# Just Another Day on Twitter: A Complete 24 Hours of Twitter Data

Jürgen Pfeffer1, Daniel Matter1, Kokil Jaidka2, Onur Varol3, Afra Mashhadi4,
Jana Lasser5, 15, Dennis Assenmacher6, Siqi Wu7, Diyi Yang8, Cornelia
Brantner9, Daniel M. Romero7, Jahna Otterbacher10, Carsten Schwemmer11,
Kenneth Joseph12, David Garcia13, Fred Morstatter14

###### Abstract

At the end of October 2022, Elon Musk concluded his acquisition of Twitter. In
the weeks and months before that, several questions were publicly discussed
that were not only of interest to the platform’s future buyers, but also of
high relevance to the Computational Social Science research community. For
example, how many active users does the platform have? What percentage of
accounts on the site are bots? And, what are the dominating topics and sub-
topical spheres on the platform? In a globally coordinated effort of $80$
scholars to shed light on these questions, and to offer a dataset that will
equip other researchers to do the same, we have collected _all_ 375 million
tweets published within a $24$-hour time period starting on September 21,
2022. To the best of our knowledge, this is the first complete 24-hour Twitter
dataset that is available for the research community. With it, the present
work aims to accomplish two goals. First, we seek to answer the aforementioned
questions and provide descriptive metrics about Twitter that can serve as
references for other researchers. Second, we create a baseline dataset for
future research that can be used to study the potential impact of the
platform’s ownership change.

## Introduction

On March 21, 2006, Twitter’s first CEO Jack Dorsey sent the first message on
the platform. In the subsequent 16 years, close to 3 trillion tweets have been
sent.111While we do not have an official source for this number, it represents
an educated guess from a collaboration of dozens of scholars of Twitter.
Roughly two-thirds of these have been either removed from the platform because
the senders deleted them or because the accounts (and all their tweets) have
been banned from the platform, have been made private by the users, or are
otherwise inaccessible via the historic search with the v2 API endpoints. We
estimate that about 900 billion public tweets were on the platform when Elon
Musk acquired Twitter in October 2022 for $44B., i.e., he paid about 5 cents
per tweet.

Besides its possible economic value, Twitter has been instrumental in studying
human behavior with social media data and the entire field of Computational
Social Science (CSS) has heavily relied on data from Twitter. At the AAAI
International Conference on Web and Social Media (ICWSM), in the past two
years alone (2021-2022), over 30 scientific papers analyzed a subset of
Twitter for a wide range of topics ranging from public and mental health
analyses to politics and partisanship. Indeed, since its emergence, Twitter
has been described as a digital socioscope (i.e., social telescope) by
researchers in fields of social science (Mejova, Weber, and Macy 2015), “a
massive antenna for social science that makes visible both the very large
(e.g., global patterns of communications) and the very small (e.g., hourly
changes in emotions)”. Beyond CSS, there is increasing use of Twitter data for
training large pre-trained language models in the field of natural language
processing and machine learning, such as Bernice (DeLucia et al. 2022), where
2.5 billion tweets are used to develop representations for Twitter-specific
languages, and TwHIN-BERT (Zhang et al. 2022) that leverages 7 billion tweets
covering over 100 distinct languages to model short, noisy, and user-generated
text.

Although Twitter data has fostered interdisciplinary research across many
fields and has become a “model organism” of big data, scholarship using
Twitter data has also been criticized for various forms of bias that can
emerge during analyses (Tufekci 2014). One major challenge giving rise to
these biases is getting access to data and knowing about data quality and
possible data biases (Ruths and Pfeffer 2014; González-Bailón et al. 2014;
Olteanu et al. 2019). While Twitter has long served as one of the most
collaborative big social media platforms in the context of data-sharing with
academic researchers, there nonetheless exists a lack of transparency in
sampling procedures and possible biases created from technical artifacts
(Morstatter et al. 2013; Pfeffer, Mayer, and Morstatter 2018). These unknown
biases may jeopardize research quality. At the same time, access to
unfiltered/unsampled Twitter data is nearly impossible to access, and thus the
above mentioned studies, as well as thousands of others, still retain unknown
and potentially significant biases in their use of sampled data.

#### Contributions.

The data collection efforts presented in this paper were driven by a desire to
address these concerns about sampling bias that exist because of the lack of a
complete sample of Twitter data. Consequently, the main contribution of this
article is to create the first _complete_ dataset of 24 hours on Twitter and
make these Tweets available via future collaborations with the authors and
contributors of this article. The dataset collected and described here can be
used by the research community to:

  * •

Promote a better understanding of the communication dynamics on the platform.
For example, it can be used to answer questions like, how many active
(posting) accounts are on the platform? And, what are the dominating languages
and topics?

  * •

Create a set of descriptive metrics that can serve as references for the
research community and provide context to past and present research papers on
Twitter.

  * •

Provide a baseline for the situation _before_ the recent sale of Twitter. With
the new ownership of Twitter, platform policies as well as the company
structures are under significant change, which will create questions about
whether previous Twitter studies will be still valuable references for future
studies.

In the following sections, we describe the data collection process and provide
some descriptive analyses of the dataset. We also discuss ethical
considerations and data availability.

## Data

#### Data Collection.

We have collected $24$ hours of Twitter data from September 20, 15:00:00 UTC
to September 21 14:59:59 UTC. The data collection was accomplished by
utilizing the Academic API (Pfeffer et al. 2022) that is free and openly
available for researchers. The technical setup of the data collection pipeline
was dominated by two major challenges: First, how can we avoid—at least to a
satisfying extent—a temporal bias in data collection? Second, how can we get a
_good_ representation of Twitter? In the following, these two aspects are
discussed in more detail.

#### What is a complete dataset?

What does _complete_ mean when we want to collect a day’s worth of Twitter
data? It has been shown previously that the availability of tweets fluctuates,
especially in the first couple of minutes (Pfeffer et al. 2022)—people might
delete their tweets because of typos, tweets might be removed because of
violations of terms of service, etc. To reduce this initial uncertainty, we
have decided to collect the data 10 minutes after the tweets were sent.
Consequently, this dataset does not include all tweets that were sent on the
collection day but instead tries to create a somewhat _stable_ representation
of Twitter.

#### Avoiding temporal collection bias.

We wanted to collect a set of tweets close to the time when they were created.
However, collecting data takes time, which can introduce possible temporal
bias, e.g., if we want to collect data from the previous hour and the data
collection job takes three hours, then the data that is collected at the end
of the collection job will be much older (with potentially more tweet
removals) than the data that is collected at the beginning. To tackle this
challenge, we have split the day into 86,400 collection tasks, each consisting
of 1 second of Twitter activity. The collection of every second of data
started exactly 10 minutes after the data creation time. Because the data
collection of a second took more than a minute during peak times, we have
distributed the workload to 80 collection processes, i.e., Academic API
tokens, in order to avoid backlogs.

#### Number of tweets.

With the above-described process, we have collected 374,937,971 tweets within
the 24 hours time span. On average, this amounts to 4,340 [2,989 – 8,955]
tweets per second. Fig. 1 plots the number of tweets per Minute (avg=260,374,
min=192,322, max=435,721). The data collection started at 15:00 UTC, when
almost the entire Twitter world is awake. Then, we can see from Japan to
Europe time zone after time zone getting off the platform. While Europe and
the Americas are sleeping, Asia keeps the number of tweets at around 200,000.
Starting at 7:00 UTC, Europe is getting active again, followed by the Americas
from East to West. Another astonishing observation of this time series is that
the first minute of every hour has on average 15.5% more tweets than the
minute before—most likely due to bot activities and other timed tweet
releases, e.g., news.

Figure 1: Tweets per minute over the 24-hour collection period, time in UTC.

## Descriptive Analyses

### Active Users

The 375 million tweets in our dataset were sent by 40,199,195 accounts. While
the publicly communicated numbers of users of a platform are often based on
the number of active and passive visitors, we can state that Twitter has (or
at least had on our observed day) 40 million active contributors who have sent
at least one tweet. Less than 100 accounts have created about 1% (=3.5M)
tweets. ${\sim}175,000$ accounts (0.44%) created 50% of all tweets.

Table 1: Distribution of user activity % Total Tweets | % Total Users | Min. no. of Tweets
---|---|---
1% | 0.00023% | 2,267
10% | 0.01199% | 465
25% | 0.07284% | 152
50% | 0.43526% | 39
75% | 1.70955% | 11
90% | 4.18836% | 3

These numbers are not surprising when we consider that $>95\%$ of active
accounts have sent one or two tweets. However, these numbers lend more nuance
to recent reports from the Pew Research Center, which reported that while the
majority of Americans use social media, approximately 97% of all tweets were
posted by 25% of the users (McClain 2021). In fact, our dataset suggests that
worldwide, the numbers may be more skewed than previously suggested.

### User metrics

Figure 2: All languages occurring in at least 1% of the tweets.

