11institutetext: Michimasa Inaba 22institutetext: The University of Electro-
Communications, Chofu, Tokyo, Japan. 22email<EMAIL_ADDRESS>33institutetext: Mariko Ukiyo44institutetext: iDEAR Human Support Service,
Shibuya, Tokyo, Japan. 55institutetext: Keiko Takamizor 66institutetext:
Japanese Organization of Mental Health and Educational Agencies, Shibuya,
Tokyo, Japan. 66email<EMAIL_ADDRESS>
# Can Large Language Models be Used to Provide Psychological Counselling?
An Analysis of GPT-4-Generated Responses Using Role-play Dialogues

Michimasa Inaba  Mariko Ukiyo and Keiko Takamizo

###### Abstract

Mental health care poses an increasingly serious challenge to modern
societies. In this context, there has been a surge in research that utilizes
information technologies to address mental health problems, including those
aiming to develop counseling dialogue systems. However, there is a need for
more evaluations of the performance of counseling dialogue systems that use
large language models. For this study, we collected counseling dialogue data
via role-playing scenarios involving expert counselors, and the utterances
were annotated with the intentions of the counselors. To determine the
feasibility of a dialogue system in real-world counseling scenarios, third-
party counselors evaluated the appropriateness of responses from human
counselors and those generated by GPT-4 in identical contexts in role-play
dialogue data. Analysis of the evaluation results showed that the responses
generated by GPT-4 were competitive with those of human counselors.

##  1 Introduction

Mental health care poses an increasingly serious challenge for modern
societies. For example, in Japan, suicide is the leading cause of death among
those aged 10 to 39 WhitePaper , and the World Health Organization has
reported that suicide is a major cause of death among young people globally
who . Consequently, text-based counselling, which refers to providing
psychological support using instant messaging apps, has become increasingly
prominent in recent years. Text-based counselling is more accessible than
telecounselling and e-mail-based counseling, especially for the younger
generation. However, the shortage of counselors proﬁcient in text-based
counseling is a major obstacle. Even those who are experienced in in-person
and e-mail-based counseling, as well as telecounselling, encounter difﬁculties
in providing text-based counseling without appropriate guidance and training.
Moreover, there is a shortage of personnel who can provide such guidance.

Researchers are currently studying the uses of natural language processing for
providing mental health support. Monitoring is an important aspect of mental
health management, and the automatic detection of mental health disorders
represents a particularly active area of current research
gkotsis2017characterisation ; chancellor2020methods ; ji2018supervised ;
nguyen-etal-2022-improving ; saha2022mental .

In the field of dialogue systems, several systems aiming to improve mental
health have been developed. For example, a system known as Woebot
fitzpatrick2017delivering  facilitates cognitive behavioral therapy. There is
also a system that provides emotional support cheng-etal-2023-pal ; liu-
etal-2021-towards , and a system that responds by rewriting less empathetic
utterances as more empathetic statements towards2021 .

Recent developments with large language models (LLM) have demonstrated their
adaptability to various tasks and domains. However, the performance of
counselling dialogue systems that utilize LLMs has not been fully evaluated.
The ChatCounselor liu2023chatcounselor  system is the most similar to the
system used in our study because this LLM-based model was constructed to
answer clients’ questions and requests as a counsellor. However, ChatCounselor
only supports single-turn question-answering and does not support multiturn
dialogue. Furthermore, in the study of ChatCounselor, the generated responses
were evaluated automatically by GPT-4 and not by expert counsellors.

In this study, we constructed a counselling dialogue system using GPT-4, and
professional counsellors evaluated the generated responses. To generate
appropriate responses, we collected counselling dialogue data via role-playing
scenarios involving expert counsellors, and the utterances were annotated with
the intentions of the counsellors. To assess the feasibility of using a
dialogue system in actual counselling situations, third-party counsellors
evaluated the appropriateness of the responses given by the human counsellors
and those generated by the GPT-4 during identical contexts in the role-play
dialogue data.

##  2 The collection of role play dialogue

In order to collect data for analysis, two counsellors participated in role-
playing scenarios, with one counsellor playing the client and the other
playing the counsellor. The dialogues were conducted in Japanese using the
messaging application LINE, and data were collected for a total of six
dialogues, one for each of the six themes shown in Table 1. Table 2 shows the
number of words and utterances, and Table 3 shows a part of the collected
role-play dialogues for Theme 1.

Table 1: List of role-play dialogue themes |  1 |  (本番になると緊張してしまいます、どうしたらいいですか？ (I get nervous at competitions. What should I do?)
---|---|---
|  2 |  パートナーのことは好きだけど束縛が強くてつらい (I love my partner, but it’s hard because she is so restrictive.)
|  3 |  友だちとケンカしてしまいました、どうしたらいいですか？ (I got into a fight with a friend. What should I do?)
|  4 |  勉強する気がありません、どうしたらいいですか？ (I don’t want to study. What should I do?)
|  5 |  男とか女とか決めつけられたくないです (I don’t want to be categorized as male or female.)
|  6 |  生きる意味ってなんだろう？ (What is the meaning of life?)
Table 2: The number of words and utterances from six role-play dialogues |  Utterances |  Words
---|---|---
Client |  1,194 |  15,962
Counsellor |  1,140 |  31,689
GPT-4 generated |  816 |  43,833
Table 3: Example of role-play dialogue and counsellors’ key points and intent annotations for Theme 1\. The abbreviation CLI stands for client, and COU stands for counsellor. The dialogues were conducted originally in Japanese and then translated into English by the authors.  Utterance |  |  Key point |  Intent
---|---|---|---
CLI: |  I am a gymnast. Right before a competition, I get nervous and don’t know what to do. |  |
COU: |  I see. |  Before expressing an opinion or asking a question, first offer an affirmative answer. |  To show that the counsellor is listening to the client.
COU: |  Sometimes when you’re nervous, you feel different than usual. |  |  To convey that it is nothing special.
|  … |  |
CLI: |  By sorting out what things make you nervous, you may be able to find a way to cope. |  Encourage concretization of the cause of the nervousness. |  Because the content touches on personal values, tell the client what is generally said to make it easier for them to accept.
CLI: |  For what? Not because it’s a competition? |  |
COU: |  For example, you may be nervous because everyone is watching you at a competition and you may wonder what they think of you, |  Because the text is long, send it once and send the rest of the text later. |  To provide concrete examples and to make it easier to imagine.

##  3 Counsellor response generation by GPT-4

To analyze whether an LLM can successfully engage in counselling dialogues, we
used the role-play dialogue data to prompt GPT-4 to generate utterances while
assuming the role of a counsellor.

In LLM prompts, chain-of-thought (CoT) prompting wei2022chain  has been found
to be effective at not only generating answers to questions and problems but
also facilitating the process of thinking about them. To perform CoT prompting
and generate higher-quality responses, we annotated the collected counsellor
utterances with the response key points and the intent of the response (Table
3). For themes one through three, the annotations were performed by the
counsellor-role speakers; for themes four through six, this was done by
counsellors who did not participate in the role-play dialogue.

Table 4: The prompt provided to GPT-4 for generation. To facilitate a process
of thinking similar to chain-of-thought prompting, the counselor’s key points
and intentions annotated in the role-play data (the texts enclosed in “[ ]”)
were added before the counselor’s utterances in the dialogue section.
Sentences highlighted in green are GPT-4-generated intentions and responses.
The prompt was originally written in Japanese and translated by the authors. #
Task
---
You provide text-based counseling. As a counselor, you should generate
appropriate responses for your client and the intentions behind them.
Intentions should be written in [ ] before the response.
# Counseling Guidelines
$\cdot$ Show empathy to close distance with the client
$\cdot$ Before expressing an opinion or asking a question, provide an
agreeable response first
$\cdot$ After offering an opinion, check how the client feels about it
$\cdot$ Summarize the client’s concerns and thoughts, and conﬁrm that what the
counselor understands is what the client wanted to talk about
$\cdot$ Communicate a positive perspective that the client may not be aware of
$\cdot$ Provide concrete examples to help the client visualize what is
happening
# Counselling Dialogue
Counsellor: [Greetings] Good morning.
Client: Good morning
Counsellor: [Expresses gratitude for the client’s decision to come for
counselling. The client may have had the courage to come to you for
counselling, and this is to put the client at ease.] Thank you for coming for
counselling today.
Counsellor: [Asks broadly at first.] Please tell me what you would like to
talk about today.
Client: OK
Client: Can I start now?
Counsellor:[rgb]0.8, 1.0, 0.8[Reassure the client that this is an environment
where they could feel comfortable [rgb]0.8, 1.0, 0.8to talk.] Sure. You can
talk about whatever you want. Please feel free to do so.

The prompt we gave to GPT-4 for response generation is shown in Table 4. The
prompt began with instructions telling GPT-4 to generate responses as a
counsellor. Moreover, counsellor guidelines that were developed under the
supervision of professional counsellors were also provided to GPT-4. Next, we
included the ongoing dialogue between the counsellor and the client in the
prompts. The provided dialogue included all text from the start of the
dialogue up until the previous client’s utterance. In addition, the annotated
key points and intentions were added before the counsellor’s utterance in the
dialogue, and the LLMs were instructed to generate them as well. Table 5 shows
the generation examples, and the texts enclosed in “[ ]” are the generated
intent.

We used GPT-4-0613 via OpenAI API, with a temperature of 0.0 and default
values for the other parameters. Numerical details of the generated utterances
are displayed in Table 2. As shown, the number of human counsellor utterances
is higher than the number of utterances generated by GPT-4 because, in the
role-play scenario, human speakers were able to send multiple consecutive
messages. In contrast, GPT-4 generated one utterance at a time, as shown in
Table 5.

Table 5: Examples of intent and response generation results by GPT-4. The score represents the average of the three professional counsellors’ evaluation scores for each generated response. |  Role-play dialogue |  GPT-4 | Score
---|---|---|---
CLI: |  I don’t have many friends, so she’s the one who’s been with me at school |  |
COU: |  I see. |  [showing empathy] I see, your relationship with her is very important to you. [expresses understanding of the client’s feelings] It’s very painful that your relationship with her has turned out this way. | 1.00
CLI: |  Yes. |  |
COU: |  From what you have told me so far, it seems to me that you feel text messages from her are bothersome and that you don’t feel you can talk about anything with her, |  [to see what the client wants to do] What do you most want to do now? | 0.33
COU: |  So I asked you if she is what you call a “friend”. |  |
CLI: |  I feel lonely being alone at school. Don’t you think so? |  |
COU: |  I wonder… Which do you think is better, forcing yourself to be with someone you don’t like very much, or being alone without forcing yourself, even though you might be lonely? |  [to empathize with the client] Sure, it’s lonely to be alone at school, but it’s important to create an environment where you feel comfortable. [to ascertain what the client would like to do] What would you like to do? | 1.66

##  4 Analysis

###  4.1 Subjective evaluation by professional counsellors

To analyze the role-play dialogues and the GPT-4-generated utterances, third-
party counsellors ranked both the utterances of the counsellors made during
the role-play dialogues and the GPT-4-generated utterances. The utterances
were rated according to a 3-point Likert scale from 0 (bad) to 2 (good), with
three professional counsellors per dialogue. The counsellors also provided
written explanations for their evaluations. A total of seven counsellors
conducted evaluations. Table 5 shows examples of generated utterances and the
average scores for each of the three counsellors’ utterances.

