# Neural Image Compression with Quantization Rectifier

Wei Luo  Bo Chen

###### Abstract

Neural image compression has been shown to outperform traditional image codecs
in terms of rate-distortion performance. However, quantization introduces
errors in the compression process, which can degrade the quality of the
compressed image. Existing approaches address the train-test mismatch problem
incurred during quantization, the random impact of quantization on the
expressiveness of image features is still unsolved. This paper presents a
novel quantization rectifier (QR) method for image compression that leverages
image feature correlation to mitigate the impact of quantization. Our method
designs a neural network architecture that predicts unquantized features from
the quantized ones, preserving feature expressiveness for better image
reconstruction quality. We develop a soft-to-predictive training technique to
integrate QR into existing neural image codecs. In evaluation, we integrate QR
into state-of-the-art neural image codecs and compare enhanced models and
baselines on the widely-used Kodak benchmark. The results show consistent
coding efficiency improvement by QR with a negligible increase in the running
time.

Machine Learning, ICML

##  1 Introduction

Neural network (NN)-based image compression methods (Ballé et al., 2016, 2018;
Cheng et al., 2020; Minnen et al., 2018) have shown superior coding efficiency
to those of the conventional compression methods, such as BPG (Bellard, 2018)
and JPEG2000 (Joint Photographic Experts Group, 2000). Quantization
discretizes image features by mapping continuous values to a limited set of
discrete values for entropy coding, compressing the image (Huffman, 1952;
Witten et al., 1987). While current quantization methods address train-test
mismatch, the random effects on feature expressiveness remain unresolved.
Quantization uniformly maps continuous values to a single discrete value,
introducing different degrees of noise depending on feature variability. It
also unpredictably alters feature expressiveness. For instance, the
quantization of features from the range of $[-0.5,0.5)$ to zero introduces
noises in the range of $(-0.5,0.5]$. More importantly, quantization alters the
expressiveness of the latent features in an unpredictable way. In this paper,
we propose a novel quantization rectifier (QR) that leverages spatial
correlation in images to mitigate the impact of quantization. Specifically, we
design a neural network architecture that predicts unquantized features from
the quantized ones. To seamlessly integrate QR into a neural image codec, we
introduce a soft-to-predictive (STP) training method. Here, we first softly
train the original image compression model end-to-end until convergence. Then,
we freeze the encoder network with hard quantization and optimize the decoder
network, along with the QR network. QR bridges the gap between original and
quantized features, preserving feature expressiveness for improved image
reconstruction quality. For evaluation, we incorporate our method into state-
of-the-art neural network-based compression methods (Ballé et al., 2016, 2018;
Cheng et al., 2020; Minnen et al., 2018). We consistently improve all baseline
models by up to 0.21 dB (PSNR) and 0.25 dB (MS-SSIM) without affecting the
bitrate. QR is lightweight, with a minimal increase (0.7-5.4%) in running time
for most baselines.The contributions of this study are summarized as follows:

  * •

We propose QR, a method that corrects quantized image features through
prediction, preserving feature expressiveness and improving coding efficiency.

  * •

We develop the STP training procedure and a hyper-parameter exploration
algorithm, enabling seamless integration of QR with existing neural image
codecs.

  * •

We extensively evaluate QR on state-of-the-art neural image codecs, which
demonstrate the superiority of QR consistently.

##  2 Related Works

Quantization plays a vital role in image compression, enabling efficient
storage and entropy coding (Huffman, 1952; Witten et al., 1987). Recent
advancements, including Additive Uniform Noises (Ballé et al., 2016, 2018) and
Straight-Through Estimator (Mentzer et al., 2018; Theis et al., 2017; Yin et
al., 2019) aim to tackle the train-test discrepancy arising from quantization.
Soft-to-hard annealing (SA) approximates quantization using a differentiable
function resembling hard rounding, but its training is fragile, requiring
empirical determination of the annealing function. The Soft-Then-Hard (STH)
strategy (Guo et al., 2021) first learns a soft latent space and then resolves
the train-test mismatch with hard quantization, partially addressing the
issue. While these approaches address the train-test discrepancy, the
expressiveness of latent features is still unpredictably affected by
quantization. Our proposed approach effectively mitigates the impact of
quantization on feature expressiveness and can be easily integrated into these
techniques.

##  3 Proposed Method

###  3.1 Formulation of Learned Compression Models

According to recent works (Ballé et al., 2018; Cheng et al., 2020; Gao et al.,
2022), the general procedure of neural image compression can be formulated as
follows:

$\displaystyle y$ $\displaystyle=g_{a}(x;\phi)$ (1) $\displaystyle\hat{y}$
$\displaystyle=Q(y)$ (2) $\displaystyle\hat{x}$
$\displaystyle=g_{s}(\hat{y};\theta),$ (3)

where $x$, $\hat{x}$, $y$, and $\hat{y}$ are the raw image, the reconstructed
image, the latent feature before quantization, and the quantized latent
feature, respectively. $\phi$ and $\theta$ are parameters of the encoder and
decoder. In the encoding process $g_{a}$, latent feature $y$ is produced from
the raw image $x$ (Eq. 1). For the quantization step $Q$, latent feature $y$
is quantized (rounded) to $\hat{y}$ (Eq. 2). During the decoding process
$g_{s}$, quantized $\hat{y}$ is fed into the decoder network to obtain the
reconstructed image $\hat{x}$ (Eq. 3). Since the quantization operation $Q$ is
not differentiable in training, the quantization $Q$ is typically approximated
by adding a uniform noise $\mathcal{U}(-0.5,0.5)$ to the input. We define a
probability model $p(\hat{y};\phi)$, which is parameterized by $\phi$ to
compute the probability mass function (PMF) of quantized feature $\hat{y}$ as
shown in Eq. 3.1.

$\displaystyle p(\hat{y};\phi)$
$\displaystyle=\prod_{i}\int^{\hat{y}^{i}+0.5}_{\hat{y}^{i}-0.5}p(\hat{y}^{i};\phi)d\hat{y}^{i}$
$\displaystyle=\prod_{i}\left(F\left(\hat{y}^{i}+0.5;\phi\right)-F\left(\hat{y}^{i}-0.5;\phi\right)\right),$
(4)

where $F(\cdot;\phi)$ is the cumulative distribution function of
$p(\cdot;\phi)$ and $i$ iterates over all symbols in $\hat{y}$. The goal of
the image compression task is to minimize the rate-distortion loss function as
shown in Eq. 5 with respect to parameters $\theta$ and $\phi$.

$\displaystyle\mathcal{L}_{\theta,\phi}$
$\displaystyle=\mathcal{R}(\hat{y})+\lambda\mathcal{D}(x,\hat{x})$ (5)
$\displaystyle=\mathbb{E}\underbrace{[-\log_{2}p(\hat{y};\phi)]}_{\text{rate}}+\underbrace{\mathcal{D}(x,\hat{x})}_{\text{distortion}},$
(6)

where the number of bits required to encode quantized $\hat{y}$ is represented
by $\mathcal{R}=\mathcal{R}(\hat{y})$. The distortion between reconstruction
image $\hat{x}$ and the original image $x$ is calculated by
$\mathcal{D}(x,\hat{x})$, which is commonly evaluated by the peak signal-to-
noise ratio (PSNR) or multiscale structural similarity (MS-SSIM) (Wang et al.,
2003) of the raw and reconstructed images. The encoder and entropy model
parameter $\phi$, and the decoder parameter $\theta$ are jointly optimized to
reduce the rate-distortion cost $\mathcal{R}+\lambda\mathcal{D}$, where
$\lambda$ controls the rate-distortion trade-off.

The hyperprior adopted in many existing works (Ballé et al., 2018; Mentzer et
al., 2018) is omitted from the above formulation for simplicity. However, such
simplification does not affect the generality of our approach.

###  3.2 Quantization Rectifier

Definition. To mitigate the random impact of quantization, we introduce the
quantization rectifier (QR). We take $y$ from Eq. 1 as the input and train the
corrected feature through a QR network, which is used to predict unquantized
features from quantized ones. The effectiveness of the QR relies on its
network design. The insight is to exploit the spatial correlation within the
image feature that recovers itself even under noise. Inspired by the success
of Model Diffusion (Ho et al., 2020) in image denoising, we design the QR
network as shown in Fig. 1. The QR network stays between the quantization step
and the decoder. It consists of convolutional layers (conv), residual blocks
(res-block), and attention layers (attn) that spatially correlate quantized
features $\hat{y}$. We then add the output of the last conv to the quantized
feature $\hat{y}$ and acquire the corrected feature $\tilde{y}$ (Eq. 7). Next,
$\tilde{y}$ replaces $\hat{y}$ in the decoding phase as shown in Eq. 8.

$\displaystyle\tilde{y}$ $\displaystyle=\text{QR}(\hat{y})$ (7)
$\displaystyle\hat{x}$ $\displaystyle=g_{s}(\tilde{y};\theta).$ (8)

A more detailed architecture of the QR is described in Appendix A. Compared to
the network in Model Diffusion (Ho et al., 2020), our network is configured
with fewer layers for efficiency while being effective.

Figure 1: Quantization Rectifier Architecture.

The QR is a versatile module, which can be seamlessly integrated into any
neural image compression method that can be broken down into an encoder, a
quantization module, and a decoder. There is no need to make significant
modifications to the encoder and decoder components of the original image
compression model.

Learning Rectifier. To facilitate the learning of the QR, we replace the loss
function in Eq. 5 with Eq. 9 that adds a _feature distance_ term to the
original formulation.

$\mathcal{L}_{\theta,\phi,\psi}=\underbrace{\mathcal{R}(\hat{y})}_{\text{rate}}+\lambda\underbrace{\mathcal{D}(x,\hat{x})}_{\text{distortion}}+\alpha\underbrace{\mathcal{D}^{f}(y,\tilde{y})}_{\text{feature
distance}},$ (9)

where $\alpha$ is the learning coefficient controlling the relative learning
rate of QR compared to that of the rate and distortion. An algorithm
describing the exploration of $\alpha$ will be further detailed in Sec. 3.3.
$\mathcal{D}^{f}(y,\tilde{y})$ is a measure of the distance between the
original and quantized image features, i.e., feature distance. The design of
the feature distance is crucial to the learning of QR, where a smaller
distance should reflect better preservation of feature expressiveness. We
consider four commonly used distance terms: L1 distance (Lasserre, 2009), L2
distance (Park & Koo, 1990), Smooth L1 distance (Girshick, 2015), and cosine
similarity (Salton & McGill, 1986; Deza & Fernández, 2009; Manning et al.,
2008; Ramos, 2003). Our empirical study shows that minimizing the L2 distance
yields the best image quality with QR. Other measures like the L1 distance do
not provide as promising results. Hence, we formulate the feature distance as
in Eq. 10.

$\mathcal{D}^{f}(y,\tilde{y})=\lVert\mathbf{{y}-\tilde{y}}\rVert_{2}.$ (10)

###  3.3 Soft-to-predictive Training

Training is an essential step that integrates the QR into the neural image
codec. We find the straightforward end-to-end training is sub-optimal due to
the inter-dependency of the training of the codec and the QR. First, the
learning of the QR relies on the stability of its input, which is the latent
feature. Second, the stability of the latent features is contingent upon the
convergence of the prediction network. Even a slight perturbation in the
latent feature would disrupt the training process of the prediction network.
Consequently, the disturbed prediction network would further affect the
stability of the latent features. In such a vicious cycle, where the latent
feature and prediction network constantly fail to converge, the overall
training process is sub-optimal.

To address the sub-optimal training issue, we develop a soft-to-predictive
(STP) training technique consisting of the soft and predictive training
phases. In the soft training phase, the image is reconstructed based on Eq. 1,
Eq. 2, and Eq. 3. Meanwhile, we learn parameters of the encoder ($\theta$),
the decoder ($\phi$), and QR ($\psi$) based on the loss function Eq. 9 softly
with additive uniform noise. Although we do not apply Eq. 7 and Eq. 8 in the
generation of $\hat{x}$, QR will still be learned to predict the feature,
which warms up the next phase. In the predictive training phase, the image is
reconstructed based on Eq. 1, Eq. 2, Eq. 7, and Eq. 8. The encoder $\theta$ is
fixed with its output being hard quantized while the decoder ($\phi$) and QR
($\psi$) are optimized according to the loss function in Eq. 11.

$\mathcal{L}_{\theta,\psi}=\mathcal{D}(x,\hat{x})+\alpha\mathcal{D}^{f}(y,\tilde{y}),$
(11)

where only $\theta$ and $\psi$ are optimized. The bitrate $\mathcal{R}$ is
omitted in Eq. 11 as its parameters are no longer optimized. $\lambda$ is also
dropped in Eq. 11 for simplicity. In the predictive training phase, the latent
feature and bitrate stay fixed, which stabilizes the training of QR.

