11institutetext:  1 Salzburg University of Applied Sciences, Dept. of
Information Technology and Digitalization
2 Salzburg University of Applied Sciences, Dept. of Health Sciences
3 Spinal Cord Injury and Tissue Regeneration Center Salzburg, Research
Institute of Experimental Neuroregeneration
4 Kardinal Schwarzenberg Klinikum, Department of Pathology and Microbiology

# MixUp-MIL: A Study on Linear & Multilinear Interpolation-Based Data
Augmentation for Whole Slide Image Classification

Michael Gadermayr1 Lukas Koller1 Maximilian Tschuchnig1 Lea Maria
Stangassinger2 Christina Kreutzer3 Sebastien Couillard-Despres3 Gertie Janneke
Oostingh2 Anton Hittmair4

###### Abstract

For classifying digital whole slide images in the absence of pixel level
annotations, multiple instance learning methods are typically applied. Such
methods are currently of very high interest in the research community due to
the generic applicability, whereby the issue of data augmentation in this
context is rarely explored. Here, we investigate the linear and multilinear
interpolation between feature vectors, a data augmentation technique which
proved to be capable of improving the generalization performance of
classification networks and recently also multiple instance learning. Recent
experiments on multiple instance learning were thus far only performed on two
rather small data sets in combination with one specific feature extraction
approach. In this case a strong dependence on the data set could be
identified. We conducted a larger study incorporating $10$ different data set
configurations, two different feature extraction approaches (supervised and
self-supervised), stain normalization and two multiple instance learning
architectures. The results showed an extraordinarily high variability in the
effect of the method used.

###### Keywords:

Histopathology Data augmentation Whole slide image classification MixUp
Interpolation Multiple Instance Learning

##  1 Introduction

The trend towards whole slide imaging for digitizing histo- or cytologic
slides in research and clinical practice boosts the demand for automated
processing techniques [16]. Due to the huge resolution of the whole slide
images (WSIs), a holistic classification based on conventional convolutional
neural network architectures is not feasible, even with state-of-the-art
hardware.

Multiple instance learning (MIL) [10, 12, 27, 30, 20, 28, 26, 18, 5]
represents a methodology for handling these huge digital WSIs corresponding to
global ”image-level” labels. In MIL, WSIs correspond to labeled bags, whereas
extracted patches correspond to unlabeled instances. Based on randomly
sampling patches, MIL can be applied to WSIs showing an arbitrary size. MIL
algorithms typically consist of a feature extraction stage, a pooling stage
and a downstream classification stage. State-of-the-art approaches mainly rely
on convolutional neural networks (CNNs) for feature extraction, often in
combination with attention [13, 12] or self-attention [19]. For training the
feature extraction stage, supervised (often pre-trained) as well as self-
supervised learning is employed [13, 12]. Most methods rely on separate
individual learning stages (separation between representation learning and
classification). Nevertheless, also end-to-end deep learning approaches have
been proposed [3, 21].

Multiple instance learning can be categorized into instance-based and
embedding-based MIL. In the case of instance-based MIL, the information per
patch is first condensed to a single scalar value, representing the
classification per patch. Downstream, these patch-based values are aggregated.
In the case of embedding-based MIL, a distinct feature vector is extracted per
patch. Downstream, all feature vectors from a WSI are aggregated. Finally, the
pooled embeddings are used to classify a global label. Embedding based MIL is
typically more effective in the case of a WSI-classfication, while instance-
based MIL can be effectively used to intuitively detect relevant regions-of-
interest (since the algorithm determines a single label per patch). State-of-
the-art models combine both, instance- and an embedding-based pathways [12,
18, 26]. Zhang et al. proposed a method including the generation of multiple
so-called pseudo-bags for each WSI to augment the amount of data [30]. Wu et
al. proposed so-called smooth attention maps, incorporating the fact that
images show spatial correlations [28]. Fillioux et al. developed an efficient
method for modelling large sequences (i.e. a large number of patches) when
using recurrent neural networks or transformers [5]. For a survey on multiple
instance learning methods for classifying digital pathology, we refer to [7].

Although the overall amount of data contained in a WSI is huge, the number of
labeled samples in MIL (represented by the number of WSIs) is often small.
Often, datasets are also imbalanced, further increasing this small data issue.
Often, datasets are also imbalanced, further increasing this small data issue
[9]. General data augmentation strategies, such as rotations, flipping, stain
augmentation and normalization are applicable to increase the amount of data
[22]. All of these augmentation approaches are performed in the image domain.
Although recently there is a very high effort on improving multiple instance
learning approaches [5, 28, 26, 18, 15, 14], there is little work on data
augmentation dedicated using characteristics unique to the MIL domain.

In this work, we consider feature-level data augmentation directly applied to
the patch-representations extracted using CNNs. These methods can be easily
combined with conventional image-based augmentation. The operations are
computationally highly efficient and can be precomputed before starting to
train the MIL downstream classification network (in the case that the tasks
are separated, which often holds true). There has been relatively little work
on such forms of data augmentation. Li et al. [13] proposed an augmentation
strategy based on sampling the patch-descriptors to generate several different
bags for each individual WSI. In this paper, we focus on the interpolations of
patch descriptors based on the idea of Zhang et al [31], which is also known
as MixUp. This method was originally proposed as data agnostic which also
shows good results if applied to raw image data [4, 1, 23, 17]. Variations
were proposed, to be applied to latent representations [25] and to balance
data sets [9].

The contribution of this paper is manifold.

  * •

We perform a large study based on our previous conference paper [6], where the
basic concepts of MixUp data augmentation [31] in WSI classification is
proposed and evaluated on two small data sets.

  * •

We investigate different feature extraction methods, including pretrained
networks and self-supervised learning.

  * •

Motivated by scores strongly depending on the data set, we investigated 10
different data set settings, derived from 3 different data sets.

  * •

We also investigated the effect of variably restricting the data set sizes.

  * •

To investigate whether stain variability plays a vital role, we analyze stain
normalization in combination with data augmentation and multiple instance
learning.

  * •

Finally, to improve the interpretability of the results, we analyzed the
variability within and between whole slide images.

##  2 Methods

In this paper, we propose and evaluate different strategies to perform data
augmentation in the MIL setting, based on the pair-wise interpolation of
feature descriptors.

We consider MIL approaches relying on separately trained feature extraction
and classification stages [12, 11, 19]. The proposed and evaluated data
augmentation methods are applied to the patch descriptors obtained by the
feature extraction stage. This is highly efficient during training since the
features are only computed once (per patch) and for augmentation only simple
arithmetic operations are applied to the feature vectors. Other generally used
image-based data augmentation strategies (such as stain-augmentation,
rotations or deformations) can be combined easily with the feature-based
approaches but require individual feature extraction during training. To avoid
the curse of meta-parameters and thereby experiments such approaches are
explicitly not considered here.

Zhang et al. [31] originally proposed the MixUp data augmentation method,
based on the idea that synthetic samples $\boldsymbol{x^{\prime}}$ are
generated such that
$\boldsymbol{x^{\prime}}=\alpha\cdot\boldsymbol{x_{i}}+(1-\alpha)\cdot\boldsymbol{x_{j}}\;,$
where $\boldsymbol{x_{i}}$ and $\boldsymbol{x_{j}}$ are randomly sampled raw
input feature vectors. Corresponding labels $\boldsymbol{y^{\prime}}$ are
generated for a new synthetic sample such that
$\boldsymbol{y^{\prime}}=\alpha\cdot\boldsymbol{y_{i}}+(1-\alpha)\cdot\boldsymbol{y_{j}}\;,$
where $\boldsymbol{y_{i}}$ and $\boldsymbol{y_{j}}$ are the corresponding one-
hot label encodings. The weight $\alpha$ is drawn from a uniform distribution
between $0$ and $1$. This can be interpreted as linear interpolation between
two feature vectors.

One single input sample (corresponding to one WSI) of a MIL approach with a
separate feature extraction stage [12] can be expressed as a P-tupel
$X=(\boldsymbol{x_{1}},...,\boldsymbol{x_{P}})$ with $\boldsymbol{x_{i}}$
being the feature vector of an individual patch and $P$ being the number of
patches per WSI. The method proposed by Zhang et al. cannot directly be
applied to this tupel representation. However, there are several options to
adapt the basic idea to the changed setting.

Figure 1: Overview of the proposed feature-based data augmentation approaches.
In the case of Inter-MixUp (a), a linear combination is applied on the pairs
of WSI descriptors with a randomly selected weight factor. In the case of
Intra-MixUp (b), patch-based descriptors from the same WSI are merged with
individual random weights.

###  2.1 Inter-MixUp

refers to the intuitive synthesis of feature vectors by linearly combining
feature vectors of a pair of WSIs (see Fig. 1 (a)). All features of a WSI with
index $w$ can be represented by $X^{(w)}$, such that
${X^{(w)}}=(\boldsymbol{x^{(w)}_{1}},\;...\;,\boldsymbol{x^{(w)}_{P}})\;.$ To
generate a novel synthetic sample ${X^{(u)}}^{\prime}$ based on two real
samples ${X^{(w)}}$ and ${X^{(v)}}$, we introduce the operation

${X^{(u)}}^{\prime}=(\alpha\cdot\boldsymbol{x_{1}^{(w)}}+(1-\alpha)\cdot\boldsymbol{x^{(v)}_{1}},\;\alpha\cdot\boldsymbol{x^{(w)}_{2}}+(1-\alpha)\cdot\boldsymbol{x^{(v)}_{2}},\;...\;,\;\alpha\cdot\boldsymbol{x^{(w)}_{P}}+(1-\alpha)\cdot\boldsymbol{x^{(v)}_{P}})$

with $\alpha$ being a uniformly sampled random weight ($\alpha\in[0,1]$). The
WSI indexes $v$ and $w$ are uniformly sampled from the set of indexes. The
index $u$ ranges from $1$ to the number of extracted WSI descriptors. This
random sampling strategy theoretically allows to sample an arbitrary number of
combinations in each epoch. However, since the new synthetic descriptors are
individually generated in each epoch, there is no benefit of increasing the
number of extracted WSI descriptors. We fix this number to the number of WSIs
in the training data set, in order to keep the number of training iterations
per epoch consistent over all experiments.

Two different configurations can be considered. The interpolation between WSIs
of the same class (V1) and the interpolation between any WSIs, including the
interpolation between different labeled WSI (V2). In the case of V2, also the
one-hot-encoded label vectors are linearly combined, such that
$\boldsymbol{y^{(u)}}^{\prime}=\alpha\cdot\boldsymbol{y^{(w)}}+(1-\alpha)\cdot\boldsymbol{y^{(v)}}$
This leads to continuous labels. The random values, $\alpha$, $v$ and $w$ are
selected individually for each individual WSI and each epoch. Before applying
the MixUp operation, the vector tupel containing all patch-level feature
vectors, is randomly shuffled. This permutation is not specific to this data
augmentation approach but is performed in each experiment.

