11institutetext:  a School of Computer Science and Engineering, NTU, Singapore
b School of Natural Sciences, University of Stirling, UK
c Temasek Laboratories, NTU, Singapore
d National Institute of Technology, Warangal, India

# Benchmarking Multimodal Sentiment Analysis

Erik Cambriaa Devamanyu Hazarikad Soujanya Poriac Amir Hussainb
R.B.V. Subramaanyamd

###### Abstract

We propose a framework for multimodal sentiment analysis and emotion
recognition using convolutional neural network-based feature extraction from
text and visual modalities. We obtain a performance improvement of 10% over
the state of the art by combining visual, text and audio features. We also
discuss some major issues frequently ignored in multimodal sentiment analysis
research: the role of speaker-independent models, importance of the modalities
and generalizability. The paper thus serve as a new benchmark for further
research in multimodal sentiment analysis and also demonstrates the different
facets of analysis to be considered while performing such tasks.

###### Keywords:

Multimodal Sentiment Analysis, Emotion Detection, CNN, Deep Learning.

##  1 Introduction

Emotion recognition and sentiment analysis have become a new trend in social
media, avidly helping users to understand opinions expressed in social
networks and user-generated content. With the advancement of communication
technology, abundance of smartphones and the rapid rise of social media, large
amount of data is uploaded by the users as videos rather than text. For
example, consumers tend to record their reviews and opinions on products using
a web camera and upload them on social media platforms such as YouTube or
Facebook to inform subscribers of their views. These videos often contain
comparisons of products from competing brands, the pros and cons of product
specifications, etc., which can aid prospective buyers in making an informed
decision.

The primary advantage of analyzing videos over textual analysis for detecting
emotions and sentiment from opinions is the surplus of behavioral cues. Video
provides multimodal data in terms of vocal and visual modalities. The vocal
modulations and facial expressions in the visual data, along with textual
data, provide important cues to better identify true affective states of the
opinion holder. Thus, a combination of text and video data helps create a
better emotion and sentiment analysis model.

Recently, a number of approaches to multimodal sentiment analysis producing
interesting results have been proposed [1, 2, 3, 4]. However, there are major
issues that remain unaddressed in this field, such as the role of speaker
dependent and independent models, the impact of each modality across datasets,
and generalization ability of a multimodal sentiment classifier. Not tackling
these issues has presented difficulties in effective comparison of different
multimodal sentiment analysis methods.

In this paper, we address some of these issues and, in particular, propose a
novel framework that outperforms the state of the art on benchmark datasets by
more than 10%. We use a deep convolutional neural network to extract features
from visual and text modalities.

The paper is organized as follows: In Section 2 we give a brief literature
review on multimodal sentiment analysis; in Section 3 we present the method;
experimental results and discussion are given in Section 4; finally, Section 6
concludes the paper.

##  2 Related Work

Text-based sentiment analysis systems can be broadly categorized into
knowledge-based and statistics-based systems [5]. While the use of knowledge
bases was initially more popular for the identification of emotions and
polarity in text, sentiment analysis researchers have recently been using
statistics-based approaches, with a special focus on supervised statistical
methods [6, 7].

In 1970, Ekman et al. [8] carried out extensive studies on facial expressions.
Their research showed that universal facial expressions are able to provide
sufficient clues to detect emotions. Recent studies on speech-based emotion
analysis [9] have focused on identifying relevant acoustic features, such as
fundamental frequency (pitch), intensity of utterance, bandwidth, and
duration.

As to fusing audio and visual modalities for emotion recognition, two of the
early works were done by De Silva et al. [10] and Chen et al. [11]. Both works
showed that a bimodal system yielded a higher accuracy than any unimodal
system. More recent research on audio-visual fusion for emotion recognition
has been conducted at either feature level [12] or decision level [13].

While there are many research papers on audio-visual fusion for emotion
recognition, only a few research works have been devoted to multimodal emotion
or sentiment analysis using textual clues along with visual and audio
modalities. Wollmer et al. [2] and Rozgic et al. [14] fused information from
audio, visual and textual modalities to extract emotion and sentiment.
Metallinou et al. [15] and Eyben et al. [16] fused audio and textual
modalities for emotion recognition. Both approaches relied on feature-level
fusion. Wu et al. [17] fused audio and textual clues at decision level.

In this paper, we propose CNN-based framework for feature extraction from
visual and text modality and a method for fusing them for multimodal sentiment
analysis and emotion recognition. Our model outperforms the state of the art.
In addition, we study the behavior of our method in the aspects rarely
addressed by other authors, such as speaker independence, generalizability of
the models and the performance of individual modalities.

##  3 Method

###  3.1 Textual Features

For feature extraction from textual data, we used a convolutional neural
network (CNN). The trained CNN features were then fed into an SVM for
classification, i.e., we used CNN as trainable feature extractor and SVM as a
classifier. (see Figure 1.)

The idea behind convolution is to take the dot product of a vector of $k$
weights $w_{k}$, known as kernel vector, with each $k$-gram in the sentence
$s(t)$ to obtain another sequence of features
$c(t)=(c_{1}(t),c_{2}(t),\ldots,c_{\mathrm{L}}(t))$:

${c}_{j}={w}_{k}^{T}\cdot{\boldsymbol{\mathrm{x}}_{i:i+k-1}}.$ (1)

We then apply a max pooling operation over the feature map and take the
maximum value $\hat{c}(t)=\mathrm{max}\\{\boldsymbol{\mathrm{c}}(t)\\}$ as the
feature corresponding to this particular kernel vector. We used varying kernel
vectors and window sizes to obtain multiple features.

For each word $x_{i}(t)$ in the vocabulary, a $d$-dimensional vector
representation, called word embedding, was given in a look-up table that had
been learned from the data [18]. The vector representation of a sentence was a
concatenation of the vectors for individual words. The convolution kernels are
then applied to word vectors instead of individual words. Similarly, one can
have look-up tables for features other than words if these features are deemed
helpful.

Figure 1: CNN for feature extraction from text modality.

We used these features to train higher layers of the CNN to represent bigger
groups of words in sentences. We denote the feature learned at a hidden neuron
$h$ in layer $l$ as $F^{l}_{h}$. Multiple features are learned in parallel at
the same CNN layer. The features learned at each layer are used to train the
next layer:

$F^{l}={\sum}_{h=1}^{n_{h}}w_{k}^{h}*F^{l-1},$ (2)

where * denotes convolution, $w_{k}$ is a weight kernel for hidden neuron $h$
and $n_{h}$ is the total number of hidden neurons. The CNN sentence model
preserves the order of words by adopting convolution kernels of gradually
increasing sizes, which span an increasing number of words and ultimately the
entire sentence.

Each word in a sentence was represented using word embeddings. We employed the
publicly available word2vec vectors, which were trained on 100 billion words
from Google News. The vectors were of dimensionality $d=300$, trained using
the continuous bag-of-words architecture [18]. Words not present in the set of
pre-trained words were initialized randomly.

Each sentence was wrapped to a window of 50 words. Our CNN had two convolution
layers. A kernel size of 3 and 4, each of them having 50 feature maps was used
in the first convolution layer and a kernel size 2 and 100 feature maps in the
second one. We used ReLU as the non-linear activation function of the network.
The convolution layers were interleaved with pooling layers of dimension 2. We
used the activation values of the 500-dimensional fully-connected layer of the
network as our feature vector in the final fusion process.