During the predictive training phase, choosing the rectifier learning
coefficient $\alpha$ in Eq. 9 is non-trivial as its optimal value varies
across different models and compression quality. Appendix B demonstrate the
learning coefficient exploration algorithm we proposed and its results.

##  4 Experiments

###  4.1 Experimental Setup

To demonstrate the effectiveness of the QR, we apply it to four baseline
neural image compression models: Factorized Prior (Ballé et al., 2016), Scale
Hyperprior (Ballé et al., 2018), Joint Hyperprior (Minnen et al., 2018), and
Attention-based Joint Hyperprior (Cheng et al., 2020), denoted by
“Factorized”, “Scale”, “Joint”, and “Attn”, respectively. The selected
baseline models capture dominant neural image compression architectures.
Accordingly, the models enhanced by the QR are represented by “Factorized+QR”,
“Scale+QR”, “Joint+QR”, and “Attn+QR”, respectively. According to CompressAI
(Gravano et al., 2021), the four baseline models are previously trained on
$256\times 256$ image patches randomly extracted and cropped from the Vimeo90K
dataset (Xue et al., 2019) with a batch size of $32$ using the Adam optimizer
(Kingma & Ba, 2014). The test is performed on the commonly used Kodak image
dataset (Hersch, 2001). We compare the performance of the four enhanced models
against the corresponding baseline models in terms of rate and distortion
trade-offs. Following many existing works (Ballé et al., 2016, 2018), the rate
is measured by bits per pixel (bpp) while the distortion is measured by either
PSNR or MS-SSIM. MS-SSIM is converted to decibels $(-10\log_{10}(1-$ MS-
SSIM$))$ to illustrate the difference clearly. For fairness, both baseline and
enhanced models are optimized with MSE or MS-SSIM, depending on the distortion
metric (PSNR or MS-SSIM). An description of the training configuration and a
more detailed illustration of metrics and is in Appendix C.

###  4.2 Coding Efficiency Improvement

(a) PSNR

(b) MS-SSIM

Figure 2: Coding efficiency of baseline models and their enhanced versions by
QR in terms of PSNR and MS-SSIM.

Fig. 2 compares the coding efficiency of the baseline models without and with
the proposed QR. Every point on curves in Fig. 2 represents the bpp and
distortion (PSNR or MS-SSIM) averaged over the Kodak image dataset (Hersch,
2001) different compression quality levels $q\in\\{1,2,3,4\\}$. For a specific
baseline model at any given quality level, the average bpp value remains the
same after applying QR. The reason is that the soft training process for the
encoder with QR is identical to the training of the baseline model. After soft
training, the encoder is fixed, so the average bpp value would not change.
Comparing the baseline models to their corresponding enhanced versions in Fig.
2, we notice QR consistently improves all baseline models at various
compression qualities in terms of both PSNR and MS-SSIM. Further, for a
relatively more complex model, e.g., Attn, QR shows improvement by a wider
margin than that of a simple model like Factorized. We speculate that a more
complex model, with more parameters, can better leverage the reduced effect of
quantization towards better image quality. Moreover, the improvement by QR is
more evident utilizing MS-SSIM than utilizing PSNR.

The image quality of all models utilizing our proposed QR method surpasses
that of the baseline models in both PSNR and MS-SSIM. Among all enhanced
models, Attn+QR demonstrates the most substantial enhancement in PSNR, with an
average quality improvement of 0.17 dB and a maximum improvement of 0.21 dB.
With MS-SSIM, Attn+QR is still the best-performing one with a 0.19 dB average
and 0.25 dB maximum improvement over Attn. While Factorized+QR exhibits a
relatively smaller improvement compared to the other enhanced models in PSNR,
its improvement is significant in MS-SSIM, with an average of 0.12 dB and a
maximum of 0.15 dB. The numerical results are summarized in Tab. 3 shown in
Appendix D.

A detailed evaluations regarding quantization error reduction of the proposed
QR component can be found in Appendix E.

###  4.3 Processing Speed

Table 1: Runtime cost increase in milliseconds after applying the quantization rectifier, evaluated on NVIDIA RTX 2080 Ti GPU (2080) and NVIDIA RTX 3090 Ti GPU (3090). HW | Attn | Joint | Scale | Factorized
---|---|---|---|---
2080 | 5.2% | 0.7% | 4.6% | 16.6%
3090 | 5.4% | 0.8% | 5.4% | 17.1%

In Tab. 1, we compare the average processing time per frame of the baseline
models and their enhanced versions by QR, including encoding and decoding, on
the Kodak dataset. During the time measurement, we factor out the time spent
in the conversion between symbol likelihoods and bits to precisely show the
impact of QR on the neural network-related computation. Tests are performed on
NVIDIA RTX 2080 Ti and NVIDIA RTX 3090 Ti GPUs. Our method slightly increases
processing time by 0.7-5.4% for most baselines (Attn, Joint, and Scale), while
Factorized is more affected as its processing time is already short due to its
simplest network architecture.

##  5 Conclusion

We introduce a Quantization Rectifier (QR) method to enhance neural image
compression. QR utilizes spatial correlation in images to predict features
before quantization, preserving their expressiveness. Our method includes a
soft-to-predictive training approach that allows seamless and optimal
integration of QR into existing neural image codecs. Experimental results
consistently demonstrate the effectiveness of QR across various state-of-the-
art neural image codecs.

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##  Appendix A Detailed Quantization Rectifier Architecture

In accordance with Sec. 3.2, the architecture presented in Fig. 3 provides a
comprehensive illustration of the Quantization Rectifier (QR) network. For the
purpose of clarity, the encoder and decoder components depicted in Fig. 1 are
omitted in this particular representation. The architecture consists of
convolutional layers (conv), which are intertwined with groups of residual
blocks (res-block) and multi-head attention layer (attn). These attention
layers spatially correlate the quantized features $\hat{y}$.

Initially, the quantized feature with a dimension of $192$ is inputted into a
conv with a dimension of $512$, employing a kernel size of $7\times 7$ and a
padding of $3$. The output of this conv is then directed to the first set of
grouped res-block, consisting of eight groups, with each group having a
dimension of $64$ and a kernel size of $3\times 3$. The resulting output from
the res-block is normalized through layer normalization (layer-norm) and
serves as the input for a multi-head attn with four heads, where each head
possesses a dimension of $32$. The output of this attn is subsequently added
to the output of the first res-block layer. This summation then serves as the
input for the second set of res-block groups, which mirror the architecture of
the first group. The output of this second set of res-block is concatenated
with the output of the initial conv and is then fed into the final group of
res-block. The concatenated output undergoes a final conv with a dimension of
$192$ and a kernel size of $1\times 1$. Lastly, the output from this conv is
added to the quantized feature $\hat{y}$, resulting in the corrected feature
$\tilde{y}$, which retains the same dimension as the original quantized
feature $\hat{y}$.

Figure 3: Detailed Architecture of the Quantization Rectifier.

##  Appendix B Learning Coeeficient Exploration in Training

Rectifier Learning Coefficient Exploration Algorithm. During the predictive
training phase as mentioned in Sec. 3.3, choosing the rectifier learning
coefficient $\alpha$ in Eq. 9 is non-trivial as its optimal value varies
across different models and compression quality. Fig. 4 compares the image
quality tested on the Kodak dataset, resulting from different rectifier
learning coefficients. An optimal coefficient, e.g., $10^{-3}$, allows the
image quality to start at a high value and converge in a few epochs, e.g., 6
epochs in Fig. 4. If the learning coefficient is too small, e.g., $10^{-6}$,
it will take a long time for the model to converge at a sufficiently good
image quality. Conversely, if the learning coefficient is too large, e.g.,
$10^{2}$, the rectifier changes too fast for the decoder to converge, which
degrades image quality. To tackle this issue, we introduce a rectifier
learning coefficient exploration method that automatically finds the optimal
learning coefficient for different models and compression quality.

One of our key findings is, there exists an optimal learning coefficient where
increasing or decreasing it would only monotonically degrade coding
efficiency. Based on this finding, we describe the exploration strategy for a
specific model and compression quality as follows: i) start the exploration at
an initial learning coefficient $\alpha=\alpha_{max}$, ii) train the codec as
specified in Sec. 3.3 using $\alpha$ until the loss (Eq. 9) stops improving
for $M$ consecutive epochs, iii) multiply the learning coefficient $\alpha$ by
0.1, iv) continue step ii) if the learning coefficient is no smaller than a
pre-defined lowest learning coefficient $\alpha_{min}$ or stop exploration
otherwise. $M$ is set to $3$, which confidently finds the non-improving loss.
Considering the efficiency of exploration, we adopt a relatively small dataset
in exploration, which remains as effective as the big one in predictive
training.

Figure 4: The impact of rectifier learning coefficient on image quality. Table 2: Rectifier Learning Coefficient Exploration: distortion measured in PSNR (dB) at compression qualities $q\in{1,2,3,4}$ using Flickr image dataset. Coefficient | Attn | Joint
---|---|---
$1$ | $2$ | $3$ | $4$ | $1$ | $2$ | $3$ | $4$
$10^{-1}$ | 28.56 | 29.85 | 31.37 | 33.47 | 28.15 | 29.71 | 31.35 | 33.11
$10^{-2}$ | 28.58 | 29.87 | 31.39 | 33.49 | 28.16 | 29.73 | 31.38 | 33.12
$10^{-3}$ | 28.61 | 29.88 | 31.41 | 33.52 | 28.16 | 29.73 | 31.39 | 33.13
$10^{-4}$ | 28.59 | 29.86 | 31.40 | 33.50 | 28.17 | 29.74 | 31.38 | 33.12
$10^{-5}$ | 28.58 | 29.84 | 31.38 | 33.49 | 28.16 | 29.74 | 31.37 | 33.11
Coefficient | Scale | Factorized
$1$ | $2$ | $3$ | $4$ | $1$ | $2$ | $3$ | $4$
$10^{-1}$ | 27.63 | 29.35 | 31.09 | 32.93 | 26.90 | 28.19 | 29.60 | 31.26
$10^{-2}$ | 27.68 | 29.40 | 31.13 | 32.96 | 26.91 | 28.21 | 29.60 | 31.27
$10^{-3}$ | 27.70 | 29.42 | 31.16 | 32.93 | 26.91 | 28.20 | 29.61 | 31.28
$10^{-4}$ | 27.73 | 29.44 | 31.15 | 32.92 | 26.90 | 28.20 | 29.59 | 31.28
$10^{-5}$ | 27.70 | 29.43 | 31.14 | 32.92 | 26.90 | 28.19 | 29.58 | 31.27

Rectifier Learning Coefficient Exploration Result. Choosing the rectifier
learning coefficients $\alpha$ in Eq. 9 is critical to the performance of the
compression model. We empirically set an $\alpha_{max}=10^{-1}$ and
$\alpha_{min}=10^{-5}$ for PSNR to cover feasible values of the learning
coefficient. We explore learning coefficients between $\alpha_{max}$ and
$\alpha_{min}$ following the exploration strategy in Sec. 3.3. The exploration
for each coefficient value usually completes with 15 rounds of iterations.
Tab. 2 shows the PSNR values for all models achieved at explored coefficients,
where the coefficient producing the best PSNR (highlighted) is selected for
predictive training. It is also observed that the PSNR performance
monotonically degrades when the coefficient increases above or decreases below
the selected value. Note that, as we round off the PSNR value to two decimal
places, some adjacent rows with different coefficients may show the same PSNR
value, e.g., 26.91 dB at $q=1$ for coefficients $10^{-2}$ and $10^{-3}$ of
Factorized. However, only the highlighted PSNR value is higher than others in
the same column. The exploration of the rectifier learning coefficient
efficiently identifies the optimal coefficient that ensures a better
predictive training result than that without exploration.

Note that we do not repeat the same exploration process for MS-SSIM as we do
for PSNR. The reason is we empirically find the distribution of optimal
coefficients is similar for PSNR and MS-SSIM on different models, except that
the optimal coefficient for MS-SSIM is roughly $10\times$ smaller than that
for PSNR. As a result, we multiply the optimal coefficient in Tab. 2 by $0.1$
for MS-SSIM during predictive training phase.

##  Appendix C Training Configuration and Detailed Experimental Metrics

Training configuration. For enhanced models, the rectifier learning
coefficient exploration and the STP training phases are conducted on the
Flickr image dataset (Flickr, 2021) and the ImageNet dataset (Deng et al.,
2009), respectively. Similar to the baseline training, the exploration and STP
training phases adopt a batch size of $32$. We fix the learning rate at
$10^{-6}$ in both phases, which yields the best performance. The training is
performed on a desktop with 2 NVIDIA RTX 3090 GPUs.

