# A Realistic Evaluation of LLMs for Quotation Attribution in Literary Texts:
A Case Study of LLaMa3

Gaspard Michel†∗
<EMAIL_ADDRESS>&Elena V. Epure†
<EMAIL_ADDRESS>Hennequin†
<EMAIL_ADDRESS>Christophe Cerisara∗
<EMAIL_ADDRESS>† Deezer Research, Paris, France
∗ Loria, Nancy, France

###### Abstract

Large Language Models (LLMs) zero-shot and few-shot performance are subject to
memorization and data contamination, complicating the assessment of their
validity. In literary tasks, the performance of LLMs is often correlated to
the degree of book memorization. In this work, we carry out a realistic
evaluation of LLMs for quotation attribution in novels, taking the instruction
fined-tuned version of Llama3 as an example. We design a task-specific
memorization measure and use it to show that Llama3’s ability to perform
quotation attribution is positively correlated to the novel degree of
memorization. However, Llama3 still performs impressively well on books it has
not memorized nor seen. Data and code will be made publicly available.

A Realistic Evaluation of LLMs for Quotation Attribution in Literary Texts: A
Case Study of LLaMa3

##  1 Introduction

In literary texts, narration is often intertwined with character dialogues to
reveal the story line. Through discourse, fictional characters present many
facets of their personalities, motivations and beliefs about the narrative
world and themselves. Quotation attribution, or the automated identification
of utterances and their attribution to fictional characters, is of crucial
importance for character analysis in digital humanities Elson et al. (2010);
Muzny et al. (2017a); Labatut and Bost (2019); Sims and Bamman (2020).

Large Language Models (LLMs) have shown promising results in a variety of
literary tasks, including Narrative Understanding Xu et al. (2023); Underwood
(2023); Wang et al. (2024a) or Character Understanding and Profiling Stammbach
et al. (2022); Soni et al. (2023); Yu et al. (2023); Zhao et al. (2024); Wang
et al. (2024b). However, the frequency with which models have encountered
information in their pretraining data is often correlated with their
performance on downstream tasks, questioning their capacity to generalize
Razeghi et al. (2022); Kandpal et al. (2023). Furthermore, evaluation
protocols often neglect the impact of data contamination on downstream
performance, making reported results unreliable Yeom et al. (2018); Shi et al.
(2024); Dong et al. (2024).

Recent works have explored the use of LLMs for quotation attribution in
novels, by extracting conversations directly with ChatGPT Zhao et al. (2024)
or by asking ChatGPT to attribute a single quote given its surrounding context
Su et al. (2023). Yet, their reported results only consider empirical or
qualitative performance, and the impact of data memorization and contamination
is not studied.

In this work, we propose a realistic evaluation methodology of LLMs for
quotation attribution that accounts for book memorization and data
contamination, taking Llama3-8b-Instruct as an example. Book memorization
occur when an LLM is capable of generating specific passages of texts in a
novel, and is linked to the number of times a novel is seen in its pretraining
data Chang et al. (2023a); Carlini et al. (2023). Closely related, data
contamination occur when an LLM has memorized benchmark evaluation data, and
is capable of generating labels without reasoning. To avoid confusion between
these two forms of memorization we will refer to data contamination as
annotation contamination in this paper. Because they pose practical issues
when evaluating LLMs performance on literary tasks, a prior analysis of book
memorization and annotation contamination should be made before reporting
downstream scores of literary tasks.

We find that Llama3 shows impressive performance on the Project Dialogism
Novel Corpus (PDNC) Vishnubhotla et al. (2022), a corpus of 28 novels
published between the 19th and 20th century. Using a task-specific measure of
memorization, we show that Llama3 has memorized some of the novels included in
PDNC, and that its performance varies significantly with the degree of
memorization. However, we could not find signs of annotation contamination,
suggesting that Llama3’s impressive performance on the quotation attribution
task is mostly due to its reasoning capabilities and memorization effect.

We also investigate the generalization capacity of Llama3 by showcasing
attribution results on a novel published in 2024 that we annotated manually
for quotation attribution. On this unseen novel, Llama3 8b performs slightly
worse than a popular baseline for quotation attribution, BookNLP+ Vishnubhotla
et al. (2023), but increasing the model size to 70b improves its results,
reaching an almost perfect accuracy. To sum up, our contributions are:

  1. 1.

We propose a structured evaluation methodology of LLMs for quotation
attribution, that could be applied to other literary tasks. Our methodology
includes: 1) analyzing book memorization with a task-specific measure, 2)
verifying downstream annotation contamination by disentangling the effect of
book memorization and 3) analyzing the effect of memorization on downstream
performance.

  2. 2.

Following the proposed evaluation methodology, we gather insights on Llama3’s
ability to perform quotation attribution. We show that its performance is not
correlated with the degree of memorization measured by the name-cloze accuracy
Chang et al. (2023b), but is instead correlated with our task-specific measure
of memorization. Besides, our results suggest that Llama3 generalizes to a
completely unseen novel, and that increasing the model size yields better
performance.

##  2 Related Work

### LLMs for literary tasks.

Most works study LLMs for tasks requiring long-range reasoning Xu et al.
(2023); Wang et al. (2024b). Their capacity of memorizing important details of
fictional characters have also been studied for character understanding
Stammbach et al. (2022); Zhao et al. (2024); Wang et al. (2024a). Here, we
propose an evaluation methodology that accounts for memorization and
annotation contamination, that can be extended to other literary tasks such as
coreference resolution or the recognition of literary entities.

### Memorization.

The zero-shot and few-shot performance of LLMs have been often attributed to
memorization Lee et al. (2022); Razeghi et al. (2022); Carlini et al. (2023).
This raises import concerns in literary studies as some novels are present
more often in the pretraining data of LLMs than others, creating discrepancies
in downstream tasks Chang et al. (2023a). Analyzing the effect of memorization
on downstream tasks could give insight on the LLMs capacity to generalize to
unseen data, and is thus of critical importance.

### Annotation Contamination.

Annotation contamination Magar and Schwartz (2022) occurs when downstream task
evaluation data (i.e. the exact annotation instance) is part of the LLMs
pretraining corpus. For this, methods such as Membership Inference Attacks
Yeom et al. (2018); Mireshghallah et al. (2022); Shi et al. (2024) have been
designed among others. Annotation contamination causes severe issues for
security and privacy Carlini et al. (2021), but also raises question on zero-
shot performance Li and Flanigan (2023).

Figure 1: Memorization results with the LOSS (Perplexity) attack on PDNC and
the unseen novel.

### Quotation Attribution.

Methods to attribute utterances of speech to their speaker in literary texts
have explored sequence labeling O’Keefe et al. (2012) or deterministic rules
Muzny et al. (2017b). BookNLP, a popular Natural Language Processing pipeline
dedicated to books also proposes a quotation attribution system, that was
recently improved Vishnubhotla et al. (2023). The current state-of-the-art
frames the task as a sequence-to-sequence problem, using BART to generate a
speaker name based on a list of candidates Su et al. (2023).

##  3 Data

We use the Project Dialogism Novel Corpus (PDNC) Vishnubhotla et al. (2022), a
dataset including 28 novels annotated with the speaker and addressee of each
quote. 22 novels were released in July 2022 ($\text{PDNC}_{1}$), and 6 novels
were added later in June 2023 ($\text{PDNC}_{2}$). This second release serves
as a basis for annotation contamination testing, as it was released after the
knowledge cutoff of Llama3 8b (March 2023). We also annotated a whole new book
published in March 2024 following the PDNC guidelines, that we will release
publicly. One of the authors annotated all quotes and a second author only a
subset of 5 chapters. The inter-annotator agreement Cohen’s $\kappa$ score is
97% on this subset, which is considered almost perfect agreement. This new
novel allows us to test for model generalization, as its text and annotations
are not part of Llama3 pretraining corpus.

##  4 Measuring Book Memorization and Annotation Contamination

Most Membership Inference Attacks (MIA) use perplexity measures of an
annotation instance to detect contamination. Our corpus is composed of quotes
and named entities coming from novels which are often used during LLMs pre-
training. An annotation instance is likely to have a lower perplexity if it is
extracted from a memorized book than an instance from a non-memorized book. We
evaluate this hypothesis by randomly sampling 20% annotations per novel in our
corpus, that we verbalize in plain text (see Appendix A). In Figure 1, we show
the LOSS attack Yeom et al. (2018) of each instance that we average to get a
single value per novel. We clearly see a difference between the unseen novel
and PDNC. This result suggests that that we need to disentangle book
memorization from annotation contamination.

### Book Memorization.

We use name-cloze accuracy as a measure of book memorization as in Chang et
al. (2023a). We prompt the model with a small book passage, where a single
masked entity appears which we ask to predict. We obtain a $4\%$ average name-
cloze accuracy on PDNC, where 13 novels show no signs of memorization at all.
These results contradict previous results on the same task Chang et al.
(2023a) and our perplexity results per novel (Figure 1). Besides, the name-
cloze accuracy on the unseen novel is higher than 0, that could be attributed
to random guessing when manually checking the results. Thus, we question the
validity of this measure for book memorization.

| $\textbf{PDNC}_{1}$ | $\textbf{PDNC}_{2}$ | Unseen Novel
---|---|---|---
SIG | $72^{+}$ | - | -
ChatGPT | $71^{+}$ | - | -
BookNLP+ |  $67.4$ (5.4) |  $66.4$ (12.0) | $96.0$
Llama3-8b |  $86.5$ (3.3) |  $84.8$ (5.0) | $92.2$
Llama3-70b | - | - | $99.4$
Table 1: Cross validation accuracy and (standard deviations) across splits for
$\textbf{PDNC}_{1}$ and accuracy averaged over novels for $\textbf{PDNC}_{2}$.
We take the reported results from Su et al. (2023) for SIG and ChatGPT.