#### Followers.

The active accounts on our day of Twitter data have a mean of 2,123 followers
(median=99). We can find six accounts with more than 100 million followers
(max=133,301,854), and 427/8,635 accounts with more than 10/1 million
followers. Exactly 50% of accounts that were active on our collection day have
less than 100 followers.

#### Following.

These accounts follow much fewer other accounts: mean=547, median=197, range:
0–4,103,801. Interestingly, there are 2,377 accounts that follow more than
100,000 other accounts. One-third of accounts follow less than 100 accounts,
but only 1.7% of accounts follow zero other accounts.

#### Listed.

Lists are a Twitter feature for users to organize accounts around topics and
filter tweets. While there is little evidence that lists are used widely on
the platform, this feature might be useful for getting an impression about the
number of _interesting_ content creators on the platform. The 40 million
active accounts in our dataset are listed (i.e., number of lists that include
a user) in 0 to 3,086,443 lists (mean=10.1, median=0). 1,692/46,139 accounts
are in lists of at least 10,000/1,000 accounts.

#### Tweets sent.

The user information of the tweet metadata also includes the number of tweets
that a user has sent—or at least how many of those tweets are still available
on Twitter. The sum of the sent tweets variable of all 40 million accounts is
${\sim}$404 billion (mean=9,704, median=1,522). If we assume that our initial
estimate of having 900 billion tweets on the platform at the time of data
collection is somewhat correct, the accounts active in our dataset have
contributed ${\sim}$45% of all of the available tweets over the entire
lifetime of Twitter.

#### Verified accounts.

At the time of our data collection, we can identify 221,246 verified accounts
among the 40 million active users.

### Tweets and retweets

79.2% of all tweets refer to other tweets, i.e. they are retweets or quotes of
or replies to other tweets. Consequently, 20.8% of the tweets in our dataset
are original tweets. The tweets with references are of the following types:
50.7% retweets, 4.3% quotes, 24.2% replies, i.e. half of all tweets are
retweets and a fourth are replies.

#### Retweeted and liked.

Studying the retweet and like numbers from the tweets’ metadata has created
little insight since the top retweeted tweets are very old tweets that have
been retweeted by chance on our collection day. Furthermore, we can see the
number of likes only for tweets that have been tweeted and retweeted. In any
case, the retweeted number is interesting—the 374 million tweets have been
retweeted 401 billion times. In other words, significant parts of historic
Twitter get retweeted on a daily basis.

### Languages

Twitter annotates a language variable for every tweet. Fig. 2 shows those
languages that were annotated on at least 1% of our dataset. Together, these
15 languages make up 92.5% of all tweets. Besides the most common languages on
Twitter, we can also find interesting language codes in this list: _und_
stands for undefined and represents tweets for which Twitter was not able to
identify a language; _qme_ and _zxx_ seem to be used by Twitter for tweets
consisting of only media or a Twitter card.

### Media

There are 112,779,266 media attachments in our data collection (76.9% photos,
20.7% videos, 2.4% animated GIFs), of which 37,803,473 have unique media keys
(83.8% photos, 10.0% videos, 6.2% animated GIFs).

### Geo-tags

We found only 0.5% of tweets to be geo-tagged. This is not surprising as
previous works have shown that the percentage of geo-tagging in Twitter has
been declining (Ajao, Hong, and Liu 2015). Fig. 3 shows the distribution of
the geo-tagged tweets across the world, with USA (20%), Brazil (11%), Japan
(8%), Saudi Arabia (6%) and India (4%) being the top five countries.

Figure 3: Choropleth map of the geo-tagged tweets across the world.

### Estimating prevalence of bot accounts

Twitter has a pivotal role in public discourse and entities that are after
power and influence often utilize this platform through social bots and other
means of automated activities. Since the early days of Twitter, researchers
have been studying bot behavior, and it has become an active research area
(Ferrara et al. 2016; Cresci 2020). The first estimation of bot prevalence on
Twitter indicates that 9-15% of Twitter accounts exhibit automated behavior
(Varol et al. 2017), while others have observed significantly higher
percentages of tweets produced by bot-likely accounts on specific discourses
(Uyheng and Carley 2021; Antenore, Camacho Rodriguez, and Panizzi 2022). One
major challenge in estimating bot prevalence is the variety of definitions,
datasets, and models used for detection (Varol 2022).

In this study, we employed BotometerLite (Yang et al. 2020), a scalable and
light-weight version of the Botometer (Sayyadiharikandeh et al. 2020), for
computing bot scores for unique accounts in our collection. In Fig. 4a, we
present the distribution of bot scores and nearly 20% of the 40 million active
accounts have scores above 0.5 suggesting bot-likely behavior.

(a)

(b)

(c)

Figure 4: BotometerLite scores distribution: (a) histogram and cumulative
distribution, (b) by account age, (c) by tweet counts in our dataset.

While identification of bots is a complex and possibly controversial
challenge, plotting the distributions of BotometerLite scores grouped by
account age in Fig. 4b suggests the proportions of accounts that show bot-like
behavior has increased dramatically in recent years. This result may also
suggest that the longevity of simpler bot accounts is limited and they are no
longer active on the platform. In Fig. 4c, we also present the distribution of
bot scores for different rates of activities in our dataset. Accounts that
have over 1,000 posts exhibit higher rates of bot-like behaviors.

It is important to mention that accounts studied in this paper were identified
due to their content creation activities. Our collection cannot capture
passive accounts that are simply used to boost follower counts without visible
activity on tweet streams. Fair assessment of bot prevalence is only possible
with complete access to Twitter’s internal database; since activity streams,
network data, and historical tweet archives can capture different sets of
accounts (Varol 2022).

### Content on Twitter

The top 500 hashtags occurred 81,468,508 times in the tweets. Via manual
inspection, we were able to identify the meaning of 95% of these top hashtags.
They can be aggregated into ten the categories.

Table 2 suggests that a large proportion of tweets referred to entertainment,
which together comprised about 30% of tweets. These included mentions of
celebrities (25.5%) and other entertainment-related tweets (5.4%) such as
mentions of South Korean boy band members, and other references to music,
movies, and TV shows. Our data collection time window occurred during
Fall/Winter 2022, when the world was discussing the protests in Iran after the
death of Mahsa Amini. Therefore, the Iranian protests also comprised a large
proportion of the hashtag volume at 16.6%.

Table 2: The categories of the top 500 hashtags in the dataset Category | Hashtags | Occurrence |
---|---|---|---
Celebrities | 159 | 20,809,742 | 25.5%
Sex | 104 | 20,529,196 | 25.2%
Iranian Protests | 15 | 13,488,295 | 16.6%
Entertainment | 45 | 4,392,227 | 5.4%
Advertisement | 32 | 4,644,540 | 5.7%
Politics | 38 | 3,858,550 | 4.7%
Finance | 30 | 3,549,107 | 4.4%
Games | 21 | 3,348,128 | 4.1%
Other | 31 | 2,672,291 | 3.3%
Unknown | 25 | 4,176,432 | 5.1%
Sum | 500 | 81,468,508 | 100.0%
Figure 5: MDS of top 10,000 hashtags based on co-usage by same accounts;
colors represent dominant language in tweets using a hashtag.

Finally, and perhaps surprisingly, the category _sex_ comprised over a quarter
of all content covered by the top hashtags, and was almost completely related
to escorts. “Other” topics reflect that on “regular” Twitter days, sports,
tech, and art may take up only about 3.3% of Twitter volume.

Fig. 5 is a hashtag visualization that attempts to provide an overview of the
entire content on Twitter. We first removed all tweets from accounts with more
than 240 tweets to reduce the noise from bots using random trending hashtags.
From the remaining tweets, we extracted the 10,000 most often used hashtags in
our dataset and created a hashtag similarity matrix with the number of
accounts that have used a pair of two hashtags on the day of data collection.
Every element in Fig. 5 represents a hashtag. The position is the result of
Multidimensional Scaling (MDS) and the color shows the dominant language that
was used in the tweets with the particular hashtag. In this figure, we can see
how languages separate the Twitter universe but that there are also topical
sub-communities within languages.

## Discussion and Potential Applications

Twitter is a social media platform with a worldwide userbase. Open access to
its data also makes it attractive to a large community of researchers,
journalists, technologists, and policymakers who are interested in examining
social and civic behavior online. Early studies of Twitter explored who says
what to whom on Twitter (Wu et al. 2011), characterizing its primary use as a
communication tool. Other early work mapped follower communities through ego
networks (Gruzd, Wellman, and Takhteyev 2011). However, Twitter has since
expanded into its own universe, with a plethora of users, uses, modalities,
communities, and real-life implications. Twitter is increasingly the source of
breaking news, and many studies from the U.S. and Europe have reported that
Twitter is one of the primary sources of news for their citizens. Twitter has
been used for political engagement and citizen activism worldwide. During the
COVID-19 pandemic, Twitter even assumed the role of the official mouthpiece
and crisis communication tool for many governments to contact their citizens,
and from which citizens could seek help and information.

