# Truth Social Dataset

Patrick Gerard, Nicholas Botzer, Tim Weninger

###### Abstract

Formally announced to the public following former President Donald Trump’s
bans and suspensions from mainstream social networks in early 2022 after his
role in the January 6 Capitol Riots, Truth Social was launched as an
“alternative” social media platform that claims to be a refuge for free
speech, offering a platform for those disaffected by the content moderation
policies of the existing, mainstream social networks. The subsequent rise of
Truth Social has been driven largely by hard-line supporters of the former
president as well as those affected by the content moderation of other social
networks. These distinct qualities combined with its status as the main
mouthpiece of the former president positions Truth Social as a particularly
influential social media platform and give rise to several research questions.
However, outside of a handful of news reports, little is known about the new
social media platform partially due to a lack of well-curated data. In the
current work, we describe a dataset of over 823,000 posts to Truth Social and
and social network with over 454,000 distinct users. In addition to the
dataset itself, we also present some basic analysis of its content, certain
temporal features, and its network.

## Introduction

The social media platform Truth Social was launched in February of 2022 about
a year after the suspension of former United States President Donald Trump
from Twitter, Facebook, and other social media platforms. Truth Social is
largely stylized after Twitter where Tweets are instead called Truths and
ReTweets are instead called ReTruths. Note that throughout the remainder of
this paper we most commonly refer to Truths as posts in order to avoid
confusion with the epistemic use of truth (as in true/false). Due to the
political and social circumstances surrounding its creation and launch, Truth
Social has positioned itself as a hub for right-wing social media users
disgruntled by mainstream platforms’ attempts to root out hateful and harmful
communities and content.

Since its inception, and largely due to the influence of the former
president’s use of the platform, Truth Social has increasingly dominated a
space of social media platforms that cater to users affiliated with the alt-
right political movement—a technology space sometimes referred to as alt-tech
that also includes Parler, Gab, Rumble and others. Truth Social, in effect,
functions as a kind of right-wing Twitter, but without the content regulation
that is typically found in mainstream social media platforms. As a result,
Truth Social, along with other alt-tech platforms, are a potential hotbed for
misinformation, conspiracy theories, and other malign social media activity.

Figure 1: Annotated illustration of Truth Social Web Interface. The Web
scraper extracted user data, and post information, including time, content
(with links or other media), quotes, ReTruths, and likes.

Despite its scope and influence, little is known about the posts and content
that is shared on the Truth Social network. The dearth of research involving
Truth Social is partially due to how new it is, but also due to the lack of a
publically available API. To ameliorate these issues, the current work
presents a large dataset of Truths, ReTruths, users, and other data collected
from a broad crawl over the Truth Social platform from its launch on February
21, 2022 until October 15, 2022. In total, this dataset contains the content
of 823,927 Truths posted by 454,458 users including the full history of the
65,536 most active users. Truth Social does not publish its usage statistics,
but we estimate that this dataset contains user data of about 20% of the total
registered users and an unknown, but larger, proportion of the total number of
posts.

The dataset represents the first of its kind and can be used to ask and answer
numerous research questions. For starters, social media’s effects on people’s
consumption of information has become a topic of increasing importance (Sharot
and Sunstein 2020). Providing users more direct agency over the information
they consume, social media—while transformative—may limit exposure to diverse
perspectives and cause the formation of like-minded users reinforcing shared
narratives (Cinelli et al. 2021). This lack of exposure to diverse information
and differing perspectives has been found to provide the scaffolding upon
which conspiracy theories (Cinelli et al. 2022) and misinformation (Del
Vicario et al. 2016) may grow. Truth Social is but the latest example of the
formation of a self-referential, insular community—commonly called an echo
chamber—that is known to lead to increased political polarization (Barberá
2020). Thus, because Truth Social is itself a right-wing echo chamber,
catering to politically polarized defectors of mainstream social media, it
provides a fertile ground for the spread of misinformation and conspiracy
theories. Therefore, with the increased understanding of conspiracy theories’
potentially damaging effects on democracy (Sternisko, Cichocka, and Van Bavel
2020), understanding Truth Social’s interaction with echo chambers and
misinformation presents an important topic for continued research.

### A Brief Overview of Truth Social

Truth Social’s announcement and ultimate launch as a social media platform can
be traced to former U.S. President Donald Trump’s ban from several major
social media platforms following his role in the January 6 United States
Capitol
attack111https://blog.twitter.com/en˙us/topics/company/2020/suspension. As it
currently stands, Truth Social is occupied largely by both users disaffected
by mainstream platforms’ moderation policies and enthusiastic followers of
Donald J. Trump, and appears generally similar to other “alt-tech” platforms.
However, Truth Social’s status as the main mouthpiece for a former President
whose influence remains momentous in the United States Republican Party
positions it as a remarkably influential alt-tech platform.

Platforms with natures similar to Truth Social—catering largely to users
disaffected by the content moderation of mainstream social networks—have been
found to be decidedly successful in drawing users over from the original
platform (Papasavva et al. 2021), specifically in the case of followers of
Donald Trump (Horta Ribeiro et al. 2020). Moreover, these types of platforms
have been shown to both harbor and instigate dangerous conspiracy theories
(Rye, Blackburn, and Beverly 2020), which, despite being birthed on seemingly
fringe platforms, have been found to ultimately jump to mainstream platforms
(Zannettou et al. 2017), nevertheless advancing the same dangerous
misinformation that mainstream social networks attempted to thwart in the
first place (Tollefson 2021).

## Dataset Collection Methodology

Collecting the posts and other activity data from Truth Social is not
straightforward because the site does not provide a public API. Instead, we
implemented a custom Web scraper and programmatically extracted the relevant
context from Truth Social’s Web interface directly. This Web interface did not
impose any crawling restrictions nor did it disallow any crawling with the
robots.txt standard. Because no API was available the most straightforward way
to collect posts was from each specific account.

We crawled the Truth Social one account at a time, starting with
@realDonaldTrump and then iteratively in a breadth first manner over the
followers of each account. Specifically, the crawling methodology proceeded as
follows:

  1. 1.

Collect information about the user’s follower count, following count, creation
date.

  2. 2.

Iterate through users following the user and users that the user follows,
create an edge for each follower-followee relationship, and add that user to
the breadth-first queue if that user has not been scraped in the past 14 days.

  3. 3.

Extract the full set of available Truths posted by the user.

This crawl began on September 4, 2022 and continued until October 14, 2022. In
that time, all content posted by 65,536 users was collected. The dataset,
therefore, has the complete set of all posts for the visited users before
September of 2022.

One particular complication of the crawling methodology was the extraction of
secondary-posts, i.e., ReTruths, Quotes, and Replies. During the initial
crawl, we collected these secondary-posts but not the original post itself.
So, at the end of the initial crawl, we additionally collected all of the
original posts from which the initially collected ReTruths were based. Due to
Truth Social’s HTTP request limitations, we were only able to to connect the
Quotes and Replies to the user to whom they were directed. Although these
efforts resulted in a full accounting of the originating user account for all
Quotes and Replies as well as the post content for all ReTruths, the opposite
is not true. In other words, although we have the original user for all
collected Quotes and Replies and the original post for all collected ReTruths,
we we may not have collected all of the ReTruths, Quotes, or Replies for a
given post. Nevertheless, as a result of our efforts the dataset is internally
consistent.

### Data Model

Figure 2: Relationship diagram between data elements in the collected schema.
Arrows represent foreign key relationships among the tables.

During this crawl, data elements were stored in a local database system. The
relational schema of this database is illustrated in Fig. 2. This dataset
contains various kinds of related data and is therefore modeled as a
relational database with foreign key dependencies. For example,
follower/followee relationships are enumerated in the follows table, with each
entry referencing the following and the followed user via a foreign key to the
corresponding user record. Likewise, truth entries are linked to the
corresponding user, and quotes, replies, media, hashtags, external_urls, and
their respective edge tables are all linked with foreign key relationships to
further contextualize the content the Truths.

