^{\prime}$ invariant mass
distribution in $B\rightarrow K\pi^{\pm}\psi^{\prime}$ decay with the
statistical significance of $6.5\sigma$. The measured mass and width were
$M=4433\pm 4(\textrm{stat})\pm 2(\textrm{syst})$ MeV and
$\Gamma=45^{+18}_{-13}(\textrm{stat})^{30}_{-13}(\textrm{syst})$ MeV
PhysRevLett.100.142001 . In an amplitude analysis of
$B^{0}\rightarrow\psi^{\prime}K^{+}\pi^{-}$ decays PhysRevD.88.074026 , the
quantum numbers of the $Z(4430)$ were set to $J^{P}=1^{+}$ and the results for
the mass and width were obtained as $4485^{+22+28}_{-22-11}$ MeV/${c^{2}}$ and
$200^{+41+26}_{-46-35}$ MeV, respectively. The Belle Collaboration then
PhysRevD.90.112009 , during the observation of a new charmonium like state
$Z^{+}_{c}(4200)$, in addition, found evidence for $Z^{+}(4430)\rightarrow
J/\psi\pi^{+}$. In the first independent confirmation by LHCb Collaboration
PhysRevLett.112.222002 , its spin-parity was assigned as $1^{+}$. The mass of
the resonance, $4475\pm 7^{+15}_{-25}$ MeV, and its width, $172\pm
13^{+37}_{-34}$ MeV, were measured. In the same mass region, an alternative
and model-independent confirmation of the existence of a $\psi(2S)\pi$
resonance is later presented by the LHCb, as well PhysRevD.92.112009 .

In theoretical side, various assignments have been made for the nature and
quark-gluon structure of $Z(4430)$ resonance. Using different phenomenological
and computational models, the mass and some other quantities of this state
have been studied. Considering a molecular structure, the $Z(4430)$ state was
studied in Liu:2007bf ; Liu:2008xz ; Lee:2007gs ; Lee:2008tz ; Braaten:2007xw
; Zhang:2009vs ; Branz:2010sh ; Close:2010wq ; Wang:2018pwi . The compact
tetraquark or diquark-antidiquark structure was assigned for this state in
Refs. Wang:2019hnw ; Wang:2019tlw ; PhysRevD.96.034026 ; Maiani:2014aja ;
Goerke:2016hxf ; Wang:2014vha ; Bracco:2008jj ; Ebert:2008kb ; Liu:2008qx  to
calculate various observables associated with the $Z(4430)$ resonance. Other
theoretical interpretations can be collected as cusp effects Bugg ;
Swanson:2014tra ; Ikeda  and hadrocharmonium Danilkin:2011sh ;
Dubynskiy:2008mq . For more information, one can see Refs. Albuquerque:2018jkn
; Olsen:2017bmm ; Guo:2017jvc ; Esposito:2016noz ; Lebed:2016hpi ; Ali:2017jda
; Chen:2016qju ; Liu:2019zoy ; Hosaka:2016pey .

In the present work, we shall adopt the compact tetraquark/diquark-antidiquark
interpretation of the state $Z(4430)$ and consider it as the radial excitation
of the ground state $Z_{c}(3900)$ with the same spin-parity. In Ref.
Maiani:2014aja , the authors have made the hypothesis that the spacing (mass
difference) in radial excitations in these channels could closely resemble
those observed in the standard P-wave charmonia. Therefore, the authors have
considered the state $Z(4430)$ as the first radial excitation of the
$Z_{c}(3900)$ state. The same scenario has been applied in different studies
Wang:2014vha ; PhysRevD.96.034026 ; Chen:2019osl ; Zhu:2016arf , as well. We
calculate the mass and current coupling of $Z(4430)$ state in the medium with
high density, comparable with the density of the neutrons star cores. We
discuss the modifications on the considered physical quantities due to the
nuclear dense medium. The medium effects are represented by various operators
that enter the nonperturbative parts of the calculations (for details see, for
instance, Ref. Azizi:2014yea  and references therein). We calculate the values
of the mass and current coupling at saturation density. These quantities are
also obtained in vacuum setting the density of the medium to zero. We compare
the obtained results at zero density limit with the existing theoretical
predictions and the experimental data.

The high energy heavy ion collision experiments are excellent platforms to
produce the heavy standard hadrons as well as the exotic particles. With
upgraded detectors at the Relativistic Heavy Ion Collider (RHIC) at the
Bookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC), now
it is possible to investigate not only the ground state particles but also
their excited states. Thus, these experiments provide opportunities to study
exotic hadrons like $Z_{c}(3900)$ and its excited state $Z(4430)$. The study
on other exotic states that are either molecular states made of various
standard hadrons or compact objects of quarks and antiquarks are now possible,
as well. Due to a large number of heavy quarks and antiquarks produced in
these experiments, many exotic states could be formed. Hence, it will be a
good opportunity to study the exotic particles and their higher states in
order to determine their nature and internal quark-gluon structure as well as
fix their quantum numbers (for more information see for instance Cho:2017dcy ;
Cho:2011ew ). At RHIC and heavy ion collision experiments at LHC it will be
also possible to investigate different phases of hadronic matter and search
for quark-gluon-plasma (QGP) as a new possible phase of matter . Some in-
medium experiments like $\bar{P}ANDA$ at Facility for Antiproton and Ion
Research (FAIR) at GSI and the Nuclotron-based Ion Collider Facility at JINR
(NICA) at Dubna aim to study the parameters of exotic states in terms of
density. These experiments can investigate the the $Z^{+}(4430)$ even further
by switching to studies of the $Z^{+}(4430)$ in formation mode. Due to the
charge of the Z, this is only possible by annihilating the antiprotons on a
neutron in a deuterium or hydrogen gas target. They can study the
$Z^{+}(4430)$ state in both production and formation experiments
Wiedner:2011mf .

The paper is organized as follows: In next section the parameters of the
$Z(4430)$ is calculated in dense medium. In section 3 the numerical analyses
of the physical quantities under study is performed. The last section includes
the summary and our concluding notes.

##  II Parameters of $Z(4430)$ in cold nuclear medium

In this section, we aim to calculate the mass and current coupling as well as
the scalar and vector self-energies of the exotic $Z(4430)$ (in what follows
we denote it by $Z$ ) state as an exited state of the $Z_{c}(3900)$ ($Z_{c}$)
in cold nuclear medium. To this end, the positively charged $Z^{+}(4430)$
state with quark content $\bar{c}cu\bar{d}$ is considered. But, the parameters
of the negatively charged state with $\bar{c}c\bar{u}d$ content do not change
as a result of Chiral limit used in the calculations. In this context, our
starting point is to consider the following density dependent two-point
correlation function:

$\Pi_{\mu\nu}(p)=i\int{d^{4}xe^{ip\cdot
x}\langle\psi_{0}|\mathcal{T}[J^{Z_{(c)}}_{\mu}(x)J^{{Z_{(c)}}\dagger}_{\nu}(0)]|\psi_{0}\rangle},$
(1)

where $|\psi_{0}\rangle$ is the ground state of nuclear matter and
$J^{Z_{(c)}}_{\mu}$ is the interpolating current of $Z$/$Z_{c}$ states that
couples to these states, simultaneously. We should note that in the present
study we choose to work at zero momentum limit of the particle’s three
momentum, $\mid\bf{p}\mid\rightarrow 0$, for which the longitudinal and
transverse polarizations become degenerate, and there is no way to distinguish
these two polarizations. For non-zero and finite three momentum in nuclear
matter, however, the longitudinal and transverse polarization states are
distinguishable and they should be calculated, separately (for more details
see for instance Kim:2019ybi ). At zero three momentum limit,
$p^{2}=p_{0}^{2}$ and the calculation goes forward in the standard way with
the above single function (see also COHEN1995221 ). For $J^{P}=1^{+}$, the
interpolating current is given by

$\displaystyle J^{Z_{(c)}}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{i\epsilon^{abc}\epsilon^{dec}}{\sqrt{2}}\Bigg{\\{}\Big{[}u_{a}^{T}(x)C\gamma_{5}c_{b}(x)\Big{]}\Big{[}\bar{d}_{d}(x)\gamma_{\mu}C\bar{c}^{T}_{e}(x)\Big{]}$
(2) $\displaystyle-$
$\displaystyle\Big{[}u_{a}^{T}(x)C\gamma_{\mu}c_{b}(x)\Big{]}\Big{[}\bar{d}_{d}(x)\gamma_{5}C\bar{c}^{T}_{e}(x)\Big{]}\Bigg{\\}},$

where, $a,b,c,d,e$ are color indices and $C$ is the charge conjugation
operator.

We need to derive the in-medium sum rules for the mass $m_{Z}$ and current
coupling $f_{Z}$ of the excited state $Z$. For this purpose, the previously
obtained expressions for the mass and current coupling of $Z_{c}$ (see Ref.
Azizi:2020itk ) are considered as input parameters. The procedure is done in
two steps: First, we derive the sum rules for the in-medium mass $m_{Z_{c}}$
and in-medium current coupling $f_{Z_{c}}$ of the ground state exotic $Z_{c}$.
For this, we use the ”ground state + continuum” approximation by including the
$Z(4430)$ state into the list of ”higher resonances” and extract sum rules and
corresponding numerical values for $Z_{c}$ as discussed in Ref. Azizi:2020itk
. At the next step, we increase the threshold and adopt the ”ground
state+radially excited state+continuum” scheme, and perform the required
standard calculations. In the second case, the parameters of both the $Z_{c}$
and $Z$ appear in the expression of the hadronic side. To extract the
parameters of $Z$, we use those of the $Z_{c}$ as inputs.

Thus, the phenomenological side of the correlation function is obtained by
saturating it with the complete sets of hadronic states with the same quantum
numbers as the interpolating current. Isolating the ground state $Z_{c}$ and
radially excited $Z$ resonances, considering the ”ground state+radially
excited state+continuum” scheme, and integrating over $x$, we obtain the
phenomenological side of the correlation function in momentum space as

$\displaystyle\Pi^{Phe}_{\mu\nu}(p)=$ $\displaystyle-$
$\displaystyle\frac{\langle\psi_{0}|J_{\mu}|Z_{c}(p)\rangle\langle
Z_{c}(p)|J^{\dagger}_{\nu}|\psi_{0}\rangle}{p^{*2}-m_{Z_{c}}^{*2}}$ (3)
$\displaystyle-$
$\displaystyle\frac{\langle\psi_{0}|J_{\mu}|Z(p)\rangle\langle
Z(p)|J^{\dagger}_{\nu}|\psi_{0}\rangle}{p^{*2}-m_{Z}^{*2}}+...,$

where $p^{*}$ is the in-medium momentum; and $m_{Z_{c}}$ and $m_{Z}$ are the
masses of $Z_{c}$ and $Z$ states, respectively. In Eq. (3), contributions
arising from higher resonances and continuum are represented by the dots. The
current meson couplings of the states under consideration are expressed in
terms of the polarization vectors $\varepsilon_{\mu}$ and
$\tilde{\varepsilon}_{\mu}$, respectively, as

$\displaystyle\langle\psi_{0}|J_{\mu}|Z_{c}(p)\rangle$ $\displaystyle=$
$\displaystyle f^{*}_{Z_{c}}m_{Z_{c}}^{*}\varepsilon_{\mu},$
$\displaystyle\langle\psi_{0}|J_{\mu}|Z(p)\rangle$ $\displaystyle=$
$\displaystyle f^{*}_{Z}m_{Z}^{*}\tilde{\varepsilon}_{\mu}.$ (4)

Using Eq. (II) in Eq. (3) and summing over the polarization vectors, we can
construct the new form of the function $\Pi^{Phe}_{\mu\nu}(p)$ as

$\displaystyle\Pi_{\mu\nu}^{Phe}(p)=$ $\displaystyle-$
$\displaystyle\frac{m^{*2}_{Z_{c}}f^{*2}_{Z_{c}}}{p^{*2}-m^{*2}_{Z_{c}}}\Big{[}-g_{\mu\nu}+\frac{p_{\mu}^{*}p^{*}_{\nu}}{m^{*2}_{Z_{c}}}\Big{]}$
(5) $\displaystyle-$
$\displaystyle\frac{m^{*2}_{Z}f^{*2}_{Z}}{p^{*2}-m^{*2}_{Z}}\Big{[}-g_{\mu\nu}+\frac{p_{\mu}^{*}p^{*}_{\nu}}{m^{*2}_{Z}}\Big{]}+....$

At this point, we should note that a particle in nuclear medium gain two kinds
of self-energies: The scalar self-energy, which is defined as the difference
between the in-medium and vacuum masses, $\Sigma_{s}=m_{Z}^{*}-m_{Z}$, and the
vector self-energy that enters the expression of the in-medium momentum,
$p^{*}_{\nu}=p_{\nu}-\Sigma_{\upsilon}u_{\nu}$, where $\Sigma_{\upsilon}$ is
the vector self-energy and $u_{\nu}$ is the four-velocity of the nuclear
medium. Our calculations take place in the rest frame of the medium,
$u_{\nu}=(1,0)$, and so $p^{\nu}\cdot u_{\nu}=p_{0}$. Consequently, the
correlation function is written as

$\displaystyle\Pi_{\mu\nu}^{Phe}(p)=$ $\displaystyle-$
$\displaystyle\frac{f^{*2}_{Z_{c}}}{p^{2}-\mu_{Z_{c}}^{2}}\Big{[}-g_{\mu\nu}m^{*2}_{Z_{c}}+p_{\mu}p_{\nu}$
$\displaystyle-$
$\displaystyle\Sigma^{Z_{c}}_{\upsilon}p_{\mu}u_{\nu}-\Sigma^{Z_{c}}_{\upsilon}p_{\nu}u_{\mu}+\Sigma_{\upsilon}^{Z_{c}2}u_{\mu}u_{\nu}\Big{]}$
$\displaystyle-$
$\displaystyle\frac{f^{*2}_{Z}}{p^{2}-\mu_{Z}^{2}}\Big{[}-g_{\mu\nu}m^{*2}_{Z}+p_{\mu}p_{\nu}-\Sigma^{Z}_{\upsilon}p_{\mu}u_{\nu}$
$\displaystyle-$
$\displaystyle\Sigma^{Z}_{\upsilon}p_{\nu}u_{\mu}+\Sigma^{Z2}_{\upsilon}u_{\mu}u_{\nu}\Big{]}+...,$

where
$\mu_{Z_{(c)}}^{2}=m^{*2}_{Z_{(c)}}-\Sigma^{Z_{(c)}2}_{\upsilon}+2p_{0}\Sigma^{Z_{(c)}}_{\upsilon}$.
After applying the Borel transformation with respect to the parameter $p^{2}$
to both sides of Eq. (II), the phenomenological side of the correlation
function yields,