###  3.2 Audio Features

We automatically extracted audio features from each annotated segment of the
videos. Audio features were also extracted in 30 Hz frame-rate; we used a
sliding window of 100 ms. To compute the features, we used the open-source
software openSMILE [19]. This toolkit automatically extracts pitch and voice
intensity. Voice normalization was performed and voice intensity was
thresholded to identify samples with and without voice. Z-standardization was
used to perform voice normalization.

The features extracted by openSMILE consist of several Low Level Descriptors
(LLD) and their statistical functionals. Some of the functionals are amplitude
mean, arithmetic mean, root quadratic mean, etc. Taking into account all
functionals of each LLD, we obtained 6373 features.

###  3.3 Visual Features

Since, the video data is very large, we only considered every tenth frame in
our training videos. The Constrained Local Model (CLM) was used to find the
outline of the face in each frame [20]. The cropped frame size was further
reduced by scaling down to a lower resolution, thus creating our new frames
for the video. In this way we could drastically reduce the amount of training
video data. The frames were then passed through a CNN architecture similar to
Figure 1.

Figure 2: Top image segments activated at two feature detectors in the first
layer of deep CNN

A video comprised of a sequence of images. To capture the temporal dependence,
we transformed each pair of consecutive images at $t$ and $t+1$ into a single
image. And provided this transformed image as out input to the multilevel CNN.
We used kernels of varying dimensions to learn Layer-1 2D features (shown in
Figure 2) from the transformed input. Similarly, the second layer also used
kernels of varying dimensions to learn 2D features. Down-sampling layer
transformed features of different kernel sizes into uniform 2D features which
was then followed by a logistic layer of neurons.

Thus, pre-processing involved scaling all video frames to half the resolution.
Each pair of consecutive video frames were converted into a single frame to
achieve temporal convolution features. All the frames were standardized to
$250\times 500$ pixels by padding with zeros.

The first convolution layer contained 100 kernels of size 10$\times$20; the
next convolution layer had 100 kernels of size $20\times 30$; this layer was
followed by a logistic layer of fully connected 300 neurons and a softmax
layer. The convolution layers were interleaved with pooling layers of
dimension $2\times 2$. The activation of the neurons in the logistic layer
were taken as the video features for the classification task.

###  3.4 Fusion

In order to fuse the information extracted from each modality, we concatenated
feature vectors extracted from each modality and sent the combined vector to a
SVM for the final decision. This scheme of fusion is called feature-level
fusion. Since the fusion involved concatenation and no overlapping merge or
combination, scaling and normalization of the features were avoided. We
discuss the results of this fusion in Section 4. The overall architecture of
the proposed method can be seen in Figure 3.

Figure 3: Overall architecture of the proposed method.

##  4 Experiments and Observations

###  4.1 Datasets

####  4.1.1 Multimodal Sentiment Analysis Datasets

For our experiments, we use the MOUD dataset, developed by Perez-Rosas et al.
[1]. They collected 80 product review and recommendation videos from YouTube.
Each video was segmented into its utterances and each utterance was labeled by
a sentiment (positive, negative and neutral). On average, each video has 6
utterances; each utterance is 5 seconds long. The dataset contains 498
utterances labeled either positive, negative or neutral. In our experiment we
did not consider neutral labels, which led to the final dataset consisting of
448 utterances. We dropped the _neutral_ label to maintain consistency with
previous work.

In a similar fashion, Zadeh et al. [21] constructed a multimodal sentiment
analysis dataset called Multimodal Opinion-Level Sentiment Intensity (MOSI),
which is bigger than MOUD, consisting of 2199 opinionated utterances, 93
videos by 89 speakers. The videos address a large array of topics, such as
movies, books, and products. In the experiment to address the generalizability
issues, we trained a model on MOSI and tested on MOUD.

####  4.1.2 Multimodal Emotion Recognition Dataset

The USC IEMOCAP database [22] was collected for the purposes of studying
multimodal expressive dyadic interactions. This dataset contains 12 hours of
video data split into 5 minutes of dyadic interaction between professional
male and female actors. Each interaction session was split into spoken
utterances. At least 3 annotators assigned to each utterance one emotion
category: _happy, sad, neutral, angry, surprised, excited, frustration,
disgust, fear_ and _other_. In this work, we considered only the utterances
with majority agreement (i.e., at least two out of three annotators labeled
the same emotion) in the emotion classes of _angry_ , _happy_ , _sad_ , and
_neutral_. We take only these four classes so as to compare with the state-of-
the-art [23] and other authors.

All the mentioned datasets already contain manually created transcripts of the
conversations or reviews. This might not be the case in a real-time video.
However, with the availability of state of the art speech-to-text softwares,
the task is efficient and trivial.

###  4.2 Speaker-Independent Experiment

Most of the research in multimodal sentiment analysis is performed on a
datasets with speaker overlap in train and test splits. As we know, each
individual is unique in his/her own way of expressing emotions and sentiments,
finding generic, person independent features for sentimental analysis is very
important. However, given this overlap, where the model has already seen the
behaviour of a certain individual, the results do not scale to true
generalization. In real world applications, the model should be robust to
person variance.

Thus, we performed person-independent experiments to emulate unseen
conditions. This time, our train/test splits of the datasets were completely
disjoint with respect to speakers. While testing, our models had to classify
emotions and sentiments from utterances by speakers they have never seen
before. Below, we enlist the procedure of this speaker-independent experiment:

  * •

IEMOCAP: As this dataset contains 10 speakers, we performed a 10 fold speaker
independent test, where in each round, one of the speaker was in the test set.
The same SVM model was used as before and macro F_score was used as a metric.

  * •

MOUD: This dataset contains videos of about 80 people reviewing various
products. Here, reviewers review products in Spanish. Each utterance in the
video has been labeled to be either _positive, negative or neutral_. In our
experiments we consider only the _positive and negative_ sentiment labels. The
speakers were divided into 5 groups and a 5-fold person independent experiment
was run, where in every fold one out of the five group was in the test set.
Finally we took average of the macro F_score to summarize the results (see
Table- 1 ).

  * •

MOSI: The MOSI dataset is a dataset rich in sentimental expressions where 93
people review topics in English. The videos are segmented with each segment’s
sentiment label scored between $+3$ to $-3$ by 5 annotators. We took the
average of these labels as the sentiment polarity thus considering two classes
_positive and negative_ as sentiment labels. Like MOUD, speakers were divided
into 5 groups and a 5-fold person independent experiment was run. During each
fold, around 75 people were in the train set and the remaining in the test
set. The train set was further split randomly into 80%–20% and shuffled to
generate train and validation splits for parameter tuning.

Modality | Source |
---|---|---
IEMOCAP | MOUD | MOSI
Unimodal | A | 51.52 | 53.70 | 57.14
V | 41.79 | 47.68 | 58.46
T | 65.13 | 48.40 | 75.16
Bimodal | T + A | 70.79 | 57.10 | 75.72
T + V | 68.55 | 49.22 | 75.06
A + V | 52.15 | 62.88 | 62.4
Multimodal | T + A + V | 71.59 | 67.90 | 76.66
Table 1: Speaker Independent: Macro F_score reported for speaker independent classification. _IEMOCAP:_ 10-fold speaker independent average. _MOUD:_ 5-fold speaker independent average. _MOSI:_ 5-fold speaker independent average. _Notes:_ A stands for Audio, V for Video, T for Text. Modality | Source | IEMOCAP | MOSI
---|---|---|---
Unimodal | Audio | 66.20 | 64.00
Video | 60.30 | 62.11
Text | 67.90 | 78.00
Bimodal | Text + Audio | 78.20 | 76.60
Text + Video | 76.30 | 78.80
Audio + Video | 73.90 | 66.65
Multimodal | Text + Audio + Video | 81.70 | 78.80
| Text + Audio + Video | 69.351 | 73.552
Table 2: Speaker Dependent: Ten-fold cross-validation results on IEMOCAP
dataset and 5-fold CV results (macro F_Score) on MOSI dataset. 1By [23]; 2by
[3].

