# zkTax: A pragmatic way to support zero-knowledge tax disclosures

Alex Berke* 1, Tobin South* 1, Robert Mahari1 2, Kent Larson1, Alex ‘Sandy’
Pentland1
1MIT Media Lab
2Harvard Law School
* These authors contributed equally.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tax returns contain key financial information of interest to third parties:
public officials are asked to share financial data for transparency, companies
seek to assess the financial status of business partners, and individuals need
to prove their income to landlords or to receive benefits. Tax returns also
contain sensitive data such that sharing them in their entirety undermines
privacy. We introduce a zero-knowledge tax disclosure system (zkTax) that
allows individuals and organizations to make provable claims about select
information in their tax returns without revealing additional information,
which can be independently verified by third parties. The system consists of
three 3distinct services that can be distributed: a tax authority provides tax
documents signed with a public key; a Redact & Prove Service enables users to
produce a redacted version of the tax documents with a zero-knowledge proof
attesting the provenance of the redacted data; a Verify Service enables anyone
to verify the proof. We implement a prototype with a user interface,
compatible with U.S. tax forms, and demonstrate how this design could be
implemented with minimal changes to existing tax infrastructure. Our system is
designed to be extensible to other contexts and jurisdictions. This work
provides a practical example of how distributed tools leveraging cryptography
can enhance existing government or financial infrastructures, providing
immediate transparency alongside privacy without system overhauls.

_Keywords_ private data sharing $\cdot$ zero-knowledge $\cdot$ financial
disclosure $\cdot$ decentralization $\cdot$ cryptography $\cdot$ prototype

##  1 Introduction

Tax documents aggregate a wealth of information about an individual’s
finances, ranging from income sources to charitable contributions. Upon
submitting these documents to government agencies, individuals declare the
information is true and accurate to the best of their knowledge. Similarly,
organizations – ranging from non-profits to corporations – conduct financial
audits and report their financial data through tax filings.

The information contained in tax returns, while primarily intended for tax-
filing purposes, is often useful across a broad range of contexts. As a key
motivating example, public officials can release tax details in the pursuit of
increased accountability and transparency. Further, data derived from tax
records can be instrumental in assessing eligibility for benefits, such as
welfare benefits for low-income households [1, 2]. Additionally, in legal
proceedings such as divorce or child support negotiations, financial
attestations are often substantiated through the use of this information [1,
2].

Yet, can these requests for financial records be satisfied in a verifiable way
without exposing other private data present in tax records? Trustworthy
mechanisms to verify the accuracy and authenticity of financial attestations
are required.

We present a system to address this problem by leveraging zero-knowledge and
public key cryptography. Zero-knowledge (ZK) in this context refers to the
ability to computationally prove a statement is true, without exposing the
inputs to the statement. The system allows individuals to produce a redacted
version of their tax documents such that the information can be verified by
any third party. System users may either send the redacted documents to
inquirers or publish the redacted versions for anyone else to publicly verify.
Simple extensions of this system allow users to prove more complex statements
about their tax returns. This allows organizations and individuals to prove
their financial information to third parties without revealing sensitive data.

###  1.1 Motivating Example

Although we present multiple use cases for our system in section 5, we start
by highlighting a use case that will follow throughout demonstrating the
system.

This system is particularly motivated by the ability to leverage cryptographic
tools to improve transparency and accountability in governance systems, of
which the common practice of public officials releasing redacted versions of
their income tax returns is central. For example, in the US, presidents have
historically released their tax returns in a tradition of transparency.
Proponents of measures to codify this act of transparency argue that public
disclosure of tax returns could expose conflicts of interest, reveal annual
tax liability and tax rates, and enable the public to observe whether the
President or candidates have engaged in tax evasion, or pursued tax avoidance
[3]. Yet, there have been recent cases where public officials have refused to
release their taxes, despite demands from their own constituents and the
general public [4]. This work is motivated by the idea that if a zero-
knowledge tax disclosure mechanism were available, this refusal to release
requested information may not be necessary or defensible. In addition, the
information disclosed could be better trusted through verification.

Consider a scenario where citizens inquire about just one field from a public
official’s tax forms, such as the total amount of tax paid. The official
provides the requested amount, but refuses to release their tax returns due to
privacy concerns. The citizens may then question whether the disclosed amount
is truthful. The ZK tax disclosure system presented here offers a solution,
allowing the official to provide verifiable proof of the specified value from
their tax form without exposing sensitive information. This balance of privacy
and transparency can potentially enhance public trust in governance systems
through the application of Privacy by Design principles.

##  2 Background and related work

The system proposed here builds on both redactable signature schemes and
recent advances in zero-knowledge proof tools. Here we elaborate on the
foundational material of this work and contrast it to alternative systems that
have been proposed.

Zero-knowledge proofs are a way to prove the validity of a statement without
revealing any additional information beyond the truth of the statement itself
[5]. Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge (zk-
SNARKs) [6] have become a flexible ecosystem of proof generation software,
with general-purpose high-level languages like circom [7], which compiles down
to arithmetic equations called ‘circuits’. These circuits can then be used by
powerful proving systems such as groth16 [8] and halo2 [9] to produce small
digitally verifiable proofs.

This work draws on these zk-SNARK advancements and their application to the
financial system. These applications have focused on the blockchain space,
such as their use in achieving anonymous transactions [10, 11]. ZK proofs have
also been used to achieve anonymous e-taxing protocols (using blockchains)
[12], provable audits (zkLedger [13]), and proof of reserves [14]. Other
related work has also used blockchains to add privacy-preserving tools to tax
systems, even without zero-knowledge proofs [15]

The works above assume a closed financial system where transactions occur on a
blockchain. This is not today’s reality. Unlike these prior works, our system
does not assume a blockchain and can be implemented to work alongside and
improve current tax and financial infrastructures.

ZK tools such as this work can play a powerful role in creating new user
centric privacy pattern selections [16]. Indeed, since the introduction of
GDPR there has been an increase in Privacy by Design research from a
regulation-centric perspective [17], but user-centric solutions still need
research. While important work such as decentralized access control in
blockchain-based systems [18] has already been done in this space, the work we
present here provides a Privacy by Design approach where less technical users
can be in control of their privacy tradeoffs while interacting with systems
designed for the general public.

Other cryptographic approaches to providing verifiable redacted data have been
taken, such as with content extractable signatures (CES) [19] and redactable
signature schemes (RSS) [20]. In terms of our use case, we might consider a
tax form as a set of messages, where each form field and its associated value
is a message that can separately have an attached signature. A verifiable
redaction scheme may then publish a subset of these signed fields. Works
studying CES and RSS improve efficiency over providing a signature for each
such message. Yet these approaches suffer from drawbacks where either the size
of the signature or key must scale with the size of the original message (e.g.
original tax document). While this is not the case for our solution with zk-
SNARKs, we note that the size of the circuit in our solution does grow with
the allowed original tax document(s) size. However, unlike with CES and RSS,
this burden does not then fall on the services that handle signing and is
instead distributed. Furthermore, unlike CES and RSS, our flexible approach
with zk-SNARKs can be easily extended beyond simple redaction to prove
relations between form field values while avoiding publishing the values
themselves.

A more similar approach to this work is the use of zk-SNARKs to redact images
by Ko et al [21]. They used a combination of zk-SNARKs to produce a “public”
and “private” version of the image, and utilized the additive properties of a
Pederson commitment scheme so that a verifier could verify that commitments on
these two versions added up to a signed commitment from the original version.
Given the complexity of their approach, signature verification was handled
outside of a zk-SNARK, unlike our implementation. This process was designed
specifically for images as represented by pixels, while our simpler approach
can leverage the structured content of a tax form.

Our system assumes the existence of a digital service that uses public key
cryptography to sign tax documents. We note this assumption is not far-fetched
given the following work. In 2005 Mexico took on an ambitious public key
infrastructure project which allows citizens to sign tax declarations [22],
and Estonia’s modern government infrastructure allows for a variety of digital
signing including with taxes [23]. In general, the global push towards Open
Banking [24] beckons a possible future where many financial providers will
allow users to access permissions and signed attestations about financial
products and balances.

##  3 zkTax System

Figure 1: Diagram of the three services that compose the ZK tax disclosure
system. (Left) An individual, Alice, retrieves tax data from a Trusted Tax
Service (TTS), which signs the tax data with a signature that can be verified
with a public key. (Middle) Alice brings the signed tax data to the Redact &
Prove service. The service allows her to select how tax data should be
redacted and then generates a redacted version and proof that the original
data was signed by the TTS. Alice can make the proof and redacted data public
for anyone to then verify by using the Verify Service.

###  3.1 System goals

To enable and enhance usability the system should achieve the following.

  * •

_Interoperability with existing tax services_ : Building on top of existing
taxation infrastructure minimizes the burden for tax administrators while
facilitating adoption.

  * •

_Non-interactive from the perspective of the tax service_ : Tax service
providers (e.g. government agencies) should sign tax data once to enable users
to generate verifiable attestations from this data; minimizing costs for
service providers and mitigating likely bottlenecks.

  * •

_Public verifiability_ : Third parties should be able to independently verify
the redacted tax information. This should be user-friendly for end-users
without a background in cryptography.

  * •

_Flexibility and extensibility in terms of what can be proved_ : The system
should be adaptable to accommodate different user desires for combinations of
attestations and future extensions.