Metrics. As mentione in 4.1, we compare the performance in terms of rate and
distortion trade-offs, measured by bpp and PSNR or MS-SSIM. To show the
performance of our approach at different compression qualities, we repeat our
experiments at compression quality levels $q\in\\{1,2,3,4\\}$ of the codec
models, where a greater value of $q$ corresponds to a greater value of
$\lambda$ in Eq. 5. To demonstrate the capability of our approach in
preserving the expressiveness of image features, we introduce a novel metric,
the quantization error $\epsilon_{Q}$. The quantization error is defined as
the L2 distance between the input to the quantization operation ($y$) and the
input to the decoder, which is either the result of quantization ($\hat{y}$)
or QR ($\tilde{y}$) depending on whether a model is enhanced by QR, as shown
in Eq. 12.

$\epsilon_{Q}=\begin{cases}\lVert{\tilde{y}-y}\rVert_{2}&\text{if using QR}\\\
\lVert{\hat{y}-y}\rVert_{2}&\text{if not using QR. }\end{cases}$ (12)

##  Appendix D Numerical Results of Coding Efficiency Improvement

As mentioned in Sec. 4.2, Tab. 3 statistically shows the benefits of QR in
terms of PSNR and MS-SSIM regarding the average and maximum values over all
compression qualities.

Table 3: Image quality improvement in PSNR and MS-SSIM. Metrics (dB) | Attn | Joint | Scale | Factorized
---|---|---|---|---
Avg | Max | Avg | Max | Avg | Max | Avg | Max
$\uparrow$ PSNR | 0.17$\pm$0.04 | 0.21 | 0.08$\pm$0.05 | 0.16 | 0.09$\pm$0.03 | 0.13 | 0.02$\pm$0.01 | 0.05
$\uparrow$ MS-SSIM | 0.19$\pm$0.05 | 0.25 | 0.12$\pm$0.01 | 0.12 | 0.11$\pm$0.05 | 0.19 | 0.12$\pm$0.03 | 0.15

##  Appendix E Evaluation of Quantization Error Reduction of QR

Quantization Error Reduction Result.

(a) Model optimized for PSNR

(b) Model optimized for MS-SSIM

Figure 5: Quantization error reduction.

Fig. 5 shows the reduction of quantization error by QR in percentage compared
against all baseline models at various compression qualities. The reduction is
generally more significant for more complex models like Attn and Joint.
Meanwhile, a lower compression quality tends to magnify the reduction.
Notably, we observe the maximum quantization error reduction of 38% and 36%
for Attn at compression quality $q=1$, when the model is optimized for PSNR
and MS-SSIM, respectively. We also notice models optimized for MS-SSIM, e.g.,
Factorized, exhibit randomness that causes the reduction to be slightly
improved with a higher compression quality.

Analysis of Quantization Rectifier. With the performance gain of one QR, a
natural idea is to embed multiple QRs into a codec. To this end, we apply
multiple sequentially connected QRs to a baseline codec and analyze the
improvements in image quality and quantization error reduction. Specifically,
we compare the performance of one, two, and three QR(s), denoted by
$1\times$QR, $2\times$QR, and $3\times$QR, respectively. Attn is used as the
baseline model for different compression qualities. As Fig. 6 shows, the
quantization error reduction is already significant when we have one QR, which
is up to 35% for PSNR and 39% for MS-SSIM models. Despite more QRs further
reduce the quantization error, the benefit of an additional QR, being less
than 2%, is rather incremental in both PSNR and MS-SSIM. Similarly in Fig. 7,
the improvement in the image quality measured by PSNR and MS-SSIM is
significant with the first QR, which is up to 0.15 dB and 0.19 dB,
respectively. After that, when we use two or three QRs, the image quality gain
from one additional QR is at most 0.01 dB. When the compression quality
changes, the above observations still apply. Given the fact that the
computation and memory (storage) overhead of QR linearly increases with its
number, affecting the training and encoding/decoding of the codec, we do not
pursue a network design with more than one QR.

(a) Model optimized for PSNR

(b) Model optimized for MS-SSIM

Figure 6: Impact of different numbers of quantization rectifiers on
quantization error reduction.

(a) Model optimized for PSNR

(b) Model optimized for MS-SSIM

Figure 7: Impact of different numbers of quantization rectifiers on image
quality.


S_{n}(p)$. The intersection of $n$ complex $(D-1)$-spheres in general position
is a complex $(D-n)$-sphere which allows us to immediately determine the
sought homology groups. But we can be significantly more concrete by employing
the Lorentz-invariance, or rather $O(D,\mathbb{C})$-invariance in our setup,
of Feynman integrals. To fully take advantage of this invariance, we first
prove the following

######  Lemma 4.17.

Let $v_{1},\ldots,v_{m}\in\mathbb{C}^{n}$ be $m\in\mathbb{N}^{\ast}$ vectors
such that

(4.97) $\det\begin{pmatrix}v_{1}v_{1}&v_{1}v_{2}&\cdots&v_{1}v_{m}\\\
v_{2}v_{1}&v_{2}v_{2}&\cdots&v_{2}v_{m}\\\ \vdots&\vdots&\ddots&\vdots\\\
v_{m}v_{1}&v_{m}v_{2}&\cdots&v_{m}v_{m}\end{pmatrix}\neq 0.$

Then there exists a complex orthogonal matrix $M\in O(n,\mathbb{C})$ such that
$(M\cdot v_{i})_{j}\neq 0$ for all $j\leq i$ and $(M\cdot v_{i})_{j}=0$ for
all $j>i$.

###### Proof.

We conduct the proof by induction on the number of vectors $m$. The base case
$m=1$ can be solved by induction on the dimension $n$. For $n=1$ there is
nothing to do. Now suppose we already now the lemma is true for $m=1$ up to
some dimension $n-1$. Let us denote $w:=v_{1}$. If
$\tilde{w}:=(w_{2},\ldots,w_{n})=0$ there is again nothing to do. By
assumption we have $w^{T}w\neq 0$. Note that this means we can not have

(4.98)
$(w_{1},\ldots,\hat{w}_{i},\ldots,w_{n})^{T}(w_{1},\ldots,\hat{w}_{i},\ldots,w_{n})=0$

for all $i\in\\{1,\ldots,n\\}$ since

(4.99)
$\sum_{i=1}^{n}(w_{1},\ldots,\hat{w}_{i},\ldots,w_{n})^{T}(w_{1},\ldots,\hat{w}_{i},\ldots,w_{n})=(n-1)w^{T}w\neq
0.$

The permutation matrices are orthogonal so that we can assume
$\tilde{w}^{T}\tilde{w}\neq 0$ without loss of generality. Thus by the
induction hypothesis we can find a complex orthogonal matrix $M_{1}\in
O(n-1,\mathbb{C})$ such that

(4.100) $M_{1}\cdot\tilde{w}=\begin{pmatrix}w_{2}^{\prime}\\\ 0\\\ \vdots\\\
0\end{pmatrix}\qquad\Rightarrow\qquad\begin{pmatrix}1&0\\\
0&M_{1}\end{pmatrix}\cdot w=\begin{pmatrix}w_{1}^{\prime}\\\ w_{2}^{\prime}\\\
0\\\ \vdots\\\ 0\end{pmatrix}$

with $w_{2}^{\prime}\neq 0$. Note that $\begin{pmatrix}1&0\\\
0&M_{1}\end{pmatrix}$ is also an orthogonal matrix. Now we define

(4.101)
$M_{2}:=\frac{1}{\sqrt{{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}}}\cdot\begin{pmatrix}w^{\prime}_{1}&w^{\prime}_{2}\\\
-w^{\prime}_{2}&w^{\prime}_{1}\end{pmatrix}.$

Note that this is well-defined as
${w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}=w^{T}w\neq 0$. Then we have

(4.102)
$M_{2}^{T}M_{2}=\frac{1}{{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}}\cdot\begin{pmatrix}{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}&w_{1}^{\prime}w_{2}^{\prime}-w_{2}^{\prime}w_{1}^{\prime}\\\
w_{2}^{\prime}w_{1}^{\prime}-w_{1}^{\prime}w_{2}^{\prime}&{w^{\prime}_{2}}^{2}+{w^{\prime}_{1}}^{2}\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$

and

(4.103) $M_{2}\cdot\begin{pmatrix}w_{1}^{\prime}\\\
w_{2}^{\prime}\end{pmatrix}=\frac{1}{\sqrt{{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}}}\cdot\begin{pmatrix}{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}\\\
w_{2}^{\prime}w_{1}^{\prime}-w_{1}^{\prime}w_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}\frac{{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}}{\sqrt{{w^{\prime}_{1}}^{2}+{w^{\prime}_{2}}^{2}}}\\\
0\end{pmatrix}\neq 0.$

We conclude that $M_{2}$ is orthogonal and that therefore the matrix
$\begin{pmatrix}M_{2}&0\\\ 0&1_{n-2}\end{pmatrix}$ is also orthogonal, where
$1_{n-2}$ is the identity matrix of size $n-2$. Hence the matrix

(4.104) $M:=\begin{pmatrix}M_{2}&0\\\
0&1_{n-2}\end{pmatrix}\cdot\begin{pmatrix}1&0\\\
0&M_{1}\end{pmatrix}\quad\in\quad O(n,\mathbb{C})$

is orthogonal and fulfills the requirement of the lemma.
Now suppose that $m>1$. Then by the induction hypothesis we can find an
orthogonal matrix $M\in O(n,\mathbb{C})$ such that for all $i<m$ we have
$(M\cdot v_{i})_{i}\neq 0$ and $(M\cdot v_{i})_{j}=0$ for all $j>i$. Now
consider the vector $v^{\prime}:=(x_{m},\ldots,x_{n})\in\mathbb{C}^{n-m+1}$
obtained by removing the first $m-1$ components of $M\cdot v_{m}$. Then there
is an orthogonal matrix $M^{\prime}\in O(n-m+1,\mathbb{C})$ such that
$(M^{\prime}\cdot v^{\prime})_{1}\neq 0$ and $(M^{\prime}\cdot
v^{\prime})_{j}=0$ for all $j>1$. We define

(4.105) $\tilde{M}:=\begin{pmatrix}1_{m-1}&0\\\ 0&M^{\prime}\end{pmatrix}\cdot
M\quad\in\quad O(n,\mathbb{C}).$

Then $\tilde{M}$ is orthogonal as

(4.106) $\tilde{M}^{T}=M^{T}\cdot\begin{pmatrix}1_{m-1}&0\\\
0&{M^{\prime}}^{T}\end{pmatrix}=M^{-1}\cdot\begin{pmatrix}1_{m-1}&0\\\
0&{M^{\prime}}^{-1}\end{pmatrix}=(\begin{pmatrix}1_{m-1}&0\\\
0&M^{\prime}\end{pmatrix}\cdot M)^{-1}=\tilde{M}^{-1}.$

Furthermore we clearly have $(\tilde{M}\cdot v_{i})_{j}\neq 0$ for all $j\leq
i$ and $(\tilde{M}\cdot v_{i})_{j}=0$ for all $j>i$. ∎

Using this lemma, we can easily compute the center and radius of the complex
sphere given by the finite part of the intersection of the
$S_{1}(p),\ldots,S_{n}(p)$. The system of equations defining
$\bigcap_{i=1}^{n}S_{i}(p)-H_{\infty}$ reads

$\displaystyle k^{2}$ $\displaystyle+\;m_{1}^{2}$ $\displaystyle=0$
$\displaystyle(k+p_{1})^{2}$ $\displaystyle+\;m_{2}^{2}$ $\displaystyle=0$
$\displaystyle\hskip 22.76228pt\vdots$ $\displaystyle\hskip 5.69046pt\vdots$
$\displaystyle\hskip 5.12128pt\vdots$
$\displaystyle(k+\sum_{j=1}^{n-1}p_{j})^{2}$ $\displaystyle+\;m_{n}^{2}$
$\displaystyle=0$

in inhomogeneous coordinates. By plugging in $k^{2}=-m_{1}^{2}$ from the first
equation into the remaining ones we see that this system is equivalent to

$\displaystyle k^{2}$ $\displaystyle+m_{1}^{2}$ $\displaystyle=0$
$\displaystyle 2kp_{1}+p_{1}^{2}$ $\displaystyle-m_{1}^{2}+m_{2}^{2}$
$\displaystyle=0$ $\displaystyle\hskip 28.45274pt\vdots$ $\displaystyle\hskip
5.12128pt\vdots$ $\displaystyle\hskip 5.69046pt\vdots$ $\displaystyle
2k\sum_{j=1}^{n-1}p_{j}+(\sum_{j=1}^{n-1}p_{j})^{2}$ $\displaystyle-
m_{1}^{2}+m_{n}^{2}$ $\displaystyle=0.$