Table | Number of Records
---|---
users.tsv | 454,458
follows.tsv | 4,002,115
truths.tsv | 823,927
quotes.tsv | 10,508
replies.tsv | 506,276
media.tsv | 184,884
hashtags.tsv | 21,599
external_urls.tsv | 173,947
truth_hashtag_edges.tsv | 213,295
truth_media_edges.tsv | 257,500
truth_external_url_edges.tsv | 252,877
truth_user_tag_edges.tsv | 145,234
Table 1: Description of the Truth Social Dataset

Tables from the database system were exported to text files in a tab-delimited
format and are available on the Zenodo data service at
https://doi.org/10.5281/zenodo.7531625 and a sample containing the first
10,000 records from each file are included as Supplemental Material. Table 1
lists the files available in the dataset and the number of records in each
file.

### FAIR Principles

The dataset presented by the present work conforms to the FAIR principles
(Wilkinson et al. 2016) and are therefore findable, accessible, interoperable,
and re-usable:

#### Findable:

We provide the dataset publically using the Zenodo data service and give it a
permanent digital object identifier (DOI)
https://doi.org/10.5281/zenodo.7531625.

#### Accessible:

The dataset is freely available on the Internet and can be accessed by anyone
with an internet connection. All of the data is provided as tab separated
value (TSV) files, a standard format for handling data tabular data.

#### Interoperable:

The dataset is easily loaded and viewed with most current database management
systems or spreadsheet systems.

#### Re-usable:

Metadata is also included in a Readme file and is linked to the DOI of this
paper for further reference.

### Limitations

Although we endeavoured to capture a complete and holistic dataset, it is
important to be aware of certain methodological and technological limitations
and possible sampling biases that may be present in the dataset. Some of the
key limitations are are follows:

  1. 1.

Access to a user’s followers is limited. While Truth Social permits clients on
its Web application to scroll through the entirety of a user’s following list,
it limits clients’ access to a user’s followers list to only 50 followers. It
is unclear why this limitation exists and how these 50 followers are selected.
It may be possible to estimate who follows whom by analyzing a complete
following lists of all users and/or the users who frequently ReTruth, Quote or
like another user. However, that analysis is not present in the current
dataset.

  2. 2.

Web Request Limits. Although scraping limits are not published by Truth
Social, we did endeavor to be responsible with the number of HTTP requests
that were issued to Truth Social. This restricted our ability to capture more
data.

  3. 3.

Sampling Bias. Recall that the crawling methodology proceeded in a breadth
first manner from the a popular user @realDonaldTrump and then proceeded to
other highly active users. As a result, this dataset likely contains the most
active users of the platform. The choice of @realDonaldTrump as the seed-user
may have also nudged the data collection towards more political users and
posts. However, the average path length of the user network was relatively
small and the post content is quite diverse, so we are confident that the
sample is moderately representative of the whole platform.

### Ethical Considerations

The dataset is collected from a publically available resource. Users who
submit content do so with the explicit intent of making their activity
publically and widely visible. This research was observational only. No
intervention or treatment was made to the population; therefore, this research
was determined to be exempt from full ethical review panel by the University
of Notre Dame institutional review board.

## Content Analysis

To get a better understanding of the collected dataset we performed some
rudimentary analysis on the content of the dataset. This analysis includes (1)
an investigation into the top-linked Web sites, (2) a look at certain temporal
artifacts of the text content of the posts, and (3) an analysis of the
follower network of the platform.

### External Link Analysis

Research has shown that misinformation campaigns often utilize and share
external links to proliferate information across and between multiple
platforms (Wilson and Starbird 2020; Golovchenko et al. 2020). For this
reason, we extracted the external Web links that users posted. We found
173,947 links in total. The domain of each link was extracted and aggregated
for further analysis.

$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$10^{7}$rumblejustthenewsbabylonbeeconservativebriefoannhannitybreitbartrsbnetworkthefederalistdcenquirerthegatewaypunditfoxsportsgamespottelegramyoutubehiphopdxfoxnewsLinked
Domain Frequency in TruthsLinked Domain Frequency in ReTruths Figure 3: Top 10
linked domains appearing in Truths (red, x-axis) and top 10 linked domains
appearing in ReTruths (blue, y-axis). Purple marks indicate that the domain is
in the top 10 in both Truths and ReTruths.

Figure 3 illustrates the top linked domains in terms of the number of Truths
in which they appear (x-axis) and the number of ReTruths in which they appear.

The top linked external Web site in terms of frequency in both Truths and
ReTruths is Rumble, a video sharing platform. This site is known for having
looser content moderation restrictions compared to many other video sharing
platforms like YouTube and has become a haven for controversial figures that
have been banned from mainstream platforms. Rumble has grown in popularity in
recent years at least partially due to its use by former President Donald
Trump to stream his political rallies and post news clips.

Other popular site include right-wing political networks like OANN, Breitbart,
FoxNews, etc. Curiously, a large number of external links point to the
Telegram social media platform. Popular in Russia, Telegram is frequently used
by supporters of Russia in the Russia-Ukraine War (Theisen et al. 2023). It
has also in been investigated as a platform that acts as a safe haven for
those that have been deplatformed from other social media sites (Rogers 2020).
Because Telegram has this association with deplatformed individuals we further
investigated the five most shared telegram channels to better understand the
type of content that was shared. In order of decreasing popularity, the five
most shared telegram channels are:

  1. 1.

RealKarliBonne. This channel shares a large variety of memes, Truths, and
messages supporting Donald Trump. The channel also shares a large amount of
video content featuring the Biden administration. Although some video messages
are supportive of Trump, the preponderance is negative of the Biden
administration and may serve as an example of affective polarization (Iyengar
et al. 2019).

  2. 2.

LauraAbolichannel. This channel does not appear to have a clear focus. Its
content contains commentary and clips from right-wing media, anti-vaccination
messages, and uplifting memes.

  3. 3.

freedomforcebattalion. This channel presents former President Donald Trump’s
messaging and other current events with a biblical perspective.

  4. 4.

realx22report. This is the official Telegram channel of X22report, a daily
show that covers financial and political issues.

  5. 5.

drawandstrikechannel This is the channel of right-wing political columnist
Brian Cates. Cates is an active user on Truth Social and draws a major
following as well in his Telegram channel. Content in the channel focuses
around various political topics and current events.

The combination of Truth Social’s politically charged user-base and external
links’ role in narrative-building and misinformation campaigns creates a
significant space for further analysis in both the information contained in
the external links and the temporal aspect of the external links.

Examinations into the information contained in the external links may provide
major insights into the spread of conspiracy theories and misinformation on
Truth Social and across the social Web. For example, to better understand the
popularization of conspiracy theories on Truth Social, one may trace the
earliest external domains that references a certain conspiracy theory and
analyze their role in the narrative spread through Truth Social.

Overall, external links in Truth Social remarkably reflect the politically
charged user-base, and therefore external agents’ influence on the user-base
cannot be overlooked. Thus, we believe that further analysis of these external
links’ role in the Truth Social network is critical to understanding how
certain narratives and conspiracy theories propagate across the network.

Feb 2022Mar 2022Apr 2022May 2022Jun 2022Jul 2022Aug 2022Sep 2022Oct
2022$0$$0.2$$0.4$$0.6$$0.8$$1$Truth Social Official Launch January 6 Committee
Public Hearings Begin FBI Search of Mar-a-LagoDateMax-Normalized
FrequencyTotalJan 6 TruthsFBI Truths Figure 4: Daily max-normalized frequency
of Truths posted in the dataset. Because Truth Social has a strong affiliation
with former United States President Donald Trump, political activity and news
stories involving the former president are commonly discussed on the site. In
this figure, posts mentioning political events like the US House of
Representative’s Special Committee on the January 6 riots at the US Capitol
(blue) and the FBI search at the former president’s residence (Mar-a-Lago)
spike surrounding these events. Dotted blue and red lines illustrate the data
from Google Trends corresponding to the event.

### Text Analysis

To better understand both the temporal nature of the dataset and the possible
interaction between external events and popular topics on Truth Social, we
next performed a text analysis on the 823,927 posts in the dataset.