###  2.2 Intra-MixUp

Intra-MixUp combinations refer to the generation of synthetic descriptors by
combining feature vectors within an individual WSI (see Fig. 1 (b)). A novel
synthetic patch-level image descriptor $\boldsymbol{x_{k}}^{\prime}$ is
created based on the randomly selected descriptors $\boldsymbol{x_{i}}$ and
$\boldsymbol{x_{j}}$, such that
$\boldsymbol{x_{k}}^{\prime}=\alpha\cdot\boldsymbol{x_{i}}+(1-\alpha)\cdot\boldsymbol{x_{j}}\;,$
with $i$ and $j$ being random indices (uniformly sampled from
$\\{1,2,...,P\\}$) and $\alpha$ being a uniformly sampled random value a
($\alpha\in[0,1]$). The index $k$ ranges from 1 to the number of extracted
descriptors per patch. The thereby obtained vector tupel
$(\boldsymbol{x_{1}}^{\prime},...,\boldsymbol{x_{P}}^{\prime})$ represents the
synthetic WSI-level descriptor. Besides performing combinations for each WSI
during training, selective interpolation can be useful to keep real samples
within the training data. This can be easily achieved by choosing
$(\boldsymbol{x_{1}}^{\prime},...,\boldsymbol{x_{P}}^{\prime})$ with a chance
of $\beta$ and $(\boldsymbol{x_{1}},...,\boldsymbol{x_{P}})$ otherwise. While
the Intra-MixUp method described before represents a linear interpolation
method, with a simple adjustment also multilinear interpolation can be
performed. This is achieved by computing $\boldsymbol{x_{k}}^{\prime}$ such
that
$\boldsymbol{x_{k}}^{\prime}=\boldsymbol{\alpha}\circ\boldsymbol{x_{i}}+(1-\boldsymbol{\alpha})\circ\boldsymbol{x_{j}}$
with $\boldsymbol{\alpha}$ being a random vector and $\circ$ being the
element-wise product. This element-wise linear (multilinear) approach enables
higher variability in the generated samples.

###  2.3 Experimental Architecture

As experimental architecture, we make use of the dual-stream MIL approach
proposed by Li et al [12]. Since this model combines both, embedding-based and
an instance-based encoding, the effect of both paths can be individually
investigated without changing any other architectural details. Since the
method represents a state-of-the-art approach, it additionally serves as well-
performing baseline. In the investigated model [12] an instance- and an
embedding-based pathway are employed in parallel and merged by weighted
addition. The embedding-based pathway contains an attention mechanism, to
higher weight patches that are similar to the so-called critical instance. The
model makes use of an individual feature extraction stage.

###  2.4 Feature Extraction

We performed two different approaches for patch-level feature extraction.
Firstly, we extract features using a ResNet18, pre-trained on the image-net
challenge data, due to the high performance in previous work and generic
applicability [8]. Secondly, we trained and applied SimCLR [2] as state-of-
the-art self-supervised learning model to particularly incorporate the
peculiarities of the data in spite of the absence of labels. As backbone, as
in the pretrained scenario, ResNet18 was applied. This model was assessed as
particularly appropriate due to the rather low dimensional output (512
dimensions).

###  2.5 Stain Normalization

We made use of the Vahadane stain normalization technique [24]. The reference
tile was chosen from the mean stain vectors of only the training data set, and
both training and testing were then aligned to the reference style.

###  2.6 Other Augmentation Techniques

For a comparison with several other augmentation methods on feature level
including random sampling, selective random sampling and random noise, we
refer to our related study [6]. Since none of the augmentation settings based
on single images lead to clear accuracy improvements, we decided to not
further investigate this category of methods.

###  2.7 Data Sets

####  2.7.1 Thyroid Paraffin & Frozen Data Sets

The goal of this data set [6] is to distinguish different nodular lesions of
the thyroid, focusing especially on benign follicular nodules (FN) and
papillary carcinomas (PC). This differentiation is crucial, due to the
different treatment options, in particular with respect to the extent of
surgical resection of the thyroid gland [29]. The data set utilized in the
experiments consists of 80 WSIs overall. One half (40) of the data set
consists of frozen and the other half (40) of paraffin sections [8]),
representing the different modalities. All images were acquired during
clinical routine at the Kardinal Schwarzenberg Hospital. Procedures were
approved by the ethics committee of the county of Salzburg (No. 1088/2021).
The mean and median age of patients at the date of dissection was 47 and 50
years, respectively. The data set comprised 13 male and 27 female patients,
corresponding to a slight gender imbalance. They were labeled by an expert
pathologist with over 20 years experience. A total of 42 (21 per modality)
slides were labeled as papillary carcinoma while 38 (19 per modality) were
labeled as benign follicular nodule. For the frozen sections, fresh tissue was
frozen at $-15^{\circ}$ Celsius, slides were cut (thickness 5 $\mu m$) and
stained immediately with hematoxylin and eosin. For the paraffin sections,
tissue was fixed in $4\;\%$ phosphate-buffered formalin for 24 hours.
Subsequently formalin fixed paraffin embedded tissue was cut (thickness 2 $\mu
m$) and stained with hematoxylin and eosin. The images were digitized with an
Olympus VS120-LD100 slide loader system. Overviews at a 2x magnification were
generated to manually define scan areas, focus points were automatically
defined and adapted if needed. Scans were performed with a 20x objective
(corresponding to a resolution of 344.57 nm/pixel). The image files were
stored in the Oympus vsi format based on lossless compression.

Beyond the original data sets denoted as ’Frozen’ and ’Paraffin’, we
considered the aggregation of both, denoted as ’All’, since both data sets
show the same biological structure. In addition, we considered stain
normalization of the original data sets (’Frozen Normalized’, ’Paraffin
Normalized’).

####  2.7.2 Camelyon Data Set

The Camelyon17 challenge data contains WSIs of hematoxylin and eosin stained
lymph node sections. We make only use of the patient-level ground truth. All
annotations were carefully prepared under supervision of expert pathologists.
For the purpose of revising the slides, additional slides stained with
cytokeratin immunohistochemistry were utilized. The data set for CAMELYON17 is
collected from 5 medical centres in the Netherlands. WSI are provided as TIFF
images. We differentiate between samples showing tumor and healthy samples
(WSIs). We make use of 161 (normal) and 200 (patient) annotated
samples111http://gigadb.org/dataset/100439. This larger data set was further
investigated to assess the effect of the data set size, with respect to both,
the number of WSIs and the number of patches per WSI. Beyond the maximum
number of training samples (WSIs), we reduced to 64, 32 and 16 samples per
class (referred to as T64, T32, T16 in the results section). In addition, we
also investigated the effect of reducing the number of patches from 2048 to
512 (P512).

###  2.8 Evaluation Details

The data set was randomly separated into training ($80\;\%$) and test data
($20\;\%$). The whole pipeline, including the separation, was repeated 32
times to achieve representative scores. Due to the almost balanced setting,
the overall classification accuracy (mean and standard deviation) is finally
reported. Adam was used as optimizer. The models were trained for 200 epochs
with an initial learning rate of 0.0002. Random shuffling of the vector tupels
(shuffling within the WSIs) was applied for all experiments. The patches were
randomly extracted from the WSI, based on uniform sampling in informative
areas. To obtain the informative areas, we applied thresholds based on patches
extracted from downscaled slides (factor $2^{-3}$). We applied thresholding of
the green color channel so ensure that large areas (at least 75 %) are covered
by tissue. In addition, we applied an entropy threshold to exclude areas
showing large blurry artifacts. Due to the high image quality (without any
blurry regions), the second threshold was omitted in the case of the thyroid
cancer data set. To obtain a representation independent of the WSI size, we
extracted 1024 samples (thyroid data sets) and 2048 samples (Camelyon data
set, due to the larger image size) patches with a size of $256\times 256$
pixel per WSI, resulting in 1024 or 2048 patch-descriptors per WSI [8].
https://gitlab.com/mgadermayr/mixupmil. We use the reference implementation of
the dual-stream MIL approach [12]. To obtain further insight into the feature
distribution, we randomly selected patch descriptor pairs and computed the
cosine distances. In detail, we selected 10,000 pairs (a) from different WSIs
with dissimilar classes only, (b) from different WSIs with similar and
dissimilar classes, (c,d) from different WSIs with similar classes only, and
(e) from the same WSI.

###  2.9 Source Code and Data

The experiments are implemented using Pytorch based on a single GPU (4 GB of
memory is sufficient). We make use of the reference implementation of the
dual-stream MIL model [12]. Source code as well as data in the form of
precomputed features is available via GitLab:
https://gitlab.com/mgadermayr/mixupmil.

##  3 Results

###  3.1 Intra-MixUp

Figure 2: Mean overall classification accuracy and standard deviation obtained
with each individual Intra-MixUp setting. The left two columns represent
embedding-based MIL, while the right column shows the scores obtained with
dual-stream MIL. Each subplot shows the accuracies obtained without
augmentation (Base), with linear (Lin) and multi-linear (MLin) augmentation
for ResNet (left) and SimCLR (right) for one individual data set.

Figure 2 shows the mean overall classification accuracy and standard deviation
obtained with each individual Intra-MixUp setting. The left two columns
represent embedding-based MIL, while the right column shows the scores
obtained with dual-stream MIL. Each subplot shows the accuracies obtained
without augmentation (Base), with linear (Lin) and multi-linear (MLin)
augmentation for ResNet (left) and SimCLR (right) for one individual data set.

Linear interpolation improved the accuracy in the case of 8 out of 16 settings
with the pretrained ResNet as feature extractor. Linear interpolation improved
the accuracy in 7 out of 16 settings with the SimCLR feature extractor.
Multilinear interpolation improved the accuracy in the case of 10 out of 16
settings with the pretrained ResNet as feature extractor and in 8 out of 16
settings with the SimCLR feature extractor. In 12 out of 15 cases where linear
interpolation improved accuracy, multilinear interpolation showed even larger
improvements.

The largest accuracy increases with linear and multilinear interpolation where
$+0.06$ (linear, Thyroid Frozen Normalized, ResNet, Embedding-based MIL) and
$+0.11$ (multilinear, Thyroid Frozen Normalized, SimCLR, Embedding-based MIL).

The largest accuracy decreases with linear and multilinear interpolation where
$-0.06$ (linear, Camelyon, Dual-Stream MIL, SimCLR) and $-0.17$ (multilinear,
Camelyon, Dual-Stream MIL, SimCLR).