$\displaystyle\mathbf{\Pi}_{\mu\nu}^{Phe}(p)$ $\displaystyle=$ $\displaystyle
f^{*2}_{Z_{c}}e^{-\mu_{Z_{c}}^{2}/M^{2}}\Big{[}-g_{\mu\nu}m^{*2}_{Z_{c}}+p_{\mu}p_{\nu}$
(7) $\displaystyle-$
$\displaystyle\Sigma^{Z_{c}}_{\upsilon}p_{\mu}u_{\nu}-\Sigma^{Z_{c}}_{\upsilon}p_{\nu}u_{\mu}+\Sigma_{\upsilon}^{Z_{c}2}u_{\mu}u_{\nu}\Big{]}$
$\displaystyle+$ $\displaystyle
f^{*2}_{Z}e^{-\mu_{Z}^{2}/M^{2}}\Big{[}-g_{\mu\nu}m^{*2}_{Z}+p_{\mu}p_{\nu}-\Sigma^{Z}_{\upsilon}p_{\mu}u_{\nu}$
$\displaystyle-$
$\displaystyle\Sigma^{Z}_{\upsilon}p_{\nu}u_{\mu}+\Sigma^{Z2}_{\upsilon}u_{\mu}u_{\nu}\Big{]}$
$\displaystyle+$ $\displaystyle...,$

where $M^{2}$ is the Borel mass parameter.

The next step is to calculate the QCD side of the correlation function. This
is done via the usage of the explicit form of the interpolating current in the
correlation function. The quark fields are contracted in the presence of the
dense medium via the Wick’s theorem. This results in the expression of the
correlation function in terms of the in-medium heavy and light propagators:

$\displaystyle\Pi_{\mu\nu}^{QCD}(p)=-\frac{i}{2}\varepsilon_{abc}\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\varepsilon_{dec}\varepsilon_{d^{\prime}e^{\prime}c^{\prime}}\int
d^{4}xe^{ipx}\Big{\\{}Tr\Big{[}\gamma_{5}\tilde{S}_{u}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle
Tr\Big{[}\gamma_{\mu}\tilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{d}^{d^{\prime}d}(-x)\Big{]}-Tr\Big{[}[\gamma_{\mu}\tilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}Tr\Big{[}\gamma_{\nu}\tilde{S}_{u}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}-$
$\displaystyle
Tr\Big{[}\gamma_{5}\tilde{S}_{u}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{5}\tilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{\nu}S_{d}^{d^{\prime}d}(-x)\Big{]}+$
$\displaystyle
Tr\Big{[}\gamma_{\nu}\tilde{S}_{u}^{aa^{\prime}}(x)\gamma_{\mu}S_{c}^{bb^{\prime}}(x)\Big{]}Tr\Big{[}\gamma_{5}\tilde{S}_{c}^{e^{\prime}e}(-x)\gamma_{5}S_{d}^{d^{\prime}d}(-x)\Big{]}\Big{\\}}_{|\psi_{0}\rangle},$

where $\tilde{S}_{q(c)}=CS_{q(c)}^{T}C$. The in-medium heavy and light quarks’
propagators are used in coordinate space and the calculations are transferred
to the momentum space by performing the Fourier integrals. To suppress the
contributions of the higher states and continuum the Borel transformations are
applied based on the standard prescriptions of the method. To further suppress
the contributions of the unwanted states, continuum subtraction supplied by
the quark-hadron duality assumption is applied. All of these procedures are
described in Ref. Azizi:2020itk . We should stress that we take into account
the non-perturbative operators up to five dimensions in the QCD side. The
contributions of the six and upper dimensional operators are expected to be
small and they are neglected. Thus, the QCD side of the correlation function
in terms of the selected Lorentz structures is written as

$\displaystyle\Pi_{\mu\nu}^{QCD}(M^{2},s_{0}^{*})=\mathbf{\Upsilon}^{QCD}_{1}(M^{2},s_{0}^{*})(-g_{\mu\nu})$
$\displaystyle+$
$\displaystyle\mathbf{\Upsilon}^{QCD}_{2}(M^{2},s_{0}^{*})p_{\mu}p_{\nu}+\mathbf{\Upsilon}^{QCD}_{3}(M^{2},s_{0}^{*})(-p_{\mu}u_{\nu})$
$\displaystyle+$
$\displaystyle\mathbf{\Upsilon}^{QCD}_{4}(M^{2},s_{0}^{*})(-p_{\nu}u_{\mu})+\mathbf{\Upsilon}^{QCD}_{5}(M^{2},s_{0}^{*})u_{\mu}u_{\nu}.$

The Borel transformed invariant functions
$\mathbf{\Upsilon}^{QCD}_{i}(M^{2},s_{0}^{*})$ in Eq. (II) are represented in
terms of the two-point spectral densities, $\rho_{i}^{QCD}(s)$, related to the
imaginary parts of the selected coefficients:

$\mathbf{\Upsilon}^{QCD}_{i}(M^{2},s_{0}^{*})=\int_{4m_{c}^{2}}^{s^{*}_{0}}ds\rho^{QCD}_{i}(s)e^{-\frac{s}{M^{2}}},$
(10)

where $s_{0}^{*}$ is the in-medium continuum threshold parameter separating
the contributions of the ground state $Z_{c}$ and the first excited state $Z$
from the higher resonances and continuum. $i$ runs from $1$ to $5$. The
spectral density corresponding to each structure is expressed in terms of
perturbative and non-perturbative contributions as follows:

$\displaystyle\rho_{i}^{QCD}(s)$ $\displaystyle=$
$\displaystyle\rho_{i}^{pert}(s)+\rho_{i}^{qq}(s)+\rho_{i}^{gg}(s)+\rho_{i}^{qgq}(s),$
(11)

where $qq$, $gg$ and $qgq$ denote the two quark, two gluon and mixed quark-
gluon condensates as the non-perturbative effects, respectively. As an
example, the spectral density corresponding to the structure $g_{\mu\nu}$ is
given in Ref. Azizi:2020itk .