Fig. 3 confirms prior reports that geotagging practices are limited in many
low- and middle-income countries (Malik et al. 2015); however, this should not
deter scholars from exploring alternative methods of triangulating the
location of users (Schwartz et al. 2013), and creating post-stratified
estimates of regional language use (Jaidka et al. 2020; Giorgi et al. 2022).
In prior studies, the difficulties in widespread data collection and analyses
have so far implied that most answers are based on smaller samples (usually
constrained by geography, for convenience) of a burgeoning Twitter population.
Fig. 5 and Table 2 also impressively illustrate that Twitter is about so much
more than US politics.

We hope that our dataset is the first step in creating alternatives for
conducting a representative and truly inclusive analysis of the Twitterverse.
Temporal snapshots are invaluable to map the national and international
migration patterns that increasingly blur geopolitical boundaries (Zagheni et
al. 2014).

The increasing popularity of Twitter has led it into issues of scale, where
its moderation can no longer check the large proportion of bots on the
platform. Our findings in Fig. 4 indicate that the infestation of bots may be
more pernicious than previously imagined. We are especially concerned that the
escalation of the war on Ukraine by Russia may reflect a spike (in our
dataset) in the online activity of bots from Russia operated either by the
Russian government or its allied intelligence agencies (Badawy, Ferrara, and
Lerman 2018). These and other bots serve to amplify trending topics and
facilitate the spread of misinformation (though, perhaps, at a rate less than
humans do (Vosoughi, Roy, and Aral 2018)). They may also misuse hashtags to
divert attention away from social or political topics (Earl, Maher, and Pan
2022; Broniatowski et al. 2018) or strategically target influential users
(Shao et al. 2018; Varol and Uluturk 2020). We hope that our work will spur
more studies on these topics, and we welcome researchers to explore our data.

By observing bursts of discussions around politically charged events and
characterizing the temporal spikes in Twitter topics, we can better
rationalize how our experience of Twitter as a political hotbed differs from
the simplified understanding of the American Twitter landscape reported in
Mukerjee, Jaidka, and Lelkes (2022), which suggested that politics is largely
a sideshow on Twitter. It is worth considering that these politically active
users may not be representative of social media users at large (McClain 2021;
Wojcieszak et al. 2022).

Twitter is also under scrutiny for how its platform governance may conflict
with users’ interests and rights (Van Dijck, Poell, and De Waal 2018).
Concerns have been raised about alleged biases in the algorithmic
amplification (and deamplification) of content, with evidence from France,
Germany, Turkey, and the United States, among other countries (Majó-Vázquez et
al. 2021; Tanash et al. 2015; Jaidka, Mukerjee, and Lelkes 2023). Other
scholars have also criticized Twitter’s use as a censorship weapon by
governments and political propagandists worldwide (Varol 2016; Elmas,
Overdorf, and Aberer 2021; Jakesch et al. 2021). They, and others, may be
interested in examining the trends in the enforcement of content moderation
policies by Twitter.

Besides answering questions of data, representativeness, access, and
censorship, we anticipate that our dataset is suited to explore the temporal
dynamics of online (mis)information in the following directions:

  * •

Content characteristics: We have provided a high-level exploration of the
topics on Twitter. However, more can be done with regard to understanding
users’ concerns and priorities. While hashtags act as signposts for the
broader Twitter community to find and engage in topics of mutual interest
(Cunha et al. 2011), tweets without hashtags may offer a different
understanding of Twitter discourse, where users may engage in more
interpersonal discussions of news, politics, and sports than the numbers
suggest (Rajadesingan, Budak, and Resnick 2021).

  * •

Patterns of information dissemination: Informational exchanges occurring on
Twitter can overcome spatio-temporal limitations as they essentially
reconfigure user connections to create newly emergent communities. However,
these communities may vanish as quickly as they are created, as the lifecycle
of a tweet determines how long it continues to circulate on Twitter timelines.
To the best of our knowledge, no prior research has reported on the average
“age” of a tweet, and we hope that a 24-hour snapshot will enable us to answer
this question empirically.

  * •

Content moderation and fake news: Prior research suggests that 0.1% of Twitter
users accounted for 80% of all fake news sources shared in the lead-up to a US
election (Grinberg et al. 2019). However, we expect there to be cross-lingual
differences in this distribution, especially for low- or under-resourced
languages with fewer open tools for fact-checking. Similarly, we expect that
the quality of moderation and hate speech will vary by geography and language,
and recommend the use of multilingual large language models to explore these
trends (with attention to persisting representativeness caveats (Wu and Dredze
2020)).

  * •

Mass mobilization: Twitter is increasingly the hotbed of protest, which has
led to some activists donning the role of “movement spilloverers” (Zhou and
Yang 2021) or serial activists (Bastos and Mercea 2016) who broker information
across different online movements, thereby acting as key coordinators,
itinerants, or gatekeepers in the exchange of information. Such users, as well
as the constant communities in which they presumably reside (Chowdhury et al.
2022), may be easier to study through temporal snapshots, as facilitated by
this dataset.

  * •

Echo chambers and filter bubbles: On Twitter, algorithms can affect the
information diets of users in over 200 countries, with an estimated 396.5
million monthly users (Kemp 2022). Recent surveys of the literature have
considered the evidence on how platforms’ designs and affordances influence
users behaviors, attitudes, and beliefs (González-Bailón and Lelkes 2022).
Studies of the structural and informational networks based on snapshots of
Twitter can offer clues to solving these puzzles without the constraints of
data selection.

## Ethics Statement and Data Availability

#### Ethics statement.

We acknowledge that privacy and ethical concerns are associated with
collecting and using social media data for research. However, we took several
steps to avoid risks to human subjects since participants no longer opt into
being part of our study, in a traditional sense (Zimmer 2020). In our
analysis, we only studied and reported population level, and aggregated
observations of our dataset. We share publicly only the tweet IDs with the
research community to account for privacy issues and Twitter’s TOS. For this
purpose, we use a data sharing and long-term archiving service provided by
GESIS - Leibniz Institute for the Social Sciences, a German infrastructure
institute for the social sciences 222https://www.gesis.org/en/data-
services/share-data.

With regards to data availability, this repository adheres to the FAIR
principles (Wilkinson et al. 2016) as follows:

  * •

Findability: In compliance with Twitter’s terms of service, only tweet IDs are
made publicly available at DOI: https://doi.org/10.7802/2516. A unique
Document Object Identifier (DOI) is associated with the dataset. Its metadata
and licenses are also readily available.

  * •

Accessibility: The dataset can be downloaded using standard APIs and
communications protocol (the REST API and OAI-PMH).

  * •

Interoperability: The data is provided in raw text format.

  * •

Reusability: The CC BY 4.0 license implies that researchers are free to use
the data with proper attribution.

Furthermore, we want to invite the broader research community to approach one
or more of the authors and collaborators (see Acknowledgments) of this paper
with research ideas about what can be done with this dataset. We will be very
happy to collaborate with you on your ideas!

## Acknowledgments

The data collection effort described in this paper could not have been
possible without the great collaboration of a large number of scholars, here
are some of them (in random order): Chris Schoenherr, Leonard Husmann, Diyi
Liu, Benedict Witzenberger, Joan Rodriguez-Amat, Florian Angermeir, Stefanie
Walter, Laura Mahrenbach, Isaac Bravo, Anahit Sargsyan, Luca Maria Aiello,
Sophie Brandt, Wienke Strathern, Bilal Çakir, David Schoch, Yuliia Holubosh,
Savvas Zannettou, Kyriaki Kalimeri.

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# Limiting Light Dark Matter with Luminous Hadronic Loops

Melissa Diamond<EMAIL_ADDRESS>Christopher V. Cappiello
<EMAIL_ADDRESS>Aaron C. Vincent<EMAIL_ADDRESS>Joseph Bramante
<EMAIL_ADDRESS>Department of Physics, Engineering Physics, and
Astronomy, Queen’s University, Kingston, Ontario, K7N 3N6, Canada,
The Arthur B. McDonald Canadian Astroparticle Physics Research Institute,
Kingston, Ontario, K7L 3N6, Canada,
and Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5,
Canada

###### Abstract

Dark matter is typically assumed not to couple to the photon at tree level.
While annihilation to photons through quark loops is often considered in
indirect detection searches, such loop-level effects are usually neglected in
direct detection, as they are typically subdominant to tree-level dark matter-
nucleus scattering. However, when dark matter is lighter than around 100 MeV,
it carries so little momentum that it is difficult to detect with nuclear
recoils at all. We show that loops of low-energy hadronic states can generate
an effective dark matter-photon coupling, and thus lead to scattering with
electrons even in the absence of tree-level dark matter-electron scattering.
For light mediators, this leads to an effective fractional electric charge
which may be very strongly constrained by astrophysical observations. Current
and upcoming searches for dark matter-electron scattering can thus set limits
on dark matter-proton interactions down to 1 MeV and below.