We calculated Krippendorff’s alpha for the dialogues of themes one through
three (counselor utterances: 157; GPT-4 generated utterances: 124) in order to
assess the agreement between the counselors’ ratings of the utterances. The
Krippendorff’s alpha was 0.24, indicating a weak correlation.

The mean rating scores for the counsellors’ utterances and GPT-4-generated
utterances were 0.99 (variance: 0.49) and 0.94 (variance: 0.61), respectively.
We also conducted a Mann-Whitney U-test at a significance level of 0.05 and
found no significant differences. The above results indicate that the
difference in response quality between counsellors and GPT-4 is minimal.
Figure 1 shows the ratios of the ratings given to utterances by counsellors
and GPT-4, respectively. As shown, a larger ratio of GPT-4 utterances was
assigned a rating of 0 and 2 compared to the counsellor’s utterances. More
than half of the counsellors’ utterances were given a score of 1; however,
this was because the counsellors’ utterances often included short utterances
that were given a score of 1, such as “I see” and “Yes.”

[scale=0.15]ratio_en.eps

Figure 1: Ratio of rating scores assigned to utterances

The evaluation results confirm that there were individual differences in the
scoring tendencies of the evaluators. Therefore, we analyzed whether the same
evaluator gave higher scores to counsellors’ utterances or GPT-4-generated
utterances when the context was identical. In cases where the counsellor made
multiple consecutive utterances before the client responded, the average
rating for all utterances was used as the rating for the counsellor’s
utterance.

The results of the comparison are shown in Table 6. We found that more
counsellors’ utterances were rated as superior compared to GPT-4 utterances
that were rated superior (34.8 % vs 30.5%). However, the difference was small,
and 34.7% of the utterances were rated as equal. We also confirmed that the
quality of GPT-4 responses was very close to that of the human counsellors’
responses.

Counsellor win | 284 (34.8%)
---|---
GPT-4 win | 249 (30.5%)
Tie | 283 (34.7%)
Table 6: Comparison of ratings for responses to the same context

We confirmed that LLM-generated responses were competitive with human
responses despite having not used few-shot examples except for given annotated
context in our prompt for GPT-4. Given that it is likely possible to further
enhance the performance of GPT-4, we believe that there is potential for
providing real-world counselling using an LLM-based system.

###  4.2 Case Study

When providing real-world counselling using a dialogue system, it is necessary
to minimize inappropriate responses as much as possible. In this section, we
analyze the responses generated by GPT-4 that were given low ratings.

The majority of responses with low ratings were phrases with inappropriate or
unnatural phrasing and wording. An example of such an utterance is shown in
Table 7. The counsellor who conducted the evaluation commented that the use of
the word “interesting” could cause discomfort for the client because it could
imply that the questions were being asked out of curiosity.

Table 7: Example of a generated response with a low rating due to use of inappropriate wording (Theme 1) Context | Councelor: |  Okay, I see.
---|---|---
| Client: |  I’m also fine when the person before me is performing.
Generated response | GPT-4: |  That is interesting. So do you mean that you get more nervous right before your performance actually starts?

There were also many responses where the GPT-4 response discusses the client’s
problem as if it were somebody else’s problem. An example of such a case is
shown in Table 8. In this example, two of the three evaluators commented that
the response sounds insincere, and one commented that the expression “It looks
difficult” should be avoided.

Table 8: Example of generated response with low rating because it treated the client’s problem as if it were somebody else’s problem (Theme 4) Context | Counsellor: |  If you don’t mind, can you tell me when you feel like you are not studying as well as you would like? (You know what you have to do, but you don’t have the motivation to do it. Or maybe you have a subject that you are not good at.)
---|---|---
| Client: |  I can’t concentrate in class, and I usually fall asleep. That’s why I can’t do my homework… I have to study for the entrance exam, but I don’t even know what to do.
Generated response | GPT-4: |  I see. You won’t be able to do your homework because you can’t concentrate in class. It looks difficult that you don’t know what to do with your exam studies.

In particular, counsellors should take great care to avoid risky responses. A
known problem with LLMs is that they can generate sentences with offensive or
discriminatory content zhuo2023red ; deshpande2023toxicity . The evaluators
and the authors confirmed that no offensive or discriminatory statements were
found among the GPT-4-generated utterances. However, a small number of risky
utterances were identified. An example of such an utterance is shown in Table
9. In response to this GPT-4-generated utterance, the evaluator wrote, “The
response that kindness causes one to suffer oneself runs the risk of
inculcating the value that one should not be kind.”

Table 9: Example of generated risky utterance (Theme 2) Context | Counsellor: |  I see. You were thinking of her and putting her first. That is your kindness, and she also made you that way.
---|---|---
| Client: |  Ah, yes. I’ve been told for some time that I’m too kind.
Generated response | GPT-4: |  You are a kind person. But it is also true that your kindness has caused you to suffer.

Although the number of risky responses identified in this study was small,
when input prompts include offensive content, LLMs tend to generate offensive
sentences in response kumar2023certifying . In the role-play dialogues
collected for this study, neither the counsellor nor GPT-4 generated responses
included such content. In the real world, clients might provide responses with
aggressive content. We plan to analyze such cases in future work.

##  5 Conclusion

We collected and annotated role-play counseling dialogue data, and
professional counselors evaluated the appropriateness of GPT-4-generated
responses. Analysis of the results demonstrated that the GPT-4-generated
responses were competitive with those of human counselor responses.
Furthermore, among the responses that received low ratings, no aggressive,
discriminatory, or high-risk responses were identiﬁed.

In this study, we generated and evaluated responses based on the context of
role-play dialogues. No experiments have demonstrated a dialogue system with a
fully automatic counseling dialogue from start to ﬁnish. In future work, we
plan to evaluate whether this system can provide fully automated,
comprehensive counseling services.

## References

  * (1) Stevie Chancellor and Munmun De Choudhury.  Methods in predictive techniques for mental health status on social media: a critical review.  NPJ digital medicine, Vol. 3, No. 1, p. 43, 2020.
  * (2) Jiale Cheng, Sahand Sabour, Hao Sun, Zhuang Chen, and Minlie Huang.  PAL: Persona-augmented emotional support conversation generation.  In Findings of the Association for Computational Linguistics: ACL 2023, pp. 535–554, 2023.
  * (3) Ameet Deshpande, Vishvak Murahari, Tanmay Rajpurohit, Ashwin Kalyan, and Karthik Narasimhan.  Toxicity in chatgpt: Analyzing persona-assigned language models, 2023.
  * (4) Kathleen Kara Fitzpatrick, Alison Darcy, and Molly Vierhile.  Delivering cognitive behavior therapy to young adults with symptoms of depression and anxiety using a fully automated conversational agent (woebot): a randomized controlled trial.  JMIR mental health, Vol. 4, No. 2, p. e7785, 2017.
  * (5) George Gkotsis, Anika Oellrich, Sumithra Velupillai, Maria Liakata, Tim JP Hubbard, Richard JB Dobson, and Rina Dutta.  Characterisation of mental health conditions in social media using informed deep learning.  Scientific reports, Vol. 7, No. 1, p. 45141, 2017.
  * (6) Shaoxiong Ji, Celina Ping Yu, Sai-fu Fung, Shirui Pan, and Guodong Long.  Supervised learning for suicidal ideation detection in online user content.  Complexity, Vol. 2018, , 2018.
  * (7) Aounon Kumar, Chirag Agarwal, Suraj Srinivas, Soheil Feizi, and Hima Lakkaraju.  Certifying llm safety against adversarial prompting, 2023.
  * (8) June M. Liu, Donghao Li, He Cao, Tianhe Ren, Zeyi Liao, and Jiamin Wu.  Chatcounselor: A large language models for mental health support, 2023.
  * (9) Siyang Liu, Chujie Zheng, Orianna Demasi, Sahand Sabour, Yu Li, Zhou Yu, Yong Jiang, and Minlie Huang.  Towards emotional support dialog systems.  In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pp. 3469–3483, 2021.
  * (10) Ministry of Health, Labour and Welfare of Japan.  White paper on suicide prevention, 2020.
  * (11) Thong Nguyen, Andrew Yates, Ayah Zirikly, Bart Desmet, and Arman Cohan.  Improving the generalizability of depression detection by leveraging clinical questionnaires.  In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 8446–8459, 2022.
  * (12) World Health Organization.  World mental health report: Transforming mental health for all, 2022.
  * (13) Tulika Saha, Saichethan Miriyala Reddy, Sriparna Saha, and Pushpak Bhattacharyya.  Mental health disorder identification from motivational conversations.  IEEE Transactions on Computational Social Systems, 2022.
  * (14) Ashish Sharma, Inna W. Lin, Adam S. Miner, David C. Atkins, and Tim Althoff.  Towards facilitating empathic conversations in online mental health support: A reinforcement learning approach.  WWW ’21, p. 194â205. Association for Computing Machinery, 2021.
  * (15) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al.  Chain-of-thought prompting elicits reasoning in large language models.  Advances in Neural Information Processing Systems, Vol. 35, pp. 24824–24837, 2022.
  * (16) Terry Yue Zhuo, Yujin Huang, Chunyang Chen, and Zhenchang Xing.  Red teaming chatgpt via jailbreaking: Bias, robustness, reliability and toxicity, 2023.


# Charge asymmetry in $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process.

Fedor Ignatov  Roman N. Lee  Budker Institute of Nuclear Physics, SB RAS,
Novosibirsk, 630090, Russia

###### Abstract

We consider the charge asymmetry in the differential cross section of the
process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$. Experimental data show large
deviations from the results of the available theoretical calculations.
Motivated by this fact, we revisit the contribution of the two photon exchange
diagrams and find the origin of discrepancy in an oversimplified account of
the pion internal structure in the theoretical calculations. We present a
natural and simple approach that accounts for the pion structure in a more
consistent way and find remarkable agreement with the experimental data.

##  1 Introduction

The $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process is a dominant channel of the hadron
production in the energy range below $\sqrt{s}<1$ GeV. The experimental total
hadron production cross-section defines, via dispersion relation, the
contribution of hadron vacuum polarization to different physical quantities,
in particular, to the effective fine structure constant $\alpha_{eff}(E)$,
Ref. [1], and to the anomalous magnetic moment of the muon
$a_{\mu}=(g_{\mu}-2)/2$, Refs. [2, 3]. There is a long-standing $3\div
4\sigma$ discrepancy between the experimental measurement and the Standard
Model prediction for $a_{\mu}$ which attracts a lot of attention in the
context of searches of New Physics. The $\pi^{+}\pi^{-}$ channel provides a
major part of the hadronic contribution to $a_{\mu}$ being $(506.0\pm
3.4)\cdot 10^{-10}$. It also determines the overall theoretical precision
$4.3\cdot 10^{-10}$ of the $g-2$ muon anomaly [4]. The final precision of the
underway Fermilab experiment [5] is expected to be $\Delta
a^{exp}_{\mu}[E989]\approx 1.6\cdot 10^{-10}$. In order to match this
experimental precision from theoretical side, it is necessary to have 0.2%
overall systematic accuracy in the $\pi\pi$ channel.