By Proposition 4.13 and Lemma 4.17 we can assume

(4.107) $p_{1}=\begin{pmatrix}(p_{1})_{0}\\\ 0\\\ \vdots\\\
0\end{pmatrix},\quad p_{2}=\begin{pmatrix}(p_{2})_{0}\\\ (p_{2})_{1}\\\ 0\\\
\vdots\\\ 0\end{pmatrix},\quad\cdots,\quad\begin{pmatrix}(p_{n})_{0}\\\
\vdots\\\ (p_{n})_{n-1}\\\ 0\\\ \vdots\\\ 0\end{pmatrix}$

with $(p_{i})_{i-1}\neq 0$ for all $i\in\\{1,\ldots,n\\}$ without loss of
generality. This allows us to compute the first $n-1$ components of $k$
recursively: By the second equation we have

(4.108)
$k_{0}=-\frac{1}{2(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})=:A_{0}(p)$

and by the $(i+2)$th equation we have

(4.109)
$k_{i}=-\frac{1}{2(p_{i+1})_{i}}(2\bar{k}^{(i)}P^{(i+2)}+(P^{(i+2)})^{2}-m_{1}^{2}+m_{i+2}^{2})=:A_{i}(p)$

where

(4.110) $\bar{k}^{(i)}:=(k_{0},\ldots,k_{i-1},0,\ldots,0).$

We denote
$A^{(n)}(p):=(A_{0}(p),\ldots,A_{n-2}(p),0,\ldots,0)\in\mathbb{C}^{D}$. Now we
can conclude that $\bigcap_{i=1}^{n}S_{i}(p)-H_{\infty}$ is given by all
solutions to

(4.111) $\bar{k}^{(n-1)}=A^{(n)}(p)$

and

(4.112) $(k_{n-1},\ldots,k_{D-1})^{2}=-m_{1}^{2}-A(p)^{2}=:r_{n}^{2}(p).$

Thus we see that $\bigcap_{i=1}^{n}S_{i}(p)-H_{\infty}$ is a complex
$(D-n)$-sphere around $A^{(n)}(p)$ with radius squared equal to
$r_{n}^{2}(p)$. The $A_{j}(p)$ are designed such that

(4.113) $2A^{(n)}(p)P^{(i)}(p)+(P^{(i)}(p))^{2}+m_{i}^{2}-m_{1}^{2}=0$

for all $i\in\\{1,\ldots,n\\}$. Indeed we have

(4.114)
$\begin{split}2A^{(n)}(p)P^{(i)}(p)&=2A_{i-1}(p)(p_{i})_{i}+2\sum_{j=0}^{i-2}A_{j}(p)(p_{i})_{j}+2A^{(n)}(p)P^{(i-1)}\\\
&=-(P^{(i)}(p))^{2}-m_{i}^{2}+m_{1}^{2}\end{split}$

for all $i\in\\{1,\ldots,n\\}$.
This discussion covers the case in which all $S_{1}(p),\ldots,S_{n}(p)$
intersect but this can easily be generalized to intersections of any subset:
Let $I\subsetneq\\{1,\ldots,n\\}$ and assume $|I|\geq 2$ (otherwise, there is
nothing to do). The defining equations for $\bigcap_{i\in
I}S_{i}(p)-H_{\infty}$ have the same form as in the case $I=\\{1,\ldots,n\\}$.
In fact, they are exactly the equations we would obtain for the cyclic graph
with $|I|$ edges obtained from shrinking every edge not in $I$ to zero length
(and thus not $C_{|I|}$ as we have defined it since the external structure is
different). Thus we can go through the same computation as above but we
replace each momentum $P^{(i)}(p)$ for $i\in I$ by the corresponding sum of
momenta. Let $I=\\{i_{1},\ldots,i_{|I|}\\}$ with $i_{1}<\cdots<i_{|I|}$. We
shall denote the external momenta $p_{I}\in(\mathbb{C}^{D})^{|I|-1}$ given by
the external structure of the graph with shrunken edges by

(4.115) $(p_{I})_{l}:=\sum_{j={i_{l}}}^{i_{l+1}-1}p_{j},\qquad
l\in\\{1,\ldots,n-1-|I|\\}.$

Then $\bigcap_{i\in I}S_{i}(p)-H_{\infty}$ is a complex $(D-|I|)$-sphere
around $A^{(|I|)}(p_{I})$ with radius squared equal to $r_{|I|}^{2}(p_{I})$.
We can also use this description of $\bigcap_{i=1}^{n}S_{i}(p)-H_{\infty}$ to
prove that the system

(4.116) $S_{1}(p)-H_{\infty},\ldots,S_{n}(p)-H_{\infty}$

has a single pinch point for $p\in L_{n}$ which is a simple pinch point.

######  Proposition 4.18.

Let $I\subset\\{1,\ldots,n\\}$ be non-empty and let $p\in L_{n,I}$ be a point
of codimension 1. Then there is a single point in
$\mathbb{C}\mathbb{P}^{D}\backslash H_{\infty}$ at which the manifolds
$S_{i}(p)$ with $i\in I$ are not in general position. Furthermore this point
is a simple pinch point.

###### Proof.

By the discussion above, we know that there can only be one finite point at
which the manifolds $S_{i}(p)$ with $i\in I$ are not in general position.
According to [FFLP65], this is a simple pinch if the corresponding solution to
the Landau equations is unique in $\alpha$ up to a homogeneous factor. We may
assume $\sum_{i\in I}\alpha_{i}=1$ without loss of generality. Write
$I=\\{i_{1},\ldots,i_{m}\\}$ with $i_{1}<\cdots<i_{m}$. Then we need to show
that $\alpha_{i_{2}},\ldots,\alpha_{i_{m}}$ are uniquely determined by the
Landau equations. This can easily be verified since
$\alpha_{i_{2}},\ldots,\alpha_{i_{m}}$ can recursively determined by the
equation

(4.117) $k=-\sum_{i\in I}\alpha_{i}P^{(i)}(p).$

∎

Notice that we have just proven the following

######  Proposition 4.19 (Vanishing Sphere for One-Loop Graphs).

Let $p\in L_{n,I}$ be a point of codimension 1. Then the system
$S_{1},\ldots,S_{n}$ has a simple pinch at $(A^{(|I|)}(p_{I}),p)$ and the
vanishing sphere can be represented by a deformation retract of

(4.118)
$\tilde{e}=\\{k\in\mathbb{C}^{D}\;|\;k=A^{(I)}(p_{I}^{\prime})\;,\;\sum_{i=|I|-1}^{D-1}k_{i}^{2}=r_{|I|}^{2}(p_{I}^{\prime})\\}$

to the real sphere within $\mathbb{C}\mathbb{P}^{D}$ for $p^{\prime}$ close to
$p$. In particular if $p^{\prime}\in(M^{D})^{n-1}$ is a configuration of
Minkowski momenta such that $r_{|I|}^{2}(p_{I}^{\prime})\in\mathbb{R}_{>0}$,
we have $\tilde{e}\subset M^{D}$.

Note that the above proof also yields an algorithm to determine the Landau
variety for any loop graph which does not require us to solve the Landau
equations:

######  Algorithm 4.20.

The Landau surface $L_{n}$ can be computed in terms of any kinematic invariant
by the following algorithm:

  1. (1)

Choose a complete set of kinematic invariants
$s_{1},\ldots,s_{\frac{n(n-1)}{2}}$.

  2. (2)

Compute $A_{i}(p)$ for $i=0,1,\ldots,n-2$ by using equations (4.108) and
(4.109).

  3. (3)

Compute $r_{n}^{2}(p)$ by using equation (4.112).

  4. (4)

Express all components of $p$ in $r_{n}^{2}(p)$ in terms of the invariants
$s_{1},\ldots,s_{\frac{n(n-1)}{2}}$.

  5. (5)

The result is a rational function in the $s_{1},\ldots,s_{\frac{n(n-1)}{2}}$
which can be solved for one of them.

######  Example 4.21.

Returning to our running example $C_{2}$, we compute $A(p)$ as well as
$r_{2}^{2}(p)$ in this case. By definition we have $A_{i}(p)=0$ for all $i\neq
0$. The entry $A_{0}(p)$ can just be read of from its definition (4.108) as

(4.119)
$A_{0}(p)=-\frac{1}{2(p_{1})_{0}}((p_{1})_{0}^{2}-m_{1}^{2}+m_{2}^{2}).$

Thus we can easily compute

(4.120)
$\begin{split}r_{2}^{2}(p)&=-m_{1}^{2}-A_{0}^{2}(p)=-\frac{4(p_{1})_{0}^{2}m_{1}^{2}}{4(p_{1})_{0}^{2}}-\frac{((p_{1})_{0}^{2}-m_{1}^{2}+m_{2}^{2})^{2}}{4(p_{1})_{0}^{2}}\\\
&=-\frac{1}{4(p_{1})_{0}^{2}}((p_{1})_{0}^{4}+m_{1}^{4}+m_{2}^{4}+2(p_{1})_{0}^{2}m_{1}^{2}+2(p_{1})_{0}^{2}m_{2}^{2}-2m_{1}^{2}m_{2}^{2})\\\
&=-\frac{\lambda(-(p_{1})_{0}^{2},m_{1}^{2},m_{2}^{2})}{4(p_{1})_{0}^{2}}.\end{split}$

Here $\lambda$ is again the Källén function (see equation (4.28)). Recall that
we required $p_{1}$ to satisfy $(p_{1})_{i}=0$ for all $i>0$. Thus
$(p_{1})_{0}^{2}=p_{1}^{2}$ and we obtain

(4.121)
$r_{2}^{2}(p)=-\frac{\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})}{4p_{1}^{2}}.$

In particular $r_{2}^{2}(p)=0$ if and only if
$\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})=0$. Thus the external momenta $p$ at
which $r_{2}^{2}$ vanishes are precisely the momenta in the Landau surface of
$C_{2}$ as computed in Example 4.3. This is of course no accident as we have
seen that $r_{n}^{2}(p)$ is in fact the radius squared of the vanishing
sphere.

A priori it might not be easy to perform step (4) in the Algorithm 4.20. But
it seems that it is possible to choose appropriate channel variables such that
$r_{n}^{2}$ can be brought into a form in which the nominator is quadratic in
each of the channel variables, making step (4) almost trivial. While we do not
prove this statement in this work, we at least see that it works out in non-
trivial examples (see Section 5).
We can also make use our formula for $r_{n}^{2}$ to proof Propositions 4.4 and
4.5 now, which stated that the $L_{n,I}$ (with $I\subset\\{1,\ldots,n\\}$) are
codimension 1 complex analytic submanifolds of $T_{n}$ intersecting in general
position:

###### Proof.

of Proposition 4.4. It suffices to show that for all $p\in T_{n}$, the
equation $\partial_{p}r_{n}^{2}(p)=0$ implies $r_{n}^{2}(p)\neq 0$. So let
$p\in T_{n}$ and suppose $\partial_{p}r_{n}^{2}(p)=0$. Then in particular
$\partial_{p_{n-1}}r_{n}^{2}(p)=0$. Since the $A_{0},\ldots,A_{n-3}$ do not
depend on $p_{n-1}$, this equation reads

(4.122) $-2A_{n-2}(p)\partial_{p_{n-1}}A_{n-2}(p)=0.$

Suppose first that $A_{n-2}(p)=0$. Then
$r_{n}^{2}(p)=r_{n-1}^{2}(p_{1},\ldots,p_{n-2})$ and
$\partial_{p}r_{n}^{2}(p)=\partial_{p}r_{n-1}^{2}(p_{1},\ldots,p_{n-2})$, so
the claim follows by induction. Now suppose $\partial_{p_{n-1}}A_{n-2}(p)=0$
and denote

(4.123) $\bar{p}_{n-1}:=((p_{n-1})_{0},\ldots,(p_{n-1})_{n-3}).$

Then

(4.124)
$\partial_{\bar{p}_{n-1}}A_{n-2}(p)=-\frac{1}{2(p_{n-1})_{n-2}}(2\bar{A}^{(n-2)}(p)+2\bar{P}^{(n)}(p))$

which implies $\bar{A}^{(n-1)}(p)=-\bar{P}^{(n)}(p)$. Furthermore we compute

(4.125)
$\begin{split}\partial_{(p_{n-1})_{n-2}}A_{n-2}(p)&=\frac{1}{2(p_{n-1})_{n-2}^{2}}(2\bar{A}^{(n-2)}(p)P^{(n)}(p)+(P^{(n)})^{2}-m_{1}^{2}+m_{n}^{2})-1\\\
&=-\frac{1}{(p_{n-1})_{n-2}}A_{n-2}(p)-1\end{split}$

which implies $A_{n-2}(p)=-(p_{n-1})_{n-2}$. This also means
$(P^{(n)}(p))^{2}=-m_{1}^{2}+m_{n}^{2}$. Together we obtain
$A(p)=-P^{(n)}(p)$. This can be used to compute

(4.126)
$r_{n}^{2}(p)=-m_{1}^{2}-A^{2}(p)=-m_{1}^{2}-(P^{(n)}(p))^{2}=-m_{n}^{2}\neq
0$

as claimed. ∎

###### Proof.

of Proposition 4.5. To show that $L_{n,I_{1}},\ldots,L_{n,I_{k}}$ intersect in
general position, we need to show that
$\partial_{p}r_{|I_{1}|}^{2}(p_{I_{1}}),\ldots,\partial_{p}r_{|I_{k}|}^{2}(p_{I_{k}})$
are linearly independent at every point $p\in\bigcap_{i=1}^{k}L_{n,I_{i}}$. We
do this by induction on $k$. By applying a cyclic permutation to the vertices
and edges of the graph $C_{n}$, we may assume without loss of generality that
$I_{k}$ contains an element $i\in I_{k}$ such that $i>j$ for all $j\in
I_{k-1}$. Since the $r_{|I_{1}|}^{2},\ldots,r_{|I_{k-1}|}^{2}$ do not depend
on $p_{i-1}$, it suffices to show
$\partial_{p_{i-1}}r_{|I_{k}|}^{2}(p_{I_{k}})\neq 0$. In the previous proof we
have seen that $\partial_{p_{i-1}}r_{|I_{k}|}^{2}(p_{I_{k}})=0$ implies
$r_{|I_{k}|}^{2}(p_{I_{k}})\neq 0$ so we are done. ∎

Finally we are in a position to state and proof Cutkosky’s Theorem in the one-
loop case.