Figure 4 provides an example illustration of our findings. Max-normalized by
posts per day, the frequency of all posts provides a backdrop for our further
evaluation into posts containing certain keywords. Overall, we found that the
daily frequency of posts was highest near the official launch of Truth Social,
before falling slightly and stabilizing.

Because of its affiliation with the former president, posts on Truth Social
largely revolve around activities related to conservative politics and news
stories. To illustrate this more concretely we considered two events that
occurred during dates covered by the data collection methodology involving the
former president: (1) the public hearings from the United States’ House of
Representative’s Select Committee on the January 6 riots at the Capitol
Building, and (2) the FBI raid of the former president’s residence commonly
called Mar-a-Lago.

Our first text analysis centers on the January 6 United States Capitol attack.
Utilizing the keywords “January 6”, “January Six”, “Jan 6”, and “Jan Six”, we
identified posts containing any of these phrases (with case-insensitive
criteria). As shown in solid blue in Fig. 4, immediately following the start
of the January 6 Committee’s Public Hearings, posts containing these phrases
spiked and continued to be elevated for approximately ten weeks. We believe
that further evaluation may find correlation between these repetitive spikes
and the ensuing broadcasts of the January 6 Committee’s public hearings, but a
thorough analysis is outside the scope of the current paper.

Next, we analyzed posts related to the FBI raid of Mar-a-Lago, utilizing the
keywords “Mar-a-Lago” and “Mar a Lago”. Illustrated in solid red in Fig. 4, we
again identified posts containing any of these phrases (with case-insensitive
criteria). Like in the January 6 example above, in this case we also found
that Truths containing these phrases spiked following the events.

Dotted blue and red lines in Fig. 4 illustrate the data from Google Trends for
the January 6 and Mar-a-Lago searches, respectively. We see that the attention
of these two events are symmetrical across Truth Social and the Web in
general. This symmetry shows that the topics discussed on Truth Social appears
to be largely representative of activity on the Web.

When evaluating these spikes, we found that they were often driven largely by
a small number of original posts that were ReTruthed many times. Some of the
most popular posts originated from the former president himself, but were
elevated and expanded by popular commentators, whose roles in narrative-
building on Truth Social is a clear topic for further inspection. For example,
following the FBI search of Mar-a-Lago, the former president posted “A
horrible thing that took place yesterday at Mar-a-Lago. We are no better than
a third world country, a banana republic. It is a continuation of Russia,
Russia, Russia, Impeachment Hoax #1, Impeachment Hoax # 2, the no collusion
Mueller Report, and more. To make matters worse it is all, in my opinion, a
coordinated attack with Radical Left Democrat state and local D.A.’s and
A.G.’s”. This set off a conspiracy-laden narrative of the events that echoed
throughout the platform. These posts typically gathered tens of thousands of
ReTruths and likes—a large number for the relatively small platform. These
findings and the dataset as a whole may provide much needed insight into the
cohesion of the Truth Social platform as well as a better understanding of the
role that popular commentators play in driving the narrative for the majority
of users.

Ultimately, external events appear to have significant influence on Truth
Social’s user network. Moreover, the near-immediate rise in posts following
certain events may point to both the interests of users and the cohesiveness
of the Truth Social network. Further work lies in both the examination of
narratives and sentiments accompanying apparent reactions to external events,
as well as the examination of the platform-wide influence of a small group of
popular commentators and if the narratives and stories that percolate within
Truth Social eventually make their way onto mainstream social media platforms.

### Network Analysis

Like Facebook, Instagram, and Twitter, the friendship network (via followers
or ReTruths) of the Truth Social platform forms a social network that can be
analyzed to find social roles, and network-based artifacts like centrality,
betweeness, cliques, and other interesting network-based phenomena.

Because the follower graph could not be reliably extracted from the Web
interface, we instead used ReTruths to construct the network. Although not the
same as the follower graph, ReTweets (the Twitter-analog to ReTruths) are
known to “more closely mirror real-world relationships and trust” (Bild et al.
2015).

$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{-5}$$10^{-3}$$10^{-1}$ReTruth
DegreePr$(x)$$10^{0}$$10^{5}$ReTruth Degree Figure 5: Degree distribution of
the ReTruth graph by distinct users (left) and by total number of ReTruths
(right). The red circle represents @realDonaldTrump who has the most ReTruths
according to both criteria.

We used two different criteria to analyze the network. First, we found the
number of unique, fully-scraped users who ReTruthed a user. Figure 5 (left)
illustrates the degree distribution of the ReTruth graph, that is, the number
of distinct users who Retruthed another user. This distribution appeared to
follow the generally followed typical power-law degree distribution found on
Twitter. The user with the highest number of distinct ReTruthing users was
ReTruthed by 1,448 users.

Rather than looking at distinct users, our next analysis looked at the total
number of ReTruths for each user. Figure 5 (right) shows that the ReTruth
multi-graph, where multiple ReTruths from the same user count as multiple
edges, has a slightly different degree distribution that more closely
resembles a log-normal distribution. The user with the highest number of
ReTruths was ReTruthed 1,0176,833 times.

## Conclusion

This paper presents a large dataset, which contains information of 454,000
users and over 823,000 posts including the complete history of the 65,536 most
active users from the Truth Social platform. In addition, this dataset covers
the ReTruths, quotes, text, media, and other information from the platform. We
also perform a preliminary analysis of this dataset.

Our preliminary analysis shows that a handful of external Web sites dominate
Truth Social posts, with Rumble appearing most frequently. Moreover, a brief
look at the most commonly linked Telegram channels finds that right-wing and
Trump-focused Telegram channels appear most often. We also analyzed the
temporal and structural nature of the posts and ReTruths in the dataset. We
have uncovered several interesting findings and avenues for further study.

Overall, Truth Social is an emerging alt-tech platform. Targeted at hard-right
users disaffected from mainstream social media platforms and working as the
main mouthpiece of former President Donald Trump, Truth Social’s unique
position in the information ecosystem cannot be overlooked. This dataset
provides researchers a means to study the Truth Social platform, permitting
research on both Truth Social itself and important socio-technical issues
including the cultivation and spread of information and narratives.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-068 LHCb-PAPER-2011-030

Measurement of the ratio of prompt $\chi_{c}$ to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in $pp$ collisions
at $\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-2.07413ptV}}$}$

The LHCb collaboration111Authors are listed on the following pages.

The prompt production of charmonium $\chi_{c}$ and $J/\psi$ states is studied
in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV at
the Large Hadron Collider. The $\chi_{c}$ and $J/\psi$ mesons are identified
through their decays $\chi_{c}\rightarrow J/\psi\gamma$ and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ using 36 pb-1 of data collected by the LHCb
detector in 2010. The ratio of the prompt production cross-sections for
$\chi_{c}$ and $J/\psi$, $\sigma(\chi_{c}\rightarrow
J/\psi\gamma)/\sigma(J/\psi)$, is determined as a function of the $J/\psi$
transverse momentum in the range $2<p_{\mathrm{T}}^{J/\psi}<15$ GeV/$c$. The
results are in excellent agreement with next-to-leading order non-relativistic
expectations and show a significant discrepancy compared with the colour
singlet model prediction at leading order, especially in the low
$p_{\mathrm{T}}^{J/\psi}$ region.

Submitted to Phys. Lett. B

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, L. Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6,
G. Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P.
Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil
Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A.
Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10, F.
Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B. Gui53, E. Gushchin30,
Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C.
Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N. Harnew52, J.
Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31,
O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A.
Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9, A. Sparkes47, P.
Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S. Stoica26, S.
Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K. Subbiah35, S.
Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S. T’Jampens4, E.
Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van Tilburg11, V.
Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, P. Urquijo53, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, R.
Waldi56, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D.
Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam

##  1 Introduction

The study of charmonium production provides an important test of the
underlying mechanisms described by Quantum Chromodynamics (QCD). At the
centre-of-mass energies of proton-proton collisions at the Large Hadron
Collider, $c\overline{}c$ pairs are expected to be produced predominantly via
Leading Order (LO) gluon-gluon interactions, followed by the formation of
bound charmonium states. The former can be calculated using perturbative QCD
and the latter is described by non-perturbative models. Other, more recent,
approaches make use of non-relativistic QCD factorization (NRQCD), which
assumes the $c\overline{}c$ pair to be a combination of colour-singlet and
colour-octet states as it evolves towards the final bound system via the
exchange of soft gluons [1]. The fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ produced through the radiative
decay of $\chi_{c}$ states is an important test of both the colour-singlet and
colour-octet production mechanisms. In addition, knowledge of this fraction is
required for the measurement of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ polarisation, since the predicted polarisation is different for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons coming from the
radiative decay of $\chi_{c}$ state compared to those that are directly
produced.