In the present paper we consider the charge asymmetry in
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ process. In contrast to the total cross section,
the charge asymmetry is appears entirely due to the radiative corrections. In
the leading order this asymmetry is made up of two contributions. The virtual
contribution comes from the interference of the Born diagram and the diagram
with double photon exchange between lepton line and pion line, see Fig. 1. The
real radiation contribution to the asymmetry comes from the interference of
initial state radiation (ISR) diagrams and final state radiation (FSR)
diagrams. It depends on the experimental cuts which restrict the momentum of
the emitted photon. Note that only the sum of these two contributions is
finite while each of them is infrared divergent. Then, similar to Ref. [6], we
write the charge-odd part of the differential cross section
$\frac{d\sigma_{a}}{dc}$ in the form

$\frac{d\sigma_{a}}{dc}=\frac{d\sigma_{a}^{V}}{dc}+\frac{d\sigma_{a}^{S}(\omega_{0})}{dc}+\frac{d\sigma_{a}^{H}(\omega_{0})}{dc}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{d\sigma_{0}}{dc}\left[\delta_{a}^{V}+\delta_{a}^{S}(\omega_{0})\right]+\frac{d\sigma_{a}^{H}(\omega_{0})}{dc}$
(1)

Here $c=\cos\theta$ with $\theta$ being the angle between the momenta of
$\pi^{-}$ and electron, $\frac{d\sigma_{a}^{V}}{dc}$ is the contribution to
the cross section which comes from the interference between the Born diagram
and the double photon exchange diagrams,
$\frac{d\sigma_{a}^{S}(\omega_{0})}{dc}$ is the real radiation contribution
which comes from the ISR-FSR interference with emitted photon having energy
below $\omega_{0}$ in center-of-mass frame, and
$\frac{d\sigma_{a}^{H}(\omega_{0})}{dc}$ is the similar contribution from the
photons with energy above $\omega_{0}$. It is this last contribution which
depends on experimental cuts.

The Born cross section is

$\frac{d\sigma_{0}}{dc}=\frac{\pi\alpha^{2}}{4s}\beta^{3}(1-c^{2})|F_{\pi}(s)|^{2},$
(2)

where $\beta=\sqrt{1-4m_{\pi}^{2}/s}$ is the velocity of $\pi$-meson.

##  2 Conventional prescription for double photon exchange radiative
corrections

The crucial issue of any channel measurement in $e^{+}e^{-}$ annihilation is
the account of radiative corrections. Of course, for hadronic channels, the
calculation of the radiative corrections necessarily relies on some model.
This is because hadrons are not point-like particles, and the internal
structure of hadrons is governed by strong interactions which do not allow for
perturbative treatment. For the process $e^{+}e^{-}\to\pi^{+}\pi^{-}(\gamma)$
the account of the pion internal structure traditionally has been reduced to
the multiplication of the amplitude in scalar QED (sQED) with point-like pion
by the pion form factor $F_{\pi}(q^{2}=s)$, see, in particular, recent paper
[6]. While this prescription is correct for the tree amplitude of
$e^{+}e^{-}\to\pi^{+}\pi^{-}(\gamma)$ when the emitted photon is soft, it
should be critically scrutinized for other kinematic regions as well as for
other radiative corrections. In particular, there is no reason to expect that
this receipt works well for the diagrams with double photon exchange between
electron and pion lines.

One possible argument in favor of this prescription is that it secures the
cancellation of infrared divergences between the virtual correction and the
real radiative correction. Technically, this cancellation is trivial as within
this approach the inclusive cross section is obtained by multiplying the
inclusive cross section in sQED, already IR finite, by $|F_{\pi}(q^{2})|^{2}$.
However, it is instructive to explain, in a different way, why this simple
prescription correctly accounts for the contribution of soft region
responsible for IR divergence.

Figure 1: Double photon exchange contribution.

Let us consider the double photon exchange amplitude in Fig. 1. Infrared
divergences come from two symmetric integration regions over loop momentum:
$(q_{1}\to 0,\ q_{2}=q-q_{1}\to q)$ and $(q_{2}\to 0,\ q_{1}=q-q_{2}\to q)$.
Let us consider, e.g., the first region. The soft photon with large wavelength
does not “see” the internal structure of the pion. The IR divergent
contributions can be depicted as in Fig. 2. Soft photon (thin wavy lines) is
attached to external legs and the divergence comes from the propagators of
slightly virtual pion and electron, corresponding to internal dashed and solid
lines, respectively. As this pion is close to mass shell, the blob corresponds
to the same factor which appears in the Born cross section — to the form
factor $F_{\pi}(q_{2}^{2})\approx F_{\pi}(q^{2})$.

Figure 2: Schematic picture of soft photon contribution. Thin wavy line
carries momentum whose components are all small.

Thus, the traditional prescription — multiplication of sQED amplitude by
$F_{\pi}(q^{2})$ — correctly accounts the contribution of integration region
responsible for IR divergence. However there is no reason to expect that hard
region is also decently described within this approach.

Let us discuss qualitatively what is going on when we gradually increase the
soft photon momentum $q_{1}$. It it clear that this photon starts to “see” the
internal structure of the pion. What is the characteristic scale of $q_{1}$
when the internal structure becomes essential? Naturally, this is the same
scale at which the pion form factor essentially deviates from $1$, i.e.,
$|q_{1}|\sim m_{\rho}\sim 800$ MeV. What is going on when we have not yet
reached this scale, $|q_{1}|\ll m_{\rho}$? We can still use the approximation
in Fig. 2, however there is a subtlety here: the form factor in these diagrams
depends on $q_{2}^{2}=(q-q_{1})^{2}$ rather than on $q^{2}$. The difference
between $F_{\pi}(q_{2}^{2})$ and $F_{\pi}(q^{2})$ might be essential when
$q^{2}$ is in the vicinity of $m_{\rho}^{2}$. In this region the form factor
changes rather rapidly with the characteristic scale $\delta
q\sim\Gamma_{\rho}\sim 150$ MeV. Of course these qualitative considerations
should be taken with a grain of salt if we recall that $\Gamma_{\rho}$ is only
about $5$ times smaller than $m_{\rho}$. However one can expect that a model
which somehow accounts these considerations will describe the asymmetry better
than the traditional approach in the vicinity of $\sqrt{s}=m_{\rho}$. We shall
see below that this is indeed the case. In the next section we consider a
natural model with these properties.

##  3 Generalized vector dominance in a loop

A natural idea to improve the description of the two-photon contribution is to
use the following modification of sQED Feynman rules:

$\displaystyle\vbox{\hbox{\includegraphics{GVDff2}}}\longrightarrow$
$\displaystyle\quad\vbox{\hbox{\includegraphics{GVDff1}}}=F_{\pi}(q^{2})\times\vbox{\hbox{\includegraphics{GVDff2}}}$
$\displaystyle\vbox{\hbox{\includegraphics{GVDsg2}}}\longrightarrow$
$\displaystyle\quad\vbox{\hbox{\includegraphics{GVDsg1}}}=F_{\pi}(q_{1}^{2})F_{\pi}(q_{2}^{2})\times\\!\\!\\!\\!\\!\\!\\!\vbox{\hbox{\includegraphics{GVDsg2}}}$
(3)

As these graphical notations hint, this modification is inspired by the
generalized vector dominance model (GVD) [7]. In particular, we represent the
pion form-factor as a sum of terms

$F_{\pi}(q^{2})=\sum_{v=1}^{n}a_{v}\frac{\Lambda_{v}}{\Lambda_{v}-q^{2}},$ (4)

where $\Lambda_{v}=m_{v}^{2}-im_{v}\Gamma_{v}$ is a complex parameter defined
via mass and width of the vector meson $v$, and $a_{v}$ are some numbers. From
the identity $F_{\pi}(0)=1$ we have $\sum_{v}a_{v}=1$. Each term in the sum
corresponds to the vector meson propagator multiplied by the coupling constant
of $\gamma v$ vertex. If we allow for sufficiently many terms in $\sum_{v}$
and treat the parameters $\Lambda_{v}$ and $a_{v}$ as fitting parameters, we
can approximate actual form factor with any precision.

In the application to double photon exchange contribution, the modification
(3) corresponds to simply multiplying the _integrand_ of sQED amplitudes by
$F_{\pi}(q_{1}^{2})F_{\pi}(q_{2}^{2})$. Such a multiplication respects gauge
invariance and clearly implements the properties that we discussed in the end
of the previous section.

Thus, we are to calculate the sum of diagrams depicted in Fig. 3.

Figure 3: Diagrams for double photon exchange contribution within our model.

##  4 Technical details and results

First, simple considerations show that there are no collinear divergences in
diagrams in Fig. 3 in the limit of zero electron mass, $m_{e}=0$. This was
already underlined in Ref. [6] for the case of sQED, and our modification
preserves this property. Thus we put electron mass to zero111This is valid up
to power corrections in $m_{e}$, which are negligible for our case.. Then the
contribution of seagull diagram (the rightmost diagram in Fig. 3) can be shown
to vanish. Thus, we are left with the box and cross-box diagrams which differ
by the replacement $t\leftrightarrow u$.

It is convenient to split the contribution into two pieces: that in the
standard sQED approach and the additional contribution

$\delta^{V}=\delta^{V}_{\text{sQED}}+\delta^{V}_{\text{FF}}$ (5)

We use standard technique of multiloop calculations, based on the reduction to
master integrals [8, 9].

Let us start with sQED contribution which is infrared divergent. To regularize
this divergence, we use dimensional regularization, $d=4-2\epsilon$.
Therefore, we can not directly compare our result of
$\delta^{V}_{\text{sQED}}$ with that of [6] as that paper used regularization
with fictitious photon mass. In order to make a comparison, we had to
calculate the soft real radiation contribution coming from ISR-FSR
interference also in dimensional regularization. Fortunately, the
corresponding formulae have been derived in Ref. [10]. Adding up these two
contributions, we obtain

$\delta_{\text{sQED}}^{V}+\delta^{S}=\frac{\alpha}{\pi}\Bigg{[}2\ln\left(\frac{1+\beta
c}{1-\beta
c}\right)\ln{\frac{\sqrt{s}}{2\omega_{0}}}+\frac{1}{2}\ln\left(\frac{1+\beta
c}{1-\beta c}\right)\ln\left(\frac{1+\beta}{1-\beta}\right)\\\ +\frac{(1+\beta
c)^{2}}{\beta^{2}\,\left(1-c^{2}\right)}\ln(1+\beta c)\ln\left(\frac{1+\beta
c}{1-\beta^{2}}\right)+2\mathrm{Li}_{2}\left(\frac{(1+c)\beta}{1+\beta}\right)+2\mathrm{Li}_{2}\left(\frac{(1+c)\beta}{1+c\beta}\right)\\\
+\frac{c\ln^{2}\left(1-\beta^{2}\right)}{2\beta\,(1-c^{2})}+\frac{\left(1+2\beta
c+\beta^{2}\right)}{\beta^{2}\,\left(1-c^{2}\right)}\left(\mathrm{Li}_{2}\left(\frac{1+2\beta
c+\beta^{2}}{\beta^{2}-1}\right)+\frac{\pi^{2}}{12}\right)\\\
+\frac{\left(1+\beta^{2}\right)c}{\beta^{2}\,\left(1-c^{2}\right)}\left(\mathrm{Li}_{2}\left(\frac{\beta-1}{\beta+1}\right)+\frac{1}{4}\ln^{2}\left(\frac{1+\beta}{1-\beta}\right)+\frac{\pi^{2}}{12}\right)\Bigg{]}-\left(c\to-c\right)$
(6)

which is to be compared with the sum of Eqs. (5) and (7) of Ref. [6]. Note
that our result is somewhat simpler. Nevertheless, we find perfect numerical
agreement with arbitrarily high precision, which unambiguously signals the
possibility to reduce the two results to each other using identities between
dilogarithms222In principle, this is a routine task if one uses symbol map
[11], but we omit this check as numerical agreement is pretty convincing..