After all these lengthy calculations, when the phenomenological and QCD
results of each structure are compensate, the sum rules for the mass, vector
self-energy and current coupling constant of $Z$ state are obtained as
follows:

$\displaystyle
m^{*2}_{Z_{c}}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}+m^{*2}_{Z}f^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\mathbf{\Upsilon}^{QCD}_{1}(M^{2},s_{0}^{*}),$
(12) $\displaystyle
f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}+f^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\mathbf{\Upsilon}^{QCD}_{2}(M^{2},s_{0}^{*}),$
(13)
$\displaystyle\Sigma^{Z_{c}}_{\upsilon}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}+\Sigma^{Z}_{\upsilon}f^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\mathbf{\Upsilon}^{QCD}_{3}(M^{2},s_{0}^{*}),$
(14)
$\displaystyle\Sigma^{Z_{c}}_{\upsilon}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}+\Sigma^{Z}_{\upsilon}f^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\mathbf{\Upsilon}^{QCD}_{4}(M^{2},s_{0}^{*}),$
(15)
$\displaystyle\Sigma^{Z_{c}2}_{\upsilon}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}+\Sigma^{Z2}_{\upsilon}f^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}$
$\displaystyle=$ $\displaystyle\mathbf{\Upsilon}^{QCD}_{5}(M^{2},s_{0}^{*}).$
(16)

The mass sum rules for $Z$ state can be derived by different ways from the
above equations. We apply derivative with respect to $-\frac{1}{M^{2}}$ to
both sides of Eq. (12) and eliminate $f^{*2}_{Z}$ by dividing both sides of
the resultant equation to both sides of the original one. As a result we get

$\mu_{Z}^{2}=\frac{\frac{\partial\mathbf{\Upsilon}^{QCD}_{1}(M^{2},s_{0}^{*})}{\partial\Big{(}-\frac{1}{M^{2}}\Big{)}}-m^{*2}_{Z_{c}}f^{*2}_{Z_{c}}\mu_{Z_{c}}^{2}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}}{\mathbf{\Upsilon}^{QCD}_{1}(M^{2},s_{0}^{*})-m^{*2}_{Z_{c}}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}},$
(17)

where, the in-medium mass of $Z$ state is found as

$m^{*2}_{Z}=\mu_{Z}^{2}+\Sigma^{Z2}_{\upsilon}-2p_{0}\Sigma^{Z}_{\upsilon}.$
(18)

By rearranging the equations (16) and (13) and dividing both sides of these
equations to each other we get $\Sigma^{Z2}_{\upsilon}$ as

$\Sigma^{Z2}_{\upsilon}=\frac{\mathbf{\Upsilon}^{QCD}_{5}(M^{2},s_{0}^{*})-\Sigma^{Z_{c}2}_{\upsilon}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}}{\mathbf{\Upsilon}^{QCD}_{2}(M^{2},s_{0}^{*})-f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}}.$
(19)

To obtain the in-medium current coupling constant, $f^{*}_{Z}$, there are
different possibilities. One way is to find it from Eq. (12), which leads to

$f_{Z}^{*2}=\frac{\mathbf{\Upsilon}^{QCD}_{1}(M^{2},s_{0}^{*})-m^{*2}_{Z_{c}}f^{*2}_{Z_{c}}e^{-\frac{\mu_{Z_{c}}^{2}}{M^{2}}}}{m^{*2}_{Z}e^{-\frac{\mu_{Z}^{2}}{M^{2}}}}.$
(20)

As it is seen, the parameters of $Z_{c}$ calculated in Ref. Azizi:2020itk  are
entered as the inputs to the sum rules for $Z$ state in the dense medium.

##  III Numerical results

The expressions obtained in Eqs. (17-20) for the physical observables in cold
nuclear matter contain quark and baryon masses, nuclear matter density and
different operators both in vacuum and dense medium as input parameters. The
operators appearing in the calculations in a dense medium are presented in
Ref. Azizi:2014yea . The numerical results for expectation values of different
operators together with the values of other input parameters are collected in
table (1).

Parameter | Value | Unit | Ref. #
---|---|---|---
$\rho^{sat}$ | $0.11^{3}$ | GeV3 | PhysRevC.45.1881
$p_{0}$ | $4.478^{+0.015}_{-0.018}$ | GeV | PhysRevD.98.030001
$m_{u}$ | $2.16^{+0.49}_{-0.26}$ | MeV | PhysRevD.98.030001
$m_{d}$ | $4.67^{+0.48}_{-0.17}$ | MeV | PhysRevD.98.030001
$m_{c}$ | $1.27\pm 0.02$ | GeV | PhysRevD.98.030001
$\langle q^{\dagger}q\rangle_{\rho}$ | $\frac{3}{2}\rho$ | GeV3 | COHEN1995221
$\langle\bar{q}q\rangle_{0}$ | $-[272(5)]^{3}$ | MeV3 | GUBLER20191 ; Bazavov:2010yq ; PhysRevD.88.014513 ; PhysRevD.90.114504 ; PhysRevD.93.054502 ; 10.1093/ptep/ptw129
$\langle\bar{q}q\rangle_{\rho}$ | $\langle\bar{q}q\rangle_{0}+\frac{\sigma_{\pi N}}{2m_{q}}\rho$ | MeV3 | PhysRevC.47.2882
$m_{q}$ | $0.00345$ | GeV | PhysRevD.98.030001
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle_{0}$ | $0.012\pm 0.004$ | GeV4 | GUBLER20191 ; SHIFMAN1979385 ; SHIFMAN1979448
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle_{\rho}$ | $\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle_{0}+\rho\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle_{N}$ | GeV4 | GUBLER20191 ; Mishra1993 ; Sait1998
$\langle\frac{\alpha_{s}}{\pi}G^{2}\rangle_{N}$ | $-\frac{8}{9}(M_{N}-\sigma_{\pi N}-\sigma_{sN})$ | GeV | GUBLER20191 ; PhysRevC.45.1881
$M_{N}$ | $939.49\pm 0.05$ | MeV | PhysRevD.98.030001
$\sigma_{\pi N}$ | $45(6)$ | MeV | PhysRevD.87.074503
$\sigma_{sN}$ | $21(6)$ | MeV | PhysRevD.87.074503
$\langle q^{\dagger}iD_{0}q\rangle_{\rho}$ | $\frac{3}{2}m_{q}\rho$ | GeV4 | GUBLER20191 ; PhysRevC.47.2882
$\langle q^{\dagger}g_{s}\sigma Gq\rangle_{\rho}$ | $m^{2}_{0}\langle q^{\dagger}q\rangle_{\rho}$ | GeV5 | GUBLER20191
$\langle\bar{q}g_{s}\sigma Gq\rangle_{0}$ | $m^{2}_{0}\langle\bar{q}q\rangle_{0}$ | GeV5 | GUBLER20191
$m^{2}_{0}$ | $0.8\pm 0.2$ | GeV2 | GUBLER20191 ; Belyaev:1982sa
$\langle\bar{q}g_{s}\sigma Gq\rangle_{\rho}$ | $\langle\bar{q}g_{s}\sigma Gq\rangle_{0}+\rho\frac{m^{2}_{0}\sigma_{\pi N}}{2m_{q}}$ | GeV5 | GUBLER20191 ; PhysRevC.47.2882
$\langle\bar{q}iD_{0}iD_{0}q\rangle_{\rho}$ | $0.3~{}GeV^{2}\rho-\frac{1}{8}\langle\bar{q}g_{s}\sigma Gq\rangle_{\rho}$ | GeV5 | GUBLER20191 ; PhysRevC.47.2882
Table 1: Input parameters and their numerical values used in calculations.