In this paper, we report the measurement of the ratio of the cross-sections
for the production of $P$-wave charmonia $\chi_{cJ}(1P)$, with $J=$ 0, 1, 2,
to the production of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in
promptly produced charmonium. The ratio is measured as a function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum in the
range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and in the rapidity
range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$.
Throughout the paper we refer to the collection of $\chi_{cJ}(1P)$ states as
$\chi_{c}$. The $\chi_{c}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates are reconstructed through their respective decays
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ using
a data sample corresponding to an integrated luminosity of 36
$\mbox{\,pb}^{-1}$ collected during 2010. Prompt (non-prompt) production
refers to charmonium states produced at the interaction point (in the decay of
$b$-hadrons); direct production refers to prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons that are not decay
products of an intermediate resonant state, such as the $\psi{(2S)}$. The
measurements are complementary to the measurements of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section [2]
and the ratio of the prompt $\chi_{c}$ production cross-sections for the $J=1$
and $J=2$ spin states [3], and extend the
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ coverage with
respect to previous experiments [4, 5].

##  2 LHCb detector and selection requirements

The LHCb detector [6] is a single-arm forward spectrometer with a pseudo-
rapidity range $2\,{<}\,\eta\,{<}\,5$. The detector consists of a silicon
vertex detector, a dipole magnet, a tracking system, two ring-imaging
Cherenkov (RICH) detectors, a calorimeter system and a muon system.

Of particular importance in this measurement are the calorimeter and muon
systems. The calorimeter system consists of a scintillating pad detector (SPD)
and a pre-shower system, followed by electromagnetic (ECAL) and hadron
calorimeters. The SPD and pre-shower are designed to distinguish between
signals from photons and electrons. The ECAL is constructed from scintillating
tiles interleaved with lead tiles. Muons are identified using hits in muon
chambers interleaved with iron filters.

The signal simulation sample used for this analysis was generated using the
Pythia $6.4$ generator [7] configured with the parameters detailed in Ref.
[8]. The EvtGen [9], Photos [10] and Geant4 [11] packages were used to decay
unstable particles, generate QED radiative corrections and simulate
interactions in the detector, respectively. The sample consists of events in
which at least one ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay takes place with no constraint on the
production mechanism.

The trigger consists of a hardware stage followed by a software stage, which
applies a full event reconstruction. For this analysis, events are selected
which have been triggered by a pair of oppositely charged muon candidates,
where either one of the muons has a transverse momentum
$p_{\mathrm{T}}\,{>}\,\mbox{${1.8}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
or one of the pair has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.56}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
and the other has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.48}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$.
The invariant mass of the candidates is required to be greater than
${2.9}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. The photons are not
involved in the trigger decision for this analysis.

Photons are reconstructed using the electromagnetic calorimeter and identified
using a likelihood-based estimator, $\mathrm{CL}_{\gamma}$, constructed from
variables that rely on calorimeter and tracking information. For example, in
order to reduce the electron background, candidate photon clusters are
required not to be matched to the trajectory of a track extrapolated from the
tracking system to the cluster position in the calorimeter. For each photon
candidate a value of $\mathrm{CL}_{\gamma}$, with a range between 0
(background-like) and 1 (signal-like), is calculated based on simulated signal
and background samples.

The photons are classified as one of two types: those that have converted to
electrons in the material after the dipole magnet and those that have not.
Converted photons are identified as clusters in the ECAL with correlated
activity in the SPD. In order to account for the different energy resolutions
of the two types of photons, the analysis is performed separately for
converted and non-converted photons and the results are combined. Photons that
convert before the magnet require a different analysis strategy and are not
considered here. The photons used to reconstruct the $\chi_{c}$ candidates are
required to have a transverse momentum
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$,
a momentum
$p^{\gamma}\,{>}\,\mbox{${5}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$ and
$\mathrm{CL}_{\gamma}\,{>}\,0.5$; the efficiency of the $\mathrm{CL}_{\gamma}$
cut for photons from $\chi_{c}$ decays is $72\%$.

All ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are
reconstructed using the decay ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$. The muon and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ identification criteria are
identical to those used in Ref. [2]: each track must be identified as a muon
with $p_{\mathrm{T}}\,{>}\,700{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and have a
track fit $\chi^{2}/\mathrm{ndf}\,{<}\,4$, where $\mathrm{ndf}$ is the number
of degrees of freedom. The two muons must originate from a vertex with a
probability of the vertex fit greater than $0.005$. In addition, the
$\mu^{+}\mu^{-}$ invariant mass is required to be in the range
${3062}\,{-}\,{\mbox{${3120}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$}}$.
The $\chi_{c}$ candidates are formed from the selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and photons.

The non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution
arising from $b$-hadron decays is taken from Ref. [2]. For the $\chi_{c}$
candidates, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-decay
time, $t_{z}$, is used to reduce the contribution from non-prompt decays, by
requiring $t_{z}\,{=}\,(z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{-}\,z_{PV})M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{/}\,p_{z}<\mbox{${0.1}\>{{\rm\,ps}}$}$, where
$M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the reconstructed dimuon
invariant mass, $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}-z_{\mathrm{PV}}$ is the $z$ separation of the reconstructed production
(primary) and decay vertices of the dimuon, and $p_{z}$ is the $z$-component
of the dimuon momentum. The $z$-axis is parallel to the beam line in the
centre-of-mass frame. Simulation studies show that, with this requirement
applied, the remaining fraction of $\chi_{c}$ from $b$-hadron decays is about
$0.1\%$. This introduces an uncertainty much smaller than any of the other
systematic or statistical uncertainties evaluated in this analysis and is not
considered further.

The distributions of the $\mu^{+}\mu^{-}$ mass of selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and the mass
difference, $\Delta
M\,{=}\,M\left(\mu^{+}\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\mu^{-}\right)$,
of the selected $\chi_{c}$ candidates for the converted and non-converted
samples are shown in Fig. 1. The total number of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates observed in the data
is $\sim 2.6$ million. The fit procedure to extract the three $\chi_{c}$
signal yields using Gaussian functions and one common function for the
combinatorial background is discussed in Ref. [3]. The total number of
$\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ candidates observed are $823$,
$38\,630$ and $26\,114$ respectively. Since the
$\chi_{c0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
branching fraction is $\sim 30$ (17) times smaller than that of the
$\chi_{c1}$ ($\chi_{c2}$), the yield of $\chi_{c0}$ is small as expected [12].

Figure 1: (a) Invariant mass of the $\mu^{+}\mu^{-}$ pair for selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates. The solid red curve
corresponds to the signal and the background is shown as a dashed purple
curve. (b) and (c) show the $\Delta
M\,{=}\,M\left(\mu^{+}\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\mu^{-}\right)$
distributions of selected $\chi_{c}$ candidates with (b) converted and (c)
non-converted photons. The upper solid blue curve corresponds to the overall
fit function described in Ref. [3]. The lower solid curves correspond to the
fitted $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ contributions from left to
right, respectively (the $\chi_{c0}$ peak is barely visible). The background
distribution is shown as a dashed purple curve.