####  4.2.1 Comparison with the Speaker Dependent Experiment

In comparison to speaker dependent experiment, speaker independent experiment
performs poor. This is due to the lack of knowledge about speakers in the
dataset. Table 2 shows the performance obtained in the speaker dependent
experiment. It can be seen that audio modality consistently performs better
than visual modality in both MOSI and IEMOCAP datasets. The text modality
plays the most important role in both emotion recognition and sentiment
analysis. The fusion of the modalities show more impact for emotion
recognition than on sentiment analysis. RMSE and TP-rate of the experiments
using different modalities on IEMOCAP and MOSI datasets are shown in Figure 4.

Figure 4: Experiments on IEMOCAP and MOSI datasets. Top left figure shows the
Root Mean Square Error (RMSE) of the models on IEMOCAP and MOSI. Top right
figure shows the dataset distribution. Bottom left and bottom right figure
present TP-rate on of the models on IEMOCAP and MOSI dataset respectively.

###  4.3 Contributions of the Modalities

As expected in all kinds of experiments, bimodal and trimodal models have
performed better than unimodal models. Overall, audio modality has performed
better than visual on all the datasets. Except the MOUD dataset, the unimodal
performance of text modality is notably better than other two modalities; see
Figure 5. Table 2 also presents the comparison with state of the art. The
present method outperformed state of the art by 12% and 5% respectively on the
IEMOCAP and MOSI datasets.111We have reimplemented the method by Poria et al.
[3] The method proposed by Poria et al. is similar to us except they used a
standard CLM based facial feature extraction method. So, our proposed CNN
based visual feature extraction algorithm has helped to outperform the method
by Poria et al.

Figure 5: Performance of the modalities on the datasets. Red line indicates the median of the F_score. Modality | Source | Macro F_Score
---|---|---
Unimodal | Audio | 41.60 $\%$
Video | 45.50 $\%$
Text | 50.89 $\%$
Bimodal | Text + Audio | 51.70 $\%$
Text + Video | 52.12 $\%$
Audio + Video | 46.35 $\%$
Multimodal | Text + Audio + Video | 52.44 $\%$
Table 3: Cross dataset results: Model (with previous configurations) trained
on MOSI dataset and tested on MOUD dataset.

###  4.4 Generalizability of the Models

To test the generalization ability of the models we have trained framework on
MOSI dataset in speaker independent fashion and tested on MOUD dataset. From
Table 3 we can see that the trained model on MOSI dataset performed poorly on
MOUD dataset. While harvesting the reason for it, we have found mainly two
major issues. First, reviews in MOUD dataset had been recorded in Spanish so
audio modality miserably fail in recognition as MOSI dataset contains reviews
in English. Second, text modality has performed very poor, too, for the same
reason. A more comprehensive study would be to perform generalizability tests
on datasets in the same language. However we were unable to do this owing to
the lack of benchmark datasets.

Also, similar experiments of cross dataset generalization was not performed on
emotion detection given the availability of only a single dataset - IEMOCAP.

###  4.5 Visualization of the Datasets

The MOSI visuals (see Figure 6) present information regarding dataset
distribution within single and multiple modalities. For the textual and audio
modalities, comprehensive clustering can be seen with substantial overlap.
However, this problem is reduced in the video and all modalities with
structured declustering but overlap is reduced only in multimodal. This forms
an intuitive explanation of the improved performance in the multimodality.

The IEMOCAP visualizations (see Figure 6) provide insight for the 4 class
distribution for uni and multimodals, where clearly, the multimodal
distribution has the least overlap (increase in red and blue visuals, apart
from the rest) with sparse distribution aiding the classification process.

Figure 6: T-SNE 2D visualization of MOSI and IEMOCAP datasets when unimodal
features and multimodal features are used.

##  5 Qualitative Analysis

In order to have a better understanding on roles of modalities for overall
classification, we have manually done some qualitative analysis. Here we show
the cases where our model successfully comprehends the semantics of the
utterances and with aid from the multiple media, correctly classifies the
emotion of the same.

While over-viewing the correctly classified utterances in the validation set,
we found that text modality often helped classification of utterances where
visual and audio cues are flat with less variance. The model, in such
situations, gathered information from the language semantics extracted by the
text modality. For example, in an utterance from the MOSI dataset - ”amazing
special effects”, there was no jest of enthusiasm in speaker’s voice and face
audio-visual classifier, which caused failure to identify the positivity of
this utterance by the audio and video unimodal classifiers. On the other
textual classifier given the presence of highly polar words, correctly
detected the polarity as positive and helped the bi and multimodal classifiers
for correct classification.

Text modality also helped in situations where the face of the reviewer was not
prominent. This result is promising since in many reviews, often the video
diverges from the face of the reviewer to other images of products or
references.

However, in some utterances, text modality misclassified due to the presence
of misleading linguistic cues. But, the overall classification was correct
given the indicative hints from the audio and video inputs. For example,
textual classifier classified this sentence - ”that like to see comic book
characters treated responsibly” as positive, possibly because of the presence
of positive phrases such as ”like to see”, ”responsibly”. However, the high
pitch of anger in the person’s voice and the frowning face helps identify this
to be a negative utterance.

The above examples demonstrates the effectiveness and robustness of our model
to capture overall video semantics of the utterances for emotion and sentiment
detection. It also shows how bi and multimodal models, given the multiple
media input, overcomes the limitations of unimodal networks.

We also explored the misclassified validation utterances and found some
interesting trends. A video is constituent of a group of utterances which have
contextual dependencies among them. Thus, our model failed to classify
utterances whose emotional polarity was highly dependent on the context
described in earlier or later part of the video. However, such interdependent
modeling was out of the scope of this paper and we therefore enlist it as a
future work.

##  6 Conclusion

We have presented a framework for multimodal sentiment analysis and multimodal
emotion recognition, which outperforms the state of the art in both tasks by a
significant margin. Apart from that, we also discuss some major aspects of
multimodal sentiment analysis problem such as the performance of speaker-
independent models and cross dataset performance of the models.