Motivated by the need for a better book memorization measure, we propose
speaker-guessing, a task-specific MIA. We give as input to the model a quote,
the character name of the quote’s addressee, and the masked referring
expression (e.g “said [MASK]”) and prompt it to predict the speaker of the
quote (see Appendix B for prompt examples). To answer this, the model needs to
infer that the quote is extracted from a specific novel, using the limited
information we provide, and then need to attribute the quote without
contextual information. This task is hard for humans: determining from which
novel the quote is taken from requires inferring that the addressee is a
specific character introduced in the novel, and predicting its speaker
requires knowing other novel characters. We calculate the speaker-guessing
accuracy per novel using $20\%$ randomly sampled quotes. In comparison to
name-cloze accuracy, the average speaker-guessing accuracy is $18\%$ , and
only three novels in PDNC have a null accuracy. We also found a null accuracy
on the unseen novel, which further validates our test. Thus, some novels are
more memorized than others, which calls for caution when analyzing annotation
contamination and quotation attribution.

### Annotation Contamination.

We propose an econometrics-inspired approach to disentangle book memorization
from annotation contamination: propensity score matching Rosenbaum and Rubin
(1983). We match each novel in $\text{PDNC}_{2}$ to another novel in
$\text{PDNC}_{1}$ that has a similar degree of memorization. This allows us to
compare unseen annotations with seen annotations while reducing the impact of
memorization. We compute the matching by defining the group treatment as being
novels in $\text{PDNC}_{2}$ from which annotated data is not seen, and use
speaker-guessing accuracy and the perplexity calculated by the LOSS attack
from Section 4 as covariates. We then calculate a propensity score, indicating
how likely annotations of a novel have not been seen by the model, given its
degree of memorization. We then match a single novel from $\text{PDNC}_{1}$ to
one of $\text{PDNC}_{2}$ using this score.

To analyze annotation contamination, we compute Min-K% Shi et al. (2024) for
various levels of K on 20% randomly sampled instances of data for each novel
in $\text{PDNC}_{2}$ and their matched novel. Min-K% averages the log-
probabilities of the K% tokens with lowest conditional probabilities in a
sequence of data, and is widely adopted to detect annotation contamination. We
test for difference in expected value between the log-probabilities of the
unseen novel annotations in $\text{PDNC}_{2}$ and the matched novel
annotations with a paired Student t-test, and find no statistically
significant differences (see Appendix C for results per novel). By controlling
for memorization, we found that Llama3-8b is unlikely to manifest signs of
contamination.

|  Attribution & Speaker-Guessing |  Attribution & Name-Cloze
---|---|---
Spearman $\rho$ | $0.41^{\star}$ | $0.10$
$\texttt{Top}_{5}$ |  $91.9$ [87.7-96.1] |  $89.5$ [86.2-96.2]
$\texttt{Bot}_{5}$ |  $83.2$ [73.2-87.7] |  $85.8$ [73.2-91.7]
$(\texttt{Top}_{5}-\texttt{Bot}_{5})$ | $2.89^{\star}$ | $0.95$
Table 2: Spearman $\rho$ correlation between attribution and memorization
accuracies (top) ; average attribution accuracy ([$\min$, $\max$]) in the
$\texttt{Top}_{5}$ and $\texttt{Bot}_{5}$ novels, and the Student-t statistic
when testing for difference in expected value.⋆ indicates a p-value $<0.05$.

##  5 Quotation Attribution

We follow previous works Vishnubhotla et al. (2023); Su et al. (2023), and
focus on major and intermediate characters. We compare to SIG and ChatGPT Su
et al. (2023), and to BookNLP+ Vishnubhotla et al. (2023). We train BookNLP+
on $\text{PDNC}_{1}$ and use the best performing model to attribute quotes on
$\text{PDNC}_{2}$ and the unseen novel. We could not reproduce the results of
Su et al. (2023), so we use their reported results.

We divide each novel in chunks of $4096$ tokens, where we modify the chunk
text by assigning a unique identifier to each quote starting from $1$ to $n$,
where $n$ is the number of quotes in the chunk. We also build a character-to-
alias list using the gold character-list from PDNC that we include in the
prompt. Given the modified text and the list of character aliases, we prompt
the model to predict the speaker of quotes $1,\dots,n$ sequentially. More
details are provided in Appendix D.

Table 1 shows surprisingly high performance for Llama3-8b, increasing the
attribution accuracy by up to $14$ points against SIG on $\text{PDNC}_{1}$. We
also see a large improvement of Llama3-8b over BookNLP+ on the unseen
$\text{PDNC}_{2}$. Since we could not find any signs of annotation
contamination, these results suggest that Llama3 is realistically good at
quotation attribution. Yet, its performance is likely to vary when considering
book memorization.

When comparing performances on the unseen novel, BookNLP+ shows better overall
accuracy than Llama3-8b, which might indicates that the latter is not able to
generalize to unseen novels. However, when testing with Llama3-70b, the
performance increases to an almost perfect accuracy. The larger size of the
model appears to improve the reasoning, yielding better attribution.

### The Effect of Memorization

We first calculate the spearman $\rho$ correlation between attribution
accuracy performance and each of our two measures of memorization, name-cloze
accuracy and speaker-guessing accuracy. We could not find a significant
relationship between name-cloze and attribution performance ($\rho=0.1$,
p-value $=0.3$), but we found a significant $\rho=0.41$ with our speaker-
guessing task, indicating a positive correlation between memorization and
quotation attribution.

Given our two measures of memorization, we extract the $\texttt{Top}_{5}$ most
memorized novels and the bottom $\texttt{Bot}_{5}$ least memorized novels, and
compare their performance in Table 2. The speaker-guessing accuracy ranges
from 37% to 78% for the $\texttt{Top}_{5}$ and from 0% to 0.01% for
$\texttt{Bot}_{5}$. When using our speaker-guessing measure, Llama3-8b
performs significantly better on the most memorized novels than on the least
memorized novels. However, we note that the attribution accuracy on least
memorized novels is still high: the average accuracy of BookNLP+ is of $0.58$,
against $0.83$ for Llama3-8b.

##  6 Conclusion

We proposed a realistic evaluation of Llama3 for quotation attribution. We
started with quantifying book memorization with a new task-specific measure,
that we used to disentangle from annotation contamination. While we could not
find signs of annotation contamination, our results suggest that Llama3
performs significantly better on memorized novels. However, we could still
find a large improvement over BookNLP+ even on novels that are the least
memorized. The impressive performance of Llama3-70b on a completely unseen
novel also suggest capabilities of generalization.

##  7 Limitations

We proposed a new, task-specific, measure of book memorization. While this
measure shows better capacity to recognize memorization on Llama3-8b, we note
that it is specific to literary texts, and that it suffers from one of the
common downsides of this kind of measures: we can not be sure that instances
of data have not been seen during pretraining. Some novels in our corpus
exhibited non-memorization, while we know that they are part of large corpora
such as The Pile or C4, indicating that we could design better test for book
memorization. Nonetheless, our measure is task-specific, and thus better
indicates how the book memorization behavior of a model will impact the
downstream task, which was validated by the positive correlations between
quotation attribution speaker-guessing accuracies. Overall, we believe that
the better way to test generalization of LLMs on a downstream task is to
provide it with completely unseen data, which we tested by annotating a
recently published novel.

The significant improvement of Llama3 over baselines such as BookNLP+ on
quotation attribution creates new possibilities to better analyze large
corpora of literary texts. However, this improvement comes with the price of
longer inference times, taking up to an hour for a single novel and limiting
its impact for the study of massive corpora such as Project Gutenberg. In
comparison, BookNLP+ can predict attributions in matter of minutes for a
single novel.

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##  Appendix A Example of a Verbalized Data Instance

We present in Figure 2 a verbalized instance of data.

Raw Data: "Q0","and what is the use of a book, without pictures or
conversations?","[ánd what is the use of a book,,́ ẃithout pictures or
conversations?]́","[[254, 284], [301, 335]]","Alice","[]","Explicit","thought
Alice","[[], []]","[[], []]","[[], []]"

text

Verbalized Data: quoteID: Q0; quoteText: and what is the use of a book,
without pictures or conversations?; subQuotationList: [’and what is the use of
a book,’, ’without pictures or conversations?’]; quoteByteSpans: [[254, 284],
[301, 335]]; speaker: Alice; addressees: []; quoteType: Explicit;
referringExpression: thought Alice; mentionTextsList: [[], []];
mentionSpansList: [[], []]; mentionEntitiesList: [[], []]

Figure 2: Example of a verbalized instance of data.

##  Appendix B Prompts used for Speaker-Guessing

We display in Figure 3 a prompt example for the name-cloze task and in Figure
4 a prompt example for speaker-guessing.

You have seen the following passage in your training data. What is the proper
name that fills in the [MASK] token in it? This name is exactly one word long,
and is a proper name (not a pronoun or any other word). You must make a guess,
even if you are uncertain.

text

Input: "Stay gold, [MASK], stay gold."

Output: <name>Ponyboy</name>

text

Input: "The door opened, and [MASK], dressed and hatted, entered with a cup of
tea."

Output: <name>Gerty</name>

text

Input: "The rabbit-hole went straight on like a tunnel for some way, and then
dipped suddenly down, so suddenly that [MASK] had not a moment to think about
stopping herself before she found herself falling down a very deep well."

Output:

Figure 3: Example of a prompt for the name-cloze task. We always give the same
two examples as input to Llama3-8b-Instruct.

You have seen the following quote in your training data. What is the name that
fills in the [SPEAKER] token in it? This name is the name of character in a
book. You must make a guess, even if you are uncertain.

text

Input: Quote: "and what is the use of a book, without pictures or
conversations?"; speaker: [SPEAKER]; Addressees: None; Referring Expression:
thought [MASK]

Output: <speaker>Alice</speaker>

text

Input: Quote: "There is no telling when Berry will be following my example and
be taking a wife unto himself."; speaker: [SPEAKER]; Addressees: None;
Referring Expression: [MASK] said <speaker>Maurice Oakley</speaker>

text

Input: Quote: "Well, I hope it will turn out all right,"; speaker: [SPEAKER];
Addressees: Marilla Cuthbert; Referring Expression: said [MASK]

Output:

Figure 4: Example of a prompt for the speaker-guessing task. We always give
the same two examples as input to Llama3-8b-Instruct.