Let us now pass to the additional contribution. This contribution is obtained
by multiplying the sQED integrand by the factor
$\left[F_{\pi}(q_{1}^{2})F_{\pi}(q_{2}^{2})-F_{\pi}(q^{2})\right]/F_{\pi}(q^{2})$.
This factor vanishes when either of $q_{1,2}$ goes to zero. Therefore, this
additional contribution is IR finite. By power counting, the additional
contribution is, of course, also UV finite. Therefore, for its calculation, it
convenient to use master integrals which remain finite at $d=4$. These master
integrals can be calculated directly using Feynman parametrization in terms of
dilogarithm function. However, it is not trivial to obtain compact explicit
expressions valid for all possible parameters and we prefer to leave a one-
fold representation for two most complicated functions ($g_{4}$ and $g_{8}$,
see below). Then we obtain

$\delta^{V}_{\text{FF}}=\mathrm{Re}\frac{2\alpha}{\pi
s\beta^{2}(1-c^{2})F_{\pi}(s)}\sum_{i,k}a_{i}a_{k}\Bigg{\\{}\frac{t^{2}\,(g_{4}-2g_{2})}{s}-\left(2T^{2}+s\,t+2\Lambda_{i}t\right)\frac{g_{8}}{s}\\\
+\frac{t^{2}g_{1}}{s-\Lambda_{i}}-t\,g_{5}-\frac{2\left(2T^{2}+s\,t+\Lambda_{i}t\right)\left(g_{7}-g_{3}\right)}{\left(s-\Lambda_{i}\right)}-\frac{\left(2T^{2}+s\,t\right)g_{6}}{s-\Lambda_{i}}\Bigg{\\}}-(t\leftrightarrow
u)\,,$ (7)

where $F_{\pi}(s)$ is defined in Eq. (4), $t=m_{\pi}^{2}-(1-\beta c)s/2,\quad
u=m_{\pi}^{2}-(1+\beta c)s/2,\ T=m_{\pi}^{2}-t,c=\cos\theta$, and the
functions $g_{1},\ldots,g_{8}$ are defined in the Appendix. We attach the
corresponding Mathematica and C++ files to the arXiv submission.

##  5 Form factor fits and numerical results for $\delta^{V}_{\text{FF}}$

Figure 4: Pion form factor function used for the $\delta^{V}_{\text{FF}}$
calculation. Points correspond to the values of the model with sum of the
Gounaris-Sakurai resonance parametrization, which is used to describe
experimental $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross-section. Dashed lines
correspond to single BWρ (see description in the text).

Figure 5: Dependence of $\delta^{V}_{\text{FF}}$ at $\theta=1$ rad with
energy. Red line function corresponds to the use of form factor with sum of
three $\rho,\rho^{\prime},\rho^{\prime\prime}$ resonances. Black dashed line
with using single BWρ form factor such that $|F(m_{\rho}^{2})|^{2}$ match
experimental value, gray dashed line by using PDG’s $m_{\rho},\Gamma_{\rho}$
values. Blue dotted line correspond to the sum
$\delta^{V}_{\text{sQED}}+\delta^{S}$ within sQED approach (without
$\ln\frac{\sqrt{s}}{2\omega_{0}}$ term in Eq. (6)).  Figure 6:
$\delta^{V}_{\text{FF}}$ correction versus $\cos\theta$ at $\sqrt{s}=750$ MeV.
Blue dotted line correspond to virtual and soft correction
$\delta^{V}_{\text{sQED}}+\delta^{S}$ from Eq. (6) (without
$\ln\frac{\sqrt{s}}{2\omega_{0}}$ term) within sQED approach.

For the numerical evaluation of the obtained correction, we use the pion form
factor measured in CMD-2 experiments, see Refs. [12, 13]. As explained in
those papers, the measured form factor is accurately described by the
Gounaris-Sakurai parametrization, which we refit by the sum of three Breit-
Wigner (BW) functions, Eq. (4) with $n=3$. The result of this fit is shown in
Fig. 4. As one can see from this figure the parametrization (4) with three BW
functions describes the form factor rather well with typical deviation being
less than 5 percent. In the same figure we also show two simpler
approximations with a single Breit-Wigner function: one, corresponding to both
$m_{\rho}$ and $\Gamma_{\rho}$ taken from PDG (gray curve), another, with the
width reduced by a factor $\approx 1.26$ (black curve), so that the peak value
of the BW function coincides with that of the actual form factor.

Using these Breit-Wigner parametrizations, we have calculated the correction
$\delta^{V}_{\text{FF}}$, Eq. (7), as a function of energy and angle. The
energy dependence of $\delta^{V}_{\text{FF}}(s)$ at $\theta$=1 rad is shown in
Fig. 5. From the left plot of Fig. 5 one can see that the value of
$\delta^{V}_{\text{FF}}$ only slightly depends on the parametrization of the
form factor up to $\sqrt{s}\approx 1$ GeV. On the right plot the energy
dependence is shown for a wide energy range. Note that in the region around
the rho meson mass, the contribution $\delta^{V}_{\text{FF}}$ is a few times
larger than the additive constant at the large logarithm in
$\delta_{\text{sQED}}^{V}+\delta^{S}$ in Eq. (6). One can observe a strong
dependence on the form factor parametrization when $\sqrt{s}>1$ GeV. This
might be considered as the indication that in this region our model loses its
validity.

A typical angular dependence of $\delta^{V}_{\text{FF}}$ is shown in Fig. 6.
Note that this dependence is quite close to linear when $|\cos\theta|<0.5$.

##  6 Comparison with the experiment

The charge asymmetry in the $\pi^{+}\pi^{-}$ channel was studied with the
CMD-3 detector [14, 15]. This analysis includes stable particle separation,
precise fiducial volume determination, theoretical precision of radiative
corrections, etc. Good understanding of the observed angular distributions is
important for investigation and controlling the systematic effects in the
fiducial volume determination. The magnitude of the obtained correction
$\delta^{V}_{\text{FF}}$ to the differential cross section is about $2.5\cdot
10^{-2}$ at $\theta$=1 rad and $m_{\rho}$ peak which should be compared to a
few ppt level of a systematic precision requirement for the
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section measurements as it was described
in the introduction. The measured asymmetry is defined as the difference
between the detected numbers of events to the forward and backward regions of
the detector,

$A=\frac{N_{\theta<\pi/2}-N_{\theta>\pi/2}}{N_{\theta<\pi/2}+N_{\theta>\pi/2}}\,.$
(8)

The following event selection criteria on pion momenta, acceptance angles and
collinearity angle were used in the experiment:

$\displaystyle p^{\pm}>0.45E_{\text{beam}},\quad
1<\theta=\tfrac{1}{2}(\pi-\theta^{+}+\theta^{-})<\pi-1\text{ rad},$
$\displaystyle\left||\phi^{+}-\phi^{-}|-\pi\right|<0.15\text{
rad},\quad|\theta^{+}+\theta^{-}-\pi|<0.25\text{ rad},$ (9)

where $E_{\text{beam}}=\sqrt{s}/2$.

These numbers were corrected for detector effects like inefficiencies,
resolution smearing, etc, so that the measured asymmetry can be directly
compared with the predictions obtained with Monte-Carlo generator. The
calculated $\delta^{V}_{\text{FF}}$ correction was introduced into the MCGPJ
generator [16], which already includes exact NLO to
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ with point-like pion and additional ISR
jets along beam axis using the structure functions approach to partially
account the higher order logarithmically enhanced effects.

Figure 7: Left plot: the measured asymmetry in $\pi^{+}\pi^{-}$ at the CMD-3
in comparison with prediction based on commonly used sQED approach (red dotted
line), and our present approach (blue line). Right plot: difference between
measured number of events and prediction based on the present calculation.

The selection criteria (9) can be roughly translated to photon energy cutoff
$\omega_{0}\lesssim 0.2E_{\text{beam}}$, which corresponds to the magnitude
$\sim 0.02$ for the first term in Eq. (6) at $\theta=1$ rad. This leads to
$\sim 0.01$ contribution to the asymmetry from the ISR-FSR interference within
these experimental cuts.

The obtained experimental result together with theoretical prediction are
shown in Fig. 7. The points correspond to the overall statistics collected to
the present moment with the CMD-3 detector at $\sqrt{s}<1$ GeV. The dotted red
line corresponds to the prediction of the conventional approach based on sQED
calculations. The discrepancy of this prediction with experimental data is
much larger than statistical precision of the latter (more than 40$\sigma$).
The theoretical result for the asymmetry obtained in the present paper is
shown by the blue line in Fig. 7. As it is seen, it agrees well with the
measured asymmetry. The relative difference between the measured asymmetry and
the theoretical result is well within statistical precision, as shown on the
right plot of this figure. The average value of this difference over the range
$\sqrt{s}=700\div 820$ MeV is $(-2.6\pm 2.2)\cdot 10^{-4}$ which is to be
compared to $(104\pm 2.2)\cdot 10^{-4}$ difference for conventional approach.

##  7 Conclusion

In the present paper we consider the charge asymmetry in the process of
$\pi^{+}\pi^{-}$ pair production in the electron-positron annihilation. We
demonstrate that the conventional approach based on sQED calculations is not
valid in the region of $\rho$ peak because of a strong energy dependence of
the pion form factor in this region. As a consequence, this approach is in
clear contradiction with the experimental data for the asymmetry obtained with
CMD-3 detector. We describe a simple and natural approach to the calculation
of the asymmetry based on generalized vector dominance and explain why it
appropriately describes the energy region around $\rho$ mass. The comparison
of the results of this approach with experimental data shows a remarkable
agreement.

The obtained result shows the importance of the appropriate choice of the
model for the calculation of the radiative corrections for the
$\pi^{+}\pi^{-}$ channel. It shows the necessity to critically revisit other
calculations withing sQED approach. We note that this approach was used in
Ref. [17] when considering the NNLO corrections for $\pi^{+}\pi^{-}$ total
cross section measurements within the ISR method.

### Acknowledgement

We are grateful to A. Bondar and A. Milstein for the interest to the work and
valuable discussions.