######  Theorem 4.22 (Cutkosky’s Theorem for One-Loop Graphs).

Let $D$ be odd. Let $I\subset E(C_{n})$ and denote $m:=|I|$. Let $p\in
L_{n,I}$ be a point of codimension 1 and $\gamma$ a simple loop around $p$.
Let $p^{\prime}\in M^{D}$ such that $r_{n}^{2}(p^{\prime})$ is small and real.
Then the discontinuity of $I(C_{n})$ around $p$ in a neighborhood of
$p^{\prime}$ is given by

(4.127) $\text{Disc}_{[\gamma]}I(C_{n})(p^{\prime})=N(2\pi
i)^{m}\int_{\tilde{e}}\mathrm{Res}^{m}[\omega_{n,D}(p^{\prime})].$

In particular for $\lambda_{i}=1$ for all $i\in I$ we obtain

(4.128) $\text{Disc}_{[\gamma]}I(C_{n})(p^{\prime})=N(2\pi
i)^{m}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}\frac{\prod_{j\in
I}\delta((k+P^{(j)}(p^{\prime}))^{2}+m_{j}^{2})}{\prod_{j\notin
I}((k+P^{(j)}(p^{\prime}))^{2}+m_{j}^{2})^{\lambda_{j}}}d^{D}k.$

###### Proof.

We have seen in Proposition 4.18 that at $p$ there is a single pinch point at
$A(p)$. So according to our considerations in Subsubsection 2.3.3 and Theorem
2.29 we thus have

(4.129)
$\text{Disc}_{[\gamma]}(I(C_{n}))(p^{\prime})=N\int_{e}\omega_{n,D}(p^{\prime})=N(2\pi
i)^{m}\int_{\tilde{e}}\text{Res}^{m}[\omega_{n,D}(p^{\prime})]$

where $\tilde{e}$ is the vanishing sphere as computed in Proposition 4.19.
Denote by

(4.130) $i:\mathbb{C}^{D}\overset{\sim}{\to}\mathbb{C}\mathbb{P}^{D}\backslash
H_{\infty}$

the natural inclusion with restricted target. The Leray residue commutes with
pull-backs (see Proposition 2.30) so that we can write

(4.131) $\text{Disc}_{[\gamma]}(I(C_{n}))(p^{\prime})=N(2\pi
i)^{m}\int_{\tilde{e}}\text{Res}^{m}[\omega_{n,D}(p^{\prime})]=(-1)^{\frac{D(D+1)}{2}}(2\pi
i)^{m}\int_{i^{-1}(\tilde{e})}\text{Res}^{m}[i^{\ast}\,\omega_{n,D}(p^{\prime})]$

by removing the plane $H_{\infty}$ at infinity, which has Lebesgue measure 0,
from the integration domain. If now $\lambda_{i}=1$ for all $i\in I$, we
obtain

(4.132) $\text{Disc}_{[\gamma]}(I(C_{n}))(p^{\prime})=N(2\pi
i)^{m}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}\frac{\prod_{j\in
I}\delta((k+P^{(j)}(p^{\prime}))^{2}+m_{j}^{2})}{\prod_{j\notin
I}((k+P^{(j)}(p^{\prime}))^{2}+m_{j}^{2})}d^{D}k,$

where we used Definition 2.31 for the $\delta$-function and the fact that
$i^{-1}(\tilde{e})\subset M^{D}$. ∎

Comparing our formula for the discontinuity with the common literate, it is
striking that we replace the propagators which go on-shell at a point in the
Landau surface by a simple $\delta$ instead of the $\delta_{+}$ which usually
appears in texts citing Cutkosky’s Theorem. The
$\delta_{+}((k+P^{(j)}(p))^{2}+m_{j}^{2})$ contains an additional factor of
$\Theta(k_{0}+P^{(j)}(p))$ forcing the energy flowing through the
corresponding edge to be positive. But this prescription does not seem to make
sense in the general case. It clearly depends on the arbitrarily chosen
orientation of the edges. If we consider singularities corresponding to a set
of edges such that removing these edges from the graph yields a graph with two
connected components $G_{1}$ and $G_{2}$, then we can make sense of this idea:
If we designate one of the connected components as the incoming part and the
other one as the outgoing part, there is a preferred orientation of the
removed edges, namely the one that points from the incoming part to the
outgoing part. The following figure illustrates this situation:

$G_{1}$$G_{2}$

But for more general cuts, there seems to be no satisfying way to agree on a
sensible orientation compatible with the $\delta_{+}$s. Curiously enough, the
$\delta_{+}$ in this sense does not appear explicitly in the original work
[Cut60] by Cutkosky. He formulates the theorem by writing $\delta_{p}$ instead
of $\delta$ and states “The subscript $p$ on the delta functions means that
only the contributions of the ‘proper’ root of $q_{i}^{2}=M_{i}^{2}$ is to be
taken”. Here $q_{i}$ denotes what we write as $k+P^{(i)}(p)$ and $M_{i}$
denotes what we write as $m_{i}$. From our discussion it is evident that in
the one-loop case the vanishing sphere over which we need to integrate lies in
a subspace of $\mathbb{C}^{D}$ with fixed $k_{0}$ (namely $k_{0}=A_{0}(p)$).
Thus the sign of all $k_{0}+P^{(j)}_{0}(p)$ is already fixed by the
$\delta$-functions alone and the $\Theta$ would be superfluous.

######  Example 4.23.

To conclude our discussion of Cutkosky’s Theorem for one-loop graphs, we
return to our running example $C_{2}$ one last time and compute its
discontinuity in the case where $\lambda_{1}=\lambda_{2}=1$. Note that in this
case the corresponding Feynman integral does not converge in $D=4$ space-time
dimensions and needs to be renormalized to make sense of it. Since we do not
discuss renormalization in this paper, the space-time dimension is assumed to
be smaller than 4 so that the integral converges.
In example 4.3 we saw that the Landau surface $L_{2}$ of $C_{2}$ consists of
all points $p_{1}\in\mathbb{C}^{D}\backslash\\{0\\}$ such that

(4.133) $-p_{1}^{2}=(m_{1}+m_{2})^{2}\qquad\text{or}\qquad-
p_{1}^{2}=(m_{1}-m_{2})^{2}$

and in example 4.21 we saw that the vanishing sphere corresponding to such a
point is the real sphere inside the complex sphere around
$(A_{0}(p),0,\ldots,0)\in\mathbb{C}^{D}$ with radius squared equal to
$-\frac{\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})}{4p_{1}^{2}}$ which needs to
be real and positive. Again we may assume $p_{1}=((p_{1})_{0},0)$. Thus it is
not difficult to see that $r_{2}^{2}(p)$ is real and positive if and only if
$(p_{1})_{0}\in i\mathbb{R}$ with either
$(\text{Im}((p_{1})_{0}))^{2}>(m_{1}+m_{2})^{2}$ or
$(\text{Im}((p_{1})_{0}))^{2}<(m_{1}-m_{2})^{2}$.
Now we use Theorem 4.22 to compute the discontinuity of $I(C_{2})$: Let
$p^{\prime}\in\mathbb{C}^{D}$ such that $-{p^{\prime}}^{2}=(m_{1}\pm
m_{2})^{2}$, let $\gamma:[0,1]\to\mathbb{C}^{D}\backslash L(C_{2})$ be a
simple loop around $p^{\prime}$ and let $p\in\mathbb{C}^{D}$ such that

(4.134) $-\frac{1}{4p^{2}}\lambda(-p^{2},m_{1}^{2},m_{2}^{2})>0.$

Then

(4.135) $\begin{split}\text{Disc}_{[\gamma]}I(C_{2})(p)&=N(2\pi
i)^{2}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}\delta(k^{2}+m_{1}^{2})\delta((k+p)^{2}+m_{2}^{2})\\\
&=-N\int_{\mathbb{R}^{D-1}}\frac{2\pi^{2}}{\sqrt{\vec{k}^{2}+m_{1}^{2}}}(\delta((i\sqrt{\vec{k}^{2}+m_{1}^{2}}+p_{0})^{2}+\vec{k}^{2}+m_{2}^{2})\\\
&\hskip
98.16191pt+\delta((-i\sqrt{\vec{k}^{2}+m_{1}^{2}}+p_{0})^{2}+\vec{k}^{2}+m_{2}^{2}))\\\
&=-N\int_{\mathbb{R}^{D-1}}\frac{2\pi^{2}}{\sqrt{\vec{k}^{2}+m_{1}^{2}}}(\delta(2ip_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p^{2}-m_{1}^{2}+m_{2}^{2})\\\
&\hskip
98.16191pt+\delta((-2ip_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p^{2}-m_{1}^{2}+m_{2}^{2})).\end{split}$

Now we compute

(4.136) $\begin{split}&\pm
2ip_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p^{2}-m_{1}^{2}+m_{2}^{2}=0\\\
\Leftrightarrow\quad&\sqrt{\vec{k}^{2}+m_{1}^{2}}=\pm\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})\\\
\Rightarrow\quad&\vec{k}^{2}=-\frac{1}{4p^{2}}(p^{2}-m_{1}^{2}+m_{2}^{2})^{2}-m_{1}^{2}=-\frac{1}{4p^{2}}\lambda(-p^{2},m_{1}^{2},m_{2}^{2})=r_{2}^{2}(p).\end{split}$

Note that the second line of (4.136) can be fulfilled for some
$\vec{k}\in\mathbb{R}^{D-1}$ if and only if

(4.137) $\pm\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})\geq m_{1}.$

Since we integrate only over non-negative $|\vec{k}|$, we can hence write

(4.138) $\delta(\pm
2ip_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p^{2}-m_{1}^{2}+m_{2}^{2})=\frac{\sqrt{\vec{k}^{2}+m_{1}^{2}}}{2|p_{0}||\vec{k}|}\Theta(\pm\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})-m_{1})\delta(|\vec{k}|-\sqrt{r_{2}^{2}(p)})$

under the integral, where $\Theta$ denotes the Heaviside-function.

######  Lemma 4.24.

We have

(4.139)
$\Theta(\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})-m_{1})+\Theta(-\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})-m_{1})=\Theta(-\frac{\lambda(-p^{2},m_{1}^{2},m_{2}^{2})}{4p^{2}})=\Theta(r_{2}^{2}(p)).$

###### Proof.

A straightforward computation reveals

(4.140) $\begin{split}&\quad\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})\geq
m_{1}\quad\text{or}\quad-\frac{i}{2p_{0}}(p^{2}-m_{1}^{2}+m_{2}^{2})\geq
m_{1}\\\
\Leftrightarrow\quad&\quad-\frac{(p^{2}-m_{1}^{2}+m_{2}^{2})^{2}}{4p^{2}}\geq
m_{1}^{2}\\\
\Leftrightarrow\quad&\quad-\frac{\lambda(-p^{2},m_{1}^{2},m_{2}^{2})}{4p^{2}}\geq
0.\end{split}$

∎

Using Lemma 4.24, the discontinuity can hence be written as

(4.141)
$\begin{split}\text{Disc}_{[\gamma]}I(C_{2})(p^{\prime})&=-N\pi^{2}\int_{\mathbb{R}^{D-1}}\frac{\delta(|\vec{k}|-\sqrt{r_{2}^{2}(p)})}{|p_{0}||\vec{k}|}\Theta(-\frac{\lambda(-p^{2},m_{1}^{2},m_{2}^{2})}{4p^{2}})\\\
&=-N\pi^{2}\frac{2\pi^{\frac{D-1}{2}}}{\Gamma(\frac{D-1}{2})}\cdot\frac{(r_{2}^{2}(p))^{\frac{D-3}{2}}}{|p_{0}|}\\\
&=-\frac{N\pi^{\frac{D+3}{2}}}{2^{D-4}\Gamma(\frac{D-1}{2})}\cdot\frac{(\lambda(-p^{2},m_{1}^{2},m_{2}^{2}))^{\frac{D-3}{2}}}{(-p^{2})^{\frac{D}{2}-1}}.\end{split}$

It should be noted that it is known by physicists that while
$-(p^{\prime})^{2}=(m_{1}+m_{2})^{2}$ yields a genuine discontinuity on the
principal branch, the point $-(p^{\prime})^{2}=(m_{1}-m_{2})^{2}$ (called the
pseudo-threshold) does not. Thus we expect the intersection index $N$ to
vanish in this case.