Our future work will focus on extracting semantics from the visual features,
relatedness of the cross-modal features and their fusion. We will also include
contextual dependency learning in our model to overcome limitations mentioned
in Section 5. Our framework is available as a demo on
http://148.204.64.164/.222Best viewed in Mozilla Firefox

## References

  * [1] Pérez-Rosas, V., Mihalcea, R., Morency, L.P.:  Utterance-level multimodal sentiment analysis.  In: ACL (1). (2013) 973–982
  * [2] Wollmer, M., Weninger, F., Knaup, T., Schuller, B., Sun, C., Sagae, K., Morency, L.P.:  Youtube movie reviews: Sentiment analysis in an audio-visual context.  IEEE Intelligent Systems 28 (2013) 46–53
  * [3] Poria, S., Cambria, E., Gelbukh, A.:  Deep convolutional neural network textual features and multiple kernel learning for utterance-level multimodal sentiment analysis.  In: Proceedings of EMNLP. (2015) 2539–2544
  * [4] Zadeh, A.:  Micro-opinion sentiment intensity analysis and summarization in online videos.  In: Proceedings of the 2015 ACM on International Conference on Multimodal Interaction, ACM (2015) 587–591
  * [5] Cambria, E.:  Affective computing and sentiment analysis.  IEEE Intelligent Systems 31 (2016) 102–107
  * [6] Pang, B., Lee, L., Vaithyanathan, S.:  Thumbs up?: sentiment classification using machine learning techniques.  In: Proceedings of ACL, Association for Computational Linguistics (2002) 79–86
  * [7] Socher, R., Perelygin, A., Wu, J.Y., Chuang, J., Manning, C.D., Ng, A.Y., Potts, C.:  Recursive deep models for semantic compositionality over a sentiment treebank.  In: Proceedings of EMNL). Volume 1631., Citeseer (2013) 1642
  * [8] Ekman, P.:  Universal facial expressions of emotion.  Culture and Personality: Contemporary Readings/Chicago (1974)
  * [9] Datcu, D., Rothkrantz, L.:  Semantic audio-visual data fusion for automatic emotion recognition.  Euromedia’2008 (2008)
  * [10] De Silva, L.C., Miyasato, T., Nakatsu, R.:  Facial emotion recognition using multi-modal information.  In: Proceedings of ICICS. Volume 1., IEEE (1997) 397–401
  * [11] Chen, L.S., Huang, T.S., Miyasato, T., Nakatsu, R.:  Multimodal human emotion/expression recognition.  In: Proceedings of the Third IEEE International Conference on Automatic Face and Gesture Recognition, IEEE (1998) 366–371
  * [12] Kessous, L., Castellano, G., Caridakis, G.:  Multimodal emotion recognition in speech-based interaction using facial expression, body gesture and acoustic analysis.  Journal on Multimodal User Interfaces 3 (2010) 33–48
  * [13] Schuller, B.:  Recognizing affect from linguistic information in 3d continuous space.  IEEE Transactions on Affective Computing 2 (2011) 192–205
  * [14] Rozgic, V., Ananthakrishnan, S., Saleem, S., Kumar, R., Prasad, R.:  Ensemble of SVM trees for multimodal emotion recognition.  In: Proceedings of APSIPA ASC, IEEE (2012) 1–4
  * [15] Metallinou, A., Lee, S., Narayanan, S.:  Audio-visual emotion recognition using gaussian mixture models for face and voice.  In: Tenth IEEE International Symposium on ISM 2008, IEEE (2008) 250–257
  * [16] Eyben, F., Wöllmer, M., Graves, A., Schuller, B., Douglas-Cowie, E., Cowie, R.:  On-line emotion recognition in a 3-d activation-valence-time continuum using acoustic and linguistic cues.  Journal on Multimodal User Interfaces 3 (2010) 7–19
  * [17] Wu, C.H., Liang, W.B.:  Emotion recognition of affective speech based on multiple classifiers using acoustic-prosodic information and semantic labels.  IEEE Transactions on Affective Computing 2 (2011) 10–21
  * [18] Mikolov, T., Chen, K., Corrado, G., Dean, J.:  Efficient estimation of word representations in vector space.  arXiv preprint arXiv:1301.3781 (2013)
  * [19] Eyben, F., Wöllmer, M., Schuller, B.:  Opensmile: the munich versatile and fast open-source audio feature extractor.  In: Proceedings of the 18th ACM international conference on Multimedia, ACM (2010) 1459–1462
  * [20] Baltrušaitis, T., Robinson, P., Morency, L.P.:  3d constrained local model for rigid and non-rigid facial tracking.  In: Computer Vision and Pattern Recognition (CVPR), IEEE (2012) 2610–2617
  * [21] Zadeh, A., Zellers, R., Pincus, E., Morency, L.P.:  Multimodal sentiment intensity analysis in videos: Facial gestures and verbal messages.  IEEE Intelligent Systems 31 (2016) 82–88
  * [22] Busso, C., Bulut, M., Lee, C.C., Kazemzadeh, A., Mower, E., Kim, S., Chang, J.N., Lee, S., Narayanan, S.S.:  Iemocap: Interactive emotional dyadic motion capture database.  Language resources and evaluation 42 (2008) 335–359
  * [23] Rozgić, V., Ananthakrishnan, S., Saleem, S., Kumar, R., Prasad, R.:  Ensemble of svm trees for multimodal emotion recognition.  In: Signal & Information Processing Association Annual Summit and Conference (APSIPA ASC), IEEE (2012) 1–4


$\displaystyle-2H_{0,3}(a+1,b+1)+3H_{2,4}(a+1,b+1)+6H_{1,4}(a+1,b+1)+H_{0,4}(a+1,b+1)\big{]}$
$\displaystyle+{2\over Z_{1}Z_{2}}[H_{1,2}(a,b+1)+H_{0,2}(a,b+1)]+{1\over
Z_{1}^{2}}H_{0,0}(a,b)\bigg{]}$

Now, from the explicit formula for the function $H_{n,m}$ in (200), setting
$a=b=(2-\eta)/2$, and reexpressing $Z_{1},Z_{2}$ as functions of
$Z_{\kappa},Z_{g}$ using (192), as well as and $Z_{2}/Z_{1}=s/(1-s)$, where
$s=Z_{\kappa}/Z_{g}$, we obtain the formula (118) for $\pi_{i}(\hbox{\bf q})$
together with the formula (120) and (119) for the amplitudes
$a_{i}(\eta,D,s)$.

##  IV IV. Calculation of the self-energy integrals

In this section we compute the integrals (125) arising in the evaluation of
the self-energy, but using a different parametrization of the propagator as in
(192), i.e. we compute (using the invariance $\hbox{\bf q}\to-\hbox{\bf q}$)

$\sigma^{i}_{\alpha\gamma}({\bf k})\simeq{2\over
d}\int_{q}\,(W_{i})_{\alpha\beta,\gamma\delta}(\hbox{\bf q})\frac{|\hbox{\bf
q}|^{4-D-2\eta}}{|{\bf k}-\hbox{\bf
q}|^{2-\eta}}\left(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf
q})\right)\quad,\quad\frac{1}{Z_{1}}=\frac{1}{Z_{g}}\quad,\quad\frac{1}{Z_{2}}=\frac{1}{Z_{\kappa}}-\frac{1}{Z_{g}}$
(238)

For convenience the result will be first parameterized as follows

$\sigma^{i}_{\alpha\gamma}({\bf
k})=\frac{2}{d}k^{2-\eta}\frac{1}{Z_{2}}(b_{i}^{(1)}\delta_{\alpha\gamma}+b_{i}^{(2)}\hat{k}_{\alpha}\hat{k}_{\gamma})$
(239)

and then at the very end will be translated in the form

$\displaystyle\sigma^{i}_{\alpha\gamma}({\bf
k})=\frac{2}{d}k^{2-\eta}\frac{1}{Z_{\kappa}}(b_{i}^{(1)}(\eta,D,s)P_{\beta\delta}^{T}({\bf
k})+b_{i}(\eta,D,s)P_{\beta\delta}^{L}({\bf k}))$ (240)

which will give the amplitudes $b_{i}^{(1)}(\eta,D,s)$ and $b_{i}(\eta,D,s)$
displayed in (II.6). Note that they differ from the $b_{i}^{(1)}$ and
$b_{i}^{(2)}$ above by some simple factors, and some linear combinations. We
denote everywhere in this calculation

$\displaystyle 2a=2-\eta\quad,\quad 2b=D+2\eta-4$ (241)