##  Appendix A Functions $g_{1},\ldots,g_{8}$

$\displaystyle g_{1}$
$\displaystyle=\frac{1}{\beta}\left[2\text{Li}_{2}\left(-\tfrac{1+\beta}{1-\beta}\right)+\tfrac{1}{2}\ln^{2}\left(\tfrac{1+\beta}{1-\beta}\right)+i\pi\ln\left(\tfrac{1+\beta}{1-\beta}\right)+\tfrac{\pi^{2}}{6}\right],$
(10) $\displaystyle g_{2}$
$\displaystyle=\frac{1}{\beta}\bigg{[}\ln{\tfrac{1+\beta_{i}}{1-\beta_{i}}}\
\ln{\tfrac{(1-\beta)\left(\beta_{i}+\beta\right)}{(1+\beta)\left(\beta_{i}-\beta\right)}}+\text{Li}_{2}\left(\tfrac{(1+\beta)\left(1-\beta_{i}\right)}{(1-\beta)\left(1+\beta_{i}\right)}\right)-\text{Li}_{2}\left(\tfrac{(1-\beta)\left(1-\beta_{i}\right)}{(1+\beta)\left(1+\beta_{i}\right)}\right)\Big{]}\,,$
(11) $\displaystyle g_{3}$
$\displaystyle=\text{Li}_{2}\left(\tfrac{t}{m_{\pi}^{2}}\right)-\text{Li}_{2}(1-p_{i})-\text{Li}_{2}(1-q_{i})+\tfrac{\pi^{2}}{6},$
(12) $\displaystyle g_{4}$
$\displaystyle=\intop_{0}^{1}dx\frac{\ln\left[\frac{B-s\bar{x}+\Lambda_{i}-\Lambda_{k}}{B+s\bar{x}+\Lambda_{i}-\Lambda_{k}}\right]+\ln\left[\frac{B-s\bar{x}+\Lambda_{k}-\Lambda_{i}}{B+s\bar{x}+\Lambda_{k}-\Lambda_{i}}\right]}{B/s},$
$\displaystyle B$
$\displaystyle=\sqrt{s^{2}\bar{x}^{2}-2\left(\Lambda_{i}+\Lambda_{k}\right)s\bar{x}+\left(\Lambda_{k}-\Lambda_{i}\right)^{2}-4m_{\pi}^{2}sx^{2}},\quad\bar{x}=1-x,$
(13) $\displaystyle g_{5}$
$\displaystyle=\ln\left(\tfrac{R+\Lambda_{i}-\Lambda_{k}+s}{R+\Lambda_{i}-\Lambda_{k}-s}\right)\ln\left(\tfrac{R-\Lambda_{i}+\Lambda_{k}-s}{R-\Lambda_{i}+\Lambda_{k}+s}\right),\quad
R=\sqrt{\left(s-\Lambda_{i}-\Lambda_{k}\right)^{2}-4\Lambda_{i}\Lambda_{k}}\,,$
(14) $\displaystyle g_{6}$
$\displaystyle=2\text{Li}_{2}\left(\tfrac{t}{m_{\pi}^{2}}\right)+\frac{1}{2}\ln^{2}\left(\tfrac{T^{2}}{m_{\pi}^{2}s}\right)+i\pi\ln\left(\tfrac{T^{2}}{m_{\pi}^{2}s}\right)+\tfrac{\pi^{2}}{6}$
(15) $\displaystyle g_{7}$
$\displaystyle=\ln\left(1-\tfrac{s}{\Lambda_{i}}\right)\ln\left(\tfrac{T^{2}}{m_{\pi}^{2}s}\right)+\text{Li}_{2}\left(1-\tfrac{\Lambda_{i}}{s}\right)+\frac{1}{2}\ln^{2}\left(\tfrac{\Lambda_{i}}{s}\right)+\tfrac{\pi^{2}}{6},$
(16) $\displaystyle g_{8}$
$\displaystyle=\intop_{0}^{\infty}dy\,s\,T\frac{\ln\left[\frac{A+\Lambda_{i}-\Lambda_{k}-s}{A+\Lambda_{i}-\Lambda_{k}+s}\right]+\ln\left[\frac{A+\Lambda_{k}-\Lambda_{i}-s}{A+\Lambda_{k}-\Lambda_{i}+s}\right]}{A\,\left(m_{\pi}^{2}+T\,y\right)},$
$\displaystyle
A=\sqrt{\left(\Lambda_{i}-\Lambda_{k}\right)^{2}-2s\,\left(\Lambda_{i}+\Lambda_{k}+2\Lambda_{i}\Lambda_{k}y\,(1+y)/m_{\pi}^{2}\right)+s^{2}}$
(17) $\displaystyle T$
$\displaystyle=m_{\pi}^{2}-t\,,\quad\beta_{i}=\sqrt{1-4m_{\pi}^{2}/\Lambda_{i}}\,,\quad
p_{i}=\frac{T\left(1+\beta_{i}\right)}{2m_{\pi}^{2}}\,,\quad
q_{i}=\frac{T\left(1-\beta_{i}\right)}{2m_{\pi}^{2}}\,.$ (18)

## References

  * [1] F. Jegerlehner, Precision measurements of sigma(hadronic) for alpha(eff)(E) at ILC energies and (g-2)(mu), Nucl. Phys. B Proc. Suppl. 162 (2006) 22–32.  arXiv:hep-ph/0608329, doi:10.1016/j.nuclphysbps.2006.09.060.
  * [2] A. Keshavarzi, D. Nomura, T. Teubner, Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis, Phys. Rev. D 97 (11) (2018) 114025.  arXiv:1802.02995, doi:10.1103/PhysRevD.97.114025.
  * [3] M. Davier, A. Hoecker, B. Malaescu, Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\alpha(m_{Z}^{2})$, Eur. Phys. J. C 80 (3) (2020) 241, [Erratum: Eur.Phys.J.C 80, 410 (2020)].  arXiv:1908.00921, doi:10.1140/epjc/s10052-020-7792-2.
  * [4] T. Aoyama, et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1–166.  arXiv:2006.04822, doi:10.1016/j.physrep.2020.07.006.
  * [5] J. Grange, et al., Muon (g-2) Technical Design Report (1 2015).  arXiv:1501.06858.
  * [6] A. B. Arbuzov, T. V. Kopylova, G. A. Seilkhanova, Forward–backward asymmetry in electron–positron annihilation into pion or kaon pairs revisited, Mod. Phys. Lett. A 35 (25) (2020) 2050210.  arXiv:2003.14054, doi:10.1142/S0217732320502107.
  * [7] J. J. Sakurai, D. Schildknecht, Generalized vector dominance and inelastic electron - proton scattering, Phys. Lett. B 40 (1972) 121–126.  doi:10.1016/0370-2693(72)90300-0.
  * [8] F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Physics Letters B 100 (1) (1981) 65–68.
URL
http://www.sciencedirect.com/science/article/B6TVN-46YSNNV-109/1/886c25adc81acf1d171b80d7b1e7cb7f

  * [9] K. G. Chetyrkin, F. V. Tkachov, Integration by parts: The algorithm to calculate $\beta$-functions in 4 loops, Nucl. Phys. B 192 (1981) 159.
  * [10] R. N. Lee, Electron-positron annihilation to photons at $O(\alpha^{3})$ revisited, Nucl. Phys. B 960 (2020) 115200.  arXiv:2006.11082, doi:10.1016/j.nuclphysb.2020.115200.
  * [11] C. Duhr, H. Gangl, J. R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075.  arXiv:1110.0458, doi:10.1007/JHEP10(2012)075.
  * [12] R. R. Akhmetshin, et al., High-statistics measurement of the pion form factor in the rho-meson energy range with the CMD-2 detector, Phys. Lett. B 648 (2007) 28–38.  arXiv:hep-ex/0610021, doi:10.1016/j.physletb.2007.01.073.
  * [13] V. M. Aul’chenko, et al., Measurement of the pion form-factor in the range 1.04-GeV to 1.38-GeV with the CMD-2 detector, JETP Lett. 82 (2005) 743–747.  arXiv:hep-ex/0603021, doi:10.1134/1.2175241.
  * [14] B. Khazin, Physics and detectors for VEPP-2000, Nucl. Phys. B Proc. Suppl. 181-182 (2008) 376–380.  doi:10.1016/j.nuclphysbps.2008.09.068.
  * [15] CMD-3 collaboration, $e^{+}e^{-}\to\pi^{+}\pi^{-}$ process analysys with the CMD-3 experiment, paper in preparation.
  * [16] A. B. Arbuzov, G. V. Fedotovich, F. V. Ignatov, E. A. Kuraev, A. L. Sibidanov, Monte-Carlo generator for e+e- annihilation into lepton and hadron pairs with precise radiative corrections, Eur. Phys. J. C 46 (2006) 689–703.  arXiv:hep-ph/0504233, doi:10.1140/epjc/s2006-02532-8.
  * [17] F. Campanario, H. Czyż, J. Gluza, T. Jeliński, G. Rodrigo, S. Tracz, D. Zhuridov, Standard model radiative corrections in the pion form factor measurements do not explain the $a_{\mu}$ anomaly, Phys. Rev. D 100 (7) (2019) 076004\.  arXiv:1903.10197, doi:10.1103/PhysRevD.100.076004.


# UltraCortex: Submillimeter Ultra-High Field 9.4 T Brain MR Image Collection
and Manual Cortical Segmentations

Lucas Mahler  Department High-field Magnetic Resonance, Max Planck Institute
for Biological Cybernetics, Tübingen, Germany  corresponding authors: Lucas
Mahler<EMAIL_ADDRESS>and Julius Steiglechner
<EMAIL_ADDRESS> Julius Steiglechner  Department of
Biomedical Magnetic Resonance, University Hospital Tübingen, Tübingen, Germany
Department High-field Magnetic Resonance, Max Planck Institute for Biological
Cybernetics, Tübingen, Germany  corresponding authors: Lucas Mahler
<EMAIL_ADDRESS>and Julius Steiglechner
<EMAIL_ADDRESS> Benjamin Bender  Department of Diagnostic
and Interventional Neuroradiology, University Hospital Tübingen, Tübingen,
Germany  Tobias Lindig  Department of Diagnostic and Interventional
Neuroradiology, University Hospital Tübingen, Tübingen, Germany  Department
High-field Magnetic Resonance, Max Planck Institute for Biological
Cybernetics, Tübingen, Germany  Dana Ramadan  Department High-field Magnetic
Resonance, Max Planck Institute for Biological Cybernetics, Tübingen, Germany
Jonas Bause  Department of Biomedical Magnetic Resonance, University Hospital
Tübingen, Tübingen, Germany  Department High-field Magnetic Resonance, Max
Planck Institute for Biological Cybernetics, Tübingen, Germany  Florian Birk
Department of Biomedical Magnetic Resonance, University Hospital Tübingen,
Tübingen, Germany  Department High-field Magnetic Resonance, Max Planck
Institute for Biological Cybernetics, Tübingen, Germany  Rahel Heule
Department High-field Magnetic Resonance, Max Planck Institute for Biological
Cybernetics, Tübingen, Germany  Department of Biomedical Magnetic Resonance,
University Hospital Tübingen, Tübingen, Germany  Center for MR Research,
University Children’s Hospital, Zurich, Switzerland  Edyta Charyasz
Department High-field Magnetic Resonance, Max Planck Institute for Biological
Cybernetics, Tübingen, Germany  Michael Erb  Department of Biomedical Magnetic
Resonance, University Hospital Tübingen, Tübingen, Germany  Department High-
field Magnetic Resonance, Max Planck Institute for Biological Cybernetics,
Tübingen, Germany  Vinod Jangir Kumar  Department High-field Magnetic
Resonance, Max Planck Institute for Biological Cybernetics, Tübingen, Germany
Gisela E Hagberg  Department of Biomedical Magnetic Resonance, University
Hospital Tübingen, Tübingen, Germany  Department High-field Magnetic
Resonance, Max Planck Institute for Biological Cybernetics, Tübingen, Germany
Pascal Martin  Department of Neurology and Epileptology, Hertie Institute for
Clinical Brain Research, University of Tübingen, Tübingen, Germany  Gabriele
Lohmann  Department of Biomedical Magnetic Resonance, University Hospital
Tübingen, Tübingen, Germany  Department High-field Magnetic Resonance, Max
Planck Institute for Biological Cybernetics, Tübingen, Germany  Klaus
Scheffler  Department of Biomedical Magnetic Resonance, University Hospital
Tübingen, Tübingen, Germany  Department High-field Magnetic Resonance, Max
Planck Institute for Biological Cybernetics, Tübingen, Germany

###### Abstract

The _UltraCortex_ repository (https://www.ultracortex.org) houses magnetic
resonance imaging data of the human brain obtained at an ultra-high field
strength of 9.4. It contains $86$ structural MR images with spatial
resolutions ranging from [range-units = single]0.60.8. Additionally, the
repository includes segmentations of $12$ brains into gray and white matter
compartments. These segmentations have been independently validated by two
expert neuroradiologists, thus establishing them as a reliable gold standard.
This resource provides researchers with access to high-quality brain imaging
data and validated segmentations, facilitating neuroimaging studies and
advancing our understanding of brain structure and function. Existing
repositories do not accommodate field strengths beyond 7, nor do they offer
validated segmentations, underscoring the significance of this new resource.