##  3 Determination of the cross-section ratio

The main contributions to the production of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ arise from direct production
and from the feed-down processes
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ and
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ where $X$
refers to any final state. The cross-section ratio for the production of
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decays
compared to all prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ can be
expressed in terms of the three $\chi_{cJ}$ $(J=0,1,2)$ signal yields,
$N_{\chi_{cJ}}$, and the prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
yield, $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, as

$\displaystyle\dfrac{\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})}$
$\displaystyle\,\approx\,\dfrac{\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{\sigma^{\mathrm{dir}}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})+\sigma(\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)+\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}$
$\displaystyle\,\,{=}\,\,\dfrac{\displaystyle{\sum_{J=0}^{J=2}}\;\dfrac{N_{\chi_{cJ}}}{\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}}\cdot\dfrac{\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}R_{2S}+\displaystyle{\sum_{J=0}^{J=2}}\;\dfrac{N_{\chi_{cJ}}}{\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}}\left[\dfrac{\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}-R_{2S}\right]}$ (1) with $\displaystyle R_{2S}$
$\displaystyle\,\,{=}\,\,\dfrac{1+f_{2S}}{1+f_{2S}\dfrac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{2S}}{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{\mathrm{dir}}}}$ (2) and $\displaystyle f_{2S}$
$\displaystyle\,\,{=}\,\,\dfrac{\sigma(\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)}{\sigma^{\mathrm{dir}}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})}.$ (3)

The total prompt $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ cross-section is
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=\sum_{J=0}^{J=2}\;\sigma_{\chi_{cJ}}\cdot{\cal
B}(\chi_{cJ}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)$
where $\sigma_{\chi_{cJ}}$ is the production cross-section for each
$\chi_{cJ}$ state and $\cal
B$($\chi_{cJ}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$)
is the corresponding branching fraction. The cross-section ratio $f_{2S}$ is
used to link the prompt $\psi{(2S)}$ contribution to the direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution and $R_{2S}$ takes
into account their efficiencies. The combination of the trigger,
reconstruction and selection efficiencies for direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $\psi{(2S)}$ decay, and
for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay
are $\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and
$\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
respectively. The efficiency to reconstruct and select a photon from a
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay,
once the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is already selected,
is $\epsilon^{\chi_{cJ}}_{\gamma}$ and the efficiency for the subsequent
selection of the $\chi_{cJ}$ is $\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$.

The efficiency terms in Eq. (1) are determined using simulated events and are
partly validated with control channels in the data. The results for the
efficiency ratios $\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, $\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
and the product
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$ are
discussed in Sect. 4.

The prompt $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$N_{\chi_{cJ}}$ yields are determined in bins of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range
$2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ using the methods
described in Refs. [2] and [3] respectively. In Ref. [2] a smaller data sample
is used to determine the non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ fractions in bins of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and rapidity.
These results are applied to the present
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample without repeating the
full analysis.

##  4 Efficiencies

The efficiencies to reconstruct and select
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ candidates are
taken from simulation. The efficiency ratio
$\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ is consistent with unity for all
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins; hence,
$R_{2S}$ is set equal to 1 in Eq. (2). The ratio of efficiencies
$\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
and the product of efficiencies
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$ for the
$\chi_{c1}$ and $\chi_{c2}$ states are shown in Fig. 2. In general these
efficiencies are the same for the two states, except at low
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ where the
reconstruction and detection efficiencies for $\chi_{c2}$ are significantly
larger than for $\chi_{c1}$. This difference arises from the effect of the
requirement
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$
which results in more photons surviving from $\chi_{c2}$ decays than from
$\chi_{c1}$ decays.

Figure 2:  (a) Ratio of the reconstruction and selection efficiency for direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ compared to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $\chi_{c}$ decays,
$\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{c}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
and (b) the photon reconstruction and selection efficiency multiplied by the
$\chi_{c}$ selection efficiency,
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$, obtained
from simulation. The efficiencies are presented separately for the $\chi_{c1}$
(red triangles) and $\chi_{c2}$ (inverted blue triangles) states, and as a
function of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$.

The photon detection efficiency obtained using simulation is validated using
candidate $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
and $B^{+}\rightarrow\chi_{c}K^{+}$ (including charge conjugate) decays
selected from the same data set as the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ candidates. The
efficiency to reconstruct and select a photon from a $\chi_{c}$ in
$B^{+}\rightarrow\chi_{c}K^{+}$ decays, $\epsilon_{\gamma}$, is evaluated
using

$\displaystyle\epsilon_{\gamma}\,{=}\,\frac{N_{B^{+}\rightarrow\chi_{c}K^{+}}}{N_{B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}}\times\frac{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})}{{\cal
B}(B^{+}\rightarrow\chi_{c}K^{+})\cdot{\cal
B}(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}\times R_{\epsilon}$ (4)

where $N_{B^{+}\rightarrow\chi_{c}K^{+}}$ and
$N_{B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ are
the measured yields of $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and ${\cal
B}$ are the known branching fractions. The factor $R_{\epsilon}=1.04\pm 0.02$
is obtained from simulation and takes into account any differences in the
acceptance, trigger, selection and reconstruction efficiencies of the $K$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\chi_{c}$ (except the photon
detection efficiency) and $B^{+}$ in
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\rightarrow\chi_{c}K^{+}$ decays. All branching fractions are taken from
Ref. [12]. The $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ branching fraction is ${\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=(1.013\pm 0.034)\times 10^{-3}$. The dominant process for
$B^{+}\rightarrow\chi_{c}K^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma K^{+}$ decays is via the $\chi_{c1}$ state, with branching
fractions ${\cal B}(B^{+}\rightarrow\chi_{c1}K^{+})=(4.6\pm 0.4)\times
10^{-4}$ and ${\cal
B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=(34.4\pm 1.5)\times 10^{-2}$; the contributions from the
$\chi_{c0}$ and $\chi_{c2}$ modes are neglected.

The $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidates
are selected keeping as many of the selection criteria in common as possible
with the main analysis. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\chi_{c}$ selection criteria are the same as for the prompt analysis, apart
from the pseudo-decay time requirement. The bachelor kaon is required to have
a well measured track ($\chi^{2}/\mathrm{ndf}<5$), a minimum impact parameter
$\chi^{2}$ with respect to all primary vertices of greater than $9$ and a
momentum greater than $5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The bachelor
is identified as a kaon by the RICH detectors by requiring the difference in
log-likelihoods between the kaon and pion hypotheses to be larger than $5$.
The $B$ candidate is formed from the $\chi_{c}$ or
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate and the bachelor
kaon. The $B$ vertex is required to be well measured
($\chi^{2}/\mathrm{ndf}<9$) and separated from the primary vertex (flight
distance $\chi^{2}>50$). The $B$ momentum vector is required to point towards
the primary vertex ($\cos\theta>0.9999$, where $\theta$ is the angle between
the $B$ momentum and the direction between the primary and $B$ vertices) and
have an impact parameter $\chi^{2}$ smaller than $9$. The combinatorial
background under the $\chi_{c}$ peak for the $B^{+}\rightarrow\chi_{c}K^{+}$
candidates is reduced by requiring the mass difference $\Delta
M_{\chi_{c}}=M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})<600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
A small number of $B^{+}\rightarrow\chi_{c}K^{+}$ candidates which form a good
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidate
are removed by requiring $|M(\mu^{+}\mu^{-}\gamma
K)-M(\mu^{+}\mu^{-}K)|>200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The $\Delta M_{B^{+}}=M(\mu^{+}\mu^{-}\gamma K)-M(\mu^{+}\mu^{-}\gamma)$ mass
distribution for the $B^{+}\rightarrow\chi_{c}K^{+}$ candidates is shown in
Fig. 3(a); $\Delta M_{B^{+}}$ is computed to improve the resolution and hence
the signal-to-background ratio. The $B^{+}\rightarrow\chi_{c}K^{+}$ yield,
$142\pm 15$ candidates, is determined from a fit that uses a Gaussian function
to describe the signal peak and a threshold function,

$\displaystyle
f(x)\,{=}\,x^{a}\\!\left(1-e^{\frac{m_{0}}{c}\left(1-x\right)}\right)+b\left(x-1\right),$
(5)

where $x\,{=}\,\Delta M_{B^{+}}\,{/}\,m_{0}$ and $m_{0}$, $a$, $b$ and $c$ are
free parameters, to model the background. The reconstructed $B^{+}$ mass
distribution for the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates is shown in Fig. 3(b). The
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield,
$8440\pm 96$ candidates, is determined from a fit that uses a Crystal Ball
function [13] to describe the signal peak and an exponential to model the
background.