###  3.2 Inter-MixUp

Figure 3 shows the mean overall classification accuracy and standard deviation
obtained with each individual Inter-MixUp setting. As shown in Fig. 2, the
left two columns represent embedding-based MIL, while the right column shows
the scores obtained with dual-stream MIL. Each subplot shows the accuracies
obtained without augmentation (Base), with Inter-MixUp V1 and Inter-MixUp V2
augmentation for ResNet (left) and SimCLR (right) for one individual data set.

Marginal improvements (below $0.02$) were obtained in 5 out of 80 cases. In
all of these cases, ResNet was used for feature extraction and a Camelyon-
based data set was employed.

Figure 3: Mean overall classification accuracy and standard deviation obtained
with each individual Inter-MixUp setting. The left two columns represent
embedding-based MIL, while the right column shows the scores obtained with
dual-stream MIL. Each subplot shows the accuracies obtained without
augmentation (Base), with linear (Lin) and multi-linear (MLin) augmentation
for ResNet (left) and SimCLR (right) for one individual data set.

Overall, SimCLR showed better accuracies compared to the pretrained ResNet’s
features. In 16 out of 20 cases, the SimCLR baseline showed the higher scores.
In the case of multilinear (intra) interpolation, also in 16 out of 20 cases,
the best scores are obtained with SimCLR.

###  3.3 Distance Analysis

In Fig. 4, the distributions of the descriptor cosine distances between (a-d)
patches from different WSIs (inter-WSI) and (e) patches within a single WSI
(intra-WSI) are provided. In each setting, the intra-WSI distances (e) show
the lowest values compared to similar values for the inter-WSI setting (a-d).
SimCLR shows higher values compared to ResNet features. The highest distances
are obtained for the Camelyon SimCLR data set which corresponds to the data
set, the self-supervised learning approach was trained on. Based on the used
common box plot variation (whiskers length is less than $1.5\times$ the
interquartile range), a large number of data points was identified as
outliers. However, these points are not considered as real outliers, but occur
due to the asymmetrical data distribution (as indicated by the violin plot in
the background).

Figure 4: Analysis of the distributions of the patch descriptors’ cosine
distances between (a) patches from different classes, (b) randomly selected
patches from different WSIs, (c,d) patches from the same class and different
WSIs (for both classes, PC and FN) and (e) patches within the WSIs.

##  4 Discussion

In this work, we proposed and examined novel data augmentation strategies
based on the idea of interpolations of feature vectors in the MIL setting.
Particularly, we focused on interpolating patch descriptors between (Intra-
MixUp) and within whole slide images (Inter-MixUp).

The Inter-MixUp approach did not show any systematic improvements.
Independently of the chosen strategy (V1, V2), the Inter-MixUp approach did
not show any systematic improvements. Since we expected that the large
variability between whole slide images might negatively influence an
interpolation, we performed experiments in combination with stain
normalization. However, even with stain normalization, we did not notice any
improvements. Due to an improved feature invariance achieved with the self-
supervised learning method (implicitly learning e.g. stain normalization), we
also expected a positive effect of Inter-MixUp in combination with the SimCLR
approach. When comparing the scores of ResNet and SimCLR, however, we noticed
that the trend is again very similar and does not strongly depend on the
feature extraction technique.

The Intra-MixUp approach showed improvements in the case of several
configurations. Particularly in the case of the thyroid cancer classification
data sets, we noticed systematic improvements. Independent of the feature
extraction method and whether stain normalization was performed, improvements
where obtained in the majority of cases. Only the combination of dual-stream
MIL with the paraffin dataset showed slightly decreased scores with both,
ResNet and SimCLR features. An explanation could be that the dual-stream
approach is supposed to perform particularly well on small and challenging
data sets due to the combined loss function. Using our proposed data
augmentation strategy, we were able to close the gap with the more basic
embedding-based MIL approach or even outperform the state-of-the-art dual
stream approach.

In the case of the Camelyon dataset, we did not notice improvements, even with
clearly reduced data (see T64, T32, T16 and P512 configurations). We expected
that the size of the training data would have a strong impact on the
performance of data augmentation, since the availability of sufficient data
was supposed to diminish positive effects of augmentation. However, whether
the number of WSIs or the number of patches per WSI were reduced, we did not
notice improvements.

Regarding the different feature extraction approaches, we noticed that SimCLR
performed slightly better on average. Particularly in the case of the Camelyon
data sets the gap was large. Considering the experiments with the proposed
MixUp-MIL data augmentation, we notice similar effects in the case of both
feature extraction methods. The potentially more invariant SimCLR features do
not make data augmentation obsolete. The same holds true for stain
normalization. We noticed very similar effects, independent whether stain
normalization was performed or not.

Based on the distance analysis, we noticed that the intra-slide distances
between pairs of patches are on average smaller for all data sets. There are
particularly less pairs showing large distances (outliers). Obviously in the
case of Inter-MixUp, interpolation between samples in a more diverse set of
data is performed, which is supposed to have negative effects. However, based
on this analysis, we are unable to explain why the Camelyon data set showed a
lower performance with interpolation although similar properties are observed
for both datasets.

Based on our analysis, we assume that the data set has a strong impact on the
performance of linear or multilinear interpolation. We assume that data set
specific properties are important such as the amount of important (relevant)
information (i.e. the number of relevant patches) in the WSIs. We assume that
if only a small number of patches is relevant for the final classification (as
in the case of the Camelyon data set), it is more likely that interpolation
washes out the features by combining relevant with non-relevant information.
For that reason we suggest to investigate more intelligent ways of selecting
samples for interpolation. For example, paired samples could be selected based
on neighborhood or feature similarity instead of a random selection of two
feature vectors. The challenge in that case would be to find a good trade-off
between a high variability in the virtual samples and not washing-out the
relevant information. Finally an effective selection strategy could not only
help the Intra-MixUp setting, but potentially also enable Inter-MixUp to
increase diversity of data and accuracy even further.

##  5 Conclusion

We proposed and investigated novel data augmentation strategies based on the
idea of interpolations of patch descriptors in the MIL setting. Based on the
experimental results, the multilinear Intra-MixUp setting proved to be highly
effective in certain cases, while the Inter-MixUp method showed inferior
scores compared to a state-of-the-art baseline. We learned that there is a
clear difference between combinations within and between WSIs with a
noticeable effect on the final classification accuracy. The negative effect in
the case of Inter-MixUp is supposedly due to the high variability between the
WSIs compared to a rather low variability within the WSIs. This effect remains
similar if the invariance of the desciptors is improved, either by performing
normalization or by applying self-supervised learning. We also did not notice
any impact of the feature extraction methods used. We found out that data set
size, the number of patches per WSI, the feature extraction method and the
stain variability are not the relevant factors that decide whether the
proposed method performs well or not. Based on our analysis and excluding
these parameters, we strongly expect that the spatial distribution of relevant
information is a crucial factor. Further we expect that a more intelligent
selection of pairs for interpolation could increase performance in the case of
data sets showing inferior performance if a random selection is applied.

### Acknowledgement

This work was partially funded by the County of Salzburg (KIAMed, AIBIA).

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OCU-PHYS-543

NITEP-111

The strong CP problem and higher dimensional gauge theories

Yuki Adachi, C.S. Lima and Nobuhito Marub,c

Department of Sciences, Matsue College of Technology, Matsue 690-8518, Japan.

a Tokyo Woman’s Christian University, Tokyo 167-8585, Japan

b Department of Mathematics and Physics, Osaka City University, Osaka
558-8585, Japan

c Namubu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),

Osaka City University, Osaka 558-8585, Japan

Abstract

We discuss a natural scenario to solve the strong CP problem in the framework
of the higher dimensional gauge theory. An axion-like field $A_{y}$ has been
built-in as the extra-space component of the higher dimensional gauge field.
The coupling of $A_{y}$ with gluons is attributed to the radiatively induced
“Chern-Simons” (CS) term. We adopt a toy model with some unknown gauge
symmetry U(1)X. The CS term is obtained in two ways: first by a concrete
1-loop calculation and next by use of the Fujikawa’s method to deal with the
chiral anomaly in 4D space-time. The obtained results are identical, which
implies that the radiative correction to the CS term is “1-loop exact” and is
also free from UV-divergence even though the theory itself is non-
renormalizable. As a novel feature of this scenario, such obtained CS term is
no longer linear in the field $A_{y}$ as in the usually discussed CS term in
5D space-time but a periodic function of $A_{y}$, since $A_{y}$ has a physical
meaning as the Wilson-loop phase. We argue how such novel feature of this
scenario causes the modification of the ordinary solutions of the strong CP
problem based on the axion fields.

##  1 Introduction

One of the attractive candidate of the physics beyond the standard model (BSM)
is the scenario based on higher dimensional theories. Among them, higher
dimensional gauge theory is of special interest. In the theory where the
extra-space component of the higher dimensional gauge field is identified with
the Higgs field, called gauge-Higgs unification (GHU) [1, 2], is known to
provide a novel solution to the gauge hierarchy problem [3], relying on higher
dimensional gauge symmetry, but without invoking supersymmetry. Five
dimensional (5D) gauge theory is also argued to be related to strongly
interacting gauge theory in 4D space-time [4] through AdS/CFT correspondence
[5], thus leading to a close relationship of GHU with “little Higgs” scenario.

In this paper we discuss a scenario where strong CP problem is solved in the
framework of higher dimensional gauge theory. In this scenario, the extra-
space component of higher dimensional gauge field, say $A_{y}$ in the case of
5D space-time, behaves as if it were the axion [6, 7] or invisible axion [8,
9] fields in the well-known solutions of the problem in the framework of
ordinary 4D space-time.

The extra-space component $A_{y}$ is a part of gauge field, not a pseudo
Nambu-Goldstone (NG) boson associated with the spontaneous breaking of some
global symmetry. However, it still has some similarity to the axion or
invisible axion fields. Namely, its coupling with fermion field is of pseudo-
scalar type from the 4D point of view, being accompanied by $\gamma_{5}$. In
addition, the higher dimensional gauge theory possesses a shift symmetry,
which is nothing but the higher dimensional local gauge symmetry $A_{y}\to
A_{y}+\partial_{y}\lambda$ where $\lambda$ is a $y$-dependent gauge
transformation parameter ($y$: extra space coordinate). This mimics the shift
symmetry under the global transformation $a\to a+$ constant ($a$: a generic
axion field) in those well-known solutions.