## Background & Summary

Here we introduce the _UltraCortex_ repository (https://www.ultracortex.org),
housing magnetic resonance imaging (MRI) data of the human brain obtained at
an ultra-high magnetic field strength of 9.4. The repository comprises $86$
$T_{1}$-weighted structural MR images of healthy human brains, acquired at
spatial resolutions ranging from [range-units = single]0.60.8 using MP-RAGE
and MP2RAGE sequences. Additionally, UltraCortex includes segmentations of
$12$ brains into gray matter (GM) and white matter (WM) compartments. These
segmentations have undergone independent validation by two expert
neuroradiologists, thus establishing them as a reliable gold standard.

While repositories containing MRI data at 7 exist [1, 2, 3], none are
currently available for field strengths beyond 7. Moreover, validated
segmentations of UHF MRI data are presently lacking, but are urgently needed
to support the development of new algorithms and tools for UHF MRI data
analysis, and enable the validation of existing methods. It should be noted
that we refer to field strengths $\geq\qty{7}{}$ as _ultra-high field_ (UHF).

MRI has made a tremendous contribution to our understanding of the human
brain. However, ultra-high resolution brain imaging is constrained by
technical challenges. Image resolution and signal-to-noise ratio (SNR) both
increase with field strength, resulting in improved image quality at a UHF of
9.4[4]. This makes it easier to visualize structures with low inherent signal,
such as small lesions or areas of low proton density. The enhanced spatial
resolution of UHF MRI is beneficial for capturing fine anatomical details
where conventional resolutions of [range-phrase = – , range-units = single]12
are insufficient. The improved tissue contrast of UHF MRI allows for clearer
classification of different tissue types, which is critical when studying
various pathological conditions. In addition, UHF MRI offers enhanced
sensitivity for functional MRI (fMRI), allowing more precise localization of
brain activity and more detailed spectroscopic imaging, which is useful for
studying metabolites in conditions such as Alzheimer’s disease and oncology.

The availability of large neuroimaging databases on open platforms such as
OpenNeuro (https://openneuro.org), Image and Data Archive
(https://ida.loni.usc.edu), and Human Connectome Project[5]
(https://humanconnectome.org) has greatly advanced the field of neuroimaging.
These repositories have not only facilitated the development of imaging
techniques, but have also provided a fertile ground for the development and
application of new algorithms for the analysis of brain images.

However, the majority of existing data analysis algorithms were designed and
optimized for MRI data acquired at lower field strengths, typically 3.0 or
1.5. In structural imaging, a spatial resolution of 1^3 has become the
standard. Consequently, when applied to UHF MRI data, many algorithms use
downsampling, resulting in a loss of the beneficial enhancements provided by
UHF MRI. This mismatch highlights a bottleneck in UHF MRI research: the
underutilization of the detailed data that UHF MRI can provide, constrained by
the limitations of existing analysis tools. The development and validation of
new algorithms rely heavily on access to high-quality public databases, but
there is currently no large public UHF MRI database with field strengths of
greater than 7 and resolutions of smaller than 1. This becomes even more
apparent when considering the availability of manually segmented MR images,
the _gold standard_ for anatomical segmentation. Only a small number of
manually segmented images are available at $\leq\qty{7}{}$[6, 7, 3], and none
are available at 9.4. This limitation has significant implications for brain
analysis, leading to inaccuracies in structural measures such as cortical
volume, thickness, and depth, and thus potentially misleading results.
Furthermore, the application of these imaging techniques to fMRI analysis is
also affected, as errors in the initial segmentation phase can cascade
throughout the analysis pipeline.

With UltraCortex we aim to provide a foundation for the development and
validation of new algorithms for UHF MRI data analysis. It can be used to
develop new algorithms that are applicable across different field strengths
and resolutions, as well as to validate and benchmark existing algorithms. The
data can also be used for image denoising, artifact removal, image quality
enhancement and super-resolution. Additionally, the data can be utilized for
educational purposes, such as to illustrate the intricate details of brain
structures. The following sections describe the initial release of
UltraCortex, including the data acquisition, processing and validation steps.
Figure 1 summarizes the dataset.

## Methods

### Demographics

Subsequent to the original measurements, we collected $86$ $T_{1}$-weighted
structural brain MR images from $78$ healthy adult volunteers ($28$ females
and $50$ males; age range: [range-phrase = – , range-units = single]20.053.0;
median age at acquisition: 29.0; interquartile range of age: 7.8). $6$
subjects were measured in two independent sessions and $1$ subject was
measured in three sessions. The demographic data, including participant sex
and age at the time of scanning, are presented in Table 1. Some of the subject
demographics and image data in this repository have been used in previous
studies [8].

Participants were selected in accordance with the guidelines and ethics of the
Max Planck Institute for Biological Cybernetics, Tübingen, Germany and the
University Hospital, Tübingen, Germany. Prior to enrollment in an MRI study,
all subjects underwent a review of inclusion and exclusion criteria by a
qualified physician. This included ensuring that, unless otherwise specified,
there was no known history of neurological or psychiatric disease and that
there were no contraindications to MRI scans. Additionally, all participants
were required to have normal or corrected-to-normal vision. Prior to
participation, written informed consent was obtained from all participants.
This consent included an agreement for open sharing of anonymized data.

Due to the more rigorous clinical criteria employed in this dataset, $21$
participants were excluded from an original collection of $109$ images from
$101$ subjects. Furhter, $2$ images were excluded due to a failed technical
validation, resulting in the final dataset.

### Data Acquisition

Data acquisition was conducted using a 9.4 whole-body MRI scanner (Siemens
Healthineers, Erlangen, Germany) at the Department for High-field Magnetic
Resonance of the Max Planck Institute for Biological Cybernetics. The scanner
was equipped with a 16-channel dual-row transmit array, operating in CP+
(circularly polarized) mode, and paired with a 31-channel receive array[9].
After $B_{0}$ shimming, whole-brain $T_{1}$-weighted magnetization-prepared
rapid acquisition gradient echo (MP-RAGE)[10] or modified MP-RAGE
(MP2RAGE)[11, 12] images were acquired. Both sequences used a TR-FOCI pulse
optimized for 9.4 for magnetization inversion[13, 12]. Nevertheless, despite
the brief echo time used for the scans, signal dropouts due to strong
$B_{1}$-field non-uniformity close to the ear canals were observed in all
subjects. It was determined that the data regarding the gradient nonlinearity
profile would not be collected, indicating that the correction of gradient
nonlinearity effects would not be conducted, as they play a relatively minor
role within the field of view. It is important to note, however, that
quantitative analysis, such as volumetric measurements, may be affected by
this. The sequence parameters were empirically optimized for the contrast
between GM and WM tissues. An example of typical images is illustrated in
Figure 2. The parameter setups differ in sequence, number of acquired sagittal
slices isotropic voxel size, acquisition matrix, inversion times TI1 and TI2,
repetition time TR, echo time TE, flip angle FA, acceleration factor R of
GRAPPA[14] technology for parallel acquisition and reconstruction, slice
partial Fourier factor $\text{PFF}=6/8$, and optional CAIPI[15] shift in the
GRAPPA technology. An overview of the sequence configurations in use is
presented in Table 2. The comprehensive image-specific scanning parameters can
be accessed via the data records.

#### MP-RAGE

A total of $18$ whole-brain MP-RAGE images were acquired with five different
sequence parameter setups and two different resolutions:  (i) $5$images with
$\text{voxel size}=\qtyproduct[product-units=power]{0.7x0.7x0.7}{\mm}$ and
(ii) $13$images with $\text{voxel size}=\qtyproduct[product-
units=power]{0.6x0.6x0.6}{\mm}$.

#### MP2RAGE

A total of $68$ whole-brain MP2RAGE images were acquired with six distinct
sequence configurations and two different resolutions:  (i) $31$images with
$\text{voxel size}=\qtyproduct[product-units=power]{0.8x0.8x0.8}{\mm}$ and
(ii) $37$images with $\text{voxel size}=\qtyproduct[product-
units=power]{0.6x0.6x0.6}{\mm}$.

### Processing

From each MP2RAGE scan, four distinct volumes were generated: INV1, INV2, UNI,
and T1. The UNI volume represents a composite of the images acquired at both
inversion times (TI1 and TI2), whereas the T1 volume is a $T_{1}$ estimation
map derived from the volumes at TI1 and TI2. Background noise was removed from
the T1 volume using a regularization term during contrast division when the
images from both inversion times were combined [16]. MP2RAGE images were
reconstructed offline using customized routines that use the $B_{1}^{+}$ map
when available to correct for the flip angle variation caused by the shorter
wavelength at 9.4[12]. The remaining $T_{1}$-weighted DICOM
(https://www.dicomstandard.org) files were converted to NIfTI-1
(https://nifti.nimh.nih.gov) format using dcm2niix[17]. The NIfTI files were
then anonymized by first using SynthStrip[18] to extract the brain mask and
then using Quickshear[19] to de-identify the images. In addition, all images
were stripped of any potentially revealing metadata in the NIfTI headers.

### Segmentation

We selected $12$ subjects whose ultra-high field 9.4 MRI scans were of high
quality for detailed manual segmentation. The objective of the segmentation
was to obtain high-precision masks of the cortical ribbon. In accordance with
FreeSurfer [20], the labels shown in Table 3 were segmented. This implies that
subcortical structures, such as the basal ganglia, were added to the “inner”
label, namely the cerebral white matter. However, the hippocampi and amygdalae
were added to the background according to the FreeSurfer ribbon file. For
cortical areas that exhibited signal dropouts, we opted to segment a
topological closed cortex, leading to guessed labels in this region.

At a high level, the segmentation procedure can be described as follows:  (i)
FreeSurfer generation of surrogate segmentations,  (ii) iterative manual
corrections interleaved with expert neuroradiologist review, and  (iii) final
validation of the individual segmentations by two neuroradiologists.  Given
the distinctive characteristics and appearance of 9.4 images, we sought to
engage the expertise of neuroradiologists with extensive experience in
interpreting UHF data acquired at 9.4. Figure 3 illustrates an examplar of the
segmentation, while Figure 4 shows the average necessity for modifying voxel
labels.

#### Initial Segmentation Using FreeSurfer

In order to optimize the efficiency of the segmentation process, we used
FreeSurfer’s v7.1.1 recon-all -hires pipeline[21] as a preliminary step.
However, this approach required adaptations due to the fact that its original
design did not account for the high field strengths of 9.4 and sub-millimeter
resolutions $<\qty{0.8}{\mm}$. The initial phase of the recon-all pipeline,
-autorecon1, includes extensive preprocessing steps such as skull stripping,
motion correction, intensity normalization, and Talairach transformation
computation. Following this phase, the generated brain mask was carefully
examined. Should it be necessary, adjustments were made to the pre-flooding
height of the watershed algorithm, which is employed to determine a boundary
between the brain and the skull. The skull stripping process was repeated
until the desired results were achieved. For minor corrections, particularly
in the dura mater, the convenient -gcut flag was utilized to remove small
portions of dura, followed by manual verification of the segmenters.