  * •

_Modularity_ : A modular design simplifies the system’s maintenance, upgrades,
and potential integration with other systems.

  * •

_Privacy by design for the taxpayer_ : The system should not reveal
unauthorized private data to any third party except the trusted tax service
and protect against re-identification attacks.

  * •

_Unforgeability_ : Individuals should be unable to fraudulently claim that
redacted tax information was sourced from real tax information handled by a
tax service.

###  3.2 Architecture

To achieve the above goals, we propose a system consisting of three distinct
services. A high-level view of this architecture is illustrated in Figure 1.
Figure 2 provides further detail of the system components and how data flows
between them. This section describes the proposed architecture at a high
level, where the specifics of implementation can vary. Section 4 describes our
prototype implementation in further detail.

| System participants | Services
---|---|---
|  Name |  Description |  Name |  Description
1 |  Trusted tax authority |  Entity that already has access to taxpayers’ tax data. E.g. in the U.S. this may be the IRS, or a company that processes taxes and sends them to the IRS on behalf of taxpayers (e.g. TurboTax). |  Trusted Tax Service (TTS) |  Returns tax data to individuals, where the data is signed and can be verified using a public key published by the TTS.
2 |  Individual taxpayer |  Individual who retrieves their signed tax data from the TTS and uses the Redact & Prove Service to redact fields from the data and produce a ZK proof about the redacted data. The full tax data is private to them. They share the redacted data. |  Redact & Prove
Service |  Shares a zk-SNARK circuit with the Verify Service. Provides an interface for taxpayers to privately select fields to redact from their tax data and produces a proof of the redacted data by using the zk-SNARK.
3 |  General
public or
auditor |  Any entity who wishes to verify a claim about the redacted tax data. They take the shared redacted tax data and proof to a Verify service. |  Verify
Service |  Shares a zk-SNARK circuit with the Redact & Prove Service. Provides interface for anyone to take shared redacted tax data and corresponding proof, output by the Redact & Prove Service, to verify.
Table 1: Overview of the system participants and their corresponding services
in the proposed tax verification system.

Table 1 summarizes the distinct participant roles that correspond to the 3
services.

In short, zk-SNARK circuit used to generate and verify proofs is publicly
shared among the Redact & Prove and Verify Services. A taxpayer receives their
signed and machine-readable tax data from the Trusted Tax Service (TTS), and
then uses the Redact & Prove Service to create a redacted version of their tax
data along with a proof of its validity. This redacted data and proof can then
be verified by any third-party using the Verify Service.

####  3.2.1 Trusted Tax Service (TTS)

Our system assumes that there is a trusted authority – which we refer to as
the Trusted Tax Service (TTS) – to which individuals submit their taxes. Such
an entity might be the government or a trusted tax preparer that submits taxes
to the government on an individual’s behalf. Possible examples in the American
tax system include the IRS, TurboTax, and H&R Block. Our proposal assumes
these services have the ability to transform the tax data they process into a
machine-readable format (e.g. JSON), have a known public key, and can return
the machine-readable version of the accepted tax data, along with a signature
and their public key, so that other entities can verify the tax data was
approved by them. We note the TTS assumed in this work is not currently
implemented, yet is reasonable in scope. The IRS published a multiyear
modernization plan in 2019 that described a taxpayer payment Application
Programming Interface (API) to support WebApps [25]. Intuit currently provides
an API [26] for third-party developers to build apps using QuickBooks data in
order to publish apps to the QuickBooks app store.

While ZK proofs may not be in the scope of software that these trusted
entities will provide in the immediate future, our proposed system allows
others to use the signed data returned from the TTS to build ZK proof systems
that enable the privacy-preserving disclosures described. Although this
approach does rely on existing entities returning signed tax data, we believe
this kind of technical progress to be more possible in the short run than
alternative solutions which have suggested complete replacement of existing
tax services (such as with a distributed ledger [27]).

In our proposed system, we assume the TTS provides a taxpayer’s complete and
unredacted tax information as JSON ($x$). The TTS uses the hash-then-sign
paradigm by using a hash function ($H$) to produce a hash of the user’s
complete tax information $H(x)=y$ and then signs $y$ using the TTS secret key
($sk$) as $S=sign(sk,y)$. The TTS returns $x$ and $S$ to the user. Note this
data is private to the taxpayer, as it still contains the unredacted tax
information, $x$. $H$ and the TTS public key, $pk$, are publicly known.

####  3.2.2 Redact & Prove Service

A taxpayer takes the tax data ($x$) and corresponding signature ($S$) output
from the TTS to a Redact & Prove Service as inputs. This service may be run by
any entity, but does require a certain level of trust because the input to the
front end of this service is private tax data ($x$). This service can and
should work by operating locally within the user’s machine, to avoid sending
private data to a server. This can be done by compiling all assets (including
files to make proofs from the zk-SNARK circuit) beforehand and serving these
files to the user’s browser.

The Redact & Prove Service operates in 2 steps. An interface allows the user
to insert their tax data from the TTS and select where redactions should be
made, producing an intermediate output: redact data. The proof generation step
then takes 2 inputs: The output ($x$, $S$) from the TTS and the redact data.
These inputs are provided to a zk-SNARK circuit that does the following. The
circuit first verifies that the input tax data, $x$, matches the signature,
$S$, by checking the hashed tax data against the TTS signature and public key:
$verify(H(x),pk,S)$. If this verification fails, an error is returned to
indicate that the tax information $x$ and signature $S$ did not match.
Otherwise, if the tax data $x$ are verified as valid, the circuit redacts all
data indicated by the redact data input from $x$ to produce $x^{\prime}$. The
circuit output is both the redacted version of the data, $x^{\prime}$, and a
proof, $\pi_{x^{\prime}}$. The Redact & Prove Service returns
$\pi_{x^{\prime}}$, $x^{\prime}$, $S$, and $pk$ so that others can check the
validity of $x^{\prime}$ (shown as the Reject/Accept output in Figure 2). The
original unredacted tax data, $x$, is the witness for the proof.

####  3.2.3 Verification service

The verification service allows for anyone to verify that the redacted tax
data, $x^{\prime}$, did indeed originate from some unredacted tax data, $x$,
that was signed by the TTS with with public key $pk$. It does so by compiling
the public circuit and verifying the compiled circuit, proof data
($\pi_{x}^{\prime}$), and redacted data ($x^{\prime}$) are compatible. Since
the circuit code includes verifying the signature with $pk$ on the full tax
data, $x$, this also means that the redacted tax data must match what the TTS
returned.

Figure 2: ZK Tax system overview with data flows.

##  4 Proof of concept implementation with U.S. tax forms

We developed a proof of concept implementation which includes a public web
app111 http://zktax.media.mit.edu , a set of reference circuits, and tax form
processing code. All code for the prototype is open source222
http://github.com/tobinsouth/zktax .

The web app is implemented using React. We implement the ZK proof and
verification by leveraging the Circom software ecosystem [7] to write and
compile R1CS circuits. We use the snarkjs javascript library [28] to handle
the trusted setup, to produce the circuit artifacts used for proof and
verification, as well as for implementing proof and verification within the
user’s browser.

###  4.1 Example Use Case

To demonstrate how this system can be used, we return to the motivating use
case where a public official wishes to make a provable statement about
specific information from their tax returns, without sharing other
information. In this example we use real data from a public official’s
released tax forms333Original forms available in the GitHub repository. . We
assume the official wishes to redact all information except “Taxable income"
(field 15 on Form 1040) and their name. The resulting redacted form is shown
in the Appendix Figure 6.

###  4.2 Data representation

Tax forms are often PDFs. We convert PDFs to and from a flat JSON
representation, where form fields map to values, with metadata for form name
and year assumed to be added by the TTS. The JSONs are converted to ASCII
strings as input for the zk-SNARK circuit.

    json: {
    // all data from the tax form

    "fname": "name",
    Ψ"lname": "name",
    "SSN": "000000000",

    "f_1": "393,229", // represents field 1
    "f_2a": "2,208", // represents field 2a

    // additional metadata
    "year": "2020",
    "form": "1040"
    }

In our interface, these machine data representations are converted into human-
readable JSONs and filled PDFs.

Our circuit uses an array of bits to represent the redaction data produced by
the taxpayer using the Redact & Prove Service, where each element in the array
indicates whether the corresponding element in the stringified JSON ASCII
array should be redacted or not.

###  4.3 zk-SNARK circuit

The zk-SNARK is implemented as an R1CS circuit with the following simple
logic: the JSON input is hashed and the circuit utilizes the TTS public key
(referred to as "pubkey") to assert that the input signature components
correspond with the hashed JSON. If the signature verification is
unsuccessful, a proof is not generated. Given a valid signature, the circuit
yields an output representing the redacted JSON. This output aligns with the
input JSON except where indicated by the redaction map bit array, which points
to the corresponding elements set to whitespace.

We use EdDSA/MiMC-7 for computing hashes and signatures [29]. The MiMC hash
function is chosen for its efficiency within zk-SNARK circuits, which is due
to its low multiplicative complexity [30]. We note consistent hash and sign
functions must be used within the circuit and TTS.

Groth16 is used as the proving system [8] requiring a circuit trusted setup
which is handled by using values computed from the Perpetual Powers of Tau
ceremony [31].