Thus, it is expected that in the higher dimensional gauge theory a solution of
the strong CP problem is naturally built-in. For concreteness, in this paper
we discuss 5D gauge theory with the extra dimension compactified on a circle
$S^{1}$. The coupling of $A_{y}$ with gluon fields, which plays a crucial role
in the solution, is attributed to the “Chern-Simons” (CS) term, which does not
exist in the original 5D theory but is shown to be induced radiatively, as we
will see below. The CS term is first derived by a concrete 1-loop calculation
and then reconfirmed by utilizing the Fujikawa’s method [10] to derive the
chiral anomaly in 4D space-time, suggesting that the 1-loop calculation of the
CS term is the exact result. This may not be surprising, since the relation of
parity non-conservation due to the CS term in odd space-time dimensions and
the chiral anomaly in even dimensions has been known for some time [11].

On the other hand, there appears some novel feature in such obtained CS term,
as the inevitable consequence of the fact that the extra space $S^{1}$ is a
non-simply-connected manifold. Namely, $A_{y}$, or more precisely its Kaluza-
Klein (KK) zero mode has a physical meaning as a phase of the Wilson-loop
exp$(ie\oint A_{y}dy)$ (in 5D QED, for instance), where the line-integral is
along the direction of the extra dimension. This is why the vacuum expectation
of $A_{y}$, which is just a constant gauge field, has a physical importance in
GHU. This in turn implies that all physical observables should be periodic in
$A_{y}$. The CS term is not an exception. The loop-induced effective
lagrangian obtained by summing up all KK modes in the internal state of the
Feynman diagram is no longer linear in $A_{y}$ as in the ordinary CS term in
the de-compactified 5D U(1) gauge theory, but some periodic function of
$A_{y}$. We argue how such novel feature of this scenario causes the
modification of the Peccei-Quinn [6] or invisible axion [8, 9] solutions of
the strong CP problem.

In the literature there are works related to the present paper [12]. In these
works, however, the CS term is not derived by concrete calculations of Feynman
diagrams, with all KK modes being taken into account in the intermediate state
of the loop diagrams. Thus, they do not address the periodicity with respect
to the extra-space component $A_{y}$ nor the non-linearity of the “Chern-
Simons” term with respect to the $A_{y}$, which are among the main issues of
the present paper.

##  2 The model

In order to discuss the mechanism to solve the strong CP problem, we adopt a
simple toy model, i.e. 5-dimensional (5D) SU(3)${}_{\rm c}\times$ U(1)X gauge
theory, where SU(3)c stands for QCD and U(1)X is some unknown gauge symmetry.
As the matter fields, in addition to two flavors of quarks, $u^{\alpha},\
d^{\alpha}$ ($\alpha=1,2,3$: color index) we introduce a colored Dirac fermion
$\psi^{\alpha}$, which carries U(1)X “charge” $g$ (the coupling constant of
QCD is denoted by $g_{s}$), while aforementioned quarks do not possess the
U(1)X interaction. The extra dimension is assumed to be a circle $S^{1}$ of
the radius $R$.

We mainly focus on a part of lagrangian, which is relevant for our purpose:

${\cal L}={\cal L}_{0}+{\cal L}_{\theta}+{\cal L}_{CS}.$ (1)

${\cal L}_{0}$ is the basic part describing the gauge interactions of matter
fermions. The 5D field of $\psi$ is supposed to satisfy an anti-periodic
boundary condition $\psi(x^{\mu},y+2\pi R)=-\psi(x^{\mu},y)$ for the
convenience of later discussions, while quarks and all gauge fields are
supposed to satisfy the periodic boundary conditions:

$\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle\sum_{n}\overline{\psi^{(n)}}\\{\gamma^{\mu}(i\partial_{\mu}+g_{s}G_{\mu}+gA_{\mu})+(i\gamma_{5})\left(\frac{n+\frac{1}{2}}{R}+g_{s}G_{y}+gA_{y}\right)-M\\}\psi^{(n)}$
(2) $\displaystyle+$
$\displaystyle\bar{u}\\{\gamma^{\mu}(i\partial_{\mu}+g_{s}G_{\mu})+(i\gamma_{5})g_{s}G_{y}-m_{u}\\}u+(u\
\to\ d),$

where $G_{M}\equiv G^{a}_{M}\frac{\lambda^{a}}{2}=(G_{\mu},G_{y})$ is 5D gluon
field ($\lambda^{a}:{\rm Gell-Mann\ matrices})$ and $A_{M}=(A_{\mu},A_{y})$ is
the 5D U(1)X gauge field. The color indices have been suppressed (for
instance, $u$ is understood to be a color triplet:
$u=(u^{1},u^{2},u^{3})^{t}$). $M,m_{u,d}$ are bulk masses of $\psi$ and
quarks, respectively, which are all assumed to be positive unless otherwise
stated.

Though we start from the 5D theory, the lagrangian given in eq.(1) and eq.(2)
are their effective 4D theories, where the fields $u,d$ and all gauge fields
should be understood to denote only their KK zero-modes, while concerning the
field $\psi$, all KK modes $\psi^{(n)}$ ($n$: integer) are included, since
they all participate in the intermediate state of the loop diagram discussed
below. One remark is that, altough we adopt the anti-periodic boundary
condition for $\psi$, actually we do not loose the generality of our
discussion by this choice. For instance, if we wish it is possible to move to
a base where all fields satisfy the periodic boundary conditions by making a
gauge transformation, $\psi\to\psi^{\prime}=e^{-i\frac{y}{2R}}\psi,\ A_{y}\to
A^{\prime}_{y}=A_{y}-\frac{1}{2gR}$, so that $\psi^{\prime}(x^{\mu},y+2\pi
R)=\psi^{\prime}(x^{\mu},y)$. We do not discuss here how to remove unnecessary
and undesirable interactions, especially those due to $G_{y}$-exchange.

Another piece in eq.(1), ${\cal L}_{\theta}$ including a parameter $\theta$,
reflects the non-trivial vacuum structure of QCD, i.e. the theta vacuum:

${\cal
L}_{\theta}=-\theta\frac{g_{s}^{2}}{32\pi^{2}}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}),$ (3)

where $G_{\mu\nu}\equiv G^{a}_{\mu\nu}\frac{\lambda^{a}}{2}$
($a=1,2,\cdots,8$) is the field strength with respect to the gluon field
$G_{\mu}$. To be precise, the $\theta$ should be replaced by
$\bar{\theta}=\theta+{\rm arg}({\rm det}M_{q})$, with $M_{q}$ being the mass
matrix for quarks, though we simply write $\bar{\theta}$ as $\theta$ in this
paper.

Finally, the “Chern-Simons (CS) term” ${\cal L}_{CS}$ does not exist in the
original action, but is induced through quantum correction as we will see
below:

${\cal
L}_{CS}=\frac{g_{s}^{2}}{32\pi^{2}}f(A_{y})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}).$ (4)

It is a remarkable feature of this scenario that $A_{y}$ in what we expect
from the 5D CS term, $A_{y}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma})$, is inevitably replaced by an odd function
$f(A_{y})$ of $A_{y}$ ($f(-A_{y})=-f(A_{y})$). It is also periodic in $A_{y}$:
$f(A_{y}+\frac{1}{gR})=f(A_{y})$, which is nothing but the periodicity of the
Wilson-loop. As was discussed in the introduction, this property is a natural
consequence of the fact that the KK zero mode of $A_{y}$ has a physical
meaning as a Wilson-loop (Aharonov-Bohm) phase when the extra space is non-
simply-connected space, i.e. $S^{1}$, and therefore all physical observables
should be periodic in $A_{y}$.

When $f(A_{y})$ is Taylor-expanded in eq.(4), the linear term of $A_{y}$,
corresponding to the ordinary CS term, is expected to arise generally even in
the theories in 4D space-time, where colored matter fields couple with some
pseudo-scalar field. In the minimal supersymmetric standard model (MSSM), for
instance, the physically remaining pseudo-scalar $A$ decays into two gluons,
$A\to gg$, through the effective operator similar to the CS term,
$A\epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma})$, while the
Higgs decay $h\to gg$ is handled by an operator $h{\rm
Tr}(G^{\mu\nu}G_{\mu\nu})$. The calculations of such effective operators in
ordinary 4D space-time have been summarized in the literature [13]. In the
next section, we will derive the “CS term” eq.(4) in our 5D model by taking
the contributions of all KK modes into account in the intermediate state of
the loop-diagram, which inevitably leads to the periodic function $f(A_{y})$.
This function may be understood as the contribution of (the imaginary part of)
the Wilson-loop (and its arbitrary powers), which is never expected in the 4D
theories.

##  3 The “Chern-Simons” term

In this section, we derive the “Chern-Simons” (CS) term. More precisely, we
determine the function $f(A_{y})$ in eq.(4) by two methods, i.e. first by the
direct calculation of the relevant 1-loop Feynman diagram and next by
utilizing the Fujikawa’s method to derive the chiral anomaly in 4D space-time
[10].

###  3.1 The derivation by the direct calculation of the 1-loop diagram

In this subsection, we perform the direct calculation of the 1-loop diagram,
contributing to the operator of the form
$f(A_{y})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}\\{(\partial_{\mu}G_{\nu}-\partial_{\nu}G_{\mu})(\partial_{\rho}G_{\sigma}-\partial_{\sigma}G_{\rho})\\}$,
which is a part of eq.(4). Its linear term in $A_{y}$ may be written in the
language of the differential form as ${\rm Tr}(AF^{2})$, ($A,\ F$ are gauge
field 1-form and field strength 2-form of a generic gauge group). Note that,
although the 5D Chern-Simons form is generally written as ${\rm
Tr}\\{AF^{2}+\frac{1}{2}A^{3}F+\frac{1}{10}A^{5}\\}$, when the gauge group
contains U(1) factor, as in the case of our model, $A^{3}F,\ A^{5}$ terms just
vanish and we do not have to worry about their contributions.

We just calculate a 1-loop triangle Feynman diagram, contributing to the
transition $A_{y}\to GG$. At the first glance, however, it seems that the
calculation provides only the term linear in $A_{y}$. To get $f(A_{y})$ with
infinite powers of $A_{y}$, the calculations of arbitrary numbers of $A_{y}$
field in the external line seem to be necessary. This, however, will be
practically impossible.

We thus need some manipulation. Let us suppose that $A_{y}$ has a vacuum
expectation value (VEV) $\bar{A}_{y}$, which really is the case as we will see
below, and define a dynamical field $\hat{A}_{y}$ so that
$A_{y}=\hat{A}_{y}+\bar{A}_{y}$. Then we calculate the triangle diagram to
describe $\hat{A}_{y}\to GG$, where in the propagators of $\psi^{(n)}$ in its
internal line we include the contribution of $g\bar{A}_{y}$ as if it behaves
as a mass (of pseudo-scalar type) provided by the VEV.