We start with

$\displaystyle\sigma^{3}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{D-1}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{T}_{\gamma\delta}(\hbox{\bf
q})(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf q}))\frac{q^{4-D-2\eta}}{({\bf k}+\hbox{\bf
q})^{2-\eta}}=\sigma^{31}_{\alpha\gamma}({\bf
k})+\sigma^{32}_{\alpha\gamma}({\bf k})$ (242)
$\displaystyle\sigma^{31}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{D-1}\frac{1}{Z_{1}}\int_{q}P^{T}_{\alpha\gamma}(\hbox{\bf
q})\frac{1}{({\bf k}+\hbox{\bf q})^{2a}q^{2b}}$ (243)
$\displaystyle\sigma^{32}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{D-1}\frac{1}{Z_{2}}k_{\beta}k_{\delta}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{T}_{\gamma\delta}(\hbox{\bf q})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}$ (244)

We use, in addition to (201)

$\displaystyle
I_{\alpha_{1}\alpha_{2}\alpha_{3}}(a,b)=-k^{D-2a-2b+3}\left[(\hat{k}_{\alpha_{1}}\delta_{\alpha_{2}\alpha_{3}}+\hat{k}_{\alpha_{2}}\delta_{\alpha_{3}\alpha_{1}}+\hat{k}_{\alpha_{3}}\delta_{\alpha_{1}\alpha_{2}})H_{1,3}(a,b)+\hat{k}_{\alpha_{1}}\hat{k}_{\alpha_{2}}\hat{k}_{\alpha_{3}}H_{0,3}(a,b)\right]$

and

$\displaystyle
I_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(a,b)=k^{D-2a-2b+4}\bigg{[}(\delta_{\alpha_{1}\alpha_{2}}\delta_{\alpha_{3}\alpha_{4}}+\delta_{\alpha_{1}\alpha_{3}}\delta_{\alpha_{2}\alpha_{4}}+\delta_{\alpha_{1}\alpha_{4}}\delta_{\alpha_{2}\alpha_{3}})H_{2,4}(a,b)$
(245)
$\displaystyle+(\hat{k}_{\alpha_{1}}\hat{k}_{\alpha_{2}}\delta_{\alpha_{3}\alpha_{4}}+\hat{k}_{\alpha_{1}}\hat{k}_{\alpha_{3}}\delta_{\alpha_{2}\alpha_{4}}+\hat{k}_{\alpha_{1}}\hat{k}_{\alpha_{4}}\delta_{\alpha_{2}\alpha_{3}}+\hat{k}_{\alpha_{2}}\hat{k}_{\alpha_{3}}\delta_{\alpha_{1}\alpha_{4}}+\hat{k}_{\alpha_{2}}\hat{k}_{\alpha_{4}}\delta_{\alpha_{1}\alpha_{3}}+\hat{k}_{\alpha_{3}}\hat{k}_{\alpha_{4}}\delta_{\alpha_{1}\alpha_{2}})H_{1,4}(a,b)$
$\displaystyle+\hat{k}_{\alpha_{1}}\hat{k}_{\alpha_{2}}\hat{k}_{\alpha_{3}}\hat{k}_{\alpha_{4}}H_{0,4}(a,b)\bigg{]}$

We have

$\displaystyle\sigma^{31}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{D-1}\frac{1}{Z_{1}}k^{D-2a-2b}\bigg{(}(H_{0,0}(a,b)-H_{1,2}(a,b+1))\delta_{\alpha\gamma}-\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(a,b+1)\bigg{)}$
(246)

and

$\displaystyle\sigma^{32}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{D-1}\frac{1}{Z_{2}}k_{\beta}k_{\delta}\bigg{(}I_{\alpha\beta\gamma\delta}(a+1,b+2)-I_{\alpha\beta}(a+1,b+1)\delta_{\gamma\delta}-\delta_{\alpha\beta}I_{\gamma\delta}(a+1,b+1)+\delta_{\alpha\beta}\delta_{\gamma\delta}I(a+1,b)\bigg{)}$
$\displaystyle={2\over
d}\frac{1}{D-1}\frac{1}{Z_{2}}k^{D-2a-2b}\bigg{(}\hat{k}_{\alpha}\hat{k}_{\gamma}\big{(}H_{0,0}(a+1,b)-2H_{1,2}(a+1,b+1)-2H_{0,2}(a+1,b+1)$
(247)
$\displaystyle+H_{0,4}(a+1,b+2)+2H_{2,4}(a+1,b+2)+5H_{1,4}(a+1,b+2)\big{)}$
$\displaystyle+\delta_{\alpha\gamma}(H_{2,4}(a+1,b+2)+H_{1,4}(a+1,b+2))\bigg{)}$

We find that

$\displaystyle\hat{k}_{\alpha}\hat{k}_{\gamma}\sigma^{32}_{\alpha\gamma}({\bf
k})={2\over
d}\frac{1}{D-1}\frac{1}{Z_{2}}k^{D-2a-2b}(D^{2}-1)H_{2,4}(a+1,b+2)={2\over
d}\frac{1}{Z_{2}}b_{3}^{\rm crumpling}k^{2-\eta}$ (248)

with $b_{3}^{\rm crumpling}=(D+1)\frac{\Sigma(\eta,D)}{D^{2}-1}$, where we
recall the definition LRReview

$\displaystyle\Sigma(\eta,D)=\frac{\left(D^{2}-1\right)\Gamma(2-\eta)\Gamma\left(\frac{D}{2}+\frac{\eta}{2}\right)\Gamma\left(\eta/2\right)}{4(4\pi)^{D/2}\Gamma\left(2-\frac{{\eta}}{2}\right)\Gamma\left(\frac{D}{2}+\eta\right)\Gamma\left(\frac{D}{2}-\frac{{\eta}}{2}+2\right)}=(D^{2}-1)H_{2,4}(a+1,b+2)$
(249)

With this we find

$\displaystyle
b_{3}^{(2)}=\frac{1}{D-1}\left(\frac{(D-2)D(D+\eta)}{D+\eta-2}+2(\eta-1)-\frac{Z_{2}}{Z_{1}}\frac{2(\eta-2)^{2}(D+2\eta-2)}{D+\eta-2}\right)\frac{\Sigma(\eta,D)}{D^{2}-1}$
(250) $\displaystyle
b_{3}^{(1)}=\frac{1}{D-1}\left(\frac{D+(5-2\eta)\eta-2}{D+\eta-2}-\frac{Z_{2}}{Z_{1}}\frac{(D+2\eta-2)\left(D^{2}+D(\eta-3)+(7-2\eta)\eta-6\right)}{D+\eta-2}\right)\frac{\Sigma(\eta,D)}{D^{2}-1}$
(251)

Note that the sum simplifies into

$\displaystyle
b_{3}^{(1)}+b_{3}^{(2)}=\frac{1}{D-1}\left(D^{2}-1-\frac{Z_{2}}{Z_{1}}(D-1)(-2+D+2\eta)\right)\frac{\Sigma(\eta,D)}{D^{2}-1}$
(252)