###  4.4. Discontinuity around Points of Higher Codimension

Aside from the somewhat pathological colinear kinematic configurations outside
of $T_{n}$ where the determinant of the matrix containing all products of
external momenta vanish, we have discussed how to compute the discontinuity
along any simple loop around a point of codimension 1 in the Landau surface of
a one-loop Feynman graph. This does however not exhaust all possible loops in
the space of non-pathological external momenta minus the Landau surface. We
have seen in Proposition 4.4 and 4.5 that the $L_{n,I}$ are codimension 1
submanifold intersecting in general position. Thus a point $p\in L_{n}$ in the
intersection of more than one $L_{n,I}$ is a point of higher codimension. But
we also know that $T_{n}\backslash L_{n}$ is spanned by simple loops (see
Proposition 2.26, so that it is possible to decompose a loop “around” $p$ into
simple loops.

######  Proposition 4.25.

Let $p\in L_{n}$ be a point of codimension $k\in\mathbb{N}^{\ast}$ and
$\gamma:[0,1]\to T$ be a simple loop around $p$. Then $\gamma$ can be
decomposed as

(4.142) $\gamma=\gamma_{1}\cdots\gamma_{k},$

where the $\gamma_{i}:[0,1]\to T$ are simple loops around the codimension 1
parts of $L_{n}$ intersecting at $p$.

###### Proof.

According to Proposition 4.5 the point $p$ lies in the intersection of $k$
codimension 1 parts of $L$ which intersect in general position. Thus locally
the intersection looks like

(4.143)
$X_{D,k}:=\mathbb{C}^{D}\backslash\\{(z_{1},\ldots,z_{D})\in\mathbb{C}^{D}\;|\;\exists
i\in\\{1,\ldots,k\\}:z_{i}=0\\}.$

Note that

(4.144) $H:X_{D,k}\times[0,1]\to
X_{D,k},\quad((z_{1},\ldots,z_{D}),t)\mapsto(z_{1},\ldots,z_{k},(1-t)z_{k+1},\ldots,(1-t)z_{D})$

is a deformation retract from $X_{D,k}$ to (a subspace isomorphic to)
$X_{k,k}$, so that

(4.145)
$\pi_{1}(X_{D,k},(z_{1},\ldots,z_{D}))\simeq\pi_{1}(X_{k,k},(z_{1},\ldots,z_{k}))$

for any $(z_{1},\ldots,z_{D})\in\mathbb{C}^{D}$. Now note that

(4.146) $\pi:X_{k,k}\to
X_{k-1,k-1},\quad(z_{1},\ldots,z_{k})\mapsto(z_{1},\ldots,z_{k-1})$

defines a Serre fibration with fiber

(4.147) $\pi^{-1}(\\{z_{1},\ldots,z_{k-1}\\})=\\{(z_{1},\ldots,z_{k})\in
X_{k,k}\;|\;z_{k}\neq 0\\}$

homotopy equivalent to the circle $S^{1}$ for any $(z_{1},\ldots,z_{k-1})\in
X_{k-1,k-1}$. Hence we obtain a short exact sequence

(4.148) $1\to\pi_{1}(S^{1})\to\pi_{1}(X_{k,k})\to\pi_{1}(X_{k-1,k-1})\to 1$

contained in the long exact sequence of the fibration. Thus

(4.149) $\pi_{1}(X_{k,k})\simeq\pi_{X_{k-1,k-1}}\oplus\pi_{1}(S^{1})$

and by induction we obtain

(4.150) $\pi_{1}(X_{k,k})\simeq\mathbb{Z}^{k}$

for all $k\in\mathbb{N}^{\ast}$, where the factors of $\mathbb{Z}$ are
generated by simple loops around the $k$ codimension 1 parts of $L_{n}$
intersecting at $p$. This proves the claim. ∎

This allows us in principal to compute the discontinuity around any loop by
Proposition 2.38: Given a loop $\gamma$ of interest, we have to find a
decomposition $\gamma=\gamma_{1}\cdots\gamma_{k}$ into simple loops whose
existence is guaranteed by Proposition 4.25. Then by applying Proposition 2.38
sufficiently often, we can reduce the computation of $\text{Disc}_{[\gamma]}$
to the computation of $\text{Disc}_{[\gamma_{i}]}$ for all
$i\in\\{1,\ldots,k\\}$. The latter case is covered by Cutkosky’s Theorem 4.22.

##  5\. More Advanced Examples

To illustrate the results from the last section and to check the results
therein by comparing them with the common literature, we look at the two
examples $C_{3}$ and $C_{4}$ in more detail. In both examples we focus the
discussion on the case where all analytic regulators $\lambda_{i}$ are set to
1.

###  5.1. The Triangle Graph $C_{3}$

First we consider the triangle graph

$C_{3}\quad=\quad$ $p_{3}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$.

In projective form, the corresponding Feynman integral reads

(5.1)
$I(C_{3})(p)=\int_{\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{(k^{2}+u^{2}m_{1}^{2})^{\lambda_{1}}((k+up_{1})^{2}+u^{2}m_{2}^{2})^{\lambda_{2}}((k+u(p_{1}+p_{2}))^{2}+u^{2}m_{3}^{2})^{\lambda_{3}}}.$

Having discussed $C_{2}$ already, we know that there are discontinuities
associated to any pair of edges put on mass-shell. The three corresponding
reduced graphs are

$G_{1}=$ $p_{2}$$p_{3}$$k$$k+p_{1}$$p_{1}$, $G_{2}=$
$p_{1}$$p_{3}$$k$$k+p_{2}$$p_{2}$, $G_{3}=$
$p_{3}$$k$$k+p_{1}+p_{2}$$p_{1}$$p_{2}$,

and they have the following Landau surfaces:

(5.2) $\begin{split}L_{2,\\{3\\}}&=\\{(p_{1},p_{2})\in
T_{3}\;|\;-p_{1}^{2}=(m_{1}\pm m_{2})^{2}\\}\\\
L_{2,\\{1\\}}&=\\{(p_{1},p_{2})\in T_{3}\;|\;-p_{2}^{2}=(m_{2}\pm
m_{3})^{2}\\}\\\ L_{2,\\{2\\}}&=\\{(p_{1},p_{2})\in
T_{3}\;|\;-(p_{1}+p_{2})^{2}=(m_{1}\pm m_{3})^{2}\\}\end{split}$

Due to the $S_{3}$-symmetry of the graph $C_{3}$ (with respect to the action
permuting the vertices), it suffices to compute the discontinuity along one of
the surfaces $L_{2,\\{1\\}},L_{2,\\{2\\}},L_{2,\\{3\\}}$. So let
$p^{\prime}\in L_{2,\\{3\\}}$, let $\gamma:[0,1]\to T_{3}\backslash L(C_{3})$
be a simple loop around $p^{\prime}$ and let $p=(p_{1},p_{2})\in
T_{3}\backslash L(C_{3})$ such that

(5.3)
$r_{2}^{2}(p_{1})=-\frac{1}{4p_{1}^{2}}\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})\in\mathbb{R}_{+}.$

Let us now restrict our attention to the case
$\lambda_{1}=\lambda_{2}=\lambda_{3}=1$. Then according to Cutkosky’s Theorem
4.22 we get

(5.4) $\begin{split}\text{Disc}_{[\gamma]}I(C_{3})(p)&=N(2\pi
i)^{2}\int_{i\mathbb{R}\times\mathbb{R}^{D-1}}\frac{\delta(k^{2}+m_{1}^{2})\delta((k+p_{1})^{2}+m_{2}^{2})}{(k+p_{1}+p_{2})^{2}+m_{3}^{2}}d^{D}k\\\
&=-N\int_{\mathbb{R}^{D-1}}\frac{2\pi^{2}}{\sqrt{\vec{k}^{2}+m_{1}^{2}}}\\\
&\hskip
56.9055pt\times\sum_{\tau\in\\{-1,1\\}}\frac{\delta(2i\tau(p_{1})_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p_{1}^{2}-m_{1}^{2}+m_{2}^{2})}{2i\tau((p_{1})_{0}+(p_{2})_{0})\sqrt{\vec{k}^{2}+m_{1}^{2}}+2\vec{k}\vec{p}_{2}+(p_{1}+p_{2})^{2}-m_{1}^{2}+m_{3}^{2}}.\end{split}$

As in our calculations regarding $C_{2}$, we have

(5.5) $\delta(\pm
2i(p_{1})_{0}\sqrt{\vec{k}^{2}+m_{1}^{2}}+p_{1}^{2}-m_{1}^{2}+m_{2}^{2})=\frac{\sqrt{\vec{k}^{2}+m_{1}^{2}}}{2|(p_{1})_{0}||\vec{k}|}\\\
\Theta(\pm\frac{i}{2(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})-m_{1})\delta(|\vec{k}|-\sqrt{r_{2}^{2}(p_{1})}).$

We set

(5.6)
$B:=(p_{1}+p_{2})^{2}-m_{1}^{2}+m_{3}^{2}-\frac{(p_{1})_{0}+(p_{2})_{0}}{(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})$

and obtain

(5.7)
$\begin{split}\text{Disc}_{[\gamma]}I(C_{3})(p)&=\frac{-\pi^{2}N}{|(p_{1})_{0}|\sqrt{r_{2}^{2}(p_{1})}}\int_{S^{D-1}}d\Omega_{D-1}\int_{-1}^{1}d\cos(\varphi)\\\
&\hskip
28.45274pt\times\frac{\sum_{\tau\in\\{-1,1\\}}\Theta(\tau\frac{i}{2(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})-m_{1})}{-\frac{(p_{1})_{0}+(p_{2})_{0}}{(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})+2\sqrt{r_{2}^{2}(p_{1})}|\vec{p}_{2}|\cos(\varphi)+(p_{1}+p_{2})^{2}-m_{1}^{2}+m_{3}^{2}}\\\
&=-\frac{\pi^{\frac{D+3}{2}}}{\Gamma(\frac{D-1}{2})|(p_{1})_{0}||\vec{p}_{2}|r_{2}^{2}(p_{1})}(\ln(B+2\sqrt{r_{2}^{2}(p_{1})}|\vec{p}_{2}|)-\ln(B-2\sqrt{r_{2}^{2}(p_{1})}|\vec{p}_{2}|)),\end{split}$

where we again used Lemma 4.24 to evaluate the sum of the two Heaviside-
functions to 1.
Now we discuss the case where all three edges are put on shell. We can easily
compute an equation for the Landau surface by the procedure from Section 4:
According to our recursive formula (4.109), we have

(5.8)
$A_{1}(p)=-\frac{1}{2(p_{2})_{1}}((p_{1}+p_{2})^{2}-m_{1}^{2}+m_{3}^{2}-\frac{(p_{1})_{0}+(p_{2})_{0}}{(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})).$

Note that $A_{1}(p)=-\frac{1}{2(p_{2})_{1}}B$. We set

(5.9) $B_{1}:=(p_{1}+p_{2})^{2}-m_{1}^{2}+m_{3}^{2}\quad\text{and}\quad
B_{2}:=\frac{(p_{1})_{0}+(p_{2})_{0}}{(p_{1})_{0}}(p_{1}^{2}-m_{1}^{2}+m_{2}^{2})$

and compute

(5.10)
$r_{3,D}^{2}(p)=r_{2,D}^{2}(p)-A_{1}^{2}(p)=-\frac{\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})}{4p_{1}^{2}}-\frac{B_{1}^{2}-2B_{1}B_{2}+B_{2}^{2}}{4(p_{2})_{1}^{2}}.$