##  Appendix C Annotation Contamination Results per Novel

Results for each novel in $\text{PDNC}_{2}$ and their matched novel are
displayed in Figure 5. We see that there is a small difference when averaging
over novels, but this difference is not statistically significant.

Figure 5: Min-K% results for various values of K for $\text{PDNC}_{2}$ and
each matched novels in $\text{PDNC}_{1}$.

##  Appendix D Method Details for Quotation Attribution

We divide novel in chapters, and build chunks of text of length $4096$ tokens
with a stride of $1024$ tokens. We assign to each chapters the quotes to be
attributed in the following way:

  * •

If a chapter contains only one chunk, then assign to it each quotes in the
chunk.

  * •

Else, we assign quotes to the first chunk in the first $3584$ tokens (hence
keeping a contextual window of $512$ tokens to the right of the chunk). For
the last chunk, we assign it all quotes after the stride (i.e all quotes
occurring after the first $1024$ tokens). For each chunk in the middle, we
assign quotes to be attributed as all quotes in between tokens $1024$ and
$3584$.

The above process ensures that a quote is not predicted twice, and allows
contextual predictions with a left contextual window of $1024$ tokens and a
right window of $512$ tokens for middle chunks.

Figure 6 displays an example prompt used for quotation attribution. The model
usually generates a list of attributions in a format similar to the one we
give in the prompt. We then parse this list to produce an attribution for each
quote. In particular, we use the character-to-alias list to replace aliases
with their canonical character name (which is our gold label). If the model
generates a name that is not a canonical character name or an alias, we
consider its predictions as wrong (i.e we dot use any lenient metrics such as
substring match).

##  Appendix E Computing Information

We used a 32-core Intel Xeon Gold 6244 CPU @ 3.60GHz CPU with 128GB RAM
equipped with 3 RTX A5000 GPUs with 24GB RAM. We use a single RTX A5000 for
all experiments. We used the 8-bit version of Llama3-8b-Instruct using the
BitsAndBytes library. The peak memory used was around 14GB of RAM. We employ a
relatively large contextual window, and ask the model to generates long
attribution lists. Thus, we observed quite large inference times, and
processing entire novels varied from 10 minutes to an hour.

Instruction: You are an excellent linguist working in the field of literature.
I will provide you with a passage of a book where all quotes have unique
identifiers marked by headers ‘|quote_id|‘. Your are tasked to build a list of
quote attributions by sequentially attributing quotes.

text

Here is an example:

|0| "It’s beautiful today" |0| said Jane. She continued,

|1| "What shall I do today?" |1|

text

Here is the valid attributions:

0: Jane

1: Jane

text

Considering these character aliases:

alias = canonical speaker name

A Gryphon = The Gryphon

The King = The King Of Hearts

text

And considering the context: CHAPTER I. Down the Rabbit-Hole

Alice was beginning to get very tired of sitting by her sister on the bank,
and of having nothing to do: once or twice she had peeped into the book her
sister was reading, but it had no pictures or conversations in it, |1| "and
what is the use of a book," |1| thought Alice |2| "without pictures or
conversations?" |2|

So she was considering in her own mind (as well as she could, for the hot day
made her feel very sleepy and stupid), whether the pleasure of making a daisy-
chain would be worth the trouble of getting up and picking the daisies, when
suddenly a White Rabbit with pink eyes ran close by her.

…

She ate a little bit, and said anxiously to herself, |33| "Which way? Which
way?" |33|, holding her hand on the top of her head to feel which way it was
growing, and she was quite surprised to find that she remained the same size:
to be sure, this generally happens when one eats cake, but Alice had got so
much into the way of expecting nothing but out-of-the-way things to happen,
that it seemed quite dull and stupid for life to go on in the common way.

So she set to work, and very soon finished off the cake.

text

Input: Attribute sequentially the quotes 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 and
33 to one of these speakers only: ’Alice’, ’The Hatter’, ’The Mock Turtle’,
’The Gryphon’, or ’The King Of Hearts’ and return a list containing their
attributed speakers formatted similarly to the example prediction. Only
generate this list and nothing else.

Constraints: The list of attribution should only contain the attributions for
the input quotes. Do not explain your reasoning and stop the generation once
the list is complete.

Figure 6: Example of a prompt used for quotation attribution.


# Semi-inclusive Diffractive Deep Inelastic Scattering at Small-$x$

Yoshitaka Hatta  Physics Department, Building 510A, Brookhaven National
Laboratory, Upton, NY 11973, USA  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973, USA  Bo-Wen Xiao  School of
Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172,
China  Feng Yuan  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA

###### Abstract

Inspired by a recent study by Iancu, Mueller, and Triantafyllopoulos
Iancu:2021rup , we propose semi-inclusive diffractive deep inelastic
scattering (SIDDIS) to investigate the gluon tomography in the nucleon and
nuclei at small-$x$. The relevant diffractive quark and gluon parton
distribution functions (DPDF) can be computed in terms of the color dipole
S-matrices in the fundamental and adjoint representations, respectively.

##  I Introduction

Nucleon tomography in terms of various ‘multi-dimensional’ parton distribution
functions is one of the ultimate goals of the current and future facilities in
high energy nuclear and particle physics Boer:2011fh ;
AbelleiraFernandez:2012cc ; Accardi:2012qut . These include the transverse
momentum dependent distributions (TMDs) and the generalized parton
distributions (GPDs), which provide different perspectives of the internal
structure of hadrons and nuclei. The so-called quantum phase space Wigner
distributions Ji:2003ak ; Belitsky:2003nz  of partons are regarded as the
mother distributions since they ingeniously encode the complete information
about how partons are distributed both in position and momentum spaces.

At small-$x$ in the gluon saturation regime, the gluon Wigner distribution is
intimately connected to the well-known color dipole S-matrix in the Color-
Glass-Condensate (CGC) formalism Mueller:1993rr ; Mueller:1999wm ;
McLerran:1993ni ; McLerran:1993ka ; McLerran:1994vd  which has been a subject
of intensive study in the last few decades Gelis:2010nm ; Kovchegov:2012mbw .
In Ref. Hatta:2016dxp , it was suggested that the diffractive dijet production
in $ep$/$eA$ collisions Hatta:2016dxp ; Altinoluk:2015dpi ; Zhou:2016rnt ;
Hagiwara:2017fye ; Mantysaari:2019csc ; Mantysaari:2019hkq  may provide a
direct probe of the gluon Wigner distribution. Recently, Iancu, Mueller, and
Triantafyllopoulos Iancu:2021rup  have considered the correction of an
additional semi-hard gluon radiation to this process, or ‘trijet’ production.
A remarkable feature is that the leading dijet can have a much larger
transverse momentum than the saturation momentum $Q_{s}$, yet, the process is
still sensitive to gluon saturation due to the third jet, whose transverse
momentum is of the order of $Q_{s}$. Another remarkable feature is that the
calculated cross section factorizes in terms of the gluon PDF of the
‘Pomeron’, or equivalently, as we shall see, the gluon diffractive parton
distribution function (DPDF). The DPDFs are important ingredients for the QCD
factorization of diffractive hard processes Collins:1997sr ; Berera:1995fj ;
Trentadue:1993ka , see, a recent phenomenology study Armesto:2019gxy  and
reference therein. At small-$x$, the DPDFs are connected to the color dipole
S-matrix Hebecker:1997gp ; Buchmuller:1998jv ; Golec-Biernat:1999qor ;
Hautmann:1999ui ; Hautmann:2000pw ; Golec-Biernat:2001gyl , and are therefore
systematically calculable including the gluon saturation effects.

Following these developments, in this paper, we will further demonstrate that
the semi-inclusive diffractive DIS (SIDDIS), see Fig. 1, can provide a unique
perspective of gluon tomography at small-$x$, where the quark and gluon DPDFs
can be systematically computed from the operator definitions consistent with
the QCD factorization Collins:1997sr  in the dipole formalism. This opens up
new opportunities to investigate the gluon Wigner distribution and gluon
saturation at the future electron-ion collider (EIC). More importantly, the
QCD factorization results in terms of DPDFs are consistent with the CGC
calculations in the kinematics that both apply.

Figure 1: Semi-inclusive diffractive DIS process, where the diffractive parton
distributions of quark and gluon can be measured through final state
particles, including hadron/jet productions. An incoming proton with momentum
$P$ diffractively scatters into the final state nucleon with momentum
$P^{\prime}$ and deposits a longitudinal momentum fraction of $x_{I\\!\\!P}$
into hard interaction with the virtual photon. The usual Bjorken $x$ is
defined as $x_{B}=\beta x_{I\\!\\!P}$ for inclusive diffractive DIS.

Compared to the usual semi-inclusive DIS (SIDIS) Mulders:1995dh ; Boer:1997nt
; Bacchetta:2006tn , the diffractive process requires a large rapidity gap,
$Y_{I\\!\\!P}\sim\ln(1/x_{I\\!\\!P})\gg 1$ with small $x_{I\\!\\!P}$
considered here, between the nucleon remnant and the hard interaction part,
which can be easily identified in experiment. Combining the methods applied in
the QCD factorization for SIDIS (see, e.g., Ji:2004wu ; Collins:2011zzd ) and
hard diffractive processes Collins:1997sr , we expect that the QCD
factorization for the SIDDIS is also valid, and we can safely extract the
relevant DPDFs from the experiment. For the transverse momentum dependent
observables, we will need the TMD fragmentation function and the associated
soft factor as well, which can be defined accordingly. Integrating over the
transverse momentum will lead to a collinear factorization for SIDDIS, where
the soft factor does not contribute.