On the other hand, if we obtain an operator
$\frac{g_{s}^{2}}{32\pi^{2}}f(A_{y})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}\\{(\partial_{\mu}G_{\nu}-\partial_{\nu}G_{\mu})(\partial_{\rho}G_{\sigma}-\partial_{\sigma}G_{\rho})\\}$,
the amplitude for the transition $\hat{A}_{y}\to GG$ is readily obtained by
replacing $A_{y}$ by $\hat{A}_{y}+\bar{A}_{y}$ and picking up a term linear in
$\hat{A}_{y}$:

$\frac{g_{s}^{2}}{32\pi^{2}}f^{\prime}(\bar{A}_{y})\hat{A}_{y}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}\\{(\partial_{\mu}G_{\nu}-\partial_{\nu}G_{\mu})(\partial_{\rho}G_{\sigma}-\partial_{\sigma}G_{\rho})\\},$
(5)

where $f^{\prime}(\bar{A}_{y})$ denotes the first derivative of
$f(\bar{A}_{y})$. Thus, calculating the triangle diagram in a way stated
above, it is able to get $f^{\prime}(\bar{A}_{y})$. Then, by replacing it by
$f^{\prime}(A_{y})$ and performing an indefinite integral, imposing a
condition that $f(A_{y})$ should vanish at $A_{y}=0$ (since $f(A_{y})$ should
be an odd function of $A_{y}$ (to preserve the P symmetry (from the viewpoint
of the ordinary 4D space-time) of the theory), we can completely determine the
function $f(A_{y})$, which is also expected to be periodic in $A_{y}$ as the
result of the summation of all KK modes’ contributions in the internal line.

Here is a remark. It should be noticed that the “CS term” inevitably vanishes
for $M=0$, since both of $M$ and the CS term are “parity-odd” from the
viewpoint of odd-dimensional space-time, such as 5D space-time.111We will see
below that in the specific limit of de-compactification, $R\to\infty$, the CS
term survives even for $M=0$ with $M$-independent coefficient, as is suggested
by the index theorem. This sounds a little strange as the scalar-type mass
term $M\bar{\psi}\psi$ is known to be invariant under the ordinary parity
transformation in 4D space-time, $P^{(4D)}:\ \psi\to\gamma^{0}\psi$. We know
that in 4D space-time the space-inversion $\vec{x}\to-\vec{x}$ caused by this
$P^{(4D)}$ is equivalent to the mirror reflection as a discrete
transformation, since they are mutually related by a space rotation. In the 5D
space-time, on the other hand, the $P^{(4D)}$ transformation causes
$(\vec{x},y)\to-(\vec{x},y)$. This is no longer a discrete transformation but
one of the space rotations, which is an automatic symmetry of the theory and
is not useful for investigating the property of the CS term. If we adopt an
alternative discrete transformation, say $P^{(5D)}:\
\psi\to\gamma^{0}\gamma_{5}\psi$, causing $(\vec{x},y)\to(-\vec{x},y)$,
$\bar{\psi}\psi$ is easily known to be odd under this transformation, while
$\bar{\psi}(i\gamma_{5})\psi$ is invariant instead. In such a sense, $M$ is a
unique parity-odd parameter in the lagrangian for $\psi$, while the CS term
itself also becomes odd under the $P^{(5D)}$. Therefore, the coefficient of
the CS term should be accompanied by some odd function of $M$ and vanishes for
$M=0$.

Now we calculate the triangle diagram for the process $\hat{A}_{y}(p+q)\to
G^{(a)}_{\nu}(p)+G^{(a)}_{\sigma}(q)$ shown in the left-hand side of Fig.1,
with $p,q$ being 4-momenta carried by $G^{(a)}_{\nu}$ and $G^{(a)}_{\sigma}$,
respectively. Actually, what we are interested in is the operator with
derivatives on the gluon fields, $\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}\\{(\partial_{\mu}G_{\nu})(\partial_{\rho}G_{\sigma})\\}$. Since
$\partial_{\mu}$ on $G^{(a)}_{\nu}(p)$ is equivalent to the multiplication of
$G^{(a)}_{\nu}(p)$ by $ip_{\mu}$ in the momentum space, for instance, we pick
up the term, which is linear in both of $p_{\mu}$ and $q_{\rho}$, by taking
derivatives of the triangle diagram with respect to $p_{\mu}$ and $q_{\rho}$
and then setting $p=q=0$. Thus, effectively what we calculate is a pentagon
diagram with 5 internal propagators, as is shown in the right-hand side of
Fig.1. The Wilson coefficient of the operator
$\hat{A}_{y}(\partial_{\mu}G^{(a)}_{\nu})(\partial_{\rho}G^{(a)}_{\sigma})$ is
known to be given as (by noting ${\rm
Tr}(T^{(a)}T^{(b)})=\frac{1}{2}\delta^{ab}$)

$\displaystyle-i\frac{gg_{s}^{2}}{2}\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{[k^{2}-(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}-M^{2}]^{5}}$
$\displaystyle\cdot{\rm
Tr}\\{i\gamma_{5}[{\ooalign{\hfil/\hfil\crcr$k$}}+i\gamma_{5}(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})+M]\gamma^{\mu}[{\ooalign{\hfil/\hfil\crcr$k$}}+i\gamma_{5}(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})+M]\gamma^{\nu}$
(10)
$\displaystyle\cdot[{\ooalign{\hfil/\hfil\crcr$k$}}+i\gamma_{5}(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})+M]\gamma^{\sigma}[{\ooalign{\hfil/\hfil\crcr$k$}}+i\gamma_{5}(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})+M]\gamma^{\rho}$
(15)
$\displaystyle\cdot[{\ooalign{\hfil/\hfil\crcr$k$}}+i\gamma_{5}(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})+M]\\}.$
(18)

Figure 1: The Feynman diagram generating the “Chern-Simons” term

By picking up terms with odd powers of $i\gamma_{5}$ inside the “Tr”, the
numerator of the integrand is simplified (after some lengthy but
straightforward arithmetic) into

$\\{(k^{2})^{2}M-2k^{2}M^{3}+M^{5}-2M(k^{2}-M^{2})(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}+M(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{4}\\}{\rm
Tr}(i\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}),$ (19)

where we have used the properties (for all different $\mu,\ \nu,\rho,\sigma$
and under the integral $\int d^{4}k$)

${\ooalign{\hfil/\hfil\crcr$k$}}\gamma^{\mu}{\ooalign{\hfil/\hfil\crcr$k$}}=-\frac{k^{2}}{2}\gamma^{\mu},\
{\ooalign{\hfil/\hfil\crcr$k$}}\gamma^{\mu}\gamma^{\nu}{\ooalign{\hfil/\hfil\crcr$k$}}=0,\
{\ooalign{\hfil/\hfil\crcr$k$}}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}{\ooalign{\hfil/\hfil\crcr$k$}}=\frac{k^{2}}{2}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho},\
{\ooalign{\hfil/\hfil\crcr$k$}}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}{\ooalign{\hfil/\hfil\crcr$k$}}=-k^{2}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma},$
(20)

and ${\rm
Tr}(i\gamma_{5}{\ooalign{\hfil/\hfil\crcr$k$}}\cdots{\ooalign{\hfil/\hfil\crcr$k$}})=-k^{2}{\rm
Tr}(i\gamma_{5}\cdots)$. Note that ${\rm
Tr}(i\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho})$ survives
only when all of $\gamma^{\mu},\gamma^{\nu},\gamma^{\sigma},\gamma^{\rho}$ are
different.

As ${\rm
Tr}(i\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho})=4\epsilon^{\mu\nu\sigma\rho}$,
we have obtained an operator
$\epsilon^{\mu\nu\sigma\rho}\hat{A}_{y}(\partial_{\mu}G^{(a)}_{\nu})(\partial_{\rho}G^{(a)}_{\sigma})=-\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}\hat{A}_{y}(\partial_{\mu}G^{(a)}_{\nu}-\partial_{\nu}G^{(a)}_{\mu})(\partial_{\rho}G^{(a)}_{\sigma}-\partial_{\sigma}G^{(a)}_{\rho})$,
which is nothing but the one in eq.(5), whose Wilson coefficient is given by

$\displaystyle\frac{g_{s}^{2}}{32\pi^{2}}f^{\prime}(\bar{A}_{y})$ (21)
$\displaystyle=$ $\displaystyle
igg_{s}^{2}M\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{(k^{2})^{2}-2k^{2}M^{2}+M^{4}-2(k^{2}-M^{2})(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}+(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{4}}{[k^{2}-(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}-M^{2}]^{5}}$
$\displaystyle=$ $\displaystyle
igg_{s}^{2}M\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{[k^{2}-(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}-M^{2}]^{3}}.$

By use of

$\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-\Delta)^{3}}=-\frac{i}{(4\pi)^{2}}\frac{1}{2}\frac{1}{\Delta},$
(22)

we find

$f^{\prime}(\bar{A}_{y})=gM\sum_{n}\frac{1}{(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}+M^{2}}.$
(23)

The KK mode sum is performed utilizing a mathematical formula,

$\sum_{n}\frac{1}{(\frac{n+\frac{1}{2}}{R}+g\bar{A}_{y})^{2}+M^{2}}=\frac{\pi
R}{M}\frac{\sinh(2\pi RM)}{\cosh(2\pi RM)+\cos(2\pi gR\bar{A}_{y})},$ (24)

to get

$f^{\prime}(\bar{A}_{y})=g(\pi R)\frac{\sinh(2\pi RM)}{\cosh(2\pi
RM)+\cos(2\pi gR\bar{A}_{y})}.$ (25)

The final task is to perform the indefinite integral of $f^{\prime}(A_{y})$,
obtained by replacing $\bar{A}_{y}$ by $A_{y}$ in eq.(25):

$f(A_{y})=g(\pi R)\ \int\ \frac{\sinh(2\pi RM)}{\cosh(2\pi RM)+\cos(2\pi
gRA_{y})}\ dA_{y}.$ (26)

Unfortunately, it is hard to perform this integral. We, however, succeed in
writing the $f(A_{y})$ in the form of KK mode-sum. Namely, the indefinite
integral of $\frac{1}{(\frac{n+\frac{1}{2}}{R}+gA_{y})^{2}+M^{2}}$, which is
obtained by replacing $\bar{A}_{y}$ by $A_{y}$ in eq.(23), is able to be
performed by the change of integral variable $A_{y}\ \to\ \alpha_{n}$, so that
$\frac{n+\frac{1}{2}}{R}+gA_{y}=M\cot\alpha_{n}$, i.e.
$\alpha_{n}=\cot^{-1}\left(\frac{\frac{n+\frac{1}{2}}{R}+gA_{y}}{M}\right)=\tan^{-1}\left(\frac{M}{\frac{n+\frac{1}{2}}{R}+gA_{y}}\right)$:

$\int\frac{1}{(\frac{n+\frac{1}{2}}{R}+gA_{y})^{2}+M^{2}}\
dA_{y}=-\frac{1}{gM}\int\ d\alpha_{n}=-\frac{1}{gM}\alpha_{n},$ (27)

where an additional constant of integral is not allowed, since its presence
contradicts with the condition that $f(A_{y})$ is an odd function of $A_{y}$.
Hence, from eq.(23) $f(A_{y})$ is found to be

$f(A_{y})=gM\sum_{n}\int\frac{1}{(\frac{n+\frac{1}{2}}{R}+gA_{y})^{2}+M^{2}}\
dA_{y}=-\sum_{n}\alpha_{n}=-\sum_{n}\tan^{-1}\left(\frac{M}{\frac{n+\frac{1}{2}}{R}+gA_{y}}\right).$
(28)

$f(A_{y})$ is an odd-function of $A_{y}$, as we easily see by replacing
$A_{y}$ by $-A_{y}$ and $n+\frac{1}{2}$ by $-n-\frac{1}{2}$ in eq.(28), in
particular $f(0)=0$. Note that if we adopted a periodic boundary condition for
5D field $\psi$, we would have obtained
$f(A_{y})=-\sum_{n}\tan^{-1}(\frac{M}{\frac{n}{R}+gA_{y}})$, which is
discontinuous at the origin $A_{y}=0$ because of the contribution of the KK
zero-mode: $\lim_{A_{y}\to\pm 0}f(A_{y})=\mp\frac{\pi}{2}$. This is the reason
why we chose the anti-periodic boundary condition for $\psi$.

Thus we finally have succeeded in getting the “Chern-Simons” term, though it
is not linear in $A_{y}$, but a periodic function of $A_{y}$:

$\displaystyle{\cal L}_{CS}$ $\displaystyle=$
$\displaystyle\frac{g_{s}^{2}}{32\pi^{2}}f(A_{y})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma})$ (29) $\displaystyle=$
$\displaystyle-\frac{g_{s}^{2}}{32\pi^{2}}\sum_{n}\tan^{-1}\left(\frac{M}{\frac{n+\frac{1}{2}}{R}+gA_{y}}\right)\
\cdot\ \epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma}).$

As we anticipated, this CS term vanishes for $M=0$.

Here appears a question whether the KK mode-sum in eq.(29) is convergent, i.e.
whether the CS term is free from UV-divergence. When we consider the UV-
divergence, it is enough to think about what happens in the de-compactified
limit, where the KK mode-sum is reduced to the integral $\int dk_{y}$
($k_{y}$: the 5-th component of the 5D momentum). In the 5D space-time the
operator linear in $A_{y}$, $gg_{s}^{2}A_{y}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma})$ in the CS term has mass dimension 5 (with gauge
couplings being included in the gauge fields, $gA_{y}$ for instance). So at
the first glance it seems that its Wilson coefficient suffers from a
logarithmic UV-divergence. Actually, however, as we have already seen, the
coefficient should be accompanied by an overall factor $M$, as it vanishes for
$M=0$, and the momentum integral behaves at the UV-region as $\int
d^{5}k/(k^{2})^{3}$, which is convergent. Thus, the predicted CS term should
be free from UV-divergences. This argument is supported by rewriting eq.(29)
as follows,

${\cal
L}_{CS}=\frac{g_{s}^{2}}{32\pi^{2}}\sum_{n=1}^{\infty}\tan^{-1}\left(\frac{8gR^{2}MA_{y}}{(2n-1)^{2}+4R^{2}[M^{2}-(gA_{y})^{2}]}\right)\
\cdot\ \epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma}),$ (30)

which should be convergent, since for large $n$, the mode-sum behaves as
$\sim\sum_{n}(1/(2n-1)^{2})\simeq(1/4)\sum_{n}(1/n^{2})$.

Interestingly, at the limit of de-compactification of 5D space-time,
$R\to\infty$, eq.(29) is found to be reduced to the ordinary 5D Chern-Simons
term. First we note that in the Taylor expansion of $f(A_{y})$ around its
origin, from eq.(25) the first derivative is reduced to

$\lim_{R\to\infty}\frac{f^{\prime}(0)}{2\pi R}=\frac{g}{2},$ (31)

where the factor $1/(2\pi R)$ is added since in general the lagrangians for
the KK zero-mode in 4D and 5D space-time, say ${\cal L}_{{\rm 4D}},\ {\cal
L}_{{\rm 5D}}$, are related by ${\cal L}_{{\rm 4D}}=(2\pi R){\cal L}_{{\rm
5D}}$. We also easily realize from eq.(25) that the higher derivatives
$f^{(2n+1)}(0)\ (n=1,2,\cdots)$, denoting $2n+1$-th derivatives, all vanish in
the limit of $RM\to\infty$ because of the exponential suppression factor
$e^{-2\pi RM}$ ($f^{(2n)}(0)\ (n=1,2,\cdots)$ identically vanish, as
$f(A_{y})$ is an odd function). Thus, in the limit of de-compactification,
only the ordinary 5D Chern-Simons term is left:

${\cal L}_{{\rm
5D-CS}}=\frac{gg_{s}^{2}}{64\pi^{2}}A_{y}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}),$ (32)

whose coefficient is independent of $M$ and also UV-finite, as we expected. To
be precise, although we have assumed that $M$ is positive, it is easy to
confirm that in general eq.(32) is accompanied by $\frac{M}{|M|}={\rm
sgn}(M)$. Let us note that the combinations $gA_{y},\ g_{s}G_{\mu}$ can be
readily re-interpreted as the corresponding 5D quantities.

Thus, in this specific limit, the periodicity in $A_{y}$ is lost. This may be
a natural result, since in the limit of de-compactification, the non-local
operator, Wilson-loop, loses its physical importance and only the local “gauge
invariant” (in the sense that gauge transformation yields a total derivative)
operator is allowed.

###  3.2 The derivation based on the Fujikawa’s method

In the previous subsection, the “Chern-Simons” term eq.(29) was derived by the
explicit calculation of 1-loop Feynman diagram. It will be natural to ask
whether it is modified by the quantum corrections at higher loops, such as
two-loop correction. From this point of view, it may be suggestive to recall
that the 1-loop contribution to the chiral anomaly is exact, i.e. there do not
exist corrections from higher loops. Since the operator eq.(29) has much
similarity to the chiral anomaly, it may be reasonable to expect that eq.(29)
is actually an exact result. In this section, we would like to demonstrate
that this really is the case, utilizing the Fujikawa’s interpretation of the
chiral anomaly [10]: it is solely due to the change of measure for the chiral
transformation in the language of fermionic path-integral. This may be a
natural consequence, since the relation of parity non-conservation due to the
CS term in odd space-time dimensions and the chiral anomaly in even dimensions
has been known for some time [11].

We start from the lagrangian for the fermionic field $\psi$, which include all
KK modes $n$ (refer to eq.(2)):

${\cal
L}_{\psi}=\sum_{n}\overline{\psi^{(n)}}\\{\gamma^{\mu}(i\partial_{\mu}+g_{s}G_{\mu})+(i\gamma_{5})(\frac{n+\frac{1}{2}}{R}+gA_{y})-M\\}\psi^{(n)},$
(33)

where the parts including the interactions due to $A_{\mu}$ and $G_{y}$ have
been ignored, since they are unnecessary for our argument below.

In the discussion on the CS term based on the symmetry under $P^{(5D)}$ we
learned that ${\cal L}_{CS}$ should vanish exactly for $M=0$. However, even
for non-vanishing $M$ we realize that the scalar-type mass term
$-M\overline{\psi^{(n)}}\psi^{(n)}$ may be eliminated by performing a suitable
chiral transformation of $\psi^{(n)}$. First, note that the mass operator for
$\psi^{(n)}$ as the whole can be written as

$(i\gamma_{5})(\frac{n+\frac{1}{2}}{R}+gA_{y})-M=(i\gamma_{5})\sqrt{(\frac{n+\frac{1}{2}}{R}+gA_{y})^{2}+M^{2}}\cdot
e^{2i\gamma_{5}\kappa_{n}},$ (34)

where $\tan(2\kappa_{n})=M/(\frac{n+\frac{1}{2}}{R}+gA_{y})$. Then, it is easy
to see that by performing a chiral transformation

$\psi^{(n)}\ \to\ \psi^{\prime(n)}=e^{i\gamma_{5}\kappa_{n}}\psi^{(n)},$ (35)

the scalar-type mass term is eliminated. Then we have to conclude that the
quantum corrections due to the exchange of $\psi^{\prime(n)}$ never yield
${\cal L}_{CS}$ in all orders of perturbation, since the transformed
lagrangian preserves $P^{(5D)}$ symmetry. We, however, have to be a little
careful. Namely, the chiral transformation leads to an additional term in the
lagrangian, originating from the change of the measure in the path integral of
$\psi^{(n)}$, i.e. from the chiral anomaly [10]:

$-\frac{g_{s}^{2}}{32\pi^{2}}(2\kappa_{n})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}).$ (36)

Summing up the contributions of all KK modes, we find

$\displaystyle{\cal L}_{{\rm anomaly}}$ $\displaystyle=$
$\displaystyle-\sum_{n}\frac{g_{s}^{2}}{32\pi^{2}}(2\kappa_{n})\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma})$ (37) $\displaystyle=$
$\displaystyle-\frac{g_{s}^{2}}{32\pi^{2}}\sum_{n}\tan^{-1}\left(\frac{M}{\frac{n+\frac{1}{2}}{R}+gA_{y}}\right)\
\cdot\ \epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma}),$

which is exactly the same as eq.(29).

This result is exact, in the sense that it is correct in all orders of
perturbation. We thus conclude that the result eq.(29) is, say, 1-loop exact.
One may wonder whether the bulk masses of quarks, $m_{u,d}$, also contribute
to the CS term and modify the result. Actually, since quarks do not have
couplings with $A_{y}$, even if chiral transformation for the quark fields,
similar to the one for $\psi$, is made, they do not produce the CS term. It is
interesting to note that this in turn means the CS term is free from UV-
divergence, even though the theory itself, being higher dimensional gauge
theory, is non-renormalizable.