Once run, the Redact & Prove setup produces a computational proof.json and a
signals.json, the latter of which contains the redacted JSON string as ASCII
and the signer public key.

####  4.3.1 Circuit size complexity

The circuit is parameterized by the maximum JSON string input size such that
the size of the circuit grows with the size of the input. For larger circuit
sizes, the compilation needs larger Powers of Tau, and produces larger
verification files. Our prototype uses a maximum JSON size of 1500. Our open
source code provides files for other input sizes.

###  4.4 Prototype web app

Our initial implementation is based on a US income tax form: the 2020 IRS Form
1040 provided 444https://www.irs.gov/pub/irs-prior/f1040--2020.pdf. The web
interface for each service is shown in the appendix. Our implementation
includes a proof-of-concept TTS to which a user may upload a completed Form
1040 which is parsed into JSON, stringified, and hashed using MiMC before an
EdDSA signature is generated from
circomlibjs555https://github.com/iden3/circomlib/blob/master/circuits/eddsa.circom
which is returned, along with other necessary outputs, for the Redact & Prove
Service.

The redact & prove service interface generates the redaction bit array based
on the user’s selection of fields and the TTS output. Note that this service
is privately used by the taxpayer, given the input information contains all of
their private tax data. The zk-SNARK circuit files, which are used for
redaction, are precompiled using snarkjs and served to the frontend. snarkjs
is also used for the groth16 proof computation. All operations are done within
the client browser. The user can download these proof artifacts and share them
as they contain no private data.

Similarly, the prototype Verify Service precompiles a verification key from
the circuit via snarkjs using groth16 that is served to the service’s
frontend. Users can provide the proof.json and signals.json output from the
proof. The verification key is then used to verify that the signals and proof
are compatible, thereby accepting or otherwise rejecting the proof.

##  5 Discussion

###  5.1 Extensible circuit logics

This prototype focuses on using zero-knowledge proofs to prove that redacted
tax data originated from official signed tax forms. While this functionality
has numerous applications, extensions to this prototype, such as range-proofs
and various logical assertions, could also be encoded into R1CS circuits. Zero
Knowledge Range Proofs (ZKRP), such as proposed by Boudot [32], could allow a
user to assert that a value, such as total taxes paid, falls within a given
range. Meanwhile, circuits can be designed to support a wide array of logical
assertions, including but not limited to, statements like ‘the value of field
A is greater than that of field B’, or ‘the charitable donations made exceed a
certain percentage of the income earned’. Services 2 and 3 have the capacity
to incorporate ZK circuits that are tailored for each of these specific
scenarios. In-line with our system goals, our implementation of a redaction
service and circuit is aimed at generalizability and does not require specific
fields to be hard-coded. The resulting flexibility enhances our ability to
adapt to different use-cases without necessitating substantial alterations to
the underlying circuitry.

###  5.2 Example Use Cases

Extending beyond the general prototype example and its application to a public
official’s disclosure, we highlight two additional contexts where zkTax allows
individuals or organizations to prove financial information from their tax
returns to third parties without sharing sensitive data.

####  5.2.1 Efficient and verifiable individual tax data attestations

Individuals may need to prove their income or other financial information to
potential landlords, employers, and lenders. To this end, they can rely on tax
authorities (e.g. IRS) as a TTS and third-party providers to redact and prove
their tax return information. The individual can freely share authenticated
financial data and associated proofs with any requester (e.g. landlord or
bank), who can confirm the authenticity using any third-party verification
service.

In this way, zkTax approaches provide a trustworthy and privacy preserving way
to share financial data, and when combined with range proofs, obviating the
need for exact numbers to be shared directly, instead proving that a given
financial figure is above or below a specific threshold.

In addition to individuals and corporations attesting their financial
information, this system could streamline the process of eligibility checking
for government benefits. Governmental systems often separate data to maintain
privacy, meaning that a welfare or benefits agency may not have access to
original tax records. For welfare recipients who file tax returns, the
proposed tool could make it easier to prove eligibility for benefits.

Not only does this approach minimize the amount of private information that is
needlessly transmitted, it also reduces the cost of verification for the
recipient, and eliminates the risk of human errors during verification.

####  5.2.2 Verifying business counterparty risk

Business transactions often involve counterparty risk —- the risk that the
other party in a deal will be unable to fulfill its contractual obligations.
Assessing counterparty risk generally involves reviewing financial data, but
this may expose proprietary business information to competitors.

A business can rely on their tax auditors (e.g., EY or KPMG) to serve as its
TTS. When they enter into a new transaction, the counterparty can ask the
business to share certain financial items from its tax return. In many cases,
it will be sufficient for the business to provide range proofs or to prove
relationships between different financial items (e.g. liquidity ratio or
accounts payable days). Using the zkTax approach, a business can assemble the
requested information and associated proofs to put its transaction partner at
ease while minimizing unnecessary disclosures that could impact its
competitiveness.

###  5.3 Implementation requires minimal system overhaul and can be
distributed

Our implementation is a single web app to provide all three services, Only (1)
the trusted tax service (TTS) that accepts and signs the taxes needs to be
operated by a specific trusted tax authority (that is already entrusted with
sensitive tax information). The (2) Redact & Prove and (3) Verify services can
be operated by one or several other entities. This flexibility is a primary
advantage of the modular system architecture we propose. We believe it is a
reasonable expectation for a trusted tax service, such as the IRS, to return
signed JSON representations of individuals’ accepted tax data to those
individuals. Yet whether or not they also choose to provide zero-knowledge
proof and verify services, such as those described in this work, does not
matter for the system’s feasibility. Any other entity can then implement
services (2) and (3).

###  5.4 Limitations

A limitation of the proposed system is its reliance on a Trusted Tax Service
(TTS) to sign tax documents. Many tax authorities currently do not have
mechanisms for digitally signing individual tax documents in a manner
compatible with our system. While the architecture is designed to minimally
impact existing tax infrastructure, this requirement may pose an adoption
hurdle. It is conceivable that interim solutions could be deployed, such as
using email signatures from tax authorities [33] as a form of less-secure but
more readily available authentication.

Second, this work does not incorporate a user study involving real tax data to
understand individual expectations and demands for privacy. Such a study would
be essential for capturing the nuances of user interaction with the system and
evaluating its practical usability. Moreover, the system’s acceptance also
hinges on the willingness of third-party requesters to rely on zero-knowledge
proofs for data verification. This aspect remains an unexplored variable that
could significantly influence the system’s possibility of adoption.

##  6 Conclusion

We present a system that leverages zero-knowledge proofs and public key
cryptography to address the challenge of verifying tax information without
exposing sensitive data. Our system comprises three distinct services: a
Trusted Tax Service (TTS), a Redact & Prove Service, and a Verification
Service. Together, these services enable taxpayers to make provable claims
about their tax data that can then be verified by third parties. The
architecture requires minimal changes to existing tax systems, offers
flexibility in its deployment, and can be extended to support a variety of
claims about financial data.

We demonstrate the feasibility of our system through a publicly available
proof-of-concept implementation. This work contributes to financial privacy
and data sovereignty for individuals, enhancing political transparency and
accountability through public officials’ financial disclosures, and allows
businesses to assess counterparty risk without requiring proprietary or
sensitive information.

## Availability

The prototype with a public web interface is available at
http://zktax.media.mit.edu All code for the prototype web app, zk-SNARK
circuits, and tax form processing code, is available at
http://github.com/tobinsouth/zktax.

## Acknowledgments

We thank Husdon Hooper for his work on the visual design of the prototype web
application.

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##  Appendix A Protoype Interfaces

Figure 3: Prototype mock TTS interface. Users can insert Form 1040 and the
service returns the JSON representation of the data signed with the TTS key.
Figure 4: Prototype Redact & Prove Service interface showing fields relevant
to the example use case provided. Figure 5: Prototype Verify Service interface
showing fields relevant to the example use case provided. Figure 6: Redacted
version of U.S. income tax return Form 1040 for Donald Trump (2020 version),
based on publicly released tax returns. All fields are "redacted" (set to
blank whitespace) except the requested field 15 and his name (in order to
verify his identity for the public release of the redacted version). Note that
the form has more pages but the fields relevant for the example are on the
first page, so only the first page is shown.


# On phenomenological study of the solution of nonlinear GLR-MQ evolution
equation beyond leading order using recent PDF data

M. Lalung<EMAIL_ADDRESS>P. Phukan<EMAIL_ADDRESS>J. K. Sarma
<EMAIL_ADDRESS>HEP Laboratory, Department of Physics, Tezpur University,
Tezpur 784028, Assam, India

###### Abstract

We present a phenomenological study of the small-x behaviour of gluon
distribution function $G(x,Q^{2})$ at next-to-leading order (NLO) and next-to-
next-to-leading order(NNLO) in light of the nonlinear Gribov-Ryskin-Levin-
Mueller-Qiu (GLR-MQ)evolution equation by keeping the trasverse size of the
gluons ($\sim 1/Q$) fixed. We consider the NLO and NNLO corrections, of the
gluon-gluon spitting function $P_{gg}(z)$ and strong coupling constant
$\alpha_{s}(Q^{2})$. We have suggested semi-analytical solutions based on
Regge like ansatz of gluon density $G(x,Q^{2})$, which are supposed to be
valid in the moderate range of photon virtuality$(Q^{2})$ and at small Bjorken
variable$(x)$. The study of the effects of nonlinearities that arise due to
gluon recombination effects at small-x is very interesting, which eventually
tames down the unusual growth of gluon densities towards small-x as predicted
by the linear DGLAP evolution equation.