The obtained in-medium sum rules for the mass, current coupling and vector
self-energy of $Z$ state contain two more auxiliary parameters, as well. These
parameters are fixed based on the standard requirements of the method: mild
variations of the physical quantities with respect to the changes in the
values of these parameters, dominance of first two resonances over the higher
states and continuum and convergence of the series of the operator product
expansion (OPE). To this end, we introduce the quantities

$\mbox{FTRC}=\frac{\Upsilon_{i}^{QCD}(M^{2},s^{*}_{0})}{\Upsilon_{i}^{QCD}(M^{2},\infty)},$
(21)

where $\mathrm{FTRC}$ stands for first two resonance contribution, and

$R=\frac{\Upsilon_{i}^{\mathrm{Dim5}}(M^{2},s^{*}_{0})}{\Upsilon_{i}^{QCD}(M^{2},s^{*}_{0})}.$
(22)

In Eq. (22) $\Upsilon_{i}^{\mathrm{Dim5}}(M^{2},s_{0})$ is the sum of all
operators with mass dimension 5, the highest dimension that we consider in the
calculations. The FTRC is used to fix upper bound for $M^{2}$, whereas
$R(M^{2})$ is necessary to fix the lower limit for the Borel parameter. To fix
the working regions, we impose $\mbox{FTRC}\geq 0.5$ and $R>0.1$ in order to
guarantee the first two resonances dominance over the higher states and
continuum and the OPE convergence. Our analyses show that all of the above
requirements are satisfied when

$\displaystyle\centering\left\\{\begin{array}[]{ll}M^{2}&\mbox{in}~{}[3-5]~{}\textrm{GeV}^{2},\\\
s_{0}^{*}&\mbox{in}~{}[21.9-23.8]~{}\textrm{GeV}^{2}.\end{array}\right.\@add@centering$
(25)

---
Figure 1: The in-medium mass of $Z$ state as functions of Borel mass parameter
$M^{2}$ at the nuclear matter saturation density and at fixed values of the
continuum threshold.
---
Figure 2: The vacuum mass, in-medium mass and vector self-energy of $Z$ state
as functions of $M^{2}$ at average value of continuum threshold and at nuclear
matter saturation density.
---
Figure 3: In-medium mass and vector self-energy as functions of
$\rho/\rho^{sat}$ at mean value of the continuum threshold and Borel
parameter.
---
Figure 4: The $\Sigma_{s}+\Sigma^{Z}_{\upsilon}$ and
$m_{Z}^{*}+\Sigma^{Z}_{\upsilon}$ as functions of $\rho/\rho^{sat}$ at mean
value of the continuum threshold and Borel parameter.
---
Figure 5: The current coupling of $Z$ state as a function of $\rho/\rho^{sat}$
at mean value of the continuum threshold and Borel parameter.
---
Figure 6: The mass $m^{*}_{Z}$ as a function of density when the uncertainties of the auxiliary and other input parameters are considered. |
---|---
|
|
Figure 7: The dependence of terms proportional to various condensates, entering the QCD side of the calculations, on $\rho/\rho^{sat}$ at average values of the auxiliary parameters.  | $m_{Z}$ | $m^{*}_{Z}$ | $f_{Z}.10^{2}$ | $f^{*}_{Z}.10^{2}$
---|---|---|---|---
PS |  $4486^{+112}_{-113}$ MeV |  $3311^{+203}_{-185}$ MeV |  $1.27^{+0.53}_{-0.36}$ GeV4 |  $1.34^{+0.78}_{-0.45}$ GeV4
Exp. PhysRevD.98.030001  |  $4478^{+15}_{-18}$ MeV | - | - | -
PhysRevD.96.034026  |  $4452^{+182}_{-228}$ MeV | - |  $1.48^{+0.31}_{-0.42}$ GeV4 | -
Wang:2014vha  |  $4.51^{+0.17}_{-0.09}$ GeV | - |  $(5.75^{+98}_{-78})/m_{Z}$ GeV4 | -
Lee:2007gs  |  $4.40\pm 0.10$ GeV | - | - | -
Bracco:2008jj  |  $4.52\pm 0.09$ GeV | - |  $(3.75\pm 0.48)/m_{Z}$ GeV4 | -
Chen:2019osl  |  $4.53^{+0.16}_{-0.10}$ GeV | - | - | -
Table 2: The vacuum and in-medium masses and current couplings of $Z$ state.
PS stands for present study.

In Fig. (1), we demonstrate the in-medium mass $m^{*}_{Z}$ as a function of
Borel mass parameter $M^{2}$ at the nuclear matter saturation density,
$\rho^{sat}=0.11^{3}$ GeV3, and at three fixed values of the continuum
threshold. As is seen, the in-medium mass of the $Z(4430)$ resonance shows
pretty good stability against variations of both of Borel mass parameter
$M^{2}$ and in-medium continuum threshold $s_{0}^{*}$. In Fig. (2), the vacuum
mass ($m_{Z}$) which is obtained at $\rho\rightarrow 0$ limit, the in-medium
mass ($m^{*}_{Z}$) and vector self-energy ($\Sigma^{Z}_{\upsilon}$) of $Z$
state are displayed as a function of $M^{2}$ at average value of continuum
threshold and at nuclear matter saturation density. From this figure we see a
considerable negative shift in the mass of the state under study due to the
medium effects. This shit is called the scalar self-energy. Strictly speaking,
the in-medium mass of the $Z$ resonance reduces to approximately $74\%$ of its
vacuum value at the nuclear matter saturation density and amounts as
$3311^{+203}_{-185}$ MeV. Thus, the negative shift in the mass due to the cold
nuclear matter is approximately $26\%$. This state gains a vector self-energy
with the value $1310^{+101}_{-86}$ MeV at nuclear matter saturation density.
At saturation density the current coupling of the state $Z$ is found as
$1.34^{+0.78}_{-0.45}$ GeV4. The numerical results for the masses and current
couplings obtained in vacuum and medium are collected in table (2). For
comparison, we demonstrate the existing values from the experiment and other
theoretical studies in the same table, as well. From this table, we see that
the vacuum mass obtained in the present study is nicely consistent with the
world experimental average PhysRevD.98.030001 . This value is also in accord
with other theoretical predictions within the errors. The vacuum current
coupling is consistent with other theoretical predictions within the presented
uncertainties. Our predictions for the in-medium mass and current coupling as
well as the vector self-energy can be tested in future in-medium experiments
and by other theoretical studies.