##  I Introduction

Although dark matter (DM) makes up most of the mass in the Universe, how (or
even whether) it interacts non-gravitationally is completely unknown Bertone
_et al._ (2005); Aghanim  _et al._ (2020); Cooley  _et al._ (2022). Direct
detection bounds on DM, as well as astrophysical and cosmological constraints,
are often presented in a “model-independent” way, e.g. as limits on dark
matter’s nonrelativistic scattering cross section with protons or electrons,
rather than limits on a larger set of model parameters. In such a framework,
limits on dark matter’s interactions with different Standard Model particles
are often treated completely independently. There are of course exceptions—DM-
proton and DM-neutron cross sections are often assumed to be identical to
avoid isospin violation—but for example, limits on DM-electron scattering are
often set assuming that DM-nucleon scattering is negligible, and vice-versa.

However, even if DM only interacts with one Standard Model particle at tree
level, interactions with other particles can be generated at loop level. A
classic example of such a loop calculation is Ref. Holdom (1986), which showed
how fermionic loops could generate an effective electromagnetic charge. This
example is commonly cited in literature on dark photons and millicharged DM
Prinz  _et al._ (1998); Davidson  _et al._ (2000); Pospelov (2009); McDermott
_et al._ (2011); Lees  _et al._ (2014); Izaguirre and Yavin (2015); Mahdawi
and Farrar (2018); Muñoz and Loeb (2018); Emken  _et al._ (2019); Caputo  _et
al._ (2021); Chiles  _et al._ (2022); Kahn and Lin (2022); Gorghetto  _et al._
(2022); Romanenko  _et al._ (2023), and explicit calculations of quark-,
lepton-, or W-loop-induced processes may be found in many references on
indirect detection, collider searches, and, to a lesser extent, direct
detection (Bern  _et al._ (1997); Bergstrom and Ullio (1997); Ullio and
Bergstrom (1998); Boudjema  _et al._ (2005); Fox and Poppitz (2009); Kopp  _et
al._ (2009); Bringmann and Weniger (2012); Bell  _et al._ (2014); Bramante
_et al._ (2016); Charles  _et al._ (2016); Kumar and Light (2017); Braaten
_et al._ (2018); Bringmann  _et al._ (2018); Abdallah  _et al._ (2018);
Fileviez Pérez and Murgui (2019); Albert  _et al._ (2020); Arina  _et al._
(2020); Rinchiuso  _et al._ (2021); Arina  _et al._ (2023); Foster  _et al._
(2023), see also Kopp  _et al._ (2014); Mohan  _et al._ (2019)). In works
focused on indirect detection or collider searches, the energies are typically
high enough that QCD is perturbative, and loops of quarks can be computed
directly.

In this work, we explicitly compute the DM-photon interaction induced by DM
couplings to hadronic states at low energy, where the relevant degrees of
freedom are not quarks, but mesons and baryons. This is in contrast with much
of the literature on kinetically mixed dark photons, which treats the mixing
between the photon and dark photon as a phenomenological parameter that is
generated at much higher energy scales. We show that such loop-level couplings
can produce detectable event rates in direct detection experiments, and/or
induce a non-negligible effective charge for the DM, in a wide range of sub-
GeV parameter space. We thus set new limits on sub-GeV dark matter’s coupling
to protons.

This paper is organized as follows. In Section II, we review the ideas of Ref.
Holdom (1986), and discuss the types of interaction that yield nonzero
results. We then introduce both our model and the effective Lagrangians used
to describe dark matter’s interactions with hadronic states, and compute the
induced interactions with photons and electrons. In Section III, we compute
new constraints on sub-GeV DM. In Section IV, we discuss the implications of
our results. Detailed descriptions of the calculations performed in this work
can be found in the Appendix A.

##  II Effective Dark Matter-electron Interactions

Suppose that DM is a fermion, and interacts with a particular charged Standard
Model fermion—in this case, a proton—via a new vector mediator that we will
refer to as $Z^{\prime}$ (the tree level diagram is shown in Fig. 1 (a)). A
proton loop then induces a mixing between the $Z^{\prime}$ and the photon, as
well as a DM-electron interaction, as shown in Fig. 1 (b). If $Z^{\prime}$ is
massless, the result is an effective charge for the DM, as derived by Ref.
Holdom (1986). Even if the mediator is massive, a mixing with the photon is
still generated, but the different momentum dependence means that the DM no
longer behaves as a truly millicharged particle.

Figure 1: Feynman diagrams utilized in this work. The tree level diagram (a)
captures DM-proton scattering. The loop level diagrams (b)-(d) show DM-
electron scattering that results from the DM-proton interaction.

The proton is, of course, not the only hadronic state that can be included in
loops like this. In fact, the more typical approach to such hadronic
corrections would be to use loops of pions (Fig. 1 (c) and (d)), whose masses
are below the QCD scale and for which the framework of chiral effective field
theory (ChEFT) can readily be applied. The use of pion loops in this context
is strikingly similar to their role in hadronic vacuum polarization, notably
in the context of muon $g-2$ (see Ref. Jegerlehner (2017) for a detailed
review).

In this work, we include both proton and meson (specifically pion and kaon)
loops in order to compute the induced mixing between the $Z^{\prime}$ and
photon. We take inspiration from Ref. Becher and Leutwyler (1999), which
showed that nucleons could be included in the framework of chiral perturbation
theory while still preserving chiral power counting.

Throughout this work, we will consider only the case of a vector mediator,
because for a scalar or axial vector, the diagrams shown in Fig. 1 (b)-(d)
vanish (see Ref. Kopp  _et al._ (2009)). An analogous interaction between DM
and electrons in the scalar case can be induced at the 2-loop level Kopp  _et
al._ (2009), or by instead mixing with the Higgs at one loop. However, these
options are suppressed compared to the one-loop mixing with the photon, so we
do not consider them.

Next, we will estimate the effective DM-electron scattering cross section that
results from couplings through hadronic loops. We focus on DM masses in the
range 1 MeV $\lesssim m_{\chi}\lesssim 100$ MeV, which is difficult to probe
using nuclear recoil searches, but where electron recoil searches have set
some bounds Essig  _et al._ (2017); Aguilar-Arevalo  _et al._ (2019); Aprile
_et al._ (2019); Barak  _et al._ (2020); Cheng  _et al._ (2021); Aprile  _et
al._ (2022); Agnes  _et al._ (2023); Arnquist  _et al._ (2023). At the
corresponding energies ($T_{\chi}\ll 1$ keV), quarks are confined into baryons
and light mesons, and their behavior is best described using ChEFT. Hadronic
loops are dominated by pions and kaons, the light pseudoscalar mesons. Proton
loops will contribute as well, and may dominate depending on the underlying
theory. We start with description of the DM quark interaction and use this to
build a consistent description of the DM-proton and DM-meson coupling. These
will then be used to estimate and compare the tree level DM-proton cross
section with the 1-loop DM-electron cross section.

Interactions between a dark fermion and quarks through a vector mediator can
be described by

$\mathcal{L}\supset\sum_{q}\alpha_{q}Z^{\prime}_{\mu}q\gamma^{\mu}\bar{q}+g_{\chi}Z^{\prime}_{\mu}\chi\gamma^{\mu}\bar{\chi}\,,$
(1)

where $\chi$ is the massive DM particle, $q$ are the quarks, $\alpha_{q}$ is
the coupling of each quark specie to $Z^{\prime}_{\mu}$, $g_{\chi}$ is the
coupling between the $Z^{\prime}$ and the DM.

We may define a resulting effective proton interaction Agrawal  _et al._
(2010):

$\mathcal{L}\supset(2\alpha_{u}+\alpha_{d})Z^{\prime}_{\mu}p\gamma^{\mu}\bar{p}\,.$
(2)

The meson interaction terms can be derived from the ChEFT Lagrangian. The
relevant lowest order interaction term in the ChEFT Lagrangian is Ecker  _et
al._ (1989)

$L\supset\frac{F^{2}}{4}Tr\left(D_{\mu}UD^{\mu}U^{{\dagger}}\right)\,,$ (3)

where $U=e^{\frac{i}{F}\pi}$ contains the light meson octet

$\pi=\begin{pmatrix}\pi^{0}+\frac{\eta_{8}}{\sqrt{3}}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{\eta_{8}}{\sqrt{3}}&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta_{8}\end{pmatrix}\,.$
(4)

$F$ is the pion decay constant. Interactions with external vector fields
(namely the photon or $Z^{\prime}$) are captured in the derivative terms

$\begin{split}&D_{\mu}U=\partial_{\mu}U-iv_{\mu}U+iUv_{\mu}\,,\\\
&D_{\mu}U^{{\dagger}}=\partial_{\mu}U^{{\dagger}}+iU^{{\dagger}}v_{\mu}-iv_{\mu}U^{{\dagger}}\,,\\\
\end{split}$ (5)

where $v_{\mu}=Z^{\prime}_{\mu}\text{diag}(\alpha_{u},\alpha_{d},\alpha_{s})$
is a matrix which represent an external vector interaction emerging from
interactions of the form depicted in (1). We can include electromagnetic
interactions here as well by including a term of the form
$v_{\mu}=eA_{\mu}\text{diag}(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$.