The study of hard diffractive processes has a long history. Theoretically,
there are three different approaches based on QCD factorization framework. For
inclusive hard diffractive process, there is a collinear factorization in
terms of the DPDFs. In the CGC approach, all diffractive processes can be
described by the dipole scattering amplitudes Golec-Biernat:1999qor ;
Hatta:2006hs ; Marquet:2007nf ; Kowalski:2008sa . Meanwhile, there is also a
generalized parton distribution (GPD) Ji:1996ek ; Mueller:1998fv ; Ji:1996nm
approach to describe the hard exclusive processes Ji:1996ek ; Mueller:1998fv ;
Ji:1996nm ; Radyushkin:1997ki ; Goeke:2001tz ; Diehl:2003ny ; Belitsky:2005qn
; Boffi:2007yc ; Boer:2011fh ; Accardi:2012qut , including deeply virtual
Compton scattering (DVCS). The consistency between the collinear GPD formalism
and the CGC/dipole formalism has been shown for the DVCS process at small-$x$
Hatta:2017cte . The investigation of SIDDIS in this paper extends this
consistency and provides a unified method that connects all the above-
mentioned approaches. Our discussions below are limited to the so-called
coherent diffractive processes. But, this can be extended to the in-coherent
diffractive process as well.

##  II Diffractive PDFs from Dipole Amplitude at Small-$x$

It has been shown that, at small-$x$, the quark and gluon TMD distribution
functions are directly related to the color dipole S-matrix in the CGC
formalism McLerran:1993ni ; McLerran:1993ka ; McLerran:1998nk ; Mueller:1999wm
; Marquet:2009ca ; Dominguez:2010xd ; Dominguez:2011wm ; Belitsky:2002sm ;
Xiao:2010sa ; Xiao:2017yya . In this section, we use the same method to
establish the connection between the DPDFs and the color dipole. The result in
the gluon case is equivalent, up to the normalization factor, to the
‘unintegrated gluon distribution of the Pomeron’ calculated in Ref.
Iancu:2021rup .

Let us begin with the standard definition of the quark DPDF Berera:1995fj
generalized to include the transverse momentum ($k_{\perp}$) dependence

$\displaystyle
2E_{P^{\prime}}\frac{df_{q}^{D}(x,k_{\perp};x_{I\\!\\!P},t)}{d^{3}P^{\prime}}=\int\frac{d\xi^{-}d^{2}\xi_{\perp}}{2(2\pi)^{6}}e^{-ix\xi^{-}P^{+}+i\vec{\xi}_{\perp}\cdot\vec{k}_{\perp}}$
$\displaystyle~{}\times\langle PS|\overline{\psi}(\xi){\cal
L}_{n}^{\dagger}(\xi)\gamma^{+}|P^{\prime}X\rangle\langle P^{\prime}X|{\cal
L}_{n}(0)\psi(0)|PS\rangle\ ,$ (1)

where the future pointing gauge link in the fundamental representation of QCD
is defined as ${\cal
L}_{n}(\xi)\equiv\exp\left(-ig\int^{\infty}_{0}d\lambda\,v\cdot A(\lambda
n+\xi)\right)$. Here, $n$ represents a light-cone vector conjugate to the
nucleon momentum $n^{2}=0$ and $n\cdot P=1$. The final state nucleon carries
momentum $P^{\prime}=P+\Delta$ with $t=\Delta^{2}$.
$x_{I\\!\\!P}=n\cdot(P-P^{\prime})$ is the momentum fraction of the incoming
nucleon carried by the Pomeron. We introduce the momentum fraction of the
Pomeron carried by the quark $\beta=x/x_{I\\!\\!P}$. Integrating over
$k_{\perp}$, we recover the collinear quark DPDF.

Just like usual TMDs, the naive definition (1) contains end-point
singularities at higher orders which will be cured by the soft factor
subtraction. This will introduce the associated TMD-like evolution and
resummation Collins:2011zzd , which are important for phenomenology
applications. We will come back to this issue in the future. In the following,
we will neglect such higher order effects. We also mention that (1) is similar
to, but different from the generalized transverse momentum dependent parton
distribution (GTMD) Meissner:2009ww  or the Wigner distribution
Belitsky:2003nz . In particular, the GTMD appears in the amplitude for the
exclusive process, whereas the TMD DPDF appears in the cross section for semi-
inclusive diffractive processes. However, there exist strong connections
between them, as we will show below.

Similarly, we can define the gluon TMD DPDF

$\displaystyle
2E_{P^{\prime}}\frac{df_{g}^{D}(x,k_{\perp};x_{I\\!\\!P},t)}{d^{3}P^{\prime}}=\int\frac{d\xi^{-}d^{2}\xi_{\perp}}{xP^{+}(2\pi)^{6}}e^{-ix\xi^{-}P^{+}+i\vec{\xi}_{\perp}\cdot\vec{k}_{\perp}}$
$\displaystyle~{}\times\\!\\!\langle PS|F^{+\mu}(\xi){\cal
L}_{n}^{\dagger}(\xi)\gamma^{+}|P^{\prime}X\rangle\\!\langle P^{\prime}X|{\cal
L}_{n}(0)F_{\mu}^{\,+}(0)|PS\rangle\ ,$ (2)

where the gauge link is in the adjoint representation.

Figure 2: The typical Feynman diagrams for the Diffractive quark (left) and
gluon (right) distribution functions computed from the dipole amplitudes at
small-$x$.

The main goal of our paper is to compute the quark and gluon DPDFs in the CGC
framework. The typical Feynman diagrams are shown in Fig. 2, where the double
lines represent gauge links. To the order of accuracy, the final state
$|X\rangle$ is saturated by a single antiquark and a gluon for the quark and
gluon DPDFs, respectively. Calculating the left diagram of Fig. 2 and its
complex conjugate, we find the following result,

$\displaystyle
x\frac{d\,f_{q}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}=\int
d^{2}k_{1\perp}d^{2}k_{2\perp}{\cal
F}_{x_{I\\!\\!P}}(k_{1\perp},\Delta_{\perp})$ $\displaystyle~{}~{}\times{\cal
F}_{x_{I\\!\\!P}}(k_{2\perp},\Delta_{\perp})\frac{N_{c}\beta}{2\pi}{\cal
T}_{q}(k_{\perp},k_{1\perp},k_{2\perp})\ ,$ (3)

for the quark DPDF at small-$x$. Here, $k_{1}$ represents the momentum for one
of the two vertical gluon lines in the left diagram of Fig. 2. For the complex
conjugate of this diagram, we use $k_{2}$. ${\cal T}_{q}$ represents the sum
of four terms ${\cal T}_{q}\equiv
T_{q}(k_{\perp},k_{1\perp},k_{2\perp})-T_{q}(k_{\perp},0,k_{2\perp})-T_{q}(k_{\perp},k_{1\perp},0)+T_{q}(k_{\perp},0,0)$
where

$\displaystyle T_{q}(k_{\perp},k_{1\perp},k_{2\perp})=$
$\displaystyle\frac{k_{1\perp}^{\prime}\cdot
k_{2\perp}^{\prime}k_{\perp}^{2}}{\left[\beta
k_{\perp}^{2}+(1-\beta)k_{1\perp}^{\prime 2}\right]\left[\beta
k_{\perp}^{2}+(1-\beta)k_{2\perp}^{\prime 2}\right]}\ ,$ (4)

with $k_{i\perp}^{\prime}=k_{\perp}-k_{i\perp}$.
$\mathcal{F}_{x_{I\\!\\!P}}(k_{i\perp},\Delta_{\perp})$ is the Fourier
transform of the dipole S-matrix in the fundamental representation,

$\displaystyle\mathcal{F}_{x}(q_{\perp},\Delta_{\perp})=\int\frac{d^{2}b_{\perp}d^{2}r_{\perp}}{(2\pi)^{4}}e^{iq_{\perp}\cdot
r_{\perp}+i\Delta_{\perp}\cdot b_{\perp}}$
$\displaystyle\times\frac{1}{N_{c}}\left\langle\text{Tr}\left[U\left(b_{\perp}+\frac{r_{\perp}}{2}\right)U^{\dagger}\left(b_{\perp}-\frac{r_{\perp}}{2}\right)\right]\right\rangle_{x}\
,$ (5)

where

$\displaystyle U(b_{\perp})={\rm
P}\exp\left(ig\int_{-\infty}^{\infty}dz^{-}A^{+}(z^{-},b_{\perp})\right),$ (6)

is the Wilson line along the lightcone in the fundamental representation.
Similarly, the diffractive gluon DPDF is represented by the right diagram in
Fig. 2 and reads

$\displaystyle
x\frac{df_{g}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}=\int
d^{2}k_{1\perp}d^{2}k_{2\perp}{\cal
G}_{x_{I\\!\\!P}}(k_{1\perp},\Delta_{\perp})$ $\displaystyle~{}~{}\times{\cal
G}_{x_{I\\!\\!P}}(k_{2\perp},\Delta_{\perp})\frac{N_{c}^{2}-1}{\pi(1-\beta)}{\cal
T}_{g}(k_{\perp},k_{1\perp},k_{2\perp})\ ,$ (7)

where we again defined ${\cal T}_{g}\equiv
T_{g}(k_{\perp},k_{1\perp},k_{2\perp})-T_{g}(k_{\perp},0,k_{2\perp})-T_{g}(k_{\perp},k_{1\perp},0)+T_{g}(k_{\perp},0,0)$
with

$\displaystyle T_{g}(k_{\perp},k_{1\perp},k_{2\perp})=\frac{1}{\left[\beta
k_{\perp}^{2}+(1-\beta)k_{1\perp}^{\prime 2}\right]}$
$\displaystyle~{}~{}~{}\times\frac{1}{\left[\beta
k_{\perp}^{2}+(1-\beta)k_{2\perp}^{\prime
2}\right]}\left[\beta(1-\beta){k_{\perp}^{2}}\frac{k_{1\perp}^{\prime
2}+k_{2\perp}^{\prime 2}}{2}\right.$
$\displaystyle~{}~{}~{}\left.+(1-\beta)^{2}(k_{1\perp}^{\prime}\cdot
k_{2\perp}^{\prime})^{2}+\beta^{2}\frac{(k_{\perp}^{2})^{2}}{2}\right]\ .$ (8)

The gluon dipole S-matrix is defined as

$\displaystyle\mathcal{G}_{x}(q_{\perp},\Delta_{\perp})=\int\frac{d^{2}b_{\perp}d^{2}r_{\perp}}{(2\pi)^{4}}e^{iq_{\perp}\cdot
r_{\perp}+i\Delta_{\perp}\cdot b_{\perp}}$
$\displaystyle\times\frac{1}{N^{2}_{c}-1}\left\langle\text{Tr}\left[\widetilde{U}\left(b_{\perp}+\frac{r_{\perp}}{2}\right)\widetilde{U}^{\dagger}\left(b_{\perp}-\frac{r_{\perp}}{2}\right)\right]\right\rangle_{x}\
,$ (9)

where $\widetilde{U}$ is the same Wilson line but in the adjoint
representation.