At this point we could end the calculation: We have successfully reduced the
description of the Landau surface $L_{3,\\{1,2,3\\}}$ to a single equation.
However, it is very useful for physicists to be able to solve this equation
for the $O(D,\mathbb{C})$-invariant products. In our case $C_{3}$ we do this
for

(5.11) $s:=-p_{1}^{2},\quad t:=-p_{2}^{2},\quad u:=-(p_{1}+p_{2})^{2}.$

Due to the symmetry of the graph, it suffices to solve for one of the
variables and we choose $s$ here. The solution solved for $t$ or $u$ can then
be obtained by simply permuting the vertices of the graph $C_{3}$. To this end
we first write

(5.12)
$B_{2}^{2}=\frac{((p_{1})_{0}+(p_{2})_{0})^{2}}{p_{1}^{2}}(\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})-4m_{1}^{2}p_{1}^{2})$

to obtain

(5.13)
$\begin{split}r_{3}^{2}(p)&=-\frac{1}{4p_{1}^{2}(p_{2})_{1}^{2}}((p_{1}+p_{2})^{2}\lambda(-p_{1}^{2},m_{1}^{2},m_{2}^{2})+p_{1}^{2}B_{1}^{2}-2p_{1}^{2}B_{1}B_{2}-4p_{1}^{2}((p_{1})_{0}+(p_{2})_{0})^{2}m_{1}^{2})\\\
&=-\frac{1}{4p_{1}^{2}(p_{2})_{1}^{2}}(-u(s+(m_{1}+m_{2})^{2})(s+(m_{1}-m_{2})^{2})-s(u+m_{1}^{2}-m_{3})^{2}\\\
&\hskip
61.17325pt+(u+m_{1}^{2}-m_{3}^{2})(s-t+u)(s+m_{1}^{2}-m_{2}^{2})-m_{1}^{2}(s-t+u)^{2})\\\
&=-\frac{1}{4(p_{1})_{0}^{2}(p_{2})_{1}^{2}}(-m_{3}^{2}s^{2}+c\cdot
s+d)\end{split}$

with

(5.14)
$c:=2m_{1}^{2}t+2m_{2}^{2}u-(t+m_{2}^{2}-m_{3}^{2})(u+m_{1}^{2}-m_{3}^{2})$

and

(5.15)
$d:=-m_{1}^{2}(t-u)^{2}-u(m_{1}^{2}-m_{2}^{2})^{2}+(m_{1}^{2}-m_{2}^{2})(-t+u)(u+m_{1}^{2}-m_{3}^{2}).$

We see that the denominator of $r_{3}^{2}(p)$ is a polynomial of degree 3 in
$s,t,u$ which is quadratic in each of the channel variables separately. In
particular we can view it as a quadratic polynomial in $s$. This allows us to
conclude that $r_{3}^{2}(p)=0$ if and only if

(5.16) $\begin{split}s&=\frac{1}{2m_{3}^{2}}(c\pm\sqrt{c^{2}+4m_{3}^{2}d})\\\
&=\frac{1}{2m_{3}^{2}}(-(t-m_{2}^{2})(u-m_{1}^{2})+m_{3}^{2}(t+u+m_{1}^{2}+m_{2}^{2})-m_{3}^{4}\\\
&\hskip
35.56593pt+\sqrt{m_{2}^{4}+(t-m_{3}^{2})^{2}-2m_{2}^{2}(t+m_{3}^{2})((u-m_{3}^{2})^{2}-2m_{1}^{2}(u+m_{3}^{2})+m_{1}^{4})}).\end{split}$

This concludes the computation of the Landau surface $L_{3}$.
Turning to the computation of the discontinuity of $I(C_{3})$ along simple
loops around codimension 1 points in $L_{3,\\{1,2,3\\}}$, assume again that
$\gamma_{1}=\gamma_{2}=\gamma_{3}=1$. Note that in this case the integral
$I(C_{3})(p)$ converges absolutely in $D=4$ space-time dimensions for $p$
close to Euklidean momenta according to the power counting criterion. Let
$p\in L_{3,\\{1,2,3\\}}$ and let $\gamma:[0,1]\to T_{3}\backslash L_{3}$ be
simple loop around $p$. We can compute the corresponding discontinuity using
Theorem 4.22. For the actual calculation it is convenient to again choose
$p_{1}$ and $p_{2}$ such that $(p_{i})_{j}=0$ for $j\geq i$ which we can do by
Lemma 4.17. By our previous calculation we have

(5.17) $\begin{split}&\qquad\quad\text{Disc}_{[\gamma]}(I(C_{3}))(p)\\\
=&\frac{-i\pi^{3}N}{|(p_{1})_{0}|\sqrt{r_{2}^{2}(p_{1})}}\int_{S^{D-1}}d\Omega_{D-1}\int_{-1}^{1}d\cos(\varphi)\delta(2\sqrt{r_{2}^{2}(p_{1})}|\vec{p}_{2}|\cos(\varphi)+B)\end{split}$

The integral over the $\delta$-function evaluates to
$\frac{1}{2\sqrt{r_{2}^{2}(p_{1})}|\vec{p}_{2}|}$ if and only if

(5.18) $\frac{A_{1}(p)}{\sqrt{r_{2}^{2}(p_{1})}}\in[-1,1].$

This is always true for any $p$ such that
$r_{3}^{2}(p)=r_{2}^{2}(p_{1})-A_{1}^{2}(p)>0$. Thus

(5.19)
$\text{Disc}_{[\gamma]}(I(C_{3}))(p)=\frac{-iN\pi^{\frac{D+5}{2}}}{\Gamma(\frac{D-1}{2})|(p_{1})_{0}||(p_{2})_{1}|r_{2}^{2}(p_{1})}.$

###  5.2. The Box Graph $C_{4}$

Finally we consider the graph

$C_{4}\quad=\quad$
$p_{4}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$$p_{3}$$k+p_{1}+p_{2}+p_{3}$

The corresponding Feynman integral in projective space reads

(5.20) $\begin{split}&\hskip 199.16928ptI(C_{4})(p)\\\
=&\int_{\mathbb{R}\mathbb{P}^{D}}\frac{u^{2\lambda-D-1}\cdot\Omega_{D}}{(k^{2}+u^{2}m_{1}^{2})^{\lambda_{1}}((k+up_{1})^{2}+u^{2}m_{2}^{2})^{\lambda_{2}}(k+u(p_{1}+p_{2}))^{2}+u^{2}m_{3}^{2})^{\lambda_{3}}((k+u(p_{1}+p_{2}+p_{3}))^{2}+u^{2}m_{4}^{2})^{\lambda_{4}}}.\end{split}$

As before, we can obtain parts of the Landau surface by considering the
reduced graphs of $C_{4}$. There are four graphs with one edge contracted to
zero length

$p_{1}$$p_{4}$$k-p_{2}-p_{3}$$p_{2}$$k-p_{3}$$p_{3}$$k$

, $p_{3}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$,
$p_{3}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$ ,
$p_{3}$$k$$p_{1}$$k+p_{1}$$p_{2}$$k+p_{1}+p_{2}$,

and 6 graphs with two edges contracted to zero length.

$k+p_{1}+p_{4}$$k$$p_{1}$$p_{4}$$p_{2}$$p_{3}$

, $k+p_{1}+p_{2}$$k$$p_{1}$$p_{2}$$p_{3}$$p_{4}$,
$p_{1}$$k-p_{1}-p_{2}$$k$$p_{2}$$p_{3}$$p_{4}$ ,
$p_{1}$$k-p_{1}-p_{2}$$k$$p_{2}$$p_{3}$$p_{4}$,
$p_{1}$$k-p_{1}-p_{2}$$k$$p_{2}$$p_{3}$$p_{4}$ ,
$p_{1}$$k-p_{1}-p_{2}$$k$$p_{2}$$p_{3}$$p_{4}$.

Using our calculations for $C_{2}$ we immediately obtain

(5.21) $\begin{split}L_{4,\\{1,2\\}}\quad&=\quad\\{p\in
T_{4}\;|\;-p_{3}^{2}=(m_{3}\pm m_{4})^{2}\\},\\\
L_{4,\\{1,3\\}}\quad&=\quad\\{p\in T_{4}\;|\;-(p_{1}+p_{2})^{2}=(m_{2}\pm
m_{4})^{2}\\},\\\ L_{4,\\{1,4\\}}\quad&=\quad\\{p\in
T_{4}\;|\;-p_{2}^{2}=(m_{2}\pm m_{3})^{2}\\},\\\
L_{4,\\{2,3\\}}\quad&=\quad\\{p\in
T_{4}\;|\;-(p_{1}+p_{2}+p_{3})^{2}=(m_{1}\pm m_{4})^{2}\\},\\\
L_{4,\\{2,4\\}}\quad&=\quad\\{p\in T_{4}\;|\;-(p_{2}+p_{3})^{2}=(m_{1}\pm
m_{3})^{2}\\},\\\ L_{4,\\{3,4\\}}\quad&=\quad\\{p\in
T_{4}\;|\;-p_{1}^{2}=(m_{1}\pm m_{2})^{2}\\}.\end{split}$

Similarly with our calculations for $C_{3}$ we obtain

(5.22) $\begin{split}L_{4,\\{1,2,3\\}}&=\\{p\in
T_{4}\;|\;r_{3}^{2}(p_{1},p_{2},p_{3}+p_{4})=0\\}-\bigcup_{\begin{subarray}{c}I\subset\\{1,2,3,4\\}\\\
|I|=2\end{subarray}}L_{4,I},\\\ L_{4,\\{1,2,4\\}}&=\\{p\in
T_{4}\;|\;r_{3}^{2}(p_{1},p_{2}+p_{3},p_{4})=0\\}-\bigcup_{\begin{subarray}{c}I\subset\\{1,2,3,4\\}\\\
|I|=2\end{subarray}}L_{4,I},\\\ L_{4,\\{1,3,4\\}}&=\\{p\in
T_{4}\;|\;r_{3}^{2}(p_{1}+p_{2},p_{3},p_{4})=0\\}-\bigcup_{\begin{subarray}{c}I\subset\\{1,2,3,4\\}\\\
|I|=2\end{subarray}}L_{4,I},\\\ L_{4,\\{2,3,4\\}}&=\\{p\in
T_{4}\;|\;r_{3}^{2}(p_{1}+p_{4},p_{2},p_{3})=0\\}-\bigcup_{\begin{subarray}{c}I\subset\\{1,2,3,4\\}\\\
|I|=2\end{subarray}}L_{4,I}.\end{split}$

Now we compute $L_{4,\\{1,2,3,4\\}}$: We have

(5.23) $r_{4}^{2}(p)=r_{3}^{2}(p)-A_{2}^{2}(p)$

with

(5.24)
$A_{2}(p)=-\frac{1}{2(p_{3})_{2}}(A_{0}(p)(p_{1}+p_{2}+p_{3})_{0}+A_{1}(p)(p_{1}+p_{2}+p_{3})_{1}+(p_{1}+p_{2}+p_{3})^{2}-m_{1}^{2}+m_{4}^{2}).$

Similarly to our example $C_{3}$ above, we have reduced the task of computing
$L_{4,\\{1,2,3,4\\}}$ to solving a single equation: $r_{4}^{2}(p)=0$. Again it
is useful to solve this equation for the channel variables. We set

(5.25)
$s:=p_{1}^{2},\;t:=p_{2}^{2},\;u:=p_{3}^{2},\;v:=(p_{1}+p_{2})^{2},\;w:=(p_{1}+p_{3})^{2},\;x:=(p_{2}+p_{3})^{2}.$

However, in this case the task is a bit more tricky and we have to distinguish
some cases. While calculating $C_{3}$, it was enough to solve for one of the
channel variables due to the $S_{3}$-symmetry of the underlying graph. Since
$C_{4}$ does not enjoy a corresponding $S_{4}$-symmetry, we have to solve for
say $s$, $v$ and $w$ to be able to cover all results. The nominator of
$r_{4}^{2}(p)$ is again quadratic in each channel variable separately. Hence
we have

(5.26) $y=\frac{1}{2a_{y}}(-b_{y}\pm\sqrt{b_{y}^{2}-4a_{y}c_{y}})$

for all $y\in\\{s,t,u,v,w,x\\}$ where the coefficients $a_{y},b_{y},c_{y}$ can
be read off from $r_{4}^{2}(p)$ directly. But $r_{4}^{2}(p)$ is a pretty messy
expression so that we do not perform the calculation by hand. Instead we used
a Mathematica program. The results obtained by solving $r_{4}^{2}(p)=0$ for
one of the channel variables are equally messy and might not seem particularly
useful. For completeness we include them anyway. For $s$ we obtain