The photon efficiency from the observation of $B^{+}\rightarrow\chi_{c}K^{+}$
and $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
is measured to be $\epsilon_{\gamma}=(11.3\pm 1.2\pm 1.2)\%$ where the first
error is statistical and is dominated by the observed yield of
$B^{+}\rightarrow\chi_{c}K^{+}$ candidates, and the second error is systematic
and is given by the uncertainty on the branching fraction $\cal
B$($B^{+}\rightarrow\chi_{c1}K^{+}$). The photon efficiency measured in data
can be compared to the photon efficiency, $(11.7\pm 0.3)\%$, obtained using
the same procedure on simulated events. The measurements are in good agreement
and the uncertainty on the difference between data and simulation is
propagated as a $\pm 14\%$ relative systematic uncertainty on the photon
efficiency in the measurement of
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$.

Figure 3: (a) Reconstructed $\Delta M_{B^{+}}=M(\mu^{+}\mu^{-}\gamma
K)-M(\mu^{+}\mu^{-}\gamma)$ mass distribution for
$B^{+}\rightarrow\chi_{c}K^{+}$ candidates and (b) the reconstructed $B^{+}$
mass distribution for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates. The LHCb data are shown as solid black points, the
full fit functions with a solid blue (upper) curve, the contribution from
signal candidates with a dashed red (lower curve) and the background with a
dashed purple curve.

##  5 Polarisation

The simulation used to calculate the efficiencies and, hence, extract the
result of Eq. (1) assumes that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\chi_{c}$ are unpolarised. The effect of polarised states is
studied by reweighting the simulated events according to different
polarisation scenarios; the results are shown in Table 1. It is also noted
that, since the $\psi{(2S)}$ decays predominantly to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi\pi$, with the $\pi\pi$ in an
$S$ wave state [14], and the $\psi{(2S)}$ polarisation should not differ
significantly from the polarisation of directly produced
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons, the effect of the
polarisation can be considered independent of the
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$
contribution [15].

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
angular distributions are calculated in the helicity frame assuming azimuthal
symmetry. This choice of reference frame provides an estimate of the effect of
polarisation on the results, pending the direct measurements of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ system is described by the
angle $\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, which is the
angle between the directions of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the laboratory frame. The
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ distribution depends
on the parameter $\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
which describes the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
polarisation; $\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}=+1,-1,0$
corresponds to pure transverse, pure longitudinal and no polarisation
respectively. The $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ system is described by three angles:
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\theta_{\chi_{c}}$ and $\phi$, where
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the angle
between the directions of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame,
$\theta_{\chi_{c}}$ is the angle between the directions of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame
and the $\chi_{c}$ in the laboratory frame, and $\phi$ is the angle between
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay plane in the
$\chi_{c}$ rest frame and the plane formed by the $\chi_{c}$ direction in the
laboratory frame and the direction of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame.
The general expressions for the angular distributions are independent of the
choice of polarisation axis (here chosen as the direction of the $\chi_{c}$ in
the laboratory frame) and are detailed in Ref. [4]. The angular distributions
of the $\chi_{c}$ states depend on $m_{\chi_{cJ}}$ which is the azimuthal
angular momentum quantum number of the $\chi_{cJ}$ state.

For each simulated event in the unpolarised sample, a weight is calculated
from the distributions of
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\theta_{\chi_{c}}$ and $\phi$ in the various polarisation hypotheses compared
to the unpolarised distributions. The weights shown in Table 1 are then the
average of these per-event weights in the simulated sample. For a given
($|m_{\chi_{c1}}|$, $|m_{\chi_{c2}}|$,
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$) polarisation
combination, the central value of the determined cross-section ratio in each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin should be
multiplied by the number in the table. The maximum effect from the possible
polarisation of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$,
$\chi_{c1}$ and $\chi_{c2}$ mesons is given separately from the systematic
uncertainties in Table 3 and Fig. 4.

Table 1:  Polarisation weights in
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins for
different combinations of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$,
$\chi_{c1}$ and $\chi_{c2}$ polarisations.
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation parameter;
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}=+1,-1,0$ corresponds
to fully transverse, fully longitudinal and no polarisation respectively.
$m_{\chi_{cJ}}$ is the azimuthal angular momentum quantum number corresponding
to total angular momentum $J$; Unpol means the $\chi_{c}$ is unpolarised.