##  I  Introduction

The study of small-x behaviour of gluons is very interesting as the gluons
become most abundant partons inside the hadrons and can explain the behaviour
of QCD observables like the hadronic cross sections through their initial
distributions. Determination of parton distribution functions (PDFs) has
always been a fascinating task which has attracted and inspired various
collaboration groups like H1, ZEUS collaboration Abramowicz  _et al._ (2015),
NNPDF Ball  _et al._ (2015), CTEQ Lai  _et al._ (1997) etc and encouraged many
researchers in this field. Moreover, parton densities in hadrons assume key
roles in the understanding standard model processes as well as in predictions
of such processes at accelerators. But, in the domain of asymptotically
small-x, gluons are expected to dominate the proton structure function.
Therefore, determination of the gluon density in the small-x region is
particularly important. Knowledge of gluon densities or say gluon distribution
functions are essential also because of the fact that gluons serve as the
basic ingredients in calculation of various high energy hadronic processes,
for instance, the mini jet productions or in the computation of inclusive
cross sections of hard and collinearly factorizable hadronic collisions.
Moreover, in the study of p-p, p-A and A-A processes at small-x, at the
relativistic heavy-ion collider (RHIC) Hirai and Kumano (2009a, b) and at the
CERN’s LHC Manashov  _et al._ (2005); Ball  _et al._ (2013), the precise
knowledge of gluon distribution is essential.

The $x$ and $Q^{2}$ dependence of the gluon density can be predicted well with
much phenomenological success through standard QCD evolution equations. The
most basic and widely studied QCD evolution equations at twist-2 level are the
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) Altarelli and Parisi (1977)
and the Ballitsky-Fadin-Kuraev-Lipatov (BFKL) Kuraev  _et al._ (1976, 1977);
Balitsky and Lipatov (1978) equations. The solution of both these equations
predict sharp growth of gluon densities at high energies towards small-x, this
has been percieved with experimental results of deep inelastic scattering
(DIS) experiments at HERA Adloff  _et al._ (2001); Adloff et al. (2000, 2001);
Chekanov et al. (2001). The basic difference between DGLAP and BFKL approach
is that the former is based on resummation of large logarithmic $Q^{2}$ and
the later is based on resummation of large logarithmic $1/x$. In recent years
DGLAP equation has been established as the standard equation for
phenomenological study of DIS experiments as well as for global fits of parton
distribution function(PDF) by various groups Cafarella  _et al._ (2006);
Baishya  _et al._ (2009); Shah and Sarma (2008); Zarrin and Boroun (2017);
Khanpour  _et al._ (2017) and phenomenological study on the nonlinear
modification on dglap equations also have been performed Boroun and Zarrin
(2013); Boroun (2009).

The growth of the quark and gluon densities even though increases abruptly
towards large-$Q^{2}$, but they remain dilute with their transverse size
proportional to $1/Q$. At very high energies, region of smaller and smaller
values of x can be achieved and the number of gluons increases. These unusual
growth of gluons have to be tamed down by means of a mechanism so as to comply
with the Froissart bound Froissart (1961) and to preserve unitarity Martin
(1963). It is a well known fact that at high energies the hadronic cross
section comply with the Froissart bound. The Froissart bound states that the
hadronic total cross section cannot grow faster than the logarithm squared of
energy, which can be mathematically expressed as
$\sigma_{total}=\frac{\pi}{m_{\pi}}(\ln{s})^{2}$,where s is the square of the
centre of mass energy and $m_{\pi}$ is the scale of the strong force. Gluon
recombination is believed to serve as the mechanism responsible for a possible
saturation of gluon densities at small-x as well as unitarization of the
physical cross sections at high energies. The pioneering finding of the
geometrical scalling in the description of HERA data Staśto  _et al._ (2001)
and in the production of comprehensive jets in the LHC data Praszalowicz
(2011) suggests that the phenomenon of saturation occurs in nature Benić  _et
al._ (2017); Zhu and Lan (2017); Mueller (1999).

Although, the DGLAP evolution equations can dileneate the available
experimental data in a fairly broad range of $x$ and $Q^{2}$ with appropriate
parametrizations, but while trying to fit the H1 data using DGLAP approach, it
fails to provide a good description simultaneously in the region of
large-$Q^{2}$ $(Q^{2}>4\,GeV^{2})$ and in the region of small-$Q^{2}$
$(1.5\,GeV^{2}<Q^{2}<4\,GeV^{2})$ Adloff  _et al._ (2001); Adloff et al.
(2000, 2001); Chekanov et al. (2001). Also, in the descripton of the ZEUS data
at $Q^{2}=1\,GeV^{2}$ the DGLAP fit to gluon distribution function can be seen
predicticting a negative distribution towards small-x, see this work Chekanov
_et al._ (2003). The gluon recombination effects at small-x introduces
nonlinear power corrections to the linear DGLAP equation due to multiple gluon
interactions. These nonlinear terms help in taming down of the unusual growth
of gluon densities in the kinematics where the QCD coupling constant
$\alpha_{s}$ is still small in the dense partonic system. Gribov, Levin and
Ryskin (GLR) Gribov  _et al._ (1983) followed by Mueller and Qiu (MQ) Mueller
and Qiu (1986); Mueller (1994, 1995) did the first perturbative QCD (pQCD)
calculations by considering the fusion of two gluon ladders into one. These
calculations, on account of nonlinear corrections in terms of the quadratic
term in gluon density, gave rise to a new evolution equation popularly known
as the GLR-MQ equation. The GLR equation sums up all the fan diagrams i.e. all
the workable 2 $\rightarrow$ 1 ladder combinations which are computed in the
double leading logarithmic approximation(DLLA). Later, Mueller and Qiu
investigated the contributions of multiparton correlation at the twist-4
approximation based on the Glauber-Mueller model into a further simplified
GLR-MQ equation.

In our previous work, we had performed phenomenological study of the gluon
distribution function $G(x,Q^{2})$ by solving the GLR-MQ evolution equation
upto next-to-next-to leading order(NNLO) at small-xLalung  _et al._ (2017);
Phukan  _et al._ (2017). In that work we studied the gluon distribution
function $G(x,Q^{2})$ with respect to the resolution scale $Q^{2}$ at fixed
values of the momentum fraction x. However, in this work, we perform a
phenomenological study of x evolution of gluon distribution function
$G(x,Q^{2})$ at fixed value of resolution scale $Q^{2}$ in light of the GLR-MQ
equation in the kinematic range of small-x and moderate $Q^{2}$. In this
kinematic range the gluons are believed to show Regge like behaviour and it is
interesting to study the higher order effects on the solution of GLR-MQ
equation. Keeping the transverse size ($\sim 1/Q$) of the partons fixed at
asymptotically small-x, the gluon density becomes so high that we can
practically ignore the gluon contribution coming from the valence quarks
$P_{gq}$. The gluon recombination in this region plays vital role. The higher
order effects can be incorporated by incorporating the higher order terms of
the gluon-gluon splitting function $P_{gg}$ and that of the strong coupling
constant $\alpha_{s}(Q^{2})$. We show comparison of our results of gluon
distribution function, $G(x,Q^{2})$ with that of various collaborations or
groups like the CT14 Dulat  _et al._ (2016), NNPDF3.0 Ball  _et al._ (2015),
PDF4LHC Rojo  _et al._ (2015), ABMP16 Alekhin  _et al._ (2017) and MMHT14
Harland-Lang  _et al._ (2015). We have also compared our results with the
recent HERA PDF data viz. HERAPDF2.0 Abramowicz  _et al._ (2015). We have
studied the sensitivity of various parameters on our results and shown a
comparison of the nonlinear growth of gluon distribution function as predicted
by the GLR-MQ equation w.r.t. the gluon distribution as predicted by the
linear DGLAP equation at asymptotic small-x.