##  4 The mechanism to solve the strong CP problem

A remarkable feature of our model is that it has some similarity to the
Peccei-Quinn (PQ) model [6, 7] or the models of invisible axion [8, 9]. The
essence of the mechanism to solve the strong CP problem in these models would
be to replace the parameter $\theta$ (or more precisely $\bar{\theta}$) by a
dynamical variable by introducing axion field $a$, which is a NG boson due to
the spontaneous breaking of global U(1)PQ symmetry of the theory. Under the
U(1)PQ transformation $a$ is shifted as $a\ \to\ a+c$ ($c$: constant): the
theory has a shift symmetry. The U(1)PQ symmetry is a sort of chiral symmetry,
and since the chiral symmetry has an anomaly
$\partial_{\mu}j^{\mu}_{5}=-\frac{g_{s}^{2}}{32\pi^{2}}\epsilon^{\mu\nu\rho\sigma}G^{a}_{\mu\nu}G^{a}_{\rho\sigma}$,
in the lagrangian there arises the following interaction term of the $a$ with
gluon fields:

$-\frac{g_{s}^{2}}{32\pi^{2}}\frac{a}{V}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}),$ (38)

where $V$ is the typical mass scale relevant for the spontaneous breaking of
the global symmetry. Now the parameter $\theta$ effectively has been replaced
by a dynamical variable $\theta+\frac{a}{V}$. Redefining the axion field as
$\frac{\hat{a}}{V}\equiv\theta+\frac{a}{V}$ ($\hat{a}=a-(-V\theta)$), the
potential for $\hat{a}$ due to QCD interaction is shown to have a minimum at
$\hat{a}=0$. Thus, the P and CP violating parameter $\theta$ is removed from
the theory and the strong CP problem is solved. (In this section, as we
discuss the 4D low-energy effective theory, the P and CP transformations are
defined in the ordinary manner: e.g. P: $u\to\gamma^{0}u$, etc. Hence the
quark masses $m_{u,d}$ are P and CP preserving, as usual.)

We realize that our model has some similarity to these models. Comparing
eq.(4) and eq.(38), we find that the role of the axion field $a$ is played by
$A_{y}$ in our model, though in our case, eq.(4), $f(A_{y})$ is not linear in
$A_{y}$ but a periodic function of $A_{y}$. We notice that $A_{y}$ also has a
shift symmetry, though it is not a NG boson, which is associated with the
local gauge symmetry, not a global symmetry: the symmetry under the local
gauge transformation $A_{y}\ \to\ A_{y}+\partial_{y}\lambda$ ($\lambda$:
$y$-dependent gauge transformation parameter).

We are now ready to discuss our mechanism to solve the strong CP problem.
First, let us focus on the combination of the theta term and the CS term,
eq.(3) and eq.(4),

${\cal L}_{\theta}+{\cal
L}_{CS}=\frac{g_{s}^{2}}{32\pi^{2}}[-\theta+f(A_{y})]\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}).$ (39)

We assume that the VEV $\bar{A}_{y}$ is tuned as

$f(\bar{A}_{y})=\theta,$ (40)

so that eq.(39) vanishes for the VEV. As is seen below, this assumption is
justified by showing that in the low-energy effective theory, the potential
for $\hat{A}_{y}=A_{y}-\bar{A}_{y}$ and pion fields, both having odd intrinsic
parities being pseudo-scalars, naturally starts from the quadratic terms of
these fields, since our model is P and CP conserving, and therefore its
minimum is expected to be at $\hat{A}_{y}=\pi^{0}=0$.

Then eq.(39) is rewritten by use of ${\cal F}(\hat{A}_{y})\equiv
f(\bar{A}_{y}+\hat{A}_{y})-f(\bar{A}_{y})=f^{\prime}(\bar{A}_{y})\hat{A}_{y}+\cdots\
({\cal F}(0)=0)$, as

${\cal L}_{\theta}+{\cal L}_{CS}=\frac{g_{s}^{2}}{32\pi^{2}}{\cal
F}(\hat{A}_{y})\epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma}).$
(41)

In this way, the lagrangian, now ignoring the field $\psi$ (and assuming that
the unnecessary field $G_{y}$ is somehow removed from the theory), reads as

$\displaystyle{\cal L}$ $\displaystyle\supset$
$\displaystyle\frac{g_{s}^{2}}{32\pi^{2}}{\cal
F}(\hat{A}_{y})\epsilon^{\mu\nu\rho\sigma}{\rm Tr}(G_{\mu\nu}G_{\rho\sigma})$
(42) $\displaystyle+$
$\displaystyle\bar{u}\\{\gamma^{\mu}(i\partial_{\mu}+g_{s}G_{\mu})-m_{u}\\}u+(u\
\to\ d).$

Now we invoke the chiral transformation of quark fields in order to eliminate
the first line of eq.(42), proportional to ${\cal F}(\hat{A}_{y})$. The
purpose of this procedure is to get the potential of $\hat{A}_{y}$ (and the
neutral pion field $\pi^{0}$) expected in the low-energy effective theory. We
perform chiral transformations of the quark fields $u,\ d$:

$u\ \to e^{i\gamma_{5}\alpha_{u}}u,\ \ \ d\ \to e^{i\gamma_{5}\alpha_{d}}d.$
(43)

Then the quark mass term in eq.(42) is modified into
$-m_{u}\bar{u}e^{2i\gamma_{5}\alpha_{u}}u+(u\ \to\ d)$. In addition, because
of the chiral anomaly, the lagrangian acquires an additional term given by

$\frac{g_{s}^{2}(\alpha_{u}+\alpha_{d})}{16\pi^{2}}\epsilon^{\mu\nu\rho\sigma}{\rm
Tr}(G_{\mu\nu}G_{\rho\sigma}).$ (44)

Now by comparing eq.(42) and eq.(44), a chiral transformation, satisfying

$\alpha_{u}+\alpha_{d}=-\frac{1}{2}{\cal F}(\hat{A}_{y})\ \ \to\ \
\alpha_{u}=-a_{u}\frac{1}{2}{\cal F}(\hat{A}_{y}),\
\alpha_{d}=-a_{d}\frac{1}{2}{\cal F}(\hat{A}_{y}),$ (45)

is known to eliminate the first line of eq.(42). Here $a_{u,d}$ are constant
coefficients satisfying $a_{u}+a_{d}=1$.

The resulting low-energy effective lagrangian for $\hat{A}_{y}$ and $\pi^{0}$
is obtained by making the following replacements [14]:

$\displaystyle\bar{u}u\ \ \to\ \ -v\cos(\frac{\pi_{0}}{F_{\pi}}),\ \ \bar{d}d\
\ \to\ \ -v\cos(\frac{\pi_{0}}{F_{\pi}}),$ (46)
$\displaystyle\bar{u}\gamma_{5}u\ \ \to\ \ -iv\sin(\frac{\pi_{0}}{F_{\pi}}),\
\ \bar{d}\gamma_{5}d\ \ \to\ \ iv\sin(\frac{\pi_{0}}{F_{\pi}}),$ (47)
$\displaystyle\bar{u}\gamma^{\mu}\gamma_{5}u\ \ \to\ \
\frac{1}{2}F_{\pi}\partial^{\mu}\pi^{0},\ \ \bar{d}\gamma^{\mu}\gamma_{5}d\ \
\to\ \ -\frac{1}{2}F_{\pi}\partial^{\mu}\pi^{0},$ (48)

where $v=|\langle\bar{u}u\rangle|=|\langle\bar{d}d\rangle|$ and $F_{\pi}$ is
the decay constant of the pions.

The relevant part of such obtained low energy effective lagrangian is given as

$\displaystyle{\cal L}_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\pi^{0})(\partial^{\mu}\pi^{0})+\frac{1}{2}(\partial_{\mu}\hat{A}_{y})(\partial^{\mu}\hat{A}_{y})+\frac{a_{u}-a_{d}}{4}F_{\pi}(\partial_{\mu}\pi^{0})(\partial^{\mu}{\cal
F}(\hat{A}_{y}))$ (49) $\displaystyle+$ $\displaystyle m_{u}\
v\cos\left[\frac{\pi^{0}}{F_{\pi}}-a_{u}{\cal F}(\hat{A}_{y})\right]+m_{d}\
v\cos\left[\frac{\pi^{0}}{F_{\pi}}+a_{d}{\cal F}(\hat{A}_{y})\right].$

To simplify the analysis we use the freedom of taking a choice of
$a_{u}=a_{d}=\frac{1}{2}$. Then the mixing in the kinetic term between
$\hat{A}_{y}$ and $\pi^{0}$ disappears and these fields become correctly
normalized fields. Thus, we finally get the effective potential for these two
pseudo-scalar fields,

$V_{eff}=-m_{u}\ v\cos\left[\frac{\pi^{0}}{F_{\pi}}-\frac{1}{2}{\cal
F}(\hat{A}_{y})\right]-m_{d}\
v\cos\left[\frac{\pi^{0}}{F_{\pi}}+\frac{1}{2}{\cal F}(\hat{A}_{y})\right].$
(50)

The quadratic part of $\pi^{0}$ and $\hat{A}_{y}$ in eq.(50) is easily
calculated by noting ${\cal
F}(\hat{A}_{y})=f^{\prime}(\bar{A}_{y})\hat{A}_{y}+\cdots$, and the mass-
squared matrix $M_{ps}^{2}$ in the basis of $\pi^{0}$ and $\hat{A}_{y}$ is
given as

$M_{ps}^{2}=\begin{pmatrix}\frac{v}{F_{\pi}^{2}}(m_{u}+m_{d})&-\frac{v}{F_{\pi}F_{A_{y}}}(m_{u}-m_{d})\cr-\frac{v}{F_{\pi}F_{A_{y}}}(m_{u}-m_{d})&\frac{v}{F_{A_{y}}^{2}}(m_{u}+m_{d})\cr\end{pmatrix},$
(51)

where $F_{A_{y}}\equiv\frac{2}{f^{\prime}(\bar{A}_{y})}$. Obviously Tr
$M_{ps}^{2}>0$ and also det
$M_{ps}^{2}=4\frac{v^{2}}{F_{\pi}^{2}F_{A_{y}}^{2}}m_{u}m_{d}>0$. Thus the two
real eigenvalues of this matrix are both positive, and we conclude that the
minimum of the potential is at $\hat{A}_{y}=\pi^{0}=0$, as we expected. Hence,
the strong CP problem is resolved.