($|m_{\chi_{c1}}|,|m_{\chi_{c2}}|,\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$) |  $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ (${\mathrm{\,Ge\kern-0.50003ptV\\!/}c}$)
---|---
2-3 | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 | 9-10 | 10-11 | 11-12 | 12-13 | 13-15
(Unpol,Unpol,-1) | 1.16 | 1.15 | 1.15 | 1.15 | 1.15 | 1.14 | 1.14 | 1.13 | 1.12 | 1.12 | 1.10 | 1.10
(Unpol,Unpol,1) | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.93 | 0.93 | 0.93 | 0.94 | 0.95 | 0.94
(Unpol,0,-1) | 1.16 | 1.14 | 1.13 | 1.11 | 1.10 | 1.09 | 1.09 | 1.08 | 1.07 | 1.06 | 1.06 | 1.07
(Unpol,0,0) | 1.00 | 0.99 | 0.98 | 0.97 | 0.96 | 0.95 | 0.95 | 0.96 | 0.95 | 0.95 | 0.96 | 0.97
(Unpol,0,1) | 0.91 | 0.91 | 0.90 | 0.89 | 0.88 | 0.87 | 0.88 | 0.89 | 0.89 | 0.88 | 0.91 | 0.92
(Unpol,1,-1) | 1.15 | 1.14 | 1.14 | 1.13 | 1.13 | 1.12 | 1.11 | 1.11 | 1.10 | 1.09 | 1.08 | 1.09
(Unpol,1,0) | 0.99 | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98
(Unpol,1,1) | 0.90 | 0.91 | 0.91 | 0.90 | 0.90 | 0.90 | 0.91 | 0.91 | 0.91 | 0.91 | 0.93 | 0.93
(Unpol,2,-1) | 1.18 | 1.17 | 1.18 | 1.20 | 1.21 | 1.21 | 1.20 | 1.19 | 1.19 | 1.19 | 1.16 | 1.15
(Unpol,2,0) | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 | 1.06 | 1.06 | 1.05 | 1.06 | 1.07 | 1.05 | 1.04
(Unpol,2,1) | 0.93 | 0.94 | 0.94 | 0.96 | 0.97 | 0.98 | 0.98 | 0.98 | 0.99 | 1.00 | 1.00 | 0.99
(0,Unpol,-1) | 1.16 | 1.15 | 1.18 | 1.21 | 1.22 | 1.23 | 1.25 | 1.25 | 1.26 | 1.22 | 1.23 | 1.25
(0,Unpol,0) | 0.99 | 1.00 | 1.02 | 1.05 | 1.07 | 1.08 | 1.10 | 1.11 | 1.12 | 1.10 | 1.12 | 1.14
(0,Unpol,1) | 0.91 | 0.93 | 0.94 | 0.97 | 0.98 | 1.00 | 1.02 | 1.04 | 1.05 | 1.03 | 1.06 | 1.08
(1,Unpol,-1) | 1.17 | 1.15 | 1.14 | 1.13 | 1.12 | 1.11 | 1.09 | 1.08 | 1.07 | 1.08 | 1.05 | 1.05
(1,Unpol,0) | 1.00 | 1.00 | 0.99 | 0.98 | 0.97 | 0.97 | 0.96 | 0.95 | 0.95 | 0.96 | 0.95 | 0.95
(1,Unpol,1) | 0.92 | 0.92 | 0.91 | 0.90 | 0.89 | 0.89 | 0.89 | 0.89 | 0.89 | 0.90 | 0.90 | 0.89
(0,0,-1) | 1.15 | 1.14 | 1.15 | 1.17 | 1.18 | 1.18 | 1.20 | 1.21 | 1.20 | 1.17 | 1.19 | 1.22
(0,0,0) | 0.99 | 0.99 | 1.00 | 1.02 | 1.02 | 1.03 | 1.05 | 1.07 | 1.07 | 1.04 | 1.08 | 1.11
(0,0,1) | 0.91 | 0.91 | 0.92 | 0.93 | 0.94 | 0.95 | 0.98 | 1.00 | 1.00 | 0.98 | 1.02 | 1.05
(0,1,-1) | 1.14 | 1.14 | 1.16 | 1.19 | 1.20 | 1.21 | 1.22 | 1.23 | 1.23 | 1.20 | 1.21 | 1.24
(0,1,0) | 0.98 | 0.99 | 1.01 | 1.03 | 1.05 | 1.06 | 1.08 | 1.09 | 1.10 | 1.07 | 1.10 | 1.12
(0,1,1) | 0.90 | 0.92 | 0.93 | 0.95 | 0.96 | 0.98 | 1.00 | 1.02 | 1.03 | 1.01 | 1.04 | 1.07
(0,2,-1) | 1.17 | 1.17 | 1.21 | 1.25 | 1.29 | 1.30 | 1.31 | 1.31 | 1.32 | 1.30 | 1.28 | 1.30
(0,2,0) | 1.01 | 1.02 | 1.05 | 1.09 | 1.12 | 1.14 | 1.16 | 1.17 | 1.19 | 1.17 | 1.17 | 1.18
(0,2,1) | 0.92 | 0.94 | 0.96 | 1.01 | 1.03 | 1.06 | 1.08 | 1.09 | 1.11 | 1.10 | 1.11 | 1.12
(1,0,-1) | 1.16 | 1.13 | 1.12 | 1.09 | 1.07 | 1.05 | 1.04 | 1.04 | 1.02 | 1.02 | 1.01 | 1.01
(1,0,0) | 1.00 | 0.99 | 0.97 | 0.94 | 0.93 | 0.92 | 0.91 | 0.91 | 0.90 | 0.91 | 0.91 | 0.92
(1,0,1) | 0.92 | 0.91 | 0.89 | 0.87 | 0.85 | 0.84 | 0.85 | 0.85 | 0.84 | 0.85 | 0.86 | 0.86
(1,1,-1) | 1.15 | 1.14 | 1.13 | 1.11 | 1.10 | 1.08 | 1.07 | 1.06 | 1.05 | 1.05 | 1.03 | 1.03
(1,1,0) | 0.99 | 0.99 | 0.98 | 0.96 | 0.95 | 0.94 | 0.94 | 0.94 | 0.93 | 0.94 | 0.94 | 0.93
(1,1,1) | 0.91 | 0.91 | 0.90 | 0.88 | 0.87 | 0.87 | 0.87 | 0.87 | 0.87 | 0.88 | 0.88 | 0.88
(1,2,-1) | 1.18 | 1.17 | 1.17 | 1.17 | 1.18 | 1.18 | 1.16 | 1.14 | 1.14 | 1.15 | 1.11 | 1.09
(1,2,0) | 1.02 | 1.01 | 1.01 | 1.02 | 1.03 | 1.03 | 1.02 | 1.01 | 1.02 | 1.03 | 1.01 | 0.99
(1,2,1) | 0.93 | 0.94 | 0.93 | 0.94 | 0.94 | 0.95 | 0.94 | 0.94 | 0.95 | 0.97 | 0.95 | 0.93

##  6 Systematic uncertainties

Table 2:  Summary of the systematic uncertainties on $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.80005ptV\\!/}c}$) | $2-3$ | $3-4$ | $4-5$ | $5-6$ | $6-7$ | $7-8$
---|---|---|---|---|---|---
Size of simulation sample | ${}^{+0.0006}_{-0.0005}$ | ${}^{+0.0006}_{-0.0005}$ | ${}^{+0.0007}_{-0.0006}$ | ${}^{+0.0009}_{-0.0009}$ | ${}^{+0.001}_{-0.001}$ | ${}^{+0.002}_{-0.002}$
Photon efficiency | ${}^{+0.011}_{-0.010}$ | ${}^{+0.013}_{-0.011}$ | ${}^{+0.013}_{-0.012}$ | ${}^{+0.016}_{-0.013}$ | ${}^{+0.016}_{-0.013}$ | ${}^{+0.017}_{-0.015}$
Non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction | ${}^{+0.002}_{-0.005}$ | ${}^{+0.003}_{-0.005}$ | ${}^{+0.003}_{-0.006}$ | ${}^{+0.004}_{-0.008}$ | ${}^{+0.005}_{-0.010}$ | ${}^{+0.006}_{-0.011}$
Fit model | ${}^{+0.003}_{-0.003}$ | ${}^{+0.003}_{-0.003}$ | ${}^{+0.002}_{-0.004}$ | ${}^{+0.003}_{-0.005}$ | ${}^{+0.002}_{-0.005}$ | ${}^{+0.003}_{-0.006}$
Simulation calibration | ${}^{+0.010}_{-0.000}$ | ${}^{+0.010}_{-0.000}$ | ${}^{+0.012}_{-0.000}$ | ${}^{+0.012}_{-0.000}$ | ${}^{+0.015}_{-0.000}$ | ${}^{+0.014}_{-0.000}$
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.80005ptV\\!/}c}$) | $8-9$ | $9-10$ | $10-11$ | $11-12$ | $12-13$ | $13-15$
Size of simulation sample | ${}^{+0.002}_{-0.002}$ | ${}^{+0.003}_{-0.003}$ | ${}^{+0.004}_{-0.004}$ | ${}^{+0.006}_{-0.006}$ | ${}^{+0.008}_{-0.008}$ | ${}^{+0.008}_{-0.008}$
Photon efficiency | ${}^{+0.018}_{-0.016}$ | ${}^{+0.020}_{-0.016}$ | ${}^{+0.019}_{-0.016}$ | ${}^{+0.019}_{-0.018}$ | ${}^{+0.021}_{-0.020}$ | ${}^{+0.023}_{-0.019}$
Non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction | ${}^{+0.009}_{-0.011}$ | ${}^{+0.012}_{-0.013}$ | ${}^{+0.011}_{-0.017}$ | ${}^{+0.019}_{-0.019}$ | ${}^{+0.022}_{-0.018}$ | ${}^{+0.018}_{-0.010}$
Fit model | ${}^{+0.002}_{-0.005}$ | ${}^{+0.002}_{-0.003}$ | ${}^{+0.006}_{-0.002}$ | ${}^{+0.001}_{-0.006}$ | ${}^{+0.003}_{-0.008}$ | ${}^{+0.002}_{-0.004}$
Simulation calibration | ${}^{+0.015}_{-0.000}$ | ${}^{+0.017}_{-0.000}$ | ${}^{+0.018}_{-0.000}$ | ${}^{+0.018}_{-0.000}$ | ${}^{+0.017}_{-0.000}$ | ${}^{+0.022}_{-0.000}$

The systematic uncertainties detailed below are measured by repeatedly
sampling from the distribution of the parameter under consideration. For each
sampled value, the cross-section ratio is calculated and the $68.3\%$
probability interval is determined from the resulting distribution.

The statistical errors from the finite number of simulated events used for the
calculation of the efficiencies are included as a systematic uncertainty in
the final results. The uncertainty is determined by sampling the efficiencies
used in Eq. 1 according to their errors. The relative systematic uncertainty
due to the limited size of the simulation sample is found to be in the range
${(0.3}\,{-}\,{3.2)\%}$ and is given for each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in Table
2.