After these modifications, the manually corrected brain masks were used as
input to complete the remaining stages of the recon-all pipeline, -autorecon2
-autorecon3, using the -hires flag. The result was a detailed cortical ribbon
mask delineating the segmented volumes of left and right gray and white
matter. This ribbon, converted to NIfTI format, served as the basis for the
subsequent manual segmentation phase.

#### Expert Validation

The manual segmentation procedure commenced with an initial training of the
segmenters. Following successful completion, the iterative labeling was
performed using ITK-Snap[22], allowing for voxel-by-voxel corrections in all
three anatomical planes. Experienced neuroradiologists and neuroscientists
performed regular assessments to verify the accuracy of the tissue
delineations. These reviews also served as guidance for the segmenters,
providing feedback on the quality of their segmentations and highlighting
areas for improvement. Special attention was given to training the segmenters
to accurately determine the boundaries between different tissue classes. The
iterative process continued until the segmentations were deemed satisfactory
by the expert reviewers.

Finally, after the manual corrections were completed and approved in a routine
review, the segmentations underwent a final validation phase. This phase
involved a thorough inspection in the axial, sagittal, and coronal planes to
ensure segmentation consistency, anatomical accuracy, and clear delineation of
tissue boundaries. The final validation was conducted by two expert
neuroradiologists, who independently reviewed the segmentations. Only those
segmentations that successfully passed these stringent validations were
considered suitable for inclusion in this publication. This rigorous
validation process ensured the high quality and reliability of the
segmentations, consistent with the standards required for scientific research
in neuroimaging.

## Data Records

### Repository and Dataset Format

The dataset described here is permanently accessible to the public via
OpenNeuro[23] under the accesssion number ds005216 or via the project page
https://ultracortex.org. The dataset is released under the Creative Commons
license CC0, which allows unrestricted reuse of dataset, including for
commercial purposes. The data used in the study were organized using the Brain
Imaging Data Structure (BIDS)[24]. Consequently, the associated metadata are
stored in files with tab-separated values (TSV) as well as with JavaScript
Object Notation (JSON), which include image-specific information about the
corresponding demographics and measurement parameters. For each subject we
provide the anonymized $T_{1}$-weighted image in NIfTI format. Furthermore, we
provide the SynthStrip-generated brain mask and skull-stripped image, the
FreeSurfer-produced ribbon segmentation that served as the basis for manual
segmentation, and the manual segmentation mask as derivative data. The
detailed data file directory structure is shown in Figure 5.

## Technical Validation

### Artifacts

The process of evaluating artifacts involved a meticulous, two-tier inspection
methodology by MRI experts. Initially, after the acquisition, a qualitative
control was conducted, examining the images for any prominent artifacts. $88$
images that passed this preliminary inspection were placed into a selection
pool. Subsequently, each image within this pool underwent a thorough visual
re-inspection, receiving a binary rating based on its quality.

Images containing significant artifacts, such as those resulting from periodic
involuntary motion, sudden involuntary movements, conscious body movements, or
hardware and sequence-related issues, were excluded from the selection. This
process yielded a final dataset of $86$ images, with the remaining $2$ images
excluded due to the presence of significant artifacts.

To quantitatively assess the severity of motion artifacts, we calculated the
Entropy Focus Criterion EFC, which leverages the Shannon entropy H of voxel
intensities as an indicator of ghosting and blurring due to head motion. The
Shannon entropy of an image with $N$ voxels is defined by the intensity value
$x_{i}$ of voxel $i$ by

$\displaystyle\text{H}=-\sum_{i=1}^{N}\frac{x_{i}}{x_{\text{max}}}\log\left(\frac{x_{i}}{x_{\text{max}}}\right)\;\text{,
where}$ (1) $\displaystyle x_{\text{max}}=\sqrt{\sum_{i=1}^{N}x_{i}^{2}}$

represents the distribution of voxel “brightness”[25]. If all the image energy
were concentrated in a single voxel, we would have the greatest possible voxel
brightness, $x_{\text{max}}$. The EFC is defined in accordance with MRIQC[26]
as H normalized by the maximum entropy, which is the case if all voxels have
the same value. This implies that $x_{i}/x_{\text{max}}=1/\sqrt{N}$, resulting
in

$\text{EFC}=\left(\sqrt{N}\log\sqrt{N}\right)\cdot\text{H}\;.$ (2)

It is preferableto have lower EFC values, with $\text{EFC}=0$ indicating
maximal energy concentration in a single voxel. The mean EFC over all subjects
and given sequence configurations is $0.531(50)$ and a corresponding histogram
including kernel density estimate is shown in Figure 6(a).

### Signal-to-Noise Ratio

To approximate the SNR within the tissue mask, we employed SynthStrip for
brain mask extraction. It is important to note that SynthStrip was not
initially designed for UHF MRI, particularly not for 9.4. Consequently, the
resulting skull-stripped image is only an approximate representation $\hat{m}$
of the actual brain mask $m$. The simplified $\text{SNR}_{\hat{m}}$ within the
approximated brain mask $\hat{m}$ was calculated in accordance with MRIQC by

$\text{SNR}_{\hat{m}}=\frac{\mu_{\hat{m}}}{\sigma_{\hat{m}}}\sqrt{\frac{n_{\hat{m}}-1}{n_{\hat{m}}}}\;,$
(3)

where $\mu_{\hat{m}}$ is the mean intensity of voxels within the brain mask,
$\sigma_{\hat{m}}$ is their standard deviation, and $n_{\hat{m}}$ is the total
number of voxels in the brain mask. The mean $\text{SNR}_{\hat{m}}$ over all
subjects is $0.437(27)$. Figure 6(b) shows a histogram of
$\text{SNR}_{\hat{m}}$ for specific sequence configurations.

### Segmentations

For the $12$ volumes with ground truth segmentations, we conducted additional
quantitative evaluations of image quality. Two key metrics were utilized: the
Coefficient of Joint Variation CJV and the Contrast to Noise Ratio CNR. The
CJV offers insight into the dispersion of tissue intensities relative to their
mean intensity and is expressed as follows:

$\text{CJV}=\frac{\sigma_{\text{WM}}+\sigma_{\text{GM}}}{\left|\mu_{\text{WM}}-\mu_{\text{GM}}\right|}$
(4)

where $\sigma_{\\{\text{WM},\text{GM}\\}}$ and
$\mu_{\\{\text{WM},\text{GM}\\}}$ represent the standard deviation and the
mean intensity of all GM and WM voxels, respectively. A lower CJV value
indicates a clearer differentiation between tissue classes, with minimal
overlap in their intensity distributions.

Furthermore, the CNR metric was employed to assess the difference in intensity
between GM and WM regions relative to the background noise. A higher CNR
indicates better differentiation between tissue classes. CNR is calculated as:

$\text{CNR}=\frac{\left|\mu_{\text{GM}}-\mu_{\text{WM}}\right|}{\sqrt{\sigma^{2}_{\text{B}}+\sigma_{\text{WM}}^{2}+\sigma_{\text{GM}}^{2}}}\;.$
(5)

Here, $\sigma_{\text{B}}$ denotes the standard deviation of all voxels outside
the gray and white matter masks.

The CNR was determinded to be $0.879(390)$, while the CJV was $1.683(1824)$.
These metrics, CJV and CNR, collectively provide a comprehensive assessment of
the segmentation quality in terms of tissue contrast and noise
characteristics, thereby ensuring the reliability of the segmentation process
in our high-resolution 9.4 MRI dataset.

## Usage Notes

In this study, we have curated a dataset optimized for automated processing to
ensure compatibility and ease of integration into various analytical
workflows. The data are provided in widely accepted and well-documented
standard formats, including NIfTI and plain text files. This choice of formats
facilitates seamless integration into various analytical environments and
eliminates reliance on proprietary software. In addition, all data processing
for this publication was performed using open-source software on conventional
computer workstations. This further enhances the accessibility and
reproducibility of our methods.

## Code Availability

All processing pipeline scripts are openly available for download. The code
used to generate processed outputs can be accessed via GitHub at
https://github.com/MPI-Neuroinformatics/ultracortex. Further, open-source
software packages have been used: dcm2niix multiple versions v1.0.<X>
(https://github.com/rordenlab/dcm2niix), FreeSurfer v7.1.1
(https://surfer.nmr.mgh.harvard.edu), ITK-Snap v3.8.0 (http://itksnap.org),
Quickshear v1.1.0 (https://github.com/nipy/quickshear), and SynthStrip as
provided by FreeSurfer v7.1.1
(https://surfer.nmr.mgh.harvard.edu/docs/synthstrip).