(5.27)
$a_{s}=\frac{1}{16}\left(m_{2}^{4}-2m_{2}^{2}\left(2m_{3}^{2}-m_{4}^{2}+t+u\right)+4m_{3}^{4}-4m_{3}^{2}\left(m_{4}^{2}+t+u-x\right)+\left(-m_{4}^{2}+t+u\right)^{2}\right)$
(5.28)
$\begin{split}b_{s}&=\frac{1}{8}(m_{2}^{2}(m_{3}^{2}(-t-3u+v+3w+2x)+m_{4}^{2}(2u-v-
w-x)-2t^{2}+t(-4u+v+2(w+x))\\\
&-2u^{2}+u(4v+w+x)-vx)+m_{3}^{2}(2m_{3}^{2}(t+u-v-w-x)+2x(2t+2u-v-w)\\\
&-(3t+u)(t+u-v-w)-2x^{2})+m_{4}^{2}\left(m_{3}^{2}(-t-3u+3v+w+2x)+t(-2v+w+x)+v(u+x)\right)\\\
&+m_{2}^{4}(t+2u-w-x)+m_{1}^{2}\left(m_{2}^{2}(-t-3u+x)+m_{4}^{2}(3t+u-x)+(t-u)\left(-2m_{3}^{2}+t+u-x\right)\right)\\\
&-m_{4}^{4}(t+v)-x\left(t^{2}+t(u-v)+uv\right)+t(t+u)(t+u-v-w))\end{split}$
(5.29)
$\begin{split}c_{s}&=\frac{1}{16}(m_{2}^{4}\left(t^{2}+4tu-2t(w+x)+4u^{2}-4u(v+w+x)+(w+x)^{2}\right)+m_{1}^{4}\left(t^{2}-2t(u+x)+(u-x)^{2}\right)\\\
&+2m_{1}^{2}(-m_{2}^{2}\left(t^{2}+t(u-w-2x)+2u^{2}-u(2v+w+3x)+x(w+x)\right)\\\
&+m_{3}^{2}\left(-t^{2}+t(v+w+x)+u^{2}-u(v+w+x)+x(w-v)\right)\\\
&+m_{4}^{2}\left(t^{2}+t(3u-v-2w-x)+v(x-u)\right)+t^{3}-t^{2}(v+w+2x)\\\
&+t\left(-u^{2}+u(v+w-2x)+x(2v+w+x)\right)+vx(u-x))\\\
&-2m_{2}^{2}(m_{4}^{2}\left(t^{2}+t(2u-v-
w-x)+v(x-w)\right)+m_{3}^{2}(x(t-u-v+w)-(t-2u+w)(t+u-v-w))\\\
&+t^{3}+t^{2}(3u-v-2(w+x))+t\left(2u^{2}-4uv-3u(w+x)+v(w+2x)+(w+x)^{2}\right)\\\
&+v\left(-2u^{2}+2u(v+w+x)-x(w+x)\right))\\\
&+\left(m_{3}^{2}(-t-u+v+w)+m_{4}^{2}(t-v)+t(t+u-v-w)+x(v-t)\right){}^{2})\end{split}$

For $v$ we obtain

(5.30)
$a_{v}=\frac{1}{16}\left(\left(-m_{3}^{2}+m_{4}^{2}+t-x\right){}^{2}-4m_{2}^{2}u\right)$
(5.31)
$\begin{split}b_{v}&=\frac{1}{8}(m_{2}^{2}(m_{3}^{2}(s-t+2u-w+x)+m_{4}^{2}(-s+t+w-x)-x(s+2(t+u)-w)+st+4su+t^{2}+4tu\\\
&-tw+2u^{2}-2uw+x^{2})+m_{4}^{2}\left(m_{3}^{2}(3s+2t+u-w)+s(-2t+u+x)+t(-2t-u+w+2x)\right)\\\
&+m_{3}^{2}\left(m_{3}^{2}(-(2s+t+u-w))+s(3t+u-2x)+x(-2t-u+w)+2t(t+u-w)\right)\\\
&-m_{4}^{4}(s+t)+m_{1}^{2}\left(m_{3}^{2}(t-u-x)-m_{4}^{2}(t+u-x)+2m_{2}^{2}u-t^{2}+t(u+2x)+x(u-x)\right)\\\
&-2m_{2}^{4}u+sx(t-u)-st(t+u)-t(t-x)(t+u-w-x))\end{split}$ (5.32)
$\begin{split}c_{v}&=\frac{1}{16}(-2m_{2}^{2}(m_{3}^{2}\left(2s^{2}+s(t+3u-3w-2x)-t^{2}+t(u+x)+(u-w)(2u-w-x)\right)\\\
&+(s+t+2u-w-x)\left(m_{4}^{2}(t-s)+s(t+u)+t(t+u-w-x)\right))\\\
&-2m_{4}^{2}\left(m_{3}^{2}\left(2s^{2}+s(t+3u-w-2x)+t(t+u-w)\right)+(s-t)(s(t+u)+t(t+u-w-x))\right)\\\
&-4m_{3}^{2}s^{2}t-4m_{3}^{2}s^{2}u+4m_{3}^{2}s^{2}x+4m_{3}^{4}s^{2}\\\
&+2m_{1}^{2}(m_{2}^{2}\left(-\left(s(t+3u-x)+t^{2}+t(u-w-2x)+(u-x)(2u-w-x)\right)\right)\\\
&+m_{3}^{2}(x(t-u+w)-(t-u)(2s+t+u-w))+m_{4}^{2}(s(3t+u-x)+t(t+3u-2w-x))\\\
&+s(t-u)(t+u-x)+t\left(x(w-2(t+u))+(t-u)(t+u-w)+x^{2}\right))-6m_{3}^{2}st^{2}\\\
&+m_{2}^{4}(s+t+2u-w-x)^{2}-8m_{3}^{2}stu+6m_{3}^{2}stw+8m_{3}^{2}stx+4m_{3}^{4}st+m_{4}^{4}(s-t)^{2}-2m_{3}^{2}su^{2}\\\
&+2m_{3}^{2}suw+8m_{3}^{2}sux+4m_{3}^{4}su-4m_{3}^{2}swx\\\
&-4m_{3}^{4}sw-4m_{3}^{2}sx^{2}-4m_{3}^{4}sx-2m_{3}^{2}t^{3}+m_{1}^{4}\left(t^{2}-2t(u+x)+(u-x)^{2}\right)-4m_{3}^{2}t^{2}u+4m_{3}^{2}t^{2}w\\\
&+2m_{3}^{2}t^{2}x+m_{3}^{4}t^{2}-2m_{3}^{2}tu^{2}+4m_{3}^{2}tuw+2m_{3}^{2}tux+2m_{3}^{4}tu-2m_{3}^{2}tw^{2}-2m_{3}^{2}twx-2m_{3}^{4}tw+m_{3}^{4}u^{2}\\\
&-2m_{3}^{4}uw+m_{3}^{4}w^{2}+(s(t+u)+t(t+u-w-x))^{2})\end{split}$

For $w$ we obtain

(5.33)
$a_{w}=\frac{1}{16}\left(-2m_{2}^{2}\left(m_{3}^{2}+t\right)+\left(t-m_{3}^{2}\right){}^{2}+m_{2}^{4}\right)$
(5.34)
$\begin{split}b_{w}&=\frac{1}{8}(m_{2}^{2}\left(m_{4}^{2}(-s+t+v)+m_{3}^{2}(3s+3u-v-x)+2st+su+2t^{2}+3tu-
tv-2tx-2uv+vx\right)\\\
&+m_{3}^{2}\left(m_{3}^{2}(-(2s+t+u-v))+s(3t+u-2x)+2t(t+u-v)+x(v-t)\right)+m_{2}^{4}(-(s+t+2u-x))\\\
&+m_{4}^{2}\left(m_{3}^{2}(s+t-v)+t(s-t+v)\right)+m_{1}^{2}\left(m_{2}^{2}(t+u-x)+m_{3}^{2}(t-u+x)+t\left(-2m_{4}^{2}-t+u+x\right)\right)\\\
&-t\left(s(t+u)+t^{2}+t(u-v-x)+vx\right))\end{split}$ (5.35)
$\begin{split}c_{w}&=\frac{1}{16}(\left(t^{2}-2(u+x)t+(u-x)^{2}\right)m_{1}^{4}+2((t-v)x^{2}+(s(u-t)-2t(t+u)+(2t+u)v)x\\\
&-\left(t^{2}+st+ut+2u^{2}+x^{2}+3su-2uv-(s+2t+3u)x\right)m_{2}^{2}\\\
&-((t-u)(2s+t+u-v)+(-t+u+v)x)m_{3}^{2}\\\ &+\left(t^{2}+3st+3ut-vt+su-
uv-(s+t-v)x\right)m_{4}^{2}+(t-u)((s+t)(t+u)-tv))m_{1}^{2}\\\
&+\left(s^{2}+2ts+4us+t^{2}+4u^{2}+x^{2}+4tu-4uv-2(s+t+2u)x\right)m_{2}^{4}+4s^{2}m_{3}^{4}+t^{2}m_{3}^{4}+u^{2}m_{3}^{4}+v^{2}m_{3}^{4}\\\
&+4stm_{3}^{4}+4sum_{3}^{4}+2tum_{3}^{4}-4svm_{3}^{4}-2tvm_{3}^{4}-2uvm_{3}^{4}-4sxm_{3}^{4}+\left(s^{2}-2(t+v)s+(t-v)^{2}\right)m_{4}^{4}\\\
&+((s+t)(t+u)-tv)^{2}+(t-v)^{2}x^{2}-2t^{3}m_{3}^{2}-6st^{2}m_{3}^{2}-2su^{2}m_{3}^{2}-2tu^{2}m_{3}^{2}-2tv^{2}m_{3}^{2}-4sx^{2}m_{3}^{2}\\\
&-4s^{2}tm_{3}^{2}-4s^{2}um_{3}^{2}-4t^{2}um_{3}^{2}-8stum_{3}^{2}+4t^{2}vm_{3}^{2}+6stvm_{3}^{2}+2suvm_{3}^{2}+4tuvm_{3}^{2}+4s^{2}xm_{3}^{2}\\\
&+2t^{2}xm_{3}^{2}+2v^{2}xm_{3}^{2}+8stxm_{3}^{2}+8suxm_{3}^{2}+2tuxm_{3}^{2}-4svxm_{3}^{2}-4tvxm_{3}^{2}-2uvxm_{3}^{2}\\\
&-2((t+u)s^{2}+(2t-u)vs-(t+v)xs+\left(2s^{2}+(t+3u-3v-2x)s+(t-v)(t+u-v)\right)m_{3}^{2}\\\
&+(t-v)((t-v)x-t(t+u-v)))m_{4}^{2}-2\left(t(t-v)(t+u-v)+s\left(t^{2}+(u-v)t+uv\right)\right)x\\\
&-2m_{2}^{2}(t^{3}+(3u-v-2x)t^{2}+(2u-x)(u-2v-x)t+(2s^{2}+(t+3u-v-2x)s-t^{2}+2u(u-v)\\\
&-(u+v)x+t(u+v+x))m_{3}^{2}+((s-t+v)x-(s-t)(s+t+2u-v))m_{4}^{2}+s^{2}(t+u)\\\
&+v\left(-2u^{2}+2(v+x)u-x^{2}\right)+s\left(2t^{2}+(4u-v-2x)t+2u^{2}+vx-u(4v+x)\right)))\end{split}$

These results agree with the results of [MT21], which cover the case
$m_{1}=\cdots=m_{4}$ and $s=t=u=(p_{1}+p_{2}+p_{3})^{2}$.141414For easier
comparison: In [MT21] our variables $v$ and $x$ are called $s$ and $t$
respectively.

##  6\. Conclusion and Outlook

We have seen that the techniques developed by Pham et. al. in works such as
[FFLP65] or [Pha11] can be applied to one-loop Feynman integrals to rigorously
state and prove Cutkosky’s Theorem on the discontinuity of Feynman integrals
on the principal branch at Minkowski momenta. This settles a claim appearing
regularly in the literature (see for example [Cut60] or [ABDG14]) in this
particular case, up to the computation of the relevant intersection index. The
latter will be done in a follow-up paper.
Of course the goal is to eventually prove Cutkosky’s Theorem in the general
multi-loop case. There seems to be no straightforward way to apply the program
to graphs with more than loop. Problems occur already at the first stage of
compactification since the naive compactification via projective space does
not work. What one could do instead is consider Feynman integrals as iterated
integrals (one iteration for each loop-momentum) and then apply the
generalization of the techniques in [Pha11] to the case where the integrand
itself is also ramified. This is a non-trivial task however and future
research has to show if this is a valid route.

## Acknowledgement

I would like to express my gratitude towards everyone who supported me during
the time this paper came to be, professionally or otherwise. In particular I
thank Dirk Kreimer, Marko Berghoff, my family, Nora Tabea Sibert and Christian
Hammermeister.

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