Next, we have

$\displaystyle\sigma^{4}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{\sqrt{D-1}}\int_{q}(P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf q})+P^{L}_{\alpha\beta}(\hbox{\bf
q})P^{T}_{\gamma\delta}(\hbox{\bf
q}))(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf q}))\frac{q^{4-D-2\eta}}{({\bf k}+\hbox{\bf
q})^{2-\eta}}=\sigma^{41}_{\alpha\gamma}({\bf
k})+\sigma^{42}_{\alpha\gamma}({\bf k})$
$\displaystyle\sigma^{41}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf
q})k_{\beta}(k_{\delta}+q_{\delta})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}$ (253) $\displaystyle\sigma^{42}_{\alpha\gamma}({\bf
k})={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}\int_{q}P^{L}_{\alpha\beta}(\hbox{\bf
q})P^{T}_{\gamma\delta}(\hbox{\bf
q})(k_{\beta}+q_{\beta})k_{\delta}\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}$ (254)

We have

$\sigma^{42}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}[k_{\beta}k_{\delta}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf q})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}+k_{\delta}\int_{q}P^{T}_{\gamma\delta}(\hbox{\bf
q})q_{\alpha}\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b}}]$ (255)

as well as

$\displaystyle\sigma^{41}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}[k_{\beta}k_{\delta}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf q})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}+k_{\beta}\int_{q}P^{T}_{\alpha\beta}(\hbox{\bf
q})q_{\gamma}\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b}}]$ (256)
$\displaystyle={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}[k_{\beta}k_{\delta}(\delta_{\alpha\beta}I_{\gamma\delta}(a+1,b+1)-I_{\alpha\beta\gamma\delta}(a+1,b+2))+k_{\beta}\delta_{\alpha\beta}I_{\gamma}(a+1,b)-k_{\beta}I_{\alpha\beta\gamma}(a+1,b+1)]$
$\displaystyle={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}k^{D-2a-2b}[\hat{k}_{\beta}\hat{k}_{\delta}\delta_{\alpha\beta}\left[\delta_{\gamma\delta}H_{1,2}(a+1,b+1)+\hat{k}_{\gamma}\hat{k}_{\delta}H_{0,2}(a+1,b+1)\right]$
(257)
$\displaystyle-\hat{k}_{\beta}\delta_{\alpha\beta}\hat{k}_{\gamma}H_{0,1}(a+1,b)$
$\displaystyle+\hat{k}_{\beta}\left[(\hat{k}_{\alpha}\delta_{\beta\gamma}+\hat{k}_{\beta}\delta_{\alpha\gamma}+\hat{k}_{\gamma}\delta_{\alpha\beta})H_{1,3}(a+1,b+1)+\hat{k}_{\alpha}\hat{k}_{\beta}\hat{k}_{\gamma}H_{0,3}(a+1,b+1)\right]$
$\displaystyle-\hat{k}_{\beta}\hat{k}_{\delta}\bigg{[}(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})H_{2,4}(a+1,b+2)$
$\displaystyle+(\hat{k}_{\alpha}\hat{k}_{\beta}\delta_{\gamma\delta}+\hat{k}_{\alpha}\hat{k}_{\gamma}\delta_{\beta\delta}+\hat{k}_{\alpha}\hat{k}_{\delta}\delta_{\beta\gamma}+\hat{k}_{\beta}\hat{k}_{\gamma}\delta_{\alpha\delta}+\hat{k}_{\beta}\hat{k}_{\delta}\delta_{\alpha\gamma}+\hat{k}_{\gamma}\hat{k}_{\delta}\delta_{\alpha\beta})H_{1,4}(a+1,b+2)$
$\displaystyle+\hat{k}_{\alpha}\hat{k}_{\beta}\hat{k}_{\gamma}\hat{k}_{\delta}H_{0,4}(a+1,b+2)\bigg{]}$

This leads to

$\displaystyle\sigma^{41}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{\sqrt{D-1}}\frac{1}{Z_{2}}k^{D-2a-2b}\bigg{[}\hat{k}_{\alpha}\hat{k}_{\gamma}[H_{1,2}(a+1,b+1)+H_{0,2}(a+1,b+1)-H_{0,1}(a+1,b)]$
(258)
$\displaystyle+\delta_{\alpha\gamma}H_{1,3}(a+1,b+1)+\hat{k}_{\alpha}\hat{k}_{\gamma}(2H_{1,3}(a+1,b+1)+H_{0,3}(a+1,b+1))$
(259)
$\displaystyle-\delta_{\alpha\gamma}(H_{2,4}(a+1,b+2)+H_{1,4}(a+1,b+2))-\hat{k}_{\alpha}\hat{k}_{\gamma}(2H_{2,4}(a+1,b+2)+5H_{1,4}(a+1,b+2)+H_{0,4}(a+1,b+2))\bigg{]}$

One observes that $\sigma^{42}_{\alpha\gamma}({\bf
k})=\sigma^{41}_{\alpha\gamma}({\bf k})$, and hence finds

$\displaystyle
b_{4}^{1}=-\frac{2}{\sqrt{D-1}}(2\eta-3)\frac{\Sigma(\eta,D)}{D^{2}-1}$ (260)
$\displaystyle
b_{4}^{2}=\frac{2}{\sqrt{D-1}}D(2\eta-3)\frac{\Sigma(\eta,D)}{D^{2}-1}$ (261)

with which we recover
$b_{4}^{1}+b_{4}^{2}=2\sqrt{D-1}(2\eta-3)\frac{\Sigma(\eta,D)}{D^{2}-1}=b_{4}^{\rm
crumpling}$. Next one has

$\displaystyle\sigma^{5}_{\alpha\gamma}({\bf k})={2\over
d}\int_{q}P^{L}_{\alpha\beta}(\hbox{\bf q})P^{L}_{\gamma\delta}(\hbox{\bf
q})(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf q}))\frac{q^{4-D-2\eta}}{({\bf k}+\hbox{\bf
q})^{2-\eta}}=\sigma^{51}_{\alpha\gamma}({\bf
k})+\sigma^{52}_{\alpha\gamma}({\bf k})$ (262)
$\displaystyle\sigma^{51}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{Z_{1}}\int_{q}P^{L}_{\alpha\gamma}(\hbox{\bf q})\frac{1}{({\bf
k}+\hbox{\bf q})^{2a}q^{2b}}={2\over d}\frac{1}{Z_{1}}I_{\alpha\gamma}(a,b+1)$
$\displaystyle={2\over
d}\frac{1}{Z_{1}}k^{D-2a-2b}\left[\delta_{\alpha\gamma}H_{1,2}(a,b+1)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(a,b+1)\right]$
(263)

as well as

$\displaystyle\sigma^{52}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{Z_{2}}\int_{q}P^{L}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf
q})(k_{\beta}+q_{\beta})(k_{\delta}+q_{\delta})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}$ (264) $\displaystyle={2\over
d}\frac{1}{Z_{2}}\bigg{[}k_{\beta}k_{\delta}\int_{q}P^{L}_{\alpha\beta}(\hbox{\bf
q})P^{L}_{\gamma\delta}(\hbox{\bf q})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}+k_{\delta}\int_{q}P^{L}_{\gamma\delta}(\hbox{\bf
q})q_{\alpha}\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}+k_{\beta}\int_{q}P^{L}_{\alpha\beta}(\hbox{\bf
q})q_{\gamma}\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b}}$
$\displaystyle+\int_{q}q_{\alpha}q_{\gamma}\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}\bigg{]}$ (265) $\displaystyle={2\over
d}\frac{1}{Z_{2}}\bigg{[}k_{\beta}k_{\delta}I_{\alpha\beta\gamma\delta}(a+1,b+2)+k_{\delta}I_{\alpha\gamma\delta}(a+1,b+1)+k_{\beta}I_{\alpha\beta\gamma}(a+1,b+1)+I_{\alpha\gamma}(a+1,b)\bigg{]}$
(266) $\displaystyle={2\over
d}\frac{1}{Z_{2}}k^{D-2a-2b}\bigg{[}\delta_{\alpha\gamma}(H_{2,4}(a+1,b+2)+H_{1,4}(a+1,b+2))$
(267)
$\displaystyle+\hat{k}_{\alpha}\hat{k}_{\gamma}(2H_{2,4}(a+1,b+2)+5H_{1,4}(a+1,b+2)+H_{0,4}(a+1,b+2))$
$\displaystyle-\delta_{\alpha\gamma}2H_{1,3}(a+1,b+1)-\hat{k}_{\alpha}\hat{k}_{\gamma}(4H_{1,3}(a+1,b+1)+2H_{0,3}(a+1,b+1))$
$\displaystyle+\delta_{\alpha\gamma}H_{1,2}(a+1,b)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(a+1,b)\bigg{]}$