The above DPDFs can be applied to semi-inclusive diffractive processes (Fig.
1) where the argument for QCD factorization should be analogous to those for
hard diffractive DIS Collins:1997sr  and non-diffractive semi-inclusive DIS
Collins:2011zzd ; Ji:2004wu . The combination of these two factorization
proofs should lay the groundwork for the QCD factorization of our process. As
an example, consider semi-inclusive quark production. To leading order, the
differential cross section can be immediately written down in terms of the
quark DPDF

$\frac{d\sigma^{\rm SIDDIS}(\ell
p\to\ell^{\prime}p^{\prime}qX)}{dx_{B}dyd^{2}k_{\perp}dY_{I\\!\\!P}dt}=\sigma_{0}e_{q}^{2}x_{B}\frac{df_{q}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}\,,$
(10)

where
$\sigma_{0}=\frac{4\pi\alpha_{em}^{2}S_{ep}}{Q^{4}}\left(1-y+\frac{y^{2}}{2}\right)$
and the usual DIS variables are defined as $x_{B}=Q^{2}/2P\cdot q$, $y=q\cdot
P/k_{\ell}\cdot P$ and $S_{ep}=(k_{\ell}+P)^{2}$. Additional soft factors will
be needed at higher orders. The formula (10) is consistent with the direct
calculation within the CGC formalism where the cross section $\gamma^{*}p\to
qXp^{\prime}$ is obtained by first considering the split $\gamma^{*}\to
q\bar{q}$ and then integrating over the phase space of the antiquark. This is
demonstrated in Appendix B using a technique developed in Marquet:2009ca . We
expect that the consistency persists to higher orders starting the $q\bar{q}g$
final state, but this has to be carefully investigated in future work.

The quark and gluon dipole S-matrices may contain nontrivial correlations
between $\Delta_{\perp}$ and $k_{\perp}$ with observable consequences.
Especially if these correlations depend on the nucleon spin, they will open up
new opportunities to explore spin-orbital correlations inside hadrons,

  * •

The $\cos(2\phi)$ correlation Hatta:2016dxp  between $\Delta_{\perp}$ and
$k_{i\perp}$ in the dipole S-matrix results in a similar correlation between
$k_{\perp}$ and $\Delta_{\perp}$ in the DPDFs. This can be observed
experimentally as $\cos 2\phi$ and higher order azimuthal asymmetries between
the recoiling proton and observed hadrons in SIDDIS in Fig. 1. Previously,
such asymmetries have been studied theoretically in exclusive processes
Hatta:2016dxp ; Mantysaari:2019hkq .

  * •

For non-diffractive processes, the leading order TMDs contain correlations
between the transverse momentum and the polarizations of the parton and/or the
nucleon states Mulders:1995dh ; Boer:1997nt ; Bacchetta:2006tn . Extending
these parameterizations to the DPDFs will provide a unique perspective of the
hadron tomography. For example, it has been shown that the TMD quark and gluon
Sivers functions at small-$x$ are related to the spin-dependent odderon
Zhou:2013gsa ; Boer:2015pni ; Boussarie:2019vmk ; Dong:2018wsp ;
Kovchegov:2021iyc . It is interesting to explore how the diffractive Sivers
functions arise from the spin-dependent dipole S-matrix.

In addition, the gluon GPDs can be expressed in terms of the dipole amplitudes
Hatta:2017cte :
$H_{g}(x_{I\\!\\!P},\Delta_{\perp})=\frac{2N_{c}}{\alpha_{s}}\int
d^{2}q_{\perp}q_{\perp}^{2}{\cal
F}_{x_{I\\!\\!P}}(q_{\perp},\Delta_{\perp})=\frac{N_{c}^{2}-1}{N_{c}\alpha_{s}}\int
d^{2}q_{\perp}q_{\perp}^{2}{\cal G}_{x_{I\\!\\!P}}(q_{\perp},\Delta_{\perp})$,
where we have set the skewness parameter in the GPD $\xi=x_{I\\!\\!P}$.
Therefore, we can rewrite the DPDFs in terms of the gluon GPD as well. In
particular, for large transverse momentum DPDFs, we have

$\left.x\frac{df_{q,g}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}\right|_{k_{\perp}\gg
Q_{s}}=\frac{\alpha_{s}^{2}}{2\pi}\frac{{\cal
C}_{q,g}}{k_{\perp}^{4}}\left(H_{g}(x_{I\\!\\!P},\Delta_{\perp})\right)^{2}\
,$ (11)

where ${\cal C}_{q}=\beta^{3}(1-\beta)^{2}/N_{c}$ and ${\cal
C}_{g}=(1+2\beta)^{2}(1-\beta)^{3}N_{c}^{2}/(N_{c}^{2}-1)$. This builds an
interesting connection between hard diffractive processes and the GPD physics.
In addition, the $1/k_{\perp}^{4}$ power behavior is very different from that
of the non-diffractive quark and gluon TMDs Marquet:2009ca ; Xiao:2017yya
which behave as $1/k_{\perp}^{2}$ at large-$k_{\perp}$ leading to
logarithmically divergent $k_{\perp}$ integrals.

##  III Unpolarized DPDFs in a saturation model

Here, we consider the kinematics of zero momentum transfer from the target
($\Delta_{\perp}=0)$ and investigate the $\beta$-dependence of DPDFs
$f^{D}_{q,g}$ in detail. To illustrate that, we evaluate the $k_{i\perp}$
integrals in (3) and (7) numerically assuming a simple Gaussian form for the
color dipole (see, e.g., Golec-Biernat:1999qor )

${\cal F}_{x_{I\\!\\!P}}(k_{i\perp})=\frac{S_{\perp}}{(2\pi)^{2}}\frac{1}{\pi
Q_{s}^{2}}e^{-k_{i\perp}^{2}/Q_{s}^{2}}\ ,$ (12)

where the (quark) saturation scale $Q_{s}$ depends on $x_{I\\!\\!P}$ and
$S_{\perp}$ represents the transverse area of the target. The same
parameterization will be used for the gluon dipole ${\cal
G}_{x_{I\\!\\!P}}(k_{i\perp})$ but with the saturation momentum for the
adjoint representation $Q_{as}$. They are related as
$Q_{as}^{2}=\frac{C_{A}}{C_{F}}Q_{s}^{2}\approx 2Q_{s}^{2}$. With these
assumptions, we find that the DPDFs depend on the ratios $k_{\perp}/Q_{s}$ and
$k_{\perp}/Q_{as}$ for the quark and gluon distributions, respectively,

$\displaystyle
x\frac{df_{q,g}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}={\cal
N}_{q,g}D_{q,g}\left(\beta,\frac{k_{\perp}}{Q_{s,as}}\right)\ ,$ (13)

where ${\cal N}_{q}=S_{\perp}^{2}N_{c}/(2\pi)^{5}$ and ${\cal
N}_{g}=S_{\perp}^{2}2(N_{c}^{2}-1)/(2\pi)^{5}$. For ordinary TMDs, relations
like (13) exhibit the phenomenon of geometric scaling, namely, distributions
$f(x,k_{\perp})$ depend on $k_{\perp}$ and $x$ only through the ratio
$k_{\perp}/Q_{s}(x)$. However, in the present problem, the extra factor
$\beta$ complicates this interpretation. In Fig. 3, we show the quark DPDF
$D_{q}(\beta,k_{\perp}/Q_{s})$ as functions of $k_{\perp}/Q_{s}$ for different
values of $\beta$. The strong falloff at large $k_{\perp}$ confirms the above
power counting analysis. On the other hand, the shape and magnitude of these
curves strongly depend on $\beta$.

Figure 3: The transverse momentum dependence of the quark diffractive
distribution for different values of $\beta=0.3,0.5,0.8$, respectively,
plotted as functions of $k_{\perp}/Q_{s}$, see, Eq. (13).  Figure 4: Same as
Fig. 3 for the gluon. Here, we plot
$\frac{k_{\perp}}{\widetilde{Q}_{as}}\frac{D(\beta,k_{\perp}/Q_{as})}{1-\beta}$
as functions of $k_{\perp}/\widetilde{Q}_{as}$ for $\beta=0.3,0.5,0.8$ (blue,
red, black curves), respectively, where
$\widetilde{Q}_{as}^{2}=(1-\beta)Q_{as}^{2}$.