Under a reasonable assumption (see the discussions below) of $F_{A_{y}}\gg
F_{\pi}$, one eigenvalue of $M_{ps}^{2}$ gives the well-known formula for the
pion mass: $m_{\pi}^{2}=\frac{v}{F_{\pi}^{2}}(m_{u}+m_{d})$. Having the det
$M_{ps}^{2}$, another eigenvalue is readily obtained, which is nothing but the
mass-squared of the axion-like particle, whose dominant component is
$\hat{A}_{y}$:

$m_{A_{y}}^{2}=\frac{v}{F_{A_{y}}^{2}}\frac{4m_{u}m_{d}}{m_{u}+m_{d}}=\left(\frac{F_{\pi}}{F_{A_{y}}}\right)^{2}\frac{4m_{u}m_{d}}{(m_{u}+m_{d})^{2}}m_{\pi}^{2}.$
(52)

The size of the mass scale $F_{A_{y}}$ is limited by astrophysical and
cosmological arguments [14, 15]. As a rough estimation, the observation of the
supernova SN1987A puts an lower bound $F_{A_{y}}>10^{10}$ GeV, while
cosmological arguments suggest $F_{A_{y}}<10^{12}$ GeV. Thus rather narrow
window is left for the allowed region: $10^{10}{\rm GeV}<F_{A_{y}}<10^{12}{\rm
GeV}$. We now investigate how the parameters of our model, such as
$M_{c},M,g$, are constrained by this condition. Here $M_{c}\equiv 1/R$ is the
“compactification mass scale” of the extra dimension. From eq.(25), depending
on the relative size of $M$ compared to $M_{c}$, $F_{A_{y}}$ behaves as

$\displaystyle F_{A_{y}}$ $\displaystyle=$
$\displaystyle\frac{2}{f^{\prime}(\bar{A}_{y})}=\frac{2M_{c}}{\pi
g}\frac{\cosh(2\pi\frac{M}{M_{c}})+\cos(2\pi
g\frac{\bar{A}_{y}}{M_{c}})}{\sinh(2\pi\frac{M}{M_{c}})}$ (53)
$\displaystyle\simeq$
$\displaystyle\begin{cases}\frac{2}{\pi}\left(\frac{M_{c}}{g}\right)\sim\frac{M_{c}}{g}&({\rm
for}\ M\gg M_{c})\\\ \frac{2}{\pi^{2}}\frac{M_{c}^{2}}{gM}\cos^{2}(\pi
g\frac{\bar{A}_{y}}{M_{c}})\sim\frac{M_{c}^{2}}{gM}&({\rm for}\ M\ll
M_{c})\end{cases},$

unless $g\frac{\bar{A}_{y}}{M_{c}}\simeq\frac{1}{2}$ accidentally.

For the case of $M\gg M_{c}$, first let us suppose $g$ is of ${\cal O}(1)$, to
get a rough idea. Then, $M_{c}$ should be of ${\cal O}(10^{10-12})$ GeV, say
the intermediate mass scale between the weak scale $M_{W}$ and the Planck
scale $M_{pl}$. However, if this model is embedded into the electro-weak
gauge-Higgs unification model (on a flat 5D space-time) [16], the
compactification scale $M_{c}$ is expected to be not so far from the weak
scale, say $M_{c}\sim$ 1 to 10 TeV. Then, extremely small gauge coupling $g$
is needed. For instance in the case of $M_{c}\sim 10$ TeV, $g\sim 10^{-7}$ is
required in order to realize $F_{A_{y}}\sim 10^{11}$ GeV.

Another possibility is to invoke the property that $A_{y}$ decouples from the
low energy theory at the limit of $M\to 0$, since in this limit the CS term
itself disappears. In fact, as is seen in eq.(53), $F_{A_{y}}\to\infty$ in
this limit. So even for $g={\cal O}(1)$, the constraint on $M_{c}$ is
considerably relaxed. For instance, assuming $M_{c}\sim 10$ TeV, $M\sim 1$ MeV
realizes $F_{A_{y}}\sim 10^{11}$ GeV. Note that even for such small $M$, the
mass of the colored fermion $\psi^{(0)}$ is estimated to be
$\sqrt{(\frac{1}{2R}+g\bar{A}_{y})^{2}+M^{2}}\sim\frac{M_{c}}{2}\sim 10$ TeV,
and therefore safely evades experimental search.

##  5 Summary and discussion

In this paper we discussed a natural scenario to solve the strong CP problem
in the framework of the higher dimensional gauge theory. It has a similarity
to the well-known Peccei-Quinn [6] or invisible axion [8, 9] solutions of the
strong CP problem. Axion-like particle $A_{y}$ has been built-in as the extra-
space component of the higher dimensional gauge field. Though $A_{y}$ is not a
NG boson associated with a global symmetry, it has a shift symmetry associated
with higher dimensional local gauge symmetry. The coupling of the axion-like
particle $A_{y}$ with gluons is attributed to the “Chern-Simons” (CS) term,
which is radiatively induced. We adopted a toy model to calculate the CS term
with some unknown gauge symmetry U(1)X. The CS term was obtained in two ways:
first by a concrete 1-loop calculation and next by use of the Fujikawa’s
method [10] to deal with the chiral anomaly in 4D space-time. The obtained
results are identical. We argued that this means that the radiative correction
to the CS term is “1-loop exact”, since the result obtained by the latter
method should be exact, valid in all orders of the perturbation. The Wilson
coefficient of the resultant CS term is also free from UV-divergence even
though the theory itself is non-renormalizable higher dimensional theory.

Though the scenario discussed in this paper has some similarity to those with
axion fields, it also has its own novel feature. The loop-induced effective
lagrangian, corresponding to the Chern-Simons term, obtained by summing up all
KK modes in the internal state of the Feynman diagram, is no longer linear in
the field $A_{y}$ as in the ordinary CS term but a periodic function of
$A_{y}$. Actually this is an inevitable consequence of the higher dimensional
gauge theory where the compactified extra space is non-simply-connected
manifold such as $S^{1}$, since in such theories $A_{y}$ has a physical
meaning as the phase of Wilson-loop. We argued how such novel feature of this
scenario causes the modification of the solutions of the strong CP problem
based on the axion fields and discussed how the mass scales of the theory,
$M_{c}=1/R$, $M$ are constrained by the astrophysical and cosmological
observations.

Here is a remark. In the previous section we discussed only the P-conserving
quadratic terms with respect to the pseudo-scalar fields $\hat{A}_{y}$ and
$\pi^{0}$ in $V_{eff}$, eq.(50). Actually, because of the fact that $f(A_{y})$
is not linear in $A_{y}$ and the presence of the VEV $\bar{A}_{y}$, there also
exist terms which are odd powers of the pseudo-scalar fields, such as
$f^{(2)}(\bar{A}_{y})\hat{A}_{y}^{2}\pi^{0}$, which violate P symmetry. Let us
note that $f^{(2n)}(0)=0$ ($n$: positive integer) as $f(A_{y})$ is an odd
function of $A_{y}$, but $f^{(2n)}(\bar{A}_{y})$ are generally non-vanishing.
Namely, such P-violating effects are due to the VEV of the P-odd field
$A_{y}$. Such higher order terms, however, are always accompanied by the
higher inverse powers of $M_{c}=1/R$, since $f(A_{y})$ is the function of
$2\pi gRA_{y}$, the phase of the Wilson-loop. Thus, these higher order terms
are strongly suppressed by the inverse powers of $M_{c}$, which is expected to
be of the order of, say, 10 TeV at least, and will be harmless. In addition,
when the bulk mass $M$ of $\psi$ is larger than $M_{c}$, such higher order
terms get exponential suppression by the powers of $e^{-2\pi RM}=e^{-2\pi
M/M_{c}}$, because of the suppression of higher order derivatives
$f^{(2n)}(\bar{A}_{y})$, as we have seen in the text for the case of the de-
compactification limit.

For the purpose to clarify the essence of the mechanism to solve the strong CP
problem in the framework of higher dimensional gauge theories, in this paper
we adopted a toy model. In order to make the model realistic, there remain
some issues to be settled. The SU(3)${}_{\rm c}\times$ U(1)X model we
discussed is a non-chiral (vector-like) theory, taking $S^{1}$ as the extra
dimension. This choice, however, has a drawback. Namely, the KK zero-mode of
the gluon’s extra-space component $G^{a}_{y}$ remains as a massless state in
the theory, which clearly contradicts with reality. A possible way out of this
problem is to adopt an orbifold $S^{1}$/$Z_{2}$ as the extra dimension,
instead of $S^{1}$, and assign the same overall $Z_{2}$-parity for all
elements of the color triplets. Then $G^{a}_{y}$ has an odd $Z_{2}$-parity and
does not have a massless KK zero-mode, while the KK zero-mode of the ordinary
gluon $G^{a}_{\mu}$, having an even $Z_{2}$-parity, remains massless. On the
other hand, because of the choice of the orbifold, fermions $u,d$ and $\psi$
all become Weyl fermions and the QCD becomes a chiral theory. We are thus
enforced to double the matter fields as $d^{\alpha}_{1},\ d^{\alpha}_{2}$, for
instance, and assign opposite overall $Z_{2}$-parities for each of them, so
that their KK zero-modes behave as $d^{\alpha}_{L},\ d^{\alpha}_{R}$.

In the case of the U(1)X gauge boson, the situation is just opposite. We need
$A_{y}$, not $A_{\mu}$ as the member of the effective low-energy theory. A
possible way to realize this situation is to embed U(1)X in some non-Abelian
gauge group. The simplest candidate for the non-Abelian gauge group will be
SU(2) [16] and we may regard the pair of $\psi^{\alpha}_{1}$,
$\psi^{\alpha}_{2}$ to form a SU(2) doublet. Then $A_{M}$ is identified with a
part of the SU(2) gauge boson, which connects the different elements of the
doublet. Since, $\psi^{\alpha}_{1}$, $\psi^{\alpha}_{2}$ have opposite overall
$Z_{2}$-parities, now $A_{\mu}$ has odd $Z_{2}$-parity and therefore its KK
zero-mode disappears, while $A_{y}$, having even $Z_{2}$-parity, possesses a
massless KK zero-mode, as is expected.

By adopting the orbifold, ordinary bulk mass term is forbidden and should be
replaced by so-called $Z_{2}$-odd bulk mass term, such as
$-\epsilon(y)M(\overline{\psi^{\alpha}_{1}}\psi^{\alpha}_{1}+\overline{\psi^{\alpha}_{2}}\psi^{\alpha}_{2})$
($\epsilon(y)$ takes either of $\pm$ 1 depending on the sign of $y$). Such
mass term is known to lead to an exponential suppression of the KK zero-mode’s
mass. One may wonder whether this implies the possibility of a too light
exotic colored fermion $\psi$, which clearly is undesirable. Fortunately, the
5D field $\psi$ obeys the anti-periodic boundary condition along the extra
dimension and therefore does not possess the KK zero-mode. In any case, more
complete analysis of such extended model is necessary, though we leave it for
further investigation.

### Acknowledgments

We would like to thank K. Hasegawa for his contribution to this work until
some stage of its progress. Thanks are also due to T. Misumi for useful and
informative discussions. This work was supported in part by Japan Society for
the Promotion of Science, Grants-in-Aid for Scientific Research, No. 15K05062.

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