The main aim in the present study, is to obtain the behavior of the physical
quantities under study with respect to the density, specially at higher
densities. In the present study, we consider the range from zero to
$5\rho^{sat}$, corresponding to the density of the cores of massive neutron
stars. Neutron stars, as natural QCD matter laboratories, are very dense and
compact that may produce strange and heavy baryons as well as the exotic
states regarding the processes that may occur in their inside. For the neutron
stars with mass $\sim 1.5M_{\bigodot}$, the corresponding core density is
approximately (2-3) $\rho^{sat}$ and for the mass $\geq 2M_{\bigodot}$ the
density is about $5\rho^{sat}$ of the nuclear matter KKim . To understand the
process occurring at higher densities with the aim of getting knowledge on the
internal structures of the neutron stars as well as analyzing of the future
data in the heavy ion collision experiments, we need to investigate the
properties of various hadrons under extreme conditions. In the present study,
we deal with cold nuclear matter ($T\simeq 0$), hence there is no phase
transition expected up to the density that we consider in the calculations.
The phase transition to QGP is expected to occur at points $\geq
5.8\rho^{sat}$ at zero temperature Alice . The QGP state of matter is studied
in current heavy ion collision experiments at the LHC/CERN and RHIC at BNL.
Presently, the QCD phase diagram of quark matter is not well known, either
experimentally or theoretically, but there are some information from the
lattice QCD and other phenomenological models. Many experiments are designed
with the aim of investigation of the hadronic properties and possible phase
transitions at finite temperatures and densities. As previously mentioned, the
FAIR at GSI and the NICA at Dubna aim to provide useful information in this
respect.

To discuss the dependence of the physical quantities on the density, in Fig.
(3) we display $m^{*}_{Z}$ and $\Sigma^{Z}_{\upsilon}$ as functions of
$\rho/\rho^{sat}$ at average values of the continuum threshold and Borel
parameter. As it is clear, the sum rules for both the in-medium mass and
vector self-energy give reliable results up to $\rho=1.6\rho^{sat}$ GeV3. To
extend our calculations for higher densities, we use some fit functions. The
best fit for the mass is the following exponential function:

$m^{*}_{Z}(\rho)=4.384e^{-0.233x}~{}\textrm{GeV},$ (26)

where $x=\rho/\rho^{sat}$. For vector self-energy we obtain

$\Sigma^{Z}_{\upsilon}(\rho)=0.099x^{2}+0.917x+0.272~{}\textrm{GeV}.$ (27)

In order to better understand how $Z(4430)$ feels the total potential, and how
much energy would be necessary to produce this state in experiments using
nuclear targets, in Fig. (4), we show the dependence of
$\Sigma_{s}+\Sigma^{Z}_{\upsilon}$, or $m_{Z}^{*}+\Sigma^{Z}_{\upsilon}$ on
$\rho/\rho^{sat}$. We observe that both of these quantities grow with
increasing in the density of the medium.

In Fig. (5), we plot the density-dependent current coupling. The following
function best describes the current coupling:

$f^{*}_{Z}(\rho)=-0.002x^{2}+0.002x+0.015\textrm{GeV}^{4}.$ (28)

The current coupling becomes zero at $x=3.2$. The vector self-energy increases
with the increasing in the density, while the mass decreases with increasing
in the density and reaches to roughly $1.4GeV$ at $x=5$ in average. Fig. (6)
shows the $m^{*}_{Z}-x$ graphic when all the uncertainties in the auxiliary
parameters as well as in other inputs are considered. As it is clear from the
presented figures, the parameters of the $Z(4430)$ state depend on the
density, considerably. In Azizi:2014yea ; COHEN1995221 ; Navarra:2004bj ;
Azizi:2015ica , which study the behavior of the nucleon, hyperon and light
$\Theta^{+}$ pentaquark with respect to density up to $\rho\simeq
1.5\rho^{sat}$, the dependence of the parameters on density are relatively
small. In the present study we consider to investigate the behavior of many
parameters of the doubly-heavy excited $Z(4430)$ tetraquark state up to
$5\rho^{sat}$, corresponding to the density of the cores of massive neutron
stars, and observe that the considered parameters depend on the density,
remarkably. These behavior can be attributed to the considerable dependence of
the various terms proportional to different condensates in the QCD side of the
present calculations on the density as depicted in Fig. (7).

##  IV Summary and conclusions

The vacuum properties of different tetraquark states have been discussed in
many theoretical studies as well as different experiments. With the
developments in the experimental side, we hope we will be able to study the
behavior of such states in in-medium experiments in near future. Thus,
theoretical studies on the spectroscopic properties of exotics can play
crucial roles in conducting the related experiments. Such studies can also
help us fix the quantum numbers and determine the nature and internal
structures of the famous candidates of charmonium-like tetraquarks. In this
accordance, we took into consideration the $Z(4430)$ state in the present
study and calculated some spectroscopic parameters of this state both in
vacuum and a medium with finite density. Based on some experimental
information, we treated this state as the first excited state in the
$Z_{c}(3900)$ channel and constructed the related sum rules based on the
standard prescriptions of the method. As we saw, the parameters of the ground
state $Z_{c}(3900)$ were entered as input parameters to the sum rules for
$Z(4430)$ state. We fixed the entering auxiliary parameters in accordance with
the requirements of the method and used the expectation values of the
different in-medium operators available in the literature to extract the
values of the parameters at saturation density and determine the behavior of
these parameters with respect to the density.

The numerical calculations showed that the mass gains a negative shift (scalar
self-energy) due to the medium effects: this shifts amounts $-26\%$ at
saturation density. This state gains a repulsive vector self-energy which
amounts $1310^{+101}_{-86}$ MeV at saturation density. We also found the
values of the current coupling as $1.34^{+0.78}_{-0.45}$ GeV4 at nuclear
matter saturation density. This value can be used in determinations of
different parameters related to the electromagnetic, weak and strong
interactions/decays of this state in medium.

We obtained the behavior of the physical quantities in the interval
$\rho\in[0,5]\rho^{sat}$, which its upper limit corresponds to the core
density of the neutron stars. We observed that the current coupling becomes
zero at $\rho=3.2\rho^{sat}$, while the vector self-energy increases with the
density. The mass exponentially reduces with the increasing in the value of
the density and reaches to roughly $1.4GeV$ at the end point. With the
progress in the experimental side we hope that the heavy ion collision
experiments at CERN and BNL as well as the FAIR at GSI and NICA at Dubna will
be able to provide useful information on the behavior of the hadronic matter
under extreme conditions as well as possible phase traditions to QGP. With
upgraded detectors at RHIC and LHC, it has become possible to measure the
properties of hadrons beyond their ground states. Thus, these experiments
provide a new platform to experimentally study the exotic hadrons that are
either in molecular structure made of standard hadrons or compact objects of
multiquarks and antiquarks. The $\bar{P}ANDA$ experiment at FAIR is also
expected to investigate the properties of the standard hadrons as well as the
exotic states like tetraquarks, including the $Z_{c}(3900)$ and $Z(4430)$
states, at finite density.

## Acknowledgments

The authors thank TUBITAK for the partial support provided under the Grant No.
119F094.