Nonetheless, the authors of Ref. Iancu:2021rup  have observed that the DPDFs
do exhibit geometric scaling if it is expressed in terms of the modified
saturation momentum $\widetilde{Q}_{(a)s}^{2}\equiv(1-\beta)Q_{(a)s}^{2}$. A
simple explanation within our approach is to look at the denominator of (4)
and (8)

$\displaystyle\frac{1}{k_{\perp}^{2}+(1-\beta)k_{i\perp}^{2}-2(1-\beta)k_{i\perp}\cdot
k_{\perp}}\,,$ (14)

and noting that, typically, $k_{i\perp}\sim Q_{(a)s}$. When $1-\beta$ is order
unity, the characteristic value of $k_{\perp}$ is $k_{i\perp}\sim Q_{(a)s}$
such that the scaling geometric variable is $k_{\perp}/Q_{(a)s}$. However,
when $1-\beta$ becomes very small, (14) is power suppressed when
$k_{\perp}^{2}\gtrsim(1-\beta)k^{2}_{i\perp}\sim(1-\beta)Q_{as}^{2}=\widetilde{Q}^{2}_{(a)s}$.
Thus the $k_{\perp}$-distribution is effectively limited to
$k_{\perp}\lesssim\widetilde{Q}_{(a)s}$ and the scaling variable becomes
$k_{\perp}/\widetilde{Q}_{(a)s}$.

To corroborate the above argument, in Fig. 4 we plot $D_{g}$ as a function of
$k_{\perp}/\widetilde{Q}_{as}$ for the same three values of $\beta$ as in Fig.
3. We divided the results by the common prefactor $1-\beta$, which naturally
arises from the large-$\beta$ analysis (see below). The three curves now agree
very well with each other and peak around the modified saturation momentum
$k_{\perp}=\widetilde{Q}_{as}$, in agreement with Iancu:2021rup .

We further integrate out $k_{\perp}$ to derive the integrated DPDFs and
compare to previous results Iancu:2021rup ; Golec-Biernat:2001gyl . Within the
Gaussian approximation for the dipole amplitudes, we can write

$\displaystyle
x\frac{d\,f_{q,g}^{D}(\beta;x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}={\cal
N}_{q,g}2\pi{\cal D}_{q,g}(\beta)Q_{s,as}^{2}\ .$ (15)

In Fig. 5, we show the numerical results for ${\cal D}_{q,g}(\beta)$. In order
to understand the different behaviors in the quark and gluon cases, we will
derive below some analytic results around the endpoint regions $\beta=0$ and
$\beta=1$. This also helps to provide simple approximate expressions for the
functions ${\cal D}_{q,g}(\beta)$ in the whole kinematic interval
$0\leq\beta\leq 1$,

$\displaystyle{\cal D}_{q}(\beta)$ $\displaystyle=$
$\displaystyle\left(b_{1}+b_{2}(1-\beta)\right)\beta(1-\beta)\ ,$ (16)
$\displaystyle{\cal D}_{g}(\beta)$ $\displaystyle=$
$\displaystyle(a_{0}+a_{1}\beta)(1-\beta)^{2}\ ,$ (17)

with $b_{1}=\frac{3\pi^{2}}{16}-1$, $b_{2}=\frac{20-3\pi^{2}}{16}$,
$a_{0}=\frac{\ln(2)}{2}$ and
$a_{1}=\frac{45\pi^{2}-272}{256}-\frac{\ln(2)}{2}$. These parameters are
determined by the endpoint behaviors. When $\beta=0$, by averaging over the
azimuthal angles of $k_{1\perp}$ and $k_{2\perp}$ in Eq. (3,7), we find

$\displaystyle{\cal T}_{q}|_{\beta\to
0}=\Theta(k_{1\perp}-k_{\perp})\times\Theta(k_{2\perp}-k_{\perp})\ ,$ (18)
$\displaystyle{\cal T}_{g}|_{\beta\to
0}=\frac{1}{8k_{\perp}^{4}}\left(k_{\perp}^{2}+k_{1\perp}^{2}+(k_{\perp}^{2}-{k}_{1\perp}^{2}){\rm
Sign}({k}_{1\perp}-k_{\perp})\right)$
$\displaystyle~{}~{}\times\left(k_{\perp}^{2}+k_{2\perp}^{2}+(k_{\perp}^{2}-{k}_{2\perp}^{2}){\rm
Sign}({k}_{2\perp}-k_{\perp})\right)\ .$ (19)

The subsequent integrals can be done analytically and this fixes the values of
$b_{1}+b_{2}$ and $a_{0}$.

On the other hand, the behavior near $\beta\to 1$ is much more complicated.
The integrand of Eq. (3) vanishes if we set $\beta=1$ naively. In order to
obtain the correct leading behavior in $1-\beta$, we first make the rescaling
$\tilde{k}_{i\perp}=\sqrt{1-\beta}k_{i\perp}$ after which $k_{\perp}$ and
$\tilde{k}_{i\perp}$ become comparable (see the argument around (14)). We then
expand the integrand ${\cal T}_{q,g}(k,k_{1\perp},k_{2\perp})$ around
$\beta=1$,

$\displaystyle{\cal T}_{q}|_{\beta\to 1}$ $\displaystyle=$
$\displaystyle\frac{\tilde{k}_{1\perp}^{2}(2k_{\perp}^{2}+\tilde{k}_{1\perp}^{2})}{(k_{\perp}^{2}+\tilde{k}_{1\perp}^{2})^{2}}\frac{\tilde{k}_{2\perp}^{2}(2k_{\perp}^{2}+\tilde{k}_{2\perp}^{2})}{(k_{\perp}^{2}+\tilde{k}_{2\perp}^{2})^{2}}\
,$ (20) $\displaystyle{\cal T}_{g}|_{\beta\to 1}$ $\displaystyle=$
$\displaystyle\frac{(1-\beta)^{2}}{2}\frac{\tilde{k}_{1\perp}^{2}(3k_{\perp}^{4}+3k_{\perp}^{2}\tilde{k}_{1\perp}^{2}+\tilde{k}_{1\perp}^{4})}{(k_{\perp}^{2}+\tilde{k}_{1\perp}^{2})^{3}}$
(21)
$\displaystyle\times\frac{\tilde{k}_{2\perp}^{2}(3k_{\perp}^{4}+3k_{\perp}^{2}\tilde{k}_{2\perp}^{2}+\tilde{k}_{2\perp}^{4})}{(k_{\perp}^{2}+\tilde{k}_{2\perp}^{2})^{3}}\
.$

The remaining integrals over $\widetilde{k}_{i\perp}$ and $k_{\perp}$ can be
performed analytically. An overall factor of $1-\beta$ comes from the final
integral $\int d^{2}k_{\perp}\sim\widetilde{Q}_{s}^{2}=(1-\beta)Q_{s}^{2}$,
and the parameters $b_{1}$ and $a_{0}+a_{1}$ can be read off from the
coefficients.

Figure 5: The integrated quark and gluon diffractive distribution functions as
functions of $\beta$. Relative sizes between these two depend on additional
factors, see, Eq. (15).

Similar parameterizations for the quark/gluon DPDFs have been derived in Ref.
Buchmuller:1998jv . Our result in Eq. (16) for the quark case agrees with
theirs, whereas there is a factor of 2 difference for the gluon case in Eq.
(17). As we explain in detail in Appendix C, we also find agreement with the
results of Refs. Golec-Biernat:1999qor ; Golec-Biernat:2001gyl , although in
the gluon case there is an overall factor 2 difference due to an inconsistent
parameterization of the dipole S-matrix in the adjoint representation in
Golec-Biernat:1999qor , see also the discussion in Marquet:2007nf . This also
explains the difference between our result and that from Ref.
Buchmuller:1998jv  for the gluon DPDF.

The above results can be regarded as inputs for the DPDFs at the initial scale
$\mu\sim Q_{s,as}$. They have also been extracted from the HERA experiments
H1:1997bdi ; ZEUS:1997fox ; Newman:2013ada . The shapes and magnitudes of the
quark/gluon diffractive PDFs calculated from the CGC/dipole formalism in Eqs.
(16,17) are very similar to those determined from these measurements. Of
course, to have a quantitative comparison, one has to evolve our results to
the relevant scales and compare them to those in Refs. H1:1997bdi ;
ZEUS:1997fox ; Newman:2013ada . We will pursue this direction in a future
publication.

In addition, we notice that the ratio between total momentum fractions carried
by quarks and gluons, $\frac{2N_{f}\int d\beta
xf^{D}_{q}(\beta;x_{I\\!\\!P})}{\int d\beta
xf^{D}_{g}(\beta;x_{I\\!\\!P})}\approx 0.33$ with $N_{f}=3$, is very close to
that determined by the ZEUS Collaboration ZEUS:2009uxs . The additional
difference can be explained by the evolution effect from $Q_{s}$ to the scales
used there. It is interesting to further notice that the total momentum
fraction is normalized,

$\int d\beta
x\left[\frac{2N_{f}df^{D}_{q}(\beta;x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}+\frac{df^{D}_{g}(\beta;x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}\right]\approx\frac{S_{\perp}^{2}}{(2\pi)^{3}}Q_{s}^{2}\
,$ (22)

where we have applied $Q_{as}^{2}/Q_{s}^{2}\approx C_{A}/C_{F}$.

##  IV Conclusion

In summary, we have established a connection between the QCD factorization for
hard diffractive processes and the small-$x$ CGC formalism by computing the
diffractive PDFs in terms of the dipole S-matrices. The transverse momentum
dependent DPDFs can be studied in semi-inclusive diffractive DIS processes.
The integrated DPDFs have been evaluated with a Gaussian assumption for the
dipole S-matrix. These results can be regarded as inputs at the initial scale
for collinear QCD factorizations applied to all other hard diffractive
processes. This provides a powerful tool to unify different frameworks at
small-$x$.

We have briefly commented on the nontrivial correlation between the momentum
transfer $\Delta_{\perp}$ and the parton transverse momentum $k_{\perp}$ in
the DPDFs as a direct consequence of the correlations in the dipole amplitude
between $\Delta_{\perp}$ and $k_{i\perp}$ Hatta:2016dxp . Such correlations
are measurable at the EIC where the proton recoil momentum $\Delta_{\perp}$
can be directly measured by Roman pots. Moreover, if these correlations depend
on the nucleon spin, they will open up new opportunities to explore the spin-
orbital correlations inside the nucleon. This may lead to a complementary
method to investigate the gluon orbital angular momentum contribution to the
proton spin, as compared to the proposals of exclusive processes in the
literature for this purpose Ji:2016jgn ; Hatta:2016aoc ; Bhattacharya:2022vvo
; Courtoy:2013oaa ; Courtoy:2014bea . We will come back to these questions in
a future publication.