Expanding out the chiral Lagrangian (3) gives the following interaction terms
between light mesons, photons, and $Z^{\prime}$.

$\begin{split}\mathcal{L}\supset&i(\alpha_{u}-\alpha_{d})Z^{\prime}_{\mu}(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial_{\mu}\pi^{-})\\\
+&i(\alpha_{u}-\alpha_{s})Z^{\prime}_{\mu}(K^{-}\partial^{\mu}K^{+}-K^{+}\partial^{\mu}K^{-})\\\
+&ieA_{\mu}(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial_{\mu}\pi^{-})\\\
+&ieA_{\mu}(K^{-}\partial^{\mu}K^{+}-K^{+}\partial{\mu}K^{-})\\\
+&2e(\alpha_{u}-\alpha_{d})Z^{\prime}_{\mu}A^{\mu}\pi^{+}\pi^{-}\\\
+&2e(\alpha_{u}-\alpha_{s})Z^{\prime}_{\mu}A^{\mu}K^{+}K^{-}\,.\\\
\end{split}$ (6)

Using these interaction terms we calculate the $\chi p$ scattering cross
section at tree level and the $\chi e$ scattering cross section at loop level
based on the diagrams shown in Fig. 1. These calculations are shown in more
detail in the Appendix A.

At tree level and low momentum exchange

$\frac{d\sigma_{\chi
p}}{d\Omega}=\frac{g_{\chi}^{2}(2\alpha_{u}+\alpha_{d})^{2}\mu_{\chi
p}^{2}}{4\pi^{2}(m_{Z}^{2}-t)^{2}}\,,$ (7)

where $m_{Z^{\prime}}$is the $Z^{\prime}$ mass.
$\mu_{ab}={m_{a}m_{b}}/({m_{a}+m_{b}})$ represents the reduced mass of two
particles with masses $m_{a}$ and $m_{b}$.

The loop terms together give a cross section of

$\frac{d\sigma_{\chi e}}{d\Omega}=\frac{d\sigma_{\chi
p}}{d\Omega}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,,$ (8)

where

$\begin{split}c_{loop}=\,\,&4(2\alpha_{u}+\alpha_{d})\ln\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{p}^{2}}\right)\\\
&+(\alpha_{u}-\alpha_{d})\ln\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{\pi}^{2}}\right)\\\
&+(\alpha_{u}-\alpha_{s})\ln\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{K}^{2}}\right)\,.\end{split}$ (9)

The first term in $c_{loop}$ comes from the proton loop as shown in Fig. 1
(b), the second term comes from a pion loop and the third from the kaon loop.
The meson loop terms get contributions from diagrams of the form depicted in
Fig. 1 (c) and (d). Loop divergences are contained in the log terms, which
depend on the mass of the particles traveling through the loops and a cutoff
term that is logarithmic in $\mu$. We follow Ref. Becher and Leutwyler (1999)
and set $\mu=m_{p}$. We use $m_{\pi}=140$ MeV, $m_{K}=494$ MeV Workman and
Others (2022), and $m_{p}=938$ MeV. $\gamma_{E}=0.577$ is the Euler Mascheroni
constant.

In the case of a heavy mediator, one can integrate over scattering angles and
give a relative total cross section of

$\sigma_{\chi e}=\sigma_{\chi
p}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,.$ (10)

The factor $\frac{e^{2}}{2304\pi^{4}}\approx 4\times 10^{-7}$. While this
suppression seems substantial, the effective cross sections are large enough
that planned and currently running electron recoil detectors should be able to
observe or rule out DM that is difficult to observe using traditional nuclear
recoil detectors.

For a light mediator, the integral of ${d\sigma_{\chi e}}/{d\Omega}$ and
${d\sigma_{\chi p}}/{d\Omega}$ over the scattering angle diverges. So we
instead report the ratio between $\bar{\sigma}_{\chi p}$ and
$\bar{\sigma}_{\chi e}$, where

$\bar{\sigma}\equiv
2\frac{d\sigma}{d\cos{\theta}}\left(\frac{q}{q_{ref}}\right)^{4}.$ (11)

Here $q$ represents the momentum exchanged between scattered particles, and
$q_{ref}$ is a reference momentum, usually taken to be $\sim\alpha m_{e}$
Emken  _et al._ (2019). $\bar{\sigma}$ is Lorentz-invariant and typically used
when discussing constraints on light mediator scattering. The resulting
relation between light mediator cross sections is

$\bar{\sigma}_{\chi e}=\bar{\sigma}_{\chi
p}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,.$ (12)

Hence, the ratio between the proton and electron cross sections does not
depend on mass of the vector mediator. As the terms shared between proton and
electron scattering diagrams divide out, the ratio between the proton and
electron cross section also does not depend on the spin of the DM, or the
Lorentz structure of its interaction with the vector mediator.

##  III Results

In direct detection literature, the interactions between DM and individual
quarks that generate the DM-nucleon cross section are typically left
unspecified. Because we also include meson loops, recasting these limits
requires a concrete choice for the couplings to individual quarks. In this
Section, we report our results for the case $\alpha_{u}=-\alpha_{d}$ and
$\alpha_{s}=0$, i.e. the case where the $Z^{\prime}$ couples to isospin.
Results for an alternative case, $\alpha_{u}=\alpha_{d}=\alpha_{s}$, are shown
in the Appendix B. The limits between these two cases differ by a factor of
$\sim 8$. We do not expect different choices of couplings to weaken the limits
much beyond this range without fine tuning.

For our constraints on the DM-proton cross section using electron-recoil
searches, we focus on SENSEI Barak  _et al._ (2020), which has reported some
of the strongest limits on DM-electron scattering while being only $\sim$100
meters underground (just under 300 meters water equivalent, or m.w.e.),
presenting a lower overburden than most direct detection experiments. Our
limits result from rescaling the reported limits of Ref. Barak  _et al._
(2020) by the ratio of the DM-proton and loop-induced DM-electron cross
sections. In the same way, we also recast projections for the DAMIC-M
experiment Battaglieri  _et al._ (2017); Settimo (2018, 2020), which has
recently released its first results Arnquist  _et al._ (2023), as well as the
upcoming Oscura experiment Aguilar-Arevalo  _et al._ (2022).

Figure 2 shows limits and projected sensitivities based on effective loop
interactions, for a heavy $Z^{\prime}$, compared to existing limits from
direct detection and cosmology. Our limit constrains DM masses from about 1
MeV to 30 MeV, and at the lowest masses is comparable to the strongest
existing Migdal effect limit, from SENSEI Barak  _et al._ (2020); Berghaus
_et al._ (2023). It is also competitive with the strongest cosmological
bounds, which come from Lyman-alpha observations Rogers  _et al._ (2022). The
projected sensitivities reach cross sections of 10-36 cm2 for masses of a few
MeV, orders of magnitude better than existing Migdal effect searches and
Lyman-alpha bounds. For comparison, Ref. Aguilar-Arevalo  _et al._ (2022)
shows projections for future Migdal effect searches from the same detectors,
but these projections only extend down to 20 MeV. We also show Migdal effect
limits from XENON10 and XENON1T Essig  _et al._ (2020), CDEX Liu  _et al._
(2019, 2022), and EDELWEISS Armengaud  _et al._ (2019, 2022), which provide
leading constraints at larger masses.

At the large cross sections we consider, DM may be stopped in the Earth before
reaching the detector. Even though the signal we use to set limits is
scattering with electrons, attenuation will be dominated by scattering with
nuclei due to the much larger cross section. To account for attenuation in the
Earth, we use the ceilings computed for SENSEI in Ref. Emken  _et al._ (2019).
These ceiling calculations are also dominated by nuclear scattering, but were
computed in a dark photon model, so we rescale them by a factor of 4 to
account for the scattering with neutrons (assuming typical spin-independent
scattering). For DAMIC-M and Oscura, we use the same ceiling, lowered by
factors of 17 and 20, respectively, to simulate the overburdens at Laboratoire
Souterrain de Modane (4800 m.w.e.) Piquemal (2012) and SNOLAB (6000 m.w.e.)
Duncan  _et al._ (2010).

Figure 2: Limits on the dark matter-nucleon cross section due to their loop-
induced coupling to electrons. Interactions via a heavy vector mediator with
$\alpha_{u}=-\alpha_{d}$ and $\alpha_{s}=0$. Our recasting of constraints from
SENSEI Barak  _et al._ (2020) is shown in red, while the regions outlined in
red dashed and dotted lines will be accessible to DAMIC-M Battaglieri  _et
al._ (2017); Settimo (2018, 2020) and Oscura Aguilar-Arevalo  _et al._ (2022)
respectively. Existing detector constrains From Migdal effect searches at
SENSEI Barak  _et al._ (2020); Berghaus  _et al._ (2023), XENON10/1T (Essig
_et al._ (2020), as shown as in Ref. Berghaus  _et al._ (2023)), CDEX Liu  _et
al._ (2019, 2022) and EDELWEISS Armengaud  _et al._ (2019, 2022) are shown in
gray, while Lyman alpha constraints Rogers  _et al._ (2022) are shown in blue.