##  II  The nonlinear evolution equation

The GLR-MQ equation is a modified version of the linear DGLAP equation
differing from the later by means of an additional term quadratic in gluon
density $[xg(x,Q^{2})]^{2}$. This term is due to the correlative interaction
between the gluons inside the hadrons. This equation can be depicted as a
balance equation where, the net growth of the gluon density $x\Delta
g(x,Q^{2})$ in a phase cell $\Delta(1/x)\Delta\ln{Q^{2}}$ is due to the
collective effects of both the emission and annihilation processes. This
collective effect occurs when the chances for recombination of two gluons into
one is as prodigious as the chances for a gluon to split into two gluons. The
emission probability of gluons by a vertex g+g $\rightarrow$ g is proportional
to $\alpha_{s}\rho$ and that of the annihilation induced by the same vertex is
proportional to $\alpha_{s}^{2}r^{2}\rho^{2}$, where $\rho$ =
$xg(x,Q^{2})/S_{\perp}$ is the density of gluons having the transverse size of
$1/Q$ and $S_{\perp}=\pi R^{2}$ is the target area where the gluons inhabit, R
being the correlation radius. At x $\sim$ 1 only the emission of gluons is
essential because $\rho<<1$, but in the region of $x\rightarrow 0$, the gluon
density $\rho$ grows up and we cannot neglect gluon recombination. In terms of
the gluon density $xg(x,Q^{2})$, the GLR-MQ equation can be mathematically
expressed as Laenen and Levin (1995)

$\frac{\partial^{2}xg(x,Q^{2})}{\partial\ln(1/x)\partial\ln(Q^{2})}=\frac{\alpha_{s}N_{c}}{\pi}xg(x,Q^{2})-\frac{\alpha_{s}^{2}\gamma}{Q^{2}R^{2}}[xg(x,Q^{2})]^{2}.$
(1)

The value of the factor $\gamma$ was calculated to be $\frac{81}{16}$ for
$N_{c}=3$ by Mueller and Qiu. Now, in the DLLA Eq. (1) can be written in terms
of $G(x,Q^{2})=xg(x,Q^{2})$ as

$\frac{\partial G(x,Q^{2})}{\partial\ln Q^{2}}=\frac{\partial
G(x,Q^{2})}{\partial\ln
Q^{2}}\bigg{|}_{DGLAP}-\frac{81}{16}\frac{\alpha_{s}^{2}(Q^{2})}{R^{2}Q^{2}}\int_{x}^{1}\frac{dz}{z}G^{2}(\frac{x}{z},Q^{2})$
(2)

The R.H.S. of Eq. (2) consists of two terms, the first term is the linear
DGLAP term while the second term is responsible for shadowing of gluons. The
standard DGLAP equation in Mellin convolution space is given by

$\frac{d}{d\ln Q^{2}}\begin{pmatrix}f_{q_{i}}(x,Q^{2})\\\
f_{g}(x,Q^{2})\end{pmatrix}=\Sigma_{j}\begin{pmatrix}P_{q_{i}q_{j}}(z)&P_{q_{i}g}(z)\\\
P_{gq_{j}}(z)&P_{gg}(z)\end{pmatrix}\otimes\begin{pmatrix}f_{q_{i}}(x,Q^{2})\\\
f_{g}(x,Q^{2})\end{pmatrix},$ (3)

where the convolution $\otimes$ reperesents the prescription $f(x)\otimes
g(x)=\int_{x}^{1}(dz/z)f(z)g(x/z)$, $f_{q}$ and $f_{g}$ are the quark and
gluon distribution function respectively and $P_{q_{i}q_{j}}$, $P_{q_{i}g}$,
$P_{gq_{j}}$ and $P_{gg}$ are the parton splitting functions Altarelli and
Parisi (1977). We neglect the contribution coming from the splitting function
$P_{gq}$ in the small-x gluon rich region. Also, one thing we can notice from
Eq. (2) is that the size of the nonlinear term depends on the correlation
radius R. When the value of R is comparable to the radius of the hadron
($R_{h}$), the shadowing corrections are negligibly small, whereas for
$R<<R_{h}$, the shadowing corrections play vital role. We expect, as R grows
up the gluon distribution function $G(x,Q^{2})$ predicted by Eq. (2) will
become steeper and steeper. So, the correlation radius here is an important
factor which can control the growth of $G(x,Q^{2})$.

In Eq. (2), if we remove the nonlinear term then the equation just yields the
linear DGLAP evolution of gluon distribution function
$\widetilde{G}(x,Q^{2})$. The gluon distribution function $G(x,Q^{2})$
predicted from Eq. (2), is expected to rise slowly with decreasing $x$ for
fixed-$Q^{2}$ in comparison to $\widetilde{G}(x,Q^{2})$. We define a parameter
$R_{T}$ such that $R_{T}=G(x,Q^{2})/\widetilde{G}(x,Q^{2})$, this parameter
quantifies the amount of taming achieved with respect to the linear DGLAP
growth of gluon distribution function as $x$ decreases. We go on defining
another parameter $R_{r}$ such that
$R_{r}={G(x,Q^{2})^{R=2}}/{G(x,Q^{2})^{R=5}}$. The value of $R=5$ $GeV^{-1}$
means that the gluons are populated around the size of the proton and $R=2$
$GeV^{-1}$ signifies the gluons concentrated on the hotspots.

##  III The Regge approximation

One of the interesting phenomenon observed at HERA was the rise of proton
structure function $F_{2}$ towards small-x corresponding to a rising cross
section $\sigma_{{\gamma}^{\ast}p}$ with increasing invariant mass $W^{2}$
($\sim\frac{Q^{2}}{x}$) of the produced hadronic state. In the framework of
Regge theory, this behaviour of structure function plays important role in
understanding the behaviour of the observables, which predicts that the total
cross section $\sigma^{tot}(ab)$ varies as
$\sum_{i}\beta_{i}s^{\alpha_{i}-1}$, where $\alpha_{i}$ is the regge
trajectory and $\beta_{i}$ are the residue functions. According to Regge
theory, at small-x, the behaviour of gluons and sea quarks are controlled by
the same singularity factor in the complex plane of angular momentum. Small-x
behaviour of the sea quarks and antiquarks as well as the valence quarks
distributions are given by the power law $q_{val}(x)\sim x^{-\alpha}$, where
the Regge intercept $\alpha=1$ corresponds to a pomeron exchange of the sea
quarks and antiquarks while that of the valence quark is given by
$\alpha=0.5$. The small-x proton structure function $F_{2}$ is related to
$\sigma^{tot}_{\gamma^{\ast}p}$ which implies $F_{2}\sim x^{1-\alpha_{i}}$. In
the Regge inspired model developed by Donnachie-Landshoff(DL)Donnachie and
Landshoff (1998); Cudell  _et al._ (1999); Donnachie and Landshoff (2002),
$\beta_{i}$ are assumed to be dependent on $Q^{2}$ and the intercept
$\alpha_{i}$ are independent of $Q^{2}$.

In DL model the HERA data could be fitted very well on adding a hard exchanged
pomeron to that of the soft pomeron in the Regge theory. In this way, the
addition of a hard pomeron could describe the rise of structure function
$F_{2}$ at small-x. The simplest fit to the small-x data corresponds to

$F_{2}(x,Q^{2})=A(Q^{2})x^{-\epsilon_{0}},$

where $\epsilon_{0}=0.437$.

Thus, in this Regge inspired model, the high energy hard hadronic processes
are predominated by a hard pomeron exchange with the intercept of
$1+\epsilon_{0}$. The Mellin transform of $F_{2}$ would have a pole at
$j=1+\epsilon_{0}$ where its origin is perturbative QCD based on summation and
resummation of small-x logarithms. The logarithms of x become large in the
small-x regime and cannot be neglected, they need to be resummed based on the
BFKL equationFadin  _et al._ (1975); Balitsky and Lipatov (1978). Resummation
of these small-x contributions is performed in accordance with the so-called
$k_{T}$ factorization scheme. Solution to the leading order BFKL equation
leads to a pole of $j=1+4N_{c}\alpha_{s}\ln 2/\pi$ in the anglular momentum
plane corresponding to a hard pomeron intercept.

Donnachie and Landshoff in their work showed also that the result of
integration of the differential equation $\frac{\partial
F_{2}(x,Q^{2})}{\partial\ln Q^{2}}\sim P_{qq}\otimes G(x,Q^{2})$ at small-x
for the gluon distribution function is described by the exchange of a hard
pomeron only as $G(x,Q^{2})=A_{G}(Q^{2})x^{-\epsilon_{0}}$Donnachie and
Landshoff (1998); Ball and Landshoff (2000); Golec-Biernat  _et al._ (2002);
Cudell  _et al._ (1999); Donnachie and Landshoff (2002); Fadin  _et al._
(1975).

Towards the small-x region of DIS processes, it is believed to have a greater
possibility in exploring the Regge limit of perturbative QCD (pQCD). Models
based on Regge ansatz provide frugal paramterizations of the parton
distribution functions, $f(x,Q^{2})=A(Q^{2})x^{-\lambda}$ , where $\lambda$ is
pomeron intercept minus one. This type of behaviour of the Regge factorization
of the structure function $F_{2}^{c\bar{c}}$ has successful experimental back
up in the description of the DIS data of ZEUS in the kinematics of $x<0.07$
and $Q^{2}<10$ $GeV^{2}$ Donnachie and Landshoff (1998).

We therefore proceed by considering a simple form of Regge like behaviour
given as

$G(x,Q^{2})=\chi(Q^{2})x^{-\lambda_{G}},$

where $\lambda_{G}=(4N_{c}\alpha_{s}\ln{2})/\pi$, as appropriate for $Q^{2}>4$
$GeV^{2}$ Kwiecinski (1993) and $N_{c}$ is the number of color charges. The
value of $\lambda_{G}$ thus depends on the choice of $\alpha_{s}$. This
implies that $G^{2}(x/z,Q^{2})=G^{2}(x,Q^{2})z^{2\lambda_{G}}$. So, the Regge
intercept $\lambda_{G}$ will play a central role in our calculations. This
type of form is believed to be valid in the region of small-x and intermediate
range of $Q^{2}$, where $Q^{2}$ must be small but not so small that
$\alpha_{s}(Q^{2})$ is too large. But, it is to note that the Regge
factorization cannot be a good ansatz in the entire kinematic region. Regge
theory is apparent to be applicable when the invariant mass
$W^{2}(=(1/x-1)/Q^{2})$ is much greater than all other variables. So, the
kinematic range which in fact we are considering viz., $10^{-5}\leq x\leq
10^{-2}$ and $5\,GeV^{2}\leq Q^{2}\leq 30\,GeV^{2}$ fall in the Regge regime.