Acknowledgments: We thank Edmond Iancu and Al Mueller for the discussions.
This material is based upon work supported by the LDRD programs of LBNL and
Brookhaven Science Associates, and by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under contract numbers DE-AC02-05CH11231
and DE- SC0012704. This work is also supported by the National Natural Science
Foundations of China under Grant No. 11575070.

##  Appendix A  Calculation of DPDFs

In this appendix we give the outline of the derivation of the quark DPDF (3)
and gluon DPDF (7). To compute the gluon DPDF from its definition of (2), we
first expand the gluonic operator $F^{+\mu}$ and the associated gauge links
${\cal L}_{n}$ which interact with the background field $A^{+}$ of the CGC.
The multiple interactions with the CGC can be resummed by the Wilson lines $U$
which constitute the dipole S-matrix. There are important differences compared
to the non-diffractive TMD gluon distributions computed in Ref. Xiao:2017yya .
First, there is no leading order correspondence between the gluon DPDF and the
dipole amplitude as was the case for the non-diffractive TMD gluon
distribution Dominguez:2010xd ; Dominguez:2011wm . This is because a final
state gluon radiation is required to guarantee the colorless exchange between
the gluonic fields and the nucleon target in the diffractive case. As a
consequence, non-zero contributions come from the diagrams shown in Fig. 2.
Second, because of the colorless exchange, the amplitude of this diagram
depends on the dipole S-matrix, whereas for the non-diffractive TMDs, it is
the amplitude squared that depends on the dipole S-matrix.

Now, let us move to the derivation. We closely follow the calculation of the
non-diffractive TMD gluon distribution at small-$x$ in Ref. Xiao:2017yya . The
major difference, as mentioned above, is the colorlesss exchange between the
nucleon target the partonic part from the gluon distribution calculations,
see, Fig. 2. Consider the right diagram in Fig. 2. We first observe that the
gluon connecting the target and the Wilson line (denoted by a double line) has
a vanishing plus-momentum. This is a consequence of the eikonal approximation
(note the $dz^{-}$ integral in (6)). Therefore, the other gluon entering the
triple gluon vertex has a plus-momentum $k_{1}^{+}=x_{I\\!\\!P}P^{+}$. The
kinematics of this splitting is

$\displaystyle k_{1}^{\mu}=\left(k_{1}^{+},0,k_{1\perp}\right)\to
k^{\mu}+q^{\mu}=(\beta
k_{1}^{+},-q^{-},k_{1\perp}-k_{\perp})+\left((1-\beta)k_{1}^{+},q^{-},k_{\perp}\right).$
(23)

Since the outgoing gluon with momentum $q^{\mu}$ is on-shell and transverse,

$\displaystyle
q^{-}=\frac{k_{\perp}^{2}}{2(1-\beta)k_{1}^{+}},\qquad\epsilon^{*}\cdot
q=0,\quad\to\quad\epsilon^{+*}=\frac{\vec{\epsilon}^{*}_{\perp}\cdot\vec{k}_{\perp}}{q^{-}},$
(24)

where $\epsilon^{\alpha}$ is the polarization vector, and we work in the gauge
$\epsilon^{-}=0$. The phase space integral for the emitted gluon is thus

$\displaystyle\frac{d^{4}q}{(2\pi)^{3}}\delta(q^{2})=\frac{d^{2}k_{\perp}}{16\pi^{3}}\frac{dx_{I\\!\\!P}}{(1-\beta)x_{I\\!\\!P}}=\frac{d^{2}k_{\perp}}{16\pi^{3}}\frac{dY_{I\\!\\!P}}{1-\beta}.$
(25)

The intermediate gluon denominator takes the form

$\displaystyle\frac{1}{k^{2}}=\frac{1}{-2\beta
k_{1}^{+}q^{-}-(\vec{k}_{1\perp}-\vec{k}_{\perp})^{2}}=-\frac{1-\beta}{\beta\vec{k}_{\perp}^{2}+(1-\beta)(\vec{k}_{1\perp}-\vec{k}_{\perp})^{2}}.$
(26)

Triple gluon vertex can be evaluated as

$\displaystyle\int d^{3}ze^{ik\cdot z}\langle q,\alpha|F^{+\nu}(z)$
$\displaystyle\sim$ $\displaystyle\langle q,\alpha|\left(\int
d^{4}xAAA\right)(k^{+}A^{\nu}-k^{\nu}A^{+})$ (27) $\displaystyle\sim$
$\displaystyle\biggl{[}k^{+}(g^{\nu\alpha}_{\perp}(-k+q)^{-}+g^{\alpha-}(-2q-k)^{\nu}+g^{-\nu}(2k+q)^{\alpha})$
$\displaystyle\qquad-k^{\nu}(g^{+\alpha}(-k+q)^{-}+g^{\alpha-}(-2q-k)^{+}+g^{-+}(2k+q)^{\alpha})\biggr{]}\epsilon^{*}_{\alpha}(q)A^{+}(k_{g})$
$\displaystyle=$
$\displaystyle\left(2k^{+}q^{-}g^{\nu\alpha}_{\perp}\epsilon^{*}_{\alpha}-2q^{-}k^{\nu}\epsilon^{+*}-2k^{\nu}k\cdot\epsilon^{*}\right)A^{+}$
$\displaystyle=$ $\displaystyle-\frac{1}{1-\beta}\left(\beta
k_{\perp}^{2}\delta^{\nu\alpha}_{\perp}+2(1-\beta)(k_{1\perp}^{\nu}-k^{\nu}_{\perp})(k_{1\perp}^{\alpha}-k^{\alpha}_{\perp})\right)\epsilon^{*}_{\alpha}A^{+},$

where $\nu$ is transverse, and we used

$\displaystyle
k\cdot\epsilon^{*}=-q^{-}\epsilon^{+*}-(k_{1\perp}-k_{\perp})\cdot\vec{\epsilon}^{*}_{\perp}=-k_{1\perp}\cdot\epsilon^{*}_{\perp}.$
(28)

Multiplying (26) by (27) and squaring, we obtain the kernel in (8).

Similarly, we can derive the TMD quark DPDF shown in left panel of Fig. 2. In
particular, the partonic part is the same as that in Eq. (41) of the published
version of Ref. Xiao:2017yya . Again, to compute the quark DPDF, we need to
make sure that the colorless exchange between the partonic part and the
nucleon target. This requires the following replacement,

$\displaystyle\frac{1}{N_{c}}{\rm Tr}\left[\langle
U(x_{1})U^{\dagger}(x_{2})U(y_{2})U^{\dagger}(y_{1})\rangle-\langle
U(x_{1})U^{\dagger}(x_{2})\rangle-\langle
U(x_{2})U^{\dagger}(y_{1})\rangle+1\right]$
$\displaystyle\longrightarrow\frac{1}{N_{c}}{\rm Tr}\left[\langle
U(x_{1})U^{\dagger}(x_{2})\rangle-1\right]\times\frac{1}{N_{c}}{\rm
Tr}\left[\langle U(y_{2})U^{\dagger}(y_{1})\rangle-1\right]\ .$ (29)

With additional kinematic variable replacements, we arrive at the result in
Eq. (3).

##  Appendix B Derivation of Eq. (10)

In this appendix, we show how the semi-inclusive quark production in
diffractive DIS calculated in the CGC/dipole formalism can be factorized into
the TMD quark DPDF in the limit of $k_{\perp}\ll Q$, where $k_{\perp}$ is
final state quark transverse momentum and $Q$ the virtuality of the photon. In
this process, $\ell+p\to\ell^{\prime}+k+X+p^{\prime}$, the incoming lepton
radiates a highly virtual photon, which interacts with the nucleon target
diffractively and produces a final state quark with momentum $k$. The
derivation here also applies to semi-inclusive hadron production in
diffractive DIS, where a final state fragmentation function will be included.
At this order, there is no difference between hadron and jet productions. At
higher orders, the TMD fragmentation and jet functions will enter the
factorization formula, respectively.

In the CGC/dipole formalism, the quark production comes from the process that
the virtual photon splits into a quark-antiquark pair and goes through
diffractive interaction with the nucleon target, $\gamma^{*}p\to q\bar{q}p$,
at the leading order. The amplitude for this process has been computed
Hatta:2016dxp , and the differential cross section for the quark production
can be derived by integrating out the phase space of the antiquark,

$\displaystyle\frac{d\sigma^{\rm SIDDIS}(\ell
p\to\ell^{\prime}qp^{\prime}X)}{dx_{B}dQ^{2}d^{2}k_{\perp}dY_{I\\!\\!P}dt}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{em}^{2}e_{q}^{2}N_{c}}{x_{B}Q^{2}}\left(1-y+\frac{y^{2}}{2}\right)\int{dz}\delta(1-\beta-\beta\frac{k_{\perp}^{2}}{\epsilon_{f}^{2}})\int
d^{2}q_{\perp}d^{2}q_{\perp}^{\prime}\mathcal{F}_{x}(q_{\perp},\Delta_{\perp})\mathcal{F}_{x}(q_{\perp}^{\prime},\Delta_{\perp})$
(30)
$\displaystyle\times\left(z^{2}+(1-z)^{2}\right)\left[\frac{k_{\perp}}{k_{\perp}^{2}+\epsilon_{f}^{2}}-\frac{k_{\perp}-q_{\perp}}{(k_{\perp}-q_{\perp})^{2}+\epsilon_{f}^{2}}\right]\cdot\left[\frac{k_{\perp}}{k_{\perp}^{2}+\epsilon_{f}^{2}}-\frac{k_{\perp}-q_{\perp}^{\prime}}{(k_{\perp}-q_{\perp}^{\prime})^{2}+\epsilon_{f}^{2}}\right]\
,$

where $\beta=x_{B}/x_{I\\!\\!P}$ and $z$ is the momentum fraction of incoming
photon carried by the final state quark and $\epsilon_{f}^{2}=z(1-z)Q^{2}$.
The Delta function in the above equation comes from the momentum conservation
along the nucleon momentum direction. To derive that, we notice that the
momentum fractions of the incoming nucleon carried by the quark and antiquark:
$x_{q}=\frac{x_{B}k_{\perp}^{2}}{zQ^{2}}$ and
$x_{\bar{q}}=\frac{x_{B}k_{\perp}^{2}}{(1-z)Q^{2}}$, respectively, where we
have applied the approximation of $\Delta_{\perp}\approx 0$ and the quark and
antiquark have balanced transverse momenta
$|k_{q\perp}|\sim|k_{\bar{q}\perp}|$. The momentum conservation leads to
$x_{I\\!\\!P}=x_{B}+x_{q}+x_{\bar{q}}$, which results in
${(1-\beta)}/{\beta}={k_{\perp}^{2}}/{\epsilon_{f}^{2}}$.