Figure 3 shows our limits and projected sensitivities for a light
$Z^{\prime}$, compared to existing limits from direct detection (cosmological
bounds exist on scattering via light mediators, e.g. Kovetz  _et al._ (2018);
Slatyer and Wu (2018) but only at higher cross section). As mentioned above,
in the case of a massless mediator, the total cross section diverges, and
limits are typically reported in terms of a reference cross section
$\overline{\sigma}$. We follow the parametrization of Ref. Emken  _et al._
(2019) (see also the Appendix A for additional discussion). Also in the light
$Z^{\prime}$ case, the scattering is typically softer, making attenuation less
of an issue, so we can constrain a much wider range of parameter space. At
large DM masses, Migdal effect bounds from SENSEI and XENON10/XENON1T are
stronger than our bounds. However, our limits are stronger than existing
Migdal effect limits for masses up to 5 MeV, and extend down well below 1 MeV.
In our projections, we show that DAMIC-M and Oscura can again probe cross
sections in the range 10-34–10-36 cm2, competitive with the Migdal effect
projections from Ref. Aguilar-Arevalo  _et al._ (2022) and surpassing them for
masses below $\sim$10 MeV if one were to extrapolate those projections to
lower mass. We again show direct detection constraints from SENSEI, XENON10,
and XENON1T for comparison. It deserves mention that models of new light
$Z^{\prime}$ mediators coupled to baryon currents will also induce SM
anomalies that can be constrained through their contribution to rare meson
decays Dror  _et al._ (2017, 2019).

Figure 3: Limits on the dark matter-nucleon cross section due to their loop-
induced coupling to electrons. Interactions via a light vector mediator with
$\alpha_{u}=-\alpha_{d}$ and $\alpha_{s}=0$. Our results for SENSEI Barak  _et
al._ (2020) are highlighted in red, and the expected reach of DAMIC-M
Battaglieri  _et al._ (2017); Settimo (2018, 2020) and Oscura Aguilar-Arevalo
_et al._ (2022) are outlined by the red dashed and dotted lines respectively.
Constraints from Migdal effect searches at SENSEI Barak  _et al._ (2020);
Berghaus  _et al._ (2023), and XENON10/1T (Essig  _et al._ (2020), as shown in
Ref. Berghaus  _et al._ (2023)) are shown in gray.

Finally, we note that an effectively massless $Z^{\prime}$ produces an
effective fractional electric charge (or “millicharge”) for the DM. We can
compute the effective charge induced by loops of hadronic states, and recast
limits on millicharged DM as limits on DM-proton interactions via an
effectively massless vector. We report our results in terms of $\epsilon$, the
DM charge in units of the electron charge, i.e. $q_{DM}=\epsilon e$.

Figure 4 shows, as a color scale, the DM charge corresponding to a given
$m_{\chi}$ and $\sigma_{n\chi}$. In gray we superimpose the same Migdal effect
limits shown in Fig. 3. In addition, we show two astrophysical bounds on
millicharged DM, which are relevant in this parameter space as a result of the
induced DM charge. First, Ref. Stebbins and Krnjaic (2019) argued that
fractionally charged DM interacting with Galactic magnetic fields in the Milky
Way would extract angular momentum from the Milky Way disk, spinning down the
disk over the course of gigayears. Although they report an order-of-magnitude
uncertainty on their limit, Fig. 4 covers more than 10 orders of magnitude in
$\epsilon$. Taking this uncertainty into account, these bounds still far
supersede those set by the tree level interactions. Second, Ref. Kadota  _et
al._ (2016) considered millicharged DM moving in galaxy clusters under the
influence of cluster magnetic fields, and argued that if the DM charge were
too large, magnetic fields would substantially alter the DM density profile.
This results in another strong bound on DM charge, also shown in Fig. 4.

Other limits on millicharged DM may not apply, or may need to be considered
more carefully in this scenario. For example, supernova cooling constraints on
millicharged particles Chang  _et al._ (2018) do not apply here because the
proton coupling is large enough to trap the DM within the proto-neutron star.
Similarly, the argument that millicharged particles would be evacuated from
the Galaxy by supernovae, put forward by Ref. Chuzhoy and Kolb (2009), depends
on the dark matter not scattering too frequently with Standard Model
particles, an assumption that may be violated in at least some of the
parameter space we consider. For this reason, and because the corresponding
limit is weaker than the other astrophysical bounds we show, we do not plot
the limit from Ref. Chuzhoy and Kolb (2009). We also note that a specific
model of DM that has a loop-induced effective electric charge through its
hadronic couplings will interact with protons differently than a typical
millicharged particle. This could even strengthen the astrophysical bounds
shown in Fig. 4, by, for example, enhancing the amount of angular momentum
extracted from the Milky Way disk.

Figure 4: Effective electric charge resulting from hadronic loop interactions
assuming $\alpha_{u}=-\alpha_{d}$ and $\alpha_{s}=0$. The white lines are
astrophysical constaints on millicharged DM: above the solid line, cluster
magnetic fields would noticeably alter the density profile of galaxy clusters
Kadota  _et al._ (2016), while above the dashed line, millicharged DM would
extract too much angular momentum from the Milky Way disk Stebbins and Krnjaic
(2019). Migdal effect constraints from SENSEI Barak  _et al._ (2020); Berghaus
_et al._ (2023) and XENON10/XENON1T (Essig  _et al._ (2020), as shown in Ref.
Berghaus  _et al._ (2023)) are shown in gray.

##  IV Conclusions

We have presented a one-loop calculation of the low energy DM-electron cross
section for DM which interacts exclusively with quarks at tree level. This
interaction should generically emerge in a wide range of DM models that
interact with quarks through a vector mediator. This has allowed us to derive
novel constraints on the DM-proton cross section using existing constraints
from SENSEI data. We have shown that currently-running and upcoming electron
recoil detectors, DAMIC-M and Oscura, should be able to probe DM-proton cross
sections that may be beyond the reach of nuclear recoil detectors. Finally we
have demonstrated that DM that interacts with quarks through a light mediator
at tree level has an effective electric charge which can be used to recast
astrophysical and cosmological constraints on the DM-electron cross section.

Standard Model loop interactions can be an effective tool in exploring DM
behavior, and are an inevitable but often-ignored part of any DM theory. While
we focused on quark scattering interactions through a vector mediator in this
work, we note that loop interactions similar to those described in this work
may be effective at bridging different DM-Standard Model interactions in
annihilation processes and with mediators not explored here.

###### Acknowledgements.

We are grateful to Kim Boddy, Humberto Gilmer, Martin Hoferichter, Matheus
Hostert, Jason Kumar, David Morrissey, Marianne Moore, and Tim Tait for
helpful discussions. This work is supported by the Natural Sciences and
Engineering Research Council of Canada, the Arthur B. McDonald Canadian
Astroparticle Research Institute, and the Canada Foundation for Innovation.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science, and Economic Development, and
by the Province of Ontario.

##  Appendix A Cross Section Calculation

###  A.1 Effective Lagrangians

We calculate the dark matter (DM) proton cross section based on tree level
interactions of the form shown in Fig. 5 and the DM-electron scattering cross
sections resulting from one loop diagrams of the form shown in Fig. 6. These
calculations are all done in the center of mass frame, as the total cross
sections are Lorentz invariant. We start with the proton tree level
scattering.

Figure 5: Tree level Feynman diagram for dark matter proton scattering

Figure 6: Dark matter-electron interaction induced by proton and pion loops.