##  IV Semi-analytical solution beyond leading order

We incorporate the higher order terms of the splitting function $P_{gg}$ and
the QCD coupling constant $\alpha_{s}(Q^{2})$. Both of these terms can be
expanded perturbatively to include higher order contributions coming from
higher twist effects. Considering the next-to-leading order (NLO) and next-to-
next-to-leading order (NNLO) terms, $\alpha_{s}$ can be written as

$\alpha_{s}({Q^{\prime}}^{2})^{NLO}=\frac{4\pi}{\beta_{0}\ln{{Q^{\prime}}^{2}}}(1-b\frac{\ln{(\ln{{Q^{\prime}}^{2}})}}{\ln{{Q^{\prime}}^{2}}}),$
$\alpha_{s}({Q^{\prime}}^{2})^{NNLO}=\frac{4\pi}{\beta_{0}(\ln{{Q^{\prime}}^{2}})^{2}}\bigg{\\{}\ln{{Q^{\prime}}^{2}}-b\ln{(\ln{{Q^{\prime}}^{2}})}b^{2}\big{(}{\ln^{2}{(\ln{{Q^{\prime}}^{2}}})}-\ln{(\ln{{Q^{\prime}}^{2}})}-1\big{)}+c\bigg{\\}},$

where $b=\frac{\beta_{1}}{\beta_{2}^{2}}$, $c=\frac{\beta_{2}}{\beta_{0}^{3}}$
$\beta_{0}=11-\frac{2}{3}N_{f}$, $\beta_{1}=102-\frac{38}{3}N_{f}$ and
$\beta_{2}=\frac{2857}{2}-\frac{6673}{28}N_{f}+\frac{325}{54}N_{f}^{2}.$

Here we take the number of flavors $N_{f}=4$ and we use the notation
${Q^{\prime}}^{2}=Q^{2}/\Lambda^{2}$, where $\Lambda$ is the QCD cut off
parameter. The splitting function $P_{gg}$ can also be expanded in powers of
$\frac{\alpha_{s}({Q^{\prime}}^{2})}{2\pi}$ as follows:

$P_{gg}(z,{Q^{\prime}}^{2})=TP_{gg}^{(0)}(z)+T^{2}P_{gg}^{(1)}(z)+T^{3}P_{gg}^{(3)}(z),$

where $T\equiv T({Q^{\prime}}^{2})=\alpha_{s}({Q^{\prime}}^{2})/2\pi$ and
$P_{gg}^{(0)}(z)$, $P_{gg}^{(1)}(z)$ and $P_{gg}^{(0)}(z)$ are the LO, NLO and
NNLO terms of the gluon-gluon splitting function respectively. Expressions of
these splitting functions are given in the Appendix. Derivations of these
higher order splitting kernels can be found in Vogt  _et al._ (2004).

In terms of the variable ${Q^{\prime}}^{2}$, Eq. (2) can be re-wriiten in the
following form:

$\frac{\partial G(x,{Q^{\prime}}^{2})}{\partial
ln{Q^{\prime}}^{2}}=\frac{\partial
G(x,{Q^{\prime}}^{2})}{\partial\ln{Q^{\prime}}^{2}}\bigg{|}_{DGLAP}-\frac{81}{16}\frac{\alpha_{s}^{2}({Q^{\prime}}^{2})}{R^{2}{Q^{\prime}}^{2}\Lambda^{2}}\int_{x}^{1}\frac{dz}{z}G^{2}(\frac{x}{z},{Q^{\prime}}^{2})$
(4)

So, we notice that choice of the QCD cut off parameter $\Lambda$ is also
important in this equation. The ${Q^{\prime}}^{2}$ dependence of $\alpha_{s}$
makes the nonlinear Eq. (4) further more complicative at NLO and NNLO. We
define two new parameters $T_{0}$ and $T_{1}$ such that $T^{2}\approx T\cdot
T_{0}$ and $T^{3}\approx T\cdot T_{1}$ respectively. These two parameters are
estimated using nonlinear model fiiting techniques in the region of 5
$GeV^{2}\leq Q^{2}\leq$ 30 $GeV^{2}$, which is the region of our interest in
this work. These parametrizations simplify the nonlinear equation which then
can be solved for $x$.

Table 1: Table showing the parameter statistics of the parameters $T_{0}$ and $T_{1}$ Parameter | Estimate | Standard Error | t-Statistic | P-Value
---|---|---|---|---
$\bf T_{0}$ | 0.0364162 | 3.41393$\times 10^{-4}$ | 106.669 | 8.104$\times 10^{-105}$
$\bf T_{1}$ | 0.00135821 | 0.164091$\times 10^{-4}$ | 82.7717 | 6.32517$\times 10^{-94}$

###  IV.0.1 On considering upto NLO terms

Now, in order to obtain the solution of Eq. (4) by considering upto the NLO
terms of the running coupling constant and the gluon-gluon splitting function,
we put the corresponding terms in Eq. (4). After few algebra and
simplifications, the GLR-MQ equation in terms of the variable
${Q^{\prime}}^{2}$ takes up the form of following partial differential
equation

$\frac{\ln({Q^{\prime}}^{2})}{\left[1-b\ln(\ln({Q^{\prime}}^{2}))/{\ln({Q^{\prime}}^{2}})\right]}\frac{\partial
G(x,{Q^{\prime}}^{2})}{\partial\ln({Q^{\prime}}^{2})}=\psi(x)G(x,{Q^{\prime}}^{2})-\phi(x)\frac{G^{2}(x,{Q^{\prime}}^{2})}{{Q^{\prime}}^{2}\Lambda^{2}}.$
(5)

The nonlinearity in Eq. (5) is definitely seen in terms of the quadratic term
in $G(x,Q^{2})$, in addition to this, the complicative functional form of the
running coupling constant in ${Q^{\prime}}^{2}$ makes it difficult to have
exact analytical solution. However, the solution to Eq. (5) have the following
functional form

$G(x,{Q^{\prime}}^{2})=\frac{e^{\frac{b\psi(x)}{\ln({Q^{\prime}}^{2})}}\ln({Q^{\prime}}^{2})^{(1+\frac{b}{\ln({Q^{\prime}}^{2})})\psi(x)}}{C+\int_{1}^{\ln({Q^{\prime}}^{2})}\frac{e^{\zeta(x,y)}\phi(x)(y-b\ln
y)dy}{y^{2}}}.$ (6)

C is a constant of integration which can be determined using suitable initial
condition of gluon distribution function for a fixed-${Q^{\prime}}^{2}$ at a
given $x_{0}(>x)$. We take input value of gluon distribution function
$G(x_{0},{Q^{\prime}}^{2})$ from PDF4LHC15 PDF data at a larger value of $x$
($x_{0}$) for a given value of ${Q^{\prime}}^{2}$. We have taken the input
value at momentum fraction $x=10^{-2}$ from PDF4LHC15, this set is based on
the LHC experimental simulations and the 2015 recommendations Rojo  _et al._
(2015) of the PDF4LHC working group. PDF4LHC15 PDFs contain combinations of
more recent CT14 Dulat  _et al._ (2016), MMHT2014 Harland-Lang  _et al._
(2015), and NNPDF3.0 Ball  _et al._ (2015) PDF ensembles and are based on an
underlying Monte Carlo combination of these three PDF groups, denoted by
MC900. Thus, the NLO x-evolution of $G(x,{Q^{\prime}}^{2})$ for
smaller-x($x<x_{0}$) at fixed-${Q^{\prime}}^{2}$ with proper initial condition
is given by

$G(x,{Q^{\prime}}^{2})=\frac{G(x_{0},{Q^{\prime}}^{2})e^{\frac{b\psi(x)}{\ln({Q^{\prime}}^{2})}}\ln({Q^{\prime}}^{2})^{(1+\frac{b}{\ln({Q^{\prime}}^{2})})\psi(x)}}{{\ln({Q^{\prime}}^{2})}^{(1+\frac{b}{\ln({Q^{\prime}}^{2})})\psi(x_{0})}e^{\frac{b\psi(x_{0})}{\ln({Q^{\prime}}^{2})}}+G(x_{0},{Q^{\prime}}^{2})\int_{1}^{\ln({Q^{\prime}}^{2})}\frac{\\{e^{\zeta(x,y)}\cdot\phi(x)-e^{\zeta(x_{0},y)}\phi(x_{0})\\}(y-b\ln
y)dy}{y^{2}}};$ (7)

$\forall\,x\leq x_{0}$, where the functions involved are given below:

$\begin{split}\small\psi(x)=&\frac{12}{\beta_{0}}\bigg{\\{}\frac{11}{12}-\frac{N_{f}}{18}+\ln(1-x)+\frac{2}{2+\lambda_{G}}-\frac{2x^{\lambda_{G}+2}}{\lambda_{G}+2}+\frac{x^{\lambda_{G}}}{\lambda_{G}}-\frac{1}{\lambda_{G}}-x+1\bigg{\\}}+\frac{2T_{0}}{\beta_{0}}\int_{x}^{1}dzP_{gg}^{(1)}(z)z^{\lambda_{G}},\\\
\zeta(x,y)&=\frac{b\psi(x)}{y}-y+\psi(x)\ln{y}+b\psi(x)\frac{\ln{y}}{y},\phi(x)=T_{0}\cdot\frac{81\pi^{2}}{2\beta_{0}R^{2}\Lambda^{2}}\bigg{(}\frac{1-x^{2\lambda_{G}}}{2\lambda_{G}}\bigg{)}\end{split}$