Similar to the non-diffractive quark production in DIS process Marquet:2009ca
, the above contribution is also dominated by the so-called aligned jet
configuration, i.e., $z\sim 1$ or $z\sim 0$. Only in this kinematics can we
find that $\epsilon_{f}^{2}\sim k_{\perp}^{2}$ and the differential cross
section does not vanish at large $Q^{2}$. This can be illustrated by rewriting
the above Delta function,

$\displaystyle\delta\left(1-\beta-\beta\frac{k_{\perp}^{2}}{\epsilon_{f}^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{\beta}{(1-\beta)^{2}}\frac{k_{\perp}^{2}}{Q^{2}}\delta\left(z(1-z)-\frac{\beta}{1-\beta}\frac{k_{\perp}^{2}}{Q^{2}}\right)$
(31) $\displaystyle=$
$\displaystyle\frac{\beta}{(1-\beta)^{2}}\frac{k_{\perp}^{2}}{Q^{2}}\left(\frac{\delta(1-z)}{z}+\frac{\delta(z)}{1-z}\right)\
,$

where the last equation comes from small $k_{\perp}^{2}/Q^{2}$ expansion.
Substituting the above expansion result into Eq. (30), we derive Eq. (10).

##  Appendix C Comparison to the diffractive structure functions calculated in
Ref. Golec-Biernat:1999qor

Integrating over the azimuthal angles in Eqs. (3) and (7), we find

$\displaystyle
x\frac{df_{q}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}$
$\displaystyle=$ $\displaystyle\frac{\pi
N_{c}\beta}{8(1-\beta)^{2}}\left[\int_{0}^{\infty}dk_{1\perp}^{2}{\cal
F}(k_{1\perp}^{2})\left\\{1-2\beta+\frac{(1-\beta)k_{1\perp}^{2}-(1-2\beta)k_{\perp}^{2}}{\sqrt{(k_{\perp}^{2}+(1-\beta)k_{1\perp}^{2})^{2}-4(1-\beta)^{2}k_{\perp}^{2}k_{1\perp}^{2}}}\right\\}\right]^{2}\
,$ (32) $\displaystyle
x\frac{df_{g}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}dt}$
$\displaystyle=$
$\displaystyle\frac{\pi(N_{c}^{2}-1)}{8(1-\beta)^{3}}\Biggl{[}\int_{0}^{\infty}dk_{1\perp}^{2}{\cal
G}(k_{1\perp}^{2})\Biggl{\\{}\beta^{2}+(1-\beta)^{2}+\frac{k_{1\perp}^{2}}{k_{\perp}^{2}}(1-\beta)$
(33)
$\displaystyle\qquad\qquad\qquad-\frac{((1-2\beta)k_{\perp}^{2}-(1-\beta)k_{1\perp}^{2})^{2}+2\beta(1-\beta)k_{\perp}^{4}}{k_{\perp}^{2}\sqrt{(k_{\perp}^{2}+(1-\beta)k_{1\perp}^{2})^{2}-4(1-\beta)^{2}k_{\perp}^{2}k_{1\perp}^{2}}}\Biggr{\\}}\Biggr{]}^{2}\
.$

We immediately recognize the same structure as in the diffractive structure
functions calculated in Ref. Golec-Biernat:1999qor . For the $q\bar{q}$
contribution to the diffractive structure function for the transversely
polarized photon, we can explicitly rewrite Eq. (20) of Ref. Golec-
Biernat:1999qor  as

$\displaystyle
F^{D}_{\\{t,q\bar{q}\\}}(Q^{2},\beta,x_{I\\!\\!P})=Q^{2}\pi(1-\beta)\int_{0}^{1}d\alpha(\alpha^{2}+(1-\alpha)^{2})\frac{df_{q}^{D}(\beta,k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}}$
(34)

after adjusting the normalization difference of the unintegrated gluon
distribution

$\displaystyle{\cal F}_{{\rm Ref.}[26]}(q_{\perp})$ $\displaystyle=$
$\displaystyle\frac{2\pi N_{c}q_{\perp}^{2}}{\alpha_{s}}{\cal F}(q_{\perp})\
.$ (35)

In (34), we have integrated over $t$ assuming the exponential form
$e^{B_{D}t}$ (as was done in Golec-Biernat:1999qor ). The parameter $\alpha$
is related to $k_{\perp}$ as

$k_{\perp}^{2}=\alpha(1-\alpha)Q^{2}\frac{1-\beta}{\beta}\ .$ (36)

Eq. (34) can then be recognized as a $k_{T}$ factorization formula. At
large-$Q^{2}$, one can take the collinear limit of this by inserting the above
constraint and expanding in $k_{\perp}^{2}/Q^{2}$

$\displaystyle 1$ $\displaystyle=$ $\displaystyle\int
dk_{\perp}^{2}\delta\left(k_{\perp}^{2}-\alpha(1-\alpha)\frac{Q^{2}(1-\beta)}{\beta}\right)\approx\frac{\beta}{Q^{2}(1-\beta)}\left(\frac{\delta(\alpha)}{1-\alpha}+\frac{\delta(1-\alpha)}{\alpha}\right)\int
dk_{\perp}^{2}\ .$ (37)

Now the integral over $\alpha$ can be easily carried out, giving

$\displaystyle F^{D}_{\\{t,q\bar{q}\\}}(Q^{2},\beta,x_{I\\!\\!P})\approx
2\beta\frac{df_{q}^{D}(\beta;x_{I\\!\\!P})}{dY_{I\\!\\!P}}\,,$ (38)

where the factor of 2 accounts for the quark and antiquark contributions. The
longitudinal diffractive structure function $F^{D}_{l,q\bar{q}}$, like the
inclusive longitudinal structure function, is power suppressed. Therefore, Eq.
(21) of Ref. Golec-Biernat:1999qor  does not have a corresponding
interpretation in terms of the DPDF.

On the other hand, the $q\bar{q}g$ contribution to the transverse structure
function, Eq. (23) of Golec-Biernat:1999qor , is related to the gluon DPDF
(33). A direct comparison is somewhat obscure because Ref. Golec-
Biernat:1999qor  did not distinguish the quark and gluon dipole amplitudes
(and hence the corresponding saturation momenta $Q_{s}$ and $Q_{as}$). Yet, we
can make the following identification in the gluon case

$\displaystyle{\cal F}_{{\rm
Ref.}[26]}(q_{\perp})\to\frac{\pi(N_{c}^{2}-1)q_{\perp}^{2}}{N_{c}\alpha_{s}}{\cal
G}(q_{\perp})\ .$ (39)

With this, Eq. (23) of Golec-Biernat:1999qor  becomes

$\displaystyle
x_{I\\!\\!P}F^{D}_{\\{t,q\bar{q}g\\}}(Q^{2},\beta,x_{I\\!\\!P})$
$\displaystyle=$
$\displaystyle\pi\beta\int_{\beta}^{1}\frac{d\xi}{\xi\beta^{\prime}}\left(\left(1-\xi\right)^{2}+\xi^{2}\right)\int^{(1-\beta^{\prime})Q^{2}}dk_{\perp}^{2}\frac{\alpha_{s}}{2\pi}\ln\frac{(1-\beta^{\prime})Q^{2}}{k_{\perp}^{2}}x^{\prime}\frac{df_{g}(\beta^{\prime},k_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}}$
(40) $\displaystyle=$
$\displaystyle\int_{\beta}^{1}d\xi\left(\left(1-\xi\right)^{2}+\xi^{2}\right)\int^{(1-\beta^{\prime})Q^{2}}\frac{d^{2}k_{\perp}}{k_{\perp}^{2}}\frac{\alpha_{s}}{2\pi^{2}}\int^{k_{\perp}^{2}}d^{2}k^{\prime}_{\perp}x^{\prime}\frac{df_{g}(\beta^{\prime},k^{\prime}_{\perp};x_{I\\!\\!P})}{dY_{I\\!\\!P}}\,,$

where $\beta^{\prime}=\beta/\xi$ and $x^{\prime}=x_{I\\!\\!P}\beta^{\prime}$.
This can be recognized as the $g\to q$ DGLAP evolution of the collinear DPDF.

We however note that (39) is an ad hoc prescription to correct for the
inconsistent treatment of the quark and gluon saturation momenta $Q_{s}$ and
$Q_{as}$ in Golec-Biernat:1999qor . This introduces ambiguities. For example,
by normalizing ${\cal G}$ to match the integrated gluon distribution with the
same saturation scale $Q_{as}=Q_{s}$ as for the quark distribution, the
unitarity of the dipole S-matrix is spoiled. As a consequence, the numerical
result for the integrated gluon DPDF differs from ours by a factor of 2. This
can be seen by comparing the curves in our Fig. 5 and those in Fig. 3 of Ref.
Golec-Biernat:2001gyl  after adjusting the other normalization factors.

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