## References

  * (1) M. Ablikim et al. [BESIII], “Observation of a Charged Charmoniumlike Structure in $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ at $\sqrt{s}$ =4.26 GeV,” Phys. Rev. Lett. 110, 252001 (2013).
  * (2) Z. Liu et al. [Belle], “Study of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ and Observation of a Charged Charmoniumlike State at Belle,” Phys. Rev. Lett. 110, 252002 (2013).
  * (3) S. S. Agaev, K. Azizi and H. Sundu, “Four-quark exotic mesons,”  Turk. J. Phys. 44, no. 2, 95 (2020).
  * (4) S. S. Agaev, K. Azizi and H. Sundu, “Strong $Z_{c}^{+}(3900)\rightarrow J/\psi\pi^{+};\eta_{c}\rho^{+}$ decays in QCD,”  Phys. Rev. D 93, no. 7, 074002 (2016).
  * (5) S. S. Agaev, K. Azizi and H. Sundu, “Treating $Z_{c}(3900)$ and $Z(4430)$ as the ground-state and first radially excited tetraquarks,” Phys. Rev. D 96, no.3, 034026 (2017).
  * (6) K. Azizi and N. Er, “Properties of $Z_{c}(3900)$ tetraquark in a cold nuclear matter,” Phys. Rev. D 101, no. 7, 074037 (2020).
  * (7) U. Ozdem and K. Azizi, “Magnetic and quadrupole moments of the $Z_{c}(3900)$,”  Phys. Rev. D 96, no. 7, 074030 (2017).
  * (8) S. Choi et al. [Belle], “Observation of a resonance-like structure in the $\pi^{\pm}\psi^{\prime}$ mass distribution in exclusive $B\to K\pi^{\pm}\psi^{\prime}$ decays,”  Phys. Rev. Lett. 100, 142001 (2008).
  * (9) K. Chilikin et al. [Belle], “Experimental constraints on the spin and parity of the $Z^{+}(4430)$,” Phys. Rev. D 88, no.7, 074026 (2013).
  * (10) K. Chilikin et al. [Belle], “Observation of a new charged charmoniumlike state in $\bar{B}^{0}\rightarrow J/\psi K^{-}\pi^{+}$ decays,” Phys. Rev. D 90, no.11, 112009 (2014).
  * (11) R. Aaij et al. [LHCb], “Observation of the resonant character of the $Z(4430)^{-}$ state,” Phys. Rev. Lett. 112, no.22, 222002 (2014).
  * (12) R. Aaij et al. [LHCb], “Model-independent confirmation of the $Z(4430)^{-}$ state,” Phys. Rev. D 92, no.11, 112009 (2015).
  * (13) X. Liu, Y. Liu, W. Deng and S. Zhu, “Is $Z^{+}(4430)$ a loosely bound molecular state?,” Phys. Rev. D 77, 034003 (2008).
  * (14) X. Liu, Y. Liu, W. Deng and S. Zhu, “$Z^{+}(4430)$ as a $D^{\prime}_{1}D^{*}(D_{1}D^{*})$ molecular state,” Phys. Rev. D 77, 094015 (2008).
  * (15) S. H. Lee, A. Mihara, F. S. Navarra and M. Nielsen, “QCD sum rules study of the meson Z+(4430),” Phys. Lett. B 661, 28-32 (2008).
  * (16) S. H. Lee, K. Morita and M. Nielsen, “Width of exotics from QCD sum rules: Tetraquarks or molecules?,” Phys. Rev. D 78, 076001 (2008), 076001.
  * (17) E. Braaten and M. Lu, “Line Shapes of the Z(4430),” Phys. Rev. D 79, 051503 (2009).
  * (18) J. Zhang and M. Huang, “{$Q\bar{q}$}{$\bar{Q}^{(^{\prime})}q$} molecular states,” Phys. Rev. D 80, 056004 (2009).
  * (19) T. Branz, T. Gutsche and V. E. Lyubovitskij, “Hidden-charm and radiative decays of the Z(4430) as a hadronic $D_{1}\bar{D^{\ast}}$ bound state,” Phys. Rev. D 82, 054025 (2010).
  * (20) F. Close, C. Downum and C. E. Thomas, “Novel Charmonium and Bottomonium Spectroscopies due to Deeply Bound Hadronic Molecules from Single Pion Exchange,” Phys. Rev. D 81, 074033 (2010).
  * (21) G. Wang, X. Liu, L. Ma, X. Liu, X. Chen, W. Deng and S. Zhu, “The strong decay patterns of $Z_{c}$ and $Z_{b}$ states in the relativized quark model,” Eur. Phys. J. C 79, no.7, 567 (2019).
  * (22) Z. Wang, “Axialvector tetraquark candidates for the $Z_{c}(3900)$, $Z_{c}(4020)$, $Z_{c}(4430)$, $Z_{c}(4600)$,” Chin. Phys. C 44, 063105 (2020).
  * (23) Z. Wang, “Analysis of the hidden-charm tetraquark mass spectrum with the QCD sum rules,” arXiv:1908.07914.
  * (24) F. Goerke, T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and P. Santorelli, “Four-quark structure of Zc(3900), Z(4430) and Xb(5568) states,” Phys. Rev. D 94, no.9, 094017 (2016).
  * (25) L. Maiani, F. Piccinini, A. Polosa and V. Riquer, “The Z(4430) and a New Paradigm for Spin Interactions in Tetraquarks,” Phys. Rev. D, 114010 89 (2014).
  * (26) Z. Wang, “Analysis of the $Z(4430)$ as the first radial excitation of the $Z_{c}(3900)$,” Commun. Theor. Phys. 63, no.3, 325-330 (2015).
  * (27) M. Bracco, S. Lee, M. Nielsen and R. Rodrigues da Silva, “The Meson $Z^{+}(4430)$ as a tetraquark state,” Phys. Lett. B 671, 240-244 (2009).
  * (28) D. Ebert, R. Faustov and V. Galkin, “Excited heavy tetraquarks with hidden charm,” Eur. Phys. J. C 58, 399-405 (2008).
  * (29) X. Liu, Q. Zhao and F. E. Close, “Search for tetraquark candidate Z(4430) in meson photoproduction,” Phys. Rev. D 77, 094005 (2008).
  * (30) D. Bugg, “How Resonances can synchronise with Thresholds,” J. Phys. G 35, 075005 (2008), 075005.
  * (31) E. S. Swanson, “$Z_{b}$ and $Z_{c}$ Exotic States as Coupled Channel Cusps,” Phys. Rev. D 91, no.3, 034009 (2015).
  * (32) Y. Ikeda et al. [HAL QCD], “Fate of the Tetraquark Candidate $Z_{c}$(3900) from Lattice QCD,” Phys. Rev. Lett. 117, no.24, 242001 (2016).
  * (33) I. Danilkin, V. Orlovsky and Y. Simonov, “Hadron interaction with heavy quarkonia,” Phys. Rev. D 85, 034012 (2012), 034012.
  * (34) S. Dubynskiy and M. Voloshin, “Hadro-Charmonium,” Phys. Lett. B 666, 344-346 (2008).
  * (35) R. M. Albuquerque, J. M. Dias, K. Khemchandani, A. Martínez Torres, F. S. Navarra, M. Nielsen and C. M. Zanetti, “QCD sum rules approach to the $X,~{}Y$ and $Z$ states,” J. Phys. G 46, no.9, 093002 (2019).