###  IV.0.2 On considering upto NNLO terms

Now considering upto NNLO terms the GLR-MQ equation upto NNLO takes up the
following form:

$\small\frac{(\ln{{Q^{\prime}}^{2}})^{2}}{\left[\ln{{Q^{\prime}}^{2}}-b\ln(\ln{{Q^{\prime}}^{2}})-b^{2}\ln(\ln{{Q^{\prime}}^{2}})+b^{2}\ln((\ln{{Q^{\prime}}^{2}})^{2})-b^{2}+c\right]}\frac{\partial
G(x,{Q^{\prime}}^{2})}{\partial\ln({Q^{\prime}}^{2})}=\gamma(x)G(x,{Q^{\prime}}^{2})-\phi(x)\frac{G^{2}(x,{Q^{\prime}}^{2})}{{Q^{\prime}}^{2}\Lambda^{2}}$
(8)

We follow the same procedure to solve this partial differential equation as in
the NLO case. Finally, after putting the initial conditions the x-evolution
solution (for $x\leq x_{0}$) of this equation for fixed- ${Q^{\prime}}^{2}$ is
given by,

$\begin{split}\small&G(x,{Q^{\prime}}^{2})\\\
&=\frac{G(x_{0},{Q^{\prime}}^{2})e^{(\frac{b}{\ln{Q^{\prime}}^{2}}-\frac{c}{\ln{Q^{\prime}}^{2}}-\frac{b^{2}\ln^{2}{(\ln{Q^{\prime}}^{2})}}{\ln{Q^{\prime}}^{2}})\gamma(x)}(\ln{Q^{\prime}}^{2})^{(1+\frac{b}{\ln{Q^{\prime}}^{2}}-\frac{b^{2}}{\ln{Q^{\prime}}^{2}})\gamma(x)}}{e^{(\frac{b}{\ln{Q^{\prime}}^{2}}-\frac{c}{\ln{Q^{\prime}}^{2}}-\frac{b^{2}\ln^{2}{(\ln{Q^{\prime}}^{2})}}{\ln{Q^{\prime}}^{2}})\gamma(x_{0})}(\ln{Q^{\prime}}^{2})^{(1+\frac{b}{\ln{Q^{\prime}}^{2}}-\frac{b^{2}}{\ln{Q^{\prime}}^{2}})\gamma(x_{0})}+G(x_{0},{Q^{\prime}}^{2}){\int_{1}^{\ln{Q^{\prime}}^{2}}\frac{\left(\phi(x)e^{\Delta(x,y)}-\phi(x_{0})e^{\Delta(x_{0},y)}\right)\eta(y)}{y^{2}}dy}},\\\
&\Delta(x,y)=\bigg{(}\frac{b}{y}-\frac{c}{y}+\ln{y}+\frac{b\ln{y}}{y}-\frac{b^{2}\ln{y}}{y}-\frac{b^{2}\ln^{2}{y}}{y}\bigg{)}\gamma(x)-y,\gamma(x)=\psi(x)+\frac{2T_{1}}{\beta_{0}}\int_{x}^{1}dzP_{gg}^{(2)}(z)z^{\lambda_{G}},\\\
&\eta(y)=-b^{2}+c+y-b\ln{y}-b^{2}\ln{y}+b^{2}\ln^{2}{y}\end{split}$ (9)

After computing all these solutions in terms of the variable
${Q^{\prime}}^{2}$, we can return back to our original variable $Q^{2}$ by
just putting $Q^{2}/\Lambda^{2}$ in place of ${Q^{\prime}}^{2}$. Thus
x-evolution of gluon distribution function $G(x,Q^{2})$ from the nonlinear
GLR-MQ equation beyond the leading orders can be obtained based on the Regge
behaviour of gluons at small-x and moderate-$Q^{2}$.

##  V Discussions

Figure 1:  Parameter fitting of $T^{2}(Q^{2})$ and $T^{3}(Q^{2})$ with respect
to $Q^{2}$.

So, we have suggested semi analytical solutions of the GLR-MQ equation at NLO
and NNLO based on the Regge like behaviour of gluons in the kinematic range of
$10^{-5}\leq x\leq 10^{-2}$ and $5\,GeV^{2}\leq Q^{2}\leq 30\,GeV^{2}$. Our
solution predicts the x-evolution of gluon distribution function $G(x,Q^{2})$
at NLO and NNLO for fixed-$Q^{2}$ which is also consistent with our previous
result at the LO Devee and Sarma (2014).

Fig. 1 shows a comparison of $T^{2}(Q^{2})$ with $T(Q^{2}).T_{0}$ and
$T^{3}(Q^{2})$ with $T(Q^{2}).T_{1}$ for $5\,GeV^{2}\leq Q^{2}\leq
30\,GeV^{2}$. We have multiplied both $T^{3}(Q^{2})$ and $T(Q^{2}).T_{1}$ by a
factor of 10, so as to represent all these variations in a single figure. We
have determined the value $T_{0}=0.0364162$ and $T_{1}=0.00135821$ for the
best fit of the data in the range of our consideration. Table I shows the
parameter statistics associated with the fitted parameters.

(a)  (b)

(c)  (d)

Figure 2: $x$-evolution of $G(x,Q^{2})$ for R= 2 $GeV^{-1}$ at four different values of $Q^{2}$ viz. $Q^{2}$ = 5, 10, 25 and 30 $GeV^{2}$. Figure showing comparison of our results with that of the global fits by various groups. Here, our LO, NLO and NNLO results are represented by dotted violet lines, dotted blue lines and dotted red lines respectively. Table 2: Table showing the various parameters for the best fit of the graphs in Fig. 2(a-d)  |  $\bf Q^{2}=5$ ($\bf GeV^{2}$) |  $\bf Q^{2}=10$ ($\bf GeV^{2}$) |  $\bf Q^{2}=25$ ($\bf GeV^{2}$) |  $\bf Q^{2}=30$ ($\bf GeV^{2}$)
---|---|---|---|---
Parameters | LO | NLO | NNLO | LO | NLO | NNLO | LO | NLO | NNLO | LO | NLO | NNLO
R ($\bf GeV^{-1}$) | $2$ | $2$ | $2$ | $2$ | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
$\bf\lambda_{G}$ | 0.36 | 0.35 | 0.39 | 0.33 | 0.32 | 0.38 | 0.325 | 0.31 | 0.36 | 0.32 | 0.31 | 0.36
$\bf\alpha_{s}$ | 0.136 | $0.132$ | $0.147$ | 0.125 | 0.121 | 0.143 | 0.123 | 0.117 | 0.136 | 0.120 | 0.117 | 0.136
$\bf\Lambda$ (GeV) | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3

In Fig. 2(a-d), we plot our NLO as well as NNLO solutions for $G(x,Q^{2})$ in
the kinematics of $10^{-5}\leq x\leq 10^{-2}$ at $Q^{2}=5,\,10,\,25$ and $30$
$GeV^{2}$. It can be observed that our NNLO result lies slightly above the NLO
result. This is due to the additional higher order gluon-gluon splitting terms
present in the splitting function $P_{gg}(z)$. The value of the Regge
intercept $\lambda_{G}$ is crucial in this phenomenological study. The strong
coupling constant $\alpha_{s}$ also enters into the picture through the
relation $\lambda_{G}=(4\alpha_{s}N_{c}/\pi)\ln{2}$ Kwiecinski (1993) which
can control the growth of $G(x,Q^{2})$. We have shown comparison of our
results with those obtained by global DGLAP fits by various collaborations
like CT14 Dulat  _et al._ (2016), NNPDF3.0 Ball  _et al._ (2015), HERAPDF2.0
Abramowicz  _et al._ (2015), PDF4LHC Rojo  _et al._ (2015), ABMP16 Alekhin
_et al._ (2017) and MMHT14 Harland-Lang  _et al._ (2015). We have used APFEL
tool Carrazza  _et al._ (2015) to generate the gluon distribution functions of
these collaborations in the kinematics of $10^{-5}\leq x\leq 10^{-2}$ for
$Q^{2}=5,\,10,\,25$ and $30$ $GeV^{2}$. We have used the LHAPDF6 Buckley  _et
al._ (2015) PDF grids to generate these data. For the best fit of results
shown in Fig. 3(a-d), all the parameters that we have considered are listed in
the Table II.

In the Fig. 3, we plot the ratio $R_{T}$ of gluon distribution function
$G(x,Q^{2})$ predicted from GLR-MQ equation to the gluon distribution function
$\widetilde{G}(x,Q^{2})$ predicted from DGLAP equation. We have shown
comparison of $R_{T}$ values for four fixed values of $Q^{2}$ viz., 5, 10, 25
and 30 $GeV^{2}$. We observe that as we go towards small-x, the $R_{T}$ value
deceases, i.e., the taming is more towards small-x for a fixed-$Q^{2}$. We
also observe that on increasing $Q^{2}$, the $R_{T}$ value also increases,
this means that taming is lesser for higher-$Q^{2}$ than for low-$Q^{2}$. This
makes sense because the transverse size of the gluons grows up as $1/Q$, the
smaller is the size of gluons lesser is the amount of shadowing. We also
observe that the taming of our NNLO solution is more as compared to the NLO
solution as x decreases.