We start the calculation by considering an underlying DM-quark interaction of
the form

$\mathcal{L}\supset\sum_{q}\alpha_{q}Z^{\prime}_{\mu}q\gamma^{\mu}\bar{q}+g_{\chi}Z^{\prime}_{\mu}\chi\gamma^{\mu}\bar{\chi}$
(13)

where $\chi$ is the massive DM particle, $q$ are the quarks, $\alpha_{q}$ is
the coupling of each quark specie to $Z^{\prime}_{\mu}$, $g_{\chi}$ is the
coupling of $\chi$ to its vector mediator. In the low energy limit quarks are
confined to light mesons and baryons. Their behavior is best described my
Chiral Effective Field Theory (ChEFT) with Baryons. At low energies (13) gives
an effective coupling to protons of the form Bishara  _et al._ (2017)

$\bra{P^{\prime}}\sum_{q}\alpha_{q}Z^{\prime}_{\mu}q\gamma^{\mu}\bar{q}\ket{P}=(2\alpha_{u}+\alpha_{d})Z^{\prime}_{\mu}P^{\prime}\gamma^{\mu}P.$
(14)

The lowest order terms in the ChEFT Lagrangian that will contribute to the
meson-$Z^{\prime}$ interaction are Ecker  _et al._ (1989)

$L\supset\frac{F^{2}}{4}Tr\left(D_{\mu}UD^{\mu}U^{{\dagger}}\right),$ (15)

where $U=e^{\frac{i}{F}\pi}$ contains the light meson octet

$\pi=\begin{pmatrix}\pi^{0}+\frac{\eta_{8}}{\sqrt{3}}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{\eta_{8}}{\sqrt{3}}&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta_{8}\end{pmatrix}.$
(16)

Here $F$ represents the pion decay constant. Interactions with external vector
fields at highest order are captured in the derivative terms

$\begin{split}&D_{\mu}U=\partial_{\mu}U-iv_{\mu}U+iUv_{\mu},\\\
&D_{\mu}U^{{\dagger}}=\partial_{\mu}U^{{\dagger}}+iU^{{\dagger}}v_{\mu}-iv_{\mu}U^{{\dagger}},\\\
\end{split}$ (17)

where $v_{\mu}$ represents an interaction with an external vector. To capture
the interaction with $Z^{\prime}$ depicted in (13), we take
$v_{\mu}=Z^{\prime}_{\mu}\text{diag}(\alpha_{u},\alpha_{d},\alpha_{s})$. We
can include electromagnetic interactions here as well by including a term of
the form $v_{\mu}=eA_{\mu}\text{diag}(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$.
Expanding out the chiral lagrangian (15) gives the following interaction terms
between light mesons, photons, and $Z^{\prime}$:

$\begin{split}\mathcal{L}\supset&i(\alpha_{u}-\alpha_{d})Z^{\prime}_{\mu}(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial^{\mu}\pi^{-})\\\
+&i(\alpha_{u}-\alpha_{s})Z^{\prime}_{\mu}(K^{-}\partial^{\mu}K^{+}-K^{+}\partial^{\mu}K^{-})\\\
+&ieA_{\mu}(\pi^{-}\partial^{\mu}\pi^{+}-\pi^{+}\partial^{\mu}\pi^{-})\\\
+&ieA_{\mu}(K^{-}\partial^{\mu}K^{+}-K^{+}\partial^{\mu}K^{-})\\\
+&2e(\alpha_{u}-\alpha_{d})Z^{\prime}_{\mu}A^{\mu}\pi^{+}\pi^{-}\\\
+&2e(\alpha_{u}-\alpha_{s})Z^{\prime}_{\mu}A^{\mu}K^{+}K^{-}~{}.\\\
\end{split}$ (18)

###  A.2 Tree Level Cross Section

Now these effective interaction terms can be used to calculate the cross
sections resulting from Fig. 5 and Fig. 6 The amplitude of the tree-level
proton scattering diagram is

$\mathcal{M}_{\text{tree}}=g_{\chi}(2\alpha_{u}+\alpha_{d})u_{\chi}(p_{1})\gamma^{\mu}\bar{u}_{\chi}(p_{3})\frac{g_{\mu\nu}}{m_{Z^{\prime}}^{2}-(p_{1}-p_{3})^{2}}u_{P}(p_{2})\gamma^{\nu}\bar{u}_{P}(p_{4}).$
(19)

Squaring this and summing over polarisation gives a cross section of

$\frac{d\sigma_{\chi
p}}{d\Omega}=\frac{g_{\chi}^{2}(2\alpha_{u}+\alpha_{d})^{2}\mu_{\chi
p}^{2}}{4\pi^{2}(m_{Z}^{2}-t)^{2}},$ (20)

where $m_{Z^{\prime}}$ is the mediator mass respectively. We use
$\mu_{ab}=\frac{m_{a}m_{b}}{m_{a}+m_{b}}$ to represent the reduced mass of two
particles $a$ and $b$ with masses $m_{a}$ and $m_{b}$. Above $\mu_{\chi
p}^{2}$ represents the reduced mass of $\chi$ and the proton. We take
$m_{p}=0.938$ GeV. We note that many model specific terms such as $g_{\chi}$
and $m_{Z^{\prime}}$ will divide out in the ratio between the tree level and
loop level cross section.

###  A.3 Loop Level Calculations

####  A.3.1 Proton Loop

Calculating the DM-electron scattering cross section will require a few
additional steps compared to the DM-proton cross section. We first calculate
the loop terms and then calculate their contribution to the overall scattering
cross section.

The proton loop piece of the diagram shown in Fig. 5 (a) can be derived by
modifying the calculation for QED vacuum polarization found in Ref. Schwartz
(2014). The loop contribution takes the form

$i\Pi_{\mu\nu}^{P}=-e(2\alpha_{u}+\alpha_{d})\int\frac{d^{4}k}{(2\pi)^{4}}\frac{Tr[\gamma^{\mu}(\cancel{k}-\cancel{q}+m_{P})\gamma^{\nu}(\cancel{k}+m_{p})]}{((q-k)^{2}-m_{P}^{2})(k^{2}-m_{P}^{2})}\,,$
(21)

where $q=p1-p3$. Taking the trace gives

$i\Pi_{\mu\nu}^{P}=-4e(2\alpha_{u}+\alpha_{d})\int\frac{d^{4}k}{(2\pi)^{4}}\frac{2k^{\mu}k^{\nu}+g^{\mu\nu}(-k^{2}+q\cdot
k+m_{P}^{2})}{((q-k)^{2}-m_{P}^{2})(k^{2}-m_{P}^{2})}\,.$ (22)

Here we have dropped $q^{\mu}q^{\nu}$ terms as these do not contribute to our
final cross section. Using the relation
$\frac{1}{AB}=\int_{0}^{1}dx\frac{1}{(A+(B-A)x)^{2}}$, shifting $k^{\mu}$ to
$k^{\mu}+q^{\mu}(1-x)$, and dropping the $q^{\mu}q^{\nu}$ terms and terms odd
in $k$ from the numerator (these will go to zero or not contribute to the
final cross section) give

$\Pi_{\mu\nu}^{P}=4ie(2\alpha_{u}+\alpha_{d})\int\frac{d^{4}k}{(2\pi)^{4}}\int_{0}^{1}dx\frac{2k^{\mu}k^{\nu}-g^{\mu\nu}(k^{2}-x(1-x)q^{2}-m_{P}^{2})}{(k^{2}+q^{2}x(1-x)-m_{P}^{2})^{2}}\,.$
(23)

This integral will diverge in 4 dimensions. We manage this divergence using
dimensional regularization by integrating over $d$ dimensions then taking the
limit $d\rightarrow 4-\epsilon$. To do these integrals, we take
$k^{\mu\nu}\rightarrow\frac{1}{d}k^{2}g^{\mu\nu}$. In $d$ dimensions the
integral takes the form

$\Pi_{\mu\nu}^{P}=-4ie(2\alpha_{u}+\alpha_{d})\mu^{d-4}g^{\mu\nu}\int\frac{d^{d}k}{(2\pi)^{d}}\int_{0}^{1}dx\frac{((1-\frac{2}{d})k^{2}-x(1-x)q^{2}-m_{P}^{2})}{(k^{2}+q^{2}x(1-x)-m_{P}^{2})^{2}}\,.$
(24)

Using the following formulas

$\begin{split}&\int\frac{d^{d}k}{(2\pi)^{d}}\frac{k^{2}}{(k^{2}-\Delta+i\epsilon)^{2}}=-\frac{d}{2}\frac{i}{(4\pi)^{d/2}}\frac{1}{\Delta^{1-\frac{d}{2}}}\Gamma\left(\frac{2-d}{2}\right)\\\
&\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-\Delta+i\epsilon)^{2}}=\frac{i}{(4\pi)^{d/2}}\frac{1}{\Delta^{2-\frac{d}{2}}}\Gamma\left(\frac{4-d}{2}\right)\,.\end{split}$
(25)

we complete the $k$ integral in Eq.(24). This gives

$\Pi_{\mu\nu}^{P}=-8e(2\alpha_{u}+\alpha_{d})\mu^{4-d}g^{\mu\nu}q^{2}\Gamma\left(2-\frac{d}{2}\right)\int_{0}^{1}dxx(1-x)\frac{1}{(m_{P}^{2}-q^{2}x(1-x))^{2-\frac{d}{2}}}\,.$
(26)

If we take the limit $p\ll m_{P}$ then the integral over $x$ is simple and
gives

$\Pi_{\mu\nu}^{P}=-\frac{e(2\alpha_{u}+\alpha_{d})}{12\pi^{2}}g^{\mu\nu}q^{2}\left(\frac{\mu}{m_{P}}\right)^{4-d}\Gamma\left(2-\frac{d}{2}\right)\,.$
(27)

This diverges in the $\epsilon=0$ limit, expanding around $\epsilon=0$ gives

$\Pi_{\mu\nu}^{P}=-\frac{e(2\alpha_{u}+\alpha_{d})}{24\pi^{2}}g^{\mu\nu}q^{2}\left(\frac{2}{\epsilon}+\text{Log}\left[\frac{4\pi
e^{\gamma_{E}}\mu^{2}}{m_{P}^{2}}\right]\right)\,.$ (28)

Keeping only the finite terms gives proton loop contribution to the scattering
diagram

$\Pi_{\mu\nu}^{P}=-\frac{e(2\alpha_{u}+\alpha_{d})}{24\pi^{2}}g^{\mu\nu}q^{2}\text{Log}\left[\frac{4\pi
e^{\gamma_{E}}\mu^{2}}{m_{P}^{2}}\right]\,.$ (29)

Here we define $\mu$ as a cutoff scale for low energy ChEFT with Baryons.
Following Ref. Becher and Leutwyler (1999) we take $\mu=m_{P}$.