The efficiency extracted from the simulation sample for reconstructing and
selecting a photon in
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decays
has been validated using $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
observed in the data, as described in Sect. 4. The relative uncertainty
between the photon efficiencies measured in the data and simulation, $\pm
14\%$, arises from the finite size of the observed
$B^{+}\rightarrow\chi_{c}K^{+}$ yield and the uncertainty on the known
$B^{+}\rightarrow\chi_{c1}K^{+}$ branching fraction, and is taken to be the
systematic error assigned to the photon efficiency in the measurement of
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$. The
relative systematic uncertainty on the cross-section ratio used in Eq. 1 is
determined by sampling the photon efficiency according to its systematic
error. It is found to be in the range ${(6.4}\,{-}\,{8.7)\%}$ and is given for
each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in
Table 2.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ yield used in Eq. 1 is
corrected for the fraction of non-prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, taken from Ref. [2]. For those
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and rapidity
bins used in this analysis and not covered by Ref. [2]
($13\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,14$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$3.5\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$;
$11\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,13$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$4\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$; and
$14\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), a linear
extrapolation is performed, allowing for asymmetric errors. The systematic
uncertainty on the cross-section ratio is determined by sampling the non-
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction according to a
bifurcated Gaussian function. The relative systematic uncertainty from the
non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction is found to
be in the range ${(1.3}\,{-}\,{10.7)\%}$ and is given for each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in Table
2.

The method used to determine the systematic uncertainty due to the fit
procedure in the extraction of the $\chi_{c}$ yields is discussed in detail in
Ref. [3]. The uncertainty includes contributions from uncertainties on the
fixed parameters, the fit range and the shape of the overall fit function. The
overall relative systematic uncertainty from the fit is found to be in the
range ${(0.4}\,{-}\,{3.2)\%}$ and is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 2.

The systematic uncertainty related to the calibration of the simulation sample
is evaluated by performing the full analysis using simulated events and
comparing to the expected cross-section ratio from simulated signal events.
The results give an underestimate of $10.9\%$ in the measurement of the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ cross-
section ratio. This deviation is caused by non-Gaussian signal shapes in the
simulation which arise from an untuned calorimeter calibration. These are not
seen in the data, which is well described by Gaussian signal shapes. This
deviation is included as a systematic error by sampling from the negative half
of a Gaussian with zero mean and a width of $10.9\%$. The relative uncertainty
on the cross-section ratio is found to be in the range ${(6.3}\,{-}\,{8.2)\%}$
and is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 2. A
second check of the procedure was performed using simulated events generated
according to the distributions observed in the data, i.e. three overlapping
Gaussians and a background shape similar to that in Fig. 1. In this case no
evidence for a deviation was observed. Other systematic uncertainties due to
the modelling of the detector in the simulation are negligible.

In summary, the overall systematic uncertainty is evaluated by simultaneously
sampling the deviation of the cross-section ratio from the central value,
using the distributions of the cross-section ratios described above. The
systematic uncertainty is then determined from the resulting distribution as
described earlier in this section. The separate systematic uncertainties are
shown in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ in Table 2 and the combined uncertainties are shown in Table 3.

##  7 Results and conclusions

The cross-section ratio,
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$,
measured in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ is given in Table 3 and shown in Fig. 4. The measurements are
consistent with, but suggest a different trend to previous results from CDF
using $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV [5] as shown in Fig. 4(a),
and from HERA-$B$ in $p\mathrm{A}$ collisions at $\sqrt{s}=41.6$ GeV, with
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ below roughly
$5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, which gave
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})=0.188\pm 0.013^{+0.024}_{-0.022}$ [4].

Table 3:  Ratio $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and in the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$. The first error is statistical and the second is systematic (apart from the polarisation). Also given is the maximum effect of the unknown polarisations on the results as described in Sect. 5.  $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ | Polarisation effects
---|---|---
$2-3$ | ${0.140}^{+0.005\;+0.015}_{-0.005\;-0.011}$ | ${}^{+0.025}_{-0.014}$
$3-4$ | ${0.160}^{+0.003\;+0.017}_{-0.004\;-0.012}$ | ${}^{+0.028}_{-0.015}$
$4-5$ | ${0.168}^{+0.003\;+0.019}_{-0.003\;-0.012}$ | ${}^{+0.035}_{-0.018}$
$5-6$ | ${0.189}^{+0.004\;+0.021}_{-0.004\;-0.015}$ | ${}^{+0.048}_{-0.025}$
$6-7$ | ${0.189}^{+0.005\;+0.022}_{-0.004\;-0.016}$ | ${}^{+0.054}_{-0.028}$
$7-8$ | ${0.211}^{+0.005\;+0.024}_{-0.005\;-0.017}$ | ${}^{+0.064}_{-0.033}$
$8-9$ | ${0.218}^{+0.007\;+0.026}_{-0.007\;-0.019}$ | ${}^{+0.068}_{-0.034}$
$9-10$ | ${0.223}^{+0.009\;+0.030}_{-0.009\;-0.019}$ | ${}^{+0.070}_{-0.034}$
$10-11$ | ${0.226}^{+0.011\;+0.030}_{-0.011\;-0.022}$ | ${}^{+0.073}_{-0.036}$
$11-12$ | ${0.233}^{+0.013\;+0.034}_{-0.013\;-0.026}$ | ${}^{+0.070}_{-0.036}$
$12-13$ | ${0.252}^{+0.018\;+0.037}_{-0.017\;-0.029}$ | ${}^{+0.071}_{-0.035}$
$13-15$ | ${0.268}^{+0.018\;+0.038}_{-0.017\;-0.025}$ | ${}^{+0.080}_{-0.037}$

Figure 4: Ratio
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in
bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the
range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$. The LHCb results, in
the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,4.5$ and assuming the production of unpolarised
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ mesons, are
shown with solid black circles and the internal error bars correspond to the
statistical error; the external error bars include the contribution from the
systematic uncertainties (apart from the polarisation). The lines surrounding
the data points show the maximum effect of the unknown
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations on
the result. The upper and lower limits correspond to the spin states as
described in the text. The CDF data points, at
$\sqrt{s}\,{=}\,\mbox{${1.8}\>{\mathrm{\,Te\kern-0.90005ptV}}$}$ in $p\bar{p}$
collisions and in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-
rapidity range $|\eta^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}|<1.0$,
are shown in (a) with open blue circles [5]. The two hatched bands in (b)
correspond to the ChiGen Monte Carlo generator prediction [16] and NLO NRQCD
[17].

Theory predictions, calculated in the LHCb rapidity range
$2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$, from
the ChiGen Monte Carlo generator [16] and from the NLO NRQCD calculations [17]
are shown as hatched bands in Fig. 4(b). The ChiGen Monte Carlo event
generator is an implementation of the leading-order colour-singlet model
described in Ref. [18]. However, since the colour-singlet model implemented in
ChiGen does not reliably predict the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section, the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$
prediction uses the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-
section measurement from Ref. [2] as the denominator in the cross-section
ratio.

Figure 4 also shows the maximum effect of the unknown
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations on
the result, shown as lines surrounding the data points. In the first
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin, the upper
limit corresponds to a spin state combination
$(|m_{\chi_{c1}}|,|m_{\chi_{c2}}|,\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ equal to $(1,2,-1)$ and the lower limit to $(0,1,1)$. For all
subsequent bins, the upper and lower limits correspond to the spin state
combinations $(0,2,-1)$ and $(1,0,1)$ respectively.

In summary, the ratio of the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ prompt
production cross-sections is measured using 36$\mbox{\,pb}^{-1}$ of data
collected by LHCb during 2010 at a centre-of-mass energy
$\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-1.00006ptV}}$}$. The results
provide a significant statistical improvement compared to previous
measurements [4, 5]. The results are in agreement with the NLO NRQCD model
[17] over the full range of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. However,
there is a significant discrepancy compared to the leading-order colour-
singlet model described by the ChiGen Monte Carlo generator [16]. At high
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, NLO
corrections fall less slowly with
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and become
important, it is therefore not unexpected that the model lies below the data.
At low $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, the
data appear to put a severe strain on the colour-singlet model.

## Acknowledgments

We would like to thank L. A. Harland-Lang, W. J. Stirling and K.-T. Chao for
supplying the theory predictions for comparison to our data and for many
helpful discussions.

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.

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