In Fig. 4(a-b), we check the sensitivity of R and $\lambda_{G}$ on our
results. In Fig. 5(a), we plot the NLO and NNLO gluon distribution functions
$G(x,Q^{2})$ in the same kinematic range that we are considering, for four
different values of $\lambda_{G}$ viz., 3, 4, 5 and 6. For reference we take
$Q^{2}=30\,GeV^{2}$ and R = 2 $GeV^{-1}$ respectively. We observe a sharp rise
of $G(x,Q^{2})$ towards small-x as we increase the value of $\lambda_{G}$ . We
notice that due to the NNLO corrections, $G(x,Q^{2})$ rises faster than that
of the gluon distribution function $G(x,Q^{2})$ when only the NLO terms were
incorporated. Finally, in Fig. 5(b) we plot the ratio ($R_{r}$) between
$G(x,Q^{2})$ at $R=2$ $GeV^{-1}$ to $G(x,Q^{2})$ at $R=5$ $GeV^{-1}$. It can
be observed that the value of $R_{r}$ decrease as x decreases for a
fixed-$Q^{2}$. This is a confirmation of the fact that the taming of
$G(x,Q^{2})$ is more when the gluons are concentrated at the hotspots(R = 2
$GeV^{-1}$) than when they are spread throughout the size of the proton (R = 5
$GeV^{-1}$). As we increase $Q^{2}$ the ratio $R_{r}$ shifts upwards as x
decreases, this means the taming will be less when $Q^{2}$ increases on
decreasing x. This is again confirmation of the fact that the size of gluons
grows as $1/Q$. Also, we observe that the taming is more in NNLO solution than
that of the NLO solution as x decreases for a fixed-$Q^{2}$. This is due to
the additional NNLO term present in the gluon-gluon splitting function
$P_{gg}$.

Figure 3: $R_{T}$ ratio of the solution of nonlinear GLR-MQ equation to the
linear DGLAP equation.

(a)  (b)

Figure 4: Sensitivity of $\lambda_{G}$ and R on our results. In Fig. (a)
$G(x,Q^{2})$ is shown as a function of x at $Q^{2}$ = 30 $GeV^{2}$ for four
different values of $\lambda_{G}$ viz., $\lambda_{G}$ = 3, 4, 5 and 6
respectively. In Fig. (b) the variation of $R_{r}$ is plotted as a function of
x for $\alpha_{s}=0.120$ for four different values of $Q^{2}$ viz., $Q^{2}$ =
5, 10, 25 and 30 respectively.

##  VI Summary

In summary, in this work, we have presented a phenomenological study of the
nonlinear effects of gluon distribution function, $G(x,Q^{2})$ in the
kinematic range of $10^{-5}\leq x\leq 10^{-2}$ and $5\,GeV^{2}\leq Q^{2}\leq
30\,GeV^{2}$ at NLO and NNLO by solving the nonlinear GLR-MQ equation. We have
employed the Regge like behaviour of gluons in our calculations. We believe
that our solutions are valid in the vicinity of saturation scale where it is
reasonable to account for the recombination effect to show up because of very
high gluon density inside the hadrons and thus our assumptions look natural
too. The gluon distribution function in our solutions increases as x decreases
which is in good agreement with the perturbative QCD fits at small-x. Through
our results we have verified the Regge like behaviour of gluons at
moderate-$Q^{2}$ and small-x. It can be observed that our results show almost
similar behaviour from the results obtained by various global paramterization
groups.

The advantage of our semi-analytical solution over the exact numerical
solutions is that the functional form of the solution makes it easier to
perform qualitative study of it with respect to various parameters in the
solution. For instance, in our phenomenology $\lambda_{G}$, R, $\alpha_{s}$
and $\Lambda$ are parameters, from the solutions we can easily control these
parameters and we can effectively study the affect of these parameters on our
results. In simple words, we can effectively manipulate or control our results
by looking upon which parameters to be fixed up, by qualitatively analyzing
the form of the solution.

We have also observed that our solutions are very sensitive to the Regge
intercept ($\lambda_{G}$) and the correlation radius of two interacting gluons
($R$). It can also be observed that compared to our NLO solution, the NNLO
solution is more sensitive to $\lambda_{G}$ and R. In our phenomenological
study we also showed that the taming of gluon distribution function becomes
more towards small-x at low-$Q^{2}$ than that of the gluon distibution
function towards small-x at a larger-$Q^{2}$.

Since the solution of the DGLAP equation predicts sharp growth of gluon
density towards small-x, therefore, the GLR-MQ equation is a better candidate
than the linear DGLAP equation to explain the nonlinear effects like the
shadowing of gluons that have been observed in the collider experiments
already. Thus, we can conclude that in kinematics, where the density of gluons
is very high, GLR-MQ equation can provide a better understanding of the
physical picture than the linear DGLAP equation.

###### Acknowledgements.

M. Lalung is grateful to CSIR, New Delhi, India for CSIR Junior Research
Fellowship and P. Phukan acknowledges DST, Govt. of India for Inspire
fellowship.

*

## Appendix A

The gluon-gluon splitting functions at LO, NLO and NNLO are given below:

$P_{gg}^{(0)}(z)=6\left(\frac{1-z}{z}+\frac{z}{(1-z)_{+}}+z(1-z)\right)+\left(\frac{11}{2}-\frac{2N_{f}}{3}\right)\delta(1-z),$

The denominator of the second term in the RHS of $P_{gg}^{(0)}(z)$ in written
terms of what is known as the ‘+ prescription’. This indicates the
cancellation of singalurity that is appearing at $z=1$ through

$\int_{0}^{1}\frac{f(z)}{(1-z)_{+}}dz=\int_{0}^{1}\frac{f(z)-f(1)}{1-z}dz.$
$\displaystyle P_{gg}^{(1)}(z)=$ $\displaystyle
C_{F}T_{f}\left\\{-16+8z+\frac{20z^{2}}{3}+\frac{4}{3z}-(6+10z){\ln
z}-2(1+z)\ln^{2}z\right\\}$
$\displaystyle+N_{c}T_{f}\left\\{2-2z+\frac{26}{9}\left(z^{2}-\frac{1}{z}\right)-\frac{4}{3}(1+z)\text{ln$z$}-\frac{20}{9}p(z)\right\\}$
$\displaystyle+N_{c}^{2}\bigg{\\{}\frac{27(1-z)}{2}+\frac{67}{9}\left(z^{2}-\frac{1}{z}\right)-\left(\frac{25}{3}-\frac{11z}{3}+\frac{44z^{2}}{3}\right)\text{ln$z$}$
$\displaystyle+4(1+z)\ln^{2}z+\left(\frac{67}{9}+\ln^{2}z-\frac{\pi^{2}}{3}\right)p(z)$
$\displaystyle-4\text{ln$z$}\ln(1-z)p(z)+2p(-z)S_{2}(z)\bigg{\\}},$

where $p(z)=\frac{1}{1-z}+\frac{1}{z}-2+z(1-z)$
and
$S_{2}(z)=\int_{\frac{z}{1+z}}^{\frac{1}{1+z}}\frac{\text{dz}}{z}\ln\left(\frac{1-z}{z}\right)\underset{z}{\overset{\text{small}}{\longrightarrow}}\frac{1}{2}\ln^{2}z-\frac{\pi^{2}}{6}+O(z)$,

$\displaystyle P_{gg}^{(2)}(z)=$ $\displaystyle
2643.52D_{0}+4425.89\delta(1-z)+3589L_{1}-20852+3968z-3363z^{2}$
$\displaystyle+4848z^{3}+L_{0}L_{1}\left(7305+8757L_{0}\right)+274.4L_{0}-7471L_{0}^{2}+72L_{0}^{3}-144L_{0}^{4}$
$\displaystyle+\frac{14214}{z}+\frac{2675.8L_{0}}{z}+N_{f}\bigg{\\{}-412.172D_{0}-528.723\delta(1-z)-320L_{1}$
$\displaystyle-350.2\,+755.7z-713.8z^{2}+559.3z^{3}+L_{0}L_{1}\left(26.15\,-808.7L_{0}\right)+1541L_{0}$
$\displaystyle+491.3L_{0}^{2}+\frac{832L_{0}^{3}}{9}+\frac{512L_{0}^{4}}{27}+\frac{182.96}{z}+\frac{157.27L_{0}}{z}\bigg{\\}}$
$\displaystyle+N_{f}^{2}\bigg{\\{}-\frac{16D_{0}}{9}+6.463\delta(1-z)-13.878\,+153.4z-187.7z^{2}$
$\displaystyle+52.75z^{3}-L_{0}L_{1}\left(115.6\,-85.25z+63.23L_{0}\right)-3.422L_{0}$
$\displaystyle+9.68L_{0}^{2}-\frac{32L_{0}^{3}}{27}-\frac{680}{243z}\bigg{\\}},$

where $D_{0}=\frac{1}{(1-z)_{+}},L_{0}=\ln z$, $L_{1}=\ln(1-z)$,
$C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}\,\text{and}\,T_{f}=\frac{1}{2}N_{f}$.
We use the following convention,

$\int_{a}^{\infty}f(x)\delta(a-x)dx=\int_{-\infty}^{a}f(x)\delta(a-x)dx=\frac{f(a)}{2}$

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