One obtains

$\displaystyle
b_{5}^{1}=\left(1-2D-2\eta+\frac{Z_{2}}{Z_{1}}(2-D-2\eta)\right)\frac{\Sigma(\eta,D)}{D^{2}-1}$
(268) $\displaystyle
b_{5}^{2}=\left(2\left(\eta\left(\frac{D}{D+\eta-2}+2\eta-7\right)+5\right)+\frac{Z_{2}}{Z_{1}}\frac{2(\eta-2)^{2}(D+2\eta-2)}{D+\eta-2}\right)\frac{\Sigma(\eta,D)}{D^{2}-1}$
(269)

One checks that

$\displaystyle
b_{5}^{1}+b_{5}^{2}|_{Z_{1}=+\infty}=\frac{-2D^{2}+4D\eta^{2}-16D\eta+15D+4\eta^{3}-24\eta^{2}+43\eta-22}{D+\eta-2}\times\frac{\Sigma(\eta,D)}{D^{2}-1}=b_{5}^{\rm
crumpling}$ (270)

Now we calculate

$\displaystyle\sigma^{2}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2}\int_{q}(P^{T}_{\alpha\gamma}(\hbox{\bf
q})P^{L}_{\beta\delta}(\hbox{\bf q})+P^{T}_{\alpha\delta}(\hbox{\bf
q})P^{L}_{\beta\gamma}(\hbox{\bf q})+P^{L}_{\alpha\gamma}(\hbox{\bf
q})P^{T}_{\beta\delta}(\hbox{\bf q})+P^{L}_{\alpha\delta}(\hbox{\bf
q})P^{T}_{\beta\gamma})(\hbox{\bf q})$ (271)
$\displaystyle\times(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf q}))\frac{1}{({\bf k}+\hbox{\bf
q})^{2a}q^{2b}}=\sigma^{21}_{\alpha\gamma}({\bf
k})+\sigma^{22}_{\alpha\gamma}({\bf k})$

First

$\displaystyle\sigma^{21}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{1}}\int_{q}(P^{T}_{\alpha\gamma}(\hbox{\bf
q})+(D-1)P^{L}_{\alpha\gamma}(\hbox{\bf q}))\frac{1}{({\bf k}+\hbox{\bf
q})^{2a}q^{2b}}={2\over
d}\frac{1}{2Z_{1}}(\delta_{\alpha\gamma}I(a,b)+(D-2)I_{\alpha\gamma}(a,b+1))$
(272) $\displaystyle={2\over
d}\frac{1}{2Z_{1}}k^{D-2a-2b}(\delta_{\alpha\gamma}H_{0,0}(a,b)+(D-2)(\delta_{\alpha\gamma}H_{1,2}(a,b+1)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(a,b+1)))$

and

$\displaystyle\sigma^{22}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{2}}\int_{q}(P^{T}_{\alpha\gamma}(\hbox{\bf
q})P^{L}_{\beta\delta}(\hbox{\bf q})+P^{T}_{\alpha\delta}(\hbox{\bf
q})P^{L}_{\beta\gamma}(\hbox{\bf q})$ (273)
$\displaystyle+P^{L}_{\alpha\gamma}(\hbox{\bf q})P^{T}_{\beta\delta}(\hbox{\bf
q})+P^{L}_{\alpha\delta}(\hbox{\bf q})P^{T}_{\beta\gamma}(\hbox{\bf
q}))(k_{\beta}+q_{\beta})(k_{\delta}+q_{\delta})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}$ $\displaystyle={2\over
d}\frac{1}{2Z_{2}}\bigg{[}k_{\beta}k_{\delta}[A_{\alpha\beta\gamma\delta}+3perm]+\int_{q}P^{T}_{\alpha\gamma}(\hbox{\bf
q})\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b-2}}$ (274)
$\displaystyle+2k_{\beta}\int_{q}P^{T}_{\alpha\gamma}(\hbox{\bf
q})q_{\beta}\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b}}+2{\rm
sym}_{\alpha\gamma}k_{\delta}\int_{q}P^{T}_{\alpha\delta}(\hbox{\bf
q})q_{\gamma}\frac{1}{({\bf k}+\hbox{\bf q})^{2a+2}q^{2b}}$

where we have defined

$\displaystyle
A_{\alpha\beta\gamma\delta}=\int_{q}P^{T}_{\alpha\gamma}(\hbox{\bf
q})P^{L}_{\beta\delta}(\hbox{\bf q})\frac{1}{({\bf k}+\hbox{\bf
q})^{2a+2}q^{2b}}=\delta_{\alpha\gamma}I_{\beta\delta}(a+1,b+1)-I_{\alpha\beta\gamma\delta}(a+1,b+2)$
(275)

One has

$\displaystyle
k_{\beta}k_{\delta}[A_{\alpha\beta\gamma\delta}+3perm]=\delta_{\alpha\gamma}k_{\beta}k_{\delta}I_{\beta\delta}(a+1,b+1)$
(276) $\displaystyle+k^{2}I_{\alpha\gamma}(a+1,b+1)+2{\rm
sym}_{\alpha\gamma}k_{\alpha}k_{\beta}I_{\beta\gamma}(a+1,b+1)-4k_{\beta}k_{\delta}I_{\alpha\beta\gamma\delta}(a+1,b+2)$

Hence we find

$\displaystyle\sigma^{22}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{2}}\big{[}\delta_{\alpha\gamma}k_{\beta}k_{\delta}I_{\beta\delta}(a+1,b+1)$
(277) $\displaystyle+k^{2}I_{\alpha\gamma}(a+1,b+1)+2{\rm
sym}_{\alpha\gamma}k_{\alpha}k_{\beta}I_{\beta\gamma}(a+1,b+1)-4k_{\beta}k_{\delta}I_{\alpha\beta\gamma\delta}(a+1,b+2)$
$\displaystyle+\delta_{\alpha\gamma}I(a+1,b-1)-I_{\alpha\gamma}(a+1,b)$
$\displaystyle+2k_{\beta}\delta_{\alpha\gamma}I_{\beta}(a+1,b)+2{\rm
sym}_{\alpha\gamma}k_{\alpha}I_{\gamma}(a+1,b)-4k_{\beta}I_{\alpha\gamma\beta}(a+1,b+1)$

which leads to

$\displaystyle\sigma^{22}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{2}}k^{D-2a-2b}\big{[}\delta_{\alpha\gamma}(H_{1,2}(a+1,b+1)+H_{0,2}(a+1,b+1))$
(278)
$\displaystyle+\delta_{\alpha\gamma}(H_{1,2}(a+1,b+1)-H_{1,2}(a+1,b)+H_{0,0}(a+1,b-1)-2H_{0,1}(a+1,b))$
$\displaystyle+\hat{k}_{\alpha}\hat{k}_{\gamma}\big{(}H_{0,2}(a+1,b+1)-H_{0,2}(a+1,b)+2H_{1,2}(a+1,b+1)+2H_{0,2}(a+1,b+1)-2H_{0,1}(a+1,b)\big{)}$
$\displaystyle+\delta_{\alpha\gamma}4H_{1,3}(a+1,b+1)+\hat{k}_{\alpha}\hat{k}_{\gamma}(8H_{1,3}(a+1,b+1)+4H_{0,3}(a+1,b+1))$
$\displaystyle-4\delta_{\alpha\gamma}(H_{2,4}(a+1,b+2)+H_{1,4}(a+1,b+2))-4\hat{k}_{\alpha}\hat{k}_{\gamma}(2H_{2,4}(a+1,b+2)+5H_{1,4}(a+1,b+2)+H_{0,4}(a+1,b+2))\bigg{]}$