  * (36) S. L. Olsen, T. Skwarnicki and D. Zieminska, “Nonstandard heavy mesons and baryons: Experimental evidence,” Rev. Mod. Phys. 90, no.1, 015003 (2018).
  * (37) F. Guo, C. Hanhart, U. Meißner, Q. Wang, Q. Zhao and B. Zou, “Hadronic molecules,” Rev. Mod. Phys. 90, no.1, 015004 (2018).
  * (38) A. Esposito, A. Pilloni and A. Polosa,“Multiquark Resonances,” Phys. Rept. 668, 1-97 (2017).
  * (39) R. F. Lebed, R. E. Mitchell and E. S. Swanson, “Heavy-Quark QCD Exotica,” Prog. Part. Nucl. Phys. 93, 143-194 (2017).
  * (40) A. Ali, J. S. Lange and S. Stone, “Exotics: Heavy Pentaquarks and Tetraquarks,” Prog. Part. Nucl. Phys. 97, 123-198 (2017).
  * (41) H. Chen, W. Chen, X. Liu and S. Zhu, “The hidden-charm pentaquark and tetraquark states,” Phys. Rept. 639, 1-121 (2016).
  * (42) Y. Liu, H. Chen, W. Chen, X. Liu and S. Zhu, “Pentaquark and Tetraquark states,” Prog. Part. Nucl. Phys. 107, 237-320 (2019).
  * (43) A. Hosaka, T. Iijima, K. Miyabayashi, Y. Sakai and S. Yasui, “Exotic hadrons with heavy flavors: X, Y, Z, and related states,” PTEP 2016, no.6, 062C01 (2016).
  * (44) H. Chen and W. Chen, “Settling the Zc(4600) in the charged charmoniumlike family,” Phys. Rev. D 99, no.7, 074022 (2019).
  * (45) R. Zhu, “Hidden charm octet tetraquarks from a diquark-antidiquark model,”Phys. Rev. D 94, no.5, 054009 (2016).
  * (46) K. Azizi and N. Er, “Properties of nucleon in nuclear matter: once more,” Eur. Phys. J. C 74, 2904 (2014).
  * (47) Cho, Sungtae and others (ExHIC Collaboration), “Exotic hadrons from heavy ion collisions,”Prog. Part. Nucl. Phys. 95, 279–322 (2017).
  * (48) S. Cho et al. [ExHIC], “Studying Exotic Hadrons in Heavy Ion Collisions,” Phys. Rev. C 84 (2011), 064910.
  * (49) Wiedner, Ulrich, “Future Prospects for Hadron Physics at PANDA,”Prog. Part. Nucl. Phys. 66, 477–518 (2011).
  * (50) Kim, HyungJoo and Gubler, Philipp, “The $\phi$ meson with finite momentum in a dense medium,”Phys. Lett. B 805, 135412 (2020).
  * (51) T. Cohen, R. Furnstahl, D. Griegel and X. Jin, “QCD sum rules and applications to nuclear physics,” Prog. Part. Nucl. Phys. 35, 221-298 (1995).
  * (52) T. D. Cohen, R. Furnstahl and D. K. Griegel, “Quark and gluon condensates in nuclear matter,” Phys. Rev. C 45, 1881-1893 (1992).
  * (53) M. Tanabashi et al. [Particle Data Group], “Review of Particle Physics,” Phys. Rev. D 98, no.3, 030001 (2018).
  * (54) P. Gubler and D. Satow, “Recent Progress in QCD Condensate Evaluations and Sum Rules,” Prog. Part. Nucl. Phys. 106, 1-67 (2019).
  * (55) A. Bazavov, et al., “Staggered chiral perturbation theory in the two-flavor case and SU(2) analysis of the MILC data,”PoS LATTICE2010, 083 (2010).
  * (56) S. Borsányi, S. Dürr, Z. Fodor, S. Krieg, A. Schafer, E. E. Scholz and K. K. Szabo, “SU(2) chiral perturbation theory low-energy constants from 2+1 flavor staggered lattice simulations,” Phys. Rev. D 88, 014513 (2013).
  * (57) S. Dürr et al. [Budapest-Marseille-Wuppertal], “Lattice QCD at the physical point meets SU(2) chiral perturbation theory,” Phys. Rev. D 90, no.11, 114504 (2014).
  * (58) P. Boyle, N. Christ, N. Garron, C. Jung, A. Jüttner, C. Kelly, R. Mawhinney, G. McGlynn, D. Murphy, S. Ohta, A. Portelli and C. Sachrajda, “Low energy constants of SU(2) partially quenched chiral perturbation theory from Nf=2+1 domain wall QCD,” Phys. Rev. D 93, no.5, 054502 (2016).
  * (59) G. Cossu, H. Fukaya, S. Hashimoto, T. Kaneko and J. Noaki, “Stochastic calculation of the Dirac spectrum on the lattice and a determination of chiral condensate in 2+1-flavor QCD,” PTEP 2016, no.9, 093B06 (2016).
  * (60) X. Jin, T. D. Cohen, R. Furnstahl and D. K. Griegel, “QCD sum rules for nucleons in nuclear matter. II.,” Phys. Rev. C 47, 2882-2900 (1993).
  * (61) M. A. Shifman, A. Vainshtein and V. I. Zakharov, “QCD and Resonance Physics. Theoretical Foundations,” Nucl. Phys. B 147, 385-447 (1979).
  * (62) M. A. Shifman, A. Vainshtein and V. I. Zakharov, “QCD and Resonance Physics: Applications,” Nucl. Phys. B 147, 448-518 (1979).
  * (63) A. Mishra, H. Mishra and S. Misra, “Gluon condensates at finite baryon densities and temperature,” Z. Phys. C 59, 159-166 (1993).
  * (64) K. Saito, K. Tsushima and A. W. Thomas, “Quark and gluon condensates in the quark - meson coupling model,” Mod. Phys. Lett. A 13, 769-778 (1998).
  * (65) P. Shanahan, A. Thomas and R. Young, “Sigma terms from an SU(3) chiral extrapolation,” Phys. Rev. D 87, 074503 (2013).
  * (66) V. Belyaev and B. Ioffe, “Determination of Baryon and Baryonic Resonance Masses from QCD Sum Rules. 1. Nonstrange Baryons,” Sov. Phys. JETP 56, 493-501 ITEP-59-1982 (1982).
  * (67) K. Kim, H. K. Lee, J. Lee, “Compact Star Matter: EoS with New Scaling Law,” Int. J. Mod. Phys. E26 (2017) 1740011.
  * (68) A. Maire “Phase diagram of QCD matter : Quark-Gluon Plasma”, CDS - ALICE PhD, CERN-THESIS-2011-263 (p.19, Fig.I.8) https://cds.cern.ch/record/2025215?ln=en.
  * (69) F. S. Navarra, M. Nielsen and K. Tsushima, “A QCD sum rule study of $\Theta^{+}$ in nuclear matter,”Phys. Lett. B 606, 335–341 (2005).
  * (70) K. Azizi, N. Er and H. Sundu, “Positive and negative parity hyperons in nuclear medium,” Phys. Rev. D 92 (2015) no.5, 054026