## References

  * [1] Forstmann, B. U. _et al._ Multi-modal ultra-high resolution structural 7-Tesla MRI data repository.  _Scientific Data_ 1, 140050, DOI: https://doi.org/10.1038/sdata.2014.50 (2014).
  * [2] Alkemade, A. _et al._ The Amsterdam Ultra-high field adult lifespan database (AHEAD): A freely available multimodal 7 Tesla submillimeter magnetic resonance imaging database.  _NeuroImage_ 221, 117200, DOI: https://doi.org/10.1016/j.neuroimage.2020.117200 (2020).
  * [3] Svanera, M., Benini, S., Bontempi, D. & Muckli, L.  CEREBRUM-7T: Fast and Fully-volumetric Brain Segmentation of 7 Tesla MR Volumes [Data set], DOI: https://doi.org/10.25493/RF12-09N (2020).
  * [4] Pohmann, R., Speck, O. & Scheffler, K.  Signal-to-noise ratio and MR tissue parameters in human brain imaging at 3, 7, and 9.4 tesla using current receive coil arrays.  _Magnetic Resonance in Medicine_ 75, 801–809, DOI: https://doi.org/10.1002/mrm.25677 (2016).
  * [5] Van Essen, D. C. _et al._ The WU-Minn Human Connectome Project: An overview.  _NeuroImage_ 80, 62–79, DOI: https://doi.org/10.1016/j.neuroimage.2013.05.041 (2013).
  * [6] Klein, A. & Tourville, J.  101 Labeled Brain Images and a Consistent Human Cortical Labeling Protocol.  _Frontiers in Neuroscience_ 6, DOI: https://doi.org/10.3389/fnins.2012.00171 (2012).
  * [7] Kuijf, H. J. _et al._ MR Brain Segmentation Challenge 2018 Data, DOI: https://doi.org/10.34894/E0U32Q (2024).
  * [8] Charyasz, E. _et al._ Functional mapping of sensorimotor activation in the human thalamus at 9.4 Tesla.  _Frontiers in Neuroscience_ 17, DOI: https://doi.org/10.3389/fnins.2023.1116002 (2023).
  * [9] Shajan, G. _et al._ A 16-channel dual-row transmit array in combination with a 31-element receive array for human brain imaging at 9.4 T.  _Magnetic Resonance in Medicine_ 71, 870–879, DOI: https://doi.org/10.1002/mrm.24726 (2014).
  * [10] Mugler III, J. P. & Brookeman, J. R.  Rapid three-dimensional T1-weighted MR imaging with the MP-RAGE sequence.  _Journal of Magnetic Resonance Imaging_ 1, 561–567, DOI: https://doi.org/10.1002/jmri.1880010509 (1991).
  * [11] Marques, J. P. _et al._ MP2RAGE, a self bias-field corrected sequence for improved segmentation and T1-mapping at high field.  _NeuroImage_ 49, 1271–1281, DOI: https://doi.org/10.1016/j.neuroimage.2009.10.002 (2010).
  * [12] Hagberg, G. _et al._ Whole brain MP2RAGE-based mapping of the longitudinal relaxation time at 9.4T.  _NeuroImage_ 144, 203–216, DOI: https://doi.org/10.1016/j.neuroimage.2016.09.047 (2017).
  * [13] Hurley, A. C. _et al._ Tailored RF pulse for magnetization inversion at ultrahigh field.  _Magnetic Resonance in Medicine_ 63, 51–58, DOI: 10.1002/mrm.22167 (2010).  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.22167.
  * [14] Griswold, M. A. _et al._ Generalized autocalibrating partially parallel acquisitions (GRAPPA).  _Magnetic Resonance in Medicine_ 47, 1202–1210, DOI: https://doi.org/10.1002/mrm.10171 (2002).
  * [15] Breuer, F. A. _et al._ Controlled aliasing in parallel imaging results in higher acceleration (CAIPIRINHA) for multi-slice imaging.  _Magnetic Resonance in Medicine_ 53, 684–691, DOI: https://doi.org/10.1002/mrm.20401 (2005).
  * [16] O’Brien, K. R. _et al._ Robust T1-Weighted Structural Brain Imaging and Morphometry at 7T Using MP2RAGE.  _PLOS ONE_ 9, e99676, DOI: https://doi.org/10.1371/journal.pone.0099676 (2014).
  * [17] Li, X., Morgan, P. S., Ashburner, J., Smith, J. & Rorden, C.  The first step for neuroimaging data analysis: DICOM to NIfTI conversion.  _Journal of Neuroscience Methods_ 264, 47–56, DOI: https://doi.org/10.1016/j.jneumeth.2016.03.001 (2016).
  * [18] Hoopes, A., Mora, J. S., Dalca, A. V., Fischl, B. & Hoffmann, M.  SynthStrip: skull-stripping for any brain image.  _NeuroImage_ 260, 119474, DOI: https://doi.org/10.1016/j.neuroimage.2022.119474 (2022).
  * [19] Schimke, N. & Hale, J.  Quickshear Defacing for Neuroimages.  In _Proceedings of the 2nd USENIX conference on Health security and privacy_ , HealthSec’11, 11 (USENIX Association, USA, 2011).
  * [20] Fischl, B. _et al._ Whole Brain Segmentation: Automated Labeling of Neuroanatomical Structures in the Human Brain.  _Neuron_ 33, 341–355, DOI: https://doi.org/10.1016/S0896-6273(02)00569-X (2002).
  * [21] Zaretskaya, N., Fischl, B., Reuter, M., Renvall, V. & Polimeni, J. R.  Advantages of cortical surface reconstruction using submillimeter 7 T MEMPRAGE.  _NeuroImage_ 165, 11–26, DOI: https://doi.org/10.1016/j.neuroimage.2017.09.060 (2018).
  * [22] Yushkevich, P. A. _et al._ User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability.  _NeuroImage_ 31, 1116–1128, DOI: https://doi.org/10.1016/j.neuroimage.2006.01.015 (2006).
  * [23] Mahler, L. _et al._ "ultracortex: Submillimeter ultra-high field 9.4t brain mr image collection and manual cortical segmentations", DOI: doi:10.18112/openneuro.ds005216.v1.0.1 (2024).
  * [24] Gorgolewski, K. J. _et al._ The brain imaging data structure, a format for organizing and describing outputs of neuroimaging experiments.  _Scientific Data_ 3, 160044, DOI: https://doi.org/10.1038/sdata.2016.44 (2016).
  * [25] Atkinson, D., Hill, D., Stoyle, P., Summers, P. & Keevil, S.  Automatic correction of motion artifacts in magnetic resonance images using an entropy focus criterion.  _IEEE Transactions on Medical Imaging_ 16, 903–910, DOI: https://doi.org/10.1109/42.650886 (1997).
  * [26] Esteban, O. _et al._ MRIQC: Advancing the automatic prediction of image quality in MRI from unseen sites.  _PLOS ONE_ 12, e0184661, DOI: https://doi.org/10.1371/journal.pone.0184661 (2017).

## Acknowledgements

We would like to express our sincere gratitude to all student assistants who
played an invaluable role in creating the manual segmentations for this
project. Their meticulous efforts, dedication, and attention to detail were
critical to the successful completion of this task. We deeply appreciate their
hard work and dedication, which greatly enhanced the overall quality and
reliability of our research. We gratefully acknowledge funding by the Max
Planck Society. This work was supported by the German Research Council (DFG)
Grant GZ: GR 833/11-1.

## Author Contributions Statement

Participant recruitment, data acquisition, and qualitative data control: J.B.,
F.B., E.C., M.E., G.H., R.H., V.K., P.M., and D.R.; Conception, data
collection, qualitative data control, and cleaning: J.S.; Design: D.R., L.M.,
and J.S.; Data processing and anonymization: L.M. and J.S.; Technical
validation: L.M.; Supervision of the segmentation process: D.R. and L.M.;
Feedback to segmentors and final validation of segmentation: B.B. and T.L.;
Writing of the manuscript: L.M. and J.S.; Supervision, participation in
discussions and suggestions for manuscript improvement: B.B., G.L., and T.L.;
Resources, supervision, and funding acquisition: K.S.; All authors contributed
to the article and approved the submitted version.

## Competing Interests

J.S. – employee of AIRAmed GmbH, unrelated to the content of this manuscript.
B.B. – co-founder, shareholder, and CTO of AIRAmed GmbH, unrelated to the
content of this manuscript. T.L. – co-founder, shareholder, and CEO of AIRAmed
GmbH, unrelated to the content of this manuscript. P.M. – has received
honoraria as an advisory board member from Biogen and Alexion, unrelated to
the content of this manuscript. The other authors declare no competing
interests.

## Figures & Tables

Figure 1: Dataset overview. The chart shows (i) from top to bottom how images were included or excluded for the subsequent steps of the pipeline, and (ii) the final distribution of the dataset across sequences, resolutions, and existing segmentations. Age Group [] | Images [n ()] | Median Age [years (IQR)] | Males [n] | Females [n]
---|---|---|---|---
$2025$ |  $16$ ($19$) |  $22$ ($2223$) | $10$ | $6$
$2530$ |  $35$ ($42$) |  $28$ ($2629$) | $24$ | $11$
$3035$ |  $17$ ($20$) |  $31$ ($3032$) | $14$ | $3$
$3540$ |  $7$ ($8$) |  $35$ ($3537$) | $3$ | $4$
$4045$ |  $7$ ($8$) |  $42$ ($4143$) | $2$ | $5$
$4550$ |  $3$ ($4$) |  $49$ ($4749$) | $0$ | $3$
$5055$ |  $1$ ($1$) |  $53$ ($5353$) | $1$ | $0$
Table 1: Age and sex characteristics of dataset. Figure 2: Example 9.4 MRI slices from subject sub-73 acquired with [product-units = power]0.6 x 0.6 x 0.6. The top row shows from left to right sagittal, coronal, and axial images and ROIs of the corresponding full resolution data in the bottom row. Sequence | Images | Voxel Size [^3] | slices | matrix | TR [] | TE [] | FA [] | R | CAIPI
---|---|---|---|---|---|---|---|---|---
MP-RAGE | 18 |  |  |  |  |  |  |  |
| 4 |  0.7 x 0.7 x 0.7 | 240 |  300 x 306 | $6500$ | $3.4$ | $7.5$ |  2 x 1 | off
| 1 |  0.7 x 0.7 x 0.7 | 256 |  300 x 306 | $6500$ | $3.4$ | $7.5$ |  2 x 1 | off
| 1 |  0.6 x 0.6 x 0.6 | 288 |  352 x 352 | $6000$ | $2.3$ | $8.0$ |  2 x 2 | on
| 4 |  0.6 x 0.6 x 0.6 | 288 |  352 x 352 | $3800$ | $2.5$ | $6.0$ |  2 x 2 | on
| 8 |  0.6 x 0.6 x 0.6 | 288 |  320 x 320 | $3800$ | $2.5$ | $6.0$ |  2 x 2 | on
MP2RAGE | 68 |  |  |  |  |  |  |  |
| 11 |  0.8 x 0.8 x 0.8 | 192 |  256 x 256 | $9000$ | $2.3$ | $4.0$ |  3 x 1 | off
| 20 |  0.8 x 0.8 x 0.8 | 236 |  256 x 256 | $6000$ | $2.3$ | $4.0$ |  3 x 1 | off
| 1 |  0.6 x 0.6 x 0.6 | 256 |  352 x 352 | $6000$ | $3.0$ | $4.0$ |  2 x 2 | on
| 6 |  0.6 x 0.6 x 0.6 | 256 |  352 x 352 | $6000$ | $3.0$ | $5.0$ |  2 x 2 | off
| 27 |  0.6 x 0.6 x 0.6 | 256 |  352 x 352 | $6000$ | $3.0$ | $5.0$ |  2 x 2 | on
| 3 |  0.6 x 0.6 x 0.6 | 260 |  261 x 315 | $6000$ | $3.0$ | $5.0$ |  2 x 2 | on
Table 2: Details of the dataset, outlining the various scanning parameter sets. MR Images acquired at the Department for High-field Magnetic Resonance of the Max Planck Institute for Biological Cybernetics, Tübingen, Germany. Structure | ID | Color | (R, G, B)
---|---|---|---
Background | 0 | black/transparent | (0, 0, 0)
Left cerebral white matter | 2 | sky blue | (86, 180, 133)
Left cerebral cortex | 3 | blueish green | (0, 158, 115)
Right cerebral white matter | 41 | reddish purple | (204, 121, 167)
Right cerebral cortex | 42 | orange | (230, 159, 0)
Table 3: UltraCortex segmentation IDs and mapping to color scheme in Figure 3.
Figure 3: Results of the manually corrected cortical segmentation of Figure 2.
Coloured labels are shown in overlay as described in Table 3. Figure 4:
Average change of voxel labels from FreeSurfer (left) to manual segmentation
(right) across all subjects. Coloured labels are shown in the sankey diagram
as described in Table 3. Cortical labels and background show the largest
differences.

.1 ./ultracortex. .2 dataset_description.json. .2 LICENSE. .2
participants.tsv. .2 README.md. .2 derivatives. .3 brainmasks. .4
sub-<participant ID>_ses-<session ID>_brainmask.nii. .3
freesurfer_segmentation. .4 sub-<participant ID>_ses-<session ID>_seg.nii. .3
manual_segmentation. .4 sub-<participant ID>_ses-<session ID>_seg.nii. .3
skullstrips. .4 sub-<participant ID>_ses-<session ID>_skullstrip.nii. .3
scanning_parameters.json. .3 scanning_parameters.tsv. .2 sub-<participant ID>.
.3 ses-<session ID>. .4 anat. .5 sub-<participant ID>_ses-<session
ID>_T1w.json. .5 sub-<participant ID>_ses-<session ID>_T1w.nii. .2 sub-….

Figure 5: Structure of files and directories.

(a) Histogram of Entropy Focus Criterion (EFC).

(b) Histogram of Signal-to-Noise Ratio (SNR).

Figure 6: Stacked histograms of image quality metrics with kernel density
estimates for MP-RAGE and MP2RAGE sequences.