####  A.3.2 Light Meson Loops

Both pions and kaons can contribute to DM-electron loop level scattering.
There are two different loop diagrams for each light meson particle. The
general form of these diagrams is displayed in Fig. 6 (b) and (c). Below, we
calculate a general expression for these loop contributions, and then add in
the specific coupling and mass values for specific mesons. These calculations
modify the vacuum polarization derivations for scalar QED shown in Schwartz
(2014). Using the interaction terms of the lagrangian shown in Eq.(3) we get
the following amplitudes for the loop in Fig. 6(b) and Fig. 1 (c) respectively

$\begin{split}i\Pi^{b}_{\mu\nu}&=-eg\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i(2k^{\mu}-q^{\mu})}{(k-q)^{2}-m^{2}+i\epsilon}\frac{i(2k^{\mu}-q^{\mu})}{k^{2}-m^{2}+i\epsilon},\\\
i\Pi^{c}_{\mu\nu}&=-eg\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\,.\end{split}$
(30)

We use $g$ as a stand in for the meson-$Z$ coupling and $m$ to represent the
meson mass. Summing these together and then preforming the same integration,
approximation and simplification steps outlined in Section A.3.1 gives the
following total loop contribution

$\Pi_{\mu\nu}^{\text{meson}}=-\frac{eg}{48\pi^{2}}g^{\mu\nu}q^{2}Log\left[\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m^{2}}\right]\,.$ (31)

The specific loop contributions from pions and kaons are then

$\begin{split}\Pi_{\mu\nu}^{\pi}&=-\frac{e(\alpha_{u}-\alpha_{d})}{48\pi^{2}}g^{\mu\nu}q^{2}Log\left[\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{\pi}^{2}}\right]\\\
\Pi_{\mu\nu}^{K}&=-\frac{e(\alpha_{u}-\alpha_{s})}{48\pi^{2}}g^{\mu\nu}q^{2}Log\left[\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{K}^{2}}\right]\,.\end{split}$ (32)

####  A.3.3 Loop Level Cross Section

Combining the loop contributions calculated in (29) and (32) gives the overall
loop contribution at low energies

$\Pi_{\mu\nu}^{\text{loop}}=-\frac{e}{48\pi^{2}}g^{\mu\nu}q^{2}c_{\text{ loop
}}\,,$ (33)

where

$c_{\text{loop}}=-4(2\alpha_{u}+\alpha_{d})\text{Log}\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{p}^{2}}\right)-(\alpha_{u}-\alpha_{d})\text{Log}\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{\pi}^{2}}\right)-(\alpha_{u}-\alpha_{s})\text{Log}\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{m_{K}^{2}}\right)\,.$ (34)

The amplitude of the full DM-electron scattering interaction is

$\mathcal{M}_{\text{loop}}=g_{\chi}eu_{\chi}(p_{1})\gamma^{\mu}\bar{u}_{\chi}(p_{3})\frac{g_{\mu\nu}}{m_{Z^{\prime}}^{2}-(p_{1}-p_{3})^{2}}\Pi_{\nu\omega}^{\text{loop}}\frac{g_{\omega\xi}}{(p_{1}-p_{3})^{2}}u_{e}(p_{2})\gamma^{\xi}\bar{u}_{e}(p_{4})\,.$
(35)

This gives a final cross section of

$\frac{d\sigma_{\chi e}}{d\Omega}=\frac{g_{\chi}^{2}e^{2}\mu_{\chi
e}^{2}}{144(4\pi)^{6}(m_{Z^{\prime}}^{2}-t)^{2}}c_{loop}^{2}\,,$ (36)
$\frac{d\sigma_{\chi e}}{d\Omega}=\frac{d\sigma_{\chi
p}}{d\Omega}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,.$ (37)

For a heavy mediator, we can perform the integration explicitly to relate the
total cross sections:

$\sigma_{\chi e}=\sigma_{\chi
p}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,.$ (38)

In the case of a light mediator, the total cross section diverges. Literature
involving light mediators often parameterizes the differential cross section
in terms of a reference cross section $\overline{\sigma}$:

$\frac{d\sigma_{\chi T}}{dq^{2}}=\frac{\overline{\sigma}_{\chi T}}{4\mu_{\chi
T}^{2}v^{2}}\left(\frac{q_{ref}}{q}\right)^{4}\,,$ (39)

where $q_{ref}$ is a reference momentum, usually taken to be $\sim\alpha
m_{e}$ Emken  _et al._ (2019). We can rewrite our differential cross sections
in terms of the momentum transfer $q$, to match direct detection literature,
by noting that

$q^{2}=\bf{p_{1}}+\bf{p_{3}}-2\bf{p_{1}}\bf{p_{3}}\,,$ (40)

so

$\frac{d\sigma}{d\cos{\theta}}=2\mu_{\chi T}^{2}v^{2}\frac{d\sigma}{dq^{2}}\,$
(41)

where $T$ denotes the target of the scattering, either proton or electron. We
can thus write

$\frac{d\sigma_{\chi e}}{dq^{2}}=\frac{d\sigma_{\chi
p}}{dq^{2}}\frac{\mu_{\chi p}^{2}}{\mu_{\chi
e}^{2}}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,,$ (42)

and given the parametrization above, we can relate the reference cross
sections:

$\overline{\sigma}_{\chi e}=\overline{\sigma}_{\chi
p}\frac{e^{2}}{2304\pi^{4}(2\alpha_{u}+\alpha_{d})^{2}}\left(\frac{\mu_{\chi
e}}{\mu_{\chi p}}\right)^{2}c_{loop}^{2}\,.$ (43)

As it turns out, the relation between the reference cross sections is exactly
the relation between the total cross sections found in the heavy mediator
case.

##  Appendix B Proton-Only Loops

In the main text, we considered the case where $\alpha_{d}=-\alpha_{u}$, and
$\alpha_{i}=0$ for all other quarks. We can also consider the case where the
couplings to all (light) quarks are equal, i.e.
$\alpha_{u}=\alpha_{d}=\alpha_{s}$. In this case, only the proton contributes,
resulting in limits that are weaker by a factor of 7—8. We show the results
below.

Figure 7: Limits on DM with a heavy vector mediator and
$\alpha_{u}=\alpha_{d}=\alpha_{s}$. Our recasting of constraints from SENSEI
Barak  _et al._ (2020) is shown in red, while the regions outlined in red
dashed and dotted lines will be accessible to DAMIC-M Battaglieri  _et al._
(2017); Settimo (2018, 2020) and Oscura Aguilar-Arevalo  _et al._ (2022)
respectively. Existing detector constrains From Migdal effect searches at
SENSEI Barak  _et al._ (2020); Berghaus  _et al._ (2023), XENON10/1T (Essig
_et al._ (2020), as shown as in Ref. Berghaus  _et al._ (2023)), CDEX Liu  _et
al._ (2019, 2022) and EDELWEISS Armengaud  _et al._ (2019, 2022) are shown in
gray, while Lyman alpha constraints Rogers  _et al._ (2022) are shown in blue.
Figure 8: Limits on DM scattering through a light vector mediator and
$\alpha_{u}=\alpha_{d}=\alpha_{s}$. Our results for SENSEI Barak  _et al._
(2020) are highlighted in red, and the expected reach of DAMIC-M Battaglieri
_et al._ (2017); Settimo (2018, 2020) and Oscura Aguilar-Arevalo  _et al._
(2022) are outlined by the red dashed and dotted lines respectively.
Constraints from Migdal effect searches at SENSEI Barak  _et al._ (2020);
Berghaus  _et al._ (2023), and XENON10/1T (Essig  _et al._ (2020), as shown in
Ref. Berghaus  _et al._ (2023)) are shown in gray. Figure 9: Effective
millicharge resulting from hadronic loop interactions assuming
$\alpha_{u}=\alpha_{d}=\alpha_{s}$. The white lines are astrophysical
constaints on millicharged DM: above the solid line, ccluster magnetic fields
would noticeably alter the density profile of galaxy clusters Kadota  _et al._
(2016), while above the dashed line, millicharged DM would extract too much
angular momentum from the Milky Way disk Stebbins and Krnjaic (2019). Migdal
effect constraints from SENSEI Barak  _et al._ (2020); Berghaus  _et al._
(2023) and XENON10/XENON1T (Essig  _et al._ (2020), as shown in Ref. Berghaus
_et al._ (2023)) are shown in gray.

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