We find

$\displaystyle
b_{2}^{1}=\left(-D^{2}-\frac{(D-2)D}{D+\eta-2}-(D+8)\eta+3D+2\eta^{2}+8+\frac{Z_{2}}{Z_{1}}\frac{(D+2\eta-2)\left(-D^{2}-(D+3)\eta+3D+\eta^{2}+2\right)}{D+\eta-2}\right)\times\frac{\Sigma(\eta,D)}{D^{2}-1}$
$\displaystyle
b_{2}^{2}=\left(\frac{D^{2}(2\eta-3)-D(\eta-1)\eta-2(\eta-3)(\eta-2)(\eta-1)}{D+\eta-2}+\frac{Z_{2}}{Z_{1}}\frac{(D-2)(\eta-2)^{2}(D+2\eta-2)}{D+\eta-2}\right)\times\frac{\Sigma(\eta,D)}{D^{2}-1}$
(279)

and we can check that

$\displaystyle
b_{2}^{1}+b_{2}^{2}|_{Z_{1}=+\infty}=-\frac{(D-1)\left(D^{2}+2\eta-4\right)}{D+\eta-2}\times\frac{\Sigma(\eta,D)}{D^{2}-1}=b_{2}^{\rm
crumpling}$ (280)

Finally, let us denote

$\displaystyle\tilde{b}_{1}=b_{1}+b_{3}+b_{5}+b_{2}$ (281)

Using the inversion formula (105), it can be extracted from

$\displaystyle\sigma^{1}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2}\int_{q}(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})(\frac{1}{Z_{1}}\delta_{\beta\delta}+\frac{1}{Z_{2}}P_{\beta\delta}^{L}({\bf
k}+\hbox{\bf q}))\frac{1}{({\bf k}+\hbox{\bf
q})^{2a}q^{2b}}=\sigma^{11}_{\alpha\gamma}({\bf
k})+\sigma^{12}_{\alpha\gamma}({\bf k})$ (282)

with

$\displaystyle\sigma^{11}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{1}}\delta_{\alpha\gamma}(D+1)k^{D-2a-2b}H_{0,0}(a,b)$ (283)

and

$\displaystyle\sigma^{12}_{\alpha\gamma}({\bf k})={2\over
d}\frac{1}{2Z_{2}}k^{D-2a-2b}(\delta_{\alpha\gamma}H_{0,0}(a,b)+\int_{q}(k_{\alpha}+q_{\alpha})(k_{\gamma}+q_{\gamma})\frac{1}{({\bf
k}+\hbox{\bf q})^{2a+2}q^{2b}})$ (284) $\displaystyle={2\over
d}\frac{1}{2Z_{2}}k^{D-2a-2b}(\delta_{\alpha\gamma}H_{0,0}(a,b)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,0}(a+1,b)-2\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,1}(a+1,b)+\delta_{\alpha\gamma}H_{1,2}(a+1,b)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(a+1,b))$
$\displaystyle={2\over
d}\frac{1}{2Z_{2}}k^{D-2a-2b}(\delta_{\alpha\gamma}H_{0,0}(a,b)+I_{\alpha\gamma}(b,a+1))$
(285) $\displaystyle={2\over
d}\frac{1}{2Z_{2}}k^{D-2a-2b}(\delta_{\alpha\gamma}H_{0,0}(a,b)+\delta_{\alpha\gamma}H_{1,2}(b,a+1)+\hat{k}_{\alpha}\hat{k}_{\gamma}H_{0,2}(b,a+1))$
(286)

We checked that these are two equivalent expressions.

Hence

$\displaystyle\tilde{b}^{1}_{1}=\left(-\frac{(D-\eta+4)(D+2\eta-4)(D+2\eta-2)}{2(D+\eta-2)}-\frac{Z_{2}}{Z_{1}}\frac{(D+1)(D-\eta+2)(D+2\eta-4)(D+2\eta-2)}{2(D+\eta-2)}\right)\times\frac{\Sigma(\eta,D)}{D^{2}-1}$
$\displaystyle\tilde{b}^{2}_{1}=\frac{1}{2}(D+2\eta-4)(D+2\eta-2)\times\frac{\Sigma(\eta,D)}{D^{2}-1}$
(287)

One finds

$\displaystyle\tilde{b}^{1}_{1}+\tilde{b}^{2}_{1}|_{Z_{1}=+\infty}=\frac{(\eta-3)(D+2\eta-4)(D+2\eta-2)}{D+\eta-2}\times\frac{\Sigma(\eta,D)}{D^{2}-1}$
(288)

which recovers exactly the $\tilde{b}_{1}^{\rm crumpling}$.

Reexpressing all the results using $Z_{\kappa}$, $Z_{g}$ and
$s=Z_{\kappa}/Z_{g}$, using the relation (281) between $\tilde{b}_{1}^{j}$ and
$b_{1}^{j}$ and expressing the $b_{i}^{(1,2)}$ in terms of $b^{1}(\eta,D,s)$
and $b(\eta,D,s)$ one finally obtains the amplitudes (II.6).

##  V Remark

Let us recall, as mentioned in Section II, that in (239), (240) an isotropic,
$\delta_{\alpha\gamma}$ component is always generated for the inverse
propagator of the vector field $\vec{t}_{\alpha}$, even in the $s\to 0$ limit
when the propagator is purely longitudinal. Here we note that this is
innocuous at the $s=0$ fixed point, up to boundary terms. Namely, for
$\vec{t}_{\alpha}=\partial_{\alpha}\vec{r}$, by integration by part we have

$\displaystyle\int d^{D}x(\partial_{\alpha}\vec{t}_{\beta})^{2}=\int
d^{D}x(\partial_{\alpha}\partial_{\beta}\vec{r})^{2}=\int
d^{D}x(\partial^{2}_{\alpha}\vec{r})^{2}+\int
d^{D}x~{}[\partial_{\alpha}(\partial_{\beta}\vec{r}\partial_{\alpha}\partial_{\beta}\vec{r})-\partial_{\beta}(\partial_{\beta}\vec{r}\partial_{\alpha}^{2}\vec{r})]$
(289) $\displaystyle=\int d^{D}x(\partial_{\alpha}\vec{t}_{\alpha})^{2}+\int
d^{D}x~{}[\partial_{\alpha}(\vec{t}_{\beta}\cdot\partial_{\alpha}\vec{t}_{\beta})-\partial_{\beta}(\vec{t}_{\beta}\cdot\partial_{\alpha}\vec{t}_{\alpha})]$
(290)